gp: [N,k,chi] = [336,4,Mod(193,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.193");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,0,-3,0,15]
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 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 / 336 Z ) × \left(\mathbb{Z}/336\mathbb{Z}\right)^\times ( Z / 3 3 6 Z ) × .
n n n
85 85 8 5
113 113 1 1 3
127 127 1 2 7
241 241 2 4 1
χ ( n ) \chi(n) χ ( n )
1 1 1
1 1 1
1 1 1
− ζ 6 -\zeta_{6} − ζ 6
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 2 − 15 T 5 + 225 T_{5}^{2} - 15T_{5} + 225 T 5 2 − 1 5 T 5 + 2 2 5
T5^2 - 15*T5 + 225
acting on S 4 n e w ( 336 , [ χ ] ) S_{4}^{\mathrm{new}}(336, [\chi]) S 4 n e w ( 3 3 6 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 T^{2} T 2
T^2
3 3 3
T 2 + 3 T + 9 T^{2} + 3T + 9 T 2 + 3 T + 9
T^2 + 3*T + 9
5 5 5
T 2 − 15 T + 225 T^{2} - 15T + 225 T 2 − 1 5 T + 2 2 5
T^2 - 15*T + 225
7 7 7
T 2 + 35 T + 343 T^{2} + 35T + 343 T 2 + 3 5 T + 3 4 3
T^2 + 35*T + 343
11 11 1 1
T 2 + 9 T + 81 T^{2} + 9T + 81 T 2 + 9 T + 8 1
T^2 + 9*T + 81
13 13 1 3
( T + 88 ) 2 (T + 88)^{2} ( T + 8 8 ) 2
(T + 88)^2
17 17 1 7
T 2 − 84 T + 7056 T^{2} - 84T + 7056 T 2 − 8 4 T + 7 0 5 6
T^2 - 84*T + 7056
19 19 1 9
T 2 − 104 T + 10816 T^{2} - 104T + 10816 T 2 − 1 0 4 T + 1 0 8 1 6
T^2 - 104*T + 10816
23 23 2 3
T 2 + 84 T + 7056 T^{2} + 84T + 7056 T 2 + 8 4 T + 7 0 5 6
T^2 + 84*T + 7056
29 29 2 9
( T − 51 ) 2 (T - 51)^{2} ( T − 5 1 ) 2
(T - 51)^2
31 31 3 1
T 2 − 185 T + 34225 T^{2} - 185T + 34225 T 2 − 1 8 5 T + 3 4 2 2 5
T^2 - 185*T + 34225
37 37 3 7
T 2 + 44 T + 1936 T^{2} + 44T + 1936 T 2 + 4 4 T + 1 9 3 6
T^2 + 44*T + 1936
41 41 4 1
( T + 168 ) 2 (T + 168)^{2} ( T + 1 6 8 ) 2
(T + 168)^2
43 43 4 3
( T + 326 ) 2 (T + 326)^{2} ( T + 3 2 6 ) 2
(T + 326)^2
47 47 4 7
T 2 + 138 T + 19044 T^{2} + 138T + 19044 T 2 + 1 3 8 T + 1 9 0 4 4
T^2 + 138*T + 19044
53 53 5 3
T 2 + 639 T + 408321 T^{2} + 639T + 408321 T 2 + 6 3 9 T + 4 0 8 3 2 1
T^2 + 639*T + 408321
59 59 5 9
T 2 − 159 T + 25281 T^{2} - 159T + 25281 T 2 − 1 5 9 T + 2 5 2 8 1
T^2 - 159*T + 25281
61 61 6 1
T 2 + 722 T + 521284 T^{2} + 722T + 521284 T 2 + 7 2 2 T + 5 2 1 2 8 4
T^2 + 722*T + 521284
67 67 6 7
T 2 + 166 T + 27556 T^{2} + 166T + 27556 T 2 + 1 6 6 T + 2 7 5 5 6
T^2 + 166*T + 27556
71 71 7 1
( T + 1086 ) 2 (T + 1086)^{2} ( T + 1 0 8 6 ) 2
(T + 1086)^2
73 73 7 3
T 2 + 218 T + 47524 T^{2} + 218T + 47524 T 2 + 2 1 8 T + 4 7 5 2 4
T^2 + 218*T + 47524
79 79 7 9
T 2 + 583 T + 339889 T^{2} + 583T + 339889 T 2 + 5 8 3 T + 3 3 9 8 8 9
T^2 + 583*T + 339889
83 83 8 3
( T − 597 ) 2 (T - 597)^{2} ( T − 5 9 7 ) 2
(T - 597)^2
89 89 8 9
T 2 − 1038 T + 1077444 T^{2} - 1038 T + 1077444 T 2 − 1 0 3 8 T + 1 0 7 7 4 4 4
T^2 - 1038*T + 1077444
97 97 9 7
( T + 169 ) 2 (T + 169)^{2} ( T + 1 6 9 ) 2
(T + 169)^2
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