Properties

Label 4002.2.a.e.1.1
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4002.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -5.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -3.00000 q^{26} +1.00000 q^{27} -1.00000 q^{29} +1.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} -4.00000 q^{38} +3.00000 q^{39} +1.00000 q^{40} +9.00000 q^{41} -6.00000 q^{43} +5.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +10.0000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{50} +4.00000 q^{51} +3.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -5.00000 q^{55} +4.00000 q^{57} +1.00000 q^{58} -7.00000 q^{59} -1.00000 q^{60} +1.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} -5.00000 q^{66} -13.0000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -5.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +3.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} -3.00000 q^{78} -2.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -6.00000 q^{83} -4.00000 q^{85} +6.00000 q^{86} -1.00000 q^{87} -5.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} -1.00000 q^{92} +5.00000 q^{93} -10.0000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +18.0000 q^{97} +7.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −3.00000 −0.588348
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 1.00000 0.182574
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.00000 0.870388
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −4.00000 −0.648886
\(39\) 3.00000 0.480384
\(40\) 1.00000 0.158114
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 5.00000 0.753778
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 4.00000 0.565685
\(51\) 4.00000 0.560112
\(52\) 3.00000 0.416025
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 1.00000 0.131306
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) −1.00000 −0.129099
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) −5.00000 −0.615457
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 3.00000 0.348743
\(75\) −4.00000 −0.461880
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 6.00000 0.646997
\(87\) −1.00000 −0.107211
\(88\) −5.00000 −0.533002
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 5.00000 0.518476
\(94\) −10.0000 −1.03142
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 7.00000 0.707107
\(99\) 5.00000 0.502519
\(100\) −4.00000 −0.400000
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) −4.00000 −0.396059
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 5.00000 0.476731
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.00000 0.0932505
\(116\) −1.00000 −0.0928477
\(117\) 3.00000 0.277350
\(118\) 7.00000 0.644402
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 14.0000 1.27273
\(122\) −1.00000 −0.0905357
\(123\) 9.00000 0.811503
\(124\) 5.00000 0.449013
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 9.00000 0.798621 0.399310 0.916816i \(-0.369250\pi\)
0.399310 + 0.916816i \(0.369250\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.00000 −0.528271
\(130\) 3.00000 0.263117
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 5.00000 0.435194
\(133\) 0 0
\(134\) 13.0000 1.12303
\(135\) −1.00000 −0.0860663
\(136\) −4.00000 −0.342997
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 1.00000 0.0851257
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 5.00000 0.419591
\(143\) 15.0000 1.25436
\(144\) 1.00000 0.0833333
\(145\) 1.00000 0.0830455
\(146\) 4.00000 0.331042
\(147\) −7.00000 −0.577350
\(148\) −3.00000 −0.246598
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 4.00000 0.326599
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) −4.00000 −0.324443
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −5.00000 −0.401610
\(156\) 3.00000 0.240192
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 2.00000 0.159111
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) 9.00000 0.702782
\(165\) −5.00000 −0.389249
\(166\) 6.00000 0.465690
\(167\) 13.0000 1.00597 0.502985 0.864295i \(-0.332235\pi\)
0.502985 + 0.864295i \(0.332235\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 4.00000 0.306786
\(171\) 4.00000 0.305888
\(172\) −6.00000 −0.457496
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) −7.00000 −0.526152
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) 1.00000 0.0737210
\(185\) 3.00000 0.220564
\(186\) −5.00000 −0.366618
\(187\) 20.0000 1.46254
\(188\) 10.0000 0.729325
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 1.00000 0.0721688
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −18.0000 −1.29232
\(195\) −3.00000 −0.214834
\(196\) −7.00000 −0.500000
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −5.00000 −0.355335
\(199\) 9.00000 0.637993 0.318997 0.947756i \(-0.396654\pi\)
0.318997 + 0.947756i \(0.396654\pi\)
\(200\) 4.00000 0.282843
\(201\) −13.0000 −0.916949
\(202\) −15.0000 −1.05540
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) −9.00000 −0.628587
\(206\) −13.0000 −0.905753
\(207\) −1.00000 −0.0695048
\(208\) 3.00000 0.208013
\(209\) 20.0000 1.38343
\(210\) 0 0
\(211\) −17.0000 −1.17033 −0.585164 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(212\) 6.00000 0.412082
\(213\) −5.00000 −0.342594
\(214\) −4.00000 −0.273434
\(215\) 6.00000 0.409197
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) −4.00000 −0.270295
\(220\) −5.00000 −0.337100
\(221\) 12.0000 0.807207
\(222\) 3.00000 0.201347
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) −6.00000 −0.399114
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 4.00000 0.264906
\(229\) 29.0000 1.91637 0.958187 0.286143i \(-0.0923732\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) −3.00000 −0.196116
\(235\) −10.0000 −0.652328
\(236\) −7.00000 −0.455661
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) 19.0000 1.22901 0.614504 0.788914i \(-0.289356\pi\)
0.614504 + 0.788914i \(0.289356\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) −14.0000 −0.899954
\(243\) 1.00000 0.0641500
\(244\) 1.00000 0.0640184
\(245\) 7.00000 0.447214
\(246\) −9.00000 −0.573819
\(247\) 12.0000 0.763542
\(248\) −5.00000 −0.317500
\(249\) −6.00000 −0.380235
\(250\) −9.00000 −0.569210
\(251\) 17.0000 1.07303 0.536515 0.843891i \(-0.319740\pi\)
0.536515 + 0.843891i \(0.319740\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) −9.00000 −0.564710
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 6.00000 0.373544
\(259\) 0 0
\(260\) −3.00000 −0.186052
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −5.00000 −0.307729
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −13.0000 −0.794101
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 1.00000 0.0608581
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) −20.0000 −1.20605
\(276\) −1.00000 −0.0601929
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) 2.00000 0.119952
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −10.0000 −0.595491
\(283\) −29.0000 −1.72387 −0.861936 0.507018i \(-0.830748\pi\)
−0.861936 + 0.507018i \(0.830748\pi\)
\(284\) −5.00000 −0.296695
\(285\) −4.00000 −0.236940
\(286\) −15.0000 −0.886969
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −1.00000 −0.0587220
\(291\) 18.0000 1.05518
\(292\) −4.00000 −0.234082
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 7.00000 0.408248
\(295\) 7.00000 0.407556
\(296\) 3.00000 0.174371
\(297\) 5.00000 0.290129
\(298\) 15.0000 0.868927
\(299\) −3.00000 −0.173494
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 20.0000 1.15087
\(303\) 15.0000 0.861727
\(304\) 4.00000 0.229416
\(305\) −1.00000 −0.0572598
\(306\) −4.00000 −0.228665
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 13.0000 0.739544
\(310\) 5.00000 0.283981
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) −3.00000 −0.169842
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −6.00000 −0.336463
\(319\) −5.00000 −0.279946
\(320\) −1.00000 −0.0559017
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) −12.0000 −0.665640
\(326\) −13.0000 −0.720003
\(327\) −14.0000 −0.774202
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 5.00000 0.275241
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −6.00000 −0.329293
\(333\) −3.00000 −0.164399
\(334\) −13.0000 −0.711328
\(335\) 13.0000 0.710266
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 4.00000 0.217571
\(339\) 6.00000 0.325875
\(340\) −4.00000 −0.216930
\(341\) 25.0000 1.35383
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 6.00000 0.323498
\(345\) 1.00000 0.0538382
\(346\) 6.00000 0.322562
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) −5.00000 −0.266501
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 7.00000 0.372046
\(355\) 5.00000 0.265372
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −1.00000 −0.0522708
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 9.00000 0.468521
\(370\) −3.00000 −0.155963
\(371\) 0 0
\(372\) 5.00000 0.259238
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) −20.0000 −1.03418
\(375\) 9.00000 0.464758
\(376\) −10.0000 −0.515711
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) −4.00000 −0.205196
\(381\) 9.00000 0.461084
\(382\) 3.00000 0.153493
\(383\) −10.0000 −0.510976 −0.255488 0.966812i \(-0.582236\pi\)
−0.255488 + 0.966812i \(0.582236\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) −6.00000 −0.304997
\(388\) 18.0000 0.913812
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 3.00000 0.151911
\(391\) −4.00000 −0.202289
\(392\) 7.00000 0.353553
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 2.00000 0.100631
\(396\) 5.00000 0.251259
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −9.00000 −0.451129
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −23.0000 −1.14857 −0.574283 0.818657i \(-0.694719\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 13.0000 0.648381
\(403\) 15.0000 0.747203
\(404\) 15.0000 0.746278
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −15.0000 −0.743522
\(408\) −4.00000 −0.198030
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 9.00000 0.444478
\(411\) 8.00000 0.394611
\(412\) 13.0000 0.640464
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 6.00000 0.294528
\(416\) −3.00000 −0.147087
\(417\) −2.00000 −0.0979404
\(418\) −20.0000 −0.978232
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 17.0000 0.827547
\(423\) 10.0000 0.486217
\(424\) −6.00000 −0.291386
\(425\) −16.0000 −0.776114
\(426\) 5.00000 0.242251
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 15.0000 0.724207
\(430\) −6.00000 −0.289346
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) −14.0000 −0.670478
\(437\) −4.00000 −0.191346
\(438\) 4.00000 0.191127
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 5.00000 0.238366
\(441\) −7.00000 −0.333333
\(442\) −12.0000 −0.570782
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −3.00000 −0.142374
\(445\) 10.0000 0.474045
\(446\) 12.0000 0.568216
\(447\) −15.0000 −0.709476
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 4.00000 0.188562
\(451\) 45.0000 2.11897
\(452\) 6.00000 0.282216
\(453\) −20.0000 −0.939682
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −29.0000 −1.35508
\(459\) 4.00000 0.186704
\(460\) 1.00000 0.0466252
\(461\) 23.0000 1.07122 0.535608 0.844466i \(-0.320082\pi\)
0.535608 + 0.844466i \(0.320082\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −5.00000 −0.231869
\(466\) −12.0000 −0.555889
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) 3.00000 0.138675
\(469\) 0 0
\(470\) 10.0000 0.461266
\(471\) 14.0000 0.645086
\(472\) 7.00000 0.322201
\(473\) −30.0000 −1.37940
\(474\) 2.00000 0.0918630
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) −19.0000 −0.869040
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 1.00000 0.0456435
\(481\) −9.00000 −0.410365
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −18.0000 −0.817338
\(486\) −1.00000 −0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 13.0000 0.587880
\(490\) −7.00000 −0.316228
\(491\) −10.0000 −0.451294 −0.225647 0.974209i \(-0.572450\pi\)
−0.225647 + 0.974209i \(0.572450\pi\)
\(492\) 9.00000 0.405751
\(493\) −4.00000 −0.180151
\(494\) −12.0000 −0.539906
\(495\) −5.00000 −0.224733
\(496\) 5.00000 0.224507
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) −38.0000 −1.70111 −0.850557 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 9.00000 0.402492
\(501\) 13.0000 0.580797
\(502\) −17.0000 −0.758747
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) 5.00000 0.222277
\(507\) −4.00000 −0.177646
\(508\) 9.00000 0.399310
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) 14.0000 0.617514
\(515\) −13.0000 −0.572848
\(516\) −6.00000 −0.264135
\(517\) 50.0000 2.19900
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 3.00000 0.131559
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 1.00000 0.0437688
\(523\) −9.00000 −0.393543 −0.196771 0.980449i \(-0.563046\pi\)
−0.196771 + 0.980449i \(0.563046\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 20.0000 0.871214
\(528\) 5.00000 0.217597
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −7.00000 −0.303774
\(532\) 0 0
\(533\) 27.0000 1.16950
\(534\) 10.0000 0.432742
\(535\) −4.00000 −0.172935
\(536\) 13.0000 0.561514
\(537\) 12.0000 0.517838
\(538\) −1.00000 −0.0431131
\(539\) −35.0000 −1.50756
\(540\) −1.00000 −0.0430331
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 13.0000 0.558398
\(543\) −2.00000 −0.0858282
\(544\) −4.00000 −0.171499
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 8.00000 0.341743
\(549\) 1.00000 0.0426790
\(550\) 20.0000 0.852803
\(551\) −4.00000 −0.170406
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −7.00000 −0.297402
\(555\) 3.00000 0.127343
\(556\) −2.00000 −0.0848189
\(557\) 25.0000 1.05928 0.529642 0.848221i \(-0.322326\pi\)
0.529642 + 0.848221i \(0.322326\pi\)
\(558\) −5.00000 −0.211667
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) 20.0000 0.844401
\(562\) −6.00000 −0.253095
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 10.0000 0.421076
\(565\) −6.00000 −0.252422
\(566\) 29.0000 1.21896
\(567\) 0 0
\(568\) 5.00000 0.209795
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 4.00000 0.167542
\(571\) −11.0000 −0.460336 −0.230168 0.973151i \(-0.573928\pi\)
−0.230168 + 0.973151i \(0.573928\pi\)
\(572\) 15.0000 0.627182
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 1.00000 0.0415945
\(579\) −16.0000 −0.664937
\(580\) 1.00000 0.0415227
\(581\) 0 0
\(582\) −18.0000 −0.746124
\(583\) 30.0000 1.24247
\(584\) 4.00000 0.165521
\(585\) −3.00000 −0.124035
\(586\) 24.0000 0.991431
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) −7.00000 −0.288675
\(589\) 20.0000 0.824086
\(590\) −7.00000 −0.288185
\(591\) 18.0000 0.740421
\(592\) −3.00000 −0.123299
\(593\) 4.00000 0.164260 0.0821302 0.996622i \(-0.473828\pi\)
0.0821302 + 0.996622i \(0.473828\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) −15.0000 −0.614424
\(597\) 9.00000 0.368345
\(598\) 3.00000 0.122679
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 4.00000 0.163299
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) −13.0000 −0.529401
\(604\) −20.0000 −0.813788
\(605\) −14.0000 −0.569181
\(606\) −15.0000 −0.609333
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 1.00000 0.0404888
\(611\) 30.0000 1.21367
\(612\) 4.00000 0.161690
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) −7.00000 −0.282497
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) −13.0000 −0.522937
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) −5.00000 −0.200805
\(621\) −1.00000 −0.0401286
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) 3.00000 0.120096
\(625\) 11.0000 0.440000
\(626\) 10.0000 0.399680
\(627\) 20.0000 0.798723
\(628\) 14.0000 0.558661
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −37.0000 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(632\) 2.00000 0.0795557
\(633\) −17.0000 −0.675689
\(634\) −3.00000 −0.119145
\(635\) −9.00000 −0.357154
\(636\) 6.00000 0.237915
\(637\) −21.0000 −0.832050
\(638\) 5.00000 0.197952
\(639\) −5.00000 −0.197797
\(640\) 1.00000 0.0395285
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) −4.00000 −0.157867
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) −16.0000 −0.629512
\(647\) 23.0000 0.904223 0.452112 0.891961i \(-0.350671\pi\)
0.452112 + 0.891961i \(0.350671\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −35.0000 −1.37387
\(650\) 12.0000 0.470679
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) −43.0000 −1.68272 −0.841360 0.540475i \(-0.818245\pi\)
−0.841360 + 0.540475i \(0.818245\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −5.00000 −0.194625
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 12.0000 0.466393
\(663\) 12.0000 0.466041
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 1.00000 0.0387202
\(668\) 13.0000 0.502985
\(669\) −12.0000 −0.463947
\(670\) −13.0000 −0.502234
\(671\) 5.00000 0.193023
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −13.0000 −0.500741
\(675\) −4.00000 −0.153960
\(676\) −4.00000 −0.153846
\(677\) 28.0000 1.07613 0.538064 0.842904i \(-0.319156\pi\)
0.538064 + 0.842904i \(0.319156\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 14.0000 0.536481
\(682\) −25.0000 −0.957299
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 4.00000 0.152944
\(685\) −8.00000 −0.305664
\(686\) 0 0
\(687\) 29.0000 1.10642
\(688\) −6.00000 −0.228748
\(689\) 18.0000 0.685745
\(690\) −1.00000 −0.0380693
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 2.00000 0.0758643
\(696\) 1.00000 0.0379049
\(697\) 36.0000 1.36360
\(698\) −1.00000 −0.0378506
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 9.00000 0.339925 0.169963 0.985451i \(-0.445635\pi\)
0.169963 + 0.985451i \(0.445635\pi\)
\(702\) −3.00000 −0.113228
\(703\) −12.0000 −0.452589
\(704\) 5.00000 0.188445
\(705\) −10.0000 −0.376622
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) −7.00000 −0.263076
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) −5.00000 −0.187647
\(711\) −2.00000 −0.0750059
\(712\) 10.0000 0.374766
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) −15.0000 −0.560968
\(716\) 12.0000 0.448461
\(717\) 19.0000 0.709568
\(718\) 24.0000 0.895672
\(719\) 1.00000 0.0372937 0.0186469 0.999826i \(-0.494064\pi\)
0.0186469 + 0.999826i \(0.494064\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −4.00000 −0.148762
\(724\) −2.00000 −0.0743294
\(725\) 4.00000 0.148556
\(726\) −14.0000 −0.519589
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) −24.0000 −0.887672
\(732\) 1.00000 0.0369611
\(733\) −23.0000 −0.849524 −0.424762 0.905305i \(-0.639642\pi\)
−0.424762 + 0.905305i \(0.639642\pi\)
\(734\) −14.0000 −0.516749
\(735\) 7.00000 0.258199
\(736\) 1.00000 0.0368605
\(737\) −65.0000 −2.39431
\(738\) −9.00000 −0.331295
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 3.00000 0.110282
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) −31.0000 −1.13728 −0.568640 0.822587i \(-0.692530\pi\)
−0.568640 + 0.822587i \(0.692530\pi\)
\(744\) −5.00000 −0.183309
\(745\) 15.0000 0.549557
\(746\) −2.00000 −0.0732252
\(747\) −6.00000 −0.219529
\(748\) 20.0000 0.731272
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) 10.0000 0.364662
\(753\) 17.0000 0.619514
\(754\) 3.00000 0.109254
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) −43.0000 −1.56286 −0.781431 0.623992i \(-0.785510\pi\)
−0.781431 + 0.623992i \(0.785510\pi\)
\(758\) −6.00000 −0.217930
\(759\) −5.00000 −0.181489
\(760\) 4.00000 0.145095
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −9.00000 −0.326036
\(763\) 0 0
\(764\) −3.00000 −0.108536
\(765\) −4.00000 −0.144620
\(766\) 10.0000 0.361315
\(767\) −21.0000 −0.758266
\(768\) 1.00000 0.0360844
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −16.0000 −0.575853
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 6.00000 0.215666
\(775\) −20.0000 −0.718421
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) −8.00000 −0.286814
\(779\) 36.0000 1.28983
\(780\) −3.00000 −0.107417
\(781\) −25.0000 −0.894570
\(782\) 4.00000 0.143040
\(783\) −1.00000 −0.0357371
\(784\) −7.00000 −0.250000
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) 45.0000 1.60408 0.802038 0.597272i \(-0.203749\pi\)
0.802038 + 0.597272i \(0.203749\pi\)
\(788\) 18.0000 0.641223
\(789\) 16.0000 0.569615
\(790\) −2.00000 −0.0711568
\(791\) 0 0
\(792\) −5.00000 −0.177667
\(793\) 3.00000 0.106533
\(794\) −38.0000 −1.34857
\(795\) −6.00000 −0.212798
\(796\) 9.00000 0.318997
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 4.00000 0.141421
\(801\) −10.0000 −0.353333
\(802\) 23.0000 0.812158
\(803\) −20.0000 −0.705785
\(804\) −13.0000 −0.458475
\(805\) 0 0
\(806\) −15.0000 −0.528352
\(807\) 1.00000 0.0352017
\(808\) −15.0000 −0.527698
\(809\) −53.0000 −1.86338 −0.931690 0.363253i \(-0.881666\pi\)
−0.931690 + 0.363253i \(0.881666\pi\)
\(810\) 1.00000 0.0351364
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −13.0000 −0.455930
\(814\) 15.0000 0.525750
\(815\) −13.0000 −0.455370
\(816\) 4.00000 0.140028
\(817\) −24.0000 −0.839654
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −8.00000 −0.279032
\(823\) −19.0000 −0.662298 −0.331149 0.943578i \(-0.607436\pi\)
−0.331149 + 0.943578i \(0.607436\pi\)
\(824\) −13.0000 −0.452876
\(825\) −20.0000 −0.696311
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) −6.00000 −0.208263
\(831\) 7.00000 0.242827
\(832\) 3.00000 0.104006
\(833\) −28.0000 −0.970143
\(834\) 2.00000 0.0692543
\(835\) −13.0000 −0.449884
\(836\) 20.0000 0.691714
\(837\) 5.00000 0.172825
\(838\) −6.00000 −0.207267
\(839\) −41.0000 −1.41548 −0.707739 0.706474i \(-0.750285\pi\)
−0.707739 + 0.706474i \(0.750285\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −15.0000 −0.516934
\(843\) 6.00000 0.206651
\(844\) −17.0000 −0.585164
\(845\) 4.00000 0.137604
\(846\) −10.0000 −0.343807
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −29.0000 −0.995277
\(850\) 16.0000 0.548795
\(851\) 3.00000 0.102839
\(852\) −5.00000 −0.171297
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) −4.00000 −0.136717
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) −15.0000 −0.512092
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) 13.0000 0.442525 0.221263 0.975214i \(-0.428982\pi\)
0.221263 + 0.975214i \(0.428982\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) −14.0000 −0.475739
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) −1.00000 −0.0339032
\(871\) −39.0000 −1.32146
\(872\) 14.0000 0.474100
\(873\) 18.0000 0.609208
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 10.0000 0.337484
\(879\) −24.0000 −0.809500
\(880\) −5.00000 −0.168550
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 7.00000 0.235702
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 12.0000 0.403604
\(885\) 7.00000 0.235302
\(886\) −18.0000 −0.604722
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 3.00000 0.100673
\(889\) 0 0
\(890\) −10.0000 −0.335201
\(891\) 5.00000 0.167506
\(892\) −12.0000 −0.401790
\(893\) 40.0000 1.33855
\(894\) 15.0000 0.501675
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 27.0000 0.901002
\(899\) −5.00000 −0.166759
\(900\) −4.00000 −0.133333
\(901\) 24.0000 0.799556
\(902\) −45.0000 −1.49834
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 2.00000 0.0664822
\(906\) 20.0000 0.664455
\(907\) 54.0000 1.79304 0.896520 0.443003i \(-0.146087\pi\)
0.896520 + 0.443003i \(0.146087\pi\)
\(908\) 14.0000 0.464606
\(909\) 15.0000 0.497519
\(910\) 0 0
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 4.00000 0.132453
\(913\) −30.0000 −0.992855
\(914\) 28.0000 0.926158
\(915\) −1.00000 −0.0330590
\(916\) 29.0000 0.958187
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) −27.0000 −0.890648 −0.445324 0.895370i \(-0.646911\pi\)
−0.445324 + 0.895370i \(0.646911\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 7.00000 0.230658
\(922\) −23.0000 −0.757465
\(923\) −15.0000 −0.493731
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) −20.0000 −0.657241
\(927\) 13.0000 0.426976
\(928\) 1.00000 0.0328266
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 5.00000 0.163956
\(931\) −28.0000 −0.917663
\(932\) 12.0000 0.393073
\(933\) 6.00000 0.196431
\(934\) 23.0000 0.752583
\(935\) −20.0000 −0.654070
\(936\) −3.00000 −0.0980581
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) −10.0000 −0.326164
\(941\) −33.0000 −1.07577 −0.537885 0.843018i \(-0.680776\pi\)
−0.537885 + 0.843018i \(0.680776\pi\)
\(942\) −14.0000 −0.456145
\(943\) −9.00000 −0.293080
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −2.00000 −0.0649570
\(949\) −12.0000 −0.389536
\(950\) 16.0000 0.519109
\(951\) 3.00000 0.0972817
\(952\) 0 0
\(953\) 10.0000 0.323932 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) −6.00000 −0.194257
\(955\) 3.00000 0.0970777
\(956\) 19.0000 0.614504
\(957\) −5.00000 −0.161627
\(958\) −15.0000 −0.484628
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −6.00000 −0.193548
\(962\) 9.00000 0.290172
\(963\) 4.00000 0.128898
\(964\) −4.00000 −0.128831
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −14.0000 −0.449977
\(969\) 16.0000 0.513994
\(970\) 18.0000 0.577945
\(971\) 7.00000 0.224641 0.112320 0.993672i \(-0.464172\pi\)
0.112320 + 0.993672i \(0.464172\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) −12.0000 −0.384308
\(976\) 1.00000 0.0320092
\(977\) −27.0000 −0.863807 −0.431903 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) −13.0000 −0.415694
\(979\) −50.0000 −1.59801
\(980\) 7.00000 0.223607
\(981\) −14.0000 −0.446986
\(982\) 10.0000 0.319113
\(983\) 17.0000 0.542216 0.271108 0.962549i \(-0.412610\pi\)
0.271108 + 0.962549i \(0.412610\pi\)
\(984\) −9.00000 −0.286910
\(985\) −18.0000 −0.573528
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 6.00000 0.190789
\(990\) 5.00000 0.158910
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −5.00000 −0.158750
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) −9.00000 −0.285319
\(996\) −6.00000 −0.190117
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 38.0000 1.20287
\(999\) −3.00000 −0.0949158
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 4002.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.e.1.1 1 1.1 even 1 trivial