gp: [N,k,chi] = [1800,4,Mod(1,1800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1800.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [1,0,0,0,0,0,12,0,0,0,-64,0,-58,0,0,0,32]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == 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
2 2 2
+ 1 +1 + 1
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 intersection of the kernels of the following linear operators acting on S 4 n e w ( Γ 0 ( 1800 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(1800)) S 4 n e w ( Γ 0 ( 1 8 0 0 ) ) :
T 7 − 12 T_{7} - 12 T 7 − 1 2
T7 - 12
T 11 + 64 T_{11} + 64 T 1 1 + 6 4
T11 + 64
T 17 − 32 T_{17} - 32 T 1 7 − 3 2
T17 - 32
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T T T
T
3 3 3
T T T
T
5 5 5
T T T
T
7 7 7
T − 12 T - 12 T − 1 2
T - 12
11 11 1 1
T + 64 T + 64 T + 6 4
T + 64
13 13 1 3
T + 58 T + 58 T + 5 8
T + 58
17 17 1 7
T − 32 T - 32 T − 3 2
T - 32
19 19 1 9
T + 136 T + 136 T + 1 3 6
T + 136
23 23 2 3
T + 128 T + 128 T + 1 2 8
T + 128
29 29 2 9
T − 144 T - 144 T − 1 4 4
T - 144
31 31 3 1
T − 20 T - 20 T − 2 0
T - 20
37 37 3 7
T − 18 T - 18 T − 1 8
T - 18
41 41 4 1
T − 288 T - 288 T − 2 8 8
T - 288
43 43 4 3
T − 200 T - 200 T − 2 0 0
T - 200
47 47 4 7
T − 384 T - 384 T − 3 8 4
T - 384
53 53 5 3
T − 496 T - 496 T − 4 9 6
T - 496
59 59 5 9
T − 128 T - 128 T − 1 2 8
T - 128
61 61 6 1
T + 458 T + 458 T + 4 5 8
T + 458
67 67 6 7
T − 496 T - 496 T − 4 9 6
T - 496
71 71 7 1
T + 512 T + 512 T + 5 1 2
T + 512
73 73 7 3
T − 602 T - 602 T − 6 0 2
T - 602
79 79 7 9
T − 1108 T - 1108 T − 1 1 0 8
T - 1108
83 83 8 3
T − 704 T - 704 T − 7 0 4
T - 704
89 89 8 9
T − 960 T - 960 T − 9 6 0
T - 960
97 97 9 7
T + 206 T + 206 T + 2 0 6
T + 206
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