# Properties

 Label 570.2.c.c Level $570$ Weight $2$ Character orbit 570.c Analytic conductor $4.551$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: 8.0.1499238400.2 Defining polynomial: $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 59 x^{4} - 66 x^{3} + 54 x^{2} - 26 x + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 59 x^{4} - 66 x^{3} + 54 x^{2} - 26 x + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{7} - 8 \nu^{6} + 34 \nu^{5} - 50 \nu^{4} + 82 \nu^{3} - 62 \nu^{2} + 46 \nu - 10$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} + 8 \nu^{6} - 34 \nu^{5} + 50 \nu^{4} - 82 \nu^{3} + 67 \nu^{2} - 51 \nu + 25$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 7 \nu^{6} + 31 \nu^{5} - 60 \nu^{4} + 118 \nu^{3} - 118 \nu^{2} + 109 \nu - 30$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{7} + 14 \nu^{6} - 57 \nu^{5} + 110 \nu^{4} - 186 \nu^{3} + 191 \nu^{2} - 138 \nu + 50$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{6} - 83 \nu^{5} + 155 \nu^{4} - 249 \nu^{3} + 229 \nu^{2} - 157 \nu + 45$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{7} + 22 \nu^{6} - 96 \nu^{5} + 170 \nu^{4} - 313 \nu^{3} + 298 \nu^{2} - 249 \nu + 85$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{4} + \beta_{2} - 3 \beta_{1} - 3$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} - 8 \beta_{1} + 13$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} - 12 \beta_{4} - 5 \beta_{3} - 13 \beta_{2} + 7 \beta_{1} + 31$$ $$\nu^{6}$$ $$=$$ $$-18 \beta_{7} + 6 \beta_{6} - 7 \beta_{5} - 35 \beta_{4} + 41 \beta_{3} + 25 \beta_{2} + 50 \beta_{1} - 43$$ $$\nu^{7}$$ $$=$$ $$32 \beta_{7} + 22 \beta_{6} - 42 \beta_{5} + 38 \beta_{4} + 70 \beta_{3} + 109 \beta_{2} + 8 \beta_{1} - 226$$

## 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
 0.5 − 1.08454i 0.5 + 0.0845405i 0.5 − 2.41267i 0.5 + 1.41267i 0.5 + 2.41267i 0.5 − 1.41267i 0.5 + 1.08454i 0.5 − 0.0845405i
1.00000i −0.500000 1.65831i −1.00000 −0.584541 2.15831i −1.65831 + 0.500000i 4.86140i 1.00000i −2.50000 + 1.65831i −2.15831 + 0.584541i
569.2 1.00000i −0.500000 1.65831i −1.00000 0.584541 2.15831i −1.65831 + 0.500000i 4.86140i 1.00000i −2.50000 + 1.65831i −2.15831 0.584541i
569.3 1.00000i −0.500000 + 1.65831i −1.00000 −1.91267 + 1.15831i 1.65831 + 0.500000i 3.21974i 1.00000i −2.50000 1.65831i 1.15831 + 1.91267i
569.4 1.00000i −0.500000 + 1.65831i −1.00000 1.91267 + 1.15831i 1.65831 + 0.500000i 3.21974i 1.00000i −2.50000 1.65831i 1.15831 1.91267i
569.5 1.00000i −0.500000 1.65831i −1.00000 −1.91267 1.15831i 1.65831 0.500000i 3.21974i 1.00000i −2.50000 + 1.65831i 1.15831 1.91267i
569.6 1.00000i −0.500000 1.65831i −1.00000 1.91267 1.15831i 1.65831 0.500000i 3.21974i 1.00000i −2.50000 + 1.65831i 1.15831 + 1.91267i
569.7 1.00000i −0.500000 + 1.65831i −1.00000 −0.584541 + 2.15831i −1.65831 0.500000i 4.86140i 1.00000i −2.50000 1.65831i −2.15831 0.584541i
569.8 1.00000i −0.500000 + 1.65831i −1.00000 0.584541 + 2.15831i −1.65831 0.500000i 4.86140i 1.00000i −2.50000 1.65831i −2.15831 + 0.584541i
 $$n$$: 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
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.c 8
3.b odd 2 1 inner 570.2.c.c 8
5.b even 2 1 570.2.c.f yes 8
15.d odd 2 1 570.2.c.f yes 8
19.b odd 2 1 570.2.c.f yes 8
57.d even 2 1 570.2.c.f yes 8
95.d odd 2 1 inner 570.2.c.c 8
285.b even 2 1 inner 570.2.c.c 8

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

## Hecke kernels

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

 $$T_{7}^{4} + 34 T_{7}^{2} + 245$$ $$T_{11}^{4} + 24 T_{11}^{2} + 100$$ $$T_{29}^{4} - 70 T_{29}^{2} + 125$$ $$T_{37} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$( 3 + T + T^{2} )^{4}$$
$5$ $$625 + 100 T^{2} + 10 T^{4} + 4 T^{6} + T^{8}$$
$7$ $$( 245 + 34 T^{2} + T^{4} )^{2}$$
$11$ $$( 100 + 24 T^{2} + T^{4} )^{2}$$
$13$ $$( -11 + T^{2} )^{4}$$
$17$ $$( 245 - 34 T^{2} + T^{4} )^{2}$$
$19$ $$( 361 + 76 T - 2 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$23$ $$( 1805 - 86 T^{2} + T^{4} )^{2}$$
$29$ $$( 125 - 70 T^{2} + T^{4} )^{2}$$
$31$ $$( 500 + 80 T^{2} + T^{4} )^{2}$$
$37$ $$( 2 + T )^{8}$$
$41$ $$( 1620 - 144 T^{2} + T^{4} )^{2}$$
$43$ $$( 20 + 16 T^{2} + T^{4} )^{2}$$
$47$ $$( 2000 - 104 T^{2} + T^{4} )^{2}$$
$53$ $$( 1 + T^{2} )^{4}$$
$59$ $$( 3125 - 130 T^{2} + T^{4} )^{2}$$
$61$ $$( -10 - 2 T + T^{2} )^{4}$$
$67$ $$( 7 + T )^{8}$$
$71$ $$( 12500 - 224 T^{2} + T^{4} )^{2}$$
$73$ $$( 405 + 126 T^{2} + T^{4} )^{2}$$
$79$ $$( 5120 + 256 T^{2} + T^{4} )^{2}$$
$83$ $$( 3920 - 136 T^{2} + T^{4} )^{2}$$
$89$ $$( 500 - 80 T^{2} + T^{4} )^{2}$$
$97$ $$( 56 + 20 T + T^{2} )^{4}$$