Properties

Label 4033.2.a.f.1.8
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 1.8
Character \(\chi\) = 4033.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.31084 q^{2} +0.814794 q^{3} +3.33999 q^{4} -0.287417 q^{5} -1.88286 q^{6} -2.20640 q^{7} -3.09651 q^{8} -2.33611 q^{9} +O(q^{10})\) \(q-2.31084 q^{2} +0.814794 q^{3} +3.33999 q^{4} -0.287417 q^{5} -1.88286 q^{6} -2.20640 q^{7} -3.09651 q^{8} -2.33611 q^{9} +0.664174 q^{10} +0.898242 q^{11} +2.72140 q^{12} +1.31308 q^{13} +5.09865 q^{14} -0.234185 q^{15} +0.475556 q^{16} +4.29513 q^{17} +5.39838 q^{18} +5.16079 q^{19} -0.959968 q^{20} -1.79776 q^{21} -2.07570 q^{22} +3.20752 q^{23} -2.52302 q^{24} -4.91739 q^{25} -3.03431 q^{26} -4.34783 q^{27} -7.36936 q^{28} +0.736008 q^{29} +0.541165 q^{30} +9.07661 q^{31} +5.09408 q^{32} +0.731883 q^{33} -9.92537 q^{34} +0.634157 q^{35} -7.80259 q^{36} +1.00000 q^{37} -11.9258 q^{38} +1.06989 q^{39} +0.889987 q^{40} +0.873864 q^{41} +4.15435 q^{42} +6.76802 q^{43} +3.00012 q^{44} +0.671437 q^{45} -7.41207 q^{46} -5.62094 q^{47} +0.387480 q^{48} -2.13179 q^{49} +11.3633 q^{50} +3.49965 q^{51} +4.38566 q^{52} -11.0462 q^{53} +10.0472 q^{54} -0.258170 q^{55} +6.83214 q^{56} +4.20498 q^{57} -1.70080 q^{58} -13.7616 q^{59} -0.782177 q^{60} -12.2127 q^{61} -20.9746 q^{62} +5.15440 q^{63} -12.7227 q^{64} -0.377400 q^{65} -1.69126 q^{66} -3.02677 q^{67} +14.3457 q^{68} +2.61347 q^{69} -1.46544 q^{70} +2.86657 q^{71} +7.23378 q^{72} +2.37897 q^{73} -2.31084 q^{74} -4.00666 q^{75} +17.2370 q^{76} -1.98188 q^{77} -2.47234 q^{78} -8.37285 q^{79} -0.136683 q^{80} +3.46575 q^{81} -2.01936 q^{82} -12.8847 q^{83} -6.00451 q^{84} -1.23449 q^{85} -15.6398 q^{86} +0.599695 q^{87} -2.78141 q^{88} +17.4249 q^{89} -1.55158 q^{90} -2.89717 q^{91} +10.7131 q^{92} +7.39556 q^{93} +12.9891 q^{94} -1.48329 q^{95} +4.15062 q^{96} +11.0293 q^{97} +4.92622 q^{98} -2.09839 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + O(q^{10}) \) \( 85q + 11q^{2} + 21q^{3} + 93q^{4} + 12q^{5} + 4q^{6} + 17q^{7} + 30q^{8} + 98q^{9} + 9q^{10} + 37q^{11} + 44q^{12} + 14q^{13} + 26q^{14} + 27q^{15} + 85q^{16} + 34q^{17} + 3q^{18} + 15q^{19} + 15q^{20} + 17q^{21} + q^{22} + 72q^{23} + 15q^{24} + 85q^{25} + 33q^{26} + 69q^{27} + 7q^{28} + 19q^{29} - 9q^{30} + 23q^{31} + 51q^{32} + 32q^{33} + 49q^{34} + 40q^{35} + 121q^{36} + 85q^{37} + 84q^{38} + 39q^{39} + 22q^{40} + 55q^{41} - 28q^{42} + 78q^{44} + 28q^{45} + 17q^{46} + 184q^{47} + 97q^{48} + 88q^{49} + 26q^{50} + 27q^{51} + 73q^{52} + 64q^{53} + 31q^{54} + 39q^{55} + 68q^{56} - 33q^{57} + 28q^{58} + 60q^{59} - 22q^{60} + 7q^{61} + 70q^{62} + 28q^{63} + 102q^{64} + 17q^{65} - 15q^{66} + 82q^{67} + 92q^{68} + 22q^{69} - 41q^{70} + 113q^{71} - 19q^{73} + 11q^{74} + 45q^{75} + 34q^{76} + 64q^{77} + 29q^{78} + 23q^{79} + 54q^{80} + 149q^{81} + 4q^{82} + 100q^{83} - 49q^{84} - 5q^{85} - 24q^{86} + 65q^{87} + 14q^{88} + 84q^{89} - 21q^{90} + 32q^{91} + 95q^{92} + 19q^{93} - 47q^{94} + 102q^{95} + 29q^{96} + 7q^{97} + 26q^{98} + 107q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.31084 −1.63401 −0.817006 0.576629i \(-0.804368\pi\)
−0.817006 + 0.576629i \(0.804368\pi\)
\(3\) 0.814794 0.470422 0.235211 0.971944i \(-0.424422\pi\)
0.235211 + 0.971944i \(0.424422\pi\)
\(4\) 3.33999 1.67000
\(5\) −0.287417 −0.128537 −0.0642683 0.997933i \(-0.520471\pi\)
−0.0642683 + 0.997933i \(0.520471\pi\)
\(6\) −1.88286 −0.768674
\(7\) −2.20640 −0.833942 −0.416971 0.908920i \(-0.636908\pi\)
−0.416971 + 0.908920i \(0.636908\pi\)
\(8\) −3.09651 −1.09478
\(9\) −2.33611 −0.778704
\(10\) 0.664174 0.210030
\(11\) 0.898242 0.270830 0.135415 0.990789i \(-0.456763\pi\)
0.135415 + 0.990789i \(0.456763\pi\)
\(12\) 2.72140 0.785602
\(13\) 1.31308 0.364182 0.182091 0.983282i \(-0.441713\pi\)
0.182091 + 0.983282i \(0.441713\pi\)
\(14\) 5.09865 1.36267
\(15\) −0.234185 −0.0604664
\(16\) 0.475556 0.118889
\(17\) 4.29513 1.04172 0.520861 0.853641i \(-0.325611\pi\)
0.520861 + 0.853641i \(0.325611\pi\)
\(18\) 5.39838 1.27241
\(19\) 5.16079 1.18397 0.591983 0.805951i \(-0.298345\pi\)
0.591983 + 0.805951i \(0.298345\pi\)
\(20\) −0.959968 −0.214655
\(21\) −1.79776 −0.392304
\(22\) −2.07570 −0.442540
\(23\) 3.20752 0.668814 0.334407 0.942429i \(-0.391464\pi\)
0.334407 + 0.942429i \(0.391464\pi\)
\(24\) −2.52302 −0.515008
\(25\) −4.91739 −0.983478
\(26\) −3.03431 −0.595077
\(27\) −4.34783 −0.836740
\(28\) −7.36936 −1.39268
\(29\) 0.736008 0.136673 0.0683366 0.997662i \(-0.478231\pi\)
0.0683366 + 0.997662i \(0.478231\pi\)
\(30\) 0.541165 0.0988028
\(31\) 9.07661 1.63021 0.815103 0.579316i \(-0.196680\pi\)
0.815103 + 0.579316i \(0.196680\pi\)
\(32\) 5.09408 0.900514
\(33\) 0.731883 0.127404
\(34\) −9.92537 −1.70219
\(35\) 0.634157 0.107192
\(36\) −7.80259 −1.30043
\(37\) 1.00000 0.164399
\(38\) −11.9258 −1.93461
\(39\) 1.06989 0.171319
\(40\) 0.889987 0.140719
\(41\) 0.873864 0.136475 0.0682373 0.997669i \(-0.478262\pi\)
0.0682373 + 0.997669i \(0.478262\pi\)
\(42\) 4.15435 0.641030
\(43\) 6.76802 1.03211 0.516056 0.856555i \(-0.327400\pi\)
0.516056 + 0.856555i \(0.327400\pi\)
\(44\) 3.00012 0.452285
\(45\) 0.671437 0.100092
\(46\) −7.41207 −1.09285
\(47\) −5.62094 −0.819898 −0.409949 0.912108i \(-0.634454\pi\)
−0.409949 + 0.912108i \(0.634454\pi\)
\(48\) 0.387480 0.0559280
\(49\) −2.13179 −0.304541
\(50\) 11.3633 1.60702
\(51\) 3.49965 0.490049
\(52\) 4.38566 0.608182
\(53\) −11.0462 −1.51731 −0.758656 0.651491i \(-0.774144\pi\)
−0.758656 + 0.651491i \(0.774144\pi\)
\(54\) 10.0472 1.36724
\(55\) −0.258170 −0.0348116
\(56\) 6.83214 0.912983
\(57\) 4.20498 0.556963
\(58\) −1.70080 −0.223326
\(59\) −13.7616 −1.79161 −0.895806 0.444446i \(-0.853401\pi\)
−0.895806 + 0.444446i \(0.853401\pi\)
\(60\) −0.782177 −0.100979
\(61\) −12.2127 −1.56368 −0.781838 0.623481i \(-0.785718\pi\)
−0.781838 + 0.623481i \(0.785718\pi\)
\(62\) −20.9746 −2.66378
\(63\) 5.15440 0.649393
\(64\) −12.7227 −1.59034
\(65\) −0.377400 −0.0468107
\(66\) −1.69126 −0.208180
\(67\) −3.02677 −0.369779 −0.184889 0.982759i \(-0.559193\pi\)
−0.184889 + 0.982759i \(0.559193\pi\)
\(68\) 14.3457 1.73967
\(69\) 2.61347 0.314624
\(70\) −1.46544 −0.175153
\(71\) 2.86657 0.340199 0.170099 0.985427i \(-0.445591\pi\)
0.170099 + 0.985427i \(0.445591\pi\)
\(72\) 7.23378 0.852509
\(73\) 2.37897 0.278437 0.139219 0.990262i \(-0.455541\pi\)
0.139219 + 0.990262i \(0.455541\pi\)
\(74\) −2.31084 −0.268630
\(75\) −4.00666 −0.462649
\(76\) 17.2370 1.97722
\(77\) −1.98188 −0.225857
\(78\) −2.47234 −0.279937
\(79\) −8.37285 −0.942020 −0.471010 0.882128i \(-0.656110\pi\)
−0.471010 + 0.882128i \(0.656110\pi\)
\(80\) −0.136683 −0.0152816
\(81\) 3.46575 0.385083
\(82\) −2.01936 −0.223001
\(83\) −12.8847 −1.41428 −0.707139 0.707075i \(-0.750014\pi\)
−0.707139 + 0.707075i \(0.750014\pi\)
\(84\) −6.00451 −0.655146
\(85\) −1.23449 −0.133899
\(86\) −15.6398 −1.68648
\(87\) 0.599695 0.0642941
\(88\) −2.78141 −0.296500
\(89\) 17.4249 1.84703 0.923517 0.383557i \(-0.125301\pi\)
0.923517 + 0.383557i \(0.125301\pi\)
\(90\) −1.55158 −0.163551
\(91\) −2.89717 −0.303706
\(92\) 10.7131 1.11692
\(93\) 7.39556 0.766884
\(94\) 12.9891 1.33972
\(95\) −1.48329 −0.152183
\(96\) 4.15062 0.423621
\(97\) 11.0293 1.11986 0.559929 0.828541i \(-0.310828\pi\)
0.559929 + 0.828541i \(0.310828\pi\)
\(98\) 4.92622 0.497624
\(99\) −2.09839 −0.210896
\(100\) −16.4240 −1.64240
\(101\) 18.0000 1.79107 0.895534 0.444994i \(-0.146794\pi\)
0.895534 + 0.444994i \(0.146794\pi\)
\(102\) −8.08713 −0.800746
\(103\) −3.63892 −0.358554 −0.179277 0.983799i \(-0.557376\pi\)
−0.179277 + 0.983799i \(0.557376\pi\)
\(104\) −4.06595 −0.398699
\(105\) 0.516707 0.0504254
\(106\) 25.5260 2.47931
\(107\) 0.0678228 0.00655668 0.00327834 0.999995i \(-0.498956\pi\)
0.00327834 + 0.999995i \(0.498956\pi\)
\(108\) −14.5217 −1.39735
\(109\) −1.00000 −0.0957826
\(110\) 0.596589 0.0568826
\(111\) 0.814794 0.0773368
\(112\) −1.04927 −0.0991465
\(113\) 20.3004 1.90970 0.954852 0.297081i \(-0.0960130\pi\)
0.954852 + 0.297081i \(0.0960130\pi\)
\(114\) −9.71704 −0.910084
\(115\) −0.921894 −0.0859670
\(116\) 2.45826 0.228244
\(117\) −3.06749 −0.283590
\(118\) 31.8010 2.92751
\(119\) −9.47679 −0.868736
\(120\) 0.725156 0.0661974
\(121\) −10.1932 −0.926651
\(122\) 28.2216 2.55507
\(123\) 0.712019 0.0642006
\(124\) 30.3158 2.72244
\(125\) 2.85042 0.254950
\(126\) −11.9110 −1.06112
\(127\) 4.10139 0.363939 0.181970 0.983304i \(-0.441753\pi\)
0.181970 + 0.983304i \(0.441753\pi\)
\(128\) 19.2120 1.69812
\(129\) 5.51454 0.485528
\(130\) 0.872111 0.0764892
\(131\) 0.864195 0.0755051 0.0377525 0.999287i \(-0.487980\pi\)
0.0377525 + 0.999287i \(0.487980\pi\)
\(132\) 2.44448 0.212765
\(133\) −11.3868 −0.987358
\(134\) 6.99438 0.604223
\(135\) 1.24964 0.107552
\(136\) −13.2999 −1.14046
\(137\) 14.4554 1.23501 0.617505 0.786567i \(-0.288144\pi\)
0.617505 + 0.786567i \(0.288144\pi\)
\(138\) −6.03931 −0.514100
\(139\) 8.07612 0.685008 0.342504 0.939516i \(-0.388725\pi\)
0.342504 + 0.939516i \(0.388725\pi\)
\(140\) 2.11808 0.179010
\(141\) −4.57991 −0.385698
\(142\) −6.62418 −0.555889
\(143\) 1.17946 0.0986314
\(144\) −1.11095 −0.0925793
\(145\) −0.211541 −0.0175675
\(146\) −5.49742 −0.454970
\(147\) −1.73697 −0.143263
\(148\) 3.33999 0.274546
\(149\) 21.3137 1.74609 0.873044 0.487641i \(-0.162143\pi\)
0.873044 + 0.487641i \(0.162143\pi\)
\(150\) 9.25876 0.755975
\(151\) 13.2986 1.08222 0.541112 0.840950i \(-0.318003\pi\)
0.541112 + 0.840950i \(0.318003\pi\)
\(152\) −15.9804 −1.29618
\(153\) −10.0339 −0.811193
\(154\) 4.57982 0.369053
\(155\) −2.60877 −0.209541
\(156\) 3.57341 0.286102
\(157\) 2.79588 0.223136 0.111568 0.993757i \(-0.464413\pi\)
0.111568 + 0.993757i \(0.464413\pi\)
\(158\) 19.3483 1.53927
\(159\) −9.00038 −0.713777
\(160\) −1.46412 −0.115749
\(161\) −7.07708 −0.557752
\(162\) −8.00879 −0.629230
\(163\) −9.71610 −0.761024 −0.380512 0.924776i \(-0.624252\pi\)
−0.380512 + 0.924776i \(0.624252\pi\)
\(164\) 2.91870 0.227912
\(165\) −0.210355 −0.0163761
\(166\) 29.7745 2.31095
\(167\) 19.3468 1.49710 0.748551 0.663077i \(-0.230750\pi\)
0.748551 + 0.663077i \(0.230750\pi\)
\(168\) 5.56679 0.429487
\(169\) −11.2758 −0.867372
\(170\) 2.85272 0.218793
\(171\) −12.0562 −0.921958
\(172\) 22.6051 1.72362
\(173\) 2.78884 0.212032 0.106016 0.994364i \(-0.466191\pi\)
0.106016 + 0.994364i \(0.466191\pi\)
\(174\) −1.38580 −0.105057
\(175\) 10.8497 0.820164
\(176\) 0.427165 0.0321987
\(177\) −11.2129 −0.842813
\(178\) −40.2662 −3.01808
\(179\) −22.2048 −1.65967 −0.829833 0.558011i \(-0.811565\pi\)
−0.829833 + 0.558011i \(0.811565\pi\)
\(180\) 2.24259 0.167153
\(181\) 0.0302952 0.00225183 0.00112591 0.999999i \(-0.499642\pi\)
0.00112591 + 0.999999i \(0.499642\pi\)
\(182\) 6.69491 0.496260
\(183\) −9.95084 −0.735587
\(184\) −9.93210 −0.732204
\(185\) −0.287417 −0.0211313
\(186\) −17.0900 −1.25310
\(187\) 3.85807 0.282130
\(188\) −18.7739 −1.36923
\(189\) 9.59307 0.697793
\(190\) 3.42766 0.248669
\(191\) 18.0105 1.30319 0.651596 0.758566i \(-0.274100\pi\)
0.651596 + 0.758566i \(0.274100\pi\)
\(192\) −10.3664 −0.748130
\(193\) 13.6294 0.981063 0.490531 0.871424i \(-0.336803\pi\)
0.490531 + 0.871424i \(0.336803\pi\)
\(194\) −25.4870 −1.82986
\(195\) −0.307503 −0.0220207
\(196\) −7.12015 −0.508582
\(197\) 6.56154 0.467490 0.233745 0.972298i \(-0.424902\pi\)
0.233745 + 0.972298i \(0.424902\pi\)
\(198\) 4.84906 0.344607
\(199\) −3.06411 −0.217209 −0.108605 0.994085i \(-0.534638\pi\)
−0.108605 + 0.994085i \(0.534638\pi\)
\(200\) 15.2267 1.07669
\(201\) −2.46619 −0.173952
\(202\) −41.5952 −2.92663
\(203\) −1.62393 −0.113978
\(204\) 11.6888 0.818379
\(205\) −0.251163 −0.0175420
\(206\) 8.40897 0.585881
\(207\) −7.49312 −0.520808
\(208\) 0.624441 0.0432972
\(209\) 4.63564 0.320654
\(210\) −1.19403 −0.0823958
\(211\) −1.57453 −0.108395 −0.0541976 0.998530i \(-0.517260\pi\)
−0.0541976 + 0.998530i \(0.517260\pi\)
\(212\) −36.8942 −2.53391
\(213\) 2.33566 0.160037
\(214\) −0.156728 −0.0107137
\(215\) −1.94524 −0.132664
\(216\) 13.4631 0.916047
\(217\) −20.0266 −1.35950
\(218\) 2.31084 0.156510
\(219\) 1.93837 0.130983
\(220\) −0.862284 −0.0581352
\(221\) 5.63983 0.379376
\(222\) −1.88286 −0.126369
\(223\) −5.10551 −0.341890 −0.170945 0.985281i \(-0.554682\pi\)
−0.170945 + 0.985281i \(0.554682\pi\)
\(224\) −11.2396 −0.750977
\(225\) 11.4876 0.765838
\(226\) −46.9111 −3.12048
\(227\) −4.90101 −0.325291 −0.162646 0.986685i \(-0.552003\pi\)
−0.162646 + 0.986685i \(0.552003\pi\)
\(228\) 14.0446 0.930125
\(229\) −13.5248 −0.893744 −0.446872 0.894598i \(-0.647462\pi\)
−0.446872 + 0.894598i \(0.647462\pi\)
\(230\) 2.13035 0.140471
\(231\) −1.61483 −0.106248
\(232\) −2.27905 −0.149627
\(233\) 13.4359 0.880215 0.440108 0.897945i \(-0.354940\pi\)
0.440108 + 0.897945i \(0.354940\pi\)
\(234\) 7.08849 0.463389
\(235\) 1.61555 0.105387
\(236\) −45.9637 −2.99198
\(237\) −6.82215 −0.443146
\(238\) 21.8994 1.41952
\(239\) 7.30707 0.472655 0.236328 0.971673i \(-0.424056\pi\)
0.236328 + 0.971673i \(0.424056\pi\)
\(240\) −0.111368 −0.00718879
\(241\) 20.0435 1.29112 0.645558 0.763711i \(-0.276625\pi\)
0.645558 + 0.763711i \(0.276625\pi\)
\(242\) 23.5548 1.51416
\(243\) 15.8674 1.01789
\(244\) −40.7903 −2.61133
\(245\) 0.612711 0.0391447
\(246\) −1.64536 −0.104905
\(247\) 6.77650 0.431179
\(248\) −28.1058 −1.78472
\(249\) −10.4984 −0.665307
\(250\) −6.58687 −0.416591
\(251\) 6.70412 0.423160 0.211580 0.977361i \(-0.432139\pi\)
0.211580 + 0.977361i \(0.432139\pi\)
\(252\) 17.2156 1.08448
\(253\) 2.88113 0.181135
\(254\) −9.47766 −0.594681
\(255\) −1.00586 −0.0629892
\(256\) −18.9506 −1.18441
\(257\) 7.55207 0.471085 0.235543 0.971864i \(-0.424313\pi\)
0.235543 + 0.971864i \(0.424313\pi\)
\(258\) −12.7432 −0.793359
\(259\) −2.20640 −0.137099
\(260\) −1.26051 −0.0781736
\(261\) −1.71940 −0.106428
\(262\) −1.99702 −0.123376
\(263\) 29.6449 1.82798 0.913992 0.405733i \(-0.132984\pi\)
0.913992 + 0.405733i \(0.132984\pi\)
\(264\) −2.26628 −0.139480
\(265\) 3.17486 0.195030
\(266\) 26.3130 1.61336
\(267\) 14.1977 0.868885
\(268\) −10.1094 −0.617528
\(269\) 21.6719 1.32136 0.660679 0.750668i \(-0.270268\pi\)
0.660679 + 0.750668i \(0.270268\pi\)
\(270\) −2.88772 −0.175741
\(271\) −12.5636 −0.763184 −0.381592 0.924331i \(-0.624624\pi\)
−0.381592 + 0.924331i \(0.624624\pi\)
\(272\) 2.04258 0.123849
\(273\) −2.36060 −0.142870
\(274\) −33.4042 −2.01802
\(275\) −4.41701 −0.266356
\(276\) 8.72895 0.525421
\(277\) −6.30526 −0.378846 −0.189423 0.981896i \(-0.560662\pi\)
−0.189423 + 0.981896i \(0.560662\pi\)
\(278\) −18.6626 −1.11931
\(279\) −21.2040 −1.26945
\(280\) −1.96367 −0.117352
\(281\) −3.68004 −0.219533 −0.109767 0.993957i \(-0.535010\pi\)
−0.109767 + 0.993957i \(0.535010\pi\)
\(282\) 10.5834 0.630235
\(283\) 24.5825 1.46128 0.730639 0.682764i \(-0.239222\pi\)
0.730639 + 0.682764i \(0.239222\pi\)
\(284\) 9.57430 0.568130
\(285\) −1.20858 −0.0715901
\(286\) −2.72555 −0.161165
\(287\) −1.92810 −0.113812
\(288\) −11.9003 −0.701234
\(289\) 1.44816 0.0851858
\(290\) 0.488838 0.0287055
\(291\) 8.98662 0.526805
\(292\) 7.94573 0.464989
\(293\) −12.3358 −0.720663 −0.360332 0.932824i \(-0.617336\pi\)
−0.360332 + 0.932824i \(0.617336\pi\)
\(294\) 4.01386 0.234093
\(295\) 3.95532 0.230288
\(296\) −3.09651 −0.179981
\(297\) −3.90541 −0.226615
\(298\) −49.2526 −2.85313
\(299\) 4.21172 0.243570
\(300\) −13.3822 −0.772622
\(301\) −14.9330 −0.860722
\(302\) −30.7310 −1.76837
\(303\) 14.6663 0.842557
\(304\) 2.45424 0.140760
\(305\) 3.51013 0.200990
\(306\) 23.1868 1.32550
\(307\) −7.42348 −0.423680 −0.211840 0.977304i \(-0.567946\pi\)
−0.211840 + 0.977304i \(0.567946\pi\)
\(308\) −6.61948 −0.377180
\(309\) −2.96497 −0.168671
\(310\) 6.02845 0.342393
\(311\) 28.1145 1.59423 0.797115 0.603828i \(-0.206359\pi\)
0.797115 + 0.603828i \(0.206359\pi\)
\(312\) −3.31291 −0.187557
\(313\) −13.4692 −0.761322 −0.380661 0.924715i \(-0.624303\pi\)
−0.380661 + 0.924715i \(0.624303\pi\)
\(314\) −6.46084 −0.364606
\(315\) −1.48146 −0.0834708
\(316\) −27.9652 −1.57317
\(317\) 14.2437 0.800008 0.400004 0.916513i \(-0.369009\pi\)
0.400004 + 0.916513i \(0.369009\pi\)
\(318\) 20.7985 1.16632
\(319\) 0.661114 0.0370153
\(320\) 3.65672 0.204417
\(321\) 0.0552616 0.00308440
\(322\) 16.3540 0.911373
\(323\) 22.1663 1.23336
\(324\) 11.5756 0.643086
\(325\) −6.45691 −0.358165
\(326\) 22.4524 1.24352
\(327\) −0.814794 −0.0450582
\(328\) −2.70593 −0.149410
\(329\) 12.4021 0.683747
\(330\) 0.486097 0.0267588
\(331\) 3.00918 0.165399 0.0826997 0.996575i \(-0.473646\pi\)
0.0826997 + 0.996575i \(0.473646\pi\)
\(332\) −43.0347 −2.36184
\(333\) −2.33611 −0.128018
\(334\) −44.7075 −2.44628
\(335\) 0.869943 0.0475301
\(336\) −0.854937 −0.0466407
\(337\) −32.7664 −1.78490 −0.892449 0.451148i \(-0.851015\pi\)
−0.892449 + 0.451148i \(0.851015\pi\)
\(338\) 26.0567 1.41730
\(339\) 16.5407 0.898366
\(340\) −4.12319 −0.223611
\(341\) 8.15299 0.441509
\(342\) 27.8599 1.50649
\(343\) 20.1484 1.08791
\(344\) −20.9572 −1.12994
\(345\) −0.751154 −0.0404407
\(346\) −6.44458 −0.346463
\(347\) 6.80346 0.365229 0.182615 0.983185i \(-0.441544\pi\)
0.182615 + 0.983185i \(0.441544\pi\)
\(348\) 2.00298 0.107371
\(349\) 3.04624 0.163061 0.0815306 0.996671i \(-0.474019\pi\)
0.0815306 + 0.996671i \(0.474019\pi\)
\(350\) −25.0720 −1.34016
\(351\) −5.70903 −0.304726
\(352\) 4.57572 0.243887
\(353\) 9.09743 0.484207 0.242104 0.970250i \(-0.422163\pi\)
0.242104 + 0.970250i \(0.422163\pi\)
\(354\) 25.9112 1.37717
\(355\) −0.823898 −0.0437280
\(356\) 58.1990 3.08454
\(357\) −7.72163 −0.408672
\(358\) 51.3118 2.71192
\(359\) −26.5807 −1.40287 −0.701437 0.712732i \(-0.747458\pi\)
−0.701437 + 0.712732i \(0.747458\pi\)
\(360\) −2.07911 −0.109579
\(361\) 7.63370 0.401774
\(362\) −0.0700075 −0.00367951
\(363\) −8.30533 −0.435917
\(364\) −9.67653 −0.507188
\(365\) −0.683755 −0.0357894
\(366\) 22.9948 1.20196
\(367\) 28.6879 1.49749 0.748747 0.662855i \(-0.230656\pi\)
0.748747 + 0.662855i \(0.230656\pi\)
\(368\) 1.52535 0.0795146
\(369\) −2.04144 −0.106273
\(370\) 0.664174 0.0345288
\(371\) 24.3724 1.26535
\(372\) 24.7011 1.28069
\(373\) −13.4992 −0.698964 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(374\) −8.91539 −0.461004
\(375\) 2.32251 0.119934
\(376\) 17.4053 0.897609
\(377\) 0.966435 0.0497739
\(378\) −22.1681 −1.14020
\(379\) −24.3828 −1.25246 −0.626231 0.779638i \(-0.715403\pi\)
−0.626231 + 0.779638i \(0.715403\pi\)
\(380\) −4.95419 −0.254145
\(381\) 3.34179 0.171205
\(382\) −41.6194 −2.12943
\(383\) 10.8263 0.553200 0.276600 0.960985i \(-0.410792\pi\)
0.276600 + 0.960985i \(0.410792\pi\)
\(384\) 15.6539 0.798833
\(385\) 0.569626 0.0290308
\(386\) −31.4953 −1.60307
\(387\) −15.8108 −0.803710
\(388\) 36.8378 1.87016
\(389\) 0.846100 0.0428990 0.0214495 0.999770i \(-0.493172\pi\)
0.0214495 + 0.999770i \(0.493172\pi\)
\(390\) 0.710591 0.0359822
\(391\) 13.7767 0.696718
\(392\) 6.60109 0.333406
\(393\) 0.704141 0.0355192
\(394\) −15.1627 −0.763885
\(395\) 2.40650 0.121084
\(396\) −7.00862 −0.352196
\(397\) 29.2462 1.46782 0.733912 0.679245i \(-0.237693\pi\)
0.733912 + 0.679245i \(0.237693\pi\)
\(398\) 7.08068 0.354923
\(399\) −9.27787 −0.464475
\(400\) −2.33850 −0.116925
\(401\) 27.7390 1.38522 0.692610 0.721312i \(-0.256461\pi\)
0.692610 + 0.721312i \(0.256461\pi\)
\(402\) 5.69898 0.284239
\(403\) 11.9183 0.593691
\(404\) 60.1198 2.99107
\(405\) −0.996112 −0.0494972
\(406\) 3.75265 0.186241
\(407\) 0.898242 0.0445242
\(408\) −10.8367 −0.536496
\(409\) −9.01240 −0.445635 −0.222817 0.974860i \(-0.571525\pi\)
−0.222817 + 0.974860i \(0.571525\pi\)
\(410\) 0.580398 0.0286638
\(411\) 11.7782 0.580975
\(412\) −12.1540 −0.598783
\(413\) 30.3637 1.49410
\(414\) 17.3154 0.851006
\(415\) 3.70327 0.181786
\(416\) 6.68891 0.327951
\(417\) 6.58038 0.322242
\(418\) −10.7122 −0.523952
\(419\) 23.4981 1.14796 0.573978 0.818871i \(-0.305399\pi\)
0.573978 + 0.818871i \(0.305399\pi\)
\(420\) 1.72580 0.0842102
\(421\) −30.7833 −1.50029 −0.750144 0.661274i \(-0.770016\pi\)
−0.750144 + 0.661274i \(0.770016\pi\)
\(422\) 3.63849 0.177119
\(423\) 13.1311 0.638458
\(424\) 34.2046 1.66112
\(425\) −21.1208 −1.02451
\(426\) −5.39734 −0.261502
\(427\) 26.9461 1.30402
\(428\) 0.226528 0.0109496
\(429\) 0.961017 0.0463984
\(430\) 4.49514 0.216775
\(431\) 11.7164 0.564359 0.282179 0.959362i \(-0.408943\pi\)
0.282179 + 0.959362i \(0.408943\pi\)
\(432\) −2.06764 −0.0994793
\(433\) 9.27250 0.445608 0.222804 0.974863i \(-0.428479\pi\)
0.222804 + 0.974863i \(0.428479\pi\)
\(434\) 46.2784 2.22143
\(435\) −0.172362 −0.00826414
\(436\) −3.33999 −0.159957
\(437\) 16.5533 0.791852
\(438\) −4.47926 −0.214028
\(439\) 27.8010 1.32687 0.663434 0.748235i \(-0.269098\pi\)
0.663434 + 0.748235i \(0.269098\pi\)
\(440\) 0.799424 0.0381111
\(441\) 4.98009 0.237147
\(442\) −13.0328 −0.619905
\(443\) 19.2136 0.912866 0.456433 0.889758i \(-0.349127\pi\)
0.456433 + 0.889758i \(0.349127\pi\)
\(444\) 2.72140 0.129152
\(445\) −5.00820 −0.237411
\(446\) 11.7980 0.558653
\(447\) 17.3663 0.821398
\(448\) 28.0714 1.32625
\(449\) −41.5434 −1.96055 −0.980277 0.197631i \(-0.936675\pi\)
−0.980277 + 0.197631i \(0.936675\pi\)
\(450\) −26.5460 −1.25139
\(451\) 0.784942 0.0369615
\(452\) 67.8033 3.18920
\(453\) 10.8356 0.509102
\(454\) 11.3255 0.531530
\(455\) 0.832696 0.0390374
\(456\) −13.0207 −0.609752
\(457\) −25.9963 −1.21605 −0.608027 0.793916i \(-0.708039\pi\)
−0.608027 + 0.793916i \(0.708039\pi\)
\(458\) 31.2537 1.46039
\(459\) −18.6745 −0.871651
\(460\) −3.07912 −0.143565
\(461\) −24.3920 −1.13605 −0.568025 0.823012i \(-0.692292\pi\)
−0.568025 + 0.823012i \(0.692292\pi\)
\(462\) 3.73161 0.173610
\(463\) 20.4853 0.952031 0.476015 0.879437i \(-0.342081\pi\)
0.476015 + 0.879437i \(0.342081\pi\)
\(464\) 0.350013 0.0162490
\(465\) −2.12561 −0.0985727
\(466\) −31.0483 −1.43828
\(467\) −8.32520 −0.385244 −0.192622 0.981273i \(-0.561699\pi\)
−0.192622 + 0.981273i \(0.561699\pi\)
\(468\) −10.2454 −0.473593
\(469\) 6.67827 0.308374
\(470\) −3.73328 −0.172203
\(471\) 2.27807 0.104968
\(472\) 42.6130 1.96142
\(473\) 6.07932 0.279527
\(474\) 15.7649 0.724106
\(475\) −25.3776 −1.16440
\(476\) −31.6524 −1.45078
\(477\) 25.8052 1.18154
\(478\) −16.8855 −0.772324
\(479\) −13.4689 −0.615408 −0.307704 0.951482i \(-0.599561\pi\)
−0.307704 + 0.951482i \(0.599561\pi\)
\(480\) −1.19296 −0.0544508
\(481\) 1.31308 0.0598711
\(482\) −46.3174 −2.10970
\(483\) −5.76636 −0.262378
\(484\) −34.0451 −1.54750
\(485\) −3.17001 −0.143943
\(486\) −36.6670 −1.66325
\(487\) −6.69308 −0.303292 −0.151646 0.988435i \(-0.548457\pi\)
−0.151646 + 0.988435i \(0.548457\pi\)
\(488\) 37.8167 1.71188
\(489\) −7.91662 −0.358002
\(490\) −1.41588 −0.0639629
\(491\) 13.0608 0.589424 0.294712 0.955586i \(-0.404776\pi\)
0.294712 + 0.955586i \(0.404776\pi\)
\(492\) 2.37814 0.107215
\(493\) 3.16125 0.142376
\(494\) −15.6594 −0.704551
\(495\) 0.603113 0.0271079
\(496\) 4.31643 0.193814
\(497\) −6.32480 −0.283706
\(498\) 24.2601 1.08712
\(499\) 11.5792 0.518357 0.259179 0.965829i \(-0.416548\pi\)
0.259179 + 0.965829i \(0.416548\pi\)
\(500\) 9.52038 0.425764
\(501\) 15.7637 0.704269
\(502\) −15.4922 −0.691449
\(503\) −8.48039 −0.378122 −0.189061 0.981965i \(-0.560544\pi\)
−0.189061 + 0.981965i \(0.560544\pi\)
\(504\) −15.9606 −0.710943
\(505\) −5.17350 −0.230218
\(506\) −6.65783 −0.295977
\(507\) −9.18748 −0.408030
\(508\) 13.6986 0.607777
\(509\) −28.4189 −1.25964 −0.629822 0.776739i \(-0.716872\pi\)
−0.629822 + 0.776739i \(0.716872\pi\)
\(510\) 2.32438 0.102925
\(511\) −5.24896 −0.232200
\(512\) 5.36764 0.237219
\(513\) −22.4382 −0.990672
\(514\) −17.4517 −0.769759
\(515\) 1.04589 0.0460873
\(516\) 18.4185 0.810830
\(517\) −5.04897 −0.222053
\(518\) 5.09865 0.224022
\(519\) 2.27233 0.0997444
\(520\) 1.16862 0.0512474
\(521\) 21.8034 0.955223 0.477611 0.878571i \(-0.341503\pi\)
0.477611 + 0.878571i \(0.341503\pi\)
\(522\) 3.97325 0.173905
\(523\) −26.4063 −1.15467 −0.577334 0.816508i \(-0.695907\pi\)
−0.577334 + 0.816508i \(0.695907\pi\)
\(524\) 2.88640 0.126093
\(525\) 8.84031 0.385823
\(526\) −68.5047 −2.98695
\(527\) 38.9852 1.69822
\(528\) 0.348051 0.0151470
\(529\) −12.7118 −0.552688
\(530\) −7.33660 −0.318682
\(531\) 32.1487 1.39513
\(532\) −38.0317 −1.64888
\(533\) 1.14745 0.0497016
\(534\) −32.8086 −1.41977
\(535\) −0.0194934 −0.000842773 0
\(536\) 9.37241 0.404826
\(537\) −18.0924 −0.780743
\(538\) −50.0803 −2.15912
\(539\) −1.91486 −0.0824789
\(540\) 4.17378 0.179611
\(541\) 12.6031 0.541849 0.270925 0.962601i \(-0.412671\pi\)
0.270925 + 0.962601i \(0.412671\pi\)
\(542\) 29.0325 1.24705
\(543\) 0.0246844 0.00105931
\(544\) 21.8797 0.938086
\(545\) 0.287417 0.0123116
\(546\) 5.45497 0.233451
\(547\) 19.3293 0.826463 0.413231 0.910626i \(-0.364400\pi\)
0.413231 + 0.910626i \(0.364400\pi\)
\(548\) 48.2810 2.06246
\(549\) 28.5302 1.21764
\(550\) 10.2070 0.435228
\(551\) 3.79838 0.161816
\(552\) −8.09262 −0.344445
\(553\) 18.4739 0.785589
\(554\) 14.5705 0.619040
\(555\) −0.234185 −0.00994061
\(556\) 26.9742 1.14396
\(557\) 30.1331 1.27678 0.638391 0.769713i \(-0.279600\pi\)
0.638391 + 0.769713i \(0.279600\pi\)
\(558\) 48.9990 2.07429
\(559\) 8.88692 0.375877
\(560\) 0.301577 0.0127440
\(561\) 3.14353 0.132720
\(562\) 8.50400 0.358720
\(563\) 24.1584 1.01815 0.509077 0.860721i \(-0.329987\pi\)
0.509077 + 0.860721i \(0.329987\pi\)
\(564\) −15.2968 −0.644114
\(565\) −5.83468 −0.245467
\(566\) −56.8062 −2.38774
\(567\) −7.64683 −0.321137
\(568\) −8.87634 −0.372443
\(569\) 41.0380 1.72040 0.860202 0.509953i \(-0.170337\pi\)
0.860202 + 0.509953i \(0.170337\pi\)
\(570\) 2.79284 0.116979
\(571\) 25.2461 1.05652 0.528259 0.849084i \(-0.322845\pi\)
0.528259 + 0.849084i \(0.322845\pi\)
\(572\) 3.93939 0.164714
\(573\) 14.6748 0.613050
\(574\) 4.45552 0.185970
\(575\) −15.7726 −0.657764
\(576\) 29.7217 1.23840
\(577\) 41.0081 1.70719 0.853595 0.520937i \(-0.174417\pi\)
0.853595 + 0.520937i \(0.174417\pi\)
\(578\) −3.34647 −0.139195
\(579\) 11.1051 0.461513
\(580\) −0.706545 −0.0293377
\(581\) 28.4288 1.17943
\(582\) −20.7667 −0.860806
\(583\) −9.92217 −0.410934
\(584\) −7.36649 −0.304828
\(585\) 0.881647 0.0364516
\(586\) 28.5060 1.17757
\(587\) −22.2761 −0.919434 −0.459717 0.888065i \(-0.652049\pi\)
−0.459717 + 0.888065i \(0.652049\pi\)
\(588\) −5.80146 −0.239248
\(589\) 46.8424 1.93011
\(590\) −9.14012 −0.376293
\(591\) 5.34630 0.219917
\(592\) 0.475556 0.0195452
\(593\) −31.4364 −1.29094 −0.645469 0.763787i \(-0.723338\pi\)
−0.645469 + 0.763787i \(0.723338\pi\)
\(594\) 9.02478 0.370291
\(595\) 2.72379 0.111664
\(596\) 71.1876 2.91596
\(597\) −2.49662 −0.102180
\(598\) −9.73261 −0.397996
\(599\) −18.0834 −0.738869 −0.369434 0.929257i \(-0.620449\pi\)
−0.369434 + 0.929257i \(0.620449\pi\)
\(600\) 12.4067 0.506500
\(601\) 42.0147 1.71382 0.856908 0.515469i \(-0.172382\pi\)
0.856908 + 0.515469i \(0.172382\pi\)
\(602\) 34.5077 1.40643
\(603\) 7.07086 0.287948
\(604\) 44.4172 1.80731
\(605\) 2.92968 0.119109
\(606\) −33.8915 −1.37675
\(607\) −10.4174 −0.422829 −0.211415 0.977396i \(-0.567807\pi\)
−0.211415 + 0.977396i \(0.567807\pi\)
\(608\) 26.2894 1.06618
\(609\) −1.32317 −0.0536175
\(610\) −8.11136 −0.328420
\(611\) −7.38072 −0.298592
\(612\) −33.5131 −1.35469
\(613\) −33.2254 −1.34196 −0.670980 0.741475i \(-0.734126\pi\)
−0.670980 + 0.741475i \(0.734126\pi\)
\(614\) 17.1545 0.692298
\(615\) −0.204646 −0.00825213
\(616\) 6.13692 0.247263
\(617\) 10.1943 0.410407 0.205203 0.978719i \(-0.434214\pi\)
0.205203 + 0.978719i \(0.434214\pi\)
\(618\) 6.85158 0.275611
\(619\) 38.9238 1.56448 0.782240 0.622977i \(-0.214077\pi\)
0.782240 + 0.622977i \(0.214077\pi\)
\(620\) −8.71325 −0.349933
\(621\) −13.9457 −0.559624
\(622\) −64.9682 −2.60499
\(623\) −38.4463 −1.54032
\(624\) 0.508791 0.0203679
\(625\) 23.7677 0.950708
\(626\) 31.1251 1.24401
\(627\) 3.77709 0.150842
\(628\) 9.33822 0.372636
\(629\) 4.29513 0.171258
\(630\) 3.42342 0.136392
\(631\) 33.2773 1.32475 0.662374 0.749173i \(-0.269549\pi\)
0.662374 + 0.749173i \(0.269549\pi\)
\(632\) 25.9266 1.03130
\(633\) −1.28292 −0.0509915
\(634\) −32.9150 −1.30722
\(635\) −1.17881 −0.0467795
\(636\) −30.0612 −1.19200
\(637\) −2.79920 −0.110908
\(638\) −1.52773 −0.0604834
\(639\) −6.69662 −0.264914
\(640\) −5.52186 −0.218271
\(641\) −27.5964 −1.08999 −0.544997 0.838438i \(-0.683469\pi\)
−0.544997 + 0.838438i \(0.683469\pi\)
\(642\) −0.127701 −0.00503995
\(643\) 5.46781 0.215629 0.107815 0.994171i \(-0.465615\pi\)
0.107815 + 0.994171i \(0.465615\pi\)
\(644\) −23.6374 −0.931443
\(645\) −1.58497 −0.0624081
\(646\) −51.2227 −2.01533
\(647\) 13.8578 0.544808 0.272404 0.962183i \(-0.412181\pi\)
0.272404 + 0.962183i \(0.412181\pi\)
\(648\) −10.7317 −0.421581
\(649\) −12.3613 −0.485223
\(650\) 14.9209 0.585246
\(651\) −16.3176 −0.639537
\(652\) −32.4517 −1.27091
\(653\) 41.5317 1.62526 0.812630 0.582780i \(-0.198035\pi\)
0.812630 + 0.582780i \(0.198035\pi\)
\(654\) 1.88286 0.0736257
\(655\) −0.248384 −0.00970516
\(656\) 0.415571 0.0162253
\(657\) −5.55753 −0.216820
\(658\) −28.6592 −1.11725
\(659\) −26.7706 −1.04284 −0.521418 0.853302i \(-0.674597\pi\)
−0.521418 + 0.853302i \(0.674597\pi\)
\(660\) −0.702584 −0.0273481
\(661\) −24.5489 −0.954843 −0.477421 0.878674i \(-0.658429\pi\)
−0.477421 + 0.878674i \(0.658429\pi\)
\(662\) −6.95374 −0.270265
\(663\) 4.59530 0.178467
\(664\) 39.8975 1.54832
\(665\) 3.27275 0.126912
\(666\) 5.39838 0.209183
\(667\) 2.36076 0.0914090
\(668\) 64.6182 2.50015
\(669\) −4.15994 −0.160833
\(670\) −2.01030 −0.0776647
\(671\) −10.9700 −0.423491
\(672\) −9.15795 −0.353276
\(673\) 17.0851 0.658581 0.329290 0.944229i \(-0.393191\pi\)
0.329290 + 0.944229i \(0.393191\pi\)
\(674\) 75.7179 2.91655
\(675\) 21.3800 0.822916
\(676\) −37.6612 −1.44851
\(677\) 32.0785 1.23288 0.616439 0.787403i \(-0.288575\pi\)
0.616439 + 0.787403i \(0.288575\pi\)
\(678\) −38.2229 −1.46794
\(679\) −24.3351 −0.933896
\(680\) 3.82261 0.146590
\(681\) −3.99331 −0.153024
\(682\) −18.8403 −0.721431
\(683\) 14.0085 0.536022 0.268011 0.963416i \(-0.413634\pi\)
0.268011 + 0.963416i \(0.413634\pi\)
\(684\) −40.2675 −1.53967
\(685\) −4.15473 −0.158744
\(686\) −46.5598 −1.77766
\(687\) −11.0199 −0.420437
\(688\) 3.21857 0.122707
\(689\) −14.5045 −0.552578
\(690\) 1.73580 0.0660807
\(691\) −31.9275 −1.21458 −0.607289 0.794481i \(-0.707743\pi\)
−0.607289 + 0.794481i \(0.707743\pi\)
\(692\) 9.31471 0.354092
\(693\) 4.62990 0.175875
\(694\) −15.7217 −0.596789
\(695\) −2.32121 −0.0880485
\(696\) −1.85696 −0.0703879
\(697\) 3.75336 0.142169
\(698\) −7.03937 −0.266444
\(699\) 10.9475 0.414072
\(700\) 36.2380 1.36967
\(701\) −39.3670 −1.48687 −0.743435 0.668808i \(-0.766805\pi\)
−0.743435 + 0.668808i \(0.766805\pi\)
\(702\) 13.1927 0.497925
\(703\) 5.16079 0.194643
\(704\) −11.4281 −0.430712
\(705\) 1.31634 0.0495763
\(706\) −21.0227 −0.791200
\(707\) −39.7153 −1.49365
\(708\) −37.4510 −1.40749
\(709\) −26.4138 −0.991992 −0.495996 0.868325i \(-0.665197\pi\)
−0.495996 + 0.868325i \(0.665197\pi\)
\(710\) 1.90390 0.0714521
\(711\) 19.5599 0.733554
\(712\) −53.9563 −2.02210
\(713\) 29.1134 1.09030
\(714\) 17.8435 0.667775
\(715\) −0.338996 −0.0126777
\(716\) −74.1639 −2.77164
\(717\) 5.95376 0.222347
\(718\) 61.4237 2.29231
\(719\) −32.4146 −1.20886 −0.604430 0.796658i \(-0.706599\pi\)
−0.604430 + 0.796658i \(0.706599\pi\)
\(720\) 0.319306 0.0118998
\(721\) 8.02893 0.299013
\(722\) −17.6403 −0.656503
\(723\) 16.3313 0.607369
\(724\) 0.101186 0.00376054
\(725\) −3.61924 −0.134415
\(726\) 19.1923 0.712293
\(727\) −45.1672 −1.67516 −0.837579 0.546316i \(-0.816030\pi\)
−0.837579 + 0.546316i \(0.816030\pi\)
\(728\) 8.97112 0.332492
\(729\) 2.53140 0.0937554
\(730\) 1.58005 0.0584802
\(731\) 29.0695 1.07517
\(732\) −33.2357 −1.22843
\(733\) 19.6790 0.726859 0.363430 0.931622i \(-0.381606\pi\)
0.363430 + 0.931622i \(0.381606\pi\)
\(734\) −66.2931 −2.44692
\(735\) 0.499233 0.0184145
\(736\) 16.3393 0.602276
\(737\) −2.71877 −0.100147
\(738\) 4.71745 0.173652
\(739\) −29.6765 −1.09167 −0.545833 0.837894i \(-0.683787\pi\)
−0.545833 + 0.837894i \(0.683787\pi\)
\(740\) −0.959968 −0.0352891
\(741\) 5.52145 0.202836
\(742\) −56.3207 −2.06760
\(743\) −18.4315 −0.676186 −0.338093 0.941113i \(-0.609782\pi\)
−0.338093 + 0.941113i \(0.609782\pi\)
\(744\) −22.9004 −0.839570
\(745\) −6.12592 −0.224436
\(746\) 31.1946 1.14212
\(747\) 30.1000 1.10130
\(748\) 12.8859 0.471156
\(749\) −0.149644 −0.00546789
\(750\) −5.36695 −0.195973
\(751\) 31.6294 1.15417 0.577087 0.816683i \(-0.304189\pi\)
0.577087 + 0.816683i \(0.304189\pi\)
\(752\) −2.67307 −0.0974769
\(753\) 5.46248 0.199064
\(754\) −2.23328 −0.0813312
\(755\) −3.82224 −0.139105
\(756\) 32.0408 1.16531
\(757\) 46.2602 1.68135 0.840677 0.541536i \(-0.182157\pi\)
0.840677 + 0.541536i \(0.182157\pi\)
\(758\) 56.3448 2.04654
\(759\) 2.34753 0.0852098
\(760\) 4.59303 0.166607
\(761\) −0.517494 −0.0187591 −0.00937957 0.999956i \(-0.502986\pi\)
−0.00937957 + 0.999956i \(0.502986\pi\)
\(762\) −7.72234 −0.279751
\(763\) 2.20640 0.0798771
\(764\) 60.1549 2.17633
\(765\) 2.88391 0.104268
\(766\) −25.0180 −0.903936
\(767\) −18.0701 −0.652472
\(768\) −15.4408 −0.557172
\(769\) 29.5329 1.06498 0.532492 0.846435i \(-0.321256\pi\)
0.532492 + 0.846435i \(0.321256\pi\)
\(770\) −1.31632 −0.0474367
\(771\) 6.15339 0.221609
\(772\) 45.5219 1.63837
\(773\) −14.7516 −0.530578 −0.265289 0.964169i \(-0.585467\pi\)
−0.265289 + 0.964169i \(0.585467\pi\)
\(774\) 36.5363 1.31327
\(775\) −44.6332 −1.60327
\(776\) −34.1524 −1.22600
\(777\) −1.79776 −0.0644944
\(778\) −1.95520 −0.0700975
\(779\) 4.50982 0.161581
\(780\) −1.02706 −0.0367745
\(781\) 2.57487 0.0921361
\(782\) −31.8358 −1.13845
\(783\) −3.20004 −0.114360
\(784\) −1.01378 −0.0362066
\(785\) −0.803583 −0.0286811
\(786\) −1.62716 −0.0580388
\(787\) 37.3630 1.33185 0.665924 0.746019i \(-0.268037\pi\)
0.665924 + 0.746019i \(0.268037\pi\)
\(788\) 21.9155 0.780707
\(789\) 24.1545 0.859923
\(790\) −5.56103 −0.197853
\(791\) −44.7909 −1.59258
\(792\) 6.49769 0.230885
\(793\) −16.0362 −0.569463
\(794\) −67.5833 −2.39844
\(795\) 2.58686 0.0917464
\(796\) −10.2341 −0.362738
\(797\) −7.09162 −0.251198 −0.125599 0.992081i \(-0.540085\pi\)
−0.125599 + 0.992081i \(0.540085\pi\)
\(798\) 21.4397 0.758957
\(799\) −24.1427 −0.854106
\(800\) −25.0496 −0.885636
\(801\) −40.7065 −1.43829
\(802\) −64.1005 −2.26347
\(803\) 2.13689 0.0754092
\(804\) −8.23706 −0.290499
\(805\) 2.03407 0.0716915
\(806\) −27.5412 −0.970099
\(807\) 17.6581 0.621596
\(808\) −55.7371 −1.96083
\(809\) −22.7574 −0.800109 −0.400055 0.916491i \(-0.631009\pi\)
−0.400055 + 0.916491i \(0.631009\pi\)
\(810\) 2.30186 0.0808791
\(811\) −39.0587 −1.37154 −0.685768 0.727820i \(-0.740534\pi\)
−0.685768 + 0.727820i \(0.740534\pi\)
\(812\) −5.42391 −0.190342
\(813\) −10.2367 −0.359018
\(814\) −2.07570 −0.0727531
\(815\) 2.79257 0.0978194
\(816\) 1.66428 0.0582614
\(817\) 34.9283 1.22199
\(818\) 20.8262 0.728172
\(819\) 6.76812 0.236497
\(820\) −0.838882 −0.0292950
\(821\) −15.3867 −0.537000 −0.268500 0.963280i \(-0.586528\pi\)
−0.268500 + 0.963280i \(0.586528\pi\)
\(822\) −27.2175 −0.949320
\(823\) 1.82397 0.0635795 0.0317897 0.999495i \(-0.489879\pi\)
0.0317897 + 0.999495i \(0.489879\pi\)
\(824\) 11.2679 0.392537
\(825\) −3.59895 −0.125299
\(826\) −70.1657 −2.44138
\(827\) 20.8658 0.725576 0.362788 0.931872i \(-0.381825\pi\)
0.362788 + 0.931872i \(0.381825\pi\)
\(828\) −25.0269 −0.869746
\(829\) −19.1267 −0.664297 −0.332148 0.943227i \(-0.607773\pi\)
−0.332148 + 0.943227i \(0.607773\pi\)
\(830\) −8.55767 −0.297041
\(831\) −5.13749 −0.178218
\(832\) −16.7059 −0.579173
\(833\) −9.15631 −0.317247
\(834\) −15.2062 −0.526548
\(835\) −5.56060 −0.192432
\(836\) 15.4830 0.535490
\(837\) −39.4635 −1.36406
\(838\) −54.3003 −1.87577
\(839\) 3.67858 0.126999 0.0634993 0.997982i \(-0.479774\pi\)
0.0634993 + 0.997982i \(0.479774\pi\)
\(840\) −1.59999 −0.0552048
\(841\) −28.4583 −0.981320
\(842\) 71.1354 2.45149
\(843\) −2.99848 −0.103273
\(844\) −5.25892 −0.181020
\(845\) 3.24086 0.111489
\(846\) −30.3440 −1.04325
\(847\) 22.4902 0.772773
\(848\) −5.25309 −0.180392
\(849\) 20.0297 0.687416
\(850\) 48.8069 1.67406
\(851\) 3.20752 0.109952
\(852\) 7.80109 0.267261
\(853\) −1.41418 −0.0484205 −0.0242103 0.999707i \(-0.507707\pi\)
−0.0242103 + 0.999707i \(0.507707\pi\)
\(854\) −62.2683 −2.13078
\(855\) 3.46514 0.118505
\(856\) −0.210014 −0.00717812
\(857\) −14.6219 −0.499476 −0.249738 0.968313i \(-0.580344\pi\)
−0.249738 + 0.968313i \(0.580344\pi\)
\(858\) −2.22076 −0.0758155
\(859\) −17.7215 −0.604651 −0.302325 0.953205i \(-0.597763\pi\)
−0.302325 + 0.953205i \(0.597763\pi\)
\(860\) −6.49708 −0.221549
\(861\) −1.57100 −0.0535396
\(862\) −27.0747 −0.922169
\(863\) 24.3375 0.828458 0.414229 0.910173i \(-0.364051\pi\)
0.414229 + 0.910173i \(0.364051\pi\)
\(864\) −22.1482 −0.753497
\(865\) −0.801560 −0.0272538
\(866\) −21.4273 −0.728129
\(867\) 1.17995 0.0400732
\(868\) −66.8888 −2.27035
\(869\) −7.52085 −0.255127
\(870\) 0.398302 0.0135037
\(871\) −3.97438 −0.134667
\(872\) 3.09651 0.104861
\(873\) −25.7657 −0.872037
\(874\) −38.2521 −1.29390
\(875\) −6.28918 −0.212613
\(876\) 6.47413 0.218741
\(877\) −17.4631 −0.589686 −0.294843 0.955546i \(-0.595267\pi\)
−0.294843 + 0.955546i \(0.595267\pi\)
\(878\) −64.2436 −2.16812
\(879\) −10.0511 −0.339016
\(880\) −0.122774 −0.00413872
\(881\) −51.4068 −1.73194 −0.865970 0.500096i \(-0.833298\pi\)
−0.865970 + 0.500096i \(0.833298\pi\)
\(882\) −11.5082 −0.387501
\(883\) 32.6339 1.09822 0.549109 0.835751i \(-0.314967\pi\)
0.549109 + 0.835751i \(0.314967\pi\)
\(884\) 18.8370 0.633557
\(885\) 3.22277 0.108332
\(886\) −44.3996 −1.49163
\(887\) −12.8326 −0.430877 −0.215439 0.976517i \(-0.569118\pi\)
−0.215439 + 0.976517i \(0.569118\pi\)
\(888\) −2.52302 −0.0846668
\(889\) −9.04931 −0.303504
\(890\) 11.5732 0.387933
\(891\) 3.11308 0.104292
\(892\) −17.0524 −0.570955
\(893\) −29.0085 −0.970731
\(894\) −40.1308 −1.34217
\(895\) 6.38203 0.213328
\(896\) −42.3895 −1.41613
\(897\) 3.43168 0.114580
\(898\) 96.0002 3.20357
\(899\) 6.68046 0.222806
\(900\) 38.3684 1.27895
\(901\) −47.4449 −1.58062
\(902\) −1.81388 −0.0603955
\(903\) −12.1673 −0.404902
\(904\) −62.8604 −2.09071
\(905\) −0.00870735 −0.000289442 0
\(906\) −25.0394 −0.831879
\(907\) 46.2552 1.53588 0.767939 0.640523i \(-0.221282\pi\)
0.767939 + 0.640523i \(0.221282\pi\)
\(908\) −16.3693 −0.543235
\(909\) −42.0500 −1.39471
\(910\) −1.92423 −0.0637875
\(911\) 57.7968 1.91489 0.957447 0.288608i \(-0.0931926\pi\)
0.957447 + 0.288608i \(0.0931926\pi\)
\(912\) 1.99970 0.0662168
\(913\) −11.5736 −0.383029
\(914\) 60.0733 1.98705
\(915\) 2.86004 0.0945499
\(916\) −45.1727 −1.49255
\(917\) −1.90676 −0.0629668
\(918\) 43.1538 1.42429
\(919\) 24.9878 0.824271 0.412135 0.911123i \(-0.364783\pi\)
0.412135 + 0.911123i \(0.364783\pi\)
\(920\) 2.85465 0.0941150
\(921\) −6.04860 −0.199308
\(922\) 56.3661 1.85632
\(923\) 3.76402 0.123894
\(924\) −5.39351 −0.177433
\(925\) −4.91739 −0.161683
\(926\) −47.3382 −1.55563
\(927\) 8.50092 0.279207
\(928\) 3.74928 0.123076
\(929\) −60.8275 −1.99568 −0.997842 0.0656572i \(-0.979086\pi\)
−0.997842 + 0.0656572i \(0.979086\pi\)
\(930\) 4.91194 0.161069
\(931\) −11.0017 −0.360566
\(932\) 44.8758 1.46996
\(933\) 22.9076 0.749960
\(934\) 19.2382 0.629494
\(935\) −1.10887 −0.0362640
\(936\) 9.49851 0.310468
\(937\) −6.01124 −0.196379 −0.0981893 0.995168i \(-0.531305\pi\)
−0.0981893 + 0.995168i \(0.531305\pi\)
\(938\) −15.4324 −0.503886
\(939\) −10.9746 −0.358142
\(940\) 5.39592 0.175996
\(941\) −10.1643 −0.331347 −0.165673 0.986181i \(-0.552980\pi\)
−0.165673 + 0.986181i \(0.552980\pi\)
\(942\) −5.26426 −0.171519
\(943\) 2.80293 0.0912761
\(944\) −6.54443 −0.213003
\(945\) −2.75721 −0.0896919
\(946\) −14.0483 −0.456751
\(947\) −49.0757 −1.59474 −0.797372 0.603488i \(-0.793777\pi\)
−0.797372 + 0.603488i \(0.793777\pi\)
\(948\) −22.7859 −0.740052
\(949\) 3.12377 0.101402
\(950\) 58.6436 1.90265
\(951\) 11.6057 0.376341
\(952\) 29.3449 0.951075
\(953\) 8.21968 0.266262 0.133131 0.991098i \(-0.457497\pi\)
0.133131 + 0.991098i \(0.457497\pi\)
\(954\) −59.6316 −1.93065
\(955\) −5.17651 −0.167508
\(956\) 24.4056 0.789332
\(957\) 0.538672 0.0174128
\(958\) 31.1244 1.00558
\(959\) −31.8945 −1.02993
\(960\) 2.97947 0.0961621
\(961\) 51.3848 1.65757
\(962\) −3.03431 −0.0978301
\(963\) −0.158442 −0.00510571
\(964\) 66.9451 2.15616
\(965\) −3.91730 −0.126102
\(966\) 13.3251 0.428730
\(967\) −53.8311 −1.73109 −0.865545 0.500831i \(-0.833028\pi\)
−0.865545 + 0.500831i \(0.833028\pi\)
\(968\) 31.5632 1.01448
\(969\) 18.0609 0.580201
\(970\) 7.32539 0.235204
\(971\) 0.515998 0.0165592 0.00827958 0.999966i \(-0.497364\pi\)
0.00827958 + 0.999966i \(0.497364\pi\)
\(972\) 52.9968 1.69987
\(973\) −17.8192 −0.571256
\(974\) 15.4667 0.495584
\(975\) −5.26105 −0.168488
\(976\) −5.80783 −0.185904
\(977\) 7.24138 0.231672 0.115836 0.993268i \(-0.463045\pi\)
0.115836 + 0.993268i \(0.463045\pi\)
\(978\) 18.2941 0.584979
\(979\) 15.6518 0.500233
\(980\) 2.04645 0.0653714
\(981\) 2.33611 0.0745863
\(982\) −30.1813 −0.963125
\(983\) −61.2524 −1.95365 −0.976824 0.214043i \(-0.931337\pi\)
−0.976824 + 0.214043i \(0.931337\pi\)
\(984\) −2.20477 −0.0702856
\(985\) −1.88589 −0.0600896
\(986\) −7.30515 −0.232643
\(987\) 10.1051 0.321650
\(988\) 22.6335 0.720066
\(989\) 21.7085 0.690291
\(990\) −1.39370 −0.0442947
\(991\) −8.97907 −0.285230 −0.142615 0.989778i \(-0.545551\pi\)
−0.142615 + 0.989778i \(0.545551\pi\)
\(992\) 46.2369 1.46802
\(993\) 2.45186 0.0778075
\(994\) 14.6156 0.463579
\(995\) 0.880677 0.0279193
\(996\) −35.0644 −1.11106
\(997\) 1.91311 0.0605888 0.0302944 0.999541i \(-0.490356\pi\)
0.0302944 + 0.999541i \(0.490356\pi\)
\(998\) −26.7577 −0.847002
\(999\) −4.34783 −0.137559
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.1.8 85
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4033.2.a.f.1.8 85 1.1 even 1 trivial