Properties

 Label 4560.2.a.bq.1.3 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1016.1 Defining polynomial: $$x^{3} - x^{2} - 6 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$0.321637$$ of defining polynomial Character $$\chi$$ $$=$$ 4560.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} +4.21819 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} +4.21819 q^{7} +1.00000 q^{9} -5.57491 q^{11} -2.21819 q^{13} +1.00000 q^{15} +0.643274 q^{17} +1.00000 q^{19} -4.21819 q^{21} -1.35673 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.86146 q^{29} -2.64327 q^{31} +5.57491 q^{33} -4.21819 q^{35} -2.21819 q^{37} +2.21819 q^{39} +3.57491 q^{41} +0.218187 q^{43} -1.00000 q^{45} +1.35673 q^{47} +10.7931 q^{49} -0.643274 q^{51} +0.643274 q^{53} +5.57491 q^{55} -1.00000 q^{57} +4.43637 q^{59} +13.0796 q^{61} +4.21819 q^{63} +2.21819 q^{65} -1.28655 q^{67} +1.35673 q^{69} -11.1498 q^{71} +10.0000 q^{73} -1.00000 q^{75} -23.5160 q^{77} +1.28655 q^{79} +1.00000 q^{81} -12.3662 q^{83} -0.643274 q^{85} -4.86146 q^{87} +12.0113 q^{89} -9.35673 q^{91} +2.64327 q^{93} -1.00000 q^{95} +5.78181 q^{97} -5.57491 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 4 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 6 q^{39} - 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} - 2 q^{51} + 2 q^{53} + 4 q^{55} - 3 q^{57} - 12 q^{59} + 14 q^{61} - 6 q^{65} - 4 q^{67} + 4 q^{69} - 8 q^{71} + 30 q^{73} - 3 q^{75} - 20 q^{77} + 4 q^{79} + 3 q^{81} - 12 q^{83} - 2 q^{85} - 2 q^{87} - 2 q^{89} - 28 q^{91} + 8 q^{93} - 3 q^{95} + 30 q^{97} - 4 q^{99} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.21819 1.59432 0.797162 0.603765i $$-0.206333\pi$$
0.797162 + 0.603765i $$0.206333\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.57491 −1.68090 −0.840450 0.541890i $$-0.817709\pi$$
−0.840450 + 0.541890i $$0.817709\pi$$
$$12$$ 0 0
$$13$$ −2.21819 −0.615214 −0.307607 0.951513i $$-0.599528\pi$$
−0.307607 + 0.951513i $$0.599528\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 0.643274 0.156017 0.0780085 0.996953i $$-0.475144\pi$$
0.0780085 + 0.996953i $$0.475144\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −4.21819 −0.920484
$$22$$ 0 0
$$23$$ −1.35673 −0.282897 −0.141448 0.989946i $$-0.545176\pi$$
−0.141448 + 0.989946i $$0.545176\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 4.86146 0.902751 0.451375 0.892334i $$-0.350934\pi$$
0.451375 + 0.892334i $$0.350934\pi$$
$$30$$ 0 0
$$31$$ −2.64327 −0.474746 −0.237373 0.971419i $$-0.576286\pi$$
−0.237373 + 0.971419i $$0.576286\pi$$
$$32$$ 0 0
$$33$$ 5.57491 0.970468
$$34$$ 0 0
$$35$$ −4.21819 −0.713004
$$36$$ 0 0
$$37$$ −2.21819 −0.364668 −0.182334 0.983237i $$-0.558365\pi$$
−0.182334 + 0.983237i $$0.558365\pi$$
$$38$$ 0 0
$$39$$ 2.21819 0.355194
$$40$$ 0 0
$$41$$ 3.57491 0.558308 0.279154 0.960246i $$-0.409946\pi$$
0.279154 + 0.960246i $$0.409946\pi$$
$$42$$ 0 0
$$43$$ 0.218187 0.0332732 0.0166366 0.999862i $$-0.494704\pi$$
0.0166366 + 0.999862i $$0.494704\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 1.35673 0.197899 0.0989494 0.995092i $$-0.468452\pi$$
0.0989494 + 0.995092i $$0.468452\pi$$
$$48$$ 0 0
$$49$$ 10.7931 1.54187
$$50$$ 0 0
$$51$$ −0.643274 −0.0900764
$$52$$ 0 0
$$53$$ 0.643274 0.0883605 0.0441803 0.999024i $$-0.485932\pi$$
0.0441803 + 0.999024i $$0.485932\pi$$
$$54$$ 0 0
$$55$$ 5.57491 0.751721
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ 4.43637 0.577567 0.288783 0.957394i $$-0.406749\pi$$
0.288783 + 0.957394i $$0.406749\pi$$
$$60$$ 0 0
$$61$$ 13.0796 1.67468 0.837339 0.546685i $$-0.184110\pi$$
0.837339 + 0.546685i $$0.184110\pi$$
$$62$$ 0 0
$$63$$ 4.21819 0.531442
$$64$$ 0 0
$$65$$ 2.21819 0.275132
$$66$$ 0 0
$$67$$ −1.28655 −0.157177 −0.0785885 0.996907i $$-0.525041\pi$$
−0.0785885 + 0.996907i $$0.525041\pi$$
$$68$$ 0 0
$$69$$ 1.35673 0.163331
$$70$$ 0 0
$$71$$ −11.1498 −1.32324 −0.661620 0.749839i $$-0.730131\pi$$
−0.661620 + 0.749839i $$0.730131\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −23.5160 −2.67990
$$78$$ 0 0
$$79$$ 1.28655 0.144748 0.0723740 0.997378i $$-0.476942\pi$$
0.0723740 + 0.997378i $$0.476942\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.3662 −1.35737 −0.678683 0.734431i $$-0.737449\pi$$
−0.678683 + 0.734431i $$0.737449\pi$$
$$84$$ 0 0
$$85$$ −0.643274 −0.0697729
$$86$$ 0 0
$$87$$ −4.86146 −0.521203
$$88$$ 0 0
$$89$$ 12.0113 1.27319 0.636597 0.771197i $$-0.280342\pi$$
0.636597 + 0.771197i $$0.280342\pi$$
$$90$$ 0 0
$$91$$ −9.35673 −0.980851
$$92$$ 0 0
$$93$$ 2.64327 0.274095
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 5.78181 0.587054 0.293527 0.955951i $$-0.405171\pi$$
0.293527 + 0.955951i $$0.405171\pi$$
$$98$$ 0 0
$$99$$ −5.57491 −0.560300
$$100$$ 0 0
$$101$$ −6.43637 −0.640443 −0.320222 0.947343i $$-0.603757\pi$$
−0.320222 + 0.947343i $$0.603757\pi$$
$$102$$ 0 0
$$103$$ 1.28655 0.126767 0.0633837 0.997989i $$-0.479811\pi$$
0.0633837 + 0.997989i $$0.479811\pi$$
$$104$$ 0 0
$$105$$ 4.21819 0.411653
$$106$$ 0 0
$$107$$ 1.28655 0.124375 0.0621877 0.998064i $$-0.480192\pi$$
0.0621877 + 0.998064i $$0.480192\pi$$
$$108$$ 0 0
$$109$$ 18.4364 1.76588 0.882942 0.469482i $$-0.155559\pi$$
0.882942 + 0.469482i $$0.155559\pi$$
$$110$$ 0 0
$$111$$ 2.21819 0.210541
$$112$$ 0 0
$$113$$ 13.0796 1.23043 0.615215 0.788359i $$-0.289069\pi$$
0.615215 + 0.788359i $$0.289069\pi$$
$$114$$ 0 0
$$115$$ 1.35673 0.126515
$$116$$ 0 0
$$117$$ −2.21819 −0.205071
$$118$$ 0 0
$$119$$ 2.71345 0.248742
$$120$$ 0 0
$$121$$ 20.0796 1.82542
$$122$$ 0 0
$$123$$ −3.57491 −0.322339
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 9.28655 0.824048 0.412024 0.911173i $$-0.364822\pi$$
0.412024 + 0.911173i $$0.364822\pi$$
$$128$$ 0 0
$$129$$ −0.218187 −0.0192103
$$130$$ 0 0
$$131$$ −2.86146 −0.250007 −0.125004 0.992156i $$-0.539894\pi$$
−0.125004 + 0.992156i $$0.539894\pi$$
$$132$$ 0 0
$$133$$ 4.21819 0.365763
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −14.8727 −1.27066 −0.635332 0.772239i $$-0.719137\pi$$
−0.635332 + 0.772239i $$0.719137\pi$$
$$138$$ 0 0
$$139$$ −0.850175 −0.0721109 −0.0360555 0.999350i $$-0.511479\pi$$
−0.0360555 + 0.999350i $$0.511479\pi$$
$$140$$ 0 0
$$141$$ −1.35673 −0.114257
$$142$$ 0 0
$$143$$ 12.3662 1.03411
$$144$$ 0 0
$$145$$ −4.86146 −0.403722
$$146$$ 0 0
$$147$$ −10.7931 −0.890200
$$148$$ 0 0
$$149$$ −20.1593 −1.65151 −0.825757 0.564026i $$-0.809252\pi$$
−0.825757 + 0.564026i $$0.809252\pi$$
$$150$$ 0 0
$$151$$ −11.5160 −0.937161 −0.468580 0.883421i $$-0.655234\pi$$
−0.468580 + 0.883421i $$0.655234\pi$$
$$152$$ 0 0
$$153$$ 0.643274 0.0520056
$$154$$ 0 0
$$155$$ 2.64327 0.212313
$$156$$ 0 0
$$157$$ 9.14982 0.730236 0.365118 0.930961i $$-0.381029\pi$$
0.365118 + 0.930961i $$0.381029\pi$$
$$158$$ 0 0
$$159$$ −0.643274 −0.0510150
$$160$$ 0 0
$$161$$ −5.72292 −0.451029
$$162$$ 0 0
$$163$$ 5.50474 0.431164 0.215582 0.976486i $$-0.430835\pi$$
0.215582 + 0.976486i $$0.430835\pi$$
$$164$$ 0 0
$$165$$ −5.57491 −0.434006
$$166$$ 0 0
$$167$$ 0.920352 0.0712190 0.0356095 0.999366i $$-0.488663\pi$$
0.0356095 + 0.999366i $$0.488663\pi$$
$$168$$ 0 0
$$169$$ −8.07965 −0.621511
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −16.6658 −1.26708 −0.633540 0.773710i $$-0.718399\pi$$
−0.633540 + 0.773710i $$0.718399\pi$$
$$174$$ 0 0
$$175$$ 4.21819 0.318865
$$176$$ 0 0
$$177$$ −4.43637 −0.333458
$$178$$ 0 0
$$179$$ 24.0226 1.79553 0.897766 0.440473i $$-0.145189\pi$$
0.897766 + 0.440473i $$0.145189\pi$$
$$180$$ 0 0
$$181$$ −9.14982 −0.680101 −0.340051 0.940407i $$-0.610444\pi$$
−0.340051 + 0.940407i $$0.610444\pi$$
$$182$$ 0 0
$$183$$ −13.0796 −0.966875
$$184$$ 0 0
$$185$$ 2.21819 0.163084
$$186$$ 0 0
$$187$$ −3.58620 −0.262249
$$188$$ 0 0
$$189$$ −4.21819 −0.306828
$$190$$ 0 0
$$191$$ 16.2884 1.17858 0.589292 0.807920i $$-0.299407\pi$$
0.589292 + 0.807920i $$0.299407\pi$$
$$192$$ 0 0
$$193$$ 19.5047 1.40398 0.701991 0.712186i $$-0.252295\pi$$
0.701991 + 0.712186i $$0.252295\pi$$
$$194$$ 0 0
$$195$$ −2.21819 −0.158848
$$196$$ 0 0
$$197$$ 9.51602 0.677988 0.338994 0.940788i $$-0.389913\pi$$
0.338994 + 0.940788i $$0.389913\pi$$
$$198$$ 0 0
$$199$$ 2.27708 0.161418 0.0807089 0.996738i $$-0.474282\pi$$
0.0807089 + 0.996738i $$0.474282\pi$$
$$200$$ 0 0
$$201$$ 1.28655 0.0907461
$$202$$ 0 0
$$203$$ 20.5066 1.43928
$$204$$ 0 0
$$205$$ −3.57491 −0.249683
$$206$$ 0 0
$$207$$ −1.35673 −0.0942990
$$208$$ 0 0
$$209$$ −5.57491 −0.385625
$$210$$ 0 0
$$211$$ −1.42690 −0.0982320 −0.0491160 0.998793i $$-0.515640\pi$$
−0.0491160 + 0.998793i $$0.515640\pi$$
$$212$$ 0 0
$$213$$ 11.1498 0.763973
$$214$$ 0 0
$$215$$ −0.218187 −0.0148802
$$216$$ 0 0
$$217$$ −11.1498 −0.756899
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −1.42690 −0.0959839
$$222$$ 0 0
$$223$$ 6.71345 0.449566 0.224783 0.974409i $$-0.427833\pi$$
0.224783 + 0.974409i $$0.427833\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 13.7931 0.915480 0.457740 0.889086i $$-0.348659\pi$$
0.457740 + 0.889086i $$0.348659\pi$$
$$228$$ 0 0
$$229$$ −14.5066 −0.958620 −0.479310 0.877646i $$-0.659113\pi$$
−0.479310 + 0.877646i $$0.659113\pi$$
$$230$$ 0 0
$$231$$ 23.5160 1.54724
$$232$$ 0 0
$$233$$ 24.2996 1.59192 0.795961 0.605347i $$-0.206966\pi$$
0.795961 + 0.605347i $$0.206966\pi$$
$$234$$ 0 0
$$235$$ −1.35673 −0.0885030
$$236$$ 0 0
$$237$$ −1.28655 −0.0835703
$$238$$ 0 0
$$239$$ 5.57491 0.360611 0.180306 0.983611i $$-0.442291\pi$$
0.180306 + 0.983611i $$0.442291\pi$$
$$240$$ 0 0
$$241$$ 24.2996 1.56528 0.782639 0.622476i $$-0.213873\pi$$
0.782639 + 0.622476i $$0.213873\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −10.7931 −0.689546
$$246$$ 0 0
$$247$$ −2.21819 −0.141140
$$248$$ 0 0
$$249$$ 12.3662 0.783676
$$250$$ 0 0
$$251$$ −14.0113 −0.884385 −0.442192 0.896920i $$-0.645799\pi$$
−0.442192 + 0.896920i $$0.645799\pi$$
$$252$$ 0 0
$$253$$ 7.56363 0.475521
$$254$$ 0 0
$$255$$ 0.643274 0.0402834
$$256$$ 0 0
$$257$$ −10.9204 −0.681193 −0.340596 0.940210i $$-0.610629\pi$$
−0.340596 + 0.940210i $$0.610629\pi$$
$$258$$ 0 0
$$259$$ −9.35673 −0.581399
$$260$$ 0 0
$$261$$ 4.86146 0.300917
$$262$$ 0 0
$$263$$ −10.6658 −0.657684 −0.328842 0.944385i $$-0.606658\pi$$
−0.328842 + 0.944385i $$0.606658\pi$$
$$264$$ 0 0
$$265$$ −0.643274 −0.0395160
$$266$$ 0 0
$$267$$ −12.0113 −0.735079
$$268$$ 0 0
$$269$$ 16.0113 0.976225 0.488113 0.872781i $$-0.337686\pi$$
0.488113 + 0.872781i $$0.337686\pi$$
$$270$$ 0 0
$$271$$ 30.7360 1.86708 0.933540 0.358473i $$-0.116702\pi$$
0.933540 + 0.358473i $$0.116702\pi$$
$$272$$ 0 0
$$273$$ 9.35673 0.566295
$$274$$ 0 0
$$275$$ −5.57491 −0.336180
$$276$$ 0 0
$$277$$ 20.2996 1.21969 0.609844 0.792522i $$-0.291232\pi$$
0.609844 + 0.792522i $$0.291232\pi$$
$$278$$ 0 0
$$279$$ −2.64327 −0.158249
$$280$$ 0 0
$$281$$ −18.5844 −1.10865 −0.554326 0.832300i $$-0.687024\pi$$
−0.554326 + 0.832300i $$0.687024\pi$$
$$282$$ 0 0
$$283$$ 25.0909 1.49150 0.745751 0.666225i $$-0.232091\pi$$
0.745751 + 0.666225i $$0.232091\pi$$
$$284$$ 0 0
$$285$$ 1.00000 0.0592349
$$286$$ 0 0
$$287$$ 15.0796 0.890123
$$288$$ 0 0
$$289$$ −16.5862 −0.975659
$$290$$ 0 0
$$291$$ −5.78181 −0.338936
$$292$$ 0 0
$$293$$ 11.7931 0.688960 0.344480 0.938794i $$-0.388055\pi$$
0.344480 + 0.938794i $$0.388055\pi$$
$$294$$ 0 0
$$295$$ −4.43637 −0.258296
$$296$$ 0 0
$$297$$ 5.57491 0.323489
$$298$$ 0 0
$$299$$ 3.00947 0.174042
$$300$$ 0 0
$$301$$ 0.920352 0.0530482
$$302$$ 0 0
$$303$$ 6.43637 0.369760
$$304$$ 0 0
$$305$$ −13.0796 −0.748938
$$306$$ 0 0
$$307$$ 8.85018 0.505106 0.252553 0.967583i $$-0.418730\pi$$
0.252553 + 0.967583i $$0.418730\pi$$
$$308$$ 0 0
$$309$$ −1.28655 −0.0731892
$$310$$ 0 0
$$311$$ 21.8709 1.24019 0.620093 0.784528i $$-0.287095\pi$$
0.620093 + 0.784528i $$0.287095\pi$$
$$312$$ 0 0
$$313$$ −1.14982 −0.0649919 −0.0324960 0.999472i $$-0.510346\pi$$
−0.0324960 + 0.999472i $$0.510346\pi$$
$$314$$ 0 0
$$315$$ −4.21819 −0.237668
$$316$$ 0 0
$$317$$ 29.1022 1.63454 0.817272 0.576252i $$-0.195486\pi$$
0.817272 + 0.576252i $$0.195486\pi$$
$$318$$ 0 0
$$319$$ −27.1022 −1.51743
$$320$$ 0 0
$$321$$ −1.28655 −0.0718081
$$322$$ 0 0
$$323$$ 0.643274 0.0357927
$$324$$ 0 0
$$325$$ −2.21819 −0.123043
$$326$$ 0 0
$$327$$ −18.4364 −1.01953
$$328$$ 0 0
$$329$$ 5.72292 0.315515
$$330$$ 0 0
$$331$$ 8.94292 0.491548 0.245774 0.969327i $$-0.420958\pi$$
0.245774 + 0.969327i $$0.420958\pi$$
$$332$$ 0 0
$$333$$ −2.21819 −0.121556
$$334$$ 0 0
$$335$$ 1.28655 0.0702917
$$336$$ 0 0
$$337$$ 20.3775 1.11003 0.555016 0.831840i $$-0.312712\pi$$
0.555016 + 0.831840i $$0.312712\pi$$
$$338$$ 0 0
$$339$$ −13.0796 −0.710389
$$340$$ 0 0
$$341$$ 14.7360 0.798000
$$342$$ 0 0
$$343$$ 16.0000 0.863919
$$344$$ 0 0
$$345$$ −1.35673 −0.0730437
$$346$$ 0 0
$$347$$ −7.07965 −0.380055 −0.190028 0.981779i $$-0.560858\pi$$
−0.190028 + 0.981779i $$0.560858\pi$$
$$348$$ 0 0
$$349$$ 36.1593 1.93556 0.967781 0.251792i $$-0.0810199\pi$$
0.967781 + 0.251792i $$0.0810199\pi$$
$$350$$ 0 0
$$351$$ 2.21819 0.118398
$$352$$ 0 0
$$353$$ 32.1593 1.71167 0.855833 0.517252i $$-0.173045\pi$$
0.855833 + 0.517252i $$0.173045\pi$$
$$354$$ 0 0
$$355$$ 11.1498 0.591771
$$356$$ 0 0
$$357$$ −2.71345 −0.143611
$$358$$ 0 0
$$359$$ 2.86146 0.151022 0.0755111 0.997145i $$-0.475941\pi$$
0.0755111 + 0.997145i $$0.475941\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −20.0796 −1.05391
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ 12.3585 0.645111 0.322555 0.946551i $$-0.395458\pi$$
0.322555 + 0.946551i $$0.395458\pi$$
$$368$$ 0 0
$$369$$ 3.57491 0.186103
$$370$$ 0 0
$$371$$ 2.71345 0.140875
$$372$$ 0 0
$$373$$ 0.495265 0.0256438 0.0128219 0.999918i $$-0.495919\pi$$
0.0128219 + 0.999918i $$0.495919\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −10.7836 −0.555385
$$378$$ 0 0
$$379$$ −2.78363 −0.142985 −0.0714927 0.997441i $$-0.522776\pi$$
−0.0714927 + 0.997441i $$0.522776\pi$$
$$380$$ 0 0
$$381$$ −9.28655 −0.475764
$$382$$ 0 0
$$383$$ 33.0131 1.68689 0.843445 0.537215i $$-0.180524\pi$$
0.843445 + 0.537215i $$0.180524\pi$$
$$384$$ 0 0
$$385$$ 23.5160 1.19849
$$386$$ 0 0
$$387$$ 0.218187 0.0110911
$$388$$ 0 0
$$389$$ 30.4589 1.54433 0.772165 0.635422i $$-0.219174\pi$$
0.772165 + 0.635422i $$0.219174\pi$$
$$390$$ 0 0
$$391$$ −0.872747 −0.0441367
$$392$$ 0 0
$$393$$ 2.86146 0.144342
$$394$$ 0 0
$$395$$ −1.28655 −0.0647333
$$396$$ 0 0
$$397$$ −15.2865 −0.767210 −0.383605 0.923497i $$-0.625318\pi$$
−0.383605 + 0.923497i $$0.625318\pi$$
$$398$$ 0 0
$$399$$ −4.21819 −0.211173
$$400$$ 0 0
$$401$$ −36.0339 −1.79944 −0.899722 0.436462i $$-0.856231\pi$$
−0.899722 + 0.436462i $$0.856231\pi$$
$$402$$ 0 0
$$403$$ 5.86328 0.292071
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 12.3662 0.612970
$$408$$ 0 0
$$409$$ −34.0226 −1.68231 −0.841154 0.540796i $$-0.818123\pi$$
−0.841154 + 0.540796i $$0.818123\pi$$
$$410$$ 0 0
$$411$$ 14.8727 0.733618
$$412$$ 0 0
$$413$$ 18.7135 0.920829
$$414$$ 0 0
$$415$$ 12.3662 0.607033
$$416$$ 0 0
$$417$$ 0.850175 0.0416333
$$418$$ 0 0
$$419$$ 8.72474 0.426231 0.213116 0.977027i $$-0.431639\pi$$
0.213116 + 0.977027i $$0.431639\pi$$
$$420$$ 0 0
$$421$$ 35.0131 1.70643 0.853217 0.521556i $$-0.174648\pi$$
0.853217 + 0.521556i $$0.174648\pi$$
$$422$$ 0 0
$$423$$ 1.35673 0.0659663
$$424$$ 0 0
$$425$$ 0.643274 0.0312034
$$426$$ 0 0
$$427$$ 55.1724 2.66998
$$428$$ 0 0
$$429$$ −12.3662 −0.597046
$$430$$ 0 0
$$431$$ 13.8633 0.667771 0.333885 0.942614i $$-0.391640\pi$$
0.333885 + 0.942614i $$0.391640\pi$$
$$432$$ 0 0
$$433$$ −30.2408 −1.45328 −0.726639 0.687019i $$-0.758919\pi$$
−0.726639 + 0.687019i $$0.758919\pi$$
$$434$$ 0 0
$$435$$ 4.86146 0.233089
$$436$$ 0 0
$$437$$ −1.35673 −0.0649010
$$438$$ 0 0
$$439$$ −31.5862 −1.50753 −0.753763 0.657146i $$-0.771764\pi$$
−0.753763 + 0.657146i $$0.771764\pi$$
$$440$$ 0 0
$$441$$ 10.7931 0.513957
$$442$$ 0 0
$$443$$ −18.2295 −0.866108 −0.433054 0.901368i $$-0.642564\pi$$
−0.433054 + 0.901368i $$0.642564\pi$$
$$444$$ 0 0
$$445$$ −12.0113 −0.569390
$$446$$ 0 0
$$447$$ 20.1593 0.953502
$$448$$ 0 0
$$449$$ −9.71164 −0.458320 −0.229160 0.973389i $$-0.573598\pi$$
−0.229160 + 0.973389i $$0.573598\pi$$
$$450$$ 0 0
$$451$$ −19.9298 −0.938459
$$452$$ 0 0
$$453$$ 11.5160 0.541070
$$454$$ 0 0
$$455$$ 9.35673 0.438650
$$456$$ 0 0
$$457$$ −19.8633 −0.929165 −0.464582 0.885530i $$-0.653796\pi$$
−0.464582 + 0.885530i $$0.653796\pi$$
$$458$$ 0 0
$$459$$ −0.643274 −0.0300255
$$460$$ 0 0
$$461$$ −20.2996 −0.945449 −0.472724 0.881210i $$-0.656729\pi$$
−0.472724 + 0.881210i $$0.656729\pi$$
$$462$$ 0 0
$$463$$ −32.2408 −1.49836 −0.749178 0.662369i $$-0.769551\pi$$
−0.749178 + 0.662369i $$0.769551\pi$$
$$464$$ 0 0
$$465$$ −2.64327 −0.122579
$$466$$ 0 0
$$467$$ 26.2295 1.21376 0.606878 0.794795i $$-0.292422\pi$$
0.606878 + 0.794795i $$0.292422\pi$$
$$468$$ 0 0
$$469$$ −5.42690 −0.250591
$$470$$ 0 0
$$471$$ −9.14982 −0.421602
$$472$$ 0 0
$$473$$ −1.21637 −0.0559288
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0.643274 0.0294535
$$478$$ 0 0
$$479$$ −33.1611 −1.51517 −0.757585 0.652737i $$-0.773621\pi$$
−0.757585 + 0.652737i $$0.773621\pi$$
$$480$$ 0 0
$$481$$ 4.92035 0.224349
$$482$$ 0 0
$$483$$ 5.72292 0.260402
$$484$$ 0 0
$$485$$ −5.78181 −0.262539
$$486$$ 0 0
$$487$$ 19.5636 0.886513 0.443256 0.896395i $$-0.353823\pi$$
0.443256 + 0.896395i $$0.353823\pi$$
$$488$$ 0 0
$$489$$ −5.50474 −0.248933
$$490$$ 0 0
$$491$$ 26.8615 1.21224 0.606120 0.795373i $$-0.292725\pi$$
0.606120 + 0.795373i $$0.292725\pi$$
$$492$$ 0 0
$$493$$ 3.12725 0.140844
$$494$$ 0 0
$$495$$ 5.57491 0.250574
$$496$$ 0 0
$$497$$ −47.0320 −2.10968
$$498$$ 0 0
$$499$$ −40.0226 −1.79166 −0.895828 0.444401i $$-0.853417\pi$$
−0.895828 + 0.444401i $$0.853417\pi$$
$$500$$ 0 0
$$501$$ −0.920352 −0.0411183
$$502$$ 0 0
$$503$$ 17.6527 0.787097 0.393549 0.919304i $$-0.371247\pi$$
0.393549 + 0.919304i $$0.371247\pi$$
$$504$$ 0 0
$$505$$ 6.43637 0.286415
$$506$$ 0 0
$$507$$ 8.07965 0.358830
$$508$$ 0 0
$$509$$ −16.8615 −0.747371 −0.373686 0.927555i $$-0.621906\pi$$
−0.373686 + 0.927555i $$0.621906\pi$$
$$510$$ 0 0
$$511$$ 42.1819 1.86602
$$512$$ 0 0
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ −1.28655 −0.0566921
$$516$$ 0 0
$$517$$ −7.56363 −0.332648
$$518$$ 0 0
$$519$$ 16.6658 0.731549
$$520$$ 0 0
$$521$$ 3.57491 0.156620 0.0783099 0.996929i $$-0.475048\pi$$
0.0783099 + 0.996929i $$0.475048\pi$$
$$522$$ 0 0
$$523$$ 28.4364 1.24344 0.621718 0.783241i $$-0.286435\pi$$
0.621718 + 0.783241i $$0.286435\pi$$
$$524$$ 0 0
$$525$$ −4.21819 −0.184097
$$526$$ 0 0
$$527$$ −1.70035 −0.0740684
$$528$$ 0 0
$$529$$ −21.1593 −0.919969
$$530$$ 0 0
$$531$$ 4.43637 0.192522
$$532$$ 0 0
$$533$$ −7.92982 −0.343479
$$534$$ 0 0
$$535$$ −1.28655 −0.0556224
$$536$$ 0 0
$$537$$ −24.0226 −1.03665
$$538$$ 0 0
$$539$$ −60.1706 −2.59173
$$540$$ 0 0
$$541$$ 0.573097 0.0246394 0.0123197 0.999924i $$-0.496078\pi$$
0.0123197 + 0.999924i $$0.496078\pi$$
$$542$$ 0 0
$$543$$ 9.14982 0.392657
$$544$$ 0 0
$$545$$ −18.4364 −0.789727
$$546$$ 0 0
$$547$$ 8.85018 0.378406 0.189203 0.981938i $$-0.439410\pi$$
0.189203 + 0.981938i $$0.439410\pi$$
$$548$$ 0 0
$$549$$ 13.0796 0.558226
$$550$$ 0 0
$$551$$ 4.86146 0.207105
$$552$$ 0 0
$$553$$ 5.42690 0.230775
$$554$$ 0 0
$$555$$ −2.21819 −0.0941568
$$556$$ 0 0
$$557$$ 8.29965 0.351667 0.175834 0.984420i $$-0.443738\pi$$
0.175834 + 0.984420i $$0.443738\pi$$
$$558$$ 0 0
$$559$$ −0.483979 −0.0204701
$$560$$ 0 0
$$561$$ 3.58620 0.151409
$$562$$ 0 0
$$563$$ −17.3793 −0.732450 −0.366225 0.930526i $$-0.619350\pi$$
−0.366225 + 0.930526i $$0.619350\pi$$
$$564$$ 0 0
$$565$$ −13.0796 −0.550265
$$566$$ 0 0
$$567$$ 4.21819 0.177147
$$568$$ 0 0
$$569$$ 19.4346 0.814739 0.407370 0.913263i $$-0.366446\pi$$
0.407370 + 0.913263i $$0.366446\pi$$
$$570$$ 0 0
$$571$$ −32.1629 −1.34598 −0.672988 0.739653i $$-0.734990\pi$$
−0.672988 + 0.739653i $$0.734990\pi$$
$$572$$ 0 0
$$573$$ −16.2884 −0.680456
$$574$$ 0 0
$$575$$ −1.35673 −0.0565794
$$576$$ 0 0
$$577$$ −31.1688 −1.29757 −0.648786 0.760971i $$-0.724723\pi$$
−0.648786 + 0.760971i $$0.724723\pi$$
$$578$$ 0 0
$$579$$ −19.5047 −0.810589
$$580$$ 0 0
$$581$$ −52.1629 −2.16408
$$582$$ 0 0
$$583$$ −3.58620 −0.148525
$$584$$ 0 0
$$585$$ 2.21819 0.0917107
$$586$$ 0 0
$$587$$ −23.5160 −0.970610 −0.485305 0.874345i $$-0.661291\pi$$
−0.485305 + 0.874345i $$0.661291\pi$$
$$588$$ 0 0
$$589$$ −2.64327 −0.108914
$$590$$ 0 0
$$591$$ −9.51602 −0.391437
$$592$$ 0 0
$$593$$ −31.0131 −1.27356 −0.636778 0.771047i $$-0.719733\pi$$
−0.636778 + 0.771047i $$0.719733\pi$$
$$594$$ 0 0
$$595$$ −2.71345 −0.111241
$$596$$ 0 0
$$597$$ −2.27708 −0.0931946
$$598$$ 0 0
$$599$$ −39.1724 −1.60054 −0.800270 0.599639i $$-0.795311\pi$$
−0.800270 + 0.599639i $$0.795311\pi$$
$$600$$ 0 0
$$601$$ −3.86328 −0.157586 −0.0787932 0.996891i $$-0.525107\pi$$
−0.0787932 + 0.996891i $$0.525107\pi$$
$$602$$ 0 0
$$603$$ −1.28655 −0.0523923
$$604$$ 0 0
$$605$$ −20.0796 −0.816354
$$606$$ 0 0
$$607$$ −8.02257 −0.325626 −0.162813 0.986657i $$-0.552057\pi$$
−0.162813 + 0.986657i $$0.552057\pi$$
$$608$$ 0 0
$$609$$ −20.5066 −0.830967
$$610$$ 0 0
$$611$$ −3.00947 −0.121750
$$612$$ 0 0
$$613$$ 19.8633 0.802270 0.401135 0.916019i $$-0.368616\pi$$
0.401135 + 0.916019i $$0.368616\pi$$
$$614$$ 0 0
$$615$$ 3.57491 0.144154
$$616$$ 0 0
$$617$$ 18.3888 0.740304 0.370152 0.928971i $$-0.379306\pi$$
0.370152 + 0.928971i $$0.379306\pi$$
$$618$$ 0 0
$$619$$ −12.4364 −0.499860 −0.249930 0.968264i $$-0.580408\pi$$
−0.249930 + 0.968264i $$0.580408\pi$$
$$620$$ 0 0
$$621$$ 1.35673 0.0544435
$$622$$ 0 0
$$623$$ 50.6658 2.02988
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 5.57491 0.222641
$$628$$ 0 0
$$629$$ −1.42690 −0.0568943
$$630$$ 0 0
$$631$$ −42.6182 −1.69661 −0.848303 0.529512i $$-0.822375\pi$$
−0.848303 + 0.529512i $$0.822375\pi$$
$$632$$ 0 0
$$633$$ 1.42690 0.0567143
$$634$$ 0 0
$$635$$ −9.28655 −0.368525
$$636$$ 0 0
$$637$$ −23.9411 −0.948581
$$638$$ 0 0
$$639$$ −11.1498 −0.441080
$$640$$ 0 0
$$641$$ −3.16111 −0.124856 −0.0624282 0.998049i $$-0.519884\pi$$
−0.0624282 + 0.998049i $$0.519884\pi$$
$$642$$ 0 0
$$643$$ −36.5368 −1.44087 −0.720435 0.693523i $$-0.756058\pi$$
−0.720435 + 0.693523i $$0.756058\pi$$
$$644$$ 0 0
$$645$$ 0.218187 0.00859109
$$646$$ 0 0
$$647$$ 17.6527 0.694001 0.347001 0.937865i $$-0.387200\pi$$
0.347001 + 0.937865i $$0.387200\pi$$
$$648$$ 0 0
$$649$$ −24.7324 −0.970831
$$650$$ 0 0
$$651$$ 11.1498 0.436996
$$652$$ 0 0
$$653$$ −5.51602 −0.215859 −0.107929 0.994159i $$-0.534422\pi$$
−0.107929 + 0.994159i $$0.534422\pi$$
$$654$$ 0 0
$$655$$ 2.86146 0.111807
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ −18.2996 −0.712853 −0.356427 0.934323i $$-0.616005\pi$$
−0.356427 + 0.934323i $$0.616005\pi$$
$$660$$ 0 0
$$661$$ 7.72292 0.300387 0.150193 0.988657i $$-0.452010\pi$$
0.150193 + 0.988657i $$0.452010\pi$$
$$662$$ 0 0
$$663$$ 1.42690 0.0554163
$$664$$ 0 0
$$665$$ −4.21819 −0.163574
$$666$$ 0 0
$$667$$ −6.59567 −0.255385
$$668$$ 0 0
$$669$$ −6.71345 −0.259557
$$670$$ 0 0
$$671$$ −72.9179 −2.81496
$$672$$ 0 0
$$673$$ 39.9637 1.54049 0.770243 0.637750i $$-0.220135\pi$$
0.770243 + 0.637750i $$0.220135\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −48.2295 −1.85361 −0.926805 0.375544i $$-0.877456\pi$$
−0.926805 + 0.375544i $$0.877456\pi$$
$$678$$ 0 0
$$679$$ 24.3888 0.935955
$$680$$ 0 0
$$681$$ −13.7931 −0.528553
$$682$$ 0 0
$$683$$ 15.5862 0.596389 0.298195 0.954505i $$-0.403616\pi$$
0.298195 + 0.954505i $$0.403616\pi$$
$$684$$ 0 0
$$685$$ 14.8727 0.568258
$$686$$ 0 0
$$687$$ 14.5066 0.553459
$$688$$ 0 0
$$689$$ −1.42690 −0.0543607
$$690$$ 0 0
$$691$$ −33.8633 −1.28822 −0.644110 0.764933i $$-0.722772\pi$$
−0.644110 + 0.764933i $$0.722772\pi$$
$$692$$ 0 0
$$693$$ −23.5160 −0.893300
$$694$$ 0 0
$$695$$ 0.850175 0.0322490
$$696$$ 0 0
$$697$$ 2.29965 0.0871054
$$698$$ 0 0
$$699$$ −24.2996 −0.919097
$$700$$ 0 0
$$701$$ 31.3317 1.18338 0.591691 0.806165i $$-0.298461\pi$$
0.591691 + 0.806165i $$0.298461\pi$$
$$702$$ 0 0
$$703$$ −2.21819 −0.0836605
$$704$$ 0 0
$$705$$ 1.35673 0.0510972
$$706$$ 0 0
$$707$$ −27.1498 −1.02107
$$708$$ 0 0
$$709$$ −0.939294 −0.0352759 −0.0176380 0.999844i $$-0.505615\pi$$
−0.0176380 + 0.999844i $$0.505615\pi$$
$$710$$ 0 0
$$711$$ 1.28655 0.0482493
$$712$$ 0 0
$$713$$ 3.58620 0.134304
$$714$$ 0 0
$$715$$ −12.3662 −0.462470
$$716$$ 0 0
$$717$$ −5.57491 −0.208199
$$718$$ 0 0
$$719$$ 33.1611 1.23670 0.618350 0.785903i $$-0.287801\pi$$
0.618350 + 0.785903i $$0.287801\pi$$
$$720$$ 0 0
$$721$$ 5.42690 0.202108
$$722$$ 0 0
$$723$$ −24.2996 −0.903714
$$724$$ 0 0
$$725$$ 4.86146 0.180550
$$726$$ 0 0
$$727$$ 32.2408 1.19574 0.597872 0.801592i $$-0.296013\pi$$
0.597872 + 0.801592i $$0.296013\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0.140354 0.00519118
$$732$$ 0 0
$$733$$ 17.5862 0.649561 0.324781 0.945789i $$-0.394710\pi$$
0.324781 + 0.945789i $$0.394710\pi$$
$$734$$ 0 0
$$735$$ 10.7931 0.398109
$$736$$ 0 0
$$737$$ 7.17240 0.264199
$$738$$ 0 0
$$739$$ −14.5731 −0.536080 −0.268040 0.963408i $$-0.586376\pi$$
−0.268040 + 0.963408i $$0.586376\pi$$
$$740$$ 0 0
$$741$$ 2.21819 0.0814871
$$742$$ 0 0
$$743$$ 27.5862 1.01204 0.506020 0.862522i $$-0.331116\pi$$
0.506020 + 0.862522i $$0.331116\pi$$
$$744$$ 0 0
$$745$$ 20.1593 0.738579
$$746$$ 0 0
$$747$$ −12.3662 −0.452455
$$748$$ 0 0
$$749$$ 5.42690 0.198295
$$750$$ 0 0
$$751$$ 11.5160 0.420226 0.210113 0.977677i $$-0.432617\pi$$
0.210113 + 0.977677i $$0.432617\pi$$
$$752$$ 0 0
$$753$$ 14.0113 0.510600
$$754$$ 0 0
$$755$$ 11.5160 0.419111
$$756$$ 0 0
$$757$$ −13.2902 −0.483040 −0.241520 0.970396i $$-0.577646\pi$$
−0.241520 + 0.970396i $$0.577646\pi$$
$$758$$ 0 0
$$759$$ −7.56363 −0.274542
$$760$$ 0 0
$$761$$ −7.84070 −0.284225 −0.142113 0.989850i $$-0.545390\pi$$
−0.142113 + 0.989850i $$0.545390\pi$$
$$762$$ 0 0
$$763$$ 77.7681 2.81539
$$764$$ 0 0
$$765$$ −0.643274 −0.0232576
$$766$$ 0 0
$$767$$ −9.84070 −0.355327
$$768$$ 0 0
$$769$$ −52.2996 −1.88597 −0.942987 0.332830i $$-0.891996\pi$$
−0.942987 + 0.332830i $$0.891996\pi$$
$$770$$ 0 0
$$771$$ 10.9204 0.393287
$$772$$ 0 0
$$773$$ −25.2426 −0.907912 −0.453956 0.891024i $$-0.649988\pi$$
−0.453956 + 0.891024i $$0.649988\pi$$
$$774$$ 0 0
$$775$$ −2.64327 −0.0949492
$$776$$ 0 0
$$777$$ 9.35673 0.335671
$$778$$ 0 0
$$779$$ 3.57491 0.128085
$$780$$ 0 0
$$781$$ 62.1593 2.22423
$$782$$ 0 0
$$783$$ −4.86146 −0.173734
$$784$$ 0 0
$$785$$ −9.14982 −0.326571
$$786$$ 0 0
$$787$$ 17.8633 0.636757 0.318379 0.947964i $$-0.396862\pi$$
0.318379 + 0.947964i $$0.396862\pi$$
$$788$$ 0 0
$$789$$ 10.6658 0.379714
$$790$$ 0 0
$$791$$ 55.1724 1.96170
$$792$$ 0 0
$$793$$ −29.0131 −1.03029
$$794$$ 0 0
$$795$$ 0.643274 0.0228146
$$796$$ 0 0
$$797$$ 6.50655 0.230474 0.115237 0.993338i $$-0.463237\pi$$
0.115237 + 0.993338i $$0.463237\pi$$
$$798$$ 0 0
$$799$$ 0.872747 0.0308756
$$800$$ 0 0
$$801$$ 12.0113 0.424398
$$802$$ 0 0
$$803$$ −55.7491 −1.96734
$$804$$ 0 0
$$805$$ 5.72292 0.201707
$$806$$ 0 0
$$807$$ −16.0113 −0.563624
$$808$$ 0 0
$$809$$ 3.84070 0.135032 0.0675160 0.997718i $$-0.478493\pi$$
0.0675160 + 0.997718i $$0.478493\pi$$
$$810$$ 0 0
$$811$$ 7.44584 0.261459 0.130729 0.991418i $$-0.458268\pi$$
0.130729 + 0.991418i $$0.458268\pi$$
$$812$$ 0 0
$$813$$ −30.7360 −1.07796
$$814$$ 0 0
$$815$$ −5.50474 −0.192822
$$816$$ 0 0
$$817$$ 0.218187 0.00763339
$$818$$ 0 0
$$819$$ −9.35673 −0.326950
$$820$$ 0 0
$$821$$ −30.2960 −1.05734 −0.528669 0.848828i $$-0.677309\pi$$
−0.528669 + 0.848828i $$0.677309\pi$$
$$822$$ 0 0
$$823$$ −32.3811 −1.12873 −0.564367 0.825524i $$-0.690880\pi$$
−0.564367 + 0.825524i $$0.690880\pi$$
$$824$$ 0 0
$$825$$ 5.57491 0.194094
$$826$$ 0 0
$$827$$ 11.9524 0.415625 0.207813 0.978169i $$-0.433366\pi$$
0.207813 + 0.978169i $$0.433366\pi$$
$$828$$ 0 0
$$829$$ −22.5767 −0.784122 −0.392061 0.919939i $$-0.628238\pi$$
−0.392061 + 0.919939i $$0.628238\pi$$
$$830$$ 0 0
$$831$$ −20.2996 −0.704187
$$832$$ 0 0
$$833$$ 6.94292 0.240558
$$834$$ 0 0
$$835$$ −0.920352 −0.0318501
$$836$$ 0 0
$$837$$ 2.64327 0.0913649
$$838$$ 0 0
$$839$$ 3.14982 0.108744 0.0543720 0.998521i $$-0.482684\pi$$
0.0543720 + 0.998521i $$0.482684\pi$$
$$840$$ 0 0
$$841$$ −5.36620 −0.185041
$$842$$ 0 0
$$843$$ 18.5844 0.640080
$$844$$ 0 0
$$845$$ 8.07965 0.277948
$$846$$ 0 0
$$847$$ 84.6997 2.91032
$$848$$ 0 0
$$849$$ −25.0909 −0.861119
$$850$$ 0 0
$$851$$ 3.00947 0.103163
$$852$$ 0 0
$$853$$ 11.5826 0.396580 0.198290 0.980143i $$-0.436461\pi$$
0.198290 + 0.980143i $$0.436461\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ 3.91088 0.133593 0.0667966 0.997767i $$-0.478722\pi$$
0.0667966 + 0.997767i $$0.478722\pi$$
$$858$$ 0 0
$$859$$ 5.01310 0.171045 0.0855224 0.996336i $$-0.472744\pi$$
0.0855224 + 0.996336i $$0.472744\pi$$
$$860$$ 0 0
$$861$$ −15.0796 −0.513913
$$862$$ 0 0
$$863$$ −40.7324 −1.38655 −0.693273 0.720675i $$-0.743832\pi$$
−0.693273 + 0.720675i $$0.743832\pi$$
$$864$$ 0 0
$$865$$ 16.6658 0.566656
$$866$$ 0 0
$$867$$ 16.5862 0.563297
$$868$$ 0 0
$$869$$ −7.17240 −0.243307
$$870$$ 0 0
$$871$$ 2.85381 0.0966975
$$872$$ 0 0
$$873$$ 5.78181 0.195685
$$874$$ 0 0
$$875$$ −4.21819 −0.142601
$$876$$ 0 0
$$877$$ 36.2371 1.22364 0.611820 0.790997i $$-0.290438\pi$$
0.611820 + 0.790997i $$0.290438\pi$$
$$878$$ 0 0
$$879$$ −11.7931 −0.397771
$$880$$ 0 0
$$881$$ −18.0226 −0.607196 −0.303598 0.952800i $$-0.598188\pi$$
−0.303598 + 0.952800i $$0.598188\pi$$
$$882$$ 0 0
$$883$$ 52.6771 1.77273 0.886363 0.462990i $$-0.153224\pi$$
0.886363 + 0.462990i $$0.153224\pi$$
$$884$$ 0 0
$$885$$ 4.43637 0.149127
$$886$$ 0 0
$$887$$ −32.8727 −1.10376 −0.551879 0.833924i $$-0.686089\pi$$
−0.551879 + 0.833924i $$0.686089\pi$$
$$888$$ 0 0
$$889$$ 39.1724 1.31380
$$890$$ 0 0
$$891$$ −5.57491 −0.186767
$$892$$ 0 0
$$893$$ 1.35673 0.0454011
$$894$$ 0 0
$$895$$ −24.0226 −0.802986
$$896$$ 0 0
$$897$$ −3.00947 −0.100483
$$898$$ 0 0
$$899$$ −12.8502 −0.428577
$$900$$ 0 0
$$901$$ 0.413802 0.0137857
$$902$$ 0 0
$$903$$ −0.920352 −0.0306274
$$904$$ 0 0
$$905$$ 9.14982 0.304150
$$906$$ 0 0
$$907$$ 26.5957 0.883095 0.441547 0.897238i $$-0.354430\pi$$
0.441547 + 0.897238i $$0.354430\pi$$
$$908$$ 0 0
$$909$$ −6.43637 −0.213481
$$910$$ 0 0
$$911$$ −6.15930 −0.204067 −0.102033 0.994781i $$-0.532535\pi$$
−0.102033 + 0.994781i $$0.532535\pi$$
$$912$$ 0 0
$$913$$ 68.9405 2.28160
$$914$$ 0 0
$$915$$ 13.0796 0.432400
$$916$$ 0 0
$$917$$ −12.0702 −0.398592
$$918$$ 0 0
$$919$$ 33.8858 1.11779 0.558895 0.829238i $$-0.311225\pi$$
0.558895 + 0.829238i $$0.311225\pi$$
$$920$$ 0 0
$$921$$ −8.85018 −0.291623
$$922$$ 0 0
$$923$$ 24.7324 0.814077
$$924$$ 0 0
$$925$$ −2.21819 −0.0729335
$$926$$ 0 0
$$927$$ 1.28655 0.0422558
$$928$$ 0 0
$$929$$ −34.8953 −1.14488 −0.572439 0.819947i $$-0.694003\pi$$
−0.572439 + 0.819947i $$0.694003\pi$$
$$930$$ 0 0
$$931$$ 10.7931 0.353730
$$932$$ 0 0
$$933$$ −21.8709 −0.716022
$$934$$ 0 0
$$935$$ 3.58620 0.117281
$$936$$ 0 0
$$937$$ −6.29602 −0.205682 −0.102841 0.994698i $$-0.532793\pi$$
−0.102841 + 0.994698i $$0.532793\pi$$
$$938$$ 0 0
$$939$$ 1.14982 0.0375231
$$940$$ 0 0
$$941$$ 10.8651 0.354192 0.177096 0.984194i $$-0.443330\pi$$
0.177096 + 0.984194i $$0.443330\pi$$
$$942$$ 0 0
$$943$$ −4.85018 −0.157943
$$944$$ 0 0
$$945$$ 4.21819 0.137218
$$946$$ 0 0
$$947$$ 9.79310 0.318233 0.159116 0.987260i $$-0.449135\pi$$
0.159116 + 0.987260i $$0.449135\pi$$
$$948$$ 0 0
$$949$$ −22.1819 −0.720054
$$950$$ 0 0
$$951$$ −29.1022 −0.943704
$$952$$ 0 0
$$953$$ 1.35310 0.0438311 0.0219155 0.999760i $$-0.493024\pi$$
0.0219155 + 0.999760i $$0.493024\pi$$
$$954$$ 0 0
$$955$$ −16.2884 −0.527079
$$956$$ 0 0
$$957$$ 27.1022 0.876090
$$958$$ 0 0
$$959$$ −62.7360 −2.02585
$$960$$ 0 0
$$961$$ −24.0131 −0.774616
$$962$$ 0 0
$$963$$ 1.28655 0.0414585
$$964$$ 0 0
$$965$$ −19.5047 −0.627880
$$966$$ 0 0
$$967$$ −21.9637 −0.706304 −0.353152 0.935566i $$-0.614890\pi$$
−0.353152 + 0.935566i $$0.614890\pi$$
$$968$$ 0 0
$$969$$ −0.643274 −0.0206649
$$970$$ 0 0
$$971$$ 13.7455 0.441114 0.220557 0.975374i $$-0.429213\pi$$
0.220557 + 0.975374i $$0.429213\pi$$
$$972$$ 0 0
$$973$$ −3.58620 −0.114968
$$974$$ 0 0
$$975$$ 2.21819 0.0710388
$$976$$ 0 0
$$977$$ 48.3698 1.54749 0.773744 0.633498i $$-0.218382\pi$$
0.773744 + 0.633498i $$0.218382\pi$$
$$978$$ 0 0
$$979$$ −66.9619 −2.14011
$$980$$ 0 0
$$981$$ 18.4364 0.588628
$$982$$ 0 0
$$983$$ −14.0665 −0.448653 −0.224327 0.974514i $$-0.572018\pi$$
−0.224327 + 0.974514i $$0.572018\pi$$
$$984$$ 0 0
$$985$$ −9.51602 −0.303206
$$986$$ 0 0
$$987$$ −5.72292 −0.182163
$$988$$ 0 0
$$989$$ −0.296019 −0.00941287
$$990$$ 0 0
$$991$$ 33.2865 1.05738 0.528691 0.848814i $$-0.322683\pi$$
0.528691 + 0.848814i $$0.322683\pi$$
$$992$$ 0 0
$$993$$ −8.94292 −0.283795
$$994$$ 0 0
$$995$$ −2.27708 −0.0721882
$$996$$ 0 0
$$997$$ −48.5957 −1.53904 −0.769520 0.638623i $$-0.779505\pi$$
−0.769520 + 0.638623i $$0.779505\pi$$
$$998$$ 0 0
$$999$$ 2.21819 0.0701803
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 4560.2.a.bq.1.3 3
4.3 odd 2 2280.2.a.t.1.1 3
12.11 even 2 6840.2.a.bn.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.t.1.1 3 4.3 odd 2
4560.2.a.bq.1.3 3 1.1 even 1 trivial
6840.2.a.bn.1.1 3 12.11 even 2