# Properties

 Label 570.2.i.b 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 $$q + (\zeta_{6} - 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} - \zeta_{6} q^{6} - q^{7} + q^{8} - \zeta_{6} q^{9} + \zeta_{6} q^{10} + 2 q^{11} + q^{12} + 3 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{14} + \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + (4 \zeta_{6} - 4) q^{17} + q^{18} + ( - 2 \zeta_{6} + 5) q^{19} - q^{20} + ( - \zeta_{6} + 1) q^{21} + (2 \zeta_{6} - 2) q^{22} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} - \zeta_{6} q^{25} - 3 q^{26} + q^{27} + \zeta_{6} q^{28} + 10 \zeta_{6} q^{29} - q^{30} + q^{31} - \zeta_{6} q^{32} + (2 \zeta_{6} - 2) q^{33} - 4 \zeta_{6} q^{34} + (\zeta_{6} - 1) q^{35} + (\zeta_{6} - 1) q^{36} - 5 q^{37} + (5 \zeta_{6} - 3) q^{38} - 3 q^{39} + ( - \zeta_{6} + 1) q^{40} + (2 \zeta_{6} - 2) q^{41} + \zeta_{6} q^{42} + ( - 5 \zeta_{6} + 5) q^{43} - 2 \zeta_{6} q^{44} - q^{45} - 6 q^{46} - \zeta_{6} q^{48} - 6 q^{49} + q^{50} - 4 \zeta_{6} q^{51} + ( - 3 \zeta_{6} + 3) q^{52} + 12 \zeta_{6} q^{53} + (\zeta_{6} - 1) q^{54} + ( - 2 \zeta_{6} + 2) q^{55} - q^{56} + (5 \zeta_{6} - 3) q^{57} - 10 q^{58} + ( - 2 \zeta_{6} + 2) q^{59} + ( - \zeta_{6} + 1) q^{60} - 5 \zeta_{6} q^{61} + (\zeta_{6} - 1) q^{62} + \zeta_{6} q^{63} + q^{64} + 3 q^{65} - 2 \zeta_{6} q^{66} + 5 \zeta_{6} q^{67} + 4 q^{68} - 6 q^{69} - \zeta_{6} q^{70} - \zeta_{6} q^{72} + (11 \zeta_{6} - 11) q^{73} + ( - 5 \zeta_{6} + 5) q^{74} + q^{75} + ( - 3 \zeta_{6} - 2) q^{76} - 2 q^{77} + ( - 3 \zeta_{6} + 3) q^{78} + ( - 11 \zeta_{6} + 11) q^{79} + \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} - 2 \zeta_{6} q^{82} + 2 q^{83} - q^{84} + 4 \zeta_{6} q^{85} + 5 \zeta_{6} q^{86} - 10 q^{87} + 2 q^{88} + ( - \zeta_{6} + 1) q^{90} - 3 \zeta_{6} q^{91} + ( - 6 \zeta_{6} + 6) q^{92} + (\zeta_{6} - 1) q^{93} + ( - 5 \zeta_{6} + 3) q^{95} + q^{96} + ( - 2 \zeta_{6} + 2) q^{97} + ( - 6 \zeta_{6} + 6) q^{98} - 2 \zeta_{6} q^{99} +O(q^{100})$$ 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 + 2 * q^11 + q^12 + 3*z * q^13 + (-z + 1) * q^14 + z * q^15 + (z - 1) * q^16 + (4*z - 4) * q^17 + q^18 + (-2*z + 5) * q^19 - q^20 + (-z + 1) * q^21 + (2*z - 2) * q^22 + 6*z * q^23 + (z - 1) * q^24 - z * q^25 - 3 * q^26 + q^27 + z * q^28 + 10*z * q^29 - q^30 + q^31 - z * q^32 + (2*z - 2) * q^33 - 4*z * q^34 + (z - 1) * q^35 + (z - 1) * q^36 - 5 * q^37 + (5*z - 3) * q^38 - 3 * q^39 + (-z + 1) * q^40 + (2*z - 2) * q^41 + z * q^42 + (-5*z + 5) * q^43 - 2*z * q^44 - q^45 - 6 * q^46 - z * q^48 - 6 * q^49 + q^50 - 4*z * q^51 + (-3*z + 3) * q^52 + 12*z * q^53 + (z - 1) * q^54 + (-2*z + 2) * q^55 - q^56 + (5*z - 3) * q^57 - 10 * q^58 + (-2*z + 2) * q^59 + (-z + 1) * q^60 - 5*z * q^61 + (z - 1) * q^62 + z * q^63 + q^64 + 3 * q^65 - 2*z * q^66 + 5*z * q^67 + 4 * q^68 - 6 * q^69 - z * q^70 - z * q^72 + (11*z - 11) * q^73 + (-5*z + 5) * q^74 + q^75 + (-3*z - 2) * q^76 - 2 * q^77 + (-3*z + 3) * q^78 + (-11*z + 11) * q^79 + z * q^80 + (z - 1) * q^81 - 2*z * q^82 + 2 * q^83 - q^84 + 4*z * q^85 + 5*z * q^86 - 10 * q^87 + 2 * q^88 + (-z + 1) * q^90 - 3*z * q^91 + (-6*z + 6) * q^92 + (z - 1) * q^93 + (-5*z + 3) * q^95 + q^96 + (-2*z + 2) * q^97 + (-6*z + 6) * q^98 - 2*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 $$2 q - q^{2} - q^{3} - q^{4} + q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - q^{9} + q^{10} + 4 q^{11} + 2 q^{12} + 3 q^{13} + q^{14} + q^{15} - q^{16} - 4 q^{17} + 2 q^{18} + 8 q^{19} - 2 q^{20} + q^{21} - 2 q^{22} + 6 q^{23} - q^{24} - q^{25} - 6 q^{26} + 2 q^{27} + q^{28} + 10 q^{29} - 2 q^{30} + 2 q^{31} - q^{32} - 2 q^{33} - 4 q^{34} - q^{35} - q^{36} - 10 q^{37} - q^{38} - 6 q^{39} + q^{40} - 2 q^{41} + q^{42} + 5 q^{43} - 2 q^{44} - 2 q^{45} - 12 q^{46} - q^{48} - 12 q^{49} + 2 q^{50} - 4 q^{51} + 3 q^{52} + 12 q^{53} - q^{54} + 2 q^{55} - 2 q^{56} - q^{57} - 20 q^{58} + 2 q^{59} + q^{60} - 5 q^{61} - q^{62} + q^{63} + 2 q^{64} + 6 q^{65} - 2 q^{66} + 5 q^{67} + 8 q^{68} - 12 q^{69} - q^{70} - q^{72} - 11 q^{73} + 5 q^{74} + 2 q^{75} - 7 q^{76} - 4 q^{77} + 3 q^{78} + 11 q^{79} + q^{80} - q^{81} - 2 q^{82} + 4 q^{83} - 2 q^{84} + 4 q^{85} + 5 q^{86} - 20 q^{87} + 4 q^{88} + q^{90} - 3 q^{91} + 6 q^{92} - q^{93} + q^{95} + 2 q^{96} + 2 q^{97} + 6 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 + q^5 - q^6 - 2 * q^7 + 2 * q^8 - q^9 + q^10 + 4 * q^11 + 2 * q^12 + 3 * q^13 + q^14 + q^15 - q^16 - 4 * q^17 + 2 * q^18 + 8 * q^19 - 2 * q^20 + q^21 - 2 * q^22 + 6 * q^23 - q^24 - q^25 - 6 * q^26 + 2 * q^27 + q^28 + 10 * q^29 - 2 * q^30 + 2 * q^31 - q^32 - 2 * q^33 - 4 * q^34 - q^35 - q^36 - 10 * q^37 - q^38 - 6 * q^39 + q^40 - 2 * q^41 + q^42 + 5 * q^43 - 2 * q^44 - 2 * q^45 - 12 * q^46 - q^48 - 12 * q^49 + 2 * q^50 - 4 * q^51 + 3 * q^52 + 12 * q^53 - q^54 + 2 * q^55 - 2 * q^56 - q^57 - 20 * q^58 + 2 * q^59 + q^60 - 5 * q^61 - q^62 + q^63 + 2 * q^64 + 6 * q^65 - 2 * q^66 + 5 * q^67 + 8 * q^68 - 12 * q^69 - q^70 - q^72 - 11 * q^73 + 5 * q^74 + 2 * q^75 - 7 * q^76 - 4 * q^77 + 3 * q^78 + 11 * q^79 + q^80 - q^81 - 2 * q^82 + 4 * q^83 - 2 * q^84 + 4 * q^85 + 5 * q^86 - 20 * q^87 + 4 * q^88 + q^90 - 3 * q^91 + 6 * q^92 - q^93 + q^95 + 2 * q^96 + 2 * q^97 + 6 * q^98 - 2 * 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.b 2
3.b odd 2 1 1710.2.l.f 2
19.c even 3 1 inner 570.2.i.b 2
57.h odd 6 1 1710.2.l.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.b 2 1.a even 1 1 trivial
570.2.i.b 2 19.c even 3 1 inner
1710.2.l.f 2 3.b odd 2 1
1710.2.l.f 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 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - 3T + 9$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$(T - 1)^{2}$$
$37$ $$(T + 5)^{2}$$
$41$ $$T^{2} + 2T + 4$$
$43$ $$T^{2} - 5T + 25$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2} - 2T + 4$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$(T - 2)^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 2T + 4$$