# Properties

 Label 539.2.e.c Level $539$ Weight $2$ Character orbit 539.e Analytic conductor $4.304$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 539.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.30393666895$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} - \zeta_{6} q^{5} - 6 \zeta_{6} q^{9} +O(q^{10})$$ q + (3*z - 3) * q^3 + (-2*z + 2) * q^4 - z * q^5 - 6*z * q^9 $$q + (3 \zeta_{6} - 3) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} - \zeta_{6} q^{5} - 6 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + 6 \zeta_{6} q^{12} + 4 q^{13} + 3 q^{15} - 4 \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{17} - 6 \zeta_{6} q^{19} - 2 q^{20} + 5 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + 9 q^{27} + 10 q^{29} + ( - \zeta_{6} + 1) q^{31} + 3 \zeta_{6} q^{33} - 12 q^{36} + 5 \zeta_{6} q^{37} + (12 \zeta_{6} - 12) q^{39} + 2 q^{41} - 8 q^{43} - 2 \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{45} + 8 \zeta_{6} q^{47} + 12 q^{48} + 6 \zeta_{6} q^{51} + ( - 8 \zeta_{6} + 8) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} - q^{55} + 18 q^{57} + ( - 3 \zeta_{6} + 3) q^{59} + ( - 6 \zeta_{6} + 6) q^{60} - 2 \zeta_{6} q^{61} - 8 q^{64} - 4 \zeta_{6} q^{65} + ( - 3 \zeta_{6} + 3) q^{67} - 4 \zeta_{6} q^{68} - 15 q^{69} + q^{71} + ( - 10 \zeta_{6} + 10) q^{73} + 12 \zeta_{6} q^{75} - 12 q^{76} - 6 \zeta_{6} q^{79} + (4 \zeta_{6} - 4) q^{80} + (9 \zeta_{6} - 9) q^{81} - 12 q^{83} - 2 q^{85} + (30 \zeta_{6} - 30) q^{87} - 15 \zeta_{6} q^{89} + 10 q^{92} + 3 \zeta_{6} q^{93} + (6 \zeta_{6} - 6) q^{95} + 5 q^{97} - 6 q^{99} +O(q^{100})$$ q + (3*z - 3) * q^3 + (-2*z + 2) * q^4 - z * q^5 - 6*z * q^9 + (-z + 1) * q^11 + 6*z * q^12 + 4 * q^13 + 3 * q^15 - 4*z * q^16 + (-2*z + 2) * q^17 - 6*z * q^19 - 2 * q^20 + 5*z * q^23 + (-4*z + 4) * q^25 + 9 * q^27 + 10 * q^29 + (-z + 1) * q^31 + 3*z * q^33 - 12 * q^36 + 5*z * q^37 + (12*z - 12) * q^39 + 2 * q^41 - 8 * q^43 - 2*z * q^44 + (6*z - 6) * q^45 + 8*z * q^47 + 12 * q^48 + 6*z * q^51 + (-8*z + 8) * q^52 + (-6*z + 6) * q^53 - q^55 + 18 * q^57 + (-3*z + 3) * q^59 + (-6*z + 6) * q^60 - 2*z * q^61 - 8 * q^64 - 4*z * q^65 + (-3*z + 3) * q^67 - 4*z * q^68 - 15 * q^69 + q^71 + (-10*z + 10) * q^73 + 12*z * q^75 - 12 * q^76 - 6*z * q^79 + (4*z - 4) * q^80 + (9*z - 9) * q^81 - 12 * q^83 - 2 * q^85 + (30*z - 30) * q^87 - 15*z * q^89 + 10 * q^92 + 3*z * q^93 + (6*z - 6) * q^95 + 5 * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 2 q^{4} - q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 2 * q^4 - q^5 - 6 * q^9 $$2 q - 3 q^{3} + 2 q^{4} - q^{5} - 6 q^{9} + q^{11} + 6 q^{12} + 8 q^{13} + 6 q^{15} - 4 q^{16} + 2 q^{17} - 6 q^{19} - 4 q^{20} + 5 q^{23} + 4 q^{25} + 18 q^{27} + 20 q^{29} + q^{31} + 3 q^{33} - 24 q^{36} + 5 q^{37} - 12 q^{39} + 4 q^{41} - 16 q^{43} - 2 q^{44} - 6 q^{45} + 8 q^{47} + 24 q^{48} + 6 q^{51} + 8 q^{52} + 6 q^{53} - 2 q^{55} + 36 q^{57} + 3 q^{59} + 6 q^{60} - 2 q^{61} - 16 q^{64} - 4 q^{65} + 3 q^{67} - 4 q^{68} - 30 q^{69} + 2 q^{71} + 10 q^{73} + 12 q^{75} - 24 q^{76} - 6 q^{79} - 4 q^{80} - 9 q^{81} - 24 q^{83} - 4 q^{85} - 30 q^{87} - 15 q^{89} + 20 q^{92} + 3 q^{93} - 6 q^{95} + 10 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 2 * q^4 - q^5 - 6 * q^9 + q^11 + 6 * q^12 + 8 * q^13 + 6 * q^15 - 4 * q^16 + 2 * q^17 - 6 * q^19 - 4 * q^20 + 5 * q^23 + 4 * q^25 + 18 * q^27 + 20 * q^29 + q^31 + 3 * q^33 - 24 * q^36 + 5 * q^37 - 12 * q^39 + 4 * q^41 - 16 * q^43 - 2 * q^44 - 6 * q^45 + 8 * q^47 + 24 * q^48 + 6 * q^51 + 8 * q^52 + 6 * q^53 - 2 * q^55 + 36 * q^57 + 3 * q^59 + 6 * q^60 - 2 * q^61 - 16 * q^64 - 4 * q^65 + 3 * q^67 - 4 * q^68 - 30 * q^69 + 2 * q^71 + 10 * q^73 + 12 * q^75 - 24 * q^76 - 6 * q^79 - 4 * q^80 - 9 * q^81 - 24 * q^83 - 4 * q^85 - 30 * q^87 - 15 * q^89 + 20 * q^92 + 3 * q^93 - 6 * q^95 + 10 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$442$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 + 2.59808i 1.00000 1.73205i −0.500000 0.866025i 0 0 0 −3.00000 5.19615i 0
177.1 0 −1.50000 2.59808i 1.00000 + 1.73205i −0.500000 + 0.866025i 0 0 0 −3.00000 + 5.19615i 0
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.e.c 2
7.b odd 2 1 539.2.e.f 2
7.c even 3 1 539.2.a.c 1
7.c even 3 1 inner 539.2.e.c 2
7.d odd 6 1 77.2.a.a 1
7.d odd 6 1 539.2.e.f 2
21.g even 6 1 693.2.a.c 1
21.h odd 6 1 4851.2.a.j 1
28.f even 6 1 1232.2.a.l 1
28.g odd 6 1 8624.2.a.a 1
35.i odd 6 1 1925.2.a.h 1
35.k even 12 2 1925.2.b.e 2
56.j odd 6 1 4928.2.a.bj 1
56.m even 6 1 4928.2.a.a 1
77.h odd 6 1 5929.2.a.f 1
77.i even 6 1 847.2.a.b 1
77.n even 30 4 847.2.f.h 4
77.p odd 30 4 847.2.f.i 4
231.k odd 6 1 7623.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 7.d odd 6 1
539.2.a.c 1 7.c even 3 1
539.2.e.c 2 1.a even 1 1 trivial
539.2.e.c 2 7.c even 3 1 inner
539.2.e.f 2 7.b odd 2 1
539.2.e.f 2 7.d odd 6 1
693.2.a.c 1 21.g even 6 1
847.2.a.b 1 77.i even 6 1
847.2.f.h 4 77.n even 30 4
847.2.f.i 4 77.p odd 30 4
1232.2.a.l 1 28.f even 6 1
1925.2.a.h 1 35.i odd 6 1
1925.2.b.e 2 35.k even 12 2
4851.2.a.j 1 21.h odd 6 1
4928.2.a.a 1 56.m even 6 1
4928.2.a.bj 1 56.j odd 6 1
5929.2.a.f 1 77.h odd 6 1
7623.2.a.j 1 231.k odd 6 1
8624.2.a.a 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(539, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{2} + 3T_{3} + 9$$ T3^2 + 3*T3 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 5T + 25$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2} - T + 1$$
$37$ $$T^{2} - 5T + 25$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$(T - 1)^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2} + 6T + 36$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$(T - 5)^{2}$$