# Properties

 Label 539.2.a.a Level $539$ Weight $2$ Character orbit 539.a Self dual yes Analytic conductor $4.304$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 7.b odd 2 1
99.2.a.d 1 21.c even 2 1
121.2.a.d 1 77.b even 2 1
121.2.c.a 4 77.l even 10 4
121.2.c.e 4 77.j odd 10 4
176.2.a.b 1 28.d even 2 1
275.2.a.b 1 35.c odd 2 1
275.2.b.a 2 35.f even 4 2
539.2.a.a 1 1.a even 1 1 trivial
539.2.e.g 2 7.c even 3 2
539.2.e.h 2 7.d odd 6 2
704.2.a.c 1 56.e even 2 1
704.2.a.h 1 56.h odd 2 1
891.2.e.b 2 63.o even 6 2
891.2.e.k 2 63.l odd 6 2
1089.2.a.b 1 231.h odd 2 1
1584.2.a.g 1 84.h odd 2 1
1859.2.a.b 1 91.b odd 2 1
1936.2.a.i 1 308.g odd 2 1
2475.2.a.a 1 105.g even 2 1
2475.2.c.a 2 105.k odd 4 2
2816.2.c.f 2 112.j even 4 2
2816.2.c.j 2 112.l odd 4 2
3025.2.a.a 1 385.h even 2 1
3179.2.a.a 1 119.d odd 2 1
3971.2.a.b 1 133.c even 2 1
4400.2.a.i 1 140.c even 2 1
4400.2.b.h 2 140.j odd 4 2
4851.2.a.t 1 3.b odd 2 1
5819.2.a.a 1 161.c even 2 1
5929.2.a.h 1 11.b odd 2 1
6336.2.a.br 1 168.i even 2 1
6336.2.a.bu 1 168.e odd 2 1
7744.2.a.k 1 616.g odd 2 1
7744.2.a.x 1 616.o even 2 1
8624.2.a.j 1 4.b odd 2 1
9251.2.a.d 1 203.c odd 2 1

## Hecke kernels

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

 $$T_{2} + 2$$ T2 + 2 $$T_{3} - 1$$ T3 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T$$
$31$ $$T + 7$$
$37$ $$T - 3$$
$41$ $$T - 8$$
$43$ $$T + 6$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T + 5$$
$61$ $$T + 12$$
$67$ $$T + 7$$
$71$ $$T + 3$$
$73$ $$T + 4$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T + 15$$
$97$ $$T - 7$$