# Properties

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

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.7278137344.1 Defining polynomial: $$x^{8} - 2 x^{7} + x^{6} + 6 x^{5} - 20 x^{4} + 18 x^{3} + 9 x^{2} - 54 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + x^{6} + 6 x^{5} - 20 x^{4} + 18 x^{3} + 9 x^{2} - 54 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} - 5 \nu^{5} + 9 \nu^{4} - 2 \nu^{3} - 15 \nu^{2} + 36 \nu - 27$$$$)/54$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} - \nu^{4} + 2 \nu^{3} - 3 \nu^{2} - \nu + 12$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - 4 \nu^{5} + 9 \nu^{4} - 7 \nu^{3} - 12 \nu^{2} + 36 \nu - 54$$$$)/54$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - \nu^{5} - 6 \nu^{4} + 20 \nu^{3} - 18 \nu^{2} - 9 \nu + 54$$$$)/27$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + \nu^{5} + 6 \nu^{4} - 11 \nu^{3} + 9 \nu^{2} + 9 \nu - 27$$$$)/18$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + \nu^{5} - 7 \nu^{4} + 14 \nu^{3} + 11 \nu^{2} - 18 \nu + 45$$$$)/18$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{5} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{2} - 3$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_{1} + 4$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} + 12 \beta_{2} - 2 \beta_{1} + 9$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{7} + 6 \beta_{6} + 2 \beta_{5} - 22 \beta_{4} + 4 \beta_{3} + 12 \beta_{2} + 3 \beta_{1} - 14$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
341.1
 −1.71731 − 0.225499i −1.71731 + 0.225499i 0.209196 − 1.71937i 0.209196 + 1.71937i 0.828750 − 1.52091i 0.828750 + 1.52091i 1.67936 − 0.423958i 1.67936 + 0.423958i
−1.00000 −1.71731 0.225499i 1.00000 1.00000i 1.71731 + 0.225499i −0.631989 −1.00000 2.89830 + 0.774501i 1.00000i
341.2 −1.00000 −1.71731 + 0.225499i 1.00000 1.00000i 1.71731 0.225499i −0.631989 −1.00000 2.89830 0.774501i 1.00000i
341.3 −1.00000 0.209196 1.71937i 1.00000 1.00000i −0.209196 + 1.71937i 0.264536 −1.00000 −2.91247 0.719371i 1.00000i
341.4 −1.00000 0.209196 + 1.71937i 1.00000 1.00000i −0.209196 1.71937i 0.264536 −1.00000 −2.91247 + 0.719371i 1.00000i
341.5 −1.00000 0.828750 1.52091i 1.00000 1.00000i −0.828750 + 1.52091i 4.83942 −1.00000 −1.62635 2.52091i 1.00000i
341.6 −1.00000 0.828750 + 1.52091i 1.00000 1.00000i −0.828750 1.52091i 4.83942 −1.00000 −1.62635 + 2.52091i 1.00000i
341.7 −1.00000 1.67936 0.423958i 1.00000 1.00000i −1.67936 + 0.423958i −2.47197 −1.00000 2.64052 1.42396i 1.00000i
341.8 −1.00000 1.67936 + 0.423958i 1.00000 1.00000i −1.67936 0.423958i −2.47197 −1.00000 2.64052 + 1.42396i 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d 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.f.c 8
3.b odd 2 1 570.2.f.d yes 8
19.b odd 2 1 570.2.f.d yes 8
57.d even 2 1 inner 570.2.f.c 8

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

## Hecke kernels

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

 $$T_{7}^{4} - 2 T_{7}^{3} - 13 T_{7}^{2} - 4 T_{7} + 2$$ $$T_{29}^{4} - 2 T_{29}^{3} - 11 T_{29}^{2} + 24 T_{29} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$81 - 54 T + 9 T^{2} + 18 T^{3} - 20 T^{4} + 6 T^{5} + T^{6} - 2 T^{7} + T^{8}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$( 2 - 4 T - 13 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$( 4 + T^{2} )^{4}$$
$13$ $$256 + 480 T^{2} + 265 T^{4} + 38 T^{6} + T^{8}$$
$17$ $$100 + 5116 T^{2} + 1069 T^{4} + 66 T^{6} + T^{8}$$
$19$ $$130321 - 82308 T + 28880 T^{2} - 7524 T^{3} + 1678 T^{4} - 396 T^{5} + 80 T^{6} - 12 T^{7} + T^{8}$$
$23$ $$15376 + 25048 T^{2} + 2761 T^{4} + 94 T^{6} + T^{8}$$
$29$ $$( -8 + 24 T - 11 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$31$ $$193600 + 42464 T^{2} + 3252 T^{4} + 100 T^{6} + T^{8}$$
$37$ $$16384 + 174080 T^{2} + 11072 T^{4} + 208 T^{6} + T^{8}$$
$41$ $$( 16 - 24 T - 2 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$43$ $$( -176 - 112 T + 88 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$47$ $$430336 + 86016 T^{2} + 5344 T^{4} + 128 T^{6} + T^{8}$$
$53$ $$( -424 - 168 T + 33 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$59$ $$( 118 + 280 T - 161 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$61$ $$( 3104 + 848 T - 140 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$67$ $$1849600 + 4202464 T^{2} + 82801 T^{4} + 510 T^{6} + T^{8}$$
$71$ $$( -5440 - 1696 T - 96 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$73$ $$( -1228 - 764 T - 119 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$79$ $$4326400 + 459904 T^{2} + 16132 T^{4} + 220 T^{6} + T^{8}$$
$83$ $$121000000 + 6119584 T^{2} + 93076 T^{4} + 540 T^{6} + T^{8}$$
$89$ $$( 11384 - 4720 T + 698 T^{2} - 44 T^{3} + T^{4} )^{2}$$
$97$ $$65536 + 38912 T^{2} + 4240 T^{4} + 120 T^{6} + T^{8}$$
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