gp: [N,k,chi] = [800,3,Mod(193,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.193");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,0,0,0,0,0,0,0,0,-14]
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 i = − 1 i = \sqrt{-1} i = − 1 .
We also show the integral q q q -expansion of the trace form .
Character values
We give the values of χ \chi χ on generators for ( Z / 800 Z ) × \left(\mathbb{Z}/800\mathbb{Z}\right)^\times ( Z / 8 0 0 Z ) × .
n n n
101 101 1 0 1
351 351 3 5 1
577 577 5 7 7
χ ( n ) \chi(n) χ ( n )
1 1 1
1 1 1
i i i
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
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 3 n e w ( 800 , [ χ ] ) S_{3}^{\mathrm{new}}(800, [\chi]) S 3 n e w ( 8 0 0 , [ χ ] ) :
T 3 T_{3} T 3
T3
T 13 2 + 14 T 13 + 98 T_{13}^{2} + 14T_{13} + 98 T 1 3 2 + 1 4 T 1 3 + 9 8
T13^2 + 14*T13 + 98
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^{2} T 2
T^2
5 5 5
T 2 T^{2} T 2
T^2
7 7 7
T 2 T^{2} T 2
T^2
11 11 1 1
T 2 T^{2} T 2
T^2
13 13 1 3
T 2 + 14 T + 98 T^{2} + 14T + 98 T 2 + 1 4 T + 9 8
T^2 + 14*T + 98
17 17 1 7
T 2 + 46 T + 1058 T^{2} + 46T + 1058 T 2 + 4 6 T + 1 0 5 8
T^2 + 46*T + 1058
19 19 1 9
T 2 T^{2} T 2
T^2
23 23 2 3
T 2 T^{2} T 2
T^2
29 29 2 9
T 2 + 1600 T^{2} + 1600 T 2 + 1 6 0 0
T^2 + 1600
31 31 3 1
T 2 T^{2} T 2
T^2
37 37 3 7
T 2 − 46 T + 1058 T^{2} - 46T + 1058 T 2 − 4 6 T + 1 0 5 8
T^2 - 46*T + 1058
41 41 4 1
( T + 80 ) 2 (T + 80)^{2} ( T + 8 0 ) 2
(T + 80)^2
43 43 4 3
T 2 T^{2} T 2
T^2
47 47 4 7
T 2 T^{2} T 2
T^2
53 53 5 3
T 2 + 146 T + 10658 T^{2} + 146T + 10658 T 2 + 1 4 6 T + 1 0 6 5 8
T^2 + 146*T + 10658
59 59 5 9
T 2 T^{2} T 2
T^2
61 61 6 1
( T + 120 ) 2 (T + 120)^{2} ( T + 1 2 0 ) 2
(T + 120)^2
67 67 6 7
T 2 T^{2} T 2
T^2
71 71 7 1
T 2 T^{2} T 2
T^2
73 73 7 3
T 2 − 14 T + 98 T^{2} - 14T + 98 T 2 − 1 4 T + 9 8
T^2 - 14*T + 98
79 79 7 9
T 2 T^{2} T 2
T^2
83 83 8 3
T 2 T^{2} T 2
T^2
89 89 8 9
T 2 + 25600 T^{2} + 25600 T 2 + 2 5 6 0 0
T^2 + 25600
97 97 9 7
T 2 + 274 T + 37538 T^{2} + 274T + 37538 T 2 + 2 7 4 T + 3 7 5 3 8
T^2 + 274*T + 37538
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