Properties

Label 175.4.a.b.1.1
Level $175$
Weight $4$
Character 175.1
Self dual yes
Analytic conductor $10.325$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3253342510\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 175.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} -7.00000 q^{4} +2.00000 q^{6} +7.00000 q^{7} -15.0000 q^{8} -23.0000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} -7.00000 q^{4} +2.00000 q^{6} +7.00000 q^{7} -15.0000 q^{8} -23.0000 q^{9} -8.00000 q^{11} -14.0000 q^{12} -28.0000 q^{13} +7.00000 q^{14} +41.0000 q^{16} -54.0000 q^{17} -23.0000 q^{18} -110.000 q^{19} +14.0000 q^{21} -8.00000 q^{22} -48.0000 q^{23} -30.0000 q^{24} -28.0000 q^{26} -100.000 q^{27} -49.0000 q^{28} -110.000 q^{29} +12.0000 q^{31} +161.000 q^{32} -16.0000 q^{33} -54.0000 q^{34} +161.000 q^{36} +246.000 q^{37} -110.000 q^{38} -56.0000 q^{39} +182.000 q^{41} +14.0000 q^{42} -128.000 q^{43} +56.0000 q^{44} -48.0000 q^{46} -324.000 q^{47} +82.0000 q^{48} +49.0000 q^{49} -108.000 q^{51} +196.000 q^{52} +162.000 q^{53} -100.000 q^{54} -105.000 q^{56} -220.000 q^{57} -110.000 q^{58} +810.000 q^{59} -488.000 q^{61} +12.0000 q^{62} -161.000 q^{63} -167.000 q^{64} -16.0000 q^{66} -244.000 q^{67} +378.000 q^{68} -96.0000 q^{69} -768.000 q^{71} +345.000 q^{72} +702.000 q^{73} +246.000 q^{74} +770.000 q^{76} -56.0000 q^{77} -56.0000 q^{78} +440.000 q^{79} +421.000 q^{81} +182.000 q^{82} +1302.00 q^{83} -98.0000 q^{84} -128.000 q^{86} -220.000 q^{87} +120.000 q^{88} +730.000 q^{89} -196.000 q^{91} +336.000 q^{92} +24.0000 q^{93} -324.000 q^{94} +322.000 q^{96} -294.000 q^{97} +49.0000 q^{98} +184.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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 \(-0.438357\pi\)
0.192450 + 0.981307i \(0.438357\pi\)
\(4\) −7.00000 −0.875000
\(5\) 0 0
\(6\) 2.00000 0.136083
\(7\) 7.00000 0.377964
\(8\) −15.0000 −0.662913
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) −8.00000 −0.219281 −0.109640 0.993971i \(-0.534970\pi\)
−0.109640 + 0.993971i \(0.534970\pi\)
\(12\) −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
\(15\) 0 0
\(16\) 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 \(-0.731184\pi\)
−0.664098 + 0.747645i \(0.731184\pi\)
\(20\) 0 0
\(21\) 14.0000 0.145479
\(22\) −8.00000 −0.0775275
\(23\) −48.0000 −0.435161 −0.217580 0.976042i \(-0.569816\pi\)
−0.217580 + 0.976042i \(0.569816\pi\)
\(24\) −30.0000 −0.255155
\(25\) 0 0
\(26\) −28.0000 −0.211202
\(27\) −100.000 −0.712778
\(28\) −49.0000 −0.330719
\(29\) −110.000 −0.704362 −0.352181 0.935932i \(-0.614560\pi\)
−0.352181 + 0.935932i \(0.614560\pi\)
\(30\) 0 0
\(31\) 12.0000 0.0695246 0.0347623 0.999396i \(-0.488933\pi\)
0.0347623 + 0.999396i \(0.488933\pi\)
\(32\) 161.000 0.889408
\(33\) −16.0000 −0.0844013
\(34\) −54.0000 −0.272380
\(35\) 0 0
\(36\) 161.000 0.745370
\(37\) 246.000 1.09303 0.546516 0.837449i \(-0.315954\pi\)
0.546516 + 0.837449i \(0.315954\pi\)
\(38\) −110.000 −0.469588
\(39\) −56.0000 −0.229928
\(40\) 0 0
\(41\) 182.000 0.693259 0.346630 0.938002i \(-0.387326\pi\)
0.346630 + 0.938002i \(0.387326\pi\)
\(42\) 14.0000 0.0514344
\(43\) −128.000 −0.453949 −0.226975 0.973901i \(-0.572883\pi\)
−0.226975 + 0.973901i \(0.572883\pi\)
\(44\) 56.0000 0.191871
\(45\) 0 0
\(46\) −48.0000 −0.153852
\(47\) −324.000 −1.00554 −0.502769 0.864421i \(-0.667685\pi\)
−0.502769 + 0.864421i \(0.667685\pi\)
\(48\) 82.0000 0.246577
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −108.000 −0.296530
\(52\) 196.000 0.522698
\(53\) 162.000 0.419857 0.209928 0.977717i \(-0.432677\pi\)
0.209928 + 0.977717i \(0.432677\pi\)
\(54\) −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 \(0.148122\pi\)
\(60\) 0 0
\(61\) −488.000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 12.0000 0.0245807
\(63\) −161.000 −0.321970
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) −16.0000 −0.0298404
\(67\) −244.000 −0.444916 −0.222458 0.974942i \(-0.571408\pi\)
−0.222458 + 0.974942i \(0.571408\pi\)
\(68\) 378.000 0.674106
\(69\) −96.0000 −0.167493
\(70\) 0 0
\(71\) −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
\(75\) 0 0
\(76\) 770.000 1.16217
\(77\) −56.0000 −0.0828804
\(78\) −56.0000 −0.0812917
\(79\) 440.000 0.626631 0.313316 0.949649i \(-0.398560\pi\)
0.313316 + 0.949649i \(0.398560\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) −128.000 −0.160495
\(87\) −220.000 −0.271109
\(88\) 120.000 0.145364
\(89\) 730.000 0.869436 0.434718 0.900567i \(-0.356848\pi\)
0.434718 + 0.900567i \(0.356848\pi\)
\(90\) 0 0
\(91\) −196.000 −0.225784
\(92\) 336.000 0.380765
\(93\) 24.0000 0.0267600
\(94\) −324.000 −0.355511
\(95\) 0 0
\(96\) 322.000 0.342333
\(97\) −294.000 −0.307744 −0.153872 0.988091i \(-0.549174\pi\)
−0.153872 + 0.988091i \(0.549174\pi\)
\(98\) 49.0000 0.0505076
\(99\) 184.000 0.186795
\(100\) 0 0
\(101\) −688.000 −0.677808 −0.338904 0.940821i \(-0.610056\pi\)
−0.338904 + 0.940821i \(0.610056\pi\)
\(102\) −108.000 −0.104839
\(103\) −1388.00 −1.32780 −0.663901 0.747820i \(-0.731101\pi\)
−0.663901 + 0.747820i \(0.731101\pi\)
\(104\) 420.000 0.396004
\(105\) 0 0
\(106\) 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 \(-0.487410\pi\)
0.0395433 + 0.999218i \(0.487410\pi\)
\(110\) 0 0
\(111\) 492.000 0.420708
\(112\) 287.000 0.242133
\(113\) −1318.00 −1.09723 −0.548615 0.836075i \(-0.684845\pi\)
−0.548615 + 0.836075i \(0.684845\pi\)
\(114\) −220.000 −0.180745
\(115\) 0 0
\(116\) 770.000 0.616316
\(117\) 644.000 0.508870
\(118\) 810.000 0.631920
\(119\) −378.000 −0.291187
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) −488.000 −0.362143
\(123\) 364.000 0.266836
\(124\) −84.0000 −0.0608341
\(125\) 0 0
\(126\) −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
\(130\) 0 0
\(131\) −1118.00 −0.745650 −0.372825 0.927902i \(-0.621611\pi\)
−0.372825 + 0.927902i \(0.621611\pi\)
\(132\) 112.000 0.0738511
\(133\) −770.000 −0.502011
\(134\) −244.000 −0.157301
\(135\) 0 0
\(136\) 810.000 0.510713
\(137\) −2274.00 −1.41811 −0.709054 0.705154i \(-0.750878\pi\)
−0.709054 + 0.705154i \(0.750878\pi\)
\(138\) −96.0000 −0.0592178
\(139\) −210.000 −0.128144 −0.0640718 0.997945i \(-0.520409\pi\)
−0.0640718 + 0.997945i \(0.520409\pi\)
\(140\) 0 0
\(141\) −648.000 −0.387032
\(142\) −768.000 −0.453867
\(143\) 224.000 0.130992
\(144\) −943.000 −0.545718
\(145\) 0 0
\(146\) 702.000 0.397931
\(147\) 98.0000 0.0549857
\(148\) −1722.00 −0.956402
\(149\) −2010.00 −1.10514 −0.552569 0.833467i \(-0.686352\pi\)
−0.552569 + 0.833467i \(0.686352\pi\)
\(150\) 0 0
\(151\) 1112.00 0.599293 0.299647 0.954050i \(-0.403131\pi\)
0.299647 + 0.954050i \(0.403131\pi\)
\(152\) 1650.00 0.880478
\(153\) 1242.00 0.656273
\(154\) −56.0000 −0.0293027
\(155\) 0 0
\(156\) 392.000 0.201187
\(157\) −124.000 −0.0630336 −0.0315168 0.999503i \(-0.510034\pi\)
−0.0315168 + 0.999503i \(0.510034\pi\)
\(158\) 440.000 0.221548
\(159\) 324.000 0.161603
\(160\) 0 0
\(161\) −336.000 −0.164475
\(162\) 421.000 0.204178
\(163\) −2008.00 −0.964900 −0.482450 0.875924i \(-0.660253\pi\)
−0.482450 + 0.875924i \(0.660253\pi\)
\(164\) −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
\(170\) 0 0
\(171\) 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
\(175\) 0 0
\(176\) −328.000 −0.140477
\(177\) 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 \(0.554764\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 24.0000 0.00946110
\(187\) 432.000 0.168936
\(188\) 2268.00 0.879845
\(189\) −700.000 −0.269405
\(190\) 0 0
\(191\) −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
\(195\) 0 0
\(196\) −343.000 −0.125000
\(197\) −3334.00 −1.20577 −0.602887 0.797826i \(-0.705983\pi\)
−0.602887 + 0.797826i \(0.705983\pi\)
\(198\) 184.000 0.0660420
\(199\) 1860.00 0.662572 0.331286 0.943530i \(-0.392517\pi\)
0.331286 + 0.943530i \(0.392517\pi\)
\(200\) 0 0
\(201\) −488.000 −0.171248
\(202\) −688.000 −0.239641
\(203\) −770.000 −0.266224
\(204\) 756.000 0.259464
\(205\) 0 0
\(206\) −1388.00 −0.469449
\(207\) 1104.00 0.370692
\(208\) −1148.00 −0.382690
\(209\) 880.000 0.291248
\(210\) 0 0
\(211\) −4268.00 −1.39252 −0.696259 0.717791i \(-0.745153\pi\)
−0.696259 + 0.717791i \(0.745153\pi\)
\(212\) −1134.00 −0.367375
\(213\) −1536.00 −0.494108
\(214\) −244.000 −0.0779416
\(215\) 0 0
\(216\) 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 \(-0.196412\pi\)
0.815591 + 0.578629i \(0.196412\pi\)
\(224\) 1127.00 0.336165
\(225\) 0 0
\(226\) −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 \(0.641474\pi\)
\(230\) 0 0
\(231\) −112.000 −0.0319007
\(232\) 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
\(235\) 0 0
\(236\) −5670.00 −1.56392
\(237\) 880.000 0.241190
\(238\) −378.000 −0.102950
\(239\) 4440.00 1.20167 0.600836 0.799372i \(-0.294834\pi\)
0.600836 + 0.799372i \(0.294834\pi\)
\(240\) 0 0
\(241\) 3302.00 0.882575 0.441287 0.897366i \(-0.354522\pi\)
0.441287 + 0.897366i \(0.354522\pi\)
\(242\) −1267.00 −0.336553
\(243\) 3542.00 0.935059
\(244\) 3416.00 0.896258
\(245\) 0 0
\(246\) 364.000 0.0943406
\(247\) 3080.00 0.793424
\(248\) −180.000 −0.0460888
\(249\) 2604.00 0.662738
\(250\) 0 0
\(251\) 1582.00 0.397829 0.198914 0.980017i \(-0.436258\pi\)
0.198914 + 0.980017i \(0.436258\pi\)
\(252\) 1127.00 0.281724
\(253\) 384.000 0.0954224
\(254\) 1776.00 0.438725
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) −2354.00 −0.571356 −0.285678 0.958326i \(-0.592219\pi\)
−0.285678 + 0.958326i \(0.592219\pi\)
\(258\) −256.000 −0.0617747
\(259\) 1722.00 0.413127
\(260\) 0 0
\(261\) 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 \(0.350028\pi\)
\(264\) 240.000 0.0559507
\(265\) 0 0
\(266\) −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 \(0.493506\pi\)
\(270\) 0 0
\(271\) 2032.00 0.455480 0.227740 0.973722i \(-0.426866\pi\)
0.227740 + 0.973722i \(0.426866\pi\)
\(272\) −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
\(280\) 0 0
\(281\) 842.000 0.178753 0.0893764 0.995998i \(-0.471513\pi\)
0.0893764 + 0.995998i \(0.471513\pi\)
\(282\) −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
\(285\) 0 0
\(286\) 224.000 0.0463126
\(287\) 1274.00 0.262027
\(288\) −3703.00 −0.757644
\(289\) −1997.00 −0.406473
\(290\) 0 0
\(291\) −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
\(295\) 0 0
\(296\) −3690.00 −0.724584
\(297\) 800.000 0.156299
\(298\) −2010.00 −0.390725
\(299\) 1344.00 0.259952
\(300\) 0 0
\(301\) −896.000 −0.171577
\(302\) 1112.00 0.211882
\(303\) −1376.00 −0.260888
\(304\) −4510.00 −0.850876
\(305\) 0 0
\(306\) 1242.00 0.232027
\(307\) −2674.00 −0.497112 −0.248556 0.968618i \(-0.579956\pi\)
−0.248556 + 0.968618i \(0.579956\pi\)
\(308\) 392.000 0.0725204
\(309\) −2776.00 −0.511072
\(310\) 0 0
\(311\) −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 \(-0.570649\pi\)
−0.220134 + 0.975470i \(0.570649\pi\)
\(314\) −124.000 −0.0222857
\(315\) 0 0
\(316\) −3080.00 −0.548302
\(317\) 3186.00 0.564491 0.282245 0.959342i \(-0.408921\pi\)
0.282245 + 0.959342i \(0.408921\pi\)
\(318\) 324.000 0.0571353
\(319\) 880.000 0.154453
\(320\) 0 0
\(321\) −488.000 −0.0848520
\(322\) −336.000 −0.0581508
\(323\) 5940.00 1.02325
\(324\) −2947.00 −0.505316
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 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
\(340\) 0 0
\(341\) −96.0000 −0.0152454
\(342\) 2530.00 0.400020
\(343\) 343.000 0.0539949
\(344\) 1920.00 0.300929
\(345\) 0 0
\(346\) −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 \(-0.630035\pi\)
−0.397248 + 0.917711i \(0.630035\pi\)
\(350\) 0 0
\(351\) 2800.00 0.425792
\(352\) −1288.00 −0.195030
\(353\) −12178.0 −1.83617 −0.918087 0.396379i \(-0.870267\pi\)
−0.918087 + 0.396379i \(0.870267\pi\)
\(354\) 1620.00 0.243226
\(355\) 0 0
\(356\) −5110.00 −0.760757
\(357\) −756.000 −0.112078
\(358\) −820.000 −0.121057
\(359\) 440.000 0.0646861 0.0323431 0.999477i \(-0.489703\pi\)
0.0323431 + 0.999477i \(0.489703\pi\)
\(360\) 0 0
\(361\) 5241.00 0.764106
\(362\) 3892.00 0.565080
\(363\) −2534.00 −0.366393
\(364\) 1372.00 0.197561
\(365\) 0 0
\(366\) −976.000 −0.139389
\(367\) 9816.00 1.39616 0.698080 0.716019i \(-0.254038\pi\)
0.698080 + 0.716019i \(0.254038\pi\)
\(368\) −1968.00 −0.278775
\(369\) −4186.00 −0.590554
\(370\) 0 0
\(371\) 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 \(0.490233\pi\)
\(374\) 432.000 0.0597278
\(375\) 0 0
\(376\) 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 \(0.586479\pi\)
\(380\) 0 0
\(381\) 3552.00 0.477623
\(382\) −5048.00 −0.676121
\(383\) −6708.00 −0.894942 −0.447471 0.894298i \(-0.647675\pi\)
−0.447471 + 0.894298i \(0.647675\pi\)
\(384\) −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 \(0.835891\pi\)
\(390\) 0 0
\(391\) 2592.00 0.335251
\(392\) −735.000 −0.0947018
\(393\) −2236.00 −0.287001
\(394\) −3334.00 −0.426306
\(395\) 0 0
\(396\) −1288.00 −0.163446
\(397\) 1356.00 0.171425 0.0857125 0.996320i \(-0.472683\pi\)
0.0857125 + 0.996320i \(0.472683\pi\)
\(398\) 1860.00 0.234255
\(399\) −1540.00 −0.193224
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) −770.000 −0.0941243
\(407\) −1968.00 −0.239681
\(408\) 1620.00 0.196573
\(409\) 5150.00 0.622619 0.311309 0.950309i \(-0.399232\pi\)
0.311309 + 0.950309i \(0.399232\pi\)
\(410\) 0 0
\(411\) −4548.00 −0.545830
\(412\) 9716.00 1.16183
\(413\) 5670.00 0.675551
\(414\) 1104.00 0.131060
\(415\) 0 0
\(416\) −4508.00 −0.531305
\(417\) −420.000 −0.0493225
\(418\) 880.000 0.102972
\(419\) 2310.00 0.269334 0.134667 0.990891i \(-0.457004\pi\)
0.134667 + 0.990891i \(0.457004\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) −1536.00 −0.174694
\(427\) −3416.00 −0.387147
\(428\) 1708.00 0.192896
\(429\) 448.000 0.0504188
\(430\) 0 0
\(431\) −4488.00 −0.501576 −0.250788 0.968042i \(-0.580690\pi\)
−0.250788 + 0.968042i \(0.580690\pi\)
\(432\) −4100.00 −0.456623
\(433\) −17038.0 −1.89098 −0.945490 0.325652i \(-0.894416\pi\)
−0.945490 + 0.325652i \(0.894416\pi\)
\(434\) 84.0000 0.00929062
\(435\) 0 0
\(436\) −630.000 −0.0692008
\(437\) 5280.00 0.577979
\(438\) 1404.00 0.153164
\(439\) 16200.0 1.76124 0.880619 0.473824i \(-0.157127\pi\)
0.880619 + 0.473824i \(0.157127\pi\)
\(440\) 0 0
\(441\) −1127.00 −0.121693
\(442\) 1512.00 0.162712
\(443\) 8772.00 0.940791 0.470395 0.882456i \(-0.344111\pi\)
0.470395 + 0.882456i \(0.344111\pi\)
\(444\) −3444.00 −0.368119
\(445\) 0 0
\(446\) 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 \(0.464294\pi\)
\(450\) 0 0
\(451\) −1456.00 −0.152019
\(452\) 9226.00 0.960076
\(453\) 2224.00 0.230668
\(454\) 2046.00 0.211506
\(455\) 0 0
\(456\) 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
\(460\) 0 0
\(461\) −9268.00 −0.936342 −0.468171 0.883638i \(-0.655087\pi\)
−0.468171 + 0.883638i \(0.655087\pi\)
\(462\) −112.000 −0.0112786
\(463\) 9392.00 0.942728 0.471364 0.881939i \(-0.343762\pi\)
0.471364 + 0.881939i \(0.343762\pi\)
\(464\) −4510.00 −0.451232
\(465\) 0 0
\(466\) −4458.00 −0.443161
\(467\) 10806.0 1.07075 0.535377 0.844613i \(-0.320170\pi\)
0.535377 + 0.844613i \(0.320170\pi\)
\(468\) −4508.00 −0.445261
\(469\) −1708.00 −0.168162
\(470\) 0 0
\(471\) −248.000 −0.0242616
\(472\) −12150.0 −1.18485
\(473\) 1024.00 0.0995424
\(474\) 880.000 0.0852737
\(475\) 0 0
\(476\) 2646.00 0.254788
\(477\) −3726.00 −0.357656
\(478\) 4440.00 0.424855
\(479\) 4940.00 0.471220 0.235610 0.971848i \(-0.424291\pi\)
0.235610 + 0.971848i \(0.424291\pi\)
\(480\) 0 0
\(481\) −6888.00 −0.652943
\(482\) 3302.00 0.312037
\(483\) −672.000 −0.0633065
\(484\) 8869.00 0.832926
\(485\) 0 0
\(486\) 3542.00 0.330593
\(487\) 5216.00 0.485338 0.242669 0.970109i \(-0.421977\pi\)
0.242669 + 0.970109i \(0.421977\pi\)
\(488\) 7320.00 0.679018
\(489\) −4016.00 −0.371390
\(490\) 0 0
\(491\) 4412.00 0.405521 0.202760 0.979228i \(-0.435009\pi\)
0.202760 + 0.979228i \(0.435009\pi\)
\(492\) −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 \(0.173586\pi\)
\(500\) 0 0
\(501\) −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
\(505\) 0 0
\(506\) 384.000 0.0337369
\(507\) −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 \(0.576975\pi\)
\(510\) 0 0
\(511\) 4914.00 0.425406
\(512\) 11521.0 0.994455
\(513\) 11000.0 0.946709
\(514\) −2354.00 −0.202005
\(515\) 0 0
\(516\) 1792.00 0.152884
\(517\) 2592.00 0.220495
\(518\) 1722.00 0.146062
\(519\) −4456.00 −0.376872
\(520\) 0 0
\(521\) −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
\(525\) 0 0
\(526\) 3872.00 0.320964
\(527\) −648.000 −0.0535623
\(528\) −656.000 −0.0540696
\(529\) −9863.00 −0.810635
\(530\) 0 0
\(531\) −18630.0 −1.52255
\(532\) 5390.00 0.439260
\(533\) −5096.00 −0.414132
\(534\) 1460.00 0.118315
\(535\) 0 0
\(536\) 3660.00 0.294940
\(537\) −1640.00 −0.131790
\(538\) 180.000 0.0144244
\(539\) −392.000 −0.0313259
\(540\) 0 0
\(541\) −1618.00 −0.128583 −0.0642914 0.997931i \(-0.520479\pi\)
−0.0642914 + 0.997931i \(0.520479\pi\)
\(542\) 2032.00 0.161037
\(543\) 7784.00 0.615181
\(544\) −8694.00 −0.685206
\(545\) 0 0
\(546\) −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
\(555\) 0 0
\(556\) 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
\(560\) 0 0
\(561\) 864.000 0.0650234
\(562\) 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
\(565\) 0 0
\(566\) 3782.00 0.280865
\(567\) 2947.00 0.218276
\(568\) 11520.0 0.851001
\(569\) −5930.00 −0.436904 −0.218452 0.975848i \(-0.570101\pi\)
−0.218452 + 0.975848i \(0.570101\pi\)
\(570\) 0 0
\(571\) −19048.0 −1.39603 −0.698016 0.716082i \(-0.745933\pi\)
−0.698016 + 0.716082i \(0.745933\pi\)
\(572\) −1568.00 −0.114618
\(573\) −10096.0 −0.736067
\(574\) 1274.00 0.0926406
\(575\) 0 0
\(576\) 3841.00 0.277850
\(577\) 14366.0 1.03651 0.518253 0.855227i \(-0.326582\pi\)
0.518253 + 0.855227i \(0.326582\pi\)
\(578\) −1997.00 −0.143710
\(579\) 5924.00 0.425204
\(580\) 0 0
\(581\) 9114.00 0.650796
\(582\) −588.000 −0.0418787
\(583\) −1296.00 −0.0920666
\(584\) −10530.0 −0.746121
\(585\) 0 0
\(586\) 4312.00 0.303971
\(587\) 3626.00 0.254959 0.127480 0.991841i \(-0.459311\pi\)
0.127480 + 0.991841i \(0.459311\pi\)
\(588\) −686.000 −0.0481125
\(589\) −1320.00 −0.0923424
\(590\) 0 0
\(591\) −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 \(0.488293\pi\)
\(594\) 800.000 0.0552599
\(595\) 0 0
\(596\) 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 \(0.613098\pi\)
\(600\) 0 0
\(601\) −25158.0 −1.70751 −0.853757 0.520671i \(-0.825682\pi\)
−0.853757 + 0.520671i \(0.825682\pi\)
\(602\) −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
\(610\) 0 0
\(611\) 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
\(615\) 0 0
\(616\) 840.000 0.0549425
\(617\) −17334.0 −1.13102 −0.565511 0.824741i \(-0.691321\pi\)
−0.565511 + 0.824741i \(0.691321\pi\)
\(618\) −2776.00 −0.180691
\(619\) 18730.0 1.21619 0.608096 0.793864i \(-0.291934\pi\)
0.608096 + 0.793864i \(0.291934\pi\)
\(620\) 0 0
\(621\) 4800.00 0.310173
\(622\) −3768.00 −0.242899
\(623\) 5110.00 0.328616
\(624\) −2296.00 −0.147297
\(625\) 0 0
\(626\) −2438.00 −0.155658
\(627\) 1760.00 0.112101
\(628\) 868.000 0.0551544
\(629\) −13284.0 −0.842079
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) −2268.00 −0.141403
\(637\) −1372.00 −0.0853385
\(638\) 880.000 0.0546074
\(639\) 17664.0 1.09355
\(640\) 0 0
\(641\) 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 \(-0.546216\pi\)
−0.144681 + 0.989478i \(0.546216\pi\)
\(644\) 2352.00 0.143916
\(645\) 0 0
\(646\) 5940.00 0.361774
\(647\) 21436.0 1.30253 0.651264 0.758851i \(-0.274239\pi\)
0.651264 + 0.758851i \(0.274239\pi\)
\(648\) −6315.00 −0.382834
\(649\) −6480.00 −0.391930
\(650\) 0 0
\(651\) 168.000 0.0101143
\(652\) 14056.0 0.844287
\(653\) −4458.00 −0.267159 −0.133580 0.991038i \(-0.542647\pi\)
−0.133580 + 0.991038i \(0.542647\pi\)
\(654\) 180.000 0.0107623
\(655\) 0 0
\(656\) 7462.00 0.444119
\(657\) −16146.0 −0.958775
\(658\) −2268.00 −0.134371
\(659\) −26640.0 −1.57473 −0.787365 0.616487i \(-0.788555\pi\)
−0.787365 + 0.616487i \(0.788555\pi\)
\(660\) 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
\(665\) 0 0
\(666\) −5658.00 −0.329194
\(667\) 5280.00 0.306510
\(668\) 20188.0 1.16931
\(669\) 10864.0 0.627842
\(670\) 0 0
\(671\) 3904.00 0.224608
\(672\) 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
\(675\) 0 0
\(676\) 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
\(680\) 0 0
\(681\) 4092.00 0.230258
\(682\) −96.0000 −0.00539007
\(683\) −18108.0 −1.01447 −0.507235 0.861808i \(-0.669332\pi\)
−0.507235 + 0.861808i \(0.669332\pi\)
\(684\) −17710.0 −0.989998
\(685\) 0 0
\(686\) 343.000 0.0190901
\(687\) −5960.00 −0.330987
\(688\) −5248.00 −0.290811
\(689\) −4536.00 −0.250810
\(690\) 0 0
\(691\) −10078.0 −0.554827 −0.277413 0.960751i \(-0.589477\pi\)
−0.277413 + 0.960751i \(0.589477\pi\)
\(692\) 15596.0 0.856750
\(693\) 1288.00 0.0706018
\(694\) −9344.00 −0.511086
\(695\) 0 0
\(696\) 3300.00 0.179722
\(697\) −9828.00 −0.534092
\(698\) −5180.00 −0.280897
\(699\) −8916.00 −0.482452
\(700\) 0 0
\(701\) 18762.0 1.01089 0.505443 0.862860i \(-0.331329\pi\)
0.505443 + 0.862860i \(0.331329\pi\)
\(702\) 2800.00 0.150540
\(703\) −27060.0 −1.45176
\(704\) 1336.00 0.0715233
\(705\) 0 0
\(706\) −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 \(0.442273\pi\)
\(710\) 0 0
\(711\) −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 \(0.459773\pi\)
\(720\) 0 0
\(721\) −9716.00 −0.501862
\(722\) 5241.00 0.270152
\(723\) 6604.00 0.339703
\(724\) −27244.0 −1.39850
\(725\) 0 0
\(726\) −2534.00 −0.129539
\(727\) 13636.0 0.695641 0.347821 0.937561i \(-0.386922\pi\)
0.347821 + 0.937561i \(0.386922\pi\)
\(728\) 2940.00 0.149675
\(729\) −4283.00 −0.217599
\(730\) 0 0
\(731\) 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
\(735\) 0 0
\(736\) −7728.00 −0.387035
\(737\) 1952.00 0.0975615
\(738\) −4186.00 −0.208792
\(739\) −5160.00 −0.256852 −0.128426 0.991719i \(-0.540992\pi\)
−0.128426 + 0.991719i \(0.540992\pi\)
\(740\) 0 0
\(741\) 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
\(745\) 0 0
\(746\) 442.000 0.0216927
\(747\) −29946.0 −1.46676
\(748\) −3024.00 −0.147819
\(749\) −1708.00 −0.0833230
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 4900.00 0.235729
\(757\) −21674.0 −1.04063 −0.520314 0.853975i \(-0.674185\pi\)
−0.520314 + 0.853975i \(0.674185\pi\)
\(758\) −3960.00 −0.189754
\(759\) 768.000 0.0367281
\(760\) 0 0
\(761\) 7422.00 0.353544 0.176772 0.984252i \(-0.443434\pi\)
0.176772 + 0.984252i \(0.443434\pi\)
\(762\) 3552.00 0.168865
\(763\) 630.000 0.0298919
\(764\) 35336.0 1.67331
\(765\) 0 0
\(766\) −6708.00 −0.316410
\(767\) −22680.0 −1.06770
\(768\) −238.000 −0.0111824
\(769\) 13790.0 0.646658 0.323329 0.946287i \(-0.395198\pi\)
0.323329 + 0.946287i \(0.395198\pi\)
\(770\) 0 0
\(771\) −4708.00 −0.219915
\(772\) −20734.0 −0.966623
\(773\) 6232.00 0.289973 0.144987 0.989434i \(-0.453686\pi\)
0.144987 + 0.989434i \(0.453686\pi\)
\(774\) 2944.00 0.136718
\(775\) 0 0
\(776\) 4410.00 0.204007
\(777\) 3444.00 0.159013
\(778\) −13350.0 −0.615194
\(779\) −20020.0 −0.920784
\(780\) 0 0
\(781\) 6144.00 0.281498
\(782\) 2592.00 0.118529
\(783\) 11000.0 0.502054
\(784\) 2009.00 0.0915179
\(785\) 0 0
\(786\) −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
\(790\) 0 0
\(791\) −9226.00 −0.414714
\(792\) −2760.00 −0.123829
\(793\) 13664.0 0.611883
\(794\) 1356.00 0.0606079
\(795\) 0 0
\(796\) −13020.0 −0.579751
\(797\) −1204.00 −0.0535105 −0.0267552 0.999642i \(-0.508517\pi\)
−0.0267552 + 0.999642i \(0.508517\pi\)
\(798\) −1540.00 −0.0683150
\(799\) 17496.0 0.774673
\(800\) 0 0
\(801\) −16790.0 −0.740631
\(802\) 6222.00 0.273948
\(803\) −5616.00 −0.246805
\(804\) 3416.00 0.149842
\(805\) 0 0
\(806\) −336.000 −0.0146837
\(807\) 360.000 0.0157033
\(808\) 10320.0 0.449327
\(809\) −7050.00 −0.306384 −0.153192 0.988196i \(-0.548955\pi\)
−0.153192 + 0.988196i \(0.548955\pi\)
\(810\) 0 0
\(811\) 23282.0 1.00807 0.504033 0.863684i \(-0.331849\pi\)
0.504033 + 0.863684i \(0.331849\pi\)
\(812\) 5390.00 0.232946
\(813\) 4064.00 0.175315
\(814\) −1968.00 −0.0847400
\(815\) 0 0
\(816\) −4428.00 −0.189964
\(817\) 14080.0 0.602934
\(818\) 5150.00 0.220129
\(819\) 4508.00 0.192335
\(820\) 0 0
\(821\) 10142.0 0.431131 0.215565 0.976489i \(-0.430841\pi\)
0.215565 + 0.976489i \(0.430841\pi\)
\(822\) −4548.00 −0.192980
\(823\) 9192.00 0.389323 0.194662 0.980870i \(-0.437639\pi\)
0.194662 + 0.980870i \(0.437639\pi\)
\(824\) 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 \(0.424343\pi\)
\(830\) 0 0
\(831\) 10852.0 0.453010
\(832\) 4676.00 0.194845
\(833\) −2646.00 −0.110058
\(834\) −420.000 −0.0174381
\(835\) 0 0
\(836\) −6160.00 −0.254842
\(837\) −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 \(0.495416\pi\)
\(840\) 0 0
\(841\) −12289.0 −0.503875
\(842\) 1262.00 0.0516525
\(843\) 1684.00 0.0688019
\(844\) 29876.0 1.21845
\(845\) 0 0
\(846\) 7452.00 0.302843
\(847\) −8869.00 −0.359790
\(848\) 6642.00 0.268971
\(849\) 7564.00 0.305767
\(850\) 0 0
\(851\) −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
\(855\) 0 0
\(856\) 3660.00 0.146140
\(857\) −28894.0 −1.15169 −0.575846 0.817558i \(-0.695327\pi\)
−0.575846 + 0.817558i \(0.695327\pi\)
\(858\) 448.000 0.0178257
\(859\) −2770.00 −0.110025 −0.0550123 0.998486i \(-0.517520\pi\)
−0.0550123 + 0.998486i \(0.517520\pi\)
\(860\) 0 0
\(861\) 2548.00 0.100854
\(862\) −4488.00 −0.177334
\(863\) −17688.0 −0.697690 −0.348845 0.937180i \(-0.613426\pi\)
−0.348845 + 0.937180i \(0.613426\pi\)
\(864\) −16100.0 −0.633950
\(865\) 0 0
\(866\) −17038.0 −0.668562
\(867\) −3994.00 −0.156451
\(868\) −588.000 −0.0229931
\(869\) −3520.00 −0.137408
\(870\) 0 0
\(871\) 6832.00 0.265779
\(872\) −1350.00 −0.0524275
\(873\) 6762.00 0.262152
\(874\) 5280.00 0.204346
\(875\) 0 0
\(876\) −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 \(-0.570127\pi\)
−0.218533 + 0.975830i \(0.570127\pi\)
\(884\) −10584.0 −0.402691
\(885\) 0 0
\(886\) 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
\(890\) 0 0
\(891\) −3368.00 −0.126636
\(892\) −38024.0 −1.42728
\(893\) 35640.0 1.33555
\(894\) −4020.00 −0.150390
\(895\) 0 0
\(896\) −10185.0 −0.379751
\(897\) 2688.00 0.100055
\(898\) 2130.00 0.0791526
\(899\) −1320.00 −0.0489705
\(900\) 0 0
\(901\) −8748.00 −0.323461
\(902\) −1456.00 −0.0537467
\(903\) −1792.00 −0.0660399
\(904\) 19770.0 0.727368
\(905\) 0 0
\(906\) 2224.00 0.0815535
\(907\) 8716.00 0.319085 0.159542 0.987191i \(-0.448998\pi\)
0.159542 + 0.987191i \(0.448998\pi\)
\(908\) −14322.0 −0.523450
\(909\) 15824.0 0.577392
\(910\) 0 0
\(911\) 7632.00 0.277563 0.138781 0.990323i \(-0.455682\pi\)
0.138781 + 0.990323i \(0.455682\pi\)
\(912\) −9020.00 −0.327502
\(913\) −10416.0 −0.377568
\(914\) −10534.0 −0.381219
\(915\) 0 0
\(916\) 20860.0 0.752439
\(917\) −7826.00 −0.281829
\(918\) 5400.00 0.194147
\(919\) −23080.0 −0.828443 −0.414221 0.910176i \(-0.635946\pi\)
−0.414221 + 0.910176i \(0.635946\pi\)
\(920\) 0 0
\(921\) −5348.00 −0.191338
\(922\) −9268.00 −0.331047
\(923\) 21504.0 0.766861
\(924\) 784.000 0.0279131
\(925\) 0 0
\(926\) 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 \(0.206650\pi\)
\(930\) 0 0
\(931\) −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
\(940\) 0 0
\(941\) 43832.0 1.51847 0.759236 0.650815i \(-0.225573\pi\)
0.759236 + 0.650815i \(0.225573\pi\)
\(942\) −248.000 −0.00857779
\(943\) −8736.00 −0.301679
\(944\) 33210.0 1.14501
\(945\) 0 0
\(946\) 1024.00 0.0351936
\(947\) 736.000 0.0252553 0.0126277 0.999920i \(-0.495980\pi\)
0.0126277 + 0.999920i \(0.495980\pi\)
\(948\) −6160.00 −0.211042
\(949\) −19656.0 −0.672351
\(950\) 0 0
\(951\) 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
\(955\) 0 0
\(956\) −31080.0 −1.05146
\(957\) 1760.00 0.0594490
\(958\) 4940.00 0.166601
\(959\) −15918.0 −0.535995
\(960\) 0 0
\(961\) −29647.0 −0.995166
\(962\) −6888.00 −0.230850
\(963\) 5612.00 0.187792
\(964\) −23114.0 −0.772253
\(965\) 0 0
\(966\) −672.000 −0.0223822
\(967\) −26224.0 −0.872086 −0.436043 0.899926i \(-0.643620\pi\)
−0.436043 + 0.899926i \(0.643620\pi\)
\(968\) 19005.0 0.631037
\(969\) 11880.0 0.393850
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) −20008.0 −0.656189
\(977\) −38394.0 −1.25725 −0.628625 0.777709i \(-0.716382\pi\)
−0.628625 + 0.777709i \(0.716382\pi\)
\(978\) −4016.00 −0.131306
\(979\) −5840.00 −0.190651
\(980\) 0 0
\(981\) −2070.00 −0.0673700
\(982\) 4412.00 0.143373
\(983\) −5388.00 −0.174822 −0.0874112 0.996172i \(-0.527859\pi\)
−0.0874112 + 0.996172i \(0.527859\pi\)
\(984\) −5460.00 −0.176889
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) −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 175.4.a.b.1.1 1
3.2 odd 2 1575.4.a.e.1.1 1
5.2 odd 4 175.4.b.b.99.2 2
5.3 odd 4 175.4.b.b.99.1 2
5.4 even 2 7.4.a.a.1.1 1
7.6 odd 2 1225.4.a.j.1.1 1
15.14 odd 2 63.4.a.b.1.1 1
20.19 odd 2 112.4.a.f.1.1 1
35.4 even 6 49.4.c.c.30.1 2
35.9 even 6 49.4.c.c.18.1 2
35.19 odd 6 49.4.c.b.18.1 2
35.24 odd 6 49.4.c.b.30.1 2
35.34 odd 2 49.4.a.b.1.1 1
40.19 odd 2 448.4.a.e.1.1 1
40.29 even 2 448.4.a.i.1.1 1
55.54 odd 2 847.4.a.b.1.1 1
60.59 even 2 1008.4.a.c.1.1 1
65.64 even 2 1183.4.a.b.1.1 1
85.84 even 2 2023.4.a.a.1.1 1
105.44 odd 6 441.4.e.h.361.1 2
105.59 even 6 441.4.e.e.226.1 2
105.74 odd 6 441.4.e.h.226.1 2
105.89 even 6 441.4.e.e.361.1 2
105.104 even 2 441.4.a.i.1.1 1
140.139 even 2 784.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.a.a.1.1 1 5.4 even 2
49.4.a.b.1.1 1 35.34 odd 2
49.4.c.b.18.1 2 35.19 odd 6
49.4.c.b.30.1 2 35.24 odd 6
49.4.c.c.18.1 2 35.9 even 6
49.4.c.c.30.1 2 35.4 even 6
63.4.a.b.1.1 1 15.14 odd 2
112.4.a.f.1.1 1 20.19 odd 2
175.4.a.b.1.1 1 1.1 even 1 trivial
175.4.b.b.99.1 2 5.3 odd 4
175.4.b.b.99.2 2 5.2 odd 4
441.4.a.i.1.1 1 105.104 even 2
441.4.e.e.226.1 2 105.59 even 6
441.4.e.e.361.1 2 105.89 even 6
441.4.e.h.226.1 2 105.74 odd 6
441.4.e.h.361.1 2 105.44 odd 6
448.4.a.e.1.1 1 40.19 odd 2
448.4.a.i.1.1 1 40.29 even 2
784.4.a.g.1.1 1 140.139 even 2
847.4.a.b.1.1 1 55.54 odd 2
1008.4.a.c.1.1 1 60.59 even 2
1183.4.a.b.1.1 1 65.64 even 2
1225.4.a.j.1.1 1 7.6 odd 2
1575.4.a.e.1.1 1 3.2 odd 2
2023.4.a.a.1.1 1 85.84 even 2