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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 61x^{3} + 71x^{2} + 882x - 567 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 99) |
| 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
\( x^{5} - 2x^{4} - 61x^{3} + 71x^{2} + 882x - 567 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{4} + 13\nu^{3} + 23\nu^{2} - 367\nu + 900 ) / 81 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{4} + 13\nu^{3} + 23\nu^{2} - 43\nu + 765 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -26\nu^{4} + 169\nu^{3} + 785\nu^{2} - 4285\nu - 936 ) / 81 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} + 68\nu^{3} - 266\nu^{2} - 2144\nu + 3636 ) / 81 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 3\beta _1 + 5 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - \beta_{2} - 23\beta _1 + 307 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{4} + 3\beta_{3} + 14\beta_{2} - 75\beta _1 + 202 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 78\beta_{4} + 62\beta_{3} - 13\beta_{2} - 1175\beta _1 + 10639 ) / 12 \)
|
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 |
|
−7.41173 | 0 | 22.9337 | −96.0875 | 0 | −129.474 | 67.1967 | 0 | 712.174 | |||||||||||||||||||||||||||||||||
| 1.2 | −6.58997 | 0 | 11.4277 | 14.6082 | 0 | 224.956 | 135.571 | 0 | −96.2674 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.533736 | 0 | −31.7151 | 51.0439 | 0 | 9.93449 | −34.0071 | 0 | 27.2440 | ||||||||||||||||||||||||||||||||||
| 1.4 | 6.41138 | 0 | 9.10581 | −62.4292 | 0 | −148.121 | −146.783 | 0 | −400.257 | ||||||||||||||||||||||||||||||||||
| 1.5 | 11.0566 | 0 | 90.2479 | −7.13545 | 0 | 60.7043 | 644.023 | 0 | −78.8937 | ||||||||||||||||||||||||||||||||||
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.x | 5 | |
| 3.b | odd | 2 | 1 | 1089.6.a.w | 5 | ||
| 11.b | odd | 2 | 1 | 99.6.a.h | ✓ | 5 | |
| 33.d | even | 2 | 1 | 99.6.a.i | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.6.a.h | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 99.6.a.i | yes | 5 | 33.d | even | 2 | 1 | |
| 1089.6.a.w | 5 | 3.b | odd | 2 | 1 | ||
| 1089.6.a.x | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{5} - 4T_{2}^{4} - 123T_{2}^{3} + 206T_{2}^{2} + 3388T_{2} - 1848 \)
|
|
\( T_{5}^{5} + 100T_{5}^{4} - 3000T_{5}^{3} - 301760T_{5}^{2} + 2506240T_{5} + 31916544 \)
|
|
\( T_{7}^{5} - 18T_{7}^{4} - 46384T_{7}^{3} - 1225952T_{7}^{2} + 278651952T_{7} - 2601713248 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 4 T^{4} + \cdots - 1848 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 100 T^{4} + \cdots + 31916544 \)
|
| $7$ |
\( T^{5} + \cdots - 2601713248 \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 12274627631264 \)
|
| $17$ |
\( T^{5} + \cdots + 10\!\cdots\!92 \)
|
| $19$ |
\( T^{5} + \cdots - 16\!\cdots\!48 \)
|
| $23$ |
\( T^{5} + \cdots + 10\!\cdots\!48 \)
|
| $29$ |
\( T^{5} + \cdots + 57\!\cdots\!64 \)
|
| $31$ |
\( T^{5} + \cdots - 46\!\cdots\!96 \)
|
| $37$ |
\( T^{5} + \cdots + 59\!\cdots\!16 \)
|
| $41$ |
\( T^{5} + \cdots - 11\!\cdots\!80 \)
|
| $43$ |
\( T^{5} + \cdots - 27\!\cdots\!96 \)
|
| $47$ |
\( T^{5} + \cdots + 96\!\cdots\!32 \)
|
| $53$ |
\( T^{5} + \cdots - 21\!\cdots\!80 \)
|
| $59$ |
\( T^{5} + \cdots - 21\!\cdots\!76 \)
|
| $61$ |
\( T^{5} + \cdots - 94\!\cdots\!16 \)
|
| $67$ |
\( T^{5} + \cdots - 44\!\cdots\!84 \)
|
| $71$ |
\( T^{5} + \cdots + 88\!\cdots\!80 \)
|
| $73$ |
\( T^{5} + \cdots - 16\!\cdots\!56 \)
|
| $79$ |
\( T^{5} + \cdots + 60\!\cdots\!20 \)
|
| $83$ |
\( T^{5} + \cdots - 53\!\cdots\!08 \)
|
| $89$ |
\( T^{5} + \cdots - 56\!\cdots\!20 \)
|
| $97$ |
\( T^{5} + \cdots + 41\!\cdots\!48 \)
|
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