Properties

Label 5054.2.a.bj.1.3
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 3x^{8} - 15x^{7} + 40x^{6} + 81x^{5} - 162x^{4} - 205x^{3} + 204x^{2} + 210x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.18756\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.18756 q^{3} +1.00000 q^{4} -0.616139 q^{5} +2.18756 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.78540 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.18756 q^{3} +1.00000 q^{4} -0.616139 q^{5} +2.18756 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.78540 q^{9} +0.616139 q^{10} +3.42998 q^{11} -2.18756 q^{12} -6.38112 q^{13} +1.00000 q^{14} +1.34784 q^{15} +1.00000 q^{16} +2.47999 q^{17} -1.78540 q^{18} -0.616139 q^{20} +2.18756 q^{21} -3.42998 q^{22} +2.70527 q^{23} +2.18756 q^{24} -4.62037 q^{25} +6.38112 q^{26} +2.65701 q^{27} -1.00000 q^{28} -6.00017 q^{29} -1.34784 q^{30} +7.72644 q^{31} -1.00000 q^{32} -7.50328 q^{33} -2.47999 q^{34} +0.616139 q^{35} +1.78540 q^{36} -4.26808 q^{37} +13.9591 q^{39} +0.616139 q^{40} +1.28851 q^{41} -2.18756 q^{42} -10.3584 q^{43} +3.42998 q^{44} -1.10005 q^{45} -2.70527 q^{46} +1.53750 q^{47} -2.18756 q^{48} +1.00000 q^{49} +4.62037 q^{50} -5.42512 q^{51} -6.38112 q^{52} +5.11208 q^{53} -2.65701 q^{54} -2.11335 q^{55} +1.00000 q^{56} +6.00017 q^{58} -12.7768 q^{59} +1.34784 q^{60} -5.54592 q^{61} -7.72644 q^{62} -1.78540 q^{63} +1.00000 q^{64} +3.93166 q^{65} +7.50328 q^{66} -4.16481 q^{67} +2.47999 q^{68} -5.91793 q^{69} -0.616139 q^{70} -0.0244707 q^{71} -1.78540 q^{72} +13.0675 q^{73} +4.26808 q^{74} +10.1073 q^{75} -3.42998 q^{77} -13.9591 q^{78} -2.96283 q^{79} -0.616139 q^{80} -11.1685 q^{81} -1.28851 q^{82} +12.9425 q^{83} +2.18756 q^{84} -1.52802 q^{85} +10.3584 q^{86} +13.1257 q^{87} -3.42998 q^{88} +12.4510 q^{89} +1.10005 q^{90} +6.38112 q^{91} +2.70527 q^{92} -16.9020 q^{93} -1.53750 q^{94} +2.18756 q^{96} +1.51714 q^{97} -1.00000 q^{98} +6.12389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 3 q^{5} + 3 q^{6} - 9 q^{7} - 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 3 q^{5} + 3 q^{6} - 9 q^{7} - 9 q^{8} + 12 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} - 12 q^{13} + 9 q^{14} + 6 q^{15} + 9 q^{16} + 12 q^{17} - 12 q^{18} + 3 q^{20} + 3 q^{21} + 3 q^{22} + 6 q^{23} + 3 q^{24} + 12 q^{26} - 24 q^{27} - 9 q^{28} + 6 q^{29} - 6 q^{30} - 9 q^{31} - 9 q^{32} + 3 q^{33} - 12 q^{34} - 3 q^{35} + 12 q^{36} - 9 q^{37} + 33 q^{39} - 3 q^{40} - 3 q^{41} - 3 q^{42} + 21 q^{43} - 3 q^{44} + 18 q^{45} - 6 q^{46} + 21 q^{47} - 3 q^{48} + 9 q^{49} - 39 q^{51} - 12 q^{52} + 24 q^{54} + 24 q^{55} + 9 q^{56} - 6 q^{58} + 9 q^{59} + 6 q^{60} + 6 q^{61} + 9 q^{62} - 12 q^{63} + 9 q^{64} + 3 q^{65} - 3 q^{66} - 27 q^{67} + 12 q^{68} - 6 q^{69} + 3 q^{70} - 9 q^{71} - 12 q^{72} + 51 q^{73} + 9 q^{74} + 3 q^{75} + 3 q^{77} - 33 q^{78} - 24 q^{79} + 3 q^{80} - 3 q^{81} + 3 q^{82} + 9 q^{83} + 3 q^{84} + 6 q^{85} - 21 q^{86} - 3 q^{87} + 3 q^{88} - 9 q^{89} - 18 q^{90} + 12 q^{91} + 6 q^{92} + 3 q^{93} - 21 q^{94} + 3 q^{96} + 12 q^{97} - 9 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.18756 −1.26299 −0.631493 0.775382i \(-0.717558\pi\)
−0.631493 + 0.775382i \(0.717558\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.616139 −0.275546 −0.137773 0.990464i \(-0.543994\pi\)
−0.137773 + 0.990464i \(0.543994\pi\)
\(6\) 2.18756 0.893066
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.78540 0.595133
\(10\) 0.616139 0.194840
\(11\) 3.42998 1.03418 0.517089 0.855931i \(-0.327015\pi\)
0.517089 + 0.855931i \(0.327015\pi\)
\(12\) −2.18756 −0.631493
\(13\) −6.38112 −1.76980 −0.884902 0.465776i \(-0.845775\pi\)
−0.884902 + 0.465776i \(0.845775\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.34784 0.348010
\(16\) 1.00000 0.250000
\(17\) 2.47999 0.601486 0.300743 0.953705i \(-0.402765\pi\)
0.300743 + 0.953705i \(0.402765\pi\)
\(18\) −1.78540 −0.420823
\(19\) 0 0
\(20\) −0.616139 −0.137773
\(21\) 2.18756 0.477364
\(22\) −3.42998 −0.731275
\(23\) 2.70527 0.564088 0.282044 0.959401i \(-0.408988\pi\)
0.282044 + 0.959401i \(0.408988\pi\)
\(24\) 2.18756 0.446533
\(25\) −4.62037 −0.924074
\(26\) 6.38112 1.25144
\(27\) 2.65701 0.511341
\(28\) −1.00000 −0.188982
\(29\) −6.00017 −1.11420 −0.557101 0.830444i \(-0.688087\pi\)
−0.557101 + 0.830444i \(0.688087\pi\)
\(30\) −1.34784 −0.246081
\(31\) 7.72644 1.38771 0.693855 0.720115i \(-0.255911\pi\)
0.693855 + 0.720115i \(0.255911\pi\)
\(32\) −1.00000 −0.176777
\(33\) −7.50328 −1.30615
\(34\) −2.47999 −0.425315
\(35\) 0.616139 0.104147
\(36\) 1.78540 0.297567
\(37\) −4.26808 −0.701669 −0.350834 0.936438i \(-0.614102\pi\)
−0.350834 + 0.936438i \(0.614102\pi\)
\(38\) 0 0
\(39\) 13.9591 2.23524
\(40\) 0.616139 0.0974202
\(41\) 1.28851 0.201232 0.100616 0.994925i \(-0.467919\pi\)
0.100616 + 0.994925i \(0.467919\pi\)
\(42\) −2.18756 −0.337547
\(43\) −10.3584 −1.57964 −0.789819 0.613340i \(-0.789826\pi\)
−0.789819 + 0.613340i \(0.789826\pi\)
\(44\) 3.42998 0.517089
\(45\) −1.10005 −0.163986
\(46\) −2.70527 −0.398871
\(47\) 1.53750 0.224267 0.112134 0.993693i \(-0.464232\pi\)
0.112134 + 0.993693i \(0.464232\pi\)
\(48\) −2.18756 −0.315746
\(49\) 1.00000 0.142857
\(50\) 4.62037 0.653419
\(51\) −5.42512 −0.759669
\(52\) −6.38112 −0.884902
\(53\) 5.11208 0.702198 0.351099 0.936338i \(-0.385808\pi\)
0.351099 + 0.936338i \(0.385808\pi\)
\(54\) −2.65701 −0.361573
\(55\) −2.11335 −0.284964
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 6.00017 0.787860
\(59\) −12.7768 −1.66339 −0.831696 0.555232i \(-0.812630\pi\)
−0.831696 + 0.555232i \(0.812630\pi\)
\(60\) 1.34784 0.174005
\(61\) −5.54592 −0.710082 −0.355041 0.934851i \(-0.615533\pi\)
−0.355041 + 0.934851i \(0.615533\pi\)
\(62\) −7.72644 −0.981259
\(63\) −1.78540 −0.224939
\(64\) 1.00000 0.125000
\(65\) 3.93166 0.487662
\(66\) 7.50328 0.923590
\(67\) −4.16481 −0.508813 −0.254406 0.967097i \(-0.581880\pi\)
−0.254406 + 0.967097i \(0.581880\pi\)
\(68\) 2.47999 0.300743
\(69\) −5.91793 −0.712435
\(70\) −0.616139 −0.0736427
\(71\) −0.0244707 −0.00290414 −0.00145207 0.999999i \(-0.500462\pi\)
−0.00145207 + 0.999999i \(0.500462\pi\)
\(72\) −1.78540 −0.210411
\(73\) 13.0675 1.52943 0.764715 0.644368i \(-0.222880\pi\)
0.764715 + 0.644368i \(0.222880\pi\)
\(74\) 4.26808 0.496155
\(75\) 10.1073 1.16709
\(76\) 0 0
\(77\) −3.42998 −0.390883
\(78\) −13.9591 −1.58055
\(79\) −2.96283 −0.333344 −0.166672 0.986012i \(-0.553302\pi\)
−0.166672 + 0.986012i \(0.553302\pi\)
\(80\) −0.616139 −0.0688865
\(81\) −11.1685 −1.24095
\(82\) −1.28851 −0.142292
\(83\) 12.9425 1.42062 0.710312 0.703887i \(-0.248554\pi\)
0.710312 + 0.703887i \(0.248554\pi\)
\(84\) 2.18756 0.238682
\(85\) −1.52802 −0.165737
\(86\) 10.3584 1.11697
\(87\) 13.1257 1.40722
\(88\) −3.42998 −0.365637
\(89\) 12.4510 1.31980 0.659899 0.751354i \(-0.270599\pi\)
0.659899 + 0.751354i \(0.270599\pi\)
\(90\) 1.10005 0.115956
\(91\) 6.38112 0.668923
\(92\) 2.70527 0.282044
\(93\) −16.9020 −1.75266
\(94\) −1.53750 −0.158581
\(95\) 0 0
\(96\) 2.18756 0.223266
\(97\) 1.51714 0.154042 0.0770211 0.997029i \(-0.475459\pi\)
0.0770211 + 0.997029i \(0.475459\pi\)
\(98\) −1.00000 −0.101015
\(99\) 6.12389 0.615474
\(100\) −4.62037 −0.462037
\(101\) −9.92823 −0.987896 −0.493948 0.869491i \(-0.664447\pi\)
−0.493948 + 0.869491i \(0.664447\pi\)
\(102\) 5.42512 0.537167
\(103\) −8.34988 −0.822738 −0.411369 0.911469i \(-0.634949\pi\)
−0.411369 + 0.911469i \(0.634949\pi\)
\(104\) 6.38112 0.625720
\(105\) −1.34784 −0.131536
\(106\) −5.11208 −0.496529
\(107\) −13.5562 −1.31053 −0.655264 0.755400i \(-0.727443\pi\)
−0.655264 + 0.755400i \(0.727443\pi\)
\(108\) 2.65701 0.255671
\(109\) −0.558302 −0.0534756 −0.0267378 0.999642i \(-0.508512\pi\)
−0.0267378 + 0.999642i \(0.508512\pi\)
\(110\) 2.11335 0.201500
\(111\) 9.33667 0.886197
\(112\) −1.00000 −0.0944911
\(113\) −10.0986 −0.949999 −0.474999 0.879986i \(-0.657552\pi\)
−0.474999 + 0.879986i \(0.657552\pi\)
\(114\) 0 0
\(115\) −1.66682 −0.155432
\(116\) −6.00017 −0.557101
\(117\) −11.3929 −1.05327
\(118\) 12.7768 1.17620
\(119\) −2.47999 −0.227340
\(120\) −1.34784 −0.123040
\(121\) 0.764783 0.0695257
\(122\) 5.54592 0.502104
\(123\) −2.81869 −0.254153
\(124\) 7.72644 0.693855
\(125\) 5.92749 0.530171
\(126\) 1.78540 0.159056
\(127\) −4.44613 −0.394530 −0.197265 0.980350i \(-0.563206\pi\)
−0.197265 + 0.980350i \(0.563206\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 22.6595 1.99506
\(130\) −3.93166 −0.344829
\(131\) 7.17183 0.626606 0.313303 0.949653i \(-0.398564\pi\)
0.313303 + 0.949653i \(0.398564\pi\)
\(132\) −7.50328 −0.653077
\(133\) 0 0
\(134\) 4.16481 0.359785
\(135\) −1.63709 −0.140898
\(136\) −2.47999 −0.212658
\(137\) −16.8073 −1.43594 −0.717971 0.696073i \(-0.754929\pi\)
−0.717971 + 0.696073i \(0.754929\pi\)
\(138\) 5.91793 0.503768
\(139\) −13.5566 −1.14986 −0.574929 0.818203i \(-0.694970\pi\)
−0.574929 + 0.818203i \(0.694970\pi\)
\(140\) 0.616139 0.0520733
\(141\) −3.36336 −0.283246
\(142\) 0.0244707 0.00205353
\(143\) −21.8871 −1.83029
\(144\) 1.78540 0.148783
\(145\) 3.69694 0.307014
\(146\) −13.0675 −1.08147
\(147\) −2.18756 −0.180427
\(148\) −4.26808 −0.350834
\(149\) 13.5990 1.11408 0.557038 0.830487i \(-0.311938\pi\)
0.557038 + 0.830487i \(0.311938\pi\)
\(150\) −10.1073 −0.825259
\(151\) 20.2325 1.64650 0.823250 0.567679i \(-0.192159\pi\)
0.823250 + 0.567679i \(0.192159\pi\)
\(152\) 0 0
\(153\) 4.42778 0.357964
\(154\) 3.42998 0.276396
\(155\) −4.76056 −0.382378
\(156\) 13.9591 1.11762
\(157\) −11.1097 −0.886648 −0.443324 0.896361i \(-0.646201\pi\)
−0.443324 + 0.896361i \(0.646201\pi\)
\(158\) 2.96283 0.235710
\(159\) −11.1830 −0.886866
\(160\) 0.616139 0.0487101
\(161\) −2.70527 −0.213205
\(162\) 11.1685 0.877484
\(163\) −10.2803 −0.805215 −0.402608 0.915373i \(-0.631896\pi\)
−0.402608 + 0.915373i \(0.631896\pi\)
\(164\) 1.28851 0.100616
\(165\) 4.62306 0.359905
\(166\) −12.9425 −1.00453
\(167\) 12.6748 0.980806 0.490403 0.871496i \(-0.336850\pi\)
0.490403 + 0.871496i \(0.336850\pi\)
\(168\) −2.18756 −0.168774
\(169\) 27.7187 2.13221
\(170\) 1.52802 0.117194
\(171\) 0 0
\(172\) −10.3584 −0.789819
\(173\) −6.24437 −0.474751 −0.237376 0.971418i \(-0.576287\pi\)
−0.237376 + 0.971418i \(0.576287\pi\)
\(174\) −13.1257 −0.995057
\(175\) 4.62037 0.349267
\(176\) 3.42998 0.258545
\(177\) 27.9499 2.10084
\(178\) −12.4510 −0.933238
\(179\) −17.7976 −1.33026 −0.665128 0.746729i \(-0.731623\pi\)
−0.665128 + 0.746729i \(0.731623\pi\)
\(180\) −1.10005 −0.0819932
\(181\) 5.92757 0.440593 0.220296 0.975433i \(-0.429298\pi\)
0.220296 + 0.975433i \(0.429298\pi\)
\(182\) −6.38112 −0.473000
\(183\) 12.1320 0.896823
\(184\) −2.70527 −0.199435
\(185\) 2.62973 0.193342
\(186\) 16.9020 1.23932
\(187\) 8.50633 0.622044
\(188\) 1.53750 0.112134
\(189\) −2.65701 −0.193269
\(190\) 0 0
\(191\) 16.3567 1.18353 0.591764 0.806111i \(-0.298432\pi\)
0.591764 + 0.806111i \(0.298432\pi\)
\(192\) −2.18756 −0.157873
\(193\) −19.9885 −1.43881 −0.719403 0.694593i \(-0.755584\pi\)
−0.719403 + 0.694593i \(0.755584\pi\)
\(194\) −1.51714 −0.108924
\(195\) −8.60072 −0.615911
\(196\) 1.00000 0.0714286
\(197\) −3.44430 −0.245396 −0.122698 0.992444i \(-0.539155\pi\)
−0.122698 + 0.992444i \(0.539155\pi\)
\(198\) −6.12389 −0.435206
\(199\) 3.13027 0.221899 0.110949 0.993826i \(-0.464611\pi\)
0.110949 + 0.993826i \(0.464611\pi\)
\(200\) 4.62037 0.326710
\(201\) 9.11076 0.642623
\(202\) 9.92823 0.698548
\(203\) 6.00017 0.421129
\(204\) −5.42512 −0.379834
\(205\) −0.793902 −0.0554485
\(206\) 8.34988 0.581764
\(207\) 4.82999 0.335708
\(208\) −6.38112 −0.442451
\(209\) 0 0
\(210\) 1.34784 0.0930097
\(211\) 16.6987 1.14959 0.574793 0.818299i \(-0.305083\pi\)
0.574793 + 0.818299i \(0.305083\pi\)
\(212\) 5.11208 0.351099
\(213\) 0.0535310 0.00366788
\(214\) 13.5562 0.926684
\(215\) 6.38221 0.435263
\(216\) −2.65701 −0.180786
\(217\) −7.72644 −0.524505
\(218\) 0.558302 0.0378130
\(219\) −28.5858 −1.93165
\(220\) −2.11335 −0.142482
\(221\) −15.8251 −1.06451
\(222\) −9.33667 −0.626636
\(223\) −14.5844 −0.976641 −0.488320 0.872664i \(-0.662390\pi\)
−0.488320 + 0.872664i \(0.662390\pi\)
\(224\) 1.00000 0.0668153
\(225\) −8.24921 −0.549947
\(226\) 10.0986 0.671750
\(227\) 4.73333 0.314162 0.157081 0.987586i \(-0.449792\pi\)
0.157081 + 0.987586i \(0.449792\pi\)
\(228\) 0 0
\(229\) 14.8611 0.982051 0.491026 0.871145i \(-0.336622\pi\)
0.491026 + 0.871145i \(0.336622\pi\)
\(230\) 1.66682 0.109907
\(231\) 7.50328 0.493679
\(232\) 6.00017 0.393930
\(233\) −4.36751 −0.286125 −0.143062 0.989714i \(-0.545695\pi\)
−0.143062 + 0.989714i \(0.545695\pi\)
\(234\) 11.3929 0.744774
\(235\) −0.947313 −0.0617959
\(236\) −12.7768 −0.831696
\(237\) 6.48135 0.421009
\(238\) 2.47999 0.160754
\(239\) −24.2217 −1.56677 −0.783385 0.621537i \(-0.786509\pi\)
−0.783385 + 0.621537i \(0.786509\pi\)
\(240\) 1.34784 0.0870026
\(241\) 17.1009 1.10157 0.550784 0.834648i \(-0.314329\pi\)
0.550784 + 0.834648i \(0.314329\pi\)
\(242\) −0.764783 −0.0491621
\(243\) 16.4608 1.05596
\(244\) −5.54592 −0.355041
\(245\) −0.616139 −0.0393637
\(246\) 2.81869 0.179713
\(247\) 0 0
\(248\) −7.72644 −0.490629
\(249\) −28.3125 −1.79423
\(250\) −5.92749 −0.374887
\(251\) 27.3096 1.72377 0.861883 0.507107i \(-0.169285\pi\)
0.861883 + 0.507107i \(0.169285\pi\)
\(252\) −1.78540 −0.112470
\(253\) 9.27904 0.583368
\(254\) 4.44613 0.278975
\(255\) 3.34263 0.209324
\(256\) 1.00000 0.0625000
\(257\) 9.96817 0.621797 0.310899 0.950443i \(-0.399370\pi\)
0.310899 + 0.950443i \(0.399370\pi\)
\(258\) −22.6595 −1.41072
\(259\) 4.26808 0.265206
\(260\) 3.93166 0.243831
\(261\) −10.7127 −0.663099
\(262\) −7.17183 −0.443077
\(263\) −24.9314 −1.53734 −0.768669 0.639647i \(-0.779080\pi\)
−0.768669 + 0.639647i \(0.779080\pi\)
\(264\) 7.50328 0.461795
\(265\) −3.14975 −0.193488
\(266\) 0 0
\(267\) −27.2371 −1.66689
\(268\) −4.16481 −0.254406
\(269\) 25.6665 1.56491 0.782456 0.622706i \(-0.213967\pi\)
0.782456 + 0.622706i \(0.213967\pi\)
\(270\) 1.63709 0.0996299
\(271\) 18.7964 1.14180 0.570900 0.821020i \(-0.306595\pi\)
0.570900 + 0.821020i \(0.306595\pi\)
\(272\) 2.47999 0.150372
\(273\) −13.9591 −0.844841
\(274\) 16.8073 1.01536
\(275\) −15.8478 −0.955658
\(276\) −5.91793 −0.356218
\(277\) 19.8745 1.19414 0.597071 0.802188i \(-0.296331\pi\)
0.597071 + 0.802188i \(0.296331\pi\)
\(278\) 13.5566 0.813073
\(279\) 13.7948 0.825872
\(280\) −0.616139 −0.0368214
\(281\) 21.2943 1.27031 0.635156 0.772384i \(-0.280936\pi\)
0.635156 + 0.772384i \(0.280936\pi\)
\(282\) 3.36336 0.200285
\(283\) −3.77466 −0.224380 −0.112190 0.993687i \(-0.535787\pi\)
−0.112190 + 0.993687i \(0.535787\pi\)
\(284\) −0.0244707 −0.00145207
\(285\) 0 0
\(286\) 21.8871 1.29421
\(287\) −1.28851 −0.0760584
\(288\) −1.78540 −0.105206
\(289\) −10.8496 −0.638214
\(290\) −3.69694 −0.217092
\(291\) −3.31883 −0.194553
\(292\) 13.0675 0.764715
\(293\) 16.9584 0.990721 0.495360 0.868688i \(-0.335036\pi\)
0.495360 + 0.868688i \(0.335036\pi\)
\(294\) 2.18756 0.127581
\(295\) 7.87226 0.458341
\(296\) 4.26808 0.248077
\(297\) 9.11349 0.528818
\(298\) −13.5990 −0.787770
\(299\) −17.2627 −0.998326
\(300\) 10.1073 0.583546
\(301\) 10.3584 0.597047
\(302\) −20.2325 −1.16425
\(303\) 21.7186 1.24770
\(304\) 0 0
\(305\) 3.41706 0.195660
\(306\) −4.42778 −0.253119
\(307\) 2.68585 0.153289 0.0766447 0.997058i \(-0.475579\pi\)
0.0766447 + 0.997058i \(0.475579\pi\)
\(308\) −3.42998 −0.195441
\(309\) 18.2658 1.03911
\(310\) 4.76056 0.270382
\(311\) 1.77079 0.100412 0.0502062 0.998739i \(-0.484012\pi\)
0.0502062 + 0.998739i \(0.484012\pi\)
\(312\) −13.9591 −0.790276
\(313\) 7.89729 0.446381 0.223191 0.974775i \(-0.428353\pi\)
0.223191 + 0.974775i \(0.428353\pi\)
\(314\) 11.1097 0.626955
\(315\) 1.10005 0.0619811
\(316\) −2.96283 −0.166672
\(317\) 11.6202 0.652656 0.326328 0.945257i \(-0.394189\pi\)
0.326328 + 0.945257i \(0.394189\pi\)
\(318\) 11.1830 0.627109
\(319\) −20.5805 −1.15229
\(320\) −0.616139 −0.0344432
\(321\) 29.6550 1.65518
\(322\) 2.70527 0.150759
\(323\) 0 0
\(324\) −11.1685 −0.620475
\(325\) 29.4832 1.63543
\(326\) 10.2803 0.569373
\(327\) 1.22132 0.0675389
\(328\) −1.28851 −0.0711461
\(329\) −1.53750 −0.0847650
\(330\) −4.62306 −0.254491
\(331\) 1.15352 0.0634032 0.0317016 0.999497i \(-0.489907\pi\)
0.0317016 + 0.999497i \(0.489907\pi\)
\(332\) 12.9425 0.710312
\(333\) −7.62023 −0.417586
\(334\) −12.6748 −0.693534
\(335\) 2.56610 0.140201
\(336\) 2.18756 0.119341
\(337\) 7.31544 0.398497 0.199249 0.979949i \(-0.436150\pi\)
0.199249 + 0.979949i \(0.436150\pi\)
\(338\) −27.7187 −1.50770
\(339\) 22.0913 1.19983
\(340\) −1.52802 −0.0828685
\(341\) 26.5016 1.43514
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 10.3584 0.558486
\(345\) 3.64627 0.196309
\(346\) 6.24437 0.335700
\(347\) −31.3704 −1.68405 −0.842025 0.539439i \(-0.818636\pi\)
−0.842025 + 0.539439i \(0.818636\pi\)
\(348\) 13.1257 0.703611
\(349\) −2.33966 −0.125239 −0.0626197 0.998037i \(-0.519946\pi\)
−0.0626197 + 0.998037i \(0.519946\pi\)
\(350\) −4.62037 −0.246969
\(351\) −16.9547 −0.904974
\(352\) −3.42998 −0.182819
\(353\) −6.64960 −0.353923 −0.176961 0.984218i \(-0.556627\pi\)
−0.176961 + 0.984218i \(0.556627\pi\)
\(354\) −27.9499 −1.48552
\(355\) 0.0150774 0.000800223 0
\(356\) 12.4510 0.659899
\(357\) 5.42512 0.287128
\(358\) 17.7976 0.940633
\(359\) 23.2605 1.22764 0.613821 0.789445i \(-0.289631\pi\)
0.613821 + 0.789445i \(0.289631\pi\)
\(360\) 1.10005 0.0579780
\(361\) 0 0
\(362\) −5.92757 −0.311546
\(363\) −1.67301 −0.0878100
\(364\) 6.38112 0.334462
\(365\) −8.05138 −0.421428
\(366\) −12.1320 −0.634150
\(367\) 10.6463 0.555733 0.277866 0.960620i \(-0.410373\pi\)
0.277866 + 0.960620i \(0.410373\pi\)
\(368\) 2.70527 0.141022
\(369\) 2.30051 0.119760
\(370\) −2.62973 −0.136713
\(371\) −5.11208 −0.265406
\(372\) −16.9020 −0.876329
\(373\) 28.8794 1.49532 0.747659 0.664083i \(-0.231178\pi\)
0.747659 + 0.664083i \(0.231178\pi\)
\(374\) −8.50633 −0.439852
\(375\) −12.9667 −0.669598
\(376\) −1.53750 −0.0792904
\(377\) 38.2878 1.97192
\(378\) 2.65701 0.136662
\(379\) 33.1311 1.70183 0.850916 0.525302i \(-0.176048\pi\)
0.850916 + 0.525302i \(0.176048\pi\)
\(380\) 0 0
\(381\) 9.72616 0.498286
\(382\) −16.3567 −0.836880
\(383\) 3.65421 0.186722 0.0933608 0.995632i \(-0.470239\pi\)
0.0933608 + 0.995632i \(0.470239\pi\)
\(384\) 2.18756 0.111633
\(385\) 2.11335 0.107706
\(386\) 19.9885 1.01739
\(387\) −18.4938 −0.940095
\(388\) 1.51714 0.0770211
\(389\) −9.25078 −0.469033 −0.234517 0.972112i \(-0.575351\pi\)
−0.234517 + 0.972112i \(0.575351\pi\)
\(390\) 8.60072 0.435515
\(391\) 6.70905 0.339291
\(392\) −1.00000 −0.0505076
\(393\) −15.6888 −0.791395
\(394\) 3.44430 0.173521
\(395\) 1.82552 0.0918517
\(396\) 6.12389 0.307737
\(397\) 19.2309 0.965172 0.482586 0.875849i \(-0.339698\pi\)
0.482586 + 0.875849i \(0.339698\pi\)
\(398\) −3.13027 −0.156906
\(399\) 0 0
\(400\) −4.62037 −0.231019
\(401\) 29.5430 1.47531 0.737654 0.675179i \(-0.235933\pi\)
0.737654 + 0.675179i \(0.235933\pi\)
\(402\) −9.11076 −0.454403
\(403\) −49.3034 −2.45597
\(404\) −9.92823 −0.493948
\(405\) 6.88138 0.341939
\(406\) −6.00017 −0.297783
\(407\) −14.6395 −0.725651
\(408\) 5.42512 0.268583
\(409\) 39.8965 1.97276 0.986378 0.164494i \(-0.0525991\pi\)
0.986378 + 0.164494i \(0.0525991\pi\)
\(410\) 0.793902 0.0392080
\(411\) 36.7668 1.81357
\(412\) −8.34988 −0.411369
\(413\) 12.7768 0.628703
\(414\) −4.82999 −0.237381
\(415\) −7.97439 −0.391447
\(416\) 6.38112 0.312860
\(417\) 29.6559 1.45225
\(418\) 0 0
\(419\) −34.5562 −1.68818 −0.844090 0.536201i \(-0.819859\pi\)
−0.844090 + 0.536201i \(0.819859\pi\)
\(420\) −1.34784 −0.0657678
\(421\) 28.8651 1.40680 0.703399 0.710795i \(-0.251665\pi\)
0.703399 + 0.710795i \(0.251665\pi\)
\(422\) −16.6987 −0.812880
\(423\) 2.74505 0.133469
\(424\) −5.11208 −0.248265
\(425\) −11.4585 −0.555818
\(426\) −0.0535310 −0.00259358
\(427\) 5.54592 0.268386
\(428\) −13.5562 −0.655264
\(429\) 47.8793 2.31164
\(430\) −6.38221 −0.307777
\(431\) 19.4879 0.938699 0.469350 0.883012i \(-0.344488\pi\)
0.469350 + 0.883012i \(0.344488\pi\)
\(432\) 2.65701 0.127835
\(433\) 1.44225 0.0693102 0.0346551 0.999399i \(-0.488967\pi\)
0.0346551 + 0.999399i \(0.488967\pi\)
\(434\) 7.72644 0.370881
\(435\) −8.08726 −0.387754
\(436\) −0.558302 −0.0267378
\(437\) 0 0
\(438\) 28.5858 1.36588
\(439\) −16.5525 −0.790006 −0.395003 0.918680i \(-0.629256\pi\)
−0.395003 + 0.918680i \(0.629256\pi\)
\(440\) 2.11335 0.100750
\(441\) 1.78540 0.0850190
\(442\) 15.8251 0.752725
\(443\) 3.08768 0.146700 0.0733500 0.997306i \(-0.476631\pi\)
0.0733500 + 0.997306i \(0.476631\pi\)
\(444\) 9.33667 0.443099
\(445\) −7.67152 −0.363665
\(446\) 14.5844 0.690589
\(447\) −29.7486 −1.40706
\(448\) −1.00000 −0.0472456
\(449\) 17.9081 0.845135 0.422568 0.906331i \(-0.361129\pi\)
0.422568 + 0.906331i \(0.361129\pi\)
\(450\) 8.24921 0.388871
\(451\) 4.41957 0.208109
\(452\) −10.0986 −0.474999
\(453\) −44.2598 −2.07951
\(454\) −4.73333 −0.222146
\(455\) −3.93166 −0.184319
\(456\) 0 0
\(457\) 31.2954 1.46394 0.731970 0.681337i \(-0.238601\pi\)
0.731970 + 0.681337i \(0.238601\pi\)
\(458\) −14.8611 −0.694415
\(459\) 6.58936 0.307565
\(460\) −1.66682 −0.0777161
\(461\) −11.6085 −0.540660 −0.270330 0.962768i \(-0.587133\pi\)
−0.270330 + 0.962768i \(0.587133\pi\)
\(462\) −7.50328 −0.349084
\(463\) 4.15132 0.192928 0.0964640 0.995336i \(-0.469247\pi\)
0.0964640 + 0.995336i \(0.469247\pi\)
\(464\) −6.00017 −0.278551
\(465\) 10.4140 0.482937
\(466\) 4.36751 0.202321
\(467\) −24.9196 −1.15314 −0.576570 0.817047i \(-0.695609\pi\)
−0.576570 + 0.817047i \(0.695609\pi\)
\(468\) −11.3929 −0.526635
\(469\) 4.16481 0.192313
\(470\) 0.947313 0.0436963
\(471\) 24.3030 1.11982
\(472\) 12.7768 0.588098
\(473\) −35.5291 −1.63363
\(474\) −6.48135 −0.297698
\(475\) 0 0
\(476\) −2.47999 −0.113670
\(477\) 9.12710 0.417901
\(478\) 24.2217 1.10787
\(479\) −18.9778 −0.867117 −0.433559 0.901125i \(-0.642742\pi\)
−0.433559 + 0.901125i \(0.642742\pi\)
\(480\) −1.34784 −0.0615201
\(481\) 27.2352 1.24182
\(482\) −17.1009 −0.778926
\(483\) 5.91793 0.269275
\(484\) 0.764783 0.0347629
\(485\) −0.934769 −0.0424457
\(486\) −16.4608 −0.746677
\(487\) −36.1508 −1.63815 −0.819074 0.573687i \(-0.805512\pi\)
−0.819074 + 0.573687i \(0.805512\pi\)
\(488\) 5.54592 0.251052
\(489\) 22.4887 1.01698
\(490\) 0.616139 0.0278343
\(491\) 3.08754 0.139339 0.0696693 0.997570i \(-0.477806\pi\)
0.0696693 + 0.997570i \(0.477806\pi\)
\(492\) −2.81869 −0.127076
\(493\) −14.8804 −0.670178
\(494\) 0 0
\(495\) −3.77317 −0.169591
\(496\) 7.72644 0.346927
\(497\) 0.0244707 0.00109766
\(498\) 28.3125 1.26871
\(499\) 30.9593 1.38593 0.692964 0.720972i \(-0.256304\pi\)
0.692964 + 0.720972i \(0.256304\pi\)
\(500\) 5.92749 0.265085
\(501\) −27.7268 −1.23874
\(502\) −27.3096 −1.21889
\(503\) 3.79268 0.169107 0.0845536 0.996419i \(-0.473054\pi\)
0.0845536 + 0.996419i \(0.473054\pi\)
\(504\) 1.78540 0.0795280
\(505\) 6.11717 0.272211
\(506\) −9.27904 −0.412504
\(507\) −60.6362 −2.69295
\(508\) −4.44613 −0.197265
\(509\) −36.0268 −1.59686 −0.798430 0.602087i \(-0.794336\pi\)
−0.798430 + 0.602087i \(0.794336\pi\)
\(510\) −3.34263 −0.148014
\(511\) −13.0675 −0.578071
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −9.96817 −0.439677
\(515\) 5.14469 0.226702
\(516\) 22.6595 0.997530
\(517\) 5.27359 0.231932
\(518\) −4.26808 −0.187529
\(519\) 13.6599 0.599604
\(520\) −3.93166 −0.172415
\(521\) −7.91028 −0.346555 −0.173278 0.984873i \(-0.555436\pi\)
−0.173278 + 0.984873i \(0.555436\pi\)
\(522\) 10.7127 0.468882
\(523\) 33.0876 1.44682 0.723409 0.690420i \(-0.242574\pi\)
0.723409 + 0.690420i \(0.242574\pi\)
\(524\) 7.17183 0.313303
\(525\) −10.1073 −0.441120
\(526\) 24.9314 1.08706
\(527\) 19.1615 0.834688
\(528\) −7.50328 −0.326538
\(529\) −15.6815 −0.681804
\(530\) 3.14975 0.136817
\(531\) −22.8116 −0.989939
\(532\) 0 0
\(533\) −8.22215 −0.356141
\(534\) 27.2371 1.17867
\(535\) 8.35251 0.361111
\(536\) 4.16481 0.179892
\(537\) 38.9333 1.68009
\(538\) −25.6665 −1.10656
\(539\) 3.42998 0.147740
\(540\) −1.63709 −0.0704490
\(541\) 9.32192 0.400781 0.200390 0.979716i \(-0.435779\pi\)
0.200390 + 0.979716i \(0.435779\pi\)
\(542\) −18.7964 −0.807374
\(543\) −12.9669 −0.556462
\(544\) −2.47999 −0.106329
\(545\) 0.343992 0.0147350
\(546\) 13.9591 0.597393
\(547\) 27.3456 1.16921 0.584607 0.811317i \(-0.301249\pi\)
0.584607 + 0.811317i \(0.301249\pi\)
\(548\) −16.8073 −0.717971
\(549\) −9.90167 −0.422593
\(550\) 15.8478 0.675752
\(551\) 0 0
\(552\) 5.91793 0.251884
\(553\) 2.96283 0.125992
\(554\) −19.8745 −0.844386
\(555\) −5.75269 −0.244188
\(556\) −13.5566 −0.574929
\(557\) −16.4963 −0.698970 −0.349485 0.936942i \(-0.613643\pi\)
−0.349485 + 0.936942i \(0.613643\pi\)
\(558\) −13.7948 −0.583980
\(559\) 66.0981 2.79565
\(560\) 0.616139 0.0260366
\(561\) −18.6081 −0.785633
\(562\) −21.2943 −0.898246
\(563\) 35.4681 1.49480 0.747401 0.664373i \(-0.231301\pi\)
0.747401 + 0.664373i \(0.231301\pi\)
\(564\) −3.36336 −0.141623
\(565\) 6.22216 0.261768
\(566\) 3.77466 0.158661
\(567\) 11.1685 0.469035
\(568\) 0.0244707 0.00102677
\(569\) 43.3131 1.81578 0.907889 0.419210i \(-0.137693\pi\)
0.907889 + 0.419210i \(0.137693\pi\)
\(570\) 0 0
\(571\) 1.18471 0.0495787 0.0247893 0.999693i \(-0.492108\pi\)
0.0247893 + 0.999693i \(0.492108\pi\)
\(572\) −21.8871 −0.915147
\(573\) −35.7811 −1.49478
\(574\) 1.28851 0.0537814
\(575\) −12.4994 −0.521260
\(576\) 1.78540 0.0743916
\(577\) 12.9555 0.539343 0.269671 0.962952i \(-0.413085\pi\)
0.269671 + 0.962952i \(0.413085\pi\)
\(578\) 10.8496 0.451286
\(579\) 43.7260 1.81719
\(580\) 3.69694 0.153507
\(581\) −12.9425 −0.536946
\(582\) 3.31883 0.137570
\(583\) 17.5343 0.726198
\(584\) −13.0675 −0.540735
\(585\) 7.01958 0.290224
\(586\) −16.9584 −0.700545
\(587\) 36.4405 1.50406 0.752030 0.659129i \(-0.229075\pi\)
0.752030 + 0.659129i \(0.229075\pi\)
\(588\) −2.18756 −0.0902133
\(589\) 0 0
\(590\) −7.87226 −0.324096
\(591\) 7.53460 0.309932
\(592\) −4.26808 −0.175417
\(593\) 31.6814 1.30100 0.650500 0.759506i \(-0.274559\pi\)
0.650500 + 0.759506i \(0.274559\pi\)
\(594\) −9.11349 −0.373931
\(595\) 1.52802 0.0626427
\(596\) 13.5990 0.557038
\(597\) −6.84763 −0.280255
\(598\) 17.2627 0.705923
\(599\) −5.21417 −0.213045 −0.106523 0.994310i \(-0.533972\pi\)
−0.106523 + 0.994310i \(0.533972\pi\)
\(600\) −10.1073 −0.412630
\(601\) 8.43231 0.343961 0.171981 0.985100i \(-0.444983\pi\)
0.171981 + 0.985100i \(0.444983\pi\)
\(602\) −10.3584 −0.422176
\(603\) −7.43585 −0.302811
\(604\) 20.2325 0.823250
\(605\) −0.471213 −0.0191575
\(606\) −21.7186 −0.882256
\(607\) −18.8481 −0.765022 −0.382511 0.923951i \(-0.624941\pi\)
−0.382511 + 0.923951i \(0.624941\pi\)
\(608\) 0 0
\(609\) −13.1257 −0.531880
\(610\) −3.41706 −0.138353
\(611\) −9.81096 −0.396909
\(612\) 4.42778 0.178982
\(613\) 14.8192 0.598542 0.299271 0.954168i \(-0.403256\pi\)
0.299271 + 0.954168i \(0.403256\pi\)
\(614\) −2.68585 −0.108392
\(615\) 1.73671 0.0700307
\(616\) 3.42998 0.138198
\(617\) 14.2858 0.575123 0.287562 0.957762i \(-0.407155\pi\)
0.287562 + 0.957762i \(0.407155\pi\)
\(618\) −18.2658 −0.734760
\(619\) −27.8881 −1.12092 −0.560459 0.828182i \(-0.689375\pi\)
−0.560459 + 0.828182i \(0.689375\pi\)
\(620\) −4.76056 −0.191189
\(621\) 7.18793 0.288442
\(622\) −1.77079 −0.0710023
\(623\) −12.4510 −0.498837
\(624\) 13.9591 0.558810
\(625\) 19.4497 0.777988
\(626\) −7.89729 −0.315639
\(627\) 0 0
\(628\) −11.1097 −0.443324
\(629\) −10.5848 −0.422044
\(630\) −1.10005 −0.0438272
\(631\) 26.3781 1.05010 0.525048 0.851073i \(-0.324047\pi\)
0.525048 + 0.851073i \(0.324047\pi\)
\(632\) 2.96283 0.117855
\(633\) −36.5293 −1.45191
\(634\) −11.6202 −0.461498
\(635\) 2.73944 0.108711
\(636\) −11.1830 −0.443433
\(637\) −6.38112 −0.252829
\(638\) 20.5805 0.814789
\(639\) −0.0436899 −0.00172835
\(640\) 0.616139 0.0243550
\(641\) −39.1472 −1.54622 −0.773110 0.634272i \(-0.781300\pi\)
−0.773110 + 0.634272i \(0.781300\pi\)
\(642\) −29.6550 −1.17039
\(643\) −6.20232 −0.244596 −0.122298 0.992493i \(-0.539026\pi\)
−0.122298 + 0.992493i \(0.539026\pi\)
\(644\) −2.70527 −0.106603
\(645\) −13.9614 −0.549731
\(646\) 0 0
\(647\) −2.11530 −0.0831610 −0.0415805 0.999135i \(-0.513239\pi\)
−0.0415805 + 0.999135i \(0.513239\pi\)
\(648\) 11.1685 0.438742
\(649\) −43.8240 −1.72024
\(650\) −29.4832 −1.15642
\(651\) 16.9020 0.662442
\(652\) −10.2803 −0.402608
\(653\) −20.2029 −0.790602 −0.395301 0.918552i \(-0.629360\pi\)
−0.395301 + 0.918552i \(0.629360\pi\)
\(654\) −1.22132 −0.0477572
\(655\) −4.41885 −0.172659
\(656\) 1.28851 0.0503079
\(657\) 23.3306 0.910215
\(658\) 1.53750 0.0599379
\(659\) 23.3082 0.907958 0.453979 0.891012i \(-0.350004\pi\)
0.453979 + 0.891012i \(0.350004\pi\)
\(660\) 4.62306 0.179953
\(661\) 28.4307 1.10583 0.552914 0.833238i \(-0.313516\pi\)
0.552914 + 0.833238i \(0.313516\pi\)
\(662\) −1.15352 −0.0448329
\(663\) 34.6184 1.34447
\(664\) −12.9425 −0.502267
\(665\) 0 0
\(666\) 7.62023 0.295278
\(667\) −16.2321 −0.628509
\(668\) 12.6748 0.490403
\(669\) 31.9041 1.23348
\(670\) −2.56610 −0.0991372
\(671\) −19.0224 −0.734351
\(672\) −2.18756 −0.0843868
\(673\) 9.48697 0.365696 0.182848 0.983141i \(-0.441468\pi\)
0.182848 + 0.983141i \(0.441468\pi\)
\(674\) −7.31544 −0.281780
\(675\) −12.2764 −0.472517
\(676\) 27.7187 1.06610
\(677\) −43.2866 −1.66364 −0.831820 0.555045i \(-0.812701\pi\)
−0.831820 + 0.555045i \(0.812701\pi\)
\(678\) −22.0913 −0.848411
\(679\) −1.51714 −0.0582225
\(680\) 1.52802 0.0585969
\(681\) −10.3544 −0.396782
\(682\) −26.5016 −1.01480
\(683\) 16.3891 0.627110 0.313555 0.949570i \(-0.398480\pi\)
0.313555 + 0.949570i \(0.398480\pi\)
\(684\) 0 0
\(685\) 10.3556 0.395668
\(686\) 1.00000 0.0381802
\(687\) −32.5096 −1.24032
\(688\) −10.3584 −0.394910
\(689\) −32.6208 −1.24275
\(690\) −3.64627 −0.138811
\(691\) 41.4257 1.57591 0.787953 0.615735i \(-0.211141\pi\)
0.787953 + 0.615735i \(0.211141\pi\)
\(692\) −6.24437 −0.237376
\(693\) −6.12389 −0.232627
\(694\) 31.3704 1.19080
\(695\) 8.35277 0.316839
\(696\) −13.1257 −0.497528
\(697\) 3.19550 0.121038
\(698\) 2.33966 0.0885576
\(699\) 9.55416 0.361372
\(700\) 4.62037 0.174634
\(701\) 46.1088 1.74150 0.870752 0.491722i \(-0.163632\pi\)
0.870752 + 0.491722i \(0.163632\pi\)
\(702\) 16.9547 0.639913
\(703\) 0 0
\(704\) 3.42998 0.129272
\(705\) 2.07230 0.0780473
\(706\) 6.64960 0.250261
\(707\) 9.92823 0.373390
\(708\) 27.9499 1.05042
\(709\) −27.0329 −1.01524 −0.507622 0.861580i \(-0.669475\pi\)
−0.507622 + 0.861580i \(0.669475\pi\)
\(710\) −0.0150774 −0.000565843 0
\(711\) −5.28983 −0.198384
\(712\) −12.4510 −0.466619
\(713\) 20.9021 0.782791
\(714\) −5.42512 −0.203030
\(715\) 13.4855 0.504330
\(716\) −17.7976 −0.665128
\(717\) 52.9862 1.97881
\(718\) −23.2605 −0.868075
\(719\) −47.8432 −1.78425 −0.892125 0.451789i \(-0.850786\pi\)
−0.892125 + 0.451789i \(0.850786\pi\)
\(720\) −1.10005 −0.0409966
\(721\) 8.34988 0.310966
\(722\) 0 0
\(723\) −37.4092 −1.39126
\(724\) 5.92757 0.220296
\(725\) 27.7230 1.02961
\(726\) 1.67301 0.0620911
\(727\) 4.90682 0.181984 0.0909919 0.995852i \(-0.470996\pi\)
0.0909919 + 0.995852i \(0.470996\pi\)
\(728\) −6.38112 −0.236500
\(729\) −2.50327 −0.0927136
\(730\) 8.05138 0.297995
\(731\) −25.6887 −0.950131
\(732\) 12.1320 0.448412
\(733\) 52.5699 1.94172 0.970858 0.239657i \(-0.0770349\pi\)
0.970858 + 0.239657i \(0.0770349\pi\)
\(734\) −10.6463 −0.392962
\(735\) 1.34784 0.0497158
\(736\) −2.70527 −0.0997177
\(737\) −14.2852 −0.526203
\(738\) −2.30051 −0.0846828
\(739\) 25.2192 0.927702 0.463851 0.885913i \(-0.346467\pi\)
0.463851 + 0.885913i \(0.346467\pi\)
\(740\) 2.62973 0.0966709
\(741\) 0 0
\(742\) 5.11208 0.187670
\(743\) −5.58183 −0.204777 −0.102389 0.994744i \(-0.532649\pi\)
−0.102389 + 0.994744i \(0.532649\pi\)
\(744\) 16.9020 0.619658
\(745\) −8.37889 −0.306979
\(746\) −28.8794 −1.05735
\(747\) 23.1075 0.845461
\(748\) 8.50633 0.311022
\(749\) 13.5562 0.495333
\(750\) 12.9667 0.473477
\(751\) −25.7524 −0.939718 −0.469859 0.882742i \(-0.655695\pi\)
−0.469859 + 0.882742i \(0.655695\pi\)
\(752\) 1.53750 0.0560668
\(753\) −59.7413 −2.17709
\(754\) −38.2878 −1.39436
\(755\) −12.4661 −0.453686
\(756\) −2.65701 −0.0966344
\(757\) −45.3483 −1.64821 −0.824106 0.566435i \(-0.808322\pi\)
−0.824106 + 0.566435i \(0.808322\pi\)
\(758\) −33.1311 −1.20338
\(759\) −20.2984 −0.736786
\(760\) 0 0
\(761\) −24.9534 −0.904560 −0.452280 0.891876i \(-0.649389\pi\)
−0.452280 + 0.891876i \(0.649389\pi\)
\(762\) −9.72616 −0.352342
\(763\) 0.558302 0.0202119
\(764\) 16.3567 0.591764
\(765\) −2.72813 −0.0986356
\(766\) −3.65421 −0.132032
\(767\) 81.5300 2.94388
\(768\) −2.18756 −0.0789366
\(769\) 12.9261 0.466126 0.233063 0.972462i \(-0.425125\pi\)
0.233063 + 0.972462i \(0.425125\pi\)
\(770\) −2.11335 −0.0761597
\(771\) −21.8059 −0.785321
\(772\) −19.9885 −0.719403
\(773\) −14.0205 −0.504281 −0.252141 0.967691i \(-0.581135\pi\)
−0.252141 + 0.967691i \(0.581135\pi\)
\(774\) 18.4938 0.664748
\(775\) −35.6990 −1.28235
\(776\) −1.51714 −0.0544621
\(777\) −9.33667 −0.334951
\(778\) 9.25078 0.331657
\(779\) 0 0
\(780\) −8.60072 −0.307955
\(781\) −0.0839340 −0.00300340
\(782\) −6.70905 −0.239915
\(783\) −15.9425 −0.569738
\(784\) 1.00000 0.0357143
\(785\) 6.84511 0.244312
\(786\) 15.6888 0.559600
\(787\) 4.13771 0.147494 0.0737468 0.997277i \(-0.476504\pi\)
0.0737468 + 0.997277i \(0.476504\pi\)
\(788\) −3.44430 −0.122698
\(789\) 54.5389 1.94164
\(790\) −1.82552 −0.0649489
\(791\) 10.0986 0.359066
\(792\) −6.12389 −0.217603
\(793\) 35.3892 1.25671
\(794\) −19.2309 −0.682480
\(795\) 6.89026 0.244372
\(796\) 3.13027 0.110949
\(797\) −53.5733 −1.89767 −0.948833 0.315780i \(-0.897734\pi\)
−0.948833 + 0.315780i \(0.897734\pi\)
\(798\) 0 0
\(799\) 3.81298 0.134894
\(800\) 4.62037 0.163355
\(801\) 22.2299 0.785456
\(802\) −29.5430 −1.04320
\(803\) 44.8212 1.58170
\(804\) 9.11076 0.321312
\(805\) 1.66682 0.0587478
\(806\) 49.3034 1.73664
\(807\) −56.1468 −1.97646
\(808\) 9.92823 0.349274
\(809\) 30.7139 1.07984 0.539922 0.841715i \(-0.318454\pi\)
0.539922 + 0.841715i \(0.318454\pi\)
\(810\) −6.88138 −0.241787
\(811\) 43.6163 1.53158 0.765788 0.643093i \(-0.222349\pi\)
0.765788 + 0.643093i \(0.222349\pi\)
\(812\) 6.00017 0.210565
\(813\) −41.1181 −1.44208
\(814\) 14.6395 0.513113
\(815\) 6.33410 0.221874
\(816\) −5.42512 −0.189917
\(817\) 0 0
\(818\) −39.8965 −1.39495
\(819\) 11.3929 0.398098
\(820\) −0.793902 −0.0277243
\(821\) −43.0095 −1.50104 −0.750521 0.660847i \(-0.770197\pi\)
−0.750521 + 0.660847i \(0.770197\pi\)
\(822\) −36.7668 −1.28239
\(823\) −3.41982 −0.119207 −0.0596037 0.998222i \(-0.518984\pi\)
−0.0596037 + 0.998222i \(0.518984\pi\)
\(824\) 8.34988 0.290882
\(825\) 34.6679 1.20698
\(826\) −12.7768 −0.444560
\(827\) 27.6387 0.961093 0.480547 0.876969i \(-0.340438\pi\)
0.480547 + 0.876969i \(0.340438\pi\)
\(828\) 4.82999 0.167854
\(829\) −25.7042 −0.892745 −0.446373 0.894847i \(-0.647284\pi\)
−0.446373 + 0.894847i \(0.647284\pi\)
\(830\) 7.97439 0.276795
\(831\) −43.4765 −1.50818
\(832\) −6.38112 −0.221226
\(833\) 2.47999 0.0859266
\(834\) −29.6559 −1.02690
\(835\) −7.80944 −0.270257
\(836\) 0 0
\(837\) 20.5292 0.709593
\(838\) 34.5562 1.19372
\(839\) −36.9844 −1.27684 −0.638422 0.769686i \(-0.720413\pi\)
−0.638422 + 0.769686i \(0.720413\pi\)
\(840\) 1.34784 0.0465049
\(841\) 7.00200 0.241448
\(842\) −28.8651 −0.994757
\(843\) −46.5825 −1.60439
\(844\) 16.6987 0.574793
\(845\) −17.0786 −0.587521
\(846\) −2.74505 −0.0943766
\(847\) −0.764783 −0.0262783
\(848\) 5.11208 0.175550
\(849\) 8.25727 0.283389
\(850\) 11.4585 0.393023
\(851\) −11.5463 −0.395803
\(852\) 0.0535310 0.00183394
\(853\) −39.1925 −1.34193 −0.670963 0.741491i \(-0.734119\pi\)
−0.670963 + 0.741491i \(0.734119\pi\)
\(854\) −5.54592 −0.189777
\(855\) 0 0
\(856\) 13.5562 0.463342
\(857\) 5.62471 0.192136 0.0960682 0.995375i \(-0.469373\pi\)
0.0960682 + 0.995375i \(0.469373\pi\)
\(858\) −47.8793 −1.63457
\(859\) −23.0096 −0.785076 −0.392538 0.919736i \(-0.628403\pi\)
−0.392538 + 0.919736i \(0.628403\pi\)
\(860\) 6.38221 0.217631
\(861\) 2.81869 0.0960607
\(862\) −19.4879 −0.663760
\(863\) −10.3667 −0.352888 −0.176444 0.984311i \(-0.556459\pi\)
−0.176444 + 0.984311i \(0.556459\pi\)
\(864\) −2.65701 −0.0903932
\(865\) 3.84740 0.130816
\(866\) −1.44225 −0.0490097
\(867\) 23.7342 0.806055
\(868\) −7.72644 −0.262252
\(869\) −10.1625 −0.344738
\(870\) 8.08726 0.274184
\(871\) 26.5762 0.900499
\(872\) 0.558302 0.0189065
\(873\) 2.70870 0.0916756
\(874\) 0 0
\(875\) −5.92749 −0.200386
\(876\) −28.5858 −0.965825
\(877\) 37.9086 1.28008 0.640040 0.768341i \(-0.278918\pi\)
0.640040 + 0.768341i \(0.278918\pi\)
\(878\) 16.5525 0.558619
\(879\) −37.0975 −1.25127
\(880\) −2.11335 −0.0712409
\(881\) 26.1648 0.881514 0.440757 0.897626i \(-0.354710\pi\)
0.440757 + 0.897626i \(0.354710\pi\)
\(882\) −1.78540 −0.0601175
\(883\) −9.63158 −0.324128 −0.162064 0.986780i \(-0.551815\pi\)
−0.162064 + 0.986780i \(0.551815\pi\)
\(884\) −15.8251 −0.532257
\(885\) −17.2210 −0.578878
\(886\) −3.08768 −0.103733
\(887\) −1.96127 −0.0658531 −0.0329266 0.999458i \(-0.510483\pi\)
−0.0329266 + 0.999458i \(0.510483\pi\)
\(888\) −9.33667 −0.313318
\(889\) 4.44613 0.149119
\(890\) 7.67152 0.257150
\(891\) −38.3079 −1.28336
\(892\) −14.5844 −0.488320
\(893\) 0 0
\(894\) 29.7486 0.994942
\(895\) 10.9658 0.366546
\(896\) 1.00000 0.0334077
\(897\) 37.7631 1.26087
\(898\) −17.9081 −0.597601
\(899\) −46.3599 −1.54619
\(900\) −8.24921 −0.274974
\(901\) 12.6779 0.422363
\(902\) −4.41957 −0.147156
\(903\) −22.6595 −0.754062
\(904\) 10.0986 0.335875
\(905\) −3.65221 −0.121403
\(906\) 44.2598 1.47043
\(907\) 43.4844 1.44387 0.721937 0.691958i \(-0.243252\pi\)
0.721937 + 0.691958i \(0.243252\pi\)
\(908\) 4.73333 0.157081
\(909\) −17.7259 −0.587930
\(910\) 3.93166 0.130333
\(911\) 4.93846 0.163618 0.0818092 0.996648i \(-0.473930\pi\)
0.0818092 + 0.996648i \(0.473930\pi\)
\(912\) 0 0
\(913\) 44.3926 1.46918
\(914\) −31.2954 −1.03516
\(915\) −7.47500 −0.247116
\(916\) 14.8611 0.491026
\(917\) −7.17183 −0.236835
\(918\) −6.58936 −0.217481
\(919\) 1.25165 0.0412883 0.0206441 0.999787i \(-0.493428\pi\)
0.0206441 + 0.999787i \(0.493428\pi\)
\(920\) 1.66682 0.0549536
\(921\) −5.87544 −0.193602
\(922\) 11.6085 0.382304
\(923\) 0.156150 0.00513975
\(924\) 7.50328 0.246840
\(925\) 19.7201 0.648394
\(926\) −4.15132 −0.136421
\(927\) −14.9079 −0.489639
\(928\) 6.00017 0.196965
\(929\) −38.7110 −1.27007 −0.635034 0.772484i \(-0.719014\pi\)
−0.635034 + 0.772484i \(0.719014\pi\)
\(930\) −10.4140 −0.341488
\(931\) 0 0
\(932\) −4.36751 −0.143062
\(933\) −3.87371 −0.126819
\(934\) 24.9196 0.815394
\(935\) −5.24108 −0.171402
\(936\) 11.3929 0.372387
\(937\) 18.5473 0.605913 0.302957 0.953004i \(-0.402026\pi\)
0.302957 + 0.953004i \(0.402026\pi\)
\(938\) −4.16481 −0.135986
\(939\) −17.2758 −0.563773
\(940\) −0.947313 −0.0308979
\(941\) −24.6283 −0.802859 −0.401429 0.915890i \(-0.631487\pi\)
−0.401429 + 0.915890i \(0.631487\pi\)
\(942\) −24.3030 −0.791835
\(943\) 3.48577 0.113512
\(944\) −12.7768 −0.415848
\(945\) 1.63709 0.0532544
\(946\) 35.5291 1.15515
\(947\) 20.7160 0.673178 0.336589 0.941652i \(-0.390727\pi\)
0.336589 + 0.941652i \(0.390727\pi\)
\(948\) 6.48135 0.210505
\(949\) −83.3851 −2.70679
\(950\) 0 0
\(951\) −25.4199 −0.824296
\(952\) 2.47999 0.0803770
\(953\) 42.6761 1.38242 0.691208 0.722656i \(-0.257079\pi\)
0.691208 + 0.722656i \(0.257079\pi\)
\(954\) −9.12710 −0.295501
\(955\) −10.0780 −0.326116
\(956\) −24.2217 −0.783385
\(957\) 45.0209 1.45532
\(958\) 18.9778 0.613144
\(959\) 16.8073 0.542735
\(960\) 1.34784 0.0435013
\(961\) 28.6979 0.925738
\(962\) −27.2352 −0.878097
\(963\) −24.2033 −0.779939
\(964\) 17.1009 0.550784
\(965\) 12.3157 0.396457
\(966\) −5.91793 −0.190406
\(967\) 46.5740 1.49772 0.748860 0.662728i \(-0.230602\pi\)
0.748860 + 0.662728i \(0.230602\pi\)
\(968\) −0.764783 −0.0245811
\(969\) 0 0
\(970\) 0.934769 0.0300136
\(971\) −16.9447 −0.543782 −0.271891 0.962328i \(-0.587649\pi\)
−0.271891 + 0.962328i \(0.587649\pi\)
\(972\) 16.4608 0.527980
\(973\) 13.5566 0.434606
\(974\) 36.1508 1.15835
\(975\) −64.4960 −2.06553
\(976\) −5.54592 −0.177520
\(977\) −10.1186 −0.323723 −0.161862 0.986813i \(-0.551750\pi\)
−0.161862 + 0.986813i \(0.551750\pi\)
\(978\) −22.4887 −0.719110
\(979\) 42.7066 1.36491
\(980\) −0.616139 −0.0196818
\(981\) −0.996792 −0.0318251
\(982\) −3.08754 −0.0985273
\(983\) −36.8594 −1.17563 −0.587816 0.808995i \(-0.700012\pi\)
−0.587816 + 0.808995i \(0.700012\pi\)
\(984\) 2.81869 0.0898565
\(985\) 2.12217 0.0676179
\(986\) 14.8804 0.473887
\(987\) 3.36336 0.107057
\(988\) 0 0
\(989\) −28.0222 −0.891055
\(990\) 3.77317 0.119919
\(991\) −50.6580 −1.60921 −0.804603 0.593813i \(-0.797622\pi\)
−0.804603 + 0.593813i \(0.797622\pi\)
\(992\) −7.72644 −0.245315
\(993\) −2.52339 −0.0800774
\(994\) −0.0244707 −0.000776163 0
\(995\) −1.92868 −0.0611432
\(996\) −28.3125 −0.897114
\(997\) −19.9246 −0.631019 −0.315510 0.948922i \(-0.602175\pi\)
−0.315510 + 0.948922i \(0.602175\pi\)
\(998\) −30.9593 −0.979999
\(999\) −11.3403 −0.358792
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 5054.2.a.bj.1.3 9
19.14 odd 18 266.2.u.c.253.1 yes 18
19.15 odd 18 266.2.u.c.225.1 18
19.18 odd 2 5054.2.a.bk.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.u.c.225.1 18 19.15 odd 18
266.2.u.c.253.1 yes 18 19.14 odd 18
5054.2.a.bj.1.3 9 1.1 even 1 trivial
5054.2.a.bk.1.7 9 19.18 odd 2