Newspace parameters
| Level: | \( N \) | \(=\) | \( 1035 = 3^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1035.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(61.0669768559\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} - x^{5} - 46x^{4} + 34x^{3} + 548x^{2} - 396x - 1152 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 345) |
| 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
\( x^{6} - x^{5} - 46x^{4} + 34x^{3} + 548x^{2} - 396x - 1152 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 16 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 34\nu^{3} + 10\nu^{2} + 212\nu + 24 ) / 12 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 22\nu + 28 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 28\nu^{2} + 56\nu + 94 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 16 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + 22\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 3\beta_{4} + 34\beta_{2} + 10\beta _1 + 366 \)
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| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 37\beta_{4} + 12\beta_{3} + 92\beta_{2} + 546\beta _1 + 318 \)
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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 |
|
−4.82618 | 0 | 15.2920 | 5.00000 | 0 | 33.3379 | −35.1925 | 0 | −24.1309 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.35475 | 0 | 10.9638 | 5.00000 | 0 | 1.54990 | −12.9067 | 0 | −21.7737 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.22853 | 0 | −6.49070 | 5.00000 | 0 | −5.36564 | 17.8023 | 0 | −6.14267 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.23364 | 0 | −3.01086 | 5.00000 | 0 | −28.6247 | −24.5943 | 0 | 11.1682 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.55160 | 0 | 4.61383 | 5.00000 | 0 | 21.0707 | −12.0263 | 0 | 17.7580 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.62423 | 0 | 23.6319 | 5.00000 | 0 | 32.0319 | 87.9174 | 0 | 28.1211 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 1035.4.a.p | 6 | |
| 3.b | odd | 2 | 1 | 345.4.a.j | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 1725.4.a.w | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 345.4.a.j | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 1035.4.a.p | 6 | 1.a | even | 1 | 1 | trivial | |
| 1725.4.a.w | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1035))\):
|
\( T_{2}^{6} - T_{2}^{5} - 46T_{2}^{4} + 34T_{2}^{3} + 548T_{2}^{2} - 396T_{2} - 1152 \)
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\( T_{7}^{6} - 54T_{7}^{5} - 258T_{7}^{4} + 47864T_{7}^{3} - 462615T_{7}^{2} - 2852618T_{7} + 5356304 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + \cdots - 1152 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( (T - 5)^{6} \)
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| $7$ |
\( T^{6} - 54 T^{5} + \cdots + 5356304 \)
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| $11$ |
\( T^{6} + 18 T^{5} + \cdots + 629703360 \)
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| $13$ |
\( T^{6} + \cdots + 6937184976 \)
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| $17$ |
\( T^{6} + \cdots - 34982581176 \)
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| $19$ |
\( T^{6} + \cdots + 187040357088 \)
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| $23$ |
\( (T + 23)^{6} \)
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| $29$ |
\( T^{6} + \cdots - 670161243660 \)
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| $31$ |
\( T^{6} + \cdots - 38815441248 \)
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| $37$ |
\( T^{6} + \cdots + 50975161849664 \)
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| $41$ |
\( T^{6} + \cdots - 2961861326820 \)
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| $43$ |
\( T^{6} + \cdots + 431387297792 \)
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| $47$ |
\( T^{6} + \cdots - 193558155475968 \)
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| $53$ |
\( T^{6} + \cdots - 34004526740460 \)
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| $59$ |
\( T^{6} + \cdots - 269708171145192 \)
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| $61$ |
\( T^{6} + \cdots - 642933076960096 \)
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| $67$ |
\( T^{6} + \cdots - 33\!\cdots\!40 \)
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| $71$ |
\( T^{6} + \cdots - 181749424122240 \)
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| $73$ |
\( T^{6} + \cdots - 5506599140016 \)
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| $79$ |
\( T^{6} + \cdots - 66\!\cdots\!84 \)
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| $83$ |
\( T^{6} + \cdots + 14\!\cdots\!80 \)
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| $89$ |
\( T^{6} + \cdots - 28\!\cdots\!36 \)
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| $97$ |
\( T^{6} + \cdots + 21\!\cdots\!12 \)
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