# Properties

 Label 570.2.c.d Level $570$ Weight $2$ Character orbit 570.c Analytic conductor $4.551$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{4} q^{3} - q^{4} + ( - \beta_{6} - \beta_{2}) q^{5} + \beta_{2} q^{6} - \beta_{7} q^{7} + \beta_1 q^{8} + 3 q^{9}+O(q^{10})$$ q - b1 * q^2 - b4 * q^3 - q^4 + (-b6 - b2) * q^5 + b2 * q^6 - b7 * q^7 + b1 * q^8 + 3 * q^9 $$q - \beta_1 q^{2} - \beta_{4} q^{3} - q^{4} + ( - \beta_{6} - \beta_{2}) q^{5} + \beta_{2} q^{6} - \beta_{7} q^{7} + \beta_1 q^{8} + 3 q^{9} + ( - \beta_{4} - \beta_{3}) q^{10} - 2 \beta_{2} q^{11} + \beta_{4} q^{12} - \beta_{5} q^{14} + (\beta_{5} + 3 \beta_1) q^{15} + q^{16} - 5 \beta_{6} q^{17} - 3 \beta_1 q^{18} + ( - 3 \beta_{3} - 1) q^{19} + (\beta_{6} + \beta_{2}) q^{20} - 3 \beta_{3} q^{21} - 2 \beta_{4} q^{22} + 4 \beta_{6} q^{23} - \beta_{2} q^{24} + (2 \beta_{7} - 1) q^{25} - 3 \beta_{4} q^{27} + \beta_{7} q^{28} - 2 \beta_{5} q^{29} + ( - \beta_{7} + 3) q^{30} - 3 \beta_{3} q^{31} - \beta_1 q^{32} + 6 \beta_1 q^{33} - 5 \beta_{3} q^{34} + ( - 3 \beta_{6} + 2 \beta_{2}) q^{35} - 3 q^{36} - 4 \beta_{4} q^{37} + (3 \beta_{6} + \beta_1) q^{38} + (\beta_{4} + \beta_{3}) q^{40} + \beta_{5} q^{41} + 3 \beta_{6} q^{42} + 4 \beta_{7} q^{43} + 2 \beta_{2} q^{44} + ( - 3 \beta_{6} - 3 \beta_{2}) q^{45} + 4 \beta_{3} q^{46} + 2 \beta_{6} q^{47} - \beta_{4} q^{48} + q^{49} + (2 \beta_{5} + \beta_1) q^{50} + 5 \beta_{5} q^{51} - 12 \beta_1 q^{53} + 3 \beta_{2} q^{54} + (2 \beta_{7} - 6) q^{55} + \beta_{5} q^{56} + ( - 3 \beta_{7} + \beta_{4}) q^{57} + 2 \beta_{7} q^{58} - 3 \beta_{5} q^{59} + ( - \beta_{5} - 3 \beta_1) q^{60} - 10 q^{61} + 3 \beta_{6} q^{62} - 3 \beta_{7} q^{63} - q^{64} + 6 q^{66} + 8 \beta_{4} q^{67} + 5 \beta_{6} q^{68} - 4 \beta_{5} q^{69} + (2 \beta_{4} - 3 \beta_{3}) q^{70} - 2 \beta_{5} q^{71} + 3 \beta_1 q^{72} - 4 \beta_{7} q^{73} + 4 \beta_{2} q^{74} + (\beta_{4} + 6 \beta_{3}) q^{75} + (3 \beta_{3} + 1) q^{76} - 6 \beta_{6} q^{77} + 3 \beta_{3} q^{79} + ( - \beta_{6} - \beta_{2}) q^{80} + 9 q^{81} - \beta_{7} q^{82} - \beta_{6} q^{83} + 3 \beta_{3} q^{84} + (5 \beta_{7} + 10) q^{85} + 4 \beta_{5} q^{86} + 6 \beta_{6} q^{87} + 2 \beta_{4} q^{88} - 5 \beta_{5} q^{89} + ( - 3 \beta_{4} - 3 \beta_{3}) q^{90} - 4 \beta_{6} q^{92} - 3 \beta_{7} q^{93} + 2 \beta_{3} q^{94} + (\beta_{6} + 3 \beta_{5} + \beta_{2} - 6 \beta_1) q^{95} + \beta_{2} q^{96} + 2 \beta_{4} q^{97} - \beta_1 q^{98} - 6 \beta_{2} q^{99}+O(q^{100})$$ q - b1 * q^2 - b4 * q^3 - q^4 + (-b6 - b2) * q^5 + b2 * q^6 - b7 * q^7 + b1 * q^8 + 3 * q^9 + (-b4 - b3) * q^10 - 2*b2 * q^11 + b4 * q^12 - b5 * q^14 + (b5 + 3*b1) * q^15 + q^16 - 5*b6 * q^17 - 3*b1 * q^18 + (-3*b3 - 1) * q^19 + (b6 + b2) * q^20 - 3*b3 * q^21 - 2*b4 * q^22 + 4*b6 * q^23 - b2 * q^24 + (2*b7 - 1) * q^25 - 3*b4 * q^27 + b7 * q^28 - 2*b5 * q^29 + (-b7 + 3) * q^30 - 3*b3 * q^31 - b1 * q^32 + 6*b1 * q^33 - 5*b3 * q^34 + (-3*b6 + 2*b2) * q^35 - 3 * q^36 - 4*b4 * q^37 + (3*b6 + b1) * q^38 + (b4 + b3) * q^40 + b5 * q^41 + 3*b6 * q^42 + 4*b7 * q^43 + 2*b2 * q^44 + (-3*b6 - 3*b2) * q^45 + 4*b3 * q^46 + 2*b6 * q^47 - b4 * q^48 + q^49 + (2*b5 + b1) * q^50 + 5*b5 * q^51 - 12*b1 * q^53 + 3*b2 * q^54 + (2*b7 - 6) * q^55 + b5 * q^56 + (-3*b7 + b4) * q^57 + 2*b7 * q^58 - 3*b5 * q^59 + (-b5 - 3*b1) * q^60 - 10 * q^61 + 3*b6 * q^62 - 3*b7 * q^63 - q^64 + 6 * q^66 + 8*b4 * q^67 + 5*b6 * q^68 - 4*b5 * q^69 + (2*b4 - 3*b3) * q^70 - 2*b5 * q^71 + 3*b1 * q^72 - 4*b7 * q^73 + 4*b2 * q^74 + (b4 + 6*b3) * q^75 + (3*b3 + 1) * q^76 - 6*b6 * q^77 + 3*b3 * q^79 + (-b6 - b2) * q^80 + 9 * q^81 - b7 * q^82 - b6 * q^83 + 3*b3 * q^84 + (5*b7 + 10) * q^85 + 4*b5 * q^86 + 6*b6 * q^87 + 2*b4 * q^88 - 5*b5 * q^89 + (-3*b4 - 3*b3) * q^90 - 4*b6 * q^92 - 3*b7 * q^93 + 2*b3 * q^94 + (b6 + 3*b5 + b2 - 6*b1) * q^95 + b2 * q^96 + 2*b4 * q^97 - b1 * q^98 - 6*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4} + 24 q^{9}+O(q^{10})$$ 8 * q - 8 * q^4 + 24 * q^9 $$8 q - 8 q^{4} + 24 q^{9} + 8 q^{16} - 8 q^{19} - 8 q^{25} + 24 q^{30} - 24 q^{36} + 8 q^{49} - 48 q^{55} - 80 q^{61} - 8 q^{64} + 48 q^{66} + 8 q^{76} + 72 q^{81} + 80 q^{85}+O(q^{100})$$ 8 * q - 8 * q^4 + 24 * q^9 + 8 * q^16 - 8 * q^19 - 8 * q^25 + 24 * q^30 - 24 * q^36 + 8 * q^49 - 48 * q^55 - 80 * q^61 - 8 * q^64 + 48 * q^66 + 8 * q^76 + 72 * q^81 + 80 * q^85

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{4} - 1$$ 2*v^4 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ -v^5 - v^3 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{5}$$ $$=$$ $$-2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -2*v^7 + v^5 + v^3 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} ) / 4$$ (b7 + b6 + b5 + b3) / 4 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_1 ) / 2$$ (b4 + b1) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{6} - \beta_{3} ) / 2$$ (b6 - b3) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} ) / 4$$ (b7 - b6 + b5 - b3) / 4 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} ) / 4$$ (b7 + b6 - b5 - b3) / 4

## Character values

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

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
1.00000i −1.73205 −1.00000 −1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 + 1.41421i
569.2 1.00000i −1.73205 −1.00000 1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 1.41421i
569.3 1.00000i 1.73205 −1.00000 −1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 + 1.41421i
569.4 1.00000i 1.73205 −1.00000 1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 1.41421i
569.5 1.00000i −1.73205 −1.00000 −1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 1.41421i
569.6 1.00000i −1.73205 −1.00000 1.41421 + 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 −1.73205 + 1.41421i
569.7 1.00000i 1.73205 −1.00000 −1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 1.41421i
569.8 1.00000i 1.73205 −1.00000 1.41421 1.73205i 1.73205i 2.44949i 1.00000i 3.00000 1.73205 + 1.41421i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 569.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner
95.d odd 2 1 inner
285.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.c.d 8
3.b odd 2 1 inner 570.2.c.d 8
5.b even 2 1 inner 570.2.c.d 8
15.d odd 2 1 inner 570.2.c.d 8
19.b odd 2 1 inner 570.2.c.d 8
57.d even 2 1 inner 570.2.c.d 8
95.d odd 2 1 inner 570.2.c.d 8
285.b even 2 1 inner 570.2.c.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.c.d 8 1.a even 1 1 trivial
570.2.c.d 8 3.b odd 2 1 inner
570.2.c.d 8 5.b even 2 1 inner
570.2.c.d 8 15.d odd 2 1 inner
570.2.c.d 8 19.b odd 2 1 inner
570.2.c.d 8 57.d even 2 1 inner
570.2.c.d 8 95.d odd 2 1 inner
570.2.c.d 8 285.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7}^{2} + 6$$ T7^2 + 6 $$T_{11}^{2} + 12$$ T11^2 + 12 $$T_{29}^{2} - 24$$ T29^2 - 24 $$T_{37}^{2} - 48$$ T37^2 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$(T^{2} - 3)^{4}$$
$5$ $$(T^{4} + 2 T^{2} + 25)^{2}$$
$7$ $$(T^{2} + 6)^{4}$$
$11$ $$(T^{2} + 12)^{4}$$
$13$ $$T^{8}$$
$17$ $$(T^{2} - 50)^{4}$$
$19$ $$(T^{2} + 2 T + 19)^{4}$$
$23$ $$(T^{2} - 32)^{4}$$
$29$ $$(T^{2} - 24)^{4}$$
$31$ $$(T^{2} + 18)^{4}$$
$37$ $$(T^{2} - 48)^{4}$$
$41$ $$(T^{2} - 6)^{4}$$
$43$ $$(T^{2} + 96)^{4}$$
$47$ $$(T^{2} - 8)^{4}$$
$53$ $$(T^{2} + 144)^{4}$$
$59$ $$(T^{2} - 54)^{4}$$
$61$ $$(T + 10)^{8}$$
$67$ $$(T^{2} - 192)^{4}$$
$71$ $$(T^{2} - 24)^{4}$$
$73$ $$(T^{2} + 96)^{4}$$
$79$ $$(T^{2} + 18)^{4}$$
$83$ $$(T^{2} - 2)^{4}$$
$89$ $$(T^{2} - 150)^{4}$$
$97$ $$(T^{2} - 12)^{4}$$