# Properties

 Label 5070.2.a.bb.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 $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.12311 q^{11} +1.00000 q^{12} -0.561553 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.12311 q^{17} -1.00000 q^{18} -0.561553 q^{19} -1.00000 q^{20} +0.561553 q^{21} +4.12311 q^{22} +4.68466 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.561553 q^{28} +2.43845 q^{29} +1.00000 q^{30} +6.68466 q^{31} -1.00000 q^{32} -4.12311 q^{33} +3.12311 q^{34} -0.561553 q^{35} +1.00000 q^{36} +4.12311 q^{37} +0.561553 q^{38} +1.00000 q^{40} -12.2462 q^{41} -0.561553 q^{42} +0.438447 q^{43} -4.12311 q^{44} -1.00000 q^{45} -4.68466 q^{46} -7.00000 q^{47} +1.00000 q^{48} -6.68466 q^{49} -1.00000 q^{50} -3.12311 q^{51} +8.56155 q^{53} -1.00000 q^{54} +4.12311 q^{55} -0.561553 q^{56} -0.561553 q^{57} -2.43845 q^{58} -6.43845 q^{59} -1.00000 q^{60} +6.00000 q^{61} -6.68466 q^{62} +0.561553 q^{63} +1.00000 q^{64} +4.12311 q^{66} +2.24621 q^{67} -3.12311 q^{68} +4.68466 q^{69} +0.561553 q^{70} +13.1231 q^{71} -1.00000 q^{72} -9.36932 q^{73} -4.12311 q^{74} +1.00000 q^{75} -0.561553 q^{76} -2.31534 q^{77} +11.5616 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.2462 q^{82} +7.12311 q^{83} +0.561553 q^{84} +3.12311 q^{85} -0.438447 q^{86} +2.43845 q^{87} +4.12311 q^{88} +18.8078 q^{89} +1.00000 q^{90} +4.68466 q^{92} +6.68466 q^{93} +7.00000 q^{94} +0.561553 q^{95} -1.00000 q^{96} +7.12311 q^{97} +6.68466 q^{98} -4.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 3q^{14} - 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 3q^{19} - 2q^{20} - 3q^{21} - 3q^{23} - 2q^{24} + 2q^{25} + 2q^{27} - 3q^{28} + 9q^{29} + 2q^{30} + q^{31} - 2q^{32} - 2q^{34} + 3q^{35} + 2q^{36} - 3q^{38} + 2q^{40} - 8q^{41} + 3q^{42} + 5q^{43} - 2q^{45} + 3q^{46} - 14q^{47} + 2q^{48} - q^{49} - 2q^{50} + 2q^{51} + 13q^{53} - 2q^{54} + 3q^{56} + 3q^{57} - 9q^{58} - 17q^{59} - 2q^{60} + 12q^{61} - q^{62} - 3q^{63} + 2q^{64} - 12q^{67} + 2q^{68} - 3q^{69} - 3q^{70} + 18q^{71} - 2q^{72} + 6q^{73} + 2q^{75} + 3q^{76} - 17q^{77} + 19q^{79} - 2q^{80} + 2q^{81} + 8q^{82} + 6q^{83} - 3q^{84} - 2q^{85} - 5q^{86} + 9q^{87} + 17q^{89} + 2q^{90} - 3q^{92} + q^{93} + 14q^{94} - 3q^{95} - 2q^{96} + 6q^{97} + q^{98} + O(q^{100})$$

## 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
$$7$$ 0.561553 0.212247 0.106124 0.994353i $$-0.466156\pi$$
0.106124 + 0.994353i $$0.466156\pi$$
$$8$$ −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
$$14$$ −0.561553 −0.150081
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −3.12311 −0.757464 −0.378732 0.925506i $$-0.623640\pi$$
−0.378732 + 0.925506i $$0.623640\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −0.561553 −0.128829 −0.0644145 0.997923i $$-0.520518\pi$$
−0.0644145 + 0.997923i $$0.520518\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0.561553 0.122541
$$22$$ 4.12311 0.879049
$$23$$ 4.68466 0.976819 0.488409 0.872615i $$-0.337577\pi$$
0.488409 + 0.872615i $$0.337577\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0.561553 0.106124
$$29$$ 2.43845 0.452808 0.226404 0.974033i $$-0.427303\pi$$
0.226404 + 0.974033i $$0.427303\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 6.68466 1.20060 0.600300 0.799775i $$-0.295048\pi$$
0.600300 + 0.799775i $$0.295048\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.12311 −0.717741
$$34$$ 3.12311 0.535608
$$35$$ −0.561553 −0.0949197
$$36$$ 1.00000 0.166667
$$37$$ 4.12311 0.677834 0.338917 0.940816i $$-0.389939\pi$$
0.338917 + 0.940816i $$0.389939\pi$$
$$38$$ 0.561553 0.0910959
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −12.2462 −1.91254 −0.956268 0.292490i $$-0.905516\pi$$
−0.956268 + 0.292490i $$0.905516\pi$$
$$42$$ −0.561553 −0.0866495
$$43$$ 0.438447 0.0668626 0.0334313 0.999441i $$-0.489357\pi$$
0.0334313 + 0.999441i $$0.489357\pi$$
$$44$$ −4.12311 −0.621582
$$45$$ −1.00000 −0.149071
$$46$$ −4.68466 −0.690715
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.68466 −0.954951
$$50$$ −1.00000 −0.141421
$$51$$ −3.12311 −0.437322
$$52$$ 0 0
$$53$$ 8.56155 1.17602 0.588010 0.808854i $$-0.299912\pi$$
0.588010 + 0.808854i $$0.299912\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 4.12311 0.555959
$$56$$ −0.561553 −0.0750407
$$57$$ −0.561553 −0.0743795
$$58$$ −2.43845 −0.320184
$$59$$ −6.43845 −0.838214 −0.419107 0.907937i $$-0.637657\pi$$
−0.419107 + 0.907937i $$0.637657\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −6.68466 −0.848952
$$63$$ 0.561553 0.0707490
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.12311 0.507519
$$67$$ 2.24621 0.274418 0.137209 0.990542i $$-0.456187\pi$$
0.137209 + 0.990542i $$0.456187\pi$$
$$68$$ −3.12311 −0.378732
$$69$$ 4.68466 0.563967
$$70$$ 0.561553 0.0671184
$$71$$ 13.1231 1.55743 0.778713 0.627380i $$-0.215873\pi$$
0.778713 + 0.627380i $$0.215873\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −9.36932 −1.09660 −0.548298 0.836283i $$-0.684724\pi$$
−0.548298 + 0.836283i $$0.684724\pi$$
$$74$$ −4.12311 −0.479301
$$75$$ 1.00000 0.115470
$$76$$ −0.561553 −0.0644145
$$77$$ −2.31534 −0.263858
$$78$$ 0 0
$$79$$ 11.5616 1.30078 0.650388 0.759602i $$-0.274606\pi$$
0.650388 + 0.759602i $$0.274606\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 12.2462 1.35237
$$83$$ 7.12311 0.781862 0.390931 0.920420i $$-0.372153\pi$$
0.390931 + 0.920420i $$0.372153\pi$$
$$84$$ 0.561553 0.0612704
$$85$$ 3.12311 0.338748
$$86$$ −0.438447 −0.0472790
$$87$$ 2.43845 0.261429
$$88$$ 4.12311 0.439525
$$89$$ 18.8078 1.99362 0.996810 0.0798174i $$-0.0254337\pi$$
0.996810 + 0.0798174i $$0.0254337\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 4.68466 0.488409
$$93$$ 6.68466 0.693167
$$94$$ 7.00000 0.721995
$$95$$ 0.561553 0.0576141
$$96$$ −1.00000 −0.102062
$$97$$ 7.12311 0.723242 0.361621 0.932325i $$-0.382223\pi$$
0.361621 + 0.932325i $$0.382223\pi$$
$$98$$ 6.68466 0.675252
$$99$$ −4.12311 −0.414388
$$100$$ 1.00000 0.100000
$$101$$ −8.87689 −0.883284 −0.441642 0.897191i $$-0.645604\pi$$
−0.441642 + 0.897191i $$0.645604\pi$$
$$102$$ 3.12311 0.309234
$$103$$ 4.56155 0.449463 0.224732 0.974421i $$-0.427849\pi$$
0.224732 + 0.974421i $$0.427849\pi$$
$$104$$ 0 0
$$105$$ −0.561553 −0.0548019
$$106$$ −8.56155 −0.831572
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 3.75379 0.359548 0.179774 0.983708i $$-0.442463\pi$$
0.179774 + 0.983708i $$0.442463\pi$$
$$110$$ −4.12311 −0.393123
$$111$$ 4.12311 0.391348
$$112$$ 0.561553 0.0530618
$$113$$ 8.68466 0.816984 0.408492 0.912762i $$-0.366055\pi$$
0.408492 + 0.912762i $$0.366055\pi$$
$$114$$ 0.561553 0.0525942
$$115$$ −4.68466 −0.436847
$$116$$ 2.43845 0.226404
$$117$$ 0 0
$$118$$ 6.43845 0.592707
$$119$$ −1.75379 −0.160770
$$120$$ 1.00000 0.0912871
$$121$$ 6.00000 0.545455
$$122$$ −6.00000 −0.543214
$$123$$ −12.2462 −1.10420
$$124$$ 6.68466 0.600300
$$125$$ −1.00000 −0.0894427
$$126$$ −0.561553 −0.0500271
$$127$$ −8.56155 −0.759715 −0.379857 0.925045i $$-0.624027\pi$$
−0.379857 + 0.925045i $$0.624027\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0.438447 0.0386031
$$130$$ 0 0
$$131$$ −2.12311 −0.185497 −0.0927483 0.995690i $$-0.529565\pi$$
−0.0927483 + 0.995690i $$0.529565\pi$$
$$132$$ −4.12311 −0.358870
$$133$$ −0.315342 −0.0273436
$$134$$ −2.24621 −0.194043
$$135$$ −1.00000 −0.0860663
$$136$$ 3.12311 0.267804
$$137$$ −11.8078 −1.00881 −0.504403 0.863469i $$-0.668287\pi$$
−0.504403 + 0.863469i $$0.668287\pi$$
$$138$$ −4.68466 −0.398785
$$139$$ 12.5616 1.06546 0.532729 0.846286i $$-0.321167\pi$$
0.532729 + 0.846286i $$0.321167\pi$$
$$140$$ −0.561553 −0.0474599
$$141$$ −7.00000 −0.589506
$$142$$ −13.1231 −1.10127
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −2.43845 −0.202502
$$146$$ 9.36932 0.775410
$$147$$ −6.68466 −0.551341
$$148$$ 4.12311 0.338917
$$149$$ 2.19224 0.179595 0.0897975 0.995960i $$-0.471378\pi$$
0.0897975 + 0.995960i $$0.471378\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 15.3693 1.25074 0.625369 0.780329i $$-0.284949\pi$$
0.625369 + 0.780329i $$0.284949\pi$$
$$152$$ 0.561553 0.0455479
$$153$$ −3.12311 −0.252488
$$154$$ 2.31534 0.186576
$$155$$ −6.68466 −0.536925
$$156$$ 0 0
$$157$$ 13.8769 1.10750 0.553748 0.832684i $$-0.313197\pi$$
0.553748 + 0.832684i $$0.313197\pi$$
$$158$$ −11.5616 −0.919788
$$159$$ 8.56155 0.678975
$$160$$ 1.00000 0.0790569
$$161$$ 2.63068 0.207327
$$162$$ −1.00000 −0.0785674
$$163$$ 14.4384 1.13091 0.565453 0.824780i $$-0.308701\pi$$
0.565453 + 0.824780i $$0.308701\pi$$
$$164$$ −12.2462 −0.956268
$$165$$ 4.12311 0.320983
$$166$$ −7.12311 −0.552860
$$167$$ −4.36932 −0.338108 −0.169054 0.985607i $$-0.554071\pi$$
−0.169054 + 0.985607i $$0.554071\pi$$
$$168$$ −0.561553 −0.0433247
$$169$$ 0 0
$$170$$ −3.12311 −0.239531
$$171$$ −0.561553 −0.0429430
$$172$$ 0.438447 0.0334313
$$173$$ 20.1771 1.53404 0.767018 0.641626i $$-0.221740\pi$$
0.767018 + 0.641626i $$0.221740\pi$$
$$174$$ −2.43845 −0.184858
$$175$$ 0.561553 0.0424494
$$176$$ −4.12311 −0.310791
$$177$$ −6.43845 −0.483943
$$178$$ −18.8078 −1.40970
$$179$$ −12.6847 −0.948096 −0.474048 0.880499i $$-0.657208\pi$$
−0.474048 + 0.880499i $$0.657208\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 2.87689 0.213838 0.106919 0.994268i $$-0.465901\pi$$
0.106919 + 0.994268i $$0.465901\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ −4.68466 −0.345358
$$185$$ −4.12311 −0.303137
$$186$$ −6.68466 −0.490143
$$187$$ 12.8769 0.941652
$$188$$ −7.00000 −0.510527
$$189$$ 0.561553 0.0408470
$$190$$ −0.561553 −0.0407393
$$191$$ 10.8769 0.787024 0.393512 0.919319i $$-0.371260\pi$$
0.393512 + 0.919319i $$0.371260\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ −7.12311 −0.511409
$$195$$ 0 0
$$196$$ −6.68466 −0.477476
$$197$$ −12.8078 −0.912515 −0.456258 0.889848i $$-0.650810\pi$$
−0.456258 + 0.889848i $$0.650810\pi$$
$$198$$ 4.12311 0.293016
$$199$$ 2.87689 0.203938 0.101969 0.994788i $$-0.467486\pi$$
0.101969 + 0.994788i $$0.467486\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 2.24621 0.158436
$$202$$ 8.87689 0.624576
$$203$$ 1.36932 0.0961072
$$204$$ −3.12311 −0.218661
$$205$$ 12.2462 0.855312
$$206$$ −4.56155 −0.317818
$$207$$ 4.68466 0.325606
$$208$$ 0 0
$$209$$ 2.31534 0.160156
$$210$$ 0.561553 0.0387508
$$211$$ −21.9309 −1.50978 −0.754892 0.655850i $$-0.772311\pi$$
−0.754892 + 0.655850i $$0.772311\pi$$
$$212$$ 8.56155 0.588010
$$213$$ 13.1231 0.899180
$$214$$ −2.00000 −0.136717
$$215$$ −0.438447 −0.0299018
$$216$$ −1.00000 −0.0680414
$$217$$ 3.75379 0.254824
$$218$$ −3.75379 −0.254239
$$219$$ −9.36932 −0.633120
$$220$$ 4.12311 0.277980
$$221$$ 0 0
$$222$$ −4.12311 −0.276725
$$223$$ −27.3002 −1.82816 −0.914078 0.405539i $$-0.867084\pi$$
−0.914078 + 0.405539i $$0.867084\pi$$
$$224$$ −0.561553 −0.0375203
$$225$$ 1.00000 0.0666667
$$226$$ −8.68466 −0.577695
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ −0.561553 −0.0371897
$$229$$ 24.2462 1.60223 0.801117 0.598507i $$-0.204239\pi$$
0.801117 + 0.598507i $$0.204239\pi$$
$$230$$ 4.68466 0.308897
$$231$$ −2.31534 −0.152338
$$232$$ −2.43845 −0.160092
$$233$$ −5.31534 −0.348220 −0.174110 0.984726i $$-0.555705\pi$$
−0.174110 + 0.984726i $$0.555705\pi$$
$$234$$ 0 0
$$235$$ 7.00000 0.456630
$$236$$ −6.43845 −0.419107
$$237$$ 11.5616 0.751004
$$238$$ 1.75379 0.113681
$$239$$ 11.3693 0.735420 0.367710 0.929941i $$-0.380142\pi$$
0.367710 + 0.929941i $$0.380142\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 28.1231 1.81157 0.905784 0.423739i $$-0.139283\pi$$
0.905784 + 0.423739i $$0.139283\pi$$
$$242$$ −6.00000 −0.385695
$$243$$ 1.00000 0.0641500
$$244$$ 6.00000 0.384111
$$245$$ 6.68466 0.427067
$$246$$ 12.2462 0.780790
$$247$$ 0 0
$$248$$ −6.68466 −0.424476
$$249$$ 7.12311 0.451408
$$250$$ 1.00000 0.0632456
$$251$$ −8.12311 −0.512726 −0.256363 0.966581i $$-0.582524\pi$$
−0.256363 + 0.966581i $$0.582524\pi$$
$$252$$ 0.561553 0.0353745
$$253$$ −19.3153 −1.21435
$$254$$ 8.56155 0.537200
$$255$$ 3.12311 0.195576
$$256$$ 1.00000 0.0625000
$$257$$ −5.56155 −0.346920 −0.173460 0.984841i $$-0.555495\pi$$
−0.173460 + 0.984841i $$0.555495\pi$$
$$258$$ −0.438447 −0.0272965
$$259$$ 2.31534 0.143868
$$260$$ 0 0
$$261$$ 2.43845 0.150936
$$262$$ 2.12311 0.131166
$$263$$ −13.0000 −0.801614 −0.400807 0.916162i $$-0.631270\pi$$
−0.400807 + 0.916162i $$0.631270\pi$$
$$264$$ 4.12311 0.253760
$$265$$ −8.56155 −0.525932
$$266$$ 0.315342 0.0193348
$$267$$ 18.8078 1.15102
$$268$$ 2.24621 0.137209
$$269$$ 21.3693 1.30291 0.651455 0.758687i $$-0.274159\pi$$
0.651455 + 0.758687i $$0.274159\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 13.1771 0.800451 0.400225 0.916417i $$-0.368932\pi$$
0.400225 + 0.916417i $$0.368932\pi$$
$$272$$ −3.12311 −0.189366
$$273$$ 0 0
$$274$$ 11.8078 0.713333
$$275$$ −4.12311 −0.248633
$$276$$ 4.68466 0.281983
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ −12.5616 −0.753392
$$279$$ 6.68466 0.400200
$$280$$ 0.561553 0.0335592
$$281$$ −16.2462 −0.969168 −0.484584 0.874745i $$-0.661029\pi$$
−0.484584 + 0.874745i $$0.661029\pi$$
$$282$$ 7.00000 0.416844
$$283$$ 15.5616 0.925038 0.462519 0.886609i $$-0.346946\pi$$
0.462519 + 0.886609i $$0.346946\pi$$
$$284$$ 13.1231 0.778713
$$285$$ 0.561553 0.0332635
$$286$$ 0 0
$$287$$ −6.87689 −0.405930
$$288$$ −1.00000 −0.0589256
$$289$$ −7.24621 −0.426248
$$290$$ 2.43845 0.143191
$$291$$ 7.12311 0.417564
$$292$$ −9.36932 −0.548298
$$293$$ −3.93087 −0.229644 −0.114822 0.993386i $$-0.536630\pi$$
−0.114822 + 0.993386i $$0.536630\pi$$
$$294$$ 6.68466 0.389857
$$295$$ 6.43845 0.374861
$$296$$ −4.12311 −0.239651
$$297$$ −4.12311 −0.239247
$$298$$ −2.19224 −0.126993
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 0.246211 0.0141914
$$302$$ −15.3693 −0.884405
$$303$$ −8.87689 −0.509964
$$304$$ −0.561553 −0.0322073
$$305$$ −6.00000 −0.343559
$$306$$ 3.12311 0.178536
$$307$$ −25.6155 −1.46196 −0.730978 0.682401i $$-0.760936\pi$$
−0.730978 + 0.682401i $$0.760936\pi$$
$$308$$ −2.31534 −0.131929
$$309$$ 4.56155 0.259498
$$310$$ 6.68466 0.379663
$$311$$ 30.7386 1.74303 0.871514 0.490371i $$-0.163139\pi$$
0.871514 + 0.490371i $$0.163139\pi$$
$$312$$ 0 0
$$313$$ 31.3693 1.77310 0.886549 0.462634i $$-0.153096\pi$$
0.886549 + 0.462634i $$0.153096\pi$$
$$314$$ −13.8769 −0.783118
$$315$$ −0.561553 −0.0316399
$$316$$ 11.5616 0.650388
$$317$$ 24.8078 1.39334 0.696671 0.717390i $$-0.254664\pi$$
0.696671 + 0.717390i $$0.254664\pi$$
$$318$$ −8.56155 −0.480108
$$319$$ −10.0540 −0.562915
$$320$$ −1.00000 −0.0559017
$$321$$ 2.00000 0.111629
$$322$$ −2.63068 −0.146602
$$323$$ 1.75379 0.0975834
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −14.4384 −0.799672
$$327$$ 3.75379 0.207585
$$328$$ 12.2462 0.676184
$$329$$ −3.93087 −0.216716
$$330$$ −4.12311 −0.226969
$$331$$ −30.7386 −1.68955 −0.844774 0.535123i $$-0.820265\pi$$
−0.844774 + 0.535123i $$0.820265\pi$$
$$332$$ 7.12311 0.390931
$$333$$ 4.12311 0.225945
$$334$$ 4.36932 0.239078
$$335$$ −2.24621 −0.122724
$$336$$ 0.561553 0.0306352
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 0 0
$$339$$ 8.68466 0.471686
$$340$$ 3.12311 0.169374
$$341$$ −27.5616 −1.49254
$$342$$ 0.561553 0.0303653
$$343$$ −7.68466 −0.414933
$$344$$ −0.438447 −0.0236395
$$345$$ −4.68466 −0.252214
$$346$$ −20.1771 −1.08473
$$347$$ 0.876894 0.0470742 0.0235371 0.999723i $$-0.492507\pi$$
0.0235371 + 0.999723i $$0.492507\pi$$
$$348$$ 2.43845 0.130714
$$349$$ −8.49242 −0.454589 −0.227294 0.973826i $$-0.572988\pi$$
−0.227294 + 0.973826i $$0.572988\pi$$
$$350$$ −0.561553 −0.0300163
$$351$$ 0 0
$$352$$ 4.12311 0.219762
$$353$$ 25.8617 1.37648 0.688241 0.725482i $$-0.258383\pi$$
0.688241 + 0.725482i $$0.258383\pi$$
$$354$$ 6.43845 0.342200
$$355$$ −13.1231 −0.696502
$$356$$ 18.8078 0.996810
$$357$$ −1.75379 −0.0928203
$$358$$ 12.6847 0.670405
$$359$$ −13.1231 −0.692611 −0.346306 0.938122i $$-0.612564\pi$$
−0.346306 + 0.938122i $$0.612564\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −18.6847 −0.983403
$$362$$ −2.87689 −0.151206
$$363$$ 6.00000 0.314918
$$364$$ 0 0
$$365$$ 9.36932 0.490412
$$366$$ −6.00000 −0.313625
$$367$$ 12.4924 0.652099 0.326050 0.945353i $$-0.394282\pi$$
0.326050 + 0.945353i $$0.394282\pi$$
$$368$$ 4.68466 0.244205
$$369$$ −12.2462 −0.637512
$$370$$ 4.12311 0.214350
$$371$$ 4.80776 0.249607
$$372$$ 6.68466 0.346583
$$373$$ −25.8078 −1.33628 −0.668138 0.744038i $$-0.732908\pi$$
−0.668138 + 0.744038i $$0.732908\pi$$
$$374$$ −12.8769 −0.665848
$$375$$ −1.00000 −0.0516398
$$376$$ 7.00000 0.360997
$$377$$ 0 0
$$378$$ −0.561553 −0.0288832
$$379$$ 17.6847 0.908400 0.454200 0.890900i $$-0.349925\pi$$
0.454200 + 0.890900i $$0.349925\pi$$
$$380$$ 0.561553 0.0288071
$$381$$ −8.56155 −0.438622
$$382$$ −10.8769 −0.556510
$$383$$ −0.192236 −0.00982280 −0.00491140 0.999988i $$-0.501563\pi$$
−0.00491140 + 0.999988i $$0.501563\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 2.31534 0.118001
$$386$$ 0 0
$$387$$ 0.438447 0.0222875
$$388$$ 7.12311 0.361621
$$389$$ −18.0540 −0.915373 −0.457686 0.889114i $$-0.651322\pi$$
−0.457686 + 0.889114i $$0.651322\pi$$
$$390$$ 0 0
$$391$$ −14.6307 −0.739905
$$392$$ 6.68466 0.337626
$$393$$ −2.12311 −0.107097
$$394$$ 12.8078 0.645246
$$395$$ −11.5616 −0.581725
$$396$$ −4.12311 −0.207194
$$397$$ 20.1231 1.00995 0.504975 0.863134i $$-0.331502\pi$$
0.504975 + 0.863134i $$0.331502\pi$$
$$398$$ −2.87689 −0.144206
$$399$$ −0.315342 −0.0157868
$$400$$ 1.00000 0.0500000
$$401$$ −0.315342 −0.0157474 −0.00787370 0.999969i $$-0.502506\pi$$
−0.00787370 + 0.999969i $$0.502506\pi$$
$$402$$ −2.24621 −0.112031
$$403$$ 0 0
$$404$$ −8.87689 −0.441642
$$405$$ −1.00000 −0.0496904
$$406$$ −1.36932 −0.0679581
$$407$$ −17.0000 −0.842659
$$408$$ 3.12311 0.154617
$$409$$ 18.8078 0.929984 0.464992 0.885315i $$-0.346057\pi$$
0.464992 + 0.885315i $$0.346057\pi$$
$$410$$ −12.2462 −0.604797
$$411$$ −11.8078 −0.582434
$$412$$ 4.56155 0.224732
$$413$$ −3.61553 −0.177909
$$414$$ −4.68466 −0.230238
$$415$$ −7.12311 −0.349660
$$416$$ 0 0
$$417$$ 12.5616 0.615142
$$418$$ −2.31534 −0.113247
$$419$$ 32.4924 1.58736 0.793679 0.608336i $$-0.208163\pi$$
0.793679 + 0.608336i $$0.208163\pi$$
$$420$$ −0.561553 −0.0274010
$$421$$ 32.4924 1.58358 0.791792 0.610791i $$-0.209148\pi$$
0.791792 + 0.610791i $$0.209148\pi$$
$$422$$ 21.9309 1.06758
$$423$$ −7.00000 −0.340352
$$424$$ −8.56155 −0.415786
$$425$$ −3.12311 −0.151493
$$426$$ −13.1231 −0.635817
$$427$$ 3.36932 0.163053
$$428$$ 2.00000 0.0966736
$$429$$ 0 0
$$430$$ 0.438447 0.0211438
$$431$$ 22.7386 1.09528 0.547641 0.836714i $$-0.315526\pi$$
0.547641 + 0.836714i $$0.315526\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −18.7386 −0.900521 −0.450261 0.892897i $$-0.648669\pi$$
−0.450261 + 0.892897i $$0.648669\pi$$
$$434$$ −3.75379 −0.180188
$$435$$ −2.43845 −0.116915
$$436$$ 3.75379 0.179774
$$437$$ −2.63068 −0.125843
$$438$$ 9.36932 0.447683
$$439$$ 25.1231 1.19906 0.599530 0.800352i $$-0.295354\pi$$
0.599530 + 0.800352i $$0.295354\pi$$
$$440$$ −4.12311 −0.196561
$$441$$ −6.68466 −0.318317
$$442$$ 0 0
$$443$$ −13.1231 −0.623498 −0.311749 0.950165i $$-0.600915\pi$$
−0.311749 + 0.950165i $$0.600915\pi$$
$$444$$ 4.12311 0.195674
$$445$$ −18.8078 −0.891574
$$446$$ 27.3002 1.29270
$$447$$ 2.19224 0.103689
$$448$$ 0.561553 0.0265309
$$449$$ 20.1771 0.952215 0.476108 0.879387i $$-0.342047\pi$$
0.476108 + 0.879387i $$0.342047\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 50.4924 2.37760
$$452$$ 8.68466 0.408492
$$453$$ 15.3693 0.722113
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0.561553 0.0262971
$$457$$ −20.2462 −0.947078 −0.473539 0.880773i $$-0.657024\pi$$
−0.473539 + 0.880773i $$0.657024\pi$$
$$458$$ −24.2462 −1.13295
$$459$$ −3.12311 −0.145774
$$460$$ −4.68466 −0.218423
$$461$$ 30.0540 1.39975 0.699877 0.714264i $$-0.253238\pi$$
0.699877 + 0.714264i $$0.253238\pi$$
$$462$$ 2.31534 0.107719
$$463$$ 7.61553 0.353924 0.176962 0.984218i $$-0.443373\pi$$
0.176962 + 0.984218i $$0.443373\pi$$
$$464$$ 2.43845 0.113202
$$465$$ −6.68466 −0.309994
$$466$$ 5.31534 0.246228
$$467$$ −17.8617 −0.826543 −0.413271 0.910608i $$-0.635614\pi$$
−0.413271 + 0.910608i $$0.635614\pi$$
$$468$$ 0 0
$$469$$ 1.26137 0.0582445
$$470$$ −7.00000 −0.322886
$$471$$ 13.8769 0.639414
$$472$$ 6.43845 0.296354
$$473$$ −1.80776 −0.0831211
$$474$$ −11.5616 −0.531040
$$475$$ −0.561553 −0.0257658
$$476$$ −1.75379 −0.0803848
$$477$$ 8.56155 0.392007
$$478$$ −11.3693 −0.520020
$$479$$ 38.2462 1.74751 0.873757 0.486363i $$-0.161677\pi$$
0.873757 + 0.486363i $$0.161677\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −28.1231 −1.28097
$$483$$ 2.63068 0.119700
$$484$$ 6.00000 0.272727
$$485$$ −7.12311 −0.323444
$$486$$ −1.00000 −0.0453609
$$487$$ −13.0540 −0.591532 −0.295766 0.955260i $$-0.595575\pi$$
−0.295766 + 0.955260i $$0.595575\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 14.4384 0.652929
$$490$$ −6.68466 −0.301982
$$491$$ −17.4384 −0.786986 −0.393493 0.919328i $$-0.628733\pi$$
−0.393493 + 0.919328i $$0.628733\pi$$
$$492$$ −12.2462 −0.552102
$$493$$ −7.61553 −0.342986
$$494$$ 0 0
$$495$$ 4.12311 0.185320
$$496$$ 6.68466 0.300150
$$497$$ 7.36932 0.330559
$$498$$ −7.12311 −0.319194
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −4.36932 −0.195207
$$502$$ 8.12311 0.362552
$$503$$ −43.0540 −1.91968 −0.959841 0.280545i $$-0.909485\pi$$
−0.959841 + 0.280545i $$0.909485\pi$$
$$504$$ −0.561553 −0.0250136
$$505$$ 8.87689 0.395017
$$506$$ 19.3153 0.858672
$$507$$ 0 0
$$508$$ −8.56155 −0.379857
$$509$$ −40.5464 −1.79719 −0.898594 0.438782i $$-0.855410\pi$$
−0.898594 + 0.438782i $$0.855410\pi$$
$$510$$ −3.12311 −0.138293
$$511$$ −5.26137 −0.232749
$$512$$ −1.00000 −0.0441942
$$513$$ −0.561553 −0.0247932
$$514$$ 5.56155 0.245310
$$515$$ −4.56155 −0.201006
$$516$$ 0.438447 0.0193016
$$517$$ 28.8617 1.26934
$$518$$ −2.31534 −0.101730
$$519$$ 20.1771 0.885676
$$520$$ 0 0
$$521$$ −6.31534 −0.276680 −0.138340 0.990385i $$-0.544177\pi$$
−0.138340 + 0.990385i $$0.544177\pi$$
$$522$$ −2.43845 −0.106728
$$523$$ −29.8078 −1.30340 −0.651701 0.758476i $$-0.725944\pi$$
−0.651701 + 0.758476i $$0.725944\pi$$
$$524$$ −2.12311 −0.0927483
$$525$$ 0.561553 0.0245082
$$526$$ 13.0000 0.566827
$$527$$ −20.8769 −0.909412
$$528$$ −4.12311 −0.179435
$$529$$ −1.05398 −0.0458250
$$530$$ 8.56155 0.371890
$$531$$ −6.43845 −0.279405
$$532$$ −0.315342 −0.0136718
$$533$$ 0 0
$$534$$ −18.8078 −0.813892
$$535$$ −2.00000 −0.0864675
$$536$$ −2.24621 −0.0970215
$$537$$ −12.6847 −0.547383
$$538$$ −21.3693 −0.921297
$$539$$ 27.5616 1.18716
$$540$$ −1.00000 −0.0430331
$$541$$ 27.3693 1.17670 0.588349 0.808607i $$-0.299778\pi$$
0.588349 + 0.808607i $$0.299778\pi$$
$$542$$ −13.1771 −0.566004
$$543$$ 2.87689 0.123459
$$544$$ 3.12311 0.133902
$$545$$ −3.75379 −0.160795
$$546$$ 0 0
$$547$$ −5.61553 −0.240103 −0.120051 0.992768i $$-0.538306\pi$$
−0.120051 + 0.992768i $$0.538306\pi$$
$$548$$ −11.8078 −0.504403
$$549$$ 6.00000 0.256074
$$550$$ 4.12311 0.175810
$$551$$ −1.36932 −0.0583349
$$552$$ −4.68466 −0.199392
$$553$$ 6.49242 0.276086
$$554$$ 1.00000 0.0424859
$$555$$ −4.12311 −0.175016
$$556$$ 12.5616 0.532729
$$557$$ 8.56155 0.362765 0.181382 0.983413i $$-0.441943\pi$$
0.181382 + 0.983413i $$0.441943\pi$$
$$558$$ −6.68466 −0.282984
$$559$$ 0 0
$$560$$ −0.561553 −0.0237299
$$561$$ 12.8769 0.543663
$$562$$ 16.2462 0.685305
$$563$$ −32.9848 −1.39015 −0.695073 0.718939i $$-0.744628\pi$$
−0.695073 + 0.718939i $$0.744628\pi$$
$$564$$ −7.00000 −0.294753
$$565$$ −8.68466 −0.365366
$$566$$ −15.5616 −0.654101
$$567$$ 0.561553 0.0235830
$$568$$ −13.1231 −0.550633
$$569$$ 26.1771 1.09740 0.548700 0.836019i $$-0.315123\pi$$
0.548700 + 0.836019i $$0.315123\pi$$
$$570$$ −0.561553 −0.0235209
$$571$$ 23.6847 0.991172 0.495586 0.868559i $$-0.334953\pi$$
0.495586 + 0.868559i $$0.334953\pi$$
$$572$$ 0 0
$$573$$ 10.8769 0.454389
$$574$$ 6.87689 0.287036
$$575$$ 4.68466 0.195364
$$576$$ 1.00000 0.0416667
$$577$$ 40.7386 1.69597 0.847986 0.530019i $$-0.177815\pi$$
0.847986 + 0.530019i $$0.177815\pi$$
$$578$$ 7.24621 0.301403
$$579$$ 0 0
$$580$$ −2.43845 −0.101251
$$581$$ 4.00000 0.165948
$$582$$ −7.12311 −0.295262
$$583$$ −35.3002 −1.46198
$$584$$ 9.36932 0.387705
$$585$$ 0 0
$$586$$ 3.93087 0.162383
$$587$$ 40.2462 1.66114 0.830569 0.556915i $$-0.188015\pi$$
0.830569 + 0.556915i $$0.188015\pi$$
$$588$$ −6.68466 −0.275671
$$589$$ −3.75379 −0.154672
$$590$$ −6.43845 −0.265067
$$591$$ −12.8078 −0.526841
$$592$$ 4.12311 0.169459
$$593$$ −33.1771 −1.36242 −0.681210 0.732088i $$-0.738546\pi$$
−0.681210 + 0.732088i $$0.738546\pi$$
$$594$$ 4.12311 0.169173
$$595$$ 1.75379 0.0718983
$$596$$ 2.19224 0.0897975
$$597$$ 2.87689 0.117743
$$598$$ 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
$$601$$ 29.9848 1.22311 0.611554 0.791203i $$-0.290545\pi$$
0.611554 + 0.791203i $$0.290545\pi$$
$$602$$ −0.246211 −0.0100348
$$603$$ 2.24621 0.0914728
$$604$$ 15.3693 0.625369
$$605$$ −6.00000 −0.243935
$$606$$ 8.87689 0.360599
$$607$$ 32.4233 1.31602 0.658010 0.753009i $$-0.271398\pi$$
0.658010 + 0.753009i $$0.271398\pi$$
$$608$$ 0.561553 0.0227740
$$609$$ 1.36932 0.0554875
$$610$$ 6.00000 0.242933
$$611$$ 0 0
$$612$$ −3.12311 −0.126244
$$613$$ −19.8769 −0.802820 −0.401410 0.915898i $$-0.631480\pi$$
−0.401410 + 0.915898i $$0.631480\pi$$
$$614$$ 25.6155 1.03376
$$615$$ 12.2462 0.493815
$$616$$ 2.31534 0.0932878
$$617$$ 29.3153 1.18019 0.590096 0.807333i $$-0.299090\pi$$
0.590096 + 0.807333i $$0.299090\pi$$
$$618$$ −4.56155 −0.183493
$$619$$ 20.5616 0.826439 0.413219 0.910632i $$-0.364404\pi$$
0.413219 + 0.910632i $$0.364404\pi$$
$$620$$ −6.68466 −0.268462
$$621$$ 4.68466 0.187989
$$622$$ −30.7386 −1.23251
$$623$$ 10.5616 0.423140
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −31.3693 −1.25377
$$627$$ 2.31534 0.0924658
$$628$$ 13.8769 0.553748
$$629$$ −12.8769 −0.513435
$$630$$ 0.561553 0.0223728
$$631$$ 13.7538 0.547530 0.273765 0.961797i $$-0.411731\pi$$
0.273765 + 0.961797i $$0.411731\pi$$
$$632$$ −11.5616 −0.459894
$$633$$ −21.9309 −0.871674
$$634$$ −24.8078 −0.985242
$$635$$ 8.56155 0.339755
$$636$$ 8.56155 0.339488
$$637$$ 0 0
$$638$$ 10.0540 0.398041
$$639$$ 13.1231 0.519142
$$640$$ 1.00000 0.0395285
$$641$$ −11.4384 −0.451792 −0.225896 0.974151i $$-0.572531\pi$$
−0.225896 + 0.974151i $$0.572531\pi$$
$$642$$ −2.00000 −0.0789337
$$643$$ −21.7538 −0.857886 −0.428943 0.903332i $$-0.641114\pi$$
−0.428943 + 0.903332i $$0.641114\pi$$
$$644$$ 2.63068 0.103663
$$645$$ −0.438447 −0.0172638
$$646$$ −1.75379 −0.0690019
$$647$$ 35.0540 1.37811 0.689057 0.724707i $$-0.258025\pi$$
0.689057 + 0.724707i $$0.258025\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 26.5464 1.04204
$$650$$ 0 0
$$651$$ 3.75379 0.147123
$$652$$ 14.4384 0.565453
$$653$$ −44.5616 −1.74383 −0.871914 0.489659i $$-0.837121\pi$$
−0.871914 + 0.489659i $$0.837121\pi$$
$$654$$ −3.75379 −0.146785
$$655$$ 2.12311 0.0829566
$$656$$ −12.2462 −0.478134
$$657$$ −9.36932 −0.365532
$$658$$ 3.93087 0.153241
$$659$$ −6.05398 −0.235829 −0.117915 0.993024i $$-0.537621\pi$$
−0.117915 + 0.993024i $$0.537621\pi$$
$$660$$ 4.12311 0.160492
$$661$$ −29.6155 −1.15191 −0.575955 0.817481i $$-0.695370\pi$$
−0.575955 + 0.817481i $$0.695370\pi$$
$$662$$ 30.7386 1.19469
$$663$$ 0 0
$$664$$ −7.12311 −0.276430
$$665$$ 0.315342 0.0122284
$$666$$ −4.12311 −0.159767
$$667$$ 11.4233 0.442312
$$668$$ −4.36932 −0.169054
$$669$$ −27.3002 −1.05549
$$670$$ 2.24621 0.0867787
$$671$$ −24.7386 −0.955024
$$672$$ −0.561553 −0.0216624
$$673$$ 25.7538 0.992736 0.496368 0.868112i $$-0.334667\pi$$
0.496368 + 0.868112i $$0.334667\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −37.1231 −1.42676 −0.713378 0.700779i $$-0.752836\pi$$
−0.713378 + 0.700779i $$0.752836\pi$$
$$678$$ −8.68466 −0.333532
$$679$$ 4.00000 0.153506
$$680$$ −3.12311 −0.119766
$$681$$ 20.0000 0.766402
$$682$$ 27.5616 1.05539
$$683$$ −17.1231 −0.655197 −0.327599 0.944817i $$-0.606239\pi$$
−0.327599 + 0.944817i $$0.606239\pi$$
$$684$$ −0.561553 −0.0214715
$$685$$ 11.8078 0.451151
$$686$$ 7.68466 0.293402
$$687$$ 24.2462 0.925051
$$688$$ 0.438447 0.0167156
$$689$$ 0 0
$$690$$ 4.68466 0.178342
$$691$$ 20.5616 0.782198 0.391099 0.920349i $$-0.372095\pi$$
0.391099 + 0.920349i $$0.372095\pi$$
$$692$$ 20.1771 0.767018
$$693$$ −2.31534 −0.0879526
$$694$$ −0.876894 −0.0332865
$$695$$ −12.5616 −0.476487
$$696$$ −2.43845 −0.0924291
$$697$$ 38.2462 1.44868
$$698$$ 8.49242 0.321443
$$699$$ −5.31534 −0.201045
$$700$$ 0.561553 0.0212247
$$701$$ −36.3002 −1.37104 −0.685520 0.728054i $$-0.740425\pi$$
−0.685520 + 0.728054i $$0.740425\pi$$
$$702$$ 0 0
$$703$$ −2.31534 −0.0873248
$$704$$ −4.12311 −0.155395
$$705$$ 7.00000 0.263635
$$706$$ −25.8617 −0.973319
$$707$$ −4.98485 −0.187474
$$708$$ −6.43845 −0.241972
$$709$$ −34.2462 −1.28614 −0.643072 0.765806i $$-0.722340\pi$$
−0.643072 + 0.765806i $$0.722340\pi$$
$$710$$ 13.1231 0.492501
$$711$$ 11.5616 0.433592
$$712$$ −18.8078 −0.704851
$$713$$ 31.3153 1.17277
$$714$$ 1.75379 0.0656339
$$715$$ 0 0
$$716$$ −12.6847 −0.474048
$$717$$ 11.3693 0.424595
$$718$$ 13.1231 0.489750
$$719$$ 44.9848 1.67765 0.838826 0.544400i $$-0.183243\pi$$
0.838826 + 0.544400i $$0.183243\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 2.56155 0.0953972
$$722$$ 18.6847 0.695371
$$723$$ 28.1231 1.04591
$$724$$ 2.87689 0.106919
$$725$$ 2.43845 0.0905617
$$726$$ −6.00000 −0.222681
$$727$$ 38.4233 1.42504 0.712521 0.701651i $$-0.247554\pi$$
0.712521 + 0.701651i $$0.247554\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −9.36932 −0.346774
$$731$$ −1.36932 −0.0506460
$$732$$ 6.00000 0.221766
$$733$$ −20.0691 −0.741270 −0.370635 0.928779i $$-0.620860\pi$$
−0.370635 + 0.928779i $$0.620860\pi$$
$$734$$ −12.4924 −0.461104
$$735$$ 6.68466 0.246567
$$736$$ −4.68466 −0.172679
$$737$$ −9.26137 −0.341147
$$738$$ 12.2462 0.450789
$$739$$ 27.5464 1.01331 0.506655 0.862149i $$-0.330882\pi$$
0.506655 + 0.862149i $$0.330882\pi$$
$$740$$ −4.12311 −0.151568
$$741$$ 0 0
$$742$$ −4.80776 −0.176499
$$743$$ −13.1771 −0.483420 −0.241710 0.970349i $$-0.577708\pi$$
−0.241710 + 0.970349i $$0.577708\pi$$
$$744$$ −6.68466 −0.245071
$$745$$ −2.19224 −0.0803173
$$746$$ 25.8078 0.944889
$$747$$ 7.12311 0.260621
$$748$$ 12.8769 0.470826
$$749$$ 1.12311 0.0410374
$$750$$ 1.00000 0.0365148
$$751$$ −23.8078 −0.868758 −0.434379 0.900730i $$-0.643032\pi$$
−0.434379 + 0.900730i $$0.643032\pi$$
$$752$$ −7.00000 −0.255264
$$753$$ −8.12311 −0.296022
$$754$$ 0 0
$$755$$ −15.3693 −0.559347
$$756$$ 0.561553 0.0204235
$$757$$ −28.4233 −1.03306 −0.516531 0.856268i $$-0.672777\pi$$
−0.516531 + 0.856268i $$0.672777\pi$$
$$758$$ −17.6847 −0.642336
$$759$$ −19.3153 −0.701102
$$760$$ −0.561553 −0.0203697
$$761$$ 17.9309 0.649994 0.324997 0.945715i $$-0.394637\pi$$
0.324997 + 0.945715i $$0.394637\pi$$
$$762$$ 8.56155 0.310152
$$763$$ 2.10795 0.0763129
$$764$$ 10.8769 0.393512
$$765$$ 3.12311 0.112916
$$766$$ 0.192236 0.00694577
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 33.3153 1.20138 0.600691 0.799481i $$-0.294892\pi$$
0.600691 + 0.799481i $$0.294892\pi$$
$$770$$ −2.31534 −0.0834391
$$771$$ −5.56155 −0.200294
$$772$$ 0 0
$$773$$ −24.1771 −0.869589 −0.434795 0.900530i $$-0.643179\pi$$
−0.434795 + 0.900530i $$0.643179\pi$$
$$774$$ −0.438447 −0.0157597
$$775$$ 6.68466 0.240120
$$776$$ −7.12311 −0.255705
$$777$$ 2.31534 0.0830624
$$778$$ 18.0540 0.647266
$$779$$ 6.87689 0.246390
$$780$$ 0 0
$$781$$ −54.1080 −1.93613
$$782$$ 14.6307 0.523192
$$783$$ 2.43845 0.0871430
$$784$$ −6.68466 −0.238738
$$785$$ −13.8769 −0.495288
$$786$$ 2.12311 0.0757287
$$787$$ −31.3153 −1.11627 −0.558136 0.829750i $$-0.688483\pi$$
−0.558136 + 0.829750i $$0.688483\pi$$
$$788$$ −12.8078 −0.456258
$$789$$ −13.0000 −0.462812
$$790$$ 11.5616 0.411342
$$791$$ 4.87689 0.173402
$$792$$ 4.12311 0.146508
$$793$$ 0 0
$$794$$ −20.1231 −0.714142
$$795$$ −8.56155 −0.303647
$$796$$ 2.87689 0.101969
$$797$$ 13.1231 0.464844 0.232422 0.972615i $$-0.425335\pi$$
0.232422 + 0.972615i $$0.425335\pi$$
$$798$$ 0.315342 0.0111630
$$799$$ 21.8617 0.773413
$$800$$ −1.00000 −0.0353553
$$801$$ 18.8078 0.664540
$$802$$ 0.315342 0.0111351
$$803$$ 38.6307 1.36325
$$804$$ 2.24621 0.0792178
$$805$$ −2.63068 −0.0927194
$$806$$ 0 0
$$807$$ 21.3693 0.752236
$$808$$ 8.87689 0.312288
$$809$$ −14.4924 −0.509526 −0.254763 0.967003i $$-0.581998\pi$$
−0.254763 + 0.967003i $$0.581998\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −47.5464 −1.66958 −0.834790 0.550569i $$-0.814411\pi$$
−0.834790 + 0.550569i $$0.814411\pi$$
$$812$$ 1.36932 0.0480536
$$813$$ 13.1771 0.462140
$$814$$ 17.0000 0.595850
$$815$$ −14.4384 −0.505757
$$816$$ −3.12311 −0.109331
$$817$$ −0.246211 −0.00861384
$$818$$ −18.8078 −0.657598
$$819$$ 0 0
$$820$$ 12.2462 0.427656
$$821$$ 8.43845 0.294504 0.147252 0.989099i $$-0.452957\pi$$
0.147252 + 0.989099i $$0.452957\pi$$
$$822$$ 11.8078 0.411843
$$823$$ 5.05398 0.176171 0.0880853 0.996113i $$-0.471925\pi$$
0.0880853 + 0.996113i $$0.471925\pi$$
$$824$$ −4.56155 −0.158909
$$825$$ −4.12311 −0.143548
$$826$$ 3.61553 0.125800
$$827$$ 28.2462 0.982217 0.491109 0.871098i $$-0.336592\pi$$
0.491109 + 0.871098i $$0.336592\pi$$
$$828$$ 4.68466 0.162803
$$829$$ 12.3845 0.430130 0.215065 0.976600i $$-0.431004\pi$$
0.215065 + 0.976600i $$0.431004\pi$$
$$830$$ 7.12311 0.247247
$$831$$ −1.00000 −0.0346896
$$832$$ 0 0
$$833$$ 20.8769 0.723342
$$834$$ −12.5616 −0.434971
$$835$$ 4.36932 0.151206
$$836$$ 2.31534 0.0800778
$$837$$ 6.68466 0.231056
$$838$$ −32.4924 −1.12243
$$839$$ −36.7386 −1.26836 −0.634179 0.773186i $$-0.718662\pi$$
−0.634179 + 0.773186i $$0.718662\pi$$
$$840$$ 0.561553 0.0193754
$$841$$ −23.0540 −0.794965
$$842$$ −32.4924 −1.11976
$$843$$ −16.2462 −0.559549
$$844$$ −21.9309 −0.754892
$$845$$ 0 0
$$846$$ 7.00000 0.240665
$$847$$ 3.36932 0.115771
$$848$$ 8.56155 0.294005
$$849$$ 15.5616 0.534071
$$850$$ 3.12311 0.107122
$$851$$ 19.3153 0.662121
$$852$$ 13.1231 0.449590
$$853$$ −2.30019 −0.0787569 −0.0393784 0.999224i $$-0.512538\pi$$
−0.0393784 + 0.999224i $$0.512538\pi$$
$$854$$ −3.36932 −0.115296
$$855$$ 0.561553 0.0192047
$$856$$ −2.00000 −0.0683586
$$857$$ −6.19224 −0.211523 −0.105761 0.994392i $$-0.533728\pi$$
−0.105761 + 0.994392i $$0.533728\pi$$
$$858$$ 0 0
$$859$$ 39.7926 1.35771 0.678853 0.734274i $$-0.262477\pi$$
0.678853 + 0.734274i $$0.262477\pi$$
$$860$$ −0.438447 −0.0149509
$$861$$ −6.87689 −0.234364
$$862$$ −22.7386 −0.774481
$$863$$ −2.43845 −0.0830057 −0.0415029 0.999138i $$-0.513215\pi$$
−0.0415029 + 0.999138i $$0.513215\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −20.1771 −0.686041
$$866$$ 18.7386 0.636765
$$867$$ −7.24621 −0.246094
$$868$$ 3.75379 0.127412
$$869$$ −47.6695 −1.61708
$$870$$ 2.43845 0.0826711
$$871$$ 0 0
$$872$$ −3.75379 −0.127119
$$873$$ 7.12311 0.241081
$$874$$ 2.63068 0.0889842
$$875$$ −0.561553 −0.0189839
$$876$$ −9.36932 −0.316560
$$877$$ −8.43845 −0.284946 −0.142473 0.989799i $$-0.545505\pi$$
−0.142473 + 0.989799i $$0.545505\pi$$
$$878$$ −25.1231 −0.847864
$$879$$ −3.93087 −0.132585
$$880$$ 4.12311 0.138990
$$881$$ −24.5616 −0.827500 −0.413750 0.910391i $$-0.635781\pi$$
−0.413750 + 0.910391i $$0.635781\pi$$
$$882$$ 6.68466 0.225084
$$883$$ −19.1771 −0.645360 −0.322680 0.946508i $$-0.604584\pi$$
−0.322680 + 0.946508i $$0.604584\pi$$
$$884$$ 0 0
$$885$$ 6.43845 0.216426
$$886$$ 13.1231 0.440879
$$887$$ −33.1080 −1.11166 −0.555828 0.831297i $$-0.687599\pi$$
−0.555828 + 0.831297i $$0.687599\pi$$
$$888$$ −4.12311 −0.138362
$$889$$ −4.80776 −0.161247
$$890$$ 18.8078 0.630438
$$891$$ −4.12311 −0.138129
$$892$$ −27.3002 −0.914078
$$893$$ 3.93087 0.131542
$$894$$ −2.19224 −0.0733193
$$895$$ 12.6847 0.424001
$$896$$ −0.561553 −0.0187602
$$897$$ 0 0
$$898$$ −20.1771 −0.673318
$$899$$ 16.3002 0.543642
$$900$$ 1.00000 0.0333333
$$901$$ −26.7386 −0.890793
$$902$$ −50.4924 −1.68121
$$903$$ 0.246211 0.00819340
$$904$$ −8.68466 −0.288847
$$905$$ −2.87689 −0.0956312
$$906$$ −15.3693 −0.510611
$$907$$ −53.9157 −1.79024 −0.895121 0.445823i $$-0.852911\pi$$
−0.895121 + 0.445823i $$0.852911\pi$$
$$908$$ 20.0000 0.663723
$$909$$ −8.87689 −0.294428
$$910$$ 0 0
$$911$$ −18.6307 −0.617262 −0.308631 0.951182i $$-0.599871\pi$$
−0.308631 + 0.951182i $$0.599871\pi$$
$$912$$ −0.561553 −0.0185949
$$913$$ −29.3693 −0.971983
$$914$$ 20.2462 0.669685
$$915$$ −6.00000 −0.198354
$$916$$ 24.2462 0.801117
$$917$$ −1.19224 −0.0393711
$$918$$ 3.12311 0.103078
$$919$$ 16.6307 0.548596 0.274298 0.961645i $$-0.411555\pi$$
0.274298 + 0.961645i $$0.411555\pi$$
$$920$$ 4.68466 0.154449
$$921$$ −25.6155 −0.844060
$$922$$ −30.0540 −0.989775
$$923$$ 0 0
$$924$$ −2.31534 −0.0761691
$$925$$ 4.12311 0.135567
$$926$$ −7.61553 −0.250262
$$927$$ 4.56155 0.149821
$$928$$ −2.43845 −0.0800460
$$929$$ 25.1231 0.824262 0.412131 0.911125i $$-0.364785\pi$$
0.412131 + 0.911125i $$0.364785\pi$$
$$930$$ 6.68466 0.219199
$$931$$ 3.75379 0.123025
$$932$$ −5.31534 −0.174110
$$933$$ 30.7386 1.00634
$$934$$ 17.8617 0.584454
$$935$$ −12.8769 −0.421119
$$936$$ 0 0
$$937$$ 53.2311 1.73898 0.869491 0.493948i $$-0.164447\pi$$
0.869491 + 0.493948i $$0.164447\pi$$
$$938$$ −1.26137 −0.0411851
$$939$$ 31.3693 1.02370
$$940$$ 7.00000 0.228315
$$941$$ 25.3693 0.827016 0.413508 0.910500i $$-0.364303\pi$$
0.413508 + 0.910500i $$0.364303\pi$$
$$942$$ −13.8769 −0.452134
$$943$$ −57.3693 −1.86820
$$944$$ −6.43845 −0.209554
$$945$$ −0.561553 −0.0182673
$$946$$ 1.80776 0.0587755
$$947$$ 43.8617 1.42532 0.712658 0.701512i $$-0.247491\pi$$
0.712658 + 0.701512i $$0.247491\pi$$
$$948$$ 11.5616 0.375502
$$949$$ 0 0
$$950$$ 0.561553 0.0182192
$$951$$ 24.8078 0.804447
$$952$$ 1.75379 0.0568406
$$953$$ 51.1771 1.65779 0.828894 0.559406i $$-0.188971\pi$$
0.828894 + 0.559406i $$0.188971\pi$$
$$954$$ −8.56155 −0.277191
$$955$$ −10.8769 −0.351968
$$956$$ 11.3693 0.367710
$$957$$ −10.0540 −0.324999
$$958$$ −38.2462 −1.23568
$$959$$ −6.63068 −0.214116
$$960$$ −1.00000 −0.0322749
$$961$$ 13.6847 0.441441
$$962$$ 0 0
$$963$$ 2.00000 0.0644491
$$964$$ 28.1231 0.905784
$$965$$ 0 0
$$966$$ −2.63068 −0.0846408
$$967$$ −15.6847 −0.504385 −0.252192 0.967677i $$-0.581152\pi$$
−0.252192 + 0.967677i $$0.581152\pi$$
$$968$$ −6.00000 −0.192847
$$969$$ 1.75379 0.0563398
$$970$$ 7.12311 0.228709
$$971$$ −4.31534 −0.138486 −0.0692430 0.997600i $$-0.522058\pi$$
−0.0692430 + 0.997600i $$0.522058\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 7.05398 0.226140
$$974$$ 13.0540 0.418276
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 48.3002 1.54526 0.772630 0.634857i $$-0.218941\pi$$
0.772630 + 0.634857i $$0.218941\pi$$
$$978$$ −14.4384 −0.461691
$$979$$ −77.5464 −2.47839
$$980$$ 6.68466 0.213534
$$981$$ 3.75379 0.119849
$$982$$ 17.4384 0.556483
$$983$$ −32.1231 −1.02457 −0.512284 0.858816i $$-0.671200\pi$$
−0.512284 + 0.858816i $$0.671200\pi$$
$$984$$ 12.2462 0.390395
$$985$$ 12.8078 0.408089
$$986$$ 7.61553 0.242528
$$987$$ −3.93087 −0.125121
$$988$$ 0 0
$$989$$ 2.05398 0.0653126
$$990$$ −4.12311 −0.131041
$$991$$ 49.4233 1.56998 0.784991 0.619507i $$-0.212667\pi$$
0.784991 + 0.619507i $$0.212667\pi$$
$$992$$ −6.68466 −0.212238
$$993$$ −30.7386 −0.975461
$$994$$ −7.36932 −0.233741
$$995$$ −2.87689 −0.0912037
$$996$$ 7.12311 0.225704
$$997$$ −37.5464 −1.18911 −0.594553 0.804056i $$-0.702671\pi$$
−0.594553 + 0.804056i $$0.702671\pi$$
$$998$$ 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.bb.1.2 2
13.4 even 6 390.2.i.g.211.2 yes 4
13.5 odd 4 5070.2.b.r.1351.4 4
13.8 odd 4 5070.2.b.r.1351.1 4
13.10 even 6 390.2.i.g.61.2 4
13.12 even 2 5070.2.a.bi.1.1 2
39.17 odd 6 1170.2.i.o.991.2 4
39.23 odd 6 1170.2.i.o.451.2 4
65.4 even 6 1950.2.i.bi.601.1 4
65.17 odd 12 1950.2.z.n.1849.4 8
65.23 odd 12 1950.2.z.n.1699.4 8
65.43 odd 12 1950.2.z.n.1849.1 8
65.49 even 6 1950.2.i.bi.451.1 4
65.62 odd 12 1950.2.z.n.1699.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.g.61.2 4 13.10 even 6
390.2.i.g.211.2 yes 4 13.4 even 6
1170.2.i.o.451.2 4 39.23 odd 6
1170.2.i.o.991.2 4 39.17 odd 6
1950.2.i.bi.451.1 4 65.49 even 6
1950.2.i.bi.601.1 4 65.4 even 6
1950.2.z.n.1699.1 8 65.62 odd 12
1950.2.z.n.1699.4 8 65.23 odd 12
1950.2.z.n.1849.1 8 65.43 odd 12
1950.2.z.n.1849.4 8 65.17 odd 12
5070.2.a.bb.1.2 2 1.1 even 1 trivial
5070.2.a.bi.1.1 2 13.12 even 2
5070.2.b.r.1351.1 4 13.8 odd 4
5070.2.b.r.1351.4 4 13.5 odd 4