# 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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

## 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.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

## Hecke kernels

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

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{3}^{2} + 2T_{3} - 4$$ T3^2 + 2*T3 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$T^{2} + 2T - 4$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} - 2T - 4$$
$19$ $$T^{2} + 4T - 16$$
$23$ $$T^{2} + 4T - 16$$
$29$ $$T^{2} - 8T - 4$$
$31$ $$T^{2} - 10T + 20$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} - 18T + 76$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 10T + 20$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 2T - 4$$
$61$ $$T^{2} - 10T + 20$$
$67$ $$T^{2} - 20T + 80$$
$71$ $$T^{2} + 12T + 16$$
$73$ $$T^{2} - 6T + 4$$
$79$ $$T^{2} - 80$$
$83$ $$T^{2} + 4T - 176$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} + 8T - 164$$