# Properties

 Label 8470.2.a.dd.1.6 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: $$x^{6} - 18x^{4} - 4x^{3} + 81x^{2} + 36x - 44$$ 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 1.6 Root $$-3.24337$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +3.24337 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.24337 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.51947 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +3.24337 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.24337 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.51947 q^{9} -1.00000 q^{10} +3.24337 q^{12} +2.14515 q^{13} -1.00000 q^{14} -3.24337 q^{15} +1.00000 q^{16} +1.43942 q^{17} +7.51947 q^{18} -0.633832 q^{19} -1.00000 q^{20} -3.24337 q^{21} -5.61769 q^{23} +3.24337 q^{24} +1.00000 q^{25} +2.14515 q^{26} +14.6583 q^{27} -1.00000 q^{28} -1.19605 q^{29} -3.24337 q^{30} +6.95889 q^{31} +1.00000 q^{32} +1.43942 q^{34} +1.00000 q^{35} +7.51947 q^{36} +4.80395 q^{37} -0.633832 q^{38} +6.95754 q^{39} -1.00000 q^{40} -8.93663 q^{41} -3.24337 q^{42} +9.70747 q^{43} -7.51947 q^{45} -5.61769 q^{46} +8.29874 q^{47} +3.24337 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.66858 q^{51} +2.14515 q^{52} +14.1537 q^{53} +14.6583 q^{54} -1.00000 q^{56} -2.05576 q^{57} -1.19605 q^{58} -1.19248 q^{59} -3.24337 q^{60} -4.19441 q^{61} +6.95889 q^{62} -7.51947 q^{63} +1.00000 q^{64} -2.14515 q^{65} -12.0099 q^{67} +1.43942 q^{68} -18.2203 q^{69} +1.00000 q^{70} -5.71591 q^{71} +7.51947 q^{72} +6.43981 q^{73} +4.80395 q^{74} +3.24337 q^{75} -0.633832 q^{76} +6.95754 q^{78} -7.69133 q^{79} -1.00000 q^{80} +24.9840 q^{81} -8.93663 q^{82} +11.5685 q^{83} -3.24337 q^{84} -1.43942 q^{85} +9.70747 q^{86} -3.87923 q^{87} +16.7855 q^{89} -7.51947 q^{90} -2.14515 q^{91} -5.61769 q^{92} +22.5703 q^{93} +8.29874 q^{94} +0.633832 q^{95} +3.24337 q^{96} -2.96529 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{7} + 6 q^{8} + 18 q^{9}+O(q^{10})$$ 6 * q + 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^7 + 6 * q^8 + 18 * q^9 $$6 q + 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{7} + 6 q^{8} + 18 q^{9} - 6 q^{10} - 6 q^{14} + 6 q^{16} - 6 q^{17} + 18 q^{18} - 6 q^{20} + 6 q^{25} - 12 q^{27} - 6 q^{28} - 12 q^{29} + 6 q^{32} - 6 q^{34} + 6 q^{35} + 18 q^{36} + 24 q^{37} + 24 q^{39} - 6 q^{40} - 12 q^{41} + 18 q^{43} - 18 q^{45} + 24 q^{47} + 6 q^{49} + 6 q^{50} + 12 q^{51} + 36 q^{53} - 12 q^{54} - 6 q^{56} + 12 q^{57} - 12 q^{58} + 30 q^{59} - 36 q^{61} - 18 q^{63} + 6 q^{64} - 12 q^{67} - 6 q^{68} + 6 q^{70} + 6 q^{71} + 18 q^{72} + 6 q^{73} + 24 q^{74} + 24 q^{78} + 24 q^{79} - 6 q^{80} + 54 q^{81} - 12 q^{82} - 24 q^{83} + 6 q^{85} + 18 q^{86} + 24 q^{87} + 36 q^{89} - 18 q^{90} + 24 q^{94} + 6 q^{98}+O(q^{100})$$ 6 * q + 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^7 + 6 * q^8 + 18 * q^9 - 6 * q^10 - 6 * q^14 + 6 * q^16 - 6 * q^17 + 18 * q^18 - 6 * q^20 + 6 * q^25 - 12 * q^27 - 6 * q^28 - 12 * q^29 + 6 * q^32 - 6 * q^34 + 6 * q^35 + 18 * q^36 + 24 * q^37 + 24 * q^39 - 6 * q^40 - 12 * q^41 + 18 * q^43 - 18 * q^45 + 24 * q^47 + 6 * q^49 + 6 * q^50 + 12 * q^51 + 36 * q^53 - 12 * q^54 - 6 * q^56 + 12 * q^57 - 12 * q^58 + 30 * q^59 - 36 * q^61 - 18 * q^63 + 6 * q^64 - 12 * q^67 - 6 * q^68 + 6 * q^70 + 6 * q^71 + 18 * q^72 + 6 * q^73 + 24 * q^74 + 24 * q^78 + 24 * q^79 - 6 * q^80 + 54 * q^81 - 12 * q^82 - 24 * q^83 + 6 * q^85 + 18 * q^86 + 24 * q^87 + 36 * q^89 - 18 * q^90 + 24 * q^94 + 6 * q^98

## 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
$$3$$ 3.24337 1.87256 0.936281 0.351252i $$-0.114244\pi$$
0.936281 + 0.351252i $$0.114244\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 3.24337 1.32410
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 7.51947 2.50649
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 3.24337 0.936281
$$13$$ 2.14515 0.594959 0.297479 0.954728i $$-0.403854\pi$$
0.297479 + 0.954728i $$0.403854\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −3.24337 −0.837435
$$16$$ 1.00000 0.250000
$$17$$ 1.43942 0.349111 0.174555 0.984647i $$-0.444151\pi$$
0.174555 + 0.984647i $$0.444151\pi$$
$$18$$ 7.51947 1.77236
$$19$$ −0.633832 −0.145411 −0.0727056 0.997353i $$-0.523163\pi$$
−0.0727056 + 0.997353i $$0.523163\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −3.24337 −0.707762
$$22$$ 0 0
$$23$$ −5.61769 −1.17137 −0.585684 0.810539i $$-0.699174\pi$$
−0.585684 + 0.810539i $$0.699174\pi$$
$$24$$ 3.24337 0.662051
$$25$$ 1.00000 0.200000
$$26$$ 2.14515 0.420699
$$27$$ 14.6583 2.82100
$$28$$ −1.00000 −0.188982
$$29$$ −1.19605 −0.222101 −0.111050 0.993815i $$-0.535421\pi$$
−0.111050 + 0.993815i $$0.535421\pi$$
$$30$$ −3.24337 −0.592156
$$31$$ 6.95889 1.24985 0.624927 0.780683i $$-0.285129\pi$$
0.624927 + 0.780683i $$0.285129\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 1.43942 0.246859
$$35$$ 1.00000 0.169031
$$36$$ 7.51947 1.25324
$$37$$ 4.80395 0.789765 0.394882 0.918732i $$-0.370785\pi$$
0.394882 + 0.918732i $$0.370785\pi$$
$$38$$ −0.633832 −0.102821
$$39$$ 6.95754 1.11410
$$40$$ −1.00000 −0.158114
$$41$$ −8.93663 −1.39567 −0.697834 0.716260i $$-0.745853\pi$$
−0.697834 + 0.716260i $$0.745853\pi$$
$$42$$ −3.24337 −0.500463
$$43$$ 9.70747 1.48038 0.740188 0.672400i $$-0.234736\pi$$
0.740188 + 0.672400i $$0.234736\pi$$
$$44$$ 0 0
$$45$$ −7.51947 −1.12094
$$46$$ −5.61769 −0.828283
$$47$$ 8.29874 1.21050 0.605248 0.796037i $$-0.293074\pi$$
0.605248 + 0.796037i $$0.293074\pi$$
$$48$$ 3.24337 0.468141
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 4.66858 0.653732
$$52$$ 2.14515 0.297479
$$53$$ 14.1537 1.94416 0.972079 0.234652i $$-0.0753952\pi$$
0.972079 + 0.234652i $$0.0753952\pi$$
$$54$$ 14.6583 1.99475
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −2.05576 −0.272291
$$58$$ −1.19605 −0.157049
$$59$$ −1.19248 −0.155248 −0.0776238 0.996983i $$-0.524733\pi$$
−0.0776238 + 0.996983i $$0.524733\pi$$
$$60$$ −3.24337 −0.418718
$$61$$ −4.19441 −0.537039 −0.268520 0.963274i $$-0.586534\pi$$
−0.268520 + 0.963274i $$0.586534\pi$$
$$62$$ 6.95889 0.883780
$$63$$ −7.51947 −0.947364
$$64$$ 1.00000 0.125000
$$65$$ −2.14515 −0.266074
$$66$$ 0 0
$$67$$ −12.0099 −1.46724 −0.733621 0.679559i $$-0.762171\pi$$
−0.733621 + 0.679559i $$0.762171\pi$$
$$68$$ 1.43942 0.174555
$$69$$ −18.2203 −2.19346
$$70$$ 1.00000 0.119523
$$71$$ −5.71591 −0.678353 −0.339177 0.940723i $$-0.610148\pi$$
−0.339177 + 0.940723i $$0.610148\pi$$
$$72$$ 7.51947 0.886178
$$73$$ 6.43981 0.753723 0.376861 0.926270i $$-0.377003\pi$$
0.376861 + 0.926270i $$0.377003\pi$$
$$74$$ 4.80395 0.558448
$$75$$ 3.24337 0.374512
$$76$$ −0.633832 −0.0727056
$$77$$ 0 0
$$78$$ 6.95754 0.787786
$$79$$ −7.69133 −0.865342 −0.432671 0.901552i $$-0.642429\pi$$
−0.432671 + 0.901552i $$0.642429\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 24.9840 2.77600
$$82$$ −8.93663 −0.986886
$$83$$ 11.5685 1.26981 0.634906 0.772590i $$-0.281039\pi$$
0.634906 + 0.772590i $$0.281039\pi$$
$$84$$ −3.24337 −0.353881
$$85$$ −1.43942 −0.156127
$$86$$ 9.70747 1.04678
$$87$$ −3.87923 −0.415897
$$88$$ 0 0
$$89$$ 16.7855 1.77926 0.889629 0.456684i $$-0.150963\pi$$
0.889629 + 0.456684i $$0.150963\pi$$
$$90$$ −7.51947 −0.792622
$$91$$ −2.14515 −0.224873
$$92$$ −5.61769 −0.585684
$$93$$ 22.5703 2.34043
$$94$$ 8.29874 0.855950
$$95$$ 0.633832 0.0650298
$$96$$ 3.24337 0.331025
$$97$$ −2.96529 −0.301080 −0.150540 0.988604i $$-0.548101\pi$$
−0.150540 + 0.988604i $$0.548101\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −15.5584 −1.54812 −0.774060 0.633113i $$-0.781777\pi$$
−0.774060 + 0.633113i $$0.781777\pi$$
$$102$$ 4.66858 0.462258
$$103$$ 0.404673 0.0398736 0.0199368 0.999801i $$-0.493653\pi$$
0.0199368 + 0.999801i $$0.493653\pi$$
$$104$$ 2.14515 0.210350
$$105$$ 3.24337 0.316521
$$106$$ 14.1537 1.37473
$$107$$ −8.17962 −0.790754 −0.395377 0.918519i $$-0.629386\pi$$
−0.395377 + 0.918519i $$0.629386\pi$$
$$108$$ 14.6583 1.41050
$$109$$ −1.34828 −0.129142 −0.0645709 0.997913i $$-0.520568\pi$$
−0.0645709 + 0.997913i $$0.520568\pi$$
$$110$$ 0 0
$$111$$ 15.5810 1.47888
$$112$$ −1.00000 −0.0944911
$$113$$ 12.6845 1.19326 0.596630 0.802516i $$-0.296506\pi$$
0.596630 + 0.802516i $$0.296506\pi$$
$$114$$ −2.05576 −0.192539
$$115$$ 5.61769 0.523852
$$116$$ −1.19605 −0.111050
$$117$$ 16.1304 1.49126
$$118$$ −1.19248 −0.109777
$$119$$ −1.43942 −0.131952
$$120$$ −3.24337 −0.296078
$$121$$ 0 0
$$122$$ −4.19441 −0.379744
$$123$$ −28.9848 −2.61347
$$124$$ 6.95889 0.624927
$$125$$ −1.00000 −0.0894427
$$126$$ −7.51947 −0.669887
$$127$$ 12.6218 1.12000 0.560001 0.828492i $$-0.310801\pi$$
0.560001 + 0.828492i $$0.310801\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 31.4850 2.77210
$$130$$ −2.14515 −0.188143
$$131$$ 4.14932 0.362528 0.181264 0.983434i $$-0.441981\pi$$
0.181264 + 0.983434i $$0.441981\pi$$
$$132$$ 0 0
$$133$$ 0.633832 0.0549602
$$134$$ −12.0099 −1.03750
$$135$$ −14.6583 −1.26159
$$136$$ 1.43942 0.123429
$$137$$ 20.8283 1.77948 0.889742 0.456463i $$-0.150884\pi$$
0.889742 + 0.456463i $$0.150884\pi$$
$$138$$ −18.2203 −1.55101
$$139$$ −18.4862 −1.56798 −0.783988 0.620776i $$-0.786818\pi$$
−0.783988 + 0.620776i $$0.786818\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 26.9159 2.26673
$$142$$ −5.71591 −0.479668
$$143$$ 0 0
$$144$$ 7.51947 0.626622
$$145$$ 1.19605 0.0993264
$$146$$ 6.43981 0.532962
$$147$$ 3.24337 0.267509
$$148$$ 4.80395 0.394882
$$149$$ −15.4372 −1.26467 −0.632333 0.774697i $$-0.717902\pi$$
−0.632333 + 0.774697i $$0.717902\pi$$
$$150$$ 3.24337 0.264820
$$151$$ −11.4606 −0.932648 −0.466324 0.884614i $$-0.654422\pi$$
−0.466324 + 0.884614i $$0.654422\pi$$
$$152$$ −0.633832 −0.0514106
$$153$$ 10.8237 0.875043
$$154$$ 0 0
$$155$$ −6.95889 −0.558951
$$156$$ 6.95754 0.557049
$$157$$ −10.2685 −0.819514 −0.409757 0.912195i $$-0.634387\pi$$
−0.409757 + 0.912195i $$0.634387\pi$$
$$158$$ −7.69133 −0.611889
$$159$$ 45.9057 3.64056
$$160$$ −1.00000 −0.0790569
$$161$$ 5.61769 0.442736
$$162$$ 24.9840 1.96293
$$163$$ −2.90989 −0.227920 −0.113960 0.993485i $$-0.536354\pi$$
−0.113960 + 0.993485i $$0.536354\pi$$
$$164$$ −8.93663 −0.697834
$$165$$ 0 0
$$166$$ 11.5685 0.897892
$$167$$ 11.7960 0.912798 0.456399 0.889775i $$-0.349139\pi$$
0.456399 + 0.889775i $$0.349139\pi$$
$$168$$ −3.24337 −0.250232
$$169$$ −8.39831 −0.646024
$$170$$ −1.43942 −0.110399
$$171$$ −4.76608 −0.364472
$$172$$ 9.70747 0.740188
$$173$$ 0.146508 0.0111388 0.00556939 0.999984i $$-0.498227\pi$$
0.00556939 + 0.999984i $$0.498227\pi$$
$$174$$ −3.87923 −0.294084
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −3.86766 −0.290711
$$178$$ 16.7855 1.25813
$$179$$ 10.2987 0.769764 0.384882 0.922966i $$-0.374242\pi$$
0.384882 + 0.922966i $$0.374242\pi$$
$$180$$ −7.51947 −0.560468
$$181$$ −19.8342 −1.47426 −0.737131 0.675750i $$-0.763820\pi$$
−0.737131 + 0.675750i $$0.763820\pi$$
$$182$$ −2.14515 −0.159009
$$183$$ −13.6040 −1.00564
$$184$$ −5.61769 −0.414141
$$185$$ −4.80395 −0.353194
$$186$$ 22.5703 1.65493
$$187$$ 0 0
$$188$$ 8.29874 0.605248
$$189$$ −14.6583 −1.06624
$$190$$ 0.633832 0.0459830
$$191$$ 5.95268 0.430720 0.215360 0.976535i $$-0.430907\pi$$
0.215360 + 0.976535i $$0.430907\pi$$
$$192$$ 3.24337 0.234070
$$193$$ −14.9671 −1.07736 −0.538678 0.842512i $$-0.681076\pi$$
−0.538678 + 0.842512i $$0.681076\pi$$
$$194$$ −2.96529 −0.212896
$$195$$ −6.95754 −0.498240
$$196$$ 1.00000 0.0714286
$$197$$ −19.1531 −1.36460 −0.682301 0.731071i $$-0.739021\pi$$
−0.682301 + 0.731071i $$0.739021\pi$$
$$198$$ 0 0
$$199$$ −0.268053 −0.0190018 −0.00950089 0.999955i $$-0.503024\pi$$
−0.00950089 + 0.999955i $$0.503024\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −38.9525 −2.74750
$$202$$ −15.5584 −1.09469
$$203$$ 1.19605 0.0839461
$$204$$ 4.66858 0.326866
$$205$$ 8.93663 0.624162
$$206$$ 0.404673 0.0281949
$$207$$ −42.2420 −2.93602
$$208$$ 2.14515 0.148740
$$209$$ 0 0
$$210$$ 3.24337 0.223814
$$211$$ −9.35228 −0.643837 −0.321919 0.946767i $$-0.604328\pi$$
−0.321919 + 0.946767i $$0.604328\pi$$
$$212$$ 14.1537 0.972079
$$213$$ −18.5388 −1.27026
$$214$$ −8.17962 −0.559147
$$215$$ −9.70747 −0.662044
$$216$$ 14.6583 0.997373
$$217$$ −6.95889 −0.472400
$$218$$ −1.34828 −0.0913171
$$219$$ 20.8867 1.41139
$$220$$ 0 0
$$221$$ 3.08778 0.207707
$$222$$ 15.5810 1.04573
$$223$$ −13.8365 −0.926559 −0.463280 0.886212i $$-0.653327\pi$$
−0.463280 + 0.886212i $$0.653327\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 7.51947 0.501298
$$226$$ 12.6845 0.843763
$$227$$ 25.8686 1.71696 0.858481 0.512846i $$-0.171409\pi$$
0.858481 + 0.512846i $$0.171409\pi$$
$$228$$ −2.05576 −0.136146
$$229$$ 11.1800 0.738796 0.369398 0.929271i $$-0.379564\pi$$
0.369398 + 0.929271i $$0.379564\pi$$
$$230$$ 5.61769 0.370419
$$231$$ 0 0
$$232$$ −1.19605 −0.0785244
$$233$$ 2.87302 0.188218 0.0941088 0.995562i $$-0.470000\pi$$
0.0941088 + 0.995562i $$0.470000\pi$$
$$234$$ 16.1304 1.05448
$$235$$ −8.29874 −0.541350
$$236$$ −1.19248 −0.0776238
$$237$$ −24.9458 −1.62041
$$238$$ −1.43942 −0.0933038
$$239$$ 8.22791 0.532219 0.266110 0.963943i $$-0.414262\pi$$
0.266110 + 0.963943i $$0.414262\pi$$
$$240$$ −3.24337 −0.209359
$$241$$ −6.54207 −0.421412 −0.210706 0.977550i $$-0.567576\pi$$
−0.210706 + 0.977550i $$0.567576\pi$$
$$242$$ 0 0
$$243$$ 37.0575 2.37724
$$244$$ −4.19441 −0.268520
$$245$$ −1.00000 −0.0638877
$$246$$ −28.9848 −1.84801
$$247$$ −1.35967 −0.0865136
$$248$$ 6.95889 0.441890
$$249$$ 37.5211 2.37780
$$250$$ −1.00000 −0.0632456
$$251$$ −11.0809 −0.699422 −0.349711 0.936858i $$-0.613720\pi$$
−0.349711 + 0.936858i $$0.613720\pi$$
$$252$$ −7.51947 −0.473682
$$253$$ 0 0
$$254$$ 12.6218 0.791960
$$255$$ −4.66858 −0.292358
$$256$$ 1.00000 0.0625000
$$257$$ 18.1347 1.13121 0.565606 0.824676i $$-0.308642\pi$$
0.565606 + 0.824676i $$0.308642\pi$$
$$258$$ 31.4850 1.96017
$$259$$ −4.80395 −0.298503
$$260$$ −2.14515 −0.133037
$$261$$ −8.99365 −0.556693
$$262$$ 4.14932 0.256346
$$263$$ −17.9268 −1.10541 −0.552706 0.833376i $$-0.686405\pi$$
−0.552706 + 0.833376i $$0.686405\pi$$
$$264$$ 0 0
$$265$$ −14.1537 −0.869454
$$266$$ 0.633832 0.0388628
$$267$$ 54.4416 3.33177
$$268$$ −12.0099 −0.733621
$$269$$ 6.06753 0.369944 0.184972 0.982744i $$-0.440781\pi$$
0.184972 + 0.982744i $$0.440781\pi$$
$$270$$ −14.6583 −0.892077
$$271$$ 21.2185 1.28893 0.644466 0.764633i $$-0.277080\pi$$
0.644466 + 0.764633i $$0.277080\pi$$
$$272$$ 1.43942 0.0872777
$$273$$ −6.95754 −0.421089
$$274$$ 20.8283 1.25829
$$275$$ 0 0
$$276$$ −18.2203 −1.09673
$$277$$ −5.01421 −0.301275 −0.150637 0.988589i $$-0.548133\pi$$
−0.150637 + 0.988589i $$0.548133\pi$$
$$278$$ −18.4862 −1.10873
$$279$$ 52.3272 3.13274
$$280$$ 1.00000 0.0597614
$$281$$ 4.76751 0.284406 0.142203 0.989838i $$-0.454581\pi$$
0.142203 + 0.989838i $$0.454581\pi$$
$$282$$ 26.9159 1.60282
$$283$$ 9.65019 0.573644 0.286822 0.957984i $$-0.407401\pi$$
0.286822 + 0.957984i $$0.407401\pi$$
$$284$$ −5.71591 −0.339177
$$285$$ 2.05576 0.121772
$$286$$ 0 0
$$287$$ 8.93663 0.527513
$$288$$ 7.51947 0.443089
$$289$$ −14.9281 −0.878122
$$290$$ 1.19605 0.0702344
$$291$$ −9.61756 −0.563791
$$292$$ 6.43981 0.376861
$$293$$ −28.5011 −1.66505 −0.832526 0.553986i $$-0.813106\pi$$
−0.832526 + 0.553986i $$0.813106\pi$$
$$294$$ 3.24337 0.189157
$$295$$ 1.19248 0.0694288
$$296$$ 4.80395 0.279224
$$297$$ 0 0
$$298$$ −15.4372 −0.894253
$$299$$ −12.0508 −0.696916
$$300$$ 3.24337 0.187256
$$301$$ −9.70747 −0.559530
$$302$$ −11.4606 −0.659482
$$303$$ −50.4617 −2.89895
$$304$$ −0.633832 −0.0363528
$$305$$ 4.19441 0.240171
$$306$$ 10.8237 0.618749
$$307$$ −0.552191 −0.0315152 −0.0157576 0.999876i $$-0.505016\pi$$
−0.0157576 + 0.999876i $$0.505016\pi$$
$$308$$ 0 0
$$309$$ 1.31251 0.0746659
$$310$$ −6.95889 −0.395238
$$311$$ −30.4747 −1.72806 −0.864032 0.503437i $$-0.832069\pi$$
−0.864032 + 0.503437i $$0.832069\pi$$
$$312$$ 6.95754 0.393893
$$313$$ −33.4678 −1.89171 −0.945856 0.324587i $$-0.894775\pi$$
−0.945856 + 0.324587i $$0.894775\pi$$
$$314$$ −10.2685 −0.579484
$$315$$ 7.51947 0.423674
$$316$$ −7.69133 −0.432671
$$317$$ −20.2125 −1.13525 −0.567623 0.823288i $$-0.692137\pi$$
−0.567623 + 0.823288i $$0.692137\pi$$
$$318$$ 45.9057 2.57426
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ −26.5296 −1.48074
$$322$$ 5.61769 0.313061
$$323$$ −0.912352 −0.0507646
$$324$$ 24.9840 1.38800
$$325$$ 2.14515 0.118992
$$326$$ −2.90989 −0.161164
$$327$$ −4.37298 −0.241826
$$328$$ −8.93663 −0.493443
$$329$$ −8.29874 −0.457524
$$330$$ 0 0
$$331$$ −19.3667 −1.06449 −0.532244 0.846591i $$-0.678651\pi$$
−0.532244 + 0.846591i $$0.678651\pi$$
$$332$$ 11.5685 0.634906
$$333$$ 36.1232 1.97954
$$334$$ 11.7960 0.645446
$$335$$ 12.0099 0.656170
$$336$$ −3.24337 −0.176941
$$337$$ 0.0549787 0.00299488 0.00149744 0.999999i $$-0.499523\pi$$
0.00149744 + 0.999999i $$0.499523\pi$$
$$338$$ −8.39831 −0.456808
$$339$$ 41.1407 2.23446
$$340$$ −1.43942 −0.0780636
$$341$$ 0 0
$$342$$ −4.76608 −0.257720
$$343$$ −1.00000 −0.0539949
$$344$$ 9.70747 0.523392
$$345$$ 18.2203 0.980945
$$346$$ 0.146508 0.00787631
$$347$$ 22.9466 1.23184 0.615920 0.787809i $$-0.288784\pi$$
0.615920 + 0.787809i $$0.288784\pi$$
$$348$$ −3.87923 −0.207949
$$349$$ −21.7380 −1.16361 −0.581804 0.813329i $$-0.697653\pi$$
−0.581804 + 0.813329i $$0.697653\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 31.4444 1.67838
$$352$$ 0 0
$$353$$ −10.3596 −0.551384 −0.275692 0.961246i $$-0.588907\pi$$
−0.275692 + 0.961246i $$0.588907\pi$$
$$354$$ −3.86766 −0.205564
$$355$$ 5.71591 0.303369
$$356$$ 16.7855 0.889629
$$357$$ −4.66858 −0.247087
$$358$$ 10.2987 0.544305
$$359$$ 33.6387 1.77538 0.887690 0.460442i $$-0.152309\pi$$
0.887690 + 0.460442i $$0.152309\pi$$
$$360$$ −7.51947 −0.396311
$$361$$ −18.5983 −0.978856
$$362$$ −19.8342 −1.04246
$$363$$ 0 0
$$364$$ −2.14515 −0.112437
$$365$$ −6.43981 −0.337075
$$366$$ −13.6040 −0.711095
$$367$$ −23.1345 −1.20761 −0.603806 0.797131i $$-0.706350\pi$$
−0.603806 + 0.797131i $$0.706350\pi$$
$$368$$ −5.61769 −0.292842
$$369$$ −67.1987 −3.49823
$$370$$ −4.80395 −0.249746
$$371$$ −14.1537 −0.734823
$$372$$ 22.5703 1.17021
$$373$$ −20.5321 −1.06311 −0.531555 0.847024i $$-0.678392\pi$$
−0.531555 + 0.847024i $$0.678392\pi$$
$$374$$ 0 0
$$375$$ −3.24337 −0.167487
$$376$$ 8.29874 0.427975
$$377$$ −2.56571 −0.132141
$$378$$ −14.6583 −0.753943
$$379$$ 21.0362 1.08056 0.540278 0.841487i $$-0.318319\pi$$
0.540278 + 0.841487i $$0.318319\pi$$
$$380$$ 0.633832 0.0325149
$$381$$ 40.9371 2.09727
$$382$$ 5.95268 0.304565
$$383$$ 11.3755 0.581263 0.290632 0.956835i $$-0.406135\pi$$
0.290632 + 0.956835i $$0.406135\pi$$
$$384$$ 3.24337 0.165513
$$385$$ 0 0
$$386$$ −14.9671 −0.761805
$$387$$ 72.9951 3.71055
$$388$$ −2.96529 −0.150540
$$389$$ 25.9812 1.31730 0.658649 0.752450i $$-0.271128\pi$$
0.658649 + 0.752450i $$0.271128\pi$$
$$390$$ −6.95754 −0.352309
$$391$$ −8.08622 −0.408938
$$392$$ 1.00000 0.0505076
$$393$$ 13.4578 0.678856
$$394$$ −19.1531 −0.964920
$$395$$ 7.69133 0.386993
$$396$$ 0 0
$$397$$ 7.16936 0.359820 0.179910 0.983683i $$-0.442419\pi$$
0.179910 + 0.983683i $$0.442419\pi$$
$$398$$ −0.268053 −0.0134363
$$399$$ 2.05576 0.102916
$$400$$ 1.00000 0.0500000
$$401$$ −26.2702 −1.31187 −0.655937 0.754816i $$-0.727726\pi$$
−0.655937 + 0.754816i $$0.727726\pi$$
$$402$$ −38.9525 −1.94278
$$403$$ 14.9279 0.743611
$$404$$ −15.5584 −0.774060
$$405$$ −24.9840 −1.24147
$$406$$ 1.19605 0.0593589
$$407$$ 0 0
$$408$$ 4.66858 0.231129
$$409$$ 36.5385 1.80671 0.903356 0.428891i $$-0.141096\pi$$
0.903356 + 0.428891i $$0.141096\pi$$
$$410$$ 8.93663 0.441349
$$411$$ 67.5541 3.33220
$$412$$ 0.404673 0.0199368
$$413$$ 1.19248 0.0586781
$$414$$ −42.2420 −2.07608
$$415$$ −11.5685 −0.567877
$$416$$ 2.14515 0.105175
$$417$$ −59.9575 −2.93613
$$418$$ 0 0
$$419$$ −17.9411 −0.876480 −0.438240 0.898858i $$-0.644398\pi$$
−0.438240 + 0.898858i $$0.644398\pi$$
$$420$$ 3.24337 0.158260
$$421$$ 36.8460 1.79576 0.897882 0.440237i $$-0.145106\pi$$
0.897882 + 0.440237i $$0.145106\pi$$
$$422$$ −9.35228 −0.455262
$$423$$ 62.4021 3.03409
$$424$$ 14.1537 0.687364
$$425$$ 1.43942 0.0698222
$$426$$ −18.5388 −0.898208
$$427$$ 4.19441 0.202982
$$428$$ −8.17962 −0.395377
$$429$$ 0 0
$$430$$ −9.70747 −0.468136
$$431$$ 31.8543 1.53437 0.767185 0.641426i $$-0.221657\pi$$
0.767185 + 0.641426i $$0.221657\pi$$
$$432$$ 14.6583 0.705249
$$433$$ 25.0278 1.20276 0.601380 0.798963i $$-0.294618\pi$$
0.601380 + 0.798963i $$0.294618\pi$$
$$434$$ −6.95889 −0.334037
$$435$$ 3.87923 0.185995
$$436$$ −1.34828 −0.0645709
$$437$$ 3.56067 0.170330
$$438$$ 20.8867 0.998005
$$439$$ −40.3537 −1.92598 −0.962988 0.269544i $$-0.913127\pi$$
−0.962988 + 0.269544i $$0.913127\pi$$
$$440$$ 0 0
$$441$$ 7.51947 0.358070
$$442$$ 3.08778 0.146871
$$443$$ −0.519900 −0.0247012 −0.0123506 0.999924i $$-0.503931\pi$$
−0.0123506 + 0.999924i $$0.503931\pi$$
$$444$$ 15.5810 0.739442
$$445$$ −16.7855 −0.795708
$$446$$ −13.8365 −0.655176
$$447$$ −50.0686 −2.36816
$$448$$ −1.00000 −0.0472456
$$449$$ 23.0974 1.09003 0.545016 0.838426i $$-0.316524\pi$$
0.545016 + 0.838426i $$0.316524\pi$$
$$450$$ 7.51947 0.354471
$$451$$ 0 0
$$452$$ 12.6845 0.596630
$$453$$ −37.1709 −1.74644
$$454$$ 25.8686 1.21407
$$455$$ 2.14515 0.100566
$$456$$ −2.05576 −0.0962696
$$457$$ −20.3786 −0.953273 −0.476637 0.879100i $$-0.658144\pi$$
−0.476637 + 0.879100i $$0.658144\pi$$
$$458$$ 11.1800 0.522407
$$459$$ 21.0995 0.984840
$$460$$ 5.61769 0.261926
$$461$$ −12.1816 −0.567354 −0.283677 0.958920i $$-0.591554\pi$$
−0.283677 + 0.958920i $$0.591554\pi$$
$$462$$ 0 0
$$463$$ −11.4172 −0.530601 −0.265301 0.964166i $$-0.585471\pi$$
−0.265301 + 0.964166i $$0.585471\pi$$
$$464$$ −1.19605 −0.0555252
$$465$$ −22.5703 −1.04667
$$466$$ 2.87302 0.133090
$$467$$ 22.7698 1.05366 0.526831 0.849970i $$-0.323380\pi$$
0.526831 + 0.849970i $$0.323380\pi$$
$$468$$ 16.1304 0.745629
$$469$$ 12.0099 0.554565
$$470$$ −8.29874 −0.382792
$$471$$ −33.3045 −1.53459
$$472$$ −1.19248 −0.0548883
$$473$$ 0 0
$$474$$ −24.9458 −1.14580
$$475$$ −0.633832 −0.0290822
$$476$$ −1.43942 −0.0659758
$$477$$ 106.428 4.87301
$$478$$ 8.22791 0.376336
$$479$$ −15.9269 −0.727719 −0.363859 0.931454i $$-0.618541\pi$$
−0.363859 + 0.931454i $$0.618541\pi$$
$$480$$ −3.24337 −0.148039
$$481$$ 10.3052 0.469878
$$482$$ −6.54207 −0.297983
$$483$$ 18.2203 0.829050
$$484$$ 0 0
$$485$$ 2.96529 0.134647
$$486$$ 37.0575 1.68096
$$487$$ 4.57953 0.207518 0.103759 0.994602i $$-0.466913\pi$$
0.103759 + 0.994602i $$0.466913\pi$$
$$488$$ −4.19441 −0.189872
$$489$$ −9.43784 −0.426794
$$490$$ −1.00000 −0.0451754
$$491$$ −28.0073 −1.26395 −0.631975 0.774989i $$-0.717756\pi$$
−0.631975 + 0.774989i $$0.717756\pi$$
$$492$$ −28.9848 −1.30674
$$493$$ −1.72162 −0.0775378
$$494$$ −1.35967 −0.0611744
$$495$$ 0 0
$$496$$ 6.95889 0.312463
$$497$$ 5.71591 0.256393
$$498$$ 37.5211 1.68136
$$499$$ 27.8295 1.24582 0.622910 0.782294i $$-0.285950\pi$$
0.622910 + 0.782294i $$0.285950\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 38.2587 1.70927
$$502$$ −11.0809 −0.494566
$$503$$ −30.4709 −1.35863 −0.679316 0.733846i $$-0.737723\pi$$
−0.679316 + 0.733846i $$0.737723\pi$$
$$504$$ −7.51947 −0.334944
$$505$$ 15.5584 0.692340
$$506$$ 0 0
$$507$$ −27.2389 −1.20972
$$508$$ 12.6218 0.560001
$$509$$ −25.7729 −1.14236 −0.571181 0.820824i $$-0.693515\pi$$
−0.571181 + 0.820824i $$0.693515\pi$$
$$510$$ −4.66858 −0.206728
$$511$$ −6.43981 −0.284880
$$512$$ 1.00000 0.0441942
$$513$$ −9.29092 −0.410204
$$514$$ 18.1347 0.799888
$$515$$ −0.404673 −0.0178320
$$516$$ 31.4850 1.38605
$$517$$ 0 0
$$518$$ −4.80395 −0.211074
$$519$$ 0.475179 0.0208581
$$520$$ −2.14515 −0.0940713
$$521$$ 3.08204 0.135027 0.0675133 0.997718i $$-0.478493\pi$$
0.0675133 + 0.997718i $$0.478493\pi$$
$$522$$ −8.99365 −0.393641
$$523$$ −10.0632 −0.440033 −0.220016 0.975496i $$-0.570611\pi$$
−0.220016 + 0.975496i $$0.570611\pi$$
$$524$$ 4.14932 0.181264
$$525$$ −3.24337 −0.141552
$$526$$ −17.9268 −0.781644
$$527$$ 10.0168 0.436338
$$528$$ 0 0
$$529$$ 8.55841 0.372105
$$530$$ −14.1537 −0.614797
$$531$$ −8.96681 −0.389126
$$532$$ 0.633832 0.0274801
$$533$$ −19.1705 −0.830365
$$534$$ 54.4416 2.35592
$$535$$ 8.17962 0.353636
$$536$$ −12.0099 −0.518748
$$537$$ 33.4027 1.44143
$$538$$ 6.06753 0.261590
$$539$$ 0 0
$$540$$ −14.6583 −0.630794
$$541$$ 21.7416 0.934744 0.467372 0.884061i $$-0.345201\pi$$
0.467372 + 0.884061i $$0.345201\pi$$
$$542$$ 21.2185 0.911413
$$543$$ −64.3296 −2.76065
$$544$$ 1.43942 0.0617147
$$545$$ 1.34828 0.0577540
$$546$$ −6.95754 −0.297755
$$547$$ 33.5075 1.43268 0.716338 0.697753i $$-0.245817\pi$$
0.716338 + 0.697753i $$0.245817\pi$$
$$548$$ 20.8283 0.889742
$$549$$ −31.5397 −1.34608
$$550$$ 0 0
$$551$$ 0.758094 0.0322959
$$552$$ −18.2203 −0.775506
$$553$$ 7.69133 0.327069
$$554$$ −5.01421 −0.213034
$$555$$ −15.5810 −0.661377
$$556$$ −18.4862 −0.783988
$$557$$ −27.9709 −1.18517 −0.592583 0.805509i $$-0.701892\pi$$
−0.592583 + 0.805509i $$0.701892\pi$$
$$558$$ 52.3272 2.21519
$$559$$ 20.8240 0.880763
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 4.76751 0.201105
$$563$$ 19.1724 0.808019 0.404009 0.914755i $$-0.367616\pi$$
0.404009 + 0.914755i $$0.367616\pi$$
$$564$$ 26.9159 1.13336
$$565$$ −12.6845 −0.533642
$$566$$ 9.65019 0.405628
$$567$$ −24.9840 −1.04923
$$568$$ −5.71591 −0.239834
$$569$$ 39.8416 1.67025 0.835124 0.550062i $$-0.185396\pi$$
0.835124 + 0.550062i $$0.185396\pi$$
$$570$$ 2.05576 0.0861061
$$571$$ 47.3425 1.98122 0.990610 0.136718i $$-0.0436555\pi$$
0.990610 + 0.136718i $$0.0436555\pi$$
$$572$$ 0 0
$$573$$ 19.3067 0.806551
$$574$$ 8.93663 0.373008
$$575$$ −5.61769 −0.234274
$$576$$ 7.51947 0.313311
$$577$$ −11.3882 −0.474097 −0.237048 0.971498i $$-0.576180\pi$$
−0.237048 + 0.971498i $$0.576180\pi$$
$$578$$ −14.9281 −0.620926
$$579$$ −48.5439 −2.01741
$$580$$ 1.19605 0.0496632
$$581$$ −11.5685 −0.479944
$$582$$ −9.61756 −0.398661
$$583$$ 0 0
$$584$$ 6.43981 0.266481
$$585$$ −16.1304 −0.666911
$$586$$ −28.5011 −1.17737
$$587$$ −22.3027 −0.920529 −0.460265 0.887782i $$-0.652245\pi$$
−0.460265 + 0.887782i $$0.652245\pi$$
$$588$$ 3.24337 0.133754
$$589$$ −4.41077 −0.181743
$$590$$ 1.19248 0.0490936
$$591$$ −62.1207 −2.55530
$$592$$ 4.80395 0.197441
$$593$$ −14.0840 −0.578361 −0.289180 0.957275i $$-0.593383\pi$$
−0.289180 + 0.957275i $$0.593383\pi$$
$$594$$ 0 0
$$595$$ 1.43942 0.0590105
$$596$$ −15.4372 −0.632333
$$597$$ −0.869397 −0.0355820
$$598$$ −12.0508 −0.492794
$$599$$ −22.5478 −0.921280 −0.460640 0.887587i $$-0.652380\pi$$
−0.460640 + 0.887587i $$0.652380\pi$$
$$600$$ 3.24337 0.132410
$$601$$ 38.4433 1.56814 0.784068 0.620675i $$-0.213141\pi$$
0.784068 + 0.620675i $$0.213141\pi$$
$$602$$ −9.70747 −0.395647
$$603$$ −90.3080 −3.67762
$$604$$ −11.4606 −0.466324
$$605$$ 0 0
$$606$$ −50.4617 −2.04987
$$607$$ 24.2061 0.982497 0.491248 0.871020i $$-0.336541\pi$$
0.491248 + 0.871020i $$0.336541\pi$$
$$608$$ −0.633832 −0.0257053
$$609$$ 3.87923 0.157194
$$610$$ 4.19441 0.169827
$$611$$ 17.8021 0.720195
$$612$$ 10.8237 0.437521
$$613$$ −26.3266 −1.06332 −0.531662 0.846957i $$-0.678432\pi$$
−0.531662 + 0.846957i $$0.678432\pi$$
$$614$$ −0.552191 −0.0222846
$$615$$ 28.9848 1.16878
$$616$$ 0 0
$$617$$ −35.7839 −1.44061 −0.720303 0.693660i $$-0.755997\pi$$
−0.720303 + 0.693660i $$0.755997\pi$$
$$618$$ 1.31251 0.0527968
$$619$$ −8.17683 −0.328654 −0.164327 0.986406i $$-0.552545\pi$$
−0.164327 + 0.986406i $$0.552545\pi$$
$$620$$ −6.95889 −0.279476
$$621$$ −82.3459 −3.30443
$$622$$ −30.4747 −1.22193
$$623$$ −16.7855 −0.672496
$$624$$ 6.95754 0.278524
$$625$$ 1.00000 0.0400000
$$626$$ −33.4678 −1.33764
$$627$$ 0 0
$$628$$ −10.2685 −0.409757
$$629$$ 6.91491 0.275716
$$630$$ 7.51947 0.299583
$$631$$ 20.1283 0.801293 0.400647 0.916233i $$-0.368785\pi$$
0.400647 + 0.916233i $$0.368785\pi$$
$$632$$ −7.69133 −0.305945
$$633$$ −30.3329 −1.20563
$$634$$ −20.2125 −0.802741
$$635$$ −12.6218 −0.500880
$$636$$ 45.9057 1.82028
$$637$$ 2.14515 0.0849941
$$638$$ 0 0
$$639$$ −42.9806 −1.70029
$$640$$ −1.00000 −0.0395285
$$641$$ 8.58218 0.338976 0.169488 0.985532i $$-0.445789\pi$$
0.169488 + 0.985532i $$0.445789\pi$$
$$642$$ −26.5296 −1.04704
$$643$$ −12.5152 −0.493550 −0.246775 0.969073i $$-0.579371\pi$$
−0.246775 + 0.969073i $$0.579371\pi$$
$$644$$ 5.61769 0.221368
$$645$$ −31.4850 −1.23972
$$646$$ −0.912352 −0.0358960
$$647$$ 16.7526 0.658611 0.329305 0.944223i $$-0.393185\pi$$
0.329305 + 0.944223i $$0.393185\pi$$
$$648$$ 24.9840 0.981464
$$649$$ 0 0
$$650$$ 2.14515 0.0841399
$$651$$ −22.5703 −0.884599
$$652$$ −2.90989 −0.113960
$$653$$ −27.3451 −1.07010 −0.535048 0.844822i $$-0.679707\pi$$
−0.535048 + 0.844822i $$0.679707\pi$$
$$654$$ −4.37298 −0.170997
$$655$$ −4.14932 −0.162127
$$656$$ −8.93663 −0.348917
$$657$$ 48.4239 1.88920
$$658$$ −8.29874 −0.323519
$$659$$ −12.7483 −0.496604 −0.248302 0.968683i $$-0.579873\pi$$
−0.248302 + 0.968683i $$0.579873\pi$$
$$660$$ 0 0
$$661$$ −0.892400 −0.0347103 −0.0173552 0.999849i $$-0.505525\pi$$
−0.0173552 + 0.999849i $$0.505525\pi$$
$$662$$ −19.3667 −0.752707
$$663$$ 10.0148 0.388944
$$664$$ 11.5685 0.448946
$$665$$ −0.633832 −0.0245790
$$666$$ 36.1232 1.39974
$$667$$ 6.71903 0.260162
$$668$$ 11.7960 0.456399
$$669$$ −44.8769 −1.73504
$$670$$ 12.0099 0.463982
$$671$$ 0 0
$$672$$ −3.24337 −0.125116
$$673$$ −16.8765 −0.650541 −0.325271 0.945621i $$-0.605455\pi$$
−0.325271 + 0.945621i $$0.605455\pi$$
$$674$$ 0.0549787 0.00211770
$$675$$ 14.6583 0.564199
$$676$$ −8.39831 −0.323012
$$677$$ 19.6250 0.754248 0.377124 0.926163i $$-0.376913\pi$$
0.377124 + 0.926163i $$0.376913\pi$$
$$678$$ 41.1407 1.58000
$$679$$ 2.96529 0.113798
$$680$$ −1.43942 −0.0551993
$$681$$ 83.9016 3.21512
$$682$$ 0 0
$$683$$ −14.3107 −0.547585 −0.273792 0.961789i $$-0.588278\pi$$
−0.273792 + 0.961789i $$0.588278\pi$$
$$684$$ −4.76608 −0.182236
$$685$$ −20.8283 −0.795810
$$686$$ −1.00000 −0.0381802
$$687$$ 36.2609 1.38344
$$688$$ 9.70747 0.370094
$$689$$ 30.3619 1.15669
$$690$$ 18.2203 0.693633
$$691$$ −6.52873 −0.248365 −0.124182 0.992259i $$-0.539631\pi$$
−0.124182 + 0.992259i $$0.539631\pi$$
$$692$$ 0.146508 0.00556939
$$693$$ 0 0
$$694$$ 22.9466 0.871042
$$695$$ 18.4862 0.701220
$$696$$ −3.87923 −0.147042
$$697$$ −12.8636 −0.487243
$$698$$ −21.7380 −0.822795
$$699$$ 9.31826 0.352449
$$700$$ −1.00000 −0.0377964
$$701$$ −0.193487 −0.00730791 −0.00365395 0.999993i $$-0.501163\pi$$
−0.00365395 + 0.999993i $$0.501163\pi$$
$$702$$ 31.4444 1.18679
$$703$$ −3.04490 −0.114841
$$704$$ 0 0
$$705$$ −26.9159 −1.01371
$$706$$ −10.3596 −0.389888
$$707$$ 15.5584 0.585134
$$708$$ −3.86766 −0.145355
$$709$$ −19.2772 −0.723970 −0.361985 0.932184i $$-0.617901\pi$$
−0.361985 + 0.932184i $$0.617901\pi$$
$$710$$ 5.71591 0.214514
$$711$$ −57.8347 −2.16897
$$712$$ 16.7855 0.629063
$$713$$ −39.0929 −1.46404
$$714$$ −4.66858 −0.174717
$$715$$ 0 0
$$716$$ 10.2987 0.384882
$$717$$ 26.6862 0.996613
$$718$$ 33.6387 1.25538
$$719$$ −41.7245 −1.55606 −0.778030 0.628227i $$-0.783781\pi$$
−0.778030 + 0.628227i $$0.783781\pi$$
$$720$$ −7.51947 −0.280234
$$721$$ −0.404673 −0.0150708
$$722$$ −18.5983 −0.692155
$$723$$ −21.2184 −0.789120
$$724$$ −19.8342 −0.737131
$$725$$ −1.19605 −0.0444201
$$726$$ 0 0
$$727$$ 2.81130 0.104265 0.0521326 0.998640i $$-0.483398\pi$$
0.0521326 + 0.998640i $$0.483398\pi$$
$$728$$ −2.14515 −0.0795047
$$729$$ 45.2392 1.67553
$$730$$ −6.43981 −0.238348
$$731$$ 13.9731 0.516816
$$732$$ −13.6040 −0.502820
$$733$$ 50.4642 1.86394 0.931969 0.362539i $$-0.118090\pi$$
0.931969 + 0.362539i $$0.118090\pi$$
$$734$$ −23.1345 −0.853911
$$735$$ −3.24337 −0.119634
$$736$$ −5.61769 −0.207071
$$737$$ 0 0
$$738$$ −67.1987 −2.47362
$$739$$ −22.9651 −0.844783 −0.422392 0.906413i $$-0.638809\pi$$
−0.422392 + 0.906413i $$0.638809\pi$$
$$740$$ −4.80395 −0.176597
$$741$$ −4.40991 −0.162002
$$742$$ −14.1537 −0.519598
$$743$$ −45.8455 −1.68191 −0.840954 0.541107i $$-0.818006\pi$$
−0.840954 + 0.541107i $$0.818006\pi$$
$$744$$ 22.5703 0.827466
$$745$$ 15.4372 0.565575
$$746$$ −20.5321 −0.751733
$$747$$ 86.9892 3.18277
$$748$$ 0 0
$$749$$ 8.17962 0.298877
$$750$$ −3.24337 −0.118431
$$751$$ −3.84194 −0.140194 −0.0700972 0.997540i $$-0.522331\pi$$
−0.0700972 + 0.997540i $$0.522331\pi$$
$$752$$ 8.29874 0.302624
$$753$$ −35.9396 −1.30971
$$754$$ −2.56571 −0.0934376
$$755$$ 11.4606 0.417093
$$756$$ −14.6583 −0.533118
$$757$$ −26.9122 −0.978140 −0.489070 0.872245i $$-0.662664\pi$$
−0.489070 + 0.872245i $$0.662664\pi$$
$$758$$ 21.0362 0.764068
$$759$$ 0 0
$$760$$ 0.633832 0.0229915
$$761$$ 0.765350 0.0277439 0.0138720 0.999904i $$-0.495584\pi$$
0.0138720 + 0.999904i $$0.495584\pi$$
$$762$$ 40.9371 1.48300
$$763$$ 1.34828 0.0488110
$$764$$ 5.95268 0.215360
$$765$$ −10.8237 −0.391331
$$766$$ 11.3755 0.411015
$$767$$ −2.55805 −0.0923659
$$768$$ 3.24337 0.117035
$$769$$ −42.2441 −1.52336 −0.761681 0.647952i $$-0.775626\pi$$
−0.761681 + 0.647952i $$0.775626\pi$$
$$770$$ 0 0
$$771$$ 58.8176 2.11826
$$772$$ −14.9671 −0.538678
$$773$$ 31.5640 1.13528 0.567639 0.823277i $$-0.307857\pi$$
0.567639 + 0.823277i $$0.307857\pi$$
$$774$$ 72.9951 2.62375
$$775$$ 6.95889 0.249971
$$776$$ −2.96529 −0.106448
$$777$$ −15.5810 −0.558966
$$778$$ 25.9812 0.931470
$$779$$ 5.66433 0.202946
$$780$$ −6.95754 −0.249120
$$781$$ 0 0
$$782$$ −8.08622 −0.289163
$$783$$ −17.5321 −0.626545
$$784$$ 1.00000 0.0357143
$$785$$ 10.2685 0.366498
$$786$$ 13.4578 0.480024
$$787$$ −20.4895 −0.730370 −0.365185 0.930935i $$-0.618994\pi$$
−0.365185 + 0.930935i $$0.618994\pi$$
$$788$$ −19.1531 −0.682301
$$789$$ −58.1432 −2.06995
$$790$$ 7.69133 0.273645
$$791$$ −12.6845 −0.451010
$$792$$ 0 0
$$793$$ −8.99766 −0.319516
$$794$$ 7.16936 0.254431
$$795$$ −45.9057 −1.62811
$$796$$ −0.268053 −0.00950089
$$797$$ −6.13739 −0.217397 −0.108699 0.994075i $$-0.534668\pi$$
−0.108699 + 0.994075i $$0.534668\pi$$
$$798$$ 2.05576 0.0727729
$$799$$ 11.9454 0.422597
$$800$$ 1.00000 0.0353553
$$801$$ 126.218 4.45969
$$802$$ −26.2702 −0.927634
$$803$$ 0 0
$$804$$ −38.9525 −1.37375
$$805$$ −5.61769 −0.197997
$$806$$ 14.9279 0.525813
$$807$$ 19.6793 0.692743
$$808$$ −15.5584 −0.547343
$$809$$ 31.7238 1.11535 0.557674 0.830060i $$-0.311694\pi$$
0.557674 + 0.830060i $$0.311694\pi$$
$$810$$ −24.9840 −0.877848
$$811$$ −9.91528 −0.348173 −0.174086 0.984730i $$-0.555697\pi$$
−0.174086 + 0.984730i $$0.555697\pi$$
$$812$$ 1.19605 0.0419731
$$813$$ 68.8196 2.41361
$$814$$ 0 0
$$815$$ 2.90989 0.101929
$$816$$ 4.66858 0.163433
$$817$$ −6.15291 −0.215263
$$818$$ 36.5385 1.27754
$$819$$ −16.1304 −0.563643
$$820$$ 8.93663 0.312081
$$821$$ 15.9145 0.555419 0.277710 0.960665i $$-0.410425\pi$$
0.277710 + 0.960665i $$0.410425\pi$$
$$822$$ 67.5541 2.35622
$$823$$ −5.51317 −0.192177 −0.0960885 0.995373i $$-0.530633\pi$$
−0.0960885 + 0.995373i $$0.530633\pi$$
$$824$$ 0.404673 0.0140975
$$825$$ 0 0
$$826$$ 1.19248 0.0414917
$$827$$ 18.8463 0.655350 0.327675 0.944790i $$-0.393735\pi$$
0.327675 + 0.944790i $$0.393735\pi$$
$$828$$ −42.2420 −1.46801
$$829$$ −23.0204 −0.799532 −0.399766 0.916617i $$-0.630909\pi$$
−0.399766 + 0.916617i $$0.630909\pi$$
$$830$$ −11.5685 −0.401550
$$831$$ −16.2630 −0.564156
$$832$$ 2.14515 0.0743699
$$833$$ 1.43942 0.0498730
$$834$$ −59.9575 −2.07616
$$835$$ −11.7960 −0.408216
$$836$$ 0 0
$$837$$ 102.006 3.52583
$$838$$ −17.9411 −0.619765
$$839$$ 41.3387 1.42717 0.713585 0.700569i $$-0.247070\pi$$
0.713585 + 0.700569i $$0.247070\pi$$
$$840$$ 3.24337 0.111907
$$841$$ −27.5695 −0.950671
$$842$$ 36.8460 1.26980
$$843$$ 15.4628 0.532567
$$844$$ −9.35228 −0.321919
$$845$$ 8.39831 0.288911
$$846$$ 62.4021 2.14543
$$847$$ 0 0
$$848$$ 14.1537 0.486040
$$849$$ 31.2992 1.07418
$$850$$ 1.43942 0.0493717
$$851$$ −26.9871 −0.925106
$$852$$ −18.5388 −0.635129
$$853$$ −19.8786 −0.680630 −0.340315 0.940312i $$-0.610534\pi$$
−0.340315 + 0.940312i $$0.610534\pi$$
$$854$$ 4.19441 0.143530
$$855$$ 4.76608 0.162997
$$856$$ −8.17962 −0.279574
$$857$$ −30.9589 −1.05753 −0.528767 0.848767i $$-0.677346\pi$$
−0.528767 + 0.848767i $$0.677346\pi$$
$$858$$ 0 0
$$859$$ −38.3155 −1.30731 −0.653654 0.756793i $$-0.726765\pi$$
−0.653654 + 0.756793i $$0.726765\pi$$
$$860$$ −9.70747 −0.331022
$$861$$ 28.9848 0.987801
$$862$$ 31.8543 1.08496
$$863$$ 33.7087 1.14746 0.573729 0.819045i $$-0.305496\pi$$
0.573729 + 0.819045i $$0.305496\pi$$
$$864$$ 14.6583 0.498686
$$865$$ −0.146508 −0.00498141
$$866$$ 25.0278 0.850480
$$867$$ −48.4173 −1.64434
$$868$$ −6.95889 −0.236200
$$869$$ 0 0
$$870$$ 3.87923 0.131518
$$871$$ −25.7631 −0.872948
$$872$$ −1.34828 −0.0456585
$$873$$ −22.2974 −0.754654
$$874$$ 3.56067 0.120442
$$875$$ 1.00000 0.0338062
$$876$$ 20.8867 0.705696
$$877$$ 32.7548 1.10605 0.553025 0.833165i $$-0.313473\pi$$
0.553025 + 0.833165i $$0.313473\pi$$
$$878$$ −40.3537 −1.36187
$$879$$ −92.4397 −3.11791
$$880$$ 0 0
$$881$$ −41.6696 −1.40389 −0.701943 0.712233i $$-0.747684\pi$$
−0.701943 + 0.712233i $$0.747684\pi$$
$$882$$ 7.51947 0.253194
$$883$$ −19.7448 −0.664467 −0.332234 0.943197i $$-0.607802\pi$$
−0.332234 + 0.943197i $$0.607802\pi$$
$$884$$ 3.08778 0.103853
$$885$$ 3.86766 0.130010
$$886$$ −0.519900 −0.0174664
$$887$$ −25.1277 −0.843704 −0.421852 0.906665i $$-0.638620\pi$$
−0.421852 + 0.906665i $$0.638620\pi$$
$$888$$ 15.5810 0.522864
$$889$$ −12.6218 −0.423321
$$890$$ −16.7855 −0.562651
$$891$$ 0 0
$$892$$ −13.8365 −0.463280
$$893$$ −5.26001 −0.176020
$$894$$ −50.0686 −1.67455
$$895$$ −10.2987 −0.344249
$$896$$ −1.00000 −0.0334077
$$897$$ −39.0853 −1.30502
$$898$$ 23.0974 0.770769
$$899$$ −8.32317 −0.277593
$$900$$ 7.51947 0.250649
$$901$$ 20.3731 0.678727
$$902$$ 0 0
$$903$$ −31.4850 −1.04775
$$904$$ 12.6845 0.421881
$$905$$ 19.8342 0.659310
$$906$$ −37.1709 −1.23492
$$907$$ 15.1970 0.504608 0.252304 0.967648i $$-0.418812\pi$$
0.252304 + 0.967648i $$0.418812\pi$$
$$908$$ 25.8686 0.858481
$$909$$ −116.991 −3.88034
$$910$$ 2.14515 0.0711112
$$911$$ −31.4976 −1.04356 −0.521781 0.853080i $$-0.674732\pi$$
−0.521781 + 0.853080i $$0.674732\pi$$
$$912$$ −2.05576 −0.0680729
$$913$$ 0 0
$$914$$ −20.3786 −0.674066
$$915$$ 13.6040 0.449736
$$916$$ 11.1800 0.369398
$$917$$ −4.14932 −0.137023
$$918$$ 21.0995 0.696387
$$919$$ 3.67799 0.121326 0.0606628 0.998158i $$-0.480679\pi$$
0.0606628 + 0.998158i $$0.480679\pi$$
$$920$$ 5.61769 0.185210
$$921$$ −1.79096 −0.0590142
$$922$$ −12.1816 −0.401180
$$923$$ −12.2615 −0.403592
$$924$$ 0 0
$$925$$ 4.80395 0.157953
$$926$$ −11.4172 −0.375192
$$927$$ 3.04293 0.0999429
$$928$$ −1.19605 −0.0392622
$$929$$ 57.4781 1.88579 0.942897 0.333084i $$-0.108089\pi$$
0.942897 + 0.333084i $$0.108089\pi$$
$$930$$ −22.5703 −0.740109
$$931$$ −0.633832 −0.0207730
$$932$$ 2.87302 0.0941088
$$933$$ −98.8410 −3.23591
$$934$$ 22.7698 0.745052
$$935$$ 0 0
$$936$$ 16.1304 0.527239
$$937$$ −18.0162 −0.588564 −0.294282 0.955719i $$-0.595081\pi$$
−0.294282 + 0.955719i $$0.595081\pi$$
$$938$$ 12.0099 0.392137
$$939$$ −108.549 −3.54235
$$940$$ −8.29874 −0.270675
$$941$$ −27.4433 −0.894625 −0.447312 0.894378i $$-0.647619\pi$$
−0.447312 + 0.894378i $$0.647619\pi$$
$$942$$ −33.3045 −1.08512
$$943$$ 50.2032 1.63484
$$944$$ −1.19248 −0.0388119
$$945$$ 14.6583 0.476835
$$946$$ 0 0
$$947$$ 0.524970 0.0170592 0.00852961 0.999964i $$-0.497285\pi$$
0.00852961 + 0.999964i $$0.497285\pi$$
$$948$$ −24.9458 −0.810204
$$949$$ 13.8144 0.448434
$$950$$ −0.633832 −0.0205642
$$951$$ −65.5567 −2.12582
$$952$$ −1.43942 −0.0466519
$$953$$ 47.1112 1.52608 0.763040 0.646351i $$-0.223706\pi$$
0.763040 + 0.646351i $$0.223706\pi$$
$$954$$ 106.428 3.44574
$$955$$ −5.95268 −0.192624
$$956$$ 8.22791 0.266110
$$957$$ 0 0
$$958$$ −15.9269 −0.514575
$$959$$ −20.8283 −0.672582
$$960$$ −3.24337 −0.104679
$$961$$ 17.4261 0.562134
$$962$$ 10.3052 0.332254
$$963$$ −61.5064 −1.98202
$$964$$ −6.54207 −0.210706
$$965$$ 14.9671 0.481808
$$966$$ 18.2203 0.586227
$$967$$ −30.5133 −0.981243 −0.490621 0.871373i $$-0.663230\pi$$
−0.490621 + 0.871373i $$0.663230\pi$$
$$968$$ 0 0
$$969$$ −2.95910 −0.0950599
$$970$$ 2.96529 0.0952099
$$971$$ 39.5470 1.26912 0.634562 0.772872i $$-0.281180\pi$$
0.634562 + 0.772872i $$0.281180\pi$$
$$972$$ 37.0575 1.18862
$$973$$ 18.4862 0.592639
$$974$$ 4.57953 0.146737
$$975$$ 6.95754 0.222820
$$976$$ −4.19441 −0.134260
$$977$$ −31.0366 −0.992950 −0.496475 0.868051i $$-0.665373\pi$$
−0.496475 + 0.868051i $$0.665373\pi$$
$$978$$ −9.43784 −0.301789
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ −10.1384 −0.323693
$$982$$ −28.0073 −0.893748
$$983$$ −51.0988 −1.62980 −0.814900 0.579602i $$-0.803208\pi$$
−0.814900 + 0.579602i $$0.803208\pi$$
$$984$$ −28.9848 −0.924003
$$985$$ 19.1531 0.610269
$$986$$ −1.72162 −0.0548275
$$987$$ −26.9159 −0.856743
$$988$$ −1.35967 −0.0432568
$$989$$ −54.5336 −1.73407
$$990$$ 0 0
$$991$$ −8.89709 −0.282625 −0.141313 0.989965i $$-0.545132\pi$$
−0.141313 + 0.989965i $$0.545132\pi$$
$$992$$ 6.95889 0.220945
$$993$$ −62.8133 −1.99332
$$994$$ 5.71591 0.181298
$$995$$ 0.268053 0.00849786
$$996$$ 37.5211 1.18890
$$997$$ −26.9013 −0.851974 −0.425987 0.904729i $$-0.640073\pi$$
−0.425987 + 0.904729i $$0.640073\pi$$
$$998$$ 27.8295 0.880928
$$999$$ 70.4179 2.22792
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.1.6 yes 6
11.10 odd 2 8470.2.a.cx.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.6 6 11.10 odd 2
8470.2.a.dd.1.6 yes 6 1.1 even 1 trivial