gp: [N,k,chi] = [1224,2,Mod(145,1224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(8))
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("1224.145");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,0,0,0,8,0,4,0,0,0,-12]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 8 \zeta_{8} ζ 8 .
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 / 1224 Z ) × \left(\mathbb{Z}/1224\mathbb{Z}\right)^\times ( Z / 1 2 2 4 Z ) × .
n n n
137 137 1 3 7
613 613 6 1 3
649 649 6 4 9
919 919 9 1 9
χ ( n ) \chi(n) χ ( n )
1 1 1
1 1 1
ζ 8 \zeta_{8} ζ 8
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 5 4 − 8 T 5 3 + 34 T 5 2 − 84 T 5 + 98 T_{5}^{4} - 8T_{5}^{3} + 34T_{5}^{2} - 84T_{5} + 98 T 5 4 − 8 T 5 3 + 3 4 T 5 2 − 8 4 T 5 + 9 8
T5^4 - 8*T5^3 + 34*T5^2 - 84*T5 + 98
acting on S 2 n e w ( 1224 , [ χ ] ) S_{2}^{\mathrm{new}}(1224, [\chi]) S 2 n e w ( 1 2 2 4 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 T^{4} T 4
T^4
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
T 4 − 8 T 3 + ⋯ + 98 T^{4} - 8 T^{3} + \cdots + 98 T 4 − 8 T 3 + ⋯ + 9 8
T^4 - 8*T^3 + 34*T^2 - 84*T + 98
7 7 7
T 4 − 4 T 3 + ⋯ + 8 T^{4} - 4 T^{3} + \cdots + 8 T 4 − 4 T 3 + ⋯ + 8
T^4 - 4*T^3 + 12*T^2 - 16*T + 8
11 11 1 1
T 4 + 12 T 3 + ⋯ + 392 T^{4} + 12 T^{3} + \cdots + 392 T 4 + 1 2 T 3 + ⋯ + 3 9 2
T^4 + 12*T^3 + 68*T^2 + 224*T + 392
13 13 1 3
T 4 + 36 T 2 + 196 T^{4} + 36T^{2} + 196 T 4 + 3 6 T 2 + 1 9 6
T^4 + 36*T^2 + 196
17 17 1 7
T 4 + 2 T 2 + 289 T^{4} + 2T^{2} + 289 T 4 + 2 T 2 + 2 8 9
T^4 + 2*T^2 + 289
19 19 1 9
T 4 − 16 T 3 + ⋯ + 784 T^{4} - 16 T^{3} + \cdots + 784 T 4 − 1 6 T 3 + ⋯ + 7 8 4
T^4 - 16*T^3 + 128*T^2 - 448*T + 784
23 23 2 3
T 4 + 4 T 3 + ⋯ + 8 T^{4} + 4 T^{3} + \cdots + 8 T 4 + 4 T 3 + ⋯ + 8
T^4 + 4*T^3 + 36*T^2 - 32*T + 8
29 29 2 9
T 4 − 12 T 3 + ⋯ + 162 T^{4} - 12 T^{3} + \cdots + 162 T 4 − 1 2 T 3 + ⋯ + 1 6 2
T^4 - 12*T^3 + 54*T^2 - 108*T + 162
31 31 3 1
T 4 − 4 T 3 + ⋯ + 8 T^{4} - 4 T^{3} + \cdots + 8 T 4 − 4 T 3 + ⋯ + 8
T^4 - 4*T^3 + 4*T^2 + 8
37 37 3 7
T 4 + 8 T 3 + ⋯ + 2 T^{4} + 8 T^{3} + \cdots + 2 T 4 + 8 T 3 + ⋯ + 2
T^4 + 8*T^3 + 114*T^2 - 28*T + 2
41 41 4 1
T 4 − 20 T 3 + ⋯ + 1250 T^{4} - 20 T^{3} + \cdots + 1250 T 4 − 2 0 T 3 + ⋯ + 1 2 5 0
T^4 - 20*T^3 + 150*T^2 - 500*T + 1250
43 43 4 3
T 4 + 16 T 3 + ⋯ + 16 T^{4} + 16 T^{3} + \cdots + 16 T 4 + 1 6 T 3 + ⋯ + 1 6
T^4 + 16*T^3 + 128*T^2 - 64*T + 16
47 47 4 7
T 4 + 48 T 2 + 64 T^{4} + 48T^{2} + 64 T 4 + 4 8 T 2 + 6 4
T^4 + 48*T^2 + 64
53 53 5 3
T 4 + 12 T 3 + ⋯ + 4 T^{4} + 12 T^{3} + \cdots + 4 T 4 + 1 2 T 3 + ⋯ + 4
T^4 + 12*T^3 + 72*T^2 + 24*T + 4
59 59 5 9
T 4 + 8 T 3 + ⋯ + 8464 T^{4} + 8 T^{3} + \cdots + 8464 T 4 + 8 T 3 + ⋯ + 8 4 6 4
T^4 + 8*T^3 + 32*T^2 - 736*T + 8464
61 61 6 1
T 4 − 8 T 3 + ⋯ + 2 T^{4} - 8 T^{3} + \cdots + 2 T 4 − 8 T 3 + ⋯ + 2
T^4 - 8*T^3 + 18*T^2 + 4*T + 2
67 67 6 7
( T 2 − 8 ) 2 (T^{2} - 8)^{2} ( T 2 − 8 ) 2
(T^2 - 8)^2
71 71 7 1
T 4 − 20 T 3 + ⋯ + 17672 T^{4} - 20 T^{3} + \cdots + 17672 T 4 − 2 0 T 3 + ⋯ + 1 7 6 7 2
T^4 - 20*T^3 + 108*T^2 - 752*T + 17672
73 73 7 3
T 4 − 4 T 3 + ⋯ + 12482 T^{4} - 4 T^{3} + \cdots + 12482 T 4 − 4 T 3 + ⋯ + 1 2 4 8 2
T^4 - 4*T^3 + 102*T^2 - 2212*T + 12482
79 79 7 9
T 4 − 12 T 3 + ⋯ + 392 T^{4} - 12 T^{3} + \cdots + 392 T 4 − 1 2 T 3 + ⋯ + 3 9 2
T^4 - 12*T^3 + 68*T^2 - 224*T + 392
83 83 8 3
T 4 − 8 T 3 + ⋯ + 8464 T^{4} - 8 T^{3} + \cdots + 8464 T 4 − 8 T 3 + ⋯ + 8 4 6 4
T^4 - 8*T^3 + 32*T^2 + 736*T + 8464
89 89 8 9
( T 2 + 18 ) 2 (T^{2} + 18)^{2} ( T 2 + 1 8 ) 2
(T^2 + 18)^2
97 97 9 7
T 4 + 8 T 3 + ⋯ + 2 T^{4} + 8 T^{3} + \cdots + 2 T 4 + 8 T 3 + ⋯ + 2
T^4 + 8*T^3 + 18*T^2 - 4*T + 2
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