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.
t1½ [0 0 0 0 0 0]
t2½ [2 0 0 0 0 0]
t3½ [2 2 0 1 0 0]
t4½ [2 2 0 0 0 0]
t5½ [2 2 2 1 1 1]
t6½ [2 2 2 1 0 0]
t7½ [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).
t1t2t3t4t5t6t7
t1 13912726954911229324966739363569/2342586064896003846482431548584275633602257/1435408128015450190691937513287077/751680445016619031983484355717/33408012190127397766275077/278401334576754702420920447/104401377484395947589083701/192
t2 3846482431548584275633602257/1435408128028545440207637342132869777/14952168198714032425664085997/7834529479453951814168837/348256885152733794797/29167522879746699791736/8789320031445718918505
t3 15450190691937513287077/751680198714032425664085997/783219200918201873871032/783211752845084675720518/87643995985149579832/29166342461853204298816/8721909311729885244280
t4 445016619031983484355717/3340804529479453951814168837/348211752845084675720518/876856833942269137505217/583701424146518329462/29696358864975982344752/291028680202152781023230
t5 12190127397766275077/27840256885152733794797/29643995985149579832/293701424146518329462/29648262571812073064/291931153261213151361642272118573723560
t6 1334576754702420920447/10440167522879746699791736/87166342461853204298816/87696358864975982344752/29193115326121315136550935271481450279424/29270848912004871421760
t7 1377484395947589083701/19289320031445718918505219093117298852442801028680202152781023230164227211857372356027084891200487142176011128058082662642490600


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.
    u1½ [0 0 0 0 0 0]
    u2½ [2 0 0 0 0 0]
    u3½ [4 0 0 0 0 0]
    u4½ [2 2 0 0 0 0]
    u5½ [2 2 0 1 0 0]
    u6½ [4 4 0 0 0 0]
    u7½ [4 4 0 2 0 0]
    u8½ [2 2 2 1 1 1]
    u9½ [2 2 2 1 0 0]
    u10½ [2 2 2 0 0 0]
    u11½ [2 2 4 0 1 1]
    u12½ [2 2 4 0 0 0]
    u13½ [2 2 6 1 1 1]
    u14½ [4 4 4 2 2 2]
    u15½ [4 4 4 2 0 0]
    u16½ [2 6 6 0 0 2]
    u17½ [4 4 4 0 0 0]
    u18½ [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.
    g1g2g3g4g5g6g7
    f1 1000000
    f2 -16320361700000
    f3 618231171506613408000+227210488606944000*a-267246255594540097273-19043390810061389*a67070371115332644380+1085208873544246090*a50325919817262092891/2-(46088807071881137*a)/2000
    f4 618231171506613408000-227210488606944000*a-267246255594540097273+19043390810061389*a67070371115332644380-1085208873544246090*a50325919817262092891/2+(46088807071881137*a)/2000
    f5 -232305640344576000-414865732608000*b138253406219347982+171744054097106*b-212246003659101076-1521538998472108*b-21025941792756411+12242688184587*b-679069732658152388-5454091892244604*b43773310206109883+324999624915189*b1065161531980287/2-(12704894680079*b)/2
    f6 -232305640344576000+414865732608000*b138253406219347982-171744054097106*b-212246003659101076+1521538998472108*b-21025941792756411-12242688184587*b-679069732658152388+5454091892244604*b43773310206109883-324999624915189*b1065161531980287/2+(12704894680079*b)/2
    f7 09548117212-5630-177684-16796663


    EIGENVALUES (return to top)

    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 43989860925451934487977969928633532006520661299368618864435203152567854537113668719476736
    f2 28996337880-448869891559603593602951573086269915955201351096326527385668719476736
    f3 438096096-786528*a94188059140405248-214034505836544*a20644074072834048-158347569659904*a4504298163535872-4831838208*a68719476736
    f4 438096096+786528*a94188059140405248+214034505836544*a20644074072834048+158347569659904*a4504298163535872+4831838208*a68719476736
    f5 4414176-23328*b-975014295552-64149387264*b8293015093248+7092043776*b-252463546368-10871635968*b68719476736
    f6 4414176+23328*b-975014295552+64149387264*b8293015093248-7092043776*b-252463546368+10871635968*b68719476736
    f7 -115200-784548495360-1062815662080-35272418918468719476736


    EIGENFORM FOURIER COEFFICIENTS
    The eigenform Fourier coefficients at the indices ui are as follows (switch to factored denominator form).
    ua(u; f1)a(u; f2)a(u; f3)a(u; f4)a(u; f5)a(u; f6)a(u; f7)
    u1 13912726954911229324966739363569/234258606489600000000
    u2 3846482431548584275633602257/143540812802276253961922699242954470/89900000
    u3 1273185684842581395234722347067/144990720491670855775303036478165520/89900000
    u4 445016619031983484355717/33408025338634084482737398380483157986848218883880234879523200000000-28209223026568228605703142400000000*a483157986848218883880234879523200000000+28209223026568228605703142400000000*a000
    u5 15450190691937513287077/75168016892422722988491598920/2924905544975348089210819557696800000000+247552802513650699141158482400000000*a24905544975348089210819557696800000000-247552802513650699141158482400000000*a000
    u6 238931060864446589481790370694277/3340808926233202559775972131550288018058420893973102154534879275029811200000000-47429390401038375315030866579558400000000*a18058420893973102154534879275029811200000000+47429390401038375315030866579558400000000*a000
    u7 6356718306955926904241951341/5762061551269113515514732196800111436770719561850261806085538560000000000+10846211138376513620237348832000000000000*a111436770719561850261806085538560000000000-10846211138376513620237348832000000000000*a000
    u8 12190127397766275077/27840721921431676109104440/292604502478552571003744671574240000000+21088490054022689088097026720000000*a2604502478552571003744671574240000000-21088490054022689088097026720000000*a-12125907806305888678654771200000000-100476340305194890926489600000000*b-12125907806305888678654771200000000+100476340305194890926489600000000*b-4157776806543360000000
    u9 1334576754702420920447/10440141475939134761300509440/29296894024926670840078363672759040000000+1513925324309992699860332497920000000*a296894024926670840078363672759040000000-1513925324309992699860332497920000000*a170529502411664191984553164800000000+1062933915860219635590758400000000*b170529502411664191984553164800000000-1062933915860219635590758400000000*b-66524428904693760000000
    u10 1377484395947589083701/1922059843800836202561180007919096816549729321044803690220800000000791909681654972932104480369022080000000010703374356721813643958681600000000001070337435672181364395868160000000000166311072261734400000000
    u11 150807173281373564010511/7240366273010495553569448000791864423033672419749093215869228800000000-233117594182986961295991035692800000000*a791864423033672419749093215869228800000000+233117594182986961295991035692800000000*a-11020879592086229499286388736000000000-152935566401380855031267328000000000*b-11020879592086229499286388736000000000+152935566401380855031267328000000000*b9812353263442329600000000
    u12 1090820713984965277377688919/928065026532102945284130881461840/2943223205363538648899808599867851466240000000-52976653635889599832969771898880000000*a43223205363538648899808599867851466240000000+52976653635889599832969771898880000000*a242261105918246462348982013132800000000+1150380061296361883820254822400000000*b242261105918246462348982013132800000000-1150380061296361883820254822400000000*b-116151652867595304960000000
    u13 3272262418907610829482205189/2784096577547038180003026782230680/29142330277070126411744442458520122576480000000+1415404302474581345353841554660359840000000*a142330277070126411744442458520122576480000000-1415404302474581345353841554660359840000000*a98228124020353346971283069337600000000+965165148514827891198708940800000000*b98228124020353346971283069337600000000-965165148514827891198708940800000000*b87679197296386375680000000
    u14 369810790745246434330583900725/192720142843396072028244172660800224439289510383052344645386210015385600000000+7190271037751172618483818431295846400000000*a224439289510383052344645386210015385600000000-7190271037751172618483818431295846400000000*a-19613800668394383160677438062592000000000-160647072637520191754669654016000000000*b-19613800668394383160677438062592000000000+160647072637520191754669654016000000000*b-883444415854333132800000000
    u15 13494821101130248089117673766917/2414046603202766142412788891386880049463705768742128489150340773605212364800000000+435458408572858220652966465486044160000000000*a49463705768742128489150340773605212364800000000-435458408572858220652966465486044160000000000*a572083781107911581064223631867904000000000+4472409628597234492786241175552000000000*b572083781107911581064223631867904000000000-4472409628597234492786241175552000000000*b-233484779592537971097600000000
    u16 878444873863123742803518491442733061/27840298020183573546453789287492422528080-3936024893458032280711494867527033044039680000000+13566021676279545626164986029506781890560000000*a-3936024893458032280711494867527033044039680000000-13566021676279545626164986029506781890560000000*a-3114087563267192904255296442571161600000000-18268308838255166832865625033932800000000*b-3114087563267192904255296442571161600000000+18268308838255166832865625033932800000000*b187632950018835606405120000000
    u17 878552105902591320830145827774226437/27840171612798856686535022883868487667840/292406722333756897075236081698109007547555840000000-6230119875909471403207087066517712199680000000*a2406722333756897075236081698109007547555840000000+6230119875909471403207087066517712199680000000*a398360677098702239144879263108300800000000-39754865972261951668032145824153600000000*b398360677098702239144879263108300800000000+39754865972261951668032145824153600000000*b3393431341854815724503040000000
    u18 47166907736321820129294279409727089843757057/5568179279409193883951374180310983246649433600/29-112151340325452273263193428510925045680013312000000000+1597781114251498211836657103258324370849792000000000*a-112151340325452273263193428510925045680013312000000000-1597781114251498211836657103258324370849792000000000*a34508362946076319850206581605561008128000000000+432548397471087937645615221911322624000000000*b34508362946076319850206581605561008128000000000-432548397471087937645615221911322624000000000*b-18937233499065791608258560000000000


    STANDARD EULER FACTORS
    The standard 2-Euler factors Q2(f,  x,   st) are given by the following table (switch to expanded form).
    fQ2(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)
    f71+(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).
    fQ2(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)
    f71+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).
    c1234258606489600/13912726954911229324966739363569
    c2899/8233210580274403161766317990
    c3171386347271814817925009/4418368140321970953739028698976047649078511651403559545543091402145750000000000000-(5057530809929307122556119*a)/113439392818696443251772692331860535366266247393960689552046100204391058375000000000000
    c4171386347271814817925009/4418368140321970953739028698976047649078511651403559545543091402145750000000000000+(5057530809929307122556119*a)/113439392818696443251772692331860535366266247393960689552046100204391058375000000000000
    c5438838105352209/1342056163286218925603818464281827305655810129920000000000000000000-(56365611095929823*b)/24437500677278760416319930416107793408686646655713280000000000000000000
    c6438838105352209/1342056163286218925603818464281827305655810129920000000000000000000+(56365611095929823*b)/24437500677278760416319930416107793408686646655713280000000000000000000
    c71/1966378962886616678400000000


    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
    Show more coefficients for all forms | cusp forms