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gp:[N,k,chi] = [21,4,Mod(1,21)]
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("21.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [2,-3]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\).
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
|
| \(3\) |
\( -1 \) |
| \(7\) |
\( -1 \) |
This newform does not admit any (nontrivial) inner twists.
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 3T_{2} - 12 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} + 3T - 12 \)
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|
| $3$ |
\( (T - 3)^{2} \)
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|
| $5$ |
\( T^{2} - 6T - 48 \)
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|
| $7$ |
\( (T - 7)^{2} \)
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|
| $11$ |
\( T^{2} + 6T - 1416 \)
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|
| $13$ |
\( T^{2} - 16T - 1988 \)
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|
| $17$ |
\( T^{2} + 6T - 48 \)
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|
| $19$ |
\( T^{2} - 64T - 7184 \)
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|
| $23$ |
\( T^{2} - 6T - 16464 \)
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|
| $29$ |
\( T^{2} + 252T + 7668 \)
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|
| $31$ |
\( T^{2} - 40T - 73472 \)
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|
| $37$ |
\( T^{2} + 248T - 3092 \)
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|
| $41$ |
\( T^{2} + 450T + 37800 \)
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|
| $43$ |
\( T^{2} - 376T + 2512 \)
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|
| $47$ |
\( T^{2} + 12T - 65856 \)
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|
| $53$ |
\( T^{2} + 1104 T + 304476 \)
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|
| $59$ |
\( T^{2} - 804T - 30144 \)
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|
| $61$ |
\( T^{2} + 428T - 28076 \)
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|
| $67$ |
\( T^{2} - 148T - 160736 \)
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| $71$ |
\( T^{2} - 954T + 214704 \)
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|
| $73$ |
\( T^{2} - 1072 T + 285244 \)
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|
| $79$ |
\( T^{2} + 572T - 84416 \)
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|
| $83$ |
\( T^{2} - 1944 T + 813456 \)
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|
| $89$ |
\( T^{2} - 366T - 253848 \)
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|
| $97$ |
\( T^{2} - 808T - 922292 \)
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