Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1089,6,Mod(1,1089)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, 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("1089.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1089.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: | \(174.657979776\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.193425.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 58x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 363) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 58x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 38 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 38 \)
|
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 |
|
−8.19585 | 0 | 35.1720 | 76.6632 | 0 | 31.4579 | −25.9969 | 0 | −628.320 | |||||||||||||||||||||||||||
| 1.2 | 0.138306 | 0 | −31.9809 | −2.75112 | 0 | −91.0659 | −8.84897 | 0 | −0.380498 | ||||||||||||||||||||||||||||
| 1.3 | 7.05754 | 0 | 17.8089 | −15.9120 | 0 | 176.608 | −100.154 | 0 | −112.300 | ||||||||||||||||||||||||||||
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 | 1089.6.a.q | 3 | |
| 3.b | odd | 2 | 1 | 363.6.a.l | yes | 3 | |
| 11.b | odd | 2 | 1 | 1089.6.a.s | 3 | ||
| 33.d | even | 2 | 1 | 363.6.a.k | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.6.a.k | ✓ | 3 | 33.d | even | 2 | 1 | |
| 363.6.a.l | yes | 3 | 3.b | odd | 2 | 1 | |
| 1089.6.a.q | 3 | 1.a | even | 1 | 1 | trivial | |
| 1089.6.a.s | 3 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{3} + T_{2}^{2} - 58T_{2} + 8 \)
|
|
\( T_{5}^{3} - 58T_{5}^{2} - 1387T_{5} - 3356 \)
|
|
\( T_{7}^{3} - 117T_{7}^{2} - 13392T_{7} + 505936 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 58T + 8 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 58 T^{2} + \cdots - 3356 \)
|
| $7$ |
\( T^{3} - 117 T^{2} + \cdots + 505936 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} - 936 T^{2} + \cdots + 83957302 \)
|
| $17$ |
\( T^{3} + \cdots + 2318045756 \)
|
| $19$ |
\( T^{3} + 2691 T^{2} + \cdots - 672786432 \)
|
| $23$ |
\( T^{3} + \cdots + 1163449088 \)
|
| $29$ |
\( T^{3} + \cdots + 3514911354 \)
|
| $31$ |
\( T^{3} + \cdots - 2168668480 \)
|
| $37$ |
\( T^{3} + \cdots - 412502233707 \)
|
| $41$ |
\( T^{3} + \cdots - 8537250382 \)
|
| $43$ |
\( T^{3} + \cdots - 1371260385792 \)
|
| $47$ |
\( T^{3} + \cdots - 247941658912 \)
|
| $53$ |
\( T^{3} + \cdots + 17336288148966 \)
|
| $59$ |
\( T^{3} + \cdots - 8060764602880 \)
|
| $61$ |
\( T^{3} + \cdots + 11020015122012 \)
|
| $67$ |
\( T^{3} + \cdots + 6879912446416 \)
|
| $71$ |
\( T^{3} + \cdots - 119298588180480 \)
|
| $73$ |
\( T^{3} + \cdots + 119944614226324 \)
|
| $79$ |
\( T^{3} + \cdots + 138393812143376 \)
|
| $83$ |
\( T^{3} + \cdots + 196802887503744 \)
|
| $89$ |
\( T^{3} + \cdots + 34917258137868 \)
|
| $97$ |
\( T^{3} + \cdots + 906624279494035 \)
|
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