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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{70}) \) |
| Defining polynomial: |
\( x^{2} - 70 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = 2\sqrt{70}\). We also show the integral \(q\)-expansion of the trace form.
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 | −12.7332 | 0 | −25.0000 | 0 | −68.7332 | 0 | −80.8656 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 20.7332 | 0 | −25.0000 | 0 | −35.2668 | 0 | 186.866 | 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.v | 2 | |
| 4.b | odd | 2 | 1 | 320.6.a.r | 2 | ||
| 8.b | even | 2 | 1 | 160.6.a.a | ✓ | 2 | |
| 8.d | odd | 2 | 1 | 160.6.a.e | yes | 2 | |
| 40.e | odd | 2 | 1 | 800.6.a.g | 2 | ||
| 40.f | even | 2 | 1 | 800.6.a.l | 2 | ||
| 40.i | odd | 4 | 2 | 800.6.c.f | 4 | ||
| 40.k | even | 4 | 2 | 800.6.c.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.6.a.a | ✓ | 2 | 8.b | even | 2 | 1 | |
| 160.6.a.e | yes | 2 | 8.d | odd | 2 | 1 | |
| 320.6.a.r | 2 | 4.b | odd | 2 | 1 | ||
| 320.6.a.v | 2 | 1.a | even | 1 | 1 | trivial | |
| 800.6.a.g | 2 | 40.e | odd | 2 | 1 | ||
| 800.6.a.l | 2 | 40.f | even | 2 | 1 | ||
| 800.6.c.f | 4 | 40.i | odd | 4 | 2 | ||
| 800.6.c.g | 4 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 8T_{3} - 264 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 8T - 264 \)
$5$
\( (T + 25)^{2} \)
$7$
\( T^{2} + 104T + 2424 \)
$11$
\( T^{2} - 320T - 2400 \)
$13$
\( T^{2} - 100T - 445500 \)
$17$
\( T^{2} - 580T - 363900 \)
$19$
\( T^{2} - 720T - 318400 \)
$23$
\( T^{2} + 1688 T - 1922184 \)
$29$
\( T^{2} + 108 T - 1789084 \)
$31$
\( T^{2} + 9840 T + 19474400 \)
$37$
\( T^{2} + 6540 T - 77115100 \)
$41$
\( T^{2} + 10620 T - 154605820 \)
$43$
\( T^{2} - 25672 T + 70895416 \)
$47$
\( T^{2} + 28296 T + 159092984 \)
$53$
\( T^{2} + 31340 T - 81043100 \)
$59$
\( T^{2} - 30800 T + 237048000 \)
$61$
\( T^{2} + 24540 T - 241966780 \)
$67$
\( T^{2} - 34584 T + 230406264 \)
$71$
\( T^{2} - 12400 T - 2464788000 \)
$73$
\( T^{2} + 7180 T - 3861863900 \)
$79$
\( T^{2} + 71840 T + 1276694400 \)
$83$
\( T^{2} + 31928 T - 1967087624 \)
$89$
\( T^{2} + 40748 T - 2597252124 \)
$97$
\( T^{2} + 190140 T + 2479136900 \)
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