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gp:[N,k,chi] = [11,8,Mod(1,11)]
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("11.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
Newform invariants
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sage:traces = [2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 \(\beta = 2\sqrt{15}\).
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
|
| \(11\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 8T_{2} - 44 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(11))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 8T - 44 \)
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|
| $3$ |
\( T^{2} + 6T - 2151 \)
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|
| $5$ |
\( T^{2} + 470T + 31225 \)
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|
| $7$ |
\( T^{2} + 1228T - 26444 \)
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|
| $11$ |
\( (T - 1331)^{2} \)
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|
| $13$ |
\( T^{2} - 344 T - 16069856 \)
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|
| $17$ |
\( T^{2} + 8468 T - 788446604 \)
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|
| $19$ |
\( T^{2} + 35280 T - 222369840 \)
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|
| $23$ |
\( T^{2} + 61486 T - 159113951 \)
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|
| $29$ |
\( T^{2} - 179040 T + 122928960 \)
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|
| $31$ |
\( T^{2} + \cdots - 23689658111 \)
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|
| $37$ |
\( T^{2} + \cdots + 191745594561 \)
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|
| $41$ |
\( T^{2} + \cdots - 264475105136 \)
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|
| $43$ |
\( T^{2} + \cdots + 9203802564 \)
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|
| $47$ |
\( T^{2} + \cdots + 677029127296 \)
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|
| $53$ |
\( T^{2} + \cdots + 388813137924 \)
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|
| $59$ |
\( T^{2} + \cdots + 207772229185 \)
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|
| $61$ |
\( T^{2} + \cdots + 9007507329744 \)
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|
| $67$ |
\( T^{2} + \cdots - 9239984638199 \)
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|
| $71$ |
\( T^{2} + \cdots - 11975092638711 \)
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|
| $73$ |
\( T^{2} + \cdots - 2090754692576 \)
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|
| $79$ |
\( T^{2} + \cdots - 2681742596060 \)
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|
| $83$ |
\( T^{2} + \cdots - 536001911676 \)
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|
| $89$ |
\( T^{2} + \cdots + 42998937114225 \)
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|
| $97$ |
\( T^{2} + \cdots - 1834997592239 \)
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