gp: [N,k,chi] = [9360,2,Mod(1,9360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("9360.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level :
N N N
= = = 9360 = 2 4 ⋅ 3 2 ⋅ 5 ⋅ 13 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 9 3 6 0 = 2 4 ⋅ 3 2 ⋅ 5 ⋅ 1 3
Weight :
k k k
= = = 2 2 2
Character orbit :
[ χ ] [\chi] [ χ ]
= = = 9360.a (trivial)
Newform invariants
sage: traces = [2,0,0,0,2,0,-4,0,0,0,-6,0,2,0,0,0,0,0,2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == 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 = 3 \beta = \sqrt{3} β = 3 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
2 2 2
− 1 -1 − 1
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 ( 9360 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(9360)) S 2 n e w ( Γ 0 ( 9 3 6 0 ) )
T 7 + 2 T_{7} + 2 T 7 + 2
T 11 2 + 6 T 11 + 6 T_{11}^{2} + 6T_{11} + 6 T 1 1 2 + 6 T 1 1 + 6
T 17 2 − 12 T_{17}^{2} - 12 T 1 7 2 − 1 2
T 19 2 − 2 T 19 − 26 T_{19}^{2} - 2T_{19} - 26 T 1 9 2 − 2 T 1 9 − 2 6
T 31 2 + 10 T 31 − 2 T_{31}^{2} + 10T_{31} - 2 T 3 1 2 + 1 0 T 3 1 − 2
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 T^{2} T 2
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 ) 2 (T + 2)^{2} ( T + 2 ) 2
11 11 1 1
T 2 + 6 T + 6 T^{2} + 6T + 6 T 2 + 6 T + 6
13 13 1 3
( T − 1 ) 2 (T - 1)^{2} ( T − 1 ) 2
17 17 1 7
T 2 − 12 T^{2} - 12 T 2 − 1 2
19 19 1 9
T 2 − 2 T − 26 T^{2} - 2T - 26 T 2 − 2 T − 2 6
23 23 2 3
T 2 − 6 T + 6 T^{2} - 6T + 6 T 2 − 6 T + 6
29 29 2 9
T 2 − 12 T + 24 T^{2} - 12T + 24 T 2 − 1 2 T + 2 4
31 31 3 1
T 2 + 10 T − 2 T^{2} + 10T - 2 T 2 + 1 0 T − 2
37 37 3 7
( T + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
41 41 4 1
T 2 − 12 T^{2} - 12 T 2 − 1 2
43 43 4 3
T 2 + 10 T − 2 T^{2} + 10T - 2 T 2 + 1 0 T − 2
47 47 4 7
( T − 6 ) 2 (T - 6)^{2} ( T − 6 ) 2
53 53 5 3
T 2 − 108 T^{2} - 108 T 2 − 1 0 8
59 59 5 9
T 2 + 6 T − 138 T^{2} + 6T - 138 T 2 + 6 T − 1 3 8
61 61 6 1
T 2 − 4 T − 104 T^{2} - 4T - 104 T 2 − 4 T − 1 0 4
67 67 6 7
T 2 − 8 T − 92 T^{2} - 8T - 92 T 2 − 8 T − 9 2
71 71 7 1
T 2 − 6 T + 6 T^{2} - 6T + 6 T 2 − 6 T + 6
73 73 7 3
( T + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
79 79 7 9
T 2 + 4 T − 104 T^{2} + 4T - 104 T 2 + 4 T − 1 0 4
83 83 8 3
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
89 89 8 9
T 2 − 12 T − 12 T^{2} - 12T - 12 T 2 − 1 2 T − 1 2
97 97 9 7
( T − 2 ) 2 (T - 2)^{2} ( T − 2 ) 2
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