# Properties

 Label 6160.2.a.r.1.1 Level $6160$ Weight $2$ Character 6160.1 Self dual yes Analytic conductor $49.188$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.1878476451$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 6160.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -4.74456 q^{13} -2.00000 q^{15} +4.74456 q^{17} -4.74456 q^{19} +2.00000 q^{21} -4.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} -2.74456 q^{29} +6.74456 q^{31} -2.00000 q^{33} -1.00000 q^{35} -10.7446 q^{37} +9.48913 q^{39} -4.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +6.74456 q^{47} +1.00000 q^{49} -9.48913 q^{51} -1.25544 q^{53} +1.00000 q^{55} +9.48913 q^{57} -2.74456 q^{59} +12.7446 q^{61} -1.00000 q^{63} -4.74456 q^{65} +4.00000 q^{67} +9.48913 q^{69} +4.00000 q^{71} -0.744563 q^{73} -2.00000 q^{75} -1.00000 q^{77} +4.74456 q^{79} -11.0000 q^{81} -8.00000 q^{83} +4.74456 q^{85} +5.48913 q^{87} -7.48913 q^{89} +4.74456 q^{91} -13.4891 q^{93} -4.74456 q^{95} -5.25544 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 4 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} - 2 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 8 q^{27} + 6 q^{29} + 2 q^{31} - 4 q^{33} - 2 q^{35} - 10 q^{37} - 4 q^{39} - 8 q^{41} - 8 q^{43} + 2 q^{45} + 2 q^{47} + 2 q^{49} + 4 q^{51} - 14 q^{53} + 2 q^{55} - 4 q^{57} + 6 q^{59} + 14 q^{61} - 2 q^{63} + 2 q^{65} + 8 q^{67} - 4 q^{69} + 8 q^{71} + 10 q^{73} - 4 q^{75} - 2 q^{77} - 2 q^{79} - 22 q^{81} - 16 q^{83} - 2 q^{85} - 12 q^{87} + 8 q^{89} - 2 q^{91} - 4 q^{93} + 2 q^{95} - 22 q^{97} + 2 q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −4.74456 −1.31590 −0.657952 0.753059i $$-0.728577\pi$$
−0.657952 + 0.753059i $$0.728577\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ 4.74456 1.15073 0.575363 0.817898i $$-0.304861\pi$$
0.575363 + 0.817898i $$0.304861\pi$$
$$18$$ 0 0
$$19$$ −4.74456 −1.08848 −0.544239 0.838930i $$-0.683181\pi$$
−0.544239 + 0.838930i $$0.683181\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −4.74456 −0.989310 −0.494655 0.869090i $$-0.664706\pi$$
−0.494655 + 0.869090i $$0.664706\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ −2.74456 −0.509652 −0.254826 0.966987i $$-0.582018\pi$$
−0.254826 + 0.966987i $$0.582018\pi$$
$$30$$ 0 0
$$31$$ 6.74456 1.21136 0.605680 0.795709i $$-0.292901\pi$$
0.605680 + 0.795709i $$0.292901\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ −10.7446 −1.76640 −0.883198 0.469001i $$-0.844614\pi$$
−0.883198 + 0.469001i $$0.844614\pi$$
$$38$$ 0 0
$$39$$ 9.48913 1.51948
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 6.74456 0.983796 0.491898 0.870653i $$-0.336303\pi$$
0.491898 + 0.870653i $$0.336303\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −9.48913 −1.32874
$$52$$ 0 0
$$53$$ −1.25544 −0.172448 −0.0862238 0.996276i $$-0.527480\pi$$
−0.0862238 + 0.996276i $$0.527480\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ 9.48913 1.25687
$$58$$ 0 0
$$59$$ −2.74456 −0.357312 −0.178656 0.983912i $$-0.557175\pi$$
−0.178656 + 0.983912i $$0.557175\pi$$
$$60$$ 0 0
$$61$$ 12.7446 1.63177 0.815887 0.578211i $$-0.196249\pi$$
0.815887 + 0.578211i $$0.196249\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −4.74456 −0.588491
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 9.48913 1.14236
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 0 0
$$73$$ −0.744563 −0.0871445 −0.0435722 0.999050i $$-0.513874\pi$$
−0.0435722 + 0.999050i $$0.513874\pi$$
$$74$$ 0 0
$$75$$ −2.00000 −0.230940
$$76$$ 0 0
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ 4.74456 0.533805 0.266903 0.963724i $$-0.414000\pi$$
0.266903 + 0.963724i $$0.414000\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ 4.74456 0.514620
$$86$$ 0 0
$$87$$ 5.48913 0.588496
$$88$$ 0 0
$$89$$ −7.48913 −0.793846 −0.396923 0.917852i $$-0.629922\pi$$
−0.396923 + 0.917852i $$0.629922\pi$$
$$90$$ 0 0
$$91$$ 4.74456 0.497365
$$92$$ 0 0
$$93$$ −13.4891 −1.39876
$$94$$ 0 0
$$95$$ −4.74456 −0.486782
$$96$$ 0 0
$$97$$ −5.25544 −0.533609 −0.266804 0.963751i $$-0.585968\pi$$
−0.266804 + 0.963751i $$0.585968\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −8.74456 −0.870117 −0.435058 0.900402i $$-0.643272\pi$$
−0.435058 + 0.900402i $$0.643272\pi$$
$$102$$ 0 0
$$103$$ −10.7446 −1.05869 −0.529347 0.848406i $$-0.677563\pi$$
−0.529347 + 0.848406i $$0.677563\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 16.2337 1.55491 0.777453 0.628941i $$-0.216511\pi$$
0.777453 + 0.628941i $$0.216511\pi$$
$$110$$ 0 0
$$111$$ 21.4891 2.03966
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 0 0
$$115$$ −4.74456 −0.442433
$$116$$ 0 0
$$117$$ −4.74456 −0.438635
$$118$$ 0 0
$$119$$ −4.74456 −0.434933
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 8.00000 0.721336
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −8.74456 −0.764016 −0.382008 0.924159i $$-0.624767\pi$$
−0.382008 + 0.924159i $$0.624767\pi$$
$$132$$ 0 0
$$133$$ 4.74456 0.411406
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ −19.4891 −1.66507 −0.832534 0.553974i $$-0.813111\pi$$
−0.832534 + 0.553974i $$0.813111\pi$$
$$138$$ 0 0
$$139$$ −3.25544 −0.276123 −0.138061 0.990424i $$-0.544087\pi$$
−0.138061 + 0.990424i $$0.544087\pi$$
$$140$$ 0 0
$$141$$ −13.4891 −1.13599
$$142$$ 0 0
$$143$$ −4.74456 −0.396760
$$144$$ 0 0
$$145$$ −2.74456 −0.227924
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ 10.7446 0.880229 0.440114 0.897942i $$-0.354938\pi$$
0.440114 + 0.897942i $$0.354938\pi$$
$$150$$ 0 0
$$151$$ 20.7446 1.68817 0.844084 0.536211i $$-0.180145\pi$$
0.844084 + 0.536211i $$0.180145\pi$$
$$152$$ 0 0
$$153$$ 4.74456 0.383575
$$154$$ 0 0
$$155$$ 6.74456 0.541736
$$156$$ 0 0
$$157$$ 23.4891 1.87464 0.937318 0.348475i $$-0.113300\pi$$
0.937318 + 0.348475i $$0.113300\pi$$
$$158$$ 0 0
$$159$$ 2.51087 0.199125
$$160$$ 0 0
$$161$$ 4.74456 0.373924
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ −2.00000 −0.155700
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 9.51087 0.731606
$$170$$ 0 0
$$171$$ −4.74456 −0.362826
$$172$$ 0 0
$$173$$ −14.2337 −1.08217 −0.541084 0.840969i $$-0.681986\pi$$
−0.541084 + 0.840969i $$0.681986\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 5.48913 0.412588
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 3.48913 0.259345 0.129672 0.991557i $$-0.458607\pi$$
0.129672 + 0.991557i $$0.458607\pi$$
$$182$$ 0 0
$$183$$ −25.4891 −1.88421
$$184$$ 0 0
$$185$$ −10.7446 −0.789956
$$186$$ 0 0
$$187$$ 4.74456 0.346957
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 9.48913 0.679530
$$196$$ 0 0
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 0 0
$$199$$ −14.7446 −1.04521 −0.522607 0.852574i $$-0.675041\pi$$
−0.522607 + 0.852574i $$0.675041\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ 2.74456 0.192631
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ 0 0
$$207$$ −4.74456 −0.329770
$$208$$ 0 0
$$209$$ −4.74456 −0.328188
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ −8.00000 −0.548151
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ −6.74456 −0.457851
$$218$$ 0 0
$$219$$ 1.48913 0.100626
$$220$$ 0 0
$$221$$ −22.5109 −1.51425
$$222$$ 0 0
$$223$$ −26.7446 −1.79095 −0.895474 0.445113i $$-0.853163\pi$$
−0.895474 + 0.445113i $$0.853163\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 2.00000 0.131590
$$232$$ 0 0
$$233$$ 20.9783 1.37433 0.687165 0.726501i $$-0.258855\pi$$
0.687165 + 0.726501i $$0.258855\pi$$
$$234$$ 0 0
$$235$$ 6.74456 0.439967
$$236$$ 0 0
$$237$$ −9.48913 −0.616385
$$238$$ 0 0
$$239$$ 3.25544 0.210577 0.105288 0.994442i $$-0.466423\pi$$
0.105288 + 0.994442i $$0.466423\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 22.5109 1.43233
$$248$$ 0 0
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ −8.23369 −0.519706 −0.259853 0.965648i $$-0.583674\pi$$
−0.259853 + 0.965648i $$0.583674\pi$$
$$252$$ 0 0
$$253$$ −4.74456 −0.298288
$$254$$ 0 0
$$255$$ −9.48913 −0.594232
$$256$$ 0 0
$$257$$ 24.2337 1.51166 0.755828 0.654770i $$-0.227235\pi$$
0.755828 + 0.654770i $$0.227235\pi$$
$$258$$ 0 0
$$259$$ 10.7446 0.667635
$$260$$ 0 0
$$261$$ −2.74456 −0.169884
$$262$$ 0 0
$$263$$ 18.9783 1.17025 0.585125 0.810943i $$-0.301046\pi$$
0.585125 + 0.810943i $$0.301046\pi$$
$$264$$ 0 0
$$265$$ −1.25544 −0.0771209
$$266$$ 0 0
$$267$$ 14.9783 0.916654
$$268$$ 0 0
$$269$$ −24.9783 −1.52295 −0.761475 0.648194i $$-0.775525\pi$$
−0.761475 + 0.648194i $$0.775525\pi$$
$$270$$ 0 0
$$271$$ 30.9783 1.88179 0.940897 0.338692i $$-0.109984\pi$$
0.940897 + 0.338692i $$0.109984\pi$$
$$272$$ 0 0
$$273$$ −9.48913 −0.574308
$$274$$ 0 0
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −7.48913 −0.449978 −0.224989 0.974361i $$-0.572235\pi$$
−0.224989 + 0.974361i $$0.572235\pi$$
$$278$$ 0 0
$$279$$ 6.74456 0.403786
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ 9.48913 0.562087
$$286$$ 0 0
$$287$$ 4.00000 0.236113
$$288$$ 0 0
$$289$$ 5.51087 0.324169
$$290$$ 0 0
$$291$$ 10.5109 0.616158
$$292$$ 0 0
$$293$$ 24.7446 1.44559 0.722796 0.691061i $$-0.242856\pi$$
0.722796 + 0.691061i $$0.242856\pi$$
$$294$$ 0 0
$$295$$ −2.74456 −0.159795
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ 22.5109 1.30184
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 17.4891 1.00472
$$304$$ 0 0
$$305$$ 12.7446 0.729752
$$306$$ 0 0
$$307$$ −21.4891 −1.22645 −0.613225 0.789909i $$-0.710128\pi$$
−0.613225 + 0.789909i $$0.710128\pi$$
$$308$$ 0 0
$$309$$ 21.4891 1.22247
$$310$$ 0 0
$$311$$ 9.25544 0.524828 0.262414 0.964955i $$-0.415481\pi$$
0.262414 + 0.964955i $$0.415481\pi$$
$$312$$ 0 0
$$313$$ 32.2337 1.82196 0.910978 0.412455i $$-0.135329\pi$$
0.910978 + 0.412455i $$0.135329\pi$$
$$314$$ 0 0
$$315$$ −1.00000 −0.0563436
$$316$$ 0 0
$$317$$ −32.2337 −1.81042 −0.905212 0.424960i $$-0.860288\pi$$
−0.905212 + 0.424960i $$0.860288\pi$$
$$318$$ 0 0
$$319$$ −2.74456 −0.153666
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ −22.5109 −1.25254
$$324$$ 0 0
$$325$$ −4.74456 −0.263181
$$326$$ 0 0
$$327$$ −32.4674 −1.79545
$$328$$ 0 0
$$329$$ −6.74456 −0.371840
$$330$$ 0 0
$$331$$ 30.9783 1.70272 0.851359 0.524583i $$-0.175779\pi$$
0.851359 + 0.524583i $$0.175779\pi$$
$$332$$ 0 0
$$333$$ −10.7446 −0.588798
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 0 0
$$339$$ −20.0000 −1.08625
$$340$$ 0 0
$$341$$ 6.74456 0.365239
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 9.48913 0.510877
$$346$$ 0 0
$$347$$ −22.9783 −1.23354 −0.616769 0.787145i $$-0.711559\pi$$
−0.616769 + 0.787145i $$0.711559\pi$$
$$348$$ 0 0
$$349$$ 19.2554 1.03072 0.515360 0.856974i $$-0.327658\pi$$
0.515360 + 0.856974i $$0.327658\pi$$
$$350$$ 0 0
$$351$$ −18.9783 −1.01298
$$352$$ 0 0
$$353$$ −2.74456 −0.146078 −0.0730392 0.997329i $$-0.523270\pi$$
−0.0730392 + 0.997329i $$0.523270\pi$$
$$354$$ 0 0
$$355$$ 4.00000 0.212298
$$356$$ 0 0
$$357$$ 9.48913 0.502218
$$358$$ 0 0
$$359$$ 12.7446 0.672632 0.336316 0.941749i $$-0.390819\pi$$
0.336316 + 0.941749i $$0.390819\pi$$
$$360$$ 0 0
$$361$$ 3.51087 0.184783
$$362$$ 0 0
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ −0.744563 −0.0389722
$$366$$ 0 0
$$367$$ −2.74456 −0.143265 −0.0716325 0.997431i $$-0.522821\pi$$
−0.0716325 + 0.997431i $$0.522821\pi$$
$$368$$ 0 0
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 1.25544 0.0651791
$$372$$ 0 0
$$373$$ 28.9783 1.50044 0.750218 0.661190i $$-0.229948\pi$$
0.750218 + 0.661190i $$0.229948\pi$$
$$374$$ 0 0
$$375$$ −2.00000 −0.103280
$$376$$ 0 0
$$377$$ 13.0217 0.670654
$$378$$ 0 0
$$379$$ 14.5109 0.745374 0.372687 0.927957i $$-0.378437\pi$$
0.372687 + 0.927957i $$0.378437\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 37.7228 1.92755 0.963773 0.266724i $$-0.0859413\pi$$
0.963773 + 0.266724i $$0.0859413\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 15.4891 0.785330 0.392665 0.919682i $$-0.371553\pi$$
0.392665 + 0.919682i $$0.371553\pi$$
$$390$$ 0 0
$$391$$ −22.5109 −1.13842
$$392$$ 0 0
$$393$$ 17.4891 0.882210
$$394$$ 0 0
$$395$$ 4.74456 0.238725
$$396$$ 0 0
$$397$$ 12.5109 0.627903 0.313951 0.949439i $$-0.398347\pi$$
0.313951 + 0.949439i $$0.398347\pi$$
$$398$$ 0 0
$$399$$ −9.48913 −0.475050
$$400$$ 0 0
$$401$$ 0.510875 0.0255119 0.0127559 0.999919i $$-0.495940\pi$$
0.0127559 + 0.999919i $$0.495940\pi$$
$$402$$ 0 0
$$403$$ −32.0000 −1.59403
$$404$$ 0 0
$$405$$ −11.0000 −0.546594
$$406$$ 0 0
$$407$$ −10.7446 −0.532588
$$408$$ 0 0
$$409$$ 1.48913 0.0736325 0.0368163 0.999322i $$-0.488278\pi$$
0.0368163 + 0.999322i $$0.488278\pi$$
$$410$$ 0 0
$$411$$ 38.9783 1.92266
$$412$$ 0 0
$$413$$ 2.74456 0.135051
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 0 0
$$417$$ 6.51087 0.318839
$$418$$ 0 0
$$419$$ −10.7446 −0.524906 −0.262453 0.964945i $$-0.584532\pi$$
−0.262453 + 0.964945i $$0.584532\pi$$
$$420$$ 0 0
$$421$$ 27.4891 1.33974 0.669869 0.742479i $$-0.266350\pi$$
0.669869 + 0.742479i $$0.266350\pi$$
$$422$$ 0 0
$$423$$ 6.74456 0.327932
$$424$$ 0 0
$$425$$ 4.74456 0.230145
$$426$$ 0 0
$$427$$ −12.7446 −0.616753
$$428$$ 0 0
$$429$$ 9.48913 0.458139
$$430$$ 0 0
$$431$$ 28.7446 1.38458 0.692288 0.721621i $$-0.256603\pi$$
0.692288 + 0.721621i $$0.256603\pi$$
$$432$$ 0 0
$$433$$ 25.2554 1.21370 0.606849 0.794817i $$-0.292433\pi$$
0.606849 + 0.794817i $$0.292433\pi$$
$$434$$ 0 0
$$435$$ 5.48913 0.263183
$$436$$ 0 0
$$437$$ 22.5109 1.07684
$$438$$ 0 0
$$439$$ 30.9783 1.47851 0.739256 0.673425i $$-0.235178\pi$$
0.739256 + 0.673425i $$0.235178\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 6.51087 0.309341 0.154670 0.987966i $$-0.450568\pi$$
0.154670 + 0.987966i $$0.450568\pi$$
$$444$$ 0 0
$$445$$ −7.48913 −0.355019
$$446$$ 0 0
$$447$$ −21.4891 −1.01640
$$448$$ 0 0
$$449$$ 40.9783 1.93388 0.966942 0.254998i $$-0.0820748\pi$$
0.966942 + 0.254998i $$0.0820748\pi$$
$$450$$ 0 0
$$451$$ −4.00000 −0.188353
$$452$$ 0 0
$$453$$ −41.4891 −1.94933
$$454$$ 0 0
$$455$$ 4.74456 0.222429
$$456$$ 0 0
$$457$$ −19.4891 −0.911663 −0.455831 0.890066i $$-0.650658\pi$$
−0.455831 + 0.890066i $$0.650658\pi$$
$$458$$ 0 0
$$459$$ 18.9783 0.885829
$$460$$ 0 0
$$461$$ −23.7228 −1.10488 −0.552441 0.833552i $$-0.686303\pi$$
−0.552441 + 0.833552i $$0.686303\pi$$
$$462$$ 0 0
$$463$$ −12.7446 −0.592290 −0.296145 0.955143i $$-0.595701\pi$$
−0.296145 + 0.955143i $$0.595701\pi$$
$$464$$ 0 0
$$465$$ −13.4891 −0.625543
$$466$$ 0 0
$$467$$ 28.9783 1.34095 0.670477 0.741931i $$-0.266090\pi$$
0.670477 + 0.741931i $$0.266090\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −46.9783 −2.16464
$$472$$ 0 0
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ −4.74456 −0.217695
$$476$$ 0 0
$$477$$ −1.25544 −0.0574825
$$478$$ 0 0
$$479$$ 18.5109 0.845783 0.422892 0.906180i $$-0.361015\pi$$
0.422892 + 0.906180i $$0.361015\pi$$
$$480$$ 0 0
$$481$$ 50.9783 2.32441
$$482$$ 0 0
$$483$$ −9.48913 −0.431770
$$484$$ 0 0
$$485$$ −5.25544 −0.238637
$$486$$ 0 0
$$487$$ −20.7446 −0.940026 −0.470013 0.882660i $$-0.655751\pi$$
−0.470013 + 0.882660i $$0.655751\pi$$
$$488$$ 0 0
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ 14.9783 0.675959 0.337979 0.941153i $$-0.390257\pi$$
0.337979 + 0.941153i $$0.390257\pi$$
$$492$$ 0 0
$$493$$ −13.0217 −0.586470
$$494$$ 0 0
$$495$$ 1.00000 0.0449467
$$496$$ 0 0
$$497$$ −4.00000 −0.179425
$$498$$ 0 0
$$499$$ −9.48913 −0.424792 −0.212396 0.977184i $$-0.568127\pi$$
−0.212396 + 0.977184i $$0.568127\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −29.4891 −1.31486 −0.657428 0.753518i $$-0.728355\pi$$
−0.657428 + 0.753518i $$0.728355\pi$$
$$504$$ 0 0
$$505$$ −8.74456 −0.389128
$$506$$ 0 0
$$507$$ −19.0217 −0.844786
$$508$$ 0 0
$$509$$ −12.5109 −0.554535 −0.277267 0.960793i $$-0.589429\pi$$
−0.277267 + 0.960793i $$0.589429\pi$$
$$510$$ 0 0
$$511$$ 0.744563 0.0329375
$$512$$ 0 0
$$513$$ −18.9783 −0.837910
$$514$$ 0 0
$$515$$ −10.7446 −0.473462
$$516$$ 0 0
$$517$$ 6.74456 0.296626
$$518$$ 0 0
$$519$$ 28.4674 1.24958
$$520$$ 0 0
$$521$$ −2.00000 −0.0876216 −0.0438108 0.999040i $$-0.513950\pi$$
−0.0438108 + 0.999040i $$0.513950\pi$$
$$522$$ 0 0
$$523$$ −5.48913 −0.240023 −0.120011 0.992773i $$-0.538293\pi$$
−0.120011 + 0.992773i $$0.538293\pi$$
$$524$$ 0 0
$$525$$ 2.00000 0.0872872
$$526$$ 0 0
$$527$$ 32.0000 1.39394
$$528$$ 0 0
$$529$$ −0.489125 −0.0212663
$$530$$ 0 0
$$531$$ −2.74456 −0.119104
$$532$$ 0 0
$$533$$ 18.9783 0.822039
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 0 0
$$537$$ −8.00000 −0.345225
$$538$$ 0 0
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ 36.2337 1.55781 0.778904 0.627143i $$-0.215776\pi$$
0.778904 + 0.627143i $$0.215776\pi$$
$$542$$ 0 0
$$543$$ −6.97825 −0.299465
$$544$$ 0 0
$$545$$ 16.2337 0.695375
$$546$$ 0 0
$$547$$ 30.9783 1.32453 0.662267 0.749268i $$-0.269594\pi$$
0.662267 + 0.749268i $$0.269594\pi$$
$$548$$ 0 0
$$549$$ 12.7446 0.543925
$$550$$ 0 0
$$551$$ 13.0217 0.554745
$$552$$ 0 0
$$553$$ −4.74456 −0.201759
$$554$$ 0 0
$$555$$ 21.4891 0.912163
$$556$$ 0 0
$$557$$ 44.9783 1.90579 0.952895 0.303301i $$-0.0980887\pi$$
0.952895 + 0.303301i $$0.0980887\pi$$
$$558$$ 0 0
$$559$$ 18.9783 0.802694
$$560$$ 0 0
$$561$$ −9.48913 −0.400631
$$562$$ 0 0
$$563$$ 17.4891 0.737079 0.368539 0.929612i $$-0.379858\pi$$
0.368539 + 0.929612i $$0.379858\pi$$
$$564$$ 0 0
$$565$$ 10.0000 0.420703
$$566$$ 0 0
$$567$$ 11.0000 0.461957
$$568$$ 0 0
$$569$$ 39.4891 1.65547 0.827735 0.561119i $$-0.189629\pi$$
0.827735 + 0.561119i $$0.189629\pi$$
$$570$$ 0 0
$$571$$ −5.48913 −0.229713 −0.114856 0.993382i $$-0.536641\pi$$
−0.114856 + 0.993382i $$0.536641\pi$$
$$572$$ 0 0
$$573$$ −32.0000 −1.33682
$$574$$ 0 0
$$575$$ −4.74456 −0.197862
$$576$$ 0 0
$$577$$ 2.74456 0.114258 0.0571288 0.998367i $$-0.481805\pi$$
0.0571288 + 0.998367i $$0.481805\pi$$
$$578$$ 0 0
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ 8.00000 0.331896
$$582$$ 0 0
$$583$$ −1.25544 −0.0519949
$$584$$ 0 0
$$585$$ −4.74456 −0.196164
$$586$$ 0 0
$$587$$ −40.9783 −1.69135 −0.845677 0.533696i $$-0.820803\pi$$
−0.845677 + 0.533696i $$0.820803\pi$$
$$588$$ 0 0
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ −20.0000 −0.822690
$$592$$ 0 0
$$593$$ −5.76631 −0.236794 −0.118397 0.992966i $$-0.537776\pi$$
−0.118397 + 0.992966i $$0.537776\pi$$
$$594$$ 0 0
$$595$$ −4.74456 −0.194508
$$596$$ 0 0
$$597$$ 29.4891 1.20691
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −10.5109 −0.428748 −0.214374 0.976752i $$-0.568771\pi$$
−0.214374 + 0.976752i $$0.568771\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ −5.48913 −0.222431
$$610$$ 0 0
$$611$$ −32.0000 −1.29458
$$612$$ 0 0
$$613$$ −11.4891 −0.464041 −0.232021 0.972711i $$-0.574534\pi$$
−0.232021 + 0.972711i $$0.574534\pi$$
$$614$$ 0 0
$$615$$ 8.00000 0.322591
$$616$$ 0 0
$$617$$ −10.0000 −0.402585 −0.201292 0.979531i $$-0.564514\pi$$
−0.201292 + 0.979531i $$0.564514\pi$$
$$618$$ 0 0
$$619$$ 10.7446 0.431860 0.215930 0.976409i $$-0.430722\pi$$
0.215930 + 0.976409i $$0.430722\pi$$
$$620$$ 0 0
$$621$$ −18.9783 −0.761571
$$622$$ 0 0
$$623$$ 7.48913 0.300045
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 9.48913 0.378959
$$628$$ 0 0
$$629$$ −50.9783 −2.03264
$$630$$ 0 0
$$631$$ −42.9783 −1.71094 −0.855469 0.517855i $$-0.826731\pi$$
−0.855469 + 0.517855i $$0.826731\pi$$
$$632$$ 0 0
$$633$$ −24.0000 −0.953914
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4.74456 −0.187986
$$638$$ 0 0
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ −11.4891 −0.453793 −0.226897 0.973919i $$-0.572858\pi$$
−0.226897 + 0.973919i $$0.572858\pi$$
$$642$$ 0 0
$$643$$ 22.4674 0.886027 0.443013 0.896515i $$-0.353909\pi$$
0.443013 + 0.896515i $$0.353909\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 0 0
$$647$$ 2.74456 0.107900 0.0539499 0.998544i $$-0.482819\pi$$
0.0539499 + 0.998544i $$0.482819\pi$$
$$648$$ 0 0
$$649$$ −2.74456 −0.107734
$$650$$ 0 0
$$651$$ 13.4891 0.528681
$$652$$ 0 0
$$653$$ −41.7228 −1.63274 −0.816370 0.577529i $$-0.804017\pi$$
−0.816370 + 0.577529i $$0.804017\pi$$
$$654$$ 0 0
$$655$$ −8.74456 −0.341678
$$656$$ 0 0
$$657$$ −0.744563 −0.0290482
$$658$$ 0 0
$$659$$ 18.5109 0.721081 0.360541 0.932744i $$-0.382592\pi$$
0.360541 + 0.932744i $$0.382592\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 0 0
$$663$$ 45.0217 1.74850
$$664$$ 0 0
$$665$$ 4.74456 0.183986
$$666$$ 0 0
$$667$$ 13.0217 0.504204
$$668$$ 0 0
$$669$$ 53.4891 2.06801
$$670$$ 0 0
$$671$$ 12.7446 0.491998
$$672$$ 0 0
$$673$$ −24.9783 −0.962841 −0.481420 0.876490i $$-0.659879\pi$$
−0.481420 + 0.876490i $$0.659879\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ 1.76631 0.0678849 0.0339424 0.999424i $$-0.489194\pi$$
0.0339424 + 0.999424i $$0.489194\pi$$
$$678$$ 0 0
$$679$$ 5.25544 0.201685
$$680$$ 0 0
$$681$$ 40.0000 1.53280
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ −19.4891 −0.744641
$$686$$ 0 0
$$687$$ 12.0000 0.457829
$$688$$ 0 0
$$689$$ 5.95650 0.226925
$$690$$ 0 0
$$691$$ −36.2337 −1.37839 −0.689197 0.724574i $$-0.742037\pi$$
−0.689197 + 0.724574i $$0.742037\pi$$
$$692$$ 0 0
$$693$$ −1.00000 −0.0379869
$$694$$ 0 0
$$695$$ −3.25544 −0.123486
$$696$$ 0 0
$$697$$ −18.9783 −0.718853
$$698$$ 0 0
$$699$$ −41.9565 −1.58694
$$700$$ 0 0
$$701$$ −12.2337 −0.462060 −0.231030 0.972947i $$-0.574210\pi$$
−0.231030 + 0.972947i $$0.574210\pi$$
$$702$$ 0 0
$$703$$ 50.9783 1.92268
$$704$$ 0 0
$$705$$ −13.4891 −0.508030
$$706$$ 0 0
$$707$$ 8.74456 0.328873
$$708$$ 0 0
$$709$$ 23.4891 0.882153 0.441076 0.897470i $$-0.354597\pi$$
0.441076 + 0.897470i $$0.354597\pi$$
$$710$$ 0 0
$$711$$ 4.74456 0.177935
$$712$$ 0 0
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ −4.74456 −0.177437
$$716$$ 0 0
$$717$$ −6.51087 −0.243153
$$718$$ 0 0
$$719$$ 49.7228 1.85435 0.927174 0.374631i $$-0.122231\pi$$
0.927174 + 0.374631i $$0.122231\pi$$
$$720$$ 0 0
$$721$$ 10.7446 0.400148
$$722$$ 0 0
$$723$$ −40.0000 −1.48762
$$724$$ 0 0
$$725$$ −2.74456 −0.101930
$$726$$ 0 0
$$727$$ 20.2337 0.750426 0.375213 0.926939i $$-0.377570\pi$$
0.375213 + 0.926939i $$0.377570\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −18.9783 −0.701936
$$732$$ 0 0
$$733$$ 18.2337 0.673477 0.336738 0.941598i $$-0.390676\pi$$
0.336738 + 0.941598i $$0.390676\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ 4.00000 0.147342
$$738$$ 0 0
$$739$$ −14.9783 −0.550984 −0.275492 0.961303i $$-0.588841\pi$$
−0.275492 + 0.961303i $$0.588841\pi$$
$$740$$ 0 0
$$741$$ −45.0217 −1.65392
$$742$$ 0 0
$$743$$ 18.9783 0.696244 0.348122 0.937449i $$-0.386819\pi$$
0.348122 + 0.937449i $$0.386819\pi$$
$$744$$ 0 0
$$745$$ 10.7446 0.393650
$$746$$ 0 0
$$747$$ −8.00000 −0.292705
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 0 0
$$753$$ 16.4674 0.600105
$$754$$ 0 0
$$755$$ 20.7446 0.754972
$$756$$ 0 0
$$757$$ −30.7446 −1.11743 −0.558715 0.829360i $$-0.688705\pi$$
−0.558715 + 0.829360i $$0.688705\pi$$
$$758$$ 0 0
$$759$$ 9.48913 0.344433
$$760$$ 0 0
$$761$$ −6.51087 −0.236019 −0.118010 0.993012i $$-0.537651\pi$$
−0.118010 + 0.993012i $$0.537651\pi$$
$$762$$ 0 0
$$763$$ −16.2337 −0.587699
$$764$$ 0 0
$$765$$ 4.74456 0.171540
$$766$$ 0 0
$$767$$ 13.0217 0.470188
$$768$$ 0 0
$$769$$ −8.00000 −0.288487 −0.144244 0.989542i $$-0.546075\pi$$
−0.144244 + 0.989542i $$0.546075\pi$$
$$770$$ 0 0
$$771$$ −48.4674 −1.74551
$$772$$ 0 0
$$773$$ −28.5109 −1.02546 −0.512732 0.858548i $$-0.671367\pi$$
−0.512732 + 0.858548i $$0.671367\pi$$
$$774$$ 0 0
$$775$$ 6.74456 0.242272
$$776$$ 0 0
$$777$$ −21.4891 −0.770918
$$778$$ 0 0
$$779$$ 18.9783 0.679966
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 0 0
$$783$$ −10.9783 −0.392331
$$784$$ 0 0
$$785$$ 23.4891 0.838363
$$786$$ 0 0
$$787$$ 17.4891 0.623420 0.311710 0.950177i $$-0.399098\pi$$
0.311710 + 0.950177i $$0.399098\pi$$
$$788$$ 0 0
$$789$$ −37.9565 −1.35129
$$790$$ 0 0
$$791$$ −10.0000 −0.355559
$$792$$ 0 0
$$793$$ −60.4674 −2.14726
$$794$$ 0 0
$$795$$ 2.51087 0.0890515
$$796$$ 0 0
$$797$$ 26.4674 0.937523 0.468761 0.883325i $$-0.344700\pi$$
0.468761 + 0.883325i $$0.344700\pi$$
$$798$$ 0 0
$$799$$ 32.0000 1.13208
$$800$$ 0 0
$$801$$ −7.48913 −0.264615
$$802$$ 0 0
$$803$$ −0.744563 −0.0262750
$$804$$ 0 0
$$805$$ 4.74456 0.167224
$$806$$ 0 0
$$807$$ 49.9565 1.75855
$$808$$ 0 0
$$809$$ −34.4674 −1.21181 −0.605904 0.795538i $$-0.707189\pi$$
−0.605904 + 0.795538i $$0.707189\pi$$
$$810$$ 0 0
$$811$$ 7.25544 0.254773 0.127386 0.991853i $$-0.459341\pi$$
0.127386 + 0.991853i $$0.459341\pi$$
$$812$$ 0 0
$$813$$ −61.9565 −2.17291
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 18.9783 0.663965
$$818$$ 0 0
$$819$$ 4.74456 0.165788
$$820$$ 0 0
$$821$$ −49.7228 −1.73534 −0.867669 0.497142i $$-0.834383\pi$$
−0.867669 + 0.497142i $$0.834383\pi$$
$$822$$ 0 0
$$823$$ −42.2337 −1.47217 −0.736087 0.676887i $$-0.763329\pi$$
−0.736087 + 0.676887i $$0.763329\pi$$
$$824$$ 0 0
$$825$$ −2.00000 −0.0696311
$$826$$ 0 0
$$827$$ −4.00000 −0.139094 −0.0695468 0.997579i $$-0.522155\pi$$
−0.0695468 + 0.997579i $$0.522155\pi$$
$$828$$ 0 0
$$829$$ −54.4674 −1.89173 −0.945865 0.324560i $$-0.894784\pi$$
−0.945865 + 0.324560i $$0.894784\pi$$
$$830$$ 0 0
$$831$$ 14.9783 0.519590
$$832$$ 0 0
$$833$$ 4.74456 0.164389
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 26.9783 0.932505
$$838$$ 0 0
$$839$$ 14.7446 0.509039 0.254519 0.967068i $$-0.418083\pi$$
0.254519 + 0.967068i $$0.418083\pi$$
$$840$$ 0 0
$$841$$ −21.4674 −0.740254
$$842$$ 0 0
$$843$$ −28.0000 −0.964371
$$844$$ 0 0
$$845$$ 9.51087 0.327184
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 0 0
$$849$$ 56.0000 1.92192
$$850$$ 0 0
$$851$$ 50.9783 1.74751
$$852$$ 0 0
$$853$$ −11.2554 −0.385379 −0.192689 0.981260i $$-0.561721\pi$$
−0.192689 + 0.981260i $$0.561721\pi$$
$$854$$ 0 0
$$855$$ −4.74456 −0.162261
$$856$$ 0 0
$$857$$ 37.2119 1.27114 0.635568 0.772045i $$-0.280766\pi$$
0.635568 + 0.772045i $$0.280766\pi$$
$$858$$ 0 0
$$859$$ −51.2119 −1.74733 −0.873664 0.486529i $$-0.838263\pi$$
−0.873664 + 0.486529i $$0.838263\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ 5.76631 0.196288 0.0981438 0.995172i $$-0.468709\pi$$
0.0981438 + 0.995172i $$0.468709\pi$$
$$864$$ 0 0
$$865$$ −14.2337 −0.483960
$$866$$ 0 0
$$867$$ −11.0217 −0.374318
$$868$$ 0 0
$$869$$ 4.74456 0.160948
$$870$$ 0 0
$$871$$ −18.9783 −0.643053
$$872$$ 0 0
$$873$$ −5.25544 −0.177870
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ 42.4674 1.43402 0.717011 0.697062i $$-0.245510\pi$$
0.717011 + 0.697062i $$0.245510\pi$$
$$878$$ 0 0
$$879$$ −49.4891 −1.66923
$$880$$ 0 0
$$881$$ −32.5109 −1.09532 −0.547660 0.836701i $$-0.684481\pi$$
−0.547660 + 0.836701i $$0.684481\pi$$
$$882$$ 0 0
$$883$$ 8.00000 0.269221 0.134611 0.990899i $$-0.457022\pi$$
0.134611 + 0.990899i $$0.457022\pi$$
$$884$$ 0 0
$$885$$ 5.48913 0.184515
$$886$$ 0 0
$$887$$ 18.5109 0.621534 0.310767 0.950486i $$-0.399414\pi$$
0.310767 + 0.950486i $$0.399414\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −11.0000 −0.368514
$$892$$ 0 0
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ −45.0217 −1.50323
$$898$$ 0 0
$$899$$ −18.5109 −0.617372
$$900$$ 0 0
$$901$$ −5.95650 −0.198440
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 0 0
$$905$$ 3.48913 0.115982
$$906$$ 0 0
$$907$$ 37.4891 1.24481 0.622403 0.782697i $$-0.286157\pi$$
0.622403 + 0.782697i $$0.286157\pi$$
$$908$$ 0 0
$$909$$ −8.74456 −0.290039
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ 0 0
$$915$$ −25.4891 −0.842644
$$916$$ 0 0
$$917$$ 8.74456 0.288771
$$918$$ 0 0
$$919$$ −25.2119 −0.831665 −0.415833 0.909441i $$-0.636510\pi$$
−0.415833 + 0.909441i $$0.636510\pi$$
$$920$$ 0 0
$$921$$ 42.9783 1.41618
$$922$$ 0 0
$$923$$ −18.9783 −0.624677
$$924$$ 0 0
$$925$$ −10.7446 −0.353279
$$926$$ 0 0
$$927$$ −10.7446 −0.352898
$$928$$ 0 0
$$929$$ 16.9783 0.557038 0.278519 0.960431i $$-0.410156\pi$$
0.278519 + 0.960431i $$0.410156\pi$$
$$930$$ 0 0
$$931$$ −4.74456 −0.155497
$$932$$ 0 0
$$933$$ −18.5109 −0.606019
$$934$$ 0 0
$$935$$ 4.74456 0.155164
$$936$$ 0 0
$$937$$ −7.25544 −0.237025 −0.118512 0.992953i $$-0.537813\pi$$
−0.118512 + 0.992953i $$0.537813\pi$$
$$938$$ 0 0
$$939$$ −64.4674 −2.10381
$$940$$ 0 0
$$941$$ −22.2337 −0.724798 −0.362399 0.932023i $$-0.618042\pi$$
−0.362399 + 0.932023i $$0.618042\pi$$
$$942$$ 0 0
$$943$$ 18.9783 0.618017
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 0 0
$$949$$ 3.53262 0.114674
$$950$$ 0 0
$$951$$ 64.4674 2.09050
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 16.0000 0.517748
$$956$$ 0 0
$$957$$ 5.48913 0.177438
$$958$$ 0 0
$$959$$ 19.4891 0.629337
$$960$$ 0 0
$$961$$ 14.4891 0.467391
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 5.02175 0.161489 0.0807443 0.996735i $$-0.474270\pi$$
0.0807443 + 0.996735i $$0.474270\pi$$
$$968$$ 0 0
$$969$$ 45.0217 1.44631
$$970$$ 0 0
$$971$$ 37.7228 1.21058 0.605291 0.796004i $$-0.293057\pi$$
0.605291 + 0.796004i $$0.293057\pi$$
$$972$$ 0 0
$$973$$ 3.25544 0.104365
$$974$$ 0 0
$$975$$ 9.48913 0.303895
$$976$$ 0 0
$$977$$ −32.9783 −1.05507 −0.527534 0.849534i $$-0.676883\pi$$
−0.527534 + 0.849534i $$0.676883\pi$$
$$978$$ 0 0
$$979$$ −7.48913 −0.239353
$$980$$ 0 0
$$981$$ 16.2337 0.518302
$$982$$ 0 0
$$983$$ 32.2337 1.02809 0.514047 0.857762i $$-0.328146\pi$$
0.514047 + 0.857762i $$0.328146\pi$$
$$984$$ 0 0
$$985$$ 10.0000 0.318626
$$986$$ 0 0
$$987$$ 13.4891 0.429364
$$988$$ 0 0
$$989$$ 18.9783 0.603473
$$990$$ 0 0
$$991$$ 21.4891 0.682625 0.341312 0.939950i $$-0.389129\pi$$
0.341312 + 0.939950i $$0.389129\pi$$
$$992$$ 0 0
$$993$$ −61.9565 −1.96613
$$994$$ 0 0
$$995$$ −14.7446 −0.467434
$$996$$ 0 0
$$997$$ −41.2119 −1.30520 −0.652598 0.757705i $$-0.726321\pi$$
−0.652598 + 0.757705i $$0.726321\pi$$
$$998$$ 0 0
$$999$$ −42.9783 −1.35977
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6160.2.a.r.1.1 2
4.3 odd 2 770.2.a.k.1.1 2
12.11 even 2 6930.2.a.bo.1.1 2
20.3 even 4 3850.2.c.y.1849.1 4
20.7 even 4 3850.2.c.y.1849.4 4
20.19 odd 2 3850.2.a.bc.1.2 2
28.27 even 2 5390.2.a.bq.1.2 2
44.43 even 2 8470.2.a.bu.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.1 2 4.3 odd 2
3850.2.a.bc.1.2 2 20.19 odd 2
3850.2.c.y.1849.1 4 20.3 even 4
3850.2.c.y.1849.4 4 20.7 even 4
5390.2.a.bq.1.2 2 28.27 even 2
6160.2.a.r.1.1 2 1.1 even 1 trivial
6930.2.a.bo.1.1 2 12.11 even 2
8470.2.a.bu.1.2 2 44.43 even 2