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gp:[N,k,chi] = [70,8,Mod(11,70)]
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("70.11");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,-8,-30]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 \(\zeta_{6}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) |
\(31\) |
\(57\) |
| \(\chi(n)\) |
\(-\zeta_{6}\) |
\(1\) |
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 kernel of the linear operator
\( T_{3}^{2} + 30T_{3} + 900 \)
acting on \(S_{8}^{\mathrm{new}}(70, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 8T + 64 \)
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|
| $3$ |
\( T^{2} + 30T + 900 \)
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|
| $5$ |
\( T^{2} - 125T + 15625 \)
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|
| $7$ |
\( T^{2} + 1372 T + 823543 \)
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|
| $11$ |
\( T^{2} + 3T + 9 \)
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|
| $13$ |
\( (T - 1745)^{2} \)
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|
| $17$ |
\( T^{2} - 3786 T + 14333796 \)
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|
| $19$ |
\( T^{2} - 1945 T + 3783025 \)
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|
| $23$ |
\( T^{2} + \cdots + 6328361601 \)
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|
| $29$ |
\( (T + 94926)^{2} \)
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|
| $31$ |
\( T^{2} + \cdots + 16288906384 \)
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|
| $37$ |
\( T^{2} + \cdots + 16449858049 \)
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| $41$ |
\( (T + 298077)^{2} \)
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| $43$ |
\( (T + 875626)^{2} \)
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|
| $47$ |
\( T^{2} + \cdots + 374004410481 \)
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| $53$ |
\( T^{2} + \cdots + 67151984769 \)
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| $59$ |
\( T^{2} + \cdots + 8279062456896 \)
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| $61$ |
\( T^{2} + \cdots + 22071262096 \)
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| $67$ |
\( T^{2} + \cdots + 3207265501456 \)
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|
| $71$ |
\( (T + 493236)^{2} \)
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| $73$ |
\( T^{2} + \cdots + 4235578034704 \)
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| $79$ |
\( T^{2} + \cdots + 34422557321476 \)
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|
| $83$ |
\( (T + 921132)^{2} \)
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|
| $89$ |
\( T^{2} + \cdots + 26245969178724 \)
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| $97$ |
\( (T + 5878306)^{2} \)
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