gp: [N,k,chi] = [1512,2,Mod(377,1512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("1512.377");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [16,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == 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 − 24 x 14 + 230 x 12 − 1052 x 10 + 2139 x 8 − 1244 x 6 + 1134 x 4 − 104 x 2 + 169 x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 x 1 6 − 2 4 x 1 4 + 2 3 0 x 1 2 − 1 0 5 2 x 1 0 + 2 1 3 9 x 8 − 1 2 4 4 x 6 + 1 1 3 4 x 4 − 1 0 4 x 2 + 1 6 9
x^16 - 24*x^14 + 230*x^12 - 1052*x^10 + 2139*x^8 - 1244*x^6 + 1134*x^4 - 104*x^2 + 169
:
β 1 \beta_{1} β 1 = = =
( − 305389 ν 14 + 7316240 ν 12 − 69689225 ν 10 + 319235369 ν 8 − 695756145 ν 6 + ⋯ + 237296969 ) / 192947092 ( - 305389 \nu^{14} + 7316240 \nu^{12} - 69689225 \nu^{10} + 319235369 \nu^{8} - 695756145 \nu^{6} + \cdots + 237296969 ) / 192947092 ( − 3 0 5 3 8 9 ν 1 4 + 7 3 1 6 2 4 0 ν 1 2 − 6 9 6 8 9 2 2 5 ν 1 0 + 3 1 9 2 3 5 3 6 9 ν 8 − 6 9 5 7 5 6 1 4 5 ν 6 + ⋯ + 2 3 7 2 9 6 9 6 9 ) / 1 9 2 9 4 7 0 9 2
(-305389*v^14 + 7316240*v^12 - 69689225*v^10 + 319235369*v^8 - 695756145*v^6 + 747498217*v^4 - 1154216584*v^2 + 237296969) / 192947092
β 2 \beta_{2} β 2 = = =
( − 2694 ν 14 + 61469 ν 12 − 552649 ν 10 + 2285409 ν 8 − 3789335 ν 6 + ⋯ − 10061233 ) / 1663337 ( - 2694 \nu^{14} + 61469 \nu^{12} - 552649 \nu^{10} + 2285409 \nu^{8} - 3789335 \nu^{6} + \cdots - 10061233 ) / 1663337 ( − 2 6 9 4 ν 1 4 + 6 1 4 6 9 ν 1 2 − 5 5 2 6 4 9 ν 1 0 + 2 2 8 5 4 0 9 ν 8 − 3 7 8 9 3 3 5 ν 6 + ⋯ − 1 0 0 6 1 2 3 3 ) / 1 6 6 3 3 3 7
(-2694*v^14 + 61469*v^12 - 552649*v^10 + 2285409*v^8 - 3789335*v^6 + 779859*v^4 - 572101*v^2 - 10061233) / 1663337
β 3 \beta_{3} β 3 = = =
( 2694 ν 14 − 61469 ν 12 + 552649 ν 10 − 2285409 ν 8 + 3789335 ν 6 + ⋯ + 81211 ) / 1663337 ( 2694 \nu^{14} - 61469 \nu^{12} + 552649 \nu^{10} - 2285409 \nu^{8} + 3789335 \nu^{6} + \cdots + 81211 ) / 1663337 ( 2 6 9 4 ν 1 4 − 6 1 4 6 9 ν 1 2 + 5 5 2 6 4 9 ν 1 0 − 2 2 8 5 4 0 9 ν 8 + 3 7 8 9 3 3 5 ν 6 + ⋯ + 8 1 2 1 1 ) / 1 6 6 3 3 3 7
(2694*v^14 - 61469*v^12 + 552649*v^10 - 2285409*v^8 + 3789335*v^6 - 779859*v^4 + 3898775*v^2 + 81211) / 1663337
β 4 \beta_{4} β 4 = = =
( − 11520 ν 14 + 293628 ν 12 − 3040296 ν 10 + 15624254 ν 8 − 39217560 ν 6 + ⋯ + 6370193 ) / 3710521 ( - 11520 \nu^{14} + 293628 \nu^{12} - 3040296 \nu^{10} + 15624254 \nu^{8} - 39217560 \nu^{6} + \cdots + 6370193 ) / 3710521 ( − 1 1 5 2 0 ν 1 4 + 2 9 3 6 2 8 ν 1 2 − 3 0 4 0 2 9 6 ν 1 0 + 1 5 6 2 4 2 5 4 ν 8 − 3 9 2 1 7 5 6 0 ν 6 + ⋯ + 6 3 7 0 1 9 3 ) / 3 7 1 0 5 2 1
(-11520*v^14 + 293628*v^12 - 3040296*v^10 + 15624254*v^8 - 39217560*v^6 + 38849348*v^4 - 16270592*v^2 + 6370193) / 3710521
β 5 \beta_{5} β 5 = = =
( − 1139534 ν 14 + 28712659 ν 12 − 295900503 ν 10 + 1529685268 ν 8 + ⋯ + 691999581 ) / 192947092 ( - 1139534 \nu^{14} + 28712659 \nu^{12} - 295900503 \nu^{10} + 1529685268 \nu^{8} + \cdots + 691999581 ) / 192947092 ( − 1 1 3 9 5 3 4 ν 1 4 + 2 8 7 1 2 6 5 9 ν 1 2 − 2 9 5 9 0 0 5 0 3 ν 1 0 + 1 5 2 9 6 8 5 2 6 8 ν 8 + ⋯ + 6 9 1 9 9 9 5 8 1 ) / 1 9 2 9 4 7 0 9 2
(-1139534*v^14 + 28712659*v^12 - 295900503*v^10 + 1529685268*v^8 - 3939691239*v^6 + 4125515812*v^4 - 1455012369*v^2 + 691999581) / 192947092
β 6 \beta_{6} β 6 = = =
( − 1429545 ν 14 + 33944011 ν 12 − 320576374 ν 10 + 1427099575 ν 8 + ⋯ − 44292248 ) / 192947092 ( - 1429545 \nu^{14} + 33944011 \nu^{12} - 320576374 \nu^{10} + 1427099575 \nu^{8} + \cdots - 44292248 ) / 192947092 ( − 1 4 2 9 5 4 5 ν 1 4 + 3 3 9 4 4 0 1 1 ν 1 2 − 3 2 0 5 7 6 3 7 4 ν 1 0 + 1 4 2 7 0 9 9 5 7 5 ν 8 + ⋯ − 4 4 2 9 2 2 4 8 ) / 1 9 2 9 4 7 0 9 2
(-1429545*v^14 + 33944011*v^12 - 320576374*v^10 + 1427099575*v^8 - 2709051014*v^6 + 1088455103*v^4 - 1464953121*v^2 - 44292248) / 192947092
β 7 \beta_{7} β 7 = = =
( − 458582 ν 15 + 16643678 ν 13 − 243643736 ν 11 + 1832626360 ν 9 + ⋯ + 2564873142 ν ) / 627078049 ( - 458582 \nu^{15} + 16643678 \nu^{13} - 243643736 \nu^{11} + 1832626360 \nu^{9} + \cdots + 2564873142 \nu ) / 627078049 ( − 4 5 8 5 8 2 ν 1 5 + 1 6 6 4 3 6 7 8 ν 1 3 − 2 4 3 6 4 3 7 3 6 ν 1 1 + 1 8 3 2 6 2 6 3 6 0 ν 9 + ⋯ + 2 5 6 4 8 7 3 1 4 2 ν ) / 6 2 7 0 7 8 0 4 9
(-458582*v^15 + 16643678*v^13 - 243643736*v^11 + 1832626360*v^9 - 7303260208*v^7 + 13830825760*v^5 - 8231666126*v^3 + 2564873142*v) / 627078049
β 8 \beta_{8} β 8 = = =
( 88700 ν 14 − 2103563 ν 12 + 19736021 ν 10 − 86408618 ν 8 + 154953525 ν 6 + ⋯ + 15626169 ) / 6653348 ( 88700 \nu^{14} - 2103563 \nu^{12} + 19736021 \nu^{10} - 86408618 \nu^{8} + 154953525 \nu^{6} + \cdots + 15626169 ) / 6653348 ( 8 8 7 0 0 ν 1 4 − 2 1 0 3 5 6 3 ν 1 2 + 1 9 7 3 6 0 2 1 ν 1 0 − 8 6 4 0 8 6 1 8 ν 8 + 1 5 4 9 5 3 5 2 5 ν 6 + ⋯ + 1 5 6 2 6 1 6 9 ) / 6 6 5 3 3 4 8
(88700*v^14 - 2103563*v^12 + 19736021*v^10 - 86408618*v^8 + 154953525*v^6 - 27036722*v^4 + 23842785*v^2 + 15626169) / 6653348
β 9 \beta_{9} β 9 = = =
( 4302859 ν 15 − 113264303 ν 13 + 1228806350 ν 11 − 6828867199 ν 9 + ⋯ − 10467201680 ν ) / 2508312196 ( 4302859 \nu^{15} - 113264303 \nu^{13} + 1228806350 \nu^{11} - 6828867199 \nu^{9} + \cdots - 10467201680 \nu ) / 2508312196 ( 4 3 0 2 8 5 9 ν 1 5 − 1 1 3 2 6 4 3 0 3 ν 1 3 + 1 2 2 8 8 0 6 3 5 0 ν 1 1 − 6 8 2 8 8 6 7 1 9 9 ν 9 + ⋯ − 1 0 4 6 7 2 0 1 6 8 0 ν ) / 2 5 0 8 3 1 2 1 9 6
(4302859*v^15 - 113264303*v^13 + 1228806350*v^11 - 6828867199*v^9 + 19868045926*v^7 - 27619863991*v^5 + 18464795563*v^3 - 10467201680*v) / 2508312196
β 10 \beta_{10} β 1 0 = = =
( − 403473 ν 15 + 10071623 ν 13 − 101747496 ν 11 + 504132793 ν 9 + ⋯ + 225114682 ν ) / 86493524 ( - 403473 \nu^{15} + 10071623 \nu^{13} - 101747496 \nu^{11} + 504132793 \nu^{9} + \cdots + 225114682 \nu ) / 86493524 ( − 4 0 3 4 7 3 ν 1 5 + 1 0 0 7 1 6 2 3 ν 1 3 − 1 0 1 7 4 7 4 9 6 ν 1 1 + 5 0 4 1 3 2 7 9 3 ν 9 + ⋯ + 2 2 5 1 1 4 6 8 2 ν ) / 8 6 4 9 3 5 2 4
(-403473*v^15 + 10071623*v^13 - 101747496*v^11 + 504132793*v^9 - 1172875960*v^7 + 854627793*v^5 + 21762817*v^3 + 225114682*v) / 86493524
β 11 \beta_{11} β 1 1 = = =
( 3300067 ν 15 − 75367115 ν 13 + 666756659 ν 11 − 2588222763 ν 9 + ⋯ + 1131065416 ν ) / 627078049 ( 3300067 \nu^{15} - 75367115 \nu^{13} + 666756659 \nu^{11} - 2588222763 \nu^{9} + \cdots + 1131065416 \nu ) / 627078049 ( 3 3 0 0 0 6 7 ν 1 5 − 7 5 3 6 7 1 1 5 ν 1 3 + 6 6 6 7 5 6 6 5 9 ν 1 1 − 2 5 8 8 2 2 2 7 6 3 ν 9 + ⋯ + 1 1 3 1 0 6 5 4 1 6 ν ) / 6 2 7 0 7 8 0 4 9
(3300067*v^15 - 75367115*v^13 + 666756659*v^11 - 2588222763*v^9 + 3036067465*v^7 + 3953470823*v^5 - 407547934*v^3 + 1131065416*v) / 627078049
β 12 \beta_{12} β 1 2 = = =
( 6600134 ν 15 − 150734230 ν 13 + 1333513318 ν 11 − 5176445526 ν 9 + ⋯ + 4770443028 ν ) / 627078049 ( 6600134 \nu^{15} - 150734230 \nu^{13} + 1333513318 \nu^{11} - 5176445526 \nu^{9} + \cdots + 4770443028 \nu ) / 627078049 ( 6 6 0 0 1 3 4 ν 1 5 − 1 5 0 7 3 4 2 3 0 ν 1 3 + 1 3 3 3 5 1 3 3 1 8 ν 1 1 − 5 1 7 6 4 4 5 5 2 6 ν 9 + ⋯ + 4 7 7 0 4 4 3 0 2 8 ν ) / 6 2 7 0 7 8 0 4 9
(6600134*v^15 - 150734230*v^13 + 1333513318*v^11 - 5176445526*v^9 + 6072134930*v^7 + 7906941646*v^5 - 815095868*v^3 + 4770443028*v) / 627078049
β 13 \beta_{13} β 1 3 = = =
( − 248063 ν 15 + 6021151 ν 13 − 58426783 ν 11 + 271352435 ν 9 + ⋯ − 8003320 ν ) / 21623381 ( - 248063 \nu^{15} + 6021151 \nu^{13} - 58426783 \nu^{11} + 271352435 \nu^{9} + \cdots - 8003320 \nu ) / 21623381 ( − 2 4 8 0 6 3 ν 1 5 + 6 0 2 1 1 5 1 ν 1 3 − 5 8 4 2 6 7 8 3 ν 1 1 + 2 7 1 3 5 2 4 3 5 ν 9 + ⋯ − 8 0 0 3 3 2 0 ν ) / 2 1 6 2 3 3 8 1
(-248063*v^15 + 6021151*v^13 - 58426783*v^11 + 271352435*v^9 - 561779405*v^7 + 316469269*v^5 - 175583178*v^3 - 8003320*v) / 21623381
β 14 \beta_{14} β 1 4 = = =
( − 10755599 ν 15 + 252302706 ν 13 − 2332684002 ν 11 + 9941641297 ν 9 + ⋯ + 1098391515 ν ) / 627078049 ( - 10755599 \nu^{15} + 252302706 \nu^{13} - 2332684002 \nu^{11} + 9941641297 \nu^{9} + \cdots + 1098391515 \nu ) / 627078049 ( − 1 0 7 5 5 5 9 9 ν 1 5 + 2 5 2 3 0 2 7 0 6 ν 1 3 − 2 3 3 2 6 8 4 0 0 2 ν 1 1 + 9 9 4 1 6 4 1 2 9 7 ν 9 + ⋯ + 1 0 9 8 3 9 1 5 1 5 ν ) / 6 2 7 0 7 8 0 4 9
(-10755599*v^15 + 252302706*v^13 - 2332684002*v^11 + 9941641297*v^9 - 16587964242*v^7 + 138651525*v^5 - 6407093879*v^3 + 1098391515*v) / 627078049
β 15 \beta_{15} β 1 5 = = =
( − 1723809 ν 15 + 41505147 ν 13 − 399318104 ν 11 + 1835444329 ν 9 + ⋯ − 140493626 ν ) / 86493524 ( - 1723809 \nu^{15} + 41505147 \nu^{13} - 399318104 \nu^{11} + 1835444329 \nu^{9} + \cdots - 140493626 \nu ) / 86493524 ( − 1 7 2 3 8 0 9 ν 1 5 + 4 1 5 0 5 1 4 7 ν 1 3 − 3 9 9 3 1 8 1 0 4 ν 1 1 + 1 8 3 5 4 4 4 3 2 9 ν 9 + ⋯ − 1 4 0 4 9 3 6 2 6 ν ) / 8 6 4 9 3 5 2 4
(-1723809*v^15 + 41505147*v^13 - 399318104*v^11 + 1835444329*v^9 - 3743153672*v^7 + 2039706321*v^5 - 1309963695*v^3 - 140493626*v) / 86493524
ν \nu ν = = =
( β 12 − 2 β 11 ) / 4 ( \beta_{12} - 2\beta_{11} ) / 4 ( β 1 2 − 2 β 1 1 ) / 4
(b12 - 2*b11) / 4
ν 2 \nu^{2} ν 2 = = =
( β 3 + β 2 + 6 ) / 2 ( \beta_{3} + \beta_{2} + 6 ) / 2 ( β 3 + β 2 + 6 ) / 2
(b3 + b2 + 6) / 2
ν 3 \nu^{3} ν 3 = = =
( 3 β 15 − β 14 + 7 β 12 − 10 β 11 − 3 β 10 + 4 β 9 − β 7 ) / 4 ( 3\beta_{15} - \beta_{14} + 7\beta_{12} - 10\beta_{11} - 3\beta_{10} + 4\beta_{9} - \beta_{7} ) / 4 ( 3 β 1 5 − β 1 4 + 7 β 1 2 − 1 0 β 1 1 − 3 β 1 0 + 4 β 9 − β 7 ) / 4
(3*b15 - b14 + 7*b12 - 10*b11 - 3*b10 + 4*b9 - b7) / 4
ν 4 \nu^{4} ν 4 = = =
( β 8 + 4 β 6 + β 5 − 4 β 4 + 13 β 3 + 5 β 2 + 2 β 1 + 29 ) / 2 ( \beta_{8} + 4\beta_{6} + \beta_{5} - 4\beta_{4} + 13\beta_{3} + 5\beta_{2} + 2\beta _1 + 29 ) / 2 ( β 8 + 4 β 6 + β 5 − 4 β 4 + 1 3 β 3 + 5 β 2 + 2 β 1 + 2 9 ) / 2
(b8 + 4*b6 + b5 - 4*b4 + 13*b3 + 5*b2 + 2*b1 + 29) / 2
ν 5 \nu^{5} ν 5 = = =
( 30 β 15 − 2 β 14 − 5 β 13 + 51 β 12 − 43 β 11 − 30 β 10 + 24 β 9 − 20 β 7 ) / 4 ( 30\beta_{15} - 2\beta_{14} - 5\beta_{13} + 51\beta_{12} - 43\beta_{11} - 30\beta_{10} + 24\beta_{9} - 20\beta_{7} ) / 4 ( 3 0 β 1 5 − 2 β 1 4 − 5 β 1 3 + 5 1 β 1 2 − 4 3 β 1 1 − 3 0 β 1 0 + 2 4 β 9 − 2 0 β 7 ) / 4
(30*b15 - 2*b14 - 5*b13 + 51*b12 - 43*b11 - 30*b10 + 24*b9 - 20*b7) / 4
ν 6 \nu^{6} ν 6 = = =
( 13 β 8 + 78 β 6 + 35 β 5 − 113 β 4 + 231 β 3 + 35 β 2 + 36 β 1 + 212 ) / 4 ( 13\beta_{8} + 78\beta_{6} + 35\beta_{5} - 113\beta_{4} + 231\beta_{3} + 35\beta_{2} + 36\beta _1 + 212 ) / 4 ( 1 3 β 8 + 7 8 β 6 + 3 5 β 5 − 1 1 3 β 4 + 2 3 1 β 3 + 3 5 β 2 + 3 6 β 1 + 2 1 2 ) / 4
(13*b8 + 78*b6 + 35*b5 - 113*b4 + 231*b3 + 35*b2 + 36*b1 + 212) / 4
ν 7 \nu^{7} ν 7 = = =
( 245 β 15 + 7 β 14 − 77 β 13 + 314 β 12 − 99 β 11 − 217 β 10 + ⋯ − 217 β 7 ) / 4 ( 245 \beta_{15} + 7 \beta_{14} - 77 \beta_{13} + 314 \beta_{12} - 99 \beta_{11} - 217 \beta_{10} + \cdots - 217 \beta_{7} ) / 4 ( 2 4 5 β 1 5 + 7 β 1 4 − 7 7 β 1 3 + 3 1 4 β 1 2 − 9 9 β 1 1 − 2 1 7 β 1 0 + ⋯ − 2 1 7 β 7 ) / 4
(245*b15 + 7*b14 - 77*b13 + 314*b12 - 99*b11 - 217*b10 + 96*b9 - 217*b7) / 4
ν 8 \nu^{8} ν 8 = = =
( 20 β 8 + 266 β 6 + 194 β 5 − 559 β 4 + 864 β 3 + 2 β 2 + 178 β 1 + 29 ) / 2 ( 20\beta_{8} + 266\beta_{6} + 194\beta_{5} - 559\beta_{4} + 864\beta_{3} + 2\beta_{2} + 178\beta _1 + 29 ) / 2 ( 2 0 β 8 + 2 6 6 β 6 + 1 9 4 β 5 − 5 5 9 β 4 + 8 6 4 β 3 + 2 β 2 + 1 7 8 β 1 + 2 9 ) / 2
(20*b8 + 266*b6 + 194*b5 - 559*b4 + 864*b3 + 2*b2 + 178*b1 + 29) / 2
ν 9 \nu^{9} ν 9 = = =
( 1728 β 15 + 88 β 14 − 771 β 13 + 1556 β 12 + 629 β 11 + ⋯ − 1868 β 7 ) / 4 ( 1728 \beta_{15} + 88 \beta_{14} - 771 \beta_{13} + 1556 \beta_{12} + 629 \beta_{11} + \cdots - 1868 \beta_{7} ) / 4 ( 1 7 2 8 β 1 5 + 8 8 β 1 4 − 7 7 1 β 1 3 + 1 5 5 6 β 1 2 + 6 2 9 β 1 1 + ⋯ − 1 8 6 8 β 7 ) / 4
(1728*b15 + 88*b14 - 771*b13 + 1556*b12 + 629*b11 - 1344*b10 - 128*b9 - 1868*b7) / 4
ν 10 \nu^{10} ν 1 0 = = =
( − 319 β 8 + 2600 β 6 + 3285 β 5 − 8921 β 4 + 11245 β 3 − 1473 β 2 + ⋯ − 8758 ) / 4 ( - 319 \beta_{8} + 2600 \beta_{6} + 3285 \beta_{5} - 8921 \beta_{4} + 11245 \beta_{3} - 1473 \beta_{2} + \cdots - 8758 ) / 4 ( − 3 1 9 β 8 + 2 6 0 0 β 6 + 3 2 8 5 β 5 − 8 9 2 1 β 4 + 1 1 2 4 5 β 3 − 1 4 7 3 β 2 + ⋯ − 8 7 5 8 ) / 4
(-319*b8 + 2600*b6 + 3285*b5 - 8921*b4 + 11245*b3 - 1473*b2 + 3398*b1 - 8758) / 4
ν 11 \nu^{11} ν 1 1 = = =
( 10560 β 15 + 286 β 14 − 5995 β 13 + 4951 β 12 + 12093 β 11 + ⋯ − 14166 β 7 ) / 4 ( 10560 \beta_{15} + 286 \beta_{14} - 5995 \beta_{13} + 4951 \beta_{12} + 12093 \beta_{11} + \cdots - 14166 \beta_{7} ) / 4 ( 1 0 5 6 0 β 1 5 + 2 8 6 β 1 4 − 5 9 9 5 β 1 3 + 4 9 5 1 β 1 2 + 1 2 0 9 3 β 1 1 + ⋯ − 1 4 1 6 6 β 7 ) / 4
(10560*b15 + 286*b14 - 5995*b13 + 4951*b12 + 12093*b11 - 7172*b10 - 7564*b9 - 14166*b7) / 4
ν 12 \nu^{12} ν 1 2 = = =
( − 3605 β 8 + 2573 β 6 + 11182 β 5 − 29241 β 4 + 31239 β 3 − 9280 β 2 + ⋯ − 56567 ) / 2 ( - 3605 \beta_{8} + 2573 \beta_{6} + 11182 \beta_{5} - 29241 \beta_{4} + 31239 \beta_{3} - 9280 \beta_{2} + \cdots - 56567 ) / 2 ( − 3 6 0 5 β 8 + 2 5 7 3 β 6 + 1 1 1 8 2 β 5 − 2 9 2 4 1 β 4 + 3 1 2 3 9 β 3 − 9 2 8 0 β 2 + ⋯ − 5 6 5 6 7 ) / 2
(-3605*b8 + 2573*b6 + 11182*b5 - 29241*b4 + 31239*b3 - 9280*b2 + 14687*b1 - 56567) / 2
ν 13 \nu^{13} ν 1 3 = = =
( 53248 β 15 − 2884 β 14 − 36868 β 13 − 10003 β 12 + 123074 β 11 + ⋯ − 97052 β 7 ) / 4 ( 53248 \beta_{15} - 2884 \beta_{14} - 36868 \beta_{13} - 10003 \beta_{12} + 123074 \beta_{11} + \cdots - 97052 \beta_{7} ) / 4 ( 5 3 2 4 8 β 1 5 − 2 8 8 4 β 1 4 − 3 6 8 6 8 β 1 3 − 1 0 0 0 3 β 1 2 + 1 2 3 0 7 4 β 1 1 + ⋯ − 9 7 0 5 2 β 7 ) / 4
(53248*b15 - 2884*b14 - 36868*b13 - 10003*b12 + 123074*b11 - 30784*b10 - 98032*b9 - 97052*b7) / 4
ν 14 \nu^{14} ν 1 4 = = =
( − 41422 β 8 − 36017 β 6 + 58447 β 5 − 148066 β 4 + 133427 β 3 + ⋯ − 517223 ) / 2 ( - 41422 \beta_{8} - 36017 \beta_{6} + 58447 \beta_{5} - 148066 \beta_{4} + 133427 \beta_{3} + \cdots - 517223 ) / 2 ( − 4 1 4 2 2 β 8 − 3 6 0 1 7 β 6 + 5 8 4 4 7 β 5 − 1 4 8 0 6 6 β 4 + 1 3 3 4 2 7 β 3 + ⋯ − 5 1 7 2 2 3 ) / 2
(-41422*b8 - 36017*b6 + 58447*b5 - 148066*b4 + 133427*b3 - 83574*b2 + 112843*b1 - 517223) / 2
ν 15 \nu^{15} ν 1 5 = = =
( 176793 β 15 − 58799 β 14 − 158604 β 13 − 357861 β 12 + 1003586 β 11 + ⋯ − 595917 β 7 ) / 4 ( 176793 \beta_{15} - 58799 \beta_{14} - 158604 \beta_{13} - 357861 \beta_{12} + 1003586 \beta_{11} + \cdots - 595917 \beta_{7} ) / 4 ( 1 7 6 7 9 3 β 1 5 − 5 8 7 9 9 β 1 4 − 1 5 8 6 0 4 β 1 3 − 3 5 7 8 6 1 β 1 2 + 1 0 0 3 5 8 6 β 1 1 + ⋯ − 5 9 5 9 1 7 β 7 ) / 4
(176793*b15 - 58799*b14 - 158604*b13 - 357861*b12 + 1003586*b11 - 72873*b10 - 927572*b9 - 595917*b7) / 4
Character values
We give the values of χ \chi χ on generators for ( Z / 1512 Z ) × \left(\mathbb{Z}/1512\mathbb{Z}\right)^\times ( Z / 1 5 1 2 Z ) × .
n n n
757 757 7 5 7
785 785 7 8 5
1081 1081 1 0 8 1
1135 1135 1 1 3 5
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 kernel of the linear operator
T 5 8 − 17 T 5 6 + 83 T 5 4 − 91 T 5 2 + 16 T_{5}^{8} - 17T_{5}^{6} + 83T_{5}^{4} - 91T_{5}^{2} + 16 T 5 8 − 1 7 T 5 6 + 8 3 T 5 4 − 9 1 T 5 2 + 1 6
T5^8 - 17*T5^6 + 83*T5^4 - 91*T5^2 + 16
acting on S 2 n e w ( 1512 , [ χ ] ) S_{2}^{\mathrm{new}}(1512, [\chi]) S 2 n e w ( 1 5 1 2 , [ χ ] ) .
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 8 − 17 T 6 + ⋯ + 16 ) 2 (T^{8} - 17 T^{6} + \cdots + 16)^{2} ( T 8 − 1 7 T 6 + ⋯ + 1 6 ) 2
(T^8 - 17*T^6 + 83*T^4 - 91*T^2 + 16)^2
7 7 7
( T 8 + T 7 + 5 T 5 + ⋯ + 2401 ) 2 (T^{8} + T^{7} + 5 T^{5} + \cdots + 2401)^{2} ( T 8 + T 7 + 5 T 5 + ⋯ + 2 4 0 1 ) 2
(T^8 + T^7 + 5*T^5 - 34*T^4 + 35*T^3 + 343*T + 2401)^2
11 11 1 1
( T 8 + 51 T 6 + ⋯ + 784 ) 2 (T^{8} + 51 T^{6} + \cdots + 784)^{2} ( T 8 + 5 1 T 6 + ⋯ + 7 8 4 ) 2
(T^8 + 51*T^6 + 779*T^4 + 3801*T^2 + 784)^2
13 13 1 3
( T 8 + 55 T 6 + ⋯ + 3136 ) 2 (T^{8} + 55 T^{6} + \cdots + 3136)^{2} ( T 8 + 5 5 T 6 + ⋯ + 3 1 3 6 ) 2
(T^8 + 55*T^6 + 956*T^4 + 5264*T^2 + 3136)^2
17 17 1 7
( T 8 − 64 T 6 + ⋯ + 4096 ) 2 (T^{8} - 64 T^{6} + \cdots + 4096)^{2} ( T 8 − 6 4 T 6 + ⋯ + 4 0 9 6 ) 2
(T^8 - 64*T^6 + 1088*T^4 - 5888*T^2 + 4096)^2
19 19 1 9
( T 8 + 52 T 6 + ⋯ + 841 ) 2 (T^{8} + 52 T^{6} + \cdots + 841)^{2} ( T 8 + 5 2 T 6 + ⋯ + 8 4 1 ) 2
(T^8 + 52*T^6 + 734*T^4 + 1700*T^2 + 841)^2
23 23 2 3
( T 8 + 163 T 6 + ⋯ + 559504 ) 2 (T^{8} + 163 T^{6} + \cdots + 559504)^{2} ( T 8 + 1 6 3 T 6 + ⋯ + 5 5 9 5 0 4 ) 2
(T^8 + 163*T^6 + 8379*T^4 + 147097*T^2 + 559504)^2
29 29 2 9
( T 8 + 60 T 6 + ⋯ + 4096 ) 2 (T^{8} + 60 T^{6} + \cdots + 4096)^{2} ( T 8 + 6 0 T 6 + ⋯ + 4 0 9 6 ) 2
(T^8 + 60*T^6 + 944*T^4 + 3648*T^2 + 4096)^2
31 31 3 1
( T 8 + 61 T 6 + ⋯ + 3844 ) 2 (T^{8} + 61 T^{6} + \cdots + 3844)^{2} ( T 8 + 6 1 T 6 + ⋯ + 3 8 4 4 ) 2
(T^8 + 61*T^6 + 791*T^4 + 3431*T^2 + 3844)^2
37 37 3 7
( T 4 + 2 T 3 + ⋯ − 161 ) 4 (T^{4} + 2 T^{3} + \cdots - 161)^{4} ( T 4 + 2 T 3 + ⋯ − 1 6 1 ) 4
(T^4 + 2*T^3 - 96*T^2 - 322*T - 161)^4
41 41 4 1
( T 8 − 193 T 6 + ⋯ + 1827904 ) 2 (T^{8} - 193 T^{6} + \cdots + 1827904)^{2} ( T 8 − 1 9 3 T 6 + ⋯ + 1 8 2 7 9 0 4 ) 2
(T^8 - 193*T^6 + 11699*T^4 - 265499*T^2 + 1827904)^2
43 43 4 3
( T 4 − 2 T 3 − 132 T 2 + ⋯ + 64 ) 4 (T^{4} - 2 T^{3} - 132 T^{2} + \cdots + 64)^{4} ( T 4 − 2 T 3 − 1 3 2 T 2 + ⋯ + 6 4 ) 4
(T^4 - 2*T^3 - 132*T^2 + 376*T + 64)^4
47 47 4 7
( T 8 − 356 T 6 + ⋯ + 30647296 ) 2 (T^{8} - 356 T^{6} + \cdots + 30647296)^{2} ( T 8 − 3 5 6 T 6 + ⋯ + 3 0 6 4 7 2 9 6 ) 2
(T^8 - 356*T^6 + 42608*T^4 - 2023360*T^2 + 30647296)^2
53 53 5 3
( T 8 + 240 T 6 + ⋯ + 1048576 ) 2 (T^{8} + 240 T^{6} + \cdots + 1048576)^{2} ( T 8 + 2 4 0 T 6 + ⋯ + 1 0 4 8 5 7 6 ) 2
(T^8 + 240*T^6 + 15104*T^4 + 233472*T^2 + 1048576)^2
59 59 5 9
( T 8 − 324 T 6 + ⋯ + 4064256 ) 2 (T^{8} - 324 T^{6} + \cdots + 4064256)^{2} ( T 8 − 3 2 4 T 6 + ⋯ + 4 0 6 4 2 5 6 ) 2
(T^8 - 324*T^6 + 27504*T^4 - 616896*T^2 + 4064256)^2
61 61 6 1
( T 8 + 171 T 6 + ⋯ + 331776 ) 2 (T^{8} + 171 T^{6} + \cdots + 331776)^{2} ( T 8 + 1 7 1 T 6 + ⋯ + 3 3 1 7 7 6 ) 2
(T^8 + 171*T^6 + 8496*T^4 + 103680*T^2 + 331776)^2
67 67 6 7
( T 4 + 7 T 3 + ⋯ − 2744 ) 4 (T^{4} + 7 T^{3} + \cdots - 2744)^{4} ( T 4 + 7 T 3 + ⋯ − 2 7 4 4 ) 4
(T^4 + 7*T^3 - 154*T^2 - 1372*T - 2744)^4
71 71 7 1
( T 8 + 227 T 6 + ⋯ + 8479744 ) 2 (T^{8} + 227 T^{6} + \cdots + 8479744)^{2} ( T 8 + 2 2 7 T 6 + ⋯ + 8 4 7 9 7 4 4 ) 2
(T^8 + 227*T^6 + 18747*T^4 + 664937*T^2 + 8479744)^2
73 73 7 3
( T 8 + 263 T 6 + ⋯ + 10291264 ) 2 (T^{8} + 263 T^{6} + \cdots + 10291264)^{2} ( T 8 + 2 6 3 T 6 + ⋯ + 1 0 2 9 1 2 6 4 ) 2
(T^8 + 263*T^6 + 23708*T^4 + 863248*T^2 + 10291264)^2
79 79 7 9
( T 4 + 11 T 3 + ⋯ + 184 ) 4 (T^{4} + 11 T^{3} + \cdots + 184)^{4} ( T 4 + 1 1 T 3 + ⋯ + 1 8 4 ) 4
(T^4 + 11*T^3 - 70*T^2 - 92*T + 184)^4
83 83 8 3
( T 8 − 532 T 6 + ⋯ + 51380224 ) 2 (T^{8} - 532 T^{6} + \cdots + 51380224)^{2} ( T 8 − 5 3 2 T 6 + ⋯ + 5 1 3 8 0 2 2 4 ) 2
(T^8 - 532*T^6 + 85040*T^4 - 3819200*T^2 + 51380224)^2
89 89 8 9
( T 8 − 361 T 6 + ⋯ + 8737936 ) 2 (T^{8} - 361 T^{6} + \cdots + 8737936)^{2} ( T 8 − 3 6 1 T 6 + ⋯ + 8 7 3 7 9 3 6 ) 2
(T^8 - 361*T^6 + 28979*T^4 - 862499*T^2 + 8737936)^2
97 97 9 7
( T 8 + 491 T 6 + ⋯ + 177209344 ) 2 (T^{8} + 491 T^{6} + \cdots + 177209344)^{2} ( T 8 + 4 9 1 T 6 + ⋯ + 1 7 7 2 0 9 3 4 4 ) 2
(T^8 + 491*T^6 + 87296*T^4 + 6602752*T^2 + 177209344)^2
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