Newspace parameters
| Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 480.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(76.9842335102\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 1320x + 6750 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 1320x + 6750 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{2} + 184\nu + 3465 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{2} + 56\nu - 3540 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 5 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 23\beta_{2} - 7\beta _1 + 7045 ) / 8 \)
|
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 | −9.00000 | 0 | −25.0000 | 0 | −173.378 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | −25.0000 | 0 | 46.3675 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −9.00000 | 0 | −25.0000 | 0 | 235.011 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 480.6.a.j | ✓ | 3 |
| 4.b | odd | 2 | 1 | 480.6.a.m | yes | 3 | |
| 8.b | even | 2 | 1 | 960.6.a.bx | 3 | ||
| 8.d | odd | 2 | 1 | 960.6.a.bs | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 480.6.a.j | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 480.6.a.m | yes | 3 | 4.b | odd | 2 | 1 | |
| 960.6.a.bs | 3 | 8.d | odd | 2 | 1 | ||
| 960.6.a.bx | 3 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 108T_{7}^{2} - 37888T_{7} + 1889280 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(480))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 9)^{3} \)
|
| $5$ |
\( (T + 25)^{3} \)
|
| $7$ |
\( T^{3} - 108 T^{2} + \cdots + 1889280 \)
|
| $11$ |
\( T^{3} + 268 T^{2} + \cdots - 55327680 \)
|
| $13$ |
\( T^{3} + 494 T^{2} + \cdots - 98115896 \)
|
| $17$ |
\( T^{3} - 1078 T^{2} + \cdots + 92349720 \)
|
| $19$ |
\( T^{3} + \cdots + 2454292224 \)
|
| $23$ |
\( T^{3} + \cdots + 3775440128 \)
|
| $29$ |
\( T^{3} + \cdots + 80648640600 \)
|
| $31$ |
\( T^{3} + \cdots + 71168716800 \)
|
| $37$ |
\( T^{3} + \cdots + 416870759304 \)
|
| $41$ |
\( T^{3} + \cdots + 3654974402424 \)
|
| $43$ |
\( T^{3} + \cdots + 801817978176 \)
|
| $47$ |
\( T^{3} + \cdots - 3535014528 \)
|
| $53$ |
\( T^{3} + \cdots + 13167420267192 \)
|
| $59$ |
\( T^{3} + \cdots - 12205496066880 \)
|
| $61$ |
\( T^{3} + \cdots - 26761916504 \)
|
| $67$ |
\( T^{3} + \cdots + 139433832000 \)
|
| $71$ |
\( T^{3} + \cdots + 12428641566720 \)
|
| $73$ |
\( T^{3} + \cdots + 72297984438040 \)
|
| $79$ |
\( T^{3} + \cdots + 2868776930048 \)
|
| $83$ |
\( T^{3} + \cdots - 11965381632576 \)
|
| $89$ |
\( T^{3} + \cdots - 570088545203880 \)
|
| $97$ |
\( T^{3} + \cdots - 10\!\cdots\!16 \)
|
| show more | |
| show less |