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gp:[N,k,chi] = [84,6,Mod(1,84)]
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("84.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
Newform invariants
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sage:traces = [2,0,-18]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 \(\beta = \sqrt{5569}\).
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 kernel of the linear operator
\( T_{5}^{2} + 6T_{5} - 5560 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(84))\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{2} \)
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|
| $3$ |
\( (T + 9)^{2} \)
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|
| $5$ |
\( T^{2} + 6T - 5560 \)
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|
| $7$ |
\( (T + 49)^{2} \)
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|
| $11$ |
\( T^{2} + 90T - 270856 \)
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|
| $13$ |
\( T^{2} - 768T - 53028 \)
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|
| $17$ |
\( T^{2} - 1926 T + 921800 \)
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|
| $19$ |
\( T^{2} - 2248 T - 1944368 \)
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|
| $23$ |
\( T^{2} - 6354 T + 10087760 \)
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|
| $29$ |
\( T^{2} - 10572 T + 19031396 \)
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|
| $31$ |
\( T^{2} + 3312 T - 26127360 \)
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|
| $37$ |
\( T^{2} - 2104 T - 124195796 \)
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|
| $41$ |
\( T^{2} - 1266 T - 5663952 \)
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|
| $43$ |
\( T^{2} + 5768 T - 312456944 \)
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|
| $47$ |
\( T^{2} - 15612 T - 85219200 \)
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|
| $53$ |
\( T^{2} - 16512 T - 236774628 \)
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|
| $59$ |
\( T^{2} + 13140 T + 29242400 \)
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|
| $61$ |
\( T^{2} + 5796 T - 379738620 \)
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|
| $67$ |
\( T^{2} + 56116 T - 401418272 \)
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|
| $71$ |
\( T^{2} + \cdots - 4430537104 \)
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|
| $73$ |
\( T^{2} + \cdots + 1653999820 \)
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|
| $79$ |
\( T^{2} + 19620 T + 62354304 \)
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|
| $83$ |
\( T^{2} + 44424 T - 76892656 \)
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|
| $89$ |
\( T^{2} + \cdots + 10266670512 \)
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|
| $97$ |
\( T^{2} + \cdots - 14004028100 \)
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