# Properties

 Label 572.2.a.d Level $572$ Weight $2$ Character orbit 572.a Self dual yes Analytic conductor $4.567$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 q^{5} + ( 3 - \beta ) q^{7} + ( 2 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 2 q^{5} + ( 3 - \beta ) q^{7} + ( 2 + \beta ) q^{9} - q^{11} - q^{13} + 2 \beta q^{15} -4 q^{17} + ( 4 - \beta ) q^{19} + ( -5 + 2 \beta ) q^{21} + ( 3 - \beta ) q^{23} - q^{25} + 5 q^{27} + ( 2 - 2 \beta ) q^{29} -\beta q^{33} + ( 6 - 2 \beta ) q^{35} + 4 q^{37} -\beta q^{39} + ( -1 + \beta ) q^{41} + ( 4 - 4 \beta ) q^{43} + ( 4 + 2 \beta ) q^{45} + ( -2 + 2 \beta ) q^{47} + ( 7 - 5 \beta ) q^{49} -4 \beta q^{51} + ( -4 + 3 \beta ) q^{53} -2 q^{55} + ( -5 + 3 \beta ) q^{57} -2 \beta q^{59} + ( 8 - 2 \beta ) q^{61} + q^{63} -2 q^{65} + 4 \beta q^{67} + ( -5 + 2 \beta ) q^{69} + ( -8 - \beta ) q^{73} -\beta q^{75} + ( -3 + \beta ) q^{77} + ( -4 + 6 \beta ) q^{79} + ( -6 + 2 \beta ) q^{81} + ( 9 - 3 \beta ) q^{83} -8 q^{85} -10 q^{87} + ( 4 + 2 \beta ) q^{89} + ( -3 + \beta ) q^{91} + ( 8 - 2 \beta ) q^{95} + ( 2 + 6 \beta ) q^{97} + ( -2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 4q^{5} + 5q^{7} + 5q^{9} + O(q^{10})$$ $$2q + q^{3} + 4q^{5} + 5q^{7} + 5q^{9} - 2q^{11} - 2q^{13} + 2q^{15} - 8q^{17} + 7q^{19} - 8q^{21} + 5q^{23} - 2q^{25} + 10q^{27} + 2q^{29} - q^{33} + 10q^{35} + 8q^{37} - q^{39} - q^{41} + 4q^{43} + 10q^{45} - 2q^{47} + 9q^{49} - 4q^{51} - 5q^{53} - 4q^{55} - 7q^{57} - 2q^{59} + 14q^{61} + 2q^{63} - 4q^{65} + 4q^{67} - 8q^{69} - 17q^{73} - q^{75} - 5q^{77} - 2q^{79} - 10q^{81} + 15q^{83} - 16q^{85} - 20q^{87} + 10q^{89} - 5q^{91} + 14q^{95} + 10q^{97} - 5q^{99} + O(q^{100})$$

## 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
 −1.79129 2.79129
0 −1.79129 0 2.00000 0 4.79129 0 0.208712 0
1.2 0 2.79129 0 2.00000 0 0.208712 0 4.79129 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.d 2
3.b odd 2 1 5148.2.a.g 2
4.b odd 2 1 2288.2.a.n 2
8.b even 2 1 9152.2.a.bl 2
8.d odd 2 1 9152.2.a.bn 2
11.b odd 2 1 6292.2.a.o 2
13.b even 2 1 7436.2.a.h 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-5 - T + T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$1 - 5 T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$( 4 + T )^{2}$$
$19$ $$7 - 7 T + T^{2}$$
$23$ $$1 - 5 T + T^{2}$$
$29$ $$-20 - 2 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$-5 + T + T^{2}$$
$43$ $$-80 - 4 T + T^{2}$$
$47$ $$-20 + 2 T + T^{2}$$
$53$ $$-41 + 5 T + T^{2}$$
$59$ $$-20 + 2 T + T^{2}$$
$61$ $$28 - 14 T + T^{2}$$
$67$ $$-80 - 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$67 + 17 T + T^{2}$$
$79$ $$-188 + 2 T + T^{2}$$
$83$ $$9 - 15 T + T^{2}$$
$89$ $$4 - 10 T + T^{2}$$
$97$ $$-164 - 10 T + T^{2}$$