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gp:[N,k,chi] = [62,6,Mod(1,62)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(62, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("62.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
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sage:traces = [2,8]
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 \(\beta = 2\sqrt{7}\).
We also show the integral \(q\)-expansion of the trace form.
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)
| \( p \) |
Sign
|
| \(2\) |
\( -1 \) |
| \(31\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 8T_{3} - 12 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(62))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 4)^{2} \)
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|
| $3$ |
\( T^{2} + 8T - 12 \)
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|
| $5$ |
\( T^{2} + 110T + 2913 \)
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|
| $7$ |
\( T^{2} + 126T - 13531 \)
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|
| $11$ |
\( T^{2} + 396T - 218844 \)
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|
| $13$ |
\( T^{2} + 632T + 79444 \)
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|
| $17$ |
\( T^{2} + 720T - 112572 \)
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|
| $19$ |
\( T^{2} + 438 T - 3682339 \)
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|
| $23$ |
\( T^{2} + 560 T - 18519900 \)
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|
| $29$ |
\( T^{2} - 4404 T - 7459128 \)
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|
| $31$ |
\( (T - 961)^{2} \)
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|
| $37$ |
\( T^{2} + 8100 T - 285052 \)
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|
| $41$ |
\( T^{2} - 1742 T - 66685071 \)
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|
| $43$ |
\( T^{2} + 8572 T - 117520056 \)
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|
| $47$ |
\( T^{2} - 26792 T + 120582816 \)
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|
| $53$ |
\( T^{2} - 33656 T - 6052704 \)
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|
| $59$ |
\( T^{2} + \cdots - 1468435731 \)
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|
| $61$ |
\( T^{2} - 26328 T + 154325684 \)
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|
| $67$ |
\( T^{2} + 60224 T + 899449744 \)
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|
| $71$ |
\( T^{2} + \cdots + 5812100253 \)
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|
| $73$ |
\( T^{2} + \cdots - 2882073900 \)
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|
| $79$ |
\( T^{2} + \cdots + 6592993416 \)
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|
| $83$ |
\( T^{2} + \cdots + 3417465444 \)
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|
| $89$ |
\( T^{2} + \cdots - 1748038584 \)
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|
| $97$ |
\( T^{2} + \cdots - 4956802759 \)
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