Newspace parameters
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.30226154915\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.431464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 187x - 569 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{3} - x^{2} - 187x - 569 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 6\nu - 121 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{2} + 6\beta _1 + 121 \)
|
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 | −14.3611 | 16.0000 | −54.3487 | 57.4445 | −82.6101 | −64.0000 | −36.7580 | 217.395 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | 4.65215 | 16.0000 | 80.1430 | −18.6086 | −38.0972 | −64.0000 | −221.357 | −320.572 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 23.7090 | 16.0000 | 42.2057 | −94.8359 | 144.707 | −64.0000 | 319.116 | −168.823 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(29\) | \(-1\) |
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 | 58.6.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 522.6.a.i | 3 | ||
| 4.b | odd | 2 | 1 | 464.6.a.f | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.6.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 464.6.a.f | 3 | 4.b | odd | 2 | 1 | ||
| 522.6.a.i | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 14T_{3}^{2} - 297T_{3} + 1584 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(58))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( T^{3} - 14 T^{2} + \cdots + 1584 \)
|
| $5$ |
\( T^{3} - 68 T^{2} + \cdots + 183834 \)
|
| $7$ |
\( T^{3} - 24 T^{2} + \cdots - 455424 \)
|
| $11$ |
\( T^{3} - 102 T^{2} + \cdots + 11144952 \)
|
| $13$ |
\( T^{3} - 220 T^{2} + \cdots + 64435938 \)
|
| $17$ |
\( T^{3} + \cdots + 4080137208 \)
|
| $19$ |
\( T^{3} - 3652 T^{2} + \cdots - 938008512 \)
|
| $23$ |
\( T^{3} + \cdots - 2582050944 \)
|
| $29$ |
\( (T - 841)^{3} \)
|
| $31$ |
\( T^{3} + \cdots - 61264891668 \)
|
| $37$ |
\( T^{3} + \cdots - 420885744288 \)
|
| $41$ |
\( T^{3} + \cdots - 18158892384 \)
|
| $43$ |
\( T^{3} + \cdots - 2396466063000 \)
|
| $47$ |
\( T^{3} + \cdots + 1441150335252 \)
|
| $53$ |
\( T^{3} + \cdots + 1095236406426 \)
|
| $59$ |
\( T^{3} + \cdots + 17372487788496 \)
|
| $61$ |
\( T^{3} + \cdots - 655741719096 \)
|
| $67$ |
\( T^{3} + \cdots - 163676706166656 \)
|
| $71$ |
\( T^{3} + \cdots + 76931026202016 \)
|
| $73$ |
\( T^{3} + \cdots - 151387711842208 \)
|
| $79$ |
\( T^{3} + \cdots - 2509585973268 \)
|
| $83$ |
\( T^{3} + \cdots + 72220786520688 \)
|
| $89$ |
\( T^{3} + \cdots - 228134244314976 \)
|
| $97$ |
\( T^{3} + \cdots - 197612904971808 \)
|
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