Properties

Label 1575.4.a.e.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
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 \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -7.00000 q^{4} +7.00000 q^{7} +15.0000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -7.00000 q^{4} +7.00000 q^{7} +15.0000 q^{8} +8.00000 q^{11} -28.0000 q^{13} -7.00000 q^{14} +41.0000 q^{16} +54.0000 q^{17} -110.000 q^{19} -8.00000 q^{22} +48.0000 q^{23} +28.0000 q^{26} -49.0000 q^{28} +110.000 q^{29} +12.0000 q^{31} -161.000 q^{32} -54.0000 q^{34} +246.000 q^{37} +110.000 q^{38} -182.000 q^{41} -128.000 q^{43} -56.0000 q^{44} -48.0000 q^{46} +324.000 q^{47} +49.0000 q^{49} +196.000 q^{52} -162.000 q^{53} +105.000 q^{56} -110.000 q^{58} -810.000 q^{59} -488.000 q^{61} -12.0000 q^{62} -167.000 q^{64} -244.000 q^{67} -378.000 q^{68} +768.000 q^{71} +702.000 q^{73} -246.000 q^{74} +770.000 q^{76} +56.0000 q^{77} +440.000 q^{79} +182.000 q^{82} -1302.00 q^{83} +128.000 q^{86} +120.000 q^{88} -730.000 q^{89} -196.000 q^{91} -336.000 q^{92} -324.000 q^{94} -294.000 q^{97} -49.0000 q^{98} +O(q^{100})\)

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 \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 15.0000 0.662913
\(9\) 0 0
\(10\) 0 0
\(11\) 8.00000 0.219281 0.109640 0.993971i \(-0.465030\pi\)
0.109640 + 0.993971i \(0.465030\pi\)
\(12\) 0 0
\(13\) −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 \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 0 0
\(19\) −110.000 −1.32820 −0.664098 0.747645i \(-0.731184\pi\)
−0.664098 + 0.747645i \(0.731184\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.00000 −0.0775275
\(23\) 48.0000 0.435161 0.217580 0.976042i \(-0.430184\pi\)
0.217580 + 0.976042i \(0.430184\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 28.0000 0.211202
\(27\) 0 0
\(28\) −49.0000 −0.330719
\(29\) 110.000 0.704362 0.352181 0.935932i \(-0.385440\pi\)
0.352181 + 0.935932i \(0.385440\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −54.0000 −0.272380
\(35\) 0 0
\(36\) 0 0
\(37\) 246.000 1.09303 0.546516 0.837449i \(-0.315954\pi\)
0.546516 + 0.837449i \(0.315954\pi\)
\(38\) 110.000 0.469588
\(39\) 0 0
\(40\) 0 0
\(41\) −182.000 −0.693259 −0.346630 0.938002i \(-0.612674\pi\)
−0.346630 + 0.938002i \(0.612674\pi\)
\(42\) 0 0
\(43\) −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 \(-0.332315\pi\)
0.502769 + 0.864421i \(0.332315\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 196.000 0.522698
\(53\) −162.000 −0.419857 −0.209928 0.977717i \(-0.567323\pi\)
−0.209928 + 0.977717i \(0.567323\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 105.000 0.250557
\(57\) 0 0
\(58\) −110.000 −0.249029
\(59\) −810.000 −1.78734 −0.893670 0.448725i \(-0.851878\pi\)
−0.893670 + 0.448725i \(0.851878\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) −244.000 −0.444916 −0.222458 0.974942i \(-0.571408\pi\)
−0.222458 + 0.974942i \(0.571408\pi\)
\(68\) −378.000 −0.674106
\(69\) 0 0
\(70\) 0 0
\(71\) 768.000 1.28373 0.641865 0.766818i \(-0.278161\pi\)
0.641865 + 0.766818i \(0.278161\pi\)
\(72\) 0 0
\(73\) 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
\(78\) 0 0
\(79\) 440.000 0.626631 0.313316 0.949649i \(-0.398560\pi\)
0.313316 + 0.949649i \(0.398560\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 182.000 0.245104
\(83\) −1302.00 −1.72184 −0.860922 0.508737i \(-0.830113\pi\)
−0.860922 + 0.508737i \(0.830113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 128.000 0.160495
\(87\) 0 0
\(88\) 120.000 0.145364
\(89\) −730.000 −0.869436 −0.434718 0.900567i \(-0.643152\pi\)
−0.434718 + 0.900567i \(0.643152\pi\)
\(90\) 0 0
\(91\) −196.000 −0.225784
\(92\) −336.000 −0.380765
\(93\) 0 0
\(94\) −324.000 −0.355511
\(95\) 0 0
\(96\) 0 0
\(97\) −294.000 −0.307744 −0.153872 0.988091i \(-0.549174\pi\)
−0.153872 + 0.988091i \(0.549174\pi\)
\(98\) −49.0000 −0.0505076
\(99\) 0 0
\(100\) 0 0
\(101\) 688.000 0.677808 0.338904 0.940821i \(-0.389944\pi\)
0.338904 + 0.940821i \(0.389944\pi\)
\(102\) 0 0
\(103\) −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 \(-0.464843\pi\)
0.110226 + 0.993907i \(0.464843\pi\)
\(108\) 0 0
\(109\) 90.0000 0.0790866 0.0395433 0.999218i \(-0.487410\pi\)
0.0395433 + 0.999218i \(0.487410\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 287.000 0.242133
\(113\) 1318.00 1.09723 0.548615 0.836075i \(-0.315155\pi\)
0.548615 + 0.836075i \(0.315155\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −770.000 −0.616316
\(117\) 0 0
\(118\) 810.000 0.631920
\(119\) 378.000 0.291187
\(120\) 0 0
\(121\) −1267.00 −0.951916
\(122\) 488.000 0.362143
\(123\) 0 0
\(124\) −84.0000 −0.0608341
\(125\) 0 0
\(126\) 0 0
\(127\) 1776.00 1.24090 0.620451 0.784245i \(-0.286950\pi\)
0.620451 + 0.784245i \(0.286950\pi\)
\(128\) 1455.00 1.00473
\(129\) 0 0
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 810.000 0.510713
\(137\) 2274.00 1.41811 0.709054 0.705154i \(-0.249122\pi\)
0.709054 + 0.705154i \(0.249122\pi\)
\(138\) 0 0
\(139\) −210.000 −0.128144 −0.0640718 0.997945i \(-0.520409\pi\)
−0.0640718 + 0.997945i \(0.520409\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −768.000 −0.453867
\(143\) −224.000 −0.130992
\(144\) 0 0
\(145\) 0 0
\(146\) −702.000 −0.397931
\(147\) 0 0
\(148\) −1722.00 −0.956402
\(149\) 2010.00 1.10514 0.552569 0.833467i \(-0.313648\pi\)
0.552569 + 0.833467i \(0.313648\pi\)
\(150\) 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
\(153\) 0 0
\(154\) −56.0000 −0.0293027
\(155\) 0 0
\(156\) 0 0
\(157\) −124.000 −0.0630336 −0.0315168 0.999503i \(-0.510034\pi\)
−0.0315168 + 0.999503i \(0.510034\pi\)
\(158\) −440.000 −0.221548
\(159\) 0 0
\(160\) 0 0
\(161\) 336.000 0.164475
\(162\) 0 0
\(163\) −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 \(-0.267076\pi\)
0.668176 + 0.744004i \(0.267076\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 0 0
\(172\) 896.000 0.397206
\(173\) 2228.00 0.979143 0.489571 0.871963i \(-0.337153\pi\)
0.489571 + 0.871963i \(0.337153\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 328.000 0.140477
\(177\) 0 0
\(178\) 730.000 0.307392
\(179\) 820.000 0.342400 0.171200 0.985236i \(-0.445236\pi\)
0.171200 + 0.985236i \(0.445236\pi\)
\(180\) 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
\(183\) 0 0
\(184\) 720.000 0.288473
\(185\) 0 0
\(186\) 0 0
\(187\) 432.000 0.168936
\(188\) −2268.00 −0.879845
\(189\) 0 0
\(190\) 0 0
\(191\) 5048.00 1.91236 0.956179 0.292782i \(-0.0945810\pi\)
0.956179 + 0.292782i \(0.0945810\pi\)
\(192\) 0 0
\(193\) 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 \(-0.294017\pi\)
0.602887 + 0.797826i \(0.294017\pi\)
\(198\) 0 0
\(199\) 1860.00 0.662572 0.331286 0.943530i \(-0.392517\pi\)
0.331286 + 0.943530i \(0.392517\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −688.000 −0.239641
\(203\) 770.000 0.266224
\(204\) 0 0
\(205\) 0 0
\(206\) 1388.00 0.469449
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) −244.000 −0.0779416
\(215\) 0 0
\(216\) 0 0
\(217\) 84.0000 0.0262778
\(218\) −90.0000 −0.0279613
\(219\) 0 0
\(220\) 0 0
\(221\) −1512.00 −0.460218
\(222\) 0 0
\(223\) 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 \(-0.596691\pi\)
−0.299114 + 0.954217i \(0.596691\pi\)
\(228\) 0 0
\(229\) −2980.00 −0.859930 −0.429965 0.902846i \(-0.641474\pi\)
−0.429965 + 0.902846i \(0.641474\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1650.00 0.466930
\(233\) 4458.00 1.25345 0.626724 0.779241i \(-0.284395\pi\)
0.626724 + 0.779241i \(0.284395\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5670.00 1.56392
\(237\) 0 0
\(238\) −378.000 −0.102950
\(239\) −4440.00 −1.20167 −0.600836 0.799372i \(-0.705166\pi\)
−0.600836 + 0.799372i \(0.705166\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 3416.00 0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 3080.00 0.793424
\(248\) 180.000 0.0460888
\(249\) 0 0
\(250\) 0 0
\(251\) −1582.00 −0.397829 −0.198914 0.980017i \(-0.563742\pi\)
−0.198914 + 0.980017i \(0.563742\pi\)
\(252\) 0 0
\(253\) 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 \(-0.407781\pi\)
0.285678 + 0.958326i \(0.407781\pi\)
\(258\) 0 0
\(259\) 1722.00 0.413127
\(260\) 0 0
\(261\) 0 0
\(262\) −1118.00 −0.263627
\(263\) −3872.00 −0.907824 −0.453912 0.891046i \(-0.649972\pi\)
−0.453912 + 0.891046i \(0.649972\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 770.000 0.177488
\(267\) 0 0
\(268\) 1708.00 0.389301
\(269\) −180.000 −0.0407985 −0.0203992 0.999792i \(-0.506494\pi\)
−0.0203992 + 0.999792i \(0.506494\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −2274.00 −0.501377
\(275\) 0 0
\(276\) 0 0
\(277\) 5426.00 1.17696 0.588478 0.808513i \(-0.299727\pi\)
0.588478 + 0.808513i \(0.299727\pi\)
\(278\) 210.000 0.0453056
\(279\) 0 0
\(280\) 0 0
\(281\) −842.000 −0.178753 −0.0893764 0.995998i \(-0.528487\pi\)
−0.0893764 + 0.995998i \(0.528487\pi\)
\(282\) 0 0
\(283\) 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
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 0 0
\(291\) 0 0
\(292\) −4914.00 −0.984829
\(293\) −4312.00 −0.859760 −0.429880 0.902886i \(-0.641444\pi\)
−0.429880 + 0.902886i \(0.641444\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3690.00 0.724584
\(297\) 0 0
\(298\) −2010.00 −0.390725
\(299\) −1344.00 −0.259952
\(300\) 0 0
\(301\) −896.000 −0.171577
\(302\) −1112.00 −0.211882
\(303\) 0 0
\(304\) −4510.00 −0.850876
\(305\) 0 0
\(306\) 0 0
\(307\) −2674.00 −0.497112 −0.248556 0.968618i \(-0.579956\pi\)
−0.248556 + 0.968618i \(0.579956\pi\)
\(308\) −392.000 −0.0725204
\(309\) 0 0
\(310\) 0 0
\(311\) 3768.00 0.687021 0.343511 0.939149i \(-0.388384\pi\)
0.343511 + 0.939149i \(0.388384\pi\)
\(312\) 0 0
\(313\) −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 \(-0.591079\pi\)
−0.282245 + 0.959342i \(0.591079\pi\)
\(318\) 0 0
\(319\) 880.000 0.154453
\(320\) 0 0
\(321\) 0 0
\(322\) −336.000 −0.0581508
\(323\) −5940.00 −1.02325
\(324\) 0 0
\(325\) 0 0
\(326\) 2008.00 0.341144
\(327\) 0 0
\(328\) −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
\(333\) 0 0
\(334\) −2884.00 −0.472471
\(335\) 0 0
\(336\) 0 0
\(337\) −814.000 −0.131577 −0.0657884 0.997834i \(-0.520956\pi\)
−0.0657884 + 0.997834i \(0.520956\pi\)
\(338\) 1413.00 0.227388
\(339\) 0 0
\(340\) 0 0
\(341\) 96.0000 0.0152454
\(342\) 0 0
\(343\) 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 \(-0.242862\pi\)
0.722784 + 0.691074i \(0.242862\pi\)
\(348\) 0 0
\(349\) −5180.00 −0.794496 −0.397248 0.917711i \(-0.630035\pi\)
−0.397248 + 0.917711i \(0.630035\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1288.00 −0.195030
\(353\) 12178.0 1.83617 0.918087 0.396379i \(-0.129733\pi\)
0.918087 + 0.396379i \(0.129733\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5110.00 0.760757
\(357\) 0 0
\(358\) −820.000 −0.121057
\(359\) −440.000 −0.0646861 −0.0323431 0.999477i \(-0.510297\pi\)
−0.0323431 + 0.999477i \(0.510297\pi\)
\(360\) 0 0
\(361\) 5241.00 0.764106
\(362\) −3892.00 −0.565080
\(363\) 0 0
\(364\) 1372.00 0.197561
\(365\) 0 0
\(366\) 0 0
\(367\) 9816.00 1.39616 0.698080 0.716019i \(-0.254038\pi\)
0.698080 + 0.716019i \(0.254038\pi\)
\(368\) 1968.00 0.278775
\(369\) 0 0
\(370\) 0 0
\(371\) −1134.00 −0.158691
\(372\) 0 0
\(373\) 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
\(378\) 0 0
\(379\) −3960.00 −0.536706 −0.268353 0.963321i \(-0.586479\pi\)
−0.268353 + 0.963321i \(0.586479\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5048.00 −0.676121
\(383\) 6708.00 0.894942 0.447471 0.894298i \(-0.352325\pi\)
0.447471 + 0.894298i \(0.352325\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2962.00 −0.390575
\(387\) 0 0
\(388\) 2058.00 0.269276
\(389\) 13350.0 1.74003 0.870015 0.493025i \(-0.164109\pi\)
0.870015 + 0.493025i \(0.164109\pi\)
\(390\) 0 0
\(391\) 2592.00 0.335251
\(392\) 735.000 0.0947018
\(393\) 0 0
\(394\) −3334.00 −0.426306
\(395\) 0 0
\(396\) 0 0
\(397\) 1356.00 0.171425 0.0857125 0.996320i \(-0.472683\pi\)
0.0857125 + 0.996320i \(0.472683\pi\)
\(398\) −1860.00 −0.234255
\(399\) 0 0
\(400\) 0 0
\(401\) −6222.00 −0.774843 −0.387421 0.921903i \(-0.626634\pi\)
−0.387421 + 0.921903i \(0.626634\pi\)
\(402\) 0 0
\(403\) −336.000 −0.0415319
\(404\) −4816.00 −0.593082
\(405\) 0 0
\(406\) −770.000 −0.0941243
\(407\) 1968.00 0.239681
\(408\) 0 0
\(409\) 5150.00 0.622619 0.311309 0.950309i \(-0.399232\pi\)
0.311309 + 0.950309i \(0.399232\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 9716.00 1.16183
\(413\) −5670.00 −0.675551
\(414\) 0 0
\(415\) 0 0
\(416\) 4508.00 0.531305
\(417\) 0 0
\(418\) 880.000 0.102972
\(419\) −2310.00 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −2430.00 −0.278328
\(425\) 0 0
\(426\) 0 0
\(427\) −3416.00 −0.387147
\(428\) −1708.00 −0.192896
\(429\) 0 0
\(430\) 0 0
\(431\) 4488.00 0.501576 0.250788 0.968042i \(-0.419310\pi\)
0.250788 + 0.968042i \(0.419310\pi\)
\(432\) 0 0
\(433\) −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
\(438\) 0 0
\(439\) 16200.0 1.76124 0.880619 0.473824i \(-0.157127\pi\)
0.880619 + 0.473824i \(0.157127\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1512.00 0.162712
\(443\) −8772.00 −0.940791 −0.470395 0.882456i \(-0.655889\pi\)
−0.470395 + 0.882456i \(0.655889\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −5432.00 −0.576710
\(447\) 0 0
\(448\) −1169.00 −0.123281
\(449\) −2130.00 −0.223877 −0.111939 0.993715i \(-0.535706\pi\)
−0.111939 + 0.993715i \(0.535706\pi\)
\(450\) 0 0
\(451\) −1456.00 −0.152019
\(452\) −9226.00 −0.960076
\(453\) 0 0
\(454\) 2046.00 0.211506
\(455\) 0 0
\(456\) 0 0
\(457\) −10534.0 −1.07825 −0.539124 0.842226i \(-0.681245\pi\)
−0.539124 + 0.842226i \(0.681245\pi\)
\(458\) 2980.00 0.304031
\(459\) 0 0
\(460\) 0 0
\(461\) 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 \(-0.343762\pi\)
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 \(-0.679830\pi\)
−0.535377 + 0.844613i \(0.679830\pi\)
\(468\) 0 0
\(469\) −1708.00 −0.168162
\(470\) 0 0
\(471\) 0 0
\(472\) −12150.0 −1.18485
\(473\) −1024.00 −0.0995424
\(474\) 0 0
\(475\) 0 0
\(476\) −2646.00 −0.254788
\(477\) 0 0
\(478\) 4440.00 0.424855
\(479\) −4940.00 −0.471220 −0.235610 0.971848i \(-0.575709\pi\)
−0.235610 + 0.971848i \(0.575709\pi\)
\(480\) 0 0
\(481\) −6888.00 −0.652943
\(482\) −3302.00 −0.312037
\(483\) 0 0
\(484\) 8869.00 0.832926
\(485\) 0 0
\(486\) 0 0
\(487\) 5216.00 0.485338 0.242669 0.970109i \(-0.421977\pi\)
0.242669 + 0.970109i \(0.421977\pi\)
\(488\) −7320.00 −0.679018
\(489\) 0 0
\(490\) 0 0
\(491\) −4412.00 −0.405521 −0.202760 0.979228i \(-0.564991\pi\)
−0.202760 + 0.979228i \(0.564991\pi\)
\(492\) 0 0
\(493\) 5940.00 0.542645
\(494\) −3080.00 −0.280518
\(495\) 0 0
\(496\) 492.000 0.0445392
\(497\) 5376.00 0.485204
\(498\) 0 0
\(499\) 19060.0 1.70991 0.854953 0.518706i \(-0.173586\pi\)
0.854953 + 0.518706i \(0.173586\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1582.00 0.140654
\(503\) 12768.0 1.13180 0.565902 0.824473i \(-0.308528\pi\)
0.565902 + 0.824473i \(0.308528\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −384.000 −0.0337369
\(507\) 0 0
\(508\) −12432.0 −1.08579
\(509\) 5500.00 0.478945 0.239473 0.970903i \(-0.423025\pi\)
0.239473 + 0.970903i \(0.423025\pi\)
\(510\) 0 0
\(511\) 4914.00 0.425406
\(512\) −11521.0 −0.994455
\(513\) 0 0
\(514\) −2354.00 −0.202005
\(515\) 0 0
\(516\) 0 0
\(517\) 2592.00 0.220495
\(518\) −1722.00 −0.146062
\(519\) 0 0
\(520\) 0 0
\(521\) 7338.00 0.617051 0.308526 0.951216i \(-0.400164\pi\)
0.308526 + 0.951216i \(0.400164\pi\)
\(522\) 0 0
\(523\) 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
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) 0 0
\(531\) 0 0
\(532\) 5390.00 0.439260
\(533\) 5096.00 0.414132
\(534\) 0 0
\(535\) 0 0
\(536\) −3660.00 −0.294940
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) −8694.00 −0.685206
\(545\) 0 0
\(546\) 0 0
\(547\) −16144.0 −1.26192 −0.630958 0.775817i \(-0.717338\pi\)
−0.630958 + 0.775817i \(0.717338\pi\)
\(548\) −15918.0 −1.24085
\(549\) 0 0
\(550\) 0 0
\(551\) −12100.0 −0.935531
\(552\) 0 0
\(553\) 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 \(-0.443355\pi\)
0.177016 + 0.984208i \(0.443355\pi\)
\(558\) 0 0
\(559\) 3584.00 0.271175
\(560\) 0 0
\(561\) 0 0
\(562\) 842.000 0.0631986
\(563\) 10078.0 0.754418 0.377209 0.926128i \(-0.376884\pi\)
0.377209 + 0.926128i \(0.376884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3782.00 −0.280865
\(567\) 0 0
\(568\) 11520.0 0.851001
\(569\) 5930.00 0.436904 0.218452 0.975848i \(-0.429899\pi\)
0.218452 + 0.975848i \(0.429899\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 1274.00 0.0926406
\(575\) 0 0
\(576\) 0 0
\(577\) 14366.0 1.03651 0.518253 0.855227i \(-0.326582\pi\)
0.518253 + 0.855227i \(0.326582\pi\)
\(578\) 1997.00 0.143710
\(579\) 0 0
\(580\) 0 0
\(581\) −9114.00 −0.650796
\(582\) 0 0
\(583\) −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 \(-0.540689\pi\)
−0.127480 + 0.991841i \(0.540689\pi\)
\(588\) 0 0
\(589\) −1320.00 −0.0923424
\(590\) 0 0
\(591\) 0 0
\(592\) 10086.0 0.700223
\(593\) −1062.00 −0.0735432 −0.0367716 0.999324i \(-0.511707\pi\)
−0.0367716 + 0.999324i \(0.511707\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14070.0 −0.966996
\(597\) 0 0
\(598\) 1344.00 0.0919068
\(599\) 10200.0 0.695761 0.347880 0.937539i \(-0.386902\pi\)
0.347880 + 0.937539i \(0.386902\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −7784.00 −0.524382
\(605\) 0 0
\(606\) 0 0
\(607\) −25664.0 −1.71609 −0.858047 0.513570i \(-0.828323\pi\)
−0.858047 + 0.513570i \(0.828323\pi\)
\(608\) 17710.0 1.18131
\(609\) 0 0
\(610\) 0 0
\(611\) −9072.00 −0.600677
\(612\) 0 0
\(613\) −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 \(-0.308679\pi\)
0.565511 + 0.824741i \(0.308679\pi\)
\(618\) 0 0
\(619\) 18730.0 1.21619 0.608096 0.793864i \(-0.291934\pi\)
0.608096 + 0.793864i \(0.291934\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3768.00 −0.242899
\(623\) −5110.00 −0.328616
\(624\) 0 0
\(625\) 0 0
\(626\) 2438.00 0.155658
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 3186.00 0.199578
\(635\) 0 0
\(636\) 0 0
\(637\) −1372.00 −0.0853385
\(638\) −880.000 −0.0546074
\(639\) 0 0
\(640\) 0 0
\(641\) −16302.0 −1.00451 −0.502255 0.864720i \(-0.667496\pi\)
−0.502255 + 0.864720i \(0.667496\pi\)
\(642\) 0 0
\(643\) −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 \(-0.725761\pi\)
−0.651264 + 0.758851i \(0.725761\pi\)
\(648\) 0 0
\(649\) −6480.00 −0.391930
\(650\) 0 0
\(651\) 0 0
\(652\) 14056.0 0.844287
\(653\) 4458.00 0.267159 0.133580 0.991038i \(-0.457353\pi\)
0.133580 + 0.991038i \(0.457353\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7462.00 −0.444119
\(657\) 0 0
\(658\) −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
\(663\) 0 0
\(664\) −19530.0 −1.14143
\(665\) 0 0
\(666\) 0 0
\(667\) 5280.00 0.306510
\(668\) −20188.0 −1.16931
\(669\) 0 0
\(670\) 0 0
\(671\) −3904.00 −0.224608
\(672\) 0 0
\(673\) −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 \(-0.709124\pi\)
−0.610729 + 0.791840i \(0.709124\pi\)
\(678\) 0 0
\(679\) −2058.00 −0.116316
\(680\) 0 0
\(681\) 0 0
\(682\) −96.0000 −0.00539007
\(683\) 18108.0 1.01447 0.507235 0.861808i \(-0.330668\pi\)
0.507235 + 0.861808i \(0.330668\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −343.000 −0.0190901
\(687\) 0 0
\(688\) −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
\(693\) 0 0
\(694\) −9344.00 −0.511086
\(695\) 0 0
\(696\) 0 0
\(697\) −9828.00 −0.534092
\(698\) 5180.00 0.280897
\(699\) 0 0
\(700\) 0 0
\(701\) −18762.0 −1.01089 −0.505443 0.862860i \(-0.668671\pi\)
−0.505443 + 0.862860i \(0.668671\pi\)
\(702\) 0 0
\(703\) −27060.0 −1.45176
\(704\) −1336.00 −0.0715233
\(705\) 0 0
\(706\) −12178.0 −0.649186
\(707\) 4816.00 0.256187
\(708\) 0 0
\(709\) 6810.00 0.360726 0.180363 0.983600i \(-0.442273\pi\)
0.180363 + 0.983600i \(0.442273\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10950.0 −0.576360
\(713\) 576.000 0.0302544
\(714\) 0 0
\(715\) 0 0
\(716\) −5740.00 −0.299600
\(717\) 0 0
\(718\) 440.000 0.0228700
\(719\) −4860.00 −0.252083 −0.126041 0.992025i \(-0.540227\pi\)
−0.126041 + 0.992025i \(0.540227\pi\)
\(720\) 0 0
\(721\) −9716.00 −0.501862
\(722\) −5241.00 −0.270152
\(723\) 0 0
\(724\) −27244.0 −1.39850
\(725\) 0 0
\(726\) 0 0
\(727\) 13636.0 0.695641 0.347821 0.937561i \(-0.386922\pi\)
0.347821 + 0.937561i \(0.386922\pi\)
\(728\) −2940.00 −0.149675
\(729\) 0 0
\(730\) 0 0
\(731\) −6912.00 −0.349726
\(732\) 0 0
\(733\) −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
\(738\) 0 0
\(739\) −5160.00 −0.256852 −0.128426 0.991719i \(-0.540992\pi\)
−0.128426 + 0.991719i \(0.540992\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1134.00 0.0561057
\(743\) −28152.0 −1.39004 −0.695018 0.718992i \(-0.744604\pi\)
−0.695018 + 0.718992i \(0.744604\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −442.000 −0.0216927
\(747\) 0 0
\(748\) −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
\(753\) 0 0
\(754\) 3080.00 0.148763
\(755\) 0 0
\(756\) 0 0
\(757\) −21674.0 −1.04063 −0.520314 0.853975i \(-0.674185\pi\)
−0.520314 + 0.853975i \(0.674185\pi\)
\(758\) 3960.00 0.189754
\(759\) 0 0
\(760\) 0 0
\(761\) −7422.00 −0.353544 −0.176772 0.984252i \(-0.556566\pi\)
−0.176772 + 0.984252i \(0.556566\pi\)
\(762\) 0 0
\(763\) 630.000 0.0298919
\(764\) −35336.0 −1.67331
\(765\) 0 0
\(766\) −6708.00 −0.316410
\(767\) 22680.0 1.06770
\(768\) 0 0
\(769\) 13790.0 0.646658 0.323329 0.946287i \(-0.395198\pi\)
0.323329 + 0.946287i \(0.395198\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −20734.0 −0.966623
\(773\) −6232.00 −0.289973 −0.144987 0.989434i \(-0.546314\pi\)
−0.144987 + 0.989434i \(0.546314\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −4410.00 −0.204007
\(777\) 0 0
\(778\) −13350.0 −0.615194
\(779\) 20020.0 0.920784
\(780\) 0 0
\(781\) 6144.00 0.281498
\(782\) −2592.00 −0.118529
\(783\) 0 0
\(784\) 2009.00 0.0915179
\(785\) 0 0
\(786\) 0 0
\(787\) 1766.00 0.0799887 0.0399943 0.999200i \(-0.487266\pi\)
0.0399943 + 0.999200i \(0.487266\pi\)
\(788\) −23338.0 −1.05505
\(789\) 0 0
\(790\) 0 0
\(791\) 9226.00 0.414714
\(792\) 0 0
\(793\) 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 \(-0.491483\pi\)
0.0267552 + 0.999642i \(0.491483\pi\)
\(798\) 0 0
\(799\) 17496.0 0.774673
\(800\) 0 0
\(801\) 0 0
\(802\) 6222.00 0.273948
\(803\) 5616.00 0.246805
\(804\) 0 0
\(805\) 0 0
\(806\) 336.000 0.0146837
\(807\) 0 0
\(808\) 10320.0 0.449327
\(809\) 7050.00 0.306384 0.153192 0.988196i \(-0.451045\pi\)
0.153192 + 0.988196i \(0.451045\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −1968.00 −0.0847400
\(815\) 0 0
\(816\) 0 0
\(817\) 14080.0 0.602934
\(818\) −5150.00 −0.220129
\(819\) 0 0
\(820\) 0 0
\(821\) −10142.0 −0.431131 −0.215565 0.976489i \(-0.569159\pi\)
−0.215565 + 0.976489i \(0.569159\pi\)
\(822\) 0 0
\(823\) 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 \(-0.939766\pi\)
−0.982149 + 0.188104i \(0.939766\pi\)
\(828\) 0 0
\(829\) 11240.0 0.470906 0.235453 0.971886i \(-0.424343\pi\)
0.235453 + 0.971886i \(0.424343\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4676.00 0.194845
\(833\) 2646.00 0.110058
\(834\) 0 0
\(835\) 0 0
\(836\) 6160.00 0.254842
\(837\) 0 0
\(838\) 2310.00 0.0952239
\(839\) −700.000 −0.0288042 −0.0144021 0.999896i \(-0.504584\pi\)
−0.0144021 + 0.999896i \(0.504584\pi\)
\(840\) 0 0
\(841\) −12289.0 −0.503875
\(842\) −1262.00 −0.0516525
\(843\) 0 0
\(844\) 29876.0 1.21845
\(845\) 0 0
\(846\) 0 0
\(847\) −8869.00 −0.359790
\(848\) −6642.00 −0.268971
\(849\) 0 0
\(850\) 0 0
\(851\) 11808.0 0.475644
\(852\) 0 0
\(853\) 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 \(-0.304673\pi\)
0.575846 + 0.817558i \(0.304673\pi\)
\(858\) 0 0
\(859\) −2770.00 −0.110025 −0.0550123 0.998486i \(-0.517520\pi\)
−0.0550123 + 0.998486i \(0.517520\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4488.00 −0.177334
\(863\) 17688.0 0.697690 0.348845 0.937180i \(-0.386574\pi\)
0.348845 + 0.937180i \(0.386574\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 17038.0 0.668562
\(867\) 0 0
\(868\) −588.000 −0.0229931
\(869\) 3520.00 0.137408
\(870\) 0 0
\(871\) 6832.00 0.265779
\(872\) 1350.00 0.0524275
\(873\) 0 0
\(874\) 5280.00 0.204346
\(875\) 0 0
\(876\) 0 0
\(877\) 33566.0 1.29241 0.646205 0.763164i \(-0.276355\pi\)
0.646205 + 0.763164i \(0.276355\pi\)
\(878\) −16200.0 −0.622692
\(879\) 0 0
\(880\) 0 0
\(881\) 16758.0 0.640853 0.320426 0.947273i \(-0.396174\pi\)
0.320426 + 0.947273i \(0.396174\pi\)
\(882\) 0 0
\(883\) −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 \(-0.902121\pi\)
−0.953094 + 0.302674i \(0.902121\pi\)
\(888\) 0 0
\(889\) 12432.0 0.469017
\(890\) 0 0
\(891\) 0 0
\(892\) −38024.0 −1.42728
\(893\) −35640.0 −1.33555
\(894\) 0 0
\(895\) 0 0
\(896\) 10185.0 0.379751
\(897\) 0 0
\(898\) 2130.00 0.0791526
\(899\) 1320.00 0.0489705
\(900\) 0 0
\(901\) −8748.00 −0.323461
\(902\) 1456.00 0.0537467
\(903\) 0 0
\(904\) 19770.0 0.727368
\(905\) 0 0
\(906\) 0 0
\(907\) 8716.00 0.319085 0.159542 0.987191i \(-0.448998\pi\)
0.159542 + 0.987191i \(0.448998\pi\)
\(908\) 14322.0 0.523450
\(909\) 0 0
\(910\) 0 0
\(911\) −7632.00 −0.277563 −0.138781 0.990323i \(-0.544318\pi\)
−0.138781 + 0.990323i \(0.544318\pi\)
\(912\) 0 0
\(913\) −10416.0 −0.377568
\(914\) 10534.0 0.381219
\(915\) 0 0
\(916\) 20860.0 0.752439
\(917\) 7826.00 0.281829
\(918\) 0 0
\(919\) −23080.0 −0.828443 −0.414221 0.910176i \(-0.635946\pi\)
−0.414221 + 0.910176i \(0.635946\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −9268.00 −0.331047
\(923\) −21504.0 −0.766861
\(924\) 0 0
\(925\) 0 0
\(926\) −9392.00 −0.333305
\(927\) 0 0
\(928\) −17710.0 −0.626465
\(929\) −45110.0 −1.59312 −0.796561 0.604558i \(-0.793350\pi\)
−0.796561 + 0.604558i \(0.793350\pi\)
\(930\) 0 0
\(931\) −5390.00 −0.189742
\(932\) −31206.0 −1.09677
\(933\) 0 0
\(934\) 10806.0 0.378569
\(935\) 0 0
\(936\) 0 0
\(937\) −16674.0 −0.581340 −0.290670 0.956823i \(-0.593878\pi\)
−0.290670 + 0.956823i \(0.593878\pi\)
\(938\) 1708.00 0.0594543
\(939\) 0 0
\(940\) 0 0
\(941\) −43832.0 −1.51847 −0.759236 0.650815i \(-0.774427\pi\)
−0.759236 + 0.650815i \(0.774427\pi\)
\(942\) 0 0
\(943\) −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 \(-0.504020\pi\)
−0.0126277 + 0.999920i \(0.504020\pi\)
\(948\) 0 0
\(949\) −19656.0 −0.672351
\(950\) 0 0
\(951\) 0 0
\(952\) 5670.00 0.193031
\(953\) 38138.0 1.29634 0.648169 0.761496i \(-0.275535\pi\)
0.648169 + 0.761496i \(0.275535\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 31080.0 1.05146
\(957\) 0 0
\(958\) 4940.00 0.166601
\(959\) 15918.0 0.535995
\(960\) 0 0
\(961\) −29647.0 −0.995166
\(962\) 6888.00 0.230850
\(963\) 0 0
\(964\) −23114.0 −0.772253
\(965\) 0 0
\(966\) 0 0
\(967\) −26224.0 −0.872086 −0.436043 0.899926i \(-0.643620\pi\)
−0.436043 + 0.899926i \(0.643620\pi\)
\(968\) −19005.0 −0.631037
\(969\) 0 0
\(970\) 0 0
\(971\) −18762.0 −0.620084 −0.310042 0.950723i \(-0.600343\pi\)
−0.310042 + 0.950723i \(0.600343\pi\)
\(972\) 0 0
\(973\) −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 \(-0.283618\pi\)
0.628625 + 0.777709i \(0.283618\pi\)
\(978\) 0 0
\(979\) −5840.00 −0.190651
\(980\) 0 0
\(981\) 0 0
\(982\) 4412.00 0.143373
\(983\) 5388.00 0.174822 0.0874112 0.996172i \(-0.472141\pi\)
0.0874112 + 0.996172i \(0.472141\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5940.00 −0.191854
\(987\) 0 0
\(988\) −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
\(993\) 0 0
\(994\) −5376.00 −0.171546
\(995\) 0 0
\(996\) 0 0
\(997\) 17096.0 0.543065 0.271532 0.962429i \(-0.412470\pi\)
0.271532 + 0.962429i \(0.412470\pi\)
\(998\) −19060.0 −0.604543
\(999\) 0 0
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 1575.4.a.e.1.1 1
3.2 odd 2 175.4.a.b.1.1 1
5.4 even 2 63.4.a.b.1.1 1
15.2 even 4 175.4.b.b.99.2 2
15.8 even 4 175.4.b.b.99.1 2
15.14 odd 2 7.4.a.a.1.1 1
20.19 odd 2 1008.4.a.c.1.1 1
21.20 even 2 1225.4.a.j.1.1 1
35.4 even 6 441.4.e.h.226.1 2
35.9 even 6 441.4.e.h.361.1 2
35.19 odd 6 441.4.e.e.361.1 2
35.24 odd 6 441.4.e.e.226.1 2
35.34 odd 2 441.4.a.i.1.1 1
60.59 even 2 112.4.a.f.1.1 1
105.44 odd 6 49.4.c.c.18.1 2
105.59 even 6 49.4.c.b.30.1 2
105.74 odd 6 49.4.c.c.30.1 2
105.89 even 6 49.4.c.b.18.1 2
105.104 even 2 49.4.a.b.1.1 1
120.29 odd 2 448.4.a.i.1.1 1
120.59 even 2 448.4.a.e.1.1 1
165.164 even 2 847.4.a.b.1.1 1
195.194 odd 2 1183.4.a.b.1.1 1
255.254 odd 2 2023.4.a.a.1.1 1
420.419 odd 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 15.14 odd 2
49.4.a.b.1.1 1 105.104 even 2
49.4.c.b.18.1 2 105.89 even 6
49.4.c.b.30.1 2 105.59 even 6
49.4.c.c.18.1 2 105.44 odd 6
49.4.c.c.30.1 2 105.74 odd 6
63.4.a.b.1.1 1 5.4 even 2
112.4.a.f.1.1 1 60.59 even 2
175.4.a.b.1.1 1 3.2 odd 2
175.4.b.b.99.1 2 15.8 even 4
175.4.b.b.99.2 2 15.2 even 4
441.4.a.i.1.1 1 35.34 odd 2
441.4.e.e.226.1 2 35.24 odd 6
441.4.e.e.361.1 2 35.19 odd 6
441.4.e.h.226.1 2 35.4 even 6
441.4.e.h.361.1 2 35.9 even 6
448.4.a.e.1.1 1 120.59 even 2
448.4.a.i.1.1 1 120.29 odd 2
784.4.a.g.1.1 1 420.419 odd 2
847.4.a.b.1.1 1 165.164 even 2
1008.4.a.c.1.1 1 20.19 odd 2
1183.4.a.b.1.1 1 195.194 odd 2
1225.4.a.j.1.1 1 21.20 even 2
1575.4.a.e.1.1 1 1.1 even 1 trivial
2023.4.a.a.1.1 1 255.254 odd 2