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gp:[N,k,chi] = [513,2,Mod(28,513)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(513, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 8]))
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("513.28");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [6,3,0,9,3]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\zeta_{18}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).
| \(n\) |
\(191\) |
\(325\) |
| \(\chi(n)\) |
\(1\) |
\(\zeta_{18}^{2}\) |
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
\( T_{2}^{6} - 3T_{2}^{5} + 3T_{2}^{3} + 9T_{2}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(513, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 9 \)
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|
| $3$ |
\( T^{6} \)
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|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 9 \)
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|
| $7$ |
\( T^{6} - 9 T^{5} + \cdots + 361 \)
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|
| $11$ |
\( T^{6} - 9 T^{5} + \cdots + 81 \)
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|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 361 \)
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|
| $17$ |
\( T^{6} + 24 T^{5} + \cdots + 12321 \)
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|
| $19$ |
\( T^{6} + 12 T^{5} + \cdots + 6859 \)
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|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 9 \)
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|
| $29$ |
\( T^{6} + 12 T^{5} + \cdots + 2601 \)
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|
| $31$ |
\( T^{6} + 57 T^{4} + \cdots + 11449 \)
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|
| $37$ |
\( (T^{3} + 27 T^{2} + \cdots + 613)^{2} \)
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| $41$ |
\( T^{6} + 6 T^{5} + \cdots + 103041 \)
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| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 16129 \)
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| $47$ |
\( T^{6} - 6 T^{5} + \cdots + 2601 \)
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| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 2601 \)
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| $59$ |
\( T^{6} - 12 T^{5} + \cdots + 3249 \)
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| $61$ |
\( T^{6} - 6 T^{5} + \cdots + 5041 \)
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| $67$ |
\( T^{6} + 3 T^{5} + \cdots + 361 \)
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| $71$ |
\( T^{6} + 24 T^{5} + \cdots + 3249 \)
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| $73$ |
\( T^{6} + 21 T^{5} + \cdots + 11449 \)
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| $79$ |
\( T^{6} - 33 T^{5} + \cdots + 1014049 \)
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| $83$ |
\( T^{6} + 27 T^{4} + \cdots + 729 \)
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| $89$ |
\( T^{6} - 15 T^{5} + \cdots + 145161 \)
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| $97$ |
\( T^{6} - 45 T^{5} + \cdots + 289 \)
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