# Properties

 Label 2205.2.a.x.1.2 Level $2205$ Weight $2$ Character 2205.1 Self dual yes Analytic conductor $17.607$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.6070136457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2205.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{5} +6.56155 q^{8} +O(q^{10})$$ $$q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{5} +6.56155 q^{8} +2.56155 q^{10} +1.56155 q^{11} -0.438447 q^{13} +7.68466 q^{16} -0.438447 q^{17} +7.12311 q^{19} +4.56155 q^{20} +4.00000 q^{22} -3.12311 q^{23} +1.00000 q^{25} -1.12311 q^{26} -6.68466 q^{29} +6.56155 q^{32} -1.12311 q^{34} +6.00000 q^{37} +18.2462 q^{38} +6.56155 q^{40} +5.12311 q^{41} +0.876894 q^{43} +7.12311 q^{44} -8.00000 q^{46} -8.68466 q^{47} +2.56155 q^{50} -2.00000 q^{52} +5.12311 q^{53} +1.56155 q^{55} -17.1231 q^{58} -4.00000 q^{59} -15.3693 q^{61} +1.43845 q^{64} -0.438447 q^{65} +10.2462 q^{67} -2.00000 q^{68} -8.00000 q^{71} +12.2462 q^{73} +15.3693 q^{74} +32.4924 q^{76} -2.43845 q^{79} +7.68466 q^{80} +13.1231 q^{82} +4.00000 q^{83} -0.438447 q^{85} +2.24621 q^{86} +10.2462 q^{88} -1.12311 q^{89} -14.2462 q^{92} -22.2462 q^{94} +7.12311 q^{95} -5.80776 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + O(q^{10})$$ $$2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + q^{10} - q^{11} - 5q^{13} + 3q^{16} - 5q^{17} + 6q^{19} + 5q^{20} + 8q^{22} + 2q^{23} + 2q^{25} + 6q^{26} - q^{29} + 9q^{32} + 6q^{34} + 12q^{37} + 20q^{38} + 9q^{40} + 2q^{41} + 10q^{43} + 6q^{44} - 16q^{46} - 5q^{47} + q^{50} - 4q^{52} + 2q^{53} - q^{55} - 26q^{58} - 8q^{59} - 6q^{61} + 7q^{64} - 5q^{65} + 4q^{67} - 4q^{68} - 16q^{71} + 8q^{73} + 6q^{74} + 32q^{76} - 9q^{79} + 3q^{80} + 18q^{82} + 8q^{83} - 5q^{85} - 12q^{86} + 4q^{88} + 6q^{89} - 12q^{92} - 28q^{94} + 6q^{95} + 9q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.56155 1.81129 0.905646 0.424035i $$-0.139387\pi$$
0.905646 + 0.424035i $$0.139387\pi$$
$$3$$ 0 0
$$4$$ 4.56155 2.28078
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 6.56155 2.31986
$$9$$ 0 0
$$10$$ 2.56155 0.810034
$$11$$ 1.56155 0.470826 0.235413 0.971895i $$-0.424356\pi$$
0.235413 + 0.971895i $$0.424356\pi$$
$$12$$ 0 0
$$13$$ −0.438447 −0.121603 −0.0608017 0.998150i $$-0.519366\pi$$
−0.0608017 + 0.998150i $$0.519366\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 7.68466 1.92116
$$17$$ −0.438447 −0.106339 −0.0531695 0.998586i $$-0.516932\pi$$
−0.0531695 + 0.998586i $$0.516932\pi$$
$$18$$ 0 0
$$19$$ 7.12311 1.63415 0.817076 0.576530i $$-0.195593\pi$$
0.817076 + 0.576530i $$0.195593\pi$$
$$20$$ 4.56155 1.01999
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −3.12311 −0.651213 −0.325606 0.945505i $$-0.605568\pi$$
−0.325606 + 0.945505i $$0.605568\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −1.12311 −0.220259
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.68466 −1.24131 −0.620655 0.784084i $$-0.713133\pi$$
−0.620655 + 0.784084i $$0.713133\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 6.56155 1.15993
$$33$$ 0 0
$$34$$ −1.12311 −0.192611
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 18.2462 2.95993
$$39$$ 0 0
$$40$$ 6.56155 1.03747
$$41$$ 5.12311 0.800095 0.400047 0.916494i $$-0.368994\pi$$
0.400047 + 0.916494i $$0.368994\pi$$
$$42$$ 0 0
$$43$$ 0.876894 0.133725 0.0668626 0.997762i $$-0.478701\pi$$
0.0668626 + 0.997762i $$0.478701\pi$$
$$44$$ 7.12311 1.07385
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ −8.68466 −1.26679 −0.633394 0.773830i $$-0.718339\pi$$
−0.633394 + 0.773830i $$0.718339\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 2.56155 0.362258
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 5.12311 0.703713 0.351856 0.936054i $$-0.385551\pi$$
0.351856 + 0.936054i $$0.385551\pi$$
$$54$$ 0 0
$$55$$ 1.56155 0.210560
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −17.1231 −2.24837
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −15.3693 −1.96784 −0.983920 0.178611i $$-0.942839\pi$$
−0.983920 + 0.178611i $$0.942839\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.43845 0.179806
$$65$$ −0.438447 −0.0543827
$$66$$ 0 0
$$67$$ 10.2462 1.25177 0.625887 0.779914i $$-0.284737\pi$$
0.625887 + 0.779914i $$0.284737\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 12.2462 1.43331 0.716655 0.697428i $$-0.245672\pi$$
0.716655 + 0.697428i $$0.245672\pi$$
$$74$$ 15.3693 1.78665
$$75$$ 0 0
$$76$$ 32.4924 3.72714
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −2.43845 −0.274347 −0.137173 0.990547i $$-0.543802\pi$$
−0.137173 + 0.990547i $$0.543802\pi$$
$$80$$ 7.68466 0.859171
$$81$$ 0 0
$$82$$ 13.1231 1.44920
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −0.438447 −0.0475563
$$86$$ 2.24621 0.242215
$$87$$ 0 0
$$88$$ 10.2462 1.09225
$$89$$ −1.12311 −0.119049 −0.0595245 0.998227i $$-0.518958\pi$$
−0.0595245 + 0.998227i $$0.518958\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −14.2462 −1.48527
$$93$$ 0 0
$$94$$ −22.2462 −2.29452
$$95$$ 7.12311 0.730815
$$96$$ 0 0
$$97$$ −5.80776 −0.589689 −0.294845 0.955545i $$-0.595268\pi$$
−0.294845 + 0.955545i $$0.595268\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 4.56155 0.456155
$$101$$ −16.2462 −1.61656 −0.808279 0.588799i $$-0.799601\pi$$
−0.808279 + 0.588799i $$0.799601\pi$$
$$102$$ 0 0
$$103$$ −5.56155 −0.547996 −0.273998 0.961730i $$-0.588346\pi$$
−0.273998 + 0.961730i $$0.588346\pi$$
$$104$$ −2.87689 −0.282103
$$105$$ 0 0
$$106$$ 13.1231 1.27463
$$107$$ −13.3693 −1.29246 −0.646230 0.763142i $$-0.723655\pi$$
−0.646230 + 0.763142i $$0.723655\pi$$
$$108$$ 0 0
$$109$$ 5.31534 0.509117 0.254559 0.967057i $$-0.418070\pi$$
0.254559 + 0.967057i $$0.418070\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ −3.12311 −0.291231
$$116$$ −30.4924 −2.83115
$$117$$ 0 0
$$118$$ −10.2462 −0.943240
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ −39.3693 −3.56433
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −6.24621 −0.554262 −0.277131 0.960832i $$-0.589384\pi$$
−0.277131 + 0.960832i $$0.589384\pi$$
$$128$$ −9.43845 −0.834249
$$129$$ 0 0
$$130$$ −1.12311 −0.0985029
$$131$$ −0.876894 −0.0766146 −0.0383073 0.999266i $$-0.512197\pi$$
−0.0383073 + 0.999266i $$0.512197\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 26.2462 2.26733
$$135$$ 0 0
$$136$$ −2.87689 −0.246692
$$137$$ 17.1231 1.46293 0.731463 0.681881i $$-0.238838\pi$$
0.731463 + 0.681881i $$0.238838\pi$$
$$138$$ 0 0
$$139$$ 15.1231 1.28273 0.641363 0.767238i $$-0.278369\pi$$
0.641363 + 0.767238i $$0.278369\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −20.4924 −1.71969
$$143$$ −0.684658 −0.0572540
$$144$$ 0 0
$$145$$ −6.68466 −0.555131
$$146$$ 31.3693 2.59614
$$147$$ 0 0
$$148$$ 27.3693 2.24974
$$149$$ −12.2462 −1.00325 −0.501624 0.865086i $$-0.667264\pi$$
−0.501624 + 0.865086i $$0.667264\pi$$
$$150$$ 0 0
$$151$$ −6.93087 −0.564026 −0.282013 0.959411i $$-0.591002\pi$$
−0.282013 + 0.959411i $$0.591002\pi$$
$$152$$ 46.7386 3.79100
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −20.2462 −1.61582 −0.807912 0.589303i $$-0.799402\pi$$
−0.807912 + 0.589303i $$0.799402\pi$$
$$158$$ −6.24621 −0.496922
$$159$$ 0 0
$$160$$ 6.56155 0.518736
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −7.12311 −0.557925 −0.278962 0.960302i $$-0.589990\pi$$
−0.278962 + 0.960302i $$0.589990\pi$$
$$164$$ 23.3693 1.82484
$$165$$ 0 0
$$166$$ 10.2462 0.795260
$$167$$ −6.93087 −0.536327 −0.268163 0.963373i $$-0.586417\pi$$
−0.268163 + 0.963373i $$0.586417\pi$$
$$168$$ 0 0
$$169$$ −12.8078 −0.985213
$$170$$ −1.12311 −0.0861383
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −4.43845 −0.337449 −0.168724 0.985663i $$-0.553965\pi$$
−0.168724 + 0.985663i $$0.553965\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ −2.87689 −0.215632
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 17.6155 1.30935 0.654676 0.755910i $$-0.272805\pi$$
0.654676 + 0.755910i $$0.272805\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −20.4924 −1.51072
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ −0.684658 −0.0500672
$$188$$ −39.6155 −2.88926
$$189$$ 0 0
$$190$$ 18.2462 1.32372
$$191$$ 13.5616 0.981280 0.490640 0.871363i $$-0.336763\pi$$
0.490640 + 0.871363i $$0.336763\pi$$
$$192$$ 0 0
$$193$$ 19.3693 1.39423 0.697117 0.716957i $$-0.254466\pi$$
0.697117 + 0.716957i $$0.254466\pi$$
$$194$$ −14.8769 −1.06810
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.12311 −0.0800180 −0.0400090 0.999199i $$-0.512739\pi$$
−0.0400090 + 0.999199i $$0.512739\pi$$
$$198$$ 0 0
$$199$$ 1.75379 0.124323 0.0621614 0.998066i $$-0.480201\pi$$
0.0621614 + 0.998066i $$0.480201\pi$$
$$200$$ 6.56155 0.463972
$$201$$ 0 0
$$202$$ −41.6155 −2.92806
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 5.12311 0.357813
$$206$$ −14.2462 −0.992581
$$207$$ 0 0
$$208$$ −3.36932 −0.233620
$$209$$ 11.1231 0.769401
$$210$$ 0 0
$$211$$ 14.0540 0.967516 0.483758 0.875202i $$-0.339272\pi$$
0.483758 + 0.875202i $$0.339272\pi$$
$$212$$ 23.3693 1.60501
$$213$$ 0 0
$$214$$ −34.2462 −2.34102
$$215$$ 0.876894 0.0598037
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 13.6155 0.922160
$$219$$ 0 0
$$220$$ 7.12311 0.480240
$$221$$ 0.192236 0.0129312
$$222$$ 0 0
$$223$$ 2.43845 0.163291 0.0816453 0.996661i $$-0.473983\pi$$
0.0816453 + 0.996661i $$0.473983\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 35.8617 2.38549
$$227$$ 11.3153 0.751026 0.375513 0.926817i $$-0.377467\pi$$
0.375513 + 0.926817i $$0.377467\pi$$
$$228$$ 0 0
$$229$$ −10.8769 −0.718765 −0.359383 0.933190i $$-0.617013\pi$$
−0.359383 + 0.933190i $$0.617013\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ −43.8617 −2.87966
$$233$$ −5.12311 −0.335626 −0.167813 0.985819i $$-0.553670\pi$$
−0.167813 + 0.985819i $$0.553670\pi$$
$$234$$ 0 0
$$235$$ −8.68466 −0.566525
$$236$$ −18.2462 −1.18773
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −19.8078 −1.28126 −0.640629 0.767851i $$-0.721326\pi$$
−0.640629 + 0.767851i $$0.721326\pi$$
$$240$$ 0 0
$$241$$ 4.24621 0.273523 0.136761 0.990604i $$-0.456331\pi$$
0.136761 + 0.990604i $$0.456331\pi$$
$$242$$ −21.9309 −1.40977
$$243$$ 0 0
$$244$$ −70.1080 −4.48820
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.12311 −0.198718
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 2.56155 0.162007
$$251$$ −8.87689 −0.560305 −0.280152 0.959956i $$-0.590385\pi$$
−0.280152 + 0.959956i $$0.590385\pi$$
$$252$$ 0 0
$$253$$ −4.87689 −0.306608
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ −27.0540 −1.69087
$$257$$ −10.4924 −0.654499 −0.327250 0.944938i $$-0.606122\pi$$
−0.327250 + 0.944938i $$0.606122\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ 0 0
$$262$$ −2.24621 −0.138771
$$263$$ 12.8769 0.794023 0.397012 0.917814i $$-0.370047\pi$$
0.397012 + 0.917814i $$0.370047\pi$$
$$264$$ 0 0
$$265$$ 5.12311 0.314710
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 46.7386 2.85502
$$269$$ −20.7386 −1.26446 −0.632228 0.774782i $$-0.717860\pi$$
−0.632228 + 0.774782i $$0.717860\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −3.36932 −0.204295
$$273$$ 0 0
$$274$$ 43.8617 2.64978
$$275$$ 1.56155 0.0941652
$$276$$ 0 0
$$277$$ −0.246211 −0.0147934 −0.00739670 0.999973i $$-0.502354\pi$$
−0.00739670 + 0.999973i $$0.502354\pi$$
$$278$$ 38.7386 2.32339
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −12.4384 −0.742016 −0.371008 0.928630i $$-0.620988\pi$$
−0.371008 + 0.928630i $$0.620988\pi$$
$$282$$ 0 0
$$283$$ 11.3153 0.672627 0.336314 0.941750i $$-0.390820\pi$$
0.336314 + 0.941750i $$0.390820\pi$$
$$284$$ −36.4924 −2.16543
$$285$$ 0 0
$$286$$ −1.75379 −0.103704
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.8078 −0.988692
$$290$$ −17.1231 −1.00550
$$291$$ 0 0
$$292$$ 55.8617 3.26906
$$293$$ −2.68466 −0.156839 −0.0784197 0.996920i $$-0.524987\pi$$
−0.0784197 + 0.996920i $$0.524987\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 39.3693 2.28830
$$297$$ 0 0
$$298$$ −31.3693 −1.81718
$$299$$ 1.36932 0.0791896
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −17.7538 −1.02162
$$303$$ 0 0
$$304$$ 54.7386 3.13948
$$305$$ −15.3693 −0.880045
$$306$$ 0 0
$$307$$ 19.3153 1.10238 0.551192 0.834378i $$-0.314173\pi$$
0.551192 + 0.834378i $$0.314173\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 31.6155 1.79275 0.896376 0.443294i $$-0.146190\pi$$
0.896376 + 0.443294i $$0.146190\pi$$
$$312$$ 0 0
$$313$$ 22.3002 1.26048 0.630241 0.776400i $$-0.282956\pi$$
0.630241 + 0.776400i $$0.282956\pi$$
$$314$$ −51.8617 −2.92673
$$315$$ 0 0
$$316$$ −11.1231 −0.625724
$$317$$ −10.4924 −0.589313 −0.294657 0.955603i $$-0.595205\pi$$
−0.294657 + 0.955603i $$0.595205\pi$$
$$318$$ 0 0
$$319$$ −10.4384 −0.584441
$$320$$ 1.43845 0.0804116
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3.12311 −0.173774
$$324$$ 0 0
$$325$$ −0.438447 −0.0243207
$$326$$ −18.2462 −1.01056
$$327$$ 0 0
$$328$$ 33.6155 1.85611
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 18.2462 1.00139
$$333$$ 0 0
$$334$$ −17.7538 −0.971444
$$335$$ 10.2462 0.559810
$$336$$ 0 0
$$337$$ −1.50758 −0.0821230 −0.0410615 0.999157i $$-0.513074\pi$$
−0.0410615 + 0.999157i $$0.513074\pi$$
$$338$$ −32.8078 −1.78451
$$339$$ 0 0
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 5.75379 0.310223
$$345$$ 0 0
$$346$$ −11.3693 −0.611218
$$347$$ −7.12311 −0.382388 −0.191194 0.981552i $$-0.561236\pi$$
−0.191194 + 0.981552i $$0.561236\pi$$
$$348$$ 0 0
$$349$$ −10.4924 −0.561646 −0.280823 0.959760i $$-0.590607\pi$$
−0.280823 + 0.959760i $$0.590607\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 10.2462 0.546125
$$353$$ 5.80776 0.309116 0.154558 0.987984i $$-0.450605\pi$$
0.154558 + 0.987984i $$0.450605\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ −5.12311 −0.271524
$$357$$ 0 0
$$358$$ −51.2311 −2.70765
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ 45.1231 2.37162
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12.2462 0.640996
$$366$$ 0 0
$$367$$ 8.68466 0.453335 0.226668 0.973972i $$-0.427217\pi$$
0.226668 + 0.973972i $$0.427217\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ 0 0
$$370$$ 15.3693 0.799013
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.63068 0.239768 0.119884 0.992788i $$-0.461748\pi$$
0.119884 + 0.992788i $$0.461748\pi$$
$$374$$ −1.75379 −0.0906863
$$375$$ 0 0
$$376$$ −56.9848 −2.93877
$$377$$ 2.93087 0.150947
$$378$$ 0 0
$$379$$ −16.4924 −0.847159 −0.423579 0.905859i $$-0.639227\pi$$
−0.423579 + 0.905859i $$0.639227\pi$$
$$380$$ 32.4924 1.66683
$$381$$ 0 0
$$382$$ 34.7386 1.77738
$$383$$ 6.24621 0.319166 0.159583 0.987184i $$-0.448985\pi$$
0.159583 + 0.987184i $$0.448985\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 49.6155 2.52536
$$387$$ 0 0
$$388$$ −26.4924 −1.34495
$$389$$ 24.9309 1.26405 0.632023 0.774950i $$-0.282225\pi$$
0.632023 + 0.774950i $$0.282225\pi$$
$$390$$ 0 0
$$391$$ 1.36932 0.0692493
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −2.87689 −0.144936
$$395$$ −2.43845 −0.122692
$$396$$ 0 0
$$397$$ −27.5616 −1.38327 −0.691637 0.722245i $$-0.743110\pi$$
−0.691637 + 0.722245i $$0.743110\pi$$
$$398$$ 4.49242 0.225185
$$399$$ 0 0
$$400$$ 7.68466 0.384233
$$401$$ −31.5616 −1.57611 −0.788054 0.615606i $$-0.788911\pi$$
−0.788054 + 0.615606i $$0.788911\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −74.1080 −3.68701
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.36932 0.464420
$$408$$ 0 0
$$409$$ −6.49242 −0.321030 −0.160515 0.987033i $$-0.551315\pi$$
−0.160515 + 0.987033i $$0.551315\pi$$
$$410$$ 13.1231 0.648104
$$411$$ 0 0
$$412$$ −25.3693 −1.24986
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ −2.87689 −0.141051
$$417$$ 0 0
$$418$$ 28.4924 1.39361
$$419$$ 26.2462 1.28221 0.641106 0.767453i $$-0.278476\pi$$
0.641106 + 0.767453i $$0.278476\pi$$
$$420$$ 0 0
$$421$$ −2.68466 −0.130842 −0.0654211 0.997858i $$-0.520839\pi$$
−0.0654211 + 0.997858i $$0.520839\pi$$
$$422$$ 36.0000 1.75245
$$423$$ 0 0
$$424$$ 33.6155 1.63251
$$425$$ −0.438447 −0.0212678
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −60.9848 −2.94781
$$429$$ 0 0
$$430$$ 2.24621 0.108322
$$431$$ 19.8078 0.954106 0.477053 0.878874i $$-0.341705\pi$$
0.477053 + 0.878874i $$0.341705\pi$$
$$432$$ 0 0
$$433$$ −8.24621 −0.396288 −0.198144 0.980173i $$-0.563491\pi$$
−0.198144 + 0.980173i $$0.563491\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 24.2462 1.16118
$$437$$ −22.2462 −1.06418
$$438$$ 0 0
$$439$$ −9.36932 −0.447173 −0.223587 0.974684i $$-0.571777\pi$$
−0.223587 + 0.974684i $$0.571777\pi$$
$$440$$ 10.2462 0.488469
$$441$$ 0 0
$$442$$ 0.492423 0.0234221
$$443$$ 2.63068 0.124988 0.0624938 0.998045i $$-0.480095\pi$$
0.0624938 + 0.998045i $$0.480095\pi$$
$$444$$ 0 0
$$445$$ −1.12311 −0.0532403
$$446$$ 6.24621 0.295767
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.80776 0.0853137 0.0426568 0.999090i $$-0.486418\pi$$
0.0426568 + 0.999090i $$0.486418\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 63.8617 3.00380
$$453$$ 0 0
$$454$$ 28.9848 1.36033
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −17.1231 −0.800985 −0.400493 0.916300i $$-0.631161\pi$$
−0.400493 + 0.916300i $$0.631161\pi$$
$$458$$ −27.8617 −1.30189
$$459$$ 0 0
$$460$$ −14.2462 −0.664233
$$461$$ −13.1231 −0.611204 −0.305602 0.952159i $$-0.598858\pi$$
−0.305602 + 0.952159i $$0.598858\pi$$
$$462$$ 0 0
$$463$$ 12.4924 0.580572 0.290286 0.956940i $$-0.406250\pi$$
0.290286 + 0.956940i $$0.406250\pi$$
$$464$$ −51.3693 −2.38476
$$465$$ 0 0
$$466$$ −13.1231 −0.607916
$$467$$ 22.4384 1.03833 0.519164 0.854675i $$-0.326243\pi$$
0.519164 + 0.854675i $$0.326243\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −22.2462 −1.02614
$$471$$ 0 0
$$472$$ −26.2462 −1.20808
$$473$$ 1.36932 0.0629613
$$474$$ 0 0
$$475$$ 7.12311 0.326831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −50.7386 −2.32073
$$479$$ 4.87689 0.222831 0.111415 0.993774i $$-0.464462\pi$$
0.111415 + 0.993774i $$0.464462\pi$$
$$480$$ 0 0
$$481$$ −2.63068 −0.119949
$$482$$ 10.8769 0.495429
$$483$$ 0 0
$$484$$ −39.0540 −1.77518
$$485$$ −5.80776 −0.263717
$$486$$ 0 0
$$487$$ −3.12311 −0.141521 −0.0707607 0.997493i $$-0.522543\pi$$
−0.0707607 + 0.997493i $$0.522543\pi$$
$$488$$ −100.847 −4.56511
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 41.1771 1.85830 0.929148 0.369708i $$-0.120542\pi$$
0.929148 + 0.369708i $$0.120542\pi$$
$$492$$ 0 0
$$493$$ 2.93087 0.132000
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 41.1771 1.84334 0.921670 0.387976i $$-0.126826\pi$$
0.921670 + 0.387976i $$0.126826\pi$$
$$500$$ 4.56155 0.203999
$$501$$ 0 0
$$502$$ −22.7386 −1.01487
$$503$$ 38.9309 1.73584 0.867921 0.496703i $$-0.165456\pi$$
0.867921 + 0.496703i $$0.165456\pi$$
$$504$$ 0 0
$$505$$ −16.2462 −0.722947
$$506$$ −12.4924 −0.555356
$$507$$ 0 0
$$508$$ −28.4924 −1.26415
$$509$$ −11.7538 −0.520978 −0.260489 0.965477i $$-0.583884\pi$$
−0.260489 + 0.965477i $$0.583884\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −50.4233 −2.22842
$$513$$ 0 0
$$514$$ −26.8769 −1.18549
$$515$$ −5.56155 −0.245071
$$516$$ 0 0
$$517$$ −13.5616 −0.596436
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −2.87689 −0.126160
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ −40.4924 −1.77061 −0.885305 0.465011i $$-0.846050\pi$$
−0.885305 + 0.465011i $$0.846050\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 32.9848 1.43821
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.2462 −0.575922
$$530$$ 13.1231 0.570031
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.24621 −0.0972942
$$534$$ 0 0
$$535$$ −13.3693 −0.578006
$$536$$ 67.2311 2.90394
$$537$$ 0 0
$$538$$ −53.1231 −2.29030
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −37.8078 −1.62548 −0.812741 0.582625i $$-0.802026\pi$$
−0.812741 + 0.582625i $$0.802026\pi$$
$$542$$ 40.9848 1.76045
$$543$$ 0 0
$$544$$ −2.87689 −0.123346
$$545$$ 5.31534 0.227684
$$546$$ 0 0
$$547$$ −2.24621 −0.0960411 −0.0480205 0.998846i $$-0.515291\pi$$
−0.0480205 + 0.998846i $$0.515291\pi$$
$$548$$ 78.1080 3.33661
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ −47.6155 −2.02849
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −0.630683 −0.0267952
$$555$$ 0 0
$$556$$ 68.9848 2.92561
$$557$$ 13.1231 0.556044 0.278022 0.960575i $$-0.410321\pi$$
0.278022 + 0.960575i $$0.410321\pi$$
$$558$$ 0 0
$$559$$ −0.384472 −0.0162614
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −31.8617 −1.34401
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ 14.0000 0.588984
$$566$$ 28.9848 1.21832
$$567$$ 0 0
$$568$$ −52.4924 −2.20253
$$569$$ 30.9848 1.29895 0.649476 0.760382i $$-0.274988\pi$$
0.649476 + 0.760382i $$0.274988\pi$$
$$570$$ 0 0
$$571$$ 40.4924 1.69456 0.847278 0.531150i $$-0.178240\pi$$
0.847278 + 0.531150i $$0.178240\pi$$
$$572$$ −3.12311 −0.130584
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3.12311 −0.130243
$$576$$ 0 0
$$577$$ 24.0540 1.00138 0.500690 0.865627i $$-0.333080\pi$$
0.500690 + 0.865627i $$0.333080\pi$$
$$578$$ −43.0540 −1.79081
$$579$$ 0 0
$$580$$ −30.4924 −1.26613
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 80.3542 3.32508
$$585$$ 0 0
$$586$$ −6.87689 −0.284082
$$587$$ −26.2462 −1.08330 −0.541649 0.840605i $$-0.682200\pi$$
−0.541649 + 0.840605i $$0.682200\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −10.2462 −0.421830
$$591$$ 0 0
$$592$$ 46.1080 1.89503
$$593$$ −27.5616 −1.13182 −0.565909 0.824468i $$-0.691475\pi$$
−0.565909 + 0.824468i $$0.691475\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −55.8617 −2.28819
$$597$$ 0 0
$$598$$ 3.50758 0.143436
$$599$$ 11.8078 0.482452 0.241226 0.970469i $$-0.422450\pi$$
0.241226 + 0.970469i $$0.422450\pi$$
$$600$$ 0 0
$$601$$ −6.49242 −0.264831 −0.132416 0.991194i $$-0.542273\pi$$
−0.132416 + 0.991194i $$0.542273\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −31.6155 −1.28642
$$605$$ −8.56155 −0.348077
$$606$$ 0 0
$$607$$ 42.0540 1.70692 0.853459 0.521160i $$-0.174500\pi$$
0.853459 + 0.521160i $$0.174500\pi$$
$$608$$ 46.7386 1.89550
$$609$$ 0 0
$$610$$ −39.3693 −1.59402
$$611$$ 3.80776 0.154046
$$612$$ 0 0
$$613$$ 40.7386 1.64542 0.822709 0.568463i $$-0.192462\pi$$
0.822709 + 0.568463i $$0.192462\pi$$
$$614$$ 49.4773 1.99674
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −32.2462 −1.29818 −0.649092 0.760710i $$-0.724851\pi$$
−0.649092 + 0.760710i $$0.724851\pi$$
$$618$$ 0 0
$$619$$ −32.1080 −1.29053 −0.645264 0.763960i $$-0.723253\pi$$
−0.645264 + 0.763960i $$0.723253\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 80.9848 3.24720
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 57.1231 2.28310
$$627$$ 0 0
$$628$$ −92.3542 −3.68533
$$629$$ −2.63068 −0.104892
$$630$$ 0 0
$$631$$ −11.8078 −0.470060 −0.235030 0.971988i $$-0.575519\pi$$
−0.235030 + 0.971988i $$0.575519\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ 0 0
$$634$$ −26.8769 −1.06742
$$635$$ −6.24621 −0.247873
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −26.7386 −1.05859
$$639$$ 0 0
$$640$$ −9.43845 −0.373087
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −1.56155 −0.0615816 −0.0307908 0.999526i $$-0.509803\pi$$
−0.0307908 + 0.999526i $$0.509803\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 36.4924 1.43467 0.717333 0.696731i $$-0.245363\pi$$
0.717333 + 0.696731i $$0.245363\pi$$
$$648$$ 0 0
$$649$$ −6.24621 −0.245185
$$650$$ −1.12311 −0.0440518
$$651$$ 0 0
$$652$$ −32.4924 −1.27250
$$653$$ 33.2311 1.30043 0.650216 0.759750i $$-0.274678\pi$$
0.650216 + 0.759750i $$0.274678\pi$$
$$654$$ 0 0
$$655$$ −0.876894 −0.0342631
$$656$$ 39.3693 1.53711
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9.17708 −0.357488 −0.178744 0.983896i $$-0.557203\pi$$
−0.178744 + 0.983896i $$0.557203\pi$$
$$660$$ 0 0
$$661$$ 5.12311 0.199266 0.0996329 0.995024i $$-0.468233\pi$$
0.0996329 + 0.995024i $$0.468233\pi$$
$$662$$ 30.7386 1.19469
$$663$$ 0 0
$$664$$ 26.2462 1.01855
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20.8769 0.808357
$$668$$ −31.6155 −1.22324
$$669$$ 0 0
$$670$$ 26.2462 1.01398
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ 31.8617 1.22818 0.614090 0.789236i $$-0.289523\pi$$
0.614090 + 0.789236i $$0.289523\pi$$
$$674$$ −3.86174 −0.148749
$$675$$ 0 0
$$676$$ −58.4233 −2.24705
$$677$$ 4.93087 0.189509 0.0947544 0.995501i $$-0.469793\pi$$
0.0947544 + 0.995501i $$0.469793\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −2.87689 −0.110324
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 6.73863 0.257847 0.128923 0.991655i $$-0.458848\pi$$
0.128923 + 0.991655i $$0.458848\pi$$
$$684$$ 0 0
$$685$$ 17.1231 0.654240
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 6.73863 0.256908
$$689$$ −2.24621 −0.0855738
$$690$$ 0 0
$$691$$ 24.4924 0.931736 0.465868 0.884854i $$-0.345742\pi$$
0.465868 + 0.884854i $$0.345742\pi$$
$$692$$ −20.2462 −0.769645
$$693$$ 0 0
$$694$$ −18.2462 −0.692617
$$695$$ 15.1231 0.573652
$$696$$ 0 0
$$697$$ −2.24621 −0.0850813
$$698$$ −26.8769 −1.01731
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −28.9309 −1.09270 −0.546352 0.837556i $$-0.683984\pi$$
−0.546352 + 0.837556i $$0.683984\pi$$
$$702$$ 0 0
$$703$$ 42.7386 1.61192
$$704$$ 2.24621 0.0846573
$$705$$ 0 0
$$706$$ 14.8769 0.559899
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 27.1771 1.02066 0.510328 0.859980i $$-0.329524\pi$$
0.510328 + 0.859980i $$0.329524\pi$$
$$710$$ −20.4924 −0.769067
$$711$$ 0 0
$$712$$ −7.36932 −0.276177
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −0.684658 −0.0256048
$$716$$ −91.2311 −3.40946
$$717$$ 0 0
$$718$$ −20.4924 −0.764770
$$719$$ 8.38447 0.312688 0.156344 0.987703i $$-0.450029\pi$$
0.156344 + 0.987703i $$0.450029\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 81.3002 3.02568
$$723$$ 0 0
$$724$$ 80.3542 2.98634
$$725$$ −6.68466 −0.248262
$$726$$ 0 0
$$727$$ −52.4924 −1.94684 −0.973418 0.229035i $$-0.926443\pi$$
−0.973418 + 0.229035i $$0.926443\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 31.3693 1.16103
$$731$$ −0.384472 −0.0142202
$$732$$ 0 0
$$733$$ −6.68466 −0.246903 −0.123452 0.992351i $$-0.539396\pi$$
−0.123452 + 0.992351i $$0.539396\pi$$
$$734$$ 22.2462 0.821123
$$735$$ 0 0
$$736$$ −20.4924 −0.755361
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 34.9309 1.28495 0.642476 0.766305i $$-0.277907\pi$$
0.642476 + 0.766305i $$0.277907\pi$$
$$740$$ 27.3693 1.00612
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 32.9848 1.21010 0.605048 0.796189i $$-0.293154\pi$$
0.605048 + 0.796189i $$0.293154\pi$$
$$744$$ 0 0
$$745$$ −12.2462 −0.448666
$$746$$ 11.8617 0.434289
$$747$$ 0 0
$$748$$ −3.12311 −0.114192
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 17.0691 0.622861 0.311431 0.950269i $$-0.399192\pi$$
0.311431 + 0.950269i $$0.399192\pi$$
$$752$$ −66.7386 −2.43371
$$753$$ 0 0
$$754$$ 7.50758 0.273410
$$755$$ −6.93087 −0.252240
$$756$$ 0 0
$$757$$ 39.3693 1.43090 0.715451 0.698663i $$-0.246221\pi$$
0.715451 + 0.698663i $$0.246221\pi$$
$$758$$ −42.2462 −1.53445
$$759$$ 0 0
$$760$$ 46.7386 1.69539
$$761$$ 48.2462 1.74892 0.874462 0.485094i $$-0.161215\pi$$
0.874462 + 0.485094i $$0.161215\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 61.8617 2.23808
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 1.75379 0.0633256
$$768$$ 0 0
$$769$$ 42.4924 1.53232 0.766158 0.642652i $$-0.222166\pi$$
0.766158 + 0.642652i $$0.222166\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 88.3542 3.17994
$$773$$ 36.9309 1.32831 0.664156 0.747594i $$-0.268791\pi$$
0.664156 + 0.747594i $$0.268791\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −38.1080 −1.36800
$$777$$ 0 0
$$778$$ 63.8617 2.28955
$$779$$ 36.4924 1.30748
$$780$$ 0 0
$$781$$ −12.4924 −0.447014
$$782$$ 3.50758 0.125431
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −20.2462 −0.722618
$$786$$ 0 0
$$787$$ 49.1771 1.75297 0.876487 0.481426i $$-0.159881\pi$$
0.876487 + 0.481426i $$0.159881\pi$$
$$788$$ −5.12311 −0.182503
$$789$$ 0 0
$$790$$ −6.24621 −0.222230
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 6.73863 0.239296
$$794$$ −70.6004 −2.50551
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 24.0540 0.852036 0.426018 0.904715i $$-0.359916\pi$$
0.426018 + 0.904715i $$0.359916\pi$$
$$798$$ 0 0
$$799$$ 3.80776 0.134709
$$800$$ 6.56155 0.231986
$$801$$ 0 0
$$802$$ −80.8466 −2.85479
$$803$$ 19.1231 0.674840
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −106.600 −3.75019
$$809$$ −16.5464 −0.581740 −0.290870 0.956763i $$-0.593945\pi$$
−0.290870 + 0.956763i $$0.593945\pi$$
$$810$$ 0 0
$$811$$ −19.6155 −0.688794 −0.344397 0.938824i $$-0.611917\pi$$
−0.344397 + 0.938824i $$0.611917\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ −7.12311 −0.249512
$$816$$ 0 0
$$817$$ 6.24621 0.218527
$$818$$ −16.6307 −0.581478
$$819$$ 0 0
$$820$$ 23.3693 0.816092
$$821$$ 21.4233 0.747678 0.373839 0.927494i $$-0.378041\pi$$
0.373839 + 0.927494i $$0.378041\pi$$
$$822$$ 0 0
$$823$$ −36.4924 −1.27205 −0.636023 0.771670i $$-0.719422\pi$$
−0.636023 + 0.771670i $$0.719422\pi$$
$$824$$ −36.4924 −1.27127
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 5.36932 0.186709 0.0933547 0.995633i $$-0.470241\pi$$
0.0933547 + 0.995633i $$0.470241\pi$$
$$828$$ 0 0
$$829$$ −34.8769 −1.21132 −0.605662 0.795722i $$-0.707092\pi$$
−0.605662 + 0.795722i $$0.707092\pi$$
$$830$$ 10.2462 0.355651
$$831$$ 0 0
$$832$$ −0.630683 −0.0218650
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −6.93087 −0.239853
$$836$$ 50.7386 1.75483
$$837$$ 0 0
$$838$$ 67.2311 2.32246
$$839$$ −28.8769 −0.996941 −0.498471 0.866907i $$-0.666105\pi$$
−0.498471 + 0.866907i $$0.666105\pi$$
$$840$$ 0 0
$$841$$ 15.6847 0.540850
$$842$$ −6.87689 −0.236993
$$843$$ 0 0
$$844$$ 64.1080 2.20669
$$845$$ −12.8078 −0.440600
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 39.3693 1.35195
$$849$$ 0 0
$$850$$ −1.12311 −0.0385222
$$851$$ −18.7386 −0.642352
$$852$$ 0 0
$$853$$ 7.26137 0.248624 0.124312 0.992243i $$-0.460328\pi$$
0.124312 + 0.992243i $$0.460328\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −87.7235 −2.99833
$$857$$ −15.7538 −0.538139 −0.269070 0.963121i $$-0.586716\pi$$
−0.269070 + 0.963121i $$0.586716\pi$$
$$858$$ 0 0
$$859$$ 16.4924 0.562714 0.281357 0.959603i $$-0.409215\pi$$
0.281357 + 0.959603i $$0.409215\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ 50.7386 1.72816
$$863$$ 25.7538 0.876669 0.438335 0.898812i $$-0.355569\pi$$
0.438335 + 0.898812i $$0.355569\pi$$
$$864$$ 0 0
$$865$$ −4.43845 −0.150912
$$866$$ −21.1231 −0.717792
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3.80776 −0.129170
$$870$$ 0 0
$$871$$ −4.49242 −0.152220
$$872$$ 34.8769 1.18108
$$873$$ 0 0
$$874$$ −56.9848 −1.92754
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −40.2462 −1.35902 −0.679509 0.733667i $$-0.737807\pi$$
−0.679509 + 0.733667i $$0.737807\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 0 0
$$880$$ 12.0000 0.404520
$$881$$ −11.8617 −0.399632 −0.199816 0.979833i $$-0.564034\pi$$
−0.199816 + 0.979833i $$0.564034\pi$$
$$882$$ 0 0
$$883$$ −8.49242 −0.285793 −0.142896 0.989738i $$-0.545642\pi$$
−0.142896 + 0.989738i $$0.545642\pi$$
$$884$$ 0.876894 0.0294931
$$885$$ 0 0
$$886$$ 6.73863 0.226389
$$887$$ 20.4924 0.688068 0.344034 0.938957i $$-0.388206\pi$$
0.344034 + 0.938957i $$0.388206\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −2.87689 −0.0964337
$$891$$ 0 0
$$892$$ 11.1231 0.372429
$$893$$ −61.8617 −2.07012
$$894$$ 0 0
$$895$$ −20.0000 −0.668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 4.63068 0.154528
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −2.24621 −0.0748321
$$902$$ 20.4924 0.682323
$$903$$ 0 0
$$904$$ 91.8617 3.05528
$$905$$ 17.6155 0.585560
$$906$$ 0 0
$$907$$ −24.1080 −0.800491 −0.400246 0.916408i $$-0.631075\pi$$
−0.400246 + 0.916408i $$0.631075\pi$$
$$908$$ 51.6155 1.71292
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 28.4924 0.943996 0.471998 0.881600i $$-0.343533\pi$$
0.471998 + 0.881600i $$0.343533\pi$$
$$912$$ 0 0
$$913$$ 6.24621 0.206719
$$914$$ −43.8617 −1.45082
$$915$$ 0 0
$$916$$ −49.6155 −1.63934
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 40.3002 1.32938 0.664690 0.747119i $$-0.268564\pi$$
0.664690 + 0.747119i $$0.268564\pi$$
$$920$$ −20.4924 −0.675615
$$921$$ 0 0
$$922$$ −33.6155 −1.10707
$$923$$ 3.50758 0.115453
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ 32.0000 1.05159
$$927$$ 0 0
$$928$$ −43.8617 −1.43983
$$929$$ 22.1080 0.725338 0.362669 0.931918i $$-0.381866\pi$$
0.362669 + 0.931918i $$0.381866\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −23.3693 −0.765487
$$933$$ 0 0
$$934$$ 57.4773 1.88071
$$935$$ −0.684658 −0.0223907
$$936$$ 0 0
$$937$$ 55.6695 1.81864 0.909322 0.416094i $$-0.136601\pi$$
0.909322 + 0.416094i $$0.136601\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −39.6155 −1.29212
$$941$$ 43.8617 1.42985 0.714926 0.699200i $$-0.246460\pi$$
0.714926 + 0.699200i $$0.246460\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ −30.7386 −1.00046
$$945$$ 0 0
$$946$$ 3.50758 0.114041
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ −5.36932 −0.174295
$$950$$ 18.2462 0.591985
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 33.1231 1.07296 0.536481 0.843912i $$-0.319753\pi$$
0.536481 + 0.843912i $$0.319753\pi$$
$$954$$ 0 0
$$955$$ 13.5616 0.438842
$$956$$ −90.3542 −2.92226
$$957$$ 0 0
$$958$$ 12.4924 0.403612
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −6.73863 −0.217262
$$963$$ 0 0
$$964$$ 19.3693 0.623844
$$965$$ 19.3693 0.623520
$$966$$ 0 0
$$967$$ −35.1231 −1.12948 −0.564741 0.825268i $$-0.691024\pi$$
−0.564741 + 0.825268i $$0.691024\pi$$
$$968$$ −56.1771 −1.80560
$$969$$ 0 0
$$970$$ −14.8769 −0.477668
$$971$$ 49.4773 1.58780 0.793901 0.608048i $$-0.208047\pi$$
0.793901 + 0.608048i $$0.208047\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −118.108 −3.78054
$$977$$ −33.2311 −1.06316 −0.531578 0.847009i $$-0.678401\pi$$
−0.531578 + 0.847009i $$0.678401\pi$$
$$978$$ 0 0
$$979$$ −1.75379 −0.0560513
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 105.477 3.36591
$$983$$ 51.4233 1.64015 0.820074 0.572257i $$-0.193932\pi$$
0.820074 + 0.572257i $$0.193932\pi$$
$$984$$ 0 0
$$985$$ −1.12311 −0.0357851
$$986$$ 7.50758 0.239090
$$987$$ 0 0
$$988$$ −14.2462 −0.453232
$$989$$ −2.73863 −0.0870835
$$990$$ 0 0
$$991$$ 12.4924 0.396835 0.198417 0.980118i $$-0.436420\pi$$
0.198417 + 0.980118i $$0.436420\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1.75379 0.0555988
$$996$$ 0 0
$$997$$ 2.68466 0.0850240 0.0425120 0.999096i $$-0.486464\pi$$
0.0425120 + 0.999096i $$0.486464\pi$$
$$998$$ 105.477 3.33882
$$999$$ 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 2205.2.a.x.1.2 2
3.2 odd 2 245.2.a.d.1.1 2
7.6 odd 2 315.2.a.e.1.2 2
12.11 even 2 3920.2.a.bs.1.2 2
15.2 even 4 1225.2.b.f.99.1 4
15.8 even 4 1225.2.b.f.99.4 4
15.14 odd 2 1225.2.a.s.1.2 2
21.2 odd 6 245.2.e.h.116.2 4
21.5 even 6 245.2.e.i.116.2 4
21.11 odd 6 245.2.e.h.226.2 4
21.17 even 6 245.2.e.i.226.2 4
21.20 even 2 35.2.a.b.1.1 2
28.27 even 2 5040.2.a.bt.1.1 2
35.13 even 4 1575.2.d.e.1324.1 4
35.27 even 4 1575.2.d.e.1324.4 4
35.34 odd 2 1575.2.a.p.1.1 2
84.83 odd 2 560.2.a.i.1.1 2
105.62 odd 4 175.2.b.b.99.1 4
105.83 odd 4 175.2.b.b.99.4 4
105.104 even 2 175.2.a.f.1.2 2
168.83 odd 2 2240.2.a.bd.1.2 2
168.125 even 2 2240.2.a.bh.1.1 2
231.230 odd 2 4235.2.a.m.1.2 2
273.272 even 2 5915.2.a.l.1.2 2
420.83 even 4 2800.2.g.t.449.2 4
420.167 even 4 2800.2.g.t.449.3 4
420.419 odd 2 2800.2.a.bi.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 21.20 even 2
175.2.a.f.1.2 2 105.104 even 2
175.2.b.b.99.1 4 105.62 odd 4
175.2.b.b.99.4 4 105.83 odd 4
245.2.a.d.1.1 2 3.2 odd 2
245.2.e.h.116.2 4 21.2 odd 6
245.2.e.h.226.2 4 21.11 odd 6
245.2.e.i.116.2 4 21.5 even 6
245.2.e.i.226.2 4 21.17 even 6
315.2.a.e.1.2 2 7.6 odd 2
560.2.a.i.1.1 2 84.83 odd 2
1225.2.a.s.1.2 2 15.14 odd 2
1225.2.b.f.99.1 4 15.2 even 4
1225.2.b.f.99.4 4 15.8 even 4
1575.2.a.p.1.1 2 35.34 odd 2
1575.2.d.e.1324.1 4 35.13 even 4
1575.2.d.e.1324.4 4 35.27 even 4
2205.2.a.x.1.2 2 1.1 even 1 trivial
2240.2.a.bd.1.2 2 168.83 odd 2
2240.2.a.bh.1.1 2 168.125 even 2
2800.2.a.bi.1.2 2 420.419 odd 2
2800.2.g.t.449.2 4 420.83 even 4
2800.2.g.t.449.3 4 420.167 even 4
3920.2.a.bs.1.2 2 12.11 even 2
4235.2.a.m.1.2 2 231.230 odd 2
5040.2.a.bt.1.1 2 28.27 even 2
5915.2.a.l.1.2 2 273.272 even 2