# Properties

 Label 4235.2.a.m.1.2 Level $4235$ Weight $2$ Character 4235.1 Self dual yes Analytic conductor $33.817$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4235 = 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4235.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.8166452560$$ 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 $$\chi$$ $$=$$ 4235.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} +1.00000 q^{5} +4.00000 q^{6} +1.00000 q^{7} +6.56155 q^{8} -0.561553 q^{9} +O(q^{10})$$ $$q+2.56155 q^{2} +1.56155 q^{3} +4.56155 q^{4} +1.00000 q^{5} +4.00000 q^{6} +1.00000 q^{7} +6.56155 q^{8} -0.561553 q^{9} +2.56155 q^{10} +7.12311 q^{12} -0.438447 q^{13} +2.56155 q^{14} +1.56155 q^{15} +7.68466 q^{16} +0.438447 q^{17} -1.43845 q^{18} +7.12311 q^{19} +4.56155 q^{20} +1.56155 q^{21} +3.12311 q^{23} +10.2462 q^{24} +1.00000 q^{25} -1.12311 q^{26} -5.56155 q^{27} +4.56155 q^{28} -6.68466 q^{29} +4.00000 q^{30} +6.56155 q^{32} +1.12311 q^{34} +1.00000 q^{35} -2.56155 q^{36} +6.00000 q^{37} +18.2462 q^{38} -0.684658 q^{39} +6.56155 q^{40} -5.12311 q^{41} +4.00000 q^{42} -0.876894 q^{43} -0.561553 q^{45} +8.00000 q^{46} -8.68466 q^{47} +12.0000 q^{48} +1.00000 q^{49} +2.56155 q^{50} +0.684658 q^{51} -2.00000 q^{52} -5.12311 q^{53} -14.2462 q^{54} +6.56155 q^{56} +11.1231 q^{57} -17.1231 q^{58} -4.00000 q^{59} +7.12311 q^{60} -15.3693 q^{61} -0.561553 q^{63} +1.43845 q^{64} -0.438447 q^{65} +10.2462 q^{67} +2.00000 q^{68} +4.87689 q^{69} +2.56155 q^{70} +8.00000 q^{71} -3.68466 q^{72} +12.2462 q^{73} +15.3693 q^{74} +1.56155 q^{75} +32.4924 q^{76} -1.75379 q^{78} +2.43845 q^{79} +7.68466 q^{80} -7.00000 q^{81} -13.1231 q^{82} -4.00000 q^{83} +7.12311 q^{84} +0.438447 q^{85} -2.24621 q^{86} -10.4384 q^{87} -1.12311 q^{89} -1.43845 q^{90} -0.438447 q^{91} +14.2462 q^{92} -22.2462 q^{94} +7.12311 q^{95} +10.2462 q^{96} +5.80776 q^{97} +2.56155 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} + 5q^{4} + 2q^{5} + 8q^{6} + 2q^{7} + 9q^{8} + 3q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} + 5q^{4} + 2q^{5} + 8q^{6} + 2q^{7} + 9q^{8} + 3q^{9} + q^{10} + 6q^{12} - 5q^{13} + q^{14} - q^{15} + 3q^{16} + 5q^{17} - 7q^{18} + 6q^{19} + 5q^{20} - q^{21} - 2q^{23} + 4q^{24} + 2q^{25} + 6q^{26} - 7q^{27} + 5q^{28} - q^{29} + 8q^{30} + 9q^{32} - 6q^{34} + 2q^{35} - q^{36} + 12q^{37} + 20q^{38} + 11q^{39} + 9q^{40} - 2q^{41} + 8q^{42} - 10q^{43} + 3q^{45} + 16q^{46} - 5q^{47} + 24q^{48} + 2q^{49} + q^{50} - 11q^{51} - 4q^{52} - 2q^{53} - 12q^{54} + 9q^{56} + 14q^{57} - 26q^{58} - 8q^{59} + 6q^{60} - 6q^{61} + 3q^{63} + 7q^{64} - 5q^{65} + 4q^{67} + 4q^{68} + 18q^{69} + q^{70} + 16q^{71} + 5q^{72} + 8q^{73} + 6q^{74} - q^{75} + 32q^{76} - 20q^{78} + 9q^{79} + 3q^{80} - 14q^{81} - 18q^{82} - 8q^{83} + 6q^{84} + 5q^{85} + 12q^{86} - 25q^{87} + 6q^{89} - 7q^{90} - 5q^{91} + 12q^{92} - 28q^{94} + 6q^{95} + 4q^{96} - 9q^{97} + q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 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
$$5$$ 1.00000 0.447214
$$6$$ 4.00000 1.63299
$$7$$ 1.00000 0.377964
$$8$$ 6.56155 2.31986
$$9$$ −0.561553 −0.187184
$$10$$ 2.56155 0.810034
$$11$$ 0 0
$$12$$ 7.12311 2.05626
$$13$$ −0.438447 −0.121603 −0.0608017 0.998150i $$-0.519366\pi$$
−0.0608017 + 0.998150i $$0.519366\pi$$
$$14$$ 2.56155 0.684604
$$15$$ 1.56155 0.403191
$$16$$ 7.68466 1.92116
$$17$$ 0.438447 0.106339 0.0531695 0.998586i $$-0.483068\pi$$
0.0531695 + 0.998586i $$0.483068\pi$$
$$18$$ −1.43845 −0.339045
$$19$$ 7.12311 1.63415 0.817076 0.576530i $$-0.195593\pi$$
0.817076 + 0.576530i $$0.195593\pi$$
$$20$$ 4.56155 1.01999
$$21$$ 1.56155 0.340759
$$22$$ 0 0
$$23$$ 3.12311 0.651213 0.325606 0.945505i $$-0.394432\pi$$
0.325606 + 0.945505i $$0.394432\pi$$
$$24$$ 10.2462 2.09150
$$25$$ 1.00000 0.200000
$$26$$ −1.12311 −0.220259
$$27$$ −5.56155 −1.07032
$$28$$ 4.56155 0.862052
$$29$$ −6.68466 −1.24131 −0.620655 0.784084i $$-0.713133\pi$$
−0.620655 + 0.784084i $$0.713133\pi$$
$$30$$ 4.00000 0.730297
$$31$$ 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
$$35$$ 1.00000 0.169031
$$36$$ −2.56155 −0.426925
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 18.2462 2.95993
$$39$$ −0.684658 −0.109633
$$40$$ 6.56155 1.03747
$$41$$ −5.12311 −0.800095 −0.400047 0.916494i $$-0.631006\pi$$
−0.400047 + 0.916494i $$0.631006\pi$$
$$42$$ 4.00000 0.617213
$$43$$ −0.876894 −0.133725 −0.0668626 0.997762i $$-0.521299\pi$$
−0.0668626 + 0.997762i $$0.521299\pi$$
$$44$$ 0 0
$$45$$ −0.561553 −0.0837114
$$46$$ 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
$$49$$ 1.00000 0.142857
$$50$$ 2.56155 0.362258
$$51$$ 0.684658 0.0958714
$$52$$ −2.00000 −0.277350
$$53$$ −5.12311 −0.703713 −0.351856 0.936054i $$-0.614449\pi$$
−0.351856 + 0.936054i $$0.614449\pi$$
$$54$$ −14.2462 −1.93866
$$55$$ 0 0
$$56$$ 6.56155 0.876824
$$57$$ 11.1231 1.47329
$$58$$ −17.1231 −2.24837
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 7.12311 0.919589
$$61$$ −15.3693 −1.96784 −0.983920 0.178611i $$-0.942839\pi$$
−0.983920 + 0.178611i $$0.942839\pi$$
$$62$$ 0 0
$$63$$ −0.561553 −0.0707490
$$64$$ 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
$$69$$ 4.87689 0.587109
$$70$$ 2.56155 0.306164
$$71$$ 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 $$-0.245672\pi$$
0.716655 + 0.697428i $$0.245672\pi$$
$$74$$ 15.3693 1.78665
$$75$$ 1.56155 0.180313
$$76$$ 32.4924 3.72714
$$77$$ 0 0
$$78$$ −1.75379 −0.198577
$$79$$ 2.43845 0.274347 0.137173 0.990547i $$-0.456198\pi$$
0.137173 + 0.990547i $$0.456198\pi$$
$$80$$ 7.68466 0.859171
$$81$$ −7.00000 −0.777778
$$82$$ −13.1231 −1.44920
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 7.12311 0.777195
$$85$$ 0.438447 0.0475563
$$86$$ −2.24621 −0.242215
$$87$$ −10.4384 −1.11912
$$88$$ 0 0
$$89$$ −1.12311 −0.119049 −0.0595245 0.998227i $$-0.518958\pi$$
−0.0595245 + 0.998227i $$0.518958\pi$$
$$90$$ −1.43845 −0.151626
$$91$$ −0.438447 −0.0459618
$$92$$ 14.2462 1.48527
$$93$$ 0 0
$$94$$ −22.2462 −2.29452
$$95$$ 7.12311 0.730815
$$96$$ 10.2462 1.04575
$$97$$ 5.80776 0.589689 0.294845 0.955545i $$-0.404732\pi$$
0.294845 + 0.955545i $$0.404732\pi$$
$$98$$ 2.56155 0.258756
$$99$$ 0 0
$$100$$ 4.56155 0.456155
$$101$$ 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
$$105$$ 1.56155 0.152392
$$106$$ −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 $$-0.581930\pi$$
−0.254559 + 0.967057i $$0.581930\pi$$
$$110$$ 0 0
$$111$$ 9.36932 0.889296
$$112$$ 7.68466 0.726132
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 28.4924 2.66856
$$115$$ 3.12311 0.291231
$$116$$ −30.4924 −2.83115
$$117$$ 0.246211 0.0227622
$$118$$ −10.2462 −0.943240
$$119$$ 0.438447 0.0401924
$$120$$ 10.2462 0.935347
$$121$$ 0 0
$$122$$ −39.3693 −3.56433
$$123$$ −8.00000 −0.721336
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ −1.43845 −0.128147
$$127$$ 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
$$130$$ −1.12311 −0.0985029
$$131$$ 0.876894 0.0766146 0.0383073 0.999266i $$-0.487803\pi$$
0.0383073 + 0.999266i $$0.487803\pi$$
$$132$$ 0 0
$$133$$ 7.12311 0.617652
$$134$$ 26.2462 2.26733
$$135$$ −5.56155 −0.478662
$$136$$ 2.87689 0.246692
$$137$$ −17.1231 −1.46293 −0.731463 0.681881i $$-0.761162\pi$$
−0.731463 + 0.681881i $$0.761162\pi$$
$$138$$ 12.4924 1.06343
$$139$$ 15.1231 1.28273 0.641363 0.767238i $$-0.278369\pi$$
0.641363 + 0.767238i $$0.278369\pi$$
$$140$$ 4.56155 0.385522
$$141$$ −13.5616 −1.14209
$$142$$ 20.4924 1.71969
$$143$$ 0 0
$$144$$ −4.31534 −0.359612
$$145$$ −6.68466 −0.555131
$$146$$ 31.3693 2.59614
$$147$$ 1.56155 0.128795
$$148$$ 27.3693 2.24974
$$149$$ −12.2462 −1.00325 −0.501624 0.865086i $$-0.667264\pi$$
−0.501624 + 0.865086i $$0.667264\pi$$
$$150$$ 4.00000 0.326599
$$151$$ 6.93087 0.564026 0.282013 0.959411i $$-0.408998\pi$$
0.282013 + 0.959411i $$0.408998\pi$$
$$152$$ 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
$$160$$ 6.56155 0.518736
$$161$$ 3.12311 0.246135
$$162$$ −17.9309 −1.40878
$$163$$ −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 $$-0.413583\pi$$
0.268163 + 0.963373i $$0.413583\pi$$
$$168$$ 10.2462 0.790512
$$169$$ −12.8078 −0.985213
$$170$$ 1.12311 0.0861383
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ 4.43845 0.337449 0.168724 0.985663i $$-0.446035\pi$$
0.168724 + 0.985663i $$0.446035\pi$$
$$174$$ −26.7386 −2.02705
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −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 $$0.231285\pi$$
$$180$$ −2.56155 −0.190927
$$181$$ −17.6155 −1.30935 −0.654676 0.755910i $$-0.727195\pi$$
−0.654676 + 0.755910i $$0.727195\pi$$
$$182$$ −1.12311 −0.0832501
$$183$$ −24.0000 −1.77413
$$184$$ 20.4924 1.51072
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −39.6155 −2.88926
$$189$$ −5.56155 −0.404543
$$190$$ 18.2462 1.32372
$$191$$ −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
$$195$$ −0.684658 −0.0490294
$$196$$ 4.56155 0.325825
$$197$$ −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 $$-0.519799\pi$$
−0.0621614 + 0.998066i $$0.519799\pi$$
$$200$$ 6.56155 0.463972
$$201$$ 16.0000 1.12855
$$202$$ 41.6155 2.92806
$$203$$ −6.68466 −0.469171
$$204$$ 3.12311 0.218661
$$205$$ −5.12311 −0.357813
$$206$$ 14.2462 0.992581
$$207$$ −1.75379 −0.121897
$$208$$ −3.36932 −0.233620
$$209$$ 0 0
$$210$$ 4.00000 0.276026
$$211$$ −14.0540 −0.967516 −0.483758 0.875202i $$-0.660728\pi$$
−0.483758 + 0.875202i $$0.660728\pi$$
$$212$$ −23.3693 −1.60501
$$213$$ 12.4924 0.855967
$$214$$ −34.2462 −2.34102
$$215$$ −0.876894 −0.0598037
$$216$$ −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 $$0.526017\pi$$
$$224$$ 6.56155 0.438412
$$225$$ −0.561553 −0.0374369
$$226$$ −35.8617 −2.38549
$$227$$ −11.3153 −0.751026 −0.375513 0.926817i $$-0.622533\pi$$
−0.375513 + 0.926817i $$0.622533\pi$$
$$228$$ 50.7386 3.36025
$$229$$ 10.8769 0.718765 0.359383 0.933190i $$-0.382987\pi$$
0.359383 + 0.933190i $$0.382987\pi$$
$$230$$ 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 $$0.553670\pi$$
$$234$$ 0.630683 0.0412290
$$235$$ −8.68466 −0.566525
$$236$$ −18.2462 −1.18773
$$237$$ 3.80776 0.247341
$$238$$ 1.12311 0.0728001
$$239$$ −19.8078 −1.28126 −0.640629 0.767851i $$-0.721326\pi$$
−0.640629 + 0.767851i $$0.721326\pi$$
$$240$$ 12.0000 0.774597
$$241$$ 4.24621 0.273523 0.136761 0.990604i $$-0.456331\pi$$
0.136761 + 0.990604i $$0.456331\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ −70.1080 −4.48820
$$245$$ 1.00000 0.0638877
$$246$$ −20.4924 −1.30655
$$247$$ −3.12311 −0.198718
$$248$$ 0 0
$$249$$ −6.24621 −0.395838
$$250$$ 2.56155 0.162007
$$251$$ −8.87689 −0.560305 −0.280152 0.959956i $$-0.590385\pi$$
−0.280152 + 0.959956i $$0.590385\pi$$
$$252$$ −2.56155 −0.161363
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ 0.684658 0.0428750
$$256$$ −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
$$259$$ 6.00000 0.372822
$$260$$ −2.00000 −0.124035
$$261$$ 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 $$0.370047\pi$$
$$264$$ 0 0
$$265$$ −5.12311 −0.314710
$$266$$ 18.2462 1.11875
$$267$$ −1.75379 −0.107330
$$268$$ 46.7386 2.85502
$$269$$ −20.7386 −1.26446 −0.632228 0.774782i $$-0.717860\pi$$
−0.632228 + 0.774782i $$0.717860\pi$$
$$270$$ −14.2462 −0.866997
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 3.36932 0.204295
$$273$$ −0.684658 −0.0414374
$$274$$ −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
$$280$$ 6.56155 0.392128
$$281$$ −12.4384 −0.742016 −0.371008 0.928630i $$-0.620988\pi$$
−0.371008 + 0.928630i $$0.620988\pi$$
$$282$$ −34.7386 −2.06866
$$283$$ 11.3153 0.672627 0.336314 0.941750i $$-0.390820\pi$$
0.336314 + 0.941750i $$0.390820\pi$$
$$284$$ 36.4924 2.16543
$$285$$ 11.1231 0.658876
$$286$$ 0 0
$$287$$ −5.12311 −0.302407
$$288$$ −3.68466 −0.217121
$$289$$ −16.8078 −0.988692
$$290$$ −17.1231 −1.00550
$$291$$ 9.06913 0.531642
$$292$$ 55.8617 3.26906
$$293$$ 2.68466 0.156839 0.0784197 0.996920i $$-0.475013\pi$$
0.0784197 + 0.996920i $$0.475013\pi$$
$$294$$ 4.00000 0.233285
$$295$$ −4.00000 −0.232889
$$296$$ 39.3693 2.28830
$$297$$ 0 0
$$298$$ −31.3693 −1.81718
$$299$$ −1.36932 −0.0791896
$$300$$ 7.12311 0.411253
$$301$$ −0.876894 −0.0505434
$$302$$ 17.7538 1.02162
$$303$$ 25.3693 1.45743
$$304$$ 54.7386 3.13948
$$305$$ −15.3693 −0.880045
$$306$$ −0.630683 −0.0360538
$$307$$ 19.3153 1.10238 0.551192 0.834378i $$-0.314173\pi$$
0.551192 + 0.834378i $$0.314173\pi$$
$$308$$ 0 0
$$309$$ 8.68466 0.494053
$$310$$ 0 0
$$311$$ 31.6155 1.79275 0.896376 0.443294i $$-0.146190\pi$$
0.896376 + 0.443294i $$0.146190\pi$$
$$312$$ −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
$$315$$ −0.561553 −0.0316399
$$316$$ 11.1231 0.625724
$$317$$ 10.4924 0.589313 0.294657 0.955603i $$-0.404795\pi$$
0.294657 + 0.955603i $$0.404795\pi$$
$$318$$ −20.4924 −1.14916
$$319$$ 0 0
$$320$$ 1.43845 0.0804116
$$321$$ −20.8769 −1.16523
$$322$$ 8.00000 0.445823
$$323$$ 3.12311 0.173774
$$324$$ −31.9309 −1.77394
$$325$$ −0.438447 −0.0243207
$$326$$ −18.2462 −1.01056
$$327$$ −8.30019 −0.459001
$$328$$ −33.6155 −1.85611
$$329$$ −8.68466 −0.478801
$$330$$ 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
$$335$$ 10.2462 0.559810
$$336$$ 12.0000 0.654654
$$337$$ 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
$$340$$ 2.00000 0.108465
$$341$$ 0 0
$$342$$ −10.2462 −0.554052
$$343$$ 1.00000 0.0539949
$$344$$ −5.75379 −0.310223
$$345$$ 4.87689 0.262563
$$346$$ 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 $$0.590607\pi$$
$$350$$ 2.56155 0.136921
$$351$$ 2.43845 0.130155
$$352$$ 0 0
$$353$$ 5.80776 0.309116 0.154558 0.987984i $$-0.450605\pi$$
0.154558 + 0.987984i $$0.450605\pi$$
$$354$$ −16.0000 −0.850390
$$355$$ 8.00000 0.424596
$$356$$ −5.12311 −0.271524
$$357$$ 0.684658 0.0362360
$$358$$ 51.2311 2.70765
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ −3.68466 −0.194199
$$361$$ 31.7386 1.67045
$$362$$ −45.1231 −2.37162
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 12.2462 0.640996
$$366$$ −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
$$370$$ 15.3693 0.799013
$$371$$ −5.12311 −0.265978
$$372$$ 0 0
$$373$$ −4.63068 −0.239768 −0.119884 0.992788i $$-0.538252\pi$$
−0.119884 + 0.992788i $$0.538252\pi$$
$$374$$ 0 0
$$375$$ 1.56155 0.0806382
$$376$$ −56.9848 −2.93877
$$377$$ 2.93087 0.150947
$$378$$ −14.2462 −0.732746
$$379$$ −16.4924 −0.847159 −0.423579 0.905859i $$-0.639227\pi$$
−0.423579 + 0.905859i $$0.639227\pi$$
$$380$$ 32.4924 1.66683
$$381$$ 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 $$0.717775\pi$$
$$390$$ −1.75379 −0.0888065
$$391$$ 1.36932 0.0692493
$$392$$ 6.56155 0.331408
$$393$$ 1.36932 0.0690729
$$394$$ −2.87689 −0.144936
$$395$$ 2.43845 0.122692
$$396$$ 0 0
$$397$$ 27.5616 1.38327 0.691637 0.722245i $$-0.256890\pi$$
0.691637 + 0.722245i $$0.256890\pi$$
$$398$$ −4.49242 −0.225185
$$399$$ 11.1231 0.556852
$$400$$ 7.68466 0.384233
$$401$$ 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
$$405$$ −7.00000 −0.347833
$$406$$ −17.1231 −0.849805
$$407$$ 0 0
$$408$$ 4.49242 0.222408
$$409$$ −6.49242 −0.321030 −0.160515 0.987033i $$-0.551315\pi$$
−0.160515 + 0.987033i $$0.551315\pi$$
$$410$$ −13.1231 −0.648104
$$411$$ −26.7386 −1.31892
$$412$$ 25.3693 1.24986
$$413$$ −4.00000 −0.196827
$$414$$ −4.49242 −0.220791
$$415$$ −4.00000 −0.196352
$$416$$ −2.87689 −0.141051
$$417$$ 23.6155 1.15646
$$418$$ 0 0
$$419$$ 26.2462 1.28221 0.641106 0.767453i $$-0.278476\pi$$
0.641106 + 0.767453i $$0.278476\pi$$
$$420$$ 7.12311 0.347572
$$421$$ −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
$$425$$ 0.438447 0.0212678
$$426$$ 32.0000 1.55041
$$427$$ −15.3693 −0.743773
$$428$$ −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 $$0.341705\pi$$
$$432$$ −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
$$435$$ −10.4384 −0.500485
$$436$$ −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
$$441$$ −0.561553 −0.0267406
$$442$$ −0.492423 −0.0234221
$$443$$ −2.63068 −0.124988 −0.0624938 0.998045i $$-0.519905\pi$$
−0.0624938 + 0.998045i $$0.519905\pi$$
$$444$$ 42.7386 2.02829
$$445$$ −1.12311 −0.0532403
$$446$$ −6.24621 −0.295767
$$447$$ −19.1231 −0.904492
$$448$$ 1.43845 0.0679602
$$449$$ −1.80776 −0.0853137 −0.0426568 0.999090i $$-0.513582\pi$$
−0.0426568 + 0.999090i $$0.513582\pi$$
$$450$$ −1.43845 −0.0678091
$$451$$ 0 0
$$452$$ −63.8617 −3.00380
$$453$$ 10.8229 0.508505
$$454$$ −28.9848 −1.36033
$$455$$ −0.438447 −0.0205547
$$456$$ 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
$$460$$ 14.2462 0.664233
$$461$$ 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 $$-0.406250\pi$$
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 $$0.326243\pi$$
$$468$$ 1.12311 0.0519156
$$469$$ 10.2462 0.473126
$$470$$ −22.2462 −1.02614
$$471$$ 31.6155 1.45677
$$472$$ −26.2462 −1.20808
$$473$$ 0 0
$$474$$ 9.75379 0.448006
$$475$$ 7.12311 0.326831
$$476$$ 2.00000 0.0916698
$$477$$ 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 $$0.535538\pi$$
$$480$$ 10.2462 0.467673
$$481$$ −2.63068 −0.119949
$$482$$ 10.8769 0.495429
$$483$$ 4.87689 0.221906
$$484$$ 0 0
$$485$$ 5.80776 0.263717
$$486$$ 14.7386 0.668558
$$487$$ −3.12311 −0.141521 −0.0707607 0.997493i $$-0.522543\pi$$
−0.0707607 + 0.997493i $$0.522543\pi$$
$$488$$ −100.847 −4.56511
$$489$$ −11.1231 −0.503004
$$490$$ 2.56155 0.115719
$$491$$ 41.1771 1.85830 0.929148 0.369708i $$-0.120542\pi$$
0.929148 + 0.369708i $$0.120542\pi$$
$$492$$ −36.4924 −1.64521
$$493$$ −2.93087 −0.132000
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000 0.358849
$$498$$ −16.0000 −0.716977
$$499$$ 41.1771 1.84334 0.921670 0.387976i $$-0.126826\pi$$
0.921670 + 0.387976i $$0.126826\pi$$
$$500$$ 4.56155 0.203999
$$501$$ 10.8229 0.483532
$$502$$ −22.7386 −1.01487
$$503$$ −38.9309 −1.73584 −0.867921 0.496703i $$-0.834544\pi$$
−0.867921 + 0.496703i $$0.834544\pi$$
$$504$$ −3.68466 −0.164128
$$505$$ 16.2462 0.722947
$$506$$ 0 0
$$507$$ −20.0000 −0.888231
$$508$$ 28.4924 1.26415
$$509$$ −11.7538 −0.520978 −0.260489 0.965477i $$-0.583884\pi$$
−0.260489 + 0.965477i $$0.583884\pi$$
$$510$$ 1.75379 0.0776591
$$511$$ 12.2462 0.541740
$$512$$ −50.4233 −2.22842
$$513$$ −39.6155 −1.74907
$$514$$ −26.8769 −1.18549
$$515$$ 5.56155 0.245071
$$516$$ −6.24621 −0.274974
$$517$$ 0 0
$$518$$ 15.3693 0.675289
$$519$$ 6.93087 0.304231
$$520$$ −2.87689 −0.126160
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 9.61553 0.420860
$$523$$ −40.4924 −1.77061 −0.885305 0.465011i $$-0.846050\pi$$
−0.885305 + 0.465011i $$0.846050\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 1.56155 0.0681518
$$526$$ 32.9848 1.43821
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.2462 −0.575922
$$530$$ −13.1231 −0.570031
$$531$$ 2.24621 0.0974773
$$532$$ 32.4924 1.40873
$$533$$ 2.24621 0.0972942
$$534$$ −4.49242 −0.194406
$$535$$ −13.3693 −0.578006
$$536$$ 67.2311 2.90394
$$537$$ 31.2311 1.34772
$$538$$ −53.1231 −2.29030
$$539$$ 0 0
$$540$$ −25.3693 −1.09172
$$541$$ 37.8078 1.62548 0.812741 0.582625i $$-0.197974\pi$$
0.812741 + 0.582625i $$0.197974\pi$$
$$542$$ 40.9848 1.76045
$$543$$ −27.5076 −1.18046
$$544$$ 2.87689 0.123346
$$545$$ −5.31534 −0.227684
$$546$$ −1.75379 −0.0750552
$$547$$ 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
$$553$$ 2.43845 0.103693
$$554$$ 0.630683 0.0267952
$$555$$ 9.36932 0.397705
$$556$$ 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
$$560$$ 7.68466 0.324736
$$561$$ 0 0
$$562$$ −31.8617 −1.34401
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ −61.8617 −2.60485
$$565$$ −14.0000 −0.588984
$$566$$ 28.9848 1.21832
$$567$$ −7.00000 −0.293972
$$568$$ 52.4924 2.20253
$$569$$ 30.9848 1.29895 0.649476 0.760382i $$-0.274988\pi$$
0.649476 + 0.760382i $$0.274988\pi$$
$$570$$ 28.4924 1.19342
$$571$$ −40.4924 −1.69456 −0.847278 0.531150i $$-0.821760\pi$$
−0.847278 + 0.531150i $$0.821760\pi$$
$$572$$ 0 0
$$573$$ −21.1771 −0.884685
$$574$$ −13.1231 −0.547748
$$575$$ 3.12311 0.130243
$$576$$ −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
$$580$$ −30.4924 −1.26613
$$581$$ −4.00000 −0.165948
$$582$$ 23.2311 0.962958
$$583$$ 0 0
$$584$$ 80.3542 3.32508
$$585$$ 0.246211 0.0101796
$$586$$ 6.87689 0.284082
$$587$$ −26.2462 −1.08330 −0.541649 0.840605i $$-0.682200\pi$$
−0.541649 + 0.840605i $$0.682200\pi$$
$$588$$ 7.12311 0.293752
$$589$$ 0 0
$$590$$ −10.2462 −0.421830
$$591$$ −1.75379 −0.0721412
$$592$$ 46.1080 1.89503
$$593$$ 27.5616 1.13182 0.565909 0.824468i $$-0.308525\pi$$
0.565909 + 0.824468i $$0.308525\pi$$
$$594$$ 0 0
$$595$$ 0.438447 0.0179746
$$596$$ −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 $$0.577550\pi$$
$$600$$ 10.2462 0.418300
$$601$$ −6.49242 −0.264831 −0.132416 0.991194i $$-0.542273\pi$$
−0.132416 + 0.991194i $$0.542273\pi$$
$$602$$ −2.24621 −0.0915487
$$603$$ −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 $$-0.174500\pi$$
0.853459 + 0.521160i $$0.174500\pi$$
$$608$$ 46.7386 1.89550
$$609$$ −10.4384 −0.422987
$$610$$ −39.3693 −1.59402
$$611$$ 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
$$615$$ −8.00000 −0.322591
$$616$$ 0 0
$$617$$ 32.2462 1.29818 0.649092 0.760710i $$-0.275149\pi$$
0.649092 + 0.760710i $$0.275149\pi$$
$$618$$ 22.2462 0.894874
$$619$$ 32.1080 1.29053 0.645264 0.763960i $$-0.276747\pi$$
0.645264 + 0.763960i $$0.276747\pi$$
$$620$$ 0 0
$$621$$ −17.3693 −0.697007
$$622$$ 80.9848 3.24720
$$623$$ −1.12311 −0.0449963
$$624$$ −5.26137 −0.210623
$$625$$ 1.00000 0.0400000
$$626$$ −57.1231 −2.28310
$$627$$ 0 0
$$628$$ 92.3542 3.68533
$$629$$ 2.63068 0.104892
$$630$$ −1.43845 −0.0573091
$$631$$ −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
$$635$$ 6.24621 0.247873
$$636$$ −36.4924 −1.44702
$$637$$ −0.438447 −0.0173719
$$638$$ 0 0
$$639$$ −4.49242 −0.177717
$$640$$ −9.43845 −0.373087
$$641$$ 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 $$0.490197\pi$$
$$644$$ 14.2462 0.561379
$$645$$ −1.36932 −0.0539168
$$646$$ 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
$$649$$ 0 0
$$650$$ −1.12311 −0.0440518
$$651$$ 0 0
$$652$$ −32.4924 −1.27250
$$653$$ −33.2311 −1.30043 −0.650216 0.759750i $$-0.725322\pi$$
−0.650216 + 0.759750i $$0.725322\pi$$
$$654$$ −21.2614 −0.831385
$$655$$ 0.876894 0.0342631
$$656$$ −39.3693 −1.53711
$$657$$ −6.87689 −0.268293
$$658$$ −22.2462 −0.867248
$$659$$ −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 $$-0.531767\pi$$
−0.0996329 + 0.995024i $$0.531767\pi$$
$$662$$ 30.7386 1.19469
$$663$$ −0.300187 −0.0116583
$$664$$ −26.2462 −1.01855
$$665$$ 7.12311 0.276222
$$666$$ −8.63068 −0.334432
$$667$$ −20.8769 −0.808357
$$668$$ 31.6155 1.22324
$$669$$ −3.80776 −0.147217
$$670$$ 26.2462 1.01398
$$671$$ 0 0
$$672$$ 10.2462 0.395256
$$673$$ −31.8617 −1.22818 −0.614090 0.789236i $$-0.710477\pi$$
−0.614090 + 0.789236i $$0.710477\pi$$
$$674$$ 3.86174 0.148749
$$675$$ −5.56155 −0.214064
$$676$$ −58.4233 −2.24705
$$677$$ −4.93087 −0.189509 −0.0947544 0.995501i $$-0.530207\pi$$
−0.0947544 + 0.995501i $$0.530207\pi$$
$$678$$ −56.0000 −2.15067
$$679$$ 5.80776 0.222882
$$680$$ 2.87689 0.110324
$$681$$ −17.6695 −0.677097
$$682$$ 0 0
$$683$$ −6.73863 −0.257847 −0.128923 0.991655i $$-0.541152\pi$$
−0.128923 + 0.991655i $$0.541152\pi$$
$$684$$ −18.2462 −0.697661
$$685$$ −17.1231 −0.654240
$$686$$ 2.56155 0.0978005
$$687$$ 16.9848 0.648012
$$688$$ −6.73863 −0.256908
$$689$$ 2.24621 0.0855738
$$690$$ 12.4924 0.475578
$$691$$ −24.4924 −0.931736 −0.465868 0.884854i $$-0.654258\pi$$
−0.465868 + 0.884854i $$0.654258\pi$$
$$692$$ 20.2462 0.769645
$$693$$ 0 0
$$694$$ −18.2462 −0.692617
$$695$$ 15.1231 0.573652
$$696$$ −68.4924 −2.59620
$$697$$ −2.24621 −0.0850813
$$698$$ −26.8769 −1.01731
$$699$$ −8.00000 −0.302588
$$700$$ 4.56155 0.172410
$$701$$ −28.9309 −1.09270 −0.546352 0.837556i $$-0.683984\pi$$
−0.546352 + 0.837556i $$0.683984\pi$$
$$702$$ 6.24621 0.235748
$$703$$ 42.7386 1.61192
$$704$$ 0 0
$$705$$ −13.5616 −0.510758
$$706$$ 14.8769 0.559899
$$707$$ 16.2462 0.611002
$$708$$ −28.4924 −1.07081
$$709$$ 27.1771 1.02066 0.510328 0.859980i $$-0.329524\pi$$
0.510328 + 0.859980i $$0.329524\pi$$
$$710$$ 20.4924 0.769067
$$711$$ −1.36932 −0.0513534
$$712$$ −7.36932 −0.276177
$$713$$ 0 0
$$714$$ 1.75379 0.0656339
$$715$$ 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 $$-0.450029\pi$$
0.156344 + 0.987703i $$0.450029\pi$$
$$720$$ −4.31534 −0.160823
$$721$$ 5.56155 0.207123
$$722$$ 81.3002 3.02568
$$723$$ 6.63068 0.246598
$$724$$ −80.3542 −2.98634
$$725$$ −6.68466 −0.248262
$$726$$ 0 0
$$727$$ 52.4924 1.94684 0.973418 0.229035i $$-0.0735572\pi$$
0.973418 + 0.229035i $$0.0735572\pi$$
$$728$$ −2.87689 −0.106625
$$729$$ 29.9848 1.11055
$$730$$ 31.3693 1.16103
$$731$$ −0.384472 −0.0142202
$$732$$ −109.477 −4.04640
$$733$$ −6.68466 −0.246903 −0.123452 0.992351i $$-0.539396\pi$$
−0.123452 + 0.992351i $$0.539396\pi$$
$$734$$ −22.2462 −0.821123
$$735$$ 1.56155 0.0575987
$$736$$ 20.4924 0.755361
$$737$$ 0 0
$$738$$ 7.36932 0.271268
$$739$$ −34.9309 −1.28495 −0.642476 0.766305i $$-0.722093\pi$$
−0.642476 + 0.766305i $$0.722093\pi$$
$$740$$ 27.3693 1.00612
$$741$$ −4.87689 −0.179157
$$742$$ −13.1231 −0.481764
$$743$$ 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
$$747$$ 2.24621 0.0821846
$$748$$ 0 0
$$749$$ −13.3693 −0.488504
$$750$$ 4.00000 0.146059
$$751$$ 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
$$755$$ 6.93087 0.252240
$$756$$ −25.3693 −0.922673
$$757$$ 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 $$-0.838785\pi$$
−0.874462 + 0.485094i $$0.838785\pi$$
$$762$$ 24.9848 0.905105
$$763$$ −5.31534 −0.192428
$$764$$ −61.8617 −2.23808
$$765$$ −0.246211 −0.00890179
$$766$$ 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
$$777$$ 9.36932 0.336122
$$778$$ −63.8617 −2.28955
$$779$$ −36.4924 −1.30748
$$780$$ −3.12311 −0.111825
$$781$$ 0 0
$$782$$ 3.50758 0.125431
$$783$$ 37.1771 1.32860
$$784$$ 7.68466 0.274452
$$785$$ 20.2462 0.722618
$$786$$ 3.50758 0.125111
$$787$$ 49.1771 1.75297 0.876487 0.481426i $$-0.159881\pi$$
0.876487 + 0.481426i $$0.159881\pi$$
$$788$$ −5.12311 −0.182503
$$789$$ 20.1080 0.715862
$$790$$ 6.24621 0.222230
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ 6.73863 0.239296
$$794$$ 70.6004 2.50551
$$795$$ −8.00000 −0.283731
$$796$$ −8.00000 −0.283552
$$797$$ 24.0540 0.852036 0.426018 0.904715i $$-0.359916\pi$$
0.426018 + 0.904715i $$0.359916\pi$$
$$798$$ 28.4924 1.00862
$$799$$ −3.80776 −0.134709
$$800$$ 6.56155 0.231986
$$801$$ 0.630683 0.0222841
$$802$$ 80.8466 2.85479
$$803$$ 0 0
$$804$$ 72.9848 2.57398
$$805$$ 3.12311 0.110075
$$806$$ 0 0
$$807$$ −32.3845 −1.13999
$$808$$ 106.600 3.75019
$$809$$ −16.5464 −0.581740 −0.290870 0.956763i $$-0.593945\pi$$
−0.290870 + 0.956763i $$0.593945\pi$$
$$810$$ −17.9309 −0.630027
$$811$$ −19.6155 −0.688794 −0.344397 0.938824i $$-0.611917\pi$$
−0.344397 + 0.938824i $$0.611917\pi$$
$$812$$ −30.4924 −1.07007
$$813$$ 24.9848 0.876257
$$814$$ 0 0
$$815$$ −7.12311 −0.249512
$$816$$ 5.26137 0.184185
$$817$$ −6.24621 −0.218527
$$818$$ −16.6307 −0.581478
$$819$$ 0.246211 0.00860332
$$820$$ −23.3693 −0.816092
$$821$$ 21.4233 0.747678 0.373839 0.927494i $$-0.378041\pi$$
0.373839 + 0.927494i $$0.378041\pi$$
$$822$$ −68.4924 −2.38895
$$823$$ −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
$$826$$ −10.2462 −0.356511
$$827$$ 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 $$-0.292908\pi$$
0.605662 + 0.795722i $$0.292908\pi$$
$$830$$ −10.2462 −0.355651
$$831$$ 0.384472 0.0133372
$$832$$ −0.630683 −0.0218650
$$833$$ 0.438447 0.0151913
$$834$$ 60.4924 2.09468
$$835$$ 6.93087 0.239853
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 67.2311 2.32246
$$839$$ −28.8769 −0.996941 −0.498471 0.866907i $$-0.666105\pi$$
−0.498471 + 0.866907i $$0.666105\pi$$
$$840$$ 10.2462 0.353528
$$841$$ 15.6847 0.540850
$$842$$ −6.87689 −0.236993
$$843$$ −19.4233 −0.668974
$$844$$ −64.1080 −2.20669
$$845$$ −12.8078 −0.440600
$$846$$ 12.4924 0.429498
$$847$$ 0 0
$$848$$ −39.3693 −1.35195
$$849$$ 17.6695 0.606416
$$850$$ 1.12311 0.0385222
$$851$$ 18.7386 0.642352
$$852$$ 56.9848 1.95227
$$853$$ 7.26137 0.248624 0.124312 0.992243i $$-0.460328\pi$$
0.124312 + 0.992243i $$0.460328\pi$$
$$854$$ −39.3693 −1.34719
$$855$$ −4.00000 −0.136797
$$856$$ −87.7235 −2.99833
$$857$$ 15.7538 0.538139 0.269070 0.963121i $$-0.413284\pi$$
0.269070 + 0.963121i $$0.413284\pi$$
$$858$$ 0 0
$$859$$ −16.4924 −0.562714 −0.281357 0.959603i $$-0.590785\pi$$
−0.281357 + 0.959603i $$0.590785\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ −8.00000 −0.272639
$$862$$ 50.7386 1.72816
$$863$$ −25.7538 −0.876669 −0.438335 0.898812i $$-0.644431\pi$$
−0.438335 + 0.898812i $$0.644431\pi$$
$$864$$ −36.4924 −1.24150
$$865$$ 4.43845 0.150912
$$866$$ 21.1231 0.717792
$$867$$ −26.2462 −0.891368
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −26.7386 −0.906525
$$871$$ −4.49242 −0.152220
$$872$$ −34.8769 −1.18108
$$873$$ −3.26137 −0.110381
$$874$$ 56.9848 1.92754
$$875$$ 1.00000 0.0338062
$$876$$ 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 $$-0.564034\pi$$
−0.199816 + 0.979833i $$0.564034\pi$$
$$882$$ −1.43845 −0.0484350
$$883$$ −8.49242 −0.285793 −0.142896 0.989738i $$-0.545642\pi$$
−0.142896 + 0.989738i $$0.545642\pi$$
$$884$$ −0.876894 −0.0294931
$$885$$ −6.24621 −0.209964
$$886$$ −6.73863 −0.226389
$$887$$ −20.4924 −0.688068 −0.344034 0.938957i $$-0.611794\pi$$
−0.344034 + 0.938957i $$0.611794\pi$$
$$888$$ 61.4773 2.06304
$$889$$ 6.24621 0.209491
$$890$$ −2.87689 −0.0964337
$$891$$ 0 0
$$892$$ −11.1231 −0.372429
$$893$$ −61.8617 −2.07012
$$894$$ −48.9848 −1.63830
$$895$$ 20.0000 0.668526
$$896$$ −9.43845 −0.315316
$$897$$ −2.13826 −0.0713944
$$898$$ −4.63068 −0.154528
$$899$$ 0 0
$$900$$ −2.56155 −0.0853851
$$901$$ −2.24621 −0.0748321
$$902$$ 0 0
$$903$$ −1.36932 −0.0455680
$$904$$ −91.8617 −3.05528
$$905$$ −17.6155 −0.585560
$$906$$ 27.7235 0.921051
$$907$$ −24.1080 −0.800491 −0.400246 0.916408i $$-0.631075\pi$$
−0.400246 + 0.916408i $$0.631075\pi$$
$$908$$ −51.6155 −1.71292
$$909$$ −9.12311 −0.302594
$$910$$ −1.12311 −0.0372306
$$911$$ −28.4924 −0.943996 −0.471998 0.881600i $$-0.656467\pi$$
−0.471998 + 0.881600i $$0.656467\pi$$
$$912$$ 85.4773 2.83044
$$913$$ 0 0
$$914$$ 43.8617 1.45082
$$915$$ −24.0000 −0.793416
$$916$$ 49.6155 1.63934
$$917$$ 0.876894 0.0289576
$$918$$ −6.24621 −0.206156
$$919$$ −40.3002 −1.32938 −0.664690 0.747119i $$-0.731436\pi$$
−0.664690 + 0.747119i $$0.731436\pi$$
$$920$$ 20.4924 0.675615
$$921$$ 30.1619 0.993869
$$922$$ 33.6155 1.10707
$$923$$ −3.50758 −0.115453
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ 32.0000 1.05159
$$927$$ −3.12311 −0.102576
$$928$$ −43.8617 −1.43983
$$929$$ 22.1080 0.725338 0.362669 0.931918i $$-0.381866\pi$$
0.362669 + 0.931918i $$0.381866\pi$$
$$930$$ 0 0
$$931$$ 7.12311 0.233450
$$932$$ −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 $$-0.136601\pi$$
0.909322 + 0.416094i $$0.136601\pi$$
$$938$$ 26.2462 0.856969
$$939$$ −34.8229 −1.13640
$$940$$ −39.6155 −1.29212
$$941$$ −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
$$945$$ −5.56155 −0.180917
$$946$$ 0 0
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 17.3693 0.564129
$$949$$ −5.36932 −0.174295
$$950$$ 18.2462 0.591985
$$951$$ 16.3845 0.531303
$$952$$ 2.87689 0.0932407
$$953$$ 33.1231 1.07296 0.536481 0.843912i $$-0.319753\pi$$
0.536481 + 0.843912i $$0.319753\pi$$
$$954$$ 7.36932 0.238590
$$955$$ −13.5616 −0.438842
$$956$$ −90.3542 −2.92226
$$957$$ 0 0
$$958$$ −12.4924 −0.403612
$$959$$ −17.1231 −0.552934
$$960$$ 2.24621 0.0724962
$$961$$ −31.0000 −1.00000
$$962$$ −6.73863 −0.217262
$$963$$ 7.50758 0.241928
$$964$$ 19.3693 0.623844
$$965$$ −19.3693 −0.623520
$$966$$ 12.4924 0.401937
$$967$$ 35.1231 1.12948 0.564741 0.825268i $$-0.308976\pi$$
0.564741 + 0.825268i $$0.308976\pi$$
$$968$$ 0 0
$$969$$ 4.87689 0.156668
$$970$$ 14.8769 0.477668
$$971$$ 49.4773 1.58780 0.793901 0.608048i $$-0.208047\pi$$
0.793901 + 0.608048i $$0.208047\pi$$
$$972$$ 26.2462 0.841848
$$973$$ 15.1231 0.484825
$$974$$ −8.00000 −0.256337
$$975$$ −0.684658 −0.0219266
$$976$$ −118.108 −3.78054
$$977$$ 33.2311 1.06316 0.531578 0.847009i $$-0.321599\pi$$
0.531578 + 0.847009i $$0.321599\pi$$
$$978$$ −28.4924 −0.911087
$$979$$ 0 0
$$980$$ 4.56155 0.145713
$$981$$ 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
$$985$$ −1.12311 −0.0357851
$$986$$ −7.50758 −0.239090
$$987$$ −13.5616 −0.431669
$$988$$ −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
$$994$$ 20.4924 0.649980
$$995$$ −1.75379 −0.0555988
$$996$$ −28.4924 −0.902817
$$997$$ 2.68466 0.0850240 0.0425120 0.999096i $$-0.486464\pi$$
0.0425120 + 0.999096i $$0.486464\pi$$
$$998$$ 105.477 3.33882
$$999$$ −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 4235.2.a.m.1.2 2
11.10 odd 2 35.2.a.b.1.1 2
33.32 even 2 315.2.a.e.1.2 2
44.43 even 2 560.2.a.i.1.1 2
55.32 even 4 175.2.b.b.99.1 4
55.43 even 4 175.2.b.b.99.4 4
55.54 odd 2 175.2.a.f.1.2 2
77.10 even 6 245.2.e.h.226.2 4
77.32 odd 6 245.2.e.i.226.2 4
77.54 even 6 245.2.e.h.116.2 4
77.65 odd 6 245.2.e.i.116.2 4
77.76 even 2 245.2.a.d.1.1 2
88.21 odd 2 2240.2.a.bh.1.1 2
88.43 even 2 2240.2.a.bd.1.2 2
132.131 odd 2 5040.2.a.bt.1.1 2
143.142 odd 2 5915.2.a.l.1.2 2
165.32 odd 4 1575.2.d.e.1324.4 4
165.98 odd 4 1575.2.d.e.1324.1 4
165.164 even 2 1575.2.a.p.1.1 2
220.43 odd 4 2800.2.g.t.449.2 4
220.87 odd 4 2800.2.g.t.449.3 4
220.219 even 2 2800.2.a.bi.1.2 2
231.230 odd 2 2205.2.a.x.1.2 2
308.307 odd 2 3920.2.a.bs.1.2 2
385.153 odd 4 1225.2.b.f.99.4 4
385.307 odd 4 1225.2.b.f.99.1 4
385.384 even 2 1225.2.a.s.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 11.10 odd 2
175.2.a.f.1.2 2 55.54 odd 2
175.2.b.b.99.1 4 55.32 even 4
175.2.b.b.99.4 4 55.43 even 4
245.2.a.d.1.1 2 77.76 even 2
245.2.e.h.116.2 4 77.54 even 6
245.2.e.h.226.2 4 77.10 even 6
245.2.e.i.116.2 4 77.65 odd 6
245.2.e.i.226.2 4 77.32 odd 6
315.2.a.e.1.2 2 33.32 even 2
560.2.a.i.1.1 2 44.43 even 2
1225.2.a.s.1.2 2 385.384 even 2
1225.2.b.f.99.1 4 385.307 odd 4
1225.2.b.f.99.4 4 385.153 odd 4
1575.2.a.p.1.1 2 165.164 even 2
1575.2.d.e.1324.1 4 165.98 odd 4
1575.2.d.e.1324.4 4 165.32 odd 4
2205.2.a.x.1.2 2 231.230 odd 2
2240.2.a.bd.1.2 2 88.43 even 2
2240.2.a.bh.1.1 2 88.21 odd 2
2800.2.a.bi.1.2 2 220.219 even 2
2800.2.g.t.449.2 4 220.43 odd 4
2800.2.g.t.449.3 4 220.87 odd 4
3920.2.a.bs.1.2 2 308.307 odd 2
4235.2.a.m.1.2 2 1.1 even 1 trivial
5040.2.a.bt.1.1 2 132.131 odd 2
5915.2.a.l.1.2 2 143.142 odd 2