# Properties

 Label 5070.2.a.be.1.2 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -0.267949 q^{11} -1.00000 q^{12} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -5.73205 q^{19} -1.00000 q^{20} -3.00000 q^{21} -0.267949 q^{22} -3.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.00000 q^{28} +1.46410 q^{29} +1.00000 q^{30} -4.92820 q^{31} +1.00000 q^{32} +0.267949 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +5.92820 q^{37} -5.73205 q^{38} -1.00000 q^{40} -4.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} -0.267949 q^{44} -1.00000 q^{45} -3.46410 q^{46} -6.46410 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -0.267949 q^{53} -1.00000 q^{54} +0.267949 q^{55} +3.00000 q^{56} +5.73205 q^{57} +1.46410 q^{58} -11.4641 q^{59} +1.00000 q^{60} -0.535898 q^{61} -4.92820 q^{62} +3.00000 q^{63} +1.00000 q^{64} +0.267949 q^{66} +1.46410 q^{67} -4.00000 q^{68} +3.46410 q^{69} -3.00000 q^{70} +12.9282 q^{71} +1.00000 q^{72} -6.92820 q^{73} +5.92820 q^{74} -1.00000 q^{75} -5.73205 q^{76} -0.803848 q^{77} +3.07180 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -9.46410 q^{83} -3.00000 q^{84} +4.00000 q^{85} -6.00000 q^{86} -1.46410 q^{87} -0.267949 q^{88} +14.1244 q^{89} -1.00000 q^{90} -3.46410 q^{92} +4.92820 q^{93} -6.46410 q^{94} +5.73205 q^{95} -1.00000 q^{96} -8.39230 q^{97} +2.00000 q^{98} -0.267949 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} + 6q^{14} + 2q^{15} + 2q^{16} - 8q^{17} + 2q^{18} - 8q^{19} - 2q^{20} - 6q^{21} - 4q^{22} - 2q^{24} + 2q^{25} - 2q^{27} + 6q^{28} - 4q^{29} + 2q^{30} + 4q^{31} + 2q^{32} + 4q^{33} - 8q^{34} - 6q^{35} + 2q^{36} - 2q^{37} - 8q^{38} - 2q^{40} - 8q^{41} - 6q^{42} - 12q^{43} - 4q^{44} - 2q^{45} - 6q^{47} - 2q^{48} + 4q^{49} + 2q^{50} + 8q^{51} - 4q^{53} - 2q^{54} + 4q^{55} + 6q^{56} + 8q^{57} - 4q^{58} - 16q^{59} + 2q^{60} - 8q^{61} + 4q^{62} + 6q^{63} + 2q^{64} + 4q^{66} - 4q^{67} - 8q^{68} - 6q^{70} + 12q^{71} + 2q^{72} - 2q^{74} - 2q^{75} - 8q^{76} - 12q^{77} + 20q^{79} - 2q^{80} + 2q^{81} - 8q^{82} - 12q^{83} - 6q^{84} + 8q^{85} - 12q^{86} + 4q^{87} - 4q^{88} + 4q^{89} - 2q^{90} - 4q^{93} - 6q^{94} + 8q^{95} - 2q^{96} + 4q^{97} + 4q^{98} - 4q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −0.267949 −0.0807897 −0.0403949 0.999184i $$-0.512862\pi$$
−0.0403949 + 0.999184i $$0.512862\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 3.00000 0.801784
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −5.73205 −1.31502 −0.657511 0.753445i $$-0.728391\pi$$
−0.657511 + 0.753445i $$0.728391\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −3.00000 −0.654654
$$22$$ −0.267949 −0.0571270
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 3.00000 0.566947
$$29$$ 1.46410 0.271877 0.135938 0.990717i $$-0.456595\pi$$
0.135938 + 0.990717i $$0.456595\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −4.92820 −0.885131 −0.442566 0.896736i $$-0.645932\pi$$
−0.442566 + 0.896736i $$0.645932\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.267949 0.0466440
$$34$$ −4.00000 −0.685994
$$35$$ −3.00000 −0.507093
$$36$$ 1.00000 0.166667
$$37$$ 5.92820 0.974591 0.487295 0.873237i $$-0.337984\pi$$
0.487295 + 0.873237i $$0.337984\pi$$
$$38$$ −5.73205 −0.929861
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ −3.00000 −0.462910
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ −0.267949 −0.0403949
$$45$$ −1.00000 −0.149071
$$46$$ −3.46410 −0.510754
$$47$$ −6.46410 −0.942886 −0.471443 0.881897i $$-0.656267\pi$$
−0.471443 + 0.881897i $$0.656267\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 2.00000 0.285714
$$50$$ 1.00000 0.141421
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ −0.267949 −0.0368057 −0.0184028 0.999831i $$-0.505858\pi$$
−0.0184028 + 0.999831i $$0.505858\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0.267949 0.0361303
$$56$$ 3.00000 0.400892
$$57$$ 5.73205 0.759229
$$58$$ 1.46410 0.192246
$$59$$ −11.4641 −1.49250 −0.746249 0.665666i $$-0.768147\pi$$
−0.746249 + 0.665666i $$0.768147\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −0.535898 −0.0686148 −0.0343074 0.999411i $$-0.510923\pi$$
−0.0343074 + 0.999411i $$0.510923\pi$$
$$62$$ −4.92820 −0.625882
$$63$$ 3.00000 0.377964
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.267949 0.0329823
$$67$$ 1.46410 0.178868 0.0894342 0.995993i $$-0.471494\pi$$
0.0894342 + 0.995993i $$0.471494\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 3.46410 0.417029
$$70$$ −3.00000 −0.358569
$$71$$ 12.9282 1.53430 0.767148 0.641470i $$-0.221675\pi$$
0.767148 + 0.641470i $$0.221675\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −6.92820 −0.810885 −0.405442 0.914121i $$-0.632883\pi$$
−0.405442 + 0.914121i $$0.632883\pi$$
$$74$$ 5.92820 0.689140
$$75$$ −1.00000 −0.115470
$$76$$ −5.73205 −0.657511
$$77$$ −0.803848 −0.0916069
$$78$$ 0 0
$$79$$ 3.07180 0.345604 0.172802 0.984957i $$-0.444718\pi$$
0.172802 + 0.984957i $$0.444718\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −4.00000 −0.441726
$$83$$ −9.46410 −1.03882 −0.519410 0.854525i $$-0.673848\pi$$
−0.519410 + 0.854525i $$0.673848\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 4.00000 0.433861
$$86$$ −6.00000 −0.646997
$$87$$ −1.46410 −0.156968
$$88$$ −0.267949 −0.0285635
$$89$$ 14.1244 1.49718 0.748589 0.663034i $$-0.230731\pi$$
0.748589 + 0.663034i $$0.230731\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −3.46410 −0.361158
$$93$$ 4.92820 0.511031
$$94$$ −6.46410 −0.666721
$$95$$ 5.73205 0.588096
$$96$$ −1.00000 −0.102062
$$97$$ −8.39230 −0.852109 −0.426055 0.904697i $$-0.640097\pi$$
−0.426055 + 0.904697i $$0.640097\pi$$
$$98$$ 2.00000 0.202031
$$99$$ −0.267949 −0.0269299
$$100$$ 1.00000 0.100000
$$101$$ −12.3923 −1.23308 −0.616540 0.787323i $$-0.711466\pi$$
−0.616540 + 0.787323i $$0.711466\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 11.5885 1.14184 0.570922 0.821004i $$-0.306586\pi$$
0.570922 + 0.821004i $$0.306586\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ −0.267949 −0.0260255
$$107$$ −12.9282 −1.24982 −0.624908 0.780698i $$-0.714864\pi$$
−0.624908 + 0.780698i $$0.714864\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.3923 0.995402 0.497701 0.867349i $$-0.334178\pi$$
0.497701 + 0.867349i $$0.334178\pi$$
$$110$$ 0.267949 0.0255480
$$111$$ −5.92820 −0.562680
$$112$$ 3.00000 0.283473
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 5.73205 0.536856
$$115$$ 3.46410 0.323029
$$116$$ 1.46410 0.135938
$$117$$ 0 0
$$118$$ −11.4641 −1.05536
$$119$$ −12.0000 −1.10004
$$120$$ 1.00000 0.0912871
$$121$$ −10.9282 −0.993473
$$122$$ −0.535898 −0.0485180
$$123$$ 4.00000 0.360668
$$124$$ −4.92820 −0.442566
$$125$$ −1.00000 −0.0894427
$$126$$ 3.00000 0.267261
$$127$$ 12.6603 1.12342 0.561708 0.827336i $$-0.310144\pi$$
0.561708 + 0.827336i $$0.310144\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 14.3205 1.25119 0.625594 0.780149i $$-0.284857\pi$$
0.625594 + 0.780149i $$0.284857\pi$$
$$132$$ 0.267949 0.0233220
$$133$$ −17.1962 −1.49110
$$134$$ 1.46410 0.126479
$$135$$ 1.00000 0.0860663
$$136$$ −4.00000 −0.342997
$$137$$ 13.4641 1.15032 0.575158 0.818042i $$-0.304941\pi$$
0.575158 + 0.818042i $$0.304941\pi$$
$$138$$ 3.46410 0.294884
$$139$$ −19.7846 −1.67811 −0.839054 0.544048i $$-0.816891\pi$$
−0.839054 + 0.544048i $$0.816891\pi$$
$$140$$ −3.00000 −0.253546
$$141$$ 6.46410 0.544376
$$142$$ 12.9282 1.08491
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −1.46410 −0.121587
$$146$$ −6.92820 −0.573382
$$147$$ −2.00000 −0.164957
$$148$$ 5.92820 0.487295
$$149$$ −18.7846 −1.53890 −0.769448 0.638710i $$-0.779468\pi$$
−0.769448 + 0.638710i $$0.779468\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 22.7846 1.85419 0.927093 0.374832i $$-0.122300\pi$$
0.927093 + 0.374832i $$0.122300\pi$$
$$152$$ −5.73205 −0.464931
$$153$$ −4.00000 −0.323381
$$154$$ −0.803848 −0.0647759
$$155$$ 4.92820 0.395843
$$156$$ 0 0
$$157$$ 21.1962 1.69164 0.845819 0.533471i $$-0.179113\pi$$
0.845819 + 0.533471i $$0.179113\pi$$
$$158$$ 3.07180 0.244379
$$159$$ 0.267949 0.0212498
$$160$$ −1.00000 −0.0790569
$$161$$ −10.3923 −0.819028
$$162$$ 1.00000 0.0785674
$$163$$ −2.92820 −0.229355 −0.114677 0.993403i $$-0.536583\pi$$
−0.114677 + 0.993403i $$0.536583\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ −0.267949 −0.0208598
$$166$$ −9.46410 −0.734557
$$167$$ 0.464102 0.0359133 0.0179566 0.999839i $$-0.494284\pi$$
0.0179566 + 0.999839i $$0.494284\pi$$
$$168$$ −3.00000 −0.231455
$$169$$ 0 0
$$170$$ 4.00000 0.306786
$$171$$ −5.73205 −0.438341
$$172$$ −6.00000 −0.457496
$$173$$ 2.12436 0.161512 0.0807559 0.996734i $$-0.474267\pi$$
0.0807559 + 0.996734i $$0.474267\pi$$
$$174$$ −1.46410 −0.110993
$$175$$ 3.00000 0.226779
$$176$$ −0.267949 −0.0201974
$$177$$ 11.4641 0.861695
$$178$$ 14.1244 1.05867
$$179$$ −9.07180 −0.678058 −0.339029 0.940776i $$-0.610098\pi$$
−0.339029 + 0.940776i $$0.610098\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −16.9282 −1.25826 −0.629132 0.777299i $$-0.716589\pi$$
−0.629132 + 0.777299i $$0.716589\pi$$
$$182$$ 0 0
$$183$$ 0.535898 0.0396147
$$184$$ −3.46410 −0.255377
$$185$$ −5.92820 −0.435850
$$186$$ 4.92820 0.361353
$$187$$ 1.07180 0.0783775
$$188$$ −6.46410 −0.471443
$$189$$ −3.00000 −0.218218
$$190$$ 5.73205 0.415847
$$191$$ −17.3205 −1.25327 −0.626634 0.779314i $$-0.715568\pi$$
−0.626634 + 0.779314i $$0.715568\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 9.85641 0.709480 0.354740 0.934965i $$-0.384569\pi$$
0.354740 + 0.934965i $$0.384569\pi$$
$$194$$ −8.39230 −0.602532
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −11.3923 −0.811668 −0.405834 0.913947i $$-0.633019\pi$$
−0.405834 + 0.913947i $$0.633019\pi$$
$$198$$ −0.267949 −0.0190423
$$199$$ 24.9282 1.76711 0.883557 0.468324i $$-0.155142\pi$$
0.883557 + 0.468324i $$0.155142\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −1.46410 −0.103270
$$202$$ −12.3923 −0.871920
$$203$$ 4.39230 0.308279
$$204$$ 4.00000 0.280056
$$205$$ 4.00000 0.279372
$$206$$ 11.5885 0.807406
$$207$$ −3.46410 −0.240772
$$208$$ 0 0
$$209$$ 1.53590 0.106240
$$210$$ 3.00000 0.207020
$$211$$ −8.07180 −0.555685 −0.277843 0.960627i $$-0.589619\pi$$
−0.277843 + 0.960627i $$0.589619\pi$$
$$212$$ −0.267949 −0.0184028
$$213$$ −12.9282 −0.885826
$$214$$ −12.9282 −0.883754
$$215$$ 6.00000 0.409197
$$216$$ −1.00000 −0.0680414
$$217$$ −14.7846 −1.00364
$$218$$ 10.3923 0.703856
$$219$$ 6.92820 0.468165
$$220$$ 0.267949 0.0180651
$$221$$ 0 0
$$222$$ −5.92820 −0.397875
$$223$$ −18.8564 −1.26272 −0.631359 0.775491i $$-0.717503\pi$$
−0.631359 + 0.775491i $$0.717503\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 1.00000 0.0666667
$$226$$ −12.0000 −0.798228
$$227$$ −6.53590 −0.433803 −0.216901 0.976194i $$-0.569595\pi$$
−0.216901 + 0.976194i $$0.569595\pi$$
$$228$$ 5.73205 0.379614
$$229$$ 4.53590 0.299741 0.149870 0.988706i $$-0.452114\pi$$
0.149870 + 0.988706i $$0.452114\pi$$
$$230$$ 3.46410 0.228416
$$231$$ 0.803848 0.0528893
$$232$$ 1.46410 0.0961230
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 6.46410 0.421671
$$236$$ −11.4641 −0.746249
$$237$$ −3.07180 −0.199535
$$238$$ −12.0000 −0.777844
$$239$$ 3.46410 0.224074 0.112037 0.993704i $$-0.464262\pi$$
0.112037 + 0.993704i $$0.464262\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −25.1962 −1.62303 −0.811513 0.584334i $$-0.801356\pi$$
−0.811513 + 0.584334i $$0.801356\pi$$
$$242$$ −10.9282 −0.702492
$$243$$ −1.00000 −0.0641500
$$244$$ −0.535898 −0.0343074
$$245$$ −2.00000 −0.127775
$$246$$ 4.00000 0.255031
$$247$$ 0 0
$$248$$ −4.92820 −0.312941
$$249$$ 9.46410 0.599763
$$250$$ −1.00000 −0.0632456
$$251$$ −19.5359 −1.23309 −0.616547 0.787318i $$-0.711469\pi$$
−0.616547 + 0.787318i $$0.711469\pi$$
$$252$$ 3.00000 0.188982
$$253$$ 0.928203 0.0583556
$$254$$ 12.6603 0.794375
$$255$$ −4.00000 −0.250490
$$256$$ 1.00000 0.0625000
$$257$$ −14.5359 −0.906724 −0.453362 0.891326i $$-0.649776\pi$$
−0.453362 + 0.891326i $$0.649776\pi$$
$$258$$ 6.00000 0.373544
$$259$$ 17.7846 1.10508
$$260$$ 0 0
$$261$$ 1.46410 0.0906256
$$262$$ 14.3205 0.884724
$$263$$ −24.5167 −1.51176 −0.755881 0.654709i $$-0.772791\pi$$
−0.755881 + 0.654709i $$0.772791\pi$$
$$264$$ 0.267949 0.0164911
$$265$$ 0.267949 0.0164600
$$266$$ −17.1962 −1.05436
$$267$$ −14.1244 −0.864397
$$268$$ 1.46410 0.0894342
$$269$$ −17.0718 −1.04089 −0.520443 0.853896i $$-0.674233\pi$$
−0.520443 + 0.853896i $$0.674233\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −24.3923 −1.48173 −0.740863 0.671656i $$-0.765583\pi$$
−0.740863 + 0.671656i $$0.765583\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 0 0
$$274$$ 13.4641 0.813396
$$275$$ −0.267949 −0.0161579
$$276$$ 3.46410 0.208514
$$277$$ −11.5885 −0.696283 −0.348141 0.937442i $$-0.613187\pi$$
−0.348141 + 0.937442i $$0.613187\pi$$
$$278$$ −19.7846 −1.18660
$$279$$ −4.92820 −0.295044
$$280$$ −3.00000 −0.179284
$$281$$ −6.92820 −0.413302 −0.206651 0.978415i $$-0.566256\pi$$
−0.206651 + 0.978415i $$0.566256\pi$$
$$282$$ 6.46410 0.384932
$$283$$ 6.39230 0.379983 0.189992 0.981786i $$-0.439154\pi$$
0.189992 + 0.981786i $$0.439154\pi$$
$$284$$ 12.9282 0.767148
$$285$$ −5.73205 −0.339537
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ −1.46410 −0.0859750
$$291$$ 8.39230 0.491966
$$292$$ −6.92820 −0.405442
$$293$$ 29.2487 1.70873 0.854364 0.519675i $$-0.173947\pi$$
0.854364 + 0.519675i $$0.173947\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 11.4641 0.667466
$$296$$ 5.92820 0.344570
$$297$$ 0.267949 0.0155480
$$298$$ −18.7846 −1.08816
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ −18.0000 −1.03750
$$302$$ 22.7846 1.31111
$$303$$ 12.3923 0.711919
$$304$$ −5.73205 −0.328756
$$305$$ 0.535898 0.0306855
$$306$$ −4.00000 −0.228665
$$307$$ 28.2487 1.61224 0.806120 0.591753i $$-0.201564\pi$$
0.806120 + 0.591753i $$0.201564\pi$$
$$308$$ −0.803848 −0.0458035
$$309$$ −11.5885 −0.659244
$$310$$ 4.92820 0.279903
$$311$$ −18.9282 −1.07332 −0.536660 0.843799i $$-0.680314\pi$$
−0.536660 + 0.843799i $$0.680314\pi$$
$$312$$ 0 0
$$313$$ −33.3205 −1.88339 −0.941693 0.336473i $$-0.890766\pi$$
−0.941693 + 0.336473i $$0.890766\pi$$
$$314$$ 21.1962 1.19617
$$315$$ −3.00000 −0.169031
$$316$$ 3.07180 0.172802
$$317$$ 30.4641 1.71103 0.855517 0.517774i $$-0.173239\pi$$
0.855517 + 0.517774i $$0.173239\pi$$
$$318$$ 0.267949 0.0150258
$$319$$ −0.392305 −0.0219649
$$320$$ −1.00000 −0.0559017
$$321$$ 12.9282 0.721582
$$322$$ −10.3923 −0.579141
$$323$$ 22.9282 1.27576
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −2.92820 −0.162178
$$327$$ −10.3923 −0.574696
$$328$$ −4.00000 −0.220863
$$329$$ −19.3923 −1.06913
$$330$$ −0.267949 −0.0147501
$$331$$ −14.3923 −0.791073 −0.395536 0.918450i $$-0.629441\pi$$
−0.395536 + 0.918450i $$0.629441\pi$$
$$332$$ −9.46410 −0.519410
$$333$$ 5.92820 0.324864
$$334$$ 0.464102 0.0253945
$$335$$ −1.46410 −0.0799924
$$336$$ −3.00000 −0.163663
$$337$$ 26.3923 1.43768 0.718840 0.695175i $$-0.244673\pi$$
0.718840 + 0.695175i $$0.244673\pi$$
$$338$$ 0 0
$$339$$ 12.0000 0.651751
$$340$$ 4.00000 0.216930
$$341$$ 1.32051 0.0715095
$$342$$ −5.73205 −0.309954
$$343$$ −15.0000 −0.809924
$$344$$ −6.00000 −0.323498
$$345$$ −3.46410 −0.186501
$$346$$ 2.12436 0.114206
$$347$$ 4.39230 0.235791 0.117896 0.993026i $$-0.462385\pi$$
0.117896 + 0.993026i $$0.462385\pi$$
$$348$$ −1.46410 −0.0784841
$$349$$ 10.5359 0.563974 0.281987 0.959418i $$-0.409007\pi$$
0.281987 + 0.959418i $$0.409007\pi$$
$$350$$ 3.00000 0.160357
$$351$$ 0 0
$$352$$ −0.267949 −0.0142817
$$353$$ 7.60770 0.404917 0.202458 0.979291i $$-0.435107\pi$$
0.202458 + 0.979291i $$0.435107\pi$$
$$354$$ 11.4641 0.609310
$$355$$ −12.9282 −0.686158
$$356$$ 14.1244 0.748589
$$357$$ 12.0000 0.635107
$$358$$ −9.07180 −0.479459
$$359$$ −0.928203 −0.0489887 −0.0244943 0.999700i $$-0.507798\pi$$
−0.0244943 + 0.999700i $$0.507798\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 13.8564 0.729285
$$362$$ −16.9282 −0.889727
$$363$$ 10.9282 0.573582
$$364$$ 0 0
$$365$$ 6.92820 0.362639
$$366$$ 0.535898 0.0280119
$$367$$ 21.3205 1.11292 0.556461 0.830874i $$-0.312159\pi$$
0.556461 + 0.830874i $$0.312159\pi$$
$$368$$ −3.46410 −0.180579
$$369$$ −4.00000 −0.208232
$$370$$ −5.92820 −0.308193
$$371$$ −0.803848 −0.0417337
$$372$$ 4.92820 0.255515
$$373$$ 9.07180 0.469720 0.234860 0.972029i $$-0.424537\pi$$
0.234860 + 0.972029i $$0.424537\pi$$
$$374$$ 1.07180 0.0554213
$$375$$ 1.00000 0.0516398
$$376$$ −6.46410 −0.333361
$$377$$ 0 0
$$378$$ −3.00000 −0.154303
$$379$$ −9.73205 −0.499902 −0.249951 0.968259i $$-0.580414\pi$$
−0.249951 + 0.968259i $$0.580414\pi$$
$$380$$ 5.73205 0.294048
$$381$$ −12.6603 −0.648604
$$382$$ −17.3205 −0.886194
$$383$$ −4.78461 −0.244482 −0.122241 0.992500i $$-0.539008\pi$$
−0.122241 + 0.992500i $$0.539008\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0.803848 0.0409679
$$386$$ 9.85641 0.501678
$$387$$ −6.00000 −0.304997
$$388$$ −8.39230 −0.426055
$$389$$ −17.8564 −0.905356 −0.452678 0.891674i $$-0.649531\pi$$
−0.452678 + 0.891674i $$0.649531\pi$$
$$390$$ 0 0
$$391$$ 13.8564 0.700749
$$392$$ 2.00000 0.101015
$$393$$ −14.3205 −0.722374
$$394$$ −11.3923 −0.573936
$$395$$ −3.07180 −0.154559
$$396$$ −0.267949 −0.0134650
$$397$$ −5.92820 −0.297528 −0.148764 0.988873i $$-0.547529\pi$$
−0.148764 + 0.988873i $$0.547529\pi$$
$$398$$ 24.9282 1.24954
$$399$$ 17.1962 0.860884
$$400$$ 1.00000 0.0500000
$$401$$ 22.1244 1.10484 0.552419 0.833567i $$-0.313705\pi$$
0.552419 + 0.833567i $$0.313705\pi$$
$$402$$ −1.46410 −0.0730228
$$403$$ 0 0
$$404$$ −12.3923 −0.616540
$$405$$ −1.00000 −0.0496904
$$406$$ 4.39230 0.217986
$$407$$ −1.58846 −0.0787369
$$408$$ 4.00000 0.198030
$$409$$ 39.0526 1.93102 0.965512 0.260357i $$-0.0838403\pi$$
0.965512 + 0.260357i $$0.0838403\pi$$
$$410$$ 4.00000 0.197546
$$411$$ −13.4641 −0.664135
$$412$$ 11.5885 0.570922
$$413$$ −34.3923 −1.69233
$$414$$ −3.46410 −0.170251
$$415$$ 9.46410 0.464574
$$416$$ 0 0
$$417$$ 19.7846 0.968857
$$418$$ 1.53590 0.0751232
$$419$$ 9.85641 0.481517 0.240758 0.970585i $$-0.422604\pi$$
0.240758 + 0.970585i $$0.422604\pi$$
$$420$$ 3.00000 0.146385
$$421$$ 21.8564 1.06522 0.532608 0.846362i $$-0.321212\pi$$
0.532608 + 0.846362i $$0.321212\pi$$
$$422$$ −8.07180 −0.392929
$$423$$ −6.46410 −0.314295
$$424$$ −0.267949 −0.0130128
$$425$$ −4.00000 −0.194029
$$426$$ −12.9282 −0.626373
$$427$$ −1.60770 −0.0778018
$$428$$ −12.9282 −0.624908
$$429$$ 0 0
$$430$$ 6.00000 0.289346
$$431$$ −13.8564 −0.667440 −0.333720 0.942672i $$-0.608304\pi$$
−0.333720 + 0.942672i $$0.608304\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 8.78461 0.422161 0.211081 0.977469i $$-0.432302\pi$$
0.211081 + 0.977469i $$0.432302\pi$$
$$434$$ −14.7846 −0.709684
$$435$$ 1.46410 0.0701983
$$436$$ 10.3923 0.497701
$$437$$ 19.8564 0.949861
$$438$$ 6.92820 0.331042
$$439$$ 13.3205 0.635753 0.317877 0.948132i $$-0.397030\pi$$
0.317877 + 0.948132i $$0.397030\pi$$
$$440$$ 0.267949 0.0127740
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −19.8564 −0.943406 −0.471703 0.881757i $$-0.656361\pi$$
−0.471703 + 0.881757i $$0.656361\pi$$
$$444$$ −5.92820 −0.281340
$$445$$ −14.1244 −0.669559
$$446$$ −18.8564 −0.892877
$$447$$ 18.7846 0.888482
$$448$$ 3.00000 0.141737
$$449$$ −6.12436 −0.289026 −0.144513 0.989503i $$-0.546162\pi$$
−0.144513 + 0.989503i $$0.546162\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 1.07180 0.0504689
$$452$$ −12.0000 −0.564433
$$453$$ −22.7846 −1.07051
$$454$$ −6.53590 −0.306745
$$455$$ 0 0
$$456$$ 5.73205 0.268428
$$457$$ −7.46410 −0.349156 −0.174578 0.984643i $$-0.555856\pi$$
−0.174578 + 0.984643i $$0.555856\pi$$
$$458$$ 4.53590 0.211949
$$459$$ 4.00000 0.186704
$$460$$ 3.46410 0.161515
$$461$$ 11.6077 0.540624 0.270312 0.962773i $$-0.412873\pi$$
0.270312 + 0.962773i $$0.412873\pi$$
$$462$$ 0.803848 0.0373984
$$463$$ 40.7846 1.89542 0.947711 0.319131i $$-0.103391\pi$$
0.947711 + 0.319131i $$0.103391\pi$$
$$464$$ 1.46410 0.0679692
$$465$$ −4.92820 −0.228540
$$466$$ −18.0000 −0.833834
$$467$$ −24.3923 −1.12874 −0.564371 0.825522i $$-0.690881\pi$$
−0.564371 + 0.825522i $$0.690881\pi$$
$$468$$ 0 0
$$469$$ 4.39230 0.202818
$$470$$ 6.46410 0.298167
$$471$$ −21.1962 −0.976667
$$472$$ −11.4641 −0.527678
$$473$$ 1.60770 0.0739219
$$474$$ −3.07180 −0.141092
$$475$$ −5.73205 −0.263005
$$476$$ −12.0000 −0.550019
$$477$$ −0.267949 −0.0122686
$$478$$ 3.46410 0.158444
$$479$$ 22.2487 1.01657 0.508285 0.861189i $$-0.330280\pi$$
0.508285 + 0.861189i $$0.330280\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −25.1962 −1.14765
$$483$$ 10.3923 0.472866
$$484$$ −10.9282 −0.496737
$$485$$ 8.39230 0.381075
$$486$$ −1.00000 −0.0453609
$$487$$ −21.0000 −0.951601 −0.475800 0.879553i $$-0.657842\pi$$
−0.475800 + 0.879553i $$0.657842\pi$$
$$488$$ −0.535898 −0.0242590
$$489$$ 2.92820 0.132418
$$490$$ −2.00000 −0.0903508
$$491$$ −5.39230 −0.243351 −0.121676 0.992570i $$-0.538827\pi$$
−0.121676 + 0.992570i $$0.538827\pi$$
$$492$$ 4.00000 0.180334
$$493$$ −5.85641 −0.263759
$$494$$ 0 0
$$495$$ 0.267949 0.0120434
$$496$$ −4.92820 −0.221283
$$497$$ 38.7846 1.73973
$$498$$ 9.46410 0.424097
$$499$$ 33.3205 1.49163 0.745815 0.666153i $$-0.232060\pi$$
0.745815 + 0.666153i $$0.232060\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −0.464102 −0.0207345
$$502$$ −19.5359 −0.871930
$$503$$ 3.73205 0.166404 0.0832020 0.996533i $$-0.473485\pi$$
0.0832020 + 0.996533i $$0.473485\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 12.3923 0.551450
$$506$$ 0.928203 0.0412637
$$507$$ 0 0
$$508$$ 12.6603 0.561708
$$509$$ 29.3205 1.29961 0.649804 0.760102i $$-0.274851\pi$$
0.649804 + 0.760102i $$0.274851\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ −20.7846 −0.919457
$$512$$ 1.00000 0.0441942
$$513$$ 5.73205 0.253076
$$514$$ −14.5359 −0.641151
$$515$$ −11.5885 −0.510648
$$516$$ 6.00000 0.264135
$$517$$ 1.73205 0.0761755
$$518$$ 17.7846 0.781411
$$519$$ −2.12436 −0.0932489
$$520$$ 0 0
$$521$$ −6.60770 −0.289488 −0.144744 0.989469i $$-0.546236\pi$$
−0.144744 + 0.989469i $$0.546236\pi$$
$$522$$ 1.46410 0.0640820
$$523$$ −41.7128 −1.82397 −0.911987 0.410219i $$-0.865452\pi$$
−0.911987 + 0.410219i $$0.865452\pi$$
$$524$$ 14.3205 0.625594
$$525$$ −3.00000 −0.130931
$$526$$ −24.5167 −1.06898
$$527$$ 19.7128 0.858704
$$528$$ 0.267949 0.0116610
$$529$$ −11.0000 −0.478261
$$530$$ 0.267949 0.0116390
$$531$$ −11.4641 −0.497500
$$532$$ −17.1962 −0.745548
$$533$$ 0 0
$$534$$ −14.1244 −0.611221
$$535$$ 12.9282 0.558935
$$536$$ 1.46410 0.0632396
$$537$$ 9.07180 0.391477
$$538$$ −17.0718 −0.736017
$$539$$ −0.535898 −0.0230828
$$540$$ 1.00000 0.0430331
$$541$$ −14.7846 −0.635640 −0.317820 0.948151i $$-0.602951\pi$$
−0.317820 + 0.948151i $$0.602951\pi$$
$$542$$ −24.3923 −1.04774
$$543$$ 16.9282 0.726459
$$544$$ −4.00000 −0.171499
$$545$$ −10.3923 −0.445157
$$546$$ 0 0
$$547$$ −5.32051 −0.227488 −0.113744 0.993510i $$-0.536284\pi$$
−0.113744 + 0.993510i $$0.536284\pi$$
$$548$$ 13.4641 0.575158
$$549$$ −0.535898 −0.0228716
$$550$$ −0.267949 −0.0114254
$$551$$ −8.39230 −0.357524
$$552$$ 3.46410 0.147442
$$553$$ 9.21539 0.391878
$$554$$ −11.5885 −0.492346
$$555$$ 5.92820 0.251638
$$556$$ −19.7846 −0.839054
$$557$$ 22.6077 0.957919 0.478959 0.877837i $$-0.341014\pi$$
0.478959 + 0.877837i $$0.341014\pi$$
$$558$$ −4.92820 −0.208627
$$559$$ 0 0
$$560$$ −3.00000 −0.126773
$$561$$ −1.07180 −0.0452513
$$562$$ −6.92820 −0.292249
$$563$$ 15.3205 0.645682 0.322841 0.946453i $$-0.395362\pi$$
0.322841 + 0.946453i $$0.395362\pi$$
$$564$$ 6.46410 0.272188
$$565$$ 12.0000 0.504844
$$566$$ 6.39230 0.268689
$$567$$ 3.00000 0.125988
$$568$$ 12.9282 0.542455
$$569$$ 16.3205 0.684191 0.342096 0.939665i $$-0.388863\pi$$
0.342096 + 0.939665i $$0.388863\pi$$
$$570$$ −5.73205 −0.240089
$$571$$ −10.8564 −0.454326 −0.227163 0.973857i $$-0.572945\pi$$
−0.227163 + 0.973857i $$0.572945\pi$$
$$572$$ 0 0
$$573$$ 17.3205 0.723575
$$574$$ −12.0000 −0.500870
$$575$$ −3.46410 −0.144463
$$576$$ 1.00000 0.0416667
$$577$$ −9.32051 −0.388018 −0.194009 0.981000i $$-0.562149\pi$$
−0.194009 + 0.981000i $$0.562149\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ −9.85641 −0.409618
$$580$$ −1.46410 −0.0607935
$$581$$ −28.3923 −1.17791
$$582$$ 8.39230 0.347872
$$583$$ 0.0717968 0.00297352
$$584$$ −6.92820 −0.286691
$$585$$ 0 0
$$586$$ 29.2487 1.20825
$$587$$ −8.92820 −0.368506 −0.184253 0.982879i $$-0.558987\pi$$
−0.184253 + 0.982879i $$0.558987\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 28.2487 1.16397
$$590$$ 11.4641 0.471970
$$591$$ 11.3923 0.468617
$$592$$ 5.92820 0.243648
$$593$$ 44.7846 1.83908 0.919542 0.392992i $$-0.128560\pi$$
0.919542 + 0.392992i $$0.128560\pi$$
$$594$$ 0.267949 0.0109941
$$595$$ 12.0000 0.491952
$$596$$ −18.7846 −0.769448
$$597$$ −24.9282 −1.02024
$$598$$ 0 0
$$599$$ 28.9282 1.18197 0.590987 0.806681i $$-0.298738\pi$$
0.590987 + 0.806681i $$0.298738\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 30.7128 1.25280 0.626401 0.779501i $$-0.284527\pi$$
0.626401 + 0.779501i $$0.284527\pi$$
$$602$$ −18.0000 −0.733625
$$603$$ 1.46410 0.0596228
$$604$$ 22.7846 0.927093
$$605$$ 10.9282 0.444295
$$606$$ 12.3923 0.503403
$$607$$ −21.4449 −0.870420 −0.435210 0.900329i $$-0.643326\pi$$
−0.435210 + 0.900329i $$0.643326\pi$$
$$608$$ −5.73205 −0.232465
$$609$$ −4.39230 −0.177985
$$610$$ 0.535898 0.0216979
$$611$$ 0 0
$$612$$ −4.00000 −0.161690
$$613$$ 45.0000 1.81753 0.908766 0.417305i $$-0.137025\pi$$
0.908766 + 0.417305i $$0.137025\pi$$
$$614$$ 28.2487 1.14003
$$615$$ −4.00000 −0.161296
$$616$$ −0.803848 −0.0323879
$$617$$ 35.4641 1.42773 0.713865 0.700283i $$-0.246943\pi$$
0.713865 + 0.700283i $$0.246943\pi$$
$$618$$ −11.5885 −0.466156
$$619$$ −2.51666 −0.101153 −0.0505766 0.998720i $$-0.516106\pi$$
−0.0505766 + 0.998720i $$0.516106\pi$$
$$620$$ 4.92820 0.197921
$$621$$ 3.46410 0.139010
$$622$$ −18.9282 −0.758952
$$623$$ 42.3731 1.69764
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −33.3205 −1.33176
$$627$$ −1.53590 −0.0613379
$$628$$ 21.1962 0.845819
$$629$$ −23.7128 −0.945492
$$630$$ −3.00000 −0.119523
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 3.07180 0.122190
$$633$$ 8.07180 0.320825
$$634$$ 30.4641 1.20988
$$635$$ −12.6603 −0.502407
$$636$$ 0.267949 0.0106249
$$637$$ 0 0
$$638$$ −0.392305 −0.0155315
$$639$$ 12.9282 0.511432
$$640$$ −1.00000 −0.0395285
$$641$$ −14.4641 −0.571298 −0.285649 0.958334i $$-0.592209\pi$$
−0.285649 + 0.958334i $$0.592209\pi$$
$$642$$ 12.9282 0.510235
$$643$$ −12.7846 −0.504176 −0.252088 0.967704i $$-0.581117\pi$$
−0.252088 + 0.967704i $$0.581117\pi$$
$$644$$ −10.3923 −0.409514
$$645$$ −6.00000 −0.236250
$$646$$ 22.9282 0.902098
$$647$$ 35.7321 1.40477 0.702386 0.711796i $$-0.252118\pi$$
0.702386 + 0.711796i $$0.252118\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 3.07180 0.120579
$$650$$ 0 0
$$651$$ 14.7846 0.579455
$$652$$ −2.92820 −0.114677
$$653$$ −1.58846 −0.0621611 −0.0310806 0.999517i $$-0.509895\pi$$
−0.0310806 + 0.999517i $$0.509895\pi$$
$$654$$ −10.3923 −0.406371
$$655$$ −14.3205 −0.559549
$$656$$ −4.00000 −0.156174
$$657$$ −6.92820 −0.270295
$$658$$ −19.3923 −0.755991
$$659$$ −39.7128 −1.54699 −0.773496 0.633801i $$-0.781494\pi$$
−0.773496 + 0.633801i $$0.781494\pi$$
$$660$$ −0.267949 −0.0104299
$$661$$ 21.6077 0.840442 0.420221 0.907422i $$-0.361953\pi$$
0.420221 + 0.907422i $$0.361953\pi$$
$$662$$ −14.3923 −0.559373
$$663$$ 0 0
$$664$$ −9.46410 −0.367278
$$665$$ 17.1962 0.666838
$$666$$ 5.92820 0.229713
$$667$$ −5.07180 −0.196381
$$668$$ 0.464102 0.0179566
$$669$$ 18.8564 0.729031
$$670$$ −1.46410 −0.0565632
$$671$$ 0.143594 0.00554337
$$672$$ −3.00000 −0.115728
$$673$$ 16.7846 0.646999 0.323500 0.946228i $$-0.395141\pi$$
0.323500 + 0.946228i $$0.395141\pi$$
$$674$$ 26.3923 1.01659
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 10.9282 0.420005 0.210002 0.977701i $$-0.432653\pi$$
0.210002 + 0.977701i $$0.432653\pi$$
$$678$$ 12.0000 0.460857
$$679$$ −25.1769 −0.966201
$$680$$ 4.00000 0.153393
$$681$$ 6.53590 0.250456
$$682$$ 1.32051 0.0505649
$$683$$ 41.3205 1.58109 0.790543 0.612407i $$-0.209799\pi$$
0.790543 + 0.612407i $$0.209799\pi$$
$$684$$ −5.73205 −0.219170
$$685$$ −13.4641 −0.514437
$$686$$ −15.0000 −0.572703
$$687$$ −4.53590 −0.173055
$$688$$ −6.00000 −0.228748
$$689$$ 0 0
$$690$$ −3.46410 −0.131876
$$691$$ 15.0526 0.572626 0.286313 0.958136i $$-0.407570\pi$$
0.286313 + 0.958136i $$0.407570\pi$$
$$692$$ 2.12436 0.0807559
$$693$$ −0.803848 −0.0305356
$$694$$ 4.39230 0.166730
$$695$$ 19.7846 0.750473
$$696$$ −1.46410 −0.0554966
$$697$$ 16.0000 0.606043
$$698$$ 10.5359 0.398790
$$699$$ 18.0000 0.680823
$$700$$ 3.00000 0.113389
$$701$$ 4.39230 0.165895 0.0829475 0.996554i $$-0.473567\pi$$
0.0829475 + 0.996554i $$0.473567\pi$$
$$702$$ 0 0
$$703$$ −33.9808 −1.28161
$$704$$ −0.267949 −0.0100987
$$705$$ −6.46410 −0.243452
$$706$$ 7.60770 0.286319
$$707$$ −37.1769 −1.39818
$$708$$ 11.4641 0.430847
$$709$$ −27.3205 −1.02604 −0.513022 0.858376i $$-0.671474\pi$$
−0.513022 + 0.858376i $$0.671474\pi$$
$$710$$ −12.9282 −0.485187
$$711$$ 3.07180 0.115201
$$712$$ 14.1244 0.529333
$$713$$ 17.0718 0.639344
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ −9.07180 −0.339029
$$717$$ −3.46410 −0.129369
$$718$$ −0.928203 −0.0346402
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 34.7654 1.29473
$$722$$ 13.8564 0.515682
$$723$$ 25.1962 0.937055
$$724$$ −16.9282 −0.629132
$$725$$ 1.46410 0.0543754
$$726$$ 10.9282 0.405584
$$727$$ 4.66025 0.172839 0.0864196 0.996259i $$-0.472457\pi$$
0.0864196 + 0.996259i $$0.472457\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 6.92820 0.256424
$$731$$ 24.0000 0.887672
$$732$$ 0.535898 0.0198074
$$733$$ 20.8564 0.770349 0.385174 0.922844i $$-0.374141\pi$$
0.385174 + 0.922844i $$0.374141\pi$$
$$734$$ 21.3205 0.786954
$$735$$ 2.00000 0.0737711
$$736$$ −3.46410 −0.127688
$$737$$ −0.392305 −0.0144507
$$738$$ −4.00000 −0.147242
$$739$$ −35.8372 −1.31829 −0.659146 0.752015i $$-0.729082\pi$$
−0.659146 + 0.752015i $$0.729082\pi$$
$$740$$ −5.92820 −0.217925
$$741$$ 0 0
$$742$$ −0.803848 −0.0295102
$$743$$ −10.9282 −0.400917 −0.200458 0.979702i $$-0.564243\pi$$
−0.200458 + 0.979702i $$0.564243\pi$$
$$744$$ 4.92820 0.180677
$$745$$ 18.7846 0.688215
$$746$$ 9.07180 0.332142
$$747$$ −9.46410 −0.346273
$$748$$ 1.07180 0.0391888
$$749$$ −38.7846 −1.41716
$$750$$ 1.00000 0.0365148
$$751$$ 21.1769 0.772757 0.386378 0.922340i $$-0.373726\pi$$
0.386378 + 0.922340i $$0.373726\pi$$
$$752$$ −6.46410 −0.235722
$$753$$ 19.5359 0.711928
$$754$$ 0 0
$$755$$ −22.7846 −0.829217
$$756$$ −3.00000 −0.109109
$$757$$ 21.7321 0.789865 0.394932 0.918710i $$-0.370768\pi$$
0.394932 + 0.918710i $$0.370768\pi$$
$$758$$ −9.73205 −0.353484
$$759$$ −0.928203 −0.0336916
$$760$$ 5.73205 0.207923
$$761$$ 7.98076 0.289302 0.144651 0.989483i $$-0.453794\pi$$
0.144651 + 0.989483i $$0.453794\pi$$
$$762$$ −12.6603 −0.458633
$$763$$ 31.1769 1.12868
$$764$$ −17.3205 −0.626634
$$765$$ 4.00000 0.144620
$$766$$ −4.78461 −0.172875
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 15.7128 0.566619 0.283309 0.959029i $$-0.408568\pi$$
0.283309 + 0.959029i $$0.408568\pi$$
$$770$$ 0.803848 0.0289687
$$771$$ 14.5359 0.523498
$$772$$ 9.85641 0.354740
$$773$$ 32.6077 1.17282 0.586409 0.810015i $$-0.300541\pi$$
0.586409 + 0.810015i $$0.300541\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ −4.92820 −0.177026
$$776$$ −8.39230 −0.301266
$$777$$ −17.7846 −0.638019
$$778$$ −17.8564 −0.640183
$$779$$ 22.9282 0.821488
$$780$$ 0 0
$$781$$ −3.46410 −0.123955
$$782$$ 13.8564 0.495504
$$783$$ −1.46410 −0.0523227
$$784$$ 2.00000 0.0714286
$$785$$ −21.1962 −0.756523
$$786$$ −14.3205 −0.510796
$$787$$ 1.46410 0.0521896 0.0260948 0.999659i $$-0.491693\pi$$
0.0260948 + 0.999659i $$0.491693\pi$$
$$788$$ −11.3923 −0.405834
$$789$$ 24.5167 0.872816
$$790$$ −3.07180 −0.109290
$$791$$ −36.0000 −1.28001
$$792$$ −0.267949 −0.00952116
$$793$$ 0 0
$$794$$ −5.92820 −0.210384
$$795$$ −0.267949 −0.00950318
$$796$$ 24.9282 0.883557
$$797$$ 10.1436 0.359305 0.179652 0.983730i $$-0.442503\pi$$
0.179652 + 0.983730i $$0.442503\pi$$
$$798$$ 17.1962 0.608737
$$799$$ 25.8564 0.914734
$$800$$ 1.00000 0.0353553
$$801$$ 14.1244 0.499060
$$802$$ 22.1244 0.781238
$$803$$ 1.85641 0.0655112
$$804$$ −1.46410 −0.0516349
$$805$$ 10.3923 0.366281
$$806$$ 0 0
$$807$$ 17.0718 0.600956
$$808$$ −12.3923 −0.435960
$$809$$ −6.78461 −0.238534 −0.119267 0.992862i $$-0.538054\pi$$
−0.119267 + 0.992862i $$0.538054\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 23.5885 0.828303 0.414151 0.910208i $$-0.364078\pi$$
0.414151 + 0.910208i $$0.364078\pi$$
$$812$$ 4.39230 0.154140
$$813$$ 24.3923 0.855475
$$814$$ −1.58846 −0.0556754
$$815$$ 2.92820 0.102570
$$816$$ 4.00000 0.140028
$$817$$ 34.3923 1.20323
$$818$$ 39.0526 1.36544
$$819$$ 0 0
$$820$$ 4.00000 0.139686
$$821$$ −14.6795 −0.512318 −0.256159 0.966635i $$-0.582457\pi$$
−0.256159 + 0.966635i $$0.582457\pi$$
$$822$$ −13.4641 −0.469614
$$823$$ 8.41154 0.293208 0.146604 0.989195i $$-0.453166\pi$$
0.146604 + 0.989195i $$0.453166\pi$$
$$824$$ 11.5885 0.403703
$$825$$ 0.267949 0.00932879
$$826$$ −34.3923 −1.19666
$$827$$ −46.4974 −1.61687 −0.808437 0.588583i $$-0.799686\pi$$
−0.808437 + 0.588583i $$0.799686\pi$$
$$828$$ −3.46410 −0.120386
$$829$$ 4.67949 0.162525 0.0812627 0.996693i $$-0.474105\pi$$
0.0812627 + 0.996693i $$0.474105\pi$$
$$830$$ 9.46410 0.328504
$$831$$ 11.5885 0.401999
$$832$$ 0 0
$$833$$ −8.00000 −0.277184
$$834$$ 19.7846 0.685085
$$835$$ −0.464102 −0.0160609
$$836$$ 1.53590 0.0531202
$$837$$ 4.92820 0.170344
$$838$$ 9.85641 0.340484
$$839$$ 19.1769 0.662061 0.331030 0.943620i $$-0.392604\pi$$
0.331030 + 0.943620i $$0.392604\pi$$
$$840$$ 3.00000 0.103510
$$841$$ −26.8564 −0.926083
$$842$$ 21.8564 0.753222
$$843$$ 6.92820 0.238620
$$844$$ −8.07180 −0.277843
$$845$$ 0 0
$$846$$ −6.46410 −0.222240
$$847$$ −32.7846 −1.12649
$$848$$ −0.267949 −0.00920141
$$849$$ −6.39230 −0.219383
$$850$$ −4.00000 −0.137199
$$851$$ −20.5359 −0.703962
$$852$$ −12.9282 −0.442913
$$853$$ 24.6410 0.843692 0.421846 0.906667i $$-0.361382\pi$$
0.421846 + 0.906667i $$0.361382\pi$$
$$854$$ −1.60770 −0.0550142
$$855$$ 5.73205 0.196032
$$856$$ −12.9282 −0.441877
$$857$$ 28.9282 0.988169 0.494084 0.869414i $$-0.335503\pi$$
0.494084 + 0.869414i $$0.335503\pi$$
$$858$$ 0 0
$$859$$ −6.07180 −0.207167 −0.103584 0.994621i $$-0.533031\pi$$
−0.103584 + 0.994621i $$0.533031\pi$$
$$860$$ 6.00000 0.204598
$$861$$ 12.0000 0.408959
$$862$$ −13.8564 −0.471951
$$863$$ 2.92820 0.0996772 0.0498386 0.998757i $$-0.484129\pi$$
0.0498386 + 0.998757i $$0.484129\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −2.12436 −0.0722303
$$866$$ 8.78461 0.298513
$$867$$ 1.00000 0.0339618
$$868$$ −14.7846 −0.501822
$$869$$ −0.823085 −0.0279213
$$870$$ 1.46410 0.0496377
$$871$$ 0 0
$$872$$ 10.3923 0.351928
$$873$$ −8.39230 −0.284036
$$874$$ 19.8564 0.671653
$$875$$ −3.00000 −0.101419
$$876$$ 6.92820 0.234082
$$877$$ −21.7128 −0.733190 −0.366595 0.930381i $$-0.619476\pi$$
−0.366595 + 0.930381i $$0.619476\pi$$
$$878$$ 13.3205 0.449545
$$879$$ −29.2487 −0.986535
$$880$$ 0.267949 0.00903257
$$881$$ −39.1051 −1.31748 −0.658742 0.752369i $$-0.728911\pi$$
−0.658742 + 0.752369i $$0.728911\pi$$
$$882$$ 2.00000 0.0673435
$$883$$ 13.3205 0.448271 0.224135 0.974558i $$-0.428044\pi$$
0.224135 + 0.974558i $$0.428044\pi$$
$$884$$ 0 0
$$885$$ −11.4641 −0.385362
$$886$$ −19.8564 −0.667089
$$887$$ 55.7321 1.87130 0.935650 0.352930i $$-0.114815\pi$$
0.935650 + 0.352930i $$0.114815\pi$$
$$888$$ −5.92820 −0.198937
$$889$$ 37.9808 1.27383
$$890$$ −14.1244 −0.473449
$$891$$ −0.267949 −0.00897664
$$892$$ −18.8564 −0.631359
$$893$$ 37.0526 1.23992
$$894$$ 18.7846 0.628251
$$895$$ 9.07180 0.303237
$$896$$ 3.00000 0.100223
$$897$$ 0 0
$$898$$ −6.12436 −0.204372
$$899$$ −7.21539 −0.240647
$$900$$ 1.00000 0.0333333
$$901$$ 1.07180 0.0357067
$$902$$ 1.07180 0.0356869
$$903$$ 18.0000 0.599002
$$904$$ −12.0000 −0.399114
$$905$$ 16.9282 0.562713
$$906$$ −22.7846 −0.756968
$$907$$ −20.9282 −0.694910 −0.347455 0.937697i $$-0.612954\pi$$
−0.347455 + 0.937697i $$0.612954\pi$$
$$908$$ −6.53590 −0.216901
$$909$$ −12.3923 −0.411027
$$910$$ 0 0
$$911$$ 12.7846 0.423573 0.211787 0.977316i $$-0.432072\pi$$
0.211787 + 0.977316i $$0.432072\pi$$
$$912$$ 5.73205 0.189807
$$913$$ 2.53590 0.0839260
$$914$$ −7.46410 −0.246891
$$915$$ −0.535898 −0.0177163
$$916$$ 4.53590 0.149870
$$917$$ 42.9615 1.41871
$$918$$ 4.00000 0.132020
$$919$$ −47.9615 −1.58210 −0.791052 0.611748i $$-0.790466\pi$$
−0.791052 + 0.611748i $$0.790466\pi$$
$$920$$ 3.46410 0.114208
$$921$$ −28.2487 −0.930827
$$922$$ 11.6077 0.382279
$$923$$ 0 0
$$924$$ 0.803848 0.0264446
$$925$$ 5.92820 0.194918
$$926$$ 40.7846 1.34027
$$927$$ 11.5885 0.380615
$$928$$ 1.46410 0.0480615
$$929$$ −57.8564 −1.89821 −0.949104 0.314964i $$-0.898007\pi$$
−0.949104 + 0.314964i $$0.898007\pi$$
$$930$$ −4.92820 −0.161602
$$931$$ −11.4641 −0.375721
$$932$$ −18.0000 −0.589610
$$933$$ 18.9282 0.619682
$$934$$ −24.3923 −0.798141
$$935$$ −1.07180 −0.0350515
$$936$$ 0 0
$$937$$ −21.7128 −0.709327 −0.354663 0.934994i $$-0.615405\pi$$
−0.354663 + 0.934994i $$0.615405\pi$$
$$938$$ 4.39230 0.143414
$$939$$ 33.3205 1.08737
$$940$$ 6.46410 0.210836
$$941$$ −60.4974 −1.97216 −0.986080 0.166273i $$-0.946827\pi$$
−0.986080 + 0.166273i $$0.946827\pi$$
$$942$$ −21.1962 −0.690608
$$943$$ 13.8564 0.451227
$$944$$ −11.4641 −0.373125
$$945$$ 3.00000 0.0975900
$$946$$ 1.60770 0.0522707
$$947$$ −34.3923 −1.11760 −0.558800 0.829303i $$-0.688738\pi$$
−0.558800 + 0.829303i $$0.688738\pi$$
$$948$$ −3.07180 −0.0997673
$$949$$ 0 0
$$950$$ −5.73205 −0.185972
$$951$$ −30.4641 −0.987866
$$952$$ −12.0000 −0.388922
$$953$$ 53.3205 1.72722 0.863610 0.504160i $$-0.168198\pi$$
0.863610 + 0.504160i $$0.168198\pi$$
$$954$$ −0.267949 −0.00867518
$$955$$ 17.3205 0.560478
$$956$$ 3.46410 0.112037
$$957$$ 0.392305 0.0126814
$$958$$ 22.2487 0.718823
$$959$$ 40.3923 1.30434
$$960$$ 1.00000 0.0322749
$$961$$ −6.71281 −0.216542
$$962$$ 0 0
$$963$$ −12.9282 −0.416606
$$964$$ −25.1962 −0.811513
$$965$$ −9.85641 −0.317289
$$966$$ 10.3923 0.334367
$$967$$ −1.14359 −0.0367755 −0.0183877 0.999831i $$-0.505853\pi$$
−0.0183877 + 0.999831i $$0.505853\pi$$
$$968$$ −10.9282 −0.351246
$$969$$ −22.9282 −0.736560
$$970$$ 8.39230 0.269461
$$971$$ −41.3923 −1.32834 −0.664171 0.747581i $$-0.731215\pi$$
−0.664171 + 0.747581i $$0.731215\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −59.3538 −1.90280
$$974$$ −21.0000 −0.672883
$$975$$ 0 0
$$976$$ −0.535898 −0.0171537
$$977$$ −5.46410 −0.174812 −0.0874060 0.996173i $$-0.527858\pi$$
−0.0874060 + 0.996173i $$0.527858\pi$$
$$978$$ 2.92820 0.0936336
$$979$$ −3.78461 −0.120957
$$980$$ −2.00000 −0.0638877
$$981$$ 10.3923 0.331801
$$982$$ −5.39230 −0.172075
$$983$$ −46.1769 −1.47281 −0.736407 0.676538i $$-0.763479\pi$$
−0.736407 + 0.676538i $$0.763479\pi$$
$$984$$ 4.00000 0.127515
$$985$$ 11.3923 0.362989
$$986$$ −5.85641 −0.186506
$$987$$ 19.3923 0.617264
$$988$$ 0 0
$$989$$ 20.7846 0.660912
$$990$$ 0.267949 0.00851598
$$991$$ 39.1769 1.24450 0.622248 0.782820i $$-0.286220\pi$$
0.622248 + 0.782820i $$0.286220\pi$$
$$992$$ −4.92820 −0.156471
$$993$$ 14.3923 0.456726
$$994$$ 38.7846 1.23017
$$995$$ −24.9282 −0.790277
$$996$$ 9.46410 0.299882
$$997$$ 61.9808 1.96295 0.981475 0.191589i $$-0.0613641\pi$$
0.981475 + 0.191589i $$0.0613641\pi$$
$$998$$ 33.3205 1.05474
$$999$$ −5.92820 −0.187560
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.be.1.2 2
13.2 odd 12 390.2.bb.a.121.2 4
13.5 odd 4 5070.2.b.p.1351.2 4
13.7 odd 12 390.2.bb.a.361.2 yes 4
13.8 odd 4 5070.2.b.p.1351.3 4
13.12 even 2 5070.2.a.ba.1.1 2
39.2 even 12 1170.2.bs.d.901.1 4
39.20 even 12 1170.2.bs.d.361.1 4
65.2 even 12 1950.2.y.d.199.1 4
65.7 even 12 1950.2.y.e.49.2 4
65.28 even 12 1950.2.y.e.199.2 4
65.33 even 12 1950.2.y.d.49.1 4
65.54 odd 12 1950.2.bc.a.901.1 4
65.59 odd 12 1950.2.bc.a.751.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.a.121.2 4 13.2 odd 12
390.2.bb.a.361.2 yes 4 13.7 odd 12
1170.2.bs.d.361.1 4 39.20 even 12
1170.2.bs.d.901.1 4 39.2 even 12
1950.2.y.d.49.1 4 65.33 even 12
1950.2.y.d.199.1 4 65.2 even 12
1950.2.y.e.49.2 4 65.7 even 12
1950.2.y.e.199.2 4 65.28 even 12
1950.2.bc.a.751.1 4 65.59 odd 12
1950.2.bc.a.901.1 4 65.54 odd 12
5070.2.a.ba.1.1 2 13.12 even 2
5070.2.a.be.1.2 2 1.1 even 1 trivial
5070.2.b.p.1351.2 4 13.5 odd 4
5070.2.b.p.1351.3 4 13.8 odd 4