gp: [N,k,chi] = [1920,2,Mod(769,1920)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1920, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1920.769");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,4,0,0,0,-2,0,0,0,0,0,-2]
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 i = − 1 i = \sqrt{-1} i = − 1 q q q trace form .
Character values
We give the values of χ \chi χ ( Z / 1920 Z ) × \left(\mathbb{Z}/1920\mathbb{Z}\right)^\times ( Z / 1 9 2 0 Z ) ×
n n n 511 511 5 1 1 641 641 6 4 1 901 901 9 0 1 1537 1537 1 5 3 7
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 ι m ( a n ) \iota_m(a_n) ι m ( a n )
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S 2 n e w ( 1920 , [ χ ] ) S_{2}^{\mathrm{new}}(1920, [\chi]) S 2 n e w ( 1 9 2 0 , [ χ ] )
T 7 2 + 16 T_{7}^{2} + 16 T 7 2 + 1 6
T 11 T_{11} T 1 1
T 19 + 6 T_{19} + 6 T 1 9 + 6
T 29 − 8 T_{29} - 8 T 2 9 − 8
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 T^{2} T 2
3 3 3
T 2 + 1 T^{2} + 1 T 2 + 1
5 5 5
T 2 − 4 T + 5 T^{2} - 4T + 5 T 2 − 4 T + 5
7 7 7
T 2 + 16 T^{2} + 16 T 2 + 1 6
11 11 1 1
T 2 T^{2} T 2
13 13 1 3
T 2 + 36 T^{2} + 36 T 2 + 3 6
17 17 1 7
T 2 + 4 T^{2} + 4 T 2 + 4
19 19 1 9
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
23 23 2 3
T 2 + 36 T^{2} + 36 T 2 + 3 6
29 29 2 9
( T − 8 ) 2 (T - 8)^{2} ( T − 8 ) 2
31 31 3 1
( T + 8 ) 2 (T + 8)^{2} ( T + 8 ) 2
37 37 3 7
T 2 + 100 T^{2} + 100 T 2 + 1 0 0
41 41 4 1
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
43 43 4 3
T 2 + 16 T^{2} + 16 T 2 + 1 6
47 47 4 7
T 2 + 4 T^{2} + 4 T 2 + 4
53 53 5 3
T 2 + 36 T^{2} + 36 T 2 + 3 6
59 59 5 9
( T − 12 ) 2 (T - 12)^{2} ( T − 1 2 ) 2
61 61 6 1
( T − 14 ) 2 (T - 14)^{2} ( T − 1 4 ) 2
67 67 6 7
T 2 + 16 T^{2} + 16 T 2 + 1 6
71 71 7 1
( T + 8 ) 2 (T + 8)^{2} ( T + 8 ) 2
73 73 7 3
T 2 + 16 T^{2} + 16 T 2 + 1 6
79 79 7 9
( T − 12 ) 2 (T - 12)^{2} ( T − 1 2 ) 2
83 83 8 3
T 2 + 64 T^{2} + 64 T 2 + 6 4
89 89 8 9
( T + 6 ) 2 (T + 6)^{2} ( T + 6 ) 2
97 97 9 7
T 2 T^{2} T 2
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