# Properties

 Label 8470.2.a.ci.1.2 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.34292$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.14637 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.68585 q^{9} -1.00000 q^{10} +1.14637 q^{12} -4.68585 q^{13} -1.00000 q^{14} +1.14637 q^{15} +1.00000 q^{16} +0.292731 q^{17} +1.68585 q^{18} -6.51806 q^{19} +1.00000 q^{20} +1.14637 q^{21} +2.85363 q^{23} -1.14637 q^{24} +1.00000 q^{25} +4.68585 q^{26} -5.37169 q^{27} +1.00000 q^{28} +1.43910 q^{29} -1.14637 q^{30} +0.978577 q^{31} -1.00000 q^{32} -0.292731 q^{34} +1.00000 q^{35} -1.68585 q^{36} +0.853635 q^{37} +6.51806 q^{38} -5.37169 q^{39} -1.00000 q^{40} +6.22533 q^{41} -1.14637 q^{42} +10.3503 q^{43} -1.68585 q^{45} -2.85363 q^{46} -9.95715 q^{47} +1.14637 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.335577 q^{51} -4.68585 q^{52} +5.43910 q^{53} +5.37169 q^{54} -1.00000 q^{56} -7.47208 q^{57} -1.43910 q^{58} -9.37169 q^{59} +1.14637 q^{60} +11.9572 q^{61} -0.978577 q^{62} -1.68585 q^{63} +1.00000 q^{64} -4.68585 q^{65} -0.585462 q^{67} +0.292731 q^{68} +3.27131 q^{69} -1.00000 q^{70} -0.335577 q^{71} +1.68585 q^{72} +3.70727 q^{73} -0.853635 q^{74} +1.14637 q^{75} -6.51806 q^{76} +5.37169 q^{78} +2.51806 q^{79} +1.00000 q^{80} -1.10038 q^{81} -6.22533 q^{82} +1.70727 q^{83} +1.14637 q^{84} +0.292731 q^{85} -10.3503 q^{86} +1.64973 q^{87} +13.0790 q^{89} +1.68585 q^{90} -4.68585 q^{91} +2.85363 q^{92} +1.12181 q^{93} +9.95715 q^{94} -6.51806 q^{95} -1.14637 q^{96} +9.10352 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} - 2q^{6} + 3q^{7} - 3q^{8} + 7q^{9} + O(q^{10})$$ $$3q - 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} - 2q^{6} + 3q^{7} - 3q^{8} + 7q^{9} - 3q^{10} + 2q^{12} - 2q^{13} - 3q^{14} + 2q^{15} + 3q^{16} - 2q^{17} - 7q^{18} + 6q^{19} + 3q^{20} + 2q^{21} + 10q^{23} - 2q^{24} + 3q^{25} + 2q^{26} + 8q^{27} + 3q^{28} - 2q^{30} - 12q^{31} - 3q^{32} + 2q^{34} + 3q^{35} + 7q^{36} + 4q^{37} - 6q^{38} + 8q^{39} - 3q^{40} - 4q^{41} - 2q^{42} - 8q^{43} + 7q^{45} - 10q^{46} + 2q^{48} + 3q^{49} - 3q^{50} + 28q^{51} - 2q^{52} + 12q^{53} - 8q^{54} - 3q^{56} + 8q^{57} - 4q^{59} + 2q^{60} + 6q^{61} + 12q^{62} + 7q^{63} + 3q^{64} - 2q^{65} + 4q^{67} - 2q^{68} - 8q^{69} - 3q^{70} - 28q^{71} - 7q^{72} + 14q^{73} - 4q^{74} + 2q^{75} + 6q^{76} - 8q^{78} - 18q^{79} + 3q^{80} + 3q^{81} + 4q^{82} + 8q^{83} + 2q^{84} - 2q^{85} + 8q^{86} + 44q^{87} + 18q^{89} - 7q^{90} - 2q^{91} + 10q^{92} + 12q^{93} + 6q^{95} - 2q^{96} - 4q^{97} - 3q^{98} + O(q^{100})$$

## 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
$$3$$ 1.14637 0.661854 0.330927 0.943656i $$-0.392639\pi$$
0.330927 + 0.943656i $$0.392639\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −1.14637 −0.468002
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −1.68585 −0.561949
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 1.14637 0.330927
$$13$$ −4.68585 −1.29962 −0.649810 0.760097i $$-0.725152\pi$$
−0.649810 + 0.760097i $$0.725152\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 1.14637 0.295990
$$16$$ 1.00000 0.250000
$$17$$ 0.292731 0.0709977 0.0354988 0.999370i $$-0.488698\pi$$
0.0354988 + 0.999370i $$0.488698\pi$$
$$18$$ 1.68585 0.397358
$$19$$ −6.51806 −1.49535 −0.747673 0.664068i $$-0.768829\pi$$
−0.747673 + 0.664068i $$0.768829\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 1.14637 0.250157
$$22$$ 0 0
$$23$$ 2.85363 0.595024 0.297512 0.954718i $$-0.403843\pi$$
0.297512 + 0.954718i $$0.403843\pi$$
$$24$$ −1.14637 −0.234001
$$25$$ 1.00000 0.200000
$$26$$ 4.68585 0.918970
$$27$$ −5.37169 −1.03378
$$28$$ 1.00000 0.188982
$$29$$ 1.43910 0.267234 0.133617 0.991033i $$-0.457341\pi$$
0.133617 + 0.991033i $$0.457341\pi$$
$$30$$ −1.14637 −0.209297
$$31$$ 0.978577 0.175758 0.0878788 0.996131i $$-0.471991\pi$$
0.0878788 + 0.996131i $$0.471991\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −0.292731 −0.0502029
$$35$$ 1.00000 0.169031
$$36$$ −1.68585 −0.280974
$$37$$ 0.853635 0.140337 0.0701683 0.997535i $$-0.477646\pi$$
0.0701683 + 0.997535i $$0.477646\pi$$
$$38$$ 6.51806 1.05737
$$39$$ −5.37169 −0.860159
$$40$$ −1.00000 −0.158114
$$41$$ 6.22533 0.972233 0.486116 0.873894i $$-0.338413\pi$$
0.486116 + 0.873894i $$0.338413\pi$$
$$42$$ −1.14637 −0.176888
$$43$$ 10.3503 1.57840 0.789201 0.614135i $$-0.210495\pi$$
0.789201 + 0.614135i $$0.210495\pi$$
$$44$$ 0 0
$$45$$ −1.68585 −0.251311
$$46$$ −2.85363 −0.420745
$$47$$ −9.95715 −1.45240 −0.726200 0.687483i $$-0.758715\pi$$
−0.726200 + 0.687483i $$0.758715\pi$$
$$48$$ 1.14637 0.165464
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 0.335577 0.0469901
$$52$$ −4.68585 −0.649810
$$53$$ 5.43910 0.747117 0.373559 0.927607i $$-0.378137\pi$$
0.373559 + 0.927607i $$0.378137\pi$$
$$54$$ 5.37169 0.730995
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −7.47208 −0.989701
$$58$$ −1.43910 −0.188963
$$59$$ −9.37169 −1.22009 −0.610045 0.792367i $$-0.708849\pi$$
−0.610045 + 0.792367i $$0.708849\pi$$
$$60$$ 1.14637 0.147995
$$61$$ 11.9572 1.53096 0.765478 0.643462i $$-0.222502\pi$$
0.765478 + 0.643462i $$0.222502\pi$$
$$62$$ −0.978577 −0.124279
$$63$$ −1.68585 −0.212397
$$64$$ 1.00000 0.125000
$$65$$ −4.68585 −0.581208
$$66$$ 0 0
$$67$$ −0.585462 −0.0715256 −0.0357628 0.999360i $$-0.511386\pi$$
−0.0357628 + 0.999360i $$0.511386\pi$$
$$68$$ 0.292731 0.0354988
$$69$$ 3.27131 0.393819
$$70$$ −1.00000 −0.119523
$$71$$ −0.335577 −0.0398256 −0.0199128 0.999802i $$-0.506339\pi$$
−0.0199128 + 0.999802i $$0.506339\pi$$
$$72$$ 1.68585 0.198679
$$73$$ 3.70727 0.433903 0.216952 0.976182i $$-0.430389\pi$$
0.216952 + 0.976182i $$0.430389\pi$$
$$74$$ −0.853635 −0.0992330
$$75$$ 1.14637 0.132371
$$76$$ −6.51806 −0.747673
$$77$$ 0 0
$$78$$ 5.37169 0.608224
$$79$$ 2.51806 0.283304 0.141652 0.989917i $$-0.454759\pi$$
0.141652 + 0.989917i $$0.454759\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −1.10038 −0.122265
$$82$$ −6.22533 −0.687472
$$83$$ 1.70727 0.187397 0.0936986 0.995601i $$-0.470131\pi$$
0.0936986 + 0.995601i $$0.470131\pi$$
$$84$$ 1.14637 0.125079
$$85$$ 0.292731 0.0317511
$$86$$ −10.3503 −1.11610
$$87$$ 1.64973 0.176870
$$88$$ 0 0
$$89$$ 13.0790 1.38637 0.693184 0.720761i $$-0.256208\pi$$
0.693184 + 0.720761i $$0.256208\pi$$
$$90$$ 1.68585 0.177704
$$91$$ −4.68585 −0.491210
$$92$$ 2.85363 0.297512
$$93$$ 1.12181 0.116326
$$94$$ 9.95715 1.02700
$$95$$ −6.51806 −0.668739
$$96$$ −1.14637 −0.117000
$$97$$ 9.10352 0.924322 0.462161 0.886796i $$-0.347074\pi$$
0.462161 + 0.886796i $$0.347074\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 10.7862 1.07327 0.536635 0.843814i $$-0.319695\pi$$
0.536635 + 0.843814i $$0.319695\pi$$
$$102$$ −0.335577 −0.0332270
$$103$$ 7.66442 0.755198 0.377599 0.925969i $$-0.376750\pi$$
0.377599 + 0.925969i $$0.376750\pi$$
$$104$$ 4.68585 0.459485
$$105$$ 1.14637 0.111874
$$106$$ −5.43910 −0.528292
$$107$$ −9.56404 −0.924591 −0.462295 0.886726i $$-0.652974\pi$$
−0.462295 + 0.886726i $$0.652974\pi$$
$$108$$ −5.37169 −0.516891
$$109$$ −3.14637 −0.301367 −0.150684 0.988582i $$-0.548147\pi$$
−0.150684 + 0.988582i $$0.548147\pi$$
$$110$$ 0 0
$$111$$ 0.978577 0.0928824
$$112$$ 1.00000 0.0944911
$$113$$ −2.58546 −0.243220 −0.121610 0.992578i $$-0.538806\pi$$
−0.121610 + 0.992578i $$0.538806\pi$$
$$114$$ 7.47208 0.699824
$$115$$ 2.85363 0.266103
$$116$$ 1.43910 0.133617
$$117$$ 7.89962 0.730320
$$118$$ 9.37169 0.862734
$$119$$ 0.292731 0.0268346
$$120$$ −1.14637 −0.104648
$$121$$ 0 0
$$122$$ −11.9572 −1.08255
$$123$$ 7.13650 0.643477
$$124$$ 0.978577 0.0878788
$$125$$ 1.00000 0.0894427
$$126$$ 1.68585 0.150187
$$127$$ 13.3717 1.18655 0.593273 0.805001i $$-0.297836\pi$$
0.593273 + 0.805001i $$0.297836\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 11.8652 1.04467
$$130$$ 4.68585 0.410976
$$131$$ 11.4391 0.999438 0.499719 0.866187i $$-0.333437\pi$$
0.499719 + 0.866187i $$0.333437\pi$$
$$132$$ 0 0
$$133$$ −6.51806 −0.565187
$$134$$ 0.585462 0.0505762
$$135$$ −5.37169 −0.462322
$$136$$ −0.292731 −0.0251015
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ −3.27131 −0.278472
$$139$$ 14.1825 1.20294 0.601471 0.798895i $$-0.294581\pi$$
0.601471 + 0.798895i $$0.294581\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ −11.4145 −0.961278
$$142$$ 0.335577 0.0281610
$$143$$ 0 0
$$144$$ −1.68585 −0.140487
$$145$$ 1.43910 0.119510
$$146$$ −3.70727 −0.306816
$$147$$ 1.14637 0.0945506
$$148$$ 0.853635 0.0701683
$$149$$ −11.5970 −0.950065 −0.475032 0.879968i $$-0.657564\pi$$
−0.475032 + 0.879968i $$0.657564\pi$$
$$150$$ −1.14637 −0.0936004
$$151$$ −1.73183 −0.140934 −0.0704671 0.997514i $$-0.522449\pi$$
−0.0704671 + 0.997514i $$0.522449\pi$$
$$152$$ 6.51806 0.528684
$$153$$ −0.493499 −0.0398971
$$154$$ 0 0
$$155$$ 0.978577 0.0786012
$$156$$ −5.37169 −0.430080
$$157$$ −12.2927 −0.981067 −0.490533 0.871422i $$-0.663198\pi$$
−0.490533 + 0.871422i $$0.663198\pi$$
$$158$$ −2.51806 −0.200326
$$159$$ 6.23519 0.494483
$$160$$ −1.00000 −0.0790569
$$161$$ 2.85363 0.224898
$$162$$ 1.10038 0.0864543
$$163$$ 23.9143 1.87311 0.936557 0.350516i $$-0.113994\pi$$
0.936557 + 0.350516i $$0.113994\pi$$
$$164$$ 6.22533 0.486116
$$165$$ 0 0
$$166$$ −1.70727 −0.132510
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ −1.14637 −0.0884440
$$169$$ 8.95715 0.689012
$$170$$ −0.292731 −0.0224514
$$171$$ 10.9884 0.840307
$$172$$ 10.3503 0.789201
$$173$$ 7.37169 0.560459 0.280230 0.959933i $$-0.409589\pi$$
0.280230 + 0.959933i $$0.409589\pi$$
$$174$$ −1.64973 −0.125066
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −10.7434 −0.807522
$$178$$ −13.0790 −0.980310
$$179$$ 1.56404 0.116902 0.0584509 0.998290i $$-0.481384\pi$$
0.0584509 + 0.998290i $$0.481384\pi$$
$$180$$ −1.68585 −0.125656
$$181$$ 14.7862 1.09905 0.549526 0.835477i $$-0.314808\pi$$
0.549526 + 0.835477i $$0.314808\pi$$
$$182$$ 4.68585 0.347338
$$183$$ 13.7073 1.01327
$$184$$ −2.85363 −0.210373
$$185$$ 0.853635 0.0627605
$$186$$ −1.12181 −0.0822549
$$187$$ 0 0
$$188$$ −9.95715 −0.726200
$$189$$ −5.37169 −0.390733
$$190$$ 6.51806 0.472870
$$191$$ −17.9572 −1.29933 −0.649667 0.760219i $$-0.725092\pi$$
−0.649667 + 0.760219i $$0.725092\pi$$
$$192$$ 1.14637 0.0827318
$$193$$ −18.6430 −1.34195 −0.670976 0.741479i $$-0.734125\pi$$
−0.670976 + 0.741479i $$0.734125\pi$$
$$194$$ −9.10352 −0.653595
$$195$$ −5.37169 −0.384675
$$196$$ 1.00000 0.0714286
$$197$$ 15.0361 1.07128 0.535639 0.844447i $$-0.320071\pi$$
0.535639 + 0.844447i $$0.320071\pi$$
$$198$$ 0 0
$$199$$ 15.3288 1.08663 0.543317 0.839528i $$-0.317168\pi$$
0.543317 + 0.839528i $$0.317168\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −0.671153 −0.0473395
$$202$$ −10.7862 −0.758917
$$203$$ 1.43910 0.101005
$$204$$ 0.335577 0.0234951
$$205$$ 6.22533 0.434796
$$206$$ −7.66442 −0.534006
$$207$$ −4.81079 −0.334373
$$208$$ −4.68585 −0.324905
$$209$$ 0 0
$$210$$ −1.14637 −0.0791067
$$211$$ −2.04285 −0.140635 −0.0703176 0.997525i $$-0.522401\pi$$
−0.0703176 + 0.997525i $$0.522401\pi$$
$$212$$ 5.43910 0.373559
$$213$$ −0.384694 −0.0263588
$$214$$ 9.56404 0.653784
$$215$$ 10.3503 0.705883
$$216$$ 5.37169 0.365497
$$217$$ 0.978577 0.0664301
$$218$$ 3.14637 0.213099
$$219$$ 4.24989 0.287181
$$220$$ 0 0
$$221$$ −1.37169 −0.0922700
$$222$$ −0.978577 −0.0656778
$$223$$ 11.0790 0.741902 0.370951 0.928652i $$-0.379032\pi$$
0.370951 + 0.928652i $$0.379032\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −1.68585 −0.112390
$$226$$ 2.58546 0.171982
$$227$$ 18.5426 1.23072 0.615358 0.788248i $$-0.289011\pi$$
0.615358 + 0.788248i $$0.289011\pi$$
$$228$$ −7.47208 −0.494850
$$229$$ −8.01469 −0.529626 −0.264813 0.964300i $$-0.585310\pi$$
−0.264813 + 0.964300i $$0.585310\pi$$
$$230$$ −2.85363 −0.188163
$$231$$ 0 0
$$232$$ −1.43910 −0.0944813
$$233$$ −27.9572 −1.83153 −0.915767 0.401710i $$-0.868416\pi$$
−0.915767 + 0.401710i $$0.868416\pi$$
$$234$$ −7.89962 −0.516414
$$235$$ −9.95715 −0.649533
$$236$$ −9.37169 −0.610045
$$237$$ 2.88661 0.187506
$$238$$ −0.292731 −0.0189749
$$239$$ 28.8108 1.86362 0.931808 0.362953i $$-0.118231\pi$$
0.931808 + 0.362953i $$0.118231\pi$$
$$240$$ 1.14637 0.0739976
$$241$$ 8.51806 0.548696 0.274348 0.961630i $$-0.411538\pi$$
0.274348 + 0.961630i $$0.411538\pi$$
$$242$$ 0 0
$$243$$ 14.8536 0.952861
$$244$$ 11.9572 0.765478
$$245$$ 1.00000 0.0638877
$$246$$ −7.13650 −0.455007
$$247$$ 30.5426 1.94338
$$248$$ −0.978577 −0.0621397
$$249$$ 1.95715 0.124030
$$250$$ −1.00000 −0.0632456
$$251$$ −23.1281 −1.45983 −0.729916 0.683537i $$-0.760441\pi$$
−0.729916 + 0.683537i $$0.760441\pi$$
$$252$$ −1.68585 −0.106198
$$253$$ 0 0
$$254$$ −13.3717 −0.839015
$$255$$ 0.335577 0.0210146
$$256$$ 1.00000 0.0625000
$$257$$ 14.8108 0.923872 0.461936 0.886913i $$-0.347155\pi$$
0.461936 + 0.886913i $$0.347155\pi$$
$$258$$ −11.8652 −0.738695
$$259$$ 0.853635 0.0530423
$$260$$ −4.68585 −0.290604
$$261$$ −2.42610 −0.150172
$$262$$ −11.4391 −0.706710
$$263$$ −18.7434 −1.15577 −0.577883 0.816119i $$-0.696121\pi$$
−0.577883 + 0.816119i $$0.696121\pi$$
$$264$$ 0 0
$$265$$ 5.43910 0.334121
$$266$$ 6.51806 0.399648
$$267$$ 14.9933 0.917573
$$268$$ −0.585462 −0.0357628
$$269$$ 12.6858 0.773470 0.386735 0.922191i $$-0.373603\pi$$
0.386735 + 0.922191i $$0.373603\pi$$
$$270$$ 5.37169 0.326911
$$271$$ −1.12181 −0.0681449 −0.0340725 0.999419i $$-0.510848\pi$$
−0.0340725 + 0.999419i $$0.510848\pi$$
$$272$$ 0.292731 0.0177494
$$273$$ −5.37169 −0.325110
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ 3.27131 0.196910
$$277$$ −30.1151 −1.80944 −0.904720 0.426007i $$-0.859920\pi$$
−0.904720 + 0.426007i $$0.859920\pi$$
$$278$$ −14.1825 −0.850609
$$279$$ −1.64973 −0.0987668
$$280$$ −1.00000 −0.0597614
$$281$$ 14.3356 0.855189 0.427594 0.903971i $$-0.359361\pi$$
0.427594 + 0.903971i $$0.359361\pi$$
$$282$$ 11.4145 0.679726
$$283$$ −3.21377 −0.191039 −0.0955194 0.995428i $$-0.530451\pi$$
−0.0955194 + 0.995428i $$0.530451\pi$$
$$284$$ −0.335577 −0.0199128
$$285$$ −7.47208 −0.442608
$$286$$ 0 0
$$287$$ 6.22533 0.367469
$$288$$ 1.68585 0.0993394
$$289$$ −16.9143 −0.994959
$$290$$ −1.43910 −0.0845067
$$291$$ 10.4360 0.611767
$$292$$ 3.70727 0.216952
$$293$$ 16.6858 0.974798 0.487399 0.873179i $$-0.337946\pi$$
0.487399 + 0.873179i $$0.337946\pi$$
$$294$$ −1.14637 −0.0668574
$$295$$ −9.37169 −0.545641
$$296$$ −0.853635 −0.0496165
$$297$$ 0 0
$$298$$ 11.5970 0.671797
$$299$$ −13.3717 −0.773305
$$300$$ 1.14637 0.0661854
$$301$$ 10.3503 0.596580
$$302$$ 1.73183 0.0996555
$$303$$ 12.3650 0.710349
$$304$$ −6.51806 −0.373836
$$305$$ 11.9572 0.684665
$$306$$ 0.493499 0.0282115
$$307$$ 6.29273 0.359145 0.179573 0.983745i $$-0.442529\pi$$
0.179573 + 0.983745i $$0.442529\pi$$
$$308$$ 0 0
$$309$$ 8.78623 0.499831
$$310$$ −0.978577 −0.0555794
$$311$$ −20.3931 −1.15639 −0.578194 0.815900i $$-0.696242\pi$$
−0.578194 + 0.815900i $$0.696242\pi$$
$$312$$ 5.37169 0.304112
$$313$$ 12.0674 0.682090 0.341045 0.940047i $$-0.389219\pi$$
0.341045 + 0.940047i $$0.389219\pi$$
$$314$$ 12.2927 0.693719
$$315$$ −1.68585 −0.0949867
$$316$$ 2.51806 0.141652
$$317$$ 28.7679 1.61577 0.807884 0.589341i $$-0.200613\pi$$
0.807884 + 0.589341i $$0.200613\pi$$
$$318$$ −6.23519 −0.349652
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ −10.9639 −0.611944
$$322$$ −2.85363 −0.159027
$$323$$ −1.90804 −0.106166
$$324$$ −1.10038 −0.0611325
$$325$$ −4.68585 −0.259924
$$326$$ −23.9143 −1.32449
$$327$$ −3.60688 −0.199461
$$328$$ −6.22533 −0.343736
$$329$$ −9.95715 −0.548956
$$330$$ 0 0
$$331$$ −3.80765 −0.209288 −0.104644 0.994510i $$-0.533370\pi$$
−0.104644 + 0.994510i $$0.533370\pi$$
$$332$$ 1.70727 0.0936986
$$333$$ −1.43910 −0.0788620
$$334$$ −8.00000 −0.437741
$$335$$ −0.585462 −0.0319872
$$336$$ 1.14637 0.0625394
$$337$$ −15.3717 −0.837349 −0.418675 0.908136i $$-0.637505\pi$$
−0.418675 + 0.908136i $$0.637505\pi$$
$$338$$ −8.95715 −0.487205
$$339$$ −2.96388 −0.160976
$$340$$ 0.292731 0.0158756
$$341$$ 0 0
$$342$$ −10.9884 −0.594187
$$343$$ 1.00000 0.0539949
$$344$$ −10.3503 −0.558049
$$345$$ 3.27131 0.176121
$$346$$ −7.37169 −0.396305
$$347$$ −3.02142 −0.162198 −0.0810992 0.996706i $$-0.525843\pi$$
−0.0810992 + 0.996706i $$0.525843\pi$$
$$348$$ 1.64973 0.0884348
$$349$$ −11.0361 −0.590750 −0.295375 0.955381i $$-0.595445\pi$$
−0.295375 + 0.955381i $$0.595445\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 25.1709 1.34352
$$352$$ 0 0
$$353$$ 23.9817 1.27642 0.638209 0.769863i $$-0.279676\pi$$
0.638209 + 0.769863i $$0.279676\pi$$
$$354$$ 10.7434 0.571004
$$355$$ −0.335577 −0.0178106
$$356$$ 13.0790 0.693184
$$357$$ 0.335577 0.0177606
$$358$$ −1.56404 −0.0826620
$$359$$ 10.5181 0.555122 0.277561 0.960708i $$-0.410474\pi$$
0.277561 + 0.960708i $$0.410474\pi$$
$$360$$ 1.68585 0.0888519
$$361$$ 23.4851 1.23606
$$362$$ −14.7862 −0.777147
$$363$$ 0 0
$$364$$ −4.68585 −0.245605
$$365$$ 3.70727 0.194047
$$366$$ −13.7073 −0.716490
$$367$$ −35.5787 −1.85719 −0.928597 0.371089i $$-0.878985\pi$$
−0.928597 + 0.371089i $$0.878985\pi$$
$$368$$ 2.85363 0.148756
$$369$$ −10.4949 −0.546345
$$370$$ −0.853635 −0.0443783
$$371$$ 5.43910 0.282384
$$372$$ 1.12181 0.0581630
$$373$$ −7.50650 −0.388672 −0.194336 0.980935i $$-0.562255\pi$$
−0.194336 + 0.980935i $$0.562255\pi$$
$$374$$ 0 0
$$375$$ 1.14637 0.0591981
$$376$$ 9.95715 0.513501
$$377$$ −6.74338 −0.347302
$$378$$ 5.37169 0.276290
$$379$$ 33.4868 1.72010 0.860050 0.510210i $$-0.170432\pi$$
0.860050 + 0.510210i $$0.170432\pi$$
$$380$$ −6.51806 −0.334369
$$381$$ 15.3288 0.785321
$$382$$ 17.9572 0.918768
$$383$$ 15.6644 0.800415 0.400207 0.916425i $$-0.368938\pi$$
0.400207 + 0.916425i $$0.368938\pi$$
$$384$$ −1.14637 −0.0585002
$$385$$ 0 0
$$386$$ 18.6430 0.948904
$$387$$ −17.4490 −0.886981
$$388$$ 9.10352 0.462161
$$389$$ −19.6216 −0.994853 −0.497427 0.867506i $$-0.665722\pi$$
−0.497427 + 0.867506i $$0.665722\pi$$
$$390$$ 5.37169 0.272006
$$391$$ 0.835347 0.0422453
$$392$$ −1.00000 −0.0505076
$$393$$ 13.1134 0.661483
$$394$$ −15.0361 −0.757509
$$395$$ 2.51806 0.126697
$$396$$ 0 0
$$397$$ 10.2499 0.514427 0.257213 0.966355i $$-0.417196\pi$$
0.257213 + 0.966355i $$0.417196\pi$$
$$398$$ −15.3288 −0.768366
$$399$$ −7.47208 −0.374072
$$400$$ 1.00000 0.0500000
$$401$$ −16.8866 −0.843277 −0.421639 0.906764i $$-0.638545\pi$$
−0.421639 + 0.906764i $$0.638545\pi$$
$$402$$ 0.671153 0.0334741
$$403$$ −4.58546 −0.228418
$$404$$ 10.7862 0.536635
$$405$$ −1.10038 −0.0546785
$$406$$ −1.43910 −0.0714212
$$407$$ 0 0
$$408$$ −0.335577 −0.0166135
$$409$$ −18.2744 −0.903613 −0.451807 0.892116i $$-0.649220\pi$$
−0.451807 + 0.892116i $$0.649220\pi$$
$$410$$ −6.22533 −0.307447
$$411$$ 11.4637 0.565460
$$412$$ 7.66442 0.377599
$$413$$ −9.37169 −0.461151
$$414$$ 4.81079 0.236437
$$415$$ 1.70727 0.0838065
$$416$$ 4.68585 0.229743
$$417$$ 16.2583 0.796173
$$418$$ 0 0
$$419$$ −37.2860 −1.82154 −0.910770 0.412914i $$-0.864511\pi$$
−0.910770 + 0.412914i $$0.864511\pi$$
$$420$$ 1.14637 0.0559369
$$421$$ −2.78623 −0.135793 −0.0678963 0.997692i $$-0.521629\pi$$
−0.0678963 + 0.997692i $$0.521629\pi$$
$$422$$ 2.04285 0.0994442
$$423$$ 16.7862 0.816174
$$424$$ −5.43910 −0.264146
$$425$$ 0.292731 0.0141995
$$426$$ 0.384694 0.0186385
$$427$$ 11.9572 0.578647
$$428$$ −9.56404 −0.462295
$$429$$ 0 0
$$430$$ −10.3503 −0.499134
$$431$$ −38.0477 −1.83269 −0.916346 0.400387i $$-0.868876\pi$$
−0.916346 + 0.400387i $$0.868876\pi$$
$$432$$ −5.37169 −0.258446
$$433$$ −10.8108 −0.519533 −0.259767 0.965671i $$-0.583646\pi$$
−0.259767 + 0.965671i $$0.583646\pi$$
$$434$$ −0.978577 −0.0469732
$$435$$ 1.64973 0.0790985
$$436$$ −3.14637 −0.150684
$$437$$ −18.6002 −0.889766
$$438$$ −4.24989 −0.203067
$$439$$ −30.9933 −1.47923 −0.739614 0.673031i $$-0.764992\pi$$
−0.739614 + 0.673031i $$0.764992\pi$$
$$440$$ 0 0
$$441$$ −1.68585 −0.0802784
$$442$$ 1.37169 0.0652448
$$443$$ 25.3717 1.20545 0.602723 0.797951i $$-0.294083\pi$$
0.602723 + 0.797951i $$0.294083\pi$$
$$444$$ 0.978577 0.0464412
$$445$$ 13.0790 0.620002
$$446$$ −11.0790 −0.524604
$$447$$ −13.2944 −0.628805
$$448$$ 1.00000 0.0472456
$$449$$ −0.886615 −0.0418419 −0.0209210 0.999781i $$-0.506660\pi$$
−0.0209210 + 0.999781i $$0.506660\pi$$
$$450$$ 1.68585 0.0794716
$$451$$ 0 0
$$452$$ −2.58546 −0.121610
$$453$$ −1.98531 −0.0932779
$$454$$ −18.5426 −0.870248
$$455$$ −4.68585 −0.219676
$$456$$ 7.47208 0.349912
$$457$$ −29.9143 −1.39933 −0.699666 0.714470i $$-0.746668\pi$$
−0.699666 + 0.714470i $$0.746668\pi$$
$$458$$ 8.01469 0.374502
$$459$$ −1.57246 −0.0733962
$$460$$ 2.85363 0.133051
$$461$$ 2.33558 0.108779 0.0543893 0.998520i $$-0.482679\pi$$
0.0543893 + 0.998520i $$0.482679\pi$$
$$462$$ 0 0
$$463$$ 0.110250 0.00512374 0.00256187 0.999997i $$-0.499185\pi$$
0.00256187 + 0.999997i $$0.499185\pi$$
$$464$$ 1.43910 0.0668084
$$465$$ 1.12181 0.0520226
$$466$$ 27.9572 1.29509
$$467$$ 36.6760 1.69716 0.848581 0.529066i $$-0.177457\pi$$
0.848581 + 0.529066i $$0.177457\pi$$
$$468$$ 7.89962 0.365160
$$469$$ −0.585462 −0.0270341
$$470$$ 9.95715 0.459289
$$471$$ −14.0920 −0.649323
$$472$$ 9.37169 0.431367
$$473$$ 0 0
$$474$$ −2.88661 −0.132587
$$475$$ −6.51806 −0.299069
$$476$$ 0.292731 0.0134173
$$477$$ −9.16948 −0.419842
$$478$$ −28.8108 −1.31777
$$479$$ 42.7434 1.95300 0.976498 0.215529i $$-0.0691474\pi$$
0.976498 + 0.215529i $$0.0691474\pi$$
$$480$$ −1.14637 −0.0523242
$$481$$ −4.00000 −0.182384
$$482$$ −8.51806 −0.387987
$$483$$ 3.27131 0.148850
$$484$$ 0 0
$$485$$ 9.10352 0.413370
$$486$$ −14.8536 −0.673775
$$487$$ −36.1396 −1.63764 −0.818822 0.574048i $$-0.805372\pi$$
−0.818822 + 0.574048i $$0.805372\pi$$
$$488$$ −11.9572 −0.541275
$$489$$ 27.4145 1.23973
$$490$$ −1.00000 −0.0451754
$$491$$ −2.54262 −0.114747 −0.0573733 0.998353i $$-0.518273\pi$$
−0.0573733 + 0.998353i $$0.518273\pi$$
$$492$$ 7.13650 0.321738
$$493$$ 0.421268 0.0189730
$$494$$ −30.5426 −1.37418
$$495$$ 0 0
$$496$$ 0.978577 0.0439394
$$497$$ −0.335577 −0.0150527
$$498$$ −1.95715 −0.0877022
$$499$$ 3.13650 0.140409 0.0702045 0.997533i $$-0.477635\pi$$
0.0702045 + 0.997533i $$0.477635\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 9.17092 0.409727
$$502$$ 23.1281 1.03226
$$503$$ 28.5855 1.27456 0.637281 0.770631i $$-0.280059\pi$$
0.637281 + 0.770631i $$0.280059\pi$$
$$504$$ 1.68585 0.0750936
$$505$$ 10.7862 0.479981
$$506$$ 0 0
$$507$$ 10.2682 0.456026
$$508$$ 13.3717 0.593273
$$509$$ 2.20077 0.0975473 0.0487737 0.998810i $$-0.484469\pi$$
0.0487737 + 0.998810i $$0.484469\pi$$
$$510$$ −0.335577 −0.0148596
$$511$$ 3.70727 0.164000
$$512$$ −1.00000 −0.0441942
$$513$$ 35.0130 1.54586
$$514$$ −14.8108 −0.653276
$$515$$ 7.66442 0.337735
$$516$$ 11.8652 0.522336
$$517$$ 0 0
$$518$$ −0.853635 −0.0375065
$$519$$ 8.45065 0.370943
$$520$$ 4.68585 0.205488
$$521$$ 37.1940 1.62950 0.814750 0.579812i $$-0.196874\pi$$
0.814750 + 0.579812i $$0.196874\pi$$
$$522$$ 2.42610 0.106187
$$523$$ −30.4078 −1.32964 −0.664820 0.747003i $$-0.731492\pi$$
−0.664820 + 0.747003i $$0.731492\pi$$
$$524$$ 11.4391 0.499719
$$525$$ 1.14637 0.0500315
$$526$$ 18.7434 0.817250
$$527$$ 0.286460 0.0124784
$$528$$ 0 0
$$529$$ −14.8568 −0.645947
$$530$$ −5.43910 −0.236259
$$531$$ 15.7992 0.685628
$$532$$ −6.51806 −0.282594
$$533$$ −29.1709 −1.26353
$$534$$ −14.9933 −0.648822
$$535$$ −9.56404 −0.413489
$$536$$ 0.585462 0.0252881
$$537$$ 1.79296 0.0773720
$$538$$ −12.6858 −0.546926
$$539$$ 0 0
$$540$$ −5.37169 −0.231161
$$541$$ −39.0607 −1.67935 −0.839675 0.543090i $$-0.817254\pi$$
−0.839675 + 0.543090i $$0.817254\pi$$
$$542$$ 1.12181 0.0481857
$$543$$ 16.9504 0.727412
$$544$$ −0.292731 −0.0125507
$$545$$ −3.14637 −0.134775
$$546$$ 5.37169 0.229887
$$547$$ 0.585462 0.0250325 0.0125163 0.999922i $$-0.496016\pi$$
0.0125163 + 0.999922i $$0.496016\pi$$
$$548$$ 10.0000 0.427179
$$549$$ −20.1579 −0.860319
$$550$$ 0 0
$$551$$ −9.38011 −0.399606
$$552$$ −3.27131 −0.139236
$$553$$ 2.51806 0.107079
$$554$$ 30.1151 1.27947
$$555$$ 0.978577 0.0415383
$$556$$ 14.1825 0.601471
$$557$$ −17.4637 −0.739959 −0.369979 0.929040i $$-0.620635\pi$$
−0.369979 + 0.929040i $$0.620635\pi$$
$$558$$ 1.64973 0.0698387
$$559$$ −48.4998 −2.05132
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ −14.3356 −0.604710
$$563$$ 18.1579 0.765265 0.382633 0.923901i $$-0.375018\pi$$
0.382633 + 0.923901i $$0.375018\pi$$
$$564$$ −11.4145 −0.480639
$$565$$ −2.58546 −0.108771
$$566$$ 3.21377 0.135085
$$567$$ −1.10038 −0.0462118
$$568$$ 0.335577 0.0140805
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 7.47208 0.312971
$$571$$ 9.03612 0.378150 0.189075 0.981963i $$-0.439451\pi$$
0.189075 + 0.981963i $$0.439451\pi$$
$$572$$ 0 0
$$573$$ −20.5855 −0.859970
$$574$$ −6.22533 −0.259840
$$575$$ 2.85363 0.119005
$$576$$ −1.68585 −0.0702436
$$577$$ 19.3963 0.807476 0.403738 0.914875i $$-0.367711\pi$$
0.403738 + 0.914875i $$0.367711\pi$$
$$578$$ 16.9143 0.703542
$$579$$ −21.3717 −0.888177
$$580$$ 1.43910 0.0597552
$$581$$ 1.70727 0.0708295
$$582$$ −10.4360 −0.432584
$$583$$ 0 0
$$584$$ −3.70727 −0.153408
$$585$$ 7.89962 0.326609
$$586$$ −16.6858 −0.689286
$$587$$ 17.2614 0.712456 0.356228 0.934399i $$-0.384063\pi$$
0.356228 + 0.934399i $$0.384063\pi$$
$$588$$ 1.14637 0.0472753
$$589$$ −6.37842 −0.262818
$$590$$ 9.37169 0.385826
$$591$$ 17.2369 0.709031
$$592$$ 0.853635 0.0350842
$$593$$ 11.0361 0.453199 0.226599 0.973988i $$-0.427239\pi$$
0.226599 + 0.973988i $$0.427239\pi$$
$$594$$ 0 0
$$595$$ 0.292731 0.0120008
$$596$$ −11.5970 −0.475032
$$597$$ 17.5725 0.719193
$$598$$ 13.3717 0.546809
$$599$$ −41.7367 −1.70531 −0.852657 0.522472i $$-0.825010\pi$$
−0.852657 + 0.522472i $$0.825010\pi$$
$$600$$ −1.14637 −0.0468002
$$601$$ −39.3106 −1.60351 −0.801756 0.597652i $$-0.796100\pi$$
−0.801756 + 0.597652i $$0.796100\pi$$
$$602$$ −10.3503 −0.421845
$$603$$ 0.986999 0.0401937
$$604$$ −1.73183 −0.0704671
$$605$$ 0 0
$$606$$ −12.3650 −0.502292
$$607$$ 6.35027 0.257749 0.128875 0.991661i $$-0.458863\pi$$
0.128875 + 0.991661i $$0.458863\pi$$
$$608$$ 6.51806 0.264342
$$609$$ 1.64973 0.0668505
$$610$$ −11.9572 −0.484131
$$611$$ 46.6577 1.88757
$$612$$ −0.493499 −0.0199485
$$613$$ 42.4507 1.71457 0.857283 0.514846i $$-0.172151\pi$$
0.857283 + 0.514846i $$0.172151\pi$$
$$614$$ −6.29273 −0.253954
$$615$$ 7.13650 0.287771
$$616$$ 0 0
$$617$$ −0.677425 −0.0272721 −0.0136360 0.999907i $$-0.504341\pi$$
−0.0136360 + 0.999907i $$0.504341\pi$$
$$618$$ −8.78623 −0.353434
$$619$$ −18.5426 −0.745291 −0.372645 0.927974i $$-0.621549\pi$$
−0.372645 + 0.927974i $$0.621549\pi$$
$$620$$ 0.978577 0.0393006
$$621$$ −15.3288 −0.615125
$$622$$ 20.3931 0.817689
$$623$$ 13.0790 0.523998
$$624$$ −5.37169 −0.215040
$$625$$ 1.00000 0.0400000
$$626$$ −12.0674 −0.482310
$$627$$ 0 0
$$628$$ −12.2927 −0.490533
$$629$$ 0.249885 0.00996358
$$630$$ 1.68585 0.0671657
$$631$$ 21.3717 0.850794 0.425397 0.905007i $$-0.360135\pi$$
0.425397 + 0.905007i $$0.360135\pi$$
$$632$$ −2.51806 −0.100163
$$633$$ −2.34185 −0.0930801
$$634$$ −28.7679 −1.14252
$$635$$ 13.3717 0.530639
$$636$$ 6.23519 0.247241
$$637$$ −4.68585 −0.185660
$$638$$ 0 0
$$639$$ 0.565731 0.0223800
$$640$$ −1.00000 −0.0395285
$$641$$ 23.8715 0.942866 0.471433 0.881902i $$-0.343737\pi$$
0.471433 + 0.881902i $$0.343737\pi$$
$$642$$ 10.9639 0.432710
$$643$$ 0.311018 0.0122654 0.00613268 0.999981i $$-0.498048\pi$$
0.00613268 + 0.999981i $$0.498048\pi$$
$$644$$ 2.85363 0.112449
$$645$$ 11.8652 0.467191
$$646$$ 1.90804 0.0750707
$$647$$ −9.62158 −0.378263 −0.189132 0.981952i $$-0.560567\pi$$
−0.189132 + 0.981952i $$0.560567\pi$$
$$648$$ 1.10038 0.0432272
$$649$$ 0 0
$$650$$ 4.68585 0.183794
$$651$$ 1.12181 0.0439671
$$652$$ 23.9143 0.936557
$$653$$ −27.0607 −1.05897 −0.529483 0.848321i $$-0.677614\pi$$
−0.529483 + 0.848321i $$0.677614\pi$$
$$654$$ 3.60688 0.141040
$$655$$ 11.4391 0.446962
$$656$$ 6.22533 0.243058
$$657$$ −6.24989 −0.243831
$$658$$ 9.95715 0.388170
$$659$$ 26.4935 1.03204 0.516020 0.856576i $$-0.327413\pi$$
0.516020 + 0.856576i $$0.327413\pi$$
$$660$$ 0 0
$$661$$ 35.0852 1.36466 0.682329 0.731046i $$-0.260967\pi$$
0.682329 + 0.731046i $$0.260967\pi$$
$$662$$ 3.80765 0.147989
$$663$$ −1.57246 −0.0610693
$$664$$ −1.70727 −0.0662549
$$665$$ −6.51806 −0.252759
$$666$$ 1.43910 0.0557639
$$667$$ 4.10666 0.159010
$$668$$ 8.00000 0.309529
$$669$$ 12.7005 0.491031
$$670$$ 0.585462 0.0226184
$$671$$ 0 0
$$672$$ −1.14637 −0.0442220
$$673$$ −32.6577 −1.25886 −0.629431 0.777057i $$-0.716712\pi$$
−0.629431 + 0.777057i $$0.716712\pi$$
$$674$$ 15.3717 0.592095
$$675$$ −5.37169 −0.206757
$$676$$ 8.95715 0.344506
$$677$$ −9.12808 −0.350821 −0.175410 0.984495i $$-0.556125\pi$$
−0.175410 + 0.984495i $$0.556125\pi$$
$$678$$ 2.96388 0.113827
$$679$$ 9.10352 0.349361
$$680$$ −0.292731 −0.0112257
$$681$$ 21.2566 0.814555
$$682$$ 0 0
$$683$$ 10.4935 0.401523 0.200761 0.979640i $$-0.435658\pi$$
0.200761 + 0.979640i $$0.435658\pi$$
$$684$$ 10.9884 0.420154
$$685$$ 10.0000 0.382080
$$686$$ −1.00000 −0.0381802
$$687$$ −9.18777 −0.350535
$$688$$ 10.3503 0.394600
$$689$$ −25.4868 −0.970969
$$690$$ −3.27131 −0.124537
$$691$$ 1.70727 0.0649476 0.0324738 0.999473i $$-0.489661\pi$$
0.0324738 + 0.999473i $$0.489661\pi$$
$$692$$ 7.37169 0.280230
$$693$$ 0 0
$$694$$ 3.02142 0.114692
$$695$$ 14.1825 0.537972
$$696$$ −1.64973 −0.0625329
$$697$$ 1.82235 0.0690263
$$698$$ 11.0361 0.417723
$$699$$ −32.0491 −1.21221
$$700$$ 1.00000 0.0377964
$$701$$ 37.3534 1.41082 0.705409 0.708800i $$-0.250763\pi$$
0.705409 + 0.708800i $$0.250763\pi$$
$$702$$ −25.1709 −0.950015
$$703$$ −5.56404 −0.209852
$$704$$ 0 0
$$705$$ −11.4145 −0.429896
$$706$$ −23.9817 −0.902564
$$707$$ 10.7862 0.405658
$$708$$ −10.7434 −0.403761
$$709$$ −8.20704 −0.308222 −0.154111 0.988054i $$-0.549251\pi$$
−0.154111 + 0.988054i $$0.549251\pi$$
$$710$$ 0.335577 0.0125940
$$711$$ −4.24506 −0.159202
$$712$$ −13.0790 −0.490155
$$713$$ 2.79250 0.104580
$$714$$ −0.335577 −0.0125586
$$715$$ 0 0
$$716$$ 1.56404 0.0584509
$$717$$ 33.0277 1.23344
$$718$$ −10.5181 −0.392530
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ −1.68585 −0.0628278
$$721$$ 7.66442 0.285438
$$722$$ −23.4851 −0.874024
$$723$$ 9.76481 0.363157
$$724$$ 14.7862 0.549526
$$725$$ 1.43910 0.0534467
$$726$$ 0 0
$$727$$ −46.9442 −1.74106 −0.870531 0.492113i $$-0.836225\pi$$
−0.870531 + 0.492113i $$0.836225\pi$$
$$728$$ 4.68585 0.173669
$$729$$ 20.3288 0.752920
$$730$$ −3.70727 −0.137212
$$731$$ 3.02984 0.112063
$$732$$ 13.7073 0.506635
$$733$$ 42.0000 1.55131 0.775653 0.631160i $$-0.217421\pi$$
0.775653 + 0.631160i $$0.217421\pi$$
$$734$$ 35.5787 1.31323
$$735$$ 1.14637 0.0422843
$$736$$ −2.85363 −0.105186
$$737$$ 0 0
$$738$$ 10.4949 0.386324
$$739$$ 0.871922 0.0320742 0.0160371 0.999871i $$-0.494895\pi$$
0.0160371 + 0.999871i $$0.494895\pi$$
$$740$$ 0.853635 0.0313802
$$741$$ 35.0130 1.28623
$$742$$ −5.43910 −0.199676
$$743$$ 24.6712 0.905097 0.452548 0.891740i $$-0.350515\pi$$
0.452548 + 0.891740i $$0.350515\pi$$
$$744$$ −1.12181 −0.0411274
$$745$$ −11.5970 −0.424882
$$746$$ 7.50650 0.274833
$$747$$ −2.87819 −0.105308
$$748$$ 0 0
$$749$$ −9.56404 −0.349462
$$750$$ −1.14637 −0.0418593
$$751$$ −3.75011 −0.136844 −0.0684218 0.997656i $$-0.521796\pi$$
−0.0684218 + 0.997656i $$0.521796\pi$$
$$752$$ −9.95715 −0.363100
$$753$$ −26.5132 −0.966196
$$754$$ 6.74338 0.245580
$$755$$ −1.73183 −0.0630277
$$756$$ −5.37169 −0.195367
$$757$$ 15.7318 0.571783 0.285891 0.958262i $$-0.407710\pi$$
0.285891 + 0.958262i $$0.407710\pi$$
$$758$$ −33.4868 −1.21629
$$759$$ 0 0
$$760$$ 6.51806 0.236435
$$761$$ −15.2614 −0.553227 −0.276613 0.960981i $$-0.589212\pi$$
−0.276613 + 0.960981i $$0.589212\pi$$
$$762$$ −15.3288 −0.555306
$$763$$ −3.14637 −0.113906
$$764$$ −17.9572 −0.649667
$$765$$ −0.493499 −0.0178425
$$766$$ −15.6644 −0.565979
$$767$$ 43.9143 1.58565
$$768$$ 1.14637 0.0413659
$$769$$ 9.63986 0.347622 0.173811 0.984779i $$-0.444392\pi$$
0.173811 + 0.984779i $$0.444392\pi$$
$$770$$ 0 0
$$771$$ 16.9786 0.611469
$$772$$ −18.6430 −0.670976
$$773$$ 12.8291 0.461430 0.230715 0.973021i $$-0.425894\pi$$
0.230715 + 0.973021i $$0.425894\pi$$
$$774$$ 17.4490 0.627190
$$775$$ 0.978577 0.0351515
$$776$$ −9.10352 −0.326797
$$777$$ 0.978577 0.0351063
$$778$$ 19.6216 0.703468
$$779$$ −40.5770 −1.45382
$$780$$ −5.37169 −0.192337
$$781$$ 0 0
$$782$$ −0.835347 −0.0298720
$$783$$ −7.73038 −0.276261
$$784$$ 1.00000 0.0357143
$$785$$ −12.2927 −0.438746
$$786$$ −13.1134 −0.467739
$$787$$ −10.8291 −0.386015 −0.193007 0.981197i $$-0.561824\pi$$
−0.193007 + 0.981197i $$0.561824\pi$$
$$788$$ 15.0361 0.535639
$$789$$ −21.4868 −0.764949
$$790$$ −2.51806 −0.0895885
$$791$$ −2.58546 −0.0919284
$$792$$ 0 0
$$793$$ −56.0294 −1.98966
$$794$$ −10.2499 −0.363755
$$795$$ 6.23519 0.221139
$$796$$ 15.3288 0.543317
$$797$$ −21.0790 −0.746655 −0.373328 0.927700i $$-0.621783\pi$$
−0.373328 + 0.927700i $$0.621783\pi$$
$$798$$ 7.47208 0.264509
$$799$$ −2.91477 −0.103117
$$800$$ −1.00000 −0.0353553
$$801$$ −22.0491 −0.779067
$$802$$ 16.8866 0.596287
$$803$$ 0 0
$$804$$ −0.671153 −0.0236698
$$805$$ 2.85363 0.100577
$$806$$ 4.58546 0.161516
$$807$$ 14.5426 0.511924
$$808$$ −10.7862 −0.379458
$$809$$ −3.62158 −0.127328 −0.0636639 0.997971i $$-0.520279\pi$$
−0.0636639 + 0.997971i $$0.520279\pi$$
$$810$$ 1.10038 0.0386636
$$811$$ −34.7188 −1.21914 −0.609571 0.792731i $$-0.708658\pi$$
−0.609571 + 0.792731i $$0.708658\pi$$
$$812$$ 1.43910 0.0505024
$$813$$ −1.28600 −0.0451020
$$814$$ 0 0
$$815$$ 23.9143 0.837682
$$816$$ 0.335577 0.0117475
$$817$$ −67.4637 −2.36025
$$818$$ 18.2744 0.638951
$$819$$ 7.89962 0.276035
$$820$$ 6.22533 0.217398
$$821$$ 37.7549 1.31766 0.658828 0.752293i $$-0.271052\pi$$
0.658828 + 0.752293i $$0.271052\pi$$
$$822$$ −11.4637 −0.399841
$$823$$ 11.1892 0.390031 0.195016 0.980800i $$-0.437524\pi$$
0.195016 + 0.980800i $$0.437524\pi$$
$$824$$ −7.66442 −0.267003
$$825$$ 0 0
$$826$$ 9.37169 0.326083
$$827$$ 7.91431 0.275207 0.137604 0.990487i $$-0.456060\pi$$
0.137604 + 0.990487i $$0.456060\pi$$
$$828$$ −4.81079 −0.167186
$$829$$ −44.7152 −1.55302 −0.776512 0.630102i $$-0.783013\pi$$
−0.776512 + 0.630102i $$0.783013\pi$$
$$830$$ −1.70727 −0.0592602
$$831$$ −34.5229 −1.19759
$$832$$ −4.68585 −0.162452
$$833$$ 0.292731 0.0101425
$$834$$ −16.2583 −0.562979
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ −5.25662 −0.181695
$$838$$ 37.2860 1.28802
$$839$$ 21.6791 0.748446 0.374223 0.927339i $$-0.377909\pi$$
0.374223 + 0.927339i $$0.377909\pi$$
$$840$$ −1.14637 −0.0395534
$$841$$ −26.9290 −0.928586
$$842$$ 2.78623 0.0960198
$$843$$ 16.4338 0.566010
$$844$$ −2.04285 −0.0703176
$$845$$ 8.95715 0.308135
$$846$$ −16.7862 −0.577122
$$847$$ 0 0
$$848$$ 5.43910 0.186779
$$849$$ −3.68415 −0.126440
$$850$$ −0.292731 −0.0100406
$$851$$ 2.43596 0.0835037
$$852$$ −0.384694 −0.0131794
$$853$$ −9.41454 −0.322348 −0.161174 0.986926i $$-0.551528\pi$$
−0.161174 + 0.986926i $$0.551528\pi$$
$$854$$ −11.9572 −0.409165
$$855$$ 10.9884 0.375797
$$856$$ 9.56404 0.326892
$$857$$ 25.8652 0.883538 0.441769 0.897129i $$-0.354351\pi$$
0.441769 + 0.897129i $$0.354351\pi$$
$$858$$ 0 0
$$859$$ −29.2860 −0.999225 −0.499613 0.866249i $$-0.666524\pi$$
−0.499613 + 0.866249i $$0.666524\pi$$
$$860$$ 10.3503 0.352941
$$861$$ 7.13650 0.243211
$$862$$ 38.0477 1.29591
$$863$$ −20.3110 −0.691395 −0.345698 0.938346i $$-0.612358\pi$$
−0.345698 + 0.938346i $$0.612358\pi$$
$$864$$ 5.37169 0.182749
$$865$$ 7.37169 0.250645
$$866$$ 10.8108 0.367366
$$867$$ −19.3900 −0.658518
$$868$$ 0.978577 0.0332151
$$869$$ 0 0
$$870$$ −1.64973 −0.0559311
$$871$$ 2.74338 0.0929560
$$872$$ 3.14637 0.106549
$$873$$ −15.3471 −0.519422
$$874$$ 18.6002 0.629160
$$875$$ 1.00000 0.0338062
$$876$$ 4.24989 0.143590
$$877$$ 38.3650 1.29549 0.647746 0.761856i $$-0.275712\pi$$
0.647746 + 0.761856i $$0.275712\pi$$
$$878$$ 30.9933 1.04597
$$879$$ 19.1281 0.645174
$$880$$ 0 0
$$881$$ 40.5426 1.36592 0.682958 0.730458i $$-0.260693\pi$$
0.682958 + 0.730458i $$0.260693\pi$$
$$882$$ 1.68585 0.0567654
$$883$$ 28.0491 0.943928 0.471964 0.881618i $$-0.343545\pi$$
0.471964 + 0.881618i $$0.343545\pi$$
$$884$$ −1.37169 −0.0461350
$$885$$ −10.7434 −0.361135
$$886$$ −25.3717 −0.852379
$$887$$ 6.65769 0.223543 0.111772 0.993734i $$-0.464347\pi$$
0.111772 + 0.993734i $$0.464347\pi$$
$$888$$ −0.978577 −0.0328389
$$889$$ 13.3717 0.448472
$$890$$ −13.0790 −0.438408
$$891$$ 0 0
$$892$$ 11.0790 0.370951
$$893$$ 64.9013 2.17184
$$894$$ 13.2944 0.444632
$$895$$ 1.56404 0.0522801
$$896$$ −1.00000 −0.0334077
$$897$$ −15.3288 −0.511815
$$898$$ 0.886615 0.0295867
$$899$$ 1.40827 0.0469683
$$900$$ −1.68585 −0.0561949
$$901$$ 1.59219 0.0530436
$$902$$ 0 0
$$903$$ 11.8652 0.394849
$$904$$ 2.58546 0.0859912
$$905$$ 14.7862 0.491511
$$906$$ 1.98531 0.0659574
$$907$$ 39.5787 1.31419 0.657095 0.753808i $$-0.271785\pi$$
0.657095 + 0.753808i $$0.271785\pi$$
$$908$$ 18.5426 0.615358
$$909$$ −18.1839 −0.603123
$$910$$ 4.68585 0.155334
$$911$$ −24.3356 −0.806274 −0.403137 0.915140i $$-0.632080\pi$$
−0.403137 + 0.915140i $$0.632080\pi$$
$$912$$ −7.47208 −0.247425
$$913$$ 0 0
$$914$$ 29.9143 0.989477
$$915$$ 13.7073 0.453148
$$916$$ −8.01469 −0.264813
$$917$$ 11.4391 0.377752
$$918$$ 1.57246 0.0518989
$$919$$ −50.6331 −1.67023 −0.835117 0.550073i $$-0.814600\pi$$
−0.835117 + 0.550073i $$0.814600\pi$$
$$920$$ −2.85363 −0.0940815
$$921$$ 7.21377 0.237702
$$922$$ −2.33558 −0.0769181
$$923$$ 1.57246 0.0517582
$$924$$ 0 0
$$925$$ 0.853635 0.0280673
$$926$$ −0.110250 −0.00362303
$$927$$ −12.9210 −0.424383
$$928$$ −1.43910 −0.0472407
$$929$$ 47.1512 1.54698 0.773490 0.633808i $$-0.218509\pi$$
0.773490 + 0.633808i $$0.218509\pi$$
$$930$$ −1.12181 −0.0367855
$$931$$ −6.51806 −0.213621
$$932$$ −27.9572 −0.915767
$$933$$ −23.3780 −0.765360
$$934$$ −36.6760 −1.20007
$$935$$ 0 0
$$936$$ −7.89962 −0.258207
$$937$$ 42.0294 1.37304 0.686520 0.727111i $$-0.259137\pi$$
0.686520 + 0.727111i $$0.259137\pi$$
$$938$$ 0.585462 0.0191160
$$939$$ 13.8337 0.451444
$$940$$ −9.95715 −0.324767
$$941$$ 36.7434 1.19780 0.598900 0.800824i $$-0.295605\pi$$
0.598900 + 0.800824i $$0.295605\pi$$
$$942$$ 14.0920 0.459141
$$943$$ 17.7648 0.578502
$$944$$ −9.37169 −0.305023
$$945$$ −5.37169 −0.174741
$$946$$ 0 0
$$947$$ 18.8782 0.613459 0.306729 0.951797i $$-0.400765\pi$$
0.306729 + 0.951797i $$0.400765\pi$$
$$948$$ 2.88661 0.0937529
$$949$$ −17.3717 −0.563909
$$950$$ 6.51806 0.211474
$$951$$ 32.9786 1.06940
$$952$$ −0.292731 −0.00948747
$$953$$ 43.2285 1.40031 0.700154 0.713992i $$-0.253115\pi$$
0.700154 + 0.713992i $$0.253115\pi$$
$$954$$ 9.16948 0.296873
$$955$$ −17.9572 −0.581080
$$956$$ 28.8108 0.931808
$$957$$ 0 0
$$958$$ −42.7434 −1.38098
$$959$$ 10.0000 0.322917
$$960$$ 1.14637 0.0369988
$$961$$ −30.0424 −0.969109
$$962$$ 4.00000 0.128965
$$963$$ 16.1235 0.519572
$$964$$ 8.51806 0.274348
$$965$$ −18.6430 −0.600139
$$966$$ −3.27131 −0.105253
$$967$$ −48.7299 −1.56705 −0.783524 0.621361i $$-0.786580\pi$$
−0.783524 + 0.621361i $$0.786580\pi$$
$$968$$ 0 0
$$969$$ −2.18731 −0.0702665
$$970$$ −9.10352 −0.292296
$$971$$ −25.9865 −0.833948 −0.416974 0.908918i $$-0.636909\pi$$
−0.416974 + 0.908918i $$0.636909\pi$$
$$972$$ 14.8536 0.476431
$$973$$ 14.1825 0.454669
$$974$$ 36.1396 1.15799
$$975$$ −5.37169 −0.172032
$$976$$ 11.9572 0.382739
$$977$$ 2.67115 0.0854578 0.0427289 0.999087i $$-0.486395\pi$$
0.0427289 + 0.999087i $$0.486395\pi$$
$$978$$ −27.4145 −0.876620
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ 5.30429 0.169353
$$982$$ 2.54262 0.0811381
$$983$$ −8.33558 −0.265864 −0.132932 0.991125i $$-0.542439\pi$$
−0.132932 + 0.991125i $$0.542439\pi$$
$$984$$ −7.13650 −0.227503
$$985$$ 15.0361 0.479090
$$986$$ −0.421268 −0.0134159
$$987$$ −11.4145 −0.363329
$$988$$ 30.5426 0.971690
$$989$$ 29.5359 0.939187
$$990$$ 0 0
$$991$$ 46.2730 1.46991 0.734955 0.678116i $$-0.237203\pi$$
0.734955 + 0.678116i $$0.237203\pi$$
$$992$$ −0.978577 −0.0310699
$$993$$ −4.36496 −0.138518
$$994$$ 0.335577 0.0106438
$$995$$ 15.3288 0.485957
$$996$$ 1.95715 0.0620148
$$997$$ −35.8715 −1.13606 −0.568030 0.823008i $$-0.692294\pi$$
−0.568030 + 0.823008i $$0.692294\pi$$
$$998$$ −3.13650 −0.0992842
$$999$$ −4.58546 −0.145078
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 8470.2.a.ci.1.2 3
11.10 odd 2 770.2.a.m.1.2 3
33.32 even 2 6930.2.a.ce.1.3 3
44.43 even 2 6160.2.a.bf.1.2 3
55.32 even 4 3850.2.c.ba.1849.5 6
55.43 even 4 3850.2.c.ba.1849.2 6
55.54 odd 2 3850.2.a.bt.1.2 3
77.76 even 2 5390.2.a.ca.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.2 3 11.10 odd 2
3850.2.a.bt.1.2 3 55.54 odd 2
3850.2.c.ba.1849.2 6 55.43 even 4
3850.2.c.ba.1849.5 6 55.32 even 4
5390.2.a.ca.1.2 3 77.76 even 2
6160.2.a.bf.1.2 3 44.43 even 2
6930.2.a.ce.1.3 3 33.32 even 2
8470.2.a.ci.1.2 3 1.1 even 1 trivial