Newspace parameters
| Level: | \( N \) | \(=\) | \( 1160 = 2^{3} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1160.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.26264663447\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.6083172.1 |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 7\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 9\beta_{2} + 3\beta _1 + 34 \)
|
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 |
|
0 | −3.11726 | 0 | 1.00000 | 0 | −0.964080 | 0 | 6.71730 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.13670 | 0 | 1.00000 | 0 | −4.33279 | 0 | 1.56547 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.689426 | 0 | 1.00000 | 0 | 4.55109 | 0 | −2.52469 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.36347 | 0 | 1.00000 | 0 | 2.16993 | 0 | 2.58601 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.57991 | 0 | 1.00000 | 0 | −2.42414 | 0 | 3.65592 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 1160.2.a.i | ✓ | 5 |
| 4.b | odd | 2 | 1 | 2320.2.a.w | 5 | ||
| 5.b | even | 2 | 1 | 5800.2.a.v | 5 | ||
| 8.b | even | 2 | 1 | 9280.2.a.cj | 5 | ||
| 8.d | odd | 2 | 1 | 9280.2.a.ch | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1160.2.a.i | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 2320.2.a.w | 5 | 4.b | odd | 2 | 1 | ||
| 5800.2.a.v | 5 | 5.b | even | 2 | 1 | ||
| 9280.2.a.ch | 5 | 8.d | odd | 2 | 1 | ||
| 9280.2.a.cj | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1160))\):
|
\( T_{3}^{5} + T_{3}^{4} - 13T_{3}^{3} - 10T_{3}^{2} + 40T_{3} + 28 \)
|
|
\( T_{7}^{5} + T_{7}^{4} - 25T_{7}^{3} - 28T_{7}^{2} + 100T_{7} + 100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots + 28 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} + T^{4} + \cdots + 100 \)
|
| $11$ |
\( T^{5} + 4 T^{4} + \cdots + 48 \)
|
| $13$ |
\( T^{5} - 13 T^{4} + \cdots + 824 \)
|
| $17$ |
\( T^{5} - 3 T^{4} + \cdots - 1964 \)
|
| $19$ |
\( T^{5} - 76 T^{3} + \cdots + 432 \)
|
| $23$ |
\( T^{5} + 7 T^{4} + \cdots - 900 \)
|
| $29$ |
\( (T + 1)^{5} \)
|
| $31$ |
\( T^{5} - 11 T^{4} + \cdots - 36 \)
|
| $37$ |
\( T^{5} - 6 T^{4} + \cdots + 16 \)
|
| $41$ |
\( T^{5} - 16 T^{4} + \cdots + 14816 \)
|
| $43$ |
\( T^{5} - 9 T^{4} + \cdots - 1156 \)
|
| $47$ |
\( T^{5} - 2 T^{4} + \cdots + 64 \)
|
| $53$ |
\( T^{5} - 7 T^{4} + \cdots - 33336 \)
|
| $59$ |
\( T^{5} - 5 T^{4} + \cdots - 16 \)
|
| $61$ |
\( T^{5} - 7 T^{4} + \cdots - 11112 \)
|
| $67$ |
\( T^{5} + 4 T^{4} + \cdots - 192 \)
|
| $71$ |
\( T^{5} - 12 T^{4} + \cdots - 30464 \)
|
| $73$ |
\( T^{5} - 7 T^{4} + \cdots + 1588 \)
|
| $79$ |
\( T^{5} - 45 T^{4} + \cdots + 36204 \)
|
| $83$ |
\( T^{5} + 22 T^{4} + \cdots + 25776 \)
|
| $89$ |
\( T^{5} - 24 T^{4} + \cdots + 88672 \)
|
| $97$ |
\( T^{5} - 17 T^{4} + \cdots - 196 \)
|
| show more | |
| show less |