Newspace parameters
| Level: | \( N \) | \(=\) | \( 5766 = 2 \cdot 3 \cdot 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5766.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(46.0417418055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{15})^+\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} + 4x + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 186) |
| 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 \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
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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 | −1.00000 | 1.00000 | −3.29456 | 1.00000 | 0.661739 | −1.00000 | 1.00000 | 3.29456 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −2.03615 | 1.00000 | 0.172909 | −1.00000 | 1.00000 | 2.03615 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | 2.03615 | 1.00000 | 2.20906 | −1.00000 | 1.00000 | −2.03615 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | 3.29456 | 1.00000 | 3.95630 | −1.00000 | 1.00000 | −3.29456 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(31\) | \( -1 \) |
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 | 5766.2.a.x | 4 | |
| 31.b | odd | 2 | 1 | 5766.2.a.ba | 4 | ||
| 31.h | odd | 30 | 2 | 186.2.m.d | ✓ | 8 | |
| 93.p | even | 30 | 2 | 558.2.ba.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 186.2.m.d | ✓ | 8 | 31.h | odd | 30 | 2 | |
| 558.2.ba.b | 8 | 93.p | even | 30 | 2 | ||
| 5766.2.a.x | 4 | 1.a | even | 1 | 1 | trivial | |
| 5766.2.a.ba | 4 | 31.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5766))\):
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\( T_{5}^{4} - 15T_{5}^{2} + 45 \)
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\( T_{7}^{4} - 7T_{7}^{3} + 14T_{7}^{2} - 8T_{7} + 1 \)
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\( T_{11}^{4} + 9T_{11}^{3} + 6T_{11}^{2} - 126T_{11} - 279 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
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| $3$ |
\( (T + 1)^{4} \)
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| $5$ |
\( T^{4} - 15T^{2} + 45 \)
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| $7$ |
\( T^{4} - 7 T^{3} + \cdots + 1 \)
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| $11$ |
\( T^{4} + 9 T^{3} + \cdots - 279 \)
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| $13$ |
\( (T^{2} + 2 T - 4)^{2} \)
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| $17$ |
\( T^{4} + 6 T^{3} + \cdots - 144 \)
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| $19$ |
\( (T^{2} - 2 T - 4)^{2} \)
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| $23$ |
\( (T + 6)^{4} \)
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| $29$ |
\( T^{4} + 15 T^{3} + \cdots + 45 \)
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| $31$ |
\( T^{4} \)
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| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 16 \)
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| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 4176 \)
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| $43$ |
\( T^{4} + 4 T^{3} + \cdots - 464 \)
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| $47$ |
\( T^{4} - 120 T^{2} + \cdots + 720 \)
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| $53$ |
\( T^{4} - 3 T^{3} + \cdots + 3501 \)
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| $59$ |
\( T^{4} - 3 T^{3} + \cdots + 801 \)
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| $61$ |
\( T^{4} - 14 T^{3} + \cdots - 1424 \)
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| $67$ |
\( T^{4} + 2 T^{3} + \cdots + 976 \)
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| $71$ |
\( T^{4} - 30 T^{3} + \cdots - 3600 \)
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| $73$ |
\( T^{4} - 26 T^{3} + \cdots + 181 \)
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| $79$ |
\( T^{4} + 10 T^{3} + \cdots - 11225 \)
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| $83$ |
\( T^{4} + 6 T^{3} + \cdots - 1629 \)
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| $89$ |
\( T^{4} - 6 T^{3} + \cdots - 144 \)
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| $97$ |
\( T^{4} - 4 T^{3} + \cdots + 7801 \)
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