Properties

Label 294.4.a.f.1.1
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3465615417\)
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 \(\chi\) \(=\) 294.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +8.00000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +8.00000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} -16.0000 q^{10} +40.0000 q^{11} +12.0000 q^{12} +4.00000 q^{13} +24.0000 q^{15} +16.0000 q^{16} -84.0000 q^{17} -18.0000 q^{18} +148.000 q^{19} +32.0000 q^{20} -80.0000 q^{22} +84.0000 q^{23} -24.0000 q^{24} -61.0000 q^{25} -8.00000 q^{26} +27.0000 q^{27} +58.0000 q^{29} -48.0000 q^{30} -136.000 q^{31} -32.0000 q^{32} +120.000 q^{33} +168.000 q^{34} +36.0000 q^{36} -222.000 q^{37} -296.000 q^{38} +12.0000 q^{39} -64.0000 q^{40} +420.000 q^{41} -164.000 q^{43} +160.000 q^{44} +72.0000 q^{45} -168.000 q^{46} +488.000 q^{47} +48.0000 q^{48} +122.000 q^{50} -252.000 q^{51} +16.0000 q^{52} +478.000 q^{53} -54.0000 q^{54} +320.000 q^{55} +444.000 q^{57} -116.000 q^{58} +548.000 q^{59} +96.0000 q^{60} +692.000 q^{61} +272.000 q^{62} +64.0000 q^{64} +32.0000 q^{65} -240.000 q^{66} -908.000 q^{67} -336.000 q^{68} +252.000 q^{69} -524.000 q^{71} -72.0000 q^{72} +440.000 q^{73} +444.000 q^{74} -183.000 q^{75} +592.000 q^{76} -24.0000 q^{78} +1216.00 q^{79} +128.000 q^{80} +81.0000 q^{81} -840.000 q^{82} -684.000 q^{83} -672.000 q^{85} +328.000 q^{86} +174.000 q^{87} -320.000 q^{88} +604.000 q^{89} -144.000 q^{90} +336.000 q^{92} -408.000 q^{93} -976.000 q^{94} +1184.00 q^{95} -96.0000 q^{96} -832.000 q^{97} +360.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 8.00000 0.715542 0.357771 0.933809i \(-0.383537\pi\)
0.357771 + 0.933809i \(0.383537\pi\)
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −16.0000 −0.505964
\(11\) 40.0000 1.09640 0.548202 0.836346i \(-0.315312\pi\)
0.548202 + 0.836346i \(0.315312\pi\)
\(12\) 12.0000 0.288675
\(13\) 4.00000 0.0853385 0.0426692 0.999089i \(-0.486414\pi\)
0.0426692 + 0.999089i \(0.486414\pi\)
\(14\) 0 0
\(15\) 24.0000 0.413118
\(16\) 16.0000 0.250000
\(17\) −84.0000 −1.19841 −0.599206 0.800595i \(-0.704517\pi\)
−0.599206 + 0.800595i \(0.704517\pi\)
\(18\) −18.0000 −0.235702
\(19\) 148.000 1.78703 0.893514 0.449036i \(-0.148232\pi\)
0.893514 + 0.449036i \(0.148232\pi\)
\(20\) 32.0000 0.357771
\(21\) 0 0
\(22\) −80.0000 −0.775275
\(23\) 84.0000 0.761531 0.380765 0.924672i \(-0.375661\pi\)
0.380765 + 0.924672i \(0.375661\pi\)
\(24\) −24.0000 −0.204124
\(25\) −61.0000 −0.488000
\(26\) −8.00000 −0.0603434
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −48.0000 −0.292119
\(31\) −136.000 −0.787946 −0.393973 0.919122i \(-0.628900\pi\)
−0.393973 + 0.919122i \(0.628900\pi\)
\(32\) −32.0000 −0.176777
\(33\) 120.000 0.633010
\(34\) 168.000 0.847405
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) −222.000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −296.000 −1.26362
\(39\) 12.0000 0.0492702
\(40\) −64.0000 −0.252982
\(41\) 420.000 1.59983 0.799914 0.600114i \(-0.204878\pi\)
0.799914 + 0.600114i \(0.204878\pi\)
\(42\) 0 0
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) 160.000 0.548202
\(45\) 72.0000 0.238514
\(46\) −168.000 −0.538484
\(47\) 488.000 1.51451 0.757257 0.653118i \(-0.226539\pi\)
0.757257 + 0.653118i \(0.226539\pi\)
\(48\) 48.0000 0.144338
\(49\) 0 0
\(50\) 122.000 0.345068
\(51\) −252.000 −0.691903
\(52\) 16.0000 0.0426692
\(53\) 478.000 1.23884 0.619418 0.785061i \(-0.287368\pi\)
0.619418 + 0.785061i \(0.287368\pi\)
\(54\) −54.0000 −0.136083
\(55\) 320.000 0.784523
\(56\) 0 0
\(57\) 444.000 1.03174
\(58\) −116.000 −0.262613
\(59\) 548.000 1.20921 0.604606 0.796525i \(-0.293331\pi\)
0.604606 + 0.796525i \(0.293331\pi\)
\(60\) 96.0000 0.206559
\(61\) 692.000 1.45248 0.726242 0.687439i \(-0.241265\pi\)
0.726242 + 0.687439i \(0.241265\pi\)
\(62\) 272.000 0.557162
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 32.0000 0.0610633
\(66\) −240.000 −0.447605
\(67\) −908.000 −1.65567 −0.827835 0.560972i \(-0.810428\pi\)
−0.827835 + 0.560972i \(0.810428\pi\)
\(68\) −336.000 −0.599206
\(69\) 252.000 0.439670
\(70\) 0 0
\(71\) −524.000 −0.875878 −0.437939 0.899005i \(-0.644291\pi\)
−0.437939 + 0.899005i \(0.644291\pi\)
\(72\) −72.0000 −0.117851
\(73\) 440.000 0.705453 0.352727 0.935726i \(-0.385255\pi\)
0.352727 + 0.935726i \(0.385255\pi\)
\(74\) 444.000 0.697486
\(75\) −183.000 −0.281747
\(76\) 592.000 0.893514
\(77\) 0 0
\(78\) −24.0000 −0.0348393
\(79\) 1216.00 1.73178 0.865890 0.500234i \(-0.166753\pi\)
0.865890 + 0.500234i \(0.166753\pi\)
\(80\) 128.000 0.178885
\(81\) 81.0000 0.111111
\(82\) −840.000 −1.13125
\(83\) −684.000 −0.904563 −0.452282 0.891875i \(-0.649390\pi\)
−0.452282 + 0.891875i \(0.649390\pi\)
\(84\) 0 0
\(85\) −672.000 −0.857513
\(86\) 328.000 0.411269
\(87\) 174.000 0.214423
\(88\) −320.000 −0.387638
\(89\) 604.000 0.719369 0.359685 0.933074i \(-0.382884\pi\)
0.359685 + 0.933074i \(0.382884\pi\)
\(90\) −144.000 −0.168655
\(91\) 0 0
\(92\) 336.000 0.380765
\(93\) −408.000 −0.454921
\(94\) −976.000 −1.07092
\(95\) 1184.00 1.27869
\(96\) −96.0000 −0.102062
\(97\) −832.000 −0.870895 −0.435447 0.900214i \(-0.643410\pi\)
−0.435447 + 0.900214i \(0.643410\pi\)
\(98\) 0 0
\(99\) 360.000 0.365468
\(100\) −244.000 −0.244000
\(101\) 464.000 0.457126 0.228563 0.973529i \(-0.426597\pi\)
0.228563 + 0.973529i \(0.426597\pi\)
\(102\) 504.000 0.489249
\(103\) −632.000 −0.604590 −0.302295 0.953214i \(-0.597753\pi\)
−0.302295 + 0.953214i \(0.597753\pi\)
\(104\) −32.0000 −0.0301717
\(105\) 0 0
\(106\) −956.000 −0.875990
\(107\) −160.000 −0.144559 −0.0722794 0.997384i \(-0.523027\pi\)
−0.0722794 + 0.997384i \(0.523027\pi\)
\(108\) 108.000 0.0962250
\(109\) −2198.00 −1.93147 −0.965735 0.259530i \(-0.916432\pi\)
−0.965735 + 0.259530i \(0.916432\pi\)
\(110\) −640.000 −0.554742
\(111\) −666.000 −0.569495
\(112\) 0 0
\(113\) 770.000 0.641022 0.320511 0.947245i \(-0.396145\pi\)
0.320511 + 0.947245i \(0.396145\pi\)
\(114\) −888.000 −0.729551
\(115\) 672.000 0.544907
\(116\) 232.000 0.185695
\(117\) 36.0000 0.0284462
\(118\) −1096.00 −0.855042
\(119\) 0 0
\(120\) −192.000 −0.146059
\(121\) 269.000 0.202104
\(122\) −1384.00 −1.02706
\(123\) 1260.00 0.923662
\(124\) −544.000 −0.393973
\(125\) −1488.00 −1.06473
\(126\) 0 0
\(127\) −184.000 −0.128562 −0.0642809 0.997932i \(-0.520475\pi\)
−0.0642809 + 0.997932i \(0.520475\pi\)
\(128\) −128.000 −0.0883883
\(129\) −492.000 −0.335800
\(130\) −64.0000 −0.0431782
\(131\) −1452.00 −0.968411 −0.484205 0.874954i \(-0.660891\pi\)
−0.484205 + 0.874954i \(0.660891\pi\)
\(132\) 480.000 0.316505
\(133\) 0 0
\(134\) 1816.00 1.17074
\(135\) 216.000 0.137706
\(136\) 672.000 0.423702
\(137\) 646.000 0.402858 0.201429 0.979503i \(-0.435442\pi\)
0.201429 + 0.979503i \(0.435442\pi\)
\(138\) −504.000 −0.310894
\(139\) −3012.00 −1.83795 −0.918973 0.394320i \(-0.870980\pi\)
−0.918973 + 0.394320i \(0.870980\pi\)
\(140\) 0 0
\(141\) 1464.00 0.874405
\(142\) 1048.00 0.619339
\(143\) 160.000 0.0935655
\(144\) 144.000 0.0833333
\(145\) 464.000 0.265746
\(146\) −880.000 −0.498831
\(147\) 0 0
\(148\) −888.000 −0.493197
\(149\) −3170.00 −1.74293 −0.871465 0.490458i \(-0.836830\pi\)
−0.871465 + 0.490458i \(0.836830\pi\)
\(150\) 366.000 0.199225
\(151\) −1880.00 −1.01319 −0.506597 0.862183i \(-0.669097\pi\)
−0.506597 + 0.862183i \(0.669097\pi\)
\(152\) −1184.00 −0.631810
\(153\) −756.000 −0.399470
\(154\) 0 0
\(155\) −1088.00 −0.563808
\(156\) 48.0000 0.0246351
\(157\) 604.000 0.307035 0.153517 0.988146i \(-0.450940\pi\)
0.153517 + 0.988146i \(0.450940\pi\)
\(158\) −2432.00 −1.22455
\(159\) 1434.00 0.715243
\(160\) −256.000 −0.126491
\(161\) 0 0
\(162\) −162.000 −0.0785674
\(163\) 1116.00 0.536269 0.268135 0.963381i \(-0.413593\pi\)
0.268135 + 0.963381i \(0.413593\pi\)
\(164\) 1680.00 0.799914
\(165\) 960.000 0.452945
\(166\) 1368.00 0.639623
\(167\) −1784.00 −0.826647 −0.413324 0.910584i \(-0.635632\pi\)
−0.413324 + 0.910584i \(0.635632\pi\)
\(168\) 0 0
\(169\) −2181.00 −0.992717
\(170\) 1344.00 0.606353
\(171\) 1332.00 0.595676
\(172\) −656.000 −0.290811
\(173\) −344.000 −0.151178 −0.0755891 0.997139i \(-0.524084\pi\)
−0.0755891 + 0.997139i \(0.524084\pi\)
\(174\) −348.000 −0.151620
\(175\) 0 0
\(176\) 640.000 0.274101
\(177\) 1644.00 0.698139
\(178\) −1208.00 −0.508671
\(179\) 1392.00 0.581246 0.290623 0.956838i \(-0.406138\pi\)
0.290623 + 0.956838i \(0.406138\pi\)
\(180\) 288.000 0.119257
\(181\) 4052.00 1.66399 0.831997 0.554781i \(-0.187198\pi\)
0.831997 + 0.554781i \(0.187198\pi\)
\(182\) 0 0
\(183\) 2076.00 0.838592
\(184\) −672.000 −0.269242
\(185\) −1776.00 −0.705806
\(186\) 816.000 0.321678
\(187\) −3360.00 −1.31394
\(188\) 1952.00 0.757257
\(189\) 0 0
\(190\) −2368.00 −0.904173
\(191\) −3108.00 −1.17742 −0.588709 0.808345i \(-0.700364\pi\)
−0.588709 + 0.808345i \(0.700364\pi\)
\(192\) 192.000 0.0721688
\(193\) 50.0000 0.0186481 0.00932404 0.999957i \(-0.497032\pi\)
0.00932404 + 0.999957i \(0.497032\pi\)
\(194\) 1664.00 0.615816
\(195\) 96.0000 0.0352549
\(196\) 0 0
\(197\) −162.000 −0.0585889 −0.0292945 0.999571i \(-0.509326\pi\)
−0.0292945 + 0.999571i \(0.509326\pi\)
\(198\) −720.000 −0.258425
\(199\) 1544.00 0.550006 0.275003 0.961443i \(-0.411321\pi\)
0.275003 + 0.961443i \(0.411321\pi\)
\(200\) 488.000 0.172534
\(201\) −2724.00 −0.955901
\(202\) −928.000 −0.323237
\(203\) 0 0
\(204\) −1008.00 −0.345952
\(205\) 3360.00 1.14474
\(206\) 1264.00 0.427510
\(207\) 756.000 0.253844
\(208\) 64.0000 0.0213346
\(209\) 5920.00 1.95931
\(210\) 0 0
\(211\) −1204.00 −0.392828 −0.196414 0.980521i \(-0.562930\pi\)
−0.196414 + 0.980521i \(0.562930\pi\)
\(212\) 1912.00 0.619418
\(213\) −1572.00 −0.505689
\(214\) 320.000 0.102218
\(215\) −1312.00 −0.416175
\(216\) −216.000 −0.0680414
\(217\) 0 0
\(218\) 4396.00 1.36576
\(219\) 1320.00 0.407294
\(220\) 1280.00 0.392262
\(221\) −336.000 −0.102271
\(222\) 1332.00 0.402694
\(223\) 2000.00 0.600583 0.300291 0.953848i \(-0.402916\pi\)
0.300291 + 0.953848i \(0.402916\pi\)
\(224\) 0 0
\(225\) −549.000 −0.162667
\(226\) −1540.00 −0.453271
\(227\) 388.000 0.113447 0.0567235 0.998390i \(-0.481935\pi\)
0.0567235 + 0.998390i \(0.481935\pi\)
\(228\) 1776.00 0.515870
\(229\) 4180.00 1.20621 0.603105 0.797662i \(-0.293930\pi\)
0.603105 + 0.797662i \(0.293930\pi\)
\(230\) −1344.00 −0.385308
\(231\) 0 0
\(232\) −464.000 −0.131306
\(233\) −1322.00 −0.371704 −0.185852 0.982578i \(-0.559505\pi\)
−0.185852 + 0.982578i \(0.559505\pi\)
\(234\) −72.0000 −0.0201145
\(235\) 3904.00 1.08370
\(236\) 2192.00 0.604606
\(237\) 3648.00 0.999844
\(238\) 0 0
\(239\) 2412.00 0.652800 0.326400 0.945232i \(-0.394164\pi\)
0.326400 + 0.945232i \(0.394164\pi\)
\(240\) 384.000 0.103280
\(241\) −4336.00 −1.15895 −0.579474 0.814991i \(-0.696742\pi\)
−0.579474 + 0.814991i \(0.696742\pi\)
\(242\) −538.000 −0.142909
\(243\) 243.000 0.0641500
\(244\) 2768.00 0.726242
\(245\) 0 0
\(246\) −2520.00 −0.653127
\(247\) 592.000 0.152502
\(248\) 1088.00 0.278581
\(249\) −2052.00 −0.522250
\(250\) 2976.00 0.752875
\(251\) 764.000 0.192125 0.0960623 0.995375i \(-0.469375\pi\)
0.0960623 + 0.995375i \(0.469375\pi\)
\(252\) 0 0
\(253\) 3360.00 0.834946
\(254\) 368.000 0.0909070
\(255\) −2016.00 −0.495086
\(256\) 256.000 0.0625000
\(257\) 4300.00 1.04368 0.521842 0.853042i \(-0.325245\pi\)
0.521842 + 0.853042i \(0.325245\pi\)
\(258\) 984.000 0.237446
\(259\) 0 0
\(260\) 128.000 0.0305316
\(261\) 522.000 0.123797
\(262\) 2904.00 0.684770
\(263\) −3860.00 −0.905011 −0.452505 0.891762i \(-0.649470\pi\)
−0.452505 + 0.891762i \(0.649470\pi\)
\(264\) −960.000 −0.223803
\(265\) 3824.00 0.886439
\(266\) 0 0
\(267\) 1812.00 0.415328
\(268\) −3632.00 −0.827835
\(269\) −2800.00 −0.634643 −0.317322 0.948318i \(-0.602783\pi\)
−0.317322 + 0.948318i \(0.602783\pi\)
\(270\) −432.000 −0.0973729
\(271\) −4880.00 −1.09387 −0.546935 0.837175i \(-0.684206\pi\)
−0.546935 + 0.837175i \(0.684206\pi\)
\(272\) −1344.00 −0.299603
\(273\) 0 0
\(274\) −1292.00 −0.284863
\(275\) −2440.00 −0.535046
\(276\) 1008.00 0.219835
\(277\) −6674.00 −1.44766 −0.723830 0.689978i \(-0.757620\pi\)
−0.723830 + 0.689978i \(0.757620\pi\)
\(278\) 6024.00 1.29962
\(279\) −1224.00 −0.262649
\(280\) 0 0
\(281\) −9402.00 −1.99600 −0.998001 0.0632056i \(-0.979868\pi\)
−0.998001 + 0.0632056i \(0.979868\pi\)
\(282\) −2928.00 −0.618297
\(283\) −9100.00 −1.91144 −0.955722 0.294270i \(-0.904924\pi\)
−0.955722 + 0.294270i \(0.904924\pi\)
\(284\) −2096.00 −0.437939
\(285\) 3552.00 0.738254
\(286\) −320.000 −0.0661608
\(287\) 0 0
\(288\) −288.000 −0.0589256
\(289\) 2143.00 0.436190
\(290\) −928.000 −0.187910
\(291\) −2496.00 −0.502811
\(292\) 1760.00 0.352727
\(293\) 5952.00 1.18676 0.593378 0.804924i \(-0.297794\pi\)
0.593378 + 0.804924i \(0.297794\pi\)
\(294\) 0 0
\(295\) 4384.00 0.865242
\(296\) 1776.00 0.348743
\(297\) 1080.00 0.211003
\(298\) 6340.00 1.23244
\(299\) 336.000 0.0649879
\(300\) −732.000 −0.140873
\(301\) 0 0
\(302\) 3760.00 0.716436
\(303\) 1392.00 0.263922
\(304\) 2368.00 0.446757
\(305\) 5536.00 1.03931
\(306\) 1512.00 0.282468
\(307\) −3004.00 −0.558460 −0.279230 0.960224i \(-0.590079\pi\)
−0.279230 + 0.960224i \(0.590079\pi\)
\(308\) 0 0
\(309\) −1896.00 −0.349060
\(310\) 2176.00 0.398673
\(311\) 688.000 0.125443 0.0627217 0.998031i \(-0.480022\pi\)
0.0627217 + 0.998031i \(0.480022\pi\)
\(312\) −96.0000 −0.0174196
\(313\) 5592.00 1.00984 0.504918 0.863167i \(-0.331523\pi\)
0.504918 + 0.863167i \(0.331523\pi\)
\(314\) −1208.00 −0.217106
\(315\) 0 0
\(316\) 4864.00 0.865890
\(317\) −2922.00 −0.517716 −0.258858 0.965915i \(-0.583346\pi\)
−0.258858 + 0.965915i \(0.583346\pi\)
\(318\) −2868.00 −0.505753
\(319\) 2320.00 0.407195
\(320\) 512.000 0.0894427
\(321\) −480.000 −0.0834610
\(322\) 0 0
\(323\) −12432.0 −2.14159
\(324\) 324.000 0.0555556
\(325\) −244.000 −0.0416452
\(326\) −2232.00 −0.379200
\(327\) −6594.00 −1.11513
\(328\) −3360.00 −0.565625
\(329\) 0 0
\(330\) −1920.00 −0.320280
\(331\) −7492.00 −1.24410 −0.622051 0.782977i \(-0.713700\pi\)
−0.622051 + 0.782977i \(0.713700\pi\)
\(332\) −2736.00 −0.452282
\(333\) −1998.00 −0.328798
\(334\) 3568.00 0.584528
\(335\) −7264.00 −1.18470
\(336\) 0 0
\(337\) 10766.0 1.74024 0.870121 0.492839i \(-0.164041\pi\)
0.870121 + 0.492839i \(0.164041\pi\)
\(338\) 4362.00 0.701957
\(339\) 2310.00 0.370094
\(340\) −2688.00 −0.428757
\(341\) −5440.00 −0.863908
\(342\) −2664.00 −0.421206
\(343\) 0 0
\(344\) 1312.00 0.205635
\(345\) 2016.00 0.314602
\(346\) 688.000 0.106899
\(347\) −3984.00 −0.616347 −0.308173 0.951330i \(-0.599718\pi\)
−0.308173 + 0.951330i \(0.599718\pi\)
\(348\) 696.000 0.107211
\(349\) −180.000 −0.0276080 −0.0138040 0.999905i \(-0.504394\pi\)
−0.0138040 + 0.999905i \(0.504394\pi\)
\(350\) 0 0
\(351\) 108.000 0.0164234
\(352\) −1280.00 −0.193819
\(353\) −10428.0 −1.57231 −0.786156 0.618028i \(-0.787932\pi\)
−0.786156 + 0.618028i \(0.787932\pi\)
\(354\) −3288.00 −0.493659
\(355\) −4192.00 −0.626727
\(356\) 2416.00 0.359685
\(357\) 0 0
\(358\) −2784.00 −0.411003
\(359\) 8684.00 1.27667 0.638334 0.769759i \(-0.279624\pi\)
0.638334 + 0.769759i \(0.279624\pi\)
\(360\) −576.000 −0.0843274
\(361\) 15045.0 2.19347
\(362\) −8104.00 −1.17662
\(363\) 807.000 0.116685
\(364\) 0 0
\(365\) 3520.00 0.504781
\(366\) −4152.00 −0.592974
\(367\) 5648.00 0.803333 0.401666 0.915786i \(-0.368431\pi\)
0.401666 + 0.915786i \(0.368431\pi\)
\(368\) 1344.00 0.190383
\(369\) 3780.00 0.533276
\(370\) 3552.00 0.499080
\(371\) 0 0
\(372\) −1632.00 −0.227460
\(373\) −2546.00 −0.353423 −0.176712 0.984263i \(-0.556546\pi\)
−0.176712 + 0.984263i \(0.556546\pi\)
\(374\) 6720.00 0.929099
\(375\) −4464.00 −0.614720
\(376\) −3904.00 −0.535461
\(377\) 232.000 0.0316939
\(378\) 0 0
\(379\) 8268.00 1.12058 0.560288 0.828298i \(-0.310690\pi\)
0.560288 + 0.828298i \(0.310690\pi\)
\(380\) 4736.00 0.639347
\(381\) −552.000 −0.0742252
\(382\) 6216.00 0.832561
\(383\) −10872.0 −1.45048 −0.725239 0.688497i \(-0.758271\pi\)
−0.725239 + 0.688497i \(0.758271\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −100.000 −0.0131862
\(387\) −1476.00 −0.193874
\(388\) −3328.00 −0.435447
\(389\) 10434.0 1.35996 0.679980 0.733230i \(-0.261988\pi\)
0.679980 + 0.733230i \(0.261988\pi\)
\(390\) −192.000 −0.0249290
\(391\) −7056.00 −0.912627
\(392\) 0 0
\(393\) −4356.00 −0.559112
\(394\) 324.000 0.0414286
\(395\) 9728.00 1.23916
\(396\) 1440.00 0.182734
\(397\) −3044.00 −0.384821 −0.192411 0.981315i \(-0.561631\pi\)
−0.192411 + 0.981315i \(0.561631\pi\)
\(398\) −3088.00 −0.388913
\(399\) 0 0
\(400\) −976.000 −0.122000
\(401\) 8910.00 1.10959 0.554793 0.831988i \(-0.312797\pi\)
0.554793 + 0.831988i \(0.312797\pi\)
\(402\) 5448.00 0.675924
\(403\) −544.000 −0.0672421
\(404\) 1856.00 0.228563
\(405\) 648.000 0.0795046
\(406\) 0 0
\(407\) −8880.00 −1.08149
\(408\) 2016.00 0.244625
\(409\) 5616.00 0.678957 0.339478 0.940614i \(-0.389749\pi\)
0.339478 + 0.940614i \(0.389749\pi\)
\(410\) −6720.00 −0.809456
\(411\) 1938.00 0.232590
\(412\) −2528.00 −0.302295
\(413\) 0 0
\(414\) −1512.00 −0.179495
\(415\) −5472.00 −0.647253
\(416\) −128.000 −0.0150859
\(417\) −9036.00 −1.06114
\(418\) −11840.0 −1.38544
\(419\) −8932.00 −1.04142 −0.520712 0.853732i \(-0.674334\pi\)
−0.520712 + 0.853732i \(0.674334\pi\)
\(420\) 0 0
\(421\) −5538.00 −0.641106 −0.320553 0.947231i \(-0.603869\pi\)
−0.320553 + 0.947231i \(0.603869\pi\)
\(422\) 2408.00 0.277772
\(423\) 4392.00 0.504838
\(424\) −3824.00 −0.437995
\(425\) 5124.00 0.584825
\(426\) 3144.00 0.357576
\(427\) 0 0
\(428\) −640.000 −0.0722794
\(429\) 480.000 0.0540201
\(430\) 2624.00 0.294280
\(431\) −6700.00 −0.748788 −0.374394 0.927270i \(-0.622149\pi\)
−0.374394 + 0.927270i \(0.622149\pi\)
\(432\) 432.000 0.0481125
\(433\) −5048.00 −0.560257 −0.280129 0.959962i \(-0.590377\pi\)
−0.280129 + 0.959962i \(0.590377\pi\)
\(434\) 0 0
\(435\) 1392.00 0.153428
\(436\) −8792.00 −0.965735
\(437\) 12432.0 1.36088
\(438\) −2640.00 −0.288000
\(439\) −1344.00 −0.146118 −0.0730588 0.997328i \(-0.523276\pi\)
−0.0730588 + 0.997328i \(0.523276\pi\)
\(440\) −2560.00 −0.277371
\(441\) 0 0
\(442\) 672.000 0.0723162
\(443\) −4392.00 −0.471039 −0.235519 0.971870i \(-0.575679\pi\)
−0.235519 + 0.971870i \(0.575679\pi\)
\(444\) −2664.00 −0.284747
\(445\) 4832.00 0.514739
\(446\) −4000.00 −0.424676
\(447\) −9510.00 −1.00628
\(448\) 0 0
\(449\) 3666.00 0.385321 0.192661 0.981265i \(-0.438288\pi\)
0.192661 + 0.981265i \(0.438288\pi\)
\(450\) 1098.00 0.115023
\(451\) 16800.0 1.75406
\(452\) 3080.00 0.320511
\(453\) −5640.00 −0.584968
\(454\) −776.000 −0.0802191
\(455\) 0 0
\(456\) −3552.00 −0.364776
\(457\) 26.0000 0.00266133 0.00133067 0.999999i \(-0.499576\pi\)
0.00133067 + 0.999999i \(0.499576\pi\)
\(458\) −8360.00 −0.852920
\(459\) −2268.00 −0.230634
\(460\) 2688.00 0.272454
\(461\) 7656.00 0.773483 0.386741 0.922188i \(-0.373601\pi\)
0.386741 + 0.922188i \(0.373601\pi\)
\(462\) 0 0
\(463\) 12608.0 1.26554 0.632768 0.774341i \(-0.281919\pi\)
0.632768 + 0.774341i \(0.281919\pi\)
\(464\) 928.000 0.0928477
\(465\) −3264.00 −0.325515
\(466\) 2644.00 0.262835
\(467\) −3068.00 −0.304005 −0.152002 0.988380i \(-0.548572\pi\)
−0.152002 + 0.988380i \(0.548572\pi\)
\(468\) 144.000 0.0142231
\(469\) 0 0
\(470\) −7808.00 −0.766290
\(471\) 1812.00 0.177267
\(472\) −4384.00 −0.427521
\(473\) −6560.00 −0.637694
\(474\) −7296.00 −0.706997
\(475\) −9028.00 −0.872070
\(476\) 0 0
\(477\) 4302.00 0.412946
\(478\) −4824.00 −0.461600
\(479\) 6456.00 0.615829 0.307915 0.951414i \(-0.400369\pi\)
0.307915 + 0.951414i \(0.400369\pi\)
\(480\) −768.000 −0.0730297
\(481\) −888.000 −0.0841774
\(482\) 8672.00 0.819500
\(483\) 0 0
\(484\) 1076.00 0.101052
\(485\) −6656.00 −0.623162
\(486\) −486.000 −0.0453609
\(487\) 11896.0 1.10690 0.553449 0.832883i \(-0.313311\pi\)
0.553449 + 0.832883i \(0.313311\pi\)
\(488\) −5536.00 −0.513531
\(489\) 3348.00 0.309615
\(490\) 0 0
\(491\) −264.000 −0.0242651 −0.0121325 0.999926i \(-0.503862\pi\)
−0.0121325 + 0.999926i \(0.503862\pi\)
\(492\) 5040.00 0.461831
\(493\) −4872.00 −0.445079
\(494\) −1184.00 −0.107835
\(495\) 2880.00 0.261508
\(496\) −2176.00 −0.196986
\(497\) 0 0
\(498\) 4104.00 0.369286
\(499\) −2628.00 −0.235762 −0.117881 0.993028i \(-0.537610\pi\)
−0.117881 + 0.993028i \(0.537610\pi\)
\(500\) −5952.00 −0.532363
\(501\) −5352.00 −0.477265
\(502\) −1528.00 −0.135853
\(503\) −13568.0 −1.20272 −0.601359 0.798979i \(-0.705374\pi\)
−0.601359 + 0.798979i \(0.705374\pi\)
\(504\) 0 0
\(505\) 3712.00 0.327093
\(506\) −6720.00 −0.590396
\(507\) −6543.00 −0.573146
\(508\) −736.000 −0.0642809
\(509\) 20656.0 1.79874 0.899372 0.437183i \(-0.144024\pi\)
0.899372 + 0.437183i \(0.144024\pi\)
\(510\) 4032.00 0.350078
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 3996.00 0.343914
\(514\) −8600.00 −0.737996
\(515\) −5056.00 −0.432610
\(516\) −1968.00 −0.167900
\(517\) 19520.0 1.66052
\(518\) 0 0
\(519\) −1032.00 −0.0872828
\(520\) −256.000 −0.0215891
\(521\) 3628.00 0.305078 0.152539 0.988297i \(-0.451255\pi\)
0.152539 + 0.988297i \(0.451255\pi\)
\(522\) −1044.00 −0.0875376
\(523\) 4852.00 0.405666 0.202833 0.979213i \(-0.434985\pi\)
0.202833 + 0.979213i \(0.434985\pi\)
\(524\) −5808.00 −0.484205
\(525\) 0 0
\(526\) 7720.00 0.639939
\(527\) 11424.0 0.944283
\(528\) 1920.00 0.158252
\(529\) −5111.00 −0.420071
\(530\) −7648.00 −0.626807
\(531\) 4932.00 0.403071
\(532\) 0 0
\(533\) 1680.00 0.136527
\(534\) −3624.00 −0.293681
\(535\) −1280.00 −0.103438
\(536\) 7264.00 0.585368
\(537\) 4176.00 0.335582
\(538\) 5600.00 0.448760
\(539\) 0 0
\(540\) 864.000 0.0688530
\(541\) −7130.00 −0.566622 −0.283311 0.959028i \(-0.591433\pi\)
−0.283311 + 0.959028i \(0.591433\pi\)
\(542\) 9760.00 0.773483
\(543\) 12156.0 0.960707
\(544\) 2688.00 0.211851
\(545\) −17584.0 −1.38205
\(546\) 0 0
\(547\) −12788.0 −0.999589 −0.499795 0.866144i \(-0.666591\pi\)
−0.499795 + 0.866144i \(0.666591\pi\)
\(548\) 2584.00 0.201429
\(549\) 6228.00 0.484161
\(550\) 4880.00 0.378334
\(551\) 8584.00 0.663685
\(552\) −2016.00 −0.155447
\(553\) 0 0
\(554\) 13348.0 1.02365
\(555\) −5328.00 −0.407497
\(556\) −12048.0 −0.918973
\(557\) 2406.00 0.183026 0.0915130 0.995804i \(-0.470830\pi\)
0.0915130 + 0.995804i \(0.470830\pi\)
\(558\) 2448.00 0.185721
\(559\) −656.000 −0.0496348
\(560\) 0 0
\(561\) −10080.0 −0.758606
\(562\) 18804.0 1.41139
\(563\) −25412.0 −1.90229 −0.951144 0.308748i \(-0.900090\pi\)
−0.951144 + 0.308748i \(0.900090\pi\)
\(564\) 5856.00 0.437202
\(565\) 6160.00 0.458678
\(566\) 18200.0 1.35160
\(567\) 0 0
\(568\) 4192.00 0.309670
\(569\) −9690.00 −0.713930 −0.356965 0.934118i \(-0.616188\pi\)
−0.356965 + 0.934118i \(0.616188\pi\)
\(570\) −7104.00 −0.522024
\(571\) 5604.00 0.410718 0.205359 0.978687i \(-0.434164\pi\)
0.205359 + 0.978687i \(0.434164\pi\)
\(572\) 640.000 0.0467828
\(573\) −9324.00 −0.679783
\(574\) 0 0
\(575\) −5124.00 −0.371627
\(576\) 576.000 0.0416667
\(577\) −21568.0 −1.55613 −0.778066 0.628183i \(-0.783799\pi\)
−0.778066 + 0.628183i \(0.783799\pi\)
\(578\) −4286.00 −0.308433
\(579\) 150.000 0.0107665
\(580\) 1856.00 0.132873
\(581\) 0 0
\(582\) 4992.00 0.355541
\(583\) 19120.0 1.35827
\(584\) −3520.00 −0.249415
\(585\) 288.000 0.0203544
\(586\) −11904.0 −0.839163
\(587\) 20300.0 1.42738 0.713689 0.700463i \(-0.247023\pi\)
0.713689 + 0.700463i \(0.247023\pi\)
\(588\) 0 0
\(589\) −20128.0 −1.40808
\(590\) −8768.00 −0.611818
\(591\) −486.000 −0.0338263
\(592\) −3552.00 −0.246598
\(593\) 13812.0 0.956477 0.478238 0.878230i \(-0.341275\pi\)
0.478238 + 0.878230i \(0.341275\pi\)
\(594\) −2160.00 −0.149202
\(595\) 0 0
\(596\) −12680.0 −0.871465
\(597\) 4632.00 0.317546
\(598\) −672.000 −0.0459534
\(599\) −21996.0 −1.50039 −0.750194 0.661218i \(-0.770040\pi\)
−0.750194 + 0.661218i \(0.770040\pi\)
\(600\) 1464.00 0.0996126
\(601\) 8368.00 0.567950 0.283975 0.958832i \(-0.408347\pi\)
0.283975 + 0.958832i \(0.408347\pi\)
\(602\) 0 0
\(603\) −8172.00 −0.551890
\(604\) −7520.00 −0.506597
\(605\) 2152.00 0.144614
\(606\) −2784.00 −0.186621
\(607\) 21504.0 1.43792 0.718962 0.695049i \(-0.244617\pi\)
0.718962 + 0.695049i \(0.244617\pi\)
\(608\) −4736.00 −0.315905
\(609\) 0 0
\(610\) −11072.0 −0.734905
\(611\) 1952.00 0.129246
\(612\) −3024.00 −0.199735
\(613\) −10270.0 −0.676674 −0.338337 0.941025i \(-0.609864\pi\)
−0.338337 + 0.941025i \(0.609864\pi\)
\(614\) 6008.00 0.394891
\(615\) 10080.0 0.660918
\(616\) 0 0
\(617\) 28358.0 1.85032 0.925162 0.379572i \(-0.123929\pi\)
0.925162 + 0.379572i \(0.123929\pi\)
\(618\) 3792.00 0.246823
\(619\) 16292.0 1.05788 0.528942 0.848658i \(-0.322589\pi\)
0.528942 + 0.848658i \(0.322589\pi\)
\(620\) −4352.00 −0.281904
\(621\) 2268.00 0.146557
\(622\) −1376.00 −0.0887019
\(623\) 0 0
\(624\) 192.000 0.0123176
\(625\) −4279.00 −0.273856
\(626\) −11184.0 −0.714062
\(627\) 17760.0 1.13121
\(628\) 2416.00 0.153517
\(629\) 18648.0 1.18211
\(630\) 0 0
\(631\) 11256.0 0.710134 0.355067 0.934841i \(-0.384458\pi\)
0.355067 + 0.934841i \(0.384458\pi\)
\(632\) −9728.00 −0.612277
\(633\) −3612.00 −0.226800
\(634\) 5844.00 0.366080
\(635\) −1472.00 −0.0919914
\(636\) 5736.00 0.357621
\(637\) 0 0
\(638\) −4640.00 −0.287930
\(639\) −4716.00 −0.291959
\(640\) −1024.00 −0.0632456
\(641\) 15518.0 0.956200 0.478100 0.878305i \(-0.341326\pi\)
0.478100 + 0.878305i \(0.341326\pi\)
\(642\) 960.000 0.0590159
\(643\) −10452.0 −0.641037 −0.320518 0.947242i \(-0.603857\pi\)
−0.320518 + 0.947242i \(0.603857\pi\)
\(644\) 0 0
\(645\) −3936.00 −0.240279
\(646\) 24864.0 1.51434
\(647\) 72.0000 0.00437498 0.00218749 0.999998i \(-0.499304\pi\)
0.00218749 + 0.999998i \(0.499304\pi\)
\(648\) −648.000 −0.0392837
\(649\) 21920.0 1.32579
\(650\) 488.000 0.0294476
\(651\) 0 0
\(652\) 4464.00 0.268135
\(653\) 11962.0 0.716859 0.358430 0.933557i \(-0.383312\pi\)
0.358430 + 0.933557i \(0.383312\pi\)
\(654\) 13188.0 0.788519
\(655\) −11616.0 −0.692938
\(656\) 6720.00 0.399957
\(657\) 3960.00 0.235151
\(658\) 0 0
\(659\) −6016.00 −0.355615 −0.177807 0.984065i \(-0.556900\pi\)
−0.177807 + 0.984065i \(0.556900\pi\)
\(660\) 3840.00 0.226472
\(661\) 26068.0 1.53393 0.766965 0.641689i \(-0.221766\pi\)
0.766965 + 0.641689i \(0.221766\pi\)
\(662\) 14984.0 0.879713
\(663\) −1008.00 −0.0590460
\(664\) 5472.00 0.319811
\(665\) 0 0
\(666\) 3996.00 0.232495
\(667\) 4872.00 0.282825
\(668\) −7136.00 −0.413324
\(669\) 6000.00 0.346746
\(670\) 14528.0 0.837710
\(671\) 27680.0 1.59251
\(672\) 0 0
\(673\) −20530.0 −1.17589 −0.587945 0.808901i \(-0.700063\pi\)
−0.587945 + 0.808901i \(0.700063\pi\)
\(674\) −21532.0 −1.23054
\(675\) −1647.00 −0.0939156
\(676\) −8724.00 −0.496359
\(677\) −10056.0 −0.570877 −0.285438 0.958397i \(-0.592139\pi\)
−0.285438 + 0.958397i \(0.592139\pi\)
\(678\) −4620.00 −0.261696
\(679\) 0 0
\(680\) 5376.00 0.303177
\(681\) 1164.00 0.0654986
\(682\) 10880.0 0.610875
\(683\) 6152.00 0.344656 0.172328 0.985040i \(-0.444871\pi\)
0.172328 + 0.985040i \(0.444871\pi\)
\(684\) 5328.00 0.297838
\(685\) 5168.00 0.288262
\(686\) 0 0
\(687\) 12540.0 0.696406
\(688\) −2624.00 −0.145406
\(689\) 1912.00 0.105720
\(690\) −4032.00 −0.222457
\(691\) −14716.0 −0.810164 −0.405082 0.914280i \(-0.632757\pi\)
−0.405082 + 0.914280i \(0.632757\pi\)
\(692\) −1376.00 −0.0755891
\(693\) 0 0
\(694\) 7968.00 0.435823
\(695\) −24096.0 −1.31513
\(696\) −1392.00 −0.0758098
\(697\) −35280.0 −1.91725
\(698\) 360.000 0.0195218
\(699\) −3966.00 −0.214604
\(700\) 0 0
\(701\) 28202.0 1.51951 0.759754 0.650211i \(-0.225319\pi\)
0.759754 + 0.650211i \(0.225319\pi\)
\(702\) −216.000 −0.0116131
\(703\) −32856.0 −1.76271
\(704\) 2560.00 0.137051
\(705\) 11712.0 0.625673
\(706\) 20856.0 1.11179
\(707\) 0 0
\(708\) 6576.00 0.349070
\(709\) 22114.0 1.17138 0.585690 0.810535i \(-0.300824\pi\)
0.585690 + 0.810535i \(0.300824\pi\)
\(710\) 8384.00 0.443163
\(711\) 10944.0 0.577260
\(712\) −4832.00 −0.254335
\(713\) −11424.0 −0.600045
\(714\) 0 0
\(715\) 1280.00 0.0669501
\(716\) 5568.00 0.290623
\(717\) 7236.00 0.376895
\(718\) −17368.0 −0.902741
\(719\) −9288.00 −0.481758 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(720\) 1152.00 0.0596285
\(721\) 0 0
\(722\) −30090.0 −1.55102
\(723\) −13008.0 −0.669119
\(724\) 16208.0 0.831997
\(725\) −3538.00 −0.181239
\(726\) −1614.00 −0.0825085
\(727\) 23848.0 1.21661 0.608304 0.793704i \(-0.291850\pi\)
0.608304 + 0.793704i \(0.291850\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −7040.00 −0.356934
\(731\) 13776.0 0.697023
\(732\) 8304.00 0.419296
\(733\) −34756.0 −1.75135 −0.875677 0.482898i \(-0.839584\pi\)
−0.875677 + 0.482898i \(0.839584\pi\)
\(734\) −11296.0 −0.568042
\(735\) 0 0
\(736\) −2688.00 −0.134621
\(737\) −36320.0 −1.81528
\(738\) −7560.00 −0.377083
\(739\) 26044.0 1.29641 0.648203 0.761468i \(-0.275521\pi\)
0.648203 + 0.761468i \(0.275521\pi\)
\(740\) −7104.00 −0.352903
\(741\) 1776.00 0.0880472
\(742\) 0 0
\(743\) 36204.0 1.78761 0.893806 0.448454i \(-0.148025\pi\)
0.893806 + 0.448454i \(0.148025\pi\)
\(744\) 3264.00 0.160839
\(745\) −25360.0 −1.24714
\(746\) 5092.00 0.249908
\(747\) −6156.00 −0.301521
\(748\) −13440.0 −0.656972
\(749\) 0 0
\(750\) 8928.00 0.434673
\(751\) −11424.0 −0.555083 −0.277542 0.960714i \(-0.589520\pi\)
−0.277542 + 0.960714i \(0.589520\pi\)
\(752\) 7808.00 0.378628
\(753\) 2292.00 0.110923
\(754\) −464.000 −0.0224110
\(755\) −15040.0 −0.724982
\(756\) 0 0
\(757\) −16622.0 −0.798067 −0.399034 0.916936i \(-0.630654\pi\)
−0.399034 + 0.916936i \(0.630654\pi\)
\(758\) −16536.0 −0.792368
\(759\) 10080.0 0.482056
\(760\) −9472.00 −0.452086
\(761\) −38524.0 −1.83508 −0.917539 0.397646i \(-0.869827\pi\)
−0.917539 + 0.397646i \(0.869827\pi\)
\(762\) 1104.00 0.0524852
\(763\) 0 0
\(764\) −12432.0 −0.588709
\(765\) −6048.00 −0.285838
\(766\) 21744.0 1.02564
\(767\) 2192.00 0.103192
\(768\) 768.000 0.0360844
\(769\) −18440.0 −0.864712 −0.432356 0.901703i \(-0.642318\pi\)
−0.432356 + 0.901703i \(0.642318\pi\)
\(770\) 0 0
\(771\) 12900.0 0.602571
\(772\) 200.000 0.00932404
\(773\) −13968.0 −0.649928 −0.324964 0.945726i \(-0.605352\pi\)
−0.324964 + 0.945726i \(0.605352\pi\)
\(774\) 2952.00 0.137090
\(775\) 8296.00 0.384518
\(776\) 6656.00 0.307908
\(777\) 0 0
\(778\) −20868.0 −0.961638
\(779\) 62160.0 2.85894
\(780\) 384.000 0.0176274
\(781\) −20960.0 −0.960317
\(782\) 14112.0 0.645325
\(783\) 1566.00 0.0714742
\(784\) 0 0
\(785\) 4832.00 0.219696
\(786\) 8712.00 0.395352
\(787\) 10916.0 0.494426 0.247213 0.968961i \(-0.420485\pi\)
0.247213 + 0.968961i \(0.420485\pi\)
\(788\) −648.000 −0.0292945
\(789\) −11580.0 −0.522508
\(790\) −19456.0 −0.876220
\(791\) 0 0
\(792\) −2880.00 −0.129213
\(793\) 2768.00 0.123953
\(794\) 6088.00 0.272110
\(795\) 11472.0 0.511786
\(796\) 6176.00 0.275003
\(797\) 12360.0 0.549327 0.274664 0.961540i \(-0.411434\pi\)
0.274664 + 0.961540i \(0.411434\pi\)
\(798\) 0 0
\(799\) −40992.0 −1.81501
\(800\) 1952.00 0.0862670
\(801\) 5436.00 0.239790
\(802\) −17820.0 −0.784596
\(803\) 17600.0 0.773463
\(804\) −10896.0 −0.477951
\(805\) 0 0
\(806\) 1088.00 0.0475474
\(807\) −8400.00 −0.366411
\(808\) −3712.00 −0.161618
\(809\) 3402.00 0.147847 0.0739233 0.997264i \(-0.476448\pi\)
0.0739233 + 0.997264i \(0.476448\pi\)
\(810\) −1296.00 −0.0562183
\(811\) 292.000 0.0126430 0.00632152 0.999980i \(-0.497988\pi\)
0.00632152 + 0.999980i \(0.497988\pi\)
\(812\) 0 0
\(813\) −14640.0 −0.631546
\(814\) 17760.0 0.764727
\(815\) 8928.00 0.383723
\(816\) −4032.00 −0.172976
\(817\) −24272.0 −1.03938
\(818\) −11232.0 −0.480095
\(819\) 0 0
\(820\) 13440.0 0.572372
\(821\) 6910.00 0.293740 0.146870 0.989156i \(-0.453080\pi\)
0.146870 + 0.989156i \(0.453080\pi\)
\(822\) −3876.00 −0.164466
\(823\) 568.000 0.0240574 0.0120287 0.999928i \(-0.496171\pi\)
0.0120287 + 0.999928i \(0.496171\pi\)
\(824\) 5056.00 0.213755
\(825\) −7320.00 −0.308909
\(826\) 0 0
\(827\) −12144.0 −0.510627 −0.255313 0.966858i \(-0.582179\pi\)
−0.255313 + 0.966858i \(0.582179\pi\)
\(828\) 3024.00 0.126922
\(829\) 14828.0 0.621228 0.310614 0.950536i \(-0.399465\pi\)
0.310614 + 0.950536i \(0.399465\pi\)
\(830\) 10944.0 0.457677
\(831\) −20022.0 −0.835807
\(832\) 256.000 0.0106673
\(833\) 0 0
\(834\) 18072.0 0.750338
\(835\) −14272.0 −0.591501
\(836\) 23680.0 0.979653
\(837\) −3672.00 −0.151640
\(838\) 17864.0 0.736398
\(839\) −22824.0 −0.939180 −0.469590 0.882885i \(-0.655598\pi\)
−0.469590 + 0.882885i \(0.655598\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) 11076.0 0.453330
\(843\) −28206.0 −1.15239
\(844\) −4816.00 −0.196414
\(845\) −17448.0 −0.710331
\(846\) −8784.00 −0.356974
\(847\) 0 0
\(848\) 7648.00 0.309709
\(849\) −27300.0 −1.10357
\(850\) −10248.0 −0.413534
\(851\) −18648.0 −0.751169
\(852\) −6288.00 −0.252844
\(853\) −41780.0 −1.67705 −0.838523 0.544866i \(-0.816580\pi\)
−0.838523 + 0.544866i \(0.816580\pi\)
\(854\) 0 0
\(855\) 10656.0 0.426231
\(856\) 1280.00 0.0511092
\(857\) −21420.0 −0.853784 −0.426892 0.904303i \(-0.640392\pi\)
−0.426892 + 0.904303i \(0.640392\pi\)
\(858\) −960.000 −0.0381980
\(859\) −18132.0 −0.720205 −0.360102 0.932913i \(-0.617258\pi\)
−0.360102 + 0.932913i \(0.617258\pi\)
\(860\) −5248.00 −0.208088
\(861\) 0 0
\(862\) 13400.0 0.529473
\(863\) 24036.0 0.948082 0.474041 0.880503i \(-0.342795\pi\)
0.474041 + 0.880503i \(0.342795\pi\)
\(864\) −864.000 −0.0340207
\(865\) −2752.00 −0.108174
\(866\) 10096.0 0.396162
\(867\) 6429.00 0.251834
\(868\) 0 0
\(869\) 48640.0 1.89873
\(870\) −2784.00 −0.108490
\(871\) −3632.00 −0.141292
\(872\) 17584.0 0.682878
\(873\) −7488.00 −0.290298
\(874\) −24864.0 −0.962285
\(875\) 0 0
\(876\) 5280.00 0.203647
\(877\) −4374.00 −0.168414 −0.0842072 0.996448i \(-0.526836\pi\)
−0.0842072 + 0.996448i \(0.526836\pi\)
\(878\) 2688.00 0.103321
\(879\) 17856.0 0.685174
\(880\) 5120.00 0.196131
\(881\) −46348.0 −1.77242 −0.886211 0.463282i \(-0.846672\pi\)
−0.886211 + 0.463282i \(0.846672\pi\)
\(882\) 0 0
\(883\) −20660.0 −0.787389 −0.393694 0.919241i \(-0.628803\pi\)
−0.393694 + 0.919241i \(0.628803\pi\)
\(884\) −1344.00 −0.0511353
\(885\) 13152.0 0.499548
\(886\) 8784.00 0.333075
\(887\) −1800.00 −0.0681376 −0.0340688 0.999419i \(-0.510847\pi\)
−0.0340688 + 0.999419i \(0.510847\pi\)
\(888\) 5328.00 0.201347
\(889\) 0 0
\(890\) −9664.00 −0.363975
\(891\) 3240.00 0.121823
\(892\) 8000.00 0.300291
\(893\) 72224.0 2.70648
\(894\) 19020.0 0.711548
\(895\) 11136.0 0.415906
\(896\) 0 0
\(897\) 1008.00 0.0375208
\(898\) −7332.00 −0.272463
\(899\) −7888.00 −0.292636
\(900\) −2196.00 −0.0813333
\(901\) −40152.0 −1.48464
\(902\) −33600.0 −1.24031
\(903\) 0 0
\(904\) −6160.00 −0.226636
\(905\) 32416.0 1.19066
\(906\) 11280.0 0.413635
\(907\) −41996.0 −1.53744 −0.768718 0.639588i \(-0.779105\pi\)
−0.768718 + 0.639588i \(0.779105\pi\)
\(908\) 1552.00 0.0567235
\(909\) 4176.00 0.152375
\(910\) 0 0
\(911\) −41308.0 −1.50230 −0.751150 0.660132i \(-0.770500\pi\)
−0.751150 + 0.660132i \(0.770500\pi\)
\(912\) 7104.00 0.257935
\(913\) −27360.0 −0.991768
\(914\) −52.0000 −0.00188185
\(915\) 16608.0 0.600048
\(916\) 16720.0 0.603105
\(917\) 0 0
\(918\) 4536.00 0.163083
\(919\) 3936.00 0.141280 0.0706402 0.997502i \(-0.477496\pi\)
0.0706402 + 0.997502i \(0.477496\pi\)
\(920\) −5376.00 −0.192654
\(921\) −9012.00 −0.322427
\(922\) −15312.0 −0.546935
\(923\) −2096.00 −0.0747461
\(924\) 0 0
\(925\) 13542.0 0.481360
\(926\) −25216.0 −0.894870
\(927\) −5688.00 −0.201530
\(928\) −1856.00 −0.0656532
\(929\) −7212.00 −0.254702 −0.127351 0.991858i \(-0.540647\pi\)
−0.127351 + 0.991858i \(0.540647\pi\)
\(930\) 6528.00 0.230174
\(931\) 0 0
\(932\) −5288.00 −0.185852
\(933\) 2064.00 0.0724248
\(934\) 6136.00 0.214964
\(935\) −26880.0 −0.940182
\(936\) −288.000 −0.0100572
\(937\) −38976.0 −1.35890 −0.679451 0.733721i \(-0.737782\pi\)
−0.679451 + 0.733721i \(0.737782\pi\)
\(938\) 0 0
\(939\) 16776.0 0.583029
\(940\) 15616.0 0.541849
\(941\) 53544.0 1.85493 0.927463 0.373916i \(-0.121985\pi\)
0.927463 + 0.373916i \(0.121985\pi\)
\(942\) −3624.00 −0.125346
\(943\) 35280.0 1.21832
\(944\) 8768.00 0.302303
\(945\) 0 0
\(946\) 13120.0 0.450918
\(947\) −21392.0 −0.734051 −0.367026 0.930211i \(-0.619624\pi\)
−0.367026 + 0.930211i \(0.619624\pi\)
\(948\) 14592.0 0.499922
\(949\) 1760.00 0.0602023
\(950\) 18056.0 0.616646
\(951\) −8766.00 −0.298903
\(952\) 0 0
\(953\) 21162.0 0.719312 0.359656 0.933085i \(-0.382894\pi\)
0.359656 + 0.933085i \(0.382894\pi\)
\(954\) −8604.00 −0.291997
\(955\) −24864.0 −0.842492
\(956\) 9648.00 0.326400
\(957\) 6960.00 0.235094
\(958\) −12912.0 −0.435457
\(959\) 0 0
\(960\) 1536.00 0.0516398
\(961\) −11295.0 −0.379141
\(962\) 1776.00 0.0595224
\(963\) −1440.00 −0.0481862
\(964\) −17344.0 −0.579474
\(965\) 400.000 0.0133435
\(966\) 0 0
\(967\) 8224.00 0.273491 0.136746 0.990606i \(-0.456336\pi\)
0.136746 + 0.990606i \(0.456336\pi\)
\(968\) −2152.00 −0.0714544
\(969\) −37296.0 −1.23645
\(970\) 13312.0 0.440642
\(971\) −8140.00 −0.269027 −0.134513 0.990912i \(-0.542947\pi\)
−0.134513 + 0.990912i \(0.542947\pi\)
\(972\) 972.000 0.0320750
\(973\) 0 0
\(974\) −23792.0 −0.782695
\(975\) −732.000 −0.0240439
\(976\) 11072.0 0.363121
\(977\) 32158.0 1.05305 0.526523 0.850161i \(-0.323495\pi\)
0.526523 + 0.850161i \(0.323495\pi\)
\(978\) −6696.00 −0.218931
\(979\) 24160.0 0.788720
\(980\) 0 0
\(981\) −19782.0 −0.643823
\(982\) 528.000 0.0171580
\(983\) −41416.0 −1.34381 −0.671905 0.740637i \(-0.734524\pi\)
−0.671905 + 0.740637i \(0.734524\pi\)
\(984\) −10080.0 −0.326564
\(985\) −1296.00 −0.0419228
\(986\) 9744.00 0.314718
\(987\) 0 0
\(988\) 2368.00 0.0762511
\(989\) −13776.0 −0.442923
\(990\) −5760.00 −0.184914
\(991\) 12296.0 0.394143 0.197071 0.980389i \(-0.436857\pi\)
0.197071 + 0.980389i \(0.436857\pi\)
\(992\) 4352.00 0.139290
\(993\) −22476.0 −0.718282
\(994\) 0 0
\(995\) 12352.0 0.393552
\(996\) −8208.00 −0.261125
\(997\) 57652.0 1.83135 0.915676 0.401918i \(-0.131656\pi\)
0.915676 + 0.401918i \(0.131656\pi\)
\(998\) 5256.00 0.166709
\(999\) −5994.00 −0.189832
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 294.4.a.f.1.1 yes 1
3.2 odd 2 882.4.a.j.1.1 1
4.3 odd 2 2352.4.a.m.1.1 1
7.2 even 3 294.4.e.f.67.1 2
7.3 odd 6 294.4.e.j.79.1 2
7.4 even 3 294.4.e.f.79.1 2
7.5 odd 6 294.4.e.j.67.1 2
7.6 odd 2 294.4.a.b.1.1 1
21.2 odd 6 882.4.g.j.361.1 2
21.5 even 6 882.4.g.c.361.1 2
21.11 odd 6 882.4.g.j.667.1 2
21.17 even 6 882.4.g.c.667.1 2
21.20 even 2 882.4.a.q.1.1 1
28.27 even 2 2352.4.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.b.1.1 1 7.6 odd 2
294.4.a.f.1.1 yes 1 1.1 even 1 trivial
294.4.e.f.67.1 2 7.2 even 3
294.4.e.f.79.1 2 7.4 even 3
294.4.e.j.67.1 2 7.5 odd 6
294.4.e.j.79.1 2 7.3 odd 6
882.4.a.j.1.1 1 3.2 odd 2
882.4.a.q.1.1 1 21.20 even 2
882.4.g.c.361.1 2 21.5 even 6
882.4.g.c.667.1 2 21.17 even 6
882.4.g.j.361.1 2 21.2 odd 6
882.4.g.j.667.1 2 21.11 odd 6
2352.4.a.m.1.1 1 4.3 odd 2
2352.4.a.z.1.1 1 28.27 even 2