# Properties

 Label 1960.2.a.p.1.2 Level $1960$ Weight $2$ Character 1960.1 Self dual yes Analytic conductor $15.651$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.6506787962$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1960.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})$$ $$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +0.828427 q^{11} +2.00000 q^{13} +0.414214 q^{15} -7.65685 q^{17} -5.65685 q^{19} -5.58579 q^{23} +1.00000 q^{25} -2.41421 q^{27} -7.82843 q^{29} +0.828427 q^{31} +0.343146 q^{33} +5.65685 q^{37} +0.828427 q^{39} +5.82843 q^{41} -6.89949 q^{43} -2.82843 q^{45} +11.6569 q^{47} -3.17157 q^{51} -5.65685 q^{53} +0.828427 q^{55} -2.34315 q^{57} -4.00000 q^{59} +6.65685 q^{61} +2.00000 q^{65} -12.8995 q^{67} -2.31371 q^{69} -12.0000 q^{71} +3.65685 q^{73} +0.414214 q^{75} -4.00000 q^{79} +7.48528 q^{81} -4.75736 q^{83} -7.65685 q^{85} -3.24264 q^{87} -5.34315 q^{89} +0.343146 q^{93} -5.65685 q^{95} +6.00000 q^{97} -2.34315 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} - 4q^{11} + 4q^{13} - 2q^{15} - 4q^{17} - 14q^{23} + 2q^{25} - 2q^{27} - 10q^{29} - 4q^{31} + 12q^{33} - 4q^{39} + 6q^{41} + 6q^{43} + 12q^{47} - 12q^{51} - 4q^{55} - 16q^{57} - 8q^{59} + 2q^{61} + 4q^{65} - 6q^{67} + 18q^{69} - 24q^{71} - 4q^{73} - 2q^{75} - 8q^{79} - 2q^{81} - 18q^{83} - 4q^{85} + 2q^{87} - 22q^{89} + 12q^{93} + 12q^{97} - 16q^{99} + O(q^{100})$$

## 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
$$3$$ 0.414214 0.239146 0.119573 0.992825i $$-0.461847\pi$$
0.119573 + 0.992825i $$0.461847\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.82843 −0.942809
$$10$$ 0 0
$$11$$ 0.828427 0.249780 0.124890 0.992171i $$-0.460142\pi$$
0.124890 + 0.992171i $$0.460142\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0.414214 0.106949
$$16$$ 0 0
$$17$$ −7.65685 −1.85706 −0.928530 0.371257i $$-0.878927\pi$$
−0.928530 + 0.371257i $$0.878927\pi$$
$$18$$ 0 0
$$19$$ −5.65685 −1.29777 −0.648886 0.760886i $$-0.724765\pi$$
−0.648886 + 0.760886i $$0.724765\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −5.58579 −1.16472 −0.582358 0.812932i $$-0.697870\pi$$
−0.582358 + 0.812932i $$0.697870\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −2.41421 −0.464616
$$28$$ 0 0
$$29$$ −7.82843 −1.45370 −0.726851 0.686795i $$-0.759017\pi$$
−0.726851 + 0.686795i $$0.759017\pi$$
$$30$$ 0 0
$$31$$ 0.828427 0.148790 0.0743950 0.997229i $$-0.476297\pi$$
0.0743950 + 0.997229i $$0.476297\pi$$
$$32$$ 0 0
$$33$$ 0.343146 0.0597340
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.65685 0.929981 0.464991 0.885316i $$-0.346058\pi$$
0.464991 + 0.885316i $$0.346058\pi$$
$$38$$ 0 0
$$39$$ 0.828427 0.132655
$$40$$ 0 0
$$41$$ 5.82843 0.910247 0.455124 0.890428i $$-0.349595\pi$$
0.455124 + 0.890428i $$0.349595\pi$$
$$42$$ 0 0
$$43$$ −6.89949 −1.05216 −0.526082 0.850434i $$-0.676339\pi$$
−0.526082 + 0.850434i $$0.676339\pi$$
$$44$$ 0 0
$$45$$ −2.82843 −0.421637
$$46$$ 0 0
$$47$$ 11.6569 1.70033 0.850163 0.526519i $$-0.176503\pi$$
0.850163 + 0.526519i $$0.176503\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −3.17157 −0.444109
$$52$$ 0 0
$$53$$ −5.65685 −0.777029 −0.388514 0.921443i $$-0.627012\pi$$
−0.388514 + 0.921443i $$0.627012\pi$$
$$54$$ 0 0
$$55$$ 0.828427 0.111705
$$56$$ 0 0
$$57$$ −2.34315 −0.310357
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 6.65685 0.852323 0.426161 0.904647i $$-0.359866\pi$$
0.426161 + 0.904647i $$0.359866\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ −12.8995 −1.57592 −0.787962 0.615724i $$-0.788864\pi$$
−0.787962 + 0.615724i $$0.788864\pi$$
$$68$$ 0 0
$$69$$ −2.31371 −0.278538
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 3.65685 0.428002 0.214001 0.976833i $$-0.431350\pi$$
0.214001 + 0.976833i $$0.431350\pi$$
$$74$$ 0 0
$$75$$ 0.414214 0.0478293
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 7.48528 0.831698
$$82$$ 0 0
$$83$$ −4.75736 −0.522188 −0.261094 0.965313i $$-0.584083\pi$$
−0.261094 + 0.965313i $$0.584083\pi$$
$$84$$ 0 0
$$85$$ −7.65685 −0.830502
$$86$$ 0 0
$$87$$ −3.24264 −0.347648
$$88$$ 0 0
$$89$$ −5.34315 −0.566372 −0.283186 0.959065i $$-0.591391\pi$$
−0.283186 + 0.959065i $$0.591391\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0.343146 0.0355826
$$94$$ 0 0
$$95$$ −5.65685 −0.580381
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ −2.34315 −0.235495
$$100$$ 0 0
$$101$$ −11.4853 −1.14283 −0.571414 0.820662i $$-0.693605\pi$$
−0.571414 + 0.820662i $$0.693605\pi$$
$$102$$ 0 0
$$103$$ 7.58579 0.747450 0.373725 0.927540i $$-0.378080\pi$$
0.373725 + 0.927540i $$0.378080\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −5.58579 −0.539998 −0.269999 0.962861i $$-0.587023\pi$$
−0.269999 + 0.962861i $$0.587023\pi$$
$$108$$ 0 0
$$109$$ 18.3137 1.75414 0.877068 0.480367i $$-0.159497\pi$$
0.877068 + 0.480367i $$0.159497\pi$$
$$110$$ 0 0
$$111$$ 2.34315 0.222402
$$112$$ 0 0
$$113$$ 11.3137 1.06430 0.532152 0.846649i $$-0.321383\pi$$
0.532152 + 0.846649i $$0.321383\pi$$
$$114$$ 0 0
$$115$$ −5.58579 −0.520877
$$116$$ 0 0
$$117$$ −5.65685 −0.522976
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 0 0
$$123$$ 2.41421 0.217682
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −4.34315 −0.385392 −0.192696 0.981259i $$-0.561723\pi$$
−0.192696 + 0.981259i $$0.561723\pi$$
$$128$$ 0 0
$$129$$ −2.85786 −0.251621
$$130$$ 0 0
$$131$$ 13.6569 1.19320 0.596602 0.802537i $$-0.296517\pi$$
0.596602 + 0.802537i $$0.296517\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.41421 −0.207782
$$136$$ 0 0
$$137$$ −4.00000 −0.341743 −0.170872 0.985293i $$-0.554658\pi$$
−0.170872 + 0.985293i $$0.554658\pi$$
$$138$$ 0 0
$$139$$ 2.48528 0.210799 0.105399 0.994430i $$-0.466388\pi$$
0.105399 + 0.994430i $$0.466388\pi$$
$$140$$ 0 0
$$141$$ 4.82843 0.406627
$$142$$ 0 0
$$143$$ 1.65685 0.138553
$$144$$ 0 0
$$145$$ −7.82843 −0.650115
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.65685 0.381504 0.190752 0.981638i $$-0.438907\pi$$
0.190752 + 0.981638i $$0.438907\pi$$
$$150$$ 0 0
$$151$$ −11.1716 −0.909130 −0.454565 0.890714i $$-0.650205\pi$$
−0.454565 + 0.890714i $$0.650205\pi$$
$$152$$ 0 0
$$153$$ 21.6569 1.75085
$$154$$ 0 0
$$155$$ 0.828427 0.0665409
$$156$$ 0 0
$$157$$ 1.31371 0.104845 0.0524227 0.998625i $$-0.483306\pi$$
0.0524227 + 0.998625i $$0.483306\pi$$
$$158$$ 0 0
$$159$$ −2.34315 −0.185824
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −15.6569 −1.22634 −0.613170 0.789951i $$-0.710106\pi$$
−0.613170 + 0.789951i $$0.710106\pi$$
$$164$$ 0 0
$$165$$ 0.343146 0.0267139
$$166$$ 0 0
$$167$$ 2.07107 0.160264 0.0801320 0.996784i $$-0.474466\pi$$
0.0801320 + 0.996784i $$0.474466\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 16.0000 1.22355
$$172$$ 0 0
$$173$$ 10.3431 0.786375 0.393187 0.919458i $$-0.371372\pi$$
0.393187 + 0.919458i $$0.371372\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.65685 −0.124537
$$178$$ 0 0
$$179$$ −6.48528 −0.484733 −0.242366 0.970185i $$-0.577924\pi$$
−0.242366 + 0.970185i $$0.577924\pi$$
$$180$$ 0 0
$$181$$ −4.17157 −0.310071 −0.155035 0.987909i $$-0.549549\pi$$
−0.155035 + 0.987909i $$0.549549\pi$$
$$182$$ 0 0
$$183$$ 2.75736 0.203830
$$184$$ 0 0
$$185$$ 5.65685 0.415900
$$186$$ 0 0
$$187$$ −6.34315 −0.463857
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −5.51472 −0.399031 −0.199516 0.979895i $$-0.563937\pi$$
−0.199516 + 0.979895i $$0.563937\pi$$
$$192$$ 0 0
$$193$$ −5.31371 −0.382489 −0.191245 0.981542i $$-0.561252\pi$$
−0.191245 + 0.981542i $$0.561252\pi$$
$$194$$ 0 0
$$195$$ 0.828427 0.0593249
$$196$$ 0 0
$$197$$ 0.343146 0.0244481 0.0122241 0.999925i $$-0.496109\pi$$
0.0122241 + 0.999925i $$0.496109\pi$$
$$198$$ 0 0
$$199$$ −23.3137 −1.65266 −0.826332 0.563183i $$-0.809577\pi$$
−0.826332 + 0.563183i $$0.809577\pi$$
$$200$$ 0 0
$$201$$ −5.34315 −0.376876
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 5.82843 0.407075
$$206$$ 0 0
$$207$$ 15.7990 1.09811
$$208$$ 0 0
$$209$$ −4.68629 −0.324158
$$210$$ 0 0
$$211$$ 26.6274 1.83311 0.916553 0.399912i $$-0.130959\pi$$
0.916553 + 0.399912i $$0.130959\pi$$
$$212$$ 0 0
$$213$$ −4.97056 −0.340577
$$214$$ 0 0
$$215$$ −6.89949 −0.470542
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 1.51472 0.102355
$$220$$ 0 0
$$221$$ −15.3137 −1.03011
$$222$$ 0 0
$$223$$ −14.9706 −1.00250 −0.501252 0.865302i $$-0.667127\pi$$
−0.501252 + 0.865302i $$0.667127\pi$$
$$224$$ 0 0
$$225$$ −2.82843 −0.188562
$$226$$ 0 0
$$227$$ 14.0000 0.929213 0.464606 0.885517i $$-0.346196\pi$$
0.464606 + 0.885517i $$0.346196\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 11.6569 0.763666 0.381833 0.924231i $$-0.375293\pi$$
0.381833 + 0.924231i $$0.375293\pi$$
$$234$$ 0 0
$$235$$ 11.6569 0.760409
$$236$$ 0 0
$$237$$ −1.65685 −0.107624
$$238$$ 0 0
$$239$$ −30.4853 −1.97193 −0.985964 0.166955i $$-0.946606\pi$$
−0.985964 + 0.166955i $$0.946606\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 10.3431 0.663513
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −11.3137 −0.719874
$$248$$ 0 0
$$249$$ −1.97056 −0.124879
$$250$$ 0 0
$$251$$ −27.4558 −1.73300 −0.866499 0.499179i $$-0.833635\pi$$
−0.866499 + 0.499179i $$0.833635\pi$$
$$252$$ 0 0
$$253$$ −4.62742 −0.290923
$$254$$ 0 0
$$255$$ −3.17157 −0.198612
$$256$$ 0 0
$$257$$ 15.3137 0.955243 0.477621 0.878566i $$-0.341499\pi$$
0.477621 + 0.878566i $$0.341499\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 22.1421 1.37056
$$262$$ 0 0
$$263$$ −15.7279 −0.969825 −0.484913 0.874563i $$-0.661149\pi$$
−0.484913 + 0.874563i $$0.661149\pi$$
$$264$$ 0 0
$$265$$ −5.65685 −0.347498
$$266$$ 0 0
$$267$$ −2.21320 −0.135446
$$268$$ 0 0
$$269$$ −6.65685 −0.405876 −0.202938 0.979192i $$-0.565049\pi$$
−0.202938 + 0.979192i $$0.565049\pi$$
$$270$$ 0 0
$$271$$ −3.31371 −0.201293 −0.100647 0.994922i $$-0.532091\pi$$
−0.100647 + 0.994922i $$0.532091\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.828427 0.0499560
$$276$$ 0 0
$$277$$ 28.6274 1.72005 0.860027 0.510248i $$-0.170446\pi$$
0.860027 + 0.510248i $$0.170446\pi$$
$$278$$ 0 0
$$279$$ −2.34315 −0.140280
$$280$$ 0 0
$$281$$ 2.68629 0.160251 0.0801254 0.996785i $$-0.474468\pi$$
0.0801254 + 0.996785i $$0.474468\pi$$
$$282$$ 0 0
$$283$$ 18.0000 1.06999 0.534994 0.844856i $$-0.320314\pi$$
0.534994 + 0.844856i $$0.320314\pi$$
$$284$$ 0 0
$$285$$ −2.34315 −0.138796
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 41.6274 2.44867
$$290$$ 0 0
$$291$$ 2.48528 0.145690
$$292$$ 0 0
$$293$$ 16.9706 0.991431 0.495715 0.868485i $$-0.334906\pi$$
0.495715 + 0.868485i $$0.334906\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ −11.1716 −0.646069
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −4.75736 −0.273303
$$304$$ 0 0
$$305$$ 6.65685 0.381170
$$306$$ 0 0
$$307$$ 4.75736 0.271517 0.135758 0.990742i $$-0.456653\pi$$
0.135758 + 0.990742i $$0.456653\pi$$
$$308$$ 0 0
$$309$$ 3.14214 0.178750
$$310$$ 0 0
$$311$$ 21.6569 1.22805 0.614024 0.789288i $$-0.289550\pi$$
0.614024 + 0.789288i $$0.289550\pi$$
$$312$$ 0 0
$$313$$ 20.9706 1.18533 0.592663 0.805450i $$-0.298077\pi$$
0.592663 + 0.805450i $$0.298077\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 0 0
$$319$$ −6.48528 −0.363106
$$320$$ 0 0
$$321$$ −2.31371 −0.129139
$$322$$ 0 0
$$323$$ 43.3137 2.41004
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 0 0
$$327$$ 7.58579 0.419495
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 26.4853 1.45576 0.727881 0.685703i $$-0.240505\pi$$
0.727881 + 0.685703i $$0.240505\pi$$
$$332$$ 0 0
$$333$$ −16.0000 −0.876795
$$334$$ 0 0
$$335$$ −12.8995 −0.704775
$$336$$ 0 0
$$337$$ 24.9706 1.36023 0.680117 0.733104i $$-0.261929\pi$$
0.680117 + 0.733104i $$0.261929\pi$$
$$338$$ 0 0
$$339$$ 4.68629 0.254524
$$340$$ 0 0
$$341$$ 0.686292 0.0371648
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −2.31371 −0.124566
$$346$$ 0 0
$$347$$ 11.3848 0.611167 0.305583 0.952165i $$-0.401149\pi$$
0.305583 + 0.952165i $$0.401149\pi$$
$$348$$ 0 0
$$349$$ 9.82843 0.526104 0.263052 0.964782i $$-0.415271\pi$$
0.263052 + 0.964782i $$0.415271\pi$$
$$350$$ 0 0
$$351$$ −4.82843 −0.257722
$$352$$ 0 0
$$353$$ 33.6569 1.79137 0.895687 0.444685i $$-0.146685\pi$$
0.895687 + 0.444685i $$0.146685\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6.48528 −0.342280 −0.171140 0.985247i $$-0.554745\pi$$
−0.171140 + 0.985247i $$0.554745\pi$$
$$360$$ 0 0
$$361$$ 13.0000 0.684211
$$362$$ 0 0
$$363$$ −4.27208 −0.224226
$$364$$ 0 0
$$365$$ 3.65685 0.191408
$$366$$ 0 0
$$367$$ 21.5858 1.12677 0.563384 0.826195i $$-0.309499\pi$$
0.563384 + 0.826195i $$0.309499\pi$$
$$368$$ 0 0
$$369$$ −16.4853 −0.858189
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 12.0000 0.621336 0.310668 0.950518i $$-0.399447\pi$$
0.310668 + 0.950518i $$0.399447\pi$$
$$374$$ 0 0
$$375$$ 0.414214 0.0213899
$$376$$ 0 0
$$377$$ −15.6569 −0.806369
$$378$$ 0 0
$$379$$ 4.68629 0.240719 0.120359 0.992730i $$-0.461595\pi$$
0.120359 + 0.992730i $$0.461595\pi$$
$$380$$ 0 0
$$381$$ −1.79899 −0.0921650
$$382$$ 0 0
$$383$$ −0.899495 −0.0459620 −0.0229810 0.999736i $$-0.507316\pi$$
−0.0229810 + 0.999736i $$0.507316\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 19.5147 0.991989
$$388$$ 0 0
$$389$$ 5.31371 0.269416 0.134708 0.990885i $$-0.456990\pi$$
0.134708 + 0.990885i $$0.456990\pi$$
$$390$$ 0 0
$$391$$ 42.7696 2.16295
$$392$$ 0 0
$$393$$ 5.65685 0.285351
$$394$$ 0 0
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ −24.6274 −1.23601 −0.618007 0.786172i $$-0.712060\pi$$
−0.618007 + 0.786172i $$0.712060\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −32.3137 −1.61367 −0.806835 0.590777i $$-0.798821\pi$$
−0.806835 + 0.590777i $$0.798821\pi$$
$$402$$ 0 0
$$403$$ 1.65685 0.0825338
$$404$$ 0 0
$$405$$ 7.48528 0.371947
$$406$$ 0 0
$$407$$ 4.68629 0.232291
$$408$$ 0 0
$$409$$ −25.1421 −1.24320 −0.621599 0.783335i $$-0.713517\pi$$
−0.621599 + 0.783335i $$0.713517\pi$$
$$410$$ 0 0
$$411$$ −1.65685 −0.0817266
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.75736 −0.233530
$$416$$ 0 0
$$417$$ 1.02944 0.0504118
$$418$$ 0 0
$$419$$ −15.3137 −0.748124 −0.374062 0.927404i $$-0.622035\pi$$
−0.374062 + 0.927404i $$0.622035\pi$$
$$420$$ 0 0
$$421$$ 27.3431 1.33262 0.666312 0.745673i $$-0.267872\pi$$
0.666312 + 0.745673i $$0.267872\pi$$
$$422$$ 0 0
$$423$$ −32.9706 −1.60308
$$424$$ 0 0
$$425$$ −7.65685 −0.371412
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0.686292 0.0331345
$$430$$ 0 0
$$431$$ −0.828427 −0.0399039 −0.0199520 0.999801i $$-0.506351\pi$$
−0.0199520 + 0.999801i $$0.506351\pi$$
$$432$$ 0 0
$$433$$ −19.3137 −0.928158 −0.464079 0.885794i $$-0.653615\pi$$
−0.464079 + 0.885794i $$0.653615\pi$$
$$434$$ 0 0
$$435$$ −3.24264 −0.155473
$$436$$ 0 0
$$437$$ 31.5980 1.51154
$$438$$ 0 0
$$439$$ −18.3431 −0.875471 −0.437735 0.899104i $$-0.644219\pi$$
−0.437735 + 0.899104i $$0.644219\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −15.5858 −0.740503 −0.370252 0.928932i $$-0.620729\pi$$
−0.370252 + 0.928932i $$0.620729\pi$$
$$444$$ 0 0
$$445$$ −5.34315 −0.253289
$$446$$ 0 0
$$447$$ 1.92893 0.0912354
$$448$$ 0 0
$$449$$ −7.48528 −0.353252 −0.176626 0.984278i $$-0.556518\pi$$
−0.176626 + 0.984278i $$0.556518\pi$$
$$450$$ 0 0
$$451$$ 4.82843 0.227362
$$452$$ 0 0
$$453$$ −4.62742 −0.217415
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 28.9706 1.35519 0.677593 0.735437i $$-0.263023\pi$$
0.677593 + 0.735437i $$0.263023\pi$$
$$458$$ 0 0
$$459$$ 18.4853 0.862819
$$460$$ 0 0
$$461$$ 1.31371 0.0611855 0.0305928 0.999532i $$-0.490261\pi$$
0.0305928 + 0.999532i $$0.490261\pi$$
$$462$$ 0 0
$$463$$ 14.8995 0.692438 0.346219 0.938154i $$-0.387465\pi$$
0.346219 + 0.938154i $$0.387465\pi$$
$$464$$ 0 0
$$465$$ 0.343146 0.0159130
$$466$$ 0 0
$$467$$ −37.8701 −1.75242 −0.876209 0.481932i $$-0.839935\pi$$
−0.876209 + 0.481932i $$0.839935\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0.544156 0.0250734
$$472$$ 0 0
$$473$$ −5.71573 −0.262809
$$474$$ 0 0
$$475$$ −5.65685 −0.259554
$$476$$ 0 0
$$477$$ 16.0000 0.732590
$$478$$ 0 0
$$479$$ −1.51472 −0.0692093 −0.0346046 0.999401i $$-0.511017\pi$$
−0.0346046 + 0.999401i $$0.511017\pi$$
$$480$$ 0 0
$$481$$ 11.3137 0.515861
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 6.00000 0.272446
$$486$$ 0 0
$$487$$ −38.2843 −1.73483 −0.867413 0.497589i $$-0.834219\pi$$
−0.867413 + 0.497589i $$0.834219\pi$$
$$488$$ 0 0
$$489$$ −6.48528 −0.293275
$$490$$ 0 0
$$491$$ 1.51472 0.0683583 0.0341791 0.999416i $$-0.489118\pi$$
0.0341791 + 0.999416i $$0.489118\pi$$
$$492$$ 0 0
$$493$$ 59.9411 2.69961
$$494$$ 0 0
$$495$$ −2.34315 −0.105317
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 44.1421 1.97607 0.988037 0.154219i $$-0.0492861\pi$$
0.988037 + 0.154219i $$0.0492861\pi$$
$$500$$ 0 0
$$501$$ 0.857864 0.0383266
$$502$$ 0 0
$$503$$ −3.92893 −0.175182 −0.0875912 0.996157i $$-0.527917\pi$$
−0.0875912 + 0.996157i $$0.527917\pi$$
$$504$$ 0 0
$$505$$ −11.4853 −0.511088
$$506$$ 0 0
$$507$$ −3.72792 −0.165563
$$508$$ 0 0
$$509$$ −33.4853 −1.48421 −0.742105 0.670284i $$-0.766172\pi$$
−0.742105 + 0.670284i $$0.766172\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 13.6569 0.602965
$$514$$ 0 0
$$515$$ 7.58579 0.334270
$$516$$ 0 0
$$517$$ 9.65685 0.424708
$$518$$ 0 0
$$519$$ 4.28427 0.188059
$$520$$ 0 0
$$521$$ 36.6274 1.60468 0.802338 0.596870i $$-0.203589\pi$$
0.802338 + 0.596870i $$0.203589\pi$$
$$522$$ 0 0
$$523$$ 27.9411 1.22178 0.610890 0.791715i $$-0.290812\pi$$
0.610890 + 0.791715i $$0.290812\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −6.34315 −0.276312
$$528$$ 0 0
$$529$$ 8.20101 0.356566
$$530$$ 0 0
$$531$$ 11.3137 0.490973
$$532$$ 0 0
$$533$$ 11.6569 0.504914
$$534$$ 0 0
$$535$$ −5.58579 −0.241495
$$536$$ 0 0
$$537$$ −2.68629 −0.115922
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −23.4853 −1.00971 −0.504856 0.863204i $$-0.668454\pi$$
−0.504856 + 0.863204i $$0.668454\pi$$
$$542$$ 0 0
$$543$$ −1.72792 −0.0741522
$$544$$ 0 0
$$545$$ 18.3137 0.784473
$$546$$ 0 0
$$547$$ −2.27208 −0.0971470 −0.0485735 0.998820i $$-0.515468\pi$$
−0.0485735 + 0.998820i $$0.515468\pi$$
$$548$$ 0 0
$$549$$ −18.8284 −0.803578
$$550$$ 0 0
$$551$$ 44.2843 1.88657
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 2.34315 0.0994610
$$556$$ 0 0
$$557$$ −17.3137 −0.733605 −0.366803 0.930299i $$-0.619548\pi$$
−0.366803 + 0.930299i $$0.619548\pi$$
$$558$$ 0 0
$$559$$ −13.7990 −0.583635
$$560$$ 0 0
$$561$$ −2.62742 −0.110930
$$562$$ 0 0
$$563$$ −4.07107 −0.171575 −0.0857875 0.996313i $$-0.527341\pi$$
−0.0857875 + 0.996313i $$0.527341\pi$$
$$564$$ 0 0
$$565$$ 11.3137 0.475971
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −36.6274 −1.53550 −0.767751 0.640749i $$-0.778624\pi$$
−0.767751 + 0.640749i $$0.778624\pi$$
$$570$$ 0 0
$$571$$ −20.9706 −0.877591 −0.438795 0.898587i $$-0.644595\pi$$
−0.438795 + 0.898587i $$0.644595\pi$$
$$572$$ 0 0
$$573$$ −2.28427 −0.0954268
$$574$$ 0 0
$$575$$ −5.58579 −0.232943
$$576$$ 0 0
$$577$$ −22.2843 −0.927706 −0.463853 0.885912i $$-0.653533\pi$$
−0.463853 + 0.885912i $$0.653533\pi$$
$$578$$ 0 0
$$579$$ −2.20101 −0.0914709
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.68629 −0.194086
$$584$$ 0 0
$$585$$ −5.65685 −0.233882
$$586$$ 0 0
$$587$$ −22.6863 −0.936363 −0.468182 0.883632i $$-0.655091\pi$$
−0.468182 + 0.883632i $$0.655091\pi$$
$$588$$ 0 0
$$589$$ −4.68629 −0.193095
$$590$$ 0 0
$$591$$ 0.142136 0.00584668
$$592$$ 0 0
$$593$$ 29.9411 1.22953 0.614767 0.788709i $$-0.289250\pi$$
0.614767 + 0.788709i $$0.289250\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −9.65685 −0.395229
$$598$$ 0 0
$$599$$ 42.6274 1.74171 0.870855 0.491541i $$-0.163566\pi$$
0.870855 + 0.491541i $$0.163566\pi$$
$$600$$ 0 0
$$601$$ −34.0000 −1.38689 −0.693444 0.720510i $$-0.743908\pi$$
−0.693444 + 0.720510i $$0.743908\pi$$
$$602$$ 0 0
$$603$$ 36.4853 1.48580
$$604$$ 0 0
$$605$$ −10.3137 −0.419312
$$606$$ 0 0
$$607$$ 27.2426 1.10574 0.552872 0.833266i $$-0.313532\pi$$
0.552872 + 0.833266i $$0.313532\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 23.3137 0.943172
$$612$$ 0 0
$$613$$ 38.9706 1.57401 0.787003 0.616949i $$-0.211632\pi$$
0.787003 + 0.616949i $$0.211632\pi$$
$$614$$ 0 0
$$615$$ 2.41421 0.0973505
$$616$$ 0 0
$$617$$ −12.6863 −0.510731 −0.255365 0.966845i $$-0.582196\pi$$
−0.255365 + 0.966845i $$0.582196\pi$$
$$618$$ 0 0
$$619$$ 39.4558 1.58586 0.792932 0.609310i $$-0.208553\pi$$
0.792932 + 0.609310i $$0.208553\pi$$
$$620$$ 0 0
$$621$$ 13.4853 0.541146
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −1.94113 −0.0775211
$$628$$ 0 0
$$629$$ −43.3137 −1.72703
$$630$$ 0 0
$$631$$ −1.51472 −0.0603000 −0.0301500 0.999545i $$-0.509598\pi$$
−0.0301500 + 0.999545i $$0.509598\pi$$
$$632$$ 0 0
$$633$$ 11.0294 0.438381
$$634$$ 0 0
$$635$$ −4.34315 −0.172352
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 33.9411 1.34269
$$640$$ 0 0
$$641$$ 14.1127 0.557418 0.278709 0.960376i $$-0.410093\pi$$
0.278709 + 0.960376i $$0.410093\pi$$
$$642$$ 0 0
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ −2.85786 −0.112528
$$646$$ 0 0
$$647$$ −20.7574 −0.816056 −0.408028 0.912969i $$-0.633783\pi$$
−0.408028 + 0.912969i $$0.633783\pi$$
$$648$$ 0 0
$$649$$ −3.31371 −0.130074
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −15.6569 −0.612700 −0.306350 0.951919i $$-0.599108\pi$$
−0.306350 + 0.951919i $$0.599108\pi$$
$$654$$ 0 0
$$655$$ 13.6569 0.533617
$$656$$ 0 0
$$657$$ −10.3431 −0.403525
$$658$$ 0 0
$$659$$ 12.6863 0.494188 0.247094 0.968992i $$-0.420524\pi$$
0.247094 + 0.968992i $$0.420524\pi$$
$$660$$ 0 0
$$661$$ −50.3137 −1.95698 −0.978488 0.206303i $$-0.933857\pi$$
−0.978488 + 0.206303i $$0.933857\pi$$
$$662$$ 0 0
$$663$$ −6.34315 −0.246347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 43.7279 1.69315
$$668$$ 0 0
$$669$$ −6.20101 −0.239745
$$670$$ 0 0
$$671$$ 5.51472 0.212893
$$672$$ 0 0
$$673$$ −5.65685 −0.218056 −0.109028 0.994039i $$-0.534774\pi$$
−0.109028 + 0.994039i $$0.534774\pi$$
$$674$$ 0 0
$$675$$ −2.41421 −0.0929231
$$676$$ 0 0
$$677$$ 22.9706 0.882830 0.441415 0.897303i $$-0.354477\pi$$
0.441415 + 0.897303i $$0.354477\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 5.79899 0.222218
$$682$$ 0 0
$$683$$ −1.92893 −0.0738085 −0.0369043 0.999319i $$-0.511750\pi$$
−0.0369043 + 0.999319i $$0.511750\pi$$
$$684$$ 0 0
$$685$$ −4.00000 −0.152832
$$686$$ 0 0
$$687$$ −5.79899 −0.221245
$$688$$ 0 0
$$689$$ −11.3137 −0.431018
$$690$$ 0 0
$$691$$ −42.7696 −1.62703 −0.813515 0.581544i $$-0.802449\pi$$
−0.813515 + 0.581544i $$0.802449\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 2.48528 0.0942721
$$696$$ 0 0
$$697$$ −44.6274 −1.69038
$$698$$ 0 0
$$699$$ 4.82843 0.182628
$$700$$ 0 0
$$701$$ −11.0000 −0.415464 −0.207732 0.978186i $$-0.566608\pi$$
−0.207732 + 0.978186i $$0.566608\pi$$
$$702$$ 0 0
$$703$$ −32.0000 −1.20690
$$704$$ 0 0
$$705$$ 4.82843 0.181849
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −35.4264 −1.33047 −0.665233 0.746636i $$-0.731668\pi$$
−0.665233 + 0.746636i $$0.731668\pi$$
$$710$$ 0 0
$$711$$ 11.3137 0.424297
$$712$$ 0 0
$$713$$ −4.62742 −0.173298
$$714$$ 0 0
$$715$$ 1.65685 0.0619628
$$716$$ 0 0
$$717$$ −12.6274 −0.471580
$$718$$ 0 0
$$719$$ −25.7990 −0.962140 −0.481070 0.876682i $$-0.659752\pi$$
−0.481070 + 0.876682i $$0.659752\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −4.14214 −0.154048
$$724$$ 0 0
$$725$$ −7.82843 −0.290740
$$726$$ 0 0
$$727$$ 38.0711 1.41198 0.705989 0.708223i $$-0.250503\pi$$
0.705989 + 0.708223i $$0.250503\pi$$
$$728$$ 0 0
$$729$$ −18.1716 −0.673021
$$730$$ 0 0
$$731$$ 52.8284 1.95393
$$732$$ 0 0
$$733$$ 7.65685 0.282812 0.141406 0.989952i $$-0.454838\pi$$
0.141406 + 0.989952i $$0.454838\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.6863 −0.393635
$$738$$ 0 0
$$739$$ 16.8284 0.619044 0.309522 0.950892i $$-0.399831\pi$$
0.309522 + 0.950892i $$0.399831\pi$$
$$740$$ 0 0
$$741$$ −4.68629 −0.172155
$$742$$ 0 0
$$743$$ −10.7574 −0.394649 −0.197325 0.980338i $$-0.563225\pi$$
−0.197325 + 0.980338i $$0.563225\pi$$
$$744$$ 0 0
$$745$$ 4.65685 0.170614
$$746$$ 0 0
$$747$$ 13.4558 0.492324
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ 0 0
$$753$$ −11.3726 −0.414440
$$754$$ 0 0
$$755$$ −11.1716 −0.406575
$$756$$ 0 0
$$757$$ 6.00000 0.218074 0.109037 0.994038i $$-0.465223\pi$$
0.109037 + 0.994038i $$0.465223\pi$$
$$758$$ 0 0
$$759$$ −1.91674 −0.0695732
$$760$$ 0 0
$$761$$ −43.9411 −1.59286 −0.796432 0.604728i $$-0.793282\pi$$
−0.796432 + 0.604728i $$0.793282\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 21.6569 0.783005
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 43.2548 1.55981 0.779905 0.625898i $$-0.215268\pi$$
0.779905 + 0.625898i $$0.215268\pi$$
$$770$$ 0 0
$$771$$ 6.34315 0.228443
$$772$$ 0 0
$$773$$ 24.3431 0.875562 0.437781 0.899082i $$-0.355765\pi$$
0.437781 + 0.899082i $$0.355765\pi$$
$$774$$ 0 0
$$775$$ 0.828427 0.0297580
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −32.9706 −1.18129
$$780$$ 0 0
$$781$$ −9.94113 −0.355721
$$782$$ 0 0
$$783$$ 18.8995 0.675413
$$784$$ 0 0
$$785$$ 1.31371 0.0468883
$$786$$ 0 0
$$787$$ −1.58579 −0.0565272 −0.0282636 0.999601i $$-0.508998\pi$$
−0.0282636 + 0.999601i $$0.508998\pi$$
$$788$$ 0 0
$$789$$ −6.51472 −0.231930
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 13.3137 0.472784
$$794$$ 0 0
$$795$$ −2.34315 −0.0831028
$$796$$ 0 0
$$797$$ −35.3137 −1.25088 −0.625438 0.780274i $$-0.715080\pi$$
−0.625438 + 0.780274i $$0.715080\pi$$
$$798$$ 0 0
$$799$$ −89.2548 −3.15761
$$800$$ 0 0
$$801$$ 15.1127 0.533981
$$802$$ 0 0
$$803$$ 3.02944 0.106907
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −2.75736 −0.0970636
$$808$$ 0 0
$$809$$ −35.9706 −1.26466 −0.632329 0.774700i $$-0.717901\pi$$
−0.632329 + 0.774700i $$0.717901\pi$$
$$810$$ 0 0
$$811$$ 52.1421 1.83096 0.915479 0.402366i $$-0.131812\pi$$
0.915479 + 0.402366i $$0.131812\pi$$
$$812$$ 0 0
$$813$$ −1.37258 −0.0481386
$$814$$ 0 0
$$815$$ −15.6569 −0.548436
$$816$$ 0 0
$$817$$ 39.0294 1.36547
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −32.6274 −1.13870 −0.569352 0.822094i $$-0.692806\pi$$
−0.569352 + 0.822094i $$0.692806\pi$$
$$822$$ 0 0
$$823$$ 30.0122 1.04616 0.523080 0.852284i $$-0.324783\pi$$
0.523080 + 0.852284i $$0.324783\pi$$
$$824$$ 0 0
$$825$$ 0.343146 0.0119468
$$826$$ 0 0
$$827$$ 13.5269 0.470377 0.235188 0.971950i $$-0.424429\pi$$
0.235188 + 0.971950i $$0.424429\pi$$
$$828$$ 0 0
$$829$$ 5.31371 0.184553 0.0922764 0.995733i $$-0.470586\pi$$
0.0922764 + 0.995733i $$0.470586\pi$$
$$830$$ 0 0
$$831$$ 11.8579 0.411345
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2.07107 0.0716723
$$836$$ 0 0
$$837$$ −2.00000 −0.0691301
$$838$$ 0 0
$$839$$ 20.1421 0.695384 0.347692 0.937609i $$-0.386966\pi$$
0.347692 + 0.937609i $$0.386966\pi$$
$$840$$ 0 0
$$841$$ 32.2843 1.11325
$$842$$ 0 0
$$843$$ 1.11270 0.0383234
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 7.45584 0.255884
$$850$$ 0 0
$$851$$ −31.5980 −1.08316
$$852$$ 0 0
$$853$$ 12.6863 0.434370 0.217185 0.976130i $$-0.430312\pi$$
0.217185 + 0.976130i $$0.430312\pi$$
$$854$$ 0 0
$$855$$ 16.0000 0.547188
$$856$$ 0 0
$$857$$ −18.9706 −0.648022 −0.324011 0.946053i $$-0.605031\pi$$
−0.324011 + 0.946053i $$0.605031\pi$$
$$858$$ 0 0
$$859$$ −30.6274 −1.04499 −0.522497 0.852641i $$-0.674999\pi$$
−0.522497 + 0.852641i $$0.674999\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −56.0122 −1.90668 −0.953339 0.301903i $$-0.902378\pi$$
−0.953339 + 0.301903i $$0.902378\pi$$
$$864$$ 0 0
$$865$$ 10.3431 0.351678
$$866$$ 0 0
$$867$$ 17.2426 0.585591
$$868$$ 0 0
$$869$$ −3.31371 −0.112410
$$870$$ 0 0
$$871$$ −25.7990 −0.874165
$$872$$ 0 0
$$873$$ −16.9706 −0.574367
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −28.2843 −0.955092 −0.477546 0.878607i $$-0.658474\pi$$
−0.477546 + 0.878607i $$0.658474\pi$$
$$878$$ 0 0
$$879$$ 7.02944 0.237097
$$880$$ 0 0
$$881$$ −26.4558 −0.891320 −0.445660 0.895202i $$-0.647031\pi$$
−0.445660 + 0.895202i $$0.647031\pi$$
$$882$$ 0 0
$$883$$ 6.68629 0.225012 0.112506 0.993651i $$-0.464112\pi$$
0.112506 + 0.993651i $$0.464112\pi$$
$$884$$ 0 0
$$885$$ −1.65685 −0.0556945
$$886$$ 0 0
$$887$$ 17.5858 0.590473 0.295236 0.955424i $$-0.404602\pi$$
0.295236 + 0.955424i $$0.404602\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 6.20101 0.207742
$$892$$ 0 0
$$893$$ −65.9411 −2.20664
$$894$$ 0 0
$$895$$ −6.48528 −0.216779
$$896$$ 0 0
$$897$$ −4.62742 −0.154505
$$898$$ 0 0
$$899$$ −6.48528 −0.216296
$$900$$ 0 0
$$901$$ 43.3137 1.44299
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4.17157 −0.138668
$$906$$ 0 0
$$907$$ 46.2132 1.53448 0.767242 0.641358i $$-0.221628\pi$$
0.767242 + 0.641358i $$0.221628\pi$$
$$908$$ 0 0
$$909$$ 32.4853 1.07747
$$910$$ 0 0
$$911$$ −18.4853 −0.612445 −0.306222 0.951960i $$-0.599065\pi$$
−0.306222 + 0.951960i $$0.599065\pi$$
$$912$$ 0 0
$$913$$ −3.94113 −0.130432
$$914$$ 0 0
$$915$$ 2.75736 0.0911555
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −33.2548 −1.09698 −0.548488 0.836159i $$-0.684796\pi$$
−0.548488 + 0.836159i $$0.684796\pi$$
$$920$$ 0 0
$$921$$ 1.97056 0.0649323
$$922$$ 0 0
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ 5.65685 0.185996
$$926$$ 0 0
$$927$$ −21.4558 −0.704702
$$928$$ 0 0
$$929$$ 9.20101 0.301875 0.150938 0.988543i $$-0.451771\pi$$
0.150938 + 0.988543i $$0.451771\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 8.97056 0.293683
$$934$$ 0 0
$$935$$ −6.34315 −0.207443
$$936$$ 0 0
$$937$$ 18.6274 0.608531 0.304266 0.952587i $$-0.401589\pi$$
0.304266 + 0.952587i $$0.401589\pi$$
$$938$$ 0 0
$$939$$ 8.68629 0.283466
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ −32.5563 −1.06018
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 11.1005 0.360718 0.180359 0.983601i $$-0.442274\pi$$
0.180359 + 0.983601i $$0.442274\pi$$
$$948$$ 0 0
$$949$$ 7.31371 0.237413
$$950$$ 0 0
$$951$$ −9.11270 −0.295499
$$952$$ 0 0
$$953$$ −54.6274 −1.76956 −0.884778 0.466013i $$-0.845690\pi$$
−0.884778 + 0.466013i $$0.845690\pi$$
$$954$$ 0 0
$$955$$ −5.51472 −0.178452
$$956$$ 0 0
$$957$$ −2.68629 −0.0868355
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.3137 −0.977862
$$962$$ 0 0
$$963$$ 15.7990 0.509115
$$964$$ 0 0
$$965$$ −5.31371 −0.171054
$$966$$ 0 0
$$967$$ −24.7574 −0.796143 −0.398072 0.917354i $$-0.630320\pi$$
−0.398072 + 0.917354i $$0.630320\pi$$
$$968$$ 0 0
$$969$$ 17.9411 0.576352
$$970$$ 0 0
$$971$$ 8.00000 0.256732 0.128366 0.991727i $$-0.459027\pi$$
0.128366 + 0.991727i $$0.459027\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0.828427 0.0265309
$$976$$ 0 0
$$977$$ 30.2843 0.968880 0.484440 0.874825i $$-0.339023\pi$$
0.484440 + 0.874825i $$0.339023\pi$$
$$978$$ 0 0
$$979$$ −4.42641 −0.141469
$$980$$ 0 0
$$981$$ −51.7990 −1.65381
$$982$$ 0 0
$$983$$ −41.3848 −1.31997 −0.659985 0.751279i $$-0.729437\pi$$
−0.659985 + 0.751279i $$0.729437\pi$$
$$984$$ 0 0
$$985$$ 0.343146 0.0109335
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 38.5391 1.22547
$$990$$ 0 0
$$991$$ 4.82843 0.153380 0.0766900 0.997055i $$-0.475565\pi$$
0.0766900 + 0.997055i $$0.475565\pi$$
$$992$$ 0 0
$$993$$ 10.9706 0.348140
$$994$$ 0 0
$$995$$ −23.3137 −0.739094
$$996$$ 0 0
$$997$$ −14.3431 −0.454252 −0.227126 0.973865i $$-0.572933\pi$$
−0.227126 + 0.973865i $$0.572933\pi$$
$$998$$ 0 0
$$999$$ −13.6569 −0.432084
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 1960.2.a.p.1.2 2
4.3 odd 2 3920.2.a.bz.1.1 2
5.4 even 2 9800.2.a.bz.1.1 2
7.2 even 3 280.2.q.d.81.1 4
7.3 odd 6 1960.2.q.q.961.2 4
7.4 even 3 280.2.q.d.121.1 yes 4
7.5 odd 6 1960.2.q.q.361.2 4
7.6 odd 2 1960.2.a.t.1.1 2
21.2 odd 6 2520.2.bi.k.361.2 4
21.11 odd 6 2520.2.bi.k.1801.2 4
28.11 odd 6 560.2.q.j.401.2 4
28.23 odd 6 560.2.q.j.81.2 4
28.27 even 2 3920.2.a.bp.1.2 2
35.2 odd 12 1400.2.bh.g.249.3 8
35.4 even 6 1400.2.q.h.401.2 4
35.9 even 6 1400.2.q.h.1201.2 4
35.18 odd 12 1400.2.bh.g.849.3 8
35.23 odd 12 1400.2.bh.g.249.2 8
35.32 odd 12 1400.2.bh.g.849.2 8
35.34 odd 2 9800.2.a.br.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.d.81.1 4 7.2 even 3
280.2.q.d.121.1 yes 4 7.4 even 3
560.2.q.j.81.2 4 28.23 odd 6
560.2.q.j.401.2 4 28.11 odd 6
1400.2.q.h.401.2 4 35.4 even 6
1400.2.q.h.1201.2 4 35.9 even 6
1400.2.bh.g.249.2 8 35.23 odd 12
1400.2.bh.g.249.3 8 35.2 odd 12
1400.2.bh.g.849.2 8 35.32 odd 12
1400.2.bh.g.849.3 8 35.18 odd 12
1960.2.a.p.1.2 2 1.1 even 1 trivial
1960.2.a.t.1.1 2 7.6 odd 2
1960.2.q.q.361.2 4 7.5 odd 6
1960.2.q.q.961.2 4 7.3 odd 6
2520.2.bi.k.361.2 4 21.2 odd 6
2520.2.bi.k.1801.2 4 21.11 odd 6
3920.2.a.bp.1.2 2 28.27 even 2
3920.2.a.bz.1.1 2 4.3 odd 2
9800.2.a.br.1.2 2 35.34 odd 2
9800.2.a.bz.1.1 2 5.4 even 2