Newspace parameters
| Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 585.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.67124851824\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 195) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(352\) | \(496\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
2.00000 | 0 | 2.00000 | 2.00000 | − | 1.00000i | 0 | 3.00000 | 0 | 0 | 4.00000 | − | 2.00000i | ||||||||||||||||||||
| 64.2 | 2.00000 | 0 | 2.00000 | 2.00000 | + | 1.00000i | 0 | 3.00000 | 0 | 0 | 4.00000 | + | 2.00000i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 585.2.h.d | 2 | |
| 3.b | odd | 2 | 1 | 195.2.h.a | ✓ | 2 | |
| 5.b | even | 2 | 1 | 585.2.h.a | 2 | ||
| 12.b | even | 2 | 1 | 3120.2.r.a | 2 | ||
| 13.b | even | 2 | 1 | 585.2.h.a | 2 | ||
| 15.d | odd | 2 | 1 | 195.2.h.b | yes | 2 | |
| 15.e | even | 4 | 1 | 975.2.b.a | 2 | ||
| 15.e | even | 4 | 1 | 975.2.b.c | 2 | ||
| 39.d | odd | 2 | 1 | 195.2.h.b | yes | 2 | |
| 60.h | even | 2 | 1 | 3120.2.r.f | 2 | ||
| 65.d | even | 2 | 1 | inner | 585.2.h.d | 2 | |
| 156.h | even | 2 | 1 | 3120.2.r.f | 2 | ||
| 195.e | odd | 2 | 1 | 195.2.h.a | ✓ | 2 | |
| 195.s | even | 4 | 1 | 975.2.b.a | 2 | ||
| 195.s | even | 4 | 1 | 975.2.b.c | 2 | ||
| 780.d | even | 2 | 1 | 3120.2.r.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.2.h.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 195.2.h.a | ✓ | 2 | 195.e | odd | 2 | 1 | |
| 195.2.h.b | yes | 2 | 15.d | odd | 2 | 1 | |
| 195.2.h.b | yes | 2 | 39.d | odd | 2 | 1 | |
| 585.2.h.a | 2 | 5.b | even | 2 | 1 | ||
| 585.2.h.a | 2 | 13.b | even | 2 | 1 | ||
| 585.2.h.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 585.2.h.d | 2 | 65.d | even | 2 | 1 | inner | |
| 975.2.b.a | 2 | 15.e | even | 4 | 1 | ||
| 975.2.b.a | 2 | 195.s | even | 4 | 1 | ||
| 975.2.b.c | 2 | 15.e | even | 4 | 1 | ||
| 975.2.b.c | 2 | 195.s | even | 4 | 1 | ||
| 3120.2.r.a | 2 | 12.b | even | 2 | 1 | ||
| 3120.2.r.a | 2 | 780.d | even | 2 | 1 | ||
| 3120.2.r.f | 2 | 60.h | even | 2 | 1 | ||
| 3120.2.r.f | 2 | 156.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 2)^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 4T + 5 \)
$7$
\( (T - 3)^{2} \)
$11$
\( T^{2} + 25 \)
$13$
\( T^{2} + 6T + 13 \)
$17$
\( T^{2} + 9 \)
$19$
\( T^{2} + 36 \)
$23$
\( T^{2} + 81 \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( (T - 3)^{2} \)
$41$
\( T^{2} + 25 \)
$43$
\( T^{2} + 36 \)
$47$
\( (T + 8)^{2} \)
$53$
\( T^{2} + 81 \)
$59$
\( T^{2} + 16 \)
$61$
\( (T + 3)^{2} \)
$67$
\( (T + 12)^{2} \)
$71$
\( T^{2} + 25 \)
$73$
\( (T + 6)^{2} \)
$79$
\( (T - 15)^{2} \)
$83$
\( (T + 4)^{2} \)
$89$
\( T^{2} + 1 \)
$97$
\( (T - 3)^{2} \)
show more
show less