Newspace parameters
| Level: | \( N \) | \(=\) | \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1560.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(92.0429796090\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 1022x^{3} - 1776x^{2} + 266445x + 1462050 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{5} - 2x^{4} - 1022x^{3} - 1776x^{2} + 266445x + 1462050 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 4\nu - 408 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 43\nu^{3} - 407\nu^{2} - 24531\nu - 132930 ) / 360 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} - 19\nu^{3} + 503\nu^{2} + 11595\nu + 43650 ) / 72 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 4\beta _1 + 408 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 15\beta_{3} - 12\beta_{2} + 523\beta _1 + 2088 \)
|
| \(\nu^{4}\) | \(=\) |
\( -129\beta_{4} - 285\beta_{3} + 1737\beta_{2} + 3670\beta _1 + 209202 \)
|
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 | 3.00000 | 0 | 5.00000 | 0 | −17.0067 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 5.00000 | 0 | −13.5671 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 5.00000 | 0 | −1.05317 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 5.00000 | 0 | 29.1777 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.00000 | 0 | 5.00000 | 0 | 29.4493 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
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 | 1560.4.a.r | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1560.4.a.r | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} - 27T_{7}^{4} - 732T_{7}^{3} + 12004T_{7}^{2} + 211680T_{7} + 208800 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1560))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} - 27 T^{4} + \cdots + 208800 \)
|
| $11$ |
\( T^{5} - 21 T^{4} + \cdots - 74197760 \)
|
| $13$ |
\( (T + 13)^{5} \)
|
| $17$ |
\( T^{5} + \cdots - 2251680096 \)
|
| $19$ |
\( T^{5} - 22 T^{4} + \cdots + 520000 \)
|
| $23$ |
\( T^{5} + \cdots - 16980748096 \)
|
| $29$ |
\( T^{5} + \cdots - 46205685376 \)
|
| $31$ |
\( T^{5} + \cdots - 45191708416 \)
|
| $37$ |
\( T^{5} + \cdots - 37164882000 \)
|
| $41$ |
\( T^{5} + \cdots + 104723512560 \)
|
| $43$ |
\( T^{5} + \cdots - 2890846363648 \)
|
| $47$ |
\( T^{5} + \cdots + 34476021760 \)
|
| $53$ |
\( T^{5} + \cdots + 1735826034000 \)
|
| $59$ |
\( T^{5} + \cdots + 6111351500800 \)
|
| $61$ |
\( T^{5} + \cdots + 8469083742512 \)
|
| $67$ |
\( T^{5} + \cdots + 43548833204224 \)
|
| $71$ |
\( T^{5} + \cdots - 4234608973824 \)
|
| $73$ |
\( T^{5} + \cdots - 5973376960512 \)
|
| $79$ |
\( T^{5} + \cdots - 87761329152 \)
|
| $83$ |
\( T^{5} + \cdots - 21140008300032 \)
|
| $89$ |
\( T^{5} + \cdots - 346089953114064 \)
|
| $97$ |
\( T^{5} + \cdots + 150763178095040 \)
|
| show more | |
| show less |