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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.172392.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 56x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| 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
\( x^{3} - x^{2} - 56x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 37 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 37 \)
|
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 | −54.0925 | 36.0000 | −49.0000 | 64.0000 | 81.0000 | −216.370 | |||||||||||||||||||||||||||
| 1.2 | 4.00000 | 9.00000 | 16.0000 | −21.8126 | 36.0000 | −49.0000 | 64.0000 | 81.0000 | −87.2505 | ||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 9.00000 | 16.0000 | 19.9051 | 36.0000 | −49.0000 | 64.0000 | 81.0000 | 79.6204 | ||||||||||||||||||||||||||||
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.p | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.6.a.p | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 56T_{5}^{2} - 331T_{5} - 23486 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{3} \)
|
| $3$ |
\( (T - 9)^{3} \)
|
| $5$ |
\( T^{3} + 56 T^{2} + \cdots - 23486 \)
|
| $7$ |
\( (T + 49)^{3} \)
|
| $11$ |
\( (T + 121)^{3} \)
|
| $13$ |
\( T^{3} + 588 T^{2} + \cdots + 4872658 \)
|
| $17$ |
\( T^{3} + \cdots - 2162039928 \)
|
| $19$ |
\( T^{3} + 34 T^{2} + \cdots + 226882908 \)
|
| $23$ |
\( T^{3} + 856 T^{2} + \cdots - 710780800 \)
|
| $29$ |
\( T^{3} + \cdots - 73919047750 \)
|
| $31$ |
\( T^{3} + \cdots - 21316648672 \)
|
| $37$ |
\( T^{3} + \cdots + 121408236330 \)
|
| $41$ |
\( T^{3} + \cdots - 585917767776 \)
|
| $43$ |
\( T^{3} + \cdots + 2767866550496 \)
|
| $47$ |
\( T^{3} + \cdots - 594955625000 \)
|
| $53$ |
\( T^{3} + \cdots - 32419763121264 \)
|
| $59$ |
\( T^{3} + \cdots - 39807964249740 \)
|
| $61$ |
\( T^{3} + \cdots - 22123077213496 \)
|
| $67$ |
\( T^{3} + \cdots - 201516727047900 \)
|
| $71$ |
\( T^{3} + \cdots - 152983564683520 \)
|
| $73$ |
\( T^{3} + \cdots - 225947762147866 \)
|
| $79$ |
\( T^{3} + \cdots - 59626597519872 \)
|
| $83$ |
\( T^{3} + \cdots + 611255858117328 \)
|
| $89$ |
\( T^{3} + \cdots - 115749662376 \)
|
| $97$ |
\( T^{3} + \cdots + 53261811815856 \)
|
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