# Properties

 Label 5070.2.b.r.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$2.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.r.1351.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +3.56155i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +3.56155i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.12311i q^{11} -1.00000 q^{12} +3.56155 q^{14} -1.00000i q^{15} +1.00000 q^{16} -5.12311 q^{17} -1.00000i q^{18} +3.56155i q^{19} +1.00000i q^{20} +3.56155i q^{21} -4.12311 q^{22} +7.68466 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -3.56155i q^{28} +6.56155 q^{29} -1.00000 q^{30} -5.68466i q^{31} -1.00000i q^{32} -4.12311i q^{33} +5.12311i q^{34} +3.56155 q^{35} -1.00000 q^{36} +4.12311i q^{37} +3.56155 q^{38} +1.00000 q^{40} +4.24621i q^{41} +3.56155 q^{42} -4.56155 q^{43} +4.12311i q^{44} -1.00000i q^{45} -7.68466i q^{46} +7.00000i q^{47} +1.00000 q^{48} -5.68466 q^{49} +1.00000i q^{50} -5.12311 q^{51} +4.43845 q^{53} -1.00000i q^{54} -4.12311 q^{55} -3.56155 q^{56} +3.56155i q^{57} -6.56155i q^{58} +10.5616i q^{59} +1.00000i q^{60} +6.00000 q^{61} -5.68466 q^{62} +3.56155i q^{63} -1.00000 q^{64} -4.12311 q^{66} -14.2462i q^{67} +5.12311 q^{68} +7.68466 q^{69} -3.56155i q^{70} +4.87689i q^{71} +1.00000i q^{72} -15.3693i q^{73} +4.12311 q^{74} -1.00000 q^{75} -3.56155i q^{76} +14.6847 q^{77} +7.43845 q^{79} -1.00000i q^{80} +1.00000 q^{81} +4.24621 q^{82} -1.12311i q^{83} -3.56155i q^{84} +5.12311i q^{85} +4.56155i q^{86} +6.56155 q^{87} +4.12311 q^{88} +1.80776i q^{89} -1.00000 q^{90} -7.68466 q^{92} -5.68466i q^{93} +7.00000 q^{94} +3.56155 q^{95} -1.00000i q^{96} -1.12311i q^{97} +5.68466i q^{98} -4.12311i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} - 4q^{12} + 6q^{14} + 4q^{16} - 4q^{17} + 6q^{23} - 4q^{25} + 4q^{27} + 18q^{29} - 4q^{30} + 6q^{35} - 4q^{36} + 6q^{38} + 4q^{40} + 6q^{42} - 10q^{43} + 4q^{48} + 2q^{49} - 4q^{51} + 26q^{53} - 6q^{56} + 24q^{61} + 2q^{62} - 4q^{64} + 4q^{68} + 6q^{69} - 4q^{75} + 34q^{77} + 38q^{79} + 4q^{81} - 16q^{82} + 18q^{87} - 4q^{90} - 6q^{92} + 28q^{94} + 6q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ 3.56155i 1.34614i 0.739579 + 0.673070i $$0.235025\pi$$
−0.739579 + 0.673070i $$0.764975\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ − 4.12311i − 1.24316i −0.783349 0.621582i $$-0.786490\pi$$
0.783349 0.621582i $$-0.213510\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 3.56155 0.951865
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −5.12311 −1.24254 −0.621268 0.783598i $$-0.713382\pi$$
−0.621268 + 0.783598i $$0.713382\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 3.56155i 0.817076i 0.912741 + 0.408538i $$0.133961\pi$$
−0.912741 + 0.408538i $$0.866039\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ 3.56155i 0.777195i
$$22$$ −4.12311 −0.879049
$$23$$ 7.68466 1.60236 0.801181 0.598422i $$-0.204205\pi$$
0.801181 + 0.598422i $$0.204205\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ − 3.56155i − 0.673070i
$$29$$ 6.56155 1.21845 0.609225 0.792998i $$-0.291481\pi$$
0.609225 + 0.792998i $$0.291481\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 5.68466i − 1.02099i −0.859879 0.510497i $$-0.829461\pi$$
0.859879 0.510497i $$-0.170539\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 4.12311i − 0.717741i
$$34$$ 5.12311i 0.878605i
$$35$$ 3.56155 0.602012
$$36$$ −1.00000 −0.166667
$$37$$ 4.12311i 0.677834i 0.940816 + 0.338917i $$0.110061\pi$$
−0.940816 + 0.338917i $$0.889939\pi$$
$$38$$ 3.56155 0.577760
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 4.24621i 0.663147i 0.943429 + 0.331573i $$0.107579\pi$$
−0.943429 + 0.331573i $$0.892421\pi$$
$$42$$ 3.56155 0.549560
$$43$$ −4.56155 −0.695630 −0.347815 0.937563i $$-0.613076\pi$$
−0.347815 + 0.937563i $$0.613076\pi$$
$$44$$ 4.12311i 0.621582i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 7.68466i − 1.13304i
$$47$$ 7.00000i 1.02105i 0.859861 + 0.510527i $$0.170550\pi$$
−0.859861 + 0.510527i $$0.829450\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −5.68466 −0.812094
$$50$$ 1.00000i 0.141421i
$$51$$ −5.12311 −0.717378
$$52$$ 0 0
$$53$$ 4.43845 0.609668 0.304834 0.952406i $$-0.401399\pi$$
0.304834 + 0.952406i $$0.401399\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ −4.12311 −0.555959
$$56$$ −3.56155 −0.475933
$$57$$ 3.56155i 0.471739i
$$58$$ − 6.56155i − 0.861574i
$$59$$ 10.5616i 1.37500i 0.726186 + 0.687499i $$0.241291\pi$$
−0.726186 + 0.687499i $$0.758709\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −5.68466 −0.721952
$$63$$ 3.56155i 0.448713i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.12311 −0.507519
$$67$$ − 14.2462i − 1.74045i −0.492653 0.870226i $$-0.663973\pi$$
0.492653 0.870226i $$-0.336027\pi$$
$$68$$ 5.12311 0.621268
$$69$$ 7.68466 0.925124
$$70$$ − 3.56155i − 0.425687i
$$71$$ 4.87689i 0.578781i 0.957211 + 0.289390i $$0.0934526\pi$$
−0.957211 + 0.289390i $$0.906547\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 15.3693i − 1.79884i −0.437083 0.899421i $$-0.643988\pi$$
0.437083 0.899421i $$-0.356012\pi$$
$$74$$ 4.12311 0.479301
$$75$$ −1.00000 −0.115470
$$76$$ − 3.56155i − 0.408538i
$$77$$ 14.6847 1.67347
$$78$$ 0 0
$$79$$ 7.43845 0.836891 0.418445 0.908242i $$-0.362575\pi$$
0.418445 + 0.908242i $$0.362575\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 4.24621 0.468916
$$83$$ − 1.12311i − 0.123277i −0.998099 0.0616384i $$-0.980367\pi$$
0.998099 0.0616384i $$-0.0196326\pi$$
$$84$$ − 3.56155i − 0.388597i
$$85$$ 5.12311i 0.555679i
$$86$$ 4.56155i 0.491885i
$$87$$ 6.56155 0.703472
$$88$$ 4.12311 0.439525
$$89$$ 1.80776i 0.191623i 0.995400 + 0.0958113i $$0.0305445\pi$$
−0.995400 + 0.0958113i $$0.969455\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −7.68466 −0.801181
$$93$$ − 5.68466i − 0.589472i
$$94$$ 7.00000 0.721995
$$95$$ 3.56155 0.365408
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 1.12311i − 0.114034i −0.998373 0.0570170i $$-0.981841\pi$$
0.998373 0.0570170i $$-0.0181589\pi$$
$$98$$ 5.68466i 0.574237i
$$99$$ − 4.12311i − 0.414388i
$$100$$ 1.00000 0.100000
$$101$$ 17.1231 1.70381 0.851906 0.523694i $$-0.175447\pi$$
0.851906 + 0.523694i $$0.175447\pi$$
$$102$$ 5.12311i 0.507263i
$$103$$ −0.438447 −0.0432015 −0.0216007 0.999767i $$-0.506876\pi$$
−0.0216007 + 0.999767i $$0.506876\pi$$
$$104$$ 0 0
$$105$$ 3.56155 0.347572
$$106$$ − 4.43845i − 0.431100i
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 20.2462i 1.93924i 0.244625 + 0.969618i $$0.421335\pi$$
−0.244625 + 0.969618i $$0.578665\pi$$
$$110$$ 4.12311i 0.393123i
$$111$$ 4.12311i 0.391348i
$$112$$ 3.56155i 0.336535i
$$113$$ −3.68466 −0.346624 −0.173312 0.984867i $$-0.555447\pi$$
−0.173312 + 0.984867i $$0.555447\pi$$
$$114$$ 3.56155 0.333570
$$115$$ − 7.68466i − 0.716598i
$$116$$ −6.56155 −0.609225
$$117$$ 0 0
$$118$$ 10.5616 0.972270
$$119$$ − 18.2462i − 1.67263i
$$120$$ 1.00000 0.0912871
$$121$$ −6.00000 −0.545455
$$122$$ − 6.00000i − 0.543214i
$$123$$ 4.24621i 0.382868i
$$124$$ 5.68466i 0.510497i
$$125$$ 1.00000i 0.0894427i
$$126$$ 3.56155 0.317288
$$127$$ 4.43845 0.393849 0.196924 0.980419i $$-0.436905\pi$$
0.196924 + 0.980419i $$0.436905\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −4.56155 −0.401622
$$130$$ 0 0
$$131$$ 6.12311 0.534978 0.267489 0.963561i $$-0.413806\pi$$
0.267489 + 0.963561i $$0.413806\pi$$
$$132$$ 4.12311i 0.358870i
$$133$$ −12.6847 −1.09990
$$134$$ −14.2462 −1.23069
$$135$$ − 1.00000i − 0.0860663i
$$136$$ − 5.12311i − 0.439303i
$$137$$ − 8.80776i − 0.752498i −0.926519 0.376249i $$-0.877214\pi$$
0.926519 0.376249i $$-0.122786\pi$$
$$138$$ − 7.68466i − 0.654162i
$$139$$ 8.43845 0.715740 0.357870 0.933771i $$-0.383503\pi$$
0.357870 + 0.933771i $$0.383503\pi$$
$$140$$ −3.56155 −0.301006
$$141$$ 7.00000i 0.589506i
$$142$$ 4.87689 0.409260
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 6.56155i − 0.544907i
$$146$$ −15.3693 −1.27197
$$147$$ −5.68466 −0.468863
$$148$$ − 4.12311i − 0.338917i
$$149$$ 22.8078i 1.86848i 0.356639 + 0.934242i $$0.383923\pi$$
−0.356639 + 0.934242i $$0.616077\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 9.36932i 0.762464i 0.924479 + 0.381232i $$0.124500\pi$$
−0.924479 + 0.381232i $$0.875500\pi$$
$$152$$ −3.56155 −0.288880
$$153$$ −5.12311 −0.414179
$$154$$ − 14.6847i − 1.18332i
$$155$$ −5.68466 −0.456603
$$156$$ 0 0
$$157$$ 22.1231 1.76562 0.882808 0.469734i $$-0.155650\pi$$
0.882808 + 0.469734i $$0.155650\pi$$
$$158$$ − 7.43845i − 0.591771i
$$159$$ 4.43845 0.351992
$$160$$ −1.00000 −0.0790569
$$161$$ 27.3693i 2.15700i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 18.5616i − 1.45385i −0.686715 0.726927i $$-0.740948\pi$$
0.686715 0.726927i $$-0.259052\pi$$
$$164$$ − 4.24621i − 0.331573i
$$165$$ −4.12311 −0.320983
$$166$$ −1.12311 −0.0871699
$$167$$ − 20.3693i − 1.57623i −0.615531 0.788113i $$-0.711058\pi$$
0.615531 0.788113i $$-0.288942\pi$$
$$168$$ −3.56155 −0.274780
$$169$$ 0 0
$$170$$ 5.12311 0.392924
$$171$$ 3.56155i 0.272359i
$$172$$ 4.56155 0.347815
$$173$$ 25.1771 1.91418 0.957089 0.289794i $$-0.0935868\pi$$
0.957089 + 0.289794i $$0.0935868\pi$$
$$174$$ − 6.56155i − 0.497430i
$$175$$ − 3.56155i − 0.269228i
$$176$$ − 4.12311i − 0.310791i
$$177$$ 10.5616i 0.793855i
$$178$$ 1.80776 0.135498
$$179$$ 0.315342 0.0235697 0.0117849 0.999931i $$-0.496249\pi$$
0.0117849 + 0.999931i $$0.496249\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ −11.1231 −0.826774 −0.413387 0.910555i $$-0.635654\pi$$
−0.413387 + 0.910555i $$0.635654\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 7.68466i 0.566521i
$$185$$ 4.12311 0.303137
$$186$$ −5.68466 −0.416819
$$187$$ 21.1231i 1.54467i
$$188$$ − 7.00000i − 0.510527i
$$189$$ 3.56155i 0.259065i
$$190$$ − 3.56155i − 0.258382i
$$191$$ 19.1231 1.38370 0.691850 0.722042i $$-0.256796\pi$$
0.691850 + 0.722042i $$0.256796\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ −1.12311 −0.0806343
$$195$$ 0 0
$$196$$ 5.68466 0.406047
$$197$$ 7.80776i 0.556280i 0.960541 + 0.278140i $$0.0897179\pi$$
−0.960541 + 0.278140i $$0.910282\pi$$
$$198$$ −4.12311 −0.293016
$$199$$ −11.1231 −0.788496 −0.394248 0.919004i $$-0.628995\pi$$
−0.394248 + 0.919004i $$0.628995\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ − 14.2462i − 1.00485i
$$202$$ − 17.1231i − 1.20478i
$$203$$ 23.3693i 1.64020i
$$204$$ 5.12311 0.358689
$$205$$ 4.24621 0.296568
$$206$$ 0.438447i 0.0305481i
$$207$$ 7.68466 0.534121
$$208$$ 0 0
$$209$$ 14.6847 1.01576
$$210$$ − 3.56155i − 0.245770i
$$211$$ 6.93087 0.477141 0.238570 0.971125i $$-0.423321\pi$$
0.238570 + 0.971125i $$0.423321\pi$$
$$212$$ −4.43845 −0.304834
$$213$$ 4.87689i 0.334159i
$$214$$ − 2.00000i − 0.136717i
$$215$$ 4.56155i 0.311095i
$$216$$ 1.00000i 0.0680414i
$$217$$ 20.2462 1.37440
$$218$$ 20.2462 1.37125
$$219$$ − 15.3693i − 1.03856i
$$220$$ 4.12311 0.277980
$$221$$ 0 0
$$222$$ 4.12311 0.276725
$$223$$ 26.3002i 1.76119i 0.473869 + 0.880595i $$0.342857\pi$$
−0.473869 + 0.880595i $$0.657143\pi$$
$$224$$ 3.56155 0.237966
$$225$$ −1.00000 −0.0666667
$$226$$ 3.68466i 0.245100i
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ − 3.56155i − 0.235870i
$$229$$ − 7.75379i − 0.512385i −0.966626 0.256192i $$-0.917532\pi$$
0.966626 0.256192i $$-0.0824681\pi$$
$$230$$ −7.68466 −0.506711
$$231$$ 14.6847 0.966180
$$232$$ 6.56155i 0.430787i
$$233$$ 17.6847 1.15856 0.579280 0.815128i $$-0.303334\pi$$
0.579280 + 0.815128i $$0.303334\pi$$
$$234$$ 0 0
$$235$$ 7.00000 0.456630
$$236$$ − 10.5616i − 0.687499i
$$237$$ 7.43845 0.483179
$$238$$ −18.2462 −1.18273
$$239$$ − 13.3693i − 0.864789i −0.901684 0.432395i $$-0.857669\pi$$
0.901684 0.432395i $$-0.142331\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 19.8769i − 1.28038i −0.768215 0.640192i $$-0.778855\pi$$
0.768215 0.640192i $$-0.221145\pi$$
$$242$$ 6.00000i 0.385695i
$$243$$ 1.00000 0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 5.68466i 0.363180i
$$246$$ 4.24621 0.270729
$$247$$ 0 0
$$248$$ 5.68466 0.360976
$$249$$ − 1.12311i − 0.0711739i
$$250$$ 1.00000 0.0632456
$$251$$ −0.123106 −0.00777036 −0.00388518 0.999992i $$-0.501237\pi$$
−0.00388518 + 0.999992i $$0.501237\pi$$
$$252$$ − 3.56155i − 0.224357i
$$253$$ − 31.6847i − 1.99200i
$$254$$ − 4.43845i − 0.278493i
$$255$$ 5.12311i 0.320821i
$$256$$ 1.00000 0.0625000
$$257$$ 1.43845 0.0897279 0.0448639 0.998993i $$-0.485715\pi$$
0.0448639 + 0.998993i $$0.485715\pi$$
$$258$$ 4.56155i 0.283990i
$$259$$ −14.6847 −0.912460
$$260$$ 0 0
$$261$$ 6.56155 0.406150
$$262$$ − 6.12311i − 0.378287i
$$263$$ −13.0000 −0.801614 −0.400807 0.916162i $$-0.631270\pi$$
−0.400807 + 0.916162i $$0.631270\pi$$
$$264$$ 4.12311 0.253760
$$265$$ − 4.43845i − 0.272652i
$$266$$ 12.6847i 0.777746i
$$267$$ 1.80776i 0.110633i
$$268$$ 14.2462i 0.870226i
$$269$$ −3.36932 −0.205431 −0.102715 0.994711i $$-0.532753\pi$$
−0.102715 + 0.994711i $$0.532753\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 32.1771i 1.95462i 0.211818 + 0.977309i $$0.432062\pi$$
−0.211818 + 0.977309i $$0.567938\pi$$
$$272$$ −5.12311 −0.310634
$$273$$ 0 0
$$274$$ −8.80776 −0.532096
$$275$$ 4.12311i 0.248633i
$$276$$ −7.68466 −0.462562
$$277$$ 1.00000 0.0600842 0.0300421 0.999549i $$-0.490436\pi$$
0.0300421 + 0.999549i $$0.490436\pi$$
$$278$$ − 8.43845i − 0.506104i
$$279$$ − 5.68466i − 0.340332i
$$280$$ 3.56155i 0.212843i
$$281$$ − 0.246211i − 0.0146877i −0.999973 0.00734387i $$-0.997662\pi$$
0.999973 0.00734387i $$-0.00233765\pi$$
$$282$$ 7.00000 0.416844
$$283$$ −11.4384 −0.679945 −0.339973 0.940435i $$-0.610418\pi$$
−0.339973 + 0.940435i $$0.610418\pi$$
$$284$$ − 4.87689i − 0.289390i
$$285$$ 3.56155 0.210968
$$286$$ 0 0
$$287$$ −15.1231 −0.892689
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 9.24621 0.543895
$$290$$ −6.56155 −0.385308
$$291$$ − 1.12311i − 0.0658376i
$$292$$ 15.3693i 0.899421i
$$293$$ − 24.9309i − 1.45648i −0.685324 0.728238i $$-0.740339\pi$$
0.685324 0.728238i $$-0.259661\pi$$
$$294$$ 5.68466i 0.331536i
$$295$$ 10.5616 0.614917
$$296$$ −4.12311 −0.239651
$$297$$ − 4.12311i − 0.239247i
$$298$$ 22.8078 1.32122
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ − 16.2462i − 0.936416i
$$302$$ 9.36932 0.539144
$$303$$ 17.1231 0.983697
$$304$$ 3.56155i 0.204269i
$$305$$ − 6.00000i − 0.343559i
$$306$$ 5.12311i 0.292868i
$$307$$ − 15.6155i − 0.891225i −0.895226 0.445613i $$-0.852986\pi$$
0.895226 0.445613i $$-0.147014\pi$$
$$308$$ −14.6847 −0.836736
$$309$$ −0.438447 −0.0249424
$$310$$ 5.68466i 0.322867i
$$311$$ 18.7386 1.06257 0.531285 0.847193i $$-0.321709\pi$$
0.531285 + 0.847193i $$0.321709\pi$$
$$312$$ 0 0
$$313$$ 6.63068 0.374788 0.187394 0.982285i $$-0.439996\pi$$
0.187394 + 0.982285i $$0.439996\pi$$
$$314$$ − 22.1231i − 1.24848i
$$315$$ 3.56155 0.200671
$$316$$ −7.43845 −0.418445
$$317$$ 4.19224i 0.235459i 0.993046 + 0.117730i $$0.0375616\pi$$
−0.993046 + 0.117730i $$0.962438\pi$$
$$318$$ − 4.43845i − 0.248896i
$$319$$ − 27.0540i − 1.51473i
$$320$$ 1.00000i 0.0559017i
$$321$$ 2.00000 0.111629
$$322$$ 27.3693 1.52523
$$323$$ − 18.2462i − 1.01525i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −18.5616 −1.02803
$$327$$ 20.2462i 1.11962i
$$328$$ −4.24621 −0.234458
$$329$$ −24.9309 −1.37448
$$330$$ 4.12311i 0.226969i
$$331$$ 18.7386i 1.02997i 0.857200 + 0.514984i $$0.172202\pi$$
−0.857200 + 0.514984i $$0.827798\pi$$
$$332$$ 1.12311i 0.0616384i
$$333$$ 4.12311i 0.225945i
$$334$$ −20.3693 −1.11456
$$335$$ −14.2462 −0.778354
$$336$$ 3.56155i 0.194299i
$$337$$ 6.00000 0.326841 0.163420 0.986557i $$-0.447747\pi$$
0.163420 + 0.986557i $$0.447747\pi$$
$$338$$ 0 0
$$339$$ −3.68466 −0.200123
$$340$$ − 5.12311i − 0.277839i
$$341$$ −23.4384 −1.26926
$$342$$ 3.56155 0.192587
$$343$$ 4.68466i 0.252948i
$$344$$ − 4.56155i − 0.245942i
$$345$$ − 7.68466i − 0.413728i
$$346$$ − 25.1771i − 1.35353i
$$347$$ 9.12311 0.489754 0.244877 0.969554i $$-0.421252\pi$$
0.244877 + 0.969554i $$0.421252\pi$$
$$348$$ −6.56155 −0.351736
$$349$$ − 24.4924i − 1.31105i −0.755174 0.655525i $$-0.772448\pi$$
0.755174 0.655525i $$-0.227552\pi$$
$$350$$ −3.56155 −0.190373
$$351$$ 0 0
$$352$$ −4.12311 −0.219762
$$353$$ − 31.8617i − 1.69583i −0.530133 0.847915i $$-0.677858\pi$$
0.530133 0.847915i $$-0.322142\pi$$
$$354$$ 10.5616 0.561340
$$355$$ 4.87689 0.258839
$$356$$ − 1.80776i − 0.0958113i
$$357$$ − 18.2462i − 0.965692i
$$358$$ − 0.315342i − 0.0166663i
$$359$$ 4.87689i 0.257393i 0.991684 + 0.128696i $$0.0410792\pi$$
−0.991684 + 0.128696i $$0.958921\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 6.31534 0.332386
$$362$$ 11.1231i 0.584617i
$$363$$ −6.00000 −0.314918
$$364$$ 0 0
$$365$$ −15.3693 −0.804467
$$366$$ − 6.00000i − 0.313625i
$$367$$ −20.4924 −1.06970 −0.534848 0.844948i $$-0.679631\pi$$
−0.534848 + 0.844948i $$0.679631\pi$$
$$368$$ 7.68466 0.400591
$$369$$ 4.24621i 0.221049i
$$370$$ − 4.12311i − 0.214350i
$$371$$ 15.8078i 0.820698i
$$372$$ 5.68466i 0.294736i
$$373$$ −5.19224 −0.268844 −0.134422 0.990924i $$-0.542918\pi$$
−0.134422 + 0.990924i $$0.542918\pi$$
$$374$$ 21.1231 1.09225
$$375$$ 1.00000i 0.0516398i
$$376$$ −7.00000 −0.360997
$$377$$ 0 0
$$378$$ 3.56155 0.183187
$$379$$ 5.31534i 0.273031i 0.990638 + 0.136515i $$0.0435903\pi$$
−0.990638 + 0.136515i $$0.956410\pi$$
$$380$$ −3.56155 −0.182704
$$381$$ 4.43845 0.227389
$$382$$ − 19.1231i − 0.978423i
$$383$$ − 20.8078i − 1.06323i −0.846987 0.531614i $$-0.821586\pi$$
0.846987 0.531614i $$-0.178414\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ − 14.6847i − 0.748399i
$$386$$ 0 0
$$387$$ −4.56155 −0.231877
$$388$$ 1.12311i 0.0570170i
$$389$$ −19.0540 −0.966075 −0.483037 0.875600i $$-0.660467\pi$$
−0.483037 + 0.875600i $$0.660467\pi$$
$$390$$ 0 0
$$391$$ −39.3693 −1.99099
$$392$$ − 5.68466i − 0.287119i
$$393$$ 6.12311 0.308870
$$394$$ 7.80776 0.393349
$$395$$ − 7.43845i − 0.374269i
$$396$$ 4.12311i 0.207194i
$$397$$ − 11.8769i − 0.596084i −0.954553 0.298042i $$-0.903666\pi$$
0.954553 0.298042i $$-0.0963336\pi$$
$$398$$ 11.1231i 0.557551i
$$399$$ −12.6847 −0.635027
$$400$$ −1.00000 −0.0500000
$$401$$ 12.6847i 0.633442i 0.948519 + 0.316721i $$0.102582\pi$$
−0.948519 + 0.316721i $$0.897418\pi$$
$$402$$ −14.2462 −0.710536
$$403$$ 0 0
$$404$$ −17.1231 −0.851906
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 23.3693 1.15980
$$407$$ 17.0000 0.842659
$$408$$ − 5.12311i − 0.253632i
$$409$$ − 1.80776i − 0.0893882i −0.999001 0.0446941i $$-0.985769\pi$$
0.999001 0.0446941i $$-0.0142313\pi$$
$$410$$ − 4.24621i − 0.209705i
$$411$$ − 8.80776i − 0.434455i
$$412$$ 0.438447 0.0216007
$$413$$ −37.6155 −1.85094
$$414$$ − 7.68466i − 0.377680i
$$415$$ −1.12311 −0.0551311
$$416$$ 0 0
$$417$$ 8.43845 0.413233
$$418$$ − 14.6847i − 0.718250i
$$419$$ −0.492423 −0.0240564 −0.0120282 0.999928i $$-0.503829\pi$$
−0.0120282 + 0.999928i $$0.503829\pi$$
$$420$$ −3.56155 −0.173786
$$421$$ − 0.492423i − 0.0239992i −0.999928 0.0119996i $$-0.996180\pi$$
0.999928 0.0119996i $$-0.00381968\pi$$
$$422$$ − 6.93087i − 0.337389i
$$423$$ 7.00000i 0.340352i
$$424$$ 4.43845i 0.215550i
$$425$$ 5.12311 0.248507
$$426$$ 4.87689 0.236286
$$427$$ 21.3693i 1.03413i
$$428$$ −2.00000 −0.0966736
$$429$$ 0 0
$$430$$ 4.56155 0.219978
$$431$$ − 26.7386i − 1.28795i −0.765045 0.643977i $$-0.777283\pi$$
0.765045 0.643977i $$-0.222717\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −30.7386 −1.47720 −0.738602 0.674141i $$-0.764514\pi$$
−0.738602 + 0.674141i $$0.764514\pi$$
$$434$$ − 20.2462i − 0.971849i
$$435$$ − 6.56155i − 0.314602i
$$436$$ − 20.2462i − 0.969618i
$$437$$ 27.3693i 1.30925i
$$438$$ −15.3693 −0.734374
$$439$$ −16.8769 −0.805490 −0.402745 0.915312i $$-0.631944\pi$$
−0.402745 + 0.915312i $$0.631944\pi$$
$$440$$ − 4.12311i − 0.196561i
$$441$$ −5.68466 −0.270698
$$442$$ 0 0
$$443$$ −4.87689 −0.231708 −0.115854 0.993266i $$-0.536961\pi$$
−0.115854 + 0.993266i $$0.536961\pi$$
$$444$$ − 4.12311i − 0.195674i
$$445$$ 1.80776 0.0856962
$$446$$ 26.3002 1.24535
$$447$$ 22.8078i 1.07877i
$$448$$ − 3.56155i − 0.168268i
$$449$$ 25.1771i 1.18818i 0.804399 + 0.594090i $$0.202488\pi$$
−0.804399 + 0.594090i $$0.797512\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ 17.5076 0.824400
$$452$$ 3.68466 0.173312
$$453$$ 9.36932i 0.440209i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ −3.56155 −0.166785
$$457$$ − 3.75379i − 0.175595i −0.996138 0.0877974i $$-0.972017\pi$$
0.996138 0.0877974i $$-0.0279828\pi$$
$$458$$ −7.75379 −0.362311
$$459$$ −5.12311 −0.239126
$$460$$ 7.68466i 0.358299i
$$461$$ − 7.05398i − 0.328536i −0.986416 0.164268i $$-0.947474\pi$$
0.986416 0.164268i $$-0.0525262\pi$$
$$462$$ − 14.6847i − 0.683192i
$$463$$ 33.6155i 1.56225i 0.624377 + 0.781123i $$0.285353\pi$$
−0.624377 + 0.781123i $$0.714647\pi$$
$$464$$ 6.56155 0.304612
$$465$$ −5.68466 −0.263620
$$466$$ − 17.6847i − 0.819226i
$$467$$ −39.8617 −1.84458 −0.922291 0.386497i $$-0.873685\pi$$
−0.922291 + 0.386497i $$0.873685\pi$$
$$468$$ 0 0
$$469$$ 50.7386 2.34289
$$470$$ − 7.00000i − 0.322886i
$$471$$ 22.1231 1.01938
$$472$$ −10.5616 −0.486135
$$473$$ 18.8078i 0.864782i
$$474$$ − 7.43845i − 0.341659i
$$475$$ − 3.56155i − 0.163415i
$$476$$ 18.2462i 0.836314i
$$477$$ 4.43845 0.203223
$$478$$ −13.3693 −0.611498
$$479$$ − 21.7538i − 0.993956i −0.867763 0.496978i $$-0.834443\pi$$
0.867763 0.496978i $$-0.165557\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −19.8769 −0.905368
$$483$$ 27.3693i 1.24535i
$$484$$ 6.00000 0.272727
$$485$$ −1.12311 −0.0509976
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 24.0540i 1.08999i 0.838439 + 0.544995i $$0.183468\pi$$
−0.838439 + 0.544995i $$0.816532\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ − 18.5616i − 0.839382i
$$490$$ 5.68466 0.256807
$$491$$ 21.5616 0.973059 0.486530 0.873664i $$-0.338263\pi$$
0.486530 + 0.873664i $$0.338263\pi$$
$$492$$ − 4.24621i − 0.191434i
$$493$$ −33.6155 −1.51397
$$494$$ 0 0
$$495$$ −4.12311 −0.185320
$$496$$ − 5.68466i − 0.255249i
$$497$$ −17.3693 −0.779120
$$498$$ −1.12311 −0.0503276
$$499$$ − 4.00000i − 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ − 20.3693i − 0.910034i
$$502$$ 0.123106i 0.00549447i
$$503$$ −5.94602 −0.265120 −0.132560 0.991175i $$-0.542320\pi$$
−0.132560 + 0.991175i $$0.542320\pi$$
$$504$$ −3.56155 −0.158644
$$505$$ − 17.1231i − 0.761968i
$$506$$ −31.6847 −1.40855
$$507$$ 0 0
$$508$$ −4.43845 −0.196924
$$509$$ 29.5464i 1.30962i 0.755793 + 0.654811i $$0.227252\pi$$
−0.755793 + 0.654811i $$0.772748\pi$$
$$510$$ 5.12311 0.226855
$$511$$ 54.7386 2.42149
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 3.56155i 0.157246i
$$514$$ − 1.43845i − 0.0634472i
$$515$$ 0.438447i 0.0193203i
$$516$$ 4.56155 0.200811
$$517$$ 28.8617 1.26934
$$518$$ 14.6847i 0.645207i
$$519$$ 25.1771 1.10515
$$520$$ 0 0
$$521$$ −18.6847 −0.818590 −0.409295 0.912402i $$-0.634225\pi$$
−0.409295 + 0.912402i $$0.634225\pi$$
$$522$$ − 6.56155i − 0.287191i
$$523$$ −9.19224 −0.401948 −0.200974 0.979597i $$-0.564411\pi$$
−0.200974 + 0.979597i $$0.564411\pi$$
$$524$$ −6.12311 −0.267489
$$525$$ − 3.56155i − 0.155439i
$$526$$ 13.0000i 0.566827i
$$527$$ 29.1231i 1.26862i
$$528$$ − 4.12311i − 0.179435i
$$529$$ 36.0540 1.56756
$$530$$ −4.43845 −0.192794
$$531$$ 10.5616i 0.458332i
$$532$$ 12.6847 0.549950
$$533$$ 0 0
$$534$$ 1.80776 0.0782296
$$535$$ − 2.00000i − 0.0864675i
$$536$$ 14.2462 0.615343
$$537$$ 0.315342 0.0136080
$$538$$ 3.36932i 0.145262i
$$539$$ 23.4384i 1.00957i
$$540$$ 1.00000i 0.0430331i
$$541$$ − 2.63068i − 0.113102i −0.998400 0.0565510i $$-0.981990\pi$$
0.998400 0.0565510i $$-0.0180103\pi$$
$$542$$ 32.1771 1.38212
$$543$$ −11.1231 −0.477338
$$544$$ 5.12311i 0.219651i
$$545$$ 20.2462 0.867252
$$546$$ 0 0
$$547$$ 35.6155 1.52281 0.761405 0.648276i $$-0.224510\pi$$
0.761405 + 0.648276i $$0.224510\pi$$
$$548$$ 8.80776i 0.376249i
$$549$$ 6.00000 0.256074
$$550$$ 4.12311 0.175810
$$551$$ 23.3693i 0.995566i
$$552$$ 7.68466i 0.327081i
$$553$$ 26.4924i 1.12657i
$$554$$ − 1.00000i − 0.0424859i
$$555$$ 4.12311 0.175016
$$556$$ −8.43845 −0.357870
$$557$$ − 4.43845i − 0.188063i −0.995569 0.0940315i $$-0.970025\pi$$
0.995569 0.0940315i $$-0.0299754\pi$$
$$558$$ −5.68466 −0.240651
$$559$$ 0 0
$$560$$ 3.56155 0.150503
$$561$$ 21.1231i 0.891818i
$$562$$ −0.246211 −0.0103858
$$563$$ −32.9848 −1.39015 −0.695073 0.718939i $$-0.744628\pi$$
−0.695073 + 0.718939i $$0.744628\pi$$
$$564$$ − 7.00000i − 0.294753i
$$565$$ 3.68466i 0.155015i
$$566$$ 11.4384i 0.480794i
$$567$$ 3.56155i 0.149571i
$$568$$ −4.87689 −0.204630
$$569$$ 19.1771 0.803945 0.401973 0.915652i $$-0.368325\pi$$
0.401973 + 0.915652i $$0.368325\pi$$
$$570$$ − 3.56155i − 0.149177i
$$571$$ −11.3153 −0.473532 −0.236766 0.971567i $$-0.576088\pi$$
−0.236766 + 0.971567i $$0.576088\pi$$
$$572$$ 0 0
$$573$$ 19.1231 0.798879
$$574$$ 15.1231i 0.631226i
$$575$$ −7.68466 −0.320472
$$576$$ −1.00000 −0.0416667
$$577$$ − 8.73863i − 0.363794i −0.983318 0.181897i $$-0.941776\pi$$
0.983318 0.181897i $$-0.0582238\pi$$
$$578$$ − 9.24621i − 0.384592i
$$579$$ 0 0
$$580$$ 6.56155i 0.272454i
$$581$$ 4.00000 0.165948
$$582$$ −1.12311 −0.0465542
$$583$$ − 18.3002i − 0.757916i
$$584$$ 15.3693 0.635987
$$585$$ 0 0
$$586$$ −24.9309 −1.02988
$$587$$ 23.7538i 0.980424i 0.871603 + 0.490212i $$0.163081\pi$$
−0.871603 + 0.490212i $$0.836919\pi$$
$$588$$ 5.68466 0.234431
$$589$$ 20.2462 0.834231
$$590$$ − 10.5616i − 0.434812i
$$591$$ 7.80776i 0.321168i
$$592$$ 4.12311i 0.169459i
$$593$$ − 12.1771i − 0.500053i −0.968239 0.250026i $$-0.919561\pi$$
0.968239 0.250026i $$-0.0804393\pi$$
$$594$$ −4.12311 −0.169173
$$595$$ −18.2462 −0.748022
$$596$$ − 22.8078i − 0.934242i
$$597$$ −11.1231 −0.455238
$$598$$ 0 0
$$599$$ −14.0000 −0.572024 −0.286012 0.958226i $$-0.592330\pi$$
−0.286012 + 0.958226i $$0.592330\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −35.9848 −1.46785 −0.733926 0.679229i $$-0.762314\pi$$
−0.733926 + 0.679229i $$0.762314\pi$$
$$602$$ −16.2462 −0.662146
$$603$$ − 14.2462i − 0.580151i
$$604$$ − 9.36932i − 0.381232i
$$605$$ 6.00000i 0.243935i
$$606$$ − 17.1231i − 0.695579i
$$607$$ −29.4233 −1.19425 −0.597127 0.802146i $$-0.703691\pi$$
−0.597127 + 0.802146i $$0.703691\pi$$
$$608$$ 3.56155 0.144440
$$609$$ 23.3693i 0.946973i
$$610$$ −6.00000 −0.242933
$$611$$ 0 0
$$612$$ 5.12311 0.207089
$$613$$ − 28.1231i − 1.13588i −0.823069 0.567941i $$-0.807740\pi$$
0.823069 0.567941i $$-0.192260\pi$$
$$614$$ −15.6155 −0.630191
$$615$$ 4.24621 0.171224
$$616$$ 14.6847i 0.591662i
$$617$$ 41.6847i 1.67816i 0.544007 + 0.839081i $$0.316906\pi$$
−0.544007 + 0.839081i $$0.683094\pi$$
$$618$$ 0.438447i 0.0176369i
$$619$$ − 16.4384i − 0.660717i −0.943856 0.330358i $$-0.892830\pi$$
0.943856 0.330358i $$-0.107170\pi$$
$$620$$ 5.68466 0.228301
$$621$$ 7.68466 0.308375
$$622$$ − 18.7386i − 0.751351i
$$623$$ −6.43845 −0.257951
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 6.63068i − 0.265015i
$$627$$ 14.6847 0.586449
$$628$$ −22.1231 −0.882808
$$629$$ − 21.1231i − 0.842233i
$$630$$ − 3.56155i − 0.141896i
$$631$$ − 30.2462i − 1.20408i −0.798465 0.602041i $$-0.794354\pi$$
0.798465 0.602041i $$-0.205646\pi$$
$$632$$ 7.43845i 0.295886i
$$633$$ 6.93087 0.275477
$$634$$ 4.19224 0.166495
$$635$$ − 4.43845i − 0.176134i
$$636$$ −4.43845 −0.175996
$$637$$ 0 0
$$638$$ −27.0540 −1.07108
$$639$$ 4.87689i 0.192927i
$$640$$ 1.00000 0.0395285
$$641$$ 15.5616 0.614644 0.307322 0.951606i $$-0.400567\pi$$
0.307322 + 0.951606i $$0.400567\pi$$
$$642$$ − 2.00000i − 0.0789337i
$$643$$ − 38.2462i − 1.50828i −0.656712 0.754142i $$-0.728053\pi$$
0.656712 0.754142i $$-0.271947\pi$$
$$644$$ − 27.3693i − 1.07850i
$$645$$ 4.56155i 0.179611i
$$646$$ −18.2462 −0.717888
$$647$$ 2.05398 0.0807501 0.0403751 0.999185i $$-0.487145\pi$$
0.0403751 + 0.999185i $$0.487145\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 43.5464 1.70935
$$650$$ 0 0
$$651$$ 20.2462 0.793512
$$652$$ 18.5616i 0.726927i
$$653$$ −40.4384 −1.58248 −0.791239 0.611507i $$-0.790564\pi$$
−0.791239 + 0.611507i $$0.790564\pi$$
$$654$$ 20.2462 0.791690
$$655$$ − 6.12311i − 0.239250i
$$656$$ 4.24621i 0.165787i
$$657$$ − 15.3693i − 0.599614i
$$658$$ 24.9309i 0.971906i
$$659$$ 31.0540 1.20969 0.604846 0.796343i $$-0.293235\pi$$
0.604846 + 0.796343i $$0.293235\pi$$
$$660$$ 4.12311 0.160492
$$661$$ − 11.6155i − 0.451792i −0.974151 0.225896i $$-0.927469\pi$$
0.974151 0.225896i $$-0.0725309\pi$$
$$662$$ 18.7386 0.728298
$$663$$ 0 0
$$664$$ 1.12311 0.0435850
$$665$$ 12.6847i 0.491890i
$$666$$ 4.12311 0.159767
$$667$$ 50.4233 1.95240
$$668$$ 20.3693i 0.788113i
$$669$$ 26.3002i 1.01682i
$$670$$ 14.2462i 0.550379i
$$671$$ − 24.7386i − 0.955024i
$$672$$ 3.56155 0.137390
$$673$$ −42.2462 −1.62847 −0.814236 0.580534i $$-0.802844\pi$$
−0.814236 + 0.580534i $$0.802844\pi$$
$$674$$ − 6.00000i − 0.231111i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −28.8769 −1.10983 −0.554915 0.831907i $$-0.687249\pi$$
−0.554915 + 0.831907i $$0.687249\pi$$
$$678$$ 3.68466i 0.141508i
$$679$$ 4.00000 0.153506
$$680$$ −5.12311 −0.196462
$$681$$ 20.0000i 0.766402i
$$682$$ 23.4384i 0.897505i
$$683$$ 8.87689i 0.339665i 0.985473 + 0.169832i $$0.0543227\pi$$
−0.985473 + 0.169832i $$0.945677\pi$$
$$684$$ − 3.56155i − 0.136179i
$$685$$ −8.80776 −0.336527
$$686$$ 4.68466 0.178861
$$687$$ − 7.75379i − 0.295825i
$$688$$ −4.56155 −0.173908
$$689$$ 0 0
$$690$$ −7.68466 −0.292550
$$691$$ 16.4384i 0.625348i 0.949860 + 0.312674i $$0.101225\pi$$
−0.949860 + 0.312674i $$0.898775\pi$$
$$692$$ −25.1771 −0.957089
$$693$$ 14.6847 0.557824
$$694$$ − 9.12311i − 0.346308i
$$695$$ − 8.43845i − 0.320089i
$$696$$ 6.56155i 0.248715i
$$697$$ − 21.7538i − 0.823984i
$$698$$ −24.4924 −0.927052
$$699$$ 17.6847 0.668895
$$700$$ 3.56155i 0.134614i
$$701$$ −17.3002 −0.653419 −0.326710 0.945125i $$-0.605940\pi$$
−0.326710 + 0.945125i $$0.605940\pi$$
$$702$$ 0 0
$$703$$ −14.6847 −0.553842
$$704$$ 4.12311i 0.155395i
$$705$$ 7.00000 0.263635
$$706$$ −31.8617 −1.19913
$$707$$ 60.9848i 2.29357i
$$708$$ − 10.5616i − 0.396927i
$$709$$ 17.7538i 0.666758i 0.942793 + 0.333379i $$0.108189\pi$$
−0.942793 + 0.333379i $$0.891811\pi$$
$$710$$ − 4.87689i − 0.183027i
$$711$$ 7.43845 0.278964
$$712$$ −1.80776 −0.0677488
$$713$$ − 43.6847i − 1.63600i
$$714$$ −18.2462 −0.682847
$$715$$ 0 0
$$716$$ −0.315342 −0.0117849
$$717$$ − 13.3693i − 0.499286i
$$718$$ 4.87689 0.182004
$$719$$ 20.9848 0.782603 0.391301 0.920263i $$-0.372025\pi$$
0.391301 + 0.920263i $$0.372025\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 1.56155i − 0.0581553i
$$722$$ − 6.31534i − 0.235033i
$$723$$ − 19.8769i − 0.739230i
$$724$$ 11.1231 0.413387
$$725$$ −6.56155 −0.243690
$$726$$ 6.00000i 0.222681i
$$727$$ 23.4233 0.868722 0.434361 0.900739i $$-0.356974\pi$$
0.434361 + 0.900739i $$0.356974\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 15.3693i 0.568844i
$$731$$ 23.3693 0.864345
$$732$$ −6.00000 −0.221766
$$733$$ − 48.9309i − 1.80730i −0.428269 0.903651i $$-0.640876\pi$$
0.428269 0.903651i $$-0.359124\pi$$
$$734$$ 20.4924i 0.756389i
$$735$$ 5.68466i 0.209682i
$$736$$ − 7.68466i − 0.283260i
$$737$$ −58.7386 −2.16367
$$738$$ 4.24621 0.156305
$$739$$ 42.5464i 1.56509i 0.622591 + 0.782547i $$0.286080\pi$$
−0.622591 + 0.782547i $$0.713920\pi$$
$$740$$ −4.12311 −0.151568
$$741$$ 0 0
$$742$$ 15.8078 0.580321
$$743$$ 32.1771i 1.18046i 0.807234 + 0.590231i $$0.200963\pi$$
−0.807234 + 0.590231i $$0.799037\pi$$
$$744$$ 5.68466 0.208410
$$745$$ 22.8078 0.835612
$$746$$ 5.19224i 0.190101i
$$747$$ − 1.12311i − 0.0410923i
$$748$$ − 21.1231i − 0.772337i
$$749$$ 7.12311i 0.260273i
$$750$$ 1.00000 0.0365148
$$751$$ 3.19224 0.116486 0.0582432 0.998302i $$-0.481450\pi$$
0.0582432 + 0.998302i $$0.481450\pi$$
$$752$$ 7.00000i 0.255264i
$$753$$ −0.123106 −0.00448622
$$754$$ 0 0
$$755$$ 9.36932 0.340984
$$756$$ − 3.56155i − 0.129532i
$$757$$ 33.4233 1.21479 0.607395 0.794400i $$-0.292215\pi$$
0.607395 + 0.794400i $$0.292215\pi$$
$$758$$ 5.31534 0.193062
$$759$$ − 31.6847i − 1.15008i
$$760$$ 3.56155i 0.129191i
$$761$$ 10.9309i 0.396244i 0.980177 + 0.198122i $$0.0634842\pi$$
−0.980177 + 0.198122i $$0.936516\pi$$
$$762$$ − 4.43845i − 0.160788i
$$763$$ −72.1080 −2.61048
$$764$$ −19.1231 −0.691850
$$765$$ 5.12311i 0.185226i
$$766$$ −20.8078 −0.751815
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 45.6847i 1.64743i 0.567003 + 0.823715i $$0.308103\pi$$
−0.567003 + 0.823715i $$0.691897\pi$$
$$770$$ −14.6847 −0.529198
$$771$$ 1.43845 0.0518044
$$772$$ 0 0
$$773$$ 21.1771i 0.761687i 0.924640 + 0.380843i $$0.124366\pi$$
−0.924640 + 0.380843i $$0.875634\pi$$
$$774$$ 4.56155i 0.163962i
$$775$$ 5.68466i 0.204199i
$$776$$ 1.12311 0.0403171
$$777$$ −14.6847 −0.526809
$$778$$ 19.0540i 0.683118i
$$779$$ −15.1231 −0.541841
$$780$$ 0 0
$$781$$ 20.1080 0.719519
$$782$$ 39.3693i 1.40784i
$$783$$ 6.56155 0.234491
$$784$$ −5.68466 −0.203024
$$785$$ − 22.1231i − 0.789607i
$$786$$ − 6.12311i − 0.218404i
$$787$$ 43.6847i 1.55719i 0.627527 + 0.778595i $$0.284067\pi$$
−0.627527 + 0.778595i $$0.715933\pi$$
$$788$$ − 7.80776i − 0.278140i
$$789$$ −13.0000 −0.462812
$$790$$ −7.43845 −0.264648
$$791$$ − 13.1231i − 0.466604i
$$792$$ 4.12311 0.146508
$$793$$ 0 0
$$794$$ −11.8769 −0.421495
$$795$$ − 4.43845i − 0.157415i
$$796$$ 11.1231 0.394248
$$797$$ −4.87689 −0.172748 −0.0863742 0.996263i $$-0.527528\pi$$
−0.0863742 + 0.996263i $$0.527528\pi$$
$$798$$ 12.6847i 0.449032i
$$799$$ − 35.8617i − 1.26870i
$$800$$ 1.00000i 0.0353553i
$$801$$ 1.80776i 0.0638742i
$$802$$ 12.6847 0.447911
$$803$$ −63.3693 −2.23625
$$804$$ 14.2462i 0.502425i
$$805$$ 27.3693 0.964642
$$806$$ 0 0
$$807$$ −3.36932 −0.118606
$$808$$ 17.1231i 0.602389i
$$809$$ 18.4924 0.650159 0.325079 0.945687i $$-0.394609\pi$$
0.325079 + 0.945687i $$0.394609\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 22.5464i 0.791711i 0.918313 + 0.395856i $$0.129552\pi$$
−0.918313 + 0.395856i $$0.870448\pi$$
$$812$$ − 23.3693i − 0.820102i
$$813$$ 32.1771i 1.12850i
$$814$$ − 17.0000i − 0.595850i
$$815$$ −18.5616 −0.650183
$$816$$ −5.12311 −0.179345
$$817$$ − 16.2462i − 0.568383i
$$818$$ −1.80776 −0.0632070
$$819$$ 0 0
$$820$$ −4.24621 −0.148284
$$821$$ 12.5616i 0.438401i 0.975680 + 0.219201i $$0.0703449\pi$$
−0.975680 + 0.219201i $$0.929655\pi$$
$$822$$ −8.80776 −0.307206
$$823$$ 32.0540 1.11733 0.558666 0.829393i $$-0.311313\pi$$
0.558666 + 0.829393i $$0.311313\pi$$
$$824$$ − 0.438447i − 0.0152740i
$$825$$ 4.12311i 0.143548i
$$826$$ 37.6155i 1.30881i
$$827$$ − 11.7538i − 0.408719i −0.978896 0.204360i $$-0.934489\pi$$
0.978896 0.204360i $$-0.0655112\pi$$
$$828$$ −7.68466 −0.267060
$$829$$ −53.6155 −1.86214 −0.931072 0.364835i $$-0.881125\pi$$
−0.931072 + 0.364835i $$0.881125\pi$$
$$830$$ 1.12311i 0.0389836i
$$831$$ 1.00000 0.0346896
$$832$$ 0 0
$$833$$ 29.1231 1.00906
$$834$$ − 8.43845i − 0.292200i
$$835$$ −20.3693 −0.704909
$$836$$ −14.6847 −0.507880
$$837$$ − 5.68466i − 0.196491i
$$838$$ 0.492423i 0.0170105i
$$839$$ − 12.7386i − 0.439786i −0.975524 0.219893i $$-0.929429\pi$$
0.975524 0.219893i $$-0.0705709\pi$$
$$840$$ 3.56155i 0.122885i
$$841$$ 14.0540 0.484620
$$842$$ −0.492423 −0.0169700
$$843$$ − 0.246211i − 0.00847997i
$$844$$ −6.93087 −0.238570
$$845$$ 0 0
$$846$$ 7.00000 0.240665
$$847$$ − 21.3693i − 0.734258i
$$848$$ 4.43845 0.152417
$$849$$ −11.4384 −0.392566
$$850$$ − 5.12311i − 0.175721i
$$851$$ 31.6847i 1.08614i
$$852$$ − 4.87689i − 0.167080i
$$853$$ − 51.3002i − 1.75648i −0.478216 0.878242i $$-0.658716\pi$$
0.478216 0.878242i $$-0.341284\pi$$
$$854$$ 21.3693 0.731243
$$855$$ 3.56155 0.121803
$$856$$ 2.00000i 0.0683586i
$$857$$ 26.8078 0.915736 0.457868 0.889020i $$-0.348613\pi$$
0.457868 + 0.889020i $$0.348613\pi$$
$$858$$ 0 0
$$859$$ −46.7926 −1.59654 −0.798272 0.602298i $$-0.794252\pi$$
−0.798272 + 0.602298i $$0.794252\pi$$
$$860$$ − 4.56155i − 0.155548i
$$861$$ −15.1231 −0.515394
$$862$$ −26.7386 −0.910721
$$863$$ − 6.56155i − 0.223358i −0.993744 0.111679i $$-0.964377\pi$$
0.993744 0.111679i $$-0.0356228\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ − 25.1771i − 0.856046i
$$866$$ 30.7386i 1.04454i
$$867$$ 9.24621 0.314018
$$868$$ −20.2462 −0.687201
$$869$$ − 30.6695i − 1.04039i
$$870$$ −6.56155 −0.222457
$$871$$ 0 0
$$872$$ −20.2462 −0.685623
$$873$$ − 1.12311i − 0.0380114i
$$874$$ 27.3693 0.925781
$$875$$ −3.56155 −0.120402
$$876$$ 15.3693i 0.519281i
$$877$$ − 12.5616i − 0.424173i −0.977251 0.212087i $$-0.931974\pi$$
0.977251 0.212087i $$-0.0680259\pi$$
$$878$$ 16.8769i 0.569568i
$$879$$ − 24.9309i − 0.840897i
$$880$$ −4.12311 −0.138990
$$881$$ 20.4384 0.688589 0.344294 0.938862i $$-0.388118\pi$$
0.344294 + 0.938862i $$0.388118\pi$$
$$882$$ 5.68466i 0.191412i
$$883$$ −26.1771 −0.880929 −0.440464 0.897770i $$-0.645186\pi$$
−0.440464 + 0.897770i $$0.645186\pi$$
$$884$$ 0 0
$$885$$ 10.5616 0.355023
$$886$$ 4.87689i 0.163842i
$$887$$ 41.1080 1.38027 0.690135 0.723681i $$-0.257551\pi$$
0.690135 + 0.723681i $$0.257551\pi$$
$$888$$ −4.12311 −0.138362
$$889$$ 15.8078i 0.530175i
$$890$$ − 1.80776i − 0.0605964i
$$891$$ − 4.12311i − 0.138129i
$$892$$ − 26.3002i − 0.880595i
$$893$$ −24.9309 −0.834280
$$894$$ 22.8078 0.762806
$$895$$ − 0.315342i − 0.0105407i
$$896$$ −3.56155 −0.118983
$$897$$ 0 0
$$898$$ 25.1771 0.840170
$$899$$ − 37.3002i − 1.24403i
$$900$$ 1.00000 0.0333333
$$901$$ −22.7386 −0.757534
$$902$$ − 17.5076i − 0.582939i
$$903$$ − 16.2462i − 0.540640i
$$904$$ − 3.68466i − 0.122550i
$$905$$ 11.1231i 0.369745i
$$906$$ 9.36932 0.311275
$$907$$ −40.9157 −1.35858 −0.679292 0.733868i $$-0.737713\pi$$
−0.679292 + 0.733868i $$0.737713\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 17.1231 0.567938
$$910$$ 0 0
$$911$$ −43.3693 −1.43689 −0.718445 0.695584i $$-0.755146\pi$$
−0.718445 + 0.695584i $$0.755146\pi$$
$$912$$ 3.56155i 0.117935i
$$913$$ −4.63068 −0.153253
$$914$$ −3.75379 −0.124164
$$915$$ − 6.00000i − 0.198354i
$$916$$ 7.75379i 0.256192i
$$917$$ 21.8078i 0.720156i
$$918$$ 5.12311i 0.169088i
$$919$$ 41.3693 1.36465 0.682324 0.731050i $$-0.260969\pi$$
0.682324 + 0.731050i $$0.260969\pi$$
$$920$$ 7.68466 0.253356
$$921$$ − 15.6155i − 0.514549i
$$922$$ −7.05398 −0.232310
$$923$$ 0 0
$$924$$ −14.6847 −0.483090
$$925$$ − 4.12311i − 0.135567i
$$926$$ 33.6155 1.10467
$$927$$ −0.438447 −0.0144005
$$928$$ − 6.56155i − 0.215394i
$$929$$ 16.8769i 0.553713i 0.960911 + 0.276856i $$0.0892927\pi$$
−0.960911 + 0.276856i $$0.910707\pi$$
$$930$$ 5.68466i 0.186407i
$$931$$ − 20.2462i − 0.663543i
$$932$$ −17.6847 −0.579280
$$933$$ 18.7386 0.613475
$$934$$ 39.8617i 1.30432i
$$935$$ 21.1231 0.690799
$$936$$ 0 0
$$937$$ −29.2311 −0.954937 −0.477468 0.878649i $$-0.658446\pi$$
−0.477468 + 0.878649i $$0.658446\pi$$
$$938$$ − 50.7386i − 1.65668i
$$939$$ 6.63068 0.216384
$$940$$ −7.00000 −0.228315
$$941$$ 0.630683i 0.0205597i 0.999947 + 0.0102798i $$0.00327223\pi$$
−0.999947 + 0.0102798i $$0.996728\pi$$
$$942$$ − 22.1231i − 0.720810i
$$943$$ 32.6307i 1.06260i
$$944$$ 10.5616i 0.343749i
$$945$$ 3.56155 0.115857
$$946$$ 18.8078 0.611493
$$947$$ 13.8617i 0.450446i 0.974307 + 0.225223i $$0.0723111\pi$$
−0.974307 + 0.225223i $$0.927689\pi$$
$$948$$ −7.43845 −0.241590
$$949$$ 0 0
$$950$$ −3.56155 −0.115552
$$951$$ 4.19224i 0.135943i
$$952$$ 18.2462 0.591363
$$953$$ −5.82292 −0.188623 −0.0943114 0.995543i $$-0.530065\pi$$
−0.0943114 + 0.995543i $$0.530065\pi$$
$$954$$ − 4.43845i − 0.143700i
$$955$$ − 19.1231i − 0.618809i
$$956$$ 13.3693i 0.432395i
$$957$$ − 27.0540i − 0.874531i
$$958$$ −21.7538 −0.702833
$$959$$ 31.3693 1.01297
$$960$$ 1.00000i 0.0322749i
$$961$$ −1.31534 −0.0424304
$$962$$ 0 0
$$963$$ 2.00000 0.0644491
$$964$$ 19.8769i 0.640192i
$$965$$ 0 0
$$966$$ 27.3693 0.880593
$$967$$ − 3.31534i − 0.106614i −0.998578 0.0533071i $$-0.983024\pi$$
0.998578 0.0533071i $$-0.0169762\pi$$
$$968$$ − 6.00000i − 0.192847i
$$969$$ − 18.2462i − 0.586153i
$$970$$ 1.12311i 0.0360607i
$$971$$ −16.6847 −0.535436 −0.267718 0.963497i $$-0.586270\pi$$
−0.267718 + 0.963497i $$0.586270\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 30.0540i 0.963486i
$$974$$ 24.0540 0.770739
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ − 5.30019i − 0.169568i −0.996399 0.0847840i $$-0.972980\pi$$
0.996399 0.0847840i $$-0.0270200\pi$$
$$978$$ −18.5616 −0.593533
$$979$$ 7.45360 0.238218
$$980$$ − 5.68466i − 0.181590i
$$981$$ 20.2462i 0.646412i
$$982$$ − 21.5616i − 0.688057i
$$983$$ 23.8769i 0.761555i 0.924667 + 0.380777i $$0.124344\pi$$
−0.924667 + 0.380777i $$0.875656\pi$$
$$984$$ −4.24621 −0.135364
$$985$$ 7.80776 0.248776
$$986$$ 33.6155i 1.07054i
$$987$$ −24.9309 −0.793558
$$988$$ 0 0
$$989$$ −35.0540 −1.11465
$$990$$ 4.12311i 0.131041i
$$991$$ −12.4233 −0.394639 −0.197319 0.980339i $$-0.563224\pi$$
−0.197319 + 0.980339i $$0.563224\pi$$
$$992$$ −5.68466 −0.180488
$$993$$ 18.7386i 0.594653i
$$994$$ 17.3693i 0.550921i
$$995$$ 11.1231i 0.352626i
$$996$$ 1.12311i 0.0355870i
$$997$$ 32.5464 1.03075 0.515377 0.856963i $$-0.327652\pi$$
0.515377 + 0.856963i $$0.327652\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 4.12311i 0.130449i
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 5070.2.b.r.1351.2 4
13.5 odd 4 5070.2.a.bb.1.1 2
13.7 odd 12 390.2.i.g.211.1 yes 4
13.8 odd 4 5070.2.a.bi.1.2 2
13.11 odd 12 390.2.i.g.61.1 4
13.12 even 2 inner 5070.2.b.r.1351.3 4
39.11 even 12 1170.2.i.o.451.1 4
39.20 even 12 1170.2.i.o.991.1 4
65.7 even 12 1950.2.z.n.1849.3 8
65.24 odd 12 1950.2.i.bi.451.2 4
65.33 even 12 1950.2.z.n.1849.2 8
65.37 even 12 1950.2.z.n.1699.2 8
65.59 odd 12 1950.2.i.bi.601.2 4
65.63 even 12 1950.2.z.n.1699.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.g.61.1 4 13.11 odd 12
390.2.i.g.211.1 yes 4 13.7 odd 12
1170.2.i.o.451.1 4 39.11 even 12
1170.2.i.o.991.1 4 39.20 even 12
1950.2.i.bi.451.2 4 65.24 odd 12
1950.2.i.bi.601.2 4 65.59 odd 12
1950.2.z.n.1699.2 8 65.37 even 12
1950.2.z.n.1699.3 8 65.63 even 12
1950.2.z.n.1849.2 8 65.33 even 12
1950.2.z.n.1849.3 8 65.7 even 12
5070.2.a.bb.1.1 2 13.5 odd 4
5070.2.a.bi.1.2 2 13.8 odd 4
5070.2.b.r.1351.2 4 1.1 even 1 trivial
5070.2.b.r.1351.3 4 13.12 even 2 inner