# Properties

 Label 570.2.a.m Level $570$ Weight $2$ Character orbit 570.a Self dual yes Analytic conductor $4.551$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.55147291521$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} + q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - 2q^{13} + 4q^{14} + q^{15} + q^{16} - 2q^{17} + q^{18} - q^{19} + q^{20} + 4q^{21} - 4q^{22} - 8q^{23} + q^{24} + q^{25} - 2q^{26} + q^{27} + 4q^{28} + 6q^{29} + q^{30} + 4q^{31} + q^{32} - 4q^{33} - 2q^{34} + 4q^{35} + q^{36} - 10q^{37} - q^{38} - 2q^{39} + q^{40} - 2q^{41} + 4q^{42} + 12q^{43} - 4q^{44} + q^{45} - 8q^{46} + q^{48} + 9q^{49} + q^{50} - 2q^{51} - 2q^{52} + 6q^{53} + q^{54} - 4q^{55} + 4q^{56} - q^{57} + 6q^{58} + q^{60} - 10q^{61} + 4q^{62} + 4q^{63} + q^{64} - 2q^{65} - 4q^{66} - 4q^{67} - 2q^{68} - 8q^{69} + 4q^{70} - 8q^{71} + q^{72} + 2q^{73} - 10q^{74} + q^{75} - q^{76} - 16q^{77} - 2q^{78} - 12q^{79} + q^{80} + q^{81} - 2q^{82} - 8q^{83} + 4q^{84} - 2q^{85} + 12q^{86} + 6q^{87} - 4q^{88} + 6q^{89} + q^{90} - 8q^{91} - 8q^{92} + 4q^{93} - q^{95} + q^{96} + 18q^{97} + 9q^{98} - 4q^{99} + O(q^{100})$$

## 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
1.00000 1.00000 1.00000 1.00000 1.00000 4.00000 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## 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 570.2.a.m 1
3.b odd 2 1 1710.2.a.f 1
4.b odd 2 1 4560.2.a.k 1
5.b even 2 1 2850.2.a.a 1
5.c odd 4 2 2850.2.d.k 2
15.d odd 2 1 8550.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.m 1 1.a even 1 1 trivial
1710.2.a.f 1 3.b odd 2 1
2850.2.a.a 1 5.b even 2 1
2850.2.d.k 2 5.c odd 4 2
4560.2.a.k 1 4.b odd 2 1
8550.2.a.t 1 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(570))$$:

 $$T_{7} - 4$$ $$T_{11} + 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$-4 + T$$
$11$ $$4 + T$$
$13$ $$2 + T$$
$17$ $$2 + T$$
$19$ $$1 + T$$
$23$ $$8 + T$$
$29$ $$-6 + T$$
$31$ $$-4 + T$$
$37$ $$10 + T$$
$41$ $$2 + T$$
$43$ $$-12 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$4 + T$$
$71$ $$8 + T$$
$73$ $$-2 + T$$
$79$ $$12 + T$$
$83$ $$8 + T$$
$89$ $$-6 + T$$
$97$ $$-18 + T$$