gp: [N,k,chi] = [63,3,Mod(10,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.10");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [2,1]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 a primitive root of unity ζ 6 \zeta_{6} ζ 6 .
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 / 63 Z ) × \left(\mathbb{Z}/63\mathbb{Z}\right)^\times ( Z / 6 3 Z ) × .
n n n
10 10 1 0
29 29 2 9
χ ( n ) \chi(n) χ ( n )
ζ 6 \zeta_{6} ζ 6
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 kernel of the linear operator
T 2 2 − T 2 + 1 T_{2}^{2} - T_{2} + 1 T 2 2 − T 2 + 1
T2^2 - T2 + 1
acting on S 3 n e w ( 63 , [ χ ] ) S_{3}^{\mathrm{new}}(63, [\chi]) S 3 n e w ( 6 3 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − T + 1 T^{2} - T + 1 T 2 − T + 1
T^2 - T + 1
3 3 3
T 2 T^{2} T 2
T^2
5 5 5
T 2 − 9 T + 27 T^{2} - 9T + 27 T 2 − 9 T + 2 7
T^2 - 9*T + 27
7 7 7
T 2 + 7 T + 49 T^{2} + 7T + 49 T 2 + 7 T + 4 9
T^2 + 7*T + 49
11 11 1 1
T 2 + 11 T + 121 T^{2} + 11T + 121 T 2 + 1 1 T + 1 2 1
T^2 + 11*T + 121
13 13 1 3
T 2 + 48 T^{2} + 48 T 2 + 4 8
T^2 + 48
17 17 1 7
T 2 + 42 T + 588 T^{2} + 42T + 588 T 2 + 4 2 T + 5 8 8
T^2 + 42*T + 588
19 19 1 9
T 2 + 6 T + 12 T^{2} + 6T + 12 T 2 + 6 T + 1 2
T^2 + 6*T + 12
23 23 2 3
T 2 − 28 T + 784 T^{2} - 28T + 784 T 2 − 2 8 T + 7 8 4
T^2 - 28*T + 784
29 29 2 9
( T + 25 ) 2 (T + 25)^{2} ( T + 2 5 ) 2
(T + 25)^2
31 31 3 1
T 2 + 57 T + 1083 T^{2} + 57T + 1083 T 2 + 5 7 T + 1 0 8 3
T^2 + 57*T + 1083
37 37 3 7
T 2 − 58 T + 3364 T^{2} - 58T + 3364 T 2 − 5 8 T + 3 3 6 4
T^2 - 58*T + 3364
41 41 4 1
T 2 + 12 T^{2} + 12 T 2 + 1 2
T^2 + 12
43 43 4 3
( T − 26 ) 2 (T - 26)^{2} ( T − 2 6 ) 2
(T - 26)^2
47 47 4 7
T 2 − 132 T + 5808 T^{2} - 132T + 5808 T 2 − 1 3 2 T + 5 8 0 8
T^2 - 132*T + 5808
53 53 5 3
T 2 − 31 T + 961 T^{2} - 31T + 961 T 2 − 3 1 T + 9 6 1
T^2 - 31*T + 961
59 59 5 9
T 2 − 15 T + 75 T^{2} - 15T + 75 T 2 − 1 5 T + 7 5
T^2 - 15*T + 75
61 61 6 1
T 2 − 24 T + 192 T^{2} - 24T + 192 T 2 − 2 4 T + 1 9 2
T^2 - 24*T + 192
67 67 6 7
T 2 − 52 T + 2704 T^{2} - 52T + 2704 T 2 − 5 2 T + 2 7 0 4
T^2 - 52*T + 2704
71 71 7 1
( T + 64 ) 2 (T + 64)^{2} ( T + 6 4 ) 2
(T + 64)^2
73 73 7 3
T 2 − 12 T + 48 T^{2} - 12T + 48 T 2 − 1 2 T + 4 8
T^2 - 12*T + 48
79 79 7 9
T 2 + 17 T + 289 T^{2} + 17T + 289 T 2 + 1 7 T + 2 8 9
T^2 + 17*T + 289
83 83 8 3
T 2 + 2883 T^{2} + 2883 T 2 + 2 8 8 3
T^2 + 2883
89 89 8 9
T 2 − 138 T + 6348 T^{2} - 138T + 6348 T 2 − 1 3 8 T + 6 3 4 8
T^2 - 138*T + 6348
97 97 9 7
T 2 + 8427 T^{2} + 8427 T 2 + 8 4 2 7
T^2 + 8427
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