gp: [N,k,chi] = [588,4,Mod(293,588)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("588.293");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,0,0,0,0,-54,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-250,0,0,0,
0,0,0,0,0,0,0,0,-646]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == 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 = − 3 \beta = \sqrt{-3} β = − 3 .
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 / 588 Z ) × \left(\mathbb{Z}/588\mathbb{Z}\right)^\times ( Z / 5 8 8 Z ) × .
n n n
197 197 1 9 7
295 295 2 9 5
493 493 4 9 3
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
1 1 1
− 1 -1 − 1
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 4 n e w ( 588 , [ χ ] ) S_{4}^{\mathrm{new}}(588, [\chi]) S 4 n e w ( 5 8 8 , [ χ ] ) :
T 5 T_{5} T 5
T5
T 13 2 + 8427 T_{13}^{2} + 8427 T 1 3 2 + 8 4 2 7
T13^2 + 8427
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 + 27 T^{2} + 27 T 2 + 2 7
T^2 + 27
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 + 8427 T^{2} + 8427 T 2 + 8 4 2 7
T^2 + 8427
17 17 1 7
T 2 T^{2} T 2
T^2
19 19 1 9
T 2 + 15987 T^{2} + 15987 T 2 + 1 5 9 8 7
T^2 + 15987
23 23 2 3
T 2 T^{2} T 2
T^2
29 29 2 9
T 2 T^{2} T 2
T^2
31 31 3 1
T 2 + 35643 T^{2} + 35643 T 2 + 3 5 6 4 3
T^2 + 35643
37 37 3 7
( T + 323 ) 2 (T + 323)^{2} ( T + 3 2 3 ) 2
(T + 323)^2
41 41 4 1
T 2 T^{2} T 2
T^2
43 43 4 3
( T + 71 ) 2 (T + 71)^{2} ( T + 7 1 ) 2
(T + 71)^2
47 47 4 7
T 2 T^{2} T 2
T^2
53 53 5 3
T 2 T^{2} T 2
T^2
59 59 5 9
T 2 T^{2} T 2
T^2
61 61 6 1
T 2 + 874800 T^{2} + 874800 T 2 + 8 7 4 8 0 0
T^2 + 874800
67 67 6 7
( T + 127 ) 2 (T + 127)^{2} ( T + 1 2 7 ) 2
(T + 127)^2
71 71 7 1
T 2 T^{2} T 2
T^2
73 73 7 3
T 2 + 1482627 T^{2} + 1482627 T 2 + 1 4 8 2 6 2 7
T^2 + 1482627
79 79 7 9
( T + 1387 ) 2 (T + 1387)^{2} ( T + 1 3 8 7 ) 2
(T + 1387)^2
83 83 8 3
T 2 T^{2} T 2
T^2
89 89 8 9
T 2 T^{2} T 2
T^2
97 97 9 7
T 2 + 1881792 T^{2} + 1881792 T 2 + 1 8 8 1 7 9 2
T^2 + 1881792
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