Newspace parameters
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(50.5209032411\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{65}) \) |
| Defining polynomial: |
\( x^{2} - x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{65})\). 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 |
|
−4.53113 | 0 | −11.4689 | 25.0000 | 0 | −49.0000 | 196.963 | 0 | −113.278 | ||||||||||||||||||||||||
| 1.2 | 3.53113 | 0 | −19.5311 | 25.0000 | 0 | −49.0000 | −181.963 | 0 | 88.2782 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 315.6.a.c | 2 | |
| 3.b | odd | 2 | 1 | 35.6.a.b | ✓ | 2 | |
| 12.b | even | 2 | 1 | 560.6.a.l | 2 | ||
| 15.d | odd | 2 | 1 | 175.6.a.d | 2 | ||
| 15.e | even | 4 | 2 | 175.6.b.d | 4 | ||
| 21.c | even | 2 | 1 | 245.6.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.6.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 175.6.a.d | 2 | 15.d | odd | 2 | 1 | ||
| 175.6.b.d | 4 | 15.e | even | 4 | 2 | ||
| 245.6.a.c | 2 | 21.c | even | 2 | 1 | ||
| 315.6.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 560.6.a.l | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} - 16 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + T - 16 \)
$3$
\( T^{2} \)
$5$
\( (T - 25)^{2} \)
$7$
\( (T + 49)^{2} \)
$11$
\( T^{2} - 601T - 62596 \)
$13$
\( T^{2} + 577T + 37586 \)
$17$
\( T^{2} + 41T - 1023346 \)
$19$
\( T^{2} - 630T - 20960 \)
$23$
\( T^{2} - 442 T - 13172224 \)
$29$
\( T^{2} + 5885 T - 5853350 \)
$31$
\( T^{2} + 396 T - 13834656 \)
$37$
\( T^{2} + 8904 T - 5978196 \)
$41$
\( T^{2} + 1774 T - 236091496 \)
$43$
\( T^{2} + 27122 T + 170086856 \)
$47$
\( T^{2} - 21289 T + 72995224 \)
$53$
\( T^{2} - 55582 T + 768990296 \)
$59$
\( T^{2} + 59600 T + 451339840 \)
$61$
\( T^{2} + 51846 T + 142927344 \)
$67$
\( T^{2} + 45344 T + 174936944 \)
$71$
\( T^{2} + 80744 T + 1172746624 \)
$73$
\( T^{2} + 13532 T - 239780284 \)
$79$
\( T^{2} + 51795 T - 4382959400 \)
$83$
\( T^{2} + 109828 T + 2765333536 \)
$89$
\( T^{2} - 37650 T - 177665240 \)
$97$
\( T^{2} + 96339 T - 838817066 \)
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