gp: [N,k,chi] = [650,2,Mod(399,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.399");
S:= CuspForms(chi, 2);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 2, names="a")
Newform invariants
sage: traces = [4,0,0,2,0,-4,0,0,2,0,-6]
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 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 / 650 Z ) × \left(\mathbb{Z}/650\mathbb{Z}\right)^\times ( Z / 6 5 0 Z ) × .
n n n
27 27 2 7
301 301 3 0 1
χ ( n ) \chi(n) χ ( n )
− 1 -1 − 1
− ζ 12 2 -\zeta_{12}^{2} − ζ 1 2 2
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 ( 650 , [ χ ] ) S_{2}^{\mathrm{new}}(650, [\chi]) S 2 n e w ( 6 5 0 , [ χ ] ) :
T 3 4 − 4 T 3 2 + 16 T_{3}^{4} - 4T_{3}^{2} + 16 T 3 4 − 4 T 3 2 + 1 6
T3^4 - 4*T3^2 + 16
T 7 4 − T 7 2 + 1 T_{7}^{4} - T_{7}^{2} + 1 T 7 4 − T 7 2 + 1
T7^4 - T7^2 + 1
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 − 4 T 2 + 16 T^{4} - 4T^{2} + 16 T 4 − 4 T 2 + 1 6
T^4 - 4*T^2 + 16
5 5 5
T 4 T^{4} T 4
T^4
7 7 7
T 4 − T 2 + 1 T^{4} - T^{2} + 1 T 4 − T 2 + 1
T^4 - T^2 + 1
11 11 1 1
( T 2 + 3 T + 9 ) 2 (T^{2} + 3 T + 9)^{2} ( T 2 + 3 T + 9 ) 2
(T^2 + 3*T + 9)^2
13 13 1 3
T 4 − T 2 + 169 T^{4} - T^{2} + 169 T 4 − T 2 + 1 6 9
T^4 - T^2 + 169
17 17 1 7
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
19 19 1 9
( T 2 − 5 T + 25 ) 2 (T^{2} - 5 T + 25)^{2} ( T 2 − 5 T + 2 5 ) 2
(T^2 - 5*T + 25)^2
23 23 2 3
T 4 T^{4} T 4
T^4
29 29 2 9
T 4 T^{4} T 4
T^4
31 31 3 1
( T + 4 ) 4 (T + 4)^{4} ( T + 4 ) 4
(T + 4)^4
37 37 3 7
T 4 − 121 T 2 + 14641 T^{4} - 121 T^{2} + 14641 T 4 − 1 2 1 T 2 + 1 4 6 4 1
T^4 - 121*T^2 + 14641
41 41 4 1
( 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
43 43 4 3
T 4 − 4 T 2 + 16 T^{4} - 4T^{2} + 16 T 4 − 4 T 2 + 1 6
T^4 - 4*T^2 + 16
47 47 4 7
( T 2 + 9 ) 2 (T^{2} + 9)^{2} ( T 2 + 9 ) 2
(T^2 + 9)^2
53 53 5 3
( T 2 + 81 ) 2 (T^{2} + 81)^{2} ( T 2 + 8 1 ) 2
(T^2 + 81)^2
59 59 5 9
T 4 T^{4} T 4
T^4
61 61 6 1
( T 2 + 8 T + 64 ) 2 (T^{2} + 8 T + 64)^{2} ( T 2 + 8 T + 6 4 ) 2
(T^2 + 8*T + 64)^2
67 67 6 7
T 4 − 256 T 2 + 65536 T^{4} - 256 T^{2} + 65536 T 4 − 2 5 6 T 2 + 6 5 5 3 6
T^4 - 256*T^2 + 65536
71 71 7 1
( 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
73 73 7 3
( T 2 + 196 ) 2 (T^{2} + 196)^{2} ( T 2 + 1 9 6 ) 2
(T^2 + 196)^2
79 79 7 9
( T − 16 ) 4 (T - 16)^{4} ( T − 1 6 ) 4
(T - 16)^4
83 83 8 3
( T 2 + 36 ) 2 (T^{2} + 36)^{2} ( T 2 + 3 6 ) 2
(T^2 + 36)^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 4 − 100 T 2 + 10000 T^{4} - 100 T^{2} + 10000 T 4 − 1 0 0 T 2 + 1 0 0 0 0
T^4 - 100*T^2 + 10000
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