gp: [N,k,chi] = [768,2,Mod(385,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("768.385");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,0,0,8,0,-2,0,0,0,0,0,4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == 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 / 768 Z ) × \left(\mathbb{Z}/768\mathbb{Z}\right)^\times ( Z / 7 6 8 Z ) × .
n n n
257 257 2 5 7
511 511 5 1 1
517 517 5 1 7
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 2 n e w ( 768 , [ χ ] ) S_{2}^{\mathrm{new}}(768, [\chi]) S 2 n e w ( 7 6 8 , [ χ ] ) :
T 5 2 + 4 T_{5}^{2} + 4 T 5 2 + 4
T5^2 + 4
T 7 − 4 T_{7} - 4 T 7 − 4
T7 - 4
T 23 T_{23} T 2 3
T23
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 + 1 T^{2} + 1 T 2 + 1
T^2 + 1
5 5 5
T 2 + 4 T^{2} + 4 T 2 + 4
T^2 + 4
7 7 7
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
(T - 4)^2
11 11 1 1
T 2 + 16 T^{2} + 16 T 2 + 1 6
T^2 + 16
13 13 1 3
T 2 + 4 T^{2} + 4 T 2 + 4
T^2 + 4
17 17 1 7
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
(T + 6)^2
19 19 1 9
T 2 + 16 T^{2} + 16 T 2 + 1 6
T^2 + 16
23 23 2 3
T 2 T^{2} T 2
T^2
29 29 2 9
T 2 + 4 T^{2} + 4 T 2 + 4
T^2 + 4
31 31 3 1
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
(T - 4)^2
37 37 3 7
T 2 + 4 T^{2} + 4 T 2 + 4
T^2 + 4
41 41 4 1
( T + 2 ) 2 (T + 2)^{2} ( T + 2 ) 2
(T + 2)^2
43 43 4 3
T 2 + 16 T^{2} + 16 T 2 + 1 6
T^2 + 16
47 47 4 7
( T − 8 ) 2 (T - 8)^{2} ( T − 8 ) 2
(T - 8)^2
53 53 5 3
T 2 + 100 T^{2} + 100 T 2 + 1 0 0
T^2 + 100
59 59 5 9
T 2 + 16 T^{2} + 16 T 2 + 1 6
T^2 + 16
61 61 6 1
T 2 + 36 T^{2} + 36 T 2 + 3 6
T^2 + 36
67 67 6 7
T 2 + 16 T^{2} + 16 T 2 + 1 6
T^2 + 16
71 71 7 1
( T − 16 ) 2 (T - 16)^{2} ( T − 1 6 ) 2
(T - 16)^2
73 73 7 3
( T − 6 ) 2 (T - 6)^{2} ( T − 6 ) 2
(T - 6)^2
79 79 7 9
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
(T - 4)^2
83 83 8 3
T 2 + 144 T^{2} + 144 T 2 + 1 4 4
T^2 + 144
89 89 8 9
( T + 10 ) 2 (T + 10)^{2} ( T + 1 0 ) 2
(T + 10)^2
97 97 9 7
( T + 14 ) 2 (T + 14)^{2} ( T + 1 4 ) 2
(T + 14)^2
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