gp: [N,k,chi] = [650,4,Mod(1,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("650.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,-4,7,8,0,-14,5,-16,23]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == 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 + 105 ) \beta = \frac{1}{2}(1 + \sqrt{105}) β = 2 1 ( 1 + 1 0 5 ) .
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
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 4 n e w ( Γ 0 ( 650 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(650)) S 4 n e w ( Γ 0 ( 6 5 0 ) ) :
T 3 2 − 7 T 3 − 14 T_{3}^{2} - 7T_{3} - 14 T 3 2 − 7 T 3 − 1 4
T3^2 - 7*T3 - 14
T 7 2 − 5 T 7 − 20 T_{7}^{2} - 5T_{7} - 20 T 7 2 − 5 T 7 − 2 0
T7^2 - 5*T7 - 20
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
(T + 2)^2
3 3 3
T 2 − 7 T − 14 T^{2} - 7T - 14 T 2 − 7 T − 1 4
T^2 - 7*T - 14
5 5 5
T 2 T^{2} T 2
T^2
7 7 7
T 2 − 5 T − 20 T^{2} - 5T - 20 T 2 − 5 T − 2 0
T^2 - 5*T - 20
11 11 1 1
T 2 + 51 T − 6 T^{2} + 51T - 6 T 2 + 5 1 T − 6
T^2 + 51*T - 6
13 13 1 3
( T + 13 ) 2 (T + 13)^{2} ( T + 1 3 ) 2
(T + 13)^2
17 17 1 7
T 2 − 63 T − 294 T^{2} - 63T - 294 T 2 − 6 3 T − 2 9 4
T^2 - 63*T - 294
19 19 1 9
T 2 − 28 T − 3584 T^{2} - 28T - 3584 T 2 − 2 8 T − 3 5 8 4
T^2 - 28*T - 3584
23 23 2 3
T 2 − 9 T − 7566 T^{2} - 9T - 7566 T 2 − 9 T − 7 5 6 6
T^2 - 9*T - 7566
29 29 2 9
T 2 + 198 T + 8121 T^{2} + 198T + 8121 T 2 + 1 9 8 T + 8 1 2 1
T^2 + 198*T + 8121
31 31 3 1
T 2 − 175 T + 4480 T^{2} - 175T + 4480 T 2 − 1 7 5 T + 4 4 8 0
T^2 - 175*T + 4480
37 37 3 7
T 2 − 632 T + 96076 T^{2} - 632T + 96076 T 2 − 6 3 2 T + 9 6 0 7 6
T^2 - 632*T + 96076
41 41 4 1
T 2 − 210 T − 1680 T^{2} - 210T - 1680 T 2 − 2 1 0 T − 1 6 8 0
T^2 - 210*T - 1680
43 43 4 3
T 2 − 23 T − 7454 T^{2} - 23T - 7454 T 2 − 2 3 T − 7 4 5 4
T^2 - 23*T - 7454
47 47 4 7
T 2 − 231 T − 78036 T^{2} - 231T - 78036 T 2 − 2 3 1 T − 7 8 0 3 6
T^2 - 231*T - 78036
53 53 5 3
T 2 + 186 T − 11931 T^{2} + 186T - 11931 T 2 + 1 8 6 T − 1 1 9 3 1
T^2 + 186*T - 11931
59 59 5 9
T 2 − 63 T − 294 T^{2} - 63T - 294 T 2 − 6 3 T − 2 9 4
T^2 - 63*T - 294
61 61 6 1
T 2 + 182 T − 213899 T^{2} + 182T - 213899 T 2 + 1 8 2 T − 2 1 3 8 9 9
T^2 + 182*T - 213899
67 67 6 7
T 2 − 695 T − 77930 T^{2} - 695T - 77930 T 2 − 6 9 5 T − 7 7 9 3 0
T^2 - 695*T - 77930
71 71 7 1
T 2 − 390 T − 909600 T^{2} - 390T - 909600 T 2 − 3 9 0 T − 9 0 9 6 0 0
T^2 - 390*T - 909600
73 73 7 3
T 2 − 1526 T + 453544 T^{2} - 1526 T + 453544 T 2 − 1 5 2 6 T + 4 5 3 5 4 4
T^2 - 1526*T + 453544
79 79 7 9
T 2 + 863 T + 5356 T^{2} + 863T + 5356 T 2 + 8 6 3 T + 5 3 5 6
T^2 + 863*T + 5356
83 83 8 3
T 2 − 315 T − 557970 T^{2} - 315T - 557970 T 2 − 3 1 5 T − 5 5 7 9 7 0
T^2 - 315*T - 557970
89 89 8 9
T 2 + 1638 T + 170856 T^{2} + 1638 T + 170856 T 2 + 1 6 3 8 T + 1 7 0 8 5 6
T^2 + 1638*T + 170856
97 97 9 7
T 2 − 98 T − 829304 T^{2} - 98T - 829304 T 2 − 9 8 T − 8 2 9 3 0 4
T^2 - 98*T - 829304
show more
show less