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gp:[N,k,chi] = [510,2,Mod(257,510)]
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("510.257");
S:= CuspForms(chi, 2);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(510, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 2, 7]))
N = Newforms(chi, 2, names="a")
Newform invariants
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sage:traces = [4,4]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\zeta_{8}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).
| \(n\) |
\(241\) |
\(307\) |
\(341\) |
| \(\chi(n)\) |
\(\zeta_{8}\) |
\(-\zeta_{8}^{2}\) |
\(-1\) |
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 \(S_{2}^{\mathrm{new}}(510, [\chi])\):
\( T_{7}^{4} + 8T_{7}^{3} + 18T_{7}^{2} - 4T_{7} + 2 \)
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\( T_{11}^{4} + 12T_{11}^{3} + 54T_{11}^{2} + 108T_{11} + 162 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 9 \)
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| $5$ |
\( (T^{2} + 4 T + 5)^{2} \)
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| $7$ |
\( T^{4} + 8 T^{3} + \cdots + 2 \)
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| $11$ |
\( T^{4} + 12 T^{3} + \cdots + 162 \)
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| $13$ |
\( T^{4} + 12 T^{3} + \cdots + 196 \)
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| $17$ |
\( (T^{2} + 6 T + 17)^{2} \)
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| $19$ |
\( T^{4} + 4 T^{3} + \cdots + 1156 \)
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| $23$ |
\( T^{4} + 8 T^{3} + \cdots + 98 \)
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| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 2 \)
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| $31$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
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| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 98 \)
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| $41$ |
\( T^{4} + 4 T^{3} + \cdots + 98 \)
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| $43$ |
\( T^{4} + 136T^{2} + 16 \)
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| $47$ |
\( T^{4} + 12 T^{3} + \cdots + 4 \)
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| $53$ |
\( T^{4} + 48T^{2} + 64 \)
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| $59$ |
\( T^{4} + 20 T^{3} + \cdots + 2116 \)
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| $61$ |
\( T^{4} - 24 T^{3} + \cdots + 1058 \)
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| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 196 \)
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| $71$ |
\( T^{4} - 16 T^{3} + \cdots + 4802 \)
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| $73$ |
\( T^{4} + 12 T^{3} + \cdots + 7938 \)
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| $79$ |
\( T^{4} + 32 T^{3} + \cdots + 28322 \)
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| $83$ |
\( T^{4} + 136T^{2} + 16 \)
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| $89$ |
\( (T^{2} + 200)^{2} \)
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| $97$ |
\( T^{4} + 20 T^{3} + \cdots + 1250 \)
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