Newspace parameters
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(74.0973247536\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 6543x^{2} + 5144x + 1444784 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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
\( x^{4} - 2x^{3} - 6543x^{2} + 5144x + 1444784 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 6123\nu - 2922 ) / 50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - \nu - 3272 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{3} + \beta _1 + 3272 \)
|
| \(\nu^{3}\) | \(=\) |
\( 50\beta_{2} + 6123\beta _1 + 2922 \)
|
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 |
|
−4.00000 | 9.00000 | 16.0000 | −61.8554 | −36.0000 | 49.0000 | −64.0000 | 81.0000 | 247.422 | ||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | 9.00000 | 16.0000 | 2.25462 | −36.0000 | 49.0000 | −64.0000 | 81.0000 | −9.01848 | |||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 9.00000 | 16.0000 | 32.5154 | −36.0000 | 49.0000 | −64.0000 | 81.0000 | −130.061 | |||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 9.00000 | 16.0000 | 97.0854 | −36.0000 | 49.0000 | −64.0000 | 81.0000 | −388.342 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 462.6.a.q | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.6.a.q | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 70T_{5}^{3} - 4707T_{5}^{2} + 206220T_{5} - 440244 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} - 70 T^{3} + \cdots - 440244 \)
|
| $7$ |
\( (T - 49)^{4} \)
|
| $11$ |
\( (T - 121)^{4} \)
|
| $13$ |
\( T^{4} + \cdots + 15696670116 \)
|
| $17$ |
\( T^{4} + \cdots + 94210971408 \)
|
| $19$ |
\( T^{4} + \cdots + 4365070299128 \)
|
| $23$ |
\( T^{4} + \cdots + 5884078580352 \)
|
| $29$ |
\( T^{4} + \cdots + 223815732228 \)
|
| $31$ |
\( T^{4} + \cdots - 201429736981344 \)
|
| $37$ |
\( T^{4} + \cdots + 12\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots - 21\!\cdots\!76 \)
|
| $43$ |
\( T^{4} + \cdots + 89\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots - 65\!\cdots\!48 \)
|
| $59$ |
\( T^{4} + \cdots - 43\!\cdots\!32 \)
|
| $61$ |
\( T^{4} + \cdots - 11\!\cdots\!28 \)
|
| $67$ |
\( T^{4} + \cdots - 21\!\cdots\!72 \)
|
| $71$ |
\( T^{4} + \cdots + 90\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 49\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 14\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots - 84\!\cdots\!32 \)
|
| $97$ |
\( T^{4} + \cdots + 16\!\cdots\!92 \)
|
| show more | |
| show less |