Properties

Label 663.4.a.a.1.1
Level $663$
Weight $4$
Character 663.1
Self dual yes
Analytic conductor $39.118$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 663.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.1182663338\)
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 \(\chi\) \(=\) 663.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -10.0000 q^{5} +12.0000 q^{6} -10.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +3.00000 q^{3} +8.00000 q^{4} -10.0000 q^{5} +12.0000 q^{6} -10.0000 q^{7} +9.00000 q^{9} -40.0000 q^{10} +18.0000 q^{11} +24.0000 q^{12} -13.0000 q^{13} -40.0000 q^{14} -30.0000 q^{15} -64.0000 q^{16} -17.0000 q^{17} +36.0000 q^{18} -74.0000 q^{19} -80.0000 q^{20} -30.0000 q^{21} +72.0000 q^{22} -132.000 q^{23} -25.0000 q^{25} -52.0000 q^{26} +27.0000 q^{27} -80.0000 q^{28} +210.000 q^{29} -120.000 q^{30} -230.000 q^{31} -256.000 q^{32} +54.0000 q^{33} -68.0000 q^{34} +100.000 q^{35} +72.0000 q^{36} -46.0000 q^{37} -296.000 q^{38} -39.0000 q^{39} -114.000 q^{41} -120.000 q^{42} +36.0000 q^{43} +144.000 q^{44} -90.0000 q^{45} -528.000 q^{46} +446.000 q^{47} -192.000 q^{48} -243.000 q^{49} -100.000 q^{50} -51.0000 q^{51} -104.000 q^{52} -754.000 q^{53} +108.000 q^{54} -180.000 q^{55} -222.000 q^{57} +840.000 q^{58} -50.0000 q^{59} -240.000 q^{60} -226.000 q^{61} -920.000 q^{62} -90.0000 q^{63} -512.000 q^{64} +130.000 q^{65} +216.000 q^{66} +582.000 q^{67} -136.000 q^{68} -396.000 q^{69} +400.000 q^{70} -370.000 q^{71} +826.000 q^{73} -184.000 q^{74} -75.0000 q^{75} -592.000 q^{76} -180.000 q^{77} -156.000 q^{78} +272.000 q^{79} +640.000 q^{80} +81.0000 q^{81} -456.000 q^{82} +162.000 q^{83} -240.000 q^{84} +170.000 q^{85} +144.000 q^{86} +630.000 q^{87} -186.000 q^{89} -360.000 q^{90} +130.000 q^{91} -1056.00 q^{92} -690.000 q^{93} +1784.00 q^{94} +740.000 q^{95} -768.000 q^{96} -790.000 q^{97} -972.000 q^{98} +162.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 3.00000 0.577350
\(4\) 8.00000 1.00000
\(5\) −10.0000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 12.0000 0.816497
\(7\) −10.0000 −0.539949 −0.269975 0.962867i \(-0.587015\pi\)
−0.269975 + 0.962867i \(0.587015\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) −40.0000 −1.26491
\(11\) 18.0000 0.493382 0.246691 0.969094i \(-0.420657\pi\)
0.246691 + 0.969094i \(0.420657\pi\)
\(12\) 24.0000 0.577350
\(13\) −13.0000 −0.277350
\(14\) −40.0000 −0.763604
\(15\) −30.0000 −0.516398
\(16\) −64.0000 −1.00000
\(17\) −17.0000 −0.242536
\(18\) 36.0000 0.471405
\(19\) −74.0000 −0.893514 −0.446757 0.894655i \(-0.647421\pi\)
−0.446757 + 0.894655i \(0.647421\pi\)
\(20\) −80.0000 −0.894427
\(21\) −30.0000 −0.311740
\(22\) 72.0000 0.697748
\(23\) −132.000 −1.19669 −0.598346 0.801238i \(-0.704175\pi\)
−0.598346 + 0.801238i \(0.704175\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) −52.0000 −0.392232
\(27\) 27.0000 0.192450
\(28\) −80.0000 −0.539949
\(29\) 210.000 1.34469 0.672345 0.740238i \(-0.265287\pi\)
0.672345 + 0.740238i \(0.265287\pi\)
\(30\) −120.000 −0.730297
\(31\) −230.000 −1.33256 −0.666278 0.745704i \(-0.732113\pi\)
−0.666278 + 0.745704i \(0.732113\pi\)
\(32\) −256.000 −1.41421
\(33\) 54.0000 0.284854
\(34\) −68.0000 −0.342997
\(35\) 100.000 0.482945
\(36\) 72.0000 0.333333
\(37\) −46.0000 −0.204388 −0.102194 0.994764i \(-0.532586\pi\)
−0.102194 + 0.994764i \(0.532586\pi\)
\(38\) −296.000 −1.26362
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) −114.000 −0.434239 −0.217120 0.976145i \(-0.569666\pi\)
−0.217120 + 0.976145i \(0.569666\pi\)
\(42\) −120.000 −0.440867
\(43\) 36.0000 0.127673 0.0638366 0.997960i \(-0.479666\pi\)
0.0638366 + 0.997960i \(0.479666\pi\)
\(44\) 144.000 0.493382
\(45\) −90.0000 −0.298142
\(46\) −528.000 −1.69238
\(47\) 446.000 1.38417 0.692083 0.721818i \(-0.256693\pi\)
0.692083 + 0.721818i \(0.256693\pi\)
\(48\) −192.000 −0.577350
\(49\) −243.000 −0.708455
\(50\) −100.000 −0.282843
\(51\) −51.0000 −0.140028
\(52\) −104.000 −0.277350
\(53\) −754.000 −1.95415 −0.977074 0.212899i \(-0.931709\pi\)
−0.977074 + 0.212899i \(0.931709\pi\)
\(54\) 108.000 0.272166
\(55\) −180.000 −0.441294
\(56\) 0 0
\(57\) −222.000 −0.515870
\(58\) 840.000 1.90168
\(59\) −50.0000 −0.110330 −0.0551648 0.998477i \(-0.517568\pi\)
−0.0551648 + 0.998477i \(0.517568\pi\)
\(60\) −240.000 −0.516398
\(61\) −226.000 −0.474366 −0.237183 0.971465i \(-0.576224\pi\)
−0.237183 + 0.971465i \(0.576224\pi\)
\(62\) −920.000 −1.88452
\(63\) −90.0000 −0.179983
\(64\) −512.000 −1.00000
\(65\) 130.000 0.248069
\(66\) 216.000 0.402845
\(67\) 582.000 1.06123 0.530617 0.847612i \(-0.321960\pi\)
0.530617 + 0.847612i \(0.321960\pi\)
\(68\) −136.000 −0.242536
\(69\) −396.000 −0.690910
\(70\) 400.000 0.682988
\(71\) −370.000 −0.618464 −0.309232 0.950987i \(-0.600072\pi\)
−0.309232 + 0.950987i \(0.600072\pi\)
\(72\) 0 0
\(73\) 826.000 1.32433 0.662164 0.749359i \(-0.269638\pi\)
0.662164 + 0.749359i \(0.269638\pi\)
\(74\) −184.000 −0.289048
\(75\) −75.0000 −0.115470
\(76\) −592.000 −0.893514
\(77\) −180.000 −0.266401
\(78\) −156.000 −0.226455
\(79\) 272.000 0.387372 0.193686 0.981064i \(-0.437956\pi\)
0.193686 + 0.981064i \(0.437956\pi\)
\(80\) 640.000 0.894427
\(81\) 81.0000 0.111111
\(82\) −456.000 −0.614107
\(83\) 162.000 0.214239 0.107119 0.994246i \(-0.465837\pi\)
0.107119 + 0.994246i \(0.465837\pi\)
\(84\) −240.000 −0.311740
\(85\) 170.000 0.216930
\(86\) 144.000 0.180557
\(87\) 630.000 0.776357
\(88\) 0 0
\(89\) −186.000 −0.221528 −0.110764 0.993847i \(-0.535330\pi\)
−0.110764 + 0.993847i \(0.535330\pi\)
\(90\) −360.000 −0.421637
\(91\) 130.000 0.149755
\(92\) −1056.00 −1.19669
\(93\) −690.000 −0.769351
\(94\) 1784.00 1.95751
\(95\) 740.000 0.799183
\(96\) −768.000 −0.816497
\(97\) −790.000 −0.826931 −0.413466 0.910520i \(-0.635682\pi\)
−0.413466 + 0.910520i \(0.635682\pi\)
\(98\) −972.000 −1.00191
\(99\) 162.000 0.164461
\(100\) −200.000 −0.200000
\(101\) 1518.00 1.49551 0.747756 0.663974i \(-0.231131\pi\)
0.747756 + 0.663974i \(0.231131\pi\)
\(102\) −204.000 −0.198030
\(103\) 1400.00 1.33928 0.669641 0.742685i \(-0.266448\pi\)
0.669641 + 0.742685i \(0.266448\pi\)
\(104\) 0 0
\(105\) 300.000 0.278829
\(106\) −3016.00 −2.76358
\(107\) −836.000 −0.755319 −0.377660 0.925944i \(-0.623271\pi\)
−0.377660 + 0.925944i \(0.623271\pi\)
\(108\) 216.000 0.192450
\(109\) 986.000 0.866437 0.433219 0.901289i \(-0.357378\pi\)
0.433219 + 0.901289i \(0.357378\pi\)
\(110\) −720.000 −0.624085
\(111\) −138.000 −0.118003
\(112\) 640.000 0.539949
\(113\) 78.0000 0.0649347 0.0324674 0.999473i \(-0.489664\pi\)
0.0324674 + 0.999473i \(0.489664\pi\)
\(114\) −888.000 −0.729551
\(115\) 1320.00 1.07035
\(116\) 1680.00 1.34469
\(117\) −117.000 −0.0924500
\(118\) −200.000 −0.156030
\(119\) 170.000 0.130957
\(120\) 0 0
\(121\) −1007.00 −0.756574
\(122\) −904.000 −0.670855
\(123\) −342.000 −0.250708
\(124\) −1840.00 −1.33256
\(125\) 1500.00 1.07331
\(126\) −360.000 −0.254535
\(127\) 2464.00 1.72161 0.860806 0.508934i \(-0.169960\pi\)
0.860806 + 0.508934i \(0.169960\pi\)
\(128\) 0 0
\(129\) 108.000 0.0737122
\(130\) 520.000 0.350823
\(131\) −2628.00 −1.75274 −0.876372 0.481635i \(-0.840043\pi\)
−0.876372 + 0.481635i \(0.840043\pi\)
\(132\) 432.000 0.284854
\(133\) 740.000 0.482452
\(134\) 2328.00 1.50081
\(135\) −270.000 −0.172133
\(136\) 0 0
\(137\) −714.000 −0.445264 −0.222632 0.974903i \(-0.571465\pi\)
−0.222632 + 0.974903i \(0.571465\pi\)
\(138\) −1584.00 −0.977094
\(139\) −356.000 −0.217234 −0.108617 0.994084i \(-0.534642\pi\)
−0.108617 + 0.994084i \(0.534642\pi\)
\(140\) 800.000 0.482945
\(141\) 1338.00 0.799148
\(142\) −1480.00 −0.874640
\(143\) −234.000 −0.136840
\(144\) −576.000 −0.333333
\(145\) −2100.00 −1.20273
\(146\) 3304.00 1.87288
\(147\) −729.000 −0.409027
\(148\) −368.000 −0.204388
\(149\) 1446.00 0.795040 0.397520 0.917594i \(-0.369871\pi\)
0.397520 + 0.917594i \(0.369871\pi\)
\(150\) −300.000 −0.163299
\(151\) −526.000 −0.283479 −0.141739 0.989904i \(-0.545269\pi\)
−0.141739 + 0.989904i \(0.545269\pi\)
\(152\) 0 0
\(153\) −153.000 −0.0808452
\(154\) −720.000 −0.376748
\(155\) 2300.00 1.19187
\(156\) −312.000 −0.160128
\(157\) −262.000 −0.133184 −0.0665920 0.997780i \(-0.521213\pi\)
−0.0665920 + 0.997780i \(0.521213\pi\)
\(158\) 1088.00 0.547827
\(159\) −2262.00 −1.12823
\(160\) 2560.00 1.26491
\(161\) 1320.00 0.646153
\(162\) 324.000 0.157135
\(163\) −2634.00 −1.26571 −0.632855 0.774270i \(-0.718117\pi\)
−0.632855 + 0.774270i \(0.718117\pi\)
\(164\) −912.000 −0.434239
\(165\) −540.000 −0.254781
\(166\) 648.000 0.302979
\(167\) 2002.00 0.927661 0.463831 0.885924i \(-0.346475\pi\)
0.463831 + 0.885924i \(0.346475\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 680.000 0.306786
\(171\) −666.000 −0.297838
\(172\) 288.000 0.127673
\(173\) 1142.00 0.501877 0.250938 0.968003i \(-0.419261\pi\)
0.250938 + 0.968003i \(0.419261\pi\)
\(174\) 2520.00 1.09794
\(175\) 250.000 0.107990
\(176\) −1152.00 −0.493382
\(177\) −150.000 −0.0636988
\(178\) −744.000 −0.313287
\(179\) −2580.00 −1.07731 −0.538654 0.842527i \(-0.681067\pi\)
−0.538654 + 0.842527i \(0.681067\pi\)
\(180\) −720.000 −0.298142
\(181\) 1874.00 0.769576 0.384788 0.923005i \(-0.374274\pi\)
0.384788 + 0.923005i \(0.374274\pi\)
\(182\) 520.000 0.211786
\(183\) −678.000 −0.273875
\(184\) 0 0
\(185\) 460.000 0.182810
\(186\) −2760.00 −1.08803
\(187\) −306.000 −0.119663
\(188\) 3568.00 1.38417
\(189\) −270.000 −0.103913
\(190\) 2960.00 1.13022
\(191\) −412.000 −0.156080 −0.0780400 0.996950i \(-0.524866\pi\)
−0.0780400 + 0.996950i \(0.524866\pi\)
\(192\) −1536.00 −0.577350
\(193\) 1082.00 0.403544 0.201772 0.979432i \(-0.435330\pi\)
0.201772 + 0.979432i \(0.435330\pi\)
\(194\) −3160.00 −1.16946
\(195\) 390.000 0.143223
\(196\) −1944.00 −0.708455
\(197\) −1194.00 −0.431822 −0.215911 0.976413i \(-0.569272\pi\)
−0.215911 + 0.976413i \(0.569272\pi\)
\(198\) 648.000 0.232583
\(199\) −2860.00 −1.01879 −0.509397 0.860532i \(-0.670132\pi\)
−0.509397 + 0.860532i \(0.670132\pi\)
\(200\) 0 0
\(201\) 1746.00 0.612703
\(202\) 6072.00 2.11497
\(203\) −2100.00 −0.726065
\(204\) −408.000 −0.140028
\(205\) 1140.00 0.388395
\(206\) 5600.00 1.89403
\(207\) −1188.00 −0.398897
\(208\) 832.000 0.277350
\(209\) −1332.00 −0.440844
\(210\) 1200.00 0.394323
\(211\) −5972.00 −1.94848 −0.974240 0.225512i \(-0.927594\pi\)
−0.974240 + 0.225512i \(0.927594\pi\)
\(212\) −6032.00 −1.95415
\(213\) −1110.00 −0.357070
\(214\) −3344.00 −1.06818
\(215\) −360.000 −0.114194
\(216\) 0 0
\(217\) 2300.00 0.719512
\(218\) 3944.00 1.22533
\(219\) 2478.00 0.764601
\(220\) −1440.00 −0.441294
\(221\) 221.000 0.0672673
\(222\) −552.000 −0.166882
\(223\) 1998.00 0.599982 0.299991 0.953942i \(-0.403016\pi\)
0.299991 + 0.953942i \(0.403016\pi\)
\(224\) 2560.00 0.763604
\(225\) −225.000 −0.0666667
\(226\) 312.000 0.0918316
\(227\) −474.000 −0.138592 −0.0692962 0.997596i \(-0.522075\pi\)
−0.0692962 + 0.997596i \(0.522075\pi\)
\(228\) −1776.00 −0.515870
\(229\) −6390.00 −1.84394 −0.921972 0.387257i \(-0.873423\pi\)
−0.921972 + 0.387257i \(0.873423\pi\)
\(230\) 5280.00 1.51371
\(231\) −540.000 −0.153807
\(232\) 0 0
\(233\) −6310.00 −1.77417 −0.887086 0.461605i \(-0.847274\pi\)
−0.887086 + 0.461605i \(0.847274\pi\)
\(234\) −468.000 −0.130744
\(235\) −4460.00 −1.23804
\(236\) −400.000 −0.110330
\(237\) 816.000 0.223649
\(238\) 680.000 0.185201
\(239\) −870.000 −0.235463 −0.117731 0.993045i \(-0.537562\pi\)
−0.117731 + 0.993045i \(0.537562\pi\)
\(240\) 1920.00 0.516398
\(241\) 6058.00 1.61921 0.809606 0.586974i \(-0.199681\pi\)
0.809606 + 0.586974i \(0.199681\pi\)
\(242\) −4028.00 −1.06996
\(243\) 243.000 0.0641500
\(244\) −1808.00 −0.474366
\(245\) 2430.00 0.633661
\(246\) −1368.00 −0.354555
\(247\) 962.000 0.247816
\(248\) 0 0
\(249\) 486.000 0.123691
\(250\) 6000.00 1.51789
\(251\) 436.000 0.109642 0.0548209 0.998496i \(-0.482541\pi\)
0.0548209 + 0.998496i \(0.482541\pi\)
\(252\) −720.000 −0.179983
\(253\) −2376.00 −0.590426
\(254\) 9856.00 2.43473
\(255\) 510.000 0.125245
\(256\) 4096.00 1.00000
\(257\) 5402.00 1.31116 0.655579 0.755127i \(-0.272425\pi\)
0.655579 + 0.755127i \(0.272425\pi\)
\(258\) 432.000 0.104245
\(259\) 460.000 0.110359
\(260\) 1040.00 0.248069
\(261\) 1890.00 0.448230
\(262\) −10512.0 −2.47875
\(263\) 6364.00 1.49210 0.746048 0.665893i \(-0.231949\pi\)
0.746048 + 0.665893i \(0.231949\pi\)
\(264\) 0 0
\(265\) 7540.00 1.74784
\(266\) 2960.00 0.682290
\(267\) −558.000 −0.127899
\(268\) 4656.00 1.06123
\(269\) −3918.00 −0.888047 −0.444024 0.896015i \(-0.646449\pi\)
−0.444024 + 0.896015i \(0.646449\pi\)
\(270\) −1080.00 −0.243432
\(271\) 2938.00 0.658564 0.329282 0.944232i \(-0.393193\pi\)
0.329282 + 0.944232i \(0.393193\pi\)
\(272\) 1088.00 0.242536
\(273\) 390.000 0.0864611
\(274\) −2856.00 −0.629698
\(275\) −450.000 −0.0986764
\(276\) −3168.00 −0.690910
\(277\) −6226.00 −1.35048 −0.675242 0.737596i \(-0.735961\pi\)
−0.675242 + 0.737596i \(0.735961\pi\)
\(278\) −1424.00 −0.307215
\(279\) −2070.00 −0.444185
\(280\) 0 0
\(281\) 1070.00 0.227156 0.113578 0.993529i \(-0.463769\pi\)
0.113578 + 0.993529i \(0.463769\pi\)
\(282\) 5352.00 1.13017
\(283\) 1152.00 0.241976 0.120988 0.992654i \(-0.461394\pi\)
0.120988 + 0.992654i \(0.461394\pi\)
\(284\) −2960.00 −0.618464
\(285\) 2220.00 0.461409
\(286\) −936.000 −0.193520
\(287\) 1140.00 0.234467
\(288\) −2304.00 −0.471405
\(289\) 289.000 0.0588235
\(290\) −8400.00 −1.70091
\(291\) −2370.00 −0.477429
\(292\) 6608.00 1.32433
\(293\) −8314.00 −1.65771 −0.828855 0.559463i \(-0.811007\pi\)
−0.828855 + 0.559463i \(0.811007\pi\)
\(294\) −2916.00 −0.578451
\(295\) 500.000 0.0986818
\(296\) 0 0
\(297\) 486.000 0.0949514
\(298\) 5784.00 1.12436
\(299\) 1716.00 0.331902
\(300\) −600.000 −0.115470
\(301\) −360.000 −0.0689371
\(302\) −2104.00 −0.400899
\(303\) 4554.00 0.863434
\(304\) 4736.00 0.893514
\(305\) 2260.00 0.424286
\(306\) −612.000 −0.114332
\(307\) 2866.00 0.532805 0.266403 0.963862i \(-0.414165\pi\)
0.266403 + 0.963862i \(0.414165\pi\)
\(308\) −1440.00 −0.266401
\(309\) 4200.00 0.773235
\(310\) 9200.00 1.68556
\(311\) −7428.00 −1.35435 −0.677176 0.735821i \(-0.736796\pi\)
−0.677176 + 0.735821i \(0.736796\pi\)
\(312\) 0 0
\(313\) −4786.00 −0.864283 −0.432142 0.901806i \(-0.642242\pi\)
−0.432142 + 0.901806i \(0.642242\pi\)
\(314\) −1048.00 −0.188351
\(315\) 900.000 0.160982
\(316\) 2176.00 0.387372
\(317\) −4402.00 −0.779940 −0.389970 0.920828i \(-0.627515\pi\)
−0.389970 + 0.920828i \(0.627515\pi\)
\(318\) −9048.00 −1.59556
\(319\) 3780.00 0.663446
\(320\) 5120.00 0.894427
\(321\) −2508.00 −0.436084
\(322\) 5280.00 0.913798
\(323\) 1258.00 0.216709
\(324\) 648.000 0.111111
\(325\) 325.000 0.0554700
\(326\) −10536.0 −1.78998
\(327\) 2958.00 0.500238
\(328\) 0 0
\(329\) −4460.00 −0.747379
\(330\) −2160.00 −0.360315
\(331\) 5470.00 0.908334 0.454167 0.890917i \(-0.349937\pi\)
0.454167 + 0.890917i \(0.349937\pi\)
\(332\) 1296.00 0.214239
\(333\) −414.000 −0.0681293
\(334\) 8008.00 1.31191
\(335\) −5820.00 −0.949196
\(336\) 1920.00 0.311740
\(337\) −3634.00 −0.587408 −0.293704 0.955896i \(-0.594888\pi\)
−0.293704 + 0.955896i \(0.594888\pi\)
\(338\) 676.000 0.108786
\(339\) 234.000 0.0374901
\(340\) 1360.00 0.216930
\(341\) −4140.00 −0.657459
\(342\) −2664.00 −0.421206
\(343\) 5860.00 0.922479
\(344\) 0 0
\(345\) 3960.00 0.617969
\(346\) 4568.00 0.709761
\(347\) −11588.0 −1.79273 −0.896364 0.443319i \(-0.853801\pi\)
−0.896364 + 0.443319i \(0.853801\pi\)
\(348\) 5040.00 0.776357
\(349\) 2434.00 0.373321 0.186661 0.982424i \(-0.440234\pi\)
0.186661 + 0.982424i \(0.440234\pi\)
\(350\) 1000.00 0.152721
\(351\) −351.000 −0.0533761
\(352\) −4608.00 −0.697748
\(353\) −8258.00 −1.24512 −0.622562 0.782570i \(-0.713908\pi\)
−0.622562 + 0.782570i \(0.713908\pi\)
\(354\) −600.000 −0.0900837
\(355\) 3700.00 0.553171
\(356\) −1488.00 −0.221528
\(357\) 510.000 0.0756080
\(358\) −10320.0 −1.52354
\(359\) −7594.00 −1.11642 −0.558212 0.829699i \(-0.688512\pi\)
−0.558212 + 0.829699i \(0.688512\pi\)
\(360\) 0 0
\(361\) −1383.00 −0.201633
\(362\) 7496.00 1.08835
\(363\) −3021.00 −0.436808
\(364\) 1040.00 0.149755
\(365\) −8260.00 −1.18452
\(366\) −2712.00 −0.387318
\(367\) −4704.00 −0.669065 −0.334532 0.942384i \(-0.608578\pi\)
−0.334532 + 0.942384i \(0.608578\pi\)
\(368\) 8448.00 1.19669
\(369\) −1026.00 −0.144746
\(370\) 1840.00 0.258533
\(371\) 7540.00 1.05514
\(372\) −5520.00 −0.769351
\(373\) 4178.00 0.579970 0.289985 0.957031i \(-0.406350\pi\)
0.289985 + 0.957031i \(0.406350\pi\)
\(374\) −1224.00 −0.169229
\(375\) 4500.00 0.619677
\(376\) 0 0
\(377\) −2730.00 −0.372950
\(378\) −1080.00 −0.146956
\(379\) 7450.00 1.00971 0.504856 0.863204i \(-0.331546\pi\)
0.504856 + 0.863204i \(0.331546\pi\)
\(380\) 5920.00 0.799183
\(381\) 7392.00 0.993973
\(382\) −1648.00 −0.220730
\(383\) −7398.00 −0.986998 −0.493499 0.869746i \(-0.664282\pi\)
−0.493499 + 0.869746i \(0.664282\pi\)
\(384\) 0 0
\(385\) 1800.00 0.238277
\(386\) 4328.00 0.570698
\(387\) 324.000 0.0425577
\(388\) −6320.00 −0.826931
\(389\) 10234.0 1.33389 0.666947 0.745106i \(-0.267601\pi\)
0.666947 + 0.745106i \(0.267601\pi\)
\(390\) 1560.00 0.202548
\(391\) 2244.00 0.290240
\(392\) 0 0
\(393\) −7884.00 −1.01195
\(394\) −4776.00 −0.610689
\(395\) −2720.00 −0.346476
\(396\) 1296.00 0.164461
\(397\) −1406.00 −0.177746 −0.0888729 0.996043i \(-0.528327\pi\)
−0.0888729 + 0.996043i \(0.528327\pi\)
\(398\) −11440.0 −1.44079
\(399\) 2220.00 0.278544
\(400\) 1600.00 0.200000
\(401\) 8550.00 1.06475 0.532377 0.846507i \(-0.321299\pi\)
0.532377 + 0.846507i \(0.321299\pi\)
\(402\) 6984.00 0.866493
\(403\) 2990.00 0.369584
\(404\) 12144.0 1.49551
\(405\) −810.000 −0.0993808
\(406\) −8400.00 −1.02681
\(407\) −828.000 −0.100841
\(408\) 0 0
\(409\) 6730.00 0.813636 0.406818 0.913509i \(-0.366638\pi\)
0.406818 + 0.913509i \(0.366638\pi\)
\(410\) 4560.00 0.549274
\(411\) −2142.00 −0.257073
\(412\) 11200.0 1.33928
\(413\) 500.000 0.0595724
\(414\) −4752.00 −0.564126
\(415\) −1620.00 −0.191621
\(416\) 3328.00 0.392232
\(417\) −1068.00 −0.125420
\(418\) −5328.00 −0.623447
\(419\) −3988.00 −0.464980 −0.232490 0.972599i \(-0.574687\pi\)
−0.232490 + 0.972599i \(0.574687\pi\)
\(420\) 2400.00 0.278829
\(421\) 14450.0 1.67280 0.836401 0.548118i \(-0.184655\pi\)
0.836401 + 0.548118i \(0.184655\pi\)
\(422\) −23888.0 −2.75557
\(423\) 4014.00 0.461389
\(424\) 0 0
\(425\) 425.000 0.0485071
\(426\) −4440.00 −0.504973
\(427\) 2260.00 0.256134
\(428\) −6688.00 −0.755319
\(429\) −702.000 −0.0790044
\(430\) −1440.00 −0.161495
\(431\) 774.000 0.0865018 0.0432509 0.999064i \(-0.486229\pi\)
0.0432509 + 0.999064i \(0.486229\pi\)
\(432\) −1728.00 −0.192450
\(433\) 11970.0 1.32850 0.664251 0.747509i \(-0.268751\pi\)
0.664251 + 0.747509i \(0.268751\pi\)
\(434\) 9200.00 1.01754
\(435\) −6300.00 −0.694395
\(436\) 7888.00 0.866437
\(437\) 9768.00 1.06926
\(438\) 9912.00 1.08131
\(439\) 12884.0 1.40073 0.700364 0.713786i \(-0.253021\pi\)
0.700364 + 0.713786i \(0.253021\pi\)
\(440\) 0 0
\(441\) −2187.00 −0.236152
\(442\) 884.000 0.0951303
\(443\) −12928.0 −1.38652 −0.693259 0.720688i \(-0.743826\pi\)
−0.693259 + 0.720688i \(0.743826\pi\)
\(444\) −1104.00 −0.118003
\(445\) 1860.00 0.198140
\(446\) 7992.00 0.848503
\(447\) 4338.00 0.459016
\(448\) 5120.00 0.539949
\(449\) 3814.00 0.400877 0.200439 0.979706i \(-0.435763\pi\)
0.200439 + 0.979706i \(0.435763\pi\)
\(450\) −900.000 −0.0942809
\(451\) −2052.00 −0.214246
\(452\) 624.000 0.0649347
\(453\) −1578.00 −0.163666
\(454\) −1896.00 −0.195999
\(455\) −1300.00 −0.133945
\(456\) 0 0
\(457\) −5390.00 −0.551715 −0.275857 0.961199i \(-0.588962\pi\)
−0.275857 + 0.961199i \(0.588962\pi\)
\(458\) −25560.0 −2.60773
\(459\) −459.000 −0.0466760
\(460\) 10560.0 1.07035
\(461\) −6634.00 −0.670230 −0.335115 0.942177i \(-0.608775\pi\)
−0.335115 + 0.942177i \(0.608775\pi\)
\(462\) −2160.00 −0.217516
\(463\) 4370.00 0.438642 0.219321 0.975653i \(-0.429616\pi\)
0.219321 + 0.975653i \(0.429616\pi\)
\(464\) −13440.0 −1.34469
\(465\) 6900.00 0.688129
\(466\) −25240.0 −2.50906
\(467\) −15164.0 −1.50258 −0.751291 0.659971i \(-0.770569\pi\)
−0.751291 + 0.659971i \(0.770569\pi\)
\(468\) −936.000 −0.0924500
\(469\) −5820.00 −0.573012
\(470\) −17840.0 −1.75085
\(471\) −786.000 −0.0768938
\(472\) 0 0
\(473\) 648.000 0.0629917
\(474\) 3264.00 0.316288
\(475\) 1850.00 0.178703
\(476\) 1360.00 0.130957
\(477\) −6786.00 −0.651383
\(478\) −3480.00 −0.332995
\(479\) 10162.0 0.969340 0.484670 0.874697i \(-0.338940\pi\)
0.484670 + 0.874697i \(0.338940\pi\)
\(480\) 7680.00 0.730297
\(481\) 598.000 0.0566870
\(482\) 24232.0 2.28991
\(483\) 3960.00 0.373056
\(484\) −8056.00 −0.756574
\(485\) 7900.00 0.739630
\(486\) 972.000 0.0907218
\(487\) −17174.0 −1.59800 −0.799002 0.601328i \(-0.794639\pi\)
−0.799002 + 0.601328i \(0.794639\pi\)
\(488\) 0 0
\(489\) −7902.00 −0.730758
\(490\) 9720.00 0.896132
\(491\) −3828.00 −0.351844 −0.175922 0.984404i \(-0.556291\pi\)
−0.175922 + 0.984404i \(0.556291\pi\)
\(492\) −2736.00 −0.250708
\(493\) −3570.00 −0.326135
\(494\) 3848.00 0.350465
\(495\) −1620.00 −0.147098
\(496\) 14720.0 1.33256
\(497\) 3700.00 0.333939
\(498\) 1944.00 0.174925
\(499\) −10894.0 −0.977319 −0.488660 0.872474i \(-0.662514\pi\)
−0.488660 + 0.872474i \(0.662514\pi\)
\(500\) 12000.0 1.07331
\(501\) 6006.00 0.535585
\(502\) 1744.00 0.155057
\(503\) 2336.00 0.207072 0.103536 0.994626i \(-0.466984\pi\)
0.103536 + 0.994626i \(0.466984\pi\)
\(504\) 0 0
\(505\) −15180.0 −1.33763
\(506\) −9504.00 −0.834989
\(507\) 507.000 0.0444116
\(508\) 19712.0 1.72161
\(509\) 4382.00 0.381589 0.190794 0.981630i \(-0.438894\pi\)
0.190794 + 0.981630i \(0.438894\pi\)
\(510\) 2040.00 0.177123
\(511\) −8260.00 −0.715070
\(512\) 16384.0 1.41421
\(513\) −1998.00 −0.171957
\(514\) 21608.0 1.85426
\(515\) −14000.0 −1.19789
\(516\) 864.000 0.0737122
\(517\) 8028.00 0.682923
\(518\) 1840.00 0.156071
\(519\) 3426.00 0.289759
\(520\) 0 0
\(521\) −14634.0 −1.23057 −0.615285 0.788305i \(-0.710959\pi\)
−0.615285 + 0.788305i \(0.710959\pi\)
\(522\) 7560.00 0.633893
\(523\) 20696.0 1.73035 0.865175 0.501470i \(-0.167207\pi\)
0.865175 + 0.501470i \(0.167207\pi\)
\(524\) −21024.0 −1.75274
\(525\) 750.000 0.0623480
\(526\) 25456.0 2.11014
\(527\) 3910.00 0.323192
\(528\) −3456.00 −0.284854
\(529\) 5257.00 0.432070
\(530\) 30160.0 2.47182
\(531\) −450.000 −0.0367765
\(532\) 5920.00 0.482452
\(533\) 1482.00 0.120436
\(534\) −2232.00 −0.180877
\(535\) 8360.00 0.675578
\(536\) 0 0
\(537\) −7740.00 −0.621984
\(538\) −15672.0 −1.25589
\(539\) −4374.00 −0.349539
\(540\) −2160.00 −0.172133
\(541\) 18034.0 1.43316 0.716582 0.697502i \(-0.245705\pi\)
0.716582 + 0.697502i \(0.245705\pi\)
\(542\) 11752.0 0.931350
\(543\) 5622.00 0.444315
\(544\) 4352.00 0.342997
\(545\) −9860.00 −0.774965
\(546\) 1560.00 0.122274
\(547\) −10836.0 −0.847009 −0.423504 0.905894i \(-0.639200\pi\)
−0.423504 + 0.905894i \(0.639200\pi\)
\(548\) −5712.00 −0.445264
\(549\) −2034.00 −0.158122
\(550\) −1800.00 −0.139550
\(551\) −15540.0 −1.20150
\(552\) 0 0
\(553\) −2720.00 −0.209161
\(554\) −24904.0 −1.90987
\(555\) 1380.00 0.105545
\(556\) −2848.00 −0.217234
\(557\) 14566.0 1.10805 0.554023 0.832502i \(-0.313092\pi\)
0.554023 + 0.832502i \(0.313092\pi\)
\(558\) −8280.00 −0.628173
\(559\) −468.000 −0.0354102
\(560\) −6400.00 −0.482945
\(561\) −918.000 −0.0690873
\(562\) 4280.00 0.321247
\(563\) −13380.0 −1.00160 −0.500799 0.865564i \(-0.666960\pi\)
−0.500799 + 0.865564i \(0.666960\pi\)
\(564\) 10704.0 0.799148
\(565\) −780.000 −0.0580794
\(566\) 4608.00 0.342206
\(567\) −810.000 −0.0599944
\(568\) 0 0
\(569\) −1582.00 −0.116557 −0.0582785 0.998300i \(-0.518561\pi\)
−0.0582785 + 0.998300i \(0.518561\pi\)
\(570\) 8880.00 0.652530
\(571\) 9680.00 0.709449 0.354725 0.934971i \(-0.384575\pi\)
0.354725 + 0.934971i \(0.384575\pi\)
\(572\) −1872.00 −0.136840
\(573\) −1236.00 −0.0901128
\(574\) 4560.00 0.331587
\(575\) 3300.00 0.239338
\(576\) −4608.00 −0.333333
\(577\) −24574.0 −1.77301 −0.886507 0.462715i \(-0.846875\pi\)
−0.886507 + 0.462715i \(0.846875\pi\)
\(578\) 1156.00 0.0831890
\(579\) 3246.00 0.232986
\(580\) −16800.0 −1.20273
\(581\) −1620.00 −0.115678
\(582\) −9480.00 −0.675187
\(583\) −13572.0 −0.964142
\(584\) 0 0
\(585\) 1170.00 0.0826898
\(586\) −33256.0 −2.34436
\(587\) 10722.0 0.753909 0.376954 0.926232i \(-0.376971\pi\)
0.376954 + 0.926232i \(0.376971\pi\)
\(588\) −5832.00 −0.409027
\(589\) 17020.0 1.19066
\(590\) 2000.00 0.139557
\(591\) −3582.00 −0.249313
\(592\) 2944.00 0.204388
\(593\) −8930.00 −0.618400 −0.309200 0.950997i \(-0.600061\pi\)
−0.309200 + 0.950997i \(0.600061\pi\)
\(594\) 1944.00 0.134282
\(595\) −1700.00 −0.117131
\(596\) 11568.0 0.795040
\(597\) −8580.00 −0.588201
\(598\) 6864.00 0.469381
\(599\) −20284.0 −1.38361 −0.691804 0.722085i \(-0.743184\pi\)
−0.691804 + 0.722085i \(0.743184\pi\)
\(600\) 0 0
\(601\) 7790.00 0.528720 0.264360 0.964424i \(-0.414839\pi\)
0.264360 + 0.964424i \(0.414839\pi\)
\(602\) −1440.00 −0.0974917
\(603\) 5238.00 0.353744
\(604\) −4208.00 −0.283479
\(605\) 10070.0 0.676700
\(606\) 18216.0 1.22108
\(607\) 13008.0 0.869816 0.434908 0.900475i \(-0.356781\pi\)
0.434908 + 0.900475i \(0.356781\pi\)
\(608\) 18944.0 1.26362
\(609\) −6300.00 −0.419194
\(610\) 9040.00 0.600031
\(611\) −5798.00 −0.383898
\(612\) −1224.00 −0.0808452
\(613\) −26478.0 −1.74459 −0.872297 0.488976i \(-0.837371\pi\)
−0.872297 + 0.488976i \(0.837371\pi\)
\(614\) 11464.0 0.753501
\(615\) 3420.00 0.224240
\(616\) 0 0
\(617\) −18290.0 −1.19340 −0.596700 0.802464i \(-0.703522\pi\)
−0.596700 + 0.802464i \(0.703522\pi\)
\(618\) 16800.0 1.09352
\(619\) 10614.0 0.689197 0.344598 0.938750i \(-0.388015\pi\)
0.344598 + 0.938750i \(0.388015\pi\)
\(620\) 18400.0 1.19187
\(621\) −3564.00 −0.230303
\(622\) −29712.0 −1.91534
\(623\) 1860.00 0.119614
\(624\) 2496.00 0.160128
\(625\) −11875.0 −0.760000
\(626\) −19144.0 −1.22228
\(627\) −3996.00 −0.254521
\(628\) −2096.00 −0.133184
\(629\) 782.000 0.0495714
\(630\) 3600.00 0.227663
\(631\) −20062.0 −1.26570 −0.632849 0.774275i \(-0.718115\pi\)
−0.632849 + 0.774275i \(0.718115\pi\)
\(632\) 0 0
\(633\) −17916.0 −1.12496
\(634\) −17608.0 −1.10300
\(635\) −24640.0 −1.53986
\(636\) −18096.0 −1.12823
\(637\) 3159.00 0.196490
\(638\) 15120.0 0.938255
\(639\) −3330.00 −0.206155
\(640\) 0 0
\(641\) −12962.0 −0.798702 −0.399351 0.916798i \(-0.630765\pi\)
−0.399351 + 0.916798i \(0.630765\pi\)
\(642\) −10032.0 −0.616716
\(643\) −21238.0 −1.30256 −0.651279 0.758838i \(-0.725767\pi\)
−0.651279 + 0.758838i \(0.725767\pi\)
\(644\) 10560.0 0.646153
\(645\) −1080.00 −0.0659302
\(646\) 5032.00 0.306473
\(647\) −20480.0 −1.24444 −0.622219 0.782843i \(-0.713769\pi\)
−0.622219 + 0.782843i \(0.713769\pi\)
\(648\) 0 0
\(649\) −900.000 −0.0544347
\(650\) 1300.00 0.0784465
\(651\) 6900.00 0.415411
\(652\) −21072.0 −1.26571
\(653\) −14558.0 −0.872433 −0.436216 0.899842i \(-0.643682\pi\)
−0.436216 + 0.899842i \(0.643682\pi\)
\(654\) 11832.0 0.707443
\(655\) 26280.0 1.56770
\(656\) 7296.00 0.434239
\(657\) 7434.00 0.441443
\(658\) −17840.0 −1.05695
\(659\) 888.000 0.0524910 0.0262455 0.999656i \(-0.491645\pi\)
0.0262455 + 0.999656i \(0.491645\pi\)
\(660\) −4320.00 −0.254781
\(661\) −31142.0 −1.83250 −0.916251 0.400605i \(-0.868800\pi\)
−0.916251 + 0.400605i \(0.868800\pi\)
\(662\) 21880.0 1.28458
\(663\) 663.000 0.0388368
\(664\) 0 0
\(665\) −7400.00 −0.431518
\(666\) −1656.00 −0.0963494
\(667\) −27720.0 −1.60918
\(668\) 16016.0 0.927661
\(669\) 5994.00 0.346400
\(670\) −23280.0 −1.34237
\(671\) −4068.00 −0.234044
\(672\) 7680.00 0.440867
\(673\) −15666.0 −0.897296 −0.448648 0.893709i \(-0.648094\pi\)
−0.448648 + 0.893709i \(0.648094\pi\)
\(674\) −14536.0 −0.830721
\(675\) −675.000 −0.0384900
\(676\) 1352.00 0.0769231
\(677\) −14098.0 −0.800340 −0.400170 0.916441i \(-0.631049\pi\)
−0.400170 + 0.916441i \(0.631049\pi\)
\(678\) 936.000 0.0530190
\(679\) 7900.00 0.446501
\(680\) 0 0
\(681\) −1422.00 −0.0800164
\(682\) −16560.0 −0.929788
\(683\) 666.000 0.0373115 0.0186558 0.999826i \(-0.494061\pi\)
0.0186558 + 0.999826i \(0.494061\pi\)
\(684\) −5328.00 −0.297838
\(685\) 7140.00 0.398256
\(686\) 23440.0 1.30458
\(687\) −19170.0 −1.06460
\(688\) −2304.00 −0.127673
\(689\) 9802.00 0.541983
\(690\) 15840.0 0.873940
\(691\) −4254.00 −0.234197 −0.117098 0.993120i \(-0.537359\pi\)
−0.117098 + 0.993120i \(0.537359\pi\)
\(692\) 9136.00 0.501877
\(693\) −1620.00 −0.0888004
\(694\) −46352.0 −2.53530
\(695\) 3560.00 0.194300
\(696\) 0 0
\(697\) 1938.00 0.105318
\(698\) 9736.00 0.527956
\(699\) −18930.0 −1.02432
\(700\) 2000.00 0.107990
\(701\) −9230.00 −0.497307 −0.248654 0.968592i \(-0.579988\pi\)
−0.248654 + 0.968592i \(0.579988\pi\)
\(702\) −1404.00 −0.0754851
\(703\) 3404.00 0.182623
\(704\) −9216.00 −0.493382
\(705\) −13380.0 −0.714780
\(706\) −33032.0 −1.76087
\(707\) −15180.0 −0.807500
\(708\) −1200.00 −0.0636988
\(709\) 1994.00 0.105622 0.0528112 0.998605i \(-0.483182\pi\)
0.0528112 + 0.998605i \(0.483182\pi\)
\(710\) 14800.0 0.782302
\(711\) 2448.00 0.129124
\(712\) 0 0
\(713\) 30360.0 1.59466
\(714\) 2040.00 0.106926
\(715\) 2340.00 0.122393
\(716\) −20640.0 −1.07731
\(717\) −2610.00 −0.135945
\(718\) −30376.0 −1.57886
\(719\) 2260.00 0.117224 0.0586118 0.998281i \(-0.481333\pi\)
0.0586118 + 0.998281i \(0.481333\pi\)
\(720\) 5760.00 0.298142
\(721\) −14000.0 −0.723145
\(722\) −5532.00 −0.285152
\(723\) 18174.0 0.934852
\(724\) 14992.0 0.769576
\(725\) −5250.00 −0.268938
\(726\) −12084.0 −0.617740
\(727\) 1136.00 0.0579531 0.0289766 0.999580i \(-0.490775\pi\)
0.0289766 + 0.999580i \(0.490775\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −33040.0 −1.67516
\(731\) −612.000 −0.0309653
\(732\) −5424.00 −0.273875
\(733\) 17434.0 0.878499 0.439249 0.898365i \(-0.355244\pi\)
0.439249 + 0.898365i \(0.355244\pi\)
\(734\) −18816.0 −0.946201
\(735\) 7290.00 0.365844
\(736\) 33792.0 1.69238
\(737\) 10476.0 0.523594
\(738\) −4104.00 −0.204702
\(739\) 25234.0 1.25609 0.628043 0.778179i \(-0.283856\pi\)
0.628043 + 0.778179i \(0.283856\pi\)
\(740\) 3680.00 0.182810
\(741\) 2886.00 0.143077
\(742\) 30160.0 1.49219
\(743\) −33546.0 −1.65637 −0.828185 0.560454i \(-0.810627\pi\)
−0.828185 + 0.560454i \(0.810627\pi\)
\(744\) 0 0
\(745\) −14460.0 −0.711105
\(746\) 16712.0 0.820201
\(747\) 1458.00 0.0714129
\(748\) −2448.00 −0.119663
\(749\) 8360.00 0.407834
\(750\) 18000.0 0.876356
\(751\) −4636.00 −0.225260 −0.112630 0.993637i \(-0.535927\pi\)
−0.112630 + 0.993637i \(0.535927\pi\)
\(752\) −28544.0 −1.38417
\(753\) 1308.00 0.0633017
\(754\) −10920.0 −0.527431
\(755\) 5260.00 0.253551
\(756\) −2160.00 −0.103913
\(757\) −11010.0 −0.528620 −0.264310 0.964438i \(-0.585144\pi\)
−0.264310 + 0.964438i \(0.585144\pi\)
\(758\) 29800.0 1.42795
\(759\) −7128.00 −0.340883
\(760\) 0 0
\(761\) 27686.0 1.31881 0.659407 0.751787i \(-0.270808\pi\)
0.659407 + 0.751787i \(0.270808\pi\)
\(762\) 29568.0 1.40569
\(763\) −9860.00 −0.467832
\(764\) −3296.00 −0.156080
\(765\) 1530.00 0.0723102
\(766\) −29592.0 −1.39583
\(767\) 650.000 0.0305999
\(768\) 12288.0 0.577350
\(769\) −21934.0 −1.02856 −0.514278 0.857623i \(-0.671940\pi\)
−0.514278 + 0.857623i \(0.671940\pi\)
\(770\) 7200.00 0.336974
\(771\) 16206.0 0.756997
\(772\) 8656.00 0.403544
\(773\) 27630.0 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(774\) 1296.00 0.0601857
\(775\) 5750.00 0.266511
\(776\) 0 0
\(777\) 1380.00 0.0637159
\(778\) 40936.0 1.88641
\(779\) 8436.00 0.387999
\(780\) 3120.00 0.143223
\(781\) −6660.00 −0.305139
\(782\) 8976.00 0.410462
\(783\) 5670.00 0.258786
\(784\) 15552.0 0.708455
\(785\) 2620.00 0.119123
\(786\) −31536.0 −1.43111
\(787\) 16886.0 0.764830 0.382415 0.923991i \(-0.375093\pi\)
0.382415 + 0.923991i \(0.375093\pi\)
\(788\) −9552.00 −0.431822
\(789\) 19092.0 0.861462
\(790\) −10880.0 −0.489991
\(791\) −780.000 −0.0350615
\(792\) 0 0
\(793\) 2938.00 0.131565
\(794\) −5624.00 −0.251371
\(795\) 22620.0 1.00912
\(796\) −22880.0 −1.01879
\(797\) 3282.00 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(798\) 8880.00 0.393921
\(799\) −7582.00 −0.335709
\(800\) 6400.00 0.282843
\(801\) −1674.00 −0.0738425
\(802\) 34200.0 1.50579
\(803\) 14868.0 0.653400
\(804\) 13968.0 0.612703
\(805\) −13200.0 −0.577936
\(806\) 11960.0 0.522671
\(807\) −11754.0 −0.512714
\(808\) 0 0
\(809\) 9830.00 0.427199 0.213600 0.976921i \(-0.431481\pi\)
0.213600 + 0.976921i \(0.431481\pi\)
\(810\) −3240.00 −0.140546
\(811\) −582.000 −0.0251995 −0.0125997 0.999921i \(-0.504011\pi\)
−0.0125997 + 0.999921i \(0.504011\pi\)
\(812\) −16800.0 −0.726065
\(813\) 8814.00 0.380222
\(814\) −3312.00 −0.142611
\(815\) 26340.0 1.13209
\(816\) 3264.00 0.140028
\(817\) −2664.00 −0.114078
\(818\) 26920.0 1.15065
\(819\) 1170.00 0.0499183
\(820\) 9120.00 0.388395
\(821\) −32650.0 −1.38793 −0.693966 0.720007i \(-0.744138\pi\)
−0.693966 + 0.720007i \(0.744138\pi\)
\(822\) −8568.00 −0.363556
\(823\) 30084.0 1.27419 0.637097 0.770783i \(-0.280135\pi\)
0.637097 + 0.770783i \(0.280135\pi\)
\(824\) 0 0
\(825\) −1350.00 −0.0569709
\(826\) 2000.00 0.0842481
\(827\) 16842.0 0.708167 0.354083 0.935214i \(-0.384793\pi\)
0.354083 + 0.935214i \(0.384793\pi\)
\(828\) −9504.00 −0.398897
\(829\) 22430.0 0.939718 0.469859 0.882742i \(-0.344305\pi\)
0.469859 + 0.882742i \(0.344305\pi\)
\(830\) −6480.00 −0.270993
\(831\) −18678.0 −0.779702
\(832\) 6656.00 0.277350
\(833\) 4131.00 0.171826
\(834\) −4272.00 −0.177371
\(835\) −20020.0 −0.829725
\(836\) −10656.0 −0.440844
\(837\) −6210.00 −0.256450
\(838\) −15952.0 −0.657581
\(839\) 13074.0 0.537979 0.268990 0.963143i \(-0.413310\pi\)
0.268990 + 0.963143i \(0.413310\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) 57800.0 2.36570
\(843\) 3210.00 0.131149
\(844\) −47776.0 −1.94848
\(845\) −1690.00 −0.0688021
\(846\) 16056.0 0.652502
\(847\) 10070.0 0.408512
\(848\) 48256.0 1.95415
\(849\) 3456.00 0.139705
\(850\) 1700.00 0.0685994
\(851\) 6072.00 0.244589
\(852\) −8880.00 −0.357070
\(853\) 47938.0 1.92423 0.962114 0.272649i \(-0.0878997\pi\)
0.962114 + 0.272649i \(0.0878997\pi\)
\(854\) 9040.00 0.362228
\(855\) 6660.00 0.266394
\(856\) 0 0
\(857\) −38.0000 −0.00151465 −0.000757325 1.00000i \(-0.500241\pi\)
−0.000757325 1.00000i \(0.500241\pi\)
\(858\) −2808.00 −0.111729
\(859\) 18256.0 0.725130 0.362565 0.931958i \(-0.381901\pi\)
0.362565 + 0.931958i \(0.381901\pi\)
\(860\) −2880.00 −0.114194
\(861\) 3420.00 0.135370
\(862\) 3096.00 0.122332
\(863\) 14994.0 0.591427 0.295714 0.955277i \(-0.404443\pi\)
0.295714 + 0.955277i \(0.404443\pi\)
\(864\) −6912.00 −0.272166
\(865\) −11420.0 −0.448892
\(866\) 47880.0 1.87879
\(867\) 867.000 0.0339618
\(868\) 18400.0 0.719512
\(869\) 4896.00 0.191122
\(870\) −25200.0 −0.982023
\(871\) −7566.00 −0.294333
\(872\) 0 0
\(873\) −7110.00 −0.275644
\(874\) 39072.0 1.51216
\(875\) −15000.0 −0.579534
\(876\) 19824.0 0.764601
\(877\) −43478.0 −1.67406 −0.837028 0.547160i \(-0.815709\pi\)
−0.837028 + 0.547160i \(0.815709\pi\)
\(878\) 51536.0 1.98093
\(879\) −24942.0 −0.957079
\(880\) 11520.0 0.441294
\(881\) −36670.0 −1.40232 −0.701160 0.713004i \(-0.747334\pi\)
−0.701160 + 0.713004i \(0.747334\pi\)
\(882\) −8748.00 −0.333969
\(883\) 22772.0 0.867881 0.433940 0.900942i \(-0.357123\pi\)
0.433940 + 0.900942i \(0.357123\pi\)
\(884\) 1768.00 0.0672673
\(885\) 1500.00 0.0569740
\(886\) −51712.0 −1.96083
\(887\) 36736.0 1.39061 0.695307 0.718713i \(-0.255269\pi\)
0.695307 + 0.718713i \(0.255269\pi\)
\(888\) 0 0
\(889\) −24640.0 −0.929583
\(890\) 7440.00 0.280213
\(891\) 1458.00 0.0548202
\(892\) 15984.0 0.599982
\(893\) −33004.0 −1.23677
\(894\) 17352.0 0.649147
\(895\) 25800.0 0.963574
\(896\) 0 0
\(897\) 5148.00 0.191624
\(898\) 15256.0 0.566926
\(899\) −48300.0 −1.79187
\(900\) −1800.00 −0.0666667
\(901\) 12818.0 0.473951
\(902\) −8208.00 −0.302989
\(903\) −1080.00 −0.0398008
\(904\) 0 0
\(905\) −18740.0 −0.688330
\(906\) −6312.00 −0.231459
\(907\) 4912.00 0.179824 0.0899120 0.995950i \(-0.471341\pi\)
0.0899120 + 0.995950i \(0.471341\pi\)
\(908\) −3792.00 −0.138592
\(909\) 13662.0 0.498504
\(910\) −5200.00 −0.189427
\(911\) −3312.00 −0.120452 −0.0602258 0.998185i \(-0.519182\pi\)
−0.0602258 + 0.998185i \(0.519182\pi\)
\(912\) 14208.0 0.515870
\(913\) 2916.00 0.105702
\(914\) −21560.0 −0.780242
\(915\) 6780.00 0.244962
\(916\) −51120.0 −1.84394
\(917\) 26280.0 0.946393
\(918\) −1836.00 −0.0660098
\(919\) −17508.0 −0.628439 −0.314220 0.949350i \(-0.601743\pi\)
−0.314220 + 0.949350i \(0.601743\pi\)
\(920\) 0 0
\(921\) 8598.00 0.307615
\(922\) −26536.0 −0.947849
\(923\) 4810.00 0.171531
\(924\) −4320.00 −0.153807
\(925\) 1150.00 0.0408776
\(926\) 17480.0 0.620333
\(927\) 12600.0 0.446428
\(928\) −53760.0 −1.90168
\(929\) −27898.0 −0.985257 −0.492628 0.870240i \(-0.663964\pi\)
−0.492628 + 0.870240i \(0.663964\pi\)
\(930\) 27600.0 0.973161
\(931\) 17982.0 0.633014
\(932\) −50480.0 −1.77417
\(933\) −22284.0 −0.781935
\(934\) −60656.0 −2.12497
\(935\) 3060.00 0.107030
\(936\) 0 0
\(937\) 13354.0 0.465588 0.232794 0.972526i \(-0.425213\pi\)
0.232794 + 0.972526i \(0.425213\pi\)
\(938\) −23280.0 −0.810361
\(939\) −14358.0 −0.498994
\(940\) −35680.0 −1.23804
\(941\) 39742.0 1.37678 0.688391 0.725340i \(-0.258317\pi\)
0.688391 + 0.725340i \(0.258317\pi\)
\(942\) −3144.00 −0.108744
\(943\) 15048.0 0.519650
\(944\) 3200.00 0.110330
\(945\) 2700.00 0.0929429
\(946\) 2592.00 0.0890837
\(947\) 2474.00 0.0848936 0.0424468 0.999099i \(-0.486485\pi\)
0.0424468 + 0.999099i \(0.486485\pi\)
\(948\) 6528.00 0.223649
\(949\) −10738.0 −0.367303
\(950\) 7400.00 0.252724
\(951\) −13206.0 −0.450299
\(952\) 0 0
\(953\) 20882.0 0.709795 0.354897 0.934905i \(-0.384516\pi\)
0.354897 + 0.934905i \(0.384516\pi\)
\(954\) −27144.0 −0.921194
\(955\) 4120.00 0.139602
\(956\) −6960.00 −0.235463
\(957\) 11340.0 0.383041
\(958\) 40648.0 1.37085
\(959\) 7140.00 0.240420
\(960\) 15360.0 0.516398
\(961\) 23109.0 0.775704
\(962\) 2392.00 0.0801675
\(963\) −7524.00 −0.251773
\(964\) 48464.0 1.61921
\(965\) −10820.0 −0.360941
\(966\) 15840.0 0.527581
\(967\) 47942.0 1.59432 0.797162 0.603766i \(-0.206334\pi\)
0.797162 + 0.603766i \(0.206334\pi\)
\(968\) 0 0
\(969\) 3774.00 0.125117
\(970\) 31600.0 1.04599
\(971\) 56208.0 1.85767 0.928837 0.370490i \(-0.120810\pi\)
0.928837 + 0.370490i \(0.120810\pi\)
\(972\) 1944.00 0.0641500
\(973\) 3560.00 0.117295
\(974\) −68696.0 −2.25992
\(975\) 975.000 0.0320256
\(976\) 14464.0 0.474366
\(977\) 23718.0 0.776669 0.388335 0.921518i \(-0.373050\pi\)
0.388335 + 0.921518i \(0.373050\pi\)
\(978\) −31608.0 −1.03345
\(979\) −3348.00 −0.109298
\(980\) 19440.0 0.633661
\(981\) 8874.00 0.288812
\(982\) −15312.0 −0.497582
\(983\) −48974.0 −1.58904 −0.794521 0.607237i \(-0.792278\pi\)
−0.794521 + 0.607237i \(0.792278\pi\)
\(984\) 0 0
\(985\) 11940.0 0.386234
\(986\) −14280.0 −0.461225
\(987\) −13380.0 −0.431500
\(988\) 7696.00 0.247816
\(989\) −4752.00 −0.152785
\(990\) −6480.00 −0.208028
\(991\) −36760.0 −1.17832 −0.589162 0.808015i \(-0.700542\pi\)
−0.589162 + 0.808015i \(0.700542\pi\)
\(992\) 58880.0 1.88452
\(993\) 16410.0 0.524427
\(994\) 14800.0 0.472261
\(995\) 28600.0 0.911237
\(996\) 3888.00 0.123691
\(997\) −6778.00 −0.215307 −0.107654 0.994188i \(-0.534334\pi\)
−0.107654 + 0.994188i \(0.534334\pi\)
\(998\) −43576.0 −1.38214
\(999\) −1242.00 −0.0393345
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 663.4.a.a.1.1 1
3.2 odd 2 1989.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
663.4.a.a.1.1 1 1.1 even 1 trivial
1989.4.a.b.1.1 1 3.2 odd 2