gp: [N,k,chi] = [525,4,Mod(1,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("525.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,-1,-6,17,0,3,14,-33,18,0,-22]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 = 1 2 ( 1 + 65 ) \beta = \frac{1}{2}(1 + \sqrt{65}) β = 2 1 ( 1 + 6 5 ) 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
3 3 3
+ 1 +1 + 1
5 5 5
+ 1 +1 + 1
7 7 7
− 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 ( 525 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(525)) S 4 n e w ( Γ 0 ( 5 2 5 ) )
T 2 2 + T 2 − 16 T_{2}^{2} + T_{2} - 16 T 2 2 + T 2 − 1 6
T 11 2 + 22 T 11 + 56 T_{11}^{2} + 22T_{11} + 56 T 1 1 2 + 2 2 T 1 1 + 5 6
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 + T − 16 T^{2} + T - 16 T 2 + T − 1 6
3 3 3
( T + 3 ) 2 (T + 3)^{2} ( T + 3 ) 2
5 5 5
T 2 T^{2} T 2
7 7 7
( T − 7 ) 2 (T - 7)^{2} ( T − 7 ) 2
11 11 1 1
T 2 + 22 T + 56 T^{2} + 22T + 56 T 2 + 2 2 T + 5 6
13 13 1 3
T 2 − 22 T + 56 T^{2} - 22T + 56 T 2 − 2 2 T + 5 6
17 17 1 7
T 2 + 116 T − 796 T^{2} + 116T - 796 T 2 + 1 1 6 T − 7 9 6
19 19 1 9
T 2 − 102 T − 584 T^{2} - 102T - 584 T 2 − 1 0 2 T − 5 8 4
23 23 2 3
T 2 + 260 T + 10400 T^{2} + 260T + 10400 T 2 + 2 6 0 T + 1 0 4 0 0
29 29 2 9
T 2 + 196 T − 27836 T^{2} + 196T - 27836 T 2 + 1 9 6 T − 2 7 8 3 6
31 31 3 1
T 2 − 150 T − 23040 T^{2} - 150T - 23040 T 2 − 1 5 0 T − 2 3 0 4 0
37 37 3 7
T 2 − 96 T − 18756 T^{2} - 96T - 18756 T 2 − 9 6 T − 1 8 7 5 6
41 41 4 1
T 2 + 176 T − 154756 T^{2} + 176T - 154756 T 2 + 1 7 6 T − 1 5 4 7 5 6
43 43 4 3
T 2 − 344 T + 12944 T^{2} - 344T + 12944 T 2 − 3 4 4 T + 1 2 9 4 4
47 47 4 7
T 2 + 560 T + 40960 T^{2} + 560T + 40960 T 2 + 5 6 0 T + 4 0 9 6 0
53 53 5 3
T 2 + 326 T − 93616 T^{2} + 326T - 93616 T 2 + 3 2 6 T − 9 3 6 1 6
59 59 5 9
T 2 + 844 T + 63424 T^{2} + 844T + 63424 T 2 + 8 4 4 T + 6 3 4 2 4
61 61 6 1
T 2 + 204 T + 1044 T^{2} + 204T + 1044 T 2 + 2 0 4 T + 1 0 4 4
67 67 6 7
T 2 − 104 T − 63856 T^{2} - 104T - 63856 T 2 − 1 0 4 T − 6 3 8 5 6
71 71 7 1
T 2 − 1670 T + 668560 T^{2} - 1670 T + 668560 T 2 − 1 6 7 0 T + 6 6 8 5 6 0
73 73 7 3
T 2 − 386 T − 625816 T^{2} - 386T - 625816 T 2 − 3 8 6 T − 6 2 5 8 1 6
79 79 7 9
T 2 + 888 T + 21376 T^{2} + 888T + 21376 T 2 + 8 8 8 T + 2 1 3 7 6
83 83 8 3
T 2 + 928 T − 542864 T^{2} + 928T - 542864 T 2 + 9 2 8 T − 5 4 2 8 6 4
89 89 8 9
T 2 − 588 T + 85396 T^{2} - 588T + 85396 T 2 − 5 8 8 T + 8 5 3 9 6
97 97 9 7
T 2 + 522 T − 1534064 T^{2} + 522 T - 1534064 T 2 + 5 2 2 T − 1 5 3 4 0 6 4
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