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gp:[N,k,chi] = [490,6,Mod(1,490)]
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("490.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,8,-5]
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 \(\beta = \frac{1}{2}(1 + \sqrt{1129})\).
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
|
| \(2\) |
\( -1 \) |
| \(5\) |
\( +1 \) |
| \(7\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 5T_{3} - 276 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 4)^{2} \)
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|
| $3$ |
\( T^{2} + 5T - 276 \)
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|
| $5$ |
\( (T + 25)^{2} \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} - 415T - 4644 \)
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|
| $13$ |
\( T^{2} + 429T + 11858 \)
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|
| $17$ |
\( T^{2} + 1319 T - 757566 \)
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|
| $19$ |
\( T^{2} + 1918 T - 2748440 \)
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|
| $23$ |
\( T^{2} + 1334 T - 2265840 \)
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|
| $29$ |
\( T^{2} + 1131 T - 2121390 \)
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|
| $31$ |
\( T^{2} - 5472 T - 49163008 \)
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|
| $37$ |
\( T^{2} + 9156 T + 13732484 \)
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|
| $41$ |
\( T^{2} - 7822 T + 11365872 \)
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|
| $43$ |
\( T^{2} - 2078 T - 364446648 \)
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|
| $47$ |
\( T^{2} - 8507 T - 572885328 \)
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|
| $53$ |
\( T^{2} + 34242 T + 9748512 \)
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|
| $59$ |
\( T^{2} + 6952 T - 48684720 \)
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|
| $61$ |
\( T^{2} - 45138 T + 402952640 \)
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|
| $67$ |
\( T^{2} + 54556 T + 649140384 \)
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|
| $71$ |
\( T^{2} + 62272 T - 106007808 \)
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|
| $73$ |
\( T^{2} + 40896 T - 405277060 \)
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|
| $79$ |
\( T^{2} - 62069 T - 653150040 \)
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|
| $83$ |
\( T^{2} + \cdots + 8986877424 \)
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|
| $89$ |
\( T^{2} + \cdots - 1135587000 \)
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|
| $97$ |
\( T^{2} + \cdots + 5804074010 \)
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