# Properties

 Label 585.2.i.b 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + (\zeta_{6} + 1) q^{6} + (2 \zeta_{6} - 2) q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ q + (-z + 1) * q^2 + (2*z - 1) * q^3 + z * q^4 + z * q^5 + (z + 1) * q^6 + (2*z - 2) * q^7 + 3 * q^8 - 3 * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + (\zeta_{6} + 1) q^{6} + (2 \zeta_{6} - 2) q^{7} + 3 q^{8} - 3 q^{9} + q^{10} + ( - \zeta_{6} + 1) q^{11} + (\zeta_{6} - 2) q^{12} + \zeta_{6} q^{13} + 2 \zeta_{6} q^{14} + (\zeta_{6} - 2) q^{15} + ( - \zeta_{6} + 1) q^{16} + 2 q^{17} + (3 \zeta_{6} - 3) q^{18} - 3 q^{19} + (\zeta_{6} - 1) q^{20} + ( - 2 \zeta_{6} - 2) q^{21} - \zeta_{6} q^{22} + (6 \zeta_{6} - 3) q^{24} + (\zeta_{6} - 1) q^{25} + q^{26} + ( - 6 \zeta_{6} + 3) q^{27} - 2 q^{28} + (5 \zeta_{6} - 5) q^{29} + (2 \zeta_{6} - 1) q^{30} + \zeta_{6} q^{31} + 5 \zeta_{6} q^{32} + (\zeta_{6} + 1) q^{33} + ( - 2 \zeta_{6} + 2) q^{34} - 2 q^{35} - 3 \zeta_{6} q^{36} - 5 q^{37} + (3 \zeta_{6} - 3) q^{38} + (\zeta_{6} - 2) q^{39} + 3 \zeta_{6} q^{40} + (2 \zeta_{6} - 4) q^{42} + ( - 8 \zeta_{6} + 8) q^{43} + q^{44} - 3 \zeta_{6} q^{45} + ( - 2 \zeta_{6} + 2) q^{47} + (\zeta_{6} + 1) q^{48} + 3 \zeta_{6} q^{49} + \zeta_{6} q^{50} + (4 \zeta_{6} - 2) q^{51} + (\zeta_{6} - 1) q^{52} + 14 q^{53} + ( - 3 \zeta_{6} - 3) q^{54} + q^{55} + (6 \zeta_{6} - 6) q^{56} + ( - 6 \zeta_{6} + 3) q^{57} + 5 \zeta_{6} q^{58} + 9 \zeta_{6} q^{59} + ( - \zeta_{6} - 1) q^{60} + ( - \zeta_{6} + 1) q^{61} + q^{62} + ( - 6 \zeta_{6} + 6) q^{63} + 7 q^{64} + (\zeta_{6} - 1) q^{65} + ( - \zeta_{6} + 2) q^{66} - 14 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} + (2 \zeta_{6} - 2) q^{70} - 9 q^{72} + 6 q^{73} + (5 \zeta_{6} - 5) q^{74} + ( - \zeta_{6} - 1) q^{75} - 3 \zeta_{6} q^{76} + 2 \zeta_{6} q^{77} + (2 \zeta_{6} - 1) q^{78} + ( - 12 \zeta_{6} + 12) q^{79} + q^{80} + 9 q^{81} + ( - 10 \zeta_{6} + 10) q^{83} + ( - 4 \zeta_{6} + 2) q^{84} + 2 \zeta_{6} q^{85} - 8 \zeta_{6} q^{86} + ( - 5 \zeta_{6} - 5) q^{87} + ( - 3 \zeta_{6} + 3) q^{88} + 2 q^{89} - 3 q^{90} - 2 q^{91} + (\zeta_{6} - 2) q^{93} - 2 \zeta_{6} q^{94} - 3 \zeta_{6} q^{95} + (5 \zeta_{6} - 10) q^{96} + (\zeta_{6} - 1) q^{97} + 3 q^{98} + (3 \zeta_{6} - 3) q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (2*z - 1) * q^3 + z * q^4 + z * q^5 + (z + 1) * q^6 + (2*z - 2) * q^7 + 3 * q^8 - 3 * q^9 + q^10 + (-z + 1) * q^11 + (z - 2) * q^12 + z * q^13 + 2*z * q^14 + (z - 2) * q^15 + (-z + 1) * q^16 + 2 * q^17 + (3*z - 3) * q^18 - 3 * q^19 + (z - 1) * q^20 + (-2*z - 2) * q^21 - z * q^22 + (6*z - 3) * q^24 + (z - 1) * q^25 + q^26 + (-6*z + 3) * q^27 - 2 * q^28 + (5*z - 5) * q^29 + (2*z - 1) * q^30 + z * q^31 + 5*z * q^32 + (z + 1) * q^33 + (-2*z + 2) * q^34 - 2 * q^35 - 3*z * q^36 - 5 * q^37 + (3*z - 3) * q^38 + (z - 2) * q^39 + 3*z * q^40 + (2*z - 4) * q^42 + (-8*z + 8) * q^43 + q^44 - 3*z * q^45 + (-2*z + 2) * q^47 + (z + 1) * q^48 + 3*z * q^49 + z * q^50 + (4*z - 2) * q^51 + (z - 1) * q^52 + 14 * q^53 + (-3*z - 3) * q^54 + q^55 + (6*z - 6) * q^56 + (-6*z + 3) * q^57 + 5*z * q^58 + 9*z * q^59 + (-z - 1) * q^60 + (-z + 1) * q^61 + q^62 + (-6*z + 6) * q^63 + 7 * q^64 + (z - 1) * q^65 + (-z + 2) * q^66 - 14*z * q^67 + 2*z * q^68 + (2*z - 2) * q^70 - 9 * q^72 + 6 * q^73 + (5*z - 5) * q^74 + (-z - 1) * q^75 - 3*z * q^76 + 2*z * q^77 + (2*z - 1) * q^78 + (-12*z + 12) * q^79 + q^80 + 9 * q^81 + (-10*z + 10) * q^83 + (-4*z + 2) * q^84 + 2*z * q^85 - 8*z * q^86 + (-5*z - 5) * q^87 + (-3*z + 3) * q^88 + 2 * q^89 - 3 * q^90 - 2 * q^91 + (z - 2) * q^93 - 2*z * q^94 - 3*z * q^95 + (5*z - 10) * q^96 + (z - 1) * q^97 + 3 * q^98 + (3*z - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + q^{5} + 3 q^{6} - 2 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q + q^2 + q^4 + q^5 + 3 * q^6 - 2 * q^7 + 6 * q^8 - 6 * q^9 $$2 q + q^{2} + q^{4} + q^{5} + 3 q^{6} - 2 q^{7} + 6 q^{8} - 6 q^{9} + 2 q^{10} + q^{11} - 3 q^{12} + q^{13} + 2 q^{14} - 3 q^{15} + q^{16} + 4 q^{17} - 3 q^{18} - 6 q^{19} - q^{20} - 6 q^{21} - q^{22} - q^{25} + 2 q^{26} - 4 q^{28} - 5 q^{29} + q^{31} + 5 q^{32} + 3 q^{33} + 2 q^{34} - 4 q^{35} - 3 q^{36} - 10 q^{37} - 3 q^{38} - 3 q^{39} + 3 q^{40} - 6 q^{42} + 8 q^{43} + 2 q^{44} - 3 q^{45} + 2 q^{47} + 3 q^{48} + 3 q^{49} + q^{50} - q^{52} + 28 q^{53} - 9 q^{54} + 2 q^{55} - 6 q^{56} + 5 q^{58} + 9 q^{59} - 3 q^{60} + q^{61} + 2 q^{62} + 6 q^{63} + 14 q^{64} - q^{65} + 3 q^{66} - 14 q^{67} + 2 q^{68} - 2 q^{70} - 18 q^{72} + 12 q^{73} - 5 q^{74} - 3 q^{75} - 3 q^{76} + 2 q^{77} + 12 q^{79} + 2 q^{80} + 18 q^{81} + 10 q^{83} + 2 q^{85} - 8 q^{86} - 15 q^{87} + 3 q^{88} + 4 q^{89} - 6 q^{90} - 4 q^{91} - 3 q^{93} - 2 q^{94} - 3 q^{95} - 15 q^{96} - q^{97} + 6 q^{98} - 3 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^4 + q^5 + 3 * q^6 - 2 * q^7 + 6 * q^8 - 6 * q^9 + 2 * q^10 + q^11 - 3 * q^12 + q^13 + 2 * q^14 - 3 * q^15 + q^16 + 4 * q^17 - 3 * q^18 - 6 * q^19 - q^20 - 6 * q^21 - q^22 - q^25 + 2 * q^26 - 4 * q^28 - 5 * q^29 + q^31 + 5 * q^32 + 3 * q^33 + 2 * q^34 - 4 * q^35 - 3 * q^36 - 10 * q^37 - 3 * q^38 - 3 * q^39 + 3 * q^40 - 6 * q^42 + 8 * q^43 + 2 * q^44 - 3 * q^45 + 2 * q^47 + 3 * q^48 + 3 * q^49 + q^50 - q^52 + 28 * q^53 - 9 * q^54 + 2 * q^55 - 6 * q^56 + 5 * q^58 + 9 * q^59 - 3 * q^60 + q^61 + 2 * q^62 + 6 * q^63 + 14 * q^64 - q^65 + 3 * q^66 - 14 * q^67 + 2 * q^68 - 2 * q^70 - 18 * q^72 + 12 * q^73 - 5 * q^74 - 3 * q^75 - 3 * q^76 + 2 * q^77 + 12 * q^79 + 2 * q^80 + 18 * q^81 + 10 * q^83 + 2 * q^85 - 8 * q^86 - 15 * q^87 + 3 * q^88 + 4 * q^89 - 6 * q^90 - 4 * q^91 - 3 * q^93 - 2 * q^94 - 3 * q^95 - 15 * q^96 - q^97 + 6 * q^98 - 3 * 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
0.500000 0.866025i 1.73205i 0.500000 + 0.866025i 0.500000 + 0.866025i 1.50000 + 0.866025i −1.00000 + 1.73205i 3.00000 −3.00000 1.00000
391.1 0.500000 + 0.866025i 1.73205i 0.500000 0.866025i 0.500000 0.866025i 1.50000 0.866025i −1.00000 1.73205i 3.00000 −3.00000 1.00000
 $$n$$: 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.b 2
3.b odd 2 1 1755.2.i.b 2
9.c even 3 1 inner 585.2.i.b 2
9.c even 3 1 5265.2.a.e 1
9.d odd 6 1 1755.2.i.b 2
9.d odd 6 1 5265.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.b 2 1.a even 1 1 trivial
585.2.i.b 2 9.c even 3 1 inner
1755.2.i.b 2 3.b odd 2 1
1755.2.i.b 2 9.d odd 6 1
5265.2.a.e 1 9.c even 3 1
5265.2.a.m 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 $$S_{2}^{\mathrm{new}}(585, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T + 3)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 5T + 25$$
$31$ $$T^{2} - T + 1$$
$37$ $$(T + 5)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$(T - 14)^{2}$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$T^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} - 12T + 144$$
$83$ $$T^{2} - 10T + 100$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + T + 1$$