Properties

Label 4025.2.a.i.1.2
Level 4025
Weight 2
Character 4025.1
Self dual yes
Analytic conductor 32.140
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 4025.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} -4.47214 q^{11} +0.618034 q^{12} +4.23607 q^{13} +1.61803 q^{14} -4.85410 q^{16} -3.23607 q^{18} -2.76393 q^{19} +1.00000 q^{21} -7.23607 q^{22} +1.00000 q^{23} -2.23607 q^{24} +6.85410 q^{26} -5.00000 q^{27} +0.618034 q^{28} +7.47214 q^{29} -9.00000 q^{31} -3.38197 q^{32} -4.47214 q^{33} -1.23607 q^{36} -7.70820 q^{37} -4.47214 q^{38} +4.23607 q^{39} +2.23607 q^{41} +1.61803 q^{42} +6.47214 q^{43} -2.76393 q^{44} +1.61803 q^{46} -5.47214 q^{47} -4.85410 q^{48} +1.00000 q^{49} +2.61803 q^{52} -6.76393 q^{53} -8.09017 q^{54} -2.23607 q^{56} -2.76393 q^{57} +12.0902 q^{58} -10.4721 q^{59} -13.4164 q^{61} -14.5623 q^{62} -2.00000 q^{63} +4.23607 q^{64} -7.23607 q^{66} -10.1803 q^{67} +1.00000 q^{69} -5.76393 q^{71} +4.47214 q^{72} -6.70820 q^{73} -12.4721 q^{74} -1.70820 q^{76} -4.47214 q^{77} +6.85410 q^{78} -2.76393 q^{79} +1.00000 q^{81} +3.61803 q^{82} +2.47214 q^{83} +0.618034 q^{84} +10.4721 q^{86} +7.47214 q^{87} +10.0000 q^{88} -8.94427 q^{89} +4.23607 q^{91} +0.618034 q^{92} -9.00000 q^{93} -8.85410 q^{94} -3.38197 q^{96} +9.70820 q^{97} +1.61803 q^{98} +8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} - 4q^{9} - q^{12} + 4q^{13} + q^{14} - 3q^{16} - 2q^{18} - 10q^{19} + 2q^{21} - 10q^{22} + 2q^{23} + 7q^{26} - 10q^{27} - q^{28} + 6q^{29} - 18q^{31} - 9q^{32} + 2q^{36} - 2q^{37} + 4q^{39} + q^{42} + 4q^{43} - 10q^{44} + q^{46} - 2q^{47} - 3q^{48} + 2q^{49} + 3q^{52} - 18q^{53} - 5q^{54} - 10q^{57} + 13q^{58} - 12q^{59} - 9q^{62} - 4q^{63} + 4q^{64} - 10q^{66} + 2q^{67} + 2q^{69} - 16q^{71} - 16q^{74} + 10q^{76} + 7q^{78} - 10q^{79} + 2q^{81} + 5q^{82} - 4q^{83} - q^{84} + 12q^{86} + 6q^{87} + 20q^{88} + 4q^{91} - q^{92} - 18q^{93} - 11q^{94} - 9q^{96} + 6q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 1.61803 0.660560
\(7\) 1.00000 0.377964
\(8\) −2.23607 −0.790569
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −4.47214 −1.34840 −0.674200 0.738549i \(-0.735511\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(12\) 0.618034 0.178411
\(13\) 4.23607 1.17487 0.587437 0.809270i \(-0.300137\pi\)
0.587437 + 0.809270i \(0.300137\pi\)
\(14\) 1.61803 0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −3.23607 −0.762749
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −7.23607 −1.54273
\(23\) 1.00000 0.208514
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) 6.85410 1.34420
\(27\) −5.00000 −0.962250
\(28\) 0.618034 0.116797
\(29\) 7.47214 1.38754 0.693770 0.720196i \(-0.255948\pi\)
0.693770 + 0.720196i \(0.255948\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −3.38197 −0.597853
\(33\) −4.47214 −0.778499
\(34\) 0 0
\(35\) 0 0
\(36\) −1.23607 −0.206011
\(37\) −7.70820 −1.26722 −0.633610 0.773652i \(-0.718428\pi\)
−0.633610 + 0.773652i \(0.718428\pi\)
\(38\) −4.47214 −0.725476
\(39\) 4.23607 0.678314
\(40\) 0 0
\(41\) 2.23607 0.349215 0.174608 0.984638i \(-0.444134\pi\)
0.174608 + 0.984638i \(0.444134\pi\)
\(42\) 1.61803 0.249668
\(43\) 6.47214 0.986991 0.493496 0.869748i \(-0.335719\pi\)
0.493496 + 0.869748i \(0.335719\pi\)
\(44\) −2.76393 −0.416678
\(45\) 0 0
\(46\) 1.61803 0.238566
\(47\) −5.47214 −0.798193 −0.399097 0.916909i \(-0.630676\pi\)
−0.399097 + 0.916909i \(0.630676\pi\)
\(48\) −4.85410 −0.700629
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 2.61803 0.363056
\(53\) −6.76393 −0.929098 −0.464549 0.885548i \(-0.653783\pi\)
−0.464549 + 0.885548i \(0.653783\pi\)
\(54\) −8.09017 −1.10093
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) −2.76393 −0.366092
\(58\) 12.0902 1.58752
\(59\) −10.4721 −1.36336 −0.681678 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) 0 0
\(61\) −13.4164 −1.71780 −0.858898 0.512148i \(-0.828850\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) −14.5623 −1.84941
\(63\) −2.00000 −0.251976
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) −7.23607 −0.890698
\(67\) −10.1803 −1.24373 −0.621863 0.783126i \(-0.713624\pi\)
−0.621863 + 0.783126i \(0.713624\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −5.76393 −0.684053 −0.342026 0.939690i \(-0.611113\pi\)
−0.342026 + 0.939690i \(0.611113\pi\)
\(72\) 4.47214 0.527046
\(73\) −6.70820 −0.785136 −0.392568 0.919723i \(-0.628413\pi\)
−0.392568 + 0.919723i \(0.628413\pi\)
\(74\) −12.4721 −1.44986
\(75\) 0 0
\(76\) −1.70820 −0.195944
\(77\) −4.47214 −0.509647
\(78\) 6.85410 0.776074
\(79\) −2.76393 −0.310967 −0.155483 0.987839i \(-0.549693\pi\)
−0.155483 + 0.987839i \(0.549693\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.61803 0.399545
\(83\) 2.47214 0.271352 0.135676 0.990753i \(-0.456679\pi\)
0.135676 + 0.990753i \(0.456679\pi\)
\(84\) 0.618034 0.0674330
\(85\) 0 0
\(86\) 10.4721 1.12924
\(87\) 7.47214 0.801097
\(88\) 10.0000 1.06600
\(89\) −8.94427 −0.948091 −0.474045 0.880500i \(-0.657207\pi\)
−0.474045 + 0.880500i \(0.657207\pi\)
\(90\) 0 0
\(91\) 4.23607 0.444061
\(92\) 0.618034 0.0644345
\(93\) −9.00000 −0.933257
\(94\) −8.85410 −0.913231
\(95\) 0 0
\(96\) −3.38197 −0.345170
\(97\) 9.70820 0.985719 0.492859 0.870109i \(-0.335952\pi\)
0.492859 + 0.870109i \(0.335952\pi\)
\(98\) 1.61803 0.163446
\(99\) 8.94427 0.898933
\(100\) 0 0
\(101\) 13.4164 1.33498 0.667491 0.744618i \(-0.267368\pi\)
0.667491 + 0.744618i \(0.267368\pi\)
\(102\) 0 0
\(103\) 2.29180 0.225817 0.112909 0.993605i \(-0.463983\pi\)
0.112909 + 0.993605i \(0.463983\pi\)
\(104\) −9.47214 −0.928819
\(105\) 0 0
\(106\) −10.9443 −1.06300
\(107\) −6.76393 −0.653894 −0.326947 0.945043i \(-0.606020\pi\)
−0.326947 + 0.945043i \(0.606020\pi\)
\(108\) −3.09017 −0.297352
\(109\) 11.4164 1.09349 0.546747 0.837298i \(-0.315866\pi\)
0.546747 + 0.837298i \(0.315866\pi\)
\(110\) 0 0
\(111\) −7.70820 −0.731630
\(112\) −4.85410 −0.458670
\(113\) 6.47214 0.608847 0.304424 0.952537i \(-0.401536\pi\)
0.304424 + 0.952537i \(0.401536\pi\)
\(114\) −4.47214 −0.418854
\(115\) 0 0
\(116\) 4.61803 0.428774
\(117\) −8.47214 −0.783249
\(118\) −16.9443 −1.55985
\(119\) 0 0
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) −21.7082 −1.96537
\(123\) 2.23607 0.201619
\(124\) −5.56231 −0.499510
\(125\) 0 0
\(126\) −3.23607 −0.288292
\(127\) −10.7082 −0.950199 −0.475100 0.879932i \(-0.657588\pi\)
−0.475100 + 0.879932i \(0.657588\pi\)
\(128\) 13.6180 1.20368
\(129\) 6.47214 0.569840
\(130\) 0 0
\(131\) −13.9443 −1.21832 −0.609158 0.793049i \(-0.708493\pi\)
−0.609158 + 0.793049i \(0.708493\pi\)
\(132\) −2.76393 −0.240569
\(133\) −2.76393 −0.239663
\(134\) −16.4721 −1.42298
\(135\) 0 0
\(136\) 0 0
\(137\) −2.29180 −0.195801 −0.0979007 0.995196i \(-0.531213\pi\)
−0.0979007 + 0.995196i \(0.531213\pi\)
\(138\) 1.61803 0.137736
\(139\) 11.4721 0.973054 0.486527 0.873666i \(-0.338264\pi\)
0.486527 + 0.873666i \(0.338264\pi\)
\(140\) 0 0
\(141\) −5.47214 −0.460837
\(142\) −9.32624 −0.782641
\(143\) −18.9443 −1.58420
\(144\) 9.70820 0.809017
\(145\) 0 0
\(146\) −10.8541 −0.898292
\(147\) 1.00000 0.0824786
\(148\) −4.76393 −0.391593
\(149\) 10.7639 0.881816 0.440908 0.897552i \(-0.354657\pi\)
0.440908 + 0.897552i \(0.354657\pi\)
\(150\) 0 0
\(151\) 7.18034 0.584328 0.292164 0.956368i \(-0.405625\pi\)
0.292164 + 0.956368i \(0.405625\pi\)
\(152\) 6.18034 0.501292
\(153\) 0 0
\(154\) −7.23607 −0.583099
\(155\) 0 0
\(156\) 2.61803 0.209610
\(157\) 11.4164 0.911129 0.455564 0.890203i \(-0.349438\pi\)
0.455564 + 0.890203i \(0.349438\pi\)
\(158\) −4.47214 −0.355784
\(159\) −6.76393 −0.536415
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 1.61803 0.127125
\(163\) 3.18034 0.249103 0.124552 0.992213i \(-0.460251\pi\)
0.124552 + 0.992213i \(0.460251\pi\)
\(164\) 1.38197 0.107913
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 13.8885 1.07473 0.537364 0.843350i \(-0.319420\pi\)
0.537364 + 0.843350i \(0.319420\pi\)
\(168\) −2.23607 −0.172516
\(169\) 4.94427 0.380329
\(170\) 0 0
\(171\) 5.52786 0.422726
\(172\) 4.00000 0.304997
\(173\) 12.4721 0.948239 0.474119 0.880461i \(-0.342766\pi\)
0.474119 + 0.880461i \(0.342766\pi\)
\(174\) 12.0902 0.916553
\(175\) 0 0
\(176\) 21.7082 1.63632
\(177\) −10.4721 −0.787134
\(178\) −14.4721 −1.08473
\(179\) −5.29180 −0.395527 −0.197764 0.980250i \(-0.563368\pi\)
−0.197764 + 0.980250i \(0.563368\pi\)
\(180\) 0 0
\(181\) −22.9443 −1.70543 −0.852717 0.522373i \(-0.825047\pi\)
−0.852717 + 0.522373i \(0.825047\pi\)
\(182\) 6.85410 0.508060
\(183\) −13.4164 −0.991769
\(184\) −2.23607 −0.164845
\(185\) 0 0
\(186\) −14.5623 −1.06776
\(187\) 0 0
\(188\) −3.38197 −0.246655
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −1.81966 −0.131666 −0.0658330 0.997831i \(-0.520970\pi\)
−0.0658330 + 0.997831i \(0.520970\pi\)
\(192\) 4.23607 0.305712
\(193\) 22.4164 1.61357 0.806784 0.590846i \(-0.201206\pi\)
0.806784 + 0.590846i \(0.201206\pi\)
\(194\) 15.7082 1.12778
\(195\) 0 0
\(196\) 0.618034 0.0441453
\(197\) 12.5279 0.892573 0.446287 0.894890i \(-0.352746\pi\)
0.446287 + 0.894890i \(0.352746\pi\)
\(198\) 14.4721 1.02849
\(199\) −19.8885 −1.40986 −0.704931 0.709276i \(-0.749022\pi\)
−0.704931 + 0.709276i \(0.749022\pi\)
\(200\) 0 0
\(201\) −10.1803 −0.718066
\(202\) 21.7082 1.52738
\(203\) 7.47214 0.524441
\(204\) 0 0
\(205\) 0 0
\(206\) 3.70820 0.258363
\(207\) −2.00000 −0.139010
\(208\) −20.5623 −1.42574
\(209\) 12.3607 0.855006
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −4.18034 −0.287107
\(213\) −5.76393 −0.394938
\(214\) −10.9443 −0.748135
\(215\) 0 0
\(216\) 11.1803 0.760726
\(217\) −9.00000 −0.610960
\(218\) 18.4721 1.25109
\(219\) −6.70820 −0.453298
\(220\) 0 0
\(221\) 0 0
\(222\) −12.4721 −0.837075
\(223\) 23.4164 1.56808 0.784039 0.620711i \(-0.213156\pi\)
0.784039 + 0.620711i \(0.213156\pi\)
\(224\) −3.38197 −0.225967
\(225\) 0 0
\(226\) 10.4721 0.696596
\(227\) −29.8885 −1.98377 −0.991886 0.127129i \(-0.959424\pi\)
−0.991886 + 0.127129i \(0.959424\pi\)
\(228\) −1.70820 −0.113129
\(229\) 19.2361 1.27116 0.635578 0.772037i \(-0.280762\pi\)
0.635578 + 0.772037i \(0.280762\pi\)
\(230\) 0 0
\(231\) −4.47214 −0.294245
\(232\) −16.7082 −1.09695
\(233\) 2.52786 0.165606 0.0828029 0.996566i \(-0.473613\pi\)
0.0828029 + 0.996566i \(0.473613\pi\)
\(234\) −13.7082 −0.896133
\(235\) 0 0
\(236\) −6.47214 −0.421300
\(237\) −2.76393 −0.179537
\(238\) 0 0
\(239\) −20.2361 −1.30896 −0.654481 0.756078i \(-0.727113\pi\)
−0.654481 + 0.756078i \(0.727113\pi\)
\(240\) 0 0
\(241\) 5.41641 0.348902 0.174451 0.984666i \(-0.444185\pi\)
0.174451 + 0.984666i \(0.444185\pi\)
\(242\) 14.5623 0.936100
\(243\) 16.0000 1.02640
\(244\) −8.29180 −0.530828
\(245\) 0 0
\(246\) 3.61803 0.230677
\(247\) −11.7082 −0.744975
\(248\) 20.1246 1.27791
\(249\) 2.47214 0.156665
\(250\) 0 0
\(251\) 21.7082 1.37021 0.685105 0.728444i \(-0.259756\pi\)
0.685105 + 0.728444i \(0.259756\pi\)
\(252\) −1.23607 −0.0778650
\(253\) −4.47214 −0.281161
\(254\) −17.3262 −1.08714
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 0.236068 0.0147255 0.00736276 0.999973i \(-0.497656\pi\)
0.00736276 + 0.999973i \(0.497656\pi\)
\(258\) 10.4721 0.651967
\(259\) −7.70820 −0.478964
\(260\) 0 0
\(261\) −14.9443 −0.925027
\(262\) −22.5623 −1.39390
\(263\) 9.05573 0.558400 0.279200 0.960233i \(-0.409931\pi\)
0.279200 + 0.960233i \(0.409931\pi\)
\(264\) 10.0000 0.615457
\(265\) 0 0
\(266\) −4.47214 −0.274204
\(267\) −8.94427 −0.547381
\(268\) −6.29180 −0.384333
\(269\) 13.1803 0.803620 0.401810 0.915723i \(-0.368381\pi\)
0.401810 + 0.915723i \(0.368381\pi\)
\(270\) 0 0
\(271\) 0.944272 0.0573604 0.0286802 0.999589i \(-0.490870\pi\)
0.0286802 + 0.999589i \(0.490870\pi\)
\(272\) 0 0
\(273\) 4.23607 0.256378
\(274\) −3.70820 −0.224021
\(275\) 0 0
\(276\) 0.618034 0.0372013
\(277\) 6.41641 0.385525 0.192762 0.981245i \(-0.438255\pi\)
0.192762 + 0.981245i \(0.438255\pi\)
\(278\) 18.5623 1.11329
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) −9.70820 −0.579143 −0.289571 0.957156i \(-0.593513\pi\)
−0.289571 + 0.957156i \(0.593513\pi\)
\(282\) −8.85410 −0.527254
\(283\) −1.05573 −0.0627565 −0.0313783 0.999508i \(-0.509990\pi\)
−0.0313783 + 0.999508i \(0.509990\pi\)
\(284\) −3.56231 −0.211384
\(285\) 0 0
\(286\) −30.6525 −1.81252
\(287\) 2.23607 0.131991
\(288\) 6.76393 0.398569
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 9.70820 0.569105
\(292\) −4.14590 −0.242620
\(293\) 2.29180 0.133888 0.0669441 0.997757i \(-0.478675\pi\)
0.0669441 + 0.997757i \(0.478675\pi\)
\(294\) 1.61803 0.0943657
\(295\) 0 0
\(296\) 17.2361 1.00183
\(297\) 22.3607 1.29750
\(298\) 17.4164 1.00891
\(299\) 4.23607 0.244978
\(300\) 0 0
\(301\) 6.47214 0.373048
\(302\) 11.6180 0.668543
\(303\) 13.4164 0.770752
\(304\) 13.4164 0.769484
\(305\) 0 0
\(306\) 0 0
\(307\) −15.4164 −0.879861 −0.439930 0.898032i \(-0.644997\pi\)
−0.439930 + 0.898032i \(0.644997\pi\)
\(308\) −2.76393 −0.157490
\(309\) 2.29180 0.130376
\(310\) 0 0
\(311\) 32.8885 1.86494 0.932469 0.361250i \(-0.117650\pi\)
0.932469 + 0.361250i \(0.117650\pi\)
\(312\) −9.47214 −0.536254
\(313\) −7.23607 −0.409007 −0.204503 0.978866i \(-0.565558\pi\)
−0.204503 + 0.978866i \(0.565558\pi\)
\(314\) 18.4721 1.04244
\(315\) 0 0
\(316\) −1.70820 −0.0960940
\(317\) 31.3050 1.75826 0.879131 0.476581i \(-0.158124\pi\)
0.879131 + 0.476581i \(0.158124\pi\)
\(318\) −10.9443 −0.613724
\(319\) −33.4164 −1.87096
\(320\) 0 0
\(321\) −6.76393 −0.377526
\(322\) 1.61803 0.0901695
\(323\) 0 0
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) 5.14590 0.285005
\(327\) 11.4164 0.631329
\(328\) −5.00000 −0.276079
\(329\) −5.47214 −0.301689
\(330\) 0 0
\(331\) −12.7082 −0.698506 −0.349253 0.937028i \(-0.613565\pi\)
−0.349253 + 0.937028i \(0.613565\pi\)
\(332\) 1.52786 0.0838524
\(333\) 15.4164 0.844814
\(334\) 22.4721 1.22962
\(335\) 0 0
\(336\) −4.85410 −0.264813
\(337\) −31.5967 −1.72118 −0.860592 0.509295i \(-0.829906\pi\)
−0.860592 + 0.509295i \(0.829906\pi\)
\(338\) 8.00000 0.435143
\(339\) 6.47214 0.351518
\(340\) 0 0
\(341\) 40.2492 2.17962
\(342\) 8.94427 0.483651
\(343\) 1.00000 0.0539949
\(344\) −14.4721 −0.780285
\(345\) 0 0
\(346\) 20.1803 1.08490
\(347\) −29.8885 −1.60450 −0.802251 0.596987i \(-0.796364\pi\)
−0.802251 + 0.596987i \(0.796364\pi\)
\(348\) 4.61803 0.247553
\(349\) 14.7082 0.787312 0.393656 0.919258i \(-0.371210\pi\)
0.393656 + 0.919258i \(0.371210\pi\)
\(350\) 0 0
\(351\) −21.1803 −1.13052
\(352\) 15.1246 0.806145
\(353\) 6.23607 0.331912 0.165956 0.986133i \(-0.446929\pi\)
0.165956 + 0.986133i \(0.446929\pi\)
\(354\) −16.9443 −0.900578
\(355\) 0 0
\(356\) −5.52786 −0.292976
\(357\) 0 0
\(358\) −8.56231 −0.452532
\(359\) 9.23607 0.487461 0.243731 0.969843i \(-0.421629\pi\)
0.243731 + 0.969843i \(0.421629\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) −37.1246 −1.95123
\(363\) 9.00000 0.472377
\(364\) 2.61803 0.137222
\(365\) 0 0
\(366\) −21.7082 −1.13471
\(367\) 31.4164 1.63992 0.819962 0.572419i \(-0.193995\pi\)
0.819962 + 0.572419i \(0.193995\pi\)
\(368\) −4.85410 −0.253038
\(369\) −4.47214 −0.232810
\(370\) 0 0
\(371\) −6.76393 −0.351166
\(372\) −5.56231 −0.288392
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.2361 0.631027
\(377\) 31.6525 1.63019
\(378\) −8.09017 −0.416113
\(379\) 21.7082 1.11508 0.557538 0.830152i \(-0.311746\pi\)
0.557538 + 0.830152i \(0.311746\pi\)
\(380\) 0 0
\(381\) −10.7082 −0.548598
\(382\) −2.94427 −0.150642
\(383\) 2.29180 0.117105 0.0585527 0.998284i \(-0.481351\pi\)
0.0585527 + 0.998284i \(0.481351\pi\)
\(384\) 13.6180 0.694942
\(385\) 0 0
\(386\) 36.2705 1.84612
\(387\) −12.9443 −0.657994
\(388\) 6.00000 0.304604
\(389\) 19.4164 0.984451 0.492225 0.870468i \(-0.336184\pi\)
0.492225 + 0.870468i \(0.336184\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.23607 −0.112938
\(393\) −13.9443 −0.703395
\(394\) 20.2705 1.02121
\(395\) 0 0
\(396\) 5.52786 0.277786
\(397\) −5.65248 −0.283690 −0.141845 0.989889i \(-0.545303\pi\)
−0.141845 + 0.989889i \(0.545303\pi\)
\(398\) −32.1803 −1.61305
\(399\) −2.76393 −0.138370
\(400\) 0 0
\(401\) 15.8885 0.793436 0.396718 0.917941i \(-0.370149\pi\)
0.396718 + 0.917941i \(0.370149\pi\)
\(402\) −16.4721 −0.821555
\(403\) −38.1246 −1.89912
\(404\) 8.29180 0.412532
\(405\) 0 0
\(406\) 12.0902 0.600025
\(407\) 34.4721 1.70872
\(408\) 0 0
\(409\) −36.1246 −1.78625 −0.893124 0.449811i \(-0.851491\pi\)
−0.893124 + 0.449811i \(0.851491\pi\)
\(410\) 0 0
\(411\) −2.29180 −0.113046
\(412\) 1.41641 0.0697814
\(413\) −10.4721 −0.515300
\(414\) −3.23607 −0.159044
\(415\) 0 0
\(416\) −14.3262 −0.702402
\(417\) 11.4721 0.561793
\(418\) 20.0000 0.978232
\(419\) 17.5279 0.856292 0.428146 0.903710i \(-0.359167\pi\)
0.428146 + 0.903710i \(0.359167\pi\)
\(420\) 0 0
\(421\) −35.4164 −1.72609 −0.863045 0.505127i \(-0.831446\pi\)
−0.863045 + 0.505127i \(0.831446\pi\)
\(422\) −19.4164 −0.945176
\(423\) 10.9443 0.532129
\(424\) 15.1246 0.734516
\(425\) 0 0
\(426\) −9.32624 −0.451858
\(427\) −13.4164 −0.649265
\(428\) −4.18034 −0.202064
\(429\) −18.9443 −0.914638
\(430\) 0 0
\(431\) −4.18034 −0.201360 −0.100680 0.994919i \(-0.532102\pi\)
−0.100680 + 0.994919i \(0.532102\pi\)
\(432\) 24.2705 1.16772
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −14.5623 −0.699013
\(435\) 0 0
\(436\) 7.05573 0.337908
\(437\) −2.76393 −0.132217
\(438\) −10.8541 −0.518629
\(439\) −32.3050 −1.54183 −0.770916 0.636937i \(-0.780201\pi\)
−0.770916 + 0.636937i \(0.780201\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −15.1803 −0.721240 −0.360620 0.932713i \(-0.617435\pi\)
−0.360620 + 0.932713i \(0.617435\pi\)
\(444\) −4.76393 −0.226086
\(445\) 0 0
\(446\) 37.8885 1.79407
\(447\) 10.7639 0.509117
\(448\) 4.23607 0.200135
\(449\) −20.8328 −0.983161 −0.491581 0.870832i \(-0.663581\pi\)
−0.491581 + 0.870832i \(0.663581\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 4.00000 0.188144
\(453\) 7.18034 0.337362
\(454\) −48.3607 −2.26968
\(455\) 0 0
\(456\) 6.18034 0.289421
\(457\) −5.52786 −0.258583 −0.129291 0.991607i \(-0.541270\pi\)
−0.129291 + 0.991607i \(0.541270\pi\)
\(458\) 31.1246 1.45436
\(459\) 0 0
\(460\) 0 0
\(461\) −30.2361 −1.40823 −0.704117 0.710084i \(-0.748657\pi\)
−0.704117 + 0.710084i \(0.748657\pi\)
\(462\) −7.23607 −0.336652
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −36.2705 −1.68382
\(465\) 0 0
\(466\) 4.09017 0.189473
\(467\) 20.4721 0.947337 0.473669 0.880703i \(-0.342929\pi\)
0.473669 + 0.880703i \(0.342929\pi\)
\(468\) −5.23607 −0.242037
\(469\) −10.1803 −0.470084
\(470\) 0 0
\(471\) 11.4164 0.526040
\(472\) 23.4164 1.07783
\(473\) −28.9443 −1.33086
\(474\) −4.47214 −0.205412
\(475\) 0 0
\(476\) 0 0
\(477\) 13.5279 0.619398
\(478\) −32.7426 −1.49761
\(479\) 17.1246 0.782443 0.391222 0.920296i \(-0.372053\pi\)
0.391222 + 0.920296i \(0.372053\pi\)
\(480\) 0 0
\(481\) −32.6525 −1.48882
\(482\) 8.76393 0.399186
\(483\) 1.00000 0.0455016
\(484\) 5.56231 0.252832
\(485\) 0 0
\(486\) 25.8885 1.17433
\(487\) 20.1246 0.911933 0.455967 0.889997i \(-0.349294\pi\)
0.455967 + 0.889997i \(0.349294\pi\)
\(488\) 30.0000 1.35804
\(489\) 3.18034 0.143820
\(490\) 0 0
\(491\) 16.2361 0.732723 0.366362 0.930472i \(-0.380603\pi\)
0.366362 + 0.930472i \(0.380603\pi\)
\(492\) 1.38197 0.0623038
\(493\) 0 0
\(494\) −18.9443 −0.852343
\(495\) 0 0
\(496\) 43.6869 1.96160
\(497\) −5.76393 −0.258548
\(498\) 4.00000 0.179244
\(499\) −23.2918 −1.04268 −0.521342 0.853348i \(-0.674568\pi\)
−0.521342 + 0.853348i \(0.674568\pi\)
\(500\) 0 0
\(501\) 13.8885 0.620494
\(502\) 35.1246 1.56769
\(503\) −33.5967 −1.49800 −0.749002 0.662567i \(-0.769467\pi\)
−0.749002 + 0.662567i \(0.769467\pi\)
\(504\) 4.47214 0.199205
\(505\) 0 0
\(506\) −7.23607 −0.321682
\(507\) 4.94427 0.219583
\(508\) −6.61803 −0.293628
\(509\) −35.1803 −1.55934 −0.779671 0.626190i \(-0.784613\pi\)
−0.779671 + 0.626190i \(0.784613\pi\)
\(510\) 0 0
\(511\) −6.70820 −0.296753
\(512\) −5.29180 −0.233867
\(513\) 13.8197 0.610153
\(514\) 0.381966 0.0168478
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 24.4721 1.07628
\(518\) −12.4721 −0.547994
\(519\) 12.4721 0.547466
\(520\) 0 0
\(521\) −41.3050 −1.80960 −0.904801 0.425834i \(-0.859981\pi\)
−0.904801 + 0.425834i \(0.859981\pi\)
\(522\) −24.1803 −1.05834
\(523\) −5.70820 −0.249602 −0.124801 0.992182i \(-0.539829\pi\)
−0.124801 + 0.992182i \(0.539829\pi\)
\(524\) −8.61803 −0.376481
\(525\) 0 0
\(526\) 14.6525 0.638878
\(527\) 0 0
\(528\) 21.7082 0.944728
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 20.9443 0.908904
\(532\) −1.70820 −0.0740600
\(533\) 9.47214 0.410284
\(534\) −14.4721 −0.626271
\(535\) 0 0
\(536\) 22.7639 0.983252
\(537\) −5.29180 −0.228358
\(538\) 21.3262 0.919439
\(539\) −4.47214 −0.192629
\(540\) 0 0
\(541\) 21.0000 0.902861 0.451430 0.892306i \(-0.350914\pi\)
0.451430 + 0.892306i \(0.350914\pi\)
\(542\) 1.52786 0.0656274
\(543\) −22.9443 −0.984633
\(544\) 0 0
\(545\) 0 0
\(546\) 6.85410 0.293328
\(547\) −20.8197 −0.890184 −0.445092 0.895485i \(-0.646829\pi\)
−0.445092 + 0.895485i \(0.646829\pi\)
\(548\) −1.41641 −0.0605059
\(549\) 26.8328 1.14520
\(550\) 0 0
\(551\) −20.6525 −0.879825
\(552\) −2.23607 −0.0951734
\(553\) −2.76393 −0.117534
\(554\) 10.3820 0.441087
\(555\) 0 0
\(556\) 7.09017 0.300690
\(557\) 31.2361 1.32351 0.661757 0.749718i \(-0.269811\pi\)
0.661757 + 0.749718i \(0.269811\pi\)
\(558\) 29.1246 1.23294
\(559\) 27.4164 1.15959
\(560\) 0 0
\(561\) 0 0
\(562\) −15.7082 −0.662611
\(563\) −39.5967 −1.66880 −0.834402 0.551156i \(-0.814187\pi\)
−0.834402 + 0.551156i \(0.814187\pi\)
\(564\) −3.38197 −0.142406
\(565\) 0 0
\(566\) −1.70820 −0.0718012
\(567\) 1.00000 0.0419961
\(568\) 12.8885 0.540791
\(569\) 14.1803 0.594471 0.297235 0.954804i \(-0.403935\pi\)
0.297235 + 0.954804i \(0.403935\pi\)
\(570\) 0 0
\(571\) 29.7082 1.24325 0.621625 0.783315i \(-0.286473\pi\)
0.621625 + 0.783315i \(0.286473\pi\)
\(572\) −11.7082 −0.489545
\(573\) −1.81966 −0.0760174
\(574\) 3.61803 0.151014
\(575\) 0 0
\(576\) −8.47214 −0.353006
\(577\) 28.7082 1.19514 0.597569 0.801817i \(-0.296133\pi\)
0.597569 + 0.801817i \(0.296133\pi\)
\(578\) −27.5066 −1.14412
\(579\) 22.4164 0.931594
\(580\) 0 0
\(581\) 2.47214 0.102561
\(582\) 15.7082 0.651126
\(583\) 30.2492 1.25279
\(584\) 15.0000 0.620704
\(585\) 0 0
\(586\) 3.70820 0.153184
\(587\) 14.8885 0.614516 0.307258 0.951626i \(-0.400589\pi\)
0.307258 + 0.951626i \(0.400589\pi\)
\(588\) 0.618034 0.0254873
\(589\) 24.8754 1.02497
\(590\) 0 0
\(591\) 12.5279 0.515327
\(592\) 37.4164 1.53780
\(593\) 10.3607 0.425462 0.212731 0.977111i \(-0.431764\pi\)
0.212731 + 0.977111i \(0.431764\pi\)
\(594\) 36.1803 1.48450
\(595\) 0 0
\(596\) 6.65248 0.272496
\(597\) −19.8885 −0.813984
\(598\) 6.85410 0.280285
\(599\) 6.47214 0.264444 0.132222 0.991220i \(-0.457789\pi\)
0.132222 + 0.991220i \(0.457789\pi\)
\(600\) 0 0
\(601\) 17.7639 0.724606 0.362303 0.932060i \(-0.381991\pi\)
0.362303 + 0.932060i \(0.381991\pi\)
\(602\) 10.4721 0.426812
\(603\) 20.3607 0.829151
\(604\) 4.43769 0.180567
\(605\) 0 0
\(606\) 21.7082 0.881836
\(607\) −29.3050 −1.18945 −0.594726 0.803929i \(-0.702739\pi\)
−0.594726 + 0.803929i \(0.702739\pi\)
\(608\) 9.34752 0.379092
\(609\) 7.47214 0.302786
\(610\) 0 0
\(611\) −23.1803 −0.937776
\(612\) 0 0
\(613\) −0.180340 −0.00728386 −0.00364193 0.999993i \(-0.501159\pi\)
−0.00364193 + 0.999993i \(0.501159\pi\)
\(614\) −24.9443 −1.00667
\(615\) 0 0
\(616\) 10.0000 0.402911
\(617\) −30.7639 −1.23851 −0.619255 0.785190i \(-0.712565\pi\)
−0.619255 + 0.785190i \(0.712565\pi\)
\(618\) 3.70820 0.149166
\(619\) 37.4853 1.50666 0.753331 0.657642i \(-0.228446\pi\)
0.753331 + 0.657642i \(0.228446\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 53.2148 2.13372
\(623\) −8.94427 −0.358345
\(624\) −20.5623 −0.823151
\(625\) 0 0
\(626\) −11.7082 −0.467954
\(627\) 12.3607 0.493638
\(628\) 7.05573 0.281554
\(629\) 0 0
\(630\) 0 0
\(631\) −23.4164 −0.932192 −0.466096 0.884734i \(-0.654340\pi\)
−0.466096 + 0.884734i \(0.654340\pi\)
\(632\) 6.18034 0.245841
\(633\) −12.0000 −0.476957
\(634\) 50.6525 2.01167
\(635\) 0 0
\(636\) −4.18034 −0.165761
\(637\) 4.23607 0.167839
\(638\) −54.0689 −2.14061
\(639\) 11.5279 0.456035
\(640\) 0 0
\(641\) 24.9443 0.985240 0.492620 0.870245i \(-0.336039\pi\)
0.492620 + 0.870245i \(0.336039\pi\)
\(642\) −10.9443 −0.431936
\(643\) 4.29180 0.169252 0.0846260 0.996413i \(-0.473030\pi\)
0.0846260 + 0.996413i \(0.473030\pi\)
\(644\) 0.618034 0.0243540
\(645\) 0 0
\(646\) 0 0
\(647\) 27.4721 1.08004 0.540021 0.841652i \(-0.318416\pi\)
0.540021 + 0.841652i \(0.318416\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 46.8328 1.83835
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 1.96556 0.0769772
\(653\) −33.4721 −1.30987 −0.654933 0.755687i \(-0.727303\pi\)
−0.654933 + 0.755687i \(0.727303\pi\)
\(654\) 18.4721 0.722318
\(655\) 0 0
\(656\) −10.8541 −0.423781
\(657\) 13.4164 0.523424
\(658\) −8.85410 −0.345169
\(659\) 11.2361 0.437695 0.218848 0.975759i \(-0.429770\pi\)
0.218848 + 0.975759i \(0.429770\pi\)
\(660\) 0 0
\(661\) −28.2918 −1.10042 −0.550212 0.835025i \(-0.685453\pi\)
−0.550212 + 0.835025i \(0.685453\pi\)
\(662\) −20.5623 −0.799177
\(663\) 0 0
\(664\) −5.52786 −0.214523
\(665\) 0 0
\(666\) 24.9443 0.966571
\(667\) 7.47214 0.289322
\(668\) 8.58359 0.332109
\(669\) 23.4164 0.905331
\(670\) 0 0
\(671\) 60.0000 2.31627
\(672\) −3.38197 −0.130462
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) −51.1246 −1.96925
\(675\) 0 0
\(676\) 3.05573 0.117528
\(677\) −18.7639 −0.721156 −0.360578 0.932729i \(-0.617421\pi\)
−0.360578 + 0.932729i \(0.617421\pi\)
\(678\) 10.4721 0.402180
\(679\) 9.70820 0.372567
\(680\) 0 0
\(681\) −29.8885 −1.14533
\(682\) 65.1246 2.49375
\(683\) −17.1803 −0.657387 −0.328694 0.944437i \(-0.606608\pi\)
−0.328694 + 0.944437i \(0.606608\pi\)
\(684\) 3.41641 0.130630
\(685\) 0 0
\(686\) 1.61803 0.0617768
\(687\) 19.2361 0.733902
\(688\) −31.4164 −1.19774
\(689\) −28.6525 −1.09157
\(690\) 0 0
\(691\) −10.8328 −0.412100 −0.206050 0.978541i \(-0.566061\pi\)
−0.206050 + 0.978541i \(0.566061\pi\)
\(692\) 7.70820 0.293022
\(693\) 8.94427 0.339765
\(694\) −48.3607 −1.83575
\(695\) 0 0
\(696\) −16.7082 −0.633323
\(697\) 0 0
\(698\) 23.7984 0.900782
\(699\) 2.52786 0.0956126
\(700\) 0 0
\(701\) −27.2361 −1.02869 −0.514346 0.857583i \(-0.671965\pi\)
−0.514346 + 0.857583i \(0.671965\pi\)
\(702\) −34.2705 −1.29346
\(703\) 21.3050 0.803531
\(704\) −18.9443 −0.713989
\(705\) 0 0
\(706\) 10.0902 0.379749
\(707\) 13.4164 0.504576
\(708\) −6.47214 −0.243238
\(709\) −46.2492 −1.73693 −0.868463 0.495754i \(-0.834892\pi\)
−0.868463 + 0.495754i \(0.834892\pi\)
\(710\) 0 0
\(711\) 5.52786 0.207311
\(712\) 20.0000 0.749532
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) 0 0
\(716\) −3.27051 −0.122225
\(717\) −20.2361 −0.755730
\(718\) 14.9443 0.557715
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 2.29180 0.0853509
\(722\) −18.3820 −0.684106
\(723\) 5.41641 0.201438
\(724\) −14.1803 −0.527008
\(725\) 0 0
\(726\) 14.5623 0.540458
\(727\) 1.23607 0.0458432 0.0229216 0.999737i \(-0.492703\pi\)
0.0229216 + 0.999737i \(0.492703\pi\)
\(728\) −9.47214 −0.351061
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) −8.29180 −0.306474
\(733\) 4.94427 0.182621 0.0913104 0.995822i \(-0.470894\pi\)
0.0913104 + 0.995822i \(0.470894\pi\)
\(734\) 50.8328 1.87627
\(735\) 0 0
\(736\) −3.38197 −0.124661
\(737\) 45.5279 1.67704
\(738\) −7.23607 −0.266363
\(739\) −4.70820 −0.173194 −0.0865970 0.996243i \(-0.527599\pi\)
−0.0865970 + 0.996243i \(0.527599\pi\)
\(740\) 0 0
\(741\) −11.7082 −0.430112
\(742\) −10.9443 −0.401777
\(743\) 34.4721 1.26466 0.632330 0.774699i \(-0.282099\pi\)
0.632330 + 0.774699i \(0.282099\pi\)
\(744\) 20.1246 0.737804
\(745\) 0 0
\(746\) −42.0689 −1.54025
\(747\) −4.94427 −0.180901
\(748\) 0 0
\(749\) −6.76393 −0.247149
\(750\) 0 0
\(751\) −35.4164 −1.29236 −0.646182 0.763184i \(-0.723635\pi\)
−0.646182 + 0.763184i \(0.723635\pi\)
\(752\) 26.5623 0.968628
\(753\) 21.7082 0.791091
\(754\) 51.2148 1.86513
\(755\) 0 0
\(756\) −3.09017 −0.112388
\(757\) −24.8328 −0.902564 −0.451282 0.892381i \(-0.649033\pi\)
−0.451282 + 0.892381i \(0.649033\pi\)
\(758\) 35.1246 1.27578
\(759\) −4.47214 −0.162328
\(760\) 0 0
\(761\) −0.347524 −0.0125977 −0.00629887 0.999980i \(-0.502005\pi\)
−0.00629887 + 0.999980i \(0.502005\pi\)
\(762\) −17.3262 −0.627663
\(763\) 11.4164 0.413302
\(764\) −1.12461 −0.0406870
\(765\) 0 0
\(766\) 3.70820 0.133983
\(767\) −44.3607 −1.60177
\(768\) 13.5623 0.489388
\(769\) −38.7639 −1.39786 −0.698932 0.715189i \(-0.746341\pi\)
−0.698932 + 0.715189i \(0.746341\pi\)
\(770\) 0 0
\(771\) 0.236068 0.00850178
\(772\) 13.8541 0.498620
\(773\) −9.34752 −0.336207 −0.168104 0.985769i \(-0.553764\pi\)
−0.168104 + 0.985769i \(0.553764\pi\)
\(774\) −20.9443 −0.752826
\(775\) 0 0
\(776\) −21.7082 −0.779279
\(777\) −7.70820 −0.276530
\(778\) 31.4164 1.12633
\(779\) −6.18034 −0.221434
\(780\) 0 0
\(781\) 25.7771 0.922377
\(782\) 0 0
\(783\) −37.3607 −1.33516
\(784\) −4.85410 −0.173361
\(785\) 0 0
\(786\) −22.5623 −0.804771
\(787\) −44.3607 −1.58129 −0.790644 0.612276i \(-0.790254\pi\)
−0.790644 + 0.612276i \(0.790254\pi\)
\(788\) 7.74265 0.275820
\(789\) 9.05573 0.322392
\(790\) 0 0
\(791\) 6.47214 0.230123
\(792\) −20.0000 −0.710669
\(793\) −56.8328 −2.01819
\(794\) −9.14590 −0.324576
\(795\) 0 0
\(796\) −12.2918 −0.435671
\(797\) 41.2361 1.46066 0.730328 0.683096i \(-0.239367\pi\)
0.730328 + 0.683096i \(0.239367\pi\)
\(798\) −4.47214 −0.158312
\(799\) 0 0
\(800\) 0 0
\(801\) 17.8885 0.632061
\(802\) 25.7082 0.907788
\(803\) 30.0000 1.05868
\(804\) −6.29180 −0.221895
\(805\) 0 0
\(806\) −61.6869 −2.17283
\(807\) 13.1803 0.463970
\(808\) −30.0000 −1.05540
\(809\) 20.8328 0.732443 0.366221 0.930528i \(-0.380651\pi\)
0.366221 + 0.930528i \(0.380651\pi\)
\(810\) 0 0
\(811\) −0.639320 −0.0224496 −0.0112248 0.999937i \(-0.503573\pi\)
−0.0112248 + 0.999937i \(0.503573\pi\)
\(812\) 4.61803 0.162061
\(813\) 0.944272 0.0331171
\(814\) 55.7771 1.95499
\(815\) 0 0
\(816\) 0 0
\(817\) −17.8885 −0.625841
\(818\) −58.4508 −2.04369
\(819\) −8.47214 −0.296040
\(820\) 0 0
\(821\) 49.4164 1.72464 0.862322 0.506360i \(-0.169009\pi\)
0.862322 + 0.506360i \(0.169009\pi\)
\(822\) −3.70820 −0.129338
\(823\) 41.5410 1.44803 0.724014 0.689785i \(-0.242295\pi\)
0.724014 + 0.689785i \(0.242295\pi\)
\(824\) −5.12461 −0.178524
\(825\) 0 0
\(826\) −16.9443 −0.589567
\(827\) −45.3050 −1.57541 −0.787704 0.616054i \(-0.788730\pi\)
−0.787704 + 0.616054i \(0.788730\pi\)
\(828\) −1.23607 −0.0429563
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 6.41641 0.222583
\(832\) 17.9443 0.622106
\(833\) 0 0
\(834\) 18.5623 0.642760
\(835\) 0 0
\(836\) 7.63932 0.264211
\(837\) 45.0000 1.55543
\(838\) 28.3607 0.979703
\(839\) −38.0689 −1.31428 −0.657142 0.753767i \(-0.728235\pi\)
−0.657142 + 0.753767i \(0.728235\pi\)
\(840\) 0 0
\(841\) 26.8328 0.925270
\(842\) −57.3050 −1.97486
\(843\) −9.70820 −0.334368
\(844\) −7.41641 −0.255283
\(845\) 0 0
\(846\) 17.7082 0.608821
\(847\) 9.00000 0.309244
\(848\) 32.8328 1.12748
\(849\) −1.05573 −0.0362325
\(850\) 0 0
\(851\) −7.70820 −0.264234
\(852\) −3.56231 −0.122043
\(853\) 8.83282 0.302430 0.151215 0.988501i \(-0.451681\pi\)
0.151215 + 0.988501i \(0.451681\pi\)
\(854\) −21.7082 −0.742839
\(855\) 0 0
\(856\) 15.1246 0.516949
\(857\) 57.5410 1.96556 0.982782 0.184769i \(-0.0591538\pi\)
0.982782 + 0.184769i \(0.0591538\pi\)
\(858\) −30.6525 −1.04646
\(859\) 38.4164 1.31075 0.655375 0.755303i \(-0.272510\pi\)
0.655375 + 0.755303i \(0.272510\pi\)
\(860\) 0 0
\(861\) 2.23607 0.0762050
\(862\) −6.76393 −0.230380
\(863\) 2.23607 0.0761166 0.0380583 0.999276i \(-0.487883\pi\)
0.0380583 + 0.999276i \(0.487883\pi\)
\(864\) 16.9098 0.575284
\(865\) 0 0
\(866\) −22.6525 −0.769762
\(867\) −17.0000 −0.577350
\(868\) −5.56231 −0.188797
\(869\) 12.3607 0.419307
\(870\) 0 0
\(871\) −43.1246 −1.46122
\(872\) −25.5279 −0.864483
\(873\) −19.4164 −0.657146
\(874\) −4.47214 −0.151272
\(875\) 0 0
\(876\) −4.14590 −0.140077
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) −52.2705 −1.76404
\(879\) 2.29180 0.0773004
\(880\) 0 0
\(881\) −23.5967 −0.794995 −0.397497 0.917603i \(-0.630121\pi\)
−0.397497 + 0.917603i \(0.630121\pi\)
\(882\) −3.23607 −0.108964
\(883\) 25.8885 0.871219 0.435609 0.900136i \(-0.356533\pi\)
0.435609 + 0.900136i \(0.356533\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24.5623 −0.825187
\(887\) 18.8885 0.634215 0.317108 0.948390i \(-0.397288\pi\)
0.317108 + 0.948390i \(0.397288\pi\)
\(888\) 17.2361 0.578405
\(889\) −10.7082 −0.359142
\(890\) 0 0
\(891\) −4.47214 −0.149822
\(892\) 14.4721 0.484563
\(893\) 15.1246 0.506126
\(894\) 17.4164 0.582492
\(895\) 0 0
\(896\) 13.6180 0.454947
\(897\) 4.23607 0.141438
\(898\) −33.7082 −1.12486
\(899\) −67.2492 −2.24289
\(900\) 0 0
\(901\) 0 0
\(902\) −16.1803 −0.538746
\(903\) 6.47214 0.215379
\(904\) −14.4721 −0.481336
\(905\) 0 0
\(906\) 11.6180 0.385983
\(907\) 54.5410 1.81100 0.905502 0.424341i \(-0.139494\pi\)
0.905502 + 0.424341i \(0.139494\pi\)
\(908\) −18.4721 −0.613019
\(909\) −26.8328 −0.889988
\(910\) 0 0
\(911\) 49.7771 1.64919 0.824594 0.565725i \(-0.191403\pi\)
0.824594 + 0.565725i \(0.191403\pi\)
\(912\) 13.4164 0.444262
\(913\) −11.0557 −0.365891
\(914\) −8.94427 −0.295850
\(915\) 0 0
\(916\) 11.8885 0.392809
\(917\) −13.9443 −0.460480
\(918\) 0 0
\(919\) −9.70820 −0.320244 −0.160122 0.987097i \(-0.551189\pi\)
−0.160122 + 0.987097i \(0.551189\pi\)
\(920\) 0 0
\(921\) −15.4164 −0.507988
\(922\) −48.9230 −1.61119
\(923\) −24.4164 −0.803676
\(924\) −2.76393 −0.0909267
\(925\) 0 0
\(926\) 0 0
\(927\) −4.58359 −0.150545
\(928\) −25.2705 −0.829545
\(929\) 20.3475 0.667581 0.333790 0.942647i \(-0.391672\pi\)
0.333790 + 0.942647i \(0.391672\pi\)
\(930\) 0 0
\(931\) −2.76393 −0.0905842
\(932\) 1.56231 0.0511750
\(933\) 32.8885 1.07672
\(934\) 33.1246 1.08387
\(935\) 0 0
\(936\) 18.9443 0.619213
\(937\) 16.8754 0.551295 0.275647 0.961259i \(-0.411108\pi\)
0.275647 + 0.961259i \(0.411108\pi\)
\(938\) −16.4721 −0.537834
\(939\) −7.23607 −0.236140
\(940\) 0 0
\(941\) −23.0132 −0.750207 −0.375104 0.926983i \(-0.622393\pi\)
−0.375104 + 0.926983i \(0.622393\pi\)
\(942\) 18.4721 0.601855
\(943\) 2.23607 0.0728164
\(944\) 50.8328 1.65447
\(945\) 0 0
\(946\) −46.8328 −1.52267
\(947\) −26.5967 −0.864278 −0.432139 0.901807i \(-0.642241\pi\)
−0.432139 + 0.901807i \(0.642241\pi\)
\(948\) −1.70820 −0.0554799
\(949\) −28.4164 −0.922436
\(950\) 0 0
\(951\) 31.3050 1.01513
\(952\) 0 0
\(953\) −47.8885 −1.55126 −0.775631 0.631187i \(-0.782568\pi\)
−0.775631 + 0.631187i \(0.782568\pi\)
\(954\) 21.8885 0.708668
\(955\) 0 0
\(956\) −12.5066 −0.404492
\(957\) −33.4164 −1.08020
\(958\) 27.7082 0.895211
\(959\) −2.29180 −0.0740060
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) −52.8328 −1.70340
\(963\) 13.5279 0.435929
\(964\) 3.34752 0.107816
\(965\) 0 0
\(966\) 1.61803 0.0520594
\(967\) −27.0689 −0.870477 −0.435238 0.900315i \(-0.643336\pi\)
−0.435238 + 0.900315i \(0.643336\pi\)
\(968\) −20.1246 −0.646830
\(969\) 0 0
\(970\) 0 0
\(971\) 20.2918 0.651195 0.325597 0.945509i \(-0.394435\pi\)
0.325597 + 0.945509i \(0.394435\pi\)
\(972\) 9.88854 0.317175
\(973\) 11.4721 0.367780
\(974\) 32.5623 1.04336
\(975\) 0 0
\(976\) 65.1246 2.08459
\(977\) −12.6525 −0.404789 −0.202394 0.979304i \(-0.564872\pi\)
−0.202394 + 0.979304i \(0.564872\pi\)
\(978\) 5.14590 0.164548
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) −22.8328 −0.728996
\(982\) 26.2705 0.838326
\(983\) −18.1803 −0.579863 −0.289931 0.957047i \(-0.593632\pi\)
−0.289931 + 0.957047i \(0.593632\pi\)
\(984\) −5.00000 −0.159394
\(985\) 0 0
\(986\) 0 0
\(987\) −5.47214 −0.174180
\(988\) −7.23607 −0.230210
\(989\) 6.47214 0.205802
\(990\) 0 0
\(991\) −31.0557 −0.986518 −0.493259 0.869883i \(-0.664194\pi\)
−0.493259 + 0.869883i \(0.664194\pi\)
\(992\) 30.4377 0.966398
\(993\) −12.7082 −0.403283
\(994\) −9.32624 −0.295810
\(995\) 0 0
\(996\) 1.52786 0.0484122
\(997\) −35.3050 −1.11812 −0.559060 0.829128i \(-0.688838\pi\)
−0.559060 + 0.829128i \(0.688838\pi\)
\(998\) −37.6869 −1.19296
\(999\) 38.5410 1.21938
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 4025.2.a.i.1.2 2
5.4 even 2 161.2.a.b.1.1 2
15.14 odd 2 1449.2.a.i.1.2 2
20.19 odd 2 2576.2.a.s.1.2 2
35.34 odd 2 1127.2.a.d.1.1 2
115.114 odd 2 3703.2.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.b.1.1 2 5.4 even 2
1127.2.a.d.1.1 2 35.34 odd 2
1449.2.a.i.1.2 2 15.14 odd 2
2576.2.a.s.1.2 2 20.19 odd 2
3703.2.a.b.1.1 2 115.114 odd 2
4025.2.a.i.1.2 2 1.1 even 1 trivial