gp: [N,k,chi] = [245,2,Mod(79,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.79");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,0,0,-2,2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 12 \zeta_{12} ζ 1 2 .
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 / 245 Z ) × \left(\mathbb{Z}/245\mathbb{Z}\right)^\times ( Z / 2 4 5 Z ) × .
n n n
101 101 1 0 1
197 197 1 9 7
χ ( n ) \chi(n) χ ( n )
− 1 + ζ 12 2 -1 + \zeta_{12}^{2} − 1 + ζ 1 2 2
− 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 ( 245 , [ χ ] ) S_{2}^{\mathrm{new}}(245, [\chi]) S 2 n e w ( 2 4 5 , [ χ ] ) :
T 2 4 − T 2 2 + 1 T_{2}^{4} - T_{2}^{2} + 1 T 2 4 − T 2 2 + 1
T2^4 - T2^2 + 1
T 3 4 − T 3 2 + 1 T_{3}^{4} - T_{3}^{2} + 1 T 3 4 − T 3 2 + 1
T3^4 - T3^2 + 1
T 31 2 − 2 T 31 + 4 T_{31}^{2} - 2T_{31} + 4 T 3 1 2 − 2 T 3 1 + 4
T31^2 - 2*T31 + 4
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 − T 2 + 1 T^{4} - T^{2} + 1 T 4 − T 2 + 1
T^4 - T^2 + 1
3 3 3
T 4 − T 2 + 1 T^{4} - T^{2} + 1 T 4 − T 2 + 1
T^4 - T^2 + 1
5 5 5
T 4 − 2 T 3 + ⋯ + 25 T^{4} - 2 T^{3} + \cdots + 25 T 4 − 2 T 3 + ⋯ + 2 5
T^4 - 2*T^3 - T^2 - 10*T + 25
7 7 7
T 4 T^{4} T 4
T^4
11 11 1 1
T 4 T^{4} T 4
T^4
13 13 1 3
( T 2 + 4 ) 2 (T^{2} + 4)^{2} ( T 2 + 4 ) 2
(T^2 + 4)^2
17 17 1 7
T 4 − 4 T 2 + 16 T^{4} - 4T^{2} + 16 T 4 − 4 T 2 + 1 6
T^4 - 4*T^2 + 16
19 19 1 9
( T 2 + 6 T + 36 ) 2 (T^{2} + 6 T + 36)^{2} ( T 2 + 6 T + 3 6 ) 2
(T^2 + 6*T + 36)^2
23 23 2 3
T 4 − 9 T 2 + 81 T^{4} - 9T^{2} + 81 T 4 − 9 T 2 + 8 1
T^4 - 9*T^2 + 81
29 29 2 9
( T + 7 ) 4 (T + 7)^{4} ( T + 7 ) 4
(T + 7)^4
31 31 3 1
( T 2 − 2 T + 4 ) 2 (T^{2} - 2 T + 4)^{2} ( T 2 − 2 T + 4 ) 2
(T^2 - 2*T + 4)^2
37 37 3 7
T 4 − 64 T 2 + 4096 T^{4} - 64T^{2} + 4096 T 4 − 6 4 T 2 + 4 0 9 6
T^4 - 64*T^2 + 4096
41 41 4 1
( T + 5 ) 4 (T + 5)^{4} ( T + 5 ) 4
(T + 5)^4
43 43 4 3
( T 2 + 49 ) 2 (T^{2} + 49)^{2} ( T 2 + 4 9 ) 2
(T^2 + 49)^2
47 47 4 7
T 4 T^{4} T 4
T^4
53 53 5 3
T 4 − 36 T 2 + 1296 T^{4} - 36T^{2} + 1296 T 4 − 3 6 T 2 + 1 2 9 6
T^4 - 36*T^2 + 1296
59 59 5 9
( T 2 + 10 T + 100 ) 2 (T^{2} + 10 T + 100)^{2} ( T 2 + 1 0 T + 1 0 0 ) 2
(T^2 + 10*T + 100)^2
61 61 6 1
( T 2 − 7 T + 49 ) 2 (T^{2} - 7 T + 49)^{2} ( T 2 − 7 T + 4 9 ) 2
(T^2 - 7*T + 49)^2
67 67 6 7
T 4 − 25 T 2 + 625 T^{4} - 25T^{2} + 625 T 4 − 2 5 T 2 + 6 2 5
T^4 - 25*T^2 + 625
71 71 7 1
( T + 2 ) 4 (T + 2)^{4} ( T + 2 ) 4
(T + 2)^4
73 73 7 3
T 4 − 36 T 2 + 1296 T^{4} - 36T^{2} + 1296 T 4 − 3 6 T 2 + 1 2 9 6
T^4 - 36*T^2 + 1296
79 79 7 9
( T 2 + 2 T + 4 ) 2 (T^{2} + 2 T + 4)^{2} ( T 2 + 2 T + 4 ) 2
(T^2 + 2*T + 4)^2
83 83 8 3
( T 2 + 121 ) 2 (T^{2} + 121)^{2} ( T 2 + 1 2 1 ) 2
(T^2 + 121)^2
89 89 8 9
( T 2 + 9 T + 81 ) 2 (T^{2} + 9 T + 81)^{2} ( T 2 + 9 T + 8 1 ) 2
(T^2 + 9*T + 81)^2
97 97 9 7
( T 2 + 256 ) 2 (T^{2} + 256)^{2} ( T 2 + 2 5 6 ) 2
(T^2 + 256)^2
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