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gp:[N,k,chi] = [294,6,Mod(1,294)]
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("294.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, 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 = [2,8,-18,32,-53,-72,0,128,162,-212,191]
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 \(\beta = \frac{1}{2}(1 + \sqrt{9601})\).
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 intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):
\( T_{5}^{2} + 53T_{5} - 1698 \)
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\( T_{11}^{2} - 191T_{11} - 50886 \)
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|
| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 4)^{2} \)
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| $3$ |
\( (T + 9)^{2} \)
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|
| $5$ |
\( T^{2} + 53T - 1698 \)
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|
| $7$ |
\( T^{2} \)
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|
| $11$ |
\( T^{2} - 191T - 50886 \)
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|
| $13$ |
\( T^{2} + 379T - 254520 \)
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|
| $17$ |
\( T^{2} + 340T - 931200 \)
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|
| $19$ |
\( T^{2} + 1769 T - 2868440 \)
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|
| $23$ |
\( T^{2} - 3236 T + 2579520 \)
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| $29$ |
\( T^{2} - 4459 T - 3960660 \)
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| $31$ |
\( T^{2} - 1994 T - 63563115 \)
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| $37$ |
\( T^{2} - 20587 T + 99713092 \)
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| $41$ |
\( T^{2} - 8814 T + 8966160 \)
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| $43$ |
\( T^{2} - 15853 T + 44661910 \)
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| $47$ |
\( T^{2} - 33912 T + 229093452 \)
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| $53$ |
\( T^{2} - 49239 T + 484607124 \)
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| $59$ |
\( T^{2} + 56735 T + 800680236 \)
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| $61$ |
\( T^{2} + \cdots + 1136874660 \)
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| $67$ |
\( T^{2} + \cdots + 1160899190 \)
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| $71$ |
\( T^{2} + \cdots - 4289674884 \)
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| $73$ |
\( T^{2} + \cdots - 1835129806 \)
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| $79$ |
\( T^{2} + \cdots - 2156463813 \)
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| $83$ |
\( T^{2} + \cdots - 7511023590 \)
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| $89$ |
\( T^{2} + \cdots + 7687683936 \)
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| $97$ |
\( T^{2} + \cdots - 5231869258 \)
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