Properties

 Label 8470.2.a.db.1.6 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $1$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.4642000.1 Defining polynomial: $$x^{6} - x^{5} - 8 x^{4} + 5 x^{3} + 14 x^{2} - 9 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.6 Root $$-1.92474$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.92474 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.92474 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.704605 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.92474 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.92474 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.704605 q^{9} +1.00000 q^{10} +1.92474 q^{12} -5.24489 q^{13} -1.00000 q^{14} +1.92474 q^{15} +1.00000 q^{16} -4.30919 q^{17} +0.704605 q^{18} -0.740633 q^{19} +1.00000 q^{20} -1.92474 q^{21} -0.856965 q^{23} +1.92474 q^{24} +1.00000 q^{25} -5.24489 q^{26} -4.41803 q^{27} -1.00000 q^{28} -8.37564 q^{29} +1.92474 q^{30} +1.10298 q^{31} +1.00000 q^{32} -4.30919 q^{34} -1.00000 q^{35} +0.704605 q^{36} -1.77921 q^{37} -0.740633 q^{38} -10.0950 q^{39} +1.00000 q^{40} -10.7200 q^{41} -1.92474 q^{42} +1.13472 q^{43} +0.704605 q^{45} -0.856965 q^{46} +8.40172 q^{47} +1.92474 q^{48} +1.00000 q^{49} +1.00000 q^{50} -8.29405 q^{51} -5.24489 q^{52} -8.80661 q^{53} -4.41803 q^{54} -1.00000 q^{56} -1.42552 q^{57} -8.37564 q^{58} -12.9838 q^{59} +1.92474 q^{60} +13.0887 q^{61} +1.10298 q^{62} -0.704605 q^{63} +1.00000 q^{64} -5.24489 q^{65} +8.93617 q^{67} -4.30919 q^{68} -1.64943 q^{69} -1.00000 q^{70} -4.50750 q^{71} +0.704605 q^{72} -3.33413 q^{73} -1.77921 q^{74} +1.92474 q^{75} -0.740633 q^{76} -10.0950 q^{78} -1.56183 q^{79} +1.00000 q^{80} -10.6173 q^{81} -10.7200 q^{82} +0.609215 q^{83} -1.92474 q^{84} -4.30919 q^{85} +1.13472 q^{86} -16.1209 q^{87} +1.46836 q^{89} +0.704605 q^{90} +5.24489 q^{91} -0.856965 q^{92} +2.12295 q^{93} +8.40172 q^{94} -0.740633 q^{95} +1.92474 q^{96} +8.41705 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + O(q^{10})$$ $$6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98} + O(q^{100})$$

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
$$3$$ 1.92474 1.11125 0.555623 0.831434i $$-0.312480\pi$$
0.555623 + 0.831434i $$0.312480\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.92474 0.785770
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0.704605 0.234868
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ 1.92474 0.555623
$$13$$ −5.24489 −1.45467 −0.727335 0.686283i $$-0.759241\pi$$
−0.727335 + 0.686283i $$0.759241\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 1.92474 0.496964
$$16$$ 1.00000 0.250000
$$17$$ −4.30919 −1.04513 −0.522566 0.852599i $$-0.675025\pi$$
−0.522566 + 0.852599i $$0.675025\pi$$
$$18$$ 0.704605 0.166077
$$19$$ −0.740633 −0.169913 −0.0849564 0.996385i $$-0.527075\pi$$
−0.0849564 + 0.996385i $$0.527075\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −1.92474 −0.420012
$$22$$ 0 0
$$23$$ −0.856965 −0.178689 −0.0893447 0.996001i $$-0.528477\pi$$
−0.0893447 + 0.996001i $$0.528477\pi$$
$$24$$ 1.92474 0.392885
$$25$$ 1.00000 0.200000
$$26$$ −5.24489 −1.02861
$$27$$ −4.41803 −0.850250
$$28$$ −1.00000 −0.188982
$$29$$ −8.37564 −1.55532 −0.777658 0.628687i $$-0.783593\pi$$
−0.777658 + 0.628687i $$0.783593\pi$$
$$30$$ 1.92474 0.351407
$$31$$ 1.10298 0.198101 0.0990506 0.995082i $$-0.468419\pi$$
0.0990506 + 0.995082i $$0.468419\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −4.30919 −0.739020
$$35$$ −1.00000 −0.169031
$$36$$ 0.704605 0.117434
$$37$$ −1.77921 −0.292500 −0.146250 0.989248i $$-0.546720\pi$$
−0.146250 + 0.989248i $$0.546720\pi$$
$$38$$ −0.740633 −0.120146
$$39$$ −10.0950 −1.61650
$$40$$ 1.00000 0.158114
$$41$$ −10.7200 −1.67418 −0.837090 0.547065i $$-0.815745\pi$$
−0.837090 + 0.547065i $$0.815745\pi$$
$$42$$ −1.92474 −0.296993
$$43$$ 1.13472 0.173044 0.0865218 0.996250i $$-0.472425\pi$$
0.0865218 + 0.996250i $$0.472425\pi$$
$$44$$ 0 0
$$45$$ 0.704605 0.105036
$$46$$ −0.856965 −0.126353
$$47$$ 8.40172 1.22552 0.612758 0.790271i $$-0.290060\pi$$
0.612758 + 0.790271i $$0.290060\pi$$
$$48$$ 1.92474 0.277812
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −8.29405 −1.16140
$$52$$ −5.24489 −0.727335
$$53$$ −8.80661 −1.20968 −0.604840 0.796347i $$-0.706763\pi$$
−0.604840 + 0.796347i $$0.706763\pi$$
$$54$$ −4.41803 −0.601217
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −1.42552 −0.188815
$$58$$ −8.37564 −1.09977
$$59$$ −12.9838 −1.69035 −0.845175 0.534490i $$-0.820504\pi$$
−0.845175 + 0.534490i $$0.820504\pi$$
$$60$$ 1.92474 0.248482
$$61$$ 13.0887 1.67584 0.837920 0.545793i $$-0.183772\pi$$
0.837920 + 0.545793i $$0.183772\pi$$
$$62$$ 1.10298 0.140079
$$63$$ −0.704605 −0.0887719
$$64$$ 1.00000 0.125000
$$65$$ −5.24489 −0.650548
$$66$$ 0 0
$$67$$ 8.93617 1.09173 0.545864 0.837874i $$-0.316202\pi$$
0.545864 + 0.837874i $$0.316202\pi$$
$$68$$ −4.30919 −0.522566
$$69$$ −1.64943 −0.198568
$$70$$ −1.00000 −0.119523
$$71$$ −4.50750 −0.534942 −0.267471 0.963566i $$-0.586188\pi$$
−0.267471 + 0.963566i $$0.586188\pi$$
$$72$$ 0.704605 0.0830385
$$73$$ −3.33413 −0.390230 −0.195115 0.980780i $$-0.562508\pi$$
−0.195115 + 0.980780i $$0.562508\pi$$
$$74$$ −1.77921 −0.206829
$$75$$ 1.92474 0.222249
$$76$$ −0.740633 −0.0849564
$$77$$ 0 0
$$78$$ −10.0950 −1.14304
$$79$$ −1.56183 −0.175719 −0.0878597 0.996133i $$-0.528003\pi$$
−0.0878597 + 0.996133i $$0.528003\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −10.6173 −1.17971
$$82$$ −10.7200 −1.18382
$$83$$ 0.609215 0.0668700 0.0334350 0.999441i $$-0.489355\pi$$
0.0334350 + 0.999441i $$0.489355\pi$$
$$84$$ −1.92474 −0.210006
$$85$$ −4.30919 −0.467397
$$86$$ 1.13472 0.122360
$$87$$ −16.1209 −1.72834
$$88$$ 0 0
$$89$$ 1.46836 0.155646 0.0778228 0.996967i $$-0.475203\pi$$
0.0778228 + 0.996967i $$0.475203\pi$$
$$90$$ 0.704605 0.0742719
$$91$$ 5.24489 0.549813
$$92$$ −0.856965 −0.0893447
$$93$$ 2.12295 0.220139
$$94$$ 8.40172 0.866571
$$95$$ −0.740633 −0.0759873
$$96$$ 1.92474 0.196442
$$97$$ 8.41705 0.854622 0.427311 0.904105i $$-0.359461\pi$$
0.427311 + 0.904105i $$0.359461\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −11.0295 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$102$$ −8.29405 −0.821233
$$103$$ 5.86953 0.578342 0.289171 0.957277i $$-0.406620\pi$$
0.289171 + 0.957277i $$0.406620\pi$$
$$104$$ −5.24489 −0.514303
$$105$$ −1.92474 −0.187835
$$106$$ −8.80661 −0.855373
$$107$$ 6.83162 0.660438 0.330219 0.943904i $$-0.392877\pi$$
0.330219 + 0.943904i $$0.392877\pi$$
$$108$$ −4.41803 −0.425125
$$109$$ −7.76784 −0.744025 −0.372012 0.928228i $$-0.621332\pi$$
−0.372012 + 0.928228i $$0.621332\pi$$
$$110$$ 0 0
$$111$$ −3.42451 −0.325040
$$112$$ −1.00000 −0.0944911
$$113$$ 4.63948 0.436446 0.218223 0.975899i $$-0.429974\pi$$
0.218223 + 0.975899i $$0.429974\pi$$
$$114$$ −1.42552 −0.133512
$$115$$ −0.856965 −0.0799124
$$116$$ −8.37564 −0.777658
$$117$$ −3.69557 −0.341656
$$118$$ −12.9838 −1.19526
$$119$$ 4.30919 0.395023
$$120$$ 1.92474 0.175703
$$121$$ 0 0
$$122$$ 13.0887 1.18500
$$123$$ −20.6331 −1.86043
$$124$$ 1.10298 0.0990506
$$125$$ 1.00000 0.0894427
$$126$$ −0.704605 −0.0627712
$$127$$ 9.42065 0.835947 0.417974 0.908459i $$-0.362740\pi$$
0.417974 + 0.908459i $$0.362740\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 2.18404 0.192294
$$130$$ −5.24489 −0.460007
$$131$$ −0.862825 −0.0753854 −0.0376927 0.999289i $$-0.512001\pi$$
−0.0376927 + 0.999289i $$0.512001\pi$$
$$132$$ 0 0
$$133$$ 0.740633 0.0642210
$$134$$ 8.93617 0.771968
$$135$$ −4.41803 −0.380243
$$136$$ −4.30919 −0.369510
$$137$$ −17.2469 −1.47350 −0.736751 0.676164i $$-0.763641\pi$$
−0.736751 + 0.676164i $$0.763641\pi$$
$$138$$ −1.64943 −0.140409
$$139$$ 20.4448 1.73411 0.867054 0.498213i $$-0.166010\pi$$
0.867054 + 0.498213i $$0.166010\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 16.1711 1.36185
$$142$$ −4.50750 −0.378261
$$143$$ 0 0
$$144$$ 0.704605 0.0587171
$$145$$ −8.37564 −0.695559
$$146$$ −3.33413 −0.275934
$$147$$ 1.92474 0.158749
$$148$$ −1.77921 −0.146250
$$149$$ 1.88298 0.154260 0.0771299 0.997021i $$-0.475424\pi$$
0.0771299 + 0.997021i $$0.475424\pi$$
$$150$$ 1.92474 0.157154
$$151$$ 5.45095 0.443592 0.221796 0.975093i $$-0.428808\pi$$
0.221796 + 0.975093i $$0.428808\pi$$
$$152$$ −0.740633 −0.0600732
$$153$$ −3.03628 −0.245468
$$154$$ 0 0
$$155$$ 1.10298 0.0885935
$$156$$ −10.0950 −0.808248
$$157$$ −11.9779 −0.955944 −0.477972 0.878375i $$-0.658628\pi$$
−0.477972 + 0.878375i $$0.658628\pi$$
$$158$$ −1.56183 −0.124252
$$159$$ −16.9504 −1.34425
$$160$$ 1.00000 0.0790569
$$161$$ 0.856965 0.0675383
$$162$$ −10.6173 −0.834178
$$163$$ −15.5543 −1.21830 −0.609152 0.793053i $$-0.708490\pi$$
−0.609152 + 0.793053i $$0.708490\pi$$
$$164$$ −10.7200 −0.837090
$$165$$ 0 0
$$166$$ 0.609215 0.0472843
$$167$$ 18.0474 1.39655 0.698274 0.715830i $$-0.253952\pi$$
0.698274 + 0.715830i $$0.253952\pi$$
$$168$$ −1.92474 −0.148497
$$169$$ 14.5088 1.11606
$$170$$ −4.30919 −0.330500
$$171$$ −0.521854 −0.0399071
$$172$$ 1.13472 0.0865218
$$173$$ −9.60665 −0.730380 −0.365190 0.930933i $$-0.618996\pi$$
−0.365190 + 0.930933i $$0.618996\pi$$
$$174$$ −16.1209 −1.22212
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −24.9904 −1.87839
$$178$$ 1.46836 0.110058
$$179$$ 13.2673 0.991643 0.495822 0.868424i $$-0.334867\pi$$
0.495822 + 0.868424i $$0.334867\pi$$
$$180$$ 0.704605 0.0525182
$$181$$ 19.5835 1.45563 0.727815 0.685774i $$-0.240536\pi$$
0.727815 + 0.685774i $$0.240536\pi$$
$$182$$ 5.24489 0.388777
$$183$$ 25.1923 1.86227
$$184$$ −0.856965 −0.0631763
$$185$$ −1.77921 −0.130810
$$186$$ 2.12295 0.155662
$$187$$ 0 0
$$188$$ 8.40172 0.612758
$$189$$ 4.41803 0.321364
$$190$$ −0.740633 −0.0537311
$$191$$ −12.7555 −0.922954 −0.461477 0.887152i $$-0.652680\pi$$
−0.461477 + 0.887152i $$0.652680\pi$$
$$192$$ 1.92474 0.138906
$$193$$ 11.8288 0.851459 0.425729 0.904851i $$-0.360018\pi$$
0.425729 + 0.904851i $$0.360018\pi$$
$$194$$ 8.41705 0.604309
$$195$$ −10.0950 −0.722919
$$196$$ 1.00000 0.0714286
$$197$$ −21.9105 −1.56106 −0.780528 0.625121i $$-0.785050\pi$$
−0.780528 + 0.625121i $$0.785050\pi$$
$$198$$ 0 0
$$199$$ 10.3121 0.731006 0.365503 0.930810i $$-0.380897\pi$$
0.365503 + 0.930810i $$0.380897\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 17.1998 1.21318
$$202$$ −11.0295 −0.776036
$$203$$ 8.37564 0.587854
$$204$$ −8.29405 −0.580700
$$205$$ −10.7200 −0.748716
$$206$$ 5.86953 0.408950
$$207$$ −0.603822 −0.0419685
$$208$$ −5.24489 −0.363667
$$209$$ 0 0
$$210$$ −1.92474 −0.132819
$$211$$ −24.8455 −1.71043 −0.855217 0.518270i $$-0.826576\pi$$
−0.855217 + 0.518270i $$0.826576\pi$$
$$212$$ −8.80661 −0.604840
$$213$$ −8.67574 −0.594452
$$214$$ 6.83162 0.467000
$$215$$ 1.13472 0.0773875
$$216$$ −4.41803 −0.300609
$$217$$ −1.10298 −0.0748752
$$218$$ −7.76784 −0.526105
$$219$$ −6.41731 −0.433641
$$220$$ 0 0
$$221$$ 22.6012 1.52032
$$222$$ −3.42451 −0.229838
$$223$$ −20.2394 −1.35533 −0.677667 0.735369i $$-0.737009\pi$$
−0.677667 + 0.735369i $$0.737009\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0.704605 0.0469737
$$226$$ 4.63948 0.308614
$$227$$ 11.8090 0.783792 0.391896 0.920009i $$-0.371819\pi$$
0.391896 + 0.920009i $$0.371819\pi$$
$$228$$ −1.42552 −0.0944075
$$229$$ 1.67906 0.110955 0.0554776 0.998460i $$-0.482332\pi$$
0.0554776 + 0.998460i $$0.482332\pi$$
$$230$$ −0.856965 −0.0565066
$$231$$ 0 0
$$232$$ −8.37564 −0.549887
$$233$$ −15.5220 −1.01688 −0.508440 0.861097i $$-0.669778\pi$$
−0.508440 + 0.861097i $$0.669778\pi$$
$$234$$ −3.69557 −0.241587
$$235$$ 8.40172 0.548067
$$236$$ −12.9838 −0.845175
$$237$$ −3.00610 −0.195267
$$238$$ 4.30919 0.279323
$$239$$ −22.0037 −1.42330 −0.711652 0.702532i $$-0.752053\pi$$
−0.711652 + 0.702532i $$0.752053\pi$$
$$240$$ 1.92474 0.124241
$$241$$ −23.8155 −1.53409 −0.767047 0.641591i $$-0.778274\pi$$
−0.767047 + 0.641591i $$0.778274\pi$$
$$242$$ 0 0
$$243$$ −7.18150 −0.460693
$$244$$ 13.0887 0.837920
$$245$$ 1.00000 0.0638877
$$246$$ −20.6331 −1.31552
$$247$$ 3.88453 0.247167
$$248$$ 1.10298 0.0700393
$$249$$ 1.17258 0.0743091
$$250$$ 1.00000 0.0632456
$$251$$ 28.5083 1.79943 0.899715 0.436477i $$-0.143774\pi$$
0.899715 + 0.436477i $$0.143774\pi$$
$$252$$ −0.704605 −0.0443859
$$253$$ 0 0
$$254$$ 9.42065 0.591104
$$255$$ −8.29405 −0.519393
$$256$$ 1.00000 0.0625000
$$257$$ 29.6194 1.84760 0.923802 0.382870i $$-0.125064\pi$$
0.923802 + 0.382870i $$0.125064\pi$$
$$258$$ 2.18404 0.135973
$$259$$ 1.77921 0.110555
$$260$$ −5.24489 −0.325274
$$261$$ −5.90151 −0.365295
$$262$$ −0.862825 −0.0533055
$$263$$ 4.09184 0.252314 0.126157 0.992010i $$-0.459736\pi$$
0.126157 + 0.992010i $$0.459736\pi$$
$$264$$ 0 0
$$265$$ −8.80661 −0.540986
$$266$$ 0.740633 0.0454111
$$267$$ 2.82620 0.172960
$$268$$ 8.93617 0.545864
$$269$$ 2.99735 0.182752 0.0913759 0.995816i $$-0.470874\pi$$
0.0913759 + 0.995816i $$0.470874\pi$$
$$270$$ −4.41803 −0.268873
$$271$$ 24.7015 1.50051 0.750255 0.661148i $$-0.229930\pi$$
0.750255 + 0.661148i $$0.229930\pi$$
$$272$$ −4.30919 −0.261283
$$273$$ 10.0950 0.610978
$$274$$ −17.2469 −1.04192
$$275$$ 0 0
$$276$$ −1.64943 −0.0992840
$$277$$ 2.67858 0.160941 0.0804703 0.996757i $$-0.474358\pi$$
0.0804703 + 0.996757i $$0.474358\pi$$
$$278$$ 20.4448 1.22620
$$279$$ 0.777166 0.0465277
$$280$$ −1.00000 −0.0597614
$$281$$ −11.6213 −0.693271 −0.346636 0.938000i $$-0.612676\pi$$
−0.346636 + 0.938000i $$0.612676\pi$$
$$282$$ 16.1711 0.962974
$$283$$ 18.1362 1.07808 0.539042 0.842279i $$-0.318786\pi$$
0.539042 + 0.842279i $$0.318786\pi$$
$$284$$ −4.50750 −0.267471
$$285$$ −1.42552 −0.0844406
$$286$$ 0 0
$$287$$ 10.7200 0.632781
$$288$$ 0.704605 0.0415192
$$289$$ 1.56912 0.0923010
$$290$$ −8.37564 −0.491834
$$291$$ 16.2006 0.949695
$$292$$ −3.33413 −0.195115
$$293$$ −22.8700 −1.33608 −0.668041 0.744125i $$-0.732867\pi$$
−0.668041 + 0.744125i $$0.732867\pi$$
$$294$$ 1.92474 0.112253
$$295$$ −12.9838 −0.755947
$$296$$ −1.77921 −0.103415
$$297$$ 0 0
$$298$$ 1.88298 0.109078
$$299$$ 4.49468 0.259934
$$300$$ 1.92474 0.111125
$$301$$ −1.13472 −0.0654044
$$302$$ 5.45095 0.313667
$$303$$ −21.2290 −1.21957
$$304$$ −0.740633 −0.0424782
$$305$$ 13.0887 0.749458
$$306$$ −3.03628 −0.173572
$$307$$ −19.9326 −1.13761 −0.568806 0.822472i $$-0.692594\pi$$
−0.568806 + 0.822472i $$0.692594\pi$$
$$308$$ 0 0
$$309$$ 11.2973 0.642681
$$310$$ 1.10298 0.0626451
$$311$$ −1.28437 −0.0728299 −0.0364149 0.999337i $$-0.511594\pi$$
−0.0364149 + 0.999337i $$0.511594\pi$$
$$312$$ −10.0950 −0.571518
$$313$$ −13.3964 −0.757210 −0.378605 0.925558i $$-0.623596\pi$$
−0.378605 + 0.925558i $$0.623596\pi$$
$$314$$ −11.9779 −0.675954
$$315$$ −0.704605 −0.0397000
$$316$$ −1.56183 −0.0878597
$$317$$ 16.6640 0.935941 0.467971 0.883744i $$-0.344985\pi$$
0.467971 + 0.883744i $$0.344985\pi$$
$$318$$ −16.9504 −0.950530
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 13.1491 0.733909
$$322$$ 0.856965 0.0477568
$$323$$ 3.19153 0.177581
$$324$$ −10.6173 −0.589853
$$325$$ −5.24489 −0.290934
$$326$$ −15.5543 −0.861471
$$327$$ −14.9510 −0.826795
$$328$$ −10.7200 −0.591912
$$329$$ −8.40172 −0.463202
$$330$$ 0 0
$$331$$ 11.1340 0.611979 0.305989 0.952035i $$-0.401013\pi$$
0.305989 + 0.952035i $$0.401013\pi$$
$$332$$ 0.609215 0.0334350
$$333$$ −1.25364 −0.0686991
$$334$$ 18.0474 0.987509
$$335$$ 8.93617 0.488235
$$336$$ −1.92474 −0.105003
$$337$$ −6.71838 −0.365974 −0.182987 0.983115i $$-0.558577\pi$$
−0.182987 + 0.983115i $$0.558577\pi$$
$$338$$ 14.5088 0.789176
$$339$$ 8.92978 0.484999
$$340$$ −4.30919 −0.233699
$$341$$ 0 0
$$342$$ −0.521854 −0.0282186
$$343$$ −1.00000 −0.0539949
$$344$$ 1.13472 0.0611802
$$345$$ −1.64943 −0.0888023
$$346$$ −9.60665 −0.516456
$$347$$ −19.4103 −1.04200 −0.520999 0.853557i $$-0.674441\pi$$
−0.520999 + 0.853557i $$0.674441\pi$$
$$348$$ −16.1209 −0.864170
$$349$$ 26.7685 1.43289 0.716444 0.697645i $$-0.245769\pi$$
0.716444 + 0.697645i $$0.245769\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 23.1720 1.23683
$$352$$ 0 0
$$353$$ −32.9268 −1.75252 −0.876259 0.481841i $$-0.839968\pi$$
−0.876259 + 0.481841i $$0.839968\pi$$
$$354$$ −24.9904 −1.32823
$$355$$ −4.50750 −0.239233
$$356$$ 1.46836 0.0778228
$$357$$ 8.29405 0.438968
$$358$$ 13.2673 0.701198
$$359$$ 36.1167 1.90617 0.953083 0.302709i $$-0.0978912\pi$$
0.953083 + 0.302709i $$0.0978912\pi$$
$$360$$ 0.704605 0.0371359
$$361$$ −18.4515 −0.971130
$$362$$ 19.5835 1.02929
$$363$$ 0 0
$$364$$ 5.24489 0.274907
$$365$$ −3.33413 −0.174516
$$366$$ 25.1923 1.31682
$$367$$ −8.35672 −0.436217 −0.218109 0.975925i $$-0.569989\pi$$
−0.218109 + 0.975925i $$0.569989\pi$$
$$368$$ −0.856965 −0.0446724
$$369$$ −7.55335 −0.393212
$$370$$ −1.77921 −0.0924968
$$371$$ 8.80661 0.457216
$$372$$ 2.12295 0.110070
$$373$$ −16.0205 −0.829509 −0.414755 0.909933i $$-0.636133\pi$$
−0.414755 + 0.909933i $$0.636133\pi$$
$$374$$ 0 0
$$375$$ 1.92474 0.0993929
$$376$$ 8.40172 0.433285
$$377$$ 43.9292 2.26247
$$378$$ 4.41803 0.227239
$$379$$ 25.3872 1.30405 0.652027 0.758196i $$-0.273919\pi$$
0.652027 + 0.758196i $$0.273919\pi$$
$$380$$ −0.740633 −0.0379937
$$381$$ 18.1323 0.928943
$$382$$ −12.7555 −0.652627
$$383$$ −0.303318 −0.0154988 −0.00774942 0.999970i $$-0.502467\pi$$
−0.00774942 + 0.999970i $$0.502467\pi$$
$$384$$ 1.92474 0.0982212
$$385$$ 0 0
$$386$$ 11.8288 0.602072
$$387$$ 0.799532 0.0406425
$$388$$ 8.41705 0.427311
$$389$$ −6.74058 −0.341761 −0.170880 0.985292i $$-0.554661\pi$$
−0.170880 + 0.985292i $$0.554661\pi$$
$$390$$ −10.0950 −0.511181
$$391$$ 3.69282 0.186754
$$392$$ 1.00000 0.0505076
$$393$$ −1.66071 −0.0837717
$$394$$ −21.9105 −1.10383
$$395$$ −1.56183 −0.0785841
$$396$$ 0 0
$$397$$ −19.9230 −0.999904 −0.499952 0.866053i $$-0.666649\pi$$
−0.499952 + 0.866053i $$0.666649\pi$$
$$398$$ 10.3121 0.516899
$$399$$ 1.42552 0.0713654
$$400$$ 1.00000 0.0500000
$$401$$ 34.0886 1.70230 0.851151 0.524920i $$-0.175905\pi$$
0.851151 + 0.524920i $$0.175905\pi$$
$$402$$ 17.1998 0.857846
$$403$$ −5.78501 −0.288172
$$404$$ −11.0295 −0.548740
$$405$$ −10.6173 −0.527580
$$406$$ 8.37564 0.415676
$$407$$ 0 0
$$408$$ −8.29405 −0.410617
$$409$$ 8.09670 0.400356 0.200178 0.979760i $$-0.435848\pi$$
0.200178 + 0.979760i $$0.435848\pi$$
$$410$$ −10.7200 −0.529422
$$411$$ −33.1957 −1.63742
$$412$$ 5.86953 0.289171
$$413$$ 12.9838 0.638892
$$414$$ −0.603822 −0.0296762
$$415$$ 0.609215 0.0299052
$$416$$ −5.24489 −0.257152
$$417$$ 39.3509 1.92702
$$418$$ 0 0
$$419$$ −24.0686 −1.17583 −0.587915 0.808923i $$-0.700051\pi$$
−0.587915 + 0.808923i $$0.700051\pi$$
$$420$$ −1.92474 −0.0939175
$$421$$ 13.7871 0.671942 0.335971 0.941872i $$-0.390936\pi$$
0.335971 + 0.941872i $$0.390936\pi$$
$$422$$ −24.8455 −1.20946
$$423$$ 5.91989 0.287835
$$424$$ −8.80661 −0.427687
$$425$$ −4.30919 −0.209026
$$426$$ −8.67574 −0.420341
$$427$$ −13.0887 −0.633408
$$428$$ 6.83162 0.330219
$$429$$ 0 0
$$430$$ 1.13472 0.0547212
$$431$$ −8.72005 −0.420030 −0.210015 0.977698i $$-0.567351\pi$$
−0.210015 + 0.977698i $$0.567351\pi$$
$$432$$ −4.41803 −0.212562
$$433$$ −8.51892 −0.409393 −0.204697 0.978825i $$-0.565621\pi$$
−0.204697 + 0.978825i $$0.565621\pi$$
$$434$$ −1.10298 −0.0529448
$$435$$ −16.1209 −0.772937
$$436$$ −7.76784 −0.372012
$$437$$ 0.634696 0.0303616
$$438$$ −6.41731 −0.306631
$$439$$ −22.5402 −1.07579 −0.537894 0.843013i $$-0.680780\pi$$
−0.537894 + 0.843013i $$0.680780\pi$$
$$440$$ 0 0
$$441$$ 0.704605 0.0335526
$$442$$ 22.6012 1.07503
$$443$$ 37.9089 1.80110 0.900552 0.434748i $$-0.143163\pi$$
0.900552 + 0.434748i $$0.143163\pi$$
$$444$$ −3.42451 −0.162520
$$445$$ 1.46836 0.0696068
$$446$$ −20.2394 −0.958365
$$447$$ 3.62424 0.171421
$$448$$ −1.00000 −0.0472456
$$449$$ −32.0551 −1.51277 −0.756387 0.654125i $$-0.773037\pi$$
−0.756387 + 0.654125i $$0.773037\pi$$
$$450$$ 0.704605 0.0332154
$$451$$ 0 0
$$452$$ 4.63948 0.218223
$$453$$ 10.4916 0.492940
$$454$$ 11.8090 0.554225
$$455$$ 5.24489 0.245884
$$456$$ −1.42552 −0.0667562
$$457$$ 24.5243 1.14720 0.573600 0.819136i $$-0.305547\pi$$
0.573600 + 0.819136i $$0.305547\pi$$
$$458$$ 1.67906 0.0784572
$$459$$ 19.0381 0.888623
$$460$$ −0.856965 −0.0399562
$$461$$ −0.188328 −0.00877132 −0.00438566 0.999990i $$-0.501396\pi$$
−0.00438566 + 0.999990i $$0.501396\pi$$
$$462$$ 0 0
$$463$$ −9.40956 −0.437299 −0.218650 0.975803i $$-0.570165\pi$$
−0.218650 + 0.975803i $$0.570165\pi$$
$$464$$ −8.37564 −0.388829
$$465$$ 2.12295 0.0984492
$$466$$ −15.5220 −0.719043
$$467$$ 1.84720 0.0854781 0.0427390 0.999086i $$-0.486392\pi$$
0.0427390 + 0.999086i $$0.486392\pi$$
$$468$$ −3.69557 −0.170828
$$469$$ −8.93617 −0.412634
$$470$$ 8.40172 0.387542
$$471$$ −23.0544 −1.06229
$$472$$ −12.9838 −0.597629
$$473$$ 0 0
$$474$$ −3.00610 −0.138075
$$475$$ −0.740633 −0.0339826
$$476$$ 4.30919 0.197511
$$477$$ −6.20518 −0.284116
$$478$$ −22.0037 −1.00643
$$479$$ 33.3506 1.52383 0.761913 0.647679i $$-0.224260\pi$$
0.761913 + 0.647679i $$0.224260\pi$$
$$480$$ 1.92474 0.0878517
$$481$$ 9.33176 0.425491
$$482$$ −23.8155 −1.08477
$$483$$ 1.64943 0.0750517
$$484$$ 0 0
$$485$$ 8.41705 0.382198
$$486$$ −7.18150 −0.325759
$$487$$ 14.1136 0.639550 0.319775 0.947493i $$-0.396393\pi$$
0.319775 + 0.947493i $$0.396393\pi$$
$$488$$ 13.0887 0.592499
$$489$$ −29.9378 −1.35384
$$490$$ 1.00000 0.0451754
$$491$$ −8.40881 −0.379484 −0.189742 0.981834i $$-0.560765\pi$$
−0.189742 + 0.981834i $$0.560765\pi$$
$$492$$ −20.6331 −0.930213
$$493$$ 36.0922 1.62551
$$494$$ 3.88453 0.174773
$$495$$ 0 0
$$496$$ 1.10298 0.0495253
$$497$$ 4.50750 0.202189
$$498$$ 1.17258 0.0525444
$$499$$ −22.8087 −1.02106 −0.510529 0.859860i $$-0.670550\pi$$
−0.510529 + 0.859860i $$0.670550\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 34.7364 1.55191
$$502$$ 28.5083 1.27239
$$503$$ −22.6508 −1.00995 −0.504975 0.863134i $$-0.668498\pi$$
−0.504975 + 0.863134i $$0.668498\pi$$
$$504$$ −0.704605 −0.0313856
$$505$$ −11.0295 −0.490808
$$506$$ 0 0
$$507$$ 27.9256 1.24022
$$508$$ 9.42065 0.417974
$$509$$ −15.0533 −0.667224 −0.333612 0.942710i $$-0.608268\pi$$
−0.333612 + 0.942710i $$0.608268\pi$$
$$510$$ −8.29405 −0.367267
$$511$$ 3.33413 0.147493
$$512$$ 1.00000 0.0441942
$$513$$ 3.27214 0.144468
$$514$$ 29.6194 1.30645
$$515$$ 5.86953 0.258643
$$516$$ 2.18404 0.0961471
$$517$$ 0 0
$$518$$ 1.77921 0.0781740
$$519$$ −18.4902 −0.811632
$$520$$ −5.24489 −0.230003
$$521$$ −7.47977 −0.327695 −0.163847 0.986486i $$-0.552390\pi$$
−0.163847 + 0.986486i $$0.552390\pi$$
$$522$$ −5.90151 −0.258302
$$523$$ −5.14535 −0.224990 −0.112495 0.993652i $$-0.535884\pi$$
−0.112495 + 0.993652i $$0.535884\pi$$
$$524$$ −0.862825 −0.0376927
$$525$$ −1.92474 −0.0840023
$$526$$ 4.09184 0.178413
$$527$$ −4.75295 −0.207042
$$528$$ 0 0
$$529$$ −22.2656 −0.968070
$$530$$ −8.80661 −0.382535
$$531$$ −9.14846 −0.397010
$$532$$ 0.740633 0.0321105
$$533$$ 56.2251 2.43538
$$534$$ 2.82620 0.122302
$$535$$ 6.83162 0.295357
$$536$$ 8.93617 0.385984
$$537$$ 25.5360 1.10196
$$538$$ 2.99735 0.129225
$$539$$ 0 0
$$540$$ −4.41803 −0.190122
$$541$$ −1.92055 −0.0825709 −0.0412855 0.999147i $$-0.513145\pi$$
−0.0412855 + 0.999147i $$0.513145\pi$$
$$542$$ 24.7015 1.06102
$$543$$ 37.6930 1.61756
$$544$$ −4.30919 −0.184755
$$545$$ −7.76784 −0.332738
$$546$$ 10.0950 0.432027
$$547$$ 0.260956 0.0111577 0.00557884 0.999984i $$-0.498224\pi$$
0.00557884 + 0.999984i $$0.498224\pi$$
$$548$$ −17.2469 −0.736751
$$549$$ 9.22238 0.393602
$$550$$ 0 0
$$551$$ 6.20327 0.264268
$$552$$ −1.64943 −0.0702044
$$553$$ 1.56183 0.0664157
$$554$$ 2.67858 0.113802
$$555$$ −3.42451 −0.145362
$$556$$ 20.4448 0.867054
$$557$$ −6.31867 −0.267731 −0.133865 0.991000i $$-0.542739\pi$$
−0.133865 + 0.991000i $$0.542739\pi$$
$$558$$ 0.777166 0.0329000
$$559$$ −5.95149 −0.251721
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −11.6213 −0.490217
$$563$$ 0.958697 0.0404043 0.0202021 0.999796i $$-0.493569\pi$$
0.0202021 + 0.999796i $$0.493569\pi$$
$$564$$ 16.1711 0.680925
$$565$$ 4.63948 0.195185
$$566$$ 18.1362 0.762321
$$567$$ 10.6173 0.445887
$$568$$ −4.50750 −0.189130
$$569$$ −7.11799 −0.298402 −0.149201 0.988807i $$-0.547670\pi$$
−0.149201 + 0.988807i $$0.547670\pi$$
$$570$$ −1.42552 −0.0597085
$$571$$ −27.1751 −1.13724 −0.568621 0.822599i $$-0.692523\pi$$
−0.568621 + 0.822599i $$0.692523\pi$$
$$572$$ 0 0
$$573$$ −24.5509 −1.02563
$$574$$ 10.7200 0.447443
$$575$$ −0.856965 −0.0357379
$$576$$ 0.704605 0.0293585
$$577$$ −32.3702 −1.34759 −0.673795 0.738918i $$-0.735337\pi$$
−0.673795 + 0.738918i $$0.735337\pi$$
$$578$$ 1.56912 0.0652667
$$579$$ 22.7674 0.946181
$$580$$ −8.37564 −0.347779
$$581$$ −0.609215 −0.0252745
$$582$$ 16.2006 0.671536
$$583$$ 0 0
$$584$$ −3.33413 −0.137967
$$585$$ −3.69557 −0.152793
$$586$$ −22.8700 −0.944753
$$587$$ −8.04888 −0.332213 −0.166106 0.986108i $$-0.553120\pi$$
−0.166106 + 0.986108i $$0.553120\pi$$
$$588$$ 1.92474 0.0793747
$$589$$ −0.816904 −0.0336599
$$590$$ −12.9838 −0.534535
$$591$$ −42.1718 −1.73472
$$592$$ −1.77921 −0.0731251
$$593$$ −39.8151 −1.63501 −0.817506 0.575920i $$-0.804644\pi$$
−0.817506 + 0.575920i $$0.804644\pi$$
$$594$$ 0 0
$$595$$ 4.30919 0.176660
$$596$$ 1.88298 0.0771299
$$597$$ 19.8481 0.812327
$$598$$ 4.49468 0.183801
$$599$$ 30.7925 1.25815 0.629074 0.777346i $$-0.283434\pi$$
0.629074 + 0.777346i $$0.283434\pi$$
$$600$$ 1.92474 0.0785770
$$601$$ 33.3140 1.35891 0.679454 0.733718i $$-0.262217\pi$$
0.679454 + 0.733718i $$0.262217\pi$$
$$602$$ −1.13472 −0.0462479
$$603$$ 6.29647 0.256412
$$604$$ 5.45095 0.221796
$$605$$ 0 0
$$606$$ −21.2290 −0.862367
$$607$$ −16.8748 −0.684927 −0.342463 0.939531i $$-0.611261\pi$$
−0.342463 + 0.939531i $$0.611261\pi$$
$$608$$ −0.740633 −0.0300366
$$609$$ 16.1209 0.653251
$$610$$ 13.0887 0.529947
$$611$$ −44.0660 −1.78272
$$612$$ −3.03628 −0.122734
$$613$$ 26.7574 1.08072 0.540360 0.841434i $$-0.318288\pi$$
0.540360 + 0.841434i $$0.318288\pi$$
$$614$$ −19.9326 −0.804413
$$615$$ −20.6331 −0.832008
$$616$$ 0 0
$$617$$ 32.5387 1.30996 0.654979 0.755647i $$-0.272677\pi$$
0.654979 + 0.755647i $$0.272677\pi$$
$$618$$ 11.2973 0.454444
$$619$$ −13.9252 −0.559701 −0.279850 0.960044i $$-0.590285\pi$$
−0.279850 + 0.960044i $$0.590285\pi$$
$$620$$ 1.10298 0.0442968
$$621$$ 3.78609 0.151931
$$622$$ −1.28437 −0.0514985
$$623$$ −1.46836 −0.0588285
$$624$$ −10.0950 −0.404124
$$625$$ 1.00000 0.0400000
$$626$$ −13.3964 −0.535429
$$627$$ 0 0
$$628$$ −11.9779 −0.477972
$$629$$ 7.66696 0.305702
$$630$$ −0.704605 −0.0280721
$$631$$ −12.8350 −0.510954 −0.255477 0.966815i $$-0.582232\pi$$
−0.255477 + 0.966815i $$0.582232\pi$$
$$632$$ −1.56183 −0.0621262
$$633$$ −47.8210 −1.90071
$$634$$ 16.6640 0.661810
$$635$$ 9.42065 0.373847
$$636$$ −16.9504 −0.672126
$$637$$ −5.24489 −0.207810
$$638$$ 0 0
$$639$$ −3.17601 −0.125641
$$640$$ 1.00000 0.0395285
$$641$$ −48.5629 −1.91812 −0.959059 0.283207i $$-0.908602\pi$$
−0.959059 + 0.283207i $$0.908602\pi$$
$$642$$ 13.1491 0.518952
$$643$$ −14.1751 −0.559009 −0.279505 0.960144i $$-0.590170\pi$$
−0.279505 + 0.960144i $$0.590170\pi$$
$$644$$ 0.856965 0.0337691
$$645$$ 2.18404 0.0859966
$$646$$ 3.19153 0.125569
$$647$$ −24.7982 −0.974917 −0.487459 0.873146i $$-0.662076\pi$$
−0.487459 + 0.873146i $$0.662076\pi$$
$$648$$ −10.6173 −0.417089
$$649$$ 0 0
$$650$$ −5.24489 −0.205721
$$651$$ −2.12295 −0.0832048
$$652$$ −15.5543 −0.609152
$$653$$ −29.0557 −1.13704 −0.568519 0.822670i $$-0.692483\pi$$
−0.568519 + 0.822670i $$0.692483\pi$$
$$654$$ −14.9510 −0.584632
$$655$$ −0.862825 −0.0337134
$$656$$ −10.7200 −0.418545
$$657$$ −2.34924 −0.0916526
$$658$$ −8.40172 −0.327533
$$659$$ 44.0463 1.71580 0.857899 0.513818i $$-0.171769\pi$$
0.857899 + 0.513818i $$0.171769\pi$$
$$660$$ 0 0
$$661$$ −0.575059 −0.0223672 −0.0111836 0.999937i $$-0.503560\pi$$
−0.0111836 + 0.999937i $$0.503560\pi$$
$$662$$ 11.1340 0.432734
$$663$$ 43.5013 1.68945
$$664$$ 0.609215 0.0236421
$$665$$ 0.740633 0.0287205
$$666$$ −1.25364 −0.0485776
$$667$$ 7.17762 0.277919
$$668$$ 18.0474 0.698274
$$669$$ −38.9556 −1.50611
$$670$$ 8.93617 0.345234
$$671$$ 0 0
$$672$$ −1.92474 −0.0742483
$$673$$ −27.7568 −1.06995 −0.534974 0.844869i $$-0.679679\pi$$
−0.534974 + 0.844869i $$0.679679\pi$$
$$674$$ −6.71838 −0.258782
$$675$$ −4.41803 −0.170050
$$676$$ 14.5088 0.558032
$$677$$ 51.1859 1.96723 0.983617 0.180272i $$-0.0576979\pi$$
0.983617 + 0.180272i $$0.0576979\pi$$
$$678$$ 8.92978 0.342946
$$679$$ −8.41705 −0.323017
$$680$$ −4.30919 −0.165250
$$681$$ 22.7292 0.870986
$$682$$ 0 0
$$683$$ −1.55522 −0.0595090 −0.0297545 0.999557i $$-0.509473\pi$$
−0.0297545 + 0.999557i $$0.509473\pi$$
$$684$$ −0.521854 −0.0199536
$$685$$ −17.2469 −0.658970
$$686$$ −1.00000 −0.0381802
$$687$$ 3.23174 0.123299
$$688$$ 1.13472 0.0432609
$$689$$ 46.1896 1.75969
$$690$$ −1.64943 −0.0627927
$$691$$ 4.60492 0.175179 0.0875896 0.996157i $$-0.472084\pi$$
0.0875896 + 0.996157i $$0.472084\pi$$
$$692$$ −9.60665 −0.365190
$$693$$ 0 0
$$694$$ −19.4103 −0.736804
$$695$$ 20.4448 0.775517
$$696$$ −16.1209 −0.611060
$$697$$ 46.1944 1.74974
$$698$$ 26.7685 1.01320
$$699$$ −29.8757 −1.13000
$$700$$ −1.00000 −0.0377964
$$701$$ −2.91484 −0.110092 −0.0550460 0.998484i $$-0.517531\pi$$
−0.0550460 + 0.998484i $$0.517531\pi$$
$$702$$ 23.1720 0.874573
$$703$$ 1.31774 0.0496996
$$704$$ 0 0
$$705$$ 16.1711 0.609038
$$706$$ −32.9268 −1.23922
$$707$$ 11.0295 0.414809
$$708$$ −24.9904 −0.939197
$$709$$ 3.62357 0.136086 0.0680430 0.997682i $$-0.478324\pi$$
0.0680430 + 0.997682i $$0.478324\pi$$
$$710$$ −4.50750 −0.169163
$$711$$ −1.10047 −0.0412709
$$712$$ 1.46836 0.0550290
$$713$$ −0.945215 −0.0353986
$$714$$ 8.29405 0.310397
$$715$$ 0 0
$$716$$ 13.2673 0.495822
$$717$$ −42.3514 −1.58164
$$718$$ 36.1167 1.34786
$$719$$ −8.72290 −0.325309 −0.162655 0.986683i $$-0.552006\pi$$
−0.162655 + 0.986683i $$0.552006\pi$$
$$720$$ 0.704605 0.0262591
$$721$$ −5.86953 −0.218593
$$722$$ −18.4515 −0.686692
$$723$$ −45.8386 −1.70476
$$724$$ 19.5835 0.727815
$$725$$ −8.37564 −0.311063
$$726$$ 0 0
$$727$$ 37.3206 1.38415 0.692073 0.721828i $$-0.256698\pi$$
0.692073 + 0.721828i $$0.256698\pi$$
$$728$$ 5.24489 0.194388
$$729$$ 18.0296 0.667761
$$730$$ −3.33413 −0.123402
$$731$$ −4.88974 −0.180854
$$732$$ 25.1923 0.931136
$$733$$ 8.80262 0.325132 0.162566 0.986698i $$-0.448023\pi$$
0.162566 + 0.986698i $$0.448023\pi$$
$$734$$ −8.35672 −0.308452
$$735$$ 1.92474 0.0709949
$$736$$ −0.856965 −0.0315881
$$737$$ 0 0
$$738$$ −7.55335 −0.278043
$$739$$ 50.6335 1.86258 0.931292 0.364274i $$-0.118683\pi$$
0.931292 + 0.364274i $$0.118683\pi$$
$$740$$ −1.77921 −0.0654051
$$741$$ 7.47670 0.274663
$$742$$ 8.80661 0.323301
$$743$$ −50.4494 −1.85081 −0.925404 0.378983i $$-0.876274\pi$$
−0.925404 + 0.378983i $$0.876274\pi$$
$$744$$ 2.12295 0.0778310
$$745$$ 1.88298 0.0689871
$$746$$ −16.0205 −0.586552
$$747$$ 0.429256 0.0157057
$$748$$ 0 0
$$749$$ −6.83162 −0.249622
$$750$$ 1.92474 0.0702814
$$751$$ −23.9655 −0.874512 −0.437256 0.899337i $$-0.644050\pi$$
−0.437256 + 0.899337i $$0.644050\pi$$
$$752$$ 8.40172 0.306379
$$753$$ 54.8710 1.99961
$$754$$ 43.9292 1.59981
$$755$$ 5.45095 0.198380
$$756$$ 4.41803 0.160682
$$757$$ −4.87670 −0.177247 −0.0886233 0.996065i $$-0.528247\pi$$
−0.0886233 + 0.996065i $$0.528247\pi$$
$$758$$ 25.3872 0.922105
$$759$$ 0 0
$$760$$ −0.740633 −0.0268656
$$761$$ −44.6872 −1.61991 −0.809955 0.586492i $$-0.800509\pi$$
−0.809955 + 0.586492i $$0.800509\pi$$
$$762$$ 18.1323 0.656862
$$763$$ 7.76784 0.281215
$$764$$ −12.7555 −0.461477
$$765$$ −3.03628 −0.109777
$$766$$ −0.303318 −0.0109593
$$767$$ 68.0986 2.45890
$$768$$ 1.92474 0.0694529
$$769$$ 52.9611 1.90983 0.954914 0.296884i $$-0.0959474\pi$$
0.954914 + 0.296884i $$0.0959474\pi$$
$$770$$ 0 0
$$771$$ 57.0094 2.05314
$$772$$ 11.8288 0.425729
$$773$$ 38.4372 1.38249 0.691246 0.722620i $$-0.257062\pi$$
0.691246 + 0.722620i $$0.257062\pi$$
$$774$$ 0.799532 0.0287386
$$775$$ 1.10298 0.0396202
$$776$$ 8.41705 0.302154
$$777$$ 3.42451 0.122854
$$778$$ −6.74058 −0.241661
$$779$$ 7.93957 0.284465
$$780$$ −10.0950 −0.361459
$$781$$ 0 0
$$782$$ 3.69282 0.132055
$$783$$ 37.0038 1.32241
$$784$$ 1.00000 0.0357143
$$785$$ −11.9779 −0.427511
$$786$$ −1.66071 −0.0592356
$$787$$ −11.9075 −0.424458 −0.212229 0.977220i $$-0.568072\pi$$
−0.212229 + 0.977220i $$0.568072\pi$$
$$788$$ −21.9105 −0.780528
$$789$$ 7.87570 0.280383
$$790$$ −1.56183 −0.0555673
$$791$$ −4.63948 −0.164961
$$792$$ 0 0
$$793$$ −68.6489 −2.43779
$$794$$ −19.9230 −0.707039
$$795$$ −16.9504 −0.601168
$$796$$ 10.3121 0.365503
$$797$$ −2.09798 −0.0743142 −0.0371571 0.999309i $$-0.511830\pi$$
−0.0371571 + 0.999309i $$0.511830\pi$$
$$798$$ 1.42552 0.0504629
$$799$$ −36.2046 −1.28083
$$800$$ 1.00000 0.0353553
$$801$$ 1.03461 0.0365562
$$802$$ 34.0886 1.20371
$$803$$ 0 0
$$804$$ 17.1998 0.606589
$$805$$ 0.856965 0.0302040
$$806$$ −5.78501 −0.203768
$$807$$ 5.76911 0.203082
$$808$$ −11.0295 −0.388018
$$809$$ −12.0315 −0.423005 −0.211503 0.977377i $$-0.567836\pi$$
−0.211503 + 0.977377i $$0.567836\pi$$
$$810$$ −10.6173 −0.373056
$$811$$ −8.52991 −0.299526 −0.149763 0.988722i $$-0.547851\pi$$
−0.149763 + 0.988722i $$0.547851\pi$$
$$812$$ 8.37564 0.293927
$$813$$ 47.5439 1.66744
$$814$$ 0 0
$$815$$ −15.5543 −0.544842
$$816$$ −8.29405 −0.290350
$$817$$ −0.840413 −0.0294023
$$818$$ 8.09670 0.283094
$$819$$ 3.69557 0.129134
$$820$$ −10.7200 −0.374358
$$821$$ 49.8308 1.73911 0.869554 0.493839i $$-0.164407\pi$$
0.869554 + 0.493839i $$0.164407\pi$$
$$822$$ −33.1957 −1.15783
$$823$$ −14.8465 −0.517516 −0.258758 0.965942i $$-0.583313\pi$$
−0.258758 + 0.965942i $$0.583313\pi$$
$$824$$ 5.86953 0.204475
$$825$$ 0 0
$$826$$ 12.9838 0.451765
$$827$$ −1.52007 −0.0528580 −0.0264290 0.999651i $$-0.508414\pi$$
−0.0264290 + 0.999651i $$0.508414\pi$$
$$828$$ −0.603822 −0.0209842
$$829$$ −10.4514 −0.362993 −0.181497 0.983392i $$-0.558094\pi$$
−0.181497 + 0.983392i $$0.558094\pi$$
$$830$$ 0.609215 0.0211462
$$831$$ 5.15556 0.178845
$$832$$ −5.24489 −0.181834
$$833$$ −4.30919 −0.149305
$$834$$ 39.3509 1.36261
$$835$$ 18.0474 0.624555
$$836$$ 0 0
$$837$$ −4.87300 −0.168435
$$838$$ −24.0686 −0.831437
$$839$$ −39.9761 −1.38013 −0.690064 0.723748i $$-0.742418\pi$$
−0.690064 + 0.723748i $$0.742418\pi$$
$$840$$ −1.92474 −0.0664097
$$841$$ 41.1513 1.41901
$$842$$ 13.7871 0.475135
$$843$$ −22.3680 −0.770395
$$844$$ −24.8455 −0.855217
$$845$$ 14.5088 0.499119
$$846$$ 5.91989 0.203530
$$847$$ 0 0
$$848$$ −8.80661 −0.302420
$$849$$ 34.9074 1.19802
$$850$$ −4.30919 −0.147804
$$851$$ 1.52472 0.0522668
$$852$$ −8.67574 −0.297226
$$853$$ −22.2228 −0.760893 −0.380446 0.924803i $$-0.624230\pi$$
−0.380446 + 0.924803i $$0.624230\pi$$
$$854$$ −13.0887 −0.447887
$$855$$ −0.521854 −0.0178470
$$856$$ 6.83162 0.233500
$$857$$ −15.8282 −0.540681 −0.270340 0.962765i $$-0.587136\pi$$
−0.270340 + 0.962765i $$0.587136\pi$$
$$858$$ 0 0
$$859$$ −20.7983 −0.709628 −0.354814 0.934937i $$-0.615456\pi$$
−0.354814 + 0.934937i $$0.615456\pi$$
$$860$$ 1.13472 0.0386937
$$861$$ 20.6331 0.703175
$$862$$ −8.72005 −0.297006
$$863$$ −18.8317 −0.641040 −0.320520 0.947242i $$-0.603858\pi$$
−0.320520 + 0.947242i $$0.603858\pi$$
$$864$$ −4.41803 −0.150304
$$865$$ −9.60665 −0.326636
$$866$$ −8.51892 −0.289485
$$867$$ 3.02013 0.102569
$$868$$ −1.10298 −0.0374376
$$869$$ 0 0
$$870$$ −16.1209 −0.546549
$$871$$ −46.8692 −1.58810
$$872$$ −7.76784 −0.263052
$$873$$ 5.93069 0.200724
$$874$$ 0.634696 0.0214689
$$875$$ −1.00000 −0.0338062
$$876$$ −6.41731 −0.216821
$$877$$ −16.6176 −0.561138 −0.280569 0.959834i $$-0.590523\pi$$
−0.280569 + 0.959834i $$0.590523\pi$$
$$878$$ −22.5402 −0.760697
$$879$$ −44.0188 −1.48472
$$880$$ 0 0
$$881$$ 21.6943 0.730901 0.365450 0.930831i $$-0.380915\pi$$
0.365450 + 0.930831i $$0.380915\pi$$
$$882$$ 0.704605 0.0237253
$$883$$ 32.8896 1.10682 0.553412 0.832907i $$-0.313325\pi$$
0.553412 + 0.832907i $$0.313325\pi$$
$$884$$ 22.6012 0.760161
$$885$$ −24.9904 −0.840044
$$886$$ 37.9089 1.27357
$$887$$ 4.02388 0.135109 0.0675544 0.997716i $$-0.478480\pi$$
0.0675544 + 0.997716i $$0.478480\pi$$
$$888$$ −3.42451 −0.114919
$$889$$ −9.42065 −0.315958
$$890$$ 1.46836 0.0492194
$$891$$ 0 0
$$892$$ −20.2394 −0.677667
$$893$$ −6.22259 −0.208231
$$894$$ 3.62424 0.121213
$$895$$ 13.2673 0.443476
$$896$$ −1.00000 −0.0334077
$$897$$ 8.65107 0.288851
$$898$$ −32.0551 −1.06969
$$899$$ −9.23816 −0.308110
$$900$$ 0.704605 0.0234868
$$901$$ 37.9493 1.26428
$$902$$ 0 0
$$903$$ −2.18404 −0.0726804
$$904$$ 4.63948 0.154307
$$905$$ 19.5835 0.650977
$$906$$ 10.4916 0.348561
$$907$$ 20.4539 0.679162 0.339581 0.940577i $$-0.389715\pi$$
0.339581 + 0.940577i $$0.389715\pi$$
$$908$$ 11.8090 0.391896
$$909$$ −7.77147 −0.257763
$$910$$ 5.24489 0.173866
$$911$$ 27.9283 0.925305 0.462653 0.886540i $$-0.346898\pi$$
0.462653 + 0.886540i $$0.346898\pi$$
$$912$$ −1.42552 −0.0472037
$$913$$ 0 0
$$914$$ 24.5243 0.811192
$$915$$ 25.1923 0.832833
$$916$$ 1.67906 0.0554776
$$917$$ 0.862825 0.0284930
$$918$$ 19.0381 0.628352
$$919$$ −13.3202 −0.439394 −0.219697 0.975568i $$-0.570507\pi$$
−0.219697 + 0.975568i $$0.570507\pi$$
$$920$$ −0.856965 −0.0282533
$$921$$ −38.3649 −1.26417
$$922$$ −0.188328 −0.00620226
$$923$$ 23.6413 0.778163
$$924$$ 0 0
$$925$$ −1.77921 −0.0585001
$$926$$ −9.40956 −0.309217
$$927$$ 4.13570 0.135834
$$928$$ −8.37564 −0.274944
$$929$$ 39.2113 1.28648 0.643240 0.765664i $$-0.277590\pi$$
0.643240 + 0.765664i $$0.277590\pi$$
$$930$$ 2.12295 0.0696141
$$931$$ −0.740633 −0.0242733
$$932$$ −15.5220 −0.508440
$$933$$ −2.47207 −0.0809319
$$934$$ 1.84720 0.0604421
$$935$$ 0 0
$$936$$ −3.69557 −0.120794
$$937$$ −21.7403 −0.710225 −0.355112 0.934824i $$-0.615557\pi$$
−0.355112 + 0.934824i $$0.615557\pi$$
$$938$$ −8.93617 −0.291776
$$939$$ −25.7846 −0.841447
$$940$$ 8.40172 0.274034
$$941$$ 16.4125 0.535033 0.267516 0.963553i $$-0.413797\pi$$
0.267516 + 0.963553i $$0.413797\pi$$
$$942$$ −23.0544 −0.751152
$$943$$ 9.18665 0.299158
$$944$$ −12.9838 −0.422587
$$945$$ 4.41803 0.143718
$$946$$ 0 0
$$947$$ −45.1297 −1.46652 −0.733259 0.679949i $$-0.762002\pi$$
−0.733259 + 0.679949i $$0.762002\pi$$
$$948$$ −3.00610 −0.0976337
$$949$$ 17.4871 0.567655
$$950$$ −0.740633 −0.0240293
$$951$$ 32.0737 1.04006
$$952$$ 4.30919 0.139662
$$953$$ −46.8181 −1.51659 −0.758294 0.651913i $$-0.773967\pi$$
−0.758294 + 0.651913i $$0.773967\pi$$
$$954$$ −6.20518 −0.200900
$$955$$ −12.7555 −0.412758
$$956$$ −22.0037 −0.711652
$$957$$ 0 0
$$958$$ 33.3506 1.07751
$$959$$ 17.2469 0.556931
$$960$$ 1.92474 0.0621206
$$961$$ −29.7834 −0.960756
$$962$$ 9.33176 0.300868
$$963$$ 4.81359 0.155116
$$964$$ −23.8155 −0.767047
$$965$$ 11.8288 0.380784
$$966$$ 1.64943 0.0530695
$$967$$ −57.9645 −1.86401 −0.932007 0.362441i $$-0.881943\pi$$
−0.932007 + 0.362441i $$0.881943\pi$$
$$968$$ 0 0
$$969$$ 6.14284 0.197337
$$970$$ 8.41705 0.270255
$$971$$ −10.6264 −0.341019 −0.170509 0.985356i $$-0.554541\pi$$
−0.170509 + 0.985356i $$0.554541\pi$$
$$972$$ −7.18150 −0.230347
$$973$$ −20.4448 −0.655432
$$974$$ 14.1136 0.452230
$$975$$ −10.0950 −0.323299
$$976$$ 13.0887 0.418960
$$977$$ −33.6202 −1.07561 −0.537803 0.843071i $$-0.680745\pi$$
−0.537803 + 0.843071i $$0.680745\pi$$
$$978$$ −29.9378 −0.957307
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ −5.47326 −0.174748
$$982$$ −8.40881 −0.268336
$$983$$ 38.3652 1.22366 0.611830 0.790989i $$-0.290434\pi$$
0.611830 + 0.790989i $$0.290434\pi$$
$$984$$ −20.6331 −0.657760
$$985$$ −21.9105 −0.698125
$$986$$ 36.0922 1.14941
$$987$$ −16.1711 −0.514731
$$988$$ 3.88453 0.123583
$$989$$ −0.972418 −0.0309211
$$990$$ 0 0
$$991$$ 55.5999 1.76619 0.883094 0.469196i $$-0.155456\pi$$
0.883094 + 0.469196i $$0.155456\pi$$
$$992$$ 1.10298 0.0350197
$$993$$ 21.4300 0.680059
$$994$$ 4.50750 0.142969
$$995$$ 10.3121 0.326916
$$996$$ 1.17258 0.0371545
$$997$$ 22.9698 0.727462 0.363731 0.931504i $$-0.381503\pi$$
0.363731 + 0.931504i $$0.381503\pi$$
$$998$$ −22.8087 −0.721997
$$999$$ 7.86060 0.248698
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.db.1.6 6
11.5 even 5 770.2.n.g.421.3 12
11.9 even 5 770.2.n.g.631.3 yes 12
11.10 odd 2 8470.2.a.cv.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.g.421.3 12 11.5 even 5
770.2.n.g.631.3 yes 12 11.9 even 5
8470.2.a.cv.1.6 6 11.10 odd 2
8470.2.a.db.1.6 6 1.1 even 1 trivial