DEGREE 3 WEIGHT 16

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DIMENSIONS

dim M₁₆(Γ₃)= 7
dim S₁₆(Γ₃) = 3

DETERMINING INDICES

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]

PULLBACK OF EISENSTEIN SERIES

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_{t_j} 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

EIGENFORM NUMBER FIELD GENERATORS

a = Sqrt[51349]

b = Sqrt[18209]

INDICES FOR COMPUTING HECKE ACTION

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]

EIGENFORM BASIS FROM PULLBACK-GENUS BASIS

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

EIGENVALUES

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)	λ_0,4(f)	λ_1,4(f)	λ_2,4(f)	λ_3,4(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

EIGENFORM FOURIER COEFFICIENTS

The eigenform Fourier coefficients at the indices u_i are as follows (switch to factored 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	13912726954911229324966739363569/234258606489600	0	0	0	0	0	0
u₂	3846482431548584275633602257/14354081280	2276253961922699242954470/899	0	0	0	0	0
u₃	1273185684842581395234722347067/144990720	491670855775303036478165520/899	0	0	0	0	0
u₄	445016619031983484355717/334080	25338634084482737398380	483157986848218883880234879523200000000-28209223026568228605703142400000000*a	483157986848218883880234879523200000000+28209223026568228605703142400000000*a	0	0	0
u₅	15450190691937513287077/751680	16892422722988491598920/29	24905544975348089210819557696800000000+247552802513650699141158482400000000*a	24905544975348089210819557696800000000-247552802513650699141158482400000000*a	0	0	0
u₆	238931060864446589481790370694277/334080	89262332025597759721315502880	18058420893973102154534879275029811200000000-47429390401038375315030866579558400000000*a	18058420893973102154534879275029811200000000+47429390401038375315030866579558400000000*a	0	0	0
u₇	6356718306955926904241951341/576	2061551269113515514732196800	111436770719561850261806085538560000000000+10846211138376513620237348832000000000000*a	111436770719561850261806085538560000000000-10846211138376513620237348832000000000000*a	0	0	0
u₈	12190127397766275077/27840	721921431676109104440/29	2604502478552571003744671574240000000+21088490054022689088097026720000000*a	2604502478552571003744671574240000000-21088490054022689088097026720000000*a	-12125907806305888678654771200000000-100476340305194890926489600000000*b	-12125907806305888678654771200000000+100476340305194890926489600000000*b	-4157776806543360000000
u₉	1334576754702420920447/10440	141475939134761300509440/29	296894024926670840078363672759040000000+1513925324309992699860332497920000000*a	296894024926670840078363672759040000000-1513925324309992699860332497920000000*a	170529502411664191984553164800000000+1062933915860219635590758400000000*b	170529502411664191984553164800000000-1062933915860219635590758400000000*b	-66524428904693760000000
u₁₀	1377484395947589083701/192	205984380083620256118000	7919096816549729321044803690220800000000	7919096816549729321044803690220800000000	1070337435672181364395868160000000000	1070337435672181364395868160000000000	166311072261734400000000
u₁₁	150807173281373564010511/72	40366273010495553569448000	791864423033672419749093215869228800000000-233117594182986961295991035692800000000*a	791864423033672419749093215869228800000000+233117594182986961295991035692800000000*a	-11020879592086229499286388736000000000-152935566401380855031267328000000000*b	-11020879592086229499286388736000000000+152935566401380855031267328000000000*b	9812353263442329600000000
u₁₂	1090820713984965277377688919/9280	65026532102945284130881461840/29	43223205363538648899808599867851466240000000-52976653635889599832969771898880000000*a	43223205363538648899808599867851466240000000+52976653635889599832969771898880000000*a	242261105918246462348982013132800000000+1150380061296361883820254822400000000*b	242261105918246462348982013132800000000-1150380061296361883820254822400000000*b	-116151652867595304960000000
u₁₃	3272262418907610829482205189/27840	96577547038180003026782230680/29	142330277070126411744442458520122576480000000+1415404302474581345353841554660359840000000*a	142330277070126411744442458520122576480000000-1415404302474581345353841554660359840000000*a	98228124020353346971283069337600000000+965165148514827891198708940800000000*b	98228124020353346971283069337600000000-965165148514827891198708940800000000*b	87679197296386375680000000
u₁₄	369810790745246434330583900725/192	720142843396072028244172660800	224439289510383052344645386210015385600000000+7190271037751172618483818431295846400000000*a	224439289510383052344645386210015385600000000-7190271037751172618483818431295846400000000*a	-19613800668394383160677438062592000000000-160647072637520191754669654016000000000*b	-19613800668394383160677438062592000000000+160647072637520191754669654016000000000*b	-883444415854333132800000000
u₁₅	13494821101130248089117673766917/24	140466032027661424127888913868800	49463705768742128489150340773605212364800000000+435458408572858220652966465486044160000000000*a	49463705768742128489150340773605212364800000000-435458408572858220652966465486044160000000000*a	572083781107911581064223631867904000000000+4472409628597234492786241175552000000000*b	572083781107911581064223631867904000000000-4472409628597234492786241175552000000000*b	-233484779592537971097600000000
u₁₆	878444873863123742803518491442733061/27840	298020183573546453789287492422528080	-3936024893458032280711494867527033044039680000000+13566021676279545626164986029506781890560000000*a	-3936024893458032280711494867527033044039680000000-13566021676279545626164986029506781890560000000*a	-3114087563267192904255296442571161600000000-18268308838255166832865625033932800000000*b	-3114087563267192904255296442571161600000000+18268308838255166832865625033932800000000*b	187632950018835606405120000000
u₁₇	878552105902591320830145827774226437/27840	171612798856686535022883868487667840/29	2406722333756897075236081698109007547555840000000-6230119875909471403207087066517712199680000000*a	2406722333756897075236081698109007547555840000000+6230119875909471403207087066517712199680000000*a	398360677098702239144879263108300800000000-39754865972261951668032145824153600000000*b	398360677098702239144879263108300800000000+39754865972261951668032145824153600000000*b	3393431341854815724503040000000
u₁₈	47166907736321820129294279409727089843757057/5568	179279409193883951374180310983246649433600/29	-112151340325452273263193428510925045680013312000000000+1597781114251498211836657103258324370849792000000000*a	-112151340325452273263193428510925045680013312000000000-1597781114251498211836657103258324370849792000000000*a	34508362946076319850206581605561008128000000000+432548397471087937645615221911322624000000000*b	34508362946076319850206581605561008128000000000-432548397471087937645615221911322624000000000*b	-18937233499065791608258560000000000

STANDARD EULER FACTORS

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

SPINOR EULER FACTORS

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

IDENTIFIED EIGENFORMS

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

UNIMODULARITY AT 2 OF APPARENT NON-LIFT EIGENFORMS

f₇ is unimodular at 2

SPECIAL VALUES

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 factored denominator form).

c₁	234258606489600/13912726954911229324966739363569
c₂	899/8233210580274403161766317990
c₃	171386347271814817925009/4418368140321970953739028698976047649078511651403559545543091402145750000000000000-(5057530809929307122556119*a)/113439392818696443251772692331860535366266247393960689552046100204391058375000000000000
c₄	171386347271814817925009/4418368140321970953739028698976047649078511651403559545543091402145750000000000000+(5057530809929307122556119*a)/113439392818696443251772692331860535366266247393960689552046100204391058375000000000000
c₅	438838105352209/1342056163286218925603818464281827305655810129920000000000000000000-(56365611095929823*b)/24437500677278760416319930416107793408686646655713280000000000000000000
c₆	438838105352209/1342056163286218925603818464281827305655810129920000000000000000000+(56365611095929823*b)/24437500677278760416319930416107793408686646655713280000000000000000000
c₇	1/1966378962886616678400000000

CONGRUENCES

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.

MORE COEFFICIENTS

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

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

2u	det(2u)	a(u; f₁)	a(u; f₂)	a(u; f₃)	a(u; f₄)	a(u; f₅)	a(u; f₆)	a(u; f₇)
[2 2 2 1 1 1]	4	12190127397766275077/27840	721921431676109104440/29	2604502478552571003744671574240000000+21088490054022689088097026720000000*a	2604502478552571003744671574240000000-21088490054022689088097026720000000*a	-12125907806305888678654771200000000-100476340305194890926489600000000*b	-12125907806305888678654771200000000+100476340305194890926489600000000*b	-4157776806543360000000
[2 2 2 1 0 0]	6	1334576754702420920447/10440	141475939134761300509440/29	296894024926670840078363672759040000000+1513925324309992699860332497920000000*a	296894024926670840078363672759040000000-1513925324309992699860332497920000000*a	170529502411664191984553164800000000+1062933915860219635590758400000000*b	170529502411664191984553164800000000-1062933915860219635590758400000000*b	-66524428904693760000000
[2 2 2 0 0 0]	8	1377484395947589083701/192	205984380083620256118000	7919096816549729321044803690220800000000	7919096816549729321044803690220800000000	1070337435672181364395868160000000000	1070337435672181364395868160000000000	166311072261734400000000
[2 2 4 1 1 1]	10	19575230692985191397083/120	4652455624367140530420480	200764871528876349392130575420678400000000+1958423427603501652858867183180800000000*a	200764871528876349392130575420678400000000-1958423427603501652858867183180800000000*a	8872616112934605998469414912000000000+64066887515654531767074816000000000*b	8872616112934605998469414912000000000-64066887515654531767074816000000000*b	-332622144523468800000000
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