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: | \(\Q(\sqrt{-2}, \sqrt{3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 4x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 363) |
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 \( x^{4} + 4x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 2 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 4\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 4\beta_1 \) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
604.1 |
|
− | 1.93185i | 0 | 0.267949 | −3.19615 | 0 | 13.7632i | − | 8.24504i | 0 | 6.17449i | ||||||||||||||||||||||||||||
604.2 | − | 0.517638i | 0 | 3.73205 | 7.19615 | 0 | − | 8.86422i | − | 4.00240i | 0 | − | 3.72500i | |||||||||||||||||||||||||||
604.3 | 0.517638i | 0 | 3.73205 | 7.19615 | 0 | 8.86422i | 4.00240i | 0 | 3.72500i | |||||||||||||||||||||||||||||||
604.4 | 1.93185i | 0 | 0.267949 | −3.19615 | 0 | − | 13.7632i | 8.24504i | 0 | − | 6.17449i | |||||||||||||||||||||||||||||
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.g | 4 | |
3.b | odd | 2 | 1 | 363.3.c.c | ✓ | 4 | |
11.b | odd | 2 | 1 | inner | 1089.3.c.g | 4 | |
33.d | even | 2 | 1 | 363.3.c.c | ✓ | 4 | |
33.f | even | 10 | 4 | 363.3.g.b | 16 | ||
33.h | odd | 10 | 4 | 363.3.g.b | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
363.3.c.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
363.3.c.c | ✓ | 4 | 33.d | even | 2 | 1 | |
363.3.g.b | 16 | 33.f | even | 10 | 4 | ||
363.3.g.b | 16 | 33.h | odd | 10 | 4 | ||
1089.3.c.g | 4 | 1.a | even | 1 | 1 | trivial | |
1089.3.c.g | 4 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 4T^{2} + 1 \)
$3$
\( T^{4} \)
$5$
\( (T^{2} - 4 T - 23)^{2} \)
$7$
\( T^{4} + 268 T^{2} + 14884 \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 444 T^{2} + 47961 \)
$17$
\( T^{4} + 1116 T^{2} + 281961 \)
$19$
\( T^{4} + 192T^{2} + 2304 \)
$23$
\( (T^{2} - 18 T - 282)^{2} \)
$29$
\( T^{4} + 2284 T^{2} + 32761 \)
$31$
\( (T^{2} + 40 T + 100)^{2} \)
$37$
\( (T^{2} - 28 T + 49)^{2} \)
$41$
\( T^{4} + 5124 T^{2} + \cdots + 4835601 \)
$43$
\( T^{4} + 516T^{2} + 4356 \)
$47$
\( (T^{2} + 8 T - 3452)^{2} \)
$53$
\( (T^{2} + 70 T + 793)^{2} \)
$59$
\( (T^{2} + 142 T + 3958)^{2} \)
$61$
\( T^{4} + 2812 T^{2} + 386884 \)
$67$
\( (T^{2} + 98 T + 1318)^{2} \)
$71$
\( (T^{2} + 2 T - 2882)^{2} \)
$73$
\( T^{4} + 24076 T^{2} + \cdots + 122456356 \)
$79$
\( T^{4} + 3648 T^{2} + 278784 \)
$83$
\( T^{4} + 10432 T^{2} + \cdots + 22886656 \)
$89$
\( (T^{2} + 168 T + 5469)^{2} \)
$97$
\( (T^{2} - 56 T - 1739)^{2} \)
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