# Properties

 Label 528.2.a.g Level $528$ Weight $2$ Character orbit 528.a Self dual yes Analytic conductor $4.216$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 528.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.21610122672$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{5} - 4 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^5 - 4 * q^7 + q^9 $$q + q^{3} - 2 q^{5} - 4 q^{7} + q^{9} - q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{17} - 4 q^{21} - 8 q^{23} - q^{25} + q^{27} - 6 q^{29} + 8 q^{31} - q^{33} + 8 q^{35} + 6 q^{37} - 2 q^{39} - 2 q^{41} - 2 q^{45} - 8 q^{47} + 9 q^{49} - 2 q^{51} + 6 q^{53} + 2 q^{55} + 4 q^{59} + 6 q^{61} - 4 q^{63} + 4 q^{65} + 4 q^{67} - 8 q^{69} - 14 q^{73} - q^{75} + 4 q^{77} + 4 q^{79} + q^{81} - 12 q^{83} + 4 q^{85} - 6 q^{87} - 6 q^{89} + 8 q^{91} + 8 q^{93} + 2 q^{97} - q^{99}+O(q^{100})$$ q + q^3 - 2 * q^5 - 4 * q^7 + q^9 - q^11 - 2 * q^13 - 2 * q^15 - 2 * q^17 - 4 * q^21 - 8 * q^23 - q^25 + q^27 - 6 * q^29 + 8 * q^31 - q^33 + 8 * q^35 + 6 * q^37 - 2 * q^39 - 2 * q^41 - 2 * q^45 - 8 * q^47 + 9 * q^49 - 2 * q^51 + 6 * q^53 + 2 * q^55 + 4 * q^59 + 6 * q^61 - 4 * q^63 + 4 * q^65 + 4 * q^67 - 8 * q^69 - 14 * q^73 - q^75 + 4 * q^77 + 4 * q^79 + q^81 - 12 * q^83 + 4 * q^85 - 6 * q^87 - 6 * q^89 + 8 * q^91 + 8 * q^93 + 2 * q^97 - 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
0 1.00000 0 −2.00000 0 −4.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 528.2.a.g 1
3.b odd 2 1 1584.2.a.o 1
4.b odd 2 1 33.2.a.a 1
8.b even 2 1 2112.2.a.j 1
8.d odd 2 1 2112.2.a.bb 1
11.b odd 2 1 5808.2.a.t 1
12.b even 2 1 99.2.a.b 1
20.d odd 2 1 825.2.a.a 1
20.e even 4 2 825.2.c.a 2
24.f even 2 1 6336.2.a.x 1
24.h odd 2 1 6336.2.a.n 1
28.d even 2 1 1617.2.a.j 1
36.f odd 6 2 891.2.e.e 2
36.h even 6 2 891.2.e.g 2
44.c even 2 1 363.2.a.b 1
44.g even 10 4 363.2.e.g 4
44.h odd 10 4 363.2.e.e 4
52.b odd 2 1 5577.2.a.a 1
60.h even 2 1 2475.2.a.g 1
60.l odd 4 2 2475.2.c.d 2
68.d odd 2 1 9537.2.a.m 1
84.h odd 2 1 4851.2.a.b 1
132.d odd 2 1 1089.2.a.j 1
220.g even 2 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 4.b odd 2 1
99.2.a.b 1 12.b even 2 1
363.2.a.b 1 44.c even 2 1
363.2.e.e 4 44.h odd 10 4
363.2.e.g 4 44.g even 10 4
528.2.a.g 1 1.a even 1 1 trivial
825.2.a.a 1 20.d odd 2 1
825.2.c.a 2 20.e even 4 2
891.2.e.e 2 36.f odd 6 2
891.2.e.g 2 36.h even 6 2
1089.2.a.j 1 132.d odd 2 1
1584.2.a.o 1 3.b odd 2 1
1617.2.a.j 1 28.d even 2 1
2112.2.a.j 1 8.b even 2 1
2112.2.a.bb 1 8.d odd 2 1
2475.2.a.g 1 60.h even 2 1
2475.2.c.d 2 60.l odd 4 2
4851.2.a.b 1 84.h odd 2 1
5577.2.a.a 1 52.b odd 2 1
5808.2.a.t 1 11.b odd 2 1
6336.2.a.n 1 24.h odd 2 1
6336.2.a.x 1 24.f even 2 1
9075.2.a.q 1 220.g even 2 1
9537.2.a.m 1 68.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7} + 4$$ T7 + 4 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T + 4$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T + 8$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T - 6$$
$41$ $$T + 2$$
$43$ $$T$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T - 4$$
$61$ $$T - 6$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T - 4$$
$83$ $$T + 12$$
$89$ $$T + 6$$
$97$ $$T - 2$$