DEGREE 3 WEIGHT 16

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 eigenvalues | standard Euler factors | spinor Euler factors | identified eigenforms | special values | congruences

DIMENSIONS
dim M163)= 7
dim S163) = 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 W3,3 and the pullback W*3,3E(6)16 ∈ M163) ⊗ M163).
The columns of the 7 x 7 matrix [a(ti x tj; W*3,3E(6)16)] are determining truncations of an M163)-basis gt1,...,gt7. Each gt is the cofficient function of e(tr(t,z)) in the Fourier expansion of W*3,3E(6)16(w,z). For convenience let gj = gtj for j=1,...,7. The basis elements g1,...,g7 are not Hecke eigenforms. The matrix [a(ti x tj; 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 Ti(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 M163) eigenform basis f1,...,f7 comprises algebraic integer linear combinations of the rational near-integer non-eigenform basis g1,...,g7 from above. Specifically fi = Σj=17 bi,jgj for each i, where the matrix (bi,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 Ti(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

EIGENFORM FOURIER COEFFICIENTS
The eigenform Fourier coefficients at the indices ui 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 (1-32768*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

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 (1-4398046511104*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

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 ci such that W*3,3E(6)16i=17 ci fi⊗fi 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