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: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.5637.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 23x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{3} - x^{2} - 23x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 15 \)
|
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 |
|
−4.45938 | 0 | 11.8861 | −5.00000 | 0 | 5.08123 | −17.3296 | 0 | 22.2969 | |||||||||||||||||||||||||||
| 1.2 | 0.258712 | 0 | −7.93307 | −5.00000 | 0 | 14.5174 | −4.12208 | 0 | −1.29356 | ||||||||||||||||||||||||||||
| 1.3 | 5.20067 | 0 | 19.0470 | −5.00000 | 0 | 24.4013 | 57.4517 | 0 | −26.0034 | ||||||||||||||||||||||||||||
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.g | yes | 3 |
| 3.b | odd | 2 | 1 | 135.4.a.f | ✓ | 3 | |
| 4.b | odd | 2 | 1 | 2160.4.a.be | 3 | ||
| 5.b | even | 2 | 1 | 675.4.a.q | 3 | ||
| 5.c | odd | 4 | 2 | 675.4.b.k | 6 | ||
| 9.c | even | 3 | 2 | 405.4.e.r | 6 | ||
| 9.d | odd | 6 | 2 | 405.4.e.t | 6 | ||
| 12.b | even | 2 | 1 | 2160.4.a.bm | 3 | ||
| 15.d | odd | 2 | 1 | 675.4.a.r | 3 | ||
| 15.e | even | 4 | 2 | 675.4.b.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.f | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 135.4.a.g | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 405.4.e.r | 6 | 9.c | even | 3 | 2 | ||
| 405.4.e.t | 6 | 9.d | odd | 6 | 2 | ||
| 675.4.a.q | 3 | 5.b | even | 2 | 1 | ||
| 675.4.a.r | 3 | 15.d | odd | 2 | 1 | ||
| 675.4.b.k | 6 | 5.c | odd | 4 | 2 | ||
| 675.4.b.l | 6 | 15.e | even | 4 | 2 | ||
| 2160.4.a.be | 3 | 4.b | odd | 2 | 1 | ||
| 2160.4.a.bm | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 23T_{2} + 6 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(135))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - T^{2} - 23T + 6 \)
$3$
\( T^{3} \)
$5$
\( (T + 5)^{3} \)
$7$
\( T^{3} - 44 T^{2} + 552 T - 1800 \)
$11$
\( T^{3} + 38 T^{2} - 2612 T - 83280 \)
$13$
\( T^{3} - 28 T^{2} - 4576 T + 100120 \)
$17$
\( T^{3} - 19 T^{2} - 11477 T + 553887 \)
$19$
\( T^{3} - 187 T^{2} + 3587 T + 525871 \)
$23$
\( T^{3} - 81 T^{2} - 23301 T + 2043981 \)
$29$
\( T^{3} + 160 T^{2} - 47768 T - 7892760 \)
$31$
\( T^{3} - 227 T^{2} - 17973 T - 246321 \)
$37$
\( T^{3} - 78 T^{2} - 99924 T + 13637080 \)
$41$
\( T^{3} - 338 T^{2} + \cdots + 12116640 \)
$43$
\( T^{3} - 22 T^{2} - 159916 T + 18464560 \)
$47$
\( T^{3} - 472 T^{2} + 54208 T + 283200 \)
$53$
\( T^{3} + 521 T^{2} + 61387 T - 939789 \)
$59$
\( T^{3} + 140 T^{2} + \cdots - 34131480 \)
$61$
\( T^{3} - 595 T^{2} - 2749 T + 1782607 \)
$67$
\( T^{3} - 878 T^{2} + \cdots + 11295000 \)
$71$
\( T^{3} - 602 T^{2} + \cdots + 280550880 \)
$73$
\( T^{3} - 1294 T^{2} + \cdots + 404091280 \)
$79$
\( T^{3} - 629 T^{2} + 97059 T - 2010303 \)
$83$
\( T^{3} - 1287 T^{2} + \cdots + 346404411 \)
$89$
\( T^{3} + 2154 T^{2} + \cdots + 74325600 \)
$97$
\( T^{3} - 1392 T^{2} + \cdots - 63595520 \)
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