Newspace parameters
| Level: | \( N \) | \(=\) | \( 1160 = 2^{3} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1160.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(68.4422156067\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1556.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{3} - x^{2} - 9x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} + 2\nu - 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 12 ) / 2 \)
|
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 |
|
0 | −10.2284 | 0 | −5.00000 | 0 | −25.3117 | 0 | 77.6202 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.18096 | 0 | −5.00000 | 0 | −0.858544 | 0 | −16.8815 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 5.40937 | 0 | −5.00000 | 0 | −15.8298 | 0 | 2.26124 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 1160.4.a.b | ✓ | 3 |
| 4.b | odd | 2 | 1 | 2320.4.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1160.4.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2320.4.a.j | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 8T_{3}^{2} - 40T_{3} - 176 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1160))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 8 T^{2} + \cdots - 176 \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} + 42 T^{2} + \cdots + 344 \)
|
| $11$ |
\( T^{3} + 70 T^{2} + \cdots - 41624 \)
|
| $13$ |
\( T^{3} - 46 T^{2} + \cdots + 159704 \)
|
| $17$ |
\( T^{3} - 226 T^{2} + \cdots - 311464 \)
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| $19$ |
\( T^{3} + 58 T^{2} + \cdots + 6072 \)
|
| $23$ |
\( T^{3} + 126 T^{2} + \cdots + 7688 \)
|
| $29$ |
\( (T + 29)^{3} \)
|
| $31$ |
\( T^{3} - 194 T^{2} + \cdots + 249864 \)
|
| $37$ |
\( T^{3} - 350 T^{2} + \cdots - 1293048 \)
|
| $41$ |
\( T^{3} - 230 T^{2} + \cdots + 9639544 \)
|
| $43$ |
\( T^{3} - 288 T^{2} + \cdots + 18447088 \)
|
| $47$ |
\( T^{3} - 188 T^{2} + \cdots - 173072 \)
|
| $53$ |
\( T^{3} + 202 T^{2} + \cdots - 18070088 \)
|
| $59$ |
\( T^{3} - 176 T^{2} + \cdots + 3281664 \)
|
| $61$ |
\( T^{3} + 474 T^{2} + \cdots - 222535688 \)
|
| $67$ |
\( T^{3} + 1750 T^{2} + \cdots + 190883464 \)
|
| $71$ |
\( T^{3} - 588 T^{2} + \cdots + 123963328 \)
|
| $73$ |
\( T^{3} + 170 T^{2} + \cdots - 243279448 \)
|
| $79$ |
\( T^{3} + 1138 T^{2} + \cdots + 12958616 \)
|
| $83$ |
\( T^{3} + 510 T^{2} + \cdots - 360723192 \)
|
| $89$ |
\( T^{3} + 1514 T^{2} + \cdots + 11526456 \)
|
| $97$ |
\( T^{3} + \cdots - 2533137368 \)
|
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