# Properties

 Label 546.2.a.i.1.2 Level $546$ Weight $2$ Character 546.1 Self dual yes Analytic conductor $4.360$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 546.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.70156 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.70156 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.70156 q^{10} -0.701562 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} -2.70156 q^{15} +1.00000 q^{16} -2.70156 q^{17} +1.00000 q^{18} -0.701562 q^{19} +2.70156 q^{20} -1.00000 q^{21} -0.701562 q^{22} +4.70156 q^{23} -1.00000 q^{24} +2.29844 q^{25} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.70156 q^{29} -2.70156 q^{30} +1.00000 q^{32} +0.701562 q^{33} -2.70156 q^{34} +2.70156 q^{35} +1.00000 q^{36} +10.7016 q^{37} -0.701562 q^{38} -1.00000 q^{39} +2.70156 q^{40} +3.40312 q^{41} -1.00000 q^{42} -10.1047 q^{43} -0.701562 q^{44} +2.70156 q^{45} +4.70156 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +2.29844 q^{50} +2.70156 q^{51} +1.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -1.89531 q^{55} +1.00000 q^{56} +0.701562 q^{57} +2.70156 q^{58} -14.8062 q^{59} -2.70156 q^{60} +1.29844 q^{61} +1.00000 q^{63} +1.00000 q^{64} +2.70156 q^{65} +0.701562 q^{66} +5.40312 q^{67} -2.70156 q^{68} -4.70156 q^{69} +2.70156 q^{70} -8.00000 q^{71} +1.00000 q^{72} -1.29844 q^{73} +10.7016 q^{74} -2.29844 q^{75} -0.701562 q^{76} -0.701562 q^{77} -1.00000 q^{78} +9.40312 q^{79} +2.70156 q^{80} +1.00000 q^{81} +3.40312 q^{82} -13.4031 q^{83} -1.00000 q^{84} -7.29844 q^{85} -10.1047 q^{86} -2.70156 q^{87} -0.701562 q^{88} -8.80625 q^{89} +2.70156 q^{90} +1.00000 q^{91} +4.70156 q^{92} -8.00000 q^{94} -1.89531 q^{95} -1.00000 q^{96} -8.80625 q^{97} +1.00000 q^{98} -0.701562 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - q^{10} + 5 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{14} + q^{15} + 2 q^{16} + q^{17} + 2 q^{18} + 5 q^{19} - q^{20} - 2 q^{21} + 5 q^{22} + 3 q^{23} - 2 q^{24} + 11 q^{25} + 2 q^{26} - 2 q^{27} + 2 q^{28} - q^{29} + q^{30} + 2 q^{32} - 5 q^{33} + q^{34} - q^{35} + 2 q^{36} + 15 q^{37} + 5 q^{38} - 2 q^{39} - q^{40} - 6 q^{41} - 2 q^{42} - q^{43} + 5 q^{44} - q^{45} + 3 q^{46} - 16 q^{47} - 2 q^{48} + 2 q^{49} + 11 q^{50} - q^{51} + 2 q^{52} - 4 q^{53} - 2 q^{54} - 23 q^{55} + 2 q^{56} - 5 q^{57} - q^{58} - 4 q^{59} + q^{60} + 9 q^{61} + 2 q^{63} + 2 q^{64} - q^{65} - 5 q^{66} - 2 q^{67} + q^{68} - 3 q^{69} - q^{70} - 16 q^{71} + 2 q^{72} - 9 q^{73} + 15 q^{74} - 11 q^{75} + 5 q^{76} + 5 q^{77} - 2 q^{78} + 6 q^{79} - q^{80} + 2 q^{81} - 6 q^{82} - 14 q^{83} - 2 q^{84} - 21 q^{85} - q^{86} + q^{87} + 5 q^{88} + 8 q^{89} - q^{90} + 2 q^{91} + 3 q^{92} - 16 q^{94} - 23 q^{95} - 2 q^{96} + 8 q^{97} + 2 q^{98} + 5 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 - q^10 + 5 * q^11 - 2 * q^12 + 2 * q^13 + 2 * q^14 + q^15 + 2 * q^16 + q^17 + 2 * q^18 + 5 * q^19 - q^20 - 2 * q^21 + 5 * q^22 + 3 * q^23 - 2 * q^24 + 11 * q^25 + 2 * q^26 - 2 * q^27 + 2 * q^28 - q^29 + q^30 + 2 * q^32 - 5 * q^33 + q^34 - q^35 + 2 * q^36 + 15 * q^37 + 5 * q^38 - 2 * q^39 - q^40 - 6 * q^41 - 2 * q^42 - q^43 + 5 * q^44 - q^45 + 3 * q^46 - 16 * q^47 - 2 * q^48 + 2 * q^49 + 11 * q^50 - q^51 + 2 * q^52 - 4 * q^53 - 2 * q^54 - 23 * q^55 + 2 * q^56 - 5 * q^57 - q^58 - 4 * q^59 + q^60 + 9 * q^61 + 2 * q^63 + 2 * q^64 - q^65 - 5 * q^66 - 2 * q^67 + q^68 - 3 * q^69 - q^70 - 16 * q^71 + 2 * q^72 - 9 * q^73 + 15 * q^74 - 11 * q^75 + 5 * q^76 + 5 * q^77 - 2 * q^78 + 6 * q^79 - q^80 + 2 * q^81 - 6 * q^82 - 14 * q^83 - 2 * q^84 - 21 * q^85 - q^86 + q^87 + 5 * q^88 + 8 * q^89 - q^90 + 2 * q^91 + 3 * q^92 - 16 * q^94 - 23 * q^95 - 2 * q^96 + 8 * q^97 + 2 * q^98 + 5 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 2.70156 1.20818 0.604088 0.796918i $$-0.293538\pi$$
0.604088 + 0.796918i $$0.293538\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.70156 0.854309
$$11$$ −0.701562 −0.211529 −0.105764 0.994391i $$-0.533729\pi$$
−0.105764 + 0.994391i $$0.533729\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ 1.00000 0.267261
$$15$$ −2.70156 −0.697540
$$16$$ 1.00000 0.250000
$$17$$ −2.70156 −0.655225 −0.327613 0.944812i $$-0.606244\pi$$
−0.327613 + 0.944812i $$0.606244\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −0.701562 −0.160949 −0.0804747 0.996757i $$-0.525644\pi$$
−0.0804747 + 0.996757i $$0.525644\pi$$
$$20$$ 2.70156 0.604088
$$21$$ −1.00000 −0.218218
$$22$$ −0.701562 −0.149574
$$23$$ 4.70156 0.980343 0.490172 0.871626i $$-0.336934\pi$$
0.490172 + 0.871626i $$0.336934\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 2.29844 0.459688
$$26$$ 1.00000 0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ 2.70156 0.501667 0.250834 0.968030i $$-0.419295\pi$$
0.250834 + 0.968030i $$0.419295\pi$$
$$30$$ −2.70156 −0.493236
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.701562 0.122126
$$34$$ −2.70156 −0.463314
$$35$$ 2.70156 0.456647
$$36$$ 1.00000 0.166667
$$37$$ 10.7016 1.75933 0.879663 0.475598i $$-0.157768\pi$$
0.879663 + 0.475598i $$0.157768\pi$$
$$38$$ −0.701562 −0.113808
$$39$$ −1.00000 −0.160128
$$40$$ 2.70156 0.427154
$$41$$ 3.40312 0.531479 0.265739 0.964045i $$-0.414384\pi$$
0.265739 + 0.964045i $$0.414384\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −10.1047 −1.54095 −0.770475 0.637470i $$-0.779981\pi$$
−0.770475 + 0.637470i $$0.779981\pi$$
$$44$$ −0.701562 −0.105764
$$45$$ 2.70156 0.402725
$$46$$ 4.70156 0.693208
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 2.29844 0.325048
$$51$$ 2.70156 0.378294
$$52$$ 1.00000 0.138675
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.89531 −0.255564
$$56$$ 1.00000 0.133631
$$57$$ 0.701562 0.0929242
$$58$$ 2.70156 0.354732
$$59$$ −14.8062 −1.92761 −0.963805 0.266609i $$-0.914097\pi$$
−0.963805 + 0.266609i $$0.914097\pi$$
$$60$$ −2.70156 −0.348770
$$61$$ 1.29844 0.166248 0.0831240 0.996539i $$-0.473510\pi$$
0.0831240 + 0.996539i $$0.473510\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 2.70156 0.335088
$$66$$ 0.701562 0.0863563
$$67$$ 5.40312 0.660097 0.330048 0.943964i $$-0.392935\pi$$
0.330048 + 0.943964i $$0.392935\pi$$
$$68$$ −2.70156 −0.327613
$$69$$ −4.70156 −0.566002
$$70$$ 2.70156 0.322898
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −1.29844 −0.151971 −0.0759853 0.997109i $$-0.524210\pi$$
−0.0759853 + 0.997109i $$0.524210\pi$$
$$74$$ 10.7016 1.24403
$$75$$ −2.29844 −0.265401
$$76$$ −0.701562 −0.0804747
$$77$$ −0.701562 −0.0799504
$$78$$ −1.00000 −0.113228
$$79$$ 9.40312 1.05793 0.528967 0.848642i $$-0.322579\pi$$
0.528967 + 0.848642i $$0.322579\pi$$
$$80$$ 2.70156 0.302044
$$81$$ 1.00000 0.111111
$$82$$ 3.40312 0.375812
$$83$$ −13.4031 −1.47118 −0.735592 0.677425i $$-0.763096\pi$$
−0.735592 + 0.677425i $$0.763096\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ −7.29844 −0.791627
$$86$$ −10.1047 −1.08962
$$87$$ −2.70156 −0.289638
$$88$$ −0.701562 −0.0747868
$$89$$ −8.80625 −0.933460 −0.466730 0.884400i $$-0.654568\pi$$
−0.466730 + 0.884400i $$0.654568\pi$$
$$90$$ 2.70156 0.284770
$$91$$ 1.00000 0.104828
$$92$$ 4.70156 0.490172
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −1.89531 −0.194455
$$96$$ −1.00000 −0.102062
$$97$$ −8.80625 −0.894139 −0.447070 0.894499i $$-0.647532\pi$$
−0.447070 + 0.894499i $$0.647532\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −0.701562 −0.0705096
$$100$$ 2.29844 0.229844
$$101$$ −3.40312 −0.338624 −0.169312 0.985563i $$-0.554154\pi$$
−0.169312 + 0.985563i $$0.554154\pi$$
$$102$$ 2.70156 0.267495
$$103$$ −3.29844 −0.325005 −0.162502 0.986708i $$-0.551957\pi$$
−0.162502 + 0.986708i $$0.551957\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ −2.70156 −0.263645
$$106$$ −2.00000 −0.194257
$$107$$ −5.40312 −0.522340 −0.261170 0.965293i $$-0.584108\pi$$
−0.261170 + 0.965293i $$0.584108\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 9.29844 0.890629 0.445314 0.895374i $$-0.353092\pi$$
0.445314 + 0.895374i $$0.353092\pi$$
$$110$$ −1.89531 −0.180711
$$111$$ −10.7016 −1.01575
$$112$$ 1.00000 0.0944911
$$113$$ 4.80625 0.452134 0.226067 0.974112i $$-0.427413\pi$$
0.226067 + 0.974112i $$0.427413\pi$$
$$114$$ 0.701562 0.0657073
$$115$$ 12.7016 1.18443
$$116$$ 2.70156 0.250834
$$117$$ 1.00000 0.0924500
$$118$$ −14.8062 −1.36303
$$119$$ −2.70156 −0.247652
$$120$$ −2.70156 −0.246618
$$121$$ −10.5078 −0.955256
$$122$$ 1.29844 0.117555
$$123$$ −3.40312 −0.306849
$$124$$ 0 0
$$125$$ −7.29844 −0.652792
$$126$$ 1.00000 0.0890871
$$127$$ −6.59688 −0.585378 −0.292689 0.956208i $$-0.594550\pi$$
−0.292689 + 0.956208i $$0.594550\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 10.1047 0.889668
$$130$$ 2.70156 0.236943
$$131$$ −7.29844 −0.637667 −0.318834 0.947811i $$-0.603291\pi$$
−0.318834 + 0.947811i $$0.603291\pi$$
$$132$$ 0.701562 0.0610631
$$133$$ −0.701562 −0.0608332
$$134$$ 5.40312 0.466759
$$135$$ −2.70156 −0.232513
$$136$$ −2.70156 −0.231657
$$137$$ −18.7016 −1.59778 −0.798891 0.601476i $$-0.794580\pi$$
−0.798891 + 0.601476i $$0.794580\pi$$
$$138$$ −4.70156 −0.400224
$$139$$ 6.80625 0.577298 0.288649 0.957435i $$-0.406794\pi$$
0.288649 + 0.957435i $$0.406794\pi$$
$$140$$ 2.70156 0.228324
$$141$$ 8.00000 0.673722
$$142$$ −8.00000 −0.671345
$$143$$ −0.701562 −0.0586676
$$144$$ 1.00000 0.0833333
$$145$$ 7.29844 0.606102
$$146$$ −1.29844 −0.107459
$$147$$ −1.00000 −0.0824786
$$148$$ 10.7016 0.879663
$$149$$ 15.4031 1.26187 0.630937 0.775834i $$-0.282671\pi$$
0.630937 + 0.775834i $$0.282671\pi$$
$$150$$ −2.29844 −0.187667
$$151$$ −4.70156 −0.382608 −0.191304 0.981531i $$-0.561272\pi$$
−0.191304 + 0.981531i $$0.561272\pi$$
$$152$$ −0.701562 −0.0569042
$$153$$ −2.70156 −0.218408
$$154$$ −0.701562 −0.0565335
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 20.1047 1.60453 0.802264 0.596969i $$-0.203628\pi$$
0.802264 + 0.596969i $$0.203628\pi$$
$$158$$ 9.40312 0.748072
$$159$$ 2.00000 0.158610
$$160$$ 2.70156 0.213577
$$161$$ 4.70156 0.370535
$$162$$ 1.00000 0.0785674
$$163$$ 5.40312 0.423205 0.211603 0.977356i $$-0.432132\pi$$
0.211603 + 0.977356i $$0.432132\pi$$
$$164$$ 3.40312 0.265739
$$165$$ 1.89531 0.147550
$$166$$ −13.4031 −1.04028
$$167$$ 3.29844 0.255241 0.127620 0.991823i $$-0.459266\pi$$
0.127620 + 0.991823i $$0.459266\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 1.00000 0.0769231
$$170$$ −7.29844 −0.559765
$$171$$ −0.701562 −0.0536498
$$172$$ −10.1047 −0.770475
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −2.70156 −0.204805
$$175$$ 2.29844 0.173746
$$176$$ −0.701562 −0.0528822
$$177$$ 14.8062 1.11291
$$178$$ −8.80625 −0.660056
$$179$$ 14.8062 1.10667 0.553335 0.832958i $$-0.313355\pi$$
0.553335 + 0.832958i $$0.313355\pi$$
$$180$$ 2.70156 0.201363
$$181$$ 8.80625 0.654563 0.327282 0.944927i $$-0.393867\pi$$
0.327282 + 0.944927i $$0.393867\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ −1.29844 −0.0959833
$$184$$ 4.70156 0.346604
$$185$$ 28.9109 2.12557
$$186$$ 0 0
$$187$$ 1.89531 0.138599
$$188$$ −8.00000 −0.583460
$$189$$ −1.00000 −0.0727393
$$190$$ −1.89531 −0.137501
$$191$$ 12.7016 0.919053 0.459526 0.888164i $$-0.348019\pi$$
0.459526 + 0.888164i $$0.348019\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 11.4031 0.820815 0.410407 0.911902i $$-0.365386\pi$$
0.410407 + 0.911902i $$0.365386\pi$$
$$194$$ −8.80625 −0.632252
$$195$$ −2.70156 −0.193463
$$196$$ 1.00000 0.0714286
$$197$$ −3.40312 −0.242463 −0.121231 0.992624i $$-0.538684\pi$$
−0.121231 + 0.992624i $$0.538684\pi$$
$$198$$ −0.701562 −0.0498578
$$199$$ 22.1047 1.56696 0.783480 0.621417i $$-0.213443\pi$$
0.783480 + 0.621417i $$0.213443\pi$$
$$200$$ 2.29844 0.162524
$$201$$ −5.40312 −0.381107
$$202$$ −3.40312 −0.239443
$$203$$ 2.70156 0.189612
$$204$$ 2.70156 0.189147
$$205$$ 9.19375 0.642119
$$206$$ −3.29844 −0.229813
$$207$$ 4.70156 0.326781
$$208$$ 1.00000 0.0693375
$$209$$ 0.492189 0.0340455
$$210$$ −2.70156 −0.186425
$$211$$ −24.7016 −1.70053 −0.850263 0.526358i $$-0.823557\pi$$
−0.850263 + 0.526358i $$0.823557\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 8.00000 0.548151
$$214$$ −5.40312 −0.369350
$$215$$ −27.2984 −1.86174
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 9.29844 0.629770
$$219$$ 1.29844 0.0877403
$$220$$ −1.89531 −0.127782
$$221$$ −2.70156 −0.181727
$$222$$ −10.7016 −0.718242
$$223$$ 9.40312 0.629680 0.314840 0.949145i $$-0.398049\pi$$
0.314840 + 0.949145i $$0.398049\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 2.29844 0.153229
$$226$$ 4.80625 0.319707
$$227$$ 21.4031 1.42058 0.710288 0.703912i $$-0.248565\pi$$
0.710288 + 0.703912i $$0.248565\pi$$
$$228$$ 0.701562 0.0464621
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 12.7016 0.837516
$$231$$ 0.701562 0.0461594
$$232$$ 2.70156 0.177366
$$233$$ −18.2094 −1.19294 −0.596468 0.802637i $$-0.703430\pi$$
−0.596468 + 0.802637i $$0.703430\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ −21.6125 −1.40984
$$236$$ −14.8062 −0.963805
$$237$$ −9.40312 −0.610799
$$238$$ −2.70156 −0.175116
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ −2.70156 −0.174385
$$241$$ −24.8062 −1.59791 −0.798955 0.601390i $$-0.794614\pi$$
−0.798955 + 0.601390i $$0.794614\pi$$
$$242$$ −10.5078 −0.675468
$$243$$ −1.00000 −0.0641500
$$244$$ 1.29844 0.0831240
$$245$$ 2.70156 0.172596
$$246$$ −3.40312 −0.216975
$$247$$ −0.701562 −0.0446393
$$248$$ 0 0
$$249$$ 13.4031 0.849388
$$250$$ −7.29844 −0.461594
$$251$$ 3.50781 0.221411 0.110706 0.993853i $$-0.464689\pi$$
0.110706 + 0.993853i $$0.464689\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −3.29844 −0.207371
$$254$$ −6.59688 −0.413925
$$255$$ 7.29844 0.457046
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 10.1047 0.629090
$$259$$ 10.7016 0.664963
$$260$$ 2.70156 0.167544
$$261$$ 2.70156 0.167222
$$262$$ −7.29844 −0.450899
$$263$$ −26.8062 −1.65294 −0.826472 0.562978i $$-0.809656\pi$$
−0.826472 + 0.562978i $$0.809656\pi$$
$$264$$ 0.701562 0.0431782
$$265$$ −5.40312 −0.331911
$$266$$ −0.701562 −0.0430155
$$267$$ 8.80625 0.538934
$$268$$ 5.40312 0.330048
$$269$$ −4.80625 −0.293042 −0.146521 0.989208i $$-0.546808\pi$$
−0.146521 + 0.989208i $$0.546808\pi$$
$$270$$ −2.70156 −0.164412
$$271$$ 12.2094 0.741667 0.370833 0.928699i $$-0.379072\pi$$
0.370833 + 0.928699i $$0.379072\pi$$
$$272$$ −2.70156 −0.163806
$$273$$ −1.00000 −0.0605228
$$274$$ −18.7016 −1.12980
$$275$$ −1.61250 −0.0972372
$$276$$ −4.70156 −0.283001
$$277$$ 27.6125 1.65907 0.829537 0.558452i $$-0.188604\pi$$
0.829537 + 0.558452i $$0.188604\pi$$
$$278$$ 6.80625 0.408212
$$279$$ 0 0
$$280$$ 2.70156 0.161449
$$281$$ 12.8062 0.763957 0.381978 0.924171i $$-0.375243\pi$$
0.381978 + 0.924171i $$0.375243\pi$$
$$282$$ 8.00000 0.476393
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 1.89531 0.112269
$$286$$ −0.701562 −0.0414842
$$287$$ 3.40312 0.200880
$$288$$ 1.00000 0.0589256
$$289$$ −9.70156 −0.570680
$$290$$ 7.29844 0.428579
$$291$$ 8.80625 0.516231
$$292$$ −1.29844 −0.0759853
$$293$$ −12.8062 −0.748149 −0.374075 0.927399i $$-0.622040\pi$$
−0.374075 + 0.927399i $$0.622040\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ −40.0000 −2.32889
$$296$$ 10.7016 0.622016
$$297$$ 0.701562 0.0407088
$$298$$ 15.4031 0.892279
$$299$$ 4.70156 0.271898
$$300$$ −2.29844 −0.132700
$$301$$ −10.1047 −0.582424
$$302$$ −4.70156 −0.270544
$$303$$ 3.40312 0.195504
$$304$$ −0.701562 −0.0402373
$$305$$ 3.50781 0.200857
$$306$$ −2.70156 −0.154438
$$307$$ −6.80625 −0.388453 −0.194227 0.980957i $$-0.562220\pi$$
−0.194227 + 0.980957i $$0.562220\pi$$
$$308$$ −0.701562 −0.0399752
$$309$$ 3.29844 0.187642
$$310$$ 0 0
$$311$$ 14.5969 0.827713 0.413856 0.910342i $$-0.364182\pi$$
0.413856 + 0.910342i $$0.364182\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 22.2094 1.25535 0.627674 0.778476i $$-0.284007\pi$$
0.627674 + 0.778476i $$0.284007\pi$$
$$314$$ 20.1047 1.13457
$$315$$ 2.70156 0.152216
$$316$$ 9.40312 0.528967
$$317$$ 7.40312 0.415801 0.207900 0.978150i $$-0.433337\pi$$
0.207900 + 0.978150i $$0.433337\pi$$
$$318$$ 2.00000 0.112154
$$319$$ −1.89531 −0.106117
$$320$$ 2.70156 0.151022
$$321$$ 5.40312 0.301573
$$322$$ 4.70156 0.262008
$$323$$ 1.89531 0.105458
$$324$$ 1.00000 0.0555556
$$325$$ 2.29844 0.127494
$$326$$ 5.40312 0.299251
$$327$$ −9.29844 −0.514205
$$328$$ 3.40312 0.187906
$$329$$ −8.00000 −0.441054
$$330$$ 1.89531 0.104334
$$331$$ 32.2094 1.77039 0.885194 0.465223i $$-0.154026\pi$$
0.885194 + 0.465223i $$0.154026\pi$$
$$332$$ −13.4031 −0.735592
$$333$$ 10.7016 0.586442
$$334$$ 3.29844 0.180482
$$335$$ 14.5969 0.797513
$$336$$ −1.00000 −0.0545545
$$337$$ −29.5078 −1.60739 −0.803696 0.595040i $$-0.797136\pi$$
−0.803696 + 0.595040i $$0.797136\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ −4.80625 −0.261040
$$340$$ −7.29844 −0.395813
$$341$$ 0 0
$$342$$ −0.701562 −0.0379361
$$343$$ 1.00000 0.0539949
$$344$$ −10.1047 −0.544808
$$345$$ −12.7016 −0.683829
$$346$$ −18.0000 −0.967686
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ −2.70156 −0.144819
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 2.29844 0.122857
$$351$$ −1.00000 −0.0533761
$$352$$ −0.701562 −0.0373934
$$353$$ 20.8062 1.10740 0.553702 0.832715i $$-0.313215\pi$$
0.553702 + 0.832715i $$0.313215\pi$$
$$354$$ 14.8062 0.786943
$$355$$ −21.6125 −1.14707
$$356$$ −8.80625 −0.466730
$$357$$ 2.70156 0.142982
$$358$$ 14.8062 0.782535
$$359$$ 26.8062 1.41478 0.707390 0.706824i $$-0.249873\pi$$
0.707390 + 0.706824i $$0.249873\pi$$
$$360$$ 2.70156 0.142385
$$361$$ −18.5078 −0.974095
$$362$$ 8.80625 0.462846
$$363$$ 10.5078 0.551517
$$364$$ 1.00000 0.0524142
$$365$$ −3.50781 −0.183607
$$366$$ −1.29844 −0.0678704
$$367$$ −29.6125 −1.54576 −0.772880 0.634552i $$-0.781184\pi$$
−0.772880 + 0.634552i $$0.781184\pi$$
$$368$$ 4.70156 0.245086
$$369$$ 3.40312 0.177160
$$370$$ 28.9109 1.50301
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ −19.4031 −1.00466 −0.502328 0.864677i $$-0.667523\pi$$
−0.502328 + 0.864677i $$0.667523\pi$$
$$374$$ 1.89531 0.0980043
$$375$$ 7.29844 0.376890
$$376$$ −8.00000 −0.412568
$$377$$ 2.70156 0.139138
$$378$$ −1.00000 −0.0514344
$$379$$ −17.6125 −0.904693 −0.452347 0.891842i $$-0.649413\pi$$
−0.452347 + 0.891842i $$0.649413\pi$$
$$380$$ −1.89531 −0.0972275
$$381$$ 6.59688 0.337968
$$382$$ 12.7016 0.649868
$$383$$ 16.9109 0.864108 0.432054 0.901848i $$-0.357789\pi$$
0.432054 + 0.901848i $$0.357789\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −1.89531 −0.0965941
$$386$$ 11.4031 0.580404
$$387$$ −10.1047 −0.513650
$$388$$ −8.80625 −0.447070
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ −2.70156 −0.136799
$$391$$ −12.7016 −0.642346
$$392$$ 1.00000 0.0505076
$$393$$ 7.29844 0.368157
$$394$$ −3.40312 −0.171447
$$395$$ 25.4031 1.27817
$$396$$ −0.701562 −0.0352548
$$397$$ 0.806248 0.0404645 0.0202322 0.999795i $$-0.493559\pi$$
0.0202322 + 0.999795i $$0.493559\pi$$
$$398$$ 22.1047 1.10801
$$399$$ 0.701562 0.0351220
$$400$$ 2.29844 0.114922
$$401$$ 4.80625 0.240013 0.120006 0.992773i $$-0.461709\pi$$
0.120006 + 0.992773i $$0.461709\pi$$
$$402$$ −5.40312 −0.269483
$$403$$ 0 0
$$404$$ −3.40312 −0.169312
$$405$$ 2.70156 0.134242
$$406$$ 2.70156 0.134076
$$407$$ −7.50781 −0.372148
$$408$$ 2.70156 0.133747
$$409$$ 5.29844 0.261991 0.130995 0.991383i $$-0.458183\pi$$
0.130995 + 0.991383i $$0.458183\pi$$
$$410$$ 9.19375 0.454047
$$411$$ 18.7016 0.922480
$$412$$ −3.29844 −0.162502
$$413$$ −14.8062 −0.728568
$$414$$ 4.70156 0.231069
$$415$$ −36.2094 −1.77745
$$416$$ 1.00000 0.0490290
$$417$$ −6.80625 −0.333303
$$418$$ 0.492189 0.0240738
$$419$$ 34.1047 1.66612 0.833061 0.553180i $$-0.186586\pi$$
0.833061 + 0.553180i $$0.186586\pi$$
$$420$$ −2.70156 −0.131823
$$421$$ 19.6125 0.955855 0.477927 0.878399i $$-0.341388\pi$$
0.477927 + 0.878399i $$0.341388\pi$$
$$422$$ −24.7016 −1.20245
$$423$$ −8.00000 −0.388973
$$424$$ −2.00000 −0.0971286
$$425$$ −6.20937 −0.301199
$$426$$ 8.00000 0.387601
$$427$$ 1.29844 0.0628358
$$428$$ −5.40312 −0.261170
$$429$$ 0.701562 0.0338717
$$430$$ −27.2984 −1.31645
$$431$$ −12.2094 −0.588105 −0.294052 0.955789i $$-0.595004\pi$$
−0.294052 + 0.955789i $$0.595004\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 14.2094 0.682859 0.341429 0.939907i $$-0.389089\pi$$
0.341429 + 0.939907i $$0.389089\pi$$
$$434$$ 0 0
$$435$$ −7.29844 −0.349933
$$436$$ 9.29844 0.445314
$$437$$ −3.29844 −0.157786
$$438$$ 1.29844 0.0620418
$$439$$ −0.492189 −0.0234909 −0.0117455 0.999931i $$-0.503739\pi$$
−0.0117455 + 0.999931i $$0.503739\pi$$
$$440$$ −1.89531 −0.0903555
$$441$$ 1.00000 0.0476190
$$442$$ −2.70156 −0.128500
$$443$$ 0.209373 0.00994760 0.00497380 0.999988i $$-0.498417\pi$$
0.00497380 + 0.999988i $$0.498417\pi$$
$$444$$ −10.7016 −0.507874
$$445$$ −23.7906 −1.12778
$$446$$ 9.40312 0.445251
$$447$$ −15.4031 −0.728543
$$448$$ 1.00000 0.0472456
$$449$$ −7.89531 −0.372603 −0.186301 0.982493i $$-0.559650\pi$$
−0.186301 + 0.982493i $$0.559650\pi$$
$$450$$ 2.29844 0.108349
$$451$$ −2.38750 −0.112423
$$452$$ 4.80625 0.226067
$$453$$ 4.70156 0.220899
$$454$$ 21.4031 1.00450
$$455$$ 2.70156 0.126651
$$456$$ 0.701562 0.0328537
$$457$$ 0.596876 0.0279207 0.0139603 0.999903i $$-0.495556\pi$$
0.0139603 + 0.999903i $$0.495556\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 2.70156 0.126098
$$460$$ 12.7016 0.592213
$$461$$ −20.3141 −0.946120 −0.473060 0.881030i $$-0.656851\pi$$
−0.473060 + 0.881030i $$0.656851\pi$$
$$462$$ 0.701562 0.0326396
$$463$$ −34.3141 −1.59471 −0.797355 0.603511i $$-0.793768\pi$$
−0.797355 + 0.603511i $$0.793768\pi$$
$$464$$ 2.70156 0.125417
$$465$$ 0 0
$$466$$ −18.2094 −0.843533
$$467$$ −4.49219 −0.207874 −0.103937 0.994584i $$-0.533144\pi$$
−0.103937 + 0.994584i $$0.533144\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 5.40312 0.249493
$$470$$ −21.6125 −0.996910
$$471$$ −20.1047 −0.926375
$$472$$ −14.8062 −0.681513
$$473$$ 7.08907 0.325956
$$474$$ −9.40312 −0.431900
$$475$$ −1.61250 −0.0739864
$$476$$ −2.70156 −0.123826
$$477$$ −2.00000 −0.0915737
$$478$$ 16.0000 0.731823
$$479$$ 7.50781 0.343041 0.171520 0.985181i $$-0.445132\pi$$
0.171520 + 0.985181i $$0.445132\pi$$
$$480$$ −2.70156 −0.123309
$$481$$ 10.7016 0.487949
$$482$$ −24.8062 −1.12989
$$483$$ −4.70156 −0.213928
$$484$$ −10.5078 −0.477628
$$485$$ −23.7906 −1.08028
$$486$$ −1.00000 −0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 1.29844 0.0587775
$$489$$ −5.40312 −0.244338
$$490$$ 2.70156 0.122044
$$491$$ 26.5969 1.20030 0.600150 0.799887i $$-0.295108\pi$$
0.600150 + 0.799887i $$0.295108\pi$$
$$492$$ −3.40312 −0.153425
$$493$$ −7.29844 −0.328705
$$494$$ −0.701562 −0.0315648
$$495$$ −1.89531 −0.0851880
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 13.4031 0.600608
$$499$$ −1.19375 −0.0534397 −0.0267198 0.999643i $$-0.508506\pi$$
−0.0267198 + 0.999643i $$0.508506\pi$$
$$500$$ −7.29844 −0.326396
$$501$$ −3.29844 −0.147363
$$502$$ 3.50781 0.156561
$$503$$ −33.4031 −1.48937 −0.744686 0.667415i $$-0.767401\pi$$
−0.744686 + 0.667415i $$0.767401\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ −9.19375 −0.409117
$$506$$ −3.29844 −0.146633
$$507$$ −1.00000 −0.0444116
$$508$$ −6.59688 −0.292689
$$509$$ 38.9109 1.72470 0.862348 0.506315i $$-0.168993\pi$$
0.862348 + 0.506315i $$0.168993\pi$$
$$510$$ 7.29844 0.323180
$$511$$ −1.29844 −0.0574395
$$512$$ 1.00000 0.0441942
$$513$$ 0.701562 0.0309747
$$514$$ −6.00000 −0.264649
$$515$$ −8.91093 −0.392663
$$516$$ 10.1047 0.444834
$$517$$ 5.61250 0.246837
$$518$$ 10.7016 0.470200
$$519$$ 18.0000 0.790112
$$520$$ 2.70156 0.118471
$$521$$ −1.29844 −0.0568856 −0.0284428 0.999595i $$-0.509055\pi$$
−0.0284428 + 0.999595i $$0.509055\pi$$
$$522$$ 2.70156 0.118244
$$523$$ −33.6125 −1.46977 −0.734886 0.678191i $$-0.762764\pi$$
−0.734886 + 0.678191i $$0.762764\pi$$
$$524$$ −7.29844 −0.318834
$$525$$ −2.29844 −0.100312
$$526$$ −26.8062 −1.16881
$$527$$ 0 0
$$528$$ 0.701562 0.0305316
$$529$$ −0.895314 −0.0389267
$$530$$ −5.40312 −0.234697
$$531$$ −14.8062 −0.642536
$$532$$ −0.701562 −0.0304166
$$533$$ 3.40312 0.147406
$$534$$ 8.80625 0.381084
$$535$$ −14.5969 −0.631078
$$536$$ 5.40312 0.233379
$$537$$ −14.8062 −0.638937
$$538$$ −4.80625 −0.207212
$$539$$ −0.701562 −0.0302184
$$540$$ −2.70156 −0.116257
$$541$$ −6.70156 −0.288123 −0.144061 0.989569i $$-0.546016\pi$$
−0.144061 + 0.989569i $$0.546016\pi$$
$$542$$ 12.2094 0.524437
$$543$$ −8.80625 −0.377912
$$544$$ −2.70156 −0.115829
$$545$$ 25.1203 1.07604
$$546$$ −1.00000 −0.0427960
$$547$$ 9.61250 0.411001 0.205500 0.978657i $$-0.434118\pi$$
0.205500 + 0.978657i $$0.434118\pi$$
$$548$$ −18.7016 −0.798891
$$549$$ 1.29844 0.0554160
$$550$$ −1.61250 −0.0687571
$$551$$ −1.89531 −0.0807431
$$552$$ −4.70156 −0.200112
$$553$$ 9.40312 0.399862
$$554$$ 27.6125 1.17314
$$555$$ −28.9109 −1.22720
$$556$$ 6.80625 0.288649
$$557$$ −24.5969 −1.04220 −0.521102 0.853495i $$-0.674479\pi$$
−0.521102 + 0.853495i $$0.674479\pi$$
$$558$$ 0 0
$$559$$ −10.1047 −0.427383
$$560$$ 2.70156 0.114162
$$561$$ −1.89531 −0.0800202
$$562$$ 12.8062 0.540199
$$563$$ 8.70156 0.366727 0.183364 0.983045i $$-0.441301\pi$$
0.183364 + 0.983045i $$0.441301\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 12.9844 0.546257
$$566$$ 4.00000 0.168133
$$567$$ 1.00000 0.0419961
$$568$$ −8.00000 −0.335673
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 1.89531 0.0793860
$$571$$ −1.19375 −0.0499569 −0.0249785 0.999688i $$-0.507952\pi$$
−0.0249785 + 0.999688i $$0.507952\pi$$
$$572$$ −0.701562 −0.0293338
$$573$$ −12.7016 −0.530615
$$574$$ 3.40312 0.142044
$$575$$ 10.8062 0.450652
$$576$$ 1.00000 0.0416667
$$577$$ −8.80625 −0.366609 −0.183304 0.983056i $$-0.558679\pi$$
−0.183304 + 0.983056i $$0.558679\pi$$
$$578$$ −9.70156 −0.403532
$$579$$ −11.4031 −0.473898
$$580$$ 7.29844 0.303051
$$581$$ −13.4031 −0.556055
$$582$$ 8.80625 0.365031
$$583$$ 1.40312 0.0581115
$$584$$ −1.29844 −0.0537297
$$585$$ 2.70156 0.111696
$$586$$ −12.8062 −0.529021
$$587$$ 35.0156 1.44525 0.722625 0.691241i $$-0.242936\pi$$
0.722625 + 0.691241i $$0.242936\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 0 0
$$590$$ −40.0000 −1.64677
$$591$$ 3.40312 0.139986
$$592$$ 10.7016 0.439831
$$593$$ −16.8062 −0.690150 −0.345075 0.938575i $$-0.612147\pi$$
−0.345075 + 0.938575i $$0.612147\pi$$
$$594$$ 0.701562 0.0287854
$$595$$ −7.29844 −0.299207
$$596$$ 15.4031 0.630937
$$597$$ −22.1047 −0.904685
$$598$$ 4.70156 0.192261
$$599$$ 11.2984 0.461642 0.230821 0.972996i $$-0.425859\pi$$
0.230821 + 0.972996i $$0.425859\pi$$
$$600$$ −2.29844 −0.0938333
$$601$$ −15.4031 −0.628307 −0.314153 0.949372i $$-0.601721\pi$$
−0.314153 + 0.949372i $$0.601721\pi$$
$$602$$ −10.1047 −0.411836
$$603$$ 5.40312 0.220032
$$604$$ −4.70156 −0.191304
$$605$$ −28.3875 −1.15412
$$606$$ 3.40312 0.138242
$$607$$ 7.50781 0.304733 0.152366 0.988324i $$-0.451311\pi$$
0.152366 + 0.988324i $$0.451311\pi$$
$$608$$ −0.701562 −0.0284521
$$609$$ −2.70156 −0.109473
$$610$$ 3.50781 0.142027
$$611$$ −8.00000 −0.323645
$$612$$ −2.70156 −0.109204
$$613$$ 38.9109 1.57160 0.785799 0.618482i $$-0.212252\pi$$
0.785799 + 0.618482i $$0.212252\pi$$
$$614$$ −6.80625 −0.274678
$$615$$ −9.19375 −0.370728
$$616$$ −0.701562 −0.0282667
$$617$$ −30.9109 −1.24443 −0.622214 0.782847i $$-0.713767\pi$$
−0.622214 + 0.782847i $$0.713767\pi$$
$$618$$ 3.29844 0.132683
$$619$$ 7.29844 0.293349 0.146674 0.989185i $$-0.453143\pi$$
0.146674 + 0.989185i $$0.453143\pi$$
$$620$$ 0 0
$$621$$ −4.70156 −0.188667
$$622$$ 14.5969 0.585281
$$623$$ −8.80625 −0.352815
$$624$$ −1.00000 −0.0400320
$$625$$ −31.2094 −1.24837
$$626$$ 22.2094 0.887665
$$627$$ −0.492189 −0.0196562
$$628$$ 20.1047 0.802264
$$629$$ −28.9109 −1.15275
$$630$$ 2.70156 0.107633
$$631$$ −7.50781 −0.298881 −0.149441 0.988771i $$-0.547747\pi$$
−0.149441 + 0.988771i $$0.547747\pi$$
$$632$$ 9.40312 0.374036
$$633$$ 24.7016 0.981799
$$634$$ 7.40312 0.294016
$$635$$ −17.8219 −0.707239
$$636$$ 2.00000 0.0793052
$$637$$ 1.00000 0.0396214
$$638$$ −1.89531 −0.0750362
$$639$$ −8.00000 −0.316475
$$640$$ 2.70156 0.106789
$$641$$ −48.8062 −1.92773 −0.963865 0.266390i $$-0.914169\pi$$
−0.963865 + 0.266390i $$0.914169\pi$$
$$642$$ 5.40312 0.213244
$$643$$ 46.3141 1.82645 0.913224 0.407458i $$-0.133585\pi$$
0.913224 + 0.407458i $$0.133585\pi$$
$$644$$ 4.70156 0.185268
$$645$$ 27.2984 1.07487
$$646$$ 1.89531 0.0745701
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 10.3875 0.407745
$$650$$ 2.29844 0.0901522
$$651$$ 0 0
$$652$$ 5.40312 0.211603
$$653$$ −9.50781 −0.372069 −0.186035 0.982543i $$-0.559564\pi$$
−0.186035 + 0.982543i $$0.559564\pi$$
$$654$$ −9.29844 −0.363598
$$655$$ −19.7172 −0.770414
$$656$$ 3.40312 0.132870
$$657$$ −1.29844 −0.0506569
$$658$$ −8.00000 −0.311872
$$659$$ −35.0156 −1.36401 −0.682007 0.731345i $$-0.738893\pi$$
−0.682007 + 0.731345i $$0.738893\pi$$
$$660$$ 1.89531 0.0737750
$$661$$ −50.4187 −1.96106 −0.980531 0.196365i $$-0.937086\pi$$
−0.980531 + 0.196365i $$0.937086\pi$$
$$662$$ 32.2094 1.25185
$$663$$ 2.70156 0.104920
$$664$$ −13.4031 −0.520142
$$665$$ −1.89531 −0.0734971
$$666$$ 10.7016 0.414677
$$667$$ 12.7016 0.491806
$$668$$ 3.29844 0.127620
$$669$$ −9.40312 −0.363546
$$670$$ 14.5969 0.563927
$$671$$ −0.910935 −0.0351662
$$672$$ −1.00000 −0.0385758
$$673$$ 42.9109 1.65409 0.827047 0.562132i $$-0.190019\pi$$
0.827047 + 0.562132i $$0.190019\pi$$
$$674$$ −29.5078 −1.13660
$$675$$ −2.29844 −0.0884669
$$676$$ 1.00000 0.0384615
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ −4.80625 −0.184583
$$679$$ −8.80625 −0.337953
$$680$$ −7.29844 −0.279882
$$681$$ −21.4031 −0.820170
$$682$$ 0 0
$$683$$ 11.5078 0.440334 0.220167 0.975462i $$-0.429340\pi$$
0.220167 + 0.975462i $$0.429340\pi$$
$$684$$ −0.701562 −0.0268249
$$685$$ −50.5234 −1.93040
$$686$$ 1.00000 0.0381802
$$687$$ −6.00000 −0.228914
$$688$$ −10.1047 −0.385238
$$689$$ −2.00000 −0.0761939
$$690$$ −12.7016 −0.483540
$$691$$ 49.6125 1.88735 0.943674 0.330876i $$-0.107344\pi$$
0.943674 + 0.330876i $$0.107344\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ −0.701562 −0.0266501
$$694$$ 4.00000 0.151838
$$695$$ 18.3875 0.697478
$$696$$ −2.70156 −0.102402
$$697$$ −9.19375 −0.348238
$$698$$ 30.0000 1.13552
$$699$$ 18.2094 0.688742
$$700$$ 2.29844 0.0868728
$$701$$ 3.19375 0.120626 0.0603132 0.998180i $$-0.480790\pi$$
0.0603132 + 0.998180i $$0.480790\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ −7.50781 −0.283162
$$704$$ −0.701562 −0.0264411
$$705$$ 21.6125 0.813974
$$706$$ 20.8062 0.783053
$$707$$ −3.40312 −0.127988
$$708$$ 14.8062 0.556453
$$709$$ 51.6125 1.93835 0.969174 0.246377i $$-0.0792402\pi$$
0.969174 + 0.246377i $$0.0792402\pi$$
$$710$$ −21.6125 −0.811103
$$711$$ 9.40312 0.352645
$$712$$ −8.80625 −0.330028
$$713$$ 0 0
$$714$$ 2.70156 0.101103
$$715$$ −1.89531 −0.0708807
$$716$$ 14.8062 0.553335
$$717$$ −16.0000 −0.597531
$$718$$ 26.8062 1.00040
$$719$$ −31.0156 −1.15669 −0.578344 0.815793i $$-0.696301\pi$$
−0.578344 + 0.815793i $$0.696301\pi$$
$$720$$ 2.70156 0.100681
$$721$$ −3.29844 −0.122840
$$722$$ −18.5078 −0.688789
$$723$$ 24.8062 0.922554
$$724$$ 8.80625 0.327282
$$725$$ 6.20937 0.230610
$$726$$ 10.5078 0.389981
$$727$$ −22.1047 −0.819817 −0.409909 0.912127i $$-0.634439\pi$$
−0.409909 + 0.912127i $$0.634439\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 1.00000 0.0370370
$$730$$ −3.50781 −0.129830
$$731$$ 27.2984 1.00967
$$732$$ −1.29844 −0.0479916
$$733$$ 20.5969 0.760763 0.380381 0.924830i $$-0.375793\pi$$
0.380381 + 0.924830i $$0.375793\pi$$
$$734$$ −29.6125 −1.09302
$$735$$ −2.70156 −0.0996486
$$736$$ 4.70156 0.173302
$$737$$ −3.79063 −0.139630
$$738$$ 3.40312 0.125271
$$739$$ 40.2094 1.47913 0.739563 0.673088i $$-0.235032\pi$$
0.739563 + 0.673088i $$0.235032\pi$$
$$740$$ 28.9109 1.06279
$$741$$ 0.701562 0.0257725
$$742$$ −2.00000 −0.0734223
$$743$$ 23.0156 0.844361 0.422181 0.906512i $$-0.361265\pi$$
0.422181 + 0.906512i $$0.361265\pi$$
$$744$$ 0 0
$$745$$ 41.6125 1.52456
$$746$$ −19.4031 −0.710399
$$747$$ −13.4031 −0.490395
$$748$$ 1.89531 0.0692995
$$749$$ −5.40312 −0.197426
$$750$$ 7.29844 0.266501
$$751$$ 34.8062 1.27010 0.635049 0.772472i $$-0.280980\pi$$
0.635049 + 0.772472i $$0.280980\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −3.50781 −0.127832
$$754$$ 2.70156 0.0983851
$$755$$ −12.7016 −0.462257
$$756$$ −1.00000 −0.0363696
$$757$$ 30.4187 1.10559 0.552794 0.833318i $$-0.313562\pi$$
0.552794 + 0.833318i $$0.313562\pi$$
$$758$$ −17.6125 −0.639715
$$759$$ 3.29844 0.119726
$$760$$ −1.89531 −0.0687503
$$761$$ 32.5969 1.18164 0.590818 0.806805i $$-0.298805\pi$$
0.590818 + 0.806805i $$0.298805\pi$$
$$762$$ 6.59688 0.238980
$$763$$ 9.29844 0.336626
$$764$$ 12.7016 0.459526
$$765$$ −7.29844 −0.263876
$$766$$ 16.9109 0.611017
$$767$$ −14.8062 −0.534623
$$768$$ −1.00000 −0.0360844
$$769$$ 50.9109 1.83590 0.917948 0.396702i $$-0.129845\pi$$
0.917948 + 0.396702i $$0.129845\pi$$
$$770$$ −1.89531 −0.0683024
$$771$$ 6.00000 0.216085
$$772$$ 11.4031 0.410407
$$773$$ −28.3141 −1.01839 −0.509193 0.860652i $$-0.670056\pi$$
−0.509193 + 0.860652i $$0.670056\pi$$
$$774$$ −10.1047 −0.363205
$$775$$ 0 0
$$776$$ −8.80625 −0.316126
$$777$$ −10.7016 −0.383916
$$778$$ 14.0000 0.501924
$$779$$ −2.38750 −0.0855412
$$780$$ −2.70156 −0.0967314
$$781$$ 5.61250 0.200831
$$782$$ −12.7016 −0.454207
$$783$$ −2.70156 −0.0965460
$$784$$ 1.00000 0.0357143
$$785$$ 54.3141 1.93855
$$786$$ 7.29844 0.260327
$$787$$ 20.9109 0.745394 0.372697 0.927953i $$-0.378433\pi$$
0.372697 + 0.927953i $$0.378433\pi$$
$$788$$ −3.40312 −0.121231
$$789$$ 26.8062 0.954328
$$790$$ 25.4031 0.903803
$$791$$ 4.80625 0.170891
$$792$$ −0.701562 −0.0249289
$$793$$ 1.29844 0.0461089
$$794$$ 0.806248 0.0286127
$$795$$ 5.40312 0.191629
$$796$$ 22.1047 0.783480
$$797$$ −40.5969 −1.43802 −0.719008 0.695002i $$-0.755403\pi$$
−0.719008 + 0.695002i $$0.755403\pi$$
$$798$$ 0.701562 0.0248350
$$799$$ 21.6125 0.764595
$$800$$ 2.29844 0.0812621
$$801$$ −8.80625 −0.311153
$$802$$ 4.80625 0.169715
$$803$$ 0.910935 0.0321462
$$804$$ −5.40312 −0.190553
$$805$$ 12.7016 0.447671
$$806$$ 0 0
$$807$$ 4.80625 0.169188
$$808$$ −3.40312 −0.119721
$$809$$ −11.6125 −0.408274 −0.204137 0.978942i $$-0.565439\pi$$
−0.204137 + 0.978942i $$0.565439\pi$$
$$810$$ 2.70156 0.0949232
$$811$$ −21.8953 −0.768848 −0.384424 0.923157i $$-0.625600\pi$$
−0.384424 + 0.923157i $$0.625600\pi$$
$$812$$ 2.70156 0.0948062
$$813$$ −12.2094 −0.428201
$$814$$ −7.50781 −0.263149
$$815$$ 14.5969 0.511306
$$816$$ 2.70156 0.0945736
$$817$$ 7.08907 0.248015
$$818$$ 5.29844 0.185256
$$819$$ 1.00000 0.0349428
$$820$$ 9.19375 0.321060
$$821$$ 28.5969 0.998038 0.499019 0.866591i $$-0.333694\pi$$
0.499019 + 0.866591i $$0.333694\pi$$
$$822$$ 18.7016 0.652292
$$823$$ 5.19375 0.181043 0.0905214 0.995895i $$-0.471147\pi$$
0.0905214 + 0.995895i $$0.471147\pi$$
$$824$$ −3.29844 −0.114907
$$825$$ 1.61250 0.0561399
$$826$$ −14.8062 −0.515175
$$827$$ 52.9109 1.83989 0.919947 0.392043i $$-0.128232\pi$$
0.919947 + 0.392043i $$0.128232\pi$$
$$828$$ 4.70156 0.163391
$$829$$ −36.3141 −1.26124 −0.630620 0.776092i $$-0.717199\pi$$
−0.630620 + 0.776092i $$0.717199\pi$$
$$830$$ −36.2094 −1.25685
$$831$$ −27.6125 −0.957867
$$832$$ 1.00000 0.0346688
$$833$$ −2.70156 −0.0936036
$$834$$ −6.80625 −0.235681
$$835$$ 8.91093 0.308376
$$836$$ 0.492189 0.0170227
$$837$$ 0 0
$$838$$ 34.1047 1.17813
$$839$$ 34.8062 1.20165 0.600823 0.799382i $$-0.294840\pi$$
0.600823 + 0.799382i $$0.294840\pi$$
$$840$$ −2.70156 −0.0932127
$$841$$ −21.7016 −0.748330
$$842$$ 19.6125 0.675891
$$843$$ −12.8062 −0.441071
$$844$$ −24.7016 −0.850263
$$845$$ 2.70156 0.0929366
$$846$$ −8.00000 −0.275046
$$847$$ −10.5078 −0.361053
$$848$$ −2.00000 −0.0686803
$$849$$ −4.00000 −0.137280
$$850$$ −6.20937 −0.212980
$$851$$ 50.3141 1.72474
$$852$$ 8.00000 0.274075
$$853$$ −22.2094 −0.760434 −0.380217 0.924897i $$-0.624151\pi$$
−0.380217 + 0.924897i $$0.624151\pi$$
$$854$$ 1.29844 0.0444316
$$855$$ −1.89531 −0.0648184
$$856$$ −5.40312 −0.184675
$$857$$ −16.8062 −0.574091 −0.287045 0.957917i $$-0.592673\pi$$
−0.287045 + 0.957917i $$0.592673\pi$$
$$858$$ 0.701562 0.0239509
$$859$$ 38.8062 1.32405 0.662026 0.749481i $$-0.269697\pi$$
0.662026 + 0.749481i $$0.269697\pi$$
$$860$$ −27.2984 −0.930869
$$861$$ −3.40312 −0.115978
$$862$$ −12.2094 −0.415853
$$863$$ −22.5969 −0.769207 −0.384603 0.923082i $$-0.625662\pi$$
−0.384603 + 0.923082i $$0.625662\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −48.6281 −1.65341
$$866$$ 14.2094 0.482854
$$867$$ 9.70156 0.329482
$$868$$ 0 0
$$869$$ −6.59688 −0.223784
$$870$$ −7.29844 −0.247440
$$871$$ 5.40312 0.183078
$$872$$ 9.29844 0.314885
$$873$$ −8.80625 −0.298046
$$874$$ −3.29844 −0.111571
$$875$$ −7.29844 −0.246732
$$876$$ 1.29844 0.0438702
$$877$$ 0.387503 0.0130850 0.00654252 0.999979i $$-0.497917\pi$$
0.00654252 + 0.999979i $$0.497917\pi$$
$$878$$ −0.492189 −0.0166106
$$879$$ 12.8062 0.431944
$$880$$ −1.89531 −0.0638910
$$881$$ −8.31406 −0.280108 −0.140054 0.990144i $$-0.544728\pi$$
−0.140054 + 0.990144i $$0.544728\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 13.8953 0.467615 0.233807 0.972283i $$-0.424882\pi$$
0.233807 + 0.972283i $$0.424882\pi$$
$$884$$ −2.70156 −0.0908634
$$885$$ 40.0000 1.34459
$$886$$ 0.209373 0.00703401
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ −10.7016 −0.359121
$$889$$ −6.59688 −0.221252
$$890$$ −23.7906 −0.797464
$$891$$ −0.701562 −0.0235032
$$892$$ 9.40312 0.314840
$$893$$ 5.61250 0.187815
$$894$$ −15.4031 −0.515158
$$895$$ 40.0000 1.33705
$$896$$ 1.00000 0.0334077
$$897$$ −4.70156 −0.156981
$$898$$ −7.89531 −0.263470
$$899$$ 0 0
$$900$$ 2.29844 0.0766146
$$901$$ 5.40312 0.180004
$$902$$ −2.38750 −0.0794952
$$903$$ 10.1047 0.336263
$$904$$ 4.80625 0.159853
$$905$$ 23.7906 0.790827
$$906$$ 4.70156 0.156199
$$907$$ −25.6125 −0.850449 −0.425225 0.905088i $$-0.639805\pi$$
−0.425225 + 0.905088i $$0.639805\pi$$
$$908$$ 21.4031 0.710288
$$909$$ −3.40312 −0.112875
$$910$$ 2.70156 0.0895559
$$911$$ 40.9109 1.35544 0.677720 0.735320i $$-0.262968\pi$$
0.677720 + 0.735320i $$0.262968\pi$$
$$912$$ 0.701562 0.0232310
$$913$$ 9.40312 0.311198
$$914$$ 0.596876 0.0197429
$$915$$ −3.50781 −0.115965
$$916$$ 6.00000 0.198246
$$917$$ −7.29844 −0.241016
$$918$$ 2.70156 0.0891648
$$919$$ −14.5969 −0.481507 −0.240753 0.970586i $$-0.577394\pi$$
−0.240753 + 0.970586i $$0.577394\pi$$
$$920$$ 12.7016 0.418758
$$921$$ 6.80625 0.224274
$$922$$ −20.3141 −0.669008
$$923$$ −8.00000 −0.263323
$$924$$ 0.701562 0.0230797
$$925$$ 24.5969 0.808740
$$926$$ −34.3141 −1.12763
$$927$$ −3.29844 −0.108335
$$928$$ 2.70156 0.0886831
$$929$$ 33.0156 1.08321 0.541604 0.840634i $$-0.317817\pi$$
0.541604 + 0.840634i $$0.317817\pi$$
$$930$$ 0 0
$$931$$ −0.701562 −0.0229928
$$932$$ −18.2094 −0.596468
$$933$$ −14.5969 −0.477880
$$934$$ −4.49219 −0.146989
$$935$$ 5.12031 0.167452
$$936$$ 1.00000 0.0326860
$$937$$ 28.8062 0.941059 0.470530 0.882384i $$-0.344063\pi$$
0.470530 + 0.882384i $$0.344063\pi$$
$$938$$ 5.40312 0.176418
$$939$$ −22.2094 −0.724775
$$940$$ −21.6125 −0.704922
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ −20.1047 −0.655046
$$943$$ 16.0000 0.521032
$$944$$ −14.8062 −0.481902
$$945$$ −2.70156 −0.0878818
$$946$$ 7.08907 0.230485
$$947$$ 4.49219 0.145977 0.0729883 0.997333i $$-0.476746\pi$$
0.0729883 + 0.997333i $$0.476746\pi$$
$$948$$ −9.40312 −0.305399
$$949$$ −1.29844 −0.0421491
$$950$$ −1.61250 −0.0523163
$$951$$ −7.40312 −0.240063
$$952$$ −2.70156 −0.0875581
$$953$$ −39.8219 −1.28996 −0.644978 0.764201i $$-0.723134\pi$$
−0.644978 + 0.764201i $$0.723134\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 34.3141 1.11038
$$956$$ 16.0000 0.517477
$$957$$ 1.89531 0.0612668
$$958$$ 7.50781 0.242566
$$959$$ −18.7016 −0.603905
$$960$$ −2.70156 −0.0871925
$$961$$ −31.0000 −1.00000
$$962$$ 10.7016 0.345032
$$963$$ −5.40312 −0.174113
$$964$$ −24.8062 −0.798955
$$965$$ 30.8062 0.991688
$$966$$ −4.70156 −0.151270
$$967$$ −51.7172 −1.66311 −0.831556 0.555441i $$-0.812550\pi$$
−0.831556 + 0.555441i $$0.812550\pi$$
$$968$$ −10.5078 −0.337734
$$969$$ −1.89531 −0.0608862
$$970$$ −23.7906 −0.763871
$$971$$ 54.8062 1.75882 0.879408 0.476069i $$-0.157939\pi$$
0.879408 + 0.476069i $$0.157939\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 6.80625 0.218198
$$974$$ −8.00000 −0.256337
$$975$$ −2.29844 −0.0736089
$$976$$ 1.29844 0.0415620
$$977$$ −41.7172 −1.33465 −0.667325 0.744766i $$-0.732561\pi$$
−0.667325 + 0.744766i $$0.732561\pi$$
$$978$$ −5.40312 −0.172773
$$979$$ 6.17813 0.197454
$$980$$ 2.70156 0.0862982
$$981$$ 9.29844 0.296876
$$982$$ 26.5969 0.848740
$$983$$ −38.1047 −1.21535 −0.607675 0.794186i $$-0.707898\pi$$
−0.607675 + 0.794186i $$0.707898\pi$$
$$984$$ −3.40312 −0.108488
$$985$$ −9.19375 −0.292937
$$986$$ −7.29844 −0.232430
$$987$$ 8.00000 0.254643
$$988$$ −0.701562 −0.0223197
$$989$$ −47.5078 −1.51066
$$990$$ −1.89531 −0.0602370
$$991$$ −38.5969 −1.22607 −0.613035 0.790056i $$-0.710052\pi$$
−0.613035 + 0.790056i $$0.710052\pi$$
$$992$$ 0 0
$$993$$ −32.2094 −1.02213
$$994$$ −8.00000 −0.253745
$$995$$ 59.7172 1.89316
$$996$$ 13.4031 0.424694
$$997$$ −12.8062 −0.405578 −0.202789 0.979222i $$-0.565001\pi$$
−0.202789 + 0.979222i $$0.565001\pi$$
$$998$$ −1.19375 −0.0377875
$$999$$ −10.7016 −0.338582
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 546.2.a.i.1.2 2
3.2 odd 2 1638.2.a.w.1.1 2
4.3 odd 2 4368.2.a.bg.1.2 2
7.6 odd 2 3822.2.a.bt.1.1 2
13.12 even 2 7098.2.a.bh.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.i.1.2 2 1.1 even 1 trivial
1638.2.a.w.1.1 2 3.2 odd 2
3822.2.a.bt.1.1 2 7.6 odd 2
4368.2.a.bg.1.2 2 4.3 odd 2
7098.2.a.bh.1.1 2 13.12 even 2