gp: [N,k,chi] = [3276,2,Mod(1945,3276)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3276, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3276.1945");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [16,0,0,0,0,0,-2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 + 36 x 14 + 472 x 12 + 2912 x 10 + 8914 x 8 + 13164 x 6 + 8828 x 4 + 2648 x 2 + 289 x^{16} + 36x^{14} + 472x^{12} + 2912x^{10} + 8914x^{8} + 13164x^{6} + 8828x^{4} + 2648x^{2} + 289 x 1 6 + 3 6 x 1 4 + 4 7 2 x 1 2 + 2 9 1 2 x 1 0 + 8 9 1 4 x 8 + 1 3 1 6 4 x 6 + 8 8 2 8 x 4 + 2 6 4 8 x 2 + 2 8 9
x^16 + 36*x^14 + 472*x^12 + 2912*x^10 + 8914*x^8 + 13164*x^6 + 8828*x^4 + 2648*x^2 + 289
:
β 1 \beta_{1} β 1 = = =
( 590 ν 14 + 20965 ν 12 + 268794 ν 10 + 1595550 ν 8 + 4545302 ν 6 + 5780895 ν 4 + ⋯ + 433464 ) / 14924 ( 590 \nu^{14} + 20965 \nu^{12} + 268794 \nu^{10} + 1595550 \nu^{8} + 4545302 \nu^{6} + 5780895 \nu^{4} + \cdots + 433464 ) / 14924 ( 5 9 0 ν 1 4 + 2 0 9 6 5 ν 1 2 + 2 6 8 7 9 4 ν 1 0 + 1 5 9 5 5 5 0 ν 8 + 4 5 4 5 3 0 2 ν 6 + 5 7 8 0 8 9 5 ν 4 + ⋯ + 4 3 3 4 6 4 ) / 1 4 9 2 4
(590*v^14 + 20965*v^12 + 268794*v^10 + 1595550*v^8 + 4545302*v^6 + 5780895*v^4 + 2750276*v^2 + 433464) / 14924
β 2 \beta_{2} β 2 = = =
( 590 ν 14 + 20965 ν 12 + 268794 ν 10 + 1595550 ν 8 + 4545302 ν 6 + 5780895 ν 4 + ⋯ + 396154 ) / 7462 ( 590 \nu^{14} + 20965 \nu^{12} + 268794 \nu^{10} + 1595550 \nu^{8} + 4545302 \nu^{6} + 5780895 \nu^{4} + \cdots + 396154 ) / 7462 ( 5 9 0 ν 1 4 + 2 0 9 6 5 ν 1 2 + 2 6 8 7 9 4 ν 1 0 + 1 5 9 5 5 5 0 ν 8 + 4 5 4 5 3 0 2 ν 6 + 5 7 8 0 8 9 5 ν 4 + ⋯ + 3 9 6 1 5 4 ) / 7 4 6 2
(590*v^14 + 20965*v^12 + 268794*v^10 + 1595550*v^8 + 4545302*v^6 + 5780895*v^4 + 2742814*v^2 + 396154) / 7462
β 3 \beta_{3} β 3 = = =
( − 2102 ν 14 − 74891 ν 12 − 964252 ν 10 − 5760993 ν 8 − 16583714 ν 6 + ⋯ − 1685303 ) / 14924 ( - 2102 \nu^{14} - 74891 \nu^{12} - 964252 \nu^{10} - 5760993 \nu^{8} - 16583714 \nu^{6} + \cdots - 1685303 ) / 14924 ( − 2 1 0 2 ν 1 4 − 7 4 8 9 1 ν 1 2 − 9 6 4 2 5 2 ν 1 0 − 5 7 6 0 9 9 3 ν 8 − 1 6 5 8 3 7 1 4 ν 6 + ⋯ − 1 6 8 5 3 0 3 ) / 1 4 9 2 4
(-2102*v^14 - 74891*v^12 - 964252*v^10 - 5760993*v^8 - 16583714*v^6 - 21486785*v^4 - 10614108*v^2 - 1685303) / 14924
β 4 \beta_{4} β 4 = = =
( − 33003 ν 15 + 23477 ν 14 − 1151473 ν 13 + 844135 ν 12 − 14284923 ν 11 + ⋯ + 40958865 ) / 2029664 ( - 33003 \nu^{15} + 23477 \nu^{14} - 1151473 \nu^{13} + 844135 \nu^{12} - 14284923 \nu^{11} + \cdots + 40958865 ) / 2029664 ( − 3 3 0 0 3 ν 1 5 + 2 3 4 7 7 ν 1 4 − 1 1 5 1 4 7 3 ν 1 3 + 8 4 4 1 3 5 ν 1 2 − 1 4 2 8 4 9 2 3 ν 1 1 + ⋯ + 4 0 9 5 8 8 6 5 ) / 2 0 2 9 6 6 4
(-33003*v^15 + 23477*v^14 - 1151473*v^13 + 844135*v^12 - 14284923*v^11 + 11049133*v^10 - 79742797*v^9 + 68039219*v^8 - 199279357*v^7 + 207895091*v^6 - 176254047*v^5 + 306043129*v^4 + 4632487*v^3 + 198119343*v^2 + 20626753*v + 40958865) / 2029664
β 5 \beta_{5} β 5 = = =
( 33003 ν 15 − 262395 ν 14 + 1151473 ν 13 − 9341041 ν 12 + 14284923 ν 11 + ⋯ − 188242343 ) / 2029664 ( 33003 \nu^{15} - 262395 \nu^{14} + 1151473 \nu^{13} - 9341041 \nu^{12} + 14284923 \nu^{11} + \cdots - 188242343 ) / 2029664 ( 3 3 0 0 3 ν 1 5 − 2 6 2 3 9 5 ν 1 4 + 1 1 5 1 4 7 3 ν 1 3 − 9 3 4 1 0 4 1 ν 1 2 + 1 4 2 8 4 9 2 3 ν 1 1 + ⋯ − 1 8 8 2 4 2 3 4 3 ) / 2 0 2 9 6 6 4
(33003*v^15 - 262395*v^14 + 1151473*v^13 - 9341041*v^12 + 14284923*v^11 - 120089139*v^10 + 79742797*v^9 - 715455829*v^8 + 199279357*v^7 - 2047490013*v^6 + 176254047*v^5 - 2616159631*v^4 - 4632487*v^3 - 1245399345*v^2 - 20626753*v - 188242343) / 2029664
β 6 \beta_{6} β 6 = = =
( 39901 ν 15 + 53805 ν 14 + 1427545 ν 13 + 1919657 ν 12 + 18511581 ν 11 + ⋯ + 48177439 ) / 1014832 ( 39901 \nu^{15} + 53805 \nu^{14} + 1427545 \nu^{13} + 1919657 \nu^{12} + 18511581 \nu^{11} + \cdots + 48177439 ) / 1014832 ( 3 9 9 0 1 ν 1 5 + 5 3 8 0 5 ν 1 4 + 1 4 2 7 5 4 5 ν 1 3 + 1 9 1 9 6 5 7 ν 1 2 + 1 8 5 1 1 5 8 1 ν 1 1 + ⋯ + 4 8 1 7 7 4 3 9 ) / 1 0 1 4 8 3 2
(39901*v^15 + 53805*v^14 + 1427545*v^13 + 1919657*v^12 + 18511581*v^11 + 24778333*v^10 + 111956893*v^9 + 148703505*v^8 + 329628295*v^7 + 431561847*v^6 + 447287131*v^5 + 567705259*v^4 + 245051627*v^3 + 288834675*v^2 + 45755883*v + 48177439) / 1014832
β 7 \beta_{7} β 7 = = =
( 39901 ν 15 − 53805 ν 14 + 1427545 ν 13 − 1919657 ν 12 + 18511581 ν 11 + ⋯ − 48177439 ) / 1014832 ( 39901 \nu^{15} - 53805 \nu^{14} + 1427545 \nu^{13} - 1919657 \nu^{12} + 18511581 \nu^{11} + \cdots - 48177439 ) / 1014832 ( 3 9 9 0 1 ν 1 5 − 5 3 8 0 5 ν 1 4 + 1 4 2 7 5 4 5 ν 1 3 − 1 9 1 9 6 5 7 ν 1 2 + 1 8 5 1 1 5 8 1 ν 1 1 + ⋯ − 4 8 1 7 7 4 3 9 ) / 1 0 1 4 8 3 2
(39901*v^15 - 53805*v^14 + 1427545*v^13 - 1919657*v^12 + 18511581*v^11 - 24778333*v^10 + 111956893*v^9 - 148703505*v^8 + 329628295*v^7 - 431561847*v^6 + 447287131*v^5 - 567705259*v^4 + 245051627*v^3 - 288834675*v^2 + 45755883*v - 48177439) / 1014832
β 8 \beta_{8} β 8 = = =
( 93265 ν 15 + 23477 ν 14 + 3323523 ν 13 + 844135 ν 12 + 42807025 ν 11 + ⋯ + 40958865 ) / 2029664 ( 93265 \nu^{15} + 23477 \nu^{14} + 3323523 \nu^{13} + 844135 \nu^{12} + 42807025 \nu^{11} + \cdots + 40958865 ) / 2029664 ( 9 3 2 6 5 ν 1 5 + 2 3 4 7 7 ν 1 4 + 3 3 2 3 5 2 3 ν 1 3 + 8 4 4 1 3 5 ν 1 2 + 4 2 8 0 7 0 2 5 ν 1 1 + ⋯ + 4 0 9 5 8 8 6 5 ) / 2 0 2 9 6 6 4
(93265*v^15 + 23477*v^14 + 3323523*v^13 + 844135*v^12 + 42807025*v^11 + 11049133*v^10 + 255924679*v^9 + 68039219*v^8 + 737592567*v^7 + 207895091*v^6 + 956800189*v^5 + 306043129*v^4 + 467944667*v^3 + 198119343*v^2 + 66804429*v + 40958865) / 2029664
β 9 \beta_{9} β 9 = = =
( 24620 ν 15 + 876290 ν 13 + 11264235 ν 11 + 67123942 ν 9 + 192338330 ν 7 + ⋯ + 18692776 ν ) / 253708 ( 24620 \nu^{15} + 876290 \nu^{13} + 11264235 \nu^{11} + 67123942 \nu^{9} + 192338330 \nu^{7} + \cdots + 18692776 \nu ) / 253708 ( 2 4 6 2 0 ν 1 5 + 8 7 6 2 9 0 ν 1 3 + 1 1 2 6 4 2 3 5 ν 1 1 + 6 7 1 2 3 9 4 2 ν 9 + 1 9 2 3 3 8 3 3 0 ν 7 + ⋯ + 1 8 6 9 2 7 7 6 ν ) / 2 5 3 7 0 8
(24620*v^15 + 876290*v^13 + 11264235*v^11 + 67123942*v^9 + 192338330*v^7 + 246827546*v^5 + 119070145*v^3 + 18692776*v) / 253708
β 10 \beta_{10} β 1 0 = = =
( 24620 ν 15 + 876290 ν 13 + 11264235 ν 11 + 67123942 ν 9 + 192338330 ν 7 + ⋯ + 18439068 ν ) / 253708 ( 24620 \nu^{15} + 876290 \nu^{13} + 11264235 \nu^{11} + 67123942 \nu^{9} + 192338330 \nu^{7} + \cdots + 18439068 \nu ) / 253708 ( 2 4 6 2 0 ν 1 5 + 8 7 6 2 9 0 ν 1 3 + 1 1 2 6 4 2 3 5 ν 1 1 + 6 7 1 2 3 9 4 2 ν 9 + 1 9 2 3 3 8 3 3 0 ν 7 + ⋯ + 1 8 4 3 9 0 6 8 ν ) / 2 5 3 7 0 8
(24620*v^15 + 876290*v^13 + 11264235*v^11 + 67123942*v^9 + 192338330*v^7 + 246827546*v^5 + 119070145*v^3 + 18439068*v) / 253708
β 11 \beta_{11} β 1 1 = = =
( − 56088 ν 15 − 70312 ν 14 − 2001743 ν 13 − 2497538 ν 12 − 25853206 ν 11 + ⋯ − 43679358 ) / 507416 ( - 56088 \nu^{15} - 70312 \nu^{14} - 2001743 \nu^{13} - 2497538 \nu^{12} - 25853206 \nu^{11} + \cdots - 43679358 ) / 507416 ( − 5 6 0 8 8 ν 1 5 − 7 0 3 1 2 ν 1 4 − 2 0 0 1 7 4 3 ν 1 3 − 2 4 9 7 5 3 8 ν 1 2 − 2 5 8 5 3 2 0 6 ν 1 1 + ⋯ − 4 3 6 7 9 3 5 8 ) / 5 0 7 4 1 6
(-56088*v^15 - 70312*v^14 - 2001743*v^13 - 2497538*v^12 - 25853206*v^11 - 31995360*v^10 - 155342727*v^9 - 189566694*v^8 - 452152838*v^7 - 537437048*v^6 - 599563459*v^5 - 674438450*v^4 - 310933340*v^3 - 308403528*v^2 - 51029543*v - 43679358) / 507416
β 12 \beta_{12} β 1 2 = = =
( − 56088 ν 15 + 70312 ν 14 − 2001743 ν 13 + 2497538 ν 12 − 25853206 ν 11 + ⋯ + 43679358 ) / 507416 ( - 56088 \nu^{15} + 70312 \nu^{14} - 2001743 \nu^{13} + 2497538 \nu^{12} - 25853206 \nu^{11} + \cdots + 43679358 ) / 507416 ( − 5 6 0 8 8 ν 1 5 + 7 0 3 1 2 ν 1 4 − 2 0 0 1 7 4 3 ν 1 3 + 2 4 9 7 5 3 8 ν 1 2 − 2 5 8 5 3 2 0 6 ν 1 1 + ⋯ + 4 3 6 7 9 3 5 8 ) / 5 0 7 4 1 6
(-56088*v^15 + 70312*v^14 - 2001743*v^13 + 2497538*v^12 - 25853206*v^11 + 31995360*v^10 - 155342727*v^9 + 189566694*v^8 - 452152838*v^7 + 537437048*v^6 - 599563459*v^5 + 674438450*v^4 - 310933340*v^3 + 308403528*v^2 - 51029543*v + 43679358) / 507416
β 13 \beta_{13} β 1 3 = = =
( 259235 ν 15 + 268345 ν 14 + 9242377 ν 13 + 9562143 ν 12 + 119142639 ν 11 + ⋯ + 205735581 ) / 2029664 ( 259235 \nu^{15} + 268345 \nu^{14} + 9242377 \nu^{13} + 9562143 \nu^{12} + 119142639 \nu^{11} + \cdots + 205735581 ) / 2029664 ( 2 5 9 2 3 5 ν 1 5 + 2 6 8 3 4 5 ν 1 4 + 9 2 4 2 3 7 7 ν 1 3 + 9 5 6 2 1 4 3 ν 1 2 + 1 1 9 1 4 2 6 3 9 ν 1 1 + ⋯ + 2 0 5 7 3 5 5 8 1 ) / 2 0 2 9 6 6 4
(259235*v^15 + 268345*v^14 + 9242377*v^13 + 9562143*v^12 + 119142639*v^11 + 123143869*v^10 + 713330057*v^9 + 735919783*v^8 + 2060989045*v^7 + 2118410919*v^6 + 2686420031*v^5 + 2739428161*v^4 + 1335890661*v^3 + 1336744935*v^2 + 215558683*v + 205735581) / 2029664
β 14 \beta_{14} β 1 4 = = =
( − 259235 ν 15 − 17527 ν 14 − 9242377 ν 13 − 623033 ν 12 − 119142639 ν 11 + ⋯ − 23465627 ) / 2029664 ( - 259235 \nu^{15} - 17527 \nu^{14} - 9242377 \nu^{13} - 623033 \nu^{12} - 119142639 \nu^{11} + \cdots - 23465627 ) / 2029664 ( − 2 5 9 2 3 5 ν 1 5 − 1 7 5 2 7 ν 1 4 − 9 2 4 2 3 7 7 ν 1 3 − 6 2 3 0 3 3 ν 1 2 − 1 1 9 1 4 2 6 3 9 ν 1 1 + ⋯ − 2 3 4 6 5 6 2 7 ) / 2 0 2 9 6 6 4
(-259235*v^15 - 17527*v^14 - 9242377*v^13 - 623033*v^12 - 119142639*v^11 - 7994403*v^10 - 713330057*v^9 - 47575265*v^8 - 2060989045*v^7 - 136974185*v^6 - 2686420031*v^5 - 182774599*v^4 - 1335890661*v^3 - 106773753*v^2 - 215558683*v - 23465627) / 2029664
β 15 \beta_{15} β 1 5 = = =
( − 39210 ν 15 − 1396175 ν 13 − 17958972 ν 11 − 107123534 ν 9 − 307406526 ν 7 + ⋯ − 29255540 ν ) / 253708 ( - 39210 \nu^{15} - 1396175 \nu^{13} - 17958972 \nu^{11} - 107123534 \nu^{9} - 307406526 \nu^{7} + \cdots - 29255540 \nu ) / 253708 ( − 3 9 2 1 0 ν 1 5 − 1 3 9 6 1 7 5 ν 1 3 − 1 7 9 5 8 9 7 2 ν 1 1 − 1 0 7 1 2 3 5 3 4 ν 9 − 3 0 7 4 0 6 5 2 6 ν 7 + ⋯ − 2 9 2 5 5 5 4 0 ν ) / 2 5 3 7 0 8
(-39210*v^15 - 1396175*v^13 - 17958972*v^11 - 107123534*v^9 - 307406526*v^7 - 395379877*v^5 - 191385598*v^3 - 29255540*v) / 253708
ν \nu ν = = =
− β 10 + β 9 -\beta_{10} + \beta_{9} − β 1 0 + β 9
-b10 + b9
ν 2 \nu^{2} ν 2 = = =
− β 2 + 2 β 1 − 5 -\beta_{2} + 2\beta _1 - 5 − β 2 + 2 β 1 − 5
-b2 + 2*b1 - 5
ν 3 \nu^{3} ν 3 = = =
3 β 15 − β 14 + β 13 + 10 β 10 − 9 β 9 + β 7 + β 6 + β 5 − β 4 3\beta_{15} - \beta_{14} + \beta_{13} + 10\beta_{10} - 9\beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} 3 β 1 5 − β 1 4 + β 1 3 + 1 0 β 1 0 − 9 β 9 + β 7 + β 6 + β 5 − β 4
3*b15 - b14 + b13 + 10*b10 - 9*b9 + b7 + b6 + b5 - b4
ν 4 \nu^{4} ν 4 = = =
− 4 β 14 − 4 β 13 − β 12 + β 11 − 4 β 7 + 4 β 6 − 4 β 5 + ⋯ + 51 - 4 \beta_{14} - 4 \beta_{13} - \beta_{12} + \beta_{11} - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 51 − 4 β 1 4 − 4 β 1 3 − β 1 2 + β 1 1 − 4 β 7 + 4 β 6 − 4 β 5 + ⋯ + 5 1
-4*b14 - 4*b13 - b12 + b11 - 4*b7 + 4*b6 - 4*b5 - 4*b4 + 2*b3 + 16*b2 - 28*b1 + 51
ν 5 \nu^{5} ν 5 = = =
− 60 β 15 + 20 β 14 − 20 β 13 + 5 β 12 + 5 β 11 − 111 β 10 + ⋯ + 2 β 3 - 60 \beta_{15} + 20 \beta_{14} - 20 \beta_{13} + 5 \beta_{12} + 5 \beta_{11} - 111 \beta_{10} + \cdots + 2 \beta_{3} − 6 0 β 1 5 + 2 0 β 1 4 − 2 0 β 1 3 + 5 β 1 2 + 5 β 1 1 − 1 1 1 β 1 0 + ⋯ + 2 β 3
-60*b15 + 20*b14 - 20*b13 + 5*b12 + 5*b11 - 111*b10 + 109*b9 - 8*b8 - 21*b7 - 21*b6 - 22*b5 + 30*b4 + 2*b3
ν 6 \nu^{6} ν 6 = = =
80 β 14 + 80 β 13 + 27 β 12 − 27 β 11 + 86 β 7 − 86 β 6 + 92 β 5 + ⋯ − 653 80 \beta_{14} + 80 \beta_{13} + 27 \beta_{12} - 27 \beta_{11} + 86 \beta_{7} - 86 \beta_{6} + 92 \beta_{5} + \cdots - 653 8 0 β 1 4 + 8 0 β 1 3 + 2 7 β 1 2 − 2 7 β 1 1 + 8 6 β 7 − 8 6 β 6 + 9 2 β 5 + ⋯ − 6 5 3
80*b14 + 80*b13 + 27*b12 - 27*b11 + 86*b7 - 86*b6 + 92*b5 + 92*b4 - 52*b3 - 255*b2 + 390*b1 - 653
ν 7 \nu^{7} ν 7 = = =
1001 β 15 − 335 β 14 + 335 β 13 − 119 β 12 − 119 β 11 + ⋯ − 66 β 3 1001 \beta_{15} - 335 \beta_{14} + 335 \beta_{13} - 119 \beta_{12} - 119 \beta_{11} + \cdots - 66 \beta_{3} 1 0 0 1 β 1 5 − 3 3 5 β 1 4 + 3 3 5 β 1 3 − 1 1 9 β 1 2 − 1 1 9 β 1 1 + ⋯ − 6 6 β 3
1001*b15 - 335*b14 + 335*b13 - 119*b12 - 119*b11 + 1408*b10 - 1499*b9 + 158*b8 + 366*b7 + 366*b6 + 401*b5 - 559*b4 - 66*b3
ν 8 \nu^{8} ν 8 = = =
− 1336 β 14 − 1336 β 13 − 520 β 12 + 520 β 11 − 1472 β 7 + 1472 β 6 + ⋯ + 9269 - 1336 \beta_{14} - 1336 \beta_{13} - 520 \beta_{12} + 520 \beta_{11} - 1472 \beta_{7} + 1472 \beta_{6} + \cdots + 9269 − 1 3 3 6 β 1 4 − 1 3 3 6 β 1 3 − 5 2 0 β 1 2 + 5 2 0 β 1 1 − 1 4 7 2 β 7 + 1 4 7 2 β 6 + ⋯ + 9 2 6 9
-1336*b14 - 1336*b13 - 520*b12 + 520*b11 - 1472*b7 + 1472*b6 - 1640*b5 - 1640*b4 + 956*b3 + 4034*b2 - 5672*b1 + 9269
ν 9 \nu^{9} ν 9 = = =
− 15930 β 15 + 5370 β 14 − 5370 β 13 + 2160 β 12 + 2160 β 11 + ⋯ + 1344 β 3 - 15930 \beta_{15} + 5370 \beta_{14} - 5370 \beta_{13} + 2160 \beta_{12} + 2160 \beta_{11} + \cdots + 1344 \beta_{3} − 1 5 9 3 0 β 1 5 + 5 3 7 0 β 1 4 − 5 3 7 0 β 1 3 + 2 1 6 0 β 1 2 + 2 1 6 0 β 1 1 + ⋯ + 1 3 4 4 β 3
-15930*b15 + 5370*b14 - 5370*b13 + 2160*b12 + 2160*b11 - 19579*b10 + 21969*b9 - 2556*b8 - 5942*b7 - 5942*b6 - 6714*b5 + 9270*b4 + 1344*b3
ν 10 \nu^{10} ν 1 0 = = =
21300 β 14 + 21300 β 13 + 8874 β 12 − 8874 β 11 + 23612 β 7 + ⋯ − 137957 21300 \beta_{14} + 21300 \beta_{13} + 8874 \beta_{12} - 8874 \beta_{11} + 23612 \beta_{7} + \cdots - 137957 2 1 3 0 0 β 1 4 + 2 1 3 0 0 β 1 3 + 8 8 7 4 β 1 2 − 8 8 7 4 β 1 1 + 2 3 6 1 2 β 7 + ⋯ − 1 3 7 9 5 7
21300*b14 + 21300*b13 + 8874*b12 - 8874*b11 + 23612*b7 - 23612*b6 + 26964*b5 + 26964*b4 - 15784*b3 - 63211*b2 + 85018*b1 - 137957
ν 11 \nu^{11} ν 1 1 = = =
249381 β 15 − 84511 β 14 + 84511 β 13 − 35838 β 12 − 35838 β 11 + ⋯ − 23412 β 3 249381 \beta_{15} - 84511 \beta_{14} + 84511 \beta_{13} - 35838 \beta_{12} - 35838 \beta_{11} + \cdots - 23412 \beta_{3} 2 4 9 3 8 1 β 1 5 − 8 4 5 1 1 β 1 4 + 8 4 5 1 1 β 1 3 − 3 5 8 3 8 β 1 2 − 3 5 8 3 8 β 1 1 + ⋯ − 2 3 4 1 2 β 3
249381*b15 - 84511*b14 + 84511*b13 - 35838*b12 - 35838*b11 + 287200*b10 - 331649*b9 + 39596*b8 + 93733*b7 + 93733*b6 + 107923*b5 - 147519*b4 - 23412*b3
ν 12 \nu^{12} ν 1 2 = = =
− 333892 β 14 − 333892 β 13 − 143761 β 12 + 143761 β 11 − 370284 β 7 + ⋯ + 2097467 - 333892 \beta_{14} - 333892 \beta_{13} - 143761 \beta_{12} + 143761 \beta_{11} - 370284 \beta_{7} + \cdots + 2097467 − 3 3 3 8 9 2 β 1 4 − 3 3 3 8 9 2 β 1 3 − 1 4 3 7 6 1 β 1 2 + 1 4 3 7 6 1 β 1 1 − 3 7 0 2 8 4 β 7 + ⋯ + 2 0 9 7 4 6 7
-333892*b14 - 333892*b13 - 143761*b12 + 143761*b11 - 370284*b7 + 370284*b6 - 428980*b5 - 428980*b4 + 250474*b3 + 984342*b2 - 1295316*b1 + 2097467
ν 13 \nu^{13} ν 1 3 = = =
− 3878186 β 15 + 1318234 β 14 − 1318234 β 13 + 572741 β 12 + ⋯ + 382610 β 3 - 3878186 \beta_{15} + 1318234 \beta_{14} - 1318234 \beta_{13} + 572741 \beta_{12} + \cdots + 382610 \beta_{3} − 3 8 7 8 1 8 6 β 1 5 + 1 3 1 8 2 3 4 β 1 4 − 1 3 1 8 2 3 4 β 1 3 + 5 7 2 7 4 1 β 1 2 + ⋯ + 3 8 2 6 1 0 β 3
-3878186*b15 + 1318234*b14 - 1318234*b13 + 572741*b12 + 572741*b11 - 4331289*b10 + 5073945*b9 - 608432*b8 - 1461339*b7 - 1461339*b6 - 1700844*b5 + 2309276*b4 + 382610*b3
ν 14 \nu^{14} ν 1 4 = = =
5196420 β 14 + 5196420 β 13 + 2273585 β 12 − 2273585 β 11 + 5757826 β 7 + ⋯ − 32193153 5196420 \beta_{14} + 5196420 \beta_{13} + 2273585 \beta_{12} - 2273585 \beta_{11} + 5757826 \beta_{7} + \cdots - 32193153 5 1 9 6 4 2 0 β 1 4 + 5 1 9 6 4 2 0 β 1 3 + 2 2 7 3 5 8 5 β 1 2 − 2 2 7 3 5 8 5 β 1 1 + 5 7 5 7 8 2 6 β 7 + ⋯ − 3 2 1 9 3 1 5 3
5196420*b14 + 5196420*b13 + 2273585*b12 - 2273585*b11 + 5757826*b7 - 5757826*b6 + 6724512*b5 + 6724512*b4 - 3913720*b3 - 15276497*b2 + 19894310*b1 - 32193153
ν 15 \nu^{15} ν 1 5 = = =
60135483 β 15 − 20472917 β 14 + 20472917 β 13 − 8998097 β 12 + ⋯ − 6075262 β 3 60135483 \beta_{15} - 20472917 \beta_{14} + 20472917 \beta_{13} - 8998097 \beta_{12} + \cdots - 6075262 \beta_{3} 6 0 1 3 5 4 8 3 β 1 5 − 2 0 4 7 2 9 1 7 β 1 4 + 2 0 4 7 2 9 1 7 β 1 3 − 8 9 9 8 0 9 7 β 1 2 + ⋯ − 6 0 7 5 2 6 2 β 3
60135483*b15 - 20472917*b14 + 20472917*b13 - 8998097*b12 - 8998097*b11 + 66206746*b10 - 78093287*b9 + 9354110*b8 + 22674458*b7 + 22674458*b6 + 26548179*b5 - 35902289*b4 - 6075262*b3
Character values
We give the values of χ \chi χ on generators for ( Z / 3276 Z ) × \left(\mathbb{Z}/3276\mathbb{Z}\right)^\times ( Z / 3 2 7 6 Z ) × .
n n n
1639 1639 1 6 3 9
2017 2017 2 0 1 7
2341 2341 2 3 4 1
2549 2549 2 5 4 9
χ ( n ) \chi(n) χ ( n )
1 1 1
− β 10 -\beta_{10} − β 1 0
− 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)
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This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 2 n e w ( 3276 , [ χ ] ) S_{2}^{\mathrm{new}}(3276, [\chi]) S 2 n e w ( 3 2 7 6 , [ χ ] ) :
T 5 16 + 187 T 5 12 + 6431 T 5 8 + 52853 T 5 4 + 4 T_{5}^{16} + 187T_{5}^{12} + 6431T_{5}^{8} + 52853T_{5}^{4} + 4 T 5 1 6 + 1 8 7 T 5 1 2 + 6 4 3 1 T 5 8 + 5 2 8 5 3 T 5 4 + 4
T5^16 + 187*T5^12 + 6431*T5^8 + 52853*T5^4 + 4
T 19 16 + 1767 T 19 12 + 920035 T 19 8 + 157342549 T 19 4 + 3262922884 T_{19}^{16} + 1767T_{19}^{12} + 920035T_{19}^{8} + 157342549T_{19}^{4} + 3262922884 T 1 9 1 6 + 1 7 6 7 T 1 9 1 2 + 9 2 0 0 3 5 T 1 9 8 + 1 5 7 3 4 2 5 4 9 T 1 9 4 + 3 2 6 2 9 2 2 8 8 4
T19^16 + 1767*T19^12 + 920035*T19^8 + 157342549*T19^4 + 3262922884
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 T^{16} T 1 6
T^16
5 5 5
T 16 + 187 T 12 + ⋯ + 4 T^{16} + 187 T^{12} + \cdots + 4 T 1 6 + 1 8 7 T 1 2 + ⋯ + 4
T^16 + 187*T^12 + 6431*T^8 + 52853*T^4 + 4
7 7 7
T 16 + 2 T 15 + ⋯ + 5764801 T^{16} + 2 T^{15} + \cdots + 5764801 T 1 6 + 2 T 1 5 + ⋯ + 5 7 6 4 8 0 1
T^16 + 2*T^15 + 2*T^14 + 30*T^13 + 68*T^12 + 118*T^11 + 550*T^10 + 1482*T^9 + 3910*T^8 + 10374*T^7 + 26950*T^6 + 40474*T^5 + 163268*T^4 + 504210*T^3 + 235298*T^2 + 1647086*T + 5764801
11 11 1 1
( T 8 + 6 T 7 + ⋯ + 1156 ) 2 (T^{8} + 6 T^{7} + \cdots + 1156)^{2} ( T 8 + 6 T 7 + ⋯ + 1 1 5 6 ) 2
(T^8 + 6*T^7 + 18*T^6 - 46*T^5 + 188*T^4 + 596*T^3 + 1250*T^2 - 1700*T + 1156)^2
13 13 1 3
T 16 + ⋯ + 815730721 T^{16} + \cdots + 815730721 T 1 6 + ⋯ + 8 1 5 7 3 0 7 2 1
T^16 - 30*T^14 + 876*T^12 - 15258*T^10 + 237270*T^8 - 2578602*T^6 + 25019436*T^4 - 144804270*T^2 + 815730721
17 17 1 7
( T 8 − 14 T 6 + 54 T 4 + ⋯ + 8 ) 2 (T^{8} - 14 T^{6} + 54 T^{4} + \cdots + 8)^{2} ( T 8 − 1 4 T 6 + 5 4 T 4 + ⋯ + 8 ) 2
(T^8 - 14*T^6 + 54*T^4 - 66*T^2 + 8)^2
19 19 1 9
T 16 + ⋯ + 3262922884 T^{16} + \cdots + 3262922884 T 1 6 + ⋯ + 3 2 6 2 9 2 2 8 8 4
T^16 + 1767*T^12 + 920035*T^8 + 157342549*T^4 + 3262922884
23 23 2 3
( T 8 + 83 T 6 + ⋯ + 484 ) 2 (T^{8} + 83 T^{6} + \cdots + 484)^{2} ( T 8 + 8 3 T 6 + ⋯ + 4 8 4 ) 2
(T^8 + 83*T^6 + 1215*T^4 + 2365*T^2 + 484)^2
29 29 2 9
( T 4 + 5 T 3 − 3 T 2 + ⋯ − 22 ) 4 (T^{4} + 5 T^{3} - 3 T^{2} + \cdots - 22)^{4} ( T 4 + 5 T 3 − 3 T 2 + ⋯ − 2 2 ) 4
(T^4 + 5*T^3 - 3*T^2 - 33*T - 22)^4
31 31 3 1
T 16 + 1347 T 12 + ⋯ + 334084 T^{16} + 1347 T^{12} + \cdots + 334084 T 1 6 + 1 3 4 7 T 1 2 + ⋯ + 3 3 4 0 8 4
T^16 + 1347*T^12 + 40567*T^8 + 312037*T^4 + 334084
37 37 3 7
( T 8 − 74 T 5 + ⋯ + 784996 ) 2 (T^{8} - 74 T^{5} + \cdots + 784996)^{2} ( T 8 − 7 4 T 5 + ⋯ + 7 8 4 9 9 6 ) 2
(T^8 - 74*T^5 + 3128*T^4 - 5180*T^3 + 2738*T^2 + 65564*T + 784996)^2
41 41 4 1
T 16 + ⋯ + 14178141184 T^{16} + \cdots + 14178141184 T 1 6 + ⋯ + 1 4 1 7 8 1 4 1 1 8 4
T^16 + 10308*T^12 + 26979136*T^8 + 5533018112*T^4 + 14178141184
43 43 4 3
( T 8 + 183 T 6 + ⋯ + 391876 ) 2 (T^{8} + 183 T^{6} + \cdots + 391876)^{2} ( T 8 + 1 8 3 T 6 + ⋯ + 3 9 1 8 7 6 ) 2
(T^8 + 183*T^6 + 9007*T^4 + 109885*T^2 + 391876)^2
47 47 4 7
T 16 + ⋯ + 1488209686084 T^{16} + \cdots + 1488209686084 T 1 6 + ⋯ + 1 4 8 8 2 0 9 6 8 6 0 8 4
T^16 + 35495*T^12 + 313380355*T^8 + 53841772341*T^4 + 1488209686084
53 53 5 3
( T 4 + 7 T 3 + ⋯ + 2074 ) 4 (T^{4} + 7 T^{3} + \cdots + 2074)^{4} ( T 4 + 7 T 3 + ⋯ + 2 0 7 4 ) 4
(T^4 + 7*T^3 - 113*T^2 - 143*T + 2074)^4
59 59 5 9
T 16 + ⋯ + 78261181504 T^{16} + \cdots + 78261181504 T 1 6 + ⋯ + 7 8 2 6 1 1 8 1 5 0 4
T^16 + 21088*T^12 + 129996400*T^8 + 231367138704*T^4 + 78261181504
61 61 6 1
( T 8 + 196 T 6 + ⋯ + 86528 ) 2 (T^{8} + 196 T^{6} + \cdots + 86528)^{2} ( T 8 + 1 9 6 T 6 + ⋯ + 8 6 5 2 8 ) 2
(T^8 + 196*T^6 + 11600*T^4 + 186272*T^2 + 86528)^2
67 67 6 7
( T 8 − 4 T 7 + ⋯ + 633616 ) 2 (T^{8} - 4 T^{7} + \cdots + 633616)^{2} ( T 8 − 4 T 7 + ⋯ + 6 3 3 6 1 6 ) 2
(T^8 - 4*T^7 + 8*T^6 + 264*T^5 + 6152*T^4 - 4560*T^3 + 3872*T^2 + 70048*T + 633616)^2
71 71 7 1
( T 8 − 2 T 7 + ⋯ + 502566724 ) 2 (T^{8} - 2 T^{7} + \cdots + 502566724)^{2} ( T 8 − 2 T 7 + ⋯ + 5 0 2 5 6 6 7 2 4 ) 2
(T^8 - 2*T^7 + 2*T^6 + 582*T^5 + 56288*T^4 + 27664*T^3 + 1458*T^2 + 1210572*T + 502566724)^2
73 73 7 3
T 16 + 1423 T 12 + ⋯ + 521284 T^{16} + 1423 T^{12} + \cdots + 521284 T 1 6 + 1 4 2 3 T 1 2 + ⋯ + 5 2 1 2 8 4
T^16 + 1423*T^12 + 431515*T^8 + 2541285*T^4 + 521284
79 79 7 9
( T 4 − T 3 − 65 T 2 + ⋯ + 724 ) 4 (T^{4} - T^{3} - 65 T^{2} + \cdots + 724)^{4} ( T 4 − T 3 − 6 5 T 2 + ⋯ + 7 2 4 ) 4
(T^4 - T^3 - 65*T^2 + 53*T + 724)^4
83 83 8 3
T 16 + ⋯ + 59341934404 T^{16} + \cdots + 59341934404 T 1 6 + ⋯ + 5 9 3 4 1 9 3 4 4 0 4
T^16 + 87363*T^12 + 57033943*T^8 + 9362370373*T^4 + 59341934404
89 89 8 9
T 16 + ⋯ + 77940681873604 T^{16} + \cdots + 77940681873604 T 1 6 + ⋯ + 7 7 9 4 0 6 8 1 8 7 3 6 0 4
T^16 + 101263*T^12 + 2004422315*T^8 + 9049264966901*T^4 + 77940681873604
97 97 9 7
T 16 + ⋯ + 408528258774724 T^{16} + \cdots + 408528258774724 T 1 6 + ⋯ + 4 0 8 5 2 8 2 5 8 7 7 4 7 2 4
T^16 + 46631*T^12 + 562817267*T^8 + 1435075611973*T^4 + 408528258774724
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