# Properties

 Label 770.2.a.k.1.2 Level $770$ Weight $2$ Character 770.1 Self dual yes Analytic conductor $6.148$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 770.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{12} +6.74456 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.74456 q^{17} +1.00000 q^{18} -6.74456 q^{19} +1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} -6.74456 q^{23} +2.00000 q^{24} +1.00000 q^{25} +6.74456 q^{26} -4.00000 q^{27} +1.00000 q^{28} +8.74456 q^{29} +2.00000 q^{30} +4.74456 q^{31} +1.00000 q^{32} -2.00000 q^{33} -6.74456 q^{34} +1.00000 q^{35} +1.00000 q^{36} +0.744563 q^{37} -6.74456 q^{38} +13.4891 q^{39} +1.00000 q^{40} -4.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -6.74456 q^{46} +4.74456 q^{47} +2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -13.4891 q^{51} +6.74456 q^{52} -12.7446 q^{53} -4.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} -13.4891 q^{57} +8.74456 q^{58} -8.74456 q^{59} +2.00000 q^{60} +1.25544 q^{61} +4.74456 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.74456 q^{65} -2.00000 q^{66} -4.00000 q^{67} -6.74456 q^{68} -13.4891 q^{69} +1.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} +10.7446 q^{73} +0.744563 q^{74} +2.00000 q^{75} -6.74456 q^{76} -1.00000 q^{77} +13.4891 q^{78} +6.74456 q^{79} +1.00000 q^{80} -11.0000 q^{81} -4.00000 q^{82} +8.00000 q^{83} +2.00000 q^{84} -6.74456 q^{85} +4.00000 q^{86} +17.4891 q^{87} -1.00000 q^{88} +15.4891 q^{89} +1.00000 q^{90} +6.74456 q^{91} -6.74456 q^{92} +9.48913 q^{93} +4.74456 q^{94} -6.74456 q^{95} +2.00000 q^{96} -16.7446 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} + 2 q^{13} + 2 q^{14} + 4 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} + 4 q^{21} - 2 q^{22} - 2 q^{23} + 4 q^{24} + 2 q^{25} + 2 q^{26} - 8 q^{27} + 2 q^{28} + 6 q^{29} + 4 q^{30} - 2 q^{31} + 2 q^{32} - 4 q^{33} - 2 q^{34} + 2 q^{35} + 2 q^{36} - 10 q^{37} - 2 q^{38} + 4 q^{39} + 2 q^{40} - 8 q^{41} + 4 q^{42} + 8 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 2 q^{47} + 4 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 2 q^{52} - 14 q^{53} - 8 q^{54} - 2 q^{55} + 2 q^{56} - 4 q^{57} + 6 q^{58} - 6 q^{59} + 4 q^{60} + 14 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} + 2 q^{65} - 4 q^{66} - 8 q^{67} - 2 q^{68} - 4 q^{69} + 2 q^{70} - 8 q^{71} + 2 q^{72} + 10 q^{73} - 10 q^{74} + 4 q^{75} - 2 q^{76} - 2 q^{77} + 4 q^{78} + 2 q^{79} + 2 q^{80} - 22 q^{81} - 8 q^{82} + 16 q^{83} + 4 q^{84} - 2 q^{85} + 8 q^{86} + 12 q^{87} - 2 q^{88} + 8 q^{89} + 2 q^{90} + 2 q^{91} - 2 q^{92} - 4 q^{93} - 2 q^{94} - 2 q^{95} + 4 q^{96} - 22 q^{97} + 2 q^{98} - 2 q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 2.00000 0.816497
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −1.00000 −0.301511
$$12$$ 2.00000 0.577350
$$13$$ 6.74456 1.87061 0.935303 0.353849i $$-0.115127\pi$$
0.935303 + 0.353849i $$0.115127\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ −6.74456 −1.63580 −0.817898 0.575363i $$-0.804861\pi$$
−0.817898 + 0.575363i $$0.804861\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −6.74456 −1.54731 −0.773654 0.633608i $$-0.781573\pi$$
−0.773654 + 0.633608i $$0.781573\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.00000 0.436436
$$22$$ −1.00000 −0.213201
$$23$$ −6.74456 −1.40634 −0.703169 0.711022i $$-0.748232\pi$$
−0.703169 + 0.711022i $$0.748232\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 1.00000 0.200000
$$26$$ 6.74456 1.32272
$$27$$ −4.00000 −0.769800
$$28$$ 1.00000 0.188982
$$29$$ 8.74456 1.62382 0.811912 0.583779i $$-0.198427\pi$$
0.811912 + 0.583779i $$0.198427\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 4.74456 0.852149 0.426074 0.904688i $$-0.359896\pi$$
0.426074 + 0.904688i $$0.359896\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.00000 −0.348155
$$34$$ −6.74456 −1.15668
$$35$$ 1.00000 0.169031
$$36$$ 1.00000 0.166667
$$37$$ 0.744563 0.122405 0.0612027 0.998125i $$-0.480506\pi$$
0.0612027 + 0.998125i $$0.480506\pi$$
$$38$$ −6.74456 −1.09411
$$39$$ 13.4891 2.15999
$$40$$ 1.00000 0.158114
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 1.00000 0.149071
$$46$$ −6.74456 −0.994432
$$47$$ 4.74456 0.692066 0.346033 0.938222i $$-0.387529\pi$$
0.346033 + 0.938222i $$0.387529\pi$$
$$48$$ 2.00000 0.288675
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −13.4891 −1.88886
$$52$$ 6.74456 0.935303
$$53$$ −12.7446 −1.75060 −0.875300 0.483580i $$-0.839336\pi$$
−0.875300 + 0.483580i $$0.839336\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ −1.00000 −0.134840
$$56$$ 1.00000 0.133631
$$57$$ −13.4891 −1.78668
$$58$$ 8.74456 1.14822
$$59$$ −8.74456 −1.13845 −0.569223 0.822183i $$-0.692756\pi$$
−0.569223 + 0.822183i $$0.692756\pi$$
$$60$$ 2.00000 0.258199
$$61$$ 1.25544 0.160742 0.0803711 0.996765i $$-0.474389\pi$$
0.0803711 + 0.996765i $$0.474389\pi$$
$$62$$ 4.74456 0.602560
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 6.74456 0.836560
$$66$$ −2.00000 −0.246183
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −6.74456 −0.817898
$$69$$ −13.4891 −1.62390
$$70$$ 1.00000 0.119523
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 10.7446 1.25756 0.628778 0.777585i $$-0.283555\pi$$
0.628778 + 0.777585i $$0.283555\pi$$
$$74$$ 0.744563 0.0865536
$$75$$ 2.00000 0.230940
$$76$$ −6.74456 −0.773654
$$77$$ −1.00000 −0.113961
$$78$$ 13.4891 1.52734
$$79$$ 6.74456 0.758823 0.379411 0.925228i $$-0.376127\pi$$
0.379411 + 0.925228i $$0.376127\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −11.0000 −1.22222
$$82$$ −4.00000 −0.441726
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 2.00000 0.218218
$$85$$ −6.74456 −0.731551
$$86$$ 4.00000 0.431331
$$87$$ 17.4891 1.87503
$$88$$ −1.00000 −0.106600
$$89$$ 15.4891 1.64184 0.820922 0.571040i $$-0.193460\pi$$
0.820922 + 0.571040i $$0.193460\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 6.74456 0.707022
$$92$$ −6.74456 −0.703169
$$93$$ 9.48913 0.983976
$$94$$ 4.74456 0.489364
$$95$$ −6.74456 −0.691978
$$96$$ 2.00000 0.204124
$$97$$ −16.7446 −1.70015 −0.850076 0.526659i $$-0.823444\pi$$
−0.850076 + 0.526659i $$0.823444\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −1.00000 −0.100504
$$100$$ 1.00000 0.100000
$$101$$ 2.74456 0.273094 0.136547 0.990634i $$-0.456399\pi$$
0.136547 + 0.990634i $$0.456399\pi$$
$$102$$ −13.4891 −1.33562
$$103$$ −0.744563 −0.0733639 −0.0366820 0.999327i $$-0.511679\pi$$
−0.0366820 + 0.999327i $$0.511679\pi$$
$$104$$ 6.74456 0.661359
$$105$$ 2.00000 0.195180
$$106$$ −12.7446 −1.23786
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ −18.2337 −1.74647 −0.873235 0.487299i $$-0.837982\pi$$
−0.873235 + 0.487299i $$0.837982\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ 1.48913 0.141342
$$112$$ 1.00000 0.0944911
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ −13.4891 −1.26337
$$115$$ −6.74456 −0.628934
$$116$$ 8.74456 0.811912
$$117$$ 6.74456 0.623535
$$118$$ −8.74456 −0.805002
$$119$$ −6.74456 −0.618273
$$120$$ 2.00000 0.182574
$$121$$ 1.00000 0.0909091
$$122$$ 1.25544 0.113662
$$123$$ −8.00000 −0.721336
$$124$$ 4.74456 0.426074
$$125$$ 1.00000 0.0894427
$$126$$ 1.00000 0.0890871
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 6.74456 0.591537
$$131$$ −2.74456 −0.239794 −0.119897 0.992786i $$-0.538256\pi$$
−0.119897 + 0.992786i $$0.538256\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ −6.74456 −0.584828
$$134$$ −4.00000 −0.345547
$$135$$ −4.00000 −0.344265
$$136$$ −6.74456 −0.578341
$$137$$ 3.48913 0.298096 0.149048 0.988830i $$-0.452379\pi$$
0.149048 + 0.988830i $$0.452379\pi$$
$$138$$ −13.4891 −1.14827
$$139$$ 14.7446 1.25062 0.625309 0.780377i $$-0.284973\pi$$
0.625309 + 0.780377i $$0.284973\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 9.48913 0.799129
$$142$$ −4.00000 −0.335673
$$143$$ −6.74456 −0.564009
$$144$$ 1.00000 0.0833333
$$145$$ 8.74456 0.726196
$$146$$ 10.7446 0.889226
$$147$$ 2.00000 0.164957
$$148$$ 0.744563 0.0612027
$$149$$ −0.744563 −0.0609969 −0.0304985 0.999535i $$-0.509709\pi$$
−0.0304985 + 0.999535i $$0.509709\pi$$
$$150$$ 2.00000 0.163299
$$151$$ −9.25544 −0.753197 −0.376598 0.926377i $$-0.622906\pi$$
−0.376598 + 0.926377i $$0.622906\pi$$
$$152$$ −6.74456 −0.547056
$$153$$ −6.74456 −0.545266
$$154$$ −1.00000 −0.0805823
$$155$$ 4.74456 0.381092
$$156$$ 13.4891 1.07999
$$157$$ 0.510875 0.0407722 0.0203861 0.999792i $$-0.493510\pi$$
0.0203861 + 0.999792i $$0.493510\pi$$
$$158$$ 6.74456 0.536569
$$159$$ −25.4891 −2.02142
$$160$$ 1.00000 0.0790569
$$161$$ −6.74456 −0.531546
$$162$$ −11.0000 −0.864242
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ −2.00000 −0.155700
$$166$$ 8.00000 0.620920
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 32.4891 2.49916
$$170$$ −6.74456 −0.517284
$$171$$ −6.74456 −0.515770
$$172$$ 4.00000 0.304997
$$173$$ 20.2337 1.53834 0.769169 0.639045i $$-0.220670\pi$$
0.769169 + 0.639045i $$0.220670\pi$$
$$174$$ 17.4891 1.32585
$$175$$ 1.00000 0.0755929
$$176$$ −1.00000 −0.0753778
$$177$$ −17.4891 −1.31456
$$178$$ 15.4891 1.16096
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −19.4891 −1.44862 −0.724308 0.689477i $$-0.757840\pi$$
−0.724308 + 0.689477i $$0.757840\pi$$
$$182$$ 6.74456 0.499940
$$183$$ 2.51087 0.185609
$$184$$ −6.74456 −0.497216
$$185$$ 0.744563 0.0547413
$$186$$ 9.48913 0.695776
$$187$$ 6.74456 0.493211
$$188$$ 4.74456 0.346033
$$189$$ −4.00000 −0.290957
$$190$$ −6.74456 −0.489302
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 2.00000 0.144338
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −16.7446 −1.20219
$$195$$ 13.4891 0.965976
$$196$$ 1.00000 0.0714286
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ 3.25544 0.230772 0.115386 0.993321i $$-0.463190\pi$$
0.115386 + 0.993321i $$0.463190\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −8.00000 −0.564276
$$202$$ 2.74456 0.193107
$$203$$ 8.74456 0.613748
$$204$$ −13.4891 −0.944428
$$205$$ −4.00000 −0.279372
$$206$$ −0.744563 −0.0518761
$$207$$ −6.74456 −0.468780
$$208$$ 6.74456 0.467651
$$209$$ 6.74456 0.466531
$$210$$ 2.00000 0.138013
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −12.7446 −0.875300
$$213$$ −8.00000 −0.548151
$$214$$ −12.0000 −0.820303
$$215$$ 4.00000 0.272798
$$216$$ −4.00000 −0.272166
$$217$$ 4.74456 0.322082
$$218$$ −18.2337 −1.23494
$$219$$ 21.4891 1.45210
$$220$$ −1.00000 −0.0674200
$$221$$ −45.4891 −3.05993
$$222$$ 1.48913 0.0999435
$$223$$ 15.2554 1.02158 0.510790 0.859706i $$-0.329353\pi$$
0.510790 + 0.859706i $$0.329353\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 1.00000 0.0666667
$$226$$ 10.0000 0.665190
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ −13.4891 −0.893339
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −6.74456 −0.444723
$$231$$ −2.00000 −0.131590
$$232$$ 8.74456 0.574109
$$233$$ −24.9783 −1.63638 −0.818190 0.574948i $$-0.805022\pi$$
−0.818190 + 0.574948i $$0.805022\pi$$
$$234$$ 6.74456 0.440906
$$235$$ 4.74456 0.309501
$$236$$ −8.74456 −0.569223
$$237$$ 13.4891 0.876213
$$238$$ −6.74456 −0.437185
$$239$$ −14.7446 −0.953746 −0.476873 0.878972i $$-0.658230\pi$$
−0.476873 + 0.878972i $$0.658230\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −10.0000 −0.641500
$$244$$ 1.25544 0.0803711
$$245$$ 1.00000 0.0638877
$$246$$ −8.00000 −0.510061
$$247$$ −45.4891 −2.89440
$$248$$ 4.74456 0.301280
$$249$$ 16.0000 1.01396
$$250$$ 1.00000 0.0632456
$$251$$ −26.2337 −1.65586 −0.827928 0.560835i $$-0.810480\pi$$
−0.827928 + 0.560835i $$0.810480\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 6.74456 0.424027
$$254$$ 0 0
$$255$$ −13.4891 −0.844722
$$256$$ 1.00000 0.0625000
$$257$$ −10.2337 −0.638360 −0.319180 0.947694i $$-0.603407\pi$$
−0.319180 + 0.947694i $$0.603407\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 0.744563 0.0462649
$$260$$ 6.74456 0.418280
$$261$$ 8.74456 0.541275
$$262$$ −2.74456 −0.169560
$$263$$ 26.9783 1.66355 0.831775 0.555113i $$-0.187325\pi$$
0.831775 + 0.555113i $$0.187325\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ −12.7446 −0.782892
$$266$$ −6.74456 −0.413536
$$267$$ 30.9783 1.89584
$$268$$ −4.00000 −0.244339
$$269$$ 20.9783 1.27907 0.639533 0.768763i $$-0.279128\pi$$
0.639533 + 0.768763i $$0.279128\pi$$
$$270$$ −4.00000 −0.243432
$$271$$ 14.9783 0.909864 0.454932 0.890526i $$-0.349664\pi$$
0.454932 + 0.890526i $$0.349664\pi$$
$$272$$ −6.74456 −0.408949
$$273$$ 13.4891 0.816399
$$274$$ 3.48913 0.210786
$$275$$ −1.00000 −0.0603023
$$276$$ −13.4891 −0.811950
$$277$$ 15.4891 0.930651 0.465326 0.885140i $$-0.345937\pi$$
0.465326 + 0.885140i $$0.345937\pi$$
$$278$$ 14.7446 0.884320
$$279$$ 4.74456 0.284050
$$280$$ 1.00000 0.0597614
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 9.48913 0.565069
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ −13.4891 −0.799027
$$286$$ −6.74456 −0.398814
$$287$$ −4.00000 −0.236113
$$288$$ 1.00000 0.0589256
$$289$$ 28.4891 1.67583
$$290$$ 8.74456 0.513498
$$291$$ −33.4891 −1.96317
$$292$$ 10.7446 0.628778
$$293$$ 13.2554 0.774391 0.387195 0.921998i $$-0.373444\pi$$
0.387195 + 0.921998i $$0.373444\pi$$
$$294$$ 2.00000 0.116642
$$295$$ −8.74456 −0.509128
$$296$$ 0.744563 0.0432768
$$297$$ 4.00000 0.232104
$$298$$ −0.744563 −0.0431314
$$299$$ −45.4891 −2.63070
$$300$$ 2.00000 0.115470
$$301$$ 4.00000 0.230556
$$302$$ −9.25544 −0.532591
$$303$$ 5.48913 0.315342
$$304$$ −6.74456 −0.386827
$$305$$ 1.25544 0.0718861
$$306$$ −6.74456 −0.385561
$$307$$ −1.48913 −0.0849889 −0.0424944 0.999097i $$-0.513530\pi$$
−0.0424944 + 0.999097i $$0.513530\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ −1.48913 −0.0847134
$$310$$ 4.74456 0.269473
$$311$$ −20.7446 −1.17632 −0.588158 0.808746i $$-0.700147\pi$$
−0.588158 + 0.808746i $$0.700147\pi$$
$$312$$ 13.4891 0.763671
$$313$$ −2.23369 −0.126256 −0.0631278 0.998005i $$-0.520108\pi$$
−0.0631278 + 0.998005i $$0.520108\pi$$
$$314$$ 0.510875 0.0288303
$$315$$ 1.00000 0.0563436
$$316$$ 6.74456 0.379411
$$317$$ 2.23369 0.125456 0.0627282 0.998031i $$-0.480020\pi$$
0.0627282 + 0.998031i $$0.480020\pi$$
$$318$$ −25.4891 −1.42936
$$319$$ −8.74456 −0.489602
$$320$$ 1.00000 0.0559017
$$321$$ −24.0000 −1.33955
$$322$$ −6.74456 −0.375860
$$323$$ 45.4891 2.53108
$$324$$ −11.0000 −0.611111
$$325$$ 6.74456 0.374121
$$326$$ −4.00000 −0.221540
$$327$$ −36.4674 −2.01665
$$328$$ −4.00000 −0.220863
$$329$$ 4.74456 0.261576
$$330$$ −2.00000 −0.110096
$$331$$ 14.9783 0.823279 0.411640 0.911347i $$-0.364956\pi$$
0.411640 + 0.911347i $$0.364956\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 0.744563 0.0408018
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 2.00000 0.109109
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 32.4891 1.76718
$$339$$ 20.0000 1.08625
$$340$$ −6.74456 −0.365775
$$341$$ −4.74456 −0.256932
$$342$$ −6.74456 −0.364704
$$343$$ 1.00000 0.0539949
$$344$$ 4.00000 0.215666
$$345$$ −13.4891 −0.726230
$$346$$ 20.2337 1.08777
$$347$$ −22.9783 −1.23354 −0.616769 0.787145i $$-0.711559\pi$$
−0.616769 + 0.787145i $$0.711559\pi$$
$$348$$ 17.4891 0.937516
$$349$$ 30.7446 1.64572 0.822859 0.568245i $$-0.192377\pi$$
0.822859 + 0.568245i $$0.192377\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −26.9783 −1.43999
$$352$$ −1.00000 −0.0533002
$$353$$ 8.74456 0.465426 0.232713 0.972545i $$-0.425240\pi$$
0.232713 + 0.972545i $$0.425240\pi$$
$$354$$ −17.4891 −0.929537
$$355$$ −4.00000 −0.212298
$$356$$ 15.4891 0.820922
$$357$$ −13.4891 −0.713920
$$358$$ −4.00000 −0.211407
$$359$$ −1.25544 −0.0662594 −0.0331297 0.999451i $$-0.510547\pi$$
−0.0331297 + 0.999451i $$0.510547\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 26.4891 1.39416
$$362$$ −19.4891 −1.02433
$$363$$ 2.00000 0.104973
$$364$$ 6.74456 0.353511
$$365$$ 10.7446 0.562396
$$366$$ 2.51087 0.131246
$$367$$ −8.74456 −0.456462 −0.228231 0.973607i $$-0.573294\pi$$
−0.228231 + 0.973607i $$0.573294\pi$$
$$368$$ −6.74456 −0.351585
$$369$$ −4.00000 −0.208232
$$370$$ 0.744563 0.0387080
$$371$$ −12.7446 −0.661665
$$372$$ 9.48913 0.491988
$$373$$ −16.9783 −0.879100 −0.439550 0.898218i $$-0.644862\pi$$
−0.439550 + 0.898218i $$0.644862\pi$$
$$374$$ 6.74456 0.348753
$$375$$ 2.00000 0.103280
$$376$$ 4.74456 0.244682
$$377$$ 58.9783 3.03753
$$378$$ −4.00000 −0.205738
$$379$$ −37.4891 −1.92569 −0.962844 0.270060i $$-0.912956\pi$$
−0.962844 + 0.270060i $$0.912956\pi$$
$$380$$ −6.74456 −0.345989
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ 19.7228 1.00779 0.503894 0.863765i $$-0.331900\pi$$
0.503894 + 0.863765i $$0.331900\pi$$
$$384$$ 2.00000 0.102062
$$385$$ −1.00000 −0.0509647
$$386$$ −2.00000 −0.101797
$$387$$ 4.00000 0.203331
$$388$$ −16.7446 −0.850076
$$389$$ −7.48913 −0.379714 −0.189857 0.981812i $$-0.560802\pi$$
−0.189857 + 0.981812i $$0.560802\pi$$
$$390$$ 13.4891 0.683048
$$391$$ 45.4891 2.30048
$$392$$ 1.00000 0.0505076
$$393$$ −5.48913 −0.276890
$$394$$ 10.0000 0.503793
$$395$$ 6.74456 0.339356
$$396$$ −1.00000 −0.0502519
$$397$$ 35.4891 1.78115 0.890574 0.454838i $$-0.150303\pi$$
0.890574 + 0.454838i $$0.150303\pi$$
$$398$$ 3.25544 0.163180
$$399$$ −13.4891 −0.675301
$$400$$ 1.00000 0.0500000
$$401$$ 23.4891 1.17299 0.586495 0.809953i $$-0.300507\pi$$
0.586495 + 0.809953i $$0.300507\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 32.0000 1.59403
$$404$$ 2.74456 0.136547
$$405$$ −11.0000 −0.546594
$$406$$ 8.74456 0.433985
$$407$$ −0.744563 −0.0369066
$$408$$ −13.4891 −0.667811
$$409$$ −21.4891 −1.06257 −0.531284 0.847194i $$-0.678290\pi$$
−0.531284 + 0.847194i $$0.678290\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 6.97825 0.344212
$$412$$ −0.744563 −0.0366820
$$413$$ −8.74456 −0.430292
$$414$$ −6.74456 −0.331477
$$415$$ 8.00000 0.392705
$$416$$ 6.74456 0.330679
$$417$$ 29.4891 1.44409
$$418$$ 6.74456 0.329887
$$419$$ −0.744563 −0.0363743 −0.0181871 0.999835i $$-0.505789\pi$$
−0.0181871 + 0.999835i $$0.505789\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 4.51087 0.219847 0.109923 0.993940i $$-0.464939\pi$$
0.109923 + 0.993940i $$0.464939\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 4.74456 0.230689
$$424$$ −12.7446 −0.618931
$$425$$ −6.74456 −0.327159
$$426$$ −8.00000 −0.387601
$$427$$ 1.25544 0.0607549
$$428$$ −12.0000 −0.580042
$$429$$ −13.4891 −0.651261
$$430$$ 4.00000 0.192897
$$431$$ −17.2554 −0.831165 −0.415583 0.909555i $$-0.636422\pi$$
−0.415583 + 0.909555i $$0.636422\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 36.7446 1.76583 0.882915 0.469532i $$-0.155577\pi$$
0.882915 + 0.469532i $$0.155577\pi$$
$$434$$ 4.74456 0.227746
$$435$$ 17.4891 0.838539
$$436$$ −18.2337 −0.873235
$$437$$ 45.4891 2.17604
$$438$$ 21.4891 1.02679
$$439$$ 14.9783 0.714873 0.357436 0.933937i $$-0.383651\pi$$
0.357436 + 0.933937i $$0.383651\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ 1.00000 0.0476190
$$442$$ −45.4891 −2.16370
$$443$$ −29.4891 −1.40107 −0.700535 0.713618i $$-0.747055\pi$$
−0.700535 + 0.713618i $$0.747055\pi$$
$$444$$ 1.48913 0.0706708
$$445$$ 15.4891 0.734255
$$446$$ 15.2554 0.722366
$$447$$ −1.48913 −0.0704332
$$448$$ 1.00000 0.0472456
$$449$$ −4.97825 −0.234938 −0.117469 0.993077i $$-0.537478\pi$$
−0.117469 + 0.993077i $$0.537478\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 4.00000 0.188353
$$452$$ 10.0000 0.470360
$$453$$ −18.5109 −0.869717
$$454$$ 20.0000 0.938647
$$455$$ 6.74456 0.316190
$$456$$ −13.4891 −0.631686
$$457$$ 3.48913 0.163214 0.0816072 0.996665i $$-0.473995\pi$$
0.0816072 + 0.996665i $$0.473995\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 26.9783 1.25924
$$460$$ −6.74456 −0.314467
$$461$$ 33.7228 1.57063 0.785314 0.619098i $$-0.212502\pi$$
0.785314 + 0.619098i $$0.212502\pi$$
$$462$$ −2.00000 −0.0930484
$$463$$ 1.25544 0.0583451 0.0291726 0.999574i $$-0.490713\pi$$
0.0291726 + 0.999574i $$0.490713\pi$$
$$464$$ 8.74456 0.405956
$$465$$ 9.48913 0.440048
$$466$$ −24.9783 −1.15710
$$467$$ 16.9783 0.785660 0.392830 0.919611i $$-0.371496\pi$$
0.392830 + 0.919611i $$0.371496\pi$$
$$468$$ 6.74456 0.311768
$$469$$ −4.00000 −0.184703
$$470$$ 4.74456 0.218850
$$471$$ 1.02175 0.0470797
$$472$$ −8.74456 −0.402501
$$473$$ −4.00000 −0.183920
$$474$$ 13.4891 0.619576
$$475$$ −6.74456 −0.309462
$$476$$ −6.74456 −0.309137
$$477$$ −12.7446 −0.583533
$$478$$ −14.7446 −0.674401
$$479$$ −41.4891 −1.89569 −0.947843 0.318737i $$-0.896741\pi$$
−0.947843 + 0.318737i $$0.896741\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 5.02175 0.228972
$$482$$ 20.0000 0.910975
$$483$$ −13.4891 −0.613776
$$484$$ 1.00000 0.0454545
$$485$$ −16.7446 −0.760331
$$486$$ −10.0000 −0.453609
$$487$$ 9.25544 0.419404 0.209702 0.977765i $$-0.432751\pi$$
0.209702 + 0.977765i $$0.432751\pi$$
$$488$$ 1.25544 0.0568310
$$489$$ −8.00000 −0.361773
$$490$$ 1.00000 0.0451754
$$491$$ 30.9783 1.39803 0.699014 0.715108i $$-0.253622\pi$$
0.699014 + 0.715108i $$0.253622\pi$$
$$492$$ −8.00000 −0.360668
$$493$$ −58.9783 −2.65625
$$494$$ −45.4891 −2.04665
$$495$$ −1.00000 −0.0449467
$$496$$ 4.74456 0.213037
$$497$$ −4.00000 −0.179425
$$498$$ 16.0000 0.716977
$$499$$ −13.4891 −0.603856 −0.301928 0.953331i $$-0.597630\pi$$
−0.301928 + 0.953331i $$0.597630\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −26.2337 −1.17087
$$503$$ 6.51087 0.290306 0.145153 0.989409i $$-0.453633\pi$$
0.145153 + 0.989409i $$0.453633\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 2.74456 0.122131
$$506$$ 6.74456 0.299832
$$507$$ 64.9783 2.88579
$$508$$ 0 0
$$509$$ −35.4891 −1.57303 −0.786514 0.617573i $$-0.788116\pi$$
−0.786514 + 0.617573i $$0.788116\pi$$
$$510$$ −13.4891 −0.597309
$$511$$ 10.7446 0.475311
$$512$$ 1.00000 0.0441942
$$513$$ 26.9783 1.19112
$$514$$ −10.2337 −0.451389
$$515$$ −0.744563 −0.0328094
$$516$$ 8.00000 0.352180
$$517$$ −4.74456 −0.208666
$$518$$ 0.744563 0.0327142
$$519$$ 40.4674 1.77632
$$520$$ 6.74456 0.295769
$$521$$ −2.00000 −0.0876216 −0.0438108 0.999040i $$-0.513950\pi$$
−0.0438108 + 0.999040i $$0.513950\pi$$
$$522$$ 8.74456 0.382739
$$523$$ −17.4891 −0.764746 −0.382373 0.924008i $$-0.624893\pi$$
−0.382373 + 0.924008i $$0.624893\pi$$
$$524$$ −2.74456 −0.119897
$$525$$ 2.00000 0.0872872
$$526$$ 26.9783 1.17631
$$527$$ −32.0000 −1.39394
$$528$$ −2.00000 −0.0870388
$$529$$ 22.4891 0.977788
$$530$$ −12.7446 −0.553588
$$531$$ −8.74456 −0.379482
$$532$$ −6.74456 −0.292414
$$533$$ −26.9783 −1.16856
$$534$$ 30.9783 1.34056
$$535$$ −12.0000 −0.518805
$$536$$ −4.00000 −0.172774
$$537$$ −8.00000 −0.345225
$$538$$ 20.9783 0.904437
$$539$$ −1.00000 −0.0430730
$$540$$ −4.00000 −0.172133
$$541$$ 1.76631 0.0759397 0.0379698 0.999279i $$-0.487911\pi$$
0.0379698 + 0.999279i $$0.487911\pi$$
$$542$$ 14.9783 0.643371
$$543$$ −38.9783 −1.67272
$$544$$ −6.74456 −0.289171
$$545$$ −18.2337 −0.781045
$$546$$ 13.4891 0.577281
$$547$$ 14.9783 0.640424 0.320212 0.947346i $$-0.396246\pi$$
0.320212 + 0.947346i $$0.396246\pi$$
$$548$$ 3.48913 0.149048
$$549$$ 1.25544 0.0535808
$$550$$ −1.00000 −0.0426401
$$551$$ −58.9783 −2.51256
$$552$$ −13.4891 −0.574135
$$553$$ 6.74456 0.286808
$$554$$ 15.4891 0.658070
$$555$$ 1.48913 0.0632098
$$556$$ 14.7446 0.625309
$$557$$ −0.978251 −0.0414498 −0.0207249 0.999785i $$-0.506597\pi$$
−0.0207249 + 0.999785i $$0.506597\pi$$
$$558$$ 4.74456 0.200853
$$559$$ 26.9783 1.14106
$$560$$ 1.00000 0.0422577
$$561$$ 13.4891 0.569511
$$562$$ 14.0000 0.590554
$$563$$ 5.48913 0.231339 0.115670 0.993288i $$-0.463099\pi$$
0.115670 + 0.993288i $$0.463099\pi$$
$$564$$ 9.48913 0.399564
$$565$$ 10.0000 0.420703
$$566$$ 28.0000 1.17693
$$567$$ −11.0000 −0.461957
$$568$$ −4.00000 −0.167836
$$569$$ 16.5109 0.692172 0.346086 0.938203i $$-0.387511\pi$$
0.346086 + 0.938203i $$0.387511\pi$$
$$570$$ −13.4891 −0.564997
$$571$$ −17.4891 −0.731897 −0.365949 0.930635i $$-0.619255\pi$$
−0.365949 + 0.930635i $$0.619255\pi$$
$$572$$ −6.74456 −0.282004
$$573$$ −32.0000 −1.33682
$$574$$ −4.00000 −0.166957
$$575$$ −6.74456 −0.281268
$$576$$ 1.00000 0.0416667
$$577$$ −8.74456 −0.364041 −0.182020 0.983295i $$-0.558264\pi$$
−0.182020 + 0.983295i $$0.558264\pi$$
$$578$$ 28.4891 1.18499
$$579$$ −4.00000 −0.166234
$$580$$ 8.74456 0.363098
$$581$$ 8.00000 0.331896
$$582$$ −33.4891 −1.38817
$$583$$ 12.7446 0.527826
$$584$$ 10.7446 0.444613
$$585$$ 6.74456 0.278853
$$586$$ 13.2554 0.547577
$$587$$ −4.97825 −0.205474 −0.102737 0.994709i $$-0.532760\pi$$
−0.102737 + 0.994709i $$0.532760\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ −32.0000 −1.31854
$$590$$ −8.74456 −0.360008
$$591$$ 20.0000 0.822690
$$592$$ 0.744563 0.0306013
$$593$$ −40.2337 −1.65220 −0.826100 0.563524i $$-0.809445\pi$$
−0.826100 + 0.563524i $$0.809445\pi$$
$$594$$ 4.00000 0.164122
$$595$$ −6.74456 −0.276500
$$596$$ −0.744563 −0.0304985
$$597$$ 6.51087 0.266472
$$598$$ −45.4891 −1.86019
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 2.00000 0.0816497
$$601$$ −33.4891 −1.36605 −0.683025 0.730395i $$-0.739336\pi$$
−0.683025 + 0.730395i $$0.739336\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −4.00000 −0.162893
$$604$$ −9.25544 −0.376598
$$605$$ 1.00000 0.0406558
$$606$$ 5.48913 0.222980
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −6.74456 −0.273528
$$609$$ 17.4891 0.708695
$$610$$ 1.25544 0.0508312
$$611$$ 32.0000 1.29458
$$612$$ −6.74456 −0.272633
$$613$$ 11.4891 0.464041 0.232021 0.972711i $$-0.425466\pi$$
0.232021 + 0.972711i $$0.425466\pi$$
$$614$$ −1.48913 −0.0600962
$$615$$ −8.00000 −0.322591
$$616$$ −1.00000 −0.0402911
$$617$$ −10.0000 −0.402585 −0.201292 0.979531i $$-0.564514\pi$$
−0.201292 + 0.979531i $$0.564514\pi$$
$$618$$ −1.48913 −0.0599014
$$619$$ 0.744563 0.0299265 0.0149632 0.999888i $$-0.495237\pi$$
0.0149632 + 0.999888i $$0.495237\pi$$
$$620$$ 4.74456 0.190546
$$621$$ 26.9783 1.08260
$$622$$ −20.7446 −0.831781
$$623$$ 15.4891 0.620559
$$624$$ 13.4891 0.539997
$$625$$ 1.00000 0.0400000
$$626$$ −2.23369 −0.0892761
$$627$$ 13.4891 0.538704
$$628$$ 0.510875 0.0203861
$$629$$ −5.02175 −0.200230
$$630$$ 1.00000 0.0398410
$$631$$ −2.97825 −0.118562 −0.0592811 0.998241i $$-0.518881\pi$$
−0.0592811 + 0.998241i $$0.518881\pi$$
$$632$$ 6.74456 0.268284
$$633$$ −24.0000 −0.953914
$$634$$ 2.23369 0.0887111
$$635$$ 0 0
$$636$$ −25.4891 −1.01071
$$637$$ 6.74456 0.267229
$$638$$ −8.74456 −0.346201
$$639$$ −4.00000 −0.158238
$$640$$ 1.00000 0.0395285
$$641$$ 11.4891 0.453793 0.226897 0.973919i $$-0.427142\pi$$
0.226897 + 0.973919i $$0.427142\pi$$
$$642$$ −24.0000 −0.947204
$$643$$ 46.4674 1.83249 0.916247 0.400613i $$-0.131203\pi$$
0.916247 + 0.400613i $$0.131203\pi$$
$$644$$ −6.74456 −0.265773
$$645$$ 8.00000 0.315000
$$646$$ 45.4891 1.78975
$$647$$ 8.74456 0.343784 0.171892 0.985116i $$-0.445012\pi$$
0.171892 + 0.985116i $$0.445012\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ 8.74456 0.343254
$$650$$ 6.74456 0.264544
$$651$$ 9.48913 0.371908
$$652$$ −4.00000 −0.156652
$$653$$ 15.7228 0.615281 0.307641 0.951503i $$-0.400461\pi$$
0.307641 + 0.951503i $$0.400461\pi$$
$$654$$ −36.4674 −1.42599
$$655$$ −2.74456 −0.107239
$$656$$ −4.00000 −0.156174
$$657$$ 10.7446 0.419185
$$658$$ 4.74456 0.184962
$$659$$ −41.4891 −1.61619 −0.808093 0.589054i $$-0.799500\pi$$
−0.808093 + 0.589054i $$0.799500\pi$$
$$660$$ −2.00000 −0.0778499
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 14.9783 0.582146
$$663$$ −90.9783 −3.53330
$$664$$ 8.00000 0.310460
$$665$$ −6.74456 −0.261543
$$666$$ 0.744563 0.0288512
$$667$$ −58.9783 −2.28365
$$668$$ 0 0
$$669$$ 30.5109 1.17962
$$670$$ −4.00000 −0.154533
$$671$$ −1.25544 −0.0484656
$$672$$ 2.00000 0.0771517
$$673$$ 20.9783 0.808652 0.404326 0.914615i $$-0.367506\pi$$
0.404326 + 0.914615i $$0.367506\pi$$
$$674$$ 26.0000 1.00148
$$675$$ −4.00000 −0.153960
$$676$$ 32.4891 1.24958
$$677$$ 36.2337 1.39257 0.696287 0.717764i $$-0.254834\pi$$
0.696287 + 0.717764i $$0.254834\pi$$
$$678$$ 20.0000 0.768095
$$679$$ −16.7446 −0.642597
$$680$$ −6.74456 −0.258642
$$681$$ 40.0000 1.53280
$$682$$ −4.74456 −0.181679
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ −6.74456 −0.257885
$$685$$ 3.48913 0.133313
$$686$$ 1.00000 0.0381802
$$687$$ −12.0000 −0.457829
$$688$$ 4.00000 0.152499
$$689$$ −85.9565 −3.27468
$$690$$ −13.4891 −0.513522
$$691$$ 1.76631 0.0671937 0.0335968 0.999435i $$-0.489304\pi$$
0.0335968 + 0.999435i $$0.489304\pi$$
$$692$$ 20.2337 0.769169
$$693$$ −1.00000 −0.0379869
$$694$$ −22.9783 −0.872242
$$695$$ 14.7446 0.559293
$$696$$ 17.4891 0.662924
$$697$$ 26.9783 1.02187
$$698$$ 30.7446 1.16370
$$699$$ −49.9565 −1.88953
$$700$$ 1.00000 0.0377964
$$701$$ 22.2337 0.839755 0.419877 0.907581i $$-0.362073\pi$$
0.419877 + 0.907581i $$0.362073\pi$$
$$702$$ −26.9783 −1.01823
$$703$$ −5.02175 −0.189399
$$704$$ −1.00000 −0.0376889
$$705$$ 9.48913 0.357381
$$706$$ 8.74456 0.329106
$$707$$ 2.74456 0.103220
$$708$$ −17.4891 −0.657282
$$709$$ 0.510875 0.0191863 0.00959315 0.999954i $$-0.496946\pi$$
0.00959315 + 0.999954i $$0.496946\pi$$
$$710$$ −4.00000 −0.150117
$$711$$ 6.74456 0.252941
$$712$$ 15.4891 0.580480
$$713$$ −32.0000 −1.19841
$$714$$ −13.4891 −0.504818
$$715$$ −6.74456 −0.252232
$$716$$ −4.00000 −0.149487
$$717$$ −29.4891 −1.10129
$$718$$ −1.25544 −0.0468525
$$719$$ 7.72281 0.288012 0.144006 0.989577i $$-0.454001\pi$$
0.144006 + 0.989577i $$0.454001\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −0.744563 −0.0277290
$$722$$ 26.4891 0.985823
$$723$$ 40.0000 1.48762
$$724$$ −19.4891 −0.724308
$$725$$ 8.74456 0.324765
$$726$$ 2.00000 0.0742270
$$727$$ 14.2337 0.527898 0.263949 0.964537i $$-0.414975\pi$$
0.263949 + 0.964537i $$0.414975\pi$$
$$728$$ 6.74456 0.249970
$$729$$ 13.0000 0.481481
$$730$$ 10.7446 0.397674
$$731$$ −26.9783 −0.997827
$$732$$ 2.51087 0.0928046
$$733$$ −16.2337 −0.599605 −0.299802 0.954001i $$-0.596921\pi$$
−0.299802 + 0.954001i $$0.596921\pi$$
$$734$$ −8.74456 −0.322768
$$735$$ 2.00000 0.0737711
$$736$$ −6.74456 −0.248608
$$737$$ 4.00000 0.147342
$$738$$ −4.00000 −0.147242
$$739$$ −30.9783 −1.13955 −0.569777 0.821800i $$-0.692970\pi$$
−0.569777 + 0.821800i $$0.692970\pi$$
$$740$$ 0.744563 0.0273707
$$741$$ −90.9783 −3.34217
$$742$$ −12.7446 −0.467868
$$743$$ 26.9783 0.989736 0.494868 0.868968i $$-0.335216\pi$$
0.494868 + 0.868968i $$0.335216\pi$$
$$744$$ 9.48913 0.347888
$$745$$ −0.744563 −0.0272787
$$746$$ −16.9783 −0.621618
$$747$$ 8.00000 0.292705
$$748$$ 6.74456 0.246606
$$749$$ −12.0000 −0.438470
$$750$$ 2.00000 0.0730297
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 4.74456 0.173016
$$753$$ −52.4674 −1.91202
$$754$$ 58.9783 2.14786
$$755$$ −9.25544 −0.336840
$$756$$ −4.00000 −0.145479
$$757$$ −19.2554 −0.699851 −0.349925 0.936778i $$-0.613793\pi$$
−0.349925 + 0.936778i $$0.613793\pi$$
$$758$$ −37.4891 −1.36167
$$759$$ 13.4891 0.489624
$$760$$ −6.74456 −0.244651
$$761$$ −29.4891 −1.06898 −0.534490 0.845175i $$-0.679496\pi$$
−0.534490 + 0.845175i $$0.679496\pi$$
$$762$$ 0 0
$$763$$ −18.2337 −0.660104
$$764$$ −16.0000 −0.578860
$$765$$ −6.74456 −0.243850
$$766$$ 19.7228 0.712614
$$767$$ −58.9783 −2.12958
$$768$$ 2.00000 0.0721688
$$769$$ −8.00000 −0.288487 −0.144244 0.989542i $$-0.546075\pi$$
−0.144244 + 0.989542i $$0.546075\pi$$
$$770$$ −1.00000 −0.0360375
$$771$$ −20.4674 −0.737115
$$772$$ −2.00000 −0.0719816
$$773$$ −51.4891 −1.85194 −0.925968 0.377603i $$-0.876748\pi$$
−0.925968 + 0.377603i $$0.876748\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 4.74456 0.170430
$$776$$ −16.7446 −0.601095
$$777$$ 1.48913 0.0534221
$$778$$ −7.48913 −0.268498
$$779$$ 26.9783 0.966596
$$780$$ 13.4891 0.482988
$$781$$ 4.00000 0.143131
$$782$$ 45.4891 1.62669
$$783$$ −34.9783 −1.25002
$$784$$ 1.00000 0.0357143
$$785$$ 0.510875 0.0182339
$$786$$ −5.48913 −0.195791
$$787$$ 5.48913 0.195666 0.0978331 0.995203i $$-0.468809\pi$$
0.0978331 + 0.995203i $$0.468809\pi$$
$$788$$ 10.0000 0.356235
$$789$$ 53.9565 1.92090
$$790$$ 6.74456 0.239961
$$791$$ 10.0000 0.355559
$$792$$ −1.00000 −0.0355335
$$793$$ 8.46738 0.300685
$$794$$ 35.4891 1.25946
$$795$$ −25.4891 −0.904006
$$796$$ 3.25544 0.115386
$$797$$ −42.4674 −1.50427 −0.752136 0.659008i $$-0.770976\pi$$
−0.752136 + 0.659008i $$0.770976\pi$$
$$798$$ −13.4891 −0.477510
$$799$$ −32.0000 −1.13208
$$800$$ 1.00000 0.0353553
$$801$$ 15.4891 0.547281
$$802$$ 23.4891 0.829430
$$803$$ −10.7446 −0.379167
$$804$$ −8.00000 −0.282138
$$805$$ −6.74456 −0.237715
$$806$$ 32.0000 1.12715
$$807$$ 41.9565 1.47694
$$808$$ 2.74456 0.0965534
$$809$$ 34.4674 1.21181 0.605904 0.795538i $$-0.292811\pi$$
0.605904 + 0.795538i $$0.292811\pi$$
$$810$$ −11.0000 −0.386501
$$811$$ −18.7446 −0.658211 −0.329105 0.944293i $$-0.606747\pi$$
−0.329105 + 0.944293i $$0.606747\pi$$
$$812$$ 8.74456 0.306874
$$813$$ 29.9565 1.05062
$$814$$ −0.744563 −0.0260969
$$815$$ −4.00000 −0.140114
$$816$$ −13.4891 −0.472214
$$817$$ −26.9783 −0.943850
$$818$$ −21.4891 −0.751350
$$819$$ 6.74456 0.235674
$$820$$ −4.00000 −0.139686
$$821$$ 7.72281 0.269528 0.134764 0.990878i $$-0.456972\pi$$
0.134764 + 0.990878i $$0.456972\pi$$
$$822$$ 6.97825 0.243394
$$823$$ 7.76631 0.270717 0.135358 0.990797i $$-0.456781\pi$$
0.135358 + 0.990797i $$0.456781\pi$$
$$824$$ −0.744563 −0.0259381
$$825$$ −2.00000 −0.0696311
$$826$$ −8.74456 −0.304262
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ −6.74456 −0.234390
$$829$$ 14.4674 0.502473 0.251236 0.967926i $$-0.419163\pi$$
0.251236 + 0.967926i $$0.419163\pi$$
$$830$$ 8.00000 0.277684
$$831$$ 30.9783 1.07462
$$832$$ 6.74456 0.233826
$$833$$ −6.74456 −0.233685
$$834$$ 29.4891 1.02112
$$835$$ 0 0
$$836$$ 6.74456 0.233266
$$837$$ −18.9783 −0.655984
$$838$$ −0.744563 −0.0257205
$$839$$ −3.25544 −0.112390 −0.0561951 0.998420i $$-0.517897\pi$$
−0.0561951 + 0.998420i $$0.517897\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 47.4674 1.63681
$$842$$ 4.51087 0.155455
$$843$$ 28.0000 0.964371
$$844$$ −12.0000 −0.413057
$$845$$ 32.4891 1.11766
$$846$$ 4.74456 0.163121
$$847$$ 1.00000 0.0343604
$$848$$ −12.7446 −0.437650
$$849$$ 56.0000 1.92192
$$850$$ −6.74456 −0.231337
$$851$$ −5.02175 −0.172143
$$852$$ −8.00000 −0.274075
$$853$$ −22.7446 −0.778759 −0.389379 0.921077i $$-0.627311\pi$$
−0.389379 + 0.921077i $$0.627311\pi$$
$$854$$ 1.25544 0.0429602
$$855$$ −6.74456 −0.230659
$$856$$ −12.0000 −0.410152
$$857$$ −43.2119 −1.47609 −0.738046 0.674751i $$-0.764251\pi$$
−0.738046 + 0.674751i $$0.764251\pi$$
$$858$$ −13.4891 −0.460511
$$859$$ −29.2119 −0.996698 −0.498349 0.866976i $$-0.666060\pi$$
−0.498349 + 0.866976i $$0.666060\pi$$
$$860$$ 4.00000 0.136399
$$861$$ −8.00000 −0.272639
$$862$$ −17.2554 −0.587723
$$863$$ −40.2337 −1.36957 −0.684785 0.728745i $$-0.740104\pi$$
−0.684785 + 0.728745i $$0.740104\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 20.2337 0.687966
$$866$$ 36.7446 1.24863
$$867$$ 56.9783 1.93508
$$868$$ 4.74456 0.161041
$$869$$ −6.74456 −0.228794
$$870$$ 17.4891 0.592937
$$871$$ −26.9783 −0.914123
$$872$$ −18.2337 −0.617471
$$873$$ −16.7446 −0.566718
$$874$$ 45.4891 1.53869
$$875$$ 1.00000 0.0338062
$$876$$ 21.4891 0.726050
$$877$$ −26.4674 −0.893740 −0.446870 0.894599i $$-0.647461\pi$$
−0.446870 + 0.894599i $$0.647461\pi$$
$$878$$ 14.9783 0.505491
$$879$$ 26.5109 0.894190
$$880$$ −1.00000 −0.0337100
$$881$$ −55.4891 −1.86948 −0.934738 0.355338i $$-0.884366\pi$$
−0.934738 + 0.355338i $$0.884366\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ −8.00000 −0.269221 −0.134611 0.990899i $$-0.542978\pi$$
−0.134611 + 0.990899i $$0.542978\pi$$
$$884$$ −45.4891 −1.52996
$$885$$ −17.4891 −0.587891
$$886$$ −29.4891 −0.990707
$$887$$ −41.4891 −1.39307 −0.696534 0.717524i $$-0.745276\pi$$
−0.696534 + 0.717524i $$0.745276\pi$$
$$888$$ 1.48913 0.0499718
$$889$$ 0 0
$$890$$ 15.4891 0.519197
$$891$$ 11.0000 0.368514
$$892$$ 15.2554 0.510790
$$893$$ −32.0000 −1.07084
$$894$$ −1.48913 −0.0498038
$$895$$ −4.00000 −0.133705
$$896$$ 1.00000 0.0334077
$$897$$ −90.9783 −3.03768
$$898$$ −4.97825 −0.166126
$$899$$ 41.4891 1.38374
$$900$$ 1.00000 0.0333333
$$901$$ 85.9565 2.86363
$$902$$ 4.00000 0.133185
$$903$$ 8.00000 0.266223
$$904$$ 10.0000 0.332595
$$905$$ −19.4891 −0.647840
$$906$$ −18.5109 −0.614983
$$907$$ −14.5109 −0.481826 −0.240913 0.970547i $$-0.577447\pi$$
−0.240913 + 0.970547i $$0.577447\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 2.74456 0.0910314
$$910$$ 6.74456 0.223580
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ −13.4891 −0.446670
$$913$$ −8.00000 −0.264761
$$914$$ 3.48913 0.115410
$$915$$ 2.51087 0.0830070
$$916$$ −6.00000 −0.198246
$$917$$ −2.74456 −0.0906334
$$918$$ 26.9783 0.890415
$$919$$ −55.2119 −1.82127 −0.910637 0.413207i $$-0.864408\pi$$
−0.910637 + 0.413207i $$0.864408\pi$$
$$920$$ −6.74456 −0.222362
$$921$$ −2.97825 −0.0981367
$$922$$ 33.7228 1.11060
$$923$$ −26.9783 −0.888000
$$924$$ −2.00000 −0.0657952
$$925$$ 0.744563 0.0244811
$$926$$ 1.25544 0.0412562
$$927$$ −0.744563 −0.0244546
$$928$$ 8.74456 0.287054
$$929$$ −28.9783 −0.950746 −0.475373 0.879784i $$-0.657687\pi$$
−0.475373 + 0.879784i $$0.657687\pi$$
$$930$$ 9.48913 0.311161
$$931$$ −6.74456 −0.221044
$$932$$ −24.9783 −0.818190
$$933$$ −41.4891 −1.35829
$$934$$ 16.9783 0.555545
$$935$$ 6.74456 0.220571
$$936$$ 6.74456 0.220453
$$937$$ −18.7446 −0.612358 −0.306179 0.951974i $$-0.599051\pi$$
−0.306179 + 0.951974i $$0.599051\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ −4.46738 −0.145787
$$940$$ 4.74456 0.154751
$$941$$ 12.2337 0.398807 0.199403 0.979917i $$-0.436100\pi$$
0.199403 + 0.979917i $$0.436100\pi$$
$$942$$ 1.02175 0.0332904
$$943$$ 26.9783 0.878533
$$944$$ −8.74456 −0.284611
$$945$$ −4.00000 −0.130120
$$946$$ −4.00000 −0.130051
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ 13.4891 0.438106
$$949$$ 72.4674 2.35239
$$950$$ −6.74456 −0.218823
$$951$$ 4.46738 0.144865
$$952$$ −6.74456 −0.218593
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ −12.7446 −0.412620
$$955$$ −16.0000 −0.517748
$$956$$ −14.7446 −0.476873
$$957$$ −17.4891 −0.565343
$$958$$ −41.4891 −1.34045
$$959$$ 3.48913 0.112670
$$960$$ 2.00000 0.0645497
$$961$$ −8.48913 −0.273843
$$962$$ 5.02175 0.161908
$$963$$ −12.0000 −0.386695
$$964$$ 20.0000 0.644157
$$965$$ −2.00000 −0.0643823
$$966$$ −13.4891 −0.434005
$$967$$ −50.9783 −1.63935 −0.819675 0.572829i $$-0.805846\pi$$
−0.819675 + 0.572829i $$0.805846\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 90.9783 2.92264
$$970$$ −16.7446 −0.537636
$$971$$ 19.7228 0.632935 0.316468 0.948603i $$-0.397503\pi$$
0.316468 + 0.948603i $$0.397503\pi$$
$$972$$ −10.0000 −0.320750
$$973$$ 14.7446 0.472689
$$974$$ 9.25544 0.296563
$$975$$ 13.4891 0.431998
$$976$$ 1.25544 0.0401856
$$977$$ 12.9783 0.415211 0.207606 0.978213i $$-0.433433\pi$$
0.207606 + 0.978213i $$0.433433\pi$$
$$978$$ −8.00000 −0.255812
$$979$$ −15.4891 −0.495035
$$980$$ 1.00000 0.0319438
$$981$$ −18.2337 −0.582157
$$982$$ 30.9783 0.988556
$$983$$ 2.23369 0.0712436 0.0356218 0.999365i $$-0.488659\pi$$
0.0356218 + 0.999365i $$0.488659\pi$$
$$984$$ −8.00000 −0.255031
$$985$$ 10.0000 0.318626
$$986$$ −58.9783 −1.87825
$$987$$ 9.48913 0.302042
$$988$$ −45.4891 −1.44720
$$989$$ −26.9783 −0.857858
$$990$$ −1.00000 −0.0317821
$$991$$ 1.48913 0.0473036 0.0236518 0.999720i $$-0.492471\pi$$
0.0236518 + 0.999720i $$0.492471\pi$$
$$992$$ 4.74456 0.150640
$$993$$ 29.9565 0.950641
$$994$$ −4.00000 −0.126872
$$995$$ 3.25544 0.103204
$$996$$ 16.0000 0.506979
$$997$$ 39.2119 1.24185 0.620927 0.783868i $$-0.286756\pi$$
0.620927 + 0.783868i $$0.286756\pi$$
$$998$$ −13.4891 −0.426991
$$999$$ −2.97825 −0.0942277
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 770.2.a.k.1.2 2
3.2 odd 2 6930.2.a.bo.1.2 2
4.3 odd 2 6160.2.a.r.1.2 2
5.2 odd 4 3850.2.c.y.1849.3 4
5.3 odd 4 3850.2.c.y.1849.2 4
5.4 even 2 3850.2.a.bc.1.1 2
7.6 odd 2 5390.2.a.bq.1.1 2
11.10 odd 2 8470.2.a.bu.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.2 2 1.1 even 1 trivial
3850.2.a.bc.1.1 2 5.4 even 2
3850.2.c.y.1849.2 4 5.3 odd 4
3850.2.c.y.1849.3 4 5.2 odd 4
5390.2.a.bq.1.1 2 7.6 odd 2
6160.2.a.r.1.2 2 4.3 odd 2
6930.2.a.bo.1.2 2 3.2 odd 2
8470.2.a.bu.1.1 2 11.10 odd 2