gp: [N,k,chi] = [528,4,Mod(65,528)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(528, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("528.65");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,8,0,0,0,0,0,10,0,0]
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 -expansion are expressed in terms of β = − 11 \beta = \sqrt{-11} β = − 1 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 / 528 Z ) × \left(\mathbb{Z}/528\mathbb{Z}\right)^\times ( Z / 5 2 8 Z ) × .
n n n
133 133 1 3 3
145 145 1 4 5
353 353 3 5 3
463 463 4 6 3
χ ( n ) \chi(n) χ ( n )
1 1 1
− 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 ( 528 , [ χ ] ) S_{4}^{\mathrm{new}}(528, [\chi]) S 4 n e w ( 5 2 8 , [ χ ] ) :
T 5 2 + 176 T_{5}^{2} + 176 T 5 2 + 1 7 6
T5^2 + 176
T 17 T_{17} T 1 7
T17
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 − 8 T + 27 T^{2} - 8T + 27 T 2 − 8 T + 2 7
T^2 - 8*T + 27
5 5 5
T 2 + 176 T^{2} + 176 T 2 + 1 7 6
T^2 + 176
7 7 7
T 2 T^{2} T 2
T^2
11 11 1 1
T 2 + 1331 T^{2} + 1331 T 2 + 1 3 3 1
T^2 + 1331
13 13 1 3
T 2 T^{2} T 2
T^2
17 17 1 7
T 2 T^{2} T 2
T^2
19 19 1 9
T 2 T^{2} T 2
T^2
23 23 2 3
T 2 + 37004 T^{2} + 37004 T 2 + 3 7 0 0 4
T^2 + 37004
29 29 2 9
T 2 T^{2} T 2
T^2
31 31 3 1
( T − 340 ) 2 (T - 340)^{2} ( T − 3 4 0 ) 2
(T - 340)^2
37 37 3 7
( T − 434 ) 2 (T - 434)^{2} ( T − 4 3 4 ) 2
(T - 434)^2
41 41 4 1
T 2 T^{2} T 2
T^2
43 43 4 3
T 2 T^{2} T 2
T^2
47 47 4 7
T 2 + 413996 T^{2} + 413996 T 2 + 4 1 3 9 9 6
T^2 + 413996
53 53 5 3
T 2 + 50864 T^{2} + 50864 T 2 + 5 0 8 6 4
T^2 + 50864
59 59 5 9
T 2 + 303116 T^{2} + 303116 T 2 + 3 0 3 1 1 6
T^2 + 303116
61 61 6 1
T 2 T^{2} T 2
T^2
67 67 6 7
( T + 416 ) 2 (T + 416)^{2} ( T + 4 1 6 ) 2
(T + 416)^2
71 71 7 1
T 2 + 1057100 T^{2} + 1057100 T 2 + 1 0 5 7 1 0 0
T^2 + 1057100
73 73 7 3
T 2 T^{2} T 2
T^2
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 + 17600 T^{2} + 17600 T 2 + 1 7 6 0 0
T^2 + 17600
97 97 9 7
( T + 34 ) 2 (T + 34)^{2} ( T + 3 4 ) 2
(T + 34)^2
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