Newspace parameters
| Level: | \( N \) | \(=\) | \( 4719 = 3 \cdot 11^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4719.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(37.6814047138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.394064.1 |
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|
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| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 6x^{2} + 5x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6x^{3} + 6x^{2} + 5x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu + 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + \nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta _1 - 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} - \beta _1 + 13 \)
|
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 |
|
−2.26881 | −1.00000 | 3.14752 | −1.60351 | 2.26881 | −4.26048 | −2.60351 | 1.00000 | 3.63807 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.864737 | −1.00000 | −1.25223 | 3.81232 | 0.864737 | 3.02012 | 2.81232 | 1.00000 | −3.29666 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.465910 | −1.00000 | −1.78293 | −0.762504 | −0.465910 | 2.44809 | −1.76250 | 1.00000 | −0.355258 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.54936 | −1.00000 | 0.400511 | −1.47818 | −1.54936 | −4.56944 | −2.47818 | 1.00000 | −2.29023 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.11828 | −1.00000 | 2.48713 | 2.03187 | −2.11828 | 1.36170 | 1.03187 | 1.00000 | 4.30408 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(11\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 4719.2.a.bd | yes | 5 |
| 11.b | odd | 2 | 1 | 4719.2.a.bb | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4719.2.a.bb | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 4719.2.a.bd | yes | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4719))\):
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\( T_{2}^{5} - T_{2}^{4} - 6T_{2}^{3} + 6T_{2}^{2} + 5T_{2} - 3 \)
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\( T_{5}^{5} - 2T_{5}^{4} - 10T_{5}^{3} + 4T_{5}^{2} + 26T_{5} + 14 \)
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\( T_{7}^{5} + 2T_{7}^{4} - 26T_{7}^{3} - 12T_{7}^{2} + 200T_{7} - 196 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} - 6 T^{3} + \cdots - 3 \)
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| $3$ |
\( (T + 1)^{5} \)
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| $5$ |
\( T^{5} - 2 T^{4} + \cdots + 14 \)
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| $7$ |
\( T^{5} + 2 T^{4} + \cdots - 196 \)
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| $11$ |
\( T^{5} \)
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| $13$ |
\( (T - 1)^{5} \)
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| $17$ |
\( T^{5} - 6 T^{4} + \cdots - 106 \)
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| $19$ |
\( T^{5} - 54 T^{3} + \cdots - 316 \)
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| $23$ |
\( T^{5} + 8 T^{4} + \cdots + 12 \)
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| $29$ |
\( T^{5} - 6 T^{4} + \cdots + 226 \)
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| $31$ |
\( T^{5} - 8 T^{4} + \cdots + 1762 \)
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| $37$ |
\( T^{5} - 14 T^{4} + \cdots + 2008 \)
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| $41$ |
\( T^{5} - 34 T^{4} + \cdots + 16708 \)
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| $43$ |
\( T^{5} + 18 T^{4} + \cdots - 1906 \)
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| $47$ |
\( T^{5} + 4 T^{4} + \cdots - 6856 \)
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| $53$ |
\( T^{5} + 2 T^{4} + \cdots + 19792 \)
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| $59$ |
\( T^{5} + 14 T^{4} + \cdots + 25248 \)
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| $61$ |
\( T^{5} + 6 T^{4} + \cdots + 4312 \)
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| $67$ |
\( T^{5} - 18 T^{4} + \cdots + 1642 \)
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| $71$ |
\( T^{5} - 2 T^{4} + \cdots + 392 \)
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| $73$ |
\( T^{5} - 12 T^{4} + \cdots + 2252 \)
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| $79$ |
\( T^{5} - 218 T^{3} + \cdots - 54 \)
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| $83$ |
\( T^{5} + 6 T^{4} + \cdots - 11088 \)
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| $89$ |
\( T^{5} + 4 T^{4} + \cdots - 15818 \)
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| $97$ |
\( T^{5} + 8 T^{4} + \cdots - 13896 \)
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