Newspace parameters
| Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 91.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.726638658394\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.316.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.81361 | −3.10278 | 1.28917 | 2.81361 | 5.62721 | −1.00000 | 1.28917 | 6.62721 | −5.10278 | |||||||||||||||||||||||||||
| 1.2 | 0.470683 | 2.24914 | −1.77846 | 0.529317 | 1.05863 | −1.00000 | −1.77846 | 2.05863 | 0.249141 | ||||||||||||||||||||||||||||
| 1.3 | 2.34292 | −1.14637 | 3.48929 | −1.34292 | −2.68585 | −1.00000 | 3.48929 | −1.68585 | −3.14637 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 91.2.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 819.2.a.i | 3 | ||
| 4.b | odd | 2 | 1 | 1456.2.a.t | 3 | ||
| 5.b | even | 2 | 1 | 2275.2.a.m | 3 | ||
| 7.b | odd | 2 | 1 | 637.2.a.j | 3 | ||
| 7.c | even | 3 | 2 | 637.2.e.j | 6 | ||
| 7.d | odd | 6 | 2 | 637.2.e.i | 6 | ||
| 8.b | even | 2 | 1 | 5824.2.a.by | 3 | ||
| 8.d | odd | 2 | 1 | 5824.2.a.bs | 3 | ||
| 13.b | even | 2 | 1 | 1183.2.a.i | 3 | ||
| 13.d | odd | 4 | 2 | 1183.2.c.f | 6 | ||
| 21.c | even | 2 | 1 | 5733.2.a.x | 3 | ||
| 91.b | odd | 2 | 1 | 8281.2.a.bg | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 637.2.a.j | 3 | 7.b | odd | 2 | 1 | ||
| 637.2.e.i | 6 | 7.d | odd | 6 | 2 | ||
| 637.2.e.j | 6 | 7.c | even | 3 | 2 | ||
| 819.2.a.i | 3 | 3.b | odd | 2 | 1 | ||
| 1183.2.a.i | 3 | 13.b | even | 2 | 1 | ||
| 1183.2.c.f | 6 | 13.d | odd | 4 | 2 | ||
| 1456.2.a.t | 3 | 4.b | odd | 2 | 1 | ||
| 2275.2.a.m | 3 | 5.b | even | 2 | 1 | ||
| 5733.2.a.x | 3 | 21.c | even | 2 | 1 | ||
| 5824.2.a.bs | 3 | 8.d | odd | 2 | 1 | ||
| 5824.2.a.by | 3 | 8.b | even | 2 | 1 | ||
| 8281.2.a.bg | 3 | 91.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - T^{2} - 4T + 2 \)
$3$
\( T^{3} + 2 T^{2} - 6 T - 8 \)
$5$
\( T^{3} - 2 T^{2} - 3 T + 2 \)
$7$
\( (T + 1)^{3} \)
$11$
\( T^{3} - 2 T^{2} - 6 T + 8 \)
$13$
\( (T - 1)^{3} \)
$17$
\( T^{3} - 4 T^{2} - 10 T - 4 \)
$19$
\( T^{3} + 4T^{2} + T - 4 \)
$23$
\( T^{3} - 10 T^{2} + T + 136 \)
$29$
\( T^{3} - 24 T^{2} + 185 T - 454 \)
$31$
\( T^{3} + 4 T^{2} - 19 T + 16 \)
$37$
\( T^{3} - 58T - 124 \)
$41$
\( T^{3} - 2 T^{2} - 28 T - 8 \)
$43$
\( T^{3} - 10 T^{2} - 71 T + 628 \)
$47$
\( T^{3} + 8 T^{2} - 79 T - 544 \)
$53$
\( T^{3} - 8 T^{2} - 35 T - 22 \)
$59$
\( T^{3} + 4 T^{2} - 156 T - 688 \)
$61$
\( (T + 2)^{3} \)
$67$
\( T^{3} + 12 T^{2} - 124 T - 976 \)
$71$
\( T^{3} + 6 T^{2} - 22 T + 16 \)
$73$
\( T^{3} + 10 T^{2} - 99 T - 274 \)
$79$
\( T^{3} + 14 T^{2} + 5 T - 16 \)
$83$
\( T^{3} + 12 T^{2} - 271 T - 3268 \)
$89$
\( T^{3} - 2 T^{2} - 95 T + 422 \)
$97$
\( T^{3} + 10 T^{2} + 29 T + 22 \)
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