Newspace parameters
| Level: | \( N \) | \(=\) | \( 935 = 5 \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 935.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.46601258899\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.381812160.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 15x^{2} - 22x + 3 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 15x^{2} - 22x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 15\nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 14 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 9\beta_{3} + 9\beta_{2} + 27\beta _1 + 4 \)
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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 |
|
−2.62765 | −0.0998927 | 4.90452 | −1.00000 | 0.262483 | 3.66691 | −7.63206 | −2.99002 | 2.62765 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.15649 | 3.40427 | 2.65043 | −1.00000 | −7.34126 | −4.12457 | −1.40265 | 8.58904 | 2.15649 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.926526 | 2.69570 | −1.14155 | −1.00000 | −2.49764 | 4.70491 | 2.91073 | 4.26682 | 0.926526 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.154862 | −1.20542 | −1.97602 | −1.00000 | 0.186674 | −4.71824 | 0.615734 | −1.54696 | 0.154862 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.71889 | −2.56128 | 0.954575 | −1.00000 | −4.40255 | 3.55712 | −1.79697 | 3.56016 | −1.71889 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.14663 | 1.76662 | 2.60803 | −1.00000 | 3.79229 | 2.91387 | 1.30522 | 0.120963 | −2.14663 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 935.2.a.h | ✓ | 6 |
| 3.b | odd | 2 | 1 | 8415.2.a.bj | 6 | ||
| 5.b | even | 2 | 1 | 4675.2.a.bh | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 935.2.a.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 4675.2.a.bh | 6 | 5.b | even | 2 | 1 | ||
| 8415.2.a.bj | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(935))\):
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\( T_{2}^{6} + 2T_{2}^{5} - 8T_{2}^{4} - 14T_{2}^{3} + 15T_{2}^{2} + 22T_{2} + 3 \)
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\( T_{7}^{6} - 6T_{7}^{5} - 30T_{7}^{4} + 236T_{7}^{3} + 16T_{7}^{2} - 2280T_{7} + 3480 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots + 3 \)
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| $3$ |
\( T^{6} - 4 T^{5} + \cdots - 5 \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} - 6 T^{5} + \cdots + 3480 \)
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| $11$ |
\( (T + 1)^{6} \)
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| $13$ |
\( T^{6} - 10 T^{5} + \cdots - 13 \)
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| $17$ |
\( (T - 1)^{6} \)
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| $19$ |
\( T^{6} + 2 T^{5} + \cdots + 56 \)
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| $23$ |
\( T^{6} - 14 T^{5} + \cdots + 24951 \)
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| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 13353 \)
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| $31$ |
\( T^{6} - 4 T^{5} + \cdots - 232 \)
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| $37$ |
\( T^{6} - 18 T^{5} + \cdots + 23627 \)
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| $41$ |
\( T^{6} - 12 T^{5} + \cdots + 33009 \)
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| $43$ |
\( T^{6} + 10 T^{5} + \cdots - 1469 \)
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| $47$ |
\( T^{6} - 20 T^{5} + \cdots + 604584 \)
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| $53$ |
\( T^{6} - 10 T^{5} + \cdots - 1752 \)
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| $59$ |
\( T^{6} - 14 T^{5} + \cdots + 10377 \)
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| $61$ |
\( T^{6} - 2 T^{5} + \cdots - 175 \)
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| $67$ |
\( T^{6} + 34 T^{5} + \cdots - 11352 \)
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| $71$ |
\( T^{6} + 18 T^{5} + \cdots - 169704 \)
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| $73$ |
\( T^{6} - 10 T^{5} + \cdots + 1066296 \)
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| $79$ |
\( T^{6} - 8 T^{5} + \cdots - 6615 \)
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| $83$ |
\( T^{6} + 30 T^{5} + \cdots - 13581 \)
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| $89$ |
\( T^{6} + 30 T^{5} + \cdots - 5571 \)
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| $97$ |
\( T^{6} - 58 T^{5} + \cdots - 228345 \)
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