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gp:[N,k,chi] = [961,2,Mod(235,961)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(961, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([26]))
N = Newforms(chi, 2, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("961.235");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [8,-6,-6,6,-4,12,-7]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
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gp:f = lf[1] \\ Warning: the index may be different
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sage:f.q_expansion() # note that sage often uses an isomorphic number field
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gp:mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).
| \(n\) |
\(3\) |
| \(\chi(n)\) |
\(1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}\) |
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
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gp:mfembed(f)
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):
\( T_{2}^{4} + 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} + 1 \)
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\( T_{3}^{8} + 6T_{3}^{7} + 20T_{3}^{6} + 64T_{3}^{5} + 144T_{3}^{4} + 64T_{3}^{3} + 256T_{3} + 256 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
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| $3$ |
\( T^{8} + 6 T^{7} + \cdots + 256 \)
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| $5$ |
\( (T^{2} + T + 1)^{4} \)
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| $7$ |
\( T^{8} + 7 T^{7} + \cdots + 1 \)
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| $11$ |
\( T^{8} - 2 T^{7} + \cdots + 256 \)
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| $13$ |
\( T^{8} + 6 T^{7} + \cdots + 256 \)
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| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 256 \)
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| $19$ |
\( T^{8} + 5 T^{7} + \cdots + 625 \)
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| $23$ |
\( (T^{4} - 16 T^{3} + \cdots + 1936)^{2} \)
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| $29$ |
\( (T^{4} + 40 T^{2} + \cdots + 400)^{2} \)
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| $31$ |
\( T^{8} \)
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| $37$ |
\( (T^{2} - 2 T + 4)^{4} \)
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| $41$ |
\( T^{8} - 7 T^{7} + \cdots + 5764801 \)
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| $43$ |
\( T^{8} - 4 T^{7} + \cdots + 256 \)
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| $47$ |
\( (T^{4} + 8 T^{3} + \cdots + 256)^{2} \)
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| $53$ |
\( T^{8} - 4 T^{7} + \cdots + 65536 \)
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| $59$ |
\( T^{8} - 5 T^{7} + \cdots + 625 \)
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| $61$ |
\( (T^{2} + 6 T - 116)^{4} \)
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| $67$ |
\( (T^{2} + 8 T + 64)^{4} \)
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| $71$ |
\( T^{8} - 27 T^{7} + \cdots + 214358881 \)
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| $73$ |
\( T^{8} + 6 T^{7} + \cdots + 256 \)
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| $79$ |
\( T^{8} - 10 T^{7} + \cdots + 160000 \)
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| $83$ |
\( T^{8} + 26 T^{7} + \cdots + 3748096 \)
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| $89$ |
\( (T^{4} - 10 T^{3} + \cdots + 400)^{2} \)
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| $97$ |
\( (T^{4} + 13 T^{3} + \cdots + 961)^{2} \)
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