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gp:[N,k,chi] = [198,4,Mod(37,198)]
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("198.37");
S:= CuspForms(chi, 4);
N := Newforms(S);
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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(198, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
Newform invariants
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sage:traces = [4,-2,0,-4,-15]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).
| \(n\) |
\(145\) |
\(155\) |
| \(\chi(n)\) |
\(-\zeta_{10}^{3}\) |
\(1\) |
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
\( T_{5}^{4} + 15T_{5}^{3} + 160T_{5}^{2} + 1550T_{5} + 24025 \)
acting on \(S_{4}^{\mathrm{new}}(198, [\chi])\).
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| $p$ |
$F_p(T)$ |
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
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|
| $3$ |
\( T^{4} \)
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|
| $5$ |
\( T^{4} + 15 T^{3} + \cdots + 24025 \)
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|
| $7$ |
\( T^{4} + 19 T^{3} + \cdots + 841 \)
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|
| $11$ |
\( T^{4} + 89 T^{3} + \cdots + 1771561 \)
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|
| $13$ |
\( T^{4} - 35 T^{3} + \cdots + 625 \)
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|
| $17$ |
\( T^{4} - 280 T^{3} + \cdots + 200364025 \)
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|
| $19$ |
\( T^{4} + 176 T^{3} + \cdots + 54479161 \)
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|
| $23$ |
\( (T^{2} - 182 T + 6281)^{2} \)
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|
| $29$ |
\( T^{4} - 98 T^{3} + \cdots + 36048016 \)
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|
| $31$ |
\( T^{4} + \cdots + 1126877761 \)
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|
| $37$ |
\( T^{4} + 39 T^{3} + \cdots + 29241 \)
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| $41$ |
\( T^{4} + \cdots + 9198536281 \)
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| $43$ |
\( (T^{2} - 546 T + 63009)^{2} \)
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|
| $47$ |
\( T^{4} + \cdots + 5377435561 \)
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|
| $53$ |
\( T^{4} + \cdots + 31116607201 \)
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| $59$ |
\( T^{4} + \cdots + 165389435761 \)
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| $61$ |
\( T^{4} + \cdots + 2555201401 \)
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| $67$ |
\( (T^{2} - 143 T - 46399)^{2} \)
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| $71$ |
\( T^{4} + \cdots + 127556836801 \)
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| $73$ |
\( T^{4} + \cdots + 101814599056 \)
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|
| $79$ |
\( T^{4} + \cdots + 39450301641 \)
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|
| $83$ |
\( T^{4} + \cdots + 424401434521 \)
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| $89$ |
\( (T^{2} - 1184 T - 561181)^{2} \)
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| $97$ |
\( T^{4} + \cdots + 330750361881 \)
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