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gp:[N,k,chi] = [225,6,Mod(1,225)]
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("225.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, 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,-1,0,141,0,0,112]
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 \(\beta = \frac{1}{2}(1 + \sqrt{409})\).
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 intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):
\( T_{2}^{2} + T_{2} - 102 \)
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\( T_{7}^{2} - 112T_{7} - 23040 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + T - 102 \)
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|
| $3$ |
\( T^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} - 112T - 23040 \)
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|
| $11$ |
\( T^{2} + 248T - 89328 \)
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| $13$ |
\( T^{2} + 876T + 165668 \)
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|
| $17$ |
\( T^{2} - 2036 T + 381924 \)
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|
| $19$ |
\( T^{2} - 1464 T - 3887920 \)
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|
| $23$ |
\( T^{2} + 3216 T + 2350080 \)
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| $29$ |
\( T^{2} + 1948 T + 529860 \)
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| $31$ |
\( T^{2} - 2672 T - 1382400 \)
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| $37$ |
\( T^{2} + 8668 T - 49300220 \)
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| $41$ |
\( T^{2} - 7628 T - 15712860 \)
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| $43$ |
\( T^{2} - 16440 T + 67149584 \)
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| $47$ |
\( T^{2} + 19360 T - 61495104 \)
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| $53$ |
\( T^{2} + 14356 T - 476289180 \)
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| $59$ |
\( T^{2} + \cdots - 1067881200 \)
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|
| $61$ |
\( T^{2} - 20220 T - 149182204 \)
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| $67$ |
\( T^{2} - 12904 T - 125898096 \)
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| $71$ |
\( T^{2} - 40976 T - 627281856 \)
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| $73$ |
\( T^{2} + 59124 T + 836113700 \)
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| $79$ |
\( T^{2} + \cdots + 1448870400 \)
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| $83$ |
\( T^{2} + \cdots + 2064089232 \)
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| $89$ |
\( T^{2} + \cdots - 2895368220 \)
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| $97$ |
\( T^{2} + \cdots - 15772164476 \)
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