# Properties

 Label 5070.2.a.bi.1.2 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

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

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.12311 q^{11} +1.00000 q^{12} +3.56155 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.12311 q^{17} +1.00000 q^{18} -3.56155 q^{19} +1.00000 q^{20} +3.56155 q^{21} -4.12311 q^{22} -7.68466 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.56155 q^{28} +6.56155 q^{29} +1.00000 q^{30} +5.68466 q^{31} +1.00000 q^{32} -4.12311 q^{33} +5.12311 q^{34} +3.56155 q^{35} +1.00000 q^{36} +4.12311 q^{37} -3.56155 q^{38} +1.00000 q^{40} -4.24621 q^{41} +3.56155 q^{42} +4.56155 q^{43} -4.12311 q^{44} +1.00000 q^{45} -7.68466 q^{46} +7.00000 q^{47} +1.00000 q^{48} +5.68466 q^{49} +1.00000 q^{50} +5.12311 q^{51} +4.43845 q^{53} +1.00000 q^{54} -4.12311 q^{55} +3.56155 q^{56} -3.56155 q^{57} +6.56155 q^{58} +10.5616 q^{59} +1.00000 q^{60} +6.00000 q^{61} +5.68466 q^{62} +3.56155 q^{63} +1.00000 q^{64} -4.12311 q^{66} +14.2462 q^{67} +5.12311 q^{68} -7.68466 q^{69} +3.56155 q^{70} -4.87689 q^{71} +1.00000 q^{72} -15.3693 q^{73} +4.12311 q^{74} +1.00000 q^{75} -3.56155 q^{76} -14.6847 q^{77} +7.43845 q^{79} +1.00000 q^{80} +1.00000 q^{81} -4.24621 q^{82} +1.12311 q^{83} +3.56155 q^{84} +5.12311 q^{85} +4.56155 q^{86} +6.56155 q^{87} -4.12311 q^{88} +1.80776 q^{89} +1.00000 q^{90} -7.68466 q^{92} +5.68466 q^{93} +7.00000 q^{94} -3.56155 q^{95} +1.00000 q^{96} +1.12311 q^{97} +5.68466 q^{98} -4.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 3q^{14} + 2q^{15} + 2q^{16} + 2q^{17} + 2q^{18} - 3q^{19} + 2q^{20} + 3q^{21} - 3q^{23} + 2q^{24} + 2q^{25} + 2q^{27} + 3q^{28} + 9q^{29} + 2q^{30} - q^{31} + 2q^{32} + 2q^{34} + 3q^{35} + 2q^{36} - 3q^{38} + 2q^{40} + 8q^{41} + 3q^{42} + 5q^{43} + 2q^{45} - 3q^{46} + 14q^{47} + 2q^{48} - q^{49} + 2q^{50} + 2q^{51} + 13q^{53} + 2q^{54} + 3q^{56} - 3q^{57} + 9q^{58} + 17q^{59} + 2q^{60} + 12q^{61} - q^{62} + 3q^{63} + 2q^{64} + 12q^{67} + 2q^{68} - 3q^{69} + 3q^{70} - 18q^{71} + 2q^{72} - 6q^{73} + 2q^{75} - 3q^{76} - 17q^{77} + 19q^{79} + 2q^{80} + 2q^{81} + 8q^{82} - 6q^{83} + 3q^{84} + 2q^{85} + 5q^{86} + 9q^{87} - 17q^{89} + 2q^{90} - 3q^{92} - q^{93} + 14q^{94} - 3q^{95} + 2q^{96} - 6q^{97} - q^{98} + O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ 3.56155 1.34614 0.673070 0.739579i $$-0.264975\pi$$
0.673070 + 0.739579i $$0.264975\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −4.12311 −1.24316 −0.621582 0.783349i $$-0.713510\pi$$
−0.621582 + 0.783349i $$0.713510\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 3.56155 0.951865
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 5.12311 1.24254 0.621268 0.783598i $$-0.286618\pi$$
0.621268 + 0.783598i $$0.286618\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −3.56155 −0.817076 −0.408538 0.912741i $$-0.633961\pi$$
−0.408538 + 0.912741i $$0.633961\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 3.56155 0.777195
$$22$$ −4.12311 −0.879049
$$23$$ −7.68466 −1.60236 −0.801181 0.598422i $$-0.795795\pi$$
−0.801181 + 0.598422i $$0.795795\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 3.56155 0.673070
$$29$$ 6.56155 1.21845 0.609225 0.792998i $$-0.291481\pi$$
0.609225 + 0.792998i $$0.291481\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 5.68466 1.02099 0.510497 0.859879i $$-0.329461\pi$$
0.510497 + 0.859879i $$0.329461\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −4.12311 −0.717741
$$34$$ 5.12311 0.878605
$$35$$ 3.56155 0.602012
$$36$$ 1.00000 0.166667
$$37$$ 4.12311 0.677834 0.338917 0.940816i $$-0.389939\pi$$
0.338917 + 0.940816i $$0.389939\pi$$
$$38$$ −3.56155 −0.577760
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −4.24621 −0.663147 −0.331573 0.943429i $$-0.607579\pi$$
−0.331573 + 0.943429i $$0.607579\pi$$
$$42$$ 3.56155 0.549560
$$43$$ 4.56155 0.695630 0.347815 0.937563i $$-0.386924\pi$$
0.347815 + 0.937563i $$0.386924\pi$$
$$44$$ −4.12311 −0.621582
$$45$$ 1.00000 0.149071
$$46$$ −7.68466 −1.13304
$$47$$ 7.00000 1.02105 0.510527 0.859861i $$-0.329450\pi$$
0.510527 + 0.859861i $$0.329450\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 5.68466 0.812094
$$50$$ 1.00000 0.141421
$$51$$ 5.12311 0.717378
$$52$$ 0 0
$$53$$ 4.43845 0.609668 0.304834 0.952406i $$-0.401399\pi$$
0.304834 + 0.952406i $$0.401399\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −4.12311 −0.555959
$$56$$ 3.56155 0.475933
$$57$$ −3.56155 −0.471739
$$58$$ 6.56155 0.861574
$$59$$ 10.5616 1.37500 0.687499 0.726186i $$-0.258709\pi$$
0.687499 + 0.726186i $$0.258709\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 5.68466 0.721952
$$63$$ 3.56155 0.448713
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.12311 −0.507519
$$67$$ 14.2462 1.74045 0.870226 0.492653i $$-0.163973\pi$$
0.870226 + 0.492653i $$0.163973\pi$$
$$68$$ 5.12311 0.621268
$$69$$ −7.68466 −0.925124
$$70$$ 3.56155 0.425687
$$71$$ −4.87689 −0.578781 −0.289390 0.957211i $$-0.593453\pi$$
−0.289390 + 0.957211i $$0.593453\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −15.3693 −1.79884 −0.899421 0.437083i $$-0.856012\pi$$
−0.899421 + 0.437083i $$0.856012\pi$$
$$74$$ 4.12311 0.479301
$$75$$ 1.00000 0.115470
$$76$$ −3.56155 −0.408538
$$77$$ −14.6847 −1.67347
$$78$$ 0 0
$$79$$ 7.43845 0.836891 0.418445 0.908242i $$-0.362575\pi$$
0.418445 + 0.908242i $$0.362575\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −4.24621 −0.468916
$$83$$ 1.12311 0.123277 0.0616384 0.998099i $$-0.480367\pi$$
0.0616384 + 0.998099i $$0.480367\pi$$
$$84$$ 3.56155 0.388597
$$85$$ 5.12311 0.555679
$$86$$ 4.56155 0.491885
$$87$$ 6.56155 0.703472
$$88$$ −4.12311 −0.439525
$$89$$ 1.80776 0.191623 0.0958113 0.995400i $$-0.469455\pi$$
0.0958113 + 0.995400i $$0.469455\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −7.68466 −0.801181
$$93$$ 5.68466 0.589472
$$94$$ 7.00000 0.721995
$$95$$ −3.56155 −0.365408
$$96$$ 1.00000 0.102062
$$97$$ 1.12311 0.114034 0.0570170 0.998373i $$-0.481841\pi$$
0.0570170 + 0.998373i $$0.481841\pi$$
$$98$$ 5.68466 0.574237
$$99$$ −4.12311 −0.414388
$$100$$ 1.00000 0.100000
$$101$$ −17.1231 −1.70381 −0.851906 0.523694i $$-0.824553\pi$$
−0.851906 + 0.523694i $$0.824553\pi$$
$$102$$ 5.12311 0.507263
$$103$$ 0.438447 0.0432015 0.0216007 0.999767i $$-0.493124\pi$$
0.0216007 + 0.999767i $$0.493124\pi$$
$$104$$ 0 0
$$105$$ 3.56155 0.347572
$$106$$ 4.43845 0.431100
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −20.2462 −1.93924 −0.969618 0.244625i $$-0.921335\pi$$
−0.969618 + 0.244625i $$0.921335\pi$$
$$110$$ −4.12311 −0.393123
$$111$$ 4.12311 0.391348
$$112$$ 3.56155 0.336535
$$113$$ −3.68466 −0.346624 −0.173312 0.984867i $$-0.555447\pi$$
−0.173312 + 0.984867i $$0.555447\pi$$
$$114$$ −3.56155 −0.333570
$$115$$ −7.68466 −0.716598
$$116$$ 6.56155 0.609225
$$117$$ 0 0
$$118$$ 10.5616 0.972270
$$119$$ 18.2462 1.67263
$$120$$ 1.00000 0.0912871
$$121$$ 6.00000 0.545455
$$122$$ 6.00000 0.543214
$$123$$ −4.24621 −0.382868
$$124$$ 5.68466 0.510497
$$125$$ 1.00000 0.0894427
$$126$$ 3.56155 0.317288
$$127$$ −4.43845 −0.393849 −0.196924 0.980419i $$-0.563095\pi$$
−0.196924 + 0.980419i $$0.563095\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.56155 0.401622
$$130$$ 0 0
$$131$$ 6.12311 0.534978 0.267489 0.963561i $$-0.413806\pi$$
0.267489 + 0.963561i $$0.413806\pi$$
$$132$$ −4.12311 −0.358870
$$133$$ −12.6847 −1.09990
$$134$$ 14.2462 1.23069
$$135$$ 1.00000 0.0860663
$$136$$ 5.12311 0.439303
$$137$$ −8.80776 −0.752498 −0.376249 0.926519i $$-0.622786\pi$$
−0.376249 + 0.926519i $$0.622786\pi$$
$$138$$ −7.68466 −0.654162
$$139$$ 8.43845 0.715740 0.357870 0.933771i $$-0.383503\pi$$
0.357870 + 0.933771i $$0.383503\pi$$
$$140$$ 3.56155 0.301006
$$141$$ 7.00000 0.589506
$$142$$ −4.87689 −0.409260
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 6.56155 0.544907
$$146$$ −15.3693 −1.27197
$$147$$ 5.68466 0.468863
$$148$$ 4.12311 0.338917
$$149$$ −22.8078 −1.86848 −0.934242 0.356639i $$-0.883923\pi$$
−0.934242 + 0.356639i $$0.883923\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 9.36932 0.762464 0.381232 0.924479i $$-0.375500\pi$$
0.381232 + 0.924479i $$0.375500\pi$$
$$152$$ −3.56155 −0.288880
$$153$$ 5.12311 0.414179
$$154$$ −14.6847 −1.18332
$$155$$ 5.68466 0.456603
$$156$$ 0 0
$$157$$ 22.1231 1.76562 0.882808 0.469734i $$-0.155650\pi$$
0.882808 + 0.469734i $$0.155650\pi$$
$$158$$ 7.43845 0.591771
$$159$$ 4.43845 0.351992
$$160$$ 1.00000 0.0790569
$$161$$ −27.3693 −2.15700
$$162$$ 1.00000 0.0785674
$$163$$ −18.5616 −1.45385 −0.726927 0.686715i $$-0.759052\pi$$
−0.726927 + 0.686715i $$0.759052\pi$$
$$164$$ −4.24621 −0.331573
$$165$$ −4.12311 −0.320983
$$166$$ 1.12311 0.0871699
$$167$$ −20.3693 −1.57623 −0.788113 0.615531i $$-0.788942\pi$$
−0.788113 + 0.615531i $$0.788942\pi$$
$$168$$ 3.56155 0.274780
$$169$$ 0 0
$$170$$ 5.12311 0.392924
$$171$$ −3.56155 −0.272359
$$172$$ 4.56155 0.347815
$$173$$ −25.1771 −1.91418 −0.957089 0.289794i $$-0.906413\pi$$
−0.957089 + 0.289794i $$0.906413\pi$$
$$174$$ 6.56155 0.497430
$$175$$ 3.56155 0.269228
$$176$$ −4.12311 −0.310791
$$177$$ 10.5616 0.793855
$$178$$ 1.80776 0.135498
$$179$$ −0.315342 −0.0235697 −0.0117849 0.999931i $$-0.503751\pi$$
−0.0117849 + 0.999931i $$0.503751\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 11.1231 0.826774 0.413387 0.910555i $$-0.364346\pi$$
0.413387 + 0.910555i $$0.364346\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ −7.68466 −0.566521
$$185$$ 4.12311 0.303137
$$186$$ 5.68466 0.416819
$$187$$ −21.1231 −1.54467
$$188$$ 7.00000 0.510527
$$189$$ 3.56155 0.259065
$$190$$ −3.56155 −0.258382
$$191$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 1.12311 0.0806343
$$195$$ 0 0
$$196$$ 5.68466 0.406047
$$197$$ −7.80776 −0.556280 −0.278140 0.960541i $$-0.589718\pi$$
−0.278140 + 0.960541i $$0.589718\pi$$
$$198$$ −4.12311 −0.293016
$$199$$ 11.1231 0.788496 0.394248 0.919004i $$-0.371005\pi$$
0.394248 + 0.919004i $$0.371005\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 14.2462 1.00485
$$202$$ −17.1231 −1.20478
$$203$$ 23.3693 1.64020
$$204$$ 5.12311 0.358689
$$205$$ −4.24621 −0.296568
$$206$$ 0.438447 0.0305481
$$207$$ −7.68466 −0.534121
$$208$$ 0 0
$$209$$ 14.6847 1.01576
$$210$$ 3.56155 0.245770
$$211$$ 6.93087 0.477141 0.238570 0.971125i $$-0.423321\pi$$
0.238570 + 0.971125i $$0.423321\pi$$
$$212$$ 4.43845 0.304834
$$213$$ −4.87689 −0.334159
$$214$$ 2.00000 0.136717
$$215$$ 4.56155 0.311095
$$216$$ 1.00000 0.0680414
$$217$$ 20.2462 1.37440
$$218$$ −20.2462 −1.37125
$$219$$ −15.3693 −1.03856
$$220$$ −4.12311 −0.277980
$$221$$ 0 0
$$222$$ 4.12311 0.276725
$$223$$ −26.3002 −1.76119 −0.880595 0.473869i $$-0.842857\pi$$
−0.880595 + 0.473869i $$0.842857\pi$$
$$224$$ 3.56155 0.237966
$$225$$ 1.00000 0.0666667
$$226$$ −3.68466 −0.245100
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ −3.56155 −0.235870
$$229$$ −7.75379 −0.512385 −0.256192 0.966626i $$-0.582468\pi$$
−0.256192 + 0.966626i $$0.582468\pi$$
$$230$$ −7.68466 −0.506711
$$231$$ −14.6847 −0.966180
$$232$$ 6.56155 0.430787
$$233$$ −17.6847 −1.15856 −0.579280 0.815128i $$-0.696666\pi$$
−0.579280 + 0.815128i $$0.696666\pi$$
$$234$$ 0 0
$$235$$ 7.00000 0.456630
$$236$$ 10.5616 0.687499
$$237$$ 7.43845 0.483179
$$238$$ 18.2462 1.18273
$$239$$ 13.3693 0.864789 0.432395 0.901684i $$-0.357669\pi$$
0.432395 + 0.901684i $$0.357669\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −19.8769 −1.28038 −0.640192 0.768215i $$-0.721145\pi$$
−0.640192 + 0.768215i $$0.721145\pi$$
$$242$$ 6.00000 0.385695
$$243$$ 1.00000 0.0641500
$$244$$ 6.00000 0.384111
$$245$$ 5.68466 0.363180
$$246$$ −4.24621 −0.270729
$$247$$ 0 0
$$248$$ 5.68466 0.360976
$$249$$ 1.12311 0.0711739
$$250$$ 1.00000 0.0632456
$$251$$ 0.123106 0.00777036 0.00388518 0.999992i $$-0.498763\pi$$
0.00388518 + 0.999992i $$0.498763\pi$$
$$252$$ 3.56155 0.224357
$$253$$ 31.6847 1.99200
$$254$$ −4.43845 −0.278493
$$255$$ 5.12311 0.320821
$$256$$ 1.00000 0.0625000
$$257$$ −1.43845 −0.0897279 −0.0448639 0.998993i $$-0.514285\pi$$
−0.0448639 + 0.998993i $$0.514285\pi$$
$$258$$ 4.56155 0.283990
$$259$$ 14.6847 0.912460
$$260$$ 0 0
$$261$$ 6.56155 0.406150
$$262$$ 6.12311 0.378287
$$263$$ −13.0000 −0.801614 −0.400807 0.916162i $$-0.631270\pi$$
−0.400807 + 0.916162i $$0.631270\pi$$
$$264$$ −4.12311 −0.253760
$$265$$ 4.43845 0.272652
$$266$$ −12.6847 −0.777746
$$267$$ 1.80776 0.110633
$$268$$ 14.2462 0.870226
$$269$$ −3.36932 −0.205431 −0.102715 0.994711i $$-0.532753\pi$$
−0.102715 + 0.994711i $$0.532753\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 32.1771 1.95462 0.977309 0.211818i $$-0.0679382\pi$$
0.977309 + 0.211818i $$0.0679382\pi$$
$$272$$ 5.12311 0.310634
$$273$$ 0 0
$$274$$ −8.80776 −0.532096
$$275$$ −4.12311 −0.248633
$$276$$ −7.68466 −0.462562
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ 8.43845 0.506104
$$279$$ 5.68466 0.340332
$$280$$ 3.56155 0.212843
$$281$$ −0.246211 −0.0146877 −0.00734387 0.999973i $$-0.502338\pi$$
−0.00734387 + 0.999973i $$0.502338\pi$$
$$282$$ 7.00000 0.416844
$$283$$ 11.4384 0.679945 0.339973 0.940435i $$-0.389582\pi$$
0.339973 + 0.940435i $$0.389582\pi$$
$$284$$ −4.87689 −0.289390
$$285$$ −3.56155 −0.210968
$$286$$ 0 0
$$287$$ −15.1231 −0.892689
$$288$$ 1.00000 0.0589256
$$289$$ 9.24621 0.543895
$$290$$ 6.56155 0.385308
$$291$$ 1.12311 0.0658376
$$292$$ −15.3693 −0.899421
$$293$$ −24.9309 −1.45648 −0.728238 0.685324i $$-0.759661\pi$$
−0.728238 + 0.685324i $$0.759661\pi$$
$$294$$ 5.68466 0.331536
$$295$$ 10.5616 0.614917
$$296$$ 4.12311 0.239651
$$297$$ −4.12311 −0.239247
$$298$$ −22.8078 −1.32122
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 16.2462 0.936416
$$302$$ 9.36932 0.539144
$$303$$ −17.1231 −0.983697
$$304$$ −3.56155 −0.204269
$$305$$ 6.00000 0.343559
$$306$$ 5.12311 0.292868
$$307$$ −15.6155 −0.891225 −0.445613 0.895226i $$-0.647014\pi$$
−0.445613 + 0.895226i $$0.647014\pi$$
$$308$$ −14.6847 −0.836736
$$309$$ 0.438447 0.0249424
$$310$$ 5.68466 0.322867
$$311$$ −18.7386 −1.06257 −0.531285 0.847193i $$-0.678291\pi$$
−0.531285 + 0.847193i $$0.678291\pi$$
$$312$$ 0 0
$$313$$ 6.63068 0.374788 0.187394 0.982285i $$-0.439996\pi$$
0.187394 + 0.982285i $$0.439996\pi$$
$$314$$ 22.1231 1.24848
$$315$$ 3.56155 0.200671
$$316$$ 7.43845 0.418445
$$317$$ −4.19224 −0.235459 −0.117730 0.993046i $$-0.537562\pi$$
−0.117730 + 0.993046i $$0.537562\pi$$
$$318$$ 4.43845 0.248896
$$319$$ −27.0540 −1.51473
$$320$$ 1.00000 0.0559017
$$321$$ 2.00000 0.111629
$$322$$ −27.3693 −1.52523
$$323$$ −18.2462 −1.01525
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −18.5616 −1.02803
$$327$$ −20.2462 −1.11962
$$328$$ −4.24621 −0.234458
$$329$$ 24.9309 1.37448
$$330$$ −4.12311 −0.226969
$$331$$ −18.7386 −1.02997 −0.514984 0.857200i $$-0.672202\pi$$
−0.514984 + 0.857200i $$0.672202\pi$$
$$332$$ 1.12311 0.0616384
$$333$$ 4.12311 0.225945
$$334$$ −20.3693 −1.11456
$$335$$ 14.2462 0.778354
$$336$$ 3.56155 0.194299
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 0 0
$$339$$ −3.68466 −0.200123
$$340$$ 5.12311 0.277839
$$341$$ −23.4384 −1.26926
$$342$$ −3.56155 −0.192587
$$343$$ −4.68466 −0.252948
$$344$$ 4.56155 0.245942
$$345$$ −7.68466 −0.413728
$$346$$ −25.1771 −1.35353
$$347$$ 9.12311 0.489754 0.244877 0.969554i $$-0.421252\pi$$
0.244877 + 0.969554i $$0.421252\pi$$
$$348$$ 6.56155 0.351736
$$349$$ −24.4924 −1.31105 −0.655525 0.755174i $$-0.727552\pi$$
−0.655525 + 0.755174i $$0.727552\pi$$
$$350$$ 3.56155 0.190373
$$351$$ 0 0
$$352$$ −4.12311 −0.219762
$$353$$ 31.8617 1.69583 0.847915 0.530133i $$-0.177858\pi$$
0.847915 + 0.530133i $$0.177858\pi$$
$$354$$ 10.5616 0.561340
$$355$$ −4.87689 −0.258839
$$356$$ 1.80776 0.0958113
$$357$$ 18.2462 0.965692
$$358$$ −0.315342 −0.0166663
$$359$$ 4.87689 0.257393 0.128696 0.991684i $$-0.458921\pi$$
0.128696 + 0.991684i $$0.458921\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −6.31534 −0.332386
$$362$$ 11.1231 0.584617
$$363$$ 6.00000 0.314918
$$364$$ 0 0
$$365$$ −15.3693 −0.804467
$$366$$ 6.00000 0.313625
$$367$$ −20.4924 −1.06970 −0.534848 0.844948i $$-0.679631\pi$$
−0.534848 + 0.844948i $$0.679631\pi$$
$$368$$ −7.68466 −0.400591
$$369$$ −4.24621 −0.221049
$$370$$ 4.12311 0.214350
$$371$$ 15.8078 0.820698
$$372$$ 5.68466 0.294736
$$373$$ −5.19224 −0.268844 −0.134422 0.990924i $$-0.542918\pi$$
−0.134422 + 0.990924i $$0.542918\pi$$
$$374$$ −21.1231 −1.09225
$$375$$ 1.00000 0.0516398
$$376$$ 7.00000 0.360997
$$377$$ 0 0
$$378$$ 3.56155 0.183187
$$379$$ −5.31534 −0.273031 −0.136515 0.990638i $$-0.543590\pi$$
−0.136515 + 0.990638i $$0.543590\pi$$
$$380$$ −3.56155 −0.182704
$$381$$ −4.43845 −0.227389
$$382$$ 19.1231 0.978423
$$383$$ 20.8078 1.06323 0.531614 0.846987i $$-0.321586\pi$$
0.531614 + 0.846987i $$0.321586\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −14.6847 −0.748399
$$386$$ 0 0
$$387$$ 4.56155 0.231877
$$388$$ 1.12311 0.0570170
$$389$$ 19.0540 0.966075 0.483037 0.875600i $$-0.339533\pi$$
0.483037 + 0.875600i $$0.339533\pi$$
$$390$$ 0 0
$$391$$ −39.3693 −1.99099
$$392$$ 5.68466 0.287119
$$393$$ 6.12311 0.308870
$$394$$ −7.80776 −0.393349
$$395$$ 7.43845 0.374269
$$396$$ −4.12311 −0.207194
$$397$$ −11.8769 −0.596084 −0.298042 0.954553i $$-0.596334\pi$$
−0.298042 + 0.954553i $$0.596334\pi$$
$$398$$ 11.1231 0.557551
$$399$$ −12.6847 −0.635027
$$400$$ 1.00000 0.0500000
$$401$$ 12.6847 0.633442 0.316721 0.948519i $$-0.397418\pi$$
0.316721 + 0.948519i $$0.397418\pi$$
$$402$$ 14.2462 0.710536
$$403$$ 0 0
$$404$$ −17.1231 −0.851906
$$405$$ 1.00000 0.0496904
$$406$$ 23.3693 1.15980
$$407$$ −17.0000 −0.842659
$$408$$ 5.12311 0.253632
$$409$$ 1.80776 0.0893882 0.0446941 0.999001i $$-0.485769\pi$$
0.0446941 + 0.999001i $$0.485769\pi$$
$$410$$ −4.24621 −0.209705
$$411$$ −8.80776 −0.434455
$$412$$ 0.438447 0.0216007
$$413$$ 37.6155 1.85094
$$414$$ −7.68466 −0.377680
$$415$$ 1.12311 0.0551311
$$416$$ 0 0
$$417$$ 8.43845 0.413233
$$418$$ 14.6847 0.718250
$$419$$ −0.492423 −0.0240564 −0.0120282 0.999928i $$-0.503829\pi$$
−0.0120282 + 0.999928i $$0.503829\pi$$
$$420$$ 3.56155 0.173786
$$421$$ 0.492423 0.0239992 0.0119996 0.999928i $$-0.496180\pi$$
0.0119996 + 0.999928i $$0.496180\pi$$
$$422$$ 6.93087 0.337389
$$423$$ 7.00000 0.340352
$$424$$ 4.43845 0.215550
$$425$$ 5.12311 0.248507
$$426$$ −4.87689 −0.236286
$$427$$ 21.3693 1.03413
$$428$$ 2.00000 0.0966736
$$429$$ 0 0
$$430$$ 4.56155 0.219978
$$431$$ 26.7386 1.28795 0.643977 0.765045i $$-0.277283\pi$$
0.643977 + 0.765045i $$0.277283\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 30.7386 1.47720 0.738602 0.674141i $$-0.235486\pi$$
0.738602 + 0.674141i $$0.235486\pi$$
$$434$$ 20.2462 0.971849
$$435$$ 6.56155 0.314602
$$436$$ −20.2462 −0.969618
$$437$$ 27.3693 1.30925
$$438$$ −15.3693 −0.734374
$$439$$ 16.8769 0.805490 0.402745 0.915312i $$-0.368056\pi$$
0.402745 + 0.915312i $$0.368056\pi$$
$$440$$ −4.12311 −0.196561
$$441$$ 5.68466 0.270698
$$442$$ 0 0
$$443$$ −4.87689 −0.231708 −0.115854 0.993266i $$-0.536961\pi$$
−0.115854 + 0.993266i $$0.536961\pi$$
$$444$$ 4.12311 0.195674
$$445$$ 1.80776 0.0856962
$$446$$ −26.3002 −1.24535
$$447$$ −22.8078 −1.07877
$$448$$ 3.56155 0.168268
$$449$$ 25.1771 1.18818 0.594090 0.804399i $$-0.297512\pi$$
0.594090 + 0.804399i $$0.297512\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 17.5076 0.824400
$$452$$ −3.68466 −0.173312
$$453$$ 9.36932 0.440209
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −3.56155 −0.166785
$$457$$ 3.75379 0.175595 0.0877974 0.996138i $$-0.472017\pi$$
0.0877974 + 0.996138i $$0.472017\pi$$
$$458$$ −7.75379 −0.362311
$$459$$ 5.12311 0.239126
$$460$$ −7.68466 −0.358299
$$461$$ 7.05398 0.328536 0.164268 0.986416i $$-0.447474\pi$$
0.164268 + 0.986416i $$0.447474\pi$$
$$462$$ −14.6847 −0.683192
$$463$$ 33.6155 1.56225 0.781123 0.624377i $$-0.214647\pi$$
0.781123 + 0.624377i $$0.214647\pi$$
$$464$$ 6.56155 0.304612
$$465$$ 5.68466 0.263620
$$466$$ −17.6847 −0.819226
$$467$$ 39.8617 1.84458 0.922291 0.386497i $$-0.126315\pi$$
0.922291 + 0.386497i $$0.126315\pi$$
$$468$$ 0 0
$$469$$ 50.7386 2.34289
$$470$$ 7.00000 0.322886
$$471$$ 22.1231 1.01938
$$472$$ 10.5616 0.486135
$$473$$ −18.8078 −0.864782
$$474$$ 7.43845 0.341659
$$475$$ −3.56155 −0.163415
$$476$$ 18.2462 0.836314
$$477$$ 4.43845 0.203223
$$478$$ 13.3693 0.611498
$$479$$ −21.7538 −0.993956 −0.496978 0.867763i $$-0.665557\pi$$
−0.496978 + 0.867763i $$0.665557\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −19.8769 −0.905368
$$483$$ −27.3693 −1.24535
$$484$$ 6.00000 0.272727
$$485$$ 1.12311 0.0509976
$$486$$ 1.00000 0.0453609
$$487$$ −24.0540 −1.08999 −0.544995 0.838439i $$-0.683468\pi$$
−0.544995 + 0.838439i $$0.683468\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −18.5616 −0.839382
$$490$$ 5.68466 0.256807
$$491$$ −21.5616 −0.973059 −0.486530 0.873664i $$-0.661737\pi$$
−0.486530 + 0.873664i $$0.661737\pi$$
$$492$$ −4.24621 −0.191434
$$493$$ 33.6155 1.51397
$$494$$ 0 0
$$495$$ −4.12311 −0.185320
$$496$$ 5.68466 0.255249
$$497$$ −17.3693 −0.779120
$$498$$ 1.12311 0.0503276
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −20.3693 −0.910034
$$502$$ 0.123106 0.00549447
$$503$$ −5.94602 −0.265120 −0.132560 0.991175i $$-0.542320\pi$$
−0.132560 + 0.991175i $$0.542320\pi$$
$$504$$ 3.56155 0.158644
$$505$$ −17.1231 −0.761968
$$506$$ 31.6847 1.40855
$$507$$ 0 0
$$508$$ −4.43845 −0.196924
$$509$$ −29.5464 −1.30962 −0.654811 0.755793i $$-0.727252\pi$$
−0.654811 + 0.755793i $$0.727252\pi$$
$$510$$ 5.12311 0.226855
$$511$$ −54.7386 −2.42149
$$512$$ 1.00000 0.0441942
$$513$$ −3.56155 −0.157246
$$514$$ −1.43845 −0.0634472
$$515$$ 0.438447 0.0193203
$$516$$ 4.56155 0.200811
$$517$$ −28.8617 −1.26934
$$518$$ 14.6847 0.645207
$$519$$ −25.1771 −1.10515
$$520$$ 0 0
$$521$$ −18.6847 −0.818590 −0.409295 0.912402i $$-0.634225\pi$$
−0.409295 + 0.912402i $$0.634225\pi$$
$$522$$ 6.56155 0.287191
$$523$$ −9.19224 −0.401948 −0.200974 0.979597i $$-0.564411\pi$$
−0.200974 + 0.979597i $$0.564411\pi$$
$$524$$ 6.12311 0.267489
$$525$$ 3.56155 0.155439
$$526$$ −13.0000 −0.566827
$$527$$ 29.1231 1.26862
$$528$$ −4.12311 −0.179435
$$529$$ 36.0540 1.56756
$$530$$ 4.43845 0.192794
$$531$$ 10.5616 0.458332
$$532$$ −12.6847 −0.549950
$$533$$ 0 0
$$534$$ 1.80776 0.0782296
$$535$$ 2.00000 0.0864675
$$536$$ 14.2462 0.615343
$$537$$ −0.315342 −0.0136080
$$538$$ −3.36932 −0.145262
$$539$$ −23.4384 −1.00957
$$540$$ 1.00000 0.0430331
$$541$$ −2.63068 −0.113102 −0.0565510 0.998400i $$-0.518010\pi$$
−0.0565510 + 0.998400i $$0.518010\pi$$
$$542$$ 32.1771 1.38212
$$543$$ 11.1231 0.477338
$$544$$ 5.12311 0.219651
$$545$$ −20.2462 −0.867252
$$546$$ 0 0
$$547$$ 35.6155 1.52281 0.761405 0.648276i $$-0.224510\pi$$
0.761405 + 0.648276i $$0.224510\pi$$
$$548$$ −8.80776 −0.376249
$$549$$ 6.00000 0.256074
$$550$$ −4.12311 −0.175810
$$551$$ −23.3693 −0.995566
$$552$$ −7.68466 −0.327081
$$553$$ 26.4924 1.12657
$$554$$ −1.00000 −0.0424859
$$555$$ 4.12311 0.175016
$$556$$ 8.43845 0.357870
$$557$$ −4.43845 −0.188063 −0.0940315 0.995569i $$-0.529975\pi$$
−0.0940315 + 0.995569i $$0.529975\pi$$
$$558$$ 5.68466 0.240651
$$559$$ 0 0
$$560$$ 3.56155 0.150503
$$561$$ −21.1231 −0.891818
$$562$$ −0.246211 −0.0103858
$$563$$ 32.9848 1.39015 0.695073 0.718939i $$-0.255372\pi$$
0.695073 + 0.718939i $$0.255372\pi$$
$$564$$ 7.00000 0.294753
$$565$$ −3.68466 −0.155015
$$566$$ 11.4384 0.480794
$$567$$ 3.56155 0.149571
$$568$$ −4.87689 −0.204630
$$569$$ −19.1771 −0.803945 −0.401973 0.915652i $$-0.631675\pi$$
−0.401973 + 0.915652i $$0.631675\pi$$
$$570$$ −3.56155 −0.149177
$$571$$ 11.3153 0.473532 0.236766 0.971567i $$-0.423912\pi$$
0.236766 + 0.971567i $$0.423912\pi$$
$$572$$ 0 0
$$573$$ 19.1231 0.798879
$$574$$ −15.1231 −0.631226
$$575$$ −7.68466 −0.320472
$$576$$ 1.00000 0.0416667
$$577$$ 8.73863 0.363794 0.181897 0.983318i $$-0.441776\pi$$
0.181897 + 0.983318i $$0.441776\pi$$
$$578$$ 9.24621 0.384592
$$579$$ 0 0
$$580$$ 6.56155 0.272454
$$581$$ 4.00000 0.165948
$$582$$ 1.12311 0.0465542
$$583$$ −18.3002 −0.757916
$$584$$ −15.3693 −0.635987
$$585$$ 0 0
$$586$$ −24.9309 −1.02988
$$587$$ −23.7538 −0.980424 −0.490212 0.871603i $$-0.663081\pi$$
−0.490212 + 0.871603i $$0.663081\pi$$
$$588$$ 5.68466 0.234431
$$589$$ −20.2462 −0.834231
$$590$$ 10.5616 0.434812
$$591$$ −7.80776 −0.321168
$$592$$ 4.12311 0.169459
$$593$$ −12.1771 −0.500053 −0.250026 0.968239i $$-0.580439\pi$$
−0.250026 + 0.968239i $$0.580439\pi$$
$$594$$ −4.12311 −0.169173
$$595$$ 18.2462 0.748022
$$596$$ −22.8078 −0.934242
$$597$$ 11.1231 0.455238
$$598$$ 0 0
$$599$$ −14.0000 −0.572024 −0.286012 0.958226i $$-0.592330\pi$$
−0.286012 + 0.958226i $$0.592330\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −35.9848 −1.46785 −0.733926 0.679229i $$-0.762314\pi$$
−0.733926 + 0.679229i $$0.762314\pi$$
$$602$$ 16.2462 0.662146
$$603$$ 14.2462 0.580151
$$604$$ 9.36932 0.381232
$$605$$ 6.00000 0.243935
$$606$$ −17.1231 −0.695579
$$607$$ −29.4233 −1.19425 −0.597127 0.802146i $$-0.703691\pi$$
−0.597127 + 0.802146i $$0.703691\pi$$
$$608$$ −3.56155 −0.144440
$$609$$ 23.3693 0.946973
$$610$$ 6.00000 0.242933
$$611$$ 0 0
$$612$$ 5.12311 0.207089
$$613$$ 28.1231 1.13588 0.567941 0.823069i $$-0.307740\pi$$
0.567941 + 0.823069i $$0.307740\pi$$
$$614$$ −15.6155 −0.630191
$$615$$ −4.24621 −0.171224
$$616$$ −14.6847 −0.591662
$$617$$ −41.6847 −1.67816 −0.839081 0.544007i $$-0.816906\pi$$
−0.839081 + 0.544007i $$0.816906\pi$$
$$618$$ 0.438447 0.0176369
$$619$$ −16.4384 −0.660717 −0.330358 0.943856i $$-0.607170\pi$$
−0.330358 + 0.943856i $$0.607170\pi$$
$$620$$ 5.68466 0.228301
$$621$$ −7.68466 −0.308375
$$622$$ −18.7386 −0.751351
$$623$$ 6.43845 0.257951
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 6.63068 0.265015
$$627$$ 14.6847 0.586449
$$628$$ 22.1231 0.882808
$$629$$ 21.1231 0.842233
$$630$$ 3.56155 0.141896
$$631$$ −30.2462 −1.20408 −0.602041 0.798465i $$-0.705646\pi$$
−0.602041 + 0.798465i $$0.705646\pi$$
$$632$$ 7.43845 0.295886
$$633$$ 6.93087 0.275477
$$634$$ −4.19224 −0.166495
$$635$$ −4.43845 −0.176134
$$636$$ 4.43845 0.175996
$$637$$ 0 0
$$638$$ −27.0540 −1.07108
$$639$$ −4.87689 −0.192927
$$640$$ 1.00000 0.0395285
$$641$$ −15.5616 −0.614644 −0.307322 0.951606i $$-0.599433\pi$$
−0.307322 + 0.951606i $$0.599433\pi$$
$$642$$ 2.00000 0.0789337
$$643$$ 38.2462 1.50828 0.754142 0.656712i $$-0.228053\pi$$
0.754142 + 0.656712i $$0.228053\pi$$
$$644$$ −27.3693 −1.07850
$$645$$ 4.56155 0.179611
$$646$$ −18.2462 −0.717888
$$647$$ −2.05398 −0.0807501 −0.0403751 0.999185i $$-0.512855\pi$$
−0.0403751 + 0.999185i $$0.512855\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −43.5464 −1.70935
$$650$$ 0 0
$$651$$ 20.2462 0.793512
$$652$$ −18.5616 −0.726927
$$653$$ −40.4384 −1.58248 −0.791239 0.611507i $$-0.790564\pi$$
−0.791239 + 0.611507i $$0.790564\pi$$
$$654$$ −20.2462 −0.791690
$$655$$ 6.12311 0.239250
$$656$$ −4.24621 −0.165787
$$657$$ −15.3693 −0.599614
$$658$$ 24.9309 0.971906
$$659$$ 31.0540 1.20969 0.604846 0.796343i $$-0.293235\pi$$
0.604846 + 0.796343i $$0.293235\pi$$
$$660$$ −4.12311 −0.160492
$$661$$ −11.6155 −0.451792 −0.225896 0.974151i $$-0.572531\pi$$
−0.225896 + 0.974151i $$0.572531\pi$$
$$662$$ −18.7386 −0.728298
$$663$$ 0 0
$$664$$ 1.12311 0.0435850
$$665$$ −12.6847 −0.491890
$$666$$ 4.12311 0.159767
$$667$$ −50.4233 −1.95240
$$668$$ −20.3693 −0.788113
$$669$$ −26.3002 −1.01682
$$670$$ 14.2462 0.550379
$$671$$ −24.7386 −0.955024
$$672$$ 3.56155 0.137390
$$673$$ 42.2462 1.62847 0.814236 0.580534i $$-0.197156\pi$$
0.814236 + 0.580534i $$0.197156\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −28.8769 −1.10983 −0.554915 0.831907i $$-0.687249\pi$$
−0.554915 + 0.831907i $$0.687249\pi$$
$$678$$ −3.68466 −0.141508
$$679$$ 4.00000 0.153506
$$680$$ 5.12311 0.196462
$$681$$ −20.0000 −0.766402
$$682$$ −23.4384 −0.897505
$$683$$ 8.87689 0.339665 0.169832 0.985473i $$-0.445677\pi$$
0.169832 + 0.985473i $$0.445677\pi$$
$$684$$ −3.56155 −0.136179
$$685$$ −8.80776 −0.336527
$$686$$ −4.68466 −0.178861
$$687$$ −7.75379 −0.295825
$$688$$ 4.56155 0.173908
$$689$$ 0 0
$$690$$ −7.68466 −0.292550
$$691$$ −16.4384 −0.625348 −0.312674 0.949860i $$-0.601225\pi$$
−0.312674 + 0.949860i $$0.601225\pi$$
$$692$$ −25.1771 −0.957089
$$693$$ −14.6847 −0.557824
$$694$$ 9.12311 0.346308
$$695$$ 8.43845 0.320089
$$696$$ 6.56155 0.248715
$$697$$ −21.7538 −0.823984
$$698$$ −24.4924 −0.927052
$$699$$ −17.6847 −0.668895
$$700$$ 3.56155 0.134614
$$701$$ 17.3002 0.653419 0.326710 0.945125i $$-0.394060\pi$$
0.326710 + 0.945125i $$0.394060\pi$$
$$702$$ 0 0
$$703$$ −14.6847 −0.553842
$$704$$ −4.12311 −0.155395
$$705$$ 7.00000 0.263635
$$706$$ 31.8617 1.19913
$$707$$ −60.9848 −2.29357
$$708$$ 10.5616 0.396927
$$709$$ 17.7538 0.666758 0.333379 0.942793i $$-0.391811\pi$$
0.333379 + 0.942793i $$0.391811\pi$$
$$710$$ −4.87689 −0.183027
$$711$$ 7.43845 0.278964
$$712$$ 1.80776 0.0677488
$$713$$ −43.6847 −1.63600
$$714$$ 18.2462 0.682847
$$715$$ 0 0
$$716$$ −0.315342 −0.0117849
$$717$$ 13.3693 0.499286
$$718$$ 4.87689 0.182004
$$719$$ −20.9848 −0.782603 −0.391301 0.920263i $$-0.627975\pi$$
−0.391301 + 0.920263i $$0.627975\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 1.56155 0.0581553
$$722$$ −6.31534 −0.235033
$$723$$ −19.8769 −0.739230
$$724$$ 11.1231 0.413387
$$725$$ 6.56155 0.243690
$$726$$ 6.00000 0.222681
$$727$$ −23.4233 −0.868722 −0.434361 0.900739i $$-0.643026\pi$$
−0.434361 + 0.900739i $$0.643026\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −15.3693 −0.568844
$$731$$ 23.3693 0.864345
$$732$$ 6.00000 0.221766
$$733$$ 48.9309 1.80730 0.903651 0.428269i $$-0.140876\pi$$
0.903651 + 0.428269i $$0.140876\pi$$
$$734$$ −20.4924 −0.756389
$$735$$ 5.68466 0.209682
$$736$$ −7.68466 −0.283260
$$737$$ −58.7386 −2.16367
$$738$$ −4.24621 −0.156305
$$739$$ 42.5464 1.56509 0.782547 0.622591i $$-0.213920\pi$$
0.782547 + 0.622591i $$0.213920\pi$$
$$740$$ 4.12311 0.151568
$$741$$ 0 0
$$742$$ 15.8078 0.580321
$$743$$ −32.1771 −1.18046 −0.590231 0.807234i $$-0.700963\pi$$
−0.590231 + 0.807234i $$0.700963\pi$$
$$744$$ 5.68466 0.208410
$$745$$ −22.8078 −0.835612
$$746$$ −5.19224 −0.190101
$$747$$ 1.12311 0.0410923
$$748$$ −21.1231 −0.772337
$$749$$ 7.12311 0.260273
$$750$$ 1.00000 0.0365148
$$751$$ −3.19224 −0.116486 −0.0582432 0.998302i $$-0.518550\pi$$
−0.0582432 + 0.998302i $$0.518550\pi$$
$$752$$ 7.00000 0.255264
$$753$$ 0.123106 0.00448622
$$754$$ 0 0
$$755$$ 9.36932 0.340984
$$756$$ 3.56155 0.129532
$$757$$ 33.4233 1.21479 0.607395 0.794400i $$-0.292215\pi$$
0.607395 + 0.794400i $$0.292215\pi$$
$$758$$ −5.31534 −0.193062
$$759$$ 31.6847 1.15008
$$760$$ −3.56155 −0.129191
$$761$$ 10.9309 0.396244 0.198122 0.980177i $$-0.436516\pi$$
0.198122 + 0.980177i $$0.436516\pi$$
$$762$$ −4.43845 −0.160788
$$763$$ −72.1080 −2.61048
$$764$$ 19.1231 0.691850
$$765$$ 5.12311 0.185226
$$766$$ 20.8078 0.751815
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −45.6847 −1.64743 −0.823715 0.567003i $$-0.808103\pi$$
−0.823715 + 0.567003i $$0.808103\pi$$
$$770$$ −14.6847 −0.529198
$$771$$ −1.43845 −0.0518044
$$772$$ 0 0
$$773$$ −21.1771 −0.761687 −0.380843 0.924640i $$-0.624366\pi$$
−0.380843 + 0.924640i $$0.624366\pi$$
$$774$$ 4.56155 0.163962
$$775$$ 5.68466 0.204199
$$776$$ 1.12311 0.0403171
$$777$$ 14.6847 0.526809
$$778$$ 19.0540 0.683118
$$779$$ 15.1231 0.541841
$$780$$ 0 0
$$781$$ 20.1080 0.719519
$$782$$ −39.3693 −1.40784
$$783$$ 6.56155 0.234491
$$784$$ 5.68466 0.203024
$$785$$ 22.1231 0.789607
$$786$$ 6.12311 0.218404
$$787$$ 43.6847 1.55719 0.778595 0.627527i $$-0.215933\pi$$
0.778595 + 0.627527i $$0.215933\pi$$
$$788$$ −7.80776 −0.278140
$$789$$ −13.0000 −0.462812
$$790$$ 7.43845 0.264648
$$791$$ −13.1231 −0.466604
$$792$$ −4.12311 −0.146508
$$793$$ 0 0
$$794$$ −11.8769 −0.421495
$$795$$ 4.43845 0.157415
$$796$$ 11.1231 0.394248
$$797$$ 4.87689 0.172748 0.0863742 0.996263i $$-0.472472\pi$$
0.0863742 + 0.996263i $$0.472472\pi$$
$$798$$ −12.6847 −0.449032
$$799$$ 35.8617 1.26870
$$800$$ 1.00000 0.0353553
$$801$$ 1.80776 0.0638742
$$802$$ 12.6847 0.447911
$$803$$ 63.3693 2.23625
$$804$$ 14.2462 0.502425
$$805$$ −27.3693 −0.964642
$$806$$ 0 0
$$807$$ −3.36932 −0.118606
$$808$$ −17.1231 −0.602389
$$809$$ 18.4924 0.650159 0.325079 0.945687i $$-0.394609\pi$$
0.325079 + 0.945687i $$0.394609\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −22.5464 −0.791711 −0.395856 0.918313i $$-0.629552\pi$$
−0.395856 + 0.918313i $$0.629552\pi$$
$$812$$ 23.3693 0.820102
$$813$$ 32.1771 1.12850
$$814$$ −17.0000 −0.595850
$$815$$ −18.5616 −0.650183
$$816$$ 5.12311 0.179345
$$817$$ −16.2462 −0.568383
$$818$$ 1.80776 0.0632070
$$819$$ 0 0
$$820$$ −4.24621 −0.148284
$$821$$ −12.5616 −0.438401 −0.219201 0.975680i $$-0.570345\pi$$
−0.219201 + 0.975680i $$0.570345\pi$$
$$822$$ −8.80776 −0.307206
$$823$$ −32.0540 −1.11733 −0.558666 0.829393i $$-0.688687\pi$$
−0.558666 + 0.829393i $$0.688687\pi$$
$$824$$ 0.438447 0.0152740
$$825$$ −4.12311 −0.143548
$$826$$ 37.6155 1.30881
$$827$$ −11.7538 −0.408719 −0.204360 0.978896i $$-0.565511\pi$$
−0.204360 + 0.978896i $$0.565511\pi$$
$$828$$ −7.68466 −0.267060
$$829$$ 53.6155 1.86214 0.931072 0.364835i $$-0.118875\pi$$
0.931072 + 0.364835i $$0.118875\pi$$
$$830$$ 1.12311 0.0389836
$$831$$ −1.00000 −0.0346896
$$832$$ 0 0
$$833$$ 29.1231 1.00906
$$834$$ 8.43845 0.292200
$$835$$ −20.3693 −0.704909
$$836$$ 14.6847 0.507880
$$837$$ 5.68466 0.196491
$$838$$ −0.492423 −0.0170105
$$839$$ −12.7386 −0.439786 −0.219893 0.975524i $$-0.570571\pi$$
−0.219893 + 0.975524i $$0.570571\pi$$
$$840$$ 3.56155 0.122885
$$841$$ 14.0540 0.484620
$$842$$ 0.492423 0.0169700
$$843$$ −0.246211 −0.00847997
$$844$$ 6.93087 0.238570
$$845$$ 0 0
$$846$$ 7.00000 0.240665
$$847$$ 21.3693 0.734258
$$848$$ 4.43845 0.152417
$$849$$ 11.4384 0.392566
$$850$$ 5.12311 0.175721
$$851$$ −31.6847 −1.08614
$$852$$ −4.87689 −0.167080
$$853$$ −51.3002 −1.75648 −0.878242 0.478216i $$-0.841284\pi$$
−0.878242 + 0.478216i $$0.841284\pi$$
$$854$$ 21.3693 0.731243
$$855$$ −3.56155 −0.121803
$$856$$ 2.00000 0.0683586
$$857$$ −26.8078 −0.915736 −0.457868 0.889020i $$-0.651387\pi$$
−0.457868 + 0.889020i $$0.651387\pi$$
$$858$$ 0 0
$$859$$ −46.7926 −1.59654 −0.798272 0.602298i $$-0.794252\pi$$
−0.798272 + 0.602298i $$0.794252\pi$$
$$860$$ 4.56155 0.155548
$$861$$ −15.1231 −0.515394
$$862$$ 26.7386 0.910721
$$863$$ 6.56155 0.223358 0.111679 0.993744i $$-0.464377\pi$$
0.111679 + 0.993744i $$0.464377\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −25.1771 −0.856046
$$866$$ 30.7386 1.04454
$$867$$ 9.24621 0.314018
$$868$$ 20.2462 0.687201
$$869$$ −30.6695 −1.04039
$$870$$ 6.56155 0.222457
$$871$$ 0 0
$$872$$ −20.2462 −0.685623
$$873$$ 1.12311 0.0380114
$$874$$ 27.3693 0.925781
$$875$$ 3.56155 0.120402
$$876$$ −15.3693 −0.519281
$$877$$ 12.5616 0.424173 0.212087 0.977251i $$-0.431974\pi$$
0.212087 + 0.977251i $$0.431974\pi$$
$$878$$ 16.8769 0.569568
$$879$$ −24.9309 −0.840897
$$880$$ −4.12311 −0.138990
$$881$$ −20.4384 −0.688589 −0.344294 0.938862i $$-0.611882\pi$$
−0.344294 + 0.938862i $$0.611882\pi$$
$$882$$ 5.68466 0.191412
$$883$$ 26.1771 0.880929 0.440464 0.897770i $$-0.354814\pi$$
0.440464 + 0.897770i $$0.354814\pi$$
$$884$$ 0 0
$$885$$ 10.5616 0.355023
$$886$$ −4.87689 −0.163842
$$887$$ 41.1080 1.38027 0.690135 0.723681i $$-0.257551\pi$$
0.690135 + 0.723681i $$0.257551\pi$$
$$888$$ 4.12311 0.138362
$$889$$ −15.8078 −0.530175
$$890$$ 1.80776 0.0605964
$$891$$ −4.12311 −0.138129
$$892$$ −26.3002 −0.880595
$$893$$ −24.9309 −0.834280
$$894$$ −22.8078 −0.762806
$$895$$ −0.315342 −0.0105407
$$896$$ 3.56155 0.118983
$$897$$ 0 0
$$898$$ 25.1771 0.840170
$$899$$ 37.3002 1.24403
$$900$$ 1.00000 0.0333333
$$901$$ 22.7386 0.757534
$$902$$ 17.5076 0.582939
$$903$$ 16.2462 0.540640
$$904$$ −3.68466 −0.122550
$$905$$ 11.1231 0.369745
$$906$$ 9.36932 0.311275
$$907$$ 40.9157 1.35858 0.679292 0.733868i $$-0.262287\pi$$
0.679292 + 0.733868i $$0.262287\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ −17.1231 −0.567938
$$910$$ 0 0
$$911$$ −43.3693 −1.43689 −0.718445 0.695584i $$-0.755146\pi$$
−0.718445 + 0.695584i $$0.755146\pi$$
$$912$$ −3.56155 −0.117935
$$913$$ −4.63068 −0.153253
$$914$$ 3.75379 0.124164
$$915$$ 6.00000 0.198354
$$916$$ −7.75379 −0.256192
$$917$$ 21.8078 0.720156
$$918$$ 5.12311 0.169088
$$919$$ 41.3693 1.36465 0.682324 0.731050i $$-0.260969\pi$$
0.682324 + 0.731050i $$0.260969\pi$$
$$920$$ −7.68466 −0.253356
$$921$$ −15.6155 −0.514549
$$922$$ 7.05398 0.232310
$$923$$ 0 0
$$924$$ −14.6847 −0.483090
$$925$$ 4.12311 0.135567
$$926$$ 33.6155 1.10467
$$927$$ 0.438447 0.0144005
$$928$$ 6.56155 0.215394
$$929$$ −16.8769 −0.553713 −0.276856 0.960911i $$-0.589293\pi$$
−0.276856 + 0.960911i $$0.589293\pi$$
$$930$$ 5.68466 0.186407
$$931$$ −20.2462 −0.663543
$$932$$ −17.6847 −0.579280
$$933$$ −18.7386 −0.613475
$$934$$ 39.8617 1.30432
$$935$$ −21.1231 −0.690799
$$936$$ 0 0
$$937$$ −29.2311 −0.954937 −0.477468 0.878649i $$-0.658446\pi$$
−0.477468 + 0.878649i $$0.658446\pi$$
$$938$$ 50.7386 1.65668
$$939$$ 6.63068 0.216384
$$940$$ 7.00000 0.228315
$$941$$ −0.630683 −0.0205597 −0.0102798 0.999947i $$-0.503272\pi$$
−0.0102798 + 0.999947i $$0.503272\pi$$
$$942$$ 22.1231 0.720810
$$943$$ 32.6307 1.06260
$$944$$ 10.5616 0.343749
$$945$$ 3.56155 0.115857
$$946$$ −18.8078 −0.611493
$$947$$ 13.8617 0.450446 0.225223 0.974307i $$-0.427689\pi$$
0.225223 + 0.974307i $$0.427689\pi$$
$$948$$ 7.43845 0.241590
$$949$$ 0 0
$$950$$ −3.56155 −0.115552
$$951$$ −4.19224 −0.135943
$$952$$ 18.2462 0.591363
$$953$$ 5.82292 0.188623 0.0943114 0.995543i $$-0.469935\pi$$
0.0943114 + 0.995543i $$0.469935\pi$$
$$954$$ 4.43845 0.143700
$$955$$ 19.1231 0.618809
$$956$$ 13.3693 0.432395
$$957$$ −27.0540 −0.874531
$$958$$ −21.7538 −0.702833
$$959$$ −31.3693 −1.01297
$$960$$ 1.00000 0.0322749
$$961$$ 1.31534 0.0424304
$$962$$ 0 0
$$963$$ 2.00000 0.0644491
$$964$$ −19.8769 −0.640192
$$965$$ 0 0
$$966$$ −27.3693 −0.880593
$$967$$ 3.31534 0.106614 0.0533071 0.998578i $$-0.483024\pi$$
0.0533071 + 0.998578i $$0.483024\pi$$
$$968$$ 6.00000 0.192847
$$969$$ −18.2462 −0.586153
$$970$$ 1.12311 0.0360607
$$971$$ −16.6847 −0.535436 −0.267718 0.963497i $$-0.586270\pi$$
−0.267718 + 0.963497i $$0.586270\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 30.0540 0.963486
$$974$$ −24.0540 −0.770739
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 5.30019 0.169568 0.0847840 0.996399i $$-0.472980\pi$$
0.0847840 + 0.996399i $$0.472980\pi$$
$$978$$ −18.5616 −0.593533
$$979$$ −7.45360 −0.238218
$$980$$ 5.68466 0.181590
$$981$$ −20.2462 −0.646412
$$982$$ −21.5616 −0.688057
$$983$$ 23.8769 0.761555 0.380777 0.924667i $$-0.375656\pi$$
0.380777 + 0.924667i $$0.375656\pi$$
$$984$$ −4.24621 −0.135364
$$985$$ −7.80776 −0.248776
$$986$$ 33.6155 1.07054
$$987$$ 24.9309 0.793558
$$988$$ 0 0
$$989$$ −35.0540 −1.11465
$$990$$ −4.12311 −0.131041
$$991$$ −12.4233 −0.394639 −0.197319 0.980339i $$-0.563224\pi$$
−0.197319 + 0.980339i $$0.563224\pi$$
$$992$$ 5.68466 0.180488
$$993$$ −18.7386 −0.594653
$$994$$ −17.3693 −0.550921
$$995$$ 11.1231 0.352626
$$996$$ 1.12311 0.0355870
$$997$$ 32.5464 1.03075 0.515377 0.856963i $$-0.327652\pi$$
0.515377 + 0.856963i $$0.327652\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 4.12311 0.130449
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.a.bi.1.2 2
13.3 even 3 390.2.i.g.61.1 4
13.5 odd 4 5070.2.b.r.1351.2 4
13.8 odd 4 5070.2.b.r.1351.3 4
13.9 even 3 390.2.i.g.211.1 yes 4
13.12 even 2 5070.2.a.bb.1.1 2
39.29 odd 6 1170.2.i.o.451.1 4
39.35 odd 6 1170.2.i.o.991.1 4
65.3 odd 12 1950.2.z.n.1699.3 8
65.9 even 6 1950.2.i.bi.601.2 4
65.22 odd 12 1950.2.z.n.1849.3 8
65.29 even 6 1950.2.i.bi.451.2 4
65.42 odd 12 1950.2.z.n.1699.2 8
65.48 odd 12 1950.2.z.n.1849.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.g.61.1 4 13.3 even 3
390.2.i.g.211.1 yes 4 13.9 even 3
1170.2.i.o.451.1 4 39.29 odd 6
1170.2.i.o.991.1 4 39.35 odd 6
1950.2.i.bi.451.2 4 65.29 even 6
1950.2.i.bi.601.2 4 65.9 even 6
1950.2.z.n.1699.2 8 65.42 odd 12
1950.2.z.n.1699.3 8 65.3 odd 12
1950.2.z.n.1849.2 8 65.48 odd 12
1950.2.z.n.1849.3 8 65.22 odd 12
5070.2.a.bb.1.1 2 13.12 even 2
5070.2.a.bi.1.2 2 1.1 even 1 trivial
5070.2.b.r.1351.2 4 13.5 odd 4
5070.2.b.r.1351.3 4 13.8 odd 4