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: | \(4\) |
| Coefficient field: | 4.4.8805372.1 |
|
|
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| Defining polynomial: |
\( x^{4} - x^{3} - 33x^{2} - 12x + 162 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 33x^{2} - 12x + 162 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 16 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 4\nu^{2} - 18\nu + 36 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 16 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 4\beta_{2} + 26\beta _1 + 28 \)
|
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 | −5.95915 | 4.00000 | 20.4534 | 11.9183 | 7.00000 | −8.00000 | 8.51152 | −40.9067 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −5.20283 | 4.00000 | −17.5606 | 10.4057 | 7.00000 | −8.00000 | 0.0694513 | 35.1211 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0.111479 | 4.00000 | 7.31483 | −0.222959 | 7.00000 | −8.00000 | −26.9876 | −14.6297 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 4.05051 | 4.00000 | −9.20761 | −8.10101 | 7.00000 | −8.00000 | −10.5934 | 18.4152 | |||||||||||||||||||||||||||||||
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.q | 4 | |
| 13.b | even | 2 | 1 | 2366.4.a.r | 4 | ||
| 13.e | even | 6 | 2 | 182.4.g.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.4.g.a | ✓ | 8 | 13.e | even | 6 | 2 | |
| 2366.4.a.q | 4 | 1.a | even | 1 | 1 | trivial | |
| 2366.4.a.r | 4 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2366))\):
|
\( T_{3}^{4} + 7T_{3}^{3} - 15T_{3}^{2} - 124T_{3} + 14 \)
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\( T_{5}^{4} - T_{5}^{3} - 432T_{5}^{2} - 485T_{5} + 24191 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
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| $3$ |
\( T^{4} + 7 T^{3} + \cdots + 14 \)
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| $5$ |
\( T^{4} - T^{3} + \cdots + 24191 \)
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| $7$ |
\( (T - 7)^{4} \)
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| $11$ |
\( T^{4} - 15 T^{3} + \cdots + 13446 \)
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| $13$ |
\( T^{4} \)
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| $17$ |
\( T^{4} + 24 T^{3} + \cdots - 7187129 \)
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| $19$ |
\( T^{4} + 21 T^{3} + \cdots + 28469644 \)
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| $23$ |
\( T^{4} + 182 T^{3} + \cdots - 4156776 \)
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| $29$ |
\( T^{4} - 133 T^{3} + \cdots + 223699827 \)
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| $31$ |
\( T^{4} - 282 T^{3} + \cdots + 179629026 \)
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| $37$ |
\( T^{4} + 730 T^{3} + \cdots + 28084048 \)
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| $41$ |
\( T^{4} + \cdots - 1871392564 \)
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| $43$ |
\( T^{4} + \cdots + 3840138794 \)
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| $47$ |
\( T^{4} - 560 T^{3} + \cdots - 133698084 \)
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| $53$ |
\( T^{4} + 106 T^{3} + \cdots - 651040857 \)
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| $59$ |
\( T^{4} - 782 T^{3} + \cdots - 404652222 \)
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| $61$ |
\( T^{4} + \cdots - 28026376196 \)
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| $67$ |
\( T^{4} + \cdots + 3346141102 \)
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| $71$ |
\( T^{4} - 82 T^{3} + \cdots + 668493488 \)
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| $73$ |
\( T^{4} + \cdots + 14950934748 \)
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| $79$ |
\( T^{4} + \cdots - 123778925184 \)
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| $83$ |
\( T^{4} + \cdots + 784513099818 \)
|
| $89$ |
\( T^{4} + \cdots - 76052207484 \)
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| $97$ |
\( T^{4} + \cdots - 276884336612 \)
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