# Properties

 Label 572.2.a.a Level $572$ Weight $2$ Character orbit 572.a Self dual yes Analytic conductor $4.567$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 3q^{5} + 2q^{7} - 2q^{9} + O(q^{10})$$ $$q + q^{3} + 3q^{5} + 2q^{7} - 2q^{9} + q^{11} + q^{13} + 3q^{15} + 2q^{19} + 2q^{21} - 3q^{23} + 4q^{25} - 5q^{27} - 6q^{29} - q^{31} + q^{33} + 6q^{35} - 7q^{37} + q^{39} + 6q^{41} + 8q^{43} - 6q^{45} + 12q^{47} - 3q^{49} - 6q^{53} + 3q^{55} + 2q^{57} + 9q^{59} + 2q^{61} - 4q^{63} + 3q^{65} - 7q^{67} - 3q^{69} - 3q^{71} + 8q^{73} + 4q^{75} + 2q^{77} - 4q^{79} + q^{81} - 12q^{83} - 6q^{87} - 15q^{89} + 2q^{91} - q^{93} + 6q^{95} - 13q^{97} - 2q^{99} + O(q^{100})$$

## 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
 0
0 1.00000 0 3.00000 0 2.00000 0 −2.00000 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
$$2$$ $$-1$$
$$11$$ $$-1$$
$$13$$ $$-1$$

## 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 572.2.a.a 1
3.b odd 2 1 5148.2.a.a 1
4.b odd 2 1 2288.2.a.g 1
8.b even 2 1 9152.2.a.i 1
8.d odd 2 1 9152.2.a.r 1
11.b odd 2 1 6292.2.a.j 1
13.b even 2 1 7436.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.a.a 1 1.a even 1 1 trivial
2288.2.a.g 1 4.b odd 2 1
5148.2.a.a 1 3.b odd 2 1
6292.2.a.j 1 11.b odd 2 1
7436.2.a.c 1 13.b even 2 1
9152.2.a.i 1 8.b even 2 1
9152.2.a.r 1 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(572))$$:

 $$T_{3} - 1$$ $$T_{5} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-3 + T$$
$7$ $$-2 + T$$
$11$ $$-1 + T$$
$13$ $$-1 + T$$
$17$ $$T$$
$19$ $$-2 + T$$
$23$ $$3 + T$$
$29$ $$6 + T$$
$31$ $$1 + T$$
$37$ $$7 + T$$
$41$ $$-6 + T$$
$43$ $$-8 + T$$
$47$ $$-12 + T$$
$53$ $$6 + T$$
$59$ $$-9 + T$$
$61$ $$-2 + T$$
$67$ $$7 + T$$
$71$ $$3 + T$$
$73$ $$-8 + T$$
$79$ $$4 + T$$
$83$ $$12 + T$$
$89$ $$15 + T$$
$97$ $$13 + T$$