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gp:[N,k,chi] = [798,2,Mod(457,798)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("798.457");
S:= CuspForms(chi, 2);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(798, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 2, names="a")
Newform invariants
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sage:traces = [4,-2,2,-2,-4,-4,0]
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 \(\zeta_{12}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).
| \(n\) |
\(115\) |
\(211\) |
\(533\) |
| \(\chi(n)\) |
\(-\zeta_{12}^{2}\) |
\(1\) |
\(1\) |
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 \(S_{2}^{\mathrm{new}}(798, [\chi])\):
\( T_{5}^{4} + 4T_{5}^{3} + 15T_{5}^{2} + 4T_{5} + 1 \)
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\( T_{11}^{2} - T_{11} + 1 \)
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\( T_{13}^{2} - 4T_{13} - 8 \)
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T^{2} + T + 1)^{2} \)
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| $3$ |
\( (T^{2} - T + 1)^{2} \)
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| $5$ |
\( T^{4} + 4 T^{3} + \cdots + 1 \)
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| $7$ |
\( T^{4} - 13T^{2} + 49 \)
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| $11$ |
\( (T^{2} - T + 1)^{2} \)
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| $13$ |
\( (T^{2} - 4 T - 8)^{2} \)
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| $17$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
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| $19$ |
\( (T^{2} - T + 1)^{2} \)
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| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
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| $29$ |
\( (T - 1)^{4} \)
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| $31$ |
\( T^{4} + 2 T^{3} + \cdots + 121 \)
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| $37$ |
\( T^{4} + 2 T^{3} + \cdots + 676 \)
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| $41$ |
\( (T^{2} - 14 T + 46)^{2} \)
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| $43$ |
\( (T^{2} + 10 T - 2)^{2} \)
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| $47$ |
\( T^{4} + 16 T^{3} + \cdots + 2704 \)
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| $53$ |
\( T^{4} + 10 T^{3} + \cdots + 169 \)
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| $59$ |
\( T^{4} + 3T^{2} + 9 \)
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| $61$ |
\( T^{4} + 16 T^{3} + \cdots + 2704 \)
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| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 144 \)
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| $71$ |
\( (T^{2} + 14 T + 22)^{2} \)
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| $73$ |
\( T^{4} + 48T^{2} + 2304 \)
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| $79$ |
\( T^{4} - 2 T^{3} + \cdots + 11449 \)
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| $83$ |
\( (T + 3)^{4} \)
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| $89$ |
\( T^{4} + 18 T^{3} + \cdots + 36 \)
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| $97$ |
\( (T^{2} - 8 T + 13)^{2} \)
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