# Properties

 Label 2850.2.d.c.799.2 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.c.799.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -8.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -2.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} -1.00000i q^{38} -6.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +8.00000i q^{43} +8.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -1.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} +1.00000 q^{76} -6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{84} -8.00000 q^{86} -2.00000i q^{87} -6.00000 q^{89} -24.0000 q^{91} +8.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{14} + 2q^{16} - 2q^{19} - 8q^{21} + 2q^{24} - 12q^{26} - 4q^{29} - 16q^{31} - 4q^{34} + 2q^{36} - 12q^{39} + 12q^{41} + 16q^{46} - 18q^{49} - 4q^{51} + 2q^{54} + 8q^{56} + 24q^{59} - 4q^{61} - 2q^{64} + 16q^{69} - 16q^{71} - 20q^{74} + 2q^{76} + 32q^{79} + 2q^{81} + 8q^{84} - 16q^{86} - 12q^{89} - 48q^{91} - 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1027$$ $$1351$$ $$1901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ − 6.00000i − 0.832050i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ − 1.00000i − 0.132453i
$$58$$ − 2.00000i − 0.262613i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ − 6.00000i − 0.679366i
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ − 2.00000i − 0.214423i
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −24.0000 −2.51588
$$92$$ 8.00000i 0.834058i
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 9.00000i − 0.909137i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 20.0000i 1.93347i 0.255774 + 0.966736i $$0.417670\pi$$
−0.255774 + 0.966736i $$0.582330\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 4.00000i 0.377964i
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 6.00000i − 0.554700i
$$118$$ 12.0000i 1.10469i
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 2.00000i − 0.181071i
$$123$$ 6.00000i 0.541002i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ − 4.00000i − 0.346844i
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 8.00000i − 0.671345i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ − 9.00000i − 0.742307i
$$148$$ − 10.0000i − 0.821995i
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ − 10.0000i − 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ 16.0000i 1.27289i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 32.0000 2.52195
$$162$$ 1.00000i 0.0785674i
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 8.00000i − 0.609994i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000i 0.901975i
$$178$$ − 6.00000i − 0.449719i
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ − 24.0000i − 1.77900i
$$183$$ − 2.00000i − 0.147844i
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ − 18.0000i − 1.26648i
$$203$$ − 8.00000i − 0.561490i
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 8.00000i 0.556038i
$$208$$ 6.00000i 0.416025i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ − 2.00000i − 0.137361i
$$213$$ − 8.00000i − 0.548151i
$$214$$ −20.0000 −1.36717
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ − 32.0000i − 2.17230i
$$218$$ − 6.00000i − 0.406371i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ − 10.0000i − 0.671156i
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000i 0.131306i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 16.0000i 1.03931i
$$238$$ − 8.00000i − 0.518563i
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 1.00000i 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ − 6.00000i − 0.381771i
$$248$$ 8.00000i 0.508001i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −40.0000 −2.48548
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ − 6.00000i − 0.367194i
$$268$$ 12.0000i 0.733017i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ − 24.0000i − 1.45255i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 12.0000i 0.719712i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ 16.0000i 0.951101i 0.879688 + 0.475551i $$0.157751\pi$$
−0.879688 + 0.475551i $$0.842249\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000i 1.41668i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 2.00000i 0.117041i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ 2.00000i 0.115857i
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ −32.0000 −1.84445
$$302$$ 0 0
$$303$$ − 18.0000i − 1.03407i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 26.0000i − 1.46031i −0.683284 0.730153i $$-0.739449\pi$$
0.683284 0.730153i $$-0.260551\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −20.0000 −1.11629
$$322$$ 32.0000i 1.78329i
$$323$$ − 2.00000i − 0.111283i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ − 6.00000i − 0.331801i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 10.0000i − 0.547997i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ − 23.0000i − 1.25104i
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 1.00000i 0.0540738i
$$343$$ − 8.00000i − 0.431959i
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 20.0000i 1.07366i 0.843692 + 0.536828i $$0.180378\pi$$
−0.843692 + 0.536828i $$0.819622\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ − 2.00000i − 0.106449i −0.998583 0.0532246i $$-0.983050\pi$$
0.998583 0.0532246i $$-0.0169499\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ − 8.00000i − 0.423405i
$$358$$ 20.0000i 1.05703i
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 18.0000i − 0.946059i
$$363$$ − 11.0000i − 0.577350i
$$364$$ 24.0000 1.25794
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 4.00000i 0.208798i 0.994535 + 0.104399i $$0.0332919\pi$$
−0.994535 + 0.104399i $$0.966708\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −8.00000 −0.415339
$$372$$ 8.00000i 0.414781i
$$373$$ − 34.0000i − 1.76045i −0.474554 0.880227i $$-0.657390\pi$$
0.474554 0.880227i $$-0.342610\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ 4.00000i 0.205738i
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ − 12.0000i − 0.613973i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 8.00000i − 0.406663i
$$388$$ 10.0000i 0.507673i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 9.00000i 0.454569i
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 38.0000i 1.90717i 0.301131 + 0.953583i $$0.402636\pi$$
−0.301131 + 0.953583i $$0.597364\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ − 48.0000i − 2.39105i
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ 0 0
$$408$$ 2.00000i 0.0990148i
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 8.00000i 0.394132i
$$413$$ 48.0000i 2.36193i
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ 40.0000 1.95413 0.977064 0.212946i $$-0.0683059\pi$$
0.977064 + 0.212946i $$0.0683059\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 0 0
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ − 8.00000i − 0.387147i
$$428$$ − 20.0000i − 0.966736i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 32.0000 1.53605
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 8.00000i 0.382692i
$$438$$ 2.00000i 0.0955637i
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ − 12.0000i − 0.570782i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 2.00000i 0.0945968i
$$448$$ − 4.00000i − 0.188982i
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 18.0000i 0.846649i
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ 48.0000 2.21643
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ − 12.0000i − 0.552345i
$$473$$ 0 0
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 20.0000i 0.914779i
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ − 14.0000i − 0.637683i
$$483$$ 32.0000i 1.45605i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ −40.0000 −1.80517 −0.902587 0.430507i $$-0.858335\pi$$
−0.902587 + 0.430507i $$0.858335\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ − 4.00000i − 0.180151i
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ − 32.0000i − 1.43540i
$$498$$ − 4.00000i − 0.179244i
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 0 0
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 23.0000i − 1.02147i
$$508$$ 8.00000i 0.354943i
$$509$$ 22.0000 0.975133 0.487566 0.873086i $$-0.337885\pi$$
0.487566 + 0.873086i $$0.337885\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 1.00000i 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ − 40.0000i − 1.75750i
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ 44.0000i 1.92399i 0.273075 + 0.961993i $$0.411959\pi$$
−0.273075 + 0.961993i $$0.588041\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 16.0000i − 0.696971i
$$528$$ 0 0
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 4.00000i 0.173422i
$$533$$ 36.0000i 1.55933i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 20.0000i 0.863064i
$$538$$ − 18.0000i − 0.776035i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ − 18.0000i − 0.772454i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 24.0000 1.02711
$$547$$ 36.0000i 1.53925i 0.638497 + 0.769624i $$0.279557\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ − 8.00000i − 0.340503i
$$553$$ 64.0000i 2.72156i
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ −48.0000 −2.03018
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 2.00000i − 0.0843649i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ 4.00000i 0.167984i
$$568$$ 8.00000i 0.335673i
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ − 12.0000i − 0.501307i
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 18.0000i 0.749350i 0.927156 + 0.374675i $$0.122246\pi$$
−0.927156 + 0.374675i $$0.877754\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ 10.0000i 0.414513i
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 20.0000i − 0.825488i −0.910847 0.412744i $$-0.864570\pi$$
0.910847 0.412744i $$-0.135430\pi$$
$$588$$ 9.00000i 0.371154i
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 10.0000i 0.410997i
$$593$$ 30.0000i 1.23195i 0.787765 + 0.615976i $$0.211238\pi$$
−0.787765 + 0.615976i $$0.788762\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −2.00000 −0.0819232
$$597$$ − 16.0000i − 0.654836i
$$598$$ 48.0000i 1.96287i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ − 32.0000i − 1.30422i
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 2.00000i 0.0808452i
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.0000i 1.04672i 0.852111 + 0.523360i $$0.175322\pi$$
−0.852111 + 0.523360i $$0.824678\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 12.0000i 0.481156i
$$623$$ − 24.0000i − 0.961540i
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ 10.0000i 0.399043i
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ − 16.0000i − 0.636446i
$$633$$ 20.0000i 0.794929i
$$634$$ 26.0000 1.03259
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ − 54.0000i − 2.13956i
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ − 20.0000i − 0.789337i
$$643$$ 40.0000i 1.57745i 0.614749 + 0.788723i $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ −32.0000 −1.26098
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 32.0000 1.25418
$$652$$ 8.00000i 0.313304i
$$653$$ − 30.0000i − 1.17399i −0.809590 0.586995i $$-0.800311\pi$$
0.809590 0.586995i $$-0.199689\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −34.0000 −1.32245 −0.661223 0.750189i $$-0.729962\pi$$
−0.661223 + 0.750189i $$0.729962\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ − 12.0000i − 0.466041i
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 16.0000i 0.619522i
$$668$$ − 8.00000i − 0.309529i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 4.00000i − 0.154303i
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 30.0000i 1.15299i 0.817099 + 0.576497i $$0.195581\pi$$
−0.817099 + 0.576497i $$0.804419\pi$$
$$678$$ 18.0000i 0.691286i
$$679$$ 40.0000 1.53506
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ − 6.00000i − 0.228914i
$$688$$ 8.00000i 0.304997i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 0 0
$$694$$ −20.0000 −0.759190
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 12.0000i 0.454532i
$$698$$ − 14.0000i − 0.529908i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 38.0000 1.43524 0.717620 0.696435i $$-0.245231\pi$$
0.717620 + 0.696435i $$0.245231\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ − 10.0000i − 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 2.00000 0.0752710
$$707$$ − 72.0000i − 2.70784i
$$708$$ − 12.0000i − 0.450988i
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 6.00000i 0.224860i
$$713$$ 64.0000i 2.39682i
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 20.0000i 0.746914i
$$718$$ − 12.0000i − 0.447836i
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 1.00000i 0.0372161i
$$723$$ − 14.0000i − 0.520666i
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ 12.0000i 0.445055i 0.974926 + 0.222528i $$0.0714308\pi$$
−0.974926 + 0.222528i $$0.928569\pi$$
$$728$$ 24.0000i 0.889499i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ 2.00000i 0.0739221i
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 0 0
$$738$$ − 6.00000i − 0.220863i
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 6.00000 0.220416
$$742$$ − 8.00000i − 0.293689i
$$743$$ 32.0000i 1.17397i 0.809599 + 0.586983i $$0.199684\pi$$
−0.809599 + 0.586983i $$0.800316\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 34.0000 1.24483
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ −80.0000 −2.92314
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 22.0000i 0.799604i 0.916602 + 0.399802i $$0.130921\pi$$
−0.916602 + 0.399802i $$0.869079\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 34.0000 1.23250 0.616250 0.787551i $$-0.288651\pi$$
0.616250 + 0.787551i $$0.288651\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ − 24.0000i − 0.868858i
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ 72.0000i 2.59977i
$$768$$ 1.00000i 0.0360844i
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 2.00000i − 0.0719816i
$$773$$ − 46.0000i − 1.65451i −0.561830 0.827253i $$-0.689903\pi$$
0.561830 0.827253i $$-0.310097\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ − 40.0000i − 1.43499i
$$778$$ − 6.00000i − 0.215110i
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 16.0000i 0.572159i
$$783$$ 2.00000i 0.0714742i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 20.0000i − 0.712923i −0.934310 0.356462i $$-0.883983\pi$$
0.934310 0.356462i $$-0.116017\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 72.0000 2.56003
$$792$$ 0 0
$$793$$ − 12.0000i − 0.426132i
$$794$$ −38.0000 −1.34857
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 6.00000i 0.211867i
$$803$$ 0 0
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ − 18.0000i − 0.633630i
$$808$$ 18.0000i 0.633238i
$$809$$ 22.0000 0.773479 0.386739 0.922189i $$-0.373601\pi$$
0.386739 + 0.922189i $$0.373601\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ 8.00000i 0.280572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ − 8.00000i − 0.279885i
$$818$$ − 18.0000i − 0.629355i
$$819$$ 24.0000 0.838628
$$820$$ 0 0
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ − 44.0000i − 1.53374i −0.641800 0.766872i $$-0.721812\pi$$
0.641800 0.766872i $$-0.278188\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ − 52.0000i − 1.80822i −0.427303 0.904109i $$-0.640536\pi$$
0.427303 0.904109i $$-0.359464\pi$$
$$828$$ − 8.00000i − 0.278019i
$$829$$ −22.0000 −0.764092 −0.382046 0.924143i $$-0.624780\pi$$
−0.382046 + 0.924143i $$0.624780\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ − 6.00000i − 0.208013i
$$833$$ − 18.0000i − 0.623663i
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ 40.0000i 1.38178i
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 22.0000i 0.758170i
$$843$$ − 2.00000i − 0.0688837i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 44.0000i − 1.51186i
$$848$$ 2.00000i 0.0686803i
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ 80.0000 2.74236
$$852$$ 8.00000i 0.274075i
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ 20.0000 0.683586
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 24.0000i 0.817443i
$$863$$ − 8.00000i − 0.272323i −0.990687 0.136162i $$-0.956523\pi$$
0.990687 0.136162i $$-0.0434766\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 13.0000i 0.441503i
$$868$$ 32.0000i 1.08615i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 72.0000 2.43963
$$872$$ 6.00000i 0.203186i
$$873$$ 10.0000i 0.338449i
$$874$$ −8.00000 −0.270604
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ − 6.00000i − 0.202606i −0.994856 0.101303i $$-0.967699\pi$$
0.994856 0.101303i $$-0.0323011\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 9.00000i 0.303046i
$$883$$ − 8.00000i − 0.269221i −0.990899 0.134611i $$-0.957022\pi$$
0.990899 0.134611i $$-0.0429784\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 16.0000i − 0.537227i −0.963248 0.268614i $$-0.913434\pi$$
0.963248 0.268614i $$-0.0865655\pi$$
$$888$$ 10.0000i 0.335578i
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ 0 0
$$894$$ −2.00000 −0.0668900
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ 48.0000i 1.60267i
$$898$$ − 30.0000i − 1.00111i
$$899$$ 16.0000 0.533630
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ 0 0
$$903$$ − 32.0000i − 1.06489i
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 60.0000i − 1.99227i −0.0878507 0.996134i $$-0.528000\pi$$
0.0878507 0.996134i $$-0.472000\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ 2.00000i 0.0660098i
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 6.00000i 0.197599i
$$923$$ − 48.0000i − 1.57994i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ 8.00000i 0.262754i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ 9.00000 0.294963
$$932$$ − 6.00000i − 0.196537i
$$933$$ 12.0000i 0.392862i
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ 48.0000i 1.56726i
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 10.0000i 0.325818i
$$943$$ − 48.0000i − 1.56310i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 44.0000i 1.42981i 0.699223 + 0.714904i $$0.253530\pi$$
−0.699223 + 0.714904i $$0.746470\pi$$
$$948$$ − 16.0000i − 0.519656i
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 26.0000 0.843108
$$952$$ 8.00000i 0.259281i
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −20.0000 −0.646846
$$957$$ 0 0
$$958$$ − 36.0000i − 1.16311i
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 60.0000i − 1.93448i
$$963$$ − 20.0000i − 0.644491i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ −32.0000 −1.02958
$$967$$ 44.0000i 1.41494i 0.706741 + 0.707472i $$0.250165\pi$$
−0.706741 + 0.707472i $$0.749835\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 48.0000i 1.53881i
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 54.0000i − 1.72761i −0.503824 0.863807i $$-0.668074\pi$$
0.503824 0.863807i $$-0.331926\pi$$
$$978$$ 8.00000i 0.255812i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ − 40.0000i − 1.27645i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ 6.00000i 0.190885i
$$989$$ 64.0000 2.03508
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ 20.0000i 0.634681i
$$994$$ 32.0000 1.01498
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ 14.0000i 0.443384i 0.975117 + 0.221692i $$0.0711580\pi$$
−0.975117 + 0.221692i $$0.928842\pi$$
$$998$$ 20.0000i 0.633089i
$$999$$ 10.0000 0.316386
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 2850.2.d.c.799.2 2
5.2 odd 4 2850.2.a.h.1.1 1
5.3 odd 4 570.2.a.i.1.1 1
5.4 even 2 inner 2850.2.d.c.799.1 2
15.2 even 4 8550.2.a.s.1.1 1
15.8 even 4 1710.2.a.e.1.1 1
20.3 even 4 4560.2.a.x.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.i.1.1 1 5.3 odd 4
1710.2.a.e.1.1 1 15.8 even 4
2850.2.a.h.1.1 1 5.2 odd 4
2850.2.d.c.799.1 2 5.4 even 2 inner
2850.2.d.c.799.2 2 1.1 even 1 trivial
4560.2.a.x.1.1 1 20.3 even 4
8550.2.a.s.1.1 1 15.2 even 4