Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,6,Mod(1,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("165.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.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: | \(26.4633302691\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 119x^{3} + 206x^{2} + 1428x - 1320 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - 2x^{4} - 119x^{3} + 206x^{2} + 1428x - 1320 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 48 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 95\nu + 66 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 115\nu^{2} - 8\nu + 992 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 48 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 95\beta _1 - 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 115\beta_{2} + 8\beta _1 + 4528 \)
|
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 |
|
−10.3866 | 9.00000 | 75.8811 | −25.0000 | −93.4793 | 40.8791 | −455.775 | 81.0000 | 259.665 | |||||||||||||||||||||||||||||||||
| 1.2 | −4.28026 | 9.00000 | −13.6794 | −25.0000 | −38.5223 | 202.127 | 195.520 | 81.0000 | 107.006 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.870481 | 9.00000 | −31.2423 | −25.0000 | −7.83433 | −236.601 | 55.0512 | 81.0000 | 21.7620 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.34733 | 9.00000 | −20.7954 | −25.0000 | 30.1260 | 42.9514 | −176.724 | 81.0000 | −83.6832 | ||||||||||||||||||||||||||||||||||
| 1.5 | 10.1900 | 9.00000 | 71.8359 | −25.0000 | 91.7099 | 134.644 | 405.928 | 81.0000 | −254.750 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(1\) |
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 | 165.6.a.f | ✓ | 5 |
| 3.b | odd | 2 | 1 | 495.6.a.j | 5 | ||
| 5.b | even | 2 | 1 | 825.6.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.f | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 495.6.a.j | 5 | 3.b | odd | 2 | 1 | ||
| 825.6.a.l | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 2T_{2}^{4} - 119T_{2}^{3} - 206T_{2}^{2} + 1428T_{2} + 1320 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots + 1320 \)
|
| $3$ |
\( (T - 9)^{5} \)
|
| $5$ |
\( (T + 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots + 11305890304 \)
|
| $11$ |
\( (T + 121)^{5} \)
|
| $13$ |
\( T^{5} + \cdots - 29790890509184 \)
|
| $17$ |
\( T^{5} + \cdots - 335880301363200 \)
|
| $19$ |
\( T^{5} + \cdots - 745265742345728 \)
|
| $23$ |
\( T^{5} + \cdots - 84\!\cdots\!08 \)
|
| $29$ |
\( T^{5} + \cdots - 23\!\cdots\!56 \)
|
| $31$ |
\( T^{5} + \cdots - 66\!\cdots\!12 \)
|
| $37$ |
\( T^{5} + \cdots + 27\!\cdots\!92 \)
|
| $41$ |
\( T^{5} + \cdots - 71\!\cdots\!76 \)
|
| $43$ |
\( T^{5} + \cdots + 57\!\cdots\!56 \)
|
| $47$ |
\( T^{5} + \cdots + 48\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots - 53\!\cdots\!16 \)
|
| $59$ |
\( T^{5} + \cdots + 14\!\cdots\!40 \)
|
| $61$ |
\( T^{5} + \cdots + 36\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots + 28\!\cdots\!60 \)
|
| $71$ |
\( T^{5} + \cdots - 26\!\cdots\!88 \)
|
| $73$ |
\( T^{5} + \cdots - 44\!\cdots\!88 \)
|
| $79$ |
\( T^{5} + \cdots + 38\!\cdots\!36 \)
|
| $83$ |
\( T^{5} + \cdots - 26\!\cdots\!08 \)
|
| $89$ |
\( T^{5} + \cdots + 28\!\cdots\!84 \)
|
| $97$ |
\( T^{5} + \cdots - 13\!\cdots\!00 \)
|
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