DEGREE 3 WEIGHT 16

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eigenvalues |

standard Euler factors |

spinor Euler factors |

identified eigenforms |

special values |

congruences

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DIMENSIONS

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     dim M₁₆(Γ₃)= 7
     dim S₁₆(Γ₃) = 3

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DETERMINING INDICES

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Introduce the notation

[a b c d e f]

=

a d e d b f e f c

We use the following set of determining Fourier coefficient indices.

t1	½ [0 0 0 0 0 0]
t2	½ [2 0 0 0 0 0]
t3	½ [2 2 0 1 0 0]
t4	½ [2 2 0 0 0 0]
t5	½ [2 2 2 1 1 1]
t6	½ [2 2 2 1 0 0]
t7	½ [2 2 2 0 0 0]

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PULLBACK OF EISENSTEIN SERIES

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Let E⁽⁶⁾₁₆ denote the monic Eisenstein series of degree 6 and weight 16.
The rescaled Eisenstein series
     E⁽⁶⁾₁₆ = 13912726954911229324966739363569/234258606489600 E⁽⁶⁾₁₆
has rational Fourier coefficients that are near-integers in that their denominator prime factors are at most 31.
Consider the Witt map W_3,3 and the pullback W^*_3,3E⁽⁶⁾₁₆ ∈ M₁₆(Γ₃) ⊗ M₁₆(Γ₃).
The columns of the 7 x 7 matrix [a(t_i x t_j; W^*_3,3E⁽⁶⁾₁₆)] are determining truncations of an M₁₆(Γ₃)-basis g_t1,...,g_t7. Each g_t is the cofficient function of e(tr(t,z)) in the Fourier expansion of W^*_3,3E⁽⁶⁾₁₆(w,z). For convenience let g_j = g_tj for j=1,...,7. The basis elements g₁,...,g₇ are not Hecke eigenforms. The matrix [a(t_i x t_j; W^*_3,3E⁽⁶⁾₁₆)] is as follows (switch to factored form).

	t1	t2	t3	t4	t5	t6	t7
t1	13912726954911229324966739363569/234258606489600	3846482431548584275633602257/14354081280	15450190691937513287077/751680	445016619031983484355717/334080	12190127397766275077/27840	1334576754702420920447/10440	1377484395947589083701/192
t2	3846482431548584275633602257/14354081280	28545440207637342132869777/14952168	198714032425664085997/783	4529479453951814168837/348	256885152733794797/29	167522879746699791736/87	89320031445718918505
t3	15450190691937513287077/751680	198714032425664085997/783	219200918201873871032/783	211752845084675720518/87	643995985149579832/29	166342461853204298816/87	21909311729885244280
t4	445016619031983484355717/334080	4529479453951814168837/348	211752845084675720518/87	6856833942269137505217/58	3701424146518329462/29	696358864975982344752/29	1028680202152781023230
t5	12190127397766275077/27840	256885152733794797/29	643995985149579832/29	3701424146518329462/29	648262571812073064/29	193115326121315136	1642272118573723560
t6	1334576754702420920447/10440	167522879746699791736/87	166342461853204298816/87	696358864975982344752/29	193115326121315136	550935271481450279424/29	270848912004871421760
t7	1377484395947589083701/192	89320031445718918505	21909311729885244280	1028680202152781023230	1642272118573723560	270848912004871421760	11128058082662642490600

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EIGENFORM NUMBER FIELD GENERATORS

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a = Sqrt[51349]

b = Sqrt[18209]

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INDICES FOR COMPUTING HECKE ACTION

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Fourier coefficients for the following indices are needed to determine the action of the Hecke operator T(2) at the determining indices t₁,...,t₇ and also the action of T_i(4) for i=0,1,2,3 at t₅.

u1	½ [0 0 0 0 0 0]
u2	½ [2 0 0 0 0 0]
u3	½ [4 0 0 0 0 0]
u4	½ [2 2 0 0 0 0]
u5	½ [2 2 0 1 0 0]
u6	½ [4 4 0 0 0 0]
u7	½ [4 4 0 2 0 0]
u8	½ [2 2 2 1 1 1]
u9	½ [2 2 2 1 0 0]
u10	½ [2 2 2 0 0 0]
u11	½ [2 2 4 0 1 1]
u12	½ [2 2 4 0 0 0]
u13	½ [2 2 6 1 1 1]
u14	½ [4 4 4 2 2 2]
u15	½ [4 4 4 2 0 0]
u16	½ [2 6 6 0 0 2]
u17	½ [4 4 4 0 0 0]
u18	½ [8 8 8 4 4 4]

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EIGENFORM BASIS FROM PULLBACK-GENUS BASIS

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The M₁₆(Γ₃) eigenform basis f₁,...,f₇ comprises algebraic integer linear combinations of the rational near-integer non-eigenform basis g₁,...,g₇ from above. Specifically f_i = Σ_j=1⁷ b_i,jg_j for each i, where the matrix (b_i,j) is as follows.

	g1	g2	g3	g4	g5	g6	g7
f1	1	0	0	0	0	0	0
f2	-16320	3617	0	0	0	0	0
f3	618231171506613408000+227210488606944000*a	-267246255594540097273-19043390810061389*a	67070371115332644380+1085208873544246090*a	50325919817262092891/2-(46088807071881137*a)/2	0	0	0
f4	618231171506613408000-227210488606944000*a	-267246255594540097273+19043390810061389*a	67070371115332644380-1085208873544246090*a	50325919817262092891/2+(46088807071881137*a)/2	0	0	0
f5	-232305640344576000-414865732608000*b	138253406219347982+171744054097106*b	-212246003659101076-1521538998472108*b	-21025941792756411+12242688184587*b	-679069732658152388-5454091892244604*b	43773310206109883+324999624915189*b	1065161531980287/2-(12704894680079*b)/2
f6	-232305640344576000+414865732608000*b	138253406219347982-171744054097106*b	-212246003659101076+1521538998472108*b	-21025941792756411-12242688184587*b	-679069732658152388+5454091892244604*b	43773310206109883-324999624915189*b	1065161531980287/2+(12704894680079*b)/2
f7	0	9548	117212	-5630	-177684	-16796	663

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EIGENVALUES

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The T(2) and T_i(4) (i=0,1,2,3) eigenvalues of the eigenforms are shown in the following table.

f	λ2(f)	λ0,4(f)	λ1,4(f)	λ2,4(f)	λ3,4(f)
f1	4398986092545	19344879779699286335320065	2066129936861886443520	31525678545371136	68719476736
f2	28996337880	-44886989155960359360	295157308626991595520	13510963265273856	68719476736
f3	438096096-786528*a	94188059140405248-214034505836544*a	20644074072834048-158347569659904*a	4504298163535872-4831838208*a	68719476736
f4	438096096+786528*a	94188059140405248+214034505836544*a	20644074072834048+158347569659904*a	4504298163535872+4831838208*a	68719476736
f5	4414176-23328*b	-975014295552-64149387264*b	8293015093248+7092043776*b	-252463546368-10871635968*b	68719476736
f6	4414176+23328*b	-975014295552+64149387264*b	8293015093248-7092043776*b	-252463546368+10871635968*b	68719476736
f7	-115200	-784548495360	-1062815662080	-352724189184	68719476736

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EIGENFORM FOURIER COEFFICIENTS

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The eigenform Fourier coefficients at the indices u_i are as follows (switch to nonfactored denominator form).

u	a(u; f1)	a(u; f2)	a(u; f3)	a(u; f4)	a(u; f5)	a(u; f6)	a(u; f7)
u1	1721*3617*9349*362903*657931*1001259881/(2^15*3^5*5^2*7*11*17*29*31)	0	0	0	0	0	0
u2	1721*9349*362903*657931*1001259881/(2^9*3^4*5*7*11*29*31)	2*3^3*5*7^4*11^2*13*9349*362903*657931/(29*31)	0	0	0	0	0
u3	331*1721*9349*362903*657931*1001259881/(2^9*3^2*5*7*29*31)	2^4*3^6*5*7^4*11^2*13*9349*362903*657931/(29*31)	0	0	0	0	0
u4	47*9349*362903*657931*4241723/(2^8*3^2*5*29)	2^2*3^4*5*7^2*11*13*9349*362903*657931	483157986848218883880234879523200000000-28209223026568228605703142400000000*a	483157986848218883880234879523200000000+28209223026568228605703142400000000*a	0	0	0
u5	419*9349*16519*362903*657931/(2^6*3^4*5*29)	2^3*3^3*5*7^2*11*13*9349*362903*657931/(29)	24905544975348089210819557696800000000+247552802513650699141158482400000000*a	24905544975348089210819557696800000000-247552802513650699141158482400000000*a	0	0	0
u6	47*9349*362903*657931*4241723*536903681/(2^8*3^2*5*29)	2^5*3^4*5*7^2*11*13*9349*362903*440347*657931	18058420893973102154534879275029811200000000-47429390401038375315030866579558400000000*a	18058420893973102154534879275029811200000000+47429390401038375315030866579558400000000*a	0	0	0
u7	11*113*331*419*9349*16519*362903*657931/(2^6*3^2)	2^6*3^6*5^2*7^2*11*13*113*9349*362903*657931	111436770719561850261806085538560000000000+10846211138376513620237348832000000000000*a	111436770719561850261806085538560000000000-10846211138376513620237348832000000000000*a	0	0	0
u8	43*127*9349*362903*657931/(2^6*3*5*29)	2^3*3^5*5*7^2*11*13*41*443*887*657931/(29)	2604502478552571003744671574240000000+21088490054022689088097026720000000*a	2604502478552571003744671574240000000-21088490054022689088097026720000000*a	-12125907806305888678654771200000000-100476340305194890926489600000000*b	-12125907806305888678654771200000000+100476340305194890926489600000000*b	-2^24*3^9*5^7*7^2*11*13*23
u9	547*1093*9349*362903*657931/(2^3*3^2*5*29)	2^8*3^4*5*7^2*11*13*257*571*2017*657931/(29)	296894024926670840078363672759040000000+1513925324309992699860332497920000000*a	296894024926670840078363672759040000000-1513925324309992699860332497920000000*a	170529502411664191984553164800000000+1062933915860219635590758400000000*b	170529502411664191984553164800000000-1062933915860219635590758400000000*b	-2^28*3^9*5^7*7^2*11*13*23
u10	43*113*127*9349*362903*657931/(2^6*3)	2^4*3^5*5^3*7^2*11*13*657931*91935989	2^14*3^11*5^8*7^2*11^2*13*23*37*97*1889*657931*883331	2^14*3^11*5^8*7^2*11^2*13*23*37*97*1889*657931*883331	2^33*3^13*5^10*7^4*11^2*13^2*19*23*373	2^33*3^13*5^10*7^4*11^2*13^2*19*23*373	2^27*3^9*5^8*7^2*11*13*23
u11	113*547*1093*9349*362903*657931/(2^3*3^2)	2^6*3^4*5^3*7^2*11*13*4451*657931*3035803	791864423033672419749093215869228800000000-233117594182986961295991035692800000000*a	791864423033672419749093215869228800000000+233117594182986961295991035692800000000*a	-11020879592086229499286388736000000000-152935566401380855031267328000000000*b	-11020879592086229499286388736000000000+152935566401380855031267328000000000*b	2^27*3^9*5^8*7^2*11*13*23*59
u12	7^2*43*127*337*5419*9349*362903*657931/(2^6*5*29)	2^4*3^7*5*7^2*11*13*657931*1216451*66274237/(29)	43223205363538648899808599867851466240000000-52976653635889599832969771898880000000*a	43223205363538648899808599867851466240000000+52976653635889599832969771898880000000*a	242261105918246462348982013132800000000+1150380061296361883820254822400000000*b	242261105918246462348982013132800000000-1150380061296361883820254822400000000*b	-2^29*3^11*5^7*7^2*11*13*23*97
u13	17*43*127*9349*362903*657931*15790321/(2^6*3*5*29)	2^3*3^5*5*7^2*11*13*657931*10968211*196499587/(29)	142330277070126411744442458520122576480000000+1415404302474581345353841554660359840000000*a	142330277070126411744442458520122576480000000-1415404302474581345353841554660359840000000*a	98228124020353346971283069337600000000+965165148514827891198708940800000000*b	98228124020353346971283069337600000000-965165148514827891198708940800000000*b	2^29*3^9*5^7*7^2*11*13*23*659
u14	5^2*29^2*43*113^3*127*9349*362903*657931/(2^6*3)	2^6*3^5*5^2*7^3*11^2*13*17*83*97*657931*38123297	224439289510383052344645386210015385600000000+7190271037751172618483818431295846400000000*a	224439289510383052344645386210015385600000000-7190271037751172618483818431295846400000000*a	-19613800668394383160677438062592000000000-160647072637520191754669654016000000000*b	-19613800668394383160677438062592000000000+160647072637520191754669654016000000000*b	-2^33*3^9*5^8*7^2*11*13*23*83
u15	7^2*113*337*547*1093*5419*9349*362903*657931/(2^3*3)	2^11*3^5*5^2*7^3*11^2*13*9323*657931*3411427949	49463705768742128489150340773605212364800000000+435458408572858220652966465486044160000000000*a	49463705768742128489150340773605212364800000000-435458408572858220652966465486044160000000000*a	572083781107911581064223631867904000000000+4472409628597234492786241175552000000000*b	572083781107911581064223631867904000000000-4472409628597234492786241175552000000000*b	-2^37*3^10*5^8*7^2*11*13*23*457
u16	43*127*9349*362903*657931*72061992889761793/(2^6*3*5*29)	2^4*3^5*5*7^2*11*13*79*1553*1789*657931*15150538260997	-3936024893458032280711494867527033044039680000000+13566021676279545626164986029506781890560000000*a	-3936024893458032280711494867527033044039680000000-13566021676279545626164986029506781890560000000*a	-3114087563267192904255296442571161600000000-18268308838255166832865625033932800000000*b	-3114087563267192904255296442571161600000000+18268308838255166832865625033932800000000*b	2^33*3^9*5^7*7^2*11*13*19*23*4639
u17	43*97*127*9349*362903*657931*742997830099873/(2^6*3*5*29)	2^7*3^5*5*7^2*11*13*3313*657931*72248614593015827/(29)	2406722333756897075236081698109007547555840000000-6230119875909471403207087066517712199680000000*a	2406722333756897075236081698109007547555840000000+6230119875909471403207087066517712199680000000*a	398360677098702239144879263108300800000000-39754865972261951668032145824153600000000*b	398360677098702239144879263108300800000000+39754865972261951668032145824153600000000*b	2^36*3^9*5^7*7^2*11*13*23*29*6871
u18	13*43*127*9349*362903*657931*18377797*16195425329525581/(2^6*3*29)	2^9*3^5*5^2*7^2*11*13*657931*14760331*847043287624775867/(29)	-112151340325452273263193428510925045680013312000000000+1597781114251498211836657103258324370849792000000000*a	-112151340325452273263193428510925045680013312000000000-1597781114251498211836657103258324370849792000000000*a	34508362946076319850206581605561008128000000000+432548397471087937645615221911322624000000000*b	34508362946076319850206581605561008128000000000-432548397471087937645615221911322624000000000*b	-2^42*3^9*5^10*7^2*11*13*23*29*4793

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STANDARD EULER FACTORS

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The standard 2-Euler factors Q₂(f,  x,   st) are given by the following table (switch to expanded form).

f	Q2(f,  x,   st)/(1-x)
f1	(1-32768*x)*(1-16384*x)*(1-8192*x)*(1-x/8192)*(1-x/16384)*(1-x/32768)
f2	(1-16384*x)*(1-8192*x)*(1-x/8192)*(1-x/16384)*(1+(295*x)/512+x^2)
f3	(1-8192*x)*(1-x/8192)*(1+(-135/1024+(3*a)/1024)*x+x^2/2)*(1+(-135/512+(3*a)/512)*x+2*x^2)
f4	(1-8192*x)*(1-x/8192)*(1+(-135/1024-(3*a)/1024)*x+x^2/2)*(1+(-135/512-(3*a)/512)*x+2*x^2)
f5	(1+(1035/4096+(27*b)/4096)*x+x^2/2)*(1+(295*x)/512+x^2)*(1+(1035/2048+(27*b)/2048)*x+2*x^2)
f6	(1+(1035/4096-(27*b)/4096)*x+x^2/2)*(1+(295*x)/512+x^2)*(1+(1035/2048-(27*b)/2048)*x+2*x^2)
f7	1+(1553*x)/1024+(801709*x^2)/524288+(33924479*x^3)/16777216+(801709*x^4)/524288+(1553*x^5)/1024+x^6

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SPINOR EULER FACTORS

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The spinor 2-Euler factors Q₂(f,  x,   spin) are given by the following table (switch to expanded form).

f	Q2(f,  x,  spin)
f1	(1-4398046511104*x)*(1-536870912*x)*(1-268435456*x)*(1-134217728*x)*(1-32768*x)*(1-16384*x)*(1-8192*x)*(1-x)
f2	(1-216*x+32768*x^2)*(1-1769472*x+2199023255552*x^2)*(1-3538944*x+8796093022208*x^2)*(1-28991029248*x+590295810358705651712*x^2)
f3	(1-268435456*x)*(1-134217728*x)*(1-32768*x)*(1-16384*x)*(1+(-4320+96*a)*x+536870912*x^2)*(1+(-35389440+786432*a)*x+36028797018963968*x^2)
f4	(1-268435456*x)*(1-134217728*x)*(1-32768*x)*(1-16384*x)*(1+(-4320-96*a)*x+536870912*x^2)*(1+(-35389440-786432*a)*x+36028797018963968*x^2)
f5	(1-1769472*x+2199023255552*x^2)*(1-3538944*x+8796093022208*x^2)*(1+(894240+23328*b)*x+(4987187888128+29302456320*b)*x^2+(3932909112089640960+102597629011034112*b)*x^3+19342813113834066795298816*x^4)
f6	(1-1769472*x+2199023255552*x^2)*(1-3538944*x+8796093022208*x^2)*(1+(894240-23328*b)*x+(4987187888128-29302456320*b)*x^2+(3932909112089640960-102597629011034112*b)*x^3+19342813113834066795298816*x^4)
f7	1+115200*x-2216899379200*x^2-261738742992076800*x^3-14607292949518362064453632*x^4-1151139165437049900155220787200*x^5-42881070384040361610309278668895027200*x^6+9800132167323027747745188694034924489932800*x^7+374144419156711147060143317175368453031918731001856*x^8

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IDENTIFIED EIGENFORMS

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Based on matching standard and spinor 2-Euler factors we identify the following eigenforms.

f₁ = E 16 : Basic Siegel Eisenstein series

f₂ = K1 16 1 : Klingen lift from degree 1

f₃ = K2 16 1 li : Klingen lift of degree 2 lift

f₄ = K2 16 2 li : Klingen lift of degree 2 lift

f₅ = M1 16 1 1 : Miyawaki lift of type 1

f₆ = M1 16 2 1 : Miyawaki lift of type 1

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UNIMODULARITY AT 2 OF APPARENT NON-LIFT EIGENFORMS

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f₇ is unimodular at 2

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SPECIAL VALUES

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The special values c_i such that W^*_3,3E⁽⁶⁾₁₆ =Σ_i=1⁷ c_i f_i⊗f_i are given in the following table (switch to nonfactored denominator form).

c1	2^15*3^5*5^2*7*11*17*29*31/(1721*3617*9349*362903*657931*1001259881)
c2	29*31/(2*3^3*5*7^4*11^2*13*3617*9349*362903*657931)
c3	(8800517546060419085631287141-10115061619858614245112238*a)/(2^13*3^19*5^15*7^5*11^4*13^2*17*23^2*37^2*97^2*1721*1889^2*51349*657931*883331^2*1001259881)
c4	(8800517546060419085631287141+10115061619858614245112238*a)/(2^13*3^19*5^15*7^5*11^4*13^2*17*23^2*37^2*97^2*1721*1889^2*51349*657931*883331^2*1001259881)
c5	(7990803060358373681-56365611095929823*b)/(2^48*3^13*5^19*7^9*11^2*13^4*19^2*23*107*131*139*373*9349*362903)
c6	(7990803060358373681+56365611095929823*b)/(2^48*3^13*5^19*7^9*11^2*13^4*19^2*23*107*131*139*373*9349*362903)
c7	1/(2^26*3^9*5^8*7^2*11*13^2*17*23*107)

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CONGRUENCES

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From Garrett's formula we can prove pairwise congruences of the eigenforms. We list congruences of cusp forms, omitting the trivial congruences within each Galois orbit and giving only one representative congruence for any pair of Galois orbits.

Eigenforms f₅, f₇ are congruent modulo a prime ideal P lying over 107. The ideal is
  P = <107,104+39y>.
Here y = Sqrt[18209]. We may need to rescale the eigenforms by algebraic integers relatively prime to P to obtain this congruence.

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MORE COEFFICIENTS

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Showing more coefficients for cusp forms | all forms | hide more coefficients

We give the positive definite coefficients of the cusp eigenforms up to determinant 70 (103 coefficients).

2u	det(2u)	a(u; f5)	a(u; f6)	a(u; f7)
[2 2 2 1 1 1]	4	-12125907806305888678654771200000000-100476340305194890926489600000000*b	-12125907806305888678654771200000000+100476340305194890926489600000000*b	-4157776806543360000000
[2 2 2 1 0 0]	6	170529502411664191984553164800000000+1062933915860219635590758400000000*b	170529502411664191984553164800000000-1062933915860219635590758400000000*b	-66524428904693760000000
[2 2 2 0 0 0]	8	1070337435672181364395868160000000000	1070337435672181364395868160000000000	166311072261734400000000
[2 2 4 1 1 1]	10	8872616112934605998469414912000000000+64066887515654531767074816000000000*b	8872616112934605998469414912000000000-64066887515654531767074816000000000*b	-332622144523468800000000
[2 2 4 0 1 1]	12	-11020879592086229499286388736000000000-152935566401380855031267328000000000*b	-11020879592086229499286388736000000000+152935566401380855031267328000000000*b	9812353263442329600000000
[2 2 4 1 0 0]	12	-33066922241292752069524979712000000000-290799681756982476423561216000000000*b	-33066922241292752069524979712000000000+290799681756982476423561216000000000*b	-17337929283285811200000000
[2 2 4 0 1 0]	14	41057604633084291355126844620800000000+71105519565455289652504166400000000*b	41057604633084291355126844620800000000-71105519565455289652504166400000000*b	12905739207510589440000000
[2 2 4 0 0 0]	16	242261105918246462348982013132800000000+1150380061296361883820254822400000000*b	242261105918246462348982013132800000000-1150380061296361883820254822400000000*b	-116151652867595304960000000
[2 2 6 1 1 1]	16	98228124020353346971283069337600000000+965165148514827891198708940800000000*b	98228124020353346971283069337600000000-965165148514827891198708940800000000*b	87679197296386375680000000
[2 2 6 1 0 0]	18	-771404934519474557724688121856000000000-5969722435806597711588753408000000000*b	-771404934519474557724688121856000000000+5969722435806597711588753408000000000*b	-113756773427026329600000000
[2 2 6 0 1 1]	20	-1746335748724554938458982842368000000000-11184127203939985619439058944000000000*b	-1746335748724554938458982842368000000000+11184127203939985619439058944000000000*b	194874998922687283200000000
[2 4 4 1 1 2]	20	509802355198072437724221014016000000000+4506205215834877392435806208000000000*b	509802355198072437724221014016000000000-4506205215834877392435806208000000000*b	-234664922961307238400000000
[2 2 6 0 1 0]	22	863119817273483010895196454912000000000+11969455567993826002441076736000000000*b	863119817273483010895196454912000000000-11969455567993826002441076736000000000*b	307675483684208640000000000
[2 2 8 1 1 1]	22	-277180457563461721467312930816000000000-3024575813466036415313215488000000000*b	-277180457563461721467312930816000000000+3024575813466036415313215488000000000*b	-354907828206541209600000000
[2 2 6 0 0 0]	24	-3474338669008679010967847593574400000000-21204357634698342084467962675200000000*b	-3474338669008679010967847593574400000000+21204357634698342084467962675200000000*b	-1259440488023662264320000000
[2 4 4 0 0 2]	24	-2172893297332920295584989970432000000000-13828928892194570924653019136000000000*b	-2172893297332920295584989970432000000000+13828928892194570924653019136000000000*b	-2403860238471109017600000000
[2 2 8 1 0 0]	24	6952535121815147542657297180262400000000+53789152089379272143163044659200000000*b	6952535121815147542657297180262400000000-53789152089379272143163044659200000000*b	1019952543966764728320000000
[2 4 4 1 1 1]	24	-3646391561962679945195504743219200000000-37440149664847922581579864473600000000*b	-3646391561962679945195504743219200000000+37440149664847922581579864473600000000*b	1294831484200959344640000000
[2 4 4 1 0 1]	26	6110374224918139468305638149324800000000+44043015826745050225061029478400000000*b	6110374224918139468305638149324800000000-44043015826745050225061029478400000000*b	-351980753334734684160000000
[2 2 8 0 1 1]	28	624735654404653738798859943936000000000-9443083755588442780505997312000000000*b	624735654404653738798859943936000000000+9443083755588442780505997312000000000*b	-6500434570422150758400000000
[2 4 4 1 0 0]	28	11287997317726983766849810857984000000000+41608204401499885126362857472000000000*b	11287997317726983766849810857984000000000-41608204401499885126362857472000000000*b	3482221231016194867200000000
[2 2 10 1 1 1]	28	-4458326494564067778787544137728000000000-38605706941758806009450594304000000000*b	-4458326494564067778787544137728000000000+38605706941758806009450594304000000000*b	-2122669793044581580800000000
[2 4 4 0 0 1]	30	-5985715466422113066998345760768000000000-59506372193770477978332954624000000000*b	-5985715466422113066998345760768000000000+59506372193770477978332954624000000000*b	-9860916096542756044800000000
[2 2 8 0 1 0]	30	13717268953508279759659147984896000000000+82347076930222349825388576768000000000*b	13717268953508279759659147984896000000000-82347076930222349825388576768000000000*b	5196223141745629593600000000
[2 2 10 1 0 0]	30	-9188376603090764330093786431488000000000-58821017788320306511802793984000000000*b	-9188376603090764330093786431488000000000+58821017788320306511802793984000000000*b	3031518225186894643200000000
[2 2 8 0 0 0]	32	-27757611541160689275501149159424000000000-252758704730023236856323244032000000000*b	-27757611541160689275501149159424000000000+252758704730023236856323244032000000000*b	6511411101191425228800000000
[2 4 4 0 0 0]	32	30388099898798240228105779150848000000000+75114842837338792731705606144000000000*b	30388099898798240228105779150848000000000-75114842837338792731705606144000000000*b	-7155367572988860825600000000
[4 4 4 2 2 2]	32	-19613800668394383160677438062592000000000-160647072637520191754669654016000000000*b	-19613800668394383160677438062592000000000+160647072637520191754669654016000000000*b	-883444415854333132800000000
[2 2 12 1 1 1]	34	8790862068517778657352356777164800000000+106605798969173000052569171558400000000*b	8790862068517778657352356777164800000000-106605798969173000052569171558400000000*b	15069379733206849290240000000
[2 4 6 1 1 2]	34	3307600382175237929531897334988800000000+77303090611703446683590747750400000000*b	3307600382175237929531897334988800000000-77303090611703446683590747750400000000*b	-10946993922840786370560000000
[2 2 10 0 1 1]	36	14044064448114679856338261927526400000000-22365541546691730976823102668800000000*b	14044064448114679856338261927526400000000+22365541546691730976823102668800000000*b	3114702865755407646720000000
[2 4 6 0 1 2]	36	113147136471575034560502326152396800000000+905686280397158851042886968934400000000*b	113147136471575034560502326152396800000000-905686280397158851042886968934400000000*b	10337185271955491389440000000
[2 2 12 1 0 0]	36	-33525424052974581176395546125926400000000-314696752429575670429653349171200000000*b	-33525424052974581176395546125926400000000+314696752429575670429653349171200000000*b	-24670925397003817451520000000
[4 4 4 2 2 1]	36	-42995331680993509358463036162048000000000-333779018027657673724630401024000000000*b	-42995331680993509358463036162048000000000+333779018027657673724630401024000000000*b	-7782360315415599513600000000
[2 2 10 0 1 0]	38	-13428468301423953506745775030272000000000-58732730814562662895801860096000000000*b	-13428468301423953506745775030272000000000+58732730814562662895801860096000000000*b	-13749601588166629785600000000
[2 4 6 1 1 1]	38	-107368755403500196669045511553024000000000-683707413144574023224655347712000000000*b	-107368755403500196669045511553024000000000+683707413144574023224655347712000000000*b	2838930003507806208000000000
[2 2 10 0 0 0]	40	111704342459773661627794323406848000000000+1003101630147869728079383363584000000000*b	111704342459773661627794323406848000000000-1003101630147869728079383363584000000000*b	-28997665937411486515200000000
[2 4 6 0 0 2]	40	-263522661031720645903990023782400000000000-2132856754422931143173540413440000000000*b	-263522661031720645903990023782400000000000+2132856754422931143173540413440000000000*b	-17369528387015540736000000000
[2 4 6 1 0 1]	40	-48589264142893178793153798340608000000000-195216856518356803616426164224000000000*b	-48589264142893178793153798340608000000000+195216856518356803616426164224000000000*b	28347389644868105011200000000
[2 2 14 1 1 1]	40	83210001162367086118755303948288000000000+533105039570790721448715485184000000000*b	83210001162367086118755303948288000000000-533105039570790721448715485184000000000*b	-27302956111064412979200000000
[2 4 6 0 1 1]	42	-31682406394765944223795186237440000000000+393997873238760725790043668480000000000*b	-31682406394765944223795186237440000000000-393997873238760725790043668480000000000*b	67228257362505419980800000000
[2 2 14 1 0 0]	42	-59708356131808389657853279862784000000000-487153877565309715050069491712000000000*b	-59708356131808389657853279862784000000000+487153877565309715050069491712000000000*b	-17761357273264186982400000000
[2 4 6 1 0 0]	42	254436171985575805112643392372736000000000+2188469721312159690283677646848000000000*b	254436171985575805112643392372736000000000-2188469721312159690283677646848000000000*b	-111337279947762617548800000000
[2 2 12 0 1 1]	44	-152363076117114390304921189613568000000000+158505340120488405921762902016000000000*b	-152363076117114390304921189613568000000000-158505340120488405921762902016000000000*b	95945023898866837094400000000
[2 4 6 0 1 0]	44	-272312975102278400028407267485286400000000-2050099776330091095117336359731200000000*b	-272312975102278400028407267485286400000000+2050099776330091095117336359731200000000*b	94938500973985207418880000000
[4 4 4 2 1 1]	44	-4002915419721681460610205745152000000000+182629815192334441543624556544000000000*b	-4002915419721681460610205745152000000000-182629815192334441543624556544000000000*b	-87031748292071443660800000000
[2 4 6 0 0 1]	46	-300888228579287527836149874897715200000000-1086374379736638527088315963801600000000*b	-300888228579287527836149874897715200000000+1086374379736638527088315963801600000000*b	24138655125783749591040000000
[2 2 12 0 1 0]	46	-321362719309076534725061481018163200000000-2240084755773558281145932159385600000000*b	-321362719309076534725061481018163200000000+2240084755773558281145932159385600000000*b	-35270320863263965839360000000
[2 2 16 1 1 1]	46	-238531821513025935983775689421619200000000-1678764267611968957358474369433600000000*b	-238531821513025935983775689421619200000000+1678764267611968957358474369433600000000*b	24513852904806222397440000000
[2 2 12 0 0 0]	48	366266674917939466147469274906624000000000+2656575823105254269463960748032000000000*b	366266674917939466147469274906624000000000-2656575823105254269463960748032000000000*b	70510572684663010099200000000
[2 4 6 0 0 0]	48	316068729146126096179926751248384000000000+2187742642784340677359810117632000000000*b	316068729146126096179926751248384000000000-2187742642784340677359810117632000000000*b	-6941158911915746918400000000
[2 2 16 1 0 0]	48	802003621404979062274651381039104000000000+6701296380855433797186745270272000000000*b	802003621404979062274651381039104000000000-6701296380855433797186745270272000000000*b	294925411640335033958400000000
[2 4 8 1 1 2]	48	148846335882521161589968223600640000000000-69804615369924871591034880000000000000*b	148846335882521161589968223600640000000000+69804615369924871591034880000000000000*b	17397468647155512115200000000
[2 6 6 1 1 3]	48	500480580018365225231774147149824000000000+2194405810615106233284408901632000000000*b	500480580018365225231774147149824000000000-2194405810615106233284408901632000000000*b	-88223367124826770636800000000
[4 4 4 2 0 0]	48	572083781107911581064223631867904000000000+4472409628597234492786241175552000000000*b	572083781107911581064223631867904000000000-4472409628597234492786241175552000000000*b	-233484779592537971097600000000
[4 4 4 1 1 -1]	50	-1192046234993578007427385660538880000000000-10125618742047366107241554903040000000000*b	-1192046234993578007427385660538880000000000+10125618742047366107241554903040000000000*b	65922382823106281472000000000
[2 2 14 0 1 1]	52	978069483705724037619480869732352000000000+3343792795493262193495869751296000000000*b	978069483705724037619480869732352000000000-3343792795493262193495869751296000000000*b	12567420908762156236800000000
[2 4 8 0 1 2]	52	858161619558078920512859069743104000000000+2258424046429139302262192996352000000000*b	858161619558078920512859069743104000000000-2258424046429139302262192996352000000000*b	202935930284141287833600000000
[2 2 18 1 1 1]	52	-491264141113779201164817472684032000000000-3968420887331505002606232600576000000000*b	-491264141113779201164817472684032000000000+3968420887331505002606232600576000000000*b	-131943596000904098611200000000
[2 4 8 1 1 1]	52	779455032273048464730159222620160000000000+6305731182472480773408659865600000000000*b	779455032273048464730159222620160000000000-6305731182472480773408659865600000000000*b	-58636793680536482611200000000
[2 6 6 0 0 3]	54	-55477418610734812428861739578163200000000-5985794996089578847312274364825600000000*b	-55477418610734812428861739578163200000000+5985794996089578847312274364825600000000*b	-1187440499900252065628160000000
[2 2 14 0 1 0]	54	662501165994125729141992786074009600000000+2742949103332531395110226414796800000000*b	662501165994125729141992786074009600000000-2742949103332531395110226414796800000000*b	-147024442882543394488320000000
[2 2 18 1 0 0]	54	-1428093296679683392611949844575027200000000-11452044663079056430063529597337600000000*b	-1428093296679683392611949844575027200000000+11452044663079056430063529597337600000000*b	-353543478694134845276160000000
[2 4 8 1 0 1]	54	-1481521462783471851720169745704550400000000-10741056146696867123137419096883200000000*b	-1481521462783471851720169745704550400000000+10741056146696867123137419096883200000000*b	433829616633055889326080000000
[4 4 4 1 1 1]	54	1266588911646306560931654613126348800000000+12382662566142646695749173012070400000000*b	1266588911646306560931654613126348800000000-12382662566142646695749173012070400000000*b	-688084187747215013314560000000
[2 2 14 0 0 0]	56	-1250274985506887526469378349885030400000000-10419360783530302785082048590643200000000*b	-1250274985506887526469378349885030400000000+10419360783530302785082048590643200000000*b	316592950330025029140480000000
[2 4 8 0 0 2]	56	1305429746693497352319104097819033600000000+3987337102556661578555152190668800000000*b	1305429746693497352319104097819033600000000-3987337102556661578555152190668800000000*b	521626562168603572961280000000
[2 4 8 1 0 0]	56	-205987773651687246578439672771379200000000-4500886865648584061855532161433600000000*b	-205987773651687246578439672771379200000000+4500886865648584061855532161433600000000*b	-301745229993928619458560000000
[2 6 6 1 1 2]	56	-2536169241649367022108840671418777600000000-18195967565468526388365136547020800000000*b	-2536169241649367022108840671418777600000000+18195967565468526388365136547020800000000*b	-1178174777869119604654080000000
[4 4 4 1 1 0]	56	46452439271074346151037377498316800000000-1121267221492698526102004013465600000000*b	46452439271074346151037377498316800000000+1121267221492698526102004013465600000000*b	-49197210054461604495360000000
[4 4 6 2 2 2]	56	-1194631122587226320647984949231616000000000-12352279520309530375566391246848000000000*b	-1194631122587226320647984949231616000000000+12352279520309530375566391246848000000000*b	701348759724237520896000000000
[2 4 8 0 1 1]	58	-1502274639973671555836960117882880000000000-8181022523368614712876762398720000000000*b	-1502274639973671555836960117882880000000000+8181022523368614712876762398720000000000*b	-884472550903055174860800000000
[2 6 6 1 0 2]	58	981532013596697556122423429431296000000000+10213112983921143921130681663488000000000*b	981532013596697556122423429431296000000000-10213112983921143921130681663488000000000*b	1164634196036571522662400000000
[2 2 20 1 1 1]	58	2534893682827495955380343935598592000000000+19511148671586732387853552582656000000000*b	2534893682827495955380343935598592000000000-19511148671586732387853552582656000000000*b	336037841325580301107200000000
[2 2 16 0 1 1]	60	-708742722317501099363948178702336000000000-382139655555941624762899365888000000000*b	-708742722317501099363948178702336000000000+382139655555941624762899365888000000000*b	-980349553573366962585600000000
[2 4 8 0 1 0]	60	434309714975311987664007116881920000000000-2432417973054526614085891522560000000000*b	434309714975311987664007116881920000000000+2432417973054526614085891522560000000000*b	-680270494425785303040000000000
[2 2 20 1 0 0]	60	762876898313856711623786700472320000000000+4650586543916380181384826716160000000000*b	762876898313856711623786700472320000000000-4650586543916380181384826716160000000000*b	-334923889763571204096000000000
[2 6 6 1 1 1]	60	2250696274342679379393674609688576000000000+12299645253852075887730578423808000000000*b	2250696274342679379393674609688576000000000-12299645253852075887730578423808000000000*b	274423497362805856665600000000
[4 4 4 1 0 0]	60	2644631686094313390181687385653248000000000+11555454759162538335454976016384000000000*b	2644631686094313390181687385653248000000000-11555454759162538335454976016384000000000*b	227783303413261192396800000000
[4 4 6 2 2 1]	60	436281342528911040938921083011072000000000+2694764024411902805124008902656000000000*b	436281342528911040938921083011072000000000-2694764024411902805124008902656000000000*b	196203491767914017587200000000
[2 4 8 0 0 1]	62	2036465528491109632451755099815936000000000+20810152568669415172581787435008000000000*b	2036465528491109632451755099815936000000000-20810152568669415172581787435008000000000*b	262114232815961977651200000000
[2 2 16 0 1 0]	62	-568882621616589933124350690459648000000000+1233141047866357265441054785536000000000*b	-568882621616589933124350690459648000000000-1233141047866357265441054785536000000000*b	1072557401367440365977600000000
[2 4 10 1 1 2]	62	3055016674866068307028582116360192000000000+23670273573494070736705453817856000000000*b	3055016674866068307028582116360192000000000-23670273573494070736705453817856000000000*b	-1106631213852424509849600000000
[2 2 16 0 0 0]	64	-964361118478976588016427734348595200000000-1274189250029280365400599140761600000000*b	-964361118478976588016427734348595200000000+1274189250029280365400599140761600000000*b	-814542137762869999042560000000
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