Newspace parameters
| Level: | \( N \) | \(=\) | \( 288 = 2^{5} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 288.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(148.330320815\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 45829x^{2} + 45830x + 94023351 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 45829x^{2} + 45830x + 94023351 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 16\nu^{3} - 24\nu^{2} - 768880\nu + 384444 ) / 15939 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 16\nu^{3} - 24\nu^{2} - 513856\nu + 256932 ) / 5313 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 192\nu^{2} - 192\nu - 4399680 ) / 77 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 3\beta _1 + 24 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 77\beta_{3} + 4\beta_{2} - 12\beta _1 + 4399776 ) / 192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 231\beta_{3} + 384452\beta_{2} - 770820\beta _1 + 13199232 ) / 384 \)
|
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 | 0 | 0 | −2138.03 | 0 | 6324.93 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −918.506 | 0 | −6357.58 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 918.506 | 0 | 6357.58 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 2138.03 | 0 | −6324.93 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 288.10.a.r | yes | 4 |
| 3.b | odd | 2 | 1 | 288.10.a.q | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 288.10.a.q | ✓ | 4 | |
| 12.b | even | 2 | 1 | inner | 288.10.a.r | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 288.10.a.q | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 288.10.a.q | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 288.10.a.r | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 288.10.a.r | yes | 4 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(288))\):
|
\( T_{5}^{4} - 5414816T_{5}^{2} + 3856474835200 \)
|
|
\( T_{7}^{4} - 80423552T_{7}^{2} + 1616944045330432 \)
|
|
\( T_{11}^{2} - 41920T_{11} - 2240912384 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 3856474835200 \)
|
| $7$ |
\( T^{4} + \cdots + 16\!\cdots\!32 \)
|
| $11$ |
\( (T^{2} - 41920 T - 2240912384)^{2} \)
|
| $13$ |
\( (T^{2} - 83828 T - 8964152540)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!28 \)
|
| $19$ |
\( T^{4} + \cdots + 13\!\cdots\!68 \)
|
| $23$ |
\( (T^{2} - 590976 T - 438012702720)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 87\!\cdots\!68 \)
|
| $31$ |
\( T^{4} + \cdots + 32\!\cdots\!72 \)
|
| $37$ |
\( (T^{2} + \cdots - 131295821054972)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 45\!\cdots\!08 \)
|
| $43$ |
\( T^{4} + \cdots + 13\!\cdots\!92 \)
|
| $47$ |
\( (T^{2} + \cdots - 735770515730432)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + \cdots + 11457366495232)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 92\!\cdots\!92)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 78\!\cdots\!08 \)
|
| $71$ |
\( (T^{2} + \cdots - 32\!\cdots\!32)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 51\!\cdots\!80)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 12\!\cdots\!32 \)
|
| $83$ |
\( (T^{2} + \cdots - 89\!\cdots\!84)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots - 19\!\cdots\!80)^{2} \)
|
| show more | |
| show less |