# Properties

 Label 3450.2.a.z.1.1 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +8.00000 q^{19} +2.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -10.0000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} -8.00000 q^{37} +8.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} +2.00000 q^{42} -12.0000 q^{43} +2.00000 q^{44} +1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +8.00000 q^{57} -10.0000 q^{58} +4.00000 q^{59} +12.0000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +4.00000 q^{67} +1.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -8.00000 q^{74} +8.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +10.0000 q^{83} +2.00000 q^{84} -12.0000 q^{86} -10.0000 q^{87} +2.00000 q^{88} +4.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 2.00000 0.426401
$$23$$ 1.00000 0.208514
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 8.00000 1.29777
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 8.00000 1.05963
$$58$$ −10.0000 −1.31306
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 12.0000 1.53644 0.768221 0.640184i $$-0.221142\pi$$
0.768221 + 0.640184i $$0.221142\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 4.00000 0.455842
$$78$$ 2.00000 0.226455
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 10.0000 1.09764 0.548821 0.835940i $$-0.315077\pi$$
0.548821 + 0.835940i $$0.315077\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ −10.0000 −1.07211
$$88$$ 2.00000 0.213201
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 1.00000 0.104257
$$93$$ 8.00000 0.829561
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 2.00000 0.197066 0.0985329 0.995134i $$-0.468585\pi$$
0.0985329 + 0.995134i $$0.468585\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 10.0000 0.966736 0.483368 0.875417i $$-0.339413\pi$$
0.483368 + 0.875417i $$0.339413\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −8.00000 −0.766261 −0.383131 0.923694i $$-0.625154\pi$$
−0.383131 + 0.923694i $$0.625154\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 2.00000 0.188982
$$113$$ −8.00000 −0.752577 −0.376288 0.926503i $$-0.622800\pi$$
−0.376288 + 0.926503i $$0.622800\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ 2.00000 0.184900
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 12.0000 1.08643
$$123$$ −6.00000 −0.541002
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 2.00000 0.174078
$$133$$ 16.0000 1.38738
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 16.0000 1.34269
$$143$$ 4.00000 0.334497
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ −3.00000 −0.247436
$$148$$ −8.00000 −0.657596
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 8.00000 0.648886
$$153$$ 0 0
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ 10.0000 0.795557
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 2.00000 0.157622
$$162$$ 1.00000 0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 10.0000 0.776151
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 2.00000 0.154303
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ −12.0000 −0.914991
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ −16.0000 −1.19590 −0.597948 0.801535i $$-0.704017\pi$$
−0.597948 + 0.801535i $$0.704017\pi$$
$$180$$ 0 0
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 12.0000 0.887066
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ −8.00000 −0.583460
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −26.0000 −1.85242 −0.926212 0.377004i $$-0.876954\pi$$
−0.926212 + 0.377004i $$0.876954\pi$$
$$198$$ 2.00000 0.142134
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 10.0000 0.703598
$$203$$ −20.0000 −1.40372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2.00000 0.139347
$$207$$ 1.00000 0.0695048
$$208$$ 2.00000 0.138675
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 16.0000 1.09630
$$214$$ 10.0000 0.683586
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 16.0000 1.08615
$$218$$ −8.00000 −0.541828
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −8.00000 −0.536925
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −8.00000 −0.532152
$$227$$ −14.0000 −0.929213 −0.464606 0.885517i $$-0.653804\pi$$
−0.464606 + 0.885517i $$0.653804\pi$$
$$228$$ 8.00000 0.529813
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ −10.0000 −0.656532
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 10.0000 0.649570
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 1.00000 0.0641500
$$244$$ 12.0000 0.768221
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 16.0000 1.01806
$$248$$ 8.00000 0.508001
$$249$$ 10.0000 0.633724
$$250$$ 0 0
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 2.00000 0.125739
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ −12.0000 −0.747087
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ −8.00000 −0.494242
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 16.0000 0.981023
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 0 0
$$273$$ 4.00000 0.242091
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ 6.00000 0.360505 0.180253 0.983620i $$-0.442309\pi$$
0.180253 + 0.983620i $$0.442309\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −16.0000 −0.954480 −0.477240 0.878773i $$-0.658363\pi$$
−0.477240 + 0.878773i $$0.658363\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ −12.0000 −0.708338
$$288$$ 1.00000 0.0589256
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 10.0000 0.585206
$$293$$ −2.00000 −0.116841 −0.0584206 0.998292i $$-0.518606\pi$$
−0.0584206 + 0.998292i $$0.518606\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ 2.00000 0.116052
$$298$$ −10.0000 −0.579284
$$299$$ 2.00000 0.115663
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ 20.0000 1.15087
$$303$$ 10.0000 0.574485
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 2.00000 0.113776
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 2.00000 0.113228
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 8.00000 0.451466
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ 10.0000 0.558146
$$322$$ 2.00000 0.111456
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ −8.00000 −0.442401
$$328$$ −6.00000 −0.331295
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 10.0000 0.548821
$$333$$ −8.00000 −0.438397
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −8.00000 −0.434500
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 8.00000 0.432590
$$343$$ −20.0000 −1.07990
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ −10.0000 −0.536056
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 2.00000 0.106600
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −16.0000 −0.845626
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 16.0000 0.840941
$$363$$ −7.00000 −0.367405
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 12.0000 0.627250
$$367$$ −18.0000 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −20.0000 −1.03835
$$372$$ 8.00000 0.414781
$$373$$ 24.0000 1.24267 0.621336 0.783544i $$-0.286590\pi$$
0.621336 + 0.783544i $$0.286590\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ −20.0000 −1.03005
$$378$$ 2.00000 0.102869
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ −16.0000 −0.818631
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −12.0000 −0.609994
$$388$$ −10.0000 −0.507673
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ −8.00000 −0.403547
$$394$$ −26.0000 −1.30986
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ −16.0000 −0.799002 −0.399501 0.916733i $$-0.630817\pi$$
−0.399501 + 0.916733i $$0.630817\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 16.0000 0.797017
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −20.0000 −0.992583
$$407$$ −16.0000 −0.793091
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 2.00000 0.0985329
$$413$$ 8.00000 0.393654
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ −20.0000 −0.979404
$$418$$ 16.0000 0.782586
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −36.0000 −1.75453 −0.877266 0.480004i $$-0.840635\pi$$
−0.877266 + 0.480004i $$0.840635\pi$$
$$422$$ 4.00000 0.194717
$$423$$ −8.00000 −0.388973
$$424$$ −10.0000 −0.485643
$$425$$ 0 0
$$426$$ 16.0000 0.775203
$$427$$ 24.0000 1.16144
$$428$$ 10.0000 0.483368
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −8.00000 −0.383131
$$437$$ 8.00000 0.382692
$$438$$ 10.0000 0.477818
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 28.0000 1.33032 0.665160 0.746701i $$-0.268363\pi$$
0.665160 + 0.746701i $$0.268363\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ −10.0000 −0.472984
$$448$$ 2.00000 0.0944911
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ −8.00000 −0.376288
$$453$$ 20.0000 0.939682
$$454$$ −14.0000 −0.657053
$$455$$ 0 0
$$456$$ 8.00000 0.374634
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −16.0000 −0.747631
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 4.00000 0.186097
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ −10.0000 −0.464238
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 8.00000 0.368621
$$472$$ 4.00000 0.184115
$$473$$ −24.0000 −1.10352
$$474$$ 10.0000 0.459315
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ −16.0000 −0.731823
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 10.0000 0.455488
$$483$$ 2.00000 0.0910032
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 40.0000 1.81257 0.906287 0.422664i $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ 12.0000 0.543214
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ 0 0
$$494$$ 16.0000 0.719874
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 32.0000 1.43540
$$498$$ 10.0000 0.448111
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ −14.0000 −0.624851
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 2.00000 0.0889108
$$507$$ −9.00000 −0.399704
$$508$$ 4.00000 0.177471
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 20.0000 0.884748
$$512$$ 1.00000 0.0441942
$$513$$ 8.00000 0.353209
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ −12.0000 −0.528271
$$517$$ −16.0000 −0.703679
$$518$$ −16.0000 −0.703000
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ −10.0000 −0.437688
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ −4.00000 −0.174408
$$527$$ 0 0
$$528$$ 2.00000 0.0870388
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 16.0000 0.693688
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −16.0000 −0.690451
$$538$$ 18.0000 0.776035
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ 28.0000 1.20270
$$543$$ 16.0000 0.686626
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 4.00000 0.171184
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 12.0000 0.512148
$$550$$ 0 0
$$551$$ −80.0000 −3.40811
$$552$$ 1.00000 0.0425628
$$553$$ 20.0000 0.850487
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −16.0000 −0.674919
$$563$$ −6.00000 −0.252870 −0.126435 0.991975i $$-0.540353\pi$$
−0.126435 + 0.991975i $$0.540353\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 2.00000 0.0839921
$$568$$ 16.0000 0.671345
$$569$$ −32.0000 −1.34151 −0.670755 0.741679i $$-0.734030\pi$$
−0.670755 + 0.741679i $$0.734030\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 4.00000 0.167248
$$573$$ −16.0000 −0.668410
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 20.0000 0.829740
$$582$$ −10.0000 −0.414513
$$583$$ −20.0000 −0.828315
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ −2.00000 −0.0826192
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 64.0000 2.63707
$$590$$ 0 0
$$591$$ −26.0000 −1.06950
$$592$$ −8.00000 −0.328798
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ −14.0000 −0.572982
$$598$$ 2.00000 0.0817861
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ −24.0000 −0.978167
$$603$$ 4.00000 0.162893
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ −12.0000 −0.487065 −0.243532 0.969893i $$-0.578306\pi$$
−0.243532 + 0.969893i $$0.578306\pi$$
$$608$$ 8.00000 0.324443
$$609$$ −20.0000 −0.810441
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 0 0
$$613$$ 44.0000 1.77714 0.888572 0.458738i $$-0.151698\pi$$
0.888572 + 0.458738i $$0.151698\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ −48.0000 −1.93241 −0.966204 0.257780i $$-0.917009\pi$$
−0.966204 + 0.257780i $$0.917009\pi$$
$$618$$ 2.00000 0.0804518
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 16.0000 0.641542
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 16.0000 0.638978
$$628$$ 8.00000 0.319235
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −2.00000 −0.0796187 −0.0398094 0.999207i $$-0.512675\pi$$
−0.0398094 + 0.999207i $$0.512675\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 4.00000 0.158986
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ −10.0000 −0.396526
$$637$$ −6.00000 −0.237729
$$638$$ −20.0000 −0.791808
$$639$$ 16.0000 0.632950
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 10.0000 0.394669
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 40.0000 1.57256 0.786281 0.617869i $$-0.212004\pi$$
0.786281 + 0.617869i $$0.212004\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ −4.00000 −0.156652
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ −8.00000 −0.312825
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 10.0000 0.390137
$$658$$ −16.0000 −0.623745
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ −28.0000 −1.08907 −0.544537 0.838737i $$-0.683295\pi$$
−0.544537 + 0.838737i $$0.683295\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 0 0
$$664$$ 10.0000 0.388075
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ −10.0000 −0.387202
$$668$$ 8.00000 0.309529
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 2.00000 0.0771517
$$673$$ 46.0000 1.77317 0.886585 0.462566i $$-0.153071\pi$$
0.886585 + 0.462566i $$0.153071\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −46.0000 −1.76792 −0.883962 0.467559i $$-0.845134\pi$$
−0.883962 + 0.467559i $$0.845134\pi$$
$$678$$ −8.00000 −0.307238
$$679$$ −20.0000 −0.767530
$$680$$ 0 0
$$681$$ −14.0000 −0.536481
$$682$$ 16.0000 0.612672
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 8.00000 0.305888
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ −16.0000 −0.610438
$$688$$ −12.0000 −0.457496
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −52.0000 −1.97817 −0.989087 0.147335i $$-0.952930\pi$$
−0.989087 + 0.147335i $$0.952930\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 4.00000 0.151947
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ 0 0
$$698$$ −6.00000 −0.227103
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −64.0000 −2.41381
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 20.0000 0.752177
$$708$$ 4.00000 0.150329
$$709$$ 28.0000 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 0 0
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −16.0000 −0.597948
$$717$$ −16.0000 −0.597531
$$718$$ 8.00000 0.298557
$$719$$ 16.0000 0.596699 0.298350 0.954457i $$-0.403564\pi$$
0.298350 + 0.954457i $$0.403564\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 45.0000 1.67473
$$723$$ 10.0000 0.371904
$$724$$ 16.0000 0.594635
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 12.0000 0.443533
$$733$$ 52.0000 1.92066 0.960332 0.278859i $$-0.0899564\pi$$
0.960332 + 0.278859i $$0.0899564\pi$$
$$734$$ −18.0000 −0.664392
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 8.00000 0.294684
$$738$$ −6.00000 −0.220863
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ −20.0000 −0.734223
$$743$$ 28.0000 1.02722 0.513610 0.858024i $$-0.328308\pi$$
0.513610 + 0.858024i $$0.328308\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ 24.0000 0.878702
$$747$$ 10.0000 0.365881
$$748$$ 0 0
$$749$$ 20.0000 0.730784
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −14.0000 −0.510188
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 8.00000 0.290765 0.145382 0.989376i $$-0.453559\pi$$
0.145382 + 0.989376i $$0.453559\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 2.00000 0.0725954
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 4.00000 0.144905
$$763$$ −16.0000 −0.579239
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ −36.0000 −1.30073
$$767$$ 8.00000 0.288863
$$768$$ 1.00000 0.0360844
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ −14.0000 −0.503871
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ −16.0000 −0.573997
$$778$$ −2.00000 −0.0717035
$$779$$ −48.0000 −1.71978
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −8.00000 −0.285351
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ −26.0000 −0.926212
$$789$$ −4.00000 −0.142404
$$790$$ 0 0
$$791$$ −16.0000 −0.568895
$$792$$ 2.00000 0.0710669
$$793$$ 24.0000 0.852265
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ −14.0000 −0.496217
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 16.0000 0.566394
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −16.0000 −0.564980
$$803$$ 20.0000 0.705785
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ 18.0000 0.633630
$$808$$ 10.0000 0.351799
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ −20.0000 −0.701862
$$813$$ 28.0000 0.982003
$$814$$ −16.0000 −0.560800
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −96.0000 −3.35861
$$818$$ −10.0000 −0.349642
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 12.0000 0.418548
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ 2.00000 0.0696733
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 6.00000 0.208640 0.104320 0.994544i $$-0.466733\pi$$
0.104320 + 0.994544i $$0.466733\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 6.00000 0.208138
$$832$$ 2.00000 0.0693375
$$833$$ 0 0
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 8.00000 0.276520
$$838$$ 30.0000 1.03633
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −36.0000 −1.24064
$$843$$ −16.0000 −0.551069
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ −14.0000 −0.481046
$$848$$ −10.0000 −0.343401
$$849$$ 16.0000 0.549119
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 16.0000 0.548151
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 24.0000 0.821263
$$855$$ 0 0
$$856$$ 10.0000 0.341793
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 4.00000 0.136558
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ −28.0000 −0.953684
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ −17.0000 −0.577350
$$868$$ 16.0000 0.543075
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −8.00000 −0.270914
$$873$$ −10.0000 −0.338449
$$874$$ 8.00000 0.270604
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 16.0000 0.539974
$$879$$ −2.00000 −0.0674583
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ −8.00000 −0.268462
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 4.00000 0.133930
$$893$$ −64.0000 −2.14168
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 2.00000 0.0667781
$$898$$ 18.0000 0.600668
$$899$$ −80.0000 −2.66815
$$900$$ 0 0
$$901$$ 0 0
$$902$$ −12.0000 −0.399556
$$903$$ −24.0000 −0.798670
$$904$$ −8.00000 −0.266076
$$905$$ 0 0
$$906$$ 20.0000 0.664455
$$907$$ −40.0000 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ −14.0000 −0.464606
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 8.00000 0.264906
$$913$$ 20.0000 0.661903
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ −16.0000 −0.528367
$$918$$ 0 0
$$919$$ 26.0000 0.857661 0.428830 0.903385i $$-0.358926\pi$$
0.428830 + 0.903385i $$0.358926\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ −30.0000 −0.987997
$$923$$ 32.0000 1.05329
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 2.00000 0.0656886
$$928$$ −10.0000 −0.328266
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −24.0000 −0.786568
$$932$$ 6.00000 0.196537
$$933$$ 16.0000 0.523816
$$934$$ 6.00000 0.196326
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −34.0000 −1.10837 −0.554184 0.832394i $$-0.686970\pi$$
−0.554184 + 0.832394i $$0.686970\pi$$
$$942$$ 8.00000 0.260654
$$943$$ −6.00000 −0.195387
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −24.0000 −0.780307
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 10.0000 0.324785
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ −8.00000 −0.259145 −0.129573 0.991570i $$-0.541361\pi$$
−0.129573 + 0.991570i $$0.541361\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ −20.0000 −0.646508
$$958$$ 24.0000 0.775405
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −16.0000 −0.515861
$$963$$ 10.0000 0.322245
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 2.00000 0.0643489
$$967$$ 56.0000 1.80084 0.900419 0.435023i $$-0.143260\pi$$
0.900419 + 0.435023i $$0.143260\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 30.0000 0.962746 0.481373 0.876516i $$-0.340138\pi$$
0.481373 + 0.876516i $$0.340138\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −40.0000 −1.28234
$$974$$ 40.0000 1.28168
$$975$$ 0 0
$$976$$ 12.0000 0.384111
$$977$$ 20.0000 0.639857 0.319928 0.947442i $$-0.396341\pi$$
0.319928 + 0.947442i $$0.396341\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −8.00000 −0.255420
$$982$$ 0 0
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −16.0000 −0.509286
$$988$$ 16.0000 0.509028
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 8.00000 0.254000
$$993$$ −12.0000 −0.380808
$$994$$ 32.0000 1.01498
$$995$$ 0 0
$$996$$ 10.0000 0.316862
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ −8.00000 −0.253109
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 3450.2.a.z.1.1 1
5.2 odd 4 3450.2.d.q.2899.2 2
5.3 odd 4 3450.2.d.q.2899.1 2
5.4 even 2 690.2.a.c.1.1 1
15.14 odd 2 2070.2.a.k.1.1 1
20.19 odd 2 5520.2.a.bg.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.c.1.1 1 5.4 even 2
2070.2.a.k.1.1 1 15.14 odd 2
3450.2.a.z.1.1 1 1.1 even 1 trivial
3450.2.d.q.2899.1 2 5.3 odd 4
3450.2.d.q.2899.2 2 5.2 odd 4
5520.2.a.bg.1.1 1 20.19 odd 2