# Properties

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

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## 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + 4 * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 4 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2 q^{13} - 4 q^{14} - q^{15} + q^{16} - 2 q^{17} - q^{18} - q^{19} - q^{20} + 4 q^{21} - q^{24} + q^{25} - 2 q^{26} + q^{27} + 4 q^{28} + 10 q^{29} + q^{30} - q^{32} + 2 q^{34} - 4 q^{35} + q^{36} + 2 q^{37} + q^{38} + 2 q^{39} + q^{40} + 2 q^{41} - 4 q^{42} - 4 q^{43} - q^{45} + q^{48} + 9 q^{49} - q^{50} - 2 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} - 4 q^{56} - q^{57} - 10 q^{58} + 8 q^{59} - q^{60} + 6 q^{61} + 4 q^{63} + q^{64} - 2 q^{65} + 12 q^{67} - 2 q^{68} + 4 q^{70} - q^{72} - 14 q^{73} - 2 q^{74} + q^{75} - q^{76} - 2 q^{78} - q^{80} + q^{81} - 2 q^{82} - 12 q^{83} + 4 q^{84} + 2 q^{85} + 4 q^{86} + 10 q^{87} + 10 q^{89} + q^{90} + 8 q^{91} + q^{95} - q^{96} + 2 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + 4 * q^7 - q^8 + q^9 + q^10 + q^12 + 2 * q^13 - 4 * q^14 - q^15 + q^16 - 2 * q^17 - q^18 - q^19 - q^20 + 4 * q^21 - q^24 + q^25 - 2 * q^26 + q^27 + 4 * q^28 + 10 * q^29 + q^30 - q^32 + 2 * q^34 - 4 * q^35 + q^36 + 2 * q^37 + q^38 + 2 * q^39 + q^40 + 2 * q^41 - 4 * q^42 - 4 * q^43 - q^45 + q^48 + 9 * q^49 - q^50 - 2 * q^51 + 2 * q^52 - 6 * q^53 - q^54 - 4 * q^56 - q^57 - 10 * q^58 + 8 * q^59 - q^60 + 6 * q^61 + 4 * q^63 + q^64 - 2 * q^65 + 12 * q^67 - 2 * q^68 + 4 * q^70 - q^72 - 14 * q^73 - 2 * q^74 + q^75 - q^76 - 2 * q^78 - q^80 + q^81 - 2 * q^82 - 12 * q^83 + 4 * q^84 + 2 * q^85 + 4 * q^86 + 10 * q^87 + 10 * q^89 + q^90 + 8 * q^91 + q^95 - q^96 + 2 * q^97 - 9 * q^98

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

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

 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 4$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T + 2$$
$19$ $$T + 1$$
$23$ $$T$$
$29$ $$T - 10$$
$31$ $$T$$
$37$ $$T - 2$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 8$$
$61$ $$T - 6$$
$67$ $$T - 12$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T$$
$83$ $$T + 12$$
$89$ $$T - 10$$
$97$ $$T - 2$$
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