gp: [N,k,chi] = [4050,2,Mod(1,4050)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4050, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4050.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,-2,0,2,0,0,4,-2,0,0,-2,0,0,-4,0,2,-2,0,-6]
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 = 6 \beta = \sqrt{6} β = 6 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
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 ( 4050 ) ) S_{2}^{\mathrm{new}}(\Gamma_0(4050)) S 2 n e w ( Γ 0 ( 4 0 5 0 ) )
T 7 2 − 4 T 7 − 2 T_{7}^{2} - 4T_{7} - 2 T 7 2 − 4 T 7 − 2
T 11 2 + 2 T 11 − 5 T_{11}^{2} + 2T_{11} - 5 T 1 1 2 + 2 T 1 1 − 5
T 13 2 − 6 T_{13}^{2} - 6 T 1 3 2 − 6
T 17 2 + 2 T 17 − 23 T_{17}^{2} + 2T_{17} - 23 T 1 7 2 + 2 T 1 7 − 2 3
T 23 2 − 4 T 23 − 20 T_{23}^{2} - 4T_{23} - 20 T 2 3 2 − 4 T 2 3 − 2 0
T 41 + 1 T_{41} + 1 T 4 1 + 1
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
3 3 3
T 2 T^{2} T 2
5 5 5
T 2 T^{2} T 2
7 7 7
T 2 − 4 T − 2 T^{2} - 4T - 2 T 2 − 4 T − 2
11 11 1 1
T 2 + 2 T − 5 T^{2} + 2T - 5 T 2 + 2 T − 5
13 13 1 3
T 2 − 6 T^{2} - 6 T 2 − 6
17 17 1 7
T 2 + 2 T − 23 T^{2} + 2T - 23 T 2 + 2 T − 2 3
19 19 1 9
T 2 + 6 T + 3 T^{2} + 6T + 3 T 2 + 6 T + 3
23 23 2 3
T 2 − 4 T − 20 T^{2} - 4T - 20 T 2 − 4 T − 2 0
29 29 2 9
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
31 31 3 1
T 2 − 8 T + 10 T^{2} - 8T + 10 T 2 − 8 T + 1 0
37 37 3 7
( T − 8 ) 2 (T - 8)^{2} ( T − 8 ) 2
41 41 4 1
( T + 1 ) 2 (T + 1)^{2} ( T + 1 ) 2
43 43 4 3
T 2 − 10 T + 19 T^{2} - 10T + 19 T 2 − 1 0 T + 1 9
47 47 4 7
T 2 + 4 T − 2 T^{2} + 4T - 2 T 2 + 4 T − 2
53 53 5 3
T 2 − 12 T + 30 T^{2} - 12T + 30 T 2 − 1 2 T + 3 0
59 59 5 9
T 2 − 2 T − 149 T^{2} - 2T - 149 T 2 − 2 T − 1 4 9
61 61 6 1
T 2 − 4 T − 2 T^{2} - 4T - 2 T 2 − 4 T − 2
67 67 6 7
T 2 − 14 T + 43 T^{2} - 14T + 43 T 2 − 1 4 T + 4 3
71 71 7 1
T 2 − 6 T^{2} - 6 T 2 − 6
73 73 7 3
T 2 − 10 T − 71 T^{2} - 10T - 71 T 2 − 1 0 T − 7 1
79 79 7 9
T 2 − 54 T^{2} - 54 T 2 − 5 4
83 83 8 3
( T + 4 ) 2 (T + 4)^{2} ( T + 4 ) 2
89 89 8 9
T 2 − 16 T + 40 T^{2} - 16T + 40 T 2 − 1 6 T + 4 0
97 97 9 7
( T − 13 ) 2 (T - 13)^{2} ( T − 1 3 ) 2
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