# Properties

 Label 539.2.a.f Level 539 Weight 2 Character orbit 539.a Self dual yes Analytic conductor 4.304 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.23607 −3.23607 3.00000 2.00000 7.23607 0 −2.23607 7.47214 −4.47214
1.2 2.23607 1.23607 3.00000 2.00000 2.76393 0 2.23607 −1.47214 4.47214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.f 2
3.b odd 2 1 4851.2.a.y 2
4.b odd 2 1 8624.2.a.ce 2
7.b odd 2 1 77.2.a.d 2
7.c even 3 2 539.2.e.j 4
7.d odd 6 2 539.2.e.i 4
11.b odd 2 1 5929.2.a.m 2
21.c even 2 1 693.2.a.h 2
28.d even 2 1 1232.2.a.m 2
35.c odd 2 1 1925.2.a.r 2
35.f even 4 2 1925.2.b.h 4
56.e even 2 1 4928.2.a.bv 2
56.h odd 2 1 4928.2.a.bm 2
77.b even 2 1 847.2.a.f 2
77.j odd 10 2 847.2.f.a 4
77.j odd 10 2 847.2.f.n 4
77.l even 10 2 847.2.f.b 4
77.l even 10 2 847.2.f.m 4
231.h odd 2 1 7623.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 7.b odd 2 1
539.2.a.f 2 1.a even 1 1 trivial
539.2.e.i 4 7.d odd 6 2
539.2.e.j 4 7.c even 3 2
693.2.a.h 2 21.c even 2 1
847.2.a.f 2 77.b even 2 1
847.2.f.a 4 77.j odd 10 2
847.2.f.b 4 77.l even 10 2
847.2.f.m 4 77.l even 10 2
847.2.f.n 4 77.j odd 10 2
1232.2.a.m 2 28.d even 2 1
1925.2.a.r 2 35.c odd 2 1
1925.2.b.h 4 35.f even 4 2
4851.2.a.y 2 3.b odd 2 1
4928.2.a.bm 2 56.h odd 2 1
4928.2.a.bv 2 56.e even 2 1
5929.2.a.m 2 11.b odd 2 1
7623.2.a.bl 2 231.h odd 2 1
8624.2.a.ce 2 4.b odd 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ $$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 - 2 T + 5 T^{2} )^{2}$$
$7$ 
$11$ $$( 1 + T )^{2}$$
$13$ $$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 - 2 T + 30 T^{2} - 34 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 4 T + 30 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 10 T + 82 T^{2} - 310 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$1 - 18 T + 158 T^{2} - 738 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$1 + 10 T + 114 T^{2} + 470 T^{3} + 2209 T^{4}$$
$53$ $$1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4}$$
$59$ $$1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4}$$
$67$ $$1 - 20 T + 214 T^{2} - 1340 T^{3} + 4489 T^{4}$$
$71$ $$1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4}$$
$73$ $$1 - 6 T + 150 T^{2} - 438 T^{3} + 5329 T^{4}$$
$79$ $$1 + 78 T^{2} + 6241 T^{4}$$
$83$ $$1 + 4 T - 10 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 2 T + 89 T^{2} )^{2}$$
$97$ $$1 + 8 T + 30 T^{2} + 776 T^{3} + 9409 T^{4}$$