gp: [N,k,chi] = [300,3,Mod(101,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.101");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [1,0,3,0,0,0,13]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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)
Character values
We give the values of χ \chi χ on generators for ( Z / 300 Z ) × \left(\mathbb{Z}/300\mathbb{Z}\right)^\times ( Z / 3 0 0 Z ) × .
n n n
101 101 1 0 1
151 151 1 5 1
277 277 2 7 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 3 n e w ( 300 , [ χ ] ) S_{3}^{\mathrm{new}}(300, [\chi]) S 3 n e w ( 3 0 0 , [ χ ] ) :
T 7 − 13 T_{7} - 13 T 7 − 1 3
T7 - 13
T 11 T_{11} T 1 1
T11
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T T T
T
3 3 3
T − 3 T - 3 T − 3
T - 3
5 5 5
T T T
T
7 7 7
T − 13 T - 13 T − 1 3
T - 13
11 11 1 1
T T T
T
13 13 1 3
T + 23 T + 23 T + 2 3
T + 23
17 17 1 7
T T T
T
19 19 1 9
T − 11 T - 11 T − 1 1
T - 11
23 23 2 3
T T T
T
29 29 2 9
T T T
T
31 31 3 1
T − 59 T - 59 T − 5 9
T - 59
37 37 3 7
T + 26 T + 26 T + 2 6
T + 26
41 41 4 1
T T T
T
43 43 4 3
T + 83 T + 83 T + 8 3
T + 83
47 47 4 7
T T T
T
53 53 5 3
T T T
T
59 59 5 9
T T T
T
61 61 6 1
T + 121 T + 121 T + 1 2 1
T + 121
67 67 6 7
T − 13 T - 13 T − 1 3
T - 13
71 71 7 1
T T T
T
73 73 7 3
T − 46 T - 46 T − 4 6
T - 46
79 79 7 9
T + 142 T + 142 T + 1 4 2
T + 142
83 83 8 3
T T T
T
89 89 8 9
T T T
T
97 97 9 7
T + 167 T + 167 T + 1 6 7
T + 167
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