# 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: $$x^{2} - x + 1$$ 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*z * q^4 + q^5 + z * q^7 $$q + 2 \zeta_{6} q^{4} + q^{5} + \zeta_{6} q^{7} + ( - 6 \zeta_{6} + 6) q^{11} + ( - 3 \zeta_{6} + 4) q^{13} + (4 \zeta_{6} - 4) q^{16} + 4 \zeta_{6} q^{19} + 2 \zeta_{6} q^{20} + (6 \zeta_{6} - 6) q^{23} + q^{25} + (2 \zeta_{6} - 2) q^{28} + (6 \zeta_{6} - 6) q^{29} + 5 q^{31} + \zeta_{6} q^{35} + (2 \zeta_{6} - 2) q^{37} - 11 \zeta_{6} q^{43} + 12 q^{44} - 6 q^{47} + ( - 6 \zeta_{6} + 6) q^{49} + (2 \zeta_{6} + 6) q^{52} + ( - 6 \zeta_{6} + 6) q^{55} + 6 \zeta_{6} q^{59} + \zeta_{6} q^{61} - 8 q^{64} + ( - 3 \zeta_{6} + 4) q^{65} + (11 \zeta_{6} - 11) q^{67} - 6 \zeta_{6} q^{71} + 5 q^{73} + (8 \zeta_{6} - 8) q^{76} + 6 q^{77} + 11 q^{79} + (4 \zeta_{6} - 4) q^{80} - 12 q^{83} + ( - 12 \zeta_{6} + 12) q^{89} + (\zeta_{6} + 3) q^{91} - 12 q^{92} + 4 \zeta_{6} q^{95} - 17 \zeta_{6} q^{97} +O(q^{100})$$ q + 2*z * q^4 + q^5 + z * q^7 + (-6*z + 6) * q^11 + (-3*z + 4) * q^13 + (4*z - 4) * q^16 + 4*z * q^19 + 2*z * q^20 + (6*z - 6) * q^23 + q^25 + (2*z - 2) * q^28 + (6*z - 6) * q^29 + 5 * q^31 + z * q^35 + (2*z - 2) * q^37 - 11*z * q^43 + 12 * q^44 - 6 * q^47 + (-6*z + 6) * q^49 + (2*z + 6) * q^52 + (-6*z + 6) * q^55 + 6*z * q^59 + z * q^61 - 8 * q^64 + (-3*z + 4) * q^65 + (11*z - 11) * q^67 - 6*z * q^71 + 5 * q^73 + (8*z - 8) * q^76 + 6 * q^77 + 11 * q^79 + (4*z - 4) * q^80 - 12 * q^83 + (-12*z + 12) * q^89 + (z + 3) * q^91 - 12 * q^92 + 4*z * q^95 - 17*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^5 + q^7 $$2 q + 2 q^{4} + 2 q^{5} + q^{7} + 6 q^{11} + 5 q^{13} - 4 q^{16} + 4 q^{19} + 2 q^{20} - 6 q^{23} + 2 q^{25} - 2 q^{28} - 6 q^{29} + 10 q^{31} + q^{35} - 2 q^{37} - 11 q^{43} + 24 q^{44} - 12 q^{47} + 6 q^{49} + 14 q^{52} + 6 q^{55} + 6 q^{59} + q^{61} - 16 q^{64} + 5 q^{65} - 11 q^{67} - 6 q^{71} + 10 q^{73} - 8 q^{76} + 12 q^{77} + 22 q^{79} - 4 q^{80} - 24 q^{83} + 12 q^{89} + 7 q^{91} - 24 q^{92} + 4 q^{95} - 17 q^{97}+O(q^{100})$$ 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

## 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

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