Properties

Label 847.4.a.b.1.1
Level $847$
Weight $4$
Character 847.1
Self dual yes
Analytic conductor $49.975$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.9746177749\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 847.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} -7.00000 q^{4} +16.0000 q^{5} -2.00000 q^{6} +7.00000 q^{7} -15.0000 q^{8} -23.0000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} -7.00000 q^{4} +16.0000 q^{5} -2.00000 q^{6} +7.00000 q^{7} -15.0000 q^{8} -23.0000 q^{9} +16.0000 q^{10} +14.0000 q^{12} -28.0000 q^{13} +7.00000 q^{14} -32.0000 q^{15} +41.0000 q^{16} -54.0000 q^{17} -23.0000 q^{18} +110.000 q^{19} -112.000 q^{20} -14.0000 q^{21} +48.0000 q^{23} +30.0000 q^{24} +131.000 q^{25} -28.0000 q^{26} +100.000 q^{27} -49.0000 q^{28} +110.000 q^{29} -32.0000 q^{30} +12.0000 q^{31} +161.000 q^{32} -54.0000 q^{34} +112.000 q^{35} +161.000 q^{36} -246.000 q^{37} +110.000 q^{38} +56.0000 q^{39} -240.000 q^{40} -182.000 q^{41} -14.0000 q^{42} -128.000 q^{43} -368.000 q^{45} +48.0000 q^{46} +324.000 q^{47} -82.0000 q^{48} +49.0000 q^{49} +131.000 q^{50} +108.000 q^{51} +196.000 q^{52} -162.000 q^{53} +100.000 q^{54} -105.000 q^{56} -220.000 q^{57} +110.000 q^{58} +810.000 q^{59} +224.000 q^{60} +488.000 q^{61} +12.0000 q^{62} -161.000 q^{63} -167.000 q^{64} -448.000 q^{65} +244.000 q^{67} +378.000 q^{68} -96.0000 q^{69} +112.000 q^{70} -768.000 q^{71} +345.000 q^{72} +702.000 q^{73} -246.000 q^{74} -262.000 q^{75} -770.000 q^{76} +56.0000 q^{78} -440.000 q^{79} +656.000 q^{80} +421.000 q^{81} -182.000 q^{82} +1302.00 q^{83} +98.0000 q^{84} -864.000 q^{85} -128.000 q^{86} -220.000 q^{87} +730.000 q^{89} -368.000 q^{90} -196.000 q^{91} -336.000 q^{92} -24.0000 q^{93} +324.000 q^{94} +1760.00 q^{95} -322.000 q^{96} +294.000 q^{97} +49.0000 q^{98} +O(q^{100})\)

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 0.353553 0.176777 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) −2.00000 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(4\) −7.00000 −0.875000
\(5\) 16.0000 1.43108 0.715542 0.698570i \(-0.246180\pi\)
0.715542 + 0.698570i \(0.246180\pi\)
\(6\) −2.00000 −0.136083
\(7\) 7.00000 0.377964
\(8\) −15.0000 −0.662913
\(9\) −23.0000 −0.851852
\(10\) 16.0000 0.505964
\(11\) 0 0
\(12\) 14.0000 0.336788
\(13\) −28.0000 −0.597369 −0.298685 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 7.00000 0.133631
\(15\) −32.0000 −0.550824
\(16\) 41.0000 0.640625
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) −23.0000 −0.301175
\(19\) 110.000 1.32820 0.664098 0.747645i \(-0.268816\pi\)
0.664098 + 0.747645i \(0.268816\pi\)
\(20\) −112.000 −1.25220
\(21\) −14.0000 −0.145479
\(22\) 0 0
\(23\) 48.0000 0.435161 0.217580 0.976042i \(-0.430184\pi\)
0.217580 + 0.976042i \(0.430184\pi\)
\(24\) 30.0000 0.255155
\(25\) 131.000 1.04800
\(26\) −28.0000 −0.211202
\(27\) 100.000 0.712778
\(28\) −49.0000 −0.330719
\(29\) 110.000 0.704362 0.352181 0.935932i \(-0.385440\pi\)
0.352181 + 0.935932i \(0.385440\pi\)
\(30\) −32.0000 −0.194746
\(31\) 12.0000 0.0695246 0.0347623 0.999396i \(-0.488933\pi\)
0.0347623 + 0.999396i \(0.488933\pi\)
\(32\) 161.000 0.889408
\(33\) 0 0
\(34\) −54.0000 −0.272380
\(35\) 112.000 0.540899
\(36\) 161.000 0.745370
\(37\) −246.000 −1.09303 −0.546516 0.837449i \(-0.684046\pi\)
−0.546516 + 0.837449i \(0.684046\pi\)
\(38\) 110.000 0.469588
\(39\) 56.0000 0.229928
\(40\) −240.000 −0.948683
\(41\) −182.000 −0.693259 −0.346630 0.938002i \(-0.612674\pi\)
−0.346630 + 0.938002i \(0.612674\pi\)
\(42\) −14.0000 −0.0514344
\(43\) −128.000 −0.453949 −0.226975 0.973901i \(-0.572883\pi\)
−0.226975 + 0.973901i \(0.572883\pi\)
\(44\) 0 0
\(45\) −368.000 −1.21907
\(46\) 48.0000 0.153852
\(47\) 324.000 1.00554 0.502769 0.864421i \(-0.332315\pi\)
0.502769 + 0.864421i \(0.332315\pi\)
\(48\) −82.0000 −0.246577
\(49\) 49.0000 0.142857
\(50\) 131.000 0.370524
\(51\) 108.000 0.296530
\(52\) 196.000 0.522698
\(53\) −162.000 −0.419857 −0.209928 0.977717i \(-0.567323\pi\)
−0.209928 + 0.977717i \(0.567323\pi\)
\(54\) 100.000 0.252005
\(55\) 0 0
\(56\) −105.000 −0.250557
\(57\) −220.000 −0.511223
\(58\) 110.000 0.249029
\(59\) 810.000 1.78734 0.893670 0.448725i \(-0.148122\pi\)
0.893670 + 0.448725i \(0.148122\pi\)
\(60\) 224.000 0.481971
\(61\) 488.000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 12.0000 0.0245807
\(63\) −161.000 −0.321970
\(64\) −167.000 −0.326172
\(65\) −448.000 −0.854886
\(66\) 0 0
\(67\) 244.000 0.444916 0.222458 0.974942i \(-0.428592\pi\)
0.222458 + 0.974942i \(0.428592\pi\)
\(68\) 378.000 0.674106
\(69\) −96.0000 −0.167493
\(70\) 112.000 0.191237
\(71\) −768.000 −1.28373 −0.641865 0.766818i \(-0.721839\pi\)
−0.641865 + 0.766818i \(0.721839\pi\)
\(72\) 345.000 0.564703
\(73\) 702.000 1.12552 0.562759 0.826621i \(-0.309740\pi\)
0.562759 + 0.826621i \(0.309740\pi\)
\(74\) −246.000 −0.386445
\(75\) −262.000 −0.403375
\(76\) −770.000 −1.16217
\(77\) 0 0
\(78\) 56.0000 0.0812917
\(79\) −440.000 −0.626631 −0.313316 0.949649i \(-0.601440\pi\)
−0.313316 + 0.949649i \(0.601440\pi\)
\(80\) 656.000 0.916788
\(81\) 421.000 0.577503
\(82\) −182.000 −0.245104
\(83\) 1302.00 1.72184 0.860922 0.508737i \(-0.169887\pi\)
0.860922 + 0.508737i \(0.169887\pi\)
\(84\) 98.0000 0.127294
\(85\) −864.000 −1.10252
\(86\) −128.000 −0.160495
\(87\) −220.000 −0.271109
\(88\) 0 0
\(89\) 730.000 0.869436 0.434718 0.900567i \(-0.356848\pi\)
0.434718 + 0.900567i \(0.356848\pi\)
\(90\) −368.000 −0.431007
\(91\) −196.000 −0.225784
\(92\) −336.000 −0.380765
\(93\) −24.0000 −0.0267600
\(94\) 324.000 0.355511
\(95\) 1760.00 1.90076
\(96\) −322.000 −0.342333
\(97\) 294.000 0.307744 0.153872 0.988091i \(-0.450826\pi\)
0.153872 + 0.988091i \(0.450826\pi\)
\(98\) 49.0000 0.0505076
\(99\) 0 0
\(100\) −917.000 −0.917000
\(101\) 688.000 0.677808 0.338904 0.940821i \(-0.389944\pi\)
0.338904 + 0.940821i \(0.389944\pi\)
\(102\) 108.000 0.104839
\(103\) 1388.00 1.32780 0.663901 0.747820i \(-0.268899\pi\)
0.663901 + 0.747820i \(0.268899\pi\)
\(104\) 420.000 0.396004
\(105\) −224.000 −0.208192
\(106\) −162.000 −0.148442
\(107\) −244.000 −0.220452 −0.110226 0.993907i \(-0.535157\pi\)
−0.110226 + 0.993907i \(0.535157\pi\)
\(108\) −700.000 −0.623681
\(109\) −90.0000 −0.0790866 −0.0395433 0.999218i \(-0.512590\pi\)
−0.0395433 + 0.999218i \(0.512590\pi\)
\(110\) 0 0
\(111\) 492.000 0.420708
\(112\) 287.000 0.242133
\(113\) 1318.00 1.09723 0.548615 0.836075i \(-0.315155\pi\)
0.548615 + 0.836075i \(0.315155\pi\)
\(114\) −220.000 −0.180745
\(115\) 768.000 0.622751
\(116\) −770.000 −0.616316
\(117\) 644.000 0.508870
\(118\) 810.000 0.631920
\(119\) −378.000 −0.291187
\(120\) 480.000 0.365148
\(121\) 0 0
\(122\) 488.000 0.362143
\(123\) 364.000 0.266836
\(124\) −84.0000 −0.0608341
\(125\) 96.0000 0.0686920
\(126\) −161.000 −0.113833
\(127\) 1776.00 1.24090 0.620451 0.784245i \(-0.286950\pi\)
0.620451 + 0.784245i \(0.286950\pi\)
\(128\) −1455.00 −1.00473
\(129\) 256.000 0.174725
\(130\) −448.000 −0.302248
\(131\) 1118.00 0.745650 0.372825 0.927902i \(-0.378389\pi\)
0.372825 + 0.927902i \(0.378389\pi\)
\(132\) 0 0
\(133\) 770.000 0.502011
\(134\) 244.000 0.157301
\(135\) 1600.00 1.02004
\(136\) 810.000 0.510713
\(137\) 2274.00 1.41811 0.709054 0.705154i \(-0.249122\pi\)
0.709054 + 0.705154i \(0.249122\pi\)
\(138\) −96.0000 −0.0592178
\(139\) 210.000 0.128144 0.0640718 0.997945i \(-0.479591\pi\)
0.0640718 + 0.997945i \(0.479591\pi\)
\(140\) −784.000 −0.473286
\(141\) −648.000 −0.387032
\(142\) −768.000 −0.453867
\(143\) 0 0
\(144\) −943.000 −0.545718
\(145\) 1760.00 1.00800
\(146\) 702.000 0.397931
\(147\) −98.0000 −0.0549857
\(148\) 1722.00 0.956402
\(149\) 2010.00 1.10514 0.552569 0.833467i \(-0.313648\pi\)
0.552569 + 0.833467i \(0.313648\pi\)
\(150\) −262.000 −0.142615
\(151\) −1112.00 −0.599293 −0.299647 0.954050i \(-0.596869\pi\)
−0.299647 + 0.954050i \(0.596869\pi\)
\(152\) −1650.00 −0.880478
\(153\) 1242.00 0.656273
\(154\) 0 0
\(155\) 192.000 0.0994956
\(156\) −392.000 −0.201187
\(157\) 124.000 0.0630336 0.0315168 0.999503i \(-0.489966\pi\)
0.0315168 + 0.999503i \(0.489966\pi\)
\(158\) −440.000 −0.221548
\(159\) 324.000 0.161603
\(160\) 2576.00 1.27282
\(161\) 336.000 0.164475
\(162\) 421.000 0.204178
\(163\) 2008.00 0.964900 0.482450 0.875924i \(-0.339747\pi\)
0.482450 + 0.875924i \(0.339747\pi\)
\(164\) 1274.00 0.606602
\(165\) 0 0
\(166\) 1302.00 0.608764
\(167\) −2884.00 −1.33635 −0.668176 0.744004i \(-0.732924\pi\)
−0.668176 + 0.744004i \(0.732924\pi\)
\(168\) 210.000 0.0964396
\(169\) −1413.00 −0.643150
\(170\) −864.000 −0.389799
\(171\) −2530.00 −1.13143
\(172\) 896.000 0.397206
\(173\) −2228.00 −0.979143 −0.489571 0.871963i \(-0.662847\pi\)
−0.489571 + 0.871963i \(0.662847\pi\)
\(174\) −220.000 −0.0958515
\(175\) 917.000 0.396107
\(176\) 0 0
\(177\) −1620.00 −0.687947
\(178\) 730.000 0.307392
\(179\) −820.000 −0.342400 −0.171200 0.985236i \(-0.554764\pi\)
−0.171200 + 0.985236i \(0.554764\pi\)
\(180\) 2576.00 1.06669
\(181\) 3892.00 1.59829 0.799144 0.601140i \(-0.205287\pi\)
0.799144 + 0.601140i \(0.205287\pi\)
\(182\) −196.000 −0.0798268
\(183\) −976.000 −0.394251
\(184\) −720.000 −0.288473
\(185\) −3936.00 −1.56422
\(186\) −24.0000 −0.00946110
\(187\) 0 0
\(188\) −2268.00 −0.879845
\(189\) 700.000 0.269405
\(190\) 1760.00 0.672020
\(191\) −5048.00 −1.91236 −0.956179 0.292782i \(-0.905419\pi\)
−0.956179 + 0.292782i \(0.905419\pi\)
\(192\) 334.000 0.125544
\(193\) 2962.00 1.10471 0.552356 0.833608i \(-0.313729\pi\)
0.552356 + 0.833608i \(0.313729\pi\)
\(194\) 294.000 0.108804
\(195\) 896.000 0.329046
\(196\) −343.000 −0.125000
\(197\) −3334.00 −1.20577 −0.602887 0.797826i \(-0.705983\pi\)
−0.602887 + 0.797826i \(0.705983\pi\)
\(198\) 0 0
\(199\) 1860.00 0.662572 0.331286 0.943530i \(-0.392517\pi\)
0.331286 + 0.943530i \(0.392517\pi\)
\(200\) −1965.00 −0.694732
\(201\) −488.000 −0.171248
\(202\) 688.000 0.239641
\(203\) 770.000 0.266224
\(204\) −756.000 −0.259464
\(205\) −2912.00 −0.992112
\(206\) 1388.00 0.469449
\(207\) −1104.00 −0.370692
\(208\) −1148.00 −0.382690
\(209\) 0 0
\(210\) −224.000 −0.0736070
\(211\) 4268.00 1.39252 0.696259 0.717791i \(-0.254847\pi\)
0.696259 + 0.717791i \(0.254847\pi\)
\(212\) 1134.00 0.367375
\(213\) 1536.00 0.494108
\(214\) −244.000 −0.0779416
\(215\) −2048.00 −0.649639
\(216\) −1500.00 −0.472510
\(217\) 84.0000 0.0262778
\(218\) −90.0000 −0.0279613
\(219\) −1404.00 −0.433212
\(220\) 0 0
\(221\) 1512.00 0.460218
\(222\) 492.000 0.148743
\(223\) −5432.00 −1.63118 −0.815591 0.578629i \(-0.803588\pi\)
−0.815591 + 0.578629i \(0.803588\pi\)
\(224\) 1127.00 0.336165
\(225\) −3013.00 −0.892741
\(226\) 1318.00 0.387929
\(227\) 2046.00 0.598228 0.299114 0.954217i \(-0.403309\pi\)
0.299114 + 0.954217i \(0.403309\pi\)
\(228\) 1540.00 0.447320
\(229\) −2980.00 −0.859930 −0.429965 0.902846i \(-0.641474\pi\)
−0.429965 + 0.902846i \(0.641474\pi\)
\(230\) 768.000 0.220176
\(231\) 0 0
\(232\) −1650.00 −0.466930
\(233\) −4458.00 −1.25345 −0.626724 0.779241i \(-0.715605\pi\)
−0.626724 + 0.779241i \(0.715605\pi\)
\(234\) 644.000 0.179913
\(235\) 5184.00 1.43901
\(236\) −5670.00 −1.56392
\(237\) 880.000 0.241190
\(238\) −378.000 −0.102950
\(239\) −4440.00 −1.20167 −0.600836 0.799372i \(-0.705166\pi\)
−0.600836 + 0.799372i \(0.705166\pi\)
\(240\) −1312.00 −0.352872
\(241\) −3302.00 −0.882575 −0.441287 0.897366i \(-0.645478\pi\)
−0.441287 + 0.897366i \(0.645478\pi\)
\(242\) 0 0
\(243\) −3542.00 −0.935059
\(244\) −3416.00 −0.896258
\(245\) 784.000 0.204441
\(246\) 364.000 0.0943406
\(247\) −3080.00 −0.793424
\(248\) −180.000 −0.0460888
\(249\) −2604.00 −0.662738
\(250\) 96.0000 0.0242863
\(251\) 1582.00 0.397829 0.198914 0.980017i \(-0.436258\pi\)
0.198914 + 0.980017i \(0.436258\pi\)
\(252\) 1127.00 0.281724
\(253\) 0 0
\(254\) 1776.00 0.438725
\(255\) 1728.00 0.424359
\(256\) −119.000 −0.0290527
\(257\) 2354.00 0.571356 0.285678 0.958326i \(-0.407781\pi\)
0.285678 + 0.958326i \(0.407781\pi\)
\(258\) 256.000 0.0617747
\(259\) −1722.00 −0.413127
\(260\) 3136.00 0.748025
\(261\) −2530.00 −0.600012
\(262\) 1118.00 0.263627
\(263\) 3872.00 0.907824 0.453912 0.891046i \(-0.350028\pi\)
0.453912 + 0.891046i \(0.350028\pi\)
\(264\) 0 0
\(265\) −2592.00 −0.600850
\(266\) 770.000 0.177488
\(267\) −1460.00 −0.334646
\(268\) −1708.00 −0.389301
\(269\) 180.000 0.0407985 0.0203992 0.999792i \(-0.493506\pi\)
0.0203992 + 0.999792i \(0.493506\pi\)
\(270\) 1600.00 0.360640
\(271\) −2032.00 −0.455480 −0.227740 0.973722i \(-0.573134\pi\)
−0.227740 + 0.973722i \(0.573134\pi\)
\(272\) −2214.00 −0.493542
\(273\) 392.000 0.0869045
\(274\) 2274.00 0.501377
\(275\) 0 0
\(276\) 672.000 0.146557
\(277\) 5426.00 1.17696 0.588478 0.808513i \(-0.299727\pi\)
0.588478 + 0.808513i \(0.299727\pi\)
\(278\) 210.000 0.0453056
\(279\) −276.000 −0.0592247
\(280\) −1680.00 −0.358569
\(281\) −842.000 −0.178753 −0.0893764 0.995998i \(-0.528487\pi\)
−0.0893764 + 0.995998i \(0.528487\pi\)
\(282\) −648.000 −0.136836
\(283\) 3782.00 0.794405 0.397202 0.917731i \(-0.369981\pi\)
0.397202 + 0.917731i \(0.369981\pi\)
\(284\) 5376.00 1.12326
\(285\) −3520.00 −0.731603
\(286\) 0 0
\(287\) −1274.00 −0.262027
\(288\) −3703.00 −0.757644
\(289\) −1997.00 −0.406473
\(290\) 1760.00 0.356382
\(291\) −588.000 −0.118451
\(292\) −4914.00 −0.984829
\(293\) 4312.00 0.859760 0.429880 0.902886i \(-0.358556\pi\)
0.429880 + 0.902886i \(0.358556\pi\)
\(294\) −98.0000 −0.0194404
\(295\) 12960.0 2.55783
\(296\) 3690.00 0.724584
\(297\) 0 0
\(298\) 2010.00 0.390725
\(299\) −1344.00 −0.259952
\(300\) 1834.00 0.352953
\(301\) −896.000 −0.171577
\(302\) −1112.00 −0.211882
\(303\) −1376.00 −0.260888
\(304\) 4510.00 0.850876
\(305\) 7808.00 1.46585
\(306\) 1242.00 0.232027
\(307\) −2674.00 −0.497112 −0.248556 0.968618i \(-0.579956\pi\)
−0.248556 + 0.968618i \(0.579956\pi\)
\(308\) 0 0
\(309\) −2776.00 −0.511072
\(310\) 192.000 0.0351770
\(311\) −3768.00 −0.687021 −0.343511 0.939149i \(-0.611616\pi\)
−0.343511 + 0.939149i \(0.611616\pi\)
\(312\) −840.000 −0.152422
\(313\) 2438.00 0.440268 0.220134 0.975470i \(-0.429351\pi\)
0.220134 + 0.975470i \(0.429351\pi\)
\(314\) 124.000 0.0222857
\(315\) −2576.00 −0.460766
\(316\) 3080.00 0.548302
\(317\) −3186.00 −0.564491 −0.282245 0.959342i \(-0.591079\pi\)
−0.282245 + 0.959342i \(0.591079\pi\)
\(318\) 324.000 0.0571353
\(319\) 0 0
\(320\) −2672.00 −0.466779
\(321\) 488.000 0.0848520
\(322\) 336.000 0.0581508
\(323\) −5940.00 −1.02325
\(324\) −2947.00 −0.505316
\(325\) −3668.00 −0.626043
\(326\) 2008.00 0.341144
\(327\) 180.000 0.0304404
\(328\) 2730.00 0.459570
\(329\) 2268.00 0.380057
\(330\) 0 0
\(331\) 8672.00 1.44005 0.720025 0.693949i \(-0.244131\pi\)
0.720025 + 0.693949i \(0.244131\pi\)
\(332\) −9114.00 −1.50661
\(333\) 5658.00 0.931101
\(334\) −2884.00 −0.472471
\(335\) 3904.00 0.636711
\(336\) −574.000 −0.0931972
\(337\) −814.000 −0.131577 −0.0657884 0.997834i \(-0.520956\pi\)
−0.0657884 + 0.997834i \(0.520956\pi\)
\(338\) −1413.00 −0.227388
\(339\) −2636.00 −0.422324
\(340\) 6048.00 0.964703
\(341\) 0 0
\(342\) −2530.00 −0.400020
\(343\) 343.000 0.0539949
\(344\) 1920.00 0.300929
\(345\) −1536.00 −0.239697
\(346\) −2228.00 −0.346179
\(347\) −9344.00 −1.44557 −0.722784 0.691074i \(-0.757138\pi\)
−0.722784 + 0.691074i \(0.757138\pi\)
\(348\) 1540.00 0.237220
\(349\) 5180.00 0.794496 0.397248 0.917711i \(-0.369965\pi\)
0.397248 + 0.917711i \(0.369965\pi\)
\(350\) 917.000 0.140045
\(351\) −2800.00 −0.425792
\(352\) 0 0
\(353\) 12178.0 1.83617 0.918087 0.396379i \(-0.129733\pi\)
0.918087 + 0.396379i \(0.129733\pi\)
\(354\) −1620.00 −0.243226
\(355\) −12288.0 −1.83712
\(356\) −5110.00 −0.760757
\(357\) 756.000 0.112078
\(358\) −820.000 −0.121057
\(359\) −440.000 −0.0646861 −0.0323431 0.999477i \(-0.510297\pi\)
−0.0323431 + 0.999477i \(0.510297\pi\)
\(360\) 5520.00 0.808138
\(361\) 5241.00 0.764106
\(362\) 3892.00 0.565080
\(363\) 0 0
\(364\) 1372.00 0.197561
\(365\) 11232.0 1.61071
\(366\) −976.000 −0.139389
\(367\) −9816.00 −1.39616 −0.698080 0.716019i \(-0.745962\pi\)
−0.698080 + 0.716019i \(0.745962\pi\)
\(368\) 1968.00 0.278775
\(369\) 4186.00 0.590554
\(370\) −3936.00 −0.553035
\(371\) −1134.00 −0.158691
\(372\) 168.000 0.0234150
\(373\) 442.000 0.0613563 0.0306781 0.999529i \(-0.490233\pi\)
0.0306781 + 0.999529i \(0.490233\pi\)
\(374\) 0 0
\(375\) −192.000 −0.0264396
\(376\) −4860.00 −0.666583
\(377\) −3080.00 −0.420764
\(378\) 700.000 0.0952490
\(379\) −3960.00 −0.536706 −0.268353 0.963321i \(-0.586479\pi\)
−0.268353 + 0.963321i \(0.586479\pi\)
\(380\) −12320.0 −1.66316
\(381\) −3552.00 −0.477623
\(382\) −5048.00 −0.676121
\(383\) 6708.00 0.894942 0.447471 0.894298i \(-0.352325\pi\)
0.447471 + 0.894298i \(0.352325\pi\)
\(384\) 2910.00 0.386720
\(385\) 0 0
\(386\) 2962.00 0.390575
\(387\) 2944.00 0.386697
\(388\) −2058.00 −0.269276
\(389\) −13350.0 −1.74003 −0.870015 0.493025i \(-0.835891\pi\)
−0.870015 + 0.493025i \(0.835891\pi\)
\(390\) 896.000 0.116335
\(391\) −2592.00 −0.335251
\(392\) −735.000 −0.0947018
\(393\) −2236.00 −0.287001
\(394\) −3334.00 −0.426306
\(395\) −7040.00 −0.896762
\(396\) 0 0
\(397\) −1356.00 −0.171425 −0.0857125 0.996320i \(-0.527317\pi\)
−0.0857125 + 0.996320i \(0.527317\pi\)
\(398\) 1860.00 0.234255
\(399\) −1540.00 −0.193224
\(400\) 5371.00 0.671375
\(401\) 6222.00 0.774843 0.387421 0.921903i \(-0.373366\pi\)
0.387421 + 0.921903i \(0.373366\pi\)
\(402\) −488.000 −0.0605453
\(403\) −336.000 −0.0415319
\(404\) −4816.00 −0.593082
\(405\) 6736.00 0.826456
\(406\) 770.000 0.0941243
\(407\) 0 0
\(408\) −1620.00 −0.196573
\(409\) −5150.00 −0.622619 −0.311309 0.950309i \(-0.600768\pi\)
−0.311309 + 0.950309i \(0.600768\pi\)
\(410\) −2912.00 −0.350764
\(411\) −4548.00 −0.545830
\(412\) −9716.00 −1.16183
\(413\) 5670.00 0.675551
\(414\) −1104.00 −0.131060
\(415\) 20832.0 2.46410
\(416\) −4508.00 −0.531305
\(417\) −420.000 −0.0493225
\(418\) 0 0
\(419\) 2310.00 0.269334 0.134667 0.990891i \(-0.457004\pi\)
0.134667 + 0.990891i \(0.457004\pi\)
\(420\) 1568.00 0.182168
\(421\) 1262.00 0.146095 0.0730476 0.997328i \(-0.476727\pi\)
0.0730476 + 0.997328i \(0.476727\pi\)
\(422\) 4268.00 0.492329
\(423\) −7452.00 −0.856569
\(424\) 2430.00 0.278328
\(425\) −7074.00 −0.807387
\(426\) 1536.00 0.174694
\(427\) 3416.00 0.387147
\(428\) 1708.00 0.192896
\(429\) 0 0
\(430\) −2048.00 −0.229682
\(431\) 4488.00 0.501576 0.250788 0.968042i \(-0.419310\pi\)
0.250788 + 0.968042i \(0.419310\pi\)
\(432\) 4100.00 0.456623
\(433\) 17038.0 1.89098 0.945490 0.325652i \(-0.105584\pi\)
0.945490 + 0.325652i \(0.105584\pi\)
\(434\) 84.0000 0.00929062
\(435\) −3520.00 −0.387979
\(436\) 630.000 0.0692008
\(437\) 5280.00 0.577979
\(438\) −1404.00 −0.153164
\(439\) −16200.0 −1.76124 −0.880619 0.473824i \(-0.842873\pi\)
−0.880619 + 0.473824i \(0.842873\pi\)
\(440\) 0 0
\(441\) −1127.00 −0.121693
\(442\) 1512.00 0.162712
\(443\) −8772.00 −0.940791 −0.470395 0.882456i \(-0.655889\pi\)
−0.470395 + 0.882456i \(0.655889\pi\)
\(444\) −3444.00 −0.368119
\(445\) 11680.0 1.24424
\(446\) −5432.00 −0.576710
\(447\) −4020.00 −0.425368
\(448\) −1169.00 −0.123281
\(449\) 2130.00 0.223877 0.111939 0.993715i \(-0.464294\pi\)
0.111939 + 0.993715i \(0.464294\pi\)
\(450\) −3013.00 −0.315632
\(451\) 0 0
\(452\) −9226.00 −0.960076
\(453\) 2224.00 0.230668
\(454\) 2046.00 0.211506
\(455\) −3136.00 −0.323116
\(456\) 3300.00 0.338896
\(457\) −10534.0 −1.07825 −0.539124 0.842226i \(-0.681245\pi\)
−0.539124 + 0.842226i \(0.681245\pi\)
\(458\) −2980.00 −0.304031
\(459\) −5400.00 −0.549129
\(460\) −5376.00 −0.544907
\(461\) 9268.00 0.936342 0.468171 0.883638i \(-0.344913\pi\)
0.468171 + 0.883638i \(0.344913\pi\)
\(462\) 0 0
\(463\) −9392.00 −0.942728 −0.471364 0.881939i \(-0.656238\pi\)
−0.471364 + 0.881939i \(0.656238\pi\)
\(464\) 4510.00 0.451232
\(465\) −384.000 −0.0382959
\(466\) −4458.00 −0.443161
\(467\) −10806.0 −1.07075 −0.535377 0.844613i \(-0.679830\pi\)
−0.535377 + 0.844613i \(0.679830\pi\)
\(468\) −4508.00 −0.445261
\(469\) 1708.00 0.168162
\(470\) 5184.00 0.508766
\(471\) −248.000 −0.0242616
\(472\) −12150.0 −1.18485
\(473\) 0 0
\(474\) 880.000 0.0852737
\(475\) 14410.0 1.39195
\(476\) 2646.00 0.254788
\(477\) 3726.00 0.357656
\(478\) −4440.00 −0.424855
\(479\) −4940.00 −0.471220 −0.235610 0.971848i \(-0.575709\pi\)
−0.235610 + 0.971848i \(0.575709\pi\)
\(480\) −5152.00 −0.489907
\(481\) 6888.00 0.652943
\(482\) −3302.00 −0.312037
\(483\) −672.000 −0.0633065
\(484\) 0 0
\(485\) 4704.00 0.440407
\(486\) −3542.00 −0.330593
\(487\) −5216.00 −0.485338 −0.242669 0.970109i \(-0.578023\pi\)
−0.242669 + 0.970109i \(0.578023\pi\)
\(488\) −7320.00 −0.679018
\(489\) −4016.00 −0.371390
\(490\) 784.000 0.0722806
\(491\) −4412.00 −0.405521 −0.202760 0.979228i \(-0.564991\pi\)
−0.202760 + 0.979228i \(0.564991\pi\)
\(492\) −2548.00 −0.233481
\(493\) −5940.00 −0.542645
\(494\) −3080.00 −0.280518
\(495\) 0 0
\(496\) 492.000 0.0445392
\(497\) −5376.00 −0.485204
\(498\) −2604.00 −0.234313
\(499\) 19060.0 1.70991 0.854953 0.518706i \(-0.173586\pi\)
0.854953 + 0.518706i \(0.173586\pi\)
\(500\) −672.000 −0.0601055
\(501\) 5768.00 0.514362
\(502\) 1582.00 0.140654
\(503\) −12768.0 −1.13180 −0.565902 0.824473i \(-0.691472\pi\)
−0.565902 + 0.824473i \(0.691472\pi\)
\(504\) 2415.00 0.213438
\(505\) 11008.0 0.969999
\(506\) 0 0
\(507\) 2826.00 0.247548
\(508\) −12432.0 −1.08579
\(509\) −5500.00 −0.478945 −0.239473 0.970903i \(-0.576975\pi\)
−0.239473 + 0.970903i \(0.576975\pi\)
\(510\) 1728.00 0.150034
\(511\) 4914.00 0.425406
\(512\) 11521.0 0.994455
\(513\) 11000.0 0.946709
\(514\) 2354.00 0.202005
\(515\) 22208.0 1.90020
\(516\) −1792.00 −0.152884
\(517\) 0 0
\(518\) −1722.00 −0.146062
\(519\) 4456.00 0.376872
\(520\) 6720.00 0.566714
\(521\) −7338.00 −0.617051 −0.308526 0.951216i \(-0.599836\pi\)
−0.308526 + 0.951216i \(0.599836\pi\)
\(522\) −2530.00 −0.212136
\(523\) 17582.0 1.46999 0.734997 0.678070i \(-0.237183\pi\)
0.734997 + 0.678070i \(0.237183\pi\)
\(524\) −7826.00 −0.652444
\(525\) −1834.00 −0.152462
\(526\) 3872.00 0.320964
\(527\) −648.000 −0.0535623
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) −2592.00 −0.212433
\(531\) −18630.0 −1.52255
\(532\) −5390.00 −0.439260
\(533\) 5096.00 0.414132
\(534\) −1460.00 −0.118315
\(535\) −3904.00 −0.315485
\(536\) −3660.00 −0.294940
\(537\) 1640.00 0.131790
\(538\) 180.000 0.0144244
\(539\) 0 0
\(540\) −11200.0 −0.892539
\(541\) 1618.00 0.128583 0.0642914 0.997931i \(-0.479521\pi\)
0.0642914 + 0.997931i \(0.479521\pi\)
\(542\) −2032.00 −0.161037
\(543\) −7784.00 −0.615181
\(544\) −8694.00 −0.685206
\(545\) −1440.00 −0.113179
\(546\) 392.000 0.0307254
\(547\) −16144.0 −1.26192 −0.630958 0.775817i \(-0.717338\pi\)
−0.630958 + 0.775817i \(0.717338\pi\)
\(548\) −15918.0 −1.24085
\(549\) −11224.0 −0.872548
\(550\) 0 0
\(551\) 12100.0 0.935531
\(552\) 1440.00 0.111033
\(553\) −3080.00 −0.236844
\(554\) 5426.00 0.416117
\(555\) 7872.00 0.602068
\(556\) −1470.00 −0.112126
\(557\) −4654.00 −0.354033 −0.177016 0.984208i \(-0.556645\pi\)
−0.177016 + 0.984208i \(0.556645\pi\)
\(558\) −276.000 −0.0209391
\(559\) 3584.00 0.271175
\(560\) 4592.00 0.346513
\(561\) 0 0
\(562\) −842.000 −0.0631986
\(563\) −10078.0 −0.754418 −0.377209 0.926128i \(-0.623116\pi\)
−0.377209 + 0.926128i \(0.623116\pi\)
\(564\) 4536.00 0.338653
\(565\) 21088.0 1.57023
\(566\) 3782.00 0.280865
\(567\) 2947.00 0.218276
\(568\) 11520.0 0.851001
\(569\) 5930.00 0.436904 0.218452 0.975848i \(-0.429899\pi\)
0.218452 + 0.975848i \(0.429899\pi\)
\(570\) −3520.00 −0.258661
\(571\) 19048.0 1.39603 0.698016 0.716082i \(-0.254067\pi\)
0.698016 + 0.716082i \(0.254067\pi\)
\(572\) 0 0
\(573\) 10096.0 0.736067
\(574\) −1274.00 −0.0926406
\(575\) 6288.00 0.456048
\(576\) 3841.00 0.277850
\(577\) −14366.0 −1.03651 −0.518253 0.855227i \(-0.673418\pi\)
−0.518253 + 0.855227i \(0.673418\pi\)
\(578\) −1997.00 −0.143710
\(579\) −5924.00 −0.425204
\(580\) −12320.0 −0.882000
\(581\) 9114.00 0.650796
\(582\) −588.000 −0.0418787
\(583\) 0 0
\(584\) −10530.0 −0.746121
\(585\) 10304.0 0.728236
\(586\) 4312.00 0.303971
\(587\) −3626.00 −0.254959 −0.127480 0.991841i \(-0.540689\pi\)
−0.127480 + 0.991841i \(0.540689\pi\)
\(588\) 686.000 0.0481125
\(589\) 1320.00 0.0923424
\(590\) 12960.0 0.904330
\(591\) 6668.00 0.464103
\(592\) −10086.0 −0.700223
\(593\) 1062.00 0.0735432 0.0367716 0.999324i \(-0.488293\pi\)
0.0367716 + 0.999324i \(0.488293\pi\)
\(594\) 0 0
\(595\) −6048.00 −0.416712
\(596\) −14070.0 −0.966996
\(597\) −3720.00 −0.255024
\(598\) −1344.00 −0.0919068
\(599\) −10200.0 −0.695761 −0.347880 0.937539i \(-0.613098\pi\)
−0.347880 + 0.937539i \(0.613098\pi\)
\(600\) 3930.00 0.267403
\(601\) 25158.0 1.70751 0.853757 0.520671i \(-0.174318\pi\)
0.853757 + 0.520671i \(0.174318\pi\)
\(602\) −896.000 −0.0606615
\(603\) −5612.00 −0.379002
\(604\) 7784.00 0.524382
\(605\) 0 0
\(606\) −1376.00 −0.0922379
\(607\) −25664.0 −1.71609 −0.858047 0.513570i \(-0.828323\pi\)
−0.858047 + 0.513570i \(0.828323\pi\)
\(608\) 17710.0 1.18131
\(609\) −1540.00 −0.102470
\(610\) 7808.00 0.518257
\(611\) −9072.00 −0.600677
\(612\) −8694.00 −0.574239
\(613\) −19018.0 −1.25307 −0.626533 0.779395i \(-0.715527\pi\)
−0.626533 + 0.779395i \(0.715527\pi\)
\(614\) −2674.00 −0.175755
\(615\) 5824.00 0.381864
\(616\) 0 0
\(617\) 17334.0 1.13102 0.565511 0.824741i \(-0.308679\pi\)
0.565511 + 0.824741i \(0.308679\pi\)
\(618\) −2776.00 −0.180691
\(619\) 18730.0 1.21619 0.608096 0.793864i \(-0.291934\pi\)
0.608096 + 0.793864i \(0.291934\pi\)
\(620\) −1344.00 −0.0870586
\(621\) 4800.00 0.310173
\(622\) −3768.00 −0.242899
\(623\) 5110.00 0.328616
\(624\) 2296.00 0.147297
\(625\) −14839.0 −0.949696
\(626\) 2438.00 0.155658
\(627\) 0 0
\(628\) −868.000 −0.0551544
\(629\) 13284.0 0.842079
\(630\) −2576.00 −0.162905
\(631\) −6928.00 −0.437083 −0.218541 0.975828i \(-0.570130\pi\)
−0.218541 + 0.975828i \(0.570130\pi\)
\(632\) 6600.00 0.415402
\(633\) −8536.00 −0.535980
\(634\) −3186.00 −0.199578
\(635\) 28416.0 1.77583
\(636\) −2268.00 −0.141403
\(637\) −1372.00 −0.0853385
\(638\) 0 0
\(639\) 17664.0 1.09355
\(640\) −23280.0 −1.43785
\(641\) 16302.0 1.00451 0.502255 0.864720i \(-0.332504\pi\)
0.502255 + 0.864720i \(0.332504\pi\)
\(642\) 488.000 0.0299997
\(643\) 4718.00 0.289362 0.144681 0.989478i \(-0.453784\pi\)
0.144681 + 0.989478i \(0.453784\pi\)
\(644\) −2352.00 −0.143916
\(645\) 4096.00 0.250046
\(646\) −5940.00 −0.361774
\(647\) −21436.0 −1.30253 −0.651264 0.758851i \(-0.725761\pi\)
−0.651264 + 0.758851i \(0.725761\pi\)
\(648\) −6315.00 −0.382834
\(649\) 0 0
\(650\) −3668.00 −0.221340
\(651\) −168.000 −0.0101143
\(652\) −14056.0 −0.844287
\(653\) 4458.00 0.267159 0.133580 0.991038i \(-0.457353\pi\)
0.133580 + 0.991038i \(0.457353\pi\)
\(654\) 180.000 0.0107623
\(655\) 17888.0 1.06709
\(656\) −7462.00 −0.444119
\(657\) −16146.0 −0.958775
\(658\) 2268.00 0.134371
\(659\) 26640.0 1.57473 0.787365 0.616487i \(-0.211445\pi\)
0.787365 + 0.616487i \(0.211445\pi\)
\(660\) 0 0
\(661\) 7432.00 0.437324 0.218662 0.975801i \(-0.429831\pi\)
0.218662 + 0.975801i \(0.429831\pi\)
\(662\) 8672.00 0.509134
\(663\) −3024.00 −0.177138
\(664\) −19530.0 −1.14143
\(665\) 12320.0 0.718420
\(666\) 5658.00 0.329194
\(667\) 5280.00 0.306510
\(668\) 20188.0 1.16931
\(669\) 10864.0 0.627842
\(670\) 3904.00 0.225111
\(671\) 0 0
\(672\) −2254.00 −0.129390
\(673\) −58.0000 −0.00332204 −0.00166102 0.999999i \(-0.500529\pi\)
−0.00166102 + 0.999999i \(0.500529\pi\)
\(674\) −814.000 −0.0465194
\(675\) 13100.0 0.746991
\(676\) 9891.00 0.562756
\(677\) 21516.0 1.22146 0.610729 0.791840i \(-0.290876\pi\)
0.610729 + 0.791840i \(0.290876\pi\)
\(678\) −2636.00 −0.149314
\(679\) 2058.00 0.116316
\(680\) 12960.0 0.730873
\(681\) −4092.00 −0.230258
\(682\) 0 0
\(683\) 18108.0 1.01447 0.507235 0.861808i \(-0.330668\pi\)
0.507235 + 0.861808i \(0.330668\pi\)
\(684\) 17710.0 0.989998
\(685\) 36384.0 2.02943
\(686\) 343.000 0.0190901
\(687\) 5960.00 0.330987
\(688\) −5248.00 −0.290811
\(689\) 4536.00 0.250810
\(690\) −1536.00 −0.0847457
\(691\) −10078.0 −0.554827 −0.277413 0.960751i \(-0.589477\pi\)
−0.277413 + 0.960751i \(0.589477\pi\)
\(692\) 15596.0 0.856750
\(693\) 0 0
\(694\) −9344.00 −0.511086
\(695\) 3360.00 0.183384
\(696\) 3300.00 0.179722
\(697\) 9828.00 0.534092
\(698\) 5180.00 0.280897
\(699\) 8916.00 0.482452
\(700\) −6419.00 −0.346593
\(701\) −18762.0 −1.01089 −0.505443 0.862860i \(-0.668671\pi\)
−0.505443 + 0.862860i \(0.668671\pi\)
\(702\) −2800.00 −0.150540
\(703\) −27060.0 −1.45176
\(704\) 0 0
\(705\) −10368.0 −0.553874
\(706\) 12178.0 0.649186
\(707\) 4816.00 0.256187
\(708\) 11340.0 0.601954
\(709\) 6810.00 0.360726 0.180363 0.983600i \(-0.442273\pi\)
0.180363 + 0.983600i \(0.442273\pi\)
\(710\) −12288.0 −0.649522
\(711\) 10120.0 0.533797
\(712\) −10950.0 −0.576360
\(713\) 576.000 0.0302544
\(714\) 756.000 0.0396255
\(715\) 0 0
\(716\) 5740.00 0.299600
\(717\) 8880.00 0.462524
\(718\) −440.000 −0.0228700
\(719\) 4860.00 0.252083 0.126041 0.992025i \(-0.459773\pi\)
0.126041 + 0.992025i \(0.459773\pi\)
\(720\) −15088.0 −0.780967
\(721\) 9716.00 0.501862
\(722\) 5241.00 0.270152
\(723\) 6604.00 0.339703
\(724\) −27244.0 −1.39850
\(725\) 14410.0 0.738171
\(726\) 0 0
\(727\) −13636.0 −0.695641 −0.347821 0.937561i \(-0.613078\pi\)
−0.347821 + 0.937561i \(0.613078\pi\)
\(728\) 2940.00 0.149675
\(729\) −4283.00 −0.217599
\(730\) 11232.0 0.569473
\(731\) 6912.00 0.349726
\(732\) 6832.00 0.344970
\(733\) −2088.00 −0.105214 −0.0526071 0.998615i \(-0.516753\pi\)
−0.0526071 + 0.998615i \(0.516753\pi\)
\(734\) −9816.00 −0.493617
\(735\) −1568.00 −0.0786892
\(736\) 7728.00 0.387035
\(737\) 0 0
\(738\) 4186.00 0.208792
\(739\) 5160.00 0.256852 0.128426 0.991719i \(-0.459008\pi\)
0.128426 + 0.991719i \(0.459008\pi\)
\(740\) 27552.0 1.36869
\(741\) 6160.00 0.305389
\(742\) −1134.00 −0.0561057
\(743\) 28152.0 1.39004 0.695018 0.718992i \(-0.255396\pi\)
0.695018 + 0.718992i \(0.255396\pi\)
\(744\) 360.000 0.0177396
\(745\) 32160.0 1.58155
\(746\) 442.000 0.0216927
\(747\) −29946.0 −1.46676
\(748\) 0 0
\(749\) −1708.00 −0.0833230
\(750\) −192.000 −0.00934780
\(751\) −16808.0 −0.816688 −0.408344 0.912828i \(-0.633894\pi\)
−0.408344 + 0.912828i \(0.633894\pi\)
\(752\) 13284.0 0.644172
\(753\) −3164.00 −0.153124
\(754\) −3080.00 −0.148763
\(755\) −17792.0 −0.857639
\(756\) −4900.00 −0.235729
\(757\) 21674.0 1.04063 0.520314 0.853975i \(-0.325815\pi\)
0.520314 + 0.853975i \(0.325815\pi\)
\(758\) −3960.00 −0.189754
\(759\) 0 0
\(760\) −26400.0 −1.26004
\(761\) −7422.00 −0.353544 −0.176772 0.984252i \(-0.556566\pi\)
−0.176772 + 0.984252i \(0.556566\pi\)
\(762\) −3552.00 −0.168865
\(763\) −630.000 −0.0298919
\(764\) 35336.0 1.67331
\(765\) 19872.0 0.939181
\(766\) 6708.00 0.316410
\(767\) −22680.0 −1.06770
\(768\) 238.000 0.0111824
\(769\) −13790.0 −0.646658 −0.323329 0.946287i \(-0.604802\pi\)
−0.323329 + 0.946287i \(0.604802\pi\)
\(770\) 0 0
\(771\) −4708.00 −0.219915
\(772\) −20734.0 −0.966623
\(773\) −6232.00 −0.289973 −0.144987 0.989434i \(-0.546314\pi\)
−0.144987 + 0.989434i \(0.546314\pi\)
\(774\) 2944.00 0.136718
\(775\) 1572.00 0.0728618
\(776\) −4410.00 −0.204007
\(777\) 3444.00 0.159013
\(778\) −13350.0 −0.615194
\(779\) −20020.0 −0.920784
\(780\) −6272.00 −0.287915
\(781\) 0 0
\(782\) −2592.00 −0.118529
\(783\) 11000.0 0.502054
\(784\) 2009.00 0.0915179
\(785\) 1984.00 0.0902064
\(786\) −2236.00 −0.101470
\(787\) 1766.00 0.0799887 0.0399943 0.999200i \(-0.487266\pi\)
0.0399943 + 0.999200i \(0.487266\pi\)
\(788\) 23338.0 1.05505
\(789\) −7744.00 −0.349422
\(790\) −7040.00 −0.317053
\(791\) 9226.00 0.414714
\(792\) 0 0
\(793\) −13664.0 −0.611883
\(794\) −1356.00 −0.0606079
\(795\) 5184.00 0.231267
\(796\) −13020.0 −0.579751
\(797\) 1204.00 0.0535105 0.0267552 0.999642i \(-0.491483\pi\)
0.0267552 + 0.999642i \(0.491483\pi\)
\(798\) −1540.00 −0.0683150
\(799\) −17496.0 −0.774673
\(800\) 21091.0 0.932099
\(801\) −16790.0 −0.740631
\(802\) 6222.00 0.273948
\(803\) 0 0
\(804\) 3416.00 0.149842
\(805\) 5376.00 0.235378
\(806\) −336.000 −0.0146837
\(807\) −360.000 −0.0157033
\(808\) −10320.0 −0.449327
\(809\) 7050.00 0.306384 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(810\) 6736.00 0.292196
\(811\) −23282.0 −1.00807 −0.504033 0.863684i \(-0.668151\pi\)
−0.504033 + 0.863684i \(0.668151\pi\)
\(812\) −5390.00 −0.232946
\(813\) 4064.00 0.175315
\(814\) 0 0
\(815\) 32128.0 1.38085
\(816\) 4428.00 0.189964
\(817\) −14080.0 −0.602934
\(818\) −5150.00 −0.220129
\(819\) 4508.00 0.192335
\(820\) 20384.0 0.868098
\(821\) −10142.0 −0.431131 −0.215565 0.976489i \(-0.569159\pi\)
−0.215565 + 0.976489i \(0.569159\pi\)
\(822\) −4548.00 −0.192980
\(823\) −9192.00 −0.389323 −0.194662 0.980870i \(-0.562361\pi\)
−0.194662 + 0.980870i \(0.562361\pi\)
\(824\) −20820.0 −0.880217
\(825\) 0 0
\(826\) 5670.00 0.238843
\(827\) 46716.0 1.96430 0.982149 0.188104i \(-0.0602344\pi\)
0.982149 + 0.188104i \(0.0602344\pi\)
\(828\) 7728.00 0.324356
\(829\) 11240.0 0.470906 0.235453 0.971886i \(-0.424343\pi\)
0.235453 + 0.971886i \(0.424343\pi\)
\(830\) 20832.0 0.871192
\(831\) −10852.0 −0.453010
\(832\) 4676.00 0.194845
\(833\) −2646.00 −0.110058
\(834\) −420.000 −0.0174381
\(835\) −46144.0 −1.91243
\(836\) 0 0
\(837\) 1200.00 0.0495556
\(838\) 2310.00 0.0952239
\(839\) 700.000 0.0288042 0.0144021 0.999896i \(-0.495416\pi\)
0.0144021 + 0.999896i \(0.495416\pi\)
\(840\) 3360.00 0.138013
\(841\) −12289.0 −0.503875
\(842\) 1262.00 0.0516525
\(843\) 1684.00 0.0688019
\(844\) −29876.0 −1.21845
\(845\) −22608.0 −0.920401
\(846\) −7452.00 −0.302843
\(847\) 0 0
\(848\) −6642.00 −0.268971
\(849\) −7564.00 −0.305767
\(850\) −7074.00 −0.285454
\(851\) −11808.0 −0.475644
\(852\) −10752.0 −0.432344
\(853\) 37492.0 1.50493 0.752463 0.658635i \(-0.228866\pi\)
0.752463 + 0.658635i \(0.228866\pi\)
\(854\) 3416.00 0.136877
\(855\) −40480.0 −1.61917
\(856\) 3660.00 0.146140
\(857\) −28894.0 −1.15169 −0.575846 0.817558i \(-0.695327\pi\)
−0.575846 + 0.817558i \(0.695327\pi\)
\(858\) 0 0
\(859\) −2770.00 −0.110025 −0.0550123 0.998486i \(-0.517520\pi\)
−0.0550123 + 0.998486i \(0.517520\pi\)
\(860\) 14336.0 0.568434
\(861\) 2548.00 0.100854
\(862\) 4488.00 0.177334
\(863\) 17688.0 0.697690 0.348845 0.937180i \(-0.386574\pi\)
0.348845 + 0.937180i \(0.386574\pi\)
\(864\) 16100.0 0.633950
\(865\) −35648.0 −1.40124
\(866\) 17038.0 0.668562
\(867\) 3994.00 0.156451
\(868\) −588.000 −0.0229931
\(869\) 0 0
\(870\) −3520.00 −0.137171
\(871\) −6832.00 −0.265779
\(872\) 1350.00 0.0524275
\(873\) −6762.00 −0.262152
\(874\) 5280.00 0.204346
\(875\) 672.000 0.0259631
\(876\) 9828.00 0.379061
\(877\) 33566.0 1.29241 0.646205 0.763164i \(-0.276355\pi\)
0.646205 + 0.763164i \(0.276355\pi\)
\(878\) −16200.0 −0.622692
\(879\) −8624.00 −0.330922
\(880\) 0 0
\(881\) −16758.0 −0.640853 −0.320426 0.947273i \(-0.603826\pi\)
−0.320426 + 0.947273i \(0.603826\pi\)
\(882\) −1127.00 −0.0430250
\(883\) 11468.0 0.437066 0.218533 0.975830i \(-0.429873\pi\)
0.218533 + 0.975830i \(0.429873\pi\)
\(884\) −10584.0 −0.402691
\(885\) −25920.0 −0.984510
\(886\) −8772.00 −0.332620
\(887\) 50356.0 1.90619 0.953094 0.302674i \(-0.0978793\pi\)
0.953094 + 0.302674i \(0.0978793\pi\)
\(888\) −7380.00 −0.278893
\(889\) 12432.0 0.469017
\(890\) 11680.0 0.439904
\(891\) 0 0
\(892\) 38024.0 1.42728
\(893\) 35640.0 1.33555
\(894\) −4020.00 −0.150390
\(895\) −13120.0 −0.490004
\(896\) −10185.0 −0.379751
\(897\) 2688.00 0.100055
\(898\) 2130.00 0.0791526
\(899\) 1320.00 0.0489705
\(900\) 21091.0 0.781148
\(901\) 8748.00 0.323461
\(902\) 0 0
\(903\) 1792.00 0.0660399
\(904\) −19770.0 −0.727368
\(905\) 62272.0 2.28728
\(906\) 2224.00 0.0815535
\(907\) −8716.00 −0.319085 −0.159542 0.987191i \(-0.551002\pi\)
−0.159542 + 0.987191i \(0.551002\pi\)
\(908\) −14322.0 −0.523450
\(909\) −15824.0 −0.577392
\(910\) −3136.00 −0.114239
\(911\) 7632.00 0.277563 0.138781 0.990323i \(-0.455682\pi\)
0.138781 + 0.990323i \(0.455682\pi\)
\(912\) −9020.00 −0.327502
\(913\) 0 0
\(914\) −10534.0 −0.381219
\(915\) −15616.0 −0.564207
\(916\) 20860.0 0.752439
\(917\) 7826.00 0.281829
\(918\) −5400.00 −0.194147
\(919\) 23080.0 0.828443 0.414221 0.910176i \(-0.364054\pi\)
0.414221 + 0.910176i \(0.364054\pi\)
\(920\) −11520.0 −0.412830
\(921\) 5348.00 0.191338
\(922\) 9268.00 0.331047
\(923\) 21504.0 0.766861
\(924\) 0 0
\(925\) −32226.0 −1.14550
\(926\) −9392.00 −0.333305
\(927\) −31924.0 −1.13109
\(928\) 17710.0 0.626465
\(929\) 45110.0 1.59312 0.796561 0.604558i \(-0.206650\pi\)
0.796561 + 0.604558i \(0.206650\pi\)
\(930\) −384.000 −0.0135396
\(931\) 5390.00 0.189742
\(932\) 31206.0 1.09677
\(933\) 7536.00 0.264435
\(934\) −10806.0 −0.378569
\(935\) 0 0
\(936\) −9660.00 −0.337337
\(937\) −16674.0 −0.581340 −0.290670 0.956823i \(-0.593878\pi\)
−0.290670 + 0.956823i \(0.593878\pi\)
\(938\) 1708.00 0.0594543
\(939\) −4876.00 −0.169459
\(940\) −36288.0 −1.25913
\(941\) −43832.0 −1.51847 −0.759236 0.650815i \(-0.774427\pi\)
−0.759236 + 0.650815i \(0.774427\pi\)
\(942\) −248.000 −0.00857779
\(943\) −8736.00 −0.301679
\(944\) 33210.0 1.14501
\(945\) 11200.0 0.385541
\(946\) 0 0
\(947\) −736.000 −0.0252553 −0.0126277 0.999920i \(-0.504020\pi\)
−0.0126277 + 0.999920i \(0.504020\pi\)
\(948\) −6160.00 −0.211042
\(949\) −19656.0 −0.672351
\(950\) 14410.0 0.492129
\(951\) 6372.00 0.217273
\(952\) 5670.00 0.193031
\(953\) −38138.0 −1.29634 −0.648169 0.761496i \(-0.724465\pi\)
−0.648169 + 0.761496i \(0.724465\pi\)
\(954\) 3726.00 0.126450
\(955\) −80768.0 −2.73674
\(956\) 31080.0 1.05146
\(957\) 0 0
\(958\) −4940.00 −0.166601
\(959\) 15918.0 0.535995
\(960\) 5344.00 0.179663
\(961\) −29647.0 −0.995166
\(962\) 6888.00 0.230850
\(963\) 5612.00 0.187792
\(964\) 23114.0 0.772253
\(965\) 47392.0 1.58094
\(966\) −672.000 −0.0223822
\(967\) −26224.0 −0.872086 −0.436043 0.899926i \(-0.643620\pi\)
−0.436043 + 0.899926i \(0.643620\pi\)
\(968\) 0 0
\(969\) 11880.0 0.393850
\(970\) 4704.00 0.155708
\(971\) 18762.0 0.620084 0.310042 0.950723i \(-0.399657\pi\)
0.310042 + 0.950723i \(0.399657\pi\)
\(972\) 24794.0 0.818177
\(973\) 1470.00 0.0484337
\(974\) −5216.00 −0.171593
\(975\) 7336.00 0.240964
\(976\) 20008.0 0.656189
\(977\) 38394.0 1.25725 0.628625 0.777709i \(-0.283618\pi\)
0.628625 + 0.777709i \(0.283618\pi\)
\(978\) −4016.00 −0.131306
\(979\) 0 0
\(980\) −5488.00 −0.178885
\(981\) 2070.00 0.0673700
\(982\) −4412.00 −0.143373
\(983\) 5388.00 0.174822 0.0874112 0.996172i \(-0.472141\pi\)
0.0874112 + 0.996172i \(0.472141\pi\)
\(984\) −5460.00 −0.176889
\(985\) −53344.0 −1.72556
\(986\) −5940.00 −0.191854
\(987\) −4536.00 −0.146284
\(988\) 21560.0 0.694246
\(989\) −6144.00 −0.197541
\(990\) 0 0
\(991\) 25472.0 0.816493 0.408247 0.912872i \(-0.366140\pi\)
0.408247 + 0.912872i \(0.366140\pi\)
\(992\) 1932.00 0.0618357
\(993\) −17344.0 −0.554275
\(994\) −5376.00 −0.171546
\(995\) 29760.0 0.948196
\(996\) 18228.0 0.579896
\(997\) 17096.0 0.543065 0.271532 0.962429i \(-0.412470\pi\)
0.271532 + 0.962429i \(0.412470\pi\)
\(998\) 19060.0 0.604543
\(999\) −24600.0 −0.779089
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 847.4.a.b.1.1 1
11.10 odd 2 7.4.a.a.1.1 1
33.32 even 2 63.4.a.b.1.1 1
44.43 even 2 112.4.a.f.1.1 1
55.32 even 4 175.4.b.b.99.1 2
55.43 even 4 175.4.b.b.99.2 2
55.54 odd 2 175.4.a.b.1.1 1
77.10 even 6 49.4.c.b.30.1 2
77.32 odd 6 49.4.c.c.30.1 2
77.54 even 6 49.4.c.b.18.1 2
77.65 odd 6 49.4.c.c.18.1 2
77.76 even 2 49.4.a.b.1.1 1
88.21 odd 2 448.4.a.i.1.1 1
88.43 even 2 448.4.a.e.1.1 1
132.131 odd 2 1008.4.a.c.1.1 1
143.142 odd 2 1183.4.a.b.1.1 1
165.164 even 2 1575.4.a.e.1.1 1
187.186 odd 2 2023.4.a.a.1.1 1
231.32 even 6 441.4.e.h.226.1 2
231.65 even 6 441.4.e.h.361.1 2
231.131 odd 6 441.4.e.e.361.1 2
231.164 odd 6 441.4.e.e.226.1 2
231.230 odd 2 441.4.a.i.1.1 1
308.307 odd 2 784.4.a.g.1.1 1
385.384 even 2 1225.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.a.a.1.1 1 11.10 odd 2
49.4.a.b.1.1 1 77.76 even 2
49.4.c.b.18.1 2 77.54 even 6
49.4.c.b.30.1 2 77.10 even 6
49.4.c.c.18.1 2 77.65 odd 6
49.4.c.c.30.1 2 77.32 odd 6
63.4.a.b.1.1 1 33.32 even 2
112.4.a.f.1.1 1 44.43 even 2
175.4.a.b.1.1 1 55.54 odd 2
175.4.b.b.99.1 2 55.32 even 4
175.4.b.b.99.2 2 55.43 even 4
441.4.a.i.1.1 1 231.230 odd 2
441.4.e.e.226.1 2 231.164 odd 6
441.4.e.e.361.1 2 231.131 odd 6
441.4.e.h.226.1 2 231.32 even 6
441.4.e.h.361.1 2 231.65 even 6
448.4.a.e.1.1 1 88.43 even 2
448.4.a.i.1.1 1 88.21 odd 2
784.4.a.g.1.1 1 308.307 odd 2
847.4.a.b.1.1 1 1.1 even 1 trivial
1008.4.a.c.1.1 1 132.131 odd 2
1183.4.a.b.1.1 1 143.142 odd 2
1225.4.a.j.1.1 1 385.384 even 2
1575.4.a.e.1.1 1 165.164 even 2
2023.4.a.a.1.1 1 187.186 odd 2