Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(1\)
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 \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} -3.00000 q^{19} -4.00000 q^{21} -3.00000 q^{22} +2.00000 q^{23} +1.00000 q^{24} -6.00000 q^{26} +5.00000 q^{27} +4.00000 q^{28} -1.00000 q^{32} -3.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} -1.00000 q^{37} +3.00000 q^{38} -6.00000 q^{39} -3.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} -2.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -3.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} -5.00000 q^{54} -4.00000 q^{56} +3.00000 q^{57} -12.0000 q^{59} +12.0000 q^{61} -8.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +9.00000 q^{67} +3.00000 q^{68} -2.00000 q^{69} -2.00000 q^{71} +2.00000 q^{72} -9.00000 q^{73} +1.00000 q^{74} -3.00000 q^{76} +12.0000 q^{77} +6.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} +3.00000 q^{82} -7.00000 q^{83} -4.00000 q^{84} +4.00000 q^{86} -3.00000 q^{88} -3.00000 q^{89} +24.0000 q^{91} +2.00000 q^{92} -4.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -9.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 2.00000 0.471405
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −3.00000 −0.639602
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 5.00000 0.962250
\(28\) 4.00000 0.755929
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −1.00000 −0.164399
\(38\) 3.00000 0.486664
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 4.00000 0.617213
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 6.00000 0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) −8.00000 −1.00791
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) 3.00000 0.363803
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 2.00000 0.235702
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 12.0000 1.36753
\(78\) 6.00000 0.679366
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −9.00000 −0.909137
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 3.00000 0.297044
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −1.00000 −0.0966736 −0.0483368 0.998831i \(-0.515392\pi\)
−0.0483368 + 0.998831i \(0.515392\pi\)
\(108\) 5.00000 0.481125
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 4.00000 0.377964
\(113\) 5.00000 0.470360 0.235180 0.971952i \(-0.424432\pi\)
0.235180 + 0.971952i \(0.424432\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) 0 0
\(117\) −12.0000 −1.10940
\(118\) 12.0000 1.10469
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −12.0000 −1.08643
\(123\) 3.00000 0.270501
\(124\) 0 0
\(125\) 0 0
\(126\) 8.00000 0.712697
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −3.00000 −0.261116
\(133\) −12.0000 −1.04053
\(134\) −9.00000 −0.777482
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −7.00000 −0.598050 −0.299025 0.954245i \(-0.596661\pi\)
−0.299025 + 0.954245i \(0.596661\pi\)
\(138\) 2.00000 0.170251
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 2.00000 0.167836
\(143\) 18.0000 1.50524
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) 9.00000 0.744845
\(147\) −9.00000 −0.742307
\(148\) −1.00000 −0.0821995
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 3.00000 0.243332
\(153\) −6.00000 −0.485071
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 24.0000 1.91541 0.957704 0.287754i \(-0.0929087\pi\)
0.957704 + 0.287754i \(0.0929087\pi\)
\(158\) 2.00000 0.159111
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) −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
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 4.00000 0.308607
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −4.00000 −0.304997
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 12.0000 0.901975
\(178\) 3.00000 0.224860
\(179\) 21.0000 1.56961 0.784807 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −24.0000 −1.77900
\(183\) −12.0000 −0.887066
\(184\) −2.00000 −0.147442
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 4.00000 0.291730
\(189\) 20.0000 1.45479
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 6.00000 0.426401
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −9.00000 −0.634811
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −4.00000 −0.278019
\(208\) 6.00000 0.416025
\(209\) −9.00000 −0.622543
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) −2.00000 −0.137361
\(213\) 2.00000 0.137038
\(214\) 1.00000 0.0683586
\(215\) 0 0
\(216\) −5.00000 −0.340207
\(217\) 0 0
\(218\) −18.0000 −1.21911
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) 18.0000 1.21081
\(222\) −1.00000 −0.0671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −5.00000 −0.332595
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) 3.00000 0.198680
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 2.00000 0.129914
\(238\) −12.0000 −0.777844
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −21.0000 −1.35273 −0.676364 0.736567i \(-0.736446\pi\)
−0.676364 + 0.736567i \(0.736446\pi\)
\(242\) 2.00000 0.128565
\(243\) −16.0000 −1.02640
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) −18.0000 −1.14531
\(248\) 0 0
\(249\) 7.00000 0.443607
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) −8.00000 −0.503953
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) −4.00000 −0.249029
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) 26.0000 1.60323 0.801614 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 3.00000 0.183597
\(268\) 9.00000 0.549762
\(269\) −28.0000 −1.70719 −0.853595 0.520937i \(-0.825583\pi\)
−0.853595 + 0.520937i \(0.825583\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 3.00000 0.181902
\(273\) −24.0000 −1.45255
\(274\) 7.00000 0.422885
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) −3.00000 −0.179928
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 4.00000 0.238197
\(283\) 7.00000 0.416107 0.208053 0.978117i \(-0.433287\pi\)
0.208053 + 0.978117i \(0.433287\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) −18.0000 −1.06436
\(287\) −12.0000 −0.708338
\(288\) 2.00000 0.117851
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) −9.00000 −0.526685
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 15.0000 0.870388
\(298\) 18.0000 1.04271
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −22.0000 −1.26596
\(303\) 10.0000 0.574485
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −29.0000 −1.65512 −0.827559 0.561379i \(-0.810271\pi\)
−0.827559 + 0.561379i \(0.810271\pi\)
\(308\) 12.0000 0.683763
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 6.00000 0.339683
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −2.00000 −0.112154
\(319\) 0 0
\(320\) 0 0
\(321\) 1.00000 0.0558146
\(322\) −8.00000 −0.445823
\(323\) −9.00000 −0.500773
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 17.0000 0.941543
\(327\) −18.0000 −0.995402
\(328\) 3.00000 0.165647
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) −7.00000 −0.384175
\(333\) 2.00000 0.109599
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) −23.0000 −1.25104
\(339\) −5.00000 −0.271563
\(340\) 0 0
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) 8.00000 0.431959
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) −31.0000 −1.66417 −0.832084 0.554650i \(-0.812852\pi\)
−0.832084 + 0.554650i \(0.812852\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) 30.0000 1.60128
\(352\) −3.00000 −0.159901
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −3.00000 −0.159000
\(357\) −12.0000 −0.635107
\(358\) −21.0000 −1.10988
\(359\) 34.0000 1.79445 0.897226 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 10.0000 0.525588
\(363\) 2.00000 0.104973
\(364\) 24.0000 1.25794
\(365\) 0 0
\(366\) 12.0000 0.627250
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 2.00000 0.104257
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) 28.0000 1.44979 0.724893 0.688862i \(-0.241889\pi\)
0.724893 + 0.688862i \(0.241889\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) −20.0000 −1.02869
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −6.00000 −0.306987
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −11.0000 −0.559885
\(387\) 8.00000 0.406663
\(388\) 2.00000 0.101535
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) −9.00000 −0.454569
\(393\) −12.0000 −0.605320
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −8.00000 −0.401004
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 9.00000 0.448879
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) 3.00000 0.148522
\(409\) 15.0000 0.741702 0.370851 0.928692i \(-0.379066\pi\)
0.370851 + 0.928692i \(0.379066\pi\)
\(410\) 0 0
\(411\) 7.00000 0.345285
\(412\) 14.0000 0.689730
\(413\) −48.0000 −2.36193
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) −3.00000 −0.146911
\(418\) 9.00000 0.440204
\(419\) 27.0000 1.31904 0.659518 0.751689i \(-0.270760\pi\)
0.659518 + 0.751689i \(0.270760\pi\)
\(420\) 0 0
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) −9.00000 −0.438113
\(423\) −8.00000 −0.388973
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 48.0000 2.32288
\(428\) −1.00000 −0.0483368
\(429\) −18.0000 −0.869048
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 5.00000 0.240563
\(433\) 5.00000 0.240285 0.120142 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) −6.00000 −0.287019
\(438\) −9.00000 −0.430037
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) −18.0000 −0.856173
\(443\) 5.00000 0.237557 0.118779 0.992921i \(-0.462102\pi\)
0.118779 + 0.992921i \(0.462102\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 18.0000 0.851371
\(448\) 4.00000 0.188982
\(449\) −37.0000 −1.74614 −0.873069 0.487597i \(-0.837874\pi\)
−0.873069 + 0.487597i \(0.837874\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 5.00000 0.235180
\(453\) −22.0000 −1.03365
\(454\) −28.0000 −1.31411
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) −33.0000 −1.54367 −0.771837 0.635820i \(-0.780662\pi\)
−0.771837 + 0.635820i \(0.780662\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 12.0000 0.558291
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −12.0000 −0.554700
\(469\) 36.0000 1.66233
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 12.0000 0.552345
\(473\) −12.0000 −0.551761
\(474\) −2.00000 −0.0918630
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 4.00000 0.183147
\(478\) 12.0000 0.548867
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 21.0000 0.956524
\(483\) −8.00000 −0.364013
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 16.0000 0.725775
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −12.0000 −0.543214
\(489\) 17.0000 0.768767
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 3.00000 0.135250
\(493\) 0 0
\(494\) 18.0000 0.809858
\(495\) 0 0
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) −7.00000 −0.313678
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 9.00000 0.401690
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 8.00000 0.356348
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −23.0000 −1.02147
\(508\) 8.00000 0.354943
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −36.0000 −1.59255
\(512\) −1.00000 −0.0441942
\(513\) −15.0000 −0.662266
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 12.0000 0.527759
\(518\) 4.00000 0.175750
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −11.0000 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(522\) 0 0
\(523\) 37.0000 1.61790 0.808949 0.587879i \(-0.200037\pi\)
0.808949 + 0.587879i \(0.200037\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −26.0000 −1.13365
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) −12.0000 −0.520266
\(533\) −18.0000 −0.779667
\(534\) −3.00000 −0.129823
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) −21.0000 −0.906217
\(538\) 28.0000 1.20717
\(539\) 27.0000 1.16297
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 28.0000 1.20270
\(543\) 10.0000 0.429141
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 24.0000 1.02711
\(547\) 5.00000 0.213785 0.106892 0.994271i \(-0.465910\pi\)
0.106892 + 0.994271i \(0.465910\pi\)
\(548\) −7.00000 −0.299025
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) 0 0
\(552\) 2.00000 0.0851257
\(553\) −8.00000 −0.340195
\(554\) −28.0000 −1.18961
\(555\) 0 0
\(556\) 3.00000 0.127228
\(557\) 32.0000 1.35588 0.677942 0.735116i \(-0.262872\pi\)
0.677942 + 0.735116i \(0.262872\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 6.00000 0.253095
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −7.00000 −0.294232
\(567\) 4.00000 0.167984
\(568\) 2.00000 0.0839181
\(569\) 19.0000 0.796521 0.398261 0.917272i \(-0.369614\pi\)
0.398261 + 0.917272i \(0.369614\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 18.0000 0.752618
\(573\) −6.00000 −0.250654
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) −15.0000 −0.624458 −0.312229 0.950007i \(-0.601076\pi\)
−0.312229 + 0.950007i \(0.601076\pi\)
\(578\) 8.00000 0.332756
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) −28.0000 −1.16164
\(582\) 2.00000 0.0829027
\(583\) −6.00000 −0.248495
\(584\) 9.00000 0.372423
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) −13.0000 −0.536567 −0.268284 0.963340i \(-0.586456\pi\)
−0.268284 + 0.963340i \(0.586456\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) −1.00000 −0.0410997
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) −15.0000 −0.615457
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −8.00000 −0.327418
\(598\) −12.0000 −0.490716
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −33.0000 −1.34610 −0.673049 0.739598i \(-0.735016\pi\)
−0.673049 + 0.739598i \(0.735016\pi\)
\(602\) 16.0000 0.652111
\(603\) −18.0000 −0.733017
\(604\) 22.0000 0.895167
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −26.0000 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(608\) 3.00000 0.121666
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) −6.00000 −0.242536
\(613\) −46.0000 −1.85792 −0.928961 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) 29.0000 1.17034
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 14.0000 0.563163
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 10.0000 0.401286
\(622\) −28.0000 −1.12270
\(623\) −12.0000 −0.480770
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 9.00000 0.359425
\(628\) 24.0000 0.957704
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) 22.0000 0.875806 0.437903 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(632\) 2.00000 0.0795557
\(633\) −9.00000 −0.357718
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 54.0000 2.13956
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −1.00000 −0.0394669
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 9.00000 0.354100
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) −17.0000 −0.665771
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 18.0000 0.702247
\(658\) −16.0000 −0.623745
\(659\) 9.00000 0.350590 0.175295 0.984516i \(-0.443912\pi\)
0.175295 + 0.984516i \(0.443912\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 3.00000 0.116598
\(663\) −18.0000 −0.699062
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) −18.0000 −0.696441
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 4.00000 0.154303
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 9.00000 0.346667
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 5.00000 0.192024
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) 17.0000 0.650487 0.325243 0.945630i \(-0.394554\pi\)
0.325243 + 0.945630i \(0.394554\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −1.00000 −0.0380418 −0.0190209 0.999819i \(-0.506055\pi\)
−0.0190209 + 0.999819i \(0.506055\pi\)
\(692\) 24.0000 0.912343
\(693\) −24.0000 −0.911685
\(694\) 31.0000 1.17674
\(695\) 0 0
\(696\) 0 0
\(697\) −9.00000 −0.340899
\(698\) 16.0000 0.605609
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) −30.0000 −1.13228
\(703\) 3.00000 0.113147
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) −40.0000 −1.50435
\(708\) 12.0000 0.450988
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 3.00000 0.112430
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 21.0000 0.784807
\(717\) 12.0000 0.448148
\(718\) −34.0000 −1.26887
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 56.0000 2.08555
\(722\) 10.0000 0.372161
\(723\) 21.0000 0.780998
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) −24.0000 −0.889499
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) −12.0000 −0.443533
\(733\) 40.0000 1.47743 0.738717 0.674016i \(-0.235432\pi\)
0.738717 + 0.674016i \(0.235432\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 27.0000 0.994558
\(738\) −6.00000 −0.220863
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 18.0000 0.661247
\(742\) 8.00000 0.293689
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −28.0000 −1.02515
\(747\) 14.0000 0.512233
\(748\) 9.00000 0.329073
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 4.00000 0.145865
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) 0 0
\(756\) 20.0000 0.727393
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) −1.00000 −0.0363216
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 8.00000 0.289809
\(763\) 72.0000 2.60658
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −72.0000 −2.59977
\(768\) −1.00000 −0.0360844
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) 11.0000 0.395899
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 4.00000 0.143499
\(778\) 12.0000 0.430221
\(779\) 9.00000 0.322458
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −6.00000 −0.214560
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 12.0000 0.427482
\(789\) −26.0000 −0.925625
\(790\) 0 0
\(791\) 20.0000 0.711118
\(792\) 6.00000 0.213201
\(793\) 72.0000 2.55679
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 16.0000 0.566749 0.283375 0.959009i \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(798\) −12.0000 −0.424795
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 5.00000 0.176556
\(803\) −27.0000 −0.952809
\(804\) −9.00000 −0.317406
\(805\) 0 0
\(806\) 0 0
\(807\) 28.0000 0.985647
\(808\) 10.0000 0.351799
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 28.0000 0.982003
\(814\) 3.00000 0.105150
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) 12.0000 0.419827
\(818\) −15.0000 −0.524463
\(819\) −48.0000 −1.67726
\(820\) 0 0
\(821\) −4.00000 −0.139601 −0.0698005 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) −7.00000 −0.244153
\(823\) 42.0000 1.46403 0.732014 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) 7.00000 0.243414 0.121707 0.992566i \(-0.461163\pi\)
0.121707 + 0.992566i \(0.461163\pi\)
\(828\) −4.00000 −0.139010
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) 6.00000 0.208013
\(833\) 27.0000 0.935495
\(834\) 3.00000 0.103882
\(835\) 0 0
\(836\) −9.00000 −0.311272
\(837\) 0 0
\(838\) −27.0000 −0.932700
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 32.0000 1.10279
\(843\) 6.00000 0.206651
\(844\) 9.00000 0.309793
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −8.00000 −0.274883
\(848\) −2.00000 −0.0686803
\(849\) −7.00000 −0.240239
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 2.00000 0.0685189
\(853\) 40.0000 1.36957 0.684787 0.728743i \(-0.259895\pi\)
0.684787 + 0.728743i \(0.259895\pi\)
\(854\) −48.0000 −1.64253
\(855\) 0 0
\(856\) 1.00000 0.0341793
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 18.0000 0.614510
\(859\) 9.00000 0.307076 0.153538 0.988143i \(-0.450933\pi\)
0.153538 + 0.988143i \(0.450933\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) −2.00000 −0.0681203
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −5.00000 −0.169907
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) 54.0000 1.82972
\(872\) −18.0000 −0.609557
\(873\) −4.00000 −0.135379
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 9.00000 0.304082
\(877\) 26.0000 0.877958 0.438979 0.898497i \(-0.355340\pi\)
0.438979 + 0.898497i \(0.355340\pi\)
\(878\) −16.0000 −0.539974
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 18.0000 0.606092
\(883\) −7.00000 −0.235569 −0.117784 0.993039i \(-0.537579\pi\)
−0.117784 + 0.993039i \(0.537579\pi\)
\(884\) 18.0000 0.605406
\(885\) 0 0
\(886\) −5.00000 −0.167978
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 8.00000 0.267860
\(893\) −12.0000 −0.401565
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) −12.0000 −0.400668
\(898\) 37.0000 1.23471
\(899\) 0 0
\(900\) 0 0
\(901\) −6.00000 −0.199889
\(902\) 9.00000 0.299667
\(903\) 16.0000 0.532447
\(904\) −5.00000 −0.166298
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 28.0000 0.929213
\(909\) 20.0000 0.663358
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 3.00000 0.0993399
\(913\) −21.0000 −0.694999
\(914\) 33.0000 1.09154
\(915\) 0 0
\(916\) 0 0
\(917\) 48.0000 1.58510
\(918\) −15.0000 −0.495074
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) 29.0000 0.955582
\(922\) −20.0000 −0.658665
\(923\) −12.0000 −0.394985
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) −6.00000 −0.197172
\(927\) −28.0000 −0.919641
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) −27.0000 −0.884889
\(932\) −26.0000 −0.851658
\(933\) −28.0000 −0.916679
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) −36.0000 −1.17544
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 24.0000 0.781962
\(943\) −6.00000 −0.195387
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 2.00000 0.0649570
\(949\) −54.0000 −1.75291
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) −12.0000 −0.388922
\(953\) 33.0000 1.06897 0.534487 0.845176i \(-0.320505\pi\)
0.534487 + 0.845176i \(0.320505\pi\)
\(954\) −4.00000 −0.129505
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) −28.0000 −0.904167
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 6.00000 0.193448
\(963\) 2.00000 0.0644491
\(964\) −21.0000 −0.676364
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 2.00000 0.0642824
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) −16.0000 −0.513200
\(973\) 12.0000 0.384702
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) 23.0000 0.735835 0.367918 0.929858i \(-0.380071\pi\)
0.367918 + 0.929858i \(0.380071\pi\)
\(978\) −17.0000 −0.543600
\(979\) −9.00000 −0.287641
\(980\) 0 0
\(981\) −36.0000 −1.14939
\(982\) 20.0000 0.638226
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) −18.0000 −0.572656
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −12.0000 −0.381193 −0.190596 0.981669i \(-0.561042\pi\)
−0.190596 + 0.981669i \(0.561042\pi\)
\(992\) 0 0
\(993\) 3.00000 0.0952021
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 7.00000 0.221803
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 16.0000 0.506471
\(999\) −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 1850.2.a.c.1.1 1
5.2 odd 4 1850.2.b.e.149.1 2
5.3 odd 4 1850.2.b.e.149.2 2
5.4 even 2 1850.2.a.m.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.c.1.1 1 1.1 even 1 trivial
1850.2.a.m.1.1 yes 1 5.4 even 2
1850.2.b.e.149.1 2 5.2 odd 4
1850.2.b.e.149.2 2 5.3 odd 4