Properties

Label 8034.2.a.ba.1.13
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: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-3.59906\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59906 q^{5} +1.00000 q^{6} +0.919823 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59906 q^{5} +1.00000 q^{6} +0.919823 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.59906 q^{10} +6.28384 q^{11} -1.00000 q^{12} -1.00000 q^{13} -0.919823 q^{14} -3.59906 q^{15} +1.00000 q^{16} -0.964091 q^{17} -1.00000 q^{18} +5.87871 q^{19} +3.59906 q^{20} -0.919823 q^{21} -6.28384 q^{22} -1.55833 q^{23} +1.00000 q^{24} +7.95327 q^{25} +1.00000 q^{26} -1.00000 q^{27} +0.919823 q^{28} +0.377820 q^{29} +3.59906 q^{30} +3.05624 q^{31} -1.00000 q^{32} -6.28384 q^{33} +0.964091 q^{34} +3.31050 q^{35} +1.00000 q^{36} +5.17705 q^{37} -5.87871 q^{38} +1.00000 q^{39} -3.59906 q^{40} +5.29346 q^{41} +0.919823 q^{42} -9.33676 q^{43} +6.28384 q^{44} +3.59906 q^{45} +1.55833 q^{46} +6.01076 q^{47} -1.00000 q^{48} -6.15393 q^{49} -7.95327 q^{50} +0.964091 q^{51} -1.00000 q^{52} +2.69483 q^{53} +1.00000 q^{54} +22.6159 q^{55} -0.919823 q^{56} -5.87871 q^{57} -0.377820 q^{58} +0.838786 q^{59} -3.59906 q^{60} +0.508838 q^{61} -3.05624 q^{62} +0.919823 q^{63} +1.00000 q^{64} -3.59906 q^{65} +6.28384 q^{66} -14.4793 q^{67} -0.964091 q^{68} +1.55833 q^{69} -3.31050 q^{70} -3.24540 q^{71} -1.00000 q^{72} +5.27039 q^{73} -5.17705 q^{74} -7.95327 q^{75} +5.87871 q^{76} +5.78001 q^{77} -1.00000 q^{78} +9.98774 q^{79} +3.59906 q^{80} +1.00000 q^{81} -5.29346 q^{82} -11.4796 q^{83} -0.919823 q^{84} -3.46983 q^{85} +9.33676 q^{86} -0.377820 q^{87} -6.28384 q^{88} +17.3823 q^{89} -3.59906 q^{90} -0.919823 q^{91} -1.55833 q^{92} -3.05624 q^{93} -6.01076 q^{94} +21.1579 q^{95} +1.00000 q^{96} +3.40393 q^{97} +6.15393 q^{98} +6.28384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) 3.59906 1.60955 0.804775 0.593579i \(-0.202286\pi\)
0.804775 + 0.593579i \(0.202286\pi\)
\(6\) 1.00000 0.408248
\(7\) 0.919823 0.347660 0.173830 0.984776i \(-0.444386\pi\)
0.173830 + 0.984776i \(0.444386\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.59906 −1.13812
\(11\) 6.28384 1.89465 0.947324 0.320277i \(-0.103776\pi\)
0.947324 + 0.320277i \(0.103776\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −0.919823 −0.245833
\(15\) −3.59906 −0.929275
\(16\) 1.00000 0.250000
\(17\) −0.964091 −0.233826 −0.116913 0.993142i \(-0.537300\pi\)
−0.116913 + 0.993142i \(0.537300\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.87871 1.34867 0.674335 0.738426i \(-0.264431\pi\)
0.674335 + 0.738426i \(0.264431\pi\)
\(20\) 3.59906 0.804775
\(21\) −0.919823 −0.200722
\(22\) −6.28384 −1.33972
\(23\) −1.55833 −0.324934 −0.162467 0.986714i \(-0.551945\pi\)
−0.162467 + 0.986714i \(0.551945\pi\)
\(24\) 1.00000 0.204124
\(25\) 7.95327 1.59065
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 0.919823 0.173830
\(29\) 0.377820 0.0701595 0.0350797 0.999385i \(-0.488831\pi\)
0.0350797 + 0.999385i \(0.488831\pi\)
\(30\) 3.59906 0.657096
\(31\) 3.05624 0.548917 0.274458 0.961599i \(-0.411501\pi\)
0.274458 + 0.961599i \(0.411501\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.28384 −1.09388
\(34\) 0.964091 0.165340
\(35\) 3.31050 0.559577
\(36\) 1.00000 0.166667
\(37\) 5.17705 0.851101 0.425551 0.904935i \(-0.360080\pi\)
0.425551 + 0.904935i \(0.360080\pi\)
\(38\) −5.87871 −0.953653
\(39\) 1.00000 0.160128
\(40\) −3.59906 −0.569062
\(41\) 5.29346 0.826700 0.413350 0.910572i \(-0.364359\pi\)
0.413350 + 0.910572i \(0.364359\pi\)
\(42\) 0.919823 0.141932
\(43\) −9.33676 −1.42384 −0.711921 0.702259i \(-0.752175\pi\)
−0.711921 + 0.702259i \(0.752175\pi\)
\(44\) 6.28384 0.947324
\(45\) 3.59906 0.536517
\(46\) 1.55833 0.229763
\(47\) 6.01076 0.876759 0.438379 0.898790i \(-0.355553\pi\)
0.438379 + 0.898790i \(0.355553\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.15393 −0.879132
\(50\) −7.95327 −1.12476
\(51\) 0.964091 0.135000
\(52\) −1.00000 −0.138675
\(53\) 2.69483 0.370163 0.185081 0.982723i \(-0.440745\pi\)
0.185081 + 0.982723i \(0.440745\pi\)
\(54\) 1.00000 0.136083
\(55\) 22.6159 3.04953
\(56\) −0.919823 −0.122916
\(57\) −5.87871 −0.778655
\(58\) −0.377820 −0.0496102
\(59\) 0.838786 0.109201 0.0546003 0.998508i \(-0.482612\pi\)
0.0546003 + 0.998508i \(0.482612\pi\)
\(60\) −3.59906 −0.464637
\(61\) 0.508838 0.0651501 0.0325750 0.999469i \(-0.489629\pi\)
0.0325750 + 0.999469i \(0.489629\pi\)
\(62\) −3.05624 −0.388143
\(63\) 0.919823 0.115887
\(64\) 1.00000 0.125000
\(65\) −3.59906 −0.446409
\(66\) 6.28384 0.773487
\(67\) −14.4793 −1.76893 −0.884465 0.466606i \(-0.845477\pi\)
−0.884465 + 0.466606i \(0.845477\pi\)
\(68\) −0.964091 −0.116913
\(69\) 1.55833 0.187601
\(70\) −3.31050 −0.395681
\(71\) −3.24540 −0.385158 −0.192579 0.981282i \(-0.561685\pi\)
−0.192579 + 0.981282i \(0.561685\pi\)
\(72\) −1.00000 −0.117851
\(73\) 5.27039 0.616852 0.308426 0.951248i \(-0.400198\pi\)
0.308426 + 0.951248i \(0.400198\pi\)
\(74\) −5.17705 −0.601819
\(75\) −7.95327 −0.918364
\(76\) 5.87871 0.674335
\(77\) 5.78001 0.658694
\(78\) −1.00000 −0.113228
\(79\) 9.98774 1.12371 0.561854 0.827236i \(-0.310088\pi\)
0.561854 + 0.827236i \(0.310088\pi\)
\(80\) 3.59906 0.402388
\(81\) 1.00000 0.111111
\(82\) −5.29346 −0.584565
\(83\) −11.4796 −1.26005 −0.630024 0.776576i \(-0.716955\pi\)
−0.630024 + 0.776576i \(0.716955\pi\)
\(84\) −0.919823 −0.100361
\(85\) −3.46983 −0.376356
\(86\) 9.33676 1.00681
\(87\) −0.377820 −0.0405066
\(88\) −6.28384 −0.669859
\(89\) 17.3823 1.84252 0.921260 0.388946i \(-0.127161\pi\)
0.921260 + 0.388946i \(0.127161\pi\)
\(90\) −3.59906 −0.379375
\(91\) −0.919823 −0.0964236
\(92\) −1.55833 −0.162467
\(93\) −3.05624 −0.316917
\(94\) −6.01076 −0.619962
\(95\) 21.1579 2.17075
\(96\) 1.00000 0.102062
\(97\) 3.40393 0.345617 0.172808 0.984955i \(-0.444716\pi\)
0.172808 + 0.984955i \(0.444716\pi\)
\(98\) 6.15393 0.621640
\(99\) 6.28384 0.631549
\(100\) 7.95327 0.795327
\(101\) 0.987205 0.0982306 0.0491153 0.998793i \(-0.484360\pi\)
0.0491153 + 0.998793i \(0.484360\pi\)
\(102\) −0.964091 −0.0954593
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −3.31050 −0.323072
\(106\) −2.69483 −0.261745
\(107\) 17.9742 1.73763 0.868814 0.495138i \(-0.164883\pi\)
0.868814 + 0.495138i \(0.164883\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.38673 −0.803303 −0.401651 0.915793i \(-0.631564\pi\)
−0.401651 + 0.915793i \(0.631564\pi\)
\(110\) −22.6159 −2.15634
\(111\) −5.17705 −0.491384
\(112\) 0.919823 0.0869151
\(113\) 5.12907 0.482502 0.241251 0.970463i \(-0.422442\pi\)
0.241251 + 0.970463i \(0.422442\pi\)
\(114\) 5.87871 0.550592
\(115\) −5.60853 −0.522998
\(116\) 0.377820 0.0350797
\(117\) −1.00000 −0.0924500
\(118\) −0.838786 −0.0772165
\(119\) −0.886793 −0.0812922
\(120\) 3.59906 0.328548
\(121\) 28.4866 2.58969
\(122\) −0.508838 −0.0460681
\(123\) −5.29346 −0.477295
\(124\) 3.05624 0.274458
\(125\) 10.6290 0.950687
\(126\) −0.919823 −0.0819443
\(127\) 4.19542 0.372284 0.186142 0.982523i \(-0.440402\pi\)
0.186142 + 0.982523i \(0.440402\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 9.33676 0.822056
\(130\) 3.59906 0.315659
\(131\) 3.43244 0.299894 0.149947 0.988694i \(-0.452090\pi\)
0.149947 + 0.988694i \(0.452090\pi\)
\(132\) −6.28384 −0.546938
\(133\) 5.40737 0.468879
\(134\) 14.4793 1.25082
\(135\) −3.59906 −0.309758
\(136\) 0.964091 0.0826701
\(137\) −17.0982 −1.46079 −0.730397 0.683023i \(-0.760665\pi\)
−0.730397 + 0.683023i \(0.760665\pi\)
\(138\) −1.55833 −0.132654
\(139\) −9.30098 −0.788899 −0.394449 0.918918i \(-0.629065\pi\)
−0.394449 + 0.918918i \(0.629065\pi\)
\(140\) 3.31050 0.279788
\(141\) −6.01076 −0.506197
\(142\) 3.24540 0.272348
\(143\) −6.28384 −0.525481
\(144\) 1.00000 0.0833333
\(145\) 1.35980 0.112925
\(146\) −5.27039 −0.436180
\(147\) 6.15393 0.507567
\(148\) 5.17705 0.425551
\(149\) −7.18823 −0.588883 −0.294442 0.955670i \(-0.595134\pi\)
−0.294442 + 0.955670i \(0.595134\pi\)
\(150\) 7.95327 0.649382
\(151\) −12.2638 −0.998013 −0.499006 0.866598i \(-0.666302\pi\)
−0.499006 + 0.866598i \(0.666302\pi\)
\(152\) −5.87871 −0.476827
\(153\) −0.964091 −0.0779422
\(154\) −5.78001 −0.465767
\(155\) 10.9996 0.883510
\(156\) 1.00000 0.0800641
\(157\) −16.6452 −1.32843 −0.664216 0.747540i \(-0.731235\pi\)
−0.664216 + 0.747540i \(0.731235\pi\)
\(158\) −9.98774 −0.794582
\(159\) −2.69483 −0.213714
\(160\) −3.59906 −0.284531
\(161\) −1.43339 −0.112967
\(162\) −1.00000 −0.0785674
\(163\) 19.0669 1.49344 0.746719 0.665140i \(-0.231628\pi\)
0.746719 + 0.665140i \(0.231628\pi\)
\(164\) 5.29346 0.413350
\(165\) −22.6159 −1.76065
\(166\) 11.4796 0.890988
\(167\) 3.57968 0.277004 0.138502 0.990362i \(-0.455771\pi\)
0.138502 + 0.990362i \(0.455771\pi\)
\(168\) 0.919823 0.0709658
\(169\) 1.00000 0.0769231
\(170\) 3.46983 0.266124
\(171\) 5.87871 0.449556
\(172\) −9.33676 −0.711921
\(173\) −1.57490 −0.119738 −0.0598689 0.998206i \(-0.519068\pi\)
−0.0598689 + 0.998206i \(0.519068\pi\)
\(174\) 0.377820 0.0286425
\(175\) 7.31559 0.553007
\(176\) 6.28384 0.473662
\(177\) −0.838786 −0.0630470
\(178\) −17.3823 −1.30286
\(179\) −8.00874 −0.598601 −0.299301 0.954159i \(-0.596753\pi\)
−0.299301 + 0.954159i \(0.596753\pi\)
\(180\) 3.59906 0.268258
\(181\) 6.76636 0.502939 0.251470 0.967865i \(-0.419086\pi\)
0.251470 + 0.967865i \(0.419086\pi\)
\(182\) 0.919823 0.0681818
\(183\) −0.508838 −0.0376144
\(184\) 1.55833 0.114882
\(185\) 18.6325 1.36989
\(186\) 3.05624 0.224094
\(187\) −6.05819 −0.443019
\(188\) 6.01076 0.438379
\(189\) −0.919823 −0.0669072
\(190\) −21.1579 −1.53495
\(191\) 3.62788 0.262504 0.131252 0.991349i \(-0.458100\pi\)
0.131252 + 0.991349i \(0.458100\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −17.8569 −1.28537 −0.642684 0.766132i \(-0.722179\pi\)
−0.642684 + 0.766132i \(0.722179\pi\)
\(194\) −3.40393 −0.244388
\(195\) 3.59906 0.257734
\(196\) −6.15393 −0.439566
\(197\) −1.10763 −0.0789151 −0.0394575 0.999221i \(-0.512563\pi\)
−0.0394575 + 0.999221i \(0.512563\pi\)
\(198\) −6.28384 −0.446573
\(199\) −9.47297 −0.671521 −0.335761 0.941947i \(-0.608993\pi\)
−0.335761 + 0.941947i \(0.608993\pi\)
\(200\) −7.95327 −0.562381
\(201\) 14.4793 1.02129
\(202\) −0.987205 −0.0694595
\(203\) 0.347528 0.0243917
\(204\) 0.964091 0.0674999
\(205\) 19.0515 1.33062
\(206\) −1.00000 −0.0696733
\(207\) −1.55833 −0.108311
\(208\) −1.00000 −0.0693375
\(209\) 36.9409 2.55525
\(210\) 3.31050 0.228446
\(211\) 16.5894 1.14206 0.571030 0.820929i \(-0.306544\pi\)
0.571030 + 0.820929i \(0.306544\pi\)
\(212\) 2.69483 0.185081
\(213\) 3.24540 0.222371
\(214\) −17.9742 −1.22869
\(215\) −33.6036 −2.29175
\(216\) 1.00000 0.0680414
\(217\) 2.81120 0.190837
\(218\) 8.38673 0.568021
\(219\) −5.27039 −0.356140
\(220\) 22.6159 1.52477
\(221\) 0.964091 0.0648518
\(222\) 5.17705 0.347461
\(223\) 10.5549 0.706811 0.353406 0.935470i \(-0.385024\pi\)
0.353406 + 0.935470i \(0.385024\pi\)
\(224\) −0.919823 −0.0614582
\(225\) 7.95327 0.530218
\(226\) −5.12907 −0.341180
\(227\) −6.01618 −0.399308 −0.199654 0.979866i \(-0.563982\pi\)
−0.199654 + 0.979866i \(0.563982\pi\)
\(228\) −5.87871 −0.389327
\(229\) −17.7551 −1.17329 −0.586645 0.809844i \(-0.699551\pi\)
−0.586645 + 0.809844i \(0.699551\pi\)
\(230\) 5.60853 0.369816
\(231\) −5.78001 −0.380297
\(232\) −0.377820 −0.0248051
\(233\) −11.1388 −0.729726 −0.364863 0.931061i \(-0.618884\pi\)
−0.364863 + 0.931061i \(0.618884\pi\)
\(234\) 1.00000 0.0653720
\(235\) 21.6331 1.41119
\(236\) 0.838786 0.0546003
\(237\) −9.98774 −0.648774
\(238\) 0.886793 0.0574822
\(239\) −11.3486 −0.734080 −0.367040 0.930205i \(-0.619629\pi\)
−0.367040 + 0.930205i \(0.619629\pi\)
\(240\) −3.59906 −0.232319
\(241\) 13.8214 0.890316 0.445158 0.895452i \(-0.353148\pi\)
0.445158 + 0.895452i \(0.353148\pi\)
\(242\) −28.4866 −1.83119
\(243\) −1.00000 −0.0641500
\(244\) 0.508838 0.0325750
\(245\) −22.1484 −1.41501
\(246\) 5.29346 0.337499
\(247\) −5.87871 −0.374054
\(248\) −3.05624 −0.194071
\(249\) 11.4796 0.727489
\(250\) −10.6290 −0.672237
\(251\) 16.6162 1.04880 0.524402 0.851471i \(-0.324289\pi\)
0.524402 + 0.851471i \(0.324289\pi\)
\(252\) 0.919823 0.0579434
\(253\) −9.79230 −0.615636
\(254\) −4.19542 −0.263244
\(255\) 3.46983 0.217289
\(256\) 1.00000 0.0625000
\(257\) −15.8529 −0.988879 −0.494440 0.869212i \(-0.664627\pi\)
−0.494440 + 0.869212i \(0.664627\pi\)
\(258\) −9.33676 −0.581281
\(259\) 4.76196 0.295894
\(260\) −3.59906 −0.223205
\(261\) 0.377820 0.0233865
\(262\) −3.43244 −0.212057
\(263\) −14.0523 −0.866502 −0.433251 0.901273i \(-0.642634\pi\)
−0.433251 + 0.901273i \(0.642634\pi\)
\(264\) 6.28384 0.386743
\(265\) 9.69886 0.595796
\(266\) −5.40737 −0.331547
\(267\) −17.3823 −1.06378
\(268\) −14.4793 −0.884465
\(269\) −21.7523 −1.32626 −0.663130 0.748505i \(-0.730772\pi\)
−0.663130 + 0.748505i \(0.730772\pi\)
\(270\) 3.59906 0.219032
\(271\) 29.6801 1.80294 0.901469 0.432844i \(-0.142490\pi\)
0.901469 + 0.432844i \(0.142490\pi\)
\(272\) −0.964091 −0.0584566
\(273\) 0.919823 0.0556702
\(274\) 17.0982 1.03294
\(275\) 49.9770 3.01373
\(276\) 1.55833 0.0938005
\(277\) 0.303986 0.0182647 0.00913237 0.999958i \(-0.497093\pi\)
0.00913237 + 0.999958i \(0.497093\pi\)
\(278\) 9.30098 0.557836
\(279\) 3.05624 0.182972
\(280\) −3.31050 −0.197840
\(281\) −31.4890 −1.87848 −0.939240 0.343262i \(-0.888468\pi\)
−0.939240 + 0.343262i \(0.888468\pi\)
\(282\) 6.01076 0.357935
\(283\) 7.53378 0.447837 0.223918 0.974608i \(-0.428115\pi\)
0.223918 + 0.974608i \(0.428115\pi\)
\(284\) −3.24540 −0.192579
\(285\) −21.1579 −1.25328
\(286\) 6.28384 0.371571
\(287\) 4.86905 0.287411
\(288\) −1.00000 −0.0589256
\(289\) −16.0705 −0.945325
\(290\) −1.35980 −0.0798502
\(291\) −3.40393 −0.199542
\(292\) 5.27039 0.308426
\(293\) 28.1598 1.64511 0.822556 0.568683i \(-0.192547\pi\)
0.822556 + 0.568683i \(0.192547\pi\)
\(294\) −6.15393 −0.358904
\(295\) 3.01885 0.175764
\(296\) −5.17705 −0.300910
\(297\) −6.28384 −0.364625
\(298\) 7.18823 0.416403
\(299\) 1.55833 0.0901206
\(300\) −7.95327 −0.459182
\(301\) −8.58816 −0.495013
\(302\) 12.2638 0.705701
\(303\) −0.987205 −0.0567135
\(304\) 5.87871 0.337167
\(305\) 1.83134 0.104862
\(306\) 0.964091 0.0551134
\(307\) 3.43777 0.196204 0.0981021 0.995176i \(-0.468723\pi\)
0.0981021 + 0.995176i \(0.468723\pi\)
\(308\) 5.78001 0.329347
\(309\) −1.00000 −0.0568880
\(310\) −10.9996 −0.624736
\(311\) −1.77167 −0.100462 −0.0502311 0.998738i \(-0.515996\pi\)
−0.0502311 + 0.998738i \(0.515996\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −27.0983 −1.53169 −0.765843 0.643028i \(-0.777678\pi\)
−0.765843 + 0.643028i \(0.777678\pi\)
\(314\) 16.6452 0.939344
\(315\) 3.31050 0.186526
\(316\) 9.98774 0.561854
\(317\) −26.5689 −1.49226 −0.746130 0.665800i \(-0.768090\pi\)
−0.746130 + 0.665800i \(0.768090\pi\)
\(318\) 2.69483 0.151118
\(319\) 2.37416 0.132928
\(320\) 3.59906 0.201194
\(321\) −17.9742 −1.00322
\(322\) 1.43339 0.0798796
\(323\) −5.66762 −0.315355
\(324\) 1.00000 0.0555556
\(325\) −7.95327 −0.441168
\(326\) −19.0669 −1.05602
\(327\) 8.38673 0.463787
\(328\) −5.29346 −0.292283
\(329\) 5.52883 0.304814
\(330\) 22.6159 1.24497
\(331\) −23.2926 −1.28028 −0.640139 0.768259i \(-0.721123\pi\)
−0.640139 + 0.768259i \(0.721123\pi\)
\(332\) −11.4796 −0.630024
\(333\) 5.17705 0.283700
\(334\) −3.57968 −0.195871
\(335\) −52.1120 −2.84718
\(336\) −0.919823 −0.0501804
\(337\) 15.3551 0.836446 0.418223 0.908344i \(-0.362653\pi\)
0.418223 + 0.908344i \(0.362653\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −5.12907 −0.278573
\(340\) −3.46983 −0.188178
\(341\) 19.2049 1.04000
\(342\) −5.87871 −0.317884
\(343\) −12.0993 −0.653300
\(344\) 9.33676 0.503404
\(345\) 5.60853 0.301953
\(346\) 1.57490 0.0846674
\(347\) −21.2368 −1.14005 −0.570025 0.821628i \(-0.693066\pi\)
−0.570025 + 0.821628i \(0.693066\pi\)
\(348\) −0.377820 −0.0202533
\(349\) 26.5667 1.42208 0.711041 0.703150i \(-0.248224\pi\)
0.711041 + 0.703150i \(0.248224\pi\)
\(350\) −7.31559 −0.391035
\(351\) 1.00000 0.0533761
\(352\) −6.28384 −0.334930
\(353\) −22.3861 −1.19149 −0.595746 0.803173i \(-0.703143\pi\)
−0.595746 + 0.803173i \(0.703143\pi\)
\(354\) 0.838786 0.0445810
\(355\) −11.6804 −0.619931
\(356\) 17.3823 0.921260
\(357\) 0.886793 0.0469341
\(358\) 8.00874 0.423275
\(359\) −8.20847 −0.433227 −0.216613 0.976257i \(-0.569501\pi\)
−0.216613 + 0.976257i \(0.569501\pi\)
\(360\) −3.59906 −0.189687
\(361\) 15.5593 0.818909
\(362\) −6.76636 −0.355632
\(363\) −28.4866 −1.49516
\(364\) −0.919823 −0.0482118
\(365\) 18.9685 0.992855
\(366\) 0.508838 0.0265974
\(367\) 30.9920 1.61777 0.808883 0.587969i \(-0.200072\pi\)
0.808883 + 0.587969i \(0.200072\pi\)
\(368\) −1.55833 −0.0812336
\(369\) 5.29346 0.275567
\(370\) −18.6325 −0.968659
\(371\) 2.47876 0.128691
\(372\) −3.05624 −0.158459
\(373\) −6.06159 −0.313857 −0.156929 0.987610i \(-0.550159\pi\)
−0.156929 + 0.987610i \(0.550159\pi\)
\(374\) 6.05819 0.313262
\(375\) −10.6290 −0.548879
\(376\) −6.01076 −0.309981
\(377\) −0.377820 −0.0194587
\(378\) 0.919823 0.0473106
\(379\) 6.33028 0.325164 0.162582 0.986695i \(-0.448018\pi\)
0.162582 + 0.986695i \(0.448018\pi\)
\(380\) 21.1579 1.08538
\(381\) −4.19542 −0.214938
\(382\) −3.62788 −0.185618
\(383\) −7.01057 −0.358224 −0.179112 0.983829i \(-0.557322\pi\)
−0.179112 + 0.983829i \(0.557322\pi\)
\(384\) 1.00000 0.0510310
\(385\) 20.8026 1.06020
\(386\) 17.8569 0.908892
\(387\) −9.33676 −0.474614
\(388\) 3.40393 0.172808
\(389\) −5.80948 −0.294552 −0.147276 0.989095i \(-0.547051\pi\)
−0.147276 + 0.989095i \(0.547051\pi\)
\(390\) −3.59906 −0.182246
\(391\) 1.50237 0.0759783
\(392\) 6.15393 0.310820
\(393\) −3.43244 −0.173144
\(394\) 1.10763 0.0558014
\(395\) 35.9465 1.80867
\(396\) 6.28384 0.315775
\(397\) −21.4430 −1.07620 −0.538098 0.842882i \(-0.680857\pi\)
−0.538098 + 0.842882i \(0.680857\pi\)
\(398\) 9.47297 0.474837
\(399\) −5.40737 −0.270707
\(400\) 7.95327 0.397663
\(401\) −4.30754 −0.215108 −0.107554 0.994199i \(-0.534302\pi\)
−0.107554 + 0.994199i \(0.534302\pi\)
\(402\) −14.4793 −0.722163
\(403\) −3.05624 −0.152242
\(404\) 0.987205 0.0491153
\(405\) 3.59906 0.178839
\(406\) −0.347528 −0.0172475
\(407\) 32.5317 1.61254
\(408\) −0.964091 −0.0477296
\(409\) 3.40095 0.168166 0.0840830 0.996459i \(-0.473204\pi\)
0.0840830 + 0.996459i \(0.473204\pi\)
\(410\) −19.0515 −0.940887
\(411\) 17.0982 0.843390
\(412\) 1.00000 0.0492665
\(413\) 0.771535 0.0379647
\(414\) 1.55833 0.0765878
\(415\) −41.3158 −2.02811
\(416\) 1.00000 0.0490290
\(417\) 9.30098 0.455471
\(418\) −36.9409 −1.80684
\(419\) 18.2249 0.890345 0.445173 0.895445i \(-0.353142\pi\)
0.445173 + 0.895445i \(0.353142\pi\)
\(420\) −3.31050 −0.161536
\(421\) 27.8199 1.35586 0.677930 0.735127i \(-0.262877\pi\)
0.677930 + 0.735127i \(0.262877\pi\)
\(422\) −16.5894 −0.807558
\(423\) 6.01076 0.292253
\(424\) −2.69483 −0.130872
\(425\) −7.66768 −0.371937
\(426\) −3.24540 −0.157240
\(427\) 0.468041 0.0226501
\(428\) 17.9742 0.868814
\(429\) 6.28384 0.303386
\(430\) 33.6036 1.62051
\(431\) 8.55540 0.412099 0.206050 0.978542i \(-0.433939\pi\)
0.206050 + 0.978542i \(0.433939\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −12.5585 −0.603525 −0.301763 0.953383i \(-0.597575\pi\)
−0.301763 + 0.953383i \(0.597575\pi\)
\(434\) −2.81120 −0.134942
\(435\) −1.35980 −0.0651974
\(436\) −8.38673 −0.401651
\(437\) −9.16098 −0.438229
\(438\) 5.27039 0.251829
\(439\) −8.35803 −0.398907 −0.199454 0.979907i \(-0.563917\pi\)
−0.199454 + 0.979907i \(0.563917\pi\)
\(440\) −22.6159 −1.07817
\(441\) −6.15393 −0.293044
\(442\) −0.964091 −0.0458571
\(443\) 11.2701 0.535459 0.267730 0.963494i \(-0.413727\pi\)
0.267730 + 0.963494i \(0.413727\pi\)
\(444\) −5.17705 −0.245692
\(445\) 62.5601 2.96563
\(446\) −10.5549 −0.499791
\(447\) 7.18823 0.339992
\(448\) 0.919823 0.0434575
\(449\) −16.6832 −0.787330 −0.393665 0.919254i \(-0.628793\pi\)
−0.393665 + 0.919254i \(0.628793\pi\)
\(450\) −7.95327 −0.374921
\(451\) 33.2633 1.56631
\(452\) 5.12907 0.241251
\(453\) 12.2638 0.576203
\(454\) 6.01618 0.282353
\(455\) −3.31050 −0.155199
\(456\) 5.87871 0.275296
\(457\) 32.0280 1.49821 0.749104 0.662452i \(-0.230484\pi\)
0.749104 + 0.662452i \(0.230484\pi\)
\(458\) 17.7551 0.829641
\(459\) 0.964091 0.0449999
\(460\) −5.60853 −0.261499
\(461\) −30.6252 −1.42636 −0.713179 0.700982i \(-0.752745\pi\)
−0.713179 + 0.700982i \(0.752745\pi\)
\(462\) 5.78001 0.268911
\(463\) −11.1986 −0.520443 −0.260221 0.965549i \(-0.583796\pi\)
−0.260221 + 0.965549i \(0.583796\pi\)
\(464\) 0.377820 0.0175399
\(465\) −10.9996 −0.510095
\(466\) 11.1388 0.515994
\(467\) 11.7793 0.545080 0.272540 0.962144i \(-0.412136\pi\)
0.272540 + 0.962144i \(0.412136\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −13.3184 −0.614987
\(470\) −21.6331 −0.997861
\(471\) 16.6452 0.766971
\(472\) −0.838786 −0.0386083
\(473\) −58.6707 −2.69768
\(474\) 9.98774 0.458752
\(475\) 46.7550 2.14527
\(476\) −0.886793 −0.0406461
\(477\) 2.69483 0.123388
\(478\) 11.3486 0.519073
\(479\) 7.23414 0.330536 0.165268 0.986249i \(-0.447151\pi\)
0.165268 + 0.986249i \(0.447151\pi\)
\(480\) 3.59906 0.164274
\(481\) −5.17705 −0.236053
\(482\) −13.8214 −0.629548
\(483\) 1.43339 0.0652214
\(484\) 28.4866 1.29485
\(485\) 12.2510 0.556288
\(486\) 1.00000 0.0453609
\(487\) 14.2025 0.643578 0.321789 0.946811i \(-0.395716\pi\)
0.321789 + 0.946811i \(0.395716\pi\)
\(488\) −0.508838 −0.0230340
\(489\) −19.0669 −0.862237
\(490\) 22.1484 1.00056
\(491\) 20.5180 0.925964 0.462982 0.886368i \(-0.346779\pi\)
0.462982 + 0.886368i \(0.346779\pi\)
\(492\) −5.29346 −0.238648
\(493\) −0.364253 −0.0164051
\(494\) 5.87871 0.264496
\(495\) 22.6159 1.01651
\(496\) 3.05624 0.137229
\(497\) −2.98519 −0.133904
\(498\) −11.4796 −0.514412
\(499\) 41.1316 1.84130 0.920651 0.390387i \(-0.127659\pi\)
0.920651 + 0.390387i \(0.127659\pi\)
\(500\) 10.6290 0.475343
\(501\) −3.57968 −0.159928
\(502\) −16.6162 −0.741616
\(503\) −24.2328 −1.08049 −0.540244 0.841508i \(-0.681668\pi\)
−0.540244 + 0.841508i \(0.681668\pi\)
\(504\) −0.919823 −0.0409722
\(505\) 3.55302 0.158107
\(506\) 9.79230 0.435321
\(507\) −1.00000 −0.0444116
\(508\) 4.19542 0.186142
\(509\) −31.4345 −1.39331 −0.696655 0.717407i \(-0.745329\pi\)
−0.696655 + 0.717407i \(0.745329\pi\)
\(510\) −3.46983 −0.153647
\(511\) 4.84782 0.214455
\(512\) −1.00000 −0.0441942
\(513\) −5.87871 −0.259552
\(514\) 15.8529 0.699243
\(515\) 3.59906 0.158594
\(516\) 9.33676 0.411028
\(517\) 37.7706 1.66115
\(518\) −4.76196 −0.209229
\(519\) 1.57490 0.0691306
\(520\) 3.59906 0.157829
\(521\) −7.44209 −0.326044 −0.163022 0.986622i \(-0.552124\pi\)
−0.163022 + 0.986622i \(0.552124\pi\)
\(522\) −0.377820 −0.0165367
\(523\) 6.33874 0.277174 0.138587 0.990350i \(-0.455744\pi\)
0.138587 + 0.990350i \(0.455744\pi\)
\(524\) 3.43244 0.149947
\(525\) −7.31559 −0.319279
\(526\) 14.0523 0.612709
\(527\) −2.94649 −0.128351
\(528\) −6.28384 −0.273469
\(529\) −20.5716 −0.894418
\(530\) −9.69886 −0.421291
\(531\) 0.838786 0.0364002
\(532\) 5.40737 0.234439
\(533\) −5.29346 −0.229285
\(534\) 17.3823 0.752206
\(535\) 64.6902 2.79680
\(536\) 14.4793 0.625412
\(537\) 8.00874 0.345602
\(538\) 21.7523 0.937807
\(539\) −38.6703 −1.66565
\(540\) −3.59906 −0.154879
\(541\) −33.7271 −1.45004 −0.725020 0.688728i \(-0.758169\pi\)
−0.725020 + 0.688728i \(0.758169\pi\)
\(542\) −29.6801 −1.27487
\(543\) −6.76636 −0.290372
\(544\) 0.964091 0.0413351
\(545\) −30.1844 −1.29296
\(546\) −0.919823 −0.0393648
\(547\) 24.8785 1.06373 0.531864 0.846830i \(-0.321492\pi\)
0.531864 + 0.846830i \(0.321492\pi\)
\(548\) −17.0982 −0.730397
\(549\) 0.508838 0.0217167
\(550\) −49.9770 −2.13103
\(551\) 2.22110 0.0946219
\(552\) −1.55833 −0.0663270
\(553\) 9.18695 0.390669
\(554\) −0.303986 −0.0129151
\(555\) −18.6325 −0.790907
\(556\) −9.30098 −0.394449
\(557\) −23.3359 −0.988775 −0.494387 0.869242i \(-0.664608\pi\)
−0.494387 + 0.869242i \(0.664608\pi\)
\(558\) −3.05624 −0.129381
\(559\) 9.33676 0.394903
\(560\) 3.31050 0.139894
\(561\) 6.05819 0.255777
\(562\) 31.4890 1.32829
\(563\) −6.00586 −0.253117 −0.126558 0.991959i \(-0.540393\pi\)
−0.126558 + 0.991959i \(0.540393\pi\)
\(564\) −6.01076 −0.253099
\(565\) 18.4598 0.776611
\(566\) −7.53378 −0.316669
\(567\) 0.919823 0.0386289
\(568\) 3.24540 0.136174
\(569\) 11.7306 0.491772 0.245886 0.969299i \(-0.420921\pi\)
0.245886 + 0.969299i \(0.420921\pi\)
\(570\) 21.1579 0.886206
\(571\) −16.8136 −0.703627 −0.351814 0.936070i \(-0.614435\pi\)
−0.351814 + 0.936070i \(0.614435\pi\)
\(572\) −6.28384 −0.262740
\(573\) −3.62788 −0.151557
\(574\) −4.86905 −0.203230
\(575\) −12.3938 −0.516858
\(576\) 1.00000 0.0416667
\(577\) −33.4762 −1.39363 −0.696817 0.717249i \(-0.745401\pi\)
−0.696817 + 0.717249i \(0.745401\pi\)
\(578\) 16.0705 0.668446
\(579\) 17.8569 0.742107
\(580\) 1.35980 0.0564626
\(581\) −10.5592 −0.438069
\(582\) 3.40393 0.141097
\(583\) 16.9339 0.701329
\(584\) −5.27039 −0.218090
\(585\) −3.59906 −0.148803
\(586\) −28.1598 −1.16327
\(587\) −29.1504 −1.20317 −0.601583 0.798810i \(-0.705463\pi\)
−0.601583 + 0.798810i \(0.705463\pi\)
\(588\) 6.15393 0.253784
\(589\) 17.9668 0.740307
\(590\) −3.01885 −0.124284
\(591\) 1.10763 0.0455616
\(592\) 5.17705 0.212775
\(593\) 5.65412 0.232187 0.116093 0.993238i \(-0.462963\pi\)
0.116093 + 0.993238i \(0.462963\pi\)
\(594\) 6.28384 0.257829
\(595\) −3.19163 −0.130844
\(596\) −7.18823 −0.294442
\(597\) 9.47297 0.387703
\(598\) −1.55833 −0.0637249
\(599\) −34.9730 −1.42896 −0.714480 0.699656i \(-0.753336\pi\)
−0.714480 + 0.699656i \(0.753336\pi\)
\(600\) 7.95327 0.324691
\(601\) 32.1974 1.31336 0.656680 0.754170i \(-0.271960\pi\)
0.656680 + 0.754170i \(0.271960\pi\)
\(602\) 8.58816 0.350027
\(603\) −14.4793 −0.589644
\(604\) −12.2638 −0.499006
\(605\) 102.525 4.16824
\(606\) 0.987205 0.0401025
\(607\) −1.13354 −0.0460088 −0.0230044 0.999735i \(-0.507323\pi\)
−0.0230044 + 0.999735i \(0.507323\pi\)
\(608\) −5.87871 −0.238413
\(609\) −0.347528 −0.0140825
\(610\) −1.83134 −0.0741489
\(611\) −6.01076 −0.243169
\(612\) −0.964091 −0.0389711
\(613\) −6.05830 −0.244693 −0.122346 0.992487i \(-0.539042\pi\)
−0.122346 + 0.992487i \(0.539042\pi\)
\(614\) −3.43777 −0.138737
\(615\) −19.0515 −0.768231
\(616\) −5.78001 −0.232883
\(617\) 24.6131 0.990885 0.495442 0.868641i \(-0.335006\pi\)
0.495442 + 0.868641i \(0.335006\pi\)
\(618\) 1.00000 0.0402259
\(619\) 30.9999 1.24599 0.622995 0.782226i \(-0.285916\pi\)
0.622995 + 0.782226i \(0.285916\pi\)
\(620\) 10.9996 0.441755
\(621\) 1.55833 0.0625337
\(622\) 1.77167 0.0710375
\(623\) 15.9886 0.640571
\(624\) 1.00000 0.0400320
\(625\) −1.51187 −0.0604748
\(626\) 27.0983 1.08306
\(627\) −36.9409 −1.47528
\(628\) −16.6452 −0.664216
\(629\) −4.99115 −0.199010
\(630\) −3.31050 −0.131894
\(631\) 20.1742 0.803122 0.401561 0.915832i \(-0.368468\pi\)
0.401561 + 0.915832i \(0.368468\pi\)
\(632\) −9.98774 −0.397291
\(633\) −16.5894 −0.659368
\(634\) 26.5689 1.05519
\(635\) 15.0996 0.599209
\(636\) −2.69483 −0.106857
\(637\) 6.15393 0.243827
\(638\) −2.37416 −0.0939939
\(639\) −3.24540 −0.128386
\(640\) −3.59906 −0.142266
\(641\) −9.99257 −0.394683 −0.197341 0.980335i \(-0.563231\pi\)
−0.197341 + 0.980335i \(0.563231\pi\)
\(642\) 17.9742 0.709384
\(643\) −10.2068 −0.402515 −0.201257 0.979538i \(-0.564503\pi\)
−0.201257 + 0.979538i \(0.564503\pi\)
\(644\) −1.43339 −0.0564834
\(645\) 33.6036 1.32314
\(646\) 5.66762 0.222989
\(647\) −6.66966 −0.262211 −0.131106 0.991368i \(-0.541853\pi\)
−0.131106 + 0.991368i \(0.541853\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 5.27080 0.206897
\(650\) 7.95327 0.311953
\(651\) −2.81120 −0.110180
\(652\) 19.0669 0.746719
\(653\) 2.46264 0.0963704 0.0481852 0.998838i \(-0.484656\pi\)
0.0481852 + 0.998838i \(0.484656\pi\)
\(654\) −8.38673 −0.327947
\(655\) 12.3536 0.482694
\(656\) 5.29346 0.206675
\(657\) 5.27039 0.205617
\(658\) −5.52883 −0.215536
\(659\) 19.7091 0.767759 0.383880 0.923383i \(-0.374588\pi\)
0.383880 + 0.923383i \(0.374588\pi\)
\(660\) −22.6159 −0.880324
\(661\) 43.5189 1.69269 0.846344 0.532637i \(-0.178799\pi\)
0.846344 + 0.532637i \(0.178799\pi\)
\(662\) 23.2926 0.905294
\(663\) −0.964091 −0.0374422
\(664\) 11.4796 0.445494
\(665\) 19.4615 0.754684
\(666\) −5.17705 −0.200606
\(667\) −0.588769 −0.0227972
\(668\) 3.57968 0.138502
\(669\) −10.5549 −0.408078
\(670\) 52.1120 2.01326
\(671\) 3.19746 0.123436
\(672\) 0.919823 0.0354829
\(673\) 30.2039 1.16428 0.582138 0.813090i \(-0.302216\pi\)
0.582138 + 0.813090i \(0.302216\pi\)
\(674\) −15.3551 −0.591457
\(675\) −7.95327 −0.306121
\(676\) 1.00000 0.0384615
\(677\) 36.4193 1.39971 0.699853 0.714287i \(-0.253249\pi\)
0.699853 + 0.714287i \(0.253249\pi\)
\(678\) 5.12907 0.196981
\(679\) 3.13101 0.120157
\(680\) 3.46983 0.133062
\(681\) 6.01618 0.230540
\(682\) −19.2049 −0.735394
\(683\) −15.1809 −0.580882 −0.290441 0.956893i \(-0.593802\pi\)
−0.290441 + 0.956893i \(0.593802\pi\)
\(684\) 5.87871 0.224778
\(685\) −61.5374 −2.35122
\(686\) 12.0993 0.461953
\(687\) 17.7551 0.677399
\(688\) −9.33676 −0.355961
\(689\) −2.69483 −0.102665
\(690\) −5.60853 −0.213513
\(691\) −21.8484 −0.831153 −0.415577 0.909558i \(-0.636420\pi\)
−0.415577 + 0.909558i \(0.636420\pi\)
\(692\) −1.57490 −0.0598689
\(693\) 5.78001 0.219565
\(694\) 21.2368 0.806137
\(695\) −33.4748 −1.26977
\(696\) 0.377820 0.0143212
\(697\) −5.10338 −0.193304
\(698\) −26.5667 −1.00556
\(699\) 11.1388 0.421307
\(700\) 7.31559 0.276503
\(701\) 46.2024 1.74504 0.872521 0.488577i \(-0.162484\pi\)
0.872521 + 0.488577i \(0.162484\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 30.4344 1.14785
\(704\) 6.28384 0.236831
\(705\) −21.6331 −0.814750
\(706\) 22.3861 0.842512
\(707\) 0.908054 0.0341509
\(708\) −0.838786 −0.0315235
\(709\) 16.4636 0.618302 0.309151 0.951013i \(-0.399955\pi\)
0.309151 + 0.951013i \(0.399955\pi\)
\(710\) 11.6804 0.438357
\(711\) 9.98774 0.374570
\(712\) −17.3823 −0.651429
\(713\) −4.76263 −0.178362
\(714\) −0.886793 −0.0331874
\(715\) −22.6159 −0.845788
\(716\) −8.00874 −0.299301
\(717\) 11.3486 0.423821
\(718\) 8.20847 0.306338
\(719\) −7.66482 −0.285850 −0.142925 0.989734i \(-0.545651\pi\)
−0.142925 + 0.989734i \(0.545651\pi\)
\(720\) 3.59906 0.134129
\(721\) 0.919823 0.0342560
\(722\) −15.5593 −0.579056
\(723\) −13.8214 −0.514024
\(724\) 6.76636 0.251470
\(725\) 3.00491 0.111599
\(726\) 28.4866 1.05724
\(727\) 49.1257 1.82197 0.910985 0.412440i \(-0.135323\pi\)
0.910985 + 0.412440i \(0.135323\pi\)
\(728\) 0.919823 0.0340909
\(729\) 1.00000 0.0370370
\(730\) −18.9685 −0.702055
\(731\) 9.00149 0.332932
\(732\) −0.508838 −0.0188072
\(733\) 23.4568 0.866398 0.433199 0.901298i \(-0.357385\pi\)
0.433199 + 0.901298i \(0.357385\pi\)
\(734\) −30.9920 −1.14393
\(735\) 22.1484 0.816955
\(736\) 1.55833 0.0574408
\(737\) −90.9857 −3.35150
\(738\) −5.29346 −0.194855
\(739\) −18.6694 −0.686764 −0.343382 0.939196i \(-0.611573\pi\)
−0.343382 + 0.939196i \(0.611573\pi\)
\(740\) 18.6325 0.684945
\(741\) 5.87871 0.215960
\(742\) −2.47876 −0.0909982
\(743\) 22.0970 0.810662 0.405331 0.914170i \(-0.367156\pi\)
0.405331 + 0.914170i \(0.367156\pi\)
\(744\) 3.05624 0.112047
\(745\) −25.8709 −0.947837
\(746\) 6.06159 0.221930
\(747\) −11.4796 −0.420016
\(748\) −6.05819 −0.221509
\(749\) 16.5330 0.604104
\(750\) 10.6290 0.388116
\(751\) −18.7085 −0.682683 −0.341341 0.939939i \(-0.610881\pi\)
−0.341341 + 0.939939i \(0.610881\pi\)
\(752\) 6.01076 0.219190
\(753\) −16.6162 −0.605527
\(754\) 0.377820 0.0137594
\(755\) −44.1382 −1.60635
\(756\) −0.919823 −0.0334536
\(757\) 42.8940 1.55901 0.779504 0.626397i \(-0.215471\pi\)
0.779504 + 0.626397i \(0.215471\pi\)
\(758\) −6.33028 −0.229926
\(759\) 9.79230 0.355438
\(760\) −21.1579 −0.767477
\(761\) −4.69061 −0.170035 −0.0850173 0.996379i \(-0.527095\pi\)
−0.0850173 + 0.996379i \(0.527095\pi\)
\(762\) 4.19542 0.151984
\(763\) −7.71430 −0.279276
\(764\) 3.62788 0.131252
\(765\) −3.46983 −0.125452
\(766\) 7.01057 0.253302
\(767\) −0.838786 −0.0302868
\(768\) −1.00000 −0.0360844
\(769\) −8.30375 −0.299441 −0.149720 0.988728i \(-0.547837\pi\)
−0.149720 + 0.988728i \(0.547837\pi\)
\(770\) −20.8026 −0.749675
\(771\) 15.8529 0.570930
\(772\) −17.8569 −0.642684
\(773\) 21.0421 0.756831 0.378416 0.925636i \(-0.376469\pi\)
0.378416 + 0.925636i \(0.376469\pi\)
\(774\) 9.33676 0.335603
\(775\) 24.3071 0.873137
\(776\) −3.40393 −0.122194
\(777\) −4.76196 −0.170835
\(778\) 5.80948 0.208280
\(779\) 31.1187 1.11494
\(780\) 3.59906 0.128867
\(781\) −20.3935 −0.729738
\(782\) −1.50237 −0.0537248
\(783\) −0.377820 −0.0135022
\(784\) −6.15393 −0.219783
\(785\) −59.9072 −2.13818
\(786\) 3.43244 0.122431
\(787\) 30.1420 1.07445 0.537224 0.843440i \(-0.319473\pi\)
0.537224 + 0.843440i \(0.319473\pi\)
\(788\) −1.10763 −0.0394575
\(789\) 14.0523 0.500275
\(790\) −35.9465 −1.27892
\(791\) 4.71783 0.167747
\(792\) −6.28384 −0.223286
\(793\) −0.508838 −0.0180694
\(794\) 21.4430 0.760985
\(795\) −9.69886 −0.343983
\(796\) −9.47297 −0.335761
\(797\) 31.5970 1.11922 0.559612 0.828755i \(-0.310950\pi\)
0.559612 + 0.828755i \(0.310950\pi\)
\(798\) 5.40737 0.191419
\(799\) −5.79492 −0.205009
\(800\) −7.95327 −0.281190
\(801\) 17.3823 0.614174
\(802\) 4.30754 0.152105
\(803\) 33.1183 1.16872
\(804\) 14.4793 0.510646
\(805\) −5.15886 −0.181826
\(806\) 3.05624 0.107651
\(807\) 21.7523 0.765716
\(808\) −0.987205 −0.0347298
\(809\) 17.1403 0.602622 0.301311 0.953526i \(-0.402576\pi\)
0.301311 + 0.953526i \(0.402576\pi\)
\(810\) −3.59906 −0.126458
\(811\) 41.2262 1.44765 0.723824 0.689984i \(-0.242383\pi\)
0.723824 + 0.689984i \(0.242383\pi\)
\(812\) 0.347528 0.0121958
\(813\) −29.6801 −1.04093
\(814\) −32.5317 −1.14024
\(815\) 68.6232 2.40376
\(816\) 0.964091 0.0337499
\(817\) −54.8881 −1.92029
\(818\) −3.40095 −0.118911
\(819\) −0.919823 −0.0321412
\(820\) 19.0515 0.665308
\(821\) 13.7131 0.478592 0.239296 0.970947i \(-0.423083\pi\)
0.239296 + 0.970947i \(0.423083\pi\)
\(822\) −17.0982 −0.596367
\(823\) 49.1486 1.71321 0.856606 0.515972i \(-0.172569\pi\)
0.856606 + 0.515972i \(0.172569\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −49.9770 −1.73998
\(826\) −0.771535 −0.0268451
\(827\) 37.5446 1.30555 0.652777 0.757551i \(-0.273604\pi\)
0.652777 + 0.757551i \(0.273604\pi\)
\(828\) −1.55833 −0.0541557
\(829\) 41.5322 1.44247 0.721236 0.692689i \(-0.243574\pi\)
0.721236 + 0.692689i \(0.243574\pi\)
\(830\) 41.3158 1.43409
\(831\) −0.303986 −0.0105451
\(832\) −1.00000 −0.0346688
\(833\) 5.93295 0.205564
\(834\) −9.30098 −0.322067
\(835\) 12.8835 0.445852
\(836\) 36.9409 1.27763
\(837\) −3.05624 −0.105639
\(838\) −18.2249 −0.629569
\(839\) −22.7145 −0.784192 −0.392096 0.919924i \(-0.628250\pi\)
−0.392096 + 0.919924i \(0.628250\pi\)
\(840\) 3.31050 0.114223
\(841\) −28.8573 −0.995078
\(842\) −27.8199 −0.958737
\(843\) 31.4890 1.08454
\(844\) 16.5894 0.571030
\(845\) 3.59906 0.123812
\(846\) −6.01076 −0.206654
\(847\) 26.2026 0.900333
\(848\) 2.69483 0.0925407
\(849\) −7.53378 −0.258559
\(850\) 7.66768 0.262999
\(851\) −8.06755 −0.276552
\(852\) 3.24540 0.111185
\(853\) 1.68429 0.0576690 0.0288345 0.999584i \(-0.490820\pi\)
0.0288345 + 0.999584i \(0.490820\pi\)
\(854\) −0.468041 −0.0160160
\(855\) 21.1579 0.723584
\(856\) −17.9742 −0.614344
\(857\) 1.23658 0.0422407 0.0211204 0.999777i \(-0.493277\pi\)
0.0211204 + 0.999777i \(0.493277\pi\)
\(858\) −6.28384 −0.214527
\(859\) −3.53572 −0.120637 −0.0603185 0.998179i \(-0.519212\pi\)
−0.0603185 + 0.998179i \(0.519212\pi\)
\(860\) −33.6036 −1.14587
\(861\) −4.86905 −0.165937
\(862\) −8.55540 −0.291398
\(863\) 3.59457 0.122360 0.0611802 0.998127i \(-0.480514\pi\)
0.0611802 + 0.998127i \(0.480514\pi\)
\(864\) 1.00000 0.0340207
\(865\) −5.66818 −0.192724
\(866\) 12.5585 0.426757
\(867\) 16.0705 0.545784
\(868\) 2.81120 0.0954183
\(869\) 62.7613 2.12903
\(870\) 1.35980 0.0461015
\(871\) 14.4793 0.490613
\(872\) 8.38673 0.284010
\(873\) 3.40393 0.115206
\(874\) 9.16098 0.309875
\(875\) 9.77680 0.330516
\(876\) −5.27039 −0.178070
\(877\) −23.0893 −0.779670 −0.389835 0.920885i \(-0.627468\pi\)
−0.389835 + 0.920885i \(0.627468\pi\)
\(878\) 8.35803 0.282070
\(879\) −28.1598 −0.949806
\(880\) 22.6159 0.762383
\(881\) −18.8867 −0.636309 −0.318155 0.948039i \(-0.603063\pi\)
−0.318155 + 0.948039i \(0.603063\pi\)
\(882\) 6.15393 0.207213
\(883\) 1.12572 0.0378834 0.0189417 0.999821i \(-0.493970\pi\)
0.0189417 + 0.999821i \(0.493970\pi\)
\(884\) 0.964091 0.0324259
\(885\) −3.01885 −0.101477
\(886\) −11.2701 −0.378627
\(887\) 35.9661 1.20762 0.603812 0.797127i \(-0.293648\pi\)
0.603812 + 0.797127i \(0.293648\pi\)
\(888\) 5.17705 0.173730
\(889\) 3.85905 0.129428
\(890\) −62.5601 −2.09702
\(891\) 6.28384 0.210516
\(892\) 10.5549 0.353406
\(893\) 35.3355 1.18246
\(894\) −7.18823 −0.240410
\(895\) −28.8240 −0.963479
\(896\) −0.919823 −0.0307291
\(897\) −1.55833 −0.0520312
\(898\) 16.6832 0.556726
\(899\) 1.15471 0.0385117
\(900\) 7.95327 0.265109
\(901\) −2.59806 −0.0865539
\(902\) −33.2633 −1.10755
\(903\) 8.58816 0.285796
\(904\) −5.12907 −0.170590
\(905\) 24.3526 0.809506
\(906\) −12.2638 −0.407437
\(907\) 23.0319 0.764760 0.382380 0.924005i \(-0.375105\pi\)
0.382380 + 0.924005i \(0.375105\pi\)
\(908\) −6.01618 −0.199654
\(909\) 0.987205 0.0327435
\(910\) 3.31050 0.109742
\(911\) 12.7623 0.422833 0.211416 0.977396i \(-0.432192\pi\)
0.211416 + 0.977396i \(0.432192\pi\)
\(912\) −5.87871 −0.194664
\(913\) −72.1358 −2.38735
\(914\) −32.0280 −1.05939
\(915\) −1.83134 −0.0605423
\(916\) −17.7551 −0.586645
\(917\) 3.15723 0.104261
\(918\) −0.964091 −0.0318198
\(919\) 12.2281 0.403368 0.201684 0.979451i \(-0.435359\pi\)
0.201684 + 0.979451i \(0.435359\pi\)
\(920\) 5.60853 0.184908
\(921\) −3.43777 −0.113279
\(922\) 30.6252 1.00859
\(923\) 3.24540 0.106824
\(924\) −5.78001 −0.190149
\(925\) 41.1744 1.35381
\(926\) 11.1986 0.368009
\(927\) 1.00000 0.0328443
\(928\) −0.377820 −0.0124026
\(929\) 37.7516 1.23859 0.619296 0.785158i \(-0.287418\pi\)
0.619296 + 0.785158i \(0.287418\pi\)
\(930\) 10.9996 0.360691
\(931\) −36.1772 −1.18566
\(932\) −11.1388 −0.364863
\(933\) 1.77167 0.0580018
\(934\) −11.7793 −0.385430
\(935\) −21.8038 −0.713061
\(936\) 1.00000 0.0326860
\(937\) 14.9497 0.488385 0.244193 0.969727i \(-0.421477\pi\)
0.244193 + 0.969727i \(0.421477\pi\)
\(938\) 13.3184 0.434861
\(939\) 27.0983 0.884319
\(940\) 21.6331 0.705594
\(941\) −37.2030 −1.21278 −0.606392 0.795166i \(-0.707384\pi\)
−0.606392 + 0.795166i \(0.707384\pi\)
\(942\) −16.6452 −0.542330
\(943\) −8.24897 −0.268623
\(944\) 0.838786 0.0273002
\(945\) −3.31050 −0.107691
\(946\) 58.6707 1.90755
\(947\) 53.3472 1.73355 0.866775 0.498699i \(-0.166189\pi\)
0.866775 + 0.498699i \(0.166189\pi\)
\(948\) −9.98774 −0.324387
\(949\) −5.27039 −0.171084
\(950\) −46.7550 −1.51693
\(951\) 26.5689 0.861557
\(952\) 0.886793 0.0287411
\(953\) −47.1799 −1.52831 −0.764153 0.645035i \(-0.776843\pi\)
−0.764153 + 0.645035i \(0.776843\pi\)
\(954\) −2.69483 −0.0872482
\(955\) 13.0570 0.422514
\(956\) −11.3486 −0.367040
\(957\) −2.37416 −0.0767457
\(958\) −7.23414 −0.233724
\(959\) −15.7273 −0.507860
\(960\) −3.59906 −0.116159
\(961\) −21.6594 −0.698690
\(962\) 5.17705 0.166915
\(963\) 17.9742 0.579209
\(964\) 13.8214 0.445158
\(965\) −64.2681 −2.06886
\(966\) −1.43339 −0.0461185
\(967\) −16.0862 −0.517296 −0.258648 0.965972i \(-0.583277\pi\)
−0.258648 + 0.965972i \(0.583277\pi\)
\(968\) −28.4866 −0.915594
\(969\) 5.66762 0.182070
\(970\) −12.2510 −0.393355
\(971\) 54.1754 1.73857 0.869286 0.494310i \(-0.164579\pi\)
0.869286 + 0.494310i \(0.164579\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −8.55525 −0.274269
\(974\) −14.2025 −0.455078
\(975\) 7.95327 0.254708
\(976\) 0.508838 0.0162875
\(977\) −50.8217 −1.62593 −0.812965 0.582312i \(-0.802148\pi\)
−0.812965 + 0.582312i \(0.802148\pi\)
\(978\) 19.0669 0.609694
\(979\) 109.228 3.49093
\(980\) −22.1484 −0.707504
\(981\) −8.38673 −0.267768
\(982\) −20.5180 −0.654755
\(983\) 10.6021 0.338153 0.169077 0.985603i \(-0.445921\pi\)
0.169077 + 0.985603i \(0.445921\pi\)
\(984\) 5.29346 0.168749
\(985\) −3.98642 −0.127018
\(986\) 0.364253 0.0116002
\(987\) −5.52883 −0.175985
\(988\) −5.87871 −0.187027
\(989\) 14.5498 0.462656
\(990\) −22.6159 −0.718782
\(991\) −11.6809 −0.371055 −0.185527 0.982639i \(-0.559399\pi\)
−0.185527 + 0.982639i \(0.559399\pi\)
\(992\) −3.05624 −0.0970357
\(993\) 23.2926 0.739169
\(994\) 2.98519 0.0946844
\(995\) −34.0938 −1.08085
\(996\) 11.4796 0.363744
\(997\) −56.4318 −1.78721 −0.893607 0.448851i \(-0.851833\pi\)
−0.893607 + 0.448851i \(0.851833\pi\)
\(998\) −41.1316 −1.30200
\(999\) −5.17705 −0.163795
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.ba.1.13 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.13 14 1.1 even 1 trivial