# 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: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{4} - q^{5} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{7} + ( - 2 \beta_{2} - 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b3 + b2 + b1) * q^4 - q^5 + (3*b3 - 2*b2 - 2*b1) * q^7 + (-2*b2 - 1) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{4} - q^{5} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{7} + ( - 2 \beta_{2} - 1) q^{8} - \beta_1 q^{10} + (\beta_{3} + 2 \beta_1 + 1) q^{11} + (4 \beta_{3} + 1) q^{13} + (\beta_{2} + 2) q^{14} + 3 \beta_1 q^{16} + (\beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{17} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{3} - \beta_{2} - \beta_1) q^{20} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{22} + (5 \beta_{3} + 2 \beta_1 + 5) q^{23} + q^{25} + (4 \beta_{2} + \beta_1) q^{26} + (5 \beta_{3} - 3 \beta_1 + 5) q^{28} + ( - 5 \beta_{3} + 4 \beta_1 - 5) q^{29} + (5 \beta_{3} - \beta_{2} - \beta_1) q^{32} + ( - 3 \beta_{2} + 4) q^{34} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{35} + ( - 3 \beta_{3} - 3) q^{37} + (\beta_{2} + 2) q^{38} + (2 \beta_{2} + 1) q^{40} + (7 \beta_{3} - 8 \beta_1 + 7) q^{41} + ( - 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{43} + (\beta_{2} - 1) q^{44} + (2 \beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{46} + 8 \beta_{2} q^{47} + ( - 6 \beta_{3} + 8 \beta_1 - 6) q^{49} + \beta_1 q^{50} + (3 \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{52} - 6 q^{53} + ( - \beta_{3} - 2 \beta_1 - 1) q^{55} + ( - 7 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{56} + (4 \beta_{3} - \beta_{2} - \beta_1) q^{58} + ( - 3 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{59} + (7 \beta_{3} - 12 \beta_{2} - 12 \beta_1) q^{61} + ( - 2 \beta_{2} + 1) q^{64} + ( - 4 \beta_{3} - 1) q^{65} + ( - \beta_{3} - 6 \beta_1 - 1) q^{67} + (5 \beta_{3} - \beta_1 + 5) q^{68} + ( - \beta_{2} - 2) q^{70} + (5 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{71} - 6 q^{73} + ( - 3 \beta_{2} - 3 \beta_1) q^{74} + (5 \beta_{3} - 3 \beta_1 + 5) q^{76} + q^{77} - 3 \beta_1 q^{80} + ( - 8 \beta_{3} - \beta_{2} - \beta_1) q^{82} + ( - 8 \beta_{2} - 4) q^{83} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{85} + ( - 3 \beta_{2} - 2) q^{86} + (3 \beta_{3} + 4 \beta_1 + 3) q^{88} + ( - 9 \beta_{3} - 9) q^{89} + ( - 9 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 12) q^{91} + (5 \beta_{2} + 3) q^{92} + ( - 8 \beta_{3} - 8 \beta_1 - 8) q^{94} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{95} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{97} + (8 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b3 + b2 + b1) * q^4 - q^5 + (3*b3 - 2*b2 - 2*b1) * q^7 + (-2*b2 - 1) * q^8 - b1 * q^10 + (b3 + 2*b1 + 1) * q^11 + (4*b3 + 1) * q^13 + (b2 + 2) * q^14 + 3*b1 * q^16 + (b3 - 4*b2 - 4*b1) * q^17 + (3*b3 - 2*b2 - 2*b1) * q^19 + (b3 - b2 - b1) * q^20 + (2*b3 + 3*b2 + 3*b1) * q^22 + (5*b3 + 2*b1 + 5) * q^23 + q^25 + (4*b2 + b1) * q^26 + (5*b3 - 3*b1 + 5) * q^28 + (-5*b3 + 4*b1 - 5) * q^29 + (5*b3 - b2 - b1) * q^32 + (-3*b2 + 4) * q^34 + (-3*b3 + 2*b2 + 2*b1) * q^35 + (-3*b3 - 3) * q^37 + (b2 + 2) * q^38 + (2*b2 + 1) * q^40 + (7*b3 - 8*b1 + 7) * q^41 + (-5*b3 + 2*b2 + 2*b1) * q^43 + (b2 - 1) * q^44 + (2*b3 + 7*b2 + 7*b1) * q^46 + 8*b2 * q^47 + (-6*b3 + 8*b1 - 6) * q^49 + b1 * q^50 + (3*b3 + b2 - 3*b1 + 4) * q^52 - 6 * q^53 + (-b3 - 2*b1 - 1) * q^55 + (-7*b3 + 4*b2 + 4*b1) * q^56 + (4*b3 - b2 - b1) * q^58 + (-3*b3 - 6*b2 - 6*b1) * q^59 + (7*b3 - 12*b2 - 12*b1) * q^61 + (-2*b2 + 1) * q^64 + (-4*b3 - 1) * q^65 + (-b3 - 6*b1 - 1) * q^67 + (5*b3 - b1 + 5) * q^68 + (-b2 - 2) * q^70 + (5*b3 - 2*b2 - 2*b1) * q^71 - 6 * q^73 + (-3*b2 - 3*b1) * q^74 + (5*b3 - 3*b1 + 5) * q^76 + q^77 - 3*b1 * q^80 + (-8*b3 - b2 - b1) * q^82 + (-8*b2 - 4) * q^83 + (-b3 + 4*b2 + 4*b1) * q^85 + (-3*b2 - 2) * q^86 + (3*b3 + 4*b1 + 3) * q^88 + (-9*b3 - 9) * q^89 + (-9*b3 - 2*b2 + 6*b1 - 12) * q^91 + (5*b2 + 3) * q^92 + (-8*b3 - 8*b1 - 8) * q^94 + (-3*b3 + 2*b2 + 2*b1) * q^95 + (-b3 + 4*b2 + 4*b1) * q^97 + (8*b3 + 2*b2 + 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{4} - 4 q^{5} - 4 q^{7}+O(q^{10})$$ 4 * q + q^2 + q^4 - 4 * q^5 - 4 * q^7 $$4 q + q^{2} + q^{4} - 4 q^{5} - 4 q^{7} - q^{10} + 4 q^{11} - 4 q^{13} + 6 q^{14} + 3 q^{16} + 2 q^{17} - 4 q^{19} - q^{20} - 7 q^{22} + 12 q^{23} + 4 q^{25} - 7 q^{26} + 7 q^{28} - 6 q^{29} - 9 q^{32} + 22 q^{34} + 4 q^{35} - 6 q^{37} + 6 q^{38} + 6 q^{41} + 8 q^{43} - 6 q^{44} - 11 q^{46} - 16 q^{47} - 4 q^{49} + q^{50} + 5 q^{52} - 24 q^{53} - 4 q^{55} + 10 q^{56} - 7 q^{58} + 12 q^{59} - 2 q^{61} + 8 q^{64} + 4 q^{65} - 8 q^{67} + 9 q^{68} - 6 q^{70} - 8 q^{71} - 24 q^{73} + 3 q^{74} + 7 q^{76} + 4 q^{77} - 3 q^{80} + 17 q^{82} - 2 q^{85} - 2 q^{86} + 10 q^{88} - 18 q^{89} - 20 q^{91} + 2 q^{92} - 24 q^{94} + 4 q^{95} - 2 q^{97} - 18 q^{98}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 1 ) / 2$$ (v^3 + 1) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_1$$ b3 + b2 + b1 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 1$$ 2*b2 - 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 $$T_{2}^{4} - T_{2}^{3} + 2T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

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