Newspace parameters
| Level: | \( N \) | \(=\) | \( 9135 = 3^{2} \cdot 5 \cdot 7 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9135.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.9433422464\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 3x^{6} - 6x^{5} + 21x^{4} + 3x^{3} - 31x^{2} + 14x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1015) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 6x^{5} + 21x^{4} + 3x^{3} - 31x^{2} + 14x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 10\nu - 5 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 13\nu^{3} + 10\nu^{2} - 17\nu + 2 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 14\nu^{3} + 9\nu^{2} - 22\nu + 5 \)
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| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 7\nu^{4} - 21\nu^{3} - 10\nu^{2} + 30\nu - 6 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{2} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + 6\beta_{2} + 2\beta _1 + 14 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 8\beta_{5} - 7\beta_{4} + 8\beta_{2} + 27\beta _1 + 4 \)
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| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} + 10\beta_{5} - 7\beta_{3} + 35\beta_{2} + 20\beta _1 + 74 \)
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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 |
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−2.54283 | 0 | 4.46600 | 1.00000 | 0 | −1.00000 | −6.27063 | 0 | −2.54283 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.38681 | 0 | 3.69684 | 1.00000 | 0 | −1.00000 | −4.05003 | 0 | −2.38681 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.23413 | 0 | −0.476927 | 1.00000 | 0 | −1.00000 | 3.05685 | 0 | −1.23413 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.454212 | 0 | −1.79369 | 1.00000 | 0 | −1.00000 | 1.72314 | 0 | −0.454212 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.0885540 | 0 | −1.99216 | 1.00000 | 0 | −1.00000 | 0.353521 | 0 | −0.0885540 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.51361 | 0 | 0.291015 | 1.00000 | 0 | −1.00000 | −2.58674 | 0 | 1.51361 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.19292 | 0 | 2.80892 | 1.00000 | 0 | −1.00000 | 1.77389 | 0 | 2.19292 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 9135.2.a.be | 7 | |
| 3.b | odd | 2 | 1 | 1015.2.a.k | ✓ | 7 | |
| 15.d | odd | 2 | 1 | 5075.2.a.y | 7 | ||
| 21.c | even | 2 | 1 | 7105.2.a.s | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1015.2.a.k | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 5075.2.a.y | 7 | 15.d | odd | 2 | 1 | ||
| 7105.2.a.s | 7 | 21.c | even | 2 | 1 | ||
| 9135.2.a.be | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9135))\):
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\( T_{2}^{7} + 3T_{2}^{6} - 6T_{2}^{5} - 21T_{2}^{4} + 3T_{2}^{3} + 31T_{2}^{2} + 14T_{2} + 1 \)
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\( T_{11}^{7} + 7T_{11}^{6} - 8T_{11}^{5} - 97T_{11}^{4} - 16T_{11}^{3} + 248T_{11}^{2} + 176T_{11} + 32 \)
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\( T_{13}^{7} - 5T_{13}^{6} - 17T_{13}^{5} + 93T_{13}^{4} + 48T_{13}^{3} - 288T_{13}^{2} - 208T_{13} + 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 3 T^{6} + \cdots + 1 \)
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| $3$ |
\( T^{7} \)
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| $5$ |
\( (T - 1)^{7} \)
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| $7$ |
\( (T + 1)^{7} \)
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| $11$ |
\( T^{7} + 7 T^{6} + \cdots + 32 \)
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| $13$ |
\( T^{7} - 5 T^{6} + \cdots + 32 \)
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| $17$ |
\( T^{7} + 14 T^{6} + \cdots + 256 \)
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| $19$ |
\( T^{7} + 9 T^{6} + \cdots + 2012 \)
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| $23$ |
\( T^{7} + 12 T^{6} + \cdots - 2752 \)
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| $29$ |
\( (T + 1)^{7} \)
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| $31$ |
\( T^{7} + 7 T^{6} + \cdots + 16556 \)
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| $37$ |
\( T^{7} - 23 T^{6} + \cdots - 18244 \)
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| $41$ |
\( T^{7} + 13 T^{6} + \cdots - 81556 \)
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| $43$ |
\( T^{7} - 31 T^{6} + \cdots + 69404 \)
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| $47$ |
\( T^{7} - 6 T^{6} + \cdots + 82096 \)
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| $53$ |
\( T^{7} + 23 T^{6} + \cdots + 283600 \)
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| $59$ |
\( T^{7} + 4 T^{6} + \cdots + 634112 \)
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| $61$ |
\( T^{7} + 4 T^{6} + \cdots - 32 \)
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| $67$ |
\( T^{7} - 13 T^{6} + \cdots - 137600 \)
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| $71$ |
\( T^{7} - 11 T^{6} + \cdots - 78512 \)
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| $73$ |
\( T^{7} - 8 T^{6} + \cdots - 3409216 \)
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| $79$ |
\( T^{7} - 97 T^{5} + \cdots + 39968 \)
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| $83$ |
\( T^{7} - 6 T^{6} + \cdots + 6935600 \)
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| $89$ |
\( T^{7} - 5 T^{6} + \cdots - 354628 \)
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| $97$ |
\( T^{7} - 4 T^{6} + \cdots - 5888 \)
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