# Properties

 Label 8470.2.a.dg.1.8 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 1.8 Root $$3.18761$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.18761 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.16083 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.18761 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.16083 q^{9} +1.00000 q^{10} +3.18761 q^{12} -5.69102 q^{13} -1.00000 q^{14} -3.18761 q^{15} +1.00000 q^{16} +0.749022 q^{17} -7.16083 q^{18} -6.78967 q^{19} -1.00000 q^{20} +3.18761 q^{21} +3.94010 q^{23} -3.18761 q^{24} +1.00000 q^{25} +5.69102 q^{26} +13.2631 q^{27} +1.00000 q^{28} +5.73837 q^{29} +3.18761 q^{30} -7.10952 q^{31} -1.00000 q^{32} -0.749022 q^{34} -1.00000 q^{35} +7.16083 q^{36} -11.6050 q^{37} +6.78967 q^{38} -18.1407 q^{39} +1.00000 q^{40} -11.4458 q^{41} -3.18761 q^{42} -2.33345 q^{43} -7.16083 q^{45} -3.94010 q^{46} -5.00025 q^{47} +3.18761 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.38759 q^{51} -5.69102 q^{52} -10.7671 q^{53} -13.2631 q^{54} -1.00000 q^{56} -21.6428 q^{57} -5.73837 q^{58} +5.22904 q^{59} -3.18761 q^{60} -7.68398 q^{61} +7.10952 q^{62} +7.16083 q^{63} +1.00000 q^{64} +5.69102 q^{65} -7.39587 q^{67} +0.749022 q^{68} +12.5595 q^{69} +1.00000 q^{70} -0.575191 q^{71} -7.16083 q^{72} +12.5704 q^{73} +11.6050 q^{74} +3.18761 q^{75} -6.78967 q^{76} +18.1407 q^{78} -6.83481 q^{79} -1.00000 q^{80} +20.7950 q^{81} +11.4458 q^{82} -10.5252 q^{83} +3.18761 q^{84} -0.749022 q^{85} +2.33345 q^{86} +18.2916 q^{87} +4.80467 q^{89} +7.16083 q^{90} -5.69102 q^{91} +3.94010 q^{92} -22.6624 q^{93} +5.00025 q^{94} +6.78967 q^{95} -3.18761 q^{96} +16.2717 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9} + O(q^{10})$$ $$8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + q^{13} - 8 q^{14} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} + 8 q^{28} - 3 q^{29} - 8 q^{31} - 8 q^{32} + 6 q^{34} - 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} - 35 q^{39} + 8 q^{40} - 11 q^{41} + 5 q^{43} - 8 q^{45} - 10 q^{46} - 15 q^{47} + 8 q^{49} - 8 q^{50} + 6 q^{51} + q^{52} - 16 q^{53} - 8 q^{56} - 38 q^{57} + 3 q^{58} - 9 q^{59} - 32 q^{61} + 8 q^{62} + 8 q^{63} + 8 q^{64} - q^{65} + 33 q^{67} - 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} - 8 q^{72} + 34 q^{73} + 6 q^{74} - 5 q^{76} + 35 q^{78} - 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} - 50 q^{83} + 6 q^{85} - 5 q^{86} + 12 q^{87} + q^{89} + 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} + 15 q^{94} + 5 q^{95} - 4 q^{97} - 8 q^{98} + O(q^{100})$$

## 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
$$3$$ 3.18761 1.84036 0.920182 0.391490i $$-0.128040\pi$$
0.920182 + 0.391490i $$0.128040\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −3.18761 −1.30133
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 7.16083 2.38694
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ 3.18761 0.920182
$$13$$ −5.69102 −1.57841 −0.789203 0.614133i $$-0.789506\pi$$
−0.789203 + 0.614133i $$0.789506\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −3.18761 −0.823036
$$16$$ 1.00000 0.250000
$$17$$ 0.749022 0.181664 0.0908322 0.995866i $$-0.471047\pi$$
0.0908322 + 0.995866i $$0.471047\pi$$
$$18$$ −7.16083 −1.68782
$$19$$ −6.78967 −1.55766 −0.778828 0.627237i $$-0.784186\pi$$
−0.778828 + 0.627237i $$0.784186\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 3.18761 0.695592
$$22$$ 0 0
$$23$$ 3.94010 0.821567 0.410783 0.911733i $$-0.365255\pi$$
0.410783 + 0.911733i $$0.365255\pi$$
$$24$$ −3.18761 −0.650667
$$25$$ 1.00000 0.200000
$$26$$ 5.69102 1.11610
$$27$$ 13.2631 2.55248
$$28$$ 1.00000 0.188982
$$29$$ 5.73837 1.06559 0.532794 0.846245i $$-0.321142\pi$$
0.532794 + 0.846245i $$0.321142\pi$$
$$30$$ 3.18761 0.581974
$$31$$ −7.10952 −1.27691 −0.638454 0.769660i $$-0.720426\pi$$
−0.638454 + 0.769660i $$0.720426\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −0.749022 −0.128456
$$35$$ −1.00000 −0.169031
$$36$$ 7.16083 1.19347
$$37$$ −11.6050 −1.90785 −0.953923 0.300052i $$-0.902996\pi$$
−0.953923 + 0.300052i $$0.902996\pi$$
$$38$$ 6.78967 1.10143
$$39$$ −18.1407 −2.90484
$$40$$ 1.00000 0.158114
$$41$$ −11.4458 −1.78754 −0.893768 0.448530i $$-0.851948\pi$$
−0.893768 + 0.448530i $$0.851948\pi$$
$$42$$ −3.18761 −0.491858
$$43$$ −2.33345 −0.355847 −0.177924 0.984044i $$-0.556938\pi$$
−0.177924 + 0.984044i $$0.556938\pi$$
$$44$$ 0 0
$$45$$ −7.16083 −1.06747
$$46$$ −3.94010 −0.580936
$$47$$ −5.00025 −0.729361 −0.364680 0.931133i $$-0.618822\pi$$
−0.364680 + 0.931133i $$0.618822\pi$$
$$48$$ 3.18761 0.460091
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 2.38759 0.334329
$$52$$ −5.69102 −0.789203
$$53$$ −10.7671 −1.47897 −0.739487 0.673171i $$-0.764932\pi$$
−0.739487 + 0.673171i $$0.764932\pi$$
$$54$$ −13.2631 −1.80488
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −21.6428 −2.86666
$$58$$ −5.73837 −0.753484
$$59$$ 5.22904 0.680763 0.340381 0.940287i $$-0.389444\pi$$
0.340381 + 0.940287i $$0.389444\pi$$
$$60$$ −3.18761 −0.411518
$$61$$ −7.68398 −0.983833 −0.491917 0.870642i $$-0.663704\pi$$
−0.491917 + 0.870642i $$0.663704\pi$$
$$62$$ 7.10952 0.902911
$$63$$ 7.16083 0.902179
$$64$$ 1.00000 0.125000
$$65$$ 5.69102 0.705884
$$66$$ 0 0
$$67$$ −7.39587 −0.903549 −0.451775 0.892132i $$-0.649209\pi$$
−0.451775 + 0.892132i $$0.649209\pi$$
$$68$$ 0.749022 0.0908322
$$69$$ 12.5595 1.51198
$$70$$ 1.00000 0.119523
$$71$$ −0.575191 −0.0682626 −0.0341313 0.999417i $$-0.510866\pi$$
−0.0341313 + 0.999417i $$0.510866\pi$$
$$72$$ −7.16083 −0.843912
$$73$$ 12.5704 1.47125 0.735627 0.677387i $$-0.236888\pi$$
0.735627 + 0.677387i $$0.236888\pi$$
$$74$$ 11.6050 1.34905
$$75$$ 3.18761 0.368073
$$76$$ −6.78967 −0.778828
$$77$$ 0 0
$$78$$ 18.1407 2.05403
$$79$$ −6.83481 −0.768976 −0.384488 0.923130i $$-0.625622\pi$$
−0.384488 + 0.923130i $$0.625622\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 20.7950 2.31055
$$82$$ 11.4458 1.26398
$$83$$ −10.5252 −1.15530 −0.577648 0.816286i $$-0.696029\pi$$
−0.577648 + 0.816286i $$0.696029\pi$$
$$84$$ 3.18761 0.347796
$$85$$ −0.749022 −0.0812428
$$86$$ 2.33345 0.251622
$$87$$ 18.2916 1.96107
$$88$$ 0 0
$$89$$ 4.80467 0.509294 0.254647 0.967034i $$-0.418041\pi$$
0.254647 + 0.967034i $$0.418041\pi$$
$$90$$ 7.16083 0.754817
$$91$$ −5.69102 −0.596581
$$92$$ 3.94010 0.410783
$$93$$ −22.6624 −2.34998
$$94$$ 5.00025 0.515736
$$95$$ 6.78967 0.696605
$$96$$ −3.18761 −0.325334
$$97$$ 16.2717 1.65214 0.826071 0.563566i $$-0.190571\pi$$
0.826071 + 0.563566i $$0.190571\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 6.38488 0.635319 0.317659 0.948205i $$-0.397103\pi$$
0.317659 + 0.948205i $$0.397103\pi$$
$$102$$ −2.38759 −0.236406
$$103$$ 2.32870 0.229454 0.114727 0.993397i $$-0.463401\pi$$
0.114727 + 0.993397i $$0.463401\pi$$
$$104$$ 5.69102 0.558051
$$105$$ −3.18761 −0.311078
$$106$$ 10.7671 1.04579
$$107$$ −1.88876 −0.182593 −0.0912967 0.995824i $$-0.529101\pi$$
−0.0912967 + 0.995824i $$0.529101\pi$$
$$108$$ 13.2631 1.27624
$$109$$ −7.76554 −0.743804 −0.371902 0.928272i $$-0.621294\pi$$
−0.371902 + 0.928272i $$0.621294\pi$$
$$110$$ 0 0
$$111$$ −36.9921 −3.51113
$$112$$ 1.00000 0.0944911
$$113$$ 7.75283 0.729325 0.364663 0.931140i $$-0.381184\pi$$
0.364663 + 0.931140i $$0.381184\pi$$
$$114$$ 21.6428 2.02703
$$115$$ −3.94010 −0.367416
$$116$$ 5.73837 0.532794
$$117$$ −40.7524 −3.76756
$$118$$ −5.22904 −0.481372
$$119$$ 0.749022 0.0686627
$$120$$ 3.18761 0.290987
$$121$$ 0 0
$$122$$ 7.68398 0.695675
$$123$$ −36.4847 −3.28972
$$124$$ −7.10952 −0.638454
$$125$$ −1.00000 −0.0894427
$$126$$ −7.16083 −0.637937
$$127$$ 15.5292 1.37799 0.688996 0.724765i $$-0.258052\pi$$
0.688996 + 0.724765i $$0.258052\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −7.43810 −0.654889
$$130$$ −5.69102 −0.499136
$$131$$ −0.957959 −0.0836973 −0.0418486 0.999124i $$-0.513325\pi$$
−0.0418486 + 0.999124i $$0.513325\pi$$
$$132$$ 0 0
$$133$$ −6.78967 −0.588739
$$134$$ 7.39587 0.638906
$$135$$ −13.2631 −1.14150
$$136$$ −0.749022 −0.0642281
$$137$$ −4.66176 −0.398281 −0.199140 0.979971i $$-0.563815\pi$$
−0.199140 + 0.979971i $$0.563815\pi$$
$$138$$ −12.5595 −1.06913
$$139$$ −4.46428 −0.378655 −0.189328 0.981914i $$-0.560631\pi$$
−0.189328 + 0.981914i $$0.560631\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −15.9388 −1.34229
$$142$$ 0.575191 0.0482690
$$143$$ 0 0
$$144$$ 7.16083 0.596736
$$145$$ −5.73837 −0.476545
$$146$$ −12.5704 −1.04033
$$147$$ 3.18761 0.262909
$$148$$ −11.6050 −0.953923
$$149$$ −15.6247 −1.28003 −0.640013 0.768364i $$-0.721071\pi$$
−0.640013 + 0.768364i $$0.721071\pi$$
$$150$$ −3.18761 −0.260267
$$151$$ 17.7850 1.44732 0.723660 0.690157i $$-0.242458\pi$$
0.723660 + 0.690157i $$0.242458\pi$$
$$152$$ 6.78967 0.550715
$$153$$ 5.36361 0.433623
$$154$$ 0 0
$$155$$ 7.10952 0.571051
$$156$$ −18.1407 −1.45242
$$157$$ −7.37542 −0.588623 −0.294311 0.955710i $$-0.595090\pi$$
−0.294311 + 0.955710i $$0.595090\pi$$
$$158$$ 6.83481 0.543748
$$159$$ −34.3212 −2.72185
$$160$$ 1.00000 0.0790569
$$161$$ 3.94010 0.310523
$$162$$ −20.7950 −1.63381
$$163$$ −10.4614 −0.819402 −0.409701 0.912220i $$-0.634367\pi$$
−0.409701 + 0.912220i $$0.634367\pi$$
$$164$$ −11.4458 −0.893768
$$165$$ 0 0
$$166$$ 10.5252 0.816918
$$167$$ 13.8928 1.07506 0.537528 0.843246i $$-0.319358\pi$$
0.537528 + 0.843246i $$0.319358\pi$$
$$168$$ −3.18761 −0.245929
$$169$$ 19.3877 1.49136
$$170$$ 0.749022 0.0574473
$$171$$ −48.6196 −3.71804
$$172$$ −2.33345 −0.177924
$$173$$ −9.79571 −0.744754 −0.372377 0.928082i $$-0.621457\pi$$
−0.372377 + 0.928082i $$0.621457\pi$$
$$174$$ −18.2916 −1.38669
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 16.6681 1.25285
$$178$$ −4.80467 −0.360126
$$179$$ 0.676043 0.0505298 0.0252649 0.999681i $$-0.491957\pi$$
0.0252649 + 0.999681i $$0.491957\pi$$
$$180$$ −7.16083 −0.533737
$$181$$ −5.50028 −0.408833 −0.204416 0.978884i $$-0.565530\pi$$
−0.204416 + 0.978884i $$0.565530\pi$$
$$182$$ 5.69102 0.421847
$$183$$ −24.4935 −1.81061
$$184$$ −3.94010 −0.290468
$$185$$ 11.6050 0.853215
$$186$$ 22.6624 1.66168
$$187$$ 0 0
$$188$$ −5.00025 −0.364680
$$189$$ 13.2631 0.964747
$$190$$ −6.78967 −0.492574
$$191$$ −12.5963 −0.911434 −0.455717 0.890125i $$-0.650617\pi$$
−0.455717 + 0.890125i $$0.650617\pi$$
$$192$$ 3.18761 0.230046
$$193$$ 2.47246 0.177971 0.0889857 0.996033i $$-0.471637\pi$$
0.0889857 + 0.996033i $$0.471637\pi$$
$$194$$ −16.2717 −1.16824
$$195$$ 18.1407 1.29908
$$196$$ 1.00000 0.0714286
$$197$$ −8.91129 −0.634903 −0.317451 0.948275i $$-0.602827\pi$$
−0.317451 + 0.948275i $$0.602827\pi$$
$$198$$ 0 0
$$199$$ 26.4542 1.87529 0.937644 0.347598i $$-0.113002\pi$$
0.937644 + 0.347598i $$0.113002\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −23.5751 −1.66286
$$202$$ −6.38488 −0.449238
$$203$$ 5.73837 0.402754
$$204$$ 2.38759 0.167164
$$205$$ 11.4458 0.799410
$$206$$ −2.32870 −0.162248
$$207$$ 28.2144 1.96103
$$208$$ −5.69102 −0.394601
$$209$$ 0 0
$$210$$ 3.18761 0.219966
$$211$$ 8.29206 0.570849 0.285424 0.958401i $$-0.407865\pi$$
0.285424 + 0.958401i $$0.407865\pi$$
$$212$$ −10.7671 −0.739487
$$213$$ −1.83348 −0.125628
$$214$$ 1.88876 0.129113
$$215$$ 2.33345 0.159140
$$216$$ −13.2631 −0.902438
$$217$$ −7.10952 −0.482626
$$218$$ 7.76554 0.525949
$$219$$ 40.0695 2.70764
$$220$$ 0 0
$$221$$ −4.26270 −0.286740
$$222$$ 36.9921 2.48275
$$223$$ 0.390611 0.0261573 0.0130786 0.999914i $$-0.495837\pi$$
0.0130786 + 0.999914i $$0.495837\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 7.16083 0.477388
$$226$$ −7.75283 −0.515711
$$227$$ 4.91748 0.326385 0.163192 0.986594i $$-0.447821\pi$$
0.163192 + 0.986594i $$0.447821\pi$$
$$228$$ −21.6428 −1.43333
$$229$$ −9.18360 −0.606869 −0.303435 0.952852i $$-0.598133\pi$$
−0.303435 + 0.952852i $$0.598133\pi$$
$$230$$ 3.94010 0.259802
$$231$$ 0 0
$$232$$ −5.73837 −0.376742
$$233$$ 20.1599 1.32072 0.660359 0.750950i $$-0.270404\pi$$
0.660359 + 0.750950i $$0.270404\pi$$
$$234$$ 40.7524 2.66407
$$235$$ 5.00025 0.326180
$$236$$ 5.22904 0.340381
$$237$$ −21.7867 −1.41520
$$238$$ −0.749022 −0.0485519
$$239$$ −9.77852 −0.632520 −0.316260 0.948673i $$-0.602427\pi$$
−0.316260 + 0.948673i $$0.602427\pi$$
$$240$$ −3.18761 −0.205759
$$241$$ 12.3008 0.792366 0.396183 0.918172i $$-0.370335\pi$$
0.396183 + 0.918172i $$0.370335\pi$$
$$242$$ 0 0
$$243$$ 26.4969 1.69978
$$244$$ −7.68398 −0.491917
$$245$$ −1.00000 −0.0638877
$$246$$ 36.4847 2.32618
$$247$$ 38.6401 2.45861
$$248$$ 7.10952 0.451455
$$249$$ −33.5503 −2.12617
$$250$$ 1.00000 0.0632456
$$251$$ −20.8441 −1.31567 −0.657833 0.753164i $$-0.728527\pi$$
−0.657833 + 0.753164i $$0.728527\pi$$
$$252$$ 7.16083 0.451090
$$253$$ 0 0
$$254$$ −15.5292 −0.974388
$$255$$ −2.38759 −0.149516
$$256$$ 1.00000 0.0625000
$$257$$ 7.52911 0.469653 0.234827 0.972037i $$-0.424548\pi$$
0.234827 + 0.972037i $$0.424548\pi$$
$$258$$ 7.43810 0.463076
$$259$$ −11.6050 −0.721098
$$260$$ 5.69102 0.352942
$$261$$ 41.0914 2.54350
$$262$$ 0.957959 0.0591829
$$263$$ −3.27610 −0.202013 −0.101007 0.994886i $$-0.532206\pi$$
−0.101007 + 0.994886i $$0.532206\pi$$
$$264$$ 0 0
$$265$$ 10.7671 0.661417
$$266$$ 6.78967 0.416301
$$267$$ 15.3154 0.937287
$$268$$ −7.39587 −0.451775
$$269$$ 1.12272 0.0684535 0.0342267 0.999414i $$-0.489103\pi$$
0.0342267 + 0.999414i $$0.489103\pi$$
$$270$$ 13.2631 0.807165
$$271$$ −9.82214 −0.596652 −0.298326 0.954464i $$-0.596428\pi$$
−0.298326 + 0.954464i $$0.596428\pi$$
$$272$$ 0.749022 0.0454161
$$273$$ −18.1407 −1.09793
$$274$$ 4.66176 0.281627
$$275$$ 0 0
$$276$$ 12.5595 0.755991
$$277$$ 0.891218 0.0535481 0.0267741 0.999642i $$-0.491477\pi$$
0.0267741 + 0.999642i $$0.491477\pi$$
$$278$$ 4.46428 0.267750
$$279$$ −50.9101 −3.04791
$$280$$ 1.00000 0.0597614
$$281$$ −17.3571 −1.03544 −0.517720 0.855550i $$-0.673219\pi$$
−0.517720 + 0.855550i $$0.673219\pi$$
$$282$$ 15.9388 0.949142
$$283$$ 2.98931 0.177696 0.0888481 0.996045i $$-0.471681\pi$$
0.0888481 + 0.996045i $$0.471681\pi$$
$$284$$ −0.575191 −0.0341313
$$285$$ 21.6428 1.28201
$$286$$ 0 0
$$287$$ −11.4458 −0.675625
$$288$$ −7.16083 −0.421956
$$289$$ −16.4390 −0.966998
$$290$$ 5.73837 0.336968
$$291$$ 51.8678 3.04054
$$292$$ 12.5704 0.735627
$$293$$ −17.5887 −1.02754 −0.513771 0.857927i $$-0.671752\pi$$
−0.513771 + 0.857927i $$0.671752\pi$$
$$294$$ −3.18761 −0.185905
$$295$$ −5.22904 −0.304446
$$296$$ 11.6050 0.674525
$$297$$ 0 0
$$298$$ 15.6247 0.905115
$$299$$ −22.4232 −1.29677
$$300$$ 3.18761 0.184036
$$301$$ −2.33345 −0.134498
$$302$$ −17.7850 −1.02341
$$303$$ 20.3525 1.16922
$$304$$ −6.78967 −0.389414
$$305$$ 7.68398 0.439984
$$306$$ −5.36361 −0.306617
$$307$$ −1.28791 −0.0735047 −0.0367524 0.999324i $$-0.511701\pi$$
−0.0367524 + 0.999324i $$0.511701\pi$$
$$308$$ 0 0
$$309$$ 7.42298 0.422279
$$310$$ −7.10952 −0.403794
$$311$$ −0.0496330 −0.00281443 −0.00140721 0.999999i $$-0.500448\pi$$
−0.00140721 + 0.999999i $$0.500448\pi$$
$$312$$ 18.1407 1.02702
$$313$$ −27.6778 −1.56444 −0.782220 0.623003i $$-0.785913\pi$$
−0.782220 + 0.623003i $$0.785913\pi$$
$$314$$ 7.37542 0.416219
$$315$$ −7.16083 −0.403467
$$316$$ −6.83481 −0.384488
$$317$$ 9.23290 0.518571 0.259286 0.965801i $$-0.416513\pi$$
0.259286 + 0.965801i $$0.416513\pi$$
$$318$$ 34.3212 1.92464
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ −6.02062 −0.336038
$$322$$ −3.94010 −0.219573
$$323$$ −5.08561 −0.282971
$$324$$ 20.7950 1.15528
$$325$$ −5.69102 −0.315681
$$326$$ 10.4614 0.579405
$$327$$ −24.7535 −1.36887
$$328$$ 11.4458 0.631989
$$329$$ −5.00025 −0.275673
$$330$$ 0 0
$$331$$ 16.5715 0.910851 0.455425 0.890274i $$-0.349487\pi$$
0.455425 + 0.890274i $$0.349487\pi$$
$$332$$ −10.5252 −0.577648
$$333$$ −83.1012 −4.55392
$$334$$ −13.8928 −0.760180
$$335$$ 7.39587 0.404080
$$336$$ 3.18761 0.173898
$$337$$ 6.46713 0.352287 0.176144 0.984364i $$-0.443638\pi$$
0.176144 + 0.984364i $$0.443638\pi$$
$$338$$ −19.3877 −1.05455
$$339$$ 24.7130 1.34222
$$340$$ −0.749022 −0.0406214
$$341$$ 0 0
$$342$$ 48.6196 2.62905
$$343$$ 1.00000 0.0539949
$$344$$ 2.33345 0.125811
$$345$$ −12.5595 −0.676179
$$346$$ 9.79571 0.526621
$$347$$ 29.5567 1.58669 0.793344 0.608774i $$-0.208338\pi$$
0.793344 + 0.608774i $$0.208338\pi$$
$$348$$ 18.2916 0.980535
$$349$$ 25.4079 1.36005 0.680027 0.733187i $$-0.261968\pi$$
0.680027 + 0.733187i $$0.261968\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ −75.4804 −4.02885
$$352$$ 0 0
$$353$$ 13.7939 0.734174 0.367087 0.930187i $$-0.380355\pi$$
0.367087 + 0.930187i $$0.380355\pi$$
$$354$$ −16.6681 −0.885900
$$355$$ 0.575191 0.0305280
$$356$$ 4.80467 0.254647
$$357$$ 2.38759 0.126364
$$358$$ −0.676043 −0.0357300
$$359$$ −22.7783 −1.20219 −0.601096 0.799177i $$-0.705269\pi$$
−0.601096 + 0.799177i $$0.705269\pi$$
$$360$$ 7.16083 0.377409
$$361$$ 27.0996 1.42629
$$362$$ 5.50028 0.289089
$$363$$ 0 0
$$364$$ −5.69102 −0.298291
$$365$$ −12.5704 −0.657965
$$366$$ 24.4935 1.28030
$$367$$ −9.71895 −0.507325 −0.253662 0.967293i $$-0.581635\pi$$
−0.253662 + 0.967293i $$0.581635\pi$$
$$368$$ 3.94010 0.205392
$$369$$ −81.9615 −4.26674
$$370$$ −11.6050 −0.603314
$$371$$ −10.7671 −0.558999
$$372$$ −22.6624 −1.17499
$$373$$ 6.44329 0.333621 0.166811 0.985989i $$-0.446653\pi$$
0.166811 + 0.985989i $$0.446653\pi$$
$$374$$ 0 0
$$375$$ −3.18761 −0.164607
$$376$$ 5.00025 0.257868
$$377$$ −32.6572 −1.68193
$$378$$ −13.2631 −0.682179
$$379$$ −28.8120 −1.47997 −0.739987 0.672621i $$-0.765169\pi$$
−0.739987 + 0.672621i $$0.765169\pi$$
$$380$$ 6.78967 0.348303
$$381$$ 49.5009 2.53601
$$382$$ 12.5963 0.644481
$$383$$ 19.2162 0.981903 0.490952 0.871187i $$-0.336649\pi$$
0.490952 + 0.871187i $$0.336649\pi$$
$$384$$ −3.18761 −0.162667
$$385$$ 0 0
$$386$$ −2.47246 −0.125845
$$387$$ −16.7094 −0.849387
$$388$$ 16.2717 0.826071
$$389$$ −1.16875 −0.0592582 −0.0296291 0.999561i $$-0.509433\pi$$
−0.0296291 + 0.999561i $$0.509433\pi$$
$$390$$ −18.1407 −0.918592
$$391$$ 2.95122 0.149249
$$392$$ −1.00000 −0.0505076
$$393$$ −3.05360 −0.154034
$$394$$ 8.91129 0.448944
$$395$$ 6.83481 0.343896
$$396$$ 0 0
$$397$$ −34.1835 −1.71562 −0.857811 0.513966i $$-0.828176\pi$$
−0.857811 + 0.513966i $$0.828176\pi$$
$$398$$ −26.4542 −1.32603
$$399$$ −21.6428 −1.08349
$$400$$ 1.00000 0.0500000
$$401$$ 24.0445 1.20072 0.600362 0.799728i $$-0.295023\pi$$
0.600362 + 0.799728i $$0.295023\pi$$
$$402$$ 23.5751 1.17582
$$403$$ 40.4605 2.01548
$$404$$ 6.38488 0.317659
$$405$$ −20.7950 −1.03331
$$406$$ −5.73837 −0.284790
$$407$$ 0 0
$$408$$ −2.38759 −0.118203
$$409$$ 5.46092 0.270025 0.135013 0.990844i $$-0.456893\pi$$
0.135013 + 0.990844i $$0.456893\pi$$
$$410$$ −11.4458 −0.565268
$$411$$ −14.8598 −0.732982
$$412$$ 2.32870 0.114727
$$413$$ 5.22904 0.257304
$$414$$ −28.2144 −1.38666
$$415$$ 10.5252 0.516664
$$416$$ 5.69102 0.279025
$$417$$ −14.2304 −0.696864
$$418$$ 0 0
$$419$$ −37.5281 −1.83337 −0.916684 0.399613i $$-0.869145\pi$$
−0.916684 + 0.399613i $$0.869145\pi$$
$$420$$ −3.18761 −0.155539
$$421$$ 9.35357 0.455865 0.227933 0.973677i $$-0.426803\pi$$
0.227933 + 0.973677i $$0.426803\pi$$
$$422$$ −8.29206 −0.403651
$$423$$ −35.8059 −1.74094
$$424$$ 10.7671 0.522896
$$425$$ 0.749022 0.0363329
$$426$$ 1.83348 0.0888325
$$427$$ −7.68398 −0.371854
$$428$$ −1.88876 −0.0912967
$$429$$ 0 0
$$430$$ −2.33345 −0.112529
$$431$$ 5.10019 0.245668 0.122834 0.992427i $$-0.460802\pi$$
0.122834 + 0.992427i $$0.460802\pi$$
$$432$$ 13.2631 0.638120
$$433$$ −15.0790 −0.724649 −0.362325 0.932052i $$-0.618017\pi$$
−0.362325 + 0.932052i $$0.618017\pi$$
$$434$$ 7.10952 0.341268
$$435$$ −18.2916 −0.877017
$$436$$ −7.76554 −0.371902
$$437$$ −26.7520 −1.27972
$$438$$ −40.0695 −1.91459
$$439$$ 8.35774 0.398893 0.199447 0.979909i $$-0.436086\pi$$
0.199447 + 0.979909i $$0.436086\pi$$
$$440$$ 0 0
$$441$$ 7.16083 0.340992
$$442$$ 4.26270 0.202756
$$443$$ 6.68524 0.317626 0.158813 0.987309i $$-0.449233\pi$$
0.158813 + 0.987309i $$0.449233\pi$$
$$444$$ −36.9921 −1.75557
$$445$$ −4.80467 −0.227763
$$446$$ −0.390611 −0.0184960
$$447$$ −49.8054 −2.35572
$$448$$ 1.00000 0.0472456
$$449$$ −7.97205 −0.376224 −0.188112 0.982148i $$-0.560237\pi$$
−0.188112 + 0.982148i $$0.560237\pi$$
$$450$$ −7.16083 −0.337565
$$451$$ 0 0
$$452$$ 7.75283 0.364663
$$453$$ 56.6915 2.66360
$$454$$ −4.91748 −0.230789
$$455$$ 5.69102 0.266799
$$456$$ 21.6428 1.01352
$$457$$ 2.34209 0.109558 0.0547791 0.998498i $$-0.482555\pi$$
0.0547791 + 0.998498i $$0.482555\pi$$
$$458$$ 9.18360 0.429121
$$459$$ 9.93433 0.463695
$$460$$ −3.94010 −0.183708
$$461$$ 0.312762 0.0145668 0.00728340 0.999973i $$-0.497682\pi$$
0.00728340 + 0.999973i $$0.497682\pi$$
$$462$$ 0 0
$$463$$ 15.8769 0.737863 0.368931 0.929457i $$-0.379724\pi$$
0.368931 + 0.929457i $$0.379724\pi$$
$$464$$ 5.73837 0.266397
$$465$$ 22.6624 1.05094
$$466$$ −20.1599 −0.933888
$$467$$ −0.774386 −0.0358343 −0.0179172 0.999839i $$-0.505704\pi$$
−0.0179172 + 0.999839i $$0.505704\pi$$
$$468$$ −40.7524 −1.88378
$$469$$ −7.39587 −0.341510
$$470$$ −5.00025 −0.230644
$$471$$ −23.5099 −1.08328
$$472$$ −5.22904 −0.240686
$$473$$ 0 0
$$474$$ 21.7867 1.00069
$$475$$ −6.78967 −0.311531
$$476$$ 0.749022 0.0343313
$$477$$ −77.1013 −3.53022
$$478$$ 9.77852 0.447259
$$479$$ −30.3787 −1.38804 −0.694019 0.719956i $$-0.744162\pi$$
−0.694019 + 0.719956i $$0.744162\pi$$
$$480$$ 3.18761 0.145494
$$481$$ 66.0441 3.01135
$$482$$ −12.3008 −0.560287
$$483$$ 12.5595 0.571476
$$484$$ 0 0
$$485$$ −16.2717 −0.738861
$$486$$ −26.4969 −1.20192
$$487$$ 9.43710 0.427636 0.213818 0.976874i $$-0.431410\pi$$
0.213818 + 0.976874i $$0.431410\pi$$
$$488$$ 7.68398 0.347838
$$489$$ −33.3469 −1.50800
$$490$$ 1.00000 0.0451754
$$491$$ −28.7129 −1.29579 −0.647897 0.761728i $$-0.724351\pi$$
−0.647897 + 0.761728i $$0.724351\pi$$
$$492$$ −36.4847 −1.64486
$$493$$ 4.29816 0.193579
$$494$$ −38.6401 −1.73850
$$495$$ 0 0
$$496$$ −7.10952 −0.319227
$$497$$ −0.575191 −0.0258008
$$498$$ 33.5503 1.50343
$$499$$ 27.6602 1.23824 0.619120 0.785297i $$-0.287490\pi$$
0.619120 + 0.785297i $$0.287490\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 44.2848 1.97850
$$502$$ 20.8441 0.930316
$$503$$ 1.65413 0.0737539 0.0368770 0.999320i $$-0.488259\pi$$
0.0368770 + 0.999320i $$0.488259\pi$$
$$504$$ −7.16083 −0.318969
$$505$$ −6.38488 −0.284123
$$506$$ 0 0
$$507$$ 61.8004 2.74465
$$508$$ 15.5292 0.688996
$$509$$ 11.0335 0.489050 0.244525 0.969643i $$-0.421368\pi$$
0.244525 + 0.969643i $$0.421368\pi$$
$$510$$ 2.38759 0.105724
$$511$$ 12.5704 0.556082
$$512$$ −1.00000 −0.0441942
$$513$$ −90.0519 −3.97589
$$514$$ −7.52911 −0.332095
$$515$$ −2.32870 −0.102615
$$516$$ −7.43810 −0.327444
$$517$$ 0 0
$$518$$ 11.6050 0.509893
$$519$$ −31.2249 −1.37062
$$520$$ −5.69102 −0.249568
$$521$$ 19.2485 0.843294 0.421647 0.906760i $$-0.361452\pi$$
0.421647 + 0.906760i $$0.361452\pi$$
$$522$$ −41.0914 −1.79852
$$523$$ −4.09540 −0.179079 −0.0895396 0.995983i $$-0.528540\pi$$
−0.0895396 + 0.995983i $$0.528540\pi$$
$$524$$ −0.957959 −0.0418486
$$525$$ 3.18761 0.139118
$$526$$ 3.27610 0.142845
$$527$$ −5.32519 −0.231969
$$528$$ 0 0
$$529$$ −7.47564 −0.325028
$$530$$ −10.7671 −0.467692
$$531$$ 37.4442 1.62494
$$532$$ −6.78967 −0.294369
$$533$$ 65.1383 2.82146
$$534$$ −15.3154 −0.662762
$$535$$ 1.88876 0.0816582
$$536$$ 7.39587 0.319453
$$537$$ 2.15496 0.0929933
$$538$$ −1.12272 −0.0484039
$$539$$ 0 0
$$540$$ −13.2631 −0.570752
$$541$$ 12.0631 0.518632 0.259316 0.965793i $$-0.416503\pi$$
0.259316 + 0.965793i $$0.416503\pi$$
$$542$$ 9.82214 0.421897
$$543$$ −17.5327 −0.752402
$$544$$ −0.749022 −0.0321140
$$545$$ 7.76554 0.332639
$$546$$ 18.1407 0.776352
$$547$$ 5.69776 0.243619 0.121809 0.992554i $$-0.461130\pi$$
0.121809 + 0.992554i $$0.461130\pi$$
$$548$$ −4.66176 −0.199140
$$549$$ −55.0237 −2.34835
$$550$$ 0 0
$$551$$ −38.9616 −1.65982
$$552$$ −12.5595 −0.534567
$$553$$ −6.83481 −0.290645
$$554$$ −0.891218 −0.0378642
$$555$$ 36.9921 1.57023
$$556$$ −4.46428 −0.189328
$$557$$ −8.87789 −0.376168 −0.188084 0.982153i $$-0.560228\pi$$
−0.188084 + 0.982153i $$0.560228\pi$$
$$558$$ 50.9101 2.15520
$$559$$ 13.2797 0.561671
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ 17.3571 0.732167
$$563$$ −20.1280 −0.848294 −0.424147 0.905593i $$-0.639426\pi$$
−0.424147 + 0.905593i $$0.639426\pi$$
$$564$$ −15.9388 −0.671145
$$565$$ −7.75283 −0.326164
$$566$$ −2.98931 −0.125650
$$567$$ 20.7950 0.873306
$$568$$ 0.575191 0.0241345
$$569$$ −24.8357 −1.04117 −0.520583 0.853811i $$-0.674285\pi$$
−0.520583 + 0.853811i $$0.674285\pi$$
$$570$$ −21.6428 −0.906516
$$571$$ 28.6955 1.20087 0.600435 0.799674i $$-0.294994\pi$$
0.600435 + 0.799674i $$0.294994\pi$$
$$572$$ 0 0
$$573$$ −40.1520 −1.67737
$$574$$ 11.4458 0.477739
$$575$$ 3.94010 0.164313
$$576$$ 7.16083 0.298368
$$577$$ −15.3853 −0.640499 −0.320249 0.947333i $$-0.603767\pi$$
−0.320249 + 0.947333i $$0.603767\pi$$
$$578$$ 16.4390 0.683771
$$579$$ 7.88122 0.327532
$$580$$ −5.73837 −0.238273
$$581$$ −10.5252 −0.436661
$$582$$ −51.8678 −2.14999
$$583$$ 0 0
$$584$$ −12.5704 −0.520167
$$585$$ 40.7524 1.68491
$$586$$ 17.5887 0.726582
$$587$$ −36.1775 −1.49321 −0.746603 0.665270i $$-0.768317\pi$$
−0.746603 + 0.665270i $$0.768317\pi$$
$$588$$ 3.18761 0.131455
$$589$$ 48.2713 1.98898
$$590$$ 5.22904 0.215276
$$591$$ −28.4057 −1.16845
$$592$$ −11.6050 −0.476961
$$593$$ −22.6239 −0.929052 −0.464526 0.885560i $$-0.653775\pi$$
−0.464526 + 0.885560i $$0.653775\pi$$
$$594$$ 0 0
$$595$$ −0.749022 −0.0307069
$$596$$ −15.6247 −0.640013
$$597$$ 84.3255 3.45121
$$598$$ 22.4232 0.916952
$$599$$ 2.93462 0.119906 0.0599528 0.998201i $$-0.480905\pi$$
0.0599528 + 0.998201i $$0.480905\pi$$
$$600$$ −3.18761 −0.130133
$$601$$ 33.3078 1.35865 0.679327 0.733836i $$-0.262272\pi$$
0.679327 + 0.733836i $$0.262272\pi$$
$$602$$ 2.33345 0.0951041
$$603$$ −52.9606 −2.15672
$$604$$ 17.7850 0.723660
$$605$$ 0 0
$$606$$ −20.3525 −0.826762
$$607$$ −27.9310 −1.13369 −0.566843 0.823826i $$-0.691835\pi$$
−0.566843 + 0.823826i $$0.691835\pi$$
$$608$$ 6.78967 0.275357
$$609$$ 18.2916 0.741215
$$610$$ −7.68398 −0.311115
$$611$$ 28.4565 1.15123
$$612$$ 5.36361 0.216811
$$613$$ −13.2976 −0.537087 −0.268543 0.963268i $$-0.586542\pi$$
−0.268543 + 0.963268i $$0.586542\pi$$
$$614$$ 1.28791 0.0519757
$$615$$ 36.4847 1.47121
$$616$$ 0 0
$$617$$ −17.0581 −0.686735 −0.343368 0.939201i $$-0.611568\pi$$
−0.343368 + 0.939201i $$0.611568\pi$$
$$618$$ −7.42298 −0.298596
$$619$$ −29.0446 −1.16740 −0.583700 0.811969i $$-0.698396\pi$$
−0.583700 + 0.811969i $$0.698396\pi$$
$$620$$ 7.10952 0.285525
$$621$$ 52.2578 2.09703
$$622$$ 0.0496330 0.00199010
$$623$$ 4.80467 0.192495
$$624$$ −18.1407 −0.726210
$$625$$ 1.00000 0.0400000
$$626$$ 27.6778 1.10623
$$627$$ 0 0
$$628$$ −7.37542 −0.294311
$$629$$ −8.69238 −0.346588
$$630$$ 7.16083 0.285294
$$631$$ 32.0364 1.27535 0.637675 0.770305i $$-0.279896\pi$$
0.637675 + 0.770305i $$0.279896\pi$$
$$632$$ 6.83481 0.271874
$$633$$ 26.4318 1.05057
$$634$$ −9.23290 −0.366685
$$635$$ −15.5292 −0.616257
$$636$$ −34.3212 −1.36093
$$637$$ −5.69102 −0.225486
$$638$$ 0 0
$$639$$ −4.11884 −0.162939
$$640$$ 1.00000 0.0395285
$$641$$ 7.75431 0.306277 0.153138 0.988205i $$-0.451062\pi$$
0.153138 + 0.988205i $$0.451062\pi$$
$$642$$ 6.02062 0.237615
$$643$$ 24.2104 0.954763 0.477382 0.878696i $$-0.341586\pi$$
0.477382 + 0.878696i $$0.341586\pi$$
$$644$$ 3.94010 0.155262
$$645$$ 7.43810 0.292875
$$646$$ 5.08561 0.200091
$$647$$ −0.861530 −0.0338702 −0.0169351 0.999857i $$-0.505391\pi$$
−0.0169351 + 0.999857i $$0.505391\pi$$
$$648$$ −20.7950 −0.816903
$$649$$ 0 0
$$650$$ 5.69102 0.223220
$$651$$ −22.6624 −0.888208
$$652$$ −10.4614 −0.409701
$$653$$ −21.7225 −0.850068 −0.425034 0.905177i $$-0.639738\pi$$
−0.425034 + 0.905177i $$0.639738\pi$$
$$654$$ 24.7535 0.967938
$$655$$ 0.957959 0.0374306
$$656$$ −11.4458 −0.446884
$$657$$ 90.0144 3.51180
$$658$$ 5.00025 0.194930
$$659$$ 35.0148 1.36398 0.681991 0.731360i $$-0.261114\pi$$
0.681991 + 0.731360i $$0.261114\pi$$
$$660$$ 0 0
$$661$$ 21.6301 0.841315 0.420657 0.907220i $$-0.361799\pi$$
0.420657 + 0.907220i $$0.361799\pi$$
$$662$$ −16.5715 −0.644069
$$663$$ −13.5878 −0.527706
$$664$$ 10.5252 0.408459
$$665$$ 6.78967 0.263292
$$666$$ 83.1012 3.22011
$$667$$ 22.6097 0.875452
$$668$$ 13.8928 0.537528
$$669$$ 1.24511 0.0481389
$$670$$ −7.39587 −0.285727
$$671$$ 0 0
$$672$$ −3.18761 −0.122965
$$673$$ 34.4741 1.32888 0.664440 0.747342i $$-0.268670\pi$$
0.664440 + 0.747342i $$0.268670\pi$$
$$674$$ −6.46713 −0.249105
$$675$$ 13.2631 0.510496
$$676$$ 19.3877 0.745682
$$677$$ 13.5315 0.520058 0.260029 0.965601i $$-0.416268\pi$$
0.260029 + 0.965601i $$0.416268\pi$$
$$678$$ −24.7130 −0.949096
$$679$$ 16.2717 0.624451
$$680$$ 0.749022 0.0287237
$$681$$ 15.6750 0.600667
$$682$$ 0 0
$$683$$ 19.6188 0.750694 0.375347 0.926884i $$-0.377524\pi$$
0.375347 + 0.926884i $$0.377524\pi$$
$$684$$ −48.6196 −1.85902
$$685$$ 4.66176 0.178117
$$686$$ −1.00000 −0.0381802
$$687$$ −29.2737 −1.11686
$$688$$ −2.33345 −0.0889618
$$689$$ 61.2757 2.33442
$$690$$ 12.5595 0.478131
$$691$$ −27.4372 −1.04376 −0.521880 0.853019i $$-0.674769\pi$$
−0.521880 + 0.853019i $$0.674769\pi$$
$$692$$ −9.79571 −0.372377
$$693$$ 0 0
$$694$$ −29.5567 −1.12196
$$695$$ 4.46428 0.169340
$$696$$ −18.2916 −0.693343
$$697$$ −8.57316 −0.324732
$$698$$ −25.4079 −0.961703
$$699$$ 64.2617 2.43060
$$700$$ 1.00000 0.0377964
$$701$$ 32.2446 1.21786 0.608930 0.793224i $$-0.291599\pi$$
0.608930 + 0.793224i $$0.291599\pi$$
$$702$$ 75.4804 2.84883
$$703$$ 78.7939 2.97177
$$704$$ 0 0
$$705$$ 15.9388 0.600290
$$706$$ −13.7939 −0.519139
$$707$$ 6.38488 0.240128
$$708$$ 16.6681 0.626426
$$709$$ 20.8485 0.782982 0.391491 0.920182i $$-0.371959\pi$$
0.391491 + 0.920182i $$0.371959\pi$$
$$710$$ −0.575191 −0.0215865
$$711$$ −48.9429 −1.83550
$$712$$ −4.80467 −0.180063
$$713$$ −28.0122 −1.04907
$$714$$ −2.38759 −0.0893531
$$715$$ 0 0
$$716$$ 0.676043 0.0252649
$$717$$ −31.1701 −1.16407
$$718$$ 22.7783 0.850078
$$719$$ −40.3188 −1.50364 −0.751818 0.659370i $$-0.770823\pi$$
−0.751818 + 0.659370i $$0.770823\pi$$
$$720$$ −7.16083 −0.266868
$$721$$ 2.32870 0.0867254
$$722$$ −27.0996 −1.00854
$$723$$ 39.2102 1.45824
$$724$$ −5.50028 −0.204416
$$725$$ 5.73837 0.213118
$$726$$ 0 0
$$727$$ 39.1899 1.45347 0.726736 0.686917i $$-0.241036\pi$$
0.726736 + 0.686917i $$0.241036\pi$$
$$728$$ 5.69102 0.210923
$$729$$ 22.0768 0.817660
$$730$$ 12.5704 0.465251
$$731$$ −1.74780 −0.0646448
$$732$$ −24.4935 −0.905306
$$733$$ 24.4077 0.901518 0.450759 0.892646i $$-0.351153\pi$$
0.450759 + 0.892646i $$0.351153\pi$$
$$734$$ 9.71895 0.358733
$$735$$ −3.18761 −0.117577
$$736$$ −3.94010 −0.145234
$$737$$ 0 0
$$738$$ 81.9615 3.01704
$$739$$ 41.5299 1.52770 0.763851 0.645392i $$-0.223306\pi$$
0.763851 + 0.645392i $$0.223306\pi$$
$$740$$ 11.6050 0.426607
$$741$$ 123.170 4.52475
$$742$$ 10.7671 0.395272
$$743$$ 22.6025 0.829207 0.414603 0.910002i $$-0.363920\pi$$
0.414603 + 0.910002i $$0.363920\pi$$
$$744$$ 22.6624 0.830842
$$745$$ 15.6247 0.572445
$$746$$ −6.44329 −0.235906
$$747$$ −75.3695 −2.75763
$$748$$ 0 0
$$749$$ −1.88876 −0.0690138
$$750$$ 3.18761 0.116395
$$751$$ −22.7802 −0.831262 −0.415631 0.909533i $$-0.636439\pi$$
−0.415631 + 0.909533i $$0.636439\pi$$
$$752$$ −5.00025 −0.182340
$$753$$ −66.4426 −2.42130
$$754$$ 32.6572 1.18930
$$755$$ −17.7850 −0.647261
$$756$$ 13.2631 0.482373
$$757$$ 6.77323 0.246177 0.123089 0.992396i $$-0.460720\pi$$
0.123089 + 0.992396i $$0.460720\pi$$
$$758$$ 28.8120 1.04650
$$759$$ 0 0
$$760$$ −6.78967 −0.246287
$$761$$ −7.24515 −0.262637 −0.131318 0.991340i $$-0.541921\pi$$
−0.131318 + 0.991340i $$0.541921\pi$$
$$762$$ −49.5009 −1.79323
$$763$$ −7.76554 −0.281132
$$764$$ −12.5963 −0.455717
$$765$$ −5.36361 −0.193922
$$766$$ −19.2162 −0.694310
$$767$$ −29.7586 −1.07452
$$768$$ 3.18761 0.115023
$$769$$ 1.45679 0.0525333 0.0262667 0.999655i $$-0.491638\pi$$
0.0262667 + 0.999655i $$0.491638\pi$$
$$770$$ 0 0
$$771$$ 23.9998 0.864333
$$772$$ 2.47246 0.0889857
$$773$$ −25.4505 −0.915390 −0.457695 0.889109i $$-0.651325\pi$$
−0.457695 + 0.889109i $$0.651325\pi$$
$$774$$ 16.7094 0.600607
$$775$$ −7.10952 −0.255382
$$776$$ −16.2717 −0.584121
$$777$$ −36.9921 −1.32708
$$778$$ 1.16875 0.0419018
$$779$$ 77.7133 2.78437
$$780$$ 18.1407 0.649542
$$781$$ 0 0
$$782$$ −2.95122 −0.105535
$$783$$ 76.1084 2.71989
$$784$$ 1.00000 0.0357143
$$785$$ 7.37542 0.263240
$$786$$ 3.05360 0.108918
$$787$$ −18.9942 −0.677069 −0.338535 0.940954i $$-0.609931\pi$$
−0.338535 + 0.940954i $$0.609931\pi$$
$$788$$ −8.91129 −0.317451
$$789$$ −10.4429 −0.371778
$$790$$ −6.83481 −0.243171
$$791$$ 7.75283 0.275659
$$792$$ 0 0
$$793$$ 43.7297 1.55289
$$794$$ 34.1835 1.21313
$$795$$ 34.3212 1.21725
$$796$$ 26.4542 0.937644
$$797$$ −44.9953 −1.59381 −0.796907 0.604102i $$-0.793532\pi$$
−0.796907 + 0.604102i $$0.793532\pi$$
$$798$$ 21.6428 0.766146
$$799$$ −3.74529 −0.132499
$$800$$ −1.00000 −0.0353553
$$801$$ 34.4054 1.21566
$$802$$ −24.0445 −0.849041
$$803$$ 0 0
$$804$$ −23.5751 −0.831430
$$805$$ −3.94010 −0.138870
$$806$$ −40.4605 −1.42516
$$807$$ 3.57879 0.125979
$$808$$ −6.38488 −0.224619
$$809$$ 31.2609 1.09907 0.549537 0.835469i $$-0.314804\pi$$
0.549537 + 0.835469i $$0.314804\pi$$
$$810$$ 20.7950 0.730661
$$811$$ −27.5975 −0.969078 −0.484539 0.874770i $$-0.661013\pi$$
−0.484539 + 0.874770i $$0.661013\pi$$
$$812$$ 5.73837 0.201377
$$813$$ −31.3091 −1.09806
$$814$$ 0 0
$$815$$ 10.4614 0.366448
$$816$$ 2.38759 0.0835822
$$817$$ 15.8433 0.554288
$$818$$ −5.46092 −0.190937
$$819$$ −40.7524 −1.42400
$$820$$ 11.4458 0.399705
$$821$$ 21.3493 0.745096 0.372548 0.928013i $$-0.378484\pi$$
0.372548 + 0.928013i $$0.378484\pi$$
$$822$$ 14.8598 0.518296
$$823$$ −52.8518 −1.84230 −0.921149 0.389210i $$-0.872748\pi$$
−0.921149 + 0.389210i $$0.872748\pi$$
$$824$$ −2.32870 −0.0811242
$$825$$ 0 0
$$826$$ −5.22904 −0.181941
$$827$$ −17.8294 −0.619990 −0.309995 0.950738i $$-0.600327\pi$$
−0.309995 + 0.950738i $$0.600327\pi$$
$$828$$ 28.2144 0.980517
$$829$$ −46.2177 −1.60521 −0.802603 0.596513i $$-0.796552\pi$$
−0.802603 + 0.596513i $$0.796552\pi$$
$$830$$ −10.5252 −0.365337
$$831$$ 2.84085 0.0985481
$$832$$ −5.69102 −0.197301
$$833$$ 0.749022 0.0259521
$$834$$ 14.2304 0.492757
$$835$$ −13.8928 −0.480780
$$836$$ 0 0
$$837$$ −94.2942 −3.25928
$$838$$ 37.5281 1.29639
$$839$$ −19.4145 −0.670263 −0.335131 0.942171i $$-0.608781\pi$$
−0.335131 + 0.942171i $$0.608781\pi$$
$$840$$ 3.18761 0.109983
$$841$$ 3.92884 0.135477
$$842$$ −9.35357 −0.322345
$$843$$ −55.3277 −1.90559
$$844$$ 8.29206 0.285424
$$845$$ −19.3877 −0.666958
$$846$$ 35.8059 1.23103
$$847$$ 0 0
$$848$$ −10.7671 −0.369743
$$849$$ 9.52875 0.327026
$$850$$ −0.749022 −0.0256912
$$851$$ −45.7247 −1.56742
$$852$$ −1.83348 −0.0628141
$$853$$ 23.3656 0.800021 0.400011 0.916510i $$-0.369006\pi$$
0.400011 + 0.916510i $$0.369006\pi$$
$$854$$ 7.68398 0.262940
$$855$$ 48.6196 1.66276
$$856$$ 1.88876 0.0645565
$$857$$ 32.8025 1.12051 0.560256 0.828320i $$-0.310703\pi$$
0.560256 + 0.828320i $$0.310703\pi$$
$$858$$ 0 0
$$859$$ −5.80773 −0.198157 −0.0990786 0.995080i $$-0.531590\pi$$
−0.0990786 + 0.995080i $$0.531590\pi$$
$$860$$ 2.33345 0.0795698
$$861$$ −36.4847 −1.24340
$$862$$ −5.10019 −0.173713
$$863$$ −44.4137 −1.51186 −0.755931 0.654652i $$-0.772815\pi$$
−0.755931 + 0.654652i $$0.772815\pi$$
$$864$$ −13.2631 −0.451219
$$865$$ 9.79571 0.333064
$$866$$ 15.0790 0.512405
$$867$$ −52.4009 −1.77963
$$868$$ −7.10952 −0.241313
$$869$$ 0 0
$$870$$ 18.2916 0.620145
$$871$$ 42.0901 1.42617
$$872$$ 7.76554 0.262974
$$873$$ 116.519 3.94357
$$874$$ 26.7520 0.904898
$$875$$ −1.00000 −0.0338062
$$876$$ 40.0695 1.35382
$$877$$ 17.3464 0.585746 0.292873 0.956151i $$-0.405389\pi$$
0.292873 + 0.956151i $$0.405389\pi$$
$$878$$ −8.35774 −0.282060
$$879$$ −56.0658 −1.89105
$$880$$ 0 0
$$881$$ 2.06132 0.0694478 0.0347239 0.999397i $$-0.488945\pi$$
0.0347239 + 0.999397i $$0.488945\pi$$
$$882$$ −7.16083 −0.241118
$$883$$ −46.5639 −1.56700 −0.783501 0.621391i $$-0.786568\pi$$
−0.783501 + 0.621391i $$0.786568\pi$$
$$884$$ −4.26270 −0.143370
$$885$$ −16.6681 −0.560292
$$886$$ −6.68524 −0.224595
$$887$$ −27.9025 −0.936873 −0.468437 0.883497i $$-0.655183\pi$$
−0.468437 + 0.883497i $$0.655183\pi$$
$$888$$ 36.9921 1.24137
$$889$$ 15.5292 0.520832
$$890$$ 4.80467 0.161053
$$891$$ 0 0
$$892$$ 0.390611 0.0130786
$$893$$ 33.9500 1.13609
$$894$$ 49.8054 1.66574
$$895$$ −0.676043 −0.0225976
$$896$$ −1.00000 −0.0334077
$$897$$ −71.4762 −2.38652
$$898$$ 7.97205 0.266031
$$899$$ −40.7971 −1.36066
$$900$$ 7.16083 0.238694
$$901$$ −8.06478 −0.268677
$$902$$ 0 0
$$903$$ −7.43810 −0.247525
$$904$$ −7.75283 −0.257855
$$905$$ 5.50028 0.182836
$$906$$ −56.6915 −1.88345
$$907$$ −7.30550 −0.242575 −0.121288 0.992617i $$-0.538702\pi$$
−0.121288 + 0.992617i $$0.538702\pi$$
$$908$$ 4.91748 0.163192
$$909$$ 45.7210 1.51647
$$910$$ −5.69102 −0.188656
$$911$$ −28.9807 −0.960174 −0.480087 0.877221i $$-0.659395\pi$$
−0.480087 + 0.877221i $$0.659395\pi$$
$$912$$ −21.6428 −0.716664
$$913$$ 0 0
$$914$$ −2.34209 −0.0774693
$$915$$ 24.4935 0.809730
$$916$$ −9.18360 −0.303435
$$917$$ −0.957959 −0.0316346
$$918$$ −9.93433 −0.327882
$$919$$ −5.06468 −0.167068 −0.0835342 0.996505i $$-0.526621\pi$$
−0.0835342 + 0.996505i $$0.526621\pi$$
$$920$$ 3.94010 0.129901
$$921$$ −4.10534 −0.135276
$$922$$ −0.312762 −0.0103003
$$923$$ 3.27342 0.107746
$$924$$ 0 0
$$925$$ −11.6050 −0.381569
$$926$$ −15.8769 −0.521748
$$927$$ 16.6754 0.547693
$$928$$ −5.73837 −0.188371
$$929$$ −23.6262 −0.775151 −0.387576 0.921838i $$-0.626687\pi$$
−0.387576 + 0.921838i $$0.626687\pi$$
$$930$$ −22.6624 −0.743128
$$931$$ −6.78967 −0.222522
$$932$$ 20.1599 0.660359
$$933$$ −0.158210 −0.00517957
$$934$$ 0.774386 0.0253387
$$935$$ 0 0
$$936$$ 40.7524 1.33203
$$937$$ 39.5066 1.29063 0.645313 0.763919i $$-0.276727\pi$$
0.645313 + 0.763919i $$0.276727\pi$$
$$938$$ 7.39587 0.241484
$$939$$ −88.2258 −2.87914
$$940$$ 5.00025 0.163090
$$941$$ −14.2738 −0.465313 −0.232656 0.972559i $$-0.574742\pi$$
−0.232656 + 0.972559i $$0.574742\pi$$
$$942$$ 23.5099 0.765995
$$943$$ −45.0976 −1.46858
$$944$$ 5.22904 0.170191
$$945$$ −13.2631 −0.431448
$$946$$ 0 0
$$947$$ 15.8228 0.514171 0.257085 0.966389i $$-0.417238\pi$$
0.257085 + 0.966389i $$0.417238\pi$$
$$948$$ −21.7867 −0.707598
$$949$$ −71.5384 −2.32223
$$950$$ 6.78967 0.220286
$$951$$ 29.4308 0.954360
$$952$$ −0.749022 −0.0242759
$$953$$ −31.9949 −1.03642 −0.518208 0.855255i $$-0.673401\pi$$
−0.518208 + 0.855255i $$0.673401\pi$$
$$954$$ 77.1013 2.49625
$$955$$ 12.5963 0.407606
$$956$$ −9.77852 −0.316260
$$957$$ 0 0
$$958$$ 30.3787 0.981491
$$959$$ −4.66176 −0.150536
$$960$$ −3.18761 −0.102880
$$961$$ 19.5453 0.630495
$$962$$ −66.0441 −2.12935
$$963$$ −13.5251 −0.435840
$$964$$ 12.3008 0.396183
$$965$$ −2.47246 −0.0795912
$$966$$ −12.5595 −0.404094
$$967$$ 24.1383 0.776237 0.388118 0.921610i $$-0.373125\pi$$
0.388118 + 0.921610i $$0.373125\pi$$
$$968$$ 0 0
$$969$$ −16.2109 −0.520770
$$970$$ 16.2717 0.522453
$$971$$ −24.3861 −0.782587 −0.391293 0.920266i $$-0.627972\pi$$
−0.391293 + 0.920266i $$0.627972\pi$$
$$972$$ 26.4969 0.849889
$$973$$ −4.46428 −0.143118
$$974$$ −9.43710 −0.302384
$$975$$ −18.1407 −0.580968
$$976$$ −7.68398 −0.245958
$$977$$ 54.4580 1.74227 0.871133 0.491046i $$-0.163385\pi$$
0.871133 + 0.491046i $$0.163385\pi$$
$$978$$ 33.3469 1.06632
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ −55.6077 −1.77542
$$982$$ 28.7129 0.916264
$$983$$ −52.4013 −1.67134 −0.835670 0.549231i $$-0.814921\pi$$
−0.835670 + 0.549231i $$0.814921\pi$$
$$984$$ 36.4847 1.16309
$$985$$ 8.91129 0.283937
$$986$$ −4.29816 −0.136881
$$987$$ −15.9388 −0.507338
$$988$$ 38.6401 1.22931
$$989$$ −9.19400 −0.292352
$$990$$ 0 0
$$991$$ 13.7527 0.436870 0.218435 0.975851i $$-0.429905\pi$$
0.218435 + 0.975851i $$0.429905\pi$$
$$992$$ 7.10952 0.225728
$$993$$ 52.8233 1.67630
$$994$$ 0.575191 0.0182440
$$995$$ −26.4542 −0.838654
$$996$$ −33.5503 −1.06308
$$997$$ −38.7130 −1.22605 −0.613026 0.790062i $$-0.710048\pi$$
−0.613026 + 0.790062i $$0.710048\pi$$
$$998$$ −27.6602 −0.875568
$$999$$ −153.918 −4.86974
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.dg.1.8 8
11.7 odd 10 770.2.n.k.71.1 16
11.8 odd 10 770.2.n.k.141.1 yes 16
11.10 odd 2 8470.2.a.dh.1.8 8

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.71.1 16 11.7 odd 10
770.2.n.k.141.1 yes 16 11.8 odd 10
8470.2.a.dg.1.8 8 1.1 even 1 trivial
8470.2.a.dh.1.8 8 11.10 odd 2