# Properties

 Label 585.2.h.c Level $585$ Weight $2$ Character orbit 585.h 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.h (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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 65) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + (\beta + 1) q^{5} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + (b + 1) * q^5 - 3 * q^8 $$q + q^{2} - q^{4} + (\beta + 1) q^{5} - 3 q^{8} + (\beta + 1) q^{10} + \beta q^{11} + (\beta + 3) q^{13} - q^{16} + 3 \beta q^{19} + ( - \beta - 1) q^{20} + \beta q^{22} + 3 \beta q^{23} + (2 \beta - 3) q^{25} + (\beta + 3) q^{26} - 6 q^{29} - 3 \beta q^{31} + 5 q^{32} + 6 q^{37} + 3 \beta q^{38} + ( - 3 \beta - 3) q^{40} - 4 \beta q^{41} + 3 \beta q^{43} - \beta q^{44} + 3 \beta q^{46} + 8 q^{47} - 7 q^{49} + (2 \beta - 3) q^{50} + ( - \beta - 3) q^{52} - 6 \beta q^{53} + (\beta - 4) q^{55} - 6 q^{58} + \beta q^{59} + 6 q^{61} - 3 \beta q^{62} + 7 q^{64} + (4 \beta - 1) q^{65} + 12 q^{67} - \beta q^{71} - 6 q^{73} + 6 q^{74} - 3 \beta q^{76} + ( - \beta - 1) q^{80} - 4 \beta q^{82} + 4 q^{83} + 3 \beta q^{86} - 3 \beta q^{88} - 4 \beta q^{89} - 3 \beta q^{92} + 8 q^{94} + (3 \beta - 12) q^{95} - 6 q^{97} - 7 q^{98} +O(q^{100})$$ q + q^2 - q^4 + (b + 1) * q^5 - 3 * q^8 + (b + 1) * q^10 + b * q^11 + (b + 3) * q^13 - q^16 + 3*b * q^19 + (-b - 1) * q^20 + b * q^22 + 3*b * q^23 + (2*b - 3) * q^25 + (b + 3) * q^26 - 6 * q^29 - 3*b * q^31 + 5 * q^32 + 6 * q^37 + 3*b * q^38 + (-3*b - 3) * q^40 - 4*b * q^41 + 3*b * q^43 - b * q^44 + 3*b * q^46 + 8 * q^47 - 7 * q^49 + (2*b - 3) * q^50 + (-b - 3) * q^52 - 6*b * q^53 + (b - 4) * q^55 - 6 * q^58 + b * q^59 + 6 * q^61 - 3*b * q^62 + 7 * q^64 + (4*b - 1) * q^65 + 12 * q^67 - b * q^71 - 6 * q^73 + 6 * q^74 - 3*b * q^76 + (-b - 1) * q^80 - 4*b * q^82 + 4 * q^83 + 3*b * q^86 - 3*b * q^88 - 4*b * q^89 - 3*b * q^92 + 8 * q^94 + (3*b - 12) * q^95 - 6 * q^97 - 7 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 6 * q^8 $$2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 6 q^{8} + 2 q^{10} + 6 q^{13} - 2 q^{16} - 2 q^{20} - 6 q^{25} + 6 q^{26} - 12 q^{29} + 10 q^{32} + 12 q^{37} - 6 q^{40} + 16 q^{47} - 14 q^{49} - 6 q^{50} - 6 q^{52} - 8 q^{55} - 12 q^{58} + 12 q^{61} + 14 q^{64} - 2 q^{65} + 24 q^{67} - 12 q^{73} + 12 q^{74} - 2 q^{80} + 8 q^{83} + 16 q^{94} - 24 q^{95} - 12 q^{97} - 14 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 6 * q^8 + 2 * q^10 + 6 * q^13 - 2 * q^16 - 2 * q^20 - 6 * q^25 + 6 * q^26 - 12 * q^29 + 10 * q^32 + 12 * q^37 - 6 * q^40 + 16 * q^47 - 14 * q^49 - 6 * q^50 - 6 * q^52 - 8 * q^55 - 12 * q^58 + 12 * q^61 + 14 * q^64 - 2 * q^65 + 24 * q^67 - 12 * q^73 + 12 * q^74 - 2 * q^80 + 8 * q^83 + 16 * q^94 - 24 * q^95 - 12 * q^97 - 14 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 − 1.00000i 1.00000i
1.00000 0 −1.00000 1.00000 2.00000i 0 0 −3.00000 0 1.00000 2.00000i
64.2 1.00000 0 −1.00000 1.00000 + 2.00000i 0 0 −3.00000 0 1.00000 + 2.00000i
 $$n$$: 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
65.d 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.h.c 2
3.b odd 2 1 65.2.d.a 2
5.b even 2 1 585.2.h.b 2
12.b even 2 1 1040.2.f.a 2
13.b even 2 1 585.2.h.b 2
15.d odd 2 1 65.2.d.b yes 2
15.e even 4 1 325.2.c.b 2
15.e even 4 1 325.2.c.e 2
39.d odd 2 1 65.2.d.b yes 2
39.f even 4 1 845.2.b.a 2
39.f even 4 1 845.2.b.b 2
39.h odd 6 2 845.2.l.a 4
39.i odd 6 2 845.2.l.b 4
39.k even 12 2 845.2.n.a 4
39.k even 12 2 845.2.n.b 4
60.h even 2 1 1040.2.f.b 2
65.d even 2 1 inner 585.2.h.c 2
156.h even 2 1 1040.2.f.b 2
195.e odd 2 1 65.2.d.a 2
195.j odd 4 1 4225.2.a.e 1
195.j odd 4 1 4225.2.a.h 1
195.n even 4 1 845.2.b.a 2
195.n even 4 1 845.2.b.b 2
195.s even 4 1 325.2.c.b 2
195.s even 4 1 325.2.c.e 2
195.u odd 4 1 4225.2.a.k 1
195.u odd 4 1 4225.2.a.m 1
195.x odd 6 2 845.2.l.a 4
195.y odd 6 2 845.2.l.b 4
195.bh even 12 2 845.2.n.a 4
195.bh even 12 2 845.2.n.b 4
780.d even 2 1 1040.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 3.b odd 2 1
65.2.d.a 2 195.e odd 2 1
65.2.d.b yes 2 15.d odd 2 1
65.2.d.b yes 2 39.d odd 2 1
325.2.c.b 2 15.e even 4 1
325.2.c.b 2 195.s even 4 1
325.2.c.e 2 15.e even 4 1
325.2.c.e 2 195.s even 4 1
585.2.h.b 2 5.b even 2 1
585.2.h.b 2 13.b even 2 1
585.2.h.c 2 1.a even 1 1 trivial
585.2.h.c 2 65.d even 2 1 inner
845.2.b.a 2 39.f even 4 1
845.2.b.a 2 195.n even 4 1
845.2.b.b 2 39.f even 4 1
845.2.b.b 2 195.n even 4 1
845.2.l.a 4 39.h odd 6 2
845.2.l.a 4 195.x odd 6 2
845.2.l.b 4 39.i odd 6 2
845.2.l.b 4 195.y odd 6 2
845.2.n.a 4 39.k even 12 2
845.2.n.a 4 195.bh even 12 2
845.2.n.b 4 39.k even 12 2
845.2.n.b 4 195.bh even 12 2
1040.2.f.a 2 12.b even 2 1
1040.2.f.a 2 780.d even 2 1
1040.2.f.b 2 60.h even 2 1
1040.2.f.b 2 156.h even 2 1
4225.2.a.e 1 195.j odd 4 1
4225.2.a.h 1 195.j odd 4 1
4225.2.a.k 1 195.u odd 4 1
4225.2.a.m 1 195.u odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

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