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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 167x^{6} - 68x^{5} + 7903x^{4} + 8528x^{3} - 88021x^{2} - 132908x - 19844 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 11^{2} \) |
| Twist minimal: | no (minimal twist has level 11) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 167x^{6} - 68x^{5} + 7903x^{4} + 8528x^{3} - 88021x^{2} - 132908x - 19844 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 42 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 41\nu^{7} + 163\nu^{6} - 4622\nu^{5} - 9774\nu^{4} + 109245\nu^{3} - 233361\nu^{2} - 617128\nu + 2845452 ) / 229888 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23 \nu^{7} - 259 \nu^{6} - 5922 \nu^{5} + 24830 \nu^{4} + 377555 \nu^{3} - 249183 \nu^{2} + \cdots - 2351116 ) / 57472 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 31 \nu^{7} + 271 \nu^{6} + 6342 \nu^{5} - 29406 \nu^{4} - 358483 \nu^{3} + 531307 \nu^{2} + \cdots + 784276 ) / 57472 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 169 \nu^{7} + 29 \nu^{6} + 25710 \nu^{5} - 5970 \nu^{4} - 1068477 \nu^{3} - 51567 \nu^{2} + \cdots + 4872884 ) / 229888 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 223 \nu^{7} + 1237 \nu^{6} - 28994 \nu^{5} - 133762 \nu^{4} + 1003499 \nu^{3} + 3123497 \nu^{2} + \cdots - 13478892 ) / 229888 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 4\beta_{6} - 4\beta_{5} - 2\beta_{4} - 2\beta_{3} + 71\beta _1 + 30 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} + 12\beta_{6} + 4\beta_{5} + 10\beta_{4} + 50\beta_{3} + 87\beta_{2} + 81\beta _1 + 3018 \)
|
| \(\nu^{5}\) | \(=\) |
\( 184\beta_{7} + 320\beta_{6} - 432\beta_{5} - 280\beta_{4} - 360\beta_{3} - 14\beta_{2} + 5659\beta _1 + 2314 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 116 \beta_{7} + 1608 \beta_{6} + 840 \beta_{5} + 1556 \beta_{4} + 6308 \beta_{3} + 7321 \beta_{2} + \cdots + 240710 \)
|
| \(\nu^{7}\) | \(=\) |
\( 15398 \beta_{7} + 21884 \beta_{6} - 40428 \beta_{5} - 30038 \beta_{4} - 42806 \beta_{3} - 4252 \beta_{2} + \cdots + 113070 \)
|
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 |
|
−10.6555 | 0 | 81.5407 | −66.1415 | 0 | 72.0972 | −527.883 | 0 | 704.774 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.89771 | 0 | 30.3738 | −64.2668 | 0 | 108.533 | 12.8433 | 0 | 507.561 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.85589 | 0 | −8.42031 | 84.8644 | 0 | −63.3549 | 196.277 | 0 | −412.092 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.39738 | 0 | −26.2526 | 16.9511 | 0 | 108.472 | 139.654 | 0 | −40.6382 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.16833 | 0 | −30.6350 | −4.24375 | 0 | −181.147 | 73.1784 | 0 | 4.95810 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.12379 | 0 | −22.2419 | 28.9123 | 0 | −88.5658 | −169.440 | 0 | 90.3160 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.86588 | 0 | 29.8720 | −73.3295 | 0 | 167.027 | −16.7383 | 0 | −576.801 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 7.98519 | 0 | 31.7633 | 7.25383 | 0 | 168.938 | −1.89018 | 0 | 57.9232 | |||||||||||||||||||||||||||||||||||||||||||
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.bb | 8 | |
| 3.b | odd | 2 | 1 | 121.6.a.i | 8 | ||
| 11.b | odd | 2 | 1 | 1089.6.a.bg | 8 | ||
| 11.d | odd | 10 | 2 | 99.6.f.a | 16 | ||
| 33.d | even | 2 | 1 | 121.6.a.g | 8 | ||
| 33.f | even | 10 | 2 | 11.6.c.a | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.6.c.a | ✓ | 16 | 33.f | even | 10 | 2 | |
| 99.6.f.a | 16 | 11.d | odd | 10 | 2 | ||
| 121.6.a.g | 8 | 33.d | even | 2 | 1 | ||
| 121.6.a.i | 8 | 3.b | odd | 2 | 1 | ||
| 1089.6.a.bb | 8 | 1.a | even | 1 | 1 | trivial | |
| 1089.6.a.bg | 8 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{8} + 8T_{2}^{7} - 139T_{2}^{6} - 1014T_{2}^{5} + 5128T_{2}^{4} + 36176T_{2}^{3} - 18176T_{2}^{2} - 253088T_{2} - 224576 \)
|
|
\( T_{5}^{8} + 70 T_{5}^{7} - 8689 T_{5}^{6} - 619710 T_{5}^{5} + 13507261 T_{5}^{4} + 777174460 T_{5}^{3} + \cdots + 399082341136 \)
|
|
\( T_{7}^{8} - 292 T_{7}^{7} - 22295 T_{7}^{6} + 12969980 T_{7}^{5} - 583123075 T_{7}^{4} + \cdots - 24\!\cdots\!20 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 8 T^{7} + \cdots - 224576 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 399082341136 \)
|
| $7$ |
\( T^{8} + \cdots - 24\!\cdots\!20 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 42\!\cdots\!56 \)
|
| $17$ |
\( T^{8} + \cdots - 25\!\cdots\!19 \)
|
| $19$ |
\( T^{8} + \cdots - 33\!\cdots\!75 \)
|
| $23$ |
\( T^{8} + \cdots - 13\!\cdots\!36 \)
|
| $29$ |
\( T^{8} + \cdots + 30\!\cdots\!80 \)
|
| $31$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots - 42\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots - 89\!\cdots\!79 \)
|
| $43$ |
\( T^{8} + \cdots - 31\!\cdots\!84 \)
|
| $47$ |
\( T^{8} + \cdots - 17\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots - 67\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots - 31\!\cdots\!75 \)
|
| $61$ |
\( T^{8} + \cdots + 25\!\cdots\!80 \)
|
| $67$ |
\( T^{8} + \cdots - 81\!\cdots\!36 \)
|
| $71$ |
\( T^{8} + \cdots + 34\!\cdots\!24 \)
|
| $73$ |
\( T^{8} + \cdots - 62\!\cdots\!71 \)
|
| $79$ |
\( T^{8} + \cdots + 29\!\cdots\!20 \)
|
| $83$ |
\( T^{8} + \cdots + 18\!\cdots\!21 \)
|
| $89$ |
\( T^{8} + \cdots - 32\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 26\!\cdots\!75 \)
|
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