eigenvalues | | standard Euler factors | | spinor Euler factors | | identified eigenforms | | special values | | congruences |
[a b c d e f] | = |
|
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] |
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 |
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] |
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 |
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 |
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 |
f | Q2(f, x, st)/(1-x) |
f1 | 1-(1879048199*x)/32768+(504403164842164231*x^2)/536870912-(19342813996539597048250369*x^3)/4398046511104+(504403164842164231*x^4)/536870912-(1879048199*x^5)/32768+x^6 |
f2 | 1-(402643747*x)/16384+(18012498724636849*x^2)/134217728+(5310870038725525799*x^3)/68719476736+(18012498724636849*x^4)/134217728-(402643747*x^5)/16384+x^6 |
f3 | 1+(-67112105/8192+(9*a)/1024)*x+(27216136309/8388608-(603992745*a)/8388608)*x^2+(-60100572981815/2147483648+(27216839061*a)/2147483648)*x^3+(27216136309/8388608-(603992745*a)/8388608)*x^4+(-67112105/8192+(9*a)/1024)*x^5+x^6 |
f4 | 1+(-67112105/8192-(9*a)/1024)*x+(27216136309/8388608+(603992745*a)/8388608)*x^2+(-60100572981815/2147483648-(27216839061*a)/2147483648)*x^3+(27216136309/8388608+(603992745*a)/8388608)*x^4+(-67112105/8192-(9*a)/1024)*x^5+x^6 |
f5 | 1+(5465/4096+(81*b)/4096)*x+(23684807/4194304+(75735*b)/4194304)*x^2+(8465101615/2147483648+(93178431*b)/2147483648)*x^3+(23684807/4194304+(75735*b)/4194304)*x^4+(5465/4096+(81*b)/4096)*x^5+x^6 |
f6 | 1+(5465/4096-(81*b)/4096)*x+(23684807/4194304-(75735*b)/4194304)*x^2+(8465101615/2147483648-(93178431*b)/2147483648)*x^3+(23684807/4194304-(75735*b)/4194304)*x^4+(5465/4096-(81*b)/4096)*x^5+x^6 |
f7 | 1+(1553*x)/1024+(801709*x^2)/524288+(33924479*x^3)/16777216+(801709*x^4)/524288+(1553*x^5)/1024+x^6 |
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 |
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 |