Properties

Label 8007.2.a.f.1.18
Level 8007
Weight 2
Character 8007.1
Self dual yes
Analytic conductor 63.936
Analytic rank 1
Dimension 48
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) = 8007.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.901374 q^{2} -1.00000 q^{3} -1.18752 q^{4} +1.97657 q^{5} +0.901374 q^{6} +4.00403 q^{7} +2.87315 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.901374 q^{2} -1.00000 q^{3} -1.18752 q^{4} +1.97657 q^{5} +0.901374 q^{6} +4.00403 q^{7} +2.87315 q^{8} +1.00000 q^{9} -1.78163 q^{10} -4.85826 q^{11} +1.18752 q^{12} -5.58466 q^{13} -3.60913 q^{14} -1.97657 q^{15} -0.214739 q^{16} -1.00000 q^{17} -0.901374 q^{18} +4.03916 q^{19} -2.34722 q^{20} -4.00403 q^{21} +4.37911 q^{22} -0.727073 q^{23} -2.87315 q^{24} -1.09318 q^{25} +5.03387 q^{26} -1.00000 q^{27} -4.75489 q^{28} +6.57485 q^{29} +1.78163 q^{30} -2.80931 q^{31} -5.55275 q^{32} +4.85826 q^{33} +0.901374 q^{34} +7.91425 q^{35} -1.18752 q^{36} +1.01262 q^{37} -3.64079 q^{38} +5.58466 q^{39} +5.67898 q^{40} +3.57212 q^{41} +3.60913 q^{42} +2.80186 q^{43} +5.76930 q^{44} +1.97657 q^{45} +0.655365 q^{46} +9.68563 q^{47} +0.214739 q^{48} +9.03230 q^{49} +0.985361 q^{50} +1.00000 q^{51} +6.63192 q^{52} -13.7314 q^{53} +0.901374 q^{54} -9.60269 q^{55} +11.5042 q^{56} -4.03916 q^{57} -5.92640 q^{58} -4.64387 q^{59} +2.34722 q^{60} +14.4603 q^{61} +2.53224 q^{62} +4.00403 q^{63} +5.43458 q^{64} -11.0385 q^{65} -4.37911 q^{66} -14.3176 q^{67} +1.18752 q^{68} +0.727073 q^{69} -7.13370 q^{70} +5.65707 q^{71} +2.87315 q^{72} -7.35379 q^{73} -0.912753 q^{74} +1.09318 q^{75} -4.79660 q^{76} -19.4526 q^{77} -5.03387 q^{78} -7.48554 q^{79} -0.424446 q^{80} +1.00000 q^{81} -3.21981 q^{82} -14.8209 q^{83} +4.75489 q^{84} -1.97657 q^{85} -2.52553 q^{86} -6.57485 q^{87} -13.9585 q^{88} +0.976320 q^{89} -1.78163 q^{90} -22.3612 q^{91} +0.863416 q^{92} +2.80931 q^{93} -8.73038 q^{94} +7.98367 q^{95} +5.55275 q^{96} -4.48034 q^{97} -8.14148 q^{98} -4.85826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} + O(q^{10}) \) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} - 20q^{10} + 5q^{11} - 45q^{12} - 8q^{13} + 4q^{14} - q^{15} + 39q^{16} - 48q^{17} - q^{18} - 6q^{19} + 6q^{20} + 13q^{21} - 35q^{22} - 8q^{23} + 6q^{24} + 13q^{25} + 17q^{26} - 48q^{27} - 38q^{28} + q^{29} + 20q^{30} - 21q^{31} - 3q^{32} - 5q^{33} + q^{34} + 19q^{35} + 45q^{36} - 58q^{37} - 14q^{38} + 8q^{39} - 54q^{40} - 3q^{41} - 4q^{42} - 33q^{43} + 2q^{44} + q^{45} - 26q^{46} + 9q^{47} - 39q^{48} + 11q^{49} + 4q^{50} + 48q^{51} - 31q^{52} - 33q^{53} + q^{54} - 21q^{55} + 6q^{57} - 55q^{58} + 77q^{59} - 6q^{60} - 29q^{61} - 46q^{62} - 13q^{63} + 24q^{64} - 49q^{65} + 35q^{66} - 44q^{67} - 45q^{68} + 8q^{69} + 4q^{70} + 22q^{71} - 6q^{72} - 63q^{73} - 16q^{74} - 13q^{75} - 46q^{76} - 30q^{77} - 17q^{78} - 46q^{79} - 14q^{80} + 48q^{81} - 75q^{82} + 11q^{83} + 38q^{84} - q^{85} + 8q^{86} - q^{87} - 116q^{88} + 10q^{89} - 20q^{90} - 67q^{91} - 64q^{92} + 21q^{93} - 16q^{94} - 8q^{95} + 3q^{96} - 96q^{97} - 46q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.901374 −0.637368 −0.318684 0.947861i \(-0.603241\pi\)
−0.318684 + 0.947861i \(0.603241\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.18752 −0.593762
\(5\) 1.97657 0.883948 0.441974 0.897028i \(-0.354278\pi\)
0.441974 + 0.897028i \(0.354278\pi\)
\(6\) 0.901374 0.367985
\(7\) 4.00403 1.51338 0.756691 0.653772i \(-0.226815\pi\)
0.756691 + 0.653772i \(0.226815\pi\)
\(8\) 2.87315 1.01581
\(9\) 1.00000 0.333333
\(10\) −1.78163 −0.563400
\(11\) −4.85826 −1.46482 −0.732410 0.680864i \(-0.761605\pi\)
−0.732410 + 0.680864i \(0.761605\pi\)
\(12\) 1.18752 0.342809
\(13\) −5.58466 −1.54891 −0.774453 0.632631i \(-0.781975\pi\)
−0.774453 + 0.632631i \(0.781975\pi\)
\(14\) −3.60913 −0.964582
\(15\) −1.97657 −0.510348
\(16\) −0.214739 −0.0536847
\(17\) −1.00000 −0.242536
\(18\) −0.901374 −0.212456
\(19\) 4.03916 0.926646 0.463323 0.886189i \(-0.346657\pi\)
0.463323 + 0.886189i \(0.346657\pi\)
\(20\) −2.34722 −0.524855
\(21\) −4.00403 −0.873752
\(22\) 4.37911 0.933630
\(23\) −0.727073 −0.151605 −0.0758026 0.997123i \(-0.524152\pi\)
−0.0758026 + 0.997123i \(0.524152\pi\)
\(24\) −2.87315 −0.586480
\(25\) −1.09318 −0.218635
\(26\) 5.03387 0.987223
\(27\) −1.00000 −0.192450
\(28\) −4.75489 −0.898589
\(29\) 6.57485 1.22092 0.610460 0.792047i \(-0.290985\pi\)
0.610460 + 0.792047i \(0.290985\pi\)
\(30\) 1.78163 0.325279
\(31\) −2.80931 −0.504567 −0.252283 0.967653i \(-0.581182\pi\)
−0.252283 + 0.967653i \(0.581182\pi\)
\(32\) −5.55275 −0.981596
\(33\) 4.85826 0.845715
\(34\) 0.901374 0.154584
\(35\) 7.91425 1.33775
\(36\) −1.18752 −0.197921
\(37\) 1.01262 0.166474 0.0832371 0.996530i \(-0.473474\pi\)
0.0832371 + 0.996530i \(0.473474\pi\)
\(38\) −3.64079 −0.590615
\(39\) 5.58466 0.894261
\(40\) 5.67898 0.897926
\(41\) 3.57212 0.557871 0.278935 0.960310i \(-0.410018\pi\)
0.278935 + 0.960310i \(0.410018\pi\)
\(42\) 3.60913 0.556902
\(43\) 2.80186 0.427280 0.213640 0.976912i \(-0.431468\pi\)
0.213640 + 0.976912i \(0.431468\pi\)
\(44\) 5.76930 0.869755
\(45\) 1.97657 0.294649
\(46\) 0.655365 0.0966283
\(47\) 9.68563 1.41279 0.706397 0.707815i \(-0.250319\pi\)
0.706397 + 0.707815i \(0.250319\pi\)
\(48\) 0.214739 0.0309949
\(49\) 9.03230 1.29033
\(50\) 0.985361 0.139351
\(51\) 1.00000 0.140028
\(52\) 6.63192 0.919682
\(53\) −13.7314 −1.88615 −0.943076 0.332579i \(-0.892081\pi\)
−0.943076 + 0.332579i \(0.892081\pi\)
\(54\) 0.901374 0.122662
\(55\) −9.60269 −1.29483
\(56\) 11.5042 1.53731
\(57\) −4.03916 −0.534999
\(58\) −5.92640 −0.778175
\(59\) −4.64387 −0.604581 −0.302290 0.953216i \(-0.597751\pi\)
−0.302290 + 0.953216i \(0.597751\pi\)
\(60\) 2.34722 0.303025
\(61\) 14.4603 1.85146 0.925728 0.378190i \(-0.123454\pi\)
0.925728 + 0.378190i \(0.123454\pi\)
\(62\) 2.53224 0.321595
\(63\) 4.00403 0.504461
\(64\) 5.43458 0.679323
\(65\) −11.0385 −1.36915
\(66\) −4.37911 −0.539031
\(67\) −14.3176 −1.74917 −0.874584 0.484874i \(-0.838866\pi\)
−0.874584 + 0.484874i \(0.838866\pi\)
\(68\) 1.18752 0.144008
\(69\) 0.727073 0.0875293
\(70\) −7.13370 −0.852641
\(71\) 5.65707 0.671371 0.335685 0.941974i \(-0.391032\pi\)
0.335685 + 0.941974i \(0.391032\pi\)
\(72\) 2.87315 0.338604
\(73\) −7.35379 −0.860696 −0.430348 0.902663i \(-0.641609\pi\)
−0.430348 + 0.902663i \(0.641609\pi\)
\(74\) −0.912753 −0.106105
\(75\) 1.09318 0.126229
\(76\) −4.79660 −0.550207
\(77\) −19.4526 −2.21683
\(78\) −5.03387 −0.569974
\(79\) −7.48554 −0.842189 −0.421095 0.907017i \(-0.638354\pi\)
−0.421095 + 0.907017i \(0.638354\pi\)
\(80\) −0.424446 −0.0474545
\(81\) 1.00000 0.111111
\(82\) −3.21981 −0.355569
\(83\) −14.8209 −1.62681 −0.813405 0.581698i \(-0.802388\pi\)
−0.813405 + 0.581698i \(0.802388\pi\)
\(84\) 4.75489 0.518801
\(85\) −1.97657 −0.214389
\(86\) −2.52553 −0.272335
\(87\) −6.57485 −0.704898
\(88\) −13.9585 −1.48798
\(89\) 0.976320 0.103490 0.0517448 0.998660i \(-0.483522\pi\)
0.0517448 + 0.998660i \(0.483522\pi\)
\(90\) −1.78163 −0.187800
\(91\) −22.3612 −2.34409
\(92\) 0.863416 0.0900174
\(93\) 2.80931 0.291312
\(94\) −8.73038 −0.900470
\(95\) 7.98367 0.819107
\(96\) 5.55275 0.566725
\(97\) −4.48034 −0.454910 −0.227455 0.973789i \(-0.573040\pi\)
−0.227455 + 0.973789i \(0.573040\pi\)
\(98\) −8.14148 −0.822414
\(99\) −4.85826 −0.488274
\(100\) 1.29817 0.129817
\(101\) 15.4458 1.53692 0.768458 0.639900i \(-0.221024\pi\)
0.768458 + 0.639900i \(0.221024\pi\)
\(102\) −0.901374 −0.0892494
\(103\) −16.2499 −1.60115 −0.800576 0.599231i \(-0.795473\pi\)
−0.800576 + 0.599231i \(0.795473\pi\)
\(104\) −16.0456 −1.57340
\(105\) −7.91425 −0.772352
\(106\) 12.3771 1.20217
\(107\) 3.01898 0.291856 0.145928 0.989295i \(-0.453383\pi\)
0.145928 + 0.989295i \(0.453383\pi\)
\(108\) 1.18752 0.114270
\(109\) −0.176188 −0.0168758 −0.00843789 0.999964i \(-0.502686\pi\)
−0.00843789 + 0.999964i \(0.502686\pi\)
\(110\) 8.65562 0.825281
\(111\) −1.01262 −0.0961139
\(112\) −0.859821 −0.0812454
\(113\) −4.38115 −0.412144 −0.206072 0.978537i \(-0.566068\pi\)
−0.206072 + 0.978537i \(0.566068\pi\)
\(114\) 3.64079 0.340991
\(115\) −1.43711 −0.134011
\(116\) −7.80779 −0.724936
\(117\) −5.58466 −0.516302
\(118\) 4.18587 0.385340
\(119\) −4.00403 −0.367049
\(120\) −5.67898 −0.518418
\(121\) 12.6027 1.14570
\(122\) −13.0342 −1.18006
\(123\) −3.57212 −0.322087
\(124\) 3.33612 0.299593
\(125\) −12.0436 −1.07721
\(126\) −3.60913 −0.321527
\(127\) 1.50887 0.133891 0.0669455 0.997757i \(-0.478675\pi\)
0.0669455 + 0.997757i \(0.478675\pi\)
\(128\) 6.20690 0.548618
\(129\) −2.80186 −0.246690
\(130\) 9.94979 0.872654
\(131\) 20.3494 1.77794 0.888969 0.457967i \(-0.151422\pi\)
0.888969 + 0.457967i \(0.151422\pi\)
\(132\) −5.76930 −0.502153
\(133\) 16.1729 1.40237
\(134\) 12.9055 1.11486
\(135\) −1.97657 −0.170116
\(136\) −2.87315 −0.246371
\(137\) −7.54344 −0.644480 −0.322240 0.946658i \(-0.604436\pi\)
−0.322240 + 0.946658i \(0.604436\pi\)
\(138\) −0.655365 −0.0557884
\(139\) −7.69508 −0.652689 −0.326344 0.945251i \(-0.605817\pi\)
−0.326344 + 0.945251i \(0.605817\pi\)
\(140\) −9.39836 −0.794307
\(141\) −9.68563 −0.815677
\(142\) −5.09914 −0.427910
\(143\) 27.1317 2.26887
\(144\) −0.214739 −0.0178949
\(145\) 12.9956 1.07923
\(146\) 6.62852 0.548580
\(147\) −9.03230 −0.744971
\(148\) −1.20251 −0.0988461
\(149\) −13.8089 −1.13127 −0.565633 0.824657i \(-0.691368\pi\)
−0.565633 + 0.824657i \(0.691368\pi\)
\(150\) −0.985361 −0.0804544
\(151\) 4.58302 0.372961 0.186480 0.982459i \(-0.440292\pi\)
0.186480 + 0.982459i \(0.440292\pi\)
\(152\) 11.6051 0.941299
\(153\) −1.00000 −0.0808452
\(154\) 17.5341 1.41294
\(155\) −5.55279 −0.446011
\(156\) −6.63192 −0.530978
\(157\) −1.00000 −0.0798087
\(158\) 6.74728 0.536785
\(159\) 13.7314 1.08897
\(160\) −10.9754 −0.867680
\(161\) −2.91122 −0.229437
\(162\) −0.901374 −0.0708187
\(163\) 10.3001 0.806762 0.403381 0.915032i \(-0.367835\pi\)
0.403381 + 0.915032i \(0.367835\pi\)
\(164\) −4.24197 −0.331243
\(165\) 9.60269 0.747568
\(166\) 13.3592 1.03688
\(167\) 16.0730 1.24377 0.621883 0.783110i \(-0.286368\pi\)
0.621883 + 0.783110i \(0.286368\pi\)
\(168\) −11.5042 −0.887569
\(169\) 18.1884 1.39911
\(170\) 1.78163 0.136645
\(171\) 4.03916 0.308882
\(172\) −3.32728 −0.253703
\(173\) −22.1149 −1.68136 −0.840681 0.541530i \(-0.817845\pi\)
−0.840681 + 0.541530i \(0.817845\pi\)
\(174\) 5.92640 0.449280
\(175\) −4.37711 −0.330879
\(176\) 1.04326 0.0786384
\(177\) 4.64387 0.349055
\(178\) −0.880030 −0.0659610
\(179\) 18.7067 1.39820 0.699102 0.715022i \(-0.253583\pi\)
0.699102 + 0.715022i \(0.253583\pi\)
\(180\) −2.34722 −0.174952
\(181\) −2.45583 −0.182541 −0.0912704 0.995826i \(-0.529093\pi\)
−0.0912704 + 0.995826i \(0.529093\pi\)
\(182\) 20.1558 1.49405
\(183\) −14.4603 −1.06894
\(184\) −2.08899 −0.154002
\(185\) 2.00152 0.147155
\(186\) −2.53224 −0.185673
\(187\) 4.85826 0.355271
\(188\) −11.5019 −0.838864
\(189\) −4.00403 −0.291251
\(190\) −7.19628 −0.522073
\(191\) −13.6758 −0.989547 −0.494773 0.869022i \(-0.664749\pi\)
−0.494773 + 0.869022i \(0.664749\pi\)
\(192\) −5.43458 −0.392207
\(193\) −14.0366 −1.01038 −0.505189 0.863009i \(-0.668577\pi\)
−0.505189 + 0.863009i \(0.668577\pi\)
\(194\) 4.03846 0.289945
\(195\) 11.0385 0.790481
\(196\) −10.7261 −0.766148
\(197\) 1.95214 0.139084 0.0695421 0.997579i \(-0.477846\pi\)
0.0695421 + 0.997579i \(0.477846\pi\)
\(198\) 4.37911 0.311210
\(199\) −6.99554 −0.495901 −0.247950 0.968773i \(-0.579757\pi\)
−0.247950 + 0.968773i \(0.579757\pi\)
\(200\) −3.14086 −0.222092
\(201\) 14.3176 1.00988
\(202\) −13.9225 −0.979581
\(203\) 26.3259 1.84772
\(204\) −1.18752 −0.0831433
\(205\) 7.06053 0.493129
\(206\) 14.6473 1.02052
\(207\) −0.727073 −0.0505350
\(208\) 1.19924 0.0831525
\(209\) −19.6233 −1.35737
\(210\) 7.13370 0.492272
\(211\) 16.7151 1.15071 0.575357 0.817902i \(-0.304863\pi\)
0.575357 + 0.817902i \(0.304863\pi\)
\(212\) 16.3064 1.11992
\(213\) −5.65707 −0.387616
\(214\) −2.72123 −0.186020
\(215\) 5.53807 0.377693
\(216\) −2.87315 −0.195493
\(217\) −11.2486 −0.763603
\(218\) 0.158812 0.0107561
\(219\) 7.35379 0.496923
\(220\) 11.4034 0.768818
\(221\) 5.58466 0.375665
\(222\) 0.912753 0.0612599
\(223\) 6.45163 0.432033 0.216017 0.976390i \(-0.430693\pi\)
0.216017 + 0.976390i \(0.430693\pi\)
\(224\) −22.2334 −1.48553
\(225\) −1.09318 −0.0728784
\(226\) 3.94906 0.262687
\(227\) 24.3973 1.61931 0.809653 0.586908i \(-0.199655\pi\)
0.809653 + 0.586908i \(0.199655\pi\)
\(228\) 4.79660 0.317662
\(229\) −15.6536 −1.03442 −0.517208 0.855860i \(-0.673029\pi\)
−0.517208 + 0.855860i \(0.673029\pi\)
\(230\) 1.29537 0.0854144
\(231\) 19.4526 1.27989
\(232\) 18.8906 1.24023
\(233\) −26.1415 −1.71259 −0.856293 0.516491i \(-0.827238\pi\)
−0.856293 + 0.516491i \(0.827238\pi\)
\(234\) 5.03387 0.329074
\(235\) 19.1443 1.24884
\(236\) 5.51471 0.358977
\(237\) 7.48554 0.486238
\(238\) 3.60913 0.233945
\(239\) −25.4998 −1.64944 −0.824722 0.565539i \(-0.808668\pi\)
−0.824722 + 0.565539i \(0.808668\pi\)
\(240\) 0.424446 0.0273979
\(241\) −7.65946 −0.493389 −0.246694 0.969093i \(-0.579344\pi\)
−0.246694 + 0.969093i \(0.579344\pi\)
\(242\) −11.3597 −0.730232
\(243\) −1.00000 −0.0641500
\(244\) −17.1720 −1.09932
\(245\) 17.8530 1.14058
\(246\) 3.21981 0.205288
\(247\) −22.5573 −1.43529
\(248\) −8.07158 −0.512546
\(249\) 14.8209 0.939239
\(250\) 10.8558 0.686580
\(251\) −7.36609 −0.464944 −0.232472 0.972603i \(-0.574681\pi\)
−0.232472 + 0.972603i \(0.574681\pi\)
\(252\) −4.75489 −0.299530
\(253\) 3.53231 0.222074
\(254\) −1.36006 −0.0853378
\(255\) 1.97657 0.123778
\(256\) −16.4639 −1.02899
\(257\) 3.39521 0.211787 0.105894 0.994377i \(-0.466230\pi\)
0.105894 + 0.994377i \(0.466230\pi\)
\(258\) 2.52553 0.157232
\(259\) 4.05458 0.251939
\(260\) 13.1084 0.812951
\(261\) 6.57485 0.406973
\(262\) −18.3425 −1.13320
\(263\) −18.2555 −1.12568 −0.562841 0.826565i \(-0.690292\pi\)
−0.562841 + 0.826565i \(0.690292\pi\)
\(264\) 13.9585 0.859088
\(265\) −27.1410 −1.66726
\(266\) −14.5779 −0.893826
\(267\) −0.976320 −0.0597498
\(268\) 17.0024 1.03859
\(269\) 19.0312 1.16035 0.580176 0.814491i \(-0.302984\pi\)
0.580176 + 0.814491i \(0.302984\pi\)
\(270\) 1.78163 0.108426
\(271\) −12.1192 −0.736190 −0.368095 0.929788i \(-0.619990\pi\)
−0.368095 + 0.929788i \(0.619990\pi\)
\(272\) 0.214739 0.0130204
\(273\) 22.3612 1.35336
\(274\) 6.79947 0.410771
\(275\) 5.31093 0.320261
\(276\) −0.863416 −0.0519716
\(277\) 32.6006 1.95878 0.979389 0.201983i \(-0.0647387\pi\)
0.979389 + 0.201983i \(0.0647387\pi\)
\(278\) 6.93615 0.416003
\(279\) −2.80931 −0.168189
\(280\) 22.7389 1.35891
\(281\) 10.6437 0.634950 0.317475 0.948267i \(-0.397165\pi\)
0.317475 + 0.948267i \(0.397165\pi\)
\(282\) 8.73038 0.519887
\(283\) −18.9562 −1.12683 −0.563416 0.826174i \(-0.690513\pi\)
−0.563416 + 0.826174i \(0.690513\pi\)
\(284\) −6.71791 −0.398634
\(285\) −7.98367 −0.472912
\(286\) −24.4559 −1.44610
\(287\) 14.3029 0.844272
\(288\) −5.55275 −0.327199
\(289\) 1.00000 0.0588235
\(290\) −11.7139 −0.687867
\(291\) 4.48034 0.262642
\(292\) 8.73281 0.511049
\(293\) −13.5918 −0.794039 −0.397019 0.917810i \(-0.629955\pi\)
−0.397019 + 0.917810i \(0.629955\pi\)
\(294\) 8.14148 0.474821
\(295\) −9.17893 −0.534418
\(296\) 2.90942 0.169107
\(297\) 4.85826 0.281905
\(298\) 12.4470 0.721033
\(299\) 4.06045 0.234822
\(300\) −1.29817 −0.0749500
\(301\) 11.2188 0.646638
\(302\) −4.13101 −0.237713
\(303\) −15.4458 −0.887339
\(304\) −0.867363 −0.0497467
\(305\) 28.5818 1.63659
\(306\) 0.901374 0.0515282
\(307\) −33.0191 −1.88450 −0.942249 0.334914i \(-0.891293\pi\)
−0.942249 + 0.334914i \(0.891293\pi\)
\(308\) 23.1005 1.31627
\(309\) 16.2499 0.924426
\(310\) 5.00515 0.284273
\(311\) −0.148991 −0.00844852 −0.00422426 0.999991i \(-0.501345\pi\)
−0.00422426 + 0.999991i \(0.501345\pi\)
\(312\) 16.0456 0.908402
\(313\) −11.2746 −0.637276 −0.318638 0.947876i \(-0.603225\pi\)
−0.318638 + 0.947876i \(0.603225\pi\)
\(314\) 0.901374 0.0508675
\(315\) 7.91425 0.445917
\(316\) 8.88926 0.500060
\(317\) 22.7598 1.27832 0.639159 0.769075i \(-0.279283\pi\)
0.639159 + 0.769075i \(0.279283\pi\)
\(318\) −12.3771 −0.694075
\(319\) −31.9423 −1.78843
\(320\) 10.7418 0.600486
\(321\) −3.01898 −0.168503
\(322\) 2.62410 0.146236
\(323\) −4.03916 −0.224745
\(324\) −1.18752 −0.0659736
\(325\) 6.10501 0.338645
\(326\) −9.28420 −0.514204
\(327\) 0.176188 0.00974324
\(328\) 10.2632 0.566692
\(329\) 38.7816 2.13810
\(330\) −8.65562 −0.476476
\(331\) 9.42962 0.518299 0.259149 0.965837i \(-0.416558\pi\)
0.259149 + 0.965837i \(0.416558\pi\)
\(332\) 17.6002 0.965937
\(333\) 1.01262 0.0554914
\(334\) −14.4878 −0.792737
\(335\) −28.2996 −1.54617
\(336\) 0.859821 0.0469071
\(337\) 0.114336 0.00622826 0.00311413 0.999995i \(-0.499009\pi\)
0.00311413 + 0.999995i \(0.499009\pi\)
\(338\) −16.3946 −0.891748
\(339\) 4.38115 0.237951
\(340\) 2.34722 0.127296
\(341\) 13.6484 0.739100
\(342\) −3.64079 −0.196872
\(343\) 8.13738 0.439377
\(344\) 8.05018 0.434036
\(345\) 1.43711 0.0773714
\(346\) 19.9338 1.07165
\(347\) −17.2859 −0.927956 −0.463978 0.885847i \(-0.653578\pi\)
−0.463978 + 0.885847i \(0.653578\pi\)
\(348\) 7.80779 0.418542
\(349\) −12.7291 −0.681374 −0.340687 0.940177i \(-0.610660\pi\)
−0.340687 + 0.940177i \(0.610660\pi\)
\(350\) 3.94542 0.210891
\(351\) 5.58466 0.298087
\(352\) 26.9767 1.43786
\(353\) −36.3196 −1.93310 −0.966550 0.256480i \(-0.917437\pi\)
−0.966550 + 0.256480i \(0.917437\pi\)
\(354\) −4.18587 −0.222476
\(355\) 11.1816 0.593457
\(356\) −1.15940 −0.0614483
\(357\) 4.00403 0.211916
\(358\) −16.8617 −0.891171
\(359\) −12.9981 −0.686012 −0.343006 0.939333i \(-0.611445\pi\)
−0.343006 + 0.939333i \(0.611445\pi\)
\(360\) 5.67898 0.299309
\(361\) −2.68521 −0.141327
\(362\) 2.21363 0.116346
\(363\) −12.6027 −0.661470
\(364\) 26.5544 1.39183
\(365\) −14.5353 −0.760811
\(366\) 13.0342 0.681307
\(367\) 1.00888 0.0526633 0.0263316 0.999653i \(-0.491617\pi\)
0.0263316 + 0.999653i \(0.491617\pi\)
\(368\) 0.156131 0.00813887
\(369\) 3.57212 0.185957
\(370\) −1.80412 −0.0937916
\(371\) −54.9810 −2.85447
\(372\) −3.33612 −0.172970
\(373\) 34.6154 1.79232 0.896160 0.443732i \(-0.146346\pi\)
0.896160 + 0.443732i \(0.146346\pi\)
\(374\) −4.37911 −0.226438
\(375\) 12.0436 0.621928
\(376\) 27.8283 1.43514
\(377\) −36.7183 −1.89109
\(378\) 3.60913 0.185634
\(379\) −16.2918 −0.836852 −0.418426 0.908251i \(-0.637418\pi\)
−0.418426 + 0.908251i \(0.637418\pi\)
\(380\) −9.48080 −0.486355
\(381\) −1.50887 −0.0773020
\(382\) 12.3270 0.630705
\(383\) −0.0693234 −0.00354226 −0.00177113 0.999998i \(-0.500564\pi\)
−0.00177113 + 0.999998i \(0.500564\pi\)
\(384\) −6.20690 −0.316744
\(385\) −38.4495 −1.95957
\(386\) 12.6522 0.643982
\(387\) 2.80186 0.142427
\(388\) 5.32051 0.270108
\(389\) −10.4815 −0.531435 −0.265718 0.964051i \(-0.585609\pi\)
−0.265718 + 0.964051i \(0.585609\pi\)
\(390\) −9.94979 −0.503827
\(391\) 0.727073 0.0367696
\(392\) 25.9512 1.31073
\(393\) −20.3494 −1.02649
\(394\) −1.75961 −0.0886478
\(395\) −14.7957 −0.744452
\(396\) 5.76930 0.289918
\(397\) 13.4453 0.674803 0.337401 0.941361i \(-0.390452\pi\)
0.337401 + 0.941361i \(0.390452\pi\)
\(398\) 6.30560 0.316071
\(399\) −16.1729 −0.809659
\(400\) 0.234747 0.0117374
\(401\) 20.3625 1.01685 0.508427 0.861105i \(-0.330227\pi\)
0.508427 + 0.861105i \(0.330227\pi\)
\(402\) −12.9055 −0.643667
\(403\) 15.6890 0.781527
\(404\) −18.3423 −0.912562
\(405\) 1.97657 0.0982165
\(406\) −23.7295 −1.17768
\(407\) −4.91959 −0.243855
\(408\) 2.87315 0.142242
\(409\) −8.39804 −0.415257 −0.207628 0.978208i \(-0.566574\pi\)
−0.207628 + 0.978208i \(0.566574\pi\)
\(410\) −6.36419 −0.314305
\(411\) 7.54344 0.372091
\(412\) 19.2972 0.950704
\(413\) −18.5942 −0.914962
\(414\) 0.655365 0.0322094
\(415\) −29.2946 −1.43802
\(416\) 31.0102 1.52040
\(417\) 7.69508 0.376830
\(418\) 17.6879 0.865144
\(419\) −17.6752 −0.863488 −0.431744 0.901996i \(-0.642102\pi\)
−0.431744 + 0.901996i \(0.642102\pi\)
\(420\) 9.39836 0.458593
\(421\) 32.2777 1.57312 0.786561 0.617513i \(-0.211860\pi\)
0.786561 + 0.617513i \(0.211860\pi\)
\(422\) −15.0666 −0.733429
\(423\) 9.68563 0.470932
\(424\) −39.4524 −1.91598
\(425\) 1.09318 0.0530268
\(426\) 5.09914 0.247054
\(427\) 57.8997 2.80196
\(428\) −3.58511 −0.173293
\(429\) −27.1317 −1.30993
\(430\) −4.99188 −0.240730
\(431\) 17.4216 0.839167 0.419583 0.907717i \(-0.362176\pi\)
0.419583 + 0.907717i \(0.362176\pi\)
\(432\) 0.214739 0.0103316
\(433\) 6.40846 0.307971 0.153986 0.988073i \(-0.450789\pi\)
0.153986 + 0.988073i \(0.450789\pi\)
\(434\) 10.1392 0.486696
\(435\) −12.9956 −0.623094
\(436\) 0.209228 0.0100202
\(437\) −2.93676 −0.140484
\(438\) −6.62852 −0.316723
\(439\) −12.0771 −0.576408 −0.288204 0.957569i \(-0.593058\pi\)
−0.288204 + 0.957569i \(0.593058\pi\)
\(440\) −27.5900 −1.31530
\(441\) 9.03230 0.430109
\(442\) −5.03387 −0.239437
\(443\) 25.8910 1.23012 0.615059 0.788481i \(-0.289132\pi\)
0.615059 + 0.788481i \(0.289132\pi\)
\(444\) 1.20251 0.0570688
\(445\) 1.92976 0.0914796
\(446\) −5.81533 −0.275364
\(447\) 13.8089 0.653137
\(448\) 21.7603 1.02808
\(449\) 15.9342 0.751979 0.375990 0.926624i \(-0.377303\pi\)
0.375990 + 0.926624i \(0.377303\pi\)
\(450\) 0.985361 0.0464503
\(451\) −17.3543 −0.817181
\(452\) 5.20272 0.244715
\(453\) −4.58302 −0.215329
\(454\) −21.9911 −1.03209
\(455\) −44.1984 −2.07205
\(456\) −11.6051 −0.543459
\(457\) −20.6003 −0.963641 −0.481820 0.876270i \(-0.660024\pi\)
−0.481820 + 0.876270i \(0.660024\pi\)
\(458\) 14.1097 0.659304
\(459\) 1.00000 0.0466760
\(460\) 1.70660 0.0795707
\(461\) 29.0359 1.35233 0.676167 0.736748i \(-0.263640\pi\)
0.676167 + 0.736748i \(0.263640\pi\)
\(462\) −17.5341 −0.815761
\(463\) −25.5814 −1.18887 −0.594434 0.804145i \(-0.702624\pi\)
−0.594434 + 0.804145i \(0.702624\pi\)
\(464\) −1.41187 −0.0655446
\(465\) 5.55279 0.257505
\(466\) 23.5633 1.09155
\(467\) 40.0020 1.85107 0.925537 0.378658i \(-0.123614\pi\)
0.925537 + 0.378658i \(0.123614\pi\)
\(468\) 6.63192 0.306561
\(469\) −57.3280 −2.64716
\(470\) −17.2562 −0.795969
\(471\) 1.00000 0.0460776
\(472\) −13.3426 −0.614141
\(473\) −13.6122 −0.625888
\(474\) −6.74728 −0.309913
\(475\) −4.41551 −0.202597
\(476\) 4.75489 0.217940
\(477\) −13.7314 −0.628717
\(478\) 22.9848 1.05130
\(479\) −31.0140 −1.41707 −0.708534 0.705677i \(-0.750643\pi\)
−0.708534 + 0.705677i \(0.750643\pi\)
\(480\) 10.9754 0.500955
\(481\) −5.65516 −0.257853
\(482\) 6.90404 0.314470
\(483\) 2.91122 0.132465
\(484\) −14.9660 −0.680273
\(485\) −8.85570 −0.402117
\(486\) 0.901374 0.0408872
\(487\) −22.8004 −1.03319 −0.516593 0.856231i \(-0.672800\pi\)
−0.516593 + 0.856231i \(0.672800\pi\)
\(488\) 41.5467 1.88073
\(489\) −10.3001 −0.465784
\(490\) −16.0922 −0.726971
\(491\) −16.6588 −0.751801 −0.375900 0.926660i \(-0.622667\pi\)
−0.375900 + 0.926660i \(0.622667\pi\)
\(492\) 4.24197 0.191243
\(493\) −6.57485 −0.296116
\(494\) 20.3326 0.914807
\(495\) −9.60269 −0.431609
\(496\) 0.603267 0.0270875
\(497\) 22.6511 1.01604
\(498\) −13.3592 −0.598641
\(499\) 10.1046 0.452344 0.226172 0.974087i \(-0.427379\pi\)
0.226172 + 0.974087i \(0.427379\pi\)
\(500\) 14.3020 0.639607
\(501\) −16.0730 −0.718089
\(502\) 6.63961 0.296340
\(503\) −11.3078 −0.504189 −0.252095 0.967703i \(-0.581119\pi\)
−0.252095 + 0.967703i \(0.581119\pi\)
\(504\) 11.5042 0.512438
\(505\) 30.5297 1.35855
\(506\) −3.18393 −0.141543
\(507\) −18.1884 −0.807777
\(508\) −1.79182 −0.0794994
\(509\) −13.7871 −0.611104 −0.305552 0.952175i \(-0.598841\pi\)
−0.305552 + 0.952175i \(0.598841\pi\)
\(510\) −1.78163 −0.0788918
\(511\) −29.4448 −1.30256
\(512\) 2.42634 0.107230
\(513\) −4.03916 −0.178333
\(514\) −3.06035 −0.134986
\(515\) −32.1191 −1.41534
\(516\) 3.32728 0.146475
\(517\) −47.0553 −2.06949
\(518\) −3.65469 −0.160578
\(519\) 22.1149 0.970735
\(520\) −31.7152 −1.39080
\(521\) −22.8461 −1.00091 −0.500453 0.865764i \(-0.666833\pi\)
−0.500453 + 0.865764i \(0.666833\pi\)
\(522\) −5.92640 −0.259392
\(523\) −34.4478 −1.50630 −0.753150 0.657849i \(-0.771466\pi\)
−0.753150 + 0.657849i \(0.771466\pi\)
\(524\) −24.1654 −1.05567
\(525\) 4.37711 0.191033
\(526\) 16.4550 0.717474
\(527\) 2.80931 0.122375
\(528\) −1.04326 −0.0454019
\(529\) −22.4714 −0.977016
\(530\) 24.4642 1.06266
\(531\) −4.64387 −0.201527
\(532\) −19.2057 −0.832674
\(533\) −19.9491 −0.864090
\(534\) 0.880030 0.0380826
\(535\) 5.96722 0.257985
\(536\) −41.1365 −1.77683
\(537\) −18.7067 −0.807254
\(538\) −17.1542 −0.739571
\(539\) −43.8812 −1.89010
\(540\) 2.34722 0.101008
\(541\) −12.9680 −0.557539 −0.278770 0.960358i \(-0.589927\pi\)
−0.278770 + 0.960358i \(0.589927\pi\)
\(542\) 10.9240 0.469224
\(543\) 2.45583 0.105390
\(544\) 5.55275 0.238072
\(545\) −0.348248 −0.0149173
\(546\) −20.1558 −0.862588
\(547\) −14.2555 −0.609523 −0.304762 0.952429i \(-0.598577\pi\)
−0.304762 + 0.952429i \(0.598577\pi\)
\(548\) 8.95802 0.382668
\(549\) 14.4603 0.617152
\(550\) −4.78714 −0.204124
\(551\) 26.5569 1.13136
\(552\) 2.08899 0.0889134
\(553\) −29.9724 −1.27455
\(554\) −29.3853 −1.24846
\(555\) −2.00152 −0.0849598
\(556\) 9.13810 0.387542
\(557\) 4.41660 0.187137 0.0935686 0.995613i \(-0.470173\pi\)
0.0935686 + 0.995613i \(0.470173\pi\)
\(558\) 2.53224 0.107198
\(559\) −15.6474 −0.661816
\(560\) −1.69950 −0.0718168
\(561\) −4.85826 −0.205116
\(562\) −9.59396 −0.404697
\(563\) 10.0813 0.424877 0.212439 0.977174i \(-0.431859\pi\)
0.212439 + 0.977174i \(0.431859\pi\)
\(564\) 11.5019 0.484318
\(565\) −8.65965 −0.364314
\(566\) 17.0867 0.718206
\(567\) 4.00403 0.168154
\(568\) 16.2536 0.681987
\(569\) 0.218235 0.00914890 0.00457445 0.999990i \(-0.498544\pi\)
0.00457445 + 0.999990i \(0.498544\pi\)
\(570\) 7.19628 0.301419
\(571\) 9.67997 0.405094 0.202547 0.979273i \(-0.435078\pi\)
0.202547 + 0.979273i \(0.435078\pi\)
\(572\) −32.2196 −1.34717
\(573\) 13.6758 0.571315
\(574\) −12.8923 −0.538112
\(575\) 0.794818 0.0331462
\(576\) 5.43458 0.226441
\(577\) −9.41424 −0.391920 −0.195960 0.980612i \(-0.562782\pi\)
−0.195960 + 0.980612i \(0.562782\pi\)
\(578\) −0.901374 −0.0374922
\(579\) 14.0366 0.583342
\(580\) −15.4326 −0.640806
\(581\) −59.3435 −2.46199
\(582\) −4.03846 −0.167400
\(583\) 66.7106 2.76287
\(584\) −21.1286 −0.874307
\(585\) −11.0385 −0.456384
\(586\) 12.2513 0.506095
\(587\) 30.7666 1.26988 0.634938 0.772563i \(-0.281026\pi\)
0.634938 + 0.772563i \(0.281026\pi\)
\(588\) 10.7261 0.442336
\(589\) −11.3472 −0.467555
\(590\) 8.27366 0.340621
\(591\) −1.95214 −0.0803003
\(592\) −0.217449 −0.00893711
\(593\) −19.2426 −0.790198 −0.395099 0.918639i \(-0.629290\pi\)
−0.395099 + 0.918639i \(0.629290\pi\)
\(594\) −4.37911 −0.179677
\(595\) −7.91425 −0.324453
\(596\) 16.3984 0.671703
\(597\) 6.99554 0.286308
\(598\) −3.65999 −0.149668
\(599\) 27.2999 1.11544 0.557721 0.830028i \(-0.311676\pi\)
0.557721 + 0.830028i \(0.311676\pi\)
\(600\) 3.14086 0.128225
\(601\) −46.6198 −1.90166 −0.950830 0.309714i \(-0.899767\pi\)
−0.950830 + 0.309714i \(0.899767\pi\)
\(602\) −10.1123 −0.412146
\(603\) −14.3176 −0.583056
\(604\) −5.44244 −0.221450
\(605\) 24.9101 1.01274
\(606\) 13.9225 0.565561
\(607\) −8.35444 −0.339096 −0.169548 0.985522i \(-0.554231\pi\)
−0.169548 + 0.985522i \(0.554231\pi\)
\(608\) −22.4284 −0.909592
\(609\) −26.3259 −1.06678
\(610\) −25.7629 −1.04311
\(611\) −54.0910 −2.18829
\(612\) 1.18752 0.0480028
\(613\) −41.5278 −1.67729 −0.838646 0.544677i \(-0.816652\pi\)
−0.838646 + 0.544677i \(0.816652\pi\)
\(614\) 29.7625 1.20112
\(615\) −7.06053 −0.284708
\(616\) −55.8904 −2.25189
\(617\) −17.6664 −0.711223 −0.355611 0.934634i \(-0.615727\pi\)
−0.355611 + 0.934634i \(0.615727\pi\)
\(618\) −14.6473 −0.589199
\(619\) 12.6051 0.506643 0.253322 0.967382i \(-0.418477\pi\)
0.253322 + 0.967382i \(0.418477\pi\)
\(620\) 6.59408 0.264824
\(621\) 0.727073 0.0291764
\(622\) 0.134297 0.00538482
\(623\) 3.90922 0.156620
\(624\) −1.19924 −0.0480081
\(625\) −18.3391 −0.733564
\(626\) 10.1626 0.406179
\(627\) 19.6233 0.783678
\(628\) 1.18752 0.0473874
\(629\) −1.01262 −0.0403759
\(630\) −7.13370 −0.284214
\(631\) −35.3696 −1.40804 −0.704021 0.710180i \(-0.748614\pi\)
−0.704021 + 0.710180i \(0.748614\pi\)
\(632\) −21.5071 −0.855507
\(633\) −16.7151 −0.664366
\(634\) −20.5151 −0.814759
\(635\) 2.98239 0.118353
\(636\) −16.3064 −0.646589
\(637\) −50.4423 −1.99860
\(638\) 28.7920 1.13989
\(639\) 5.65707 0.223790
\(640\) 12.2684 0.484950
\(641\) −29.4882 −1.16471 −0.582356 0.812934i \(-0.697869\pi\)
−0.582356 + 0.812934i \(0.697869\pi\)
\(642\) 2.72123 0.107398
\(643\) −2.17328 −0.0857057 −0.0428528 0.999081i \(-0.513645\pi\)
−0.0428528 + 0.999081i \(0.513645\pi\)
\(644\) 3.45715 0.136231
\(645\) −5.53807 −0.218061
\(646\) 3.64079 0.143245
\(647\) −36.4755 −1.43400 −0.717000 0.697073i \(-0.754485\pi\)
−0.717000 + 0.697073i \(0.754485\pi\)
\(648\) 2.87315 0.112868
\(649\) 22.5611 0.885602
\(650\) −5.50290 −0.215842
\(651\) 11.2486 0.440866
\(652\) −12.2316 −0.479025
\(653\) 36.1841 1.41599 0.707996 0.706217i \(-0.249600\pi\)
0.707996 + 0.706217i \(0.249600\pi\)
\(654\) −0.158812 −0.00621003
\(655\) 40.2221 1.57161
\(656\) −0.767071 −0.0299491
\(657\) −7.35379 −0.286899
\(658\) −34.9568 −1.36276
\(659\) 37.6293 1.46583 0.732914 0.680321i \(-0.238160\pi\)
0.732914 + 0.680321i \(0.238160\pi\)
\(660\) −11.4034 −0.443877
\(661\) 25.3661 0.986625 0.493313 0.869852i \(-0.335786\pi\)
0.493313 + 0.869852i \(0.335786\pi\)
\(662\) −8.49962 −0.330347
\(663\) −5.58466 −0.216890
\(664\) −42.5828 −1.65253
\(665\) 31.9669 1.23962
\(666\) −0.912753 −0.0353684
\(667\) −4.78040 −0.185098
\(668\) −19.0871 −0.738501
\(669\) −6.45163 −0.249434
\(670\) 25.5086 0.985482
\(671\) −70.2520 −2.71205
\(672\) 22.2334 0.857672
\(673\) −30.9145 −1.19167 −0.595834 0.803108i \(-0.703178\pi\)
−0.595834 + 0.803108i \(0.703178\pi\)
\(674\) −0.103059 −0.00396969
\(675\) 1.09318 0.0420764
\(676\) −21.5992 −0.830738
\(677\) 25.9296 0.996557 0.498279 0.867017i \(-0.333966\pi\)
0.498279 + 0.867017i \(0.333966\pi\)
\(678\) −3.94906 −0.151663
\(679\) −17.9394 −0.688452
\(680\) −5.67898 −0.217779
\(681\) −24.3973 −0.934907
\(682\) −12.3023 −0.471079
\(683\) 33.1170 1.26719 0.633593 0.773667i \(-0.281579\pi\)
0.633593 + 0.773667i \(0.281579\pi\)
\(684\) −4.79660 −0.183402
\(685\) −14.9101 −0.569687
\(686\) −7.33483 −0.280045
\(687\) 15.6536 0.597221
\(688\) −0.601668 −0.0229384
\(689\) 76.6851 2.92147
\(690\) −1.29537 −0.0493140
\(691\) 32.8021 1.24785 0.623926 0.781484i \(-0.285537\pi\)
0.623926 + 0.781484i \(0.285537\pi\)
\(692\) 26.2619 0.998329
\(693\) −19.4526 −0.738945
\(694\) 15.5811 0.591450
\(695\) −15.2099 −0.576943
\(696\) −18.8906 −0.716045
\(697\) −3.57212 −0.135304
\(698\) 11.4737 0.434286
\(699\) 26.1415 0.988762
\(700\) 5.19793 0.196463
\(701\) 12.0851 0.456450 0.228225 0.973608i \(-0.426708\pi\)
0.228225 + 0.973608i \(0.426708\pi\)
\(702\) −5.03387 −0.189991
\(703\) 4.09014 0.154263
\(704\) −26.4026 −0.995086
\(705\) −19.1443 −0.721017
\(706\) 32.7376 1.23210
\(707\) 61.8456 2.32594
\(708\) −5.51471 −0.207256
\(709\) −47.9036 −1.79906 −0.899528 0.436863i \(-0.856089\pi\)
−0.899528 + 0.436863i \(0.856089\pi\)
\(710\) −10.0788 −0.378251
\(711\) −7.48554 −0.280730
\(712\) 2.80512 0.105126
\(713\) 2.04257 0.0764949
\(714\) −3.60913 −0.135068
\(715\) 53.6277 2.00556
\(716\) −22.2147 −0.830201
\(717\) 25.4998 0.952306
\(718\) 11.7161 0.437242
\(719\) −36.8186 −1.37310 −0.686551 0.727082i \(-0.740876\pi\)
−0.686551 + 0.727082i \(0.740876\pi\)
\(720\) −0.424446 −0.0158182
\(721\) −65.0653 −2.42316
\(722\) 2.42038 0.0900773
\(723\) 7.65946 0.284858
\(724\) 2.91636 0.108386
\(725\) −7.18747 −0.266936
\(726\) 11.3597 0.421600
\(727\) 13.9091 0.515860 0.257930 0.966164i \(-0.416960\pi\)
0.257930 + 0.966164i \(0.416960\pi\)
\(728\) −64.2471 −2.38115
\(729\) 1.00000 0.0370370
\(730\) 13.1017 0.484917
\(731\) −2.80186 −0.103631
\(732\) 17.1720 0.634695
\(733\) −25.3253 −0.935412 −0.467706 0.883884i \(-0.654919\pi\)
−0.467706 + 0.883884i \(0.654919\pi\)
\(734\) −0.909382 −0.0335659
\(735\) −17.8530 −0.658516
\(736\) 4.03725 0.148815
\(737\) 69.5584 2.56222
\(738\) −3.21981 −0.118523
\(739\) −25.0658 −0.922061 −0.461030 0.887384i \(-0.652520\pi\)
−0.461030 + 0.887384i \(0.652520\pi\)
\(740\) −2.37685 −0.0873748
\(741\) 22.5573 0.828664
\(742\) 49.5584 1.81935
\(743\) 28.7396 1.05435 0.527176 0.849756i \(-0.323251\pi\)
0.527176 + 0.849756i \(0.323251\pi\)
\(744\) 8.07158 0.295918
\(745\) −27.2942 −0.999981
\(746\) −31.2015 −1.14237
\(747\) −14.8209 −0.542270
\(748\) −5.76930 −0.210947
\(749\) 12.0881 0.441689
\(750\) −10.8558 −0.396397
\(751\) −31.2776 −1.14133 −0.570667 0.821182i \(-0.693315\pi\)
−0.570667 + 0.821182i \(0.693315\pi\)
\(752\) −2.07988 −0.0758454
\(753\) 7.36609 0.268435
\(754\) 33.0970 1.20532
\(755\) 9.05865 0.329678
\(756\) 4.75489 0.172934
\(757\) −38.7372 −1.40793 −0.703964 0.710236i \(-0.748588\pi\)
−0.703964 + 0.710236i \(0.748588\pi\)
\(758\) 14.6850 0.533383
\(759\) −3.53231 −0.128215
\(760\) 22.9383 0.832060
\(761\) 21.3683 0.774601 0.387300 0.921954i \(-0.373408\pi\)
0.387300 + 0.921954i \(0.373408\pi\)
\(762\) 1.36006 0.0492698
\(763\) −0.705464 −0.0255395
\(764\) 16.2403 0.587555
\(765\) −1.97657 −0.0714630
\(766\) 0.0624864 0.00225772
\(767\) 25.9345 0.936439
\(768\) 16.4639 0.594090
\(769\) 37.8457 1.36475 0.682376 0.731001i \(-0.260947\pi\)
0.682376 + 0.731001i \(0.260947\pi\)
\(770\) 34.6574 1.24897
\(771\) −3.39521 −0.122275
\(772\) 16.6688 0.599924
\(773\) 21.6879 0.780060 0.390030 0.920802i \(-0.372465\pi\)
0.390030 + 0.920802i \(0.372465\pi\)
\(774\) −2.52553 −0.0907782
\(775\) 3.07107 0.110316
\(776\) −12.8727 −0.462103
\(777\) −4.05458 −0.145457
\(778\) 9.44779 0.338720
\(779\) 14.4283 0.516949
\(780\) −13.1084 −0.469358
\(781\) −27.4835 −0.983437
\(782\) −0.655365 −0.0234358
\(783\) −6.57485 −0.234966
\(784\) −1.93958 −0.0692708
\(785\) −1.97657 −0.0705468
\(786\) 18.3425 0.654254
\(787\) −31.9571 −1.13915 −0.569573 0.821941i \(-0.692892\pi\)
−0.569573 + 0.821941i \(0.692892\pi\)
\(788\) −2.31821 −0.0825829
\(789\) 18.2555 0.649913
\(790\) 13.3365 0.474490
\(791\) −17.5423 −0.623732
\(792\) −13.9585 −0.495995
\(793\) −80.7560 −2.86773
\(794\) −12.1193 −0.430098
\(795\) 27.1410 0.962593
\(796\) 8.30737 0.294447
\(797\) −10.4559 −0.370368 −0.185184 0.982704i \(-0.559288\pi\)
−0.185184 + 0.982704i \(0.559288\pi\)
\(798\) 14.5779 0.516051
\(799\) −9.68563 −0.342653
\(800\) 6.07013 0.214611
\(801\) 0.976320 0.0344966
\(802\) −18.3542 −0.648110
\(803\) 35.7266 1.26077
\(804\) −17.0024 −0.599630
\(805\) −5.75424 −0.202810
\(806\) −14.1417 −0.498120
\(807\) −19.0312 −0.669929
\(808\) 44.3782 1.56122
\(809\) 4.86640 0.171094 0.0855468 0.996334i \(-0.472736\pi\)
0.0855468 + 0.996334i \(0.472736\pi\)
\(810\) −1.78163 −0.0626001
\(811\) 26.6349 0.935278 0.467639 0.883920i \(-0.345105\pi\)
0.467639 + 0.883920i \(0.345105\pi\)
\(812\) −31.2627 −1.09711
\(813\) 12.1192 0.425039
\(814\) 4.43439 0.155425
\(815\) 20.3588 0.713136
\(816\) −0.214739 −0.00751736
\(817\) 11.3172 0.395937
\(818\) 7.56978 0.264671
\(819\) −22.3612 −0.781363
\(820\) −8.38455 −0.292801
\(821\) −33.4488 −1.16737 −0.583686 0.811980i \(-0.698390\pi\)
−0.583686 + 0.811980i \(0.698390\pi\)
\(822\) −6.79947 −0.237159
\(823\) 12.5234 0.436538 0.218269 0.975889i \(-0.429959\pi\)
0.218269 + 0.975889i \(0.429959\pi\)
\(824\) −46.6885 −1.62647
\(825\) −5.31093 −0.184903
\(826\) 16.7604 0.583168
\(827\) −17.6604 −0.614113 −0.307057 0.951691i \(-0.599344\pi\)
−0.307057 + 0.951691i \(0.599344\pi\)
\(828\) 0.863416 0.0300058
\(829\) −13.7180 −0.476446 −0.238223 0.971210i \(-0.576565\pi\)
−0.238223 + 0.971210i \(0.576565\pi\)
\(830\) 26.4054 0.916545
\(831\) −32.6006 −1.13090
\(832\) −30.3503 −1.05221
\(833\) −9.03230 −0.312950
\(834\) −6.93615 −0.240179
\(835\) 31.7694 1.09943
\(836\) 23.3031 0.805955
\(837\) 2.80931 0.0971039
\(838\) 15.9319 0.550360
\(839\) −15.2862 −0.527737 −0.263869 0.964559i \(-0.584999\pi\)
−0.263869 + 0.964559i \(0.584999\pi\)
\(840\) −22.7389 −0.784565
\(841\) 14.2287 0.490644
\(842\) −29.0943 −1.00266
\(843\) −10.6437 −0.366589
\(844\) −19.8496 −0.683251
\(845\) 35.9507 1.23674
\(846\) −8.73038 −0.300157
\(847\) 50.4616 1.73388
\(848\) 2.94866 0.101257
\(849\) 18.9562 0.650576
\(850\) −0.985361 −0.0337976
\(851\) −0.736251 −0.0252383
\(852\) 6.71791 0.230152
\(853\) −30.1917 −1.03375 −0.516873 0.856062i \(-0.672904\pi\)
−0.516873 + 0.856062i \(0.672904\pi\)
\(854\) −52.1893 −1.78588
\(855\) 7.98367 0.273036
\(856\) 8.67399 0.296471
\(857\) −34.8504 −1.19047 −0.595234 0.803553i \(-0.702941\pi\)
−0.595234 + 0.803553i \(0.702941\pi\)
\(858\) 24.4559 0.834909
\(859\) −30.8469 −1.05248 −0.526242 0.850335i \(-0.676399\pi\)
−0.526242 + 0.850335i \(0.676399\pi\)
\(860\) −6.57659 −0.224260
\(861\) −14.3029 −0.487441
\(862\) −15.7033 −0.534858
\(863\) 42.6497 1.45181 0.725906 0.687794i \(-0.241421\pi\)
0.725906 + 0.687794i \(0.241421\pi\)
\(864\) 5.55275 0.188908
\(865\) −43.7116 −1.48624
\(866\) −5.77642 −0.196291
\(867\) −1.00000 −0.0339618
\(868\) 13.3580 0.453398
\(869\) 36.3667 1.23366
\(870\) 11.7139 0.397140
\(871\) 79.9587 2.70930
\(872\) −0.506216 −0.0171426
\(873\) −4.48034 −0.151637
\(874\) 2.64712 0.0895402
\(875\) −48.2229 −1.63023
\(876\) −8.73281 −0.295054
\(877\) 18.3259 0.618823 0.309411 0.950928i \(-0.399868\pi\)
0.309411 + 0.950928i \(0.399868\pi\)
\(878\) 10.8860 0.367384
\(879\) 13.5918 0.458439
\(880\) 2.06207 0.0695123
\(881\) 41.7070 1.40515 0.702573 0.711612i \(-0.252035\pi\)
0.702573 + 0.711612i \(0.252035\pi\)
\(882\) −8.14148 −0.274138
\(883\) −49.1414 −1.65374 −0.826870 0.562393i \(-0.809881\pi\)
−0.826870 + 0.562393i \(0.809881\pi\)
\(884\) −6.63192 −0.223056
\(885\) 9.17893 0.308547
\(886\) −23.3375 −0.784038
\(887\) 28.3542 0.952040 0.476020 0.879434i \(-0.342079\pi\)
0.476020 + 0.879434i \(0.342079\pi\)
\(888\) −2.90942 −0.0976338
\(889\) 6.04159 0.202628
\(890\) −1.73944 −0.0583061
\(891\) −4.85826 −0.162758
\(892\) −7.66146 −0.256525
\(893\) 39.1218 1.30916
\(894\) −12.4470 −0.416289
\(895\) 36.9751 1.23594
\(896\) 24.8526 0.830268
\(897\) −4.06045 −0.135575
\(898\) −14.3626 −0.479288
\(899\) −18.4708 −0.616035
\(900\) 1.29817 0.0432724
\(901\) 13.7314 0.457459
\(902\) 15.6427 0.520845
\(903\) −11.2188 −0.373337
\(904\) −12.5877 −0.418661
\(905\) −4.85412 −0.161357
\(906\) 4.13101 0.137244
\(907\) −24.9896 −0.829765 −0.414882 0.909875i \(-0.636177\pi\)
−0.414882 + 0.909875i \(0.636177\pi\)
\(908\) −28.9724 −0.961483
\(909\) 15.4458 0.512305
\(910\) 39.8393 1.32066
\(911\) 33.8947 1.12298 0.561490 0.827483i \(-0.310228\pi\)
0.561490 + 0.827483i \(0.310228\pi\)
\(912\) 0.867363 0.0287213
\(913\) 72.0040 2.38298
\(914\) 18.5686 0.614194
\(915\) −28.5818 −0.944886
\(916\) 18.5890 0.614197
\(917\) 81.4798 2.69070
\(918\) −0.901374 −0.0297498
\(919\) 3.06773 0.101195 0.0505975 0.998719i \(-0.483887\pi\)
0.0505975 + 0.998719i \(0.483887\pi\)
\(920\) −4.12903 −0.136130
\(921\) 33.0191 1.08802
\(922\) −26.1722 −0.861935
\(923\) −31.5928 −1.03989
\(924\) −23.1005 −0.759950
\(925\) −1.10697 −0.0363971
\(926\) 23.0584 0.757746
\(927\) −16.2499 −0.533718
\(928\) −36.5085 −1.19845
\(929\) 28.3020 0.928560 0.464280 0.885688i \(-0.346313\pi\)
0.464280 + 0.885688i \(0.346313\pi\)
\(930\) −5.00515 −0.164125
\(931\) 36.4829 1.19568
\(932\) 31.0436 1.01687
\(933\) 0.148991 0.00487776
\(934\) −36.0568 −1.17981
\(935\) 9.60269 0.314041
\(936\) −16.0456 −0.524466
\(937\) −41.6790 −1.36159 −0.680797 0.732472i \(-0.738366\pi\)
−0.680797 + 0.732472i \(0.738366\pi\)
\(938\) 51.6740 1.68722
\(939\) 11.2746 0.367931
\(940\) −22.7343 −0.741512
\(941\) −36.9478 −1.20446 −0.602231 0.798322i \(-0.705721\pi\)
−0.602231 + 0.798322i \(0.705721\pi\)
\(942\) −0.901374 −0.0293684
\(943\) −2.59719 −0.0845761
\(944\) 0.997219 0.0324567
\(945\) −7.91425 −0.257451
\(946\) 12.2697 0.398921
\(947\) 34.3275 1.11549 0.557746 0.830011i \(-0.311666\pi\)
0.557746 + 0.830011i \(0.311666\pi\)
\(948\) −8.88926 −0.288710
\(949\) 41.0684 1.33314
\(950\) 3.98003 0.129129
\(951\) −22.7598 −0.738037
\(952\) −11.5042 −0.372853
\(953\) −24.5438 −0.795051 −0.397525 0.917591i \(-0.630131\pi\)
−0.397525 + 0.917591i \(0.630131\pi\)
\(954\) 12.3771 0.400724
\(955\) −27.0312 −0.874708
\(956\) 30.2816 0.979377
\(957\) 31.9423 1.03255
\(958\) 27.9553 0.903194
\(959\) −30.2042 −0.975345
\(960\) −10.7418 −0.346691
\(961\) −23.1078 −0.745412
\(962\) 5.09741 0.164347
\(963\) 3.01898 0.0972852
\(964\) 9.09579 0.292956
\(965\) −27.7443 −0.893121
\(966\) −2.62410 −0.0844291
\(967\) −3.74119 −0.120309 −0.0601543 0.998189i \(-0.519159\pi\)
−0.0601543 + 0.998189i \(0.519159\pi\)
\(968\) 36.2095 1.16382
\(969\) 4.03916 0.129756
\(970\) 7.98230 0.256296
\(971\) −8.28895 −0.266005 −0.133003 0.991116i \(-0.542462\pi\)
−0.133003 + 0.991116i \(0.542462\pi\)
\(972\) 1.18752 0.0380899
\(973\) −30.8114 −0.987768
\(974\) 20.5517 0.658520
\(975\) −6.10501 −0.195517
\(976\) −3.10519 −0.0993948
\(977\) 37.4629 1.19854 0.599272 0.800545i \(-0.295457\pi\)
0.599272 + 0.800545i \(0.295457\pi\)
\(978\) 9.28420 0.296876
\(979\) −4.74322 −0.151594
\(980\) −21.2008 −0.677235
\(981\) −0.176188 −0.00562526
\(982\) 15.0158 0.479174
\(983\) 27.2958 0.870601 0.435301 0.900285i \(-0.356642\pi\)
0.435301 + 0.900285i \(0.356642\pi\)
\(984\) −10.2632 −0.327180
\(985\) 3.85854 0.122943
\(986\) 5.92640 0.188735
\(987\) −38.7816 −1.23443
\(988\) 26.7874 0.852219
\(989\) −2.03716 −0.0647778
\(990\) 8.65562 0.275094
\(991\) 58.7636 1.86669 0.933343 0.358986i \(-0.116877\pi\)
0.933343 + 0.358986i \(0.116877\pi\)
\(992\) 15.5994 0.495281
\(993\) −9.42962 −0.299240
\(994\) −20.4171 −0.647592
\(995\) −13.8272 −0.438351
\(996\) −17.6002 −0.557684
\(997\) −0.968736 −0.0306802 −0.0153401 0.999882i \(-0.504883\pi\)
−0.0153401 + 0.999882i \(0.504883\pi\)
\(998\) −9.10804 −0.288310
\(999\) −1.01262 −0.0320380
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 8007.2.a.f.1.18 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8007.2.a.f.1.18 48 1.1 even 1 trivial