# Properties

 Label 570.2.i.e 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - \zeta_{6} q^{6} - 5 q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (-z + 1) * q^3 - z * q^4 + (z - 1) * q^5 - z * q^6 - 5 * q^7 - q^8 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{5} - \zeta_{6} q^{6} - 5 q^{7} - q^{8} - \zeta_{6} q^{9} + \zeta_{6} q^{10} + q^{11} - q^{12} - 6 \zeta_{6} q^{13} + (5 \zeta_{6} - 5) q^{14} + \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} + (4 \zeta_{6} - 4) q^{17} - q^{18} + (5 \zeta_{6} - 3) q^{19} + q^{20} + (5 \zeta_{6} - 5) q^{21} + ( - \zeta_{6} + 1) q^{22} - 7 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} - \zeta_{6} q^{25} - 6 q^{26} - q^{27} + 5 \zeta_{6} q^{28} - 6 \zeta_{6} q^{29} + q^{30} + \zeta_{6} q^{32} + ( - \zeta_{6} + 1) q^{33} + 4 \zeta_{6} q^{34} + ( - 5 \zeta_{6} + 5) q^{35} + (\zeta_{6} - 1) q^{36} + 7 q^{37} + (3 \zeta_{6} + 2) q^{38} - 6 q^{39} + ( - \zeta_{6} + 1) q^{40} + ( - 5 \zeta_{6} + 5) q^{41} + 5 \zeta_{6} q^{42} + (6 \zeta_{6} - 6) q^{43} - \zeta_{6} q^{44} + q^{45} - 7 q^{46} + 8 \zeta_{6} q^{47} + \zeta_{6} q^{48} + 18 q^{49} - q^{50} + 4 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{52} - 11 \zeta_{6} q^{53} + (\zeta_{6} - 1) q^{54} + (\zeta_{6} - 1) q^{55} + 5 q^{56} + (3 \zeta_{6} + 2) q^{57} - 6 q^{58} + ( - 8 \zeta_{6} + 8) q^{59} + ( - \zeta_{6} + 1) q^{60} + 4 \zeta_{6} q^{61} + 5 \zeta_{6} q^{63} + q^{64} + 6 q^{65} - \zeta_{6} q^{66} - 12 \zeta_{6} q^{67} + 4 q^{68} - 7 q^{69} - 5 \zeta_{6} q^{70} + (2 \zeta_{6} - 2) q^{71} + \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + ( - 7 \zeta_{6} + 7) q^{74} - q^{75} + ( - 2 \zeta_{6} + 5) q^{76} - 5 q^{77} + (6 \zeta_{6} - 6) q^{78} + (10 \zeta_{6} - 10) q^{79} - \zeta_{6} q^{80} + (\zeta_{6} - 1) q^{81} - 5 \zeta_{6} q^{82} - 10 q^{83} + 5 q^{84} - 4 \zeta_{6} q^{85} + 6 \zeta_{6} q^{86} - 6 q^{87} - q^{88} - 13 \zeta_{6} q^{89} + ( - \zeta_{6} + 1) q^{90} + 30 \zeta_{6} q^{91} + (7 \zeta_{6} - 7) q^{92} + 8 q^{94} + ( - 3 \zeta_{6} - 2) q^{95} + q^{96} + (2 \zeta_{6} - 2) q^{97} + ( - 18 \zeta_{6} + 18) q^{98} - \zeta_{6} q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (-z + 1) * q^3 - z * q^4 + (z - 1) * q^5 - z * q^6 - 5 * q^7 - q^8 - z * q^9 + z * q^10 + q^11 - q^12 - 6*z * q^13 + (5*z - 5) * q^14 + z * q^15 + (z - 1) * q^16 + (4*z - 4) * q^17 - q^18 + (5*z - 3) * q^19 + q^20 + (5*z - 5) * q^21 + (-z + 1) * q^22 - 7*z * q^23 + (z - 1) * q^24 - z * q^25 - 6 * q^26 - q^27 + 5*z * q^28 - 6*z * q^29 + q^30 + z * q^32 + (-z + 1) * q^33 + 4*z * q^34 + (-5*z + 5) * q^35 + (z - 1) * q^36 + 7 * q^37 + (3*z + 2) * q^38 - 6 * q^39 + (-z + 1) * q^40 + (-5*z + 5) * q^41 + 5*z * q^42 + (6*z - 6) * q^43 - z * q^44 + q^45 - 7 * q^46 + 8*z * q^47 + z * q^48 + 18 * q^49 - q^50 + 4*z * q^51 + (6*z - 6) * q^52 - 11*z * q^53 + (z - 1) * q^54 + (z - 1) * q^55 + 5 * q^56 + (3*z + 2) * q^57 - 6 * q^58 + (-8*z + 8) * q^59 + (-z + 1) * q^60 + 4*z * q^61 + 5*z * q^63 + q^64 + 6 * q^65 - z * q^66 - 12*z * q^67 + 4 * q^68 - 7 * q^69 - 5*z * q^70 + (2*z - 2) * q^71 + z * q^72 + (-2*z + 2) * q^73 + (-7*z + 7) * q^74 - q^75 + (-2*z + 5) * q^76 - 5 * q^77 + (6*z - 6) * q^78 + (10*z - 10) * q^79 - z * q^80 + (z - 1) * q^81 - 5*z * q^82 - 10 * q^83 + 5 * q^84 - 4*z * q^85 + 6*z * q^86 - 6 * q^87 - q^88 - 13*z * q^89 + (-z + 1) * q^90 + 30*z * q^91 + (7*z - 7) * q^92 + 8 * q^94 + (-3*z - 2) * q^95 + q^96 + (2*z - 2) * q^97 + (-18*z + 18) * q^98 - z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 10 q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 - q^4 - q^5 - q^6 - 10 * q^7 - 2 * q^8 - q^9 $$2 q + q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 10 q^{7} - 2 q^{8} - q^{9} + q^{10} + 2 q^{11} - 2 q^{12} - 6 q^{13} - 5 q^{14} + q^{15} - q^{16} - 4 q^{17} - 2 q^{18} - q^{19} + 2 q^{20} - 5 q^{21} + q^{22} - 7 q^{23} - q^{24} - q^{25} - 12 q^{26} - 2 q^{27} + 5 q^{28} - 6 q^{29} + 2 q^{30} + q^{32} + q^{33} + 4 q^{34} + 5 q^{35} - q^{36} + 14 q^{37} + 7 q^{38} - 12 q^{39} + q^{40} + 5 q^{41} + 5 q^{42} - 6 q^{43} - q^{44} + 2 q^{45} - 14 q^{46} + 8 q^{47} + q^{48} + 36 q^{49} - 2 q^{50} + 4 q^{51} - 6 q^{52} - 11 q^{53} - q^{54} - q^{55} + 10 q^{56} + 7 q^{57} - 12 q^{58} + 8 q^{59} + q^{60} + 4 q^{61} + 5 q^{63} + 2 q^{64} + 12 q^{65} - q^{66} - 12 q^{67} + 8 q^{68} - 14 q^{69} - 5 q^{70} - 2 q^{71} + q^{72} + 2 q^{73} + 7 q^{74} - 2 q^{75} + 8 q^{76} - 10 q^{77} - 6 q^{78} - 10 q^{79} - q^{80} - q^{81} - 5 q^{82} - 20 q^{83} + 10 q^{84} - 4 q^{85} + 6 q^{86} - 12 q^{87} - 2 q^{88} - 13 q^{89} + q^{90} + 30 q^{91} - 7 q^{92} + 16 q^{94} - 7 q^{95} + 2 q^{96} - 2 q^{97} + 18 q^{98} - q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 - q^5 - q^6 - 10 * q^7 - 2 * q^8 - q^9 + q^10 + 2 * q^11 - 2 * q^12 - 6 * q^13 - 5 * q^14 + q^15 - q^16 - 4 * q^17 - 2 * q^18 - q^19 + 2 * q^20 - 5 * q^21 + q^22 - 7 * q^23 - q^24 - q^25 - 12 * q^26 - 2 * q^27 + 5 * q^28 - 6 * q^29 + 2 * q^30 + q^32 + q^33 + 4 * q^34 + 5 * q^35 - q^36 + 14 * q^37 + 7 * q^38 - 12 * q^39 + q^40 + 5 * q^41 + 5 * q^42 - 6 * q^43 - q^44 + 2 * q^45 - 14 * q^46 + 8 * q^47 + q^48 + 36 * q^49 - 2 * q^50 + 4 * q^51 - 6 * q^52 - 11 * q^53 - q^54 - q^55 + 10 * q^56 + 7 * q^57 - 12 * q^58 + 8 * q^59 + q^60 + 4 * q^61 + 5 * q^63 + 2 * q^64 + 12 * q^65 - q^66 - 12 * q^67 + 8 * q^68 - 14 * q^69 - 5 * q^70 - 2 * q^71 + q^72 + 2 * q^73 + 7 * q^74 - 2 * q^75 + 8 * q^76 - 10 * q^77 - 6 * q^78 - 10 * q^79 - q^80 - q^81 - 5 * q^82 - 20 * q^83 + 10 * q^84 - 4 * q^85 + 6 * q^86 - 12 * q^87 - 2 * q^88 - 13 * q^89 + q^90 + 30 * q^91 - 7 * q^92 + 16 * q^94 - 7 * q^95 + 2 * q^96 - 2 * q^97 + 18 * q^98 - 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 −5.00000 −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 −5.00000 −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.e 2
3.b odd 2 1 1710.2.l.c 2
19.c even 3 1 inner 570.2.i.e 2
57.h odd 6 1 1710.2.l.c 2

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

 $$T_{7} + 5$$ T7 + 5 $$T_{11} - 1$$ T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$(T + 5)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 6T + 36$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2} + 7T + 49$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2}$$
$37$ $$(T - 7)^{2}$$
$41$ $$T^{2} - 5T + 25$$
$43$ $$T^{2} + 6T + 36$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} + 11T + 121$$
$59$ $$T^{2} - 8T + 64$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} + 12T + 144$$
$71$ $$T^{2} + 2T + 4$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} + 10T + 100$$
$83$ $$(T + 10)^{2}$$
$89$ $$T^{2} + 13T + 169$$
$97$ $$T^{2} + 2T + 4$$