gp: [N,k,chi] = [588,4,Mod(361,588)]
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("588.361");
S:= CuspForms(chi, 4);
N := Newforms(S);
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
Newform invariants
sage: traces = [2,0,-3,0,-14]
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 / 588 Z ) × \left(\mathbb{Z}/588\mathbb{Z}\right)^\times ( Z / 5 8 8 Z ) × .
n n n
197 197 1 9 7
295 295 2 9 5
493 493 4 9 3
χ ( n ) \chi(n) χ ( n )
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 + 14 T 5 + 196 T_{5}^{2} + 14T_{5} + 196 T 5 2 + 1 4 T 5 + 1 9 6
T5^2 + 14*T5 + 196
acting on S 4 n e w ( 588 , [ χ ] ) S_{4}^{\mathrm{new}}(588, [\chi]) S 4 n e w ( 5 8 8 , [ 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 + 14 T + 196 T^{2} + 14T + 196 T 2 + 1 4 T + 1 9 6
T^2 + 14*T + 196
7 7 7
T 2 T^{2} T 2
T^2
11 11 1 1
T 2 + 4 T + 16 T^{2} + 4T + 16 T 2 + 4 T + 1 6
T^2 + 4*T + 16
13 13 1 3
( T − 54 ) 2 (T - 54)^{2} ( T − 5 4 ) 2
(T - 54)^2
17 17 1 7
T 2 − 14 T + 196 T^{2} - 14T + 196 T 2 − 1 4 T + 1 9 6
T^2 - 14*T + 196
19 19 1 9
T 2 + 92 T + 8464 T^{2} + 92T + 8464 T 2 + 9 2 T + 8 4 6 4
T^2 + 92*T + 8464
23 23 2 3
T 2 − 152 T + 23104 T^{2} - 152T + 23104 T 2 − 1 5 2 T + 2 3 1 0 4
T^2 - 152*T + 23104
29 29 2 9
( T + 106 ) 2 (T + 106)^{2} ( T + 1 0 6 ) 2
(T + 106)^2
31 31 3 1
T 2 − 144 T + 20736 T^{2} - 144T + 20736 T 2 − 1 4 4 T + 2 0 7 3 6
T^2 - 144*T + 20736
37 37 3 7
T 2 + 158 T + 24964 T^{2} + 158T + 24964 T 2 + 1 5 8 T + 2 4 9 6 4
T^2 + 158*T + 24964
41 41 4 1
( T + 390 ) 2 (T + 390)^{2} ( T + 3 9 0 ) 2
(T + 390)^2
43 43 4 3
( T + 508 ) 2 (T + 508)^{2} ( T + 5 0 8 ) 2
(T + 508)^2
47 47 4 7
T 2 − 528 T + 278784 T^{2} - 528T + 278784 T 2 − 5 2 8 T + 2 7 8 7 8 4
T^2 - 528*T + 278784
53 53 5 3
T 2 + 606 T + 367236 T^{2} + 606T + 367236 T 2 + 6 0 6 T + 3 6 7 2 3 6
T^2 + 606*T + 367236
59 59 5 9
T 2 − 364 T + 132496 T^{2} - 364T + 132496 T 2 − 3 6 4 T + 1 3 2 4 9 6
T^2 - 364*T + 132496
61 61 6 1
T 2 + 678 T + 459684 T^{2} + 678T + 459684 T 2 + 6 7 8 T + 4 5 9 6 8 4
T^2 + 678*T + 459684
67 67 6 7
T 2 + 844 T + 712336 T^{2} + 844T + 712336 T 2 + 8 4 4 T + 7 1 2 3 3 6
T^2 + 844*T + 712336
71 71 7 1
( T + 8 ) 2 (T + 8)^{2} ( T + 8 ) 2
(T + 8)^2
73 73 7 3
T 2 − 422 T + 178084 T^{2} - 422T + 178084 T 2 − 4 2 2 T + 1 7 8 0 8 4
T^2 - 422*T + 178084
79 79 7 9
T 2 + 384 T + 147456 T^{2} + 384T + 147456 T 2 + 3 8 4 T + 1 4 7 4 5 6
T^2 + 384*T + 147456
83 83 8 3
( T + 548 ) 2 (T + 548)^{2} ( T + 5 4 8 ) 2
(T + 548)^2
89 89 8 9
T 2 + 1194 T + 1425636 T^{2} + 1194 T + 1425636 T 2 + 1 1 9 4 T + 1 4 2 5 6 3 6
T^2 + 1194*T + 1425636
97 97 9 7
( T + 1502 ) 2 (T + 1502)^{2} ( T + 1 5 0 2 ) 2
(T + 1502)^2
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