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: | \(0\) |
| 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: |
\( x^{8} - 202x^{6} + 12577x^{4} - 277608x^{2} + 1411344 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3\cdot 11 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 202x^{6} + 12577x^{4} - 277608x^{2} + 1411344 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 101\nu^{2} - 1188 ) / 112 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 202\nu^{5} + 11389\nu^{3} - 157620\nu ) / 12096 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{7} - 794\nu^{5} + 29081\nu^{3} - 186756\nu ) / 12096 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 94\nu^{5} - 5567\nu^{3} + 375900\nu ) / 12096 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{4} + 157\nu^{2} - 4044 ) / 56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} - 162\nu^{4} + 6229\nu^{2} - 44244 ) / 224 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 2\beta_{2} + 51 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{5} + 2\beta_{4} - 6\beta_{3} + 77\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 101\beta_{6} - 314\beta_{2} + 3963 \)
|
| \(\nu^{5}\) | \(=\) |
\( -516\beta_{5} + 314\beta_{4} - 1054\beta_{3} + 7149\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 224\beta_{7} + 10133\beta_{6} - 38410\beta_{2} + 368571 \)
|
| \(\nu^{7}\) | \(=\) |
\( -58676\beta_{5} + 40650\beta_{4} - 132478\beta_{3} + 724765\beta_1 \)
|
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 |
|
−10.4248 | 0 | 76.6770 | 85.6712 | 0 | −69.6430 | −465.749 | 0 | −893.107 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.01340 | 0 | 17.1878 | −4.85934 | 0 | 231.657 | 103.884 | 0 | 34.0805 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.08271 | 0 | 4.99939 | 96.9680 | 0 | −145.693 | 164.237 | 0 | −589.829 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.67129 | 0 | −24.8642 | −49.7799 | 0 | 17.0422 | 151.901 | 0 | 132.977 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.67129 | 0 | −24.8642 | −49.7799 | 0 | −17.0422 | −151.901 | 0 | −132.977 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6.08271 | 0 | 4.99939 | 96.9680 | 0 | 145.693 | −164.237 | 0 | 589.829 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.01340 | 0 | 17.1878 | −4.85934 | 0 | −231.657 | −103.884 | 0 | −34.0805 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.4248 | 0 | 76.6770 | 85.6712 | 0 | 69.6430 | 465.749 | 0 | 893.107 | |||||||||||||||||||||||||||||||||||||||||||
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 | 1089.6.a.bd | 8 | |
| 3.b | odd | 2 | 1 | 121.6.a.h | ✓ | 8 | |
| 11.b | odd | 2 | 1 | inner | 1089.6.a.bd | 8 | |
| 33.d | even | 2 | 1 | 121.6.a.h | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.6.a.h | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 121.6.a.h | ✓ | 8 | 33.d | even | 2 | 1 | |
| 1089.6.a.bd | 8 | 1.a | even | 1 | 1 | trivial | |
| 1089.6.a.bd | 8 | 11.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{8} - 202T_{2}^{6} + 12577T_{2}^{4} - 277608T_{2}^{2} + 1411344 \)
|
|
\( T_{5}^{4} - 128T_{5}^{3} - 1430T_{5}^{2} + 409728T_{5} + 2009529 \)
|
|
\( T_{7}^{8} - 80032T_{7}^{6} + 1525515112T_{7}^{4} - 5961242249088T_{7}^{2} + 1604635993336464 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 202 T^{6} + \cdots + 1411344 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 128 T^{3} + \cdots + 2009529)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 16\!\cdots\!64 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 38\!\cdots\!61 \)
|
| $17$ |
\( T^{8} + \cdots + 10\!\cdots\!89 \)
|
| $19$ |
\( T^{8} + \cdots + 33\!\cdots\!16 \)
|
| $23$ |
\( (T^{4} - 5264 T^{3} + \cdots + 977147869140)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 35\!\cdots\!25 \)
|
| $31$ |
\( (T^{4} + \cdots - 75038313868332)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots - 13\!\cdots\!83)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 36\!\cdots\!21 \)
|
| $43$ |
\( T^{8} + \cdots + 24\!\cdots\!00 \)
|
| $47$ |
\( (T^{4} + \cdots + 98\!\cdots\!32)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots - 33\!\cdots\!47)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots - 29\!\cdots\!76)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 84\!\cdots\!00 \)
|
| $67$ |
\( (T^{4} + \cdots - 24\!\cdots\!80)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots - 25\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 49\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 90\!\cdots\!04 \)
|
| $83$ |
\( T^{8} + \cdots + 66\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} + \cdots - 14\!\cdots\!99)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 18\!\cdots\!65)^{2} \)
|
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