# Properties

 Label 570.2.i.a 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} - \zeta_{6} q^{6} - q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (z - 1) * q^3 - z * q^4 + (-z + 1) * q^5 - z * q^6 - q^7 + q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} - \zeta_{6} q^{6} - q^{7} + q^{8} - \zeta_{6} q^{9} + \zeta_{6} q^{10} - 3 q^{11} + q^{12} - 2 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{14} + \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + q^{18} + (3 \zeta_{6} - 5) q^{19} - q^{20} + ( - \zeta_{6} + 1) q^{21} + ( - 3 \zeta_{6} + 3) q^{22} - 9 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} - \zeta_{6} q^{25} + 2 q^{26} + q^{27} + \zeta_{6} q^{28} - q^{30} - 4 q^{31} - \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{33} + 6 \zeta_{6} q^{34} + (\zeta_{6} - 1) q^{35} + (\zeta_{6} - 1) q^{36} + 5 q^{37} + ( - 5 \zeta_{6} + 2) q^{38} + 2 q^{39} + ( - \zeta_{6} + 1) q^{40} + ( - 3 \zeta_{6} + 3) q^{41} + \zeta_{6} q^{42} + ( - 10 \zeta_{6} + 10) q^{43} + 3 \zeta_{6} q^{44} - q^{45} + 9 q^{46} - \zeta_{6} q^{48} - 6 q^{49} + q^{50} + 6 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{52} - 3 \zeta_{6} q^{53} + (\zeta_{6} - 1) q^{54} + (3 \zeta_{6} - 3) q^{55} - q^{56} + ( - 5 \zeta_{6} + 2) q^{57} + ( - 12 \zeta_{6} + 12) q^{59} + ( - \zeta_{6} + 1) q^{60} + 10 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + \zeta_{6} q^{63} + q^{64} - 2 q^{65} + 3 \zeta_{6} q^{66} + 10 \zeta_{6} q^{67} - 6 q^{68} + 9 q^{69} - \zeta_{6} q^{70} - \zeta_{6} q^{72} + ( - 4 \zeta_{6} + 4) q^{73} + (5 \zeta_{6} - 5) q^{74} + q^{75} + (2 \zeta_{6} + 3) q^{76} + 3 q^{77} + (2 \zeta_{6} - 2) q^{78} + (14 \zeta_{6} - 14) q^{79} + \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} + 3 \zeta_{6} q^{82} - 18 q^{83} - q^{84} - 6 \zeta_{6} q^{85} + 10 \zeta_{6} q^{86} - 3 q^{88} - 15 \zeta_{6} q^{89} + ( - \zeta_{6} + 1) q^{90} + 2 \zeta_{6} q^{91} + (9 \zeta_{6} - 9) q^{92} + ( - 4 \zeta_{6} + 4) q^{93} + (5 \zeta_{6} - 2) q^{95} + q^{96} + (8 \zeta_{6} - 8) q^{97} + ( - 6 \zeta_{6} + 6) q^{98} + 3 \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (z - 1) * q^3 - z * q^4 + (-z + 1) * q^5 - z * q^6 - q^7 + q^8 - z * q^9 + z * q^10 - 3 * q^11 + q^12 - 2*z * q^13 + (-z + 1) * q^14 + z * q^15 + (z - 1) * q^16 + (-6*z + 6) * q^17 + q^18 + (3*z - 5) * q^19 - q^20 + (-z + 1) * q^21 + (-3*z + 3) * q^22 - 9*z * q^23 + (z - 1) * q^24 - z * q^25 + 2 * q^26 + q^27 + z * q^28 - q^30 - 4 * q^31 - z * q^32 + (-3*z + 3) * q^33 + 6*z * q^34 + (z - 1) * q^35 + (z - 1) * q^36 + 5 * q^37 + (-5*z + 2) * q^38 + 2 * q^39 + (-z + 1) * q^40 + (-3*z + 3) * q^41 + z * q^42 + (-10*z + 10) * q^43 + 3*z * q^44 - q^45 + 9 * q^46 - z * q^48 - 6 * q^49 + q^50 + 6*z * q^51 + (2*z - 2) * q^52 - 3*z * q^53 + (z - 1) * q^54 + (3*z - 3) * q^55 - q^56 + (-5*z + 2) * q^57 + (-12*z + 12) * q^59 + (-z + 1) * q^60 + 10*z * q^61 + (-4*z + 4) * q^62 + z * q^63 + q^64 - 2 * q^65 + 3*z * q^66 + 10*z * q^67 - 6 * q^68 + 9 * q^69 - z * q^70 - z * q^72 + (-4*z + 4) * q^73 + (5*z - 5) * q^74 + q^75 + (2*z + 3) * q^76 + 3 * q^77 + (2*z - 2) * q^78 + (14*z - 14) * q^79 + z * q^80 + (z - 1) * q^81 + 3*z * q^82 - 18 * q^83 - q^84 - 6*z * q^85 + 10*z * q^86 - 3 * q^88 - 15*z * q^89 + (-z + 1) * q^90 + 2*z * q^91 + (9*z - 9) * q^92 + (-4*z + 4) * q^93 + (5*z - 2) * q^95 + q^96 + (8*z - 8) * q^97 + (-6*z + 6) * q^98 + 3*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} + q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 + q^5 - q^6 - 2 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{3} - q^{4} + q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - q^{9} + q^{10} - 6 q^{11} + 2 q^{12} - 2 q^{13} + q^{14} + q^{15} - q^{16} + 6 q^{17} + 2 q^{18} - 7 q^{19} - 2 q^{20} + q^{21} + 3 q^{22} - 9 q^{23} - q^{24} - q^{25} + 4 q^{26} + 2 q^{27} + q^{28} - 2 q^{30} - 8 q^{31} - q^{32} + 3 q^{33} + 6 q^{34} - q^{35} - q^{36} + 10 q^{37} - q^{38} + 4 q^{39} + q^{40} + 3 q^{41} + q^{42} + 10 q^{43} + 3 q^{44} - 2 q^{45} + 18 q^{46} - q^{48} - 12 q^{49} + 2 q^{50} + 6 q^{51} - 2 q^{52} - 3 q^{53} - q^{54} - 3 q^{55} - 2 q^{56} - q^{57} + 12 q^{59} + q^{60} + 10 q^{61} + 4 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + 3 q^{66} + 10 q^{67} - 12 q^{68} + 18 q^{69} - q^{70} - q^{72} + 4 q^{73} - 5 q^{74} + 2 q^{75} + 8 q^{76} + 6 q^{77} - 2 q^{78} - 14 q^{79} + q^{80} - q^{81} + 3 q^{82} - 36 q^{83} - 2 q^{84} - 6 q^{85} + 10 q^{86} - 6 q^{88} - 15 q^{89} + q^{90} + 2 q^{91} - 9 q^{92} + 4 q^{93} + q^{95} + 2 q^{96} - 8 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 + q^5 - q^6 - 2 * q^7 + 2 * q^8 - q^9 + q^10 - 6 * q^11 + 2 * q^12 - 2 * q^13 + q^14 + q^15 - q^16 + 6 * q^17 + 2 * q^18 - 7 * q^19 - 2 * q^20 + q^21 + 3 * q^22 - 9 * q^23 - q^24 - q^25 + 4 * q^26 + 2 * q^27 + q^28 - 2 * q^30 - 8 * q^31 - q^32 + 3 * q^33 + 6 * q^34 - q^35 - q^36 + 10 * q^37 - q^38 + 4 * q^39 + q^40 + 3 * q^41 + q^42 + 10 * q^43 + 3 * q^44 - 2 * q^45 + 18 * q^46 - q^48 - 12 * q^49 + 2 * q^50 + 6 * q^51 - 2 * q^52 - 3 * q^53 - q^54 - 3 * q^55 - 2 * q^56 - q^57 + 12 * q^59 + q^60 + 10 * q^61 + 4 * q^62 + q^63 + 2 * q^64 - 4 * q^65 + 3 * q^66 + 10 * q^67 - 12 * q^68 + 18 * q^69 - q^70 - q^72 + 4 * q^73 - 5 * q^74 + 2 * q^75 + 8 * q^76 + 6 * q^77 - 2 * q^78 - 14 * q^79 + q^80 - q^81 + 3 * q^82 - 36 * q^83 - 2 * q^84 - 6 * q^85 + 10 * q^86 - 6 * q^88 - 15 * q^89 + q^90 + 2 * q^91 - 9 * q^92 + 4 * q^93 + q^95 + 2 * q^96 - 8 * q^97 + 6 * q^98 + 3 * q^99

## 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.a 2
3.b odd 2 1 1710.2.l.g 2
19.c even 3 1 inner 570.2.i.a 2
57.h odd 6 1 1710.2.l.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.a 2 1.a even 1 1 trivial
570.2.i.a 2 19.c even 3 1 inner
1710.2.l.g 2 3.b odd 2 1
1710.2.l.g 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 $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2} + 9T + 81$$
$29$ $$T^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T - 5)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 10T + 100$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} - 10T + 100$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$(T + 18)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$T^{2} + 8T + 64$$