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gp:[N,k,chi] = [213,2,Mod(25,213)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(213, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 2, names="a")
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magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("213.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
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sage:traces = [4]
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 a primitive root of unity \(\zeta_{10}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/213\mathbb{Z}\right)^\times\).
| \(n\) |
\(7\) |
\(143\) |
| \(\chi(n)\) |
\(-\zeta_{10}^{3}\) |
\(1\) |
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)
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} - 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(213, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
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|
| $3$ |
\( T^{4} - T^{3} + T^{2} + \cdots + 1 \)
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| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 121 \)
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| $7$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
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| $11$ |
\( T^{4} + 13 T^{3} + \cdots + 961 \)
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| $13$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( T^{4} - 14 T^{3} + \cdots + 841 \)
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| $23$ |
\( (T^{2} - 2 T - 19)^{2} \)
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| $29$ |
\( T^{4} - 16 T^{3} + \cdots + 3481 \)
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| $31$ |
\( T^{4} + 13 T^{3} + \cdots + 121 \)
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| $37$ |
\( (T^{2} + 7 T + 11)^{2} \)
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| $41$ |
\( (T^{2} + 5 T - 25)^{2} \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
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| $47$ |
\( T^{4} + 15 T^{3} + \cdots + 2025 \)
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| $53$ |
\( T^{4} - 9 T^{3} + \cdots + 17161 \)
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| $59$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
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| $61$ |
\( T^{4} + 5 T^{3} + \cdots + 3025 \)
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| $67$ |
\( T^{4} - 16 T^{3} + \cdots + 256 \)
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| $71$ |
\( T^{4} + 19 T^{3} + \cdots + 5041 \)
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| $73$ |
\( T^{4} - 20 T^{3} + \cdots + 3025 \)
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| $79$ |
\( T^{4} + 27 T^{3} + \cdots + 961 \)
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| $83$ |
\( T^{4} + 17 T^{3} + \cdots + 5041 \)
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| $89$ |
\( T^{4} - 14 T^{3} + \cdots + 961 \)
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| $97$ |
\( (T^{2} - 6 T - 71)^{2} \)
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