Newspace parameters
| Level: | \( N \) | \(=\) | \( 4760 = 2^{3} \cdot 5 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4760.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(38.0087913621\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1891397.1 |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + \nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 8\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{4} + 2\beta_{3} + 8\beta_{2} + 14\beta _1 + 19 \)
|
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 |
|
0 | −2.08153 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.33275 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.164718 | 0 | −1.00000 | 0 | −1.00000 | 0 | −2.97287 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.661653 | 0 | −1.00000 | 0 | −1.00000 | 0 | −2.56222 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.37221 | 0 | −1.00000 | 0 | −1.00000 | 0 | −1.11703 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.21238 | 0 | −1.00000 | 0 | −1.00000 | 0 | 7.31937 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 4760.2.a.n | ✓ | 5 |
| 4.b | odd | 2 | 1 | 9520.2.a.bp | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4760.2.a.n | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 9520.2.a.bp | 5 | 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4760))\):
|
\( T_{3}^{5} - 3T_{3}^{4} - 4T_{3}^{3} + 12T_{3}^{2} - 4T_{3} - 1 \)
|
|
\( T_{13}^{5} - 9T_{13}^{4} + 6T_{13}^{3} + 61T_{13}^{2} + 60T_{13} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( (T + 1)^{5} \)
|
| $11$ |
\( T^{5} - 4 T^{4} + \cdots + 125 \)
|
| $13$ |
\( T^{5} - 9 T^{4} + \cdots + 16 \)
|
| $17$ |
\( (T - 1)^{5} \)
|
| $19$ |
\( T^{5} - 35 T^{3} + \cdots - 111 \)
|
| $23$ |
\( T^{5} - 10 T^{4} + \cdots + 9 \)
|
| $29$ |
\( T^{5} + 6 T^{4} + \cdots + 1401 \)
|
| $31$ |
\( T^{5} - 4 T^{4} + \cdots - 315 \)
|
| $37$ |
\( T^{5} - 8 T^{4} + \cdots - 1332 \)
|
| $41$ |
\( T^{5} + 10 T^{4} + \cdots + 121 \)
|
| $43$ |
\( T^{5} + 14 T^{4} + \cdots - 8673 \)
|
| $47$ |
\( T^{5} - 12 T^{4} + \cdots + 9 \)
|
| $53$ |
\( T^{5} + 8 T^{4} + \cdots - 711 \)
|
| $59$ |
\( T^{5} + 4 T^{4} + \cdots + 78651 \)
|
| $61$ |
\( T^{5} - 25 T^{4} + \cdots - 2932 \)
|
| $67$ |
\( T^{5} - 8 T^{4} + \cdots - 92369 \)
|
| $71$ |
\( T^{5} - 27 T^{4} + \cdots + 28 \)
|
| $73$ |
\( T^{5} - 3 T^{4} + \cdots + 14660 \)
|
| $79$ |
\( T^{5} - 18 T^{4} + \cdots + 23604 \)
|
| $83$ |
\( T^{5} - 26 T^{4} + \cdots + 1764 \)
|
| $89$ |
\( T^{5} + 5 T^{4} + \cdots + 4672 \)
|
| $97$ |
\( T^{5} - 18 T^{4} + \cdots - 28 \)
|
| show more | |
| show less |