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gp:[N,k,chi] = [360,10,Mod(1,360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("360.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
Newform invariants
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sage:traces = [2,0,0,0,1250,0,-908]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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 \(\beta = 24\sqrt{46}\).
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
|
| \(2\) |
\( +1 \) |
| \(3\) |
\( -1 \) |
| \(5\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(360))\):
\( T_{7}^{2} + 908T_{7} - 4271708 \)
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\( T_{11}^{2} + 25120T_{11} - 318008576 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( (T - 625)^{2} \)
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|
| $7$ |
\( T^{2} + 908 T - 4271708 \)
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|
| $11$ |
\( T^{2} + 25120 T - 318008576 \)
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|
| $13$ |
\( T^{2} + \cdots - 2558426108 \)
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|
| $17$ |
\( T^{2} + \cdots + 7111617700 \)
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|
| $19$ |
\( T^{2} + \cdots - 273876705776 \)
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|
| $23$ |
\( T^{2} + \cdots + 363587809732 \)
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| $29$ |
\( T^{2} + \cdots + 480965491876 \)
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| $31$ |
\( T^{2} + \cdots + 20138081829520 \)
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| $37$ |
\( T^{2} + \cdots - 155824381229436 \)
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| $41$ |
\( T^{2} + \cdots - 149515623312732 \)
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| $43$ |
\( T^{2} + \cdots + 69358444830148 \)
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| $47$ |
\( T^{2} + \cdots - 11\!\cdots\!84 \)
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| $53$ |
\( T^{2} + \cdots - 31\!\cdots\!84 \)
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| $59$ |
\( T^{2} + \cdots - 58\!\cdots\!52 \)
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| $61$ |
\( T^{2} + \cdots - 10\!\cdots\!60 \)
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| $67$ |
\( T^{2} + \cdots - 32\!\cdots\!48 \)
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| $71$ |
\( T^{2} + \cdots - 23\!\cdots\!40 \)
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| $73$ |
\( T^{2} + \cdots + 12\!\cdots\!12 \)
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| $79$ |
\( T^{2} + \cdots + 36\!\cdots\!80 \)
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| $83$ |
\( T^{2} + \cdots - 33\!\cdots\!32 \)
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| $89$ |
\( T^{2} + \cdots - 11\!\cdots\!36 \)
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| $97$ |
\( T^{2} + \cdots + 98\!\cdots\!96 \)
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