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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 129x^{3} - 11x^{2} + 3044x - 528 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 121) |
| 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
\( x^{5} - x^{4} - 129x^{3} - 11x^{2} + 3044x - 528 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 14\nu^{3} + 67\nu^{2} - 980\nu - 104 ) / 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 52 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{4} - 2\nu^{3} - 321\nu^{2} - 220\nu + 3832 ) / 40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 3\beta_{2} + 82\beta _1 + 68 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{4} + 109\beta_{3} + 2\beta_{2} + 235\beta _1 + 4332 \)
|
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 |
|
−9.94807 | 0 | 66.9641 | −4.78869 | 0 | 15.5561 | −347.825 | 0 | 47.6382 | |||||||||||||||||||||||||||||||||
| 1.2 | −7.11644 | 0 | 18.6437 | −72.6055 | 0 | 4.73719 | 95.0493 | 0 | 516.693 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.826212 | 0 | −31.3174 | 88.9492 | 0 | 178.151 | 52.3136 | 0 | −73.4909 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.18551 | 0 | −14.4815 | −96.8810 | 0 | −111.515 | −194.549 | 0 | −405.496 | ||||||||||||||||||||||||||||||||||
| 1.5 | 9.70521 | 0 | 62.1911 | 56.3260 | 0 | −188.929 | 293.011 | 0 | 546.656 | ||||||||||||||||||||||||||||||||||
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.v | 5 | |
| 3.b | odd | 2 | 1 | 121.6.a.f | yes | 5 | |
| 11.b | odd | 2 | 1 | 1089.6.a.y | 5 | ||
| 33.d | even | 2 | 1 | 121.6.a.e | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.6.a.e | ✓ | 5 | 33.d | even | 2 | 1 | |
| 121.6.a.f | yes | 5 | 3.b | odd | 2 | 1 | |
| 1089.6.a.v | 5 | 1.a | even | 1 | 1 | trivial | |
| 1089.6.a.y | 5 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{5} + 4T_{2}^{4} - 123T_{2}^{3} - 394T_{2}^{2} + 2636T_{2} + 2376 \)
|
|
\( T_{5}^{5} + 29T_{5}^{4} - 12462T_{5}^{3} - 232958T_{5}^{2} + 34414781T_{5} + 168762609 \)
|
|
\( T_{7}^{5} + 102T_{7}^{4} - 34864T_{7}^{3} - 3085712T_{7}^{2} + 73776432T_{7} - 276594208 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 4 T^{4} + \cdots + 2376 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 29 T^{4} + \cdots + 168762609 \)
|
| $7$ |
\( T^{5} + 102 T^{4} + \cdots - 276594208 \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} + \cdots - 8862177946831 \)
|
| $17$ |
\( T^{5} + \cdots - 65999437473915 \)
|
| $19$ |
\( T^{5} + \cdots - 428775941034144 \)
|
| $23$ |
\( T^{5} + \cdots - 13\!\cdots\!12 \)
|
| $29$ |
\( T^{5} + \cdots - 54\!\cdots\!17 \)
|
| $31$ |
\( T^{5} + \cdots + 72\!\cdots\!16 \)
|
| $37$ |
\( T^{5} + \cdots + 75\!\cdots\!31 \)
|
| $41$ |
\( T^{5} + \cdots - 24\!\cdots\!15 \)
|
| $43$ |
\( T^{5} + \cdots - 13\!\cdots\!60 \)
|
| $47$ |
\( T^{5} + \cdots - 22\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots - 43\!\cdots\!23 \)
|
| $59$ |
\( T^{5} + \cdots - 42\!\cdots\!72 \)
|
| $61$ |
\( T^{5} + \cdots - 34\!\cdots\!16 \)
|
| $67$ |
\( T^{5} + \cdots - 10\!\cdots\!16 \)
|
| $71$ |
\( T^{5} + \cdots + 19\!\cdots\!12 \)
|
| $73$ |
\( T^{5} + \cdots - 13\!\cdots\!08 \)
|
| $79$ |
\( T^{5} + \cdots + 19\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 54\!\cdots\!44 \)
|
| $89$ |
\( T^{5} + \cdots + 56\!\cdots\!51 \)
|
| $97$ |
\( T^{5} + \cdots + 48\!\cdots\!53 \)
|
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