Properties

Label 8034.2.a.w.1.12
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-0.782524\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.21809 q^{5} -1.00000 q^{6} -4.81342 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.21809 q^{5} -1.00000 q^{6} -4.81342 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.21809 q^{10} -4.29400 q^{11} +1.00000 q^{12} +1.00000 q^{13} +4.81342 q^{14} +4.21809 q^{15} +1.00000 q^{16} -2.64843 q^{17} -1.00000 q^{18} +0.655968 q^{19} +4.21809 q^{20} -4.81342 q^{21} +4.29400 q^{22} +1.52610 q^{23} -1.00000 q^{24} +12.7923 q^{25} -1.00000 q^{26} +1.00000 q^{27} -4.81342 q^{28} -0.992673 q^{29} -4.21809 q^{30} +1.93587 q^{31} -1.00000 q^{32} -4.29400 q^{33} +2.64843 q^{34} -20.3034 q^{35} +1.00000 q^{36} -7.13733 q^{37} -0.655968 q^{38} +1.00000 q^{39} -4.21809 q^{40} +7.02108 q^{41} +4.81342 q^{42} -4.82571 q^{43} -4.29400 q^{44} +4.21809 q^{45} -1.52610 q^{46} +1.34179 q^{47} +1.00000 q^{48} +16.1690 q^{49} -12.7923 q^{50} -2.64843 q^{51} +1.00000 q^{52} -1.36163 q^{53} -1.00000 q^{54} -18.1125 q^{55} +4.81342 q^{56} +0.655968 q^{57} +0.992673 q^{58} -14.8662 q^{59} +4.21809 q^{60} -9.04935 q^{61} -1.93587 q^{62} -4.81342 q^{63} +1.00000 q^{64} +4.21809 q^{65} +4.29400 q^{66} +3.79388 q^{67} -2.64843 q^{68} +1.52610 q^{69} +20.3034 q^{70} +4.83297 q^{71} -1.00000 q^{72} +1.21359 q^{73} +7.13733 q^{74} +12.7923 q^{75} +0.655968 q^{76} +20.6688 q^{77} -1.00000 q^{78} -14.6709 q^{79} +4.21809 q^{80} +1.00000 q^{81} -7.02108 q^{82} +8.38742 q^{83} -4.81342 q^{84} -11.1713 q^{85} +4.82571 q^{86} -0.992673 q^{87} +4.29400 q^{88} +12.2086 q^{89} -4.21809 q^{90} -4.81342 q^{91} +1.52610 q^{92} +1.93587 q^{93} -1.34179 q^{94} +2.76693 q^{95} -1.00000 q^{96} +2.87820 q^{97} -16.1690 q^{98} -4.29400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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
\(5\) 4.21809 1.88639 0.943193 0.332244i \(-0.107806\pi\)
0.943193 + 0.332244i \(0.107806\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.81342 −1.81930 −0.909651 0.415374i \(-0.863651\pi\)
−0.909651 + 0.415374i \(0.863651\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −4.21809 −1.33388
\(11\) −4.29400 −1.29469 −0.647345 0.762197i \(-0.724121\pi\)
−0.647345 + 0.762197i \(0.724121\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 4.81342 1.28644
\(15\) 4.21809 1.08911
\(16\) 1.00000 0.250000
\(17\) −2.64843 −0.642338 −0.321169 0.947022i \(-0.604076\pi\)
−0.321169 + 0.947022i \(0.604076\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.655968 0.150489 0.0752446 0.997165i \(-0.476026\pi\)
0.0752446 + 0.997165i \(0.476026\pi\)
\(20\) 4.21809 0.943193
\(21\) −4.81342 −1.05037
\(22\) 4.29400 0.915485
\(23\) 1.52610 0.318215 0.159107 0.987261i \(-0.449138\pi\)
0.159107 + 0.987261i \(0.449138\pi\)
\(24\) −1.00000 −0.204124
\(25\) 12.7923 2.55846
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −4.81342 −0.909651
\(29\) −0.992673 −0.184335 −0.0921674 0.995744i \(-0.529379\pi\)
−0.0921674 + 0.995744i \(0.529379\pi\)
\(30\) −4.21809 −0.770114
\(31\) 1.93587 0.347692 0.173846 0.984773i \(-0.444380\pi\)
0.173846 + 0.984773i \(0.444380\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.29400 −0.747490
\(34\) 2.64843 0.454202
\(35\) −20.3034 −3.43191
\(36\) 1.00000 0.166667
\(37\) −7.13733 −1.17337 −0.586685 0.809815i \(-0.699567\pi\)
−0.586685 + 0.809815i \(0.699567\pi\)
\(38\) −0.655968 −0.106412
\(39\) 1.00000 0.160128
\(40\) −4.21809 −0.666938
\(41\) 7.02108 1.09651 0.548254 0.836312i \(-0.315293\pi\)
0.548254 + 0.836312i \(0.315293\pi\)
\(42\) 4.81342 0.742727
\(43\) −4.82571 −0.735913 −0.367957 0.929843i \(-0.619943\pi\)
−0.367957 + 0.929843i \(0.619943\pi\)
\(44\) −4.29400 −0.647345
\(45\) 4.21809 0.628796
\(46\) −1.52610 −0.225012
\(47\) 1.34179 0.195721 0.0978603 0.995200i \(-0.468800\pi\)
0.0978603 + 0.995200i \(0.468800\pi\)
\(48\) 1.00000 0.144338
\(49\) 16.1690 2.30986
\(50\) −12.7923 −1.80910
\(51\) −2.64843 −0.370854
\(52\) 1.00000 0.138675
\(53\) −1.36163 −0.187034 −0.0935168 0.995618i \(-0.529811\pi\)
−0.0935168 + 0.995618i \(0.529811\pi\)
\(54\) −1.00000 −0.136083
\(55\) −18.1125 −2.44229
\(56\) 4.81342 0.643220
\(57\) 0.655968 0.0868850
\(58\) 0.992673 0.130344
\(59\) −14.8662 −1.93542 −0.967710 0.252067i \(-0.918890\pi\)
−0.967710 + 0.252067i \(0.918890\pi\)
\(60\) 4.21809 0.544553
\(61\) −9.04935 −1.15865 −0.579325 0.815097i \(-0.696684\pi\)
−0.579325 + 0.815097i \(0.696684\pi\)
\(62\) −1.93587 −0.245856
\(63\) −4.81342 −0.606434
\(64\) 1.00000 0.125000
\(65\) 4.21809 0.523190
\(66\) 4.29400 0.528555
\(67\) 3.79388 0.463496 0.231748 0.972776i \(-0.425556\pi\)
0.231748 + 0.972776i \(0.425556\pi\)
\(68\) −2.64843 −0.321169
\(69\) 1.52610 0.183721
\(70\) 20.3034 2.42672
\(71\) 4.83297 0.573568 0.286784 0.957995i \(-0.407414\pi\)
0.286784 + 0.957995i \(0.407414\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.21359 0.142040 0.0710199 0.997475i \(-0.477375\pi\)
0.0710199 + 0.997475i \(0.477375\pi\)
\(74\) 7.13733 0.829698
\(75\) 12.7923 1.47713
\(76\) 0.655968 0.0752446
\(77\) 20.6688 2.35543
\(78\) −1.00000 −0.113228
\(79\) −14.6709 −1.65061 −0.825305 0.564688i \(-0.808997\pi\)
−0.825305 + 0.564688i \(0.808997\pi\)
\(80\) 4.21809 0.471597
\(81\) 1.00000 0.111111
\(82\) −7.02108 −0.775348
\(83\) 8.38742 0.920639 0.460319 0.887753i \(-0.347735\pi\)
0.460319 + 0.887753i \(0.347735\pi\)
\(84\) −4.81342 −0.525187
\(85\) −11.1713 −1.21170
\(86\) 4.82571 0.520369
\(87\) −0.992673 −0.106426
\(88\) 4.29400 0.457742
\(89\) 12.2086 1.29410 0.647052 0.762446i \(-0.276002\pi\)
0.647052 + 0.762446i \(0.276002\pi\)
\(90\) −4.21809 −0.444626
\(91\) −4.81342 −0.504584
\(92\) 1.52610 0.159107
\(93\) 1.93587 0.200740
\(94\) −1.34179 −0.138395
\(95\) 2.76693 0.283881
\(96\) −1.00000 −0.102062
\(97\) 2.87820 0.292237 0.146118 0.989267i \(-0.453322\pi\)
0.146118 + 0.989267i \(0.453322\pi\)
\(98\) −16.1690 −1.63332
\(99\) −4.29400 −0.431564
\(100\) 12.7923 1.27923
\(101\) −14.6408 −1.45681 −0.728405 0.685147i \(-0.759738\pi\)
−0.728405 + 0.685147i \(0.759738\pi\)
\(102\) 2.64843 0.262233
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −20.3034 −1.98141
\(106\) 1.36163 0.132253
\(107\) −13.4887 −1.30401 −0.652003 0.758216i \(-0.726071\pi\)
−0.652003 + 0.758216i \(0.726071\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.10949 −0.202052 −0.101026 0.994884i \(-0.532213\pi\)
−0.101026 + 0.994884i \(0.532213\pi\)
\(110\) 18.1125 1.72696
\(111\) −7.13733 −0.677446
\(112\) −4.81342 −0.454825
\(113\) −3.48546 −0.327884 −0.163942 0.986470i \(-0.552421\pi\)
−0.163942 + 0.986470i \(0.552421\pi\)
\(114\) −0.655968 −0.0614370
\(115\) 6.43724 0.600276
\(116\) −0.992673 −0.0921674
\(117\) 1.00000 0.0924500
\(118\) 14.8662 1.36855
\(119\) 12.7480 1.16861
\(120\) −4.21809 −0.385057
\(121\) 7.43847 0.676225
\(122\) 9.04935 0.819289
\(123\) 7.02108 0.633069
\(124\) 1.93587 0.173846
\(125\) 32.8685 2.93985
\(126\) 4.81342 0.428814
\(127\) −18.7851 −1.66691 −0.833456 0.552585i \(-0.813641\pi\)
−0.833456 + 0.552585i \(0.813641\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.82571 −0.424880
\(130\) −4.21809 −0.369951
\(131\) −2.17667 −0.190176 −0.0950881 0.995469i \(-0.530313\pi\)
−0.0950881 + 0.995469i \(0.530313\pi\)
\(132\) −4.29400 −0.373745
\(133\) −3.15745 −0.273785
\(134\) −3.79388 −0.327741
\(135\) 4.21809 0.363035
\(136\) 2.64843 0.227101
\(137\) 10.6655 0.911213 0.455606 0.890181i \(-0.349422\pi\)
0.455606 + 0.890181i \(0.349422\pi\)
\(138\) −1.52610 −0.129911
\(139\) −7.36838 −0.624978 −0.312489 0.949921i \(-0.601163\pi\)
−0.312489 + 0.949921i \(0.601163\pi\)
\(140\) −20.3034 −1.71595
\(141\) 1.34179 0.112999
\(142\) −4.83297 −0.405574
\(143\) −4.29400 −0.359083
\(144\) 1.00000 0.0833333
\(145\) −4.18718 −0.347727
\(146\) −1.21359 −0.100437
\(147\) 16.1690 1.33360
\(148\) −7.13733 −0.586685
\(149\) −4.93093 −0.403958 −0.201979 0.979390i \(-0.564737\pi\)
−0.201979 + 0.979390i \(0.564737\pi\)
\(150\) −12.7923 −1.04449
\(151\) −9.49514 −0.772703 −0.386352 0.922352i \(-0.626265\pi\)
−0.386352 + 0.922352i \(0.626265\pi\)
\(152\) −0.655968 −0.0532060
\(153\) −2.64843 −0.214113
\(154\) −20.6688 −1.66554
\(155\) 8.16567 0.655882
\(156\) 1.00000 0.0800641
\(157\) 12.2902 0.980861 0.490430 0.871480i \(-0.336840\pi\)
0.490430 + 0.871480i \(0.336840\pi\)
\(158\) 14.6709 1.16716
\(159\) −1.36163 −0.107984
\(160\) −4.21809 −0.333469
\(161\) −7.34578 −0.578928
\(162\) −1.00000 −0.0785674
\(163\) 4.10109 0.321223 0.160611 0.987018i \(-0.448653\pi\)
0.160611 + 0.987018i \(0.448653\pi\)
\(164\) 7.02108 0.548254
\(165\) −18.1125 −1.41006
\(166\) −8.38742 −0.650990
\(167\) −7.18868 −0.556277 −0.278139 0.960541i \(-0.589717\pi\)
−0.278139 + 0.960541i \(0.589717\pi\)
\(168\) 4.81342 0.371363
\(169\) 1.00000 0.0769231
\(170\) 11.1713 0.856800
\(171\) 0.655968 0.0501631
\(172\) −4.82571 −0.367957
\(173\) −14.0660 −1.06942 −0.534709 0.845036i \(-0.679579\pi\)
−0.534709 + 0.845036i \(0.679579\pi\)
\(174\) 0.992673 0.0752543
\(175\) −61.5746 −4.65460
\(176\) −4.29400 −0.323673
\(177\) −14.8662 −1.11742
\(178\) −12.2086 −0.915070
\(179\) −21.1002 −1.57710 −0.788551 0.614969i \(-0.789169\pi\)
−0.788551 + 0.614969i \(0.789169\pi\)
\(180\) 4.21809 0.314398
\(181\) −18.8679 −1.40244 −0.701219 0.712946i \(-0.747360\pi\)
−0.701219 + 0.712946i \(0.747360\pi\)
\(182\) 4.81342 0.356794
\(183\) −9.04935 −0.668947
\(184\) −1.52610 −0.112506
\(185\) −30.1059 −2.21343
\(186\) −1.93587 −0.141945
\(187\) 11.3724 0.831629
\(188\) 1.34179 0.0978603
\(189\) −4.81342 −0.350125
\(190\) −2.76693 −0.200734
\(191\) −1.99582 −0.144412 −0.0722062 0.997390i \(-0.523004\pi\)
−0.0722062 + 0.997390i \(0.523004\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0177 −1.00902 −0.504508 0.863407i \(-0.668326\pi\)
−0.504508 + 0.863407i \(0.668326\pi\)
\(194\) −2.87820 −0.206643
\(195\) 4.21809 0.302064
\(196\) 16.1690 1.15493
\(197\) 0.236467 0.0168476 0.00842378 0.999965i \(-0.497319\pi\)
0.00842378 + 0.999965i \(0.497319\pi\)
\(198\) 4.29400 0.305162
\(199\) −21.3648 −1.51451 −0.757255 0.653119i \(-0.773460\pi\)
−0.757255 + 0.653119i \(0.773460\pi\)
\(200\) −12.7923 −0.904551
\(201\) 3.79388 0.267599
\(202\) 14.6408 1.03012
\(203\) 4.77815 0.335361
\(204\) −2.64843 −0.185427
\(205\) 29.6155 2.06844
\(206\) 1.00000 0.0696733
\(207\) 1.52610 0.106072
\(208\) 1.00000 0.0693375
\(209\) −2.81673 −0.194837
\(210\) 20.3034 1.40107
\(211\) −8.26635 −0.569079 −0.284539 0.958664i \(-0.591841\pi\)
−0.284539 + 0.958664i \(0.591841\pi\)
\(212\) −1.36163 −0.0935168
\(213\) 4.83297 0.331150
\(214\) 13.4887 0.922072
\(215\) −20.3553 −1.38822
\(216\) −1.00000 −0.0680414
\(217\) −9.31815 −0.632557
\(218\) 2.10949 0.142872
\(219\) 1.21359 0.0820067
\(220\) −18.1125 −1.22114
\(221\) −2.64843 −0.178153
\(222\) 7.13733 0.479026
\(223\) −2.67211 −0.178938 −0.0894689 0.995990i \(-0.528517\pi\)
−0.0894689 + 0.995990i \(0.528517\pi\)
\(224\) 4.81342 0.321610
\(225\) 12.7923 0.852819
\(226\) 3.48546 0.231849
\(227\) 12.4207 0.824392 0.412196 0.911095i \(-0.364762\pi\)
0.412196 + 0.911095i \(0.364762\pi\)
\(228\) 0.655968 0.0434425
\(229\) −24.6135 −1.62651 −0.813253 0.581911i \(-0.802305\pi\)
−0.813253 + 0.581911i \(0.802305\pi\)
\(230\) −6.43724 −0.424459
\(231\) 20.6688 1.35991
\(232\) 0.992673 0.0651722
\(233\) 5.52696 0.362083 0.181042 0.983475i \(-0.442053\pi\)
0.181042 + 0.983475i \(0.442053\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 5.65980 0.369205
\(236\) −14.8662 −0.967710
\(237\) −14.6709 −0.952980
\(238\) −12.7480 −0.826330
\(239\) 14.7857 0.956410 0.478205 0.878248i \(-0.341288\pi\)
0.478205 + 0.878248i \(0.341288\pi\)
\(240\) 4.21809 0.272276
\(241\) −15.9625 −1.02823 −0.514117 0.857720i \(-0.671880\pi\)
−0.514117 + 0.857720i \(0.671880\pi\)
\(242\) −7.43847 −0.478163
\(243\) 1.00000 0.0641500
\(244\) −9.04935 −0.579325
\(245\) 68.2023 4.35729
\(246\) −7.02108 −0.447647
\(247\) 0.655968 0.0417382
\(248\) −1.93587 −0.122928
\(249\) 8.38742 0.531531
\(250\) −32.8685 −2.07879
\(251\) −7.52069 −0.474702 −0.237351 0.971424i \(-0.576279\pi\)
−0.237351 + 0.971424i \(0.576279\pi\)
\(252\) −4.81342 −0.303217
\(253\) −6.55309 −0.411990
\(254\) 18.7851 1.17869
\(255\) −11.1713 −0.699574
\(256\) 1.00000 0.0625000
\(257\) −27.5314 −1.71736 −0.858681 0.512511i \(-0.828715\pi\)
−0.858681 + 0.512511i \(0.828715\pi\)
\(258\) 4.82571 0.300435
\(259\) 34.3550 2.13471
\(260\) 4.21809 0.261595
\(261\) −0.992673 −0.0614449
\(262\) 2.17667 0.134475
\(263\) −4.79598 −0.295733 −0.147866 0.989007i \(-0.547241\pi\)
−0.147866 + 0.989007i \(0.547241\pi\)
\(264\) 4.29400 0.264278
\(265\) −5.74346 −0.352818
\(266\) 3.15745 0.193596
\(267\) 12.2086 0.747151
\(268\) 3.79388 0.231748
\(269\) −18.4173 −1.12292 −0.561462 0.827503i \(-0.689761\pi\)
−0.561462 + 0.827503i \(0.689761\pi\)
\(270\) −4.21809 −0.256705
\(271\) 27.1420 1.64876 0.824380 0.566037i \(-0.191524\pi\)
0.824380 + 0.566037i \(0.191524\pi\)
\(272\) −2.64843 −0.160585
\(273\) −4.81342 −0.291321
\(274\) −10.6655 −0.644325
\(275\) −54.9301 −3.31241
\(276\) 1.52610 0.0918606
\(277\) 30.3796 1.82533 0.912667 0.408704i \(-0.134019\pi\)
0.912667 + 0.408704i \(0.134019\pi\)
\(278\) 7.36838 0.441926
\(279\) 1.93587 0.115897
\(280\) 20.3034 1.21336
\(281\) −22.8883 −1.36540 −0.682700 0.730698i \(-0.739195\pi\)
−0.682700 + 0.730698i \(0.739195\pi\)
\(282\) −1.34179 −0.0799026
\(283\) 17.8384 1.06038 0.530192 0.847877i \(-0.322120\pi\)
0.530192 + 0.847877i \(0.322120\pi\)
\(284\) 4.83297 0.286784
\(285\) 2.76693 0.163899
\(286\) 4.29400 0.253910
\(287\) −33.7954 −1.99488
\(288\) −1.00000 −0.0589256
\(289\) −9.98583 −0.587402
\(290\) 4.18718 0.245880
\(291\) 2.87820 0.168723
\(292\) 1.21359 0.0710199
\(293\) −15.8528 −0.926131 −0.463065 0.886324i \(-0.653251\pi\)
−0.463065 + 0.886324i \(0.653251\pi\)
\(294\) −16.1690 −0.942996
\(295\) −62.7071 −3.65095
\(296\) 7.13733 0.414849
\(297\) −4.29400 −0.249163
\(298\) 4.93093 0.285641
\(299\) 1.52610 0.0882568
\(300\) 12.7923 0.738563
\(301\) 23.2282 1.33885
\(302\) 9.49514 0.546384
\(303\) −14.6408 −0.841089
\(304\) 0.655968 0.0376223
\(305\) −38.1710 −2.18566
\(306\) 2.64843 0.151401
\(307\) 28.0252 1.59948 0.799742 0.600343i \(-0.204969\pi\)
0.799742 + 0.600343i \(0.204969\pi\)
\(308\) 20.6688 1.17772
\(309\) −1.00000 −0.0568880
\(310\) −8.16567 −0.463779
\(311\) −10.6744 −0.605288 −0.302644 0.953104i \(-0.597869\pi\)
−0.302644 + 0.953104i \(0.597869\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 15.8503 0.895912 0.447956 0.894056i \(-0.352152\pi\)
0.447956 + 0.894056i \(0.352152\pi\)
\(314\) −12.2902 −0.693573
\(315\) −20.3034 −1.14397
\(316\) −14.6709 −0.825305
\(317\) 22.0671 1.23941 0.619705 0.784835i \(-0.287252\pi\)
0.619705 + 0.784835i \(0.287252\pi\)
\(318\) 1.36163 0.0763561
\(319\) 4.26254 0.238657
\(320\) 4.21809 0.235798
\(321\) −13.4887 −0.752868
\(322\) 7.34578 0.409364
\(323\) −1.73728 −0.0966650
\(324\) 1.00000 0.0555556
\(325\) 12.7923 0.709588
\(326\) −4.10109 −0.227139
\(327\) −2.10949 −0.116655
\(328\) −7.02108 −0.387674
\(329\) −6.45861 −0.356075
\(330\) 18.1125 0.997060
\(331\) −5.84180 −0.321095 −0.160547 0.987028i \(-0.551326\pi\)
−0.160547 + 0.987028i \(0.551326\pi\)
\(332\) 8.38742 0.460319
\(333\) −7.13733 −0.391123
\(334\) 7.18868 0.393347
\(335\) 16.0029 0.874332
\(336\) −4.81342 −0.262594
\(337\) 6.31009 0.343732 0.171866 0.985120i \(-0.445020\pi\)
0.171866 + 0.985120i \(0.445020\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −3.48546 −0.189304
\(340\) −11.1713 −0.605849
\(341\) −8.31263 −0.450154
\(342\) −0.655968 −0.0354707
\(343\) −44.1343 −2.38303
\(344\) 4.82571 0.260185
\(345\) 6.43724 0.346569
\(346\) 14.0660 0.756193
\(347\) 11.8496 0.636122 0.318061 0.948070i \(-0.396968\pi\)
0.318061 + 0.948070i \(0.396968\pi\)
\(348\) −0.992673 −0.0532129
\(349\) 2.98569 0.159820 0.0799102 0.996802i \(-0.474537\pi\)
0.0799102 + 0.996802i \(0.474537\pi\)
\(350\) 61.5746 3.29130
\(351\) 1.00000 0.0533761
\(352\) 4.29400 0.228871
\(353\) −33.7345 −1.79550 −0.897752 0.440501i \(-0.854801\pi\)
−0.897752 + 0.440501i \(0.854801\pi\)
\(354\) 14.8662 0.790132
\(355\) 20.3859 1.08197
\(356\) 12.2086 0.647052
\(357\) 12.7480 0.674696
\(358\) 21.1002 1.11518
\(359\) −6.51521 −0.343860 −0.171930 0.985109i \(-0.555000\pi\)
−0.171930 + 0.985109i \(0.555000\pi\)
\(360\) −4.21809 −0.222313
\(361\) −18.5697 −0.977353
\(362\) 18.8679 0.991673
\(363\) 7.43847 0.390418
\(364\) −4.81342 −0.252292
\(365\) 5.11902 0.267942
\(366\) 9.04935 0.473017
\(367\) 25.6512 1.33898 0.669490 0.742821i \(-0.266513\pi\)
0.669490 + 0.742821i \(0.266513\pi\)
\(368\) 1.52610 0.0795536
\(369\) 7.02108 0.365503
\(370\) 30.1059 1.56513
\(371\) 6.55407 0.340271
\(372\) 1.93587 0.100370
\(373\) −14.8679 −0.769830 −0.384915 0.922952i \(-0.625769\pi\)
−0.384915 + 0.922952i \(0.625769\pi\)
\(374\) −11.3724 −0.588051
\(375\) 32.8685 1.69732
\(376\) −1.34179 −0.0691977
\(377\) −0.992673 −0.0511253
\(378\) 4.81342 0.247576
\(379\) −8.72164 −0.448001 −0.224000 0.974589i \(-0.571912\pi\)
−0.224000 + 0.974589i \(0.571912\pi\)
\(380\) 2.76693 0.141940
\(381\) −18.7851 −0.962392
\(382\) 1.99582 0.102115
\(383\) −4.66833 −0.238541 −0.119270 0.992862i \(-0.538055\pi\)
−0.119270 + 0.992862i \(0.538055\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 87.1830 4.44326
\(386\) 14.0177 0.713481
\(387\) −4.82571 −0.245304
\(388\) 2.87820 0.146118
\(389\) −19.0163 −0.964166 −0.482083 0.876126i \(-0.660120\pi\)
−0.482083 + 0.876126i \(0.660120\pi\)
\(390\) −4.21809 −0.213591
\(391\) −4.04178 −0.204401
\(392\) −16.1690 −0.816658
\(393\) −2.17667 −0.109798
\(394\) −0.236467 −0.0119130
\(395\) −61.8833 −3.11369
\(396\) −4.29400 −0.215782
\(397\) 12.2106 0.612833 0.306416 0.951898i \(-0.400870\pi\)
0.306416 + 0.951898i \(0.400870\pi\)
\(398\) 21.3648 1.07092
\(399\) −3.15745 −0.158070
\(400\) 12.7923 0.639614
\(401\) 25.4544 1.27113 0.635567 0.772046i \(-0.280767\pi\)
0.635567 + 0.772046i \(0.280767\pi\)
\(402\) −3.79388 −0.189221
\(403\) 1.93587 0.0964325
\(404\) −14.6408 −0.728405
\(405\) 4.21809 0.209599
\(406\) −4.77815 −0.237136
\(407\) 30.6477 1.51915
\(408\) 2.64843 0.131117
\(409\) −17.5734 −0.868951 −0.434475 0.900684i \(-0.643066\pi\)
−0.434475 + 0.900684i \(0.643066\pi\)
\(410\) −29.6155 −1.46261
\(411\) 10.6655 0.526089
\(412\) −1.00000 −0.0492665
\(413\) 71.5575 3.52111
\(414\) −1.52610 −0.0750039
\(415\) 35.3789 1.73668
\(416\) −1.00000 −0.0490290
\(417\) −7.36838 −0.360831
\(418\) 2.81673 0.137771
\(419\) 25.0377 1.22317 0.611585 0.791179i \(-0.290532\pi\)
0.611585 + 0.791179i \(0.290532\pi\)
\(420\) −20.3034 −0.990706
\(421\) 37.5028 1.82778 0.913888 0.405966i \(-0.133065\pi\)
0.913888 + 0.405966i \(0.133065\pi\)
\(422\) 8.26635 0.402400
\(423\) 1.34179 0.0652402
\(424\) 1.36163 0.0661264
\(425\) −33.8794 −1.64339
\(426\) −4.83297 −0.234158
\(427\) 43.5583 2.10793
\(428\) −13.4887 −0.652003
\(429\) −4.29400 −0.207316
\(430\) 20.3553 0.981618
\(431\) −9.35222 −0.450480 −0.225240 0.974303i \(-0.572317\pi\)
−0.225240 + 0.974303i \(0.572317\pi\)
\(432\) 1.00000 0.0481125
\(433\) −1.48956 −0.0715835 −0.0357917 0.999359i \(-0.511395\pi\)
−0.0357917 + 0.999359i \(0.511395\pi\)
\(434\) 9.31815 0.447285
\(435\) −4.18718 −0.200760
\(436\) −2.10949 −0.101026
\(437\) 1.00107 0.0478879
\(438\) −1.21359 −0.0579875
\(439\) 35.5882 1.69853 0.849265 0.527967i \(-0.177045\pi\)
0.849265 + 0.527967i \(0.177045\pi\)
\(440\) 18.1125 0.863479
\(441\) 16.1690 0.769953
\(442\) 2.64843 0.125973
\(443\) −3.05759 −0.145270 −0.0726352 0.997359i \(-0.523141\pi\)
−0.0726352 + 0.997359i \(0.523141\pi\)
\(444\) −7.13733 −0.338723
\(445\) 51.4968 2.44118
\(446\) 2.67211 0.126528
\(447\) −4.93093 −0.233225
\(448\) −4.81342 −0.227413
\(449\) −5.76893 −0.272253 −0.136126 0.990691i \(-0.543465\pi\)
−0.136126 + 0.990691i \(0.543465\pi\)
\(450\) −12.7923 −0.603034
\(451\) −30.1485 −1.41964
\(452\) −3.48546 −0.163942
\(453\) −9.49514 −0.446120
\(454\) −12.4207 −0.582933
\(455\) −20.3034 −0.951840
\(456\) −0.655968 −0.0307185
\(457\) 28.6429 1.33986 0.669929 0.742425i \(-0.266324\pi\)
0.669929 + 0.742425i \(0.266324\pi\)
\(458\) 24.6135 1.15011
\(459\) −2.64843 −0.123618
\(460\) 6.43724 0.300138
\(461\) 8.15612 0.379868 0.189934 0.981797i \(-0.439173\pi\)
0.189934 + 0.981797i \(0.439173\pi\)
\(462\) −20.6688 −0.961602
\(463\) −10.4360 −0.485001 −0.242501 0.970151i \(-0.577968\pi\)
−0.242501 + 0.970151i \(0.577968\pi\)
\(464\) −0.992673 −0.0460837
\(465\) 8.16567 0.378674
\(466\) −5.52696 −0.256031
\(467\) −29.0299 −1.34334 −0.671671 0.740850i \(-0.734423\pi\)
−0.671671 + 0.740850i \(0.734423\pi\)
\(468\) 1.00000 0.0462250
\(469\) −18.2615 −0.843239
\(470\) −5.65980 −0.261067
\(471\) 12.2902 0.566300
\(472\) 14.8662 0.684274
\(473\) 20.7216 0.952780
\(474\) 14.6709 0.673858
\(475\) 8.39132 0.385020
\(476\) 12.7480 0.584303
\(477\) −1.36163 −0.0623445
\(478\) −14.7857 −0.676284
\(479\) 12.8742 0.588238 0.294119 0.955769i \(-0.404974\pi\)
0.294119 + 0.955769i \(0.404974\pi\)
\(480\) −4.21809 −0.192529
\(481\) −7.13733 −0.325434
\(482\) 15.9625 0.727071
\(483\) −7.34578 −0.334244
\(484\) 7.43847 0.338112
\(485\) 12.1405 0.551272
\(486\) −1.00000 −0.0453609
\(487\) 1.48082 0.0671023 0.0335511 0.999437i \(-0.489318\pi\)
0.0335511 + 0.999437i \(0.489318\pi\)
\(488\) 9.04935 0.409645
\(489\) 4.10109 0.185458
\(490\) −68.2023 −3.08107
\(491\) 11.9383 0.538769 0.269384 0.963033i \(-0.413180\pi\)
0.269384 + 0.963033i \(0.413180\pi\)
\(492\) 7.02108 0.316535
\(493\) 2.62902 0.118405
\(494\) −0.655968 −0.0295134
\(495\) −18.1125 −0.814096
\(496\) 1.93587 0.0869231
\(497\) −23.2631 −1.04349
\(498\) −8.38742 −0.375849
\(499\) −41.2431 −1.84629 −0.923147 0.384447i \(-0.874392\pi\)
−0.923147 + 0.384447i \(0.874392\pi\)
\(500\) 32.8685 1.46993
\(501\) −7.18868 −0.321167
\(502\) 7.52069 0.335665
\(503\) −13.4839 −0.601217 −0.300608 0.953748i \(-0.597190\pi\)
−0.300608 + 0.953748i \(0.597190\pi\)
\(504\) 4.81342 0.214407
\(505\) −61.7560 −2.74811
\(506\) 6.55309 0.291321
\(507\) 1.00000 0.0444116
\(508\) −18.7851 −0.833456
\(509\) 16.7375 0.741877 0.370939 0.928657i \(-0.379036\pi\)
0.370939 + 0.928657i \(0.379036\pi\)
\(510\) 11.1713 0.494674
\(511\) −5.84151 −0.258413
\(512\) −1.00000 −0.0441942
\(513\) 0.655968 0.0289617
\(514\) 27.5314 1.21436
\(515\) −4.21809 −0.185871
\(516\) −4.82571 −0.212440
\(517\) −5.76166 −0.253398
\(518\) −34.3550 −1.50947
\(519\) −14.0660 −0.617429
\(520\) −4.21809 −0.184975
\(521\) 30.8017 1.34945 0.674723 0.738071i \(-0.264263\pi\)
0.674723 + 0.738071i \(0.264263\pi\)
\(522\) 0.992673 0.0434481
\(523\) 7.33470 0.320724 0.160362 0.987058i \(-0.448734\pi\)
0.160362 + 0.987058i \(0.448734\pi\)
\(524\) −2.17667 −0.0950881
\(525\) −61.5746 −2.68734
\(526\) 4.79598 0.209115
\(527\) −5.12701 −0.223336
\(528\) −4.29400 −0.186873
\(529\) −20.6710 −0.898739
\(530\) 5.74346 0.249480
\(531\) −14.8662 −0.645140
\(532\) −3.15745 −0.136893
\(533\) 7.02108 0.304117
\(534\) −12.2086 −0.528316
\(535\) −56.8967 −2.45986
\(536\) −3.79388 −0.163870
\(537\) −21.1002 −0.910541
\(538\) 18.4173 0.794026
\(539\) −69.4298 −2.99055
\(540\) 4.21809 0.181518
\(541\) 15.0975 0.649091 0.324545 0.945870i \(-0.394789\pi\)
0.324545 + 0.945870i \(0.394789\pi\)
\(542\) −27.1420 −1.16585
\(543\) −18.8679 −0.809698
\(544\) 2.64843 0.113550
\(545\) −8.89800 −0.381149
\(546\) 4.81342 0.205995
\(547\) −9.07339 −0.387950 −0.193975 0.981006i \(-0.562138\pi\)
−0.193975 + 0.981006i \(0.562138\pi\)
\(548\) 10.6655 0.455606
\(549\) −9.04935 −0.386217
\(550\) 54.9301 2.34223
\(551\) −0.651161 −0.0277404
\(552\) −1.52610 −0.0649553
\(553\) 70.6174 3.00296
\(554\) −30.3796 −1.29071
\(555\) −30.1059 −1.27792
\(556\) −7.36838 −0.312489
\(557\) 25.0527 1.06152 0.530758 0.847523i \(-0.321907\pi\)
0.530758 + 0.847523i \(0.321907\pi\)
\(558\) −1.93587 −0.0819518
\(559\) −4.82571 −0.204106
\(560\) −20.3034 −0.857977
\(561\) 11.3724 0.480141
\(562\) 22.8883 0.965484
\(563\) 27.3887 1.15430 0.577149 0.816639i \(-0.304165\pi\)
0.577149 + 0.816639i \(0.304165\pi\)
\(564\) 1.34179 0.0564997
\(565\) −14.7020 −0.618516
\(566\) −17.8384 −0.749805
\(567\) −4.81342 −0.202145
\(568\) −4.83297 −0.202787
\(569\) 7.91850 0.331961 0.165980 0.986129i \(-0.446921\pi\)
0.165980 + 0.986129i \(0.446921\pi\)
\(570\) −2.76693 −0.115894
\(571\) 19.1442 0.801161 0.400581 0.916262i \(-0.368808\pi\)
0.400581 + 0.916262i \(0.368808\pi\)
\(572\) −4.29400 −0.179541
\(573\) −1.99582 −0.0833766
\(574\) 33.7954 1.41059
\(575\) 19.5223 0.814138
\(576\) 1.00000 0.0416667
\(577\) 35.7241 1.48721 0.743607 0.668617i \(-0.233113\pi\)
0.743607 + 0.668617i \(0.233113\pi\)
\(578\) 9.98583 0.415356
\(579\) −14.0177 −0.582555
\(580\) −4.18718 −0.173863
\(581\) −40.3722 −1.67492
\(582\) −2.87820 −0.119305
\(583\) 5.84682 0.242151
\(584\) −1.21359 −0.0502186
\(585\) 4.21809 0.174397
\(586\) 15.8528 0.654873
\(587\) −29.4458 −1.21536 −0.607680 0.794182i \(-0.707900\pi\)
−0.607680 + 0.794182i \(0.707900\pi\)
\(588\) 16.1690 0.666799
\(589\) 1.26987 0.0523239
\(590\) 62.7071 2.58161
\(591\) 0.236467 0.00972694
\(592\) −7.13733 −0.293343
\(593\) −43.7704 −1.79743 −0.898717 0.438528i \(-0.855500\pi\)
−0.898717 + 0.438528i \(0.855500\pi\)
\(594\) 4.29400 0.176185
\(595\) 53.7722 2.20444
\(596\) −4.93093 −0.201979
\(597\) −21.3648 −0.874403
\(598\) −1.52610 −0.0624070
\(599\) −37.2570 −1.52228 −0.761140 0.648588i \(-0.775360\pi\)
−0.761140 + 0.648588i \(0.775360\pi\)
\(600\) −12.7923 −0.522243
\(601\) 27.2474 1.11145 0.555723 0.831368i \(-0.312442\pi\)
0.555723 + 0.831368i \(0.312442\pi\)
\(602\) −23.2282 −0.946709
\(603\) 3.79388 0.154499
\(604\) −9.49514 −0.386352
\(605\) 31.3761 1.27562
\(606\) 14.6408 0.594740
\(607\) 5.08465 0.206380 0.103190 0.994662i \(-0.467095\pi\)
0.103190 + 0.994662i \(0.467095\pi\)
\(608\) −0.655968 −0.0266030
\(609\) 4.77815 0.193620
\(610\) 38.1710 1.54550
\(611\) 1.34179 0.0542831
\(612\) −2.64843 −0.107056
\(613\) 38.3291 1.54810 0.774048 0.633127i \(-0.218229\pi\)
0.774048 + 0.633127i \(0.218229\pi\)
\(614\) −28.0252 −1.13101
\(615\) 29.6155 1.19421
\(616\) −20.6688 −0.832772
\(617\) 13.7858 0.554996 0.277498 0.960726i \(-0.410495\pi\)
0.277498 + 0.960726i \(0.410495\pi\)
\(618\) 1.00000 0.0402259
\(619\) 35.3882 1.42237 0.711187 0.703003i \(-0.248158\pi\)
0.711187 + 0.703003i \(0.248158\pi\)
\(620\) 8.16567 0.327941
\(621\) 1.52610 0.0612404
\(622\) 10.6744 0.428004
\(623\) −58.7649 −2.35437
\(624\) 1.00000 0.0400320
\(625\) 74.6810 2.98724
\(626\) −15.8503 −0.633505
\(627\) −2.81673 −0.112489
\(628\) 12.2902 0.490430
\(629\) 18.9027 0.753701
\(630\) 20.3034 0.808908
\(631\) 13.8617 0.551827 0.275914 0.961182i \(-0.411020\pi\)
0.275914 + 0.961182i \(0.411020\pi\)
\(632\) 14.6709 0.583578
\(633\) −8.26635 −0.328558
\(634\) −22.0671 −0.876395
\(635\) −79.2374 −3.14444
\(636\) −1.36163 −0.0539919
\(637\) 16.1690 0.640640
\(638\) −4.26254 −0.168756
\(639\) 4.83297 0.191189
\(640\) −4.21809 −0.166735
\(641\) 39.4048 1.55640 0.778198 0.628019i \(-0.216134\pi\)
0.778198 + 0.628019i \(0.216134\pi\)
\(642\) 13.4887 0.532358
\(643\) 15.6956 0.618975 0.309488 0.950903i \(-0.399842\pi\)
0.309488 + 0.950903i \(0.399842\pi\)
\(644\) −7.34578 −0.289464
\(645\) −20.3553 −0.801488
\(646\) 1.73728 0.0683525
\(647\) 4.10390 0.161341 0.0806705 0.996741i \(-0.474294\pi\)
0.0806705 + 0.996741i \(0.474294\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 63.8357 2.50577
\(650\) −12.7923 −0.501754
\(651\) −9.31815 −0.365207
\(652\) 4.10109 0.160611
\(653\) 34.4890 1.34966 0.674830 0.737973i \(-0.264217\pi\)
0.674830 + 0.737973i \(0.264217\pi\)
\(654\) 2.10949 0.0824875
\(655\) −9.18137 −0.358746
\(656\) 7.02108 0.274127
\(657\) 1.21359 0.0473466
\(658\) 6.45861 0.251783
\(659\) 3.43753 0.133907 0.0669535 0.997756i \(-0.478672\pi\)
0.0669535 + 0.997756i \(0.478672\pi\)
\(660\) −18.1125 −0.705028
\(661\) −21.8422 −0.849565 −0.424782 0.905296i \(-0.639649\pi\)
−0.424782 + 0.905296i \(0.639649\pi\)
\(662\) 5.84180 0.227048
\(663\) −2.64843 −0.102856
\(664\) −8.38742 −0.325495
\(665\) −13.3184 −0.516465
\(666\) 7.13733 0.276566
\(667\) −1.51492 −0.0586580
\(668\) −7.18868 −0.278139
\(669\) −2.67211 −0.103310
\(670\) −16.0029 −0.618246
\(671\) 38.8579 1.50009
\(672\) 4.81342 0.185682
\(673\) −29.9645 −1.15505 −0.577523 0.816374i \(-0.695981\pi\)
−0.577523 + 0.816374i \(0.695981\pi\)
\(674\) −6.31009 −0.243055
\(675\) 12.7923 0.492375
\(676\) 1.00000 0.0384615
\(677\) −32.0957 −1.23354 −0.616770 0.787144i \(-0.711559\pi\)
−0.616770 + 0.787144i \(0.711559\pi\)
\(678\) 3.48546 0.133858
\(679\) −13.8540 −0.531667
\(680\) 11.1713 0.428400
\(681\) 12.4207 0.475963
\(682\) 8.31263 0.318307
\(683\) −48.1248 −1.84144 −0.920722 0.390218i \(-0.872400\pi\)
−0.920722 + 0.390218i \(0.872400\pi\)
\(684\) 0.655968 0.0250815
\(685\) 44.9879 1.71890
\(686\) 44.1343 1.68506
\(687\) −24.6135 −0.939063
\(688\) −4.82571 −0.183978
\(689\) −1.36163 −0.0518738
\(690\) −6.43724 −0.245062
\(691\) 35.3486 1.34472 0.672362 0.740222i \(-0.265280\pi\)
0.672362 + 0.740222i \(0.265280\pi\)
\(692\) −14.0660 −0.534709
\(693\) 20.6688 0.785145
\(694\) −11.8496 −0.449806
\(695\) −31.0805 −1.17895
\(696\) 0.992673 0.0376272
\(697\) −18.5948 −0.704329
\(698\) −2.98569 −0.113010
\(699\) 5.52696 0.209049
\(700\) −61.5746 −2.32730
\(701\) −14.7957 −0.558826 −0.279413 0.960171i \(-0.590140\pi\)
−0.279413 + 0.960171i \(0.590140\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −4.68186 −0.176580
\(704\) −4.29400 −0.161836
\(705\) 5.65980 0.213160
\(706\) 33.7345 1.26961
\(707\) 70.4721 2.65038
\(708\) −14.8662 −0.558708
\(709\) 28.9327 1.08659 0.543296 0.839541i \(-0.317176\pi\)
0.543296 + 0.839541i \(0.317176\pi\)
\(710\) −20.3859 −0.765070
\(711\) −14.6709 −0.550203
\(712\) −12.2086 −0.457535
\(713\) 2.95434 0.110641
\(714\) −12.7480 −0.477082
\(715\) −18.1125 −0.677369
\(716\) −21.1002 −0.788551
\(717\) 14.7857 0.552184
\(718\) 6.51521 0.243146
\(719\) −38.8020 −1.44707 −0.723534 0.690288i \(-0.757484\pi\)
−0.723534 + 0.690288i \(0.757484\pi\)
\(720\) 4.21809 0.157199
\(721\) 4.81342 0.179261
\(722\) 18.5697 0.691093
\(723\) −15.9625 −0.593651
\(724\) −18.8679 −0.701219
\(725\) −12.6985 −0.471612
\(726\) −7.43847 −0.276068
\(727\) 16.6426 0.617238 0.308619 0.951186i \(-0.400133\pi\)
0.308619 + 0.951186i \(0.400133\pi\)
\(728\) 4.81342 0.178397
\(729\) 1.00000 0.0370370
\(730\) −5.11902 −0.189464
\(731\) 12.7805 0.472705
\(732\) −9.04935 −0.334473
\(733\) −40.2474 −1.48657 −0.743285 0.668975i \(-0.766733\pi\)
−0.743285 + 0.668975i \(0.766733\pi\)
\(734\) −25.6512 −0.946802
\(735\) 68.2023 2.51568
\(736\) −1.52610 −0.0562529
\(737\) −16.2909 −0.600084
\(738\) −7.02108 −0.258449
\(739\) −40.9021 −1.50461 −0.752305 0.658815i \(-0.771058\pi\)
−0.752305 + 0.658815i \(0.771058\pi\)
\(740\) −30.1059 −1.10672
\(741\) 0.655968 0.0240976
\(742\) −6.55407 −0.240608
\(743\) −10.1196 −0.371253 −0.185627 0.982620i \(-0.559432\pi\)
−0.185627 + 0.982620i \(0.559432\pi\)
\(744\) −1.93587 −0.0709724
\(745\) −20.7991 −0.762020
\(746\) 14.8679 0.544352
\(747\) 8.38742 0.306880
\(748\) 11.3724 0.415815
\(749\) 64.9270 2.37238
\(750\) −32.8685 −1.20019
\(751\) 2.57695 0.0940342 0.0470171 0.998894i \(-0.485028\pi\)
0.0470171 + 0.998894i \(0.485028\pi\)
\(752\) 1.34179 0.0489302
\(753\) −7.52069 −0.274069
\(754\) 0.992673 0.0361510
\(755\) −40.0513 −1.45762
\(756\) −4.81342 −0.175062
\(757\) 39.8064 1.44679 0.723394 0.690435i \(-0.242581\pi\)
0.723394 + 0.690435i \(0.242581\pi\)
\(758\) 8.72164 0.316784
\(759\) −6.55309 −0.237862
\(760\) −2.76693 −0.100367
\(761\) 43.5365 1.57820 0.789098 0.614267i \(-0.210548\pi\)
0.789098 + 0.614267i \(0.210548\pi\)
\(762\) 18.7851 0.680514
\(763\) 10.1538 0.367594
\(764\) −1.99582 −0.0722062
\(765\) −11.1713 −0.403899
\(766\) 4.66833 0.168674
\(767\) −14.8662 −0.536789
\(768\) 1.00000 0.0360844
\(769\) 3.22205 0.116190 0.0580950 0.998311i \(-0.481497\pi\)
0.0580950 + 0.998311i \(0.481497\pi\)
\(770\) −87.1830 −3.14186
\(771\) −27.5314 −0.991519
\(772\) −14.0177 −0.504508
\(773\) 31.7493 1.14194 0.570971 0.820970i \(-0.306567\pi\)
0.570971 + 0.820970i \(0.306567\pi\)
\(774\) 4.82571 0.173456
\(775\) 24.7642 0.889555
\(776\) −2.87820 −0.103321
\(777\) 34.3550 1.23248
\(778\) 19.0163 0.681768
\(779\) 4.60560 0.165013
\(780\) 4.21809 0.151032
\(781\) −20.7528 −0.742594
\(782\) 4.04178 0.144534
\(783\) −0.992673 −0.0354752
\(784\) 16.1690 0.577465
\(785\) 51.8410 1.85028
\(786\) 2.17667 0.0776391
\(787\) −20.9982 −0.748504 −0.374252 0.927327i \(-0.622101\pi\)
−0.374252 + 0.927327i \(0.622101\pi\)
\(788\) 0.236467 0.00842378
\(789\) −4.79598 −0.170741
\(790\) 61.8833 2.20171
\(791\) 16.7770 0.596520
\(792\) 4.29400 0.152581
\(793\) −9.04935 −0.321352
\(794\) −12.2106 −0.433338
\(795\) −5.74346 −0.203699
\(796\) −21.3648 −0.757255
\(797\) 31.6900 1.12252 0.561258 0.827641i \(-0.310318\pi\)
0.561258 + 0.827641i \(0.310318\pi\)
\(798\) 3.15745 0.111772
\(799\) −3.55364 −0.125719
\(800\) −12.7923 −0.452275
\(801\) 12.2086 0.431368
\(802\) −25.4544 −0.898827
\(803\) −5.21115 −0.183898
\(804\) 3.79388 0.133800
\(805\) −30.9851 −1.09208
\(806\) −1.93587 −0.0681881
\(807\) −18.4173 −0.648320
\(808\) 14.6408 0.515060
\(809\) −15.3005 −0.537937 −0.268969 0.963149i \(-0.586683\pi\)
−0.268969 + 0.963149i \(0.586683\pi\)
\(810\) −4.21809 −0.148209
\(811\) 35.3356 1.24080 0.620400 0.784286i \(-0.286970\pi\)
0.620400 + 0.784286i \(0.286970\pi\)
\(812\) 4.77815 0.167680
\(813\) 27.1420 0.951912
\(814\) −30.6477 −1.07420
\(815\) 17.2988 0.605950
\(816\) −2.64843 −0.0927135
\(817\) −3.16551 −0.110747
\(818\) 17.5734 0.614441
\(819\) −4.81342 −0.168195
\(820\) 29.6155 1.03422
\(821\) −47.5746 −1.66037 −0.830183 0.557491i \(-0.811764\pi\)
−0.830183 + 0.557491i \(0.811764\pi\)
\(822\) −10.6655 −0.372001
\(823\) −33.6003 −1.17123 −0.585616 0.810589i \(-0.699147\pi\)
−0.585616 + 0.810589i \(0.699147\pi\)
\(824\) 1.00000 0.0348367
\(825\) −54.9301 −1.91242
\(826\) −71.5575 −2.48980
\(827\) 45.2481 1.57343 0.786715 0.617316i \(-0.211780\pi\)
0.786715 + 0.617316i \(0.211780\pi\)
\(828\) 1.52610 0.0530358
\(829\) 2.31906 0.0805443 0.0402721 0.999189i \(-0.487178\pi\)
0.0402721 + 0.999189i \(0.487178\pi\)
\(830\) −35.3789 −1.22802
\(831\) 30.3796 1.05386
\(832\) 1.00000 0.0346688
\(833\) −42.8225 −1.48371
\(834\) 7.36838 0.255146
\(835\) −30.3225 −1.04935
\(836\) −2.81673 −0.0974185
\(837\) 1.93587 0.0669134
\(838\) −25.0377 −0.864911
\(839\) −15.1801 −0.524076 −0.262038 0.965058i \(-0.584395\pi\)
−0.262038 + 0.965058i \(0.584395\pi\)
\(840\) 20.3034 0.700535
\(841\) −28.0146 −0.966021
\(842\) −37.5028 −1.29243
\(843\) −22.8883 −0.788315
\(844\) −8.26635 −0.284539
\(845\) 4.21809 0.145107
\(846\) −1.34179 −0.0461318
\(847\) −35.8045 −1.23026
\(848\) −1.36163 −0.0467584
\(849\) 17.8384 0.612213
\(850\) 33.8794 1.16205
\(851\) −10.8923 −0.373384
\(852\) 4.83297 0.165575
\(853\) 7.81605 0.267617 0.133808 0.991007i \(-0.457279\pi\)
0.133808 + 0.991007i \(0.457279\pi\)
\(854\) −43.5583 −1.49053
\(855\) 2.76693 0.0946270
\(856\) 13.4887 0.461036
\(857\) −49.9702 −1.70695 −0.853475 0.521134i \(-0.825509\pi\)
−0.853475 + 0.521134i \(0.825509\pi\)
\(858\) 4.29400 0.146595
\(859\) −28.9793 −0.988761 −0.494380 0.869246i \(-0.664605\pi\)
−0.494380 + 0.869246i \(0.664605\pi\)
\(860\) −20.3553 −0.694109
\(861\) −33.7954 −1.15174
\(862\) 9.35222 0.318538
\(863\) −40.7436 −1.38693 −0.693463 0.720492i \(-0.743916\pi\)
−0.693463 + 0.720492i \(0.743916\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −59.3317 −2.01734
\(866\) 1.48956 0.0506172
\(867\) −9.98583 −0.339137
\(868\) −9.31815 −0.316279
\(869\) 62.9971 2.13703
\(870\) 4.18718 0.141959
\(871\) 3.79388 0.128551
\(872\) 2.10949 0.0714362
\(873\) 2.87820 0.0974123
\(874\) −1.00107 −0.0338618
\(875\) −158.210 −5.34847
\(876\) 1.21359 0.0410034
\(877\) −42.0415 −1.41964 −0.709820 0.704384i \(-0.751224\pi\)
−0.709820 + 0.704384i \(0.751224\pi\)
\(878\) −35.5882 −1.20104
\(879\) −15.8528 −0.534702
\(880\) −18.1125 −0.610572
\(881\) −24.3898 −0.821714 −0.410857 0.911700i \(-0.634770\pi\)
−0.410857 + 0.911700i \(0.634770\pi\)
\(882\) −16.1690 −0.544439
\(883\) 12.4278 0.418230 0.209115 0.977891i \(-0.432942\pi\)
0.209115 + 0.977891i \(0.432942\pi\)
\(884\) −2.64843 −0.0890763
\(885\) −62.7071 −2.10788
\(886\) 3.05759 0.102722
\(887\) 2.20287 0.0739651 0.0369826 0.999316i \(-0.488225\pi\)
0.0369826 + 0.999316i \(0.488225\pi\)
\(888\) 7.13733 0.239513
\(889\) 90.4208 3.03262
\(890\) −51.4968 −1.72618
\(891\) −4.29400 −0.143855
\(892\) −2.67211 −0.0894689
\(893\) 0.880173 0.0294539
\(894\) 4.93093 0.164915
\(895\) −89.0025 −2.97503
\(896\) 4.81342 0.160805
\(897\) 1.52610 0.0509551
\(898\) 5.76893 0.192512
\(899\) −1.92168 −0.0640918
\(900\) 12.7923 0.426409
\(901\) 3.60617 0.120139
\(902\) 30.1485 1.00384
\(903\) 23.2282 0.772984
\(904\) 3.48546 0.115925
\(905\) −79.5864 −2.64554
\(906\) 9.49514 0.315455
\(907\) 18.8620 0.626303 0.313152 0.949703i \(-0.398615\pi\)
0.313152 + 0.949703i \(0.398615\pi\)
\(908\) 12.4207 0.412196
\(909\) −14.6408 −0.485603
\(910\) 20.3034 0.673052
\(911\) 7.57458 0.250957 0.125479 0.992096i \(-0.459953\pi\)
0.125479 + 0.992096i \(0.459953\pi\)
\(912\) 0.655968 0.0217213
\(913\) −36.0156 −1.19194
\(914\) −28.6429 −0.947423
\(915\) −38.1710 −1.26189
\(916\) −24.6135 −0.813253
\(917\) 10.4772 0.345988
\(918\) 2.64843 0.0874111
\(919\) −35.9284 −1.18517 −0.592584 0.805508i \(-0.701892\pi\)
−0.592584 + 0.805508i \(0.701892\pi\)
\(920\) −6.43724 −0.212230
\(921\) 28.0252 0.923463
\(922\) −8.15612 −0.268607
\(923\) 4.83297 0.159079
\(924\) 20.6688 0.679955
\(925\) −91.3028 −3.00202
\(926\) 10.4360 0.342948
\(927\) −1.00000 −0.0328443
\(928\) 0.992673 0.0325861
\(929\) −16.4262 −0.538927 −0.269464 0.963011i \(-0.586846\pi\)
−0.269464 + 0.963011i \(0.586846\pi\)
\(930\) −8.16567 −0.267763
\(931\) 10.6063 0.347609
\(932\) 5.52696 0.181042
\(933\) −10.6744 −0.349463
\(934\) 29.0299 0.949886
\(935\) 47.9696 1.56877
\(936\) −1.00000 −0.0326860
\(937\) 51.2234 1.67340 0.836698 0.547665i \(-0.184483\pi\)
0.836698 + 0.547665i \(0.184483\pi\)
\(938\) 18.2615 0.596260
\(939\) 15.8503 0.517255
\(940\) 5.65980 0.184602
\(941\) 22.0325 0.718238 0.359119 0.933292i \(-0.383077\pi\)
0.359119 + 0.933292i \(0.383077\pi\)
\(942\) −12.2902 −0.400435
\(943\) 10.7149 0.348925
\(944\) −14.8662 −0.483855
\(945\) −20.3034 −0.660471
\(946\) −20.7216 −0.673717
\(947\) −36.1189 −1.17371 −0.586854 0.809693i \(-0.699634\pi\)
−0.586854 + 0.809693i \(0.699634\pi\)
\(948\) −14.6709 −0.476490
\(949\) 1.21359 0.0393947
\(950\) −8.39132 −0.272250
\(951\) 22.0671 0.715574
\(952\) −12.7480 −0.413165
\(953\) −25.1622 −0.815083 −0.407541 0.913187i \(-0.633614\pi\)
−0.407541 + 0.913187i \(0.633614\pi\)
\(954\) 1.36163 0.0440842
\(955\) −8.41855 −0.272418
\(956\) 14.7857 0.478205
\(957\) 4.26254 0.137788
\(958\) −12.8742 −0.415947
\(959\) −51.3374 −1.65777
\(960\) 4.21809 0.136138
\(961\) −27.2524 −0.879110
\(962\) 7.13733 0.230117
\(963\) −13.4887 −0.434669
\(964\) −15.9625 −0.514117
\(965\) −59.1279 −1.90339
\(966\) 7.34578 0.236347
\(967\) −1.46713 −0.0471797 −0.0235898 0.999722i \(-0.507510\pi\)
−0.0235898 + 0.999722i \(0.507510\pi\)
\(968\) −7.43847 −0.239082
\(969\) −1.73728 −0.0558096
\(970\) −12.1405 −0.389808
\(971\) 40.7250 1.30693 0.653464 0.756957i \(-0.273315\pi\)
0.653464 + 0.756957i \(0.273315\pi\)
\(972\) 1.00000 0.0320750
\(973\) 35.4671 1.13702
\(974\) −1.48082 −0.0474485
\(975\) 12.7923 0.409681
\(976\) −9.04935 −0.289663
\(977\) −15.3214 −0.490176 −0.245088 0.969501i \(-0.578817\pi\)
−0.245088 + 0.969501i \(0.578817\pi\)
\(978\) −4.10109 −0.131139
\(979\) −52.4236 −1.67547
\(980\) 68.2023 2.17864
\(981\) −2.10949 −0.0673507
\(982\) −11.9383 −0.380967
\(983\) 15.3087 0.488272 0.244136 0.969741i \(-0.421496\pi\)
0.244136 + 0.969741i \(0.421496\pi\)
\(984\) −7.02108 −0.223824
\(985\) 0.997438 0.0317810
\(986\) −2.62902 −0.0837251
\(987\) −6.45861 −0.205580
\(988\) 0.655968 0.0208691
\(989\) −7.36453 −0.234178
\(990\) 18.1125 0.575653
\(991\) 56.9074 1.80772 0.903862 0.427825i \(-0.140720\pi\)
0.903862 + 0.427825i \(0.140720\pi\)
\(992\) −1.93587 −0.0614639
\(993\) −5.84180 −0.185384
\(994\) 23.2631 0.737862
\(995\) −90.1186 −2.85695
\(996\) 8.38742 0.265766
\(997\) 58.7078 1.85930 0.929648 0.368449i \(-0.120111\pi\)
0.929648 + 0.368449i \(0.120111\pi\)
\(998\) 41.2431 1.30553
\(999\) −7.13733 −0.225815
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.w.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.12 12 1.1 even 1 trivial