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gp:[N,k,chi] = [2883,1,Mod(338,2883)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2883, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 16]))
N = Newforms(chi, 1, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2883.338");
S:= CuspForms(chi, 1);
N := Newforms(S);
Newform invariants
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sage:traces = [8,0,-1,-2,0,0,-3]
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)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2883\mathbb{Z}\right)^\times\).
| \(n\) |
\(962\) |
\(964\) |
| \(\chi(n)\) |
\(-1\) |
\(\zeta_{30}^{4}\) |
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 \(S_{1}^{\mathrm{new}}(2883, [\chi])\):
\( T_{2} \)
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\( T_{7}^{8} + 3T_{7}^{7} + 5T_{7}^{6} + 8T_{7}^{5} + 9T_{7}^{4} + 2T_{7}^{3} + 2T_{7} + 1 \)
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\( T_{13}^{8} - 3T_{13}^{7} + 5T_{13}^{6} - 8T_{13}^{5} + 9T_{13}^{4} - 2T_{13}^{3} - 2T_{13} + 1 \)
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{8} \)
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| $3$ |
\( T^{8} + T^{7} - T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{8} \)
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| $7$ |
\( T^{8} + 3 T^{7} + \cdots + 1 \)
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| $11$ |
\( T^{8} \)
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| $13$ |
\( T^{8} - 3 T^{7} + \cdots + 1 \)
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| $17$ |
\( T^{8} \)
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| $19$ |
\( T^{8} + 3 T^{7} + \cdots + 1 \)
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| $23$ |
\( T^{8} \)
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| $29$ |
\( T^{8} \)
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| $31$ |
\( T^{8} \)
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| $37$ |
\( (T^{4} + T^{3} + 2 T^{2} + \cdots + 1)^{2} \)
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| $41$ |
\( T^{8} \)
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| $43$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
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| $47$ |
\( T^{8} \)
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| $53$ |
\( T^{8} \)
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| $59$ |
\( T^{8} \)
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| $61$ |
\( (T^{2} - T - 1)^{4} \)
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| $67$ |
\( (T^{4} - T^{3} + 2 T^{2} + \cdots + 1)^{2} \)
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| $71$ |
\( T^{8} \)
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| $73$ |
\( T^{8} - 3 T^{7} + \cdots + 1 \)
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| $79$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
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| $83$ |
\( T^{8} \)
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| $89$ |
\( T^{8} \)
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| $97$ |
\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
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