# Properties

 Label 585.2.h.d 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: $$1$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 2 q^{4} + ( - i + 2) q^{5} + 3 q^{7}+O(q^{10})$$ q + 2 * q^2 + 2 * q^4 + (-i + 2) * q^5 + 3 * q^7 $$q + 2 q^{2} + 2 q^{4} + ( - i + 2) q^{5} + 3 q^{7} + ( - 2 i + 4) q^{10} + 5 i q^{11} + ( - 2 i - 3) q^{13} + 6 q^{14} - 4 q^{16} - 3 i q^{17} - 6 i q^{19} + ( - 2 i + 4) q^{20} + 10 i q^{22} + 9 i q^{23} + ( - 4 i + 3) q^{25} + ( - 4 i - 6) q^{26} + 6 q^{28} - 8 q^{32} - 6 i q^{34} + ( - 3 i + 6) q^{35} + 3 q^{37} - 12 i q^{38} - 5 i q^{41} + 6 i q^{43} + 10 i q^{44} + 18 i q^{46} - 8 q^{47} + 2 q^{49} + ( - 8 i + 6) q^{50} + ( - 4 i - 6) q^{52} + 9 i q^{53} + (10 i + 5) q^{55} - 4 i q^{59} - 3 q^{61} - 8 q^{64} + ( - i - 8) q^{65} - 12 q^{67} - 6 i q^{68} + ( - 6 i + 12) q^{70} - 5 i q^{71} - 6 q^{73} + 6 q^{74} - 12 i q^{76} + 15 i q^{77} + 15 q^{79} + (4 i - 8) q^{80} - 10 i q^{82} - 4 q^{83} + ( - 6 i - 3) q^{85} + 12 i q^{86} + i q^{89} + ( - 6 i - 9) q^{91} + 18 i q^{92} - 16 q^{94} + ( - 12 i - 6) q^{95} + 3 q^{97} + 4 q^{98} +O(q^{100})$$ q + 2 * q^2 + 2 * q^4 + (-i + 2) * q^5 + 3 * q^7 + (-2*i + 4) * q^10 + 5*i * q^11 + (-2*i - 3) * q^13 + 6 * q^14 - 4 * q^16 - 3*i * q^17 - 6*i * q^19 + (-2*i + 4) * q^20 + 10*i * q^22 + 9*i * q^23 + (-4*i + 3) * q^25 + (-4*i - 6) * q^26 + 6 * q^28 - 8 * q^32 - 6*i * q^34 + (-3*i + 6) * q^35 + 3 * q^37 - 12*i * q^38 - 5*i * q^41 + 6*i * q^43 + 10*i * q^44 + 18*i * q^46 - 8 * q^47 + 2 * q^49 + (-8*i + 6) * q^50 + (-4*i - 6) * q^52 + 9*i * q^53 + (10*i + 5) * q^55 - 4*i * q^59 - 3 * q^61 - 8 * q^64 + (-i - 8) * q^65 - 12 * q^67 - 6*i * q^68 + (-6*i + 12) * q^70 - 5*i * q^71 - 6 * q^73 + 6 * q^74 - 12*i * q^76 + 15*i * q^77 + 15 * q^79 + (4*i - 8) * q^80 - 10*i * q^82 - 4 * q^83 + (-6*i - 3) * q^85 + 12*i * q^86 + i * q^89 + (-6*i - 9) * q^91 + 18*i * q^92 - 16 * q^94 + (-12*i - 6) * q^95 + 3 * q^97 + 4 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7}+O(q^{10})$$ 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 6 * q^7 $$2 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} + 8 q^{10} - 6 q^{13} + 12 q^{14} - 8 q^{16} + 8 q^{20} + 6 q^{25} - 12 q^{26} + 12 q^{28} - 16 q^{32} + 12 q^{35} + 6 q^{37} - 16 q^{47} + 4 q^{49} + 12 q^{50} - 12 q^{52} + 10 q^{55} - 6 q^{61} - 16 q^{64} - 16 q^{65} - 24 q^{67} + 24 q^{70} - 12 q^{73} + 12 q^{74} + 30 q^{79} - 16 q^{80} - 8 q^{83} - 6 q^{85} - 18 q^{91} - 32 q^{94} - 12 q^{95} + 6 q^{97} + 8 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 6 * q^7 + 8 * q^10 - 6 * q^13 + 12 * q^14 - 8 * q^16 + 8 * q^20 + 6 * q^25 - 12 * q^26 + 12 * q^28 - 16 * q^32 + 12 * q^35 + 6 * q^37 - 16 * q^47 + 4 * q^49 + 12 * q^50 - 12 * q^52 + 10 * q^55 - 6 * q^61 - 16 * q^64 - 16 * q^65 - 24 * q^67 + 24 * q^70 - 12 * q^73 + 12 * q^74 + 30 * q^79 - 16 * q^80 - 8 * q^83 - 6 * q^85 - 18 * q^91 - 32 * q^94 - 12 * q^95 + 6 * q^97 + 8 * 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
2.00000 0 2.00000 2.00000 1.00000i 0 3.00000 0 0 4.00000 2.00000i
64.2 2.00000 0 2.00000 2.00000 + 1.00000i 0 3.00000 0 0 4.00000 + 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.d 2
3.b odd 2 1 195.2.h.a 2
5.b even 2 1 585.2.h.a 2
12.b even 2 1 3120.2.r.a 2
13.b even 2 1 585.2.h.a 2
15.d odd 2 1 195.2.h.b yes 2
15.e even 4 1 975.2.b.a 2
15.e even 4 1 975.2.b.c 2
39.d odd 2 1 195.2.h.b yes 2
60.h even 2 1 3120.2.r.f 2
65.d even 2 1 inner 585.2.h.d 2
156.h even 2 1 3120.2.r.f 2
195.e odd 2 1 195.2.h.a 2
195.s even 4 1 975.2.b.a 2
195.s even 4 1 975.2.b.c 2
780.d even 2 1 3120.2.r.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.h.a 2 3.b odd 2 1
195.2.h.a 2 195.e odd 2 1
195.2.h.b yes 2 15.d odd 2 1
195.2.h.b yes 2 39.d odd 2 1
585.2.h.a 2 5.b even 2 1
585.2.h.a 2 13.b even 2 1
585.2.h.d 2 1.a even 1 1 trivial
585.2.h.d 2 65.d even 2 1 inner
975.2.b.a 2 15.e even 4 1
975.2.b.a 2 195.s even 4 1
975.2.b.c 2 15.e even 4 1
975.2.b.c 2 195.s even 4 1
3120.2.r.a 2 12.b even 2 1
3120.2.r.a 2 780.d even 2 1
3120.2.r.f 2 60.h even 2 1
3120.2.r.f 2 156.h even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$(T - 3)^{2}$$
$11$ $$T^{2} + 25$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$T^{2} + 9$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 81$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} + 25$$
$43$ $$T^{2} + 36$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 81$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 3)^{2}$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2} + 25$$
$73$ $$(T + 6)^{2}$$
$79$ $$(T - 15)^{2}$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + 1$$
$97$ $$(T - 3)^{2}$$