Properties

Label 2009.2.a.n.1.2
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0419457661\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.633117.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 4 x^{2} + 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.20098\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.20098 q^{2} +0.200978 q^{3} -0.557652 q^{4} +3.21704 q^{5} -0.241370 q^{6} +3.07168 q^{8} -2.95961 q^{9} +O(q^{10})\) \(q-1.20098 q^{2} +0.200978 q^{3} -0.557652 q^{4} +3.21704 q^{5} -0.241370 q^{6} +3.07168 q^{8} -2.95961 q^{9} -3.86360 q^{10} +4.57695 q^{11} -0.112076 q^{12} -0.703013 q^{13} +0.646554 q^{15} -2.57372 q^{16} -4.25337 q^{17} +3.55442 q^{18} -8.04736 q^{19} -1.79399 q^{20} -5.49681 q^{22} -5.34842 q^{23} +0.617340 q^{24} +5.34937 q^{25} +0.844303 q^{26} -1.19775 q^{27} -5.39204 q^{29} -0.776497 q^{30} -7.61900 q^{31} -3.05239 q^{32} +0.919865 q^{33} +5.10820 q^{34} +1.65043 q^{36} +5.19272 q^{37} +9.66470 q^{38} -0.141290 q^{39} +9.88174 q^{40} +1.00000 q^{41} +10.3241 q^{43} -2.55235 q^{44} -9.52119 q^{45} +6.42333 q^{46} -12.1160 q^{47} -0.517260 q^{48} -6.42448 q^{50} -0.854832 q^{51} +0.392037 q^{52} -12.2837 q^{53} +1.43847 q^{54} +14.7242 q^{55} -1.61734 q^{57} +6.47572 q^{58} +7.73023 q^{59} -0.360553 q^{60} +2.48971 q^{61} +9.15025 q^{62} +8.81329 q^{64} -2.26162 q^{65} -1.10474 q^{66} -3.09767 q^{67} +2.37190 q^{68} -1.07491 q^{69} +5.11581 q^{71} -9.09098 q^{72} -4.13640 q^{73} -6.23634 q^{74} +1.07511 q^{75} +4.48763 q^{76} +0.169686 q^{78} -13.7414 q^{79} -8.27977 q^{80} +8.63810 q^{81} -1.20098 q^{82} -4.90626 q^{83} -13.6833 q^{85} -12.3990 q^{86} -1.08368 q^{87} +14.0589 q^{88} -5.97567 q^{89} +11.4347 q^{90} +2.98256 q^{92} -1.53125 q^{93} +14.5511 q^{94} -25.8887 q^{95} -0.613462 q^{96} -1.45550 q^{97} -13.5460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + 2q^{11} + 2q^{12} - 5q^{13} - 5q^{15} - q^{16} - 13q^{17} + 21q^{18} + 23q^{20} + q^{22} + 2q^{23} - 2q^{24} + 22q^{25} - 10q^{27} - 5q^{29} - 33q^{30} - 17q^{31} - 12q^{32} - 3q^{33} + 8q^{34} + 15q^{36} - 7q^{37} + 3q^{38} + 5q^{39} - 7q^{40} + 5q^{41} + q^{43} - 47q^{44} + 23q^{45} - 24q^{46} - 9q^{47} + 19q^{48} + 2q^{50} + 5q^{51} - 20q^{52} + 5q^{53} - 2q^{54} - 33q^{55} - 3q^{57} - 27q^{58} - 7q^{59} - 16q^{60} - 22q^{61} + 28q^{62} - 3q^{64} - 31q^{65} + 42q^{66} - 3q^{67} - 17q^{68} + 22q^{69} - 24q^{71} - 12q^{72} - 40q^{73} - 5q^{74} - 24q^{75} + 19q^{76} + 30q^{78} - 42q^{79} - 24q^{80} + 9q^{81} - q^{82} + 12q^{83} - 23q^{85} + 16q^{86} + 32q^{87} + 26q^{88} - 8q^{89} + 59q^{90} + 12q^{92} - 11q^{93} + 23q^{94} - 17q^{95} + 17q^{96} - 16q^{97} - 45q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.20098 −0.849220 −0.424610 0.905376i \(-0.639589\pi\)
−0.424610 + 0.905376i \(0.639589\pi\)
\(3\) 0.200978 0.116035 0.0580173 0.998316i \(-0.481522\pi\)
0.0580173 + 0.998316i \(0.481522\pi\)
\(4\) −0.557652 −0.278826
\(5\) 3.21704 1.43871 0.719353 0.694645i \(-0.244438\pi\)
0.719353 + 0.694645i \(0.244438\pi\)
\(6\) −0.241370 −0.0985388
\(7\) 0 0
\(8\) 3.07168 1.08600
\(9\) −2.95961 −0.986536
\(10\) −3.86360 −1.22178
\(11\) 4.57695 1.38000 0.690001 0.723809i \(-0.257610\pi\)
0.690001 + 0.723809i \(0.257610\pi\)
\(12\) −0.112076 −0.0323535
\(13\) −0.703013 −0.194981 −0.0974904 0.995236i \(-0.531082\pi\)
−0.0974904 + 0.995236i \(0.531082\pi\)
\(14\) 0 0
\(15\) 0.646554 0.166940
\(16\) −2.57372 −0.643430
\(17\) −4.25337 −1.03159 −0.515796 0.856711i \(-0.672504\pi\)
−0.515796 + 0.856711i \(0.672504\pi\)
\(18\) 3.55442 0.837786
\(19\) −8.04736 −1.84619 −0.923095 0.384571i \(-0.874349\pi\)
−0.923095 + 0.384571i \(0.874349\pi\)
\(20\) −1.79399 −0.401149
\(21\) 0 0
\(22\) −5.49681 −1.17192
\(23\) −5.34842 −1.11522 −0.557611 0.830102i \(-0.688282\pi\)
−0.557611 + 0.830102i \(0.688282\pi\)
\(24\) 0.617340 0.126014
\(25\) 5.34937 1.06987
\(26\) 0.844303 0.165581
\(27\) −1.19775 −0.230507
\(28\) 0 0
\(29\) −5.39204 −1.00128 −0.500638 0.865657i \(-0.666901\pi\)
−0.500638 + 0.865657i \(0.666901\pi\)
\(30\) −0.776497 −0.141768
\(31\) −7.61900 −1.36841 −0.684206 0.729288i \(-0.739851\pi\)
−0.684206 + 0.729288i \(0.739851\pi\)
\(32\) −3.05239 −0.539591
\(33\) 0.919865 0.160128
\(34\) 5.10820 0.876049
\(35\) 0 0
\(36\) 1.65043 0.275072
\(37\) 5.19272 0.853678 0.426839 0.904328i \(-0.359627\pi\)
0.426839 + 0.904328i \(0.359627\pi\)
\(38\) 9.66470 1.56782
\(39\) −0.141290 −0.0226245
\(40\) 9.88174 1.56244
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 10.3241 1.57441 0.787205 0.616692i \(-0.211528\pi\)
0.787205 + 0.616692i \(0.211528\pi\)
\(44\) −2.55235 −0.384781
\(45\) −9.52119 −1.41934
\(46\) 6.42333 0.947068
\(47\) −12.1160 −1.76730 −0.883651 0.468147i \(-0.844922\pi\)
−0.883651 + 0.468147i \(0.844922\pi\)
\(48\) −0.517260 −0.0746601
\(49\) 0 0
\(50\) −6.42448 −0.908559
\(51\) −0.854832 −0.119700
\(52\) 0.392037 0.0543657
\(53\) −12.2837 −1.68730 −0.843648 0.536897i \(-0.819596\pi\)
−0.843648 + 0.536897i \(0.819596\pi\)
\(54\) 1.43847 0.195751
\(55\) 14.7242 1.98542
\(56\) 0 0
\(57\) −1.61734 −0.214222
\(58\) 6.47572 0.850303
\(59\) 7.73023 1.00639 0.503195 0.864173i \(-0.332158\pi\)
0.503195 + 0.864173i \(0.332158\pi\)
\(60\) −0.360553 −0.0465471
\(61\) 2.48971 0.318774 0.159387 0.987216i \(-0.449048\pi\)
0.159387 + 0.987216i \(0.449048\pi\)
\(62\) 9.15025 1.16208
\(63\) 0 0
\(64\) 8.81329 1.10166
\(65\) −2.26162 −0.280520
\(66\) −1.10474 −0.135984
\(67\) −3.09767 −0.378440 −0.189220 0.981935i \(-0.560596\pi\)
−0.189220 + 0.981935i \(0.560596\pi\)
\(68\) 2.37190 0.287635
\(69\) −1.07491 −0.129404
\(70\) 0 0
\(71\) 5.11581 0.607135 0.303568 0.952810i \(-0.401822\pi\)
0.303568 + 0.952810i \(0.401822\pi\)
\(72\) −9.09098 −1.07138
\(73\) −4.13640 −0.484129 −0.242065 0.970260i \(-0.577825\pi\)
−0.242065 + 0.970260i \(0.577825\pi\)
\(74\) −6.23634 −0.724960
\(75\) 1.07511 0.124142
\(76\) 4.48763 0.514766
\(77\) 0 0
\(78\) 0.169686 0.0192132
\(79\) −13.7414 −1.54603 −0.773015 0.634388i \(-0.781252\pi\)
−0.773015 + 0.634388i \(0.781252\pi\)
\(80\) −8.27977 −0.925706
\(81\) 8.63810 0.959789
\(82\) −1.20098 −0.132626
\(83\) −4.90626 −0.538532 −0.269266 0.963066i \(-0.586781\pi\)
−0.269266 + 0.963066i \(0.586781\pi\)
\(84\) 0 0
\(85\) −13.6833 −1.48416
\(86\) −12.3990 −1.33702
\(87\) −1.08368 −0.116183
\(88\) 14.0589 1.49869
\(89\) −5.97567 −0.633420 −0.316710 0.948522i \(-0.602578\pi\)
−0.316710 + 0.948522i \(0.602578\pi\)
\(90\) 11.4347 1.20533
\(91\) 0 0
\(92\) 2.98256 0.310953
\(93\) −1.53125 −0.158783
\(94\) 14.5511 1.50083
\(95\) −25.8887 −2.65613
\(96\) −0.613462 −0.0626112
\(97\) −1.45550 −0.147783 −0.0738916 0.997266i \(-0.523542\pi\)
−0.0738916 + 0.997266i \(0.523542\pi\)
\(98\) 0 0
\(99\) −13.5460 −1.36142
\(100\) −2.98309 −0.298309
\(101\) −1.92425 −0.191470 −0.0957348 0.995407i \(-0.530520\pi\)
−0.0957348 + 0.995407i \(0.530520\pi\)
\(102\) 1.02663 0.101652
\(103\) 5.35990 0.528127 0.264063 0.964505i \(-0.414937\pi\)
0.264063 + 0.964505i \(0.414937\pi\)
\(104\) −2.15943 −0.211750
\(105\) 0 0
\(106\) 14.7524 1.43288
\(107\) 2.83031 0.273617 0.136808 0.990598i \(-0.456316\pi\)
0.136808 + 0.990598i \(0.456316\pi\)
\(108\) 0.667927 0.0642713
\(109\) 11.3960 1.09154 0.545768 0.837936i \(-0.316238\pi\)
0.545768 + 0.837936i \(0.316238\pi\)
\(110\) −17.6835 −1.68605
\(111\) 1.04362 0.0990561
\(112\) 0 0
\(113\) 18.4852 1.73894 0.869470 0.493986i \(-0.164461\pi\)
0.869470 + 0.493986i \(0.164461\pi\)
\(114\) 1.94239 0.181921
\(115\) −17.2061 −1.60448
\(116\) 3.00688 0.279182
\(117\) 2.08064 0.192356
\(118\) −9.28384 −0.854647
\(119\) 0 0
\(120\) 1.98601 0.181297
\(121\) 9.94845 0.904405
\(122\) −2.99008 −0.270709
\(123\) 0.200978 0.0181216
\(124\) 4.24875 0.381549
\(125\) 1.12395 0.100530
\(126\) 0 0
\(127\) 9.98152 0.885717 0.442858 0.896592i \(-0.353964\pi\)
0.442858 + 0.896592i \(0.353964\pi\)
\(128\) −4.47979 −0.395961
\(129\) 2.07491 0.182686
\(130\) 2.71616 0.238223
\(131\) 11.2796 0.985506 0.492753 0.870169i \(-0.335991\pi\)
0.492753 + 0.870169i \(0.335991\pi\)
\(132\) −0.512965 −0.0446479
\(133\) 0 0
\(134\) 3.72023 0.321379
\(135\) −3.85321 −0.331632
\(136\) −13.0650 −1.12031
\(137\) 9.73838 0.832006 0.416003 0.909363i \(-0.363431\pi\)
0.416003 + 0.909363i \(0.363431\pi\)
\(138\) 1.29095 0.109893
\(139\) 4.85307 0.411632 0.205816 0.978591i \(-0.434015\pi\)
0.205816 + 0.978591i \(0.434015\pi\)
\(140\) 0 0
\(141\) −2.43505 −0.205068
\(142\) −6.14398 −0.515591
\(143\) −3.21765 −0.269074
\(144\) 7.61720 0.634767
\(145\) −17.3464 −1.44054
\(146\) 4.96773 0.411132
\(147\) 0 0
\(148\) −2.89573 −0.238028
\(149\) −6.47268 −0.530263 −0.265131 0.964212i \(-0.585415\pi\)
−0.265131 + 0.964212i \(0.585415\pi\)
\(150\) −1.29118 −0.105424
\(151\) −16.6124 −1.35190 −0.675949 0.736948i \(-0.736266\pi\)
−0.675949 + 0.736948i \(0.736266\pi\)
\(152\) −24.7189 −2.00497
\(153\) 12.5883 1.01770
\(154\) 0 0
\(155\) −24.5107 −1.96874
\(156\) 0.0787907 0.00630831
\(157\) −21.5294 −1.71824 −0.859119 0.511777i \(-0.828988\pi\)
−0.859119 + 0.511777i \(0.828988\pi\)
\(158\) 16.5031 1.31292
\(159\) −2.46875 −0.195785
\(160\) −9.81967 −0.776313
\(161\) 0 0
\(162\) −10.3742 −0.815072
\(163\) −0.00553783 −0.000433757 0 −0.000216878 1.00000i \(-0.500069\pi\)
−0.000216878 1.00000i \(0.500069\pi\)
\(164\) −0.557652 −0.0435453
\(165\) 2.95925 0.230377
\(166\) 5.89231 0.457332
\(167\) −22.4914 −1.74044 −0.870220 0.492663i \(-0.836024\pi\)
−0.870220 + 0.492663i \(0.836024\pi\)
\(168\) 0 0
\(169\) −12.5058 −0.961983
\(170\) 16.4333 1.26038
\(171\) 23.8170 1.82133
\(172\) −5.75725 −0.438986
\(173\) −6.07961 −0.462224 −0.231112 0.972927i \(-0.574236\pi\)
−0.231112 + 0.972927i \(0.574236\pi\)
\(174\) 1.30148 0.0986646
\(175\) 0 0
\(176\) −11.7798 −0.887934
\(177\) 1.55361 0.116776
\(178\) 7.17665 0.537913
\(179\) −7.91618 −0.591683 −0.295842 0.955237i \(-0.595600\pi\)
−0.295842 + 0.955237i \(0.595600\pi\)
\(180\) 5.30951 0.395748
\(181\) 2.20401 0.163823 0.0819115 0.996640i \(-0.473898\pi\)
0.0819115 + 0.996640i \(0.473898\pi\)
\(182\) 0 0
\(183\) 0.500376 0.0369888
\(184\) −16.4286 −1.21114
\(185\) 16.7052 1.22819
\(186\) 1.83900 0.134842
\(187\) −19.4674 −1.42360
\(188\) 6.75652 0.492770
\(189\) 0 0
\(190\) 31.0918 2.25563
\(191\) −1.75601 −0.127061 −0.0635303 0.997980i \(-0.520236\pi\)
−0.0635303 + 0.997980i \(0.520236\pi\)
\(192\) 1.77128 0.127831
\(193\) 2.06376 0.148553 0.0742763 0.997238i \(-0.476335\pi\)
0.0742763 + 0.997238i \(0.476335\pi\)
\(194\) 1.74802 0.125500
\(195\) −0.454536 −0.0325500
\(196\) 0 0
\(197\) −21.2674 −1.51524 −0.757621 0.652695i \(-0.773638\pi\)
−0.757621 + 0.652695i \(0.773638\pi\)
\(198\) 16.2684 1.15615
\(199\) 6.50285 0.460975 0.230488 0.973075i \(-0.425968\pi\)
0.230488 + 0.973075i \(0.425968\pi\)
\(200\) 16.4316 1.16189
\(201\) −0.622563 −0.0439122
\(202\) 2.31098 0.162600
\(203\) 0 0
\(204\) 0.476699 0.0333756
\(205\) 3.21704 0.224688
\(206\) −6.43713 −0.448496
\(207\) 15.8292 1.10021
\(208\) 1.80936 0.125456
\(209\) −36.8323 −2.54775
\(210\) 0 0
\(211\) −10.4931 −0.722377 −0.361188 0.932493i \(-0.617629\pi\)
−0.361188 + 0.932493i \(0.617629\pi\)
\(212\) 6.85003 0.470462
\(213\) 1.02816 0.0704487
\(214\) −3.39914 −0.232361
\(215\) 33.2131 2.26511
\(216\) −3.67911 −0.250331
\(217\) 0 0
\(218\) −13.6863 −0.926953
\(219\) −0.831325 −0.0561757
\(220\) −8.21101 −0.553586
\(221\) 2.99017 0.201141
\(222\) −1.25337 −0.0841204
\(223\) −14.4757 −0.969366 −0.484683 0.874690i \(-0.661065\pi\)
−0.484683 + 0.874690i \(0.661065\pi\)
\(224\) 0 0
\(225\) −15.8321 −1.05547
\(226\) −22.2003 −1.47674
\(227\) 17.5499 1.16483 0.582414 0.812892i \(-0.302108\pi\)
0.582414 + 0.812892i \(0.302108\pi\)
\(228\) 0.901914 0.0597307
\(229\) 0.189800 0.0125423 0.00627117 0.999980i \(-0.498004\pi\)
0.00627117 + 0.999980i \(0.498004\pi\)
\(230\) 20.6641 1.36255
\(231\) 0 0
\(232\) −16.5626 −1.08739
\(233\) 2.03466 0.133295 0.0666476 0.997777i \(-0.478770\pi\)
0.0666476 + 0.997777i \(0.478770\pi\)
\(234\) −2.49881 −0.163352
\(235\) −38.9777 −2.54263
\(236\) −4.31078 −0.280608
\(237\) −2.76172 −0.179393
\(238\) 0 0
\(239\) 11.6053 0.750684 0.375342 0.926886i \(-0.377525\pi\)
0.375342 + 0.926886i \(0.377525\pi\)
\(240\) −1.66405 −0.107414
\(241\) 4.33539 0.279267 0.139633 0.990203i \(-0.455408\pi\)
0.139633 + 0.990203i \(0.455408\pi\)
\(242\) −11.9479 −0.768038
\(243\) 5.32931 0.341876
\(244\) −1.38839 −0.0888826
\(245\) 0 0
\(246\) −0.241370 −0.0153892
\(247\) 5.65740 0.359972
\(248\) −23.4032 −1.48610
\(249\) −0.986049 −0.0624883
\(250\) −1.34984 −0.0853716
\(251\) −23.3501 −1.47385 −0.736923 0.675976i \(-0.763722\pi\)
−0.736923 + 0.675976i \(0.763722\pi\)
\(252\) 0 0
\(253\) −24.4794 −1.53901
\(254\) −11.9876 −0.752168
\(255\) −2.75003 −0.172214
\(256\) −12.2465 −0.765403
\(257\) 19.7199 1.23009 0.615046 0.788491i \(-0.289138\pi\)
0.615046 + 0.788491i \(0.289138\pi\)
\(258\) −2.49192 −0.155140
\(259\) 0 0
\(260\) 1.26120 0.0782163
\(261\) 15.9583 0.987795
\(262\) −13.5466 −0.836911
\(263\) 1.67120 0.103050 0.0515252 0.998672i \(-0.483592\pi\)
0.0515252 + 0.998672i \(0.483592\pi\)
\(264\) 2.82553 0.173900
\(265\) −39.5172 −2.42752
\(266\) 0 0
\(267\) −1.20098 −0.0734986
\(268\) 1.72742 0.105519
\(269\) 24.4059 1.48805 0.744027 0.668150i \(-0.232914\pi\)
0.744027 + 0.668150i \(0.232914\pi\)
\(270\) 4.62762 0.281628
\(271\) −7.83315 −0.475830 −0.237915 0.971286i \(-0.576464\pi\)
−0.237915 + 0.971286i \(0.576464\pi\)
\(272\) 10.9470 0.663757
\(273\) 0 0
\(274\) −11.6956 −0.706555
\(275\) 24.4838 1.47643
\(276\) 0.599428 0.0360813
\(277\) −6.53052 −0.392381 −0.196190 0.980566i \(-0.562857\pi\)
−0.196190 + 0.980566i \(0.562857\pi\)
\(278\) −5.82843 −0.349566
\(279\) 22.5493 1.34999
\(280\) 0 0
\(281\) −2.23112 −0.133097 −0.0665487 0.997783i \(-0.521199\pi\)
−0.0665487 + 0.997783i \(0.521199\pi\)
\(282\) 2.92444 0.174148
\(283\) 18.2949 1.08752 0.543759 0.839242i \(-0.317001\pi\)
0.543759 + 0.839242i \(0.317001\pi\)
\(284\) −2.85285 −0.169285
\(285\) −5.20306 −0.308202
\(286\) 3.86433 0.228503
\(287\) 0 0
\(288\) 9.03387 0.532326
\(289\) 1.09112 0.0641835
\(290\) 20.8327 1.22334
\(291\) −0.292522 −0.0171480
\(292\) 2.30667 0.134988
\(293\) 7.37440 0.430817 0.215409 0.976524i \(-0.430892\pi\)
0.215409 + 0.976524i \(0.430892\pi\)
\(294\) 0 0
\(295\) 24.8685 1.44790
\(296\) 15.9504 0.927098
\(297\) −5.48203 −0.318100
\(298\) 7.77355 0.450309
\(299\) 3.76001 0.217447
\(300\) −0.599535 −0.0346142
\(301\) 0 0
\(302\) 19.9511 1.14806
\(303\) −0.386731 −0.0222171
\(304\) 20.7116 1.18789
\(305\) 8.00949 0.458622
\(306\) −15.1183 −0.864254
\(307\) 11.6587 0.665398 0.332699 0.943033i \(-0.392041\pi\)
0.332699 + 0.943033i \(0.392041\pi\)
\(308\) 0 0
\(309\) 1.07722 0.0612810
\(310\) 29.4368 1.67190
\(311\) 19.2508 1.09161 0.545806 0.837911i \(-0.316223\pi\)
0.545806 + 0.837911i \(0.316223\pi\)
\(312\) −0.433998 −0.0245703
\(313\) 3.73012 0.210839 0.105419 0.994428i \(-0.466382\pi\)
0.105419 + 0.994428i \(0.466382\pi\)
\(314\) 25.8564 1.45916
\(315\) 0 0
\(316\) 7.66293 0.431074
\(317\) −19.6027 −1.10100 −0.550499 0.834836i \(-0.685562\pi\)
−0.550499 + 0.834836i \(0.685562\pi\)
\(318\) 2.96491 0.166264
\(319\) −24.6791 −1.38176
\(320\) 28.3527 1.58497
\(321\) 0.568830 0.0317490
\(322\) 0 0
\(323\) 34.2284 1.90452
\(324\) −4.81706 −0.267614
\(325\) −3.76068 −0.208605
\(326\) 0.00665081 0.000368355 0
\(327\) 2.29034 0.126656
\(328\) 3.07168 0.169605
\(329\) 0 0
\(330\) −3.55399 −0.195641
\(331\) −23.4828 −1.29073 −0.645366 0.763873i \(-0.723295\pi\)
−0.645366 + 0.763873i \(0.723295\pi\)
\(332\) 2.73599 0.150157
\(333\) −15.3684 −0.842184
\(334\) 27.0117 1.47802
\(335\) −9.96534 −0.544465
\(336\) 0 0
\(337\) 1.20201 0.0654778 0.0327389 0.999464i \(-0.489577\pi\)
0.0327389 + 0.999464i \(0.489577\pi\)
\(338\) 15.0192 0.816934
\(339\) 3.71511 0.201777
\(340\) 7.63051 0.413822
\(341\) −34.8718 −1.88841
\(342\) −28.6037 −1.54671
\(343\) 0 0
\(344\) 31.7123 1.70981
\(345\) −3.45804 −0.186175
\(346\) 7.30148 0.392530
\(347\) −3.01800 −0.162015 −0.0810074 0.996713i \(-0.525814\pi\)
−0.0810074 + 0.996713i \(0.525814\pi\)
\(348\) 0.604317 0.0323948
\(349\) 30.2795 1.62082 0.810412 0.585861i \(-0.199243\pi\)
0.810412 + 0.585861i \(0.199243\pi\)
\(350\) 0 0
\(351\) 0.842033 0.0449444
\(352\) −13.9706 −0.744637
\(353\) −4.59065 −0.244336 −0.122168 0.992509i \(-0.538985\pi\)
−0.122168 + 0.992509i \(0.538985\pi\)
\(354\) −1.86585 −0.0991686
\(355\) 16.4578 0.873489
\(356\) 3.33235 0.176614
\(357\) 0 0
\(358\) 9.50716 0.502469
\(359\) 18.4050 0.971381 0.485691 0.874131i \(-0.338568\pi\)
0.485691 + 0.874131i \(0.338568\pi\)
\(360\) −29.2461 −1.54140
\(361\) 45.7600 2.40842
\(362\) −2.64697 −0.139122
\(363\) 1.99942 0.104942
\(364\) 0 0
\(365\) −13.3070 −0.696519
\(366\) −0.600940 −0.0314116
\(367\) −17.0270 −0.888802 −0.444401 0.895828i \(-0.646583\pi\)
−0.444401 + 0.895828i \(0.646583\pi\)
\(368\) 13.7653 0.717567
\(369\) −2.95961 −0.154071
\(370\) −20.0626 −1.04300
\(371\) 0 0
\(372\) 0.853905 0.0442729
\(373\) 26.3381 1.36373 0.681867 0.731476i \(-0.261168\pi\)
0.681867 + 0.731476i \(0.261168\pi\)
\(374\) 23.3800 1.20895
\(375\) 0.225890 0.0116649
\(376\) −37.2165 −1.91930
\(377\) 3.79067 0.195230
\(378\) 0 0
\(379\) −25.4631 −1.30795 −0.653975 0.756516i \(-0.726900\pi\)
−0.653975 + 0.756516i \(0.726900\pi\)
\(380\) 14.4369 0.740597
\(381\) 2.00606 0.102774
\(382\) 2.10893 0.107902
\(383\) −26.6465 −1.36157 −0.680786 0.732483i \(-0.738361\pi\)
−0.680786 + 0.732483i \(0.738361\pi\)
\(384\) −0.900338 −0.0459452
\(385\) 0 0
\(386\) −2.47853 −0.126154
\(387\) −30.5553 −1.55321
\(388\) 0.811660 0.0412058
\(389\) −33.7017 −1.70874 −0.854372 0.519662i \(-0.826058\pi\)
−0.854372 + 0.519662i \(0.826058\pi\)
\(390\) 0.545888 0.0276421
\(391\) 22.7488 1.15045
\(392\) 0 0
\(393\) 2.26695 0.114353
\(394\) 25.5417 1.28677
\(395\) −44.2067 −2.22428
\(396\) 7.55394 0.379600
\(397\) 24.7242 1.24087 0.620437 0.784256i \(-0.286955\pi\)
0.620437 + 0.784256i \(0.286955\pi\)
\(398\) −7.80978 −0.391469
\(399\) 0 0
\(400\) −13.7678 −0.688389
\(401\) 14.7100 0.734581 0.367291 0.930106i \(-0.380285\pi\)
0.367291 + 0.930106i \(0.380285\pi\)
\(402\) 0.747684 0.0372911
\(403\) 5.35626 0.266814
\(404\) 1.07306 0.0533867
\(405\) 27.7892 1.38085
\(406\) 0 0
\(407\) 23.7668 1.17808
\(408\) −2.62577 −0.129995
\(409\) −18.8735 −0.933233 −0.466616 0.884460i \(-0.654527\pi\)
−0.466616 + 0.884460i \(0.654527\pi\)
\(410\) −3.86360 −0.190810
\(411\) 1.95720 0.0965414
\(412\) −2.98896 −0.147256
\(413\) 0 0
\(414\) −19.0105 −0.934317
\(415\) −15.7837 −0.774789
\(416\) 2.14587 0.105210
\(417\) 0.975359 0.0477635
\(418\) 44.2348 2.16360
\(419\) −28.7795 −1.40597 −0.702985 0.711205i \(-0.748150\pi\)
−0.702985 + 0.711205i \(0.748150\pi\)
\(420\) 0 0
\(421\) 36.0120 1.75512 0.877559 0.479468i \(-0.159171\pi\)
0.877559 + 0.479468i \(0.159171\pi\)
\(422\) 12.6020 0.613456
\(423\) 35.8586 1.74351
\(424\) −37.7316 −1.83241
\(425\) −22.7528 −1.10368
\(426\) −1.23480 −0.0598264
\(427\) 0 0
\(428\) −1.57833 −0.0762915
\(429\) −0.646677 −0.0312219
\(430\) −39.8881 −1.92358
\(431\) 9.23519 0.444843 0.222422 0.974951i \(-0.428604\pi\)
0.222422 + 0.974951i \(0.428604\pi\)
\(432\) 3.08267 0.148315
\(433\) 18.1355 0.871537 0.435768 0.900059i \(-0.356477\pi\)
0.435768 + 0.900059i \(0.356477\pi\)
\(434\) 0 0
\(435\) −3.48625 −0.167153
\(436\) −6.35499 −0.304349
\(437\) 43.0406 2.05891
\(438\) 0.998402 0.0477055
\(439\) 6.79012 0.324075 0.162037 0.986785i \(-0.448193\pi\)
0.162037 + 0.986785i \(0.448193\pi\)
\(440\) 45.2282 2.15617
\(441\) 0 0
\(442\) −3.59113 −0.170813
\(443\) 34.2284 1.62624 0.813119 0.582097i \(-0.197768\pi\)
0.813119 + 0.582097i \(0.197768\pi\)
\(444\) −0.581978 −0.0276194
\(445\) −19.2240 −0.911306
\(446\) 17.3850 0.823204
\(447\) −1.30086 −0.0615288
\(448\) 0 0
\(449\) −2.19148 −0.103422 −0.0517112 0.998662i \(-0.516468\pi\)
−0.0517112 + 0.998662i \(0.516468\pi\)
\(450\) 19.0139 0.896326
\(451\) 4.57695 0.215520
\(452\) −10.3083 −0.484862
\(453\) −3.33872 −0.156867
\(454\) −21.0770 −0.989195
\(455\) 0 0
\(456\) −4.96796 −0.232646
\(457\) 15.2932 0.715387 0.357694 0.933839i \(-0.383563\pi\)
0.357694 + 0.933839i \(0.383563\pi\)
\(458\) −0.227946 −0.0106512
\(459\) 5.09446 0.237789
\(460\) 9.59502 0.447370
\(461\) 7.37440 0.343460 0.171730 0.985144i \(-0.445064\pi\)
0.171730 + 0.985144i \(0.445064\pi\)
\(462\) 0 0
\(463\) −25.3329 −1.17732 −0.588661 0.808380i \(-0.700345\pi\)
−0.588661 + 0.808380i \(0.700345\pi\)
\(464\) 13.8776 0.644251
\(465\) −4.92610 −0.228442
\(466\) −2.44358 −0.113197
\(467\) −9.05902 −0.419201 −0.209601 0.977787i \(-0.567216\pi\)
−0.209601 + 0.977787i \(0.567216\pi\)
\(468\) −1.16028 −0.0536338
\(469\) 0 0
\(470\) 46.8114 2.15925
\(471\) −4.32694 −0.199375
\(472\) 23.7448 1.09294
\(473\) 47.2528 2.17269
\(474\) 3.31676 0.152344
\(475\) −43.0483 −1.97519
\(476\) 0 0
\(477\) 36.3549 1.66458
\(478\) −13.9377 −0.637496
\(479\) −15.7297 −0.718708 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(480\) −1.97353 −0.0900791
\(481\) −3.65055 −0.166451
\(482\) −5.20670 −0.237159
\(483\) 0 0
\(484\) −5.54778 −0.252172
\(485\) −4.68239 −0.212617
\(486\) −6.40039 −0.290327
\(487\) −11.5758 −0.524549 −0.262275 0.964993i \(-0.584473\pi\)
−0.262275 + 0.964993i \(0.584473\pi\)
\(488\) 7.64759 0.346190
\(489\) −0.00111298 −5.03308e−5 0
\(490\) 0 0
\(491\) −15.2590 −0.688628 −0.344314 0.938855i \(-0.611889\pi\)
−0.344314 + 0.938855i \(0.611889\pi\)
\(492\) −0.112076 −0.00505276
\(493\) 22.9343 1.03291
\(494\) −6.79441 −0.305695
\(495\) −43.5780 −1.95869
\(496\) 19.6092 0.880478
\(497\) 0 0
\(498\) 1.18422 0.0530663
\(499\) 8.35644 0.374086 0.187043 0.982352i \(-0.440110\pi\)
0.187043 + 0.982352i \(0.440110\pi\)
\(500\) −0.626776 −0.0280303
\(501\) −4.52028 −0.201951
\(502\) 28.0430 1.25162
\(503\) −2.53984 −0.113246 −0.0566230 0.998396i \(-0.518033\pi\)
−0.0566230 + 0.998396i \(0.518033\pi\)
\(504\) 0 0
\(505\) −6.19038 −0.275468
\(506\) 29.3992 1.30696
\(507\) −2.51338 −0.111623
\(508\) −5.56622 −0.246961
\(509\) 6.48190 0.287305 0.143653 0.989628i \(-0.454115\pi\)
0.143653 + 0.989628i \(0.454115\pi\)
\(510\) 3.30273 0.146247
\(511\) 0 0
\(512\) 23.6673 1.04596
\(513\) 9.63871 0.425560
\(514\) −23.6831 −1.04462
\(515\) 17.2430 0.759819
\(516\) −1.15708 −0.0509376
\(517\) −55.4543 −2.43888
\(518\) 0 0
\(519\) −1.22187 −0.0536340
\(520\) −6.94700 −0.304646
\(521\) 1.36749 0.0599107 0.0299553 0.999551i \(-0.490463\pi\)
0.0299553 + 0.999551i \(0.490463\pi\)
\(522\) −19.1656 −0.838855
\(523\) −12.9198 −0.564944 −0.282472 0.959276i \(-0.591154\pi\)
−0.282472 + 0.959276i \(0.591154\pi\)
\(524\) −6.29011 −0.274785
\(525\) 0 0
\(526\) −2.00707 −0.0875123
\(527\) 32.4064 1.41164
\(528\) −2.36747 −0.103031
\(529\) 5.60555 0.243720
\(530\) 47.4593 2.06150
\(531\) −22.8785 −0.992841
\(532\) 0 0
\(533\) −0.703013 −0.0304509
\(534\) 1.44235 0.0624165
\(535\) 9.10525 0.393654
\(536\) −9.51506 −0.410988
\(537\) −1.59098 −0.0686557
\(538\) −29.3109 −1.26368
\(539\) 0 0
\(540\) 2.14875 0.0924676
\(541\) −4.47397 −0.192351 −0.0961756 0.995364i \(-0.530661\pi\)
−0.0961756 + 0.995364i \(0.530661\pi\)
\(542\) 9.40744 0.404084
\(543\) 0.442958 0.0190091
\(544\) 12.9829 0.556638
\(545\) 36.6613 1.57040
\(546\) 0 0
\(547\) −22.3133 −0.954048 −0.477024 0.878890i \(-0.658285\pi\)
−0.477024 + 0.878890i \(0.658285\pi\)
\(548\) −5.43063 −0.231985
\(549\) −7.36855 −0.314482
\(550\) −29.4045 −1.25381
\(551\) 43.3917 1.84855
\(552\) −3.30179 −0.140534
\(553\) 0 0
\(554\) 7.84300 0.333217
\(555\) 3.35738 0.142513
\(556\) −2.70633 −0.114774
\(557\) 15.3755 0.651482 0.325741 0.945459i \(-0.394386\pi\)
0.325741 + 0.945459i \(0.394386\pi\)
\(558\) −27.0812 −1.14644
\(559\) −7.25797 −0.306979
\(560\) 0 0
\(561\) −3.91252 −0.165187
\(562\) 2.67952 0.113029
\(563\) 4.13910 0.174442 0.0872212 0.996189i \(-0.472201\pi\)
0.0872212 + 0.996189i \(0.472201\pi\)
\(564\) 1.35791 0.0571783
\(565\) 59.4677 2.50182
\(566\) −21.9717 −0.923541
\(567\) 0 0
\(568\) 15.7142 0.659351
\(569\) 20.9479 0.878182 0.439091 0.898443i \(-0.355301\pi\)
0.439091 + 0.898443i \(0.355301\pi\)
\(570\) 6.24875 0.261731
\(571\) −20.5728 −0.860947 −0.430473 0.902603i \(-0.641653\pi\)
−0.430473 + 0.902603i \(0.641653\pi\)
\(572\) 1.79433 0.0750248
\(573\) −0.352919 −0.0147434
\(574\) 0 0
\(575\) −28.6107 −1.19315
\(576\) −26.0839 −1.08683
\(577\) 26.3061 1.09514 0.547568 0.836761i \(-0.315554\pi\)
0.547568 + 0.836761i \(0.315554\pi\)
\(578\) −1.31041 −0.0545059
\(579\) 0.414770 0.0172372
\(580\) 9.67327 0.401661
\(581\) 0 0
\(582\) 0.351313 0.0145624
\(583\) −56.2218 −2.32847
\(584\) −12.7057 −0.525766
\(585\) 6.69352 0.276743
\(586\) −8.85649 −0.365858
\(587\) 0.835878 0.0345004 0.0172502 0.999851i \(-0.494509\pi\)
0.0172502 + 0.999851i \(0.494509\pi\)
\(588\) 0 0
\(589\) 61.3128 2.52635
\(590\) −29.8665 −1.22959
\(591\) −4.27428 −0.175820
\(592\) −13.3646 −0.549282
\(593\) 5.90293 0.242404 0.121202 0.992628i \(-0.461325\pi\)
0.121202 + 0.992628i \(0.461325\pi\)
\(594\) 6.58380 0.270137
\(595\) 0 0
\(596\) 3.60951 0.147851
\(597\) 1.30693 0.0534890
\(598\) −4.51568 −0.184660
\(599\) −20.7273 −0.846896 −0.423448 0.905920i \(-0.639180\pi\)
−0.423448 + 0.905920i \(0.639180\pi\)
\(600\) 3.30238 0.134819
\(601\) −4.69241 −0.191407 −0.0957037 0.995410i \(-0.530510\pi\)
−0.0957037 + 0.995410i \(0.530510\pi\)
\(602\) 0 0
\(603\) 9.16789 0.373345
\(604\) 9.26395 0.376945
\(605\) 32.0046 1.30117
\(606\) 0.464455 0.0188672
\(607\) −27.8857 −1.13184 −0.565922 0.824459i \(-0.691480\pi\)
−0.565922 + 0.824459i \(0.691480\pi\)
\(608\) 24.5637 0.996188
\(609\) 0 0
\(610\) −9.61922 −0.389471
\(611\) 8.51771 0.344590
\(612\) −7.01989 −0.283762
\(613\) −15.6076 −0.630383 −0.315192 0.949028i \(-0.602069\pi\)
−0.315192 + 0.949028i \(0.602069\pi\)
\(614\) −14.0019 −0.565069
\(615\) 0.646554 0.0260716
\(616\) 0 0
\(617\) 12.2414 0.492820 0.246410 0.969166i \(-0.420749\pi\)
0.246410 + 0.969166i \(0.420749\pi\)
\(618\) −1.29372 −0.0520410
\(619\) −23.4809 −0.943778 −0.471889 0.881658i \(-0.656428\pi\)
−0.471889 + 0.881658i \(0.656428\pi\)
\(620\) 13.6684 0.548937
\(621\) 6.40606 0.257066
\(622\) −23.1198 −0.927019
\(623\) 0 0
\(624\) 0.363641 0.0145573
\(625\) −23.1311 −0.925243
\(626\) −4.47979 −0.179048
\(627\) −7.40248 −0.295627
\(628\) 12.0059 0.479090
\(629\) −22.0865 −0.880648
\(630\) 0 0
\(631\) −0.0226990 −0.000903631 0 −0.000451816 1.00000i \(-0.500144\pi\)
−0.000451816 1.00000i \(0.500144\pi\)
\(632\) −42.2093 −1.67899
\(633\) −2.10889 −0.0838207
\(634\) 23.5424 0.934989
\(635\) 32.1110 1.27429
\(636\) 1.37670 0.0545899
\(637\) 0 0
\(638\) 29.6390 1.17342
\(639\) −15.1408 −0.598961
\(640\) −14.4117 −0.569671
\(641\) 38.0777 1.50398 0.751989 0.659175i \(-0.229095\pi\)
0.751989 + 0.659175i \(0.229095\pi\)
\(642\) −0.683152 −0.0269619
\(643\) −24.2224 −0.955239 −0.477619 0.878567i \(-0.658500\pi\)
−0.477619 + 0.878567i \(0.658500\pi\)
\(644\) 0 0
\(645\) 6.67509 0.262831
\(646\) −41.1075 −1.61735
\(647\) 44.1321 1.73501 0.867507 0.497426i \(-0.165721\pi\)
0.867507 + 0.497426i \(0.165721\pi\)
\(648\) 26.5335 1.04234
\(649\) 35.3809 1.38882
\(650\) 4.51649 0.177151
\(651\) 0 0
\(652\) 0.00308819 0.000120943 0
\(653\) −3.76579 −0.147367 −0.0736833 0.997282i \(-0.523475\pi\)
−0.0736833 + 0.997282i \(0.523475\pi\)
\(654\) −2.75064 −0.107559
\(655\) 36.2871 1.41785
\(656\) −2.57372 −0.100487
\(657\) 12.2421 0.477611
\(658\) 0 0
\(659\) −21.1980 −0.825757 −0.412878 0.910786i \(-0.635477\pi\)
−0.412878 + 0.910786i \(0.635477\pi\)
\(660\) −1.65023 −0.0642351
\(661\) 11.3027 0.439624 0.219812 0.975542i \(-0.429456\pi\)
0.219812 + 0.975542i \(0.429456\pi\)
\(662\) 28.2023 1.09612
\(663\) 0.600958 0.0233393
\(664\) −15.0705 −0.584848
\(665\) 0 0
\(666\) 18.4571 0.715199
\(667\) 28.8389 1.11664
\(668\) 12.5424 0.485280
\(669\) −2.90930 −0.112480
\(670\) 11.9682 0.462370
\(671\) 11.3953 0.439909
\(672\) 0 0
\(673\) 18.3138 0.705946 0.352973 0.935634i \(-0.385171\pi\)
0.352973 + 0.935634i \(0.385171\pi\)
\(674\) −1.44359 −0.0556050
\(675\) −6.40721 −0.246614
\(676\) 6.97387 0.268226
\(677\) 15.9963 0.614788 0.307394 0.951582i \(-0.400543\pi\)
0.307394 + 0.951582i \(0.400543\pi\)
\(678\) −4.46177 −0.171353
\(679\) 0 0
\(680\) −42.0307 −1.61180
\(681\) 3.52714 0.135160
\(682\) 41.8802 1.60368
\(683\) 6.75437 0.258449 0.129224 0.991615i \(-0.458751\pi\)
0.129224 + 0.991615i \(0.458751\pi\)
\(684\) −13.2816 −0.507835
\(685\) 31.3288 1.19701
\(686\) 0 0
\(687\) 0.0381456 0.00145534
\(688\) −26.5713 −1.01302
\(689\) 8.63560 0.328990
\(690\) 4.15303 0.158103
\(691\) −8.11399 −0.308671 −0.154335 0.988019i \(-0.549324\pi\)
−0.154335 + 0.988019i \(0.549324\pi\)
\(692\) 3.39031 0.128880
\(693\) 0 0
\(694\) 3.62455 0.137586
\(695\) 15.6125 0.592217
\(696\) −3.32872 −0.126175
\(697\) −4.25337 −0.161108
\(698\) −36.3650 −1.37644
\(699\) 0.408922 0.0154668
\(700\) 0 0
\(701\) −49.0400 −1.85221 −0.926107 0.377260i \(-0.876866\pi\)
−0.926107 + 0.377260i \(0.876866\pi\)
\(702\) −1.01126 −0.0381677
\(703\) −41.7877 −1.57605
\(704\) 40.3380 1.52029
\(705\) −7.83366 −0.295033
\(706\) 5.51327 0.207495
\(707\) 0 0
\(708\) −0.866372 −0.0325602
\(709\) 10.8982 0.409292 0.204646 0.978836i \(-0.434396\pi\)
0.204646 + 0.978836i \(0.434396\pi\)
\(710\) −19.7654 −0.741784
\(711\) 40.6692 1.52521
\(712\) −18.3554 −0.687897
\(713\) 40.7496 1.52608
\(714\) 0 0
\(715\) −10.3513 −0.387118
\(716\) 4.41448 0.164977
\(717\) 2.33241 0.0871053
\(718\) −22.1040 −0.824916
\(719\) −0.0450043 −0.00167838 −0.000839188 1.00000i \(-0.500267\pi\)
−0.000839188 1.00000i \(0.500267\pi\)
\(720\) 24.5049 0.913243
\(721\) 0 0
\(722\) −54.9567 −2.04528
\(723\) 0.871316 0.0324046
\(724\) −1.22907 −0.0456782
\(725\) −28.8440 −1.07124
\(726\) −2.40126 −0.0891190
\(727\) 29.3527 1.08863 0.544315 0.838881i \(-0.316790\pi\)
0.544315 + 0.838881i \(0.316790\pi\)
\(728\) 0 0
\(729\) −24.8432 −0.920120
\(730\) 15.9814 0.591498
\(731\) −43.9121 −1.62415
\(732\) −0.279036 −0.0103135
\(733\) −19.0948 −0.705281 −0.352640 0.935759i \(-0.614716\pi\)
−0.352640 + 0.935759i \(0.614716\pi\)
\(734\) 20.4490 0.754788
\(735\) 0 0
\(736\) 16.3254 0.601764
\(737\) −14.1779 −0.522249
\(738\) 3.55442 0.130840
\(739\) −29.0981 −1.07039 −0.535196 0.844728i \(-0.679762\pi\)
−0.535196 + 0.844728i \(0.679762\pi\)
\(740\) −9.31570 −0.342452
\(741\) 1.13701 0.0417692
\(742\) 0 0
\(743\) −10.9674 −0.402354 −0.201177 0.979555i \(-0.564477\pi\)
−0.201177 + 0.979555i \(0.564477\pi\)
\(744\) −4.70351 −0.172439
\(745\) −20.8229 −0.762892
\(746\) −31.6314 −1.15811
\(747\) 14.5206 0.531281
\(748\) 10.8561 0.396937
\(749\) 0 0
\(750\) −0.271289 −0.00990606
\(751\) 33.9703 1.23959 0.619797 0.784762i \(-0.287215\pi\)
0.619797 + 0.784762i \(0.287215\pi\)
\(752\) 31.1832 1.13713
\(753\) −4.69286 −0.171017
\(754\) −4.55251 −0.165793
\(755\) −53.4428 −1.94498
\(756\) 0 0
\(757\) 7.52333 0.273440 0.136720 0.990610i \(-0.456344\pi\)
0.136720 + 0.990610i \(0.456344\pi\)
\(758\) 30.5806 1.11074
\(759\) −4.91982 −0.178578
\(760\) −79.5219 −2.88456
\(761\) 23.8739 0.865428 0.432714 0.901531i \(-0.357556\pi\)
0.432714 + 0.901531i \(0.357556\pi\)
\(762\) −2.40924 −0.0872775
\(763\) 0 0
\(764\) 0.979244 0.0354278
\(765\) 40.4971 1.46418
\(766\) 32.0018 1.15627
\(767\) −5.43446 −0.196227
\(768\) −2.46126 −0.0888132
\(769\) −2.21793 −0.0799805 −0.0399903 0.999200i \(-0.512733\pi\)
−0.0399903 + 0.999200i \(0.512733\pi\)
\(770\) 0 0
\(771\) 3.96325 0.142733
\(772\) −1.15086 −0.0414204
\(773\) −18.4207 −0.662547 −0.331274 0.943535i \(-0.607478\pi\)
−0.331274 + 0.943535i \(0.607478\pi\)
\(774\) 36.6962 1.31902
\(775\) −40.7569 −1.46403
\(776\) −4.47082 −0.160493
\(777\) 0 0
\(778\) 40.4750 1.45110
\(779\) −8.04736 −0.288327
\(780\) 0.253473 0.00907580
\(781\) 23.4148 0.837848
\(782\) −27.3208 −0.976989
\(783\) 6.45831 0.230801
\(784\) 0 0
\(785\) −69.2612 −2.47204
\(786\) −2.72256 −0.0971106
\(787\) −18.9009 −0.673745 −0.336873 0.941550i \(-0.609369\pi\)
−0.336873 + 0.941550i \(0.609369\pi\)
\(788\) 11.8598 0.422489
\(789\) 0.335873 0.0119574
\(790\) 53.0913 1.88890
\(791\) 0 0
\(792\) −41.6089 −1.47851
\(793\) −1.75030 −0.0621548
\(794\) −29.6933 −1.05377
\(795\) −7.94208 −0.281677
\(796\) −3.62633 −0.128532
\(797\) 1.21120 0.0429028 0.0214514 0.999770i \(-0.493171\pi\)
0.0214514 + 0.999770i \(0.493171\pi\)
\(798\) 0 0
\(799\) 51.5338 1.82313
\(800\) −16.3284 −0.577295
\(801\) 17.6857 0.624892
\(802\) −17.6664 −0.623821
\(803\) −18.9321 −0.668099
\(804\) 0.347174 0.0122439
\(805\) 0 0
\(806\) −6.43275 −0.226584
\(807\) 4.90504 0.172666
\(808\) −5.91067 −0.207937
\(809\) −1.52068 −0.0534643 −0.0267322 0.999643i \(-0.508510\pi\)
−0.0267322 + 0.999643i \(0.508510\pi\)
\(810\) −33.3742 −1.17265
\(811\) 27.6060 0.969379 0.484689 0.874686i \(-0.338933\pi\)
0.484689 + 0.874686i \(0.338933\pi\)
\(812\) 0 0
\(813\) −1.57429 −0.0552127
\(814\) −28.5434 −1.00045
\(815\) −0.0178155 −0.000624048 0
\(816\) 2.20010 0.0770188
\(817\) −83.0817 −2.90666
\(818\) 22.6666 0.792519
\(819\) 0 0
\(820\) −1.79399 −0.0626489
\(821\) 50.0892 1.74813 0.874063 0.485813i \(-0.161476\pi\)
0.874063 + 0.485813i \(0.161476\pi\)
\(822\) −2.35055 −0.0819849
\(823\) −10.0645 −0.350826 −0.175413 0.984495i \(-0.556126\pi\)
−0.175413 + 0.984495i \(0.556126\pi\)
\(824\) 16.4639 0.573548
\(825\) 4.92070 0.171317
\(826\) 0 0
\(827\) −7.95416 −0.276593 −0.138297 0.990391i \(-0.544163\pi\)
−0.138297 + 0.990391i \(0.544163\pi\)
\(828\) −8.82720 −0.306766
\(829\) −50.9978 −1.77123 −0.885614 0.464422i \(-0.846262\pi\)
−0.885614 + 0.464422i \(0.846262\pi\)
\(830\) 18.9558 0.657966
\(831\) −1.31249 −0.0455297
\(832\) −6.19586 −0.214803
\(833\) 0 0
\(834\) −1.17138 −0.0405617
\(835\) −72.3560 −2.50398
\(836\) 20.5396 0.710378
\(837\) 9.12565 0.315429
\(838\) 34.5635 1.19398
\(839\) 10.8839 0.375753 0.187877 0.982193i \(-0.439839\pi\)
0.187877 + 0.982193i \(0.439839\pi\)
\(840\) 0 0
\(841\) 0.0740626 0.00255388
\(842\) −43.2496 −1.49048
\(843\) −0.448405 −0.0154439
\(844\) 5.85152 0.201418
\(845\) −40.2316 −1.38401
\(846\) −43.0654 −1.48062
\(847\) 0 0
\(848\) 31.6148 1.08566
\(849\) 3.67686 0.126190
\(850\) 27.3257 0.937263
\(851\) −27.7728 −0.952040
\(852\) −0.573358 −0.0196429
\(853\) 13.4544 0.460670 0.230335 0.973111i \(-0.426018\pi\)
0.230335 + 0.973111i \(0.426018\pi\)
\(854\) 0 0
\(855\) 76.6204 2.62036
\(856\) 8.69383 0.297149
\(857\) −39.6908 −1.35581 −0.677906 0.735149i \(-0.737112\pi\)
−0.677906 + 0.735149i \(0.737112\pi\)
\(858\) 0.776645 0.0265142
\(859\) −23.5262 −0.802705 −0.401352 0.915924i \(-0.631460\pi\)
−0.401352 + 0.915924i \(0.631460\pi\)
\(860\) −18.5213 −0.631572
\(861\) 0 0
\(862\) −11.0913 −0.377770
\(863\) −39.1998 −1.33438 −0.667189 0.744888i \(-0.732503\pi\)
−0.667189 + 0.744888i \(0.732503\pi\)
\(864\) 3.65599 0.124379
\(865\) −19.5584 −0.665005
\(866\) −21.7803 −0.740126
\(867\) 0.219291 0.00744751
\(868\) 0 0
\(869\) −62.8937 −2.13352
\(870\) 4.18690 0.141949
\(871\) 2.17770 0.0737886
\(872\) 35.0048 1.18541
\(873\) 4.30770 0.145793
\(874\) −51.6908 −1.74847
\(875\) 0 0
\(876\) 0.463590 0.0156633
\(877\) 24.7835 0.836879 0.418439 0.908245i \(-0.362577\pi\)
0.418439 + 0.908245i \(0.362577\pi\)
\(878\) −8.15478 −0.275211
\(879\) 1.48209 0.0499897
\(880\) −37.8961 −1.27748
\(881\) −33.8534 −1.14055 −0.570275 0.821454i \(-0.693163\pi\)
−0.570275 + 0.821454i \(0.693163\pi\)
\(882\) 0 0
\(883\) −9.96707 −0.335419 −0.167709 0.985837i \(-0.553637\pi\)
−0.167709 + 0.985837i \(0.553637\pi\)
\(884\) −1.66748 −0.0560833
\(885\) 4.99802 0.168007
\(886\) −41.1075 −1.38103
\(887\) 26.9208 0.903910 0.451955 0.892041i \(-0.350727\pi\)
0.451955 + 0.892041i \(0.350727\pi\)
\(888\) 3.20567 0.107575
\(889\) 0 0
\(890\) 23.0876 0.773898
\(891\) 39.5361 1.32451
\(892\) 8.07242 0.270285
\(893\) 97.5018 3.26277
\(894\) 1.56231 0.0522515
\(895\) −25.4667 −0.851258
\(896\) 0 0
\(897\) 0.755678 0.0252313
\(898\) 2.63192 0.0878284
\(899\) 41.0819 1.37016
\(900\) 8.82878 0.294293
\(901\) 52.2471 1.74060
\(902\) −5.49681 −0.183024
\(903\) 0 0
\(904\) 56.7806 1.88850
\(905\) 7.09041 0.235693
\(906\) 4.00973 0.133214
\(907\) −26.0776 −0.865894 −0.432947 0.901419i \(-0.642526\pi\)
−0.432947 + 0.901419i \(0.642526\pi\)
\(908\) −9.78674 −0.324785
\(909\) 5.69501 0.188892
\(910\) 0 0
\(911\) −34.1407 −1.13113 −0.565566 0.824703i \(-0.691342\pi\)
−0.565566 + 0.824703i \(0.691342\pi\)
\(912\) 4.16258 0.137837
\(913\) −22.4557 −0.743175
\(914\) −18.3668 −0.607521
\(915\) 1.60973 0.0532160
\(916\) −0.105842 −0.00349713
\(917\) 0 0
\(918\) −6.11834 −0.201935
\(919\) 11.7633 0.388034 0.194017 0.980998i \(-0.437848\pi\)
0.194017 + 0.980998i \(0.437848\pi\)
\(920\) −52.8517 −1.74247
\(921\) 2.34314 0.0772091
\(922\) −8.85649 −0.291673
\(923\) −3.59648 −0.118380
\(924\) 0 0
\(925\) 27.7778 0.913328
\(926\) 30.4243 0.999805
\(927\) −15.8632 −0.521016
\(928\) 16.4586 0.540280
\(929\) 21.9176 0.719093 0.359547 0.933127i \(-0.382931\pi\)
0.359547 + 0.933127i \(0.382931\pi\)
\(930\) 5.91613 0.193998
\(931\) 0 0
\(932\) −1.13463 −0.0371662
\(933\) 3.86898 0.126665
\(934\) 10.8797 0.355994
\(935\) −62.6276 −2.04814
\(936\) 6.39108 0.208899
\(937\) 17.5365 0.572892 0.286446 0.958096i \(-0.407526\pi\)
0.286446 + 0.958096i \(0.407526\pi\)
\(938\) 0 0
\(939\) 0.749671 0.0244646
\(940\) 21.7360 0.708951
\(941\) 47.1279 1.53633 0.768163 0.640255i \(-0.221171\pi\)
0.768163 + 0.640255i \(0.221171\pi\)
\(942\) 5.19656 0.169313
\(943\) −5.34842 −0.174168
\(944\) −19.8955 −0.647542
\(945\) 0 0
\(946\) −56.7496 −1.84509
\(947\) 15.4167 0.500976 0.250488 0.968120i \(-0.419409\pi\)
0.250488 + 0.968120i \(0.419409\pi\)
\(948\) 1.54008 0.0500194
\(949\) 2.90794 0.0943959
\(950\) 51.7001 1.67737
\(951\) −3.93971 −0.127754
\(952\) 0 0
\(953\) 24.6552 0.798660 0.399330 0.916807i \(-0.369243\pi\)
0.399330 + 0.916807i \(0.369243\pi\)
\(954\) −43.6615 −1.41359
\(955\) −5.64917 −0.182803
\(956\) −6.47172 −0.209310
\(957\) −4.95995 −0.160332
\(958\) 18.8910 0.610341
\(959\) 0 0
\(960\) 5.69827 0.183911
\(961\) 27.0492 0.872554
\(962\) 4.38423 0.141353
\(963\) −8.37662 −0.269933
\(964\) −2.41764 −0.0778669
\(965\) 6.63920 0.213723
\(966\) 0 0
\(967\) −28.2925 −0.909826 −0.454913 0.890536i \(-0.650330\pi\)
−0.454913 + 0.890536i \(0.650330\pi\)
\(968\) 30.5585 0.982187
\(969\) 6.87914 0.220990
\(970\) 5.62345 0.180558
\(971\) −48.5600 −1.55837 −0.779183 0.626797i \(-0.784366\pi\)
−0.779183 + 0.626797i \(0.784366\pi\)
\(972\) −2.97190 −0.0953239
\(973\) 0 0
\(974\) 13.9023 0.445458
\(975\) −0.755813 −0.0242054
\(976\) −6.40780 −0.205109
\(977\) 45.7403 1.46336 0.731682 0.681647i \(-0.238736\pi\)
0.731682 + 0.681647i \(0.238736\pi\)
\(978\) 0.00133667 4.27419e−5 0
\(979\) −27.3504 −0.874121
\(980\) 0 0
\(981\) −33.7276 −1.07684
\(982\) 18.3257 0.584797
\(983\) −15.9534 −0.508834 −0.254417 0.967095i \(-0.581884\pi\)
−0.254417 + 0.967095i \(0.581884\pi\)
\(984\) 0.617340 0.0196801
\(985\) −68.4183 −2.17999
\(986\) −27.5436 −0.877167
\(987\) 0 0
\(988\) −3.15486 −0.100370
\(989\) −55.2175 −1.75582
\(990\) 52.3362 1.66335
\(991\) −40.8830 −1.29869 −0.649346 0.760493i \(-0.724957\pi\)
−0.649346 + 0.760493i \(0.724957\pi\)
\(992\) 23.2561 0.738383
\(993\) −4.71953 −0.149770
\(994\) 0 0
\(995\) 20.9200 0.663208
\(996\) 0.549873 0.0174234
\(997\) −4.18063 −0.132402 −0.0662010 0.997806i \(-0.521088\pi\)
−0.0662010 + 0.997806i \(0.521088\pi\)
\(998\) −10.0359 −0.317681
\(999\) −6.21957 −0.196779
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 2009.2.a.n.1.2 5
7.6 odd 2 287.2.a.e.1.2 5
21.20 even 2 2583.2.a.r.1.4 5
28.27 even 2 4592.2.a.bb.1.4 5
35.34 odd 2 7175.2.a.n.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.a.e.1.2 5 7.6 odd 2
2009.2.a.n.1.2 5 1.1 even 1 trivial
2583.2.a.r.1.4 5 21.20 even 2
4592.2.a.bb.1.4 5 28.27 even 2
7175.2.a.n.1.4 5 35.34 odd 2