# Properties

 Label 585.2.j.e Level $585$ Weight $2$ Character orbit 585.j Analytic conductor $4.671$ Analytic rank $0$ Dimension $4$ 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.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
406.1
 −0.309017 − 0.535233i 0.809017 + 1.40126i −0.309017 + 0.535233i 0.809017 − 1.40126i
−0.309017 0.535233i 0 0.809017 1.40126i −1.00000 0 −2.11803 + 3.66854i −2.23607 0 0.309017 + 0.535233i
406.2 0.809017 + 1.40126i 0 −0.309017 + 0.535233i −1.00000 0 0.118034 0.204441i 2.23607 0 −0.809017 1.40126i
451.1 −0.309017 + 0.535233i 0 0.809017 + 1.40126i −1.00000 0 −2.11803 3.66854i −2.23607 0 0.309017 0.535233i
451.2 0.809017 1.40126i 0 −0.309017 0.535233i −1.00000 0 0.118034 + 0.204441i 2.23607 0 −0.809017 + 1.40126i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.e 4
3.b odd 2 1 65.2.e.a 4
12.b even 2 1 1040.2.q.n 4
13.c even 3 1 inner 585.2.j.e 4
13.c even 3 1 7605.2.a.ba 2
13.e even 6 1 7605.2.a.bf 2
15.d odd 2 1 325.2.e.b 4
15.e even 4 2 325.2.o.a 8
39.d odd 2 1 845.2.e.g 4
39.f even 4 2 845.2.m.e 8
39.h odd 6 1 845.2.a.b 2
39.h odd 6 1 845.2.e.g 4
39.i odd 6 1 65.2.e.a 4
39.i odd 6 1 845.2.a.e 2
39.k even 12 2 845.2.c.c 4
39.k even 12 2 845.2.m.e 8
156.p even 6 1 1040.2.q.n 4
195.x odd 6 1 325.2.e.b 4
195.x odd 6 1 4225.2.a.u 2
195.y odd 6 1 4225.2.a.y 2
195.bl even 12 2 325.2.o.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 3.b odd 2 1
65.2.e.a 4 39.i odd 6 1
325.2.e.b 4 15.d odd 2 1
325.2.e.b 4 195.x odd 6 1
325.2.o.a 8 15.e even 4 2
325.2.o.a 8 195.bl even 12 2
585.2.j.e 4 1.a even 1 1 trivial
585.2.j.e 4 13.c even 3 1 inner
845.2.a.b 2 39.h odd 6 1
845.2.a.e 2 39.i odd 6 1
845.2.c.c 4 39.k even 12 2
845.2.e.g 4 39.d odd 2 1
845.2.e.g 4 39.h odd 6 1
845.2.m.e 8 39.f even 4 2
845.2.m.e 8 39.k even 12 2
1040.2.q.n 4 12.b even 2 1
1040.2.q.n 4 156.p even 6 1
4225.2.a.u 2 195.x odd 6 1
4225.2.a.y 2 195.y odd 6 1
7605.2.a.ba 2 13.c even 3 1
7605.2.a.bf 2 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + 2 T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2} - T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$1 + 4 T + 17 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$( 13 + 2 T + T^{2} )^{2}$$
$17$ $$361 + 38 T + 23 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$961 - 372 T + 113 T^{2} - 12 T^{3} + T^{4}$$
$29$ $$121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$( 9 + 3 T + T^{2} )^{2}$$
$41$ $$5041 + 426 T + 107 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$121 - 88 T + 53 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$( -64 + 8 T + T^{2} )^{2}$$
$53$ $$( 6 + T )^{4}$$
$59$ $$81 + 108 T + 153 T^{2} - 12 T^{3} + T^{4}$$
$61$ $$32041 - 358 T + 183 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$841 - 232 T + 93 T^{2} + 8 T^{3} + T^{4}$$
$71$ $$121 + 88 T + 53 T^{2} + 8 T^{3} + T^{4}$$
$73$ $$( 6 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$( -80 + T^{2} )^{2}$$
$89$ $$( 81 + 9 T + T^{2} )^{2}$$
$97$ $$361 - 38 T + 23 T^{2} + 2 T^{3} + T^{4}$$