gp: [N,k,chi] = [7605,2,Mod(1,7605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7605, 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("7605.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,1,0,5,2,0,3,9,0,1,7]
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 β = 1 2 ( 1 + 17 ) \beta = \frac{1}{2}(1 + \sqrt{17}) β = 2 1 ( 1 + 1 7 ) q q q trace form .
For each embedding ι m \iota_m ι m ι m ( a n ) \iota_m(a_n) ι m ( a n )
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
p p p
Sign
3 3 3
− 1 -1 − 1
5 5 5
− 1 -1 − 1
13 13 1 3
− 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 ( 7605 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(7605)) S 2 n e w ( Γ 0 ( 7 6 0 5 ) )
T 2 2 − T 2 − 4 T_{2}^{2} - T_{2} - 4 T 2 2 − T 2 − 4
T 7 2 − 3 T 7 − 2 T_{7}^{2} - 3T_{7} - 2 T 7 2 − 3 T 7 − 2
T 11 2 − 7 T 11 + 8 T_{11}^{2} - 7T_{11} + 8 T 1 1 2 − 7 T 1 1 + 8
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − T − 4 T^{2} - T - 4 T 2 − T − 4
3 3 3
T 2 T^{2} T 2
5 5 5
( T − 1 ) 2 (T - 1)^{2} ( T − 1 ) 2
7 7 7
T 2 − 3 T − 2 T^{2} - 3T - 2 T 2 − 3 T − 2
11 11 1 1
T 2 − 7 T + 8 T^{2} - 7T + 8 T 2 − 7 T + 8
13 13 1 3
T 2 T^{2} T 2
17 17 1 7
T 2 + 5 T + 2 T^{2} + 5T + 2 T 2 + 5 T + 2
19 19 1 9
T 2 + 6 T − 8 T^{2} + 6T - 8 T 2 + 6 T − 8
23 23 2 3
T 2 − 7 T + 8 T^{2} - 7T + 8 T 2 − 7 T + 8
29 29 2 9
T 2 − 68 T^{2} - 68 T 2 − 6 8
31 31 3 1
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
37 37 3 7
T 2 + 7 T + 8 T^{2} + 7T + 8 T 2 + 7 T + 8
41 41 4 1
T 2 − 5 T + 2 T^{2} - 5T + 2 T 2 − 5 T + 2
43 43 4 3
T 2 + 6 T − 8 T^{2} + 6T - 8 T 2 + 6 T − 8
47 47 4 7
( T + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
53 53 5 3
T 2 + T − 106 T^{2} + T - 106 T 2 + T − 1 0 6
59 59 5 9
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
61 61 6 1
T 2 + 21 T + 106 T^{2} + 21T + 106 T 2 + 2 1 T + 1 0 6
67 67 6 7
T 2 − 14 T + 32 T^{2} - 14T + 32 T 2 − 1 4 T + 3 2
71 71 7 1
T 2 + 5 T − 32 T^{2} + 5T - 32 T 2 + 5 T − 3 2
73 73 7 3
T 2 + 14 T + 32 T^{2} + 14T + 32 T 2 + 1 4 T + 3 2
79 79 7 9
T 2 + 7 T + 8 T^{2} + 7T + 8 T 2 + 7 T + 8
83 83 8 3
T 2 + 2 T − 152 T^{2} + 2T - 152 T 2 + 2 T − 1 5 2
89 89 8 9
T 2 − 15 T + 18 T^{2} - 15T + 18 T 2 − 1 5 T + 1 8
97 97 9 7
T 2 + T − 208 T^{2} + T - 208 T 2 + T − 2 0 8
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