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gp:[N,k,chi] = [325,6,Mod(1,325)]
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("325.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, 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 = [2,5]
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\).
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
|
| \(5\) |
\( +1 \) |
| \(13\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 5T_{2} + 2 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(325))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 5T + 2 \)
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|
| $3$ |
\( T^{2} - 28T + 43 \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} - 36T - 20501 \)
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|
| $11$ |
\( T^{2} + 376T + 5356 \)
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|
| $13$ |
\( (T - 169)^{2} \)
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|
| $17$ |
\( T^{2} - 2630 T + 1659593 \)
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|
| $19$ |
\( T^{2} + 312T + 21004 \)
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|
| $23$ |
\( T^{2} - 2624 T - 6800144 \)
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|
| $29$ |
\( T^{2} + 812T + 155044 \)
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|
| $31$ |
\( T^{2} - 7720 T + 14046608 \)
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|
| $37$ |
\( T^{2} - 16858 T + 56659241 \)
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|
| $41$ |
\( T^{2} - 7840 T - 827392 \)
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|
| $43$ |
\( T^{2} + 2420 T - 93138877 \)
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|
| $47$ |
\( T^{2} + 9972 T - 372552629 \)
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|
| $53$ |
\( T^{2} - 43720 T + 280413712 \)
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|
| $59$ |
\( T^{2} + 38936 T + 59682572 \)
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|
| $61$ |
\( T^{2} - 1984 T - 366282304 \)
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|
| $67$ |
\( T^{2} + \cdots + 1222318028 \)
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|
| $71$ |
\( T^{2} + \cdots + 1078437091 \)
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|
| $73$ |
\( T^{2} + 74412 T - 79726364 \)
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|
| $79$ |
\( T^{2} + 55296 T + 361555696 \)
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|
| $83$ |
\( T^{2} - 75712 T + 68289536 \)
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|
| $89$ |
\( T^{2} + \cdots + 1045666948 \)
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|
| $97$ |
\( T^{2} + \cdots - 14352168316 \)
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