# Properties

 Label 585.2.h.b Level $585$ Weight $2$ Character orbit 585.h Analytic conductor $4.671$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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.

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

## 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.b 2
3.b odd 2 1 65.2.d.b yes 2
5.b even 2 1 585.2.h.c 2
12.b even 2 1 1040.2.f.b 2
13.b even 2 1 585.2.h.c 2
15.d odd 2 1 65.2.d.a 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.a 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.b 4
39.i odd 6 2 845.2.l.a 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.a 2
65.d even 2 1 inner 585.2.h.b 2
156.h even 2 1 1040.2.f.a 2
195.e odd 2 1 65.2.d.b yes 2
195.j odd 4 1 4225.2.a.k 1
195.j odd 4 1 4225.2.a.m 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.e 1
195.u odd 4 1 4225.2.a.h 1
195.x odd 6 2 845.2.l.b 4
195.y odd 6 2 845.2.l.a 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.b 2

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

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