Newspace parameters
| Level: | \( N \) | \(=\) | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5415.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(43.2389926945\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1397493.1 |
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 285) |
| 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
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 2\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18 \)
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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 |
|
−1.68091 | −1.00000 | 0.825466 | 1.00000 | 1.68091 | −2.40254 | 1.97429 | 1.00000 | −1.68091 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.05432 | −1.00000 | −0.888399 | 1.00000 | 1.05432 | 1.62517 | 3.04531 | 1.00000 | −1.05432 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.415466 | −1.00000 | −1.82739 | 1.00000 | −0.415466 | −4.56248 | −1.59015 | 1.00000 | 0.415466 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.801527 | −1.00000 | −1.35755 | 1.00000 | −0.801527 | −0.782248 | −2.69117 | 1.00000 | 0.801527 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.11662 | −1.00000 | 2.48009 | 1.00000 | −2.11662 | 0.335802 | 1.01617 | 1.00000 | 2.11662 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.40162 | −1.00000 | 3.76778 | 1.00000 | −2.40162 | −0.213698 | 4.24555 | 1.00000 | 2.40162 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 5415.2.a.bh | 6 | |
| 19.b | odd | 2 | 1 | 5415.2.a.bb | 6 | ||
| 19.f | odd | 18 | 2 | 285.2.u.a | ✓ | 12 | |
| 57.j | even | 18 | 2 | 855.2.bs.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.2.u.a | ✓ | 12 | 19.f | odd | 18 | 2 | |
| 855.2.bs.a | 12 | 57.j | even | 18 | 2 | ||
| 5415.2.a.bb | 6 | 19.b | odd | 2 | 1 | ||
| 5415.2.a.bh | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
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\( T_{2}^{6} - 3T_{2}^{5} - 3T_{2}^{4} + 12T_{2}^{3} - 9T_{2} + 3 \)
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\( T_{7}^{6} + 6T_{7}^{5} + 3T_{7}^{4} - 19T_{7}^{3} - 12T_{7}^{2} + 3T_{7} + 1 \)
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\( T_{11}^{6} + 12T_{11}^{5} + 51T_{11}^{4} + 87T_{11}^{3} + 36T_{11}^{2} - 27T_{11} + 3 \)
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\( T_{13}^{6} + 3T_{13}^{5} - 15T_{13}^{4} - 44T_{13}^{3} - 3T_{13}^{2} + 6T_{13} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 3 \)
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| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 1 \)
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| $11$ |
\( T^{6} + 12 T^{5} + \cdots + 3 \)
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| $13$ |
\( T^{6} + 3 T^{5} + \cdots + 1 \)
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| $17$ |
\( (T^{3} + 9 T^{2} + 18 T + 9)^{2} \)
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| $19$ |
\( T^{6} \)
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| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 267 \)
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| $29$ |
\( T^{6} - 9 T^{5} + \cdots + 999 \)
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| $31$ |
\( T^{6} - 54 T^{4} + \cdots - 683 \)
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| $37$ |
\( T^{6} - 6 T^{5} + \cdots + 5833 \)
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| $41$ |
\( T^{6} + 6 T^{5} + \cdots + 4647 \)
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| $43$ |
\( T^{6} - 132 T^{4} + \cdots + 757 \)
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| $47$ |
\( T^{6} + 9 T^{5} + \cdots + 2109 \)
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| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 4539 \)
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| $59$ |
\( T^{6} - 12 T^{5} + \cdots - 54321 \)
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| $61$ |
\( T^{6} + 33 T^{5} + \cdots + 17317 \)
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| $67$ |
\( T^{6} - 12 T^{5} + \cdots - 25721 \)
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| $71$ |
\( T^{6} - 6 T^{5} + \cdots - 6471 \)
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| $73$ |
\( T^{6} + 21 T^{5} + \cdots - 7361 \)
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| $79$ |
\( T^{6} + 6 T^{5} + \cdots - 97829 \)
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| $83$ |
\( T^{6} - 3 T^{5} + \cdots + 52599 \)
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| $89$ |
\( T^{6} + 6 T^{5} + \cdots - 44277 \)
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| $97$ |
\( T^{6} + 12 T^{5} + \cdots - 214073 \)
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