Newspace parameters
| Level: | \( N \) | \(=\) | \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4020.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(32.0998616126\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1257629.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 9x^{3} + 9x^{2} + 17x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{5} - x^{4} - 9x^{3} + 9x^{2} + 17x - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 9\nu^{2} - 6\nu + 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 6\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 8\beta_{2} + 21 \)
|
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 | −1.00000 | 0 | 1.00000 | 0 | −4.78538 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | 1.00000 | 0 | −1.20813 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | 1.00000 | 0 | −0.300647 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 1.00000 | 0 | 0.397192 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 1.00000 | 0 | 2.89696 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(67\) | \(1\) |
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 | 4020.2.a.e | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4020.2.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} + 3T_{7}^{4} - 12T_{7}^{3} - 16T_{7}^{2} + 3T_{7} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} + 3 T^{4} + \cdots + 2 \)
|
| $11$ |
\( T^{5} - 3 T^{4} + \cdots - 118 \)
|
| $13$ |
\( T^{5} + 11 T^{4} + \cdots + 6 \)
|
| $17$ |
\( T^{5} + 11 T^{4} + \cdots - 353 \)
|
| $19$ |
\( T^{5} - 6 T^{4} + \cdots - 53 \)
|
| $23$ |
\( T^{5} - 3 T^{4} + \cdots - 1119 \)
|
| $29$ |
\( T^{5} + 2 T^{4} + \cdots + 8977 \)
|
| $31$ |
\( T^{5} - T^{4} + \cdots + 604 \)
|
| $37$ |
\( T^{5} + 13 T^{4} + \cdots - 1299 \)
|
| $41$ |
\( T^{5} + 13 T^{4} + \cdots - 2778 \)
|
| $43$ |
\( T^{5} + 9 T^{4} + \cdots + 38882 \)
|
| $47$ |
\( T^{5} + 12 T^{4} + \cdots - 33 \)
|
| $53$ |
\( T^{5} + 19 T^{4} + \cdots - 43248 \)
|
| $59$ |
\( T^{5} - 6 T^{4} + \cdots + 3251 \)
|
| $61$ |
\( T^{5} + 3 T^{4} + \cdots + 1228 \)
|
| $67$ |
\( (T + 1)^{5} \)
|
| $71$ |
\( T^{5} - 6 T^{4} + \cdots - 1524 \)
|
| $73$ |
\( T^{5} + 21 T^{4} + \cdots - 3837 \)
|
| $79$ |
\( T^{5} + T^{4} + \cdots + 37164 \)
|
| $83$ |
\( T^{5} + 8 T^{4} + \cdots + 160516 \)
|
| $89$ |
\( T^{5} + 29 T^{4} + \cdots - 2913 \)
|
| $97$ |
\( T^{5} + 13 T^{4} + \cdots + 240446 \)
|
| show more | |
| show less |