Newspace parameters
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.4633302691\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.18257.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 26x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 a root \(\nu\) of
\( x^{3} - x^{2} - 26x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 17 \)
|
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 |
|
−4.47894 | −9.00000 | −11.9391 | 25.0000 | 40.3105 | −168.818 | 196.801 | 81.0000 | −111.974 | |||||||||||||||||||||||||||
| 1.2 | 0.694797 | −9.00000 | −31.5173 | 25.0000 | −6.25317 | 83.1683 | −44.1316 | 81.0000 | 17.3699 | ||||||||||||||||||||||||||||
| 1.3 | 5.78415 | −9.00000 | 1.45634 | 25.0000 | −52.0573 | 17.6498 | −176.669 | 81.0000 | 144.604 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 165.6.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 495.6.a.b | 3 | ||
| 5.b | even | 2 | 1 | 825.6.a.g | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 495.6.a.b | 3 | 3.b | odd | 2 | 1 | ||
| 825.6.a.g | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2T_{2}^{2} - 25T_{2} + 18 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 2 T^{2} - 25 T + 18 \)
$3$
\( (T + 9)^{3} \)
$5$
\( (T - 25)^{3} \)
$7$
\( T^{3} + 68 T^{2} - 15552 T + 247808 \)
$11$
\( (T - 121)^{3} \)
$13$
\( T^{3} - 290 T^{2} + \cdots + 132063592 \)
$17$
\( T^{3} - 434 T^{2} + \cdots + 2547052488 \)
$19$
\( T^{3} + 2856 T^{2} + \cdots + 137703680 \)
$23$
\( T^{3} + 640 T^{2} + \cdots - 15777349632 \)
$29$
\( T^{3} + 4538 T^{2} + \cdots - 44413548456 \)
$31$
\( T^{3} + 14968 T^{2} + \cdots + 121645522944 \)
$37$
\( T^{3} + 6190 T^{2} + \cdots + 28013661736 \)
$41$
\( T^{3} + 8926 T^{2} + \cdots + 119305168392 \)
$43$
\( T^{3} + 33592 T^{2} + \cdots - 38997547520 \)
$47$
\( T^{3} + 24640 T^{2} + \cdots - 679997104128 \)
$53$
\( T^{3} + 22934 T^{2} + \cdots - 4393759072056 \)
$59$
\( T^{3} + 13756 T^{2} + \cdots - 20798004639936 \)
$61$
\( T^{3} - 24602 T^{2} + \cdots + 43064794504 \)
$67$
\( T^{3} - 16868 T^{2} + \cdots + 1826752720192 \)
$71$
\( T^{3} - 4856 T^{2} + \cdots + 34155066048000 \)
$73$
\( T^{3} - 1910 T^{2} + \cdots - 163103734088 \)
$79$
\( T^{3} + 36844 T^{2} + \cdots - 70772253539328 \)
$83$
\( T^{3} + 48796 T^{2} + \cdots - 70386077185728 \)
$89$
\( T^{3} + 188978 T^{2} + \cdots - 16088649675432 \)
$97$
\( T^{3} + \cdots - 148869121092488 \)
show more
show less