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: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 947x - 3456 \)
|
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 947x - 3456 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 5\nu - 630 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{2} + 5\beta _1 + 1265 ) / 2 \)
|
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 | −73.8745 | 64.0000 | −176.475 | 590.996 | 245.299 | −512.000 | 3270.45 | 1411.80 | |||||||||||||||||||||||||||
| 1.2 | −8.00000 | −0.563391 | 64.0000 | 266.067 | 4.50713 | −1155.64 | −512.000 | −2186.68 | −2128.54 | ||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 48.4379 | 64.0000 | 62.4075 | −387.504 | −593.661 | −512.000 | 159.234 | −499.260 | ||||||||||||||||||||||||||||
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.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 558.8.a.g | 3 | ||
| 4.b | odd | 2 | 1 | 496.8.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 62.8.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 496.8.a.a | 3 | 4.b | odd | 2 | 1 | ||
| 558.8.a.g | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 26T_{3}^{2} - 3564T_{3} - 2016 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(62))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{3} \)
|
| $3$ |
\( T^{3} + 26 T^{2} + \cdots - 2016 \)
|
| $5$ |
\( T^{3} - 152 T^{2} + \cdots + 2930298 \)
|
| $7$ |
\( T^{3} + 1504 T^{2} + \cdots - 168289456 \)
|
| $11$ |
\( T^{3} + \cdots + 121900627296 \)
|
| $13$ |
\( T^{3} + \cdots + 103204066296 \)
|
| $17$ |
\( T^{3} + \cdots + 6570154100232 \)
|
| $19$ |
\( T^{3} + \cdots + 539939504716 \)
|
| $23$ |
\( T^{3} + \cdots + 116422844777568 \)
|
| $29$ |
\( T^{3} + \cdots - 739019343333672 \)
|
| $31$ |
\( (T - 29791)^{3} \)
|
| $37$ |
\( T^{3} + \cdots + 30\!\cdots\!92 \)
|
| $41$ |
\( T^{3} + \cdots - 37\!\cdots\!22 \)
|
| $43$ |
\( T^{3} + \cdots - 16\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots + 54\!\cdots\!52 \)
|
| $53$ |
\( T^{3} + \cdots + 87\!\cdots\!32 \)
|
| $59$ |
\( T^{3} + \cdots - 62\!\cdots\!48 \)
|
| $61$ |
\( T^{3} + \cdots - 85\!\cdots\!96 \)
|
| $67$ |
\( T^{3} + \cdots + 17\!\cdots\!64 \)
|
| $71$ |
\( T^{3} + \cdots - 13\!\cdots\!72 \)
|
| $73$ |
\( T^{3} + \cdots - 13\!\cdots\!56 \)
|
| $79$ |
\( T^{3} + \cdots + 17\!\cdots\!72 \)
|
| $83$ |
\( T^{3} + \cdots - 17\!\cdots\!36 \)
|
| $89$ |
\( T^{3} + \cdots - 93\!\cdots\!48 \)
|
| $97$ |
\( T^{3} + \cdots - 27\!\cdots\!22 \)
|
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