Newspace parameters
| Level: | \( N \) | \(=\) | \( 5046 = 2 \cdot 3 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5046.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(40.2925128599\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.21246449.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 16x^{4} + 21x^{3} + 80x^{2} - 43x - 97 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 174) |
| 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
\( x^{6} - 2x^{5} - 16x^{4} + 21x^{3} + 80x^{2} - 43x - 97 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{3} - 11\nu^{2} + 9\nu + 38 ) / 14 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} - 9\nu^{3} + 66\nu^{2} + 30\nu - 81 ) / 28 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} - 7\nu^{4} - 27\nu^{3} + 72\nu^{2} + 20\nu - 173 ) / 28 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 9\nu^{2} + 11\nu + 15 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 6 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - 3\beta_{3} + 8\beta _1 + 5 \)
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| \(\nu^{4}\) | \(=\) |
\( 11\beta_{4} - 15\beta_{3} - 9\beta_{2} + 14\beta _1 + 49 \)
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| \(\nu^{5}\) | \(=\) |
\( -9\beta_{5} + 20\beta_{4} - 38\beta_{3} + 3\beta_{2} + 74\beta _1 + 73 \)
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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 |
|
−1.00000 | 1.00000 | 1.00000 | −2.63351 | −1.00000 | 0.205383 | −1.00000 | 1.00000 | 2.63351 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | −1.39213 | −1.00000 | −2.65367 | −1.00000 | 1.00000 | 1.39213 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | −1.36084 | −1.00000 | −3.76841 | −1.00000 | 1.00000 | 1.36084 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | 1.14515 | −1.00000 | 4.45561 | −1.00000 | 1.00000 | −1.14515 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.00000 | 1.00000 | 4.07856 | −1.00000 | −1.45236 | −1.00000 | 1.00000 | −4.07856 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.00000 | 1.00000 | 4.16278 | −1.00000 | 4.21345 | −1.00000 | 1.00000 | −4.16278 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 5046.2.a.bo | 6 | |
| 29.b | even | 2 | 1 | 5046.2.a.bq | 6 | ||
| 29.d | even | 7 | 2 | 174.2.g.c | ✓ | 12 | |
| 87.j | odd | 14 | 2 | 522.2.k.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 174.2.g.c | ✓ | 12 | 29.d | even | 7 | 2 | |
| 522.2.k.g | 12 | 87.j | odd | 14 | 2 | ||
| 5046.2.a.bo | 6 | 1.a | even | 1 | 1 | trivial | |
| 5046.2.a.bq | 6 | 29.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5046))\):
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\( T_{5}^{6} - 4T_{5}^{5} - 15T_{5}^{4} + 42T_{5}^{3} + 90T_{5}^{2} - 46T_{5} - 97 \)
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\( T_{7}^{6} - T_{7}^{5} - 30T_{7}^{4} + T_{7}^{3} + 238T_{7}^{2} + 224T_{7} - 56 \)
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\( T_{11}^{6} + T_{11}^{5} - 32T_{11}^{4} - 49T_{11}^{3} + 236T_{11}^{2} + 492T_{11} + 232 \)
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\( T_{17}^{6} - 3T_{17}^{5} - 50T_{17}^{4} + 133T_{17}^{3} - 8T_{17}^{2} - 52T_{17} + 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( (T - 1)^{6} \)
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| $5$ |
\( T^{6} - 4 T^{5} + \cdots - 97 \)
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| $7$ |
\( T^{6} - T^{5} + \cdots - 56 \)
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| $11$ |
\( T^{6} + T^{5} + \cdots + 232 \)
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| $13$ |
\( T^{6} + T^{5} + \cdots - 1000 \)
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| $17$ |
\( T^{6} - 3 T^{5} + \cdots + 8 \)
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| $19$ |
\( T^{6} + 12 T^{5} + \cdots - 64 \)
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| $23$ |
\( (T^{3} - 28 T + 56)^{2} \)
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| $29$ |
\( T^{6} \)
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| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 1784 \)
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| $37$ |
\( T^{6} + 7 T^{5} + \cdots + 3464 \)
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| $41$ |
\( T^{6} - 19 T^{5} + \cdots - 13208 \)
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| $43$ |
\( T^{6} + 8 T^{5} + \cdots - 448 \)
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| $47$ |
\( T^{6} - 16 T^{5} + \cdots - 64 \)
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| $53$ |
\( T^{6} - 20 T^{5} + \cdots - 23183 \)
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| $59$ |
\( T^{6} - 5 T^{5} + \cdots + 17576 \)
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| $61$ |
\( T^{6} - 7 T^{5} + \cdots - 1352 \)
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| $67$ |
\( (T^{3} - 2 T^{2} + \cdots + 232)^{2} \)
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| $71$ |
\( T^{6} - 26 T^{5} + \cdots + 118784 \)
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| $73$ |
\( T^{6} + 16 T^{5} + \cdots + 74467 \)
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| $79$ |
\( T^{6} + 2 T^{5} + \cdots - 3584 \)
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| $83$ |
\( T^{6} - 21 T^{5} + \cdots + 120344 \)
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| $89$ |
\( T^{6} - 17 T^{5} + \cdots - 50624 \)
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| $97$ |
\( T^{6} + 34 T^{5} + \cdots + 94081 \)
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