# Properties

 Label 570.2.i.i 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: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## 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
 1.32288 + 2.29129i −1.32288 − 2.29129i 1.32288 − 2.29129i −1.32288 + 2.29129i
0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.64575 −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 2.64575 −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 −2.64575 −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 2.64575 −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.i 4
3.b odd 2 1 1710.2.l.j 4
19.c even 3 1 inner 570.2.i.i 4
57.h odd 6 1 1710.2.l.j 4

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

 $$T_{7}^{2} - 7$$ T7^2 - 7 $$T_{11}^{2} + 2T_{11} - 27$$ T11^2 + 2*T11 - 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$(T^{2} + 2 T - 27)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 6 T^{3} + 34 T^{2} - 12 T + 4$$
$19$ $$T^{4} + 12 T^{3} + 67 T^{2} + \cdots + 361$$
$23$ $$(T^{2} + T + 1)^{2}$$
$29$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$31$ $$(T^{2} - 4 T - 24)^{2}$$
$37$ $$(T^{2} - 6 T - 19)^{2}$$
$41$ $$T^{4} - 12 T^{3} + 115 T^{2} + \cdots + 841$$
$43$ $$(T^{2} + 6 T + 36)^{2}$$
$47$ $$T^{4} + 4 T^{3} + 124 T^{2} + \cdots + 11664$$
$53$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$59$ $$T^{4} + 28T^{2} + 784$$
$61$ $$T^{4} - 2 T^{3} + 66 T^{2} + \cdots + 3844$$
$67$ $$T^{4} - 14 T^{3} + 154 T^{2} + \cdots + 1764$$
$71$ $$T^{4} - 22 T^{3} + 370 T^{2} + \cdots + 12996$$
$73$ $$T^{4} + 22 T^{3} + 370 T^{2} + \cdots + 12996$$
$79$ $$T^{4} + 28T^{2} + 784$$
$83$ $$(T^{2} - 12 T - 76)^{2}$$
$89$ $$T^{4} + 175 T^{2} + 30625$$
$97$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$