gp: [N,k,chi] = [11,4,Mod(1,11)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("11.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = []
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == 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 β = 3 \beta = \sqrt{3} β = 3 q q q trace form .
For each embedding ι m \iota_m ι m ι m ( a n ) \iota_m(a_n) ι m ( a n )
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
p p p
Sign
11 11 1 1
+ 1 +1 + 1
This newform does not admit any (nontrivial ) inner twists .
This newform subspace is the entire newspace S 4 n e w ( Γ 0 ( 11 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(11)) S 4 n e w ( Γ 0 ( 1 1 ) )
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 − 2 T − 2 T^{2} - 2T - 2 T 2 − 2 T − 2
3 3 3
T 2 + 2 T − 47 T^{2} + 2T - 47 T 2 + 2 T − 4 7
5 5 5
T 2 − 2 T − 191 T^{2} - 2T - 191 T 2 − 2 T − 1 9 1
7 7 7
T 2 − 20 T + 52 T^{2} - 20T + 52 T 2 − 2 0 T + 5 2
11 11 1 1
( T + 11 ) 2 (T + 11)^{2} ( T + 1 1 ) 2
13 13 1 3
T 2 − 80 T + 400 T^{2} - 80T + 400 T 2 − 8 0 T + 4 0 0
17 17 1 7
T 2 + 124 T + 3412 T^{2} + 124T + 3412 T 2 + 1 2 4 T + 3 4 1 2
19 19 1 9
T 2 − 72 T − 9504 T^{2} - 72T - 9504 T 2 − 7 2 T − 9 5 0 4
23 23 2 3
T 2 + 98 T − 1487 T^{2} + 98T - 1487 T 2 + 9 8 T − 1 4 8 7
29 29 2 9
T 2 − 144 T − 4224 T^{2} - 144T - 4224 T 2 − 1 4 4 T − 4 2 2 4
31 31 3 1
T 2 + 34 T − 2063 T^{2} + 34T - 2063 T 2 + 3 4 T − 2 0 6 3
37 37 3 7
T 2 − 54 T + 537 T^{2} - 54T + 537 T 2 − 5 4 T + 5 3 7
41 41 4 1
T 2 − 536 T + 71776 T^{2} - 536T + 71776 T 2 − 5 3 6 T + 7 1 7 7 6
43 43 4 3
T 2 + 60 T + 132 T^{2} + 60T + 132 T 2 + 6 0 T + 1 3 2
47 47 4 7
T 2 + 272 T − 24704 T^{2} + 272T - 24704 T 2 + 2 7 2 T − 2 4 7 0 4
53 53 5 3
T 2 + 492 T + 51108 T^{2} + 492T + 51108 T 2 + 4 9 2 T + 5 1 1 0 8
59 59 5 9
T 2 − 634 T + 48217 T^{2} - 634T + 48217 T 2 − 6 3 4 T + 4 8 2 1 7
61 61 6 1
T 2 − 840 T + 74832 T^{2} - 840T + 74832 T 2 − 8 4 0 T + 7 4 8 3 2
67 67 6 7
T 2 − 754 T + 140929 T^{2} - 754T + 140929 T 2 − 7 5 4 T + 1 4 0 9 2 9
71 71 7 1
T 2 + 678 T + 97593 T^{2} + 678T + 97593 T 2 + 6 7 8 T + 9 7 5 9 3
73 73 7 3
T 2 + 400 T − 617072 T^{2} + 400T - 617072 T 2 + 4 0 0 T − 6 1 7 0 7 2
79 79 7 9
T 2 − 316 T − 1266044 T^{2} - 316 T - 1266044 T 2 − 3 1 6 T − 1 2 6 6 0 4 4
83 83 8 3
T 2 − 468 T + 11556 T^{2} - 468T + 11556 T 2 − 4 6 8 T + 1 1 5 5 6
89 89 8 9
T 2 + 1842 T + 525489 T^{2} + 1842 T + 525489 T 2 + 1 8 4 2 T + 5 2 5 4 8 9
97 97 9 7
T 2 − 2194 T + 1141201 T^{2} - 2194 T + 1141201 T 2 − 2 1 9 4 T + 1 1 4 1 2 0 1
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