Newspace parameters
| Level: | \( N \) | \(=\) | \( 6069 = 3 \cdot 7 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6069.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.4612089867\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{18})^+\) |
|
|
|
| Defining polynomial: |
\( x^{3} - 3x - 1 \)
|
| 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
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 |
|
−1.87939 | 1.00000 | 1.53209 | −0.347296 | −1.87939 | −1.00000 | 0.879385 | 1.00000 | 0.652704 | |||||||||||||||||||||||||||
| 1.2 | 0.347296 | 1.00000 | −1.87939 | −1.53209 | 0.347296 | −1.00000 | −1.34730 | 1.00000 | −0.532089 | ||||||||||||||||||||||||||||
| 1.3 | 1.53209 | 1.00000 | 0.347296 | 1.87939 | 1.53209 | −1.00000 | −2.53209 | 1.00000 | 2.87939 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 6069.2.a.p | yes | 3 |
| 17.b | even | 2 | 1 | 6069.2.a.n | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6069.2.a.n | ✓ | 3 | 17.b | even | 2 | 1 | |
| 6069.2.a.p | yes | 3 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):
|
\( T_{2}^{3} - 3T_{2} + 1 \)
|
|
\( T_{5}^{3} - 3T_{5} - 1 \)
|
|
\( T_{11}^{3} + 6T_{11}^{2} + 3T_{11} - 19 \)
|
|
\( T_{23}^{3} - 27T_{23} + 27 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 3T + 1 \)
|
| $3$ |
\( (T - 1)^{3} \)
|
| $5$ |
\( T^{3} - 3T - 1 \)
|
| $7$ |
\( (T + 1)^{3} \)
|
| $11$ |
\( T^{3} + 6 T^{2} + \cdots - 19 \)
|
| $13$ |
\( T^{3} - 3 T^{2} + \cdots + 17 \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} - 3 T^{2} + \cdots + 17 \)
|
| $23$ |
\( T^{3} - 27T + 27 \)
|
| $29$ |
\( T^{3} - 3 T^{2} + \cdots - 37 \)
|
| $31$ |
\( T^{3} + 3 T^{2} + \cdots - 3 \)
|
| $37$ |
\( T^{3} - 3 T^{2} + \cdots + 111 \)
|
| $41$ |
\( T^{3} + 15 T^{2} + \cdots + 19 \)
|
| $43$ |
\( T^{3} + 3 T^{2} + \cdots - 71 \)
|
| $47$ |
\( T^{3} + 6 T^{2} + \cdots - 213 \)
|
| $53$ |
\( T^{3} - 3 T^{2} + \cdots + 57 \)
|
| $59$ |
\( T^{3} - 9 T^{2} + \cdots + 1143 \)
|
| $61$ |
\( T^{3} + 9 T^{2} + \cdots - 37 \)
|
| $67$ |
\( T^{3} + 3 T^{2} + \cdots - 379 \)
|
| $71$ |
\( T^{3} - 3 T^{2} + \cdots - 213 \)
|
| $73$ |
\( T^{3} + 15 T^{2} + \cdots - 51 \)
|
| $79$ |
\( (T + 7)^{3} \)
|
| $83$ |
\( T^{3} - 6 T^{2} + \cdots + 51 \)
|
| $89$ |
\( (T - 1)^{3} \)
|
| $97$ |
\( T^{3} + 3 T^{2} + \cdots - 543 \)
|
| show more | |
| show less |