gp: [N,k,chi] = [7200,2,Mod(6049,7200)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7200, 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("7200.6049");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,8,0,
10,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,-4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == 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 / 7200 Z ) × \left(\mathbb{Z}/7200\mathbb{Z}\right)^\times ( Z / 7 2 0 0 Z ) × .
n n n
577 577 5 7 7
901 901 9 0 1
6401 6401 6 4 0 1
6751 6751 6 7 5 1
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 2 n e w ( 7200 , [ χ ] ) S_{2}^{\mathrm{new}}(7200, [\chi]) S 2 n e w ( 7 2 0 0 , [ χ ] ) :
T 7 2 + 9 T_{7}^{2} + 9 T 7 2 + 9
T7^2 + 9
T 11 T_{11} T 1 1
T11
T 13 2 + 25 T_{13}^{2} + 25 T 1 3 2 + 2 5
T13^2 + 25
T 17 T_{17} T 1 7
T17
T 19 − 5 T_{19} - 5 T 1 9 − 5
T19 - 5
T 29 − 4 T_{29} - 4 T 2 9 − 4
T29 - 4
T 31 − 5 T_{31} - 5 T 3 1 − 5
T31 - 5
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 + 9 T^{2} + 9 T 2 + 9
T^2 + 9
11 11 1 1
T 2 T^{2} T 2
T^2
13 13 1 3
T 2 + 25 T^{2} + 25 T 2 + 2 5
T^2 + 25
17 17 1 7
T 2 T^{2} T 2
T^2
19 19 1 9
( T − 5 ) 2 (T - 5)^{2} ( T − 5 ) 2
(T - 5)^2
23 23 2 3
T 2 + 16 T^{2} + 16 T 2 + 1 6
T^2 + 16
29 29 2 9
( T − 4 ) 2 (T - 4)^{2} ( T − 4 ) 2
(T - 4)^2
31 31 3 1
( T − 5 ) 2 (T - 5)^{2} ( T − 5 ) 2
(T - 5)^2
37 37 3 7
T 2 + 100 T^{2} + 100 T 2 + 1 0 0
T^2 + 100
41 41 4 1
( T − 10 ) 2 (T - 10)^{2} ( T − 1 0 ) 2
(T - 10)^2
43 43 4 3
T 2 + 1 T^{2} + 1 T 2 + 1
T^2 + 1
47 47 4 7
T 2 + 4 T^{2} + 4 T 2 + 4
T^2 + 4
53 53 5 3
T 2 + 100 T^{2} + 100 T 2 + 1 0 0
T^2 + 100
59 59 5 9
( T − 10 ) 2 (T - 10)^{2} ( T − 1 0 ) 2
(T - 10)^2
61 61 6 1
( T + 5 ) 2 (T + 5)^{2} ( T + 5 ) 2
(T + 5)^2
67 67 6 7
T 2 + 9 T^{2} + 9 T 2 + 9
T^2 + 9
71 71 7 1
( T + 10 ) 2 (T + 10)^{2} ( T + 1 0 ) 2
(T + 10)^2
73 73 7 3
T 2 + 100 T^{2} + 100 T 2 + 1 0 0
T^2 + 100
79 79 7 9
T 2 T^{2} T 2
T^2
83 83 8 3
T 2 + 196 T^{2} + 196 T 2 + 1 9 6
T^2 + 196
89 89 8 9
( T − 16 ) 2 (T - 16)^{2} ( T − 1 6 ) 2
(T - 16)^2
97 97 9 7
T 2 + 25 T^{2} + 25 T 2 + 2 5
T^2 + 25
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