gp: [N,k,chi] = [231,4,Mod(1,231)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("231.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Newform invariants
sage: traces = [2,-3,6,-3,25]
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 β = 1 2 ( 1 + 17 ) \beta = \frac{1}{2}(1 + \sqrt{17}) β = 2 1 ( 1 + 1 7 ) 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
3 3 3
− 1 -1 − 1
7 7 7
− 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 intersection of the kernels of the following linear operators acting on S 4 n e w ( Γ 0 ( 231 ) ) S_{4}^{\mathrm{new}}(\Gamma_0(231)) S 4 n e w ( Γ 0 ( 2 3 1 ) )
T 2 2 + 3 T 2 − 2 T_{2}^{2} + 3T_{2} - 2 T 2 2 + 3 T 2 − 2
T 5 2 − 25 T 5 + 118 T_{5}^{2} - 25T_{5} + 118 T 5 2 − 2 5 T 5 + 1 1 8
p p p F p ( T ) F_p(T) F p ( T )
2 2 2
T 2 + 3 T − 2 T^{2} + 3T - 2 T 2 + 3 T − 2
3 3 3
( T − 3 ) 2 (T - 3)^{2} ( T − 3 ) 2
5 5 5
T 2 − 25 T + 118 T^{2} - 25T + 118 T 2 − 2 5 T + 1 1 8
7 7 7
( T − 7 ) 2 (T - 7)^{2} ( T − 7 ) 2
11 11 1 1
( T + 11 ) 2 (T + 11)^{2} ( T + 1 1 ) 2
13 13 1 3
T 2 − 77 T + 1478 T^{2} - 77T + 1478 T 2 − 7 7 T + 1 4 7 8
17 17 1 7
T 2 − 74 T + 1352 T^{2} - 74T + 1352 T 2 − 7 4 T + 1 3 5 2
19 19 1 9
T 2 + 101 T + 1832 T^{2} + 101T + 1832 T 2 + 1 0 1 T + 1 8 3 2
23 23 2 3
T 2 − 58 T − 27736 T^{2} - 58T - 27736 T 2 − 5 8 T − 2 7 7 3 6
29 29 2 9
T 2 − 91 T − 52198 T^{2} - 91T - 52198 T 2 − 9 1 T − 5 2 1 9 8
31 31 3 1
T 2 − 152 T − 7552 T^{2} - 152T - 7552 T 2 − 1 5 2 T − 7 5 5 2
37 37 3 7
T 2 − 619 T + 82934 T^{2} - 619T + 82934 T 2 − 6 1 9 T + 8 2 9 3 4
41 41 4 1
T 2 − 138 T − 148664 T^{2} - 138T - 148664 T 2 − 1 3 8 T − 1 4 8 6 6 4
43 43 4 3
T 2 + 230 T + 832 T^{2} + 230T + 832 T 2 + 2 3 0 T + 8 3 2
47 47 4 7
T 2 − 149 T − 217324 T^{2} - 149T - 217324 T 2 − 1 4 9 T − 2 1 7 3 2 4
53 53 5 3
T 2 − 1228 T + 376724 T^{2} - 1228 T + 376724 T 2 − 1 2 2 8 T + 3 7 6 7 2 4
59 59 5 9
T 2 − 649 T − 13228 T^{2} - 649T - 13228 T 2 − 6 4 9 T − 1 3 2 2 8
61 61 6 1
T 2 + 412 T − 170812 T^{2} + 412T - 170812 T 2 + 4 1 2 T − 1 7 0 8 1 2
67 67 6 7
T 2 + 1243 T + 374324 T^{2} + 1243 T + 374324 T 2 + 1 2 4 3 T + 3 7 4 3 2 4
71 71 7 1
T 2 − 960 T − 30992 T^{2} - 960T - 30992 T 2 − 9 6 0 T − 3 0 9 9 2
73 73 7 3
T 2 + 741 T + 133186 T^{2} + 741T + 133186 T 2 + 7 4 1 T + 1 3 3 1 8 6
79 79 7 9
T 2 + 492 T − 226784 T^{2} + 492T - 226784 T 2 + 4 9 2 T − 2 2 6 7 8 4
83 83 8 3
T 2 + 1744 T + 562096 T^{2} + 1744 T + 562096 T 2 + 1 7 4 4 T + 5 6 2 0 9 6
89 89 8 9
T 2 + 1552 T + 586876 T^{2} + 1552 T + 586876 T 2 + 1 5 5 2 T + 5 8 6 8 7 6
97 97 9 7
T 2 − 440 T − 1303508 T^{2} - 440 T - 1303508 T 2 − 4 4 0 T − 1 3 0 3 5 0 8
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