Properties

Label 74.2.a.b.1.2
Level $74$
Weight $2$
Character 74.1
Self dual yes
Analytic conductor $0.591$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 74.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -2.85410 q^{5} +0.618034 q^{6} +1.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -2.85410 q^{5} +0.618034 q^{6} +1.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} -2.85410 q^{10} -3.61803 q^{11} +0.618034 q^{12} +3.85410 q^{13} +1.23607 q^{14} -1.76393 q^{15} +1.00000 q^{16} +4.47214 q^{17} -2.61803 q^{18} -4.47214 q^{19} -2.85410 q^{20} +0.763932 q^{21} -3.61803 q^{22} -3.85410 q^{23} +0.618034 q^{24} +3.14590 q^{25} +3.85410 q^{26} -3.47214 q^{27} +1.23607 q^{28} +6.32624 q^{29} -1.76393 q^{30} +9.61803 q^{31} +1.00000 q^{32} -2.23607 q^{33} +4.47214 q^{34} -3.52786 q^{35} -2.61803 q^{36} -1.00000 q^{37} -4.47214 q^{38} +2.38197 q^{39} -2.85410 q^{40} +7.38197 q^{41} +0.763932 q^{42} -0.763932 q^{43} -3.61803 q^{44} +7.47214 q^{45} -3.85410 q^{46} +3.23607 q^{47} +0.618034 q^{48} -5.47214 q^{49} +3.14590 q^{50} +2.76393 q^{51} +3.85410 q^{52} -8.47214 q^{53} -3.47214 q^{54} +10.3262 q^{55} +1.23607 q^{56} -2.76393 q^{57} +6.32624 q^{58} -9.23607 q^{59} -1.76393 q^{60} +8.38197 q^{61} +9.61803 q^{62} -3.23607 q^{63} +1.00000 q^{64} -11.0000 q^{65} -2.23607 q^{66} -10.0902 q^{67} +4.47214 q^{68} -2.38197 q^{69} -3.52786 q^{70} -14.9443 q^{71} -2.61803 q^{72} -4.09017 q^{73} -1.00000 q^{74} +1.94427 q^{75} -4.47214 q^{76} -4.47214 q^{77} +2.38197 q^{78} +11.5623 q^{79} -2.85410 q^{80} +5.70820 q^{81} +7.38197 q^{82} -5.52786 q^{83} +0.763932 q^{84} -12.7639 q^{85} -0.763932 q^{86} +3.90983 q^{87} -3.61803 q^{88} -10.4721 q^{89} +7.47214 q^{90} +4.76393 q^{91} -3.85410 q^{92} +5.94427 q^{93} +3.23607 q^{94} +12.7639 q^{95} +0.618034 q^{96} +8.47214 q^{97} -5.47214 q^{98} +9.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{7} + 2 q^{8} - 3 q^{9} + q^{10} - 5 q^{11} - q^{12} + q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} - 3 q^{18} + q^{20} + 6 q^{21} - 5 q^{22} - q^{23} - q^{24} + 13 q^{25} + q^{26} + 2 q^{27} - 2 q^{28} - 3 q^{29} - 8 q^{30} + 17 q^{31} + 2 q^{32} - 16 q^{35} - 3 q^{36} - 2 q^{37} + 7 q^{39} + q^{40} + 17 q^{41} + 6 q^{42} - 6 q^{43} - 5 q^{44} + 6 q^{45} - q^{46} + 2 q^{47} - q^{48} - 2 q^{49} + 13 q^{50} + 10 q^{51} + q^{52} - 8 q^{53} + 2 q^{54} + 5 q^{55} - 2 q^{56} - 10 q^{57} - 3 q^{58} - 14 q^{59} - 8 q^{60} + 19 q^{61} + 17 q^{62} - 2 q^{63} + 2 q^{64} - 22 q^{65} - 9 q^{67} - 7 q^{69} - 16 q^{70} - 12 q^{71} - 3 q^{72} + 3 q^{73} - 2 q^{74} - 14 q^{75} + 7 q^{78} + 3 q^{79} + q^{80} - 2 q^{81} + 17 q^{82} - 20 q^{83} + 6 q^{84} - 30 q^{85} - 6 q^{86} + 19 q^{87} - 5 q^{88} - 12 q^{89} + 6 q^{90} + 14 q^{91} - q^{92} - 6 q^{93} + 2 q^{94} + 30 q^{95} - q^{96} + 8 q^{97} - 2 q^{98} + 10 q^{99}+O(q^{100}) \) 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
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.85410 −1.27639 −0.638197 0.769873i \(-0.720319\pi\)
−0.638197 + 0.769873i \(0.720319\pi\)
\(6\) 0.618034 0.252311
\(7\) 1.23607 0.467190 0.233595 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.61803 −0.872678
\(10\) −2.85410 −0.902546
\(11\) −3.61803 −1.09088 −0.545439 0.838150i \(-0.683637\pi\)
−0.545439 + 0.838150i \(0.683637\pi\)
\(12\) 0.618034 0.178411
\(13\) 3.85410 1.06894 0.534468 0.845189i \(-0.320512\pi\)
0.534468 + 0.845189i \(0.320512\pi\)
\(14\) 1.23607 0.330353
\(15\) −1.76393 −0.455445
\(16\) 1.00000 0.250000
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) −2.61803 −0.617077
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) −2.85410 −0.638197
\(21\) 0.763932 0.166704
\(22\) −3.61803 −0.771367
\(23\) −3.85410 −0.803636 −0.401818 0.915720i \(-0.631622\pi\)
−0.401818 + 0.915720i \(0.631622\pi\)
\(24\) 0.618034 0.126156
\(25\) 3.14590 0.629180
\(26\) 3.85410 0.755852
\(27\) −3.47214 −0.668213
\(28\) 1.23607 0.233595
\(29\) 6.32624 1.17475 0.587376 0.809314i \(-0.300161\pi\)
0.587376 + 0.809314i \(0.300161\pi\)
\(30\) −1.76393 −0.322048
\(31\) 9.61803 1.72745 0.863725 0.503964i \(-0.168125\pi\)
0.863725 + 0.503964i \(0.168125\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.23607 −0.389249
\(34\) 4.47214 0.766965
\(35\) −3.52786 −0.596318
\(36\) −2.61803 −0.436339
\(37\) −1.00000 −0.164399
\(38\) −4.47214 −0.725476
\(39\) 2.38197 0.381420
\(40\) −2.85410 −0.451273
\(41\) 7.38197 1.15287 0.576435 0.817143i \(-0.304444\pi\)
0.576435 + 0.817143i \(0.304444\pi\)
\(42\) 0.763932 0.117877
\(43\) −0.763932 −0.116499 −0.0582493 0.998302i \(-0.518552\pi\)
−0.0582493 + 0.998302i \(0.518552\pi\)
\(44\) −3.61803 −0.545439
\(45\) 7.47214 1.11388
\(46\) −3.85410 −0.568256
\(47\) 3.23607 0.472029 0.236015 0.971750i \(-0.424159\pi\)
0.236015 + 0.971750i \(0.424159\pi\)
\(48\) 0.618034 0.0892055
\(49\) −5.47214 −0.781734
\(50\) 3.14590 0.444897
\(51\) 2.76393 0.387028
\(52\) 3.85410 0.534468
\(53\) −8.47214 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) −3.47214 −0.472498
\(55\) 10.3262 1.39239
\(56\) 1.23607 0.165177
\(57\) −2.76393 −0.366092
\(58\) 6.32624 0.830676
\(59\) −9.23607 −1.20243 −0.601217 0.799086i \(-0.705317\pi\)
−0.601217 + 0.799086i \(0.705317\pi\)
\(60\) −1.76393 −0.227723
\(61\) 8.38197 1.07320 0.536600 0.843836i \(-0.319708\pi\)
0.536600 + 0.843836i \(0.319708\pi\)
\(62\) 9.61803 1.22149
\(63\) −3.23607 −0.407706
\(64\) 1.00000 0.125000
\(65\) −11.0000 −1.36438
\(66\) −2.23607 −0.275241
\(67\) −10.0902 −1.23271 −0.616355 0.787468i \(-0.711391\pi\)
−0.616355 + 0.787468i \(0.711391\pi\)
\(68\) 4.47214 0.542326
\(69\) −2.38197 −0.286755
\(70\) −3.52786 −0.421660
\(71\) −14.9443 −1.77356 −0.886779 0.462193i \(-0.847063\pi\)
−0.886779 + 0.462193i \(0.847063\pi\)
\(72\) −2.61803 −0.308538
\(73\) −4.09017 −0.478718 −0.239359 0.970931i \(-0.576937\pi\)
−0.239359 + 0.970931i \(0.576937\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.94427 0.224505
\(76\) −4.47214 −0.512989
\(77\) −4.47214 −0.509647
\(78\) 2.38197 0.269705
\(79\) 11.5623 1.30086 0.650431 0.759566i \(-0.274589\pi\)
0.650431 + 0.759566i \(0.274589\pi\)
\(80\) −2.85410 −0.319098
\(81\) 5.70820 0.634245
\(82\) 7.38197 0.815202
\(83\) −5.52786 −0.606762 −0.303381 0.952869i \(-0.598115\pi\)
−0.303381 + 0.952869i \(0.598115\pi\)
\(84\) 0.763932 0.0833518
\(85\) −12.7639 −1.38444
\(86\) −0.763932 −0.0823769
\(87\) 3.90983 0.419178
\(88\) −3.61803 −0.385684
\(89\) −10.4721 −1.11004 −0.555022 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 7.47214 0.787632
\(91\) 4.76393 0.499396
\(92\) −3.85410 −0.401818
\(93\) 5.94427 0.616392
\(94\) 3.23607 0.333775
\(95\) 12.7639 1.30955
\(96\) 0.618034 0.0630778
\(97\) 8.47214 0.860215 0.430108 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) −5.47214 −0.552769
\(99\) 9.47214 0.951985
\(100\) 3.14590 0.314590
\(101\) 12.4721 1.24102 0.620512 0.784197i \(-0.286925\pi\)
0.620512 + 0.784197i \(0.286925\pi\)
\(102\) 2.76393 0.273670
\(103\) −16.2705 −1.60318 −0.801590 0.597873i \(-0.796013\pi\)
−0.801590 + 0.597873i \(0.796013\pi\)
\(104\) 3.85410 0.377926
\(105\) −2.18034 −0.212779
\(106\) −8.47214 −0.822887
\(107\) 8.32624 0.804928 0.402464 0.915436i \(-0.368154\pi\)
0.402464 + 0.915436i \(0.368154\pi\)
\(108\) −3.47214 −0.334106
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 10.3262 0.984568
\(111\) −0.618034 −0.0586612
\(112\) 1.23607 0.116797
\(113\) 10.9443 1.02955 0.514775 0.857325i \(-0.327875\pi\)
0.514775 + 0.857325i \(0.327875\pi\)
\(114\) −2.76393 −0.258866
\(115\) 11.0000 1.02576
\(116\) 6.32624 0.587376
\(117\) −10.0902 −0.932837
\(118\) −9.23607 −0.850249
\(119\) 5.52786 0.506738
\(120\) −1.76393 −0.161024
\(121\) 2.09017 0.190015
\(122\) 8.38197 0.758868
\(123\) 4.56231 0.411369
\(124\) 9.61803 0.863725
\(125\) 5.29180 0.473313
\(126\) −3.23607 −0.288292
\(127\) 0.472136 0.0418953 0.0209476 0.999781i \(-0.493332\pi\)
0.0209476 + 0.999781i \(0.493332\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.472136 −0.0415693
\(130\) −11.0000 −0.964764
\(131\) 8.65248 0.755970 0.377985 0.925812i \(-0.376617\pi\)
0.377985 + 0.925812i \(0.376617\pi\)
\(132\) −2.23607 −0.194625
\(133\) −5.52786 −0.479327
\(134\) −10.0902 −0.871658
\(135\) 9.90983 0.852902
\(136\) 4.47214 0.383482
\(137\) 19.3262 1.65115 0.825576 0.564291i \(-0.190850\pi\)
0.825576 + 0.564291i \(0.190850\pi\)
\(138\) −2.38197 −0.202766
\(139\) −1.85410 −0.157263 −0.0786314 0.996904i \(-0.525055\pi\)
−0.0786314 + 0.996904i \(0.525055\pi\)
\(140\) −3.52786 −0.298159
\(141\) 2.00000 0.168430
\(142\) −14.9443 −1.25410
\(143\) −13.9443 −1.16608
\(144\) −2.61803 −0.218169
\(145\) −18.0557 −1.49945
\(146\) −4.09017 −0.338505
\(147\) −3.38197 −0.278940
\(148\) −1.00000 −0.0821995
\(149\) −6.18034 −0.506313 −0.253157 0.967425i \(-0.581469\pi\)
−0.253157 + 0.967425i \(0.581469\pi\)
\(150\) 1.94427 0.158749
\(151\) 17.7082 1.44107 0.720537 0.693417i \(-0.243896\pi\)
0.720537 + 0.693417i \(0.243896\pi\)
\(152\) −4.47214 −0.362738
\(153\) −11.7082 −0.946552
\(154\) −4.47214 −0.360375
\(155\) −27.4508 −2.20491
\(156\) 2.38197 0.190710
\(157\) −7.52786 −0.600789 −0.300394 0.953815i \(-0.597118\pi\)
−0.300394 + 0.953815i \(0.597118\pi\)
\(158\) 11.5623 0.919848
\(159\) −5.23607 −0.415247
\(160\) −2.85410 −0.225637
\(161\) −4.76393 −0.375450
\(162\) 5.70820 0.448479
\(163\) −12.4721 −0.976893 −0.488447 0.872594i \(-0.662436\pi\)
−0.488447 + 0.872594i \(0.662436\pi\)
\(164\) 7.38197 0.576435
\(165\) 6.38197 0.496835
\(166\) −5.52786 −0.429045
\(167\) 7.14590 0.552966 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(168\) 0.763932 0.0589386
\(169\) 1.85410 0.142623
\(170\) −12.7639 −0.978949
\(171\) 11.7082 0.895349
\(172\) −0.763932 −0.0582493
\(173\) 8.47214 0.644125 0.322062 0.946718i \(-0.395624\pi\)
0.322062 + 0.946718i \(0.395624\pi\)
\(174\) 3.90983 0.296403
\(175\) 3.88854 0.293946
\(176\) −3.61803 −0.272720
\(177\) −5.70820 −0.429055
\(178\) −10.4721 −0.784920
\(179\) 18.6525 1.39415 0.697076 0.716997i \(-0.254484\pi\)
0.697076 + 0.716997i \(0.254484\pi\)
\(180\) 7.47214 0.556940
\(181\) 5.52786 0.410883 0.205441 0.978669i \(-0.434137\pi\)
0.205441 + 0.978669i \(0.434137\pi\)
\(182\) 4.76393 0.353126
\(183\) 5.18034 0.382942
\(184\) −3.85410 −0.284128
\(185\) 2.85410 0.209838
\(186\) 5.94427 0.435855
\(187\) −16.1803 −1.18322
\(188\) 3.23607 0.236015
\(189\) −4.29180 −0.312182
\(190\) 12.7639 0.925993
\(191\) 4.09017 0.295954 0.147977 0.988991i \(-0.452724\pi\)
0.147977 + 0.988991i \(0.452724\pi\)
\(192\) 0.618034 0.0446028
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 8.47214 0.608264
\(195\) −6.79837 −0.486842
\(196\) −5.47214 −0.390867
\(197\) −16.4721 −1.17359 −0.586796 0.809735i \(-0.699611\pi\)
−0.586796 + 0.809735i \(0.699611\pi\)
\(198\) 9.47214 0.673155
\(199\) 20.9443 1.48470 0.742350 0.670012i \(-0.233711\pi\)
0.742350 + 0.670012i \(0.233711\pi\)
\(200\) 3.14590 0.222449
\(201\) −6.23607 −0.439858
\(202\) 12.4721 0.877536
\(203\) 7.81966 0.548833
\(204\) 2.76393 0.193514
\(205\) −21.0689 −1.47151
\(206\) −16.2705 −1.13362
\(207\) 10.0902 0.701315
\(208\) 3.85410 0.267234
\(209\) 16.1803 1.11922
\(210\) −2.18034 −0.150458
\(211\) −22.2705 −1.53317 −0.766583 0.642146i \(-0.778044\pi\)
−0.766583 + 0.642146i \(0.778044\pi\)
\(212\) −8.47214 −0.581869
\(213\) −9.23607 −0.632845
\(214\) 8.32624 0.569170
\(215\) 2.18034 0.148698
\(216\) −3.47214 −0.236249
\(217\) 11.8885 0.807047
\(218\) −14.9443 −1.01215
\(219\) −2.52786 −0.170817
\(220\) 10.3262 0.696195
\(221\) 17.2361 1.15942
\(222\) −0.618034 −0.0414797
\(223\) −8.18034 −0.547796 −0.273898 0.961759i \(-0.588313\pi\)
−0.273898 + 0.961759i \(0.588313\pi\)
\(224\) 1.23607 0.0825883
\(225\) −8.23607 −0.549071
\(226\) 10.9443 0.728002
\(227\) −17.7082 −1.17533 −0.587667 0.809103i \(-0.699954\pi\)
−0.587667 + 0.809103i \(0.699954\pi\)
\(228\) −2.76393 −0.183046
\(229\) 17.1246 1.13163 0.565813 0.824534i \(-0.308562\pi\)
0.565813 + 0.824534i \(0.308562\pi\)
\(230\) 11.0000 0.725319
\(231\) −2.76393 −0.181853
\(232\) 6.32624 0.415338
\(233\) 13.5623 0.888496 0.444248 0.895904i \(-0.353471\pi\)
0.444248 + 0.895904i \(0.353471\pi\)
\(234\) −10.0902 −0.659615
\(235\) −9.23607 −0.602495
\(236\) −9.23607 −0.601217
\(237\) 7.14590 0.464176
\(238\) 5.52786 0.358318
\(239\) 3.14590 0.203491 0.101746 0.994810i \(-0.467557\pi\)
0.101746 + 0.994810i \(0.467557\pi\)
\(240\) −1.76393 −0.113861
\(241\) −10.4721 −0.674570 −0.337285 0.941403i \(-0.609509\pi\)
−0.337285 + 0.941403i \(0.609509\pi\)
\(242\) 2.09017 0.134361
\(243\) 13.9443 0.894525
\(244\) 8.38197 0.536600
\(245\) 15.6180 0.997800
\(246\) 4.56231 0.290882
\(247\) −17.2361 −1.09670
\(248\) 9.61803 0.610746
\(249\) −3.41641 −0.216506
\(250\) 5.29180 0.334683
\(251\) −3.05573 −0.192876 −0.0964379 0.995339i \(-0.530745\pi\)
−0.0964379 + 0.995339i \(0.530745\pi\)
\(252\) −3.23607 −0.203853
\(253\) 13.9443 0.876669
\(254\) 0.472136 0.0296244
\(255\) −7.88854 −0.494000
\(256\) 1.00000 0.0625000
\(257\) −18.9443 −1.18171 −0.590856 0.806777i \(-0.701210\pi\)
−0.590856 + 0.806777i \(0.701210\pi\)
\(258\) −0.472136 −0.0293939
\(259\) −1.23607 −0.0768055
\(260\) −11.0000 −0.682191
\(261\) −16.5623 −1.02518
\(262\) 8.65248 0.534552
\(263\) −8.76393 −0.540407 −0.270204 0.962803i \(-0.587091\pi\)
−0.270204 + 0.962803i \(0.587091\pi\)
\(264\) −2.23607 −0.137620
\(265\) 24.1803 1.48539
\(266\) −5.52786 −0.338935
\(267\) −6.47214 −0.396088
\(268\) −10.0902 −0.616355
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 9.90983 0.603093
\(271\) −4.94427 −0.300343 −0.150172 0.988660i \(-0.547983\pi\)
−0.150172 + 0.988660i \(0.547983\pi\)
\(272\) 4.47214 0.271163
\(273\) 2.94427 0.178195
\(274\) 19.3262 1.16754
\(275\) −11.3820 −0.686358
\(276\) −2.38197 −0.143378
\(277\) −7.79837 −0.468559 −0.234279 0.972169i \(-0.575273\pi\)
−0.234279 + 0.972169i \(0.575273\pi\)
\(278\) −1.85410 −0.111202
\(279\) −25.1803 −1.50751
\(280\) −3.52786 −0.210830
\(281\) −5.88854 −0.351281 −0.175641 0.984454i \(-0.556200\pi\)
−0.175641 + 0.984454i \(0.556200\pi\)
\(282\) 2.00000 0.119098
\(283\) 11.2361 0.667915 0.333957 0.942588i \(-0.391616\pi\)
0.333957 + 0.942588i \(0.391616\pi\)
\(284\) −14.9443 −0.886779
\(285\) 7.88854 0.467277
\(286\) −13.9443 −0.824542
\(287\) 9.12461 0.538609
\(288\) −2.61803 −0.154269
\(289\) 3.00000 0.176471
\(290\) −18.0557 −1.06027
\(291\) 5.23607 0.306944
\(292\) −4.09017 −0.239359
\(293\) 18.6525 1.08969 0.544845 0.838537i \(-0.316589\pi\)
0.544845 + 0.838537i \(0.316589\pi\)
\(294\) −3.38197 −0.197240
\(295\) 26.3607 1.53478
\(296\) −1.00000 −0.0581238
\(297\) 12.5623 0.728939
\(298\) −6.18034 −0.358017
\(299\) −14.8541 −0.859035
\(300\) 1.94427 0.112253
\(301\) −0.944272 −0.0544269
\(302\) 17.7082 1.01899
\(303\) 7.70820 0.442825
\(304\) −4.47214 −0.256495
\(305\) −23.9230 −1.36983
\(306\) −11.7082 −0.669313
\(307\) −6.14590 −0.350765 −0.175382 0.984500i \(-0.556116\pi\)
−0.175382 + 0.984500i \(0.556116\pi\)
\(308\) −4.47214 −0.254824
\(309\) −10.0557 −0.572050
\(310\) −27.4508 −1.55910
\(311\) 2.03444 0.115363 0.0576813 0.998335i \(-0.481629\pi\)
0.0576813 + 0.998335i \(0.481629\pi\)
\(312\) 2.38197 0.134852
\(313\) 5.81966 0.328947 0.164473 0.986382i \(-0.447408\pi\)
0.164473 + 0.986382i \(0.447408\pi\)
\(314\) −7.52786 −0.424822
\(315\) 9.23607 0.520393
\(316\) 11.5623 0.650431
\(317\) 3.05573 0.171627 0.0858134 0.996311i \(-0.472651\pi\)
0.0858134 + 0.996311i \(0.472651\pi\)
\(318\) −5.23607 −0.293624
\(319\) −22.8885 −1.28151
\(320\) −2.85410 −0.159549
\(321\) 5.14590 0.287216
\(322\) −4.76393 −0.265484
\(323\) −20.0000 −1.11283
\(324\) 5.70820 0.317122
\(325\) 12.1246 0.672552
\(326\) −12.4721 −0.690768
\(327\) −9.23607 −0.510756
\(328\) 7.38197 0.407601
\(329\) 4.00000 0.220527
\(330\) 6.38197 0.351316
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −5.52786 −0.303381
\(333\) 2.61803 0.143467
\(334\) 7.14590 0.391006
\(335\) 28.7984 1.57342
\(336\) 0.763932 0.0416759
\(337\) −17.0344 −0.927925 −0.463963 0.885855i \(-0.653573\pi\)
−0.463963 + 0.885855i \(0.653573\pi\)
\(338\) 1.85410 0.100850
\(339\) 6.76393 0.367366
\(340\) −12.7639 −0.692221
\(341\) −34.7984 −1.88444
\(342\) 11.7082 0.633107
\(343\) −15.4164 −0.832408
\(344\) −0.763932 −0.0411885
\(345\) 6.79837 0.366012
\(346\) 8.47214 0.455465
\(347\) 12.7639 0.685204 0.342602 0.939481i \(-0.388692\pi\)
0.342602 + 0.939481i \(0.388692\pi\)
\(348\) 3.90983 0.209589
\(349\) 12.1803 0.651999 0.325999 0.945370i \(-0.394299\pi\)
0.325999 + 0.945370i \(0.394299\pi\)
\(350\) 3.88854 0.207851
\(351\) −13.3820 −0.714277
\(352\) −3.61803 −0.192842
\(353\) 29.7082 1.58121 0.790604 0.612328i \(-0.209767\pi\)
0.790604 + 0.612328i \(0.209767\pi\)
\(354\) −5.70820 −0.303388
\(355\) 42.6525 2.26376
\(356\) −10.4721 −0.555022
\(357\) 3.41641 0.180815
\(358\) 18.6525 0.985814
\(359\) 4.47214 0.236030 0.118015 0.993012i \(-0.462347\pi\)
0.118015 + 0.993012i \(0.462347\pi\)
\(360\) 7.47214 0.393816
\(361\) 1.00000 0.0526316
\(362\) 5.52786 0.290538
\(363\) 1.29180 0.0678017
\(364\) 4.76393 0.249698
\(365\) 11.6738 0.611033
\(366\) 5.18034 0.270781
\(367\) −27.1246 −1.41589 −0.707947 0.706266i \(-0.750378\pi\)
−0.707947 + 0.706266i \(0.750378\pi\)
\(368\) −3.85410 −0.200909
\(369\) −19.3262 −1.00608
\(370\) 2.85410 0.148378
\(371\) −10.4721 −0.543686
\(372\) 5.94427 0.308196
\(373\) 14.2918 0.740001 0.370001 0.929032i \(-0.379357\pi\)
0.370001 + 0.929032i \(0.379357\pi\)
\(374\) −16.1803 −0.836665
\(375\) 3.27051 0.168888
\(376\) 3.23607 0.166887
\(377\) 24.3820 1.25574
\(378\) −4.29180 −0.220746
\(379\) 16.9098 0.868600 0.434300 0.900768i \(-0.356996\pi\)
0.434300 + 0.900768i \(0.356996\pi\)
\(380\) 12.7639 0.654776
\(381\) 0.291796 0.0149492
\(382\) 4.09017 0.209271
\(383\) −17.8885 −0.914062 −0.457031 0.889451i \(-0.651087\pi\)
−0.457031 + 0.889451i \(0.651087\pi\)
\(384\) 0.618034 0.0315389
\(385\) 12.7639 0.650510
\(386\) 4.00000 0.203595
\(387\) 2.00000 0.101666
\(388\) 8.47214 0.430108
\(389\) −0.145898 −0.00739732 −0.00369866 0.999993i \(-0.501177\pi\)
−0.00369866 + 0.999993i \(0.501177\pi\)
\(390\) −6.79837 −0.344249
\(391\) −17.2361 −0.871665
\(392\) −5.47214 −0.276385
\(393\) 5.34752 0.269747
\(394\) −16.4721 −0.829854
\(395\) −33.0000 −1.66041
\(396\) 9.47214 0.475993
\(397\) −10.6525 −0.534632 −0.267316 0.963609i \(-0.586137\pi\)
−0.267316 + 0.963609i \(0.586137\pi\)
\(398\) 20.9443 1.04984
\(399\) −3.41641 −0.171034
\(400\) 3.14590 0.157295
\(401\) −9.23607 −0.461227 −0.230614 0.973045i \(-0.574073\pi\)
−0.230614 + 0.973045i \(0.574073\pi\)
\(402\) −6.23607 −0.311027
\(403\) 37.0689 1.84653
\(404\) 12.4721 0.620512
\(405\) −16.2918 −0.809546
\(406\) 7.81966 0.388083
\(407\) 3.61803 0.179339
\(408\) 2.76393 0.136835
\(409\) −3.81966 −0.188870 −0.0944350 0.995531i \(-0.530104\pi\)
−0.0944350 + 0.995531i \(0.530104\pi\)
\(410\) −21.0689 −1.04052
\(411\) 11.9443 0.589167
\(412\) −16.2705 −0.801590
\(413\) −11.4164 −0.561765
\(414\) 10.0902 0.495905
\(415\) 15.7771 0.774467
\(416\) 3.85410 0.188963
\(417\) −1.14590 −0.0561149
\(418\) 16.1803 0.791406
\(419\) 9.56231 0.467149 0.233575 0.972339i \(-0.424958\pi\)
0.233575 + 0.972339i \(0.424958\pi\)
\(420\) −2.18034 −0.106390
\(421\) −1.96556 −0.0957954 −0.0478977 0.998852i \(-0.515252\pi\)
−0.0478977 + 0.998852i \(0.515252\pi\)
\(422\) −22.2705 −1.08411
\(423\) −8.47214 −0.411929
\(424\) −8.47214 −0.411443
\(425\) 14.0689 0.682441
\(426\) −9.23607 −0.447489
\(427\) 10.3607 0.501388
\(428\) 8.32624 0.402464
\(429\) −8.61803 −0.416083
\(430\) 2.18034 0.105145
\(431\) −32.3607 −1.55876 −0.779380 0.626552i \(-0.784466\pi\)
−0.779380 + 0.626552i \(0.784466\pi\)
\(432\) −3.47214 −0.167053
\(433\) −36.3262 −1.74573 −0.872864 0.487964i \(-0.837740\pi\)
−0.872864 + 0.487964i \(0.837740\pi\)
\(434\) 11.8885 0.570668
\(435\) −11.1591 −0.535036
\(436\) −14.9443 −0.715701
\(437\) 17.2361 0.824513
\(438\) −2.52786 −0.120786
\(439\) −16.7984 −0.801743 −0.400871 0.916134i \(-0.631293\pi\)
−0.400871 + 0.916134i \(0.631293\pi\)
\(440\) 10.3262 0.492284
\(441\) 14.3262 0.682202
\(442\) 17.2361 0.819836
\(443\) −18.2705 −0.868058 −0.434029 0.900899i \(-0.642908\pi\)
−0.434029 + 0.900899i \(0.642908\pi\)
\(444\) −0.618034 −0.0293306
\(445\) 29.8885 1.41685
\(446\) −8.18034 −0.387350
\(447\) −3.81966 −0.180664
\(448\) 1.23607 0.0583987
\(449\) −19.5279 −0.921577 −0.460788 0.887510i \(-0.652433\pi\)
−0.460788 + 0.887510i \(0.652433\pi\)
\(450\) −8.23607 −0.388252
\(451\) −26.7082 −1.25764
\(452\) 10.9443 0.514775
\(453\) 10.9443 0.514207
\(454\) −17.7082 −0.831087
\(455\) −13.5967 −0.637425
\(456\) −2.76393 −0.129433
\(457\) −29.2361 −1.36761 −0.683803 0.729667i \(-0.739675\pi\)
−0.683803 + 0.729667i \(0.739675\pi\)
\(458\) 17.1246 0.800181
\(459\) −15.5279 −0.724779
\(460\) 11.0000 0.512878
\(461\) −21.0557 −0.980663 −0.490332 0.871536i \(-0.663124\pi\)
−0.490332 + 0.871536i \(0.663124\pi\)
\(462\) −2.76393 −0.128590
\(463\) 15.5623 0.723242 0.361621 0.932325i \(-0.382223\pi\)
0.361621 + 0.932325i \(0.382223\pi\)
\(464\) 6.32624 0.293688
\(465\) −16.9656 −0.786759
\(466\) 13.5623 0.628262
\(467\) 20.3607 0.942180 0.471090 0.882085i \(-0.343861\pi\)
0.471090 + 0.882085i \(0.343861\pi\)
\(468\) −10.0902 −0.466418
\(469\) −12.4721 −0.575910
\(470\) −9.23607 −0.426028
\(471\) −4.65248 −0.214375
\(472\) −9.23607 −0.425124
\(473\) 2.76393 0.127086
\(474\) 7.14590 0.328222
\(475\) −14.0689 −0.645525
\(476\) 5.52786 0.253369
\(477\) 22.1803 1.01557
\(478\) 3.14590 0.143890
\(479\) 16.4377 0.751057 0.375529 0.926811i \(-0.377461\pi\)
0.375529 + 0.926811i \(0.377461\pi\)
\(480\) −1.76393 −0.0805121
\(481\) −3.85410 −0.175732
\(482\) −10.4721 −0.476993
\(483\) −2.94427 −0.133969
\(484\) 2.09017 0.0950077
\(485\) −24.1803 −1.09797
\(486\) 13.9443 0.632525
\(487\) −25.3050 −1.14668 −0.573338 0.819319i \(-0.694352\pi\)
−0.573338 + 0.819319i \(0.694352\pi\)
\(488\) 8.38197 0.379434
\(489\) −7.70820 −0.348577
\(490\) 15.6180 0.705551
\(491\) 27.4508 1.23884 0.619420 0.785060i \(-0.287368\pi\)
0.619420 + 0.785060i \(0.287368\pi\)
\(492\) 4.56231 0.205685
\(493\) 28.2918 1.27420
\(494\) −17.2361 −0.775487
\(495\) −27.0344 −1.21511
\(496\) 9.61803 0.431862
\(497\) −18.4721 −0.828589
\(498\) −3.41641 −0.153093
\(499\) 23.7082 1.06132 0.530662 0.847583i \(-0.321943\pi\)
0.530662 + 0.847583i \(0.321943\pi\)
\(500\) 5.29180 0.236656
\(501\) 4.41641 0.197311
\(502\) −3.05573 −0.136384
\(503\) 7.90983 0.352682 0.176341 0.984329i \(-0.443574\pi\)
0.176341 + 0.984329i \(0.443574\pi\)
\(504\) −3.23607 −0.144146
\(505\) −35.5967 −1.58403
\(506\) 13.9443 0.619898
\(507\) 1.14590 0.0508911
\(508\) 0.472136 0.0209476
\(509\) −4.29180 −0.190231 −0.0951153 0.995466i \(-0.530322\pi\)
−0.0951153 + 0.995466i \(0.530322\pi\)
\(510\) −7.88854 −0.349311
\(511\) −5.05573 −0.223652
\(512\) 1.00000 0.0441942
\(513\) 15.5279 0.685572
\(514\) −18.9443 −0.835596
\(515\) 46.4377 2.04629
\(516\) −0.472136 −0.0207846
\(517\) −11.7082 −0.514926
\(518\) −1.23607 −0.0543097
\(519\) 5.23607 0.229838
\(520\) −11.0000 −0.482382
\(521\) 25.4164 1.11351 0.556757 0.830676i \(-0.312046\pi\)
0.556757 + 0.830676i \(0.312046\pi\)
\(522\) −16.5623 −0.724912
\(523\) 34.1803 1.49460 0.747301 0.664486i \(-0.231349\pi\)
0.747301 + 0.664486i \(0.231349\pi\)
\(524\) 8.65248 0.377985
\(525\) 2.40325 0.104887
\(526\) −8.76393 −0.382126
\(527\) 43.0132 1.87368
\(528\) −2.23607 −0.0973124
\(529\) −8.14590 −0.354169
\(530\) 24.1803 1.05033
\(531\) 24.1803 1.04934
\(532\) −5.52786 −0.239663
\(533\) 28.4508 1.23234
\(534\) −6.47214 −0.280077
\(535\) −23.7639 −1.02740
\(536\) −10.0902 −0.435829
\(537\) 11.5279 0.497464
\(538\) 4.00000 0.172452
\(539\) 19.7984 0.852776
\(540\) 9.90983 0.426451
\(541\) 5.32624 0.228993 0.114496 0.993424i \(-0.463475\pi\)
0.114496 + 0.993424i \(0.463475\pi\)
\(542\) −4.94427 −0.212375
\(543\) 3.41641 0.146612
\(544\) 4.47214 0.191741
\(545\) 42.6525 1.82703
\(546\) 2.94427 0.126003
\(547\) 42.0689 1.79874 0.899368 0.437193i \(-0.144027\pi\)
0.899368 + 0.437193i \(0.144027\pi\)
\(548\) 19.3262 0.825576
\(549\) −21.9443 −0.936559
\(550\) −11.3820 −0.485329
\(551\) −28.2918 −1.20527
\(552\) −2.38197 −0.101383
\(553\) 14.2918 0.607749
\(554\) −7.79837 −0.331321
\(555\) 1.76393 0.0748747
\(556\) −1.85410 −0.0786314
\(557\) −0.562306 −0.0238257 −0.0119128 0.999929i \(-0.503792\pi\)
−0.0119128 + 0.999929i \(0.503792\pi\)
\(558\) −25.1803 −1.06597
\(559\) −2.94427 −0.124529
\(560\) −3.52786 −0.149079
\(561\) −10.0000 −0.422200
\(562\) −5.88854 −0.248393
\(563\) −27.8885 −1.17536 −0.587681 0.809093i \(-0.699959\pi\)
−0.587681 + 0.809093i \(0.699959\pi\)
\(564\) 2.00000 0.0842152
\(565\) −31.2361 −1.31411
\(566\) 11.2361 0.472287
\(567\) 7.05573 0.296313
\(568\) −14.9443 −0.627048
\(569\) −21.8885 −0.917615 −0.458808 0.888536i \(-0.651723\pi\)
−0.458808 + 0.888536i \(0.651723\pi\)
\(570\) 7.88854 0.330415
\(571\) 9.56231 0.400170 0.200085 0.979779i \(-0.435878\pi\)
0.200085 + 0.979779i \(0.435878\pi\)
\(572\) −13.9443 −0.583039
\(573\) 2.52786 0.105603
\(574\) 9.12461 0.380854
\(575\) −12.1246 −0.505631
\(576\) −2.61803 −0.109085
\(577\) −20.6525 −0.859774 −0.429887 0.902883i \(-0.641447\pi\)
−0.429887 + 0.902883i \(0.641447\pi\)
\(578\) 3.00000 0.124784
\(579\) 2.47214 0.102738
\(580\) −18.0557 −0.749723
\(581\) −6.83282 −0.283473
\(582\) 5.23607 0.217042
\(583\) 30.6525 1.26950
\(584\) −4.09017 −0.169252
\(585\) 28.7984 1.19067
\(586\) 18.6525 0.770527
\(587\) −30.9443 −1.27721 −0.638603 0.769536i \(-0.720488\pi\)
−0.638603 + 0.769536i \(0.720488\pi\)
\(588\) −3.38197 −0.139470
\(589\) −43.0132 −1.77233
\(590\) 26.3607 1.08525
\(591\) −10.1803 −0.418763
\(592\) −1.00000 −0.0410997
\(593\) −2.56231 −0.105221 −0.0526106 0.998615i \(-0.516754\pi\)
−0.0526106 + 0.998615i \(0.516754\pi\)
\(594\) 12.5623 0.515438
\(595\) −15.7771 −0.646798
\(596\) −6.18034 −0.253157
\(597\) 12.9443 0.529774
\(598\) −14.8541 −0.607429
\(599\) 38.3607 1.56737 0.783687 0.621155i \(-0.213336\pi\)
0.783687 + 0.621155i \(0.213336\pi\)
\(600\) 1.94427 0.0793746
\(601\) 35.6869 1.45570 0.727850 0.685737i \(-0.240520\pi\)
0.727850 + 0.685737i \(0.240520\pi\)
\(602\) −0.944272 −0.0384856
\(603\) 26.4164 1.07576
\(604\) 17.7082 0.720537
\(605\) −5.96556 −0.242534
\(606\) 7.70820 0.313124
\(607\) 5.96556 0.242135 0.121067 0.992644i \(-0.461368\pi\)
0.121067 + 0.992644i \(0.461368\pi\)
\(608\) −4.47214 −0.181369
\(609\) 4.83282 0.195836
\(610\) −23.9230 −0.968613
\(611\) 12.4721 0.504569
\(612\) −11.7082 −0.473276
\(613\) −36.1803 −1.46131 −0.730655 0.682747i \(-0.760785\pi\)
−0.730655 + 0.682747i \(0.760785\pi\)
\(614\) −6.14590 −0.248028
\(615\) −13.0213 −0.525069
\(616\) −4.47214 −0.180187
\(617\) −11.0902 −0.446473 −0.223237 0.974764i \(-0.571662\pi\)
−0.223237 + 0.974764i \(0.571662\pi\)
\(618\) −10.0557 −0.404501
\(619\) 18.2705 0.734354 0.367177 0.930151i \(-0.380324\pi\)
0.367177 + 0.930151i \(0.380324\pi\)
\(620\) −27.4508 −1.10245
\(621\) 13.3820 0.537000
\(622\) 2.03444 0.0815737
\(623\) −12.9443 −0.518601
\(624\) 2.38197 0.0953550
\(625\) −30.8328 −1.23331
\(626\) 5.81966 0.232600
\(627\) 10.0000 0.399362
\(628\) −7.52786 −0.300394
\(629\) −4.47214 −0.178316
\(630\) 9.23607 0.367974
\(631\) 26.3951 1.05077 0.525387 0.850864i \(-0.323921\pi\)
0.525387 + 0.850864i \(0.323921\pi\)
\(632\) 11.5623 0.459924
\(633\) −13.7639 −0.547067
\(634\) 3.05573 0.121358
\(635\) −1.34752 −0.0534749
\(636\) −5.23607 −0.207624
\(637\) −21.0902 −0.835623
\(638\) −22.8885 −0.906166
\(639\) 39.1246 1.54775
\(640\) −2.85410 −0.112818
\(641\) −22.5066 −0.888956 −0.444478 0.895790i \(-0.646611\pi\)
−0.444478 + 0.895790i \(0.646611\pi\)
\(642\) 5.14590 0.203092
\(643\) −33.2361 −1.31070 −0.655351 0.755324i \(-0.727479\pi\)
−0.655351 + 0.755324i \(0.727479\pi\)
\(644\) −4.76393 −0.187725
\(645\) 1.34752 0.0530587
\(646\) −20.0000 −0.786889
\(647\) 18.9098 0.743422 0.371711 0.928348i \(-0.378771\pi\)
0.371711 + 0.928348i \(0.378771\pi\)
\(648\) 5.70820 0.224239
\(649\) 33.4164 1.31171
\(650\) 12.1246 0.475566
\(651\) 7.34752 0.287972
\(652\) −12.4721 −0.488447
\(653\) −4.72949 −0.185079 −0.0925396 0.995709i \(-0.529498\pi\)
−0.0925396 + 0.995709i \(0.529498\pi\)
\(654\) −9.23607 −0.361159
\(655\) −24.6950 −0.964915
\(656\) 7.38197 0.288217
\(657\) 10.7082 0.417767
\(658\) 4.00000 0.155936
\(659\) −15.4508 −0.601880 −0.300940 0.953643i \(-0.597300\pi\)
−0.300940 + 0.953643i \(0.597300\pi\)
\(660\) 6.38197 0.248418
\(661\) 1.67376 0.0651018 0.0325509 0.999470i \(-0.489637\pi\)
0.0325509 + 0.999470i \(0.489637\pi\)
\(662\) 28.0000 1.08825
\(663\) 10.6525 0.413708
\(664\) −5.52786 −0.214523
\(665\) 15.7771 0.611809
\(666\) 2.61803 0.101447
\(667\) −24.3820 −0.944073
\(668\) 7.14590 0.276483
\(669\) −5.05573 −0.195466
\(670\) 28.7984 1.11258
\(671\) −30.3262 −1.17073
\(672\) 0.763932 0.0294693
\(673\) 17.8541 0.688225 0.344113 0.938928i \(-0.388180\pi\)
0.344113 + 0.938928i \(0.388180\pi\)
\(674\) −17.0344 −0.656142
\(675\) −10.9230 −0.420426
\(676\) 1.85410 0.0713116
\(677\) 3.34752 0.128656 0.0643279 0.997929i \(-0.479510\pi\)
0.0643279 + 0.997929i \(0.479510\pi\)
\(678\) 6.76393 0.259767
\(679\) 10.4721 0.401884
\(680\) −12.7639 −0.489474
\(681\) −10.9443 −0.419385
\(682\) −34.7984 −1.33250
\(683\) −35.4164 −1.35517 −0.677586 0.735444i \(-0.736974\pi\)
−0.677586 + 0.735444i \(0.736974\pi\)
\(684\) 11.7082 0.447674
\(685\) −55.1591 −2.10752
\(686\) −15.4164 −0.588601
\(687\) 10.5836 0.403789
\(688\) −0.763932 −0.0291246
\(689\) −32.6525 −1.24396
\(690\) 6.79837 0.258810
\(691\) −35.7771 −1.36102 −0.680512 0.732737i \(-0.738243\pi\)
−0.680512 + 0.732737i \(0.738243\pi\)
\(692\) 8.47214 0.322062
\(693\) 11.7082 0.444758
\(694\) 12.7639 0.484512
\(695\) 5.29180 0.200729
\(696\) 3.90983 0.148202
\(697\) 33.0132 1.25046
\(698\) 12.1803 0.461033
\(699\) 8.38197 0.317035
\(700\) 3.88854 0.146973
\(701\) 3.97871 0.150274 0.0751370 0.997173i \(-0.476061\pi\)
0.0751370 + 0.997173i \(0.476061\pi\)
\(702\) −13.3820 −0.505070
\(703\) 4.47214 0.168670
\(704\) −3.61803 −0.136360
\(705\) −5.70820 −0.214983
\(706\) 29.7082 1.11808
\(707\) 15.4164 0.579794
\(708\) −5.70820 −0.214527
\(709\) 43.2148 1.62297 0.811483 0.584377i \(-0.198661\pi\)
0.811483 + 0.584377i \(0.198661\pi\)
\(710\) 42.6525 1.60072
\(711\) −30.2705 −1.13523
\(712\) −10.4721 −0.392460
\(713\) −37.0689 −1.38824
\(714\) 3.41641 0.127856
\(715\) 39.7984 1.48837
\(716\) 18.6525 0.697076
\(717\) 1.94427 0.0726102
\(718\) 4.47214 0.166899
\(719\) 0.583592 0.0217643 0.0108822 0.999941i \(-0.496536\pi\)
0.0108822 + 0.999941i \(0.496536\pi\)
\(720\) 7.47214 0.278470
\(721\) −20.1115 −0.748990
\(722\) 1.00000 0.0372161
\(723\) −6.47214 −0.240701
\(724\) 5.52786 0.205441
\(725\) 19.9017 0.739131
\(726\) 1.29180 0.0479430
\(727\) −23.1459 −0.858434 −0.429217 0.903201i \(-0.641210\pi\)
−0.429217 + 0.903201i \(0.641210\pi\)
\(728\) 4.76393 0.176563
\(729\) −8.50658 −0.315058
\(730\) 11.6738 0.432065
\(731\) −3.41641 −0.126360
\(732\) 5.18034 0.191471
\(733\) 36.4721 1.34713 0.673565 0.739128i \(-0.264762\pi\)
0.673565 + 0.739128i \(0.264762\pi\)
\(734\) −27.1246 −1.00119
\(735\) 9.65248 0.356037
\(736\) −3.85410 −0.142064
\(737\) 36.5066 1.34474
\(738\) −19.3262 −0.711409
\(739\) −22.0902 −0.812600 −0.406300 0.913740i \(-0.633181\pi\)
−0.406300 + 0.913740i \(0.633181\pi\)
\(740\) 2.85410 0.104919
\(741\) −10.6525 −0.391328
\(742\) −10.4721 −0.384444
\(743\) 10.0689 0.369392 0.184696 0.982796i \(-0.440870\pi\)
0.184696 + 0.982796i \(0.440870\pi\)
\(744\) 5.94427 0.217928
\(745\) 17.6393 0.646255
\(746\) 14.2918 0.523260
\(747\) 14.4721 0.529508
\(748\) −16.1803 −0.591612
\(749\) 10.2918 0.376054
\(750\) 3.27051 0.119422
\(751\) −18.9443 −0.691286 −0.345643 0.938366i \(-0.612339\pi\)
−0.345643 + 0.938366i \(0.612339\pi\)
\(752\) 3.23607 0.118007
\(753\) −1.88854 −0.0688224
\(754\) 24.3820 0.887939
\(755\) −50.5410 −1.83938
\(756\) −4.29180 −0.156091
\(757\) −10.8541 −0.394499 −0.197250 0.980353i \(-0.563201\pi\)
−0.197250 + 0.980353i \(0.563201\pi\)
\(758\) 16.9098 0.614193
\(759\) 8.61803 0.312815
\(760\) 12.7639 0.462996
\(761\) −25.8541 −0.937210 −0.468605 0.883408i \(-0.655243\pi\)
−0.468605 + 0.883408i \(0.655243\pi\)
\(762\) 0.291796 0.0105707
\(763\) −18.4721 −0.668736
\(764\) 4.09017 0.147977
\(765\) 33.4164 1.20817
\(766\) −17.8885 −0.646339
\(767\) −35.5967 −1.28532
\(768\) 0.618034 0.0223014
\(769\) −11.8885 −0.428712 −0.214356 0.976756i \(-0.568765\pi\)
−0.214356 + 0.976756i \(0.568765\pi\)
\(770\) 12.7639 0.459980
\(771\) −11.7082 −0.421661
\(772\) 4.00000 0.143963
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 2.00000 0.0718885
\(775\) 30.2574 1.08688
\(776\) 8.47214 0.304132
\(777\) −0.763932 −0.0274059
\(778\) −0.145898 −0.00523070
\(779\) −33.0132 −1.18282
\(780\) −6.79837 −0.243421
\(781\) 54.0689 1.93474
\(782\) −17.2361 −0.616361
\(783\) −21.9656 −0.784985
\(784\) −5.47214 −0.195433
\(785\) 21.4853 0.766843
\(786\) 5.34752 0.190740
\(787\) −34.4721 −1.22880 −0.614399 0.788995i \(-0.710602\pi\)
−0.614399 + 0.788995i \(0.710602\pi\)
\(788\) −16.4721 −0.586796
\(789\) −5.41641 −0.192829
\(790\) −33.0000 −1.17409
\(791\) 13.5279 0.480995
\(792\) 9.47214 0.336578
\(793\) 32.3050 1.14718
\(794\) −10.6525 −0.378042
\(795\) 14.9443 0.530019
\(796\) 20.9443 0.742350
\(797\) 12.7295 0.450902 0.225451 0.974255i \(-0.427614\pi\)
0.225451 + 0.974255i \(0.427614\pi\)
\(798\) −3.41641 −0.120940
\(799\) 14.4721 0.511987
\(800\) 3.14590 0.111224
\(801\) 27.4164 0.968711
\(802\) −9.23607 −0.326137
\(803\) 14.7984 0.522223
\(804\) −6.23607 −0.219929
\(805\) 13.5967 0.479222
\(806\) 37.0689 1.30570
\(807\) 2.47214 0.0870233
\(808\) 12.4721 0.438768
\(809\) 27.1246 0.953651 0.476825 0.878998i \(-0.341787\pi\)
0.476825 + 0.878998i \(0.341787\pi\)
\(810\) −16.2918 −0.572435
\(811\) 53.1033 1.86471 0.932355 0.361544i \(-0.117750\pi\)
0.932355 + 0.361544i \(0.117750\pi\)
\(812\) 7.81966 0.274416
\(813\) −3.05573 −0.107169
\(814\) 3.61803 0.126812
\(815\) 35.5967 1.24690
\(816\) 2.76393 0.0967570
\(817\) 3.41641 0.119525
\(818\) −3.81966 −0.133551
\(819\) −12.4721 −0.435812
\(820\) −21.0689 −0.735757
\(821\) 21.4164 0.747438 0.373719 0.927542i \(-0.378082\pi\)
0.373719 + 0.927542i \(0.378082\pi\)
\(822\) 11.9443 0.416604
\(823\) −33.8885 −1.18128 −0.590640 0.806935i \(-0.701125\pi\)
−0.590640 + 0.806935i \(0.701125\pi\)
\(824\) −16.2705 −0.566810
\(825\) −7.03444 −0.244908
\(826\) −11.4164 −0.397228
\(827\) 56.0689 1.94971 0.974853 0.222849i \(-0.0715356\pi\)
0.974853 + 0.222849i \(0.0715356\pi\)
\(828\) 10.0902 0.350658
\(829\) 31.2016 1.08368 0.541839 0.840483i \(-0.317728\pi\)
0.541839 + 0.840483i \(0.317728\pi\)
\(830\) 15.7771 0.547631
\(831\) −4.81966 −0.167192
\(832\) 3.85410 0.133617
\(833\) −24.4721 −0.847909
\(834\) −1.14590 −0.0396792
\(835\) −20.3951 −0.705802
\(836\) 16.1803 0.559609
\(837\) −33.3951 −1.15430
\(838\) 9.56231 0.330324
\(839\) −5.34752 −0.184617 −0.0923085 0.995730i \(-0.529425\pi\)
−0.0923085 + 0.995730i \(0.529425\pi\)
\(840\) −2.18034 −0.0752289
\(841\) 11.0213 0.380044
\(842\) −1.96556 −0.0677376
\(843\) −3.63932 −0.125345
\(844\) −22.2705 −0.766583
\(845\) −5.29180 −0.182043
\(846\) −8.47214 −0.291278
\(847\) 2.58359 0.0887733
\(848\) −8.47214 −0.290934
\(849\) 6.94427 0.238327
\(850\) 14.0689 0.482559
\(851\) 3.85410 0.132117
\(852\) −9.23607 −0.316422
\(853\) −12.7426 −0.436300 −0.218150 0.975915i \(-0.570002\pi\)
−0.218150 + 0.975915i \(0.570002\pi\)
\(854\) 10.3607 0.354535
\(855\) −33.4164 −1.14282
\(856\) 8.32624 0.284585
\(857\) −26.9443 −0.920399 −0.460199 0.887816i \(-0.652222\pi\)
−0.460199 + 0.887816i \(0.652222\pi\)
\(858\) −8.61803 −0.294215
\(859\) 26.5836 0.907020 0.453510 0.891251i \(-0.350172\pi\)
0.453510 + 0.891251i \(0.350172\pi\)
\(860\) 2.18034 0.0743490
\(861\) 5.63932 0.192188
\(862\) −32.3607 −1.10221
\(863\) −7.41641 −0.252457 −0.126229 0.992001i \(-0.540287\pi\)
−0.126229 + 0.992001i \(0.540287\pi\)
\(864\) −3.47214 −0.118124
\(865\) −24.1803 −0.822156
\(866\) −36.3262 −1.23442
\(867\) 1.85410 0.0629686
\(868\) 11.8885 0.403523
\(869\) −41.8328 −1.41908
\(870\) −11.1591 −0.378327
\(871\) −38.8885 −1.31769
\(872\) −14.9443 −0.506077
\(873\) −22.1803 −0.750691
\(874\) 17.2361 0.583019
\(875\) 6.54102 0.221127
\(876\) −2.52786 −0.0854086
\(877\) 36.8328 1.24376 0.621878 0.783114i \(-0.286370\pi\)
0.621878 + 0.783114i \(0.286370\pi\)
\(878\) −16.7984 −0.566918
\(879\) 11.5279 0.388825
\(880\) 10.3262 0.348097
\(881\) −22.7426 −0.766219 −0.383110 0.923703i \(-0.625147\pi\)
−0.383110 + 0.923703i \(0.625147\pi\)
\(882\) 14.3262 0.482390
\(883\) 29.3050 0.986190 0.493095 0.869975i \(-0.335865\pi\)
0.493095 + 0.869975i \(0.335865\pi\)
\(884\) 17.2361 0.579712
\(885\) 16.2918 0.547643
\(886\) −18.2705 −0.613810
\(887\) −7.88854 −0.264871 −0.132436 0.991192i \(-0.542280\pi\)
−0.132436 + 0.991192i \(0.542280\pi\)
\(888\) −0.618034 −0.0207399
\(889\) 0.583592 0.0195731
\(890\) 29.8885 1.00187
\(891\) −20.6525 −0.691884
\(892\) −8.18034 −0.273898
\(893\) −14.4721 −0.484292
\(894\) −3.81966 −0.127749
\(895\) −53.2361 −1.77949
\(896\) 1.23607 0.0412941
\(897\) −9.18034 −0.306523
\(898\) −19.5279 −0.651653
\(899\) 60.8460 2.02933
\(900\) −8.23607 −0.274536
\(901\) −37.8885 −1.26225
\(902\) −26.7082 −0.889286
\(903\) −0.583592 −0.0194207
\(904\) 10.9443 0.364001
\(905\) −15.7771 −0.524448
\(906\) 10.9443 0.363599
\(907\) −20.1115 −0.667790 −0.333895 0.942610i \(-0.608363\pi\)
−0.333895 + 0.942610i \(0.608363\pi\)
\(908\) −17.7082 −0.587667
\(909\) −32.6525 −1.08301
\(910\) −13.5967 −0.450728
\(911\) −21.8885 −0.725200 −0.362600 0.931945i \(-0.618111\pi\)
−0.362600 + 0.931945i \(0.618111\pi\)
\(912\) −2.76393 −0.0915229
\(913\) 20.0000 0.661903
\(914\) −29.2361 −0.967043
\(915\) −14.7852 −0.488784
\(916\) 17.1246 0.565813
\(917\) 10.6950 0.353182
\(918\) −15.5279 −0.512496
\(919\) 29.8885 0.985932 0.492966 0.870049i \(-0.335913\pi\)
0.492966 + 0.870049i \(0.335913\pi\)
\(920\) 11.0000 0.362659
\(921\) −3.79837 −0.125161
\(922\) −21.0557 −0.693433
\(923\) −57.5967 −1.89582
\(924\) −2.76393 −0.0909267
\(925\) −3.14590 −0.103436
\(926\) 15.5623 0.511409
\(927\) 42.5967 1.39906
\(928\) 6.32624 0.207669
\(929\) 28.4508 0.933442 0.466721 0.884405i \(-0.345435\pi\)
0.466721 + 0.884405i \(0.345435\pi\)
\(930\) −16.9656 −0.556323
\(931\) 24.4721 0.802042
\(932\) 13.5623 0.444248
\(933\) 1.25735 0.0411639
\(934\) 20.3607 0.666222
\(935\) 46.1803 1.51026
\(936\) −10.0902 −0.329808
\(937\) 57.0476 1.86366 0.931832 0.362890i \(-0.118210\pi\)
0.931832 + 0.362890i \(0.118210\pi\)
\(938\) −12.4721 −0.407230
\(939\) 3.59675 0.117375
\(940\) −9.23607 −0.301247
\(941\) −3.81966 −0.124517 −0.0622587 0.998060i \(-0.519830\pi\)
−0.0622587 + 0.998060i \(0.519830\pi\)
\(942\) −4.65248 −0.151586
\(943\) −28.4508 −0.926487
\(944\) −9.23607 −0.300608
\(945\) 12.2492 0.398467
\(946\) 2.76393 0.0898632
\(947\) −34.8328 −1.13191 −0.565957 0.824435i \(-0.691493\pi\)
−0.565957 + 0.824435i \(0.691493\pi\)
\(948\) 7.14590 0.232088
\(949\) −15.7639 −0.511719
\(950\) −14.0689 −0.456455
\(951\) 1.88854 0.0612402
\(952\) 5.52786 0.179159
\(953\) −44.4508 −1.43990 −0.719952 0.694024i \(-0.755836\pi\)
−0.719952 + 0.694024i \(0.755836\pi\)
\(954\) 22.1803 0.718115
\(955\) −11.6738 −0.377754
\(956\) 3.14590 0.101746
\(957\) −14.1459 −0.457272
\(958\) 16.4377 0.531078
\(959\) 23.8885 0.771401
\(960\) −1.76393 −0.0569307
\(961\) 61.5066 1.98408
\(962\) −3.85410 −0.124261
\(963\) −21.7984 −0.702443
\(964\) −10.4721 −0.337285
\(965\) −11.4164 −0.367507
\(966\) −2.94427 −0.0947304
\(967\) 11.7295 0.377195 0.188597 0.982054i \(-0.439606\pi\)
0.188597 + 0.982054i \(0.439606\pi\)
\(968\) 2.09017 0.0671806
\(969\) −12.3607 −0.397082
\(970\) −24.1803 −0.776384
\(971\) 26.6738 0.856002 0.428001 0.903778i \(-0.359218\pi\)
0.428001 + 0.903778i \(0.359218\pi\)
\(972\) 13.9443 0.447263
\(973\) −2.29180 −0.0734716
\(974\) −25.3050 −0.810823
\(975\) 7.49342 0.239982
\(976\) 8.38197 0.268300
\(977\) 52.4721 1.67873 0.839366 0.543566i \(-0.182926\pi\)
0.839366 + 0.543566i \(0.182926\pi\)
\(978\) −7.70820 −0.246481
\(979\) 37.8885 1.21092
\(980\) 15.6180 0.498900
\(981\) 39.1246 1.24915
\(982\) 27.4508 0.875992
\(983\) 39.7771 1.26869 0.634346 0.773049i \(-0.281269\pi\)
0.634346 + 0.773049i \(0.281269\pi\)
\(984\) 4.56231 0.145441
\(985\) 47.0132 1.49796
\(986\) 28.2918 0.900994
\(987\) 2.47214 0.0786890
\(988\) −17.2361 −0.548352
\(989\) 2.94427 0.0936224
\(990\) −27.0344 −0.859211
\(991\) −54.1033 −1.71865 −0.859324 0.511431i \(-0.829116\pi\)
−0.859324 + 0.511431i \(0.829116\pi\)
\(992\) 9.61803 0.305373
\(993\) 17.3050 0.549156
\(994\) −18.4721 −0.585901
\(995\) −59.7771 −1.89506
\(996\) −3.41641 −0.108253
\(997\) 53.7771 1.70314 0.851569 0.524243i \(-0.175652\pi\)
0.851569 + 0.524243i \(0.175652\pi\)
\(998\) 23.7082 0.750470
\(999\) 3.47214 0.109854
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 74.2.a.b.1.2 2
3.2 odd 2 666.2.a.i.1.2 2
4.3 odd 2 592.2.a.g.1.1 2
5.2 odd 4 1850.2.b.j.149.3 4
5.3 odd 4 1850.2.b.j.149.2 4
5.4 even 2 1850.2.a.t.1.1 2
7.6 odd 2 3626.2.a.s.1.1 2
8.3 odd 2 2368.2.a.u.1.2 2
8.5 even 2 2368.2.a.y.1.1 2
11.10 odd 2 8954.2.a.j.1.2 2
12.11 even 2 5328.2.a.bc.1.2 2
37.36 even 2 2738.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.2 2 1.1 even 1 trivial
592.2.a.g.1.1 2 4.3 odd 2
666.2.a.i.1.2 2 3.2 odd 2
1850.2.a.t.1.1 2 5.4 even 2
1850.2.b.j.149.2 4 5.3 odd 4
1850.2.b.j.149.3 4 5.2 odd 4
2368.2.a.u.1.2 2 8.3 odd 2
2368.2.a.y.1.1 2 8.5 even 2
2738.2.a.g.1.2 2 37.36 even 2
3626.2.a.s.1.1 2 7.6 odd 2
5328.2.a.bc.1.2 2 12.11 even 2
8954.2.a.j.1.2 2 11.10 odd 2