# Properties

 Label 585.2.i.a Level $585$ Weight $2$ Character orbit 585.i 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.i (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 2) q^{3} + 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10})$$ q + (-z + 2) * q^3 + 2*z * q^4 - z * q^5 + (-4*z + 4) * q^7 + (-3*z + 3) * q^9 $$q + ( - \zeta_{6} + 2) q^{3} + 2 \zeta_{6} q^{4} - \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + (2 \zeta_{6} + 2) q^{12} - \zeta_{6} q^{13} + ( - \zeta_{6} - 1) q^{15} + (4 \zeta_{6} - 4) q^{16} - 3 q^{17} + 2 q^{19} + ( - 2 \zeta_{6} + 2) q^{20} + ( - 8 \zeta_{6} + 4) q^{21} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + 8 q^{28} + (6 \zeta_{6} - 6) q^{29} + 4 \zeta_{6} q^{31} - 4 q^{35} + 6 q^{36} + 8 q^{37} + ( - \zeta_{6} - 1) q^{39} + 12 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 3 q^{45} + (8 \zeta_{6} - 4) q^{48} - 9 \zeta_{6} q^{49} + (3 \zeta_{6} - 6) q^{51} + ( - 2 \zeta_{6} + 2) q^{52} - 3 q^{53} + ( - 2 \zeta_{6} + 4) q^{57} - 6 \zeta_{6} q^{59} + ( - 4 \zeta_{6} + 2) q^{60} + (11 \zeta_{6} - 11) q^{61} - 12 \zeta_{6} q^{63} - 8 q^{64} + (\zeta_{6} - 1) q^{65} + 4 \zeta_{6} q^{67} - 6 \zeta_{6} q^{68} + ( - 3 \zeta_{6} - 3) q^{69} + 6 q^{71} + 8 q^{73} + (2 \zeta_{6} - 1) q^{75} + 4 \zeta_{6} q^{76} + ( - 7 \zeta_{6} + 7) q^{79} + 4 q^{80} - 9 \zeta_{6} q^{81} + (6 \zeta_{6} - 6) q^{83} + ( - 8 \zeta_{6} + 16) q^{84} + 3 \zeta_{6} q^{85} + (12 \zeta_{6} - 6) q^{87} - 18 q^{89} - 4 q^{91} + ( - 6 \zeta_{6} + 6) q^{92} + (4 \zeta_{6} + 4) q^{93} - 2 \zeta_{6} q^{95} + (14 \zeta_{6} - 14) q^{97} +O(q^{100})$$ q + (-z + 2) * q^3 + 2*z * q^4 - z * q^5 + (-4*z + 4) * q^7 + (-3*z + 3) * q^9 + (2*z + 2) * q^12 - z * q^13 + (-z - 1) * q^15 + (4*z - 4) * q^16 - 3 * q^17 + 2 * q^19 + (-2*z + 2) * q^20 + (-8*z + 4) * q^21 - 3*z * q^23 + (z - 1) * q^25 + (-6*z + 3) * q^27 + 8 * q^28 + (6*z - 6) * q^29 + 4*z * q^31 - 4 * q^35 + 6 * q^36 + 8 * q^37 + (-z - 1) * q^39 + 12*z * q^41 + (-z + 1) * q^43 - 3 * q^45 + (8*z - 4) * q^48 - 9*z * q^49 + (3*z - 6) * q^51 + (-2*z + 2) * q^52 - 3 * q^53 + (-2*z + 4) * q^57 - 6*z * q^59 + (-4*z + 2) * q^60 + (11*z - 11) * q^61 - 12*z * q^63 - 8 * q^64 + (z - 1) * q^65 + 4*z * q^67 - 6*z * q^68 + (-3*z - 3) * q^69 + 6 * q^71 + 8 * q^73 + (2*z - 1) * q^75 + 4*z * q^76 + (-7*z + 7) * q^79 + 4 * q^80 - 9*z * q^81 + (6*z - 6) * q^83 + (-8*z + 16) * q^84 + 3*z * q^85 + (12*z - 6) * q^87 - 18 * q^89 - 4 * q^91 + (-6*z + 6) * q^92 + (4*z + 4) * q^93 - 2*z * q^95 + (14*z - 14) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 2 q^{4} - q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 2 * q^4 - q^5 + 4 * q^7 + 3 * q^9 $$2 q + 3 q^{3} + 2 q^{4} - q^{5} + 4 q^{7} + 3 q^{9} + 6 q^{12} - q^{13} - 3 q^{15} - 4 q^{16} - 6 q^{17} + 4 q^{19} + 2 q^{20} - 3 q^{23} - q^{25} + 16 q^{28} - 6 q^{29} + 4 q^{31} - 8 q^{35} + 12 q^{36} + 16 q^{37} - 3 q^{39} + 12 q^{41} + q^{43} - 6 q^{45} - 9 q^{49} - 9 q^{51} + 2 q^{52} - 6 q^{53} + 6 q^{57} - 6 q^{59} - 11 q^{61} - 12 q^{63} - 16 q^{64} - q^{65} + 4 q^{67} - 6 q^{68} - 9 q^{69} + 12 q^{71} + 16 q^{73} + 4 q^{76} + 7 q^{79} + 8 q^{80} - 9 q^{81} - 6 q^{83} + 24 q^{84} + 3 q^{85} - 36 q^{89} - 8 q^{91} + 6 q^{92} + 12 q^{93} - 2 q^{95} - 14 q^{97}+O(q^{100})$$ 2 * q + 3 * q^3 + 2 * q^4 - q^5 + 4 * q^7 + 3 * q^9 + 6 * q^12 - q^13 - 3 * q^15 - 4 * q^16 - 6 * q^17 + 4 * q^19 + 2 * q^20 - 3 * q^23 - q^25 + 16 * q^28 - 6 * q^29 + 4 * q^31 - 8 * q^35 + 12 * q^36 + 16 * q^37 - 3 * q^39 + 12 * q^41 + q^43 - 6 * q^45 - 9 * q^49 - 9 * q^51 + 2 * q^52 - 6 * q^53 + 6 * q^57 - 6 * q^59 - 11 * q^61 - 12 * q^63 - 16 * q^64 - q^65 + 4 * q^67 - 6 * q^68 - 9 * q^69 + 12 * q^71 + 16 * q^73 + 4 * q^76 + 7 * q^79 + 8 * q^80 - 9 * q^81 - 6 * q^83 + 24 * q^84 + 3 * q^85 - 36 * q^89 - 8 * q^91 + 6 * q^92 + 12 * q^93 - 2 * q^95 - 14 * q^97

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
196.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 0.866025i 1.00000 + 1.73205i −0.500000 0.866025i 0 2.00000 3.46410i 0 1.50000 2.59808i 0
391.1 0 1.50000 + 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i 0 2.00000 + 3.46410i 0 1.50000 + 2.59808i 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
9.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.i.a 2
3.b odd 2 1 1755.2.i.d 2
9.c even 3 1 inner 585.2.i.a 2
9.c even 3 1 5265.2.a.i 1
9.d odd 6 1 1755.2.i.d 2
9.d odd 6 1 5265.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.a 2 1.a even 1 1 trivial
585.2.i.a 2 9.c even 3 1 inner
1755.2.i.d 2 3.b odd 2 1
1755.2.i.d 2 9.d odd 6 1
5265.2.a.g 1 9.d odd 6 1
5265.2.a.i 1 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{2} - 4T_{7} + 16$$ T7^2 - 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} - 4T + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + T + 1$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 12T + 144$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 6)^{2}$$
$73$ $$(T - 8)^{2}$$
$79$ $$T^{2} - 7T + 49$$
$83$ $$T^{2} + 6T + 36$$
$89$ $$(T + 18)^{2}$$
$97$ $$T^{2} + 14T + 196$$