Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1089,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1089.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(64.2530799963\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.22606886592.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 34x^{4} + 289x^{2} - 192 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 34x^{4} + 289x^{2} - 192 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 17\nu ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 17\nu^{2} ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 25\nu^{2} - 88 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 28\nu^{3} + 171\nu ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{2} + 17\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 17\beta_{4} + 25\beta_{3} + 187 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 224\beta_{2} + 305\beta_1 \)
|
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 |
|
−4.48234 | 0 | 12.0913 | −18.8550 | 0 | −18.7894 | −18.3387 | 0 | 84.5143 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.63095 | 0 | 5.18381 | 2.10518 | 0 | −9.81287 | 10.2255 | 0 | −7.64382 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.851385 | 0 | −7.27514 | 9.74979 | 0 | 25.6645 | 13.0050 | 0 | −8.30082 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.851385 | 0 | −7.27514 | 9.74979 | 0 | −25.6645 | −13.0050 | 0 | 8.30082 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.63095 | 0 | 5.18381 | 2.10518 | 0 | 9.81287 | −10.2255 | 0 | 7.64382 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.48234 | 0 | 12.0913 | −18.8550 | 0 | 18.7894 | 18.3387 | 0 | −84.5143 | |||||||||||||||||||||||||||||||||||||
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 | 1089.4.a.bj | 6 | |
| 3.b | odd | 2 | 1 | 121.4.a.h | ✓ | 6 | |
| 11.b | odd | 2 | 1 | inner | 1089.4.a.bj | 6 | |
| 12.b | even | 2 | 1 | 1936.4.a.bq | 6 | ||
| 33.d | even | 2 | 1 | 121.4.a.h | ✓ | 6 | |
| 33.f | even | 10 | 4 | 121.4.c.j | 24 | ||
| 33.h | odd | 10 | 4 | 121.4.c.j | 24 | ||
| 132.d | odd | 2 | 1 | 1936.4.a.bq | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.4.a.h | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 121.4.a.h | ✓ | 6 | 33.d | even | 2 | 1 | |
| 121.4.c.j | 24 | 33.f | even | 10 | 4 | ||
| 121.4.c.j | 24 | 33.h | odd | 10 | 4 | ||
| 1089.4.a.bj | 6 | 1.a | even | 1 | 1 | trivial | |
| 1089.4.a.bj | 6 | 11.b | odd | 2 | 1 | inner | |
| 1936.4.a.bq | 6 | 12.b | even | 2 | 1 | ||
| 1936.4.a.bq | 6 | 132.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{6} - 34T_{2}^{4} + 289T_{2}^{2} - 192 \)
|
|
\( T_{5}^{3} + 7T_{5}^{2} - 203T_{5} + 387 \)
|
|
\( T_{7}^{6} - 1108T_{7}^{4} + 329956T_{7}^{2} - 22391472 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 34 T^{4} + \cdots - 192 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} + 7 T^{2} + \cdots + 387)^{2} \)
|
| $7$ |
\( T^{6} - 1108 T^{4} + \cdots - 22391472 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 1413 T^{4} + \cdots - 1701027 \)
|
| $17$ |
\( T^{6} + \cdots - 1769575107 \)
|
| $19$ |
\( T^{6} + \cdots - 6415522608 \)
|
| $23$ |
\( (T^{3} + 256 T^{2} + \cdots + 182244)^{2} \)
|
| $29$ |
\( T^{6} + \cdots - 164674006563 \)
|
| $31$ |
\( (T^{3} + 8 T^{2} + \cdots - 2748)^{2} \)
|
| $37$ |
\( (T^{3} - 185 T^{2} + \cdots - 305197)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 1007763271707 \)
|
| $43$ |
\( T^{6} + \cdots - 851862941073408 \)
|
| $47$ |
\( (T^{3} + 724 T^{2} + \cdots + 4790028)^{2} \)
|
| $53$ |
\( (T^{3} - 411 T^{2} + \cdots + 49829337)^{2} \)
|
| $59$ |
\( (T^{3} + 732 T^{2} + \cdots - 171154944)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 575173695909888 \)
|
| $67$ |
\( (T^{3} + 608 T^{2} + \cdots - 45615564)^{2} \)
|
| $71$ |
\( (T^{3} + 548 T^{2} + \cdots - 13675008)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 30\!\cdots\!72 \)
|
| $79$ |
\( T^{6} + \cdots - 373629568354992 \)
|
| $83$ |
\( T^{6} + \cdots - 36\!\cdots\!52 \)
|
| $89$ |
\( (T^{3} + 173 T^{2} + \cdots + 198698571)^{2} \)
|
| $97$ |
\( (T^{3} - 867 T^{2} + \cdots + 30520411)^{2} \)
|
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