gp: [N,k,chi] = [960,4,Mod(671,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.671");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [16,0,0,0,80]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the q q q -expansion are expressed in terms of a basis 1 , β 1 , … , β 15 1,\beta_1,\ldots,\beta_{15} 1 , β 1 , … , β 1 5 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 16 + 3472 x 12 + 2676096 x 8 + 573099300 x 4 + 31755240000 x^{16} + 3472x^{12} + 2676096x^{8} + 573099300x^{4} + 31755240000 x 1 6 + 3 4 7 2 x 1 2 + 2 6 7 6 0 9 6 x 8 + 5 7 3 0 9 9 3 0 0 x 4 + 3 1 7 5 5 2 4 0 0 0 0
x^16 + 3472*x^12 + 2676096*x^8 + 573099300*x^4 + 31755240000
:
β 1 \beta_{1} β 1 = = =
( − 9491 ν 14 − 33546752 ν 10 − 27708714936 ν 6 − 8590329053100 ν 2 ) / 14972808163500 ( -9491\nu^{14} - 33546752\nu^{10} - 27708714936\nu^{6} - 8590329053100\nu^{2} ) / 14972808163500 ( − 9 4 9 1 ν 1 4 − 3 3 5 4 6 7 5 2 ν 1 0 − 2 7 7 0 8 7 1 4 9 3 6 ν 6 − 8 5 9 0 3 2 9 0 5 3 1 0 0 ν 2 ) / 1 4 9 7 2 8 0 8 1 6 3 5 0 0
(-9491*v^14 - 33546752*v^10 - 27708714936*v^6 - 8590329053100*v^2) / 14972808163500
β 2 \beta_{2} β 2 = = =
( 4231 ν 12 + 13092190 ν 8 + 6863423634 ν 4 + 625475850885 ) / 13611643785 ( 4231\nu^{12} + 13092190\nu^{8} + 6863423634\nu^{4} + 625475850885 ) / 13611643785 ( 4 2 3 1 ν 1 2 + 1 3 0 9 2 1 9 0 ν 8 + 6 8 6 3 4 2 3 6 3 4 ν 4 + 6 2 5 4 7 5 8 5 0 8 8 5 ) / 1 3 6 1 1 6 4 3 7 8 5
(4231*v^12 + 13092190*v^8 + 6863423634*v^4 + 625475850885) / 13611643785
β 3 \beta_{3} β 3 = = =
( − 28473 ν 15 + 350020 ν 14 − 100640256 ν 11 + 621724840 ν 10 + ⋯ + 179673697962000 ν ) / 179673697962000 ( - 28473 \nu^{15} + 350020 \nu^{14} - 100640256 \nu^{11} + 621724840 \nu^{10} + \cdots + 179673697962000 \nu ) / 179673697962000 ( − 2 8 4 7 3 ν 1 5 + 3 5 0 0 2 0 ν 1 4 − 1 0 0 6 4 0 2 5 6 ν 1 1 + 6 2 1 7 2 4 8 4 0 ν 1 0 + ⋯ + 1 7 9 6 7 3 6 9 7 9 6 2 0 0 0 ν ) / 1 7 9 6 7 3 6 9 7 9 6 2 0 0 0
(-28473*v^15 + 350020*v^14 - 100640256*v^11 + 621724840*v^10 - 83126144808*v^7 - 905609065680*v^6 - 25770987159300*v^3 - 608813123886000*v^2 + 179673697962000*v) / 179673697962000
β 4 \beta_{4} β 4 = = =
( − 28473 ν 15 + 1672642 ν 14 − 100640256 ν 11 + 5623847224 ν 10 + ⋯ + 179673697962000 ν ) / 359347395924000 ( - 28473 \nu^{15} + 1672642 \nu^{14} - 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots + 179673697962000 \nu ) / 359347395924000 ( − 2 8 4 7 3 ν 1 5 + 1 6 7 2 6 4 2 ν 1 4 − 1 0 0 6 4 0 2 5 6 ν 1 1 + 5 6 2 3 8 4 7 2 2 4 ν 1 0 + ⋯ + 1 7 9 6 7 3 6 9 7 9 6 2 0 0 0 ν ) / 3 5 9 3 4 7 3 9 5 9 2 4 0 0 0
(-28473*v^15 + 1672642*v^14 - 100640256*v^11 + 5623847224*v^10 - 83126144808*v^7 + 3828864053232*v^6 - 25770987159300*v^3 + 472992545097000*v^2 + 179673697962000*v) / 359347395924000
β 5 \beta_{5} β 5 = = =
( 9491 ν 15 − 6020080 ν 12 + 33546752 ν 11 − 20452693360 ν 8 + ⋯ − 17 ⋯ 00 ) / 29945616327000 ( 9491 \nu^{15} - 6020080 \nu^{12} + 33546752 \nu^{11} - 20452693360 \nu^{8} + \cdots - 17\!\cdots\!00 ) / 29945616327000 ( 9 4 9 1 ν 1 5 − 6 0 2 0 0 8 0 ν 1 2 + 3 3 5 4 6 7 5 2 ν 1 1 − 2 0 4 5 2 6 9 3 3 6 0 ν 8 + ⋯ − 1 7 ⋯ 0 0 ) / 2 9 9 4 5 6 1 6 3 2 7 0 0 0
(9491*v^15 - 6020080*v^12 + 33546752*v^11 - 20452693360*v^8 + 27708714936*v^7 - 14231099195280*v^4 + 8590329053100*v^3 + 59891232654000*v - 1708827842106000) / 29945616327000
β 6 \beta_{6} β 6 = = =
( − 9491 ν 15 − 3010040 ν 12 − 33546752 ν 11 − 10226346680 ν 8 + ⋯ − 854413921053000 ) / 14972808163500 ( - 9491 \nu^{15} - 3010040 \nu^{12} - 33546752 \nu^{11} - 10226346680 \nu^{8} + \cdots - 854413921053000 ) / 14972808163500 ( − 9 4 9 1 ν 1 5 − 3 0 1 0 0 4 0 ν 1 2 − 3 3 5 4 6 7 5 2 ν 1 1 − 1 0 2 2 6 3 4 6 6 8 0 ν 8 + ⋯ − 8 5 4 4 1 3 9 2 1 0 5 3 0 0 0 ) / 1 4 9 7 2 8 0 8 1 6 3 5 0 0
(-9491*v^15 - 3010040*v^12 - 33546752*v^11 - 10226346680*v^8 - 27708714936*v^7 - 7115549597640*v^4 - 8590329053100*v^3 - 59891232654000*v - 854413921053000) / 14972808163500
β 7 \beta_{7} β 7 = = =
( − 9491 ν 15 + 1739994 ν 14 − 33546752 ν 11 + 5643774168 ν 10 + ⋯ + 59891232654000 ν ) / 119782465308000 ( - 9491 \nu^{15} + 1739994 \nu^{14} - 33546752 \nu^{11} + 5643774168 \nu^{10} + \cdots + 59891232654000 \nu ) / 119782465308000 ( − 9 4 9 1 ν 1 5 + 1 7 3 9 9 9 4 ν 1 4 − 3 3 5 4 6 7 5 2 ν 1 1 + 5 6 4 3 7 7 4 1 6 8 ν 1 0 + ⋯ + 5 9 8 9 1 2 3 2 6 5 4 0 0 0 ν ) / 1 1 9 7 8 2 4 6 5 3 0 8 0 0 0
(-9491*v^15 + 1739994*v^14 - 33546752*v^11 + 5643774168*v^10 - 27708714936*v^7 + 3443420134224*v^6 - 8590329053100*v^3 + 371256740469000*v^2 + 59891232654000*v) / 119782465308000
β 8 \beta_{8} β 8 = = =
( 28473 ν 15 + 1672642 ν 14 + 100640256 ν 11 + 5623847224 ν 10 + ⋯ − 179673697962000 ν ) / 119782465308000 ( 28473 \nu^{15} + 1672642 \nu^{14} + 100640256 \nu^{11} + 5623847224 \nu^{10} + \cdots - 179673697962000 \nu ) / 119782465308000 ( 2 8 4 7 3 ν 1 5 + 1 6 7 2 6 4 2 ν 1 4 + 1 0 0 6 4 0 2 5 6 ν 1 1 + 5 6 2 3 8 4 7 2 2 4 ν 1 0 + ⋯ − 1 7 9 6 7 3 6 9 7 9 6 2 0 0 0 ν ) / 1 1 9 7 8 2 4 6 5 3 0 8 0 0 0
(28473*v^15 + 1672642*v^14 + 100640256*v^11 + 5623847224*v^10 + 83126144808*v^7 + 3828864053232*v^6 + 25770987159300*v^3 + 472992545097000*v^2 - 179673697962000*v) / 119782465308000
β 9 \beta_{9} β 9 = = =
( 836321 ν 15 + 9030120 ν 13 − 1782000 ν 12 + 2811923612 ν 11 + ⋯ − 35 ⋯ 00 ) / 179673697962000 ( 836321 \nu^{15} + 9030120 \nu^{13} - 1782000 \nu^{12} + 2811923612 \nu^{11} + \cdots - 35\!\cdots\!00 ) / 179673697962000 ( 8 3 6 3 2 1 ν 1 5 + 9 0 3 0 1 2 0 ν 1 3 − 1 7 8 2 0 0 0 ν 1 2 + 2 8 1 1 9 2 3 6 1 2 ν 1 1 + ⋯ − 3 5 ⋯ 0 0 ) / 1 7 9 6 7 3 6 9 7 9 6 2 0 0 0
(836321*v^15 + 9030120*v^13 - 1782000*v^12 + 2811923612*v^11 + 30679040040*v^9 - 6929663400*v^8 + 1914432026616*v^7 + 21346648792920*v^5 - 9453130790400*v^4 + 236496272548500*v^3 + 2473404914178000*v - 3587675500530000) / 179673697962000
β 10 \beta_{10} β 1 0 = = =
( − 836321 ν 15 − 9030120 ν 13 + 34338480 ν 12 − 2811923612 ν 11 + ⋯ + 66 ⋯ 00 ) / 179673697962000 ( - 836321 \nu^{15} - 9030120 \nu^{13} + 34338480 \nu^{12} - 2811923612 \nu^{11} + \cdots + 66\!\cdots\!00 ) / 179673697962000 ( − 8 3 6 3 2 1 ν 1 5 − 9 0 3 0 1 2 0 ν 1 3 + 3 4 3 3 8 4 8 0 ν 1 2 − 2 8 1 1 9 2 3 6 1 2 ν 1 1 + ⋯ + 6 6 ⋯ 0 0 ) / 1 7 9 6 7 3 6 9 7 9 6 2 0 0 0
(-836321*v^15 - 9030120*v^13 + 34338480*v^12 - 2811923612*v^11 - 30679040040*v^9 + 115786496760*v^8 - 1914432026616*v^7 - 21346648792920*v^5 + 75933464381280*v^4 - 236496272548500*v^3 - 2473404914178000*v + 6665291552106000) / 179673697962000
β 11 \beta_{11} β 1 1 = = =
( − 8826043 ν 15 − 5359602 ν 14 + 75804960 ν 13 − 30043408096 ν 11 + ⋯ + 31 ⋯ 00 ν ) / 10 ⋯ 00 ( - 8826043 \nu^{15} - 5359602 \nu^{14} + 75804960 \nu^{13} - 30043408096 \nu^{11} + \cdots + 31\!\cdots\!00 \nu ) / 10\!\cdots\!00 ( − 8 8 2 6 0 4 3 ν 1 5 − 5 3 5 9 6 0 2 ν 1 4 + 7 5 8 0 4 9 6 0 ν 1 3 − 3 0 0 4 3 4 0 8 0 9 6 ν 1 1 + ⋯ + 3 1 ⋯ 0 0 ν ) / 1 0 ⋯ 0 0
(-8826043*v^15 - 5359602*v^14 + 75804960*v^13 - 30043408096*v^11 - 18079224744*v^10 + 259291647120*v^9 - 21549917073528*v^7 - 12484105897392*v^6 + 189679451924160*v^5 - 3465446821411500*v^3 - 1728229481202600*v^2 + 31454432763534000*v) / 1078042187772000
β 12 \beta_{12} β 1 2 = = =
( − 4455731 ν 15 − 37902480 ν 13 − 83773800 ν 12 − 15172664432 ν 11 + ⋯ − 12 ⋯ 00 ) / 539021093886000 ( - 4455731 \nu^{15} - 37902480 \nu^{13} - 83773800 \nu^{12} - 15172664432 \nu^{11} + \cdots - 12\!\cdots\!00 ) / 539021093886000 ( − 4 4 5 5 7 3 1 ν 1 5 − 3 7 9 0 2 4 8 0 ν 1 3 − 8 3 7 7 3 8 0 0 ν 1 2 − 1 5 1 7 2 6 6 4 4 3 2 ν 1 1 + ⋯ − 1 2 ⋯ 0 0 ) / 5 3 9 0 2 1 0 9 3 8 8 6 0 0 0
(-4455731*v^15 - 37902480*v^13 - 83773800*v^12 - 15172664432*v^11 - 129645823560*v^9 - 259225362000*v^8 - 10899647753976*v^7 - 94839725962080*v^5 - 135895787953200*v^4 - 1771379891444700*v^3 - 15996726928710000*v - 12114911300580000) / 539021093886000
β 13 \beta_{13} β 1 3 = = =
( − 1701115 ν 15 − 56946 ν 14 + 18060240 ν 13 − 5724487480 ν 11 + ⋯ + 51 ⋯ 00 ν ) / 179673697962000 ( - 1701115 \nu^{15} - 56946 \nu^{14} + 18060240 \nu^{13} - 5724487480 \nu^{11} + \cdots + 51\!\cdots\!00 \nu ) / 179673697962000 ( − 1 7 0 1 1 1 5 ν 1 5 − 5 6 9 4 6 ν 1 4 + 1 8 0 6 0 2 4 0 ν 1 3 − 5 7 2 4 4 8 7 4 8 0 ν 1 1 + ⋯ + 5 1 ⋯ 0 0 ν ) / 1 7 9 6 7 3 6 9 7 9 6 2 0 0 0
(-1701115*v^15 - 56946*v^14 + 18060240*v^13 - 5724487480*v^11 - 201280512*v^10 + 61358080080*v^9 - 3911990198040*v^7 - 166252289616*v^6 + 42693297585840*v^5 - 498763532256300*v^3 - 51541974318600*v^2 + 5126483526318000*v) / 179673697962000
β 14 \beta_{14} β 1 4 = = =
( − 6244177 ν 15 − 85255500 ν 13 − 20426779144 ν 11 − 276078217800 ν 9 + ⋯ − 15 ⋯ 00 ν ) / 539021093886000 ( - 6244177 \nu^{15} - 85255500 \nu^{13} - 20426779144 \nu^{11} - 276078217800 \nu^{9} + \cdots - 15\!\cdots\!00 \nu ) / 539021093886000 ( − 6 2 4 4 1 7 7 ν 1 5 − 8 5 2 5 5 5 0 0 ν 1 3 − 2 0 4 2 6 7 7 9 1 4 4 ν 1 1 − 2 7 6 0 7 8 2 1 7 8 0 0 ν 9 + ⋯ − 1 5 ⋯ 0 0 ν ) / 5 3 9 0 2 1 0 9 3 8 8 6 0 0 0
(-6244177*v^15 - 85255500*v^13 - 20426779144*v^11 - 276078217800*v^9 - 12702558354192*v^7 - 165228007150800*v^5 - 1301547587591700*v^3 - 15555499512498000*v) / 539021093886000
β 15 \beta_{15} β 1 5 = = =
( − 3564376 ν 15 + 85419 ν 14 + 58165140 ν 13 − 11387166772 ν 11 + ⋯ + 70 ⋯ 00 ν ) / 269510546943000 ( - 3564376 \nu^{15} + 85419 \nu^{14} + 58165140 \nu^{13} - 11387166772 \nu^{11} + \cdots + 70\!\cdots\!00 \nu ) / 269510546943000 ( − 3 5 6 4 3 7 6 ν 1 5 + 8 5 4 1 9 ν 1 4 + 5 8 1 6 5 1 4 0 ν 1 3 − 1 1 3 8 7 1 6 6 7 7 2 ν 1 1 + ⋯ + 7 0 ⋯ 0 0 ν ) / 2 6 9 5 1 0 5 4 6 9 4 3 0 0 0
(-3564376*v^15 + 85419*v^14 + 58165140*v^13 - 11387166772*v^11 + 301920768*v^10 + 184041097680*v^9 - 6460505405496*v^7 + 249378434424*v^6 + 101188060772040*v^5 - 437432846990400*v^3 + 77312961477900*v^2 + 7057242582192000*v) / 269510546943000
ν \nu ν = = =
( − 2 β 8 − β 6 + β 5 + 6 β 4 ) / 12 ( -2\beta_{8} - \beta_{6} + \beta_{5} + 6\beta_{4} ) / 12 ( − 2 β 8 − β 6 + β 5 + 6 β 4 ) / 1 2
(-2*b8 - b6 + b5 + 6*b4) / 12
ν 2 \nu^{2} ν 2 = = =
( 2 β 8 − 6 β 7 + 6 β 3 − 75 β 1 ) / 12 ( 2\beta_{8} - 6\beta_{7} + 6\beta_{3} - 75\beta_1 ) / 12 ( 2 β 8 − 6 β 7 + 6 β 3 − 7 5 β 1 ) / 1 2
(2*b8 - 6*b7 + 6*b3 - 75*b1) / 12
ν 3 \nu^{3} ν 3 = = =
( − 12 β 13 + 18 β 12 + 18 β 11 − 12 β 10 + 12 β 9 + 74 β 8 + ⋯ − 9 ) / 12 ( - 12 \beta_{13} + 18 \beta_{12} + 18 \beta_{11} - 12 \beta_{10} + 12 \beta_{9} + 74 \beta_{8} + \cdots - 9 ) / 12 ( − 1 2 β 1 3 + 1 8 β 1 2 + 1 8 β 1 1 − 1 2 β 1 0 + 1 2 β 9 + 7 4 β 8 + ⋯ − 9 ) / 1 2
(-12*b13 + 18*b12 + 18*b11 - 12*b10 + 12*b9 + 74*b8 - 43*b6 + 31*b5 - 204*b4 + 9*b2 - 3*b1 - 9) / 12
ν 4 \nu^{4} ν 4 = = =
( − 174 β 10 − 174 β 9 − 43 β 6 − 86 β 5 + 18 β 2 − 5208 ) / 6 ( -174\beta_{10} - 174\beta_{9} - 43\beta_{6} - 86\beta_{5} + 18\beta_{2} - 5208 ) / 6 ( − 1 7 4 β 1 0 − 1 7 4 β 9 − 4 3 β 6 − 8 6 β 5 + 1 8 β 2 − 5 2 0 8 ) / 6
(-174*b10 - 174*b9 - 43*b6 - 86*b5 + 18*b2 - 5208) / 6
ν 5 \nu^{5} ν 5 = = =
( 36 β 15 − 72 β 14 − 858 β 13 − 1116 β 12 + 1116 β 11 + 894 β 10 + ⋯ + 558 ) / 12 ( 36 \beta_{15} - 72 \beta_{14} - 858 \beta_{13} - 1116 \beta_{12} + 1116 \beta_{11} + 894 \beta_{10} + \cdots + 558 ) / 12 ( 3 6 β 1 5 − 7 2 β 1 4 − 8 5 8 β 1 3 − 1 1 1 6 β 1 2 + 1 1 1 6 β 1 1 + 8 9 4 β 1 0 + ⋯ + 5 5 8 ) / 1 2
(36*b15 - 72*b14 - 858*b13 - 1116*b12 + 1116*b11 + 894*b10 - 894*b9 + 3194*b8 + 2062*b6 - 1168*b5 - 8466*b4 - 558*b2 - 111*b1 + 558) / 12
ν 6 \nu^{6} ν 6 = = =
( − 2633 β 8 + 6009 β 7 − 5220 β 4 − 4344 β 3 + 28020 β 1 ) / 3 ( -2633\beta_{8} + 6009\beta_{7} - 5220\beta_{4} - 4344\beta_{3} + 28020\beta_1 ) / 3 ( − 2 6 3 3 β 8 + 6 0 0 9 β 7 − 5 2 2 0 β 4 − 4 3 4 4 β 3 + 2 8 0 2 0 β 1 ) / 3
(-2633*b8 + 6009*b7 - 5220*b4 - 4344*b3 + 28020*b1) / 3
ν 7 \nu^{7} ν 7 = = =
( − 1665 β 15 − 3330 β 14 + 24261 β 13 − 29394 β 12 − 29394 β 11 + ⋯ + 14697 ) / 6 ( - 1665 \beta_{15} - 3330 \beta_{14} + 24261 \beta_{13} - 29394 \beta_{12} - 29394 \beta_{11} + \cdots + 14697 ) / 6 ( − 1 6 6 5 β 1 5 − 3 3 3 0 β 1 4 + 2 4 2 6 1 β 1 3 − 2 9 3 9 4 β 1 2 − 2 9 3 9 4 β 1 1 + ⋯ + 1 4 6 9 7 ) / 6
(-1665*b15 - 3330*b14 + 24261*b13 - 29394*b12 - 29394*b11 + 25926*b10 - 25926*b9 - 74257*b8 + 50924*b6 - 24998*b5 + 193377*b4 - 14697*b2 + 1734*b1 + 14697) / 6
ν 8 \nu^{8} ν 8 = = =
( 212937 β 10 + 212937 β 9 + 36209 β 6 + 72418 β 5 − 53874 β 2 + 5026944 ) / 3 ( 212937\beta_{10} + 212937\beta_{9} + 36209\beta_{6} + 72418\beta_{5} - 53874\beta_{2} + 5026944 ) / 3 ( 2 1 2 9 3 7 β 1 0 + 2 1 2 9 3 7 β 9 + 3 6 2 0 9 β 6 + 7 2 4 1 8 β 5 − 5 3 8 7 4 β 2 + 5 0 2 6 9 4 4 ) / 3
(212937*b10 + 212937*b9 + 36209*b6 + 72418*b5 - 53874*b2 + 5026944) / 3
ν 9 \nu^{9} ν 9 = = =
( − 107748 β 15 + 215496 β 14 + 1275864 β 13 + 1493118 β 12 − 1493118 β 11 + ⋯ − 746559 ) / 6 ( - 107748 \beta_{15} + 215496 \beta_{14} + 1275864 \beta_{13} + 1493118 \beta_{12} - 1493118 \beta_{11} + \cdots - 746559 ) / 6 ( − 1 0 7 7 4 8 β 1 5 + 2 1 5 4 9 6 β 1 4 + 1 2 7 5 8 6 4 β 1 3 + 1 4 9 3 1 1 8 β 1 2 − 1 4 9 3 1 1 8 β 1 1 + ⋯ − 7 4 6 5 5 9 ) / 6
(-107748*b15 + 215496*b14 + 1275864*b13 + 1493118*b12 - 1493118*b11 - 1383612*b10 + 1383612*b9 - 3576682*b8 - 2534021*b6 + 1150409*b5 + 9236928*b4 + 746559*b2 + 54753*b1 - 746559) / 6
ν 10 \nu^{10} ν 1 0 = = =
( 7202168 β 8 − 16563174 β 7 + 17223930 β 4 + 10472874 β 3 − 62245920 β 1 ) / 3 ( 7202168\beta_{8} - 16563174\beta_{7} + 17223930\beta_{4} + 10472874\beta_{3} - 62245920\beta_1 ) / 3 ( 7 2 0 2 1 6 8 β 8 − 1 6 5 6 3 1 7 4 β 7 + 1 7 2 2 3 9 3 0 β 4 + 1 0 4 7 2 8 7 4 β 3 − 6 2 2 4 5 9 2 0 β 1 ) / 3
(7202168*b8 - 16563174*b7 + 17223930*b4 + 10472874*b3 - 62245920*b1) / 3
ν 11 \nu^{11} ν 1 1 = = =
( 6090300 β 15 + 12180600 β 14 − 65205726 β 13 + 75017844 β 12 + 75017844 β 11 + ⋯ − 37508922 ) / 6 ( 6090300 \beta_{15} + 12180600 \beta_{14} - 65205726 \beta_{13} + 75017844 \beta_{12} + 75017844 \beta_{11} + \cdots - 37508922 ) / 6 ( 6 0 9 0 3 0 0 β 1 5 + 1 2 1 8 0 6 0 0 β 1 4 − 6 5 2 0 5 7 2 6 β 1 3 + 7 5 0 1 7 8 4 4 β 1 2 + 7 5 0 1 7 8 4 4 β 1 1 + ⋯ − 3 7 5 0 8 9 2 2 ) / 6
(6090300*b15 + 12180600*b14 - 65205726*b13 + 75017844*b12 + 75017844*b11 - 71296026*b10 + 71296026*b9 + 175246942*b8 - 126316634*b6 + 55020608*b5 - 450722982*b4 + 37508922*b2 - 1860909*b1 - 37508922) / 6
ν 12 \nu^{12} ν 1 2 = = =
( − 517772112 β 10 − 517772112 β 9 − 77166509 β 6 − 154333018 β 5 + ⋯ − 11774469009 ) / 3 ( - 517772112 \beta_{10} - 517772112 \beta_{9} - 77166509 \beta_{6} - 154333018 \beta_{5} + \cdots - 11774469009 ) / 3 ( − 5 1 7 7 7 2 1 1 2 β 1 0 − 5 1 7 7 7 2 1 1 2 β 9 − 7 7 1 6 6 5 0 9 β 6 − 1 5 4 3 3 3 0 1 8 β 5 + ⋯ − 1 1 7 7 4 4 6 9 0 0 9 ) / 3
(-517772112*b10 - 517772112*b9 - 77166509*b6 - 154333018*b5 + 161756739*b2 - 11774469009) / 3
ν 13 \nu^{13} ν 1 3 = = =
( 161756739 β 15 − 323513478 β 14 − 1645330287 β 13 − 1876829814 β 12 + ⋯ + 938414907 ) / 3 ( 161756739 \beta_{15} - 323513478 \beta_{14} - 1645330287 \beta_{13} - 1876829814 \beta_{12} + \cdots + 938414907 ) / 3 ( 1 6 1 7 5 6 7 3 9 β 1 5 − 3 2 3 5 1 3 4 7 8 β 1 4 − 1 6 4 5 3 3 0 2 8 7 β 1 3 − 1 8 7 6 8 2 9 8 1 4 β 1 2 + ⋯ + 9 3 8 4 1 4 9 0 7 ) / 3
(161756739*b15 - 323513478*b14 - 1645330287*b13 - 1876829814*b12 + 1876829814*b11 + 1807087026*b10 - 1807087026*b9 + 4330053971*b8 + 3149448868*b6 - 1342361842*b5 - 11113332099*b4 - 938414907*b2 - 34871394*b1 + 938414907) / 3
ν 14 \nu^{14} ν 1 4 = = =
( − 18222259178 β 8 + 42358499214 β 7 − 45639807840 β 4 − 25692734454 β 3 + 150447844695 β 1 ) / 3 ( - 18222259178 \beta_{8} + 42358499214 \beta_{7} - 45639807840 \beta_{4} - 25692734454 \beta_{3} + 150447844695 \beta_1 ) / 3 ( − 1 8 2 2 2 2 5 9 1 7 8 β 8 + 4 2 3 5 8 4 9 9 2 1 4 β 7 − 4 5 6 3 9 8 0 7 8 4 0 β 4 − 2 5 6 9 2 7 3 4 4 5 4 β 3 + 1 5 0 4 4 7 8 4 4 6 9 5 β 1 ) / 3
(-18222259178*b8 + 42358499214*b7 - 45639807840*b4 - 25692734454*b3 + 150447844695*b1) / 3
ν 15 \nu^{15} ν 1 5 = = =
( − 8332882380 β 15 − 16665764760 β 14 + 82538255208 β 13 − 93743968122 β 12 + ⋯ + 46871984061 ) / 3 ( - 8332882380 \beta_{15} - 16665764760 \beta_{14} + 82538255208 \beta_{13} - 93743968122 \beta_{12} + \cdots + 46871984061 ) / 3 ( − 8 3 3 2 8 8 2 3 8 0 β 1 5 − 1 6 6 6 5 7 6 4 7 6 0 β 1 4 + 8 2 5 3 8 2 5 5 2 0 8 β 1 3 − 9 3 7 4 3 9 6 8 1 2 2 β 1 2 + ⋯ + 4 6 8 7 1 9 8 4 0 6 1 ) / 3
(-8332882380*b15 - 16665764760*b14 + 82538255208*b13 - 93743968122*b12 - 93743968122*b11 + 90871137588*b10 - 90871137588*b9 - 214906258526*b8 + 157055139247*b6 - 66184001659*b5 + 550974807456*b4 - 46871984061*b2 + 1436415267*b1 + 46871984061) / 3
Character values
We give the values of χ \chi χ on generators for ( Z / 960 Z ) × \left(\mathbb{Z}/960\mathbb{Z}\right)^\times ( Z / 9 6 0 Z ) × .
n n n
511 511 5 1 1
577 577 5 7 7
641 641 6 4 1
901 901 9 0 1
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
1 1 1
− 1 -1 − 1
− 1 -1 − 1
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
Refresh table
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 4 n e w ( 960 , [ χ ] ) S_{4}^{\mathrm{new}}(960, [\chi]) S 4 n e w ( 9 6 0 , [ χ ] ) :
T 7 8 + 1228 T 7 6 + 355728 T 7 4 + 29741776 T 7 2 + 746382400 T_{7}^{8} + 1228T_{7}^{6} + 355728T_{7}^{4} + 29741776T_{7}^{2} + 746382400 T 7 8 + 1 2 2 8 T 7 6 + 3 5 5 7 2 8 T 7 4 + 2 9 7 4 1 7 7 6 T 7 2 + 7 4 6 3 8 2 4 0 0
T7^8 + 1228*T7^6 + 355728*T7^4 + 29741776*T7^2 + 746382400
T 29 4 − 48 T 29 3 − 37488 T 29 2 + 1914544 T 29 + 39321360 T_{29}^{4} - 48T_{29}^{3} - 37488T_{29}^{2} + 1914544T_{29} + 39321360 T 2 9 4 − 4 8 T 2 9 3 − 3 7 4 8 8 T 2 9 2 + 1 9 1 4 5 4 4 T 2 9 + 3 9 3 2 1 3 6 0
T29^4 - 48*T29^3 - 37488*T29^2 + 1914544*T29 + 39321360
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 16 T^{16} T 1 6
T^16
3 3 3
T 16 + ⋯ + 282429536481 T^{16} + \cdots + 282429536481 T 1 6 + ⋯ + 2 8 2 4 2 9 5 3 6 4 8 1
T^16 + 16*T^14 + 1068*T^12 - 8064*T^10 + 425574*T^8 - 5878656*T^6 + 567578988*T^4 + 6198727824*T^2 + 282429536481
5 5 5
( T − 5 ) 16 (T - 5)^{16} ( T − 5 ) 1 6
(T - 5)^16
7 7 7
( T 8 + 1228 T 6 + ⋯ + 746382400 ) 2 (T^{8} + 1228 T^{6} + \cdots + 746382400)^{2} ( T 8 + 1 2 2 8 T 6 + ⋯ + 7 4 6 3 8 2 4 0 0 ) 2
(T^8 + 1228*T^6 + 355728*T^4 + 29741776*T^2 + 746382400)^2
11 11 1 1
( T 8 + 5328 T 6 + ⋯ + 20666937600 ) 2 (T^{8} + 5328 T^{6} + \cdots + 20666937600)^{2} ( T 8 + 5 3 2 8 T 6 + ⋯ + 2 0 6 6 6 9 3 7 6 0 0 ) 2
(T^8 + 5328*T^6 + 6443040*T^4 + 1295435776*T^2 + 20666937600)^2
13 13 1 3
( T 8 + ⋯ + 5700677875200 ) 2 (T^{8} + \cdots + 5700677875200)^{2} ( T 8 + ⋯ + 5 7 0 0 6 7 7 8 7 5 2 0 0 ) 2
(T^8 + 11100*T^6 + 33417936*T^4 + 26308811840*T^2 + 5700677875200)^2
17 17 1 7
( T 8 + ⋯ + 430656937920000 ) 2 (T^{8} + \cdots + 430656937920000)^{2} ( T 8 + ⋯ + 4 3 0 6 5 6 9 3 7 9 2 0 0 0 0 ) 2
(T^8 + 24972*T^6 + 204866640*T^4 + 615178049600*T^2 + 430656937920000)^2
19 19 1 9
( T 8 + ⋯ + 420208715520000 ) 2 (T^{8} + \cdots + 420208715520000)^{2} ( T 8 + ⋯ + 4 2 0 2 0 8 7 1 5 5 2 0 0 0 0 ) 2
(T^8 - 22452*T^6 + 170524080*T^4 - 491994718400*T^2 + 420208715520000)^2
23 23 2 3
( T 8 + ⋯ + 26 ⋯ 00 ) 2 (T^{8} + \cdots + 26\!\cdots\!00)^{2} ( T 8 + ⋯ + 2 6 ⋯ 0 0 ) 2
(T^8 - 54228*T^6 + 722739600*T^4 - 2534842350000*T^2 + 2630632950000000)^2
29 29 2 9
( T 4 − 48 T 3 + ⋯ + 39321360 ) 4 (T^{4} - 48 T^{3} + \cdots + 39321360)^{4} ( T 4 − 4 8 T 3 + ⋯ + 3 9 3 2 1 3 6 0 ) 4
(T^4 - 48*T^3 - 37488*T^2 + 1914544*T + 39321360)^4
31 31 3 1
( T 8 + ⋯ + 21 ⋯ 00 ) 2 (T^{8} + \cdots + 21\!\cdots\!00)^{2} ( T 8 + ⋯ + 2 1 ⋯ 0 0 ) 2
(T^8 + 115300*T^6 + 3827042304*T^4 + 49051753171200*T^2 + 210462939653760000)^2
37 37 3 7
( T 8 + ⋯ + 35 ⋯ 00 ) 2 (T^{8} + \cdots + 35\!\cdots\!00)^{2} ( T 8 + ⋯ + 3 5 ⋯ 0 0 ) 2
(T^8 + 312204*T^6 + 28084899984*T^4 + 655059569514560*T^2 + 3580109939665036800)^2
41 41 4 1
( T 8 + ⋯ + 11 ⋯ 00 ) 2 (T^{8} + \cdots + 11\!\cdots\!00)^{2} ( T 8 + ⋯ + 1 1 ⋯ 0 0 ) 2
(T^8 + 354048*T^6 + 31346984640*T^4 + 566830807366400*T^2 + 1100247971328000000)^2
43 43 4 3
( T 8 + ⋯ + 769198518000000 ) 2 (T^{8} + \cdots + 769198518000000)^{2} ( T 8 + ⋯ + 7 6 9 1 9 8 5 1 8 0 0 0 0 0 0 ) 2
(T^8 - 136776*T^6 + 4800178944*T^4 - 12264362766000*T^2 + 769198518000000)^2
47 47 4 7
( T 8 + ⋯ + 16 ⋯ 00 ) 2 (T^{8} + \cdots + 16\!\cdots\!00)^{2} ( T 8 + ⋯ + 1 6 ⋯ 0 0 ) 2
(T^8 - 240132*T^6 + 15098390160*T^4 - 318069644807600*T^2 + 1619598725086320000)^2
53 53 5 3
( T 4 − 240 T 3 + ⋯ − 1424598000 ) 4 (T^{4} - 240 T^{3} + \cdots - 1424598000)^{4} ( T 4 − 2 4 0 T 3 + ⋯ − 1 4 2 4 5 9 8 0 0 0 ) 4
(T^4 - 240*T^3 - 125208*T^2 + 34562720*T - 1424598000)^4
59 59 5 9
( T 8 + ⋯ + 71 ⋯ 00 ) 2 (T^{8} + \cdots + 71\!\cdots\!00)^{2} ( T 8 + ⋯ + 7 1 ⋯ 0 0 ) 2
(T^8 + 784416*T^6 + 148492728864*T^4 + 2027858972588800*T^2 + 7113919566126240000)^2
61 61 6 1
( T 8 + ⋯ + 13 ⋯ 00 ) 2 (T^{8} + \cdots + 13\!\cdots\!00)^{2} ( T 8 + ⋯ + 1 3 ⋯ 0 0 ) 2
(T^8 + 996240*T^6 + 163089475200*T^4 + 8813159912160000*T^2 + 130601565388800000000)^2
67 67 6 7
( T 8 + ⋯ + 63 ⋯ 00 ) 2 (T^{8} + \cdots + 63\!\cdots\!00)^{2} ( T 8 + ⋯ + 6 3 ⋯ 0 0 ) 2
(T^8 - 1839864*T^6 + 1008444966144*T^4 - 149273005485656240*T^2 + 6326348137507031356800)^2
71 71 7 1
( T 8 + ⋯ + 14 ⋯ 00 ) 2 (T^{8} + \cdots + 14\!\cdots\!00)^{2} ( T 8 + ⋯ + 1 4 ⋯ 0 0 ) 2
(T^8 - 1526832*T^6 + 578265123840*T^4 - 66391674608921600*T^2 + 1492639422100992000000)^2
73 73 7 3
( T 4 − 196 T 3 + ⋯ − 801702800 ) 4 (T^{4} - 196 T^{3} + \cdots - 801702800)^{4} ( T 4 − 1 9 6 T 3 + ⋯ − 8 0 1 7 0 2 8 0 0 ) 4
(T^4 - 196*T^3 - 784176*T^2 - 80103760*T - 801702800)^4
79 79 7 9
( T 8 + ⋯ + 56 ⋯ 00 ) 2 (T^{8} + \cdots + 56\!\cdots\!00)^{2} ( T 8 + ⋯ + 5 6 ⋯ 0 0 ) 2
(T^8 + 1968580*T^6 + 1179370743744*T^4 + 235521627919360000*T^2 + 5665551824419600000000)^2
83 83 8 3
( T 8 + ⋯ + 19 ⋯ 00 ) 2 (T^{8} + \cdots + 19\!\cdots\!00)^{2} ( T 8 + ⋯ + 1 9 ⋯ 0 0 ) 2
(T^8 + 1240296*T^6 + 493626060000*T^4 + 65687100981250000*T^2 + 1909613525765625000000)^2
89 89 8 9
( T 8 + ⋯ + 16 ⋯ 00 ) 2 (T^{8} + \cdots + 16\!\cdots\!00)^{2} ( T 8 + ⋯ + 1 6 ⋯ 0 0 ) 2
(T^8 + 4551072*T^6 + 5063578767360*T^4 + 183483160220057600*T^2 + 1656555228979200000000)^2
97 97 9 7
( T 4 − 380 T 3 + ⋯ + 191786509040 ) 4 (T^{4} - 380 T^{3} + \cdots + 191786509040)^{4} ( T 4 − 3 8 0 T 3 + ⋯ + 1 9 1 7 8 6 5 0 9 0 4 0 ) 4
(T^4 - 380*T^3 - 1235568*T^2 + 54862928*T + 191786509040)^4
show more
show less