Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1089,3,Mod(604,1089)]
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, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1089.604");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1089.c (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(29.6731007888\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.523388583936.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 28x^{6} + 262x^{4} + 948x^{2} + 1089 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 11 \) |
| Twist minimal: | no (minimal twist has level 121) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 28x^{6} + 262x^{4} + 948x^{2} + 1089 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 13\nu^{5} + 41\nu^{3} + 489\nu ) / 108 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 13\nu^{4} + 5\nu^{2} + 201 ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 22\nu^{4} - 121\nu^{2} - 114 ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 22\nu^{4} + 139\nu^{2} + 240 ) / 18 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 22\nu^{5} + 139\nu^{3} + 240\nu ) / 18 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} - 53\nu^{5} - 404\nu^{3} - 777\nu ) / 54 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 2\beta_{2} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -14\beta_{5} - 16\beta_{4} + 4\beta_{3} + 63 \)
|
| \(\nu^{5}\) | \(=\) |
\( -20\beta_{7} - 18\beta_{6} - 28\beta_{2} + 79\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 187\beta_{5} + 213\beta_{4} - 88\beta_{3} - 653 \)
|
| \(\nu^{7}\) | \(=\) |
\( 301\beta_{7} + 275\beta_{6} + 338\beta_{2} - 866\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).
| \(n\) | \(244\) | \(848\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 604.1 |
|
− | 3.57626i | 0 | −8.78965 | 3.44471 | 0 | 3.88348i | 17.1290i | 0 | − | 12.3192i | ||||||||||||||||||||||||||||||||||||||||
| 604.2 | − | 2.89067i | 0 | −4.35597 | 1.96778 | 0 | − | 9.76707i | 1.02900i | 0 | − | 5.68820i | ||||||||||||||||||||||||||||||||||||||||
| 604.3 | − | 2.16205i | 0 | −0.674453 | −7.64086 | 0 | 6.05557i | − | 7.18999i | 0 | 16.5199i | |||||||||||||||||||||||||||||||||||||||||
| 604.4 | − | 1.47646i | 0 | 1.82008 | 4.22837 | 0 | 2.20298i | − | 8.59309i | 0 | − | 6.24301i | ||||||||||||||||||||||||||||||||||||||||
| 604.5 | 1.47646i | 0 | 1.82008 | 4.22837 | 0 | − | 2.20298i | 8.59309i | 0 | 6.24301i | ||||||||||||||||||||||||||||||||||||||||||
| 604.6 | 2.16205i | 0 | −0.674453 | −7.64086 | 0 | − | 6.05557i | 7.18999i | 0 | − | 16.5199i | |||||||||||||||||||||||||||||||||||||||||
| 604.7 | 2.89067i | 0 | −4.35597 | 1.96778 | 0 | 9.76707i | − | 1.02900i | 0 | 5.68820i | ||||||||||||||||||||||||||||||||||||||||||
| 604.8 | 3.57626i | 0 | −8.78965 | 3.44471 | 0 | − | 3.88348i | − | 17.1290i | 0 | 12.3192i | |||||||||||||||||||||||||||||||||||||||||
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 | 1089.3.c.k | 8 | |
| 3.b | odd | 2 | 1 | 121.3.b.c | ✓ | 8 | |
| 11.b | odd | 2 | 1 | inner | 1089.3.c.k | 8 | |
| 33.d | even | 2 | 1 | 121.3.b.c | ✓ | 8 | |
| 33.f | even | 10 | 4 | 121.3.d.f | 32 | ||
| 33.h | odd | 10 | 4 | 121.3.d.f | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.3.b.c | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 121.3.b.c | ✓ | 8 | 33.d | even | 2 | 1 | |
| 121.3.d.f | 32 | 33.f | even | 10 | 4 | ||
| 121.3.d.f | 32 | 33.h | odd | 10 | 4 | ||
| 1089.3.c.k | 8 | 1.a | even | 1 | 1 | trivial | |
| 1089.3.c.k | 8 | 11.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 28T_{2}^{6} + 262T_{2}^{4} + 948T_{2}^{2} + 1089 \)
acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 28 T^{6} + \cdots + 1089 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 2 T^{3} + \cdots - 219)^{2} \)
|
| $7$ |
\( T^{8} + 152 T^{6} + \cdots + 256036 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + 580 T^{6} + \cdots + 5803281 \)
|
| $17$ |
\( T^{8} + 600 T^{6} + \cdots + 36469521 \)
|
| $19$ |
\( T^{8} + \cdots + 1961249796 \)
|
| $23$ |
\( (T^{4} - 6 T^{3} + \cdots - 162)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 36874368729 \)
|
| $31$ |
\( (T^{4} + 58 T^{3} + \cdots - 13458)^{2} \)
|
| $37$ |
\( (T^{4} + 2 T^{3} + \cdots + 291069)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 534239660889 \)
|
| $43$ |
\( T^{8} + \cdots + 20286952842816 \)
|
| $47$ |
\( (T^{4} - 122 T^{3} + \cdots - 3669186)^{2} \)
|
| $53$ |
\( (T^{4} + 134 T^{3} + \cdots + 730761)^{2} \)
|
| $59$ |
\( (T^{4} - 28 T^{3} + \cdots + 233952)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 87785731315216 \)
|
| $67$ |
\( (T^{4} - 142 T^{3} + \cdots - 23025234)^{2} \)
|
| $71$ |
\( (T^{4} + 136 T^{3} + \cdots + 10616856)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 4549825513024 \)
|
| $79$ |
\( T^{8} + \cdots + 163631576196 \)
|
| $83$ |
\( T^{8} + \cdots + 75\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} - 12 T^{3} + \cdots - 6209127)^{2} \)
|
| $97$ |
\( (T^{4} - 76 T^{3} + \cdots - 65182079)^{2} \)
|
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