# Properties

 Label 1225.2.a.s.1.2 Level $1225$ Weight $2$ Character 1225.1 Self dual yes Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.78167424761$$ 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 $$\chi$$ $$=$$ 1225.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} +4.00000 q^{6} +6.56155 q^{8} -0.561553 q^{9} +O(q^{10})$$ $$q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} +4.00000 q^{6} +6.56155 q^{8} -0.561553 q^{9} -1.56155 q^{11} +7.12311 q^{12} +0.438447 q^{13} +7.68466 q^{16} -0.438447 q^{17} -1.43845 q^{18} +7.12311 q^{19} -4.00000 q^{22} -3.12311 q^{23} +10.2462 q^{24} +1.12311 q^{26} -5.56155 q^{27} +6.68466 q^{29} +6.56155 q^{32} -2.43845 q^{33} -1.12311 q^{34} -2.56155 q^{36} -6.00000 q^{37} +18.2462 q^{38} +0.684658 q^{39} -5.12311 q^{41} -0.876894 q^{43} -7.12311 q^{44} -8.00000 q^{46} -8.68466 q^{47} +12.0000 q^{48} -0.684658 q^{51} +2.00000 q^{52} +5.12311 q^{53} -14.2462 q^{54} +11.1231 q^{57} +17.1231 q^{58} +4.00000 q^{59} -15.3693 q^{61} +1.43845 q^{64} -6.24621 q^{66} -10.2462 q^{67} -2.00000 q^{68} -4.87689 q^{69} +8.00000 q^{71} -3.68466 q^{72} -12.2462 q^{73} -15.3693 q^{74} +32.4924 q^{76} +1.75379 q^{78} -2.43845 q^{79} -7.00000 q^{81} -13.1231 q^{82} +4.00000 q^{83} -2.24621 q^{86} +10.4384 q^{87} -10.2462 q^{88} +1.12311 q^{89} -14.2462 q^{92} -22.2462 q^{94} +10.2462 q^{96} +5.80776 q^{97} +0.876894 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 3q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 3q^{9} + q^{11} + 6q^{12} + 5q^{13} + 3q^{16} - 5q^{17} - 7q^{18} + 6q^{19} - 8q^{22} + 2q^{23} + 4q^{24} - 6q^{26} - 7q^{27} + q^{29} + 9q^{32} - 9q^{33} + 6q^{34} - q^{36} - 12q^{37} + 20q^{38} - 11q^{39} - 2q^{41} - 10q^{43} - 6q^{44} - 16q^{46} - 5q^{47} + 24q^{48} + 11q^{51} + 4q^{52} + 2q^{53} - 12q^{54} + 14q^{57} + 26q^{58} + 8q^{59} - 6q^{61} + 7q^{64} + 4q^{66} - 4q^{67} - 4q^{68} - 18q^{69} + 16q^{71} + 5q^{72} - 8q^{73} - 6q^{74} + 32q^{76} + 20q^{78} - 9q^{79} - 14q^{81} - 18q^{82} + 8q^{83} + 12q^{86} + 25q^{87} - 4q^{88} - 6q^{89} - 12q^{92} - 28q^{94} + 4q^{96} - 9q^{97} + 10q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.56155 1.81129 0.905646 0.424035i $$-0.139387\pi$$
0.905646 + 0.424035i $$0.139387\pi$$
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 4.56155 2.28078
$$5$$ 0 0
$$6$$ 4.00000 1.63299
$$7$$ 0 0
$$8$$ 6.56155 2.31986
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ −1.56155 −0.470826 −0.235413 0.971895i $$-0.575644\pi$$
−0.235413 + 0.971895i $$0.575644\pi$$
$$12$$ 7.12311 2.05626
$$13$$ 0.438447 0.121603 0.0608017 0.998150i $$-0.480634\pi$$
0.0608017 + 0.998150i $$0.480634\pi$$
$$14$$ 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 $$0.516932\pi$$
$$18$$ −1.43845 −0.339045
$$19$$ 7.12311 1.63415 0.817076 0.576530i $$-0.195593\pi$$
0.817076 + 0.576530i $$0.195593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ −3.12311 −0.651213 −0.325606 0.945505i $$-0.605568\pi$$
−0.325606 + 0.945505i $$0.605568\pi$$
$$24$$ 10.2462 2.09150
$$25$$ 0 0
$$26$$ 1.12311 0.220259
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ 6.68466 1.24131 0.620655 0.784084i $$-0.286867\pi$$
0.620655 + 0.784084i $$0.286867\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 6.56155 1.15993
$$33$$ −2.43845 −0.424479
$$34$$ −1.12311 −0.192611
$$35$$ 0 0
$$36$$ −2.56155 −0.426925
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 18.2462 2.95993
$$39$$ 0.684658 0.109633
$$40$$ 0 0
$$41$$ −5.12311 −0.800095 −0.400047 0.916494i $$-0.631006\pi$$
−0.400047 + 0.916494i $$0.631006\pi$$
$$42$$ 0 0
$$43$$ −0.876894 −0.133725 −0.0668626 0.997762i $$-0.521299\pi$$
−0.0668626 + 0.997762i $$0.521299\pi$$
$$44$$ −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 $$0.718339\pi$$
$$48$$ 12.0000 1.73205
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −0.684658 −0.0958714
$$52$$ 2.00000 0.277350
$$53$$ 5.12311 0.703713 0.351856 0.936054i $$-0.385551\pi$$
0.351856 + 0.936054i $$0.385551\pi$$
$$54$$ −14.2462 −1.93866
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 11.1231 1.47329
$$58$$ 17.1231 2.24837
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 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 0
$$66$$ −6.24621 −0.768855
$$67$$ −10.2462 −1.25177 −0.625887 0.779914i $$-0.715263\pi$$
−0.625887 + 0.779914i $$0.715263\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −4.87689 −0.587109
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ −3.68466 −0.434241
$$73$$ −12.2462 −1.43331 −0.716655 0.697428i $$-0.754328\pi$$
−0.716655 + 0.697428i $$0.754328\pi$$
$$74$$ −15.3693 −1.78665
$$75$$ 0 0
$$76$$ 32.4924 3.72714
$$77$$ 0 0
$$78$$ 1.75379 0.198577
$$79$$ −2.43845 −0.274347 −0.137173 0.990547i $$-0.543802\pi$$
−0.137173 + 0.990547i $$0.543802\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ −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 0
$$86$$ −2.24621 −0.242215
$$87$$ 10.4384 1.11912
$$88$$ −10.2462 −1.09225
$$89$$ 1.12311 0.119049 0.0595245 0.998227i $$-0.481042\pi$$
0.0595245 + 0.998227i $$0.481042\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −14.2462 −1.48527
$$93$$ 0 0
$$94$$ −22.2462 −2.29452
$$95$$ 0 0
$$96$$ 10.2462 1.04575
$$97$$ 5.80776 0.589689 0.294845 0.955545i $$-0.404732\pi$$
0.294845 + 0.955545i $$0.404732\pi$$
$$98$$ 0 0
$$99$$ 0.876894 0.0881312
$$100$$ 0 0
$$101$$ 16.2462 1.61656 0.808279 0.588799i $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ −1.75379 −0.173651
$$103$$ 5.56155 0.547996 0.273998 0.961730i $$-0.411654\pi$$
0.273998 + 0.961730i $$0.411654\pi$$
$$104$$ 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 $$0.723655\pi$$
$$108$$ −25.3693 −2.44116
$$109$$ 5.31534 0.509117 0.254559 0.967057i $$-0.418070\pi$$
0.254559 + 0.967057i $$0.418070\pi$$
$$110$$ 0 0
$$111$$ −9.36932 −0.889296
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 28.4924 2.66856
$$115$$ 0 0
$$116$$ 30.4924 2.83115
$$117$$ −0.246211 −0.0227622
$$118$$ 10.2462 0.943240
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ −39.3693 −3.56433
$$123$$ −8.00000 −0.721336
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.24621 0.554262 0.277131 0.960832i $$-0.410616\pi$$
0.277131 + 0.960832i $$0.410616\pi$$
$$128$$ −9.43845 −0.834249
$$129$$ −1.36932 −0.120562
$$130$$ 0 0
$$131$$ 0.876894 0.0766146 0.0383073 0.999266i $$-0.487803\pi$$
0.0383073 + 0.999266i $$0.487803\pi$$
$$132$$ −11.1231 −0.968142
$$133$$ 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 $$0.238838\pi$$
$$138$$ −12.4924 −1.06343
$$139$$ 15.1231 1.28273 0.641363 0.767238i $$-0.278369\pi$$
0.641363 + 0.767238i $$0.278369\pi$$
$$140$$ 0 0
$$141$$ −13.5616 −1.14209
$$142$$ 20.4924 1.71969
$$143$$ −0.684658 −0.0572540
$$144$$ −4.31534 −0.359612
$$145$$ 0 0
$$146$$ −31.3693 −2.59614
$$147$$ 0 0
$$148$$ −27.3693 −2.24974
$$149$$ 12.2462 1.00325 0.501624 0.865086i $$-0.332736\pi$$
0.501624 + 0.865086i $$0.332736\pi$$
$$150$$ 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
$$153$$ 0.246211 0.0199050
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.12311 0.250049
$$157$$ 20.2462 1.61582 0.807912 0.589303i $$-0.200598\pi$$
0.807912 + 0.589303i $$0.200598\pi$$
$$158$$ −6.24621 −0.496922
$$159$$ 8.00000 0.634441
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −17.9309 −1.40878
$$163$$ 7.12311 0.557925 0.278962 0.960302i $$-0.410010\pi$$
0.278962 + 0.960302i $$0.410010\pi$$
$$164$$ −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
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −4.43845 −0.337449 −0.168724 0.985663i $$-0.553965\pi$$
−0.168724 + 0.985663i $$0.553965\pi$$
$$174$$ 26.7386 2.02705
$$175$$ 0 0
$$176$$ −12.0000 −0.904534
$$177$$ 6.24621 0.469494
$$178$$ 2.87689 0.215632
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ 17.6155 1.30935 0.654676 0.755910i $$-0.272805\pi$$
0.654676 + 0.755910i $$0.272805\pi$$
$$182$$ 0 0
$$183$$ −24.0000 −1.77413
$$184$$ −20.4924 −1.51072
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.684658 0.0500672
$$188$$ −39.6155 −2.88926
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −13.5616 −0.981280 −0.490640 0.871363i $$-0.663237\pi$$
−0.490640 + 0.871363i $$0.663237\pi$$
$$192$$ 2.24621 0.162106
$$193$$ −19.3693 −1.39423 −0.697117 0.716957i $$-0.745534\pi$$
−0.697117 + 0.716957i $$0.745534\pi$$
$$194$$ 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 $$0.512739\pi$$
$$198$$ 2.24621 0.159631
$$199$$ 1.75379 0.124323 0.0621614 0.998066i $$-0.480201\pi$$
0.0621614 + 0.998066i $$0.480201\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 41.6155 2.92806
$$203$$ 0 0
$$204$$ −3.12311 −0.218661
$$205$$ 0 0
$$206$$ 14.2462 0.992581
$$207$$ 1.75379 0.121897
$$208$$ 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
$$213$$ 12.4924 0.855967
$$214$$ −34.2462 −2.34102
$$215$$ 0 0
$$216$$ −36.4924 −2.48299
$$217$$ 0 0
$$218$$ 13.6155 0.922160
$$219$$ −19.1231 −1.29222
$$220$$ 0 0
$$221$$ −0.192236 −0.0129312
$$222$$ −24.0000 −1.61077
$$223$$ −2.43845 −0.163291 −0.0816453 0.996661i $$-0.526017\pi$$
−0.0816453 + 0.996661i $$0.526017\pi$$
$$224$$ 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 $$0.377467\pi$$
$$228$$ 50.7386 3.36025
$$229$$ −10.8769 −0.718765 −0.359383 0.933190i $$-0.617013\pi$$
−0.359383 + 0.933190i $$0.617013\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 43.8617 2.87966
$$233$$ −5.12311 −0.335626 −0.167813 0.985819i $$-0.553670\pi$$
−0.167813 + 0.985819i $$0.553670\pi$$
$$234$$ −0.630683 −0.0412290
$$235$$ 0 0
$$236$$ 18.2462 1.18773
$$237$$ −3.80776 −0.247341
$$238$$ 0 0
$$239$$ 19.8078 1.28126 0.640629 0.767851i $$-0.278674\pi$$
0.640629 + 0.767851i $$0.278674\pi$$
$$240$$ 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
$$243$$ 5.75379 0.369106
$$244$$ −70.1080 −4.48820
$$245$$ 0 0
$$246$$ −20.4924 −1.30655
$$247$$ 3.12311 0.198718
$$248$$ 0 0
$$249$$ 6.24621 0.395838
$$250$$ 0 0
$$251$$ 8.87689 0.560305 0.280152 0.959956i $$-0.409615\pi$$
0.280152 + 0.959956i $$0.409615\pi$$
$$252$$ 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 $$0.606122\pi$$
$$258$$ −3.50758 −0.218372
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −3.75379 −0.232354
$$262$$ 2.24621 0.138771
$$263$$ 12.8769 0.794023 0.397012 0.917814i $$-0.370047\pi$$
0.397012 + 0.917814i $$0.370047\pi$$
$$264$$ −16.0000 −0.984732
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 1.75379 0.107330
$$268$$ −46.7386 −2.85502
$$269$$ 20.7386 1.26446 0.632228 0.774782i $$-0.282140\pi$$
0.632228 + 0.774782i $$0.282140\pi$$
$$270$$ 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
$$275$$ 0 0
$$276$$ −22.2462 −1.33906
$$277$$ 0.246211 0.0147934 0.00739670 0.999973i $$-0.497646\pi$$
0.00739670 + 0.999973i $$0.497646\pi$$
$$278$$ 38.7386 2.32339
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.4384 0.742016 0.371008 0.928630i $$-0.379012\pi$$
0.371008 + 0.928630i $$0.379012\pi$$
$$282$$ −34.7386 −2.06866
$$283$$ −11.3153 −0.672627 −0.336314 0.941750i $$-0.609180\pi$$
−0.336314 + 0.941750i $$0.609180\pi$$
$$284$$ 36.4924 2.16543
$$285$$ 0 0
$$286$$ −1.75379 −0.103704
$$287$$ 0 0
$$288$$ −3.68466 −0.217121
$$289$$ −16.8078 −0.988692
$$290$$ 0 0
$$291$$ 9.06913 0.531642
$$292$$ −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
$$295$$ 0 0
$$296$$ −39.3693 −2.28830
$$297$$ 8.68466 0.503935
$$298$$ 31.3693 1.81718
$$299$$ −1.36932 −0.0791896
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −17.7538 −1.02162
$$303$$ 25.3693 1.45743
$$304$$ 54.7386 3.13948
$$305$$ 0 0
$$306$$ 0.630683 0.0360538
$$307$$ −19.3153 −1.10238 −0.551192 0.834378i $$-0.685827\pi$$
−0.551192 + 0.834378i $$0.685827\pi$$
$$308$$ 0 0
$$309$$ 8.68466 0.494053
$$310$$ 0 0
$$311$$ −31.6155 −1.79275 −0.896376 0.443294i $$-0.853810\pi$$
−0.896376 + 0.443294i $$0.853810\pi$$
$$312$$ 4.49242 0.254333
$$313$$ −22.3002 −1.26048 −0.630241 0.776400i $$-0.717044\pi$$
−0.630241 + 0.776400i $$0.717044\pi$$
$$314$$ 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 $$0.595205\pi$$
$$318$$ 20.4924 1.14916
$$319$$ −10.4384 −0.584441
$$320$$ 0 0
$$321$$ −20.8769 −1.16523
$$322$$ 0 0
$$323$$ −3.12311 −0.173774
$$324$$ −31.9309 −1.77394
$$325$$ 0 0
$$326$$ 18.2462 1.01056
$$327$$ 8.30019 0.459001
$$328$$ −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
$$333$$ 3.36932 0.184637
$$334$$ −17.7538 −0.971444
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.50758 0.0821230 0.0410615 0.999157i $$-0.486926\pi$$
0.0410615 + 0.999157i $$0.486926\pi$$
$$338$$ −32.8078 −1.78451
$$339$$ 21.8617 1.18737
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −10.2462 −0.554052
$$343$$ 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 $$0.561236\pi$$
$$348$$ 47.6155 2.55246
$$349$$ −10.4924 −0.561646 −0.280823 0.959760i $$-0.590607\pi$$
−0.280823 + 0.959760i $$0.590607\pi$$
$$350$$ 0 0
$$351$$ −2.43845 −0.130155
$$352$$ −10.2462 −0.546125
$$353$$ 5.80776 0.309116 0.154558 0.987984i $$-0.450605\pi$$
0.154558 + 0.987984i $$0.450605\pi$$
$$354$$ 16.0000 0.850390
$$355$$ 0 0
$$356$$ 5.12311 0.271524
$$357$$ 0 0
$$358$$ 51.2311 2.70765
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ 45.1231 2.37162
$$363$$ −13.3693 −0.701707
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −61.4773 −3.21347
$$367$$ −8.68466 −0.453335 −0.226668 0.973972i $$-0.572783\pi$$
−0.226668 + 0.973972i $$0.572783\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ 2.87689 0.149765
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4.63068 −0.239768 −0.119884 0.992788i $$-0.538252\pi$$
−0.119884 + 0.992788i $$0.538252\pi$$
$$374$$ 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 $$0.639227\pi$$
$$380$$ 0 0
$$381$$ 9.75379 0.499702
$$382$$ −34.7386 −1.77738
$$383$$ 6.24621 0.319166 0.159583 0.987184i $$-0.448985\pi$$
0.159583 + 0.987184i $$0.448985\pi$$
$$384$$ −14.7386 −0.752128
$$385$$ 0 0
$$386$$ −49.6155 −2.52536
$$387$$ 0.492423 0.0250312
$$388$$ 26.4924 1.34495
$$389$$ −24.9309 −1.26405 −0.632023 0.774950i $$-0.717775\pi$$
−0.632023 + 0.774950i $$0.717775\pi$$
$$390$$ 0 0
$$391$$ 1.36932 0.0692493
$$392$$ 0 0
$$393$$ 1.36932 0.0690729
$$394$$ −2.87689 −0.144936
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 27.5616 1.38327 0.691637 0.722245i $$-0.256890\pi$$
0.691637 + 0.722245i $$0.256890\pi$$
$$398$$ 4.49242 0.225185
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 31.5616 1.57611 0.788054 0.615606i $$-0.211089\pi$$
0.788054 + 0.615606i $$0.211089\pi$$
$$402$$ −40.9848 −2.04414
$$403$$ 0 0
$$404$$ 74.1080 3.68701
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.36932 0.464420
$$408$$ −4.49242 −0.222408
$$409$$ −6.49242 −0.321030 −0.160515 0.987033i $$-0.551315\pi$$
−0.160515 + 0.987033i $$0.551315\pi$$
$$410$$ 0 0
$$411$$ 26.7386 1.31892
$$412$$ 25.3693 1.24986
$$413$$ 0 0
$$414$$ 4.49242 0.220791
$$415$$ 0 0
$$416$$ 2.87689 0.141051
$$417$$ 23.6155 1.15646
$$418$$ −28.4924 −1.39361
$$419$$ −26.2462 −1.28221 −0.641106 0.767453i $$-0.721524\pi$$
−0.641106 + 0.767453i $$0.721524\pi$$
$$420$$ 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
$$423$$ 4.87689 0.237123
$$424$$ 33.6155 1.63251
$$425$$ 0 0
$$426$$ 32.0000 1.55041
$$427$$ 0 0
$$428$$ −60.9848 −2.94781
$$429$$ −1.06913 −0.0516181
$$430$$ 0 0
$$431$$ −19.8078 −0.954106 −0.477053 0.878874i $$-0.658295\pi$$
−0.477053 + 0.878874i $$0.658295\pi$$
$$432$$ −42.7386 −2.05626
$$433$$ 8.24621 0.396288 0.198144 0.980173i $$-0.436509\pi$$
0.198144 + 0.980173i $$0.436509\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 24.2462 1.16118
$$437$$ −22.2462 −1.06418
$$438$$ −48.9848 −2.34059
$$439$$ −9.36932 −0.447173 −0.223587 0.974684i $$-0.571777\pi$$
−0.223587 + 0.974684i $$0.571777\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −0.492423 −0.0234221
$$443$$ 2.63068 0.124988 0.0624938 0.998045i $$-0.480095\pi$$
0.0624938 + 0.998045i $$0.480095\pi$$
$$444$$ −42.7386 −2.02829
$$445$$ 0 0
$$446$$ −6.24621 −0.295767
$$447$$ 19.1231 0.904492
$$448$$ 0 0
$$449$$ −1.80776 −0.0853137 −0.0426568 0.999090i $$-0.513582\pi$$
−0.0426568 + 0.999090i $$0.513582\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 63.8617 3.00380
$$453$$ −10.8229 −0.508505
$$454$$ 28.9848 1.36033
$$455$$ 0 0
$$456$$ 72.9848 3.41783
$$457$$ 17.1231 0.800985 0.400493 0.916300i $$-0.368839\pi$$
0.400493 + 0.916300i $$0.368839\pi$$
$$458$$ −27.8617 −1.30189
$$459$$ 2.43845 0.113817
$$460$$ 0 0
$$461$$ 13.1231 0.611204 0.305602 0.952159i $$-0.401142\pi$$
0.305602 + 0.952159i $$0.401142\pi$$
$$462$$ 0 0
$$463$$ −12.4924 −0.580572 −0.290286 0.956940i $$-0.593750\pi$$
−0.290286 + 0.956940i $$0.593750\pi$$
$$464$$ 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 $$0.326243\pi$$
$$468$$ −1.12311 −0.0519156
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 31.6155 1.45677
$$472$$ 26.2462 1.20808
$$473$$ 1.36932 0.0629613
$$474$$ −9.75379 −0.448006
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2.87689 −0.131724
$$478$$ 50.7386 2.32073
$$479$$ −4.87689 −0.222831 −0.111415 0.993774i $$-0.535538\pi$$
−0.111415 + 0.993774i $$0.535538\pi$$
$$480$$ 0 0
$$481$$ −2.63068 −0.119949
$$482$$ 10.8769 0.495429
$$483$$ 0 0
$$484$$ −39.0540 −1.77518
$$485$$ 0 0
$$486$$ 14.7386 0.668558
$$487$$ 3.12311 0.141521 0.0707607 0.997493i $$-0.477457\pi$$
0.0707607 + 0.997493i $$0.477457\pi$$
$$488$$ −100.847 −4.56511
$$489$$ 11.1231 0.503004
$$490$$ 0 0
$$491$$ −41.1771 −1.85830 −0.929148 0.369708i $$-0.879458\pi$$
−0.929148 + 0.369708i $$0.879458\pi$$
$$492$$ −36.4924 −1.64521
$$493$$ −2.93087 −0.132000
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 16.0000 0.716977
$$499$$ 41.1771 1.84334 0.921670 0.387976i $$-0.126826\pi$$
0.921670 + 0.387976i $$0.126826\pi$$
$$500$$ 0 0
$$501$$ −10.8229 −0.483532
$$502$$ 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
$$505$$ 0 0
$$506$$ 12.4924 0.555356
$$507$$ −20.0000 −0.888231
$$508$$ 28.4924 1.26415
$$509$$ 11.7538 0.520978 0.260489 0.965477i $$-0.416116\pi$$
0.260489 + 0.965477i $$0.416116\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −50.4233 −2.22842
$$513$$ −39.6155 −1.74907
$$514$$ −26.8769 −1.18549
$$515$$ 0 0
$$516$$ −6.24621 −0.274974
$$517$$ 13.5616 0.596436
$$518$$ 0 0
$$519$$ −6.93087 −0.304231
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ −9.61553 −0.420860
$$523$$ 40.4924 1.77061 0.885305 0.465011i $$-0.153950\pi$$
0.885305 + 0.465011i $$0.153950\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ 32.9848 1.43821
$$527$$ 0 0
$$528$$ −18.7386 −0.815494
$$529$$ −13.2462 −0.575922
$$530$$ 0 0
$$531$$ −2.24621 −0.0974773
$$532$$ 0 0
$$533$$ −2.24621 −0.0972942
$$534$$ 4.49242 0.194406
$$535$$ 0 0
$$536$$ −67.2311 −2.90394
$$537$$ 31.2311 1.34772
$$538$$ 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
$$543$$ 27.5076 1.18046
$$544$$ −2.87689 −0.123346
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.24621 0.0960411 0.0480205 0.998846i $$-0.484709\pi$$
0.0480205 + 0.998846i $$0.484709\pi$$
$$548$$ 78.1080 3.33661
$$549$$ 8.63068 0.368349
$$550$$ 0 0
$$551$$ 47.6155 2.02849
$$552$$ −32.0000 −1.36201
$$553$$ 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
$$561$$ 1.06913 0.0451387
$$562$$ 31.8617 1.34401
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ −61.8617 −2.60485
$$565$$ 0 0
$$566$$ −28.9848 −1.21832
$$567$$ 0 0
$$568$$ 52.4924 2.20253
$$569$$ −30.9848 −1.29895 −0.649476 0.760382i $$-0.725012\pi$$
−0.649476 + 0.760382i $$0.725012\pi$$
$$570$$ 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
$$573$$ −21.1771 −0.884685
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −0.807764 −0.0336568
$$577$$ −24.0540 −1.00138 −0.500690 0.865627i $$-0.666920\pi$$
−0.500690 + 0.865627i $$0.666920\pi$$
$$578$$ −43.0540 −1.79081
$$579$$ −30.2462 −1.25699
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 23.2311 0.962958
$$583$$ −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
$$590$$ 0 0
$$591$$ −1.75379 −0.0721412
$$592$$ −46.1080 −1.89503
$$593$$ −27.5616 −1.13182 −0.565909 0.824468i $$-0.691475\pi$$
−0.565909 + 0.824468i $$0.691475\pi$$
$$594$$ 22.2462 0.912773
$$595$$ 0 0
$$596$$ 55.8617 2.28819
$$597$$ 2.73863 0.112085
$$598$$ −3.50758 −0.143436
$$599$$ −11.8078 −0.482452 −0.241226 0.970469i $$-0.577550\pi$$
−0.241226 + 0.970469i $$0.577550\pi$$
$$600$$ 0 0
$$601$$ −6.49242 −0.264831 −0.132416 0.991194i $$-0.542273\pi$$
−0.132416 + 0.991194i $$0.542273\pi$$
$$602$$ 0 0
$$603$$ 5.75379 0.234312
$$604$$ −31.6155 −1.28642
$$605$$ 0 0
$$606$$ 64.9848 2.63983
$$607$$ −42.0540 −1.70692 −0.853459 0.521160i $$-0.825500\pi$$
−0.853459 + 0.521160i $$0.825500\pi$$
$$608$$ 46.7386 1.89550
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.80776 −0.154046
$$612$$ 1.12311 0.0453989
$$613$$ −40.7386 −1.64542 −0.822709 0.568463i $$-0.807538\pi$$
−0.822709 + 0.568463i $$0.807538\pi$$
$$614$$ −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 $$0.724851\pi$$
$$618$$ 22.2462 0.894874
$$619$$ −32.1080 −1.29053 −0.645264 0.763960i $$-0.723253\pi$$
−0.645264 + 0.763960i $$0.723253\pi$$
$$620$$ 0 0
$$621$$ 17.3693 0.697007
$$622$$ −80.9848 −3.24720
$$623$$ 0 0
$$624$$ 5.26137 0.210623
$$625$$ 0 0
$$626$$ −57.1231 −2.28310
$$627$$ −17.3693 −0.693664
$$628$$ 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
$$633$$ 21.9460 0.872276
$$634$$ −26.8769 −1.06742
$$635$$ 0 0
$$636$$ 36.4924 1.44702
$$637$$ 0 0
$$638$$ −26.7386 −1.05859
$$639$$ −4.49242 −0.177717
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ −53.4773 −2.11058
$$643$$ 1.56155 0.0615816 0.0307908 0.999526i $$-0.490197\pi$$
0.0307908 + 0.999526i $$0.490197\pi$$
$$644$$ 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 $$0.245363\pi$$
$$648$$ −45.9309 −1.80433
$$649$$ −6.24621 −0.245185
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 32.4924 1.27250
$$653$$ 33.2311 1.30043 0.650216 0.759750i $$-0.274678\pi$$
0.650216 + 0.759750i $$0.274678\pi$$
$$654$$ 21.2614 0.831385
$$655$$ 0 0
$$656$$ −39.3693 −1.53711
$$657$$ 6.87689 0.268293
$$658$$ 0 0
$$659$$ 9.17708 0.357488 0.178744 0.983896i $$-0.442797\pi$$
0.178744 + 0.983896i $$0.442797\pi$$
$$660$$ 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
$$663$$ −0.300187 −0.0116583
$$664$$ 26.2462 1.01855
$$665$$ 0 0
$$666$$ 8.63068 0.334432
$$667$$ −20.8769 −0.808357
$$668$$ −31.6155 −1.22324
$$669$$ −3.80776 −0.147217
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ −31.8617 −1.22818 −0.614090 0.789236i $$-0.710477\pi$$
−0.614090 + 0.789236i $$0.710477\pi$$
$$674$$ 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 $$0.469793\pi$$
$$678$$ 56.0000 2.15067
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 17.6695 0.677097
$$682$$ 0 0
$$683$$ 6.73863 0.257847 0.128923 0.991655i $$-0.458848\pi$$
0.128923 + 0.991655i $$0.458848\pi$$
$$684$$ −18.2462 −0.697661
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −16.9848 −0.648012
$$688$$ −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
$$695$$ 0 0
$$696$$ 68.4924 2.59620
$$697$$ 2.24621 0.0850813
$$698$$ −26.8769 −1.01731
$$699$$ −8.00000 −0.302588
$$700$$ 0 0
$$701$$ 28.9309 1.09270 0.546352 0.837556i $$-0.316016\pi$$
0.546352 + 0.837556i $$0.316016\pi$$
$$702$$ −6.24621 −0.235748
$$703$$ −42.7386 −1.61192
$$704$$ −2.24621 −0.0846573
$$705$$ 0 0
$$706$$ 14.8769 0.559899
$$707$$ 0 0
$$708$$ 28.4924 1.07081
$$709$$ 27.1771 1.02066 0.510328 0.859980i $$-0.329524\pi$$
0.510328 + 0.859980i $$0.329524\pi$$
$$710$$ 0 0
$$711$$ 1.36932 0.0513534
$$712$$ 7.36932 0.276177
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 91.2311 3.40946
$$717$$ 30.9309 1.15513
$$718$$ 20.4924 0.764770
$$719$$ −8.38447 −0.312688 −0.156344 0.987703i $$-0.549971\pi$$
−0.156344 + 0.987703i $$0.549971\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 81.3002 3.02568
$$723$$ 6.63068 0.246598
$$724$$ 80.3542 2.98634
$$725$$ 0 0
$$726$$ −34.2462 −1.27100
$$727$$ 52.4924 1.94684 0.973418 0.229035i $$-0.0735572\pi$$
0.973418 + 0.229035i $$0.0735572\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 0.384472 0.0142202
$$732$$ −109.477 −4.04640
$$733$$ 6.68466 0.246903 0.123452 0.992351i $$-0.460604\pi$$
0.123452 + 0.992351i $$0.460604\pi$$
$$734$$ −22.2462 −0.821123
$$735$$ 0 0
$$736$$ −20.4924 −0.755361
$$737$$ 16.0000 0.589368
$$738$$ 7.36932 0.271268
$$739$$ 34.9309 1.28495 0.642476 0.766305i $$-0.277907\pi$$
0.642476 + 0.766305i $$0.277907\pi$$
$$740$$ 0 0
$$741$$ 4.87689 0.179157
$$742$$ 0 0
$$743$$ 32.9848 1.21010 0.605048 0.796189i $$-0.293154\pi$$
0.605048 + 0.796189i $$0.293154\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −11.8617 −0.434289
$$747$$ −2.24621 −0.0821846
$$748$$ 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
$$753$$ 13.8617 0.505150
$$754$$ 7.50758 0.273410
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −39.3693 −1.43090 −0.715451 0.698663i $$-0.753779\pi$$
−0.715451 + 0.698663i $$0.753779\pi$$
$$758$$ −42.2462 −1.53445
$$759$$ 7.61553 0.276426
$$760$$ 0 0
$$761$$ −48.2462 −1.74892 −0.874462 0.485094i $$-0.838785\pi$$
−0.874462 + 0.485094i $$0.838785\pi$$
$$762$$ 24.9848 0.905105
$$763$$ 0 0
$$764$$ −61.8617 −2.23808
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 1.75379 0.0633256
$$768$$ −42.2462 −1.52443
$$769$$ 42.4924 1.53232 0.766158 0.642652i $$-0.222166\pi$$
0.766158 + 0.642652i $$0.222166\pi$$
$$770$$ 0 0
$$771$$ −16.3845 −0.590072
$$772$$ −88.3542 −3.17994
$$773$$ 36.9309 1.32831 0.664156 0.747594i $$-0.268791\pi$$
0.664156 + 0.747594i $$0.268791\pi$$
$$774$$ 1.26137 0.0453389
$$775$$ 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
$$783$$ −37.1771 −1.32860
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 3.50758 0.125111
$$787$$ −49.1771 −1.75297 −0.876487 0.481426i $$-0.840119\pi$$
−0.876487 + 0.481426i $$0.840119\pi$$
$$788$$ −5.12311 −0.182503
$$789$$ 20.1080 0.715862
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 5.75379 0.204452
$$793$$ −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
$$800$$ 0 0
$$801$$ −0.630683 −0.0222841
$$802$$ 80.8466 2.85479
$$803$$ 19.1231 0.674840
$$804$$ −72.9848 −2.57398
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 32.3845 1.13999
$$808$$ 106.600 3.75019
$$809$$ 16.5464 0.581740 0.290870 0.956763i $$-0.406055\pi$$
0.290870 + 0.956763i $$0.406055\pi$$
$$810$$ 0 0
$$811$$ −19.6155 −0.688794 −0.344397 0.938824i $$-0.611917\pi$$
−0.344397 + 0.938824i $$0.611917\pi$$
$$812$$ 0 0
$$813$$ 24.9848 0.876257
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −5.26137 −0.184185
$$817$$ −6.24621 −0.218527
$$818$$ −16.6307 −0.581478
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −21.4233 −0.747678 −0.373839 0.927494i $$-0.621959\pi$$
−0.373839 + 0.927494i $$0.621959\pi$$
$$822$$ 68.4924 2.38895
$$823$$ 36.4924 1.27205 0.636023 0.771670i $$-0.280578\pi$$
0.636023 + 0.771670i $$0.280578\pi$$
$$824$$ 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 $$0.470241\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −34.8769 −1.21132 −0.605662 0.795722i $$-0.707092\pi$$
−0.605662 + 0.795722i $$0.707092\pi$$
$$830$$ 0 0
$$831$$ 0.384472 0.0133372
$$832$$ 0.630683 0.0218650
$$833$$ 0 0
$$834$$ 60.4924 2.09468
$$835$$ 0 0
$$836$$ −50.7386 −1.75483
$$837$$ 0 0
$$838$$ −67.2311 −2.32246
$$839$$ 28.8769 0.996941 0.498471 0.866907i $$-0.333895\pi$$
0.498471 + 0.866907i $$0.333895\pi$$
$$840$$ 0 0
$$841$$ 15.6847 0.540850
$$842$$ −6.87689 −0.236993
$$843$$ 19.4233 0.668974
$$844$$ 64.1080 2.20669
$$845$$ 0 0
$$846$$ 12.4924 0.429498
$$847$$ 0 0
$$848$$ 39.3693 1.35195
$$849$$ −17.6695 −0.606416
$$850$$ 0 0
$$851$$ 18.7386 0.642352
$$852$$ 56.9848 1.95227
$$853$$ −7.26137 −0.248624 −0.124312 0.992243i $$-0.539672\pi$$
−0.124312 + 0.992243i $$0.539672\pi$$
$$854$$ 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 $$0.586716\pi$$
$$858$$ −2.73863 −0.0934954
$$859$$ 16.4924 0.562714 0.281357 0.959603i $$-0.409215\pi$$
0.281357 + 0.959603i $$0.409215\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −50.7386 −1.72816
$$863$$ 25.7538 0.876669 0.438335 0.898812i $$-0.355569\pi$$
0.438335 + 0.898812i $$0.355569\pi$$
$$864$$ −36.4924 −1.24150
$$865$$ 0 0
$$866$$ 21.1231 0.717792
$$867$$ −26.2462 −0.891368
$$868$$ 0 0
$$869$$ 3.80776 0.129170
$$870$$ 0 0
$$871$$ −4.49242 −0.152220
$$872$$ 34.8769 1.18108
$$873$$ −3.26137 −0.110381
$$874$$ −56.9848 −1.92754
$$875$$ 0 0
$$876$$ −87.2311 −2.94726
$$877$$ 40.2462 1.35902 0.679509 0.733667i $$-0.262193\pi$$
0.679509 + 0.733667i $$0.262193\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ −4.19224 −0.141401
$$880$$ 0 0
$$881$$ 11.8617 0.399632 0.199816 0.979833i $$-0.435966\pi$$
0.199816 + 0.979833i $$0.435966\pi$$
$$882$$ 0 0
$$883$$ 8.49242 0.285793 0.142896 0.989738i $$-0.454358\pi$$
0.142896 + 0.989738i $$0.454358\pi$$
$$884$$ −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 $$0.388206\pi$$
$$888$$ −61.4773 −2.06304
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 10.9309 0.366198
$$892$$ −11.1231 −0.372429
$$893$$ −61.8617 −2.07012
$$894$$ 48.9848 1.63830
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −2.13826 −0.0713944
$$898$$ −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
$$905$$ 0 0
$$906$$ −27.7235 −0.921051
$$907$$ 24.1080 0.800491 0.400246 0.916408i $$-0.368925\pi$$
0.400246 + 0.916408i $$0.368925\pi$$
$$908$$ 51.6155 1.71292
$$909$$ −9.12311 −0.302594
$$910$$ 0 0
$$911$$ −28.4924 −0.943996 −0.471998 0.881600i $$-0.656467\pi$$
−0.471998 + 0.881600i $$0.656467\pi$$
$$912$$ 85.4773 2.83044
$$913$$ −6.24621 −0.206719
$$914$$ 43.8617 1.45082
$$915$$ 0 0
$$916$$ −49.6155 −1.63934
$$917$$ 0 0
$$918$$ 6.24621 0.206156
$$919$$ 40.3002 1.32938 0.664690 0.747119i $$-0.268564\pi$$
0.664690 + 0.747119i $$0.268564\pi$$
$$920$$ 0 0
$$921$$ −30.1619 −0.993869
$$922$$ 33.6155 1.10707
$$923$$ 3.50758 0.115453
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ −3.12311 −0.102576
$$928$$ 43.8617 1.43983
$$929$$ −22.1080 −0.725338 −0.362669 0.931918i $$-0.618134\pi$$
−0.362669 + 0.931918i $$0.618134\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −23.3693 −0.765487
$$933$$ −49.3693 −1.61628
$$934$$ 57.4773 1.88071
$$935$$ 0 0
$$936$$ −1.61553 −0.0528052
$$937$$ −55.6695 −1.81864 −0.909322 0.416094i $$-0.863399\pi$$
−0.909322 + 0.416094i $$0.863399\pi$$
$$938$$ 0 0
$$939$$ −34.8229 −1.13640
$$940$$ 0 0
$$941$$ −43.8617 −1.42985 −0.714926 0.699200i $$-0.753540\pi$$
−0.714926 + 0.699200i $$0.753540\pi$$
$$942$$ 80.9848 2.63863
$$943$$ 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 $$0.520702\pi$$
$$948$$ −17.3693 −0.564129
$$949$$ −5.36932 −0.174295
$$950$$ 0 0
$$951$$ −16.3845 −0.531303
$$952$$ 0 0
$$953$$ 33.1231 1.07296 0.536481 0.843912i $$-0.319753\pi$$
0.536481 + 0.843912i $$0.319753\pi$$
$$954$$ −7.36932 −0.238590
$$955$$ 0 0
$$956$$ 90.3542 2.92226
$$957$$ −16.3002 −0.526910
$$958$$ −12.4924 −0.403612
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −6.73863 −0.217262
$$963$$ 7.50758 0.241928
$$964$$ 19.3693 0.623844
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 35.1231 1.12948 0.564741 0.825268i $$-0.308976\pi$$
0.564741 + 0.825268i $$0.308976\pi$$
$$968$$ −56.1771 −1.80560
$$969$$ −4.87689 −0.156668
$$970$$ 0 0
$$971$$ −49.4773 −1.58780 −0.793901 0.608048i $$-0.791953\pi$$
−0.793901 + 0.608048i $$0.791953\pi$$
$$972$$ 26.2462 0.841848
$$973$$ 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 $$0.678401\pi$$
$$978$$ 28.4924 0.911087
$$979$$ −1.75379 −0.0560513
$$980$$ 0 0
$$981$$ −2.98485 −0.0952988
$$982$$ −105.477 −3.36591
$$983$$ 51.4233 1.64015 0.820074 0.572257i $$-0.193932\pi$$
0.820074 + 0.572257i $$0.193932\pi$$
$$984$$ −52.4924 −1.67340
$$985$$ 0 0
$$986$$ −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
$$993$$ 18.7386 0.594653
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 28.4924 0.902817
$$997$$ −2.68466 −0.0850240 −0.0425120 0.999096i $$-0.513536\pi$$
−0.0425120 + 0.999096i $$0.513536\pi$$
$$998$$ 105.477 3.33882
$$999$$ 33.3693 1.05576
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 1225.2.a.s.1.2 2
5.2 odd 4 1225.2.b.f.99.4 4
5.3 odd 4 1225.2.b.f.99.1 4
5.4 even 2 245.2.a.d.1.1 2
7.6 odd 2 175.2.a.f.1.2 2
15.14 odd 2 2205.2.a.x.1.2 2
20.19 odd 2 3920.2.a.bs.1.2 2
21.20 even 2 1575.2.a.p.1.1 2
28.27 even 2 2800.2.a.bi.1.2 2
35.4 even 6 245.2.e.h.226.2 4
35.9 even 6 245.2.e.h.116.2 4
35.13 even 4 175.2.b.b.99.1 4
35.19 odd 6 245.2.e.i.116.2 4
35.24 odd 6 245.2.e.i.226.2 4
35.27 even 4 175.2.b.b.99.4 4
35.34 odd 2 35.2.a.b.1.1 2
105.62 odd 4 1575.2.d.e.1324.1 4
105.83 odd 4 1575.2.d.e.1324.4 4
105.104 even 2 315.2.a.e.1.2 2
140.27 odd 4 2800.2.g.t.449.2 4
140.83 odd 4 2800.2.g.t.449.3 4
140.139 even 2 560.2.a.i.1.1 2
280.69 odd 2 2240.2.a.bh.1.1 2
280.139 even 2 2240.2.a.bd.1.2 2
385.384 even 2 4235.2.a.m.1.2 2
420.419 odd 2 5040.2.a.bt.1.1 2
455.454 odd 2 5915.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 35.34 odd 2
175.2.a.f.1.2 2 7.6 odd 2
175.2.b.b.99.1 4 35.13 even 4
175.2.b.b.99.4 4 35.27 even 4
245.2.a.d.1.1 2 5.4 even 2
245.2.e.h.116.2 4 35.9 even 6
245.2.e.h.226.2 4 35.4 even 6
245.2.e.i.116.2 4 35.19 odd 6
245.2.e.i.226.2 4 35.24 odd 6
315.2.a.e.1.2 2 105.104 even 2
560.2.a.i.1.1 2 140.139 even 2
1225.2.a.s.1.2 2 1.1 even 1 trivial
1225.2.b.f.99.1 4 5.3 odd 4
1225.2.b.f.99.4 4 5.2 odd 4
1575.2.a.p.1.1 2 21.20 even 2
1575.2.d.e.1324.1 4 105.62 odd 4
1575.2.d.e.1324.4 4 105.83 odd 4
2205.2.a.x.1.2 2 15.14 odd 2
2240.2.a.bd.1.2 2 280.139 even 2
2240.2.a.bh.1.1 2 280.69 odd 2
2800.2.a.bi.1.2 2 28.27 even 2
2800.2.g.t.449.2 4 140.27 odd 4
2800.2.g.t.449.3 4 140.83 odd 4
3920.2.a.bs.1.2 2 20.19 odd 2
4235.2.a.m.1.2 2 385.384 even 2
5040.2.a.bt.1.1 2 420.419 odd 2
5915.2.a.l.1.2 2 455.454 odd 2