Newspace parameters
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(51.3228223402\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.39180.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 36x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 160) |
| 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
\( x^{3} - x^{2} - 36x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 8\nu^{2} - 16\nu - 189 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 4\beta _1 + 193 ) / 8 \)
|
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 |
|
0 | −18.4715 | 0 | 25.0000 | 0 | −121.899 | 0 | 98.1968 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.755735 | 0 | 25.0000 | 0 | 172.424 | 0 | −242.429 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 29.2272 | 0 | 25.0000 | 0 | −44.5253 | 0 | 611.232 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
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 | 320.6.a.y | 3 | |
| 4.b | odd | 2 | 1 | 320.6.a.x | 3 | ||
| 8.b | even | 2 | 1 | 160.6.a.f | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 160.6.a.g | yes | 3 | |
| 40.e | odd | 2 | 1 | 800.6.a.n | 3 | ||
| 40.f | even | 2 | 1 | 800.6.a.o | 3 | ||
| 40.i | odd | 4 | 2 | 800.6.c.k | 6 | ||
| 40.k | even | 4 | 2 | 800.6.c.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.6.a.f | ✓ | 3 | 8.b | even | 2 | 1 | |
| 160.6.a.g | yes | 3 | 8.d | odd | 2 | 1 | |
| 320.6.a.x | 3 | 4.b | odd | 2 | 1 | ||
| 320.6.a.y | 3 | 1.a | even | 1 | 1 | trivial | |
| 800.6.a.n | 3 | 40.e | odd | 2 | 1 | ||
| 800.6.a.o | 3 | 40.f | even | 2 | 1 | ||
| 800.6.c.j | 6 | 40.k | even | 4 | 2 | ||
| 800.6.c.k | 6 | 40.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 10T_{3}^{2} - 548T_{3} - 408 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} - 10 T^{2} - 548 T - 408 \)
$5$
\( (T - 25)^{3} \)
$7$
\( T^{3} - 6 T^{2} - 23268 T - 935848 \)
$11$
\( T^{3} + 396 T^{2} + \cdots - 59934400 \)
$13$
\( T^{3} - 354 T^{2} + \cdots - 28863000 \)
$17$
\( T^{3} - 1158 T^{2} + \cdots + 2743838200 \)
$19$
\( T^{3} - 3192 T^{2} + \cdots - 126323200 \)
$23$
\( T^{3} + 6126 T^{2} + \cdots + 5283148104 \)
$29$
\( T^{3} + 426 T^{2} + \cdots - 159249002312 \)
$31$
\( T^{3} + 3276 T^{2} + \cdots - 229217617600 \)
$37$
\( T^{3} - 11562 T^{2} + \cdots + 1078137168200 \)
$41$
\( T^{3} - 12450 T^{2} + \cdots - 1203781400 \)
$43$
\( T^{3} - 26346 T^{2} + \cdots - 352818354968 \)
$47$
\( T^{3} + 36762 T^{2} + \cdots + 1628197032152 \)
$53$
\( T^{3} - 21162 T^{2} + \cdots + 18581204112200 \)
$59$
\( T^{3} - 35040 T^{2} + \cdots - 887454720000 \)
$61$
\( T^{3} - 24138 T^{2} + \cdots + 47677792189640 \)
$67$
\( T^{3} + 9570 T^{2} + \cdots + 57239443872504 \)
$71$
\( T^{3} + 88092 T^{2} + \cdots + 11664981864000 \)
$73$
\( T^{3} + \cdots + 165394983407000 \)
$79$
\( T^{3} - 92952 T^{2} + \cdots - 1808931596800 \)
$83$
\( T^{3} + \cdots - 187637358980920 \)
$89$
\( T^{3} + \cdots - 175016035497384 \)
$97$
\( T^{3} - 170910 T^{2} + \cdots + 83009979455000 \)
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