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gp:[N,k,chi] = [15,10,Mod(1,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
Newform invariants
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sage:traces = [2,31]
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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{241})\).
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
|
| \(3\) |
\( +1 \) |
| \(5\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 31T_{2} - 302 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(15))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} - 31T - 302 \)
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|
| $3$ |
\( (T + 81)^{2} \)
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|
| $5$ |
\( (T - 625)^{2} \)
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|
| $7$ |
\( T^{2} - 14112 T + 22579200 \)
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|
| $11$ |
\( T^{2} + \cdots - 2924934128 \)
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|
| $13$ |
\( T^{2} + \cdots - 15338329532 \)
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|
| $17$ |
\( T^{2} + \cdots - 24505657916 \)
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|
| $19$ |
\( T^{2} + \cdots - 15492203120 \)
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|
| $23$ |
\( T^{2} + \cdots - 239117414400 \)
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|
| $29$ |
\( T^{2} + \cdots - 13616383922300 \)
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|
| $31$ |
\( T^{2} + \cdots + 16415447040000 \)
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|
| $37$ |
\( T^{2} + \cdots + 99286893737380 \)
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|
| $41$ |
\( T^{2} + \cdots + 38475315093220 \)
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|
| $43$ |
\( T^{2} + \cdots - 254550637865456 \)
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|
| $47$ |
\( T^{2} + \cdots - 18\!\cdots\!24 \)
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|
| $53$ |
\( T^{2} + \cdots - 14677210114460 \)
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|
| $59$ |
\( T^{2} + \cdots - 99\!\cdots\!20 \)
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|
| $61$ |
\( T^{2} + \cdots + 28\!\cdots\!96 \)
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|
| $67$ |
\( T^{2} + \cdots + 20\!\cdots\!64 \)
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|
| $71$ |
\( T^{2} + \cdots - 147594805309376 \)
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|
| $73$ |
\( T^{2} + \cdots + 12\!\cdots\!60 \)
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|
| $79$ |
\( T^{2} + \cdots - 19\!\cdots\!00 \)
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|
| $83$ |
\( T^{2} + \cdots + 60\!\cdots\!92 \)
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|
| $89$ |
\( T^{2} + \cdots + 13\!\cdots\!20 \)
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| $97$ |
\( T^{2} + \cdots + 68\!\cdots\!64 \)
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