gp: [N,k,chi] = [357,2,Mod(1,357)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(357, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("357.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [3,1,3,3,2,1,3]
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
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
3 3 3
− 1 -1 − 1
7 7 7
− 1 -1 − 1
17 17 1 7
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
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 ( 357 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(357)) S 2 n e w ( Γ 0 ( 3 5 7 ) ) :
T 2 3 − T 2 2 − 4 T 2 + 2 T_{2}^{3} - T_{2}^{2} - 4T_{2} + 2 T 2 3 − T 2 2 − 4 T 2 + 2
T2^3 - T2^2 - 4*T2 + 2
T 11 3 − 6 T 11 2 + 5 T 11 + 4 T_{11}^{3} - 6T_{11}^{2} + 5T_{11} + 4 T 1 1 3 − 6 T 1 1 2 + 5 T 1 1 + 4
T11^3 - 6*T11^2 + 5*T11 + 4
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 3 − T 2 − 4 T + 2 T^{3} - T^{2} - 4T + 2 T 3 − T 2 − 4 T + 2
T^3 - T^2 - 4*T + 2
3 3 3
( T − 1 ) 3 (T - 1)^{3} ( T − 1 ) 3
(T - 1)^3
5 5 5
T 3 − 2 T 2 + ⋯ + 2 T^{3} - 2 T^{2} + \cdots + 2 T 3 − 2 T 2 + ⋯ + 2
T^3 - 2*T^2 - 3*T + 2
7 7 7
( T − 1 ) 3 (T - 1)^{3} ( T − 1 ) 3
(T - 1)^3
11 11 1 1
T 3 − 6 T 2 + ⋯ + 4 T^{3} - 6 T^{2} + \cdots + 4 T 3 − 6 T 2 + ⋯ + 4
T^3 - 6*T^2 + 5*T + 4
13 13 1 3
T 3 + 2 T 2 + ⋯ − 62 T^{3} + 2 T^{2} + \cdots - 62 T 3 + 2 T 2 + ⋯ − 6 2
T^3 + 2*T^2 - 23*T - 62
17 17 1 7
( T − 1 ) 3 (T - 1)^{3} ( T − 1 ) 3
(T - 1)^3
19 19 1 9
T 3 + 4 T 2 + T − 4 T^{3} + 4T^{2} + T - 4 T 3 + 4 T 2 + T − 4
T^3 + 4*T^2 + T - 4
23 23 2 3
T 3 − 6 T 2 + ⋯ + 8 T^{3} - 6 T^{2} + \cdots + 8 T 3 − 6 T 2 + ⋯ + 8
T^3 - 6*T^2 + 5*T + 8
29 29 2 9
T 3 − 6 T 2 + ⋯ + 64 T^{3} - 6 T^{2} + \cdots + 64 T 3 − 6 T 2 + ⋯ + 6 4
T^3 - 6*T^2 - 16*T + 64
31 31 3 1
T 3 + 14 T 2 + ⋯ + 64 T^{3} + 14 T^{2} + \cdots + 64 T 3 + 1 4 T 2 + ⋯ + 6 4
T^3 + 14*T^2 + 58*T + 64
37 37 3 7
T 3 + 4 T 2 + ⋯ + 68 T^{3} + 4 T^{2} + \cdots + 68 T 3 + 4 T 2 + ⋯ + 6 8
T^3 + 4*T^2 - 50*T + 68
41 41 4 1
T 3 − 14 T 2 + ⋯ − 82 T^{3} - 14 T^{2} + \cdots - 82 T 3 − 1 4 T 2 + ⋯ − 8 2
T^3 - 14*T^2 + 61*T - 82
43 43 4 3
T 3 − 2 T 2 + ⋯ − 124 T^{3} - 2 T^{2} + \cdots - 124 T 3 − 2 T 2 + ⋯ − 1 2 4
T^3 - 2*T^2 - 59*T - 124
47 47 4 7
T 3 + 2 T 2 + ⋯ − 352 T^{3} + 2 T^{2} + \cdots - 352 T 3 + 2 T 2 + ⋯ − 3 5 2
T^3 + 2*T^2 - 94*T - 352
53 53 5 3
T 3 + 4 T 2 + ⋯ − 668 T^{3} + 4 T^{2} + \cdots - 668 T 3 + 4 T 2 + ⋯ − 6 6 8
T^3 + 4*T^2 - 178*T - 668
59 59 5 9
T 3 + 2 T 2 + ⋯ − 152 T^{3} + 2 T^{2} + \cdots - 152 T 3 + 2 T 2 + ⋯ − 1 5 2
T^3 + 2*T^2 - 78*T - 152
61 61 6 1
T 3 + 12 T 2 + ⋯ − 108 T^{3} + 12 T^{2} + \cdots - 108 T 3 + 1 2 T 2 + ⋯ − 1 0 8
T^3 + 12*T^2 - 18*T - 108
67 67 6 7
T 3 − 12 T 2 + ⋯ + 688 T^{3} - 12 T^{2} + \cdots + 688 T 3 − 1 2 T 2 + ⋯ + 6 8 8
T^3 - 12*T^2 - 108*T + 688
71 71 7 1
T 3 − 172 T + 352 T^{3} - 172T + 352 T 3 − 1 7 2 T + 3 5 2
T^3 - 172*T + 352
73 73 7 3
T 3 + 22 T 2 + ⋯ − 232 T^{3} + 22 T^{2} + \cdots - 232 T 3 + 2 2 T 2 + ⋯ − 2 3 2
T^3 + 22*T^2 + 100*T - 232
79 79 7 9
T 3 − 2 T 2 + ⋯ + 32 T^{3} - 2 T^{2} + \cdots + 32 T 3 − 2 T 2 + ⋯ + 3 2
T^3 - 2*T^2 - 14*T + 32
83 83 8 3
T 3 − 24 T 2 + ⋯ − 272 T^{3} - 24 T^{2} + \cdots - 272 T 3 − 2 4 T 2 + ⋯ − 2 7 2
T^3 - 24*T^2 + 164*T - 272
89 89 8 9
T 3 + 10 T 2 + ⋯ − 656 T^{3} + 10 T^{2} + \cdots - 656 T 3 + 1 0 T 2 + ⋯ − 6 5 6
T^3 + 10*T^2 - 64*T - 656
97 97 9 7
T 3 + 6 T 2 + ⋯ − 64 T^{3} + 6 T^{2} + \cdots - 64 T 3 + 6 T 2 + ⋯ − 6 4
T^3 + 6*T^2 - 16*T - 64
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