# Properties

 Label 585.2.b.g Level $585$ Weight $2$ Character orbit 585.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5089536.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 16 x^{2} - 24 x + 18$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + ( -2 + \beta_{3} ) q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} +O(q^{10})$$ $$q -\beta_{5} q^{2} + ( -2 + \beta_{3} ) q^{4} + \beta_{4} q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} -\beta_{1} q^{10} + ( -\beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{13} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{14} + ( 1 - 2 \beta_{1} ) q^{16} + ( -2 + 2 \beta_{3} ) q^{17} + ( -\beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{19} + ( -\beta_{2} - 2 \beta_{4} ) q^{20} + ( -5 - \beta_{1} + 2 \beta_{3} ) q^{22} + ( -3 - \beta_{1} ) q^{23} - q^{25} + ( -3 - 2 \beta_{1} + \beta_{2} + 4 \beta_{4} + \beta_{5} ) q^{26} + ( \beta_{2} + 7 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 1 - \beta_{3} ) q^{29} + ( \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{31} + ( -6 \beta_{4} + \beta_{5} ) q^{32} + ( 2 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{34} + ( 1 - \beta_{3} ) q^{35} + ( -\beta_{2} - 7 \beta_{4} ) q^{37} + ( 3 - \beta_{1} ) q^{38} + ( -1 + \beta_{1} + \beta_{3} ) q^{40} + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( 3 - \beta_{1} + 2 \beta_{3} ) q^{43} + ( -\beta_{2} + 2 \beta_{4} + 5 \beta_{5} ) q^{44} + ( -\beta_{2} - 4 \beta_{4} + 3 \beta_{5} ) q^{46} + ( -\beta_{2} - \beta_{4} ) q^{47} + ( 1 + 2 \beta_{1} ) q^{49} + \beta_{5} q^{50} + ( 3 - 3 \beta_{1} - 2 \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{52} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( -2 - \beta_{1} - \beta_{3} ) q^{55} + ( 7 - 4 \beta_{1} - \beta_{3} ) q^{56} + ( -\beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{58} + ( -\beta_{2} - 4 \beta_{4} - \beta_{5} ) q^{59} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{61} + ( -3 - 5 \beta_{1} ) q^{62} + ( 6 + 2 \beta_{1} - \beta_{3} ) q^{64} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{65} + ( \beta_{2} + 7 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 14 - 4 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -\beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{70} + ( \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{71} + ( \beta_{2} - 5 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -1 + 8 \beta_{1} + \beta_{3} ) q^{74} + ( -3 \beta_{2} - \beta_{5} ) q^{76} + ( -4 + 4 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 6 + 4 \beta_{1} - 2 \beta_{3} ) q^{79} + ( \beta_{4} + 2 \beta_{5} ) q^{80} + ( -6 - 4 \beta_{1} ) q^{82} + ( \beta_{2} - 5 \beta_{4} - 2 \beta_{5} ) q^{83} + ( -2 \beta_{2} - 2 \beta_{4} ) q^{85} + ( \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{86} + ( 9 - 3 \beta_{1} ) q^{88} + ( -2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{89} + ( 5 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 5 + 3 \beta_{1} - 2 \beta_{3} ) q^{92} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{94} + ( -2 + \beta_{1} - \beta_{3} ) q^{95} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{97} + ( 2 \beta_{2} + 8 \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} + O(q^{10})$$ $$6 q - 10 q^{4} + 2 q^{10} - 8 q^{13} - 8 q^{14} + 10 q^{16} - 8 q^{17} - 24 q^{22} - 16 q^{23} - 6 q^{25} - 14 q^{26} + 4 q^{29} + 4 q^{35} + 20 q^{38} - 6 q^{40} + 24 q^{43} + 2 q^{49} + 20 q^{52} - 12 q^{53} - 12 q^{55} + 48 q^{56} - 12 q^{61} - 8 q^{62} + 30 q^{64} - 2 q^{65} + 88 q^{68} - 20 q^{74} - 36 q^{77} + 24 q^{79} - 28 q^{82} + 60 q^{88} + 32 q^{91} + 20 q^{92} - 8 q^{94} - 16 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 16 x^{2} - 24 x + 18$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 24 \nu^{4} - 6 \nu^{3} - \nu^{2} + 6 \nu + 285$$$$)/131$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} - 11 \nu^{4} + 101 \nu^{3} - 136 \nu^{2} + 292 \nu - 147$$$$)/393$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{5} - 13 \nu^{4} + 36 \nu^{3} + 6 \nu^{2} - 36 \nu - 7$$$$)/131$$ $$\beta_{4}$$ $$=$$ $$($$$$23 \nu^{5} - 28 \nu^{4} + 7 \nu^{3} + 154 \nu^{2} + 386 \nu - 267$$$$)/393$$ $$\beta_{5}$$ $$=$$ $$($$$$-23 \nu^{5} + 28 \nu^{4} - 7 \nu^{3} - 23 \nu^{2} - 386 \nu + 267$$$$)/131$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 3 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 6 \beta_{1} - 13$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{5} - 12 \beta_{4} + 9 \beta_{3} - 9 \beta_{2} + 7 \beta_{1} - 12$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 1.66044 + 1.66044i 0.675970 − 0.675970i −1.33641 − 1.33641i −1.33641 + 1.33641i 0.675970 + 0.675970i 1.66044 − 1.66044i
2.51414i 0 −4.32088 1.00000i 0 3.32088i 5.83502i 0 2.51414
181.2 2.08613i 0 −2.35194 1.00000i 0 1.35194i 0.734191i 0 −2.08613
181.3 0.571993i 0 1.67282 1.00000i 0 2.67282i 2.10083i 0 0.571993
181.4 0.571993i 0 1.67282 1.00000i 0 2.67282i 2.10083i 0 0.571993
181.5 2.08613i 0 −2.35194 1.00000i 0 1.35194i 0.734191i 0 −2.08613
181.6 2.51414i 0 −4.32088 1.00000i 0 3.32088i 5.83502i 0 2.51414
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 181.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.b.g 6
3.b odd 2 1 65.2.c.a 6
12.b even 2 1 1040.2.k.d 6
13.b even 2 1 inner 585.2.b.g 6
13.d odd 4 1 7605.2.a.bs 3
13.d odd 4 1 7605.2.a.cc 3
15.d odd 2 1 325.2.c.g 6
15.e even 4 1 325.2.d.e 6
15.e even 4 1 325.2.d.f 6
39.d odd 2 1 65.2.c.a 6
39.f even 4 1 845.2.a.i 3
39.f even 4 1 845.2.a.k 3
39.h odd 6 2 845.2.m.h 12
39.i odd 6 2 845.2.m.h 12
39.k even 12 2 845.2.e.i 6
39.k even 12 2 845.2.e.k 6
156.h even 2 1 1040.2.k.d 6
195.e odd 2 1 325.2.c.g 6
195.n even 4 1 4225.2.a.bc 3
195.n even 4 1 4225.2.a.be 3
195.s even 4 1 325.2.d.e 6
195.s even 4 1 325.2.d.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a 6 3.b odd 2 1
65.2.c.a 6 39.d odd 2 1
325.2.c.g 6 15.d odd 2 1
325.2.c.g 6 195.e odd 2 1
325.2.d.e 6 15.e even 4 1
325.2.d.e 6 195.s even 4 1
325.2.d.f 6 15.e even 4 1
325.2.d.f 6 195.s even 4 1
585.2.b.g 6 1.a even 1 1 trivial
585.2.b.g 6 13.b even 2 1 inner
845.2.a.i 3 39.f even 4 1
845.2.a.k 3 39.f even 4 1
845.2.e.i 6 39.k even 12 2
845.2.e.k 6 39.k even 12 2
845.2.m.h 12 39.h odd 6 2
845.2.m.h 12 39.i odd 6 2
1040.2.k.d 6 12.b even 2 1
1040.2.k.d 6 156.h even 2 1
4225.2.a.bc 3 195.n even 4 1
4225.2.a.be 3 195.n even 4 1
7605.2.a.bs 3 13.d odd 4 1
7605.2.a.cc 3 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$:

 $$T_{2}^{6} + 11 T_{2}^{4} + 31 T_{2}^{2} + 9$$ $$T_{7}^{6} + 20 T_{7}^{4} + 112 T_{7}^{2} + 144$$ $$T_{17}^{3} + 4 T_{17}^{2} - 32 T_{17} - 96$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 31 T^{2} + 11 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$144 + 112 T^{2} + 20 T^{4} + T^{6}$$
$11$ $$2916 + 684 T^{2} + 48 T^{4} + T^{6}$$
$13$ $$2197 + 1352 T + 507 T^{2} + 160 T^{3} + 39 T^{4} + 8 T^{5} + T^{6}$$
$17$ $$( -96 - 32 T + 4 T^{2} + T^{3} )^{2}$$
$19$ $$36 + 196 T^{2} + 44 T^{4} + T^{6}$$
$23$ $$( 6 + 16 T + 8 T^{2} + T^{3} )^{2}$$
$29$ $$( 12 - 8 T - 2 T^{2} + T^{3} )^{2}$$
$31$ $$36 + 604 T^{2} + 56 T^{4} + T^{6}$$
$37$ $$51984 + 6256 T^{2} + 152 T^{4} + T^{6}$$
$41$ $$5184 + 2224 T^{2} + 92 T^{4} + T^{6}$$
$43$ $$( -2 + 12 T - 12 T^{2} + T^{3} )^{2}$$
$47$ $$144 + 112 T^{2} + 20 T^{4} + T^{6}$$
$53$ $$( 72 - 60 T + 6 T^{2} + T^{3} )^{2}$$
$59$ $$324 + 468 T^{2} + 84 T^{4} + T^{6}$$
$61$ $$( -76 - 12 T + 6 T^{2} + T^{3} )^{2}$$
$67$ $$11664 + 3168 T^{2} + 156 T^{4} + T^{6}$$
$71$ $$36 + 196 T^{2} + 44 T^{4} + T^{6}$$
$73$ $$266256 + 23344 T^{2} + 296 T^{4} + T^{6}$$
$79$ $$( 32 - 48 T - 12 T^{2} + T^{3} )^{2}$$
$83$ $$1296 + 5760 T^{2} + 156 T^{4} + T^{6}$$
$89$ $$82944 + 9216 T^{2} + 192 T^{4} + T^{6}$$
$97$ $$576 + 1456 T^{2} + 140 T^{4} + T^{6}$$