# Properties

 Label 2450.2.c.q.99.2 Level $2450$ Weight $2$ Character 2450.99 Analytic conductor $19.563$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5633484952$$ 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 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.99 Dual form 2450.2.c.q.99.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +3.00000i q^{22} +9.00000i q^{23} -2.00000 q^{24} +1.00000 q^{26} -4.00000i q^{27} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} +3.00000 q^{41} +2.00000i q^{43} -3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} -2.00000i q^{48} +12.0000 q^{51} +1.00000i q^{52} +9.00000i q^{53} +4.00000 q^{54} -2.00000i q^{57} -6.00000i q^{58} +8.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +18.0000 q^{69} +1.00000i q^{72} -4.00000i q^{73} -7.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +3.00000i q^{82} -2.00000 q^{86} +12.0000i q^{87} -3.00000i q^{88} -6.00000 q^{89} -9.00000i q^{92} -16.0000i q^{93} +9.00000 q^{94} +2.00000 q^{96} +10.0000i q^{97} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 4q^{6} - 2q^{9} + 6q^{11} + 2q^{16} + 2q^{19} - 4q^{24} + 2q^{26} - 12q^{29} + 16q^{31} - 12q^{34} + 2q^{36} - 4q^{39} + 6q^{41} - 6q^{44} - 18q^{46} + 24q^{51} + 8q^{54} + 16q^{61} - 2q^{64} + 12q^{66} + 36q^{69} - 14q^{74} - 2q^{76} + 20q^{79} - 22q^{81} - 4q^{86} - 12q^{89} + 18q^{94} + 4q^{96} - 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 0 0
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 2.00000i 0.577350i
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.00000i 0.639602i
$$23$$ 9.00000i 1.87663i 0.345782 + 0.938315i $$0.387614\pi$$
−0.345782 + 0.938315i $$0.612386\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 6.00000i − 1.04447i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 7.00000i 1.15079i 0.817875 + 0.575396i $$0.195152\pi$$
−0.817875 + 0.575396i $$0.804848\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ −9.00000 −1.32698
$$47$$ − 9.00000i − 1.31278i −0.754420 0.656392i $$-0.772082\pi$$
0.754420 0.656392i $$-0.227918\pi$$
$$48$$ − 2.00000i − 0.288675i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 12.0000 1.68034
$$52$$ 1.00000i 0.138675i
$$53$$ 9.00000i 1.23625i 0.786082 + 0.618123i $$0.212106\pi$$
−0.786082 + 0.618123i $$0.787894\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ − 6.00000i − 0.787839i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ 18.0000 2.16695
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ −7.00000 −0.813733
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 3.00000i 0.331295i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 12.0000i 1.28654i
$$88$$ − 3.00000i − 0.319801i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 9.00000i − 0.938315i
$$93$$ − 16.0000i − 1.65912i
$$94$$ 9.00000 0.928279
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 12.0000i 1.18818i
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 4.00000i 0.384900i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ 14.0000 1.32882
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 1.00000i 0.0924500i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 8.00000i 0.724286i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1.00000i 0.0887357i 0.999015 + 0.0443678i $$0.0141274\pi$$
−0.999015 + 0.0443678i $$0.985873\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 6.00000i 0.522233i
$$133$$ 0 0
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 18.0000i 1.53226i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −18.0000 −1.51587
$$142$$ 0 0
$$143$$ − 3.00000i − 0.250873i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ − 7.00000i − 0.575396i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 23.0000i − 1.83560i −0.397043 0.917800i $$-0.629964\pi$$
0.397043 0.917800i $$-0.370036\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ 18.0000 1.42749
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 11.0000i − 0.864242i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 3.00000i − 0.232147i −0.993241 0.116073i $$-0.962969\pi$$
0.993241 0.116073i $$-0.0370308\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ − 2.00000i − 0.152499i
$$173$$ 9.00000i 0.684257i 0.939653 + 0.342129i $$0.111148\pi$$
−0.939653 + 0.342129i $$0.888852\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ − 6.00000i − 0.449719i
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ − 16.0000i − 1.18275i
$$184$$ 9.00000 0.663489
$$185$$ 0 0
$$186$$ 16.0000 1.17318
$$187$$ 18.0000i 1.31629i
$$188$$ 9.00000i 0.656392i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 2.00000i 0.144338i
$$193$$ − 16.0000i − 1.15171i −0.817554 0.575853i $$-0.804670\pi$$
0.817554 0.575853i $$-0.195330\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 15.0000i − 1.06871i −0.845262 0.534353i $$-0.820555\pi$$
0.845262 0.534353i $$-0.179445\pi$$
$$198$$ − 3.00000i − 0.213201i
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 12.0000i 0.844317i
$$203$$ 0 0
$$204$$ −12.0000 −0.840168
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ − 9.00000i − 0.625543i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ − 9.00000i − 0.618123i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ 0 0
$$218$$ 16.0000i 1.08366i
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 14.0000i 0.939618i
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 2.00000i 0.132453i
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 20.0000i − 1.29914i
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −0.0644157 −0.0322078 0.999481i $$-0.510254\pi$$
−0.0322078 + 0.999481i $$0.510254\pi$$
$$242$$ − 2.00000i − 0.128565i
$$243$$ 10.0000i 0.641500i
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ − 1.00000i − 0.0636285i
$$248$$ − 8.00000i − 0.508001i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 0 0
$$253$$ 27.0000i 1.69748i
$$254$$ −1.00000 −0.0627456
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 3.00000i 0.185341i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000i 0.734388i
$$268$$ 8.00000i 0.488678i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ −18.0000 −1.08347
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −27.0000 −1.61068 −0.805342 0.592810i $$-0.798019\pi$$
−0.805342 + 0.592810i $$0.798019\pi$$
$$282$$ − 18.0000i − 1.07188i
$$283$$ 14.0000i 0.832214i 0.909316 + 0.416107i $$0.136606\pi$$
−0.909316 + 0.416107i $$0.863394\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 20.0000 1.17242
$$292$$ 4.00000i 0.234082i
$$293$$ − 9.00000i − 0.525786i −0.964825 0.262893i $$-0.915323\pi$$
0.964825 0.262893i $$-0.0846766\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 7.00000 0.406867
$$297$$ − 12.0000i − 0.696311i
$$298$$ 6.00000i 0.347571i
$$299$$ 9.00000 0.520483
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 10.0000i − 0.575435i
$$303$$ − 24.0000i − 1.37876i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 14.0000i − 0.799022i −0.916728 0.399511i $$-0.869180\pi$$
0.916728 0.399511i $$-0.130820\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 28.0000i − 1.58265i −0.611393 0.791327i $$-0.709391\pi$$
0.611393 0.791327i $$-0.290609\pi$$
$$314$$ 23.0000 1.29797
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 18.0000i 1.00939i
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 24.0000 1.33955
$$322$$ 0 0
$$323$$ 6.00000i 0.333849i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ − 32.0000i − 1.76960i
$$328$$ − 3.00000i − 0.165647i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 0 0
$$333$$ − 7.00000i − 0.383598i
$$334$$ 3.00000 0.164153
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ 12.0000i 0.652714i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ − 1.00000i − 0.0540738i
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ −9.00000 −0.483843
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ − 12.0000i − 0.643268i
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 3.00000i 0.159901i
$$353$$ 12.0000i 0.638696i 0.947638 + 0.319348i $$0.103464\pi$$
−0.947638 + 0.319348i $$0.896536\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 3.00000i 0.158555i
$$359$$ −18.0000 −0.950004 −0.475002 0.879985i $$-0.657553\pi$$
−0.475002 + 0.879985i $$0.657553\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 2.00000i 0.105118i
$$363$$ 4.00000i 0.209946i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 16.0000 0.836333
$$367$$ 19.0000i 0.991792i 0.868382 + 0.495896i $$0.165160\pi$$
−0.868382 + 0.495896i $$0.834840\pi$$
$$368$$ 9.00000i 0.469157i
$$369$$ −3.00000 −0.156174
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 16.0000i 0.829561i
$$373$$ 2.00000i 0.103556i 0.998659 + 0.0517780i $$0.0164888\pi$$
−0.998659 + 0.0517780i $$0.983511\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ 6.00000i 0.309016i
$$378$$ 0 0
$$379$$ −23.0000 −1.18143 −0.590715 0.806880i $$-0.701154\pi$$
−0.590715 + 0.806880i $$0.701154\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 12.0000i 0.613973i
$$383$$ 21.0000i 1.07305i 0.843884 + 0.536525i $$0.180263\pi$$
−0.843884 + 0.536525i $$0.819737\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ 16.0000 0.814379
$$387$$ − 2.00000i − 0.101666i
$$388$$ − 10.0000i − 0.507673i
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ −54.0000 −2.73090
$$392$$ 0 0
$$393$$ − 6.00000i − 0.302660i
$$394$$ 15.0000 0.755689
$$395$$ 0 0
$$396$$ 3.00000 0.150756
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 16.0000i 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ − 16.0000i − 0.798007i
$$403$$ − 8.00000i − 0.398508i
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 21.0000i 1.04093i
$$408$$ − 12.0000i − 0.594089i
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ −24.0000 −1.18383
$$412$$ 4.00000i 0.197066i
$$413$$ 0 0
$$414$$ 9.00000 0.442326
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ − 8.00000i − 0.391762i
$$418$$ 3.00000i 0.146735i
$$419$$ 9.00000 0.439679 0.219839 0.975536i $$-0.429447\pi$$
0.219839 + 0.975536i $$0.429447\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 23.0000i 1.11962i
$$423$$ 9.00000i 0.437595i
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 12.0000i − 0.580042i
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ − 4.00000i − 0.192450i
$$433$$ − 40.0000i − 1.92228i −0.276066 0.961139i $$-0.589031\pi$$
0.276066 0.961139i $$-0.410969\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 9.00000i 0.430528i
$$438$$ − 8.00000i − 0.382255i
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 6.00000i 0.285391i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ −14.0000 −0.664411
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ − 12.0000i − 0.567581i
$$448$$ 0 0
$$449$$ −21.0000 −0.991051 −0.495526 0.868593i $$-0.665025\pi$$
−0.495526 + 0.868593i $$0.665025\pi$$
$$450$$ 0 0
$$451$$ 9.00000 0.423793
$$452$$ 0 0
$$453$$ 20.0000i 0.939682i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ − 14.0000i − 0.654892i −0.944870 0.327446i $$-0.893812\pi$$
0.944870 0.327446i $$-0.106188\pi$$
$$458$$ 4.00000i 0.186908i
$$459$$ 24.0000 1.12022
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ − 1.00000i − 0.0464739i −0.999730 0.0232370i $$-0.992603\pi$$
0.999730 0.0232370i $$-0.00739722\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −46.0000 −2.11957
$$472$$ 0 0
$$473$$ 6.00000i 0.275880i
$$474$$ 20.0000 0.918630
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 9.00000i − 0.412082i
$$478$$ 6.00000i 0.274434i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 7.00000 0.319173
$$482$$ − 1.00000i − 0.0455488i
$$483$$ 0 0
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ 16.0000i 0.725029i 0.931978 + 0.362515i $$0.118082\pi$$
−0.931978 + 0.362515i $$0.881918\pi$$
$$488$$ − 8.00000i − 0.362143i
$$489$$ 40.0000 1.80886
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ − 36.0000i − 1.62136i
$$494$$ 1.00000 0.0449921
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −6.00000 −0.268060
$$502$$ − 15.0000i − 0.669483i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −27.0000 −1.20030
$$507$$ − 24.0000i − 1.06588i
$$508$$ − 1.00000i − 0.0443678i
$$509$$ 42.0000 1.86162 0.930809 0.365507i $$-0.119104\pi$$
0.930809 + 0.365507i $$0.119104\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ − 27.0000i − 1.18746i
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −15.0000 −0.657162 −0.328581 0.944476i $$-0.606570\pi$$
−0.328581 + 0.944476i $$0.606570\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ − 28.0000i − 1.22435i −0.790721 0.612177i $$-0.790294\pi$$
0.790721 0.612177i $$-0.209706\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 48.0000i 2.09091i
$$528$$ − 6.00000i − 0.261116i
$$529$$ −58.0000 −2.52174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 3.00000i − 0.129944i
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ −8.00000 −0.345547
$$537$$ − 6.00000i − 0.258919i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ − 4.00000i − 0.171656i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ − 18.0000i − 0.766131i
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 9.00000i 0.381342i 0.981654 + 0.190671i $$0.0610664\pi$$
−0.981654 + 0.190671i $$0.938934\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 36.0000 1.51992
$$562$$ − 27.0000i − 1.13893i
$$563$$ 42.0000i 1.77009i 0.465506 + 0.885044i $$0.345872\pi$$
−0.465506 + 0.885044i $$0.654128\pi$$
$$564$$ 18.0000 0.757937
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −21.0000 −0.880366 −0.440183 0.897908i $$-0.645086\pi$$
−0.440183 + 0.897908i $$0.645086\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 3.00000i 0.125436i
$$573$$ − 24.0000i − 1.00261i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 44.0000i − 1.83174i −0.401470 0.915872i $$-0.631501\pi$$
0.401470 0.915872i $$-0.368499\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −32.0000 −1.32987
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 20.0000i 0.829027i
$$583$$ 27.0000i 1.11823i
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ 24.0000i 0.990586i 0.868726 + 0.495293i $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −30.0000 −1.23404
$$592$$ 7.00000i 0.287698i
$$593$$ 24.0000i 0.985562i 0.870153 + 0.492781i $$0.164020\pi$$
−0.870153 + 0.492781i $$0.835980\pi$$
$$594$$ 12.0000 0.492366
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 32.0000i − 1.30967i
$$598$$ 9.00000i 0.368037i
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 8.00000i 0.325785i
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 24.0000 0.974933
$$607$$ 1.00000i 0.0405887i 0.999794 + 0.0202944i $$0.00646034\pi$$
−0.999794 + 0.0202944i $$0.993540\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −9.00000 −0.364101
$$612$$ 6.00000i 0.242536i
$$613$$ 29.0000i 1.17130i 0.810564 + 0.585649i $$0.199160\pi$$
−0.810564 + 0.585649i $$0.800840\pi$$
$$614$$ 14.0000 0.564994
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ −23.0000 −0.924448 −0.462224 0.886763i $$-0.652948\pi$$
−0.462224 + 0.886763i $$0.652948\pi$$
$$620$$ 0 0
$$621$$ 36.0000 1.44463
$$622$$ − 24.0000i − 0.962312i
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 28.0000 1.11911
$$627$$ − 6.00000i − 0.239617i
$$628$$ 23.0000i 0.917800i
$$629$$ −42.0000 −1.67465
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ − 46.0000i − 1.82834i
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ −18.0000 −0.713746
$$637$$ 0 0
$$638$$ − 18.0000i − 0.712627i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 27.0000 1.06644 0.533218 0.845978i $$-0.320983\pi$$
0.533218 + 0.845978i $$0.320983\pi$$
$$642$$ 24.0000i 0.947204i
$$643$$ 2.00000i 0.0788723i 0.999222 + 0.0394362i $$0.0125562\pi$$
−0.999222 + 0.0394362i $$0.987444\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −6.00000 −0.236067
$$647$$ − 33.0000i − 1.29736i −0.761060 0.648682i $$-0.775321\pi$$
0.761060 0.648682i $$-0.224679\pi$$
$$648$$ 11.0000i 0.432121i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 20.0000i − 0.783260i
$$653$$ − 9.00000i − 0.352197i −0.984373 0.176099i $$-0.943652\pi$$
0.984373 0.176099i $$-0.0563478\pi$$
$$654$$ 32.0000 1.25130
$$655$$ 0 0
$$656$$ 3.00000 0.117130
$$657$$ 4.00000i 0.156055i
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ −28.0000 −1.08907 −0.544537 0.838737i $$-0.683295\pi$$
−0.544537 + 0.838737i $$0.683295\pi$$
$$662$$ − 7.00000i − 0.272063i
$$663$$ − 12.0000i − 0.466041i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 7.00000 0.271244
$$667$$ − 54.0000i − 2.09089i
$$668$$ 3.00000i 0.116073i
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 9.00000i 0.345898i 0.984931 + 0.172949i $$0.0553296\pi$$
−0.984931 + 0.172949i $$0.944670\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 24.0000 0.919682
$$682$$ 24.0000i 0.919007i
$$683$$ 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
$$686$$ 0 0
$$687$$ − 8.00000i − 0.305219i
$$688$$ 2.00000i 0.0762493i
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ − 9.00000i − 0.342129i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 12.0000 0.454859
$$697$$ 18.0000i 0.681799i
$$698$$ − 26.0000i − 0.984115i
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 7.00000i 0.264010i
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ −12.0000 −0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 6.00000i 0.224860i
$$713$$ 72.0000i 2.69642i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −3.00000 −0.112115
$$717$$ − 12.0000i − 0.448148i
$$718$$ − 18.0000i − 0.671754i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 18.0000i − 0.669891i
$$723$$ 2.00000i 0.0743808i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −4.00000 −0.148454
$$727$$ 1.00000i 0.0370879i 0.999828 + 0.0185440i $$0.00590307\pi$$
−0.999828 + 0.0185440i $$0.994097\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 16.0000i 0.591377i
$$733$$ − 43.0000i − 1.58824i −0.607760 0.794121i $$-0.707932\pi$$
0.607760 0.794121i $$-0.292068\pi$$
$$734$$ −19.0000 −0.701303
$$735$$ 0 0
$$736$$ −9.00000 −0.331744
$$737$$ − 24.0000i − 0.884051i
$$738$$ − 3.00000i − 0.110432i
$$739$$ −35.0000 −1.28750 −0.643748 0.765238i $$-0.722621\pi$$
−0.643748 + 0.765238i $$0.722621\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ − 45.0000i − 1.65089i −0.564483 0.825445i $$-0.690924\pi$$
0.564483 0.825445i $$-0.309076\pi$$
$$744$$ −16.0000 −0.586588
$$745$$ 0 0
$$746$$ −2.00000 −0.0732252
$$747$$ 0 0
$$748$$ − 18.0000i − 0.658145i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ − 9.00000i − 0.328196i
$$753$$ 30.0000i 1.09326i
$$754$$ −6.00000 −0.218507
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 38.0000i − 1.38113i −0.723269 0.690567i $$-0.757361\pi$$
0.723269 0.690567i $$-0.242639\pi$$
$$758$$ − 23.0000i − 0.835398i
$$759$$ 54.0000 1.96008
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 2.00000i 0.0724524i
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −21.0000 −0.758761
$$767$$ 0 0
$$768$$ − 2.00000i − 0.0721688i
$$769$$ −23.0000 −0.829401 −0.414701 0.909958i $$-0.636114\pi$$
−0.414701 + 0.909958i $$0.636114\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 16.0000i 0.575853i
$$773$$ − 51.0000i − 1.83434i −0.398493 0.917171i $$-0.630467\pi$$
0.398493 0.917171i $$-0.369533\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 12.0000i 0.430221i
$$779$$ 3.00000 0.107486
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 54.0000i − 1.93104i
$$783$$ 24.0000i 0.857690i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 6.00000 0.214013
$$787$$ 22.0000i 0.784215i 0.919919 + 0.392108i $$0.128254\pi$$
−0.919919 + 0.392108i $$0.871746\pi$$
$$788$$ 15.0000i 0.534353i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 3.00000i 0.106600i
$$793$$ − 8.00000i − 0.284088i
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 0 0
$$799$$ 54.0000 1.91038
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ − 27.0000i − 0.953403i
$$803$$ − 12.0000i − 0.423471i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ − 12.0000i − 0.422159i
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ −25.0000 −0.877869 −0.438934 0.898519i $$-0.644644\pi$$
−0.438934 + 0.898519i $$0.644644\pi$$
$$812$$ 0 0
$$813$$ 32.0000i 1.12229i
$$814$$ −21.0000 −0.736050
$$815$$ 0 0
$$816$$ 12.0000 0.420084
$$817$$ 2.00000i 0.0699711i
$$818$$ − 26.0000i − 0.909069i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ − 24.0000i − 0.837096i
$$823$$ − 4.00000i − 0.139431i −0.997567 0.0697156i $$-0.977791\pi$$
0.997567 0.0697156i $$-0.0222092\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 6.00000i − 0.208640i −0.994544 0.104320i $$-0.966733\pi$$
0.994544 0.104320i $$-0.0332667\pi$$
$$828$$ 9.00000i 0.312772i
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 20.0000 0.693792
$$832$$ 1.00000i 0.0346688i
$$833$$ 0 0
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ −3.00000 −0.103757
$$837$$ − 32.0000i − 1.10608i
$$838$$ 9.00000i 0.310900i
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 2.00000i 0.0689246i
$$843$$ 54.0000i 1.85986i
$$844$$ −23.0000 −0.791693
$$845$$ 0 0
$$846$$ −9.00000 −0.309426
$$847$$ 0 0
$$848$$ 9.00000i 0.309061i
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ −63.0000 −2.15961
$$852$$ 0 0
$$853$$ − 19.0000i − 0.650548i −0.945620 0.325274i $$-0.894544\pi$$
0.945620 0.325274i $$-0.105456\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ − 6.00000i − 0.204837i
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 12.0000i 0.408722i
$$863$$ − 3.00000i − 0.102121i −0.998696 0.0510606i $$-0.983740\pi$$
0.998696 0.0510606i $$-0.0162602\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ 40.0000 1.35926
$$867$$ 38.0000i 1.29055i
$$868$$ 0 0
$$869$$ 30.0000 1.01768
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ − 16.0000i − 0.541828i
$$873$$ − 10.0000i − 0.338449i
$$874$$ −9.00000 −0.304430
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 13.0000i 0.438979i 0.975615 + 0.219489i $$0.0704391\pi$$
−0.975615 + 0.219489i $$0.929561\pi$$
$$878$$ − 26.0000i − 0.877457i
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ 0 0
$$883$$ 8.00000i 0.269221i 0.990899 + 0.134611i $$0.0429784\pi$$
−0.990899 + 0.134611i $$0.957022\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ − 14.0000i − 0.469809i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −33.0000 −1.10554
$$892$$ − 8.00000i − 0.267860i
$$893$$ − 9.00000i − 0.301174i
$$894$$ 12.0000 0.401340
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 18.0000i − 0.601003i
$$898$$ − 21.0000i − 0.700779i
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ −54.0000 −1.79900
$$902$$ 9.00000i 0.299667i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −20.0000 −0.664455
$$907$$ 10.0000i 0.332045i 0.986122 + 0.166022i $$0.0530924\pi$$
−0.986122 + 0.166022i $$0.946908\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ − 2.00000i − 0.0662266i
$$913$$ 0 0
$$914$$ 14.0000 0.463079
$$915$$ 0 0
$$916$$ −4.00000 −0.132164
$$917$$ 0 0
$$918$$ 24.0000i 0.792118i
$$919$$ 22.0000 0.725713 0.362857 0.931845i $$-0.381802\pi$$
0.362857 + 0.931845i $$0.381802\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ − 30.0000i − 0.987997i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 1.00000 0.0328620
$$927$$ 4.00000i 0.131377i
$$928$$ − 6.00000i − 0.196960i
$$929$$ −57.0000 −1.87011 −0.935055 0.354504i $$-0.884650\pi$$
−0.935055 + 0.354504i $$0.884650\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000i 0.196537i
$$933$$ 48.0000i 1.57145i
$$934$$ −6.00000 −0.196326
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ 10.0000i 0.326686i 0.986569 + 0.163343i $$0.0522277\pi$$
−0.986569 + 0.163343i $$0.947772\pi$$
$$938$$ 0 0
$$939$$ −56.0000 −1.82749
$$940$$ 0 0
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ − 46.0000i − 1.49876i
$$943$$ 27.0000i 0.879241i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ − 6.00000i − 0.194974i −0.995237 0.0974869i $$-0.968920\pi$$
0.995237 0.0974869i $$-0.0310804\pi$$
$$948$$ 20.0000i 0.649570i
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ 36.0000i 1.16615i 0.812417 + 0.583077i $$0.198151\pi$$
−0.812417 + 0.583077i $$0.801849\pi$$
$$954$$ 9.00000 0.291386
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ 36.0000i 1.16371i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 7.00000i 0.225689i
$$963$$ − 12.0000i − 0.386695i
$$964$$ 1.00000 0.0322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 2.00000i 0.0642824i
$$969$$ 12.0000 0.385496
$$970$$ 0 0
$$971$$ −45.0000 −1.44412 −0.722059 0.691831i $$-0.756804\pi$$
−0.722059 + 0.691831i $$0.756804\pi$$
$$972$$ − 10.0000i − 0.320750i
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 40.0000i 1.27906i
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 36.0000i 1.14881i
$$983$$ 3.00000i 0.0956851i 0.998855 + 0.0478426i $$0.0152346\pi$$
−0.998855 + 0.0478426i $$0.984765\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ 1.00000i 0.0318142i
$$989$$ −18.0000 −0.572367
$$990$$ 0 0
$$991$$ 44.0000 1.39771 0.698853 0.715265i $$-0.253694\pi$$
0.698853 + 0.715265i $$0.253694\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 14.0000i 0.444277i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 22.0000i 0.696747i 0.937356 + 0.348373i $$0.113266\pi$$
−0.937356 + 0.348373i $$0.886734\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 28.0000 0.885881
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 2450.2.c.q.99.2 2
5.2 odd 4 490.2.a.a.1.1 1
5.3 odd 4 2450.2.a.bf.1.1 1
5.4 even 2 inner 2450.2.c.q.99.1 2
7.2 even 3 350.2.j.d.249.2 4
7.4 even 3 350.2.j.d.149.1 4
7.6 odd 2 2450.2.c.e.99.2 2
15.2 even 4 4410.2.a.x.1.1 1
20.7 even 4 3920.2.a.bh.1.1 1
35.2 odd 12 70.2.e.d.11.1 2
35.4 even 6 350.2.j.d.149.2 4
35.9 even 6 350.2.j.d.249.1 4
35.12 even 12 490.2.e.g.361.1 2
35.13 even 4 2450.2.a.v.1.1 1
35.17 even 12 490.2.e.g.471.1 2
35.18 odd 12 350.2.e.b.51.1 2
35.23 odd 12 350.2.e.b.151.1 2
35.27 even 4 490.2.a.d.1.1 1
35.32 odd 12 70.2.e.d.51.1 yes 2
35.34 odd 2 2450.2.c.e.99.1 2
105.2 even 12 630.2.k.d.361.1 2
105.32 even 12 630.2.k.d.541.1 2
105.62 odd 4 4410.2.a.bg.1.1 1
140.27 odd 4 3920.2.a.e.1.1 1
140.67 even 12 560.2.q.b.401.1 2
140.107 even 12 560.2.q.b.81.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 35.2 odd 12
70.2.e.d.51.1 yes 2 35.32 odd 12
350.2.e.b.51.1 2 35.18 odd 12
350.2.e.b.151.1 2 35.23 odd 12
350.2.j.d.149.1 4 7.4 even 3
350.2.j.d.149.2 4 35.4 even 6
350.2.j.d.249.1 4 35.9 even 6
350.2.j.d.249.2 4 7.2 even 3
490.2.a.a.1.1 1 5.2 odd 4
490.2.a.d.1.1 1 35.27 even 4
490.2.e.g.361.1 2 35.12 even 12
490.2.e.g.471.1 2 35.17 even 12
560.2.q.b.81.1 2 140.107 even 12
560.2.q.b.401.1 2 140.67 even 12
630.2.k.d.361.1 2 105.2 even 12
630.2.k.d.541.1 2 105.32 even 12
2450.2.a.v.1.1 1 35.13 even 4
2450.2.a.bf.1.1 1 5.3 odd 4
2450.2.c.e.99.1 2 35.34 odd 2
2450.2.c.e.99.2 2 7.6 odd 2
2450.2.c.q.99.1 2 5.4 even 2 inner
2450.2.c.q.99.2 2 1.1 even 1 trivial
3920.2.a.e.1.1 1 140.27 odd 4
3920.2.a.bh.1.1 1 20.7 even 4
4410.2.a.x.1.1 1 15.2 even 4
4410.2.a.bg.1.1 1 105.62 odd 4