Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(58.2193265921\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−4.47214 | −9.00000 | −12.0000 | −98.0000 | 40.2492 | −237.023 | 196.774 | 81.0000 | 438.269 | ||||||||||||||||||||||||
| 1.2 | 4.47214 | −9.00000 | −12.0000 | −98.0000 | −40.2492 | 237.023 | −196.774 | 81.0000 | −438.269 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.6.a.h | ✓ | 2 |
| 3.b | odd | 2 | 1 | 1089.6.a.l | 2 | ||
| 11.b | odd | 2 | 1 | inner | 363.6.a.h | ✓ | 2 |
| 33.d | even | 2 | 1 | 1089.6.a.l | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.6.a.h | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 363.6.a.h | ✓ | 2 | 11.b | odd | 2 | 1 | inner |
| 1089.6.a.l | 2 | 3.b | odd | 2 | 1 | ||
| 1089.6.a.l | 2 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 20 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 20 \)
|
| $3$ |
\( (T + 9)^{2} \)
|
| $5$ |
\( (T + 98)^{2} \)
|
| $7$ |
\( T^{2} - 56180 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} - 882000 \)
|
| $17$ |
\( T^{2} - 106580 \)
|
| $19$ |
\( T^{2} - 188180 \)
|
| $23$ |
\( (T - 376)^{2} \)
|
| $29$ |
\( T^{2} - 12576980 \)
|
| $31$ |
\( (T - 3720)^{2} \)
|
| $37$ |
\( (T + 9478)^{2} \)
|
| $41$ |
\( T^{2} - 55577780 \)
|
| $43$ |
\( T^{2} - 36828980 \)
|
| $47$ |
\( (T - 5188)^{2} \)
|
| $53$ |
\( (T + 34494)^{2} \)
|
| $59$ |
\( (T + 16440)^{2} \)
|
| $61$ |
\( T^{2} - 2365747520 \)
|
| $67$ |
\( (T - 21108)^{2} \)
|
| $71$ |
\( (T + 54868)^{2} \)
|
| $73$ |
\( T^{2} - 181683920 \)
|
| $79$ |
\( T^{2} - 1744338420 \)
|
| $83$ |
\( T^{2} - 5587155920 \)
|
| $89$ |
\( (T - 5666)^{2} \)
|
| $97$ |
\( (T + 25918)^{2} \)
|
| show more | |
| show less |