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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
| Defining polynomial: |
\( x^{4} - x^{3} - 41x^{2} - 30x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 41x^{2} - 30x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( -2\nu^{3} + 4\nu^{2} + 80\nu + 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( -3\nu^{3} + 4\nu^{2} + 128\nu + 46 \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\nu^{3} + 4\nu^{2} + 120\nu + 48 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 2 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{3} + 3\beta _1 + 84 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{3} + 5\beta_{2} + \beta _1 + 54 \)
|
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 | −35.1436 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 5.00000 | 0 | −14.8428 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 5.00000 | 0 | 4.92462 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 5.00000 | 0 | 11.0618 | 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.n | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1560.4.a.n | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 34T_{7}^{3} - 223T_{7}^{2} - 5616T_{7} + 28416 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1560))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T - 3)^{4} \)
$5$
\( (T - 5)^{4} \)
$7$
\( T^{4} + 34 T^{3} - 223 T^{2} + \cdots + 28416 \)
$11$
\( T^{4} + 38 T^{3} - 1759 T^{2} + \cdots + 794640 \)
$13$
\( (T - 13)^{4} \)
$17$
\( T^{4} + 50 T^{3} - 10091 T^{2} + \cdots + 2634756 \)
$19$
\( T^{4} + 180 T^{3} + 4868 T^{2} + \cdots + 1578560 \)
$23$
\( T^{4} + 170 T^{3} + \cdots - 115845696 \)
$29$
\( T^{4} + 84 T^{3} + \cdots + 1641072688 \)
$31$
\( T^{4} + 408 T^{3} + 49948 T^{2} + \cdots + 9081600 \)
$37$
\( T^{4} + 38 T^{3} + \cdots - 153758476 \)
$41$
\( T^{4} + 90 T^{3} + \cdots + 1993594900 \)
$43$
\( T^{4} + 732 T^{3} + \cdots + 2881893632 \)
$47$
\( T^{4} + 520 T^{3} + \cdots - 2001440000 \)
$53$
\( T^{4} + 338 T^{3} + \cdots - 9572486780 \)
$59$
\( T^{4} + 232 T^{3} + \cdots + 923054080 \)
$61$
\( T^{4} - 326 T^{3} + \cdots + 13972040964 \)
$67$
\( T^{4} + 1040 T^{3} + \cdots - 15293380352 \)
$71$
\( T^{4} + 702 T^{3} + \cdots - 18758062080 \)
$73$
\( T^{4} + 368 T^{3} + \cdots + 10624585104 \)
$79$
\( T^{4} + 678 T^{3} + \cdots - 203520914688 \)
$83$
\( T^{4} + 1888 T^{3} + \cdots - 460820907008 \)
$89$
\( T^{4} + 2070 T^{3} + \cdots - 796693724300 \)
$97$
\( T^{4} - 1126 T^{3} + \cdots - 13823634476 \)
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