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gp:[N,k,chi] = [210,8,Mod(1,210)]
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("210.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2,16,-54,128,-250,-432,686]
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 \(\beta = 6\sqrt{60349}\).
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
|
| \(2\) |
\( -1 \) |
| \(3\) |
\( +1 \) |
| \(5\) |
\( +1 \) |
| \(7\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} - 4092T_{11} + 2013552 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(210))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( (T - 8)^{2} \)
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|
| $3$ |
\( (T + 27)^{2} \)
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|
| $5$ |
\( (T + 125)^{2} \)
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|
| $7$ |
\( (T - 343)^{2} \)
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|
| $11$ |
\( T^{2} - 4092 T + 2013552 \)
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|
| $13$ |
\( T^{2} + 7280 T - 93206036 \)
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|
| $17$ |
\( T^{2} + 13860 T - 91019196 \)
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|
| $19$ |
\( T^{2} + \cdots - 2361914192 \)
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|
| $23$ |
\( T^{2} + 9096 T - 535492080 \)
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|
| $29$ |
\( T^{2} + \cdots - 16522618716 \)
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|
| $31$ |
\( T^{2} + \cdots - 5428053488 \)
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|
| $37$ |
\( T^{2} + \cdots + 43566183316 \)
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|
| $41$ |
\( T^{2} + \cdots - 183781616508 \)
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|
| $43$ |
\( T^{2} + \cdots + 292614706096 \)
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|
| $47$ |
\( T^{2} + \cdots + 373986171072 \)
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|
| $53$ |
\( T^{2} + \cdots + 632733407532 \)
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|
| $59$ |
\( T^{2} + \cdots + 831021676848 \)
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|
| $61$ |
\( T^{2} + \cdots - 644491113500 \)
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|
| $67$ |
\( T^{2} + \cdots - 4354237287536 \)
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|
| $71$ |
\( T^{2} + \cdots + 6324026961168 \)
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|
| $73$ |
\( T^{2} + \cdots - 12046067776820 \)
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|
| $79$ |
\( T^{2} + \cdots + 28730309848000 \)
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|
| $83$ |
\( T^{2} + \cdots - 1805868690192 \)
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|
| $89$ |
\( T^{2} + \cdots + 36727973263620 \)
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|
| $97$ |
\( T^{2} + \cdots - 76369512323300 \)
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