Properties

Label 2001.2.a.a.1.1
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
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 \(\chi\) \(=\) 2001.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -3.00000 q^{15} -1.00000 q^{16} -1.00000 q^{18} +8.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} +3.00000 q^{30} +7.00000 q^{31} -5.00000 q^{32} -3.00000 q^{33} -1.00000 q^{36} -5.00000 q^{37} -8.00000 q^{38} -3.00000 q^{39} +9.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} -3.00000 q^{44} +3.00000 q^{45} +1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -4.00000 q^{50} -3.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +9.00000 q^{55} -8.00000 q^{57} -1.00000 q^{58} +1.00000 q^{59} +3.00000 q^{60} +7.00000 q^{61} -7.00000 q^{62} +7.00000 q^{64} +9.00000 q^{65} +3.00000 q^{66} -9.00000 q^{67} +1.00000 q^{69} +3.00000 q^{71} +3.00000 q^{72} -12.0000 q^{73} +5.00000 q^{74} -4.00000 q^{75} -8.00000 q^{76} +3.00000 q^{78} -14.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} +14.0000 q^{83} -2.00000 q^{86} -1.00000 q^{87} +9.00000 q^{88} -10.0000 q^{89} -3.00000 q^{90} +1.00000 q^{92} -7.00000 q^{93} +6.00000 q^{94} +24.0000 q^{95} +5.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 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
\(15\) −3.00000 −0.774597
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −1.00000 −0.208514
\(24\) −3.00000 −0.612372
\(25\) 4.00000 0.800000
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 3.00000 0.547723
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −5.00000 −0.883883
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) −8.00000 −1.29777
\(39\) −3.00000 −0.480384
\(40\) 9.00000 1.42302
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −3.00000 −0.452267
\(45\) 3.00000 0.447214
\(46\) 1.00000 0.147442
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −3.00000 −0.416025
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) −1.00000 −0.131306
\(59\) 1.00000 0.130189 0.0650945 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 3.00000 0.387298
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) −7.00000 −0.889001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 9.00000 1.11631
\(66\) 3.00000 0.369274
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 3.00000 0.353553
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 5.00000 0.581238
\(75\) −4.00000 −0.461880
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −1.00000 −0.107211
\(88\) 9.00000 0.959403
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −7.00000 −0.725866
\(94\) 6.00000 0.618853
\(95\) 24.0000 2.46235
\(96\) 5.00000 0.510310
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 7.00000 0.707107
\(99\) 3.00000 0.301511
\(100\) −4.00000 −0.400000
\(101\) 5.00000 0.497519 0.248759 0.968565i \(-0.419977\pi\)
0.248759 + 0.968565i \(0.419977\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 9.00000 0.882523
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −9.00000 −0.858116
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 8.00000 0.749269
\(115\) −3.00000 −0.279751
\(116\) −1.00000 −0.0928477
\(117\) 3.00000 0.277350
\(118\) −1.00000 −0.0920575
\(119\) 0 0
\(120\) −9.00000 −0.821584
\(121\) −2.00000 −0.181818
\(122\) −7.00000 −0.633750
\(123\) −3.00000 −0.270501
\(124\) −7.00000 −0.628619
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 3.00000 0.265165
\(129\) −2.00000 −0.176090
\(130\) −9.00000 −0.789352
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) 9.00000 0.777482
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) −3.00000 −0.251754
\(143\) 9.00000 0.752618
\(144\) −1.00000 −0.0833333
\(145\) 3.00000 0.249136
\(146\) 12.0000 0.993127
\(147\) 7.00000 0.577350
\(148\) 5.00000 0.410997
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 4.00000 0.326599
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 24.0000 1.94666
\(153\) 0 0
\(154\) 0 0
\(155\) 21.0000 1.68676
\(156\) 3.00000 0.240192
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 14.0000 1.11378
\(159\) 2.00000 0.158610
\(160\) −15.0000 −1.18585
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −17.0000 −1.33154 −0.665771 0.746156i \(-0.731897\pi\)
−0.665771 + 0.746156i \(0.731897\pi\)
\(164\) −3.00000 −0.234261
\(165\) −9.00000 −0.700649
\(166\) −14.0000 −1.08661
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −2.00000 −0.152499
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −1.00000 −0.0751646
\(178\) 10.0000 0.749532
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −3.00000 −0.223607
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) −3.00000 −0.221163
\(185\) −15.0000 −1.10282
\(186\) 7.00000 0.513265
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −7.00000 −0.505181
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −14.0000 −1.00514
\(195\) −9.00000 −0.644503
\(196\) 7.00000 0.500000
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −3.00000 −0.213201
\(199\) 21.0000 1.48865 0.744325 0.667817i \(-0.232771\pi\)
0.744325 + 0.667817i \(0.232771\pi\)
\(200\) 12.0000 0.848528
\(201\) 9.00000 0.634811
\(202\) −5.00000 −0.351799
\(203\) 0 0
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) −1.00000 −0.0696733
\(207\) −1.00000 −0.0695048
\(208\) −3.00000 −0.208013
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 2.00000 0.137361
\(213\) −3.00000 −0.205557
\(214\) −8.00000 −0.546869
\(215\) 6.00000 0.409197
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 12.0000 0.810885
\(220\) −9.00000 −0.606780
\(221\) 0 0
\(222\) −5.00000 −0.335578
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 6.00000 0.399114
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 8.00000 0.529813
\(229\) −21.0000 −1.38772 −0.693860 0.720110i \(-0.744091\pi\)
−0.693860 + 0.720110i \(0.744091\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −28.0000 −1.83434 −0.917170 0.398495i \(-0.869533\pi\)
−0.917170 + 0.398495i \(0.869533\pi\)
\(234\) −3.00000 −0.196116
\(235\) −18.0000 −1.17419
\(236\) −1.00000 −0.0650945
\(237\) 14.0000 0.909398
\(238\) 0 0
\(239\) 19.0000 1.22901 0.614504 0.788914i \(-0.289356\pi\)
0.614504 + 0.788914i \(0.289356\pi\)
\(240\) 3.00000 0.193649
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) −7.00000 −0.448129
\(245\) −21.0000 −1.34164
\(246\) 3.00000 0.191273
\(247\) 24.0000 1.52708
\(248\) 21.0000 1.33350
\(249\) −14.0000 −0.887214
\(250\) 3.00000 0.189737
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 5.00000 0.313728
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) −9.00000 −0.558156
\(261\) 1.00000 0.0618984
\(262\) −12.0000 −0.741362
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −9.00000 −0.553912
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 9.00000 0.549762
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 3.00000 0.182574
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 12.0000 0.723627
\(276\) −1.00000 −0.0601929
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) −18.0000 −1.07957
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −6.00000 −0.357295
\(283\) 23.0000 1.36721 0.683604 0.729853i \(-0.260412\pi\)
0.683604 + 0.729853i \(0.260412\pi\)
\(284\) −3.00000 −0.178017
\(285\) −24.0000 −1.42164
\(286\) −9.00000 −0.532181
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −17.0000 −1.00000
\(290\) −3.00000 −0.176166
\(291\) −14.0000 −0.820695
\(292\) 12.0000 0.702247
\(293\) 20.0000 1.16841 0.584206 0.811605i \(-0.301406\pi\)
0.584206 + 0.811605i \(0.301406\pi\)
\(294\) −7.00000 −0.408248
\(295\) 3.00000 0.174667
\(296\) −15.0000 −0.871857
\(297\) −3.00000 −0.174078
\(298\) 3.00000 0.173785
\(299\) −3.00000 −0.173494
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −5.00000 −0.287242
\(304\) −8.00000 −0.458831
\(305\) 21.0000 1.20246
\(306\) 0 0
\(307\) 29.0000 1.65512 0.827559 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) −21.0000 −1.19272
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) −9.00000 −0.509525
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −7.00000 −0.393159 −0.196580 0.980488i \(-0.562983\pi\)
−0.196580 + 0.980488i \(0.562983\pi\)
\(318\) −2.00000 −0.112154
\(319\) 3.00000 0.167968
\(320\) 21.0000 1.17394
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 12.0000 0.665640
\(326\) 17.0000 0.941543
\(327\) −2.00000 −0.110600
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 9.00000 0.495434
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −14.0000 −0.768350
\(333\) −5.00000 −0.273998
\(334\) 3.00000 0.164153
\(335\) −27.0000 −1.47517
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) 4.00000 0.217571
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 21.0000 1.13721
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) 6.00000 0.323498
\(345\) 3.00000 0.161515
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 1.00000 0.0536056
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) −15.0000 −0.799503
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 1.00000 0.0531494
\(355\) 9.00000 0.477670
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 9.00000 0.474342
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −36.0000 −1.88433
\(366\) 7.00000 0.365896
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 1.00000 0.0521286
\(369\) 3.00000 0.156174
\(370\) 15.0000 0.779813
\(371\) 0 0
\(372\) 7.00000 0.362933
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) −18.0000 −0.928279
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) −24.0000 −1.23117
\(381\) 5.00000 0.256158
\(382\) −3.00000 −0.153493
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 2.00000 0.101666
\(388\) −14.0000 −0.710742
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 9.00000 0.455733
\(391\) 0 0
\(392\) −21.0000 −1.06066
\(393\) −12.0000 −0.605320
\(394\) −10.0000 −0.503793
\(395\) −42.0000 −2.11325
\(396\) −3.00000 −0.150756
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −21.0000 −1.05263
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) −9.00000 −0.448879
\(403\) 21.0000 1.04608
\(404\) −5.00000 −0.248759
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) −15.0000 −0.743522
\(408\) 0 0
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) −9.00000 −0.444478
\(411\) −8.00000 −0.394611
\(412\) −1.00000 −0.0492665
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 42.0000 2.06170
\(416\) −15.0000 −0.735436
\(417\) −18.0000 −0.881464
\(418\) −24.0000 −1.17388
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) 3.00000 0.146038
\(423\) −6.00000 −0.291730
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 3.00000 0.145350
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) −9.00000 −0.434524
\(430\) −6.00000 −0.289346
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) −2.00000 −0.0957826
\(437\) −8.00000 −0.382692
\(438\) −12.0000 −0.573382
\(439\) 38.0000 1.81364 0.906821 0.421517i \(-0.138502\pi\)
0.906821 + 0.421517i \(0.138502\pi\)
\(440\) 27.0000 1.28717
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 30.0000 1.42534 0.712672 0.701498i \(-0.247485\pi\)
0.712672 + 0.701498i \(0.247485\pi\)
\(444\) −5.00000 −0.237289
\(445\) −30.0000 −1.42214
\(446\) −8.00000 −0.378811
\(447\) 3.00000 0.141895
\(448\) 0 0
\(449\) 7.00000 0.330350 0.165175 0.986264i \(-0.447181\pi\)
0.165175 + 0.986264i \(0.447181\pi\)
\(450\) −4.00000 −0.188562
\(451\) 9.00000 0.423793
\(452\) 6.00000 0.282216
\(453\) 8.00000 0.375873
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) −24.0000 −1.12390
\(457\) −16.0000 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) 21.0000 0.981266
\(459\) 0 0
\(460\) 3.00000 0.139876
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −21.0000 −0.973852
\(466\) 28.0000 1.29707
\(467\) −25.0000 −1.15686 −0.578431 0.815731i \(-0.696335\pi\)
−0.578431 + 0.815731i \(0.696335\pi\)
\(468\) −3.00000 −0.138675
\(469\) 0 0
\(470\) 18.0000 0.830278
\(471\) −2.00000 −0.0921551
\(472\) 3.00000 0.138086
\(473\) 6.00000 0.275880
\(474\) −14.0000 −0.643041
\(475\) 32.0000 1.46826
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −19.0000 −0.869040
\(479\) −23.0000 −1.05090 −0.525448 0.850825i \(-0.676102\pi\)
−0.525448 + 0.850825i \(0.676102\pi\)
\(480\) 15.0000 0.684653
\(481\) −15.0000 −0.683941
\(482\) −12.0000 −0.546585
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 42.0000 1.90712
\(486\) 1.00000 0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 21.0000 0.950625
\(489\) 17.0000 0.768767
\(490\) 21.0000 0.948683
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 3.00000 0.135250
\(493\) 0 0
\(494\) −24.0000 −1.07981
\(495\) 9.00000 0.404520
\(496\) −7.00000 −0.314309
\(497\) 0 0
\(498\) 14.0000 0.627355
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 3.00000 0.134164
\(501\) 3.00000 0.134030
\(502\) 9.00000 0.401690
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) 15.0000 0.667491
\(506\) 3.00000 0.133366
\(507\) 4.00000 0.177646
\(508\) 5.00000 0.221839
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) −8.00000 −0.353209
\(514\) 6.00000 0.264649
\(515\) 3.00000 0.132196
\(516\) 2.00000 0.0880451
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 27.0000 1.18403
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 3.00000 0.130558
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) 1.00000 0.0433963
\(532\) 0 0
\(533\) 9.00000 0.389833
\(534\) −10.0000 −0.432742
\(535\) 24.0000 1.03761
\(536\) −27.0000 −1.16622
\(537\) −20.0000 −0.863064
\(538\) 21.0000 0.905374
\(539\) −21.0000 −0.904534
\(540\) 3.00000 0.129099
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 15.0000 0.644305
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −8.00000 −0.341743
\(549\) 7.00000 0.298753
\(550\) −12.0000 −0.511682
\(551\) 8.00000 0.340811
\(552\) 3.00000 0.127688
\(553\) 0 0
\(554\) 17.0000 0.722261
\(555\) 15.0000 0.636715
\(556\) −18.0000 −0.763370
\(557\) −27.0000 −1.14403 −0.572013 0.820244i \(-0.693837\pi\)
−0.572013 + 0.820244i \(0.693837\pi\)
\(558\) −7.00000 −0.296334
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) −6.00000 −0.252646
\(565\) −18.0000 −0.757266
\(566\) −23.0000 −0.966762
\(567\) 0 0
\(568\) 9.00000 0.377632
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 24.0000 1.00525
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) −9.00000 −0.376309
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 7.00000 0.291667
\(577\) −28.0000 −1.16566 −0.582828 0.812596i \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) 17.0000 0.707107
\(579\) −4.00000 −0.166234
\(580\) −3.00000 −0.124568
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) −6.00000 −0.248495
\(584\) −36.0000 −1.48969
\(585\) 9.00000 0.372104
\(586\) −20.0000 −0.826192
\(587\) 40.0000 1.65098 0.825488 0.564419i \(-0.190900\pi\)
0.825488 + 0.564419i \(0.190900\pi\)
\(588\) −7.00000 −0.288675
\(589\) 56.0000 2.30744
\(590\) −3.00000 −0.123508
\(591\) −10.0000 −0.411345
\(592\) 5.00000 0.205499
\(593\) −16.0000 −0.657041 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 3.00000 0.122885
\(597\) −21.0000 −0.859473
\(598\) 3.00000 0.122679
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) −12.0000 −0.489898
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 0 0
\(603\) −9.00000 −0.366508
\(604\) 8.00000 0.325515
\(605\) −6.00000 −0.243935
\(606\) 5.00000 0.203111
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) −21.0000 −0.850265
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −29.0000 −1.17034
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 1.00000 0.0402259
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) −21.0000 −0.843380
\(621\) 1.00000 0.0401286
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) 3.00000 0.120096
\(625\) −29.0000 −1.16000
\(626\) 26.0000 1.03917
\(627\) −24.0000 −0.958468
\(628\) −2.00000 −0.0798087
\(629\) 0 0
\(630\) 0 0
\(631\) 23.0000 0.915616 0.457808 0.889051i \(-0.348635\pi\)
0.457808 + 0.889051i \(0.348635\pi\)
\(632\) −42.0000 −1.67067
\(633\) 3.00000 0.119239
\(634\) 7.00000 0.278006
\(635\) −15.0000 −0.595257
\(636\) −2.00000 −0.0793052
\(637\) −21.0000 −0.832050
\(638\) −3.00000 −0.118771
\(639\) 3.00000 0.118678
\(640\) 9.00000 0.355756
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 8.00000 0.315735
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) −17.0000 −0.668339 −0.334169 0.942513i \(-0.608456\pi\)
−0.334169 + 0.942513i \(0.608456\pi\)
\(648\) 3.00000 0.117851
\(649\) 3.00000 0.117760
\(650\) −12.0000 −0.470679
\(651\) 0 0
\(652\) 17.0000 0.665771
\(653\) 23.0000 0.900060 0.450030 0.893014i \(-0.351413\pi\)
0.450030 + 0.893014i \(0.351413\pi\)
\(654\) 2.00000 0.0782062
\(655\) 36.0000 1.40664
\(656\) −3.00000 −0.117130
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 9.00000 0.350325
\(661\) −44.0000 −1.71140 −0.855701 0.517471i \(-0.826874\pi\)
−0.855701 + 0.517471i \(0.826874\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 42.0000 1.62992
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) −1.00000 −0.0387202
\(668\) 3.00000 0.116073
\(669\) −8.00000 −0.309298
\(670\) 27.0000 1.04310
\(671\) 21.0000 0.810696
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 29.0000 1.11704
\(675\) −4.00000 −0.153960
\(676\) 4.00000 0.153846
\(677\) 4.00000 0.153732 0.0768662 0.997041i \(-0.475509\pi\)
0.0768662 + 0.997041i \(0.475509\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) −21.0000 −0.804132
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) −8.00000 −0.305888
\(685\) 24.0000 0.916993
\(686\) 0 0
\(687\) 21.0000 0.801200
\(688\) −2.00000 −0.0762493
\(689\) −6.00000 −0.228582
\(690\) −3.00000 −0.114208
\(691\) 34.0000 1.29342 0.646710 0.762736i \(-0.276144\pi\)
0.646710 + 0.762736i \(0.276144\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 54.0000 2.04834
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) −25.0000 −0.946264
\(699\) 28.0000 1.05906
\(700\) 0 0
\(701\) −11.0000 −0.415464 −0.207732 0.978186i \(-0.566608\pi\)
−0.207732 + 0.978186i \(0.566608\pi\)
\(702\) 3.00000 0.113228
\(703\) −40.0000 −1.50863
\(704\) 21.0000 0.791467
\(705\) 18.0000 0.677919
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) 1.00000 0.0375823
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) −9.00000 −0.337764
\(711\) −14.0000 −0.525041
\(712\) −30.0000 −1.12430
\(713\) −7.00000 −0.262152
\(714\) 0 0
\(715\) 27.0000 1.00974
\(716\) −20.0000 −0.747435
\(717\) −19.0000 −0.709568
\(718\) 24.0000 0.895672
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) −45.0000 −1.67473
\(723\) −12.0000 −0.446285
\(724\) −10.0000 −0.371647
\(725\) 4.00000 0.148556
\(726\) −2.00000 −0.0742270
\(727\) −34.0000 −1.26099 −0.630495 0.776193i \(-0.717148\pi\)
−0.630495 + 0.776193i \(0.717148\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 36.0000 1.33242
\(731\) 0 0
\(732\) 7.00000 0.258727
\(733\) −25.0000 −0.923396 −0.461698 0.887037i \(-0.652760\pi\)
−0.461698 + 0.887037i \(0.652760\pi\)
\(734\) 34.0000 1.25496
\(735\) 21.0000 0.774597
\(736\) 5.00000 0.184302
\(737\) −27.0000 −0.994558
\(738\) −3.00000 −0.110432
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 15.0000 0.551411
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) −21.0000 −0.769897
\(745\) −9.00000 −0.329734
\(746\) −6.00000 −0.219676
\(747\) 14.0000 0.512233
\(748\) 0 0
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 6.00000 0.218797
\(753\) 9.00000 0.327978
\(754\) −3.00000 −0.109254
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) −5.00000 −0.181728 −0.0908640 0.995863i \(-0.528963\pi\)
−0.0908640 + 0.995863i \(0.528963\pi\)
\(758\) 14.0000 0.508503
\(759\) 3.00000 0.108893
\(760\) 72.0000 2.61171
\(761\) 50.0000 1.81250 0.906249 0.422744i \(-0.138933\pi\)
0.906249 + 0.422744i \(0.138933\pi\)
\(762\) −5.00000 −0.181131
\(763\) 0 0
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 3.00000 0.108324
\(768\) 17.0000 0.613435
\(769\) 27.0000 0.973645 0.486822 0.873501i \(-0.338156\pi\)
0.486822 + 0.873501i \(0.338156\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −4.00000 −0.143963
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 28.0000 1.00579
\(776\) 42.0000 1.50771
\(777\) 0 0
\(778\) −20.0000 −0.717035
\(779\) 24.0000 0.859889
\(780\) 9.00000 0.322252
\(781\) 9.00000 0.322045
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 7.00000 0.250000
\(785\) 6.00000 0.214149
\(786\) 12.0000 0.428026
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) −10.0000 −0.356235
\(789\) −16.0000 −0.569615
\(790\) 42.0000 1.49429
\(791\) 0 0
\(792\) 9.00000 0.319801
\(793\) 21.0000 0.745732
\(794\) 2.00000 0.0709773
\(795\) 6.00000 0.212798
\(796\) −21.0000 −0.744325
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −20.0000 −0.707107
\(801\) −10.0000 −0.353333
\(802\) 27.0000 0.953403
\(803\) −36.0000 −1.27041
\(804\) −9.00000 −0.317406
\(805\) 0 0
\(806\) −21.0000 −0.739693
\(807\) 21.0000 0.739235
\(808\) 15.0000 0.527698
\(809\) −31.0000 −1.08990 −0.544951 0.838468i \(-0.683452\pi\)
−0.544951 + 0.838468i \(0.683452\pi\)
\(810\) −3.00000 −0.105409
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 15.0000 0.526073
\(814\) 15.0000 0.525750
\(815\) −51.0000 −1.78645
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 38.0000 1.32864
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 8.00000 0.279032
\(823\) −9.00000 −0.313720 −0.156860 0.987621i \(-0.550137\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) 3.00000 0.104510
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −35.0000 −1.21707 −0.608535 0.793527i \(-0.708242\pi\)
−0.608535 + 0.793527i \(0.708242\pi\)
\(828\) 1.00000 0.0347524
\(829\) 24.0000 0.833554 0.416777 0.909009i \(-0.363160\pi\)
0.416777 + 0.909009i \(0.363160\pi\)
\(830\) −42.0000 −1.45784
\(831\) 17.0000 0.589723
\(832\) 21.0000 0.728044
\(833\) 0 0
\(834\) 18.0000 0.623289
\(835\) −9.00000 −0.311458
\(836\) −24.0000 −0.830057
\(837\) −7.00000 −0.241955
\(838\) −6.00000 −0.207267
\(839\) 33.0000 1.13929 0.569643 0.821892i \(-0.307081\pi\)
0.569643 + 0.821892i \(0.307081\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −9.00000 −0.310160
\(843\) 18.0000 0.619953
\(844\) 3.00000 0.103264
\(845\) −12.0000 −0.412813
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −23.0000 −0.789358
\(850\) 0 0
\(851\) 5.00000 0.171398
\(852\) 3.00000 0.102778
\(853\) 40.0000 1.36957 0.684787 0.728743i \(-0.259895\pi\)
0.684787 + 0.728743i \(0.259895\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) 24.0000 0.820303
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 9.00000 0.307255
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) 21.0000 0.714848 0.357424 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 5.00000 0.170103
\(865\) 54.0000 1.83606
\(866\) 14.0000 0.475739
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) −42.0000 −1.42475
\(870\) 3.00000 0.101710
\(871\) −27.0000 −0.914860
\(872\) 6.00000 0.203186
\(873\) 14.0000 0.473828
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) −38.0000 −1.28244
\(879\) −20.0000 −0.674583
\(880\) −9.00000 −0.303390
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 7.00000 0.235702
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) −3.00000 −0.100844
\(886\) −30.0000 −1.00787
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 15.0000 0.503367
\(889\) 0 0
\(890\) 30.0000 1.00560
\(891\) 3.00000 0.100504
\(892\) −8.00000 −0.267860
\(893\) −48.0000 −1.60626
\(894\) −3.00000 −0.100335
\(895\) 60.0000 2.00558
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) −7.00000 −0.233593
\(899\) 7.00000 0.233463
\(900\) −4.00000 −0.133333
\(901\) 0 0
\(902\) −9.00000 −0.299667
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 30.0000 0.997234
\(906\) −8.00000 −0.265782
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −6.00000 −0.199117
\(909\) 5.00000 0.165840
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 8.00000 0.264906
\(913\) 42.0000 1.39000
\(914\) 16.0000 0.529233
\(915\) −21.0000 −0.694239
\(916\) 21.0000 0.693860
\(917\) 0 0
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) −9.00000 −0.296721
\(921\) −29.0000 −0.955582
\(922\) 3.00000 0.0987997
\(923\) 9.00000 0.296239
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) −4.00000 −0.131448
\(927\) 1.00000 0.0328443
\(928\) −5.00000 −0.164133
\(929\) −38.0000 −1.24674 −0.623370 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 21.0000 0.688617
\(931\) −56.0000 −1.83533
\(932\) 28.0000 0.917170
\(933\) −6.00000 −0.196431
\(934\) 25.0000 0.818025
\(935\) 0 0
\(936\) 9.00000 0.294174
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 18.0000 0.587095
\(941\) 11.0000 0.358590 0.179295 0.983795i \(-0.442618\pi\)
0.179295 + 0.983795i \(0.442618\pi\)
\(942\) 2.00000 0.0651635
\(943\) −3.00000 −0.0976934
\(944\) −1.00000 −0.0325472
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) −14.0000 −0.454699
\(949\) −36.0000 −1.16861
\(950\) −32.0000 −1.03822
\(951\) 7.00000 0.226991
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 2.00000 0.0647524
\(955\) 9.00000 0.291233
\(956\) −19.0000 −0.614504
\(957\) −3.00000 −0.0969762
\(958\) 23.0000 0.743096
\(959\) 0 0
\(960\) −21.0000 −0.677772
\(961\) 18.0000 0.580645
\(962\) 15.0000 0.483619
\(963\) 8.00000 0.257796
\(964\) −12.0000 −0.386494
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −6.00000 −0.192847
\(969\) 0 0
\(970\) −42.0000 −1.34854
\(971\) 1.00000 0.0320915 0.0160458 0.999871i \(-0.494892\pi\)
0.0160458 + 0.999871i \(0.494892\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) −12.0000 −0.384308
\(976\) −7.00000 −0.224065
\(977\) −7.00000 −0.223950 −0.111975 0.993711i \(-0.535718\pi\)
−0.111975 + 0.993711i \(0.535718\pi\)
\(978\) −17.0000 −0.543600
\(979\) −30.0000 −0.958804
\(980\) 21.0000 0.670820
\(981\) 2.00000 0.0638551
\(982\) −30.0000 −0.957338
\(983\) −49.0000 −1.56286 −0.781429 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) −9.00000 −0.286910
\(985\) 30.0000 0.955879
\(986\) 0 0
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) −2.00000 −0.0635963
\(990\) −9.00000 −0.286039
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −35.0000 −1.11125
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) 63.0000 1.99723
\(996\) 14.0000 0.443607
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −14.0000 −0.443162
\(999\) 5.00000 0.158193
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 2001.2.a.a.1.1 1
3.2 odd 2 6003.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.a.1.1 1 1.1 even 1 trivial
6003.2.a.c.1.1 1 3.2 odd 2