Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,10,Mod(1,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(186.958008527\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Embeddings
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
36.0000 | −81.0000 | 784.000 | −1314.00 | −2916.00 | 4480.00 | 9792.00 | 6561.00 | −47304.0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.10.a.b | 1 | |
| 11.b | odd | 2 | 1 | 3.10.a.a | ✓ | 1 | |
| 33.d | even | 2 | 1 | 9.10.a.c | 1 | ||
| 44.c | even | 2 | 1 | 48.10.a.e | 1 | ||
| 55.d | odd | 2 | 1 | 75.10.a.d | 1 | ||
| 55.e | even | 4 | 2 | 75.10.b.a | 2 | ||
| 77.b | even | 2 | 1 | 147.10.a.a | 1 | ||
| 88.b | odd | 2 | 1 | 192.10.a.m | 1 | ||
| 88.g | even | 2 | 1 | 192.10.a.f | 1 | ||
| 99.g | even | 6 | 2 | 81.10.c.a | 2 | ||
| 99.h | odd | 6 | 2 | 81.10.c.e | 2 | ||
| 132.d | odd | 2 | 1 | 144.10.a.l | 1 | ||
| 165.d | even | 2 | 1 | 225.10.a.a | 1 | ||
| 165.l | odd | 4 | 2 | 225.10.b.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.10.a.a | ✓ | 1 | 11.b | odd | 2 | 1 | |
| 9.10.a.c | 1 | 33.d | even | 2 | 1 | ||
| 48.10.a.e | 1 | 44.c | even | 2 | 1 | ||
| 75.10.a.d | 1 | 55.d | odd | 2 | 1 | ||
| 75.10.b.a | 2 | 55.e | even | 4 | 2 | ||
| 81.10.c.a | 2 | 99.g | even | 6 | 2 | ||
| 81.10.c.e | 2 | 99.h | odd | 6 | 2 | ||
| 144.10.a.l | 1 | 132.d | odd | 2 | 1 | ||
| 147.10.a.a | 1 | 77.b | even | 2 | 1 | ||
| 192.10.a.f | 1 | 88.g | even | 2 | 1 | ||
| 192.10.a.m | 1 | 88.b | odd | 2 | 1 | ||
| 225.10.a.a | 1 | 165.d | even | 2 | 1 | ||
| 225.10.b.a | 2 | 165.l | odd | 4 | 2 | ||
| 363.10.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 36 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(363))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T - 36 \)
|
| $3$ |
\( T + 81 \)
|
| $5$ |
\( T + 1314 \)
|
| $7$ |
\( T - 4480 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T - 151522 \)
|
| $17$ |
\( T + 108162 \)
|
| $19$ |
\( T + 593084 \)
|
| $23$ |
\( T + 969480 \)
|
| $29$ |
\( T - 6642522 \)
|
| $31$ |
\( T - 7070600 \)
|
| $37$ |
\( T + 7472410 \)
|
| $41$ |
\( T - 4350150 \)
|
| $43$ |
\( T - 4358716 \)
|
| $47$ |
\( T - 28309248 \)
|
| $53$ |
\( T - 16111710 \)
|
| $59$ |
\( T + 86075964 \)
|
| $61$ |
\( T + 32213918 \)
|
| $67$ |
\( T - 99531452 \)
|
| $71$ |
\( T + 44170488 \)
|
| $73$ |
\( T - 23560630 \)
|
| $79$ |
\( T - 401754760 \)
|
| $83$ |
\( T - 744528708 \)
|
| $89$ |
\( T - 769871034 \)
|
| $97$ |
\( T - 907130882 \)
|
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