Newspace parameters
| Level: | \( N \) | \(=\) | \( 1062 = 2 \cdot 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1062.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(62.6600284261\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.45581.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 37x + 90 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 354) |
| 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
\( x^{3} - x^{2} - 37x + 90 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2\nu - 26 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2\beta _1 + 26 \)
|
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 |
|
2.00000 | 0 | 4.00000 | −13.3177 | 0 | 35.2646 | 8.00000 | 0 | −26.6354 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | −3.86257 | 0 | 4.92301 | 8.00000 | 0 | −7.72514 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | 13.1803 | 0 | −27.1877 | 8.00000 | 0 | 26.3605 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(59\) | \( -1 \) |
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 | 1062.4.a.j | 3 | |
| 3.b | odd | 2 | 1 | 354.4.a.e | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 354.4.a.e | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 1062.4.a.j | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 4T_{5}^{2} - 175T_{5} - 678 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1062))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 4 T^{2} + \cdots - 678 \)
|
| $7$ |
\( T^{3} - 13 T^{2} + \cdots + 4720 \)
|
| $11$ |
\( T^{3} + 54 T^{2} + \cdots - 35260 \)
|
| $13$ |
\( T^{3} + 52 T^{2} + \cdots - 90938 \)
|
| $17$ |
\( T^{3} + 89 T^{2} + \cdots - 38950 \)
|
| $19$ |
\( T^{3} + 163 T^{2} + \cdots - 1015500 \)
|
| $23$ |
\( T^{3} - 31 T^{2} + \cdots - 720336 \)
|
| $29$ |
\( T^{3} - 127 T^{2} + \cdots + 2149250 \)
|
| $31$ |
\( T^{3} + 55 T^{2} + \cdots - 61892 \)
|
| $37$ |
\( T^{3} + 401 T^{2} + \cdots - 2727486 \)
|
| $41$ |
\( T^{3} + 255 T^{2} + \cdots + 909850 \)
|
| $43$ |
\( T^{3} + 400 T^{2} + \cdots - 26264676 \)
|
| $47$ |
\( T^{3} + 1247 T^{2} + \cdots + 54812280 \)
|
| $53$ |
\( T^{3} + 496 T^{2} + \cdots + 3999106 \)
|
| $59$ |
\( (T - 59)^{3} \)
|
| $61$ |
\( T^{3} - 1063 T^{2} + \cdots - 31375986 \)
|
| $67$ |
\( T^{3} + 72 T^{2} + \cdots - 1091780 \)
|
| $71$ |
\( T^{3} + 2234 T^{2} + \cdots + 67438172 \)
|
| $73$ |
\( T^{3} + 577 T^{2} + \cdots - 8907498 \)
|
| $79$ |
\( T^{3} - 1164 T^{2} + \cdots + 973809000 \)
|
| $83$ |
\( T^{3} + 85 T^{2} + \cdots - 159266284 \)
|
| $89$ |
\( T^{3} - 467 T^{2} + \cdots + 761230 \)
|
| $97$ |
\( T^{3} + \cdots + 1537407402 \)
|
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