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: | \(3\) |
| Coefficient field: | 3.3.7117.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 19x - 22 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{3}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{3} - x^{2} - 19x - 22 \)
:
| \(\beta_{1}\) | \(=\) |
\( -12\nu^{2} + 84\nu + 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( 1272\nu^{2} - 3144\nu - 15488 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 106\beta _1 + 1920 ) / 5760 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{2} + 262\beta _1 + 74880 ) / 5760 \)
|
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 | −262.608 | 0 | −625.000 | 0 | −1790.86 | 0 | 49280.0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −13.6876 | 0 | −625.000 | 0 | 6441.91 | 0 | −19495.7 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 192.296 | 0 | −625.000 | 0 | −10171.1 | 0 | 17294.6 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 320.10.a.u | 3 | |
| 4.b | odd | 2 | 1 | 320.10.a.v | 3 | ||
| 8.b | even | 2 | 1 | 40.10.a.d | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 80.10.a.k | 3 | ||
| 24.h | odd | 2 | 1 | 360.10.a.j | 3 | ||
| 40.e | odd | 2 | 1 | 400.10.a.x | 3 | ||
| 40.f | even | 2 | 1 | 200.10.a.f | 3 | ||
| 40.i | odd | 4 | 2 | 200.10.c.f | 6 | ||
| 40.k | even | 4 | 2 | 400.10.c.r | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.10.a.d | ✓ | 3 | 8.b | even | 2 | 1 | |
| 80.10.a.k | 3 | 8.d | odd | 2 | 1 | ||
| 200.10.a.f | 3 | 40.f | even | 2 | 1 | ||
| 200.10.c.f | 6 | 40.i | odd | 4 | 2 | ||
| 320.10.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
| 320.10.a.v | 3 | 4.b | odd | 2 | 1 | ||
| 360.10.a.j | 3 | 24.h | odd | 2 | 1 | ||
| 400.10.a.x | 3 | 40.e | odd | 2 | 1 | ||
| 400.10.c.r | 6 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 84T_{3}^{2} - 49536T_{3} - 691200 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 84 T^{2} + \cdots - 691200 \)
|
| $5$ |
\( (T + 625)^{3} \)
|
| $7$ |
\( T^{3} + \cdots - 117339254144 \)
|
| $11$ |
\( T^{3} + \cdots + 46675499449024 \)
|
| $13$ |
\( T^{3} + \cdots + 10\!\cdots\!88 \)
|
| $17$ |
\( T^{3} + \cdots + 17\!\cdots\!00 \)
|
| $19$ |
\( T^{3} + \cdots - 13\!\cdots\!96 \)
|
| $23$ |
\( T^{3} + \cdots - 59\!\cdots\!96 \)
|
| $29$ |
\( T^{3} + \cdots - 81\!\cdots\!08 \)
|
| $31$ |
\( T^{3} + \cdots - 55\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 28\!\cdots\!04 \)
|
| $41$ |
\( T^{3} + \cdots - 62\!\cdots\!56 \)
|
| $43$ |
\( T^{3} + \cdots + 20\!\cdots\!92 \)
|
| $47$ |
\( T^{3} + \cdots - 19\!\cdots\!24 \)
|
| $53$ |
\( T^{3} + \cdots - 76\!\cdots\!84 \)
|
| $59$ |
\( T^{3} + \cdots - 34\!\cdots\!36 \)
|
| $61$ |
\( T^{3} + \cdots - 17\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots - 46\!\cdots\!72 \)
|
| $71$ |
\( T^{3} + \cdots - 54\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots + 89\!\cdots\!24 \)
|
| $79$ |
\( T^{3} + \cdots - 67\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 65\!\cdots\!36 \)
|
| $89$ |
\( T^{3} + \cdots + 71\!\cdots\!96 \)
|
| $97$ |
\( T^{3} + \cdots + 77\!\cdots\!04 \)
|
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