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: |
\( x^{8} - 229x^{6} + 16722x^{4} - 418240x^{2} + 3026848 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{8} - 229x^{6} + 16722x^{4} - 418240x^{2} + 3026848 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 57 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 249\nu^{5} - 17478\nu^{3} + 273592\nu ) / 4224 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 249\nu^{5} + 21702\nu^{3} - 624184\nu ) / 4224 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 249\nu^{4} - 17478\nu^{2} + 261976 ) / 1056 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 161\nu^{4} + 6830\nu^{2} - 70312 ) / 176 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} - 571\nu^{5} + 30786\nu^{3} - 387816\nu ) / 352 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 57 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 83\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 12\beta_{5} + 121\beta_{2} + 4719 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 123\beta_{4} + 195\beta_{3} + 7749\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 498\beta_{6} + 1932\beta_{5} + 12651\beta_{2} + 440761 \)
|
| \(\nu^{7}\) | \(=\) |
\( 498\beta_{7} + 13149\beta_{4} + 26853\beta_{3} + 752419\beta_1 \)
|
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 |
|
−10.2702 | 0 | 73.4766 | −96.0074 | 0 | 231.048 | −425.972 | 0 | 986.013 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.12913 | 0 | 51.3411 | 42.6812 | 0 | −149.019 | −176.567 | 0 | −389.642 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.27195 | 0 | −4.20653 | 35.7371 | 0 | 128.331 | 190.879 | 0 | −188.404 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.51979 | 0 | −19.6111 | −88.3672 | 0 | −157.360 | 181.660 | 0 | 311.034 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.51979 | 0 | −19.6111 | 88.3672 | 0 | −157.360 | −181.660 | 0 | 311.034 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.27195 | 0 | −4.20653 | −35.7371 | 0 | 128.331 | −190.879 | 0 | −188.404 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 9.12913 | 0 | 51.3411 | −42.6812 | 0 | −149.019 | 176.567 | 0 | −389.642 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.2702 | 0 | 73.4766 | 96.0074 | 0 | 231.048 | 425.972 | 0 | 986.013 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.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.bf | yes | 8 |
| 3.b | odd | 2 | 1 | inner | 1089.6.a.bf | yes | 8 |
| 11.b | odd | 2 | 1 | 1089.6.a.be | ✓ | 8 | |
| 33.d | even | 2 | 1 | 1089.6.a.be | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1089.6.a.be | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 1089.6.a.be | ✓ | 8 | 33.d | even | 2 | 1 | |
| 1089.6.a.bf | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 1089.6.a.bf | yes | 8 | 3.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{8} - 229T_{2}^{6} + 16722T_{2}^{4} - 418240T_{2}^{2} + 3026848 \)
|
|
\( T_{5}^{8} - 20125T_{5}^{6} + 127064259T_{5}^{4} - 262655010847T_{5}^{2} + 167456866563712 \)
|
|
\( T_{7}^{4} - 53T_{7}^{3} - 57006T_{7}^{2} + 657028T_{7} + 695298376 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 229 T^{6} + \cdots + 3026848 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 167456866563712 \)
|
| $7$ |
\( (T^{4} - 53 T^{3} + \cdots + 695298376)^{2} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( (T^{4} - 809 T^{3} + \cdots - 7604252450)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 27\!\cdots\!12 \)
|
| $19$ |
\( (T^{4} + \cdots + 4234032000000)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 56\!\cdots\!88 \)
|
| $29$ |
\( T^{8} + \cdots + 13\!\cdots\!08 \)
|
| $31$ |
\( (T^{4} + \cdots + 182895361654720)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots - 38\!\cdots\!77)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 97\!\cdots\!72 \)
|
| $43$ |
\( (T^{4} + \cdots + 42\!\cdots\!96)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 19\!\cdots\!12 \)
|
| $53$ |
\( T^{8} + \cdots + 13\!\cdots\!32 \)
|
| $59$ |
\( T^{8} + \cdots + 48\!\cdots\!12 \)
|
| $61$ |
\( (T^{4} + \cdots - 10\!\cdots\!16)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 64\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 83\!\cdots\!00 \)
|
| $73$ |
\( (T^{4} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 18\!\cdots\!12)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 68\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots + 99\!\cdots\!88 \)
|
| $97$ |
\( (T^{4} + \cdots + 16\!\cdots\!25)^{2} \)
|
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