Newspace parameters
| Level: | \( N \) | \(=\) | \( 62 = 2 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 62.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(19.3678715800\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 919x^{2} - 316x + 185892 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - 2x^{3} - 919x^{2} - 316x + 185892 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 24\nu^{2} - 451\nu + 9654 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 22\nu^{2} - 453\nu + 8738 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - 6\beta_{2} + \beta _1 + 458 \)
|
| \(\nu^{3}\) | \(=\) |
\( 48\beta_{3} - 132\beta_{2} + 475\beta _1 + 1338 \)
|
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 |
|
8.00000 | −90.4899 | 64.0000 | 149.027 | −723.919 | 673.892 | 512.000 | 6001.42 | 1192.22 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −31.2462 | 64.0000 | −62.2536 | −249.970 | 431.967 | 512.000 | −1210.68 | −498.029 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 2.83188 | 64.0000 | 91.0828 | 22.6550 | −1506.18 | 512.000 | −2178.98 | 728.663 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 50.9042 | 64.0000 | −511.857 | 407.234 | −547.674 | 512.000 | 404.237 | −4094.85 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(31\) | \( +1 \) |
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 | 62.8.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 558.8.a.j | 4 | ||
| 4.b | odd | 2 | 1 | 496.8.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 62.8.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 496.8.a.b | 4 | 4.b | odd | 2 | 1 | ||
| 558.8.a.j | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 68T_{3}^{3} - 3570T_{3}^{2} - 134388T_{3} + 407592 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(62))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} + 68 T^{3} + \cdots + 407592 \)
|
| $5$ |
\( T^{4} + 334 T^{3} + \cdots + 432529620 \)
|
| $7$ |
\( T^{4} + \cdots + 240127196940 \)
|
| $11$ |
\( T^{4} + \cdots - 382278707309352 \)
|
| $13$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 27\!\cdots\!12 \)
|
| $19$ |
\( T^{4} + \cdots - 26\!\cdots\!88 \)
|
| $23$ |
\( T^{4} + \cdots - 19\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots + 24\!\cdots\!88 \)
|
| $31$ |
\( (T + 29791)^{4} \)
|
| $37$ |
\( T^{4} + \cdots + 19\!\cdots\!04 \)
|
| $41$ |
\( T^{4} + \cdots + 13\!\cdots\!48 \)
|
| $43$ |
\( T^{4} + \cdots - 34\!\cdots\!28 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!52 \)
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| $53$ |
\( T^{4} + \cdots + 14\!\cdots\!28 \)
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| $59$ |
\( T^{4} + \cdots + 29\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots + 37\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots + 70\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 94\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots + 63\!\cdots\!48 \)
|
| $79$ |
\( T^{4} + \cdots + 13\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots - 38\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 12\!\cdots\!68 \)
|
| $97$ |
\( T^{4} + \cdots - 32\!\cdots\!60 \)
|
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