Properties

Label 43.10.a.a.1.3
Level 43
Weight 10
Character 43.1
Self dual yes
Analytic conductor 22.147
Analytic rank 1
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.1465409550\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(32.6728\) of \(x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} - 11474166224 x^{9} + 47465836576 x^{8} + 5986976782464 x^{7} - 32493903147264 x^{6} - 1516975415483904 x^{5} + 10892588268404224 x^{4} + 139803541742443008 x^{3} - 1349125586394823680 x^{2} + 2103623681144094720 x + 529838441422848000\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-34.6728 q^{2} +96.4953 q^{3} +690.205 q^{4} +1855.36 q^{5} -3345.76 q^{6} -11539.9 q^{7} -6178.86 q^{8} -10371.7 q^{9} +O(q^{10})\) \(q-34.6728 q^{2} +96.4953 q^{3} +690.205 q^{4} +1855.36 q^{5} -3345.76 q^{6} -11539.9 q^{7} -6178.86 q^{8} -10371.7 q^{9} -64330.7 q^{10} +75330.5 q^{11} +66601.5 q^{12} -125614. q^{13} +400120. q^{14} +179034. q^{15} -139146. q^{16} +122856. q^{17} +359615. q^{18} +424932. q^{19} +1.28058e6 q^{20} -1.11354e6 q^{21} -2.61192e6 q^{22} -884779. q^{23} -596230. q^{24} +1.48925e6 q^{25} +4.35541e6 q^{26} -2.90013e6 q^{27} -7.96488e6 q^{28} +2.04931e6 q^{29} -6.20760e6 q^{30} -6.30828e6 q^{31} +7.98817e6 q^{32} +7.26904e6 q^{33} -4.25977e6 q^{34} -2.14107e7 q^{35} -7.15857e6 q^{36} -1.74953e7 q^{37} -1.47336e7 q^{38} -1.21212e7 q^{39} -1.14640e7 q^{40} +2.62550e6 q^{41} +3.86097e7 q^{42} -3.41880e6 q^{43} +5.19935e7 q^{44} -1.92432e7 q^{45} +3.06778e7 q^{46} -2.82101e7 q^{47} -1.34270e7 q^{48} +9.28153e7 q^{49} -5.16363e7 q^{50} +1.18550e7 q^{51} -8.66997e7 q^{52} -6.08384e7 q^{53} +1.00556e8 q^{54} +1.39765e8 q^{55} +7.13033e7 q^{56} +4.10039e7 q^{57} -7.10552e7 q^{58} +1.13153e7 q^{59} +1.23570e8 q^{60} -7.50606e7 q^{61} +2.18726e8 q^{62} +1.19688e8 q^{63} -2.05730e8 q^{64} -2.33060e8 q^{65} -2.52038e8 q^{66} -1.92184e8 q^{67} +8.47959e7 q^{68} -8.53770e7 q^{69} +7.42368e8 q^{70} -3.76388e8 q^{71} +6.40850e7 q^{72} -1.95344e8 q^{73} +6.06612e8 q^{74} +1.43705e8 q^{75} +2.93290e8 q^{76} -8.69306e8 q^{77} +4.20276e8 q^{78} -3.67969e8 q^{79} -2.58167e8 q^{80} -7.57036e7 q^{81} -9.10335e7 q^{82} +4.46611e8 q^{83} -7.68573e8 q^{84} +2.27943e8 q^{85} +1.18539e8 q^{86} +1.97748e8 q^{87} -4.65457e8 q^{88} +7.20754e8 q^{89} +6.67216e8 q^{90} +1.44958e9 q^{91} -6.10678e8 q^{92} -6.08719e8 q^{93} +9.78124e8 q^{94} +7.88402e8 q^{95} +7.70821e8 q^{96} -1.39219e9 q^{97} -3.21817e9 q^{98} -7.81303e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −34.6728 −1.53234 −0.766168 0.642640i \(-0.777839\pi\)
−0.766168 + 0.642640i \(0.777839\pi\)
\(3\) 96.4953 0.687797 0.343899 0.939007i \(-0.388252\pi\)
0.343899 + 0.939007i \(0.388252\pi\)
\(4\) 690.205 1.34806
\(5\) 1855.36 1.32759 0.663795 0.747915i \(-0.268945\pi\)
0.663795 + 0.747915i \(0.268945\pi\)
\(6\) −3345.76 −1.05394
\(7\) −11539.9 −1.81660 −0.908302 0.418315i \(-0.862621\pi\)
−0.908302 + 0.418315i \(0.862621\pi\)
\(8\) −6178.86 −0.533339
\(9\) −10371.7 −0.526935
\(10\) −64330.7 −2.03431
\(11\) 75330.5 1.55133 0.775665 0.631145i \(-0.217415\pi\)
0.775665 + 0.631145i \(0.217415\pi\)
\(12\) 66601.5 0.927189
\(13\) −125614. −1.21982 −0.609908 0.792472i \(-0.708794\pi\)
−0.609908 + 0.792472i \(0.708794\pi\)
\(14\) 400120. 2.78365
\(15\) 179034. 0.913112
\(16\) −139146. −0.530801
\(17\) 122856. 0.356761 0.178380 0.983962i \(-0.442914\pi\)
0.178380 + 0.983962i \(0.442914\pi\)
\(18\) 359615. 0.807442
\(19\) 424932. 0.748045 0.374023 0.927420i \(-0.377978\pi\)
0.374023 + 0.927420i \(0.377978\pi\)
\(20\) 1.28058e6 1.78966
\(21\) −1.11354e6 −1.24946
\(22\) −2.61192e6 −2.37716
\(23\) −884779. −0.659264 −0.329632 0.944109i \(-0.606925\pi\)
−0.329632 + 0.944109i \(0.606925\pi\)
\(24\) −596230. −0.366829
\(25\) 1.48925e6 0.762494
\(26\) 4.35541e6 1.86917
\(27\) −2.90013e6 −1.05022
\(28\) −7.96488e6 −2.44888
\(29\) 2.04931e6 0.538042 0.269021 0.963134i \(-0.413300\pi\)
0.269021 + 0.963134i \(0.413300\pi\)
\(30\) −6.20760e6 −1.39920
\(31\) −6.30828e6 −1.22683 −0.613414 0.789762i \(-0.710204\pi\)
−0.613414 + 0.789762i \(0.710204\pi\)
\(32\) 7.98817e6 1.34671
\(33\) 7.26904e6 1.06700
\(34\) −4.25977e6 −0.546677
\(35\) −2.14107e7 −2.41170
\(36\) −7.15857e6 −0.710338
\(37\) −1.74953e7 −1.53467 −0.767333 0.641248i \(-0.778417\pi\)
−0.767333 + 0.641248i \(0.778417\pi\)
\(38\) −1.47336e7 −1.14626
\(39\) −1.21212e7 −0.838986
\(40\) −1.14640e7 −0.708055
\(41\) 2.62550e6 0.145106 0.0725529 0.997365i \(-0.476885\pi\)
0.0725529 + 0.997365i \(0.476885\pi\)
\(42\) 3.86097e7 1.91459
\(43\) −3.41880e6 −0.152499
\(44\) 5.19935e7 2.09128
\(45\) −1.92432e7 −0.699554
\(46\) 3.06778e7 1.01021
\(47\) −2.82101e7 −0.843266 −0.421633 0.906767i \(-0.638543\pi\)
−0.421633 + 0.906767i \(0.638543\pi\)
\(48\) −1.34270e7 −0.365084
\(49\) 9.28153e7 2.30005
\(50\) −5.16363e7 −1.16840
\(51\) 1.18550e7 0.245379
\(52\) −8.66997e7 −1.64438
\(53\) −6.08384e7 −1.05910 −0.529549 0.848279i \(-0.677639\pi\)
−0.529549 + 0.848279i \(0.677639\pi\)
\(54\) 1.00556e8 1.60929
\(55\) 1.39765e8 2.05953
\(56\) 7.13033e7 0.968865
\(57\) 4.10039e7 0.514503
\(58\) −7.10552e7 −0.824461
\(59\) 1.13153e7 0.121571 0.0607857 0.998151i \(-0.480639\pi\)
0.0607857 + 0.998151i \(0.480639\pi\)
\(60\) 1.23570e8 1.23093
\(61\) −7.50606e7 −0.694109 −0.347055 0.937845i \(-0.612818\pi\)
−0.347055 + 0.937845i \(0.612818\pi\)
\(62\) 2.18726e8 1.87991
\(63\) 1.19688e8 0.957233
\(64\) −2.05730e8 −1.53280
\(65\) −2.33060e8 −1.61942
\(66\) −2.52038e8 −1.63500
\(67\) −1.92184e8 −1.16514 −0.582572 0.812779i \(-0.697954\pi\)
−0.582572 + 0.812779i \(0.697954\pi\)
\(68\) 8.47959e7 0.480933
\(69\) −8.53770e7 −0.453440
\(70\) 7.42368e8 3.69554
\(71\) −3.76388e8 −1.75781 −0.878907 0.476993i \(-0.841727\pi\)
−0.878907 + 0.476993i \(0.841727\pi\)
\(72\) 6.40850e7 0.281035
\(73\) −1.95344e8 −0.805095 −0.402548 0.915399i \(-0.631875\pi\)
−0.402548 + 0.915399i \(0.631875\pi\)
\(74\) 6.06612e8 2.35163
\(75\) 1.43705e8 0.524441
\(76\) 2.93290e8 1.00841
\(77\) −8.69306e8 −2.81815
\(78\) 4.20276e8 1.28561
\(79\) −3.67969e8 −1.06289 −0.531446 0.847092i \(-0.678351\pi\)
−0.531446 + 0.847092i \(0.678351\pi\)
\(80\) −2.58167e8 −0.704686
\(81\) −7.57036e7 −0.195404
\(82\) −9.10335e7 −0.222351
\(83\) 4.46611e8 1.03295 0.516474 0.856303i \(-0.327244\pi\)
0.516474 + 0.856303i \(0.327244\pi\)
\(84\) −7.68573e8 −1.68434
\(85\) 2.27943e8 0.473632
\(86\) 1.18539e8 0.233679
\(87\) 1.97748e8 0.370063
\(88\) −4.65457e8 −0.827384
\(89\) 7.20754e8 1.21768 0.608838 0.793294i \(-0.291636\pi\)
0.608838 + 0.793294i \(0.291636\pi\)
\(90\) 6.67216e8 1.07195
\(91\) 1.44958e9 2.21592
\(92\) −6.10678e8 −0.888725
\(93\) −6.08719e8 −0.843809
\(94\) 9.78124e8 1.29217
\(95\) 7.88402e8 0.993097
\(96\) 7.70821e8 0.926260
\(97\) −1.39219e9 −1.59671 −0.798356 0.602185i \(-0.794297\pi\)
−0.798356 + 0.602185i \(0.794297\pi\)
\(98\) −3.21817e9 −3.52445
\(99\) −7.81303e8 −0.817450
\(100\) 1.02788e9 1.02788
\(101\) −6.37735e8 −0.609810 −0.304905 0.952383i \(-0.598625\pi\)
−0.304905 + 0.952383i \(0.598625\pi\)
\(102\) −4.11048e8 −0.376003
\(103\) 2.27677e9 1.99321 0.996604 0.0823449i \(-0.0262409\pi\)
0.996604 + 0.0823449i \(0.0262409\pi\)
\(104\) 7.76154e8 0.650575
\(105\) −2.06603e9 −1.65876
\(106\) 2.10944e9 1.62290
\(107\) 3.35952e8 0.247771 0.123885 0.992297i \(-0.460465\pi\)
0.123885 + 0.992297i \(0.460465\pi\)
\(108\) −2.00168e9 −1.41576
\(109\) −7.49612e8 −0.508648 −0.254324 0.967119i \(-0.581853\pi\)
−0.254324 + 0.967119i \(0.581853\pi\)
\(110\) −4.84606e9 −3.15589
\(111\) −1.68822e9 −1.05554
\(112\) 1.60573e9 0.964256
\(113\) 5.38431e8 0.310654 0.155327 0.987863i \(-0.450357\pi\)
0.155327 + 0.987863i \(0.450357\pi\)
\(114\) −1.42172e9 −0.788392
\(115\) −1.64159e9 −0.875232
\(116\) 1.41444e9 0.725310
\(117\) 1.30283e9 0.642764
\(118\) −3.92333e8 −0.186288
\(119\) −1.41775e9 −0.648093
\(120\) −1.10622e9 −0.486998
\(121\) 3.31674e9 1.40662
\(122\) 2.60256e9 1.06361
\(123\) 2.53348e8 0.0998033
\(124\) −4.35401e9 −1.65383
\(125\) −8.60665e8 −0.315311
\(126\) −4.14991e9 −1.46680
\(127\) 2.77771e9 0.947481 0.473741 0.880664i \(-0.342903\pi\)
0.473741 + 0.880664i \(0.342903\pi\)
\(128\) 3.04328e9 1.00207
\(129\) −3.29898e8 −0.104888
\(130\) 8.08086e9 2.48149
\(131\) 1.62508e9 0.482117 0.241059 0.970511i \(-0.422505\pi\)
0.241059 + 0.970511i \(0.422505\pi\)
\(132\) 5.01712e9 1.43838
\(133\) −4.90366e9 −1.35890
\(134\) 6.66355e9 1.78539
\(135\) −5.38080e9 −1.39426
\(136\) −7.59111e8 −0.190274
\(137\) 5.41054e9 1.31219 0.656097 0.754677i \(-0.272206\pi\)
0.656097 + 0.754677i \(0.272206\pi\)
\(138\) 2.96026e9 0.694823
\(139\) 5.60140e8 0.127271 0.0636356 0.997973i \(-0.479730\pi\)
0.0636356 + 0.997973i \(0.479730\pi\)
\(140\) −1.47777e10 −3.25111
\(141\) −2.72214e9 −0.579996
\(142\) 1.30504e10 2.69356
\(143\) −9.46260e9 −1.89234
\(144\) 1.44318e9 0.279698
\(145\) 3.80221e9 0.714298
\(146\) 6.77313e9 1.23368
\(147\) 8.95624e9 1.58197
\(148\) −1.20753e10 −2.06882
\(149\) 6.58990e9 1.09532 0.547660 0.836701i \(-0.315519\pi\)
0.547660 + 0.836701i \(0.315519\pi\)
\(150\) −4.98266e9 −0.803620
\(151\) −8.64538e9 −1.35328 −0.676641 0.736314i \(-0.736565\pi\)
−0.676641 + 0.736314i \(0.736565\pi\)
\(152\) −2.62559e9 −0.398962
\(153\) −1.27422e9 −0.187990
\(154\) 3.01413e10 4.31836
\(155\) −1.17042e10 −1.62872
\(156\) −8.36611e9 −1.13100
\(157\) −4.75777e9 −0.624964 −0.312482 0.949924i \(-0.601160\pi\)
−0.312482 + 0.949924i \(0.601160\pi\)
\(158\) 1.27585e10 1.62871
\(159\) −5.87062e9 −0.728445
\(160\) 1.48210e10 1.78787
\(161\) 1.02102e10 1.19762
\(162\) 2.62486e9 0.299425
\(163\) 6.41636e9 0.711941 0.355971 0.934497i \(-0.384150\pi\)
0.355971 + 0.934497i \(0.384150\pi\)
\(164\) 1.81213e9 0.195611
\(165\) 1.34867e10 1.41654
\(166\) −1.54853e10 −1.58282
\(167\) 9.98739e9 0.993637 0.496819 0.867854i \(-0.334501\pi\)
0.496819 + 0.867854i \(0.334501\pi\)
\(168\) 6.88043e9 0.666383
\(169\) 5.17449e9 0.487952
\(170\) −7.90342e9 −0.725763
\(171\) −4.40725e9 −0.394171
\(172\) −2.35967e9 −0.205577
\(173\) −3.01680e8 −0.0256058 −0.0128029 0.999918i \(-0.504075\pi\)
−0.0128029 + 0.999918i \(0.504075\pi\)
\(174\) −6.85649e9 −0.567062
\(175\) −1.71857e10 −1.38515
\(176\) −1.04820e10 −0.823448
\(177\) 1.09187e9 0.0836164
\(178\) −2.49906e10 −1.86589
\(179\) −6.66970e9 −0.485588 −0.242794 0.970078i \(-0.578064\pi\)
−0.242794 + 0.970078i \(0.578064\pi\)
\(180\) −1.32817e10 −0.943037
\(181\) −1.63412e10 −1.13170 −0.565849 0.824509i \(-0.691451\pi\)
−0.565849 + 0.824509i \(0.691451\pi\)
\(182\) −5.02609e10 −3.39554
\(183\) −7.24299e9 −0.477406
\(184\) 5.46692e9 0.351611
\(185\) −3.24602e10 −2.03741
\(186\) 2.11060e10 1.29300
\(187\) 9.25482e9 0.553453
\(188\) −1.94707e10 −1.13677
\(189\) 3.34672e10 1.90784
\(190\) −2.73361e10 −1.52176
\(191\) 1.79686e10 0.976930 0.488465 0.872584i \(-0.337557\pi\)
0.488465 + 0.872584i \(0.337557\pi\)
\(192\) −1.98519e10 −1.05426
\(193\) −2.76589e10 −1.43492 −0.717459 0.696601i \(-0.754695\pi\)
−0.717459 + 0.696601i \(0.754695\pi\)
\(194\) 4.82713e10 2.44670
\(195\) −2.24892e10 −1.11383
\(196\) 6.40616e10 3.10060
\(197\) 8.40437e9 0.397564 0.198782 0.980044i \(-0.436301\pi\)
0.198782 + 0.980044i \(0.436301\pi\)
\(198\) 2.70900e10 1.25261
\(199\) −6.68199e9 −0.302042 −0.151021 0.988531i \(-0.548256\pi\)
−0.151021 + 0.988531i \(0.548256\pi\)
\(200\) −9.20183e9 −0.406667
\(201\) −1.85448e10 −0.801383
\(202\) 2.21121e10 0.934434
\(203\) −2.36488e10 −0.977409
\(204\) 8.18240e9 0.330785
\(205\) 4.87125e9 0.192641
\(206\) −7.89422e10 −3.05427
\(207\) 9.17663e9 0.347389
\(208\) 1.74788e10 0.647480
\(209\) 3.20103e10 1.16046
\(210\) 7.16350e10 2.54178
\(211\) 5.14442e10 1.78676 0.893378 0.449305i \(-0.148328\pi\)
0.893378 + 0.449305i \(0.148328\pi\)
\(212\) −4.19909e10 −1.42772
\(213\) −3.63196e10 −1.20902
\(214\) −1.16484e10 −0.379668
\(215\) −6.34312e9 −0.202455
\(216\) 1.79195e10 0.560124
\(217\) 7.27969e10 2.22866
\(218\) 2.59912e10 0.779420
\(219\) −1.88498e10 −0.553742
\(220\) 9.64668e10 2.77636
\(221\) −1.54325e10 −0.435183
\(222\) 5.85352e10 1.61744
\(223\) −3.33216e10 −0.902307 −0.451153 0.892446i \(-0.648987\pi\)
−0.451153 + 0.892446i \(0.648987\pi\)
\(224\) −9.21826e10 −2.44643
\(225\) −1.54460e10 −0.401785
\(226\) −1.86689e10 −0.476027
\(227\) 5.99444e10 1.49841 0.749207 0.662336i \(-0.230435\pi\)
0.749207 + 0.662336i \(0.230435\pi\)
\(228\) 2.83011e10 0.693579
\(229\) 3.62871e10 0.871952 0.435976 0.899958i \(-0.356403\pi\)
0.435976 + 0.899958i \(0.356403\pi\)
\(230\) 5.69184e10 1.34115
\(231\) −8.38839e10 −1.93832
\(232\) −1.26624e10 −0.286958
\(233\) −4.31801e10 −0.959804 −0.479902 0.877322i \(-0.659328\pi\)
−0.479902 + 0.877322i \(0.659328\pi\)
\(234\) −4.51728e10 −0.984931
\(235\) −5.23400e10 −1.11951
\(236\) 7.80986e9 0.163885
\(237\) −3.55073e10 −0.731054
\(238\) 4.91573e10 0.993097
\(239\) 9.43512e9 0.187050 0.0935248 0.995617i \(-0.470187\pi\)
0.0935248 + 0.995617i \(0.470187\pi\)
\(240\) −2.49119e10 −0.484681
\(241\) −1.65553e10 −0.316126 −0.158063 0.987429i \(-0.550525\pi\)
−0.158063 + 0.987429i \(0.550525\pi\)
\(242\) −1.15001e11 −2.15542
\(243\) 4.97783e10 0.915823
\(244\) −5.18072e10 −0.935698
\(245\) 1.72206e11 3.05352
\(246\) −8.78430e9 −0.152932
\(247\) −5.33776e10 −0.912478
\(248\) 3.89780e10 0.654315
\(249\) 4.30959e10 0.710458
\(250\) 2.98417e10 0.483163
\(251\) 5.72536e10 0.910481 0.455240 0.890369i \(-0.349553\pi\)
0.455240 + 0.890369i \(0.349553\pi\)
\(252\) 8.26091e10 1.29040
\(253\) −6.66509e10 −1.02274
\(254\) −9.63111e10 −1.45186
\(255\) 2.19954e10 0.325762
\(256\) −1.85561e8 −0.00270026
\(257\) −1.07273e11 −1.53388 −0.766938 0.641721i \(-0.778221\pi\)
−0.766938 + 0.641721i \(0.778221\pi\)
\(258\) 1.14385e10 0.160724
\(259\) 2.01894e11 2.78788
\(260\) −1.60859e11 −2.18306
\(261\) −2.12547e10 −0.283513
\(262\) −5.63459e10 −0.738766
\(263\) −2.57565e10 −0.331960 −0.165980 0.986129i \(-0.553079\pi\)
−0.165980 + 0.986129i \(0.553079\pi\)
\(264\) −4.49144e10 −0.569072
\(265\) −1.12877e11 −1.40605
\(266\) 1.70024e11 2.08230
\(267\) 6.95493e10 0.837514
\(268\) −1.32646e11 −1.57068
\(269\) 1.90103e10 0.221363 0.110681 0.993856i \(-0.464697\pi\)
0.110681 + 0.993856i \(0.464697\pi\)
\(270\) 1.86567e11 2.13648
\(271\) −6.40499e10 −0.721368 −0.360684 0.932688i \(-0.617457\pi\)
−0.360684 + 0.932688i \(0.617457\pi\)
\(272\) −1.70950e10 −0.189369
\(273\) 1.39877e11 1.52411
\(274\) −1.87599e11 −2.01072
\(275\) 1.12186e11 1.18288
\(276\) −5.89276e10 −0.611262
\(277\) 9.43276e10 0.962675 0.481338 0.876535i \(-0.340151\pi\)
0.481338 + 0.876535i \(0.340151\pi\)
\(278\) −1.94216e10 −0.195022
\(279\) 6.54274e10 0.646459
\(280\) 1.32293e11 1.28626
\(281\) 9.83265e10 0.940789 0.470394 0.882456i \(-0.344112\pi\)
0.470394 + 0.882456i \(0.344112\pi\)
\(282\) 9.43843e10 0.888749
\(283\) 2.70064e10 0.250281 0.125140 0.992139i \(-0.460062\pi\)
0.125140 + 0.992139i \(0.460062\pi\)
\(284\) −2.59785e11 −2.36963
\(285\) 7.60771e10 0.683049
\(286\) 3.28095e11 2.89970
\(287\) −3.02980e10 −0.263600
\(288\) −8.28506e10 −0.709626
\(289\) −1.03494e11 −0.872722
\(290\) −1.31833e11 −1.09455
\(291\) −1.34340e11 −1.09821
\(292\) −1.34827e11 −1.08531
\(293\) −1.02895e11 −0.815628 −0.407814 0.913065i \(-0.633709\pi\)
−0.407814 + 0.913065i \(0.633709\pi\)
\(294\) −3.10538e11 −2.42411
\(295\) 2.09939e10 0.161397
\(296\) 1.08101e11 0.818497
\(297\) −2.18469e11 −1.62924
\(298\) −2.28491e11 −1.67840
\(299\) 1.11141e11 0.804181
\(300\) 9.91859e10 0.706976
\(301\) 3.94526e10 0.277030
\(302\) 2.99760e11 2.07368
\(303\) −6.15384e10 −0.419425
\(304\) −5.91277e10 −0.397063
\(305\) −1.39265e11 −0.921492
\(306\) 4.41809e10 0.288064
\(307\) 9.14285e10 0.587434 0.293717 0.955892i \(-0.405108\pi\)
0.293717 + 0.955892i \(0.405108\pi\)
\(308\) −5.99999e11 −3.79903
\(309\) 2.19698e11 1.37092
\(310\) 4.05816e11 2.49575
\(311\) 9.16357e10 0.555447 0.277724 0.960661i \(-0.410420\pi\)
0.277724 + 0.960661i \(0.410420\pi\)
\(312\) 7.48951e10 0.447464
\(313\) −7.07526e10 −0.416670 −0.208335 0.978057i \(-0.566804\pi\)
−0.208335 + 0.978057i \(0.566804\pi\)
\(314\) 1.64965e11 0.957655
\(315\) 2.22064e11 1.27081
\(316\) −2.53974e11 −1.43284
\(317\) 2.08700e11 1.16079 0.580397 0.814334i \(-0.302897\pi\)
0.580397 + 0.814334i \(0.302897\pi\)
\(318\) 2.03551e11 1.11622
\(319\) 1.54375e11 0.834680
\(320\) −3.81703e11 −2.03493
\(321\) 3.24177e10 0.170416
\(322\) −3.54018e11 −1.83516
\(323\) 5.22055e10 0.266873
\(324\) −5.22509e10 −0.263416
\(325\) −1.87071e11 −0.930102
\(326\) −2.22473e11 −1.09093
\(327\) −7.23341e10 −0.349847
\(328\) −1.62226e10 −0.0773905
\(329\) 3.25541e11 1.53188
\(330\) −4.67622e11 −2.17061
\(331\) 1.72342e11 0.789160 0.394580 0.918862i \(-0.370890\pi\)
0.394580 + 0.918862i \(0.370890\pi\)
\(332\) 3.08253e11 1.39247
\(333\) 1.81456e11 0.808670
\(334\) −3.46291e11 −1.52259
\(335\) −3.56570e11 −1.54683
\(336\) 1.54946e11 0.663212
\(337\) 1.35583e11 0.572624 0.286312 0.958136i \(-0.407571\pi\)
0.286312 + 0.958136i \(0.407571\pi\)
\(338\) −1.79414e11 −0.747707
\(339\) 5.19561e10 0.213667
\(340\) 1.57327e11 0.638482
\(341\) −4.75206e11 −1.90321
\(342\) 1.52812e11 0.604003
\(343\) −6.05402e11 −2.36168
\(344\) 2.11243e10 0.0813334
\(345\) −1.58405e11 −0.601982
\(346\) 1.04601e10 0.0392368
\(347\) −5.06071e10 −0.187382 −0.0936912 0.995601i \(-0.529867\pi\)
−0.0936912 + 0.995601i \(0.529867\pi\)
\(348\) 1.36487e11 0.498866
\(349\) 1.37248e11 0.495212 0.247606 0.968861i \(-0.420356\pi\)
0.247606 + 0.968861i \(0.420356\pi\)
\(350\) 5.95877e11 2.12251
\(351\) 3.64299e11 1.28108
\(352\) 6.01753e11 2.08918
\(353\) −4.03352e11 −1.38261 −0.691303 0.722565i \(-0.742963\pi\)
−0.691303 + 0.722565i \(0.742963\pi\)
\(354\) −3.78582e10 −0.128129
\(355\) −6.98336e11 −2.33366
\(356\) 4.97468e11 1.64150
\(357\) −1.36806e11 −0.445756
\(358\) 2.31257e11 0.744084
\(359\) 5.00707e11 1.59096 0.795479 0.605981i \(-0.207219\pi\)
0.795479 + 0.605981i \(0.207219\pi\)
\(360\) 1.18901e11 0.373099
\(361\) −1.42121e11 −0.440428
\(362\) 5.66596e11 1.73414
\(363\) 3.20050e11 0.967471
\(364\) 1.00050e12 2.98719
\(365\) −3.62434e11 −1.06884
\(366\) 2.51135e11 0.731547
\(367\) 3.22474e11 0.927891 0.463946 0.885864i \(-0.346433\pi\)
0.463946 + 0.885864i \(0.346433\pi\)
\(368\) 1.23114e11 0.349938
\(369\) −2.72308e10 −0.0764613
\(370\) 1.12549e12 3.12199
\(371\) 7.02068e11 1.92396
\(372\) −4.20141e11 −1.13750
\(373\) −4.15302e11 −1.11090 −0.555449 0.831551i \(-0.687454\pi\)
−0.555449 + 0.831551i \(0.687454\pi\)
\(374\) −3.20891e11 −0.848077
\(375\) −8.30501e10 −0.216870
\(376\) 1.74306e11 0.449746
\(377\) −2.57422e11 −0.656312
\(378\) −1.16040e12 −2.92345
\(379\) −1.91892e11 −0.477727 −0.238863 0.971053i \(-0.576775\pi\)
−0.238863 + 0.971053i \(0.576775\pi\)
\(380\) 5.44159e11 1.33875
\(381\) 2.68036e11 0.651675
\(382\) −6.23021e11 −1.49699
\(383\) 3.56315e11 0.846134 0.423067 0.906098i \(-0.360953\pi\)
0.423067 + 0.906098i \(0.360953\pi\)
\(384\) 2.93662e11 0.689219
\(385\) −1.61288e12 −3.74135
\(386\) 9.59013e11 2.19878
\(387\) 3.54587e10 0.0803569
\(388\) −9.60898e11 −2.15246
\(389\) −1.15303e11 −0.255310 −0.127655 0.991819i \(-0.540745\pi\)
−0.127655 + 0.991819i \(0.540745\pi\)
\(390\) 7.79765e11 1.70676
\(391\) −1.08701e11 −0.235200
\(392\) −5.73493e11 −1.22671
\(393\) 1.56812e11 0.331599
\(394\) −2.91403e11 −0.609202
\(395\) −6.82716e11 −1.41108
\(396\) −5.39259e11 −1.10197
\(397\) 5.04803e11 1.01992 0.509958 0.860199i \(-0.329661\pi\)
0.509958 + 0.860199i \(0.329661\pi\)
\(398\) 2.31684e11 0.462830
\(399\) −4.73180e11 −0.934649
\(400\) −2.07223e11 −0.404733
\(401\) 3.05024e10 0.0589093 0.0294547 0.999566i \(-0.490623\pi\)
0.0294547 + 0.999566i \(0.490623\pi\)
\(402\) 6.43001e11 1.22799
\(403\) 7.92412e11 1.49650
\(404\) −4.40168e11 −0.822057
\(405\) −1.40458e11 −0.259416
\(406\) 8.19969e11 1.49772
\(407\) −1.31793e12 −2.38077
\(408\) −7.32506e10 −0.130870
\(409\) −6.41840e11 −1.13415 −0.567076 0.823665i \(-0.691926\pi\)
−0.567076 + 0.823665i \(0.691926\pi\)
\(410\) −1.68900e11 −0.295191
\(411\) 5.22091e11 0.902523
\(412\) 1.57144e12 2.68696
\(413\) −1.30577e11 −0.220847
\(414\) −3.18180e11 −0.532318
\(415\) 8.28626e11 1.37133
\(416\) −1.00343e12 −1.64273
\(417\) 5.40509e10 0.0875368
\(418\) −1.10989e12 −1.77822
\(419\) −8.21896e11 −1.30273 −0.651364 0.758765i \(-0.725803\pi\)
−0.651364 + 0.758765i \(0.725803\pi\)
\(420\) −1.42598e12 −2.23611
\(421\) 1.03580e12 1.60696 0.803479 0.595333i \(-0.202980\pi\)
0.803479 + 0.595333i \(0.202980\pi\)
\(422\) −1.78372e12 −2.73791
\(423\) 2.92586e11 0.444346
\(424\) 3.75912e11 0.564858
\(425\) 1.82963e11 0.272028
\(426\) 1.25930e12 1.85263
\(427\) 8.66191e11 1.26092
\(428\) 2.31875e11 0.334009
\(429\) −9.13096e11 −1.30154
\(430\) 2.19934e11 0.310230
\(431\) −7.76516e11 −1.08393 −0.541967 0.840400i \(-0.682320\pi\)
−0.541967 + 0.840400i \(0.682320\pi\)
\(432\) 4.03543e11 0.557459
\(433\) −2.89671e11 −0.396013 −0.198007 0.980201i \(-0.563447\pi\)
−0.198007 + 0.980201i \(0.563447\pi\)
\(434\) −2.52407e12 −3.41506
\(435\) 3.66895e11 0.491292
\(436\) −5.17386e11 −0.685686
\(437\) −3.75971e11 −0.493160
\(438\) 6.53575e11 0.848519
\(439\) 6.84069e11 0.879042 0.439521 0.898232i \(-0.355148\pi\)
0.439521 + 0.898232i \(0.355148\pi\)
\(440\) −8.63591e11 −1.09843
\(441\) −9.62650e11 −1.21198
\(442\) 5.35089e11 0.666846
\(443\) −6.61169e11 −0.815635 −0.407817 0.913063i \(-0.633710\pi\)
−0.407817 + 0.913063i \(0.633710\pi\)
\(444\) −1.16521e12 −1.42293
\(445\) 1.33726e12 1.61657
\(446\) 1.15535e12 1.38264
\(447\) 6.35894e11 0.753358
\(448\) 2.37410e12 2.78450
\(449\) −3.55659e11 −0.412976 −0.206488 0.978449i \(-0.566203\pi\)
−0.206488 + 0.978449i \(0.566203\pi\)
\(450\) 5.35555e11 0.615669
\(451\) 1.97780e11 0.225107
\(452\) 3.71628e11 0.418779
\(453\) −8.34238e11 −0.930783
\(454\) −2.07844e12 −2.29608
\(455\) 2.68949e12 2.94184
\(456\) −2.53357e11 −0.274405
\(457\) −6.19771e11 −0.664674 −0.332337 0.943161i \(-0.607837\pi\)
−0.332337 + 0.943161i \(0.607837\pi\)
\(458\) −1.25818e12 −1.33612
\(459\) −3.56299e11 −0.374678
\(460\) −1.13303e12 −1.17986
\(461\) −1.38169e10 −0.0142481 −0.00712406 0.999975i \(-0.502268\pi\)
−0.00712406 + 0.999975i \(0.502268\pi\)
\(462\) 2.90849e12 2.97015
\(463\) −6.83393e11 −0.691124 −0.345562 0.938396i \(-0.612312\pi\)
−0.345562 + 0.938396i \(0.612312\pi\)
\(464\) −2.85154e11 −0.285593
\(465\) −1.12940e12 −1.12023
\(466\) 1.49718e12 1.47074
\(467\) −1.72900e12 −1.68217 −0.841085 0.540902i \(-0.818083\pi\)
−0.841085 + 0.540902i \(0.818083\pi\)
\(468\) 8.99220e11 0.866482
\(469\) 2.21778e12 2.11661
\(470\) 1.81478e12 1.71547
\(471\) −4.59102e11 −0.429848
\(472\) −6.99155e10 −0.0648387
\(473\) −2.57540e11 −0.236575
\(474\) 1.23114e12 1.12022
\(475\) 6.32828e11 0.570380
\(476\) −9.78535e11 −0.873665
\(477\) 6.30995e11 0.558076
\(478\) −3.27142e11 −0.286623
\(479\) 1.45121e12 1.25956 0.629782 0.776772i \(-0.283144\pi\)
0.629782 + 0.776772i \(0.283144\pi\)
\(480\) 1.43015e12 1.22969
\(481\) 2.19766e12 1.87201
\(482\) 5.74019e11 0.484411
\(483\) 9.85240e11 0.823721
\(484\) 2.28923e12 1.89621
\(485\) −2.58302e12 −2.11978
\(486\) −1.72595e12 −1.40335
\(487\) −1.71558e12 −1.38207 −0.691036 0.722820i \(-0.742845\pi\)
−0.691036 + 0.722820i \(0.742845\pi\)
\(488\) 4.63789e11 0.370195
\(489\) 6.19148e11 0.489671
\(490\) −5.97087e12 −4.67903
\(491\) −3.70516e11 −0.287701 −0.143850 0.989599i \(-0.545948\pi\)
−0.143850 + 0.989599i \(0.545948\pi\)
\(492\) 1.74862e11 0.134540
\(493\) 2.51770e11 0.191952
\(494\) 1.85075e12 1.39822
\(495\) −1.44960e12 −1.08524
\(496\) 8.77775e11 0.651202
\(497\) 4.34347e12 3.19325
\(498\) −1.49426e12 −1.08866
\(499\) −2.23176e12 −1.61137 −0.805684 0.592345i \(-0.798202\pi\)
−0.805684 + 0.592345i \(0.798202\pi\)
\(500\) −5.94035e11 −0.425057
\(501\) 9.63736e11 0.683421
\(502\) −1.98514e12 −1.39516
\(503\) 1.98853e12 1.38509 0.692544 0.721376i \(-0.256490\pi\)
0.692544 + 0.721376i \(0.256490\pi\)
\(504\) −7.39534e11 −0.510529
\(505\) −1.18323e12 −0.809577
\(506\) 2.31097e12 1.56718
\(507\) 4.99314e11 0.335612
\(508\) 1.91719e12 1.27726
\(509\) 1.54768e11 0.102200 0.0511000 0.998694i \(-0.483727\pi\)
0.0511000 + 0.998694i \(0.483727\pi\)
\(510\) −7.62643e11 −0.499178
\(511\) 2.25425e12 1.46254
\(512\) −1.55172e12 −0.997930
\(513\) −1.23236e12 −0.785613
\(514\) 3.71945e12 2.35042
\(515\) 4.22424e12 2.64616
\(516\) −2.27697e11 −0.141395
\(517\) −2.12508e12 −1.30818
\(518\) −7.00023e12 −4.27197
\(519\) −2.91107e10 −0.0176116
\(520\) 1.44005e12 0.863697
\(521\) 5.11185e11 0.303955 0.151977 0.988384i \(-0.451436\pi\)
0.151977 + 0.988384i \(0.451436\pi\)
\(522\) 7.36961e11 0.434438
\(523\) −2.71654e12 −1.58766 −0.793830 0.608139i \(-0.791916\pi\)
−0.793830 + 0.608139i \(0.791916\pi\)
\(524\) 1.12163e12 0.649921
\(525\) −1.65834e12 −0.952701
\(526\) 8.93050e11 0.508675
\(527\) −7.75012e11 −0.437684
\(528\) −1.01146e12 −0.566365
\(529\) −1.01832e12 −0.565371
\(530\) 3.91377e12 2.15454
\(531\) −1.17358e11 −0.0640602
\(532\) −3.38453e12 −1.83188
\(533\) −3.29801e11 −0.177002
\(534\) −2.41147e12 −1.28335
\(535\) 6.23312e11 0.328938
\(536\) 1.18748e12 0.621417
\(537\) −6.43595e11 −0.333986
\(538\) −6.59141e11 −0.339202
\(539\) 6.99183e12 3.56814
\(540\) −3.71385e12 −1.87954
\(541\) 6.13789e11 0.308057 0.154029 0.988066i \(-0.450775\pi\)
0.154029 + 0.988066i \(0.450775\pi\)
\(542\) 2.22079e12 1.10538
\(543\) −1.57685e12 −0.778378
\(544\) 9.81396e11 0.480451
\(545\) −1.39080e12 −0.675276
\(546\) −4.84994e12 −2.33544
\(547\) −1.23334e12 −0.589033 −0.294517 0.955646i \(-0.595159\pi\)
−0.294517 + 0.955646i \(0.595159\pi\)
\(548\) 3.73438e12 1.76891
\(549\) 7.78503e11 0.365750
\(550\) −3.88979e12 −1.81257
\(551\) 8.70815e11 0.402480
\(552\) 5.27532e11 0.241837
\(553\) 4.24632e12 1.93085
\(554\) −3.27060e12 −1.47514
\(555\) −3.13225e12 −1.40132
\(556\) 3.86611e11 0.171569
\(557\) −2.90504e11 −0.127880 −0.0639401 0.997954i \(-0.520367\pi\)
−0.0639401 + 0.997954i \(0.520367\pi\)
\(558\) −2.26855e12 −0.990592
\(559\) 4.29451e11 0.186020
\(560\) 2.97922e12 1.28014
\(561\) 8.93047e11 0.380664
\(562\) −3.40926e12 −1.44161
\(563\) 3.39497e12 1.42412 0.712062 0.702117i \(-0.247762\pi\)
0.712062 + 0.702117i \(0.247762\pi\)
\(564\) −1.87883e12 −0.781867
\(565\) 9.98986e11 0.412421
\(566\) −9.36387e11 −0.383514
\(567\) 8.73610e11 0.354972
\(568\) 2.32565e12 0.937511
\(569\) −3.90600e11 −0.156217 −0.0781084 0.996945i \(-0.524888\pi\)
−0.0781084 + 0.996945i \(0.524888\pi\)
\(570\) −2.63781e12 −1.04666
\(571\) −2.37747e12 −0.935950 −0.467975 0.883742i \(-0.655016\pi\)
−0.467975 + 0.883742i \(0.655016\pi\)
\(572\) −6.53113e12 −2.55098
\(573\) 1.73388e12 0.671929
\(574\) 1.05052e12 0.403924
\(575\) −1.31765e12 −0.502685
\(576\) 2.13376e12 0.807689
\(577\) −3.52891e12 −1.32541 −0.662704 0.748881i \(-0.730591\pi\)
−0.662704 + 0.748881i \(0.730591\pi\)
\(578\) 3.58844e12 1.33730
\(579\) −2.66895e12 −0.986933
\(580\) 2.62430e12 0.962914
\(581\) −5.15384e12 −1.87646
\(582\) 4.65795e12 1.68283
\(583\) −4.58299e12 −1.64301
\(584\) 1.20700e12 0.429388
\(585\) 2.41722e12 0.853327
\(586\) 3.56768e12 1.24982
\(587\) 2.71792e12 0.944854 0.472427 0.881370i \(-0.343378\pi\)
0.472427 + 0.881370i \(0.343378\pi\)
\(588\) 6.18164e12 2.13258
\(589\) −2.68059e12 −0.917723
\(590\) −7.27919e11 −0.247314
\(591\) 8.10982e11 0.273443
\(592\) 2.43441e12 0.814603
\(593\) −3.41879e12 −1.13534 −0.567671 0.823255i \(-0.692155\pi\)
−0.567671 + 0.823255i \(0.692155\pi\)
\(594\) 7.57492e12 2.49654
\(595\) −2.63043e12 −0.860401
\(596\) 4.54838e12 1.47655
\(597\) −6.44781e11 −0.207744
\(598\) −3.85357e12 −1.23228
\(599\) 4.17007e12 1.32350 0.661749 0.749726i \(-0.269815\pi\)
0.661749 + 0.749726i \(0.269815\pi\)
\(600\) −8.87933e11 −0.279705
\(601\) −1.37919e12 −0.431210 −0.215605 0.976481i \(-0.569172\pi\)
−0.215605 + 0.976481i \(0.569172\pi\)
\(602\) −1.36793e12 −0.424503
\(603\) 1.99326e12 0.613956
\(604\) −5.96708e12 −1.82430
\(605\) 6.15376e12 1.86742
\(606\) 2.13371e12 0.642701
\(607\) −5.87777e12 −1.75737 −0.878685 0.477402i \(-0.841578\pi\)
−0.878685 + 0.477402i \(0.841578\pi\)
\(608\) 3.39443e12 1.00740
\(609\) −2.28199e12 −0.672259
\(610\) 4.82870e12 1.41204
\(611\) 3.54360e12 1.02863
\(612\) −8.79475e11 −0.253421
\(613\) 1.36479e12 0.390387 0.195193 0.980765i \(-0.437467\pi\)
0.195193 + 0.980765i \(0.437467\pi\)
\(614\) −3.17008e12 −0.900146
\(615\) 4.70053e11 0.132498
\(616\) 5.37132e12 1.50303
\(617\) −1.18056e12 −0.327949 −0.163974 0.986465i \(-0.552431\pi\)
−0.163974 + 0.986465i \(0.552431\pi\)
\(618\) −7.61755e12 −2.10071
\(619\) 1.80437e12 0.493989 0.246994 0.969017i \(-0.420557\pi\)
0.246994 + 0.969017i \(0.420557\pi\)
\(620\) −8.07826e12 −2.19561
\(621\) 2.56598e12 0.692373
\(622\) −3.17727e12 −0.851132
\(623\) −8.31742e12 −2.21204
\(624\) 1.68662e12 0.445335
\(625\) −4.50553e12 −1.18110
\(626\) 2.45319e12 0.638479
\(627\) 3.08885e12 0.798164
\(628\) −3.28383e12 −0.842486
\(629\) −2.14941e12 −0.547509
\(630\) −7.69960e12 −1.94731
\(631\) 7.87738e12 1.97811 0.989053 0.147563i \(-0.0471428\pi\)
0.989053 + 0.147563i \(0.0471428\pi\)
\(632\) 2.27363e12 0.566882
\(633\) 4.96412e12 1.22893
\(634\) −7.23621e12 −1.77873
\(635\) 5.15366e12 1.25787
\(636\) −4.05193e12 −0.981984
\(637\) −1.16589e13 −2.80564
\(638\) −5.35263e12 −1.27901
\(639\) 3.90377e12 0.926254
\(640\) 5.64639e12 1.33033
\(641\) −7.06685e12 −1.65335 −0.826674 0.562681i \(-0.809770\pi\)
−0.826674 + 0.562681i \(0.809770\pi\)
\(642\) −1.12401e12 −0.261135
\(643\) 1.18635e12 0.273694 0.136847 0.990592i \(-0.456303\pi\)
0.136847 + 0.990592i \(0.456303\pi\)
\(644\) 7.04716e12 1.61446
\(645\) −6.12081e11 −0.139248
\(646\) −1.81011e12 −0.408940
\(647\) −1.63713e12 −0.367294 −0.183647 0.982992i \(-0.558790\pi\)
−0.183647 + 0.982992i \(0.558790\pi\)
\(648\) 4.67761e11 0.104217
\(649\) 8.52386e11 0.188597
\(650\) 6.48627e12 1.42523
\(651\) 7.02455e12 1.53287
\(652\) 4.42860e12 0.959737
\(653\) 2.02628e12 0.436105 0.218052 0.975937i \(-0.430030\pi\)
0.218052 + 0.975937i \(0.430030\pi\)
\(654\) 2.50803e12 0.536083
\(655\) 3.01510e12 0.640054
\(656\) −3.65329e11 −0.0770223
\(657\) 2.02604e12 0.424233
\(658\) −1.12874e13 −2.34736
\(659\) 2.90498e12 0.600011 0.300005 0.953937i \(-0.403011\pi\)
0.300005 + 0.953937i \(0.403011\pi\)
\(660\) 9.30859e12 1.90957
\(661\) 3.38252e12 0.689181 0.344591 0.938753i \(-0.388018\pi\)
0.344591 + 0.938753i \(0.388018\pi\)
\(662\) −5.97558e12 −1.20926
\(663\) −1.48916e12 −0.299317
\(664\) −2.75955e12 −0.550911
\(665\) −9.09807e12 −1.80406
\(666\) −6.29158e12 −1.23915
\(667\) −1.81318e12 −0.354712
\(668\) 6.89334e12 1.33948
\(669\) −3.21538e12 −0.620604
\(670\) 1.23633e13 2.37027
\(671\) −5.65435e12 −1.07679
\(672\) −8.89518e12 −1.68265
\(673\) −6.52234e12 −1.22556 −0.612781 0.790252i \(-0.709949\pi\)
−0.612781 + 0.790252i \(0.709949\pi\)
\(674\) −4.70103e12 −0.877453
\(675\) −4.31901e12 −0.800787
\(676\) 3.57146e12 0.657787
\(677\) 7.85574e11 0.143727 0.0718635 0.997414i \(-0.477105\pi\)
0.0718635 + 0.997414i \(0.477105\pi\)
\(678\) −1.80146e12 −0.327410
\(679\) 1.60658e13 2.90059
\(680\) −1.40843e12 −0.252606
\(681\) 5.78435e12 1.03061
\(682\) 1.64767e13 2.91636
\(683\) 1.01729e13 1.78876 0.894380 0.447308i \(-0.147617\pi\)
0.894380 + 0.447308i \(0.147617\pi\)
\(684\) −3.04190e12 −0.531365
\(685\) 1.00385e13 1.74205
\(686\) 2.09910e13 3.61889
\(687\) 3.50153e12 0.599726
\(688\) 4.75714e11 0.0809464
\(689\) 7.64218e12 1.29191
\(690\) 5.49236e12 0.922439
\(691\) 5.01808e12 0.837310 0.418655 0.908145i \(-0.362502\pi\)
0.418655 + 0.908145i \(0.362502\pi\)
\(692\) −2.08221e11 −0.0345181
\(693\) 9.01615e12 1.48498
\(694\) 1.75469e12 0.287133
\(695\) 1.03926e12 0.168964
\(696\) −1.22186e12 −0.197369
\(697\) 3.22559e11 0.0517680
\(698\) −4.75877e12 −0.758832
\(699\) −4.16668e12 −0.660150
\(700\) −1.18617e13 −1.86726
\(701\) −5.01880e12 −0.784999 −0.392499 0.919752i \(-0.628389\pi\)
−0.392499 + 0.919752i \(0.628389\pi\)
\(702\) −1.26313e13 −1.96304
\(703\) −7.43432e12 −1.14800
\(704\) −1.54977e13 −2.37788
\(705\) −5.05056e12 −0.769996
\(706\) 1.39854e13 2.11862
\(707\) 7.35939e12 1.10778
\(708\) 7.53614e11 0.112720
\(709\) −7.51726e12 −1.11725 −0.558627 0.829419i \(-0.688672\pi\)
−0.558627 + 0.829419i \(0.688672\pi\)
\(710\) 2.42133e13 3.57595
\(711\) 3.81645e12 0.560075
\(712\) −4.45343e12 −0.649434
\(713\) 5.58144e12 0.808804
\(714\) 4.74344e12 0.683049
\(715\) −1.75566e13 −2.51225
\(716\) −4.60346e12 −0.654600
\(717\) 9.10444e11 0.128652
\(718\) −1.73609e13 −2.43788
\(719\) −8.63730e11 −0.120531 −0.0602654 0.998182i \(-0.519195\pi\)
−0.0602654 + 0.998182i \(0.519195\pi\)
\(720\) 2.67762e12 0.371324
\(721\) −2.62737e13 −3.62087
\(722\) 4.92773e12 0.674884
\(723\) −1.59751e12 −0.217430
\(724\) −1.12788e13 −1.52559
\(725\) 3.05192e12 0.410253
\(726\) −1.10970e13 −1.48249
\(727\) 1.36082e13 1.80674 0.903369 0.428865i \(-0.141086\pi\)
0.903369 + 0.428865i \(0.141086\pi\)
\(728\) −8.95672e12 −1.18184
\(729\) 6.29344e12 0.825305
\(730\) 1.25666e13 1.63782
\(731\) −4.20021e11 −0.0544055
\(732\) −4.99915e12 −0.643570
\(733\) −3.19686e12 −0.409031 −0.204515 0.978863i \(-0.565562\pi\)
−0.204515 + 0.978863i \(0.565562\pi\)
\(734\) −1.11811e13 −1.42184
\(735\) 1.66171e13 2.10020
\(736\) −7.06777e12 −0.887834
\(737\) −1.44773e13 −1.80752
\(738\) 9.44169e11 0.117164
\(739\) −8.36687e12 −1.03196 −0.515980 0.856601i \(-0.672572\pi\)
−0.515980 + 0.856601i \(0.672572\pi\)
\(740\) −2.24042e13 −2.74654
\(741\) −5.15068e12 −0.627600
\(742\) −2.43427e13 −2.94816
\(743\) −1.08223e13 −1.30278 −0.651389 0.758744i \(-0.725814\pi\)
−0.651389 + 0.758744i \(0.725814\pi\)
\(744\) 3.76119e12 0.450036
\(745\) 1.22267e13 1.45413
\(746\) 1.43997e13 1.70227
\(747\) −4.63210e12 −0.544296
\(748\) 6.38772e12 0.746086
\(749\) −3.87684e12 −0.450101
\(750\) 2.87958e12 0.332318
\(751\) −4.70796e12 −0.540074 −0.270037 0.962850i \(-0.587036\pi\)
−0.270037 + 0.962850i \(0.587036\pi\)
\(752\) 3.92533e12 0.447607
\(753\) 5.52470e12 0.626226
\(754\) 8.92556e12 1.00569
\(755\) −1.60403e13 −1.79660
\(756\) 2.30992e13 2.57187
\(757\) 2.12827e12 0.235557 0.117778 0.993040i \(-0.462423\pi\)
0.117778 + 0.993040i \(0.462423\pi\)
\(758\) 6.65342e12 0.732038
\(759\) −6.43149e12 −0.703435
\(760\) −4.87143e12 −0.529657
\(761\) −3.42283e12 −0.369960 −0.184980 0.982742i \(-0.559222\pi\)
−0.184980 + 0.982742i \(0.559222\pi\)
\(762\) −9.29357e12 −0.998585
\(763\) 8.65044e12 0.924013
\(764\) 1.24020e13 1.31696
\(765\) −2.36415e12 −0.249573
\(766\) −1.23544e13 −1.29656
\(767\) −1.42136e12 −0.148295
\(768\) −1.79057e10 −0.00185723
\(769\) 1.30449e13 1.34515 0.672576 0.740028i \(-0.265188\pi\)
0.672576 + 0.740028i \(0.265188\pi\)
\(770\) 5.59230e13 5.73300
\(771\) −1.03513e13 −1.05500
\(772\) −1.90903e13 −1.93435
\(773\) −9.97333e12 −1.00469 −0.502346 0.864667i \(-0.667529\pi\)
−0.502346 + 0.864667i \(0.667529\pi\)
\(774\) −1.22945e12 −0.123134
\(775\) −9.39458e12 −0.935448
\(776\) 8.60216e12 0.851589
\(777\) 1.94818e13 1.91750
\(778\) 3.99788e12 0.391220
\(779\) 1.11566e12 0.108546
\(780\) −1.55222e13 −1.50150
\(781\) −2.83535e13 −2.72695
\(782\) 3.76896e12 0.360405
\(783\) −5.94326e12 −0.565063
\(784\) −1.29149e13 −1.22087
\(785\) −8.82738e12 −0.829695
\(786\) −5.43712e12 −0.508121
\(787\) 1.67432e13 1.55580 0.777899 0.628389i \(-0.216286\pi\)
0.777899 + 0.628389i \(0.216286\pi\)
\(788\) 5.80073e12 0.535939
\(789\) −2.48538e12 −0.228321
\(790\) 2.36717e13 2.16226
\(791\) −6.21344e12 −0.564336
\(792\) 4.82756e12 0.435978
\(793\) 9.42869e12 0.846686
\(794\) −1.75029e13 −1.56286
\(795\) −1.08921e13 −0.967076
\(796\) −4.61194e12 −0.407169
\(797\) 4.49012e12 0.394180 0.197090 0.980385i \(-0.436851\pi\)
0.197090 + 0.980385i \(0.436851\pi\)
\(798\) 1.64065e13 1.43220
\(799\) −3.46579e12 −0.300844
\(800\) 1.18963e13 1.02685
\(801\) −7.47542e12 −0.641637
\(802\) −1.05760e12 −0.0902689
\(803\) −1.47154e13 −1.24897
\(804\) −1.27997e13 −1.08031
\(805\) 1.89437e13 1.58995
\(806\) −2.74751e13 −2.29315
\(807\) 1.83441e12 0.152252
\(808\) 3.94047e12 0.325235
\(809\) 1.95010e13 1.60062 0.800311 0.599585i \(-0.204668\pi\)
0.800311 + 0.599585i \(0.204668\pi\)
\(810\) 4.87006e12 0.397513
\(811\) 1.69412e13 1.37515 0.687575 0.726114i \(-0.258675\pi\)
0.687575 + 0.726114i \(0.258675\pi\)
\(812\) −1.63225e13 −1.31760
\(813\) −6.18051e12 −0.496155
\(814\) 4.56964e13 3.64815
\(815\) 1.19047e13 0.945166
\(816\) −1.64959e12 −0.130247
\(817\) −1.45276e12 −0.114076
\(818\) 2.22544e13 1.73790
\(819\) −1.50345e13 −1.16765
\(820\) 3.36216e12 0.259691
\(821\) 1.04174e13 0.800230 0.400115 0.916465i \(-0.368970\pi\)
0.400115 + 0.916465i \(0.368970\pi\)
\(822\) −1.81024e13 −1.38297
\(823\) 1.76032e12 0.133750 0.0668748 0.997761i \(-0.478697\pi\)
0.0668748 + 0.997761i \(0.478697\pi\)
\(824\) −1.40679e13 −1.06305
\(825\) 1.08254e13 0.813580
\(826\) 4.52747e12 0.338412
\(827\) −4.72692e12 −0.351401 −0.175701 0.984444i \(-0.556219\pi\)
−0.175701 + 0.984444i \(0.556219\pi\)
\(828\) 6.33375e12 0.468300
\(829\) −1.20464e13 −0.885854 −0.442927 0.896558i \(-0.646060\pi\)
−0.442927 + 0.896558i \(0.646060\pi\)
\(830\) −2.87308e13 −2.10134
\(831\) 9.10217e12 0.662125
\(832\) 2.58426e13 1.86974
\(833\) 1.14029e13 0.820568
\(834\) −1.87410e12 −0.134136
\(835\) 1.85302e13 1.31914
\(836\) 2.20937e13 1.56437
\(837\) 1.82949e13 1.28844
\(838\) 2.84975e13 1.99622
\(839\) −1.76324e13 −1.22852 −0.614260 0.789103i \(-0.710546\pi\)
−0.614260 + 0.789103i \(0.710546\pi\)
\(840\) 1.27657e13 0.884683
\(841\) −1.03075e13 −0.710511
\(842\) −3.59139e13 −2.46240
\(843\) 9.48804e12 0.647072
\(844\) 3.55070e13 2.40865
\(845\) 9.60056e12 0.647800
\(846\) −1.01448e13 −0.680888
\(847\) −3.82748e13 −2.55528
\(848\) 8.46544e12 0.562171
\(849\) 2.60599e12 0.172142
\(850\) −6.34384e12 −0.416838
\(851\) 1.54795e13 1.01175
\(852\) −2.50680e13 −1.62983
\(853\) 2.53288e12 0.163811 0.0819056 0.996640i \(-0.473899\pi\)
0.0819056 + 0.996640i \(0.473899\pi\)
\(854\) −3.00333e13 −1.93216
\(855\) −8.17705e12 −0.523298
\(856\) −2.07580e12 −0.132146
\(857\) 1.48511e13 0.940471 0.470235 0.882541i \(-0.344169\pi\)
0.470235 + 0.882541i \(0.344169\pi\)
\(858\) 3.16596e13 1.99440
\(859\) −1.42907e13 −0.895538 −0.447769 0.894149i \(-0.647781\pi\)
−0.447769 + 0.894149i \(0.647781\pi\)
\(860\) −4.37805e12 −0.272921
\(861\) −2.92361e12 −0.181303
\(862\) 2.69240e13 1.66095
\(863\) 1.21047e13 0.742860 0.371430 0.928461i \(-0.378868\pi\)
0.371430 + 0.928461i \(0.378868\pi\)
\(864\) −2.31668e13 −1.41434
\(865\) −5.59725e11 −0.0339940
\(866\) 1.00437e13 0.606826
\(867\) −9.98670e12 −0.600256
\(868\) 5.02447e13 3.00436
\(869\) −2.77193e13 −1.64890
\(870\) −1.27213e13 −0.752825
\(871\) 2.41410e13 1.42126
\(872\) 4.63175e12 0.271282
\(873\) 1.44394e13 0.841364
\(874\) 1.30360e13 0.755686
\(875\) 9.93198e12 0.572795
\(876\) −1.30102e13 −0.746475
\(877\) 3.78272e11 0.0215927 0.0107963 0.999942i \(-0.496563\pi\)
0.0107963 + 0.999942i \(0.496563\pi\)
\(878\) −2.37186e13 −1.34699
\(879\) −9.92892e12 −0.560986
\(880\) −1.94479e13 −1.09320
\(881\) −2.34147e13 −1.30948 −0.654738 0.755856i \(-0.727221\pi\)
−0.654738 + 0.755856i \(0.727221\pi\)
\(882\) 3.33778e13 1.85716
\(883\) −5.94939e12 −0.329344 −0.164672 0.986348i \(-0.552657\pi\)
−0.164672 + 0.986348i \(0.552657\pi\)
\(884\) −1.06516e13 −0.586650
\(885\) 2.02582e12 0.111008
\(886\) 2.29246e13 1.24983
\(887\) −1.93804e12 −0.105125 −0.0525626 0.998618i \(-0.516739\pi\)
−0.0525626 + 0.998618i \(0.516739\pi\)
\(888\) 1.04312e13 0.562960
\(889\) −3.20545e13 −1.72120
\(890\) −4.63666e13 −2.47714
\(891\) −5.70279e12 −0.303136
\(892\) −2.29987e13 −1.21636
\(893\) −1.19874e13 −0.630801
\(894\) −2.20483e13 −1.15440
\(895\) −1.23747e13 −0.644661
\(896\) −3.51191e13 −1.82036
\(897\) 1.07246e13 0.553114
\(898\) 1.23317e13 0.632819
\(899\) −1.29276e13 −0.660084
\(900\) −1.06609e13 −0.541628
\(901\) −7.47437e12 −0.377845
\(902\) −6.85760e12 −0.344939
\(903\) 3.80699e12 0.190540
\(904\) −3.32689e12 −0.165684
\(905\) −3.03189e13 −1.50243
\(906\) 2.89254e13 1.42627
\(907\) 2.29706e13 1.12704 0.563521 0.826102i \(-0.309446\pi\)
0.563521 + 0.826102i \(0.309446\pi\)
\(908\) 4.13739e13 2.01995
\(909\) 6.61438e12 0.321330
\(910\) −9.32522e13 −4.50788
\(911\) −7.65595e11 −0.0368270 −0.0184135 0.999830i \(-0.505862\pi\)
−0.0184135 + 0.999830i \(0.505862\pi\)
\(912\) −5.70554e12 −0.273099
\(913\) 3.36435e13 1.60244
\(914\) 2.14892e13 1.01850
\(915\) −1.34384e13 −0.633799
\(916\) 2.50455e13 1.17544
\(917\) −1.87532e13 −0.875816
\(918\) 1.23539e13 0.574132
\(919\) −2.41913e13 −1.11876 −0.559382 0.828910i \(-0.688962\pi\)
−0.559382 + 0.828910i \(0.688962\pi\)
\(920\) 1.01431e13 0.466795
\(921\) 8.82241e12 0.404035
\(922\) 4.79072e11 0.0218329
\(923\) 4.72798e13 2.14421
\(924\) −5.78970e13 −2.61296
\(925\) −2.60548e13 −1.17017
\(926\) 2.36952e13 1.05904
\(927\) −2.36139e13 −1.05029
\(928\) 1.63702e13 0.724583
\(929\) 2.91719e13 1.28497 0.642486 0.766298i \(-0.277903\pi\)
0.642486 + 0.766298i \(0.277903\pi\)
\(930\) 3.91593e13 1.71657
\(931\) 3.94402e13 1.72054
\(932\) −2.98031e13 −1.29387
\(933\) 8.84241e12 0.382035
\(934\) 5.99494e13 2.57765
\(935\) 1.71711e13 0.734759
\(936\) −8.05001e12 −0.342811
\(937\) −2.58239e13 −1.09444 −0.547222 0.836988i \(-0.684315\pi\)
−0.547222 + 0.836988i \(0.684315\pi\)
\(938\) −7.68966e13 −3.24335
\(939\) −6.82729e12 −0.286585
\(940\) −3.61253e13 −1.50916
\(941\) −2.51385e12 −0.104517 −0.0522584 0.998634i \(-0.516642\pi\)
−0.0522584 + 0.998634i \(0.516642\pi\)
\(942\) 1.59184e13 0.658672
\(943\) −2.32299e12 −0.0956630
\(944\) −1.57448e12 −0.0645302
\(945\) 6.20938e13 2.53282
\(946\) 8.92964e12 0.362513
\(947\) 2.13070e13 0.860888 0.430444 0.902617i \(-0.358357\pi\)
0.430444 + 0.902617i \(0.358357\pi\)
\(948\) −2.45073e13 −0.985502
\(949\) 2.45380e13 0.982068
\(950\) −2.19419e13 −0.874014
\(951\) 2.01385e13 0.798391
\(952\) 8.76005e12 0.345653
\(953\) 2.12669e13 0.835193 0.417596 0.908633i \(-0.362873\pi\)
0.417596 + 0.908633i \(0.362873\pi\)
\(954\) −2.18784e13 −0.855161
\(955\) 3.33382e13 1.29696
\(956\) 6.51216e12 0.252153
\(957\) 1.48965e13 0.574090
\(958\) −5.03175e13 −1.93008
\(959\) −6.24370e13 −2.38374
\(960\) −3.68325e13 −1.39962
\(961\) 1.33548e13 0.505106
\(962\) −7.61992e13 −2.86855
\(963\) −3.48438e12 −0.130559
\(964\) −1.14265e13 −0.426155
\(965\) −5.13173e13 −1.90498
\(966\) −3.41611e13 −1.26222
\(967\) 2.66718e13 0.980918 0.490459 0.871464i \(-0.336829\pi\)
0.490459 + 0.871464i \(0.336829\pi\)
\(968\) −2.04937e13 −0.750206
\(969\) 5.03758e12 0.183555
\(970\) 8.95607e13 3.24821
\(971\) −2.90609e13 −1.04911 −0.524557 0.851375i \(-0.675769\pi\)
−0.524557 + 0.851375i \(0.675769\pi\)
\(972\) 3.43572e13 1.23458
\(973\) −6.46396e12 −0.231201
\(974\) 5.94840e13 2.11780
\(975\) −1.80514e13 −0.639722
\(976\) 1.04444e13 0.368434
\(977\) −4.44178e13 −1.55967 −0.779834 0.625987i \(-0.784696\pi\)
−0.779834 + 0.625987i \(0.784696\pi\)
\(978\) −2.14676e13 −0.750341
\(979\) 5.42948e13 1.88902
\(980\) 1.18857e14 4.11632
\(981\) 7.77473e12 0.268025
\(982\) 1.28468e13 0.440854
\(983\) 4.71077e13 1.60917 0.804583 0.593840i \(-0.202389\pi\)
0.804583 + 0.593840i \(0.202389\pi\)
\(984\) −1.56540e12 −0.0532290
\(985\) 1.55932e13 0.527802
\(986\) −8.72958e12 −0.294135
\(987\) 3.14132e13 1.05362
\(988\) −3.68414e13 −1.23007
\(989\) 3.02488e12 0.100537
\(990\) 5.02617e13 1.66295
\(991\) −3.94920e13 −1.30070 −0.650350 0.759634i \(-0.725378\pi\)
−0.650350 + 0.759634i \(0.725378\pi\)
\(992\) −5.03917e13 −1.65218
\(993\) 1.66302e13 0.542782
\(994\) −1.50600e14 −4.89314
\(995\) −1.23975e13 −0.400988
\(996\) 2.97450e13 0.957737
\(997\) 1.48982e13 0.477537 0.238768 0.971077i \(-0.423256\pi\)
0.238768 + 0.971077i \(0.423256\pi\)
\(998\) 7.73814e13 2.46916
\(999\) 5.07387e13 1.61174
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 43.10.a.a.1.3 15
3.2 odd 2 387.10.a.c.1.13 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.3 15 1.1 even 1 trivial
387.10.a.c.1.13 15 3.2 odd 2