# Properties

 Label 585.2.a.k Level $585$ Weight $2$ Character orbit 585.a Self dual yes Analytic conductor $4.671$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} + q^{5} + 2 q^{7} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 + q^5 + 2 * q^7 - b * q^8 $$q + \beta q^{2} + q^{4} + q^{5} + 2 q^{7} - \beta q^{8} + \beta q^{10} + (\beta + 3) q^{11} + q^{13} + 2 \beta q^{14} - 5 q^{16} + 2 \beta q^{17} + ( - 3 \beta - 1) q^{19} + q^{20} + (3 \beta + 3) q^{22} + (\beta - 3) q^{23} + q^{25} + \beta q^{26} + 2 q^{28} + ( - 2 \beta + 6) q^{29} + (3 \beta + 5) q^{31} - 3 \beta q^{32} + 6 q^{34} + 2 q^{35} - 4 q^{37} + ( - \beta - 9) q^{38} - \beta q^{40} - 2 \beta q^{41} + ( - 3 \beta + 5) q^{43} + (\beta + 3) q^{44} + ( - 3 \beta + 3) q^{46} - 6 q^{47} - 3 q^{49} + \beta q^{50} + q^{52} - 6 \beta q^{53} + (\beta + 3) q^{55} - 2 \beta q^{56} + (6 \beta - 6) q^{58} + ( - 7 \beta + 3) q^{59} + ( - 6 \beta + 2) q^{61} + (5 \beta + 9) q^{62} + q^{64} + q^{65} + (6 \beta - 4) q^{67} + 2 \beta q^{68} + 2 \beta q^{70} + ( - \beta - 3) q^{71} - 4 q^{73} - 4 \beta q^{74} + ( - 3 \beta - 1) q^{76} + (2 \beta + 6) q^{77} + ( - 6 \beta + 2) q^{79} - 5 q^{80} - 6 q^{82} + 6 q^{83} + 2 \beta q^{85} + (5 \beta - 9) q^{86} + ( - 3 \beta - 3) q^{88} + (4 \beta + 6) q^{89} + 2 q^{91} + (\beta - 3) q^{92} - 6 \beta q^{94} + ( - 3 \beta - 1) q^{95} + 2 q^{97} - 3 \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 + q^5 + 2 * q^7 - b * q^8 + b * q^10 + (b + 3) * q^11 + q^13 + 2*b * q^14 - 5 * q^16 + 2*b * q^17 + (-3*b - 1) * q^19 + q^20 + (3*b + 3) * q^22 + (b - 3) * q^23 + q^25 + b * q^26 + 2 * q^28 + (-2*b + 6) * q^29 + (3*b + 5) * q^31 - 3*b * q^32 + 6 * q^34 + 2 * q^35 - 4 * q^37 + (-b - 9) * q^38 - b * q^40 - 2*b * q^41 + (-3*b + 5) * q^43 + (b + 3) * q^44 + (-3*b + 3) * q^46 - 6 * q^47 - 3 * q^49 + b * q^50 + q^52 - 6*b * q^53 + (b + 3) * q^55 - 2*b * q^56 + (6*b - 6) * q^58 + (-7*b + 3) * q^59 + (-6*b + 2) * q^61 + (5*b + 9) * q^62 + q^64 + q^65 + (6*b - 4) * q^67 + 2*b * q^68 + 2*b * q^70 + (-b - 3) * q^71 - 4 * q^73 - 4*b * q^74 + (-3*b - 1) * q^76 + (2*b + 6) * q^77 + (-6*b + 2) * q^79 - 5 * q^80 - 6 * q^82 + 6 * q^83 + 2*b * q^85 + (5*b - 9) * q^86 + (-3*b - 3) * q^88 + (4*b + 6) * q^89 + 2 * q^91 + (b - 3) * q^92 - 6*b * q^94 + (-3*b - 1) * q^95 + 2 * q^97 - 3*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 $$2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{11} + 2 q^{13} - 10 q^{16} - 2 q^{19} + 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{28} + 12 q^{29} + 10 q^{31} + 12 q^{34} + 4 q^{35} - 8 q^{37} - 18 q^{38} + 10 q^{43} + 6 q^{44} + 6 q^{46} - 12 q^{47} - 6 q^{49} + 2 q^{52} + 6 q^{55} - 12 q^{58} + 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 6 q^{71} - 8 q^{73} - 2 q^{76} + 12 q^{77} + 4 q^{79} - 10 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 6 q^{92} - 2 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 + 6 * q^11 + 2 * q^13 - 10 * q^16 - 2 * q^19 + 2 * q^20 + 6 * q^22 - 6 * q^23 + 2 * q^25 + 4 * q^28 + 12 * q^29 + 10 * q^31 + 12 * q^34 + 4 * q^35 - 8 * q^37 - 18 * q^38 + 10 * q^43 + 6 * q^44 + 6 * q^46 - 12 * q^47 - 6 * q^49 + 2 * q^52 + 6 * q^55 - 12 * q^58 + 6 * q^59 + 4 * q^61 + 18 * q^62 + 2 * q^64 + 2 * q^65 - 8 * q^67 - 6 * q^71 - 8 * q^73 - 2 * q^76 + 12 * q^77 + 4 * q^79 - 10 * q^80 - 12 * q^82 + 12 * q^83 - 18 * q^86 - 6 * q^88 + 12 * q^89 + 4 * q^91 - 6 * q^92 - 2 * q^95 + 4 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 0 1.00000 1.00000 0 2.00000 1.73205 0 −1.73205
1.2 1.73205 0 1.00000 1.00000 0 2.00000 −1.73205 0 1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.a.k 2
3.b odd 2 1 65.2.a.c 2
4.b odd 2 1 9360.2.a.cm 2
5.b even 2 1 2925.2.a.z 2
5.c odd 4 2 2925.2.c.v 4
12.b even 2 1 1040.2.a.h 2
13.b even 2 1 7605.2.a.be 2
15.d odd 2 1 325.2.a.g 2
15.e even 4 2 325.2.b.e 4
21.c even 2 1 3185.2.a.k 2
24.f even 2 1 4160.2.a.bj 2
24.h odd 2 1 4160.2.a.y 2
33.d even 2 1 7865.2.a.h 2
39.d odd 2 1 845.2.a.d 2
39.f even 4 2 845.2.c.e 4
39.h odd 6 2 845.2.e.f 4
39.i odd 6 2 845.2.e.e 4
39.k even 12 2 845.2.m.a 4
39.k even 12 2 845.2.m.c 4
60.h even 2 1 5200.2.a.ca 2
195.e odd 2 1 4225.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.c 2 3.b odd 2 1
325.2.a.g 2 15.d odd 2 1
325.2.b.e 4 15.e even 4 2
585.2.a.k 2 1.a even 1 1 trivial
845.2.a.d 2 39.d odd 2 1
845.2.c.e 4 39.f even 4 2
845.2.e.e 4 39.i odd 6 2
845.2.e.f 4 39.h odd 6 2
845.2.m.a 4 39.k even 12 2
845.2.m.c 4 39.k even 12 2
1040.2.a.h 2 12.b even 2 1
2925.2.a.z 2 5.b even 2 1
2925.2.c.v 4 5.c odd 4 2
3185.2.a.k 2 21.c even 2 1
4160.2.a.y 2 24.h odd 2 1
4160.2.a.bj 2 24.f even 2 1
4225.2.a.w 2 195.e odd 2 1
5200.2.a.ca 2 60.h even 2 1
7605.2.a.be 2 13.b even 2 1
7865.2.a.h 2 33.d even 2 1
9360.2.a.cm 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(585))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{7} - 2$$ T7 - 2 $$T_{11}^{2} - 6T_{11} + 6$$ T11^2 - 6*T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} - 6T + 6$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 12$$
$19$ $$T^{2} + 2T - 26$$
$23$ $$T^{2} + 6T + 6$$
$29$ $$T^{2} - 12T + 24$$
$31$ $$T^{2} - 10T - 2$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 12$$
$43$ $$T^{2} - 10T - 2$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} - 108$$
$59$ $$T^{2} - 6T - 138$$
$61$ $$T^{2} - 4T - 104$$
$67$ $$T^{2} + 8T - 92$$
$71$ $$T^{2} + 6T + 6$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 4T - 104$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 12T - 12$$
$97$ $$(T - 2)^{2}$$