# Properties

 Label 8470.2.a.dd.1.5 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: 6.6.745749504.1 Defining polynomial: $$x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-2.23874$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.23874 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23874 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.01195 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.23874 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.23874 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.01195 q^{9} -1.00000 q^{10} +2.23874 q^{12} +5.12829 q^{13} -1.00000 q^{14} -2.23874 q^{15} +1.00000 q^{16} -6.65568 q^{17} +2.01195 q^{18} -1.15751 q^{19} -1.00000 q^{20} -2.23874 q^{21} +3.87761 q^{23} +2.23874 q^{24} +1.00000 q^{25} +5.12829 q^{26} -2.21199 q^{27} -1.00000 q^{28} +5.89442 q^{29} -2.23874 q^{30} -6.64373 q^{31} +1.00000 q^{32} -6.65568 q^{34} +1.00000 q^{35} +2.01195 q^{36} +11.8944 q^{37} -1.15751 q^{38} +11.4809 q^{39} -1.00000 q^{40} +10.4700 q^{41} -2.23874 q^{42} +1.77464 q^{43} -2.01195 q^{45} +3.87761 q^{46} +8.71479 q^{47} +2.23874 q^{48} +1.00000 q^{49} +1.00000 q^{50} -14.9003 q^{51} +5.12829 q^{52} +8.28476 q^{53} -2.21199 q^{54} -1.00000 q^{56} -2.59135 q^{57} +5.89442 q^{58} -10.2614 q^{59} -2.23874 q^{60} -12.8132 q^{61} -6.64373 q^{62} -2.01195 q^{63} +1.00000 q^{64} -5.12829 q^{65} +14.9682 q^{67} -6.65568 q^{68} +8.68095 q^{69} +1.00000 q^{70} +7.76716 q^{71} +2.01195 q^{72} -2.54037 q^{73} +11.8944 q^{74} +2.23874 q^{75} -1.15751 q^{76} +11.4809 q^{78} +10.2605 q^{79} -1.00000 q^{80} -10.9879 q^{81} +10.4700 q^{82} -6.86423 q^{83} -2.23874 q^{84} +6.65568 q^{85} +1.77464 q^{86} +13.1961 q^{87} +15.1923 q^{89} -2.01195 q^{90} -5.12829 q^{91} +3.87761 q^{92} -14.8736 q^{93} +8.71479 q^{94} +1.15751 q^{95} +2.23874 q^{96} -0.406766 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} + O(q^{10})$$ $$6q + 6q^{2} + 6q^{4} - 6q^{5} - 6q^{7} + 6q^{8} + 18q^{9} - 6q^{10} - 6q^{14} + 6q^{16} - 6q^{17} + 18q^{18} - 6q^{20} + 6q^{25} - 12q^{27} - 6q^{28} - 12q^{29} + 6q^{32} - 6q^{34} + 6q^{35} + 18q^{36} + 24q^{37} + 24q^{39} - 6q^{40} - 12q^{41} + 18q^{43} - 18q^{45} + 24q^{47} + 6q^{49} + 6q^{50} + 12q^{51} + 36q^{53} - 12q^{54} - 6q^{56} + 12q^{57} - 12q^{58} + 30q^{59} - 36q^{61} - 18q^{63} + 6q^{64} - 12q^{67} - 6q^{68} + 6q^{70} + 6q^{71} + 18q^{72} + 6q^{73} + 24q^{74} + 24q^{78} + 24q^{79} - 6q^{80} + 54q^{81} - 12q^{82} - 24q^{83} + 6q^{85} + 18q^{86} + 24q^{87} + 36q^{89} - 18q^{90} + 24q^{94} + 6q^{98} + O(q^{100})$$

## 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
$$3$$ 2.23874 1.29254 0.646268 0.763111i $$-0.276329\pi$$
0.646268 + 0.763111i $$0.276329\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 2.23874 0.913961
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 2.01195 0.670649
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 2.23874 0.646268
$$13$$ 5.12829 1.42233 0.711167 0.703024i $$-0.248167\pi$$
0.711167 + 0.703024i $$0.248167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −2.23874 −0.578040
$$16$$ 1.00000 0.250000
$$17$$ −6.65568 −1.61424 −0.807119 0.590388i $$-0.798975\pi$$
−0.807119 + 0.590388i $$0.798975\pi$$
$$18$$ 2.01195 0.474221
$$19$$ −1.15751 −0.265550 −0.132775 0.991146i $$-0.542389\pi$$
−0.132775 + 0.991146i $$0.542389\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −2.23874 −0.488533
$$22$$ 0 0
$$23$$ 3.87761 0.808537 0.404269 0.914640i $$-0.367526\pi$$
0.404269 + 0.914640i $$0.367526\pi$$
$$24$$ 2.23874 0.456981
$$25$$ 1.00000 0.200000
$$26$$ 5.12829 1.00574
$$27$$ −2.21199 −0.425697
$$28$$ −1.00000 −0.188982
$$29$$ 5.89442 1.09457 0.547283 0.836948i $$-0.315662\pi$$
0.547283 + 0.836948i $$0.315662\pi$$
$$30$$ −2.23874 −0.408736
$$31$$ −6.64373 −1.19325 −0.596624 0.802521i $$-0.703492\pi$$
−0.596624 + 0.802521i $$0.703492\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −6.65568 −1.14144
$$35$$ 1.00000 0.169031
$$36$$ 2.01195 0.335325
$$37$$ 11.8944 1.95543 0.977715 0.209937i $$-0.0673259\pi$$
0.977715 + 0.209937i $$0.0673259\pi$$
$$38$$ −1.15751 −0.187772
$$39$$ 11.4809 1.83842
$$40$$ −1.00000 −0.158114
$$41$$ 10.4700 1.63514 0.817570 0.575829i $$-0.195321\pi$$
0.817570 + 0.575829i $$0.195321\pi$$
$$42$$ −2.23874 −0.345445
$$43$$ 1.77464 0.270630 0.135315 0.990803i $$-0.456795\pi$$
0.135315 + 0.990803i $$0.456795\pi$$
$$44$$ 0 0
$$45$$ −2.01195 −0.299924
$$46$$ 3.87761 0.571722
$$47$$ 8.71479 1.27118 0.635591 0.772026i $$-0.280756\pi$$
0.635591 + 0.772026i $$0.280756\pi$$
$$48$$ 2.23874 0.323134
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −14.9003 −2.08646
$$52$$ 5.12829 0.711167
$$53$$ 8.28476 1.13800 0.568999 0.822338i $$-0.307331\pi$$
0.568999 + 0.822338i $$0.307331\pi$$
$$54$$ −2.21199 −0.301014
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −2.59135 −0.343233
$$58$$ 5.89442 0.773975
$$59$$ −10.2614 −1.33593 −0.667963 0.744194i $$-0.732834\pi$$
−0.667963 + 0.744194i $$0.732834\pi$$
$$60$$ −2.23874 −0.289020
$$61$$ −12.8132 −1.64056 −0.820280 0.571962i $$-0.806182\pi$$
−0.820280 + 0.571962i $$0.806182\pi$$
$$62$$ −6.64373 −0.843754
$$63$$ −2.01195 −0.253482
$$64$$ 1.00000 0.125000
$$65$$ −5.12829 −0.636087
$$66$$ 0 0
$$67$$ 14.9682 1.82865 0.914327 0.404977i $$-0.132720\pi$$
0.914327 + 0.404977i $$0.132720\pi$$
$$68$$ −6.65568 −0.807119
$$69$$ 8.68095 1.04506
$$70$$ 1.00000 0.119523
$$71$$ 7.76716 0.921793 0.460896 0.887454i $$-0.347528\pi$$
0.460896 + 0.887454i $$0.347528\pi$$
$$72$$ 2.01195 0.237110
$$73$$ −2.54037 −0.297328 −0.148664 0.988888i $$-0.547497\pi$$
−0.148664 + 0.988888i $$0.547497\pi$$
$$74$$ 11.8944 1.38270
$$75$$ 2.23874 0.258507
$$76$$ −1.15751 −0.132775
$$77$$ 0 0
$$78$$ 11.4809 1.29996
$$79$$ 10.2605 1.15439 0.577197 0.816605i $$-0.304146\pi$$
0.577197 + 0.816605i $$0.304146\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ −10.9879 −1.22088
$$82$$ 10.4700 1.15622
$$83$$ −6.86423 −0.753448 −0.376724 0.926326i $$-0.622949\pi$$
−0.376724 + 0.926326i $$0.622949\pi$$
$$84$$ −2.23874 −0.244266
$$85$$ 6.65568 0.721910
$$86$$ 1.77464 0.191364
$$87$$ 13.1961 1.41477
$$88$$ 0 0
$$89$$ 15.1923 1.61038 0.805188 0.593019i $$-0.202064\pi$$
0.805188 + 0.593019i $$0.202064\pi$$
$$90$$ −2.01195 −0.212078
$$91$$ −5.12829 −0.537591
$$92$$ 3.87761 0.404269
$$93$$ −14.8736 −1.54232
$$94$$ 8.71479 0.898862
$$95$$ 1.15751 0.118758
$$96$$ 2.23874 0.228490
$$97$$ −0.406766 −0.0413008 −0.0206504 0.999787i $$-0.506574\pi$$
−0.0206504 + 0.999787i $$0.506574\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 0.964155 0.0959370 0.0479685 0.998849i $$-0.484725\pi$$
0.0479685 + 0.998849i $$0.484725\pi$$
$$102$$ −14.9003 −1.47535
$$103$$ 12.4021 1.22202 0.611010 0.791623i $$-0.290764\pi$$
0.611010 + 0.791623i $$0.290764\pi$$
$$104$$ 5.12829 0.502871
$$105$$ 2.23874 0.218478
$$106$$ 8.28476 0.804687
$$107$$ 11.3466 1.09691 0.548457 0.836179i $$-0.315215\pi$$
0.548457 + 0.836179i $$0.315215\pi$$
$$108$$ −2.21199 −0.212849
$$109$$ −9.81672 −0.940271 −0.470135 0.882594i $$-0.655795\pi$$
−0.470135 + 0.882594i $$0.655795\pi$$
$$110$$ 0 0
$$111$$ 26.6285 2.52746
$$112$$ −1.00000 −0.0944911
$$113$$ −15.4263 −1.45118 −0.725591 0.688126i $$-0.758433\pi$$
−0.725591 + 0.688126i $$0.758433\pi$$
$$114$$ −2.59135 −0.242702
$$115$$ −3.87761 −0.361589
$$116$$ 5.89442 0.547283
$$117$$ 10.3179 0.953887
$$118$$ −10.2614 −0.944643
$$119$$ 6.65568 0.610125
$$120$$ −2.23874 −0.204368
$$121$$ 0 0
$$122$$ −12.8132 −1.16005
$$123$$ 23.4396 2.11348
$$124$$ −6.64373 −0.596624
$$125$$ −1.00000 −0.0894427
$$126$$ −2.01195 −0.179239
$$127$$ 15.5058 1.37592 0.687961 0.725748i $$-0.258506\pi$$
0.687961 + 0.725748i $$0.258506\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 3.97295 0.349798
$$130$$ −5.12829 −0.449781
$$131$$ −16.5157 −1.44299 −0.721493 0.692421i $$-0.756544\pi$$
−0.721493 + 0.692421i $$0.756544\pi$$
$$132$$ 0 0
$$133$$ 1.15751 0.100368
$$134$$ 14.9682 1.29305
$$135$$ 2.21199 0.190378
$$136$$ −6.65568 −0.570720
$$137$$ 13.8267 1.18129 0.590646 0.806931i $$-0.298873\pi$$
0.590646 + 0.806931i $$0.298873\pi$$
$$138$$ 8.68095 0.738971
$$139$$ 1.19890 0.101689 0.0508445 0.998707i $$-0.483809\pi$$
0.0508445 + 0.998707i $$0.483809\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 19.5101 1.64305
$$142$$ 7.76716 0.651806
$$143$$ 0 0
$$144$$ 2.01195 0.167662
$$145$$ −5.89442 −0.489505
$$146$$ −2.54037 −0.210243
$$147$$ 2.23874 0.184648
$$148$$ 11.8944 0.977715
$$149$$ −5.37555 −0.440382 −0.220191 0.975457i $$-0.570668\pi$$
−0.220191 + 0.975457i $$0.570668\pi$$
$$150$$ 2.23874 0.182792
$$151$$ 1.95930 0.159445 0.0797226 0.996817i $$-0.474597\pi$$
0.0797226 + 0.996817i $$0.474597\pi$$
$$152$$ −1.15751 −0.0938861
$$153$$ −13.3909 −1.08259
$$154$$ 0 0
$$155$$ 6.64373 0.533637
$$156$$ 11.4809 0.919208
$$157$$ 13.1054 1.04593 0.522964 0.852355i $$-0.324826\pi$$
0.522964 + 0.852355i $$0.324826\pi$$
$$158$$ 10.2605 0.816280
$$159$$ 18.5474 1.47090
$$160$$ −1.00000 −0.0790569
$$161$$ −3.87761 −0.305598
$$162$$ −10.9879 −0.863292
$$163$$ −21.5818 −1.69041 −0.845207 0.534438i $$-0.820523\pi$$
−0.845207 + 0.534438i $$0.820523\pi$$
$$164$$ 10.4700 0.817570
$$165$$ 0 0
$$166$$ −6.86423 −0.532768
$$167$$ 0.900461 0.0696797 0.0348399 0.999393i $$-0.488908\pi$$
0.0348399 + 0.999393i $$0.488908\pi$$
$$168$$ −2.23874 −0.172722
$$169$$ 13.2994 1.02303
$$170$$ 6.65568 0.510467
$$171$$ −2.32884 −0.178091
$$172$$ 1.77464 0.135315
$$173$$ −14.9963 −1.14015 −0.570075 0.821592i $$-0.693086\pi$$
−0.570075 + 0.821592i $$0.693086\pi$$
$$174$$ 13.1961 1.00039
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −22.9727 −1.72673
$$178$$ 15.1923 1.13871
$$179$$ 10.7148 0.800861 0.400430 0.916327i $$-0.368861\pi$$
0.400430 + 0.916327i $$0.368861\pi$$
$$180$$ −2.01195 −0.149962
$$181$$ −6.20078 −0.460900 −0.230450 0.973084i $$-0.574020\pi$$
−0.230450 + 0.973084i $$0.574020\pi$$
$$182$$ −5.12829 −0.380134
$$183$$ −28.6854 −2.12048
$$184$$ 3.87761 0.285861
$$185$$ −11.8944 −0.874495
$$186$$ −14.8736 −1.09058
$$187$$ 0 0
$$188$$ 8.71479 0.635591
$$189$$ 2.21199 0.160899
$$190$$ 1.15751 0.0839743
$$191$$ −0.133153 −0.00963464 −0.00481732 0.999988i $$-0.501533\pi$$
−0.00481732 + 0.999988i $$0.501533\pi$$
$$192$$ 2.23874 0.161567
$$193$$ −12.7468 −0.917531 −0.458766 0.888557i $$-0.651708\pi$$
−0.458766 + 0.888557i $$0.651708\pi$$
$$194$$ −0.406766 −0.0292041
$$195$$ −11.4809 −0.822165
$$196$$ 1.00000 0.0714286
$$197$$ 4.39162 0.312890 0.156445 0.987687i $$-0.449997\pi$$
0.156445 + 0.987687i $$0.449997\pi$$
$$198$$ 0 0
$$199$$ −0.430314 −0.0305041 −0.0152521 0.999884i $$-0.504855\pi$$
−0.0152521 + 0.999884i $$0.504855\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 33.5098 2.36360
$$202$$ 0.964155 0.0678377
$$203$$ −5.89442 −0.413707
$$204$$ −14.9003 −1.04323
$$205$$ −10.4700 −0.731257
$$206$$ 12.4021 0.864098
$$207$$ 7.80155 0.542245
$$208$$ 5.12829 0.355583
$$209$$ 0 0
$$210$$ 2.23874 0.154488
$$211$$ −8.33737 −0.573968 −0.286984 0.957935i $$-0.592653\pi$$
−0.286984 + 0.957935i $$0.592653\pi$$
$$212$$ 8.28476 0.568999
$$213$$ 17.3886 1.19145
$$214$$ 11.3466 0.775635
$$215$$ −1.77464 −0.121029
$$216$$ −2.21199 −0.150507
$$217$$ 6.64373 0.451006
$$218$$ −9.81672 −0.664872
$$219$$ −5.68723 −0.384308
$$220$$ 0 0
$$221$$ −34.1323 −2.29599
$$222$$ 26.6285 1.78719
$$223$$ 1.13218 0.0758166 0.0379083 0.999281i $$-0.487931\pi$$
0.0379083 + 0.999281i $$0.487931\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 2.01195 0.134130
$$226$$ −15.4263 −1.02614
$$227$$ −10.2741 −0.681915 −0.340957 0.940079i $$-0.610751\pi$$
−0.340957 + 0.940079i $$0.610751\pi$$
$$228$$ −2.59135 −0.171616
$$229$$ −9.23127 −0.610019 −0.305010 0.952349i $$-0.598660\pi$$
−0.305010 + 0.952349i $$0.598660\pi$$
$$230$$ −3.87761 −0.255682
$$231$$ 0 0
$$232$$ 5.89442 0.386987
$$233$$ −6.68548 −0.437980 −0.218990 0.975727i $$-0.570276\pi$$
−0.218990 + 0.975727i $$0.570276\pi$$
$$234$$ 10.3179 0.674500
$$235$$ −8.71479 −0.568490
$$236$$ −10.2614 −0.667963
$$237$$ 22.9705 1.49210
$$238$$ 6.65568 0.431423
$$239$$ −22.4534 −1.45239 −0.726193 0.687490i $$-0.758712\pi$$
−0.726193 + 0.687490i $$0.758712\pi$$
$$240$$ −2.23874 −0.144510
$$241$$ −28.6046 −1.84258 −0.921292 0.388872i $$-0.872865\pi$$
−0.921292 + 0.388872i $$0.872865\pi$$
$$242$$ 0 0
$$243$$ −17.9631 −1.15233
$$244$$ −12.8132 −0.820280
$$245$$ −1.00000 −0.0638877
$$246$$ 23.4396 1.49445
$$247$$ −5.93603 −0.377701
$$248$$ −6.64373 −0.421877
$$249$$ −15.3672 −0.973858
$$250$$ −1.00000 −0.0632456
$$251$$ 25.4351 1.60545 0.802725 0.596349i $$-0.203383\pi$$
0.802725 + 0.596349i $$0.203383\pi$$
$$252$$ −2.01195 −0.126741
$$253$$ 0 0
$$254$$ 15.5058 0.972924
$$255$$ 14.9003 0.933094
$$256$$ 1.00000 0.0625000
$$257$$ −24.9567 −1.55676 −0.778379 0.627795i $$-0.783958\pi$$
−0.778379 + 0.627795i $$0.783958\pi$$
$$258$$ 3.97295 0.247345
$$259$$ −11.8944 −0.739083
$$260$$ −5.12829 −0.318043
$$261$$ 11.8593 0.734070
$$262$$ −16.5157 −1.02035
$$263$$ −0.333627 −0.0205723 −0.0102862 0.999947i $$-0.503274\pi$$
−0.0102862 + 0.999947i $$0.503274\pi$$
$$264$$ 0 0
$$265$$ −8.28476 −0.508929
$$266$$ 1.15751 0.0709712
$$267$$ 34.0115 2.08147
$$268$$ 14.9682 0.914327
$$269$$ 1.82597 0.111331 0.0556656 0.998449i $$-0.482272\pi$$
0.0556656 + 0.998449i $$0.482272\pi$$
$$270$$ 2.21199 0.134617
$$271$$ 13.3284 0.809642 0.404821 0.914396i $$-0.367334\pi$$
0.404821 + 0.914396i $$0.367334\pi$$
$$272$$ −6.65568 −0.403560
$$273$$ −11.4809 −0.694856
$$274$$ 13.8267 0.835299
$$275$$ 0 0
$$276$$ 8.68095 0.522532
$$277$$ −15.4834 −0.930306 −0.465153 0.885230i $$-0.654001\pi$$
−0.465153 + 0.885230i $$0.654001\pi$$
$$278$$ 1.19890 0.0719050
$$279$$ −13.3668 −0.800252
$$280$$ 1.00000 0.0597614
$$281$$ 28.3105 1.68886 0.844432 0.535663i $$-0.179938\pi$$
0.844432 + 0.535663i $$0.179938\pi$$
$$282$$ 19.5101 1.16181
$$283$$ 13.5891 0.807791 0.403895 0.914805i $$-0.367656\pi$$
0.403895 + 0.914805i $$0.367656\pi$$
$$284$$ 7.76716 0.460896
$$285$$ 2.59135 0.153498
$$286$$ 0 0
$$287$$ −10.4700 −0.618025
$$288$$ 2.01195 0.118555
$$289$$ 27.2980 1.60577
$$290$$ −5.89442 −0.346132
$$291$$ −0.910642 −0.0533828
$$292$$ −2.54037 −0.148664
$$293$$ 1.83423 0.107157 0.0535785 0.998564i $$-0.482937\pi$$
0.0535785 + 0.998564i $$0.482937\pi$$
$$294$$ 2.23874 0.130566
$$295$$ 10.2614 0.597445
$$296$$ 11.8944 0.691349
$$297$$ 0 0
$$298$$ −5.37555 −0.311397
$$299$$ 19.8855 1.15001
$$300$$ 2.23874 0.129254
$$301$$ −1.77464 −0.102288
$$302$$ 1.95930 0.112745
$$303$$ 2.15849 0.124002
$$304$$ −1.15751 −0.0663875
$$305$$ 12.8132 0.733681
$$306$$ −13.3909 −0.765506
$$307$$ 8.45358 0.482471 0.241236 0.970467i $$-0.422447\pi$$
0.241236 + 0.970467i $$0.422447\pi$$
$$308$$ 0 0
$$309$$ 27.7652 1.57950
$$310$$ 6.64373 0.377338
$$311$$ 16.9898 0.963402 0.481701 0.876336i $$-0.340019\pi$$
0.481701 + 0.876336i $$0.340019\pi$$
$$312$$ 11.4809 0.649978
$$313$$ 18.0862 1.02230 0.511148 0.859493i $$-0.329221\pi$$
0.511148 + 0.859493i $$0.329221\pi$$
$$314$$ 13.1054 0.739583
$$315$$ 2.01195 0.113360
$$316$$ 10.2605 0.577197
$$317$$ −0.393194 −0.0220840 −0.0110420 0.999939i $$-0.503515\pi$$
−0.0110420 + 0.999939i $$0.503515\pi$$
$$318$$ 18.5474 1.04009
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 25.4020 1.41780
$$322$$ −3.87761 −0.216091
$$323$$ 7.70398 0.428661
$$324$$ −10.9879 −0.610439
$$325$$ 5.12829 0.284467
$$326$$ −21.5818 −1.19530
$$327$$ −21.9771 −1.21533
$$328$$ 10.4700 0.578109
$$329$$ −8.71479 −0.480462
$$330$$ 0 0
$$331$$ −34.5395 −1.89846 −0.949232 0.314577i $$-0.898137\pi$$
−0.949232 + 0.314577i $$0.898137\pi$$
$$332$$ −6.86423 −0.376724
$$333$$ 23.9309 1.31141
$$334$$ 0.900461 0.0492710
$$335$$ −14.9682 −0.817799
$$336$$ −2.23874 −0.122133
$$337$$ 2.36075 0.128598 0.0642991 0.997931i $$-0.479519\pi$$
0.0642991 + 0.997931i $$0.479519\pi$$
$$338$$ 13.2994 0.723392
$$339$$ −34.5354 −1.87570
$$340$$ 6.65568 0.360955
$$341$$ 0 0
$$342$$ −2.32884 −0.125929
$$343$$ −1.00000 −0.0539949
$$344$$ 1.77464 0.0956820
$$345$$ −8.68095 −0.467367
$$346$$ −14.9963 −0.806208
$$347$$ −6.85646 −0.368074 −0.184037 0.982919i $$-0.558917\pi$$
−0.184037 + 0.982919i $$0.558917\pi$$
$$348$$ 13.1961 0.707383
$$349$$ −8.34033 −0.446448 −0.223224 0.974767i $$-0.571658\pi$$
−0.223224 + 0.974767i $$0.571658\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ −11.3437 −0.605484
$$352$$ 0 0
$$353$$ 19.7691 1.05220 0.526101 0.850422i $$-0.323653\pi$$
0.526101 + 0.850422i $$0.323653\pi$$
$$354$$ −22.9727 −1.22098
$$355$$ −7.76716 −0.412238
$$356$$ 15.1923 0.805188
$$357$$ 14.9003 0.788608
$$358$$ 10.7148 0.566294
$$359$$ 0.257703 0.0136010 0.00680051 0.999977i $$-0.497835\pi$$
0.00680051 + 0.999977i $$0.497835\pi$$
$$360$$ −2.01195 −0.106039
$$361$$ −17.6602 −0.929483
$$362$$ −6.20078 −0.325906
$$363$$ 0 0
$$364$$ −5.12829 −0.268796
$$365$$ 2.54037 0.132969
$$366$$ −28.6854 −1.49941
$$367$$ 25.6755 1.34025 0.670125 0.742248i $$-0.266241\pi$$
0.670125 + 0.742248i $$0.266241\pi$$
$$368$$ 3.87761 0.202134
$$369$$ 21.0651 1.09661
$$370$$ −11.8944 −0.618361
$$371$$ −8.28476 −0.430123
$$372$$ −14.8736 −0.771159
$$373$$ −5.70957 −0.295631 −0.147815 0.989015i $$-0.547224\pi$$
−0.147815 + 0.989015i $$0.547224\pi$$
$$374$$ 0 0
$$375$$ −2.23874 −0.115608
$$376$$ 8.71479 0.449431
$$377$$ 30.2283 1.55684
$$378$$ 2.21199 0.113772
$$379$$ 2.16686 0.111304 0.0556520 0.998450i $$-0.482276\pi$$
0.0556520 + 0.998450i $$0.482276\pi$$
$$380$$ 1.15751 0.0593788
$$381$$ 34.7135 1.77843
$$382$$ −0.133153 −0.00681272
$$383$$ 10.3043 0.526523 0.263262 0.964725i $$-0.415202\pi$$
0.263262 + 0.964725i $$0.415202\pi$$
$$384$$ 2.23874 0.114245
$$385$$ 0 0
$$386$$ −12.7468 −0.648793
$$387$$ 3.57048 0.181498
$$388$$ −0.406766 −0.0206504
$$389$$ 5.86030 0.297129 0.148564 0.988903i $$-0.452535\pi$$
0.148564 + 0.988903i $$0.452535\pi$$
$$390$$ −11.4809 −0.581358
$$391$$ −25.8081 −1.30517
$$392$$ 1.00000 0.0505076
$$393$$ −36.9744 −1.86511
$$394$$ 4.39162 0.221246
$$395$$ −10.2605 −0.516261
$$396$$ 0 0
$$397$$ −9.65824 −0.484733 −0.242367 0.970185i $$-0.577924\pi$$
−0.242367 + 0.970185i $$0.577924\pi$$
$$398$$ −0.430314 −0.0215697
$$399$$ 2.59135 0.129730
$$400$$ 1.00000 0.0500000
$$401$$ −3.76869 −0.188199 −0.0940996 0.995563i $$-0.529997\pi$$
−0.0940996 + 0.995563i $$0.529997\pi$$
$$402$$ 33.5098 1.67132
$$403$$ −34.0710 −1.69720
$$404$$ 0.964155 0.0479685
$$405$$ 10.9879 0.545994
$$406$$ −5.89442 −0.292535
$$407$$ 0 0
$$408$$ −14.9003 −0.737676
$$409$$ 7.59552 0.375575 0.187787 0.982210i $$-0.439868\pi$$
0.187787 + 0.982210i $$0.439868\pi$$
$$410$$ −10.4700 −0.517077
$$411$$ 30.9543 1.52686
$$412$$ 12.4021 0.611010
$$413$$ 10.2614 0.504933
$$414$$ 7.80155 0.383425
$$415$$ 6.86423 0.336952
$$416$$ 5.12829 0.251435
$$417$$ 2.68401 0.131437
$$418$$ 0 0
$$419$$ −10.0288 −0.489937 −0.244968 0.969531i $$-0.578778\pi$$
−0.244968 + 0.969531i $$0.578778\pi$$
$$420$$ 2.23874 0.109239
$$421$$ −4.21566 −0.205459 −0.102729 0.994709i $$-0.532758\pi$$
−0.102729 + 0.994709i $$0.532758\pi$$
$$422$$ −8.33737 −0.405857
$$423$$ 17.5337 0.852518
$$424$$ 8.28476 0.402343
$$425$$ −6.65568 −0.322848
$$426$$ 17.3886 0.842483
$$427$$ 12.8132 0.620073
$$428$$ 11.3466 0.548457
$$429$$ 0 0
$$430$$ −1.77464 −0.0855806
$$431$$ −23.7780 −1.14534 −0.572672 0.819784i $$-0.694093\pi$$
−0.572672 + 0.819784i $$0.694093\pi$$
$$432$$ −2.21199 −0.106424
$$433$$ −33.5935 −1.61440 −0.807199 0.590279i $$-0.799018\pi$$
−0.807199 + 0.590279i $$0.799018\pi$$
$$434$$ 6.64373 0.318909
$$435$$ −13.1961 −0.632702
$$436$$ −9.81672 −0.470135
$$437$$ −4.48835 −0.214707
$$438$$ −5.68723 −0.271747
$$439$$ −27.7198 −1.32299 −0.661496 0.749949i $$-0.730078\pi$$
−0.661496 + 0.749949i $$0.730078\pi$$
$$440$$ 0 0
$$441$$ 2.01195 0.0958071
$$442$$ −34.1323 −1.62351
$$443$$ 28.5238 1.35521 0.677603 0.735427i $$-0.263019\pi$$
0.677603 + 0.735427i $$0.263019\pi$$
$$444$$ 26.6285 1.26373
$$445$$ −15.1923 −0.720182
$$446$$ 1.13218 0.0536104
$$447$$ −12.0344 −0.569210
$$448$$ −1.00000 −0.0472456
$$449$$ −0.982968 −0.0463891 −0.0231946 0.999731i $$-0.507384\pi$$
−0.0231946 + 0.999731i $$0.507384\pi$$
$$450$$ 2.01195 0.0948442
$$451$$ 0 0
$$452$$ −15.4263 −0.725591
$$453$$ 4.38635 0.206089
$$454$$ −10.2741 −0.482187
$$455$$ 5.12829 0.240418
$$456$$ −2.59135 −0.121351
$$457$$ 4.98742 0.233302 0.116651 0.993173i $$-0.462784\pi$$
0.116651 + 0.993173i $$0.462784\pi$$
$$458$$ −9.23127 −0.431349
$$459$$ 14.7223 0.687177
$$460$$ −3.87761 −0.180794
$$461$$ 1.28781 0.0599791 0.0299896 0.999550i $$-0.490453\pi$$
0.0299896 + 0.999550i $$0.490453\pi$$
$$462$$ 0 0
$$463$$ −17.4015 −0.808717 −0.404358 0.914601i $$-0.632505\pi$$
−0.404358 + 0.914601i $$0.632505\pi$$
$$464$$ 5.89442 0.273641
$$465$$ 14.8736 0.689745
$$466$$ −6.68548 −0.309699
$$467$$ 6.97072 0.322566 0.161283 0.986908i $$-0.448437\pi$$
0.161283 + 0.986908i $$0.448437\pi$$
$$468$$ 10.3179 0.476943
$$469$$ −14.9682 −0.691166
$$470$$ −8.71479 −0.401983
$$471$$ 29.3397 1.35190
$$472$$ −10.2614 −0.472321
$$473$$ 0 0
$$474$$ 22.9705 1.05507
$$475$$ −1.15751 −0.0531100
$$476$$ 6.65568 0.305062
$$477$$ 16.6685 0.763198
$$478$$ −22.4534 −1.02699
$$479$$ −26.7035 −1.22011 −0.610056 0.792358i $$-0.708853\pi$$
−0.610056 + 0.792358i $$0.708853\pi$$
$$480$$ −2.23874 −0.102184
$$481$$ 60.9981 2.78127
$$482$$ −28.6046 −1.30290
$$483$$ −8.68095 −0.394997
$$484$$ 0 0
$$485$$ 0.406766 0.0184703
$$486$$ −17.9631 −0.814822
$$487$$ 4.32910 0.196170 0.0980851 0.995178i $$-0.468728\pi$$
0.0980851 + 0.995178i $$0.468728\pi$$
$$488$$ −12.8132 −0.580026
$$489$$ −48.3159 −2.18492
$$490$$ −1.00000 −0.0451754
$$491$$ 12.8945 0.581923 0.290961 0.956735i $$-0.406025\pi$$
0.290961 + 0.956735i $$0.406025\pi$$
$$492$$ 23.4396 1.05674
$$493$$ −39.2313 −1.76689
$$494$$ −5.93603 −0.267075
$$495$$ 0 0
$$496$$ −6.64373 −0.298312
$$497$$ −7.76716 −0.348405
$$498$$ −15.3672 −0.688622
$$499$$ −3.01676 −0.135049 −0.0675243 0.997718i $$-0.521510\pi$$
−0.0675243 + 0.997718i $$0.521510\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 2.01590 0.0900636
$$502$$ 25.4351 1.13523
$$503$$ −6.17711 −0.275424 −0.137712 0.990472i $$-0.543975\pi$$
−0.137712 + 0.990472i $$0.543975\pi$$
$$504$$ −2.01195 −0.0896193
$$505$$ −0.964155 −0.0429043
$$506$$ 0 0
$$507$$ 29.7739 1.32230
$$508$$ 15.5058 0.687961
$$509$$ 33.6703 1.49241 0.746205 0.665717i $$-0.231874\pi$$
0.746205 + 0.665717i $$0.231874\pi$$
$$510$$ 14.9003 0.659797
$$511$$ 2.54037 0.112380
$$512$$ 1.00000 0.0441942
$$513$$ 2.56039 0.113044
$$514$$ −24.9567 −1.10079
$$515$$ −12.4021 −0.546504
$$516$$ 3.97295 0.174899
$$517$$ 0 0
$$518$$ −11.8944 −0.522611
$$519$$ −33.5729 −1.47369
$$520$$ −5.12829 −0.224891
$$521$$ 24.5687 1.07637 0.538187 0.842826i $$-0.319110\pi$$
0.538187 + 0.842826i $$0.319110\pi$$
$$522$$ 11.8593 0.519066
$$523$$ 44.4679 1.94444 0.972222 0.234059i $$-0.0752010\pi$$
0.972222 + 0.234059i $$0.0752010\pi$$
$$524$$ −16.5157 −0.721493
$$525$$ −2.23874 −0.0977065
$$526$$ −0.333627 −0.0145468
$$527$$ 44.2185 1.92619
$$528$$ 0 0
$$529$$ −7.96415 −0.346268
$$530$$ −8.28476 −0.359867
$$531$$ −20.6455 −0.895939
$$532$$ 1.15751 0.0501842
$$533$$ 53.6933 2.32571
$$534$$ 34.0115 1.47182
$$535$$ −11.3466 −0.490555
$$536$$ 14.9682 0.646527
$$537$$ 23.9876 1.03514
$$538$$ 1.82597 0.0787231
$$539$$ 0 0
$$540$$ 2.21199 0.0951888
$$541$$ −1.30850 −0.0562569 −0.0281284 0.999604i $$-0.508955\pi$$
−0.0281284 + 0.999604i $$0.508955\pi$$
$$542$$ 13.3284 0.572503
$$543$$ −13.8819 −0.595730
$$544$$ −6.65568 −0.285360
$$545$$ 9.81672 0.420502
$$546$$ −11.4809 −0.491338
$$547$$ −17.4553 −0.746335 −0.373168 0.927764i $$-0.621728\pi$$
−0.373168 + 0.927764i $$0.621728\pi$$
$$548$$ 13.8267 0.590646
$$549$$ −25.7795 −1.10024
$$550$$ 0 0
$$551$$ −6.82282 −0.290662
$$552$$ 8.68095 0.369486
$$553$$ −10.2605 −0.436320
$$554$$ −15.4834 −0.657826
$$555$$ −26.6285 −1.13032
$$556$$ 1.19890 0.0508445
$$557$$ −0.219818 −0.00931400 −0.00465700 0.999989i $$-0.501482\pi$$
−0.00465700 + 0.999989i $$0.501482\pi$$
$$558$$ −13.3668 −0.565863
$$559$$ 9.10086 0.384925
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 28.3105 1.19421
$$563$$ 22.4246 0.945084 0.472542 0.881308i $$-0.343336\pi$$
0.472542 + 0.881308i $$0.343336\pi$$
$$564$$ 19.5101 0.821525
$$565$$ 15.4263 0.648988
$$566$$ 13.5891 0.571194
$$567$$ 10.9879 0.461449
$$568$$ 7.76716 0.325903
$$569$$ 3.86730 0.162126 0.0810629 0.996709i $$-0.474169\pi$$
0.0810629 + 0.996709i $$0.474169\pi$$
$$570$$ 2.59135 0.108540
$$571$$ 25.4899 1.06672 0.533360 0.845888i $$-0.320929\pi$$
0.533360 + 0.845888i $$0.320929\pi$$
$$572$$ 0 0
$$573$$ −0.298095 −0.0124531
$$574$$ −10.4700 −0.437010
$$575$$ 3.87761 0.161707
$$576$$ 2.01195 0.0838312
$$577$$ −33.6011 −1.39883 −0.699415 0.714716i $$-0.746556\pi$$
−0.699415 + 0.714716i $$0.746556\pi$$
$$578$$ 27.2980 1.13545
$$579$$ −28.5366 −1.18594
$$580$$ −5.89442 −0.244752
$$581$$ 6.86423 0.284776
$$582$$ −0.910642 −0.0377473
$$583$$ 0 0
$$584$$ −2.54037 −0.105121
$$585$$ −10.3179 −0.426591
$$586$$ 1.83423 0.0757715
$$587$$ 38.8396 1.60308 0.801542 0.597939i $$-0.204013\pi$$
0.801542 + 0.597939i $$0.204013\pi$$
$$588$$ 2.23874 0.0923240
$$589$$ 7.69015 0.316867
$$590$$ 10.2614 0.422457
$$591$$ 9.83168 0.404421
$$592$$ 11.8944 0.488857
$$593$$ −29.8393 −1.22535 −0.612677 0.790333i $$-0.709907\pi$$
−0.612677 + 0.790333i $$0.709907\pi$$
$$594$$ 0 0
$$595$$ −6.65568 −0.272856
$$596$$ −5.37555 −0.220191
$$597$$ −0.963359 −0.0394277
$$598$$ 19.8855 0.813179
$$599$$ 26.4186 1.07944 0.539718 0.841846i $$-0.318531\pi$$
0.539718 + 0.841846i $$0.318531\pi$$
$$600$$ 2.23874 0.0913961
$$601$$ −31.2101 −1.27309 −0.636543 0.771241i $$-0.719636\pi$$
−0.636543 + 0.771241i $$0.719636\pi$$
$$602$$ −1.77464 −0.0723288
$$603$$ 30.1152 1.22639
$$604$$ 1.95930 0.0797226
$$605$$ 0 0
$$606$$ 2.15849 0.0876827
$$607$$ −29.3084 −1.18959 −0.594795 0.803878i $$-0.702767\pi$$
−0.594795 + 0.803878i $$0.702767\pi$$
$$608$$ −1.15751 −0.0469431
$$609$$ −13.1961 −0.534731
$$610$$ 12.8132 0.518791
$$611$$ 44.6920 1.80804
$$612$$ −13.3909 −0.541294
$$613$$ −20.1964 −0.815725 −0.407863 0.913043i $$-0.633726\pi$$
−0.407863 + 0.913043i $$0.633726\pi$$
$$614$$ 8.45358 0.341159
$$615$$ −23.4396 −0.945176
$$616$$ 0 0
$$617$$ −2.08426 −0.0839091 −0.0419546 0.999120i $$-0.513358\pi$$
−0.0419546 + 0.999120i $$0.513358\pi$$
$$618$$ 27.7652 1.11688
$$619$$ −3.03990 −0.122184 −0.0610919 0.998132i $$-0.519458\pi$$
−0.0610919 + 0.998132i $$0.519458\pi$$
$$620$$ 6.64373 0.266819
$$621$$ −8.57723 −0.344192
$$622$$ 16.9898 0.681228
$$623$$ −15.1923 −0.608665
$$624$$ 11.4809 0.459604
$$625$$ 1.00000 0.0400000
$$626$$ 18.0862 0.722872
$$627$$ 0 0
$$628$$ 13.1054 0.522964
$$629$$ −79.1654 −3.15653
$$630$$ 2.01195 0.0801579
$$631$$ −11.3562 −0.452081 −0.226041 0.974118i $$-0.572578\pi$$
−0.226041 + 0.974118i $$0.572578\pi$$
$$632$$ 10.2605 0.408140
$$633$$ −18.6652 −0.741874
$$634$$ −0.393194 −0.0156157
$$635$$ −15.5058 −0.615331
$$636$$ 18.5474 0.735452
$$637$$ 5.12829 0.203190
$$638$$ 0 0
$$639$$ 15.6271 0.618200
$$640$$ −1.00000 −0.0395285
$$641$$ −34.4272 −1.35979 −0.679896 0.733308i $$-0.737975\pi$$
−0.679896 + 0.733308i $$0.737975\pi$$
$$642$$ 25.4020 1.00254
$$643$$ −31.4442 −1.24004 −0.620020 0.784586i $$-0.712875\pi$$
−0.620020 + 0.784586i $$0.712875\pi$$
$$644$$ −3.87761 −0.152799
$$645$$ −3.97295 −0.156435
$$646$$ 7.70398 0.303109
$$647$$ −48.9448 −1.92422 −0.962110 0.272662i $$-0.912096\pi$$
−0.962110 + 0.272662i $$0.912096\pi$$
$$648$$ −10.9879 −0.431646
$$649$$ 0 0
$$650$$ 5.12829 0.201148
$$651$$ 14.8736 0.582941
$$652$$ −21.5818 −0.845207
$$653$$ −17.0563 −0.667463 −0.333732 0.942668i $$-0.608308\pi$$
−0.333732 + 0.942668i $$0.608308\pi$$
$$654$$ −21.9771 −0.859371
$$655$$ 16.5157 0.645323
$$656$$ 10.4700 0.408785
$$657$$ −5.11110 −0.199403
$$658$$ −8.71479 −0.339738
$$659$$ 39.3755 1.53385 0.766926 0.641736i $$-0.221786\pi$$
0.766926 + 0.641736i $$0.221786\pi$$
$$660$$ 0 0
$$661$$ −47.5548 −1.84967 −0.924833 0.380374i $$-0.875795\pi$$
−0.924833 + 0.380374i $$0.875795\pi$$
$$662$$ −34.5395 −1.34242
$$663$$ −76.4132 −2.96764
$$664$$ −6.86423 −0.266384
$$665$$ −1.15751 −0.0448861
$$666$$ 23.9309 0.927305
$$667$$ 22.8562 0.884997
$$668$$ 0.900461 0.0348399
$$669$$ 2.53466 0.0979956
$$670$$ −14.9682 −0.578271
$$671$$ 0 0
$$672$$ −2.23874 −0.0863612
$$673$$ 30.7041 1.18356 0.591779 0.806100i $$-0.298426\pi$$
0.591779 + 0.806100i $$0.298426\pi$$
$$674$$ 2.36075 0.0909326
$$675$$ −2.21199 −0.0851395
$$676$$ 13.2994 0.511516
$$677$$ −30.7722 −1.18267 −0.591335 0.806426i $$-0.701399\pi$$
−0.591335 + 0.806426i $$0.701399\pi$$
$$678$$ −34.5354 −1.32632
$$679$$ 0.406766 0.0156102
$$680$$ 6.65568 0.255234
$$681$$ −23.0010 −0.881400
$$682$$ 0 0
$$683$$ −10.3871 −0.397451 −0.198726 0.980055i $$-0.563680\pi$$
−0.198726 + 0.980055i $$0.563680\pi$$
$$684$$ −2.32884 −0.0890455
$$685$$ −13.8267 −0.528289
$$686$$ −1.00000 −0.0381802
$$687$$ −20.6664 −0.788472
$$688$$ 1.77464 0.0676574
$$689$$ 42.4867 1.61861
$$690$$ −8.68095 −0.330478
$$691$$ 20.9815 0.798175 0.399088 0.916913i $$-0.369327\pi$$
0.399088 + 0.916913i $$0.369327\pi$$
$$692$$ −14.9963 −0.570075
$$693$$ 0 0
$$694$$ −6.85646 −0.260267
$$695$$ −1.19890 −0.0454767
$$696$$ 13.1961 0.500195
$$697$$ −69.6850 −2.63951
$$698$$ −8.34033 −0.315686
$$699$$ −14.9670 −0.566105
$$700$$ −1.00000 −0.0377964
$$701$$ 30.6295 1.15686 0.578431 0.815731i $$-0.303665\pi$$
0.578431 + 0.815731i $$0.303665\pi$$
$$702$$ −11.3437 −0.428142
$$703$$ −13.7679 −0.519264
$$704$$ 0 0
$$705$$ −19.5101 −0.734794
$$706$$ 19.7691 0.744019
$$707$$ −0.964155 −0.0362608
$$708$$ −22.9727 −0.863367
$$709$$ 41.2618 1.54962 0.774810 0.632194i $$-0.217845\pi$$
0.774810 + 0.632194i $$0.217845\pi$$
$$710$$ −7.76716 −0.291496
$$711$$ 20.6436 0.774194
$$712$$ 15.1923 0.569354
$$713$$ −25.7618 −0.964786
$$714$$ 14.9003 0.557630
$$715$$ 0 0
$$716$$ 10.7148 0.400430
$$717$$ −50.2672 −1.87726
$$718$$ 0.257703 0.00961738
$$719$$ 26.9646 1.00561 0.502804 0.864400i $$-0.332302\pi$$
0.502804 + 0.864400i $$0.332302\pi$$
$$720$$ −2.01195 −0.0749809
$$721$$ −12.4021 −0.461880
$$722$$ −17.6602 −0.657244
$$723$$ −64.0382 −2.38161
$$724$$ −6.20078 −0.230450
$$725$$ 5.89442 0.218913
$$726$$ 0 0
$$727$$ −29.0208 −1.07632 −0.538161 0.842842i $$-0.680881\pi$$
−0.538161 + 0.842842i $$0.680881\pi$$
$$728$$ −5.12829 −0.190067
$$729$$ −7.25091 −0.268552
$$730$$ 2.54037 0.0940235
$$731$$ −11.8114 −0.436861
$$732$$ −28.6854 −1.06024
$$733$$ 28.4461 1.05068 0.525339 0.850893i $$-0.323938\pi$$
0.525339 + 0.850893i $$0.323938\pi$$
$$734$$ 25.6755 0.947700
$$735$$ −2.23874 −0.0825771
$$736$$ 3.87761 0.142931
$$737$$ 0 0
$$738$$ 21.0651 0.775417
$$739$$ −24.4968 −0.901128 −0.450564 0.892744i $$-0.648777\pi$$
−0.450564 + 0.892744i $$0.648777\pi$$
$$740$$ −11.8944 −0.437247
$$741$$ −13.2892 −0.488192
$$742$$ −8.28476 −0.304143
$$743$$ −6.04973 −0.221943 −0.110972 0.993824i $$-0.535396\pi$$
−0.110972 + 0.993824i $$0.535396\pi$$
$$744$$ −14.8736 −0.545291
$$745$$ 5.37555 0.196945
$$746$$ −5.70957 −0.209042
$$747$$ −13.8105 −0.505299
$$748$$ 0 0
$$749$$ −11.3466 −0.414595
$$750$$ −2.23874 −0.0817472
$$751$$ 5.08525 0.185564 0.0927818 0.995686i $$-0.470424\pi$$
0.0927818 + 0.995686i $$0.470424\pi$$
$$752$$ 8.71479 0.317796
$$753$$ 56.9426 2.07510
$$754$$ 30.2283 1.10085
$$755$$ −1.95930 −0.0713061
$$756$$ 2.21199 0.0804493
$$757$$ 25.4296 0.924254 0.462127 0.886814i $$-0.347086\pi$$
0.462127 + 0.886814i $$0.347086\pi$$
$$758$$ 2.16686 0.0787039
$$759$$ 0 0
$$760$$ 1.15751 0.0419871
$$761$$ 41.6911 1.51130 0.755650 0.654976i $$-0.227321\pi$$
0.755650 + 0.654976i $$0.227321\pi$$
$$762$$ 34.7135 1.25754
$$763$$ 9.81672 0.355389
$$764$$ −0.133153 −0.00481732
$$765$$ 13.3909 0.484148
$$766$$ 10.3043 0.372308
$$767$$ −52.6237 −1.90013
$$768$$ 2.23874 0.0807835
$$769$$ −14.8389 −0.535106 −0.267553 0.963543i $$-0.586215\pi$$
−0.267553 + 0.963543i $$0.586215\pi$$
$$770$$ 0 0
$$771$$ −55.8716 −2.01216
$$772$$ −12.7468 −0.458766
$$773$$ 25.3422 0.911497 0.455749 0.890109i $$-0.349372\pi$$
0.455749 + 0.890109i $$0.349372\pi$$
$$774$$ 3.57048 0.128338
$$775$$ −6.64373 −0.238650
$$776$$ −0.406766 −0.0146020
$$777$$ −26.6285 −0.955291
$$778$$ 5.86030 0.210102
$$779$$ −12.1191 −0.434211
$$780$$ −11.4809 −0.411082
$$781$$ 0 0
$$782$$ −25.8081 −0.922896
$$783$$ −13.0384 −0.465954
$$784$$ 1.00000 0.0357143
$$785$$ −13.1054 −0.467753
$$786$$ −36.9744 −1.31883
$$787$$ 17.7718 0.633496 0.316748 0.948510i $$-0.397409\pi$$
0.316748 + 0.948510i $$0.397409\pi$$
$$788$$ 4.39162 0.156445
$$789$$ −0.746903 −0.0265905
$$790$$ −10.2605 −0.365051
$$791$$ 15.4263 0.548495
$$792$$ 0 0
$$793$$ −65.7098 −2.33342
$$794$$ −9.65824 −0.342758
$$795$$ −18.5474 −0.657808
$$796$$ −0.430314 −0.0152521
$$797$$ −3.36014 −0.119022 −0.0595112 0.998228i $$-0.518954\pi$$
−0.0595112 + 0.998228i $$0.518954\pi$$
$$798$$ 2.59135 0.0917329
$$799$$ −58.0028 −2.05199
$$800$$ 1.00000 0.0353553
$$801$$ 30.5661 1.08000
$$802$$ −3.76869 −0.133077
$$803$$ 0 0
$$804$$ 33.5098 1.18180
$$805$$ 3.87761 0.136668
$$806$$ −34.0710 −1.20010
$$807$$ 4.08787 0.143900
$$808$$ 0.964155 0.0339189
$$809$$ −17.3076 −0.608502 −0.304251 0.952592i $$-0.598406\pi$$
−0.304251 + 0.952592i $$0.598406\pi$$
$$810$$ 10.9879 0.386076
$$811$$ −12.3583 −0.433960 −0.216980 0.976176i $$-0.569621\pi$$
−0.216980 + 0.976176i $$0.569621\pi$$
$$812$$ −5.89442 −0.206853
$$813$$ 29.8388 1.04649
$$814$$ 0 0
$$815$$ 21.5818 0.755977
$$816$$ −14.9003 −0.521615
$$817$$ −2.05415 −0.0718657
$$818$$ 7.59552 0.265571
$$819$$ −10.3179 −0.360535
$$820$$ −10.4700 −0.365628
$$821$$ −5.55685 −0.193935 −0.0969677 0.995288i $$-0.530914\pi$$
−0.0969677 + 0.995288i $$0.530914\pi$$
$$822$$ 30.9543 1.07965
$$823$$ −52.8246 −1.84135 −0.920676 0.390329i $$-0.872361\pi$$
−0.920676 + 0.390329i $$0.872361\pi$$
$$824$$ 12.4021 0.432049
$$825$$ 0 0
$$826$$ 10.2614 0.357041
$$827$$ −49.4908 −1.72097 −0.860483 0.509480i $$-0.829838\pi$$
−0.860483 + 0.509480i $$0.829838\pi$$
$$828$$ 7.80155 0.271123
$$829$$ −48.2163 −1.67462 −0.837311 0.546727i $$-0.815873\pi$$
−0.837311 + 0.546727i $$0.815873\pi$$
$$830$$ 6.86423 0.238261
$$831$$ −34.6632 −1.20245
$$832$$ 5.12829 0.177792
$$833$$ −6.65568 −0.230606
$$834$$ 2.68401 0.0929398
$$835$$ −0.900461 −0.0311617
$$836$$ 0 0
$$837$$ 14.6959 0.507963
$$838$$ −10.0288 −0.346438
$$839$$ 22.3658 0.772154 0.386077 0.922467i $$-0.373830\pi$$
0.386077 + 0.922467i $$0.373830\pi$$
$$840$$ 2.23874 0.0772438
$$841$$ 5.74413 0.198073
$$842$$ −4.21566 −0.145281
$$843$$ 63.3798 2.18292
$$844$$ −8.33737 −0.286984
$$845$$ −13.2994 −0.457513
$$846$$ 17.5337 0.602821
$$847$$ 0 0
$$848$$ 8.28476 0.284500
$$849$$ 30.4225 1.04410
$$850$$ −6.65568 −0.228288
$$851$$ 46.1219 1.58104
$$852$$ 17.3886 0.595725
$$853$$ 24.9471 0.854173 0.427087 0.904211i $$-0.359540\pi$$
0.427087 + 0.904211i $$0.359540\pi$$
$$854$$ 12.8132 0.438458
$$855$$ 2.32884 0.0796447
$$856$$ 11.3466 0.387818
$$857$$ −19.7163 −0.673496 −0.336748 0.941595i $$-0.609327\pi$$
−0.336748 + 0.941595i $$0.609327\pi$$
$$858$$ 0 0
$$859$$ 15.4209 0.526156 0.263078 0.964775i $$-0.415262\pi$$
0.263078 + 0.964775i $$0.415262\pi$$
$$860$$ −1.77464 −0.0605146
$$861$$ −23.4396 −0.798819
$$862$$ −23.7780 −0.809881
$$863$$ −14.2128 −0.483810 −0.241905 0.970300i $$-0.577772\pi$$
−0.241905 + 0.970300i $$0.577772\pi$$
$$864$$ −2.21199 −0.0752534
$$865$$ 14.9963 0.509891
$$866$$ −33.5935 −1.14155
$$867$$ 61.1132 2.07551
$$868$$ 6.64373 0.225503
$$869$$ 0 0
$$870$$ −13.1961 −0.447388
$$871$$ 76.7612 2.60095
$$872$$ −9.81672 −0.332436
$$873$$ −0.818392 −0.0276984
$$874$$ −4.48835 −0.151821
$$875$$ 1.00000 0.0338062
$$876$$ −5.68723 −0.192154
$$877$$ 6.27739 0.211972 0.105986 0.994368i $$-0.466200\pi$$
0.105986 + 0.994368i $$0.466200\pi$$
$$878$$ −27.7198 −0.935496
$$879$$ 4.10637 0.138504
$$880$$ 0 0
$$881$$ 0.955481 0.0321910 0.0160955 0.999870i $$-0.494876\pi$$
0.0160955 + 0.999870i $$0.494876\pi$$
$$882$$ 2.01195 0.0677458
$$883$$ 8.50497 0.286215 0.143107 0.989707i $$-0.454291\pi$$
0.143107 + 0.989707i $$0.454291\pi$$
$$884$$ −34.1323 −1.14799
$$885$$ 22.9727 0.772219
$$886$$ 28.5238 0.958276
$$887$$ −2.18620 −0.0734056 −0.0367028 0.999326i $$-0.511685\pi$$
−0.0367028 + 0.999326i $$0.511685\pi$$
$$888$$ 26.6285 0.893593
$$889$$ −15.5058 −0.520049
$$890$$ −15.1923 −0.509246
$$891$$ 0 0
$$892$$ 1.13218 0.0379083
$$893$$ −10.0874 −0.337563
$$894$$ −12.0344 −0.402492
$$895$$ −10.7148 −0.358156
$$896$$ −1.00000 −0.0334077
$$897$$ 44.5185 1.48643
$$898$$ −0.982968 −0.0328021
$$899$$ −39.1609 −1.30609
$$900$$ 2.01195 0.0670649
$$901$$ −55.1407 −1.83700
$$902$$ 0 0
$$903$$ −3.97295 −0.132211
$$904$$ −15.4263 −0.513070
$$905$$ 6.20078 0.206121
$$906$$ 4.38635 0.145727
$$907$$ 0.266857 0.00886082 0.00443041 0.999990i $$-0.498590\pi$$
0.00443041 + 0.999990i $$0.498590\pi$$
$$908$$ −10.2741 −0.340957
$$909$$ 1.93983 0.0643401
$$910$$ 5.12829 0.170001
$$911$$ −20.3552 −0.674397 −0.337198 0.941434i $$-0.609479\pi$$
−0.337198 + 0.941434i $$0.609479\pi$$
$$912$$ −2.59135 −0.0858082
$$913$$ 0 0
$$914$$ 4.98742 0.164969
$$915$$ 28.6854 0.948309
$$916$$ −9.23127 −0.305010
$$917$$ 16.5157 0.545398
$$918$$ 14.7223 0.485908
$$919$$ 8.19185 0.270224 0.135112 0.990830i $$-0.456861\pi$$
0.135112 + 0.990830i $$0.456861\pi$$
$$920$$ −3.87761 −0.127841
$$921$$ 18.9254 0.623612
$$922$$ 1.28781 0.0424116
$$923$$ 39.8323 1.31110
$$924$$ 0 0
$$925$$ 11.8944 0.391086
$$926$$ −17.4015 −0.571849
$$927$$ 24.9525 0.819547
$$928$$ 5.89442 0.193494
$$929$$ −36.0873 −1.18398 −0.591992 0.805943i $$-0.701658\pi$$
−0.591992 + 0.805943i $$0.701658\pi$$
$$930$$ 14.8736 0.487724
$$931$$ −1.15751 −0.0379357
$$932$$ −6.68548 −0.218990
$$933$$ 38.0357 1.24523
$$934$$ 6.97072 0.228089
$$935$$ 0 0
$$936$$ 10.3179 0.337250
$$937$$ 42.0240 1.37286 0.686432 0.727194i $$-0.259176\pi$$
0.686432 + 0.727194i $$0.259176\pi$$
$$938$$ −14.9682 −0.488728
$$939$$ 40.4904 1.32135
$$940$$ −8.71479 −0.284245
$$941$$ −57.0071 −1.85838 −0.929189 0.369605i $$-0.879493\pi$$
−0.929189 + 0.369605i $$0.879493\pi$$
$$942$$ 29.3397 0.955938
$$943$$ 40.5986 1.32207
$$944$$ −10.2614 −0.333982
$$945$$ −2.21199 −0.0719560
$$946$$ 0 0
$$947$$ 42.6517 1.38599 0.692997 0.720940i $$-0.256290\pi$$
0.692997 + 0.720940i $$0.256290\pi$$
$$948$$ 22.9705 0.746048
$$949$$ −13.0278 −0.422900
$$950$$ −1.15751 −0.0375544
$$951$$ −0.880257 −0.0285443
$$952$$ 6.65568 0.215712
$$953$$ −11.5905 −0.375453 −0.187726 0.982221i $$-0.560112\pi$$
−0.187726 + 0.982221i $$0.560112\pi$$
$$954$$ 16.6685 0.539663
$$955$$ 0.133153 0.00430874
$$956$$ −22.4534 −0.726193
$$957$$ 0 0
$$958$$ −26.7035 −0.862750
$$959$$ −13.8267 −0.446486
$$960$$ −2.23874 −0.0722550
$$961$$ 13.1391 0.423843
$$962$$ 60.9981 1.96666
$$963$$ 22.8287 0.735645
$$964$$ −28.6046 −0.921292
$$965$$ 12.7468 0.410332
$$966$$ −8.68095 −0.279305
$$967$$ −38.6220 −1.24200 −0.621000 0.783811i $$-0.713273\pi$$
−0.621000 + 0.783811i $$0.713273\pi$$
$$968$$ 0 0
$$969$$ 17.2472 0.554060
$$970$$ 0.406766 0.0130605
$$971$$ −53.2173 −1.70782 −0.853912 0.520418i $$-0.825776\pi$$
−0.853912 + 0.520418i $$0.825776\pi$$
$$972$$ −17.9631 −0.576166
$$973$$ −1.19890 −0.0384348
$$974$$ 4.32910 0.138713
$$975$$ 11.4809 0.367683
$$976$$ −12.8132 −0.410140
$$977$$ 3.80768 0.121819 0.0609093 0.998143i $$-0.480600\pi$$
0.0609093 + 0.998143i $$0.480600\pi$$
$$978$$ −48.3159 −1.54497
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ −19.7507 −0.630592
$$982$$ 12.8945 0.411481
$$983$$ 27.5427 0.878474 0.439237 0.898371i $$-0.355249\pi$$
0.439237 + 0.898371i $$0.355249\pi$$
$$984$$ 23.4396 0.747227
$$985$$ −4.39162 −0.139929
$$986$$ −39.2313 −1.24938
$$987$$ −19.5101 −0.621014
$$988$$ −5.93603 −0.188850
$$989$$ 6.88134 0.218814
$$990$$ 0 0
$$991$$ −59.3959 −1.88677 −0.943386 0.331697i $$-0.892379\pi$$
−0.943386 + 0.331697i $$0.892379\pi$$
$$992$$ −6.64373 −0.210939
$$993$$ −77.3249 −2.45383
$$994$$ −7.76716 −0.246360
$$995$$ 0.430314 0.0136419
$$996$$ −15.3672 −0.486929
$$997$$ 32.6231 1.03318 0.516592 0.856232i $$-0.327201\pi$$
0.516592 + 0.856232i $$0.327201\pi$$
$$998$$ −3.01676 −0.0954937
$$999$$ −26.3103 −0.832422
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.dd.1.5 yes 6
11.10 odd 2 8470.2.a.cx.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.5 6 11.10 odd 2
8470.2.a.dd.1.5 yes 6 1.1 even 1 trivial