# Properties

 Label 585.2.c.b Level $585$ Weight $2$ Character orbit 585.c Analytic conductor $4.671$ Analytic rank $0$ Dimension $6$ 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{5} ) q^{2} + ( -2 - \beta_{1} - \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -4 \beta_{3} + 3 \beta_{4} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{5} ) q^{2} + ( -2 - \beta_{1} - \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -4 \beta_{3} + 3 \beta_{4} ) q^{8} + ( 1 + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{10} + ( 2 + \beta_{1} ) q^{11} -\beta_{3} q^{13} + ( -1 + \beta_{1} + \beta_{2} ) q^{14} + ( 3 + 2 \beta_{1} + 4 \beta_{2} ) q^{16} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{17} + \beta_{1} q^{19} + ( 4 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{20} + ( 5 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{22} + ( -5 \beta_{3} + \beta_{4} ) q^{23} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{25} + ( 1 + \beta_{1} ) q^{26} + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{28} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{29} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( 5 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} ) q^{32} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{35} + ( -\beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{37} + ( 3 \beta_{3} - \beta_{4} ) q^{38} + ( -6 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{40} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( \beta_{3} + 3 \beta_{4} + 2 \beta_{5} ) q^{43} + ( -8 - 3 \beta_{1} - 4 \beta_{2} ) q^{44} + ( 6 + 5 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{47} + ( 3 + 2 \beta_{1} ) q^{49} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{50} + ( 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{52} + ( -2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{53} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{55} + ( -7 + \beta_{1} - \beta_{2} ) q^{56} + ( -3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{58} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{61} + ( -3 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{62} + ( -12 - \beta_{1} - 3 \beta_{2} ) q^{64} + ( -\beta_{2} - \beta_{5} ) q^{65} + ( -3 \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{67} + ( 4 \beta_{3} - 2 \beta_{4} ) q^{68} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{70} + ( 2 - \beta_{1} - 6 \beta_{2} ) q^{71} + ( -9 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{73} + ( 1 + \beta_{1} + 5 \beta_{2} ) q^{74} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{76} + 2 \beta_{4} q^{77} + ( 6 - 4 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -10 - \beta_{1} - 6 \beta_{2} - 10 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{80} + ( 10 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{82} + ( -7 \beta_{3} - \beta_{4} - \beta_{5} ) q^{83} + ( -2 + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{85} + ( 8 - \beta_{1} + 8 \beta_{2} ) q^{86} + ( -11 \beta_{3} + 9 \beta_{4} + 4 \beta_{5} ) q^{88} + ( 2 + 4 \beta_{1} - 4 \beta_{2} ) q^{89} + ( -1 + \beta_{1} - \beta_{2} ) q^{91} + ( 13 \beta_{3} - 7 \beta_{4} - 6 \beta_{5} ) q^{92} + ( -5 - 3 \beta_{1} + \beta_{2} ) q^{94} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{95} + ( 6 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} ) q^{97} + ( 9 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} + O(q^{10})$$ $$6 q - 10 q^{4} + 2 q^{10} + 12 q^{11} - 8 q^{14} + 10 q^{16} + 20 q^{20} + 2 q^{25} + 6 q^{26} + 12 q^{29} - 20 q^{31} + 20 q^{34} - 8 q^{35} - 34 q^{40} + 8 q^{41} - 40 q^{44} + 32 q^{46} + 18 q^{49} - 16 q^{50} - 16 q^{55} - 40 q^{56} - 16 q^{59} + 12 q^{61} - 66 q^{64} + 2 q^{65} - 20 q^{70} + 24 q^{71} - 4 q^{74} - 20 q^{76} + 32 q^{79} - 48 q^{80} - 12 q^{85} + 32 q^{86} + 20 q^{89} - 4 q^{91} - 32 q^{94} - 16 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
469.1
 0.403032 − 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 + 0.403032i
2.67513i 0 −5.15633 −1.67513 + 1.48119i 0 0.806063i 8.44358i 0 3.96239 + 4.48119i
469.2 1.53919i 0 −0.369102 −0.539189 2.17009i 0 1.70928i 2.51026i 0 −3.34017 + 0.829914i
469.3 1.21432i 0 0.525428 2.21432 + 0.311108i 0 2.90321i 3.06668i 0 0.377784 2.68889i
469.4 1.21432i 0 0.525428 2.21432 0.311108i 0 2.90321i 3.06668i 0 0.377784 + 2.68889i
469.5 1.53919i 0 −0.369102 −0.539189 + 2.17009i 0 1.70928i 2.51026i 0 −3.34017 0.829914i
469.6 2.67513i 0 −5.15633 −1.67513 1.48119i 0 0.806063i 8.44358i 0 3.96239 4.48119i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 469.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.c.b 6
3.b odd 2 1 65.2.b.a 6
5.b even 2 1 inner 585.2.c.b 6
5.c odd 4 1 2925.2.a.bf 3
5.c odd 4 1 2925.2.a.bj 3
12.b even 2 1 1040.2.d.c 6
15.d odd 2 1 65.2.b.a 6
15.e even 4 1 325.2.a.j 3
15.e even 4 1 325.2.a.k 3
39.d odd 2 1 845.2.b.c 6
39.f even 4 1 845.2.d.a 6
39.f even 4 1 845.2.d.b 6
39.h odd 6 2 845.2.n.g 12
39.i odd 6 2 845.2.n.f 12
39.k even 12 2 845.2.l.d 12
39.k even 12 2 845.2.l.e 12
60.h even 2 1 1040.2.d.c 6
60.l odd 4 1 5200.2.a.cb 3
60.l odd 4 1 5200.2.a.cj 3
195.e odd 2 1 845.2.b.c 6
195.n even 4 1 845.2.d.a 6
195.n even 4 1 845.2.d.b 6
195.s even 4 1 4225.2.a.ba 3
195.s even 4 1 4225.2.a.bh 3
195.x odd 6 2 845.2.n.f 12
195.y odd 6 2 845.2.n.g 12
195.bh even 12 2 845.2.l.d 12
195.bh even 12 2 845.2.l.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 3.b odd 2 1
65.2.b.a 6 15.d odd 2 1
325.2.a.j 3 15.e even 4 1
325.2.a.k 3 15.e even 4 1
585.2.c.b 6 1.a even 1 1 trivial
585.2.c.b 6 5.b even 2 1 inner
845.2.b.c 6 39.d odd 2 1
845.2.b.c 6 195.e odd 2 1
845.2.d.a 6 39.f even 4 1
845.2.d.a 6 195.n even 4 1
845.2.d.b 6 39.f even 4 1
845.2.d.b 6 195.n even 4 1
845.2.l.d 12 39.k even 12 2
845.2.l.d 12 195.bh even 12 2
845.2.l.e 12 39.k even 12 2
845.2.l.e 12 195.bh even 12 2
845.2.n.f 12 39.i odd 6 2
845.2.n.f 12 195.x odd 6 2
845.2.n.g 12 39.h odd 6 2
845.2.n.g 12 195.y odd 6 2
1040.2.d.c 6 12.b even 2 1
1040.2.d.c 6 60.h even 2 1
2925.2.a.bf 3 5.c odd 4 1
2925.2.a.bj 3 5.c odd 4 1
4225.2.a.ba 3 195.s even 4 1
4225.2.a.bh 3 195.s even 4 1
5200.2.a.cb 3 60.l odd 4 1
5200.2.a.cj 3 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 11 T_{2}^{4} + 31 T_{2}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 + 31 T^{2} + 11 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$125 - 5 T^{2} - 16 T^{3} - T^{4} + T^{6}$$
$7$ $$16 + 32 T^{2} + 12 T^{4} + T^{6}$$
$11$ $$( 2 + 8 T - 6 T^{2} + T^{3} )^{2}$$
$13$ $$( 1 + T^{2} )^{3}$$
$17$ $$64 + 112 T^{2} + 44 T^{4} + T^{6}$$
$19$ $$( 2 - 4 T + T^{3} )^{2}$$
$23$ $$7396 + 1436 T^{2} + 72 T^{4} + T^{6}$$
$29$ $$( 108 - 36 T - 6 T^{2} + T^{3} )^{2}$$
$31$ $$( -26 + 20 T + 10 T^{2} + T^{3} )^{2}$$
$37$ $$2704 + 784 T^{2} + 56 T^{4} + T^{6}$$
$41$ $$( -32 - 32 T - 4 T^{2} + T^{3} )^{2}$$
$43$ $$77284 + 5452 T^{2} + 128 T^{4} + T^{6}$$
$47$ $$400 + 384 T^{2} + 44 T^{4} + T^{6}$$
$53$ $$92416 + 6464 T^{2} + 144 T^{4} + T^{6}$$
$59$ $$( -262 - 40 T + 8 T^{2} + T^{3} )^{2}$$
$61$ $$( -4 - 16 T - 6 T^{2} + T^{3} )^{2}$$
$67$ $$364816 + 15680 T^{2} + 220 T^{4} + T^{6}$$
$71$ $$( 754 - 88 T - 12 T^{2} + T^{3} )^{2}$$
$73$ $$55696 + 15568 T^{2} + 248 T^{4} + T^{6}$$
$79$ $$( 16 + 24 T - 16 T^{2} + T^{3} )^{2}$$
$83$ $$99856 + 9200 T^{2} + 180 T^{4} + T^{6}$$
$89$ $$( 200 - 52 T - 10 T^{2} + T^{3} )^{2}$$
$97$ $$40000 + 12656 T^{2} + 364 T^{4} + T^{6}$$