# Properties

 Label 1050.2.a.m.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1050.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +8.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -2.00000 q^{38} -2.00000 q^{39} +6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} -2.00000 q^{44} +4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -12.0000 q^{67} +8.00000 q^{68} -14.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} -2.00000 q^{76} -2.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +16.0000 q^{83} -1.00000 q^{84} +8.00000 q^{86} +6.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -6.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## 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
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 8.00000 1.94029 0.970143 0.242536i $$-0.0779791\pi$$
0.970143 + 0.242536i $$0.0779791\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ −2.00000 −0.426401
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.00000 0.348155
$$34$$ 8.00000 1.37199
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 2.00000 0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 2.00000 0.264906
$$58$$ −6.00000 −0.787839
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 8.00000 0.970143
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ −2.00000 −0.227921
$$78$$ −2.00000 −0.226455
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 6.00000 0.643268
$$88$$ −2.00000 −0.213201
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ −6.00000 −0.622171
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ −8.00000 −0.792118
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 1.00000 0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 2.00000 0.184900
$$118$$ −8.00000 −0.736460
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 10.0000 0.905357
$$123$$ −6.00000 −0.541002
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 2.00000 0.174078
$$133$$ −2.00000 −0.173422
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 8.00000 0.685994
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ −2.00000 −0.169638 −0.0848189 0.996396i $$-0.527031\pi$$
−0.0848189 + 0.996396i $$0.527031\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ −14.0000 −1.17485
$$143$$ −4.00000 −0.334497
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ −1.00000 −0.0824786
$$148$$ 8.00000 0.657596
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 8.00000 0.646762
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 8.00000 0.609994
$$173$$ −16.0000 −1.21646 −0.608229 0.793762i $$-0.708120\pi$$
−0.608229 + 0.793762i $$0.708120\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 8.00000 0.601317
$$178$$ 10.0000 0.749532
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −6.00000 −0.439941
$$187$$ −16.0000 −1.17004
$$188$$ 4.00000 0.291730
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 14.0000 1.01300 0.506502 0.862239i $$-0.330938\pi$$
0.506502 + 0.862239i $$0.330938\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ −26.0000 −1.84309 −0.921546 0.388270i $$-0.873073\pi$$
−0.921546 + 0.388270i $$0.873073\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ −10.0000 −0.703598
$$203$$ −6.00000 −0.421117
$$204$$ −8.00000 −0.560112
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 14.0000 0.959264
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 6.00000 0.407307
$$218$$ −6.00000 −0.406371
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 16.0000 1.07628
$$222$$ −8.00000 −0.536925
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 2.00000 0.132453
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 2.00000 0.131590
$$232$$ −6.00000 −0.393919
$$233$$ 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
$$236$$ −8.00000 −0.520756
$$237$$ −4.00000 −0.259828
$$238$$ 8.00000 0.518563
$$239$$ −22.0000 −1.42306 −0.711531 0.702655i $$-0.751998\pi$$
−0.711531 + 0.702655i $$0.751998\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ −4.00000 −0.254514
$$248$$ 6.00000 0.381000
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −20.0000 −1.24757 −0.623783 0.781598i $$-0.714405\pi$$
−0.623783 + 0.781598i $$0.714405\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ −20.0000 −1.23560
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ −10.0000 −0.611990
$$268$$ −12.0000 −0.733017
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ 8.00000 0.485071
$$273$$ −2.00000 −0.121046
$$274$$ −14.0000 −0.845771
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ −2.00000 −0.119952
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ −4.00000 −0.238197
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −14.0000 −0.830747
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 6.00000 0.354169
$$288$$ 1.00000 0.0589256
$$289$$ 47.0000 2.76471
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ −10.0000 −0.585206
$$293$$ −16.0000 −0.934730 −0.467365 0.884064i $$-0.654797\pi$$
−0.467365 + 0.884064i $$0.654797\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ 2.00000 0.116052
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 8.00000 0.460348
$$303$$ 10.0000 0.574485
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 8.00000 0.457330
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ −2.00000 −0.113961
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 2.00000 0.112154
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ −16.0000 −0.890264
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 6.00000 0.331801
$$328$$ 6.00000 0.331295
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 16.0000 0.878114
$$333$$ 8.00000 0.438397
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ −2.00000 −0.108148
$$343$$ 1.00000 0.0539949
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −16.0000 −0.860165
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ −2.00000 −0.106600
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ −8.00000 −0.423405
$$358$$ −2.00000 −0.105703
$$359$$ 14.0000 0.738892 0.369446 0.929252i $$-0.379548\pi$$
0.369446 + 0.929252i $$0.379548\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 22.0000 1.15629
$$363$$ 7.00000 0.367405
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ −6.00000 −0.311086
$$373$$ 36.0000 1.86401 0.932005 0.362446i $$-0.118058\pi$$
0.932005 + 0.362446i $$0.118058\pi$$
$$374$$ −16.0000 −0.827340
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ −12.0000 −0.618031
$$378$$ −1.00000 −0.0514344
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 14.0000 0.716302
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000 0.406663
$$388$$ 10.0000 0.507673
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1.00000 0.0505076
$$393$$ 20.0000 1.00887
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −26.0000 −1.30326
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 12.0000 0.597763
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ −16.0000 −0.793091
$$408$$ −8.00000 −0.396059
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 2.00000 0.0979404
$$418$$ 4.00000 0.195646
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 4.00000 0.194487
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ 14.0000 0.678302
$$427$$ 10.0000 0.483934
$$428$$ −12.0000 −0.580042
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 6.00000 0.288009
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ 0 0
$$438$$ 10.0000 0.477818
$$439$$ 6.00000 0.286364 0.143182 0.989696i $$-0.454267\pi$$
0.143182 + 0.989696i $$0.454267\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 16.0000 0.761042
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 18.0000 0.851371
$$448$$ 1.00000 0.0472456
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 6.00000 0.282216
$$453$$ −8.00000 −0.375873
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 2.00000 0.0936586
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ 26.0000 1.21490
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 2.00000 0.0930484
$$463$$ −12.0000 −0.557687 −0.278844 0.960337i $$-0.589951\pi$$
−0.278844 + 0.960337i $$0.589951\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 16.0000 0.740392 0.370196 0.928954i $$-0.379291\pi$$
0.370196 + 0.928954i $$0.379291\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ −8.00000 −0.368230
$$473$$ −16.0000 −0.735681
$$474$$ −4.00000 −0.183726
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ −2.00000 −0.0915737
$$478$$ −22.0000 −1.00626
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 36.0000 1.63132 0.815658 0.578535i $$-0.196375\pi$$
0.815658 + 0.578535i $$0.196375\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −42.0000 −1.89543 −0.947717 0.319113i $$-0.896615\pi$$
−0.947717 + 0.319113i $$0.896615\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −48.0000 −2.16181
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ −14.0000 −0.627986
$$498$$ −16.0000 −0.716977
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 12.0000 0.532414
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 1.00000 0.0441942
$$513$$ 2.00000 0.0883022
$$514$$ −20.0000 −0.882162
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ −8.00000 −0.351840
$$518$$ 8.00000 0.351500
$$519$$ 16.0000 0.702322
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −12.0000 −0.524723 −0.262362 0.964970i $$-0.584501\pi$$
−0.262362 + 0.964970i $$0.584501\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 48.0000 2.09091
$$528$$ 2.00000 0.0870388
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ −2.00000 −0.0867110
$$533$$ 12.0000 0.519778
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 2.00000 0.0863064
$$538$$ 14.0000 0.603583
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 2.00000 0.0859074
$$543$$ −22.0000 −0.944110
$$544$$ 8.00000 0.342997
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ −2.00000 −0.0848189
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 16.0000 0.675521
$$562$$ 22.0000 0.928014
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 1.00000 0.0419961
$$568$$ −14.0000 −0.587427
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ −14.0000 −0.584858
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 47.0000 1.95494
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 16.0000 0.663792
$$582$$ −10.0000 −0.414513
$$583$$ 4.00000 0.165663
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ −16.0000 −0.660954
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 8.00000 0.328798
$$593$$ −32.0000 −1.31408 −0.657041 0.753855i $$-0.728192\pi$$
−0.657041 + 0.753855i $$0.728192\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 26.0000 1.06411
$$598$$ 0 0
$$599$$ 14.0000 0.572024 0.286012 0.958226i $$-0.407670\pi$$
0.286012 + 0.958226i $$0.407670\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −12.0000 −0.488678
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 6.00000 0.243132
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 8.00000 0.323381
$$613$$ 32.0000 1.29247 0.646234 0.763139i $$-0.276343\pi$$
0.646234 + 0.763139i $$0.276343\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −18.0000 −0.723481 −0.361741 0.932279i $$-0.617817\pi$$
−0.361741 + 0.932279i $$0.617817\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −8.00000 −0.320771
$$623$$ 10.0000 0.400642
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ −4.00000 −0.159745
$$628$$ −10.0000 −0.399043
$$629$$ 64.0000 2.55185
$$630$$ 0 0
$$631$$ 12.0000 0.477712 0.238856 0.971055i $$-0.423228\pi$$
0.238856 + 0.971055i $$0.423228\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 8.00000 0.317971
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ 2.00000 0.0792429
$$638$$ 12.0000 0.475085
$$639$$ −14.0000 −0.553831
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −16.0000 −0.629512
$$647$$ 16.0000 0.629025 0.314512 0.949253i $$-0.398159\pi$$
0.314512 + 0.949253i $$0.398159\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ −6.00000 −0.235159
$$652$$ 0 0
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ −10.0000 −0.390137
$$658$$ 4.00000 0.155936
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 0 0
$$663$$ −16.0000 −0.621389
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 0 0
$$668$$ −8.00000 −0.309529
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ −20.0000 −0.772091
$$672$$ −1.00000 −0.0385758
$$673$$ 12.0000 0.462566 0.231283 0.972887i $$-0.425708\pi$$
0.231283 + 0.972887i $$0.425708\pi$$
$$674$$ 32.0000 1.23259
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ −12.0000 −0.459504
$$683$$ 28.0000 1.07139 0.535695 0.844411i $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ −26.0000 −0.991962
$$688$$ 8.00000 0.304997
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ −16.0000 −0.608229
$$693$$ −2.00000 −0.0759737
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 48.0000 1.81813
$$698$$ 18.0000 0.681310
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −16.0000 −0.603451
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −10.0000 −0.376089
$$708$$ 8.00000 0.300658
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 10.0000 0.374766
$$713$$ 0 0
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ −2.00000 −0.0747435
$$717$$ 22.0000 0.821605
$$718$$ 14.0000 0.522475
$$719$$ −20.0000 −0.745874 −0.372937 0.927857i $$-0.621649\pi$$
−0.372937 + 0.927857i $$0.621649\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −15.0000 −0.558242
$$723$$ 10.0000 0.371904
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 64.0000 2.36713
$$732$$ −10.0000 −0.369611
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24.0000 0.884051
$$738$$ 6.00000 0.220863
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ −2.00000 −0.0734223
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 0 0
$$746$$ 36.0000 1.31805
$$747$$ 16.0000 0.585409
$$748$$ −16.0000 −0.585018
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 4.00000 0.145865
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ −4.00000 −0.145382 −0.0726912 0.997354i $$-0.523159\pi$$
−0.0726912 + 0.997354i $$0.523159\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ −6.00000 −0.217215
$$764$$ 14.0000 0.506502
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −16.0000 −0.577727
$$768$$ −1.00000 −0.0360844
$$769$$ −42.0000 −1.51456 −0.757279 0.653091i $$-0.773472\pi$$
−0.757279 + 0.653091i $$0.773472\pi$$
$$770$$ 0 0
$$771$$ 20.0000 0.720282
$$772$$ 0 0
$$773$$ −32.0000 −1.15096 −0.575480 0.817816i $$-0.695185\pi$$
−0.575480 + 0.817816i $$0.695185\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ −8.00000 −0.286998
$$778$$ 6.00000 0.215110
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 28.0000 1.00192
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ −2.00000 −0.0710669
$$793$$ 20.0000 0.710221
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −26.0000 −0.921546
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 2.00000 0.0707992
$$799$$ 32.0000 1.13208
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 18.0000 0.635602
$$803$$ 20.0000 0.705785
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 12.0000 0.422682
$$807$$ −14.0000 −0.492823
$$808$$ −10.0000 −0.351799
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 14.0000 0.491606 0.245803 0.969320i $$-0.420948\pi$$
0.245803 + 0.969320i $$0.420948\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ −2.00000 −0.0701431
$$814$$ −16.0000 −0.560800
$$815$$ 0 0
$$816$$ −8.00000 −0.280056
$$817$$ −16.0000 −0.559769
$$818$$ 10.0000 0.349642
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 14.0000 0.488306
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 2.00000 0.0693375
$$833$$ 8.00000 0.277184
$$834$$ 2.00000 0.0692543
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ −6.00000 −0.207390
$$838$$ 12.0000 0.414533
$$839$$ 44.0000 1.51905 0.759524 0.650479i $$-0.225432\pi$$
0.759524 + 0.650479i $$0.225432\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ −22.0000 −0.757720
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ −7.00000 −0.240523
$$848$$ −2.00000 −0.0686803
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 14.0000 0.479632
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 20.0000 0.683187 0.341593 0.939848i $$-0.389033\pi$$
0.341593 + 0.939848i $$0.389033\pi$$
$$858$$ 4.00000 0.136558
$$859$$ −6.00000 −0.204717 −0.102359 0.994748i $$-0.532639\pi$$
−0.102359 + 0.994748i $$0.532639\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ −6.00000 −0.204361
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ −47.0000 −1.59620
$$868$$ 6.00000 0.203653
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ −6.00000 −0.203186
$$873$$ 10.0000 0.338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ −48.0000 −1.62084 −0.810422 0.585846i $$-0.800762\pi$$
−0.810422 + 0.585846i $$0.800762\pi$$
$$878$$ 6.00000 0.202490
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 16.0000 0.538138
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ −8.00000 −0.268462
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ −16.0000 −0.535720
$$893$$ −8.00000 −0.267710
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ −36.0000 −1.20067
$$900$$ 0 0
$$901$$ −16.0000 −0.533037
$$902$$ −12.0000 −0.399556
$$903$$ −8.00000 −0.266223
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ −32.0000 −1.05905
$$914$$ −28.0000 −0.926158
$$915$$ 0 0
$$916$$ 26.0000 0.859064
$$917$$ −20.0000 −0.660458
$$918$$ −8.00000 −0.264039
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 18.0000 0.592798
$$923$$ −28.0000 −0.921631
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ −12.0000 −0.394344
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 6.00000 0.196537
$$933$$ 8.00000 0.261908
$$934$$ 16.0000 0.523536
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 10.0000 0.325818
$$943$$ 0 0
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ −4.00000 −0.129914
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 8.00000 0.259281
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ −22.0000 −0.711531
$$957$$ −12.0000 −0.387905
$$958$$ 8.00000 0.258468
$$959$$ −14.0000 −0.452084
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 16.0000 0.515861
$$963$$ −12.0000 −0.386695
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −24.0000 −0.771788 −0.385894 0.922543i $$-0.626107\pi$$
−0.385894 + 0.922543i $$0.626107\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 16.0000 0.513994
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −2.00000 −0.0641171
$$974$$ 36.0000 1.15351
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ −42.0000 −1.34027
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −48.0000 −1.52863
$$987$$ −4.00000 −0.127321
$$988$$ −4.00000 −0.127257
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −60.0000 −1.90596 −0.952981 0.303029i $$-0.902002\pi$$
−0.952981 + 0.303029i $$0.902002\pi$$
$$992$$ 6.00000 0.190500
$$993$$ 0 0
$$994$$ −14.0000 −0.444053
$$995$$ 0 0
$$996$$ −16.0000 −0.506979
$$997$$ −38.0000 −1.20347 −0.601736 0.798695i $$-0.705524\pi$$
−0.601736 + 0.798695i $$0.705524\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ −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 1050.2.a.m.1.1 1
3.2 odd 2 3150.2.a.q.1.1 1
4.3 odd 2 8400.2.a.ca.1.1 1
5.2 odd 4 210.2.g.a.169.2 yes 2
5.3 odd 4 210.2.g.a.169.1 2
5.4 even 2 1050.2.a.g.1.1 1
7.6 odd 2 7350.2.a.co.1.1 1
15.2 even 4 630.2.g.d.379.1 2
15.8 even 4 630.2.g.d.379.2 2
15.14 odd 2 3150.2.a.be.1.1 1
20.3 even 4 1680.2.t.d.1009.2 2
20.7 even 4 1680.2.t.d.1009.1 2
20.19 odd 2 8400.2.a.bd.1.1 1
35.2 odd 12 1470.2.n.g.949.2 4
35.3 even 12 1470.2.n.c.79.2 4
35.12 even 12 1470.2.n.c.949.2 4
35.13 even 4 1470.2.g.e.589.1 2
35.17 even 12 1470.2.n.c.79.1 4
35.18 odd 12 1470.2.n.g.79.2 4
35.23 odd 12 1470.2.n.g.949.1 4
35.27 even 4 1470.2.g.e.589.2 2
35.32 odd 12 1470.2.n.g.79.1 4
35.33 even 12 1470.2.n.c.949.1 4
35.34 odd 2 7350.2.a.g.1.1 1
60.23 odd 4 5040.2.t.k.1009.2 2
60.47 odd 4 5040.2.t.k.1009.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.g.a.169.1 2 5.3 odd 4
210.2.g.a.169.2 yes 2 5.2 odd 4
630.2.g.d.379.1 2 15.2 even 4
630.2.g.d.379.2 2 15.8 even 4
1050.2.a.g.1.1 1 5.4 even 2
1050.2.a.m.1.1 1 1.1 even 1 trivial
1470.2.g.e.589.1 2 35.13 even 4
1470.2.g.e.589.2 2 35.27 even 4
1470.2.n.c.79.1 4 35.17 even 12
1470.2.n.c.79.2 4 35.3 even 12
1470.2.n.c.949.1 4 35.33 even 12
1470.2.n.c.949.2 4 35.12 even 12
1470.2.n.g.79.1 4 35.32 odd 12
1470.2.n.g.79.2 4 35.18 odd 12
1470.2.n.g.949.1 4 35.23 odd 12
1470.2.n.g.949.2 4 35.2 odd 12
1680.2.t.d.1009.1 2 20.7 even 4
1680.2.t.d.1009.2 2 20.3 even 4
3150.2.a.q.1.1 1 3.2 odd 2
3150.2.a.be.1.1 1 15.14 odd 2
5040.2.t.k.1009.1 2 60.47 odd 4
5040.2.t.k.1009.2 2 60.23 odd 4
7350.2.a.g.1.1 1 35.34 odd 2
7350.2.a.co.1.1 1 7.6 odd 2
8400.2.a.bd.1.1 1 20.19 odd 2
8400.2.a.ca.1.1 1 4.3 odd 2