Properties

Label 1850.2.a.t.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} -1.23607 q^{7} -1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} -1.23607 q^{7} -1.00000 q^{8} -2.61803 q^{9} -3.61803 q^{11} -0.618034 q^{12} -3.85410 q^{13} +1.23607 q^{14} +1.00000 q^{16} -4.47214 q^{17} +2.61803 q^{18} -4.47214 q^{19} +0.763932 q^{21} +3.61803 q^{22} +3.85410 q^{23} +0.618034 q^{24} +3.85410 q^{26} +3.47214 q^{27} -1.23607 q^{28} +6.32624 q^{29} +9.61803 q^{31} -1.00000 q^{32} +2.23607 q^{33} +4.47214 q^{34} -2.61803 q^{36} +1.00000 q^{37} +4.47214 q^{38} +2.38197 q^{39} +7.38197 q^{41} -0.763932 q^{42} +0.763932 q^{43} -3.61803 q^{44} -3.85410 q^{46} -3.23607 q^{47} -0.618034 q^{48} -5.47214 q^{49} +2.76393 q^{51} -3.85410 q^{52} +8.47214 q^{53} -3.47214 q^{54} +1.23607 q^{56} +2.76393 q^{57} -6.32624 q^{58} -9.23607 q^{59} +8.38197 q^{61} -9.61803 q^{62} +3.23607 q^{63} +1.00000 q^{64} -2.23607 q^{66} +10.0902 q^{67} -4.47214 q^{68} -2.38197 q^{69} -14.9443 q^{71} +2.61803 q^{72} +4.09017 q^{73} -1.00000 q^{74} -4.47214 q^{76} +4.47214 q^{77} -2.38197 q^{78} +11.5623 q^{79} +5.70820 q^{81} -7.38197 q^{82} +5.52786 q^{83} +0.763932 q^{84} -0.763932 q^{86} -3.90983 q^{87} +3.61803 q^{88} -10.4721 q^{89} +4.76393 q^{91} +3.85410 q^{92} -5.94427 q^{93} +3.23607 q^{94} +0.618034 q^{96} -8.47214 q^{97} +5.47214 q^{98} +9.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} - 5 q^{11} + q^{12} - q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{18} + 6 q^{21} + 5 q^{22} + q^{23} - q^{24} + q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} + 17 q^{31} - 2 q^{32} - 3 q^{36} + 2 q^{37} + 7 q^{39} + 17 q^{41} - 6 q^{42} + 6 q^{43} - 5 q^{44} - q^{46} - 2 q^{47} + q^{48} - 2 q^{49} + 10 q^{51} - q^{52} + 8 q^{53} + 2 q^{54} - 2 q^{56} + 10 q^{57} + 3 q^{58} - 14 q^{59} + 19 q^{61} - 17 q^{62} + 2 q^{63} + 2 q^{64} + 9 q^{67} - 7 q^{69} - 12 q^{71} + 3 q^{72} - 3 q^{73} - 2 q^{74} - 7 q^{78} + 3 q^{79} - 2 q^{81} - 17 q^{82} + 20 q^{83} + 6 q^{84} - 6 q^{86} - 19 q^{87} + 5 q^{88} - 12 q^{89} + 14 q^{91} + q^{92} + 6 q^{93} + 2 q^{94} - q^{96} - 8 q^{97} + 2 q^{98} + 10 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.618034 −0.356822 −0.178411 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) −1.23607 −0.467190 −0.233595 0.972334i \(-0.575049\pi\)
−0.233595 + 0.972334i \(0.575049\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) −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 \(-0.679488\pi\)
−0.534468 + 0.845189i \(0.679488\pi\)
\(14\) 1.23607 0.330353
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 2.61803 0.617077
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) 0.763932 0.166704
\(22\) 3.61803 0.771367
\(23\) 3.85410 0.803636 0.401818 0.915720i \(-0.368378\pi\)
0.401818 + 0.915720i \(0.368378\pi\)
\(24\) 0.618034 0.126156
\(25\) 0 0
\(26\) 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 \(0.300161\pi\)
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) −2.61803 −0.436339
\(37\) 1.00000 0.164399
\(38\) 4.47214 0.725476
\(39\) 2.38197 0.381420
\(40\) 0 0
\(41\) 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 \(-0.481448\pi\)
0.0582493 + 0.998302i \(0.481448\pi\)
\(44\) −3.61803 −0.545439
\(45\) 0 0
\(46\) −3.85410 −0.568256
\(47\) −3.23607 −0.472029 −0.236015 0.971750i \(-0.575841\pi\)
−0.236015 + 0.971750i \(0.575841\pi\)
\(48\) −0.618034 −0.0892055
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) 2.76393 0.387028
\(52\) −3.85410 −0.534468
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) 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 \(0.705317\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) −2.23607 −0.275241
\(67\) 10.0902 1.23271 0.616355 0.787468i \(-0.288609\pi\)
0.616355 + 0.787468i \(0.288609\pi\)
\(68\) −4.47214 −0.542326
\(69\) −2.38197 −0.286755
\(70\) 0 0
\(71\) −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 \(-0.423063\pi\)
0.239359 + 0.970931i \(0.423063\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −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 \(0.274589\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) −7.38197 −0.815202
\(83\) 5.52786 0.606762 0.303381 0.952869i \(-0.401885\pi\)
0.303381 + 0.952869i \(0.401885\pi\)
\(84\) 0.763932 0.0833518
\(85\) 0 0
\(86\) −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 \(0.687290\pi\)
\(90\) 0 0
\(91\) 4.76393 0.499396
\(92\) 3.85410 0.401818
\(93\) −5.94427 −0.616392
\(94\) 3.23607 0.333775
\(95\) 0 0
\(96\) 0.618034 0.0630778
\(97\) −8.47214 −0.860215 −0.430108 0.902778i \(-0.641524\pi\)
−0.430108 + 0.902778i \(0.641524\pi\)
\(98\) 5.47214 0.552769
\(99\) 9.47214 0.951985
\(100\) 0 0
\(101\) 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 \(-0.203987\pi\)
0.801590 + 0.597873i \(0.203987\pi\)
\(104\) 3.85410 0.377926
\(105\) 0 0
\(106\) −8.47214 −0.822887
\(107\) −8.32624 −0.804928 −0.402464 0.915436i \(-0.631846\pi\)
−0.402464 + 0.915436i \(0.631846\pi\)
\(108\) 3.47214 0.334106
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) −0.618034 −0.0586612
\(112\) −1.23607 −0.116797
\(113\) −10.9443 −1.02955 −0.514775 0.857325i \(-0.672125\pi\)
−0.514775 + 0.857325i \(0.672125\pi\)
\(114\) −2.76393 −0.258866
\(115\) 0 0
\(116\) 6.32624 0.587376
\(117\) 10.0902 0.932837
\(118\) 9.23607 0.850249
\(119\) 5.52786 0.506738
\(120\) 0 0
\(121\) 2.09017 0.190015
\(122\) −8.38197 −0.758868
\(123\) −4.56231 −0.411369
\(124\) 9.61803 0.863725
\(125\) 0 0
\(126\) −3.23607 −0.288292
\(127\) −0.472136 −0.0418953 −0.0209476 0.999781i \(-0.506668\pi\)
−0.0209476 + 0.999781i \(0.506668\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.472136 −0.0415693
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 4.47214 0.383482
\(137\) −19.3262 −1.65115 −0.825576 0.564291i \(-0.809150\pi\)
−0.825576 + 0.564291i \(0.809150\pi\)
\(138\) 2.38197 0.202766
\(139\) −1.85410 −0.157263 −0.0786314 0.996904i \(-0.525055\pi\)
−0.0786314 + 0.996904i \(0.525055\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 14.9443 1.25410
\(143\) 13.9443 1.16608
\(144\) −2.61803 −0.218169
\(145\) 0 0
\(146\) −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 \(0.581469\pi\)
\(150\) 0 0
\(151\) 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
\(155\) 0 0
\(156\) 2.38197 0.190710
\(157\) 7.52786 0.600789 0.300394 0.953815i \(-0.402882\pi\)
0.300394 + 0.953815i \(0.402882\pi\)
\(158\) −11.5623 −0.919848
\(159\) −5.23607 −0.415247
\(160\) 0 0
\(161\) −4.76393 −0.375450
\(162\) −5.70820 −0.448479
\(163\) 12.4721 0.976893 0.488447 0.872594i \(-0.337564\pi\)
0.488447 + 0.872594i \(0.337564\pi\)
\(164\) 7.38197 0.576435
\(165\) 0 0
\(166\) −5.52786 −0.429045
\(167\) −7.14590 −0.552966 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(168\) −0.763932 −0.0589386
\(169\) 1.85410 0.142623
\(170\) 0 0
\(171\) 11.7082 0.895349
\(172\) 0.763932 0.0582493
\(173\) −8.47214 −0.644125 −0.322062 0.946718i \(-0.604376\pi\)
−0.322062 + 0.946718i \(0.604376\pi\)
\(174\) 3.90983 0.296403
\(175\) 0 0
\(176\) −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 \(0.254484\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 5.94427 0.435855
\(187\) 16.1803 1.18322
\(188\) −3.23607 −0.236015
\(189\) −4.29180 −0.312182
\(190\) 0 0
\(191\) 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 \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 8.47214 0.608264
\(195\) 0 0
\(196\) −5.47214 −0.390867
\(197\) 16.4721 1.17359 0.586796 0.809735i \(-0.300389\pi\)
0.586796 + 0.809735i \(0.300389\pi\)
\(198\) −9.47214 −0.673155
\(199\) 20.9443 1.48470 0.742350 0.670012i \(-0.233711\pi\)
0.742350 + 0.670012i \(0.233711\pi\)
\(200\) 0 0
\(201\) −6.23607 −0.439858
\(202\) −12.4721 −0.877536
\(203\) −7.81966 −0.548833
\(204\) 2.76393 0.193514
\(205\) 0 0
\(206\) −16.2705 −1.13362
\(207\) −10.0902 −0.701315
\(208\) −3.85410 −0.267234
\(209\) 16.1803 1.11922
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) −3.47214 −0.236249
\(217\) −11.8885 −0.807047
\(218\) 14.9443 1.01215
\(219\) −2.52786 −0.170817
\(220\) 0 0
\(221\) 17.2361 1.15942
\(222\) 0.618034 0.0414797
\(223\) 8.18034 0.547796 0.273898 0.961759i \(-0.411687\pi\)
0.273898 + 0.961759i \(0.411687\pi\)
\(224\) 1.23607 0.0825883
\(225\) 0 0
\(226\) 10.9443 0.728002
\(227\) 17.7082 1.17533 0.587667 0.809103i \(-0.300046\pi\)
0.587667 + 0.809103i \(0.300046\pi\)
\(228\) 2.76393 0.183046
\(229\) 17.1246 1.13163 0.565813 0.824534i \(-0.308562\pi\)
0.565813 + 0.824534i \(0.308562\pi\)
\(230\) 0 0
\(231\) −2.76393 −0.181853
\(232\) −6.32624 −0.415338
\(233\) −13.5623 −0.888496 −0.444248 0.895904i \(-0.646529\pi\)
−0.444248 + 0.895904i \(0.646529\pi\)
\(234\) −10.0902 −0.659615
\(235\) 0 0
\(236\) −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 \(0.467557\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) 4.56231 0.290882
\(247\) 17.2361 1.09670
\(248\) −9.61803 −0.610746
\(249\) −3.41641 −0.216506
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.9443 1.18171 0.590856 0.806777i \(-0.298790\pi\)
0.590856 + 0.806777i \(0.298790\pi\)
\(258\) 0.472136 0.0293939
\(259\) −1.23607 −0.0768055
\(260\) 0 0
\(261\) −16.5623 −1.02518
\(262\) −8.65248 −0.534552
\(263\) 8.76393 0.540407 0.270204 0.962803i \(-0.412909\pi\)
0.270204 + 0.962803i \(0.412909\pi\)
\(264\) −2.23607 −0.137620
\(265\) 0 0
\(266\) −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 \(0.461088\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) −2.38197 −0.143378
\(277\) 7.79837 0.468559 0.234279 0.972169i \(-0.424727\pi\)
0.234279 + 0.972169i \(0.424727\pi\)
\(278\) 1.85410 0.111202
\(279\) −25.1803 −1.50751
\(280\) 0 0
\(281\) −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 \(-0.608384\pi\)
−0.333957 + 0.942588i \(0.608384\pi\)
\(284\) −14.9443 −0.886779
\(285\) 0 0
\(286\) −13.9443 −0.824542
\(287\) −9.12461 −0.538609
\(288\) 2.61803 0.154269
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 5.23607 0.306944
\(292\) 4.09017 0.239359
\(293\) −18.6525 −1.08969 −0.544845 0.838537i \(-0.683411\pi\)
−0.544845 + 0.838537i \(0.683411\pi\)
\(294\) −3.38197 −0.197240
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −12.5623 −0.728939
\(298\) 6.18034 0.358017
\(299\) −14.8541 −0.859035
\(300\) 0 0
\(301\) −0.944272 −0.0544269
\(302\) −17.7082 −1.01899
\(303\) −7.70820 −0.442825
\(304\) −4.47214 −0.256495
\(305\) 0 0
\(306\) −11.7082 −0.669313
\(307\) 6.14590 0.350765 0.175382 0.984500i \(-0.443884\pi\)
0.175382 + 0.984500i \(0.443884\pi\)
\(308\) 4.47214 0.254824
\(309\) −10.0557 −0.572050
\(310\) 0 0
\(311\) 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 \(-0.552592\pi\)
−0.164473 + 0.986382i \(0.552592\pi\)
\(314\) −7.52786 −0.424822
\(315\) 0 0
\(316\) 11.5623 0.650431
\(317\) −3.05573 −0.171627 −0.0858134 0.996311i \(-0.527349\pi\)
−0.0858134 + 0.996311i \(0.527349\pi\)
\(318\) 5.23607 0.293624
\(319\) −22.8885 −1.28151
\(320\) 0 0
\(321\) 5.14590 0.287216
\(322\) 4.76393 0.265484
\(323\) 20.0000 1.11283
\(324\) 5.70820 0.317122
\(325\) 0 0
\(326\) −12.4721 −0.690768
\(327\) 9.23607 0.510756
\(328\) −7.38197 −0.407601
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 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
\(335\) 0 0
\(336\) 0.763932 0.0416759
\(337\) 17.0344 0.927925 0.463963 0.885855i \(-0.346427\pi\)
0.463963 + 0.885855i \(0.346427\pi\)
\(338\) −1.85410 −0.100850
\(339\) 6.76393 0.367366
\(340\) 0 0
\(341\) −34.7984 −1.88444
\(342\) −11.7082 −0.633107
\(343\) 15.4164 0.832408
\(344\) −0.763932 −0.0411885
\(345\) 0 0
\(346\) 8.47214 0.455465
\(347\) −12.7639 −0.685204 −0.342602 0.939481i \(-0.611308\pi\)
−0.342602 + 0.939481i \(0.611308\pi\)
\(348\) −3.90983 −0.209589
\(349\) 12.1803 0.651999 0.325999 0.945370i \(-0.394299\pi\)
0.325999 + 0.945370i \(0.394299\pi\)
\(350\) 0 0
\(351\) −13.3820 −0.714277
\(352\) 3.61803 0.192842
\(353\) −29.7082 −1.58121 −0.790604 0.612328i \(-0.790233\pi\)
−0.790604 + 0.612328i \(0.790233\pi\)
\(354\) −5.70820 −0.303388
\(355\) 0 0
\(356\) −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 \(0.462347\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.52786 −0.290538
\(363\) −1.29180 −0.0678017
\(364\) 4.76393 0.249698
\(365\) 0 0
\(366\) 5.18034 0.270781
\(367\) 27.1246 1.41589 0.707947 0.706266i \(-0.249622\pi\)
0.707947 + 0.706266i \(0.249622\pi\)
\(368\) 3.85410 0.200909
\(369\) −19.3262 −1.00608
\(370\) 0 0
\(371\) −10.4721 −0.543686
\(372\) −5.94427 −0.308196
\(373\) −14.2918 −0.740001 −0.370001 0.929032i \(-0.620643\pi\)
−0.370001 + 0.929032i \(0.620643\pi\)
\(374\) −16.1803 −0.836665
\(375\) 0 0
\(376\) 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 \(0.356996\pi\)
\(380\) 0 0
\(381\) 0.291796 0.0149492
\(382\) −4.09017 −0.209271
\(383\) 17.8885 0.914062 0.457031 0.889451i \(-0.348913\pi\)
0.457031 + 0.889451i \(0.348913\pi\)
\(384\) 0.618034 0.0315389
\(385\) 0 0
\(386\) 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 \(0.501177\pi\)
\(390\) 0 0
\(391\) −17.2361 −0.871665
\(392\) 5.47214 0.276385
\(393\) −5.34752 −0.269747
\(394\) −16.4721 −0.829854
\(395\) 0 0
\(396\) 9.47214 0.475993
\(397\) 10.6525 0.534632 0.267316 0.963609i \(-0.413863\pi\)
0.267316 + 0.963609i \(0.413863\pi\)
\(398\) −20.9443 −1.04984
\(399\) −3.41641 −0.171034
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) 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 \(0.530104\pi\)
\(410\) 0 0
\(411\) 11.9443 0.589167
\(412\) 16.2705 0.801590
\(413\) 11.4164 0.561765
\(414\) 10.0902 0.495905
\(415\) 0 0
\(416\) 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 \(0.424958\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −9.23607 −0.447489
\(427\) −10.3607 −0.501388
\(428\) −8.32624 −0.402464
\(429\) −8.61803 −0.416083
\(430\) 0 0
\(431\) −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 \(-0.162260\pi\)
0.872864 + 0.487964i \(0.162260\pi\)
\(434\) 11.8885 0.570668
\(435\) 0 0
\(436\) −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 \(0.631293\pi\)
\(440\) 0 0
\(441\) 14.3262 0.682202
\(442\) −17.2361 −0.819836
\(443\) 18.2705 0.868058 0.434029 0.900899i \(-0.357092\pi\)
0.434029 + 0.900899i \(0.357092\pi\)
\(444\) −0.618034 −0.0293306
\(445\) 0 0
\(446\) −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 \(0.652433\pi\)
\(450\) 0 0
\(451\) −26.7082 −1.25764
\(452\) −10.9443 −0.514775
\(453\) −10.9443 −0.514207
\(454\) −17.7082 −0.831087
\(455\) 0 0
\(456\) −2.76393 −0.129433
\(457\) 29.2361 1.36761 0.683803 0.729667i \(-0.260325\pi\)
0.683803 + 0.729667i \(0.260325\pi\)
\(458\) −17.1246 −0.800181
\(459\) −15.5279 −0.724779
\(460\) 0 0
\(461\) −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 \(-0.617777\pi\)
−0.361621 + 0.932325i \(0.617777\pi\)
\(464\) 6.32624 0.293688
\(465\) 0 0
\(466\) 13.5623 0.628262
\(467\) −20.3607 −0.942180 −0.471090 0.882085i \(-0.656139\pi\)
−0.471090 + 0.882085i \(0.656139\pi\)
\(468\) 10.0902 0.466418
\(469\) −12.4721 −0.575910
\(470\) 0 0
\(471\) −4.65248 −0.214375
\(472\) 9.23607 0.425124
\(473\) −2.76393 −0.127086
\(474\) 7.14590 0.328222
\(475\) 0 0
\(476\) 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 \(0.377461\pi\)
\(480\) 0 0
\(481\) −3.85410 −0.175732
\(482\) 10.4721 0.476993
\(483\) 2.94427 0.133969
\(484\) 2.09017 0.0950077
\(485\) 0 0
\(486\) 13.9443 0.632525
\(487\) 25.3050 1.14668 0.573338 0.819319i \(-0.305648\pi\)
0.573338 + 0.819319i \(0.305648\pi\)
\(488\) −8.38197 −0.379434
\(489\) −7.70820 −0.348577
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) 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 \(0.321943\pi\)
\(500\) 0 0
\(501\) 4.41641 0.197311
\(502\) 3.05573 0.136384
\(503\) −7.90983 −0.352682 −0.176341 0.984329i \(-0.556426\pi\)
−0.176341 + 0.984329i \(0.556426\pi\)
\(504\) −3.23607 −0.144146
\(505\) 0 0
\(506\) 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 \(0.530322\pi\)
\(510\) 0 0
\(511\) −5.05573 −0.223652
\(512\) −1.00000 −0.0441942
\(513\) −15.5279 −0.685572
\(514\) −18.9443 −0.835596
\(515\) 0 0
\(516\) −0.472136 −0.0207846
\(517\) 11.7082 0.514926
\(518\) 1.23607 0.0543097
\(519\) 5.23607 0.229838
\(520\) 0 0
\(521\) 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 \(-0.768651\pi\)
−0.747301 + 0.664486i \(0.768651\pi\)
\(524\) 8.65248 0.377985
\(525\) 0 0
\(526\) −8.76393 −0.382126
\(527\) −43.0132 −1.87368
\(528\) 2.23607 0.0973124
\(529\) −8.14590 −0.354169
\(530\) 0 0
\(531\) 24.1803 1.04934
\(532\) 5.52786 0.239663
\(533\) −28.4508 −1.23234
\(534\) −6.47214 −0.280077
\(535\) 0 0
\(536\) −10.0902 −0.435829
\(537\) −11.5279 −0.497464
\(538\) −4.00000 −0.172452
\(539\) 19.7984 0.852776
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) 2.94427 0.126003
\(547\) −42.0689 −1.79874 −0.899368 0.437193i \(-0.855973\pi\)
−0.899368 + 0.437193i \(0.855973\pi\)
\(548\) −19.3262 −0.825576
\(549\) −21.9443 −0.936559
\(550\) 0 0
\(551\) −28.2918 −1.20527
\(552\) 2.38197 0.101383
\(553\) −14.2918 −0.607749
\(554\) −7.79837 −0.331321
\(555\) 0 0
\(556\) −1.85410 −0.0786314
\(557\) 0.562306 0.0238257 0.0119128 0.999929i \(-0.496208\pi\)
0.0119128 + 0.999929i \(0.496208\pi\)
\(558\) 25.1803 1.06597
\(559\) −2.94427 −0.124529
\(560\) 0 0
\(561\) −10.0000 −0.422200
\(562\) 5.88854 0.248393
\(563\) 27.8885 1.17536 0.587681 0.809093i \(-0.300041\pi\)
0.587681 + 0.809093i \(0.300041\pi\)
\(564\) 2.00000 0.0842152
\(565\) 0 0
\(566\) 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 \(0.651723\pi\)
\(570\) 0 0
\(571\) 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
\(575\) 0 0
\(576\) −2.61803 −0.109085
\(577\) 20.6525 0.859774 0.429887 0.902883i \(-0.358553\pi\)
0.429887 + 0.902883i \(0.358553\pi\)
\(578\) −3.00000 −0.124784
\(579\) 2.47214 0.102738
\(580\) 0 0
\(581\) −6.83282 −0.283473
\(582\) −5.23607 −0.217042
\(583\) −30.6525 −1.26950
\(584\) −4.09017 −0.169252
\(585\) 0 0
\(586\) 18.6525 0.770527
\(587\) 30.9443 1.27721 0.638603 0.769536i \(-0.279512\pi\)
0.638603 + 0.769536i \(0.279512\pi\)
\(588\) 3.38197 0.139470
\(589\) −43.0132 −1.77233
\(590\) 0 0
\(591\) −10.1803 −0.418763
\(592\) 1.00000 0.0410997
\(593\) 2.56231 0.105221 0.0526106 0.998615i \(-0.483246\pi\)
0.0526106 + 0.998615i \(0.483246\pi\)
\(594\) 12.5623 0.515438
\(595\) 0 0
\(596\) −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 \(0.213336\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) 7.70820 0.313124
\(607\) −5.96556 −0.242135 −0.121067 0.992644i \(-0.538632\pi\)
−0.121067 + 0.992644i \(0.538632\pi\)
\(608\) 4.47214 0.181369
\(609\) 4.83282 0.195836
\(610\) 0 0
\(611\) 12.4721 0.504569
\(612\) 11.7082 0.473276
\(613\) 36.1803 1.46131 0.730655 0.682747i \(-0.239215\pi\)
0.730655 + 0.682747i \(0.239215\pi\)
\(614\) −6.14590 −0.248028
\(615\) 0 0
\(616\) −4.47214 −0.180187
\(617\) 11.0902 0.446473 0.223237 0.974764i \(-0.428338\pi\)
0.223237 + 0.974764i \(0.428338\pi\)
\(618\) 10.0557 0.404501
\(619\) 18.2705 0.734354 0.367177 0.930151i \(-0.380324\pi\)
0.367177 + 0.930151i \(0.380324\pi\)
\(620\) 0 0
\(621\) 13.3820 0.537000
\(622\) −2.03444 −0.0815737
\(623\) 12.9443 0.518601
\(624\) 2.38197 0.0953550
\(625\) 0 0
\(626\) 5.81966 0.232600
\(627\) −10.0000 −0.399362
\(628\) 7.52786 0.300394
\(629\) −4.47214 −0.178316
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) −5.23607 −0.207624
\(637\) 21.0902 0.835623
\(638\) 22.8885 0.906166
\(639\) 39.1246 1.54775
\(640\) 0 0
\(641\) −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 \(-0.272521\pi\)
0.655351 + 0.755324i \(0.272521\pi\)
\(644\) −4.76393 −0.187725
\(645\) 0 0
\(646\) −20.0000 −0.786889
\(647\) −18.9098 −0.743422 −0.371711 0.928348i \(-0.621229\pi\)
−0.371711 + 0.928348i \(0.621229\pi\)
\(648\) −5.70820 −0.224239
\(649\) 33.4164 1.31171
\(650\) 0 0
\(651\) 7.34752 0.287972
\(652\) 12.4721 0.488447
\(653\) 4.72949 0.185079 0.0925396 0.995709i \(-0.470502\pi\)
0.0925396 + 0.995709i \(0.470502\pi\)
\(654\) −9.23607 −0.361159
\(655\) 0 0
\(656\) 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 \(0.597300\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) 2.61803 0.101447
\(667\) 24.3820 0.944073
\(668\) −7.14590 −0.276483
\(669\) −5.05573 −0.195466
\(670\) 0 0
\(671\) −30.3262 −1.17073
\(672\) −0.763932 −0.0294693
\(673\) −17.8541 −0.688225 −0.344113 0.938928i \(-0.611820\pi\)
−0.344113 + 0.938928i \(0.611820\pi\)
\(674\) −17.0344 −0.656142
\(675\) 0 0
\(676\) 1.85410 0.0713116
\(677\) −3.34752 −0.128656 −0.0643279 0.997929i \(-0.520490\pi\)
−0.0643279 + 0.997929i \(0.520490\pi\)
\(678\) −6.76393 −0.259767
\(679\) 10.4721 0.401884
\(680\) 0 0
\(681\) −10.9443 −0.419385
\(682\) 34.7984 1.33250
\(683\) 35.4164 1.35517 0.677586 0.735444i \(-0.263026\pi\)
0.677586 + 0.735444i \(0.263026\pi\)
\(684\) 11.7082 0.447674
\(685\) 0 0
\(686\) −15.4164 −0.588601
\(687\) −10.5836 −0.403789
\(688\) 0.763932 0.0291246
\(689\) −32.6525 −1.24396
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) 3.90983 0.148202
\(697\) −33.0132 −1.25046
\(698\) −12.1803 −0.461033
\(699\) 8.38197 0.317035
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) 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 \(0.198661\pi\)
\(710\) 0 0
\(711\) −30.2705 −1.13523
\(712\) 10.4721 0.392460
\(713\) 37.0689 1.38824
\(714\) 3.41641 0.127856
\(715\) 0 0
\(716\) 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 \(0.496536\pi\)
\(720\) 0 0
\(721\) −20.1115 −0.748990
\(722\) −1.00000 −0.0372161
\(723\) 6.47214 0.240701
\(724\) 5.52786 0.205441
\(725\) 0 0
\(726\) 1.29180 0.0479430
\(727\) 23.1459 0.858434 0.429217 0.903201i \(-0.358790\pi\)
0.429217 + 0.903201i \(0.358790\pi\)
\(728\) −4.76393 −0.176563
\(729\) −8.50658 −0.315058
\(730\) 0 0
\(731\) −3.41641 −0.126360
\(732\) −5.18034 −0.191471
\(733\) −36.4721 −1.34713 −0.673565 0.739128i \(-0.735238\pi\)
−0.673565 + 0.739128i \(0.735238\pi\)
\(734\) −27.1246 −1.00119
\(735\) 0 0
\(736\) −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 \(0.633181\pi\)
\(740\) 0 0
\(741\) −10.6525 −0.391328
\(742\) 10.4721 0.384444
\(743\) −10.0689 −0.369392 −0.184696 0.982796i \(-0.559130\pi\)
−0.184696 + 0.982796i \(0.559130\pi\)
\(744\) 5.94427 0.217928
\(745\) 0 0
\(746\) 14.2918 0.523260
\(747\) −14.4721 −0.529508
\(748\) 16.1803 0.591612
\(749\) 10.2918 0.376054
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) −4.29180 −0.156091
\(757\) 10.8541 0.394499 0.197250 0.980353i \(-0.436799\pi\)
0.197250 + 0.980353i \(0.436799\pi\)
\(758\) −16.9098 −0.614193
\(759\) 8.61803 0.312815
\(760\) 0 0
\(761\) −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
\(765\) 0 0
\(766\) −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 \(0.568765\pi\)
\(770\) 0 0
\(771\) −11.7082 −0.421661
\(772\) −4.00000 −0.143963
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) 8.47214 0.304132
\(777\) 0.763932 0.0274059
\(778\) 0.145898 0.00523070
\(779\) −33.0132 −1.18282
\(780\) 0 0
\(781\) 54.0689 1.93474
\(782\) 17.2361 0.616361
\(783\) 21.9656 0.784985
\(784\) −5.47214 −0.195433
\(785\) 0 0
\(786\) 5.34752 0.190740
\(787\) 34.4721 1.22880 0.614399 0.788995i \(-0.289398\pi\)
0.614399 + 0.788995i \(0.289398\pi\)
\(788\) 16.4721 0.586796
\(789\) −5.41641 −0.192829
\(790\) 0 0
\(791\) 13.5279 0.480995
\(792\) −9.47214 −0.336578
\(793\) −32.3050 −1.14718
\(794\) −10.6525 −0.378042
\(795\) 0 0
\(796\) 20.9443 0.742350
\(797\) −12.7295 −0.450902 −0.225451 0.974255i \(-0.572386\pi\)
−0.225451 + 0.974255i \(0.572386\pi\)
\(798\) 3.41641 0.120940
\(799\) 14.4721 0.511987
\(800\) 0 0
\(801\) 27.4164 0.968711
\(802\) 9.23607 0.326137
\(803\) −14.7984 −0.522223
\(804\) −6.23607 −0.219929
\(805\) 0 0
\(806\) 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 \(0.341787\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) 2.76393 0.0967570
\(817\) −3.41641 −0.119525
\(818\) 3.81966 0.133551
\(819\) −12.4721 −0.435812
\(820\) 0 0
\(821\) 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 \(-0.298875\pi\)
0.590640 + 0.806935i \(0.298875\pi\)
\(824\) −16.2705 −0.566810
\(825\) 0 0
\(826\) −11.4164 −0.397228
\(827\) −56.0689 −1.94971 −0.974853 0.222849i \(-0.928464\pi\)
−0.974853 + 0.222849i \(0.928464\pi\)
\(828\) −10.0902 −0.350658
\(829\) 31.2016 1.08368 0.541839 0.840483i \(-0.317728\pi\)
0.541839 + 0.840483i \(0.317728\pi\)
\(830\) 0 0
\(831\) −4.81966 −0.167192
\(832\) −3.85410 −0.133617
\(833\) 24.4721 0.847909
\(834\) −1.14590 −0.0396792
\(835\) 0 0
\(836\) 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 \(0.529425\pi\)
\(840\) 0 0
\(841\) 11.0213 0.380044
\(842\) 1.96556 0.0677376
\(843\) 3.63932 0.125345
\(844\) −22.2705 −0.766583
\(845\) 0 0
\(846\) −8.47214 −0.291278
\(847\) −2.58359 −0.0887733
\(848\) 8.47214 0.290934
\(849\) 6.94427 0.238327
\(850\) 0 0
\(851\) 3.85410 0.132117
\(852\) 9.23607 0.316422
\(853\) 12.7426 0.436300 0.218150 0.975915i \(-0.429998\pi\)
0.218150 + 0.975915i \(0.429998\pi\)
\(854\) 10.3607 0.354535
\(855\) 0 0
\(856\) 8.32624 0.284585
\(857\) 26.9443 0.920399 0.460199 0.887816i \(-0.347778\pi\)
0.460199 + 0.887816i \(0.347778\pi\)
\(858\) 8.61803 0.294215
\(859\) 26.5836 0.907020 0.453510 0.891251i \(-0.350172\pi\)
0.453510 + 0.891251i \(0.350172\pi\)
\(860\) 0 0
\(861\) 5.63932 0.192188
\(862\) 32.3607 1.10221
\(863\) 7.41641 0.252457 0.126229 0.992001i \(-0.459713\pi\)
0.126229 + 0.992001i \(0.459713\pi\)
\(864\) −3.47214 −0.118124
\(865\) 0 0
\(866\) −36.3262 −1.23442
\(867\) −1.85410 −0.0629686
\(868\) −11.8885 −0.403523
\(869\) −41.8328 −1.41908
\(870\) 0 0
\(871\) −38.8885 −1.31769
\(872\) 14.9443 0.506077
\(873\) 22.1803 0.750691
\(874\) 17.2361 0.583019
\(875\) 0 0
\(876\) −2.52786 −0.0854086
\(877\) −36.8328 −1.24376 −0.621878 0.783114i \(-0.713630\pi\)
−0.621878 + 0.783114i \(0.713630\pi\)
\(878\) 16.7984 0.566918
\(879\) 11.5279 0.388825
\(880\) 0 0
\(881\) −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 \(-0.664135\pi\)
−0.493095 + 0.869975i \(0.664135\pi\)
\(884\) 17.2361 0.579712
\(885\) 0 0
\(886\) −18.2705 −0.613810
\(887\) 7.88854 0.264871 0.132436 0.991192i \(-0.457720\pi\)
0.132436 + 0.991192i \(0.457720\pi\)
\(888\) 0.618034 0.0207399
\(889\) 0.583592 0.0195731
\(890\) 0 0
\(891\) −20.6525 −0.691884
\(892\) 8.18034 0.273898
\(893\) 14.4721 0.484292
\(894\) −3.81966 −0.127749
\(895\) 0 0
\(896\) 1.23607 0.0412941
\(897\) 9.18034 0.306523
\(898\) 19.5279 0.651653
\(899\) 60.8460 2.02933
\(900\) 0 0
\(901\) −37.8885 −1.26225
\(902\) 26.7082 0.889286
\(903\) 0.583592 0.0194207
\(904\) 10.9443 0.364001
\(905\) 0 0
\(906\) 10.9443 0.363599
\(907\) 20.1115 0.667790 0.333895 0.942610i \(-0.391637\pi\)
0.333895 + 0.942610i \(0.391637\pi\)
\(908\) 17.7082 0.587667
\(909\) −32.6525 −1.08301
\(910\) 0 0
\(911\) −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
\(915\) 0 0
\(916\) 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 \(0.335913\pi\)
\(920\) 0 0
\(921\) −3.79837 −0.125161
\(922\) 21.0557 0.693433
\(923\) 57.5967 1.89582
\(924\) −2.76393 −0.0909267
\(925\) 0 0
\(926\) 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 \(0.345435\pi\)
\(930\) 0 0
\(931\) 24.4721 0.802042
\(932\) −13.5623 −0.444248
\(933\) −1.25735 −0.0411639
\(934\) 20.3607 0.666222
\(935\) 0 0
\(936\) −10.0902 −0.329808
\(937\) −57.0476 −1.86366 −0.931832 0.362890i \(-0.881790\pi\)
−0.931832 + 0.362890i \(0.881790\pi\)
\(938\) 12.4721 0.407230
\(939\) 3.59675 0.117375
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) 2.76393 0.0898632
\(947\) 34.8328 1.13191 0.565957 0.824435i \(-0.308507\pi\)
0.565957 + 0.824435i \(0.308507\pi\)
\(948\) −7.14590 −0.232088
\(949\) −15.7639 −0.511719
\(950\) 0 0
\(951\) 1.88854 0.0612402
\(952\) −5.52786 −0.179159
\(953\) 44.4508 1.43990 0.719952 0.694024i \(-0.244164\pi\)
0.719952 + 0.694024i \(0.244164\pi\)
\(954\) 22.1803 0.718115
\(955\) 0 0
\(956\) 3.14590 0.101746
\(957\) 14.1459 0.457272
\(958\) −16.4377 −0.531078
\(959\) 23.8885 0.771401
\(960\) 0 0
\(961\) 61.5066 1.98408
\(962\) 3.85410 0.124261
\(963\) 21.7984 0.702443
\(964\) −10.4721 −0.337285
\(965\) 0 0
\(966\) −2.94427 −0.0947304
\(967\) −11.7295 −0.377195 −0.188597 0.982054i \(-0.560394\pi\)
−0.188597 + 0.982054i \(0.560394\pi\)
\(968\) −2.09017 −0.0671806
\(969\) −12.3607 −0.397082
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) 8.38197 0.268300
\(977\) −52.4721 −1.67873 −0.839366 0.543566i \(-0.817074\pi\)
−0.839366 + 0.543566i \(0.817074\pi\)
\(978\) 7.70820 0.246481
\(979\) 37.8885 1.21092
\(980\) 0 0
\(981\) 39.1246 1.24915
\(982\) −27.4508 −0.875992
\(983\) −39.7771 −1.26869 −0.634346 0.773049i \(-0.718731\pi\)
−0.634346 + 0.773049i \(0.718731\pi\)
\(984\) 4.56231 0.145441
\(985\) 0 0
\(986\) 28.2918 0.900994
\(987\) −2.47214 −0.0786890
\(988\) 17.2361 0.548352
\(989\) 2.94427 0.0936224
\(990\) 0 0
\(991\) −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
\(995\) 0 0
\(996\) −3.41641 −0.108253
\(997\) −53.7771 −1.70314 −0.851569 0.524243i \(-0.824348\pi\)
−0.851569 + 0.524243i \(0.824348\pi\)
\(998\) −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 1850.2.a.t.1.1 2
5.2 odd 4 1850.2.b.j.149.2 4
5.3 odd 4 1850.2.b.j.149.3 4
5.4 even 2 74.2.a.b.1.2 2
15.14 odd 2 666.2.a.i.1.2 2
20.19 odd 2 592.2.a.g.1.1 2
35.34 odd 2 3626.2.a.s.1.1 2
40.19 odd 2 2368.2.a.u.1.2 2
40.29 even 2 2368.2.a.y.1.1 2
55.54 odd 2 8954.2.a.j.1.2 2
60.59 even 2 5328.2.a.bc.1.2 2
185.184 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 5.4 even 2
592.2.a.g.1.1 2 20.19 odd 2
666.2.a.i.1.2 2 15.14 odd 2
1850.2.a.t.1.1 2 1.1 even 1 trivial
1850.2.b.j.149.2 4 5.2 odd 4
1850.2.b.j.149.3 4 5.3 odd 4
2368.2.a.u.1.2 2 40.19 odd 2
2368.2.a.y.1.1 2 40.29 even 2
2738.2.a.g.1.2 2 185.184 even 2
3626.2.a.s.1.1 2 35.34 odd 2
5328.2.a.bc.1.2 2 60.59 even 2
8954.2.a.j.1.2 2 55.54 odd 2