Newspace parameters
Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1089.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(29.6731007888\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-11})\) |
Defining polynomial: |
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu + 5 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} + 19\nu - 9 ) / 3 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} + 44\nu - 21 ) / 3 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 2\beta _1 - 9 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -8\beta_{3} + 19\beta_{2} + 3\beta _1 - 14 ) / 2 \)
|
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
604.1 |
|
− | 3.31662i | 0 | −7.00000 | −9.38083 | 0 | 9.89949i | 9.94987i | 0 | 31.1127i | |||||||||||||||||||||||||||||
604.2 | − | 3.31662i | 0 | −7.00000 | 9.38083 | 0 | − | 9.89949i | 9.94987i | 0 | − | 31.1127i | ||||||||||||||||||||||||||||
604.3 | 3.31662i | 0 | −7.00000 | −9.38083 | 0 | − | 9.89949i | − | 9.94987i | 0 | − | 31.1127i | ||||||||||||||||||||||||||||
604.4 | 3.31662i | 0 | −7.00000 | 9.38083 | 0 | 9.89949i | − | 9.94987i | 0 | 31.1127i | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
33.d | even | 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.b | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 1089.3.c.b | ✓ | 4 |
11.b | odd | 2 | 1 | inner | 1089.3.c.b | ✓ | 4 |
33.d | even | 2 | 1 | inner | 1089.3.c.b | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1089.3.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
1089.3.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
1089.3.c.b | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
1089.3.c.b | ✓ | 4 | 33.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 11 \)
acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 11)^{2} \)
$3$
\( T^{4} \)
$5$
\( (T^{2} - 88)^{2} \)
$7$
\( (T^{2} + 98)^{2} \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 162)^{2} \)
$17$
\( T^{4} \)
$19$
\( (T^{2} + 18)^{2} \)
$23$
\( (T^{2} - 792)^{2} \)
$29$
\( (T^{2} + 176)^{2} \)
$31$
\( (T + 44)^{4} \)
$37$
\( (T - 44)^{4} \)
$41$
\( (T^{2} + 1584)^{2} \)
$43$
\( (T^{2} + 882)^{2} \)
$47$
\( (T^{2} - 88)^{2} \)
$53$
\( (T^{2} - 4312)^{2} \)
$59$
\( (T^{2} - 352)^{2} \)
$61$
\( (T^{2} + 4802)^{2} \)
$67$
\( (T + 88)^{4} \)
$71$
\( (T^{2} - 2200)^{2} \)
$73$
\( (T^{2} + 1682)^{2} \)
$79$
\( (T^{2} + 162)^{2} \)
$83$
\( (T^{2} + 8624)^{2} \)
$89$
\( (T^{2} - 12672)^{2} \)
$97$
\( (T - 70)^{4} \)
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