Newspace parameters
| Level: | \( N \) | \(=\) | \( 2366 = 2 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2366.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(18.8926051182\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.153664.1 |
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|
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| Defining polynomial: |
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 3\nu^{2} + 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 4\nu^{3} + 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 5 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} + 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).
| \(n\) | \(339\) | \(2199\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 1.00000i | −2.80194 | −1.00000 | − | 0.554958i | 2.80194i | 1.00000i | 1.00000i | 4.85086 | −0.554958 | ||||||||||||||||||||||||||||||||||
| 337.2 | − | 1.00000i | −1.44504 | −1.00000 | − | 2.24698i | 1.44504i | 1.00000i | 1.00000i | −0.911854 | −2.24698 | |||||||||||||||||||||||||||||||||||
| 337.3 | − | 1.00000i | 0.246980 | −1.00000 | 0.801938i | − | 0.246980i | 1.00000i | 1.00000i | −2.93900 | 0.801938 | |||||||||||||||||||||||||||||||||||
| 337.4 | 1.00000i | −2.80194 | −1.00000 | 0.554958i | − | 2.80194i | − | 1.00000i | − | 1.00000i | 4.85086 | −0.554958 | ||||||||||||||||||||||||||||||||||
| 337.5 | 1.00000i | −1.44504 | −1.00000 | 2.24698i | − | 1.44504i | − | 1.00000i | − | 1.00000i | −0.911854 | −2.24698 | ||||||||||||||||||||||||||||||||||
| 337.6 | 1.00000i | 0.246980 | −1.00000 | − | 0.801938i | 0.246980i | − | 1.00000i | − | 1.00000i | −2.93900 | 0.801938 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2366.2.d.n | 6 | |
| 13.b | even | 2 | 1 | inner | 2366.2.d.n | 6 | |
| 13.d | odd | 4 | 1 | 2366.2.a.v | ✓ | 3 | |
| 13.d | odd | 4 | 1 | 2366.2.a.ba | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2366.2.a.v | ✓ | 3 | 13.d | odd | 4 | 1 | |
| 2366.2.a.ba | yes | 3 | 13.d | odd | 4 | 1 | |
| 2366.2.d.n | 6 | 1.a | even | 1 | 1 | trivial | |
| 2366.2.d.n | 6 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2366, [\chi])\):
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\( T_{3}^{3} + 4T_{3}^{2} + 3T_{3} - 1 \)
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\( T_{5}^{6} + 6T_{5}^{4} + 5T_{5}^{2} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
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| $3$ |
\( (T^{3} + 4 T^{2} + 3 T - 1)^{2} \)
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| $5$ |
\( T^{6} + 6 T^{4} + \cdots + 1 \)
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| $7$ |
\( (T^{2} + 1)^{3} \)
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| $11$ |
\( T^{6} + 45 T^{4} + \cdots + 1681 \)
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| $13$ |
\( T^{6} \)
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| $17$ |
\( (T^{3} - 2 T^{2} - 57 T + 71)^{2} \)
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| $19$ |
\( T^{6} + 61 T^{4} + \cdots + 5041 \)
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| $23$ |
\( (T^{3} - 7 T^{2} + 49)^{2} \)
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| $29$ |
\( (T^{3} + 3 T^{2} - 46 T + 1)^{2} \)
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| $31$ |
\( T^{6} + 62 T^{4} + \cdots + 5041 \)
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| $37$ |
\( T^{6} + 42 T^{4} + \cdots + 49 \)
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| $41$ |
\( T^{6} + 245 T^{4} + \cdots + 117649 \)
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| $43$ |
\( (T^{3} - 17 T^{2} + \cdots - 71)^{2} \)
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| $47$ |
\( T^{6} + 153 T^{4} + \cdots + 841 \)
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| $53$ |
\( (T^{3} - 49 T + 91)^{2} \)
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| $59$ |
\( T^{6} + 101 T^{4} + \cdots + 28561 \)
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| $61$ |
\( (T^{3} - 3 T^{2} - 46 T - 1)^{2} \)
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| $67$ |
\( T^{6} + 189 T^{4} + \cdots + 41209 \)
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| $71$ |
\( T^{6} + 314 T^{4} + \cdots + 707281 \)
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| $73$ |
\( T^{6} + 33 T^{4} + \cdots + 841 \)
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| $79$ |
\( (T^{3} - 10 T^{2} + \cdots + 125)^{2} \)
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| $83$ |
\( T^{6} + 201 T^{4} + \cdots + 32761 \)
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| $89$ |
\( T^{6} + 234 T^{4} + \cdots + 57121 \)
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| $97$ |
\( T^{6} + 321 T^{4} + \cdots + 212521 \)
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