# Properties

 Label 572.2.a.b Level $572$ Weight $2$ Character orbit 572.a Self dual yes Analytic conductor $4.567$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + ( -2 + \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + ( -2 + \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} - q^{11} + q^{13} + ( -2 + 2 \beta ) q^{17} + ( -3 - \beta ) q^{19} - q^{21} + ( -4 + \beta ) q^{23} -5 q^{25} + ( -7 - 4 \beta ) q^{27} + ( 4 - 4 \beta ) q^{29} + 2 \beta q^{31} + ( 1 + \beta ) q^{33} -4 \beta q^{37} + ( -1 - \beta ) q^{39} + ( 4 - \beta ) q^{41} + ( -2 - 2 \beta ) q^{43} -6 q^{47} -3 \beta q^{49} + ( -4 - 2 \beta ) q^{51} + ( -7 + \beta ) q^{53} + ( 6 + 5 \beta ) q^{57} + ( 10 + 2 \beta ) q^{59} + ( -4 + 6 \beta ) q^{61} + ( 7 - 2 \beta ) q^{63} + ( -6 + 2 \beta ) q^{67} + ( 1 + 2 \beta ) q^{69} + ( -8 + 2 \beta ) q^{71} + ( 1 - 5 \beta ) q^{73} + ( 5 + 5 \beta ) q^{75} + ( 2 - \beta ) q^{77} + ( -6 - 4 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( 2 - 5 \beta ) q^{83} + ( 8 + 4 \beta ) q^{87} + ( -2 + \beta ) q^{91} + ( -6 - 4 \beta ) q^{93} + ( 12 - 4 \beta ) q^{97} + ( -1 - 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 3q^{7} + 5q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 3q^{7} + 5q^{9} - 2q^{11} + 2q^{13} - 2q^{17} - 7q^{19} - 2q^{21} - 7q^{23} - 10q^{25} - 18q^{27} + 4q^{29} + 2q^{31} + 3q^{33} - 4q^{37} - 3q^{39} + 7q^{41} - 6q^{43} - 12q^{47} - 3q^{49} - 10q^{51} - 13q^{53} + 17q^{57} + 22q^{59} - 2q^{61} + 12q^{63} - 10q^{67} + 4q^{69} - 14q^{71} - 3q^{73} + 15q^{75} + 3q^{77} - 16q^{79} + 38q^{81} - q^{83} + 20q^{87} - 3q^{91} - 16q^{93} + 20q^{97} - 5q^{99} + O(q^{100})$$

## 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
 2.30278 −1.30278
0 −3.30278 0 0 0 0.302776 0 7.90833 0
1.2 0 0.302776 0 0 0 −3.30278 0 −2.90833 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.b 2
3.b odd 2 1 5148.2.a.h 2
4.b odd 2 1 2288.2.a.r 2
8.b even 2 1 9152.2.a.bq 2
8.d odd 2 1 9152.2.a.bg 2
11.b odd 2 1 6292.2.a.k 2
13.b even 2 1 7436.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.a.b 2 1.a even 1 1 trivial
2288.2.a.r 2 4.b odd 2 1
5148.2.a.h 2 3.b odd 2 1
6292.2.a.k 2 11.b odd 2 1
7436.2.a.e 2 13.b even 2 1
9152.2.a.bg 2 8.d odd 2 1
9152.2.a.bq 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(572))$$:

 $$T_{3}^{2} + 3 T_{3} - 1$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 + 3 T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-12 + 2 T + T^{2}$$
$19$ $$9 + 7 T + T^{2}$$
$23$ $$9 + 7 T + T^{2}$$
$29$ $$-48 - 4 T + T^{2}$$
$31$ $$-12 - 2 T + T^{2}$$
$37$ $$-48 + 4 T + T^{2}$$
$41$ $$9 - 7 T + T^{2}$$
$43$ $$-4 + 6 T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$39 + 13 T + T^{2}$$
$59$ $$108 - 22 T + T^{2}$$
$61$ $$-116 + 2 T + T^{2}$$
$67$ $$12 + 10 T + T^{2}$$
$71$ $$36 + 14 T + T^{2}$$
$73$ $$-79 + 3 T + T^{2}$$
$79$ $$12 + 16 T + T^{2}$$
$83$ $$-81 + T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$48 - 20 T + T^{2}$$