# Properties

 Label 8016.2.a.be.1.3 Level 8016 Weight 2 Character 8016.1 Self dual yes Analytic conductor 64.008 Analytic rank 0 Dimension 11 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8016 = 2^{4} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8016.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.0080822603$$ Analytic rank: $$0$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 4008) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.80306$$ of defining polynomial Character $$\chi$$ $$=$$ 8016.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.80306 q^{5} +4.10861 q^{7} +1.00000 q^{9} +2.59144 q^{11} -3.46748 q^{13} -1.80306 q^{15} +1.68470 q^{17} +2.49774 q^{19} +4.10861 q^{21} +8.91932 q^{23} -1.74898 q^{25} +1.00000 q^{27} -4.42158 q^{29} +2.95084 q^{31} +2.59144 q^{33} -7.40807 q^{35} +0.918154 q^{37} -3.46748 q^{39} -1.83849 q^{41} -0.850997 q^{43} -1.80306 q^{45} +6.74458 q^{47} +9.88072 q^{49} +1.68470 q^{51} +4.65455 q^{53} -4.67252 q^{55} +2.49774 q^{57} -1.23850 q^{59} -7.97208 q^{61} +4.10861 q^{63} +6.25208 q^{65} +1.65708 q^{67} +8.91932 q^{69} +6.24057 q^{71} -4.76409 q^{73} -1.74898 q^{75} +10.6472 q^{77} +10.7193 q^{79} +1.00000 q^{81} +3.69008 q^{83} -3.03761 q^{85} -4.42158 q^{87} -12.0739 q^{89} -14.2466 q^{91} +2.95084 q^{93} -4.50357 q^{95} +14.8662 q^{97} +2.59144 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + O(q^{10})$$ $$11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + q^{11} + 10q^{13} + 10q^{15} + 17q^{17} - 2q^{19} + q^{21} + 3q^{23} + 21q^{25} + 11q^{27} + 17q^{29} + 15q^{31} + q^{33} - 11q^{35} + 4q^{37} + 10q^{39} + 16q^{41} - 10q^{43} + 10q^{45} + 16q^{47} + 22q^{49} + 17q^{51} + 42q^{53} + 5q^{55} - 2q^{57} + 2q^{59} + 12q^{61} + q^{63} + 10q^{65} + q^{67} + 3q^{69} + 9q^{71} + 24q^{73} + 21q^{75} + 22q^{77} + 30q^{79} + 11q^{81} - 16q^{83} + 25q^{85} + 17q^{87} + 37q^{89} - q^{91} + 15q^{93} - 5q^{95} + 4q^{97} + q^{99} + O(q^{100})$$

## 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 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.80306 −0.806352 −0.403176 0.915122i $$-0.632094\pi$$
−0.403176 + 0.915122i $$0.632094\pi$$
$$6$$ 0 0
$$7$$ 4.10861 1.55291 0.776455 0.630172i $$-0.217016\pi$$
0.776455 + 0.630172i $$0.217016\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.59144 0.781349 0.390675 0.920529i $$-0.372242\pi$$
0.390675 + 0.920529i $$0.372242\pi$$
$$12$$ 0 0
$$13$$ −3.46748 −0.961707 −0.480853 0.876801i $$-0.659673\pi$$
−0.480853 + 0.876801i $$0.659673\pi$$
$$14$$ 0 0
$$15$$ −1.80306 −0.465548
$$16$$ 0 0
$$17$$ 1.68470 0.408600 0.204300 0.978908i $$-0.434508\pi$$
0.204300 + 0.978908i $$0.434508\pi$$
$$18$$ 0 0
$$19$$ 2.49774 0.573021 0.286510 0.958077i $$-0.407505\pi$$
0.286510 + 0.958077i $$0.407505\pi$$
$$20$$ 0 0
$$21$$ 4.10861 0.896573
$$22$$ 0 0
$$23$$ 8.91932 1.85981 0.929904 0.367803i $$-0.119890\pi$$
0.929904 + 0.367803i $$0.119890\pi$$
$$24$$ 0 0
$$25$$ −1.74898 −0.349796
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −4.42158 −0.821067 −0.410534 0.911845i $$-0.634658\pi$$
−0.410534 + 0.911845i $$0.634658\pi$$
$$30$$ 0 0
$$31$$ 2.95084 0.529987 0.264993 0.964250i $$-0.414630\pi$$
0.264993 + 0.964250i $$0.414630\pi$$
$$32$$ 0 0
$$33$$ 2.59144 0.451112
$$34$$ 0 0
$$35$$ −7.40807 −1.25219
$$36$$ 0 0
$$37$$ 0.918154 0.150944 0.0754718 0.997148i $$-0.475954\pi$$
0.0754718 + 0.997148i $$0.475954\pi$$
$$38$$ 0 0
$$39$$ −3.46748 −0.555242
$$40$$ 0 0
$$41$$ −1.83849 −0.287125 −0.143562 0.989641i $$-0.545856\pi$$
−0.143562 + 0.989641i $$0.545856\pi$$
$$42$$ 0 0
$$43$$ −0.850997 −0.129776 −0.0648879 0.997893i $$-0.520669\pi$$
−0.0648879 + 0.997893i $$0.520669\pi$$
$$44$$ 0 0
$$45$$ −1.80306 −0.268784
$$46$$ 0 0
$$47$$ 6.74458 0.983798 0.491899 0.870652i $$-0.336303\pi$$
0.491899 + 0.870652i $$0.336303\pi$$
$$48$$ 0 0
$$49$$ 9.88072 1.41153
$$50$$ 0 0
$$51$$ 1.68470 0.235905
$$52$$ 0 0
$$53$$ 4.65455 0.639351 0.319676 0.947527i $$-0.396426\pi$$
0.319676 + 0.947527i $$0.396426\pi$$
$$54$$ 0 0
$$55$$ −4.67252 −0.630043
$$56$$ 0 0
$$57$$ 2.49774 0.330834
$$58$$ 0 0
$$59$$ −1.23850 −0.161239 −0.0806194 0.996745i $$-0.525690\pi$$
−0.0806194 + 0.996745i $$0.525690\pi$$
$$60$$ 0 0
$$61$$ −7.97208 −1.02072 −0.510360 0.859961i $$-0.670488\pi$$
−0.510360 + 0.859961i $$0.670488\pi$$
$$62$$ 0 0
$$63$$ 4.10861 0.517637
$$64$$ 0 0
$$65$$ 6.25208 0.775475
$$66$$ 0 0
$$67$$ 1.65708 0.202445 0.101223 0.994864i $$-0.467725\pi$$
0.101223 + 0.994864i $$0.467725\pi$$
$$68$$ 0 0
$$69$$ 8.91932 1.07376
$$70$$ 0 0
$$71$$ 6.24057 0.740620 0.370310 0.928908i $$-0.379251\pi$$
0.370310 + 0.928908i $$0.379251\pi$$
$$72$$ 0 0
$$73$$ −4.76409 −0.557594 −0.278797 0.960350i $$-0.589936\pi$$
−0.278797 + 0.960350i $$0.589936\pi$$
$$74$$ 0 0
$$75$$ −1.74898 −0.201955
$$76$$ 0 0
$$77$$ 10.6472 1.21337
$$78$$ 0 0
$$79$$ 10.7193 1.20601 0.603007 0.797736i $$-0.293969\pi$$
0.603007 + 0.797736i $$0.293969\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 3.69008 0.405039 0.202519 0.979278i $$-0.435087\pi$$
0.202519 + 0.979278i $$0.435087\pi$$
$$84$$ 0 0
$$85$$ −3.03761 −0.329475
$$86$$ 0 0
$$87$$ −4.42158 −0.474044
$$88$$ 0 0
$$89$$ −12.0739 −1.27983 −0.639916 0.768445i $$-0.721031\pi$$
−0.639916 + 0.768445i $$0.721031\pi$$
$$90$$ 0 0
$$91$$ −14.2466 −1.49344
$$92$$ 0 0
$$93$$ 2.95084 0.305988
$$94$$ 0 0
$$95$$ −4.50357 −0.462057
$$96$$ 0 0
$$97$$ 14.8662 1.50944 0.754719 0.656048i $$-0.227773\pi$$
0.754719 + 0.656048i $$0.227773\pi$$
$$98$$ 0 0
$$99$$ 2.59144 0.260450
$$100$$ 0 0
$$101$$ 1.27675 0.127041 0.0635206 0.997981i $$-0.479767\pi$$
0.0635206 + 0.997981i $$0.479767\pi$$
$$102$$ 0 0
$$103$$ −5.26805 −0.519077 −0.259538 0.965733i $$-0.583570\pi$$
−0.259538 + 0.965733i $$0.583570\pi$$
$$104$$ 0 0
$$105$$ −7.40807 −0.722954
$$106$$ 0 0
$$107$$ −7.81076 −0.755094 −0.377547 0.925990i $$-0.623232\pi$$
−0.377547 + 0.925990i $$0.623232\pi$$
$$108$$ 0 0
$$109$$ −2.01235 −0.192748 −0.0963739 0.995345i $$-0.530724\pi$$
−0.0963739 + 0.995345i $$0.530724\pi$$
$$110$$ 0 0
$$111$$ 0.918154 0.0871473
$$112$$ 0 0
$$113$$ 8.20964 0.772298 0.386149 0.922436i $$-0.373805\pi$$
0.386149 + 0.922436i $$0.373805\pi$$
$$114$$ 0 0
$$115$$ −16.0821 −1.49966
$$116$$ 0 0
$$117$$ −3.46748 −0.320569
$$118$$ 0 0
$$119$$ 6.92178 0.634519
$$120$$ 0 0
$$121$$ −4.28443 −0.389494
$$122$$ 0 0
$$123$$ −1.83849 −0.165771
$$124$$ 0 0
$$125$$ 12.1688 1.08841
$$126$$ 0 0
$$127$$ −9.38074 −0.832406 −0.416203 0.909272i $$-0.636639\pi$$
−0.416203 + 0.909272i $$0.636639\pi$$
$$128$$ 0 0
$$129$$ −0.850997 −0.0749261
$$130$$ 0 0
$$131$$ −19.8572 −1.73493 −0.867465 0.497498i $$-0.834252\pi$$
−0.867465 + 0.497498i $$0.834252\pi$$
$$132$$ 0 0
$$133$$ 10.2623 0.889850
$$134$$ 0 0
$$135$$ −1.80306 −0.155183
$$136$$ 0 0
$$137$$ 5.44295 0.465023 0.232511 0.972594i $$-0.425306\pi$$
0.232511 + 0.972594i $$0.425306\pi$$
$$138$$ 0 0
$$139$$ 13.0858 1.10992 0.554960 0.831877i $$-0.312734\pi$$
0.554960 + 0.831877i $$0.312734\pi$$
$$140$$ 0 0
$$141$$ 6.74458 0.567996
$$142$$ 0 0
$$143$$ −8.98578 −0.751429
$$144$$ 0 0
$$145$$ 7.97238 0.662070
$$146$$ 0 0
$$147$$ 9.88072 0.814948
$$148$$ 0 0
$$149$$ −0.961230 −0.0787470 −0.0393735 0.999225i $$-0.512536\pi$$
−0.0393735 + 0.999225i $$0.512536\pi$$
$$150$$ 0 0
$$151$$ 8.62272 0.701707 0.350853 0.936430i $$-0.385892\pi$$
0.350853 + 0.936430i $$0.385892\pi$$
$$152$$ 0 0
$$153$$ 1.68470 0.136200
$$154$$ 0 0
$$155$$ −5.32054 −0.427356
$$156$$ 0 0
$$157$$ 3.40121 0.271446 0.135723 0.990747i $$-0.456664\pi$$
0.135723 + 0.990747i $$0.456664\pi$$
$$158$$ 0 0
$$159$$ 4.65455 0.369130
$$160$$ 0 0
$$161$$ 36.6461 2.88811
$$162$$ 0 0
$$163$$ 2.70291 0.211708 0.105854 0.994382i $$-0.466242\pi$$
0.105854 + 0.994382i $$0.466242\pi$$
$$164$$ 0 0
$$165$$ −4.67252 −0.363755
$$166$$ 0 0
$$167$$ −1.00000 −0.0773823
$$168$$ 0 0
$$169$$ −0.976559 −0.0751200
$$170$$ 0 0
$$171$$ 2.49774 0.191007
$$172$$ 0 0
$$173$$ 13.7489 1.04531 0.522654 0.852545i $$-0.324942\pi$$
0.522654 + 0.852545i $$0.324942\pi$$
$$174$$ 0 0
$$175$$ −7.18588 −0.543201
$$176$$ 0 0
$$177$$ −1.23850 −0.0930912
$$178$$ 0 0
$$179$$ −0.148844 −0.0111251 −0.00556257 0.999985i $$-0.501771\pi$$
−0.00556257 + 0.999985i $$0.501771\pi$$
$$180$$ 0 0
$$181$$ 7.10022 0.527755 0.263877 0.964556i $$-0.414999\pi$$
0.263877 + 0.964556i $$0.414999\pi$$
$$182$$ 0 0
$$183$$ −7.97208 −0.589313
$$184$$ 0 0
$$185$$ −1.65549 −0.121714
$$186$$ 0 0
$$187$$ 4.36580 0.319259
$$188$$ 0 0
$$189$$ 4.10861 0.298858
$$190$$ 0 0
$$191$$ 5.56545 0.402702 0.201351 0.979519i $$-0.435467\pi$$
0.201351 + 0.979519i $$0.435467\pi$$
$$192$$ 0 0
$$193$$ −16.6812 −1.20074 −0.600371 0.799721i $$-0.704981\pi$$
−0.600371 + 0.799721i $$0.704981\pi$$
$$194$$ 0 0
$$195$$ 6.25208 0.447721
$$196$$ 0 0
$$197$$ 10.0656 0.717145 0.358573 0.933502i $$-0.383264\pi$$
0.358573 + 0.933502i $$0.383264\pi$$
$$198$$ 0 0
$$199$$ 14.2270 1.00853 0.504263 0.863550i $$-0.331764\pi$$
0.504263 + 0.863550i $$0.331764\pi$$
$$200$$ 0 0
$$201$$ 1.65708 0.116882
$$202$$ 0 0
$$203$$ −18.1666 −1.27504
$$204$$ 0 0
$$205$$ 3.31491 0.231524
$$206$$ 0 0
$$207$$ 8.91932 0.619936
$$208$$ 0 0
$$209$$ 6.47275 0.447729
$$210$$ 0 0
$$211$$ −19.4006 −1.33560 −0.667798 0.744343i $$-0.732763\pi$$
−0.667798 + 0.744343i $$0.732763\pi$$
$$212$$ 0 0
$$213$$ 6.24057 0.427597
$$214$$ 0 0
$$215$$ 1.53440 0.104645
$$216$$ 0 0
$$217$$ 12.1239 0.823022
$$218$$ 0 0
$$219$$ −4.76409 −0.321927
$$220$$ 0 0
$$221$$ −5.84167 −0.392953
$$222$$ 0 0
$$223$$ 5.50496 0.368640 0.184320 0.982866i $$-0.440992\pi$$
0.184320 + 0.982866i $$0.440992\pi$$
$$224$$ 0 0
$$225$$ −1.74898 −0.116599
$$226$$ 0 0
$$227$$ 24.7959 1.64576 0.822880 0.568215i $$-0.192366\pi$$
0.822880 + 0.568215i $$0.192366\pi$$
$$228$$ 0 0
$$229$$ −22.8293 −1.50860 −0.754302 0.656528i $$-0.772024\pi$$
−0.754302 + 0.656528i $$0.772024\pi$$
$$230$$ 0 0
$$231$$ 10.6472 0.700537
$$232$$ 0 0
$$233$$ 12.2217 0.800672 0.400336 0.916368i $$-0.368893\pi$$
0.400336 + 0.916368i $$0.368893\pi$$
$$234$$ 0 0
$$235$$ −12.1609 −0.793288
$$236$$ 0 0
$$237$$ 10.7193 0.696293
$$238$$ 0 0
$$239$$ −7.79185 −0.504013 −0.252007 0.967726i $$-0.581090\pi$$
−0.252007 + 0.967726i $$0.581090\pi$$
$$240$$ 0 0
$$241$$ −22.1396 −1.42613 −0.713067 0.701096i $$-0.752695\pi$$
−0.713067 + 0.701096i $$0.752695\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −17.8155 −1.13819
$$246$$ 0 0
$$247$$ −8.66087 −0.551078
$$248$$ 0 0
$$249$$ 3.69008 0.233849
$$250$$ 0 0
$$251$$ 16.1687 1.02056 0.510279 0.860009i $$-0.329542\pi$$
0.510279 + 0.860009i $$0.329542\pi$$
$$252$$ 0 0
$$253$$ 23.1139 1.45316
$$254$$ 0 0
$$255$$ −3.03761 −0.190223
$$256$$ 0 0
$$257$$ 22.3740 1.39565 0.697825 0.716268i $$-0.254151\pi$$
0.697825 + 0.716268i $$0.254151\pi$$
$$258$$ 0 0
$$259$$ 3.77234 0.234402
$$260$$ 0 0
$$261$$ −4.42158 −0.273689
$$262$$ 0 0
$$263$$ 20.4376 1.26023 0.630117 0.776500i $$-0.283007\pi$$
0.630117 + 0.776500i $$0.283007\pi$$
$$264$$ 0 0
$$265$$ −8.39242 −0.515542
$$266$$ 0 0
$$267$$ −12.0739 −0.738911
$$268$$ 0 0
$$269$$ 16.8017 1.02442 0.512209 0.858861i $$-0.328827\pi$$
0.512209 + 0.858861i $$0.328827\pi$$
$$270$$ 0 0
$$271$$ 8.49263 0.515890 0.257945 0.966160i $$-0.416955\pi$$
0.257945 + 0.966160i $$0.416955\pi$$
$$272$$ 0 0
$$273$$ −14.2466 −0.862241
$$274$$ 0 0
$$275$$ −4.53238 −0.273313
$$276$$ 0 0
$$277$$ −12.1608 −0.730670 −0.365335 0.930876i $$-0.619046\pi$$
−0.365335 + 0.930876i $$0.619046\pi$$
$$278$$ 0 0
$$279$$ 2.95084 0.176662
$$280$$ 0 0
$$281$$ −14.5701 −0.869178 −0.434589 0.900629i $$-0.643106\pi$$
−0.434589 + 0.900629i $$0.643106\pi$$
$$282$$ 0 0
$$283$$ −8.16542 −0.485384 −0.242692 0.970103i $$-0.578030\pi$$
−0.242692 + 0.970103i $$0.578030\pi$$
$$284$$ 0 0
$$285$$ −4.50357 −0.266769
$$286$$ 0 0
$$287$$ −7.55366 −0.445879
$$288$$ 0 0
$$289$$ −14.1618 −0.833046
$$290$$ 0 0
$$291$$ 14.8662 0.871474
$$292$$ 0 0
$$293$$ 29.2346 1.70791 0.853953 0.520350i $$-0.174199\pi$$
0.853953 + 0.520350i $$0.174199\pi$$
$$294$$ 0 0
$$295$$ 2.23309 0.130015
$$296$$ 0 0
$$297$$ 2.59144 0.150371
$$298$$ 0 0
$$299$$ −30.9276 −1.78859
$$300$$ 0 0
$$301$$ −3.49642 −0.201530
$$302$$ 0 0
$$303$$ 1.27675 0.0733473
$$304$$ 0 0
$$305$$ 14.3741 0.823061
$$306$$ 0 0
$$307$$ −23.8192 −1.35943 −0.679716 0.733475i $$-0.737897\pi$$
−0.679716 + 0.733475i $$0.737897\pi$$
$$308$$ 0 0
$$309$$ −5.26805 −0.299689
$$310$$ 0 0
$$311$$ 16.4270 0.931492 0.465746 0.884918i $$-0.345786\pi$$
0.465746 + 0.884918i $$0.345786\pi$$
$$312$$ 0 0
$$313$$ 12.4875 0.705832 0.352916 0.935655i $$-0.385190\pi$$
0.352916 + 0.935655i $$0.385190\pi$$
$$314$$ 0 0
$$315$$ −7.40807 −0.417398
$$316$$ 0 0
$$317$$ −3.38861 −0.190323 −0.0951617 0.995462i $$-0.530337\pi$$
−0.0951617 + 0.995462i $$0.530337\pi$$
$$318$$ 0 0
$$319$$ −11.4583 −0.641540
$$320$$ 0 0
$$321$$ −7.81076 −0.435954
$$322$$ 0 0
$$323$$ 4.20794 0.234136
$$324$$ 0 0
$$325$$ 6.06455 0.336401
$$326$$ 0 0
$$327$$ −2.01235 −0.111283
$$328$$ 0 0
$$329$$ 27.7109 1.52775
$$330$$ 0 0
$$331$$ −7.83764 −0.430795 −0.215398 0.976526i $$-0.569105\pi$$
−0.215398 + 0.976526i $$0.569105\pi$$
$$332$$ 0 0
$$333$$ 0.918154 0.0503145
$$334$$ 0 0
$$335$$ −2.98782 −0.163242
$$336$$ 0 0
$$337$$ −19.0047 −1.03525 −0.517627 0.855606i $$-0.673185\pi$$
−0.517627 + 0.855606i $$0.673185\pi$$
$$338$$ 0 0
$$339$$ 8.20964 0.445887
$$340$$ 0 0
$$341$$ 7.64694 0.414105
$$342$$ 0 0
$$343$$ 11.8358 0.639071
$$344$$ 0 0
$$345$$ −16.0821 −0.865829
$$346$$ 0 0
$$347$$ −25.2208 −1.35392 −0.676961 0.736019i $$-0.736703\pi$$
−0.676961 + 0.736019i $$0.736703\pi$$
$$348$$ 0 0
$$349$$ 1.06370 0.0569387 0.0284693 0.999595i $$-0.490937\pi$$
0.0284693 + 0.999595i $$0.490937\pi$$
$$350$$ 0 0
$$351$$ −3.46748 −0.185081
$$352$$ 0 0
$$353$$ 19.0049 1.01153 0.505764 0.862672i $$-0.331211\pi$$
0.505764 + 0.862672i $$0.331211\pi$$
$$354$$ 0 0
$$355$$ −11.2521 −0.597201
$$356$$ 0 0
$$357$$ 6.92178 0.366340
$$358$$ 0 0
$$359$$ −22.4615 −1.18547 −0.592736 0.805397i $$-0.701952\pi$$
−0.592736 + 0.805397i $$0.701952\pi$$
$$360$$ 0 0
$$361$$ −12.7613 −0.671647
$$362$$ 0 0
$$363$$ −4.28443 −0.224874
$$364$$ 0 0
$$365$$ 8.58993 0.449618
$$366$$ 0 0
$$367$$ −6.29449 −0.328569 −0.164285 0.986413i $$-0.552532\pi$$
−0.164285 + 0.986413i $$0.552532\pi$$
$$368$$ 0 0
$$369$$ −1.83849 −0.0957082
$$370$$ 0 0
$$371$$ 19.1237 0.992855
$$372$$ 0 0
$$373$$ −11.9702 −0.619794 −0.309897 0.950770i $$-0.600295\pi$$
−0.309897 + 0.950770i $$0.600295\pi$$
$$374$$ 0 0
$$375$$ 12.1688 0.628394
$$376$$ 0 0
$$377$$ 15.3318 0.789626
$$378$$ 0 0
$$379$$ 18.5730 0.954032 0.477016 0.878895i $$-0.341718\pi$$
0.477016 + 0.878895i $$0.341718\pi$$
$$380$$ 0 0
$$381$$ −9.38074 −0.480590
$$382$$ 0 0
$$383$$ 29.3120 1.49777 0.748886 0.662698i $$-0.230589\pi$$
0.748886 + 0.662698i $$0.230589\pi$$
$$384$$ 0 0
$$385$$ −19.1976 −0.978400
$$386$$ 0 0
$$387$$ −0.850997 −0.0432586
$$388$$ 0 0
$$389$$ 15.9476 0.808573 0.404287 0.914632i $$-0.367520\pi$$
0.404287 + 0.914632i $$0.367520\pi$$
$$390$$ 0 0
$$391$$ 15.0264 0.759917
$$392$$ 0 0
$$393$$ −19.8572 −1.00166
$$394$$ 0 0
$$395$$ −19.3275 −0.972473
$$396$$ 0 0
$$397$$ −22.9256 −1.15060 −0.575302 0.817941i $$-0.695116\pi$$
−0.575302 + 0.817941i $$0.695116\pi$$
$$398$$ 0 0
$$399$$ 10.2623 0.513755
$$400$$ 0 0
$$401$$ 21.0031 1.04884 0.524421 0.851459i $$-0.324282\pi$$
0.524421 + 0.851459i $$0.324282\pi$$
$$402$$ 0 0
$$403$$ −10.2320 −0.509692
$$404$$ 0 0
$$405$$ −1.80306 −0.0895947
$$406$$ 0 0
$$407$$ 2.37934 0.117940
$$408$$ 0 0
$$409$$ 34.4707 1.70447 0.852234 0.523160i $$-0.175247\pi$$
0.852234 + 0.523160i $$0.175247\pi$$
$$410$$ 0 0
$$411$$ 5.44295 0.268481
$$412$$ 0 0
$$413$$ −5.08851 −0.250389
$$414$$ 0 0
$$415$$ −6.65343 −0.326604
$$416$$ 0 0
$$417$$ 13.0858 0.640812
$$418$$ 0 0
$$419$$ 7.80419 0.381260 0.190630 0.981662i $$-0.438947\pi$$
0.190630 + 0.981662i $$0.438947\pi$$
$$420$$ 0 0
$$421$$ −28.5378 −1.39085 −0.695424 0.718599i $$-0.744783\pi$$
−0.695424 + 0.718599i $$0.744783\pi$$
$$422$$ 0 0
$$423$$ 6.74458 0.327933
$$424$$ 0 0
$$425$$ −2.94650 −0.142926
$$426$$ 0 0
$$427$$ −32.7542 −1.58509
$$428$$ 0 0
$$429$$ −8.98578 −0.433838
$$430$$ 0 0
$$431$$ −10.9732 −0.528559 −0.264279 0.964446i $$-0.585134\pi$$
−0.264279 + 0.964446i $$0.585134\pi$$
$$432$$ 0 0
$$433$$ 18.8851 0.907558 0.453779 0.891114i $$-0.350076\pi$$
0.453779 + 0.891114i $$0.350076\pi$$
$$434$$ 0 0
$$435$$ 7.97238 0.382246
$$436$$ 0 0
$$437$$ 22.2781 1.06571
$$438$$ 0 0
$$439$$ 25.2125 1.20333 0.601665 0.798749i $$-0.294504\pi$$
0.601665 + 0.798749i $$0.294504\pi$$
$$440$$ 0 0
$$441$$ 9.88072 0.470510
$$442$$ 0 0
$$443$$ 20.2876 0.963893 0.481947 0.876201i $$-0.339930\pi$$
0.481947 + 0.876201i $$0.339930\pi$$
$$444$$ 0 0
$$445$$ 21.7700 1.03200
$$446$$ 0 0
$$447$$ −0.961230 −0.0454646
$$448$$ 0 0
$$449$$ −26.3700 −1.24448 −0.622240 0.782827i $$-0.713777\pi$$
−0.622240 + 0.782827i $$0.713777\pi$$
$$450$$ 0 0
$$451$$ −4.76435 −0.224345
$$452$$ 0 0
$$453$$ 8.62272 0.405131
$$454$$ 0 0
$$455$$ 25.6874 1.20424
$$456$$ 0 0
$$457$$ −20.0483 −0.937818 −0.468909 0.883247i $$-0.655353\pi$$
−0.468909 + 0.883247i $$0.655353\pi$$
$$458$$ 0 0
$$459$$ 1.68470 0.0786351
$$460$$ 0 0
$$461$$ −28.0956 −1.30854 −0.654272 0.756260i $$-0.727025\pi$$
−0.654272 + 0.756260i $$0.727025\pi$$
$$462$$ 0 0
$$463$$ 23.8772 1.10967 0.554834 0.831961i $$-0.312782\pi$$
0.554834 + 0.831961i $$0.312782\pi$$
$$464$$ 0 0
$$465$$ −5.32054 −0.246734
$$466$$ 0 0
$$467$$ 33.4385 1.54735 0.773674 0.633584i $$-0.218417\pi$$
0.773674 + 0.633584i $$0.218417\pi$$
$$468$$ 0 0
$$469$$ 6.80832 0.314379
$$470$$ 0 0
$$471$$ 3.40121 0.156720
$$472$$ 0 0
$$473$$ −2.20531 −0.101400
$$474$$ 0 0
$$475$$ −4.36849 −0.200440
$$476$$ 0 0
$$477$$ 4.65455 0.213117
$$478$$ 0 0
$$479$$ 3.13119 0.143068 0.0715340 0.997438i $$-0.477211\pi$$
0.0715340 + 0.997438i $$0.477211\pi$$
$$480$$ 0 0
$$481$$ −3.18368 −0.145163
$$482$$ 0 0
$$483$$ 36.6461 1.66745
$$484$$ 0 0
$$485$$ −26.8047 −1.21714
$$486$$ 0 0
$$487$$ 26.2378 1.18895 0.594475 0.804114i $$-0.297360\pi$$
0.594475 + 0.804114i $$0.297360\pi$$
$$488$$ 0 0
$$489$$ 2.70291 0.122230
$$490$$ 0 0
$$491$$ 5.20065 0.234702 0.117351 0.993090i $$-0.462560\pi$$
0.117351 + 0.993090i $$0.462560\pi$$
$$492$$ 0 0
$$493$$ −7.44904 −0.335488
$$494$$ 0 0
$$495$$ −4.67252 −0.210014
$$496$$ 0 0
$$497$$ 25.6401 1.15012
$$498$$ 0 0
$$499$$ 33.0631 1.48011 0.740053 0.672548i $$-0.234800\pi$$
0.740053 + 0.672548i $$0.234800\pi$$
$$500$$ 0 0
$$501$$ −1.00000 −0.0446767
$$502$$ 0 0
$$503$$ −14.7807 −0.659040 −0.329520 0.944149i $$-0.606887\pi$$
−0.329520 + 0.944149i $$0.606887\pi$$
$$504$$ 0 0
$$505$$ −2.30205 −0.102440
$$506$$ 0 0
$$507$$ −0.976559 −0.0433705
$$508$$ 0 0
$$509$$ 31.0505 1.37629 0.688145 0.725573i $$-0.258425\pi$$
0.688145 + 0.725573i $$0.258425\pi$$
$$510$$ 0 0
$$511$$ −19.5738 −0.865894
$$512$$ 0 0
$$513$$ 2.49774 0.110278
$$514$$ 0 0
$$515$$ 9.49861 0.418559
$$516$$ 0 0
$$517$$ 17.4782 0.768689
$$518$$ 0 0
$$519$$ 13.7489 0.603509
$$520$$ 0 0
$$521$$ 3.63406 0.159211 0.0796056 0.996826i $$-0.474634\pi$$
0.0796056 + 0.996826i $$0.474634\pi$$
$$522$$ 0 0
$$523$$ −26.1659 −1.14416 −0.572078 0.820199i $$-0.693863\pi$$
−0.572078 + 0.820199i $$0.693863\pi$$
$$524$$ 0 0
$$525$$ −7.18588 −0.313617
$$526$$ 0 0
$$527$$ 4.97128 0.216553
$$528$$ 0 0
$$529$$ 56.5543 2.45888
$$530$$ 0 0
$$531$$ −1.23850 −0.0537462
$$532$$ 0 0
$$533$$ 6.37495 0.276130
$$534$$ 0 0
$$535$$ 14.0833 0.608872
$$536$$ 0 0
$$537$$ −0.148844 −0.00642310
$$538$$ 0 0
$$539$$ 25.6053 1.10290
$$540$$ 0 0
$$541$$ 0.260024 0.0111793 0.00558966 0.999984i $$-0.498221\pi$$
0.00558966 + 0.999984i $$0.498221\pi$$
$$542$$ 0 0
$$543$$ 7.10022 0.304699
$$544$$ 0 0
$$545$$ 3.62838 0.155423
$$546$$ 0 0
$$547$$ 27.8527 1.19090 0.595448 0.803394i $$-0.296975\pi$$
0.595448 + 0.803394i $$0.296975\pi$$
$$548$$ 0 0
$$549$$ −7.97208 −0.340240
$$550$$ 0 0
$$551$$ −11.0440 −0.470489
$$552$$ 0 0
$$553$$ 44.0414 1.87283
$$554$$ 0 0
$$555$$ −1.65549 −0.0702714
$$556$$ 0 0
$$557$$ 8.13555 0.344714 0.172357 0.985035i $$-0.444862\pi$$
0.172357 + 0.985035i $$0.444862\pi$$
$$558$$ 0 0
$$559$$ 2.95082 0.124806
$$560$$ 0 0
$$561$$ 4.36580 0.184324
$$562$$ 0 0
$$563$$ 44.9622 1.89493 0.947466 0.319856i $$-0.103634\pi$$
0.947466 + 0.319856i $$0.103634\pi$$
$$564$$ 0 0
$$565$$ −14.8025 −0.622745
$$566$$ 0 0
$$567$$ 4.10861 0.172546
$$568$$ 0 0
$$569$$ 8.14797 0.341581 0.170790 0.985307i $$-0.445368\pi$$
0.170790 + 0.985307i $$0.445368\pi$$
$$570$$ 0 0
$$571$$ −5.71162 −0.239024 −0.119512 0.992833i $$-0.538133\pi$$
−0.119512 + 0.992833i $$0.538133\pi$$
$$572$$ 0 0
$$573$$ 5.56545 0.232500
$$574$$ 0 0
$$575$$ −15.5997 −0.650553
$$576$$ 0 0
$$577$$ 8.28848 0.345054 0.172527 0.985005i $$-0.444807\pi$$
0.172527 + 0.985005i $$0.444807\pi$$
$$578$$ 0 0
$$579$$ −16.6812 −0.693249
$$580$$ 0 0
$$581$$ 15.1611 0.628989
$$582$$ 0 0
$$583$$ 12.0620 0.499556
$$584$$ 0 0
$$585$$ 6.25208 0.258492
$$586$$ 0 0
$$587$$ −32.2682 −1.33185 −0.665925 0.746019i $$-0.731963\pi$$
−0.665925 + 0.746019i $$0.731963\pi$$
$$588$$ 0 0
$$589$$ 7.37043 0.303693
$$590$$ 0 0
$$591$$ 10.0656 0.414044
$$592$$ 0 0
$$593$$ −26.2913 −1.07965 −0.539826 0.841776i $$-0.681510\pi$$
−0.539826 + 0.841776i $$0.681510\pi$$
$$594$$ 0 0
$$595$$ −12.4804 −0.511646
$$596$$ 0 0
$$597$$ 14.2270 0.582273
$$598$$ 0 0
$$599$$ −1.95363 −0.0798233 −0.0399117 0.999203i $$-0.512708\pi$$
−0.0399117 + 0.999203i $$0.512708\pi$$
$$600$$ 0 0
$$601$$ −0.855179 −0.0348835 −0.0174417 0.999848i $$-0.505552\pi$$
−0.0174417 + 0.999848i $$0.505552\pi$$
$$602$$ 0 0
$$603$$ 1.65708 0.0674817
$$604$$ 0 0
$$605$$ 7.72508 0.314069
$$606$$ 0 0
$$607$$ 17.8511 0.724553 0.362276 0.932071i $$-0.382000\pi$$
0.362276 + 0.932071i $$0.382000\pi$$
$$608$$ 0 0
$$609$$ −18.1666 −0.736147
$$610$$ 0 0
$$611$$ −23.3867 −0.946125
$$612$$ 0 0
$$613$$ 21.0354 0.849611 0.424806 0.905285i $$-0.360342\pi$$
0.424806 + 0.905285i $$0.360342\pi$$
$$614$$ 0 0
$$615$$ 3.31491 0.133670
$$616$$ 0 0
$$617$$ −29.7077 −1.19599 −0.597993 0.801501i $$-0.704035\pi$$
−0.597993 + 0.801501i $$0.704035\pi$$
$$618$$ 0 0
$$619$$ 7.12602 0.286419 0.143209 0.989692i $$-0.454258\pi$$
0.143209 + 0.989692i $$0.454258\pi$$
$$620$$ 0 0
$$621$$ 8.91932 0.357920
$$622$$ 0 0
$$623$$ −49.6070 −1.98746
$$624$$ 0 0
$$625$$ −13.1962 −0.527847
$$626$$ 0 0
$$627$$ 6.47275 0.258497
$$628$$ 0 0
$$629$$ 1.54681 0.0616755
$$630$$ 0 0
$$631$$ 10.8881 0.433447 0.216724 0.976233i $$-0.430463\pi$$
0.216724 + 0.976233i $$0.430463\pi$$
$$632$$ 0 0
$$633$$ −19.4006 −0.771106
$$634$$ 0 0
$$635$$ 16.9140 0.671213
$$636$$ 0 0
$$637$$ −34.2612 −1.35748
$$638$$ 0 0
$$639$$ 6.24057 0.246873
$$640$$ 0 0
$$641$$ 6.00715 0.237268 0.118634 0.992938i $$-0.462148\pi$$
0.118634 + 0.992938i $$0.462148\pi$$
$$642$$ 0 0
$$643$$ −22.8736 −0.902045 −0.451023 0.892513i $$-0.648941\pi$$
−0.451023 + 0.892513i $$0.648941\pi$$
$$644$$ 0 0
$$645$$ 1.53440 0.0604169
$$646$$ 0 0
$$647$$ −19.1587 −0.753205 −0.376602 0.926375i $$-0.622908\pi$$
−0.376602 + 0.926375i $$0.622908\pi$$
$$648$$ 0 0
$$649$$ −3.20950 −0.125984
$$650$$ 0 0
$$651$$ 12.1239 0.475172
$$652$$ 0 0
$$653$$ −29.0547 −1.13700 −0.568499 0.822684i $$-0.692475\pi$$
−0.568499 + 0.822684i $$0.692475\pi$$
$$654$$ 0 0
$$655$$ 35.8037 1.39897
$$656$$ 0 0
$$657$$ −4.76409 −0.185865
$$658$$ 0 0
$$659$$ −18.7083 −0.728771 −0.364386 0.931248i $$-0.618721\pi$$
−0.364386 + 0.931248i $$0.618721\pi$$
$$660$$ 0 0
$$661$$ −6.65833 −0.258979 −0.129490 0.991581i $$-0.541334\pi$$
−0.129490 + 0.991581i $$0.541334\pi$$
$$662$$ 0 0
$$663$$ −5.84167 −0.226872
$$664$$ 0 0
$$665$$ −18.5034 −0.717533
$$666$$ 0 0
$$667$$ −39.4375 −1.52703
$$668$$ 0 0
$$669$$ 5.50496 0.212834
$$670$$ 0 0
$$671$$ −20.6592 −0.797539
$$672$$ 0 0
$$673$$ −25.7547 −0.992772 −0.496386 0.868102i $$-0.665340\pi$$
−0.496386 + 0.868102i $$0.665340\pi$$
$$674$$ 0 0
$$675$$ −1.74898 −0.0673182
$$676$$ 0 0
$$677$$ 11.7352 0.451019 0.225510 0.974241i $$-0.427595\pi$$
0.225510 + 0.974241i $$0.427595\pi$$
$$678$$ 0 0
$$679$$ 61.0797 2.34402
$$680$$ 0 0
$$681$$ 24.7959 0.950180
$$682$$ 0 0
$$683$$ 30.4318 1.16444 0.582220 0.813031i $$-0.302184\pi$$
0.582220 + 0.813031i $$0.302184\pi$$
$$684$$ 0 0
$$685$$ −9.81397 −0.374972
$$686$$ 0 0
$$687$$ −22.8293 −0.870993
$$688$$ 0 0
$$689$$ −16.1396 −0.614868
$$690$$ 0 0
$$691$$ 35.5839 1.35368 0.676838 0.736132i $$-0.263350\pi$$
0.676838 + 0.736132i $$0.263350\pi$$
$$692$$ 0 0
$$693$$ 10.6472 0.404455
$$694$$ 0 0
$$695$$ −23.5944 −0.894986
$$696$$ 0 0
$$697$$ −3.09731 −0.117319
$$698$$ 0 0
$$699$$ 12.2217 0.462268
$$700$$ 0 0
$$701$$ −11.4767 −0.433468 −0.216734 0.976231i $$-0.569540\pi$$
−0.216734 + 0.976231i $$0.569540\pi$$
$$702$$ 0 0
$$703$$ 2.29331 0.0864938
$$704$$ 0 0
$$705$$ −12.1609 −0.458005
$$706$$ 0 0
$$707$$ 5.24567 0.197284
$$708$$ 0 0
$$709$$ 2.41968 0.0908729 0.0454365 0.998967i $$-0.485532\pi$$
0.0454365 + 0.998967i $$0.485532\pi$$
$$710$$ 0 0
$$711$$ 10.7193 0.402005
$$712$$ 0 0
$$713$$ 26.3195 0.985673
$$714$$ 0 0
$$715$$ 16.2019 0.605916
$$716$$ 0 0
$$717$$ −7.79185 −0.290992
$$718$$ 0 0
$$719$$ −26.9803 −1.00620 −0.503098 0.864229i $$-0.667806\pi$$
−0.503098 + 0.864229i $$0.667806\pi$$
$$720$$ 0 0
$$721$$ −21.6444 −0.806080
$$722$$ 0 0
$$723$$ −22.1396 −0.823379
$$724$$ 0 0
$$725$$ 7.73325 0.287206
$$726$$ 0 0
$$727$$ −8.04133 −0.298237 −0.149118 0.988819i $$-0.547644\pi$$
−0.149118 + 0.988819i $$0.547644\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −1.43368 −0.0530264
$$732$$ 0 0
$$733$$ −48.1812 −1.77961 −0.889807 0.456336i $$-0.849161\pi$$
−0.889807 + 0.456336i $$0.849161\pi$$
$$734$$ 0 0
$$735$$ −17.8155 −0.657135
$$736$$ 0 0
$$737$$ 4.29424 0.158180
$$738$$ 0 0
$$739$$ −21.9605 −0.807829 −0.403915 0.914797i $$-0.632351\pi$$
−0.403915 + 0.914797i $$0.632351\pi$$
$$740$$ 0 0
$$741$$ −8.66087 −0.318165
$$742$$ 0 0
$$743$$ −4.55455 −0.167090 −0.0835451 0.996504i $$-0.526624\pi$$
−0.0835451 + 0.996504i $$0.526624\pi$$
$$744$$ 0 0
$$745$$ 1.73315 0.0634979
$$746$$ 0 0
$$747$$ 3.69008 0.135013
$$748$$ 0 0
$$749$$ −32.0914 −1.17259
$$750$$ 0 0
$$751$$ −24.3222 −0.887530 −0.443765 0.896143i $$-0.646357\pi$$
−0.443765 + 0.896143i $$0.646357\pi$$
$$752$$ 0 0
$$753$$ 16.1687 0.589219
$$754$$ 0 0
$$755$$ −15.5473 −0.565823
$$756$$ 0 0
$$757$$ 9.86234 0.358453 0.179226 0.983808i $$-0.442641\pi$$
0.179226 + 0.983808i $$0.442641\pi$$
$$758$$ 0 0
$$759$$ 23.1139 0.838982
$$760$$ 0 0
$$761$$ −1.04767 −0.0379781 −0.0189891 0.999820i $$-0.506045\pi$$
−0.0189891 + 0.999820i $$0.506045\pi$$
$$762$$ 0 0
$$763$$ −8.26796 −0.299320
$$764$$ 0 0
$$765$$ −3.03761 −0.109825
$$766$$ 0 0
$$767$$ 4.29447 0.155064
$$768$$ 0 0
$$769$$ −11.2840 −0.406912 −0.203456 0.979084i $$-0.565217\pi$$
−0.203456 + 0.979084i $$0.565217\pi$$
$$770$$ 0 0
$$771$$ 22.3740 0.805779
$$772$$ 0 0
$$773$$ 2.32588 0.0836561 0.0418280 0.999125i $$-0.486682\pi$$
0.0418280 + 0.999125i $$0.486682\pi$$
$$774$$ 0 0
$$775$$ −5.16096 −0.185387
$$776$$ 0 0
$$777$$ 3.77234 0.135332
$$778$$ 0 0
$$779$$ −4.59208 −0.164528
$$780$$ 0 0
$$781$$ 16.1721 0.578683
$$782$$ 0 0
$$783$$ −4.42158 −0.158015
$$784$$ 0 0
$$785$$ −6.13259 −0.218882
$$786$$ 0 0
$$787$$ 9.04798 0.322526 0.161263 0.986912i $$-0.448443\pi$$
0.161263 + 0.986912i $$0.448443\pi$$
$$788$$ 0 0
$$789$$ 20.4376 0.727596
$$790$$ 0 0
$$791$$ 33.7303 1.19931
$$792$$ 0 0
$$793$$ 27.6431 0.981634
$$794$$ 0 0
$$795$$ −8.39242 −0.297649
$$796$$ 0 0
$$797$$ 5.67319 0.200955 0.100477 0.994939i $$-0.467963\pi$$
0.100477 + 0.994939i $$0.467963\pi$$
$$798$$ 0 0
$$799$$ 11.3626 0.401979
$$800$$ 0 0
$$801$$ −12.0739 −0.426611
$$802$$ 0 0
$$803$$ −12.3459 −0.435676
$$804$$ 0 0
$$805$$ −66.0750 −2.32884
$$806$$ 0 0
$$807$$ 16.8017 0.591448
$$808$$ 0 0
$$809$$ −37.7932 −1.32874 −0.664369 0.747405i $$-0.731300\pi$$
−0.664369 + 0.747405i $$0.731300\pi$$
$$810$$ 0 0
$$811$$ −17.2974 −0.607394 −0.303697 0.952769i $$-0.598221\pi$$
−0.303697 + 0.952769i $$0.598221\pi$$
$$812$$ 0 0
$$813$$ 8.49263 0.297849
$$814$$ 0 0
$$815$$ −4.87350 −0.170711
$$816$$ 0 0
$$817$$ −2.12557 −0.0743643
$$818$$ 0 0
$$819$$ −14.2466 −0.497815
$$820$$ 0 0
$$821$$ 13.1038 0.457325 0.228662 0.973506i $$-0.426565\pi$$
0.228662 + 0.973506i $$0.426565\pi$$
$$822$$ 0 0
$$823$$ 5.47594 0.190879 0.0954396 0.995435i $$-0.469574\pi$$
0.0954396 + 0.995435i $$0.469574\pi$$
$$824$$ 0 0
$$825$$ −4.53238 −0.157797
$$826$$ 0 0
$$827$$ 2.81367 0.0978411 0.0489205 0.998803i $$-0.484422\pi$$
0.0489205 + 0.998803i $$0.484422\pi$$
$$828$$ 0 0
$$829$$ 11.7545 0.408250 0.204125 0.978945i $$-0.434565\pi$$
0.204125 + 0.978945i $$0.434565\pi$$
$$830$$ 0 0
$$831$$ −12.1608 −0.421852
$$832$$ 0 0
$$833$$ 16.6460 0.576751
$$834$$ 0 0
$$835$$ 1.80306 0.0623974
$$836$$ 0 0
$$837$$ 2.95084 0.101996
$$838$$ 0 0
$$839$$ 35.1585 1.21381 0.606903 0.794776i $$-0.292412\pi$$
0.606903 + 0.794776i $$0.292412\pi$$
$$840$$ 0 0
$$841$$ −9.44960 −0.325848
$$842$$ 0 0
$$843$$ −14.5701 −0.501820
$$844$$ 0 0
$$845$$ 1.76079 0.0605732
$$846$$ 0 0
$$847$$ −17.6031 −0.604849
$$848$$ 0 0
$$849$$ −8.16542 −0.280236
$$850$$ 0 0
$$851$$ 8.18931 0.280726
$$852$$ 0 0
$$853$$ −5.19930 −0.178021 −0.0890104 0.996031i $$-0.528370\pi$$
−0.0890104 + 0.996031i $$0.528370\pi$$
$$854$$ 0 0
$$855$$ −4.50357 −0.154019
$$856$$ 0 0
$$857$$ 1.20592 0.0411933 0.0205966 0.999788i $$-0.493443\pi$$
0.0205966 + 0.999788i $$0.493443\pi$$
$$858$$ 0 0
$$859$$ 39.2254 1.33835 0.669176 0.743104i $$-0.266647\pi$$
0.669176 + 0.743104i $$0.266647\pi$$
$$860$$ 0 0
$$861$$ −7.55366 −0.257428
$$862$$ 0 0
$$863$$ −17.4008 −0.592329 −0.296164 0.955137i $$-0.595708\pi$$
−0.296164 + 0.955137i $$0.595708\pi$$
$$864$$ 0 0
$$865$$ −24.7900 −0.842886
$$866$$ 0 0
$$867$$ −14.1618 −0.480959
$$868$$ 0 0
$$869$$ 27.7784 0.942319
$$870$$ 0 0
$$871$$ −5.74591 −0.194693
$$872$$ 0 0
$$873$$ 14.8662 0.503146
$$874$$ 0 0
$$875$$ 49.9969 1.69020
$$876$$ 0 0
$$877$$ −51.8185 −1.74979 −0.874893 0.484316i $$-0.839068\pi$$
−0.874893 + 0.484316i $$0.839068\pi$$
$$878$$ 0 0
$$879$$ 29.2346 0.986060
$$880$$ 0 0
$$881$$ −55.7905 −1.87963 −0.939814 0.341686i $$-0.889002\pi$$
−0.939814 + 0.341686i $$0.889002\pi$$
$$882$$ 0 0
$$883$$ −34.6270 −1.16529 −0.582645 0.812727i $$-0.697982\pi$$
−0.582645 + 0.812727i $$0.697982\pi$$
$$884$$ 0 0
$$885$$ 2.23309 0.0750643
$$886$$ 0 0
$$887$$ 3.25650 0.109343 0.0546713 0.998504i $$-0.482589\pi$$
0.0546713 + 0.998504i $$0.482589\pi$$
$$888$$ 0 0
$$889$$ −38.5418 −1.29265
$$890$$ 0 0
$$891$$ 2.59144 0.0868166
$$892$$ 0 0
$$893$$ 16.8462 0.563736
$$894$$ 0 0
$$895$$ 0.268375 0.00897079
$$896$$ 0 0
$$897$$ −30.9276 −1.03264
$$898$$ 0 0
$$899$$ −13.0474 −0.435155
$$900$$ 0 0
$$901$$ 7.84151 0.261239
$$902$$ 0 0
$$903$$ −3.49642 −0.116354
$$904$$ 0 0
$$905$$ −12.8021 −0.425556
$$906$$ 0 0
$$907$$ −1.93813 −0.0643547 −0.0321773 0.999482i $$-0.510244\pi$$
−0.0321773 + 0.999482i $$0.510244\pi$$
$$908$$ 0 0
$$909$$ 1.27675 0.0423471
$$910$$ 0 0
$$911$$ 22.6161 0.749304 0.374652 0.927165i $$-0.377762\pi$$
0.374652 + 0.927165i $$0.377762\pi$$
$$912$$ 0 0
$$913$$ 9.56262 0.316477
$$914$$ 0 0
$$915$$ 14.3741 0.475194
$$916$$ 0 0
$$917$$ −81.5855 −2.69419
$$918$$ 0 0
$$919$$ 0.152045 0.00501549 0.00250774 0.999997i $$-0.499202\pi$$
0.00250774 + 0.999997i $$0.499202\pi$$
$$920$$ 0 0
$$921$$ −23.8192 −0.784869
$$922$$ 0 0
$$923$$ −21.6391 −0.712259
$$924$$ 0 0
$$925$$ −1.60583 −0.0527994
$$926$$ 0 0
$$927$$ −5.26805 −0.173026
$$928$$ 0 0
$$929$$ −40.4885 −1.32838 −0.664192 0.747562i $$-0.731224\pi$$
−0.664192 + 0.747562i $$0.731224\pi$$
$$930$$ 0 0
$$931$$ 24.6795 0.808836
$$932$$ 0 0
$$933$$ 16.4270 0.537797
$$934$$ 0 0
$$935$$ −7.87180 −0.257435
$$936$$ 0 0
$$937$$ −49.3072 −1.61079 −0.805397 0.592735i $$-0.798048\pi$$
−0.805397 + 0.592735i $$0.798048\pi$$
$$938$$ 0 0
$$939$$ 12.4875 0.407513
$$940$$ 0 0
$$941$$ −22.8417 −0.744618 −0.372309 0.928109i $$-0.621434\pi$$
−0.372309 + 0.928109i $$0.621434\pi$$
$$942$$ 0 0
$$943$$ −16.3981 −0.533996
$$944$$ 0 0
$$945$$ −7.40807 −0.240985
$$946$$ 0 0
$$947$$ −20.8527 −0.677622 −0.338811 0.940854i $$-0.610025\pi$$
−0.338811 + 0.940854i $$0.610025\pi$$
$$948$$ 0 0
$$949$$ 16.5194 0.536242
$$950$$ 0 0
$$951$$ −3.38861 −0.109883
$$952$$ 0 0
$$953$$ −20.6396 −0.668581 −0.334291 0.942470i $$-0.608497\pi$$
−0.334291 + 0.942470i $$0.608497\pi$$
$$954$$ 0 0
$$955$$ −10.0348 −0.324719
$$956$$ 0 0
$$957$$ −11.4583 −0.370393
$$958$$ 0 0
$$959$$ 22.3630 0.722139
$$960$$ 0 0
$$961$$ −22.2925 −0.719114
$$962$$ 0 0
$$963$$ −7.81076 −0.251698
$$964$$ 0 0
$$965$$ 30.0773 0.968222
$$966$$ 0 0
$$967$$ −43.8774 −1.41100 −0.705501 0.708709i $$-0.749278\pi$$
−0.705501 + 0.708709i $$0.749278\pi$$
$$968$$ 0 0
$$969$$ 4.20794 0.135179
$$970$$ 0 0
$$971$$ −9.54975 −0.306466 −0.153233 0.988190i $$-0.548969\pi$$
−0.153233 + 0.988190i $$0.548969\pi$$
$$972$$ 0 0
$$973$$ 53.7643 1.72361
$$974$$ 0 0
$$975$$ 6.06455 0.194221
$$976$$ 0 0
$$977$$ 11.9627 0.382721 0.191360 0.981520i $$-0.438710\pi$$
0.191360 + 0.981520i $$0.438710\pi$$
$$978$$ 0 0
$$979$$ −31.2888 −0.999995
$$980$$ 0 0
$$981$$ −2.01235 −0.0642493
$$982$$ 0 0
$$983$$ 37.2845 1.18919 0.594595 0.804025i $$-0.297312\pi$$
0.594595 + 0.804025i $$0.297312\pi$$
$$984$$ 0 0
$$985$$ −18.1489 −0.578272
$$986$$ 0 0
$$987$$ 27.7109 0.882047
$$988$$ 0 0
$$989$$ −7.59032 −0.241358
$$990$$ 0 0
$$991$$ 46.0389 1.46247 0.731237 0.682123i $$-0.238943\pi$$
0.731237 + 0.682123i $$0.238943\pi$$
$$992$$ 0 0
$$993$$ −7.83764 −0.248720
$$994$$ 0 0
$$995$$ −25.6521 −0.813228
$$996$$ 0 0
$$997$$ −6.30573 −0.199705 −0.0998523 0.995002i $$-0.531837\pi$$
−0.0998523 + 0.995002i $$0.531837\pi$$
$$998$$ 0 0
$$999$$ 0.918154 0.0290491
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 8016.2.a.be.1.3 11
4.3 odd 2 4008.2.a.k.1.3 11

By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.k.1.3 11 4.3 odd 2
8016.2.a.be.1.3 11 1.1 even 1 trivial