# Properties

 Label 570.4.i.i Level $570$ Weight $4$ Character orbit 570.i Analytic conductor $33.631$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 570.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.6310887033$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{481})$$ Defining polynomial: $$x^{4} - x^{3} + 121 x^{2} + 120 x + 14400$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + 3 \beta_{2} q^{3} + ( -4 + 4 \beta_{2} ) q^{4} + 5 \beta_{2} q^{5} + ( -6 + 6 \beta_{2} ) q^{6} + ( 2 - \beta_{3} ) q^{7} -8 q^{8} + ( -9 + 9 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + 3 \beta_{2} q^{3} + ( -4 + 4 \beta_{2} ) q^{4} + 5 \beta_{2} q^{5} + ( -6 + 6 \beta_{2} ) q^{6} + ( 2 - \beta_{3} ) q^{7} -8 q^{8} + ( -9 + 9 \beta_{2} ) q^{9} + ( -10 + 10 \beta_{2} ) q^{10} + ( 3 - 3 \beta_{3} ) q^{11} -12 q^{12} + ( 54 + \beta_{1} - 55 \beta_{2} + \beta_{3} ) q^{13} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{14} + ( -15 + 15 \beta_{2} ) q^{15} -16 \beta_{2} q^{16} + ( 10 \beta_{1} - 8 \beta_{2} ) q^{17} -18 q^{18} + ( 5 - \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{19} -20 q^{20} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{21} + 6 \beta_{1} q^{22} + ( 28 + 2 \beta_{1} - 30 \beta_{2} + 2 \beta_{3} ) q^{23} -24 \beta_{2} q^{24} + ( -25 + 25 \beta_{2} ) q^{25} + ( 108 + 2 \beta_{3} ) q^{26} -27 q^{27} + ( -8 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{28} + ( -155 + 3 \beta_{1} + 152 \beta_{2} + 3 \beta_{3} ) q^{29} -30 q^{30} + ( 187 + 2 \beta_{3} ) q^{31} + ( 32 - 32 \beta_{2} ) q^{32} + 9 \beta_{1} q^{33} + ( -4 + 20 \beta_{1} - 16 \beta_{2} + 20 \beta_{3} ) q^{34} + ( 5 \beta_{1} + 5 \beta_{2} ) q^{35} -36 \beta_{2} q^{36} + ( -130 - 3 \beta_{3} ) q^{37} + ( 6 + 14 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 162 + 3 \beta_{3} ) q^{39} -40 \beta_{2} q^{40} + ( -6 \beta_{1} - 208 \beta_{2} ) q^{41} + ( -12 + 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{42} + ( 5 \beta_{1} + 37 \beta_{2} ) q^{43} + ( -12 + 12 \beta_{1} + 12 \beta_{3} ) q^{44} -45 q^{45} + ( 56 + 4 \beta_{3} ) q^{46} + ( 180 + 30 \beta_{1} - 210 \beta_{2} + 30 \beta_{3} ) q^{47} + ( 48 - 48 \beta_{2} ) q^{48} + ( -219 - 3 \beta_{3} ) q^{49} -50 q^{50} + ( -6 + 30 \beta_{1} - 24 \beta_{2} + 30 \beta_{3} ) q^{51} + ( -4 \beta_{1} + 220 \beta_{2} ) q^{52} + ( -18 - 24 \beta_{1} + 42 \beta_{2} - 24 \beta_{3} ) q^{53} -54 \beta_{2} q^{54} + 15 \beta_{1} q^{55} + ( -16 + 8 \beta_{3} ) q^{56} + ( 9 + 21 \beta_{1} - 15 \beta_{2} - 3 \beta_{3} ) q^{57} + ( -310 + 6 \beta_{3} ) q^{58} + ( -9 \beta_{1} - 100 \beta_{2} ) q^{59} -60 \beta_{2} q^{60} + ( -461 - 22 \beta_{1} + 483 \beta_{2} - 22 \beta_{3} ) q^{61} + ( -4 \beta_{1} + 378 \beta_{2} ) q^{62} + ( -18 + 9 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{63} + 64 q^{64} + ( 270 + 5 \beta_{3} ) q^{65} + ( -18 + 18 \beta_{1} + 18 \beta_{3} ) q^{66} + ( 44 + \beta_{1} - 45 \beta_{2} + \beta_{3} ) q^{67} + ( -8 + 40 \beta_{3} ) q^{68} + ( 84 + 6 \beta_{3} ) q^{69} + ( -20 + 10 \beta_{1} + 10 \beta_{2} + 10 \beta_{3} ) q^{70} + ( 45 \beta_{1} + 246 \beta_{2} ) q^{71} + ( 72 - 72 \beta_{2} ) q^{72} + ( 43 \beta_{1} + 95 \beta_{2} ) q^{73} + ( 6 \beta_{1} - 266 \beta_{2} ) q^{74} -75 q^{75} + ( -8 + 32 \beta_{1} - 12 \beta_{2} + 28 \beta_{3} ) q^{76} + ( 366 - 6 \beta_{3} ) q^{77} + ( -6 \beta_{1} + 330 \beta_{2} ) q^{78} + ( -38 \beta_{1} - 921 \beta_{2} ) q^{79} + ( 80 - 80 \beta_{2} ) q^{80} -81 \beta_{2} q^{81} + ( 428 - 12 \beta_{1} - 416 \beta_{2} - 12 \beta_{3} ) q^{82} + ( -166 + 58 \beta_{3} ) q^{83} + ( -24 + 12 \beta_{3} ) q^{84} + ( -10 + 50 \beta_{1} - 40 \beta_{2} + 50 \beta_{3} ) q^{85} + ( -84 + 10 \beta_{1} + 74 \beta_{2} + 10 \beta_{3} ) q^{86} + ( -465 + 9 \beta_{3} ) q^{87} + ( -24 + 24 \beta_{3} ) q^{88} + ( -379 + 59 \beta_{1} + 320 \beta_{2} + 59 \beta_{3} ) q^{89} -90 \beta_{2} q^{90} + ( -12 - 53 \beta_{1} + 65 \beta_{2} - 53 \beta_{3} ) q^{91} + ( -8 \beta_{1} + 120 \beta_{2} ) q^{92} + ( -6 \beta_{1} + 567 \beta_{2} ) q^{93} + ( 360 + 60 \beta_{3} ) q^{94} + ( 15 + 35 \beta_{1} - 25 \beta_{2} - 5 \beta_{3} ) q^{95} + 96 q^{96} + ( 14 \beta_{1} - 218 \beta_{2} ) q^{97} + ( 6 \beta_{1} - 444 \beta_{2} ) q^{98} + ( -27 + 27 \beta_{1} + 27 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} - 12q^{6} + 6q^{7} - 32q^{8} - 18q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 6q^{3} - 8q^{4} + 10q^{5} - 12q^{6} + 6q^{7} - 32q^{8} - 18q^{9} - 20q^{10} + 6q^{11} - 48q^{12} + 109q^{13} + 6q^{14} - 30q^{15} - 32q^{16} - 6q^{17} - 72q^{18} - q^{19} - 80q^{20} + 9q^{21} + 6q^{22} + 58q^{23} - 48q^{24} - 50q^{25} + 436q^{26} - 108q^{27} - 12q^{28} - 307q^{29} - 120q^{30} + 752q^{31} + 64q^{32} + 9q^{33} + 12q^{34} + 15q^{35} - 72q^{36} - 526q^{37} + 14q^{38} + 654q^{39} - 80q^{40} - 422q^{41} - 18q^{42} + 79q^{43} - 12q^{44} - 180q^{45} + 232q^{46} + 390q^{47} + 96q^{48} - 882q^{49} - 200q^{50} + 18q^{51} + 436q^{52} - 60q^{53} - 108q^{54} + 15q^{55} - 48q^{56} + 21q^{57} - 1228q^{58} - 209q^{59} - 120q^{60} - 944q^{61} + 752q^{62} - 27q^{63} + 256q^{64} + 1090q^{65} - 18q^{66} + 89q^{67} + 48q^{68} + 348q^{69} - 30q^{70} + 537q^{71} + 144q^{72} + 233q^{73} - 526q^{74} - 300q^{75} + 32q^{76} + 1452q^{77} + 654q^{78} - 1880q^{79} + 160q^{80} - 162q^{81} + 844q^{82} - 548q^{83} - 72q^{84} + 30q^{85} - 158q^{86} - 1842q^{87} - 48q^{88} - 699q^{89} - 180q^{90} - 77q^{91} + 232q^{92} + 1128q^{93} + 1560q^{94} + 35q^{95} + 384q^{96} - 422q^{97} - 882q^{98} - 27q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 121 x^{2} + 120 x + 14400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 121 \nu^{2} - 121 \nu + 14400$$$$)/14520$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 241$$$$)/121$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 120 \beta_{2} + \beta_{1} - 121$$ $$\nu^{3}$$ $$=$$ $$121 \beta_{3} - 241$$

## Character values

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

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$ $$1$$

## 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
 −5.23293 − 9.06370i 5.73293 + 9.92972i −5.23293 + 9.06370i 5.73293 − 9.92972i
1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i −3.00000 + 5.19615i −9.46586 −8.00000 −4.50000 + 7.79423i −5.00000 + 8.66025i
121.2 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i −3.00000 + 5.19615i 12.4659 −8.00000 −4.50000 + 7.79423i −5.00000 + 8.66025i
391.1 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 2.50000 4.33013i −3.00000 5.19615i −9.46586 −8.00000 −4.50000 7.79423i −5.00000 8.66025i
391.2 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 2.50000 4.33013i −3.00000 5.19615i 12.4659 −8.00000 −4.50000 7.79423i −5.00000 8.66025i
 $$n$$: 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 570.4.i.i 4
19.c even 3 1 inner 570.4.i.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.4.i.i 4 1.a even 1 1 trivial
570.4.i.i 4 19.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 3 T_{7} - 118$$ acting on $$S_{4}^{\mathrm{new}}(570, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{2}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$( 25 - 5 T + T^{2} )^{2}$$
$7$ $$( -118 - 3 T + T^{2} )^{2}$$
$11$ $$( -1080 - 3 T + T^{2} )^{2}$$
$13$ $$8122500 - 310650 T + 9031 T^{2} - 109 T^{3} + T^{4}$$
$17$ $$144384256 - 72096 T + 12052 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$47045881 + 6859 T - 13338 T^{2} + T^{3} + T^{4}$$
$23$ $$129600 - 20880 T + 3004 T^{2} - 58 T^{3} + T^{4}$$
$29$ $$505350400 + 6901360 T + 71769 T^{2} + 307 T^{3} + T^{4}$$
$31$ $$( 34863 - 376 T + T^{2} )^{2}$$
$37$ $$( 16210 + 263 T + T^{2} )^{2}$$
$41$ $$1615396864 + 16961024 T + 137892 T^{2} + 422 T^{3} + T^{4}$$
$43$ $$2090916 + 114234 T + 7687 T^{2} - 79 T^{3} + T^{4}$$
$47$ $$4928040000 + 27378000 T + 222300 T^{2} - 390 T^{3} + T^{4}$$
$53$ $$4673636496 - 4101840 T + 71964 T^{2} + 60 T^{3} + T^{4}$$
$59$ $$1392400 + 246620 T + 42501 T^{2} + 209 T^{3} + T^{4}$$
$61$ $$27087563889 + 155366352 T + 726553 T^{2} + 944 T^{3} + T^{4}$$
$67$ $$3459600 - 165540 T + 6061 T^{2} - 89 T^{3} + T^{4}$$
$71$ $$29382759396 + 92049318 T + 459783 T^{2} - 537 T^{3} + T^{4}$$
$73$ $$43584912900 + 48643410 T + 263059 T^{2} - 233 T^{3} + T^{4}$$
$79$ $$504041781681 + 1334722920 T + 2824441 T^{2} + 1880 T^{3} + T^{4}$$
$83$ $$( -385752 + 274 T + T^{2} )^{2}$$
$89$ $$87876673600 - 207211560 T + 785041 T^{2} + 699 T^{3} + T^{4}$$
$97$ $$438986304 + 8841744 T + 157132 T^{2} + 422 T^{3} + T^{4}$$