gp: [N,k,chi] = [33,4,Mod(1,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,1,-6]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 β = 1 2 ( 1 + 97 ) \beta = \frac{1}{2}(1 + \sqrt{97}) β = 2 1 ( 1 + 9 7 ) .
We also show the integral q q q -expansion of the trace form .
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
p p p
Sign
3 3 3
+ 1 +1 + 1
11 11 1 1
+ 1 +1 + 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace can be constructed as the kernel of the linear operator
T 2 2 − T 2 − 24 T_{2}^{2} - T_{2} - 24 T 2 2 − T 2 − 2 4
T2^2 - T2 - 24
acting on S 4 n e w ( Γ 0 ( 33 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(33)) S 4 n e w ( Γ 0 ( 3 3 ) ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − T − 24 T^{2} - T - 24 T 2 − T − 2 4
T^2 - T - 24
3 3 3
( T + 3 ) 2 (T + 3)^{2} ( T + 3 ) 2
(T + 3)^2
5 5 5
T 2 + 14 T − 48 T^{2} + 14T - 48 T 2 + 1 4 T − 4 8
T^2 + 14*T - 48
7 7 7
T 2 − 24 T − 244 T^{2} - 24T - 244 T 2 − 2 4 T − 2 4 4
T^2 - 24*T - 244
11 11 1 1
( T + 11 ) 2 (T + 11)^{2} ( T + 1 1 ) 2
(T + 11)^2
13 13 1 3
T 2 − 30 T + 128 T^{2} - 30T + 128 T 2 − 3 0 T + 1 2 8
T^2 - 30*T + 128
17 17 1 7
T 2 − 106 T − 1944 T^{2} - 106T - 1944 T 2 − 1 0 6 T − 1 9 4 4
T^2 - 106*T - 1944
19 19 1 9
T 2 − 50 T + 528 T^{2} - 50T + 528 T 2 − 5 0 T + 5 2 8
T^2 - 50*T + 528
23 23 2 3
T 2 − 134 T + 2064 T^{2} - 134T + 2064 T 2 − 1 3 4 T + 2 0 6 4
T^2 - 134*T + 2064
29 29 2 9
T 2 + 198 T + 8928 T^{2} + 198T + 8928 T 2 + 1 9 8 T + 8 9 2 8
T^2 + 198*T + 8928
31 31 3 1
T 2 − 360 T + 30848 T^{2} - 360T + 30848 T 2 − 3 6 0 T + 3 0 8 4 8
T^2 - 360*T + 30848
37 37 3 7
T 2 + 328 T − 38676 T^{2} + 328T - 38676 T 2 + 3 2 8 T − 3 8 6 7 6
T^2 + 328*T - 38676
41 41 4 1
T 2 + 782 T + 148128 T^{2} + 782T + 148128 T 2 + 7 8 2 T + 1 4 8 1 2 8
T^2 + 782*T + 148128
43 43 4 3
T 2 − 386 T + 20856 T^{2} - 386T + 20856 T 2 − 3 8 6 T + 2 0 8 5 6
T^2 - 386*T + 20856
47 47 4 7
T 2 − 266 T − 115104 T^{2} - 266T - 115104 T 2 − 2 6 6 T − 1 1 5 1 0 4
T^2 - 266*T - 115104
53 53 5 3
T 2 + 522 T − 2592 T^{2} + 522T - 2592 T 2 + 5 2 2 T − 2 5 9 2
T^2 + 522*T - 2592
59 59 5 9
T 2 + 172 T − 235104 T^{2} + 172T - 235104 T 2 + 1 7 2 T − 2 3 5 1 0 4
T^2 + 172*T - 235104
61 61 6 1
T 2 + 778 T + 123288 T^{2} + 778T + 123288 T 2 + 7 7 8 T + 1 2 3 2 8 8
T^2 + 778*T + 123288
67 67 6 7
T 2 + 776 T − 72944 T^{2} + 776T - 72944 T 2 + 7 7 6 T − 7 2 9 4 4
T^2 + 776*T - 72944
71 71 7 1
T 2 − 630 T + 28512 T^{2} - 630T + 28512 T 2 − 6 3 0 T + 2 8 5 1 2
T^2 - 630*T + 28512
73 73 7 3
T 2 − 1296 T + 400892 T^{2} - 1296 T + 400892 T 2 − 1 2 9 6 T + 4 0 0 8 9 2
T^2 - 1296*T + 400892
79 79 7 9
T 2 − 652 T − 396572 T^{2} - 652T - 396572 T 2 − 6 5 2 T − 3 9 6 5 7 2
T^2 - 652*T - 396572
83 83 8 3
T 2 + 324 T − 563904 T^{2} + 324T - 563904 T 2 + 3 2 4 T − 5 6 3 9 0 4
T^2 + 324*T - 563904
89 89 8 9
T 2 + 756 T + 17172 T^{2} + 756T + 17172 T 2 + 7 5 6 T + 1 7 1 7 2
T^2 + 756*T + 17172
97 97 9 7
T 2 + 452 T − 842876 T^{2} + 452T - 842876 T 2 + 4 5 2 T − 8 4 2 8 7 6
T^2 + 452*T - 842876
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