Newspace parameters
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.96525785077\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 |
|
−2.00000 | 0 | −4.00000 | 5.00000 | 0 | 0 | 24.0000 | 0 | −10.0000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
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 | 135.4.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 135.4.a.d | yes | 1 | |
| 4.b | odd | 2 | 1 | 2160.4.a.n | 1 | ||
| 5.b | even | 2 | 1 | 675.4.a.i | 1 | ||
| 5.c | odd | 4 | 2 | 675.4.b.d | 2 | ||
| 9.c | even | 3 | 2 | 405.4.e.j | 2 | ||
| 9.d | odd | 6 | 2 | 405.4.e.e | 2 | ||
| 12.b | even | 2 | 1 | 2160.4.a.d | 1 | ||
| 15.d | odd | 2 | 1 | 675.4.a.b | 1 | ||
| 15.e | even | 4 | 2 | 675.4.b.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 135.4.a.d | yes | 1 | 3.b | odd | 2 | 1 | |
| 405.4.e.e | 2 | 9.d | odd | 6 | 2 | ||
| 405.4.e.j | 2 | 9.c | even | 3 | 2 | ||
| 675.4.a.b | 1 | 15.d | odd | 2 | 1 | ||
| 675.4.a.i | 1 | 5.b | even | 2 | 1 | ||
| 675.4.b.c | 2 | 15.e | even | 4 | 2 | ||
| 675.4.b.d | 2 | 5.c | odd | 4 | 2 | ||
| 2160.4.a.d | 1 | 12.b | even | 2 | 1 | ||
| 2160.4.a.n | 1 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 2 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 2 \)
$3$
\( T \)
$5$
\( T - 5 \)
$7$
\( T \)
$11$
\( T - 10 \)
$13$
\( T + 80 \)
$17$
\( T - 7 \)
$19$
\( T + 113 \)
$23$
\( T + 81 \)
$29$
\( T + 220 \)
$31$
\( T + 189 \)
$37$
\( T - 170 \)
$41$
\( T + 130 \)
$43$
\( T - 10 \)
$47$
\( T - 160 \)
$53$
\( T - 631 \)
$59$
\( T + 560 \)
$61$
\( T - 229 \)
$67$
\( T - 750 \)
$71$
\( T - 890 \)
$73$
\( T + 890 \)
$79$
\( T + 27 \)
$83$
\( T - 429 \)
$89$
\( T + 750 \)
$97$
\( T + 1480 \)
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