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gp:[N,k,chi] = [588,6,Mod(1,588)]
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("588.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,0,18,0,47,0,0,0,162,0,-407]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{7081})\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( -1 \) |
| \(7\) |
\( +1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 47T_{5} - 1218 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(588))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T - 9)^{2} \)
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|
| $5$ |
\( T^{2} - 47T - 1218 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} + 407T + 39642 \)
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|
| $13$ |
\( T^{2} - 449T + 6144 \)
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|
| $17$ |
\( T^{2} + 1868 T + 844032 \)
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|
| $19$ |
\( T^{2} + 1463 T - 4069328 \)
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|
| $23$ |
\( T^{2} + 44 T - 17702016 \)
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|
| $29$ |
\( T^{2} - 767 T - 20885268 \)
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|
| $31$ |
\( T^{2} + 11170 T + 20967261 \)
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|
| $37$ |
\( T^{2} + 3113 T + 1642012 \)
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|
| $41$ |
\( T^{2} + 7842 T + 15310512 \)
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|
| $43$ |
\( T^{2} + 12629 T + 39092230 \)
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|
| $47$ |
\( T^{2} - 9576 T + 20630700 \)
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|
| $53$ |
\( T^{2} - 13395 T - 49605804 \)
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|
| $59$ |
\( T^{2} + 47521 T - 204224580 \)
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|
| $61$ |
\( T^{2} + 63652 T + 670173876 \)
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|
| $67$ |
\( T^{2} - 44541 T + 421580414 \)
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|
| $71$ |
\( T^{2} + \cdots + 3516363900 \)
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|
| $73$ |
\( T^{2} - 6039 T - 989305390 \)
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|
| $79$ |
\( T^{2} + \cdots - 4649155173 \)
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|
| $83$ |
\( T^{2} + \cdots - 6214225254 \)
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|
| $89$ |
\( T^{2} + \cdots + 1725081120 \)
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|
| $97$ |
\( T^{2} + \cdots + 6255893750 \)
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