Properties

Label 8034.2.a.w.1.2
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.829811\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.54947 q^{5} -1.00000 q^{6} +1.31288 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.54947 q^{5} -1.00000 q^{6} +1.31288 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.54947 q^{10} +3.27526 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.31288 q^{14} -3.54947 q^{15} +1.00000 q^{16} +2.46483 q^{17} -1.00000 q^{18} +1.21778 q^{19} -3.54947 q^{20} +1.31288 q^{21} -3.27526 q^{22} -4.22010 q^{23} -1.00000 q^{24} +7.59874 q^{25} -1.00000 q^{26} +1.00000 q^{27} +1.31288 q^{28} -7.93838 q^{29} +3.54947 q^{30} -3.85461 q^{31} -1.00000 q^{32} +3.27526 q^{33} -2.46483 q^{34} -4.66005 q^{35} +1.00000 q^{36} -5.26848 q^{37} -1.21778 q^{38} +1.00000 q^{39} +3.54947 q^{40} +9.66923 q^{41} -1.31288 q^{42} -6.54065 q^{43} +3.27526 q^{44} -3.54947 q^{45} +4.22010 q^{46} -8.05401 q^{47} +1.00000 q^{48} -5.27633 q^{49} -7.59874 q^{50} +2.46483 q^{51} +1.00000 q^{52} +0.0283729 q^{53} -1.00000 q^{54} -11.6254 q^{55} -1.31288 q^{56} +1.21778 q^{57} +7.93838 q^{58} -11.0105 q^{59} -3.54947 q^{60} +2.25319 q^{61} +3.85461 q^{62} +1.31288 q^{63} +1.00000 q^{64} -3.54947 q^{65} -3.27526 q^{66} +14.9179 q^{67} +2.46483 q^{68} -4.22010 q^{69} +4.66005 q^{70} -0.0351683 q^{71} -1.00000 q^{72} +7.18909 q^{73} +5.26848 q^{74} +7.59874 q^{75} +1.21778 q^{76} +4.30004 q^{77} -1.00000 q^{78} +4.16443 q^{79} -3.54947 q^{80} +1.00000 q^{81} -9.66923 q^{82} -8.86586 q^{83} +1.31288 q^{84} -8.74884 q^{85} +6.54065 q^{86} -7.93838 q^{87} -3.27526 q^{88} -7.73254 q^{89} +3.54947 q^{90} +1.31288 q^{91} -4.22010 q^{92} -3.85461 q^{93} +8.05401 q^{94} -4.32247 q^{95} -1.00000 q^{96} +9.04389 q^{97} +5.27633 q^{98} +3.27526 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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
\(5\) −3.54947 −1.58737 −0.793686 0.608328i \(-0.791841\pi\)
−0.793686 + 0.608328i \(0.791841\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.31288 0.496224 0.248112 0.968731i \(-0.420190\pi\)
0.248112 + 0.968731i \(0.420190\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.54947 1.12244
\(11\) 3.27526 0.987528 0.493764 0.869596i \(-0.335621\pi\)
0.493764 + 0.869596i \(0.335621\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −1.31288 −0.350883
\(15\) −3.54947 −0.916469
\(16\) 1.00000 0.250000
\(17\) 2.46483 0.597809 0.298904 0.954283i \(-0.403379\pi\)
0.298904 + 0.954283i \(0.403379\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.21778 0.279378 0.139689 0.990195i \(-0.455390\pi\)
0.139689 + 0.990195i \(0.455390\pi\)
\(20\) −3.54947 −0.793686
\(21\) 1.31288 0.286495
\(22\) −3.27526 −0.698288
\(23\) −4.22010 −0.879951 −0.439975 0.898010i \(-0.645013\pi\)
−0.439975 + 0.898010i \(0.645013\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.59874 1.51975
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 1.31288 0.248112
\(29\) −7.93838 −1.47412 −0.737060 0.675827i \(-0.763787\pi\)
−0.737060 + 0.675827i \(0.763787\pi\)
\(30\) 3.54947 0.648042
\(31\) −3.85461 −0.692308 −0.346154 0.938178i \(-0.612512\pi\)
−0.346154 + 0.938178i \(0.612512\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.27526 0.570150
\(34\) −2.46483 −0.422715
\(35\) −4.66005 −0.787692
\(36\) 1.00000 0.166667
\(37\) −5.26848 −0.866133 −0.433066 0.901362i \(-0.642568\pi\)
−0.433066 + 0.901362i \(0.642568\pi\)
\(38\) −1.21778 −0.197550
\(39\) 1.00000 0.160128
\(40\) 3.54947 0.561221
\(41\) 9.66923 1.51008 0.755040 0.655679i \(-0.227617\pi\)
0.755040 + 0.655679i \(0.227617\pi\)
\(42\) −1.31288 −0.202583
\(43\) −6.54065 −0.997440 −0.498720 0.866763i \(-0.666196\pi\)
−0.498720 + 0.866763i \(0.666196\pi\)
\(44\) 3.27526 0.493764
\(45\) −3.54947 −0.529124
\(46\) 4.22010 0.622219
\(47\) −8.05401 −1.17480 −0.587399 0.809298i \(-0.699848\pi\)
−0.587399 + 0.809298i \(0.699848\pi\)
\(48\) 1.00000 0.144338
\(49\) −5.27633 −0.753762
\(50\) −7.59874 −1.07462
\(51\) 2.46483 0.345145
\(52\) 1.00000 0.138675
\(53\) 0.0283729 0.00389732 0.00194866 0.999998i \(-0.499380\pi\)
0.00194866 + 0.999998i \(0.499380\pi\)
\(54\) −1.00000 −0.136083
\(55\) −11.6254 −1.56757
\(56\) −1.31288 −0.175442
\(57\) 1.21778 0.161299
\(58\) 7.93838 1.04236
\(59\) −11.0105 −1.43345 −0.716725 0.697356i \(-0.754360\pi\)
−0.716725 + 0.697356i \(0.754360\pi\)
\(60\) −3.54947 −0.458235
\(61\) 2.25319 0.288491 0.144245 0.989542i \(-0.453925\pi\)
0.144245 + 0.989542i \(0.453925\pi\)
\(62\) 3.85461 0.489535
\(63\) 1.31288 0.165408
\(64\) 1.00000 0.125000
\(65\) −3.54947 −0.440258
\(66\) −3.27526 −0.403157
\(67\) 14.9179 1.82251 0.911253 0.411847i \(-0.135116\pi\)
0.911253 + 0.411847i \(0.135116\pi\)
\(68\) 2.46483 0.298904
\(69\) −4.22010 −0.508040
\(70\) 4.66005 0.556982
\(71\) −0.0351683 −0.00417371 −0.00208685 0.999998i \(-0.500664\pi\)
−0.00208685 + 0.999998i \(0.500664\pi\)
\(72\) −1.00000 −0.117851
\(73\) 7.18909 0.841419 0.420710 0.907195i \(-0.361781\pi\)
0.420710 + 0.907195i \(0.361781\pi\)
\(74\) 5.26848 0.612448
\(75\) 7.59874 0.877427
\(76\) 1.21778 0.139689
\(77\) 4.30004 0.490035
\(78\) −1.00000 −0.113228
\(79\) 4.16443 0.468535 0.234268 0.972172i \(-0.424731\pi\)
0.234268 + 0.972172i \(0.424731\pi\)
\(80\) −3.54947 −0.396843
\(81\) 1.00000 0.111111
\(82\) −9.66923 −1.06779
\(83\) −8.86586 −0.973155 −0.486577 0.873638i \(-0.661755\pi\)
−0.486577 + 0.873638i \(0.661755\pi\)
\(84\) 1.31288 0.143247
\(85\) −8.74884 −0.948945
\(86\) 6.54065 0.705297
\(87\) −7.93838 −0.851084
\(88\) −3.27526 −0.349144
\(89\) −7.73254 −0.819648 −0.409824 0.912165i \(-0.634410\pi\)
−0.409824 + 0.912165i \(0.634410\pi\)
\(90\) 3.54947 0.374147
\(91\) 1.31288 0.137628
\(92\) −4.22010 −0.439975
\(93\) −3.85461 −0.399704
\(94\) 8.05401 0.830707
\(95\) −4.32247 −0.443476
\(96\) −1.00000 −0.102062
\(97\) 9.04389 0.918268 0.459134 0.888367i \(-0.348160\pi\)
0.459134 + 0.888367i \(0.348160\pi\)
\(98\) 5.27633 0.532990
\(99\) 3.27526 0.329176
\(100\) 7.59874 0.759874
\(101\) 6.18777 0.615707 0.307853 0.951434i \(-0.400389\pi\)
0.307853 + 0.951434i \(0.400389\pi\)
\(102\) −2.46483 −0.244054
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −4.66005 −0.454774
\(106\) −0.0283729 −0.00275582
\(107\) −6.37624 −0.616414 −0.308207 0.951319i \(-0.599729\pi\)
−0.308207 + 0.951319i \(0.599729\pi\)
\(108\) 1.00000 0.0962250
\(109\) 17.7480 1.69995 0.849975 0.526823i \(-0.176617\pi\)
0.849975 + 0.526823i \(0.176617\pi\)
\(110\) 11.6254 1.10844
\(111\) −5.26848 −0.500062
\(112\) 1.31288 0.124056
\(113\) 0.866788 0.0815406 0.0407703 0.999169i \(-0.487019\pi\)
0.0407703 + 0.999169i \(0.487019\pi\)
\(114\) −1.21778 −0.114055
\(115\) 14.9791 1.39681
\(116\) −7.93838 −0.737060
\(117\) 1.00000 0.0924500
\(118\) 11.0105 1.01360
\(119\) 3.23604 0.296647
\(120\) 3.54947 0.324021
\(121\) −0.272670 −0.0247882
\(122\) −2.25319 −0.203994
\(123\) 9.66923 0.871845
\(124\) −3.85461 −0.346154
\(125\) −9.22417 −0.825035
\(126\) −1.31288 −0.116961
\(127\) −11.5081 −1.02118 −0.510590 0.859824i \(-0.670573\pi\)
−0.510590 + 0.859824i \(0.670573\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.54065 −0.575872
\(130\) 3.54947 0.311309
\(131\) −6.55057 −0.572326 −0.286163 0.958181i \(-0.592380\pi\)
−0.286163 + 0.958181i \(0.592380\pi\)
\(132\) 3.27526 0.285075
\(133\) 1.59880 0.138634
\(134\) −14.9179 −1.28871
\(135\) −3.54947 −0.305490
\(136\) −2.46483 −0.211357
\(137\) −4.06930 −0.347664 −0.173832 0.984775i \(-0.555615\pi\)
−0.173832 + 0.984775i \(0.555615\pi\)
\(138\) 4.22010 0.359238
\(139\) 2.50932 0.212838 0.106419 0.994321i \(-0.466062\pi\)
0.106419 + 0.994321i \(0.466062\pi\)
\(140\) −4.66005 −0.393846
\(141\) −8.05401 −0.678270
\(142\) 0.0351683 0.00295126
\(143\) 3.27526 0.273891
\(144\) 1.00000 0.0833333
\(145\) 28.1771 2.33998
\(146\) −7.18909 −0.594973
\(147\) −5.27633 −0.435185
\(148\) −5.26848 −0.433066
\(149\) −3.28234 −0.268900 −0.134450 0.990920i \(-0.542927\pi\)
−0.134450 + 0.990920i \(0.542927\pi\)
\(150\) −7.59874 −0.620435
\(151\) −21.5380 −1.75274 −0.876369 0.481641i \(-0.840041\pi\)
−0.876369 + 0.481641i \(0.840041\pi\)
\(152\) −1.21778 −0.0987749
\(153\) 2.46483 0.199270
\(154\) −4.30004 −0.346507
\(155\) 13.6818 1.09895
\(156\) 1.00000 0.0800641
\(157\) −1.17946 −0.0941310 −0.0470655 0.998892i \(-0.514987\pi\)
−0.0470655 + 0.998892i \(0.514987\pi\)
\(158\) −4.16443 −0.331305
\(159\) 0.0283729 0.00225012
\(160\) 3.54947 0.280610
\(161\) −5.54050 −0.436653
\(162\) −1.00000 −0.0785674
\(163\) 25.1785 1.97213 0.986067 0.166350i \(-0.0531982\pi\)
0.986067 + 0.166350i \(0.0531982\pi\)
\(164\) 9.66923 0.755040
\(165\) −11.6254 −0.905039
\(166\) 8.86586 0.688124
\(167\) 4.03583 0.312302 0.156151 0.987733i \(-0.450091\pi\)
0.156151 + 0.987733i \(0.450091\pi\)
\(168\) −1.31288 −0.101291
\(169\) 1.00000 0.0769231
\(170\) 8.74884 0.671005
\(171\) 1.21778 0.0931259
\(172\) −6.54065 −0.498720
\(173\) 8.06124 0.612885 0.306442 0.951889i \(-0.400861\pi\)
0.306442 + 0.951889i \(0.400861\pi\)
\(174\) 7.93838 0.601807
\(175\) 9.97628 0.754136
\(176\) 3.27526 0.246882
\(177\) −11.0105 −0.827603
\(178\) 7.73254 0.579579
\(179\) 10.6146 0.793376 0.396688 0.917954i \(-0.370159\pi\)
0.396688 + 0.917954i \(0.370159\pi\)
\(180\) −3.54947 −0.264562
\(181\) −7.32852 −0.544724 −0.272362 0.962195i \(-0.587805\pi\)
−0.272362 + 0.962195i \(0.587805\pi\)
\(182\) −1.31288 −0.0973175
\(183\) 2.25319 0.166560
\(184\) 4.22010 0.311110
\(185\) 18.7003 1.37487
\(186\) 3.85461 0.282633
\(187\) 8.07296 0.590353
\(188\) −8.05401 −0.587399
\(189\) 1.31288 0.0954983
\(190\) 4.32247 0.313585
\(191\) −7.81942 −0.565794 −0.282897 0.959150i \(-0.591295\pi\)
−0.282897 + 0.959150i \(0.591295\pi\)
\(192\) 1.00000 0.0721688
\(193\) 20.7634 1.49458 0.747292 0.664496i \(-0.231354\pi\)
0.747292 + 0.664496i \(0.231354\pi\)
\(194\) −9.04389 −0.649314
\(195\) −3.54947 −0.254183
\(196\) −5.27633 −0.376881
\(197\) −25.9541 −1.84916 −0.924578 0.380992i \(-0.875582\pi\)
−0.924578 + 0.380992i \(0.875582\pi\)
\(198\) −3.27526 −0.232763
\(199\) −22.0696 −1.56447 −0.782236 0.622983i \(-0.785921\pi\)
−0.782236 + 0.622983i \(0.785921\pi\)
\(200\) −7.59874 −0.537312
\(201\) 14.9179 1.05222
\(202\) −6.18777 −0.435370
\(203\) −10.4222 −0.731494
\(204\) 2.46483 0.172573
\(205\) −34.3206 −2.39706
\(206\) 1.00000 0.0696733
\(207\) −4.22010 −0.293317
\(208\) 1.00000 0.0693375
\(209\) 3.98854 0.275893
\(210\) 4.66005 0.321574
\(211\) 6.54556 0.450615 0.225307 0.974288i \(-0.427661\pi\)
0.225307 + 0.974288i \(0.427661\pi\)
\(212\) 0.0283729 0.00194866
\(213\) −0.0351683 −0.00240969
\(214\) 6.37624 0.435870
\(215\) 23.2159 1.58331
\(216\) −1.00000 −0.0680414
\(217\) −5.06065 −0.343540
\(218\) −17.7480 −1.20205
\(219\) 7.18909 0.485794
\(220\) −11.6254 −0.783787
\(221\) 2.46483 0.165802
\(222\) 5.26848 0.353597
\(223\) −25.7961 −1.72744 −0.863718 0.503975i \(-0.831870\pi\)
−0.863718 + 0.503975i \(0.831870\pi\)
\(224\) −1.31288 −0.0877208
\(225\) 7.59874 0.506583
\(226\) −0.866788 −0.0576579
\(227\) −19.8609 −1.31822 −0.659108 0.752048i \(-0.729066\pi\)
−0.659108 + 0.752048i \(0.729066\pi\)
\(228\) 1.21778 0.0806494
\(229\) 10.8734 0.718537 0.359268 0.933234i \(-0.383026\pi\)
0.359268 + 0.933234i \(0.383026\pi\)
\(230\) −14.9791 −0.987693
\(231\) 4.30004 0.282922
\(232\) 7.93838 0.521180
\(233\) 10.1426 0.664461 0.332230 0.943198i \(-0.392199\pi\)
0.332230 + 0.943198i \(0.392199\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 28.5875 1.86484
\(236\) −11.0105 −0.716725
\(237\) 4.16443 0.270509
\(238\) −3.23604 −0.209761
\(239\) −2.00408 −0.129633 −0.0648165 0.997897i \(-0.520646\pi\)
−0.0648165 + 0.997897i \(0.520646\pi\)
\(240\) −3.54947 −0.229117
\(241\) 9.96182 0.641697 0.320849 0.947130i \(-0.396032\pi\)
0.320849 + 0.947130i \(0.396032\pi\)
\(242\) 0.272670 0.0175279
\(243\) 1.00000 0.0641500
\(244\) 2.25319 0.144245
\(245\) 18.7282 1.19650
\(246\) −9.66923 −0.616488
\(247\) 1.21778 0.0774854
\(248\) 3.85461 0.244768
\(249\) −8.86586 −0.561851
\(250\) 9.22417 0.583388
\(251\) −20.2587 −1.27872 −0.639359 0.768909i \(-0.720800\pi\)
−0.639359 + 0.768909i \(0.720800\pi\)
\(252\) 1.31288 0.0827040
\(253\) −13.8219 −0.868976
\(254\) 11.5081 0.722084
\(255\) −8.74884 −0.547874
\(256\) 1.00000 0.0625000
\(257\) −28.9701 −1.80710 −0.903551 0.428480i \(-0.859049\pi\)
−0.903551 + 0.428480i \(0.859049\pi\)
\(258\) 6.54065 0.407203
\(259\) −6.91691 −0.429796
\(260\) −3.54947 −0.220129
\(261\) −7.93838 −0.491373
\(262\) 6.55057 0.404696
\(263\) 24.8430 1.53189 0.765943 0.642909i \(-0.222272\pi\)
0.765943 + 0.642909i \(0.222272\pi\)
\(264\) −3.27526 −0.201578
\(265\) −0.100709 −0.00618650
\(266\) −1.59880 −0.0980290
\(267\) −7.73254 −0.473224
\(268\) 14.9179 0.911253
\(269\) −11.9152 −0.726484 −0.363242 0.931695i \(-0.618330\pi\)
−0.363242 + 0.931695i \(0.618330\pi\)
\(270\) 3.54947 0.216014
\(271\) 9.02697 0.548349 0.274175 0.961680i \(-0.411595\pi\)
0.274175 + 0.961680i \(0.411595\pi\)
\(272\) 2.46483 0.149452
\(273\) 1.31288 0.0794594
\(274\) 4.06930 0.245836
\(275\) 24.8879 1.50079
\(276\) −4.22010 −0.254020
\(277\) 3.05604 0.183620 0.0918098 0.995777i \(-0.470735\pi\)
0.0918098 + 0.995777i \(0.470735\pi\)
\(278\) −2.50932 −0.150499
\(279\) −3.85461 −0.230769
\(280\) 4.66005 0.278491
\(281\) −32.8450 −1.95937 −0.979684 0.200549i \(-0.935727\pi\)
−0.979684 + 0.200549i \(0.935727\pi\)
\(282\) 8.05401 0.479609
\(283\) −22.0832 −1.31271 −0.656356 0.754451i \(-0.727903\pi\)
−0.656356 + 0.754451i \(0.727903\pi\)
\(284\) −0.0351683 −0.00208685
\(285\) −4.32247 −0.256041
\(286\) −3.27526 −0.193670
\(287\) 12.6946 0.749338
\(288\) −1.00000 −0.0589256
\(289\) −10.9246 −0.642625
\(290\) −28.1771 −1.65461
\(291\) 9.04389 0.530162
\(292\) 7.18909 0.420710
\(293\) 16.6660 0.973636 0.486818 0.873503i \(-0.338157\pi\)
0.486818 + 0.873503i \(0.338157\pi\)
\(294\) 5.27633 0.307722
\(295\) 39.0816 2.27542
\(296\) 5.26848 0.306224
\(297\) 3.27526 0.190050
\(298\) 3.28234 0.190141
\(299\) −4.22010 −0.244054
\(300\) 7.59874 0.438714
\(301\) −8.58713 −0.494954
\(302\) 21.5380 1.23937
\(303\) 6.18777 0.355478
\(304\) 1.21778 0.0698444
\(305\) −7.99762 −0.457942
\(306\) −2.46483 −0.140905
\(307\) 14.8288 0.846325 0.423162 0.906054i \(-0.360920\pi\)
0.423162 + 0.906054i \(0.360920\pi\)
\(308\) 4.30004 0.245018
\(309\) −1.00000 −0.0568880
\(310\) −13.6818 −0.777075
\(311\) −26.2560 −1.48884 −0.744420 0.667711i \(-0.767274\pi\)
−0.744420 + 0.667711i \(0.767274\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −30.8949 −1.74628 −0.873141 0.487467i \(-0.837921\pi\)
−0.873141 + 0.487467i \(0.837921\pi\)
\(314\) 1.17946 0.0665607
\(315\) −4.66005 −0.262564
\(316\) 4.16443 0.234268
\(317\) −5.89232 −0.330946 −0.165473 0.986214i \(-0.552915\pi\)
−0.165473 + 0.986214i \(0.552915\pi\)
\(318\) −0.0283729 −0.00159108
\(319\) −26.0003 −1.45574
\(320\) −3.54947 −0.198421
\(321\) −6.37624 −0.355887
\(322\) 5.54050 0.308760
\(323\) 3.00162 0.167014
\(324\) 1.00000 0.0555556
\(325\) 7.59874 0.421502
\(326\) −25.1785 −1.39451
\(327\) 17.7480 0.981467
\(328\) −9.66923 −0.533894
\(329\) −10.5740 −0.582963
\(330\) 11.6254 0.639959
\(331\) −9.61858 −0.528685 −0.264342 0.964429i \(-0.585155\pi\)
−0.264342 + 0.964429i \(0.585155\pi\)
\(332\) −8.86586 −0.486577
\(333\) −5.26848 −0.288711
\(334\) −4.03583 −0.220831
\(335\) −52.9505 −2.89299
\(336\) 1.31288 0.0716237
\(337\) −21.8295 −1.18913 −0.594566 0.804047i \(-0.702676\pi\)
−0.594566 + 0.804047i \(0.702676\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0.866788 0.0470775
\(340\) −8.74884 −0.474472
\(341\) −12.6248 −0.683673
\(342\) −1.21778 −0.0658500
\(343\) −16.1174 −0.870258
\(344\) 6.54065 0.352648
\(345\) 14.9791 0.806448
\(346\) −8.06124 −0.433375
\(347\) −7.45575 −0.400246 −0.200123 0.979771i \(-0.564134\pi\)
−0.200123 + 0.979771i \(0.564134\pi\)
\(348\) −7.93838 −0.425542
\(349\) 1.75232 0.0937996 0.0468998 0.998900i \(-0.485066\pi\)
0.0468998 + 0.998900i \(0.485066\pi\)
\(350\) −9.97628 −0.533254
\(351\) 1.00000 0.0533761
\(352\) −3.27526 −0.174572
\(353\) −3.17439 −0.168956 −0.0844779 0.996425i \(-0.526922\pi\)
−0.0844779 + 0.996425i \(0.526922\pi\)
\(354\) 11.0105 0.585204
\(355\) 0.124829 0.00662522
\(356\) −7.73254 −0.409824
\(357\) 3.23604 0.171269
\(358\) −10.6146 −0.561002
\(359\) −30.4195 −1.60548 −0.802740 0.596330i \(-0.796625\pi\)
−0.802740 + 0.596330i \(0.796625\pi\)
\(360\) 3.54947 0.187074
\(361\) −17.5170 −0.921948
\(362\) 7.32852 0.385178
\(363\) −0.272670 −0.0143115
\(364\) 1.31288 0.0688139
\(365\) −25.5175 −1.33564
\(366\) −2.25319 −0.117776
\(367\) −30.9444 −1.61528 −0.807642 0.589673i \(-0.799257\pi\)
−0.807642 + 0.589673i \(0.799257\pi\)
\(368\) −4.22010 −0.219988
\(369\) 9.66923 0.503360
\(370\) −18.7003 −0.972183
\(371\) 0.0372504 0.00193394
\(372\) −3.85461 −0.199852
\(373\) −1.37337 −0.0711103 −0.0355551 0.999368i \(-0.511320\pi\)
−0.0355551 + 0.999368i \(0.511320\pi\)
\(374\) −8.07296 −0.417443
\(375\) −9.22417 −0.476334
\(376\) 8.05401 0.415354
\(377\) −7.93838 −0.408847
\(378\) −1.31288 −0.0675275
\(379\) 23.2957 1.19662 0.598311 0.801264i \(-0.295839\pi\)
0.598311 + 0.801264i \(0.295839\pi\)
\(380\) −4.32247 −0.221738
\(381\) −11.5081 −0.589579
\(382\) 7.81942 0.400077
\(383\) 34.3225 1.75380 0.876899 0.480675i \(-0.159608\pi\)
0.876899 + 0.480675i \(0.159608\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −15.2629 −0.777868
\(386\) −20.7634 −1.05683
\(387\) −6.54065 −0.332480
\(388\) 9.04389 0.459134
\(389\) −10.1277 −0.513494 −0.256747 0.966479i \(-0.582651\pi\)
−0.256747 + 0.966479i \(0.582651\pi\)
\(390\) 3.54947 0.179734
\(391\) −10.4018 −0.526042
\(392\) 5.27633 0.266495
\(393\) −6.55057 −0.330433
\(394\) 25.9541 1.30755
\(395\) −14.7815 −0.743740
\(396\) 3.27526 0.164588
\(397\) −2.07368 −0.104075 −0.0520375 0.998645i \(-0.516572\pi\)
−0.0520375 + 0.998645i \(0.516572\pi\)
\(398\) 22.0696 1.10625
\(399\) 1.59880 0.0800403
\(400\) 7.59874 0.379937
\(401\) −0.176916 −0.00883477 −0.00441739 0.999990i \(-0.501406\pi\)
−0.00441739 + 0.999990i \(0.501406\pi\)
\(402\) −14.9179 −0.744035
\(403\) −3.85461 −0.192012
\(404\) 6.18777 0.307853
\(405\) −3.54947 −0.176375
\(406\) 10.4222 0.517244
\(407\) −17.2556 −0.855330
\(408\) −2.46483 −0.122027
\(409\) 10.6175 0.525003 0.262502 0.964932i \(-0.415453\pi\)
0.262502 + 0.964932i \(0.415453\pi\)
\(410\) 34.3206 1.69498
\(411\) −4.06930 −0.200724
\(412\) −1.00000 −0.0492665
\(413\) −14.4556 −0.711312
\(414\) 4.22010 0.207406
\(415\) 31.4691 1.54476
\(416\) −1.00000 −0.0490290
\(417\) 2.50932 0.122882
\(418\) −3.98854 −0.195086
\(419\) 27.1084 1.32433 0.662167 0.749356i \(-0.269637\pi\)
0.662167 + 0.749356i \(0.269637\pi\)
\(420\) −4.66005 −0.227387
\(421\) −38.7035 −1.88629 −0.943146 0.332378i \(-0.892149\pi\)
−0.943146 + 0.332378i \(0.892149\pi\)
\(422\) −6.54556 −0.318633
\(423\) −8.05401 −0.391599
\(424\) −0.0283729 −0.00137791
\(425\) 18.7296 0.908519
\(426\) 0.0351683 0.00170391
\(427\) 2.95817 0.143156
\(428\) −6.37624 −0.308207
\(429\) 3.27526 0.158131
\(430\) −23.2159 −1.11957
\(431\) 31.7096 1.52740 0.763698 0.645574i \(-0.223382\pi\)
0.763698 + 0.645574i \(0.223382\pi\)
\(432\) 1.00000 0.0481125
\(433\) 4.72182 0.226916 0.113458 0.993543i \(-0.463807\pi\)
0.113458 + 0.993543i \(0.463807\pi\)
\(434\) 5.06065 0.242919
\(435\) 28.1771 1.35099
\(436\) 17.7480 0.849975
\(437\) −5.13914 −0.245839
\(438\) −7.18909 −0.343508
\(439\) 10.2850 0.490876 0.245438 0.969412i \(-0.421068\pi\)
0.245438 + 0.969412i \(0.421068\pi\)
\(440\) 11.6254 0.554221
\(441\) −5.27633 −0.251254
\(442\) −2.46483 −0.117240
\(443\) 3.63714 0.172806 0.0864029 0.996260i \(-0.472463\pi\)
0.0864029 + 0.996260i \(0.472463\pi\)
\(444\) −5.26848 −0.250031
\(445\) 27.4464 1.30109
\(446\) 25.7961 1.22148
\(447\) −3.28234 −0.155249
\(448\) 1.31288 0.0620280
\(449\) −19.6202 −0.925934 −0.462967 0.886375i \(-0.653215\pi\)
−0.462967 + 0.886375i \(0.653215\pi\)
\(450\) −7.59874 −0.358208
\(451\) 31.6692 1.49125
\(452\) 0.866788 0.0407703
\(453\) −21.5380 −1.01194
\(454\) 19.8609 0.932119
\(455\) −4.66005 −0.218466
\(456\) −1.21778 −0.0570277
\(457\) −19.0474 −0.891001 −0.445500 0.895282i \(-0.646974\pi\)
−0.445500 + 0.895282i \(0.646974\pi\)
\(458\) −10.8734 −0.508082
\(459\) 2.46483 0.115048
\(460\) 14.9791 0.698404
\(461\) 31.1260 1.44968 0.724841 0.688916i \(-0.241913\pi\)
0.724841 + 0.688916i \(0.241913\pi\)
\(462\) −4.30004 −0.200056
\(463\) 18.9293 0.879721 0.439860 0.898066i \(-0.355028\pi\)
0.439860 + 0.898066i \(0.355028\pi\)
\(464\) −7.93838 −0.368530
\(465\) 13.6818 0.634479
\(466\) −10.1426 −0.469845
\(467\) −29.5896 −1.36924 −0.684621 0.728899i \(-0.740032\pi\)
−0.684621 + 0.728899i \(0.740032\pi\)
\(468\) 1.00000 0.0462250
\(469\) 19.5854 0.904371
\(470\) −28.5875 −1.31864
\(471\) −1.17946 −0.0543466
\(472\) 11.0105 0.506801
\(473\) −21.4223 −0.985000
\(474\) −4.16443 −0.191279
\(475\) 9.25359 0.424584
\(476\) 3.23604 0.148324
\(477\) 0.0283729 0.00129911
\(478\) 2.00408 0.0916644
\(479\) −38.3540 −1.75244 −0.876219 0.481913i \(-0.839942\pi\)
−0.876219 + 0.481913i \(0.839942\pi\)
\(480\) 3.54947 0.162010
\(481\) −5.26848 −0.240222
\(482\) −9.96182 −0.453749
\(483\) −5.54050 −0.252101
\(484\) −0.272670 −0.0123941
\(485\) −32.1010 −1.45763
\(486\) −1.00000 −0.0453609
\(487\) −27.2762 −1.23600 −0.618000 0.786178i \(-0.712057\pi\)
−0.618000 + 0.786178i \(0.712057\pi\)
\(488\) −2.25319 −0.101997
\(489\) 25.1785 1.13861
\(490\) −18.7282 −0.846053
\(491\) −12.2897 −0.554625 −0.277312 0.960780i \(-0.589444\pi\)
−0.277312 + 0.960780i \(0.589444\pi\)
\(492\) 9.66923 0.435923
\(493\) −19.5668 −0.881242
\(494\) −1.21778 −0.0547905
\(495\) −11.6254 −0.522525
\(496\) −3.85461 −0.173077
\(497\) −0.0461719 −0.00207109
\(498\) 8.86586 0.397289
\(499\) −18.6348 −0.834210 −0.417105 0.908858i \(-0.636955\pi\)
−0.417105 + 0.908858i \(0.636955\pi\)
\(500\) −9.22417 −0.412517
\(501\) 4.03583 0.180308
\(502\) 20.2587 0.904190
\(503\) 16.0880 0.717330 0.358665 0.933466i \(-0.383232\pi\)
0.358665 + 0.933466i \(0.383232\pi\)
\(504\) −1.31288 −0.0584805
\(505\) −21.9633 −0.977355
\(506\) 13.8219 0.614459
\(507\) 1.00000 0.0444116
\(508\) −11.5081 −0.510590
\(509\) −17.7238 −0.785595 −0.392797 0.919625i \(-0.628493\pi\)
−0.392797 + 0.919625i \(0.628493\pi\)
\(510\) 8.74884 0.387405
\(511\) 9.43845 0.417532
\(512\) −1.00000 −0.0441942
\(513\) 1.21778 0.0537663
\(514\) 28.9701 1.27781
\(515\) 3.54947 0.156408
\(516\) −6.54065 −0.287936
\(517\) −26.3790 −1.16015
\(518\) 6.91691 0.303911
\(519\) 8.06124 0.353849
\(520\) 3.54947 0.155655
\(521\) 8.24006 0.361004 0.180502 0.983575i \(-0.442228\pi\)
0.180502 + 0.983575i \(0.442228\pi\)
\(522\) 7.93838 0.347454
\(523\) 30.8351 1.34833 0.674163 0.738583i \(-0.264505\pi\)
0.674163 + 0.738583i \(0.264505\pi\)
\(524\) −6.55057 −0.286163
\(525\) 9.97628 0.435400
\(526\) −24.8430 −1.08321
\(527\) −9.50094 −0.413868
\(528\) 3.27526 0.142537
\(529\) −5.19079 −0.225687
\(530\) 0.100709 0.00437452
\(531\) −11.0105 −0.477817
\(532\) 1.59880 0.0693169
\(533\) 9.66923 0.418821
\(534\) 7.73254 0.334620
\(535\) 22.6323 0.978478
\(536\) −14.9179 −0.644353
\(537\) 10.6146 0.458056
\(538\) 11.9152 0.513701
\(539\) −17.2814 −0.744361
\(540\) −3.54947 −0.152745
\(541\) 7.33425 0.315324 0.157662 0.987493i \(-0.449604\pi\)
0.157662 + 0.987493i \(0.449604\pi\)
\(542\) −9.02697 −0.387741
\(543\) −7.32852 −0.314497
\(544\) −2.46483 −0.105679
\(545\) −62.9960 −2.69845
\(546\) −1.31288 −0.0561863
\(547\) −25.4282 −1.08723 −0.543615 0.839335i \(-0.682945\pi\)
−0.543615 + 0.839335i \(0.682945\pi\)
\(548\) −4.06930 −0.173832
\(549\) 2.25319 0.0961636
\(550\) −24.8879 −1.06122
\(551\) −9.66720 −0.411836
\(552\) 4.22010 0.179619
\(553\) 5.46742 0.232498
\(554\) −3.05604 −0.129839
\(555\) 18.7003 0.793784
\(556\) 2.50932 0.106419
\(557\) 27.2315 1.15384 0.576918 0.816802i \(-0.304255\pi\)
0.576918 + 0.816802i \(0.304255\pi\)
\(558\) 3.85461 0.163178
\(559\) −6.54065 −0.276640
\(560\) −4.66005 −0.196923
\(561\) 8.07296 0.340841
\(562\) 32.8450 1.38548
\(563\) 11.4854 0.484050 0.242025 0.970270i \(-0.422188\pi\)
0.242025 + 0.970270i \(0.422188\pi\)
\(564\) −8.05401 −0.339135
\(565\) −3.07664 −0.129435
\(566\) 22.0832 0.928227
\(567\) 1.31288 0.0551360
\(568\) 0.0351683 0.00147563
\(569\) −38.6617 −1.62078 −0.810392 0.585888i \(-0.800746\pi\)
−0.810392 + 0.585888i \(0.800746\pi\)
\(570\) 4.32247 0.181048
\(571\) 42.8225 1.79207 0.896034 0.443986i \(-0.146436\pi\)
0.896034 + 0.443986i \(0.146436\pi\)
\(572\) 3.27526 0.136946
\(573\) −7.81942 −0.326661
\(574\) −12.6946 −0.529862
\(575\) −32.0674 −1.33730
\(576\) 1.00000 0.0416667
\(577\) 30.4377 1.26714 0.633568 0.773687i \(-0.281590\pi\)
0.633568 + 0.773687i \(0.281590\pi\)
\(578\) 10.9246 0.454404
\(579\) 20.7634 0.862899
\(580\) 28.1771 1.16999
\(581\) −11.6399 −0.482902
\(582\) −9.04389 −0.374881
\(583\) 0.0929287 0.00384872
\(584\) −7.18909 −0.297487
\(585\) −3.54947 −0.146753
\(586\) −16.6660 −0.688465
\(587\) −16.5649 −0.683705 −0.341853 0.939754i \(-0.611054\pi\)
−0.341853 + 0.939754i \(0.611054\pi\)
\(588\) −5.27633 −0.217592
\(589\) −4.69406 −0.193415
\(590\) −39.0816 −1.60896
\(591\) −25.9541 −1.06761
\(592\) −5.26848 −0.216533
\(593\) 44.9535 1.84602 0.923009 0.384778i \(-0.125722\pi\)
0.923009 + 0.384778i \(0.125722\pi\)
\(594\) −3.27526 −0.134386
\(595\) −11.4862 −0.470889
\(596\) −3.28234 −0.134450
\(597\) −22.0696 −0.903248
\(598\) 4.22010 0.172573
\(599\) 0.224550 0.00917488 0.00458744 0.999989i \(-0.498540\pi\)
0.00458744 + 0.999989i \(0.498540\pi\)
\(600\) −7.59874 −0.310217
\(601\) −31.9340 −1.30261 −0.651307 0.758815i \(-0.725779\pi\)
−0.651307 + 0.758815i \(0.725779\pi\)
\(602\) 8.58713 0.349985
\(603\) 14.9179 0.607502
\(604\) −21.5380 −0.876369
\(605\) 0.967835 0.0393481
\(606\) −6.18777 −0.251361
\(607\) 33.0320 1.34073 0.670363 0.742033i \(-0.266138\pi\)
0.670363 + 0.742033i \(0.266138\pi\)
\(608\) −1.21778 −0.0493875
\(609\) −10.4222 −0.422328
\(610\) 7.99762 0.323814
\(611\) −8.05401 −0.325830
\(612\) 2.46483 0.0996348
\(613\) −16.7730 −0.677454 −0.338727 0.940885i \(-0.609996\pi\)
−0.338727 + 0.940885i \(0.609996\pi\)
\(614\) −14.8288 −0.598442
\(615\) −34.3206 −1.38394
\(616\) −4.30004 −0.173254
\(617\) 6.69410 0.269494 0.134747 0.990880i \(-0.456978\pi\)
0.134747 + 0.990880i \(0.456978\pi\)
\(618\) 1.00000 0.0402259
\(619\) 32.8906 1.32199 0.660993 0.750392i \(-0.270135\pi\)
0.660993 + 0.750392i \(0.270135\pi\)
\(620\) 13.6818 0.549475
\(621\) −4.22010 −0.169347
\(622\) 26.2560 1.05277
\(623\) −10.1519 −0.406729
\(624\) 1.00000 0.0400320
\(625\) −5.25281 −0.210112
\(626\) 30.8949 1.23481
\(627\) 3.98854 0.159287
\(628\) −1.17946 −0.0470655
\(629\) −12.9859 −0.517782
\(630\) 4.66005 0.185661
\(631\) −28.0303 −1.11587 −0.557935 0.829884i \(-0.688406\pi\)
−0.557935 + 0.829884i \(0.688406\pi\)
\(632\) −4.16443 −0.165652
\(633\) 6.54556 0.260163
\(634\) 5.89232 0.234014
\(635\) 40.8477 1.62099
\(636\) 0.0283729 0.00112506
\(637\) −5.27633 −0.209056
\(638\) 26.0003 1.02936
\(639\) −0.0351683 −0.00139124
\(640\) 3.54947 0.140305
\(641\) 37.9363 1.49839 0.749197 0.662347i \(-0.230440\pi\)
0.749197 + 0.662347i \(0.230440\pi\)
\(642\) 6.37624 0.251650
\(643\) 2.71039 0.106887 0.0534436 0.998571i \(-0.482980\pi\)
0.0534436 + 0.998571i \(0.482980\pi\)
\(644\) −5.54050 −0.218326
\(645\) 23.2159 0.914124
\(646\) −3.00162 −0.118097
\(647\) −27.0949 −1.06521 −0.532605 0.846364i \(-0.678787\pi\)
−0.532605 + 0.846364i \(0.678787\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −36.0624 −1.41557
\(650\) −7.59874 −0.298047
\(651\) −5.06065 −0.198343
\(652\) 25.1785 0.986067
\(653\) 16.1167 0.630696 0.315348 0.948976i \(-0.397879\pi\)
0.315348 + 0.948976i \(0.397879\pi\)
\(654\) −17.7480 −0.694002
\(655\) 23.2511 0.908495
\(656\) 9.66923 0.377520
\(657\) 7.18909 0.280473
\(658\) 10.5740 0.412217
\(659\) −3.02984 −0.118026 −0.0590130 0.998257i \(-0.518795\pi\)
−0.0590130 + 0.998257i \(0.518795\pi\)
\(660\) −11.6254 −0.452520
\(661\) 19.0713 0.741789 0.370894 0.928675i \(-0.379051\pi\)
0.370894 + 0.928675i \(0.379051\pi\)
\(662\) 9.61858 0.373837
\(663\) 2.46483 0.0957260
\(664\) 8.86586 0.344062
\(665\) −5.67491 −0.220064
\(666\) 5.26848 0.204149
\(667\) 33.5007 1.29715
\(668\) 4.03583 0.156151
\(669\) −25.7961 −0.997336
\(670\) 52.9505 2.04566
\(671\) 7.37977 0.284893
\(672\) −1.31288 −0.0506456
\(673\) −24.5603 −0.946732 −0.473366 0.880866i \(-0.656961\pi\)
−0.473366 + 0.880866i \(0.656961\pi\)
\(674\) 21.8295 0.840843
\(675\) 7.59874 0.292476
\(676\) 1.00000 0.0384615
\(677\) −40.8652 −1.57058 −0.785290 0.619129i \(-0.787486\pi\)
−0.785290 + 0.619129i \(0.787486\pi\)
\(678\) −0.866788 −0.0332888
\(679\) 11.8736 0.455667
\(680\) 8.74884 0.335503
\(681\) −19.8609 −0.761072
\(682\) 12.6248 0.483430
\(683\) 10.5681 0.404377 0.202188 0.979347i \(-0.435195\pi\)
0.202188 + 0.979347i \(0.435195\pi\)
\(684\) 1.21778 0.0465630
\(685\) 14.4439 0.551872
\(686\) 16.1174 0.615366
\(687\) 10.8734 0.414847
\(688\) −6.54065 −0.249360
\(689\) 0.0283729 0.00108092
\(690\) −14.9791 −0.570245
\(691\) 26.3114 1.00093 0.500467 0.865755i \(-0.333162\pi\)
0.500467 + 0.865755i \(0.333162\pi\)
\(692\) 8.06124 0.306442
\(693\) 4.30004 0.163345
\(694\) 7.45575 0.283016
\(695\) −8.90677 −0.337853
\(696\) 7.93838 0.300904
\(697\) 23.8330 0.902739
\(698\) −1.75232 −0.0663264
\(699\) 10.1426 0.383627
\(700\) 9.97628 0.377068
\(701\) 42.9229 1.62117 0.810587 0.585618i \(-0.199148\pi\)
0.810587 + 0.585618i \(0.199148\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −6.41585 −0.241978
\(704\) 3.27526 0.123441
\(705\) 28.5875 1.07667
\(706\) 3.17439 0.119470
\(707\) 8.12384 0.305528
\(708\) −11.0105 −0.413802
\(709\) −31.6761 −1.18962 −0.594811 0.803865i \(-0.702773\pi\)
−0.594811 + 0.803865i \(0.702773\pi\)
\(710\) −0.124829 −0.00468474
\(711\) 4.16443 0.156178
\(712\) 7.73254 0.289789
\(713\) 16.2668 0.609197
\(714\) −3.23604 −0.121106
\(715\) −11.6254 −0.434767
\(716\) 10.6146 0.396688
\(717\) −2.00408 −0.0748437
\(718\) 30.4195 1.13525
\(719\) −1.13312 −0.0422584 −0.0211292 0.999777i \(-0.506726\pi\)
−0.0211292 + 0.999777i \(0.506726\pi\)
\(720\) −3.54947 −0.132281
\(721\) −1.31288 −0.0488944
\(722\) 17.5170 0.651916
\(723\) 9.96182 0.370484
\(724\) −7.32852 −0.272362
\(725\) −60.3217 −2.24029
\(726\) 0.272670 0.0101197
\(727\) −9.95634 −0.369260 −0.184630 0.982808i \(-0.559109\pi\)
−0.184630 + 0.982808i \(0.559109\pi\)
\(728\) −1.31288 −0.0486588
\(729\) 1.00000 0.0370370
\(730\) 25.5175 0.944444
\(731\) −16.1216 −0.596279
\(732\) 2.25319 0.0832802
\(733\) −8.82241 −0.325863 −0.162932 0.986637i \(-0.552095\pi\)
−0.162932 + 0.986637i \(0.552095\pi\)
\(734\) 30.9444 1.14218
\(735\) 18.7282 0.690800
\(736\) 4.22010 0.155555
\(737\) 48.8599 1.79978
\(738\) −9.66923 −0.355929
\(739\) 12.5656 0.462234 0.231117 0.972926i \(-0.425762\pi\)
0.231117 + 0.972926i \(0.425762\pi\)
\(740\) 18.7003 0.687437
\(741\) 1.21778 0.0447362
\(742\) −0.0372504 −0.00136751
\(743\) 6.11064 0.224178 0.112089 0.993698i \(-0.464246\pi\)
0.112089 + 0.993698i \(0.464246\pi\)
\(744\) 3.85461 0.141317
\(745\) 11.6506 0.426844
\(746\) 1.37337 0.0502826
\(747\) −8.86586 −0.324385
\(748\) 8.07296 0.295177
\(749\) −8.37126 −0.305879
\(750\) 9.22417 0.336819
\(751\) −27.1339 −0.990132 −0.495066 0.868855i \(-0.664856\pi\)
−0.495066 + 0.868855i \(0.664856\pi\)
\(752\) −8.05401 −0.293699
\(753\) −20.2587 −0.738268
\(754\) 7.93838 0.289099
\(755\) 76.4485 2.78225
\(756\) 1.31288 0.0477492
\(757\) −37.5185 −1.36363 −0.681816 0.731524i \(-0.738809\pi\)
−0.681816 + 0.731524i \(0.738809\pi\)
\(758\) −23.2957 −0.846139
\(759\) −13.8219 −0.501704
\(760\) 4.32247 0.156793
\(761\) 31.2389 1.13241 0.566205 0.824264i \(-0.308411\pi\)
0.566205 + 0.824264i \(0.308411\pi\)
\(762\) 11.5081 0.416895
\(763\) 23.3011 0.843556
\(764\) −7.81942 −0.282897
\(765\) −8.74884 −0.316315
\(766\) −34.3225 −1.24012
\(767\) −11.0105 −0.397568
\(768\) 1.00000 0.0360844
\(769\) −39.8353 −1.43650 −0.718249 0.695786i \(-0.755056\pi\)
−0.718249 + 0.695786i \(0.755056\pi\)
\(770\) 15.2629 0.550036
\(771\) −28.9701 −1.04333
\(772\) 20.7634 0.747292
\(773\) −4.17643 −0.150216 −0.0751079 0.997175i \(-0.523930\pi\)
−0.0751079 + 0.997175i \(0.523930\pi\)
\(774\) 6.54065 0.235099
\(775\) −29.2902 −1.05213
\(776\) −9.04389 −0.324657
\(777\) −6.91691 −0.248143
\(778\) 10.1277 0.363095
\(779\) 11.7750 0.421883
\(780\) −3.54947 −0.127091
\(781\) −0.115185 −0.00412165
\(782\) 10.4018 0.371968
\(783\) −7.93838 −0.283695
\(784\) −5.27633 −0.188440
\(785\) 4.18645 0.149421
\(786\) 6.55057 0.233651
\(787\) 33.3947 1.19039 0.595197 0.803580i \(-0.297074\pi\)
0.595197 + 0.803580i \(0.297074\pi\)
\(788\) −25.9541 −0.924578
\(789\) 24.8430 0.884435
\(790\) 14.7815 0.525903
\(791\) 1.13799 0.0404624
\(792\) −3.27526 −0.116381
\(793\) 2.25319 0.0800130
\(794\) 2.07368 0.0735921
\(795\) −0.100709 −0.00357178
\(796\) −22.0696 −0.782236
\(797\) 47.3175 1.67607 0.838036 0.545616i \(-0.183704\pi\)
0.838036 + 0.545616i \(0.183704\pi\)
\(798\) −1.59880 −0.0565970
\(799\) −19.8517 −0.702304
\(800\) −7.59874 −0.268656
\(801\) −7.73254 −0.273216
\(802\) 0.176916 0.00624713
\(803\) 23.5461 0.830925
\(804\) 14.9179 0.526112
\(805\) 19.6658 0.693130
\(806\) 3.85461 0.135773
\(807\) −11.9152 −0.419435
\(808\) −6.18777 −0.217685
\(809\) 10.8583 0.381759 0.190880 0.981613i \(-0.438866\pi\)
0.190880 + 0.981613i \(0.438866\pi\)
\(810\) 3.54947 0.124716
\(811\) −22.0538 −0.774414 −0.387207 0.921993i \(-0.626560\pi\)
−0.387207 + 0.921993i \(0.626560\pi\)
\(812\) −10.4222 −0.365747
\(813\) 9.02697 0.316590
\(814\) 17.2556 0.604810
\(815\) −89.3704 −3.13051
\(816\) 2.46483 0.0862863
\(817\) −7.96507 −0.278663
\(818\) −10.6175 −0.371233
\(819\) 1.31288 0.0458759
\(820\) −34.3206 −1.19853
\(821\) 32.6102 1.13810 0.569052 0.822301i \(-0.307310\pi\)
0.569052 + 0.822301i \(0.307310\pi\)
\(822\) 4.06930 0.141933
\(823\) 10.5809 0.368826 0.184413 0.982849i \(-0.440962\pi\)
0.184413 + 0.982849i \(0.440962\pi\)
\(824\) 1.00000 0.0348367
\(825\) 24.8879 0.866484
\(826\) 14.4556 0.502974
\(827\) −13.4362 −0.467222 −0.233611 0.972330i \(-0.575054\pi\)
−0.233611 + 0.972330i \(0.575054\pi\)
\(828\) −4.22010 −0.146658
\(829\) −49.3413 −1.71369 −0.856847 0.515572i \(-0.827580\pi\)
−0.856847 + 0.515572i \(0.827580\pi\)
\(830\) −31.4691 −1.09231
\(831\) 3.05604 0.106013
\(832\) 1.00000 0.0346688
\(833\) −13.0053 −0.450606
\(834\) −2.50932 −0.0868907
\(835\) −14.3251 −0.495739
\(836\) 3.98854 0.137947
\(837\) −3.85461 −0.133235
\(838\) −27.1084 −0.936446
\(839\) 23.3668 0.806712 0.403356 0.915043i \(-0.367844\pi\)
0.403356 + 0.915043i \(0.367844\pi\)
\(840\) 4.66005 0.160787
\(841\) 34.0179 1.17303
\(842\) 38.7035 1.33381
\(843\) −32.8450 −1.13124
\(844\) 6.54556 0.225307
\(845\) −3.54947 −0.122106
\(846\) 8.05401 0.276902
\(847\) −0.357985 −0.0123005
\(848\) 0.0283729 0.000974331 0
\(849\) −22.0832 −0.757894
\(850\) −18.7296 −0.642420
\(851\) 22.2335 0.762154
\(852\) −0.0351683 −0.00120485
\(853\) 7.07693 0.242309 0.121155 0.992634i \(-0.461340\pi\)
0.121155 + 0.992634i \(0.461340\pi\)
\(854\) −2.95817 −0.101227
\(855\) −4.32247 −0.147825
\(856\) 6.37624 0.217935
\(857\) 43.0705 1.47126 0.735629 0.677384i \(-0.236887\pi\)
0.735629 + 0.677384i \(0.236887\pi\)
\(858\) −3.27526 −0.111816
\(859\) −24.9053 −0.849759 −0.424879 0.905250i \(-0.639684\pi\)
−0.424879 + 0.905250i \(0.639684\pi\)
\(860\) 23.2159 0.791654
\(861\) 12.6946 0.432630
\(862\) −31.7096 −1.08003
\(863\) 26.4828 0.901486 0.450743 0.892654i \(-0.351159\pi\)
0.450743 + 0.892654i \(0.351159\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −28.6131 −0.972876
\(866\) −4.72182 −0.160454
\(867\) −10.9246 −0.371019
\(868\) −5.06065 −0.171770
\(869\) 13.6396 0.462692
\(870\) −28.1771 −0.955292
\(871\) 14.9179 0.505472
\(872\) −17.7480 −0.601023
\(873\) 9.04389 0.306089
\(874\) 5.13914 0.173834
\(875\) −12.1103 −0.409402
\(876\) 7.18909 0.242897
\(877\) 23.3655 0.788998 0.394499 0.918896i \(-0.370918\pi\)
0.394499 + 0.918896i \(0.370918\pi\)
\(878\) −10.2850 −0.347102
\(879\) 16.6660 0.562129
\(880\) −11.6254 −0.391894
\(881\) 34.9825 1.17859 0.589296 0.807917i \(-0.299405\pi\)
0.589296 + 0.807917i \(0.299405\pi\)
\(882\) 5.27633 0.177663
\(883\) −42.0769 −1.41600 −0.708000 0.706212i \(-0.750402\pi\)
−0.708000 + 0.706212i \(0.750402\pi\)
\(884\) 2.46483 0.0829012
\(885\) 39.0816 1.31371
\(886\) −3.63714 −0.122192
\(887\) 44.2447 1.48559 0.742796 0.669518i \(-0.233499\pi\)
0.742796 + 0.669518i \(0.233499\pi\)
\(888\) 5.26848 0.176799
\(889\) −15.1088 −0.506734
\(890\) −27.4464 −0.920007
\(891\) 3.27526 0.109725
\(892\) −25.7961 −0.863718
\(893\) −9.80800 −0.328212
\(894\) 3.28234 0.109778
\(895\) −37.6764 −1.25938
\(896\) −1.31288 −0.0438604
\(897\) −4.22010 −0.140905
\(898\) 19.6202 0.654735
\(899\) 30.5993 1.02054
\(900\) 7.59874 0.253291
\(901\) 0.0699344 0.00232985
\(902\) −31.6692 −1.05447
\(903\) −8.58713 −0.285762
\(904\) −0.866788 −0.0288289
\(905\) 26.0124 0.864680
\(906\) 21.5380 0.715552
\(907\) 41.0442 1.36285 0.681424 0.731889i \(-0.261361\pi\)
0.681424 + 0.731889i \(0.261361\pi\)
\(908\) −19.8609 −0.659108
\(909\) 6.18777 0.205236
\(910\) 4.66005 0.154479
\(911\) −37.0432 −1.22730 −0.613649 0.789579i \(-0.710299\pi\)
−0.613649 + 0.789579i \(0.710299\pi\)
\(912\) 1.21778 0.0403247
\(913\) −29.0380 −0.961017
\(914\) 19.0474 0.630033
\(915\) −7.99762 −0.264393
\(916\) 10.8734 0.359268
\(917\) −8.60015 −0.284002
\(918\) −2.46483 −0.0813515
\(919\) 51.8898 1.71169 0.855843 0.517236i \(-0.173039\pi\)
0.855843 + 0.517236i \(0.173039\pi\)
\(920\) −14.9791 −0.493847
\(921\) 14.8288 0.488626
\(922\) −31.1260 −1.02508
\(923\) −0.0351683 −0.00115758
\(924\) 4.30004 0.141461
\(925\) −40.0338 −1.31630
\(926\) −18.9293 −0.622057
\(927\) −1.00000 −0.0328443
\(928\) 7.93838 0.260590
\(929\) 44.6433 1.46470 0.732350 0.680928i \(-0.238423\pi\)
0.732350 + 0.680928i \(0.238423\pi\)
\(930\) −13.6818 −0.448644
\(931\) −6.42541 −0.210584
\(932\) 10.1426 0.332230
\(933\) −26.2560 −0.859582
\(934\) 29.5896 0.968201
\(935\) −28.6547 −0.937110
\(936\) −1.00000 −0.0326860
\(937\) −29.4637 −0.962536 −0.481268 0.876573i \(-0.659824\pi\)
−0.481268 + 0.876573i \(0.659824\pi\)
\(938\) −19.5854 −0.639487
\(939\) −30.8949 −1.00822
\(940\) 28.5875 0.932420
\(941\) 42.9192 1.39913 0.699563 0.714571i \(-0.253378\pi\)
0.699563 + 0.714571i \(0.253378\pi\)
\(942\) 1.17946 0.0384288
\(943\) −40.8051 −1.32880
\(944\) −11.0105 −0.358363
\(945\) −4.66005 −0.151591
\(946\) 21.4223 0.696500
\(947\) −23.1573 −0.752510 −0.376255 0.926516i \(-0.622788\pi\)
−0.376255 + 0.926516i \(0.622788\pi\)
\(948\) 4.16443 0.135255
\(949\) 7.18909 0.233368
\(950\) −9.25359 −0.300226
\(951\) −5.89232 −0.191072
\(952\) −3.23604 −0.104881
\(953\) −14.7472 −0.477708 −0.238854 0.971055i \(-0.576772\pi\)
−0.238854 + 0.971055i \(0.576772\pi\)
\(954\) −0.0283729 −0.000918608 0
\(955\) 27.7548 0.898125
\(956\) −2.00408 −0.0648165
\(957\) −26.0003 −0.840469
\(958\) 38.3540 1.23916
\(959\) −5.34253 −0.172519
\(960\) −3.54947 −0.114559
\(961\) −16.1420 −0.520710
\(962\) 5.26848 0.169863
\(963\) −6.37624 −0.205471
\(964\) 9.96182 0.320849
\(965\) −73.6992 −2.37246
\(966\) 5.54050 0.178263
\(967\) −4.33633 −0.139447 −0.0697235 0.997566i \(-0.522212\pi\)
−0.0697235 + 0.997566i \(0.522212\pi\)
\(968\) 0.272670 0.00876396
\(969\) 3.00162 0.0964259
\(970\) 32.1010 1.03070
\(971\) −24.3935 −0.782824 −0.391412 0.920216i \(-0.628013\pi\)
−0.391412 + 0.920216i \(0.628013\pi\)
\(972\) 1.00000 0.0320750
\(973\) 3.29445 0.105615
\(974\) 27.2762 0.873984
\(975\) 7.59874 0.243355
\(976\) 2.25319 0.0721227
\(977\) −22.6323 −0.724072 −0.362036 0.932164i \(-0.617918\pi\)
−0.362036 + 0.932164i \(0.617918\pi\)
\(978\) −25.1785 −0.805120
\(979\) −25.3261 −0.809425
\(980\) 18.7282 0.598250
\(981\) 17.7480 0.566650
\(982\) 12.2897 0.392179
\(983\) −17.5959 −0.561221 −0.280610 0.959822i \(-0.590537\pi\)
−0.280610 + 0.959822i \(0.590537\pi\)
\(984\) −9.66923 −0.308244
\(985\) 92.1235 2.93530
\(986\) 19.5668 0.623132
\(987\) −10.5740 −0.336574
\(988\) 1.21778 0.0387427
\(989\) 27.6022 0.877698
\(990\) 11.6254 0.369481
\(991\) −32.3462 −1.02751 −0.513755 0.857937i \(-0.671746\pi\)
−0.513755 + 0.857937i \(0.671746\pi\)
\(992\) 3.85461 0.122384
\(993\) −9.61858 −0.305236
\(994\) 0.0461719 0.00146448
\(995\) 78.3353 2.48340
\(996\) −8.86586 −0.280926
\(997\) 3.79023 0.120038 0.0600189 0.998197i \(-0.480884\pi\)
0.0600189 + 0.998197i \(0.480884\pi\)
\(998\) 18.6348 0.589876
\(999\) −5.26848 −0.166687
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.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.2 12 1.1 even 1 trivial