Newspace parameters
| Level: | \( N \) | \(=\) | \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1710.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(100.893266110\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.15357.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 26x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 190) |
| 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
\( x^{3} - x^{2} - 26x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 17 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 17 \)
|
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 | −5.00000 | 0 | −7.77504 | −8.00000 | 0 | 10.0000 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 10.8926 | −8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | −5.00000 | 0 | 25.8824 | −8.00000 | 0 | 10.0000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 1710.4.a.w | 3 | |
| 3.b | odd | 2 | 1 | 190.4.a.i | ✓ | 3 | |
| 12.b | even | 2 | 1 | 1520.4.a.n | 3 | ||
| 15.d | odd | 2 | 1 | 950.4.a.j | 3 | ||
| 15.e | even | 4 | 2 | 950.4.b.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.4.a.i | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 950.4.a.j | 3 | 15.d | odd | 2 | 1 | ||
| 950.4.b.n | 6 | 15.e | even | 4 | 2 | ||
| 1520.4.a.n | 3 | 12.b | even | 2 | 1 | ||
| 1710.4.a.w | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1710))\):
|
\( T_{7}^{3} - 29T_{7}^{2} - 4T_{7} + 2192 \)
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\( T_{11}^{3} + 12T_{11}^{2} - 3072T_{11} + 43776 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( (T + 5)^{3} \)
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| $7$ |
\( T^{3} - 29 T^{2} + \cdots + 2192 \)
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| $11$ |
\( T^{3} + 12 T^{2} + \cdots + 43776 \)
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| $13$ |
\( T^{3} - 59 T^{2} + \cdots + 156372 \)
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| $17$ |
\( T^{3} + 17 T^{2} + \cdots - 3236 \)
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| $19$ |
\( (T + 19)^{3} \)
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| $23$ |
\( T^{3} + 77 T^{2} + \cdots - 1541424 \)
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| $29$ |
\( T^{3} + 51 T^{2} + \cdots + 1398340 \)
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| $31$ |
\( T^{3} + 94 T^{2} + \cdots - 1494912 \)
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| $37$ |
\( T^{3} + 144 T^{2} + \cdots + 2909128 \)
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| $41$ |
\( T^{3} + 102 T^{2} + \cdots - 17262136 \)
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| $43$ |
\( T^{3} - 26 T^{2} + \cdots + 96 \)
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| $47$ |
\( T^{3} - 204 T^{2} + \cdots - 1641600 \)
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| $53$ |
\( T^{3} - 45 T^{2} + \cdots + 1647396 \)
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| $59$ |
\( T^{3} - 669 T^{2} + \cdots + 8508080 \)
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| $61$ |
\( T^{3} + 608 T^{2} + \cdots - 9793824 \)
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| $67$ |
\( T^{3} + 385 T^{2} + \cdots - 20641692 \)
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| $71$ |
\( T^{3} - 252 T^{2} + \cdots - 166046976 \)
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| $73$ |
\( T^{3} + 41 T^{2} + \cdots + 51931164 \)
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| $79$ |
\( T^{3} + 1154 T^{2} + \cdots + 14064480 \)
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| $83$ |
\( T^{3} + \cdots - 1321554048 \)
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| $89$ |
\( T^{3} + \cdots + 1561003000 \)
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| $97$ |
\( T^{3} + 2800 T^{2} + \cdots + 639684024 \)
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