Newspace parameters
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(164.811467572\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 353x^{2} + 354x + 2313 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{2}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 160) |
| 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
\( x^{4} - 2x^{3} - 353x^{2} + 354x + 2313 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 16\nu^{3} - 24\nu^{2} - 5452\nu + 2730 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 16\nu^{2} - 16\nu - 2832 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 6 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 4\beta _1 + 8520 ) / 48 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{3} + 18\beta_{2} + 2738\beta _1 + 25536 ) / 96 \)
|
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 | −223.725 | 0 | 625.000 | 0 | −7001.03 | 0 | 30370.1 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −31.5423 | 0 | 625.000 | 0 | −5763.01 | 0 | −18688.1 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 31.5423 | 0 | 625.000 | 0 | 5763.01 | 0 | −18688.1 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 223.725 | 0 | 625.000 | 0 | 7001.03 | 0 | 30370.1 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.10.a.z | 4 | |
| 4.b | odd | 2 | 1 | inner | 320.10.a.z | 4 | |
| 8.b | even | 2 | 1 | 160.10.a.c | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 160.10.a.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.10.a.c | ✓ | 4 | 8.b | even | 2 | 1 | |
| 160.10.a.c | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 320.10.a.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 320.10.a.z | 4 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 51048T_{3}^{2} + 49798800 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 51048 T^{2} + 49798800 \)
|
| $5$ |
\( (T - 625)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + 100628 T - 11203050908)^{2} \)
|
| $17$ |
\( (T^{2} + 221020 T + 11848483396)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots - 3437979067260)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 58121226872060)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 18406476732380)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 55\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 96\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + \cdots + 19\!\cdots\!96)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots - 19\!\cdots\!80)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots - 172635344399196)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 34\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 70\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + \cdots + 26\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 14\!\cdots\!76)^{2} \)
|
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