# Properties

 Label 570.2.i.d Level $570$ Weight $2$ Character orbit 570.i Analytic conductor $4.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{5} + \zeta_{6} q^{6} - q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{5} + \zeta_{6} q^{6} - q^{7} - q^{8} -\zeta_{6} q^{9} -\zeta_{6} q^{10} + 6 q^{11} + q^{12} -5 \zeta_{6} q^{13} + ( -1 + \zeta_{6} ) q^{14} + \zeta_{6} q^{15} + ( -1 + \zeta_{6} ) q^{16} - q^{18} + ( 5 - 2 \zeta_{6} ) q^{19} - q^{20} + ( 1 - \zeta_{6} ) q^{21} + ( 6 - 6 \zeta_{6} ) q^{22} -6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} -\zeta_{6} q^{25} -5 q^{26} + q^{27} + \zeta_{6} q^{28} -6 \zeta_{6} q^{29} + q^{30} + 5 q^{31} + \zeta_{6} q^{32} + ( -6 + 6 \zeta_{6} ) q^{33} + ( -1 + \zeta_{6} ) q^{35} + ( -1 + \zeta_{6} ) q^{36} + 11 q^{37} + ( 3 - 5 \zeta_{6} ) q^{38} + 5 q^{39} + ( -1 + \zeta_{6} ) q^{40} + ( -6 + 6 \zeta_{6} ) q^{41} -\zeta_{6} q^{42} + ( 1 - \zeta_{6} ) q^{43} -6 \zeta_{6} q^{44} - q^{45} -6 q^{46} -12 \zeta_{6} q^{47} -\zeta_{6} q^{48} -6 q^{49} - q^{50} + ( -5 + 5 \zeta_{6} ) q^{52} + 12 \zeta_{6} q^{53} + ( 1 - \zeta_{6} ) q^{54} + ( 6 - 6 \zeta_{6} ) q^{55} + q^{56} + ( -3 + 5 \zeta_{6} ) q^{57} -6 q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} + 7 \zeta_{6} q^{61} + ( 5 - 5 \zeta_{6} ) q^{62} + \zeta_{6} q^{63} + q^{64} -5 q^{65} + 6 \zeta_{6} q^{66} + \zeta_{6} q^{67} + 6 q^{69} + \zeta_{6} q^{70} + \zeta_{6} q^{72} + ( 1 - \zeta_{6} ) q^{73} + ( 11 - 11 \zeta_{6} ) q^{74} + q^{75} + ( -2 - 3 \zeta_{6} ) q^{76} -6 q^{77} + ( 5 - 5 \zeta_{6} ) q^{78} + ( 7 - 7 \zeta_{6} ) q^{79} + \zeta_{6} q^{80} + ( -1 + \zeta_{6} ) q^{81} + 6 \zeta_{6} q^{82} -6 q^{83} - q^{84} -\zeta_{6} q^{86} + 6 q^{87} -6 q^{88} + 12 \zeta_{6} q^{89} + ( -1 + \zeta_{6} ) q^{90} + 5 \zeta_{6} q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + ( -5 + 5 \zeta_{6} ) q^{93} -12 q^{94} + ( 3 - 5 \zeta_{6} ) q^{95} - q^{96} + ( -14 + 14 \zeta_{6} ) q^{97} + ( -6 + 6 \zeta_{6} ) q^{98} -6 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2q^{7} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 2q^{7} - 2q^{8} - q^{9} - q^{10} + 12q^{11} + 2q^{12} - 5q^{13} - q^{14} + q^{15} - q^{16} - 2q^{18} + 8q^{19} - 2q^{20} + q^{21} + 6q^{22} - 6q^{23} + q^{24} - q^{25} - 10q^{26} + 2q^{27} + q^{28} - 6q^{29} + 2q^{30} + 10q^{31} + q^{32} - 6q^{33} - q^{35} - q^{36} + 22q^{37} + q^{38} + 10q^{39} - q^{40} - 6q^{41} - q^{42} + q^{43} - 6q^{44} - 2q^{45} - 12q^{46} - 12q^{47} - q^{48} - 12q^{49} - 2q^{50} - 5q^{52} + 12q^{53} + q^{54} + 6q^{55} + 2q^{56} - q^{57} - 12q^{58} - 6q^{59} + q^{60} + 7q^{61} + 5q^{62} + q^{63} + 2q^{64} - 10q^{65} + 6q^{66} + q^{67} + 12q^{69} + q^{70} + q^{72} + q^{73} + 11q^{74} + 2q^{75} - 7q^{76} - 12q^{77} + 5q^{78} + 7q^{79} + q^{80} - q^{81} + 6q^{82} - 12q^{83} - 2q^{84} - q^{86} + 12q^{87} - 12q^{88} + 12q^{89} - q^{90} + 5q^{91} - 6q^{92} - 5q^{93} - 24q^{94} + q^{95} - 2q^{96} - 14q^{97} - 6q^{98} - 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −1.00000 −1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i
391.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −1.00000 −1.00000 −0.500000 0.866025i −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.i.d 2
3.b odd 2 1 1710.2.l.a 2
19.c even 3 1 inner 570.2.i.d 2
57.h odd 6 1 1710.2.l.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.d 2 1.a even 1 1 trivial
570.2.i.d 2 19.c even 3 1 inner
1710.2.l.a 2 3.b odd 2 1
1710.2.l.a 2 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7} + 1$$ $$T_{11} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$( -11 + T )^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$1 - T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1 - T + T^{2}$$
$79$ $$49 - 7 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$144 - 12 T + T^{2}$$
$97$ $$196 + 14 T + T^{2}$$