# Properties

 Label 570.2.i.g Level $570$ Weight $2$ Character orbit 570.i Analytic conductor $4.551$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{73})$$ Defining polynomial: $$x^{4} - x^{3} + 19x^{2} + 18x + 324$$ x^4 - x^3 + 19*x^2 + 18*x + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{2} + (\beta_{2} - 1) q^{3} - \beta_{2} q^{4} + (\beta_{2} - 1) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^2 + (b2 - 1) * q^3 - b2 * q^4 + (b2 - 1) * q^5 + b2 * q^6 - q^7 - q^8 - b2 * q^9 $$q + ( - \beta_{2} + 1) q^{2} + (\beta_{2} - 1) q^{3} - \beta_{2} q^{4} + (\beta_{2} - 1) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} - \beta_{2} q^{9} + \beta_{2} q^{10} + (\beta_{3} - 1) q^{11} + q^{12} + (\beta_{2} - \beta_1) q^{13} + (\beta_{2} - 1) q^{14} - \beta_{2} q^{15} + (\beta_{2} - 1) q^{16} - q^{18} + ( - \beta_1 + 1) q^{19} + q^{20} + ( - \beta_{2} + 1) q^{21} + (\beta_{3} + \beta_1 - 1) q^{22} - \beta_1 q^{23} + ( - \beta_{2} + 1) q^{24} - \beta_{2} q^{25} + \beta_{3} q^{26} + q^{27} + \beta_{2} q^{28} + 6 \beta_{2} q^{29} - q^{30} + (\beta_{3} - 2) q^{31} + \beta_{2} q^{32} + ( - \beta_{3} - \beta_1 + 1) q^{33} + ( - \beta_{2} + 1) q^{35} + (\beta_{2} - 1) q^{36} - q^{37} + (\beta_{3} - \beta_{2}) q^{38} - \beta_{3} q^{39} + ( - \beta_{2} + 1) q^{40} + (\beta_{3} + \beta_1 - 1) q^{41} - \beta_{2} q^{42} + ( - \beta_{3} + 5 \beta_{2} - \beta_1 - 4) q^{43} + \beta_1 q^{44} + q^{45} + (\beta_{3} - 1) q^{46} - \beta_{2} q^{48} - 6 q^{49} - q^{50} + (\beta_{3} - \beta_{2} + \beta_1) q^{52} + ( - 6 \beta_{2} + \beta_1) q^{53} + ( - \beta_{2} + 1) q^{54} + ( - \beta_{3} - \beta_1 + 1) q^{55} + q^{56} + ( - \beta_{3} + \beta_{2}) q^{57} + 6 q^{58} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 4) q^{59} + (\beta_{2} - 1) q^{60} + (\beta_{2} + \beta_1) q^{61} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{62} + \beta_{2} q^{63} + q^{64} - \beta_{3} q^{65} - \beta_1 q^{66} + (7 \beta_{2} - \beta_1) q^{67} + ( - \beta_{3} + 1) q^{69} - \beta_{2} q^{70} + (2 \beta_{3} + 2 \beta_1 - 2) q^{71} + \beta_{2} q^{72} + ( - \beta_{3} - 7 \beta_{2} - \beta_1 + 8) q^{73} + (\beta_{2} - 1) q^{74} + q^{75} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{76} + ( - \beta_{3} + 1) q^{77} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{78} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{79} - \beta_{2} q^{80} + (\beta_{2} - 1) q^{81} + \beta_1 q^{82} + 6 q^{83} - q^{84} + (5 \beta_{2} - \beta_1) q^{86} - 6 q^{87} + ( - \beta_{3} + 1) q^{88} + ( - 6 \beta_{2} + 3 \beta_1) q^{89} + ( - \beta_{2} + 1) q^{90} + ( - \beta_{2} + \beta_1) q^{91} + (\beta_{3} + \beta_1 - 1) q^{92} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{3} + \beta_{2}) q^{95} - q^{96} + ( - 10 \beta_{2} + 10) q^{97} + (6 \beta_{2} - 6) q^{98} + \beta_1 q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^2 + (b2 - 1) * q^3 - b2 * q^4 + (b2 - 1) * q^5 + b2 * q^6 - q^7 - q^8 - b2 * q^9 + b2 * q^10 + (b3 - 1) * q^11 + q^12 + (b2 - b1) * q^13 + (b2 - 1) * q^14 - b2 * q^15 + (b2 - 1) * q^16 - q^18 + (-b1 + 1) * q^19 + q^20 + (-b2 + 1) * q^21 + (b3 + b1 - 1) * q^22 - b1 * q^23 + (-b2 + 1) * q^24 - b2 * q^25 + b3 * q^26 + q^27 + b2 * q^28 + 6*b2 * q^29 - q^30 + (b3 - 2) * q^31 + b2 * q^32 + (-b3 - b1 + 1) * q^33 + (-b2 + 1) * q^35 + (b2 - 1) * q^36 - q^37 + (b3 - b2) * q^38 - b3 * q^39 + (-b2 + 1) * q^40 + (b3 + b1 - 1) * q^41 - b2 * q^42 + (-b3 + 5*b2 - b1 - 4) * q^43 + b1 * q^44 + q^45 + (b3 - 1) * q^46 - b2 * q^48 - 6 * q^49 - q^50 + (b3 - b2 + b1) * q^52 + (-6*b2 + b1) * q^53 + (-b2 + 1) * q^54 + (-b3 - b1 + 1) * q^55 + q^56 + (-b3 + b2) * q^57 + 6 * q^58 + (2*b3 - 6*b2 + 2*b1 + 4) * q^59 + (b2 - 1) * q^60 + (b2 + b1) * q^61 + (b3 + b2 + b1 - 2) * q^62 + b2 * q^63 + q^64 - b3 * q^65 - b1 * q^66 + (7*b2 - b1) * q^67 + (-b3 + 1) * q^69 - b2 * q^70 + (2*b3 + 2*b1 - 2) * q^71 + b2 * q^72 + (-b3 - 7*b2 - b1 + 8) * q^73 + (b2 - 1) * q^74 + q^75 + (b3 - b2 + b1 - 1) * q^76 + (-b3 + 1) * q^77 + (-b3 + b2 - b1) * q^78 + (-3*b3 - b2 - 3*b1 + 4) * q^79 - b2 * q^80 + (b2 - 1) * q^81 + b1 * q^82 + 6 * q^83 - q^84 + (5*b2 - b1) * q^86 - 6 * q^87 + (-b3 + 1) * q^88 + (-6*b2 + 3*b1) * q^89 + (-b2 + 1) * q^90 + (-b2 + b1) * q^91 + (b3 + b1 - 1) * q^92 + (-b3 - b2 - b1 + 2) * q^93 + (-b3 + b2) * q^95 - q^96 + (-10*b2 + 10) * q^97 + (6*b2 - 6) * q^98 + b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 4 * q^8 - 2 * q^9 $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 4 q^{8} - 2 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} + q^{13} - 2 q^{14} - 2 q^{15} - 2 q^{16} - 4 q^{18} + 3 q^{19} + 4 q^{20} + 2 q^{21} - q^{22} - q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{26} + 4 q^{27} + 2 q^{28} + 12 q^{29} - 4 q^{30} - 6 q^{31} + 2 q^{32} + q^{33} + 2 q^{35} - 2 q^{36} - 4 q^{37} - 2 q^{39} + 2 q^{40} - q^{41} - 2 q^{42} - 9 q^{43} + q^{44} + 4 q^{45} - 2 q^{46} - 2 q^{48} - 24 q^{49} - 4 q^{50} + q^{52} - 11 q^{53} + 2 q^{54} + q^{55} + 4 q^{56} + 24 q^{58} + 10 q^{59} - 2 q^{60} + 3 q^{61} - 3 q^{62} + 2 q^{63} + 4 q^{64} - 2 q^{65} - q^{66} + 13 q^{67} + 2 q^{69} - 2 q^{70} - 2 q^{71} + 2 q^{72} + 15 q^{73} - 2 q^{74} + 4 q^{75} - 3 q^{76} + 2 q^{77} - q^{78} + 5 q^{79} - 2 q^{80} - 2 q^{81} + q^{82} + 24 q^{83} - 4 q^{84} + 9 q^{86} - 24 q^{87} + 2 q^{88} - 9 q^{89} + 2 q^{90} - q^{91} - q^{92} + 3 q^{93} - 4 q^{96} + 20 q^{97} - 12 q^{98} + q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 4 * q^8 - 2 * q^9 + 2 * q^10 - 2 * q^11 + 4 * q^12 + q^13 - 2 * q^14 - 2 * q^15 - 2 * q^16 - 4 * q^18 + 3 * q^19 + 4 * q^20 + 2 * q^21 - q^22 - q^23 + 2 * q^24 - 2 * q^25 + 2 * q^26 + 4 * q^27 + 2 * q^28 + 12 * q^29 - 4 * q^30 - 6 * q^31 + 2 * q^32 + q^33 + 2 * q^35 - 2 * q^36 - 4 * q^37 - 2 * q^39 + 2 * q^40 - q^41 - 2 * q^42 - 9 * q^43 + q^44 + 4 * q^45 - 2 * q^46 - 2 * q^48 - 24 * q^49 - 4 * q^50 + q^52 - 11 * q^53 + 2 * q^54 + q^55 + 4 * q^56 + 24 * q^58 + 10 * q^59 - 2 * q^60 + 3 * q^61 - 3 * q^62 + 2 * q^63 + 4 * q^64 - 2 * q^65 - q^66 + 13 * q^67 + 2 * q^69 - 2 * q^70 - 2 * q^71 + 2 * q^72 + 15 * q^73 - 2 * q^74 + 4 * q^75 - 3 * q^76 + 2 * q^77 - q^78 + 5 * q^79 - 2 * q^80 - 2 * q^81 + q^82 + 24 * q^83 - 4 * q^84 + 9 * q^86 - 24 * q^87 + 2 * q^88 - 9 * q^89 + 2 * q^90 - q^91 - q^92 + 3 * q^93 - 4 * q^96 + 20 * q^97 - 12 * q^98 + q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 19x^{2} + 18x + 324$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 19\nu^{2} - 19\nu + 324 ) / 342$$ (-v^3 + 19*v^2 - 19*v + 324) / 342 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 37 ) / 19$$ (v^3 + 37) / 19
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 18\beta_{2} + \beta _1 - 19$$ b3 + 18*b2 + b1 - 19 $$\nu^{3}$$ $$=$$ $$19\beta_{3} - 37$$ 19*b3 - 37

## Character values

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

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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
 2.38600 − 4.13267i −1.88600 + 3.26665i 2.38600 + 4.13267i −1.88600 − 3.26665i
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
121.2 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
391.2 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.g 4
3.b odd 2 1 1710.2.l.l 4
19.c even 3 1 inner 570.2.i.g 4
57.h odd 6 1 1710.2.l.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.g 4 1.a even 1 1 trivial
570.2.i.g 4 19.c even 3 1 inner
1710.2.l.l 4 3.b odd 2 1
1710.2.l.l 4 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 $$T_{11}^{2} + T_{11} - 18$$ T11^2 + T11 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$(T + 1)^{4}$$
$11$ $$(T^{2} + T - 18)^{2}$$
$13$ $$T^{4} - T^{3} + 19 T^{2} + 18 T + 324$$
$17$ $$T^{4}$$
$19$ $$T^{4} - 3 T^{3} + 22 T^{2} - 57 T + 361$$
$23$ $$T^{4} + T^{3} + 19 T^{2} - 18 T + 324$$
$29$ $$(T^{2} - 6 T + 36)^{2}$$
$31$ $$(T^{2} + 3 T - 16)^{2}$$
$37$ $$(T + 1)^{4}$$
$41$ $$T^{4} + T^{3} + 19 T^{2} - 18 T + 324$$
$43$ $$T^{4} + 9 T^{3} + 79 T^{2} + 18 T + 4$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 11 T^{3} + 109 T^{2} + \cdots + 144$$
$59$ $$T^{4} - 10 T^{3} + 148 T^{2} + \cdots + 2304$$
$61$ $$T^{4} - 3 T^{3} + 25 T^{2} + 48 T + 256$$
$67$ $$T^{4} - 13 T^{3} + 145 T^{2} + \cdots + 576$$
$71$ $$T^{4} + 2 T^{3} + 76 T^{2} + \cdots + 5184$$
$73$ $$T^{4} - 15 T^{3} + 187 T^{2} + \cdots + 1444$$
$79$ $$T^{4} - 5 T^{3} + 183 T^{2} + \cdots + 24964$$
$83$ $$(T - 6)^{4}$$
$89$ $$T^{4} + 9 T^{3} + 225 T^{2} + \cdots + 20736$$
$97$ $$(T^{2} - 10 T + 100)^{2}$$