# Properties

 Label 2070.2.a.z.1.3 Level $2070$ Weight $2$ Character 2070.1 Self dual yes Analytic conductor $16.529$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.5290332184$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.11903$$ of defining polynomial Character $$\chi$$ $$=$$ 2070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.33763 q^{11} -3.72833 q^{13} -4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} +4.50973 q^{19} +1.00000 q^{20} +4.33763 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.72833 q^{26} +4.50973 q^{28} +8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +1.11903 q^{34} +4.50973 q^{35} -0.781399 q^{37} -4.50973 q^{38} -1.00000 q^{40} -3.90043 q^{41} +8.00000 q^{43} -4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +13.3376 q^{49} -1.00000 q^{50} -3.72833 q^{52} +6.00000 q^{53} -4.33763 q^{55} -4.50973 q^{56} -8.23805 q^{58} +2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} +1.00000 q^{64} -3.72833 q^{65} +2.43720 q^{67} -1.11903 q^{68} -4.50973 q^{70} -7.11903 q^{71} -9.45665 q^{73} +0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -14.9133 q^{79} +1.00000 q^{80} +3.90043 q^{82} -2.78140 q^{83} -1.11903 q^{85} -8.00000 q^{86} +4.33763 q^{88} +7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} -11.4567 q^{94} +4.50973 q^{95} -0.642920 q^{97} -13.3376 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} - 3q^{10} - 3q^{11} - q^{13} - 3q^{14} + 3q^{16} + 7q^{17} + 3q^{19} + 3q^{20} + 3q^{22} + 3q^{23} + 3q^{25} + q^{26} + 3q^{28} + 4q^{29} - 5q^{31} - 3q^{32} - 7q^{34} + 3q^{35} - 2q^{37} - 3q^{38} - 3q^{40} - q^{41} + 24q^{43} - 3q^{44} - 3q^{46} + 14q^{47} + 30q^{49} - 3q^{50} - q^{52} + 18q^{53} - 3q^{55} - 3q^{56} - 4q^{58} - 14q^{59} + q^{61} + 5q^{62} + 3q^{64} - q^{65} + 8q^{67} + 7q^{68} - 3q^{70} - 11q^{71} - 8q^{73} + 2q^{74} + 3q^{76} + 24q^{77} - 4q^{79} + 3q^{80} + q^{82} - 8q^{83} + 7q^{85} - 24q^{86} + 3q^{88} - 18q^{89} + q^{91} + 3q^{92} - 14q^{94} + 3q^{95} - 33q^{97} - 30q^{98} + O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.50973 1.70452 0.852258 0.523122i $$-0.175233\pi$$
0.852258 + 0.523122i $$0.175233\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ −4.33763 −1.30784 −0.653922 0.756562i $$-0.726878\pi$$
−0.653922 + 0.756562i $$0.726878\pi$$
$$12$$ 0 0
$$13$$ −3.72833 −1.03405 −0.517026 0.855970i $$-0.672961\pi$$
−0.517026 + 0.855970i $$0.672961\pi$$
$$14$$ −4.50973 −1.20527
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.11903 −0.271404 −0.135702 0.990750i $$-0.543329\pi$$
−0.135702 + 0.990750i $$0.543329\pi$$
$$18$$ 0 0
$$19$$ 4.50973 1.03460 0.517301 0.855803i $$-0.326937\pi$$
0.517301 + 0.855803i $$0.326937\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 4.33763 0.924785
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 3.72833 0.731185
$$27$$ 0 0
$$28$$ 4.50973 0.852258
$$29$$ 8.23805 1.52977 0.764884 0.644168i $$-0.222796\pi$$
0.764884 + 0.644168i $$0.222796\pi$$
$$30$$ 0 0
$$31$$ 1.72833 0.310417 0.155208 0.987882i $$-0.450395\pi$$
0.155208 + 0.987882i $$0.450395\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 1.11903 0.191911
$$35$$ 4.50973 0.762283
$$36$$ 0 0
$$37$$ −0.781399 −0.128461 −0.0642306 0.997935i $$-0.520459\pi$$
−0.0642306 + 0.997935i $$0.520459\pi$$
$$38$$ −4.50973 −0.731574
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −3.90043 −0.609144 −0.304572 0.952489i $$-0.598513\pi$$
−0.304572 + 0.952489i $$0.598513\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −4.33763 −0.653922
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 11.4567 1.67112 0.835562 0.549396i $$-0.185142\pi$$
0.835562 + 0.549396i $$0.185142\pi$$
$$48$$ 0 0
$$49$$ 13.3376 1.90538
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −3.72833 −0.517026
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −4.33763 −0.584886
$$56$$ −4.50973 −0.602637
$$57$$ 0 0
$$58$$ −8.23805 −1.08171
$$59$$ 2.23805 0.291370 0.145685 0.989331i $$-0.453461\pi$$
0.145685 + 0.989331i $$0.453461\pi$$
$$60$$ 0 0
$$61$$ 3.55623 0.455329 0.227664 0.973740i $$-0.426891\pi$$
0.227664 + 0.973740i $$0.426891\pi$$
$$62$$ −1.72833 −0.219498
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −3.72833 −0.462442
$$66$$ 0 0
$$67$$ 2.43720 0.297752 0.148876 0.988856i $$-0.452435\pi$$
0.148876 + 0.988856i $$0.452435\pi$$
$$68$$ −1.11903 −0.135702
$$69$$ 0 0
$$70$$ −4.50973 −0.539015
$$71$$ −7.11903 −0.844873 −0.422437 0.906393i $$-0.638825\pi$$
−0.422437 + 0.906393i $$0.638825\pi$$
$$72$$ 0 0
$$73$$ −9.45665 −1.10682 −0.553409 0.832910i $$-0.686673\pi$$
−0.553409 + 0.832910i $$0.686673\pi$$
$$74$$ 0.781399 0.0908357
$$75$$ 0 0
$$76$$ 4.50973 0.517301
$$77$$ −19.5615 −2.22924
$$78$$ 0 0
$$79$$ −14.9133 −1.67788 −0.838939 0.544225i $$-0.816824\pi$$
−0.838939 + 0.544225i $$0.816824\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ 3.90043 0.430730
$$83$$ −2.78140 −0.305298 −0.152649 0.988280i $$-0.548780\pi$$
−0.152649 + 0.988280i $$0.548780\pi$$
$$84$$ 0 0
$$85$$ −1.11903 −0.121375
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 4.33763 0.462393
$$89$$ 7.69471 0.815637 0.407819 0.913063i $$-0.366290\pi$$
0.407819 + 0.913063i $$0.366290\pi$$
$$90$$ 0 0
$$91$$ −16.8137 −1.76256
$$92$$ 1.00000 0.104257
$$93$$ 0 0
$$94$$ −11.4567 −1.18166
$$95$$ 4.50973 0.462688
$$96$$ 0 0
$$97$$ −0.642920 −0.0652786 −0.0326393 0.999467i $$-0.510391\pi$$
−0.0326393 + 0.999467i $$0.510391\pi$$
$$98$$ −13.3376 −1.34730
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 8.23805 0.819717 0.409858 0.912149i $$-0.365578\pi$$
0.409858 + 0.912149i $$0.365578\pi$$
$$102$$ 0 0
$$103$$ 12.3376 1.21566 0.607831 0.794066i $$-0.292040\pi$$
0.607831 + 0.794066i $$0.292040\pi$$
$$104$$ 3.72833 0.365593
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 15.9328 1.54028 0.770139 0.637876i $$-0.220187\pi$$
0.770139 + 0.637876i $$0.220187\pi$$
$$108$$ 0 0
$$109$$ −1.49027 −0.142742 −0.0713712 0.997450i $$-0.522737\pi$$
−0.0713712 + 0.997450i $$0.522737\pi$$
$$110$$ 4.33763 0.413577
$$111$$ 0 0
$$112$$ 4.50973 0.426129
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 8.23805 0.764884
$$117$$ 0 0
$$118$$ −2.23805 −0.206030
$$119$$ −5.04650 −0.462612
$$120$$ 0 0
$$121$$ 7.81502 0.710456
$$122$$ −3.55623 −0.321966
$$123$$ 0 0
$$124$$ 1.72833 0.155208
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −0.675256 −0.0599193 −0.0299597 0.999551i $$-0.509538\pi$$
−0.0299597 + 0.999551i $$0.509538\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 3.72833 0.326996
$$131$$ 13.6947 1.19651 0.598256 0.801305i $$-0.295861\pi$$
0.598256 + 0.801305i $$0.295861\pi$$
$$132$$ 0 0
$$133$$ 20.3376 1.76350
$$134$$ −2.43720 −0.210542
$$135$$ 0 0
$$136$$ 1.11903 0.0959557
$$137$$ −7.52918 −0.643261 −0.321631 0.946865i $$-0.604231\pi$$
−0.321631 + 0.946865i $$0.604231\pi$$
$$138$$ 0 0
$$139$$ 4.67526 0.396550 0.198275 0.980146i $$-0.436466\pi$$
0.198275 + 0.980146i $$0.436466\pi$$
$$140$$ 4.50973 0.381141
$$141$$ 0 0
$$142$$ 7.11903 0.597415
$$143$$ 16.1721 1.35238
$$144$$ 0 0
$$145$$ 8.23805 0.684133
$$146$$ 9.45665 0.782638
$$147$$ 0 0
$$148$$ −0.781399 −0.0642306
$$149$$ −7.52918 −0.616814 −0.308407 0.951254i $$-0.599796\pi$$
−0.308407 + 0.951254i $$0.599796\pi$$
$$150$$ 0 0
$$151$$ −13.3571 −1.08698 −0.543492 0.839414i $$-0.682898\pi$$
−0.543492 + 0.839414i $$0.682898\pi$$
$$152$$ −4.50973 −0.365787
$$153$$ 0 0
$$154$$ 19.5615 1.57631
$$155$$ 1.72833 0.138823
$$156$$ 0 0
$$157$$ 16.2381 1.29594 0.647969 0.761667i $$-0.275619\pi$$
0.647969 + 0.761667i $$0.275619\pi$$
$$158$$ 14.9133 1.18644
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 4.50973 0.355416
$$162$$ 0 0
$$163$$ −3.29112 −0.257781 −0.128890 0.991659i $$-0.541142\pi$$
−0.128890 + 0.991659i $$0.541142\pi$$
$$164$$ −3.90043 −0.304572
$$165$$ 0 0
$$166$$ 2.78140 0.215878
$$167$$ −22.9133 −1.77309 −0.886543 0.462647i $$-0.846900\pi$$
−0.886543 + 0.462647i $$0.846900\pi$$
$$168$$ 0 0
$$169$$ 0.900425 0.0692635
$$170$$ 1.11903 0.0858254
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ −0.575681 −0.0437683 −0.0218841 0.999761i $$-0.506966\pi$$
−0.0218841 + 0.999761i $$0.506966\pi$$
$$174$$ 0 0
$$175$$ 4.50973 0.340903
$$176$$ −4.33763 −0.326961
$$177$$ 0 0
$$178$$ −7.69471 −0.576743
$$179$$ −5.01945 −0.375171 −0.187586 0.982248i $$-0.560066\pi$$
−0.187586 + 0.982248i $$0.560066\pi$$
$$180$$ 0 0
$$181$$ −11.5292 −0.856957 −0.428479 0.903552i $$-0.640950\pi$$
−0.428479 + 0.903552i $$0.640950\pi$$
$$182$$ 16.8137 1.24632
$$183$$ 0 0
$$184$$ −1.00000 −0.0737210
$$185$$ −0.781399 −0.0574496
$$186$$ 0 0
$$187$$ 4.85392 0.354954
$$188$$ 11.4567 0.835562
$$189$$ 0 0
$$190$$ −4.50973 −0.327170
$$191$$ 18.7142 1.35411 0.677055 0.735933i $$-0.263256\pi$$
0.677055 + 0.735933i $$0.263256\pi$$
$$192$$ 0 0
$$193$$ 23.4956 1.69125 0.845624 0.533780i $$-0.179229\pi$$
0.845624 + 0.533780i $$0.179229\pi$$
$$194$$ 0.642920 0.0461590
$$195$$ 0 0
$$196$$ 13.3376 0.952688
$$197$$ 18.1385 1.29231 0.646157 0.763205i $$-0.276375\pi$$
0.646157 + 0.763205i $$0.276375\pi$$
$$198$$ 0 0
$$199$$ −23.2575 −1.64868 −0.824340 0.566094i $$-0.808454\pi$$
−0.824340 + 0.566094i $$0.808454\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −8.23805 −0.579627
$$203$$ 37.1514 2.60751
$$204$$ 0 0
$$205$$ −3.90043 −0.272418
$$206$$ −12.3376 −0.859603
$$207$$ 0 0
$$208$$ −3.72833 −0.258513
$$209$$ −19.5615 −1.35310
$$210$$ 0 0
$$211$$ 4.34420 0.299067 0.149533 0.988757i $$-0.452223\pi$$
0.149533 + 0.988757i $$0.452223\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −15.9328 −1.08914
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 7.79428 0.529110
$$218$$ 1.49027 0.100934
$$219$$ 0 0
$$220$$ −4.33763 −0.292443
$$221$$ 4.17210 0.280646
$$222$$ 0 0
$$223$$ 12.4761 0.835462 0.417731 0.908571i $$-0.362825\pi$$
0.417731 + 0.908571i $$0.362825\pi$$
$$224$$ −4.50973 −0.301319
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −15.9328 −1.05749 −0.528747 0.848779i $$-0.677338\pi$$
−0.528747 + 0.848779i $$0.677338\pi$$
$$228$$ 0 0
$$229$$ −3.56280 −0.235436 −0.117718 0.993047i $$-0.537558\pi$$
−0.117718 + 0.993047i $$0.537558\pi$$
$$230$$ −1.00000 −0.0659380
$$231$$ 0 0
$$232$$ −8.23805 −0.540855
$$233$$ 27.4956 1.80129 0.900647 0.434552i $$-0.143093\pi$$
0.900647 + 0.434552i $$0.143093\pi$$
$$234$$ 0 0
$$235$$ 11.4567 0.747350
$$236$$ 2.23805 0.145685
$$237$$ 0 0
$$238$$ 5.04650 0.327116
$$239$$ −10.0389 −0.649363 −0.324681 0.945823i $$-0.605257\pi$$
−0.324681 + 0.945823i $$0.605257\pi$$
$$240$$ 0 0
$$241$$ −23.6947 −1.52631 −0.763155 0.646215i $$-0.776351\pi$$
−0.763155 + 0.646215i $$0.776351\pi$$
$$242$$ −7.81502 −0.502368
$$243$$ 0 0
$$244$$ 3.55623 0.227664
$$245$$ 13.3376 0.852110
$$246$$ 0 0
$$247$$ −16.8137 −1.06983
$$248$$ −1.72833 −0.109749
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −12.4425 −0.785363 −0.392681 0.919675i $$-0.628452\pi$$
−0.392681 + 0.919675i $$0.628452\pi$$
$$252$$ 0 0
$$253$$ −4.33763 −0.272704
$$254$$ 0.675256 0.0423693
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −5.45665 −0.340377 −0.170188 0.985412i $$-0.554438\pi$$
−0.170188 + 0.985412i $$0.554438\pi$$
$$258$$ 0 0
$$259$$ −3.52389 −0.218964
$$260$$ −3.72833 −0.231221
$$261$$ 0 0
$$262$$ −13.6947 −0.846062
$$263$$ −0.138479 −0.00853895 −0.00426948 0.999991i $$-0.501359\pi$$
−0.00426948 + 0.999991i $$0.501359\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ −20.3376 −1.24698
$$267$$ 0 0
$$268$$ 2.43720 0.148876
$$269$$ −14.6753 −0.894766 −0.447383 0.894342i $$-0.647644\pi$$
−0.447383 + 0.894342i $$0.647644\pi$$
$$270$$ 0 0
$$271$$ −8.31058 −0.504832 −0.252416 0.967619i $$-0.581225\pi$$
−0.252416 + 0.967619i $$0.581225\pi$$
$$272$$ −1.11903 −0.0678510
$$273$$ 0 0
$$274$$ 7.52918 0.454854
$$275$$ −4.33763 −0.261569
$$276$$ 0 0
$$277$$ 12.9133 0.775886 0.387943 0.921683i $$-0.373186\pi$$
0.387943 + 0.921683i $$0.373186\pi$$
$$278$$ −4.67526 −0.280403
$$279$$ 0 0
$$280$$ −4.50973 −0.269508
$$281$$ −2.67526 −0.159592 −0.0797962 0.996811i $$-0.525427\pi$$
−0.0797962 + 0.996811i $$0.525427\pi$$
$$282$$ 0 0
$$283$$ 0.742495 0.0441367 0.0220684 0.999756i $$-0.492975\pi$$
0.0220684 + 0.999756i $$0.492975\pi$$
$$284$$ −7.11903 −0.422437
$$285$$ 0 0
$$286$$ −16.1721 −0.956276
$$287$$ −17.5898 −1.03830
$$288$$ 0 0
$$289$$ −15.7478 −0.926340
$$290$$ −8.23805 −0.483755
$$291$$ 0 0
$$292$$ −9.45665 −0.553409
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ 2.23805 0.130305
$$296$$ 0.781399 0.0454179
$$297$$ 0 0
$$298$$ 7.52918 0.436154
$$299$$ −3.72833 −0.215615
$$300$$ 0 0
$$301$$ 36.0778 2.07949
$$302$$ 13.3571 0.768614
$$303$$ 0 0
$$304$$ 4.50973 0.258651
$$305$$ 3.55623 0.203629
$$306$$ 0 0
$$307$$ 30.5084 1.74121 0.870604 0.491984i $$-0.163728\pi$$
0.870604 + 0.491984i $$0.163728\pi$$
$$308$$ −19.5615 −1.11462
$$309$$ 0 0
$$310$$ −1.72833 −0.0981624
$$311$$ −5.56280 −0.315437 −0.157719 0.987484i $$-0.550414\pi$$
−0.157719 + 0.987484i $$0.550414\pi$$
$$312$$ 0 0
$$313$$ 4.07252 0.230193 0.115096 0.993354i $$-0.463282\pi$$
0.115096 + 0.993354i $$0.463282\pi$$
$$314$$ −16.2381 −0.916366
$$315$$ 0 0
$$316$$ −14.9133 −0.838939
$$317$$ 6.16553 0.346291 0.173145 0.984896i $$-0.444607\pi$$
0.173145 + 0.984896i $$0.444607\pi$$
$$318$$ 0 0
$$319$$ −35.7336 −2.00070
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −4.50973 −0.251317
$$323$$ −5.04650 −0.280795
$$324$$ 0 0
$$325$$ −3.72833 −0.206810
$$326$$ 3.29112 0.182279
$$327$$ 0 0
$$328$$ 3.90043 0.215365
$$329$$ 51.6664 2.84846
$$330$$ 0 0
$$331$$ 27.5886 1.51640 0.758202 0.652019i $$-0.226078\pi$$
0.758202 + 0.652019i $$0.226078\pi$$
$$332$$ −2.78140 −0.152649
$$333$$ 0 0
$$334$$ 22.9133 1.25376
$$335$$ 2.43720 0.133159
$$336$$ 0 0
$$337$$ −17.4230 −0.949093 −0.474547 0.880230i $$-0.657388\pi$$
−0.474547 + 0.880230i $$0.657388\pi$$
$$338$$ −0.900425 −0.0489767
$$339$$ 0 0
$$340$$ −1.11903 −0.0606877
$$341$$ −7.49684 −0.405977
$$342$$ 0 0
$$343$$ 28.5810 1.54323
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 0.575681 0.0309488
$$347$$ −4.88097 −0.262024 −0.131012 0.991381i $$-0.541823\pi$$
−0.131012 + 0.991381i $$0.541823\pi$$
$$348$$ 0 0
$$349$$ 24.0389 1.28677 0.643387 0.765542i $$-0.277529\pi$$
0.643387 + 0.765542i $$0.277529\pi$$
$$350$$ −4.50973 −0.241055
$$351$$ 0 0
$$352$$ 4.33763 0.231196
$$353$$ 14.3442 0.763464 0.381732 0.924273i $$-0.375328\pi$$
0.381732 + 0.924273i $$0.375328\pi$$
$$354$$ 0 0
$$355$$ −7.11903 −0.377839
$$356$$ 7.69471 0.407819
$$357$$ 0 0
$$358$$ 5.01945 0.265286
$$359$$ −26.7814 −1.41347 −0.706734 0.707479i $$-0.749832\pi$$
−0.706734 + 0.707479i $$0.749832\pi$$
$$360$$ 0 0
$$361$$ 1.33763 0.0704015
$$362$$ 11.5292 0.605960
$$363$$ 0 0
$$364$$ −16.8137 −0.881279
$$365$$ −9.45665 −0.494984
$$366$$ 0 0
$$367$$ −20.4761 −1.06884 −0.534422 0.845218i $$-0.679471\pi$$
−0.534422 + 0.845218i $$0.679471\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 0 0
$$370$$ 0.781399 0.0406230
$$371$$ 27.0584 1.40480
$$372$$ 0 0
$$373$$ −3.89386 −0.201616 −0.100808 0.994906i $$-0.532143\pi$$
−0.100808 + 0.994906i $$0.532143\pi$$
$$374$$ −4.85392 −0.250990
$$375$$ 0 0
$$376$$ −11.4567 −0.590832
$$377$$ −30.7142 −1.58186
$$378$$ 0 0
$$379$$ −30.3765 −1.56034 −0.780169 0.625569i $$-0.784867\pi$$
−0.780169 + 0.625569i $$0.784867\pi$$
$$380$$ 4.50973 0.231344
$$381$$ 0 0
$$382$$ −18.7142 −0.957500
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −19.5615 −0.996947
$$386$$ −23.4956 −1.19589
$$387$$ 0 0
$$388$$ −0.642920 −0.0326393
$$389$$ 18.6818 0.947206 0.473603 0.880738i $$-0.342953\pi$$
0.473603 + 0.880738i $$0.342953\pi$$
$$390$$ 0 0
$$391$$ −1.11903 −0.0565916
$$392$$ −13.3376 −0.673652
$$393$$ 0 0
$$394$$ −18.1385 −0.913803
$$395$$ −14.9133 −0.750370
$$396$$ 0 0
$$397$$ −28.5757 −1.43417 −0.717086 0.696985i $$-0.754525\pi$$
−0.717086 + 0.696985i $$0.754525\pi$$
$$398$$ 23.2575 1.16579
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −12.1061 −0.604552 −0.302276 0.953220i $$-0.597746\pi$$
−0.302276 + 0.953220i $$0.597746\pi$$
$$402$$ 0 0
$$403$$ −6.44377 −0.320987
$$404$$ 8.23805 0.409858
$$405$$ 0 0
$$406$$ −37.1514 −1.84379
$$407$$ 3.38942 0.168007
$$408$$ 0 0
$$409$$ 25.2911 1.25057 0.625283 0.780398i $$-0.284984\pi$$
0.625283 + 0.780398i $$0.284984\pi$$
$$410$$ 3.90043 0.192628
$$411$$ 0 0
$$412$$ 12.3376 0.607831
$$413$$ 10.0930 0.496644
$$414$$ 0 0
$$415$$ −2.78140 −0.136533
$$416$$ 3.72833 0.182796
$$417$$ 0 0
$$418$$ 19.5615 0.956785
$$419$$ 17.3505 0.847628 0.423814 0.905749i $$-0.360691\pi$$
0.423814 + 0.905749i $$0.360691\pi$$
$$420$$ 0 0
$$421$$ 21.4230 1.04409 0.522047 0.852916i $$-0.325168\pi$$
0.522047 + 0.852916i $$0.325168\pi$$
$$422$$ −4.34420 −0.211472
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ −1.11903 −0.0542808
$$426$$ 0 0
$$427$$ 16.0376 0.776115
$$428$$ 15.9328 0.770139
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ −22.5822 −1.08775 −0.543874 0.839167i $$-0.683043\pi$$
−0.543874 + 0.839167i $$0.683043\pi$$
$$432$$ 0 0
$$433$$ 1.01417 0.0487378 0.0243689 0.999703i $$-0.492242\pi$$
0.0243689 + 0.999703i $$0.492242\pi$$
$$434$$ −7.79428 −0.374138
$$435$$ 0 0
$$436$$ −1.49027 −0.0713712
$$437$$ 4.50973 0.215729
$$438$$ 0 0
$$439$$ −26.7478 −1.27660 −0.638301 0.769787i $$-0.720362\pi$$
−0.638301 + 0.769787i $$0.720362\pi$$
$$440$$ 4.33763 0.206788
$$441$$ 0 0
$$442$$ −4.17210 −0.198446
$$443$$ −10.2044 −0.484827 −0.242414 0.970173i $$-0.577939\pi$$
−0.242414 + 0.970173i $$0.577939\pi$$
$$444$$ 0 0
$$445$$ 7.69471 0.364764
$$446$$ −12.4761 −0.590761
$$447$$ 0 0
$$448$$ 4.50973 0.213065
$$449$$ −38.7867 −1.83046 −0.915228 0.402936i $$-0.867990\pi$$
−0.915228 + 0.402936i $$0.867990\pi$$
$$450$$ 0 0
$$451$$ 16.9186 0.796665
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 15.9328 0.747762
$$455$$ −16.8137 −0.788240
$$456$$ 0 0
$$457$$ 34.9522 1.63500 0.817498 0.575932i $$-0.195361\pi$$
0.817498 + 0.575932i $$0.195361\pi$$
$$458$$ 3.56280 0.166479
$$459$$ 0 0
$$460$$ 1.00000 0.0466252
$$461$$ −16.3700 −0.762425 −0.381213 0.924487i $$-0.624493\pi$$
−0.381213 + 0.924487i $$0.624493\pi$$
$$462$$ 0 0
$$463$$ 29.2186 1.35790 0.678952 0.734183i $$-0.262435\pi$$
0.678952 + 0.734183i $$0.262435\pi$$
$$464$$ 8.23805 0.382442
$$465$$ 0 0
$$466$$ −27.4956 −1.27371
$$467$$ −24.2770 −1.12340 −0.561702 0.827340i $$-0.689853\pi$$
−0.561702 + 0.827340i $$0.689853\pi$$
$$468$$ 0 0
$$469$$ 10.9911 0.507523
$$470$$ −11.4567 −0.528456
$$471$$ 0 0
$$472$$ −2.23805 −0.103015
$$473$$ −34.7010 −1.59555
$$474$$ 0 0
$$475$$ 4.50973 0.206920
$$476$$ −5.04650 −0.231306
$$477$$ 0 0
$$478$$ 10.0389 0.459169
$$479$$ −24.6080 −1.12437 −0.562185 0.827012i $$-0.690039\pi$$
−0.562185 + 0.827012i $$0.690039\pi$$
$$480$$ 0 0
$$481$$ 2.91331 0.132835
$$482$$ 23.6947 1.07926
$$483$$ 0 0
$$484$$ 7.81502 0.355228
$$485$$ −0.642920 −0.0291935
$$486$$ 0 0
$$487$$ −30.2381 −1.37022 −0.685108 0.728441i $$-0.740245\pi$$
−0.685108 + 0.728441i $$0.740245\pi$$
$$488$$ −3.55623 −0.160983
$$489$$ 0 0
$$490$$ −13.3376 −0.602533
$$491$$ −12.3311 −0.556493 −0.278246 0.960510i $$-0.589753\pi$$
−0.278246 + 0.960510i $$0.589753\pi$$
$$492$$ 0 0
$$493$$ −9.21860 −0.415185
$$494$$ 16.8137 0.756486
$$495$$ 0 0
$$496$$ 1.72833 0.0776042
$$497$$ −32.1049 −1.44010
$$498$$ 0 0
$$499$$ −26.9133 −1.20481 −0.602403 0.798192i $$-0.705790\pi$$
−0.602403 + 0.798192i $$0.705790\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 12.4425 0.555335
$$503$$ −20.5097 −0.914483 −0.457242 0.889342i $$-0.651163\pi$$
−0.457242 + 0.889342i $$0.651163\pi$$
$$504$$ 0 0
$$505$$ 8.23805 0.366589
$$506$$ 4.33763 0.192831
$$507$$ 0 0
$$508$$ −0.675256 −0.0299597
$$509$$ 36.7142 1.62733 0.813663 0.581336i $$-0.197470\pi$$
0.813663 + 0.581336i $$0.197470\pi$$
$$510$$ 0 0
$$511$$ −42.6469 −1.88659
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 5.45665 0.240683
$$515$$ 12.3376 0.543661
$$516$$ 0 0
$$517$$ −49.6947 −2.18557
$$518$$ 3.52389 0.154831
$$519$$ 0 0
$$520$$ 3.72833 0.163498
$$521$$ 4.91331 0.215256 0.107628 0.994191i $$-0.465674\pi$$
0.107628 + 0.994191i $$0.465674\pi$$
$$522$$ 0 0
$$523$$ −0.344196 −0.0150506 −0.00752531 0.999972i $$-0.502395\pi$$
−0.00752531 + 0.999972i $$0.502395\pi$$
$$524$$ 13.6947 0.598256
$$525$$ 0 0
$$526$$ 0.138479 0.00603795
$$527$$ −1.93404 −0.0842483
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ 20.3376 0.881748
$$533$$ 14.5421 0.629887
$$534$$ 0 0
$$535$$ 15.9328 0.688833
$$536$$ −2.43720 −0.105271
$$537$$ 0 0
$$538$$ 14.6753 0.632695
$$539$$ −57.8537 −2.49193
$$540$$ 0 0
$$541$$ −6.13191 −0.263631 −0.131816 0.991274i $$-0.542081\pi$$
−0.131816 + 0.991274i $$0.542081\pi$$
$$542$$ 8.31058 0.356970
$$543$$ 0 0
$$544$$ 1.11903 0.0479779
$$545$$ −1.49027 −0.0638363
$$546$$ 0 0
$$547$$ −9.18498 −0.392721 −0.196361 0.980532i $$-0.562912\pi$$
−0.196361 + 0.980532i $$0.562912\pi$$
$$548$$ −7.52918 −0.321631
$$549$$ 0 0
$$550$$ 4.33763 0.184957
$$551$$ 37.1514 1.58270
$$552$$ 0 0
$$553$$ −67.2549 −2.85997
$$554$$ −12.9133 −0.548634
$$555$$ 0 0
$$556$$ 4.67526 0.198275
$$557$$ 4.30529 0.182421 0.0912105 0.995832i $$-0.470926\pi$$
0.0912105 + 0.995832i $$0.470926\pi$$
$$558$$ 0 0
$$559$$ −29.8266 −1.26153
$$560$$ 4.50973 0.190571
$$561$$ 0 0
$$562$$ 2.67526 0.112849
$$563$$ −11.1256 −0.468888 −0.234444 0.972130i $$-0.575327\pi$$
−0.234444 + 0.972130i $$0.575327\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ −0.742495 −0.0312094
$$567$$ 0 0
$$568$$ 7.11903 0.298708
$$569$$ 16.0389 0.672386 0.336193 0.941793i $$-0.390861\pi$$
0.336193 + 0.941793i $$0.390861\pi$$
$$570$$ 0 0
$$571$$ 17.9004 0.749109 0.374555 0.927205i $$-0.377796\pi$$
0.374555 + 0.927205i $$0.377796\pi$$
$$572$$ 16.1721 0.676189
$$573$$ 0 0
$$574$$ 17.5898 0.734186
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −9.12559 −0.379903 −0.189952 0.981793i $$-0.560833\pi$$
−0.189952 + 0.981793i $$0.560833\pi$$
$$578$$ 15.7478 0.655021
$$579$$ 0 0
$$580$$ 8.23805 0.342067
$$581$$ −12.5433 −0.520386
$$582$$ 0 0
$$583$$ −26.0258 −1.07788
$$584$$ 9.45665 0.391319
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 33.6340 1.38823 0.694113 0.719866i $$-0.255797\pi$$
0.694113 + 0.719866i $$0.255797\pi$$
$$588$$ 0 0
$$589$$ 7.79428 0.321158
$$590$$ −2.23805 −0.0921392
$$591$$ 0 0
$$592$$ −0.781399 −0.0321153
$$593$$ 17.4567 0.716859 0.358429 0.933557i $$-0.383312\pi$$
0.358429 + 0.933557i $$0.383312\pi$$
$$594$$ 0 0
$$595$$ −5.04650 −0.206886
$$596$$ −7.52918 −0.308407
$$597$$ 0 0
$$598$$ 3.72833 0.152463
$$599$$ 11.5951 0.473764 0.236882 0.971538i $$-0.423874\pi$$
0.236882 + 0.971538i $$0.423874\pi$$
$$600$$ 0 0
$$601$$ −31.6611 −1.29148 −0.645741 0.763556i $$-0.723452\pi$$
−0.645741 + 0.763556i $$0.723452\pi$$
$$602$$ −36.0778 −1.47042
$$603$$ 0 0
$$604$$ −13.3571 −0.543492
$$605$$ 7.81502 0.317726
$$606$$ 0 0
$$607$$ −36.0778 −1.46435 −0.732177 0.681115i $$-0.761495\pi$$
−0.732177 + 0.681115i $$0.761495\pi$$
$$608$$ −4.50973 −0.182894
$$609$$ 0 0
$$610$$ −3.55623 −0.143988
$$611$$ −42.7142 −1.72803
$$612$$ 0 0
$$613$$ −32.0389 −1.29404 −0.647020 0.762473i $$-0.723985\pi$$
−0.647020 + 0.762473i $$0.723985\pi$$
$$614$$ −30.5084 −1.23122
$$615$$ 0 0
$$616$$ 19.5615 0.788156
$$617$$ 13.3960 0.539302 0.269651 0.962958i $$-0.413092\pi$$
0.269651 + 0.962958i $$0.413092\pi$$
$$618$$ 0 0
$$619$$ 37.1309 1.49242 0.746208 0.665713i $$-0.231872\pi$$
0.746208 + 0.665713i $$0.231872\pi$$
$$620$$ 1.72833 0.0694113
$$621$$ 0 0
$$622$$ 5.56280 0.223048
$$623$$ 34.7010 1.39027
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −4.07252 −0.162771
$$627$$ 0 0
$$628$$ 16.2381 0.647969
$$629$$ 0.874406 0.0348648
$$630$$ 0 0
$$631$$ 11.1125 0.442380 0.221190 0.975231i $$-0.429006\pi$$
0.221190 + 0.975231i $$0.429006\pi$$
$$632$$ 14.9133 0.593220
$$633$$ 0 0
$$634$$ −6.16553 −0.244864
$$635$$ −0.675256 −0.0267967
$$636$$ 0 0
$$637$$ −49.7270 −1.97026
$$638$$ 35.7336 1.41471
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −12.3831 −0.489103 −0.244552 0.969636i $$-0.578641\pi$$
−0.244552 + 0.969636i $$0.578641\pi$$
$$642$$ 0 0
$$643$$ −37.4956 −1.47868 −0.739340 0.673332i $$-0.764862\pi$$
−0.739340 + 0.673332i $$0.764862\pi$$
$$644$$ 4.50973 0.177708
$$645$$ 0 0
$$646$$ 5.04650 0.198552
$$647$$ −14.5691 −0.572771 −0.286385 0.958114i $$-0.592454\pi$$
−0.286385 + 0.958114i $$0.592454\pi$$
$$648$$ 0 0
$$649$$ −9.70784 −0.381066
$$650$$ 3.72833 0.146237
$$651$$ 0 0
$$652$$ −3.29112 −0.128890
$$653$$ 4.41672 0.172840 0.0864198 0.996259i $$-0.472457\pi$$
0.0864198 + 0.996259i $$0.472457\pi$$
$$654$$ 0 0
$$655$$ 13.6947 0.535097
$$656$$ −3.90043 −0.152286
$$657$$ 0 0
$$658$$ −51.6664 −2.01416
$$659$$ 31.8655 1.24130 0.620652 0.784086i $$-0.286868\pi$$
0.620652 + 0.784086i $$0.286868\pi$$
$$660$$ 0 0
$$661$$ 33.1190 1.28818 0.644090 0.764949i $$-0.277236\pi$$
0.644090 + 0.764949i $$0.277236\pi$$
$$662$$ −27.5886 −1.07226
$$663$$ 0 0
$$664$$ 2.78140 0.107939
$$665$$ 20.3376 0.788659
$$666$$ 0 0
$$667$$ 8.23805 0.318979
$$668$$ −22.9133 −0.886543
$$669$$ 0 0
$$670$$ −2.43720 −0.0941574
$$671$$ −15.4256 −0.595499
$$672$$ 0 0
$$673$$ 19.3505 0.745907 0.372954 0.927850i $$-0.378345\pi$$
0.372954 + 0.927850i $$0.378345\pi$$
$$674$$ 17.4230 0.671110
$$675$$ 0 0
$$676$$ 0.900425 0.0346317
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ −2.89939 −0.111268
$$680$$ 1.11903 0.0429127
$$681$$ 0 0
$$682$$ 7.49684 0.287069
$$683$$ −38.3495 −1.46740 −0.733701 0.679472i $$-0.762209\pi$$
−0.733701 + 0.679472i $$0.762209\pi$$
$$684$$ 0 0
$$685$$ −7.52918 −0.287675
$$686$$ −28.5810 −1.09123
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ −22.3700 −0.852228
$$690$$ 0 0
$$691$$ −21.0195 −0.799618 −0.399809 0.916599i $$-0.630923\pi$$
−0.399809 + 0.916599i $$0.630923\pi$$
$$692$$ −0.575681 −0.0218841
$$693$$ 0 0
$$694$$ 4.88097 0.185279
$$695$$ 4.67526 0.177343
$$696$$ 0 0
$$697$$ 4.36468 0.165324
$$698$$ −24.0389 −0.909886
$$699$$ 0 0
$$700$$ 4.50973 0.170452
$$701$$ −19.3169 −0.729589 −0.364794 0.931088i $$-0.618861\pi$$
−0.364794 + 0.931088i $$0.618861\pi$$
$$702$$ 0 0
$$703$$ −3.52389 −0.132906
$$704$$ −4.33763 −0.163481
$$705$$ 0 0
$$706$$ −14.3442 −0.539851
$$707$$ 37.1514 1.39722
$$708$$ 0 0
$$709$$ 12.2315 0.459363 0.229682 0.973266i $$-0.426232\pi$$
0.229682 + 0.973266i $$0.426232\pi$$
$$710$$ 7.11903 0.267172
$$711$$ 0 0
$$712$$ −7.69471 −0.288371
$$713$$ 1.72833 0.0647264
$$714$$ 0 0
$$715$$ 16.1721 0.604802
$$716$$ −5.01945 −0.187586
$$717$$ 0 0
$$718$$ 26.7814 0.999473
$$719$$ 40.6416 1.51568 0.757839 0.652442i $$-0.226255\pi$$
0.757839 + 0.652442i $$0.226255\pi$$
$$720$$ 0 0
$$721$$ 55.6393 2.07212
$$722$$ −1.33763 −0.0497814
$$723$$ 0 0
$$724$$ −11.5292 −0.428479
$$725$$ 8.23805 0.305954
$$726$$ 0 0
$$727$$ −23.4501 −0.869716 −0.434858 0.900499i $$-0.643201\pi$$
−0.434858 + 0.900499i $$0.643201\pi$$
$$728$$ 16.8137 0.623158
$$729$$ 0 0
$$730$$ 9.45665 0.350006
$$731$$ −8.95221 −0.331110
$$732$$ 0 0
$$733$$ −12.5150 −0.462252 −0.231126 0.972924i $$-0.574241\pi$$
−0.231126 + 0.972924i $$0.574241\pi$$
$$734$$ 20.4761 0.755787
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ −10.5717 −0.389413
$$738$$ 0 0
$$739$$ −21.3505 −0.785391 −0.392696 0.919668i $$-0.628457\pi$$
−0.392696 + 0.919668i $$0.628457\pi$$
$$740$$ −0.781399 −0.0287248
$$741$$ 0 0
$$742$$ −27.0584 −0.993343
$$743$$ −24.9858 −0.916641 −0.458321 0.888787i $$-0.651549\pi$$
−0.458321 + 0.888787i $$0.651549\pi$$
$$744$$ 0 0
$$745$$ −7.52918 −0.275848
$$746$$ 3.89386 0.142564
$$747$$ 0 0
$$748$$ 4.85392 0.177477
$$749$$ 71.8524 2.62543
$$750$$ 0 0
$$751$$ −33.6275 −1.22708 −0.613542 0.789662i $$-0.710256\pi$$
−0.613542 + 0.789662i $$0.710256\pi$$
$$752$$ 11.4567 0.417781
$$753$$ 0 0
$$754$$ 30.7142 1.11854
$$755$$ −13.3571 −0.486114
$$756$$ 0 0
$$757$$ −37.1230 −1.34926 −0.674630 0.738156i $$-0.735697\pi$$
−0.674630 + 0.738156i $$0.735697\pi$$
$$758$$ 30.3765 1.10333
$$759$$ 0 0
$$760$$ −4.50973 −0.163585
$$761$$ 3.87337 0.140410 0.0702048 0.997533i $$-0.477635\pi$$
0.0702048 + 0.997533i $$0.477635\pi$$
$$762$$ 0 0
$$763$$ −6.72073 −0.243307
$$764$$ 18.7142 0.677055
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −8.34420 −0.301291
$$768$$ 0 0
$$769$$ 23.1645 0.835333 0.417667 0.908600i $$-0.362848\pi$$
0.417667 + 0.908600i $$0.362848\pi$$
$$770$$ 19.5615 0.704948
$$771$$ 0 0
$$772$$ 23.4956 0.845624
$$773$$ 8.78140 0.315845 0.157922 0.987452i $$-0.449520\pi$$
0.157922 + 0.987452i $$0.449520\pi$$
$$774$$ 0 0
$$775$$ 1.72833 0.0620834
$$776$$ 0.642920 0.0230795
$$777$$ 0 0
$$778$$ −18.6818 −0.669776
$$779$$ −17.5898 −0.630222
$$780$$ 0 0
$$781$$ 30.8797 1.10496
$$782$$ 1.11903 0.0400163
$$783$$ 0 0
$$784$$ 13.3376 0.476344
$$785$$ 16.2381 0.579561
$$786$$ 0 0
$$787$$ 49.6275 1.76903 0.884514 0.466513i $$-0.154490\pi$$
0.884514 + 0.466513i $$0.154490\pi$$
$$788$$ 18.1385 0.646157
$$789$$ 0 0
$$790$$ 14.9133 0.530592
$$791$$ 27.0584 0.962084
$$792$$ 0 0
$$793$$ −13.2588 −0.470833
$$794$$ 28.5757 1.01411
$$795$$ 0 0
$$796$$ −23.2575 −0.824340
$$797$$ −18.3311 −0.649319 −0.324660 0.945831i $$-0.605250\pi$$
−0.324660 + 0.945831i $$0.605250\pi$$
$$798$$ 0 0
$$799$$ −12.8203 −0.453550
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 12.1061 0.427483
$$803$$ 41.0195 1.44755
$$804$$ 0 0
$$805$$ 4.50973 0.158947
$$806$$ 6.44377 0.226972
$$807$$ 0 0
$$808$$ −8.23805 −0.289814
$$809$$ 1.93933 0.0681832 0.0340916 0.999419i $$-0.489146\pi$$
0.0340916 + 0.999419i $$0.489146\pi$$
$$810$$ 0 0
$$811$$ −5.41775 −0.190243 −0.0951215 0.995466i $$-0.530324\pi$$
−0.0951215 + 0.995466i $$0.530324\pi$$
$$812$$ 37.1514 1.30376
$$813$$ 0 0
$$814$$ −3.38942 −0.118799
$$815$$ −3.29112 −0.115283
$$816$$ 0 0
$$817$$ 36.0778 1.26220
$$818$$ −25.2911 −0.884283
$$819$$ 0 0
$$820$$ −3.90043 −0.136209
$$821$$ −37.8655 −1.32152 −0.660758 0.750599i $$-0.729765\pi$$
−0.660758 + 0.750599i $$0.729765\pi$$
$$822$$ 0 0
$$823$$ 43.7336 1.52446 0.762229 0.647308i $$-0.224105\pi$$
0.762229 + 0.647308i $$0.224105\pi$$
$$824$$ −12.3376 −0.429802
$$825$$ 0 0
$$826$$ −10.0930 −0.351181
$$827$$ −28.1991 −0.980581 −0.490290 0.871559i $$-0.663109\pi$$
−0.490290 + 0.871559i $$0.663109\pi$$
$$828$$ 0 0
$$829$$ 1.12559 0.0390935 0.0195468 0.999809i $$-0.493778\pi$$
0.0195468 + 0.999809i $$0.493778\pi$$
$$830$$ 2.78140 0.0965438
$$831$$ 0 0
$$832$$ −3.72833 −0.129256
$$833$$ −14.9252 −0.517126
$$834$$ 0 0
$$835$$ −22.9133 −0.792948
$$836$$ −19.5615 −0.676549
$$837$$ 0 0
$$838$$ −17.3505 −0.599364
$$839$$ −39.9328 −1.37863 −0.689316 0.724461i $$-0.742089\pi$$
−0.689316 + 0.724461i $$0.742089\pi$$
$$840$$ 0 0
$$841$$ 38.8655 1.34019
$$842$$ −21.4230 −0.738287
$$843$$ 0 0
$$844$$ 4.34420 0.149533
$$845$$ 0.900425 0.0309756
$$846$$ 0 0
$$847$$ 35.2436 1.21098
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 1.11903 0.0383823
$$851$$ −0.781399 −0.0267860
$$852$$ 0 0
$$853$$ 47.9921 1.64322 0.821610 0.570050i $$-0.193076\pi$$
0.821610 + 0.570050i $$0.193076\pi$$
$$854$$ −16.0376 −0.548796
$$855$$ 0 0
$$856$$ −15.9328 −0.544571
$$857$$ 43.4283 1.48348 0.741742 0.670686i $$-0.234000\pi$$
0.741742 + 0.670686i $$0.234000\pi$$
$$858$$ 0 0
$$859$$ 32.5433 1.11036 0.555182 0.831729i $$-0.312648\pi$$
0.555182 + 0.831729i $$0.312648\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ 22.5822 0.769154
$$863$$ −46.1036 −1.56938 −0.784692 0.619886i $$-0.787179\pi$$
−0.784692 + 0.619886i $$0.787179\pi$$
$$864$$ 0 0
$$865$$ −0.575681 −0.0195738
$$866$$ −1.01417 −0.0344628
$$867$$ 0 0
$$868$$ 7.79428 0.264555
$$869$$ 64.6884 2.19440
$$870$$ 0 0
$$871$$ −9.08669 −0.307891
$$872$$ 1.49027 0.0504670
$$873$$ 0 0
$$874$$ −4.50973 −0.152544
$$875$$ 4.50973 0.152457
$$876$$ 0 0
$$877$$ −24.0996 −0.813785 −0.406892 0.913476i $$-0.633388\pi$$
−0.406892 + 0.913476i $$0.633388\pi$$
$$878$$ 26.7478 0.902694
$$879$$ 0 0
$$880$$ −4.33763 −0.146221
$$881$$ −2.34420 −0.0789780 −0.0394890 0.999220i $$-0.512573\pi$$
−0.0394890 + 0.999220i $$0.512573\pi$$
$$882$$ 0 0
$$883$$ −41.0505 −1.38146 −0.690730 0.723113i $$-0.742711\pi$$
−0.690730 + 0.723113i $$0.742711\pi$$
$$884$$ 4.17210 0.140323
$$885$$ 0 0
$$886$$ 10.2044 0.342825
$$887$$ −54.7788 −1.83929 −0.919647 0.392747i $$-0.871525\pi$$
−0.919647 + 0.392747i $$0.871525\pi$$
$$888$$ 0 0
$$889$$ −3.04522 −0.102133
$$890$$ −7.69471 −0.257927
$$891$$ 0 0
$$892$$ 12.4761 0.417731
$$893$$ 51.6664 1.72895
$$894$$ 0 0
$$895$$ −5.01945 −0.167782
$$896$$ −4.50973 −0.150659
$$897$$ 0 0
$$898$$ 38.7867 1.29433
$$899$$ 14.2381 0.474866
$$900$$ 0 0
$$901$$ −6.71416 −0.223681
$$902$$ −16.9186 −0.563328
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ −11.5292 −0.383243
$$906$$ 0 0
$$907$$ −10.1061 −0.335569 −0.167784 0.985824i $$-0.553661\pi$$
−0.167784 + 0.985824i $$0.553661\pi$$
$$908$$ −15.9328 −0.528747
$$909$$ 0 0
$$910$$ 16.8137 0.557370
$$911$$ 25.4178 0.842128 0.421064 0.907031i $$-0.361657\pi$$
0.421064 + 0.907031i $$0.361657\pi$$
$$912$$ 0 0
$$913$$ 12.0647 0.399282
$$914$$ −34.9522 −1.15612
$$915$$ 0 0
$$916$$ −3.56280 −0.117718
$$917$$ 61.7594 2.03947
$$918$$ 0 0
$$919$$ 23.6017 0.778548 0.389274 0.921122i $$-0.372726\pi$$
0.389274 + 0.921122i $$0.372726\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ 0 0
$$922$$ 16.3700 0.539116
$$923$$ 26.5421 0.873643
$$924$$ 0 0
$$925$$ −0.781399 −0.0256922
$$926$$ −29.2186 −0.960183
$$927$$ 0 0
$$928$$ −8.23805 −0.270427
$$929$$ 7.08669 0.232507 0.116253 0.993220i $$-0.462912\pi$$
0.116253 + 0.993220i $$0.462912\pi$$
$$930$$ 0 0
$$931$$ 60.1490 1.97131
$$932$$ 27.4956 0.900647
$$933$$ 0 0
$$934$$ 24.2770 0.794366
$$935$$ 4.85392 0.158740
$$936$$ 0 0
$$937$$ 27.3169 0.892404 0.446202 0.894932i $$-0.352776\pi$$
0.446202 + 0.894932i $$0.352776\pi$$
$$938$$ −10.9911 −0.358873
$$939$$ 0 0
$$940$$ 11.4567 0.373675
$$941$$ −55.8979 −1.82222 −0.911109 0.412165i $$-0.864773\pi$$
−0.911109 + 0.412165i $$0.864773\pi$$
$$942$$ 0 0
$$943$$ −3.90043 −0.127015
$$944$$ 2.23805 0.0728424
$$945$$ 0 0
$$946$$ 34.7010 1.12823
$$947$$ −37.5939 −1.22164 −0.610818 0.791771i $$-0.709159\pi$$
−0.610818 + 0.791771i $$0.709159\pi$$
$$948$$ 0 0
$$949$$ 35.2575 1.14451
$$950$$ −4.50973 −0.146315
$$951$$ 0 0
$$952$$ 5.04650 0.163558
$$953$$ 29.3828 0.951804 0.475902 0.879498i $$-0.342122\pi$$
0.475902 + 0.879498i $$0.342122\pi$$
$$954$$ 0 0
$$955$$ 18.7142 0.605576
$$956$$ −10.0389 −0.324681
$$957$$ 0 0
$$958$$ 24.6080 0.795049
$$959$$ −33.9545 −1.09645
$$960$$ 0 0
$$961$$ −28.0129 −0.903641
$$962$$ −2.91331 −0.0939289
$$963$$ 0 0
$$964$$ −23.6947 −0.763155
$$965$$ 23.4956 0.756349
$$966$$ 0 0
$$967$$ −49.2292 −1.58310 −0.791552 0.611102i $$-0.790726\pi$$
−0.791552 + 0.611102i $$0.790726\pi$$
$$968$$ −7.81502 −0.251184
$$969$$ 0 0
$$970$$ 0.642920 0.0206429
$$971$$ −9.62347 −0.308832 −0.154416 0.988006i $$-0.549350\pi$$
−0.154416 + 0.988006i $$0.549350\pi$$
$$972$$ 0 0
$$973$$ 21.0841 0.675926
$$974$$ 30.2381 0.968890
$$975$$ 0 0
$$976$$ 3.55623 0.113832
$$977$$ −18.9858 −0.607411 −0.303705 0.952766i $$-0.598224\pi$$
−0.303705 + 0.952766i $$0.598224\pi$$
$$978$$ 0 0
$$979$$ −33.3768 −1.06673
$$980$$ 13.3376 0.426055
$$981$$ 0 0
$$982$$ 12.3311 0.393500
$$983$$ −4.33763 −0.138349 −0.0691744 0.997605i $$-0.522037\pi$$
−0.0691744 + 0.997605i $$0.522037\pi$$
$$984$$ 0 0
$$985$$ 18.1385 0.577940
$$986$$ 9.21860 0.293580
$$987$$ 0 0
$$988$$ −16.8137 −0.534916
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 1.96766 0.0625049 0.0312524 0.999512i $$-0.490050\pi$$
0.0312524 + 0.999512i $$0.490050\pi$$
$$992$$ −1.72833 −0.0548744
$$993$$ 0 0
$$994$$ 32.1049 1.01830
$$995$$ −23.2575 −0.737312
$$996$$ 0 0
$$997$$ 3.96110 0.125449 0.0627246 0.998031i $$-0.480021\pi$$
0.0627246 + 0.998031i $$0.480021\pi$$
$$998$$ 26.9133 0.851926
$$999$$ 0 0
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 2070.2.a.z.1.3 3
3.2 odd 2 230.2.a.d.1.1 3
12.11 even 2 1840.2.a.r.1.3 3
15.2 even 4 1150.2.b.j.599.6 6
15.8 even 4 1150.2.b.j.599.1 6
15.14 odd 2 1150.2.a.q.1.3 3
24.5 odd 2 7360.2.a.bz.1.3 3
24.11 even 2 7360.2.a.ce.1.1 3
60.59 even 2 9200.2.a.cf.1.1 3
69.68 even 2 5290.2.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 3.2 odd 2
1150.2.a.q.1.3 3 15.14 odd 2
1150.2.b.j.599.1 6 15.8 even 4
1150.2.b.j.599.6 6 15.2 even 4
1840.2.a.r.1.3 3 12.11 even 2
2070.2.a.z.1.3 3 1.1 even 1 trivial
5290.2.a.r.1.1 3 69.68 even 2
7360.2.a.bz.1.3 3 24.5 odd 2
7360.2.a.ce.1.1 3 24.11 even 2
9200.2.a.cf.1.1 3 60.59 even 2