# Properties

 Label 2166.2.a.u.1.2 Level $2166$ Weight $2$ Character 2166.1 Self dual yes Analytic conductor $17.296$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2166 = 2 \cdot 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2166.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.2955970778$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.53209$$ of defining polynomial Character $$\chi$$ $$=$$ 2166.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.18479 q^{5} +1.00000 q^{6} +0.532089 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.18479 q^{5} +1.00000 q^{6} +0.532089 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.18479 q^{10} +1.87939 q^{11} +1.00000 q^{12} +3.87939 q^{13} +0.532089 q^{14} +1.18479 q^{15} +1.00000 q^{16} +1.16250 q^{17} +1.00000 q^{18} +1.18479 q^{20} +0.532089 q^{21} +1.87939 q^{22} -6.70233 q^{23} +1.00000 q^{24} -3.59627 q^{25} +3.87939 q^{26} +1.00000 q^{27} +0.532089 q^{28} +4.02229 q^{29} +1.18479 q^{30} +1.95811 q^{31} +1.00000 q^{32} +1.87939 q^{33} +1.16250 q^{34} +0.630415 q^{35} +1.00000 q^{36} -6.88713 q^{37} +3.87939 q^{39} +1.18479 q^{40} +8.98545 q^{41} +0.532089 q^{42} +2.42602 q^{43} +1.87939 q^{44} +1.18479 q^{45} -6.70233 q^{46} +2.04189 q^{47} +1.00000 q^{48} -6.71688 q^{49} -3.59627 q^{50} +1.16250 q^{51} +3.87939 q^{52} -12.9709 q^{53} +1.00000 q^{54} +2.22668 q^{55} +0.532089 q^{56} +4.02229 q^{58} +2.68004 q^{59} +1.18479 q^{60} -11.3473 q^{61} +1.95811 q^{62} +0.532089 q^{63} +1.00000 q^{64} +4.59627 q^{65} +1.87939 q^{66} +11.1702 q^{67} +1.16250 q^{68} -6.70233 q^{69} +0.630415 q^{70} -6.07192 q^{71} +1.00000 q^{72} -0.327696 q^{73} -6.88713 q^{74} -3.59627 q^{75} +1.00000 q^{77} +3.87939 q^{78} +16.1334 q^{79} +1.18479 q^{80} +1.00000 q^{81} +8.98545 q^{82} +11.5740 q^{83} +0.532089 q^{84} +1.37733 q^{85} +2.42602 q^{86} +4.02229 q^{87} +1.87939 q^{88} +3.55943 q^{89} +1.18479 q^{90} +2.06418 q^{91} -6.70233 q^{92} +1.95811 q^{93} +2.04189 q^{94} +1.00000 q^{96} -5.87939 q^{97} -6.71688 q^{98} +1.87939 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + 3q^{12} + 6q^{13} - 3q^{14} + 3q^{16} + 6q^{17} + 3q^{18} - 3q^{21} + 6q^{23} + 3q^{24} + 3q^{25} + 6q^{26} + 3q^{27} - 3q^{28} + 6q^{29} + 9q^{31} + 3q^{32} + 6q^{34} + 9q^{35} + 3q^{36} + 9q^{37} + 6q^{39} + 9q^{41} - 3q^{42} + 15q^{43} + 6q^{46} + 3q^{47} + 3q^{48} - 12q^{49} + 3q^{50} + 6q^{51} + 6q^{52} - 3q^{53} + 3q^{54} - 3q^{56} + 6q^{58} - 12q^{59} - 33q^{61} + 9q^{62} - 3q^{63} + 3q^{64} + 12q^{67} + 6q^{68} + 6q^{69} + 9q^{70} + 15q^{71} + 3q^{72} + 3q^{73} + 9q^{74} + 3q^{75} + 3q^{77} + 6q^{78} + 15q^{79} + 3q^{81} + 9q^{82} + 27q^{83} - 3q^{84} - 27q^{85} + 15q^{86} + 6q^{87} - 15q^{89} - 3q^{91} + 6q^{92} + 9q^{93} + 3q^{94} + 3q^{96} - 12q^{97} - 12q^{98} + O(q^{100})$$

## 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
$$5$$ 1.18479 0.529855 0.264928 0.964268i $$-0.414652\pi$$
0.264928 + 0.964268i $$0.414652\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0.532089 0.201111 0.100555 0.994931i $$-0.467938\pi$$
0.100555 + 0.994931i $$0.467938\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.18479 0.374664
$$11$$ 1.87939 0.566656 0.283328 0.959023i $$-0.408561\pi$$
0.283328 + 0.959023i $$0.408561\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 3.87939 1.07595 0.537974 0.842961i $$-0.319190\pi$$
0.537974 + 0.842961i $$0.319190\pi$$
$$14$$ 0.532089 0.142207
$$15$$ 1.18479 0.305912
$$16$$ 1.00000 0.250000
$$17$$ 1.16250 0.281949 0.140974 0.990013i $$-0.454977\pi$$
0.140974 + 0.990013i $$0.454977\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0
$$20$$ 1.18479 0.264928
$$21$$ 0.532089 0.116111
$$22$$ 1.87939 0.400686
$$23$$ −6.70233 −1.39753 −0.698767 0.715350i $$-0.746267\pi$$
−0.698767 + 0.715350i $$0.746267\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −3.59627 −0.719253
$$26$$ 3.87939 0.760810
$$27$$ 1.00000 0.192450
$$28$$ 0.532089 0.100555
$$29$$ 4.02229 0.746920 0.373460 0.927646i $$-0.378171\pi$$
0.373460 + 0.927646i $$0.378171\pi$$
$$30$$ 1.18479 0.216313
$$31$$ 1.95811 0.351687 0.175844 0.984418i $$-0.443735\pi$$
0.175844 + 0.984418i $$0.443735\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.87939 0.327159
$$34$$ 1.16250 0.199368
$$35$$ 0.630415 0.106560
$$36$$ 1.00000 0.166667
$$37$$ −6.88713 −1.13224 −0.566118 0.824324i $$-0.691555\pi$$
−0.566118 + 0.824324i $$0.691555\pi$$
$$38$$ 0 0
$$39$$ 3.87939 0.621199
$$40$$ 1.18479 0.187332
$$41$$ 8.98545 1.40329 0.701646 0.712526i $$-0.252449\pi$$
0.701646 + 0.712526i $$0.252449\pi$$
$$42$$ 0.532089 0.0821031
$$43$$ 2.42602 0.369965 0.184982 0.982742i $$-0.440777\pi$$
0.184982 + 0.982742i $$0.440777\pi$$
$$44$$ 1.87939 0.283328
$$45$$ 1.18479 0.176618
$$46$$ −6.70233 −0.988205
$$47$$ 2.04189 0.297840 0.148920 0.988849i $$-0.452420\pi$$
0.148920 + 0.988849i $$0.452420\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.71688 −0.959554
$$50$$ −3.59627 −0.508589
$$51$$ 1.16250 0.162783
$$52$$ 3.87939 0.537974
$$53$$ −12.9709 −1.78169 −0.890845 0.454307i $$-0.849887\pi$$
−0.890845 + 0.454307i $$0.849887\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 2.22668 0.300246
$$56$$ 0.532089 0.0711034
$$57$$ 0 0
$$58$$ 4.02229 0.528152
$$59$$ 2.68004 0.348912 0.174456 0.984665i $$-0.444183\pi$$
0.174456 + 0.984665i $$0.444183\pi$$
$$60$$ 1.18479 0.152956
$$61$$ −11.3473 −1.45287 −0.726436 0.687234i $$-0.758825\pi$$
−0.726436 + 0.687234i $$0.758825\pi$$
$$62$$ 1.95811 0.248680
$$63$$ 0.532089 0.0670369
$$64$$ 1.00000 0.125000
$$65$$ 4.59627 0.570097
$$66$$ 1.87939 0.231336
$$67$$ 11.1702 1.36466 0.682331 0.731043i $$-0.260966\pi$$
0.682331 + 0.731043i $$0.260966\pi$$
$$68$$ 1.16250 0.140974
$$69$$ −6.70233 −0.806866
$$70$$ 0.630415 0.0753490
$$71$$ −6.07192 −0.720604 −0.360302 0.932836i $$-0.617326\pi$$
−0.360302 + 0.932836i $$0.617326\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −0.327696 −0.0383539 −0.0191770 0.999816i $$-0.506105\pi$$
−0.0191770 + 0.999816i $$0.506105\pi$$
$$74$$ −6.88713 −0.800612
$$75$$ −3.59627 −0.415261
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 3.87939 0.439254
$$79$$ 16.1334 1.81515 0.907575 0.419890i $$-0.137931\pi$$
0.907575 + 0.419890i $$0.137931\pi$$
$$80$$ 1.18479 0.132464
$$81$$ 1.00000 0.111111
$$82$$ 8.98545 0.992277
$$83$$ 11.5740 1.27041 0.635205 0.772344i $$-0.280916\pi$$
0.635205 + 0.772344i $$0.280916\pi$$
$$84$$ 0.532089 0.0580557
$$85$$ 1.37733 0.149392
$$86$$ 2.42602 0.261605
$$87$$ 4.02229 0.431235
$$88$$ 1.87939 0.200343
$$89$$ 3.55943 0.377299 0.188649 0.982044i $$-0.439589\pi$$
0.188649 + 0.982044i $$0.439589\pi$$
$$90$$ 1.18479 0.124888
$$91$$ 2.06418 0.216385
$$92$$ −6.70233 −0.698767
$$93$$ 1.95811 0.203047
$$94$$ 2.04189 0.210605
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −5.87939 −0.596961 −0.298481 0.954416i $$-0.596480\pi$$
−0.298481 + 0.954416i $$0.596480\pi$$
$$98$$ −6.71688 −0.678507
$$99$$ 1.87939 0.188885
$$100$$ −3.59627 −0.359627
$$101$$ 12.6459 1.25831 0.629157 0.777278i $$-0.283400\pi$$
0.629157 + 0.777278i $$0.283400\pi$$
$$102$$ 1.16250 0.115105
$$103$$ −2.96316 −0.291969 −0.145985 0.989287i $$-0.546635\pi$$
−0.145985 + 0.989287i $$0.546635\pi$$
$$104$$ 3.87939 0.380405
$$105$$ 0.630415 0.0615222
$$106$$ −12.9709 −1.25985
$$107$$ −19.1138 −1.84780 −0.923901 0.382632i $$-0.875018\pi$$
−0.923901 + 0.382632i $$0.875018\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 16.4115 1.57193 0.785967 0.618268i $$-0.212166\pi$$
0.785967 + 0.618268i $$0.212166\pi$$
$$110$$ 2.22668 0.212306
$$111$$ −6.88713 −0.653697
$$112$$ 0.532089 0.0502777
$$113$$ 5.73648 0.539643 0.269821 0.962910i $$-0.413035\pi$$
0.269821 + 0.962910i $$0.413035\pi$$
$$114$$ 0 0
$$115$$ −7.94087 −0.740490
$$116$$ 4.02229 0.373460
$$117$$ 3.87939 0.358649
$$118$$ 2.68004 0.246718
$$119$$ 0.618555 0.0567029
$$120$$ 1.18479 0.108156
$$121$$ −7.46791 −0.678901
$$122$$ −11.3473 −1.02734
$$123$$ 8.98545 0.810191
$$124$$ 1.95811 0.175844
$$125$$ −10.1848 −0.910956
$$126$$ 0.532089 0.0474022
$$127$$ 1.88713 0.167455 0.0837277 0.996489i $$-0.473317\pi$$
0.0837277 + 0.996489i $$0.473317\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 2.42602 0.213599
$$130$$ 4.59627 0.403119
$$131$$ 10.7023 0.935067 0.467534 0.883975i $$-0.345143\pi$$
0.467534 + 0.883975i $$0.345143\pi$$
$$132$$ 1.87939 0.163579
$$133$$ 0 0
$$134$$ 11.1702 0.964962
$$135$$ 1.18479 0.101971
$$136$$ 1.16250 0.0996839
$$137$$ −11.2567 −0.961726 −0.480863 0.876796i $$-0.659677\pi$$
−0.480863 + 0.876796i $$0.659677\pi$$
$$138$$ −6.70233 −0.570541
$$139$$ −12.1702 −1.03227 −0.516133 0.856508i $$-0.672629\pi$$
−0.516133 + 0.856508i $$0.672629\pi$$
$$140$$ 0.630415 0.0532798
$$141$$ 2.04189 0.171958
$$142$$ −6.07192 −0.509544
$$143$$ 7.29086 0.609692
$$144$$ 1.00000 0.0833333
$$145$$ 4.76558 0.395760
$$146$$ −0.327696 −0.0271203
$$147$$ −6.71688 −0.553999
$$148$$ −6.88713 −0.566118
$$149$$ −16.3550 −1.33986 −0.669928 0.742426i $$-0.733675\pi$$
−0.669928 + 0.742426i $$0.733675\pi$$
$$150$$ −3.59627 −0.293634
$$151$$ −20.5226 −1.67010 −0.835052 0.550170i $$-0.814563\pi$$
−0.835052 + 0.550170i $$0.814563\pi$$
$$152$$ 0 0
$$153$$ 1.16250 0.0939829
$$154$$ 1.00000 0.0805823
$$155$$ 2.31996 0.186343
$$156$$ 3.87939 0.310599
$$157$$ −3.19934 −0.255335 −0.127668 0.991817i $$-0.540749\pi$$
−0.127668 + 0.991817i $$0.540749\pi$$
$$158$$ 16.1334 1.28351
$$159$$ −12.9709 −1.02866
$$160$$ 1.18479 0.0936661
$$161$$ −3.56624 −0.281059
$$162$$ 1.00000 0.0785674
$$163$$ −24.0077 −1.88043 −0.940216 0.340580i $$-0.889377\pi$$
−0.940216 + 0.340580i $$0.889377\pi$$
$$164$$ 8.98545 0.701646
$$165$$ 2.22668 0.173347
$$166$$ 11.5740 0.898315
$$167$$ −3.66044 −0.283254 −0.141627 0.989920i $$-0.545233\pi$$
−0.141627 + 0.989920i $$0.545233\pi$$
$$168$$ 0.532089 0.0410515
$$169$$ 2.04963 0.157664
$$170$$ 1.37733 0.105636
$$171$$ 0 0
$$172$$ 2.42602 0.184982
$$173$$ 5.33275 0.405441 0.202721 0.979237i $$-0.435022\pi$$
0.202721 + 0.979237i $$0.435022\pi$$
$$174$$ 4.02229 0.304929
$$175$$ −1.91353 −0.144650
$$176$$ 1.87939 0.141664
$$177$$ 2.68004 0.201445
$$178$$ 3.55943 0.266791
$$179$$ 25.2567 1.88778 0.943888 0.330267i $$-0.107139\pi$$
0.943888 + 0.330267i $$0.107139\pi$$
$$180$$ 1.18479 0.0883092
$$181$$ 17.3063 1.28637 0.643185 0.765711i $$-0.277613\pi$$
0.643185 + 0.765711i $$0.277613\pi$$
$$182$$ 2.06418 0.153007
$$183$$ −11.3473 −0.838816
$$184$$ −6.70233 −0.494103
$$185$$ −8.15982 −0.599922
$$186$$ 1.95811 0.143576
$$187$$ 2.18479 0.159768
$$188$$ 2.04189 0.148920
$$189$$ 0.532089 0.0387038
$$190$$ 0 0
$$191$$ −15.1780 −1.09824 −0.549120 0.835743i $$-0.685037\pi$$
−0.549120 + 0.835743i $$0.685037\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −13.0077 −0.936318 −0.468159 0.883644i $$-0.655082\pi$$
−0.468159 + 0.883644i $$0.655082\pi$$
$$194$$ −5.87939 −0.422115
$$195$$ 4.59627 0.329145
$$196$$ −6.71688 −0.479777
$$197$$ −8.94862 −0.637562 −0.318781 0.947828i $$-0.603274\pi$$
−0.318781 + 0.947828i $$0.603274\pi$$
$$198$$ 1.87939 0.133562
$$199$$ −4.50475 −0.319333 −0.159667 0.987171i $$-0.551042\pi$$
−0.159667 + 0.987171i $$0.551042\pi$$
$$200$$ −3.59627 −0.254294
$$201$$ 11.1702 0.787888
$$202$$ 12.6459 0.889762
$$203$$ 2.14022 0.150214
$$204$$ 1.16250 0.0813915
$$205$$ 10.6459 0.743542
$$206$$ −2.96316 −0.206453
$$207$$ −6.70233 −0.465844
$$208$$ 3.87939 0.268987
$$209$$ 0 0
$$210$$ 0.630415 0.0435028
$$211$$ 6.40373 0.440851 0.220426 0.975404i $$-0.429255\pi$$
0.220426 + 0.975404i $$0.429255\pi$$
$$212$$ −12.9709 −0.890845
$$213$$ −6.07192 −0.416041
$$214$$ −19.1138 −1.30659
$$215$$ 2.87433 0.196028
$$216$$ 1.00000 0.0680414
$$217$$ 1.04189 0.0707280
$$218$$ 16.4115 1.11153
$$219$$ −0.327696 −0.0221436
$$220$$ 2.22668 0.150123
$$221$$ 4.50980 0.303362
$$222$$ −6.88713 −0.462234
$$223$$ −3.24897 −0.217567 −0.108784 0.994065i $$-0.534696\pi$$
−0.108784 + 0.994065i $$0.534696\pi$$
$$224$$ 0.532089 0.0355517
$$225$$ −3.59627 −0.239751
$$226$$ 5.73648 0.381585
$$227$$ 11.7023 0.776711 0.388356 0.921510i $$-0.373043\pi$$
0.388356 + 0.921510i $$0.373043\pi$$
$$228$$ 0 0
$$229$$ −22.9067 −1.51372 −0.756860 0.653578i $$-0.773267\pi$$
−0.756860 + 0.653578i $$0.773267\pi$$
$$230$$ −7.94087 −0.523606
$$231$$ 1.00000 0.0657952
$$232$$ 4.02229 0.264076
$$233$$ −3.19934 −0.209596 −0.104798 0.994494i $$-0.533420\pi$$
−0.104798 + 0.994494i $$0.533420\pi$$
$$234$$ 3.87939 0.253603
$$235$$ 2.41921 0.157812
$$236$$ 2.68004 0.174456
$$237$$ 16.1334 1.04798
$$238$$ 0.618555 0.0400950
$$239$$ 6.17293 0.399294 0.199647 0.979868i $$-0.436020\pi$$
0.199647 + 0.979868i $$0.436020\pi$$
$$240$$ 1.18479 0.0764780
$$241$$ −1.53714 −0.0990160 −0.0495080 0.998774i $$-0.515765\pi$$
−0.0495080 + 0.998774i $$0.515765\pi$$
$$242$$ −7.46791 −0.480056
$$243$$ 1.00000 0.0641500
$$244$$ −11.3473 −0.726436
$$245$$ −7.95811 −0.508425
$$246$$ 8.98545 0.572892
$$247$$ 0 0
$$248$$ 1.95811 0.124340
$$249$$ 11.5740 0.733471
$$250$$ −10.1848 −0.644143
$$251$$ 17.9436 1.13259 0.566294 0.824203i $$-0.308377\pi$$
0.566294 + 0.824203i $$0.308377\pi$$
$$252$$ 0.532089 0.0335184
$$253$$ −12.5963 −0.791920
$$254$$ 1.88713 0.118409
$$255$$ 1.37733 0.0862515
$$256$$ 1.00000 0.0625000
$$257$$ −29.4124 −1.83470 −0.917348 0.398087i $$-0.869674\pi$$
−0.917348 + 0.398087i $$0.869674\pi$$
$$258$$ 2.42602 0.151038
$$259$$ −3.66456 −0.227705
$$260$$ 4.59627 0.285048
$$261$$ 4.02229 0.248973
$$262$$ 10.7023 0.661192
$$263$$ 10.6827 0.658726 0.329363 0.944203i $$-0.393166\pi$$
0.329363 + 0.944203i $$0.393166\pi$$
$$264$$ 1.87939 0.115668
$$265$$ −15.3678 −0.944038
$$266$$ 0 0
$$267$$ 3.55943 0.217834
$$268$$ 11.1702 0.682331
$$269$$ 20.5767 1.25458 0.627291 0.778785i $$-0.284164\pi$$
0.627291 + 0.778785i $$0.284164\pi$$
$$270$$ 1.18479 0.0721042
$$271$$ 1.87670 0.114001 0.0570006 0.998374i $$-0.481846\pi$$
0.0570006 + 0.998374i $$0.481846\pi$$
$$272$$ 1.16250 0.0704871
$$273$$ 2.06418 0.124930
$$274$$ −11.2567 −0.680043
$$275$$ −6.75877 −0.407569
$$276$$ −6.70233 −0.403433
$$277$$ 4.87258 0.292765 0.146382 0.989228i $$-0.453237\pi$$
0.146382 + 0.989228i $$0.453237\pi$$
$$278$$ −12.1702 −0.729923
$$279$$ 1.95811 0.117229
$$280$$ 0.630415 0.0376745
$$281$$ 0.403733 0.0240847 0.0120424 0.999927i $$-0.496167\pi$$
0.0120424 + 0.999927i $$0.496167\pi$$
$$282$$ 2.04189 0.121593
$$283$$ 17.5895 1.04558 0.522792 0.852460i $$-0.324890\pi$$
0.522792 + 0.852460i $$0.324890\pi$$
$$284$$ −6.07192 −0.360302
$$285$$ 0 0
$$286$$ 7.29086 0.431118
$$287$$ 4.78106 0.282217
$$288$$ 1.00000 0.0589256
$$289$$ −15.6486 −0.920505
$$290$$ 4.76558 0.279844
$$291$$ −5.87939 −0.344656
$$292$$ −0.327696 −0.0191770
$$293$$ 13.1753 0.769709 0.384855 0.922977i $$-0.374252\pi$$
0.384855 + 0.922977i $$0.374252\pi$$
$$294$$ −6.71688 −0.391736
$$295$$ 3.17530 0.184873
$$296$$ −6.88713 −0.400306
$$297$$ 1.87939 0.109053
$$298$$ −16.3550 −0.947422
$$299$$ −26.0009 −1.50367
$$300$$ −3.59627 −0.207631
$$301$$ 1.29086 0.0744039
$$302$$ −20.5226 −1.18094
$$303$$ 12.6459 0.726488
$$304$$ 0 0
$$305$$ −13.4442 −0.769812
$$306$$ 1.16250 0.0664559
$$307$$ −16.9017 −0.964629 −0.482315 0.875998i $$-0.660204\pi$$
−0.482315 + 0.875998i $$0.660204\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ −2.96316 −0.168568
$$310$$ 2.31996 0.131765
$$311$$ −2.38144 −0.135039 −0.0675197 0.997718i $$-0.521509\pi$$
−0.0675197 + 0.997718i $$0.521509\pi$$
$$312$$ 3.87939 0.219627
$$313$$ −29.8631 −1.68796 −0.843981 0.536374i $$-0.819794\pi$$
−0.843981 + 0.536374i $$0.819794\pi$$
$$314$$ −3.19934 −0.180549
$$315$$ 0.630415 0.0355199
$$316$$ 16.1334 0.907575
$$317$$ 7.02229 0.394411 0.197206 0.980362i $$-0.436813\pi$$
0.197206 + 0.980362i $$0.436813\pi$$
$$318$$ −12.9709 −0.727372
$$319$$ 7.55943 0.423247
$$320$$ 1.18479 0.0662319
$$321$$ −19.1138 −1.06683
$$322$$ −3.56624 −0.198739
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −13.9513 −0.773879
$$326$$ −24.0077 −1.32967
$$327$$ 16.4115 0.907557
$$328$$ 8.98545 0.496139
$$329$$ 1.08647 0.0598988
$$330$$ 2.22668 0.122575
$$331$$ −27.1985 −1.49497 −0.747483 0.664281i $$-0.768738\pi$$
−0.747483 + 0.664281i $$0.768738\pi$$
$$332$$ 11.5740 0.635205
$$333$$ −6.88713 −0.377412
$$334$$ −3.66044 −0.200291
$$335$$ 13.2344 0.723074
$$336$$ 0.532089 0.0290278
$$337$$ 11.2172 0.611039 0.305520 0.952186i $$-0.401170\pi$$
0.305520 + 0.952186i $$0.401170\pi$$
$$338$$ 2.04963 0.111485
$$339$$ 5.73648 0.311563
$$340$$ 1.37733 0.0746960
$$341$$ 3.68004 0.199286
$$342$$ 0 0
$$343$$ −7.29860 −0.394087
$$344$$ 2.42602 0.130802
$$345$$ −7.94087 −0.427522
$$346$$ 5.33275 0.286690
$$347$$ 1.93582 0.103920 0.0519602 0.998649i $$-0.483453\pi$$
0.0519602 + 0.998649i $$0.483453\pi$$
$$348$$ 4.02229 0.215617
$$349$$ 14.3969 0.770650 0.385325 0.922781i $$-0.374089\pi$$
0.385325 + 0.922781i $$0.374089\pi$$
$$350$$ −1.91353 −0.102283
$$351$$ 3.87939 0.207066
$$352$$ 1.87939 0.100172
$$353$$ 15.7237 0.836887 0.418444 0.908243i $$-0.362576\pi$$
0.418444 + 0.908243i $$0.362576\pi$$
$$354$$ 2.68004 0.142443
$$355$$ −7.19396 −0.381816
$$356$$ 3.55943 0.188649
$$357$$ 0.618555 0.0327374
$$358$$ 25.2567 1.33486
$$359$$ 15.5175 0.818984 0.409492 0.912314i $$-0.365706\pi$$
0.409492 + 0.912314i $$0.365706\pi$$
$$360$$ 1.18479 0.0624440
$$361$$ 0 0
$$362$$ 17.3063 0.909601
$$363$$ −7.46791 −0.391964
$$364$$ 2.06418 0.108192
$$365$$ −0.388252 −0.0203220
$$366$$ −11.3473 −0.593133
$$367$$ −15.7442 −0.821842 −0.410921 0.911671i $$-0.634793\pi$$
−0.410921 + 0.911671i $$0.634793\pi$$
$$368$$ −6.70233 −0.349383
$$369$$ 8.98545 0.467764
$$370$$ −8.15982 −0.424209
$$371$$ −6.90167 −0.358317
$$372$$ 1.95811 0.101523
$$373$$ 29.6287 1.53411 0.767057 0.641579i $$-0.221720\pi$$
0.767057 + 0.641579i $$0.221720\pi$$
$$374$$ 2.18479 0.112973
$$375$$ −10.1848 −0.525940
$$376$$ 2.04189 0.105302
$$377$$ 15.6040 0.803647
$$378$$ 0.532089 0.0273677
$$379$$ −4.72462 −0.242688 −0.121344 0.992611i $$-0.538720\pi$$
−0.121344 + 0.992611i $$0.538720\pi$$
$$380$$ 0 0
$$381$$ 1.88713 0.0966804
$$382$$ −15.1780 −0.776573
$$383$$ 5.24123 0.267814 0.133907 0.990994i $$-0.457248\pi$$
0.133907 + 0.990994i $$0.457248\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 1.18479 0.0603826
$$386$$ −13.0077 −0.662077
$$387$$ 2.42602 0.123322
$$388$$ −5.87939 −0.298481
$$389$$ −15.0770 −0.764433 −0.382216 0.924073i $$-0.624839\pi$$
−0.382216 + 0.924073i $$0.624839\pi$$
$$390$$ 4.59627 0.232741
$$391$$ −7.79149 −0.394033
$$392$$ −6.71688 −0.339254
$$393$$ 10.7023 0.539861
$$394$$ −8.94862 −0.450825
$$395$$ 19.1147 0.961767
$$396$$ 1.87939 0.0944427
$$397$$ −14.5449 −0.729987 −0.364993 0.931010i $$-0.618929\pi$$
−0.364993 + 0.931010i $$0.618929\pi$$
$$398$$ −4.50475 −0.225803
$$399$$ 0 0
$$400$$ −3.59627 −0.179813
$$401$$ −2.85473 −0.142559 −0.0712793 0.997456i $$-0.522708\pi$$
−0.0712793 + 0.997456i $$0.522708\pi$$
$$402$$ 11.1702 0.557121
$$403$$ 7.59627 0.378397
$$404$$ 12.6459 0.629157
$$405$$ 1.18479 0.0588728
$$406$$ 2.14022 0.106217
$$407$$ −12.9436 −0.641589
$$408$$ 1.16250 0.0575525
$$409$$ −34.2276 −1.69245 −0.846223 0.532828i $$-0.821129\pi$$
−0.846223 + 0.532828i $$0.821129\pi$$
$$410$$ 10.6459 0.525763
$$411$$ −11.2567 −0.555253
$$412$$ −2.96316 −0.145985
$$413$$ 1.42602 0.0701700
$$414$$ −6.70233 −0.329402
$$415$$ 13.7128 0.673133
$$416$$ 3.87939 0.190203
$$417$$ −12.1702 −0.595979
$$418$$ 0 0
$$419$$ −7.91891 −0.386864 −0.193432 0.981114i $$-0.561962\pi$$
−0.193432 + 0.981114i $$0.561962\pi$$
$$420$$ 0.630415 0.0307611
$$421$$ −16.2098 −0.790016 −0.395008 0.918678i $$-0.629258\pi$$
−0.395008 + 0.918678i $$0.629258\pi$$
$$422$$ 6.40373 0.311729
$$423$$ 2.04189 0.0992800
$$424$$ −12.9709 −0.629923
$$425$$ −4.18067 −0.202792
$$426$$ −6.07192 −0.294185
$$427$$ −6.03777 −0.292188
$$428$$ −19.1138 −0.923901
$$429$$ 7.29086 0.352006
$$430$$ 2.87433 0.138613
$$431$$ −25.4320 −1.22502 −0.612508 0.790464i $$-0.709839\pi$$
−0.612508 + 0.790464i $$0.709839\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −11.8479 −0.569375 −0.284687 0.958620i $$-0.591890\pi$$
−0.284687 + 0.958620i $$0.591890\pi$$
$$434$$ 1.04189 0.0500123
$$435$$ 4.76558 0.228492
$$436$$ 16.4115 0.785967
$$437$$ 0 0
$$438$$ −0.327696 −0.0156579
$$439$$ −25.8598 −1.23422 −0.617110 0.786877i $$-0.711697\pi$$
−0.617110 + 0.786877i $$0.711697\pi$$
$$440$$ 2.22668 0.106153
$$441$$ −6.71688 −0.319851
$$442$$ 4.50980 0.214509
$$443$$ 0.923029 0.0438544 0.0219272 0.999760i $$-0.493020\pi$$
0.0219272 + 0.999760i $$0.493020\pi$$
$$444$$ −6.88713 −0.326849
$$445$$ 4.21719 0.199914
$$446$$ −3.24897 −0.153843
$$447$$ −16.3550 −0.773567
$$448$$ 0.532089 0.0251388
$$449$$ −9.34461 −0.440999 −0.220500 0.975387i $$-0.570769\pi$$
−0.220500 + 0.975387i $$0.570769\pi$$
$$450$$ −3.59627 −0.169530
$$451$$ 16.8871 0.795184
$$452$$ 5.73648 0.269821
$$453$$ −20.5226 −0.964236
$$454$$ 11.7023 0.549218
$$455$$ 2.44562 0.114653
$$456$$ 0 0
$$457$$ 30.9009 1.44548 0.722740 0.691120i $$-0.242882\pi$$
0.722740 + 0.691120i $$0.242882\pi$$
$$458$$ −22.9067 −1.07036
$$459$$ 1.16250 0.0542610
$$460$$ −7.94087 −0.370245
$$461$$ −29.6878 −1.38270 −0.691349 0.722521i $$-0.742983\pi$$
−0.691349 + 0.722521i $$0.742983\pi$$
$$462$$ 1.00000 0.0465242
$$463$$ −11.8307 −0.549819 −0.274909 0.961470i $$-0.588648\pi$$
−0.274909 + 0.961470i $$0.588648\pi$$
$$464$$ 4.02229 0.186730
$$465$$ 2.31996 0.107585
$$466$$ −3.19934 −0.148207
$$467$$ 38.3182 1.77315 0.886577 0.462580i $$-0.153076\pi$$
0.886577 + 0.462580i $$0.153076\pi$$
$$468$$ 3.87939 0.179325
$$469$$ 5.94356 0.274448
$$470$$ 2.41921 0.111590
$$471$$ −3.19934 −0.147418
$$472$$ 2.68004 0.123359
$$473$$ 4.55943 0.209643
$$474$$ 16.1334 0.741032
$$475$$ 0 0
$$476$$ 0.618555 0.0283514
$$477$$ −12.9709 −0.593897
$$478$$ 6.17293 0.282343
$$479$$ 37.2841 1.70355 0.851776 0.523906i $$-0.175526\pi$$
0.851776 + 0.523906i $$0.175526\pi$$
$$480$$ 1.18479 0.0540781
$$481$$ −26.7178 −1.21823
$$482$$ −1.53714 −0.0700149
$$483$$ −3.56624 −0.162269
$$484$$ −7.46791 −0.339451
$$485$$ −6.96585 −0.316303
$$486$$ 1.00000 0.0453609
$$487$$ −32.9786 −1.49441 −0.747203 0.664596i $$-0.768603\pi$$
−0.747203 + 0.664596i $$0.768603\pi$$
$$488$$ −11.3473 −0.513668
$$489$$ −24.0077 −1.08567
$$490$$ −7.95811 −0.359511
$$491$$ 17.6810 0.797931 0.398966 0.916966i $$-0.369369\pi$$
0.398966 + 0.916966i $$0.369369\pi$$
$$492$$ 8.98545 0.405095
$$493$$ 4.67593 0.210593
$$494$$ 0 0
$$495$$ 2.22668 0.100082
$$496$$ 1.95811 0.0879218
$$497$$ −3.23080 −0.144921
$$498$$ 11.5740 0.518642
$$499$$ 27.1239 1.21423 0.607117 0.794613i $$-0.292326\pi$$
0.607117 + 0.794613i $$0.292326\pi$$
$$500$$ −10.1848 −0.455478
$$501$$ −3.66044 −0.163537
$$502$$ 17.9436 0.800860
$$503$$ 0.571290 0.0254725 0.0127363 0.999919i $$-0.495946\pi$$
0.0127363 + 0.999919i $$0.495946\pi$$
$$504$$ 0.532089 0.0237011
$$505$$ 14.9828 0.666724
$$506$$ −12.5963 −0.559972
$$507$$ 2.04963 0.0910273
$$508$$ 1.88713 0.0837277
$$509$$ −40.5509 −1.79739 −0.898693 0.438579i $$-0.855482\pi$$
−0.898693 + 0.438579i $$0.855482\pi$$
$$510$$ 1.37733 0.0609890
$$511$$ −0.174363 −0.00771338
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −29.4124 −1.29733
$$515$$ −3.51073 −0.154701
$$516$$ 2.42602 0.106800
$$517$$ 3.83750 0.168773
$$518$$ −3.66456 −0.161012
$$519$$ 5.33275 0.234082
$$520$$ 4.59627 0.201560
$$521$$ −18.8999 −0.828020 −0.414010 0.910272i $$-0.635872\pi$$
−0.414010 + 0.910272i $$0.635872\pi$$
$$522$$ 4.02229 0.176051
$$523$$ 4.74186 0.207347 0.103673 0.994611i $$-0.466940\pi$$
0.103673 + 0.994611i $$0.466940\pi$$
$$524$$ 10.7023 0.467534
$$525$$ −1.91353 −0.0835134
$$526$$ 10.6827 0.465789
$$527$$ 2.27631 0.0991577
$$528$$ 1.87939 0.0817897
$$529$$ 21.9213 0.953099
$$530$$ −15.3678 −0.667536
$$531$$ 2.68004 0.116304
$$532$$ 0 0
$$533$$ 34.8580 1.50987
$$534$$ 3.55943 0.154032
$$535$$ −22.6459 −0.979067
$$536$$ 11.1702 0.482481
$$537$$ 25.2567 1.08991
$$538$$ 20.5767 0.887123
$$539$$ −12.6236 −0.543737
$$540$$ 1.18479 0.0509854
$$541$$ −28.3610 −1.21934 −0.609668 0.792657i $$-0.708697\pi$$
−0.609668 + 0.792657i $$0.708697\pi$$
$$542$$ 1.87670 0.0806110
$$543$$ 17.3063 0.742686
$$544$$ 1.16250 0.0498419
$$545$$ 19.4442 0.832898
$$546$$ 2.06418 0.0883387
$$547$$ 6.88032 0.294181 0.147091 0.989123i $$-0.453009\pi$$
0.147091 + 0.989123i $$0.453009\pi$$
$$548$$ −11.2567 −0.480863
$$549$$ −11.3473 −0.484291
$$550$$ −6.75877 −0.288195
$$551$$ 0 0
$$552$$ −6.70233 −0.285270
$$553$$ 8.58441 0.365046
$$554$$ 4.87258 0.207016
$$555$$ −8.15982 −0.346365
$$556$$ −12.1702 −0.516133
$$557$$ −14.1625 −0.600085 −0.300042 0.953926i $$-0.597001\pi$$
−0.300042 + 0.953926i $$0.597001\pi$$
$$558$$ 1.95811 0.0828934
$$559$$ 9.41147 0.398063
$$560$$ 0.630415 0.0266399
$$561$$ 2.18479 0.0922420
$$562$$ 0.403733 0.0170305
$$563$$ 36.2550 1.52796 0.763982 0.645237i $$-0.223242\pi$$
0.763982 + 0.645237i $$0.223242\pi$$
$$564$$ 2.04189 0.0859790
$$565$$ 6.79654 0.285933
$$566$$ 17.5895 0.739340
$$567$$ 0.532089 0.0223456
$$568$$ −6.07192 −0.254772
$$569$$ 30.2918 1.26990 0.634949 0.772554i $$-0.281021\pi$$
0.634949 + 0.772554i $$0.281021\pi$$
$$570$$ 0 0
$$571$$ 3.39094 0.141906 0.0709532 0.997480i $$-0.477396\pi$$
0.0709532 + 0.997480i $$0.477396\pi$$
$$572$$ 7.29086 0.304846
$$573$$ −15.1780 −0.634069
$$574$$ 4.78106 0.199558
$$575$$ 24.1034 1.00518
$$576$$ 1.00000 0.0416667
$$577$$ −22.3027 −0.928474 −0.464237 0.885711i $$-0.653671\pi$$
−0.464237 + 0.885711i $$0.653671\pi$$
$$578$$ −15.6486 −0.650895
$$579$$ −13.0077 −0.540583
$$580$$ 4.76558 0.197880
$$581$$ 6.15839 0.255493
$$582$$ −5.87939 −0.243708
$$583$$ −24.3773 −1.00961
$$584$$ −0.327696 −0.0135602
$$585$$ 4.59627 0.190032
$$586$$ 13.1753 0.544267
$$587$$ −28.7502 −1.18665 −0.593324 0.804964i $$-0.702185\pi$$
−0.593324 + 0.804964i $$0.702185\pi$$
$$588$$ −6.71688 −0.277000
$$589$$ 0 0
$$590$$ 3.17530 0.130725
$$591$$ −8.94862 −0.368097
$$592$$ −6.88713 −0.283059
$$593$$ 11.7760 0.483583 0.241791 0.970328i $$-0.422265\pi$$
0.241791 + 0.970328i $$0.422265\pi$$
$$594$$ 1.87939 0.0771121
$$595$$ 0.732860 0.0300443
$$596$$ −16.3550 −0.669928
$$597$$ −4.50475 −0.184367
$$598$$ −26.0009 −1.06326
$$599$$ 21.6313 0.883833 0.441916 0.897056i $$-0.354299\pi$$
0.441916 + 0.897056i $$0.354299\pi$$
$$600$$ −3.59627 −0.146817
$$601$$ 26.1935 1.06845 0.534227 0.845341i $$-0.320603\pi$$
0.534227 + 0.845341i $$0.320603\pi$$
$$602$$ 1.29086 0.0526115
$$603$$ 11.1702 0.454888
$$604$$ −20.5226 −0.835052
$$605$$ −8.84793 −0.359719
$$606$$ 12.6459 0.513704
$$607$$ 13.2249 0.536783 0.268392 0.963310i $$-0.413508\pi$$
0.268392 + 0.963310i $$0.413508\pi$$
$$608$$ 0 0
$$609$$ 2.14022 0.0867259
$$610$$ −13.4442 −0.544339
$$611$$ 7.92127 0.320460
$$612$$ 1.16250 0.0469914
$$613$$ −5.62267 −0.227098 −0.113549 0.993532i $$-0.536222\pi$$
−0.113549 + 0.993532i $$0.536222\pi$$
$$614$$ −16.9017 −0.682096
$$615$$ 10.6459 0.429284
$$616$$ 1.00000 0.0402911
$$617$$ 9.47834 0.381584 0.190792 0.981631i $$-0.438894\pi$$
0.190792 + 0.981631i $$0.438894\pi$$
$$618$$ −2.96316 −0.119196
$$619$$ −14.7493 −0.592823 −0.296412 0.955060i $$-0.595790\pi$$
−0.296412 + 0.955060i $$0.595790\pi$$
$$620$$ 2.31996 0.0931716
$$621$$ −6.70233 −0.268955
$$622$$ −2.38144 −0.0954872
$$623$$ 1.89393 0.0758788
$$624$$ 3.87939 0.155300
$$625$$ 5.91447 0.236579
$$626$$ −29.8631 −1.19357
$$627$$ 0 0
$$628$$ −3.19934 −0.127668
$$629$$ −8.00631 −0.319232
$$630$$ 0.630415 0.0251163
$$631$$ 25.0009 0.995271 0.497636 0.867386i $$-0.334202\pi$$
0.497636 + 0.867386i $$0.334202\pi$$
$$632$$ 16.1334 0.641753
$$633$$ 6.40373 0.254526
$$634$$ 7.02229 0.278891
$$635$$ 2.23585 0.0887271
$$636$$ −12.9709 −0.514330
$$637$$ −26.0574 −1.03243
$$638$$ 7.55943 0.299281
$$639$$ −6.07192 −0.240201
$$640$$ 1.18479 0.0468330
$$641$$ 43.7306 1.72726 0.863628 0.504130i $$-0.168187\pi$$
0.863628 + 0.504130i $$0.168187\pi$$
$$642$$ −19.1138 −0.754362
$$643$$ −33.4534 −1.31927 −0.659636 0.751585i $$-0.729290\pi$$
−0.659636 + 0.751585i $$0.729290\pi$$
$$644$$ −3.56624 −0.140529
$$645$$ 2.87433 0.113177
$$646$$ 0 0
$$647$$ −32.1266 −1.26303 −0.631514 0.775365i $$-0.717566\pi$$
−0.631514 + 0.775365i $$0.717566\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 5.03684 0.197713
$$650$$ −13.9513 −0.547215
$$651$$ 1.04189 0.0408349
$$652$$ −24.0077 −0.940216
$$653$$ 2.89630 0.113341 0.0566704 0.998393i $$-0.481952\pi$$
0.0566704 + 0.998393i $$0.481952\pi$$
$$654$$ 16.4115 0.641739
$$655$$ 12.6800 0.495450
$$656$$ 8.98545 0.350823
$$657$$ −0.327696 −0.0127846
$$658$$ 1.08647 0.0423549
$$659$$ −18.1935 −0.708717 −0.354358 0.935110i $$-0.615301\pi$$
−0.354358 + 0.935110i $$0.615301\pi$$
$$660$$ 2.22668 0.0866735
$$661$$ −21.4192 −0.833111 −0.416555 0.909110i $$-0.636763\pi$$
−0.416555 + 0.909110i $$0.636763\pi$$
$$662$$ −27.1985 −1.05710
$$663$$ 4.50980 0.175146
$$664$$ 11.5740 0.449157
$$665$$ 0 0
$$666$$ −6.88713 −0.266871
$$667$$ −26.9587 −1.04385
$$668$$ −3.66044 −0.141627
$$669$$ −3.24897 −0.125612
$$670$$ 13.2344 0.511290
$$671$$ −21.3259 −0.823279
$$672$$ 0.532089 0.0205258
$$673$$ −40.6914 −1.56854 −0.784269 0.620421i $$-0.786962\pi$$
−0.784269 + 0.620421i $$0.786962\pi$$
$$674$$ 11.2172 0.432070
$$675$$ −3.59627 −0.138420
$$676$$ 2.04963 0.0788319
$$677$$ 0.136096 0.00523061 0.00261530 0.999997i $$-0.499168\pi$$
0.00261530 + 0.999997i $$0.499168\pi$$
$$678$$ 5.73648 0.220308
$$679$$ −3.12836 −0.120055
$$680$$ 1.37733 0.0528180
$$681$$ 11.7023 0.448434
$$682$$ 3.68004 0.140916
$$683$$ 43.6459 1.67006 0.835032 0.550202i $$-0.185449\pi$$
0.835032 + 0.550202i $$0.185449\pi$$
$$684$$ 0 0
$$685$$ −13.3369 −0.509576
$$686$$ −7.29860 −0.278662
$$687$$ −22.9067 −0.873946
$$688$$ 2.42602 0.0924912
$$689$$ −50.3191 −1.91701
$$690$$ −7.94087 −0.302304
$$691$$ 15.8675 0.603629 0.301815 0.953367i $$-0.402408\pi$$
0.301815 + 0.953367i $$0.402408\pi$$
$$692$$ 5.33275 0.202721
$$693$$ 1.00000 0.0379869
$$694$$ 1.93582 0.0734828
$$695$$ −14.4192 −0.546952
$$696$$ 4.02229 0.152464
$$697$$ 10.4456 0.395656
$$698$$ 14.3969 0.544932
$$699$$ −3.19934 −0.121010
$$700$$ −1.91353 −0.0723248
$$701$$ 19.2098 0.725543 0.362771 0.931878i $$-0.381831\pi$$
0.362771 + 0.931878i $$0.381831\pi$$
$$702$$ 3.87939 0.146418
$$703$$ 0 0
$$704$$ 1.87939 0.0708320
$$705$$ 2.41921 0.0911129
$$706$$ 15.7237 0.591769
$$707$$ 6.72874 0.253060
$$708$$ 2.68004 0.100722
$$709$$ 33.0259 1.24031 0.620157 0.784478i $$-0.287069\pi$$
0.620157 + 0.784478i $$0.287069\pi$$
$$710$$ −7.19396 −0.269985
$$711$$ 16.1334 0.605050
$$712$$ 3.55943 0.133395
$$713$$ −13.1239 −0.491494
$$714$$ 0.618555 0.0231489
$$715$$ 8.63816 0.323049
$$716$$ 25.2567 0.943888
$$717$$ 6.17293 0.230532
$$718$$ 15.5175 0.579109
$$719$$ 39.4448 1.47104 0.735521 0.677501i $$-0.236937\pi$$
0.735521 + 0.677501i $$0.236937\pi$$
$$720$$ 1.18479 0.0441546
$$721$$ −1.57667 −0.0587181
$$722$$ 0 0
$$723$$ −1.53714 −0.0571669
$$724$$ 17.3063 0.643185
$$725$$ −14.4652 −0.537225
$$726$$ −7.46791 −0.277160
$$727$$ 18.2517 0.676917 0.338458 0.940981i $$-0.390095\pi$$
0.338458 + 0.940981i $$0.390095\pi$$
$$728$$ 2.06418 0.0765035
$$729$$ 1.00000 0.0370370
$$730$$ −0.388252 −0.0143698
$$731$$ 2.82026 0.104311
$$732$$ −11.3473 −0.419408
$$733$$ 15.3928 0.568546 0.284273 0.958743i $$-0.408248\pi$$
0.284273 + 0.958743i $$0.408248\pi$$
$$734$$ −15.7442 −0.581130
$$735$$ −7.95811 −0.293539
$$736$$ −6.70233 −0.247051
$$737$$ 20.9932 0.773294
$$738$$ 8.98545 0.330759
$$739$$ 15.1771 0.558297 0.279148 0.960248i $$-0.409948\pi$$
0.279148 + 0.960248i $$0.409948\pi$$
$$740$$ −8.15982 −0.299961
$$741$$ 0 0
$$742$$ −6.90167 −0.253368
$$743$$ 9.86659 0.361970 0.180985 0.983486i $$-0.442071\pi$$
0.180985 + 0.983486i $$0.442071\pi$$
$$744$$ 1.95811 0.0717878
$$745$$ −19.3773 −0.709930
$$746$$ 29.6287 1.08478
$$747$$ 11.5740 0.423470
$$748$$ 2.18479 0.0798839
$$749$$ −10.1702 −0.371613
$$750$$ −10.1848 −0.371896
$$751$$ 9.75877 0.356103 0.178051 0.984021i $$-0.443021\pi$$
0.178051 + 0.984021i $$0.443021\pi$$
$$752$$ 2.04189 0.0744600
$$753$$ 17.9436 0.653900
$$754$$ 15.6040 0.568264
$$755$$ −24.3150 −0.884914
$$756$$ 0.532089 0.0193519
$$757$$ −43.5039 −1.58118 −0.790589 0.612348i $$-0.790225\pi$$
−0.790589 + 0.612348i $$0.790225\pi$$
$$758$$ −4.72462 −0.171606
$$759$$ −12.5963 −0.457216
$$760$$ 0 0
$$761$$ 44.7137 1.62087 0.810435 0.585828i $$-0.199231\pi$$
0.810435 + 0.585828i $$0.199231\pi$$
$$762$$ 1.88713 0.0683634
$$763$$ 8.73236 0.316133
$$764$$ −15.1780 −0.549120
$$765$$ 1.37733 0.0497973
$$766$$ 5.24123 0.189373
$$767$$ 10.3969 0.375411
$$768$$ 1.00000 0.0360844
$$769$$ 22.1019 0.797017 0.398508 0.917165i $$-0.369528\pi$$
0.398508 + 0.917165i $$0.369528\pi$$
$$770$$ 1.18479 0.0426970
$$771$$ −29.4124 −1.05926
$$772$$ −13.0077 −0.468159
$$773$$ −42.1789 −1.51707 −0.758535 0.651632i $$-0.774085\pi$$
−0.758535 + 0.651632i $$0.774085\pi$$
$$774$$ 2.42602 0.0872016
$$775$$ −7.04189 −0.252952
$$776$$ −5.87939 −0.211058
$$777$$ −3.66456 −0.131465
$$778$$ −15.0770 −0.540536
$$779$$ 0 0
$$780$$ 4.59627 0.164573
$$781$$ −11.4115 −0.408335
$$782$$ −7.79149 −0.278623
$$783$$ 4.02229 0.143745
$$784$$ −6.71688 −0.239889
$$785$$ −3.79055 −0.135291
$$786$$ 10.7023 0.381740
$$787$$ 28.4793 1.01518 0.507588 0.861600i $$-0.330537\pi$$
0.507588 + 0.861600i $$0.330537\pi$$
$$788$$ −8.94862 −0.318781
$$789$$ 10.6827 0.380315
$$790$$ 19.1147 0.680072
$$791$$ 3.05232 0.108528
$$792$$ 1.87939 0.0667810
$$793$$ −44.0205 −1.56321
$$794$$ −14.5449 −0.516179
$$795$$ −15.3678 −0.545041
$$796$$ −4.50475 −0.159667
$$797$$ −16.6081 −0.588290 −0.294145 0.955761i $$-0.595035\pi$$
−0.294145 + 0.955761i $$0.595035\pi$$
$$798$$ 0 0
$$799$$ 2.37370 0.0839756
$$800$$ −3.59627 −0.127147
$$801$$ 3.55943 0.125766
$$802$$ −2.85473 −0.100804
$$803$$ −0.615867 −0.0217335
$$804$$ 11.1702 0.393944
$$805$$ −4.22525 −0.148921
$$806$$ 7.59627 0.267567
$$807$$ 20.5767 0.724333
$$808$$ 12.6459 0.444881
$$809$$ −13.6783 −0.480903 −0.240452 0.970661i $$-0.577296\pi$$
−0.240452 + 0.970661i $$0.577296\pi$$
$$810$$ 1.18479 0.0416294
$$811$$ 0.618231 0.0217090 0.0108545 0.999941i $$-0.496545\pi$$
0.0108545 + 0.999941i $$0.496545\pi$$
$$812$$ 2.14022 0.0751068
$$813$$ 1.87670 0.0658186
$$814$$ −12.9436 −0.453672
$$815$$ −28.4442 −0.996357
$$816$$ 1.16250 0.0406958
$$817$$ 0 0
$$818$$ −34.2276 −1.19674
$$819$$ 2.06418 0.0721282
$$820$$ 10.6459 0.371771
$$821$$ 6.11205 0.213312 0.106656 0.994296i $$-0.465986\pi$$
0.106656 + 0.994296i $$0.465986\pi$$
$$822$$ −11.2567 −0.392623
$$823$$ 30.6546 1.06855 0.534276 0.845310i $$-0.320584\pi$$
0.534276 + 0.845310i $$0.320584\pi$$
$$824$$ −2.96316 −0.103227
$$825$$ −6.75877 −0.235310
$$826$$ 1.42602 0.0496177
$$827$$ 50.7134 1.76348 0.881738 0.471739i $$-0.156373\pi$$
0.881738 + 0.471739i $$0.156373\pi$$
$$828$$ −6.70233 −0.232922
$$829$$ 13.0000 0.451509 0.225754 0.974184i $$-0.427515\pi$$
0.225754 + 0.974184i $$0.427515\pi$$
$$830$$ 13.7128 0.475977
$$831$$ 4.87258 0.169028
$$832$$ 3.87939 0.134493
$$833$$ −7.80840 −0.270545
$$834$$ −12.1702 −0.421421
$$835$$ −4.33687 −0.150083
$$836$$ 0 0
$$837$$ 1.95811 0.0676822
$$838$$ −7.91891 −0.273554
$$839$$ 21.4979 0.742191 0.371096 0.928595i $$-0.378982\pi$$
0.371096 + 0.928595i $$0.378982\pi$$
$$840$$ 0.630415 0.0217514
$$841$$ −12.8212 −0.442110
$$842$$ −16.2098 −0.558626
$$843$$ 0.403733 0.0139053
$$844$$ 6.40373 0.220426
$$845$$ 2.42839 0.0835390
$$846$$ 2.04189 0.0702016
$$847$$ −3.97359 −0.136534
$$848$$ −12.9709 −0.445423
$$849$$ 17.5895 0.603669
$$850$$ −4.18067 −0.143396
$$851$$ 46.1598 1.58234
$$852$$ −6.07192 −0.208021
$$853$$ −46.2276 −1.58280 −0.791402 0.611296i $$-0.790648\pi$$
−0.791402 + 0.611296i $$0.790648\pi$$
$$854$$ −6.03777 −0.206608
$$855$$ 0 0
$$856$$ −19.1138 −0.653296
$$857$$ 43.6623 1.49148 0.745738 0.666239i $$-0.232097\pi$$
0.745738 + 0.666239i $$0.232097\pi$$
$$858$$ 7.29086 0.248906
$$859$$ 20.1652 0.688027 0.344014 0.938965i $$-0.388213\pi$$
0.344014 + 0.938965i $$0.388213\pi$$
$$860$$ 2.87433 0.0980139
$$861$$ 4.78106 0.162938
$$862$$ −25.4320 −0.866218
$$863$$ 32.6932 1.11289 0.556444 0.830885i $$-0.312165\pi$$
0.556444 + 0.830885i $$0.312165\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 6.31820 0.214825
$$866$$ −11.8479 −0.402609
$$867$$ −15.6486 −0.531454
$$868$$ 1.04189 0.0353640
$$869$$ 30.3209 1.02857
$$870$$ 4.76558 0.161568
$$871$$ 43.3337 1.46831
$$872$$ 16.4115 0.555763
$$873$$ −5.87939 −0.198987
$$874$$ 0 0
$$875$$ −5.41921 −0.183203
$$876$$ −0.327696 −0.0110718
$$877$$ −30.2968 −1.02305 −0.511526 0.859268i $$-0.670920\pi$$
−0.511526 + 0.859268i $$0.670920\pi$$
$$878$$ −25.8598 −0.872725
$$879$$ 13.1753 0.444392
$$880$$ 2.22668 0.0750614
$$881$$ 23.5577 0.793678 0.396839 0.917888i $$-0.370107\pi$$
0.396839 + 0.917888i $$0.370107\pi$$
$$882$$ −6.71688 −0.226169
$$883$$ 36.8289 1.23939 0.619696 0.784842i $$-0.287256\pi$$
0.619696 + 0.784842i $$0.287256\pi$$
$$884$$ 4.50980 0.151681
$$885$$ 3.17530 0.106736
$$886$$ 0.923029 0.0310098
$$887$$ −4.30810 −0.144652 −0.0723258 0.997381i $$-0.523042\pi$$
−0.0723258 + 0.997381i $$0.523042\pi$$
$$888$$ −6.88713 −0.231117
$$889$$ 1.00412 0.0336771
$$890$$ 4.21719 0.141360
$$891$$ 1.87939 0.0629618
$$892$$ −3.24897 −0.108784
$$893$$ 0 0
$$894$$ −16.3550 −0.546994
$$895$$ 29.9240 1.00025
$$896$$ 0.532089 0.0177758
$$897$$ −26.0009 −0.868146
$$898$$ −9.34461 −0.311834
$$899$$ 7.87609 0.262682
$$900$$ −3.59627 −0.119876
$$901$$ −15.0787 −0.502345
$$902$$ 16.8871 0.562280
$$903$$ 1.29086 0.0429571
$$904$$ 5.73648 0.190793
$$905$$ 20.5044 0.681590
$$906$$ −20.5226 −0.681817
$$907$$ −23.3405 −0.775008 −0.387504 0.921868i $$-0.626663\pi$$
−0.387504 + 0.921868i $$0.626663\pi$$
$$908$$ 11.7023 0.388356
$$909$$ 12.6459 0.419438
$$910$$ 2.44562 0.0810716
$$911$$ −22.5631 −0.747547 −0.373774 0.927520i $$-0.621936\pi$$
−0.373774 + 0.927520i $$0.621936\pi$$
$$912$$ 0 0
$$913$$ 21.7520 0.719885
$$914$$ 30.9009 1.02211
$$915$$ −13.4442 −0.444451
$$916$$ −22.9067 −0.756860
$$917$$ 5.69459 0.188052
$$918$$ 1.16250 0.0383683
$$919$$ 4.25578 0.140385 0.0701926 0.997533i $$-0.477639\pi$$
0.0701926 + 0.997533i $$0.477639\pi$$
$$920$$ −7.94087 −0.261803
$$921$$ −16.9017 −0.556929
$$922$$ −29.6878 −0.977715
$$923$$ −23.5553 −0.775333
$$924$$ 1.00000 0.0328976
$$925$$ 24.7679 0.814365
$$926$$ −11.8307 −0.388781
$$927$$ −2.96316 −0.0973231
$$928$$ 4.02229 0.132038
$$929$$ 59.5245 1.95293 0.976467 0.215666i $$-0.0691924\pi$$
0.976467 + 0.215666i $$0.0691924\pi$$
$$930$$ 2.31996 0.0760743
$$931$$ 0 0
$$932$$ −3.19934 −0.104798
$$933$$ −2.38144 −0.0779650
$$934$$ 38.3182 1.25381
$$935$$ 2.58853 0.0846538
$$936$$ 3.87939 0.126802
$$937$$ 33.5090 1.09469 0.547345 0.836907i $$-0.315638\pi$$
0.547345 + 0.836907i $$0.315638\pi$$
$$938$$ 5.94356 0.194064
$$939$$ −29.8631 −0.974545
$$940$$ 2.41921 0.0789061
$$941$$ 5.52435 0.180089 0.0900443 0.995938i $$-0.471299\pi$$
0.0900443 + 0.995938i $$0.471299\pi$$
$$942$$ −3.19934 −0.104240
$$943$$ −60.2235 −1.96115
$$944$$ 2.68004 0.0872280
$$945$$ 0.630415 0.0205074
$$946$$ 4.55943 0.148240
$$947$$ −14.9050 −0.484347 −0.242173 0.970233i $$-0.577860\pi$$
−0.242173 + 0.970233i $$0.577860\pi$$
$$948$$ 16.1334 0.523989
$$949$$ −1.27126 −0.0412668
$$950$$ 0 0
$$951$$ 7.02229 0.227713
$$952$$ 0.618555 0.0200475
$$953$$ 46.3661 1.50194 0.750972 0.660334i $$-0.229585\pi$$
0.750972 + 0.660334i $$0.229585\pi$$
$$954$$ −12.9709 −0.419949
$$955$$ −17.9828 −0.581909
$$956$$ 6.17293 0.199647
$$957$$ 7.55943 0.244362
$$958$$ 37.2841 1.20459
$$959$$ −5.98957 −0.193413
$$960$$ 1.18479 0.0382390
$$961$$ −27.1658 −0.876316
$$962$$ −26.7178 −0.861417
$$963$$ −19.1138 −0.615934
$$964$$ −1.53714 −0.0495080
$$965$$ −15.4115 −0.496113
$$966$$ −3.56624 −0.114742
$$967$$ −23.2431 −0.747448 −0.373724 0.927540i $$-0.621919\pi$$
−0.373724 + 0.927540i $$0.621919\pi$$
$$968$$ −7.46791 −0.240028
$$969$$ 0 0
$$970$$ −6.96585 −0.223660
$$971$$ −4.93045 −0.158226 −0.0791128 0.996866i $$-0.525209\pi$$
−0.0791128 + 0.996866i $$0.525209\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −6.47565 −0.207600
$$974$$ −32.9786 −1.05670
$$975$$ −13.9513 −0.446799
$$976$$ −11.3473 −0.363218
$$977$$ −32.7442 −1.04758 −0.523790 0.851847i $$-0.675482\pi$$
−0.523790 + 0.851847i $$0.675482\pi$$
$$978$$ −24.0077 −0.767683
$$979$$ 6.68954 0.213799
$$980$$ −7.95811 −0.254213
$$981$$ 16.4115 0.523978
$$982$$ 17.6810 0.564223
$$983$$ 25.4406 0.811428 0.405714 0.914000i $$-0.367023\pi$$
0.405714 + 0.914000i $$0.367023\pi$$
$$984$$ 8.98545 0.286446
$$985$$ −10.6023 −0.337816
$$986$$ 4.67593 0.148912
$$987$$ 1.08647 0.0345826
$$988$$ 0 0
$$989$$ −16.2600 −0.517038
$$990$$ 2.22668 0.0707686
$$991$$ −10.1533 −0.322531 −0.161266 0.986911i $$-0.551558\pi$$
−0.161266 + 0.986911i $$0.551558\pi$$
$$992$$ 1.95811 0.0621701
$$993$$ −27.1985 −0.863119
$$994$$ −3.23080 −0.102475
$$995$$ −5.33719 −0.169200
$$996$$ 11.5740 0.366736
$$997$$ 55.7698 1.76625 0.883124 0.469140i $$-0.155436\pi$$
0.883124 + 0.469140i $$0.155436\pi$$
$$998$$ 27.1239 0.858592
$$999$$ −6.88713 −0.217899
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 2166.2.a.u.1.2 3
3.2 odd 2 6498.2.a.bn.1.2 3
19.2 odd 18 114.2.i.d.61.1 yes 6
19.10 odd 18 114.2.i.d.43.1 6
19.18 odd 2 2166.2.a.o.1.2 3
57.2 even 18 342.2.u.a.289.1 6
57.29 even 18 342.2.u.a.271.1 6
57.56 even 2 6498.2.a.bs.1.2 3
76.59 even 18 912.2.bo.f.289.1 6
76.67 even 18 912.2.bo.f.385.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.43.1 6 19.10 odd 18
114.2.i.d.61.1 yes 6 19.2 odd 18
342.2.u.a.271.1 6 57.29 even 18
342.2.u.a.289.1 6 57.2 even 18
912.2.bo.f.289.1 6 76.59 even 18
912.2.bo.f.385.1 6 76.67 even 18
2166.2.a.o.1.2 3 19.18 odd 2
2166.2.a.u.1.2 3 1.1 even 1 trivial
6498.2.a.bn.1.2 3 3.2 odd 2
6498.2.a.bs.1.2 3 57.56 even 2