gp: [N,k,chi] = [45,6,Mod(1,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
sage: traces = [1,2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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)
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
5 5 5
− 1 -1 − 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 2 − 2 T_{2} - 2 T 2 − 2
T2 - 2
acting on S 6 n e w ( Γ 0 ( 45 ) ) S_{6}^{\mathrm{new}}(\Gamma_0(45)) S 6 n e w ( Γ 0 ( 4 5 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T − 2 T - 2 T − 2
T - 2
3 3 3
T T T
T
5 5 5
T − 25 T - 25 T − 2 5
T - 25
7 7 7
T + 132 T + 132 T + 1 3 2
T + 132
11 11 1 1
T + 472 T + 472 T + 4 7 2
T + 472
13 13 1 3
T + 686 T + 686 T + 6 8 6
T + 686
17 17 1 7
T − 1562 T - 1562 T − 1 5 6 2
T - 1562
19 19 1 9
T + 2180 T + 2180 T + 2 1 8 0
T + 2180
23 23 2 3
T + 264 T + 264 T + 2 6 4
T + 264
29 29 2 9
T + 170 T + 170 T + 1 7 0
T + 170
31 31 3 1
T − 7272 T - 7272 T − 7 2 7 2
T - 7272
37 37 3 7
T + 142 T + 142 T + 1 4 2
T + 142
41 41 4 1
T − 16198 T - 16198 T − 1 6 1 9 8
T - 16198
43 43 4 3
T + 10316 T + 10316 T + 1 0 3 1 6
T + 10316
47 47 4 7
T + 18568 T + 18568 T + 1 8 5 6 8
T + 18568
53 53 5 3
T + 21514 T + 21514 T + 2 1 5 1 4
T + 21514
59 59 5 9
T + 34600 T + 34600 T + 3 4 6 0 0
T + 34600
61 61 6 1
T + 35738 T + 35738 T + 3 5 7 3 8
T + 35738
67 67 6 7
T + 5772 T + 5772 T + 5 7 7 2
T + 5772
71 71 7 1
T − 69088 T - 69088 T − 6 9 0 8 8
T - 69088
73 73 7 3
T + 70526 T + 70526 T + 7 0 5 2 6
T + 70526
79 79 7 9
T − 47640 T - 47640 T − 4 7 6 4 0
T - 47640
83 83 8 3
T + 74004 T + 74004 T + 7 4 0 0 4
T + 74004
89 89 8 9
T − 90030 T - 90030 T − 9 0 0 3 0
T - 90030
97 97 9 7
T + 33502 T + 33502 T + 3 3 5 0 2
T + 33502
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