# 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: $$x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 2 q^{4} + ( 2 - i ) q^{5} + 3 q^{7} +O(q^{10})$$ $$q + 2 q^{2} + 2 q^{4} + ( 2 - i ) q^{5} + 3 q^{7} + ( 4 - 2 i ) q^{10} + 5 i q^{11} + ( -3 - 2 i ) q^{13} + 6 q^{14} -4 q^{16} -3 i q^{17} -6 i q^{19} + ( 4 - 2 i ) q^{20} + 10 i q^{22} + 9 i q^{23} + ( 3 - 4 i ) q^{25} + ( -6 - 4 i ) q^{26} + 6 q^{28} -8 q^{32} -6 i q^{34} + ( 6 - 3 i ) q^{35} + 3 q^{37} -12 i q^{38} -5 i q^{41} + 6 i q^{43} + 10 i q^{44} + 18 i q^{46} -8 q^{47} + 2 q^{49} + ( 6 - 8 i ) q^{50} + ( -6 - 4 i ) q^{52} + 9 i q^{53} + ( 5 + 10 i ) q^{55} -4 i q^{59} -3 q^{61} -8 q^{64} + ( -8 - i ) q^{65} -12 q^{67} -6 i q^{68} + ( 12 - 6 i ) q^{70} -5 i q^{71} -6 q^{73} + 6 q^{74} -12 i q^{76} + 15 i q^{77} + 15 q^{79} + ( -8 + 4 i ) q^{80} -10 i q^{82} -4 q^{83} + ( -3 - 6 i ) q^{85} + 12 i q^{86} + i q^{89} + ( -9 - 6 i ) q^{91} + 18 i q^{92} -16 q^{94} + ( -6 - 12 i ) q^{95} + 3 q^{97} + 4 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} + O(q^{10})$$ $$2 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} + 8 q^{10} - 6 q^{13} + 12 q^{14} - 8 q^{16} + 8 q^{20} + 6 q^{25} - 12 q^{26} + 12 q^{28} - 16 q^{32} + 12 q^{35} + 6 q^{37} - 16 q^{47} + 4 q^{49} + 12 q^{50} - 12 q^{52} + 10 q^{55} - 6 q^{61} - 16 q^{64} - 16 q^{65} - 24 q^{67} + 24 q^{70} - 12 q^{73} + 12 q^{74} + 30 q^{79} - 16 q^{80} - 8 q^{83} - 6 q^{85} - 18 q^{91} - 32 q^{94} - 12 q^{95} + 6 q^{97} + 8 q^{98} + O(q^{100})$$

## 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

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