Properties

Label 4033.2.a.f.1.2
Level 4033
Weight 2
Character 4033.1
Self dual yes
Analytic conductor 32.204
Analytic rank 0
Dimension 85
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4033 = 37 \cdot 109 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4033.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.2036671352\)
Analytic rank: \(0\)
Dimension: \(85\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) = 4033.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.68964 q^{2} -0.345637 q^{3} +5.23416 q^{4} +3.86855 q^{5} +0.929638 q^{6} -2.16399 q^{7} -8.69874 q^{8} -2.88054 q^{9} +O(q^{10})\) \(q-2.68964 q^{2} -0.345637 q^{3} +5.23416 q^{4} +3.86855 q^{5} +0.929638 q^{6} -2.16399 q^{7} -8.69874 q^{8} -2.88054 q^{9} -10.4050 q^{10} -2.11365 q^{11} -1.80912 q^{12} -3.53651 q^{13} +5.82035 q^{14} -1.33711 q^{15} +12.9281 q^{16} -1.96634 q^{17} +7.74760 q^{18} -0.950502 q^{19} +20.2486 q^{20} +0.747953 q^{21} +5.68496 q^{22} +0.909001 q^{23} +3.00660 q^{24} +9.96568 q^{25} +9.51193 q^{26} +2.03253 q^{27} -11.3267 q^{28} -8.86822 q^{29} +3.59635 q^{30} -5.20464 q^{31} -17.3746 q^{32} +0.730555 q^{33} +5.28875 q^{34} -8.37149 q^{35} -15.0772 q^{36} +1.00000 q^{37} +2.55651 q^{38} +1.22235 q^{39} -33.6515 q^{40} +0.661852 q^{41} -2.01173 q^{42} +2.54695 q^{43} -11.0632 q^{44} -11.1435 q^{45} -2.44488 q^{46} +3.71734 q^{47} -4.46844 q^{48} -2.31716 q^{49} -26.8041 q^{50} +0.679640 q^{51} -18.5106 q^{52} +8.37049 q^{53} -5.46677 q^{54} -8.17677 q^{55} +18.8240 q^{56} +0.328528 q^{57} +23.8523 q^{58} -3.52961 q^{59} -6.99867 q^{60} -6.66154 q^{61} +13.9986 q^{62} +6.23344 q^{63} +20.8751 q^{64} -13.6812 q^{65} -1.96493 q^{66} +11.9472 q^{67} -10.2922 q^{68} -0.314184 q^{69} +22.5163 q^{70} +1.05772 q^{71} +25.0570 q^{72} -2.24645 q^{73} -2.68964 q^{74} -3.44451 q^{75} -4.97508 q^{76} +4.57391 q^{77} -3.28767 q^{78} +9.45818 q^{79} +50.0132 q^{80} +7.93909 q^{81} -1.78014 q^{82} +0.586152 q^{83} +3.91491 q^{84} -7.60689 q^{85} -6.85039 q^{86} +3.06518 q^{87} +18.3861 q^{88} +9.11543 q^{89} +29.9720 q^{90} +7.65295 q^{91} +4.75786 q^{92} +1.79891 q^{93} -9.99830 q^{94} -3.67706 q^{95} +6.00529 q^{96} +4.64541 q^{97} +6.23232 q^{98} +6.08845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + O(q^{10}) \) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + 9q^{10} + 37q^{11} + 44q^{12} + 14q^{13} + 26q^{14} + 27q^{15} + 85q^{16} + 34q^{17} + 3q^{18} + 15q^{19} + 15q^{20} + 17q^{21} + q^{22} + 72q^{23} + 15q^{24} + 85q^{25} + 33q^{26} + 69q^{27} + 7q^{28} + 19q^{29} - 9q^{30} + 23q^{31} + 51q^{32} + 32q^{33} + 49q^{34} + 40q^{35} + 121q^{36} + 85q^{37} + 84q^{38} + 39q^{39} + 22q^{40} + 55q^{41} - 28q^{42} + 78q^{44} + 28q^{45} + 17q^{46} + 184q^{47} + 97q^{48} + 88q^{49} + 26q^{50} + 27q^{51} + 73q^{52} + 64q^{53} + 31q^{54} + 39q^{55} + 68q^{56} - 33q^{57} + 28q^{58} + 60q^{59} - 22q^{60} + 7q^{61} + 70q^{62} + 28q^{63} + 102q^{64} + 17q^{65} - 15q^{66} + 82q^{67} + 92q^{68} + 22q^{69} - 41q^{70} + 113q^{71} - 19q^{73} + 11q^{74} + 45q^{75} + 34q^{76} + 64q^{77} + 29q^{78} + 23q^{79} + 54q^{80} + 149q^{81} + 4q^{82} + 100q^{83} - 49q^{84} - 5q^{85} - 24q^{86} + 65q^{87} + 14q^{88} + 84q^{89} - 21q^{90} + 32q^{91} + 95q^{92} + 19q^{93} - 47q^{94} + 102q^{95} + 29q^{96} + 7q^{97} + 26q^{98} + 107q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.68964 −1.90186 −0.950931 0.309402i \(-0.899871\pi\)
−0.950931 + 0.309402i \(0.899871\pi\)
\(3\) −0.345637 −0.199553 −0.0997767 0.995010i \(-0.531813\pi\)
−0.0997767 + 0.995010i \(0.531813\pi\)
\(4\) 5.23416 2.61708
\(5\) 3.86855 1.73007 0.865034 0.501713i \(-0.167297\pi\)
0.865034 + 0.501713i \(0.167297\pi\)
\(6\) 0.929638 0.379523
\(7\) −2.16399 −0.817910 −0.408955 0.912554i \(-0.634107\pi\)
−0.408955 + 0.912554i \(0.634107\pi\)
\(8\) −8.69874 −3.07547
\(9\) −2.88054 −0.960178
\(10\) −10.4050 −3.29035
\(11\) −2.11365 −0.637290 −0.318645 0.947874i \(-0.603228\pi\)
−0.318645 + 0.947874i \(0.603228\pi\)
\(12\) −1.80912 −0.522248
\(13\) −3.53651 −0.980850 −0.490425 0.871483i \(-0.663158\pi\)
−0.490425 + 0.871483i \(0.663158\pi\)
\(14\) 5.82035 1.55555
\(15\) −1.33711 −0.345241
\(16\) 12.9281 3.23204
\(17\) −1.96634 −0.476908 −0.238454 0.971154i \(-0.576641\pi\)
−0.238454 + 0.971154i \(0.576641\pi\)
\(18\) 7.74760 1.82613
\(19\) −0.950502 −0.218060 −0.109030 0.994038i \(-0.534774\pi\)
−0.109030 + 0.994038i \(0.534774\pi\)
\(20\) 20.2486 4.52773
\(21\) 0.747953 0.163217
\(22\) 5.68496 1.21204
\(23\) 0.909001 0.189540 0.0947699 0.995499i \(-0.469788\pi\)
0.0947699 + 0.995499i \(0.469788\pi\)
\(24\) 3.00660 0.613720
\(25\) 9.96568 1.99314
\(26\) 9.51193 1.86544
\(27\) 2.03253 0.391160
\(28\) −11.3267 −2.14054
\(29\) −8.86822 −1.64679 −0.823393 0.567471i \(-0.807922\pi\)
−0.823393 + 0.567471i \(0.807922\pi\)
\(30\) 3.59635 0.656601
\(31\) −5.20464 −0.934781 −0.467390 0.884051i \(-0.654806\pi\)
−0.467390 + 0.884051i \(0.654806\pi\)
\(32\) −17.3746 −3.07142
\(33\) 0.730555 0.127173
\(34\) 5.28875 0.907014
\(35\) −8.37149 −1.41504
\(36\) −15.0772 −2.51287
\(37\) 1.00000 0.164399
\(38\) 2.55651 0.414720
\(39\) 1.22235 0.195732
\(40\) −33.6515 −5.32077
\(41\) 0.661852 0.103364 0.0516820 0.998664i \(-0.483542\pi\)
0.0516820 + 0.998664i \(0.483542\pi\)
\(42\) −2.01173 −0.310416
\(43\) 2.54695 0.388407 0.194203 0.980961i \(-0.437788\pi\)
0.194203 + 0.980961i \(0.437788\pi\)
\(44\) −11.0632 −1.66784
\(45\) −11.1435 −1.66117
\(46\) −2.44488 −0.360479
\(47\) 3.71734 0.542230 0.271115 0.962547i \(-0.412608\pi\)
0.271115 + 0.962547i \(0.412608\pi\)
\(48\) −4.46844 −0.644964
\(49\) −2.31716 −0.331023
\(50\) −26.8041 −3.79067
\(51\) 0.679640 0.0951686
\(52\) −18.5106 −2.56697
\(53\) 8.37049 1.14977 0.574887 0.818233i \(-0.305046\pi\)
0.574887 + 0.818233i \(0.305046\pi\)
\(54\) −5.46677 −0.743933
\(55\) −8.17677 −1.10255
\(56\) 18.8240 2.51546
\(57\) 0.328528 0.0435146
\(58\) 23.8523 3.13196
\(59\) −3.52961 −0.459516 −0.229758 0.973248i \(-0.573793\pi\)
−0.229758 + 0.973248i \(0.573793\pi\)
\(60\) −6.99867 −0.903524
\(61\) −6.66154 −0.852923 −0.426462 0.904506i \(-0.640240\pi\)
−0.426462 + 0.904506i \(0.640240\pi\)
\(62\) 13.9986 1.77783
\(63\) 6.23344 0.785340
\(64\) 20.8751 2.60938
\(65\) −13.6812 −1.69694
\(66\) −1.96493 −0.241866
\(67\) 11.9472 1.45958 0.729792 0.683670i \(-0.239617\pi\)
0.729792 + 0.683670i \(0.239617\pi\)
\(68\) −10.2922 −1.24811
\(69\) −0.314184 −0.0378233
\(70\) 22.5163 2.69121
\(71\) 1.05772 0.125528 0.0627641 0.998028i \(-0.480008\pi\)
0.0627641 + 0.998028i \(0.480008\pi\)
\(72\) 25.0570 2.95300
\(73\) −2.24645 −0.262927 −0.131464 0.991321i \(-0.541968\pi\)
−0.131464 + 0.991321i \(0.541968\pi\)
\(74\) −2.68964 −0.312664
\(75\) −3.44451 −0.397737
\(76\) −4.97508 −0.570681
\(77\) 4.57391 0.521246
\(78\) −3.28767 −0.372255
\(79\) 9.45818 1.06413 0.532064 0.846704i \(-0.321417\pi\)
0.532064 + 0.846704i \(0.321417\pi\)
\(80\) 50.0132 5.59164
\(81\) 7.93909 0.882121
\(82\) −1.78014 −0.196584
\(83\) 0.586152 0.0643385 0.0321693 0.999482i \(-0.489758\pi\)
0.0321693 + 0.999482i \(0.489758\pi\)
\(84\) 3.91491 0.427152
\(85\) −7.60689 −0.825083
\(86\) −6.85039 −0.738696
\(87\) 3.06518 0.328622
\(88\) 18.3861 1.95996
\(89\) 9.11543 0.966234 0.483117 0.875556i \(-0.339505\pi\)
0.483117 + 0.875556i \(0.339505\pi\)
\(90\) 29.9720 3.15933
\(91\) 7.65295 0.802247
\(92\) 4.75786 0.496041
\(93\) 1.79891 0.186539
\(94\) −9.99830 −1.03125
\(95\) −3.67706 −0.377259
\(96\) 6.00529 0.612912
\(97\) 4.64541 0.471670 0.235835 0.971793i \(-0.424217\pi\)
0.235835 + 0.971793i \(0.424217\pi\)
\(98\) 6.23232 0.629560
\(99\) 6.08845 0.611912
\(100\) 52.1620 5.21620
\(101\) 16.5113 1.64294 0.821468 0.570254i \(-0.193155\pi\)
0.821468 + 0.570254i \(0.193155\pi\)
\(102\) −1.82799 −0.180998
\(103\) 9.99313 0.984652 0.492326 0.870411i \(-0.336147\pi\)
0.492326 + 0.870411i \(0.336147\pi\)
\(104\) 30.7631 3.01657
\(105\) 2.89350 0.282376
\(106\) −22.5136 −2.18671
\(107\) −7.15332 −0.691538 −0.345769 0.938320i \(-0.612382\pi\)
−0.345769 + 0.938320i \(0.612382\pi\)
\(108\) 10.6386 1.02370
\(109\) −1.00000 −0.0957826
\(110\) 21.9926 2.09691
\(111\) −0.345637 −0.0328064
\(112\) −27.9763 −2.64351
\(113\) 11.1017 1.04436 0.522180 0.852835i \(-0.325119\pi\)
0.522180 + 0.852835i \(0.325119\pi\)
\(114\) −0.883623 −0.0827589
\(115\) 3.51651 0.327917
\(116\) −46.4177 −4.30978
\(117\) 10.1870 0.941791
\(118\) 9.49337 0.873936
\(119\) 4.25514 0.390068
\(120\) 11.6312 1.06178
\(121\) −6.53248 −0.593862
\(122\) 17.9172 1.62214
\(123\) −0.228760 −0.0206266
\(124\) −27.2419 −2.44640
\(125\) 19.2100 1.71819
\(126\) −16.7657 −1.49361
\(127\) 16.4682 1.46131 0.730657 0.682745i \(-0.239214\pi\)
0.730657 + 0.682745i \(0.239214\pi\)
\(128\) −21.3973 −1.89127
\(129\) −0.880320 −0.0775079
\(130\) 36.7974 3.22734
\(131\) −4.62509 −0.404096 −0.202048 0.979376i \(-0.564760\pi\)
−0.202048 + 0.979376i \(0.564760\pi\)
\(132\) 3.82385 0.332823
\(133\) 2.05687 0.178354
\(134\) −32.1337 −2.77593
\(135\) 7.86294 0.676734
\(136\) 17.1047 1.46672
\(137\) 4.42448 0.378009 0.189005 0.981976i \(-0.439474\pi\)
0.189005 + 0.981976i \(0.439474\pi\)
\(138\) 0.845042 0.0719347
\(139\) 18.0338 1.52961 0.764804 0.644263i \(-0.222836\pi\)
0.764804 + 0.644263i \(0.222836\pi\)
\(140\) −43.8178 −3.70328
\(141\) −1.28485 −0.108204
\(142\) −2.84489 −0.238737
\(143\) 7.47494 0.625086
\(144\) −37.2400 −3.10333
\(145\) −34.3071 −2.84905
\(146\) 6.04215 0.500052
\(147\) 0.800895 0.0660567
\(148\) 5.23416 0.430246
\(149\) −13.6491 −1.11818 −0.559088 0.829108i \(-0.688849\pi\)
−0.559088 + 0.829108i \(0.688849\pi\)
\(150\) 9.26448 0.756442
\(151\) −1.80394 −0.146802 −0.0734012 0.997302i \(-0.523385\pi\)
−0.0734012 + 0.997302i \(0.523385\pi\)
\(152\) 8.26816 0.670637
\(153\) 5.66412 0.457917
\(154\) −12.3022 −0.991338
\(155\) −20.1344 −1.61724
\(156\) 6.39796 0.512247
\(157\) −22.8056 −1.82008 −0.910041 0.414518i \(-0.863950\pi\)
−0.910041 + 0.414518i \(0.863950\pi\)
\(158\) −25.4391 −2.02383
\(159\) −2.89315 −0.229441
\(160\) −67.2144 −5.31377
\(161\) −1.96707 −0.155027
\(162\) −21.3533 −1.67767
\(163\) 1.30217 0.101994 0.0509970 0.998699i \(-0.483760\pi\)
0.0509970 + 0.998699i \(0.483760\pi\)
\(164\) 3.46424 0.270512
\(165\) 2.82619 0.220019
\(166\) −1.57654 −0.122363
\(167\) 14.7007 1.13757 0.568786 0.822486i \(-0.307413\pi\)
0.568786 + 0.822486i \(0.307413\pi\)
\(168\) −6.50625 −0.501968
\(169\) −0.493128 −0.0379329
\(170\) 20.4598 1.56920
\(171\) 2.73795 0.209377
\(172\) 13.3312 1.01649
\(173\) 6.25440 0.475513 0.237757 0.971325i \(-0.423588\pi\)
0.237757 + 0.971325i \(0.423588\pi\)
\(174\) −8.24423 −0.624994
\(175\) −21.5656 −1.63021
\(176\) −27.3256 −2.05974
\(177\) 1.21996 0.0916980
\(178\) −24.5172 −1.83764
\(179\) 24.0229 1.79556 0.897779 0.440446i \(-0.145180\pi\)
0.897779 + 0.440446i \(0.145180\pi\)
\(180\) −58.3269 −4.34743
\(181\) −6.76865 −0.503110 −0.251555 0.967843i \(-0.580942\pi\)
−0.251555 + 0.967843i \(0.580942\pi\)
\(182\) −20.5837 −1.52576
\(183\) 2.30247 0.170204
\(184\) −7.90716 −0.582923
\(185\) 3.86855 0.284421
\(186\) −4.83843 −0.354771
\(187\) 4.15616 0.303929
\(188\) 19.4572 1.41906
\(189\) −4.39837 −0.319934
\(190\) 9.88998 0.717494
\(191\) 7.10824 0.514334 0.257167 0.966367i \(-0.417211\pi\)
0.257167 + 0.966367i \(0.417211\pi\)
\(192\) −7.21519 −0.520711
\(193\) 12.9026 0.928747 0.464373 0.885640i \(-0.346280\pi\)
0.464373 + 0.885640i \(0.346280\pi\)
\(194\) −12.4945 −0.897052
\(195\) 4.72871 0.338630
\(196\) −12.1284 −0.866314
\(197\) 17.3765 1.23803 0.619013 0.785381i \(-0.287533\pi\)
0.619013 + 0.785381i \(0.287533\pi\)
\(198\) −16.3757 −1.16377
\(199\) 1.04875 0.0743437 0.0371718 0.999309i \(-0.488165\pi\)
0.0371718 + 0.999309i \(0.488165\pi\)
\(200\) −86.6888 −6.12983
\(201\) −4.12939 −0.291265
\(202\) −44.4095 −3.12464
\(203\) 19.1907 1.34692
\(204\) 3.55735 0.249064
\(205\) 2.56041 0.178827
\(206\) −26.8779 −1.87267
\(207\) −2.61841 −0.181992
\(208\) −45.7204 −3.17014
\(209\) 2.00903 0.138967
\(210\) −7.78246 −0.537041
\(211\) −8.82102 −0.607264 −0.303632 0.952789i \(-0.598199\pi\)
−0.303632 + 0.952789i \(0.598199\pi\)
\(212\) 43.8125 3.00905
\(213\) −0.365587 −0.0250496
\(214\) 19.2399 1.31521
\(215\) 9.85302 0.671970
\(216\) −17.6804 −1.20300
\(217\) 11.2628 0.764567
\(218\) 2.68964 0.182165
\(219\) 0.776456 0.0524681
\(220\) −42.7985 −2.88548
\(221\) 6.95398 0.467775
\(222\) 0.929638 0.0623932
\(223\) −2.80122 −0.187584 −0.0937919 0.995592i \(-0.529899\pi\)
−0.0937919 + 0.995592i \(0.529899\pi\)
\(224\) 37.5984 2.51215
\(225\) −28.7065 −1.91377
\(226\) −29.8596 −1.98623
\(227\) 1.65494 0.109843 0.0549213 0.998491i \(-0.482509\pi\)
0.0549213 + 0.998491i \(0.482509\pi\)
\(228\) 1.71957 0.113881
\(229\) −10.7610 −0.711107 −0.355554 0.934656i \(-0.615708\pi\)
−0.355554 + 0.934656i \(0.615708\pi\)
\(230\) −9.45816 −0.623653
\(231\) −1.58091 −0.104016
\(232\) 77.1423 5.06464
\(233\) 28.3296 1.85593 0.927967 0.372662i \(-0.121555\pi\)
0.927967 + 0.372662i \(0.121555\pi\)
\(234\) −27.3994 −1.79116
\(235\) 14.3807 0.938094
\(236\) −18.4745 −1.20259
\(237\) −3.26909 −0.212350
\(238\) −11.4448 −0.741856
\(239\) −7.79201 −0.504023 −0.252011 0.967724i \(-0.581092\pi\)
−0.252011 + 0.967724i \(0.581092\pi\)
\(240\) −17.2864 −1.11583
\(241\) −22.8239 −1.47022 −0.735110 0.677948i \(-0.762869\pi\)
−0.735110 + 0.677948i \(0.762869\pi\)
\(242\) 17.5700 1.12944
\(243\) −8.84163 −0.567191
\(244\) −34.8676 −2.23217
\(245\) −8.96405 −0.572692
\(246\) 0.615283 0.0392290
\(247\) 3.36146 0.213884
\(248\) 45.2738 2.87489
\(249\) −0.202596 −0.0128390
\(250\) −51.6680 −3.26777
\(251\) 10.4314 0.658425 0.329212 0.944256i \(-0.393217\pi\)
0.329212 + 0.944256i \(0.393217\pi\)
\(252\) 32.6269 2.05530
\(253\) −1.92131 −0.120792
\(254\) −44.2935 −2.77922
\(255\) 2.62922 0.164648
\(256\) 15.8008 0.987551
\(257\) −11.7357 −0.732055 −0.366027 0.930604i \(-0.619282\pi\)
−0.366027 + 0.930604i \(0.619282\pi\)
\(258\) 2.36775 0.147409
\(259\) −2.16399 −0.134464
\(260\) −71.6094 −4.44103
\(261\) 25.5452 1.58121
\(262\) 12.4398 0.768536
\(263\) −18.8468 −1.16214 −0.581072 0.813852i \(-0.697367\pi\)
−0.581072 + 0.813852i \(0.697367\pi\)
\(264\) −6.35491 −0.391118
\(265\) 32.3816 1.98919
\(266\) −5.53225 −0.339204
\(267\) −3.15063 −0.192815
\(268\) 62.5336 3.81985
\(269\) 16.1673 0.985740 0.492870 0.870103i \(-0.335948\pi\)
0.492870 + 0.870103i \(0.335948\pi\)
\(270\) −21.1485 −1.28706
\(271\) 17.3848 1.05605 0.528025 0.849229i \(-0.322933\pi\)
0.528025 + 0.849229i \(0.322933\pi\)
\(272\) −25.4211 −1.54138
\(273\) −2.64514 −0.160091
\(274\) −11.9003 −0.718922
\(275\) −21.0640 −1.27021
\(276\) −1.64449 −0.0989867
\(277\) −26.4921 −1.59176 −0.795879 0.605455i \(-0.792991\pi\)
−0.795879 + 0.605455i \(0.792991\pi\)
\(278\) −48.5045 −2.90911
\(279\) 14.9921 0.897557
\(280\) 72.8214 4.35191
\(281\) 7.41753 0.442493 0.221246 0.975218i \(-0.428987\pi\)
0.221246 + 0.975218i \(0.428987\pi\)
\(282\) 3.45578 0.205789
\(283\) −2.64722 −0.157361 −0.0786805 0.996900i \(-0.525071\pi\)
−0.0786805 + 0.996900i \(0.525071\pi\)
\(284\) 5.53628 0.328518
\(285\) 1.27093 0.0752833
\(286\) −20.1049 −1.18883
\(287\) −1.43224 −0.0845425
\(288\) 50.0481 2.94911
\(289\) −13.1335 −0.772559
\(290\) 92.2739 5.41851
\(291\) −1.60562 −0.0941234
\(292\) −11.7583 −0.688102
\(293\) 0.472409 0.0275984 0.0137992 0.999905i \(-0.495607\pi\)
0.0137992 + 0.999905i \(0.495607\pi\)
\(294\) −2.15412 −0.125631
\(295\) −13.6545 −0.794994
\(296\) −8.69874 −0.505604
\(297\) −4.29606 −0.249282
\(298\) 36.7111 2.12662
\(299\) −3.21469 −0.185910
\(300\) −18.0291 −1.04091
\(301\) −5.51157 −0.317682
\(302\) 4.85194 0.279198
\(303\) −5.70691 −0.327854
\(304\) −12.2882 −0.704778
\(305\) −25.7705 −1.47562
\(306\) −15.2344 −0.870895
\(307\) 24.7340 1.41165 0.705823 0.708389i \(-0.250578\pi\)
0.705823 + 0.708389i \(0.250578\pi\)
\(308\) 23.9406 1.36414
\(309\) −3.45399 −0.196491
\(310\) 54.1543 3.07576
\(311\) 21.0280 1.19239 0.596195 0.802839i \(-0.296678\pi\)
0.596195 + 0.802839i \(0.296678\pi\)
\(312\) −10.6329 −0.601968
\(313\) −1.89149 −0.106913 −0.0534567 0.998570i \(-0.517024\pi\)
−0.0534567 + 0.998570i \(0.517024\pi\)
\(314\) 61.3388 3.46155
\(315\) 24.1144 1.35869
\(316\) 49.5057 2.78491
\(317\) 24.1853 1.35838 0.679190 0.733962i \(-0.262331\pi\)
0.679190 + 0.733962i \(0.262331\pi\)
\(318\) 7.78152 0.436366
\(319\) 18.7443 1.04948
\(320\) 80.7562 4.51441
\(321\) 2.47245 0.137999
\(322\) 5.29070 0.294839
\(323\) 1.86901 0.103995
\(324\) 41.5545 2.30858
\(325\) −35.2437 −1.95497
\(326\) −3.50237 −0.193979
\(327\) 0.345637 0.0191138
\(328\) −5.75728 −0.317893
\(329\) −8.04427 −0.443495
\(330\) −7.60143 −0.418445
\(331\) −30.0361 −1.65093 −0.825467 0.564451i \(-0.809088\pi\)
−0.825467 + 0.564451i \(0.809088\pi\)
\(332\) 3.06801 0.168379
\(333\) −2.88054 −0.157852
\(334\) −39.5395 −2.16351
\(335\) 46.2184 2.52518
\(336\) 9.66965 0.527522
\(337\) 28.0745 1.52932 0.764658 0.644436i \(-0.222908\pi\)
0.764658 + 0.644436i \(0.222908\pi\)
\(338\) 1.32634 0.0721432
\(339\) −3.83715 −0.208406
\(340\) −39.8157 −2.15931
\(341\) 11.0008 0.595726
\(342\) −7.36411 −0.398205
\(343\) 20.1622 1.08866
\(344\) −22.1553 −1.19453
\(345\) −1.21544 −0.0654369
\(346\) −16.8221 −0.904361
\(347\) −32.0287 −1.71939 −0.859696 0.510807i \(-0.829347\pi\)
−0.859696 + 0.510807i \(0.829347\pi\)
\(348\) 16.0437 0.860031
\(349\) −27.4101 −1.46723 −0.733614 0.679566i \(-0.762168\pi\)
−0.733614 + 0.679566i \(0.762168\pi\)
\(350\) 58.0037 3.10043
\(351\) −7.18805 −0.383670
\(352\) 36.7238 1.95738
\(353\) −13.0182 −0.692889 −0.346445 0.938070i \(-0.612611\pi\)
−0.346445 + 0.938070i \(0.612611\pi\)
\(354\) −3.28126 −0.174397
\(355\) 4.09184 0.217172
\(356\) 47.7117 2.52871
\(357\) −1.47073 −0.0778394
\(358\) −64.6130 −3.41490
\(359\) 23.8608 1.25932 0.629662 0.776869i \(-0.283193\pi\)
0.629662 + 0.776869i \(0.283193\pi\)
\(360\) 96.9343 5.10889
\(361\) −18.0965 −0.952450
\(362\) 18.2052 0.956846
\(363\) 2.25786 0.118507
\(364\) 40.0568 2.09955
\(365\) −8.69051 −0.454882
\(366\) −6.19283 −0.323704
\(367\) 12.3113 0.642645 0.321323 0.946970i \(-0.395873\pi\)
0.321323 + 0.946970i \(0.395873\pi\)
\(368\) 11.7517 0.612599
\(369\) −1.90649 −0.0992478
\(370\) −10.4050 −0.540931
\(371\) −18.1136 −0.940412
\(372\) 9.41581 0.488187
\(373\) −23.7004 −1.22716 −0.613580 0.789633i \(-0.710271\pi\)
−0.613580 + 0.789633i \(0.710271\pi\)
\(374\) −11.1786 −0.578030
\(375\) −6.63968 −0.342872
\(376\) −32.3361 −1.66761
\(377\) 31.3625 1.61525
\(378\) 11.8300 0.608471
\(379\) −18.9519 −0.973496 −0.486748 0.873542i \(-0.661817\pi\)
−0.486748 + 0.873542i \(0.661817\pi\)
\(380\) −19.2464 −0.987317
\(381\) −5.69201 −0.291610
\(382\) −19.1186 −0.978193
\(383\) 4.69227 0.239764 0.119882 0.992788i \(-0.461748\pi\)
0.119882 + 0.992788i \(0.461748\pi\)
\(384\) 7.39568 0.377409
\(385\) 17.6944 0.901791
\(386\) −34.7032 −1.76635
\(387\) −7.33659 −0.372940
\(388\) 24.3148 1.23440
\(389\) −8.92481 −0.452506 −0.226253 0.974069i \(-0.572648\pi\)
−0.226253 + 0.974069i \(0.572648\pi\)
\(390\) −12.7185 −0.644027
\(391\) −1.78741 −0.0903930
\(392\) 20.1564 1.01805
\(393\) 1.59860 0.0806388
\(394\) −46.7366 −2.35456
\(395\) 36.5894 1.84101
\(396\) 31.8679 1.60142
\(397\) −11.7872 −0.591581 −0.295791 0.955253i \(-0.595583\pi\)
−0.295791 + 0.955253i \(0.595583\pi\)
\(398\) −2.82075 −0.141391
\(399\) −0.710931 −0.0355911
\(400\) 128.838 6.44189
\(401\) 1.28910 0.0643748 0.0321874 0.999482i \(-0.489753\pi\)
0.0321874 + 0.999482i \(0.489753\pi\)
\(402\) 11.1066 0.553946
\(403\) 18.4062 0.916880
\(404\) 86.4229 4.29970
\(405\) 30.7128 1.52613
\(406\) −51.6161 −2.56166
\(407\) −2.11365 −0.104770
\(408\) −5.91201 −0.292688
\(409\) 23.1148 1.14296 0.571478 0.820618i \(-0.306370\pi\)
0.571478 + 0.820618i \(0.306370\pi\)
\(410\) −6.88658 −0.340104
\(411\) −1.52926 −0.0754330
\(412\) 52.3057 2.57691
\(413\) 7.63803 0.375843
\(414\) 7.04258 0.346124
\(415\) 2.26756 0.111310
\(416\) 61.4453 3.01260
\(417\) −6.23315 −0.305239
\(418\) −5.40357 −0.264297
\(419\) −11.3461 −0.554293 −0.277146 0.960828i \(-0.589389\pi\)
−0.277146 + 0.960828i \(0.589389\pi\)
\(420\) 15.1450 0.739002
\(421\) 6.08285 0.296460 0.148230 0.988953i \(-0.452642\pi\)
0.148230 + 0.988953i \(0.452642\pi\)
\(422\) 23.7254 1.15493
\(423\) −10.7079 −0.520637
\(424\) −72.8126 −3.53609
\(425\) −19.5959 −0.950543
\(426\) 0.983297 0.0476409
\(427\) 14.4155 0.697615
\(428\) −37.4417 −1.80981
\(429\) −2.58361 −0.124738
\(430\) −26.5011 −1.27799
\(431\) −15.7949 −0.760813 −0.380406 0.924819i \(-0.624216\pi\)
−0.380406 + 0.924819i \(0.624216\pi\)
\(432\) 26.2768 1.26424
\(433\) 16.0145 0.769610 0.384805 0.922998i \(-0.374269\pi\)
0.384805 + 0.922998i \(0.374269\pi\)
\(434\) −30.2928 −1.45410
\(435\) 11.8578 0.568538
\(436\) −5.23416 −0.250671
\(437\) −0.864007 −0.0413310
\(438\) −2.08839 −0.0997870
\(439\) 31.3650 1.49697 0.748485 0.663151i \(-0.230781\pi\)
0.748485 + 0.663151i \(0.230781\pi\)
\(440\) 71.1275 3.39087
\(441\) 6.67466 0.317841
\(442\) −18.7037 −0.889644
\(443\) 16.3845 0.778451 0.389226 0.921142i \(-0.372743\pi\)
0.389226 + 0.921142i \(0.372743\pi\)
\(444\) −1.80912 −0.0858570
\(445\) 35.2635 1.67165
\(446\) 7.53428 0.356759
\(447\) 4.71762 0.223136
\(448\) −45.1734 −2.13424
\(449\) 7.43781 0.351012 0.175506 0.984478i \(-0.443844\pi\)
0.175506 + 0.984478i \(0.443844\pi\)
\(450\) 77.2102 3.63972
\(451\) −1.39892 −0.0658728
\(452\) 58.1081 2.73318
\(453\) 0.623507 0.0292949
\(454\) −4.45121 −0.208905
\(455\) 29.6058 1.38794
\(456\) −2.85778 −0.133828
\(457\) −37.8000 −1.76821 −0.884104 0.467290i \(-0.845231\pi\)
−0.884104 + 0.467290i \(0.845231\pi\)
\(458\) 28.9432 1.35243
\(459\) −3.99665 −0.186548
\(460\) 18.4060 0.858185
\(461\) 13.0769 0.609051 0.304526 0.952504i \(-0.401502\pi\)
0.304526 + 0.952504i \(0.401502\pi\)
\(462\) 4.25209 0.197825
\(463\) 5.67804 0.263881 0.131940 0.991258i \(-0.457879\pi\)
0.131940 + 0.991258i \(0.457879\pi\)
\(464\) −114.650 −5.32247
\(465\) 6.95919 0.322725
\(466\) −76.1964 −3.52973
\(467\) −15.5874 −0.721301 −0.360651 0.932701i \(-0.617445\pi\)
−0.360651 + 0.932701i \(0.617445\pi\)
\(468\) 53.3206 2.46474
\(469\) −25.8536 −1.19381
\(470\) −38.6789 −1.78413
\(471\) 7.88244 0.363204
\(472\) 30.7031 1.41323
\(473\) −5.38337 −0.247528
\(474\) 8.79269 0.403861
\(475\) −9.47240 −0.434623
\(476\) 22.2721 1.02084
\(477\) −24.1115 −1.10399
\(478\) 20.9577 0.958582
\(479\) 13.5956 0.621200 0.310600 0.950541i \(-0.399470\pi\)
0.310600 + 0.950541i \(0.399470\pi\)
\(480\) 23.2318 1.06038
\(481\) −3.53651 −0.161251
\(482\) 61.3882 2.79616
\(483\) 0.679890 0.0309361
\(484\) −34.1921 −1.55418
\(485\) 17.9710 0.816022
\(486\) 23.7808 1.07872
\(487\) −13.8809 −0.629005 −0.314502 0.949257i \(-0.601838\pi\)
−0.314502 + 0.949257i \(0.601838\pi\)
\(488\) 57.9470 2.62314
\(489\) −0.450079 −0.0203533
\(490\) 24.1101 1.08918
\(491\) −6.06555 −0.273734 −0.136867 0.990589i \(-0.543703\pi\)
−0.136867 + 0.990589i \(0.543703\pi\)
\(492\) −1.19737 −0.0539816
\(493\) 17.4379 0.785366
\(494\) −9.04110 −0.406778
\(495\) 23.5535 1.05865
\(496\) −67.2863 −3.02124
\(497\) −2.28889 −0.102671
\(498\) 0.544909 0.0244180
\(499\) −10.4959 −0.469859 −0.234929 0.972012i \(-0.575486\pi\)
−0.234929 + 0.972012i \(0.575486\pi\)
\(500\) 100.548 4.49665
\(501\) −5.08109 −0.227006
\(502\) −28.0567 −1.25223
\(503\) −14.9886 −0.668310 −0.334155 0.942518i \(-0.608451\pi\)
−0.334155 + 0.942518i \(0.608451\pi\)
\(504\) −54.2231 −2.41529
\(505\) 63.8748 2.84239
\(506\) 5.16763 0.229729
\(507\) 0.170443 0.00756964
\(508\) 86.1971 3.82438
\(509\) −31.3789 −1.39085 −0.695423 0.718600i \(-0.744783\pi\)
−0.695423 + 0.718600i \(0.744783\pi\)
\(510\) −7.07166 −0.313138
\(511\) 4.86129 0.215051
\(512\) 0.296039 0.0130832
\(513\) −1.93192 −0.0852965
\(514\) 31.5649 1.39227
\(515\) 38.6589 1.70352
\(516\) −4.60774 −0.202845
\(517\) −7.85716 −0.345557
\(518\) 5.82035 0.255731
\(519\) −2.16175 −0.0948903
\(520\) 119.009 5.21888
\(521\) −7.70180 −0.337422 −0.168711 0.985666i \(-0.553960\pi\)
−0.168711 + 0.985666i \(0.553960\pi\)
\(522\) −68.7074 −3.00724
\(523\) 20.3564 0.890122 0.445061 0.895500i \(-0.353182\pi\)
0.445061 + 0.895500i \(0.353182\pi\)
\(524\) −24.2085 −1.05755
\(525\) 7.45387 0.325313
\(526\) 50.6911 2.21024
\(527\) 10.2341 0.445805
\(528\) 9.44472 0.411029
\(529\) −22.1737 −0.964075
\(530\) −87.0950 −3.78316
\(531\) 10.1672 0.441217
\(532\) 10.7660 0.466766
\(533\) −2.34064 −0.101385
\(534\) 8.47406 0.366708
\(535\) −27.6730 −1.19641
\(536\) −103.926 −4.48890
\(537\) −8.30321 −0.358310
\(538\) −43.4843 −1.87474
\(539\) 4.89767 0.210957
\(540\) 41.1559 1.77107
\(541\) 37.7833 1.62443 0.812216 0.583357i \(-0.198261\pi\)
0.812216 + 0.583357i \(0.198261\pi\)
\(542\) −46.7588 −2.00846
\(543\) 2.33949 0.100397
\(544\) 34.1644 1.46478
\(545\) −3.86855 −0.165710
\(546\) 7.11448 0.304472
\(547\) −38.8381 −1.66060 −0.830299 0.557318i \(-0.811830\pi\)
−0.830299 + 0.557318i \(0.811830\pi\)
\(548\) 23.1585 0.989281
\(549\) 19.1888 0.818958
\(550\) 56.6545 2.41576
\(551\) 8.42926 0.359098
\(552\) 2.73300 0.116324
\(553\) −20.4674 −0.870362
\(554\) 71.2543 3.02731
\(555\) −1.33711 −0.0567573
\(556\) 94.3920 4.00311
\(557\) 29.7144 1.25904 0.629519 0.776985i \(-0.283252\pi\)
0.629519 + 0.776985i \(0.283252\pi\)
\(558\) −40.3235 −1.70703
\(559\) −9.00731 −0.380969
\(560\) −108.228 −4.57346
\(561\) −1.43652 −0.0606500
\(562\) −19.9505 −0.841561
\(563\) 12.0044 0.505924 0.252962 0.967476i \(-0.418595\pi\)
0.252962 + 0.967476i \(0.418595\pi\)
\(564\) −6.72511 −0.283178
\(565\) 42.9475 1.80681
\(566\) 7.12007 0.299279
\(567\) −17.1801 −0.721496
\(568\) −9.20082 −0.386058
\(569\) 4.82179 0.202140 0.101070 0.994879i \(-0.467773\pi\)
0.101070 + 0.994879i \(0.467773\pi\)
\(570\) −3.41834 −0.143179
\(571\) 17.7041 0.740892 0.370446 0.928854i \(-0.379205\pi\)
0.370446 + 0.928854i \(0.379205\pi\)
\(572\) 39.1251 1.63590
\(573\) −2.45687 −0.102637
\(574\) 3.85221 0.160788
\(575\) 9.05881 0.377779
\(576\) −60.1314 −2.50547
\(577\) −30.7159 −1.27872 −0.639359 0.768908i \(-0.720800\pi\)
−0.639359 + 0.768908i \(0.720800\pi\)
\(578\) 35.3244 1.46930
\(579\) −4.45960 −0.185335
\(580\) −179.569 −7.45621
\(581\) −1.26843 −0.0526231
\(582\) 4.31855 0.179010
\(583\) −17.6923 −0.732740
\(584\) 19.5413 0.808624
\(585\) 39.4090 1.62936
\(586\) −1.27061 −0.0524884
\(587\) 27.1975 1.12256 0.561281 0.827625i \(-0.310309\pi\)
0.561281 + 0.827625i \(0.310309\pi\)
\(588\) 4.19202 0.172876
\(589\) 4.94702 0.203838
\(590\) 36.7256 1.51197
\(591\) −6.00597 −0.247052
\(592\) 12.9281 0.531343
\(593\) −25.5638 −1.04978 −0.524889 0.851171i \(-0.675893\pi\)
−0.524889 + 0.851171i \(0.675893\pi\)
\(594\) 11.5548 0.474101
\(595\) 16.4612 0.674844
\(596\) −71.4415 −2.92636
\(597\) −0.362485 −0.0148355
\(598\) 8.64635 0.353575
\(599\) 24.3202 0.993696 0.496848 0.867838i \(-0.334491\pi\)
0.496848 + 0.867838i \(0.334491\pi\)
\(600\) 29.9628 1.22323
\(601\) −15.9277 −0.649705 −0.324853 0.945765i \(-0.605315\pi\)
−0.324853 + 0.945765i \(0.605315\pi\)
\(602\) 14.8242 0.604187
\(603\) −34.4143 −1.40146
\(604\) −9.44210 −0.384194
\(605\) −25.2712 −1.02742
\(606\) 15.3495 0.623533
\(607\) −22.7920 −0.925097 −0.462548 0.886594i \(-0.653065\pi\)
−0.462548 + 0.886594i \(0.653065\pi\)
\(608\) 16.5146 0.669754
\(609\) −6.63301 −0.268783
\(610\) 69.3134 2.80642
\(611\) −13.1464 −0.531846
\(612\) 29.6469 1.19841
\(613\) 23.7478 0.959163 0.479581 0.877497i \(-0.340789\pi\)
0.479581 + 0.877497i \(0.340789\pi\)
\(614\) −66.5256 −2.68476
\(615\) −0.884971 −0.0356855
\(616\) −39.7873 −1.60307
\(617\) 27.1045 1.09119 0.545594 0.838050i \(-0.316304\pi\)
0.545594 + 0.838050i \(0.316304\pi\)
\(618\) 9.28999 0.373698
\(619\) −44.0418 −1.77019 −0.885094 0.465413i \(-0.845906\pi\)
−0.885094 + 0.465413i \(0.845906\pi\)
\(620\) −105.387 −4.23244
\(621\) 1.84757 0.0741404
\(622\) −56.5579 −2.26776
\(623\) −19.7257 −0.790293
\(624\) 15.8027 0.632613
\(625\) 24.4864 0.979457
\(626\) 5.08743 0.203335
\(627\) −0.694394 −0.0277314
\(628\) −119.368 −4.76330
\(629\) −1.96634 −0.0784032
\(630\) −64.8590 −2.58404
\(631\) −20.6647 −0.822648 −0.411324 0.911489i \(-0.634933\pi\)
−0.411324 + 0.911489i \(0.634933\pi\)
\(632\) −82.2742 −3.27269
\(633\) 3.04887 0.121182
\(634\) −65.0497 −2.58345
\(635\) 63.7080 2.52817
\(636\) −15.1432 −0.600467
\(637\) 8.19465 0.324684
\(638\) −50.4155 −1.99597
\(639\) −3.04680 −0.120530
\(640\) −82.7764 −3.27202
\(641\) 12.7501 0.503598 0.251799 0.967780i \(-0.418978\pi\)
0.251799 + 0.967780i \(0.418978\pi\)
\(642\) −6.65000 −0.262455
\(643\) 36.9270 1.45626 0.728128 0.685441i \(-0.240390\pi\)
0.728128 + 0.685441i \(0.240390\pi\)
\(644\) −10.2959 −0.405717
\(645\) −3.40556 −0.134094
\(646\) −5.02697 −0.197783
\(647\) 38.0762 1.49693 0.748465 0.663174i \(-0.230791\pi\)
0.748465 + 0.663174i \(0.230791\pi\)
\(648\) −69.0600 −2.71293
\(649\) 7.46036 0.292845
\(650\) 94.7928 3.71808
\(651\) −3.89283 −0.152572
\(652\) 6.81578 0.266927
\(653\) −0.00788604 −0.000308605 0 −0.000154302 1.00000i \(-0.500049\pi\)
−0.000154302 1.00000i \(0.500049\pi\)
\(654\) −0.929638 −0.0363517
\(655\) −17.8924 −0.699114
\(656\) 8.55652 0.334076
\(657\) 6.47098 0.252457
\(658\) 21.6362 0.843467
\(659\) −50.6543 −1.97321 −0.986606 0.163123i \(-0.947843\pi\)
−0.986606 + 0.163123i \(0.947843\pi\)
\(660\) 14.7927 0.575807
\(661\) 16.1565 0.628416 0.314208 0.949354i \(-0.398261\pi\)
0.314208 + 0.949354i \(0.398261\pi\)
\(662\) 80.7863 3.13985
\(663\) −2.40355 −0.0933462
\(664\) −5.09878 −0.197871
\(665\) 7.95712 0.308564
\(666\) 7.74760 0.300214
\(667\) −8.06122 −0.312132
\(668\) 76.9457 2.97712
\(669\) 0.968206 0.0374330
\(670\) −124.311 −4.80254
\(671\) 14.0802 0.543559
\(672\) −12.9954 −0.501307
\(673\) −0.534697 −0.0206111 −0.0103055 0.999947i \(-0.503280\pi\)
−0.0103055 + 0.999947i \(0.503280\pi\)
\(674\) −75.5104 −2.90855
\(675\) 20.2555 0.779636
\(676\) −2.58111 −0.0992735
\(677\) 12.1931 0.468617 0.234309 0.972162i \(-0.424717\pi\)
0.234309 + 0.972162i \(0.424717\pi\)
\(678\) 10.3206 0.396359
\(679\) −10.0526 −0.385784
\(680\) 66.1704 2.53752
\(681\) −0.572010 −0.0219195
\(682\) −29.5882 −1.13299
\(683\) −42.0787 −1.61009 −0.805047 0.593210i \(-0.797860\pi\)
−0.805047 + 0.593210i \(0.797860\pi\)
\(684\) 14.3309 0.547956
\(685\) 17.1163 0.653982
\(686\) −54.2291 −2.07048
\(687\) 3.71940 0.141904
\(688\) 32.9274 1.25534
\(689\) −29.6023 −1.12776
\(690\) 3.26909 0.124452
\(691\) 31.3963 1.19437 0.597187 0.802102i \(-0.296285\pi\)
0.597187 + 0.802102i \(0.296285\pi\)
\(692\) 32.7365 1.24446
\(693\) −13.1753 −0.500489
\(694\) 86.1457 3.27005
\(695\) 69.7647 2.64633
\(696\) −26.6632 −1.01067
\(697\) −1.30143 −0.0492951
\(698\) 73.7233 2.79047
\(699\) −9.79175 −0.370358
\(700\) −112.878 −4.26638
\(701\) 36.8293 1.39102 0.695511 0.718515i \(-0.255178\pi\)
0.695511 + 0.718515i \(0.255178\pi\)
\(702\) 19.3333 0.729687
\(703\) −0.950502 −0.0358489
\(704\) −44.1226 −1.66293
\(705\) −4.97050 −0.187200
\(706\) 35.0143 1.31778
\(707\) −35.7303 −1.34377
\(708\) 6.38548 0.239981
\(709\) −14.0304 −0.526922 −0.263461 0.964670i \(-0.584864\pi\)
−0.263461 + 0.964670i \(0.584864\pi\)
\(710\) −11.0056 −0.413032
\(711\) −27.2446 −1.02175
\(712\) −79.2927 −2.97162
\(713\) −4.73102 −0.177178
\(714\) 3.95574 0.148040
\(715\) 28.9172 1.08144
\(716\) 125.740 4.69912
\(717\) 2.69320 0.100580
\(718\) −64.1769 −2.39506
\(719\) 16.2660 0.606620 0.303310 0.952892i \(-0.401908\pi\)
0.303310 + 0.952892i \(0.401908\pi\)
\(720\) −144.065 −5.36897
\(721\) −21.6250 −0.805357
\(722\) 48.6732 1.81143
\(723\) 7.88879 0.293387
\(724\) −35.4282 −1.31668
\(725\) −88.3778 −3.28227
\(726\) −6.07284 −0.225384
\(727\) −24.5893 −0.911965 −0.455982 0.889989i \(-0.650712\pi\)
−0.455982 + 0.889989i \(0.650712\pi\)
\(728\) −66.5710 −2.46729
\(729\) −20.7613 −0.768936
\(730\) 23.3743 0.865123
\(731\) −5.00818 −0.185234
\(732\) 12.0515 0.445437
\(733\) −5.04851 −0.186471 −0.0932355 0.995644i \(-0.529721\pi\)
−0.0932355 + 0.995644i \(0.529721\pi\)
\(734\) −33.1130 −1.22222
\(735\) 3.09830 0.114283
\(736\) −15.7935 −0.582156
\(737\) −25.2522 −0.930178
\(738\) 5.12777 0.188756
\(739\) 25.2991 0.930643 0.465322 0.885142i \(-0.345939\pi\)
0.465322 + 0.885142i \(0.345939\pi\)
\(740\) 20.2486 0.744354
\(741\) −1.16184 −0.0426813
\(742\) 48.7191 1.78854
\(743\) 14.3827 0.527651 0.263825 0.964570i \(-0.415016\pi\)
0.263825 + 0.964570i \(0.415016\pi\)
\(744\) −15.6483 −0.573694
\(745\) −52.8022 −1.93452
\(746\) 63.7455 2.33389
\(747\) −1.68843 −0.0617765
\(748\) 21.7540 0.795406
\(749\) 15.4797 0.565616
\(750\) 17.8583 0.652095
\(751\) 22.8332 0.833195 0.416597 0.909091i \(-0.363223\pi\)
0.416597 + 0.909091i \(0.363223\pi\)
\(752\) 48.0583 1.75250
\(753\) −3.60548 −0.131391
\(754\) −84.3538 −3.07199
\(755\) −6.97862 −0.253978
\(756\) −23.0218 −0.837294
\(757\) −36.8537 −1.33947 −0.669735 0.742601i \(-0.733592\pi\)
−0.669735 + 0.742601i \(0.733592\pi\)
\(758\) 50.9739 1.85146
\(759\) 0.664075 0.0241044
\(760\) 31.9858 1.16025
\(761\) 4.39081 0.159167 0.0795834 0.996828i \(-0.474641\pi\)
0.0795834 + 0.996828i \(0.474641\pi\)
\(762\) 15.3094 0.554603
\(763\) 2.16399 0.0783416
\(764\) 37.2057 1.34606
\(765\) 21.9119 0.792227
\(766\) −12.6205 −0.455998
\(767\) 12.4825 0.450716
\(768\) −5.46134 −0.197069
\(769\) −20.7518 −0.748329 −0.374165 0.927362i \(-0.622071\pi\)
−0.374165 + 0.927362i \(0.622071\pi\)
\(770\) −47.5916 −1.71508
\(771\) 4.05630 0.146084
\(772\) 67.5341 2.43061
\(773\) 2.51774 0.0905567 0.0452784 0.998974i \(-0.485583\pi\)
0.0452784 + 0.998974i \(0.485583\pi\)
\(774\) 19.7328 0.709280
\(775\) −51.8678 −1.86315
\(776\) −40.4092 −1.45061
\(777\) 0.747953 0.0268327
\(778\) 24.0045 0.860604
\(779\) −0.629092 −0.0225396
\(780\) 24.7508 0.886222
\(781\) −2.23565 −0.0799979
\(782\) 4.80748 0.171915
\(783\) −18.0249 −0.644158
\(784\) −29.9566 −1.06988
\(785\) −88.2245 −3.14887
\(786\) −4.29966 −0.153364
\(787\) −0.718531 −0.0256128 −0.0128064 0.999918i \(-0.504077\pi\)
−0.0128064 + 0.999918i \(0.504077\pi\)
\(788\) 90.9516 3.24002
\(789\) 6.51415 0.231910
\(790\) −98.4124 −3.50136
\(791\) −24.0239 −0.854193
\(792\) −52.9618 −1.88192
\(793\) 23.5586 0.836590
\(794\) 31.7033 1.12511
\(795\) −11.1923 −0.396949
\(796\) 5.48931 0.194564
\(797\) −9.08434 −0.321784 −0.160892 0.986972i \(-0.551437\pi\)
−0.160892 + 0.986972i \(0.551437\pi\)
\(798\) 1.91215 0.0676893
\(799\) −7.30956 −0.258594
\(800\) −173.149 −6.12176
\(801\) −26.2573 −0.927757
\(802\) −3.46723 −0.122432
\(803\) 4.74822 0.167561
\(804\) −21.6139 −0.762264
\(805\) −7.60969 −0.268206
\(806\) −49.5062 −1.74378
\(807\) −5.58803 −0.196708
\(808\) −143.627 −5.05280
\(809\) −1.93279 −0.0679533 −0.0339767 0.999423i \(-0.510817\pi\)
−0.0339767 + 0.999423i \(0.510817\pi\)
\(810\) −82.6063 −2.90249
\(811\) 54.6206 1.91799 0.958995 0.283423i \(-0.0914699\pi\)
0.958995 + 0.283423i \(0.0914699\pi\)
\(812\) 100.447 3.52501
\(813\) −6.00881 −0.210738
\(814\) 5.68496 0.199258
\(815\) 5.03752 0.176457
\(816\) 8.78648 0.307588
\(817\) −2.42088 −0.0846960
\(818\) −62.1706 −2.17374
\(819\) −22.0446 −0.770301
\(820\) 13.4016 0.468004
\(821\) −14.3807 −0.501891 −0.250945 0.968001i \(-0.580741\pi\)
−0.250945 + 0.968001i \(0.580741\pi\)
\(822\) 4.11317 0.143463
\(823\) 33.5316 1.16884 0.584418 0.811452i \(-0.301323\pi\)
0.584418 + 0.811452i \(0.301323\pi\)
\(824\) −86.9276 −3.02827
\(825\) 7.28048 0.253474
\(826\) −20.5435 −0.714801
\(827\) 55.6060 1.93361 0.966805 0.255514i \(-0.0822446\pi\)
0.966805 + 0.255514i \(0.0822446\pi\)
\(828\) −13.7052 −0.476288
\(829\) 19.9074 0.691411 0.345706 0.938343i \(-0.387640\pi\)
0.345706 + 0.938343i \(0.387640\pi\)
\(830\) −6.09891 −0.211696
\(831\) 9.15666 0.317641
\(832\) −73.8248 −2.55941
\(833\) 4.55633 0.157867
\(834\) 16.7649 0.580522
\(835\) 56.8703 1.96808
\(836\) 10.5156 0.363689
\(837\) −10.5786 −0.365649
\(838\) 30.5169 1.05419
\(839\) 47.4290 1.63743 0.818716 0.574198i \(-0.194686\pi\)
0.818716 + 0.574198i \(0.194686\pi\)
\(840\) −25.1698 −0.868439
\(841\) 49.6453 1.71191
\(842\) −16.3607 −0.563826
\(843\) −2.56377 −0.0883010
\(844\) −46.1707 −1.58926
\(845\) −1.90769 −0.0656265
\(846\) 28.8005 0.990180
\(847\) 14.1362 0.485726
\(848\) 108.215 3.71611
\(849\) 0.914977 0.0314019
\(850\) 52.7060 1.80780
\(851\) 0.909001 0.0311601
\(852\) −1.91354 −0.0655568
\(853\) −14.5863 −0.499427 −0.249713 0.968320i \(-0.580336\pi\)
−0.249713 + 0.968320i \(0.580336\pi\)
\(854\) −38.7725 −1.32677
\(855\) 10.5919 0.362236
\(856\) 62.2249 2.12680
\(857\) 16.1067 0.550194 0.275097 0.961416i \(-0.411290\pi\)
0.275097 + 0.961416i \(0.411290\pi\)
\(858\) 6.94899 0.237235
\(859\) −6.26434 −0.213737 −0.106868 0.994273i \(-0.534082\pi\)
−0.106868 + 0.994273i \(0.534082\pi\)
\(860\) 51.5723 1.75860
\(861\) 0.495035 0.0168707
\(862\) 42.4826 1.44696
\(863\) −10.2778 −0.349859 −0.174930 0.984581i \(-0.555970\pi\)
−0.174930 + 0.984581i \(0.555970\pi\)
\(864\) −35.3143 −1.20142
\(865\) 24.1955 0.822670
\(866\) −43.0733 −1.46369
\(867\) 4.53942 0.154167
\(868\) 58.9512 2.00093
\(869\) −19.9913 −0.678158
\(870\) −31.8932 −1.08128
\(871\) −42.2514 −1.43163
\(872\) 8.69874 0.294576
\(873\) −13.3813 −0.452887
\(874\) 2.32387 0.0786060
\(875\) −41.5702 −1.40533
\(876\) 4.06410 0.137313
\(877\) 22.5972 0.763054 0.381527 0.924358i \(-0.375398\pi\)
0.381527 + 0.924358i \(0.375398\pi\)
\(878\) −84.3606 −2.84703
\(879\) −0.163282 −0.00550736
\(880\) −105.710 −3.56350
\(881\) 24.1769 0.814541 0.407270 0.913308i \(-0.366481\pi\)
0.407270 + 0.913308i \(0.366481\pi\)
\(882\) −17.9524 −0.604490
\(883\) 22.8981 0.770583 0.385292 0.922795i \(-0.374101\pi\)
0.385292 + 0.922795i \(0.374101\pi\)
\(884\) 36.3983 1.22421
\(885\) 4.71948 0.158644
\(886\) −44.0684 −1.48051
\(887\) 42.6153 1.43088 0.715440 0.698674i \(-0.246226\pi\)
0.715440 + 0.698674i \(0.246226\pi\)
\(888\) 3.00660 0.100895
\(889\) −35.6369 −1.19522
\(890\) −94.8462 −3.17925
\(891\) −16.7805 −0.562167
\(892\) −14.6621 −0.490922
\(893\) −3.53334 −0.118239
\(894\) −12.6887 −0.424374
\(895\) 92.9339 3.10644
\(896\) 46.3034 1.54689
\(897\) 1.11111 0.0370990
\(898\) −20.0050 −0.667577
\(899\) 46.1559 1.53938
\(900\) −150.255 −5.00848
\(901\) −16.4592 −0.548337
\(902\) 3.76260 0.125281
\(903\) 1.90500 0.0633945
\(904\) −96.5707 −3.21190
\(905\) −26.1849 −0.870414
\(906\) −1.67701 −0.0557149
\(907\) 8.16789 0.271210 0.135605 0.990763i \(-0.456702\pi\)
0.135605 + 0.990763i \(0.456702\pi\)
\(908\) 8.66225 0.287467
\(909\) −47.5614 −1.57751
\(910\) −79.6290 −2.63968
\(911\) 22.9932 0.761800 0.380900 0.924616i \(-0.375614\pi\)
0.380900 + 0.924616i \(0.375614\pi\)
\(912\) 4.24726 0.140641
\(913\) −1.23892 −0.0410023
\(914\) 101.668 3.36289
\(915\) 8.90724 0.294464
\(916\) −56.3248 −1.86103
\(917\) 10.0086 0.330515
\(918\) 10.7495 0.354788
\(919\) −8.88718 −0.293161 −0.146580 0.989199i \(-0.546827\pi\)
−0.146580 + 0.989199i \(0.546827\pi\)
\(920\) −30.5892 −1.00850
\(921\) −8.54899 −0.281699
\(922\) −35.1721 −1.15833
\(923\) −3.74063 −0.123124
\(924\) −8.27475 −0.272219
\(925\) 9.96568 0.327670
\(926\) −15.2719 −0.501865
\(927\) −28.7856 −0.945442
\(928\) 154.081 5.05797
\(929\) 40.2339 1.32003 0.660015 0.751252i \(-0.270550\pi\)
0.660015 + 0.751252i \(0.270550\pi\)
\(930\) −18.7177 −0.613778
\(931\) 2.20246 0.0721828
\(932\) 148.282 4.85713
\(933\) −7.26806 −0.237946
\(934\) 41.9246 1.37182
\(935\) 16.0783 0.525817
\(936\) −88.6143 −2.89645
\(937\) 12.7307 0.415895 0.207947 0.978140i \(-0.433322\pi\)
0.207947 + 0.978140i \(0.433322\pi\)
\(938\) 69.5369 2.27046
\(939\) 0.653768 0.0213349
\(940\) 75.2710 2.45507
\(941\) 40.1734 1.30961 0.654807 0.755796i \(-0.272750\pi\)
0.654807 + 0.755796i \(0.272750\pi\)
\(942\) −21.2009 −0.690764
\(943\) 0.601624 0.0195916
\(944\) −45.6313 −1.48517
\(945\) −17.0153 −0.553508
\(946\) 14.4793 0.470764
\(947\) −23.2868 −0.756718 −0.378359 0.925659i \(-0.623511\pi\)
−0.378359 + 0.925659i \(0.623511\pi\)
\(948\) −17.1110 −0.555739
\(949\) 7.94459 0.257892
\(950\) 25.4773 0.826594
\(951\) −8.35932 −0.271069
\(952\) −37.0143 −1.19964
\(953\) 39.9622 1.29450 0.647251 0.762277i \(-0.275919\pi\)
0.647251 + 0.762277i \(0.275919\pi\)
\(954\) 64.8512 2.09963
\(955\) 27.4986 0.889834
\(956\) −40.7846 −1.31907
\(957\) −6.47872 −0.209427
\(958\) −36.5673 −1.18144
\(959\) −9.57453 −0.309178
\(960\) −27.9123 −0.900866
\(961\) −3.91172 −0.126185
\(962\) 9.51193 0.306677
\(963\) 20.6054 0.664000
\(964\) −119.464 −3.84768
\(965\) 49.9142 1.60680
\(966\) −1.82866 −0.0588362
\(967\) −54.5352 −1.75373 −0.876867 0.480733i \(-0.840370\pi\)
−0.876867 + 0.480733i \(0.840370\pi\)
\(968\) 56.8243 1.82640
\(969\) −0.645999 −0.0207525
\(970\) −48.3355 −1.55196
\(971\) 19.7653 0.634298 0.317149 0.948376i \(-0.397275\pi\)
0.317149 + 0.948376i \(0.397275\pi\)
\(972\) −46.2785 −1.48438
\(973\) −39.0250 −1.25108
\(974\) 37.3347 1.19628
\(975\) 12.1815 0.390121
\(976\) −86.1214 −2.75668
\(977\) −16.9922 −0.543629 −0.271815 0.962350i \(-0.587624\pi\)
−0.271815 + 0.962350i \(0.587624\pi\)
\(978\) 1.21055 0.0387091
\(979\) −19.2668 −0.615771
\(980\) −46.9193 −1.49878
\(981\) 2.88054 0.0919684
\(982\) 16.3141 0.520605
\(983\) −22.0993 −0.704857 −0.352428 0.935839i \(-0.614644\pi\)
−0.352428 + 0.935839i \(0.614644\pi\)
\(984\) 1.98993 0.0634365
\(985\) 67.2220 2.14187
\(986\) −46.9018 −1.49366
\(987\) 2.78040 0.0885010
\(988\) 17.5944 0.559753
\(989\) 2.31518 0.0736185
\(990\) −63.3503 −2.01341
\(991\) 10.9833 0.348895 0.174447 0.984667i \(-0.444186\pi\)
0.174447 + 0.984667i \(0.444186\pi\)
\(992\) 90.4284 2.87110
\(993\) 10.3816 0.329449
\(994\) 6.15630 0.195266
\(995\) 4.05713 0.128620
\(996\) −1.06042 −0.0336006
\(997\) −49.1972 −1.55809 −0.779046 0.626967i \(-0.784296\pi\)
−0.779046 + 0.626967i \(0.784296\pi\)
\(998\) 28.2301 0.893607
\(999\) 2.03253 0.0643064
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 4033.2.a.f.1.2 85
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4033.2.a.f.1.2 85 1.1 even 1 trivial