gp: [N,k,chi] = [3344,2,Mod(1,3344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("3344.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,-1,0,-1,0,-1]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 -expansion are expressed in terms of β = 1 2 ( 1 + 13 ) \beta = \frac{1}{2}(1 + \sqrt{13}) β = 2 1 ( 1 + 1 3 ) .
We also show the integral q q q -expansion of the trace form .
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
11 11 1 1
− 1 -1 − 1
19 19 1 9
− 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 ( 3344 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(3344)) S 2 n e w ( Γ 0 ( 3 3 4 4 ) ) :
T 3 2 + T 3 − 3 T_{3}^{2} + T_{3} - 3 T 3 2 + T 3 − 3
T3^2 + T3 - 3
T 5 2 + T 5 − 3 T_{5}^{2} + T_{5} - 3 T 5 2 + T 5 − 3
T5^2 + T5 - 3
T 7 2 + T 7 − 3 T_{7}^{2} + T_{7} - 3 T 7 2 + T 7 − 3
T7^2 + T7 - 3
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 2 + T − 3 T^{2} + T - 3 T 2 + T − 3
T^2 + T - 3
5 5 5
T 2 + T − 3 T^{2} + T - 3 T 2 + T − 3
T^2 + T - 3
7 7 7
T 2 + T − 3 T^{2} + T - 3 T 2 + T − 3
T^2 + T - 3
11 11 1 1
( T − 1 ) 2 (T - 1)^{2} ( T − 1 ) 2
(T - 1)^2
13 13 1 3
T 2 + 7 T + 9 T^{2} + 7T + 9 T 2 + 7 T + 9
T^2 + 7*T + 9
17 17 1 7
T 2 − 2 T − 12 T^{2} - 2T - 12 T 2 − 2 T − 1 2
T^2 - 2*T - 12
19 19 1 9
( T − 1 ) 2 (T - 1)^{2} ( T − 1 ) 2
(T - 1)^2
23 23 2 3
T 2 − 2 T − 12 T^{2} - 2T - 12 T 2 − 2 T − 1 2
T^2 - 2*T - 12
29 29 2 9
T 2 + 15 T + 53 T^{2} + 15T + 53 T 2 + 1 5 T + 5 3
T^2 + 15*T + 53
31 31 3 1
T 2 − 5 T − 23 T^{2} - 5T - 23 T 2 − 5 T − 2 3
T^2 - 5*T - 23
37 37 3 7
( T − 8 ) 2 (T - 8)^{2} ( T − 8 ) 2
(T - 8)^2
41 41 4 1
T 2 + 11 T + 27 T^{2} + 11T + 27 T 2 + 1 1 T + 2 7
T^2 + 11*T + 27
43 43 4 3
T 2 − T − 3 T^{2} - T - 3 T 2 − T − 3
T^2 - T - 3
47 47 4 7
T 2 − 8 T − 36 T^{2} - 8T - 36 T 2 − 8 T − 3 6
T^2 - 8*T - 36
53 53 5 3
T 2 − 6 T − 4 T^{2} - 6T - 4 T 2 − 6 T − 4
T^2 - 6*T - 4
59 59 5 9
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
(T - 4)^2
61 61 6 1
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
(T + 2)^2
67 67 6 7
T 2 + 7 T − 69 T^{2} + 7T - 69 T 2 + 7 T − 6 9
T^2 + 7*T - 69
71 71 7 1
T 2 − 15 T + 27 T^{2} - 15T + 27 T 2 − 1 5 T + 2 7
T^2 - 15*T + 27
73 73 7 3
T 2 + 26 T + 156 T^{2} + 26T + 156 T 2 + 2 6 T + 1 5 6
T^2 + 26*T + 156
79 79 7 9
T 2 − 2 T − 116 T^{2} - 2T - 116 T 2 − 2 T − 1 1 6
T^2 - 2*T - 116
83 83 8 3
T 2 + 11 T + 1 T^{2} + 11T + 1 T 2 + 1 1 T + 1
T^2 + 11*T + 1
89 89 8 9
T 2 − 6 T − 108 T^{2} - 6T - 108 T 2 − 6 T − 1 0 8
T^2 - 6*T - 108
97 97 9 7
T 2 − 208 T^{2} - 208 T 2 − 2 0 8
T^2 - 208
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