# Properties

 Label 1170.2.a.o.1.1 Level $1170$ Weight $2$ Character 1170.1 Self dual yes Analytic conductor $9.342$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1170.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.65685 q^{11} -1.00000 q^{13} +2.82843 q^{14} +1.00000 q^{16} -0.828427 q^{17} +2.82843 q^{19} -1.00000 q^{20} +5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{25} +1.00000 q^{26} -2.82843 q^{28} +8.82843 q^{29} +4.00000 q^{31} -1.00000 q^{32} +0.828427 q^{34} +2.82843 q^{35} -11.6569 q^{37} -2.82843 q^{38} +1.00000 q^{40} +7.65685 q^{41} +9.65685 q^{43} -5.65685 q^{44} -8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -13.3137 q^{53} +5.65685 q^{55} +2.82843 q^{56} -8.82843 q^{58} +2.34315 q^{59} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +5.65685 q^{67} -0.828427 q^{68} -2.82843 q^{70} +5.65685 q^{71} -14.4853 q^{73} +11.6569 q^{74} +2.82843 q^{76} +16.0000 q^{77} +2.34315 q^{79} -1.00000 q^{80} -7.65685 q^{82} -6.34315 q^{83} +0.828427 q^{85} -9.65685 q^{86} +5.65685 q^{88} +15.6569 q^{89} +2.82843 q^{91} +8.48528 q^{92} -8.00000 q^{94} -2.82843 q^{95} -3.17157 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + 2q^{10} - 2q^{13} + 2q^{16} + 4q^{17} - 2q^{20} + 2q^{25} + 2q^{26} + 12q^{29} + 8q^{31} - 2q^{32} - 4q^{34} - 12q^{37} + 2q^{40} + 4q^{41} + 8q^{43} + 16q^{47} + 2q^{49} - 2q^{50} - 2q^{52} - 4q^{53} - 12q^{58} + 16q^{59} + 12q^{61} - 8q^{62} + 2q^{64} + 2q^{65} + 4q^{68} - 12q^{73} + 12q^{74} + 32q^{77} + 16q^{79} - 2q^{80} - 4q^{82} - 24q^{83} - 4q^{85} - 8q^{86} + 20q^{89} - 16q^{94} - 12q^{97} - 2q^{98} + O(q^{100})$$

## 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
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ −5.65685 −1.70561 −0.852803 0.522233i $$-0.825099\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 2.82843 0.755929
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −0.828427 −0.200923 −0.100462 0.994941i $$-0.532032\pi$$
−0.100462 + 0.994941i $$0.532032\pi$$
$$18$$ 0 0
$$19$$ 2.82843 0.648886 0.324443 0.945905i $$-0.394823\pi$$
0.324443 + 0.945905i $$0.394823\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 5.65685 1.20605
$$23$$ 8.48528 1.76930 0.884652 0.466252i $$-0.154396\pi$$
0.884652 + 0.466252i $$0.154396\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ −2.82843 −0.534522
$$29$$ 8.82843 1.63940 0.819699 0.572795i $$-0.194141\pi$$
0.819699 + 0.572795i $$0.194141\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0.828427 0.142074
$$35$$ 2.82843 0.478091
$$36$$ 0 0
$$37$$ −11.6569 −1.91638 −0.958188 0.286141i $$-0.907627\pi$$
−0.958188 + 0.286141i $$0.907627\pi$$
$$38$$ −2.82843 −0.458831
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 7.65685 1.19580 0.597900 0.801571i $$-0.296002\pi$$
0.597900 + 0.801571i $$0.296002\pi$$
$$42$$ 0 0
$$43$$ 9.65685 1.47266 0.736328 0.676625i $$-0.236558\pi$$
0.736328 + 0.676625i $$0.236558\pi$$
$$44$$ −5.65685 −0.852803
$$45$$ 0 0
$$46$$ −8.48528 −1.25109
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ −13.3137 −1.82878 −0.914389 0.404836i $$-0.867329\pi$$
−0.914389 + 0.404836i $$0.867329\pi$$
$$54$$ 0 0
$$55$$ 5.65685 0.762770
$$56$$ 2.82843 0.377964
$$57$$ 0 0
$$58$$ −8.82843 −1.15923
$$59$$ 2.34315 0.305052 0.152526 0.988299i $$-0.451259\pi$$
0.152526 + 0.988299i $$0.451259\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 5.65685 0.691095 0.345547 0.938401i $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ −0.828427 −0.100462
$$69$$ 0 0
$$70$$ −2.82843 −0.338062
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ −14.4853 −1.69537 −0.847687 0.530497i $$-0.822005\pi$$
−0.847687 + 0.530497i $$0.822005\pi$$
$$74$$ 11.6569 1.35508
$$75$$ 0 0
$$76$$ 2.82843 0.324443
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ 2.34315 0.263624 0.131812 0.991275i $$-0.457920\pi$$
0.131812 + 0.991275i $$0.457920\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ −7.65685 −0.845558
$$83$$ −6.34315 −0.696251 −0.348125 0.937448i $$-0.613182\pi$$
−0.348125 + 0.937448i $$0.613182\pi$$
$$84$$ 0 0
$$85$$ 0.828427 0.0898555
$$86$$ −9.65685 −1.04133
$$87$$ 0 0
$$88$$ 5.65685 0.603023
$$89$$ 15.6569 1.65962 0.829812 0.558044i $$-0.188448\pi$$
0.829812 + 0.558044i $$0.188448\pi$$
$$90$$ 0 0
$$91$$ 2.82843 0.296500
$$92$$ 8.48528 0.884652
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −2.82843 −0.290191
$$96$$ 0 0
$$97$$ −3.17157 −0.322024 −0.161012 0.986952i $$-0.551476\pi$$
−0.161012 + 0.986952i $$0.551476\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −16.1421 −1.60620 −0.803101 0.595843i $$-0.796818\pi$$
−0.803101 + 0.595843i $$0.796818\pi$$
$$102$$ 0 0
$$103$$ −1.65685 −0.163255 −0.0816274 0.996663i $$-0.526012\pi$$
−0.0816274 + 0.996663i $$0.526012\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 13.3137 1.29314
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ 8.82843 0.845610 0.422805 0.906221i $$-0.361046\pi$$
0.422805 + 0.906221i $$0.361046\pi$$
$$110$$ −5.65685 −0.539360
$$111$$ 0 0
$$112$$ −2.82843 −0.267261
$$113$$ −6.48528 −0.610084 −0.305042 0.952339i $$-0.598670\pi$$
−0.305042 + 0.952339i $$0.598670\pi$$
$$114$$ 0 0
$$115$$ −8.48528 −0.791257
$$116$$ 8.82843 0.819699
$$117$$ 0 0
$$118$$ −2.34315 −0.215704
$$119$$ 2.34315 0.214796
$$120$$ 0 0
$$121$$ 21.0000 1.90909
$$122$$ −6.00000 −0.543214
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 9.65685 0.856907 0.428454 0.903564i $$-0.359059\pi$$
0.428454 + 0.903564i $$0.359059\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −1.00000 −0.0877058
$$131$$ 6.14214 0.536641 0.268320 0.963330i $$-0.413531\pi$$
0.268320 + 0.963330i $$0.413531\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ −5.65685 −0.488678
$$135$$ 0 0
$$136$$ 0.828427 0.0710370
$$137$$ 17.3137 1.47921 0.739605 0.673041i $$-0.235012\pi$$
0.739605 + 0.673041i $$0.235012\pi$$
$$138$$ 0 0
$$139$$ 6.34315 0.538019 0.269009 0.963138i $$-0.413304\pi$$
0.269009 + 0.963138i $$0.413304\pi$$
$$140$$ 2.82843 0.239046
$$141$$ 0 0
$$142$$ −5.65685 −0.474713
$$143$$ 5.65685 0.473050
$$144$$ 0 0
$$145$$ −8.82843 −0.733161
$$146$$ 14.4853 1.19881
$$147$$ 0 0
$$148$$ −11.6569 −0.958188
$$149$$ −3.65685 −0.299581 −0.149791 0.988718i $$-0.547860\pi$$
−0.149791 + 0.988718i $$0.547860\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ −2.82843 −0.229416
$$153$$ 0 0
$$154$$ −16.0000 −1.28932
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 5.31371 0.424080 0.212040 0.977261i $$-0.431989\pi$$
0.212040 + 0.977261i $$0.431989\pi$$
$$158$$ −2.34315 −0.186411
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ −24.0000 −1.89146
$$162$$ 0 0
$$163$$ 11.3137 0.886158 0.443079 0.896483i $$-0.353886\pi$$
0.443079 + 0.896483i $$0.353886\pi$$
$$164$$ 7.65685 0.597900
$$165$$ 0 0
$$166$$ 6.34315 0.492324
$$167$$ −8.97056 −0.694163 −0.347081 0.937835i $$-0.612827\pi$$
−0.347081 + 0.937835i $$0.612827\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −0.828427 −0.0635375
$$171$$ 0 0
$$172$$ 9.65685 0.736328
$$173$$ 9.31371 0.708108 0.354054 0.935225i $$-0.384803\pi$$
0.354054 + 0.935225i $$0.384803\pi$$
$$174$$ 0 0
$$175$$ −2.82843 −0.213809
$$176$$ −5.65685 −0.426401
$$177$$ 0 0
$$178$$ −15.6569 −1.17353
$$179$$ −7.51472 −0.561676 −0.280838 0.959755i $$-0.590612\pi$$
−0.280838 + 0.959755i $$0.590612\pi$$
$$180$$ 0 0
$$181$$ −7.65685 −0.569129 −0.284565 0.958657i $$-0.591849\pi$$
−0.284565 + 0.958657i $$0.591849\pi$$
$$182$$ −2.82843 −0.209657
$$183$$ 0 0
$$184$$ −8.48528 −0.625543
$$185$$ 11.6569 0.857029
$$186$$ 0 0
$$187$$ 4.68629 0.342696
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 2.82843 0.205196
$$191$$ −11.3137 −0.818631 −0.409316 0.912393i $$-0.634232\pi$$
−0.409316 + 0.912393i $$0.634232\pi$$
$$192$$ 0 0
$$193$$ 2.48528 0.178894 0.0894472 0.995992i $$-0.471490\pi$$
0.0894472 + 0.995992i $$0.471490\pi$$
$$194$$ 3.17157 0.227706
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 13.3137 0.948562 0.474281 0.880373i $$-0.342708\pi$$
0.474281 + 0.880373i $$0.342708\pi$$
$$198$$ 0 0
$$199$$ 10.3431 0.733206 0.366603 0.930377i $$-0.380521\pi$$
0.366603 + 0.930377i $$0.380521\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 16.1421 1.13576
$$203$$ −24.9706 −1.75259
$$204$$ 0 0
$$205$$ −7.65685 −0.534778
$$206$$ 1.65685 0.115439
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 0.686292 0.0472463 0.0236231 0.999721i $$-0.492480\pi$$
0.0236231 + 0.999721i $$0.492480\pi$$
$$212$$ −13.3137 −0.914389
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ −9.65685 −0.658592
$$216$$ 0 0
$$217$$ −11.3137 −0.768025
$$218$$ −8.82843 −0.597937
$$219$$ 0 0
$$220$$ 5.65685 0.381385
$$221$$ 0.828427 0.0557260
$$222$$ 0 0
$$223$$ 10.8284 0.725125 0.362563 0.931959i $$-0.381902\pi$$
0.362563 + 0.931959i $$0.381902\pi$$
$$224$$ 2.82843 0.188982
$$225$$ 0 0
$$226$$ 6.48528 0.431394
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ 4.14214 0.273720 0.136860 0.990590i $$-0.456299\pi$$
0.136860 + 0.990590i $$0.456299\pi$$
$$230$$ 8.48528 0.559503
$$231$$ 0 0
$$232$$ −8.82843 −0.579615
$$233$$ −5.51472 −0.361281 −0.180641 0.983549i $$-0.557817\pi$$
−0.180641 + 0.983549i $$0.557817\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 2.34315 0.152526
$$237$$ 0 0
$$238$$ −2.34315 −0.151884
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 5.31371 0.342286 0.171143 0.985246i $$-0.445254\pi$$
0.171143 + 0.985246i $$0.445254\pi$$
$$242$$ −21.0000 −1.34993
$$243$$ 0 0
$$244$$ 6.00000 0.384111
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ −2.82843 −0.179969
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ −10.8284 −0.683484 −0.341742 0.939794i $$-0.611017\pi$$
−0.341742 + 0.939794i $$0.611017\pi$$
$$252$$ 0 0
$$253$$ −48.0000 −3.01773
$$254$$ −9.65685 −0.605925
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 4.82843 0.301189 0.150595 0.988596i $$-0.451881\pi$$
0.150595 + 0.988596i $$0.451881\pi$$
$$258$$ 0 0
$$259$$ 32.9706 2.04869
$$260$$ 1.00000 0.0620174
$$261$$ 0 0
$$262$$ −6.14214 −0.379462
$$263$$ −16.4853 −1.01653 −0.508263 0.861202i $$-0.669712\pi$$
−0.508263 + 0.861202i $$0.669712\pi$$
$$264$$ 0 0
$$265$$ 13.3137 0.817855
$$266$$ 8.00000 0.490511
$$267$$ 0 0
$$268$$ 5.65685 0.345547
$$269$$ 14.4853 0.883183 0.441592 0.897216i $$-0.354414\pi$$
0.441592 + 0.897216i $$0.354414\pi$$
$$270$$ 0 0
$$271$$ 7.31371 0.444276 0.222138 0.975015i $$-0.428696\pi$$
0.222138 + 0.975015i $$0.428696\pi$$
$$272$$ −0.828427 −0.0502308
$$273$$ 0 0
$$274$$ −17.3137 −1.04596
$$275$$ −5.65685 −0.341121
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −6.34315 −0.380437
$$279$$ 0 0
$$280$$ −2.82843 −0.169031
$$281$$ −8.34315 −0.497710 −0.248855 0.968541i $$-0.580054\pi$$
−0.248855 + 0.968541i $$0.580054\pi$$
$$282$$ 0 0
$$283$$ −17.6569 −1.04959 −0.524796 0.851228i $$-0.675858\pi$$
−0.524796 + 0.851228i $$0.675858\pi$$
$$284$$ 5.65685 0.335673
$$285$$ 0 0
$$286$$ −5.65685 −0.334497
$$287$$ −21.6569 −1.27836
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ 8.82843 0.518423
$$291$$ 0 0
$$292$$ −14.4853 −0.847687
$$293$$ 16.6274 0.971384 0.485692 0.874130i $$-0.338568\pi$$
0.485692 + 0.874130i $$0.338568\pi$$
$$294$$ 0 0
$$295$$ −2.34315 −0.136423
$$296$$ 11.6569 0.677541
$$297$$ 0 0
$$298$$ 3.65685 0.211836
$$299$$ −8.48528 −0.490716
$$300$$ 0 0
$$301$$ −27.3137 −1.57434
$$302$$ −12.0000 −0.690522
$$303$$ 0 0
$$304$$ 2.82843 0.162221
$$305$$ −6.00000 −0.343559
$$306$$ 0 0
$$307$$ −21.6569 −1.23602 −0.618011 0.786169i $$-0.712061\pi$$
−0.618011 + 0.786169i $$0.712061\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −30.9706 −1.75056 −0.875280 0.483617i $$-0.839323\pi$$
−0.875280 + 0.483617i $$0.839323\pi$$
$$314$$ −5.31371 −0.299870
$$315$$ 0 0
$$316$$ 2.34315 0.131812
$$317$$ −25.3137 −1.42176 −0.710880 0.703314i $$-0.751703\pi$$
−0.710880 + 0.703314i $$0.751703\pi$$
$$318$$ 0 0
$$319$$ −49.9411 −2.79617
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 24.0000 1.33747
$$323$$ −2.34315 −0.130376
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ −11.3137 −0.626608
$$327$$ 0 0
$$328$$ −7.65685 −0.422779
$$329$$ −22.6274 −1.24749
$$330$$ 0 0
$$331$$ −8.48528 −0.466393 −0.233197 0.972430i $$-0.574919\pi$$
−0.233197 + 0.972430i $$0.574919\pi$$
$$332$$ −6.34315 −0.348125
$$333$$ 0 0
$$334$$ 8.97056 0.490847
$$335$$ −5.65685 −0.309067
$$336$$ 0 0
$$337$$ 10.9706 0.597605 0.298802 0.954315i $$-0.403413\pi$$
0.298802 + 0.954315i $$0.403413\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 0.828427 0.0449278
$$341$$ −22.6274 −1.22534
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ −9.65685 −0.520663
$$345$$ 0 0
$$346$$ −9.31371 −0.500708
$$347$$ 9.65685 0.518407 0.259204 0.965823i $$-0.416540\pi$$
0.259204 + 0.965823i $$0.416540\pi$$
$$348$$ 0 0
$$349$$ 12.1421 0.649954 0.324977 0.945722i $$-0.394644\pi$$
0.324977 + 0.945722i $$0.394644\pi$$
$$350$$ 2.82843 0.151186
$$351$$ 0 0
$$352$$ 5.65685 0.301511
$$353$$ −5.31371 −0.282820 −0.141410 0.989951i $$-0.545164\pi$$
−0.141410 + 0.989951i $$0.545164\pi$$
$$354$$ 0 0
$$355$$ −5.65685 −0.300235
$$356$$ 15.6569 0.829812
$$357$$ 0 0
$$358$$ 7.51472 0.397165
$$359$$ 28.2843 1.49279 0.746393 0.665505i $$-0.231784\pi$$
0.746393 + 0.665505i $$0.231784\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 7.65685 0.402435
$$363$$ 0 0
$$364$$ 2.82843 0.148250
$$365$$ 14.4853 0.758194
$$366$$ 0 0
$$367$$ 25.6569 1.33928 0.669638 0.742687i $$-0.266449\pi$$
0.669638 + 0.742687i $$0.266449\pi$$
$$368$$ 8.48528 0.442326
$$369$$ 0 0
$$370$$ −11.6569 −0.606011
$$371$$ 37.6569 1.95505
$$372$$ 0 0
$$373$$ −2.68629 −0.139091 −0.0695455 0.997579i $$-0.522155\pi$$
−0.0695455 + 0.997579i $$0.522155\pi$$
$$374$$ −4.68629 −0.242322
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ −8.82843 −0.454687
$$378$$ 0 0
$$379$$ −7.51472 −0.386005 −0.193003 0.981198i $$-0.561823\pi$$
−0.193003 + 0.981198i $$0.561823\pi$$
$$380$$ −2.82843 −0.145095
$$381$$ 0 0
$$382$$ 11.3137 0.578860
$$383$$ 29.6569 1.51539 0.757697 0.652606i $$-0.226324\pi$$
0.757697 + 0.652606i $$0.226324\pi$$
$$384$$ 0 0
$$385$$ −16.0000 −0.815436
$$386$$ −2.48528 −0.126497
$$387$$ 0 0
$$388$$ −3.17157 −0.161012
$$389$$ 6.48528 0.328817 0.164408 0.986392i $$-0.447429\pi$$
0.164408 + 0.986392i $$0.447429\pi$$
$$390$$ 0 0
$$391$$ −7.02944 −0.355494
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ −13.3137 −0.670735
$$395$$ −2.34315 −0.117896
$$396$$ 0 0
$$397$$ 30.2843 1.51992 0.759962 0.649968i $$-0.225218\pi$$
0.759962 + 0.649968i $$0.225218\pi$$
$$398$$ −10.3431 −0.518455
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 26.9706 1.34685 0.673423 0.739258i $$-0.264823\pi$$
0.673423 + 0.739258i $$0.264823\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ −16.1421 −0.803101
$$405$$ 0 0
$$406$$ 24.9706 1.23927
$$407$$ 65.9411 3.26858
$$408$$ 0 0
$$409$$ −3.65685 −0.180820 −0.0904099 0.995905i $$-0.528818\pi$$
−0.0904099 + 0.995905i $$0.528818\pi$$
$$410$$ 7.65685 0.378145
$$411$$ 0 0
$$412$$ −1.65685 −0.0816274
$$413$$ −6.62742 −0.326114
$$414$$ 0 0
$$415$$ 6.34315 0.311373
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 16.0000 0.782586
$$419$$ 10.8284 0.529003 0.264502 0.964385i $$-0.414793\pi$$
0.264502 + 0.964385i $$0.414793\pi$$
$$420$$ 0 0
$$421$$ −24.1421 −1.17662 −0.588308 0.808637i $$-0.700206\pi$$
−0.588308 + 0.808637i $$0.700206\pi$$
$$422$$ −0.686292 −0.0334081
$$423$$ 0 0
$$424$$ 13.3137 0.646571
$$425$$ −0.828427 −0.0401846
$$426$$ 0 0
$$427$$ −16.9706 −0.821263
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ 9.65685 0.465695
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ −22.9706 −1.10389 −0.551947 0.833879i $$-0.686115\pi$$
−0.551947 + 0.833879i $$0.686115\pi$$
$$434$$ 11.3137 0.543075
$$435$$ 0 0
$$436$$ 8.82843 0.422805
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ 22.6274 1.07995 0.539974 0.841682i $$-0.318434\pi$$
0.539974 + 0.841682i $$0.318434\pi$$
$$440$$ −5.65685 −0.269680
$$441$$ 0 0
$$442$$ −0.828427 −0.0394043
$$443$$ −30.3431 −1.44165 −0.720823 0.693119i $$-0.756236\pi$$
−0.720823 + 0.693119i $$0.756236\pi$$
$$444$$ 0 0
$$445$$ −15.6569 −0.742206
$$446$$ −10.8284 −0.512741
$$447$$ 0 0
$$448$$ −2.82843 −0.133631
$$449$$ −26.2843 −1.24043 −0.620216 0.784431i $$-0.712955\pi$$
−0.620216 + 0.784431i $$0.712955\pi$$
$$450$$ 0 0
$$451$$ −43.3137 −2.03956
$$452$$ −6.48528 −0.305042
$$453$$ 0 0
$$454$$ 4.00000 0.187729
$$455$$ −2.82843 −0.132599
$$456$$ 0 0
$$457$$ 20.8284 0.974313 0.487156 0.873315i $$-0.338034\pi$$
0.487156 + 0.873315i $$0.338034\pi$$
$$458$$ −4.14214 −0.193549
$$459$$ 0 0
$$460$$ −8.48528 −0.395628
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ −3.79899 −0.176554 −0.0882770 0.996096i $$-0.528136\pi$$
−0.0882770 + 0.996096i $$0.528136\pi$$
$$464$$ 8.82843 0.409849
$$465$$ 0 0
$$466$$ 5.51472 0.255464
$$467$$ −7.31371 −0.338438 −0.169219 0.985578i $$-0.554125\pi$$
−0.169219 + 0.985578i $$0.554125\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ −2.34315 −0.107852
$$473$$ −54.6274 −2.51177
$$474$$ 0 0
$$475$$ 2.82843 0.129777
$$476$$ 2.34315 0.107398
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ 11.3137 0.516937 0.258468 0.966020i $$-0.416782\pi$$
0.258468 + 0.966020i $$0.416782\pi$$
$$480$$ 0 0
$$481$$ 11.6569 0.531507
$$482$$ −5.31371 −0.242033
$$483$$ 0 0
$$484$$ 21.0000 0.954545
$$485$$ 3.17157 0.144014
$$486$$ 0 0
$$487$$ 16.4853 0.747019 0.373510 0.927626i $$-0.378154\pi$$
0.373510 + 0.927626i $$0.378154\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 0 0
$$490$$ 1.00000 0.0451754
$$491$$ 38.1421 1.72133 0.860665 0.509171i $$-0.170048\pi$$
0.860665 + 0.509171i $$0.170048\pi$$
$$492$$ 0 0
$$493$$ −7.31371 −0.329393
$$494$$ 2.82843 0.127257
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ −16.0000 −0.717698
$$498$$ 0 0
$$499$$ 0.485281 0.0217242 0.0108621 0.999941i $$-0.496542\pi$$
0.0108621 + 0.999941i $$0.496542\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 10.8284 0.483296
$$503$$ 23.5147 1.04847 0.524235 0.851574i $$-0.324351\pi$$
0.524235 + 0.851574i $$0.324351\pi$$
$$504$$ 0 0
$$505$$ 16.1421 0.718316
$$506$$ 48.0000 2.13386
$$507$$ 0 0
$$508$$ 9.65685 0.428454
$$509$$ 37.3137 1.65390 0.826951 0.562275i $$-0.190074\pi$$
0.826951 + 0.562275i $$0.190074\pi$$
$$510$$ 0 0
$$511$$ 40.9706 1.81243
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −4.82843 −0.212973
$$515$$ 1.65685 0.0730097
$$516$$ 0 0
$$517$$ −45.2548 −1.99031
$$518$$ −32.9706 −1.44864
$$519$$ 0 0
$$520$$ −1.00000 −0.0438529
$$521$$ −26.9706 −1.18160 −0.590801 0.806817i $$-0.701188\pi$$
−0.590801 + 0.806817i $$0.701188\pi$$
$$522$$ 0 0
$$523$$ −10.6274 −0.464704 −0.232352 0.972632i $$-0.574642\pi$$
−0.232352 + 0.972632i $$0.574642\pi$$
$$524$$ 6.14214 0.268320
$$525$$ 0 0
$$526$$ 16.4853 0.718792
$$527$$ −3.31371 −0.144347
$$528$$ 0 0
$$529$$ 49.0000 2.13043
$$530$$ −13.3137 −0.578311
$$531$$ 0 0
$$532$$ −8.00000 −0.346844
$$533$$ −7.65685 −0.331655
$$534$$ 0 0
$$535$$ −4.00000 −0.172935
$$536$$ −5.65685 −0.244339
$$537$$ 0 0
$$538$$ −14.4853 −0.624505
$$539$$ −5.65685 −0.243658
$$540$$ 0 0
$$541$$ 14.4853 0.622771 0.311385 0.950284i $$-0.399207\pi$$
0.311385 + 0.950284i $$0.399207\pi$$
$$542$$ −7.31371 −0.314151
$$543$$ 0 0
$$544$$ 0.828427 0.0355185
$$545$$ −8.82843 −0.378168
$$546$$ 0 0
$$547$$ 0.686292 0.0293437 0.0146719 0.999892i $$-0.495330\pi$$
0.0146719 + 0.999892i $$0.495330\pi$$
$$548$$ 17.3137 0.739605
$$549$$ 0 0
$$550$$ 5.65685 0.241209
$$551$$ 24.9706 1.06378
$$552$$ 0 0
$$553$$ −6.62742 −0.281826
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ 6.34315 0.269009
$$557$$ −10.6863 −0.452793 −0.226396 0.974035i $$-0.572694\pi$$
−0.226396 + 0.974035i $$0.572694\pi$$
$$558$$ 0 0
$$559$$ −9.65685 −0.408441
$$560$$ 2.82843 0.119523
$$561$$ 0 0
$$562$$ 8.34315 0.351934
$$563$$ 30.3431 1.27881 0.639406 0.768870i $$-0.279181\pi$$
0.639406 + 0.768870i $$0.279181\pi$$
$$564$$ 0 0
$$565$$ 6.48528 0.272838
$$566$$ 17.6569 0.742173
$$567$$ 0 0
$$568$$ −5.65685 −0.237356
$$569$$ −31.6569 −1.32712 −0.663562 0.748121i $$-0.730956\pi$$
−0.663562 + 0.748121i $$0.730956\pi$$
$$570$$ 0 0
$$571$$ 20.9706 0.877591 0.438795 0.898587i $$-0.355405\pi$$
0.438795 + 0.898587i $$0.355405\pi$$
$$572$$ 5.65685 0.236525
$$573$$ 0 0
$$574$$ 21.6569 0.903940
$$575$$ 8.48528 0.353861
$$576$$ 0 0
$$577$$ −23.4558 −0.976480 −0.488240 0.872710i $$-0.662361\pi$$
−0.488240 + 0.872710i $$0.662361\pi$$
$$578$$ 16.3137 0.678561
$$579$$ 0 0
$$580$$ −8.82843 −0.366580
$$581$$ 17.9411 0.744323
$$582$$ 0 0
$$583$$ 75.3137 3.11918
$$584$$ 14.4853 0.599405
$$585$$ 0 0
$$586$$ −16.6274 −0.686872
$$587$$ −2.62742 −0.108445 −0.0542226 0.998529i $$-0.517268\pi$$
−0.0542226 + 0.998529i $$0.517268\pi$$
$$588$$ 0 0
$$589$$ 11.3137 0.466173
$$590$$ 2.34315 0.0964658
$$591$$ 0 0
$$592$$ −11.6569 −0.479094
$$593$$ 0.343146 0.0140913 0.00704565 0.999975i $$-0.497757\pi$$
0.00704565 + 0.999975i $$0.497757\pi$$
$$594$$ 0 0
$$595$$ −2.34315 −0.0960596
$$596$$ −3.65685 −0.149791
$$597$$ 0 0
$$598$$ 8.48528 0.346989
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 29.3137 1.19573 0.597866 0.801596i $$-0.296016\pi$$
0.597866 + 0.801596i $$0.296016\pi$$
$$602$$ 27.3137 1.11322
$$603$$ 0 0
$$604$$ 12.0000 0.488273
$$605$$ −21.0000 −0.853771
$$606$$ 0 0
$$607$$ 28.9706 1.17588 0.587939 0.808905i $$-0.299939\pi$$
0.587939 + 0.808905i $$0.299939\pi$$
$$608$$ −2.82843 −0.114708
$$609$$ 0 0
$$610$$ 6.00000 0.242933
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ 22.2843 0.900053 0.450027 0.893015i $$-0.351414\pi$$
0.450027 + 0.893015i $$0.351414\pi$$
$$614$$ 21.6569 0.874000
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ −34.8284 −1.39987 −0.699936 0.714205i $$-0.746788\pi$$
−0.699936 + 0.714205i $$0.746788\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ −44.2843 −1.77421
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 30.9706 1.23783
$$627$$ 0 0
$$628$$ 5.31371 0.212040
$$629$$ 9.65685 0.385044
$$630$$ 0 0
$$631$$ −33.6569 −1.33986 −0.669929 0.742425i $$-0.733676\pi$$
−0.669929 + 0.742425i $$0.733676\pi$$
$$632$$ −2.34315 −0.0932053
$$633$$ 0 0
$$634$$ 25.3137 1.00534
$$635$$ −9.65685 −0.383221
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 49.9411 1.97719
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 4.62742 0.182772 0.0913860 0.995816i $$-0.470870\pi$$
0.0913860 + 0.995816i $$0.470870\pi$$
$$642$$ 0 0
$$643$$ 39.5980 1.56159 0.780796 0.624786i $$-0.214814\pi$$
0.780796 + 0.624786i $$0.214814\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 2.34315 0.0921898
$$647$$ −8.48528 −0.333591 −0.166795 0.985992i $$-0.553342\pi$$
−0.166795 + 0.985992i $$0.553342\pi$$
$$648$$ 0 0
$$649$$ −13.2548 −0.520298
$$650$$ 1.00000 0.0392232
$$651$$ 0 0
$$652$$ 11.3137 0.443079
$$653$$ 42.2843 1.65471 0.827356 0.561678i $$-0.189844\pi$$
0.827356 + 0.561678i $$0.189844\pi$$
$$654$$ 0 0
$$655$$ −6.14214 −0.239993
$$656$$ 7.65685 0.298950
$$657$$ 0 0
$$658$$ 22.6274 0.882109
$$659$$ 7.51472 0.292732 0.146366 0.989231i $$-0.453242\pi$$
0.146366 + 0.989231i $$0.453242\pi$$
$$660$$ 0 0
$$661$$ −8.14214 −0.316692 −0.158346 0.987384i $$-0.550616\pi$$
−0.158346 + 0.987384i $$0.550616\pi$$
$$662$$ 8.48528 0.329790
$$663$$ 0 0
$$664$$ 6.34315 0.246162
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ 74.9117 2.90059
$$668$$ −8.97056 −0.347081
$$669$$ 0 0
$$670$$ 5.65685 0.218543
$$671$$ −33.9411 −1.31028
$$672$$ 0 0
$$673$$ 32.6274 1.25769 0.628847 0.777529i $$-0.283527\pi$$
0.628847 + 0.777529i $$0.283527\pi$$
$$674$$ −10.9706 −0.422570
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −12.3431 −0.474386 −0.237193 0.971463i $$-0.576227\pi$$
−0.237193 + 0.971463i $$0.576227\pi$$
$$678$$ 0 0
$$679$$ 8.97056 0.344259
$$680$$ −0.828427 −0.0317687
$$681$$ 0 0
$$682$$ 22.6274 0.866449
$$683$$ −33.6569 −1.28784 −0.643922 0.765091i $$-0.722694\pi$$
−0.643922 + 0.765091i $$0.722694\pi$$
$$684$$ 0 0
$$685$$ −17.3137 −0.661523
$$686$$ −16.9706 −0.647939
$$687$$ 0 0
$$688$$ 9.65685 0.368164
$$689$$ 13.3137 0.507212
$$690$$ 0 0
$$691$$ −27.7990 −1.05752 −0.528762 0.848770i $$-0.677343\pi$$
−0.528762 + 0.848770i $$0.677343\pi$$
$$692$$ 9.31371 0.354054
$$693$$ 0 0
$$694$$ −9.65685 −0.366569
$$695$$ −6.34315 −0.240609
$$696$$ 0 0
$$697$$ −6.34315 −0.240264
$$698$$ −12.1421 −0.459587
$$699$$ 0 0
$$700$$ −2.82843 −0.106904
$$701$$ −0.142136 −0.00536839 −0.00268419 0.999996i $$-0.500854\pi$$
−0.00268419 + 0.999996i $$0.500854\pi$$
$$702$$ 0 0
$$703$$ −32.9706 −1.24351
$$704$$ −5.65685 −0.213201
$$705$$ 0 0
$$706$$ 5.31371 0.199984
$$707$$ 45.6569 1.71710
$$708$$ 0 0
$$709$$ −7.17157 −0.269334 −0.134667 0.990891i $$-0.542996\pi$$
−0.134667 + 0.990891i $$0.542996\pi$$
$$710$$ 5.65685 0.212298
$$711$$ 0 0
$$712$$ −15.6569 −0.586765
$$713$$ 33.9411 1.27111
$$714$$ 0 0
$$715$$ −5.65685 −0.211554
$$716$$ −7.51472 −0.280838
$$717$$ 0 0
$$718$$ −28.2843 −1.05556
$$719$$ 29.6569 1.10601 0.553007 0.833177i $$-0.313480\pi$$
0.553007 + 0.833177i $$0.313480\pi$$
$$720$$ 0 0
$$721$$ 4.68629 0.174527
$$722$$ 11.0000 0.409378
$$723$$ 0 0
$$724$$ −7.65685 −0.284565
$$725$$ 8.82843 0.327880
$$726$$ 0 0
$$727$$ −45.9411 −1.70386 −0.851931 0.523654i $$-0.824568\pi$$
−0.851931 + 0.523654i $$0.824568\pi$$
$$728$$ −2.82843 −0.104828
$$729$$ 0 0
$$730$$ −14.4853 −0.536124
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −0.343146 −0.0126744 −0.00633719 0.999980i $$-0.502017\pi$$
−0.00633719 + 0.999980i $$0.502017\pi$$
$$734$$ −25.6569 −0.947012
$$735$$ 0 0
$$736$$ −8.48528 −0.312772
$$737$$ −32.0000 −1.17874
$$738$$ 0 0
$$739$$ −14.1421 −0.520227 −0.260113 0.965578i $$-0.583760\pi$$
−0.260113 + 0.965578i $$0.583760\pi$$
$$740$$ 11.6569 0.428514
$$741$$ 0 0
$$742$$ −37.6569 −1.38243
$$743$$ 36.2843 1.33114 0.665570 0.746335i $$-0.268188\pi$$
0.665570 + 0.746335i $$0.268188\pi$$
$$744$$ 0 0
$$745$$ 3.65685 0.133977
$$746$$ 2.68629 0.0983521
$$747$$ 0 0
$$748$$ 4.68629 0.171348
$$749$$ −11.3137 −0.413394
$$750$$ 0 0
$$751$$ 11.3137 0.412843 0.206422 0.978463i $$-0.433818\pi$$
0.206422 + 0.978463i $$0.433818\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ 8.82843 0.321512
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ 19.9411 0.724773 0.362386 0.932028i $$-0.381962\pi$$
0.362386 + 0.932028i $$0.381962\pi$$
$$758$$ 7.51472 0.272947
$$759$$ 0 0
$$760$$ 2.82843 0.102598
$$761$$ −27.6569 −1.00256 −0.501280 0.865285i $$-0.667137\pi$$
−0.501280 + 0.865285i $$0.667137\pi$$
$$762$$ 0 0
$$763$$ −24.9706 −0.903995
$$764$$ −11.3137 −0.409316
$$765$$ 0 0
$$766$$ −29.6569 −1.07155
$$767$$ −2.34315 −0.0846061
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 16.0000 0.576600
$$771$$ 0 0
$$772$$ 2.48528 0.0894472
$$773$$ 53.3137 1.91756 0.958780 0.284148i $$-0.0917107\pi$$
0.958780 + 0.284148i $$0.0917107\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 3.17157 0.113853
$$777$$ 0 0
$$778$$ −6.48528 −0.232509
$$779$$ 21.6569 0.775937
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 7.02944 0.251372
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −5.31371 −0.189654
$$786$$ 0 0
$$787$$ −24.0000 −0.855508 −0.427754 0.903895i $$-0.640695\pi$$
−0.427754 + 0.903895i $$0.640695\pi$$
$$788$$ 13.3137 0.474281
$$789$$ 0 0
$$790$$ 2.34315 0.0833654
$$791$$ 18.3431 0.652207
$$792$$ 0 0
$$793$$ −6.00000 −0.213066
$$794$$ −30.2843 −1.07475
$$795$$ 0 0
$$796$$ 10.3431 0.366603
$$797$$ −16.6274 −0.588973 −0.294487 0.955656i $$-0.595149\pi$$
−0.294487 + 0.955656i $$0.595149\pi$$
$$798$$ 0 0
$$799$$ −6.62742 −0.234461
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −26.9706 −0.952364
$$803$$ 81.9411 2.89164
$$804$$ 0 0
$$805$$ 24.0000 0.845889
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 16.1421 0.567878
$$809$$ −13.3137 −0.468085 −0.234043 0.972226i $$-0.575196\pi$$
−0.234043 + 0.972226i $$0.575196\pi$$
$$810$$ 0 0
$$811$$ −1.85786 −0.0652384 −0.0326192 0.999468i $$-0.510385\pi$$
−0.0326192 + 0.999468i $$0.510385\pi$$
$$812$$ −24.9706 −0.876295
$$813$$ 0 0
$$814$$ −65.9411 −2.31124
$$815$$ −11.3137 −0.396302
$$816$$ 0 0
$$817$$ 27.3137 0.955586
$$818$$ 3.65685 0.127859
$$819$$ 0 0
$$820$$ −7.65685 −0.267389
$$821$$ −34.2843 −1.19653 −0.598265 0.801299i $$-0.704143\pi$$
−0.598265 + 0.801299i $$0.704143\pi$$
$$822$$ 0 0
$$823$$ −52.9706 −1.84644 −0.923219 0.384275i $$-0.874452\pi$$
−0.923219 + 0.384275i $$0.874452\pi$$
$$824$$ 1.65685 0.0577193
$$825$$ 0 0
$$826$$ 6.62742 0.230597
$$827$$ −9.65685 −0.335802 −0.167901 0.985804i $$-0.553699\pi$$
−0.167901 + 0.985804i $$0.553699\pi$$
$$828$$ 0 0
$$829$$ −53.3137 −1.85166 −0.925831 0.377938i $$-0.876633\pi$$
−0.925831 + 0.377938i $$0.876633\pi$$
$$830$$ −6.34315 −0.220174
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ −0.828427 −0.0287033
$$834$$ 0 0
$$835$$ 8.97056 0.310439
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ −10.8284 −0.374062
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 48.9411 1.68763
$$842$$ 24.1421 0.831993
$$843$$ 0 0
$$844$$ 0.686292 0.0236231
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ −59.3970 −2.04090
$$848$$ −13.3137 −0.457195
$$849$$ 0 0
$$850$$ 0.828427 0.0284148
$$851$$ −98.9117 −3.39065
$$852$$ 0 0
$$853$$ 38.2843 1.31083 0.655414 0.755270i $$-0.272494\pi$$
0.655414 + 0.755270i $$0.272494\pi$$
$$854$$ 16.9706 0.580721
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ 20.8284 0.711486 0.355743 0.934584i $$-0.384228\pi$$
0.355743 + 0.934584i $$0.384228\pi$$
$$858$$ 0 0
$$859$$ 37.9411 1.29453 0.647267 0.762263i $$-0.275912\pi$$
0.647267 + 0.762263i $$0.275912\pi$$
$$860$$ −9.65685 −0.329296
$$861$$ 0 0
$$862$$ −16.0000 −0.544962
$$863$$ 28.2843 0.962808 0.481404 0.876499i $$-0.340127\pi$$
0.481404 + 0.876499i $$0.340127\pi$$
$$864$$ 0 0
$$865$$ −9.31371 −0.316676
$$866$$ 22.9706 0.780571
$$867$$ 0 0
$$868$$ −11.3137 −0.384012
$$869$$ −13.2548 −0.449639
$$870$$ 0 0
$$871$$ −5.65685 −0.191675
$$872$$ −8.82843 −0.298968
$$873$$ 0 0
$$874$$ −24.0000 −0.811812
$$875$$ 2.82843 0.0956183
$$876$$ 0 0
$$877$$ −51.2548 −1.73075 −0.865376 0.501122i $$-0.832921\pi$$
−0.865376 + 0.501122i $$0.832921\pi$$
$$878$$ −22.6274 −0.763638
$$879$$ 0 0
$$880$$ 5.65685 0.190693
$$881$$ 10.2843 0.346486 0.173243 0.984879i $$-0.444575\pi$$
0.173243 + 0.984879i $$0.444575\pi$$
$$882$$ 0 0
$$883$$ 31.3137 1.05379 0.526895 0.849930i $$-0.323356\pi$$
0.526895 + 0.849930i $$0.323356\pi$$
$$884$$ 0.828427 0.0278630
$$885$$ 0 0
$$886$$ 30.3431 1.01940
$$887$$ 40.4853 1.35936 0.679681 0.733508i $$-0.262118\pi$$
0.679681 + 0.733508i $$0.262118\pi$$
$$888$$ 0 0
$$889$$ −27.3137 −0.916072
$$890$$ 15.6569 0.524819
$$891$$ 0 0
$$892$$ 10.8284 0.362563
$$893$$ 22.6274 0.757198
$$894$$ 0 0
$$895$$ 7.51472 0.251189
$$896$$ 2.82843 0.0944911
$$897$$ 0 0
$$898$$ 26.2843 0.877117
$$899$$ 35.3137 1.17778
$$900$$ 0 0
$$901$$ 11.0294 0.367444
$$902$$ 43.3137 1.44219
$$903$$ 0 0
$$904$$ 6.48528 0.215697
$$905$$ 7.65685 0.254522
$$906$$ 0 0
$$907$$ 8.28427 0.275075 0.137537 0.990497i $$-0.456081\pi$$
0.137537 + 0.990497i $$0.456081\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ 0 0
$$910$$ 2.82843 0.0937614
$$911$$ −24.9706 −0.827312 −0.413656 0.910433i $$-0.635748\pi$$
−0.413656 + 0.910433i $$0.635748\pi$$
$$912$$ 0 0
$$913$$ 35.8823 1.18753
$$914$$ −20.8284 −0.688943
$$915$$ 0 0
$$916$$ 4.14214 0.136860
$$917$$ −17.3726 −0.573693
$$918$$ 0 0
$$919$$ 41.9411 1.38351 0.691755 0.722132i $$-0.256838\pi$$
0.691755 + 0.722132i $$0.256838\pi$$
$$920$$ 8.48528 0.279751
$$921$$ 0 0
$$922$$ 14.0000 0.461065
$$923$$ −5.65685 −0.186198
$$924$$ 0 0
$$925$$ −11.6569 −0.383275
$$926$$ 3.79899 0.124843
$$927$$ 0 0
$$928$$ −8.82843 −0.289807
$$929$$ 33.5980 1.10231 0.551157 0.834402i $$-0.314187\pi$$
0.551157 + 0.834402i $$0.314187\pi$$
$$930$$ 0 0
$$931$$ 2.82843 0.0926980
$$932$$ −5.51472 −0.180641
$$933$$ 0 0
$$934$$ 7.31371 0.239312
$$935$$ −4.68629 −0.153258
$$936$$ 0 0
$$937$$ 16.6274 0.543194 0.271597 0.962411i $$-0.412448\pi$$
0.271597 + 0.962411i $$0.412448\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ −54.9706 −1.79199 −0.895995 0.444065i $$-0.853536\pi$$
−0.895995 + 0.444065i $$0.853536\pi$$
$$942$$ 0 0
$$943$$ 64.9706 2.11573
$$944$$ 2.34315 0.0762629
$$945$$ 0 0
$$946$$ 54.6274 1.77609
$$947$$ −30.3431 −0.986020 −0.493010 0.870024i $$-0.664103\pi$$
−0.493010 + 0.870024i $$0.664103\pi$$
$$948$$ 0 0
$$949$$ 14.4853 0.470212
$$950$$ −2.82843 −0.0917663
$$951$$ 0 0
$$952$$ −2.34315 −0.0759418
$$953$$ 27.8579 0.902405 0.451202 0.892422i $$-0.350995\pi$$
0.451202 + 0.892422i $$0.350995\pi$$
$$954$$ 0 0
$$955$$ 11.3137 0.366103
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ −11.3137 −0.365529
$$959$$ −48.9706 −1.58134
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −11.6569 −0.375832
$$963$$ 0 0
$$964$$ 5.31371 0.171143
$$965$$ −2.48528 −0.0800040
$$966$$ 0 0
$$967$$ −7.51472 −0.241657 −0.120829 0.992673i $$-0.538555\pi$$
−0.120829 + 0.992673i $$0.538555\pi$$
$$968$$ −21.0000 −0.674966
$$969$$ 0 0
$$970$$ −3.17157 −0.101833
$$971$$ −15.5147 −0.497891 −0.248946 0.968517i $$-0.580084\pi$$
−0.248946 + 0.968517i $$0.580084\pi$$
$$972$$ 0 0
$$973$$ −17.9411 −0.575166
$$974$$ −16.4853 −0.528222
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 8.34315 0.266921 0.133460 0.991054i $$-0.457391\pi$$
0.133460 + 0.991054i $$0.457391\pi$$
$$978$$ 0 0
$$979$$ −88.5685 −2.83066
$$980$$ −1.00000 −0.0319438
$$981$$ 0 0
$$982$$ −38.1421 −1.21716
$$983$$ 2.34315 0.0747347 0.0373674 0.999302i $$-0.488103\pi$$
0.0373674 + 0.999302i $$0.488103\pi$$
$$984$$ 0 0
$$985$$ −13.3137 −0.424210
$$986$$ 7.31371 0.232916
$$987$$ 0 0
$$988$$ −2.82843 −0.0899843
$$989$$ 81.9411 2.60558
$$990$$ 0 0
$$991$$ −42.9117 −1.36313 −0.681567 0.731755i $$-0.738701\pi$$
−0.681567 + 0.731755i $$0.738701\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 16.0000 0.507489
$$995$$ −10.3431 −0.327900
$$996$$ 0 0
$$997$$ 61.3137 1.94182 0.970912 0.239435i $$-0.0769623\pi$$
0.970912 + 0.239435i $$0.0769623\pi$$
$$998$$ −0.485281 −0.0153613
$$999$$ 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 1170.2.a.o.1.1 2
3.2 odd 2 390.2.a.h.1.1 2
4.3 odd 2 9360.2.a.ch.1.2 2
5.2 odd 4 5850.2.e.bk.5149.1 4
5.3 odd 4 5850.2.e.bk.5149.4 4
5.4 even 2 5850.2.a.cl.1.2 2
12.11 even 2 3120.2.a.bc.1.2 2
15.2 even 4 1950.2.e.o.1249.3 4
15.8 even 4 1950.2.e.o.1249.2 4
15.14 odd 2 1950.2.a.bd.1.2 2
39.5 even 4 5070.2.b.q.1351.1 4
39.8 even 4 5070.2.b.q.1351.4 4
39.38 odd 2 5070.2.a.bc.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 3.2 odd 2
1170.2.a.o.1.1 2 1.1 even 1 trivial
1950.2.a.bd.1.2 2 15.14 odd 2
1950.2.e.o.1249.2 4 15.8 even 4
1950.2.e.o.1249.3 4 15.2 even 4
3120.2.a.bc.1.2 2 12.11 even 2
5070.2.a.bc.1.2 2 39.38 odd 2
5070.2.b.q.1351.1 4 39.5 even 4
5070.2.b.q.1351.4 4 39.8 even 4
5850.2.a.cl.1.2 2 5.4 even 2
5850.2.e.bk.5149.1 4 5.2 odd 4
5850.2.e.bk.5149.4 4 5.3 odd 4
9360.2.a.ch.1.2 2 4.3 odd 2