# Properties

 Label 7098.2.a.bg.1.1 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.6778153547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.56155 q^{10} +5.56155 q^{11} -1.00000 q^{12} +1.00000 q^{14} +3.56155 q^{15} +1.00000 q^{16} +6.68466 q^{17} -1.00000 q^{18} +1.56155 q^{19} -3.56155 q^{20} +1.00000 q^{21} -5.56155 q^{22} +6.68466 q^{23} +1.00000 q^{24} +7.68466 q^{25} -1.00000 q^{27} -1.00000 q^{28} +1.56155 q^{29} -3.56155 q^{30} +6.24621 q^{31} -1.00000 q^{32} -5.56155 q^{33} -6.68466 q^{34} +3.56155 q^{35} +1.00000 q^{36} +10.6847 q^{37} -1.56155 q^{38} +3.56155 q^{40} -4.00000 q^{41} -1.00000 q^{42} +6.43845 q^{43} +5.56155 q^{44} -3.56155 q^{45} -6.68466 q^{46} -10.2462 q^{47} -1.00000 q^{48} +1.00000 q^{49} -7.68466 q^{50} -6.68466 q^{51} +4.87689 q^{53} +1.00000 q^{54} -19.8078 q^{55} +1.00000 q^{56} -1.56155 q^{57} -1.56155 q^{58} +4.24621 q^{59} +3.56155 q^{60} -1.56155 q^{61} -6.24621 q^{62} -1.00000 q^{63} +1.00000 q^{64} +5.56155 q^{66} +1.12311 q^{67} +6.68466 q^{68} -6.68466 q^{69} -3.56155 q^{70} -9.36932 q^{71} -1.00000 q^{72} +11.5616 q^{73} -10.6847 q^{74} -7.68466 q^{75} +1.56155 q^{76} -5.56155 q^{77} +16.0000 q^{79} -3.56155 q^{80} +1.00000 q^{81} +4.00000 q^{82} +2.00000 q^{83} +1.00000 q^{84} -23.8078 q^{85} -6.43845 q^{86} -1.56155 q^{87} -5.56155 q^{88} -8.00000 q^{89} +3.56155 q^{90} +6.68466 q^{92} -6.24621 q^{93} +10.2462 q^{94} -5.56155 q^{95} +1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +5.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 3q^{10} + 7q^{11} - 2q^{12} + 2q^{14} + 3q^{15} + 2q^{16} + q^{17} - 2q^{18} - q^{19} - 3q^{20} + 2q^{21} - 7q^{22} + q^{23} + 2q^{24} + 3q^{25} - 2q^{27} - 2q^{28} - q^{29} - 3q^{30} - 4q^{31} - 2q^{32} - 7q^{33} - q^{34} + 3q^{35} + 2q^{36} + 9q^{37} + q^{38} + 3q^{40} - 8q^{41} - 2q^{42} + 17q^{43} + 7q^{44} - 3q^{45} - q^{46} - 4q^{47} - 2q^{48} + 2q^{49} - 3q^{50} - q^{51} + 18q^{53} + 2q^{54} - 19q^{55} + 2q^{56} + q^{57} + q^{58} - 8q^{59} + 3q^{60} + q^{61} + 4q^{62} - 2q^{63} + 2q^{64} + 7q^{66} - 6q^{67} + q^{68} - q^{69} - 3q^{70} + 6q^{71} - 2q^{72} + 19q^{73} - 9q^{74} - 3q^{75} - q^{76} - 7q^{77} + 32q^{79} - 3q^{80} + 2q^{81} + 8q^{82} + 4q^{83} + 2q^{84} - 27q^{85} - 17q^{86} + q^{87} - 7q^{88} - 16q^{89} + 3q^{90} + q^{92} + 4q^{93} + 4q^{94} - 7q^{95} + 2q^{96} - 20q^{97} - 2q^{98} + 7q^{99} + O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.56155 −1.59277 −0.796387 0.604787i $$-0.793258\pi$$
−0.796387 + 0.604787i $$0.793258\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.56155 1.12626
$$11$$ 5.56155 1.67687 0.838436 0.545001i $$-0.183471\pi$$
0.838436 + 0.545001i $$0.183471\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 1.00000 0.267261
$$15$$ 3.56155 0.919589
$$16$$ 1.00000 0.250000
$$17$$ 6.68466 1.62127 0.810634 0.585553i $$-0.199123\pi$$
0.810634 + 0.585553i $$0.199123\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.56155 0.358245 0.179122 0.983827i $$-0.442674\pi$$
0.179122 + 0.983827i $$0.442674\pi$$
$$20$$ −3.56155 −0.796387
$$21$$ 1.00000 0.218218
$$22$$ −5.56155 −1.18573
$$23$$ 6.68466 1.39385 0.696924 0.717145i $$-0.254552\pi$$
0.696924 + 0.717145i $$0.254552\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 7.68466 1.53693
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 1.56155 0.289973 0.144987 0.989434i $$-0.453686\pi$$
0.144987 + 0.989434i $$0.453686\pi$$
$$30$$ −3.56155 −0.650248
$$31$$ 6.24621 1.12185 0.560926 0.827866i $$-0.310445\pi$$
0.560926 + 0.827866i $$0.310445\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −5.56155 −0.968142
$$34$$ −6.68466 −1.14641
$$35$$ 3.56155 0.602012
$$36$$ 1.00000 0.166667
$$37$$ 10.6847 1.75655 0.878274 0.478159i $$-0.158696\pi$$
0.878274 + 0.478159i $$0.158696\pi$$
$$38$$ −1.56155 −0.253317
$$39$$ 0 0
$$40$$ 3.56155 0.563131
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 6.43845 0.981854 0.490927 0.871201i $$-0.336658\pi$$
0.490927 + 0.871201i $$0.336658\pi$$
$$44$$ 5.56155 0.838436
$$45$$ −3.56155 −0.530925
$$46$$ −6.68466 −0.985599
$$47$$ −10.2462 −1.49456 −0.747282 0.664507i $$-0.768641\pi$$
−0.747282 + 0.664507i $$0.768641\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ −7.68466 −1.08677
$$51$$ −6.68466 −0.936039
$$52$$ 0 0
$$53$$ 4.87689 0.669893 0.334946 0.942237i $$-0.391282\pi$$
0.334946 + 0.942237i $$0.391282\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −19.8078 −2.67088
$$56$$ 1.00000 0.133631
$$57$$ −1.56155 −0.206833
$$58$$ −1.56155 −0.205042
$$59$$ 4.24621 0.552810 0.276405 0.961041i $$-0.410857\pi$$
0.276405 + 0.961041i $$0.410857\pi$$
$$60$$ 3.56155 0.459794
$$61$$ −1.56155 −0.199936 −0.0999682 0.994991i $$-0.531874\pi$$
−0.0999682 + 0.994991i $$0.531874\pi$$
$$62$$ −6.24621 −0.793270
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 5.56155 0.684580
$$67$$ 1.12311 0.137209 0.0686046 0.997644i $$-0.478145\pi$$
0.0686046 + 0.997644i $$0.478145\pi$$
$$68$$ 6.68466 0.810634
$$69$$ −6.68466 −0.804738
$$70$$ −3.56155 −0.425687
$$71$$ −9.36932 −1.11193 −0.555967 0.831205i $$-0.687652\pi$$
−0.555967 + 0.831205i $$0.687652\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 11.5616 1.35318 0.676589 0.736361i $$-0.263458\pi$$
0.676589 + 0.736361i $$0.263458\pi$$
$$74$$ −10.6847 −1.24207
$$75$$ −7.68466 −0.887348
$$76$$ 1.56155 0.179122
$$77$$ −5.56155 −0.633798
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ −3.56155 −0.398194
$$81$$ 1.00000 0.111111
$$82$$ 4.00000 0.441726
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ 1.00000 0.109109
$$85$$ −23.8078 −2.58231
$$86$$ −6.43845 −0.694276
$$87$$ −1.56155 −0.167416
$$88$$ −5.56155 −0.592864
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 3.56155 0.375421
$$91$$ 0 0
$$92$$ 6.68466 0.696924
$$93$$ −6.24621 −0.647702
$$94$$ 10.2462 1.05682
$$95$$ −5.56155 −0.570603
$$96$$ 1.00000 0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 5.56155 0.558957
$$100$$ 7.68466 0.768466
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 6.68466 0.661880
$$103$$ −1.80776 −0.178124 −0.0890621 0.996026i $$-0.528387\pi$$
−0.0890621 + 0.996026i $$0.528387\pi$$
$$104$$ 0 0
$$105$$ −3.56155 −0.347572
$$106$$ −4.87689 −0.473686
$$107$$ −4.87689 −0.471467 −0.235734 0.971818i $$-0.575749\pi$$
−0.235734 + 0.971818i $$0.575749\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −12.9309 −1.23855 −0.619276 0.785173i $$-0.712574\pi$$
−0.619276 + 0.785173i $$0.712574\pi$$
$$110$$ 19.8078 1.88860
$$111$$ −10.6847 −1.01414
$$112$$ −1.00000 −0.0944911
$$113$$ 20.2462 1.90460 0.952302 0.305158i $$-0.0987093\pi$$
0.952302 + 0.305158i $$0.0987093\pi$$
$$114$$ 1.56155 0.146253
$$115$$ −23.8078 −2.22009
$$116$$ 1.56155 0.144987
$$117$$ 0 0
$$118$$ −4.24621 −0.390895
$$119$$ −6.68466 −0.612782
$$120$$ −3.56155 −0.325124
$$121$$ 19.9309 1.81190
$$122$$ 1.56155 0.141376
$$123$$ 4.00000 0.360668
$$124$$ 6.24621 0.560926
$$125$$ −9.56155 −0.855211
$$126$$ 1.00000 0.0890871
$$127$$ −10.2462 −0.909204 −0.454602 0.890695i $$-0.650219\pi$$
−0.454602 + 0.890695i $$0.650219\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −6.43845 −0.566874
$$130$$ 0 0
$$131$$ −1.56155 −0.136434 −0.0682168 0.997671i $$-0.521731\pi$$
−0.0682168 + 0.997671i $$0.521731\pi$$
$$132$$ −5.56155 −0.484071
$$133$$ −1.56155 −0.135404
$$134$$ −1.12311 −0.0970215
$$135$$ 3.56155 0.306530
$$136$$ −6.68466 −0.573205
$$137$$ 6.68466 0.571109 0.285554 0.958362i $$-0.407822\pi$$
0.285554 + 0.958362i $$0.407822\pi$$
$$138$$ 6.68466 0.569036
$$139$$ 10.2462 0.869072 0.434536 0.900654i $$-0.356912\pi$$
0.434536 + 0.900654i $$0.356912\pi$$
$$140$$ 3.56155 0.301006
$$141$$ 10.2462 0.862887
$$142$$ 9.36932 0.786256
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −5.56155 −0.461862
$$146$$ −11.5616 −0.956841
$$147$$ −1.00000 −0.0824786
$$148$$ 10.6847 0.878274
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 7.68466 0.627450
$$151$$ 16.6847 1.35778 0.678889 0.734241i $$-0.262462\pi$$
0.678889 + 0.734241i $$0.262462\pi$$
$$152$$ −1.56155 −0.126659
$$153$$ 6.68466 0.540423
$$154$$ 5.56155 0.448163
$$155$$ −22.2462 −1.78686
$$156$$ 0 0
$$157$$ −11.8078 −0.942362 −0.471181 0.882037i $$-0.656172\pi$$
−0.471181 + 0.882037i $$0.656172\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ −4.87689 −0.386763
$$160$$ 3.56155 0.281565
$$161$$ −6.68466 −0.526825
$$162$$ −1.00000 −0.0785674
$$163$$ 9.12311 0.714577 0.357288 0.933994i $$-0.383701\pi$$
0.357288 + 0.933994i $$0.383701\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 19.8078 1.54203
$$166$$ −2.00000 −0.155230
$$167$$ −11.8078 −0.913712 −0.456856 0.889541i $$-0.651025\pi$$
−0.456856 + 0.889541i $$0.651025\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 0 0
$$170$$ 23.8078 1.82597
$$171$$ 1.56155 0.119415
$$172$$ 6.43845 0.490927
$$173$$ 3.75379 0.285395 0.142698 0.989766i $$-0.454422\pi$$
0.142698 + 0.989766i $$0.454422\pi$$
$$174$$ 1.56155 0.118381
$$175$$ −7.68466 −0.580906
$$176$$ 5.56155 0.419218
$$177$$ −4.24621 −0.319165
$$178$$ 8.00000 0.599625
$$179$$ −11.1231 −0.831380 −0.415690 0.909506i $$-0.636460\pi$$
−0.415690 + 0.909506i $$0.636460\pi$$
$$180$$ −3.56155 −0.265462
$$181$$ 13.3693 0.993733 0.496867 0.867827i $$-0.334484\pi$$
0.496867 + 0.867827i $$0.334484\pi$$
$$182$$ 0 0
$$183$$ 1.56155 0.115433
$$184$$ −6.68466 −0.492800
$$185$$ −38.0540 −2.79778
$$186$$ 6.24621 0.457994
$$187$$ 37.1771 2.71866
$$188$$ −10.2462 −0.747282
$$189$$ 1.00000 0.0727393
$$190$$ 5.56155 0.403477
$$191$$ −24.0540 −1.74048 −0.870242 0.492624i $$-0.836038\pi$$
−0.870242 + 0.492624i $$0.836038\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −12.0000 −0.863779 −0.431889 0.901927i $$-0.642153\pi$$
−0.431889 + 0.901927i $$0.642153\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −5.56155 −0.395242
$$199$$ −4.93087 −0.349540 −0.174770 0.984609i $$-0.555918\pi$$
−0.174770 + 0.984609i $$0.555918\pi$$
$$200$$ −7.68466 −0.543387
$$201$$ −1.12311 −0.0792178
$$202$$ 6.00000 0.422159
$$203$$ −1.56155 −0.109600
$$204$$ −6.68466 −0.468020
$$205$$ 14.2462 0.994999
$$206$$ 1.80776 0.125953
$$207$$ 6.68466 0.464616
$$208$$ 0 0
$$209$$ 8.68466 0.600730
$$210$$ 3.56155 0.245770
$$211$$ 7.80776 0.537509 0.268754 0.963209i $$-0.413388\pi$$
0.268754 + 0.963209i $$0.413388\pi$$
$$212$$ 4.87689 0.334946
$$213$$ 9.36932 0.641975
$$214$$ 4.87689 0.333378
$$215$$ −22.9309 −1.56387
$$216$$ 1.00000 0.0680414
$$217$$ −6.24621 −0.424020
$$218$$ 12.9309 0.875789
$$219$$ −11.5616 −0.781257
$$220$$ −19.8078 −1.33544
$$221$$ 0 0
$$222$$ 10.6847 0.717107
$$223$$ 1.75379 0.117442 0.0587212 0.998274i $$-0.481298\pi$$
0.0587212 + 0.998274i $$0.481298\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 7.68466 0.512311
$$226$$ −20.2462 −1.34676
$$227$$ 17.6155 1.16918 0.584592 0.811328i $$-0.301255\pi$$
0.584592 + 0.811328i $$0.301255\pi$$
$$228$$ −1.56155 −0.103416
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 23.8078 1.56984
$$231$$ 5.56155 0.365923
$$232$$ −1.56155 −0.102521
$$233$$ −2.00000 −0.131024 −0.0655122 0.997852i $$-0.520868\pi$$
−0.0655122 + 0.997852i $$0.520868\pi$$
$$234$$ 0 0
$$235$$ 36.4924 2.38050
$$236$$ 4.24621 0.276405
$$237$$ −16.0000 −1.03931
$$238$$ 6.68466 0.433302
$$239$$ −14.2462 −0.921511 −0.460755 0.887527i $$-0.652421\pi$$
−0.460755 + 0.887527i $$0.652421\pi$$
$$240$$ 3.56155 0.229897
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ −19.9309 −1.28120
$$243$$ −1.00000 −0.0641500
$$244$$ −1.56155 −0.0999682
$$245$$ −3.56155 −0.227539
$$246$$ −4.00000 −0.255031
$$247$$ 0 0
$$248$$ −6.24621 −0.396635
$$249$$ −2.00000 −0.126745
$$250$$ 9.56155 0.604726
$$251$$ −22.0540 −1.39203 −0.696017 0.718025i $$-0.745046\pi$$
−0.696017 + 0.718025i $$0.745046\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 37.1771 2.33730
$$254$$ 10.2462 0.642904
$$255$$ 23.8078 1.49090
$$256$$ 1.00000 0.0625000
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ 6.43845 0.400840
$$259$$ −10.6847 −0.663912
$$260$$ 0 0
$$261$$ 1.56155 0.0966577
$$262$$ 1.56155 0.0964731
$$263$$ −23.3693 −1.44101 −0.720507 0.693448i $$-0.756091\pi$$
−0.720507 + 0.693448i $$0.756091\pi$$
$$264$$ 5.56155 0.342290
$$265$$ −17.3693 −1.06699
$$266$$ 1.56155 0.0957449
$$267$$ 8.00000 0.489592
$$268$$ 1.12311 0.0686046
$$269$$ 31.3693 1.91262 0.956311 0.292353i $$-0.0944382\pi$$
0.956311 + 0.292353i $$0.0944382\pi$$
$$270$$ −3.56155 −0.216749
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 6.68466 0.405317
$$273$$ 0 0
$$274$$ −6.68466 −0.403835
$$275$$ 42.7386 2.57724
$$276$$ −6.68466 −0.402369
$$277$$ 10.8769 0.653529 0.326765 0.945106i $$-0.394042\pi$$
0.326765 + 0.945106i $$0.394042\pi$$
$$278$$ −10.2462 −0.614527
$$279$$ 6.24621 0.373951
$$280$$ −3.56155 −0.212843
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ −10.2462 −0.610153
$$283$$ 4.87689 0.289901 0.144951 0.989439i $$-0.453698\pi$$
0.144951 + 0.989439i $$0.453698\pi$$
$$284$$ −9.36932 −0.555967
$$285$$ 5.56155 0.329438
$$286$$ 0 0
$$287$$ 4.00000 0.236113
$$288$$ −1.00000 −0.0589256
$$289$$ 27.6847 1.62851
$$290$$ 5.56155 0.326586
$$291$$ 10.0000 0.586210
$$292$$ 11.5616 0.676589
$$293$$ −7.75379 −0.452981 −0.226491 0.974013i $$-0.572725\pi$$
−0.226491 + 0.974013i $$0.572725\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −15.1231 −0.880501
$$296$$ −10.6847 −0.621033
$$297$$ −5.56155 −0.322714
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ −7.68466 −0.443674
$$301$$ −6.43845 −0.371106
$$302$$ −16.6847 −0.960094
$$303$$ 6.00000 0.344691
$$304$$ 1.56155 0.0895612
$$305$$ 5.56155 0.318454
$$306$$ −6.68466 −0.382136
$$307$$ 26.2462 1.49795 0.748975 0.662598i $$-0.230546\pi$$
0.748975 + 0.662598i $$0.230546\pi$$
$$308$$ −5.56155 −0.316899
$$309$$ 1.80776 0.102840
$$310$$ 22.2462 1.26350
$$311$$ 0.492423 0.0279227 0.0139614 0.999903i $$-0.495556\pi$$
0.0139614 + 0.999903i $$0.495556\pi$$
$$312$$ 0 0
$$313$$ 32.2462 1.82266 0.911332 0.411673i $$-0.135055\pi$$
0.911332 + 0.411673i $$0.135055\pi$$
$$314$$ 11.8078 0.666351
$$315$$ 3.56155 0.200671
$$316$$ 16.0000 0.900070
$$317$$ 3.36932 0.189240 0.0946198 0.995513i $$-0.469836\pi$$
0.0946198 + 0.995513i $$0.469836\pi$$
$$318$$ 4.87689 0.273483
$$319$$ 8.68466 0.486248
$$320$$ −3.56155 −0.199097
$$321$$ 4.87689 0.272202
$$322$$ 6.68466 0.372521
$$323$$ 10.4384 0.580811
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −9.12311 −0.505282
$$327$$ 12.9309 0.715079
$$328$$ 4.00000 0.220863
$$329$$ 10.2462 0.564892
$$330$$ −19.8078 −1.09038
$$331$$ 1.12311 0.0617315 0.0308657 0.999524i $$-0.490174\pi$$
0.0308657 + 0.999524i $$0.490174\pi$$
$$332$$ 2.00000 0.109764
$$333$$ 10.6847 0.585516
$$334$$ 11.8078 0.646092
$$335$$ −4.00000 −0.218543
$$336$$ 1.00000 0.0545545
$$337$$ −20.0540 −1.09241 −0.546205 0.837652i $$-0.683928\pi$$
−0.546205 + 0.837652i $$0.683928\pi$$
$$338$$ 0 0
$$339$$ −20.2462 −1.09962
$$340$$ −23.8078 −1.29116
$$341$$ 34.7386 1.88120
$$342$$ −1.56155 −0.0844391
$$343$$ −1.00000 −0.0539949
$$344$$ −6.43845 −0.347138
$$345$$ 23.8078 1.28177
$$346$$ −3.75379 −0.201805
$$347$$ −24.4924 −1.31482 −0.657411 0.753532i $$-0.728348\pi$$
−0.657411 + 0.753532i $$0.728348\pi$$
$$348$$ −1.56155 −0.0837080
$$349$$ 3.36932 0.180355 0.0901777 0.995926i $$-0.471256\pi$$
0.0901777 + 0.995926i $$0.471256\pi$$
$$350$$ 7.68466 0.410762
$$351$$ 0 0
$$352$$ −5.56155 −0.296432
$$353$$ 18.2462 0.971148 0.485574 0.874196i $$-0.338611\pi$$
0.485574 + 0.874196i $$0.338611\pi$$
$$354$$ 4.24621 0.225684
$$355$$ 33.3693 1.77106
$$356$$ −8.00000 −0.423999
$$357$$ 6.68466 0.353790
$$358$$ 11.1231 0.587874
$$359$$ 23.6155 1.24638 0.623190 0.782071i $$-0.285836\pi$$
0.623190 + 0.782071i $$0.285836\pi$$
$$360$$ 3.56155 0.187710
$$361$$ −16.5616 −0.871661
$$362$$ −13.3693 −0.702676
$$363$$ −19.9309 −1.04610
$$364$$ 0 0
$$365$$ −41.1771 −2.15531
$$366$$ −1.56155 −0.0816237
$$367$$ 0.630683 0.0329214 0.0164607 0.999865i $$-0.494760\pi$$
0.0164607 + 0.999865i $$0.494760\pi$$
$$368$$ 6.68466 0.348462
$$369$$ −4.00000 −0.208232
$$370$$ 38.0540 1.97833
$$371$$ −4.87689 −0.253196
$$372$$ −6.24621 −0.323851
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ −37.1771 −1.92238
$$375$$ 9.56155 0.493756
$$376$$ 10.2462 0.528408
$$377$$ 0 0
$$378$$ −1.00000 −0.0514344
$$379$$ 18.8769 0.969641 0.484820 0.874614i $$-0.338885\pi$$
0.484820 + 0.874614i $$0.338885\pi$$
$$380$$ −5.56155 −0.285302
$$381$$ 10.2462 0.524929
$$382$$ 24.0540 1.23071
$$383$$ −17.5616 −0.897353 −0.448677 0.893694i $$-0.648105\pi$$
−0.448677 + 0.893694i $$0.648105\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 19.8078 1.00950
$$386$$ 12.0000 0.610784
$$387$$ 6.43845 0.327285
$$388$$ −10.0000 −0.507673
$$389$$ −27.1231 −1.37520 −0.687598 0.726092i $$-0.741335\pi$$
−0.687598 + 0.726092i $$0.741335\pi$$
$$390$$ 0 0
$$391$$ 44.6847 2.25980
$$392$$ −1.00000 −0.0505076
$$393$$ 1.56155 0.0787699
$$394$$ −6.00000 −0.302276
$$395$$ −56.9848 −2.86722
$$396$$ 5.56155 0.279479
$$397$$ 15.3693 0.771364 0.385682 0.922632i $$-0.373966\pi$$
0.385682 + 0.922632i $$0.373966\pi$$
$$398$$ 4.93087 0.247162
$$399$$ 1.56155 0.0781754
$$400$$ 7.68466 0.384233
$$401$$ −30.9848 −1.54731 −0.773655 0.633608i $$-0.781573\pi$$
−0.773655 + 0.633608i $$0.781573\pi$$
$$402$$ 1.12311 0.0560154
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ −3.56155 −0.176975
$$406$$ 1.56155 0.0774986
$$407$$ 59.4233 2.94550
$$408$$ 6.68466 0.330940
$$409$$ −13.8078 −0.682750 −0.341375 0.939927i $$-0.610893\pi$$
−0.341375 + 0.939927i $$0.610893\pi$$
$$410$$ −14.2462 −0.703570
$$411$$ −6.68466 −0.329730
$$412$$ −1.80776 −0.0890621
$$413$$ −4.24621 −0.208942
$$414$$ −6.68466 −0.328533
$$415$$ −7.12311 −0.349660
$$416$$ 0 0
$$417$$ −10.2462 −0.501759
$$418$$ −8.68466 −0.424781
$$419$$ −20.6847 −1.01051 −0.505256 0.862970i $$-0.668602\pi$$
−0.505256 + 0.862970i $$0.668602\pi$$
$$420$$ −3.56155 −0.173786
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ −7.80776 −0.380076
$$423$$ −10.2462 −0.498188
$$424$$ −4.87689 −0.236843
$$425$$ 51.3693 2.49178
$$426$$ −9.36932 −0.453945
$$427$$ 1.56155 0.0755688
$$428$$ −4.87689 −0.235734
$$429$$ 0 0
$$430$$ 22.9309 1.10582
$$431$$ −29.8617 −1.43839 −0.719195 0.694809i $$-0.755489\pi$$
−0.719195 + 0.694809i $$0.755489\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −23.3693 −1.12306 −0.561529 0.827457i $$-0.689787\pi$$
−0.561529 + 0.827457i $$0.689787\pi$$
$$434$$ 6.24621 0.299828
$$435$$ 5.56155 0.266656
$$436$$ −12.9309 −0.619276
$$437$$ 10.4384 0.499339
$$438$$ 11.5616 0.552432
$$439$$ 8.93087 0.426247 0.213124 0.977025i $$-0.431636\pi$$
0.213124 + 0.977025i $$0.431636\pi$$
$$440$$ 19.8078 0.944298
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 6.63068 0.315033 0.157517 0.987516i $$-0.449651\pi$$
0.157517 + 0.987516i $$0.449651\pi$$
$$444$$ −10.6847 −0.507071
$$445$$ 28.4924 1.35067
$$446$$ −1.75379 −0.0830443
$$447$$ 10.0000 0.472984
$$448$$ −1.00000 −0.0472456
$$449$$ −9.31534 −0.439618 −0.219809 0.975543i $$-0.570543\pi$$
−0.219809 + 0.975543i $$0.570543\pi$$
$$450$$ −7.68466 −0.362258
$$451$$ −22.2462 −1.04753
$$452$$ 20.2462 0.952302
$$453$$ −16.6847 −0.783914
$$454$$ −17.6155 −0.826738
$$455$$ 0 0
$$456$$ 1.56155 0.0731264
$$457$$ 16.0000 0.748448 0.374224 0.927338i $$-0.377909\pi$$
0.374224 + 0.927338i $$0.377909\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −6.68466 −0.312013
$$460$$ −23.8078 −1.11004
$$461$$ −36.9309 −1.72004 −0.860021 0.510259i $$-0.829550\pi$$
−0.860021 + 0.510259i $$0.829550\pi$$
$$462$$ −5.56155 −0.258747
$$463$$ −30.5464 −1.41961 −0.709806 0.704397i $$-0.751217\pi$$
−0.709806 + 0.704397i $$0.751217\pi$$
$$464$$ 1.56155 0.0724933
$$465$$ 22.2462 1.03164
$$466$$ 2.00000 0.0926482
$$467$$ −1.56155 −0.0722600 −0.0361300 0.999347i $$-0.511503\pi$$
−0.0361300 + 0.999347i $$0.511503\pi$$
$$468$$ 0 0
$$469$$ −1.12311 −0.0518602
$$470$$ −36.4924 −1.68327
$$471$$ 11.8078 0.544073
$$472$$ −4.24621 −0.195448
$$473$$ 35.8078 1.64644
$$474$$ 16.0000 0.734904
$$475$$ 12.0000 0.550598
$$476$$ −6.68466 −0.306391
$$477$$ 4.87689 0.223298
$$478$$ 14.2462 0.651607
$$479$$ 12.3002 0.562010 0.281005 0.959706i $$-0.409332\pi$$
0.281005 + 0.959706i $$0.409332\pi$$
$$480$$ −3.56155 −0.162562
$$481$$ 0 0
$$482$$ −6.00000 −0.273293
$$483$$ 6.68466 0.304162
$$484$$ 19.9309 0.905949
$$485$$ 35.6155 1.61722
$$486$$ 1.00000 0.0453609
$$487$$ 13.7538 0.623244 0.311622 0.950206i $$-0.399128\pi$$
0.311622 + 0.950206i $$0.399128\pi$$
$$488$$ 1.56155 0.0706882
$$489$$ −9.12311 −0.412561
$$490$$ 3.56155 0.160895
$$491$$ 10.2462 0.462405 0.231203 0.972906i $$-0.425734\pi$$
0.231203 + 0.972906i $$0.425734\pi$$
$$492$$ 4.00000 0.180334
$$493$$ 10.4384 0.470124
$$494$$ 0 0
$$495$$ −19.8078 −0.890293
$$496$$ 6.24621 0.280463
$$497$$ 9.36932 0.420271
$$498$$ 2.00000 0.0896221
$$499$$ 1.50758 0.0674884 0.0337442 0.999431i $$-0.489257\pi$$
0.0337442 + 0.999431i $$0.489257\pi$$
$$500$$ −9.56155 −0.427606
$$501$$ 11.8078 0.527532
$$502$$ 22.0540 0.984317
$$503$$ 35.6155 1.58802 0.794009 0.607906i $$-0.207990\pi$$
0.794009 + 0.607906i $$0.207990\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 21.3693 0.950922
$$506$$ −37.1771 −1.65272
$$507$$ 0 0
$$508$$ −10.2462 −0.454602
$$509$$ −2.68466 −0.118995 −0.0594977 0.998228i $$-0.518950\pi$$
−0.0594977 + 0.998228i $$0.518950\pi$$
$$510$$ −23.8078 −1.05423
$$511$$ −11.5616 −0.511453
$$512$$ −1.00000 −0.0441942
$$513$$ −1.56155 −0.0689442
$$514$$ −26.0000 −1.14681
$$515$$ 6.43845 0.283712
$$516$$ −6.43845 −0.283437
$$517$$ −56.9848 −2.50619
$$518$$ 10.6847 0.469457
$$519$$ −3.75379 −0.164773
$$520$$ 0 0
$$521$$ 8.05398 0.352851 0.176426 0.984314i $$-0.443547\pi$$
0.176426 + 0.984314i $$0.443547\pi$$
$$522$$ −1.56155 −0.0683473
$$523$$ −8.49242 −0.371348 −0.185674 0.982611i $$-0.559447\pi$$
−0.185674 + 0.982611i $$0.559447\pi$$
$$524$$ −1.56155 −0.0682168
$$525$$ 7.68466 0.335386
$$526$$ 23.3693 1.01895
$$527$$ 41.7538 1.81882
$$528$$ −5.56155 −0.242036
$$529$$ 21.6847 0.942811
$$530$$ 17.3693 0.754475
$$531$$ 4.24621 0.184270
$$532$$ −1.56155 −0.0677019
$$533$$ 0 0
$$534$$ −8.00000 −0.346194
$$535$$ 17.3693 0.750941
$$536$$ −1.12311 −0.0485108
$$537$$ 11.1231 0.479997
$$538$$ −31.3693 −1.35243
$$539$$ 5.56155 0.239553
$$540$$ 3.56155 0.153265
$$541$$ −6.19224 −0.266225 −0.133113 0.991101i $$-0.542497\pi$$
−0.133113 + 0.991101i $$0.542497\pi$$
$$542$$ 8.00000 0.343629
$$543$$ −13.3693 −0.573732
$$544$$ −6.68466 −0.286602
$$545$$ 46.0540 1.97274
$$546$$ 0 0
$$547$$ 28.9848 1.23930 0.619651 0.784877i $$-0.287274\pi$$
0.619651 + 0.784877i $$0.287274\pi$$
$$548$$ 6.68466 0.285554
$$549$$ −1.56155 −0.0666455
$$550$$ −42.7386 −1.82238
$$551$$ 2.43845 0.103881
$$552$$ 6.68466 0.284518
$$553$$ −16.0000 −0.680389
$$554$$ −10.8769 −0.462115
$$555$$ 38.0540 1.61530
$$556$$ 10.2462 0.434536
$$557$$ 16.2462 0.688374 0.344187 0.938901i $$-0.388155\pi$$
0.344187 + 0.938901i $$0.388155\pi$$
$$558$$ −6.24621 −0.264423
$$559$$ 0 0
$$560$$ 3.56155 0.150503
$$561$$ −37.1771 −1.56962
$$562$$ −6.00000 −0.253095
$$563$$ −38.0540 −1.60378 −0.801892 0.597469i $$-0.796173\pi$$
−0.801892 + 0.597469i $$0.796173\pi$$
$$564$$ 10.2462 0.431443
$$565$$ −72.1080 −3.03360
$$566$$ −4.87689 −0.204991
$$567$$ −1.00000 −0.0419961
$$568$$ 9.36932 0.393128
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ −5.56155 −0.232948
$$571$$ 18.2462 0.763580 0.381790 0.924249i $$-0.375308\pi$$
0.381790 + 0.924249i $$0.375308\pi$$
$$572$$ 0 0
$$573$$ 24.0540 1.00487
$$574$$ −4.00000 −0.166957
$$575$$ 51.3693 2.14225
$$576$$ 1.00000 0.0416667
$$577$$ −7.75379 −0.322794 −0.161397 0.986890i $$-0.551600\pi$$
−0.161397 + 0.986890i $$0.551600\pi$$
$$578$$ −27.6847 −1.15153
$$579$$ 12.0000 0.498703
$$580$$ −5.56155 −0.230931
$$581$$ −2.00000 −0.0829740
$$582$$ −10.0000 −0.414513
$$583$$ 27.1231 1.12332
$$584$$ −11.5616 −0.478420
$$585$$ 0 0
$$586$$ 7.75379 0.320306
$$587$$ 37.1231 1.53223 0.766117 0.642701i $$-0.222186\pi$$
0.766117 + 0.642701i $$0.222186\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 9.75379 0.401898
$$590$$ 15.1231 0.622608
$$591$$ −6.00000 −0.246807
$$592$$ 10.6847 0.439137
$$593$$ −28.0000 −1.14982 −0.574911 0.818216i $$-0.694963\pi$$
−0.574911 + 0.818216i $$0.694963\pi$$
$$594$$ 5.56155 0.228193
$$595$$ 23.8078 0.976023
$$596$$ −10.0000 −0.409616
$$597$$ 4.93087 0.201807
$$598$$ 0 0
$$599$$ 30.3002 1.23803 0.619016 0.785378i $$-0.287532\pi$$
0.619016 + 0.785378i $$0.287532\pi$$
$$600$$ 7.68466 0.313725
$$601$$ −31.8617 −1.29967 −0.649834 0.760076i $$-0.725161\pi$$
−0.649834 + 0.760076i $$0.725161\pi$$
$$602$$ 6.43845 0.262412
$$603$$ 1.12311 0.0457364
$$604$$ 16.6847 0.678889
$$605$$ −70.9848 −2.88594
$$606$$ −6.00000 −0.243733
$$607$$ −28.5464 −1.15866 −0.579331 0.815092i $$-0.696686\pi$$
−0.579331 + 0.815092i $$0.696686\pi$$
$$608$$ −1.56155 −0.0633293
$$609$$ 1.56155 0.0632773
$$610$$ −5.56155 −0.225181
$$611$$ 0 0
$$612$$ 6.68466 0.270211
$$613$$ 1.80776 0.0730149 0.0365075 0.999333i $$-0.488377\pi$$
0.0365075 + 0.999333i $$0.488377\pi$$
$$614$$ −26.2462 −1.05921
$$615$$ −14.2462 −0.574463
$$616$$ 5.56155 0.224081
$$617$$ −6.19224 −0.249290 −0.124645 0.992201i $$-0.539779\pi$$
−0.124645 + 0.992201i $$0.539779\pi$$
$$618$$ −1.80776 −0.0727189
$$619$$ −30.9309 −1.24322 −0.621608 0.783328i $$-0.713520\pi$$
−0.621608 + 0.783328i $$0.713520\pi$$
$$620$$ −22.2462 −0.893429
$$621$$ −6.68466 −0.268246
$$622$$ −0.492423 −0.0197443
$$623$$ 8.00000 0.320513
$$624$$ 0 0
$$625$$ −4.36932 −0.174773
$$626$$ −32.2462 −1.28882
$$627$$ −8.68466 −0.346832
$$628$$ −11.8078 −0.471181
$$629$$ 71.4233 2.84783
$$630$$ −3.56155 −0.141896
$$631$$ 12.6847 0.504968 0.252484 0.967601i $$-0.418752\pi$$
0.252484 + 0.967601i $$0.418752\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ −7.80776 −0.310331
$$634$$ −3.36932 −0.133813
$$635$$ 36.4924 1.44816
$$636$$ −4.87689 −0.193381
$$637$$ 0 0
$$638$$ −8.68466 −0.343829
$$639$$ −9.36932 −0.370644
$$640$$ 3.56155 0.140783
$$641$$ −30.1080 −1.18919 −0.594596 0.804024i $$-0.702688\pi$$
−0.594596 + 0.804024i $$0.702688\pi$$
$$642$$ −4.87689 −0.192476
$$643$$ 0.192236 0.00758105 0.00379052 0.999993i $$-0.498793\pi$$
0.00379052 + 0.999993i $$0.498793\pi$$
$$644$$ −6.68466 −0.263412
$$645$$ 22.9309 0.902902
$$646$$ −10.4384 −0.410695
$$647$$ −10.6307 −0.417935 −0.208968 0.977923i $$-0.567010\pi$$
−0.208968 + 0.977923i $$0.567010\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 23.6155 0.926991
$$650$$ 0 0
$$651$$ 6.24621 0.244808
$$652$$ 9.12311 0.357288
$$653$$ −29.5616 −1.15683 −0.578416 0.815742i $$-0.696329\pi$$
−0.578416 + 0.815742i $$0.696329\pi$$
$$654$$ −12.9309 −0.505637
$$655$$ 5.56155 0.217308
$$656$$ −4.00000 −0.156174
$$657$$ 11.5616 0.451059
$$658$$ −10.2462 −0.399439
$$659$$ 12.8769 0.501613 0.250806 0.968037i $$-0.419304\pi$$
0.250806 + 0.968037i $$0.419304\pi$$
$$660$$ 19.8078 0.771016
$$661$$ −44.7386 −1.74013 −0.870066 0.492936i $$-0.835924\pi$$
−0.870066 + 0.492936i $$0.835924\pi$$
$$662$$ −1.12311 −0.0436507
$$663$$ 0 0
$$664$$ −2.00000 −0.0776151
$$665$$ 5.56155 0.215668
$$666$$ −10.6847 −0.414022
$$667$$ 10.4384 0.404178
$$668$$ −11.8078 −0.456856
$$669$$ −1.75379 −0.0678054
$$670$$ 4.00000 0.154533
$$671$$ −8.68466 −0.335268
$$672$$ −1.00000 −0.0385758
$$673$$ −37.8078 −1.45738 −0.728691 0.684843i $$-0.759871\pi$$
−0.728691 + 0.684843i $$0.759871\pi$$
$$674$$ 20.0540 0.772450
$$675$$ −7.68466 −0.295783
$$676$$ 0 0
$$677$$ 31.3693 1.20562 0.602810 0.797884i $$-0.294048\pi$$
0.602810 + 0.797884i $$0.294048\pi$$
$$678$$ 20.2462 0.777551
$$679$$ 10.0000 0.383765
$$680$$ 23.8078 0.912986
$$681$$ −17.6155 −0.675029
$$682$$ −34.7386 −1.33021
$$683$$ 0.684658 0.0261977 0.0130989 0.999914i $$-0.495830\pi$$
0.0130989 + 0.999914i $$0.495830\pi$$
$$684$$ 1.56155 0.0597075
$$685$$ −23.8078 −0.909648
$$686$$ 1.00000 0.0381802
$$687$$ −2.00000 −0.0763048
$$688$$ 6.43845 0.245463
$$689$$ 0 0
$$690$$ −23.8078 −0.906346
$$691$$ −36.4924 −1.38824 −0.694119 0.719861i $$-0.744206\pi$$
−0.694119 + 0.719861i $$0.744206\pi$$
$$692$$ 3.75379 0.142698
$$693$$ −5.56155 −0.211266
$$694$$ 24.4924 0.929720
$$695$$ −36.4924 −1.38424
$$696$$ 1.56155 0.0591905
$$697$$ −26.7386 −1.01280
$$698$$ −3.36932 −0.127531
$$699$$ 2.00000 0.0756469
$$700$$ −7.68466 −0.290453
$$701$$ 3.61553 0.136557 0.0682783 0.997666i $$-0.478249\pi$$
0.0682783 + 0.997666i $$0.478249\pi$$
$$702$$ 0 0
$$703$$ 16.6847 0.629274
$$704$$ 5.56155 0.209609
$$705$$ −36.4924 −1.37438
$$706$$ −18.2462 −0.686705
$$707$$ 6.00000 0.225653
$$708$$ −4.24621 −0.159582
$$709$$ 31.7538 1.19254 0.596269 0.802784i $$-0.296649\pi$$
0.596269 + 0.802784i $$0.296649\pi$$
$$710$$ −33.3693 −1.25233
$$711$$ 16.0000 0.600047
$$712$$ 8.00000 0.299813
$$713$$ 41.7538 1.56369
$$714$$ −6.68466 −0.250167
$$715$$ 0 0
$$716$$ −11.1231 −0.415690
$$717$$ 14.2462 0.532035
$$718$$ −23.6155 −0.881324
$$719$$ −13.7538 −0.512930 −0.256465 0.966554i $$-0.582558\pi$$
−0.256465 + 0.966554i $$0.582558\pi$$
$$720$$ −3.56155 −0.132731
$$721$$ 1.80776 0.0673247
$$722$$ 16.5616 0.616357
$$723$$ −6.00000 −0.223142
$$724$$ 13.3693 0.496867
$$725$$ 12.0000 0.445669
$$726$$ 19.9309 0.739704
$$727$$ 48.9309 1.81475 0.907373 0.420327i $$-0.138085\pi$$
0.907373 + 0.420327i $$0.138085\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 41.1771 1.52403
$$731$$ 43.0388 1.59185
$$732$$ 1.56155 0.0577167
$$733$$ 22.8769 0.844977 0.422489 0.906368i $$-0.361157\pi$$
0.422489 + 0.906368i $$0.361157\pi$$
$$734$$ −0.630683 −0.0232789
$$735$$ 3.56155 0.131370
$$736$$ −6.68466 −0.246400
$$737$$ 6.24621 0.230082
$$738$$ 4.00000 0.147242
$$739$$ −17.6155 −0.647998 −0.323999 0.946057i $$-0.605027\pi$$
−0.323999 + 0.946057i $$0.605027\pi$$
$$740$$ −38.0540 −1.39889
$$741$$ 0 0
$$742$$ 4.87689 0.179036
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 6.24621 0.228997
$$745$$ 35.6155 1.30485
$$746$$ 6.00000 0.219676
$$747$$ 2.00000 0.0731762
$$748$$ 37.1771 1.35933
$$749$$ 4.87689 0.178198
$$750$$ −9.56155 −0.349139
$$751$$ −2.24621 −0.0819654 −0.0409827 0.999160i $$-0.513049\pi$$
−0.0409827 + 0.999160i $$0.513049\pi$$
$$752$$ −10.2462 −0.373641
$$753$$ 22.0540 0.803692
$$754$$ 0 0
$$755$$ −59.4233 −2.16264
$$756$$ 1.00000 0.0363696
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −18.8769 −0.685640
$$759$$ −37.1771 −1.34944
$$760$$ 5.56155 0.201739
$$761$$ −39.1231 −1.41821 −0.709106 0.705102i $$-0.750901\pi$$
−0.709106 + 0.705102i $$0.750901\pi$$
$$762$$ −10.2462 −0.371181
$$763$$ 12.9309 0.468129
$$764$$ −24.0540 −0.870242
$$765$$ −23.8078 −0.860772
$$766$$ 17.5616 0.634525
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 8.43845 0.304298 0.152149 0.988358i $$-0.451381\pi$$
0.152149 + 0.988358i $$0.451381\pi$$
$$770$$ −19.8078 −0.713822
$$771$$ −26.0000 −0.936367
$$772$$ −12.0000 −0.431889
$$773$$ −12.9309 −0.465091 −0.232546 0.972586i $$-0.574705\pi$$
−0.232546 + 0.972586i $$0.574705\pi$$
$$774$$ −6.43845 −0.231425
$$775$$ 48.0000 1.72421
$$776$$ 10.0000 0.358979
$$777$$ 10.6847 0.383310
$$778$$ 27.1231 0.972410
$$779$$ −6.24621 −0.223794
$$780$$ 0 0
$$781$$ −52.1080 −1.86457
$$782$$ −44.6847 −1.59792
$$783$$ −1.56155 −0.0558053
$$784$$ 1.00000 0.0357143
$$785$$ 42.0540 1.50097
$$786$$ −1.56155 −0.0556987
$$787$$ 17.0691 0.608449 0.304224 0.952600i $$-0.401603\pi$$
0.304224 + 0.952600i $$0.401603\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 23.3693 0.831970
$$790$$ 56.9848 2.02743
$$791$$ −20.2462 −0.719872
$$792$$ −5.56155 −0.197621
$$793$$ 0 0
$$794$$ −15.3693 −0.545437
$$795$$ 17.3693 0.616026
$$796$$ −4.93087 −0.174770
$$797$$ −3.75379 −0.132966 −0.0664830 0.997788i $$-0.521178\pi$$
−0.0664830 + 0.997788i $$0.521178\pi$$
$$798$$ −1.56155 −0.0552784
$$799$$ −68.4924 −2.42309
$$800$$ −7.68466 −0.271694
$$801$$ −8.00000 −0.282666
$$802$$ 30.9848 1.09411
$$803$$ 64.3002 2.26910
$$804$$ −1.12311 −0.0396089
$$805$$ 23.8078 0.839113
$$806$$ 0 0
$$807$$ −31.3693 −1.10425
$$808$$ 6.00000 0.211079
$$809$$ −55.3693 −1.94668 −0.973341 0.229364i $$-0.926335\pi$$
−0.973341 + 0.229364i $$0.926335\pi$$
$$810$$ 3.56155 0.125140
$$811$$ 2.05398 0.0721248 0.0360624 0.999350i $$-0.488518\pi$$
0.0360624 + 0.999350i $$0.488518\pi$$
$$812$$ −1.56155 −0.0547998
$$813$$ 8.00000 0.280572
$$814$$ −59.4233 −2.08279
$$815$$ −32.4924 −1.13816
$$816$$ −6.68466 −0.234010
$$817$$ 10.0540 0.351744
$$818$$ 13.8078 0.482777
$$819$$ 0 0
$$820$$ 14.2462 0.497499
$$821$$ 30.8769 1.07761 0.538806 0.842430i $$-0.318876\pi$$
0.538806 + 0.842430i $$0.318876\pi$$
$$822$$ 6.68466 0.233154
$$823$$ 30.7386 1.07148 0.535741 0.844383i $$-0.320032\pi$$
0.535741 + 0.844383i $$0.320032\pi$$
$$824$$ 1.80776 0.0629764
$$825$$ −42.7386 −1.48797
$$826$$ 4.24621 0.147745
$$827$$ 22.0540 0.766892 0.383446 0.923563i $$-0.374737\pi$$
0.383446 + 0.923563i $$0.374737\pi$$
$$828$$ 6.68466 0.232308
$$829$$ −42.0540 −1.46059 −0.730297 0.683129i $$-0.760619\pi$$
−0.730297 + 0.683129i $$0.760619\pi$$
$$830$$ 7.12311 0.247247
$$831$$ −10.8769 −0.377315
$$832$$ 0 0
$$833$$ 6.68466 0.231610
$$834$$ 10.2462 0.354797
$$835$$ 42.0540 1.45534
$$836$$ 8.68466 0.300365
$$837$$ −6.24621 −0.215901
$$838$$ 20.6847 0.714540
$$839$$ 25.7538 0.889120 0.444560 0.895749i $$-0.353360\pi$$
0.444560 + 0.895749i $$0.353360\pi$$
$$840$$ 3.56155 0.122885
$$841$$ −26.5616 −0.915916
$$842$$ −10.0000 −0.344623
$$843$$ −6.00000 −0.206651
$$844$$ 7.80776 0.268754
$$845$$ 0 0
$$846$$ 10.2462 0.352272
$$847$$ −19.9309 −0.684833
$$848$$ 4.87689 0.167473
$$849$$ −4.87689 −0.167375
$$850$$ −51.3693 −1.76195
$$851$$ 71.4233 2.44836
$$852$$ 9.36932 0.320988
$$853$$ 10.8769 0.372418 0.186209 0.982510i $$-0.440380\pi$$
0.186209 + 0.982510i $$0.440380\pi$$
$$854$$ −1.56155 −0.0534352
$$855$$ −5.56155 −0.190201
$$856$$ 4.87689 0.166689
$$857$$ 38.0000 1.29806 0.649028 0.760765i $$-0.275176\pi$$
0.649028 + 0.760765i $$0.275176\pi$$
$$858$$ 0 0
$$859$$ 0.384472 0.0131180 0.00655901 0.999978i $$-0.497912\pi$$
0.00655901 + 0.999978i $$0.497912\pi$$
$$860$$ −22.9309 −0.781936
$$861$$ −4.00000 −0.136320
$$862$$ 29.8617 1.01709
$$863$$ 42.7386 1.45484 0.727420 0.686192i $$-0.240719\pi$$
0.727420 + 0.686192i $$0.240719\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −13.3693 −0.454570
$$866$$ 23.3693 0.794122
$$867$$ −27.6847 −0.940220
$$868$$ −6.24621 −0.212010
$$869$$ 88.9848 3.01860
$$870$$ −5.56155 −0.188554
$$871$$ 0 0
$$872$$ 12.9309 0.437895
$$873$$ −10.0000 −0.338449
$$874$$ −10.4384 −0.353086
$$875$$ 9.56155 0.323239
$$876$$ −11.5616 −0.390629
$$877$$ 22.9848 0.776143 0.388072 0.921629i $$-0.373141\pi$$
0.388072 + 0.921629i $$0.373141\pi$$
$$878$$ −8.93087 −0.301402
$$879$$ 7.75379 0.261529
$$880$$ −19.8078 −0.667720
$$881$$ −14.3002 −0.481786 −0.240893 0.970552i $$-0.577440\pi$$
−0.240893 + 0.970552i $$0.577440\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 14.4384 0.485892 0.242946 0.970040i $$-0.421886\pi$$
0.242946 + 0.970040i $$0.421886\pi$$
$$884$$ 0 0
$$885$$ 15.1231 0.508358
$$886$$ −6.63068 −0.222762
$$887$$ −37.8617 −1.27127 −0.635636 0.771989i $$-0.719262\pi$$
−0.635636 + 0.771989i $$0.719262\pi$$
$$888$$ 10.6847 0.358554
$$889$$ 10.2462 0.343647
$$890$$ −28.4924 −0.955068
$$891$$ 5.56155 0.186319
$$892$$ 1.75379 0.0587212
$$893$$ −16.0000 −0.535420
$$894$$ −10.0000 −0.334450
$$895$$ 39.6155 1.32420
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 9.31534 0.310857
$$899$$ 9.75379 0.325307
$$900$$ 7.68466 0.256155
$$901$$ 32.6004 1.08608
$$902$$ 22.2462 0.740718
$$903$$ 6.43845 0.214258
$$904$$ −20.2462 −0.673379
$$905$$ −47.6155 −1.58279
$$906$$ 16.6847 0.554311
$$907$$ 24.4924 0.813258 0.406629 0.913593i $$-0.366704\pi$$
0.406629 + 0.913593i $$0.366704\pi$$
$$908$$ 17.6155 0.584592
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −4.43845 −0.147052 −0.0735262 0.997293i $$-0.523425\pi$$
−0.0735262 + 0.997293i $$0.523425\pi$$
$$912$$ −1.56155 −0.0517082
$$913$$ 11.1231 0.368121
$$914$$ −16.0000 −0.529233
$$915$$ −5.56155 −0.183859
$$916$$ 2.00000 0.0660819
$$917$$ 1.56155 0.0515670
$$918$$ 6.68466 0.220627
$$919$$ −17.8617 −0.589204 −0.294602 0.955620i $$-0.595187\pi$$
−0.294602 + 0.955620i $$0.595187\pi$$
$$920$$ 23.8078 0.784919
$$921$$ −26.2462 −0.864842
$$922$$ 36.9309 1.21625
$$923$$ 0 0
$$924$$ 5.56155 0.182962
$$925$$ 82.1080 2.69969
$$926$$ 30.5464 1.00382
$$927$$ −1.80776 −0.0593748
$$928$$ −1.56155 −0.0512605
$$929$$ 40.8769 1.34113 0.670564 0.741852i $$-0.266052\pi$$
0.670564 + 0.741852i $$0.266052\pi$$
$$930$$ −22.2462 −0.729482
$$931$$ 1.56155 0.0511778
$$932$$ −2.00000 −0.0655122
$$933$$ −0.492423 −0.0161212
$$934$$ 1.56155 0.0510956
$$935$$ −132.408 −4.33021
$$936$$ 0 0
$$937$$ −19.8617 −0.648855 −0.324427 0.945911i $$-0.605172\pi$$
−0.324427 + 0.945911i $$0.605172\pi$$
$$938$$ 1.12311 0.0366707
$$939$$ −32.2462 −1.05232
$$940$$ 36.4924 1.19025
$$941$$ 26.4924 0.863628 0.431814 0.901963i $$-0.357874\pi$$
0.431814 + 0.901963i $$0.357874\pi$$
$$942$$ −11.8078 −0.384718
$$943$$ −26.7386 −0.870730
$$944$$ 4.24621 0.138202
$$945$$ −3.56155 −0.115857
$$946$$ −35.8078 −1.16421
$$947$$ 27.8078 0.903631 0.451815 0.892111i $$-0.350777\pi$$
0.451815 + 0.892111i $$0.350777\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ 0 0
$$950$$ −12.0000 −0.389331
$$951$$ −3.36932 −0.109258
$$952$$ 6.68466 0.216651
$$953$$ −0.738634 −0.0239267 −0.0119633 0.999928i $$-0.503808\pi$$
−0.0119633 + 0.999928i $$0.503808\pi$$
$$954$$ −4.87689 −0.157895
$$955$$ 85.6695 2.77220
$$956$$ −14.2462 −0.460755
$$957$$ −8.68466 −0.280735
$$958$$ −12.3002 −0.397401
$$959$$ −6.68466 −0.215859
$$960$$ 3.56155 0.114949
$$961$$ 8.01515 0.258553
$$962$$ 0 0
$$963$$ −4.87689 −0.157156
$$964$$ 6.00000 0.193247
$$965$$ 42.7386 1.37581
$$966$$ −6.68466 −0.215075
$$967$$ −2.93087 −0.0942504 −0.0471252 0.998889i $$-0.515006\pi$$
−0.0471252 + 0.998889i $$0.515006\pi$$
$$968$$ −19.9309 −0.640602
$$969$$ −10.4384 −0.335331
$$970$$ −35.6155 −1.14355
$$971$$ 4.00000 0.128366 0.0641831 0.997938i $$-0.479556\pi$$
0.0641831 + 0.997938i $$0.479556\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −10.2462 −0.328478
$$974$$ −13.7538 −0.440700
$$975$$ 0 0
$$976$$ −1.56155 −0.0499841
$$977$$ 46.7926 1.49703 0.748514 0.663119i $$-0.230768\pi$$
0.748514 + 0.663119i $$0.230768\pi$$
$$978$$ 9.12311 0.291725
$$979$$ −44.4924 −1.42198
$$980$$ −3.56155 −0.113770
$$981$$ −12.9309 −0.412851
$$982$$ −10.2462 −0.326970
$$983$$ −2.43845 −0.0777744 −0.0388872 0.999244i $$-0.512381\pi$$
−0.0388872 + 0.999244i $$0.512381\pi$$
$$984$$ −4.00000 −0.127515
$$985$$ −21.3693 −0.680883
$$986$$ −10.4384 −0.332428
$$987$$ −10.2462 −0.326140
$$988$$ 0 0
$$989$$ 43.0388 1.36855
$$990$$ 19.8078 0.629532
$$991$$ −30.2462 −0.960803 −0.480401 0.877049i $$-0.659509\pi$$
−0.480401 + 0.877049i $$0.659509\pi$$
$$992$$ −6.24621 −0.198317
$$993$$ −1.12311 −0.0356407
$$994$$ −9.36932 −0.297177
$$995$$ 17.5616 0.556739
$$996$$ −2.00000 −0.0633724
$$997$$ −34.6307 −1.09676 −0.548382 0.836228i $$-0.684756\pi$$
−0.548382 + 0.836228i $$0.684756\pi$$
$$998$$ −1.50758 −0.0477215
$$999$$ −10.6847 −0.338048
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 7098.2.a.bg.1.1 2
13.5 odd 4 546.2.c.e.337.4 yes 4
13.8 odd 4 546.2.c.e.337.1 4
13.12 even 2 7098.2.a.bv.1.2 2
39.5 even 4 1638.2.c.h.883.1 4
39.8 even 4 1638.2.c.h.883.4 4
52.31 even 4 4368.2.h.n.337.4 4
52.47 even 4 4368.2.h.n.337.1 4
91.34 even 4 3822.2.c.h.883.2 4
91.83 even 4 3822.2.c.h.883.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.e.337.1 4 13.8 odd 4
546.2.c.e.337.4 yes 4 13.5 odd 4
1638.2.c.h.883.1 4 39.5 even 4
1638.2.c.h.883.4 4 39.8 even 4
3822.2.c.h.883.2 4 91.34 even 4
3822.2.c.h.883.3 4 91.83 even 4
4368.2.h.n.337.1 4 52.47 even 4
4368.2.h.n.337.4 4 52.31 even 4
7098.2.a.bg.1.1 2 1.1 even 1 trivial
7098.2.a.bv.1.2 2 13.12 even 2