Properties

Label 8034.2.a.w.1.6
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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 1.6
Root \(3.10048\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.18783 q^{5} -1.00000 q^{6} +0.107097 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.18783 q^{5} -1.00000 q^{6} +0.107097 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.18783 q^{10} +0.456412 q^{11} +1.00000 q^{12} +1.00000 q^{13} -0.107097 q^{14} -1.18783 q^{15} +1.00000 q^{16} -3.46810 q^{17} -1.00000 q^{18} -0.368926 q^{19} -1.18783 q^{20} +0.107097 q^{21} -0.456412 q^{22} +5.15956 q^{23} -1.00000 q^{24} -3.58905 q^{25} -1.00000 q^{26} +1.00000 q^{27} +0.107097 q^{28} -3.44062 q^{29} +1.18783 q^{30} +0.0661943 q^{31} -1.00000 q^{32} +0.456412 q^{33} +3.46810 q^{34} -0.127214 q^{35} +1.00000 q^{36} +4.66735 q^{37} +0.368926 q^{38} +1.00000 q^{39} +1.18783 q^{40} +0.724266 q^{41} -0.107097 q^{42} +3.85350 q^{43} +0.456412 q^{44} -1.18783 q^{45} -5.15956 q^{46} -11.5878 q^{47} +1.00000 q^{48} -6.98853 q^{49} +3.58905 q^{50} -3.46810 q^{51} +1.00000 q^{52} -7.67799 q^{53} -1.00000 q^{54} -0.542142 q^{55} -0.107097 q^{56} -0.368926 q^{57} +3.44062 q^{58} +4.84255 q^{59} -1.18783 q^{60} +6.69692 q^{61} -0.0661943 q^{62} +0.107097 q^{63} +1.00000 q^{64} -1.18783 q^{65} -0.456412 q^{66} +2.46560 q^{67} -3.46810 q^{68} +5.15956 q^{69} +0.127214 q^{70} +6.86345 q^{71} -1.00000 q^{72} -7.04406 q^{73} -4.66735 q^{74} -3.58905 q^{75} -0.368926 q^{76} +0.0488806 q^{77} -1.00000 q^{78} +0.265300 q^{79} -1.18783 q^{80} +1.00000 q^{81} -0.724266 q^{82} +0.324525 q^{83} +0.107097 q^{84} +4.11953 q^{85} -3.85350 q^{86} -3.44062 q^{87} -0.456412 q^{88} -4.77088 q^{89} +1.18783 q^{90} +0.107097 q^{91} +5.15956 q^{92} +0.0661943 q^{93} +11.5878 q^{94} +0.438223 q^{95} -1.00000 q^{96} +3.46515 q^{97} +6.98853 q^{98} +0.456412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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
\(5\) −1.18783 −0.531216 −0.265608 0.964081i \(-0.585573\pi\)
−0.265608 + 0.964081i \(0.585573\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.107097 0.0404790 0.0202395 0.999795i \(-0.493557\pi\)
0.0202395 + 0.999795i \(0.493557\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.18783 0.375626
\(11\) 0.456412 0.137613 0.0688067 0.997630i \(-0.478081\pi\)
0.0688067 + 0.997630i \(0.478081\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −0.107097 −0.0286230
\(15\) −1.18783 −0.306698
\(16\) 1.00000 0.250000
\(17\) −3.46810 −0.841138 −0.420569 0.907261i \(-0.638170\pi\)
−0.420569 + 0.907261i \(0.638170\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.368926 −0.0846375 −0.0423187 0.999104i \(-0.513475\pi\)
−0.0423187 + 0.999104i \(0.513475\pi\)
\(20\) −1.18783 −0.265608
\(21\) 0.107097 0.0233706
\(22\) −0.456412 −0.0973074
\(23\) 5.15956 1.07584 0.537922 0.842995i \(-0.319210\pi\)
0.537922 + 0.842995i \(0.319210\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.58905 −0.717810
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0.107097 0.0202395
\(29\) −3.44062 −0.638907 −0.319453 0.947602i \(-0.603499\pi\)
−0.319453 + 0.947602i \(0.603499\pi\)
\(30\) 1.18783 0.216868
\(31\) 0.0661943 0.0118888 0.00594442 0.999982i \(-0.498108\pi\)
0.00594442 + 0.999982i \(0.498108\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.456412 0.0794511
\(34\) 3.46810 0.594775
\(35\) −0.127214 −0.0215031
\(36\) 1.00000 0.166667
\(37\) 4.66735 0.767308 0.383654 0.923477i \(-0.374666\pi\)
0.383654 + 0.923477i \(0.374666\pi\)
\(38\) 0.368926 0.0598477
\(39\) 1.00000 0.160128
\(40\) 1.18783 0.187813
\(41\) 0.724266 0.113111 0.0565556 0.998399i \(-0.481988\pi\)
0.0565556 + 0.998399i \(0.481988\pi\)
\(42\) −0.107097 −0.0165255
\(43\) 3.85350 0.587653 0.293826 0.955859i \(-0.405071\pi\)
0.293826 + 0.955859i \(0.405071\pi\)
\(44\) 0.456412 0.0688067
\(45\) −1.18783 −0.177072
\(46\) −5.15956 −0.760736
\(47\) −11.5878 −1.69025 −0.845127 0.534565i \(-0.820475\pi\)
−0.845127 + 0.534565i \(0.820475\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.98853 −0.998361
\(50\) 3.58905 0.507568
\(51\) −3.46810 −0.485631
\(52\) 1.00000 0.138675
\(53\) −7.67799 −1.05465 −0.527326 0.849663i \(-0.676806\pi\)
−0.527326 + 0.849663i \(0.676806\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.542142 −0.0731024
\(56\) −0.107097 −0.0143115
\(57\) −0.368926 −0.0488655
\(58\) 3.44062 0.451775
\(59\) 4.84255 0.630447 0.315223 0.949018i \(-0.397921\pi\)
0.315223 + 0.949018i \(0.397921\pi\)
\(60\) −1.18783 −0.153349
\(61\) 6.69692 0.857452 0.428726 0.903435i \(-0.358963\pi\)
0.428726 + 0.903435i \(0.358963\pi\)
\(62\) −0.0661943 −0.00840668
\(63\) 0.107097 0.0134930
\(64\) 1.00000 0.125000
\(65\) −1.18783 −0.147333
\(66\) −0.456412 −0.0561804
\(67\) 2.46560 0.301222 0.150611 0.988593i \(-0.451876\pi\)
0.150611 + 0.988593i \(0.451876\pi\)
\(68\) −3.46810 −0.420569
\(69\) 5.15956 0.621139
\(70\) 0.127214 0.0152050
\(71\) 6.86345 0.814541 0.407271 0.913308i \(-0.366481\pi\)
0.407271 + 0.913308i \(0.366481\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.04406 −0.824445 −0.412222 0.911083i \(-0.635247\pi\)
−0.412222 + 0.911083i \(0.635247\pi\)
\(74\) −4.66735 −0.542569
\(75\) −3.58905 −0.414428
\(76\) −0.368926 −0.0423187
\(77\) 0.0488806 0.00557046
\(78\) −1.00000 −0.113228
\(79\) 0.265300 0.0298486 0.0149243 0.999889i \(-0.495249\pi\)
0.0149243 + 0.999889i \(0.495249\pi\)
\(80\) −1.18783 −0.132804
\(81\) 1.00000 0.111111
\(82\) −0.724266 −0.0799818
\(83\) 0.324525 0.0356213 0.0178106 0.999841i \(-0.494330\pi\)
0.0178106 + 0.999841i \(0.494330\pi\)
\(84\) 0.107097 0.0116853
\(85\) 4.11953 0.446826
\(86\) −3.85350 −0.415533
\(87\) −3.44062 −0.368873
\(88\) −0.456412 −0.0486537
\(89\) −4.77088 −0.505713 −0.252856 0.967504i \(-0.581370\pi\)
−0.252856 + 0.967504i \(0.581370\pi\)
\(90\) 1.18783 0.125209
\(91\) 0.107097 0.0112269
\(92\) 5.15956 0.537922
\(93\) 0.0661943 0.00686403
\(94\) 11.5878 1.19519
\(95\) 0.438223 0.0449608
\(96\) −1.00000 −0.102062
\(97\) 3.46515 0.351833 0.175916 0.984405i \(-0.443711\pi\)
0.175916 + 0.984405i \(0.443711\pi\)
\(98\) 6.98853 0.705948
\(99\) 0.456412 0.0458711
\(100\) −3.58905 −0.358905
\(101\) −9.70265 −0.965450 −0.482725 0.875772i \(-0.660353\pi\)
−0.482725 + 0.875772i \(0.660353\pi\)
\(102\) 3.46810 0.343393
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −0.127214 −0.0124148
\(106\) 7.67799 0.745752
\(107\) −10.1786 −0.984006 −0.492003 0.870593i \(-0.663735\pi\)
−0.492003 + 0.870593i \(0.663735\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.55701 −0.628047 −0.314024 0.949415i \(-0.601677\pi\)
−0.314024 + 0.949415i \(0.601677\pi\)
\(110\) 0.542142 0.0516912
\(111\) 4.66735 0.443006
\(112\) 0.107097 0.0101198
\(113\) −5.75203 −0.541105 −0.270553 0.962705i \(-0.587206\pi\)
−0.270553 + 0.962705i \(0.587206\pi\)
\(114\) 0.368926 0.0345531
\(115\) −6.12871 −0.571505
\(116\) −3.44062 −0.319453
\(117\) 1.00000 0.0924500
\(118\) −4.84255 −0.445793
\(119\) −0.371425 −0.0340485
\(120\) 1.18783 0.108434
\(121\) −10.7917 −0.981063
\(122\) −6.69692 −0.606310
\(123\) 0.724266 0.0653048
\(124\) 0.0661943 0.00594442
\(125\) 10.2024 0.912528
\(126\) −0.107097 −0.00954100
\(127\) 6.20597 0.550690 0.275345 0.961345i \(-0.411208\pi\)
0.275345 + 0.961345i \(0.411208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.85350 0.339281
\(130\) 1.18783 0.104180
\(131\) −4.57249 −0.399500 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(132\) 0.456412 0.0397256
\(133\) −0.0395111 −0.00342605
\(134\) −2.46560 −0.212996
\(135\) −1.18783 −0.102233
\(136\) 3.46810 0.297387
\(137\) 6.01069 0.513528 0.256764 0.966474i \(-0.417344\pi\)
0.256764 + 0.966474i \(0.417344\pi\)
\(138\) −5.15956 −0.439211
\(139\) −4.90383 −0.415937 −0.207969 0.978136i \(-0.566685\pi\)
−0.207969 + 0.978136i \(0.566685\pi\)
\(140\) −0.127214 −0.0107516
\(141\) −11.5878 −0.975869
\(142\) −6.86345 −0.575968
\(143\) 0.456412 0.0381671
\(144\) 1.00000 0.0833333
\(145\) 4.08688 0.339397
\(146\) 7.04406 0.582971
\(147\) −6.98853 −0.576404
\(148\) 4.66735 0.383654
\(149\) −20.4378 −1.67433 −0.837164 0.546952i \(-0.815788\pi\)
−0.837164 + 0.546952i \(0.815788\pi\)
\(150\) 3.58905 0.293045
\(151\) 11.0469 0.898987 0.449494 0.893284i \(-0.351604\pi\)
0.449494 + 0.893284i \(0.351604\pi\)
\(152\) 0.368926 0.0299239
\(153\) −3.46810 −0.280379
\(154\) −0.0488806 −0.00393891
\(155\) −0.0786279 −0.00631554
\(156\) 1.00000 0.0800641
\(157\) −7.83563 −0.625351 −0.312676 0.949860i \(-0.601225\pi\)
−0.312676 + 0.949860i \(0.601225\pi\)
\(158\) −0.265300 −0.0211061
\(159\) −7.67799 −0.608904
\(160\) 1.18783 0.0939066
\(161\) 0.552576 0.0435491
\(162\) −1.00000 −0.0785674
\(163\) 18.6201 1.45844 0.729221 0.684279i \(-0.239883\pi\)
0.729221 + 0.684279i \(0.239883\pi\)
\(164\) 0.724266 0.0565556
\(165\) −0.542142 −0.0422057
\(166\) −0.324525 −0.0251880
\(167\) −5.85397 −0.452994 −0.226497 0.974012i \(-0.572727\pi\)
−0.226497 + 0.974012i \(0.572727\pi\)
\(168\) −0.107097 −0.00826275
\(169\) 1.00000 0.0769231
\(170\) −4.11953 −0.315954
\(171\) −0.368926 −0.0282125
\(172\) 3.85350 0.293826
\(173\) 16.2765 1.23748 0.618740 0.785596i \(-0.287644\pi\)
0.618740 + 0.785596i \(0.287644\pi\)
\(174\) 3.44062 0.260833
\(175\) −0.384378 −0.0290563
\(176\) 0.456412 0.0344033
\(177\) 4.84255 0.363989
\(178\) 4.77088 0.357593
\(179\) −26.4246 −1.97507 −0.987533 0.157415i \(-0.949684\pi\)
−0.987533 + 0.157415i \(0.949684\pi\)
\(180\) −1.18783 −0.0885360
\(181\) 2.44713 0.181894 0.0909468 0.995856i \(-0.471011\pi\)
0.0909468 + 0.995856i \(0.471011\pi\)
\(182\) −0.107097 −0.00793859
\(183\) 6.69692 0.495050
\(184\) −5.15956 −0.380368
\(185\) −5.54404 −0.407606
\(186\) −0.0661943 −0.00485360
\(187\) −1.58288 −0.115752
\(188\) −11.5878 −0.845127
\(189\) 0.107097 0.00779020
\(190\) −0.438223 −0.0317921
\(191\) 9.01356 0.652198 0.326099 0.945336i \(-0.394266\pi\)
0.326099 + 0.945336i \(0.394266\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.1229 −1.01659 −0.508293 0.861184i \(-0.669723\pi\)
−0.508293 + 0.861184i \(0.669723\pi\)
\(194\) −3.46515 −0.248783
\(195\) −1.18783 −0.0850626
\(196\) −6.98853 −0.499181
\(197\) 1.74941 0.124640 0.0623201 0.998056i \(-0.480150\pi\)
0.0623201 + 0.998056i \(0.480150\pi\)
\(198\) −0.456412 −0.0324358
\(199\) 18.2966 1.29701 0.648507 0.761209i \(-0.275394\pi\)
0.648507 + 0.761209i \(0.275394\pi\)
\(200\) 3.58905 0.253784
\(201\) 2.46560 0.173910
\(202\) 9.70265 0.682676
\(203\) −0.368481 −0.0258623
\(204\) −3.46810 −0.242816
\(205\) −0.860308 −0.0600865
\(206\) 1.00000 0.0696733
\(207\) 5.15956 0.358615
\(208\) 1.00000 0.0693375
\(209\) −0.168382 −0.0116473
\(210\) 0.127214 0.00877861
\(211\) 11.2374 0.773617 0.386808 0.922160i \(-0.373578\pi\)
0.386808 + 0.922160i \(0.373578\pi\)
\(212\) −7.67799 −0.527326
\(213\) 6.86345 0.470276
\(214\) 10.1786 0.695797
\(215\) −4.57732 −0.312170
\(216\) −1.00000 −0.0680414
\(217\) 0.00708924 0.000481249 0
\(218\) 6.55701 0.444096
\(219\) −7.04406 −0.475993
\(220\) −0.542142 −0.0365512
\(221\) −3.46810 −0.233290
\(222\) −4.66735 −0.313252
\(223\) 1.37443 0.0920386 0.0460193 0.998941i \(-0.485346\pi\)
0.0460193 + 0.998941i \(0.485346\pi\)
\(224\) −0.107097 −0.00715575
\(225\) −3.58905 −0.239270
\(226\) 5.75203 0.382619
\(227\) −4.63666 −0.307746 −0.153873 0.988091i \(-0.549175\pi\)
−0.153873 + 0.988091i \(0.549175\pi\)
\(228\) −0.368926 −0.0244327
\(229\) −12.1523 −0.803049 −0.401524 0.915848i \(-0.631520\pi\)
−0.401524 + 0.915848i \(0.631520\pi\)
\(230\) 6.12871 0.404115
\(231\) 0.0488806 0.00321611
\(232\) 3.44062 0.225888
\(233\) −14.8168 −0.970682 −0.485341 0.874325i \(-0.661305\pi\)
−0.485341 + 0.874325i \(0.661305\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 13.7644 0.897890
\(236\) 4.84255 0.315223
\(237\) 0.265300 0.0172331
\(238\) 0.371425 0.0240759
\(239\) 7.26585 0.469989 0.234994 0.971997i \(-0.424493\pi\)
0.234994 + 0.971997i \(0.424493\pi\)
\(240\) −1.18783 −0.0766744
\(241\) 0.440354 0.0283657 0.0141829 0.999899i \(-0.495485\pi\)
0.0141829 + 0.999899i \(0.495485\pi\)
\(242\) 10.7917 0.693716
\(243\) 1.00000 0.0641500
\(244\) 6.69692 0.428726
\(245\) 8.30122 0.530345
\(246\) −0.724266 −0.0461775
\(247\) −0.368926 −0.0234742
\(248\) −0.0661943 −0.00420334
\(249\) 0.324525 0.0205660
\(250\) −10.2024 −0.645254
\(251\) −26.3082 −1.66056 −0.830280 0.557346i \(-0.811820\pi\)
−0.830280 + 0.557346i \(0.811820\pi\)
\(252\) 0.107097 0.00674651
\(253\) 2.35489 0.148050
\(254\) −6.20597 −0.389397
\(255\) 4.11953 0.257975
\(256\) 1.00000 0.0625000
\(257\) 21.8953 1.36579 0.682895 0.730517i \(-0.260721\pi\)
0.682895 + 0.730517i \(0.260721\pi\)
\(258\) −3.85350 −0.239908
\(259\) 0.499862 0.0310599
\(260\) −1.18783 −0.0736664
\(261\) −3.44062 −0.212969
\(262\) 4.57249 0.282489
\(263\) −24.7060 −1.52344 −0.761719 0.647908i \(-0.775644\pi\)
−0.761719 + 0.647908i \(0.775644\pi\)
\(264\) −0.456412 −0.0280902
\(265\) 9.12018 0.560248
\(266\) 0.0395111 0.00242258
\(267\) −4.77088 −0.291973
\(268\) 2.46560 0.150611
\(269\) −23.3163 −1.42162 −0.710811 0.703383i \(-0.751672\pi\)
−0.710811 + 0.703383i \(0.751672\pi\)
\(270\) 1.18783 0.0722893
\(271\) 3.22302 0.195784 0.0978922 0.995197i \(-0.468790\pi\)
0.0978922 + 0.995197i \(0.468790\pi\)
\(272\) −3.46810 −0.210285
\(273\) 0.107097 0.00648183
\(274\) −6.01069 −0.363119
\(275\) −1.63808 −0.0987802
\(276\) 5.15956 0.310569
\(277\) −17.0469 −1.02425 −0.512125 0.858911i \(-0.671142\pi\)
−0.512125 + 0.858911i \(0.671142\pi\)
\(278\) 4.90383 0.294112
\(279\) 0.0661943 0.00396295
\(280\) 0.127214 0.00760250
\(281\) 18.5875 1.10884 0.554418 0.832238i \(-0.312941\pi\)
0.554418 + 0.832238i \(0.312941\pi\)
\(282\) 11.5878 0.690044
\(283\) −31.2994 −1.86056 −0.930279 0.366852i \(-0.880436\pi\)
−0.930279 + 0.366852i \(0.880436\pi\)
\(284\) 6.86345 0.407271
\(285\) 0.438223 0.0259581
\(286\) −0.456412 −0.0269882
\(287\) 0.0775670 0.00457864
\(288\) −1.00000 −0.0589256
\(289\) −4.97227 −0.292486
\(290\) −4.08688 −0.239990
\(291\) 3.46515 0.203131
\(292\) −7.04406 −0.412222
\(293\) −9.83799 −0.574742 −0.287371 0.957819i \(-0.592781\pi\)
−0.287371 + 0.957819i \(0.592781\pi\)
\(294\) 6.98853 0.407579
\(295\) −5.75215 −0.334903
\(296\) −4.66735 −0.271284
\(297\) 0.456412 0.0264837
\(298\) 20.4378 1.18393
\(299\) 5.15956 0.298385
\(300\) −3.58905 −0.207214
\(301\) 0.412700 0.0237876
\(302\) −11.0469 −0.635680
\(303\) −9.70265 −0.557403
\(304\) −0.368926 −0.0211594
\(305\) −7.95483 −0.455492
\(306\) 3.46810 0.198258
\(307\) 27.2497 1.55523 0.777613 0.628744i \(-0.216431\pi\)
0.777613 + 0.628744i \(0.216431\pi\)
\(308\) 0.0488806 0.00278523
\(309\) −1.00000 −0.0568880
\(310\) 0.0786279 0.00446576
\(311\) −13.5283 −0.767119 −0.383559 0.923516i \(-0.625302\pi\)
−0.383559 + 0.923516i \(0.625302\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 31.8819 1.80207 0.901035 0.433745i \(-0.142808\pi\)
0.901035 + 0.433745i \(0.142808\pi\)
\(314\) 7.83563 0.442190
\(315\) −0.127214 −0.00716770
\(316\) 0.265300 0.0149243
\(317\) −20.7024 −1.16276 −0.581382 0.813631i \(-0.697488\pi\)
−0.581382 + 0.813631i \(0.697488\pi\)
\(318\) 7.67799 0.430560
\(319\) −1.57034 −0.0879221
\(320\) −1.18783 −0.0664020
\(321\) −10.1786 −0.568116
\(322\) −0.552576 −0.0307939
\(323\) 1.27947 0.0711918
\(324\) 1.00000 0.0555556
\(325\) −3.58905 −0.199085
\(326\) −18.6201 −1.03127
\(327\) −6.55701 −0.362603
\(328\) −0.724266 −0.0399909
\(329\) −1.24102 −0.0684199
\(330\) 0.542142 0.0298439
\(331\) −7.09103 −0.389758 −0.194879 0.980827i \(-0.562431\pi\)
−0.194879 + 0.980827i \(0.562431\pi\)
\(332\) 0.324525 0.0178106
\(333\) 4.66735 0.255769
\(334\) 5.85397 0.320315
\(335\) −2.92873 −0.160014
\(336\) 0.107097 0.00584265
\(337\) −31.5584 −1.71910 −0.859549 0.511054i \(-0.829255\pi\)
−0.859549 + 0.511054i \(0.829255\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −5.75203 −0.312407
\(340\) 4.11953 0.223413
\(341\) 0.0302119 0.00163606
\(342\) 0.368926 0.0199492
\(343\) −1.49814 −0.0808918
\(344\) −3.85350 −0.207767
\(345\) −6.12871 −0.329959
\(346\) −16.2765 −0.875030
\(347\) 22.8790 1.22821 0.614104 0.789225i \(-0.289517\pi\)
0.614104 + 0.789225i \(0.289517\pi\)
\(348\) −3.44062 −0.184436
\(349\) −6.93530 −0.371238 −0.185619 0.982622i \(-0.559429\pi\)
−0.185619 + 0.982622i \(0.559429\pi\)
\(350\) 0.384378 0.0205459
\(351\) 1.00000 0.0533761
\(352\) −0.456412 −0.0243268
\(353\) −26.6699 −1.41950 −0.709748 0.704456i \(-0.751191\pi\)
−0.709748 + 0.704456i \(0.751191\pi\)
\(354\) −4.84255 −0.257379
\(355\) −8.15264 −0.432697
\(356\) −4.77088 −0.252856
\(357\) −0.371425 −0.0196579
\(358\) 26.4246 1.39658
\(359\) −15.3548 −0.810397 −0.405198 0.914229i \(-0.632798\pi\)
−0.405198 + 0.914229i \(0.632798\pi\)
\(360\) 1.18783 0.0626044
\(361\) −18.8639 −0.992836
\(362\) −2.44713 −0.128618
\(363\) −10.7917 −0.566417
\(364\) 0.107097 0.00561343
\(365\) 8.36718 0.437958
\(366\) −6.69692 −0.350053
\(367\) 12.6547 0.660571 0.330285 0.943881i \(-0.392855\pi\)
0.330285 + 0.943881i \(0.392855\pi\)
\(368\) 5.15956 0.268961
\(369\) 0.724266 0.0377038
\(370\) 5.54404 0.288221
\(371\) −0.822293 −0.0426913
\(372\) 0.0661943 0.00343201
\(373\) −5.12658 −0.265444 −0.132722 0.991153i \(-0.542372\pi\)
−0.132722 + 0.991153i \(0.542372\pi\)
\(374\) 1.58288 0.0818489
\(375\) 10.2024 0.526848
\(376\) 11.5878 0.597595
\(377\) −3.44062 −0.177201
\(378\) −0.107097 −0.00550850
\(379\) −15.8405 −0.813674 −0.406837 0.913501i \(-0.633368\pi\)
−0.406837 + 0.913501i \(0.633368\pi\)
\(380\) 0.438223 0.0224804
\(381\) 6.20597 0.317941
\(382\) −9.01356 −0.461174
\(383\) −34.9706 −1.78691 −0.893457 0.449149i \(-0.851727\pi\)
−0.893457 + 0.449149i \(0.851727\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.0580620 −0.00295912
\(386\) 14.1229 0.718834
\(387\) 3.85350 0.195884
\(388\) 3.46515 0.175916
\(389\) −33.1833 −1.68246 −0.841229 0.540678i \(-0.818168\pi\)
−0.841229 + 0.540678i \(0.818168\pi\)
\(390\) 1.18783 0.0601483
\(391\) −17.8939 −0.904933
\(392\) 6.98853 0.352974
\(393\) −4.57249 −0.230652
\(394\) −1.74941 −0.0881339
\(395\) −0.315133 −0.0158560
\(396\) 0.456412 0.0229356
\(397\) −21.0514 −1.05654 −0.528270 0.849077i \(-0.677159\pi\)
−0.528270 + 0.849077i \(0.677159\pi\)
\(398\) −18.2966 −0.917127
\(399\) −0.0395111 −0.00197803
\(400\) −3.58905 −0.179452
\(401\) 20.3736 1.01741 0.508703 0.860942i \(-0.330125\pi\)
0.508703 + 0.860942i \(0.330125\pi\)
\(402\) −2.46560 −0.122973
\(403\) 0.0661943 0.00329737
\(404\) −9.70265 −0.482725
\(405\) −1.18783 −0.0590240
\(406\) 0.368481 0.0182874
\(407\) 2.13024 0.105592
\(408\) 3.46810 0.171697
\(409\) 32.4487 1.60448 0.802242 0.596999i \(-0.203640\pi\)
0.802242 + 0.596999i \(0.203640\pi\)
\(410\) 0.860308 0.0424876
\(411\) 6.01069 0.296485
\(412\) −1.00000 −0.0492665
\(413\) 0.518625 0.0255199
\(414\) −5.15956 −0.253579
\(415\) −0.385482 −0.0189226
\(416\) −1.00000 −0.0490290
\(417\) −4.90383 −0.240141
\(418\) 0.168382 0.00823585
\(419\) −36.4708 −1.78172 −0.890858 0.454281i \(-0.849896\pi\)
−0.890858 + 0.454281i \(0.849896\pi\)
\(420\) −0.127214 −0.00620741
\(421\) 33.3994 1.62779 0.813893 0.581015i \(-0.197344\pi\)
0.813893 + 0.581015i \(0.197344\pi\)
\(422\) −11.2374 −0.547030
\(423\) −11.5878 −0.563418
\(424\) 7.67799 0.372876
\(425\) 12.4472 0.603777
\(426\) −6.86345 −0.332535
\(427\) 0.717223 0.0347088
\(428\) −10.1786 −0.492003
\(429\) 0.456412 0.0220358
\(430\) 4.57732 0.220738
\(431\) 15.1652 0.730481 0.365241 0.930913i \(-0.380987\pi\)
0.365241 + 0.930913i \(0.380987\pi\)
\(432\) 1.00000 0.0481125
\(433\) −1.62897 −0.0782834 −0.0391417 0.999234i \(-0.512462\pi\)
−0.0391417 + 0.999234i \(0.512462\pi\)
\(434\) −0.00708924 −0.000340294 0
\(435\) 4.08688 0.195951
\(436\) −6.55701 −0.314024
\(437\) −1.90350 −0.0910567
\(438\) 7.04406 0.336578
\(439\) 1.42189 0.0678629 0.0339314 0.999424i \(-0.489197\pi\)
0.0339314 + 0.999424i \(0.489197\pi\)
\(440\) 0.542142 0.0258456
\(441\) −6.98853 −0.332787
\(442\) 3.46810 0.164961
\(443\) −11.0750 −0.526190 −0.263095 0.964770i \(-0.584743\pi\)
−0.263095 + 0.964770i \(0.584743\pi\)
\(444\) 4.66735 0.221503
\(445\) 5.66702 0.268642
\(446\) −1.37443 −0.0650811
\(447\) −20.4378 −0.966674
\(448\) 0.107097 0.00505988
\(449\) 40.1535 1.89496 0.947480 0.319816i \(-0.103621\pi\)
0.947480 + 0.319816i \(0.103621\pi\)
\(450\) 3.58905 0.169189
\(451\) 0.330564 0.0155656
\(452\) −5.75203 −0.270553
\(453\) 11.0469 0.519030
\(454\) 4.63666 0.217609
\(455\) −0.127214 −0.00596389
\(456\) 0.368926 0.0172766
\(457\) −7.61902 −0.356403 −0.178201 0.983994i \(-0.557028\pi\)
−0.178201 + 0.983994i \(0.557028\pi\)
\(458\) 12.1523 0.567841
\(459\) −3.46810 −0.161877
\(460\) −6.12871 −0.285753
\(461\) −19.5827 −0.912057 −0.456029 0.889965i \(-0.650729\pi\)
−0.456029 + 0.889965i \(0.650729\pi\)
\(462\) −0.0488806 −0.00227413
\(463\) 16.2709 0.756173 0.378087 0.925770i \(-0.376582\pi\)
0.378087 + 0.925770i \(0.376582\pi\)
\(464\) −3.44062 −0.159727
\(465\) −0.0786279 −0.00364628
\(466\) 14.8168 0.686376
\(467\) 23.5047 1.08767 0.543834 0.839193i \(-0.316972\pi\)
0.543834 + 0.839193i \(0.316972\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0.264060 0.0121932
\(470\) −13.7644 −0.634904
\(471\) −7.83563 −0.361047
\(472\) −4.84255 −0.222897
\(473\) 1.75878 0.0808689
\(474\) −0.265300 −0.0121856
\(475\) 1.32409 0.0607536
\(476\) −0.371425 −0.0170242
\(477\) −7.67799 −0.351551
\(478\) −7.26585 −0.332332
\(479\) 30.9214 1.41283 0.706417 0.707796i \(-0.250310\pi\)
0.706417 + 0.707796i \(0.250310\pi\)
\(480\) 1.18783 0.0542170
\(481\) 4.66735 0.212813
\(482\) −0.440354 −0.0200576
\(483\) 0.552576 0.0251431
\(484\) −10.7917 −0.490531
\(485\) −4.11603 −0.186899
\(486\) −1.00000 −0.0453609
\(487\) −21.0594 −0.954291 −0.477145 0.878824i \(-0.658328\pi\)
−0.477145 + 0.878824i \(0.658328\pi\)
\(488\) −6.69692 −0.303155
\(489\) 18.6201 0.842031
\(490\) −8.30122 −0.375011
\(491\) −11.0866 −0.500333 −0.250166 0.968203i \(-0.580485\pi\)
−0.250166 + 0.968203i \(0.580485\pi\)
\(492\) 0.724266 0.0326524
\(493\) 11.9324 0.537409
\(494\) 0.368926 0.0165988
\(495\) −0.542142 −0.0243675
\(496\) 0.0661943 0.00297221
\(497\) 0.735058 0.0329719
\(498\) −0.324525 −0.0145423
\(499\) −17.8682 −0.799889 −0.399944 0.916539i \(-0.630971\pi\)
−0.399944 + 0.916539i \(0.630971\pi\)
\(500\) 10.2024 0.456264
\(501\) −5.85397 −0.261536
\(502\) 26.3082 1.17419
\(503\) 35.9018 1.60078 0.800391 0.599478i \(-0.204625\pi\)
0.800391 + 0.599478i \(0.204625\pi\)
\(504\) −0.107097 −0.00477050
\(505\) 11.5251 0.512862
\(506\) −2.35489 −0.104687
\(507\) 1.00000 0.0444116
\(508\) 6.20597 0.275345
\(509\) 8.78750 0.389499 0.194750 0.980853i \(-0.437611\pi\)
0.194750 + 0.980853i \(0.437611\pi\)
\(510\) −4.11953 −0.182416
\(511\) −0.754401 −0.0333727
\(512\) −1.00000 −0.0441942
\(513\) −0.368926 −0.0162885
\(514\) −21.8953 −0.965759
\(515\) 1.18783 0.0523422
\(516\) 3.85350 0.169641
\(517\) −5.28881 −0.232602
\(518\) −0.499862 −0.0219627
\(519\) 16.2765 0.714459
\(520\) 1.18783 0.0520900
\(521\) 33.1245 1.45121 0.725606 0.688110i \(-0.241559\pi\)
0.725606 + 0.688110i \(0.241559\pi\)
\(522\) 3.44062 0.150592
\(523\) 3.48372 0.152332 0.0761662 0.997095i \(-0.475732\pi\)
0.0761662 + 0.997095i \(0.475732\pi\)
\(524\) −4.57249 −0.199750
\(525\) −0.384378 −0.0167756
\(526\) 24.7060 1.07723
\(527\) −0.229569 −0.0100002
\(528\) 0.456412 0.0198628
\(529\) 3.62110 0.157439
\(530\) −9.12018 −0.396155
\(531\) 4.84255 0.210149
\(532\) −0.0395111 −0.00171302
\(533\) 0.724266 0.0313714
\(534\) 4.77088 0.206456
\(535\) 12.0905 0.522720
\(536\) −2.46560 −0.106498
\(537\) −26.4246 −1.14030
\(538\) 23.3163 1.00524
\(539\) −3.18965 −0.137388
\(540\) −1.18783 −0.0511163
\(541\) −28.6991 −1.23387 −0.616936 0.787013i \(-0.711626\pi\)
−0.616936 + 0.787013i \(0.711626\pi\)
\(542\) −3.22302 −0.138440
\(543\) 2.44713 0.105016
\(544\) 3.46810 0.148694
\(545\) 7.78864 0.333629
\(546\) −0.107097 −0.00458335
\(547\) 15.6681 0.669920 0.334960 0.942232i \(-0.391277\pi\)
0.334960 + 0.942232i \(0.391277\pi\)
\(548\) 6.01069 0.256764
\(549\) 6.69692 0.285817
\(550\) 1.63808 0.0698482
\(551\) 1.26933 0.0540755
\(552\) −5.15956 −0.219606
\(553\) 0.0284130 0.00120824
\(554\) 17.0469 0.724254
\(555\) −5.54404 −0.235332
\(556\) −4.90383 −0.207969
\(557\) −30.3738 −1.28698 −0.643489 0.765456i \(-0.722514\pi\)
−0.643489 + 0.765456i \(0.722514\pi\)
\(558\) −0.0661943 −0.00280223
\(559\) 3.85350 0.162986
\(560\) −0.127214 −0.00537578
\(561\) −1.58288 −0.0668294
\(562\) −18.5875 −0.784066
\(563\) 2.56468 0.108088 0.0540442 0.998539i \(-0.482789\pi\)
0.0540442 + 0.998539i \(0.482789\pi\)
\(564\) −11.5878 −0.487934
\(565\) 6.83246 0.287444
\(566\) 31.2994 1.31561
\(567\) 0.107097 0.00449767
\(568\) −6.86345 −0.287984
\(569\) −31.4569 −1.31874 −0.659371 0.751818i \(-0.729177\pi\)
−0.659371 + 0.751818i \(0.729177\pi\)
\(570\) −0.438223 −0.0183552
\(571\) 29.6702 1.24166 0.620830 0.783945i \(-0.286796\pi\)
0.620830 + 0.783945i \(0.286796\pi\)
\(572\) 0.456412 0.0190835
\(573\) 9.01356 0.376547
\(574\) −0.0775670 −0.00323759
\(575\) −18.5179 −0.772251
\(576\) 1.00000 0.0416667
\(577\) −14.8406 −0.617821 −0.308911 0.951091i \(-0.599964\pi\)
−0.308911 + 0.951091i \(0.599964\pi\)
\(578\) 4.97227 0.206819
\(579\) −14.1229 −0.586926
\(580\) 4.08688 0.169699
\(581\) 0.0347558 0.00144192
\(582\) −3.46515 −0.143635
\(583\) −3.50433 −0.145134
\(584\) 7.04406 0.291485
\(585\) −1.18783 −0.0491109
\(586\) 9.83799 0.406404
\(587\) 39.7071 1.63889 0.819445 0.573158i \(-0.194282\pi\)
0.819445 + 0.573158i \(0.194282\pi\)
\(588\) −6.98853 −0.288202
\(589\) −0.0244208 −0.00100624
\(590\) 5.75215 0.236812
\(591\) 1.74941 0.0719610
\(592\) 4.66735 0.191827
\(593\) −22.7522 −0.934321 −0.467160 0.884173i \(-0.654723\pi\)
−0.467160 + 0.884173i \(0.654723\pi\)
\(594\) −0.456412 −0.0187268
\(595\) 0.441191 0.0180871
\(596\) −20.4378 −0.837164
\(597\) 18.2966 0.748831
\(598\) −5.15956 −0.210990
\(599\) −37.4816 −1.53145 −0.765727 0.643165i \(-0.777621\pi\)
−0.765727 + 0.643165i \(0.777621\pi\)
\(600\) 3.58905 0.146522
\(601\) −21.9244 −0.894315 −0.447158 0.894455i \(-0.647564\pi\)
−0.447158 + 0.894455i \(0.647564\pi\)
\(602\) −0.412700 −0.0168204
\(603\) 2.46560 0.100407
\(604\) 11.0469 0.449494
\(605\) 12.8187 0.521156
\(606\) 9.70265 0.394143
\(607\) −21.6305 −0.877954 −0.438977 0.898498i \(-0.644659\pi\)
−0.438977 + 0.898498i \(0.644659\pi\)
\(608\) 0.368926 0.0149619
\(609\) −0.368481 −0.0149316
\(610\) 7.95483 0.322082
\(611\) −11.5878 −0.468792
\(612\) −3.46810 −0.140190
\(613\) −16.8235 −0.679496 −0.339748 0.940516i \(-0.610342\pi\)
−0.339748 + 0.940516i \(0.610342\pi\)
\(614\) −27.2497 −1.09971
\(615\) −0.860308 −0.0346910
\(616\) −0.0488806 −0.00196945
\(617\) −20.9447 −0.843204 −0.421602 0.906781i \(-0.638532\pi\)
−0.421602 + 0.906781i \(0.638532\pi\)
\(618\) 1.00000 0.0402259
\(619\) −18.8028 −0.755748 −0.377874 0.925857i \(-0.623345\pi\)
−0.377874 + 0.925857i \(0.623345\pi\)
\(620\) −0.0786279 −0.00315777
\(621\) 5.15956 0.207046
\(622\) 13.5283 0.542435
\(623\) −0.510950 −0.0204708
\(624\) 1.00000 0.0400320
\(625\) 5.82652 0.233061
\(626\) −31.8819 −1.27426
\(627\) −0.168382 −0.00672454
\(628\) −7.83563 −0.312676
\(629\) −16.1869 −0.645412
\(630\) 0.127214 0.00506833
\(631\) −33.1528 −1.31979 −0.659896 0.751357i \(-0.729400\pi\)
−0.659896 + 0.751357i \(0.729400\pi\)
\(632\) −0.265300 −0.0105531
\(633\) 11.2374 0.446648
\(634\) 20.7024 0.822198
\(635\) −7.37166 −0.292535
\(636\) −7.67799 −0.304452
\(637\) −6.98853 −0.276896
\(638\) 1.57034 0.0621703
\(639\) 6.86345 0.271514
\(640\) 1.18783 0.0469533
\(641\) 33.1896 1.31091 0.655456 0.755233i \(-0.272476\pi\)
0.655456 + 0.755233i \(0.272476\pi\)
\(642\) 10.1786 0.401719
\(643\) −23.2804 −0.918089 −0.459045 0.888413i \(-0.651808\pi\)
−0.459045 + 0.888413i \(0.651808\pi\)
\(644\) 0.552576 0.0217746
\(645\) −4.57732 −0.180232
\(646\) −1.27947 −0.0503402
\(647\) 5.02505 0.197555 0.0987776 0.995110i \(-0.468507\pi\)
0.0987776 + 0.995110i \(0.468507\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 2.21020 0.0867579
\(650\) 3.58905 0.140774
\(651\) 0.00708924 0.000277849 0
\(652\) 18.6201 0.729221
\(653\) 47.7700 1.86938 0.934692 0.355460i \(-0.115676\pi\)
0.934692 + 0.355460i \(0.115676\pi\)
\(654\) 6.55701 0.256399
\(655\) 5.43136 0.212221
\(656\) 0.724266 0.0282778
\(657\) −7.04406 −0.274815
\(658\) 1.24102 0.0483802
\(659\) −43.9098 −1.71048 −0.855241 0.518231i \(-0.826591\pi\)
−0.855241 + 0.518231i \(0.826591\pi\)
\(660\) −0.542142 −0.0211028
\(661\) 21.7286 0.845145 0.422572 0.906329i \(-0.361127\pi\)
0.422572 + 0.906329i \(0.361127\pi\)
\(662\) 7.09103 0.275601
\(663\) −3.46810 −0.134690
\(664\) −0.324525 −0.0125940
\(665\) 0.0469326 0.00181997
\(666\) −4.66735 −0.180856
\(667\) −17.7521 −0.687363
\(668\) −5.85397 −0.226497
\(669\) 1.37443 0.0531385
\(670\) 2.92873 0.113147
\(671\) 3.05655 0.117997
\(672\) −0.107097 −0.00413138
\(673\) −48.7646 −1.87974 −0.939869 0.341536i \(-0.889053\pi\)
−0.939869 + 0.341536i \(0.889053\pi\)
\(674\) 31.5584 1.21559
\(675\) −3.58905 −0.138143
\(676\) 1.00000 0.0384615
\(677\) −27.9698 −1.07497 −0.537483 0.843275i \(-0.680625\pi\)
−0.537483 + 0.843275i \(0.680625\pi\)
\(678\) 5.75203 0.220905
\(679\) 0.371109 0.0142419
\(680\) −4.11953 −0.157977
\(681\) −4.63666 −0.177677
\(682\) −0.0302119 −0.00115687
\(683\) −14.1838 −0.542728 −0.271364 0.962477i \(-0.587475\pi\)
−0.271364 + 0.962477i \(0.587475\pi\)
\(684\) −0.368926 −0.0141062
\(685\) −7.13970 −0.272794
\(686\) 1.49814 0.0571991
\(687\) −12.1523 −0.463641
\(688\) 3.85350 0.146913
\(689\) −7.67799 −0.292508
\(690\) 6.12871 0.233316
\(691\) 8.76863 0.333574 0.166787 0.985993i \(-0.446661\pi\)
0.166787 + 0.985993i \(0.446661\pi\)
\(692\) 16.2765 0.618740
\(693\) 0.0488806 0.00185682
\(694\) −22.8790 −0.868475
\(695\) 5.82494 0.220952
\(696\) 3.44062 0.130416
\(697\) −2.51183 −0.0951422
\(698\) 6.93530 0.262505
\(699\) −14.8168 −0.560424
\(700\) −0.384378 −0.0145281
\(701\) −32.7726 −1.23780 −0.618901 0.785469i \(-0.712422\pi\)
−0.618901 + 0.785469i \(0.712422\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −1.72191 −0.0649430
\(704\) 0.456412 0.0172017
\(705\) 13.7644 0.518397
\(706\) 26.6699 1.00373
\(707\) −1.03913 −0.0390805
\(708\) 4.84255 0.181994
\(709\) 25.5291 0.958764 0.479382 0.877606i \(-0.340861\pi\)
0.479382 + 0.877606i \(0.340861\pi\)
\(710\) 8.15264 0.305963
\(711\) 0.265300 0.00994953
\(712\) 4.77088 0.178796
\(713\) 0.341534 0.0127905
\(714\) 0.371425 0.0139002
\(715\) −0.542142 −0.0202750
\(716\) −26.4246 −0.987533
\(717\) 7.26585 0.271348
\(718\) 15.3548 0.573037
\(719\) −3.14371 −0.117240 −0.0586202 0.998280i \(-0.518670\pi\)
−0.0586202 + 0.998280i \(0.518670\pi\)
\(720\) −1.18783 −0.0442680
\(721\) −0.107097 −0.00398852
\(722\) 18.8639 0.702041
\(723\) 0.440354 0.0163770
\(724\) 2.44713 0.0909468
\(725\) 12.3485 0.458613
\(726\) 10.7917 0.400517
\(727\) −39.5438 −1.46660 −0.733300 0.679905i \(-0.762021\pi\)
−0.733300 + 0.679905i \(0.762021\pi\)
\(728\) −0.107097 −0.00396930
\(729\) 1.00000 0.0370370
\(730\) −8.36718 −0.309683
\(731\) −13.3643 −0.494297
\(732\) 6.69692 0.247525
\(733\) 52.9505 1.95577 0.977885 0.209141i \(-0.0670669\pi\)
0.977885 + 0.209141i \(0.0670669\pi\)
\(734\) −12.6547 −0.467094
\(735\) 8.30122 0.306195
\(736\) −5.15956 −0.190184
\(737\) 1.12533 0.0414521
\(738\) −0.724266 −0.0266606
\(739\) 44.1961 1.62578 0.812891 0.582416i \(-0.197893\pi\)
0.812891 + 0.582416i \(0.197893\pi\)
\(740\) −5.54404 −0.203803
\(741\) −0.368926 −0.0135528
\(742\) 0.822293 0.0301873
\(743\) −12.6310 −0.463388 −0.231694 0.972789i \(-0.574427\pi\)
−0.231694 + 0.972789i \(0.574427\pi\)
\(744\) −0.0661943 −0.00242680
\(745\) 24.2767 0.889429
\(746\) 5.12658 0.187697
\(747\) 0.324525 0.0118738
\(748\) −1.58288 −0.0578759
\(749\) −1.09011 −0.0398316
\(750\) −10.2024 −0.372538
\(751\) 10.2830 0.375232 0.187616 0.982242i \(-0.439924\pi\)
0.187616 + 0.982242i \(0.439924\pi\)
\(752\) −11.5878 −0.422564
\(753\) −26.3082 −0.958725
\(754\) 3.44062 0.125300
\(755\) −13.1219 −0.477556
\(756\) 0.107097 0.00389510
\(757\) 25.5769 0.929609 0.464805 0.885413i \(-0.346125\pi\)
0.464805 + 0.885413i \(0.346125\pi\)
\(758\) 15.8405 0.575354
\(759\) 2.35489 0.0854770
\(760\) −0.438223 −0.0158960
\(761\) 38.5641 1.39795 0.698974 0.715147i \(-0.253640\pi\)
0.698974 + 0.715147i \(0.253640\pi\)
\(762\) −6.20597 −0.224818
\(763\) −0.702239 −0.0254228
\(764\) 9.01356 0.326099
\(765\) 4.11953 0.148942
\(766\) 34.9706 1.26354
\(767\) 4.84255 0.174854
\(768\) 1.00000 0.0360844
\(769\) −7.99743 −0.288395 −0.144197 0.989549i \(-0.546060\pi\)
−0.144197 + 0.989549i \(0.546060\pi\)
\(770\) 0.0580620 0.00209241
\(771\) 21.8953 0.788539
\(772\) −14.1229 −0.508293
\(773\) 35.3627 1.27191 0.635953 0.771727i \(-0.280607\pi\)
0.635953 + 0.771727i \(0.280607\pi\)
\(774\) −3.85350 −0.138511
\(775\) −0.237575 −0.00853393
\(776\) −3.46515 −0.124392
\(777\) 0.499862 0.0179324
\(778\) 33.1833 1.18968
\(779\) −0.267201 −0.00957346
\(780\) −1.18783 −0.0425313
\(781\) 3.13256 0.112092
\(782\) 17.8939 0.639884
\(783\) −3.44062 −0.122958
\(784\) −6.98853 −0.249590
\(785\) 9.30743 0.332196
\(786\) 4.57249 0.163095
\(787\) 54.0602 1.92704 0.963518 0.267642i \(-0.0862445\pi\)
0.963518 + 0.267642i \(0.0862445\pi\)
\(788\) 1.74941 0.0623201
\(789\) −24.7060 −0.879557
\(790\) 0.315133 0.0112119
\(791\) −0.616028 −0.0219034
\(792\) −0.456412 −0.0162179
\(793\) 6.69692 0.237814
\(794\) 21.0514 0.747086
\(795\) 9.12018 0.323459
\(796\) 18.2966 0.648507
\(797\) 6.52244 0.231037 0.115518 0.993305i \(-0.463147\pi\)
0.115518 + 0.993305i \(0.463147\pi\)
\(798\) 0.0395111 0.00139868
\(799\) 40.1877 1.42174
\(800\) 3.58905 0.126892
\(801\) −4.77088 −0.168571
\(802\) −20.3736 −0.719415
\(803\) −3.21499 −0.113455
\(804\) 2.46560 0.0869552
\(805\) −0.656369 −0.0231340
\(806\) −0.0661943 −0.00233159
\(807\) −23.3163 −0.820773
\(808\) 9.70265 0.341338
\(809\) 7.00356 0.246232 0.123116 0.992392i \(-0.460711\pi\)
0.123116 + 0.992392i \(0.460711\pi\)
\(810\) 1.18783 0.0417363
\(811\) 17.6302 0.619080 0.309540 0.950886i \(-0.399825\pi\)
0.309540 + 0.950886i \(0.399825\pi\)
\(812\) −0.368481 −0.0129312
\(813\) 3.22302 0.113036
\(814\) −2.13024 −0.0746647
\(815\) −22.1176 −0.774747
\(816\) −3.46810 −0.121408
\(817\) −1.42166 −0.0497374
\(818\) −32.4487 −1.13454
\(819\) 0.107097 0.00374229
\(820\) −0.860308 −0.0300433
\(821\) −3.77869 −0.131877 −0.0659386 0.997824i \(-0.521004\pi\)
−0.0659386 + 0.997824i \(0.521004\pi\)
\(822\) −6.01069 −0.209647
\(823\) −1.72699 −0.0601990 −0.0300995 0.999547i \(-0.509582\pi\)
−0.0300995 + 0.999547i \(0.509582\pi\)
\(824\) 1.00000 0.0348367
\(825\) −1.63808 −0.0570308
\(826\) −0.518625 −0.0180453
\(827\) −12.1844 −0.423692 −0.211846 0.977303i \(-0.567948\pi\)
−0.211846 + 0.977303i \(0.567948\pi\)
\(828\) 5.15956 0.179307
\(829\) 21.8809 0.759955 0.379977 0.924996i \(-0.375932\pi\)
0.379977 + 0.924996i \(0.375932\pi\)
\(830\) 0.385482 0.0133803
\(831\) −17.0469 −0.591351
\(832\) 1.00000 0.0346688
\(833\) 24.2369 0.839760
\(834\) 4.90383 0.169806
\(835\) 6.95355 0.240638
\(836\) −0.168382 −0.00582363
\(837\) 0.0661943 0.00228801
\(838\) 36.4708 1.25986
\(839\) −38.0183 −1.31254 −0.656269 0.754527i \(-0.727866\pi\)
−0.656269 + 0.754527i \(0.727866\pi\)
\(840\) 0.127214 0.00438930
\(841\) −17.1622 −0.591798
\(842\) −33.3994 −1.15102
\(843\) 18.5875 0.640187
\(844\) 11.2374 0.386808
\(845\) −1.18783 −0.0408628
\(846\) 11.5878 0.398397
\(847\) −1.15576 −0.0397125
\(848\) −7.67799 −0.263663
\(849\) −31.2994 −1.07419
\(850\) −12.4472 −0.426935
\(851\) 24.0815 0.825504
\(852\) 6.86345 0.235138
\(853\) −27.6896 −0.948075 −0.474037 0.880505i \(-0.657204\pi\)
−0.474037 + 0.880505i \(0.657204\pi\)
\(854\) −0.717223 −0.0245429
\(855\) 0.438223 0.0149869
\(856\) 10.1786 0.347899
\(857\) −18.4397 −0.629889 −0.314945 0.949110i \(-0.601986\pi\)
−0.314945 + 0.949110i \(0.601986\pi\)
\(858\) −0.456412 −0.0155816
\(859\) −39.0274 −1.33160 −0.665800 0.746131i \(-0.731909\pi\)
−0.665800 + 0.746131i \(0.731909\pi\)
\(860\) −4.57732 −0.156085
\(861\) 0.0775670 0.00264348
\(862\) −15.1652 −0.516528
\(863\) 30.3765 1.03403 0.517013 0.855977i \(-0.327044\pi\)
0.517013 + 0.855977i \(0.327044\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −19.3338 −0.657368
\(866\) 1.62897 0.0553547
\(867\) −4.97227 −0.168867
\(868\) 0.00708924 0.000240625 0
\(869\) 0.121086 0.00410757
\(870\) −4.08688 −0.138558
\(871\) 2.46560 0.0835438
\(872\) 6.55701 0.222048
\(873\) 3.46515 0.117278
\(874\) 1.90350 0.0643868
\(875\) 1.09265 0.0369382
\(876\) −7.04406 −0.237997
\(877\) −30.7986 −1.04000 −0.519998 0.854168i \(-0.674067\pi\)
−0.519998 + 0.854168i \(0.674067\pi\)
\(878\) −1.42189 −0.0479863
\(879\) −9.83799 −0.331827
\(880\) −0.542142 −0.0182756
\(881\) −8.55855 −0.288345 −0.144172 0.989553i \(-0.546052\pi\)
−0.144172 + 0.989553i \(0.546052\pi\)
\(882\) 6.98853 0.235316
\(883\) −27.6244 −0.929635 −0.464818 0.885406i \(-0.653880\pi\)
−0.464818 + 0.885406i \(0.653880\pi\)
\(884\) −3.46810 −0.116645
\(885\) −5.75215 −0.193356
\(886\) 11.0750 0.372072
\(887\) 11.4518 0.384515 0.192258 0.981344i \(-0.438419\pi\)
0.192258 + 0.981344i \(0.438419\pi\)
\(888\) −4.66735 −0.156626
\(889\) 0.664643 0.0222914
\(890\) −5.66702 −0.189959
\(891\) 0.456412 0.0152904
\(892\) 1.37443 0.0460193
\(893\) 4.27504 0.143059
\(894\) 20.4378 0.683541
\(895\) 31.3880 1.04919
\(896\) −0.107097 −0.00357788
\(897\) 5.15956 0.172273
\(898\) −40.1535 −1.33994
\(899\) −0.227749 −0.00759586
\(900\) −3.58905 −0.119635
\(901\) 26.6280 0.887109
\(902\) −0.330564 −0.0110066
\(903\) 0.412700 0.0137338
\(904\) 5.75203 0.191310
\(905\) −2.90678 −0.0966247
\(906\) −11.0469 −0.367010
\(907\) 30.9403 1.02736 0.513678 0.857983i \(-0.328283\pi\)
0.513678 + 0.857983i \(0.328283\pi\)
\(908\) −4.63666 −0.153873
\(909\) −9.70265 −0.321817
\(910\) 0.127214 0.00421711
\(911\) −18.7957 −0.622731 −0.311365 0.950290i \(-0.600786\pi\)
−0.311365 + 0.950290i \(0.600786\pi\)
\(912\) −0.368926 −0.0122164
\(913\) 0.148117 0.00490196
\(914\) 7.61902 0.252015
\(915\) −7.95483 −0.262978
\(916\) −12.1523 −0.401524
\(917\) −0.489702 −0.0161714
\(918\) 3.46810 0.114464
\(919\) −45.6953 −1.50735 −0.753675 0.657248i \(-0.771721\pi\)
−0.753675 + 0.657248i \(0.771721\pi\)
\(920\) 6.12871 0.202058
\(921\) 27.2497 0.897910
\(922\) 19.5827 0.644922
\(923\) 6.86345 0.225913
\(924\) 0.0488806 0.00160805
\(925\) −16.7514 −0.550781
\(926\) −16.2709 −0.534695
\(927\) −1.00000 −0.0328443
\(928\) 3.44062 0.112944
\(929\) −8.51591 −0.279398 −0.139699 0.990194i \(-0.544613\pi\)
−0.139699 + 0.990194i \(0.544613\pi\)
\(930\) 0.0786279 0.00257831
\(931\) 2.57825 0.0844988
\(932\) −14.8168 −0.485341
\(933\) −13.5283 −0.442896
\(934\) −23.5047 −0.769098
\(935\) 1.88020 0.0614892
\(936\) −1.00000 −0.0326860
\(937\) −22.2829 −0.727951 −0.363976 0.931409i \(-0.618581\pi\)
−0.363976 + 0.931409i \(0.618581\pi\)
\(938\) −0.264060 −0.00862187
\(939\) 31.8819 1.04043
\(940\) 13.7644 0.448945
\(941\) −58.3034 −1.90064 −0.950318 0.311281i \(-0.899242\pi\)
−0.950318 + 0.311281i \(0.899242\pi\)
\(942\) 7.83563 0.255299
\(943\) 3.73690 0.121690
\(944\) 4.84255 0.157612
\(945\) −0.127214 −0.00413827
\(946\) −1.75878 −0.0571829
\(947\) 54.0157 1.75527 0.877637 0.479325i \(-0.159118\pi\)
0.877637 + 0.479325i \(0.159118\pi\)
\(948\) 0.265300 0.00861655
\(949\) −7.04406 −0.228660
\(950\) −1.32409 −0.0429593
\(951\) −20.7024 −0.671322
\(952\) 0.371425 0.0120380
\(953\) −52.8072 −1.71059 −0.855296 0.518139i \(-0.826625\pi\)
−0.855296 + 0.518139i \(0.826625\pi\)
\(954\) 7.67799 0.248584
\(955\) −10.7066 −0.346458
\(956\) 7.26585 0.234994
\(957\) −1.57034 −0.0507618
\(958\) −30.9214 −0.999025
\(959\) 0.643730 0.0207871
\(960\) −1.18783 −0.0383372
\(961\) −30.9956 −0.999859
\(962\) −4.66735 −0.150482
\(963\) −10.1786 −0.328002
\(964\) 0.440354 0.0141829
\(965\) 16.7756 0.540026
\(966\) −0.552576 −0.0177789
\(967\) 51.2680 1.64867 0.824333 0.566105i \(-0.191550\pi\)
0.824333 + 0.566105i \(0.191550\pi\)
\(968\) 10.7917 0.346858
\(969\) 1.27947 0.0411026
\(970\) 4.11603 0.132158
\(971\) −3.19475 −0.102524 −0.0512621 0.998685i \(-0.516324\pi\)
−0.0512621 + 0.998685i \(0.516324\pi\)
\(972\) 1.00000 0.0320750
\(973\) −0.525188 −0.0168367
\(974\) 21.0594 0.674785
\(975\) −3.58905 −0.114942
\(976\) 6.69692 0.214363
\(977\) 36.6736 1.17329 0.586646 0.809843i \(-0.300448\pi\)
0.586646 + 0.809843i \(0.300448\pi\)
\(978\) −18.6201 −0.595406
\(979\) −2.17749 −0.0695928
\(980\) 8.30122 0.265173
\(981\) −6.55701 −0.209349
\(982\) 11.0866 0.353789
\(983\) −49.2075 −1.56948 −0.784739 0.619827i \(-0.787203\pi\)
−0.784739 + 0.619827i \(0.787203\pi\)
\(984\) −0.724266 −0.0230887
\(985\) −2.07801 −0.0662108
\(986\) −11.9324 −0.380005
\(987\) −1.24102 −0.0395022
\(988\) −0.368926 −0.0117371
\(989\) 19.8824 0.632222
\(990\) 0.542142 0.0172304
\(991\) 49.6099 1.57591 0.787956 0.615732i \(-0.211140\pi\)
0.787956 + 0.615732i \(0.211140\pi\)
\(992\) −0.0661943 −0.00210167
\(993\) −7.09103 −0.225027
\(994\) −0.735058 −0.0233146
\(995\) −21.7334 −0.688994
\(996\) 0.324525 0.0102830
\(997\) 14.7221 0.466252 0.233126 0.972446i \(-0.425104\pi\)
0.233126 + 0.972446i \(0.425104\pi\)
\(998\) 17.8682 0.565607
\(999\) 4.66735 0.147669
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.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.6 12 1.1 even 1 trivial