Newspace parameters
| Level: | \( N \) | \(=\) | \( 2366 = 2 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2366.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.8926051182\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6052921.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 11x^{4} + 7x^{3} + 33x^{2} - 9x - 27 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - x^{5} - 11x^{4} + 7x^{3} + 33x^{2} - 9x - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 4\nu + 12 ) / 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - 8\nu^{3} + 31\nu^{2} + 12\nu - 36 ) / 9 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{5} - 4\nu^{4} - 35\nu^{3} + 19\nu^{2} + 60\nu ) / 9 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} - 7\nu^{4} - 32\nu^{3} + 52\nu^{2} + 39\nu - 72 ) / 9 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} - 3\beta_{2} + 5\beta _1 + 4 \)
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| \(\nu^{4}\) | \(=\) |
\( 9\beta_{5} - 8\beta_{4} - 4\beta_{3} + 8\beta_{2} + 9\beta _1 + 24 \)
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| \(\nu^{5}\) | \(=\) |
\( 13\beta_{5} - \beta_{4} - 39\beta_{3} - 23\beta_{2} + 33\beta _1 + 40 \)
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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 | −2.46434 | 1.00000 | −3.19361 | 2.46434 | −1.00000 | −1.00000 | 3.07299 | 3.19361 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.52377 | 1.00000 | 1.45506 | 1.52377 | −1.00000 | −1.00000 | −0.678139 | −1.45506 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.05140 | 1.00000 | −2.26985 | 1.05140 | −1.00000 | −1.00000 | −1.89456 | 2.26985 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.21736 | 1.00000 | 3.44059 | −1.21736 | −1.00000 | −1.00000 | −1.51803 | −3.44059 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.96881 | 1.00000 | −2.90010 | −1.96881 | −1.00000 | −1.00000 | 0.876202 | 2.90010 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.85334 | 1.00000 | −0.532083 | −2.85334 | −1.00000 | −1.00000 | 5.14154 | 0.532083 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 2366.2.a.be | ✓ | 6 |
| 13.b | even | 2 | 1 | 2366.2.a.bg | yes | 6 | |
| 13.d | odd | 4 | 2 | 2366.2.d.q | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2366.2.a.be | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2366.2.a.bg | yes | 6 | 13.b | even | 2 | 1 | |
| 2366.2.d.q | 12 | 13.d | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):
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\( T_{3}^{6} - T_{3}^{5} - 11T_{3}^{4} + 7T_{3}^{3} + 33T_{3}^{2} - 9T_{3} - 27 \)
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\( T_{5}^{6} + 4T_{5}^{5} - 11T_{5}^{4} - 57T_{5}^{3} - 14T_{5}^{2} + 112T_{5} + 56 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( T^{6} - T^{5} + \cdots - 27 \)
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| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 56 \)
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| $7$ |
\( (T + 1)^{6} \)
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| $11$ |
\( T^{6} + 6 T^{5} + \cdots + 97 \)
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| $13$ |
\( T^{6} \)
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| $17$ |
\( T^{6} + 9 T^{5} + \cdots + 7 \)
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| $19$ |
\( T^{6} + 10 T^{5} + \cdots + 5011 \)
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| $23$ |
\( T^{6} - 21 T^{5} + \cdots + 14776 \)
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| $29$ |
\( T^{6} + T^{5} + \cdots - 56 \)
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| $31$ |
\( T^{6} + 20 T^{5} + \cdots - 1576 \)
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| $37$ |
\( T^{6} + 16 T^{5} + \cdots + 16072 \)
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| $41$ |
\( T^{6} + 2 T^{5} + \cdots - 43211 \)
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| $43$ |
\( T^{6} - 2 T^{5} + \cdots - 455461 \)
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| $47$ |
\( T^{6} - 5 T^{5} + \cdots - 23192 \)
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| $53$ |
\( T^{6} - 28 T^{5} + \cdots - 7288 \)
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| $59$ |
\( T^{6} - 12 T^{5} + \cdots + 4493 \)
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| $61$ |
\( T^{6} + 27 T^{5} + \cdots + 4936 \)
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| $67$ |
\( T^{6} + 16 T^{5} + \cdots + 3977 \)
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| $71$ |
\( T^{6} - 257 T^{4} + \cdots - 30584 \)
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| $73$ |
\( T^{6} + 38 T^{5} + \cdots + 207523 \)
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| $79$ |
\( T^{6} - 6 T^{5} + \cdots + 1576 \)
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| $83$ |
\( T^{6} - 6 T^{5} + \cdots - 31549 \)
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| $89$ |
\( T^{6} - T^{5} + \cdots - 923 \)
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| $97$ |
\( T^{6} + 16 T^{5} + \cdots + 30079 \)
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