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gp:[N,k,chi] = [600,6,Mod(1,600)]
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("600.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,0,-18,0,0,0,8]
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 \(\beta = 4\sqrt{2161}\).
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 kernel of the linear operator
\( T_{7}^{2} - 8T_{7} - 34560 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(600))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T + 9)^{2} \)
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|
| $5$ |
\( T^{2} \)
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|
| $7$ |
\( T^{2} - 8T - 34560 \)
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|
| $11$ |
\( T^{2} - 488T - 78768 \)
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|
| $13$ |
\( T^{2} + 612T - 217548 \)
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|
| $17$ |
\( T^{2} + 860T + 150324 \)
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|
| $19$ |
\( T^{2} + 96T - 4181392 \)
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|
| $23$ |
\( T^{2} + 2984 T - 3617280 \)
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|
| $29$ |
\( T^{2} - 2236 T - 12580476 \)
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|
| $31$ |
\( T^{2} - 9352 T - 11362560 \)
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|
| $37$ |
\( T^{2} + 10612 T - 5073900 \)
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|
| $41$ |
\( T^{2} - 17156 T + 66805188 \)
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|
| $43$ |
\( T^{2} + 440 T - 292602864 \)
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|
| $47$ |
\( T^{2} - 16728 T + 40878080 \)
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|
| $53$ |
\( T^{2} + 31484 T + 15321540 \)
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|
| $59$ |
\( T^{2} + 61464 T + 843632208 \)
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|
| $61$ |
\( T^{2} - 51596 T + 360023268 \)
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|
| $67$ |
\( T^{2} + \cdots - 1210108912 \)
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|
| $71$ |
\( T^{2} + \cdots + 2265987648 \)
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|
| $73$ |
\( T^{2} + 58852 T + 281555076 \)
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|
| $79$ |
\( T^{2} + \cdots + 2779010944 \)
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|
| $83$ |
\( T^{2} + \cdots + 4662096912 \)
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|
| $89$ |
\( T^{2} + \cdots + 2197457860 \)
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|
| $97$ |
\( T^{2} + \cdots - 7187636924 \)
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