Newspace parameters
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.8957735523\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
| Defining polynomial: |
\( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 26\nu^{3} + 30\nu^{2} + 141\nu - 36 ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 19\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 21\nu^{2} + 3\nu + 30 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 20\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} + 22\beta_{3} + 38\beta _1 + 255 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 28\beta_{4} + 40\beta_{3} + 24\beta_{2} + 425\beta _1 + 468 \)
|
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 |
|
−4.11734 | 0 | 8.95250 | 5.00000 | 0 | −20.0229 | −3.92177 | 0 | −20.5867 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.57457 | 0 | 4.77759 | 5.00000 | 0 | 14.1597 | 11.5188 | 0 | −17.8729 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.149150 | 0 | −7.97775 | 5.00000 | 0 | 20.1424 | 2.38308 | 0 | −0.745751 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.07326 | 0 | −3.70159 | 5.00000 | 0 | −4.66112 | −24.2604 | 0 | 10.3663 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.53444 | 0 | 12.5612 | 5.00000 | 0 | −2.63618 | 20.6823 | 0 | 22.6722 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.23336 | 0 | 19.3881 | 5.00000 | 0 | 33.0180 | 59.5981 | 0 | 26.1668 | |||||||||||||||||||||||||||||||||||||
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 | 405.4.a.l | yes | 6 |
| 3.b | odd | 2 | 1 | 405.4.a.k | ✓ | 6 | |
| 5.b | even | 2 | 1 | 2025.4.a.y | 6 | ||
| 9.c | even | 3 | 2 | 405.4.e.w | 12 | ||
| 9.d | odd | 6 | 2 | 405.4.e.x | 12 | ||
| 15.d | odd | 2 | 1 | 2025.4.a.z | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.a.k | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 405.4.a.l | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 405.4.e.w | 12 | 9.c | even | 3 | 2 | ||
| 405.4.e.x | 12 | 9.d | odd | 6 | 2 | ||
| 2025.4.a.y | 6 | 5.b | even | 2 | 1 | ||
| 2025.4.a.z | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 4T_{2}^{5} - 33T_{2}^{4} + 110T_{2}^{3} + 286T_{2}^{2} - 684T_{2} - 108 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - 4 T^{5} - 33 T^{4} + 110 T^{3} + \cdots - 108 \)
$3$
\( T^{6} \)
$5$
\( (T - 5)^{6} \)
$7$
\( T^{6} - 40 T^{5} - 263 T^{4} + \cdots - 2316924 \)
$11$
\( T^{6} - 88 T^{5} + 1518 T^{4} + \cdots + 1197108 \)
$13$
\( T^{6} - 20 T^{5} + \cdots - 185793728 \)
$17$
\( T^{6} - 124 T^{5} + \cdots + 105966288 \)
$19$
\( T^{6} + 46 T^{5} + \cdots - 36821611175 \)
$23$
\( T^{6} - 210 T^{5} + \cdots + 332569842768 \)
$29$
\( T^{6} - 296 T^{5} + \cdots + 635447099088 \)
$31$
\( T^{6} + 104 T^{5} + \cdots - 69859854216 \)
$37$
\( T^{6} + 204 T^{5} + \cdots + 12008297128192 \)
$41$
\( T^{6} - 344 T^{5} + \cdots - 86213802377403 \)
$43$
\( T^{6} + \cdots - 180465347194400 \)
$47$
\( T^{6} - 238 T^{5} + \cdots - 49451750433900 \)
$53$
\( T^{6} - 850 T^{5} + \cdots + 15\!\cdots\!00 \)
$59$
\( T^{6} + \cdots - 168320389359483 \)
$61$
\( T^{6} + 364 T^{5} + \cdots - 20\!\cdots\!36 \)
$67$
\( T^{6} - 88 T^{5} + \cdots + 19\!\cdots\!00 \)
$71$
\( T^{6} - 1364 T^{5} + \cdots - 11\!\cdots\!84 \)
$73$
\( T^{6} - 836 T^{5} + \cdots - 10\!\cdots\!36 \)
$79$
\( T^{6} + 680 T^{5} + \cdots - 15\!\cdots\!72 \)
$83$
\( T^{6} - 2148 T^{5} + \cdots + 41\!\cdots\!32 \)
$89$
\( T^{6} - 3000 T^{5} + \cdots + 16\!\cdots\!48 \)
$97$
\( T^{6} + 612 T^{5} + \cdots - 97\!\cdots\!24 \)
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