Properties

 Label 370.2.a.e.1.2 Level $370$ Weight $2$ Character 370.1 Self dual yes Analytic conductor $2.954$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$2.95446487479$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 370.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} -4.73205 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} -4.73205 q^{7} -1.00000 q^{8} -2.46410 q^{9} -1.00000 q^{10} -5.46410 q^{11} +0.732051 q^{12} -5.46410 q^{13} +4.73205 q^{14} +0.732051 q^{15} +1.00000 q^{16} +5.46410 q^{17} +2.46410 q^{18} +6.19615 q^{19} +1.00000 q^{20} -3.46410 q^{21} +5.46410 q^{22} -8.00000 q^{23} -0.732051 q^{24} +1.00000 q^{25} +5.46410 q^{26} -4.00000 q^{27} -4.73205 q^{28} +4.92820 q^{29} -0.732051 q^{30} +0.732051 q^{31} -1.00000 q^{32} -4.00000 q^{33} -5.46410 q^{34} -4.73205 q^{35} -2.46410 q^{36} +1.00000 q^{37} -6.19615 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +3.46410 q^{42} +6.92820 q^{43} -5.46410 q^{44} -2.46410 q^{45} +8.00000 q^{46} -4.73205 q^{47} +0.732051 q^{48} +15.3923 q^{49} -1.00000 q^{50} +4.00000 q^{51} -5.46410 q^{52} -6.00000 q^{53} +4.00000 q^{54} -5.46410 q^{55} +4.73205 q^{56} +4.53590 q^{57} -4.92820 q^{58} -10.1962 q^{59} +0.732051 q^{60} -4.92820 q^{61} -0.732051 q^{62} +11.6603 q^{63} +1.00000 q^{64} -5.46410 q^{65} +4.00000 q^{66} -3.66025 q^{67} +5.46410 q^{68} -5.85641 q^{69} +4.73205 q^{70} +2.92820 q^{71} +2.46410 q^{72} -0.928203 q^{73} -1.00000 q^{74} +0.732051 q^{75} +6.19615 q^{76} +25.8564 q^{77} +4.00000 q^{78} +8.73205 q^{79} +1.00000 q^{80} +4.46410 q^{81} +2.00000 q^{82} -8.73205 q^{83} -3.46410 q^{84} +5.46410 q^{85} -6.92820 q^{86} +3.60770 q^{87} +5.46410 q^{88} -2.00000 q^{89} +2.46410 q^{90} +25.8564 q^{91} -8.00000 q^{92} +0.535898 q^{93} +4.73205 q^{94} +6.19615 q^{95} -0.732051 q^{96} -2.00000 q^{97} -15.3923 q^{98} +13.4641 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - 4 q^{13} + 6 q^{14} - 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{20} + 4 q^{22} - 16 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} - 8 q^{27} - 6 q^{28} - 4 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 8 q^{33} - 4 q^{34} - 6 q^{35} + 2 q^{36} + 2 q^{37} - 2 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 4 q^{44} + 2 q^{45} + 16 q^{46} - 6 q^{47} - 2 q^{48} + 10 q^{49} - 2 q^{50} + 8 q^{51} - 4 q^{52} - 12 q^{53} + 8 q^{54} - 4 q^{55} + 6 q^{56} + 16 q^{57} + 4 q^{58} - 10 q^{59} - 2 q^{60} + 4 q^{61} + 2 q^{62} + 6 q^{63} + 2 q^{64} - 4 q^{65} + 8 q^{66} + 10 q^{67} + 4 q^{68} + 16 q^{69} + 6 q^{70} - 8 q^{71} - 2 q^{72} + 12 q^{73} - 2 q^{74} - 2 q^{75} + 2 q^{76} + 24 q^{77} + 8 q^{78} + 14 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 14 q^{83} + 4 q^{85} + 28 q^{87} + 4 q^{88} - 4 q^{89} - 2 q^{90} + 24 q^{91} - 16 q^{92} + 8 q^{93} + 6 q^{94} + 2 q^{95} + 2 q^{96} - 4 q^{97} - 10 q^{98} + 20 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 - 4 * q^13 + 6 * q^14 - 2 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 2 * q^19 + 2 * q^20 + 4 * q^22 - 16 * q^23 + 2 * q^24 + 2 * q^25 + 4 * q^26 - 8 * q^27 - 6 * q^28 - 4 * q^29 + 2 * q^30 - 2 * q^31 - 2 * q^32 - 8 * q^33 - 4 * q^34 - 6 * q^35 + 2 * q^36 + 2 * q^37 - 2 * q^38 - 8 * q^39 - 2 * q^40 - 4 * q^41 - 4 * q^44 + 2 * q^45 + 16 * q^46 - 6 * q^47 - 2 * q^48 + 10 * q^49 - 2 * q^50 + 8 * q^51 - 4 * q^52 - 12 * q^53 + 8 * q^54 - 4 * q^55 + 6 * q^56 + 16 * q^57 + 4 * q^58 - 10 * q^59 - 2 * q^60 + 4 * q^61 + 2 * q^62 + 6 * q^63 + 2 * q^64 - 4 * q^65 + 8 * q^66 + 10 * q^67 + 4 * q^68 + 16 * q^69 + 6 * q^70 - 8 * q^71 - 2 * q^72 + 12 * q^73 - 2 * q^74 - 2 * q^75 + 2 * q^76 + 24 * q^77 + 8 * q^78 + 14 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 14 * q^83 + 4 * q^85 + 28 * q^87 + 4 * q^88 - 4 * q^89 - 2 * q^90 + 24 * q^91 - 16 * q^92 + 8 * q^93 + 6 * q^94 + 2 * q^95 + 2 * q^96 - 4 * q^97 - 10 * q^98 + 20 * 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
$$3$$ 0.732051 0.422650 0.211325 0.977416i $$-0.432222\pi$$
0.211325 + 0.977416i $$0.432222\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −0.732051 −0.298858
$$7$$ −4.73205 −1.78855 −0.894274 0.447521i $$-0.852307\pi$$
−0.894274 + 0.447521i $$0.852307\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.46410 −0.821367
$$10$$ −1.00000 −0.316228
$$11$$ −5.46410 −1.64749 −0.823744 0.566961i $$-0.808119\pi$$
−0.823744 + 0.566961i $$0.808119\pi$$
$$12$$ 0.732051 0.211325
$$13$$ −5.46410 −1.51547 −0.757735 0.652563i $$-0.773694\pi$$
−0.757735 + 0.652563i $$0.773694\pi$$
$$14$$ 4.73205 1.26469
$$15$$ 0.732051 0.189015
$$16$$ 1.00000 0.250000
$$17$$ 5.46410 1.32524 0.662620 0.748956i $$-0.269445\pi$$
0.662620 + 0.748956i $$0.269445\pi$$
$$18$$ 2.46410 0.580794
$$19$$ 6.19615 1.42149 0.710747 0.703447i $$-0.248357\pi$$
0.710747 + 0.703447i $$0.248357\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −3.46410 −0.755929
$$22$$ 5.46410 1.16495
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ −0.732051 −0.149429
$$25$$ 1.00000 0.200000
$$26$$ 5.46410 1.07160
$$27$$ −4.00000 −0.769800
$$28$$ −4.73205 −0.894274
$$29$$ 4.92820 0.915144 0.457572 0.889172i $$-0.348719\pi$$
0.457572 + 0.889172i $$0.348719\pi$$
$$30$$ −0.732051 −0.133654
$$31$$ 0.732051 0.131480 0.0657401 0.997837i $$-0.479059\pi$$
0.0657401 + 0.997837i $$0.479059\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.00000 −0.696311
$$34$$ −5.46410 −0.937086
$$35$$ −4.73205 −0.799863
$$36$$ −2.46410 −0.410684
$$37$$ 1.00000 0.164399
$$38$$ −6.19615 −1.00515
$$39$$ −4.00000 −0.640513
$$40$$ −1.00000 −0.158114
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 3.46410 0.534522
$$43$$ 6.92820 1.05654 0.528271 0.849076i $$-0.322841\pi$$
0.528271 + 0.849076i $$0.322841\pi$$
$$44$$ −5.46410 −0.823744
$$45$$ −2.46410 −0.367327
$$46$$ 8.00000 1.17954
$$47$$ −4.73205 −0.690241 −0.345120 0.938558i $$-0.612162\pi$$
−0.345120 + 0.938558i $$0.612162\pi$$
$$48$$ 0.732051 0.105662
$$49$$ 15.3923 2.19890
$$50$$ −1.00000 −0.141421
$$51$$ 4.00000 0.560112
$$52$$ −5.46410 −0.757735
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 4.00000 0.544331
$$55$$ −5.46410 −0.736779
$$56$$ 4.73205 0.632347
$$57$$ 4.53590 0.600794
$$58$$ −4.92820 −0.647105
$$59$$ −10.1962 −1.32743 −0.663713 0.747987i $$-0.731020\pi$$
−0.663713 + 0.747987i $$0.731020\pi$$
$$60$$ 0.732051 0.0945074
$$61$$ −4.92820 −0.630992 −0.315496 0.948927i $$-0.602171\pi$$
−0.315496 + 0.948927i $$0.602171\pi$$
$$62$$ −0.732051 −0.0929705
$$63$$ 11.6603 1.46905
$$64$$ 1.00000 0.125000
$$65$$ −5.46410 −0.677738
$$66$$ 4.00000 0.492366
$$67$$ −3.66025 −0.447171 −0.223586 0.974684i $$-0.571776\pi$$
−0.223586 + 0.974684i $$0.571776\pi$$
$$68$$ 5.46410 0.662620
$$69$$ −5.85641 −0.705028
$$70$$ 4.73205 0.565588
$$71$$ 2.92820 0.347514 0.173757 0.984789i $$-0.444409\pi$$
0.173757 + 0.984789i $$0.444409\pi$$
$$72$$ 2.46410 0.290397
$$73$$ −0.928203 −0.108638 −0.0543190 0.998524i $$-0.517299\pi$$
−0.0543190 + 0.998524i $$0.517299\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0.732051 0.0845299
$$76$$ 6.19615 0.710747
$$77$$ 25.8564 2.94661
$$78$$ 4.00000 0.452911
$$79$$ 8.73205 0.982432 0.491216 0.871038i $$-0.336552\pi$$
0.491216 + 0.871038i $$0.336552\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 4.46410 0.496011
$$82$$ 2.00000 0.220863
$$83$$ −8.73205 −0.958467 −0.479234 0.877687i $$-0.659085\pi$$
−0.479234 + 0.877687i $$0.659085\pi$$
$$84$$ −3.46410 −0.377964
$$85$$ 5.46410 0.592665
$$86$$ −6.92820 −0.747087
$$87$$ 3.60770 0.386786
$$88$$ 5.46410 0.582475
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 2.46410 0.259739
$$91$$ 25.8564 2.71049
$$92$$ −8.00000 −0.834058
$$93$$ 0.535898 0.0555701
$$94$$ 4.73205 0.488074
$$95$$ 6.19615 0.635712
$$96$$ −0.732051 −0.0747146
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −15.3923 −1.55486
$$99$$ 13.4641 1.35319
$$100$$ 1.00000 0.100000
$$101$$ −9.46410 −0.941713 −0.470857 0.882210i $$-0.656055\pi$$
−0.470857 + 0.882210i $$0.656055\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ −6.53590 −0.644001 −0.322001 0.946739i $$-0.604355\pi$$
−0.322001 + 0.946739i $$0.604355\pi$$
$$104$$ 5.46410 0.535799
$$105$$ −3.46410 −0.338062
$$106$$ 6.00000 0.582772
$$107$$ 3.26795 0.315925 0.157962 0.987445i $$-0.449508\pi$$
0.157962 + 0.987445i $$0.449508\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 5.46410 0.520982
$$111$$ 0.732051 0.0694832
$$112$$ −4.73205 −0.447137
$$113$$ −10.5359 −0.991134 −0.495567 0.868570i $$-0.665040\pi$$
−0.495567 + 0.868570i $$0.665040\pi$$
$$114$$ −4.53590 −0.424826
$$115$$ −8.00000 −0.746004
$$116$$ 4.92820 0.457572
$$117$$ 13.4641 1.24476
$$118$$ 10.1962 0.938632
$$119$$ −25.8564 −2.37025
$$120$$ −0.732051 −0.0668268
$$121$$ 18.8564 1.71422
$$122$$ 4.92820 0.446179
$$123$$ −1.46410 −0.132014
$$124$$ 0.732051 0.0657401
$$125$$ 1.00000 0.0894427
$$126$$ −11.6603 −1.03878
$$127$$ 3.66025 0.324795 0.162398 0.986725i $$-0.448077\pi$$
0.162398 + 0.986725i $$0.448077\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 5.07180 0.446547
$$130$$ 5.46410 0.479233
$$131$$ −18.5885 −1.62408 −0.812041 0.583601i $$-0.801643\pi$$
−0.812041 + 0.583601i $$0.801643\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ −29.3205 −2.54241
$$134$$ 3.66025 0.316198
$$135$$ −4.00000 −0.344265
$$136$$ −5.46410 −0.468543
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 5.85641 0.498530
$$139$$ 6.92820 0.587643 0.293821 0.955860i $$-0.405073\pi$$
0.293821 + 0.955860i $$0.405073\pi$$
$$140$$ −4.73205 −0.399931
$$141$$ −3.46410 −0.291730
$$142$$ −2.92820 −0.245729
$$143$$ 29.8564 2.49672
$$144$$ −2.46410 −0.205342
$$145$$ 4.92820 0.409265
$$146$$ 0.928203 0.0768186
$$147$$ 11.2679 0.929365
$$148$$ 1.00000 0.0821995
$$149$$ 4.39230 0.359832 0.179916 0.983682i $$-0.442417\pi$$
0.179916 + 0.983682i $$0.442417\pi$$
$$150$$ −0.732051 −0.0597717
$$151$$ −12.3923 −1.00847 −0.504236 0.863566i $$-0.668226\pi$$
−0.504236 + 0.863566i $$0.668226\pi$$
$$152$$ −6.19615 −0.502574
$$153$$ −13.4641 −1.08851
$$154$$ −25.8564 −2.08357
$$155$$ 0.732051 0.0587997
$$156$$ −4.00000 −0.320256
$$157$$ 3.07180 0.245156 0.122578 0.992459i $$-0.460884\pi$$
0.122578 + 0.992459i $$0.460884\pi$$
$$158$$ −8.73205 −0.694685
$$159$$ −4.39230 −0.348332
$$160$$ −1.00000 −0.0790569
$$161$$ 37.8564 2.98350
$$162$$ −4.46410 −0.350733
$$163$$ −11.3205 −0.886691 −0.443345 0.896351i $$-0.646208\pi$$
−0.443345 + 0.896351i $$0.646208\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ −4.00000 −0.311400
$$166$$ 8.73205 0.677739
$$167$$ −1.46410 −0.113296 −0.0566478 0.998394i $$-0.518041\pi$$
−0.0566478 + 0.998394i $$0.518041\pi$$
$$168$$ 3.46410 0.267261
$$169$$ 16.8564 1.29665
$$170$$ −5.46410 −0.419077
$$171$$ −15.2679 −1.16757
$$172$$ 6.92820 0.528271
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ −3.60770 −0.273499
$$175$$ −4.73205 −0.357709
$$176$$ −5.46410 −0.411872
$$177$$ −7.46410 −0.561036
$$178$$ 2.00000 0.149906
$$179$$ −0.339746 −0.0253938 −0.0126969 0.999919i $$-0.504042\pi$$
−0.0126969 + 0.999919i $$0.504042\pi$$
$$180$$ −2.46410 −0.183663
$$181$$ 5.46410 0.406143 0.203072 0.979164i $$-0.434908\pi$$
0.203072 + 0.979164i $$0.434908\pi$$
$$182$$ −25.8564 −1.91660
$$183$$ −3.60770 −0.266688
$$184$$ 8.00000 0.589768
$$185$$ 1.00000 0.0735215
$$186$$ −0.535898 −0.0392940
$$187$$ −29.8564 −2.18332
$$188$$ −4.73205 −0.345120
$$189$$ 18.9282 1.37682
$$190$$ −6.19615 −0.449516
$$191$$ −8.73205 −0.631829 −0.315915 0.948788i $$-0.602311\pi$$
−0.315915 + 0.948788i $$0.602311\pi$$
$$192$$ 0.732051 0.0528312
$$193$$ 15.8564 1.14137 0.570685 0.821169i $$-0.306678\pi$$
0.570685 + 0.821169i $$0.306678\pi$$
$$194$$ 2.00000 0.143592
$$195$$ −4.00000 −0.286446
$$196$$ 15.3923 1.09945
$$197$$ −22.7846 −1.62334 −0.811668 0.584119i $$-0.801440\pi$$
−0.811668 + 0.584119i $$0.801440\pi$$
$$198$$ −13.4641 −0.956852
$$199$$ −12.0526 −0.854383 −0.427192 0.904161i $$-0.640497\pi$$
−0.427192 + 0.904161i $$0.640497\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −2.67949 −0.188997
$$202$$ 9.46410 0.665892
$$203$$ −23.3205 −1.63678
$$204$$ 4.00000 0.280056
$$205$$ −2.00000 −0.139686
$$206$$ 6.53590 0.455378
$$207$$ 19.7128 1.37014
$$208$$ −5.46410 −0.378867
$$209$$ −33.8564 −2.34190
$$210$$ 3.46410 0.239046
$$211$$ −17.8564 −1.22929 −0.614643 0.788806i $$-0.710700\pi$$
−0.614643 + 0.788806i $$0.710700\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 2.14359 0.146877
$$214$$ −3.26795 −0.223392
$$215$$ 6.92820 0.472500
$$216$$ 4.00000 0.272166
$$217$$ −3.46410 −0.235159
$$218$$ −2.00000 −0.135457
$$219$$ −0.679492 −0.0459158
$$220$$ −5.46410 −0.368390
$$221$$ −29.8564 −2.00836
$$222$$ −0.732051 −0.0491320
$$223$$ 16.0526 1.07496 0.537479 0.843277i $$-0.319377\pi$$
0.537479 + 0.843277i $$0.319377\pi$$
$$224$$ 4.73205 0.316173
$$225$$ −2.46410 −0.164273
$$226$$ 10.5359 0.700838
$$227$$ 24.3923 1.61897 0.809487 0.587138i $$-0.199745\pi$$
0.809487 + 0.587138i $$0.199745\pi$$
$$228$$ 4.53590 0.300397
$$229$$ −11.8564 −0.783493 −0.391747 0.920073i $$-0.628129\pi$$
−0.391747 + 0.920073i $$0.628129\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 18.9282 1.24538
$$232$$ −4.92820 −0.323552
$$233$$ 28.9282 1.89515 0.947575 0.319534i $$-0.103526\pi$$
0.947575 + 0.319534i $$0.103526\pi$$
$$234$$ −13.4641 −0.880176
$$235$$ −4.73205 −0.308685
$$236$$ −10.1962 −0.663713
$$237$$ 6.39230 0.415225
$$238$$ 25.8564 1.67602
$$239$$ −20.7321 −1.34104 −0.670522 0.741889i $$-0.733930\pi$$
−0.670522 + 0.741889i $$0.733930\pi$$
$$240$$ 0.732051 0.0472537
$$241$$ 4.92820 0.317453 0.158727 0.987323i $$-0.449261\pi$$
0.158727 + 0.987323i $$0.449261\pi$$
$$242$$ −18.8564 −1.21214
$$243$$ 15.2679 0.979439
$$244$$ −4.92820 −0.315496
$$245$$ 15.3923 0.983378
$$246$$ 1.46410 0.0933477
$$247$$ −33.8564 −2.15423
$$248$$ −0.732051 −0.0464853
$$249$$ −6.39230 −0.405096
$$250$$ −1.00000 −0.0632456
$$251$$ 19.2679 1.21618 0.608091 0.793867i $$-0.291936\pi$$
0.608091 + 0.793867i $$0.291936\pi$$
$$252$$ 11.6603 0.734527
$$253$$ 43.7128 2.74820
$$254$$ −3.66025 −0.229665
$$255$$ 4.00000 0.250490
$$256$$ 1.00000 0.0625000
$$257$$ −18.5359 −1.15624 −0.578119 0.815953i $$-0.696213\pi$$
−0.578119 + 0.815953i $$0.696213\pi$$
$$258$$ −5.07180 −0.315756
$$259$$ −4.73205 −0.294035
$$260$$ −5.46410 −0.338869
$$261$$ −12.1436 −0.751670
$$262$$ 18.5885 1.14840
$$263$$ −8.05256 −0.496542 −0.248271 0.968691i $$-0.579862\pi$$
−0.248271 + 0.968691i $$0.579862\pi$$
$$264$$ 4.00000 0.246183
$$265$$ −6.00000 −0.368577
$$266$$ 29.3205 1.79776
$$267$$ −1.46410 −0.0896016
$$268$$ −3.66025 −0.223586
$$269$$ 20.3923 1.24334 0.621670 0.783279i $$-0.286454\pi$$
0.621670 + 0.783279i $$0.286454\pi$$
$$270$$ 4.00000 0.243432
$$271$$ −24.7846 −1.50556 −0.752779 0.658273i $$-0.771287\pi$$
−0.752779 + 0.658273i $$0.771287\pi$$
$$272$$ 5.46410 0.331310
$$273$$ 18.9282 1.14559
$$274$$ −2.00000 −0.120824
$$275$$ −5.46410 −0.329498
$$276$$ −5.85641 −0.352514
$$277$$ 22.2487 1.33680 0.668398 0.743804i $$-0.266980\pi$$
0.668398 + 0.743804i $$0.266980\pi$$
$$278$$ −6.92820 −0.415526
$$279$$ −1.80385 −0.107994
$$280$$ 4.73205 0.282794
$$281$$ −8.92820 −0.532612 −0.266306 0.963889i $$-0.585803\pi$$
−0.266306 + 0.963889i $$0.585803\pi$$
$$282$$ 3.46410 0.206284
$$283$$ 16.3923 0.974421 0.487211 0.873284i $$-0.338014\pi$$
0.487211 + 0.873284i $$0.338014\pi$$
$$284$$ 2.92820 0.173757
$$285$$ 4.53590 0.268683
$$286$$ −29.8564 −1.76545
$$287$$ 9.46410 0.558648
$$288$$ 2.46410 0.145199
$$289$$ 12.8564 0.756259
$$290$$ −4.92820 −0.289394
$$291$$ −1.46410 −0.0858272
$$292$$ −0.928203 −0.0543190
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −11.2679 −0.657160
$$295$$ −10.1962 −0.593643
$$296$$ −1.00000 −0.0581238
$$297$$ 21.8564 1.26824
$$298$$ −4.39230 −0.254439
$$299$$ 43.7128 2.52798
$$300$$ 0.732051 0.0422650
$$301$$ −32.7846 −1.88967
$$302$$ 12.3923 0.713097
$$303$$ −6.92820 −0.398015
$$304$$ 6.19615 0.355374
$$305$$ −4.92820 −0.282188
$$306$$ 13.4641 0.769691
$$307$$ 18.5885 1.06090 0.530450 0.847716i $$-0.322023\pi$$
0.530450 + 0.847716i $$0.322023\pi$$
$$308$$ 25.8564 1.47331
$$309$$ −4.78461 −0.272187
$$310$$ −0.732051 −0.0415777
$$311$$ 2.87564 0.163063 0.0815314 0.996671i $$-0.474019\pi$$
0.0815314 + 0.996671i $$0.474019\pi$$
$$312$$ 4.00000 0.226455
$$313$$ −23.8564 −1.34844 −0.674222 0.738529i $$-0.735521\pi$$
−0.674222 + 0.738529i $$0.735521\pi$$
$$314$$ −3.07180 −0.173352
$$315$$ 11.6603 0.656981
$$316$$ 8.73205 0.491216
$$317$$ −4.14359 −0.232727 −0.116364 0.993207i $$-0.537124\pi$$
−0.116364 + 0.993207i $$0.537124\pi$$
$$318$$ 4.39230 0.246308
$$319$$ −26.9282 −1.50769
$$320$$ 1.00000 0.0559017
$$321$$ 2.39230 0.133525
$$322$$ −37.8564 −2.10966
$$323$$ 33.8564 1.88382
$$324$$ 4.46410 0.248006
$$325$$ −5.46410 −0.303094
$$326$$ 11.3205 0.626985
$$327$$ 1.46410 0.0809650
$$328$$ 2.00000 0.110432
$$329$$ 22.3923 1.23453
$$330$$ 4.00000 0.220193
$$331$$ −33.1244 −1.82068 −0.910340 0.413862i $$-0.864180\pi$$
−0.910340 + 0.413862i $$0.864180\pi$$
$$332$$ −8.73205 −0.479234
$$333$$ −2.46410 −0.135032
$$334$$ 1.46410 0.0801121
$$335$$ −3.66025 −0.199981
$$336$$ −3.46410 −0.188982
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ −16.8564 −0.916868
$$339$$ −7.71281 −0.418902
$$340$$ 5.46410 0.296333
$$341$$ −4.00000 −0.216612
$$342$$ 15.2679 0.825596
$$343$$ −39.7128 −2.14429
$$344$$ −6.92820 −0.373544
$$345$$ −5.85641 −0.315298
$$346$$ −10.0000 −0.537603
$$347$$ 30.9282 1.66031 0.830156 0.557530i $$-0.188251\pi$$
0.830156 + 0.557530i $$0.188251\pi$$
$$348$$ 3.60770 0.193393
$$349$$ −15.3205 −0.820088 −0.410044 0.912066i $$-0.634487\pi$$
−0.410044 + 0.912066i $$0.634487\pi$$
$$350$$ 4.73205 0.252939
$$351$$ 21.8564 1.16661
$$352$$ 5.46410 0.291238
$$353$$ −11.8564 −0.631053 −0.315526 0.948917i $$-0.602181\pi$$
−0.315526 + 0.948917i $$0.602181\pi$$
$$354$$ 7.46410 0.396713
$$355$$ 2.92820 0.155413
$$356$$ −2.00000 −0.106000
$$357$$ −18.9282 −1.00179
$$358$$ 0.339746 0.0179561
$$359$$ −12.3923 −0.654041 −0.327020 0.945017i $$-0.606045\pi$$
−0.327020 + 0.945017i $$0.606045\pi$$
$$360$$ 2.46410 0.129870
$$361$$ 19.3923 1.02065
$$362$$ −5.46410 −0.287187
$$363$$ 13.8038 0.724514
$$364$$ 25.8564 1.35524
$$365$$ −0.928203 −0.0485844
$$366$$ 3.60770 0.188577
$$367$$ −4.33975 −0.226533 −0.113266 0.993565i $$-0.536131\pi$$
−0.113266 + 0.993565i $$0.536131\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ 4.92820 0.256552
$$370$$ −1.00000 −0.0519875
$$371$$ 28.3923 1.47406
$$372$$ 0.535898 0.0277850
$$373$$ −10.7846 −0.558406 −0.279203 0.960232i $$-0.590070\pi$$
−0.279203 + 0.960232i $$0.590070\pi$$
$$374$$ 29.8564 1.54384
$$375$$ 0.732051 0.0378029
$$376$$ 4.73205 0.244037
$$377$$ −26.9282 −1.38687
$$378$$ −18.9282 −0.973562
$$379$$ 32.3923 1.66388 0.831940 0.554865i $$-0.187230\pi$$
0.831940 + 0.554865i $$0.187230\pi$$
$$380$$ 6.19615 0.317856
$$381$$ 2.67949 0.137275
$$382$$ 8.73205 0.446771
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ −0.732051 −0.0373573
$$385$$ 25.8564 1.31776
$$386$$ −15.8564 −0.807070
$$387$$ −17.0718 −0.867808
$$388$$ −2.00000 −0.101535
$$389$$ −11.8564 −0.601144 −0.300572 0.953759i $$-0.597178\pi$$
−0.300572 + 0.953759i $$0.597178\pi$$
$$390$$ 4.00000 0.202548
$$391$$ −43.7128 −2.21065
$$392$$ −15.3923 −0.777429
$$393$$ −13.6077 −0.686417
$$394$$ 22.7846 1.14787
$$395$$ 8.73205 0.439357
$$396$$ 13.4641 0.676597
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 12.0526 0.604140
$$399$$ −21.4641 −1.07455
$$400$$ 1.00000 0.0500000
$$401$$ −32.9282 −1.64436 −0.822178 0.569230i $$-0.807241\pi$$
−0.822178 + 0.569230i $$0.807241\pi$$
$$402$$ 2.67949 0.133641
$$403$$ −4.00000 −0.199254
$$404$$ −9.46410 −0.470857
$$405$$ 4.46410 0.221823
$$406$$ 23.3205 1.15738
$$407$$ −5.46410 −0.270845
$$408$$ −4.00000 −0.198030
$$409$$ 3.07180 0.151891 0.0759453 0.997112i $$-0.475803\pi$$
0.0759453 + 0.997112i $$0.475803\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 1.46410 0.0722188
$$412$$ −6.53590 −0.322001
$$413$$ 48.2487 2.37416
$$414$$ −19.7128 −0.968832
$$415$$ −8.73205 −0.428640
$$416$$ 5.46410 0.267900
$$417$$ 5.07180 0.248367
$$418$$ 33.8564 1.65597
$$419$$ 14.2487 0.696095 0.348048 0.937477i $$-0.386845\pi$$
0.348048 + 0.937477i $$0.386845\pi$$
$$420$$ −3.46410 −0.169031
$$421$$ −0.143594 −0.00699832 −0.00349916 0.999994i $$-0.501114\pi$$
−0.00349916 + 0.999994i $$0.501114\pi$$
$$422$$ 17.8564 0.869236
$$423$$ 11.6603 0.566941
$$424$$ 6.00000 0.291386
$$425$$ 5.46410 0.265048
$$426$$ −2.14359 −0.103857
$$427$$ 23.3205 1.12856
$$428$$ 3.26795 0.157962
$$429$$ 21.8564 1.05524
$$430$$ −6.92820 −0.334108
$$431$$ −2.19615 −0.105785 −0.0528925 0.998600i $$-0.516844\pi$$
−0.0528925 + 0.998600i $$0.516844\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 3.46410 0.166282
$$435$$ 3.60770 0.172976
$$436$$ 2.00000 0.0957826
$$437$$ −49.5692 −2.37122
$$438$$ 0.679492 0.0324674
$$439$$ 13.5167 0.645115 0.322558 0.946550i $$-0.395457\pi$$
0.322558 + 0.946550i $$0.395457\pi$$
$$440$$ 5.46410 0.260491
$$441$$ −37.9282 −1.80610
$$442$$ 29.8564 1.42012
$$443$$ −14.8756 −0.706763 −0.353382 0.935479i $$-0.614968\pi$$
−0.353382 + 0.935479i $$0.614968\pi$$
$$444$$ 0.732051 0.0347416
$$445$$ −2.00000 −0.0948091
$$446$$ −16.0526 −0.760111
$$447$$ 3.21539 0.152083
$$448$$ −4.73205 −0.223568
$$449$$ −21.7128 −1.02469 −0.512345 0.858779i $$-0.671223\pi$$
−0.512345 + 0.858779i $$0.671223\pi$$
$$450$$ 2.46410 0.116159
$$451$$ 10.9282 0.514589
$$452$$ −10.5359 −0.495567
$$453$$ −9.07180 −0.426230
$$454$$ −24.3923 −1.14479
$$455$$ 25.8564 1.21217
$$456$$ −4.53590 −0.212413
$$457$$ −31.8564 −1.49018 −0.745090 0.666964i $$-0.767593\pi$$
−0.745090 + 0.666964i $$0.767593\pi$$
$$458$$ 11.8564 0.554013
$$459$$ −21.8564 −1.02017
$$460$$ −8.00000 −0.373002
$$461$$ 14.7846 0.688588 0.344294 0.938862i $$-0.388118\pi$$
0.344294 + 0.938862i $$0.388118\pi$$
$$462$$ −18.9282 −0.880620
$$463$$ 18.9282 0.879668 0.439834 0.898079i $$-0.355037\pi$$
0.439834 + 0.898079i $$0.355037\pi$$
$$464$$ 4.92820 0.228786
$$465$$ 0.535898 0.0248517
$$466$$ −28.9282 −1.34007
$$467$$ −25.1769 −1.16505 −0.582524 0.812813i $$-0.697935\pi$$
−0.582524 + 0.812813i $$0.697935\pi$$
$$468$$ 13.4641 0.622378
$$469$$ 17.3205 0.799787
$$470$$ 4.73205 0.218273
$$471$$ 2.24871 0.103615
$$472$$ 10.1962 0.469316
$$473$$ −37.8564 −1.74064
$$474$$ −6.39230 −0.293608
$$475$$ 6.19615 0.284299
$$476$$ −25.8564 −1.18513
$$477$$ 14.7846 0.676941
$$478$$ 20.7321 0.948262
$$479$$ 4.05256 0.185166 0.0925831 0.995705i $$-0.470488\pi$$
0.0925831 + 0.995705i $$0.470488\pi$$
$$480$$ −0.732051 −0.0334134
$$481$$ −5.46410 −0.249142
$$482$$ −4.92820 −0.224474
$$483$$ 27.7128 1.26098
$$484$$ 18.8564 0.857109
$$485$$ −2.00000 −0.0908153
$$486$$ −15.2679 −0.692568
$$487$$ 4.39230 0.199034 0.0995172 0.995036i $$-0.468270\pi$$
0.0995172 + 0.995036i $$0.468270\pi$$
$$488$$ 4.92820 0.223089
$$489$$ −8.28719 −0.374760
$$490$$ −15.3923 −0.695353
$$491$$ −14.9282 −0.673700 −0.336850 0.941558i $$-0.609362\pi$$
−0.336850 + 0.941558i $$0.609362\pi$$
$$492$$ −1.46410 −0.0660068
$$493$$ 26.9282 1.21279
$$494$$ 33.8564 1.52327
$$495$$ 13.4641 0.605166
$$496$$ 0.732051 0.0328701
$$497$$ −13.8564 −0.621545
$$498$$ 6.39230 0.286446
$$499$$ 9.41154 0.421319 0.210659 0.977560i $$-0.432439\pi$$
0.210659 + 0.977560i $$0.432439\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −1.07180 −0.0478843
$$502$$ −19.2679 −0.859971
$$503$$ −18.9282 −0.843967 −0.421983 0.906604i $$-0.638666\pi$$
−0.421983 + 0.906604i $$0.638666\pi$$
$$504$$ −11.6603 −0.519389
$$505$$ −9.46410 −0.421147
$$506$$ −43.7128 −1.94327
$$507$$ 12.3397 0.548027
$$508$$ 3.66025 0.162398
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ 4.39230 0.194304
$$512$$ −1.00000 −0.0441942
$$513$$ −24.7846 −1.09427
$$514$$ 18.5359 0.817583
$$515$$ −6.53590 −0.288006
$$516$$ 5.07180 0.223273
$$517$$ 25.8564 1.13716
$$518$$ 4.73205 0.207914
$$519$$ 7.32051 0.321335
$$520$$ 5.46410 0.239617
$$521$$ 26.5359 1.16256 0.581279 0.813704i $$-0.302552\pi$$
0.581279 + 0.813704i $$0.302552\pi$$
$$522$$ 12.1436 0.531511
$$523$$ 36.7846 1.60848 0.804239 0.594306i $$-0.202573\pi$$
0.804239 + 0.594306i $$0.202573\pi$$
$$524$$ −18.5885 −0.812041
$$525$$ −3.46410 −0.151186
$$526$$ 8.05256 0.351108
$$527$$ 4.00000 0.174243
$$528$$ −4.00000 −0.174078
$$529$$ 41.0000 1.78261
$$530$$ 6.00000 0.260623
$$531$$ 25.1244 1.09030
$$532$$ −29.3205 −1.27121
$$533$$ 10.9282 0.473353
$$534$$ 1.46410 0.0633579
$$535$$ 3.26795 0.141286
$$536$$ 3.66025 0.158099
$$537$$ −0.248711 −0.0107327
$$538$$ −20.3923 −0.879175
$$539$$ −84.1051 −3.62266
$$540$$ −4.00000 −0.172133
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 24.7846 1.06459
$$543$$ 4.00000 0.171656
$$544$$ −5.46410 −0.234271
$$545$$ 2.00000 0.0856706
$$546$$ −18.9282 −0.810052
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 12.1436 0.518276
$$550$$ 5.46410 0.232990
$$551$$ 30.5359 1.30087
$$552$$ 5.85641 0.249265
$$553$$ −41.3205 −1.75713
$$554$$ −22.2487 −0.945257
$$555$$ 0.732051 0.0310738
$$556$$ 6.92820 0.293821
$$557$$ 44.1051 1.86879 0.934397 0.356234i $$-0.115939\pi$$
0.934397 + 0.356234i $$0.115939\pi$$
$$558$$ 1.80385 0.0763630
$$559$$ −37.8564 −1.60116
$$560$$ −4.73205 −0.199966
$$561$$ −21.8564 −0.922778
$$562$$ 8.92820 0.376614
$$563$$ 30.9282 1.30347 0.651734 0.758447i $$-0.274042\pi$$
0.651734 + 0.758447i $$0.274042\pi$$
$$564$$ −3.46410 −0.145865
$$565$$ −10.5359 −0.443249
$$566$$ −16.3923 −0.689020
$$567$$ −21.1244 −0.887140
$$568$$ −2.92820 −0.122865
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ −4.53590 −0.189988
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 29.8564 1.24836
$$573$$ −6.39230 −0.267042
$$574$$ −9.46410 −0.395024
$$575$$ −8.00000 −0.333623
$$576$$ −2.46410 −0.102671
$$577$$ −24.3923 −1.01546 −0.507732 0.861515i $$-0.669516\pi$$
−0.507732 + 0.861515i $$0.669516\pi$$
$$578$$ −12.8564 −0.534756
$$579$$ 11.6077 0.482399
$$580$$ 4.92820 0.204633
$$581$$ 41.3205 1.71426
$$582$$ 1.46410 0.0606890
$$583$$ 32.7846 1.35780
$$584$$ 0.928203 0.0384093
$$585$$ 13.4641 0.556672
$$586$$ −6.00000 −0.247858
$$587$$ 44.7846 1.84846 0.924229 0.381838i $$-0.124709\pi$$
0.924229 + 0.381838i $$0.124709\pi$$
$$588$$ 11.2679 0.464682
$$589$$ 4.53590 0.186898
$$590$$ 10.1962 0.419769
$$591$$ −16.6795 −0.686103
$$592$$ 1.00000 0.0410997
$$593$$ −34.7846 −1.42843 −0.714216 0.699925i $$-0.753217\pi$$
−0.714216 + 0.699925i $$0.753217\pi$$
$$594$$ −21.8564 −0.896779
$$595$$ −25.8564 −1.06001
$$596$$ 4.39230 0.179916
$$597$$ −8.82309 −0.361105
$$598$$ −43.7128 −1.78755
$$599$$ 9.46410 0.386693 0.193346 0.981131i $$-0.438066\pi$$
0.193346 + 0.981131i $$0.438066\pi$$
$$600$$ −0.732051 −0.0298858
$$601$$ 27.6077 1.12614 0.563071 0.826409i $$-0.309620\pi$$
0.563071 + 0.826409i $$0.309620\pi$$
$$602$$ 32.7846 1.33620
$$603$$ 9.01924 0.367292
$$604$$ −12.3923 −0.504236
$$605$$ 18.8564 0.766622
$$606$$ 6.92820 0.281439
$$607$$ 0.784610 0.0318463 0.0159232 0.999873i $$-0.494931\pi$$
0.0159232 + 0.999873i $$0.494931\pi$$
$$608$$ −6.19615 −0.251287
$$609$$ −17.0718 −0.691784
$$610$$ 4.92820 0.199537
$$611$$ 25.8564 1.04604
$$612$$ −13.4641 −0.544254
$$613$$ 16.9282 0.683724 0.341862 0.939750i $$-0.388943\pi$$
0.341862 + 0.939750i $$0.388943\pi$$
$$614$$ −18.5885 −0.750169
$$615$$ −1.46410 −0.0590383
$$616$$ −25.8564 −1.04178
$$617$$ −0.928203 −0.0373681 −0.0186840 0.999825i $$-0.505948\pi$$
−0.0186840 + 0.999825i $$0.505948\pi$$
$$618$$ 4.78461 0.192465
$$619$$ −17.8564 −0.717710 −0.358855 0.933393i $$-0.616833\pi$$
−0.358855 + 0.933393i $$0.616833\pi$$
$$620$$ 0.732051 0.0293999
$$621$$ 32.0000 1.28412
$$622$$ −2.87564 −0.115303
$$623$$ 9.46410 0.379171
$$624$$ −4.00000 −0.160128
$$625$$ 1.00000 0.0400000
$$626$$ 23.8564 0.953494
$$627$$ −24.7846 −0.989802
$$628$$ 3.07180 0.122578
$$629$$ 5.46410 0.217868
$$630$$ −11.6603 −0.464556
$$631$$ 30.9808 1.23332 0.616662 0.787228i $$-0.288484\pi$$
0.616662 + 0.787228i $$0.288484\pi$$
$$632$$ −8.73205 −0.347342
$$633$$ −13.0718 −0.519557
$$634$$ 4.14359 0.164563
$$635$$ 3.66025 0.145253
$$636$$ −4.39230 −0.174166
$$637$$ −84.1051 −3.33237
$$638$$ 26.9282 1.06610
$$639$$ −7.21539 −0.285436
$$640$$ −1.00000 −0.0395285
$$641$$ −6.53590 −0.258152 −0.129076 0.991635i $$-0.541201\pi$$
−0.129076 + 0.991635i $$0.541201\pi$$
$$642$$ −2.39230 −0.0944167
$$643$$ 34.5359 1.36196 0.680981 0.732301i $$-0.261553\pi$$
0.680981 + 0.732301i $$0.261553\pi$$
$$644$$ 37.8564 1.49175
$$645$$ 5.07180 0.199702
$$646$$ −33.8564 −1.33206
$$647$$ −35.7128 −1.40402 −0.702008 0.712169i $$-0.747713\pi$$
−0.702008 + 0.712169i $$0.747713\pi$$
$$648$$ −4.46410 −0.175366
$$649$$ 55.7128 2.18692
$$650$$ 5.46410 0.214320
$$651$$ −2.53590 −0.0993897
$$652$$ −11.3205 −0.443345
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −1.46410 −0.0572509
$$655$$ −18.5885 −0.726311
$$656$$ −2.00000 −0.0780869
$$657$$ 2.28719 0.0892317
$$658$$ −22.3923 −0.872943
$$659$$ −6.92820 −0.269884 −0.134942 0.990853i $$-0.543085\pi$$
−0.134942 + 0.990853i $$0.543085\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ −31.0718 −1.20855 −0.604276 0.796775i $$-0.706538\pi$$
−0.604276 + 0.796775i $$0.706538\pi$$
$$662$$ 33.1244 1.28741
$$663$$ −21.8564 −0.848832
$$664$$ 8.73205 0.338869
$$665$$ −29.3205 −1.13700
$$666$$ 2.46410 0.0954820
$$667$$ −39.4256 −1.52657
$$668$$ −1.46410 −0.0566478
$$669$$ 11.7513 0.454331
$$670$$ 3.66025 0.141408
$$671$$ 26.9282 1.03955
$$672$$ 3.46410 0.133631
$$673$$ −32.9282 −1.26929 −0.634644 0.772804i $$-0.718853\pi$$
−0.634644 + 0.772804i $$0.718853\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ −4.00000 −0.153960
$$676$$ 16.8564 0.648323
$$677$$ −4.14359 −0.159251 −0.0796256 0.996825i $$-0.525372\pi$$
−0.0796256 + 0.996825i $$0.525372\pi$$
$$678$$ 7.71281 0.296209
$$679$$ 9.46410 0.363199
$$680$$ −5.46410 −0.209539
$$681$$ 17.8564 0.684259
$$682$$ 4.00000 0.153168
$$683$$ −4.78461 −0.183078 −0.0915390 0.995801i $$-0.529179\pi$$
−0.0915390 + 0.995801i $$0.529179\pi$$
$$684$$ −15.2679 −0.583785
$$685$$ 2.00000 0.0764161
$$686$$ 39.7128 1.51624
$$687$$ −8.67949 −0.331143
$$688$$ 6.92820 0.264135
$$689$$ 32.7846 1.24899
$$690$$ 5.85641 0.222950
$$691$$ −0.392305 −0.0149240 −0.00746199 0.999972i $$-0.502375\pi$$
−0.00746199 + 0.999972i $$0.502375\pi$$
$$692$$ 10.0000 0.380143
$$693$$ −63.7128 −2.42025
$$694$$ −30.9282 −1.17402
$$695$$ 6.92820 0.262802
$$696$$ −3.60770 −0.136749
$$697$$ −10.9282 −0.413935
$$698$$ 15.3205 0.579890
$$699$$ 21.1769 0.800984
$$700$$ −4.73205 −0.178855
$$701$$ −11.8564 −0.447810 −0.223905 0.974611i $$-0.571881\pi$$
−0.223905 + 0.974611i $$0.571881\pi$$
$$702$$ −21.8564 −0.824917
$$703$$ 6.19615 0.233692
$$704$$ −5.46410 −0.205936
$$705$$ −3.46410 −0.130466
$$706$$ 11.8564 0.446222
$$707$$ 44.7846 1.68430
$$708$$ −7.46410 −0.280518
$$709$$ 43.8564 1.64706 0.823531 0.567271i $$-0.192001\pi$$
0.823531 + 0.567271i $$0.192001\pi$$
$$710$$ −2.92820 −0.109894
$$711$$ −21.5167 −0.806938
$$712$$ 2.00000 0.0749532
$$713$$ −5.85641 −0.219324
$$714$$ 18.9282 0.708370
$$715$$ 29.8564 1.11657
$$716$$ −0.339746 −0.0126969
$$717$$ −15.1769 −0.566792
$$718$$ 12.3923 0.462477
$$719$$ −12.3923 −0.462155 −0.231077 0.972935i $$-0.574225\pi$$
−0.231077 + 0.972935i $$0.574225\pi$$
$$720$$ −2.46410 −0.0918316
$$721$$ 30.9282 1.15183
$$722$$ −19.3923 −0.721707
$$723$$ 3.60770 0.134172
$$724$$ 5.46410 0.203072
$$725$$ 4.92820 0.183029
$$726$$ −13.8038 −0.512309
$$727$$ 8.78461 0.325803 0.162902 0.986642i $$-0.447915\pi$$
0.162902 + 0.986642i $$0.447915\pi$$
$$728$$ −25.8564 −0.958302
$$729$$ −2.21539 −0.0820515
$$730$$ 0.928203 0.0343543
$$731$$ 37.8564 1.40017
$$732$$ −3.60770 −0.133344
$$733$$ 27.8564 1.02890 0.514450 0.857520i $$-0.327996\pi$$
0.514450 + 0.857520i $$0.327996\pi$$
$$734$$ 4.33975 0.160183
$$735$$ 11.2679 0.415625
$$736$$ 8.00000 0.294884
$$737$$ 20.0000 0.736709
$$738$$ −4.92820 −0.181410
$$739$$ 6.92820 0.254858 0.127429 0.991848i $$-0.459327\pi$$
0.127429 + 0.991848i $$0.459327\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ −24.7846 −0.910485
$$742$$ −28.3923 −1.04231
$$743$$ 17.1244 0.628232 0.314116 0.949385i $$-0.398292\pi$$
0.314116 + 0.949385i $$0.398292\pi$$
$$744$$ −0.535898 −0.0196470
$$745$$ 4.39230 0.160922
$$746$$ 10.7846 0.394853
$$747$$ 21.5167 0.787253
$$748$$ −29.8564 −1.09166
$$749$$ −15.4641 −0.565046
$$750$$ −0.732051 −0.0267307
$$751$$ −20.3923 −0.744126 −0.372063 0.928208i $$-0.621349\pi$$
−0.372063 + 0.928208i $$0.621349\pi$$
$$752$$ −4.73205 −0.172560
$$753$$ 14.1051 0.514019
$$754$$ 26.9282 0.980667
$$755$$ −12.3923 −0.451002
$$756$$ 18.9282 0.688412
$$757$$ 1.71281 0.0622532 0.0311266 0.999515i $$-0.490090\pi$$
0.0311266 + 0.999515i $$0.490090\pi$$
$$758$$ −32.3923 −1.17654
$$759$$ 32.0000 1.16153
$$760$$ −6.19615 −0.224758
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ −2.67949 −0.0970678
$$763$$ −9.46410 −0.342623
$$764$$ −8.73205 −0.315915
$$765$$ −13.4641 −0.486796
$$766$$ 8.00000 0.289052
$$767$$ 55.7128 2.01167
$$768$$ 0.732051 0.0264156
$$769$$ 7.07180 0.255016 0.127508 0.991838i $$-0.459302\pi$$
0.127508 + 0.991838i $$0.459302\pi$$
$$770$$ −25.8564 −0.931800
$$771$$ −13.5692 −0.488683
$$772$$ 15.8564 0.570685
$$773$$ 2.78461 0.100155 0.0500777 0.998745i $$-0.484053\pi$$
0.0500777 + 0.998745i $$0.484053\pi$$
$$774$$ 17.0718 0.613633
$$775$$ 0.732051 0.0262960
$$776$$ 2.00000 0.0717958
$$777$$ −3.46410 −0.124274
$$778$$ 11.8564 0.425073
$$779$$ −12.3923 −0.444000
$$780$$ −4.00000 −0.143223
$$781$$ −16.0000 −0.572525
$$782$$ 43.7128 1.56317
$$783$$ −19.7128 −0.704478
$$784$$ 15.3923 0.549725
$$785$$ 3.07180 0.109637
$$786$$ 13.6077 0.485370
$$787$$ 22.1962 0.791207 0.395604 0.918421i $$-0.370535\pi$$
0.395604 + 0.918421i $$0.370535\pi$$
$$788$$ −22.7846 −0.811668
$$789$$ −5.89488 −0.209863
$$790$$ −8.73205 −0.310672
$$791$$ 49.8564 1.77269
$$792$$ −13.4641 −0.478426
$$793$$ 26.9282 0.956249
$$794$$ 22.0000 0.780751
$$795$$ −4.39230 −0.155779
$$796$$ −12.0526 −0.427192
$$797$$ 10.5359 0.373201 0.186600 0.982436i $$-0.440253\pi$$
0.186600 + 0.982436i $$0.440253\pi$$
$$798$$ 21.4641 0.759821
$$799$$ −25.8564 −0.914734
$$800$$ −1.00000 −0.0353553
$$801$$ 4.92820 0.174129
$$802$$ 32.9282 1.16274
$$803$$ 5.07180 0.178980
$$804$$ −2.67949 −0.0944984
$$805$$ 37.8564 1.33426
$$806$$ 4.00000 0.140894
$$807$$ 14.9282 0.525498
$$808$$ 9.46410 0.332946
$$809$$ 43.5692 1.53181 0.765906 0.642952i $$-0.222291\pi$$
0.765906 + 0.642952i $$0.222291\pi$$
$$810$$ −4.46410 −0.156853
$$811$$ −41.8564 −1.46978 −0.734889 0.678188i $$-0.762766\pi$$
−0.734889 + 0.678188i $$0.762766\pi$$
$$812$$ −23.3205 −0.818389
$$813$$ −18.1436 −0.636324
$$814$$ 5.46410 0.191517
$$815$$ −11.3205 −0.396540
$$816$$ 4.00000 0.140028
$$817$$ 42.9282 1.50187
$$818$$ −3.07180 −0.107403
$$819$$ −63.7128 −2.22631
$$820$$ −2.00000 −0.0698430
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ −1.46410 −0.0510664
$$823$$ 48.4449 1.68868 0.844341 0.535806i $$-0.179992\pi$$
0.844341 + 0.535806i $$0.179992\pi$$
$$824$$ 6.53590 0.227689
$$825$$ −4.00000 −0.139262
$$826$$ −48.2487 −1.67879
$$827$$ −16.3923 −0.570016 −0.285008 0.958525i $$-0.591996\pi$$
−0.285008 + 0.958525i $$0.591996\pi$$
$$828$$ 19.7128 0.685068
$$829$$ −51.5692 −1.79107 −0.895537 0.444988i $$-0.853208\pi$$
−0.895537 + 0.444988i $$0.853208\pi$$
$$830$$ 8.73205 0.303094
$$831$$ 16.2872 0.564996
$$832$$ −5.46410 −0.189434
$$833$$ 84.1051 2.91407
$$834$$ −5.07180 −0.175622
$$835$$ −1.46410 −0.0506673
$$836$$ −33.8564 −1.17095
$$837$$ −2.92820 −0.101214
$$838$$ −14.2487 −0.492214
$$839$$ −8.78461 −0.303278 −0.151639 0.988436i $$-0.548455\pi$$
−0.151639 + 0.988436i $$0.548455\pi$$
$$840$$ 3.46410 0.119523
$$841$$ −4.71281 −0.162511
$$842$$ 0.143594 0.00494856
$$843$$ −6.53590 −0.225108
$$844$$ −17.8564 −0.614643
$$845$$ 16.8564 0.579878
$$846$$ −11.6603 −0.400888
$$847$$ −89.2295 −3.06596
$$848$$ −6.00000 −0.206041
$$849$$ 12.0000 0.411839
$$850$$ −5.46410 −0.187417
$$851$$ −8.00000 −0.274236
$$852$$ 2.14359 0.0734383
$$853$$ −30.0000 −1.02718 −0.513590 0.858036i $$-0.671685\pi$$
−0.513590 + 0.858036i $$0.671685\pi$$
$$854$$ −23.3205 −0.798011
$$855$$ −15.2679 −0.522153
$$856$$ −3.26795 −0.111696
$$857$$ −31.8564 −1.08819 −0.544097 0.839022i $$-0.683128\pi$$
−0.544097 + 0.839022i $$0.683128\pi$$
$$858$$ −21.8564 −0.746165
$$859$$ 44.4449 1.51644 0.758220 0.651999i $$-0.226069\pi$$
0.758220 + 0.651999i $$0.226069\pi$$
$$860$$ 6.92820 0.236250
$$861$$ 6.92820 0.236113
$$862$$ 2.19615 0.0748012
$$863$$ −20.4449 −0.695951 −0.347976 0.937504i $$-0.613131\pi$$
−0.347976 + 0.937504i $$0.613131\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 10.0000 0.340010
$$866$$ 34.0000 1.15537
$$867$$ 9.41154 0.319633
$$868$$ −3.46410 −0.117579
$$869$$ −47.7128 −1.61855
$$870$$ −3.60770 −0.122312
$$871$$ 20.0000 0.677674
$$872$$ −2.00000 −0.0677285
$$873$$ 4.92820 0.166794
$$874$$ 49.5692 1.67670
$$875$$ −4.73205 −0.159973
$$876$$ −0.679492 −0.0229579
$$877$$ −9.21539 −0.311182 −0.155591 0.987822i $$-0.549728\pi$$
−0.155591 + 0.987822i $$0.549728\pi$$
$$878$$ −13.5167 −0.456165
$$879$$ 4.39230 0.148149
$$880$$ −5.46410 −0.184195
$$881$$ 12.6795 0.427183 0.213591 0.976923i $$-0.431484\pi$$
0.213591 + 0.976923i $$0.431484\pi$$
$$882$$ 37.9282 1.27711
$$883$$ 14.2487 0.479507 0.239754 0.970834i $$-0.422933\pi$$
0.239754 + 0.970834i $$0.422933\pi$$
$$884$$ −29.8564 −1.00418
$$885$$ −7.46410 −0.250903
$$886$$ 14.8756 0.499757
$$887$$ 9.80385 0.329181 0.164590 0.986362i $$-0.447370\pi$$
0.164590 + 0.986362i $$0.447370\pi$$
$$888$$ −0.732051 −0.0245660
$$889$$ −17.3205 −0.580911
$$890$$ 2.00000 0.0670402
$$891$$ −24.3923 −0.817173
$$892$$ 16.0526 0.537479
$$893$$ −29.3205 −0.981173
$$894$$ −3.21539 −0.107539
$$895$$ −0.339746 −0.0113565
$$896$$ 4.73205 0.158087
$$897$$ 32.0000 1.06845
$$898$$ 21.7128 0.724566
$$899$$ 3.60770 0.120323
$$900$$ −2.46410 −0.0821367
$$901$$ −32.7846 −1.09221
$$902$$ −10.9282 −0.363869
$$903$$ −24.0000 −0.798670
$$904$$ 10.5359 0.350419
$$905$$ 5.46410 0.181633
$$906$$ 9.07180 0.301390
$$907$$ 54.2487 1.80130 0.900649 0.434546i $$-0.143091\pi$$
0.900649 + 0.434546i $$0.143091\pi$$
$$908$$ 24.3923 0.809487
$$909$$ 23.3205 0.773492
$$910$$ −25.8564 −0.857132
$$911$$ 37.9090 1.25598 0.627990 0.778221i $$-0.283878\pi$$
0.627990 + 0.778221i $$0.283878\pi$$
$$912$$ 4.53590 0.150199
$$913$$ 47.7128 1.57906
$$914$$ 31.8564 1.05372
$$915$$ −3.60770 −0.119267
$$916$$ −11.8564 −0.391747
$$917$$ 87.9615 2.90475
$$918$$ 21.8564 0.721369
$$919$$ −16.7321 −0.551939 −0.275970 0.961166i $$-0.588999\pi$$
−0.275970 + 0.961166i $$0.588999\pi$$
$$920$$ 8.00000 0.263752
$$921$$ 13.6077 0.448389
$$922$$ −14.7846 −0.486905
$$923$$ −16.0000 −0.526646
$$924$$ 18.9282 0.622692
$$925$$ 1.00000 0.0328798
$$926$$ −18.9282 −0.622019
$$927$$ 16.1051 0.528961
$$928$$ −4.92820 −0.161776
$$929$$ 11.8564 0.388996 0.194498 0.980903i $$-0.437692\pi$$
0.194498 + 0.980903i $$0.437692\pi$$
$$930$$ −0.535898 −0.0175728
$$931$$ 95.3731 3.12573
$$932$$ 28.9282 0.947575
$$933$$ 2.10512 0.0689185
$$934$$ 25.1769 0.823814
$$935$$ −29.8564 −0.976409
$$936$$ −13.4641 −0.440088
$$937$$ −23.8564 −0.779355 −0.389677 0.920951i $$-0.627413\pi$$
−0.389677 + 0.920951i $$0.627413\pi$$
$$938$$ −17.3205 −0.565535
$$939$$ −17.4641 −0.569919
$$940$$ −4.73205 −0.154342
$$941$$ 7.60770 0.248004 0.124002 0.992282i $$-0.460427\pi$$
0.124002 + 0.992282i $$0.460427\pi$$
$$942$$ −2.24871 −0.0732670
$$943$$ 16.0000 0.521032
$$944$$ −10.1962 −0.331856
$$945$$ 18.9282 0.615734
$$946$$ 37.8564 1.23082
$$947$$ 17.1769 0.558175 0.279087 0.960266i $$-0.409968\pi$$
0.279087 + 0.960266i $$0.409968\pi$$
$$948$$ 6.39230 0.207612
$$949$$ 5.07180 0.164637
$$950$$ −6.19615 −0.201030
$$951$$ −3.03332 −0.0983622
$$952$$ 25.8564 0.838011
$$953$$ 15.8564 0.513639 0.256820 0.966459i $$-0.417325\pi$$
0.256820 + 0.966459i $$0.417325\pi$$
$$954$$ −14.7846 −0.478669
$$955$$ −8.73205 −0.282563
$$956$$ −20.7321 −0.670522
$$957$$ −19.7128 −0.637225
$$958$$ −4.05256 −0.130932
$$959$$ −9.46410 −0.305612
$$960$$ 0.732051 0.0236268
$$961$$ −30.4641 −0.982713
$$962$$ 5.46410 0.176170
$$963$$ −8.05256 −0.259490
$$964$$ 4.92820 0.158727
$$965$$ 15.8564 0.510436
$$966$$ −27.7128 −0.891645
$$967$$ −44.3923 −1.42756 −0.713780 0.700370i $$-0.753018\pi$$
−0.713780 + 0.700370i $$0.753018\pi$$
$$968$$ −18.8564 −0.606068
$$969$$ 24.7846 0.796196
$$970$$ 2.00000 0.0642161
$$971$$ 1.07180 0.0343956 0.0171978 0.999852i $$-0.494526\pi$$
0.0171978 + 0.999852i $$0.494526\pi$$
$$972$$ 15.2679 0.489720
$$973$$ −32.7846 −1.05103
$$974$$ −4.39230 −0.140739
$$975$$ −4.00000 −0.128103
$$976$$ −4.92820 −0.157748
$$977$$ −34.0000 −1.08776 −0.543878 0.839164i $$-0.683045\pi$$
−0.543878 + 0.839164i $$0.683045\pi$$
$$978$$ 8.28719 0.264995
$$979$$ 10.9282 0.349267
$$980$$ 15.3923 0.491689
$$981$$ −4.92820 −0.157345
$$982$$ 14.9282 0.476378
$$983$$ 20.4449 0.652090 0.326045 0.945354i $$-0.394284\pi$$
0.326045 + 0.945354i $$0.394284\pi$$
$$984$$ 1.46410 0.0466739
$$985$$ −22.7846 −0.725978
$$986$$ −26.9282 −0.857569
$$987$$ 16.3923 0.521773
$$988$$ −33.8564 −1.07712
$$989$$ −55.4256 −1.76243
$$990$$ −13.4641 −0.427917
$$991$$ 44.0526 1.39938 0.699688 0.714449i $$-0.253322\pi$$
0.699688 + 0.714449i $$0.253322\pi$$
$$992$$ −0.732051 −0.0232426
$$993$$ −24.2487 −0.769510
$$994$$ 13.8564 0.439499
$$995$$ −12.0526 −0.382092
$$996$$ −6.39230 −0.202548
$$997$$ −31.8564 −1.00890 −0.504451 0.863440i $$-0.668305\pi$$
−0.504451 + 0.863440i $$0.668305\pi$$
$$998$$ −9.41154 −0.297917
$$999$$ −4.00000 −0.126554
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 370.2.a.e.1.2 2
3.2 odd 2 3330.2.a.bd.1.1 2
4.3 odd 2 2960.2.a.q.1.1 2
5.2 odd 4 1850.2.b.l.149.1 4
5.3 odd 4 1850.2.b.l.149.4 4
5.4 even 2 1850.2.a.x.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.2 2 1.1 even 1 trivial
1850.2.a.x.1.1 2 5.4 even 2
1850.2.b.l.149.1 4 5.2 odd 4
1850.2.b.l.149.4 4 5.3 odd 4
2960.2.a.q.1.1 2 4.3 odd 2
3330.2.a.bd.1.1 2 3.2 odd 2