# Properties

 Label 539.2.e.j Level 539 Weight 2 Character orbit 539.e Analytic conductor 4.304 Analytic rank 0 Dimension 4 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + 3 \beta_{1} q^{4} + ( -2 - 2 \beta_{1} ) q^{5} + ( 5 + \beta_{3} ) q^{6} -\beta_{3} q^{8} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + 3 \beta_{1} q^{4} + ( -2 - 2 \beta_{1} ) q^{5} + ( 5 + \beta_{3} ) q^{6} -\beta_{3} q^{8} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{9} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{10} -\beta_{1} q^{11} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{12} + ( -1 + \beta_{3} ) q^{13} + ( -2 - 2 \beta_{3} ) q^{15} + ( 1 + \beta_{1} ) q^{16} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( -10 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{18} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{19} + 6 q^{20} + \beta_{3} q^{22} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{24} -\beta_{1} q^{25} + ( 5 + 5 \beta_{1} + \beta_{2} ) q^{26} + ( -10 - 2 \beta_{3} ) q^{27} + ( 4 - 2 \beta_{3} ) q^{29} + ( -10 - 10 \beta_{1} + 2 \beta_{2} ) q^{30} + ( 5 \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{32} + ( -1 - \beta_{1} + \beta_{2} ) q^{33} + ( 5 - \beta_{3} ) q^{34} + ( 9 + 6 \beta_{3} ) q^{36} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -10 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{38} -4 \beta_{1} q^{39} -2 \beta_{2} q^{40} + ( 9 - \beta_{3} ) q^{41} + 8 q^{43} + ( 3 + 3 \beta_{1} ) q^{44} + ( 6 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{45} + ( 10 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{46} + ( 5 + 5 \beta_{1} - \beta_{2} ) q^{47} + ( 1 + \beta_{3} ) q^{48} + \beta_{3} q^{50} + ( -4 - 4 \beta_{1} ) q^{51} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{52} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -10 - 10 \beta_{1} + 10 \beta_{2} ) q^{54} -2 q^{55} -8 q^{57} + ( -10 - 10 \beta_{1} - 4 \beta_{2} ) q^{58} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + ( -6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{60} + ( -5 - 5 \beta_{1} - \beta_{2} ) q^{61} + ( -5 - 5 \beta_{3} ) q^{62} -13 q^{64} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{66} + ( 10 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{68} + ( 12 + 4 \beta_{3} ) q^{69} + ( -6 + 2 \beta_{3} ) q^{71} + ( 10 + 10 \beta_{1} - 3 \beta_{2} ) q^{72} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{73} + ( -10 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{74} + ( -1 - \beta_{1} + \beta_{2} ) q^{75} + ( -6 + 6 \beta_{3} ) q^{76} + 4 \beta_{3} q^{78} + 4 \beta_{2} q^{79} -2 \beta_{1} q^{80} + ( 11 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{81} + ( -5 - 5 \beta_{1} - 9 \beta_{2} ) q^{82} + ( -2 - 6 \beta_{3} ) q^{83} + ( 2 - 2 \beta_{3} ) q^{85} -8 \beta_{2} q^{86} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( -\beta_{2} - \beta_{3} ) q^{88} + ( 2 + 2 \beta_{1} ) q^{89} + ( -20 - 6 \beta_{3} ) q^{90} + ( -6 - 6 \beta_{3} ) q^{92} + ( 10 + 10 \beta_{1} - 6 \beta_{2} ) q^{93} + ( 5 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{94} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 15 + 15 \beta_{1} - 3 \beta_{2} ) q^{96} + ( -4 + 6 \beta_{3} ) q^{97} + ( -3 - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 6q^{4} - 4q^{5} + 20q^{6} - 6q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 6q^{4} - 4q^{5} + 20q^{6} - 6q^{9} + 2q^{11} + 6q^{12} - 4q^{13} - 8q^{15} + 2q^{16} - 2q^{17} + 20q^{18} + 4q^{19} + 24q^{20} + 4q^{23} - 10q^{24} + 2q^{25} + 10q^{26} - 40q^{27} + 16q^{29} - 20q^{30} - 10q^{31} - 2q^{33} + 20q^{34} + 36q^{36} + 8q^{37} + 20q^{38} + 8q^{39} + 36q^{41} + 32q^{43} + 6q^{44} - 12q^{45} - 20q^{46} + 10q^{47} + 4q^{48} - 8q^{51} + 6q^{52} - 8q^{53} - 20q^{54} - 8q^{55} - 32q^{57} - 20q^{58} + 2q^{59} + 12q^{60} - 10q^{61} - 20q^{62} - 52q^{64} + 4q^{65} + 10q^{66} - 20q^{67} - 6q^{68} + 48q^{69} - 24q^{71} + 20q^{72} - 6q^{73} + 20q^{74} - 2q^{75} - 24q^{76} + 4q^{80} - 22q^{81} - 10q^{82} - 8q^{83} + 8q^{85} - 12q^{87} + 4q^{89} - 80q^{90} - 24q^{92} + 20q^{93} - 10q^{94} + 8q^{95} + 30q^{96} - 16q^{97} - 12q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu^{2} + 6 \nu - 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 3 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
−1.11803 1.93649i −0.618034 + 1.07047i −1.50000 + 2.59808i −1.00000 1.73205i 2.76393 0 2.23607 0.736068 + 1.27491i −2.23607 + 3.87298i
67.2 1.11803 + 1.93649i 1.61803 2.80252i −1.50000 + 2.59808i −1.00000 1.73205i 7.23607 0 −2.23607 −3.73607 6.47106i 2.23607 3.87298i
177.1 −1.11803 + 1.93649i −0.618034 1.07047i −1.50000 2.59808i −1.00000 + 1.73205i 2.76393 0 2.23607 0.736068 1.27491i −2.23607 3.87298i
177.2 1.11803 1.93649i 1.61803 + 2.80252i −1.50000 2.59808i −1.00000 + 1.73205i 7.23607 0 −2.23607 −3.73607 + 6.47106i 2.23607 + 3.87298i
 $$n$$: 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.j 4
7.b odd 2 1 539.2.e.i 4
7.c even 3 1 539.2.a.f 2
7.c even 3 1 inner 539.2.e.j 4
7.d odd 6 1 77.2.a.d 2
7.d odd 6 1 539.2.e.i 4
21.g even 6 1 693.2.a.h 2
21.h odd 6 1 4851.2.a.y 2
28.f even 6 1 1232.2.a.m 2
28.g odd 6 1 8624.2.a.ce 2
35.i odd 6 1 1925.2.a.r 2
35.k even 12 2 1925.2.b.h 4
56.j odd 6 1 4928.2.a.bm 2
56.m even 6 1 4928.2.a.bv 2
77.h odd 6 1 5929.2.a.m 2
77.i even 6 1 847.2.a.f 2
77.n even 30 2 847.2.f.b 4
77.n even 30 2 847.2.f.m 4
77.p odd 30 2 847.2.f.a 4
77.p odd 30 2 847.2.f.n 4
231.k odd 6 1 7623.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 7.d odd 6 1
539.2.a.f 2 7.c even 3 1
539.2.e.i 4 7.b odd 2 1
539.2.e.i 4 7.d odd 6 1
539.2.e.j 4 1.a even 1 1 trivial
539.2.e.j 4 7.c even 3 1 inner
693.2.a.h 2 21.g even 6 1
847.2.a.f 2 77.i even 6 1
847.2.f.a 4 77.p odd 30 2
847.2.f.b 4 77.n even 30 2
847.2.f.m 4 77.n even 30 2
847.2.f.n 4 77.p odd 30 2
1232.2.a.m 2 28.f even 6 1
1925.2.a.r 2 35.i odd 6 1
1925.2.b.h 4 35.k even 12 2
4851.2.a.y 2 21.h odd 6 1
4928.2.a.bm 2 56.j odd 6 1
4928.2.a.bv 2 56.m even 6 1
5929.2.a.m 2 77.h odd 6 1
7623.2.a.bl 2 231.k odd 6 1
8624.2.a.ce 2 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 $$S_{2}^{\mathrm{new}}(539, [\chi])$$:

 $$T_{2}^{4} + 5 T_{2}^{2} + 25$$ $$T_{3}^{4} - 2 T_{3}^{3} + 8 T_{3}^{2} + 8 T_{3} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} )( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} )$$
$3$ $$1 - 2 T + 2 T^{2} + 8 T^{3} - 17 T^{4} + 24 T^{5} + 18 T^{6} - 54 T^{7} + 81 T^{8}$$
$5$ $$( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} )^{2}$$
$7$ 
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 + 2 T - 26 T^{2} - 8 T^{3} + 543 T^{4} - 136 T^{5} - 7514 T^{6} + 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 - 4 T - 6 T^{2} + 64 T^{3} - 181 T^{4} + 1216 T^{5} - 2166 T^{6} - 27436 T^{7} + 130321 T^{8}$$
$23$ $$1 - 4 T - 14 T^{2} + 64 T^{3} + 3 T^{4} + 1472 T^{5} - 7406 T^{6} - 48668 T^{7} + 279841 T^{8}$$
$29$ $$( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} )^{2}$$
$31$ $$1 + 10 T + 18 T^{2} + 200 T^{3} + 2663 T^{4} + 6200 T^{5} + 17298 T^{6} + 297910 T^{7} + 923521 T^{8}$$
$37$ $$1 - 8 T - 6 T^{2} + 32 T^{3} + 1163 T^{4} + 1184 T^{5} - 8214 T^{6} - 405224 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 - 18 T + 158 T^{2} - 738 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{4}$$
$47$ $$1 - 10 T - 14 T^{2} - 200 T^{3} + 6087 T^{4} - 9400 T^{5} - 30926 T^{6} - 1038230 T^{7} + 4879681 T^{8}$$
$53$ $$1 + 8 T - 38 T^{2} - 32 T^{3} + 4203 T^{4} - 1696 T^{5} - 106742 T^{6} + 1191016 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 2 T - 110 T^{2} + 8 T^{3} + 9279 T^{4} + 472 T^{5} - 382910 T^{6} - 410758 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 10 T - 42 T^{2} + 200 T^{3} + 10343 T^{4} + 12200 T^{5} - 156282 T^{6} + 2269810 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 20 T + 186 T^{2} + 1600 T^{3} + 14507 T^{4} + 107200 T^{5} + 834954 T^{6} + 6015260 T^{7} + 20151121 T^{8}$$
$71$ $$( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 + 6 T - 114 T^{2} + 24 T^{3} + 14543 T^{4} + 1752 T^{5} - 607506 T^{6} + 2334102 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 78 T^{2} - 157 T^{4} - 486798 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 + 4 T - 10 T^{2} + 332 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 8 T + 30 T^{2} + 776 T^{3} + 9409 T^{4} )^{2}$$