# Properties

 Label 539.2.a.i Level $539$ Weight $2$ Character orbit 539.a Self dual yes Analytic conductor $4.304$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 539.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.30393666895$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} + ( -3 + \beta_{2} ) q^{8} + \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} + ( -3 + \beta_{2} ) q^{8} + \beta_{2} q^{9} + ( 3 - \beta_{2} ) q^{10} - q^{11} + 3 q^{12} + ( 4 - \beta_{1} ) q^{13} + ( -3 + 2 \beta_{1} ) q^{15} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{16} + ( 1 + \beta_{2} ) q^{17} + ( 3 + \beta_{1} - \beta_{2} ) q^{18} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{19} + ( -5 + \beta_{1} + 2 \beta_{2} ) q^{20} -\beta_{2} q^{22} + ( 4 - \beta_{2} ) q^{23} + ( -2 \beta_{1} + \beta_{2} ) q^{24} + ( 2 - 3 \beta_{1} ) q^{25} + ( -\beta_{1} + 3 \beta_{2} ) q^{26} + ( -2 \beta_{1} + \beta_{2} ) q^{27} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{29} + ( 2 \beta_{1} - \beta_{2} ) q^{30} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{31} -3 \beta_{1} q^{32} -\beta_{1} q^{33} + ( 3 + \beta_{1} ) q^{34} + ( -3 + 3 \beta_{2} ) q^{36} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -3 + \beta_{1} + 6 \beta_{2} ) q^{38} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{39} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{40} + ( 2 - \beta_{1} - 4 \beta_{2} ) q^{41} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} ) q^{44} + ( 3 - \beta_{2} ) q^{45} + ( -3 - \beta_{1} + 5 \beta_{2} ) q^{46} + ( 1 + \beta_{2} ) q^{47} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{48} + ( -3 \beta_{1} - \beta_{2} ) q^{50} + ( 2 \beta_{1} + \beta_{2} ) q^{51} + ( 1 + 4 \beta_{1} - 4 \beta_{2} ) q^{52} + ( 6 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{54} + ( -1 + \beta_{1} - \beta_{2} ) q^{55} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{57} + ( -3 - 4 \beta_{1} - 4 \beta_{2} ) q^{58} + ( -3 + \beta_{1} - 6 \beta_{2} ) q^{59} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{60} + ( 8 - 2 \beta_{2} ) q^{61} + ( -3 - 4 \beta_{1} ) q^{62} + ( -2 - \beta_{1} + \beta_{2} ) q^{64} + ( 7 - 6 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -\beta_{1} - \beta_{2} ) q^{66} + ( -6 + 2 \beta_{1} ) q^{67} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{68} + ( 3 \beta_{1} - \beta_{2} ) q^{69} + ( 1 + 4 \beta_{1} - 3 \beta_{2} ) q^{71} + ( 3 + \beta_{1} - 4 \beta_{2} ) q^{72} + ( 5 + 5 \beta_{1} ) q^{73} + ( 6 + 4 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -9 + 2 \beta_{1} - 3 \beta_{2} ) q^{75} + ( 12 + 3 \beta_{1} - 6 \beta_{2} ) q^{76} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{78} + ( 3 \beta_{1} - \beta_{2} ) q^{79} + ( -2 - 3 \beta_{1} + 3 \beta_{2} ) q^{80} + ( -6 + \beta_{1} - 4 \beta_{2} ) q^{81} + ( -12 - 5 \beta_{1} + 5 \beta_{2} ) q^{82} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{83} + ( 4 - \beta_{1} ) q^{85} + ( -6 - 3 \beta_{1} + 2 \beta_{2} ) q^{86} + ( -9 - 3 \beta_{1} - 4 \beta_{2} ) q^{87} + ( 3 - \beta_{2} ) q^{88} + ( -\beta_{1} + \beta_{2} ) q^{89} + ( -3 - \beta_{1} + 4 \beta_{2} ) q^{90} + ( 7 + 4 \beta_{1} - 7 \beta_{2} ) q^{92} + ( -9 + \beta_{1} - 4 \beta_{2} ) q^{93} + ( 3 + \beta_{1} ) q^{94} + ( -6 + \beta_{1} + 4 \beta_{2} ) q^{95} + ( -9 - 3 \beta_{2} ) q^{96} + ( -4 + 3 \beta_{1} ) q^{97} -\beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{3} + 4q^{4} + 2q^{5} + q^{6} - 9q^{8} + O(q^{10})$$ $$3q + q^{3} + 4q^{4} + 2q^{5} + q^{6} - 9q^{8} + 9q^{10} - 3q^{11} + 9q^{12} + 11q^{13} - 7q^{15} + 2q^{16} + 3q^{17} + 10q^{18} + 11q^{19} - 14q^{20} + 12q^{23} - 2q^{24} + 3q^{25} - q^{26} - 2q^{27} - 9q^{29} + 2q^{30} + 3q^{31} - 3q^{32} - q^{33} + 10q^{34} - 9q^{36} - 4q^{37} - 8q^{38} - 5q^{39} + 3q^{40} + 5q^{41} + 2q^{43} - 4q^{44} + 9q^{45} - 10q^{46} + 3q^{47} - 10q^{48} - 3q^{50} + 2q^{51} + 7q^{52} + 17q^{53} + 8q^{54} - 2q^{55} + 20q^{57} - 13q^{58} - 8q^{59} + 6q^{60} + 24q^{61} - 13q^{62} - 7q^{64} + 15q^{65} - q^{66} - 16q^{67} - 5q^{68} + 3q^{69} + 7q^{71} + 10q^{72} + 20q^{73} + 22q^{74} - 25q^{75} + 39q^{76} - 6q^{78} + 3q^{79} - 9q^{80} - 17q^{81} - 41q^{82} + 11q^{83} + 11q^{85} - 21q^{86} - 30q^{87} + 9q^{88} - q^{89} - 10q^{90} + 25q^{92} - 26q^{93} + 10q^{94} - 17q^{95} - 27q^{96} - 9q^{97} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.713538 −1.91223 2.19869
−2.49086 0.713538 4.20440 −2.20440 −1.77733 0 −5.49086 −2.49086 5.49086
1.2 0.656620 −1.91223 −1.56885 3.56885 −1.25561 0 −2.34338 0.656620 2.34338
1.3 1.83424 2.19869 1.36445 0.635552 4.03293 0 −1.16576 1.83424 1.16576
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.a.i 3
3.b odd 2 1 4851.2.a.bn 3
4.b odd 2 1 8624.2.a.ck 3
7.b odd 2 1 539.2.a.h 3
7.c even 3 2 539.2.e.l 6
7.d odd 6 2 77.2.e.b 6
11.b odd 2 1 5929.2.a.w 3
21.c even 2 1 4851.2.a.bo 3
21.g even 6 2 693.2.i.g 6
28.d even 2 1 8624.2.a.cl 3
28.f even 6 2 1232.2.q.k 6
77.b even 2 1 5929.2.a.v 3
77.i even 6 2 847.2.e.d 6
77.n even 30 8 847.2.n.d 24
77.p odd 30 8 847.2.n.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.b 6 7.d odd 6 2
539.2.a.h 3 7.b odd 2 1
539.2.a.i 3 1.a even 1 1 trivial
539.2.e.l 6 7.c even 3 2
693.2.i.g 6 21.g even 6 2
847.2.e.d 6 77.i even 6 2
847.2.n.d 24 77.n even 30 8
847.2.n.e 24 77.p odd 30 8
1232.2.q.k 6 28.f even 6 2
4851.2.a.bn 3 3.b odd 2 1
4851.2.a.bo 3 21.c even 2 1
5929.2.a.v 3 77.b even 2 1
5929.2.a.w 3 11.b odd 2 1
8624.2.a.ck 3 4.b odd 2 1
8624.2.a.cl 3 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(539))$$:

 $$T_{2}^{3} - 5 T_{2} + 3$$ $$T_{3}^{3} - T_{3}^{2} - 4 T_{3} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 5 T + T^{3}$$
$3$ $$3 - 4 T - T^{2} + T^{3}$$
$5$ $$5 - 7 T - 2 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$-35 + 36 T - 11 T^{2} + T^{3}$$
$17$ $$7 - 2 T - 3 T^{2} + T^{3}$$
$19$ $$57 + 20 T - 11 T^{2} + T^{3}$$
$23$ $$-47 + 43 T - 12 T^{2} + T^{3}$$
$29$ $$-53 - 20 T + 9 T^{2} + T^{3}$$
$31$ $$107 - 44 T - 3 T^{2} + T^{3}$$
$37$ $$-152 - 36 T + 4 T^{2} + T^{3}$$
$41$ $$109 - 80 T - 5 T^{2} + T^{3}$$
$43$ $$41 - 25 T - 2 T^{2} + T^{3}$$
$47$ $$7 - 2 T - 3 T^{2} + T^{3}$$
$53$ $$-21 + 74 T - 17 T^{2} + T^{3}$$
$59$ $$-1323 - 157 T + 8 T^{2} + T^{3}$$
$61$ $$-376 + 172 T - 24 T^{2} + T^{3}$$
$67$ $$72 + 68 T + 16 T^{2} + T^{3}$$
$71$ $$419 - 86 T - 7 T^{2} + T^{3}$$
$73$ $$625 + 25 T - 20 T^{2} + T^{3}$$
$79$ $$141 - 38 T - 3 T^{2} + T^{3}$$
$83$ $$3 + 16 T - 11 T^{2} + T^{3}$$
$89$ $$-3 - 8 T + T^{2} + T^{3}$$
$97$ $$-47 - 12 T + 9 T^{2} + T^{3}$$