# Properties

 Label 570.2.i.f 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + ( - \beta_1 + 1) q^{3} - \beta_1 q^{4} + (\beta_1 - 1) q^{5} + \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + q^{8} - \beta_1 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + (-b1 + 1) * q^3 - b1 * q^4 + (b1 - 1) * q^5 + b1 * q^6 + (-b3 + 1) * q^7 + q^8 - b1 * q^9 $$q + (\beta_1 - 1) q^{2} + ( - \beta_1 + 1) q^{3} - \beta_1 q^{4} + (\beta_1 - 1) q^{5} + \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + q^{8} - \beta_1 q^{9} - \beta_1 q^{10} - 3 q^{11} - q^{12} - 2 \beta_{2} q^{13} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{14} + \beta_1 q^{15} + (\beta_1 - 1) q^{16} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{17} + q^{18} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{19} + q^{20} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{21} + ( - 3 \beta_1 + 3) q^{22} + (2 \beta_{2} + \beta_1) q^{23} + ( - \beta_1 + 1) q^{24} - \beta_1 q^{25} + 2 \beta_{3} q^{26} - q^{27} + (\beta_{2} - \beta_1) q^{28} + ( - 3 \beta_{2} - 2 \beta_1) q^{29} - q^{30} + ( - 2 \beta_{3} + 4) q^{31} - \beta_1 q^{32} + (3 \beta_1 - 3) q^{33} + ( - \beta_{2} + 4 \beta_1) q^{34} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{35} + (\beta_1 - 1) q^{36} + ( - 2 \beta_{3} - 5) q^{37} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{38} - 2 \beta_{3} q^{39} + (\beta_1 - 1) q^{40} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{41} + ( - \beta_{2} + \beta_1) q^{42} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{43} + 3 \beta_1 q^{44} + q^{45} + ( - 2 \beta_{3} - 1) q^{46} + ( - 2 \beta_{2} + 6 \beta_1) q^{47} + \beta_1 q^{48} - 2 \beta_{3} q^{49} + q^{50} + (\beta_{2} - 4 \beta_1) q^{51} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{52} + (\beta_{2} + 5 \beta_1) q^{53} + ( - \beta_1 + 1) q^{54} + ( - 3 \beta_1 + 3) q^{55} + ( - \beta_{3} + 1) q^{56} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{57} + (3 \beta_{3} + 2) q^{58} + ( - 6 \beta_1 + 6) q^{59} + ( - \beta_1 + 1) q^{60} + ( - \beta_{2} - 10 \beta_1) q^{61} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{62} + (\beta_{2} - \beta_1) q^{63} + q^{64} + 2 \beta_{3} q^{65} - 3 \beta_1 q^{66} + 3 \beta_{2} q^{67} + (\beta_{3} - 4) q^{68} + (2 \beta_{3} + 1) q^{69} + (\beta_{2} - \beta_1) q^{70} + (3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 4) q^{71} - \beta_1 q^{72} + ( - \beta_{3} + \beta_{2} + 8 \beta_1 - 8) q^{73} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 5) q^{74} - q^{75} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{76} + (3 \beta_{3} - 3) q^{77} + (2 \beta_{3} - 2 \beta_{2}) q^{78} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{79} - \beta_1 q^{80} + (\beta_1 - 1) q^{81} + ( - 3 \beta_{2} + 3 \beta_1) q^{82} + (4 \beta_{3} + 2) q^{83} + (\beta_{3} - 1) q^{84} + ( - \beta_{2} + 4 \beta_1) q^{85} + (4 \beta_{2} + 2 \beta_1) q^{86} + ( - 3 \beta_{3} - 2) q^{87} - 3 q^{88} + (\beta_{2} + 3 \beta_1) q^{89} + (\beta_1 - 1) q^{90} + ( - 2 \beta_{2} + 12 \beta_1) q^{91} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{92} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{93} + (2 \beta_{3} - 6) q^{94} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{95} - q^{96} + (\beta_{3} - \beta_{2} - 16 \beta_1 + 16) q^{97} + (2 \beta_{3} - 2 \beta_{2}) q^{98} + 3 \beta_1 q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + (-b1 + 1) * q^3 - b1 * q^4 + (b1 - 1) * q^5 + b1 * q^6 + (-b3 + 1) * q^7 + q^8 - b1 * q^9 - b1 * q^10 - 3 * q^11 - q^12 - 2*b2 * q^13 + (b3 - b2 + b1 - 1) * q^14 + b1 * q^15 + (b1 - 1) * q^16 + (-b3 + b2 - 4*b1 + 4) * q^17 + q^18 + (-b3 - b2 + b1 - 1) * q^19 + q^20 + (-b3 + b2 - b1 + 1) * q^21 + (-3*b1 + 3) * q^22 + (2*b2 + b1) * q^23 + (-b1 + 1) * q^24 - b1 * q^25 + 2*b3 * q^26 - q^27 + (b2 - b1) * q^28 + (-3*b2 - 2*b1) * q^29 - q^30 + (-2*b3 + 4) * q^31 - b1 * q^32 + (3*b1 - 3) * q^33 + (-b2 + 4*b1) * q^34 + (b3 - b2 + b1 - 1) * q^35 + (b1 - 1) * q^36 + (-2*b3 - 5) * q^37 + (2*b3 - b2 - b1) * q^38 - 2*b3 * q^39 + (b1 - 1) * q^40 + (-3*b3 + 3*b2 - 3*b1 + 3) * q^41 + (-b2 + b1) * q^42 + (4*b3 - 4*b2 - 2*b1 + 2) * q^43 + 3*b1 * q^44 + q^45 + (-2*b3 - 1) * q^46 + (-2*b2 + 6*b1) * q^47 + b1 * q^48 - 2*b3 * q^49 + q^50 + (b2 - 4*b1) * q^51 + (-2*b3 + 2*b2) * q^52 + (b2 + 5*b1) * q^53 + (-b1 + 1) * q^54 + (-3*b1 + 3) * q^55 + (-b3 + 1) * q^56 + (-2*b3 + b2 + b1) * q^57 + (3*b3 + 2) * q^58 + (-6*b1 + 6) * q^59 + (-b1 + 1) * q^60 + (-b2 - 10*b1) * q^61 + (2*b3 - 2*b2 + 4*b1 - 4) * q^62 + (b2 - b1) * q^63 + q^64 + 2*b3 * q^65 - 3*b1 * q^66 + 3*b2 * q^67 + (b3 - 4) * q^68 + (2*b3 + 1) * q^69 + (b2 - b1) * q^70 + (3*b3 - 3*b2 + 4*b1 - 4) * q^71 - b1 * q^72 + (-b3 + b2 + 8*b1 - 8) * q^73 + (2*b3 - 2*b2 - 5*b1 + 5) * q^74 - q^75 + (-b3 + 2*b2 + 1) * q^76 + (3*b3 - 3) * q^77 + (2*b3 - 2*b2) * q^78 + (-2*b3 + 2*b2 + 2*b1 - 2) * q^79 - b1 * q^80 + (b1 - 1) * q^81 + (-3*b2 + 3*b1) * q^82 + (4*b3 + 2) * q^83 + (b3 - 1) * q^84 + (-b2 + 4*b1) * q^85 + (4*b2 + 2*b1) * q^86 + (-3*b3 - 2) * q^87 - 3 * q^88 + (b2 + 3*b1) * q^89 + (b1 - 1) * q^90 + (-2*b2 + 12*b1) * q^91 + (2*b3 - 2*b2 - b1 + 1) * q^92 + (-2*b3 + 2*b2 - 4*b1 + 4) * q^93 + (2*b3 - 6) * q^94 + (2*b3 - b2 - b1) * q^95 - q^96 + (b3 - b2 - 16*b1 + 16) * q^97 + (2*b3 - 2*b2) * q^98 + 3*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 $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} - 2 q^{9} - 2 q^{10} - 12 q^{11} - 4 q^{12} - 2 q^{14} + 2 q^{15} - 2 q^{16} + 8 q^{17} + 4 q^{18} - 2 q^{19} + 4 q^{20} + 2 q^{21} + 6 q^{22} + 2 q^{23} + 2 q^{24} - 2 q^{25} - 4 q^{27} - 2 q^{28} - 4 q^{29} - 4 q^{30} + 16 q^{31} - 2 q^{32} - 6 q^{33} + 8 q^{34} - 2 q^{35} - 2 q^{36} - 20 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 2 q^{42} + 4 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{46} + 12 q^{47} + 2 q^{48} + 4 q^{50} - 8 q^{51} + 10 q^{53} + 2 q^{54} + 6 q^{55} + 4 q^{56} + 2 q^{57} + 8 q^{58} + 12 q^{59} + 2 q^{60} - 20 q^{61} - 8 q^{62} - 2 q^{63} + 4 q^{64} - 6 q^{66} - 16 q^{68} + 4 q^{69} - 2 q^{70} - 8 q^{71} - 2 q^{72} - 16 q^{73} + 10 q^{74} - 4 q^{75} + 4 q^{76} - 12 q^{77} - 4 q^{79} - 2 q^{80} - 2 q^{81} + 6 q^{82} + 8 q^{83} - 4 q^{84} + 8 q^{85} + 4 q^{86} - 8 q^{87} - 12 q^{88} + 6 q^{89} - 2 q^{90} + 24 q^{91} + 2 q^{92} + 8 q^{93} - 24 q^{94} - 2 q^{95} - 4 q^{96} + 32 q^{97} + 6 q^{99}+O(q^{100})$$ 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 - 12 * q^11 - 4 * q^12 - 2 * q^14 + 2 * q^15 - 2 * q^16 + 8 * q^17 + 4 * q^18 - 2 * q^19 + 4 * q^20 + 2 * q^21 + 6 * q^22 + 2 * q^23 + 2 * q^24 - 2 * q^25 - 4 * q^27 - 2 * q^28 - 4 * q^29 - 4 * q^30 + 16 * q^31 - 2 * q^32 - 6 * q^33 + 8 * q^34 - 2 * q^35 - 2 * q^36 - 20 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 2 * q^42 + 4 * q^43 + 6 * q^44 + 4 * q^45 - 4 * q^46 + 12 * q^47 + 2 * q^48 + 4 * q^50 - 8 * q^51 + 10 * q^53 + 2 * q^54 + 6 * q^55 + 4 * q^56 + 2 * q^57 + 8 * q^58 + 12 * q^59 + 2 * q^60 - 20 * q^61 - 8 * q^62 - 2 * q^63 + 4 * q^64 - 6 * q^66 - 16 * q^68 + 4 * q^69 - 2 * q^70 - 8 * q^71 - 2 * q^72 - 16 * q^73 + 10 * q^74 - 4 * q^75 + 4 * q^76 - 12 * q^77 - 4 * q^79 - 2 * q^80 - 2 * q^81 + 6 * q^82 + 8 * q^83 - 4 * q^84 + 8 * q^85 + 4 * q^86 - 8 * q^87 - 12 * q^88 + 6 * q^89 - 2 * q^90 + 24 * q^91 + 2 * q^92 + 8 * q^93 - 24 * q^94 - 2 * q^95 - 4 * q^96 + 32 * q^97 + 6 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 2$$ (v^3 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu ) / 2$$ (-v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 4\beta_{2} ) / 3$$ (-2*b3 + 4*b2) / 3

## Character values

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

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

## 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
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.44949 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 3.44949 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 −1.44949 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 3.44949 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.f 4
3.b odd 2 1 1710.2.l.n 4
19.c even 3 1 inner 570.2.i.f 4
57.h odd 6 1 1710.2.l.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.f 4 1.a even 1 1 trivial
570.2.i.f 4 19.c even 3 1 inner
1710.2.l.n 4 3.b odd 2 1
1710.2.l.n 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 $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7}^{2} - 2T_{7} - 5$$ T7^2 - 2*T7 - 5 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$(T^{2} - 2 T - 5)^{2}$$
$11$ $$(T + 3)^{4}$$
$13$ $$T^{4} + 24T^{2} + 576$$
$17$ $$T^{4} - 8 T^{3} + 54 T^{2} - 80 T + 100$$
$19$ $$T^{4} + 2 T^{3} - 15 T^{2} + 38 T + 361$$
$23$ $$T^{4} - 2 T^{3} + 27 T^{2} + 46 T + 529$$
$29$ $$T^{4} + 4 T^{3} + 66 T^{2} + \cdots + 2500$$
$31$ $$(T^{2} - 8 T - 8)^{2}$$
$37$ $$(T^{2} + 10 T + 1)^{2}$$
$41$ $$T^{4} - 6 T^{3} + 81 T^{2} + \cdots + 2025$$
$43$ $$T^{4} - 4 T^{3} + 108 T^{2} + \cdots + 8464$$
$47$ $$T^{4} - 12 T^{3} + 132 T^{2} + \cdots + 144$$
$53$ $$T^{4} - 10 T^{3} + 81 T^{2} + \cdots + 361$$
$59$ $$(T^{2} - 6 T + 36)^{2}$$
$61$ $$T^{4} + 20 T^{3} + 306 T^{2} + \cdots + 8836$$
$67$ $$T^{4} + 54T^{2} + 2916$$
$71$ $$T^{4} + 8 T^{3} + 102 T^{2} + \cdots + 1444$$
$73$ $$T^{4} + 16 T^{3} + 198 T^{2} + \cdots + 3364$$
$79$ $$T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400$$
$83$ $$(T^{2} - 4 T - 92)^{2}$$
$89$ $$T^{4} - 6 T^{3} + 33 T^{2} - 18 T + 9$$
$97$ $$T^{4} - 32 T^{3} + 774 T^{2} + \cdots + 62500$$