Properties

Label 8034.2.a.z.1.10
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.34598\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.34598 q^{5} +1.00000 q^{6} -3.24510 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.34598 q^{5} +1.00000 q^{6} -3.24510 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.34598 q^{10} -3.77351 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.24510 q^{14} -1.34598 q^{15} +1.00000 q^{16} -7.24113 q^{17} -1.00000 q^{18} +0.661018 q^{19} +1.34598 q^{20} +3.24510 q^{21} +3.77351 q^{22} -3.88709 q^{23} +1.00000 q^{24} -3.18835 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.24510 q^{28} -2.76061 q^{29} +1.34598 q^{30} +7.79350 q^{31} -1.00000 q^{32} +3.77351 q^{33} +7.24113 q^{34} -4.36783 q^{35} +1.00000 q^{36} -5.09830 q^{37} -0.661018 q^{38} -1.00000 q^{39} -1.34598 q^{40} -11.1873 q^{41} -3.24510 q^{42} +0.261647 q^{43} -3.77351 q^{44} +1.34598 q^{45} +3.88709 q^{46} +0.430909 q^{47} -1.00000 q^{48} +3.53068 q^{49} +3.18835 q^{50} +7.24113 q^{51} +1.00000 q^{52} -6.98136 q^{53} +1.00000 q^{54} -5.07905 q^{55} +3.24510 q^{56} -0.661018 q^{57} +2.76061 q^{58} +5.79920 q^{59} -1.34598 q^{60} +12.2715 q^{61} -7.79350 q^{62} -3.24510 q^{63} +1.00000 q^{64} +1.34598 q^{65} -3.77351 q^{66} -15.9624 q^{67} -7.24113 q^{68} +3.88709 q^{69} +4.36783 q^{70} +1.48957 q^{71} -1.00000 q^{72} -7.24146 q^{73} +5.09830 q^{74} +3.18835 q^{75} +0.661018 q^{76} +12.2454 q^{77} +1.00000 q^{78} +9.27347 q^{79} +1.34598 q^{80} +1.00000 q^{81} +11.1873 q^{82} +3.12239 q^{83} +3.24510 q^{84} -9.74638 q^{85} -0.261647 q^{86} +2.76061 q^{87} +3.77351 q^{88} +11.5122 q^{89} -1.34598 q^{90} -3.24510 q^{91} -3.88709 q^{92} -7.79350 q^{93} -0.430909 q^{94} +0.889714 q^{95} +1.00000 q^{96} -4.13251 q^{97} -3.53068 q^{98} -3.77351 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + 3q^{10} + 5q^{11} - 14q^{12} + 14q^{13} - 4q^{14} + 3q^{15} + 14q^{16} - 7q^{17} - 14q^{18} + 31q^{19} - 3q^{20} - 4q^{21} - 5q^{22} + 14q^{23} + 14q^{24} + 11q^{25} - 14q^{26} - 14q^{27} + 4q^{28} + 9q^{29} - 3q^{30} + 23q^{31} - 14q^{32} - 5q^{33} + 7q^{34} - 2q^{35} + 14q^{36} + 12q^{37} - 31q^{38} - 14q^{39} + 3q^{40} - 5q^{41} + 4q^{42} - 2q^{43} + 5q^{44} - 3q^{45} - 14q^{46} - 11q^{47} - 14q^{48} + 52q^{49} - 11q^{50} + 7q^{51} + 14q^{52} + 6q^{53} + 14q^{54} + 30q^{55} - 4q^{56} - 31q^{57} - 9q^{58} - 4q^{59} + 3q^{60} + 12q^{61} - 23q^{62} + 4q^{63} + 14q^{64} - 3q^{65} + 5q^{66} + 24q^{67} - 7q^{68} - 14q^{69} + 2q^{70} + 20q^{71} - 14q^{72} + 2q^{73} - 12q^{74} - 11q^{75} + 31q^{76} - 28q^{77} + 14q^{78} + 59q^{79} - 3q^{80} + 14q^{81} + 5q^{82} + q^{83} - 4q^{84} - 29q^{85} + 2q^{86} - 9q^{87} - 5q^{88} + 6q^{89} + 3q^{90} + 4q^{91} + 14q^{92} - 23q^{93} + 11q^{94} - 58q^{95} + 14q^{96} - 6q^{97} - 52q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.34598 0.601938 0.300969 0.953634i \(-0.402690\pi\)
0.300969 + 0.953634i \(0.402690\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.24510 −1.22653 −0.613266 0.789876i \(-0.710145\pi\)
−0.613266 + 0.789876i \(0.710145\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.34598 −0.425635
\(11\) −3.77351 −1.13776 −0.568878 0.822422i \(-0.692622\pi\)
−0.568878 + 0.822422i \(0.692622\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 3.24510 0.867290
\(15\) −1.34598 −0.347529
\(16\) 1.00000 0.250000
\(17\) −7.24113 −1.75623 −0.878116 0.478448i \(-0.841199\pi\)
−0.878116 + 0.478448i \(0.841199\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.661018 0.151648 0.0758240 0.997121i \(-0.475841\pi\)
0.0758240 + 0.997121i \(0.475841\pi\)
\(20\) 1.34598 0.300969
\(21\) 3.24510 0.708139
\(22\) 3.77351 0.804515
\(23\) −3.88709 −0.810514 −0.405257 0.914203i \(-0.632818\pi\)
−0.405257 + 0.914203i \(0.632818\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.18835 −0.637670
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −3.24510 −0.613266
\(29\) −2.76061 −0.512632 −0.256316 0.966593i \(-0.582509\pi\)
−0.256316 + 0.966593i \(0.582509\pi\)
\(30\) 1.34598 0.245740
\(31\) 7.79350 1.39975 0.699877 0.714263i \(-0.253238\pi\)
0.699877 + 0.714263i \(0.253238\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.77351 0.656883
\(34\) 7.24113 1.24184
\(35\) −4.36783 −0.738297
\(36\) 1.00000 0.166667
\(37\) −5.09830 −0.838156 −0.419078 0.907950i \(-0.637647\pi\)
−0.419078 + 0.907950i \(0.637647\pi\)
\(38\) −0.661018 −0.107231
\(39\) −1.00000 −0.160128
\(40\) −1.34598 −0.212817
\(41\) −11.1873 −1.74716 −0.873582 0.486677i \(-0.838209\pi\)
−0.873582 + 0.486677i \(0.838209\pi\)
\(42\) −3.24510 −0.500730
\(43\) 0.261647 0.0399007 0.0199504 0.999801i \(-0.493649\pi\)
0.0199504 + 0.999801i \(0.493649\pi\)
\(44\) −3.77351 −0.568878
\(45\) 1.34598 0.200646
\(46\) 3.88709 0.573120
\(47\) 0.430909 0.0628545 0.0314273 0.999506i \(-0.489995\pi\)
0.0314273 + 0.999506i \(0.489995\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.53068 0.504383
\(50\) 3.18835 0.450901
\(51\) 7.24113 1.01396
\(52\) 1.00000 0.138675
\(53\) −6.98136 −0.958963 −0.479481 0.877552i \(-0.659175\pi\)
−0.479481 + 0.877552i \(0.659175\pi\)
\(54\) 1.00000 0.136083
\(55\) −5.07905 −0.684859
\(56\) 3.24510 0.433645
\(57\) −0.661018 −0.0875540
\(58\) 2.76061 0.362485
\(59\) 5.79920 0.754992 0.377496 0.926011i \(-0.376785\pi\)
0.377496 + 0.926011i \(0.376785\pi\)
\(60\) −1.34598 −0.173765
\(61\) 12.2715 1.57120 0.785599 0.618736i \(-0.212355\pi\)
0.785599 + 0.618736i \(0.212355\pi\)
\(62\) −7.79350 −0.989776
\(63\) −3.24510 −0.408844
\(64\) 1.00000 0.125000
\(65\) 1.34598 0.166948
\(66\) −3.77351 −0.464487
\(67\) −15.9624 −1.95012 −0.975061 0.221939i \(-0.928762\pi\)
−0.975061 + 0.221939i \(0.928762\pi\)
\(68\) −7.24113 −0.878116
\(69\) 3.88709 0.467950
\(70\) 4.36783 0.522055
\(71\) 1.48957 0.176779 0.0883897 0.996086i \(-0.471828\pi\)
0.0883897 + 0.996086i \(0.471828\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.24146 −0.847548 −0.423774 0.905768i \(-0.639295\pi\)
−0.423774 + 0.905768i \(0.639295\pi\)
\(74\) 5.09830 0.592666
\(75\) 3.18835 0.368159
\(76\) 0.661018 0.0758240
\(77\) 12.2454 1.39549
\(78\) 1.00000 0.113228
\(79\) 9.27347 1.04335 0.521674 0.853145i \(-0.325308\pi\)
0.521674 + 0.853145i \(0.325308\pi\)
\(80\) 1.34598 0.150485
\(81\) 1.00000 0.111111
\(82\) 11.1873 1.23543
\(83\) 3.12239 0.342727 0.171363 0.985208i \(-0.445183\pi\)
0.171363 + 0.985208i \(0.445183\pi\)
\(84\) 3.24510 0.354070
\(85\) −9.74638 −1.05714
\(86\) −0.261647 −0.0282141
\(87\) 2.76061 0.295968
\(88\) 3.77351 0.402257
\(89\) 11.5122 1.22029 0.610144 0.792290i \(-0.291112\pi\)
0.610144 + 0.792290i \(0.291112\pi\)
\(90\) −1.34598 −0.141878
\(91\) −3.24510 −0.340179
\(92\) −3.88709 −0.405257
\(93\) −7.79350 −0.808149
\(94\) −0.430909 −0.0444449
\(95\) 0.889714 0.0912828
\(96\) 1.00000 0.102062
\(97\) −4.13251 −0.419593 −0.209797 0.977745i \(-0.567280\pi\)
−0.209797 + 0.977745i \(0.567280\pi\)
\(98\) −3.53068 −0.356653
\(99\) −3.77351 −0.379252
\(100\) −3.18835 −0.318835
\(101\) −10.6527 −1.05999 −0.529993 0.848002i \(-0.677805\pi\)
−0.529993 + 0.848002i \(0.677805\pi\)
\(102\) −7.24113 −0.716979
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 4.36783 0.426256
\(106\) 6.98136 0.678089
\(107\) −13.5536 −1.31027 −0.655137 0.755510i \(-0.727389\pi\)
−0.655137 + 0.755510i \(0.727389\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −13.6544 −1.30786 −0.653928 0.756557i \(-0.726880\pi\)
−0.653928 + 0.756557i \(0.726880\pi\)
\(110\) 5.07905 0.484268
\(111\) 5.09830 0.483909
\(112\) −3.24510 −0.306633
\(113\) −1.82889 −0.172048 −0.0860239 0.996293i \(-0.527416\pi\)
−0.0860239 + 0.996293i \(0.527416\pi\)
\(114\) 0.661018 0.0619100
\(115\) −5.23192 −0.487879
\(116\) −2.76061 −0.256316
\(117\) 1.00000 0.0924500
\(118\) −5.79920 −0.533860
\(119\) 23.4982 2.15408
\(120\) 1.34598 0.122870
\(121\) 3.23936 0.294487
\(122\) −12.2715 −1.11101
\(123\) 11.1873 1.00873
\(124\) 7.79350 0.699877
\(125\) −11.0213 −0.985777
\(126\) 3.24510 0.289097
\(127\) 5.03432 0.446724 0.223362 0.974736i \(-0.428297\pi\)
0.223362 + 0.974736i \(0.428297\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.261647 −0.0230367
\(130\) −1.34598 −0.118050
\(131\) 17.9193 1.56562 0.782809 0.622262i \(-0.213786\pi\)
0.782809 + 0.622262i \(0.213786\pi\)
\(132\) 3.77351 0.328442
\(133\) −2.14507 −0.186001
\(134\) 15.9624 1.37894
\(135\) −1.34598 −0.115843
\(136\) 7.24113 0.620922
\(137\) 3.26677 0.279099 0.139550 0.990215i \(-0.455435\pi\)
0.139550 + 0.990215i \(0.455435\pi\)
\(138\) −3.88709 −0.330891
\(139\) 10.4650 0.887629 0.443814 0.896119i \(-0.353625\pi\)
0.443814 + 0.896119i \(0.353625\pi\)
\(140\) −4.36783 −0.369149
\(141\) −0.430909 −0.0362891
\(142\) −1.48957 −0.125002
\(143\) −3.77351 −0.315557
\(144\) 1.00000 0.0833333
\(145\) −3.71571 −0.308573
\(146\) 7.24146 0.599307
\(147\) −3.53068 −0.291206
\(148\) −5.09830 −0.419078
\(149\) 18.9960 1.55621 0.778104 0.628135i \(-0.216181\pi\)
0.778104 + 0.628135i \(0.216181\pi\)
\(150\) −3.18835 −0.260328
\(151\) −2.86092 −0.232818 −0.116409 0.993201i \(-0.537138\pi\)
−0.116409 + 0.993201i \(0.537138\pi\)
\(152\) −0.661018 −0.0536157
\(153\) −7.24113 −0.585411
\(154\) −12.2454 −0.986764
\(155\) 10.4899 0.842566
\(156\) −1.00000 −0.0800641
\(157\) −8.29324 −0.661872 −0.330936 0.943653i \(-0.607364\pi\)
−0.330936 + 0.943653i \(0.607364\pi\)
\(158\) −9.27347 −0.737758
\(159\) 6.98136 0.553658
\(160\) −1.34598 −0.106409
\(161\) 12.6140 0.994122
\(162\) −1.00000 −0.0785674
\(163\) 14.7826 1.15787 0.578933 0.815375i \(-0.303469\pi\)
0.578933 + 0.815375i \(0.303469\pi\)
\(164\) −11.1873 −0.873582
\(165\) 5.07905 0.395403
\(166\) −3.12239 −0.242344
\(167\) 1.14854 0.0888765 0.0444382 0.999012i \(-0.485850\pi\)
0.0444382 + 0.999012i \(0.485850\pi\)
\(168\) −3.24510 −0.250365
\(169\) 1.00000 0.0769231
\(170\) 9.74638 0.747513
\(171\) 0.661018 0.0505493
\(172\) 0.261647 0.0199504
\(173\) 17.4991 1.33044 0.665218 0.746650i \(-0.268339\pi\)
0.665218 + 0.746650i \(0.268339\pi\)
\(174\) −2.76061 −0.209281
\(175\) 10.3465 0.782123
\(176\) −3.77351 −0.284439
\(177\) −5.79920 −0.435895
\(178\) −11.5122 −0.862874
\(179\) −3.80501 −0.284400 −0.142200 0.989838i \(-0.545418\pi\)
−0.142200 + 0.989838i \(0.545418\pi\)
\(180\) 1.34598 0.100323
\(181\) 13.2055 0.981556 0.490778 0.871285i \(-0.336713\pi\)
0.490778 + 0.871285i \(0.336713\pi\)
\(182\) 3.24510 0.240543
\(183\) −12.2715 −0.907132
\(184\) 3.88709 0.286560
\(185\) −6.86219 −0.504518
\(186\) 7.79350 0.571447
\(187\) 27.3245 1.99816
\(188\) 0.430909 0.0314273
\(189\) 3.24510 0.236046
\(190\) −0.889714 −0.0645467
\(191\) −14.5232 −1.05086 −0.525430 0.850837i \(-0.676095\pi\)
−0.525430 + 0.850837i \(0.676095\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.8956 1.36013 0.680066 0.733151i \(-0.261951\pi\)
0.680066 + 0.733151i \(0.261951\pi\)
\(194\) 4.13251 0.296697
\(195\) −1.34598 −0.0963873
\(196\) 3.53068 0.252192
\(197\) −4.69606 −0.334580 −0.167290 0.985908i \(-0.553502\pi\)
−0.167290 + 0.985908i \(0.553502\pi\)
\(198\) 3.77351 0.268172
\(199\) −25.7438 −1.82493 −0.912465 0.409156i \(-0.865823\pi\)
−0.912465 + 0.409156i \(0.865823\pi\)
\(200\) 3.18835 0.225450
\(201\) 15.9624 1.12590
\(202\) 10.6527 0.749523
\(203\) 8.95845 0.628760
\(204\) 7.24113 0.506980
\(205\) −15.0578 −1.05169
\(206\) 1.00000 0.0696733
\(207\) −3.88709 −0.270171
\(208\) 1.00000 0.0693375
\(209\) −2.49436 −0.172538
\(210\) −4.36783 −0.301409
\(211\) 16.6079 1.14334 0.571668 0.820485i \(-0.306296\pi\)
0.571668 + 0.820485i \(0.306296\pi\)
\(212\) −6.98136 −0.479481
\(213\) −1.48957 −0.102064
\(214\) 13.5536 0.926503
\(215\) 0.352170 0.0240178
\(216\) 1.00000 0.0680414
\(217\) −25.2907 −1.71684
\(218\) 13.6544 0.924794
\(219\) 7.24146 0.489332
\(220\) −5.07905 −0.342429
\(221\) −7.24113 −0.487091
\(222\) −5.09830 −0.342176
\(223\) 4.74985 0.318074 0.159037 0.987273i \(-0.449161\pi\)
0.159037 + 0.987273i \(0.449161\pi\)
\(224\) 3.24510 0.216822
\(225\) −3.18835 −0.212557
\(226\) 1.82889 0.121656
\(227\) 14.8843 0.987904 0.493952 0.869489i \(-0.335552\pi\)
0.493952 + 0.869489i \(0.335552\pi\)
\(228\) −0.661018 −0.0437770
\(229\) −23.6580 −1.56337 −0.781683 0.623676i \(-0.785639\pi\)
−0.781683 + 0.623676i \(0.785639\pi\)
\(230\) 5.23192 0.344983
\(231\) −12.2454 −0.805689
\(232\) 2.76061 0.181243
\(233\) −24.5541 −1.60859 −0.804297 0.594228i \(-0.797458\pi\)
−0.804297 + 0.594228i \(0.797458\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0.579993 0.0378345
\(236\) 5.79920 0.377496
\(237\) −9.27347 −0.602377
\(238\) −23.4982 −1.52316
\(239\) −0.194543 −0.0125839 −0.00629195 0.999980i \(-0.502003\pi\)
−0.00629195 + 0.999980i \(0.502003\pi\)
\(240\) −1.34598 −0.0868823
\(241\) 5.77889 0.372251 0.186126 0.982526i \(-0.440407\pi\)
0.186126 + 0.982526i \(0.440407\pi\)
\(242\) −3.23936 −0.208234
\(243\) −1.00000 −0.0641500
\(244\) 12.2715 0.785599
\(245\) 4.75221 0.303608
\(246\) −11.1873 −0.713277
\(247\) 0.661018 0.0420596
\(248\) −7.79350 −0.494888
\(249\) −3.12239 −0.197873
\(250\) 11.0213 0.697049
\(251\) −25.3501 −1.60008 −0.800042 0.599944i \(-0.795190\pi\)
−0.800042 + 0.599944i \(0.795190\pi\)
\(252\) −3.24510 −0.204422
\(253\) 14.6680 0.922167
\(254\) −5.03432 −0.315881
\(255\) 9.74638 0.610342
\(256\) 1.00000 0.0625000
\(257\) 21.3251 1.33022 0.665111 0.746745i \(-0.268384\pi\)
0.665111 + 0.746745i \(0.268384\pi\)
\(258\) 0.261647 0.0162894
\(259\) 16.5445 1.02803
\(260\) 1.34598 0.0834738
\(261\) −2.76061 −0.170877
\(262\) −17.9193 −1.10706
\(263\) −24.8870 −1.53460 −0.767300 0.641288i \(-0.778400\pi\)
−0.767300 + 0.641288i \(0.778400\pi\)
\(264\) −3.77351 −0.232243
\(265\) −9.39673 −0.577237
\(266\) 2.14507 0.131523
\(267\) −11.5122 −0.704534
\(268\) −15.9624 −0.975061
\(269\) 27.1724 1.65673 0.828365 0.560189i \(-0.189271\pi\)
0.828365 + 0.560189i \(0.189271\pi\)
\(270\) 1.34598 0.0819134
\(271\) −9.83001 −0.597130 −0.298565 0.954389i \(-0.596508\pi\)
−0.298565 + 0.954389i \(0.596508\pi\)
\(272\) −7.24113 −0.439058
\(273\) 3.24510 0.196402
\(274\) −3.26677 −0.197353
\(275\) 12.0313 0.725513
\(276\) 3.88709 0.233975
\(277\) −1.13804 −0.0683780 −0.0341890 0.999415i \(-0.510885\pi\)
−0.0341890 + 0.999415i \(0.510885\pi\)
\(278\) −10.4650 −0.627648
\(279\) 7.79350 0.466585
\(280\) 4.36783 0.261028
\(281\) −1.48374 −0.0885125 −0.0442562 0.999020i \(-0.514092\pi\)
−0.0442562 + 0.999020i \(0.514092\pi\)
\(282\) 0.430909 0.0256602
\(283\) 24.5874 1.46157 0.730784 0.682609i \(-0.239155\pi\)
0.730784 + 0.682609i \(0.239155\pi\)
\(284\) 1.48957 0.0883897
\(285\) −0.889714 −0.0527021
\(286\) 3.77351 0.223132
\(287\) 36.3040 2.14295
\(288\) −1.00000 −0.0589256
\(289\) 35.4339 2.08435
\(290\) 3.71571 0.218194
\(291\) 4.13251 0.242252
\(292\) −7.24146 −0.423774
\(293\) −23.1487 −1.35236 −0.676182 0.736735i \(-0.736367\pi\)
−0.676182 + 0.736735i \(0.736367\pi\)
\(294\) 3.53068 0.205914
\(295\) 7.80559 0.454459
\(296\) 5.09830 0.296333
\(297\) 3.77351 0.218961
\(298\) −18.9960 −1.10041
\(299\) −3.88709 −0.224796
\(300\) 3.18835 0.184080
\(301\) −0.849070 −0.0489396
\(302\) 2.86092 0.164627
\(303\) 10.6527 0.611983
\(304\) 0.661018 0.0379120
\(305\) 16.5171 0.945765
\(306\) 7.24113 0.413948
\(307\) 19.4854 1.11209 0.556046 0.831152i \(-0.312318\pi\)
0.556046 + 0.831152i \(0.312318\pi\)
\(308\) 12.2454 0.697747
\(309\) 1.00000 0.0568880
\(310\) −10.4899 −0.595784
\(311\) −24.2475 −1.37495 −0.687474 0.726209i \(-0.741280\pi\)
−0.687474 + 0.726209i \(0.741280\pi\)
\(312\) 1.00000 0.0566139
\(313\) 17.0379 0.963042 0.481521 0.876435i \(-0.340085\pi\)
0.481521 + 0.876435i \(0.340085\pi\)
\(314\) 8.29324 0.468014
\(315\) −4.36783 −0.246099
\(316\) 9.27347 0.521674
\(317\) −17.1424 −0.962816 −0.481408 0.876497i \(-0.659874\pi\)
−0.481408 + 0.876497i \(0.659874\pi\)
\(318\) −6.98136 −0.391495
\(319\) 10.4172 0.583250
\(320\) 1.34598 0.0752423
\(321\) 13.5536 0.756487
\(322\) −12.6140 −0.702950
\(323\) −4.78652 −0.266329
\(324\) 1.00000 0.0555556
\(325\) −3.18835 −0.176858
\(326\) −14.7826 −0.818735
\(327\) 13.6544 0.755091
\(328\) 11.1873 0.617716
\(329\) −1.39834 −0.0770931
\(330\) −5.07905 −0.279592
\(331\) −8.77403 −0.482264 −0.241132 0.970492i \(-0.577519\pi\)
−0.241132 + 0.970492i \(0.577519\pi\)
\(332\) 3.12239 0.171363
\(333\) −5.09830 −0.279385
\(334\) −1.14854 −0.0628452
\(335\) −21.4850 −1.17385
\(336\) 3.24510 0.177035
\(337\) −4.00567 −0.218203 −0.109101 0.994031i \(-0.534797\pi\)
−0.109101 + 0.994031i \(0.534797\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 1.82889 0.0993319
\(340\) −9.74638 −0.528572
\(341\) −29.4088 −1.59258
\(342\) −0.661018 −0.0357438
\(343\) 11.2583 0.607890
\(344\) −0.261647 −0.0141070
\(345\) 5.23192 0.281677
\(346\) −17.4991 −0.940760
\(347\) −15.7245 −0.844137 −0.422069 0.906564i \(-0.638696\pi\)
−0.422069 + 0.906564i \(0.638696\pi\)
\(348\) 2.76061 0.147984
\(349\) −9.84845 −0.527175 −0.263588 0.964635i \(-0.584906\pi\)
−0.263588 + 0.964635i \(0.584906\pi\)
\(350\) −10.3465 −0.553045
\(351\) −1.00000 −0.0533761
\(352\) 3.77351 0.201129
\(353\) −3.60939 −0.192108 −0.0960541 0.995376i \(-0.530622\pi\)
−0.0960541 + 0.995376i \(0.530622\pi\)
\(354\) 5.79920 0.308224
\(355\) 2.00492 0.106410
\(356\) 11.5122 0.610144
\(357\) −23.4982 −1.24366
\(358\) 3.80501 0.201101
\(359\) 6.53877 0.345103 0.172552 0.985000i \(-0.444799\pi\)
0.172552 + 0.985000i \(0.444799\pi\)
\(360\) −1.34598 −0.0709391
\(361\) −18.5631 −0.977003
\(362\) −13.2055 −0.694065
\(363\) −3.23936 −0.170022
\(364\) −3.24510 −0.170090
\(365\) −9.74682 −0.510172
\(366\) 12.2715 0.641439
\(367\) 30.8684 1.61132 0.805659 0.592379i \(-0.201811\pi\)
0.805659 + 0.592379i \(0.201811\pi\)
\(368\) −3.88709 −0.202628
\(369\) −11.1873 −0.582388
\(370\) 6.86219 0.356748
\(371\) 22.6552 1.17620
\(372\) −7.79350 −0.404074
\(373\) 11.4169 0.591146 0.295573 0.955320i \(-0.404489\pi\)
0.295573 + 0.955320i \(0.404489\pi\)
\(374\) −27.3245 −1.41291
\(375\) 11.0213 0.569138
\(376\) −0.430909 −0.0222224
\(377\) −2.76061 −0.142179
\(378\) −3.24510 −0.166910
\(379\) −6.20355 −0.318655 −0.159327 0.987226i \(-0.550933\pi\)
−0.159327 + 0.987226i \(0.550933\pi\)
\(380\) 0.889714 0.0456414
\(381\) −5.03432 −0.257916
\(382\) 14.5232 0.743070
\(383\) 2.81254 0.143714 0.0718570 0.997415i \(-0.477107\pi\)
0.0718570 + 0.997415i \(0.477107\pi\)
\(384\) 1.00000 0.0510310
\(385\) 16.4820 0.840002
\(386\) −18.8956 −0.961759
\(387\) 0.261647 0.0133002
\(388\) −4.13251 −0.209797
\(389\) −15.6657 −0.794284 −0.397142 0.917757i \(-0.629998\pi\)
−0.397142 + 0.917757i \(0.629998\pi\)
\(390\) 1.34598 0.0681561
\(391\) 28.1469 1.42345
\(392\) −3.53068 −0.178326
\(393\) −17.9193 −0.903910
\(394\) 4.69606 0.236584
\(395\) 12.4819 0.628031
\(396\) −3.77351 −0.189626
\(397\) −3.89885 −0.195678 −0.0978388 0.995202i \(-0.531193\pi\)
−0.0978388 + 0.995202i \(0.531193\pi\)
\(398\) 25.7438 1.29042
\(399\) 2.14507 0.107388
\(400\) −3.18835 −0.159418
\(401\) −28.2737 −1.41192 −0.705960 0.708251i \(-0.749484\pi\)
−0.705960 + 0.708251i \(0.749484\pi\)
\(402\) −15.9624 −0.796134
\(403\) 7.79350 0.388222
\(404\) −10.6527 −0.529993
\(405\) 1.34598 0.0668820
\(406\) −8.95845 −0.444600
\(407\) 19.2385 0.953616
\(408\) −7.24113 −0.358489
\(409\) −8.55426 −0.422981 −0.211491 0.977380i \(-0.567832\pi\)
−0.211491 + 0.977380i \(0.567832\pi\)
\(410\) 15.0578 0.743654
\(411\) −3.26677 −0.161138
\(412\) −1.00000 −0.0492665
\(413\) −18.8190 −0.926023
\(414\) 3.88709 0.191040
\(415\) 4.20266 0.206300
\(416\) −1.00000 −0.0490290
\(417\) −10.4650 −0.512473
\(418\) 2.49436 0.122003
\(419\) 7.01647 0.342777 0.171388 0.985204i \(-0.445175\pi\)
0.171388 + 0.985204i \(0.445175\pi\)
\(420\) 4.36783 0.213128
\(421\) −18.0352 −0.878982 −0.439491 0.898247i \(-0.644841\pi\)
−0.439491 + 0.898247i \(0.644841\pi\)
\(422\) −16.6079 −0.808461
\(423\) 0.430909 0.0209515
\(424\) 6.98136 0.339045
\(425\) 23.0873 1.11990
\(426\) 1.48957 0.0721699
\(427\) −39.8221 −1.92713
\(428\) −13.5536 −0.655137
\(429\) 3.77351 0.182187
\(430\) −0.352170 −0.0169831
\(431\) 28.1776 1.35727 0.678634 0.734477i \(-0.262572\pi\)
0.678634 + 0.734477i \(0.262572\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 18.6955 0.898451 0.449225 0.893418i \(-0.351700\pi\)
0.449225 + 0.893418i \(0.351700\pi\)
\(434\) 25.2907 1.21399
\(435\) 3.71571 0.178155
\(436\) −13.6544 −0.653928
\(437\) −2.56944 −0.122913
\(438\) −7.24146 −0.346010
\(439\) −6.18598 −0.295241 −0.147620 0.989044i \(-0.547161\pi\)
−0.147620 + 0.989044i \(0.547161\pi\)
\(440\) 5.07905 0.242134
\(441\) 3.53068 0.168128
\(442\) 7.24113 0.344425
\(443\) 32.4240 1.54051 0.770255 0.637736i \(-0.220129\pi\)
0.770255 + 0.637736i \(0.220129\pi\)
\(444\) 5.09830 0.241955
\(445\) 15.4951 0.734538
\(446\) −4.74985 −0.224912
\(447\) −18.9960 −0.898478
\(448\) −3.24510 −0.153317
\(449\) −25.2808 −1.19308 −0.596538 0.802585i \(-0.703457\pi\)
−0.596538 + 0.802585i \(0.703457\pi\)
\(450\) 3.18835 0.150300
\(451\) 42.2154 1.98785
\(452\) −1.82889 −0.0860239
\(453\) 2.86092 0.134418
\(454\) −14.8843 −0.698553
\(455\) −4.36783 −0.204767
\(456\) 0.661018 0.0309550
\(457\) 16.3853 0.766470 0.383235 0.923651i \(-0.374810\pi\)
0.383235 + 0.923651i \(0.374810\pi\)
\(458\) 23.6580 1.10547
\(459\) 7.24113 0.337987
\(460\) −5.23192 −0.243940
\(461\) 17.8016 0.829104 0.414552 0.910026i \(-0.363938\pi\)
0.414552 + 0.910026i \(0.363938\pi\)
\(462\) 12.2454 0.569708
\(463\) −13.5211 −0.628378 −0.314189 0.949360i \(-0.601733\pi\)
−0.314189 + 0.949360i \(0.601733\pi\)
\(464\) −2.76061 −0.128158
\(465\) −10.4899 −0.486456
\(466\) 24.5541 1.13745
\(467\) 30.5160 1.41211 0.706057 0.708155i \(-0.250472\pi\)
0.706057 + 0.708155i \(0.250472\pi\)
\(468\) 1.00000 0.0462250
\(469\) 51.7997 2.39189
\(470\) −0.579993 −0.0267531
\(471\) 8.29324 0.382132
\(472\) −5.79920 −0.266930
\(473\) −0.987326 −0.0453973
\(474\) 9.27347 0.425945
\(475\) −2.10756 −0.0967014
\(476\) 23.4982 1.07704
\(477\) −6.98136 −0.319654
\(478\) 0.194543 0.00889817
\(479\) 8.25272 0.377076 0.188538 0.982066i \(-0.439625\pi\)
0.188538 + 0.982066i \(0.439625\pi\)
\(480\) 1.34598 0.0614351
\(481\) −5.09830 −0.232463
\(482\) −5.77889 −0.263221
\(483\) −12.6140 −0.573957
\(484\) 3.23936 0.147244
\(485\) −5.56226 −0.252569
\(486\) 1.00000 0.0453609
\(487\) −3.27770 −0.148527 −0.0742634 0.997239i \(-0.523661\pi\)
−0.0742634 + 0.997239i \(0.523661\pi\)
\(488\) −12.2715 −0.555503
\(489\) −14.7826 −0.668494
\(490\) −4.75221 −0.214683
\(491\) 7.49528 0.338257 0.169129 0.985594i \(-0.445905\pi\)
0.169129 + 0.985594i \(0.445905\pi\)
\(492\) 11.1873 0.504363
\(493\) 19.9899 0.900300
\(494\) −0.661018 −0.0297406
\(495\) −5.07905 −0.228286
\(496\) 7.79350 0.349939
\(497\) −4.83381 −0.216826
\(498\) 3.12239 0.139918
\(499\) 14.3424 0.642054 0.321027 0.947070i \(-0.395972\pi\)
0.321027 + 0.947070i \(0.395972\pi\)
\(500\) −11.0213 −0.492888
\(501\) −1.14854 −0.0513129
\(502\) 25.3501 1.13143
\(503\) 36.3950 1.62277 0.811386 0.584511i \(-0.198714\pi\)
0.811386 + 0.584511i \(0.198714\pi\)
\(504\) 3.24510 0.144548
\(505\) −14.3383 −0.638046
\(506\) −14.6680 −0.652070
\(507\) −1.00000 −0.0444116
\(508\) 5.03432 0.223362
\(509\) −9.94797 −0.440936 −0.220468 0.975394i \(-0.570758\pi\)
−0.220468 + 0.975394i \(0.570758\pi\)
\(510\) −9.74638 −0.431577
\(511\) 23.4993 1.03955
\(512\) −1.00000 −0.0441942
\(513\) −0.661018 −0.0291847
\(514\) −21.3251 −0.940609
\(515\) −1.34598 −0.0593108
\(516\) −0.261647 −0.0115184
\(517\) −1.62604 −0.0715131
\(518\) −16.5445 −0.726924
\(519\) −17.4991 −0.768127
\(520\) −1.34598 −0.0590249
\(521\) −8.19107 −0.358857 −0.179429 0.983771i \(-0.557425\pi\)
−0.179429 + 0.983771i \(0.557425\pi\)
\(522\) 2.76061 0.120828
\(523\) 22.4531 0.981805 0.490903 0.871214i \(-0.336667\pi\)
0.490903 + 0.871214i \(0.336667\pi\)
\(524\) 17.9193 0.782809
\(525\) −10.3465 −0.451559
\(526\) 24.8870 1.08513
\(527\) −56.4338 −2.45829
\(528\) 3.77351 0.164221
\(529\) −7.89055 −0.343067
\(530\) 9.39673 0.408168
\(531\) 5.79920 0.251664
\(532\) −2.14507 −0.0930006
\(533\) −11.1873 −0.484576
\(534\) 11.5122 0.498181
\(535\) −18.2428 −0.788704
\(536\) 15.9624 0.689472
\(537\) 3.80501 0.164198
\(538\) −27.1724 −1.17149
\(539\) −13.3231 −0.573865
\(540\) −1.34598 −0.0579215
\(541\) −29.5197 −1.26915 −0.634576 0.772860i \(-0.718825\pi\)
−0.634576 + 0.772860i \(0.718825\pi\)
\(542\) 9.83001 0.422235
\(543\) −13.2055 −0.566702
\(544\) 7.24113 0.310461
\(545\) −18.3785 −0.787249
\(546\) −3.24510 −0.138878
\(547\) 22.5800 0.965450 0.482725 0.875772i \(-0.339647\pi\)
0.482725 + 0.875772i \(0.339647\pi\)
\(548\) 3.26677 0.139550
\(549\) 12.2715 0.523733
\(550\) −12.0313 −0.513015
\(551\) −1.82481 −0.0777396
\(552\) −3.88709 −0.165445
\(553\) −30.0934 −1.27970
\(554\) 1.13804 0.0483505
\(555\) 6.86219 0.291284
\(556\) 10.4650 0.443814
\(557\) 22.8533 0.968327 0.484163 0.874978i \(-0.339124\pi\)
0.484163 + 0.874978i \(0.339124\pi\)
\(558\) −7.79350 −0.329925
\(559\) 0.261647 0.0110665
\(560\) −4.36783 −0.184574
\(561\) −27.3245 −1.15364
\(562\) 1.48374 0.0625878
\(563\) 13.3416 0.562283 0.281142 0.959666i \(-0.409287\pi\)
0.281142 + 0.959666i \(0.409287\pi\)
\(564\) −0.430909 −0.0181445
\(565\) −2.46165 −0.103562
\(566\) −24.5874 −1.03348
\(567\) −3.24510 −0.136281
\(568\) −1.48957 −0.0625010
\(569\) 19.8928 0.833950 0.416975 0.908918i \(-0.363090\pi\)
0.416975 + 0.908918i \(0.363090\pi\)
\(570\) 0.889714 0.0372660
\(571\) −31.0386 −1.29893 −0.649463 0.760393i \(-0.725006\pi\)
−0.649463 + 0.760393i \(0.725006\pi\)
\(572\) −3.77351 −0.157778
\(573\) 14.5232 0.606714
\(574\) −36.3040 −1.51530
\(575\) 12.3934 0.516841
\(576\) 1.00000 0.0416667
\(577\) 6.32970 0.263509 0.131754 0.991282i \(-0.457939\pi\)
0.131754 + 0.991282i \(0.457939\pi\)
\(578\) −35.4339 −1.47386
\(579\) −18.8956 −0.785273
\(580\) −3.71571 −0.154286
\(581\) −10.1325 −0.420366
\(582\) −4.13251 −0.171298
\(583\) 26.3442 1.09107
\(584\) 7.24146 0.299654
\(585\) 1.34598 0.0556492
\(586\) 23.1487 0.956266
\(587\) 6.56457 0.270949 0.135474 0.990781i \(-0.456744\pi\)
0.135474 + 0.990781i \(0.456744\pi\)
\(588\) −3.53068 −0.145603
\(589\) 5.15165 0.212270
\(590\) −7.80559 −0.321351
\(591\) 4.69606 0.193170
\(592\) −5.09830 −0.209539
\(593\) −39.5198 −1.62288 −0.811442 0.584434i \(-0.801317\pi\)
−0.811442 + 0.584434i \(0.801317\pi\)
\(594\) −3.77351 −0.154829
\(595\) 31.6280 1.29662
\(596\) 18.9960 0.778104
\(597\) 25.7438 1.05362
\(598\) 3.88709 0.158955
\(599\) 22.9693 0.938501 0.469250 0.883065i \(-0.344524\pi\)
0.469250 + 0.883065i \(0.344524\pi\)
\(600\) −3.18835 −0.130164
\(601\) 10.4639 0.426833 0.213416 0.976961i \(-0.431541\pi\)
0.213416 + 0.976961i \(0.431541\pi\)
\(602\) 0.849070 0.0346055
\(603\) −15.9624 −0.650040
\(604\) −2.86092 −0.116409
\(605\) 4.36010 0.177263
\(606\) −10.6527 −0.432737
\(607\) 3.58571 0.145539 0.0727697 0.997349i \(-0.476816\pi\)
0.0727697 + 0.997349i \(0.476816\pi\)
\(608\) −0.661018 −0.0268078
\(609\) −8.95845 −0.363015
\(610\) −16.5171 −0.668757
\(611\) 0.430909 0.0174327
\(612\) −7.24113 −0.292705
\(613\) −24.5872 −0.993068 −0.496534 0.868017i \(-0.665394\pi\)
−0.496534 + 0.868017i \(0.665394\pi\)
\(614\) −19.4854 −0.786367
\(615\) 15.0578 0.607191
\(616\) −12.2454 −0.493382
\(617\) −43.5235 −1.75219 −0.876094 0.482140i \(-0.839860\pi\)
−0.876094 + 0.482140i \(0.839860\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 18.6399 0.749202 0.374601 0.927186i \(-0.377780\pi\)
0.374601 + 0.927186i \(0.377780\pi\)
\(620\) 10.4899 0.421283
\(621\) 3.88709 0.155983
\(622\) 24.2475 0.972235
\(623\) −37.3582 −1.49672
\(624\) −1.00000 −0.0400320
\(625\) 1.10733 0.0442934
\(626\) −17.0379 −0.680973
\(627\) 2.49436 0.0996151
\(628\) −8.29324 −0.330936
\(629\) 36.9175 1.47200
\(630\) 4.36783 0.174018
\(631\) 33.4063 1.32989 0.664943 0.746895i \(-0.268456\pi\)
0.664943 + 0.746895i \(0.268456\pi\)
\(632\) −9.27347 −0.368879
\(633\) −16.6079 −0.660106
\(634\) 17.1424 0.680813
\(635\) 6.77607 0.268900
\(636\) 6.98136 0.276829
\(637\) 3.53068 0.139891
\(638\) −10.4172 −0.412420
\(639\) 1.48957 0.0589265
\(640\) −1.34598 −0.0532043
\(641\) −35.5136 −1.40270 −0.701352 0.712815i \(-0.747420\pi\)
−0.701352 + 0.712815i \(0.747420\pi\)
\(642\) −13.5536 −0.534917
\(643\) 37.5268 1.47991 0.739956 0.672655i \(-0.234846\pi\)
0.739956 + 0.672655i \(0.234846\pi\)
\(644\) 12.6140 0.497061
\(645\) −0.352170 −0.0138667
\(646\) 4.78652 0.188323
\(647\) −1.85892 −0.0730817 −0.0365409 0.999332i \(-0.511634\pi\)
−0.0365409 + 0.999332i \(0.511634\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −21.8833 −0.858996
\(650\) 3.18835 0.125057
\(651\) 25.2907 0.991221
\(652\) 14.7826 0.578933
\(653\) −17.1112 −0.669612 −0.334806 0.942287i \(-0.608671\pi\)
−0.334806 + 0.942287i \(0.608671\pi\)
\(654\) −13.6544 −0.533930
\(655\) 24.1190 0.942406
\(656\) −11.1873 −0.436791
\(657\) −7.24146 −0.282516
\(658\) 1.39834 0.0545131
\(659\) 48.2990 1.88146 0.940731 0.339155i \(-0.110141\pi\)
0.940731 + 0.339155i \(0.110141\pi\)
\(660\) 5.07905 0.197702
\(661\) 12.4202 0.483091 0.241545 0.970389i \(-0.422346\pi\)
0.241545 + 0.970389i \(0.422346\pi\)
\(662\) 8.77403 0.341012
\(663\) 7.24113 0.281222
\(664\) −3.12239 −0.121172
\(665\) −2.88721 −0.111961
\(666\) 5.09830 0.197555
\(667\) 10.7307 0.415495
\(668\) 1.14854 0.0444382
\(669\) −4.74985 −0.183640
\(670\) 21.4850 0.830039
\(671\) −46.3064 −1.78764
\(672\) −3.24510 −0.125182
\(673\) −11.3026 −0.435682 −0.217841 0.975984i \(-0.569901\pi\)
−0.217841 + 0.975984i \(0.569901\pi\)
\(674\) 4.00567 0.154293
\(675\) 3.18835 0.122720
\(676\) 1.00000 0.0384615
\(677\) 8.16619 0.313852 0.156926 0.987610i \(-0.449842\pi\)
0.156926 + 0.987610i \(0.449842\pi\)
\(678\) −1.82889 −0.0702382
\(679\) 13.4104 0.514645
\(680\) 9.74638 0.373757
\(681\) −14.8843 −0.570366
\(682\) 29.4088 1.12612
\(683\) −34.0204 −1.30176 −0.650878 0.759183i \(-0.725599\pi\)
−0.650878 + 0.759183i \(0.725599\pi\)
\(684\) 0.661018 0.0252747
\(685\) 4.39700 0.168001
\(686\) −11.2583 −0.429843
\(687\) 23.6580 0.902610
\(688\) 0.261647 0.00997519
\(689\) −6.98136 −0.265968
\(690\) −5.23192 −0.199176
\(691\) 19.5621 0.744177 0.372088 0.928197i \(-0.378642\pi\)
0.372088 + 0.928197i \(0.378642\pi\)
\(692\) 17.4991 0.665218
\(693\) 12.2454 0.465165
\(694\) 15.7245 0.596895
\(695\) 14.0856 0.534298
\(696\) −2.76061 −0.104641
\(697\) 81.0088 3.06843
\(698\) 9.84845 0.372769
\(699\) 24.5541 0.928722
\(700\) 10.3465 0.391062
\(701\) 35.7782 1.35132 0.675661 0.737212i \(-0.263858\pi\)
0.675661 + 0.737212i \(0.263858\pi\)
\(702\) 1.00000 0.0377426
\(703\) −3.37007 −0.127105
\(704\) −3.77351 −0.142219
\(705\) −0.579993 −0.0218438
\(706\) 3.60939 0.135841
\(707\) 34.5692 1.30011
\(708\) −5.79920 −0.217947
\(709\) 7.42880 0.278994 0.139497 0.990222i \(-0.455451\pi\)
0.139497 + 0.990222i \(0.455451\pi\)
\(710\) −2.00492 −0.0752435
\(711\) 9.27347 0.347782
\(712\) −11.5122 −0.431437
\(713\) −30.2940 −1.13452
\(714\) 23.4982 0.879398
\(715\) −5.07905 −0.189946
\(716\) −3.80501 −0.142200
\(717\) 0.194543 0.00726532
\(718\) −6.53877 −0.244025
\(719\) 10.3305 0.385264 0.192632 0.981271i \(-0.438298\pi\)
0.192632 + 0.981271i \(0.438298\pi\)
\(720\) 1.34598 0.0501615
\(721\) 3.24510 0.120854
\(722\) 18.5631 0.690845
\(723\) −5.77889 −0.214919
\(724\) 13.2055 0.490778
\(725\) 8.80178 0.326890
\(726\) 3.23936 0.120224
\(727\) −13.0139 −0.482657 −0.241328 0.970443i \(-0.577583\pi\)
−0.241328 + 0.970443i \(0.577583\pi\)
\(728\) 3.24510 0.120271
\(729\) 1.00000 0.0370370
\(730\) 9.74682 0.360746
\(731\) −1.89462 −0.0700750
\(732\) −12.2715 −0.453566
\(733\) −3.85705 −0.142463 −0.0712317 0.997460i \(-0.522693\pi\)
−0.0712317 + 0.997460i \(0.522693\pi\)
\(734\) −30.8684 −1.13937
\(735\) −4.75221 −0.175288
\(736\) 3.88709 0.143280
\(737\) 60.2344 2.21876
\(738\) 11.1873 0.411811
\(739\) 19.4194 0.714356 0.357178 0.934036i \(-0.383739\pi\)
0.357178 + 0.934036i \(0.383739\pi\)
\(740\) −6.86219 −0.252259
\(741\) −0.661018 −0.0242831
\(742\) −22.6552 −0.831699
\(743\) −44.8989 −1.64718 −0.823591 0.567184i \(-0.808033\pi\)
−0.823591 + 0.567184i \(0.808033\pi\)
\(744\) 7.79350 0.285724
\(745\) 25.5681 0.936742
\(746\) −11.4169 −0.418003
\(747\) 3.12239 0.114242
\(748\) 27.3245 0.999081
\(749\) 43.9827 1.60709
\(750\) −11.0213 −0.402442
\(751\) −24.8939 −0.908390 −0.454195 0.890902i \(-0.650073\pi\)
−0.454195 + 0.890902i \(0.650073\pi\)
\(752\) 0.430909 0.0157136
\(753\) 25.3501 0.923809
\(754\) 2.76061 0.100535
\(755\) −3.85072 −0.140142
\(756\) 3.24510 0.118023
\(757\) −31.0682 −1.12919 −0.564596 0.825367i \(-0.690968\pi\)
−0.564596 + 0.825367i \(0.690968\pi\)
\(758\) 6.20355 0.225323
\(759\) −14.6680 −0.532413
\(760\) −0.889714 −0.0322733
\(761\) 52.5673 1.90556 0.952781 0.303659i \(-0.0982084\pi\)
0.952781 + 0.303659i \(0.0982084\pi\)
\(762\) 5.03432 0.182374
\(763\) 44.3100 1.60413
\(764\) −14.5232 −0.525430
\(765\) −9.74638 −0.352381
\(766\) −2.81254 −0.101621
\(767\) 5.79920 0.209397
\(768\) −1.00000 −0.0360844
\(769\) 47.6484 1.71824 0.859122 0.511771i \(-0.171010\pi\)
0.859122 + 0.511771i \(0.171010\pi\)
\(770\) −16.4820 −0.593971
\(771\) −21.3251 −0.768004
\(772\) 18.8956 0.680066
\(773\) 52.2759 1.88023 0.940117 0.340853i \(-0.110716\pi\)
0.940117 + 0.340853i \(0.110716\pi\)
\(774\) −0.261647 −0.00940470
\(775\) −24.8484 −0.892582
\(776\) 4.13251 0.148349
\(777\) −16.5445 −0.593531
\(778\) 15.6657 0.561644
\(779\) −7.39502 −0.264954
\(780\) −1.34598 −0.0481936
\(781\) −5.62091 −0.201132
\(782\) −28.1469 −1.00653
\(783\) 2.76061 0.0986561
\(784\) 3.53068 0.126096
\(785\) −11.1625 −0.398406
\(786\) 17.9193 0.639161
\(787\) −25.6614 −0.914730 −0.457365 0.889279i \(-0.651207\pi\)
−0.457365 + 0.889279i \(0.651207\pi\)
\(788\) −4.69606 −0.167290
\(789\) 24.8870 0.886002
\(790\) −12.4819 −0.444085
\(791\) 5.93494 0.211022
\(792\) 3.77351 0.134086
\(793\) 12.2715 0.435772
\(794\) 3.89885 0.138365
\(795\) 9.39673 0.333268
\(796\) −25.7438 −0.912465
\(797\) −34.2164 −1.21201 −0.606003 0.795462i \(-0.707228\pi\)
−0.606003 + 0.795462i \(0.707228\pi\)
\(798\) −2.14507 −0.0759347
\(799\) −3.12027 −0.110387
\(800\) 3.18835 0.112725
\(801\) 11.5122 0.406763
\(802\) 28.2737 0.998379
\(803\) 27.3257 0.964303
\(804\) 15.9624 0.562952
\(805\) 16.9781 0.598400
\(806\) −7.79350 −0.274514
\(807\) −27.1724 −0.956514
\(808\) 10.6527 0.374761
\(809\) 14.0727 0.494772 0.247386 0.968917i \(-0.420429\pi\)
0.247386 + 0.968917i \(0.420429\pi\)
\(810\) −1.34598 −0.0472927
\(811\) −28.9741 −1.01742 −0.508709 0.860938i \(-0.669877\pi\)
−0.508709 + 0.860938i \(0.669877\pi\)
\(812\) 8.95845 0.314380
\(813\) 9.83001 0.344753
\(814\) −19.2385 −0.674309
\(815\) 19.8971 0.696964
\(816\) 7.24113 0.253490
\(817\) 0.172953 0.00605087
\(818\) 8.55426 0.299093
\(819\) −3.24510 −0.113393
\(820\) −15.0578 −0.525843
\(821\) −0.124312 −0.00433852 −0.00216926 0.999998i \(-0.500690\pi\)
−0.00216926 + 0.999998i \(0.500690\pi\)
\(822\) 3.26677 0.113942
\(823\) 17.7601 0.619079 0.309540 0.950887i \(-0.399825\pi\)
0.309540 + 0.950887i \(0.399825\pi\)
\(824\) 1.00000 0.0348367
\(825\) −12.0313 −0.418875
\(826\) 18.8190 0.654797
\(827\) 12.4195 0.431869 0.215935 0.976408i \(-0.430720\pi\)
0.215935 + 0.976408i \(0.430720\pi\)
\(828\) −3.88709 −0.135086
\(829\) 39.5635 1.37410 0.687049 0.726611i \(-0.258906\pi\)
0.687049 + 0.726611i \(0.258906\pi\)
\(830\) −4.20266 −0.145876
\(831\) 1.13804 0.0394780
\(832\) 1.00000 0.0346688
\(833\) −25.5661 −0.885814
\(834\) 10.4650 0.362373
\(835\) 1.54590 0.0534982
\(836\) −2.49436 −0.0862692
\(837\) −7.79350 −0.269383
\(838\) −7.01647 −0.242380
\(839\) −17.5349 −0.605371 −0.302685 0.953091i \(-0.597883\pi\)
−0.302685 + 0.953091i \(0.597883\pi\)
\(840\) −4.36783 −0.150704
\(841\) −21.3790 −0.737209
\(842\) 18.0352 0.621534
\(843\) 1.48374 0.0511027
\(844\) 16.6079 0.571668
\(845\) 1.34598 0.0463030
\(846\) −0.430909 −0.0148150
\(847\) −10.5121 −0.361199
\(848\) −6.98136 −0.239741
\(849\) −24.5874 −0.843837
\(850\) −23.0873 −0.791886
\(851\) 19.8175 0.679337
\(852\) −1.48957 −0.0510318
\(853\) 44.9350 1.53854 0.769272 0.638922i \(-0.220619\pi\)
0.769272 + 0.638922i \(0.220619\pi\)
\(854\) 39.8221 1.36268
\(855\) 0.889714 0.0304276
\(856\) 13.5536 0.463252
\(857\) −18.4177 −0.629138 −0.314569 0.949235i \(-0.601860\pi\)
−0.314569 + 0.949235i \(0.601860\pi\)
\(858\) −3.77351 −0.128825
\(859\) 33.2889 1.13580 0.567902 0.823096i \(-0.307755\pi\)
0.567902 + 0.823096i \(0.307755\pi\)
\(860\) 0.352170 0.0120089
\(861\) −36.3040 −1.23724
\(862\) −28.1776 −0.959733
\(863\) −39.3854 −1.34070 −0.670348 0.742047i \(-0.733855\pi\)
−0.670348 + 0.742047i \(0.733855\pi\)
\(864\) 1.00000 0.0340207
\(865\) 23.5534 0.800840
\(866\) −18.6955 −0.635301
\(867\) −35.4339 −1.20340
\(868\) −25.2907 −0.858422
\(869\) −34.9935 −1.18707
\(870\) −3.71571 −0.125974
\(871\) −15.9624 −0.540866
\(872\) 13.6544 0.462397
\(873\) −4.13251 −0.139864
\(874\) 2.56944 0.0869125
\(875\) 35.7653 1.20909
\(876\) 7.24146 0.244666
\(877\) 53.7415 1.81472 0.907361 0.420352i \(-0.138094\pi\)
0.907361 + 0.420352i \(0.138094\pi\)
\(878\) 6.18598 0.208767
\(879\) 23.1487 0.780788
\(880\) −5.07905 −0.171215
\(881\) 22.6595 0.763418 0.381709 0.924282i \(-0.375336\pi\)
0.381709 + 0.924282i \(0.375336\pi\)
\(882\) −3.53068 −0.118884
\(883\) 31.9862 1.07642 0.538210 0.842811i \(-0.319101\pi\)
0.538210 + 0.842811i \(0.319101\pi\)
\(884\) −7.24113 −0.243546
\(885\) −7.80559 −0.262382
\(886\) −32.4240 −1.08930
\(887\) −51.3286 −1.72345 −0.861723 0.507378i \(-0.830615\pi\)
−0.861723 + 0.507378i \(0.830615\pi\)
\(888\) −5.09830 −0.171088
\(889\) −16.3369 −0.547921
\(890\) −15.4951 −0.519397
\(891\) −3.77351 −0.126417
\(892\) 4.74985 0.159037
\(893\) 0.284839 0.00953176
\(894\) 18.9960 0.635320
\(895\) −5.12145 −0.171191
\(896\) 3.24510 0.108411
\(897\) 3.88709 0.129786
\(898\) 25.2808 0.843632
\(899\) −21.5148 −0.717559
\(900\) −3.18835 −0.106278
\(901\) 50.5529 1.68416
\(902\) −42.2154 −1.40562
\(903\) 0.849070 0.0282553
\(904\) 1.82889 0.0608281
\(905\) 17.7743 0.590836
\(906\) −2.86092 −0.0950476
\(907\) 21.3592 0.709220 0.354610 0.935014i \(-0.384614\pi\)
0.354610 + 0.935014i \(0.384614\pi\)
\(908\) 14.8843 0.493952
\(909\) −10.6527 −0.353328
\(910\) 4.36783 0.144792
\(911\) 24.3445 0.806570 0.403285 0.915074i \(-0.367868\pi\)
0.403285 + 0.915074i \(0.367868\pi\)
\(912\) −0.661018 −0.0218885
\(913\) −11.7824 −0.389939
\(914\) −16.3853 −0.541976
\(915\) −16.5171 −0.546038
\(916\) −23.6580 −0.781683
\(917\) −58.1500 −1.92028
\(918\) −7.24113 −0.238993
\(919\) 18.3753 0.606146 0.303073 0.952967i \(-0.401987\pi\)
0.303073 + 0.952967i \(0.401987\pi\)
\(920\) 5.23192 0.172491
\(921\) −19.4854 −0.642066
\(922\) −17.8016 −0.586265
\(923\) 1.48957 0.0490298
\(924\) −12.2454 −0.402845
\(925\) 16.2552 0.534467
\(926\) 13.5211 0.444330
\(927\) −1.00000 −0.0328443
\(928\) 2.76061 0.0906214
\(929\) −27.2635 −0.894487 −0.447243 0.894412i \(-0.647594\pi\)
−0.447243 + 0.894412i \(0.647594\pi\)
\(930\) 10.4899 0.343976
\(931\) 2.33385 0.0764887
\(932\) −24.5541 −0.804297
\(933\) 24.2475 0.793826
\(934\) −30.5160 −0.998515
\(935\) 36.7780 1.20277
\(936\) −1.00000 −0.0326860
\(937\) −18.9150 −0.617925 −0.308962 0.951074i \(-0.599982\pi\)
−0.308962 + 0.951074i \(0.599982\pi\)
\(938\) −51.7997 −1.69132
\(939\) −17.0379 −0.556012
\(940\) 0.579993 0.0189173
\(941\) 12.5725 0.409853 0.204927 0.978777i \(-0.434304\pi\)
0.204927 + 0.978777i \(0.434304\pi\)
\(942\) −8.29324 −0.270208
\(943\) 43.4861 1.41610
\(944\) 5.79920 0.188748
\(945\) 4.36783 0.142085
\(946\) 0.987326 0.0321007
\(947\) −18.5027 −0.601258 −0.300629 0.953741i \(-0.597197\pi\)
−0.300629 + 0.953741i \(0.597197\pi\)
\(948\) −9.27347 −0.301188
\(949\) −7.24146 −0.235068
\(950\) 2.10756 0.0683782
\(951\) 17.1424 0.555882
\(952\) −23.4982 −0.761581
\(953\) 41.8571 1.35589 0.677943 0.735115i \(-0.262872\pi\)
0.677943 + 0.735115i \(0.262872\pi\)
\(954\) 6.98136 0.226030
\(955\) −19.5478 −0.632553
\(956\) −0.194543 −0.00629195
\(957\) −10.4172 −0.336739
\(958\) −8.25272 −0.266633
\(959\) −10.6010 −0.342325
\(960\) −1.34598 −0.0434412
\(961\) 29.7387 0.959313
\(962\) 5.09830 0.164376
\(963\) −13.5536 −0.436758
\(964\) 5.77889 0.186126
\(965\) 25.4330 0.818716
\(966\) 12.6140 0.405849
\(967\) 12.5222 0.402687 0.201343 0.979521i \(-0.435469\pi\)
0.201343 + 0.979521i \(0.435469\pi\)
\(968\) −3.23936 −0.104117
\(969\) 4.78652 0.153765
\(970\) 5.56226 0.178593
\(971\) 36.4400 1.16942 0.584708 0.811244i \(-0.301209\pi\)
0.584708 + 0.811244i \(0.301209\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −33.9600 −1.08871
\(974\) 3.27770 0.105024
\(975\) 3.18835 0.102109
\(976\) 12.2715 0.392800
\(977\) −53.3142 −1.70567 −0.852836 0.522179i \(-0.825119\pi\)
−0.852836 + 0.522179i \(0.825119\pi\)
\(978\) 14.7826 0.472697
\(979\) −43.4413 −1.38839
\(980\) 4.75221 0.151804
\(981\) −13.6544 −0.435952
\(982\) −7.49528 −0.239184
\(983\) −0.0934276 −0.00297988 −0.00148994 0.999999i \(-0.500474\pi\)
−0.00148994 + 0.999999i \(0.500474\pi\)
\(984\) −11.1873 −0.356638
\(985\) −6.32078 −0.201397
\(986\) −19.9899 −0.636609
\(987\) 1.39834 0.0445097
\(988\) 0.661018 0.0210298
\(989\) −1.01704 −0.0323401
\(990\) 5.07905 0.161423
\(991\) 13.1691 0.418331 0.209165 0.977880i \(-0.432925\pi\)
0.209165 + 0.977880i \(0.432925\pi\)
\(992\) −7.79350 −0.247444
\(993\) 8.77403 0.278435
\(994\) 4.83381 0.153319
\(995\) −34.6505 −1.09850
\(996\) −3.12239 −0.0989367
\(997\) 45.6533 1.44586 0.722928 0.690924i \(-0.242796\pi\)
0.722928 + 0.690924i \(0.242796\pi\)
\(998\) −14.3424 −0.454000
\(999\) 5.09830 0.161303
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 8034.2.a.z.1.10 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.z.1.10 14 1.1 even 1 trivial