# Properties

 Label 8034.2.a.o.1.4 Level 8034 Weight 2 Character 8034.1 Self dual yes Analytic conductor 64.152 Analytic rank 1 Dimension 7 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8034 = 2 \cdot 3 \cdot 13 \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8034.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1518129839$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 4 x^{5} + 14 x^{4} + 3 x^{3} - 12 x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.86678$$ of defining polynomial Character $$\chi$$ $$=$$ 8034.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.72171 q^{5} +1.00000 q^{6} +2.39003 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.72171 q^{5} +1.00000 q^{6} +2.39003 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.72171 q^{10} -4.25377 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.39003 q^{14} -1.72171 q^{15} +1.00000 q^{16} +6.64067 q^{17} -1.00000 q^{18} +1.39003 q^{19} +1.72171 q^{20} -2.39003 q^{21} +4.25377 q^{22} +0.660198 q^{23} +1.00000 q^{24} -2.03571 q^{25} -1.00000 q^{26} -1.00000 q^{27} +2.39003 q^{28} -7.21542 q^{29} +1.72171 q^{30} -7.06840 q^{31} -1.00000 q^{32} +4.25377 q^{33} -6.64067 q^{34} +4.11494 q^{35} +1.00000 q^{36} +0.542704 q^{37} -1.39003 q^{38} -1.00000 q^{39} -1.72171 q^{40} +6.29089 q^{41} +2.39003 q^{42} -12.5305 q^{43} -4.25377 q^{44} +1.72171 q^{45} -0.660198 q^{46} -5.06122 q^{47} -1.00000 q^{48} -1.28777 q^{49} +2.03571 q^{50} -6.64067 q^{51} +1.00000 q^{52} +3.16708 q^{53} +1.00000 q^{54} -7.32377 q^{55} -2.39003 q^{56} -1.39003 q^{57} +7.21542 q^{58} +0.644743 q^{59} -1.72171 q^{60} +7.01661 q^{61} +7.06840 q^{62} +2.39003 q^{63} +1.00000 q^{64} +1.72171 q^{65} -4.25377 q^{66} -8.17546 q^{67} +6.64067 q^{68} -0.660198 q^{69} -4.11494 q^{70} +1.21505 q^{71} -1.00000 q^{72} -1.21748 q^{73} -0.542704 q^{74} +2.03571 q^{75} +1.39003 q^{76} -10.1666 q^{77} +1.00000 q^{78} -5.16356 q^{79} +1.72171 q^{80} +1.00000 q^{81} -6.29089 q^{82} -3.97674 q^{83} -2.39003 q^{84} +11.4333 q^{85} +12.5305 q^{86} +7.21542 q^{87} +4.25377 q^{88} -12.3981 q^{89} -1.72171 q^{90} +2.39003 q^{91} +0.660198 q^{92} +7.06840 q^{93} +5.06122 q^{94} +2.39323 q^{95} +1.00000 q^{96} -18.4985 q^{97} +1.28777 q^{98} -4.25377 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 7q^{2} - 7q^{3} + 7q^{4} + 2q^{5} + 7q^{6} - 9q^{7} - 7q^{8} + 7q^{9} + O(q^{10})$$ $$7q - 7q^{2} - 7q^{3} + 7q^{4} + 2q^{5} + 7q^{6} - 9q^{7} - 7q^{8} + 7q^{9} - 2q^{10} - 7q^{12} + 7q^{13} + 9q^{14} - 2q^{15} + 7q^{16} + 3q^{17} - 7q^{18} - 16q^{19} + 2q^{20} + 9q^{21} + 6q^{23} + 7q^{24} + 15q^{25} - 7q^{26} - 7q^{27} - 9q^{28} - 5q^{29} + 2q^{30} - 16q^{31} - 7q^{32} - 3q^{34} - 10q^{35} + 7q^{36} + 17q^{37} + 16q^{38} - 7q^{39} - 2q^{40} + 12q^{41} - 9q^{42} - 22q^{43} + 2q^{45} - 6q^{46} - 7q^{48} - 2q^{49} - 15q^{50} - 3q^{51} + 7q^{52} + 2q^{53} + 7q^{54} - 16q^{55} + 9q^{56} + 16q^{57} + 5q^{58} - 3q^{59} - 2q^{60} - 6q^{61} + 16q^{62} - 9q^{63} + 7q^{64} + 2q^{65} + q^{67} + 3q^{68} - 6q^{69} + 10q^{70} + 15q^{71} - 7q^{72} + 17q^{73} - 17q^{74} - 15q^{75} - 16q^{76} - 10q^{77} + 7q^{78} - 27q^{79} + 2q^{80} + 7q^{81} - 12q^{82} + 12q^{83} + 9q^{84} + 15q^{85} + 22q^{86} + 5q^{87} - 9q^{89} - 2q^{90} - 9q^{91} + 6q^{92} + 16q^{93} - 12q^{95} + 7q^{96} - 3q^{97} + 2q^{98} + O(q^{100})$$

## 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
$$5$$ 1.72171 0.769972 0.384986 0.922922i $$-0.374206\pi$$
0.384986 + 0.922922i $$0.374206\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 2.39003 0.903346 0.451673 0.892184i $$-0.350827\pi$$
0.451673 + 0.892184i $$0.350827\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.72171 −0.544453
$$11$$ −4.25377 −1.28256 −0.641281 0.767306i $$-0.721597\pi$$
−0.641281 + 0.767306i $$0.721597\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ −2.39003 −0.638762
$$15$$ −1.72171 −0.444544
$$16$$ 1.00000 0.250000
$$17$$ 6.64067 1.61060 0.805299 0.592869i $$-0.202005\pi$$
0.805299 + 0.592869i $$0.202005\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.39003 0.318894 0.159447 0.987206i $$-0.449029\pi$$
0.159447 + 0.987206i $$0.449029\pi$$
$$20$$ 1.72171 0.384986
$$21$$ −2.39003 −0.521547
$$22$$ 4.25377 0.906908
$$23$$ 0.660198 0.137661 0.0688304 0.997628i $$-0.478073\pi$$
0.0688304 + 0.997628i $$0.478073\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −2.03571 −0.407143
$$26$$ −1.00000 −0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 2.39003 0.451673
$$29$$ −7.21542 −1.33987 −0.669935 0.742420i $$-0.733678\pi$$
−0.669935 + 0.742420i $$0.733678\pi$$
$$30$$ 1.72171 0.314340
$$31$$ −7.06840 −1.26952 −0.634761 0.772708i $$-0.718902\pi$$
−0.634761 + 0.772708i $$0.718902\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.25377 0.740487
$$34$$ −6.64067 −1.13887
$$35$$ 4.11494 0.695551
$$36$$ 1.00000 0.166667
$$37$$ 0.542704 0.0892199 0.0446100 0.999004i $$-0.485795\pi$$
0.0446100 + 0.999004i $$0.485795\pi$$
$$38$$ −1.39003 −0.225492
$$39$$ −1.00000 −0.160128
$$40$$ −1.72171 −0.272226
$$41$$ 6.29089 0.982472 0.491236 0.871027i $$-0.336545\pi$$
0.491236 + 0.871027i $$0.336545\pi$$
$$42$$ 2.39003 0.368789
$$43$$ −12.5305 −1.91088 −0.955441 0.295182i $$-0.904620\pi$$
−0.955441 + 0.295182i $$0.904620\pi$$
$$44$$ −4.25377 −0.641281
$$45$$ 1.72171 0.256657
$$46$$ −0.660198 −0.0973408
$$47$$ −5.06122 −0.738254 −0.369127 0.929379i $$-0.620343\pi$$
−0.369127 + 0.929379i $$0.620343\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −1.28777 −0.183967
$$50$$ 2.03571 0.287893
$$51$$ −6.64067 −0.929879
$$52$$ 1.00000 0.138675
$$53$$ 3.16708 0.435032 0.217516 0.976057i $$-0.430205\pi$$
0.217516 + 0.976057i $$0.430205\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −7.32377 −0.987536
$$56$$ −2.39003 −0.319381
$$57$$ −1.39003 −0.184114
$$58$$ 7.21542 0.947431
$$59$$ 0.644743 0.0839384 0.0419692 0.999119i $$-0.486637\pi$$
0.0419692 + 0.999119i $$0.486637\pi$$
$$60$$ −1.72171 −0.222272
$$61$$ 7.01661 0.898384 0.449192 0.893435i $$-0.351712\pi$$
0.449192 + 0.893435i $$0.351712\pi$$
$$62$$ 7.06840 0.897688
$$63$$ 2.39003 0.301115
$$64$$ 1.00000 0.125000
$$65$$ 1.72171 0.213552
$$66$$ −4.25377 −0.523603
$$67$$ −8.17546 −0.998792 −0.499396 0.866374i $$-0.666445\pi$$
−0.499396 + 0.866374i $$0.666445\pi$$
$$68$$ 6.64067 0.805299
$$69$$ −0.660198 −0.0794784
$$70$$ −4.11494 −0.491829
$$71$$ 1.21505 0.144200 0.0720998 0.997397i $$-0.477030\pi$$
0.0720998 + 0.997397i $$0.477030\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −1.21748 −0.142495 −0.0712475 0.997459i $$-0.522698\pi$$
−0.0712475 + 0.997459i $$0.522698\pi$$
$$74$$ −0.542704 −0.0630880
$$75$$ 2.03571 0.235064
$$76$$ 1.39003 0.159447
$$77$$ −10.1666 −1.15860
$$78$$ 1.00000 0.113228
$$79$$ −5.16356 −0.580946 −0.290473 0.956883i $$-0.593813\pi$$
−0.290473 + 0.956883i $$0.593813\pi$$
$$80$$ 1.72171 0.192493
$$81$$ 1.00000 0.111111
$$82$$ −6.29089 −0.694713
$$83$$ −3.97674 −0.436504 −0.218252 0.975892i $$-0.570035\pi$$
−0.218252 + 0.975892i $$0.570035\pi$$
$$84$$ −2.39003 −0.260773
$$85$$ 11.4333 1.24012
$$86$$ 12.5305 1.35120
$$87$$ 7.21542 0.773574
$$88$$ 4.25377 0.453454
$$89$$ −12.3981 −1.31420 −0.657099 0.753804i $$-0.728217\pi$$
−0.657099 + 0.753804i $$0.728217\pi$$
$$90$$ −1.72171 −0.181484
$$91$$ 2.39003 0.250543
$$92$$ 0.660198 0.0688304
$$93$$ 7.06840 0.732959
$$94$$ 5.06122 0.522025
$$95$$ 2.39323 0.245540
$$96$$ 1.00000 0.102062
$$97$$ −18.4985 −1.87824 −0.939120 0.343589i $$-0.888357\pi$$
−0.939120 + 0.343589i $$0.888357\pi$$
$$98$$ 1.28777 0.130084
$$99$$ −4.25377 −0.427520
$$100$$ −2.03571 −0.203571
$$101$$ 18.3579 1.82668 0.913338 0.407201i $$-0.133495\pi$$
0.913338 + 0.407201i $$0.133495\pi$$
$$102$$ 6.64067 0.657524
$$103$$ 1.00000 0.0985329
$$104$$ −1.00000 −0.0980581
$$105$$ −4.11494 −0.401577
$$106$$ −3.16708 −0.307614
$$107$$ −14.1501 −1.36794 −0.683971 0.729509i $$-0.739749\pi$$
−0.683971 + 0.729509i $$0.739749\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 1.16806 0.111880 0.0559400 0.998434i $$-0.482184\pi$$
0.0559400 + 0.998434i $$0.482184\pi$$
$$110$$ 7.32377 0.698294
$$111$$ −0.542704 −0.0515112
$$112$$ 2.39003 0.225836
$$113$$ −6.56362 −0.617453 −0.308726 0.951151i $$-0.599903\pi$$
−0.308726 + 0.951151i $$0.599903\pi$$
$$114$$ 1.39003 0.130188
$$115$$ 1.13667 0.105995
$$116$$ −7.21542 −0.669935
$$117$$ 1.00000 0.0924500
$$118$$ −0.644743 −0.0593534
$$119$$ 15.8714 1.45493
$$120$$ 1.72171 0.157170
$$121$$ 7.09460 0.644963
$$122$$ −7.01661 −0.635254
$$123$$ −6.29089 −0.567230
$$124$$ −7.06840 −0.634761
$$125$$ −12.1135 −1.08346
$$126$$ −2.39003 −0.212921
$$127$$ −18.3701 −1.63008 −0.815042 0.579402i $$-0.803286\pi$$
−0.815042 + 0.579402i $$0.803286\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 12.5305 1.10325
$$130$$ −1.72171 −0.151004
$$131$$ 19.5981 1.71230 0.856149 0.516730i $$-0.172851\pi$$
0.856149 + 0.516730i $$0.172851\pi$$
$$132$$ 4.25377 0.370244
$$133$$ 3.32221 0.288072
$$134$$ 8.17546 0.706253
$$135$$ −1.72171 −0.148181
$$136$$ −6.64067 −0.569433
$$137$$ 9.23862 0.789308 0.394654 0.918830i $$-0.370864\pi$$
0.394654 + 0.918830i $$0.370864\pi$$
$$138$$ 0.660198 0.0561997
$$139$$ −16.8128 −1.42604 −0.713020 0.701144i $$-0.752673\pi$$
−0.713020 + 0.701144i $$0.752673\pi$$
$$140$$ 4.11494 0.347776
$$141$$ 5.06122 0.426231
$$142$$ −1.21505 −0.101965
$$143$$ −4.25377 −0.355718
$$144$$ 1.00000 0.0833333
$$145$$ −12.4229 −1.03166
$$146$$ 1.21748 0.100759
$$147$$ 1.28777 0.106213
$$148$$ 0.542704 0.0446100
$$149$$ 3.72254 0.304962 0.152481 0.988306i $$-0.451274\pi$$
0.152481 + 0.988306i $$0.451274\pi$$
$$150$$ −2.03571 −0.166215
$$151$$ −15.0853 −1.22762 −0.613812 0.789452i $$-0.710365\pi$$
−0.613812 + 0.789452i $$0.710365\pi$$
$$152$$ −1.39003 −0.112746
$$153$$ 6.64067 0.536866
$$154$$ 10.1666 0.819251
$$155$$ −12.1697 −0.977497
$$156$$ −1.00000 −0.0800641
$$157$$ −22.4610 −1.79258 −0.896292 0.443464i $$-0.853749\pi$$
−0.896292 + 0.443464i $$0.853749\pi$$
$$158$$ 5.16356 0.410791
$$159$$ −3.16708 −0.251166
$$160$$ −1.72171 −0.136113
$$161$$ 1.57789 0.124355
$$162$$ −1.00000 −0.0785674
$$163$$ 15.9747 1.25123 0.625616 0.780131i $$-0.284848\pi$$
0.625616 + 0.780131i $$0.284848\pi$$
$$164$$ 6.29089 0.491236
$$165$$ 7.32377 0.570154
$$166$$ 3.97674 0.308655
$$167$$ 12.6294 0.977291 0.488645 0.872482i $$-0.337491\pi$$
0.488645 + 0.872482i $$0.337491\pi$$
$$168$$ 2.39003 0.184395
$$169$$ 1.00000 0.0769231
$$170$$ −11.4333 −0.876894
$$171$$ 1.39003 0.106298
$$172$$ −12.5305 −0.955441
$$173$$ −0.502612 −0.0382129 −0.0191064 0.999817i $$-0.506082\pi$$
−0.0191064 + 0.999817i $$0.506082\pi$$
$$174$$ −7.21542 −0.546999
$$175$$ −4.86541 −0.367791
$$176$$ −4.25377 −0.320640
$$177$$ −0.644743 −0.0484619
$$178$$ 12.3981 0.929278
$$179$$ 2.79727 0.209078 0.104539 0.994521i $$-0.466663\pi$$
0.104539 + 0.994521i $$0.466663\pi$$
$$180$$ 1.72171 0.128329
$$181$$ −21.0300 −1.56315 −0.781574 0.623813i $$-0.785583\pi$$
−0.781574 + 0.623813i $$0.785583\pi$$
$$182$$ −2.39003 −0.177161
$$183$$ −7.01661 −0.518682
$$184$$ −0.660198 −0.0486704
$$185$$ 0.934379 0.0686969
$$186$$ −7.06840 −0.518281
$$187$$ −28.2479 −2.06569
$$188$$ −5.06122 −0.369127
$$189$$ −2.39003 −0.173849
$$190$$ −2.39323 −0.173623
$$191$$ −14.3576 −1.03888 −0.519441 0.854506i $$-0.673860\pi$$
−0.519441 + 0.854506i $$0.673860\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 8.89416 0.640216 0.320108 0.947381i $$-0.396281\pi$$
0.320108 + 0.947381i $$0.396281\pi$$
$$194$$ 18.4985 1.32812
$$195$$ −1.72171 −0.123294
$$196$$ −1.28777 −0.0919833
$$197$$ 18.5418 1.32105 0.660525 0.750804i $$-0.270334\pi$$
0.660525 + 0.750804i $$0.270334\pi$$
$$198$$ 4.25377 0.302303
$$199$$ 8.13646 0.576779 0.288389 0.957513i $$-0.406880\pi$$
0.288389 + 0.957513i $$0.406880\pi$$
$$200$$ 2.03571 0.143947
$$201$$ 8.17546 0.576653
$$202$$ −18.3579 −1.29166
$$203$$ −17.2451 −1.21037
$$204$$ −6.64067 −0.464940
$$205$$ 10.8311 0.756476
$$206$$ −1.00000 −0.0696733
$$207$$ 0.660198 0.0458869
$$208$$ 1.00000 0.0693375
$$209$$ −5.91287 −0.409001
$$210$$ 4.11494 0.283958
$$211$$ 15.3277 1.05520 0.527602 0.849491i $$-0.323091\pi$$
0.527602 + 0.849491i $$0.323091\pi$$
$$212$$ 3.16708 0.217516
$$213$$ −1.21505 −0.0832537
$$214$$ 14.1501 0.967281
$$215$$ −21.5739 −1.47133
$$216$$ 1.00000 0.0680414
$$217$$ −16.8937 −1.14682
$$218$$ −1.16806 −0.0791112
$$219$$ 1.21748 0.0822696
$$220$$ −7.32377 −0.493768
$$221$$ 6.64067 0.446700
$$222$$ 0.542704 0.0364239
$$223$$ 15.4713 1.03604 0.518018 0.855370i $$-0.326670\pi$$
0.518018 + 0.855370i $$0.326670\pi$$
$$224$$ −2.39003 −0.159690
$$225$$ −2.03571 −0.135714
$$226$$ 6.56362 0.436605
$$227$$ 12.8564 0.853312 0.426656 0.904414i $$-0.359692\pi$$
0.426656 + 0.904414i $$0.359692\pi$$
$$228$$ −1.39003 −0.0920569
$$229$$ 18.6495 1.23239 0.616196 0.787593i $$-0.288673\pi$$
0.616196 + 0.787593i $$0.288673\pi$$
$$230$$ −1.13667 −0.0749497
$$231$$ 10.1666 0.668916
$$232$$ 7.21542 0.473715
$$233$$ −24.7604 −1.62210 −0.811052 0.584974i $$-0.801105\pi$$
−0.811052 + 0.584974i $$0.801105\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ −8.71395 −0.568435
$$236$$ 0.644743 0.0419692
$$237$$ 5.16356 0.335409
$$238$$ −15.8714 −1.02879
$$239$$ 6.41094 0.414689 0.207345 0.978268i $$-0.433518\pi$$
0.207345 + 0.978268i $$0.433518\pi$$
$$240$$ −1.72171 −0.111136
$$241$$ −9.82599 −0.632948 −0.316474 0.948601i $$-0.602499\pi$$
−0.316474 + 0.948601i $$0.602499\pi$$
$$242$$ −7.09460 −0.456058
$$243$$ −1.00000 −0.0641500
$$244$$ 7.01661 0.449192
$$245$$ −2.21716 −0.141649
$$246$$ 6.29089 0.401093
$$247$$ 1.39003 0.0884454
$$248$$ 7.06840 0.448844
$$249$$ 3.97674 0.252016
$$250$$ 12.1135 0.766123
$$251$$ 0.529566 0.0334259 0.0167130 0.999860i $$-0.494680\pi$$
0.0167130 + 0.999860i $$0.494680\pi$$
$$252$$ 2.39003 0.150558
$$253$$ −2.80833 −0.176558
$$254$$ 18.3701 1.15264
$$255$$ −11.4333 −0.715981
$$256$$ 1.00000 0.0625000
$$257$$ −16.9954 −1.06014 −0.530071 0.847953i $$-0.677835\pi$$
−0.530071 + 0.847953i $$0.677835\pi$$
$$258$$ −12.5305 −0.780114
$$259$$ 1.29708 0.0805965
$$260$$ 1.72171 0.106776
$$261$$ −7.21542 −0.446623
$$262$$ −19.5981 −1.21078
$$263$$ −20.0780 −1.23806 −0.619030 0.785367i $$-0.712474\pi$$
−0.619030 + 0.785367i $$0.712474\pi$$
$$264$$ −4.25377 −0.261802
$$265$$ 5.45280 0.334963
$$266$$ −3.32221 −0.203698
$$267$$ 12.3981 0.758752
$$268$$ −8.17546 −0.499396
$$269$$ 9.40162 0.573227 0.286613 0.958046i $$-0.407470\pi$$
0.286613 + 0.958046i $$0.407470\pi$$
$$270$$ 1.72171 0.104780
$$271$$ −19.3124 −1.17315 −0.586573 0.809896i $$-0.699523\pi$$
−0.586573 + 0.809896i $$0.699523\pi$$
$$272$$ 6.64067 0.402650
$$273$$ −2.39003 −0.144651
$$274$$ −9.23862 −0.558125
$$275$$ 8.65947 0.522186
$$276$$ −0.660198 −0.0397392
$$277$$ 30.6289 1.84031 0.920155 0.391555i $$-0.128063\pi$$
0.920155 + 0.391555i $$0.128063\pi$$
$$278$$ 16.8128 1.00836
$$279$$ −7.06840 −0.423174
$$280$$ −4.11494 −0.245914
$$281$$ −18.9263 −1.12905 −0.564524 0.825416i $$-0.690940\pi$$
−0.564524 + 0.825416i $$0.690940\pi$$
$$282$$ −5.06122 −0.301391
$$283$$ 28.5045 1.69442 0.847208 0.531262i $$-0.178282\pi$$
0.847208 + 0.531262i $$0.178282\pi$$
$$284$$ 1.21505 0.0720998
$$285$$ −2.39323 −0.141762
$$286$$ 4.25377 0.251531
$$287$$ 15.0354 0.887512
$$288$$ −1.00000 −0.0589256
$$289$$ 27.0985 1.59403
$$290$$ 12.4229 0.729495
$$291$$ 18.4985 1.08440
$$292$$ −1.21748 −0.0712475
$$293$$ 1.45581 0.0850493 0.0425247 0.999095i $$-0.486460\pi$$
0.0425247 + 0.999095i $$0.486460\pi$$
$$294$$ −1.28777 −0.0751040
$$295$$ 1.11006 0.0646302
$$296$$ −0.542704 −0.0315440
$$297$$ 4.25377 0.246829
$$298$$ −3.72254 −0.215641
$$299$$ 0.660198 0.0381802
$$300$$ 2.03571 0.117532
$$301$$ −29.9482 −1.72619
$$302$$ 15.0853 0.868061
$$303$$ −18.3579 −1.05463
$$304$$ 1.39003 0.0797236
$$305$$ 12.0806 0.691731
$$306$$ −6.64067 −0.379622
$$307$$ −11.7328 −0.669626 −0.334813 0.942285i $$-0.608673\pi$$
−0.334813 + 0.942285i $$0.608673\pi$$
$$308$$ −10.1666 −0.579298
$$309$$ −1.00000 −0.0568880
$$310$$ 12.1697 0.691195
$$311$$ −18.1845 −1.03115 −0.515574 0.856845i $$-0.672421\pi$$
−0.515574 + 0.856845i $$0.672421\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −4.10088 −0.231796 −0.115898 0.993261i $$-0.536975\pi$$
−0.115898 + 0.993261i $$0.536975\pi$$
$$314$$ 22.4610 1.26755
$$315$$ 4.11494 0.231850
$$316$$ −5.16356 −0.290473
$$317$$ 5.55377 0.311931 0.155965 0.987763i $$-0.450151\pi$$
0.155965 + 0.987763i $$0.450151\pi$$
$$318$$ 3.16708 0.177601
$$319$$ 30.6928 1.71846
$$320$$ 1.72171 0.0962465
$$321$$ 14.1501 0.789782
$$322$$ −1.57789 −0.0879324
$$323$$ 9.23071 0.513611
$$324$$ 1.00000 0.0555556
$$325$$ −2.03571 −0.112921
$$326$$ −15.9747 −0.884755
$$327$$ −1.16806 −0.0645940
$$328$$ −6.29089 −0.347356
$$329$$ −12.0965 −0.666899
$$330$$ −7.32377 −0.403160
$$331$$ 20.6613 1.13565 0.567825 0.823149i $$-0.307785\pi$$
0.567825 + 0.823149i $$0.307785\pi$$
$$332$$ −3.97674 −0.218252
$$333$$ 0.542704 0.0297400
$$334$$ −12.6294 −0.691049
$$335$$ −14.0758 −0.769042
$$336$$ −2.39003 −0.130387
$$337$$ 34.9396 1.90328 0.951640 0.307216i $$-0.0993976\pi$$
0.951640 + 0.307216i $$0.0993976\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 6.56362 0.356487
$$340$$ 11.4333 0.620058
$$341$$ 30.0674 1.62824
$$342$$ −1.39003 −0.0751641
$$343$$ −19.8080 −1.06953
$$344$$ 12.5305 0.675599
$$345$$ −1.13667 −0.0611962
$$346$$ 0.502612 0.0270206
$$347$$ 30.1347 1.61771 0.808857 0.588005i $$-0.200086\pi$$
0.808857 + 0.588005i $$0.200086\pi$$
$$348$$ 7.21542 0.386787
$$349$$ −1.84054 −0.0985216 −0.0492608 0.998786i $$-0.515687\pi$$
−0.0492608 + 0.998786i $$0.515687\pi$$
$$350$$ 4.86541 0.260067
$$351$$ −1.00000 −0.0533761
$$352$$ 4.25377 0.226727
$$353$$ −4.55407 −0.242389 −0.121194 0.992629i $$-0.538672\pi$$
−0.121194 + 0.992629i $$0.538672\pi$$
$$354$$ 0.644743 0.0342677
$$355$$ 2.09196 0.111030
$$356$$ −12.3981 −0.657099
$$357$$ −15.8714 −0.840003
$$358$$ −2.79727 −0.147841
$$359$$ 0.674029 0.0355739 0.0177869 0.999842i $$-0.494338\pi$$
0.0177869 + 0.999842i $$0.494338\pi$$
$$360$$ −1.72171 −0.0907421
$$361$$ −17.0678 −0.898306
$$362$$ 21.0300 1.10531
$$363$$ −7.09460 −0.372370
$$364$$ 2.39003 0.125272
$$365$$ −2.09615 −0.109717
$$366$$ 7.01661 0.366764
$$367$$ 19.6521 1.02583 0.512916 0.858439i $$-0.328565\pi$$
0.512916 + 0.858439i $$0.328565\pi$$
$$368$$ 0.660198 0.0344152
$$369$$ 6.29089 0.327491
$$370$$ −0.934379 −0.0485760
$$371$$ 7.56942 0.392985
$$372$$ 7.06840 0.366480
$$373$$ −25.0095 −1.29494 −0.647472 0.762090i $$-0.724174\pi$$
−0.647472 + 0.762090i $$0.724174\pi$$
$$374$$ 28.2479 1.46066
$$375$$ 12.1135 0.625536
$$376$$ 5.06122 0.261012
$$377$$ −7.21542 −0.371613
$$378$$ 2.39003 0.122930
$$379$$ −15.4257 −0.792363 −0.396181 0.918172i $$-0.629665\pi$$
−0.396181 + 0.918172i $$0.629665\pi$$
$$380$$ 2.39323 0.122770
$$381$$ 18.3701 0.941129
$$382$$ 14.3576 0.734600
$$383$$ −22.4939 −1.14938 −0.574692 0.818370i $$-0.694878\pi$$
−0.574692 + 0.818370i $$0.694878\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −17.5040 −0.892087
$$386$$ −8.89416 −0.452701
$$387$$ −12.5305 −0.636961
$$388$$ −18.4985 −0.939120
$$389$$ 13.6967 0.694451 0.347226 0.937782i $$-0.387124\pi$$
0.347226 + 0.937782i $$0.387124\pi$$
$$390$$ 1.72171 0.0871822
$$391$$ 4.38415 0.221716
$$392$$ 1.28777 0.0650420
$$393$$ −19.5981 −0.988595
$$394$$ −18.5418 −0.934124
$$395$$ −8.89016 −0.447312
$$396$$ −4.25377 −0.213760
$$397$$ −26.2443 −1.31716 −0.658582 0.752509i $$-0.728843\pi$$
−0.658582 + 0.752509i $$0.728843\pi$$
$$398$$ −8.13646 −0.407844
$$399$$ −3.32221 −0.166318
$$400$$ −2.03571 −0.101786
$$401$$ 11.9508 0.596796 0.298398 0.954441i $$-0.403548\pi$$
0.298398 + 0.954441i $$0.403548\pi$$
$$402$$ −8.17546 −0.407755
$$403$$ −7.06840 −0.352102
$$404$$ 18.3579 0.913338
$$405$$ 1.72171 0.0855525
$$406$$ 17.2451 0.855858
$$407$$ −2.30854 −0.114430
$$408$$ 6.64067 0.328762
$$409$$ 6.19764 0.306454 0.153227 0.988191i $$-0.451033\pi$$
0.153227 + 0.988191i $$0.451033\pi$$
$$410$$ −10.8311 −0.534909
$$411$$ −9.23862 −0.455707
$$412$$ 1.00000 0.0492665
$$413$$ 1.54095 0.0758254
$$414$$ −0.660198 −0.0324469
$$415$$ −6.84679 −0.336096
$$416$$ −1.00000 −0.0490290
$$417$$ 16.8128 0.823325
$$418$$ 5.91287 0.289208
$$419$$ 14.6735 0.716850 0.358425 0.933559i $$-0.383314\pi$$
0.358425 + 0.933559i $$0.383314\pi$$
$$420$$ −4.11494 −0.200788
$$421$$ 34.3333 1.67330 0.836650 0.547738i $$-0.184511\pi$$
0.836650 + 0.547738i $$0.184511\pi$$
$$422$$ −15.3277 −0.746142
$$423$$ −5.06122 −0.246085
$$424$$ −3.16708 −0.153807
$$425$$ −13.5185 −0.655744
$$426$$ 1.21505 0.0588692
$$427$$ 16.7699 0.811552
$$428$$ −14.1501 −0.683971
$$429$$ 4.25377 0.205374
$$430$$ 21.5739 1.04038
$$431$$ −7.64446 −0.368221 −0.184110 0.982906i $$-0.558940\pi$$
−0.184110 + 0.982906i $$0.558940\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −12.6684 −0.608807 −0.304403 0.952543i $$-0.598457\pi$$
−0.304403 + 0.952543i $$0.598457\pi$$
$$434$$ 16.8937 0.810923
$$435$$ 12.4229 0.595630
$$436$$ 1.16806 0.0559400
$$437$$ 0.917693 0.0438992
$$438$$ −1.21748 −0.0581734
$$439$$ 31.1964 1.48892 0.744461 0.667666i $$-0.232706\pi$$
0.744461 + 0.667666i $$0.232706\pi$$
$$440$$ 7.32377 0.349147
$$441$$ −1.28777 −0.0613222
$$442$$ −6.64067 −0.315864
$$443$$ −5.89484 −0.280072 −0.140036 0.990146i $$-0.544722\pi$$
−0.140036 + 0.990146i $$0.544722\pi$$
$$444$$ −0.542704 −0.0257556
$$445$$ −21.3460 −1.01190
$$446$$ −15.4713 −0.732588
$$447$$ −3.72254 −0.176070
$$448$$ 2.39003 0.112918
$$449$$ 21.1771 0.999411 0.499706 0.866195i $$-0.333441\pi$$
0.499706 + 0.866195i $$0.333441\pi$$
$$450$$ 2.03571 0.0959645
$$451$$ −26.7600 −1.26008
$$452$$ −6.56362 −0.308726
$$453$$ 15.0853 0.708769
$$454$$ −12.8564 −0.603382
$$455$$ 4.11494 0.192911
$$456$$ 1.39003 0.0650940
$$457$$ 17.4405 0.815833 0.407916 0.913019i $$-0.366255\pi$$
0.407916 + 0.913019i $$0.366255\pi$$
$$458$$ −18.6495 −0.871432
$$459$$ −6.64067 −0.309960
$$460$$ 1.13667 0.0529975
$$461$$ 17.9659 0.836754 0.418377 0.908274i $$-0.362599\pi$$
0.418377 + 0.908274i $$0.362599\pi$$
$$462$$ −10.1666 −0.472995
$$463$$ 6.96127 0.323518 0.161759 0.986830i $$-0.448283\pi$$
0.161759 + 0.986830i $$0.448283\pi$$
$$464$$ −7.21542 −0.334967
$$465$$ 12.1697 0.564358
$$466$$ 24.7604 1.14700
$$467$$ −19.2176 −0.889285 −0.444642 0.895708i $$-0.646669\pi$$
−0.444642 + 0.895708i $$0.646669\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ −19.5396 −0.902254
$$470$$ 8.71395 0.401944
$$471$$ 22.4610 1.03495
$$472$$ −0.644743 −0.0296767
$$473$$ 53.3019 2.45082
$$474$$ −5.16356 −0.237170
$$475$$ −2.82970 −0.129836
$$476$$ 15.8714 0.727464
$$477$$ 3.16708 0.145011
$$478$$ −6.41094 −0.293229
$$479$$ 8.05490 0.368038 0.184019 0.982923i $$-0.441089\pi$$
0.184019 + 0.982923i $$0.441089\pi$$
$$480$$ 1.72171 0.0785850
$$481$$ 0.542704 0.0247452
$$482$$ 9.82599 0.447562
$$483$$ −1.57789 −0.0717965
$$484$$ 7.09460 0.322482
$$485$$ −31.8491 −1.44619
$$486$$ 1.00000 0.0453609
$$487$$ −37.2525 −1.68807 −0.844036 0.536286i $$-0.819827\pi$$
−0.844036 + 0.536286i $$0.819827\pi$$
$$488$$ −7.01661 −0.317627
$$489$$ −15.9747 −0.722399
$$490$$ 2.21716 0.100161
$$491$$ −33.0779 −1.49279 −0.746393 0.665505i $$-0.768216\pi$$
−0.746393 + 0.665505i $$0.768216\pi$$
$$492$$ −6.29089 −0.283615
$$493$$ −47.9152 −2.15799
$$494$$ −1.39003 −0.0625403
$$495$$ −7.32377 −0.329179
$$496$$ −7.06840 −0.317381
$$497$$ 2.90400 0.130262
$$498$$ −3.97674 −0.178202
$$499$$ 20.3427 0.910666 0.455333 0.890321i $$-0.349520\pi$$
0.455333 + 0.890321i $$0.349520\pi$$
$$500$$ −12.1135 −0.541730
$$501$$ −12.6294 −0.564239
$$502$$ −0.529566 −0.0236357
$$503$$ 10.9249 0.487119 0.243559 0.969886i $$-0.421685\pi$$
0.243559 + 0.969886i $$0.421685\pi$$
$$504$$ −2.39003 −0.106460
$$505$$ 31.6069 1.40649
$$506$$ 2.80833 0.124846
$$507$$ −1.00000 −0.0444116
$$508$$ −18.3701 −0.815042
$$509$$ −5.17386 −0.229327 −0.114664 0.993404i $$-0.536579\pi$$
−0.114664 + 0.993404i $$0.536579\pi$$
$$510$$ 11.4333 0.506275
$$511$$ −2.90981 −0.128722
$$512$$ −1.00000 −0.0441942
$$513$$ −1.39003 −0.0613712
$$514$$ 16.9954 0.749633
$$515$$ 1.72171 0.0758676
$$516$$ 12.5305 0.551624
$$517$$ 21.5293 0.946856
$$518$$ −1.29708 −0.0569903
$$519$$ 0.502612 0.0220622
$$520$$ −1.72171 −0.0755020
$$521$$ −24.0410 −1.05326 −0.526629 0.850095i $$-0.676544\pi$$
−0.526629 + 0.850095i $$0.676544\pi$$
$$522$$ 7.21542 0.315810
$$523$$ −36.8960 −1.61335 −0.806676 0.590994i $$-0.798736\pi$$
−0.806676 + 0.590994i $$0.798736\pi$$
$$524$$ 19.5981 0.856149
$$525$$ 4.86541 0.212344
$$526$$ 20.0780 0.875441
$$527$$ −46.9389 −2.04469
$$528$$ 4.25377 0.185122
$$529$$ −22.5641 −0.981050
$$530$$ −5.45280 −0.236854
$$531$$ 0.644743 0.0279795
$$532$$ 3.32221 0.144036
$$533$$ 6.29089 0.272489
$$534$$ −12.3981 −0.536519
$$535$$ −24.3624 −1.05328
$$536$$ 8.17546 0.353126
$$537$$ −2.79727 −0.120711
$$538$$ −9.40162 −0.405333
$$539$$ 5.47787 0.235948
$$540$$ −1.72171 −0.0740906
$$541$$ 0.662284 0.0284738 0.0142369 0.999899i $$-0.495468\pi$$
0.0142369 + 0.999899i $$0.495468\pi$$
$$542$$ 19.3124 0.829540
$$543$$ 21.0300 0.902484
$$544$$ −6.64067 −0.284716
$$545$$ 2.01106 0.0861446
$$546$$ 2.39003 0.102284
$$547$$ 7.57319 0.323806 0.161903 0.986807i $$-0.448237\pi$$
0.161903 + 0.986807i $$0.448237\pi$$
$$548$$ 9.23862 0.394654
$$549$$ 7.01661 0.299461
$$550$$ −8.65947 −0.369241
$$551$$ −10.0296 −0.427277
$$552$$ 0.660198 0.0280999
$$553$$ −12.3411 −0.524795
$$554$$ −30.6289 −1.30130
$$555$$ −0.934379 −0.0396622
$$556$$ −16.8128 −0.713020
$$557$$ 17.0626 0.722965 0.361483 0.932379i $$-0.382271\pi$$
0.361483 + 0.932379i $$0.382271\pi$$
$$558$$ 7.06840 0.299229
$$559$$ −12.5305 −0.529983
$$560$$ 4.11494 0.173888
$$561$$ 28.2479 1.19263
$$562$$ 18.9263 0.798358
$$563$$ 30.7787 1.29717 0.648583 0.761144i $$-0.275362\pi$$
0.648583 + 0.761144i $$0.275362\pi$$
$$564$$ 5.06122 0.213116
$$565$$ −11.3006 −0.475422
$$566$$ −28.5045 −1.19813
$$567$$ 2.39003 0.100372
$$568$$ −1.21505 −0.0509823
$$569$$ 12.0641 0.505753 0.252876 0.967499i $$-0.418623\pi$$
0.252876 + 0.967499i $$0.418623\pi$$
$$570$$ 2.39323 0.100241
$$571$$ −8.17714 −0.342203 −0.171101 0.985253i $$-0.554733\pi$$
−0.171101 + 0.985253i $$0.554733\pi$$
$$572$$ −4.25377 −0.177859
$$573$$ 14.3576 0.599799
$$574$$ −15.0354 −0.627566
$$575$$ −1.34397 −0.0560476
$$576$$ 1.00000 0.0416667
$$577$$ 0.163804 0.00681926 0.00340963 0.999994i $$-0.498915\pi$$
0.00340963 + 0.999994i $$0.498915\pi$$
$$578$$ −27.0985 −1.12715
$$579$$ −8.89416 −0.369629
$$580$$ −12.4229 −0.515831
$$581$$ −9.50452 −0.394314
$$582$$ −18.4985 −0.766788
$$583$$ −13.4721 −0.557955
$$584$$ 1.21748 0.0503796
$$585$$ 1.72171 0.0711840
$$586$$ −1.45581 −0.0601390
$$587$$ −3.42990 −0.141567 −0.0707835 0.997492i $$-0.522550\pi$$
−0.0707835 + 0.997492i $$0.522550\pi$$
$$588$$ 1.28777 0.0531066
$$589$$ −9.82528 −0.404844
$$590$$ −1.11006 −0.0457005
$$591$$ −18.5418 −0.762709
$$592$$ 0.542704 0.0223050
$$593$$ 5.84030 0.239832 0.119916 0.992784i $$-0.461737\pi$$
0.119916 + 0.992784i $$0.461737\pi$$
$$594$$ −4.25377 −0.174534
$$595$$ 27.3259 1.12025
$$596$$ 3.72254 0.152481
$$597$$ −8.13646 −0.333003
$$598$$ −0.660198 −0.0269975
$$599$$ −45.4221 −1.85590 −0.927948 0.372709i $$-0.878429\pi$$
−0.927948 + 0.372709i $$0.878429\pi$$
$$600$$ −2.03571 −0.0831077
$$601$$ −38.8873 −1.58625 −0.793123 0.609062i $$-0.791546\pi$$
−0.793123 + 0.609062i $$0.791546\pi$$
$$602$$ 29.9482 1.22060
$$603$$ −8.17546 −0.332931
$$604$$ −15.0853 −0.613812
$$605$$ 12.2148 0.496604
$$606$$ 18.3579 0.745738
$$607$$ −20.0091 −0.812143 −0.406072 0.913841i $$-0.633102\pi$$
−0.406072 + 0.913841i $$0.633102\pi$$
$$608$$ −1.39003 −0.0563731
$$609$$ 17.2451 0.698805
$$610$$ −12.0806 −0.489128
$$611$$ −5.06122 −0.204755
$$612$$ 6.64067 0.268433
$$613$$ −43.4642 −1.75550 −0.877751 0.479117i $$-0.840957\pi$$
−0.877751 + 0.479117i $$0.840957\pi$$
$$614$$ 11.7328 0.473497
$$615$$ −10.8311 −0.436752
$$616$$ 10.1666 0.409626
$$617$$ −38.4997 −1.54994 −0.774969 0.631999i $$-0.782235\pi$$
−0.774969 + 0.631999i $$0.782235\pi$$
$$618$$ 1.00000 0.0402259
$$619$$ −26.8467 −1.07906 −0.539530 0.841966i $$-0.681398\pi$$
−0.539530 + 0.841966i $$0.681398\pi$$
$$620$$ −12.1697 −0.488749
$$621$$ −0.660198 −0.0264928
$$622$$ 18.1845 0.729131
$$623$$ −29.6318 −1.18717
$$624$$ −1.00000 −0.0400320
$$625$$ −10.6773 −0.427092
$$626$$ 4.10088 0.163904
$$627$$ 5.91287 0.236137
$$628$$ −22.4610 −0.896292
$$629$$ 3.60391 0.143697
$$630$$ −4.11494 −0.163943
$$631$$ −41.5903 −1.65568 −0.827841 0.560963i $$-0.810431\pi$$
−0.827841 + 0.560963i $$0.810431\pi$$
$$632$$ 5.16356 0.205395
$$633$$ −15.3277 −0.609223
$$634$$ −5.55377 −0.220568
$$635$$ −31.6280 −1.25512
$$636$$ −3.16708 −0.125583
$$637$$ −1.28777 −0.0510232
$$638$$ −30.6928 −1.21514
$$639$$ 1.21505 0.0480665
$$640$$ −1.72171 −0.0680566
$$641$$ −7.32445 −0.289298 −0.144649 0.989483i $$-0.546205\pi$$
−0.144649 + 0.989483i $$0.546205\pi$$
$$642$$ −14.1501 −0.558460
$$643$$ −28.6820 −1.13111 −0.565554 0.824711i $$-0.691338\pi$$
−0.565554 + 0.824711i $$0.691338\pi$$
$$644$$ 1.57789 0.0621776
$$645$$ 21.5739 0.849471
$$646$$ −9.23071 −0.363178
$$647$$ −47.4197 −1.86426 −0.932130 0.362124i $$-0.882052\pi$$
−0.932130 + 0.362124i $$0.882052\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −2.74259 −0.107656
$$650$$ 2.03571 0.0798473
$$651$$ 16.8937 0.662116
$$652$$ 15.9747 0.625616
$$653$$ 30.5387 1.19507 0.597536 0.801842i $$-0.296147\pi$$
0.597536 + 0.801842i $$0.296147\pi$$
$$654$$ 1.16806 0.0456749
$$655$$ 33.7423 1.31842
$$656$$ 6.29089 0.245618
$$657$$ −1.21748 −0.0474984
$$658$$ 12.0965 0.471569
$$659$$ −14.5640 −0.567334 −0.283667 0.958923i $$-0.591551\pi$$
−0.283667 + 0.958923i $$0.591551\pi$$
$$660$$ 7.32377 0.285077
$$661$$ −42.6428 −1.65861 −0.829307 0.558793i $$-0.811265\pi$$
−0.829307 + 0.558793i $$0.811265\pi$$
$$662$$ −20.6613 −0.803026
$$663$$ −6.64067 −0.257902
$$664$$ 3.97674 0.154327
$$665$$ 5.71988 0.221807
$$666$$ −0.542704 −0.0210293
$$667$$ −4.76360 −0.184447
$$668$$ 12.6294 0.488645
$$669$$ −15.4713 −0.598155
$$670$$ 14.0758 0.543795
$$671$$ −29.8471 −1.15223
$$672$$ 2.39003 0.0921973
$$673$$ −13.8465 −0.533742 −0.266871 0.963732i $$-0.585990\pi$$
−0.266871 + 0.963732i $$0.585990\pi$$
$$674$$ −34.9396 −1.34582
$$675$$ 2.03571 0.0783547
$$676$$ 1.00000 0.0384615
$$677$$ −36.1681 −1.39005 −0.695027 0.718984i $$-0.744607\pi$$
−0.695027 + 0.718984i $$0.744607\pi$$
$$678$$ −6.56362 −0.252074
$$679$$ −44.2120 −1.69670
$$680$$ −11.4333 −0.438447
$$681$$ −12.8564 −0.492660
$$682$$ −30.0674 −1.15134
$$683$$ −9.23341 −0.353307 −0.176653 0.984273i $$-0.556527\pi$$
−0.176653 + 0.984273i $$0.556527\pi$$
$$684$$ 1.39003 0.0531490
$$685$$ 15.9062 0.607746
$$686$$ 19.8080 0.756273
$$687$$ −18.6495 −0.711522
$$688$$ −12.5305 −0.477721
$$689$$ 3.16708 0.120656
$$690$$ 1.13667 0.0432722
$$691$$ 1.26275 0.0480373 0.0240186 0.999712i $$-0.492354\pi$$
0.0240186 + 0.999712i $$0.492354\pi$$
$$692$$ −0.502612 −0.0191064
$$693$$ −10.1666 −0.386199
$$694$$ −30.1347 −1.14390
$$695$$ −28.9467 −1.09801
$$696$$ −7.21542 −0.273500
$$697$$ 41.7757 1.58237
$$698$$ 1.84054 0.0696653
$$699$$ 24.7604 0.936523
$$700$$ −4.86541 −0.183895
$$701$$ 48.9768 1.84983 0.924913 0.380178i $$-0.124137\pi$$
0.924913 + 0.380178i $$0.124137\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ 0.754373 0.0284517
$$704$$ −4.25377 −0.160320
$$705$$ 8.71395 0.328186
$$706$$ 4.55407 0.171395
$$707$$ 43.8758 1.65012
$$708$$ −0.644743 −0.0242309
$$709$$ −38.6968 −1.45329 −0.726644 0.687014i $$-0.758921\pi$$
−0.726644 + 0.687014i $$0.758921\pi$$
$$710$$ −2.09196 −0.0785099
$$711$$ −5.16356 −0.193649
$$712$$ 12.3981 0.464639
$$713$$ −4.66654 −0.174763
$$714$$ 15.8714 0.593971
$$715$$ −7.32377 −0.273893
$$716$$ 2.79727 0.104539
$$717$$ −6.41094 −0.239421
$$718$$ −0.674029 −0.0251545
$$719$$ −6.34028 −0.236452 −0.118226 0.992987i $$-0.537721\pi$$
−0.118226 + 0.992987i $$0.537721\pi$$
$$720$$ 1.72171 0.0641643
$$721$$ 2.39003 0.0890093
$$722$$ 17.0678 0.635199
$$723$$ 9.82599 0.365432
$$724$$ −21.0300 −0.781574
$$725$$ 14.6885 0.545518
$$726$$ 7.09460 0.263305
$$727$$ 7.41722 0.275089 0.137545 0.990496i $$-0.456079\pi$$
0.137545 + 0.990496i $$0.456079\pi$$
$$728$$ −2.39003 −0.0885803
$$729$$ 1.00000 0.0370370
$$730$$ 2.09615 0.0775818
$$731$$ −83.2108 −3.07766
$$732$$ −7.01661 −0.259341
$$733$$ −2.41675 −0.0892648 −0.0446324 0.999003i $$-0.514212\pi$$
−0.0446324 + 0.999003i $$0.514212\pi$$
$$734$$ −19.6521 −0.725373
$$735$$ 2.21716 0.0817812
$$736$$ −0.660198 −0.0243352
$$737$$ 34.7766 1.28101
$$738$$ −6.29089 −0.231571
$$739$$ 18.7670 0.690356 0.345178 0.938537i $$-0.387818\pi$$
0.345178 + 0.938537i $$0.387818\pi$$
$$740$$ 0.934379 0.0343484
$$741$$ −1.39003 −0.0510640
$$742$$ −7.56942 −0.277882
$$743$$ 42.8482 1.57195 0.785973 0.618261i $$-0.212162\pi$$
0.785973 + 0.618261i $$0.212162\pi$$
$$744$$ −7.06840 −0.259140
$$745$$ 6.40913 0.234812
$$746$$ 25.0095 0.915663
$$747$$ −3.97674 −0.145501
$$748$$ −28.2479 −1.03285
$$749$$ −33.8191 −1.23572
$$750$$ −12.1135 −0.442321
$$751$$ −27.4564 −1.00190 −0.500949 0.865477i $$-0.667016\pi$$
−0.500949 + 0.865477i $$0.667016\pi$$
$$752$$ −5.06122 −0.184564
$$753$$ −0.529566 −0.0192985
$$754$$ 7.21542 0.262770
$$755$$ −25.9725 −0.945236
$$756$$ −2.39003 −0.0869245
$$757$$ −12.1256 −0.440713 −0.220357 0.975419i $$-0.570722\pi$$
−0.220357 + 0.975419i $$0.570722\pi$$
$$758$$ 15.4257 0.560285
$$759$$ 2.80833 0.101936
$$760$$ −2.39323 −0.0868114
$$761$$ −46.8246 −1.69739 −0.848696 0.528881i $$-0.822612\pi$$
−0.848696 + 0.528881i $$0.822612\pi$$
$$762$$ −18.3701 −0.665479
$$763$$ 2.79170 0.101066
$$764$$ −14.3576 −0.519441
$$765$$ 11.4333 0.413372
$$766$$ 22.4939 0.812737
$$767$$ 0.644743 0.0232803
$$768$$ −1.00000 −0.0360844
$$769$$ −2.71689 −0.0979736 −0.0489868 0.998799i $$-0.515599\pi$$
−0.0489868 + 0.998799i $$0.515599\pi$$
$$770$$ 17.5040 0.630801
$$771$$ 16.9954 0.612073
$$772$$ 8.89416 0.320108
$$773$$ 44.9781 1.61775 0.808875 0.587981i $$-0.200077\pi$$
0.808875 + 0.587981i $$0.200077\pi$$
$$774$$ 12.5305 0.450399
$$775$$ 14.3892 0.516877
$$776$$ 18.4985 0.664058
$$777$$ −1.29708 −0.0465324
$$778$$ −13.6967 −0.491051
$$779$$ 8.74451 0.313305
$$780$$ −1.72171 −0.0616471
$$781$$ −5.16854 −0.184945
$$782$$ −4.38415 −0.156777
$$783$$ 7.21542 0.257858
$$784$$ −1.28777 −0.0459916
$$785$$ −38.6714 −1.38024
$$786$$ 19.5981 0.699042
$$787$$ 17.0228 0.606797 0.303399 0.952864i $$-0.401879\pi$$
0.303399 + 0.952864i $$0.401879\pi$$
$$788$$ 18.5418 0.660525
$$789$$ 20.0780 0.714794
$$790$$ 8.89016 0.316298
$$791$$ −15.6872 −0.557773
$$792$$ 4.25377 0.151151
$$793$$ 7.01661 0.249167
$$794$$ 26.2443 0.931376
$$795$$ −5.45280 −0.193391
$$796$$ 8.13646 0.288389
$$797$$ 29.5539 1.04685 0.523427 0.852071i $$-0.324653\pi$$
0.523427 + 0.852071i $$0.324653\pi$$
$$798$$ 3.32221 0.117605
$$799$$ −33.6099 −1.18903
$$800$$ 2.03571 0.0719734
$$801$$ −12.3981 −0.438066
$$802$$ −11.9508 −0.421999
$$803$$ 5.17888 0.182759
$$804$$ 8.17546 0.288326
$$805$$ 2.71667 0.0957500
$$806$$ 7.06840 0.248974
$$807$$ −9.40162 −0.330953
$$808$$ −18.3579 −0.645828
$$809$$ 42.2616 1.48584 0.742920 0.669380i $$-0.233440\pi$$
0.742920 + 0.669380i $$0.233440\pi$$
$$810$$ −1.72171 −0.0604947
$$811$$ −36.5413 −1.28314 −0.641570 0.767065i $$-0.721717\pi$$
−0.641570 + 0.767065i $$0.721717\pi$$
$$812$$ −17.2451 −0.605183
$$813$$ 19.3124 0.677317
$$814$$ 2.30854 0.0809143
$$815$$ 27.5037 0.963414
$$816$$ −6.64067 −0.232470
$$817$$ −17.4177 −0.609369
$$818$$ −6.19764 −0.216696
$$819$$ 2.39003 0.0835143
$$820$$ 10.8311 0.378238
$$821$$ 27.6549 0.965162 0.482581 0.875851i $$-0.339699\pi$$
0.482581 + 0.875851i $$0.339699\pi$$
$$822$$ 9.23862 0.322234
$$823$$ 1.46514 0.0510716 0.0255358 0.999674i $$-0.491871\pi$$
0.0255358 + 0.999674i $$0.491871\pi$$
$$824$$ −1.00000 −0.0348367
$$825$$ −8.65947 −0.301484
$$826$$ −1.54095 −0.0536167
$$827$$ 26.9118 0.935814 0.467907 0.883778i $$-0.345008\pi$$
0.467907 + 0.883778i $$0.345008\pi$$
$$828$$ 0.660198 0.0229435
$$829$$ 4.73394 0.164416 0.0822082 0.996615i $$-0.473803\pi$$
0.0822082 + 0.996615i $$0.473803\pi$$
$$830$$ 6.84679 0.237656
$$831$$ −30.6289 −1.06250
$$832$$ 1.00000 0.0346688
$$833$$ −8.55163 −0.296296
$$834$$ −16.8128 −0.582178
$$835$$ 21.7441 0.752487
$$836$$ −5.91287 −0.204501
$$837$$ 7.06840 0.244320
$$838$$ −14.6735 −0.506889
$$839$$ 24.6854 0.852236 0.426118 0.904668i $$-0.359881\pi$$
0.426118 + 0.904668i $$0.359881\pi$$
$$840$$ 4.11494 0.141979
$$841$$ 23.0623 0.795250
$$842$$ −34.3333 −1.18320
$$843$$ 18.9263 0.651857
$$844$$ 15.3277 0.527602
$$845$$ 1.72171 0.0592286
$$846$$ 5.06122 0.174008
$$847$$ 16.9563 0.582625
$$848$$ 3.16708 0.108758
$$849$$ −28.5045 −0.978271
$$850$$ 13.5185 0.463681
$$851$$ 0.358292 0.0122821
$$852$$ −1.21505 −0.0416268
$$853$$ 28.3671 0.971271 0.485635 0.874161i $$-0.338588\pi$$
0.485635 + 0.874161i $$0.338588\pi$$
$$854$$ −16.7699 −0.573854
$$855$$ 2.39323 0.0818466
$$856$$ 14.1501 0.483641
$$857$$ 25.3169 0.864807 0.432404 0.901680i $$-0.357666\pi$$
0.432404 + 0.901680i $$0.357666\pi$$
$$858$$ −4.25377 −0.145221
$$859$$ 54.4404 1.85748 0.928741 0.370729i $$-0.120892\pi$$
0.928741 + 0.370729i $$0.120892\pi$$
$$860$$ −21.5739 −0.735663
$$861$$ −15.0354 −0.512405
$$862$$ 7.64446 0.260371
$$863$$ −38.2577 −1.30231 −0.651153 0.758947i $$-0.725714\pi$$
−0.651153 + 0.758947i $$0.725714\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −0.865352 −0.0294228
$$866$$ 12.6684 0.430491
$$867$$ −27.0985 −0.920312
$$868$$ −16.8937 −0.573409
$$869$$ 21.9646 0.745099
$$870$$ −12.4229 −0.421174
$$871$$ −8.17546 −0.277015
$$872$$ −1.16806 −0.0395556
$$873$$ −18.4985 −0.626080
$$874$$ −0.917693 −0.0310414
$$875$$ −28.9515 −0.978740
$$876$$ 1.21748 0.0411348
$$877$$ −11.5755 −0.390875 −0.195438 0.980716i $$-0.562613\pi$$
−0.195438 + 0.980716i $$0.562613\pi$$
$$878$$ −31.1964 −1.05283
$$879$$ −1.45581 −0.0491033
$$880$$ −7.32377 −0.246884
$$881$$ −50.0849 −1.68740 −0.843701 0.536813i $$-0.819628\pi$$
−0.843701 + 0.536813i $$0.819628\pi$$
$$882$$ 1.28777 0.0433613
$$883$$ −9.54160 −0.321100 −0.160550 0.987028i $$-0.551327\pi$$
−0.160550 + 0.987028i $$0.551327\pi$$
$$884$$ 6.64067 0.223350
$$885$$ −1.11006 −0.0373143
$$886$$ 5.89484 0.198041
$$887$$ 14.7887 0.496557 0.248278 0.968689i $$-0.420135\pi$$
0.248278 + 0.968689i $$0.420135\pi$$
$$888$$ 0.542704 0.0182119
$$889$$ −43.9051 −1.47253
$$890$$ 21.3460 0.715518
$$891$$ −4.25377 −0.142507
$$892$$ 15.4713 0.518018
$$893$$ −7.03523 −0.235425
$$894$$ 3.72254 0.124500
$$895$$ 4.81609 0.160984
$$896$$ −2.39003 −0.0798452
$$897$$ −0.660198 −0.0220434
$$898$$ −21.1771 −0.706691
$$899$$ 51.0015 1.70099
$$900$$ −2.03571 −0.0678571
$$901$$ 21.0315 0.700662
$$902$$ 26.7600 0.891011
$$903$$ 29.9482 0.996615
$$904$$ 6.56362 0.218303
$$905$$ −36.2076 −1.20358
$$906$$ −15.0853 −0.501175
$$907$$ 19.9866 0.663644 0.331822 0.943342i $$-0.392337\pi$$
0.331822 + 0.943342i $$0.392337\pi$$
$$908$$ 12.8564 0.426656
$$909$$ 18.3579 0.608892
$$910$$ −4.11494 −0.136409
$$911$$ −22.1720 −0.734590 −0.367295 0.930104i $$-0.619716\pi$$
−0.367295 + 0.930104i $$0.619716\pi$$
$$912$$ −1.39003 −0.0460284
$$913$$ 16.9162 0.559843
$$914$$ −17.4405 −0.576881
$$915$$ −12.0806 −0.399371
$$916$$ 18.6495 0.616196
$$917$$ 46.8401 1.54680
$$918$$ 6.64067 0.219175
$$919$$ 19.1791 0.632659 0.316329 0.948649i $$-0.397550\pi$$
0.316329 + 0.948649i $$0.397550\pi$$
$$920$$ −1.13667 −0.0374749
$$921$$ 11.7328 0.386609
$$922$$ −17.9659 −0.591674
$$923$$ 1.21505 0.0399938
$$924$$ 10.1666 0.334458
$$925$$ −1.10479 −0.0363253
$$926$$ −6.96127 −0.228762
$$927$$ 1.00000 0.0328443
$$928$$ 7.21542 0.236858
$$929$$ −21.8824 −0.717938 −0.358969 0.933350i $$-0.616872\pi$$
−0.358969 + 0.933350i $$0.616872\pi$$
$$930$$ −12.1697 −0.399062
$$931$$ −1.79003 −0.0586659
$$932$$ −24.7604 −0.811052
$$933$$ 18.1845 0.595333
$$934$$ 19.2176 0.628819
$$935$$ −48.6347 −1.59052
$$936$$ −1.00000 −0.0326860
$$937$$ 18.7388 0.612170 0.306085 0.952004i $$-0.400981\pi$$
0.306085 + 0.952004i $$0.400981\pi$$
$$938$$ 19.5396 0.637990
$$939$$ 4.10088 0.133827
$$940$$ −8.71395 −0.284218
$$941$$ −28.1297 −0.917003 −0.458502 0.888694i $$-0.651614\pi$$
−0.458502 + 0.888694i $$0.651614\pi$$
$$942$$ −22.4610 −0.731819
$$943$$ 4.15323 0.135248
$$944$$ 0.644743 0.0209846
$$945$$ −4.11494 −0.133859
$$946$$ −53.3019 −1.73299
$$947$$ −26.4796 −0.860472 −0.430236 0.902716i $$-0.641570\pi$$
−0.430236 + 0.902716i $$0.641570\pi$$
$$948$$ 5.16356 0.167705
$$949$$ −1.21748 −0.0395210
$$950$$ 2.82970 0.0918076
$$951$$ −5.55377 −0.180093
$$952$$ −15.8714 −0.514394
$$953$$ 44.9734 1.45683 0.728415 0.685136i $$-0.240257\pi$$
0.728415 + 0.685136i $$0.240257\pi$$
$$954$$ −3.16708 −0.102538
$$955$$ −24.7197 −0.799910
$$956$$ 6.41094 0.207345
$$957$$ −30.6928 −0.992156
$$958$$ −8.05490 −0.260242
$$959$$ 22.0806 0.713018
$$960$$ −1.72171 −0.0555680
$$961$$ 18.9623 0.611688
$$962$$ −0.542704 −0.0174975
$$963$$ −14.1501 −0.455981
$$964$$ −9.82599 −0.316474
$$965$$ 15.3132 0.492948
$$966$$ 1.57789 0.0507678
$$967$$ −2.34236 −0.0753253 −0.0376626 0.999291i $$-0.511991\pi$$
−0.0376626 + 0.999291i $$0.511991\pi$$
$$968$$ −7.09460 −0.228029
$$969$$ −9.23071 −0.296533
$$970$$ 31.8491 1.02261
$$971$$ 37.9469 1.21777 0.608887 0.793257i $$-0.291616\pi$$
0.608887 + 0.793257i $$0.291616\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −40.1830 −1.28821
$$974$$ 37.2525 1.19365
$$975$$ 2.03571 0.0651950
$$976$$ 7.01661 0.224596
$$977$$ −1.93752 −0.0619867 −0.0309933 0.999520i $$-0.509867\pi$$
−0.0309933 + 0.999520i $$0.509867\pi$$
$$978$$ 15.9747 0.510814
$$979$$ 52.7388 1.68554
$$980$$ −2.21716 −0.0708246
$$981$$ 1.16806 0.0372934
$$982$$ 33.0779 1.05556
$$983$$ −16.3449 −0.521320 −0.260660 0.965431i $$-0.583940\pi$$
−0.260660 + 0.965431i $$0.583940\pi$$
$$984$$ 6.29089 0.200546
$$985$$ 31.9237 1.01717
$$986$$ 47.9152 1.52593
$$987$$ 12.0965 0.385034
$$988$$ 1.39003 0.0442227
$$989$$ −8.27260 −0.263053
$$990$$ 7.32377 0.232765
$$991$$ 5.85837 0.186097 0.0930486 0.995662i $$-0.470339\pi$$
0.0930486 + 0.995662i $$0.470339\pi$$
$$992$$ 7.06840 0.224422
$$993$$ −20.6613 −0.655668
$$994$$ −2.90400 −0.0921092
$$995$$ 14.0086 0.444104
$$996$$ 3.97674 0.126008
$$997$$ −37.3175 −1.18186 −0.590929 0.806724i $$-0.701239\pi$$
−0.590929 + 0.806724i $$0.701239\pi$$
$$998$$ −20.3427 −0.643938
$$999$$ −0.542704 −0.0171704
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 8034.2.a.o.1.4 7

By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.o.1.4 7 1.1 even 1 trivial