gp: [N,k,chi] = [4650,2,Mod(1,4650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4650.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4650, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
Newform invariants
sage: traces = [2,-2,-2,2,0,2,-1,-2,2,0,7,-2,4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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 + 33 ) \beta = \frac{1}{2}(1 + \sqrt{33}) β = 2 1 ( 1 + 3 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
3 3 3
+ 1 +1 + 1
5 5 5
+ 1 +1 + 1
31 31 3 1
+ 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 ( 4650 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(4650)) S 2 n e w ( Γ 0 ( 4 6 5 0 ) ) :
T 7 2 + T 7 − 8 T_{7}^{2} + T_{7} - 8 T 7 2 + T 7 − 8
T7^2 + T7 - 8
T 11 2 − 7 T 11 + 4 T_{11}^{2} - 7T_{11} + 4 T 1 1 2 − 7 T 1 1 + 4
T11^2 - 7*T11 + 4
T 13 − 2 T_{13} - 2 T 1 3 − 2
T13 - 2
T 19 2 + 7 T 19 + 4 T_{19}^{2} + 7T_{19} + 4 T 1 9 2 + 7 T 1 9 + 4
T19^2 + 7*T19 + 4
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
(T + 1)^2
3 3 3
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
(T + 1)^2
5 5 5
T 2 T^{2} T 2
T^2
7 7 7
T 2 + T − 8 T^{2} + T - 8 T 2 + T − 8
T^2 + T - 8
11 11 1 1
T 2 − 7 T + 4 T^{2} - 7T + 4 T 2 − 7 T + 4
T^2 - 7*T + 4
13 13 1 3
( T − 2 ) 2 (T - 2)^{2} ( T − 2 ) 2
(T - 2)^2
17 17 1 7
T 2 + 2 T − 32 T^{2} + 2T - 32 T 2 + 2 T − 3 2
T^2 + 2*T - 32
19 19 1 9
T 2 + 7 T + 4 T^{2} + 7T + 4 T 2 + 7 T + 4
T^2 + 7*T + 4
23 23 2 3
T 2 + T − 8 T^{2} + T - 8 T 2 + T − 8
T^2 + T - 8
29 29 2 9
T 2 + 6 T − 24 T^{2} + 6T - 24 T 2 + 6 T − 2 4
T^2 + 6*T - 24
31 31 3 1
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
(T + 1)^2
37 37 3 7
T 2 + 10 T − 8 T^{2} + 10T - 8 T 2 + 1 0 T − 8
T^2 + 10*T - 8
41 41 4 1
T 2 + 10 T − 8 T^{2} + 10T - 8 T 2 + 1 0 T − 8
T^2 + 10*T - 8
43 43 4 3
T 2 + 7 T + 4 T^{2} + 7T + 4 T 2 + 7 T + 4
T^2 + 7*T + 4
47 47 4 7
T 2 − 2 T − 32 T^{2} - 2T - 32 T 2 − 2 T − 3 2
T^2 - 2*T - 32
53 53 5 3
T 2 − 3 T − 6 T^{2} - 3T - 6 T 2 − 3 T − 6
T^2 - 3*T - 6
59 59 5 9
T 2 + 6 T − 24 T^{2} + 6T - 24 T 2 + 6 T − 2 4
T^2 + 6*T - 24
61 61 6 1
T 2 − 132 T^{2} - 132 T 2 − 1 3 2
T^2 - 132
67 67 6 7
T 2 + 10 T − 8 T^{2} + 10T - 8 T 2 + 1 0 T − 8
T^2 + 10*T - 8
71 71 7 1
T 2 − T − 8 T^{2} - T - 8 T 2 − T − 8
T^2 - T - 8
73 73 7 3
T 2 + T − 74 T^{2} + T - 74 T 2 + T − 7 4
T^2 + T - 74
79 79 7 9
T 2 + 15 T + 48 T^{2} + 15T + 48 T 2 + 1 5 T + 4 8
T^2 + 15*T + 48
83 83 8 3
( T − 12 ) 2 (T - 12)^{2} ( T − 1 2 ) 2
(T - 12)^2
89 89 8 9
T 2 − 3 T − 6 T^{2} - 3T - 6 T 2 − 3 T − 6
T^2 - 3*T - 6
97 97 9 7
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
(T + 2)^2
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