gp: [N,k,chi] = [1682,2,Mod(1681,1682)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1682.1681");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [16,0,0,-16,-10,-2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == 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 + 37 x 14 + 548 x 12 + 4119 x 10 + 16415 x 8 + 33099 x 6 + 30128 x 4 + 10537 x 2 + 961 x^{16} + 37x^{14} + 548x^{12} + 4119x^{10} + 16415x^{8} + 33099x^{6} + 30128x^{4} + 10537x^{2} + 961 x 1 6 + 3 7 x 1 4 + 5 4 8 x 1 2 + 4 1 1 9 x 1 0 + 1 6 4 1 5 x 8 + 3 3 0 9 9 x 6 + 3 0 1 2 8 x 4 + 1 0 5 3 7 x 2 + 9 6 1
x^16 + 37*x^14 + 548*x^12 + 4119*x^10 + 16415*x^8 + 33099*x^6 + 30128*x^4 + 10537*x^2 + 961
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( 1284 ν 14 + 2104 ν 12 − 607273 ν 10 − 8639916 ν 8 − 44272376 ν 6 + ⋯ + 6710203 ) / 6652257 ( 1284 \nu^{14} + 2104 \nu^{12} - 607273 \nu^{10} - 8639916 \nu^{8} - 44272376 \nu^{6} + \cdots + 6710203 ) / 6652257 ( 1 2 8 4 ν 1 4 + 2 1 0 4 ν 1 2 − 6 0 7 2 7 3 ν 1 0 − 8 6 3 9 9 1 6 ν 8 − 4 4 2 7 2 3 7 6 ν 6 + ⋯ + 6 7 1 0 2 0 3 ) / 6 6 5 2 2 5 7
(1284*v^14 + 2104*v^12 - 607273*v^10 - 8639916*v^8 - 44272376*v^6 - 83894220*v^4 - 40149880*v^2 + 6710203) / 6652257
β 3 \beta_{3} β 3 = = =
( 27407 ν 14 + 991285 ν 12 + 14477445 ν 10 + 104632823 ν 8 + 356706384 ν 6 + ⋯ − 465992667 ) / 73174827 ( 27407 \nu^{14} + 991285 \nu^{12} + 14477445 \nu^{10} + 104632823 \nu^{8} + 356706384 \nu^{6} + \cdots - 465992667 ) / 73174827 ( 2 7 4 0 7 ν 1 4 + 9 9 1 2 8 5 ν 1 2 + 1 4 4 7 7 4 4 5 ν 1 0 + 1 0 4 6 3 2 8 2 3 ν 8 + 3 5 6 7 0 6 3 8 4 ν 6 + ⋯ − 4 6 5 9 9 2 6 6 7 ) / 7 3 1 7 4 8 2 7
(27407*v^14 + 991285*v^12 + 14477445*v^10 + 104632823*v^8 + 356706384*v^6 + 344485276*v^4 - 567920218*v^2 - 465992667) / 73174827
β 4 \beta_{4} β 4 = = =
( 83062 ν 14 + 2028858 ν 12 + 15594884 ν 10 + 19187494 ν 8 − 260579504 ν 6 + ⋯ − 272137079 ) / 73174827 ( 83062 \nu^{14} + 2028858 \nu^{12} + 15594884 \nu^{10} + 19187494 \nu^{8} - 260579504 \nu^{6} + \cdots - 272137079 ) / 73174827 ( 8 3 0 6 2 ν 1 4 + 2 0 2 8 8 5 8 ν 1 2 + 1 5 5 9 4 8 8 4 ν 1 0 + 1 9 1 8 7 4 9 4 ν 8 − 2 6 0 5 7 9 5 0 4 ν 6 + ⋯ − 2 7 2 1 3 7 0 7 9 ) / 7 3 1 7 4 8 2 7
(83062*v^14 + 2028858*v^12 + 15594884*v^10 + 19187494*v^8 - 260579504*v^6 - 1083527461*v^4 - 1360564353*v^2 - 272137079) / 73174827
β 5 \beta_{5} β 5 = = =
( − 111310 ν 14 − 2075146 ν 12 − 2234878 ν 10 + 170890658 ν 8 + 1234571776 ν 6 + ⋯ + 563561575 ) / 73174827 ( - 111310 \nu^{14} - 2075146 \nu^{12} - 2234878 \nu^{10} + 170890658 \nu^{8} + 1234571776 \nu^{6} + \cdots + 563561575 ) / 73174827 ( − 1 1 1 3 1 0 ν 1 4 − 2 0 7 5 1 4 6 ν 1 2 − 2 2 3 4 8 7 8 ν 1 0 + 1 7 0 8 9 0 6 5 8 ν 8 + 1 2 3 4 5 7 1 7 7 6 ν 6 + ⋯ + 5 6 3 5 6 1 5 7 5 ) / 7 3 1 7 4 8 2 7
(-111310*v^14 - 2075146*v^12 - 2234878*v^10 + 170890658*v^8 + 1234571776*v^6 + 2929200301*v^4 + 2317036540*v^2 + 563561575) / 73174827
β 6 \beta_{6} β 6 = = =
( 97 ν 14 + 3198 ν 12 + 40629 ν 10 + 249235 ν 8 + 759007 ν 6 + 1071058 ν 4 + ⋯ + 73501 ) / 35299 ( 97 \nu^{14} + 3198 \nu^{12} + 40629 \nu^{10} + 249235 \nu^{8} + 759007 \nu^{6} + 1071058 \nu^{4} + \cdots + 73501 ) / 35299 ( 9 7 ν 1 4 + 3 1 9 8 ν 1 2 + 4 0 6 2 9 ν 1 0 + 2 4 9 2 3 5 ν 8 + 7 5 9 0 0 7 ν 6 + 1 0 7 1 0 5 8 ν 4 + ⋯ + 7 3 5 0 1 ) / 3 5 2 9 9
(97*v^14 + 3198*v^12 + 40629*v^10 + 249235*v^8 + 759007*v^6 + 1071058*v^4 + 571977*v^2 + 73501) / 35299
β 7 \beta_{7} β 7 = = =
( 210476 ν 14 + 7093859 ν 12 + 95469962 ν 10 + 648806909 ν 8 + 2301318694 ν 6 + ⋯ + 372298774 ) / 73174827 ( 210476 \nu^{14} + 7093859 \nu^{12} + 95469962 \nu^{10} + 648806909 \nu^{8} + 2301318694 \nu^{6} + \cdots + 372298774 ) / 73174827 ( 2 1 0 4 7 6 ν 1 4 + 7 0 9 3 8 5 9 ν 1 2 + 9 5 4 6 9 9 6 2 ν 1 0 + 6 4 8 8 0 6 9 0 9 ν 8 + 2 3 0 1 3 1 8 6 9 4 ν 6 + ⋯ + 3 7 2 2 9 8 7 7 4 ) / 7 3 1 7 4 8 2 7
(210476*v^14 + 7093859*v^12 + 95469962*v^10 + 648806909*v^8 + 2301318694*v^6 + 3913685887*v^4 + 2487776512*v^2 + 372298774) / 73174827
β 8 \beta_{8} β 8 = = =
( − 297522 ν 14 − 8735498 ν 12 − 96829366 ν 10 − 496888416 ν 8 − 1147603841 ν 6 + ⋯ + 167012926 ) / 73174827 ( - 297522 \nu^{14} - 8735498 \nu^{12} - 96829366 \nu^{10} - 496888416 \nu^{8} - 1147603841 \nu^{6} + \cdots + 167012926 ) / 73174827 ( − 2 9 7 5 2 2 ν 1 4 − 8 7 3 5 4 9 8 ν 1 2 − 9 6 8 2 9 3 6 6 ν 1 0 − 4 9 6 8 8 8 4 1 6 ν 8 − 1 1 4 7 6 0 3 8 4 1 ν 6 + ⋯ + 1 6 7 0 1 2 9 2 6 ) / 7 3 1 7 4 8 2 7
(-297522*v^14 - 8735498*v^12 - 96829366*v^10 - 496888416*v^8 - 1147603841*v^6 - 910206261*v^4 + 136745381*v^2 + 167012926) / 73174827
β 9 \beta_{9} β 9 = = =
( 1284 ν 15 + 2104 ν 13 − 607273 ν 11 − 8639916 ν 9 − 44272376 ν 7 + ⋯ + 6710203 ν ) / 6652257 ( 1284 \nu^{15} + 2104 \nu^{13} - 607273 \nu^{11} - 8639916 \nu^{9} - 44272376 \nu^{7} + \cdots + 6710203 \nu ) / 6652257 ( 1 2 8 4 ν 1 5 + 2 1 0 4 ν 1 3 − 6 0 7 2 7 3 ν 1 1 − 8 6 3 9 9 1 6 ν 9 − 4 4 2 7 2 3 7 6 ν 7 + ⋯ + 6 7 1 0 2 0 3 ν ) / 6 6 5 2 2 5 7
(1284*v^15 + 2104*v^13 - 607273*v^11 - 8639916*v^9 - 44272376*v^7 - 83894220*v^5 - 40149880*v^3 + 6710203*v) / 6652257
β 10 \beta_{10} β 1 0 = = =
( − 836497 ν 15 − 30535640 ν 13 − 456007590 ν 11 − 3538203628 ν 9 + ⋯ − 7632738708 ν ) / 2268419637 ( - 836497 \nu^{15} - 30535640 \nu^{13} - 456007590 \nu^{11} - 3538203628 \nu^{9} + \cdots - 7632738708 \nu ) / 2268419637 ( − 8 3 6 4 9 7 ν 1 5 − 3 0 5 3 5 6 4 0 ν 1 3 − 4 5 6 0 0 7 5 9 0 ν 1 1 − 3 5 3 8 2 0 3 6 2 8 ν 9 + ⋯ − 7 6 3 2 7 3 8 7 0 8 ν ) / 2 2 6 8 4 1 9 6 3 7
(-836497*v^15 - 30535640*v^13 - 456007590*v^11 - 3538203628*v^9 - 14938958478*v^7 - 32809597349*v^5 - 32225636488*v^3 - 7632738708*v) / 2268419637
β 11 \beta_{11} β 1 1 = = =
( 1126355 ν 15 + 25028414 ν 13 + 126772847 ν 11 − 793255129 ν 9 + ⋯ − 16490385368 ν ) / 2268419637 ( 1126355 \nu^{15} + 25028414 \nu^{13} + 126772847 \nu^{11} - 793255129 \nu^{9} + \cdots - 16490385368 \nu ) / 2268419637 ( 1 1 2 6 3 5 5 ν 1 5 + 2 5 0 2 8 4 1 4 ν 1 3 + 1 2 6 7 7 2 8 4 7 ν 1 1 − 7 9 3 2 5 5 1 2 9 ν 9 + ⋯ − 1 6 4 9 0 3 8 5 3 6 8 ν ) / 2 2 6 8 4 1 9 6 3 7
(1126355*v^15 + 25028414*v^13 + 126772847*v^11 - 793255129*v^9 - 9396663905*v^7 - 29467116938*v^5 - 36805700960*v^3 - 16490385368*v) / 2268419637
β 12 \beta_{12} β 1 2 = = =
( 2276840 ν 15 + 95943751 ν 13 + 1564112238 ν 11 + 12441794609 ν 9 + ⋯ + 4712034027 ν ) / 2268419637 ( 2276840 \nu^{15} + 95943751 \nu^{13} + 1564112238 \nu^{11} + 12441794609 \nu^{9} + \cdots + 4712034027 \nu ) / 2268419637 ( 2 2 7 6 8 4 0 ν 1 5 + 9 5 9 4 3 7 5 1 ν 1 3 + 1 5 6 4 1 1 2 2 3 8 ν 1 1 + 1 2 4 4 1 7 9 4 6 0 9 ν 9 + ⋯ + 4 7 1 2 0 3 4 0 2 7 ν ) / 2 2 6 8 4 1 9 6 3 7
(2276840*v^15 + 95943751*v^13 + 1564112238*v^11 + 12441794609*v^9 + 49540927182*v^7 + 89294419090*v^5 + 54847293839*v^3 + 4712034027*v) / 2268419637
β 13 \beta_{13} β 1 3 = = =
( 2371 ν 15 + 84720 ν 13 + 1200170 ν 11 + 8506650 ν 9 + 31193680 ν 7 + ⋯ + 7251940 ν ) / 1094269 ( 2371 \nu^{15} + 84720 \nu^{13} + 1200170 \nu^{11} + 8506650 \nu^{9} + 31193680 \nu^{7} + \cdots + 7251940 \nu ) / 1094269 ( 2 3 7 1 ν 1 5 + 8 4 7 2 0 ν 1 3 + 1 2 0 0 1 7 0 ν 1 1 + 8 5 0 6 6 5 0 ν 9 + 3 1 1 9 3 6 8 0 ν 7 + ⋯ + 7 2 5 1 9 4 0 ν ) / 1 0 9 4 2 6 9
(2371*v^15 + 84720*v^13 + 1200170*v^11 + 8506650*v^9 + 31193680*v^7 + 54948512*v^5 + 38230690*v^3 + 7251940*v) / 1094269
β 14 \beta_{14} β 1 4 = = =
( 5002886 ν 15 + 210639250 ν 13 + 3491805708 ν 11 + 28921496975 ν 9 + ⋯ + 68484475212 ν ) / 2268419637 ( 5002886 \nu^{15} + 210639250 \nu^{13} + 3491805708 \nu^{11} + 28921496975 \nu^{9} + \cdots + 68484475212 \nu ) / 2268419637 ( 5 0 0 2 8 8 6 ν 1 5 + 2 1 0 6 3 9 2 5 0 ν 1 3 + 3 4 9 1 8 0 5 7 0 8 ν 1 1 + 2 8 9 2 1 4 9 6 9 7 5 ν 9 + ⋯ + 6 8 4 8 4 4 7 5 2 1 2 ν ) / 2 2 6 8 4 1 9 6 3 7
(5002886*v^15 + 210639250*v^13 + 3491805708*v^11 + 28921496975*v^9 + 124791972708*v^7 + 265298765335*v^5 + 240357553952*v^3 + 68484475212*v) / 2268419637
β 15 \beta_{15} β 1 5 = = =
( 10751328 ν 15 + 371223239 ν 13 + 5113668292 ν 11 + 35594343402 ν 9 + ⋯ + 45353098205 ν ) / 2268419637 ( 10751328 \nu^{15} + 371223239 \nu^{13} + 5113668292 \nu^{11} + 35594343402 \nu^{9} + \cdots + 45353098205 \nu ) / 2268419637 ( 1 0 7 5 1 3 2 8 ν 1 5 + 3 7 1 2 2 3 2 3 9 ν 1 3 + 5 1 1 3 6 6 8 2 9 2 ν 1 1 + 3 5 5 9 4 3 4 3 4 0 2 ν 9 + ⋯ + 4 5 3 5 3 0 9 8 2 0 5 ν ) / 2 2 6 8 4 1 9 6 3 7
(10751328*v^15 + 371223239*v^13 + 5113668292*v^11 + 35594343402*v^9 + 130305161843*v^7 + 236091715791*v^5 + 179487884122*v^3 + 45353098205*v) / 2268419637
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
β 5 + β 4 + 2 β 2 − 6 \beta_{5} + \beta_{4} + 2\beta_{2} - 6 β 5 + β 4 + 2 β 2 − 6
b5 + b4 + 2*b2 - 6
ν 3 \nu^{3} ν 3 = = =
β 15 − 2 β 13 + β 12 − β 11 + 3 β 10 + β 9 − 7 β 1 \beta_{15} - 2\beta_{13} + \beta_{12} - \beta_{11} + 3\beta_{10} + \beta_{9} - 7\beta_1 β 1 5 − 2 β 1 3 + β 1 2 − β 1 1 + 3 β 1 0 + β 9 − 7 β 1
b15 - 2*b13 + b12 - b11 + 3*b10 + b9 - 7*b1
ν 4 \nu^{4} ν 4 = = =
− 9 β 5 − 8 β 4 − 2 β 3 − 20 β 2 + 47 -9\beta_{5} - 8\beta_{4} - 2\beta_{3} - 20\beta_{2} + 47 − 9 β 5 − 8 β 4 − 2 β 3 − 2 0 β 2 + 4 7
-9*b5 - 8*b4 - 2*b3 - 20*b2 + 47
ν 5 \nu^{5} ν 5 = = =
− 10 β 15 + 23 β 13 − 14 β 12 + 8 β 11 − 27 β 10 − 12 β 9 + 56 β 1 -10\beta_{15} + 23\beta_{13} - 14\beta_{12} + 8\beta_{11} - 27\beta_{10} - 12\beta_{9} + 56\beta_1 − 1 0 β 1 5 + 2 3 β 1 3 − 1 4 β 1 2 + 8 β 1 1 − 2 7 β 1 0 − 1 2 β 9 + 5 6 β 1
-10*b15 + 23*b13 - 14*b12 + 8*b11 - 27*b10 - 12*b9 + 56*b1
ν 6 \nu^{6} ν 6 = = =
− 5 β 8 − 2 β 7 − 2 β 6 + 78 β 5 + 57 β 4 + 26 β 3 + 182 β 2 − 381 -5\beta_{8} - 2\beta_{7} - 2\beta_{6} + 78\beta_{5} + 57\beta_{4} + 26\beta_{3} + 182\beta_{2} - 381 − 5 β 8 − 2 β 7 − 2 β 6 + 7 8 β 5 + 5 7 β 4 + 2 6 β 3 + 1 8 2 β 2 − 3 8 1
-5*b8 - 2*b7 - 2*b6 + 78*b5 + 57*b4 + 26*b3 + 182*b2 - 381
ν 7 \nu^{7} ν 7 = = =
88 β 15 − 5 β 14 − 214 β 13 + 148 β 12 − 75 β 11 + 211 β 10 + ⋯ − 459 β 1 88 \beta_{15} - 5 \beta_{14} - 214 \beta_{13} + 148 \beta_{12} - 75 \beta_{11} + 211 \beta_{10} + \cdots - 459 \beta_1 8 8 β 1 5 − 5 β 1 4 − 2 1 4 β 1 3 + 1 4 8 β 1 2 − 7 5 β 1 1 + 2 1 1 β 1 0 + ⋯ − 4 5 9 β 1
88*b15 - 5*b14 - 214*b13 + 148*b12 - 75*b11 + 211*b10 + 125*b9 - 459*b1
ν 8 \nu^{8} ν 8 = = =
105 β 8 + 47 β 7 + 41 β 6 − 682 β 5 − 392 β 4 − 273 β 3 − 1612 β 2 + 3118 105\beta_{8} + 47\beta_{7} + 41\beta_{6} - 682\beta_{5} - 392\beta_{4} - 273\beta_{3} - 1612\beta_{2} + 3118 1 0 5 β 8 + 4 7 β 7 + 4 1 β 6 − 6 8 2 β 5 − 3 9 2 β 4 − 2 7 3 β 3 − 1 6 1 2 β 2 + 3 1 1 8
105*b8 + 47*b7 + 41*b6 - 682*b5 - 392*b4 - 273*b3 - 1612*b2 + 3118
ν 9 \nu^{9} ν 9 = = =
− 764 β 15 + 105 β 14 + 1905 β 13 − 1454 β 12 + 746 β 11 + ⋯ + 3800 β 1 - 764 \beta_{15} + 105 \beta_{14} + 1905 \beta_{13} - 1454 \beta_{12} + 746 \beta_{11} + \cdots + 3800 \beta_1 − 7 6 4 β 1 5 + 1 0 5 β 1 4 + 1 9 0 5 β 1 3 − 1 4 5 4 β 1 2 + 7 4 6 β 1 1 + ⋯ + 3 8 0 0 β 1
-764*b15 + 105*b14 + 1905*b13 - 1454*b12 + 746*b11 - 1593*b10 - 1226*b9 + 3800*b1
ν 10 \nu^{10} ν 1 0 = = =
− 1476 β 8 − 672 β 7 − 596 β 6 + 6000 β 5 + 2608 β 4 + 2680 β 3 + ⋯ − 25693 - 1476 \beta_{8} - 672 \beta_{7} - 596 \beta_{6} + 6000 \beta_{5} + 2608 \beta_{4} + 2680 \beta_{3} + \cdots - 25693 − 1 4 7 6 β 8 − 6 7 2 β 7 − 5 9 6 β 6 + 6 0 0 0 β 5 + 2 6 0 8 β 4 + 2 6 8 0 β 3 + ⋯ − 2 5 6 9 3
-1476*b8 - 672*b7 - 596*b6 + 6000*b5 + 2608*b4 + 2680*b3 + 14155*b2 - 25693
ν 11 \nu^{11} ν 1 1 = = =
6688 β 15 − 1476 β 14 − 16884 β 13 + 13872 β 12 − 7384 β 11 + ⋯ − 31693 β 1 6688 \beta_{15} - 1476 \beta_{14} - 16884 \beta_{13} + 13872 \beta_{12} - 7384 \beta_{11} + \cdots - 31693 \beta_1 6 6 8 8 β 1 5 − 1 4 7 6 β 1 4 − 1 6 8 8 4 β 1 3 + 1 3 8 7 2 β 1 2 − 7 3 8 4 β 1 1 + ⋯ − 3 1 6 9 3 β 1
6688*b15 - 1476*b14 - 16884*b13 + 13872*b12 - 7384*b11 + 11824*b10 + 11623*b9 - 31693*b1
ν 12 \nu^{12} ν 1 2 = = =
17603 β 8 + 7880 β 7 + 7509 β 6 − 52949 β 5 − 16546 β 4 − 25495 β 3 + ⋯ + 213109 17603 \beta_{8} + 7880 \beta_{7} + 7509 \beta_{6} - 52949 \beta_{5} - 16546 \beta_{4} - 25495 \beta_{3} + \cdots + 213109 1 7 6 0 3 β 8 + 7 8 8 0 β 7 + 7 5 0 9 β 6 − 5 2 9 4 9 β 5 − 1 6 5 4 6 β 4 − 2 5 4 9 5 β 3 + ⋯ + 2 1 3 1 0 9
17603*b8 + 7880*b7 + 7509*b6 - 52949*b5 - 16546*b4 - 25495*b3 - 124034*b2 + 213109
ν 13 \nu^{13} ν 1 3 = = =
− 59273 β 15 + 17603 β 14 + 150533 β 13 − 130248 β 12 + 71737 β 11 + ⋯ + 266058 β 1 - 59273 \beta_{15} + 17603 \beta_{14} + 150533 \beta_{13} - 130248 \beta_{12} + 71737 \beta_{11} + \cdots + 266058 \beta_1 − 5 9 2 7 3 β 1 5 + 1 7 6 0 3 β 1 4 + 1 5 0 5 3 3 β 1 3 − 1 3 0 2 4 8 β 1 2 + 7 1 7 3 7 β 1 1 + ⋯ + 2 6 6 0 5 8 β 1
-59273*b15 + 17603*b14 + 150533*b13 - 130248*b12 + 71737*b11 - 86424*b10 - 107859*b9 + 266058*b1
ν 14 \nu^{14} ν 1 4 = = =
− 192790 β 8 − 83439 β 7 − 87260 β 6 + 468043 β 5 + 96772 β 4 + ⋯ − 1778929 - 192790 \beta_{8} - 83439 \beta_{7} - 87260 \beta_{6} + 468043 \beta_{5} + 96772 \beta_{4} + \cdots - 1778929 − 1 9 2 7 9 0 β 8 − 8 3 4 3 9 β 7 − 8 7 2 6 0 β 6 + 4 6 8 0 4 3 β 5 + 9 6 7 7 2 β 4 + ⋯ − 1 7 7 8 9 2 9
-192790*b8 - 83439*b7 - 87260*b6 + 468043*b5 + 96772*b4 + 238107*b3 + 1087237*b2 - 1778929
ν 15 \nu^{15} ν 1 5 = = =
531490 β 15 − 192790 β 14 − 1351947 β 13 + 1209998 β 12 − 684646 β 11 + ⋯ − 2246972 β 1 531490 \beta_{15} - 192790 \beta_{14} - 1351947 \beta_{13} + 1209998 \beta_{12} - 684646 \beta_{11} + \cdots - 2246972 \beta_1 5 3 1 4 9 0 β 1 5 − 1 9 2 7 9 0 β 1 4 − 1 3 5 1 9 4 7 β 1 3 + 1 2 0 9 9 9 8 β 1 2 − 6 8 4 6 4 6 β 1 1 + ⋯ − 2 2 4 6 9 7 2 β 1
531490*b15 - 192790*b14 - 1351947*b13 + 1209998*b12 - 684646*b11 + 619644*b10 + 986644*b9 - 2246972*b1
Character values
We give the values of χ \chi χ on generators for ( Z / 1682 Z ) × \left(\mathbb{Z}/1682\mathbb{Z}\right)^\times ( Z / 1 6 8 2 Z ) × .
n n n
843 843 8 4 3
χ ( n ) \chi(n) χ ( n )
− 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 2 n e w ( 1682 , [ χ ] ) S_{2}^{\mathrm{new}}(1682, [\chi]) S 2 n e w ( 1 6 8 2 , [ χ ] ) :
T 3 16 + 37 T 3 14 + 548 T 3 12 + 4119 T 3 10 + 16415 T 3 8 + 33099 T 3 6 + 30128 T 3 4 + 10537 T 3 2 + 961 T_{3}^{16} + 37T_{3}^{14} + 548T_{3}^{12} + 4119T_{3}^{10} + 16415T_{3}^{8} + 33099T_{3}^{6} + 30128T_{3}^{4} + 10537T_{3}^{2} + 961 T 3 1 6 + 3 7 T 3 1 4 + 5 4 8 T 3 1 2 + 4 1 1 9 T 3 1 0 + 1 6 4 1 5 T 3 8 + 3 3 0 9 9 T 3 6 + 3 0 1 2 8 T 3 4 + 1 0 5 3 7 T 3 2 + 9 6 1
T3^16 + 37*T3^14 + 548*T3^12 + 4119*T3^10 + 16415*T3^8 + 33099*T3^6 + 30128*T3^4 + 10537*T3^2 + 961
T 5 8 + 5 T 5 7 − 30 T 5 6 − 160 T 5 5 + 265 T 5 4 + 1650 T 5 3 − 300 T 5 2 − 5400 T 5 − 3600 T_{5}^{8} + 5T_{5}^{7} - 30T_{5}^{6} - 160T_{5}^{5} + 265T_{5}^{4} + 1650T_{5}^{3} - 300T_{5}^{2} - 5400T_{5} - 3600 T 5 8 + 5 T 5 7 − 3 0 T 5 6 − 1 6 0 T 5 5 + 2 6 5 T 5 4 + 1 6 5 0 T 5 3 − 3 0 0 T 5 2 − 5 4 0 0 T 5 − 3 6 0 0
T5^8 + 5*T5^7 - 30*T5^6 - 160*T5^5 + 265*T5^4 + 1650*T5^3 - 300*T5^2 - 5400*T5 - 3600
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T 2 + 1 ) 8 (T^{2} + 1)^{8} ( T 2 + 1 ) 8
(T^2 + 1)^8
3 3 3
T 16 + 37 T 14 + ⋯ + 961 T^{16} + 37 T^{14} + \cdots + 961 T 1 6 + 3 7 T 1 4 + ⋯ + 9 6 1
T^16 + 37*T^14 + 548*T^12 + 4119*T^10 + 16415*T^8 + 33099*T^6 + 30128*T^4 + 10537*T^2 + 961
5 5 5
( T 8 + 5 T 7 + ⋯ − 3600 ) 2 (T^{8} + 5 T^{7} + \cdots - 3600)^{2} ( T 8 + 5 T 7 + ⋯ − 3 6 0 0 ) 2
(T^8 + 5*T^7 - 30*T^6 - 160*T^5 + 265*T^4 + 1650*T^3 - 300*T^2 - 5400*T - 3600)^2
7 7 7
( T 8 − 7 T 7 + ⋯ + 976 ) 2 (T^{8} - 7 T^{7} + \cdots + 976)^{2} ( T 8 − 7 T 7 + ⋯ + 9 7 6 ) 2
(T^8 - 7*T^7 - 17*T^6 + 221*T^5 - 285*T^4 - 1424*T^3 + 4528*T^2 - 4232*T + 976)^2
11 11 1 1
T 16 + 93 T 14 + ⋯ + 2653641 T^{16} + 93 T^{14} + \cdots + 2653641 T 1 6 + 9 3 T 1 4 + ⋯ + 2 6 5 3 6 4 1
T^16 + 93*T^14 + 3488*T^12 + 68526*T^10 + 766690*T^8 + 4922301*T^6 + 16956828*T^4 + 24837408*T^2 + 2653641
13 13 1 3
( T 8 + 13 T 7 + ⋯ − 464 ) 2 (T^{8} + 13 T^{7} + \cdots - 464)^{2} ( T 8 + 1 3 T 7 + ⋯ − 4 6 4 ) 2
(T^8 + 13*T^7 + 28*T^6 - 284*T^5 - 1545*T^4 - 1954*T^3 + 1108*T^2 + 1208*T - 464)^2
17 17 1 7
T 16 + 147 T 14 + ⋯ + 6305121 T^{16} + 147 T^{14} + \cdots + 6305121 T 1 6 + 1 4 7 T 1 4 + ⋯ + 6 3 0 5 1 2 1
T^16 + 147*T^14 + 7983*T^12 + 196209*T^10 + 2187810*T^8 + 11122839*T^6 + 24331833*T^4 + 21314502*T^2 + 6305121
19 19 1 9
T 16 + 93 T 14 + ⋯ + 2595321 T^{16} + 93 T^{14} + \cdots + 2595321 T 1 6 + 9 3 T 1 4 + ⋯ + 2 5 9 5 3 2 1
T^16 + 93*T^14 + 3368*T^12 + 61191*T^10 + 605635*T^8 + 3294471*T^6 + 9282348*T^4 + 11226573*T^2 + 2595321
23 23 2 3
( T 8 − 12 T 7 + ⋯ − 144 ) 2 (T^{8} - 12 T^{7} + \cdots - 144)^{2} ( T 8 − 1 2 T 7 + ⋯ − 1 4 4 ) 2
(T^8 - 12*T^7 + 8*T^6 + 246*T^5 - 65*T^4 - 1914*T^3 - 2712*T^2 - 1152*T - 144)^2
29 29 2 9
T 16 T^{16} T 1 6
T^16
31 31 3 1
T 16 + ⋯ + 701190400 T^{16} + \cdots + 701190400 T 1 6 + ⋯ + 7 0 1 1 9 0 4 0 0
T^16 + 205*T^14 + 16670*T^12 + 692040*T^10 + 15660665*T^8 + 188376900*T^6 + 1046943200*T^4 + 1672595200*T^2 + 701190400
37 37 3 7
T 16 + ⋯ + 2590402816 T^{16} + \cdots + 2590402816 T 1 6 + ⋯ + 2 5 9 0 4 0 2 8 1 6
T^16 + 162*T^14 + 10663*T^12 + 369114*T^10 + 7306905*T^8 + 84645264*T^6 + 559620448*T^4 + 1920175872*T^2 + 2590402816
41 41 4 1
T 16 + 168 T 14 + ⋯ + 77841 T^{16} + 168 T^{14} + \cdots + 77841 T 1 6 + 1 6 8 T 1 4 + ⋯ + 7 7 8 4 1
T^16 + 168*T^14 + 10988*T^12 + 349176*T^10 + 5412790*T^8 + 34153176*T^6 + 31206348*T^4 + 7448328*T^2 + 77841
43 43 4 3
T 16 + ⋯ + 13610668861696 T^{16} + \cdots + 13610668861696 T 1 6 + ⋯ + 1 3 6 1 0 6 6 8 8 6 1 6 9 6
T^16 + 408*T^14 + 70678*T^12 + 6773436*T^10 + 391201905*T^8 + 13848513876*T^6 + 289935197728*T^4 + 3207077453568*T^2 + 13610668861696
47 47 4 7
T 16 + ⋯ + 6106750361856 T^{16} + \cdots + 6106750361856 T 1 6 + ⋯ + 6 1 0 6 7 5 0 3 6 1 8 5 6
T^16 + 393*T^14 + 64658*T^12 + 5775216*T^10 + 303487225*T^8 + 9501323316*T^6 + 171123149088*T^4 + 1612477214208*T^2 + 6106750361856
53 53 5 3
( T 8 − 4 T 7 + ⋯ − 924624 ) 2 (T^{8} - 4 T^{7} + \cdots - 924624)^{2} ( T 8 − 4 T 7 + ⋯ − 9 2 4 6 2 4 ) 2
(T^8 - 4*T^7 - 213*T^6 + 758*T^5 + 11335*T^4 - 52428*T^3 - 102648*T^2 + 769464*T - 924624)^2
59 59 5 9
( T 8 − 8 T 7 + ⋯ + 5024961 ) 2 (T^{8} - 8 T^{7} + \cdots + 5024961)^{2} ( T 8 − 8 T 7 + ⋯ + 5 0 2 4 9 6 1 ) 2
(T^8 - 8*T^7 - 217*T^6 + 1169*T^5 + 16630*T^4 - 48891*T^3 - 519177*T^2 + 504927*T + 5024961)^2
61 61 6 1
T 16 + ⋯ + 173602222336 T^{16} + \cdots + 173602222336 T 1 6 + ⋯ + 1 7 3 6 0 2 2 2 2 3 3 6
T^16 + 337*T^14 + 43718*T^12 + 2903844*T^10 + 107726705*T^8 + 2240699604*T^6 + 24425370848*T^4 + 117188119552*T^2 + 173602222336
67 67 6 7
( T 8 + 34 T 7 + ⋯ + 967471 ) 2 (T^{8} + 34 T^{7} + \cdots + 967471)^{2} ( T 8 + 3 4 T 7 + ⋯ + 9 6 7 4 7 1 ) 2
(T^8 + 34*T^7 + 337*T^6 - 473*T^5 - 28290*T^4 - 173287*T^3 - 336623*T^2 + 201221*T + 967471)^2
71 71 7 1
( T 8 − 11 T 7 + ⋯ + 6717456 ) 2 (T^{8} - 11 T^{7} + \cdots + 6717456)^{2} ( T 8 − 1 1 T 7 + ⋯ + 6 7 1 7 4 5 6 ) 2
(T^8 - 11*T^7 - 238*T^6 + 2252*T^5 + 17635*T^4 - 113292*T^3 - 613608*T^2 + 1403496*T + 6717456)^2
73 73 7 3
T 16 + 293 T 14 + ⋯ + 1745041 T^{16} + 293 T^{14} + \cdots + 1745041 T 1 6 + 2 9 3 T 1 4 + ⋯ + 1 7 4 5 0 4 1
T^16 + 293*T^14 + 27578*T^12 + 1118616*T^10 + 20957405*T^8 + 196626936*T^6 + 904041578*T^4 + 1625837813*T^2 + 1745041
79 79 7 9
T 16 + ⋯ + 12606362695936 T^{16} + \cdots + 12606362695936 T 1 6 + ⋯ + 1 2 6 0 6 3 6 2 6 9 5 9 3 6
T^16 + 568*T^14 + 125378*T^12 + 13772961*T^10 + 811210805*T^8 + 26275155156*T^6 + 458097301088*T^4 + 3921853884928*T^2 + 12606362695936
83 83 8 3
( T 8 − 10 T 7 + ⋯ + 110386845 ) 2 (T^{8} - 10 T^{7} + \cdots + 110386845)^{2} ( T 8 − 1 0 T 7 + ⋯ + 1 1 0 3 8 6 8 4 5 ) 2
(T^8 - 10*T^7 - 435*T^6 + 3500*T^5 + 67630*T^4 - 353550*T^3 - 4639200*T^2 + 10060920*T + 110386845)^2
89 89 8 9
T 16 + ⋯ + 6276723504921 T^{16} + \cdots + 6276723504921 T 1 6 + ⋯ + 6 2 7 6 7 2 3 5 0 4 9 2 1
T^16 + 512*T^14 + 107898*T^12 + 12104414*T^10 + 778880035*T^8 + 28673559534*T^6 + 559296390123*T^4 + 4615409856087*T^2 + 6276723504921
97 97 9 7
T 16 + ⋯ + 24 ⋯ 76 T^{16} + \cdots + 24\!\cdots\!76 T 1 6 + ⋯ + 2 4 ⋯ 7 6
T^16 + 1017*T^14 + 419578*T^12 + 90728184*T^10 + 11075022105*T^8 + 763576277424*T^6 + 28049003392288*T^4 + 476707997969472*T^2 + 2486595616082176
show more
show less