# Properties

 Label 8470.2.a.cz.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.10784448.1 Defining polynomial: $$x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.0816388$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.34292 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.19656 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.34292 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.19656 q^{9} -1.00000 q^{10} -1.34292 q^{12} -4.31157 q^{13} +1.00000 q^{14} -1.34292 q^{15} +1.00000 q^{16} +2.63565 q^{17} +1.19656 q^{18} -1.56877 q^{19} +1.00000 q^{20} +1.34292 q^{21} +5.45794 q^{23} +1.34292 q^{24} +1.00000 q^{25} +4.31157 q^{26} +5.63565 q^{27} -1.00000 q^{28} -2.83920 q^{29} +1.34292 q^{30} +1.14637 q^{31} -1.00000 q^{32} -2.63565 q^{34} -1.00000 q^{35} -1.19656 q^{36} -0.839198 q^{37} +1.56877 q^{38} +5.79011 q^{39} -1.00000 q^{40} -3.47854 q^{41} -1.34292 q^{42} +2.01075 q^{43} -1.19656 q^{45} -5.45794 q^{46} -8.39442 q^{47} -1.34292 q^{48} +1.00000 q^{49} -1.00000 q^{50} -3.53948 q^{51} -4.31157 q^{52} +1.40604 q^{53} -5.63565 q^{54} +1.00000 q^{56} +2.10674 q^{57} +2.83920 q^{58} +2.32408 q^{59} -1.34292 q^{60} +5.24387 q^{61} -1.14637 q^{62} +1.19656 q^{63} +1.00000 q^{64} -4.31157 q^{65} -1.60636 q^{67} +2.63565 q^{68} -7.32959 q^{69} +1.00000 q^{70} -13.8057 q^{71} +1.19656 q^{72} +1.14830 q^{73} +0.839198 q^{74} -1.34292 q^{75} -1.56877 q^{76} -5.79011 q^{78} -9.65408 q^{79} +1.00000 q^{80} -3.97858 q^{81} +3.47854 q^{82} +2.51880 q^{83} +1.34292 q^{84} +2.63565 q^{85} -2.01075 q^{86} +3.81282 q^{87} +2.19326 q^{89} +1.19656 q^{90} +4.31157 q^{91} +5.45794 q^{92} -1.53948 q^{93} +8.39442 q^{94} -1.56877 q^{95} +1.34292 q^{96} -7.56372 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} - 6 q^{8} + 2 q^{9} + O(q^{10})$$ $$6 q - 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} - 6 q^{8} + 2 q^{9} - 6 q^{10} + 4 q^{12} + 6 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} - 2 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{23} - 4 q^{24} + 6 q^{25} + 16 q^{27} - 6 q^{28} - 8 q^{29} - 4 q^{30} + 4 q^{31} - 6 q^{32} + 2 q^{34} - 6 q^{35} + 2 q^{36} + 4 q^{37} - 6 q^{40} - 12 q^{41} + 4 q^{42} + 6 q^{43} + 2 q^{45} - 4 q^{46} + 16 q^{47} + 4 q^{48} + 6 q^{49} - 6 q^{50} + 12 q^{53} - 16 q^{54} + 6 q^{56} - 8 q^{57} + 8 q^{58} + 22 q^{59} + 4 q^{60} + 4 q^{61} - 4 q^{62} - 2 q^{63} + 6 q^{64} + 20 q^{67} - 2 q^{68} + 12 q^{69} + 6 q^{70} + 14 q^{71} - 2 q^{72} - 18 q^{73} - 4 q^{74} + 4 q^{75} - 32 q^{79} + 6 q^{80} + 6 q^{81} + 12 q^{82} + 16 q^{83} - 4 q^{84} - 2 q^{85} - 6 q^{86} + 4 q^{87} + 4 q^{89} - 2 q^{90} + 4 q^{92} + 12 q^{93} - 16 q^{94} - 4 q^{96} + 4 q^{97} - 6 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.34292 −0.775337 −0.387669 0.921799i $$-0.626719\pi$$
−0.387669 + 0.921799i $$0.626719\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.34292 0.548246
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −1.19656 −0.398853
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ −1.34292 −0.387669
$$13$$ −4.31157 −1.19582 −0.597908 0.801565i $$-0.704001\pi$$
−0.597908 + 0.801565i $$0.704001\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −1.34292 −0.346741
$$16$$ 1.00000 0.250000
$$17$$ 2.63565 0.639240 0.319620 0.947546i $$-0.396445\pi$$
0.319620 + 0.947546i $$0.396445\pi$$
$$18$$ 1.19656 0.282031
$$19$$ −1.56877 −0.359901 −0.179951 0.983676i $$-0.557594\pi$$
−0.179951 + 0.983676i $$0.557594\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 1.34292 0.293050
$$22$$ 0 0
$$23$$ 5.45794 1.13806 0.569030 0.822317i $$-0.307319\pi$$
0.569030 + 0.822317i $$0.307319\pi$$
$$24$$ 1.34292 0.274123
$$25$$ 1.00000 0.200000
$$26$$ 4.31157 0.845569
$$27$$ 5.63565 1.08458
$$28$$ −1.00000 −0.188982
$$29$$ −2.83920 −0.527226 −0.263613 0.964629i $$-0.584914\pi$$
−0.263613 + 0.964629i $$0.584914\pi$$
$$30$$ 1.34292 0.245183
$$31$$ 1.14637 0.205893 0.102947 0.994687i $$-0.467173\pi$$
0.102947 + 0.994687i $$0.467173\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −2.63565 −0.452011
$$35$$ −1.00000 −0.169031
$$36$$ −1.19656 −0.199426
$$37$$ −0.839198 −0.137963 −0.0689816 0.997618i $$-0.521975\pi$$
−0.0689816 + 0.997618i $$0.521975\pi$$
$$38$$ 1.56877 0.254489
$$39$$ 5.79011 0.927160
$$40$$ −1.00000 −0.158114
$$41$$ −3.47854 −0.543256 −0.271628 0.962402i $$-0.587562\pi$$
−0.271628 + 0.962402i $$0.587562\pi$$
$$42$$ −1.34292 −0.207218
$$43$$ 2.01075 0.306637 0.153318 0.988177i $$-0.451004\pi$$
0.153318 + 0.988177i $$0.451004\pi$$
$$44$$ 0 0
$$45$$ −1.19656 −0.178372
$$46$$ −5.45794 −0.804729
$$47$$ −8.39442 −1.22445 −0.612226 0.790683i $$-0.709726\pi$$
−0.612226 + 0.790683i $$0.709726\pi$$
$$48$$ −1.34292 −0.193834
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ −3.53948 −0.495626
$$52$$ −4.31157 −0.597908
$$53$$ 1.40604 0.193134 0.0965672 0.995326i $$-0.469214\pi$$
0.0965672 + 0.995326i $$0.469214\pi$$
$$54$$ −5.63565 −0.766915
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 2.10674 0.279045
$$58$$ 2.83920 0.372805
$$59$$ 2.32408 0.302569 0.151285 0.988490i $$-0.451659\pi$$
0.151285 + 0.988490i $$0.451659\pi$$
$$60$$ −1.34292 −0.173371
$$61$$ 5.24387 0.671409 0.335704 0.941967i $$-0.391026\pi$$
0.335704 + 0.941967i $$0.391026\pi$$
$$62$$ −1.14637 −0.145589
$$63$$ 1.19656 0.150752
$$64$$ 1.00000 0.125000
$$65$$ −4.31157 −0.534785
$$66$$ 0 0
$$67$$ −1.60636 −0.196248 −0.0981241 0.995174i $$-0.531284\pi$$
−0.0981241 + 0.995174i $$0.531284\pi$$
$$68$$ 2.63565 0.319620
$$69$$ −7.32959 −0.882379
$$70$$ 1.00000 0.119523
$$71$$ −13.8057 −1.63844 −0.819220 0.573480i $$-0.805593\pi$$
−0.819220 + 0.573480i $$0.805593\pi$$
$$72$$ 1.19656 0.141016
$$73$$ 1.14830 0.134398 0.0671990 0.997740i $$-0.478594\pi$$
0.0671990 + 0.997740i $$0.478594\pi$$
$$74$$ 0.839198 0.0975548
$$75$$ −1.34292 −0.155067
$$76$$ −1.56877 −0.179951
$$77$$ 0 0
$$78$$ −5.79011 −0.655601
$$79$$ −9.65408 −1.08617 −0.543084 0.839678i $$-0.682744\pi$$
−0.543084 + 0.839678i $$0.682744\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −3.97858 −0.442064
$$82$$ 3.47854 0.384140
$$83$$ 2.51880 0.276475 0.138237 0.990399i $$-0.455856\pi$$
0.138237 + 0.990399i $$0.455856\pi$$
$$84$$ 1.34292 0.146525
$$85$$ 2.63565 0.285877
$$86$$ −2.01075 −0.216825
$$87$$ 3.81282 0.408778
$$88$$ 0 0
$$89$$ 2.19326 0.232485 0.116242 0.993221i $$-0.462915\pi$$
0.116242 + 0.993221i $$0.462915\pi$$
$$90$$ 1.19656 0.126128
$$91$$ 4.31157 0.451976
$$92$$ 5.45794 0.569030
$$93$$ −1.53948 −0.159637
$$94$$ 8.39442 0.865818
$$95$$ −1.56877 −0.160953
$$96$$ 1.34292 0.137062
$$97$$ −7.56372 −0.767979 −0.383990 0.923337i $$-0.625450\pi$$
−0.383990 + 0.923337i $$0.625450\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −17.6449 −1.75574 −0.877868 0.478902i $$-0.841035\pi$$
−0.877868 + 0.478902i $$0.841035\pi$$
$$102$$ 3.53948 0.350461
$$103$$ 10.1528 1.00039 0.500194 0.865913i $$-0.333262\pi$$
0.500194 + 0.865913i $$0.333262\pi$$
$$104$$ 4.31157 0.422785
$$105$$ 1.34292 0.131056
$$106$$ −1.40604 −0.136567
$$107$$ −15.4232 −1.49102 −0.745508 0.666497i $$-0.767793\pi$$
−0.745508 + 0.666497i $$0.767793\pi$$
$$108$$ 5.63565 0.542291
$$109$$ −8.49656 −0.813823 −0.406911 0.913468i $$-0.633394\pi$$
−0.406911 + 0.913468i $$0.633394\pi$$
$$110$$ 0 0
$$111$$ 1.12698 0.106968
$$112$$ −1.00000 −0.0944911
$$113$$ 2.72299 0.256158 0.128079 0.991764i $$-0.459119\pi$$
0.128079 + 0.991764i $$0.459119\pi$$
$$114$$ −2.10674 −0.197314
$$115$$ 5.45794 0.508956
$$116$$ −2.83920 −0.263613
$$117$$ 5.15905 0.476954
$$118$$ −2.32408 −0.213949
$$119$$ −2.63565 −0.241610
$$120$$ 1.34292 0.122592
$$121$$ 0 0
$$122$$ −5.24387 −0.474758
$$123$$ 4.67141 0.421207
$$124$$ 1.14637 0.102947
$$125$$ 1.00000 0.0894427
$$126$$ −1.19656 −0.106598
$$127$$ 9.21928 0.818079 0.409040 0.912517i $$-0.365864\pi$$
0.409040 + 0.912517i $$0.365864\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −2.70028 −0.237747
$$130$$ 4.31157 0.378150
$$131$$ −17.5357 −1.53210 −0.766052 0.642779i $$-0.777781\pi$$
−0.766052 + 0.642779i $$0.777781\pi$$
$$132$$ 0 0
$$133$$ 1.56877 0.136030
$$134$$ 1.60636 0.138768
$$135$$ 5.63565 0.485040
$$136$$ −2.63565 −0.226005
$$137$$ 9.12927 0.779966 0.389983 0.920822i $$-0.372481\pi$$
0.389983 + 0.920822i $$0.372481\pi$$
$$138$$ 7.32959 0.623936
$$139$$ −0.657844 −0.0557976 −0.0278988 0.999611i $$-0.508882\pi$$
−0.0278988 + 0.999611i $$0.508882\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 11.2731 0.949363
$$142$$ 13.8057 1.15855
$$143$$ 0 0
$$144$$ −1.19656 −0.0997131
$$145$$ −2.83920 −0.235783
$$146$$ −1.14830 −0.0950337
$$147$$ −1.34292 −0.110762
$$148$$ −0.839198 −0.0689816
$$149$$ 20.5982 1.68747 0.843737 0.536757i $$-0.180351\pi$$
0.843737 + 0.536757i $$0.180351\pi$$
$$150$$ 1.34292 0.109649
$$151$$ 4.29219 0.349293 0.174647 0.984631i $$-0.444122\pi$$
0.174647 + 0.984631i $$0.444122\pi$$
$$152$$ 1.56877 0.127244
$$153$$ −3.15371 −0.254962
$$154$$ 0 0
$$155$$ 1.14637 0.0920783
$$156$$ 5.79011 0.463580
$$157$$ 10.9039 0.870226 0.435113 0.900376i $$-0.356708\pi$$
0.435113 + 0.900376i $$0.356708\pi$$
$$158$$ 9.65408 0.768037
$$159$$ −1.88820 −0.149744
$$160$$ −1.00000 −0.0790569
$$161$$ −5.45794 −0.430146
$$162$$ 3.97858 0.312587
$$163$$ 12.3525 0.967526 0.483763 0.875199i $$-0.339270\pi$$
0.483763 + 0.875199i $$0.339270\pi$$
$$164$$ −3.47854 −0.271628
$$165$$ 0 0
$$166$$ −2.51880 −0.195497
$$167$$ 6.93334 0.536518 0.268259 0.963347i $$-0.413552\pi$$
0.268259 + 0.963347i $$0.413552\pi$$
$$168$$ −1.34292 −0.103609
$$169$$ 5.58967 0.429975
$$170$$ −2.63565 −0.202145
$$171$$ 1.87713 0.143548
$$172$$ 2.01075 0.153318
$$173$$ 11.5843 0.880741 0.440370 0.897816i $$-0.354847\pi$$
0.440370 + 0.897816i $$0.354847\pi$$
$$174$$ −3.81282 −0.289049
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −3.12106 −0.234593
$$178$$ −2.19326 −0.164392
$$179$$ 20.4825 1.53093 0.765466 0.643476i $$-0.222508\pi$$
0.765466 + 0.643476i $$0.222508\pi$$
$$180$$ −1.19656 −0.0891861
$$181$$ −8.01175 −0.595509 −0.297754 0.954642i $$-0.596238\pi$$
−0.297754 + 0.954642i $$0.596238\pi$$
$$182$$ −4.31157 −0.319595
$$183$$ −7.04211 −0.520568
$$184$$ −5.45794 −0.402365
$$185$$ −0.839198 −0.0616990
$$186$$ 1.53948 0.112880
$$187$$ 0 0
$$188$$ −8.39442 −0.612226
$$189$$ −5.63565 −0.409934
$$190$$ 1.56877 0.113811
$$191$$ 7.72371 0.558868 0.279434 0.960165i $$-0.409853\pi$$
0.279434 + 0.960165i $$0.409853\pi$$
$$192$$ −1.34292 −0.0969171
$$193$$ 16.3855 1.17945 0.589727 0.807603i $$-0.299235\pi$$
0.589727 + 0.807603i $$0.299235\pi$$
$$194$$ 7.56372 0.543043
$$195$$ 5.79011 0.414639
$$196$$ 1.00000 0.0714286
$$197$$ −5.51971 −0.393263 −0.196631 0.980477i $$-0.563000\pi$$
−0.196631 + 0.980477i $$0.563000\pi$$
$$198$$ 0 0
$$199$$ 4.27443 0.303006 0.151503 0.988457i $$-0.451589\pi$$
0.151503 + 0.988457i $$0.451589\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 2.15722 0.152159
$$202$$ 17.6449 1.24149
$$203$$ 2.83920 0.199273
$$204$$ −3.53948 −0.247813
$$205$$ −3.47854 −0.242952
$$206$$ −10.1528 −0.707381
$$207$$ −6.53074 −0.453918
$$208$$ −4.31157 −0.298954
$$209$$ 0 0
$$210$$ −1.34292 −0.0926705
$$211$$ 19.6106 1.35005 0.675026 0.737794i $$-0.264132\pi$$
0.675026 + 0.737794i $$0.264132\pi$$
$$212$$ 1.40604 0.0965672
$$213$$ 18.5400 1.27034
$$214$$ 15.4232 1.05431
$$215$$ 2.01075 0.137132
$$216$$ −5.63565 −0.383458
$$217$$ −1.14637 −0.0778204
$$218$$ 8.49656 0.575459
$$219$$ −1.54207 −0.104204
$$220$$ 0 0
$$221$$ −11.3638 −0.764413
$$222$$ −1.12698 −0.0756378
$$223$$ 4.01943 0.269161 0.134580 0.990903i $$-0.457031\pi$$
0.134580 + 0.990903i $$0.457031\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −1.19656 −0.0797705
$$226$$ −2.72299 −0.181131
$$227$$ −20.4787 −1.35922 −0.679610 0.733574i $$-0.737851\pi$$
−0.679610 + 0.733574i $$0.737851\pi$$
$$228$$ 2.10674 0.139522
$$229$$ 8.74872 0.578132 0.289066 0.957309i $$-0.406655\pi$$
0.289066 + 0.957309i $$0.406655\pi$$
$$230$$ −5.45794 −0.359886
$$231$$ 0 0
$$232$$ 2.83920 0.186402
$$233$$ −14.1462 −0.926748 −0.463374 0.886163i $$-0.653361\pi$$
−0.463374 + 0.886163i $$0.653361\pi$$
$$234$$ −5.15905 −0.337257
$$235$$ −8.39442 −0.547591
$$236$$ 2.32408 0.151285
$$237$$ 12.9647 0.842147
$$238$$ 2.63565 0.170844
$$239$$ −2.12134 −0.137218 −0.0686091 0.997644i $$-0.521856\pi$$
−0.0686091 + 0.997644i $$0.521856\pi$$
$$240$$ −1.34292 −0.0866853
$$241$$ −17.4357 −1.12313 −0.561566 0.827432i $$-0.689801\pi$$
−0.561566 + 0.827432i $$0.689801\pi$$
$$242$$ 0 0
$$243$$ −11.5640 −0.741833
$$244$$ 5.24387 0.335704
$$245$$ 1.00000 0.0638877
$$246$$ −4.67141 −0.297838
$$247$$ 6.76388 0.430376
$$248$$ −1.14637 −0.0727943
$$249$$ −3.38256 −0.214361
$$250$$ −1.00000 −0.0632456
$$251$$ 0.936840 0.0591328 0.0295664 0.999563i $$-0.490587\pi$$
0.0295664 + 0.999563i $$0.490587\pi$$
$$252$$ 1.19656 0.0753760
$$253$$ 0 0
$$254$$ −9.21928 −0.578469
$$255$$ −3.53948 −0.221651
$$256$$ 1.00000 0.0625000
$$257$$ 13.7242 0.856095 0.428047 0.903756i $$-0.359202\pi$$
0.428047 + 0.903756i $$0.359202\pi$$
$$258$$ 2.70028 0.168112
$$259$$ 0.839198 0.0521452
$$260$$ −4.31157 −0.267392
$$261$$ 3.39726 0.210285
$$262$$ 17.5357 1.08336
$$263$$ 18.9341 1.16753 0.583763 0.811924i $$-0.301580\pi$$
0.583763 + 0.811924i $$0.301580\pi$$
$$264$$ 0 0
$$265$$ 1.40604 0.0863723
$$266$$ −1.56877 −0.0961877
$$267$$ −2.94538 −0.180254
$$268$$ −1.60636 −0.0981241
$$269$$ 12.9639 0.790422 0.395211 0.918590i $$-0.370672\pi$$
0.395211 + 0.918590i $$0.370672\pi$$
$$270$$ −5.63565 −0.342975
$$271$$ −23.5085 −1.42804 −0.714020 0.700125i $$-0.753127\pi$$
−0.714020 + 0.700125i $$0.753127\pi$$
$$272$$ 2.63565 0.159810
$$273$$ −5.79011 −0.350434
$$274$$ −9.12927 −0.551519
$$275$$ 0 0
$$276$$ −7.32959 −0.441190
$$277$$ −2.44560 −0.146942 −0.0734710 0.997297i $$-0.523408\pi$$
−0.0734710 + 0.997297i $$0.523408\pi$$
$$278$$ 0.657844 0.0394549
$$279$$ −1.37169 −0.0821211
$$280$$ 1.00000 0.0597614
$$281$$ 18.0116 1.07448 0.537242 0.843428i $$-0.319466\pi$$
0.537242 + 0.843428i $$0.319466\pi$$
$$282$$ −11.2731 −0.671301
$$283$$ 9.78930 0.581914 0.290957 0.956736i $$-0.406026\pi$$
0.290957 + 0.956736i $$0.406026\pi$$
$$284$$ −13.8057 −0.819220
$$285$$ 2.10674 0.124793
$$286$$ 0 0
$$287$$ 3.47854 0.205332
$$288$$ 1.19656 0.0705078
$$289$$ −10.0533 −0.591372
$$290$$ 2.83920 0.166723
$$291$$ 10.1575 0.595443
$$292$$ 1.14830 0.0671990
$$293$$ 8.44836 0.493558 0.246779 0.969072i $$-0.420628\pi$$
0.246779 + 0.969072i $$0.420628\pi$$
$$294$$ 1.34292 0.0783209
$$295$$ 2.32408 0.135313
$$296$$ 0.839198 0.0487774
$$297$$ 0 0
$$298$$ −20.5982 −1.19322
$$299$$ −23.5323 −1.36091
$$300$$ −1.34292 −0.0775337
$$301$$ −2.01075 −0.115898
$$302$$ −4.29219 −0.246988
$$303$$ 23.6958 1.36129
$$304$$ −1.56877 −0.0899753
$$305$$ 5.24387 0.300263
$$306$$ 3.15371 0.180286
$$307$$ 9.34689 0.533455 0.266728 0.963772i $$-0.414058\pi$$
0.266728 + 0.963772i $$0.414058\pi$$
$$308$$ 0 0
$$309$$ −13.6345 −0.775638
$$310$$ −1.14637 −0.0651092
$$311$$ −27.6729 −1.56919 −0.784593 0.620011i $$-0.787128\pi$$
−0.784593 + 0.620011i $$0.787128\pi$$
$$312$$ −5.79011 −0.327801
$$313$$ 5.34524 0.302131 0.151065 0.988524i $$-0.451730\pi$$
0.151065 + 0.988524i $$0.451730\pi$$
$$314$$ −10.9039 −0.615343
$$315$$ 1.19656 0.0674184
$$316$$ −9.65408 −0.543084
$$317$$ 7.13917 0.400976 0.200488 0.979696i $$-0.435747\pi$$
0.200488 + 0.979696i $$0.435747\pi$$
$$318$$ 1.88820 0.105885
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 20.7122 1.15604
$$322$$ 5.45794 0.304159
$$323$$ −4.13474 −0.230063
$$324$$ −3.97858 −0.221032
$$325$$ −4.31157 −0.239163
$$326$$ −12.3525 −0.684144
$$327$$ 11.4102 0.630987
$$328$$ 3.47854 0.192070
$$329$$ 8.39442 0.462799
$$330$$ 0 0
$$331$$ 5.83207 0.320559 0.160280 0.987072i $$-0.448760\pi$$
0.160280 + 0.987072i $$0.448760\pi$$
$$332$$ 2.51880 0.138237
$$333$$ 1.00415 0.0550270
$$334$$ −6.93334 −0.379375
$$335$$ −1.60636 −0.0877649
$$336$$ 1.34292 0.0732625
$$337$$ −22.2848 −1.21393 −0.606964 0.794729i $$-0.707613\pi$$
−0.606964 + 0.794729i $$0.707613\pi$$
$$338$$ −5.58967 −0.304038
$$339$$ −3.65677 −0.198609
$$340$$ 2.63565 0.142938
$$341$$ 0 0
$$342$$ −1.87713 −0.101503
$$343$$ −1.00000 −0.0539949
$$344$$ −2.01075 −0.108412
$$345$$ −7.32959 −0.394612
$$346$$ −11.5843 −0.622778
$$347$$ −16.6253 −0.892495 −0.446248 0.894910i $$-0.647240\pi$$
−0.446248 + 0.894910i $$0.647240\pi$$
$$348$$ 3.81282 0.204389
$$349$$ −3.23934 −0.173398 −0.0866989 0.996235i $$-0.527632\pi$$
−0.0866989 + 0.996235i $$0.527632\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −24.2985 −1.29696
$$352$$ 0 0
$$353$$ −24.6580 −1.31241 −0.656205 0.754582i $$-0.727839\pi$$
−0.656205 + 0.754582i $$0.727839\pi$$
$$354$$ 3.12106 0.165882
$$355$$ −13.8057 −0.732732
$$356$$ 2.19326 0.116242
$$357$$ 3.53948 0.187329
$$358$$ −20.4825 −1.08253
$$359$$ −20.7493 −1.09511 −0.547554 0.836771i $$-0.684441\pi$$
−0.547554 + 0.836771i $$0.684441\pi$$
$$360$$ 1.19656 0.0630641
$$361$$ −16.5390 −0.870471
$$362$$ 8.01175 0.421088
$$363$$ 0 0
$$364$$ 4.31157 0.225988
$$365$$ 1.14830 0.0601046
$$366$$ 7.04211 0.368097
$$367$$ 2.12160 0.110746 0.0553732 0.998466i $$-0.482365\pi$$
0.0553732 + 0.998466i $$0.482365\pi$$
$$368$$ 5.45794 0.284515
$$369$$ 4.16227 0.216679
$$370$$ 0.839198 0.0436278
$$371$$ −1.40604 −0.0729979
$$372$$ −1.53948 −0.0798184
$$373$$ 7.19996 0.372800 0.186400 0.982474i $$-0.440318\pi$$
0.186400 + 0.982474i $$0.440318\pi$$
$$374$$ 0 0
$$375$$ −1.34292 −0.0693482
$$376$$ 8.39442 0.432909
$$377$$ 12.2414 0.630465
$$378$$ 5.63565 0.289867
$$379$$ −23.1300 −1.18811 −0.594055 0.804424i $$-0.702474\pi$$
−0.594055 + 0.804424i $$0.702474\pi$$
$$380$$ −1.56877 −0.0804764
$$381$$ −12.3808 −0.634287
$$382$$ −7.72371 −0.395180
$$383$$ 31.4479 1.60691 0.803455 0.595365i $$-0.202993\pi$$
0.803455 + 0.595365i $$0.202993\pi$$
$$384$$ 1.34292 0.0685308
$$385$$ 0 0
$$386$$ −16.3855 −0.834000
$$387$$ −2.40598 −0.122303
$$388$$ −7.56372 −0.383990
$$389$$ 17.5618 0.890420 0.445210 0.895426i $$-0.353129\pi$$
0.445210 + 0.895426i $$0.353129\pi$$
$$390$$ −5.79011 −0.293194
$$391$$ 14.3852 0.727493
$$392$$ −1.00000 −0.0505076
$$393$$ 23.5491 1.18790
$$394$$ 5.51971 0.278079
$$395$$ −9.65408 −0.485749
$$396$$ 0 0
$$397$$ 32.7839 1.64538 0.822689 0.568492i $$-0.192473\pi$$
0.822689 + 0.568492i $$0.192473\pi$$
$$398$$ −4.27443 −0.214258
$$399$$ −2.10674 −0.105469
$$400$$ 1.00000 0.0500000
$$401$$ 9.25146 0.461996 0.230998 0.972954i $$-0.425801\pi$$
0.230998 + 0.972954i $$0.425801\pi$$
$$402$$ −2.15722 −0.107592
$$403$$ −4.94264 −0.246210
$$404$$ −17.6449 −0.877868
$$405$$ −3.97858 −0.197697
$$406$$ −2.83920 −0.140907
$$407$$ 0 0
$$408$$ 3.53948 0.175230
$$409$$ 0.0174947 0.000865057 0 0.000432529 1.00000i $$-0.499862\pi$$
0.000432529 1.00000i $$0.499862\pi$$
$$410$$ 3.47854 0.171793
$$411$$ −12.2599 −0.604737
$$412$$ 10.1528 0.500194
$$413$$ −2.32408 −0.114360
$$414$$ 6.53074 0.320968
$$415$$ 2.51880 0.123643
$$416$$ 4.31157 0.211392
$$417$$ 0.883434 0.0432620
$$418$$ 0 0
$$419$$ 32.4579 1.58567 0.792837 0.609433i $$-0.208603\pi$$
0.792837 + 0.609433i $$0.208603\pi$$
$$420$$ 1.34292 0.0655279
$$421$$ 9.15329 0.446104 0.223052 0.974807i $$-0.428398\pi$$
0.223052 + 0.974807i $$0.428398\pi$$
$$422$$ −19.6106 −0.954631
$$423$$ 10.0444 0.488376
$$424$$ −1.40604 −0.0682833
$$425$$ 2.63565 0.127848
$$426$$ −18.5400 −0.898268
$$427$$ −5.24387 −0.253769
$$428$$ −15.4232 −0.745508
$$429$$ 0 0
$$430$$ −2.01075 −0.0969670
$$431$$ 26.0024 1.25249 0.626247 0.779625i $$-0.284590\pi$$
0.626247 + 0.779625i $$0.284590\pi$$
$$432$$ 5.63565 0.271146
$$433$$ 23.6006 1.13417 0.567085 0.823659i $$-0.308071\pi$$
0.567085 + 0.823659i $$0.308071\pi$$
$$434$$ 1.14637 0.0550273
$$435$$ 3.81282 0.182811
$$436$$ −8.49656 −0.406911
$$437$$ −8.56227 −0.409589
$$438$$ 1.54207 0.0736832
$$439$$ −31.7894 −1.51722 −0.758612 0.651543i $$-0.774122\pi$$
−0.758612 + 0.651543i $$0.774122\pi$$
$$440$$ 0 0
$$441$$ −1.19656 −0.0569789
$$442$$ 11.3638 0.540522
$$443$$ 36.1071 1.71550 0.857750 0.514067i $$-0.171862\pi$$
0.857750 + 0.514067i $$0.171862\pi$$
$$444$$ 1.12698 0.0534840
$$445$$ 2.19326 0.103970
$$446$$ −4.01943 −0.190326
$$447$$ −27.6619 −1.30836
$$448$$ −1.00000 −0.0472456
$$449$$ −4.52269 −0.213439 −0.106719 0.994289i $$-0.534035\pi$$
−0.106719 + 0.994289i $$0.534035\pi$$
$$450$$ 1.19656 0.0564063
$$451$$ 0 0
$$452$$ 2.72299 0.128079
$$453$$ −5.76408 −0.270820
$$454$$ 20.4787 0.961114
$$455$$ 4.31157 0.202130
$$456$$ −2.10674 −0.0986572
$$457$$ 15.1593 0.709120 0.354560 0.935033i $$-0.384631\pi$$
0.354560 + 0.935033i $$0.384631\pi$$
$$458$$ −8.74872 −0.408801
$$459$$ 14.8536 0.693308
$$460$$ 5.45794 0.254478
$$461$$ 27.6126 1.28605 0.643024 0.765846i $$-0.277680\pi$$
0.643024 + 0.765846i $$0.277680\pi$$
$$462$$ 0 0
$$463$$ 13.3557 0.620692 0.310346 0.950624i $$-0.399555\pi$$
0.310346 + 0.950624i $$0.399555\pi$$
$$464$$ −2.83920 −0.131806
$$465$$ −1.53948 −0.0713917
$$466$$ 14.1462 0.655310
$$467$$ 26.3287 1.21835 0.609173 0.793038i $$-0.291502\pi$$
0.609173 + 0.793038i $$0.291502\pi$$
$$468$$ 5.15905 0.238477
$$469$$ 1.60636 0.0741749
$$470$$ 8.39442 0.387206
$$471$$ −14.6431 −0.674719
$$472$$ −2.32408 −0.106974
$$473$$ 0 0
$$474$$ −12.9647 −0.595488
$$475$$ −1.56877 −0.0719803
$$476$$ −2.63565 −0.120805
$$477$$ −1.68241 −0.0770322
$$478$$ 2.12134 0.0970279
$$479$$ −8.18258 −0.373872 −0.186936 0.982372i $$-0.559856\pi$$
−0.186936 + 0.982372i $$0.559856\pi$$
$$480$$ 1.34292 0.0612958
$$481$$ 3.61826 0.164979
$$482$$ 17.4357 0.794174
$$483$$ 7.32959 0.333508
$$484$$ 0 0
$$485$$ −7.56372 −0.343451
$$486$$ 11.5640 0.524555
$$487$$ −9.09993 −0.412357 −0.206179 0.978514i $$-0.566103\pi$$
−0.206179 + 0.978514i $$0.566103\pi$$
$$488$$ −5.24387 −0.237379
$$489$$ −16.5885 −0.750159
$$490$$ −1.00000 −0.0451754
$$491$$ 3.80239 0.171600 0.0857998 0.996312i $$-0.472655\pi$$
0.0857998 + 0.996312i $$0.472655\pi$$
$$492$$ 4.67141 0.210603
$$493$$ −7.48314 −0.337024
$$494$$ −6.76388 −0.304321
$$495$$ 0 0
$$496$$ 1.14637 0.0514733
$$497$$ 13.8057 0.619272
$$498$$ 3.38256 0.151576
$$499$$ −22.9856 −1.02898 −0.514488 0.857497i $$-0.672018\pi$$
−0.514488 + 0.857497i $$0.672018\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −9.31094 −0.415982
$$502$$ −0.936840 −0.0418132
$$503$$ −13.4287 −0.598756 −0.299378 0.954135i $$-0.596779\pi$$
−0.299378 + 0.954135i $$0.596779\pi$$
$$504$$ −1.19656 −0.0532989
$$505$$ −17.6449 −0.785189
$$506$$ 0 0
$$507$$ −7.50650 −0.333375
$$508$$ 9.21928 0.409040
$$509$$ 36.9453 1.63757 0.818785 0.574100i $$-0.194648\pi$$
0.818785 + 0.574100i $$0.194648\pi$$
$$510$$ 3.53948 0.156731
$$511$$ −1.14830 −0.0507977
$$512$$ −1.00000 −0.0441942
$$513$$ −8.84106 −0.390342
$$514$$ −13.7242 −0.605350
$$515$$ 10.1528 0.447387
$$516$$ −2.70028 −0.118873
$$517$$ 0 0
$$518$$ −0.839198 −0.0368722
$$519$$ −15.5569 −0.682871
$$520$$ 4.31157 0.189075
$$521$$ 12.8367 0.562388 0.281194 0.959651i $$-0.409270\pi$$
0.281194 + 0.959651i $$0.409270\pi$$
$$522$$ −3.39726 −0.148694
$$523$$ 8.66972 0.379100 0.189550 0.981871i $$-0.439297\pi$$
0.189550 + 0.981871i $$0.439297\pi$$
$$524$$ −17.5357 −0.766052
$$525$$ 1.34292 0.0586100
$$526$$ −18.9341 −0.825565
$$527$$ 3.02142 0.131615
$$528$$ 0 0
$$529$$ 6.78911 0.295179
$$530$$ −1.40604 −0.0610745
$$531$$ −2.78090 −0.120681
$$532$$ 1.56877 0.0680149
$$533$$ 14.9980 0.649634
$$534$$ 2.94538 0.127459
$$535$$ −15.4232 −0.666803
$$536$$ 1.60636 0.0693842
$$537$$ −27.5064 −1.18699
$$538$$ −12.9639 −0.558913
$$539$$ 0 0
$$540$$ 5.63565 0.242520
$$541$$ 38.6634 1.66227 0.831136 0.556070i $$-0.187691\pi$$
0.831136 + 0.556070i $$0.187691\pi$$
$$542$$ 23.5085 1.00978
$$543$$ 10.7592 0.461720
$$544$$ −2.63565 −0.113003
$$545$$ −8.49656 −0.363953
$$546$$ 5.79011 0.247794
$$547$$ 6.20768 0.265421 0.132711 0.991155i $$-0.457632\pi$$
0.132711 + 0.991155i $$0.457632\pi$$
$$548$$ 9.12927 0.389983
$$549$$ −6.27459 −0.267793
$$550$$ 0 0
$$551$$ 4.45406 0.189749
$$552$$ 7.32959 0.311968
$$553$$ 9.65408 0.410533
$$554$$ 2.44560 0.103904
$$555$$ 1.12698 0.0478376
$$556$$ −0.657844 −0.0278988
$$557$$ 36.9739 1.56663 0.783317 0.621623i $$-0.213526\pi$$
0.783317 + 0.621623i $$0.213526\pi$$
$$558$$ 1.37169 0.0580684
$$559$$ −8.66950 −0.366681
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −18.0116 −0.759776
$$563$$ −26.0483 −1.09780 −0.548901 0.835887i $$-0.684954\pi$$
−0.548901 + 0.835887i $$0.684954\pi$$
$$564$$ 11.2731 0.474681
$$565$$ 2.72299 0.114557
$$566$$ −9.78930 −0.411475
$$567$$ 3.97858 0.167085
$$568$$ 13.8057 0.579276
$$569$$ 18.6278 0.780920 0.390460 0.920620i $$-0.372316\pi$$
0.390460 + 0.920620i $$0.372316\pi$$
$$570$$ −2.10674 −0.0882417
$$571$$ −34.4156 −1.44025 −0.720123 0.693846i $$-0.755915\pi$$
−0.720123 + 0.693846i $$0.755915\pi$$
$$572$$ 0 0
$$573$$ −10.3724 −0.433311
$$574$$ −3.47854 −0.145191
$$575$$ 5.45794 0.227612
$$576$$ −1.19656 −0.0498566
$$577$$ 24.5630 1.02257 0.511285 0.859411i $$-0.329170\pi$$
0.511285 + 0.859411i $$0.329170\pi$$
$$578$$ 10.0533 0.418163
$$579$$ −22.0045 −0.914474
$$580$$ −2.83920 −0.117891
$$581$$ −2.51880 −0.104498
$$582$$ −10.1575 −0.421042
$$583$$ 0 0
$$584$$ −1.14830 −0.0475169
$$585$$ 5.15905 0.213300
$$586$$ −8.44836 −0.348999
$$587$$ 18.3415 0.757037 0.378518 0.925594i $$-0.376434\pi$$
0.378518 + 0.925594i $$0.376434\pi$$
$$588$$ −1.34292 −0.0553812
$$589$$ −1.79839 −0.0741013
$$590$$ −2.32408 −0.0956809
$$591$$ 7.41254 0.304911
$$592$$ −0.839198 −0.0344908
$$593$$ 10.9987 0.451664 0.225832 0.974166i $$-0.427490\pi$$
0.225832 + 0.974166i $$0.427490\pi$$
$$594$$ 0 0
$$595$$ −2.63565 −0.108051
$$596$$ 20.5982 0.843737
$$597$$ −5.74023 −0.234932
$$598$$ 23.5323 0.962308
$$599$$ 4.76034 0.194502 0.0972511 0.995260i $$-0.468995\pi$$
0.0972511 + 0.995260i $$0.468995\pi$$
$$600$$ 1.34292 0.0548246
$$601$$ 16.5137 0.673608 0.336804 0.941575i $$-0.390654\pi$$
0.336804 + 0.941575i $$0.390654\pi$$
$$602$$ 2.01075 0.0819521
$$603$$ 1.92210 0.0782741
$$604$$ 4.29219 0.174647
$$605$$ 0 0
$$606$$ −23.6958 −0.962576
$$607$$ −3.14655 −0.127714 −0.0638572 0.997959i $$-0.520340\pi$$
−0.0638572 + 0.997959i $$0.520340\pi$$
$$608$$ 1.56877 0.0636222
$$609$$ −3.81282 −0.154503
$$610$$ −5.24387 −0.212318
$$611$$ 36.1932 1.46422
$$612$$ −3.15371 −0.127481
$$613$$ 45.9924 1.85761 0.928807 0.370564i $$-0.120836\pi$$
0.928807 + 0.370564i $$0.120836\pi$$
$$614$$ −9.34689 −0.377210
$$615$$ 4.67141 0.188369
$$616$$ 0 0
$$617$$ 45.5056 1.83198 0.915992 0.401196i $$-0.131405\pi$$
0.915992 + 0.401196i $$0.131405\pi$$
$$618$$ 13.6345 0.548459
$$619$$ −22.5870 −0.907849 −0.453925 0.891040i $$-0.649976\pi$$
−0.453925 + 0.891040i $$0.649976\pi$$
$$620$$ 1.14637 0.0460391
$$621$$ 30.7591 1.23432
$$622$$ 27.6729 1.10958
$$623$$ −2.19326 −0.0878710
$$624$$ 5.79011 0.231790
$$625$$ 1.00000 0.0400000
$$626$$ −5.34524 −0.213639
$$627$$ 0 0
$$628$$ 10.9039 0.435113
$$629$$ −2.21184 −0.0881916
$$630$$ −1.19656 −0.0476720
$$631$$ 14.8009 0.589213 0.294606 0.955619i $$-0.404811\pi$$
0.294606 + 0.955619i $$0.404811\pi$$
$$632$$ 9.65408 0.384019
$$633$$ −26.3356 −1.04675
$$634$$ −7.13917 −0.283533
$$635$$ 9.21928 0.365856
$$636$$ −1.88820 −0.0748721
$$637$$ −4.31157 −0.170831
$$638$$ 0 0
$$639$$ 16.5194 0.653496
$$640$$ −1.00000 −0.0395285
$$641$$ −34.5333 −1.36398 −0.681992 0.731360i $$-0.738886\pi$$
−0.681992 + 0.731360i $$0.738886\pi$$
$$642$$ −20.7122 −0.817444
$$643$$ −16.2144 −0.639434 −0.319717 0.947513i $$-0.603588\pi$$
−0.319717 + 0.947513i $$0.603588\pi$$
$$644$$ −5.45794 −0.215073
$$645$$ −2.70028 −0.106324
$$646$$ 4.13474 0.162679
$$647$$ −43.3962 −1.70608 −0.853041 0.521844i $$-0.825244\pi$$
−0.853041 + 0.521844i $$0.825244\pi$$
$$648$$ 3.97858 0.156293
$$649$$ 0 0
$$650$$ 4.31157 0.169114
$$651$$ 1.53948 0.0603370
$$652$$ 12.3525 0.483763
$$653$$ 7.75135 0.303334 0.151667 0.988432i $$-0.451536\pi$$
0.151667 + 0.988432i $$0.451536\pi$$
$$654$$ −11.4102 −0.446175
$$655$$ −17.5357 −0.685178
$$656$$ −3.47854 −0.135814
$$657$$ −1.37400 −0.0536050
$$658$$ −8.39442 −0.327248
$$659$$ −6.80255 −0.264990 −0.132495 0.991184i $$-0.542299\pi$$
−0.132495 + 0.991184i $$0.542299\pi$$
$$660$$ 0 0
$$661$$ −17.3537 −0.674980 −0.337490 0.941329i $$-0.609578\pi$$
−0.337490 + 0.941329i $$0.609578\pi$$
$$662$$ −5.83207 −0.226670
$$663$$ 15.2607 0.592678
$$664$$ −2.51880 −0.0977486
$$665$$ 1.56877 0.0608344
$$666$$ −1.00415 −0.0389100
$$667$$ −15.4962 −0.600014
$$668$$ 6.93334 0.268259
$$669$$ −5.39779 −0.208690
$$670$$ 1.60636 0.0620592
$$671$$ 0 0
$$672$$ −1.34292 −0.0518044
$$673$$ 47.8170 1.84321 0.921605 0.388130i $$-0.126879\pi$$
0.921605 + 0.388130i $$0.126879\pi$$
$$674$$ 22.2848 0.858377
$$675$$ 5.63565 0.216916
$$676$$ 5.58967 0.214987
$$677$$ 7.12176 0.273711 0.136856 0.990591i $$-0.456300\pi$$
0.136856 + 0.990591i $$0.456300\pi$$
$$678$$ 3.65677 0.140437
$$679$$ 7.56372 0.290269
$$680$$ −2.63565 −0.101073
$$681$$ 27.5013 1.05385
$$682$$ 0 0
$$683$$ 17.8302 0.682252 0.341126 0.940018i $$-0.389192\pi$$
0.341126 + 0.940018i $$0.389192\pi$$
$$684$$ 1.87713 0.0717738
$$685$$ 9.12927 0.348812
$$686$$ 1.00000 0.0381802
$$687$$ −11.7489 −0.448247
$$688$$ 2.01075 0.0766591
$$689$$ −6.06225 −0.230953
$$690$$ 7.32959 0.279033
$$691$$ −15.7399 −0.598772 −0.299386 0.954132i $$-0.596782\pi$$
−0.299386 + 0.954132i $$0.596782\pi$$
$$692$$ 11.5843 0.440370
$$693$$ 0 0
$$694$$ 16.6253 0.631089
$$695$$ −0.657844 −0.0249535
$$696$$ −3.81282 −0.144525
$$697$$ −9.16822 −0.347271
$$698$$ 3.23934 0.122611
$$699$$ 18.9973 0.718542
$$700$$ −1.00000 −0.0377964
$$701$$ 19.1512 0.723329 0.361665 0.932308i $$-0.382209\pi$$
0.361665 + 0.932308i $$0.382209\pi$$
$$702$$ 24.2985 0.917089
$$703$$ 1.31651 0.0496532
$$704$$ 0 0
$$705$$ 11.2731 0.424568
$$706$$ 24.6580 0.928014
$$707$$ 17.6449 0.663606
$$708$$ −3.12106 −0.117297
$$709$$ 6.08007 0.228342 0.114171 0.993461i $$-0.463579\pi$$
0.114171 + 0.993461i $$0.463579\pi$$
$$710$$ 13.8057 0.518120
$$711$$ 11.5517 0.433221
$$712$$ −2.19326 −0.0821958
$$713$$ 6.25679 0.234319
$$714$$ −3.53948 −0.132462
$$715$$ 0 0
$$716$$ 20.4825 0.765466
$$717$$ 2.84880 0.106390
$$718$$ 20.7493 0.774358
$$719$$ 27.3905 1.02149 0.510746 0.859732i $$-0.329369\pi$$
0.510746 + 0.859732i $$0.329369\pi$$
$$720$$ −1.19656 −0.0445931
$$721$$ −10.1528 −0.378111
$$722$$ 16.5390 0.615516
$$723$$ 23.4148 0.870805
$$724$$ −8.01175 −0.297754
$$725$$ −2.83920 −0.105445
$$726$$ 0 0
$$727$$ 28.3641 1.05197 0.525984 0.850495i $$-0.323697\pi$$
0.525984 + 0.850495i $$0.323697\pi$$
$$728$$ −4.31157 −0.159798
$$729$$ 27.4653 1.01724
$$730$$ −1.14830 −0.0425004
$$731$$ 5.29964 0.196014
$$732$$ −7.04211 −0.260284
$$733$$ 48.3933 1.78745 0.893724 0.448618i $$-0.148084\pi$$
0.893724 + 0.448618i $$0.148084\pi$$
$$734$$ −2.12160 −0.0783096
$$735$$ −1.34292 −0.0495345
$$736$$ −5.45794 −0.201182
$$737$$ 0 0
$$738$$ −4.16227 −0.153215
$$739$$ −6.07467 −0.223460 −0.111730 0.993739i $$-0.535639\pi$$
−0.111730 + 0.993739i $$0.535639\pi$$
$$740$$ −0.839198 −0.0308495
$$741$$ −9.08337 −0.333686
$$742$$ 1.40604 0.0516173
$$743$$ −32.1862 −1.18080 −0.590398 0.807112i $$-0.701029\pi$$
−0.590398 + 0.807112i $$0.701029\pi$$
$$744$$ 1.53948 0.0564401
$$745$$ 20.5982 0.754661
$$746$$ −7.19996 −0.263609
$$747$$ −3.01389 −0.110273
$$748$$ 0 0
$$749$$ 15.4232 0.563551
$$750$$ 1.34292 0.0490366
$$751$$ −24.5881 −0.897234 −0.448617 0.893724i $$-0.648083\pi$$
−0.448617 + 0.893724i $$0.648083\pi$$
$$752$$ −8.39442 −0.306113
$$753$$ −1.25810 −0.0458479
$$754$$ −12.2414 −0.445806
$$755$$ 4.29219 0.156209
$$756$$ −5.63565 −0.204967
$$757$$ −25.5181 −0.927470 −0.463735 0.885974i $$-0.653491\pi$$
−0.463735 + 0.885974i $$0.653491\pi$$
$$758$$ 23.1300 0.840121
$$759$$ 0 0
$$760$$ 1.56877 0.0569054
$$761$$ 13.0436 0.472830 0.236415 0.971652i $$-0.424028\pi$$
0.236415 + 0.971652i $$0.424028\pi$$
$$762$$ 12.3808 0.448509
$$763$$ 8.49656 0.307596
$$764$$ 7.72371 0.279434
$$765$$ −3.15371 −0.114023
$$766$$ −31.4479 −1.13626
$$767$$ −10.0204 −0.361817
$$768$$ −1.34292 −0.0484586
$$769$$ 48.0699 1.73344 0.866722 0.498792i $$-0.166223\pi$$
0.866722 + 0.498792i $$0.166223\pi$$
$$770$$ 0 0
$$771$$ −18.4306 −0.663762
$$772$$ 16.3855 0.589727
$$773$$ −40.5127 −1.45714 −0.728570 0.684972i $$-0.759815\pi$$
−0.728570 + 0.684972i $$0.759815\pi$$
$$774$$ 2.40598 0.0864811
$$775$$ 1.14637 0.0411787
$$776$$ 7.56372 0.271522
$$777$$ −1.12698 −0.0404301
$$778$$ −17.5618 −0.629622
$$779$$ 5.45704 0.195519
$$780$$ 5.79011 0.207319
$$781$$ 0 0
$$782$$ −14.3852 −0.514415
$$783$$ −16.0007 −0.571820
$$784$$ 1.00000 0.0357143
$$785$$ 10.9039 0.389177
$$786$$ −23.5491 −0.839970
$$787$$ −36.3703 −1.29646 −0.648231 0.761444i $$-0.724491\pi$$
−0.648231 + 0.761444i $$0.724491\pi$$
$$788$$ −5.51971 −0.196631
$$789$$ −25.4270 −0.905226
$$790$$ 9.65408 0.343477
$$791$$ −2.72299 −0.0968185
$$792$$ 0 0
$$793$$ −22.6093 −0.802881
$$794$$ −32.7839 −1.16346
$$795$$ −1.88820 −0.0669677
$$796$$ 4.27443 0.151503
$$797$$ 30.7299 1.08851 0.544254 0.838920i $$-0.316813\pi$$
0.544254 + 0.838920i $$0.316813\pi$$
$$798$$ 2.10674 0.0745779
$$799$$ −22.1248 −0.782719
$$800$$ −1.00000 −0.0353553
$$801$$ −2.62436 −0.0927272
$$802$$ −9.25146 −0.326680
$$803$$ 0 0
$$804$$ 2.15722 0.0760793
$$805$$ −5.45794 −0.192367
$$806$$ 4.94264 0.174097
$$807$$ −17.4095 −0.612843
$$808$$ 17.6449 0.620747
$$809$$ 45.6988 1.60668 0.803342 0.595517i $$-0.203053\pi$$
0.803342 + 0.595517i $$0.203053\pi$$
$$810$$ 3.97858 0.139793
$$811$$ 9.43169 0.331191 0.165596 0.986194i $$-0.447045\pi$$
0.165596 + 0.986194i $$0.447045\pi$$
$$812$$ 2.83920 0.0996363
$$813$$ 31.5701 1.10721
$$814$$ 0 0
$$815$$ 12.3525 0.432691
$$816$$ −3.53948 −0.123907
$$817$$ −3.15441 −0.110359
$$818$$ −0.0174947 −0.000611688 0
$$819$$ −5.15905 −0.180272
$$820$$ −3.47854 −0.121476
$$821$$ 12.2170 0.426376 0.213188 0.977011i $$-0.431615\pi$$
0.213188 + 0.977011i $$0.431615\pi$$
$$822$$ 12.2599 0.427613
$$823$$ 8.03161 0.279964 0.139982 0.990154i $$-0.455295\pi$$
0.139982 + 0.990154i $$0.455295\pi$$
$$824$$ −10.1528 −0.353691
$$825$$ 0 0
$$826$$ 2.32408 0.0808651
$$827$$ −9.90168 −0.344315 −0.172158 0.985069i $$-0.555074\pi$$
−0.172158 + 0.985069i $$0.555074\pi$$
$$828$$ −6.53074 −0.226959
$$829$$ −20.2243 −0.702420 −0.351210 0.936297i $$-0.614230\pi$$
−0.351210 + 0.936297i $$0.614230\pi$$
$$830$$ −2.51880 −0.0874290
$$831$$ 3.28426 0.113930
$$832$$ −4.31157 −0.149477
$$833$$ 2.63565 0.0913200
$$834$$ −0.883434 −0.0305908
$$835$$ 6.93334 0.239938
$$836$$ 0 0
$$837$$ 6.46052 0.223308
$$838$$ −32.4579 −1.12124
$$839$$ 35.7528 1.23432 0.617162 0.786836i $$-0.288282\pi$$
0.617162 + 0.786836i $$0.288282\pi$$
$$840$$ −1.34292 −0.0463352
$$841$$ −20.9390 −0.722033
$$842$$ −9.15329 −0.315443
$$843$$ −24.1883 −0.833088
$$844$$ 19.6106 0.675026
$$845$$ 5.58967 0.192291
$$846$$ −10.0444 −0.345334
$$847$$ 0 0
$$848$$ 1.40604 0.0482836
$$849$$ −13.1463 −0.451179
$$850$$ −2.63565 −0.0904022
$$851$$ −4.58029 −0.157010
$$852$$ 18.5400 0.635171
$$853$$ −23.0500 −0.789218 −0.394609 0.918849i $$-0.629120\pi$$
−0.394609 + 0.918849i $$0.629120\pi$$
$$854$$ 5.24387 0.179442
$$855$$ 1.87713 0.0641964
$$856$$ 15.4232 0.527154
$$857$$ −27.3702 −0.934948 −0.467474 0.884007i $$-0.654836\pi$$
−0.467474 + 0.884007i $$0.654836\pi$$
$$858$$ 0 0
$$859$$ 43.4968 1.48409 0.742045 0.670350i $$-0.233856\pi$$
0.742045 + 0.670350i $$0.233856\pi$$
$$860$$ 2.01075 0.0685660
$$861$$ −4.67141 −0.159201
$$862$$ −26.0024 −0.885647
$$863$$ −17.9594 −0.611345 −0.305672 0.952137i $$-0.598881\pi$$
−0.305672 + 0.952137i $$0.598881\pi$$
$$864$$ −5.63565 −0.191729
$$865$$ 11.5843 0.393879
$$866$$ −23.6006 −0.801980
$$867$$ 13.5008 0.458513
$$868$$ −1.14637 −0.0389102
$$869$$ 0 0
$$870$$ −3.81282 −0.129267
$$871$$ 6.92595 0.234677
$$872$$ 8.49656 0.287730
$$873$$ 9.05042 0.306310
$$874$$ 8.56227 0.289623
$$875$$ −1.00000 −0.0338062
$$876$$ −1.54207 −0.0521019
$$877$$ −32.8608 −1.10963 −0.554816 0.831973i $$-0.687211\pi$$
−0.554816 + 0.831973i $$0.687211\pi$$
$$878$$ 31.7894 1.07284
$$879$$ −11.3455 −0.382674
$$880$$ 0 0
$$881$$ −24.9270 −0.839812 −0.419906 0.907568i $$-0.637937\pi$$
−0.419906 + 0.907568i $$0.637937\pi$$
$$882$$ 1.19656 0.0402902
$$883$$ −18.1957 −0.612334 −0.306167 0.951978i $$-0.599047\pi$$
−0.306167 + 0.951978i $$0.599047\pi$$
$$884$$ −11.3638 −0.382207
$$885$$ −3.12106 −0.104913
$$886$$ −36.1071 −1.21304
$$887$$ 38.9219 1.30687 0.653434 0.756983i $$-0.273328\pi$$
0.653434 + 0.756983i $$0.273328\pi$$
$$888$$ −1.12698 −0.0378189
$$889$$ −9.21928 −0.309205
$$890$$ −2.19326 −0.0735182
$$891$$ 0 0
$$892$$ 4.01943 0.134580
$$893$$ 13.1689 0.440682
$$894$$ 27.6619 0.925151
$$895$$ 20.4825 0.684654
$$896$$ 1.00000 0.0334077
$$897$$ 31.6021 1.05516
$$898$$ 4.52269 0.150924
$$899$$ −3.25476 −0.108552
$$900$$ −1.19656 −0.0398853
$$901$$ 3.70583 0.123459
$$902$$ 0 0
$$903$$ 2.70028 0.0898598
$$904$$ −2.72299 −0.0905655
$$905$$ −8.01175 −0.266320
$$906$$ 5.76408 0.191499
$$907$$ −47.4377 −1.57514 −0.787572 0.616223i $$-0.788662\pi$$
−0.787572 + 0.616223i $$0.788662\pi$$
$$908$$ −20.4787 −0.679610
$$909$$ 21.1132 0.700280
$$910$$ −4.31157 −0.142927
$$911$$ −10.2118 −0.338333 −0.169167 0.985587i $$-0.554108\pi$$
−0.169167 + 0.985587i $$0.554108\pi$$
$$912$$ 2.10674 0.0697612
$$913$$ 0 0
$$914$$ −15.1593 −0.501423
$$915$$ −7.04211 −0.232805
$$916$$ 8.74872 0.289066
$$917$$ 17.5357 0.579081
$$918$$ −14.8536 −0.490243
$$919$$ 33.7533 1.11342 0.556710 0.830707i $$-0.312064\pi$$
0.556710 + 0.830707i $$0.312064\pi$$
$$920$$ −5.45794 −0.179943
$$921$$ −12.5522 −0.413608
$$922$$ −27.6126 −0.909373
$$923$$ 59.5244 1.95927
$$924$$ 0 0
$$925$$ −0.839198 −0.0275927
$$926$$ −13.3557 −0.438895
$$927$$ −12.1484 −0.399007
$$928$$ 2.83920 0.0932012
$$929$$ 41.2273 1.35262 0.676312 0.736615i $$-0.263577\pi$$
0.676312 + 0.736615i $$0.263577\pi$$
$$930$$ 1.53948 0.0504816
$$931$$ −1.56877 −0.0514145
$$932$$ −14.1462 −0.463374
$$933$$ 37.1626 1.21665
$$934$$ −26.3287 −0.861500
$$935$$ 0 0
$$936$$ −5.15905 −0.168629
$$937$$ 32.2808 1.05457 0.527283 0.849690i $$-0.323211\pi$$
0.527283 + 0.849690i $$0.323211\pi$$
$$938$$ −1.60636 −0.0524496
$$939$$ −7.17825 −0.234253
$$940$$ −8.39442 −0.273796
$$941$$ 47.3632 1.54400 0.771998 0.635625i $$-0.219257\pi$$
0.771998 + 0.635625i $$0.219257\pi$$
$$942$$ 14.6431 0.477098
$$943$$ −18.9857 −0.618258
$$944$$ 2.32408 0.0756424
$$945$$ −5.63565 −0.183328
$$946$$ 0 0
$$947$$ 25.4907 0.828335 0.414167 0.910201i $$-0.364073\pi$$
0.414167 + 0.910201i $$0.364073\pi$$
$$948$$ 12.9647 0.421073
$$949$$ −4.95097 −0.160715
$$950$$ 1.56877 0.0508977
$$951$$ −9.58736 −0.310891
$$952$$ 2.63565 0.0854220
$$953$$ 39.8135 1.28969 0.644843 0.764315i $$-0.276923\pi$$
0.644843 + 0.764315i $$0.276923\pi$$
$$954$$ 1.68241 0.0544700
$$955$$ 7.72371 0.249934
$$956$$ −2.12134 −0.0686091
$$957$$ 0 0
$$958$$ 8.18258 0.264367
$$959$$ −9.12927 −0.294800
$$960$$ −1.34292 −0.0433427
$$961$$ −29.6858 −0.957608
$$962$$ −3.61826 −0.116657
$$963$$ 18.4547 0.594695
$$964$$ −17.4357 −0.561566
$$965$$ 16.3855 0.527468
$$966$$ −7.32959 −0.235826
$$967$$ −29.9522 −0.963198 −0.481599 0.876392i $$-0.659944\pi$$
−0.481599 + 0.876392i $$0.659944\pi$$
$$968$$ 0 0
$$969$$ 5.55264 0.178377
$$970$$ 7.56372 0.242856
$$971$$ 8.45499 0.271333 0.135667 0.990755i $$-0.456682\pi$$
0.135667 + 0.990755i $$0.456682\pi$$
$$972$$ −11.5640 −0.370917
$$973$$ 0.657844 0.0210895
$$974$$ 9.09993 0.291581
$$975$$ 5.79011 0.185432
$$976$$ 5.24387 0.167852
$$977$$ −4.10946 −0.131473 −0.0657366 0.997837i $$-0.520940\pi$$
−0.0657366 + 0.997837i $$0.520940\pi$$
$$978$$ 16.5885 0.530442
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ 10.1666 0.324595
$$982$$ −3.80239 −0.121339
$$983$$ 20.8447 0.664842 0.332421 0.943131i $$-0.392135\pi$$
0.332421 + 0.943131i $$0.392135\pi$$
$$984$$ −4.67141 −0.148919
$$985$$ −5.51971 −0.175873
$$986$$ 7.48314 0.238312
$$987$$ −11.2731 −0.358825
$$988$$ 6.76388 0.215188
$$989$$ 10.9746 0.348971
$$990$$ 0 0
$$991$$ 29.5690 0.939290 0.469645 0.882855i $$-0.344382\pi$$
0.469645 + 0.882855i $$0.344382\pi$$
$$992$$ −1.14637 −0.0363971
$$993$$ −7.83202 −0.248542
$$994$$ −13.8057 −0.437891
$$995$$ 4.27443 0.135508
$$996$$ −3.38256 −0.107181
$$997$$ −57.0730 −1.80752 −0.903760 0.428040i $$-0.859204\pi$$
−0.903760 + 0.428040i $$0.859204\pi$$
$$998$$ 22.9856 0.727596
$$999$$ −4.72943 −0.149632
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 8470.2.a.cz.1.1 6
11.10 odd 2 8470.2.a.df.1.2 yes 6

By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cz.1.1 6 1.1 even 1 trivial
8470.2.a.df.1.2 yes 6 11.10 odd 2