# Properties

 Label 165.2.a.a.1.1 Level $165$ Weight $2$ Character 165.1 Self dual yes Analytic conductor $1.318$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.31753163335$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 165.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -1.00000 q^{11} -3.82843 q^{12} -5.65685 q^{13} -2.00000 q^{14} +1.00000 q^{15} +3.00000 q^{16} -1.17157 q^{17} -2.41421 q^{18} -6.82843 q^{19} -3.82843 q^{20} -0.828427 q^{21} +2.41421 q^{22} -4.00000 q^{23} +4.41421 q^{24} +1.00000 q^{25} +13.6569 q^{26} -1.00000 q^{27} +3.17157 q^{28} -4.82843 q^{29} -2.41421 q^{30} +1.58579 q^{32} +1.00000 q^{33} +2.82843 q^{34} -0.828427 q^{35} +3.82843 q^{36} +11.6569 q^{37} +16.4853 q^{38} +5.65685 q^{39} +4.41421 q^{40} +4.82843 q^{41} +2.00000 q^{42} -8.82843 q^{43} -3.82843 q^{44} -1.00000 q^{45} +9.65685 q^{46} -4.00000 q^{47} -3.00000 q^{48} -6.31371 q^{49} -2.41421 q^{50} +1.17157 q^{51} -21.6569 q^{52} +9.31371 q^{53} +2.41421 q^{54} +1.00000 q^{55} -3.65685 q^{56} +6.82843 q^{57} +11.6569 q^{58} -4.00000 q^{59} +3.82843 q^{60} -11.6569 q^{61} +0.828427 q^{63} -9.82843 q^{64} +5.65685 q^{65} -2.41421 q^{66} -5.65685 q^{67} -4.48528 q^{68} +4.00000 q^{69} +2.00000 q^{70} +2.34315 q^{71} -4.41421 q^{72} +11.3137 q^{73} -28.1421 q^{74} -1.00000 q^{75} -26.1421 q^{76} -0.828427 q^{77} -13.6569 q^{78} +8.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} -11.6569 q^{82} -10.0000 q^{83} -3.17157 q^{84} +1.17157 q^{85} +21.3137 q^{86} +4.82843 q^{87} +4.41421 q^{88} +3.65685 q^{89} +2.41421 q^{90} -4.68629 q^{91} -15.3137 q^{92} +9.65685 q^{94} +6.82843 q^{95} -1.58579 q^{96} +11.6569 q^{97} +15.2426 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + 2q^{10} - 2q^{11} - 2q^{12} - 4q^{14} + 2q^{15} + 6q^{16} - 8q^{17} - 2q^{18} - 8q^{19} - 2q^{20} + 4q^{21} + 2q^{22} - 8q^{23} + 6q^{24} + 2q^{25} + 16q^{26} - 2q^{27} + 12q^{28} - 4q^{29} - 2q^{30} + 6q^{32} + 2q^{33} + 4q^{35} + 2q^{36} + 12q^{37} + 16q^{38} + 6q^{40} + 4q^{41} + 4q^{42} - 12q^{43} - 2q^{44} - 2q^{45} + 8q^{46} - 8q^{47} - 6q^{48} + 10q^{49} - 2q^{50} + 8q^{51} - 32q^{52} - 4q^{53} + 2q^{54} + 2q^{55} + 4q^{56} + 8q^{57} + 12q^{58} - 8q^{59} + 2q^{60} - 12q^{61} - 4q^{63} - 14q^{64} - 2q^{66} + 8q^{68} + 8q^{69} + 4q^{70} + 16q^{71} - 6q^{72} - 28q^{74} - 2q^{75} - 24q^{76} + 4q^{77} - 16q^{78} - 6q^{80} + 2q^{81} - 12q^{82} - 20q^{83} - 12q^{84} + 8q^{85} + 20q^{86} + 4q^{87} + 6q^{88} - 4q^{89} + 2q^{90} - 32q^{91} - 8q^{92} + 8q^{94} + 8q^{95} - 6q^{96} + 12q^{97} + 22q^{98} - 2q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.41421 −1.70711 −0.853553 0.521005i $$-0.825557\pi$$
−0.853553 + 0.521005i $$0.825557\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 3.82843 1.91421
$$5$$ −1.00000 −0.447214
$$6$$ 2.41421 0.985599
$$7$$ 0.828427 0.313116 0.156558 0.987669i $$-0.449960\pi$$
0.156558 + 0.987669i $$0.449960\pi$$
$$8$$ −4.41421 −1.56066
$$9$$ 1.00000 0.333333
$$10$$ 2.41421 0.763441
$$11$$ −1.00000 −0.301511
$$12$$ −3.82843 −1.10517
$$13$$ −5.65685 −1.56893 −0.784465 0.620174i $$-0.787062\pi$$
−0.784465 + 0.620174i $$0.787062\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 1.00000 0.258199
$$16$$ 3.00000 0.750000
$$17$$ −1.17157 −0.284148 −0.142074 0.989856i $$-0.545377\pi$$
−0.142074 + 0.989856i $$0.545377\pi$$
$$18$$ −2.41421 −0.569036
$$19$$ −6.82843 −1.56655 −0.783274 0.621676i $$-0.786452\pi$$
−0.783274 + 0.621676i $$0.786452\pi$$
$$20$$ −3.82843 −0.856062
$$21$$ −0.828427 −0.180778
$$22$$ 2.41421 0.514712
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 4.41421 0.901048
$$25$$ 1.00000 0.200000
$$26$$ 13.6569 2.67833
$$27$$ −1.00000 −0.192450
$$28$$ 3.17157 0.599371
$$29$$ −4.82843 −0.896616 −0.448308 0.893879i $$-0.647973\pi$$
−0.448308 + 0.893879i $$0.647973\pi$$
$$30$$ −2.41421 −0.440773
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.58579 0.280330
$$33$$ 1.00000 0.174078
$$34$$ 2.82843 0.485071
$$35$$ −0.828427 −0.140030
$$36$$ 3.82843 0.638071
$$37$$ 11.6569 1.91638 0.958188 0.286141i $$-0.0923726\pi$$
0.958188 + 0.286141i $$0.0923726\pi$$
$$38$$ 16.4853 2.67427
$$39$$ 5.65685 0.905822
$$40$$ 4.41421 0.697948
$$41$$ 4.82843 0.754074 0.377037 0.926198i $$-0.376943\pi$$
0.377037 + 0.926198i $$0.376943\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −8.82843 −1.34632 −0.673161 0.739496i $$-0.735064\pi$$
−0.673161 + 0.739496i $$0.735064\pi$$
$$44$$ −3.82843 −0.577157
$$45$$ −1.00000 −0.149071
$$46$$ 9.65685 1.42383
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ −3.00000 −0.433013
$$49$$ −6.31371 −0.901958
$$50$$ −2.41421 −0.341421
$$51$$ 1.17157 0.164053
$$52$$ −21.6569 −3.00327
$$53$$ 9.31371 1.27934 0.639668 0.768651i $$-0.279072\pi$$
0.639668 + 0.768651i $$0.279072\pi$$
$$54$$ 2.41421 0.328533
$$55$$ 1.00000 0.134840
$$56$$ −3.65685 −0.488668
$$57$$ 6.82843 0.904447
$$58$$ 11.6569 1.53062
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 3.82843 0.494248
$$61$$ −11.6569 −1.49251 −0.746254 0.665662i $$-0.768149\pi$$
−0.746254 + 0.665662i $$0.768149\pi$$
$$62$$ 0 0
$$63$$ 0.828427 0.104372
$$64$$ −9.82843 −1.22855
$$65$$ 5.65685 0.701646
$$66$$ −2.41421 −0.297169
$$67$$ −5.65685 −0.691095 −0.345547 0.938401i $$-0.612307\pi$$
−0.345547 + 0.938401i $$0.612307\pi$$
$$68$$ −4.48528 −0.543920
$$69$$ 4.00000 0.481543
$$70$$ 2.00000 0.239046
$$71$$ 2.34315 0.278080 0.139040 0.990287i $$-0.455598\pi$$
0.139040 + 0.990287i $$0.455598\pi$$
$$72$$ −4.41421 −0.520220
$$73$$ 11.3137 1.32417 0.662085 0.749429i $$-0.269672\pi$$
0.662085 + 0.749429i $$0.269672\pi$$
$$74$$ −28.1421 −3.27146
$$75$$ −1.00000 −0.115470
$$76$$ −26.1421 −2.99871
$$77$$ −0.828427 −0.0944080
$$78$$ −13.6569 −1.54633
$$79$$ 8.48528 0.954669 0.477334 0.878722i $$-0.341603\pi$$
0.477334 + 0.878722i $$0.341603\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ −11.6569 −1.28728
$$83$$ −10.0000 −1.09764 −0.548821 0.835940i $$-0.684923\pi$$
−0.548821 + 0.835940i $$0.684923\pi$$
$$84$$ −3.17157 −0.346047
$$85$$ 1.17157 0.127075
$$86$$ 21.3137 2.29832
$$87$$ 4.82843 0.517662
$$88$$ 4.41421 0.470557
$$89$$ 3.65685 0.387626 0.193813 0.981039i $$-0.437915\pi$$
0.193813 + 0.981039i $$0.437915\pi$$
$$90$$ 2.41421 0.254480
$$91$$ −4.68629 −0.491257
$$92$$ −15.3137 −1.59656
$$93$$ 0 0
$$94$$ 9.65685 0.996028
$$95$$ 6.82843 0.700582
$$96$$ −1.58579 −0.161849
$$97$$ 11.6569 1.18357 0.591787 0.806094i $$-0.298423\pi$$
0.591787 + 0.806094i $$0.298423\pi$$
$$98$$ 15.2426 1.53974
$$99$$ −1.00000 −0.100504
$$100$$ 3.82843 0.382843
$$101$$ −0.828427 −0.0824316 −0.0412158 0.999150i $$-0.513123\pi$$
−0.0412158 + 0.999150i $$0.513123\pi$$
$$102$$ −2.82843 −0.280056
$$103$$ −3.31371 −0.326509 −0.163255 0.986584i $$-0.552199\pi$$
−0.163255 + 0.986584i $$0.552199\pi$$
$$104$$ 24.9706 2.44857
$$105$$ 0.828427 0.0808462
$$106$$ −22.4853 −2.18396
$$107$$ 17.3137 1.67378 0.836890 0.547372i $$-0.184372\pi$$
0.836890 + 0.547372i $$0.184372\pi$$
$$108$$ −3.82843 −0.368391
$$109$$ 17.3137 1.65835 0.829176 0.558987i $$-0.188810\pi$$
0.829176 + 0.558987i $$0.188810\pi$$
$$110$$ −2.41421 −0.230186
$$111$$ −11.6569 −1.10642
$$112$$ 2.48528 0.234837
$$113$$ −18.9706 −1.78460 −0.892300 0.451442i $$-0.850910\pi$$
−0.892300 + 0.451442i $$0.850910\pi$$
$$114$$ −16.4853 −1.54399
$$115$$ 4.00000 0.373002
$$116$$ −18.4853 −1.71632
$$117$$ −5.65685 −0.522976
$$118$$ 9.65685 0.888985
$$119$$ −0.970563 −0.0889713
$$120$$ −4.41421 −0.402961
$$121$$ 1.00000 0.0909091
$$122$$ 28.1421 2.54787
$$123$$ −4.82843 −0.435365
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ −14.4853 −1.28536 −0.642680 0.766134i $$-0.722178\pi$$
−0.642680 + 0.766134i $$0.722178\pi$$
$$128$$ 20.5563 1.81694
$$129$$ 8.82843 0.777300
$$130$$ −13.6569 −1.19779
$$131$$ 3.31371 0.289520 0.144760 0.989467i $$-0.453759\pi$$
0.144760 + 0.989467i $$0.453759\pi$$
$$132$$ 3.82843 0.333222
$$133$$ −5.65685 −0.490511
$$134$$ 13.6569 1.17977
$$135$$ 1.00000 0.0860663
$$136$$ 5.17157 0.443459
$$137$$ −13.3137 −1.13747 −0.568733 0.822522i $$-0.692566\pi$$
−0.568733 + 0.822522i $$0.692566\pi$$
$$138$$ −9.65685 −0.822046
$$139$$ 0.485281 0.0411610 0.0205805 0.999788i $$-0.493449\pi$$
0.0205805 + 0.999788i $$0.493449\pi$$
$$140$$ −3.17157 −0.268047
$$141$$ 4.00000 0.336861
$$142$$ −5.65685 −0.474713
$$143$$ 5.65685 0.473050
$$144$$ 3.00000 0.250000
$$145$$ 4.82843 0.400979
$$146$$ −27.3137 −2.26050
$$147$$ 6.31371 0.520746
$$148$$ 44.6274 3.66835
$$149$$ 1.51472 0.124091 0.0620453 0.998073i $$-0.480238\pi$$
0.0620453 + 0.998073i $$0.480238\pi$$
$$150$$ 2.41421 0.197120
$$151$$ 16.4853 1.34155 0.670777 0.741659i $$-0.265961\pi$$
0.670777 + 0.741659i $$0.265961\pi$$
$$152$$ 30.1421 2.44485
$$153$$ −1.17157 −0.0947161
$$154$$ 2.00000 0.161165
$$155$$ 0 0
$$156$$ 21.6569 1.73394
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −20.4853 −1.62972
$$159$$ −9.31371 −0.738625
$$160$$ −1.58579 −0.125367
$$161$$ −3.31371 −0.261157
$$162$$ −2.41421 −0.189679
$$163$$ −7.31371 −0.572854 −0.286427 0.958102i $$-0.592468\pi$$
−0.286427 + 0.958102i $$0.592468\pi$$
$$164$$ 18.4853 1.44346
$$165$$ −1.00000 −0.0778499
$$166$$ 24.1421 1.87379
$$167$$ −13.3137 −1.03025 −0.515123 0.857116i $$-0.672254\pi$$
−0.515123 + 0.857116i $$0.672254\pi$$
$$168$$ 3.65685 0.282132
$$169$$ 19.0000 1.46154
$$170$$ −2.82843 −0.216930
$$171$$ −6.82843 −0.522183
$$172$$ −33.7990 −2.57715
$$173$$ 2.82843 0.215041 0.107521 0.994203i $$-0.465709\pi$$
0.107521 + 0.994203i $$0.465709\pi$$
$$174$$ −11.6569 −0.883704
$$175$$ 0.828427 0.0626232
$$176$$ −3.00000 −0.226134
$$177$$ 4.00000 0.300658
$$178$$ −8.82843 −0.661719
$$179$$ −17.6569 −1.31974 −0.659868 0.751382i $$-0.729388\pi$$
−0.659868 + 0.751382i $$0.729388\pi$$
$$180$$ −3.82843 −0.285354
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 11.3137 0.838628
$$183$$ 11.6569 0.861699
$$184$$ 17.6569 1.30168
$$185$$ −11.6569 −0.857029
$$186$$ 0 0
$$187$$ 1.17157 0.0856739
$$188$$ −15.3137 −1.11687
$$189$$ −0.828427 −0.0602592
$$190$$ −16.4853 −1.19597
$$191$$ 5.65685 0.409316 0.204658 0.978834i $$-0.434392\pi$$
0.204658 + 0.978834i $$0.434392\pi$$
$$192$$ 9.82843 0.709306
$$193$$ 13.6569 0.983042 0.491521 0.870866i $$-0.336441\pi$$
0.491521 + 0.870866i $$0.336441\pi$$
$$194$$ −28.1421 −2.02049
$$195$$ −5.65685 −0.405096
$$196$$ −24.1716 −1.72654
$$197$$ −8.48528 −0.604551 −0.302276 0.953221i $$-0.597746\pi$$
−0.302276 + 0.953221i $$0.597746\pi$$
$$198$$ 2.41421 0.171571
$$199$$ −21.6569 −1.53521 −0.767607 0.640921i $$-0.778553\pi$$
−0.767607 + 0.640921i $$0.778553\pi$$
$$200$$ −4.41421 −0.312132
$$201$$ 5.65685 0.399004
$$202$$ 2.00000 0.140720
$$203$$ −4.00000 −0.280745
$$204$$ 4.48528 0.314033
$$205$$ −4.82843 −0.337232
$$206$$ 8.00000 0.557386
$$207$$ −4.00000 −0.278019
$$208$$ −16.9706 −1.17670
$$209$$ 6.82843 0.472332
$$210$$ −2.00000 −0.138013
$$211$$ 1.17157 0.0806544 0.0403272 0.999187i $$-0.487160\pi$$
0.0403272 + 0.999187i $$0.487160\pi$$
$$212$$ 35.6569 2.44892
$$213$$ −2.34315 −0.160550
$$214$$ −41.7990 −2.85732
$$215$$ 8.82843 0.602094
$$216$$ 4.41421 0.300349
$$217$$ 0 0
$$218$$ −41.7990 −2.83098
$$219$$ −11.3137 −0.764510
$$220$$ 3.82843 0.258113
$$221$$ 6.62742 0.445808
$$222$$ 28.1421 1.88878
$$223$$ −6.34315 −0.424768 −0.212384 0.977186i $$-0.568123\pi$$
−0.212384 + 0.977186i $$0.568123\pi$$
$$224$$ 1.31371 0.0877758
$$225$$ 1.00000 0.0666667
$$226$$ 45.7990 3.04650
$$227$$ −14.0000 −0.929213 −0.464606 0.885517i $$-0.653804\pi$$
−0.464606 + 0.885517i $$0.653804\pi$$
$$228$$ 26.1421 1.73131
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ −9.65685 −0.636754
$$231$$ 0.828427 0.0545065
$$232$$ 21.3137 1.39931
$$233$$ −18.8284 −1.23349 −0.616746 0.787163i $$-0.711549\pi$$
−0.616746 + 0.787163i $$0.711549\pi$$
$$234$$ 13.6569 0.892776
$$235$$ 4.00000 0.260931
$$236$$ −15.3137 −0.996838
$$237$$ −8.48528 −0.551178
$$238$$ 2.34315 0.151884
$$239$$ 17.6569 1.14213 0.571063 0.820906i $$-0.306531\pi$$
0.571063 + 0.820906i $$0.306531\pi$$
$$240$$ 3.00000 0.193649
$$241$$ −12.3431 −0.795092 −0.397546 0.917582i $$-0.630138\pi$$
−0.397546 + 0.917582i $$0.630138\pi$$
$$242$$ −2.41421 −0.155192
$$243$$ −1.00000 −0.0641500
$$244$$ −44.6274 −2.85698
$$245$$ 6.31371 0.403368
$$246$$ 11.6569 0.743214
$$247$$ 38.6274 2.45780
$$248$$ 0 0
$$249$$ 10.0000 0.633724
$$250$$ 2.41421 0.152688
$$251$$ 20.9706 1.32365 0.661825 0.749658i $$-0.269782\pi$$
0.661825 + 0.749658i $$0.269782\pi$$
$$252$$ 3.17157 0.199790
$$253$$ 4.00000 0.251478
$$254$$ 34.9706 2.19425
$$255$$ −1.17157 −0.0733667
$$256$$ −29.9706 −1.87316
$$257$$ −16.3431 −1.01946 −0.509729 0.860335i $$-0.670254\pi$$
−0.509729 + 0.860335i $$0.670254\pi$$
$$258$$ −21.3137 −1.32693
$$259$$ 9.65685 0.600048
$$260$$ 21.6569 1.34310
$$261$$ −4.82843 −0.298872
$$262$$ −8.00000 −0.494242
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ −4.41421 −0.271676
$$265$$ −9.31371 −0.572137
$$266$$ 13.6569 0.837355
$$267$$ −3.65685 −0.223796
$$268$$ −21.6569 −1.32290
$$269$$ 20.6274 1.25768 0.628838 0.777536i $$-0.283531\pi$$
0.628838 + 0.777536i $$0.283531\pi$$
$$270$$ −2.41421 −0.146924
$$271$$ −11.7990 −0.716738 −0.358369 0.933580i $$-0.616667\pi$$
−0.358369 + 0.933580i $$0.616667\pi$$
$$272$$ −3.51472 −0.213111
$$273$$ 4.68629 0.283627
$$274$$ 32.1421 1.94178
$$275$$ −1.00000 −0.0603023
$$276$$ 15.3137 0.921777
$$277$$ −2.34315 −0.140786 −0.0703930 0.997519i $$-0.522425\pi$$
−0.0703930 + 0.997519i $$0.522425\pi$$
$$278$$ −1.17157 −0.0702663
$$279$$ 0 0
$$280$$ 3.65685 0.218539
$$281$$ −11.1716 −0.666440 −0.333220 0.942849i $$-0.608135\pi$$
−0.333220 + 0.942849i $$0.608135\pi$$
$$282$$ −9.65685 −0.575057
$$283$$ −8.82843 −0.524796 −0.262398 0.964960i $$-0.584513\pi$$
−0.262398 + 0.964960i $$0.584513\pi$$
$$284$$ 8.97056 0.532305
$$285$$ −6.82843 −0.404481
$$286$$ −13.6569 −0.807547
$$287$$ 4.00000 0.236113
$$288$$ 1.58579 0.0934434
$$289$$ −15.6274 −0.919260
$$290$$ −11.6569 −0.684514
$$291$$ −11.6569 −0.683337
$$292$$ 43.3137 2.53474
$$293$$ −6.82843 −0.398921 −0.199460 0.979906i $$-0.563919\pi$$
−0.199460 + 0.979906i $$0.563919\pi$$
$$294$$ −15.2426 −0.888969
$$295$$ 4.00000 0.232889
$$296$$ −51.4558 −2.99081
$$297$$ 1.00000 0.0580259
$$298$$ −3.65685 −0.211836
$$299$$ 22.6274 1.30858
$$300$$ −3.82843 −0.221034
$$301$$ −7.31371 −0.421555
$$302$$ −39.7990 −2.29017
$$303$$ 0.828427 0.0475919
$$304$$ −20.4853 −1.17491
$$305$$ 11.6569 0.667470
$$306$$ 2.82843 0.161690
$$307$$ −3.17157 −0.181011 −0.0905056 0.995896i $$-0.528848\pi$$
−0.0905056 + 0.995896i $$0.528848\pi$$
$$308$$ −3.17157 −0.180717
$$309$$ 3.31371 0.188510
$$310$$ 0 0
$$311$$ −3.31371 −0.187903 −0.0939516 0.995577i $$-0.529950\pi$$
−0.0939516 + 0.995577i $$0.529950\pi$$
$$312$$ −24.9706 −1.41368
$$313$$ 15.6569 0.884978 0.442489 0.896774i $$-0.354096\pi$$
0.442489 + 0.896774i $$0.354096\pi$$
$$314$$ −43.4558 −2.45236
$$315$$ −0.828427 −0.0466766
$$316$$ 32.4853 1.82744
$$317$$ −26.2843 −1.47627 −0.738136 0.674652i $$-0.764294\pi$$
−0.738136 + 0.674652i $$0.764294\pi$$
$$318$$ 22.4853 1.26091
$$319$$ 4.82843 0.270340
$$320$$ 9.82843 0.549426
$$321$$ −17.3137 −0.966357
$$322$$ 8.00000 0.445823
$$323$$ 8.00000 0.445132
$$324$$ 3.82843 0.212690
$$325$$ −5.65685 −0.313786
$$326$$ 17.6569 0.977923
$$327$$ −17.3137 −0.957450
$$328$$ −21.3137 −1.17685
$$329$$ −3.31371 −0.182691
$$330$$ 2.41421 0.132898
$$331$$ 6.34315 0.348651 0.174325 0.984688i $$-0.444226\pi$$
0.174325 + 0.984688i $$0.444226\pi$$
$$332$$ −38.2843 −2.10112
$$333$$ 11.6569 0.638792
$$334$$ 32.1421 1.75874
$$335$$ 5.65685 0.309067
$$336$$ −2.48528 −0.135583
$$337$$ 3.31371 0.180509 0.0902546 0.995919i $$-0.471232\pi$$
0.0902546 + 0.995919i $$0.471232\pi$$
$$338$$ −45.8701 −2.49500
$$339$$ 18.9706 1.03034
$$340$$ 4.48528 0.243249
$$341$$ 0 0
$$342$$ 16.4853 0.891422
$$343$$ −11.0294 −0.595534
$$344$$ 38.9706 2.10115
$$345$$ −4.00000 −0.215353
$$346$$ −6.82843 −0.367099
$$347$$ −29.3137 −1.57364 −0.786821 0.617181i $$-0.788275\pi$$
−0.786821 + 0.617181i $$0.788275\pi$$
$$348$$ 18.4853 0.990915
$$349$$ 10.9706 0.587241 0.293620 0.955922i $$-0.405140\pi$$
0.293620 + 0.955922i $$0.405140\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 5.65685 0.301941
$$352$$ −1.58579 −0.0845227
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ −9.65685 −0.513256
$$355$$ −2.34315 −0.124361
$$356$$ 14.0000 0.741999
$$357$$ 0.970563 0.0513676
$$358$$ 42.6274 2.25293
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 4.41421 0.232649
$$361$$ 27.6274 1.45407
$$362$$ 33.7990 1.77644
$$363$$ −1.00000 −0.0524864
$$364$$ −17.9411 −0.940370
$$365$$ −11.3137 −0.592187
$$366$$ −28.1421 −1.47101
$$367$$ 9.65685 0.504084 0.252042 0.967716i $$-0.418898\pi$$
0.252042 + 0.967716i $$0.418898\pi$$
$$368$$ −12.0000 −0.625543
$$369$$ 4.82843 0.251358
$$370$$ 28.1421 1.46304
$$371$$ 7.71573 0.400581
$$372$$ 0 0
$$373$$ 10.6274 0.550267 0.275133 0.961406i $$-0.411278\pi$$
0.275133 + 0.961406i $$0.411278\pi$$
$$374$$ −2.82843 −0.146254
$$375$$ 1.00000 0.0516398
$$376$$ 17.6569 0.910583
$$377$$ 27.3137 1.40673
$$378$$ 2.00000 0.102869
$$379$$ −23.3137 −1.19754 −0.598772 0.800919i $$-0.704345\pi$$
−0.598772 + 0.800919i $$0.704345\pi$$
$$380$$ 26.1421 1.34106
$$381$$ 14.4853 0.742103
$$382$$ −13.6569 −0.698745
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ −20.5563 −1.04901
$$385$$ 0.828427 0.0422206
$$386$$ −32.9706 −1.67816
$$387$$ −8.82843 −0.448774
$$388$$ 44.6274 2.26561
$$389$$ −23.6569 −1.19945 −0.599725 0.800206i $$-0.704723\pi$$
−0.599725 + 0.800206i $$0.704723\pi$$
$$390$$ 13.6569 0.691542
$$391$$ 4.68629 0.236996
$$392$$ 27.8701 1.40765
$$393$$ −3.31371 −0.167154
$$394$$ 20.4853 1.03203
$$395$$ −8.48528 −0.426941
$$396$$ −3.82843 −0.192386
$$397$$ −14.9706 −0.751351 −0.375676 0.926751i $$-0.622589\pi$$
−0.375676 + 0.926751i $$0.622589\pi$$
$$398$$ 52.2843 2.62077
$$399$$ 5.65685 0.283197
$$400$$ 3.00000 0.150000
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ −13.6569 −0.681142
$$403$$ 0 0
$$404$$ −3.17157 −0.157792
$$405$$ −1.00000 −0.0496904
$$406$$ 9.65685 0.479262
$$407$$ −11.6569 −0.577809
$$408$$ −5.17157 −0.256031
$$409$$ −19.6569 −0.971969 −0.485984 0.873967i $$-0.661539\pi$$
−0.485984 + 0.873967i $$0.661539\pi$$
$$410$$ 11.6569 0.575691
$$411$$ 13.3137 0.656717
$$412$$ −12.6863 −0.625009
$$413$$ −3.31371 −0.163057
$$414$$ 9.65685 0.474608
$$415$$ 10.0000 0.490881
$$416$$ −8.97056 −0.439818
$$417$$ −0.485281 −0.0237643
$$418$$ −16.4853 −0.806321
$$419$$ −36.9706 −1.80613 −0.903065 0.429504i $$-0.858689\pi$$
−0.903065 + 0.429504i $$0.858689\pi$$
$$420$$ 3.17157 0.154757
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ −2.82843 −0.137686
$$423$$ −4.00000 −0.194487
$$424$$ −41.1127 −1.99661
$$425$$ −1.17157 −0.0568296
$$426$$ 5.65685 0.274075
$$427$$ −9.65685 −0.467328
$$428$$ 66.2843 3.20397
$$429$$ −5.65685 −0.273115
$$430$$ −21.3137 −1.02784
$$431$$ 21.6569 1.04317 0.521587 0.853198i $$-0.325340\pi$$
0.521587 + 0.853198i $$0.325340\pi$$
$$432$$ −3.00000 −0.144338
$$433$$ −15.6569 −0.752420 −0.376210 0.926534i $$-0.622773\pi$$
−0.376210 + 0.926534i $$0.622773\pi$$
$$434$$ 0 0
$$435$$ −4.82843 −0.231505
$$436$$ 66.2843 3.17444
$$437$$ 27.3137 1.30659
$$438$$ 27.3137 1.30510
$$439$$ −20.4853 −0.977709 −0.488855 0.872365i $$-0.662585\pi$$
−0.488855 + 0.872365i $$0.662585\pi$$
$$440$$ −4.41421 −0.210439
$$441$$ −6.31371 −0.300653
$$442$$ −16.0000 −0.761042
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ −44.6274 −2.11792
$$445$$ −3.65685 −0.173352
$$446$$ 15.3137 0.725125
$$447$$ −1.51472 −0.0716437
$$448$$ −8.14214 −0.384680
$$449$$ 30.9706 1.46159 0.730796 0.682596i $$-0.239149\pi$$
0.730796 + 0.682596i $$0.239149\pi$$
$$450$$ −2.41421 −0.113807
$$451$$ −4.82843 −0.227362
$$452$$ −72.6274 −3.41611
$$453$$ −16.4853 −0.774546
$$454$$ 33.7990 1.58627
$$455$$ 4.68629 0.219697
$$456$$ −30.1421 −1.41153
$$457$$ −23.3137 −1.09057 −0.545285 0.838251i $$-0.683578\pi$$
−0.545285 + 0.838251i $$0.683578\pi$$
$$458$$ 4.82843 0.225618
$$459$$ 1.17157 0.0546843
$$460$$ 15.3137 0.714005
$$461$$ 0.142136 0.00661992 0.00330996 0.999995i $$-0.498946\pi$$
0.00330996 + 0.999995i $$0.498946\pi$$
$$462$$ −2.00000 −0.0930484
$$463$$ 4.97056 0.231002 0.115501 0.993307i $$-0.463153\pi$$
0.115501 + 0.993307i $$0.463153\pi$$
$$464$$ −14.4853 −0.672462
$$465$$ 0 0
$$466$$ 45.4558 2.10570
$$467$$ −22.6274 −1.04707 −0.523536 0.852004i $$-0.675387\pi$$
−0.523536 + 0.852004i $$0.675387\pi$$
$$468$$ −21.6569 −1.00109
$$469$$ −4.68629 −0.216393
$$470$$ −9.65685 −0.445437
$$471$$ −18.0000 −0.829396
$$472$$ 17.6569 0.812723
$$473$$ 8.82843 0.405932
$$474$$ 20.4853 0.940920
$$475$$ −6.82843 −0.313310
$$476$$ −3.71573 −0.170310
$$477$$ 9.31371 0.426445
$$478$$ −42.6274 −1.94973
$$479$$ 36.9706 1.68923 0.844614 0.535376i $$-0.179830\pi$$
0.844614 + 0.535376i $$0.179830\pi$$
$$480$$ 1.58579 0.0723809
$$481$$ −65.9411 −3.00666
$$482$$ 29.7990 1.35731
$$483$$ 3.31371 0.150779
$$484$$ 3.82843 0.174019
$$485$$ −11.6569 −0.529310
$$486$$ 2.41421 0.109511
$$487$$ −12.9706 −0.587752 −0.293876 0.955844i $$-0.594945\pi$$
−0.293876 + 0.955844i $$0.594945\pi$$
$$488$$ 51.4558 2.32930
$$489$$ 7.31371 0.330737
$$490$$ −15.2426 −0.688592
$$491$$ 14.3431 0.647297 0.323649 0.946177i $$-0.395090\pi$$
0.323649 + 0.946177i $$0.395090\pi$$
$$492$$ −18.4853 −0.833381
$$493$$ 5.65685 0.254772
$$494$$ −93.2548 −4.19573
$$495$$ 1.00000 0.0449467
$$496$$ 0 0
$$497$$ 1.94113 0.0870714
$$498$$ −24.1421 −1.08183
$$499$$ −22.3431 −1.00022 −0.500108 0.865963i $$-0.666706\pi$$
−0.500108 + 0.865963i $$0.666706\pi$$
$$500$$ −3.82843 −0.171212
$$501$$ 13.3137 0.594813
$$502$$ −50.6274 −2.25961
$$503$$ 17.3137 0.771980 0.385990 0.922503i $$-0.373860\pi$$
0.385990 + 0.922503i $$0.373860\pi$$
$$504$$ −3.65685 −0.162889
$$505$$ 0.828427 0.0368645
$$506$$ −9.65685 −0.429300
$$507$$ −19.0000 −0.843820
$$508$$ −55.4558 −2.46046
$$509$$ 18.6863 0.828255 0.414128 0.910219i $$-0.364087\pi$$
0.414128 + 0.910219i $$0.364087\pi$$
$$510$$ 2.82843 0.125245
$$511$$ 9.37258 0.414619
$$512$$ 31.2426 1.38074
$$513$$ 6.82843 0.301482
$$514$$ 39.4558 1.74032
$$515$$ 3.31371 0.146019
$$516$$ 33.7990 1.48792
$$517$$ 4.00000 0.175920
$$518$$ −23.3137 −1.02435
$$519$$ −2.82843 −0.124154
$$520$$ −24.9706 −1.09503
$$521$$ −32.6274 −1.42943 −0.714717 0.699414i $$-0.753444\pi$$
−0.714717 + 0.699414i $$0.753444\pi$$
$$522$$ 11.6569 0.510207
$$523$$ −9.51472 −0.416050 −0.208025 0.978124i $$-0.566703\pi$$
−0.208025 + 0.978124i $$0.566703\pi$$
$$524$$ 12.6863 0.554203
$$525$$ −0.828427 −0.0361555
$$526$$ −43.4558 −1.89476
$$527$$ 0 0
$$528$$ 3.00000 0.130558
$$529$$ −7.00000 −0.304348
$$530$$ 22.4853 0.976698
$$531$$ −4.00000 −0.173585
$$532$$ −21.6569 −0.938944
$$533$$ −27.3137 −1.18309
$$534$$ 8.82843 0.382043
$$535$$ −17.3137 −0.748537
$$536$$ 24.9706 1.07856
$$537$$ 17.6569 0.761950
$$538$$ −49.7990 −2.14699
$$539$$ 6.31371 0.271951
$$540$$ 3.82843 0.164749
$$541$$ 17.3137 0.744374 0.372187 0.928158i $$-0.378608\pi$$
0.372187 + 0.928158i $$0.378608\pi$$
$$542$$ 28.4853 1.22355
$$543$$ 14.0000 0.600798
$$544$$ −1.85786 −0.0796553
$$545$$ −17.3137 −0.741638
$$546$$ −11.3137 −0.484182
$$547$$ −8.14214 −0.348133 −0.174066 0.984734i $$-0.555691\pi$$
−0.174066 + 0.984734i $$0.555691\pi$$
$$548$$ −50.9706 −2.17735
$$549$$ −11.6569 −0.497502
$$550$$ 2.41421 0.102942
$$551$$ 32.9706 1.40459
$$552$$ −17.6569 −0.751526
$$553$$ 7.02944 0.298922
$$554$$ 5.65685 0.240337
$$555$$ 11.6569 0.494806
$$556$$ 1.85786 0.0787910
$$557$$ 5.17157 0.219127 0.109563 0.993980i $$-0.465055\pi$$
0.109563 + 0.993980i $$0.465055\pi$$
$$558$$ 0 0
$$559$$ 49.9411 2.11228
$$560$$ −2.48528 −0.105022
$$561$$ −1.17157 −0.0494638
$$562$$ 26.9706 1.13768
$$563$$ 31.6569 1.33418 0.667089 0.744978i $$-0.267540\pi$$
0.667089 + 0.744978i $$0.267540\pi$$
$$564$$ 15.3137 0.644823
$$565$$ 18.9706 0.798098
$$566$$ 21.3137 0.895882
$$567$$ 0.828427 0.0347907
$$568$$ −10.3431 −0.433989
$$569$$ −35.4558 −1.48639 −0.743193 0.669077i $$-0.766690\pi$$
−0.743193 + 0.669077i $$0.766690\pi$$
$$570$$ 16.4853 0.690492
$$571$$ −16.4853 −0.689888 −0.344944 0.938623i $$-0.612102\pi$$
−0.344944 + 0.938623i $$0.612102\pi$$
$$572$$ 21.6569 0.905519
$$573$$ −5.65685 −0.236318
$$574$$ −9.65685 −0.403069
$$575$$ −4.00000 −0.166812
$$576$$ −9.82843 −0.409518
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 37.7279 1.56927
$$579$$ −13.6569 −0.567559
$$580$$ 18.4853 0.767560
$$581$$ −8.28427 −0.343689
$$582$$ 28.1421 1.16653
$$583$$ −9.31371 −0.385734
$$584$$ −49.9411 −2.06658
$$585$$ 5.65685 0.233882
$$586$$ 16.4853 0.681001
$$587$$ 14.6274 0.603738 0.301869 0.953349i $$-0.402389\pi$$
0.301869 + 0.953349i $$0.402389\pi$$
$$588$$ 24.1716 0.996819
$$589$$ 0 0
$$590$$ −9.65685 −0.397566
$$591$$ 8.48528 0.349038
$$592$$ 34.9706 1.43728
$$593$$ −22.8284 −0.937451 −0.468726 0.883344i $$-0.655287\pi$$
−0.468726 + 0.883344i $$0.655287\pi$$
$$594$$ −2.41421 −0.0990564
$$595$$ 0.970563 0.0397892
$$596$$ 5.79899 0.237536
$$597$$ 21.6569 0.886356
$$598$$ −54.6274 −2.23388
$$599$$ 27.3137 1.11601 0.558004 0.829838i $$-0.311567\pi$$
0.558004 + 0.829838i $$0.311567\pi$$
$$600$$ 4.41421 0.180210
$$601$$ −5.31371 −0.216751 −0.108375 0.994110i $$-0.534565\pi$$
−0.108375 + 0.994110i $$0.534565\pi$$
$$602$$ 17.6569 0.719640
$$603$$ −5.65685 −0.230365
$$604$$ 63.1127 2.56802
$$605$$ −1.00000 −0.0406558
$$606$$ −2.00000 −0.0812444
$$607$$ 1.51472 0.0614805 0.0307403 0.999527i $$-0.490214\pi$$
0.0307403 + 0.999527i $$0.490214\pi$$
$$608$$ −10.8284 −0.439151
$$609$$ 4.00000 0.162088
$$610$$ −28.1421 −1.13944
$$611$$ 22.6274 0.915407
$$612$$ −4.48528 −0.181307
$$613$$ −45.9411 −1.85554 −0.927772 0.373147i $$-0.878279\pi$$
−0.927772 + 0.373147i $$0.878279\pi$$
$$614$$ 7.65685 0.309005
$$615$$ 4.82843 0.194701
$$616$$ 3.65685 0.147339
$$617$$ 0.343146 0.0138145 0.00690726 0.999976i $$-0.497801\pi$$
0.00690726 + 0.999976i $$0.497801\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −14.3431 −0.576500 −0.288250 0.957555i $$-0.593073\pi$$
−0.288250 + 0.957555i $$0.593073\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 8.00000 0.320771
$$623$$ 3.02944 0.121372
$$624$$ 16.9706 0.679366
$$625$$ 1.00000 0.0400000
$$626$$ −37.7990 −1.51075
$$627$$ −6.82843 −0.272701
$$628$$ 68.9117 2.74988
$$629$$ −13.6569 −0.544534
$$630$$ 2.00000 0.0796819
$$631$$ 45.6569 1.81757 0.908785 0.417264i $$-0.137011\pi$$
0.908785 + 0.417264i $$0.137011\pi$$
$$632$$ −37.4558 −1.48991
$$633$$ −1.17157 −0.0465658
$$634$$ 63.4558 2.52015
$$635$$ 14.4853 0.574831
$$636$$ −35.6569 −1.41389
$$637$$ 35.7157 1.41511
$$638$$ −11.6569 −0.461499
$$639$$ 2.34315 0.0926934
$$640$$ −20.5563 −0.812561
$$641$$ 6.97056 0.275321 0.137660 0.990479i $$-0.456042\pi$$
0.137660 + 0.990479i $$0.456042\pi$$
$$642$$ 41.7990 1.64967
$$643$$ 37.9411 1.49625 0.748126 0.663557i $$-0.230954\pi$$
0.748126 + 0.663557i $$0.230954\pi$$
$$644$$ −12.6863 −0.499910
$$645$$ −8.82843 −0.347619
$$646$$ −19.3137 −0.759888
$$647$$ 4.68629 0.184237 0.0921186 0.995748i $$-0.470636\pi$$
0.0921186 + 0.995748i $$0.470636\pi$$
$$648$$ −4.41421 −0.173407
$$649$$ 4.00000 0.157014
$$650$$ 13.6569 0.535666
$$651$$ 0 0
$$652$$ −28.0000 −1.09656
$$653$$ 6.97056 0.272779 0.136390 0.990655i $$-0.456450\pi$$
0.136390 + 0.990655i $$0.456450\pi$$
$$654$$ 41.7990 1.63447
$$655$$ −3.31371 −0.129477
$$656$$ 14.4853 0.565555
$$657$$ 11.3137 0.441390
$$658$$ 8.00000 0.311872
$$659$$ −15.3137 −0.596537 −0.298269 0.954482i $$-0.596409\pi$$
−0.298269 + 0.954482i $$0.596409\pi$$
$$660$$ −3.82843 −0.149021
$$661$$ 9.31371 0.362261 0.181131 0.983459i $$-0.442024\pi$$
0.181131 + 0.983459i $$0.442024\pi$$
$$662$$ −15.3137 −0.595184
$$663$$ −6.62742 −0.257388
$$664$$ 44.1421 1.71305
$$665$$ 5.65685 0.219363
$$666$$ −28.1421 −1.09049
$$667$$ 19.3137 0.747830
$$668$$ −50.9706 −1.97211
$$669$$ 6.34315 0.245240
$$670$$ −13.6569 −0.527610
$$671$$ 11.6569 0.450008
$$672$$ −1.31371 −0.0506774
$$673$$ 18.3431 0.707076 0.353538 0.935420i $$-0.384978\pi$$
0.353538 + 0.935420i $$0.384978\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ −1.00000 −0.0384900
$$676$$ 72.7401 2.79770
$$677$$ 29.4558 1.13208 0.566040 0.824378i $$-0.308475\pi$$
0.566040 + 0.824378i $$0.308475\pi$$
$$678$$ −45.7990 −1.75890
$$679$$ 9.65685 0.370596
$$680$$ −5.17157 −0.198321
$$681$$ 14.0000 0.536481
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ −26.1421 −0.999570
$$685$$ 13.3137 0.508691
$$686$$ 26.6274 1.01664
$$687$$ 2.00000 0.0763048
$$688$$ −26.4853 −1.00974
$$689$$ −52.6863 −2.00719
$$690$$ 9.65685 0.367630
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 10.8284 0.411635
$$693$$ −0.828427 −0.0314693
$$694$$ 70.7696 2.68638
$$695$$ −0.485281 −0.0184078
$$696$$ −21.3137 −0.807894
$$697$$ −5.65685 −0.214269
$$698$$ −26.4853 −1.00248
$$699$$ 18.8284 0.712157
$$700$$ 3.17157 0.119874
$$701$$ 36.1421 1.36507 0.682535 0.730853i $$-0.260878\pi$$
0.682535 + 0.730853i $$0.260878\pi$$
$$702$$ −13.6569 −0.515445
$$703$$ −79.5980 −3.00209
$$704$$ 9.82843 0.370423
$$705$$ −4.00000 −0.150649
$$706$$ −62.7696 −2.36236
$$707$$ −0.686292 −0.0258106
$$708$$ 15.3137 0.575524
$$709$$ 6.68629 0.251109 0.125554 0.992087i $$-0.459929\pi$$
0.125554 + 0.992087i $$0.459929\pi$$
$$710$$ 5.65685 0.212298
$$711$$ 8.48528 0.318223
$$712$$ −16.1421 −0.604952
$$713$$ 0 0
$$714$$ −2.34315 −0.0876900
$$715$$ −5.65685 −0.211554
$$716$$ −67.5980 −2.52626
$$717$$ −17.6569 −0.659407
$$718$$ −28.9706 −1.08117
$$719$$ −47.5980 −1.77511 −0.887553 0.460706i $$-0.847596\pi$$
−0.887553 + 0.460706i $$0.847596\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ −2.74517 −0.102235
$$722$$ −66.6985 −2.48226
$$723$$ 12.3431 0.459047
$$724$$ −53.5980 −1.99195
$$725$$ −4.82843 −0.179323
$$726$$ 2.41421 0.0895999
$$727$$ 33.9411 1.25881 0.629403 0.777079i $$-0.283299\pi$$
0.629403 + 0.777079i $$0.283299\pi$$
$$728$$ 20.6863 0.766685
$$729$$ 1.00000 0.0370370
$$730$$ 27.3137 1.01093
$$731$$ 10.3431 0.382555
$$732$$ 44.6274 1.64948
$$733$$ −6.34315 −0.234289 −0.117145 0.993115i $$-0.537374\pi$$
−0.117145 + 0.993115i $$0.537374\pi$$
$$734$$ −23.3137 −0.860525
$$735$$ −6.31371 −0.232885
$$736$$ −6.34315 −0.233811
$$737$$ 5.65685 0.208373
$$738$$ −11.6569 −0.429095
$$739$$ −15.1127 −0.555930 −0.277965 0.960591i $$-0.589660\pi$$
−0.277965 + 0.960591i $$0.589660\pi$$
$$740$$ −44.6274 −1.64054
$$741$$ −38.6274 −1.41901
$$742$$ −18.6274 −0.683834
$$743$$ 36.3431 1.33330 0.666650 0.745371i $$-0.267727\pi$$
0.666650 + 0.745371i $$0.267727\pi$$
$$744$$ 0 0
$$745$$ −1.51472 −0.0554950
$$746$$ −25.6569 −0.939364
$$747$$ −10.0000 −0.365881
$$748$$ 4.48528 0.163998
$$749$$ 14.3431 0.524087
$$750$$ −2.41421 −0.0881546
$$751$$ 20.2843 0.740184 0.370092 0.928995i $$-0.379326\pi$$
0.370092 + 0.928995i $$0.379326\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ −20.9706 −0.764210
$$754$$ −65.9411 −2.40143
$$755$$ −16.4853 −0.599961
$$756$$ −3.17157 −0.115349
$$757$$ −36.6274 −1.33125 −0.665623 0.746288i $$-0.731834\pi$$
−0.665623 + 0.746288i $$0.731834\pi$$
$$758$$ 56.2843 2.04434
$$759$$ −4.00000 −0.145191
$$760$$ −30.1421 −1.09337
$$761$$ −28.8284 −1.04503 −0.522515 0.852630i $$-0.675006\pi$$
−0.522515 + 0.852630i $$0.675006\pi$$
$$762$$ −34.9706 −1.26685
$$763$$ 14.3431 0.519257
$$764$$ 21.6569 0.783517
$$765$$ 1.17157 0.0423583
$$766$$ 19.3137 0.697833
$$767$$ 22.6274 0.817029
$$768$$ 29.9706 1.08147
$$769$$ 10.6863 0.385358 0.192679 0.981262i $$-0.438282\pi$$
0.192679 + 0.981262i $$0.438282\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ 16.3431 0.588584
$$772$$ 52.2843 1.88175
$$773$$ 3.65685 0.131528 0.0657640 0.997835i $$-0.479052\pi$$
0.0657640 + 0.997835i $$0.479052\pi$$
$$774$$ 21.3137 0.766105
$$775$$ 0 0
$$776$$ −51.4558 −1.84716
$$777$$ −9.65685 −0.346438
$$778$$ 57.1127 2.04759
$$779$$ −32.9706 −1.18129
$$780$$ −21.6569 −0.775440
$$781$$ −2.34315 −0.0838443
$$782$$ −11.3137 −0.404577
$$783$$ 4.82843 0.172554
$$784$$ −18.9411 −0.676469
$$785$$ −18.0000 −0.642448
$$786$$ 8.00000 0.285351
$$787$$ −20.1421 −0.717990 −0.358995 0.933340i $$-0.616880\pi$$
−0.358995 + 0.933340i $$0.616880\pi$$
$$788$$ −32.4853 −1.15724
$$789$$ −18.0000 −0.640817
$$790$$ 20.4853 0.728834
$$791$$ −15.7157 −0.558787
$$792$$ 4.41421 0.156852
$$793$$ 65.9411 2.34164
$$794$$ 36.1421 1.28264
$$795$$ 9.31371 0.330323
$$796$$ −82.9117 −2.93873
$$797$$ 34.9706 1.23872 0.619360 0.785107i $$-0.287392\pi$$
0.619360 + 0.785107i $$0.287392\pi$$
$$798$$ −13.6569 −0.483447
$$799$$ 4.68629 0.165789
$$800$$ 1.58579 0.0560660
$$801$$ 3.65685 0.129209
$$802$$ 16.1421 0.569999
$$803$$ −11.3137 −0.399252
$$804$$ 21.6569 0.763778
$$805$$ 3.31371 0.116793
$$806$$ 0 0
$$807$$ −20.6274 −0.726119
$$808$$ 3.65685 0.128648
$$809$$ 28.4264 0.999419 0.499710 0.866193i $$-0.333440\pi$$
0.499710 + 0.866193i $$0.333440\pi$$
$$810$$ 2.41421 0.0848268
$$811$$ 0.485281 0.0170405 0.00852027 0.999964i $$-0.497288\pi$$
0.00852027 + 0.999964i $$0.497288\pi$$
$$812$$ −15.3137 −0.537406
$$813$$ 11.7990 0.413809
$$814$$ 28.1421 0.986381
$$815$$ 7.31371 0.256188
$$816$$ 3.51472 0.123040
$$817$$ 60.2843 2.10908
$$818$$ 47.4558 1.65925
$$819$$ −4.68629 −0.163752
$$820$$ −18.4853 −0.645534
$$821$$ −12.8284 −0.447715 −0.223858 0.974622i $$-0.571865\pi$$
−0.223858 + 0.974622i $$0.571865\pi$$
$$822$$ −32.1421 −1.12109
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 14.6274 0.509570
$$825$$ 1.00000 0.0348155
$$826$$ 8.00000 0.278356
$$827$$ 41.3137 1.43662 0.718309 0.695724i $$-0.244916\pi$$
0.718309 + 0.695724i $$0.244916\pi$$
$$828$$ −15.3137 −0.532188
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ −24.1421 −0.837986
$$831$$ 2.34315 0.0812828
$$832$$ 55.5980 1.92751
$$833$$ 7.39697 0.256290
$$834$$ 1.17157 0.0405683
$$835$$ 13.3137 0.460740
$$836$$ 26.1421 0.904145
$$837$$ 0 0
$$838$$ 89.2548 3.08326
$$839$$ −22.6274 −0.781185 −0.390593 0.920564i $$-0.627730\pi$$
−0.390593 + 0.920564i $$0.627730\pi$$
$$840$$ −3.65685 −0.126173
$$841$$ −5.68629 −0.196079
$$842$$ 14.4853 0.499196
$$843$$ 11.1716 0.384769
$$844$$ 4.48528 0.154390
$$845$$ −19.0000 −0.653620
$$846$$ 9.65685 0.332009
$$847$$ 0.828427 0.0284651
$$848$$ 27.9411 0.959502
$$849$$ 8.82843 0.302991
$$850$$ 2.82843 0.0970143
$$851$$ −46.6274 −1.59837
$$852$$ −8.97056 −0.307326
$$853$$ −8.68629 −0.297413 −0.148706 0.988881i $$-0.547511\pi$$
−0.148706 + 0.988881i $$0.547511\pi$$
$$854$$ 23.3137 0.797779
$$855$$ 6.82843 0.233527
$$856$$ −76.4264 −2.61220
$$857$$ −28.4853 −0.973039 −0.486519 0.873670i $$-0.661734\pi$$
−0.486519 + 0.873670i $$0.661734\pi$$
$$858$$ 13.6569 0.466237
$$859$$ −52.9706 −1.80733 −0.903666 0.428238i $$-0.859135\pi$$
−0.903666 + 0.428238i $$0.859135\pi$$
$$860$$ 33.7990 1.15254
$$861$$ −4.00000 −0.136320
$$862$$ −52.2843 −1.78081
$$863$$ −20.6863 −0.704170 −0.352085 0.935968i $$-0.614527\pi$$
−0.352085 + 0.935968i $$0.614527\pi$$
$$864$$ −1.58579 −0.0539496
$$865$$ −2.82843 −0.0961694
$$866$$ 37.7990 1.28446
$$867$$ 15.6274 0.530735
$$868$$ 0 0
$$869$$ −8.48528 −0.287843
$$870$$ 11.6569 0.395204
$$871$$ 32.0000 1.08428
$$872$$ −76.4264 −2.58812
$$873$$ 11.6569 0.394525
$$874$$ −65.9411 −2.23049
$$875$$ −0.828427 −0.0280059
$$876$$ −43.3137 −1.46343
$$877$$ 2.62742 0.0887216 0.0443608 0.999016i $$-0.485875\pi$$
0.0443608 + 0.999016i $$0.485875\pi$$
$$878$$ 49.4558 1.66905
$$879$$ 6.82843 0.230317
$$880$$ 3.00000 0.101130
$$881$$ −46.9706 −1.58248 −0.791239 0.611507i $$-0.790564\pi$$
−0.791239 + 0.611507i $$0.790564\pi$$
$$882$$ 15.2426 0.513246
$$883$$ −5.37258 −0.180802 −0.0904009 0.995905i $$-0.528815\pi$$
−0.0904009 + 0.995905i $$0.528815\pi$$
$$884$$ 25.3726 0.853372
$$885$$ −4.00000 −0.134459
$$886$$ −28.9706 −0.973285
$$887$$ −15.6569 −0.525706 −0.262853 0.964836i $$-0.584663\pi$$
−0.262853 + 0.964836i $$0.584663\pi$$
$$888$$ 51.4558 1.72675
$$889$$ −12.0000 −0.402467
$$890$$ 8.82843 0.295930
$$891$$ −1.00000 −0.0335013
$$892$$ −24.2843 −0.813098
$$893$$ 27.3137 0.914018
$$894$$ 3.65685 0.122304
$$895$$ 17.6569 0.590204
$$896$$ 17.0294 0.568914
$$897$$ −22.6274 −0.755507
$$898$$ −74.7696 −2.49509
$$899$$ 0 0
$$900$$ 3.82843 0.127614
$$901$$ −10.9117 −0.363521
$$902$$ 11.6569 0.388131
$$903$$ 7.31371 0.243385
$$904$$ 83.7401 2.78515
$$905$$ 14.0000 0.465376
$$906$$ 39.7990 1.32223
$$907$$ 40.9706 1.36041 0.680203 0.733024i $$-0.261892\pi$$
0.680203 + 0.733024i $$0.261892\pi$$
$$908$$ −53.5980 −1.77871
$$909$$ −0.828427 −0.0274772
$$910$$ −11.3137 −0.375046
$$911$$ −48.9706 −1.62247 −0.811234 0.584722i $$-0.801203\pi$$
−0.811234 + 0.584722i $$0.801203\pi$$
$$912$$ 20.4853 0.678335
$$913$$ 10.0000 0.330952
$$914$$ 56.2843 1.86172
$$915$$ −11.6569 −0.385364
$$916$$ −7.65685 −0.252990
$$917$$ 2.74517 0.0906534
$$918$$ −2.82843 −0.0933520
$$919$$ 11.5147 0.379836 0.189918 0.981800i $$-0.439178\pi$$
0.189918 + 0.981800i $$0.439178\pi$$
$$920$$ −17.6569 −0.582129
$$921$$ 3.17157 0.104507
$$922$$ −0.343146 −0.0113009
$$923$$ −13.2548 −0.436288
$$924$$ 3.17157 0.104337
$$925$$ 11.6569 0.383275
$$926$$ −12.0000 −0.394344
$$927$$ −3.31371 −0.108836
$$928$$ −7.65685 −0.251349
$$929$$ −45.5980 −1.49602 −0.748011 0.663687i $$-0.768991\pi$$
−0.748011 + 0.663687i $$0.768991\pi$$
$$930$$ 0 0
$$931$$ 43.1127 1.41296
$$932$$ −72.0833 −2.36117
$$933$$ 3.31371 0.108486
$$934$$ 54.6274 1.78746
$$935$$ −1.17157 −0.0383145
$$936$$ 24.9706 0.816188
$$937$$ 11.0294 0.360316 0.180158 0.983638i $$-0.442339\pi$$
0.180158 + 0.983638i $$0.442339\pi$$
$$938$$ 11.3137 0.369406
$$939$$ −15.6569 −0.510942
$$940$$ 15.3137 0.499478
$$941$$ −34.7696 −1.13346 −0.566728 0.823905i $$-0.691791\pi$$
−0.566728 + 0.823905i $$0.691791\pi$$
$$942$$ 43.4558 1.41587
$$943$$ −19.3137 −0.628941
$$944$$ −12.0000 −0.390567
$$945$$ 0.828427 0.0269487
$$946$$ −21.3137 −0.692968
$$947$$ 6.62742 0.215362 0.107681 0.994185i $$-0.465657\pi$$
0.107681 + 0.994185i $$0.465657\pi$$
$$948$$ −32.4853 −1.05507
$$949$$ −64.0000 −2.07753
$$950$$ 16.4853 0.534853
$$951$$ 26.2843 0.852326
$$952$$ 4.28427 0.138854
$$953$$ −11.7990 −0.382207 −0.191103 0.981570i $$-0.561207\pi$$
−0.191103 + 0.981570i $$0.561207\pi$$
$$954$$ −22.4853 −0.727988
$$955$$ −5.65685 −0.183052
$$956$$ 67.5980 2.18627
$$957$$ −4.82843 −0.156081
$$958$$ −89.2548 −2.88369
$$959$$ −11.0294 −0.356159
$$960$$ −9.82843 −0.317211
$$961$$ −31.0000 −1.00000
$$962$$ 159.196 5.13268
$$963$$ 17.3137 0.557926
$$964$$ −47.2548 −1.52198
$$965$$ −13.6569 −0.439630
$$966$$ −8.00000 −0.257396
$$967$$ 11.4558 0.368395 0.184198 0.982889i $$-0.441031\pi$$
0.184198 + 0.982889i $$0.441031\pi$$
$$968$$ −4.41421 −0.141878
$$969$$ −8.00000 −0.256997
$$970$$ 28.1421 0.903590
$$971$$ −34.6274 −1.11125 −0.555623 0.831434i $$-0.687520\pi$$
−0.555623 + 0.831434i $$0.687520\pi$$
$$972$$ −3.82843 −0.122797
$$973$$ 0.402020 0.0128882
$$974$$ 31.3137 1.00336
$$975$$ 5.65685 0.181164
$$976$$ −34.9706 −1.11938
$$977$$ 2.68629 0.0859421 0.0429710 0.999076i $$-0.486318\pi$$
0.0429710 + 0.999076i $$0.486318\pi$$
$$978$$ −17.6569 −0.564604
$$979$$ −3.65685 −0.116874
$$980$$ 24.1716 0.772133
$$981$$ 17.3137 0.552784
$$982$$ −34.6274 −1.10501
$$983$$ −30.6274 −0.976863 −0.488431 0.872602i $$-0.662431\pi$$
−0.488431 + 0.872602i $$0.662431\pi$$
$$984$$ 21.3137 0.679456
$$985$$ 8.48528 0.270364
$$986$$ −13.6569 −0.434923
$$987$$ 3.31371 0.105477
$$988$$ 147.882 4.70476
$$989$$ 35.3137 1.12291
$$990$$ −2.41421 −0.0767287
$$991$$ −30.6274 −0.972912 −0.486456 0.873705i $$-0.661711\pi$$
−0.486456 + 0.873705i $$0.661711\pi$$
$$992$$ 0 0
$$993$$ −6.34315 −0.201294
$$994$$ −4.68629 −0.148640
$$995$$ 21.6569 0.686568
$$996$$ 38.2843 1.21308
$$997$$ −39.3137 −1.24508 −0.622539 0.782589i $$-0.713899\pi$$
−0.622539 + 0.782589i $$0.713899\pi$$
$$998$$ 53.9411 1.70748
$$999$$ −11.6569 −0.368807
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 165.2.a.a.1.1 2
3.2 odd 2 495.2.a.d.1.2 2
4.3 odd 2 2640.2.a.bb.1.1 2
5.2 odd 4 825.2.c.e.199.1 4
5.3 odd 4 825.2.c.e.199.4 4
5.4 even 2 825.2.a.g.1.2 2
7.6 odd 2 8085.2.a.ba.1.1 2
11.10 odd 2 1815.2.a.k.1.2 2
12.11 even 2 7920.2.a.cg.1.1 2
15.2 even 4 2475.2.c.m.199.4 4
15.8 even 4 2475.2.c.m.199.1 4
15.14 odd 2 2475.2.a.m.1.1 2
33.32 even 2 5445.2.a.m.1.1 2
55.54 odd 2 9075.2.a.v.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 1.1 even 1 trivial
495.2.a.d.1.2 2 3.2 odd 2
825.2.a.g.1.2 2 5.4 even 2
825.2.c.e.199.1 4 5.2 odd 4
825.2.c.e.199.4 4 5.3 odd 4
1815.2.a.k.1.2 2 11.10 odd 2
2475.2.a.m.1.1 2 15.14 odd 2
2475.2.c.m.199.1 4 15.8 even 4
2475.2.c.m.199.4 4 15.2 even 4
2640.2.a.bb.1.1 2 4.3 odd 2
5445.2.a.m.1.1 2 33.32 even 2
7920.2.a.cg.1.1 2 12.11 even 2
8085.2.a.ba.1.1 2 7.6 odd 2
9075.2.a.v.1.1 2 55.54 odd 2