# Properties

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

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## 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + (\zeta_{6} - 2) q^{3} - 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + (4 \zeta_{6} - 2) q^{6} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^2 + (z - 2) * q^3 - 2*z * q^4 + z * q^5 + (4*z - 2) * q^6 + (-3*z + 3) * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{2} + (\zeta_{6} - 2) q^{3} - 2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + (4 \zeta_{6} - 2) q^{6} + ( - 3 \zeta_{6} + 3) q^{9} + 2 q^{10} + (4 \zeta_{6} - 4) q^{11} + (2 \zeta_{6} + 2) q^{12} + \zeta_{6} q^{13} + ( - \zeta_{6} - 1) q^{15} + ( - 4 \zeta_{6} + 4) q^{16} + 7 q^{17} - 6 \zeta_{6} q^{18} + 4 q^{19} + ( - 2 \zeta_{6} + 2) q^{20} + 8 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 2 q^{26} + (6 \zeta_{6} - 3) q^{27} + ( - 10 \zeta_{6} + 10) q^{29} + (2 \zeta_{6} - 4) q^{30} + 6 \zeta_{6} q^{31} - 8 \zeta_{6} q^{32} + ( - 8 \zeta_{6} + 4) q^{33} + ( - 14 \zeta_{6} + 14) q^{34} - 6 q^{36} - 10 q^{37} + ( - 8 \zeta_{6} + 8) q^{38} + ( - \zeta_{6} - 1) q^{39} - 2 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + 8 q^{44} + 3 q^{45} + 6 q^{46} + ( - 2 \zeta_{6} + 2) q^{47} + (8 \zeta_{6} - 4) q^{48} + 7 \zeta_{6} q^{49} + 2 \zeta_{6} q^{50} + (7 \zeta_{6} - 14) q^{51} + ( - 2 \zeta_{6} + 2) q^{52} - 5 q^{53} + (6 \zeta_{6} + 6) q^{54} - 4 q^{55} + (4 \zeta_{6} - 8) q^{57} - 20 \zeta_{6} q^{58} - 10 \zeta_{6} q^{59} + (4 \zeta_{6} - 2) q^{60} + ( - 5 \zeta_{6} + 5) q^{61} + 12 q^{62} - 8 q^{64} + (\zeta_{6} - 1) q^{65} + ( - 8 \zeta_{6} - 8) q^{66} + 12 \zeta_{6} q^{67} - 14 \zeta_{6} q^{68} + ( - 3 \zeta_{6} - 3) q^{69} + 8 q^{71} - 14 q^{73} + (20 \zeta_{6} - 20) q^{74} + ( - 2 \zeta_{6} + 1) q^{75} - 8 \zeta_{6} q^{76} + (2 \zeta_{6} - 4) q^{78} + (9 \zeta_{6} - 9) q^{79} + 4 q^{80} - 9 \zeta_{6} q^{81} - 4 q^{82} + (6 \zeta_{6} - 6) q^{83} + 7 \zeta_{6} q^{85} + 2 \zeta_{6} q^{86} + (20 \zeta_{6} - 10) q^{87} + 12 q^{89} + ( - 6 \zeta_{6} + 6) q^{90} + ( - 6 \zeta_{6} + 6) q^{92} + ( - 6 \zeta_{6} - 6) q^{93} - 4 \zeta_{6} q^{94} + 4 \zeta_{6} q^{95} + (8 \zeta_{6} + 8) q^{96} + (8 \zeta_{6} - 8) q^{97} + 14 q^{98} + 12 \zeta_{6} q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + (z - 2) * q^3 - 2*z * q^4 + z * q^5 + (4*z - 2) * q^6 + (-3*z + 3) * q^9 + 2 * q^10 + (4*z - 4) * q^11 + (2*z + 2) * q^12 + z * q^13 + (-z - 1) * q^15 + (-4*z + 4) * q^16 + 7 * q^17 - 6*z * q^18 + 4 * q^19 + (-2*z + 2) * q^20 + 8*z * q^22 + 3*z * q^23 + (z - 1) * q^25 + 2 * q^26 + (6*z - 3) * q^27 + (-10*z + 10) * q^29 + (2*z - 4) * q^30 + 6*z * q^31 - 8*z * q^32 + (-8*z + 4) * q^33 + (-14*z + 14) * q^34 - 6 * q^36 - 10 * q^37 + (-8*z + 8) * q^38 + (-z - 1) * q^39 - 2*z * q^41 + (z - 1) * q^43 + 8 * q^44 + 3 * q^45 + 6 * q^46 + (-2*z + 2) * q^47 + (8*z - 4) * q^48 + 7*z * q^49 + 2*z * q^50 + (7*z - 14) * q^51 + (-2*z + 2) * q^52 - 5 * q^53 + (6*z + 6) * q^54 - 4 * q^55 + (4*z - 8) * q^57 - 20*z * q^58 - 10*z * q^59 + (4*z - 2) * q^60 + (-5*z + 5) * q^61 + 12 * q^62 - 8 * q^64 + (z - 1) * q^65 + (-8*z - 8) * q^66 + 12*z * q^67 - 14*z * q^68 + (-3*z - 3) * q^69 + 8 * q^71 - 14 * q^73 + (20*z - 20) * q^74 + (-2*z + 1) * q^75 - 8*z * q^76 + (2*z - 4) * q^78 + (9*z - 9) * q^79 + 4 * q^80 - 9*z * q^81 - 4 * q^82 + (6*z - 6) * q^83 + 7*z * q^85 + 2*z * q^86 + (20*z - 10) * q^87 + 12 * q^89 + (-6*z + 6) * q^90 + (-6*z + 6) * q^92 + (-6*z - 6) * q^93 - 4*z * q^94 + 4*z * q^95 + (8*z + 8) * q^96 + (8*z - 8) * q^97 + 14 * q^98 + 12*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 - 2 * q^4 + q^5 + 3 * q^9 $$2 q + 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{5} + 3 q^{9} + 4 q^{10} - 4 q^{11} + 6 q^{12} + q^{13} - 3 q^{15} + 4 q^{16} + 14 q^{17} - 6 q^{18} + 8 q^{19} + 2 q^{20} + 8 q^{22} + 3 q^{23} - q^{25} + 4 q^{26} + 10 q^{29} - 6 q^{30} + 6 q^{31} - 8 q^{32} + 14 q^{34} - 12 q^{36} - 20 q^{37} + 8 q^{38} - 3 q^{39} - 2 q^{41} - q^{43} + 16 q^{44} + 6 q^{45} + 12 q^{46} + 2 q^{47} + 7 q^{49} + 2 q^{50} - 21 q^{51} + 2 q^{52} - 10 q^{53} + 18 q^{54} - 8 q^{55} - 12 q^{57} - 20 q^{58} - 10 q^{59} + 5 q^{61} + 24 q^{62} - 16 q^{64} - q^{65} - 24 q^{66} + 12 q^{67} - 14 q^{68} - 9 q^{69} + 16 q^{71} - 28 q^{73} - 20 q^{74} - 8 q^{76} - 6 q^{78} - 9 q^{79} + 8 q^{80} - 9 q^{81} - 8 q^{82} - 6 q^{83} + 7 q^{85} + 2 q^{86} + 24 q^{89} + 6 q^{90} + 6 q^{92} - 18 q^{93} - 4 q^{94} + 4 q^{95} + 24 q^{96} - 8 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 - 2 * q^4 + q^5 + 3 * q^9 + 4 * q^10 - 4 * q^11 + 6 * q^12 + q^13 - 3 * q^15 + 4 * q^16 + 14 * q^17 - 6 * q^18 + 8 * q^19 + 2 * q^20 + 8 * q^22 + 3 * q^23 - q^25 + 4 * q^26 + 10 * q^29 - 6 * q^30 + 6 * q^31 - 8 * q^32 + 14 * q^34 - 12 * q^36 - 20 * q^37 + 8 * q^38 - 3 * q^39 - 2 * q^41 - q^43 + 16 * q^44 + 6 * q^45 + 12 * q^46 + 2 * q^47 + 7 * q^49 + 2 * q^50 - 21 * q^51 + 2 * q^52 - 10 * q^53 + 18 * q^54 - 8 * q^55 - 12 * q^57 - 20 * q^58 - 10 * q^59 + 5 * q^61 + 24 * q^62 - 16 * q^64 - q^65 - 24 * q^66 + 12 * q^67 - 14 * q^68 - 9 * q^69 + 16 * q^71 - 28 * q^73 - 20 * q^74 - 8 * q^76 - 6 * q^78 - 9 * q^79 + 8 * q^80 - 9 * q^81 - 8 * q^82 - 6 * q^83 + 7 * q^85 + 2 * q^86 + 24 * q^89 + 6 * q^90 + 6 * q^92 - 18 * q^93 - 4 * q^94 + 4 * q^95 + 24 * q^96 - 8 * q^97 + 28 * q^98 + 12 * q^99

## 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
1.00000 1.73205i −1.50000 + 0.866025i −1.00000 1.73205i 0.500000 + 0.866025i 3.46410i 0 0 1.50000 2.59808i 2.00000
391.1 1.00000 + 1.73205i −1.50000 0.866025i −1.00000 + 1.73205i 0.500000 0.866025i 3.46410i 0 0 1.50000 + 2.59808i 2.00000
 $$n$$: 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.d 2
3.b odd 2 1 1755.2.i.a 2
9.c even 3 1 inner 585.2.i.d 2
9.c even 3 1 5265.2.a.a 1
9.d odd 6 1 1755.2.i.a 2
9.d odd 6 1 5265.2.a.p 1

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T - 7)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$T^{2} - 6T + 36$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 2T + 4$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$(T + 5)^{2}$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T + 14)^{2}$$
$79$ $$T^{2} + 9T + 81$$
$83$ $$T^{2} + 6T + 36$$
$89$ $$(T - 12)^{2}$$
$97$ $$T^{2} + 8T + 64$$
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