# Properties

 Label 177.2.a.b.1.2 Level $177$ Weight $2$ Character 177.1 Self dual yes Analytic conductor $1.413$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 177.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.41335211578$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 177.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -2.23607 q^{5} -0.618034 q^{6} -2.38197 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -2.23607 q^{5} -0.618034 q^{6} -2.38197 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.38197 q^{10} +2.23607 q^{11} +1.61803 q^{12} -6.23607 q^{13} -1.47214 q^{14} +2.23607 q^{15} +1.85410 q^{16} +1.85410 q^{17} +0.618034 q^{18} +3.09017 q^{19} +3.61803 q^{20} +2.38197 q^{21} +1.38197 q^{22} -4.61803 q^{23} +2.23607 q^{24} -3.85410 q^{26} -1.00000 q^{27} +3.85410 q^{28} +6.38197 q^{29} +1.38197 q^{30} -10.5623 q^{31} +5.61803 q^{32} -2.23607 q^{33} +1.14590 q^{34} +5.32624 q^{35} -1.61803 q^{36} -0.145898 q^{37} +1.90983 q^{38} +6.23607 q^{39} +5.00000 q^{40} +8.09017 q^{41} +1.47214 q^{42} -8.70820 q^{43} -3.61803 q^{44} -2.23607 q^{45} -2.85410 q^{46} -10.8541 q^{47} -1.85410 q^{48} -1.32624 q^{49} -1.85410 q^{51} +10.0902 q^{52} +6.23607 q^{53} -0.618034 q^{54} -5.00000 q^{55} +5.32624 q^{56} -3.09017 q^{57} +3.94427 q^{58} -1.00000 q^{59} -3.61803 q^{60} -3.14590 q^{61} -6.52786 q^{62} -2.38197 q^{63} -0.236068 q^{64} +13.9443 q^{65} -1.38197 q^{66} +10.7082 q^{67} -3.00000 q^{68} +4.61803 q^{69} +3.29180 q^{70} -7.94427 q^{71} -2.23607 q^{72} +0.854102 q^{73} -0.0901699 q^{74} -5.00000 q^{76} -5.32624 q^{77} +3.85410 q^{78} -3.00000 q^{79} -4.14590 q^{80} +1.00000 q^{81} +5.00000 q^{82} -1.61803 q^{83} -3.85410 q^{84} -4.14590 q^{85} -5.38197 q^{86} -6.38197 q^{87} -5.00000 q^{88} -13.7984 q^{89} -1.38197 q^{90} +14.8541 q^{91} +7.47214 q^{92} +10.5623 q^{93} -6.70820 q^{94} -6.90983 q^{95} -5.61803 q^{96} +3.00000 q^{97} -0.819660 q^{98} +2.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 7q^{7} + 2q^{9} - 5q^{10} + q^{12} - 8q^{13} + 6q^{14} - 3q^{16} - 3q^{17} - q^{18} - 5q^{19} + 5q^{20} + 7q^{21} + 5q^{22} - 7q^{23} - q^{26} - 2q^{27} + q^{28} + 15q^{29} + 5q^{30} - q^{31} + 9q^{32} + 9q^{34} - 5q^{35} - q^{36} - 7q^{37} + 15q^{38} + 8q^{39} + 10q^{40} + 5q^{41} - 6q^{42} - 4q^{43} - 5q^{44} + q^{46} - 15q^{47} + 3q^{48} + 13q^{49} + 3q^{51} + 9q^{52} + 8q^{53} + q^{54} - 10q^{55} - 5q^{56} + 5q^{57} - 10q^{58} - 2q^{59} - 5q^{60} - 13q^{61} - 22q^{62} - 7q^{63} + 4q^{64} + 10q^{65} - 5q^{66} + 8q^{67} - 6q^{68} + 7q^{69} + 20q^{70} + 2q^{71} - 5q^{73} + 11q^{74} - 10q^{76} + 5q^{77} + q^{78} - 6q^{79} - 15q^{80} + 2q^{81} + 10q^{82} - q^{83} - q^{84} - 15q^{85} - 13q^{86} - 15q^{87} - 10q^{88} - 3q^{89} - 5q^{90} + 23q^{91} + 6q^{92} + q^{93} - 25q^{95} - 9q^{96} + 6q^{97} - 24q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034 0.437016 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.61803 −0.809017
$$5$$ −2.23607 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$6$$ −0.618034 −0.252311
$$7$$ −2.38197 −0.900299 −0.450149 0.892953i $$-0.648629\pi$$
−0.450149 + 0.892953i $$0.648629\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 1.00000 0.333333
$$10$$ −1.38197 −0.437016
$$11$$ 2.23607 0.674200 0.337100 0.941469i $$-0.390554\pi$$
0.337100 + 0.941469i $$0.390554\pi$$
$$12$$ 1.61803 0.467086
$$13$$ −6.23607 −1.72957 −0.864787 0.502139i $$-0.832547\pi$$
−0.864787 + 0.502139i $$0.832547\pi$$
$$14$$ −1.47214 −0.393445
$$15$$ 2.23607 0.577350
$$16$$ 1.85410 0.463525
$$17$$ 1.85410 0.449686 0.224843 0.974395i $$-0.427813\pi$$
0.224843 + 0.974395i $$0.427813\pi$$
$$18$$ 0.618034 0.145672
$$19$$ 3.09017 0.708934 0.354467 0.935069i $$-0.384662\pi$$
0.354467 + 0.935069i $$0.384662\pi$$
$$20$$ 3.61803 0.809017
$$21$$ 2.38197 0.519788
$$22$$ 1.38197 0.294636
$$23$$ −4.61803 −0.962927 −0.481463 0.876466i $$-0.659895\pi$$
−0.481463 + 0.876466i $$0.659895\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −3.85410 −0.755852
$$27$$ −1.00000 −0.192450
$$28$$ 3.85410 0.728357
$$29$$ 6.38197 1.18510 0.592551 0.805533i $$-0.298121\pi$$
0.592551 + 0.805533i $$0.298121\pi$$
$$30$$ 1.38197 0.252311
$$31$$ −10.5623 −1.89705 −0.948523 0.316708i $$-0.897422\pi$$
−0.948523 + 0.316708i $$0.897422\pi$$
$$32$$ 5.61803 0.993137
$$33$$ −2.23607 −0.389249
$$34$$ 1.14590 0.196520
$$35$$ 5.32624 0.900299
$$36$$ −1.61803 −0.269672
$$37$$ −0.145898 −0.0239855 −0.0119927 0.999928i $$-0.503818\pi$$
−0.0119927 + 0.999928i $$0.503818\pi$$
$$38$$ 1.90983 0.309815
$$39$$ 6.23607 0.998570
$$40$$ 5.00000 0.790569
$$41$$ 8.09017 1.26347 0.631736 0.775183i $$-0.282343\pi$$
0.631736 + 0.775183i $$0.282343\pi$$
$$42$$ 1.47214 0.227156
$$43$$ −8.70820 −1.32799 −0.663994 0.747738i $$-0.731140\pi$$
−0.663994 + 0.747738i $$0.731140\pi$$
$$44$$ −3.61803 −0.545439
$$45$$ −2.23607 −0.333333
$$46$$ −2.85410 −0.420814
$$47$$ −10.8541 −1.58323 −0.791617 0.611018i $$-0.790760\pi$$
−0.791617 + 0.611018i $$0.790760\pi$$
$$48$$ −1.85410 −0.267617
$$49$$ −1.32624 −0.189463
$$50$$ 0 0
$$51$$ −1.85410 −0.259626
$$52$$ 10.0902 1.39925
$$53$$ 6.23607 0.856590 0.428295 0.903639i $$-0.359114\pi$$
0.428295 + 0.903639i $$0.359114\pi$$
$$54$$ −0.618034 −0.0841038
$$55$$ −5.00000 −0.674200
$$56$$ 5.32624 0.711748
$$57$$ −3.09017 −0.409303
$$58$$ 3.94427 0.517908
$$59$$ −1.00000 −0.130189
$$60$$ −3.61803 −0.467086
$$61$$ −3.14590 −0.402791 −0.201395 0.979510i $$-0.564548\pi$$
−0.201395 + 0.979510i $$0.564548\pi$$
$$62$$ −6.52786 −0.829040
$$63$$ −2.38197 −0.300100
$$64$$ −0.236068 −0.0295085
$$65$$ 13.9443 1.72957
$$66$$ −1.38197 −0.170108
$$67$$ 10.7082 1.30822 0.654108 0.756401i $$-0.273044\pi$$
0.654108 + 0.756401i $$0.273044\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 4.61803 0.555946
$$70$$ 3.29180 0.393445
$$71$$ −7.94427 −0.942812 −0.471406 0.881916i $$-0.656253\pi$$
−0.471406 + 0.881916i $$0.656253\pi$$
$$72$$ −2.23607 −0.263523
$$73$$ 0.854102 0.0999651 0.0499825 0.998750i $$-0.484083\pi$$
0.0499825 + 0.998750i $$0.484083\pi$$
$$74$$ −0.0901699 −0.0104820
$$75$$ 0 0
$$76$$ −5.00000 −0.573539
$$77$$ −5.32624 −0.606981
$$78$$ 3.85410 0.436391
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ −4.14590 −0.463525
$$81$$ 1.00000 0.111111
$$82$$ 5.00000 0.552158
$$83$$ −1.61803 −0.177602 −0.0888012 0.996049i $$-0.528304\pi$$
−0.0888012 + 0.996049i $$0.528304\pi$$
$$84$$ −3.85410 −0.420517
$$85$$ −4.14590 −0.449686
$$86$$ −5.38197 −0.580352
$$87$$ −6.38197 −0.684219
$$88$$ −5.00000 −0.533002
$$89$$ −13.7984 −1.46262 −0.731312 0.682043i $$-0.761092\pi$$
−0.731312 + 0.682043i $$0.761092\pi$$
$$90$$ −1.38197 −0.145672
$$91$$ 14.8541 1.55713
$$92$$ 7.47214 0.779024
$$93$$ 10.5623 1.09526
$$94$$ −6.70820 −0.691898
$$95$$ −6.90983 −0.708934
$$96$$ −5.61803 −0.573388
$$97$$ 3.00000 0.304604 0.152302 0.988334i $$-0.451331\pi$$
0.152302 + 0.988334i $$0.451331\pi$$
$$98$$ −0.819660 −0.0827982
$$99$$ 2.23607 0.224733
$$100$$ 0 0
$$101$$ 3.70820 0.368980 0.184490 0.982834i $$-0.440937\pi$$
0.184490 + 0.982834i $$0.440937\pi$$
$$102$$ −1.14590 −0.113461
$$103$$ −3.23607 −0.318859 −0.159430 0.987209i $$-0.550966\pi$$
−0.159430 + 0.987209i $$0.550966\pi$$
$$104$$ 13.9443 1.36735
$$105$$ −5.32624 −0.519788
$$106$$ 3.85410 0.374343
$$107$$ 0.909830 0.0879566 0.0439783 0.999032i $$-0.485997\pi$$
0.0439783 + 0.999032i $$0.485997\pi$$
$$108$$ 1.61803 0.155695
$$109$$ −4.14590 −0.397105 −0.198553 0.980090i $$-0.563624\pi$$
−0.198553 + 0.980090i $$0.563624\pi$$
$$110$$ −3.09017 −0.294636
$$111$$ 0.145898 0.0138480
$$112$$ −4.41641 −0.417311
$$113$$ −9.00000 −0.846649 −0.423324 0.905978i $$-0.639137\pi$$
−0.423324 + 0.905978i $$0.639137\pi$$
$$114$$ −1.90983 −0.178872
$$115$$ 10.3262 0.962927
$$116$$ −10.3262 −0.958767
$$117$$ −6.23607 −0.576525
$$118$$ −0.618034 −0.0568946
$$119$$ −4.41641 −0.404851
$$120$$ −5.00000 −0.456435
$$121$$ −6.00000 −0.545455
$$122$$ −1.94427 −0.176026
$$123$$ −8.09017 −0.729466
$$124$$ 17.0902 1.53474
$$125$$ 11.1803 1.00000
$$126$$ −1.47214 −0.131148
$$127$$ 15.9443 1.41483 0.707413 0.706801i $$-0.249862\pi$$
0.707413 + 0.706801i $$0.249862\pi$$
$$128$$ −11.3820 −1.00603
$$129$$ 8.70820 0.766715
$$130$$ 8.61803 0.755852
$$131$$ −20.6525 −1.80442 −0.902208 0.431302i $$-0.858054\pi$$
−0.902208 + 0.431302i $$0.858054\pi$$
$$132$$ 3.61803 0.314909
$$133$$ −7.36068 −0.638252
$$134$$ 6.61803 0.571711
$$135$$ 2.23607 0.192450
$$136$$ −4.14590 −0.355508
$$137$$ 12.2361 1.04540 0.522699 0.852517i $$-0.324925\pi$$
0.522699 + 0.852517i $$0.324925\pi$$
$$138$$ 2.85410 0.242957
$$139$$ 11.7639 0.997804 0.498902 0.866658i $$-0.333737\pi$$
0.498902 + 0.866658i $$0.333737\pi$$
$$140$$ −8.61803 −0.728357
$$141$$ 10.8541 0.914080
$$142$$ −4.90983 −0.412024
$$143$$ −13.9443 −1.16608
$$144$$ 1.85410 0.154508
$$145$$ −14.2705 −1.18510
$$146$$ 0.527864 0.0436863
$$147$$ 1.32624 0.109386
$$148$$ 0.236068 0.0194047
$$149$$ 19.0902 1.56393 0.781964 0.623324i $$-0.214218\pi$$
0.781964 + 0.623324i $$0.214218\pi$$
$$150$$ 0 0
$$151$$ 2.56231 0.208517 0.104259 0.994550i $$-0.466753\pi$$
0.104259 + 0.994550i $$0.466753\pi$$
$$152$$ −6.90983 −0.560461
$$153$$ 1.85410 0.149895
$$154$$ −3.29180 −0.265260
$$155$$ 23.6180 1.89705
$$156$$ −10.0902 −0.807860
$$157$$ 9.00000 0.718278 0.359139 0.933284i $$-0.383070\pi$$
0.359139 + 0.933284i $$0.383070\pi$$
$$158$$ −1.85410 −0.147504
$$159$$ −6.23607 −0.494552
$$160$$ −12.5623 −0.993137
$$161$$ 11.0000 0.866921
$$162$$ 0.618034 0.0485573
$$163$$ −18.5623 −1.45391 −0.726956 0.686684i $$-0.759066\pi$$
−0.726956 + 0.686684i $$0.759066\pi$$
$$164$$ −13.0902 −1.02217
$$165$$ 5.00000 0.389249
$$166$$ −1.00000 −0.0776151
$$167$$ 7.03444 0.544341 0.272171 0.962249i $$-0.412258\pi$$
0.272171 + 0.962249i $$0.412258\pi$$
$$168$$ −5.32624 −0.410928
$$169$$ 25.8885 1.99143
$$170$$ −2.56231 −0.196520
$$171$$ 3.09017 0.236311
$$172$$ 14.0902 1.07437
$$173$$ 12.3820 0.941383 0.470692 0.882298i $$-0.344004\pi$$
0.470692 + 0.882298i $$0.344004\pi$$
$$174$$ −3.94427 −0.299014
$$175$$ 0 0
$$176$$ 4.14590 0.312509
$$177$$ 1.00000 0.0751646
$$178$$ −8.52786 −0.639190
$$179$$ −0.527864 −0.0394544 −0.0197272 0.999805i $$-0.506280\pi$$
−0.0197272 + 0.999805i $$0.506280\pi$$
$$180$$ 3.61803 0.269672
$$181$$ 22.2705 1.65535 0.827677 0.561205i $$-0.189662\pi$$
0.827677 + 0.561205i $$0.189662\pi$$
$$182$$ 9.18034 0.680492
$$183$$ 3.14590 0.232551
$$184$$ 10.3262 0.761260
$$185$$ 0.326238 0.0239855
$$186$$ 6.52786 0.478646
$$187$$ 4.14590 0.303178
$$188$$ 17.5623 1.28086
$$189$$ 2.38197 0.173263
$$190$$ −4.27051 −0.309815
$$191$$ 16.4164 1.18785 0.593925 0.804521i $$-0.297578\pi$$
0.593925 + 0.804521i $$0.297578\pi$$
$$192$$ 0.236068 0.0170367
$$193$$ 8.00000 0.575853 0.287926 0.957653i $$-0.407034\pi$$
0.287926 + 0.957653i $$0.407034\pi$$
$$194$$ 1.85410 0.133117
$$195$$ −13.9443 −0.998570
$$196$$ 2.14590 0.153278
$$197$$ −20.6525 −1.47143 −0.735714 0.677292i $$-0.763153\pi$$
−0.735714 + 0.677292i $$0.763153\pi$$
$$198$$ 1.38197 0.0982120
$$199$$ −16.5623 −1.17407 −0.587035 0.809561i $$-0.699705\pi$$
−0.587035 + 0.809561i $$0.699705\pi$$
$$200$$ 0 0
$$201$$ −10.7082 −0.755298
$$202$$ 2.29180 0.161250
$$203$$ −15.2016 −1.06694
$$204$$ 3.00000 0.210042
$$205$$ −18.0902 −1.26347
$$206$$ −2.00000 −0.139347
$$207$$ −4.61803 −0.320976
$$208$$ −11.5623 −0.801702
$$209$$ 6.90983 0.477963
$$210$$ −3.29180 −0.227156
$$211$$ −2.14590 −0.147730 −0.0738649 0.997268i $$-0.523533\pi$$
−0.0738649 + 0.997268i $$0.523533\pi$$
$$212$$ −10.0902 −0.692996
$$213$$ 7.94427 0.544333
$$214$$ 0.562306 0.0384384
$$215$$ 19.4721 1.32799
$$216$$ 2.23607 0.152145
$$217$$ 25.1591 1.70791
$$218$$ −2.56231 −0.173541
$$219$$ −0.854102 −0.0577149
$$220$$ 8.09017 0.545439
$$221$$ −11.5623 −0.777765
$$222$$ 0.0901699 0.00605181
$$223$$ −9.52786 −0.638033 −0.319016 0.947749i $$-0.603353\pi$$
−0.319016 + 0.947749i $$0.603353\pi$$
$$224$$ −13.3820 −0.894120
$$225$$ 0 0
$$226$$ −5.56231 −0.369999
$$227$$ −22.1459 −1.46987 −0.734937 0.678135i $$-0.762789\pi$$
−0.734937 + 0.678135i $$0.762789\pi$$
$$228$$ 5.00000 0.331133
$$229$$ −9.14590 −0.604378 −0.302189 0.953248i $$-0.597717\pi$$
−0.302189 + 0.953248i $$0.597717\pi$$
$$230$$ 6.38197 0.420814
$$231$$ 5.32624 0.350441
$$232$$ −14.2705 −0.936905
$$233$$ −8.29180 −0.543214 −0.271607 0.962408i $$-0.587555\pi$$
−0.271607 + 0.962408i $$0.587555\pi$$
$$234$$ −3.85410 −0.251951
$$235$$ 24.2705 1.58323
$$236$$ 1.61803 0.105325
$$237$$ 3.00000 0.194871
$$238$$ −2.72949 −0.176927
$$239$$ −1.47214 −0.0952246 −0.0476123 0.998866i $$-0.515161\pi$$
−0.0476123 + 0.998866i $$0.515161\pi$$
$$240$$ 4.14590 0.267617
$$241$$ −23.4164 −1.50838 −0.754192 0.656654i $$-0.771971\pi$$
−0.754192 + 0.656654i $$0.771971\pi$$
$$242$$ −3.70820 −0.238372
$$243$$ −1.00000 −0.0641500
$$244$$ 5.09017 0.325865
$$245$$ 2.96556 0.189463
$$246$$ −5.00000 −0.318788
$$247$$ −19.2705 −1.22615
$$248$$ 23.6180 1.49975
$$249$$ 1.61803 0.102539
$$250$$ 6.90983 0.437016
$$251$$ 15.1803 0.958175 0.479087 0.877767i $$-0.340968\pi$$
0.479087 + 0.877767i $$0.340968\pi$$
$$252$$ 3.85410 0.242786
$$253$$ −10.3262 −0.649205
$$254$$ 9.85410 0.618301
$$255$$ 4.14590 0.259626
$$256$$ −6.56231 −0.410144
$$257$$ −19.4164 −1.21116 −0.605581 0.795784i $$-0.707059\pi$$
−0.605581 + 0.795784i $$0.707059\pi$$
$$258$$ 5.38197 0.335067
$$259$$ 0.347524 0.0215941
$$260$$ −22.5623 −1.39925
$$261$$ 6.38197 0.395034
$$262$$ −12.7639 −0.788558
$$263$$ 1.61803 0.0997722 0.0498861 0.998755i $$-0.484114\pi$$
0.0498861 + 0.998755i $$0.484114\pi$$
$$264$$ 5.00000 0.307729
$$265$$ −13.9443 −0.856590
$$266$$ −4.54915 −0.278926
$$267$$ 13.7984 0.844447
$$268$$ −17.3262 −1.05837
$$269$$ −11.4721 −0.699468 −0.349734 0.936849i $$-0.613728\pi$$
−0.349734 + 0.936849i $$0.613728\pi$$
$$270$$ 1.38197 0.0841038
$$271$$ −7.76393 −0.471625 −0.235813 0.971799i $$-0.575775\pi$$
−0.235813 + 0.971799i $$0.575775\pi$$
$$272$$ 3.43769 0.208441
$$273$$ −14.8541 −0.899011
$$274$$ 7.56231 0.456856
$$275$$ 0 0
$$276$$ −7.47214 −0.449770
$$277$$ −5.47214 −0.328789 −0.164394 0.986395i $$-0.552567\pi$$
−0.164394 + 0.986395i $$0.552567\pi$$
$$278$$ 7.27051 0.436056
$$279$$ −10.5623 −0.632349
$$280$$ −11.9098 −0.711748
$$281$$ 9.70820 0.579143 0.289571 0.957156i $$-0.406487\pi$$
0.289571 + 0.957156i $$0.406487\pi$$
$$282$$ 6.70820 0.399468
$$283$$ −23.2705 −1.38329 −0.691644 0.722238i $$-0.743113\pi$$
−0.691644 + 0.722238i $$0.743113\pi$$
$$284$$ 12.8541 0.762751
$$285$$ 6.90983 0.409303
$$286$$ −8.61803 −0.509595
$$287$$ −19.2705 −1.13750
$$288$$ 5.61803 0.331046
$$289$$ −13.5623 −0.797783
$$290$$ −8.81966 −0.517908
$$291$$ −3.00000 −0.175863
$$292$$ −1.38197 −0.0808734
$$293$$ 19.3820 1.13231 0.566153 0.824300i $$-0.308431\pi$$
0.566153 + 0.824300i $$0.308431\pi$$
$$294$$ 0.819660 0.0478035
$$295$$ 2.23607 0.130189
$$296$$ 0.326238 0.0189622
$$297$$ −2.23607 −0.129750
$$298$$ 11.7984 0.683461
$$299$$ 28.7984 1.66545
$$300$$ 0 0
$$301$$ 20.7426 1.19559
$$302$$ 1.58359 0.0911255
$$303$$ −3.70820 −0.213031
$$304$$ 5.72949 0.328609
$$305$$ 7.03444 0.402791
$$306$$ 1.14590 0.0655066
$$307$$ −25.8885 −1.47754 −0.738769 0.673959i $$-0.764592\pi$$
−0.738769 + 0.673959i $$0.764592\pi$$
$$308$$ 8.61803 0.491058
$$309$$ 3.23607 0.184093
$$310$$ 14.5967 0.829040
$$311$$ 27.4508 1.55659 0.778297 0.627896i $$-0.216084\pi$$
0.778297 + 0.627896i $$0.216084\pi$$
$$312$$ −13.9443 −0.789439
$$313$$ −20.7984 −1.17559 −0.587797 0.809009i $$-0.700005\pi$$
−0.587797 + 0.809009i $$0.700005\pi$$
$$314$$ 5.56231 0.313899
$$315$$ 5.32624 0.300100
$$316$$ 4.85410 0.273065
$$317$$ −29.1803 −1.63893 −0.819466 0.573128i $$-0.805730\pi$$
−0.819466 + 0.573128i $$0.805730\pi$$
$$318$$ −3.85410 −0.216127
$$319$$ 14.2705 0.798995
$$320$$ 0.527864 0.0295085
$$321$$ −0.909830 −0.0507818
$$322$$ 6.79837 0.378859
$$323$$ 5.72949 0.318797
$$324$$ −1.61803 −0.0898908
$$325$$ 0 0
$$326$$ −11.4721 −0.635383
$$327$$ 4.14590 0.229269
$$328$$ −18.0902 −0.998863
$$329$$ 25.8541 1.42538
$$330$$ 3.09017 0.170108
$$331$$ −11.1246 −0.611464 −0.305732 0.952118i $$-0.598901\pi$$
−0.305732 + 0.952118i $$0.598901\pi$$
$$332$$ 2.61803 0.143683
$$333$$ −0.145898 −0.00799516
$$334$$ 4.34752 0.237886
$$335$$ −23.9443 −1.30822
$$336$$ 4.41641 0.240935
$$337$$ −14.9098 −0.812190 −0.406095 0.913831i $$-0.633110\pi$$
−0.406095 + 0.913831i $$0.633110\pi$$
$$338$$ 16.0000 0.870285
$$339$$ 9.00000 0.488813
$$340$$ 6.70820 0.363803
$$341$$ −23.6180 −1.27899
$$342$$ 1.90983 0.103272
$$343$$ 19.8328 1.07087
$$344$$ 19.4721 1.04987
$$345$$ −10.3262 −0.555946
$$346$$ 7.65248 0.411400
$$347$$ 31.0344 1.66602 0.833008 0.553261i $$-0.186617\pi$$
0.833008 + 0.553261i $$0.186617\pi$$
$$348$$ 10.3262 0.553544
$$349$$ −27.0000 −1.44528 −0.722638 0.691226i $$-0.757071\pi$$
−0.722638 + 0.691226i $$0.757071\pi$$
$$350$$ 0 0
$$351$$ 6.23607 0.332857
$$352$$ 12.5623 0.669573
$$353$$ −16.0344 −0.853427 −0.426714 0.904387i $$-0.640329\pi$$
−0.426714 + 0.904387i $$0.640329\pi$$
$$354$$ 0.618034 0.0328481
$$355$$ 17.7639 0.942812
$$356$$ 22.3262 1.18329
$$357$$ 4.41641 0.233741
$$358$$ −0.326238 −0.0172422
$$359$$ −21.8885 −1.15523 −0.577617 0.816308i $$-0.696017\pi$$
−0.577617 + 0.816308i $$0.696017\pi$$
$$360$$ 5.00000 0.263523
$$361$$ −9.45085 −0.497413
$$362$$ 13.7639 0.723416
$$363$$ 6.00000 0.314918
$$364$$ −24.0344 −1.25975
$$365$$ −1.90983 −0.0999651
$$366$$ 1.94427 0.101629
$$367$$ 29.8328 1.55726 0.778630 0.627483i $$-0.215915\pi$$
0.778630 + 0.627483i $$0.215915\pi$$
$$368$$ −8.56231 −0.446341
$$369$$ 8.09017 0.421157
$$370$$ 0.201626 0.0104820
$$371$$ −14.8541 −0.771187
$$372$$ −17.0902 −0.886084
$$373$$ 34.3262 1.77735 0.888673 0.458542i $$-0.151628\pi$$
0.888673 + 0.458542i $$0.151628\pi$$
$$374$$ 2.56231 0.132494
$$375$$ −11.1803 −0.577350
$$376$$ 24.2705 1.25166
$$377$$ −39.7984 −2.04972
$$378$$ 1.47214 0.0757185
$$379$$ 22.4164 1.15145 0.575727 0.817642i $$-0.304719\pi$$
0.575727 + 0.817642i $$0.304719\pi$$
$$380$$ 11.1803 0.573539
$$381$$ −15.9443 −0.816850
$$382$$ 10.1459 0.519109
$$383$$ −18.2361 −0.931820 −0.465910 0.884832i $$-0.654273\pi$$
−0.465910 + 0.884832i $$0.654273\pi$$
$$384$$ 11.3820 0.580834
$$385$$ 11.9098 0.606981
$$386$$ 4.94427 0.251657
$$387$$ −8.70820 −0.442663
$$388$$ −4.85410 −0.246430
$$389$$ 32.4721 1.64640 0.823201 0.567750i $$-0.192186\pi$$
0.823201 + 0.567750i $$0.192186\pi$$
$$390$$ −8.61803 −0.436391
$$391$$ −8.56231 −0.433014
$$392$$ 2.96556 0.149783
$$393$$ 20.6525 1.04178
$$394$$ −12.7639 −0.643038
$$395$$ 6.70820 0.337526
$$396$$ −3.61803 −0.181813
$$397$$ 3.00000 0.150566 0.0752828 0.997162i $$-0.476014\pi$$
0.0752828 + 0.997162i $$0.476014\pi$$
$$398$$ −10.2361 −0.513088
$$399$$ 7.36068 0.368495
$$400$$ 0 0
$$401$$ −17.0902 −0.853442 −0.426721 0.904383i $$-0.640331\pi$$
−0.426721 + 0.904383i $$0.640331\pi$$
$$402$$ −6.61803 −0.330078
$$403$$ 65.8673 3.28108
$$404$$ −6.00000 −0.298511
$$405$$ −2.23607 −0.111111
$$406$$ −9.39512 −0.466272
$$407$$ −0.326238 −0.0161710
$$408$$ 4.14590 0.205253
$$409$$ −10.5836 −0.523325 −0.261662 0.965159i $$-0.584271\pi$$
−0.261662 + 0.965159i $$0.584271\pi$$
$$410$$ −11.1803 −0.552158
$$411$$ −12.2361 −0.603561
$$412$$ 5.23607 0.257963
$$413$$ 2.38197 0.117209
$$414$$ −2.85410 −0.140271
$$415$$ 3.61803 0.177602
$$416$$ −35.0344 −1.71770
$$417$$ −11.7639 −0.576082
$$418$$ 4.27051 0.208877
$$419$$ −31.3050 −1.52935 −0.764673 0.644418i $$-0.777100\pi$$
−0.764673 + 0.644418i $$0.777100\pi$$
$$420$$ 8.61803 0.420517
$$421$$ 1.00000 0.0487370 0.0243685 0.999703i $$-0.492242\pi$$
0.0243685 + 0.999703i $$0.492242\pi$$
$$422$$ −1.32624 −0.0645603
$$423$$ −10.8541 −0.527744
$$424$$ −13.9443 −0.677194
$$425$$ 0 0
$$426$$ 4.90983 0.237882
$$427$$ 7.49342 0.362632
$$428$$ −1.47214 −0.0711584
$$429$$ 13.9443 0.673236
$$430$$ 12.0344 0.580352
$$431$$ 12.3820 0.596418 0.298209 0.954501i $$-0.403611\pi$$
0.298209 + 0.954501i $$0.403611\pi$$
$$432$$ −1.85410 −0.0892055
$$433$$ 3.67376 0.176550 0.0882749 0.996096i $$-0.471865\pi$$
0.0882749 + 0.996096i $$0.471865\pi$$
$$434$$ 15.5492 0.746383
$$435$$ 14.2705 0.684219
$$436$$ 6.70820 0.321265
$$437$$ −14.2705 −0.682651
$$438$$ −0.527864 −0.0252223
$$439$$ −34.6180 −1.65223 −0.826114 0.563503i $$-0.809453\pi$$
−0.826114 + 0.563503i $$0.809453\pi$$
$$440$$ 11.1803 0.533002
$$441$$ −1.32624 −0.0631542
$$442$$ −7.14590 −0.339896
$$443$$ 3.38197 0.160682 0.0803410 0.996767i $$-0.474399\pi$$
0.0803410 + 0.996767i $$0.474399\pi$$
$$444$$ −0.236068 −0.0112033
$$445$$ 30.8541 1.46262
$$446$$ −5.88854 −0.278831
$$447$$ −19.0902 −0.902934
$$448$$ 0.562306 0.0265665
$$449$$ 6.88854 0.325090 0.162545 0.986701i $$-0.448030\pi$$
0.162545 + 0.986701i $$0.448030\pi$$
$$450$$ 0 0
$$451$$ 18.0902 0.851833
$$452$$ 14.5623 0.684953
$$453$$ −2.56231 −0.120388
$$454$$ −13.6869 −0.642359
$$455$$ −33.2148 −1.55713
$$456$$ 6.90983 0.323582
$$457$$ 33.6869 1.57581 0.787904 0.615798i $$-0.211166\pi$$
0.787904 + 0.615798i $$0.211166\pi$$
$$458$$ −5.65248 −0.264123
$$459$$ −1.85410 −0.0865421
$$460$$ −16.7082 −0.779024
$$461$$ 23.7426 1.10581 0.552903 0.833246i $$-0.313520\pi$$
0.552903 + 0.833246i $$0.313520\pi$$
$$462$$ 3.29180 0.153148
$$463$$ 4.14590 0.192676 0.0963381 0.995349i $$-0.469287\pi$$
0.0963381 + 0.995349i $$0.469287\pi$$
$$464$$ 11.8328 0.549325
$$465$$ −23.6180 −1.09526
$$466$$ −5.12461 −0.237393
$$467$$ 14.8328 0.686381 0.343190 0.939266i $$-0.388492\pi$$
0.343190 + 0.939266i $$0.388492\pi$$
$$468$$ 10.0902 0.466418
$$469$$ −25.5066 −1.17778
$$470$$ 15.0000 0.691898
$$471$$ −9.00000 −0.414698
$$472$$ 2.23607 0.102923
$$473$$ −19.4721 −0.895330
$$474$$ 1.85410 0.0851617
$$475$$ 0 0
$$476$$ 7.14590 0.327532
$$477$$ 6.23607 0.285530
$$478$$ −0.909830 −0.0416147
$$479$$ −14.9098 −0.681248 −0.340624 0.940200i $$-0.610638\pi$$
−0.340624 + 0.940200i $$0.610638\pi$$
$$480$$ 12.5623 0.573388
$$481$$ 0.909830 0.0414847
$$482$$ −14.4721 −0.659188
$$483$$ −11.0000 −0.500517
$$484$$ 9.70820 0.441282
$$485$$ −6.70820 −0.304604
$$486$$ −0.618034 −0.0280346
$$487$$ −5.74265 −0.260224 −0.130112 0.991499i $$-0.541534\pi$$
−0.130112 + 0.991499i $$0.541534\pi$$
$$488$$ 7.03444 0.318434
$$489$$ 18.5623 0.839416
$$490$$ 1.83282 0.0827982
$$491$$ −11.5066 −0.519285 −0.259642 0.965705i $$-0.583605\pi$$
−0.259642 + 0.965705i $$0.583605\pi$$
$$492$$ 13.0902 0.590150
$$493$$ 11.8328 0.532923
$$494$$ −11.9098 −0.535849
$$495$$ −5.00000 −0.224733
$$496$$ −19.5836 −0.879329
$$497$$ 18.9230 0.848812
$$498$$ 1.00000 0.0448111
$$499$$ −14.4164 −0.645367 −0.322684 0.946507i $$-0.604585\pi$$
−0.322684 + 0.946507i $$0.604585\pi$$
$$500$$ −18.0902 −0.809017
$$501$$ −7.03444 −0.314276
$$502$$ 9.38197 0.418738
$$503$$ 3.79837 0.169361 0.0846806 0.996408i $$-0.473013\pi$$
0.0846806 + 0.996408i $$0.473013\pi$$
$$504$$ 5.32624 0.237249
$$505$$ −8.29180 −0.368980
$$506$$ −6.38197 −0.283713
$$507$$ −25.8885 −1.14975
$$508$$ −25.7984 −1.14462
$$509$$ 34.9230 1.54793 0.773967 0.633226i $$-0.218270\pi$$
0.773967 + 0.633226i $$0.218270\pi$$
$$510$$ 2.56231 0.113461
$$511$$ −2.03444 −0.0899984
$$512$$ 18.7082 0.826794
$$513$$ −3.09017 −0.136434
$$514$$ −12.0000 −0.529297
$$515$$ 7.23607 0.318859
$$516$$ −14.0902 −0.620285
$$517$$ −24.2705 −1.06742
$$518$$ 0.214782 0.00943697
$$519$$ −12.3820 −0.543508
$$520$$ −31.1803 −1.36735
$$521$$ 23.5066 1.02984 0.514921 0.857238i $$-0.327821\pi$$
0.514921 + 0.857238i $$0.327821\pi$$
$$522$$ 3.94427 0.172636
$$523$$ 14.0557 0.614614 0.307307 0.951610i $$-0.400572\pi$$
0.307307 + 0.951610i $$0.400572\pi$$
$$524$$ 33.4164 1.45980
$$525$$ 0 0
$$526$$ 1.00000 0.0436021
$$527$$ −19.5836 −0.853075
$$528$$ −4.14590 −0.180427
$$529$$ −1.67376 −0.0727723
$$530$$ −8.61803 −0.374343
$$531$$ −1.00000 −0.0433963
$$532$$ 11.9098 0.516357
$$533$$ −50.4508 −2.18527
$$534$$ 8.52786 0.369037
$$535$$ −2.03444 −0.0879566
$$536$$ −23.9443 −1.03424
$$537$$ 0.527864 0.0227790
$$538$$ −7.09017 −0.305679
$$539$$ −2.96556 −0.127736
$$540$$ −3.61803 −0.155695
$$541$$ −26.1246 −1.12318 −0.561592 0.827414i $$-0.689811\pi$$
−0.561592 + 0.827414i $$0.689811\pi$$
$$542$$ −4.79837 −0.206108
$$543$$ −22.2705 −0.955719
$$544$$ 10.4164 0.446600
$$545$$ 9.27051 0.397105
$$546$$ −9.18034 −0.392882
$$547$$ 12.4377 0.531797 0.265899 0.964001i $$-0.414331\pi$$
0.265899 + 0.964001i $$0.414331\pi$$
$$548$$ −19.7984 −0.845745
$$549$$ −3.14590 −0.134264
$$550$$ 0 0
$$551$$ 19.7214 0.840158
$$552$$ −10.3262 −0.439514
$$553$$ 7.14590 0.303874
$$554$$ −3.38197 −0.143686
$$555$$ −0.326238 −0.0138480
$$556$$ −19.0344 −0.807240
$$557$$ −25.4164 −1.07693 −0.538464 0.842649i $$-0.680995\pi$$
−0.538464 + 0.842649i $$0.680995\pi$$
$$558$$ −6.52786 −0.276347
$$559$$ 54.3050 2.29685
$$560$$ 9.87539 0.417311
$$561$$ −4.14590 −0.175040
$$562$$ 6.00000 0.253095
$$563$$ −20.5967 −0.868049 −0.434025 0.900901i $$-0.642907\pi$$
−0.434025 + 0.900901i $$0.642907\pi$$
$$564$$ −17.5623 −0.739506
$$565$$ 20.1246 0.846649
$$566$$ −14.3820 −0.604519
$$567$$ −2.38197 −0.100033
$$568$$ 17.7639 0.745358
$$569$$ −11.5066 −0.482381 −0.241190 0.970478i $$-0.577538\pi$$
−0.241190 + 0.970478i $$0.577538\pi$$
$$570$$ 4.27051 0.178872
$$571$$ 1.58359 0.0662713 0.0331356 0.999451i $$-0.489451\pi$$
0.0331356 + 0.999451i $$0.489451\pi$$
$$572$$ 22.5623 0.943377
$$573$$ −16.4164 −0.685805
$$574$$ −11.9098 −0.497107
$$575$$ 0 0
$$576$$ −0.236068 −0.00983617
$$577$$ 12.5279 0.521542 0.260771 0.965401i $$-0.416023\pi$$
0.260771 + 0.965401i $$0.416023\pi$$
$$578$$ −8.38197 −0.348644
$$579$$ −8.00000 −0.332469
$$580$$ 23.0902 0.958767
$$581$$ 3.85410 0.159895
$$582$$ −1.85410 −0.0768550
$$583$$ 13.9443 0.577513
$$584$$ −1.90983 −0.0790293
$$585$$ 13.9443 0.576525
$$586$$ 11.9787 0.494836
$$587$$ −15.3607 −0.634003 −0.317002 0.948425i $$-0.602676\pi$$
−0.317002 + 0.948425i $$0.602676\pi$$
$$588$$ −2.14590 −0.0884953
$$589$$ −32.6393 −1.34488
$$590$$ 1.38197 0.0568946
$$591$$ 20.6525 0.849529
$$592$$ −0.270510 −0.0111179
$$593$$ 16.0902 0.660744 0.330372 0.943851i $$-0.392826\pi$$
0.330372 + 0.943851i $$0.392826\pi$$
$$594$$ −1.38197 −0.0567028
$$595$$ 9.87539 0.404851
$$596$$ −30.8885 −1.26524
$$597$$ 16.5623 0.677850
$$598$$ 17.7984 0.727830
$$599$$ 2.65248 0.108377 0.0541886 0.998531i $$-0.482743\pi$$
0.0541886 + 0.998531i $$0.482743\pi$$
$$600$$ 0 0
$$601$$ 23.3820 0.953770 0.476885 0.878966i $$-0.341766\pi$$
0.476885 + 0.878966i $$0.341766\pi$$
$$602$$ 12.8197 0.522490
$$603$$ 10.7082 0.436072
$$604$$ −4.14590 −0.168694
$$605$$ 13.4164 0.545455
$$606$$ −2.29180 −0.0930979
$$607$$ −13.6525 −0.554137 −0.277068 0.960850i $$-0.589363\pi$$
−0.277068 + 0.960850i $$0.589363\pi$$
$$608$$ 17.3607 0.704069
$$609$$ 15.2016 0.616001
$$610$$ 4.34752 0.176026
$$611$$ 67.6869 2.73832
$$612$$ −3.00000 −0.121268
$$613$$ −38.6525 −1.56116 −0.780579 0.625057i $$-0.785076\pi$$
−0.780579 + 0.625057i $$0.785076\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 18.0902 0.729466
$$616$$ 11.9098 0.479861
$$617$$ 22.1459 0.891560 0.445780 0.895142i $$-0.352926\pi$$
0.445780 + 0.895142i $$0.352926\pi$$
$$618$$ 2.00000 0.0804518
$$619$$ 12.1246 0.487329 0.243665 0.969860i $$-0.421650\pi$$
0.243665 + 0.969860i $$0.421650\pi$$
$$620$$ −38.2148 −1.53474
$$621$$ 4.61803 0.185315
$$622$$ 16.9656 0.680257
$$623$$ 32.8673 1.31680
$$624$$ 11.5623 0.462863
$$625$$ −25.0000 −1.00000
$$626$$ −12.8541 −0.513753
$$627$$ −6.90983 −0.275952
$$628$$ −14.5623 −0.581099
$$629$$ −0.270510 −0.0107859
$$630$$ 3.29180 0.131148
$$631$$ −40.4164 −1.60895 −0.804476 0.593985i $$-0.797554\pi$$
−0.804476 + 0.593985i $$0.797554\pi$$
$$632$$ 6.70820 0.266838
$$633$$ 2.14590 0.0852918
$$634$$ −18.0344 −0.716239
$$635$$ −35.6525 −1.41483
$$636$$ 10.0902 0.400101
$$637$$ 8.27051 0.327690
$$638$$ 8.81966 0.349174
$$639$$ −7.94427 −0.314271
$$640$$ 25.4508 1.00603
$$641$$ −27.9443 −1.10373 −0.551866 0.833933i $$-0.686084\pi$$
−0.551866 + 0.833933i $$0.686084\pi$$
$$642$$ −0.562306 −0.0221924
$$643$$ −19.8541 −0.782969 −0.391485 0.920185i $$-0.628038\pi$$
−0.391485 + 0.920185i $$0.628038\pi$$
$$644$$ −17.7984 −0.701354
$$645$$ −19.4721 −0.766715
$$646$$ 3.54102 0.139320
$$647$$ 44.9443 1.76694 0.883471 0.468486i $$-0.155200\pi$$
0.883471 + 0.468486i $$0.155200\pi$$
$$648$$ −2.23607 −0.0878410
$$649$$ −2.23607 −0.0877733
$$650$$ 0 0
$$651$$ −25.1591 −0.986061
$$652$$ 30.0344 1.17624
$$653$$ 40.0902 1.56885 0.784425 0.620224i $$-0.212958\pi$$
0.784425 + 0.620224i $$0.212958\pi$$
$$654$$ 2.56231 0.100194
$$655$$ 46.1803 1.80442
$$656$$ 15.0000 0.585652
$$657$$ 0.854102 0.0333217
$$658$$ 15.9787 0.622915
$$659$$ 1.85410 0.0722256 0.0361128 0.999348i $$-0.488502\pi$$
0.0361128 + 0.999348i $$0.488502\pi$$
$$660$$ −8.09017 −0.314909
$$661$$ 30.7426 1.19575 0.597875 0.801589i $$-0.296012\pi$$
0.597875 + 0.801589i $$0.296012\pi$$
$$662$$ −6.87539 −0.267220
$$663$$ 11.5623 0.449043
$$664$$ 3.61803 0.140407
$$665$$ 16.4590 0.638252
$$666$$ −0.0901699 −0.00349401
$$667$$ −29.4721 −1.14117
$$668$$ −11.3820 −0.440381
$$669$$ 9.52786 0.368369
$$670$$ −14.7984 −0.571711
$$671$$ −7.03444 −0.271562
$$672$$ 13.3820 0.516221
$$673$$ −8.47214 −0.326577 −0.163288 0.986578i $$-0.552210\pi$$
−0.163288 + 0.986578i $$0.552210\pi$$
$$674$$ −9.21478 −0.354940
$$675$$ 0 0
$$676$$ −41.8885 −1.61110
$$677$$ −27.8197 −1.06920 −0.534598 0.845106i $$-0.679537\pi$$
−0.534598 + 0.845106i $$0.679537\pi$$
$$678$$ 5.56231 0.213619
$$679$$ −7.14590 −0.274234
$$680$$ 9.27051 0.355508
$$681$$ 22.1459 0.848633
$$682$$ −14.5967 −0.558938
$$683$$ 39.0000 1.49229 0.746147 0.665782i $$-0.231902\pi$$
0.746147 + 0.665782i $$0.231902\pi$$
$$684$$ −5.00000 −0.191180
$$685$$ −27.3607 −1.04540
$$686$$ 12.2574 0.467988
$$687$$ 9.14590 0.348938
$$688$$ −16.1459 −0.615557
$$689$$ −38.8885 −1.48154
$$690$$ −6.38197 −0.242957
$$691$$ 35.1246 1.33620 0.668102 0.744070i $$-0.267107\pi$$
0.668102 + 0.744070i $$0.267107\pi$$
$$692$$ −20.0344 −0.761595
$$693$$ −5.32624 −0.202327
$$694$$ 19.1803 0.728076
$$695$$ −26.3050 −0.997804
$$696$$ 14.2705 0.540922
$$697$$ 15.0000 0.568166
$$698$$ −16.6869 −0.631609
$$699$$ 8.29180 0.313625
$$700$$ 0 0
$$701$$ 18.2016 0.687466 0.343733 0.939067i $$-0.388308\pi$$
0.343733 + 0.939067i $$0.388308\pi$$
$$702$$ 3.85410 0.145464
$$703$$ −0.450850 −0.0170041
$$704$$ −0.527864 −0.0198946
$$705$$ −24.2705 −0.914080
$$706$$ −9.90983 −0.372961
$$707$$ −8.83282 −0.332192
$$708$$ −1.61803 −0.0608094
$$709$$ −27.7082 −1.04060 −0.520302 0.853983i $$-0.674181\pi$$
−0.520302 + 0.853983i $$0.674181\pi$$
$$710$$ 10.9787 0.412024
$$711$$ −3.00000 −0.112509
$$712$$ 30.8541 1.15631
$$713$$ 48.7771 1.82672
$$714$$ 2.72949 0.102149
$$715$$ 31.1803 1.16608
$$716$$ 0.854102 0.0319193
$$717$$ 1.47214 0.0549779
$$718$$ −13.5279 −0.504855
$$719$$ 14.6180 0.545161 0.272580 0.962133i $$-0.412123\pi$$
0.272580 + 0.962133i $$0.412123\pi$$
$$720$$ −4.14590 −0.154508
$$721$$ 7.70820 0.287069
$$722$$ −5.84095 −0.217378
$$723$$ 23.4164 0.870866
$$724$$ −36.0344 −1.33921
$$725$$ 0 0
$$726$$ 3.70820 0.137624
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ −33.2148 −1.23102
$$729$$ 1.00000 0.0370370
$$730$$ −1.18034 −0.0436863
$$731$$ −16.1459 −0.597178
$$732$$ −5.09017 −0.188138
$$733$$ 20.5279 0.758214 0.379107 0.925353i $$-0.376231\pi$$
0.379107 + 0.925353i $$0.376231\pi$$
$$734$$ 18.4377 0.680548
$$735$$ −2.96556 −0.109386
$$736$$ −25.9443 −0.956319
$$737$$ 23.9443 0.881999
$$738$$ 5.00000 0.184053
$$739$$ 48.1033 1.76951 0.884755 0.466057i $$-0.154326\pi$$
0.884755 + 0.466057i $$0.154326\pi$$
$$740$$ −0.527864 −0.0194047
$$741$$ 19.2705 0.707920
$$742$$ −9.18034 −0.337021
$$743$$ −24.6180 −0.903148 −0.451574 0.892234i $$-0.649137\pi$$
−0.451574 + 0.892234i $$0.649137\pi$$
$$744$$ −23.6180 −0.865879
$$745$$ −42.6869 −1.56393
$$746$$ 21.2148 0.776728
$$747$$ −1.61803 −0.0592008
$$748$$ −6.70820 −0.245276
$$749$$ −2.16718 −0.0791872
$$750$$ −6.90983 −0.252311
$$751$$ −4.87539 −0.177905 −0.0889527 0.996036i $$-0.528352\pi$$
−0.0889527 + 0.996036i $$0.528352\pi$$
$$752$$ −20.1246 −0.733869
$$753$$ −15.1803 −0.553202
$$754$$ −24.5967 −0.895761
$$755$$ −5.72949 −0.208517
$$756$$ −3.85410 −0.140172
$$757$$ −41.3820 −1.50405 −0.752027 0.659133i $$-0.770924\pi$$
−0.752027 + 0.659133i $$0.770924\pi$$
$$758$$ 13.8541 0.503204
$$759$$ 10.3262 0.374819
$$760$$ 15.4508 0.560461
$$761$$ 30.7082 1.11317 0.556586 0.830790i $$-0.312111\pi$$
0.556586 + 0.830790i $$0.312111\pi$$
$$762$$ −9.85410 −0.356976
$$763$$ 9.87539 0.357513
$$764$$ −26.5623 −0.960991
$$765$$ −4.14590 −0.149895
$$766$$ −11.2705 −0.407220
$$767$$ 6.23607 0.225171
$$768$$ 6.56231 0.236797
$$769$$ 10.9787 0.395903 0.197951 0.980212i $$-0.436571\pi$$
0.197951 + 0.980212i $$0.436571\pi$$
$$770$$ 7.36068 0.265260
$$771$$ 19.4164 0.699265
$$772$$ −12.9443 −0.465875
$$773$$ 28.6525 1.03056 0.515279 0.857023i $$-0.327688\pi$$
0.515279 + 0.857023i $$0.327688\pi$$
$$774$$ −5.38197 −0.193451
$$775$$ 0 0
$$776$$ −6.70820 −0.240810
$$777$$ −0.347524 −0.0124674
$$778$$ 20.0689 0.719504
$$779$$ 25.0000 0.895718
$$780$$ 22.5623 0.807860
$$781$$ −17.7639 −0.635643
$$782$$ −5.29180 −0.189234
$$783$$ −6.38197 −0.228073
$$784$$ −2.45898 −0.0878207
$$785$$ −20.1246 −0.718278
$$786$$ 12.7639 0.455274
$$787$$ −4.29180 −0.152986 −0.0764930 0.997070i $$-0.524372\pi$$
−0.0764930 + 0.997070i $$0.524372\pi$$
$$788$$ 33.4164 1.19041
$$789$$ −1.61803 −0.0576035
$$790$$ 4.14590 0.147504
$$791$$ 21.4377 0.762237
$$792$$ −5.00000 −0.177667
$$793$$ 19.6180 0.696657
$$794$$ 1.85410 0.0657996
$$795$$ 13.9443 0.494552
$$796$$ 26.7984 0.949843
$$797$$ −1.23607 −0.0437838 −0.0218919 0.999760i $$-0.506969\pi$$
−0.0218919 + 0.999760i $$0.506969\pi$$
$$798$$ 4.54915 0.161038
$$799$$ −20.1246 −0.711958
$$800$$ 0 0
$$801$$ −13.7984 −0.487542
$$802$$ −10.5623 −0.372968
$$803$$ 1.90983 0.0673964
$$804$$ 17.3262 0.611049
$$805$$ −24.5967 −0.866921
$$806$$ 40.7082 1.43389
$$807$$ 11.4721 0.403838
$$808$$ −8.29180 −0.291704
$$809$$ 7.67376 0.269795 0.134898 0.990860i $$-0.456929\pi$$
0.134898 + 0.990860i $$0.456929\pi$$
$$810$$ −1.38197 −0.0485573
$$811$$ 14.6525 0.514518 0.257259 0.966342i $$-0.417181\pi$$
0.257259 + 0.966342i $$0.417181\pi$$
$$812$$ 24.5967 0.863177
$$813$$ 7.76393 0.272293
$$814$$ −0.201626 −0.00706699
$$815$$ 41.5066 1.45391
$$816$$ −3.43769 −0.120343
$$817$$ −26.9098 −0.941456
$$818$$ −6.54102 −0.228701
$$819$$ 14.8541 0.519044
$$820$$ 29.2705 1.02217
$$821$$ −40.9230 −1.42822 −0.714111 0.700032i $$-0.753169\pi$$
−0.714111 + 0.700032i $$0.753169\pi$$
$$822$$ −7.56231 −0.263766
$$823$$ 11.2918 0.393607 0.196804 0.980443i $$-0.436944\pi$$
0.196804 + 0.980443i $$0.436944\pi$$
$$824$$ 7.23607 0.252080
$$825$$ 0 0
$$826$$ 1.47214 0.0512222
$$827$$ −2.81966 −0.0980492 −0.0490246 0.998798i $$-0.515611\pi$$
−0.0490246 + 0.998798i $$0.515611\pi$$
$$828$$ 7.47214 0.259675
$$829$$ −45.6869 −1.58677 −0.793386 0.608719i $$-0.791684\pi$$
−0.793386 + 0.608719i $$0.791684\pi$$
$$830$$ 2.23607 0.0776151
$$831$$ 5.47214 0.189826
$$832$$ 1.47214 0.0510371
$$833$$ −2.45898 −0.0851986
$$834$$ −7.27051 −0.251757
$$835$$ −15.7295 −0.544341
$$836$$ −11.1803 −0.386680
$$837$$ 10.5623 0.365087
$$838$$ −19.3475 −0.668349
$$839$$ −24.8197 −0.856870 −0.428435 0.903573i $$-0.640935\pi$$
−0.428435 + 0.903573i $$0.640935\pi$$
$$840$$ 11.9098 0.410928
$$841$$ 11.7295 0.404465
$$842$$ 0.618034 0.0212989
$$843$$ −9.70820 −0.334368
$$844$$ 3.47214 0.119516
$$845$$ −57.8885 −1.99143
$$846$$ −6.70820 −0.230633
$$847$$ 14.2918 0.491072
$$848$$ 11.5623 0.397051
$$849$$ 23.2705 0.798642
$$850$$ 0 0
$$851$$ 0.673762 0.0230963
$$852$$ −12.8541 −0.440374
$$853$$ −3.96556 −0.135778 −0.0678891 0.997693i $$-0.521626\pi$$
−0.0678891 + 0.997693i $$0.521626\pi$$
$$854$$ 4.63119 0.158476
$$855$$ −6.90983 −0.236311
$$856$$ −2.03444 −0.0695358
$$857$$ −28.7771 −0.983007 −0.491503 0.870876i $$-0.663552\pi$$
−0.491503 + 0.870876i $$0.663552\pi$$
$$858$$ 8.61803 0.294215
$$859$$ 15.5279 0.529804 0.264902 0.964275i $$-0.414660\pi$$
0.264902 + 0.964275i $$0.414660\pi$$
$$860$$ −31.5066 −1.07437
$$861$$ 19.2705 0.656737
$$862$$ 7.65248 0.260644
$$863$$ 19.2016 0.653631 0.326815 0.945088i $$-0.394024\pi$$
0.326815 + 0.945088i $$0.394024\pi$$
$$864$$ −5.61803 −0.191129
$$865$$ −27.6869 −0.941383
$$866$$ 2.27051 0.0771551
$$867$$ 13.5623 0.460600
$$868$$ −40.7082 −1.38173
$$869$$ −6.70820 −0.227560
$$870$$ 8.81966 0.299014
$$871$$ −66.7771 −2.26266
$$872$$ 9.27051 0.313939
$$873$$ 3.00000 0.101535
$$874$$ −8.81966 −0.298329
$$875$$ −26.6312 −0.900299
$$876$$ 1.38197 0.0466923
$$877$$ −16.1115 −0.544045 −0.272023 0.962291i $$-0.587693\pi$$
−0.272023 + 0.962291i $$0.587693\pi$$
$$878$$ −21.3951 −0.722050
$$879$$ −19.3820 −0.653737
$$880$$ −9.27051 −0.312509
$$881$$ −15.7082 −0.529223 −0.264611 0.964355i $$-0.585244\pi$$
−0.264611 + 0.964355i $$0.585244\pi$$
$$882$$ −0.819660 −0.0275994
$$883$$ −23.4164 −0.788025 −0.394012 0.919105i $$-0.628913\pi$$
−0.394012 + 0.919105i $$0.628913\pi$$
$$884$$ 18.7082 0.629225
$$885$$ −2.23607 −0.0751646
$$886$$ 2.09017 0.0702206
$$887$$ −22.5279 −0.756412 −0.378206 0.925722i $$-0.623459\pi$$
−0.378206 + 0.925722i $$0.623459\pi$$
$$888$$ −0.326238 −0.0109478
$$889$$ −37.9787 −1.27377
$$890$$ 19.0689 0.639190
$$891$$ 2.23607 0.0749111
$$892$$ 15.4164 0.516180
$$893$$ −33.5410 −1.12241
$$894$$ −11.7984 −0.394597
$$895$$ 1.18034 0.0394544
$$896$$ 27.1115 0.905730
$$897$$ −28.7984 −0.961550
$$898$$ 4.25735 0.142070
$$899$$ −67.4083 −2.24819
$$900$$ 0 0
$$901$$ 11.5623 0.385196
$$902$$ 11.1803 0.372265
$$903$$ −20.7426 −0.690272
$$904$$ 20.1246 0.669335
$$905$$ −49.7984 −1.65535
$$906$$ −1.58359 −0.0526113
$$907$$ −19.9443 −0.662239 −0.331119 0.943589i $$-0.607426\pi$$
−0.331119 + 0.943589i $$0.607426\pi$$
$$908$$ 35.8328 1.18915
$$909$$ 3.70820 0.122993
$$910$$ −20.5279 −0.680492
$$911$$ −45.1033 −1.49434 −0.747170 0.664633i $$-0.768588\pi$$
−0.747170 + 0.664633i $$0.768588\pi$$
$$912$$ −5.72949 −0.189722
$$913$$ −3.61803 −0.119739
$$914$$ 20.8197 0.688653
$$915$$ −7.03444 −0.232551
$$916$$ 14.7984 0.488952
$$917$$ 49.1935 1.62451
$$918$$ −1.14590 −0.0378203
$$919$$ −13.5967 −0.448515 −0.224258 0.974530i $$-0.571996\pi$$
−0.224258 + 0.974530i $$0.571996\pi$$
$$920$$ −23.0902 −0.761260
$$921$$ 25.8885 0.853057
$$922$$ 14.6738 0.483255
$$923$$ 49.5410 1.63066
$$924$$ −8.61803 −0.283513
$$925$$ 0 0
$$926$$ 2.56231 0.0842026
$$927$$ −3.23607 −0.106286
$$928$$ 35.8541 1.17697
$$929$$ −48.7082 −1.59806 −0.799032 0.601288i $$-0.794654\pi$$
−0.799032 + 0.601288i $$0.794654\pi$$
$$930$$ −14.5967 −0.478646
$$931$$ −4.09830 −0.134316
$$932$$ 13.4164 0.439469
$$933$$ −27.4508 −0.898700
$$934$$ 9.16718 0.299959
$$935$$ −9.27051 −0.303178
$$936$$ 13.9443 0.455783
$$937$$ 51.7214 1.68966 0.844832 0.535032i $$-0.179701\pi$$
0.844832 + 0.535032i $$0.179701\pi$$
$$938$$ −15.7639 −0.514711
$$939$$ 20.7984 0.678729
$$940$$ −39.2705 −1.28086
$$941$$ 40.3050 1.31390 0.656952 0.753932i $$-0.271845\pi$$
0.656952 + 0.753932i $$0.271845\pi$$
$$942$$ −5.56231 −0.181230
$$943$$ −37.3607 −1.21663
$$944$$ −1.85410 −0.0603459
$$945$$ −5.32624 −0.173263
$$946$$ −12.0344 −0.391273
$$947$$ 25.0344 0.813510 0.406755 0.913537i $$-0.366660\pi$$
0.406755 + 0.913537i $$0.366660\pi$$
$$948$$ −4.85410 −0.157654
$$949$$ −5.32624 −0.172897
$$950$$ 0 0
$$951$$ 29.1803 0.946237
$$952$$ 9.87539 0.320063
$$953$$ 4.94427 0.160161 0.0800803 0.996788i $$-0.474482\pi$$
0.0800803 + 0.996788i $$0.474482\pi$$
$$954$$ 3.85410 0.124781
$$955$$ −36.7082 −1.18785
$$956$$ 2.38197 0.0770383
$$957$$ −14.2705 −0.461300
$$958$$ −9.21478 −0.297716
$$959$$ −29.1459 −0.941170
$$960$$ −0.527864 −0.0170367
$$961$$ 80.5623 2.59878
$$962$$ 0.562306 0.0181295
$$963$$ 0.909830 0.0293189
$$964$$ 37.8885 1.22031
$$965$$ −17.8885 −0.575853
$$966$$ −6.79837 −0.218734
$$967$$ 15.2705 0.491066 0.245533 0.969388i $$-0.421037\pi$$
0.245533 + 0.969388i $$0.421037\pi$$
$$968$$ 13.4164 0.431220
$$969$$ −5.72949 −0.184058
$$970$$ −4.14590 −0.133117
$$971$$ 46.7984 1.50183 0.750916 0.660398i $$-0.229612\pi$$
0.750916 + 0.660398i $$0.229612\pi$$
$$972$$ 1.61803 0.0518985
$$973$$ −28.0213 −0.898321
$$974$$ −3.54915 −0.113722
$$975$$ 0 0
$$976$$ −5.83282 −0.186704
$$977$$ −20.8885 −0.668284 −0.334142 0.942523i $$-0.608446\pi$$
−0.334142 + 0.942523i $$0.608446\pi$$
$$978$$ 11.4721 0.366838
$$979$$ −30.8541 −0.986101
$$980$$ −4.79837 −0.153278
$$981$$ −4.14590 −0.132368
$$982$$ −7.11146 −0.226936
$$983$$ −9.40325 −0.299917 −0.149959 0.988692i $$-0.547914\pi$$
−0.149959 + 0.988692i $$0.547914\pi$$
$$984$$ 18.0902 0.576694
$$985$$ 46.1803 1.47143
$$986$$ 7.31308 0.232896
$$987$$ −25.8541 −0.822945
$$988$$ 31.1803 0.991979
$$989$$ 40.2148 1.27876
$$990$$ −3.09017 −0.0982120
$$991$$ 23.7426 0.754210 0.377105 0.926171i $$-0.376920\pi$$
0.377105 + 0.926171i $$0.376920\pi$$
$$992$$ −59.3394 −1.88403
$$993$$ 11.1246 0.353029
$$994$$ 11.6950 0.370944
$$995$$ 37.0344 1.17407
$$996$$ −2.61803 −0.0829556
$$997$$ 53.2148 1.68533 0.842665 0.538439i $$-0.180986\pi$$
0.842665 + 0.538439i $$0.180986\pi$$
$$998$$ −8.90983 −0.282036
$$999$$ 0.145898 0.00461601
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 177.2.a.b.1.2 2
3.2 odd 2 531.2.a.b.1.1 2
4.3 odd 2 2832.2.a.o.1.1 2
5.4 even 2 4425.2.a.t.1.1 2
7.6 odd 2 8673.2.a.k.1.2 2
12.11 even 2 8496.2.a.bb.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
177.2.a.b.1.2 2 1.1 even 1 trivial
531.2.a.b.1.1 2 3.2 odd 2
2832.2.a.o.1.1 2 4.3 odd 2
4425.2.a.t.1.1 2 5.4 even 2
8496.2.a.bb.1.2 2 12.11 even 2
8673.2.a.k.1.2 2 7.6 odd 2