Properties

Label 129.2.a.a.1.1
Level 129
Weight 2
Character 129.1
Self dual yes
Analytic conductor 1.030
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 129.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.03007018607\)
Analytic rank: \(1\)
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
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 129.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} +2.00000 q^{12} +3.00000 q^{13} +2.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} +4.00000 q^{20} +2.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{28} -5.00000 q^{31} +5.00000 q^{33} +4.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -3.00000 q^{39} -7.00000 q^{41} -1.00000 q^{43} +10.0000 q^{44} -2.00000 q^{45} -8.00000 q^{47} -4.00000 q^{48} -3.00000 q^{49} +3.00000 q^{51} -6.00000 q^{52} +3.00000 q^{53} +10.0000 q^{55} -2.00000 q^{57} +12.0000 q^{59} -4.00000 q^{60} -8.00000 q^{61} -2.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} -15.0000 q^{67} +6.00000 q^{68} +1.00000 q^{69} -14.0000 q^{71} +12.0000 q^{73} +1.00000 q^{75} -4.00000 q^{76} +10.0000 q^{77} -16.0000 q^{79} -8.00000 q^{80} +1.00000 q^{81} +15.0000 q^{83} -4.00000 q^{84} +6.00000 q^{85} +10.0000 q^{89} -6.00000 q^{91} +2.00000 q^{92} +5.00000 q^{93} -4.00000 q^{95} +11.0000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 2.00000 0.577350
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 4.00000 1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 4.00000 0.894427
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 10.0000 1.50756
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −4.00000 −0.577350
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −6.00000 −0.832050
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 10.0000 1.34840
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −4.00000 −0.516398
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) −8.00000 −1.00000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −15.0000 −1.83254 −0.916271 0.400559i \(-0.868816\pi\)
−0.916271 + 0.400559i \(0.868816\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 10.0000 1.13961
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −8.00000 −0.894427
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.0000 1.64646 0.823232 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) −4.00000 −0.436436
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 2.00000 0.208514
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 2.00000 0.200000
\(101\) −9.00000 −0.895533 −0.447767 0.894150i \(-0.647781\pi\)
−0.447767 + 0.894150i \(0.647781\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 2.00000 0.192450
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −8.00000 −0.755929
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 7.00000 0.631169
\(124\) 10.0000 0.898027
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) −10.0000 −0.870388
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) −8.00000 −0.676123
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) −15.0000 −1.25436
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) −16.0000 −1.31519
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 10.0000 0.803219
\(156\) 6.00000 0.480384
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 14.0000 1.09322
\(165\) −10.0000 −0.778499
\(166\) 0 0
\(167\) −1.00000 −0.0773823 −0.0386912 0.999251i \(-0.512319\pi\)
−0.0386912 + 0.999251i \(0.512319\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 2.00000 0.152499
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −20.0000 −1.50756
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −26.0000 −1.94333 −0.971666 0.236360i \(-0.924046\pi\)
−0.971666 + 0.236360i \(0.924046\pi\)
\(180\) 4.00000 0.298142
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) −16.0000 −1.17634
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) 16.0000 1.16692
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 8.00000 0.577350
\(193\) −21.0000 −1.51161 −0.755807 0.654795i \(-0.772755\pi\)
−0.755807 + 0.654795i \(0.772755\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 6.00000 0.428571
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 15.0000 1.05802
\(202\) 0 0
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 14.0000 0.977802
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 12.0000 0.832050
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) −6.00000 −0.412082
\(213\) 14.0000 0.959264
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) −12.0000 −0.810885
\(220\) −20.0000 −1.34840
\(221\) −9.00000 −0.605406
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 4.00000 0.264906
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) −10.0000 −0.657952
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 16.0000 1.04372
\(236\) −24.0000 −1.56227
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 8.00000 0.516398
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 16.0000 1.02430
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) −15.0000 −0.950586
\(250\) 0 0
\(251\) 25.0000 1.57799 0.788993 0.614402i \(-0.210603\pi\)
0.788993 + 0.614402i \(0.210603\pi\)
\(252\) 4.00000 0.251976
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 16.0000 1.00000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 12.0000 0.744208
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 30.0000 1.83254
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) −17.0000 −1.03268 −0.516338 0.856385i \(-0.672705\pi\)
−0.516338 + 0.856385i \(0.672705\pi\)
\(272\) −12.0000 −0.727607
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) −2.00000 −0.120386
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) 1.00000 0.0594438 0.0297219 0.999558i \(-0.490538\pi\)
0.0297219 + 0.999558i \(0.490538\pi\)
\(284\) 28.0000 1.66149
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 14.0000 0.826394
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −11.0000 −0.644831
\(292\) −24.0000 −1.40449
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) −2.00000 −0.115470
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) 9.00000 0.517036
\(304\) 8.00000 0.458831
\(305\) 16.0000 0.916157
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) −20.0000 −1.13961
\(309\) −5.00000 −0.284440
\(310\) 0 0
\(311\) −5.00000 −0.283524 −0.141762 0.989901i \(-0.545277\pi\)
−0.141762 + 0.989901i \(0.545277\pi\)
\(312\) 0 0
\(313\) 12.0000 0.678280 0.339140 0.940736i \(-0.389864\pi\)
0.339140 + 0.940736i \(0.389864\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 32.0000 1.80014
\(317\) 5.00000 0.280828 0.140414 0.990093i \(-0.455157\pi\)
0.140414 + 0.990093i \(0.455157\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 16.0000 0.894427
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) −2.00000 −0.111111
\(325\) −3.00000 −0.166410
\(326\) 0 0
\(327\) −11.0000 −0.608301
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −30.0000 −1.64646
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) 8.00000 0.436436
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) −12.0000 −0.650791
\(341\) 25.0000 1.35383
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −2.00000 −0.107676
\(346\) 0 0
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 0 0
\(353\) −27.0000 −1.43706 −0.718532 0.695493i \(-0.755186\pi\)
−0.718532 + 0.695493i \(0.755186\pi\)
\(354\) 0 0
\(355\) 28.0000 1.48609
\(356\) −20.0000 −1.06000
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) −5.00000 −0.263890 −0.131945 0.991257i \(-0.542122\pi\)
−0.131945 + 0.991257i \(0.542122\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 12.0000 0.628971
\(365\) −24.0000 −1.25622
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −4.00000 −0.208514
\(369\) −7.00000 −0.364405
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) −10.0000 −0.518476
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −13.0000 −0.667765 −0.333883 0.942615i \(-0.608359\pi\)
−0.333883 + 0.942615i \(0.608359\pi\)
\(380\) 8.00000 0.410391
\(381\) 3.00000 0.153695
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) −20.0000 −1.01929
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) −22.0000 −1.11688
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −16.0000 −0.807093
\(394\) 0 0
\(395\) 32.0000 1.61009
\(396\) 10.0000 0.502519
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) −4.00000 −0.200000
\(401\) −23.0000 −1.14857 −0.574283 0.818657i \(-0.694719\pi\)
−0.574283 + 0.818657i \(0.694719\pi\)
\(402\) 0 0
\(403\) −15.0000 −0.747203
\(404\) 18.0000 0.895533
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −10.0000 −0.492665
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) −30.0000 −1.47264
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) 34.0000 1.66101 0.830504 0.557012i \(-0.188052\pi\)
0.830504 + 0.557012i \(0.188052\pi\)
\(420\) 8.00000 0.390360
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 0 0
\(429\) 15.0000 0.724207
\(430\) 0 0
\(431\) 31.0000 1.49322 0.746609 0.665263i \(-0.231681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(432\) −4.00000 −0.192450
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −22.0000 −1.05361
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 16.0000 0.759326
\(445\) −20.0000 −0.948091
\(446\) 0 0
\(447\) 16.0000 0.756774
\(448\) 16.0000 0.755929
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) 35.0000 1.64809
\(452\) 8.00000 0.376288
\(453\) 2.00000 0.0939682
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) −4.00000 −0.186501
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) −10.0000 −0.463739
\(466\) 0 0
\(467\) −14.0000 −0.647843 −0.323921 0.946084i \(-0.605001\pi\)
−0.323921 + 0.946084i \(0.605001\pi\)
\(468\) −6.00000 −0.277350
\(469\) 30.0000 1.38527
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) 5.00000 0.229900
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) −12.0000 −0.550019
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 13.0000 0.593985 0.296993 0.954880i \(-0.404016\pi\)
0.296993 + 0.954880i \(0.404016\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 0 0
\(483\) −2.00000 −0.0910032
\(484\) −28.0000 −1.27273
\(485\) −22.0000 −0.998969
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) −14.0000 −0.631169
\(493\) 0 0
\(494\) 0 0
\(495\) 10.0000 0.449467
\(496\) −20.0000 −0.898027
\(497\) 28.0000 1.25597
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) −24.0000 −1.07331
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 4.00000 0.177646
\(508\) 6.00000 0.266207
\(509\) −39.0000 −1.72864 −0.864322 0.502938i \(-0.832252\pi\)
−0.864322 + 0.502938i \(0.832252\pi\)
\(510\) 0 0
\(511\) −24.0000 −1.06170
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) −10.0000 −0.440653
\(516\) −2.00000 −0.0880451
\(517\) 40.0000 1.75920
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) −6.00000 −0.262362 −0.131181 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) −32.0000 −1.39793
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 15.0000 0.653410
\(528\) 20.0000 0.870388
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 8.00000 0.346844
\(533\) −21.0000 −0.909611
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 26.0000 1.12198
\(538\) 0 0
\(539\) 15.0000 0.646096
\(540\) −4.00000 −0.172133
\(541\) −3.00000 −0.128980 −0.0644900 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) −22.0000 −0.942376
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) −36.0000 −1.53784
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 10.0000 0.424094
\(557\) −43.0000 −1.82197 −0.910984 0.412441i \(-0.864676\pi\)
−0.910984 + 0.412441i \(0.864676\pi\)
\(558\) 0 0
\(559\) −3.00000 −0.126886
\(560\) 16.0000 0.676123
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 1.00000 0.0421450 0.0210725 0.999778i \(-0.493292\pi\)
0.0210725 + 0.999778i \(0.493292\pi\)
\(564\) −16.0000 −0.673722
\(565\) 8.00000 0.336563
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) 27.0000 1.13190 0.565949 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 30.0000 1.25436
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) −8.00000 −0.333333
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 21.0000 0.872730
\(580\) 0 0
\(581\) −30.0000 −1.24461
\(582\) 0 0
\(583\) −15.0000 −0.621237
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) −6.00000 −0.247436
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 32.0000 1.31519
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 32.0000 1.31077
\(597\) 14.0000 0.572982
\(598\) 0 0
\(599\) 11.0000 0.449448 0.224724 0.974422i \(-0.427852\pi\)
0.224724 + 0.974422i \(0.427852\pi\)
\(600\) 0 0
\(601\) −12.0000 −0.489490 −0.244745 0.969587i \(-0.578704\pi\)
−0.244745 + 0.969587i \(0.578704\pi\)
\(602\) 0 0
\(603\) −15.0000 −0.610847
\(604\) 4.00000 0.162758
\(605\) −28.0000 −1.13836
\(606\) 0 0
\(607\) 36.0000 1.46119 0.730597 0.682808i \(-0.239242\pi\)
0.730597 + 0.682808i \(0.239242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 6.00000 0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) −14.0000 −0.564534
\(616\) 0 0
\(617\) −25.0000 −1.00646 −0.503231 0.864152i \(-0.667856\pi\)
−0.503231 + 0.864152i \(0.667856\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) −20.0000 −0.803219
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −20.0000 −0.801283
\(624\) −12.0000 −0.480384
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 10.0000 0.399362
\(628\) 12.0000 0.478852
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) 0 0
\(633\) 6.00000 0.238479
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) 6.00000 0.237915
\(637\) −9.00000 −0.356593
\(638\) 0 0
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) −4.00000 −0.157622
\(645\) −2.00000 −0.0787499
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) −60.0000 −2.35521
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) −32.0000 −1.25322
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) −32.0000 −1.25034
\(656\) −28.0000 −1.09322
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 1.00000 0.0389545 0.0194772 0.999810i \(-0.493800\pi\)
0.0194772 + 0.999810i \(0.493800\pi\)
\(660\) 20.0000 0.778499
\(661\) −1.00000 −0.0388955 −0.0194477 0.999811i \(-0.506191\pi\)
−0.0194477 + 0.999811i \(0.506191\pi\)
\(662\) 0 0
\(663\) 9.00000 0.349531
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 0 0
\(668\) 2.00000 0.0773823
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 8.00000 0.307692
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 0 0
\(679\) −22.0000 −0.844283
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) −4.00000 −0.152944
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) −5.00000 −0.190762
\(688\) −4.00000 −0.152499
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 20.0000 0.760286
\(693\) 10.0000 0.379869
\(694\) 0 0
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 21.0000 0.795432
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) −4.00000 −0.151186
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 40.0000 1.50756
\(705\) −16.0000 −0.602595
\(706\) 0 0
\(707\) 18.0000 0.676960
\(708\) 24.0000 0.901975
\(709\) 31.0000 1.16423 0.582115 0.813107i \(-0.302225\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 5.00000 0.187251
\(714\) 0 0
\(715\) 30.0000 1.12194
\(716\) 52.0000 1.94333
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) −8.00000 −0.298142
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) 4.00000 0.148762
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) −16.0000 −0.591377
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 75.0000 2.76266
\(738\) 0 0
\(739\) −54.0000 −1.98642 −0.993211 0.116326i \(-0.962888\pi\)
−0.993211 + 0.116326i \(0.962888\pi\)
\(740\) 32.0000 1.17634
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 32.0000 1.17239
\(746\) 0 0
\(747\) 15.0000 0.548821
\(748\) −30.0000 −1.09691
\(749\) 0 0
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) −32.0000 −1.16692
\(753\) −25.0000 −0.911051
\(754\) 0 0
\(755\) 4.00000 0.145575
\(756\) −4.00000 −0.145479
\(757\) −40.0000 −1.45382 −0.726912 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) 0 0
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) −22.0000 −0.796453
\(764\) 16.0000 0.578860
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 36.0000 1.29988
\(768\) −16.0000 −0.577350
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 42.0000 1.51161
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) −14.0000 −0.501602
\(780\) −12.0000 −0.429669
\(781\) 70.0000 2.50480
\(782\) 0 0
\(783\) 0 0
\(784\) −12.0000 −0.428571
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 28.0000 0.997459
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 28.0000 0.992434
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) −60.0000 −2.11735
\(804\) −30.0000 −1.05802
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 17.0000 0.596216
\(814\) 0 0
\(815\) −32.0000 −1.12091
\(816\) 12.0000 0.420084
\(817\) −2.00000 −0.0699711
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) −28.0000 −0.977802
\(821\) 53.0000 1.84971 0.924856 0.380317i \(-0.124185\pi\)
0.924856 + 0.380317i \(0.124185\pi\)
\(822\) 0 0
\(823\) −9.00000 −0.313720 −0.156860 0.987621i \(-0.550137\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) 0 0
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 2.00000 0.0695048
\(829\) 12.0000 0.416777 0.208389 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) −24.0000 −0.832050
\(833\) 9.00000 0.311832
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) 20.0000 0.691714
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) −38.0000 −1.31191 −0.655953 0.754802i \(-0.727733\pi\)
−0.655953 + 0.754802i \(0.727733\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −3.00000 −0.103325
\(844\) 12.0000 0.413057
\(845\) 8.00000 0.275208
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) 12.0000 0.412082
\(849\) −1.00000 −0.0343199
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) −28.0000 −0.959264
\(853\) −5.00000 −0.171197 −0.0855984 0.996330i \(-0.527280\pi\)
−0.0855984 + 0.996330i \(0.527280\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 14.0000 0.478231 0.239115 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −4.00000 −0.136399
\(861\) −14.0000 −0.477119
\(862\) 0 0
\(863\) 2.00000 0.0680808 0.0340404 0.999420i \(-0.489163\pi\)
0.0340404 + 0.999420i \(0.489163\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) −20.0000 −0.678844
\(869\) 80.0000 2.71381
\(870\) 0 0
\(871\) −45.0000 −1.52477
\(872\) 0 0
\(873\) 11.0000 0.372294
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 24.0000 0.810885
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 40.0000 1.34840
\(881\) −3.00000 −0.101073 −0.0505363 0.998722i \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) 0 0
\(883\) 15.0000 0.504790 0.252395 0.967624i \(-0.418782\pi\)
0.252395 + 0.967624i \(0.418782\pi\)
\(884\) 18.0000 0.605406
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) −58.0000 −1.94745 −0.973725 0.227728i \(-0.926870\pi\)
−0.973725 + 0.227728i \(0.926870\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) −28.0000 −0.937509
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) 52.0000 1.73817
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) 0 0
\(899\) 0 0
\(900\) 2.00000 0.0666667
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) −2.00000 −0.0665558
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −21.0000 −0.697294 −0.348647 0.937254i \(-0.613359\pi\)
−0.348647 + 0.937254i \(0.613359\pi\)
\(908\) 16.0000 0.530979
\(909\) −9.00000 −0.298511
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −8.00000 −0.264906
\(913\) −75.0000 −2.48214
\(914\) 0 0
\(915\) −16.0000 −0.528944
\(916\) −10.0000 −0.330409
\(917\) −32.0000 −1.05673
\(918\) 0 0
\(919\) −37.0000 −1.22052 −0.610259 0.792202i \(-0.708935\pi\)
−0.610259 + 0.792202i \(0.708935\pi\)
\(920\) 0 0
\(921\) 7.00000 0.230658
\(922\) 0 0
\(923\) −42.0000 −1.38245
\(924\) 20.0000 0.657952
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) 5.00000 0.164222
\(928\) 0 0
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 44.0000 1.44127
\(933\) 5.00000 0.163693
\(934\) 0 0
\(935\) −30.0000 −0.981105
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) −32.0000 −1.04372
\(941\) −15.0000 −0.488986 −0.244493 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 0 0
\(943\) 7.00000 0.227951
\(944\) 48.0000 1.56227
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −13.0000 −0.422443 −0.211222 0.977438i \(-0.567744\pi\)
−0.211222 + 0.977438i \(0.567744\pi\)
\(948\) −32.0000 −1.03931
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −5.00000 −0.162136
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) −32.0000 −1.03495
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) −16.0000 −0.516398
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0 0
\(964\) 8.00000 0.257663
\(965\) 42.0000 1.35203
\(966\) 0 0
\(967\) 53.0000 1.70437 0.852183 0.523245i \(-0.175279\pi\)
0.852183 + 0.523245i \(0.175279\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 2.00000 0.0641500
\(973\) 10.0000 0.320585
\(974\) 0 0
\(975\) 3.00000 0.0960769
\(976\) −32.0000 −1.02430
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) 0 0
\(979\) −50.0000 −1.59801
\(980\) −12.0000 −0.383326
\(981\) 11.0000 0.351203
\(982\) 0 0
\(983\) 10.0000 0.318950 0.159475 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(984\) 0 0
\(985\) 28.0000 0.892154
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) −12.0000 −0.381771
\(989\) 1.00000 0.0317982
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 28.0000 0.887660
\(996\) 30.0000 0.950586
\(997\) 58.0000 1.83688 0.918439 0.395562i \(-0.129450\pi\)
0.918439 + 0.395562i \(0.129450\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 129.2.a.a.1.1 1
3.2 odd 2 387.2.a.c.1.1 1
4.3 odd 2 2064.2.a.k.1.1 1
5.4 even 2 3225.2.a.g.1.1 1
7.6 odd 2 6321.2.a.e.1.1 1
8.3 odd 2 8256.2.a.s.1.1 1
8.5 even 2 8256.2.a.bm.1.1 1
12.11 even 2 6192.2.a.v.1.1 1
15.14 odd 2 9675.2.a.m.1.1 1
43.42 odd 2 5547.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
129.2.a.a.1.1 1 1.1 even 1 trivial
387.2.a.c.1.1 1 3.2 odd 2
2064.2.a.k.1.1 1 4.3 odd 2
3225.2.a.g.1.1 1 5.4 even 2
5547.2.a.c.1.1 1 43.42 odd 2
6192.2.a.v.1.1 1 12.11 even 2
6321.2.a.e.1.1 1 7.6 odd 2
8256.2.a.s.1.1 1 8.3 odd 2
8256.2.a.bm.1.1 1 8.5 even 2
9675.2.a.m.1.1 1 15.14 odd 2