gp: [N,k,chi] = [525,2,Mod(1,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, 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("525.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,2,6,0,0,-2,0,2,0,4]
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 = 5 \beta = \sqrt{5} β = 5 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
7 7 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 ( 525 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(525)) S 2 n e w ( Γ 0 ( 5 2 5 ) )
T 2 2 − 5 T_{2}^{2} - 5 T 2 2 − 5
T 11 2 − 4 T 11 − 16 T_{11}^{2} - 4T_{11} - 16 T 1 1 2 − 4 T 1 1 − 1 6
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − 5 T^{2} - 5 T 2 − 5
3 3 3
( T − 1 ) 2 (T - 1)^{2} ( T − 1 ) 2
5 5 5
T 2 T^{2} T 2
7 7 7
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
11 11 1 1
T 2 − 4 T − 16 T^{2} - 4T - 16 T 2 − 4 T − 1 6
13 13 1 3
T 2 − 20 T^{2} - 20 T 2 − 2 0
17 17 1 7
( T − 2 ) 2 (T - 2)^{2} ( T − 2 ) 2
19 19 1 9
T 2 − 4 T − 16 T^{2} - 4T - 16 T 2 − 4 T − 1 6
23 23 2 3
( T + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
29 29 2 9
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
31 31 3 1
T 2 − 12 T + 16 T^{2} - 12T + 16 T 2 − 1 2 T + 1 6
37 37 3 7
T 2 + 4 T − 76 T^{2} + 4T - 76 T 2 + 4 T − 7 6
41 41 4 1
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
43 43 4 3
T 2 − 80 T^{2} - 80 T 2 − 8 0
47 47 4 7
T 2 + 8 T − 64 T^{2} + 8T - 64 T 2 + 8 T − 6 4
53 53 5 3
T 2 − 16 T + 44 T^{2} - 16T + 44 T 2 − 1 6 T + 4 4
59 59 5 9
T 2 − 80 T^{2} - 80 T 2 − 8 0
61 61 6 1
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
67 67 6 7
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
71 71 7 1
T 2 − 20 T + 80 T^{2} - 20T + 80 T 2 − 2 0 T + 8 0
73 73 7 3
T 2 − 16 T + 44 T^{2} - 16T + 44 T 2 − 1 6 T + 4 4
79 79 7 9
T 2 − 8 T − 64 T^{2} - 8T - 64 T 2 − 8 T − 6 4
83 83 8 3
T 2 − 16 T − 16 T^{2} - 16T - 16 T 2 − 1 6 T − 1 6
89 89 8 9
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
97 97 9 7
T 2 + 8 T − 4 T^{2} + 8T - 4 T 2 + 8 T − 4
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