Newspace parameters
| Level: | \( N \) | \(=\) | \( 2366 = 2 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2366.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(139.598519074\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 95x^{3} + 122x^{2} + 2074x - 2668 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 182) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 95x^{3} + 122x^{2} + 2074x - 2668 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 37 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 52\nu + 61 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 3\nu^{3} - 62\nu^{2} - 129\nu + 511 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 37 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{3} + 3\beta_{2} + 52\beta _1 - 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} - 27\beta_{3} + 177\beta_{2} - 27\beta _1 + 1855 \)
|
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 |
|
−2.00000 | −8.06998 | 4.00000 | −9.92174 | 16.1400 | 7.00000 | −8.00000 | 38.1246 | 19.8435 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −5.70204 | 4.00000 | 9.31361 | 11.4041 | 7.00000 | −8.00000 | 5.51321 | −18.6272 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 1.28617 | 4.00000 | −10.0966 | −2.57235 | 7.00000 | −8.00000 | −25.3458 | 20.1932 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 6.12082 | 4.00000 | 0.00519876 | −12.2416 | 7.00000 | −8.00000 | 10.4644 | −0.0103975 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 7.36502 | 4.00000 | 16.6995 | −14.7300 | 7.00000 | −8.00000 | 27.2435 | −33.3991 | ||||||||||||||||||||||||||||||||||
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 | 2366.4.a.s | 5 | |
| 13.b | even | 2 | 1 | 2366.4.a.t | 5 | ||
| 13.c | even | 3 | 2 | 182.4.g.b | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.4.g.b | ✓ | 10 | 13.c | even | 3 | 2 | |
| 2366.4.a.s | 5 | 1.a | even | 1 | 1 | trivial | |
| 2366.4.a.t | 5 | 13.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2366))\):
|
\( T_{3}^{5} - T_{3}^{4} - 95T_{3}^{3} + 122T_{3}^{2} + 2074T_{3} - 2668 \)
|
|
\( T_{5}^{5} - 6T_{5}^{4} - 265T_{5}^{3} + 509T_{5}^{2} + 15578T_{5} - 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( T^{5} - T^{4} + \cdots - 2668 \)
|
| $5$ |
\( T^{5} - 6 T^{4} + \cdots - 81 \)
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| $7$ |
\( (T - 7)^{5} \)
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| $11$ |
\( T^{5} + 9 T^{4} + \cdots - 17878068 \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} + \cdots + 2516871123 \)
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| $19$ |
\( T^{5} + 49 T^{4} + \cdots + 637349312 \)
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| $23$ |
\( T^{5} + \cdots + 17531905104 \)
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| $29$ |
\( T^{5} + 196 T^{4} + \cdots + 146086551 \)
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| $31$ |
\( T^{5} + \cdots + 9196721532 \)
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| $37$ |
\( T^{5} + \cdots + 4856825008 \)
|
| $41$ |
\( T^{5} + \cdots + 27700535772 \)
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| $43$ |
\( T^{5} + \cdots - 79438805108 \)
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| $47$ |
\( T^{5} + \cdots - 510478904208 \)
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| $53$ |
\( T^{5} + \cdots + 7326385988883 \)
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| $59$ |
\( T^{5} + \cdots + 55437062123244 \)
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| $61$ |
\( T^{5} + \cdots + 77742611388 \)
|
| $67$ |
\( T^{5} + \cdots + 732135144212 \)
|
| $71$ |
\( T^{5} + \cdots - 57137783691456 \)
|
| $73$ |
\( T^{5} + \cdots + 1638605790124 \)
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| $79$ |
\( T^{5} + \cdots - 62909566918592 \)
|
| $83$ |
\( T^{5} + \cdots + 12429069998436 \)
|
| $89$ |
\( T^{5} + \cdots + 325640751383016 \)
|
| $97$ |
\( T^{5} + \cdots + 10\!\cdots\!24 \)
|
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