gp: [N,k,chi] = [9610,2,Mod(1,9610)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9610, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9610.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [6,6,0,6,6,0,8,6,22,6,0]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 5 1,\beta_1,\ldots,\beta_{5} 1 , β 1 , … , β 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 6 − 2 x 5 − 13 x 4 + 14 x 3 + 26 x 2 − 28 x + 1 x^{6} - 2x^{5} - 13x^{4} + 14x^{3} + 26x^{2} - 28x + 1 x 6 − 2 x 5 − 1 3 x 4 + 1 4 x 3 + 2 6 x 2 − 2 8 x + 1
x^6 - 2*x^5 - 13*x^4 + 14*x^3 + 26*x^2 - 28*x + 1
:
β 1 \beta_{1} β 1 = = =
( − 22 ν 5 + 15 ν 4 + 310 ν 3 + 95 ν 2 − 513 ν − 25 ) / 31 ( -22\nu^{5} + 15\nu^{4} + 310\nu^{3} + 95\nu^{2} - 513\nu - 25 ) / 31 ( − 2 2 ν 5 + 1 5 ν 4 + 3 1 0 ν 3 + 9 5 ν 2 − 5 1 3 ν − 2 5 ) / 3 1
(-22*v^5 + 15*v^4 + 310*v^3 + 95*v^2 - 513*v - 25) / 31
β 2 \beta_{2} β 2 = = =
( − 28 ν 5 + 36 ν 4 + 372 ν 3 − 82 ν 2 − 636 ν + 157 ) / 31 ( -28\nu^{5} + 36\nu^{4} + 372\nu^{3} - 82\nu^{2} - 636\nu + 157 ) / 31 ( − 2 8 ν 5 + 3 6 ν 4 + 3 7 2 ν 3 − 8 2 ν 2 − 6 3 6 ν + 1 5 7 ) / 3 1
(-28*v^5 + 36*v^4 + 372*v^3 - 82*v^2 - 636*v + 157) / 31
β 3 \beta_{3} β 3 = = =
( − 29 ν 5 + 24 ν 4 + 403 ν 3 + 59 ν 2 − 641 ν + 84 ) / 31 ( -29\nu^{5} + 24\nu^{4} + 403\nu^{3} + 59\nu^{2} - 641\nu + 84 ) / 31 ( − 2 9 ν 5 + 2 4 ν 4 + 4 0 3 ν 3 + 5 9 ν 2 − 6 4 1 ν + 8 4 ) / 3 1
(-29*v^5 + 24*v^4 + 403*v^3 + 59*v^2 - 641*v + 84) / 31
β 4 \beta_{4} β 4 = = =
( 41 ν 5 − 35 ν 4 − 589 ν 3 − 46 ν 2 + 1104 ν − 169 ) / 31 ( 41\nu^{5} - 35\nu^{4} - 589\nu^{3} - 46\nu^{2} + 1104\nu - 169 ) / 31 ( 4 1 ν 5 − 3 5 ν 4 − 5 8 9 ν 3 − 4 6 ν 2 + 1 1 0 4 ν − 1 6 9 ) / 3 1
(41*v^5 - 35*v^4 - 589*v^3 - 46*v^2 + 1104*v - 169) / 31
β 5 \beta_{5} β 5 = = =
( − 44 ν 5 + 30 ν 4 + 620 ν 3 + 190 ν 2 − 964 ν − 81 ) / 31 ( -44\nu^{5} + 30\nu^{4} + 620\nu^{3} + 190\nu^{2} - 964\nu - 81 ) / 31 ( − 4 4 ν 5 + 3 0 ν 4 + 6 2 0 ν 3 + 1 9 0 ν 2 − 9 6 4 ν − 8 1 ) / 3 1
(-44*v^5 + 30*v^4 + 620*v^3 + 190*v^2 - 964*v - 81) / 31
ν \nu ν = = =
( β 5 − 2 β 1 + 1 ) / 2 ( \beta_{5} - 2\beta _1 + 1 ) / 2 ( β 5 − 2 β 1 + 1 ) / 2
(b5 - 2*b1 + 1) / 2
ν 2 \nu^{2} ν 2 = = =
( 2 β 5 − 4 β 3 + β 2 + 11 ) / 2 ( 2\beta_{5} - 4\beta_{3} + \beta_{2} + 11 ) / 2 ( 2 β 5 − 4 β 3 + β 2 + 1 1 ) / 2
(2*b5 - 4*b3 + b2 + 11) / 2
ν 3 \nu^{3} ν 3 = = =
( 12 β 5 − 3 β 4 − 9 β 3 + β 2 − 19 β 1 + 19 ) / 2 ( 12\beta_{5} - 3\beta_{4} - 9\beta_{3} + \beta_{2} - 19\beta _1 + 19 ) / 2 ( 1 2 β 5 − 3 β 4 − 9 β 3 + β 2 − 1 9 β 1 + 1 9 ) / 2
(12*b5 - 3*b4 - 9*b3 + b2 - 19*b1 + 19) / 2
ν 4 \nu^{4} ν 4 = = =
( 39 β 5 − 4 β 4 − 60 β 3 + 17 β 2 − 28 β 1 + 134 ) / 2 ( 39\beta_{5} - 4\beta_{4} - 60\beta_{3} + 17\beta_{2} - 28\beta _1 + 134 ) / 2 ( 3 9 β 5 − 4 β 4 − 6 0 β 3 + 1 7 β 2 − 2 8 β 1 + 1 3 4 ) / 2
(39*b5 - 4*b4 - 60*b3 + 17*b2 - 28*b1 + 134) / 2
ν 5 \nu^{5} ν 5 = = =
( 181 β 5 − 45 β 4 − 185 β 3 + 30 β 2 − 243 β 1 + 381 ) / 2 ( 181\beta_{5} - 45\beta_{4} - 185\beta_{3} + 30\beta_{2} - 243\beta _1 + 381 ) / 2 ( 1 8 1 β 5 − 4 5 β 4 − 1 8 5 β 3 + 3 0 β 2 − 2 4 3 β 1 + 3 8 1 ) / 2
(181*b5 - 45*b4 - 185*b3 + 30*b2 - 243*b1 + 381) / 2
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
p p p
Sign
2 2 2
− 1 -1 − 1
5 5 5
− 1 -1 − 1
31 31 3 1
− 1 -1 − 1
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 2 n e w ( Γ 0 ( 9610 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(9610)) S 2 n e w ( Γ 0 ( 9 6 1 0 ) ) :
T 3 6 − 20 T 3 4 + 116 T 3 2 − 200 T_{3}^{6} - 20T_{3}^{4} + 116T_{3}^{2} - 200 T 3 6 − 2 0 T 3 4 + 1 1 6 T 3 2 − 2 0 0
T3^6 - 20*T3^4 + 116*T3^2 - 200
T 7 3 − 4 T 7 2 − 12 T 7 + 40 T_{7}^{3} - 4T_{7}^{2} - 12T_{7} + 40 T 7 3 − 4 T 7 2 − 1 2 T 7 + 4 0
T7^3 - 4*T7^2 - 12*T7 + 40
T 11 6 − 18 T 11 4 + 56 T 11 2 − 8 T_{11}^{6} - 18T_{11}^{4} + 56T_{11}^{2} - 8 T 1 1 6 − 1 8 T 1 1 4 + 5 6 T 1 1 2 − 8
T11^6 - 18*T11^4 + 56*T11^2 - 8
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T − 1 ) 6 (T - 1)^{6} ( T − 1 ) 6
(T - 1)^6
3 3 3
T 6 − 20 T 4 + ⋯ − 200 T^{6} - 20 T^{4} + \cdots - 200 T 6 − 2 0 T 4 + ⋯ − 2 0 0
T^6 - 20*T^4 + 116*T^2 - 200
5 5 5
( T − 1 ) 6 (T - 1)^{6} ( T − 1 ) 6
(T - 1)^6
7 7 7
( T 3 − 4 T 2 − 12 T + 40 ) 2 (T^{3} - 4 T^{2} - 12 T + 40)^{2} ( T 3 − 4 T 2 − 1 2 T + 4 0 ) 2
(T^3 - 4*T^2 - 12*T + 40)^2
11 11 1 1
T 6 − 18 T 4 + ⋯ − 8 T^{6} - 18 T^{4} + \cdots - 8 T 6 − 1 8 T 4 + ⋯ − 8
T^6 - 18*T^4 + 56*T^2 - 8
13 13 1 3
T 6 − 58 T 4 + ⋯ − 5000 T^{6} - 58 T^{4} + \cdots - 5000 T 6 − 5 8 T 4 + ⋯ − 5 0 0 0
T^6 - 58*T^4 + 1000*T^2 - 5000
17 17 1 7
T 6 T^{6} T 6
T^6
19 19 1 9
( T 3 + 6 T 2 + ⋯ − 200 ) 2 (T^{3} + 6 T^{2} + \cdots - 200)^{2} ( T 3 + 6 T 2 + ⋯ − 2 0 0 ) 2
(T^3 + 6*T^2 - 40*T - 200)^2
23 23 2 3
T 6 − 70 T 4 + ⋯ − 5000 T^{6} - 70 T^{4} + \cdots - 5000 T 6 − 7 0 T 4 + ⋯ − 5 0 0 0
T^6 - 70*T^4 + 1356*T^2 - 5000
29 29 2 9
T 6 − 60 T 4 + ⋯ − 200 T^{6} - 60 T^{4} + \cdots - 200 T 6 − 6 0 T 4 + ⋯ − 2 0 0
T^6 - 60*T^4 + 836*T^2 - 200
31 31 3 1
T 6 T^{6} T 6
T^6
37 37 3 7
T 6 − 210 T 4 + ⋯ − 192200 T^{6} - 210 T^{4} + \cdots - 192200 T 6 − 2 1 0 T 4 + ⋯ − 1 9 2 2 0 0
T^6 - 210*T^4 + 11736*T^2 - 192200
41 41 4 1
( T 3 − 16 T 2 + ⋯ − 40 ) 2 (T^{3} - 16 T^{2} + \cdots - 40)^{2} ( T 3 − 1 6 T 2 + ⋯ − 4 0 ) 2
(T^3 - 16*T^2 + 68*T - 40)^2
43 43 4 3
T 6 − 28 T 4 + ⋯ − 8 T^{6} - 28 T^{4} + \cdots - 8 T 6 − 2 8 T 4 + ⋯ − 8
T^6 - 28*T^4 + 36*T^2 - 8
47 47 4 7
( T 3 + 6 T 2 + ⋯ − 200 ) 2 (T^{3} + 6 T^{2} + \cdots - 200)^{2} ( T 3 + 6 T 2 + ⋯ − 2 0 0 ) 2
(T^3 + 6*T^2 - 40*T - 200)^2
53 53 5 3
T 6 − 122 T 4 + ⋯ − 5000 T^{6} - 122 T^{4} + \cdots - 5000 T 6 − 1 2 2 T 4 + ⋯ − 5 0 0 0
T^6 - 122*T^4 + 3800*T^2 - 5000
59 59 5 9
( T 3 − 2 T 2 − 16 T − 8 ) 2 (T^{3} - 2 T^{2} - 16 T - 8)^{2} ( T 3 − 2 T 2 − 1 6 T − 8 ) 2
(T^3 - 2*T^2 - 16*T - 8)^2
61 61 6 1
T 6 − 340 T 4 + ⋯ − 1065800 T^{6} - 340 T^{4} + \cdots - 1065800 T 6 − 3 4 0 T 4 + ⋯ − 1 0 6 5 8 0 0
T^6 - 340*T^4 + 34356*T^2 - 1065800
67 67 6 7
( T 3 + 4 T 2 − 12 T − 40 ) 2 (T^{3} + 4 T^{2} - 12 T - 40)^{2} ( T 3 + 4 T 2 − 1 2 T − 4 0 ) 2
(T^3 + 4*T^2 - 12*T - 40)^2
71 71 7 1
( T 3 − 4 T 2 − 64 T − 64 ) 2 (T^{3} - 4 T^{2} - 64 T - 64)^{2} ( T 3 − 4 T 2 − 6 4 T − 6 4 ) 2
(T^3 - 4*T^2 - 64*T - 64)^2
73 73 7 3
T 6 − 208 T 4 + ⋯ − 86528 T^{6} - 208 T^{4} + \cdots - 86528 T 6 − 2 0 8 T 4 + ⋯ − 8 6 5 2 8
T^6 - 208*T^4 + 10816*T^2 - 86528
79 79 7 9
T 6 − 552 T 4 + ⋯ − 6083072 T^{6} - 552 T^{4} + \cdots - 6083072 T 6 − 5 5 2 T 4 + ⋯ − 6 0 8 3 0 7 2
T^6 - 552*T^4 + 100736*T^2 - 6083072
83 83 8 3
T 6 − 396 T 4 + ⋯ − 773768 T^{6} - 396 T^{4} + \cdots - 773768 T 6 − 3 9 6 T 4 + ⋯ − 7 7 3 7 6 8
T^6 - 396*T^4 + 43172*T^2 - 773768
89 89 8 9
T 6 − 470 T 4 + ⋯ − 1065800 T^{6} - 470 T^{4} + \cdots - 1065800 T 6 − 4 7 0 T 4 + ⋯ − 1 0 6 5 8 0 0
T^6 - 470*T^4 + 52556*T^2 - 1065800
97 97 9 7
( T 3 − 14 T 2 + ⋯ − 40 ) 2 (T^{3} - 14 T^{2} + \cdots - 40)^{2} ( T 3 − 1 4 T 2 + ⋯ − 4 0 ) 2
(T^3 - 14*T^2 + 48*T - 40)^2
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