Properties

Label 8034.2.a.bc.1.11
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 15
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-1.40985\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.40985 q^{5} -1.00000 q^{6} +4.86981 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.40985 q^{5} -1.00000 q^{6} +4.86981 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.40985 q^{10} -2.33063 q^{11} -1.00000 q^{12} -1.00000 q^{13} +4.86981 q^{14} -1.40985 q^{15} +1.00000 q^{16} +5.38125 q^{17} +1.00000 q^{18} +5.46412 q^{19} +1.40985 q^{20} -4.86981 q^{21} -2.33063 q^{22} -1.00935 q^{23} -1.00000 q^{24} -3.01231 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.86981 q^{28} -2.01110 q^{29} -1.40985 q^{30} +0.445300 q^{31} +1.00000 q^{32} +2.33063 q^{33} +5.38125 q^{34} +6.86572 q^{35} +1.00000 q^{36} +9.09660 q^{37} +5.46412 q^{38} +1.00000 q^{39} +1.40985 q^{40} -0.952506 q^{41} -4.86981 q^{42} -3.56484 q^{43} -2.33063 q^{44} +1.40985 q^{45} -1.00935 q^{46} +0.884177 q^{47} -1.00000 q^{48} +16.7150 q^{49} -3.01231 q^{50} -5.38125 q^{51} -1.00000 q^{52} -7.50659 q^{53} -1.00000 q^{54} -3.28585 q^{55} +4.86981 q^{56} -5.46412 q^{57} -2.01110 q^{58} +4.52096 q^{59} -1.40985 q^{60} +6.62115 q^{61} +0.445300 q^{62} +4.86981 q^{63} +1.00000 q^{64} -1.40985 q^{65} +2.33063 q^{66} +2.00724 q^{67} +5.38125 q^{68} +1.00935 q^{69} +6.86572 q^{70} -9.19397 q^{71} +1.00000 q^{72} -6.35844 q^{73} +9.09660 q^{74} +3.01231 q^{75} +5.46412 q^{76} -11.3497 q^{77} +1.00000 q^{78} +3.70385 q^{79} +1.40985 q^{80} +1.00000 q^{81} -0.952506 q^{82} +14.4610 q^{83} -4.86981 q^{84} +7.58678 q^{85} -3.56484 q^{86} +2.01110 q^{87} -2.33063 q^{88} +2.48255 q^{89} +1.40985 q^{90} -4.86981 q^{91} -1.00935 q^{92} -0.445300 q^{93} +0.884177 q^{94} +7.70361 q^{95} -1.00000 q^{96} +6.09866 q^{97} +16.7150 q^{98} -2.33063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.40985 0.630506 0.315253 0.949008i \(-0.397911\pi\)
0.315253 + 0.949008i \(0.397911\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.86981 1.84061 0.920307 0.391196i \(-0.127939\pi\)
0.920307 + 0.391196i \(0.127939\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.40985 0.445835
\(11\) −2.33063 −0.702712 −0.351356 0.936242i \(-0.614279\pi\)
−0.351356 + 0.936242i \(0.614279\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 4.86981 1.30151
\(15\) −1.40985 −0.364023
\(16\) 1.00000 0.250000
\(17\) 5.38125 1.30515 0.652573 0.757726i \(-0.273690\pi\)
0.652573 + 0.757726i \(0.273690\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.46412 1.25355 0.626777 0.779199i \(-0.284374\pi\)
0.626777 + 0.779199i \(0.284374\pi\)
\(20\) 1.40985 0.315253
\(21\) −4.86981 −1.06268
\(22\) −2.33063 −0.496892
\(23\) −1.00935 −0.210464 −0.105232 0.994448i \(-0.533558\pi\)
−0.105232 + 0.994448i \(0.533558\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.01231 −0.602462
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.86981 0.920307
\(29\) −2.01110 −0.373452 −0.186726 0.982412i \(-0.559788\pi\)
−0.186726 + 0.982412i \(0.559788\pi\)
\(30\) −1.40985 −0.257403
\(31\) 0.445300 0.0799783 0.0399891 0.999200i \(-0.487268\pi\)
0.0399891 + 0.999200i \(0.487268\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.33063 0.405711
\(34\) 5.38125 0.922877
\(35\) 6.86572 1.16052
\(36\) 1.00000 0.166667
\(37\) 9.09660 1.49547 0.747736 0.663996i \(-0.231141\pi\)
0.747736 + 0.663996i \(0.231141\pi\)
\(38\) 5.46412 0.886397
\(39\) 1.00000 0.160128
\(40\) 1.40985 0.222918
\(41\) −0.952506 −0.148756 −0.0743782 0.997230i \(-0.523697\pi\)
−0.0743782 + 0.997230i \(0.523697\pi\)
\(42\) −4.86981 −0.751428
\(43\) −3.56484 −0.543633 −0.271817 0.962349i \(-0.587624\pi\)
−0.271817 + 0.962349i \(0.587624\pi\)
\(44\) −2.33063 −0.351356
\(45\) 1.40985 0.210169
\(46\) −1.00935 −0.148820
\(47\) 0.884177 0.128971 0.0644853 0.997919i \(-0.479459\pi\)
0.0644853 + 0.997919i \(0.479459\pi\)
\(48\) −1.00000 −0.144338
\(49\) 16.7150 2.38786
\(50\) −3.01231 −0.426005
\(51\) −5.38125 −0.753526
\(52\) −1.00000 −0.138675
\(53\) −7.50659 −1.03111 −0.515555 0.856857i \(-0.672414\pi\)
−0.515555 + 0.856857i \(0.672414\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.28585 −0.443064
\(56\) 4.86981 0.650756
\(57\) −5.46412 −0.723740
\(58\) −2.01110 −0.264071
\(59\) 4.52096 0.588579 0.294289 0.955716i \(-0.404917\pi\)
0.294289 + 0.955716i \(0.404917\pi\)
\(60\) −1.40985 −0.182011
\(61\) 6.62115 0.847752 0.423876 0.905720i \(-0.360669\pi\)
0.423876 + 0.905720i \(0.360669\pi\)
\(62\) 0.445300 0.0565532
\(63\) 4.86981 0.613538
\(64\) 1.00000 0.125000
\(65\) −1.40985 −0.174871
\(66\) 2.33063 0.286881
\(67\) 2.00724 0.245224 0.122612 0.992455i \(-0.460873\pi\)
0.122612 + 0.992455i \(0.460873\pi\)
\(68\) 5.38125 0.652573
\(69\) 1.00935 0.121511
\(70\) 6.86572 0.820611
\(71\) −9.19397 −1.09112 −0.545562 0.838070i \(-0.683684\pi\)
−0.545562 + 0.838070i \(0.683684\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.35844 −0.744199 −0.372099 0.928193i \(-0.621362\pi\)
−0.372099 + 0.928193i \(0.621362\pi\)
\(74\) 9.09660 1.05746
\(75\) 3.01231 0.347832
\(76\) 5.46412 0.626777
\(77\) −11.3497 −1.29342
\(78\) 1.00000 0.113228
\(79\) 3.70385 0.416715 0.208358 0.978053i \(-0.433188\pi\)
0.208358 + 0.978053i \(0.433188\pi\)
\(80\) 1.40985 0.157627
\(81\) 1.00000 0.111111
\(82\) −0.952506 −0.105187
\(83\) 14.4610 1.58730 0.793648 0.608377i \(-0.208179\pi\)
0.793648 + 0.608377i \(0.208179\pi\)
\(84\) −4.86981 −0.531340
\(85\) 7.58678 0.822902
\(86\) −3.56484 −0.384407
\(87\) 2.01110 0.215613
\(88\) −2.33063 −0.248446
\(89\) 2.48255 0.263150 0.131575 0.991306i \(-0.457997\pi\)
0.131575 + 0.991306i \(0.457997\pi\)
\(90\) 1.40985 0.148612
\(91\) −4.86981 −0.510495
\(92\) −1.00935 −0.105232
\(93\) −0.445300 −0.0461755
\(94\) 0.884177 0.0911959
\(95\) 7.70361 0.790373
\(96\) −1.00000 −0.102062
\(97\) 6.09866 0.619225 0.309613 0.950863i \(-0.399801\pi\)
0.309613 + 0.950863i \(0.399801\pi\)
\(98\) 16.7150 1.68847
\(99\) −2.33063 −0.234237
\(100\) −3.01231 −0.301231
\(101\) 7.35330 0.731681 0.365840 0.930678i \(-0.380782\pi\)
0.365840 + 0.930678i \(0.380782\pi\)
\(102\) −5.38125 −0.532823
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −6.86572 −0.670026
\(106\) −7.50659 −0.729104
\(107\) −0.00520205 −0.000502901 0 −0.000251450 1.00000i \(-0.500080\pi\)
−0.000251450 1.00000i \(0.500080\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −12.1170 −1.16060 −0.580299 0.814404i \(-0.697064\pi\)
−0.580299 + 0.814404i \(0.697064\pi\)
\(110\) −3.28585 −0.313294
\(111\) −9.09660 −0.863411
\(112\) 4.86981 0.460154
\(113\) −3.32058 −0.312373 −0.156187 0.987728i \(-0.549920\pi\)
−0.156187 + 0.987728i \(0.549920\pi\)
\(114\) −5.46412 −0.511761
\(115\) −1.42303 −0.132699
\(116\) −2.01110 −0.186726
\(117\) −1.00000 −0.0924500
\(118\) 4.52096 0.416188
\(119\) 26.2057 2.40227
\(120\) −1.40985 −0.128702
\(121\) −5.56816 −0.506196
\(122\) 6.62115 0.599451
\(123\) 0.952506 0.0858845
\(124\) 0.445300 0.0399891
\(125\) −11.2962 −1.01036
\(126\) 4.86981 0.433837
\(127\) −3.48520 −0.309261 −0.154631 0.987972i \(-0.549419\pi\)
−0.154631 + 0.987972i \(0.549419\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.56484 0.313867
\(130\) −1.40985 −0.123652
\(131\) 0.539376 0.0471255 0.0235628 0.999722i \(-0.492499\pi\)
0.0235628 + 0.999722i \(0.492499\pi\)
\(132\) 2.33063 0.202855
\(133\) 26.6092 2.30731
\(134\) 2.00724 0.173399
\(135\) −1.40985 −0.121341
\(136\) 5.38125 0.461439
\(137\) 1.94980 0.166583 0.0832915 0.996525i \(-0.473457\pi\)
0.0832915 + 0.996525i \(0.473457\pi\)
\(138\) 1.00935 0.0859214
\(139\) 10.6357 0.902110 0.451055 0.892496i \(-0.351048\pi\)
0.451055 + 0.892496i \(0.351048\pi\)
\(140\) 6.86572 0.580259
\(141\) −0.884177 −0.0744612
\(142\) −9.19397 −0.771541
\(143\) 2.33063 0.194897
\(144\) 1.00000 0.0833333
\(145\) −2.83536 −0.235464
\(146\) −6.35844 −0.526228
\(147\) −16.7150 −1.37863
\(148\) 9.09660 0.747736
\(149\) −5.69498 −0.466551 −0.233276 0.972411i \(-0.574944\pi\)
−0.233276 + 0.972411i \(0.574944\pi\)
\(150\) 3.01231 0.245954
\(151\) −4.77259 −0.388388 −0.194194 0.980963i \(-0.562209\pi\)
−0.194194 + 0.980963i \(0.562209\pi\)
\(152\) 5.46412 0.443198
\(153\) 5.38125 0.435048
\(154\) −11.3497 −0.914587
\(155\) 0.627809 0.0504268
\(156\) 1.00000 0.0800641
\(157\) 4.58778 0.366145 0.183073 0.983099i \(-0.441396\pi\)
0.183073 + 0.983099i \(0.441396\pi\)
\(158\) 3.70385 0.294662
\(159\) 7.50659 0.595311
\(160\) 1.40985 0.111459
\(161\) −4.91533 −0.387383
\(162\) 1.00000 0.0785674
\(163\) −3.09389 −0.242332 −0.121166 0.992632i \(-0.538663\pi\)
−0.121166 + 0.992632i \(0.538663\pi\)
\(164\) −0.952506 −0.0743782
\(165\) 3.28585 0.255803
\(166\) 14.4610 1.12239
\(167\) −11.7557 −0.909680 −0.454840 0.890573i \(-0.650304\pi\)
−0.454840 + 0.890573i \(0.650304\pi\)
\(168\) −4.86981 −0.375714
\(169\) 1.00000 0.0769231
\(170\) 7.58678 0.581880
\(171\) 5.46412 0.417851
\(172\) −3.56484 −0.271817
\(173\) 3.99859 0.304007 0.152004 0.988380i \(-0.451427\pi\)
0.152004 + 0.988380i \(0.451427\pi\)
\(174\) 2.01110 0.152461
\(175\) −14.6694 −1.10890
\(176\) −2.33063 −0.175678
\(177\) −4.52096 −0.339816
\(178\) 2.48255 0.186075
\(179\) −12.3417 −0.922465 −0.461233 0.887279i \(-0.652593\pi\)
−0.461233 + 0.887279i \(0.652593\pi\)
\(180\) 1.40985 0.105084
\(181\) 22.4793 1.67087 0.835437 0.549586i \(-0.185215\pi\)
0.835437 + 0.549586i \(0.185215\pi\)
\(182\) −4.86981 −0.360974
\(183\) −6.62115 −0.489450
\(184\) −1.00935 −0.0744102
\(185\) 12.8249 0.942904
\(186\) −0.445300 −0.0326510
\(187\) −12.5417 −0.917141
\(188\) 0.884177 0.0644853
\(189\) −4.86981 −0.354226
\(190\) 7.70361 0.558878
\(191\) −19.7321 −1.42776 −0.713882 0.700266i \(-0.753065\pi\)
−0.713882 + 0.700266i \(0.753065\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 17.3031 1.24550 0.622751 0.782420i \(-0.286015\pi\)
0.622751 + 0.782420i \(0.286015\pi\)
\(194\) 6.09866 0.437858
\(195\) 1.40985 0.100962
\(196\) 16.7150 1.19393
\(197\) 4.26407 0.303803 0.151901 0.988396i \(-0.451460\pi\)
0.151901 + 0.988396i \(0.451460\pi\)
\(198\) −2.33063 −0.165631
\(199\) −20.1794 −1.43048 −0.715238 0.698881i \(-0.753682\pi\)
−0.715238 + 0.698881i \(0.753682\pi\)
\(200\) −3.01231 −0.213003
\(201\) −2.00724 −0.141580
\(202\) 7.35330 0.517376
\(203\) −9.79368 −0.687382
\(204\) −5.38125 −0.376763
\(205\) −1.34289 −0.0937918
\(206\) −1.00000 −0.0696733
\(207\) −1.00935 −0.0701546
\(208\) −1.00000 −0.0693375
\(209\) −12.7348 −0.880887
\(210\) −6.86572 −0.473780
\(211\) −0.460401 −0.0316953 −0.0158477 0.999874i \(-0.505045\pi\)
−0.0158477 + 0.999874i \(0.505045\pi\)
\(212\) −7.50659 −0.515555
\(213\) 9.19397 0.629961
\(214\) −0.00520205 −0.000355605 0
\(215\) −5.02591 −0.342764
\(216\) −1.00000 −0.0680414
\(217\) 2.16853 0.147209
\(218\) −12.1170 −0.820666
\(219\) 6.35844 0.429663
\(220\) −3.28585 −0.221532
\(221\) −5.38125 −0.361982
\(222\) −9.09660 −0.610524
\(223\) 19.9215 1.33404 0.667021 0.745039i \(-0.267569\pi\)
0.667021 + 0.745039i \(0.267569\pi\)
\(224\) 4.86981 0.325378
\(225\) −3.01231 −0.200821
\(226\) −3.32058 −0.220881
\(227\) −20.1728 −1.33892 −0.669458 0.742850i \(-0.733473\pi\)
−0.669458 + 0.742850i \(0.733473\pi\)
\(228\) −5.46412 −0.361870
\(229\) −16.5160 −1.09140 −0.545702 0.837979i \(-0.683737\pi\)
−0.545702 + 0.837979i \(0.683737\pi\)
\(230\) −1.42303 −0.0938321
\(231\) 11.3497 0.746757
\(232\) −2.01110 −0.132035
\(233\) −11.7843 −0.772016 −0.386008 0.922495i \(-0.626146\pi\)
−0.386008 + 0.922495i \(0.626146\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 1.24656 0.0813167
\(236\) 4.52096 0.294289
\(237\) −3.70385 −0.240591
\(238\) 26.2057 1.69866
\(239\) 8.32974 0.538806 0.269403 0.963028i \(-0.413174\pi\)
0.269403 + 0.963028i \(0.413174\pi\)
\(240\) −1.40985 −0.0910057
\(241\) −10.9822 −0.707426 −0.353713 0.935354i \(-0.615081\pi\)
−0.353713 + 0.935354i \(0.615081\pi\)
\(242\) −5.56816 −0.357935
\(243\) −1.00000 −0.0641500
\(244\) 6.62115 0.423876
\(245\) 23.5658 1.50556
\(246\) 0.952506 0.0607295
\(247\) −5.46412 −0.347673
\(248\) 0.445300 0.0282766
\(249\) −14.4610 −0.916426
\(250\) −11.2962 −0.714434
\(251\) 8.73482 0.551337 0.275669 0.961253i \(-0.411101\pi\)
0.275669 + 0.961253i \(0.411101\pi\)
\(252\) 4.86981 0.306769
\(253\) 2.35242 0.147895
\(254\) −3.48520 −0.218681
\(255\) −7.58678 −0.475103
\(256\) 1.00000 0.0625000
\(257\) −9.60029 −0.598850 −0.299425 0.954120i \(-0.596795\pi\)
−0.299425 + 0.954120i \(0.596795\pi\)
\(258\) 3.56484 0.221937
\(259\) 44.2987 2.75259
\(260\) −1.40985 −0.0874355
\(261\) −2.01110 −0.124484
\(262\) 0.539376 0.0333228
\(263\) −22.0405 −1.35908 −0.679538 0.733640i \(-0.737820\pi\)
−0.679538 + 0.733640i \(0.737820\pi\)
\(264\) 2.33063 0.143440
\(265\) −10.5832 −0.650120
\(266\) 26.6092 1.63151
\(267\) −2.48255 −0.151930
\(268\) 2.00724 0.122612
\(269\) 13.0683 0.796788 0.398394 0.917214i \(-0.369568\pi\)
0.398394 + 0.917214i \(0.369568\pi\)
\(270\) −1.40985 −0.0858010
\(271\) 9.85282 0.598516 0.299258 0.954172i \(-0.403261\pi\)
0.299258 + 0.954172i \(0.403261\pi\)
\(272\) 5.38125 0.326286
\(273\) 4.86981 0.294734
\(274\) 1.94980 0.117792
\(275\) 7.02059 0.423357
\(276\) 1.00935 0.0607556
\(277\) −28.2551 −1.69769 −0.848843 0.528644i \(-0.822701\pi\)
−0.848843 + 0.528644i \(0.822701\pi\)
\(278\) 10.6357 0.637888
\(279\) 0.445300 0.0266594
\(280\) 6.86572 0.410305
\(281\) 1.71433 0.102268 0.0511341 0.998692i \(-0.483716\pi\)
0.0511341 + 0.998692i \(0.483716\pi\)
\(282\) −0.884177 −0.0526520
\(283\) 16.3986 0.974793 0.487396 0.873181i \(-0.337947\pi\)
0.487396 + 0.873181i \(0.337947\pi\)
\(284\) −9.19397 −0.545562
\(285\) −7.70361 −0.456322
\(286\) 2.33063 0.137813
\(287\) −4.63852 −0.273803
\(288\) 1.00000 0.0589256
\(289\) 11.9579 0.703404
\(290\) −2.83536 −0.166498
\(291\) −6.09866 −0.357510
\(292\) −6.35844 −0.372099
\(293\) 22.5655 1.31829 0.659146 0.752015i \(-0.270918\pi\)
0.659146 + 0.752015i \(0.270918\pi\)
\(294\) −16.7150 −0.974841
\(295\) 6.37389 0.371102
\(296\) 9.09660 0.528729
\(297\) 2.33063 0.135237
\(298\) −5.69498 −0.329901
\(299\) 1.00935 0.0583721
\(300\) 3.01231 0.173916
\(301\) −17.3601 −1.00062
\(302\) −4.77259 −0.274632
\(303\) −7.35330 −0.422436
\(304\) 5.46412 0.313389
\(305\) 9.33486 0.534513
\(306\) 5.38125 0.307626
\(307\) 26.4261 1.50822 0.754109 0.656749i \(-0.228069\pi\)
0.754109 + 0.656749i \(0.228069\pi\)
\(308\) −11.3497 −0.646711
\(309\) 1.00000 0.0568880
\(310\) 0.627809 0.0356571
\(311\) −5.10993 −0.289758 −0.144879 0.989449i \(-0.546279\pi\)
−0.144879 + 0.989449i \(0.546279\pi\)
\(312\) 1.00000 0.0566139
\(313\) 26.4548 1.49531 0.747657 0.664085i \(-0.231179\pi\)
0.747657 + 0.664085i \(0.231179\pi\)
\(314\) 4.58778 0.258904
\(315\) 6.86572 0.386840
\(316\) 3.70385 0.208358
\(317\) −6.08045 −0.341512 −0.170756 0.985313i \(-0.554621\pi\)
−0.170756 + 0.985313i \(0.554621\pi\)
\(318\) 7.50659 0.420949
\(319\) 4.68714 0.262429
\(320\) 1.40985 0.0788133
\(321\) 0.00520205 0.000290350 0
\(322\) −4.91533 −0.273921
\(323\) 29.4038 1.63607
\(324\) 1.00000 0.0555556
\(325\) 3.01231 0.167093
\(326\) −3.09389 −0.171354
\(327\) 12.1170 0.670071
\(328\) −0.952506 −0.0525933
\(329\) 4.30577 0.237385
\(330\) 3.28585 0.180880
\(331\) 3.03497 0.166817 0.0834085 0.996515i \(-0.473419\pi\)
0.0834085 + 0.996515i \(0.473419\pi\)
\(332\) 14.4610 0.793648
\(333\) 9.09660 0.498491
\(334\) −11.7557 −0.643241
\(335\) 2.82992 0.154615
\(336\) −4.86981 −0.265670
\(337\) 13.8840 0.756310 0.378155 0.925742i \(-0.376559\pi\)
0.378155 + 0.925742i \(0.376559\pi\)
\(338\) 1.00000 0.0543928
\(339\) 3.32058 0.180349
\(340\) 7.58678 0.411451
\(341\) −1.03783 −0.0562017
\(342\) 5.46412 0.295466
\(343\) 47.3104 2.55452
\(344\) −3.56484 −0.192203
\(345\) 1.42303 0.0766136
\(346\) 3.99859 0.214966
\(347\) 25.4955 1.36867 0.684336 0.729167i \(-0.260092\pi\)
0.684336 + 0.729167i \(0.260092\pi\)
\(348\) 2.01110 0.107806
\(349\) −19.0588 −1.02019 −0.510096 0.860117i \(-0.670390\pi\)
−0.510096 + 0.860117i \(0.670390\pi\)
\(350\) −14.6694 −0.784111
\(351\) 1.00000 0.0533761
\(352\) −2.33063 −0.124223
\(353\) −28.7383 −1.52959 −0.764793 0.644276i \(-0.777159\pi\)
−0.764793 + 0.644276i \(0.777159\pi\)
\(354\) −4.52096 −0.240286
\(355\) −12.9622 −0.687960
\(356\) 2.48255 0.131575
\(357\) −26.2057 −1.38695
\(358\) −12.3417 −0.652281
\(359\) −8.50496 −0.448874 −0.224437 0.974489i \(-0.572054\pi\)
−0.224437 + 0.974489i \(0.572054\pi\)
\(360\) 1.40985 0.0743058
\(361\) 10.8566 0.571398
\(362\) 22.4793 1.18149
\(363\) 5.56816 0.292252
\(364\) −4.86981 −0.255247
\(365\) −8.96447 −0.469222
\(366\) −6.62115 −0.346093
\(367\) −9.47425 −0.494552 −0.247276 0.968945i \(-0.579535\pi\)
−0.247276 + 0.968945i \(0.579535\pi\)
\(368\) −1.00935 −0.0526159
\(369\) −0.952506 −0.0495855
\(370\) 12.8249 0.666734
\(371\) −36.5556 −1.89787
\(372\) −0.445300 −0.0230877
\(373\) −4.61157 −0.238778 −0.119389 0.992848i \(-0.538094\pi\)
−0.119389 + 0.992848i \(0.538094\pi\)
\(374\) −12.5417 −0.648517
\(375\) 11.2962 0.583333
\(376\) 0.884177 0.0455980
\(377\) 2.01110 0.103577
\(378\) −4.86981 −0.250476
\(379\) 21.9061 1.12524 0.562620 0.826716i \(-0.309793\pi\)
0.562620 + 0.826716i \(0.309793\pi\)
\(380\) 7.70361 0.395187
\(381\) 3.48520 0.178552
\(382\) −19.7321 −1.00958
\(383\) −8.61706 −0.440311 −0.220156 0.975465i \(-0.570657\pi\)
−0.220156 + 0.975465i \(0.570657\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −16.0015 −0.815510
\(386\) 17.3031 0.880703
\(387\) −3.56484 −0.181211
\(388\) 6.09866 0.309613
\(389\) 31.6874 1.60661 0.803307 0.595566i \(-0.203072\pi\)
0.803307 + 0.595566i \(0.203072\pi\)
\(390\) 1.40985 0.0713907
\(391\) −5.43156 −0.274686
\(392\) 16.7150 0.844237
\(393\) −0.539376 −0.0272079
\(394\) 4.26407 0.214821
\(395\) 5.22188 0.262741
\(396\) −2.33063 −0.117119
\(397\) −32.9613 −1.65428 −0.827141 0.561994i \(-0.810034\pi\)
−0.827141 + 0.561994i \(0.810034\pi\)
\(398\) −20.1794 −1.01150
\(399\) −26.6092 −1.33213
\(400\) −3.01231 −0.150616
\(401\) 23.0372 1.15042 0.575211 0.818005i \(-0.304920\pi\)
0.575211 + 0.818005i \(0.304920\pi\)
\(402\) −2.00724 −0.100112
\(403\) −0.445300 −0.0221820
\(404\) 7.35330 0.365840
\(405\) 1.40985 0.0700562
\(406\) −9.79368 −0.486052
\(407\) −21.2008 −1.05089
\(408\) −5.38125 −0.266412
\(409\) 2.78238 0.137580 0.0687900 0.997631i \(-0.478086\pi\)
0.0687900 + 0.997631i \(0.478086\pi\)
\(410\) −1.34289 −0.0663208
\(411\) −1.94980 −0.0961768
\(412\) −1.00000 −0.0492665
\(413\) 22.0162 1.08335
\(414\) −1.00935 −0.0496068
\(415\) 20.3879 1.00080
\(416\) −1.00000 −0.0490290
\(417\) −10.6357 −0.520833
\(418\) −12.7348 −0.622881
\(419\) 6.97312 0.340659 0.170330 0.985387i \(-0.445517\pi\)
0.170330 + 0.985387i \(0.445517\pi\)
\(420\) −6.86572 −0.335013
\(421\) 4.97348 0.242393 0.121196 0.992629i \(-0.461327\pi\)
0.121196 + 0.992629i \(0.461327\pi\)
\(422\) −0.460401 −0.0224120
\(423\) 0.884177 0.0429902
\(424\) −7.50659 −0.364552
\(425\) −16.2100 −0.786301
\(426\) 9.19397 0.445450
\(427\) 32.2437 1.56038
\(428\) −0.00520205 −0.000251450 0
\(429\) −2.33063 −0.112524
\(430\) −5.02591 −0.242371
\(431\) −10.6617 −0.513554 −0.256777 0.966471i \(-0.582661\pi\)
−0.256777 + 0.966471i \(0.582661\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 36.5166 1.75487 0.877437 0.479692i \(-0.159252\pi\)
0.877437 + 0.479692i \(0.159252\pi\)
\(434\) 2.16853 0.104093
\(435\) 2.83536 0.135945
\(436\) −12.1170 −0.580299
\(437\) −5.51520 −0.263828
\(438\) 6.35844 0.303818
\(439\) 28.5065 1.36054 0.680271 0.732961i \(-0.261862\pi\)
0.680271 + 0.732961i \(0.261862\pi\)
\(440\) −3.28585 −0.156647
\(441\) 16.7150 0.795954
\(442\) −5.38125 −0.255960
\(443\) −18.8447 −0.895340 −0.447670 0.894199i \(-0.647746\pi\)
−0.447670 + 0.894199i \(0.647746\pi\)
\(444\) −9.09660 −0.431706
\(445\) 3.50004 0.165918
\(446\) 19.9215 0.943310
\(447\) 5.69498 0.269363
\(448\) 4.86981 0.230077
\(449\) 26.6224 1.25639 0.628195 0.778056i \(-0.283794\pi\)
0.628195 + 0.778056i \(0.283794\pi\)
\(450\) −3.01231 −0.142002
\(451\) 2.21994 0.104533
\(452\) −3.32058 −0.156187
\(453\) 4.77259 0.224236
\(454\) −20.1728 −0.946756
\(455\) −6.86572 −0.321870
\(456\) −5.46412 −0.255881
\(457\) −2.05556 −0.0961550 −0.0480775 0.998844i \(-0.515309\pi\)
−0.0480775 + 0.998844i \(0.515309\pi\)
\(458\) −16.5160 −0.771740
\(459\) −5.38125 −0.251175
\(460\) −1.42303 −0.0663493
\(461\) 12.0356 0.560553 0.280277 0.959919i \(-0.409574\pi\)
0.280277 + 0.959919i \(0.409574\pi\)
\(462\) 11.3497 0.528037
\(463\) −26.5610 −1.23440 −0.617198 0.786808i \(-0.711732\pi\)
−0.617198 + 0.786808i \(0.711732\pi\)
\(464\) −2.01110 −0.0933631
\(465\) −0.627809 −0.0291139
\(466\) −11.7843 −0.545898
\(467\) −25.8182 −1.19472 −0.597361 0.801973i \(-0.703784\pi\)
−0.597361 + 0.801973i \(0.703784\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 9.77489 0.451363
\(470\) 1.24656 0.0574996
\(471\) −4.58778 −0.211394
\(472\) 4.52096 0.208094
\(473\) 8.30833 0.382017
\(474\) −3.70385 −0.170123
\(475\) −16.4596 −0.755219
\(476\) 26.2057 1.20113
\(477\) −7.50659 −0.343703
\(478\) 8.32974 0.380993
\(479\) −4.91205 −0.224437 −0.112219 0.993684i \(-0.535796\pi\)
−0.112219 + 0.993684i \(0.535796\pi\)
\(480\) −1.40985 −0.0643508
\(481\) −9.09660 −0.414769
\(482\) −10.9822 −0.500226
\(483\) 4.91533 0.223655
\(484\) −5.56816 −0.253098
\(485\) 8.59822 0.390425
\(486\) −1.00000 −0.0453609
\(487\) −33.5943 −1.52230 −0.761152 0.648573i \(-0.775366\pi\)
−0.761152 + 0.648573i \(0.775366\pi\)
\(488\) 6.62115 0.299725
\(489\) 3.09389 0.139910
\(490\) 23.5658 1.06459
\(491\) 20.7294 0.935503 0.467752 0.883860i \(-0.345064\pi\)
0.467752 + 0.883860i \(0.345064\pi\)
\(492\) 0.952506 0.0429423
\(493\) −10.8222 −0.487410
\(494\) −5.46412 −0.245842
\(495\) −3.28585 −0.147688
\(496\) 0.445300 0.0199946
\(497\) −44.7729 −2.00834
\(498\) −14.4610 −0.648011
\(499\) 1.53739 0.0688231 0.0344116 0.999408i \(-0.489044\pi\)
0.0344116 + 0.999408i \(0.489044\pi\)
\(500\) −11.2962 −0.505181
\(501\) 11.7557 0.525204
\(502\) 8.73482 0.389854
\(503\) −23.1448 −1.03198 −0.515988 0.856596i \(-0.672575\pi\)
−0.515988 + 0.856596i \(0.672575\pi\)
\(504\) 4.86981 0.216919
\(505\) 10.3671 0.461329
\(506\) 2.35242 0.104578
\(507\) −1.00000 −0.0444116
\(508\) −3.48520 −0.154631
\(509\) −15.4322 −0.684020 −0.342010 0.939696i \(-0.611108\pi\)
−0.342010 + 0.939696i \(0.611108\pi\)
\(510\) −7.58678 −0.335948
\(511\) −30.9644 −1.36978
\(512\) 1.00000 0.0441942
\(513\) −5.46412 −0.241247
\(514\) −9.60029 −0.423451
\(515\) −1.40985 −0.0621256
\(516\) 3.56484 0.156933
\(517\) −2.06069 −0.0906291
\(518\) 44.2987 1.94637
\(519\) −3.99859 −0.175519
\(520\) −1.40985 −0.0618262
\(521\) −15.5549 −0.681471 −0.340736 0.940159i \(-0.610676\pi\)
−0.340736 + 0.940159i \(0.610676\pi\)
\(522\) −2.01110 −0.0880236
\(523\) 25.8172 1.12891 0.564455 0.825464i \(-0.309086\pi\)
0.564455 + 0.825464i \(0.309086\pi\)
\(524\) 0.539376 0.0235628
\(525\) 14.6694 0.640224
\(526\) −22.0405 −0.961012
\(527\) 2.39627 0.104383
\(528\) 2.33063 0.101428
\(529\) −21.9812 −0.955705
\(530\) −10.5832 −0.459705
\(531\) 4.52096 0.196193
\(532\) 26.6092 1.15365
\(533\) 0.952506 0.0412576
\(534\) −2.48255 −0.107431
\(535\) −0.00733413 −0.000317082 0
\(536\) 2.00724 0.0866997
\(537\) 12.3417 0.532586
\(538\) 13.0683 0.563414
\(539\) −38.9566 −1.67798
\(540\) −1.40985 −0.0606705
\(541\) −23.3074 −1.00206 −0.501032 0.865429i \(-0.667046\pi\)
−0.501032 + 0.865429i \(0.667046\pi\)
\(542\) 9.85282 0.423215
\(543\) −22.4793 −0.964680
\(544\) 5.38125 0.230719
\(545\) −17.0832 −0.731764
\(546\) 4.86981 0.208409
\(547\) 40.4603 1.72996 0.864978 0.501810i \(-0.167332\pi\)
0.864978 + 0.501810i \(0.167332\pi\)
\(548\) 1.94980 0.0832915
\(549\) 6.62115 0.282584
\(550\) 7.02059 0.299359
\(551\) −10.9889 −0.468143
\(552\) 1.00935 0.0429607
\(553\) 18.0370 0.767012
\(554\) −28.2551 −1.20045
\(555\) −12.8249 −0.544386
\(556\) 10.6357 0.451055
\(557\) 35.6694 1.51136 0.755680 0.654941i \(-0.227307\pi\)
0.755680 + 0.654941i \(0.227307\pi\)
\(558\) 0.445300 0.0188511
\(559\) 3.56484 0.150777
\(560\) 6.86572 0.290130
\(561\) 12.5417 0.529512
\(562\) 1.71433 0.0723146
\(563\) −11.0723 −0.466641 −0.233321 0.972400i \(-0.574959\pi\)
−0.233321 + 0.972400i \(0.574959\pi\)
\(564\) −0.884177 −0.0372306
\(565\) −4.68153 −0.196953
\(566\) 16.3986 0.689283
\(567\) 4.86981 0.204513
\(568\) −9.19397 −0.385771
\(569\) 9.70010 0.406649 0.203325 0.979111i \(-0.434825\pi\)
0.203325 + 0.979111i \(0.434825\pi\)
\(570\) −7.70361 −0.322669
\(571\) −34.8614 −1.45890 −0.729451 0.684033i \(-0.760224\pi\)
−0.729451 + 0.684033i \(0.760224\pi\)
\(572\) 2.33063 0.0974486
\(573\) 19.7321 0.824320
\(574\) −4.63852 −0.193608
\(575\) 3.04047 0.126796
\(576\) 1.00000 0.0416667
\(577\) −3.62525 −0.150921 −0.0754605 0.997149i \(-0.524043\pi\)
−0.0754605 + 0.997149i \(0.524043\pi\)
\(578\) 11.9579 0.497382
\(579\) −17.3031 −0.719091
\(580\) −2.83536 −0.117732
\(581\) 70.4221 2.92160
\(582\) −6.09866 −0.252798
\(583\) 17.4951 0.724573
\(584\) −6.35844 −0.263114
\(585\) −1.40985 −0.0582903
\(586\) 22.5655 0.932174
\(587\) −29.6395 −1.22335 −0.611676 0.791109i \(-0.709504\pi\)
−0.611676 + 0.791109i \(0.709504\pi\)
\(588\) −16.7150 −0.689316
\(589\) 2.43317 0.100257
\(590\) 6.37389 0.262409
\(591\) −4.26407 −0.175401
\(592\) 9.09660 0.373868
\(593\) −9.58200 −0.393485 −0.196743 0.980455i \(-0.563036\pi\)
−0.196743 + 0.980455i \(0.563036\pi\)
\(594\) 2.33063 0.0956270
\(595\) 36.9462 1.51465
\(596\) −5.69498 −0.233276
\(597\) 20.1794 0.825886
\(598\) 1.00935 0.0412753
\(599\) −2.64077 −0.107899 −0.0539496 0.998544i \(-0.517181\pi\)
−0.0539496 + 0.998544i \(0.517181\pi\)
\(600\) 3.01231 0.122977
\(601\) 36.7215 1.49790 0.748952 0.662625i \(-0.230558\pi\)
0.748952 + 0.662625i \(0.230558\pi\)
\(602\) −17.3601 −0.707544
\(603\) 2.00724 0.0817413
\(604\) −4.77259 −0.194194
\(605\) −7.85029 −0.319160
\(606\) −7.35330 −0.298707
\(607\) −13.5559 −0.550217 −0.275108 0.961413i \(-0.588714\pi\)
−0.275108 + 0.961413i \(0.588714\pi\)
\(608\) 5.46412 0.221599
\(609\) 9.79368 0.396860
\(610\) 9.33486 0.377957
\(611\) −0.884177 −0.0357700
\(612\) 5.38125 0.217524
\(613\) 6.82023 0.275467 0.137733 0.990469i \(-0.456018\pi\)
0.137733 + 0.990469i \(0.456018\pi\)
\(614\) 26.4261 1.06647
\(615\) 1.34289 0.0541507
\(616\) −11.3497 −0.457294
\(617\) 27.9917 1.12690 0.563452 0.826149i \(-0.309473\pi\)
0.563452 + 0.826149i \(0.309473\pi\)
\(618\) 1.00000 0.0402259
\(619\) −18.5850 −0.746995 −0.373498 0.927631i \(-0.621842\pi\)
−0.373498 + 0.927631i \(0.621842\pi\)
\(620\) 0.627809 0.0252134
\(621\) 1.00935 0.0405038
\(622\) −5.10993 −0.204890
\(623\) 12.0896 0.484358
\(624\) 1.00000 0.0400320
\(625\) −0.864429 −0.0345772
\(626\) 26.4548 1.05735
\(627\) 12.7348 0.508581
\(628\) 4.58778 0.183073
\(629\) 48.9511 1.95181
\(630\) 6.86572 0.273537
\(631\) 12.6504 0.503602 0.251801 0.967779i \(-0.418977\pi\)
0.251801 + 0.967779i \(0.418977\pi\)
\(632\) 3.70385 0.147331
\(633\) 0.460401 0.0182993
\(634\) −6.08045 −0.241485
\(635\) −4.91362 −0.194991
\(636\) 7.50659 0.297656
\(637\) −16.7150 −0.662274
\(638\) 4.68714 0.185566
\(639\) −9.19397 −0.363708
\(640\) 1.40985 0.0557294
\(641\) 8.11914 0.320687 0.160343 0.987061i \(-0.448740\pi\)
0.160343 + 0.987061i \(0.448740\pi\)
\(642\) 0.00520205 0.000205308 0
\(643\) 20.6268 0.813442 0.406721 0.913552i \(-0.366672\pi\)
0.406721 + 0.913552i \(0.366672\pi\)
\(644\) −4.91533 −0.193691
\(645\) 5.02591 0.197895
\(646\) 29.4038 1.15688
\(647\) 48.7601 1.91696 0.958478 0.285166i \(-0.0920487\pi\)
0.958478 + 0.285166i \(0.0920487\pi\)
\(648\) 1.00000 0.0392837
\(649\) −10.5367 −0.413601
\(650\) 3.01231 0.118153
\(651\) −2.16853 −0.0849913
\(652\) −3.09389 −0.121166
\(653\) 16.8596 0.659766 0.329883 0.944022i \(-0.392991\pi\)
0.329883 + 0.944022i \(0.392991\pi\)
\(654\) 12.1170 0.473812
\(655\) 0.760442 0.0297129
\(656\) −0.952506 −0.0371891
\(657\) −6.35844 −0.248066
\(658\) 4.30577 0.167857
\(659\) −22.1441 −0.862613 −0.431306 0.902206i \(-0.641947\pi\)
−0.431306 + 0.902206i \(0.641947\pi\)
\(660\) 3.28585 0.127902
\(661\) −12.0035 −0.466881 −0.233441 0.972371i \(-0.574998\pi\)
−0.233441 + 0.972371i \(0.574998\pi\)
\(662\) 3.03497 0.117957
\(663\) 5.38125 0.208991
\(664\) 14.4610 0.561194
\(665\) 37.5151 1.45477
\(666\) 9.09660 0.352486
\(667\) 2.02990 0.0785982
\(668\) −11.7557 −0.454840
\(669\) −19.9215 −0.770210
\(670\) 2.82992 0.109329
\(671\) −15.4315 −0.595725
\(672\) −4.86981 −0.187857
\(673\) −8.44454 −0.325513 −0.162757 0.986666i \(-0.552039\pi\)
−0.162757 + 0.986666i \(0.552039\pi\)
\(674\) 13.8840 0.534792
\(675\) 3.01231 0.115944
\(676\) 1.00000 0.0384615
\(677\) 24.4985 0.941554 0.470777 0.882252i \(-0.343974\pi\)
0.470777 + 0.882252i \(0.343974\pi\)
\(678\) 3.32058 0.127526
\(679\) 29.6993 1.13975
\(680\) 7.58678 0.290940
\(681\) 20.1728 0.773023
\(682\) −1.03783 −0.0397406
\(683\) 44.0120 1.68407 0.842037 0.539420i \(-0.181356\pi\)
0.842037 + 0.539420i \(0.181356\pi\)
\(684\) 5.46412 0.208926
\(685\) 2.74894 0.105032
\(686\) 47.3104 1.80632
\(687\) 16.5160 0.630123
\(688\) −3.56484 −0.135908
\(689\) 7.50659 0.285978
\(690\) 1.42303 0.0541740
\(691\) −39.9976 −1.52158 −0.760790 0.648998i \(-0.775188\pi\)
−0.760790 + 0.648998i \(0.775188\pi\)
\(692\) 3.99859 0.152004
\(693\) −11.3497 −0.431141
\(694\) 25.4955 0.967797
\(695\) 14.9948 0.568786
\(696\) 2.01110 0.0762306
\(697\) −5.12567 −0.194149
\(698\) −19.0588 −0.721385
\(699\) 11.7843 0.445724
\(700\) −14.6694 −0.554450
\(701\) −43.4942 −1.64276 −0.821378 0.570385i \(-0.806794\pi\)
−0.821378 + 0.570385i \(0.806794\pi\)
\(702\) 1.00000 0.0377426
\(703\) 49.7049 1.87466
\(704\) −2.33063 −0.0878390
\(705\) −1.24656 −0.0469482
\(706\) −28.7383 −1.08158
\(707\) 35.8092 1.34674
\(708\) −4.52096 −0.169908
\(709\) 30.5582 1.14764 0.573819 0.818982i \(-0.305461\pi\)
0.573819 + 0.818982i \(0.305461\pi\)
\(710\) −12.9622 −0.486461
\(711\) 3.70385 0.138905
\(712\) 2.48255 0.0930376
\(713\) −0.449463 −0.0168325
\(714\) −26.2057 −0.980723
\(715\) 3.28585 0.122884
\(716\) −12.3417 −0.461233
\(717\) −8.32974 −0.311080
\(718\) −8.50496 −0.317402
\(719\) 13.7827 0.514008 0.257004 0.966410i \(-0.417265\pi\)
0.257004 + 0.966410i \(0.417265\pi\)
\(720\) 1.40985 0.0525422
\(721\) −4.86981 −0.181361
\(722\) 10.8566 0.404039
\(723\) 10.9822 0.408433
\(724\) 22.4793 0.835437
\(725\) 6.05807 0.224991
\(726\) 5.56816 0.206654
\(727\) −1.87934 −0.0697007 −0.0348504 0.999393i \(-0.511095\pi\)
−0.0348504 + 0.999393i \(0.511095\pi\)
\(728\) −4.86981 −0.180487
\(729\) 1.00000 0.0370370
\(730\) −8.96447 −0.331790
\(731\) −19.1833 −0.709520
\(732\) −6.62115 −0.244725
\(733\) 13.4501 0.496791 0.248395 0.968659i \(-0.420097\pi\)
0.248395 + 0.968659i \(0.420097\pi\)
\(734\) −9.47425 −0.349701
\(735\) −23.5658 −0.869236
\(736\) −1.00935 −0.0372051
\(737\) −4.67814 −0.172322
\(738\) −0.952506 −0.0350622
\(739\) −22.0485 −0.811069 −0.405534 0.914080i \(-0.632915\pi\)
−0.405534 + 0.914080i \(0.632915\pi\)
\(740\) 12.8249 0.471452
\(741\) 5.46412 0.200729
\(742\) −36.5556 −1.34200
\(743\) 7.21093 0.264543 0.132272 0.991214i \(-0.457773\pi\)
0.132272 + 0.991214i \(0.457773\pi\)
\(744\) −0.445300 −0.0163255
\(745\) −8.02909 −0.294163
\(746\) −4.61157 −0.168842
\(747\) 14.4610 0.529099
\(748\) −12.5417 −0.458571
\(749\) −0.0253330 −0.000925647 0
\(750\) 11.2962 0.412479
\(751\) −41.0774 −1.49894 −0.749469 0.662040i \(-0.769691\pi\)
−0.749469 + 0.662040i \(0.769691\pi\)
\(752\) 0.884177 0.0322426
\(753\) −8.73482 −0.318315
\(754\) 2.01110 0.0732400
\(755\) −6.72866 −0.244881
\(756\) −4.86981 −0.177113
\(757\) 6.35811 0.231089 0.115545 0.993302i \(-0.463139\pi\)
0.115545 + 0.993302i \(0.463139\pi\)
\(758\) 21.9061 0.795665
\(759\) −2.35242 −0.0853874
\(760\) 7.70361 0.279439
\(761\) 15.9103 0.576747 0.288373 0.957518i \(-0.406886\pi\)
0.288373 + 0.957518i \(0.406886\pi\)
\(762\) 3.48520 0.126255
\(763\) −59.0074 −2.13621
\(764\) −19.7321 −0.713882
\(765\) 7.58678 0.274301
\(766\) −8.61706 −0.311347
\(767\) −4.52096 −0.163242
\(768\) −1.00000 −0.0360844
\(769\) 22.9658 0.828167 0.414083 0.910239i \(-0.364102\pi\)
0.414083 + 0.910239i \(0.364102\pi\)
\(770\) −16.0015 −0.576653
\(771\) 9.60029 0.345746
\(772\) 17.3031 0.622751
\(773\) 15.0931 0.542861 0.271431 0.962458i \(-0.412503\pi\)
0.271431 + 0.962458i \(0.412503\pi\)
\(774\) −3.56484 −0.128136
\(775\) −1.34138 −0.0481839
\(776\) 6.09866 0.218929
\(777\) −44.2987 −1.58921
\(778\) 31.6874 1.13605
\(779\) −5.20460 −0.186474
\(780\) 1.40985 0.0504809
\(781\) 21.4278 0.766746
\(782\) −5.43156 −0.194232
\(783\) 2.01110 0.0718709
\(784\) 16.7150 0.596966
\(785\) 6.46811 0.230857
\(786\) −0.539376 −0.0192389
\(787\) 17.0156 0.606539 0.303269 0.952905i \(-0.401922\pi\)
0.303269 + 0.952905i \(0.401922\pi\)
\(788\) 4.26407 0.151901
\(789\) 22.0405 0.784663
\(790\) 5.22188 0.185786
\(791\) −16.1706 −0.574959
\(792\) −2.33063 −0.0828154
\(793\) −6.62115 −0.235124
\(794\) −32.9613 −1.16975
\(795\) 10.5832 0.375347
\(796\) −20.1794 −0.715238
\(797\) −8.33831 −0.295358 −0.147679 0.989035i \(-0.547180\pi\)
−0.147679 + 0.989035i \(0.547180\pi\)
\(798\) −26.6092 −0.941955
\(799\) 4.75798 0.168325
\(800\) −3.01231 −0.106501
\(801\) 2.48255 0.0877167
\(802\) 23.0372 0.813472
\(803\) 14.8192 0.522957
\(804\) −2.00724 −0.0707900
\(805\) −6.92990 −0.244247
\(806\) −0.445300 −0.0156850
\(807\) −13.0683 −0.460026
\(808\) 7.35330 0.258688
\(809\) 3.28317 0.115430 0.0577151 0.998333i \(-0.481619\pi\)
0.0577151 + 0.998333i \(0.481619\pi\)
\(810\) 1.40985 0.0495372
\(811\) 33.7600 1.18547 0.592736 0.805397i \(-0.298048\pi\)
0.592736 + 0.805397i \(0.298048\pi\)
\(812\) −9.79368 −0.343691
\(813\) −9.85282 −0.345553
\(814\) −21.2008 −0.743089
\(815\) −4.36193 −0.152792
\(816\) −5.38125 −0.188382
\(817\) −19.4787 −0.681473
\(818\) 2.78238 0.0972837
\(819\) −4.86981 −0.170165
\(820\) −1.34289 −0.0468959
\(821\) −15.4035 −0.537587 −0.268793 0.963198i \(-0.586625\pi\)
−0.268793 + 0.963198i \(0.586625\pi\)
\(822\) −1.94980 −0.0680072
\(823\) 5.10990 0.178120 0.0890599 0.996026i \(-0.471614\pi\)
0.0890599 + 0.996026i \(0.471614\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −7.02059 −0.244425
\(826\) 22.0162 0.766041
\(827\) −33.9926 −1.18204 −0.591020 0.806657i \(-0.701274\pi\)
−0.591020 + 0.806657i \(0.701274\pi\)
\(828\) −1.00935 −0.0350773
\(829\) −14.5425 −0.505082 −0.252541 0.967586i \(-0.581266\pi\)
−0.252541 + 0.967586i \(0.581266\pi\)
\(830\) 20.3879 0.707673
\(831\) 28.2551 0.980160
\(832\) −1.00000 −0.0346688
\(833\) 89.9478 3.11651
\(834\) −10.6357 −0.368285
\(835\) −16.5738 −0.573559
\(836\) −12.7348 −0.440444
\(837\) −0.445300 −0.0153918
\(838\) 6.97312 0.240882
\(839\) −38.8377 −1.34083 −0.670414 0.741987i \(-0.733883\pi\)
−0.670414 + 0.741987i \(0.733883\pi\)
\(840\) −6.86572 −0.236890
\(841\) −24.9555 −0.860533
\(842\) 4.97348 0.171398
\(843\) −1.71433 −0.0590446
\(844\) −0.460401 −0.0158477
\(845\) 1.40985 0.0485005
\(846\) 0.884177 0.0303986
\(847\) −27.1159 −0.931712
\(848\) −7.50659 −0.257777
\(849\) −16.3986 −0.562797
\(850\) −16.2100 −0.555999
\(851\) −9.18164 −0.314743
\(852\) 9.19397 0.314980
\(853\) 37.5994 1.28738 0.643690 0.765286i \(-0.277403\pi\)
0.643690 + 0.765286i \(0.277403\pi\)
\(854\) 32.2437 1.10336
\(855\) 7.70361 0.263458
\(856\) −0.00520205 −0.000177802 0
\(857\) −53.1598 −1.81590 −0.907952 0.419073i \(-0.862355\pi\)
−0.907952 + 0.419073i \(0.862355\pi\)
\(858\) −2.33063 −0.0795665
\(859\) 0.402154 0.0137213 0.00686067 0.999976i \(-0.497816\pi\)
0.00686067 + 0.999976i \(0.497816\pi\)
\(860\) −5.02591 −0.171382
\(861\) 4.63852 0.158080
\(862\) −10.6617 −0.363137
\(863\) 25.1477 0.856039 0.428019 0.903770i \(-0.359211\pi\)
0.428019 + 0.903770i \(0.359211\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 5.63743 0.191678
\(866\) 36.5166 1.24088
\(867\) −11.9579 −0.406111
\(868\) 2.16853 0.0736046
\(869\) −8.63230 −0.292831
\(870\) 2.83536 0.0961277
\(871\) −2.00724 −0.0680128
\(872\) −12.1170 −0.410333
\(873\) 6.09866 0.206408
\(874\) −5.51520 −0.186554
\(875\) −55.0103 −1.85969
\(876\) 6.35844 0.214832
\(877\) −24.7664 −0.836302 −0.418151 0.908378i \(-0.637322\pi\)
−0.418151 + 0.908378i \(0.637322\pi\)
\(878\) 28.5065 0.962048
\(879\) −22.5655 −0.761117
\(880\) −3.28585 −0.110766
\(881\) 3.38245 0.113958 0.0569788 0.998375i \(-0.481853\pi\)
0.0569788 + 0.998375i \(0.481853\pi\)
\(882\) 16.7150 0.562825
\(883\) 17.7294 0.596643 0.298322 0.954465i \(-0.403573\pi\)
0.298322 + 0.954465i \(0.403573\pi\)
\(884\) −5.38125 −0.180991
\(885\) −6.37389 −0.214256
\(886\) −18.8447 −0.633101
\(887\) −23.9345 −0.803641 −0.401820 0.915719i \(-0.631622\pi\)
−0.401820 + 0.915719i \(0.631622\pi\)
\(888\) −9.09660 −0.305262
\(889\) −16.9722 −0.569231
\(890\) 3.50004 0.117322
\(891\) −2.33063 −0.0780791
\(892\) 19.9215 0.667021
\(893\) 4.83125 0.161671
\(894\) 5.69498 0.190469
\(895\) −17.4001 −0.581620
\(896\) 4.86981 0.162689
\(897\) −1.00935 −0.0337012
\(898\) 26.6224 0.888402
\(899\) −0.895544 −0.0298681
\(900\) −3.01231 −0.100410
\(901\) −40.3948 −1.34575
\(902\) 2.21994 0.0739159
\(903\) 17.3601 0.577708
\(904\) −3.32058 −0.110441
\(905\) 31.6926 1.05350
\(906\) 4.77259 0.158559
\(907\) 17.8828 0.593789 0.296894 0.954910i \(-0.404049\pi\)
0.296894 + 0.954910i \(0.404049\pi\)
\(908\) −20.1728 −0.669458
\(909\) 7.35330 0.243894
\(910\) −6.86572 −0.227596
\(911\) −29.8178 −0.987906 −0.493953 0.869489i \(-0.664449\pi\)
−0.493953 + 0.869489i \(0.664449\pi\)
\(912\) −5.46412 −0.180935
\(913\) −33.7032 −1.11541
\(914\) −2.05556 −0.0679919
\(915\) −9.33486 −0.308601
\(916\) −16.5160 −0.545702
\(917\) 2.62666 0.0867399
\(918\) −5.38125 −0.177608
\(919\) −6.38995 −0.210785 −0.105392 0.994431i \(-0.533610\pi\)
−0.105392 + 0.994431i \(0.533610\pi\)
\(920\) −1.42303 −0.0469161
\(921\) −26.4261 −0.870770
\(922\) 12.0356 0.396371
\(923\) 9.19397 0.302623
\(924\) 11.3497 0.373379
\(925\) −27.4018 −0.900965
\(926\) −26.5610 −0.872850
\(927\) −1.00000 −0.0328443
\(928\) −2.01110 −0.0660177
\(929\) −38.2288 −1.25425 −0.627124 0.778920i \(-0.715768\pi\)
−0.627124 + 0.778920i \(0.715768\pi\)
\(930\) −0.627809 −0.0205867
\(931\) 91.3329 2.99331
\(932\) −11.7843 −0.386008
\(933\) 5.10993 0.167292
\(934\) −25.8182 −0.844796
\(935\) −17.6820 −0.578263
\(936\) −1.00000 −0.0326860
\(937\) 6.86973 0.224424 0.112212 0.993684i \(-0.464206\pi\)
0.112212 + 0.993684i \(0.464206\pi\)
\(938\) 9.77489 0.319161
\(939\) −26.4548 −0.863320
\(940\) 1.24656 0.0406583
\(941\) −11.5963 −0.378027 −0.189014 0.981974i \(-0.560529\pi\)
−0.189014 + 0.981974i \(0.560529\pi\)
\(942\) −4.58778 −0.149478
\(943\) 0.961410 0.0313078
\(944\) 4.52096 0.147145
\(945\) −6.86572 −0.223342
\(946\) 8.30833 0.270127
\(947\) 8.74704 0.284241 0.142120 0.989849i \(-0.454608\pi\)
0.142120 + 0.989849i \(0.454608\pi\)
\(948\) −3.70385 −0.120295
\(949\) 6.35844 0.206404
\(950\) −16.4596 −0.534020
\(951\) 6.08045 0.197172
\(952\) 26.2057 0.849331
\(953\) −8.36637 −0.271013 −0.135507 0.990776i \(-0.543266\pi\)
−0.135507 + 0.990776i \(0.543266\pi\)
\(954\) −7.50659 −0.243035
\(955\) −27.8194 −0.900214
\(956\) 8.32974 0.269403
\(957\) −4.68714 −0.151514
\(958\) −4.91205 −0.158701
\(959\) 9.49518 0.306615
\(960\) −1.40985 −0.0455029
\(961\) −30.8017 −0.993603
\(962\) −9.09660 −0.293286
\(963\) −0.00520205 −0.000167634 0
\(964\) −10.9822 −0.353713
\(965\) 24.3948 0.785296
\(966\) 4.91533 0.158148
\(967\) −3.64903 −0.117345 −0.0586725 0.998277i \(-0.518687\pi\)
−0.0586725 + 0.998277i \(0.518687\pi\)
\(968\) −5.56816 −0.178967
\(969\) −29.4038 −0.944586
\(970\) 8.59822 0.276072
\(971\) −26.8560 −0.861850 −0.430925 0.902388i \(-0.641813\pi\)
−0.430925 + 0.902388i \(0.641813\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 51.7939 1.66044
\(974\) −33.5943 −1.07643
\(975\) −3.01231 −0.0964712
\(976\) 6.62115 0.211938
\(977\) −1.16535 −0.0372828 −0.0186414 0.999826i \(-0.505934\pi\)
−0.0186414 + 0.999826i \(0.505934\pi\)
\(978\) 3.09389 0.0989315
\(979\) −5.78592 −0.184919
\(980\) 23.5658 0.752781
\(981\) −12.1170 −0.386866
\(982\) 20.7294 0.661501
\(983\) −36.4004 −1.16099 −0.580496 0.814263i \(-0.697141\pi\)
−0.580496 + 0.814263i \(0.697141\pi\)
\(984\) 0.952506 0.0303648
\(985\) 6.01172 0.191549
\(986\) −10.8222 −0.344651
\(987\) −4.30577 −0.137054
\(988\) −5.46412 −0.173837
\(989\) 3.59817 0.114415
\(990\) −3.28585 −0.104431
\(991\) −41.3784 −1.31443 −0.657214 0.753704i \(-0.728265\pi\)
−0.657214 + 0.753704i \(0.728265\pi\)
\(992\) 0.445300 0.0141383
\(993\) −3.03497 −0.0963119
\(994\) −44.7729 −1.42011
\(995\) −28.4499 −0.901924
\(996\) −14.4610 −0.458213
\(997\) 5.37352 0.170181 0.0850906 0.996373i \(-0.472882\pi\)
0.0850906 + 0.996373i \(0.472882\pi\)
\(998\) 1.53739 0.0486653
\(999\) −9.09660 −0.287804
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.bc.1.11 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.11 15 1.1 even 1 trivial