# Properties

 Label 585.2.j.a Level $585$ Weight $2$ Character orbit 585.j Analytic conductor $4.671$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{4} + q^{5} + \zeta_{6} q^{7} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{4} + q^{5} + \zeta_{6} q^{7} + ( 6 - 6 \zeta_{6} ) q^{11} + ( 4 - 3 \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + 4 \zeta_{6} q^{19} + 2 \zeta_{6} q^{20} + ( -6 + 6 \zeta_{6} ) q^{23} + q^{25} + ( -2 + 2 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + 5 q^{31} + \zeta_{6} q^{35} + ( -2 + 2 \zeta_{6} ) q^{37} -11 \zeta_{6} q^{43} + 12 q^{44} -6 q^{47} + ( 6 - 6 \zeta_{6} ) q^{49} + ( 6 + 2 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{55} + 6 \zeta_{6} q^{59} + \zeta_{6} q^{61} -8 q^{64} + ( 4 - 3 \zeta_{6} ) q^{65} + ( -11 + 11 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{71} + 5 q^{73} + ( -8 + 8 \zeta_{6} ) q^{76} + 6 q^{77} + 11 q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} -12 q^{83} + ( 12 - 12 \zeta_{6} ) q^{89} + ( 3 + \zeta_{6} ) q^{91} -12 q^{92} + 4 \zeta_{6} q^{95} -17 \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{5} + q^{7} + O(q^{10})$$ $$2 q + 2 q^{4} + 2 q^{5} + q^{7} + 6 q^{11} + 5 q^{13} - 4 q^{16} + 4 q^{19} + 2 q^{20} - 6 q^{23} + 2 q^{25} - 2 q^{28} - 6 q^{29} + 10 q^{31} + q^{35} - 2 q^{37} - 11 q^{43} + 24 q^{44} - 12 q^{47} + 6 q^{49} + 14 q^{52} + 6 q^{55} + 6 q^{59} + q^{61} - 16 q^{64} + 5 q^{65} - 11 q^{67} - 6 q^{71} + 10 q^{73} - 8 q^{76} + 12 q^{77} + 22 q^{79} - 4 q^{80} - 24 q^{83} + 12 q^{89} + 7 q^{91} - 24 q^{92} + 4 q^{95} - 17 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
406.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 1.00000 1.73205i 1.00000 0 0.500000 0.866025i 0 0 0
451.1 0 0 1.00000 + 1.73205i 1.00000 0 0.500000 + 0.866025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.a 2
3.b odd 2 1 195.2.i.b 2
13.c even 3 1 inner 585.2.j.a 2
13.c even 3 1 7605.2.a.l 1
13.e even 6 1 7605.2.a.k 1
15.d odd 2 1 975.2.i.d 2
15.e even 4 2 975.2.bb.b 4
39.h odd 6 1 2535.2.a.i 1
39.i odd 6 1 195.2.i.b 2
39.i odd 6 1 2535.2.a.h 1
195.x odd 6 1 975.2.i.d 2
195.bl even 12 2 975.2.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.b 2 3.b odd 2 1
195.2.i.b 2 39.i odd 6 1
585.2.j.a 2 1.a even 1 1 trivial
585.2.j.a 2 13.c even 3 1 inner
975.2.i.d 2 15.d odd 2 1
975.2.i.d 2 195.x odd 6 1
975.2.bb.b 4 15.e even 4 2
975.2.bb.b 4 195.bl even 12 2
2535.2.a.h 1 39.i odd 6 1
2535.2.a.i 1 39.h odd 6 1
7605.2.a.k 1 13.e even 6 1
7605.2.a.l 1 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$13 - 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$121 + 11 T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$( -5 + T )^{2}$$
$79$ $$( -11 + T )^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$144 - 12 T + T^{2}$$
$97$ $$289 + 17 T + T^{2}$$