# Properties

 Label 585.2.b.b Level $585$ Weight $2$ Character orbit 585.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + i q^{5} + 4 i q^{7} + 3 i q^{8}+O(q^{10})$$ q + i * q^2 + q^4 + i * q^5 + 4*i * q^7 + 3*i * q^8 $$q + i q^{2} + q^{4} + i q^{5} + 4 i q^{7} + 3 i q^{8} - q^{10} + ( - 2 i + 3) q^{13} - 4 q^{14} - q^{16} - 4 q^{17} - 4 i q^{19} + i q^{20} - 8 q^{23} - q^{25} + (3 i + 2) q^{26} + 4 i q^{28} + 8 q^{29} + 4 i q^{31} + 5 i q^{32} - 4 i q^{34} - 4 q^{35} + 4 i q^{37} + 4 q^{38} - 3 q^{40} - 2 i q^{41} + 8 q^{43} - 8 i q^{46} + 8 i q^{47} - 9 q^{49} - i q^{50} + ( - 2 i + 3) q^{52} + 12 q^{53} - 12 q^{56} + 8 i q^{58} - 2 q^{61} - 4 q^{62} - 7 q^{64} + (3 i + 2) q^{65} - 12 i q^{67} - 4 q^{68} - 4 i q^{70} - 4 i q^{71} + 4 i q^{73} - 4 q^{74} - 4 i q^{76} - 8 q^{79} - i q^{80} + 2 q^{82} - 12 i q^{83} - 4 i q^{85} + 8 i q^{86} - 18 i q^{89} + (12 i + 8) q^{91} - 8 q^{92} - 8 q^{94} + 4 q^{95} + 4 i q^{97} - 9 i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 + i * q^5 + 4*i * q^7 + 3*i * q^8 - q^10 + (-2*i + 3) * q^13 - 4 * q^14 - q^16 - 4 * q^17 - 4*i * q^19 + i * q^20 - 8 * q^23 - q^25 + (3*i + 2) * q^26 + 4*i * q^28 + 8 * q^29 + 4*i * q^31 + 5*i * q^32 - 4*i * q^34 - 4 * q^35 + 4*i * q^37 + 4 * q^38 - 3 * q^40 - 2*i * q^41 + 8 * q^43 - 8*i * q^46 + 8*i * q^47 - 9 * q^49 - i * q^50 + (-2*i + 3) * q^52 + 12 * q^53 - 12 * q^56 + 8*i * q^58 - 2 * q^61 - 4 * q^62 - 7 * q^64 + (3*i + 2) * q^65 - 12*i * q^67 - 4 * q^68 - 4*i * q^70 - 4*i * q^71 + 4*i * q^73 - 4 * q^74 - 4*i * q^76 - 8 * q^79 - i * q^80 + 2 * q^82 - 12*i * q^83 - 4*i * q^85 + 8*i * q^86 - 18*i * q^89 + (12*i + 8) * q^91 - 8 * q^92 - 8 * q^94 + 4 * q^95 + 4*i * q^97 - 9*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} - 2 q^{10} + 6 q^{13} - 8 q^{14} - 2 q^{16} - 8 q^{17} - 16 q^{23} - 2 q^{25} + 4 q^{26} + 16 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} - 18 q^{49} + 6 q^{52} + 24 q^{53} - 24 q^{56} - 4 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{65} - 8 q^{68} - 8 q^{74} - 16 q^{79} + 4 q^{82} + 16 q^{91} - 16 q^{92} - 16 q^{94} + 8 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^10 + 6 * q^13 - 8 * q^14 - 2 * q^16 - 8 * q^17 - 16 * q^23 - 2 * q^25 + 4 * q^26 + 16 * q^29 - 8 * q^35 + 8 * q^38 - 6 * q^40 + 16 * q^43 - 18 * q^49 + 6 * q^52 + 24 * q^53 - 24 * q^56 - 4 * q^61 - 8 * q^62 - 14 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^74 - 16 * q^79 + 4 * q^82 + 16 * q^91 - 16 * q^92 - 16 * q^94 + 8 * q^95

## Character values

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

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 1.00000i 0 4.00000i 3.00000i 0 −1.00000
181.2 1.00000i 0 1.00000 1.00000i 0 4.00000i 3.00000i 0 −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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.b.b 2
3.b odd 2 1 585.2.b.c yes 2
13.b even 2 1 inner 585.2.b.b 2
13.d odd 4 1 7605.2.a.e 1
13.d odd 4 1 7605.2.a.o 1
39.d odd 2 1 585.2.b.c yes 2
39.f even 4 1 7605.2.a.b 1
39.f even 4 1 7605.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.b.b 2 1.a even 1 1 trivial
585.2.b.b 2 13.b even 2 1 inner
585.2.b.c yes 2 3.b odd 2 1
585.2.b.c yes 2 39.d odd 2 1
7605.2.a.b 1 39.f even 4 1
7605.2.a.e 1 13.d odd 4 1
7605.2.a.o 1 13.d odd 4 1
7605.2.a.r 1 39.f even 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{17} + 4$$ T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T + 4)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 8)^{2}$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 16$$
$41$ $$T^{2} + 4$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2} + 16$$
$73$ $$T^{2} + 16$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 324$$
$97$ $$T^{2} + 16$$