# Properties

 Label 539.2.e.g 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} - 2 q^{6} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + 2*z * q^2 + (z - 1) * q^3 + (2*z - 2) * q^4 + z * q^5 - 2 * q^6 + 2*z * q^9 $$q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} - 2 q^{6} + 2 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + (\zeta_{6} - 1) q^{11} - 2 \zeta_{6} q^{12} - 4 q^{13} - q^{15} + 4 \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + (4 \zeta_{6} - 4) q^{18} - 2 q^{20} - 2 q^{22} + \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 8 \zeta_{6} q^{26} - 5 q^{27} - 2 \zeta_{6} q^{30} + ( - 7 \zeta_{6} + 7) q^{31} + (8 \zeta_{6} - 8) q^{32} - \zeta_{6} q^{33} - 4 q^{34} - 4 q^{36} - 3 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} + 8 q^{41} - 6 q^{43} - 2 \zeta_{6} q^{44} + (2 \zeta_{6} - 2) q^{45} + (2 \zeta_{6} - 2) q^{46} + 8 \zeta_{6} q^{47} - 4 q^{48} + 8 q^{50} - 2 \zeta_{6} q^{51} + ( - 8 \zeta_{6} + 8) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} - 10 \zeta_{6} q^{54} - q^{55} + ( - 5 \zeta_{6} + 5) q^{59} + ( - 2 \zeta_{6} + 2) q^{60} + 12 \zeta_{6} q^{61} + 14 q^{62} - 8 q^{64} - 4 \zeta_{6} q^{65} + ( - 2 \zeta_{6} + 2) q^{66} + ( - 7 \zeta_{6} + 7) q^{67} - 4 \zeta_{6} q^{68} - q^{69} - 3 q^{71} + ( - 4 \zeta_{6} + 4) q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + 4 \zeta_{6} q^{75} + 8 q^{78} + 10 \zeta_{6} q^{79} + (4 \zeta_{6} - 4) q^{80} + (\zeta_{6} - 1) q^{81} + 16 \zeta_{6} q^{82} + 6 q^{83} - 2 q^{85} - 12 \zeta_{6} q^{86} + 15 \zeta_{6} q^{89} - 4 q^{90} - 2 q^{92} + 7 \zeta_{6} q^{93} + (16 \zeta_{6} - 16) q^{94} - 8 \zeta_{6} q^{96} + 7 q^{97} - 2 q^{99} +O(q^{100})$$ q + 2*z * q^2 + (z - 1) * q^3 + (2*z - 2) * q^4 + z * q^5 - 2 * q^6 + 2*z * q^9 + (2*z - 2) * q^10 + (z - 1) * q^11 - 2*z * q^12 - 4 * q^13 - q^15 + 4*z * q^16 + (2*z - 2) * q^17 + (4*z - 4) * q^18 - 2 * q^20 - 2 * q^22 + z * q^23 + (-4*z + 4) * q^25 - 8*z * q^26 - 5 * q^27 - 2*z * q^30 + (-7*z + 7) * q^31 + (8*z - 8) * q^32 - z * q^33 - 4 * q^34 - 4 * q^36 - 3*z * q^37 + (-4*z + 4) * q^39 + 8 * q^41 - 6 * q^43 - 2*z * q^44 + (2*z - 2) * q^45 + (2*z - 2) * q^46 + 8*z * q^47 - 4 * q^48 + 8 * q^50 - 2*z * q^51 + (-8*z + 8) * q^52 + (-6*z + 6) * q^53 - 10*z * q^54 - q^55 + (-5*z + 5) * q^59 + (-2*z + 2) * q^60 + 12*z * q^61 + 14 * q^62 - 8 * q^64 - 4*z * q^65 + (-2*z + 2) * q^66 + (-7*z + 7) * q^67 - 4*z * q^68 - q^69 - 3 * q^71 + (-4*z + 4) * q^73 + (-6*z + 6) * q^74 + 4*z * q^75 + 8 * q^78 + 10*z * q^79 + (4*z - 4) * q^80 + (z - 1) * q^81 + 16*z * q^82 + 6 * q^83 - 2 * q^85 - 12*z * q^86 + 15*z * q^89 - 4 * q^90 - 2 * q^92 + 7*z * q^93 + (16*z - 16) * q^94 - 8*z * q^96 + 7 * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - q^3 - 2 * q^4 + q^5 - 4 * q^6 + 2 * q^9 $$2 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{9} - 2 q^{10} - q^{11} - 2 q^{12} - 8 q^{13} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 4 q^{20} - 4 q^{22} + q^{23} + 4 q^{25} - 8 q^{26} - 10 q^{27} - 2 q^{30} + 7 q^{31} - 8 q^{32} - q^{33} - 8 q^{34} - 8 q^{36} - 3 q^{37} + 4 q^{39} + 16 q^{41} - 12 q^{43} - 2 q^{44} - 2 q^{45} - 2 q^{46} + 8 q^{47} - 8 q^{48} + 16 q^{50} - 2 q^{51} + 8 q^{52} + 6 q^{53} - 10 q^{54} - 2 q^{55} + 5 q^{59} + 2 q^{60} + 12 q^{61} + 28 q^{62} - 16 q^{64} - 4 q^{65} + 2 q^{66} + 7 q^{67} - 4 q^{68} - 2 q^{69} - 6 q^{71} + 4 q^{73} + 6 q^{74} + 4 q^{75} + 16 q^{78} + 10 q^{79} - 4 q^{80} - q^{81} + 16 q^{82} + 12 q^{83} - 4 q^{85} - 12 q^{86} + 15 q^{89} - 8 q^{90} - 4 q^{92} + 7 q^{93} - 16 q^{94} - 8 q^{96} + 14 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - q^3 - 2 * q^4 + q^5 - 4 * q^6 + 2 * q^9 - 2 * q^10 - q^11 - 2 * q^12 - 8 * q^13 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 4 * q^20 - 4 * q^22 + q^23 + 4 * q^25 - 8 * q^26 - 10 * q^27 - 2 * q^30 + 7 * q^31 - 8 * q^32 - q^33 - 8 * q^34 - 8 * q^36 - 3 * q^37 + 4 * q^39 + 16 * q^41 - 12 * q^43 - 2 * q^44 - 2 * q^45 - 2 * q^46 + 8 * q^47 - 8 * q^48 + 16 * q^50 - 2 * q^51 + 8 * q^52 + 6 * q^53 - 10 * q^54 - 2 * q^55 + 5 * q^59 + 2 * q^60 + 12 * q^61 + 28 * q^62 - 16 * q^64 - 4 * q^65 + 2 * q^66 + 7 * q^67 - 4 * q^68 - 2 * q^69 - 6 * q^71 + 4 * q^73 + 6 * q^74 + 4 * q^75 + 16 * q^78 + 10 * q^79 - 4 * q^80 - q^81 + 16 * q^82 + 12 * q^83 - 4 * q^85 - 12 * q^86 + 15 * q^89 - 8 * q^90 - 4 * q^92 + 7 * q^93 - 16 * q^94 - 8 * q^96 + 14 * q^97 - 4 * 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
1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i −2.00000 0 0 1.00000 + 1.73205i −1.00000 + 1.73205i
177.1 1.00000 1.73205i −0.500000 0.866025i −1.00000 1.73205i 0.500000 0.866025i −2.00000 0 0 1.00000 1.73205i −1.00000 1.73205i
 $$n$$: 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.g 2
7.b odd 2 1 539.2.e.h 2
7.c even 3 1 539.2.a.a 1
7.c even 3 1 inner 539.2.e.g 2
7.d odd 6 1 11.2.a.a 1
7.d odd 6 1 539.2.e.h 2
21.g even 6 1 99.2.a.d 1
21.h odd 6 1 4851.2.a.t 1
28.f even 6 1 176.2.a.b 1
28.g odd 6 1 8624.2.a.j 1
35.i odd 6 1 275.2.a.b 1
35.k even 12 2 275.2.b.a 2
56.j odd 6 1 704.2.a.h 1
56.m even 6 1 704.2.a.c 1
63.i even 6 1 891.2.e.b 2
63.k odd 6 1 891.2.e.k 2
63.s even 6 1 891.2.e.b 2
63.t odd 6 1 891.2.e.k 2
77.h odd 6 1 5929.2.a.h 1
77.i even 6 1 121.2.a.d 1
77.n even 30 4 121.2.c.a 4
77.p odd 30 4 121.2.c.e 4
84.j odd 6 1 1584.2.a.g 1
91.s odd 6 1 1859.2.a.b 1
105.p even 6 1 2475.2.a.a 1
105.w odd 12 2 2475.2.c.a 2
112.v even 12 2 2816.2.c.f 2
112.x odd 12 2 2816.2.c.j 2
119.h odd 6 1 3179.2.a.a 1
133.o even 6 1 3971.2.a.b 1
140.s even 6 1 4400.2.a.i 1
140.x odd 12 2 4400.2.b.h 2
161.g even 6 1 5819.2.a.a 1
168.ba even 6 1 6336.2.a.br 1
168.be odd 6 1 6336.2.a.bu 1
203.i odd 6 1 9251.2.a.d 1
231.k odd 6 1 1089.2.a.b 1
308.m odd 6 1 1936.2.a.i 1
385.o even 6 1 3025.2.a.a 1
616.s even 6 1 7744.2.a.x 1
616.z odd 6 1 7744.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 7.d odd 6 1
99.2.a.d 1 21.g even 6 1
121.2.a.d 1 77.i even 6 1
121.2.c.a 4 77.n even 30 4
121.2.c.e 4 77.p odd 30 4
176.2.a.b 1 28.f even 6 1
275.2.a.b 1 35.i odd 6 1
275.2.b.a 2 35.k even 12 2
539.2.a.a 1 7.c even 3 1
539.2.e.g 2 1.a even 1 1 trivial
539.2.e.g 2 7.c even 3 1 inner
539.2.e.h 2 7.b odd 2 1
539.2.e.h 2 7.d odd 6 1
704.2.a.c 1 56.m even 6 1
704.2.a.h 1 56.j odd 6 1
891.2.e.b 2 63.i even 6 1
891.2.e.b 2 63.s even 6 1
891.2.e.k 2 63.k odd 6 1
891.2.e.k 2 63.t odd 6 1
1089.2.a.b 1 231.k odd 6 1
1584.2.a.g 1 84.j odd 6 1
1859.2.a.b 1 91.s odd 6 1
1936.2.a.i 1 308.m odd 6 1
2475.2.a.a 1 105.p even 6 1
2475.2.c.a 2 105.w odd 12 2
2816.2.c.f 2 112.v even 12 2
2816.2.c.j 2 112.x odd 12 2
3025.2.a.a 1 385.o even 6 1
3179.2.a.a 1 119.h odd 6 1
3971.2.a.b 1 133.o even 6 1
4400.2.a.i 1 140.s even 6 1
4400.2.b.h 2 140.x odd 12 2
4851.2.a.t 1 21.h odd 6 1
5819.2.a.a 1 161.g even 6 1
5929.2.a.h 1 77.h odd 6 1
6336.2.a.br 1 168.ba even 6 1
6336.2.a.bu 1 168.be odd 6 1
7744.2.a.k 1 616.z odd 6 1
7744.2.a.x 1 616.s even 6 1
8624.2.a.j 1 28.g odd 6 1
9251.2.a.d 1 203.i odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + T + 1$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} - T + 1$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 7T + 49$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T - 8)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 5T + 25$$
$61$ $$T^{2} - 12T + 144$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 15T + 225$$
$97$ $$(T - 7)^{2}$$