Properties

Label 1850.2.a.m.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
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: \(1\)
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
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 \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 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 \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −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 \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 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 \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 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 \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −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 \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) −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 \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 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 \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) −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 \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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 \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 1.00000 0.0966736 0.0483368 0.998831i \(-0.484608\pi\)
0.0483368 + 0.998831i \(0.484608\pi\)
\(108\) −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 \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) −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 \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 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 \(-0.403339\pi\)
0.299025 + 0.954245i \(0.403339\pi\)
\(138\) −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 \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) −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 \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) −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 \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 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 \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) −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 \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −5.00000 −0.332595
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) −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 \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 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 \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 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 \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) 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 \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 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 \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) −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 \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 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 \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) −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 \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 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 \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 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 \(-0.187148\pi\)
0.832084 + 0.554650i \(0.187148\pi\)
\(348\) 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 \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −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 \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −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 \(-0.758111\pi\)
−0.724893 + 0.688862i \(0.758111\pi\)
\(374\) −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 \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 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 \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 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 \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 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 \(-0.537898\pi\)
−0.118779 + 0.992921i \(0.537898\pi\)
\(444\) 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 \(-0.219338\pi\)
0.771837 + 0.635820i \(0.219338\pi\)
\(458\) 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 \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 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 \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 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 \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 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 \(-0.799963\pi\)
−0.808949 + 0.587879i \(0.799963\pi\)
\(524\) 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 \(-0.534090\pi\)
−0.106892 + 0.994271i \(0.534090\pi\)
\(548\) 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 \(-0.737128\pi\)
−0.677942 + 0.735116i \(0.737128\pi\)
\(558\) 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 \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −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 \(-0.398924\pi\)
0.312229 + 0.950007i \(0.398924\pi\)
\(578\) −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 \(-0.413544\pi\)
0.268284 + 0.963340i \(0.413544\pi\)
\(588\) 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 \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) −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 \(-0.323072\pi\)
0.527654 + 0.849460i \(0.323072\pi\)
\(608\) −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 \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) 29.0000 1.17034
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −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 \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 9.00000 0.354100
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 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 \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 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 \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 9.00000 0.346667
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 8.00000 0.307465 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(678\) −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 \(-0.605446\pi\)
−0.325243 + 0.945630i \(0.605446\pi\)
\(684\) 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 \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 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 \(-0.764568\pi\)
−0.738717 + 0.674016i \(0.764568\pi\)
\(734\) 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 \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 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 \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 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 \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) −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 \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −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 \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(798\) 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 \(-0.761419\pi\)
−0.732014 + 0.681290i \(0.761419\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) −7.00000 −0.243414 −0.121707 0.992566i \(-0.538837\pi\)
−0.121707 + 0.992566i \(0.538837\pi\)
\(828\) 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 \(-0.740105\pi\)
−0.684787 + 0.728743i \(0.740105\pi\)
\(854\) −48.0000 −1.64253
\(855\) 0 0
\(856\) 1.00000 0.0341793
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) −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 \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) −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 \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) 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 \(-0.462421\pi\)
0.117784 + 0.993039i \(0.462421\pi\)
\(884\) 18.0000 0.605406
\(885\) 0 0
\(886\) −5.00000 −0.167978
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 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 \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −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 \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 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 \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −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 \(-0.679495\pi\)
−0.534487 + 0.845176i \(0.679495\pi\)
\(954\) −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 \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) −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 \(-0.619929\pi\)
−0.367918 + 0.929858i \(0.619929\pi\)
\(978\) 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 \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) −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 \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −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.m.1.1 yes 1
5.2 odd 4 1850.2.b.e.149.2 2
5.3 odd 4 1850.2.b.e.149.1 2
5.4 even 2 1850.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.c.1.1 1 5.4 even 2
1850.2.a.m.1.1 yes 1 1.1 even 1 trivial
1850.2.b.e.149.1 2 5.3 odd 4
1850.2.b.e.149.2 2 5.2 odd 4