Properties

Label 8470.2.a.w.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
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 \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} -3.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +4.00000 q^{34} +1.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -3.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} -10.0000 q^{43} +2.00000 q^{45} -3.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +1.00000 q^{52} +6.00000 q^{53} +5.00000 q^{54} -1.00000 q^{56} +3.00000 q^{57} +2.00000 q^{58} +3.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -2.00000 q^{67} +4.00000 q^{68} +3.00000 q^{69} +1.00000 q^{70} -12.0000 q^{71} -2.00000 q^{72} -6.00000 q^{73} +8.00000 q^{74} -1.00000 q^{75} -3.00000 q^{76} -1.00000 q^{78} +17.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -3.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} -10.0000 q^{86} -2.00000 q^{87} +8.00000 q^{89} +2.00000 q^{90} -1.00000 q^{91} -3.00000 q^{92} -4.00000 q^{94} +3.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −2.00000 −0.471405
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 5.00000 0.962250
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −3.00000 −0.486664
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 1.00000 0.154303
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) −3.00000 −0.442326
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) 1.00000 0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 3.00000 0.397360
\(58\) 2.00000 0.262613
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 4.00000 0.485071
\(69\) 3.00000 0.361158
\(70\) 1.00000 0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −2.00000 −0.235702
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 8.00000 0.929981
\(75\) −1.00000 −0.115470
\(76\) −3.00000 −0.344124
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 17.0000 1.91265 0.956325 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 1.00000 0.109109
\(85\) −4.00000 −0.433861
\(86\) −10.0000 −1.07833
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 2.00000 0.210819
\(91\) −1.00000 −0.104828
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 3.00000 0.307794
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 17.0000 1.69156 0.845782 0.533529i \(-0.179135\pi\)
0.845782 + 0.533529i \(0.179135\pi\)
\(102\) −4.00000 −0.396059
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 1.00000 0.0980581
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 5.00000 0.481125
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −1.00000 −0.0944911
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 3.00000 0.280976
\(115\) 3.00000 0.279751
\(116\) 2.00000 0.185695
\(117\) −2.00000 −0.184900
\(118\) 3.00000 0.276172
\(119\) −4.00000 −0.366679
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) −14.0000 −1.26750
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.0000 0.880451
\(130\) −1.00000 −0.0877058
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 0 0
\(133\) 3.00000 0.260133
\(134\) −2.00000 −0.172774
\(135\) −5.00000 −0.430331
\(136\) 4.00000 0.342997
\(137\) −7.00000 −0.598050 −0.299025 0.954245i \(-0.596661\pi\)
−0.299025 + 0.954245i \(0.596661\pi\)
\(138\) 3.00000 0.255377
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 1.00000 0.0845154
\(141\) 4.00000 0.336861
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −2.00000 −0.166091
\(146\) −6.00000 −0.496564
\(147\) −1.00000 −0.0824786
\(148\) 8.00000 0.657596
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 3.00000 0.244137 0.122068 0.992522i \(-0.461047\pi\)
0.122068 + 0.992522i \(0.461047\pi\)
\(152\) −3.00000 −0.243332
\(153\) −8.00000 −0.646762
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 17.0000 1.35245
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 3.00000 0.236433
\(162\) 1.00000 0.0785674
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.0000 −0.923077
\(170\) −4.00000 −0.306786
\(171\) 6.00000 0.458831
\(172\) −10.0000 −0.762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −2.00000 −0.151620
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 8.00000 0.599625
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 2.00000 0.149071
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 14.0000 1.03491
\(184\) −3.00000 −0.221163
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) −4.00000 −0.291730
\(189\) −5.00000 −0.363696
\(190\) 3.00000 0.217643
\(191\) −25.0000 −1.80894 −0.904468 0.426541i \(-0.859732\pi\)
−0.904468 + 0.426541i \(0.859732\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 1.00000 0.0719816 0.0359908 0.999352i \(-0.488541\pi\)
0.0359908 + 0.999352i \(0.488541\pi\)
\(194\) −2.00000 −0.143592
\(195\) 1.00000 0.0716115
\(196\) 1.00000 0.0714286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 1.00000 0.0707107
\(201\) 2.00000 0.141069
\(202\) 17.0000 1.19612
\(203\) −2.00000 −0.140372
\(204\) −4.00000 −0.280056
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) 6.00000 0.417029
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000 0.822226
\(214\) −8.00000 −0.546869
\(215\) 10.0000 0.681994
\(216\) 5.00000 0.340207
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −8.00000 −0.536925
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.00000 −0.133333
\(226\) −3.00000 −0.199557
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 3.00000 0.198680
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −19.0000 −1.24473 −0.622366 0.782727i \(-0.713828\pi\)
−0.622366 + 0.782727i \(0.713828\pi\)
\(234\) −2.00000 −0.130744
\(235\) 4.00000 0.260931
\(236\) 3.00000 0.195283
\(237\) −17.0000 −1.10427
\(238\) −4.00000 −0.259281
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 1.00000 0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) −14.0000 −0.896258
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) −3.00000 −0.190885
\(248\) 0 0
\(249\) 3.00000 0.190117
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −17.0000 −1.06667
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 10.0000 0.622573
\(259\) −8.00000 −0.497096
\(260\) −1.00000 −0.0620174
\(261\) −4.00000 −0.247594
\(262\) −15.0000 −0.926703
\(263\) −23.0000 −1.41824 −0.709120 0.705087i \(-0.750908\pi\)
−0.709120 + 0.705087i \(0.750908\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 3.00000 0.183942
\(267\) −8.00000 −0.489592
\(268\) −2.00000 −0.122169
\(269\) 17.0000 1.03651 0.518254 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) −5.00000 −0.304290
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 1.00000 0.0605228
\(274\) −7.00000 −0.422885
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −3.00000 −0.179928
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 4.00000 0.238197
\(283\) 1.00000 0.0594438 0.0297219 0.999558i \(-0.490538\pi\)
0.0297219 + 0.999558i \(0.490538\pi\)
\(284\) −12.0000 −0.712069
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) −2.00000 −0.117851
\(289\) −1.00000 −0.0588235
\(290\) −2.00000 −0.117444
\(291\) 2.00000 0.117242
\(292\) −6.00000 −0.351123
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −3.00000 −0.174667
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −16.0000 −0.926855
\(299\) −3.00000 −0.173494
\(300\) −1.00000 −0.0577350
\(301\) 10.0000 0.576390
\(302\) 3.00000 0.172631
\(303\) −17.0000 −0.976624
\(304\) −3.00000 −0.172062
\(305\) 14.0000 0.801638
\(306\) −8.00000 −0.457330
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) −13.0000 −0.733632
\(315\) −2.00000 −0.112687
\(316\) 17.0000 0.956325
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 8.00000 0.446516
\(322\) 3.00000 0.167183
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 6.00000 0.332309
\(327\) 4.00000 0.221201
\(328\) 6.00000 0.331295
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) −3.00000 −0.164646
\(333\) −16.0000 −0.876795
\(334\) 6.00000 0.328305
\(335\) 2.00000 0.109272
\(336\) 1.00000 0.0545545
\(337\) −7.00000 −0.381314 −0.190657 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(338\) −12.0000 −0.652714
\(339\) 3.00000 0.162938
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) −1.00000 −0.0539949
\(344\) −10.0000 −0.539164
\(345\) −3.00000 −0.161515
\(346\) 6.00000 0.322562
\(347\) −14.0000 −0.751559 −0.375780 0.926709i \(-0.622625\pi\)
−0.375780 + 0.926709i \(0.622625\pi\)
\(348\) −2.00000 −0.107211
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −3.00000 −0.159448
\(355\) 12.0000 0.636894
\(356\) 8.00000 0.423999
\(357\) 4.00000 0.211702
\(358\) 2.00000 0.105703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 2.00000 0.105409
\(361\) −10.0000 −0.526316
\(362\) −7.00000 −0.367912
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 6.00000 0.314054
\(366\) 14.0000 0.731792
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −3.00000 −0.156386
\(369\) −12.0000 −0.624695
\(370\) −8.00000 −0.415900
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −4.00000 −0.206284
\(377\) 2.00000 0.103005
\(378\) −5.00000 −0.257172
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 3.00000 0.153897
\(381\) 17.0000 0.870936
\(382\) −25.0000 −1.27911
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 1.00000 0.0508987
\(387\) 20.0000 1.01666
\(388\) −2.00000 −0.101535
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 1.00000 0.0506370
\(391\) −12.0000 −0.606866
\(392\) 1.00000 0.0505076
\(393\) 15.0000 0.756650
\(394\) −10.0000 −0.503793
\(395\) −17.0000 −0.855363
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −14.0000 −0.701757
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 2.00000 0.0997509
\(403\) 0 0
\(404\) 17.0000 0.845782
\(405\) −1.00000 −0.0496904
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) −4.00000 −0.198030
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) −6.00000 −0.296319
\(411\) 7.00000 0.345285
\(412\) −8.00000 −0.394132
\(413\) −3.00000 −0.147620
\(414\) 6.00000 0.294884
\(415\) 3.00000 0.147264
\(416\) 1.00000 0.0490290
\(417\) 3.00000 0.146911
\(418\) 0 0
\(419\) −13.0000 −0.635092 −0.317546 0.948243i \(-0.602859\pi\)
−0.317546 + 0.948243i \(0.602859\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) −10.0000 −0.486792
\(423\) 8.00000 0.388973
\(424\) 6.00000 0.291386
\(425\) 4.00000 0.194029
\(426\) 12.0000 0.581402
\(427\) 14.0000 0.677507
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 10.0000 0.482243
\(431\) −17.0000 −0.818861 −0.409431 0.912341i \(-0.634273\pi\)
−0.409431 + 0.912341i \(0.634273\pi\)
\(432\) 5.00000 0.240563
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) −4.00000 −0.191565
\(437\) 9.00000 0.430528
\(438\) 6.00000 0.286691
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 4.00000 0.190261
\(443\) −8.00000 −0.380091 −0.190046 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) −8.00000 −0.379663
\(445\) −8.00000 −0.379236
\(446\) −2.00000 −0.0947027
\(447\) 16.0000 0.756774
\(448\) −1.00000 −0.0472456
\(449\) −31.0000 −1.46298 −0.731490 0.681852i \(-0.761175\pi\)
−0.731490 + 0.681852i \(0.761175\pi\)
\(450\) −2.00000 −0.0942809
\(451\) 0 0
\(452\) −3.00000 −0.141108
\(453\) −3.00000 −0.140952
\(454\) 24.0000 1.12638
\(455\) 1.00000 0.0468807
\(456\) 3.00000 0.140488
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 6.00000 0.280362
\(459\) 20.0000 0.933520
\(460\) 3.00000 0.139876
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 41.0000 1.90543 0.952716 0.303863i \(-0.0982765\pi\)
0.952716 + 0.303863i \(0.0982765\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −19.0000 −0.880158
\(467\) 11.0000 0.509019 0.254510 0.967070i \(-0.418086\pi\)
0.254510 + 0.967070i \(0.418086\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 2.00000 0.0923514
\(470\) 4.00000 0.184506
\(471\) 13.0000 0.599008
\(472\) 3.00000 0.138086
\(473\) 0 0
\(474\) −17.0000 −0.780836
\(475\) −3.00000 −0.137649
\(476\) −4.00000 −0.183340
\(477\) −12.0000 −0.549442
\(478\) 15.0000 0.686084
\(479\) 22.0000 1.00521 0.502603 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) 1.00000 0.0456435
\(481\) 8.00000 0.364769
\(482\) −10.0000 −0.455488
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) −16.0000 −0.725775
\(487\) 3.00000 0.135943 0.0679715 0.997687i \(-0.478347\pi\)
0.0679715 + 0.997687i \(0.478347\pi\)
\(488\) −14.0000 −0.633750
\(489\) −6.00000 −0.271329
\(490\) −1.00000 −0.0451754
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) −6.00000 −0.270501
\(493\) 8.00000 0.360302
\(494\) −3.00000 −0.134976
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 3.00000 0.134433
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −6.00000 −0.268060
\(502\) −12.0000 −0.535586
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 2.00000 0.0890871
\(505\) −17.0000 −0.756490
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) −17.0000 −0.754253
\(509\) 29.0000 1.28540 0.642701 0.766117i \(-0.277814\pi\)
0.642701 + 0.766117i \(0.277814\pi\)
\(510\) 4.00000 0.177123
\(511\) 6.00000 0.265424
\(512\) 1.00000 0.0441942
\(513\) −15.0000 −0.662266
\(514\) 14.0000 0.617514
\(515\) 8.00000 0.352522
\(516\) 10.0000 0.440225
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) −1.00000 −0.0438529
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) −4.00000 −0.175075
\(523\) 17.0000 0.743358 0.371679 0.928361i \(-0.378782\pi\)
0.371679 + 0.928361i \(0.378782\pi\)
\(524\) −15.0000 −0.655278
\(525\) 1.00000 0.0436436
\(526\) −23.0000 −1.00285
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −6.00000 −0.260623
\(531\) −6.00000 −0.260378
\(532\) 3.00000 0.130066
\(533\) 6.00000 0.259889
\(534\) −8.00000 −0.346194
\(535\) 8.00000 0.345870
\(536\) −2.00000 −0.0863868
\(537\) −2.00000 −0.0863064
\(538\) 17.0000 0.732922
\(539\) 0 0
\(540\) −5.00000 −0.215166
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 20.0000 0.859074
\(543\) 7.00000 0.300399
\(544\) 4.00000 0.171499
\(545\) 4.00000 0.171341
\(546\) 1.00000 0.0427960
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −7.00000 −0.299025
\(549\) 28.0000 1.19501
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 3.00000 0.127688
\(553\) −17.0000 −0.722914
\(554\) 22.0000 0.934690
\(555\) 8.00000 0.339581
\(556\) −3.00000 −0.127228
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −1.00000 −0.0421825
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 4.00000 0.168430
\(565\) 3.00000 0.126211
\(566\) 1.00000 0.0420331
\(567\) −1.00000 −0.0419961
\(568\) −12.0000 −0.503509
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) −3.00000 −0.125656
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 25.0000 1.04439
\(574\) −6.00000 −0.250435
\(575\) −3.00000 −0.125109
\(576\) −2.00000 −0.0833333
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −1.00000 −0.0415586
\(580\) −2.00000 −0.0830455
\(581\) 3.00000 0.124461
\(582\) 2.00000 0.0829027
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 2.00000 0.0826898
\(586\) −21.0000 −0.867502
\(587\) −11.0000 −0.454019 −0.227009 0.973893i \(-0.572895\pi\)
−0.227009 + 0.973893i \(0.572895\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) −3.00000 −0.123508
\(591\) 10.0000 0.411345
\(592\) 8.00000 0.328798
\(593\) 40.0000 1.64260 0.821302 0.570494i \(-0.193248\pi\)
0.821302 + 0.570494i \(0.193248\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) −16.0000 −0.655386
\(597\) 14.0000 0.572982
\(598\) −3.00000 −0.122679
\(599\) −1.00000 −0.0408589 −0.0204294 0.999791i \(-0.506503\pi\)
−0.0204294 + 0.999791i \(0.506503\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 10.0000 0.407570
\(603\) 4.00000 0.162893
\(604\) 3.00000 0.122068
\(605\) 0 0
\(606\) −17.0000 −0.690578
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −3.00000 −0.121666
\(609\) 2.00000 0.0810441
\(610\) 14.0000 0.566843
\(611\) −4.00000 −0.161823
\(612\) −8.00000 −0.323381
\(613\) 32.0000 1.29247 0.646234 0.763139i \(-0.276343\pi\)
0.646234 + 0.763139i \(0.276343\pi\)
\(614\) 12.0000 0.484281
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 8.00000 0.321807
\(619\) 25.0000 1.00483 0.502417 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) −20.0000 −0.801927
\(623\) −8.00000 −0.320513
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) 32.0000 1.27592
\(630\) −2.00000 −0.0796819
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 17.0000 0.676224
\(633\) 10.0000 0.397464
\(634\) −12.0000 −0.476581
\(635\) 17.0000 0.674624
\(636\) −6.00000 −0.237915
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) −1.00000 −0.0395285
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 8.00000 0.315735
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 3.00000 0.118217
\(645\) −10.0000 −0.393750
\(646\) −12.0000 −0.472134
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 4.00000 0.156412
\(655\) 15.0000 0.586098
\(656\) 6.00000 0.234261
\(657\) 12.0000 0.468165
\(658\) 4.00000 0.155936
\(659\) −2.00000 −0.0779089 −0.0389545 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) 0 0
\(661\) −45.0000 −1.75030 −0.875149 0.483854i \(-0.839236\pi\)
−0.875149 + 0.483854i \(0.839236\pi\)
\(662\) 6.00000 0.233197
\(663\) −4.00000 −0.155347
\(664\) −3.00000 −0.116423
\(665\) −3.00000 −0.116335
\(666\) −16.0000 −0.619987
\(667\) −6.00000 −0.232321
\(668\) 6.00000 0.232147
\(669\) 2.00000 0.0773245
\(670\) 2.00000 0.0772667
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 15.0000 0.578208 0.289104 0.957298i \(-0.406643\pi\)
0.289104 + 0.957298i \(0.406643\pi\)
\(674\) −7.00000 −0.269630
\(675\) 5.00000 0.192450
\(676\) −12.0000 −0.461538
\(677\) −23.0000 −0.883962 −0.441981 0.897024i \(-0.645724\pi\)
−0.441981 + 0.897024i \(0.645724\pi\)
\(678\) 3.00000 0.115214
\(679\) 2.00000 0.0767530
\(680\) −4.00000 −0.153393
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −34.0000 −1.30097 −0.650487 0.759517i \(-0.725435\pi\)
−0.650487 + 0.759517i \(0.725435\pi\)
\(684\) 6.00000 0.229416
\(685\) 7.00000 0.267456
\(686\) −1.00000 −0.0381802
\(687\) −6.00000 −0.228914
\(688\) −10.0000 −0.381246
\(689\) 6.00000 0.228582
\(690\) −3.00000 −0.114208
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −14.0000 −0.531433
\(695\) 3.00000 0.113796
\(696\) −2.00000 −0.0758098
\(697\) 24.0000 0.909065
\(698\) 15.0000 0.567758
\(699\) 19.0000 0.718646
\(700\) −1.00000 −0.0377964
\(701\) −16.0000 −0.604312 −0.302156 0.953259i \(-0.597706\pi\)
−0.302156 + 0.953259i \(0.597706\pi\)
\(702\) 5.00000 0.188713
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 2.00000 0.0752710
\(707\) −17.0000 −0.639351
\(708\) −3.00000 −0.112747
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 12.0000 0.450352
\(711\) −34.0000 −1.27510
\(712\) 8.00000 0.299813
\(713\) 0 0
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) −15.0000 −0.560185
\(718\) −24.0000 −0.895672
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 2.00000 0.0745356
\(721\) 8.00000 0.297936
\(722\) −10.0000 −0.372161
\(723\) 10.0000 0.371904
\(724\) −7.00000 −0.260153
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 13.0000 0.481481
\(730\) 6.00000 0.222070
\(731\) −40.0000 −1.47945
\(732\) 14.0000 0.517455
\(733\) 11.0000 0.406294 0.203147 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) −18.0000 −0.664392
\(735\) 1.00000 0.0368856
\(736\) −3.00000 −0.110581
\(737\) 0 0
\(738\) −12.0000 −0.441726
\(739\) 18.0000 0.662141 0.331070 0.943606i \(-0.392590\pi\)
0.331070 + 0.943606i \(0.392590\pi\)
\(740\) −8.00000 −0.294086
\(741\) 3.00000 0.110208
\(742\) −6.00000 −0.220267
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 16.0000 0.586195
\(746\) −16.0000 −0.585802
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 1.00000 0.0365148
\(751\) 31.0000 1.13121 0.565603 0.824678i \(-0.308643\pi\)
0.565603 + 0.824678i \(0.308643\pi\)
\(752\) −4.00000 −0.145865
\(753\) 12.0000 0.437304
\(754\) 2.00000 0.0728357
\(755\) −3.00000 −0.109181
\(756\) −5.00000 −0.181848
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 3.00000 0.108821
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 17.0000 0.615845
\(763\) 4.00000 0.144810
\(764\) −25.0000 −0.904468
\(765\) 8.00000 0.289241
\(766\) 28.0000 1.01168
\(767\) 3.00000 0.108324
\(768\) −1.00000 −0.0360844
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 1.00000 0.0359908
\(773\) −15.0000 −0.539513 −0.269756 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 20.0000 0.718885
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 8.00000 0.286998
\(778\) −18.0000 −0.645331
\(779\) −18.0000 −0.644917
\(780\) 1.00000 0.0358057
\(781\) 0 0
\(782\) −12.0000 −0.429119
\(783\) 10.0000 0.357371
\(784\) 1.00000 0.0357143
\(785\) 13.0000 0.463990
\(786\) 15.0000 0.535032
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −10.0000 −0.356235
\(789\) 23.0000 0.818822
\(790\) −17.0000 −0.604833
\(791\) 3.00000 0.106668
\(792\) 0 0
\(793\) −14.0000 −0.497155
\(794\) −22.0000 −0.780751
\(795\) 6.00000 0.212798
\(796\) −14.0000 −0.496217
\(797\) −21.0000 −0.743858 −0.371929 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) −3.00000 −0.106199
\(799\) −16.0000 −0.566039
\(800\) 1.00000 0.0353553
\(801\) −16.0000 −0.565332
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) −17.0000 −0.598428
\(808\) 17.0000 0.598058
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −48.0000 −1.68551 −0.842754 0.538299i \(-0.819067\pi\)
−0.842754 + 0.538299i \(0.819067\pi\)
\(812\) −2.00000 −0.0701862
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) −6.00000 −0.210171
\(816\) −4.00000 −0.140028
\(817\) 30.0000 1.04957
\(818\) 6.00000 0.209785
\(819\) 2.00000 0.0698857
\(820\) −6.00000 −0.209529
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 7.00000 0.244153
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 6.00000 0.208514
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 3.00000 0.104132
\(831\) −22.0000 −0.763172
\(832\) 1.00000 0.0346688
\(833\) 4.00000 0.138592
\(834\) 3.00000 0.103882
\(835\) −6.00000 −0.207639
\(836\) 0 0
\(837\) 0 0
\(838\) −13.0000 −0.449078
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −25.0000 −0.862069
\(842\) −32.0000 −1.10279
\(843\) 1.00000 0.0344418
\(844\) −10.0000 −0.344214
\(845\) 12.0000 0.412813
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −1.00000 −0.0343199
\(850\) 4.00000 0.137199
\(851\) −24.0000 −0.822709
\(852\) 12.0000 0.411113
\(853\) −43.0000 −1.47229 −0.736146 0.676823i \(-0.763356\pi\)
−0.736146 + 0.676823i \(0.763356\pi\)
\(854\) 14.0000 0.479070
\(855\) −6.00000 −0.205196
\(856\) −8.00000 −0.273434
\(857\) 24.0000 0.819824 0.409912 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 10.0000 0.340997
\(861\) 6.00000 0.204479
\(862\) −17.0000 −0.579022
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 5.00000 0.170103
\(865\) −6.00000 −0.204006
\(866\) −20.0000 −0.679628
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 2.00000 0.0678064
\(871\) −2.00000 −0.0677674
\(872\) −4.00000 −0.135457
\(873\) 4.00000 0.135379
\(874\) 9.00000 0.304430
\(875\) 1.00000 0.0338062
\(876\) 6.00000 0.202721
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −24.0000 −0.809961
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) 28.0000 0.943344 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 4.00000 0.134535
\(885\) 3.00000 0.100844
\(886\) −8.00000 −0.268765
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −8.00000 −0.268462
\(889\) 17.0000 0.570162
\(890\) −8.00000 −0.268161
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 12.0000 0.401565
\(894\) 16.0000 0.535120
\(895\) −2.00000 −0.0668526
\(896\) −1.00000 −0.0334077
\(897\) 3.00000 0.100167
\(898\) −31.0000 −1.03448
\(899\) 0 0
\(900\) −2.00000 −0.0666667
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) −10.0000 −0.332779
\(904\) −3.00000 −0.0997785
\(905\) 7.00000 0.232688
\(906\) −3.00000 −0.0996683
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 24.0000 0.796468
\(909\) −34.0000 −1.12771
\(910\) 1.00000 0.0331497
\(911\) −39.0000 −1.29213 −0.646064 0.763283i \(-0.723586\pi\)
−0.646064 + 0.763283i \(0.723586\pi\)
\(912\) 3.00000 0.0993399
\(913\) 0 0
\(914\) 5.00000 0.165385
\(915\) −14.0000 −0.462826
\(916\) 6.00000 0.198246
\(917\) 15.0000 0.495344
\(918\) 20.0000 0.660098
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 3.00000 0.0989071
\(921\) −12.0000 −0.395413
\(922\) 2.00000 0.0658665
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 41.0000 1.34734
\(927\) 16.0000 0.525509
\(928\) 2.00000 0.0656532
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −19.0000 −0.622366
\(933\) 20.0000 0.654771
\(934\) 11.0000 0.359931
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 54.0000 1.76410 0.882052 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) 2.00000 0.0653023
\(939\) −18.0000 −0.587408
\(940\) 4.00000 0.130466
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 13.0000 0.423563
\(943\) −18.0000 −0.586161
\(944\) 3.00000 0.0976417
\(945\) 5.00000 0.162650
\(946\) 0 0
\(947\) 38.0000 1.23483 0.617417 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) −17.0000 −0.552134
\(949\) −6.00000 −0.194768
\(950\) −3.00000 −0.0973329
\(951\) 12.0000 0.389127
\(952\) −4.00000 −0.129641
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) −12.0000 −0.388514
\(955\) 25.0000 0.808981
\(956\) 15.0000 0.485135
\(957\) 0 0
\(958\) 22.0000 0.710788
\(959\) 7.00000 0.226042
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 8.00000 0.257930
\(963\) 16.0000 0.515593
\(964\) −10.0000 −0.322078
\(965\) −1.00000 −0.0321911
\(966\) −3.00000 −0.0965234
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 2.00000 0.0642161
\(971\) −3.00000 −0.0962746 −0.0481373 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) −16.0000 −0.513200
\(973\) 3.00000 0.0961756
\(974\) 3.00000 0.0961262
\(975\) −1.00000 −0.0320256
\(976\) −14.0000 −0.448129
\(977\) −3.00000 −0.0959785 −0.0479893 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) −6.00000 −0.191859
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 8.00000 0.255420
\(982\) 6.00000 0.191468
\(983\) −32.0000 −1.02064 −0.510321 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) −6.00000 −0.191273
\(985\) 10.0000 0.318626
\(986\) 8.00000 0.254772
\(987\) −4.00000 −0.127321
\(988\) −3.00000 −0.0954427
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 0 0
\(993\) −6.00000 −0.190404
\(994\) 12.0000 0.380617
\(995\) 14.0000 0.443830
\(996\) 3.00000 0.0950586
\(997\) 27.0000 0.855099 0.427549 0.903992i \(-0.359377\pi\)
0.427549 + 0.903992i \(0.359377\pi\)
\(998\) −4.00000 −0.126618
\(999\) 40.0000 1.26554
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 8470.2.a.w.1.1 yes 1
11.10 odd 2 8470.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.g.1.1 1 11.10 odd 2
8470.2.a.w.1.1 yes 1 1.1 even 1 trivial