Properties

Label 2450.4.a.c.1.1
Level $2450$
Weight $4$
Character 2450.1
Self dual yes
Analytic conductor $144.555$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.554679514\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2450.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -7.00000 q^{3} +4.00000 q^{4} +14.0000 q^{6} -8.00000 q^{8} +22.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -7.00000 q^{3} +4.00000 q^{4} +14.0000 q^{6} -8.00000 q^{8} +22.0000 q^{9} -37.0000 q^{11} -28.0000 q^{12} +51.0000 q^{13} +16.0000 q^{16} +41.0000 q^{17} -44.0000 q^{18} +108.000 q^{19} +74.0000 q^{22} +70.0000 q^{23} +56.0000 q^{24} -102.000 q^{26} +35.0000 q^{27} -249.000 q^{29} +134.000 q^{31} -32.0000 q^{32} +259.000 q^{33} -82.0000 q^{34} +88.0000 q^{36} +334.000 q^{37} -216.000 q^{38} -357.000 q^{39} -206.000 q^{41} +376.000 q^{43} -148.000 q^{44} -140.000 q^{46} -287.000 q^{47} -112.000 q^{48} -287.000 q^{51} +204.000 q^{52} +6.00000 q^{53} -70.0000 q^{54} -756.000 q^{57} +498.000 q^{58} +2.00000 q^{59} +940.000 q^{61} -268.000 q^{62} +64.0000 q^{64} -518.000 q^{66} -106.000 q^{67} +164.000 q^{68} -490.000 q^{69} +456.000 q^{71} -176.000 q^{72} +650.000 q^{73} -668.000 q^{74} +432.000 q^{76} +714.000 q^{78} -1239.00 q^{79} -839.000 q^{81} +412.000 q^{82} +428.000 q^{83} -752.000 q^{86} +1743.00 q^{87} +296.000 q^{88} +220.000 q^{89} +280.000 q^{92} -938.000 q^{93} +574.000 q^{94} +224.000 q^{96} -1055.00 q^{97} -814.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −7.00000 −1.34715 −0.673575 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 14.0000 0.952579
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) −37.0000 −1.01417 −0.507087 0.861895i \(-0.669278\pi\)
−0.507087 + 0.861895i \(0.669278\pi\)
\(12\) −28.0000 −0.673575
\(13\) 51.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 41.0000 0.584939 0.292469 0.956275i \(-0.405523\pi\)
0.292469 + 0.956275i \(0.405523\pi\)
\(18\) −44.0000 −0.576161
\(19\) 108.000 1.30405 0.652024 0.758199i \(-0.273920\pi\)
0.652024 + 0.758199i \(0.273920\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 74.0000 0.717130
\(23\) 70.0000 0.634609 0.317305 0.948324i \(-0.397222\pi\)
0.317305 + 0.948324i \(0.397222\pi\)
\(24\) 56.0000 0.476290
\(25\) 0 0
\(26\) −102.000 −0.769379
\(27\) 35.0000 0.249472
\(28\) 0 0
\(29\) −249.000 −1.59442 −0.797209 0.603703i \(-0.793691\pi\)
−0.797209 + 0.603703i \(0.793691\pi\)
\(30\) 0 0
\(31\) 134.000 0.776358 0.388179 0.921584i \(-0.373104\pi\)
0.388179 + 0.921584i \(0.373104\pi\)
\(32\) −32.0000 −0.176777
\(33\) 259.000 1.36625
\(34\) −82.0000 −0.413614
\(35\) 0 0
\(36\) 88.0000 0.407407
\(37\) 334.000 1.48403 0.742017 0.670381i \(-0.233869\pi\)
0.742017 + 0.670381i \(0.233869\pi\)
\(38\) −216.000 −0.922101
\(39\) −357.000 −1.46579
\(40\) 0 0
\(41\) −206.000 −0.784678 −0.392339 0.919821i \(-0.628334\pi\)
−0.392339 + 0.919821i \(0.628334\pi\)
\(42\) 0 0
\(43\) 376.000 1.33348 0.666738 0.745292i \(-0.267690\pi\)
0.666738 + 0.745292i \(0.267690\pi\)
\(44\) −148.000 −0.507087
\(45\) 0 0
\(46\) −140.000 −0.448736
\(47\) −287.000 −0.890708 −0.445354 0.895355i \(-0.646922\pi\)
−0.445354 + 0.895355i \(0.646922\pi\)
\(48\) −112.000 −0.336788
\(49\) 0 0
\(50\) 0 0
\(51\) −287.000 −0.788001
\(52\) 204.000 0.544033
\(53\) 6.00000 0.0155503 0.00777513 0.999970i \(-0.497525\pi\)
0.00777513 + 0.999970i \(0.497525\pi\)
\(54\) −70.0000 −0.176404
\(55\) 0 0
\(56\) 0 0
\(57\) −756.000 −1.75675
\(58\) 498.000 1.12742
\(59\) 2.00000 0.00441318 0.00220659 0.999998i \(-0.499298\pi\)
0.00220659 + 0.999998i \(0.499298\pi\)
\(60\) 0 0
\(61\) 940.000 1.97303 0.986514 0.163679i \(-0.0523361\pi\)
0.986514 + 0.163679i \(0.0523361\pi\)
\(62\) −268.000 −0.548968
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −518.000 −0.966082
\(67\) −106.000 −0.193283 −0.0966415 0.995319i \(-0.530810\pi\)
−0.0966415 + 0.995319i \(0.530810\pi\)
\(68\) 164.000 0.292469
\(69\) −490.000 −0.854914
\(70\) 0 0
\(71\) 456.000 0.762215 0.381107 0.924531i \(-0.375543\pi\)
0.381107 + 0.924531i \(0.375543\pi\)
\(72\) −176.000 −0.288081
\(73\) 650.000 1.04215 0.521074 0.853512i \(-0.325532\pi\)
0.521074 + 0.853512i \(0.325532\pi\)
\(74\) −668.000 −1.04937
\(75\) 0 0
\(76\) 432.000 0.652024
\(77\) 0 0
\(78\) 714.000 1.03647
\(79\) −1239.00 −1.76454 −0.882268 0.470747i \(-0.843984\pi\)
−0.882268 + 0.470747i \(0.843984\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 412.000 0.554851
\(83\) 428.000 0.566013 0.283007 0.959118i \(-0.408668\pi\)
0.283007 + 0.959118i \(0.408668\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −752.000 −0.942910
\(87\) 1743.00 2.14792
\(88\) 296.000 0.358565
\(89\) 220.000 0.262022 0.131011 0.991381i \(-0.458178\pi\)
0.131011 + 0.991381i \(0.458178\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 280.000 0.317305
\(93\) −938.000 −1.04587
\(94\) 574.000 0.629825
\(95\) 0 0
\(96\) 224.000 0.238145
\(97\) −1055.00 −1.10432 −0.552160 0.833738i \(-0.686196\pi\)
−0.552160 + 0.833738i \(0.686196\pi\)
\(98\) 0 0
\(99\) −814.000 −0.826364
\(100\) 0 0
\(101\) −1960.00 −1.93096 −0.965482 0.260471i \(-0.916122\pi\)
−0.965482 + 0.260471i \(0.916122\pi\)
\(102\) 574.000 0.557201
\(103\) 1825.00 1.74585 0.872925 0.487854i \(-0.162220\pi\)
0.872925 + 0.487854i \(0.162220\pi\)
\(104\) −408.000 −0.384689
\(105\) 0 0
\(106\) −12.0000 −0.0109957
\(107\) 144.000 0.130103 0.0650514 0.997882i \(-0.479279\pi\)
0.0650514 + 0.997882i \(0.479279\pi\)
\(108\) 140.000 0.124736
\(109\) 1681.00 1.47716 0.738581 0.674165i \(-0.235496\pi\)
0.738581 + 0.674165i \(0.235496\pi\)
\(110\) 0 0
\(111\) −2338.00 −1.99922
\(112\) 0 0
\(113\) −798.000 −0.664332 −0.332166 0.943221i \(-0.607779\pi\)
−0.332166 + 0.943221i \(0.607779\pi\)
\(114\) 1512.00 1.24221
\(115\) 0 0
\(116\) −996.000 −0.797209
\(117\) 1122.00 0.886572
\(118\) −4.00000 −0.00312059
\(119\) 0 0
\(120\) 0 0
\(121\) 38.0000 0.0285500
\(122\) −1880.00 −1.39514
\(123\) 1442.00 1.05708
\(124\) 536.000 0.388179
\(125\) 0 0
\(126\) 0 0
\(127\) −434.000 −0.303238 −0.151619 0.988439i \(-0.548449\pi\)
−0.151619 + 0.988439i \(0.548449\pi\)
\(128\) −128.000 −0.0883883
\(129\) −2632.00 −1.79639
\(130\) 0 0
\(131\) 1290.00 0.860365 0.430183 0.902742i \(-0.358449\pi\)
0.430183 + 0.902742i \(0.358449\pi\)
\(132\) 1036.00 0.683123
\(133\) 0 0
\(134\) 212.000 0.136672
\(135\) 0 0
\(136\) −328.000 −0.206807
\(137\) −192.000 −0.119735 −0.0598674 0.998206i \(-0.519068\pi\)
−0.0598674 + 0.998206i \(0.519068\pi\)
\(138\) 980.000 0.604516
\(139\) −1402.00 −0.855511 −0.427756 0.903894i \(-0.640696\pi\)
−0.427756 + 0.903894i \(0.640696\pi\)
\(140\) 0 0
\(141\) 2009.00 1.19992
\(142\) −912.000 −0.538967
\(143\) −1887.00 −1.10349
\(144\) 352.000 0.203704
\(145\) 0 0
\(146\) −1300.00 −0.736909
\(147\) 0 0
\(148\) 1336.00 0.742017
\(149\) −302.000 −0.166046 −0.0830228 0.996548i \(-0.526457\pi\)
−0.0830228 + 0.996548i \(0.526457\pi\)
\(150\) 0 0
\(151\) −3167.00 −1.70680 −0.853400 0.521257i \(-0.825463\pi\)
−0.853400 + 0.521257i \(0.825463\pi\)
\(152\) −864.000 −0.461050
\(153\) 902.000 0.476617
\(154\) 0 0
\(155\) 0 0
\(156\) −1428.00 −0.732894
\(157\) −470.000 −0.238918 −0.119459 0.992839i \(-0.538116\pi\)
−0.119459 + 0.992839i \(0.538116\pi\)
\(158\) 2478.00 1.24772
\(159\) −42.0000 −0.0209485
\(160\) 0 0
\(161\) 0 0
\(162\) 1678.00 0.813803
\(163\) 2390.00 1.14846 0.574231 0.818693i \(-0.305301\pi\)
0.574231 + 0.818693i \(0.305301\pi\)
\(164\) −824.000 −0.392339
\(165\) 0 0
\(166\) −856.000 −0.400232
\(167\) −2631.00 −1.21912 −0.609560 0.792740i \(-0.708654\pi\)
−0.609560 + 0.792740i \(0.708654\pi\)
\(168\) 0 0
\(169\) 404.000 0.183887
\(170\) 0 0
\(171\) 2376.00 1.06256
\(172\) 1504.00 0.666738
\(173\) −2243.00 −0.985735 −0.492867 0.870104i \(-0.664051\pi\)
−0.492867 + 0.870104i \(0.664051\pi\)
\(174\) −3486.00 −1.51881
\(175\) 0 0
\(176\) −592.000 −0.253544
\(177\) −14.0000 −0.00594522
\(178\) −440.000 −0.185277
\(179\) 52.0000 0.0217132 0.0108566 0.999941i \(-0.496544\pi\)
0.0108566 + 0.999941i \(0.496544\pi\)
\(180\) 0 0
\(181\) −2462.00 −1.01104 −0.505522 0.862814i \(-0.668700\pi\)
−0.505522 + 0.862814i \(0.668700\pi\)
\(182\) 0 0
\(183\) −6580.00 −2.65797
\(184\) −560.000 −0.224368
\(185\) 0 0
\(186\) 1876.00 0.739543
\(187\) −1517.00 −0.593230
\(188\) −1148.00 −0.445354
\(189\) 0 0
\(190\) 0 0
\(191\) 3159.00 1.19674 0.598370 0.801220i \(-0.295815\pi\)
0.598370 + 0.801220i \(0.295815\pi\)
\(192\) −448.000 −0.168394
\(193\) −2060.00 −0.768301 −0.384150 0.923271i \(-0.625506\pi\)
−0.384150 + 0.923271i \(0.625506\pi\)
\(194\) 2110.00 0.780872
\(195\) 0 0
\(196\) 0 0
\(197\) −1738.00 −0.628565 −0.314283 0.949329i \(-0.601764\pi\)
−0.314283 + 0.949329i \(0.601764\pi\)
\(198\) 1628.00 0.584328
\(199\) 894.000 0.318462 0.159231 0.987241i \(-0.449099\pi\)
0.159231 + 0.987241i \(0.449099\pi\)
\(200\) 0 0
\(201\) 742.000 0.260381
\(202\) 3920.00 1.36540
\(203\) 0 0
\(204\) −1148.00 −0.394000
\(205\) 0 0
\(206\) −3650.00 −1.23450
\(207\) 1540.00 0.517089
\(208\) 816.000 0.272016
\(209\) −3996.00 −1.32253
\(210\) 0 0
\(211\) −4083.00 −1.33216 −0.666079 0.745881i \(-0.732029\pi\)
−0.666079 + 0.745881i \(0.732029\pi\)
\(212\) 24.0000 0.00777513
\(213\) −3192.00 −1.02682
\(214\) −288.000 −0.0919966
\(215\) 0 0
\(216\) −280.000 −0.0882018
\(217\) 0 0
\(218\) −3362.00 −1.04451
\(219\) −4550.00 −1.40393
\(220\) 0 0
\(221\) 2091.00 0.636452
\(222\) 4676.00 1.41366
\(223\) 377.000 0.113210 0.0566049 0.998397i \(-0.481972\pi\)
0.0566049 + 0.998397i \(0.481972\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1596.00 0.469754
\(227\) −2551.00 −0.745885 −0.372942 0.927855i \(-0.621651\pi\)
−0.372942 + 0.927855i \(0.621651\pi\)
\(228\) −3024.00 −0.878374
\(229\) −74.0000 −0.0213540 −0.0106770 0.999943i \(-0.503399\pi\)
−0.0106770 + 0.999943i \(0.503399\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1992.00 0.563712
\(233\) −1888.00 −0.530845 −0.265423 0.964132i \(-0.585512\pi\)
−0.265423 + 0.964132i \(0.585512\pi\)
\(234\) −2244.00 −0.626901
\(235\) 0 0
\(236\) 8.00000 0.00220659
\(237\) 8673.00 2.37710
\(238\) 0 0
\(239\) 4997.00 1.35242 0.676211 0.736708i \(-0.263621\pi\)
0.676211 + 0.736708i \(0.263621\pi\)
\(240\) 0 0
\(241\) 3830.00 1.02370 0.511851 0.859074i \(-0.328960\pi\)
0.511851 + 0.859074i \(0.328960\pi\)
\(242\) −76.0000 −0.0201879
\(243\) 4928.00 1.30095
\(244\) 3760.00 0.986514
\(245\) 0 0
\(246\) −2884.00 −0.747468
\(247\) 5508.00 1.41889
\(248\) −1072.00 −0.274484
\(249\) −2996.00 −0.762505
\(250\) 0 0
\(251\) 3390.00 0.852490 0.426245 0.904608i \(-0.359836\pi\)
0.426245 + 0.904608i \(0.359836\pi\)
\(252\) 0 0
\(253\) −2590.00 −0.643604
\(254\) 868.000 0.214422
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 7170.00 1.74028 0.870141 0.492803i \(-0.164028\pi\)
0.870141 + 0.492803i \(0.164028\pi\)
\(258\) 5264.00 1.27024
\(259\) 0 0
\(260\) 0 0
\(261\) −5478.00 −1.29916
\(262\) −2580.00 −0.608370
\(263\) 7672.00 1.79877 0.899384 0.437160i \(-0.144016\pi\)
0.899384 + 0.437160i \(0.144016\pi\)
\(264\) −2072.00 −0.483041
\(265\) 0 0
\(266\) 0 0
\(267\) −1540.00 −0.352983
\(268\) −424.000 −0.0966415
\(269\) 54.0000 0.0122395 0.00611977 0.999981i \(-0.498052\pi\)
0.00611977 + 0.999981i \(0.498052\pi\)
\(270\) 0 0
\(271\) −2932.00 −0.657219 −0.328609 0.944466i \(-0.606580\pi\)
−0.328609 + 0.944466i \(0.606580\pi\)
\(272\) 656.000 0.146235
\(273\) 0 0
\(274\) 384.000 0.0846653
\(275\) 0 0
\(276\) −1960.00 −0.427457
\(277\) −3254.00 −0.705826 −0.352913 0.935656i \(-0.614809\pi\)
−0.352913 + 0.935656i \(0.614809\pi\)
\(278\) 2804.00 0.604938
\(279\) 2948.00 0.632588
\(280\) 0 0
\(281\) 3327.00 0.706307 0.353153 0.935565i \(-0.385109\pi\)
0.353153 + 0.935565i \(0.385109\pi\)
\(282\) −4018.00 −0.848470
\(283\) 4627.00 0.971896 0.485948 0.873988i \(-0.338474\pi\)
0.485948 + 0.873988i \(0.338474\pi\)
\(284\) 1824.00 0.381107
\(285\) 0 0
\(286\) 3774.00 0.780284
\(287\) 0 0
\(288\) −704.000 −0.144040
\(289\) −3232.00 −0.657847
\(290\) 0 0
\(291\) 7385.00 1.48769
\(292\) 2600.00 0.521074
\(293\) −4083.00 −0.814100 −0.407050 0.913406i \(-0.633443\pi\)
−0.407050 + 0.913406i \(0.633443\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2672.00 −0.524685
\(297\) −1295.00 −0.253008
\(298\) 604.000 0.117412
\(299\) 3570.00 0.690496
\(300\) 0 0
\(301\) 0 0
\(302\) 6334.00 1.20689
\(303\) 13720.0 2.60130
\(304\) 1728.00 0.326012
\(305\) 0 0
\(306\) −1804.00 −0.337019
\(307\) −4089.00 −0.760168 −0.380084 0.924952i \(-0.624105\pi\)
−0.380084 + 0.924952i \(0.624105\pi\)
\(308\) 0 0
\(309\) −12775.0 −2.35192
\(310\) 0 0
\(311\) 4008.00 0.730781 0.365390 0.930854i \(-0.380935\pi\)
0.365390 + 0.930854i \(0.380935\pi\)
\(312\) 2856.00 0.518234
\(313\) 7355.00 1.32821 0.664104 0.747640i \(-0.268813\pi\)
0.664104 + 0.747640i \(0.268813\pi\)
\(314\) 940.000 0.168940
\(315\) 0 0
\(316\) −4956.00 −0.882268
\(317\) 1684.00 0.298369 0.149184 0.988809i \(-0.452335\pi\)
0.149184 + 0.988809i \(0.452335\pi\)
\(318\) 84.0000 0.0148128
\(319\) 9213.00 1.61702
\(320\) 0 0
\(321\) −1008.00 −0.175268
\(322\) 0 0
\(323\) 4428.00 0.762788
\(324\) −3356.00 −0.575446
\(325\) 0 0
\(326\) −4780.00 −0.812085
\(327\) −11767.0 −1.98996
\(328\) 1648.00 0.277426
\(329\) 0 0
\(330\) 0 0
\(331\) −1460.00 −0.242444 −0.121222 0.992625i \(-0.538681\pi\)
−0.121222 + 0.992625i \(0.538681\pi\)
\(332\) 1712.00 0.283007
\(333\) 7348.00 1.20921
\(334\) 5262.00 0.862047
\(335\) 0 0
\(336\) 0 0
\(337\) −7514.00 −1.21458 −0.607290 0.794480i \(-0.707744\pi\)
−0.607290 + 0.794480i \(0.707744\pi\)
\(338\) −808.000 −0.130028
\(339\) 5586.00 0.894955
\(340\) 0 0
\(341\) −4958.00 −0.787363
\(342\) −4752.00 −0.751341
\(343\) 0 0
\(344\) −3008.00 −0.471455
\(345\) 0 0
\(346\) 4486.00 0.697020
\(347\) 2862.00 0.442767 0.221384 0.975187i \(-0.428943\pi\)
0.221384 + 0.975187i \(0.428943\pi\)
\(348\) 6972.00 1.07396
\(349\) 6368.00 0.976708 0.488354 0.872646i \(-0.337597\pi\)
0.488354 + 0.872646i \(0.337597\pi\)
\(350\) 0 0
\(351\) 1785.00 0.271442
\(352\) 1184.00 0.179282
\(353\) −3635.00 −0.548078 −0.274039 0.961719i \(-0.588360\pi\)
−0.274039 + 0.961719i \(0.588360\pi\)
\(354\) 28.0000 0.00420391
\(355\) 0 0
\(356\) 880.000 0.131011
\(357\) 0 0
\(358\) −104.000 −0.0153535
\(359\) 7116.00 1.04615 0.523075 0.852286i \(-0.324785\pi\)
0.523075 + 0.852286i \(0.324785\pi\)
\(360\) 0 0
\(361\) 4805.00 0.700539
\(362\) 4924.00 0.714916
\(363\) −266.000 −0.0384611
\(364\) 0 0
\(365\) 0 0
\(366\) 13160.0 1.87947
\(367\) −319.000 −0.0453724 −0.0226862 0.999743i \(-0.507222\pi\)
−0.0226862 + 0.999743i \(0.507222\pi\)
\(368\) 1120.00 0.158652
\(369\) −4532.00 −0.639367
\(370\) 0 0
\(371\) 0 0
\(372\) −3752.00 −0.522936
\(373\) 11652.0 1.61747 0.808737 0.588171i \(-0.200152\pi\)
0.808737 + 0.588171i \(0.200152\pi\)
\(374\) 3034.00 0.419477
\(375\) 0 0
\(376\) 2296.00 0.314913
\(377\) −12699.0 −1.73483
\(378\) 0 0
\(379\) 7748.00 1.05010 0.525050 0.851071i \(-0.324047\pi\)
0.525050 + 0.851071i \(0.324047\pi\)
\(380\) 0 0
\(381\) 3038.00 0.408508
\(382\) −6318.00 −0.846223
\(383\) 8680.00 1.15803 0.579017 0.815315i \(-0.303436\pi\)
0.579017 + 0.815315i \(0.303436\pi\)
\(384\) 896.000 0.119072
\(385\) 0 0
\(386\) 4120.00 0.543271
\(387\) 8272.00 1.08654
\(388\) −4220.00 −0.552160
\(389\) −1711.00 −0.223011 −0.111505 0.993764i \(-0.535567\pi\)
−0.111505 + 0.993764i \(0.535567\pi\)
\(390\) 0 0
\(391\) 2870.00 0.371208
\(392\) 0 0
\(393\) −9030.00 −1.15904
\(394\) 3476.00 0.444463
\(395\) 0 0
\(396\) −3256.00 −0.413182
\(397\) 1589.00 0.200881 0.100440 0.994943i \(-0.467975\pi\)
0.100440 + 0.994943i \(0.467975\pi\)
\(398\) −1788.00 −0.225187
\(399\) 0 0
\(400\) 0 0
\(401\) −5147.00 −0.640970 −0.320485 0.947254i \(-0.603846\pi\)
−0.320485 + 0.947254i \(0.603846\pi\)
\(402\) −1484.00 −0.184117
\(403\) 6834.00 0.844729
\(404\) −7840.00 −0.965482
\(405\) 0 0
\(406\) 0 0
\(407\) −12358.0 −1.50507
\(408\) 2296.00 0.278600
\(409\) 9100.00 1.10016 0.550081 0.835111i \(-0.314597\pi\)
0.550081 + 0.835111i \(0.314597\pi\)
\(410\) 0 0
\(411\) 1344.00 0.161301
\(412\) 7300.00 0.872925
\(413\) 0 0
\(414\) −3080.00 −0.365637
\(415\) 0 0
\(416\) −1632.00 −0.192345
\(417\) 9814.00 1.15250
\(418\) 7992.00 0.935171
\(419\) −2618.00 −0.305245 −0.152623 0.988285i \(-0.548772\pi\)
−0.152623 + 0.988285i \(0.548772\pi\)
\(420\) 0 0
\(421\) −3695.00 −0.427751 −0.213876 0.976861i \(-0.568609\pi\)
−0.213876 + 0.976861i \(0.568609\pi\)
\(422\) 8166.00 0.941978
\(423\) −6314.00 −0.725762
\(424\) −48.0000 −0.00549784
\(425\) 0 0
\(426\) 6384.00 0.726070
\(427\) 0 0
\(428\) 576.000 0.0650514
\(429\) 13209.0 1.48657
\(430\) 0 0
\(431\) 15779.0 1.76345 0.881726 0.471762i \(-0.156382\pi\)
0.881726 + 0.471762i \(0.156382\pi\)
\(432\) 560.000 0.0623681
\(433\) 7238.00 0.803317 0.401658 0.915790i \(-0.368434\pi\)
0.401658 + 0.915790i \(0.368434\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6724.00 0.738581
\(437\) 7560.00 0.827560
\(438\) 9100.00 0.992728
\(439\) 2646.00 0.287669 0.143834 0.989602i \(-0.454057\pi\)
0.143834 + 0.989602i \(0.454057\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4182.00 −0.450039
\(443\) −5688.00 −0.610034 −0.305017 0.952347i \(-0.598662\pi\)
−0.305017 + 0.952347i \(0.598662\pi\)
\(444\) −9352.00 −0.999609
\(445\) 0 0
\(446\) −754.000 −0.0800514
\(447\) 2114.00 0.223689
\(448\) 0 0
\(449\) −3285.00 −0.345276 −0.172638 0.984985i \(-0.555229\pi\)
−0.172638 + 0.984985i \(0.555229\pi\)
\(450\) 0 0
\(451\) 7622.00 0.795800
\(452\) −3192.00 −0.332166
\(453\) 22169.0 2.29932
\(454\) 5102.00 0.527420
\(455\) 0 0
\(456\) 6048.00 0.621104
\(457\) −14834.0 −1.51839 −0.759196 0.650862i \(-0.774408\pi\)
−0.759196 + 0.650862i \(0.774408\pi\)
\(458\) 148.000 0.0150995
\(459\) 1435.00 0.145926
\(460\) 0 0
\(461\) 9972.00 1.00747 0.503734 0.863859i \(-0.331959\pi\)
0.503734 + 0.863859i \(0.331959\pi\)
\(462\) 0 0
\(463\) 9096.00 0.913017 0.456509 0.889719i \(-0.349100\pi\)
0.456509 + 0.889719i \(0.349100\pi\)
\(464\) −3984.00 −0.398605
\(465\) 0 0
\(466\) 3776.00 0.375364
\(467\) 15867.0 1.57224 0.786121 0.618072i \(-0.212086\pi\)
0.786121 + 0.618072i \(0.212086\pi\)
\(468\) 4488.00 0.443286
\(469\) 0 0
\(470\) 0 0
\(471\) 3290.00 0.321858
\(472\) −16.0000 −0.00156030
\(473\) −13912.0 −1.35238
\(474\) −17346.0 −1.68086
\(475\) 0 0
\(476\) 0 0
\(477\) 132.000 0.0126706
\(478\) −9994.00 −0.956307
\(479\) −242.000 −0.0230841 −0.0115420 0.999933i \(-0.503674\pi\)
−0.0115420 + 0.999933i \(0.503674\pi\)
\(480\) 0 0
\(481\) 17034.0 1.61473
\(482\) −7660.00 −0.723866
\(483\) 0 0
\(484\) 152.000 0.0142750
\(485\) 0 0
\(486\) −9856.00 −0.919912
\(487\) 3558.00 0.331064 0.165532 0.986204i \(-0.447066\pi\)
0.165532 + 0.986204i \(0.447066\pi\)
\(488\) −7520.00 −0.697571
\(489\) −16730.0 −1.54715
\(490\) 0 0
\(491\) 1473.00 0.135388 0.0676941 0.997706i \(-0.478436\pi\)
0.0676941 + 0.997706i \(0.478436\pi\)
\(492\) 5768.00 0.528540
\(493\) −10209.0 −0.932637
\(494\) −11016.0 −1.00331
\(495\) 0 0
\(496\) 2144.00 0.194090
\(497\) 0 0
\(498\) 5992.00 0.539173
\(499\) 603.000 0.0540962 0.0270481 0.999634i \(-0.491389\pi\)
0.0270481 + 0.999634i \(0.491389\pi\)
\(500\) 0 0
\(501\) 18417.0 1.64234
\(502\) −6780.00 −0.602801
\(503\) 18387.0 1.62989 0.814946 0.579537i \(-0.196766\pi\)
0.814946 + 0.579537i \(0.196766\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 5180.00 0.455097
\(507\) −2828.00 −0.247724
\(508\) −1736.00 −0.151619
\(509\) −9018.00 −0.785296 −0.392648 0.919689i \(-0.628441\pi\)
−0.392648 + 0.919689i \(0.628441\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 3780.00 0.325324
\(514\) −14340.0 −1.23056
\(515\) 0 0
\(516\) −10528.0 −0.898196
\(517\) 10619.0 0.903333
\(518\) 0 0
\(519\) 15701.0 1.32793
\(520\) 0 0
\(521\) −4624.00 −0.388831 −0.194416 0.980919i \(-0.562281\pi\)
−0.194416 + 0.980919i \(0.562281\pi\)
\(522\) 10956.0 0.918642
\(523\) −5876.00 −0.491280 −0.245640 0.969361i \(-0.578998\pi\)
−0.245640 + 0.969361i \(0.578998\pi\)
\(524\) 5160.00 0.430183
\(525\) 0 0
\(526\) −15344.0 −1.27192
\(527\) 5494.00 0.454122
\(528\) 4144.00 0.341561
\(529\) −7267.00 −0.597271
\(530\) 0 0
\(531\) 44.0000 0.00359593
\(532\) 0 0
\(533\) −10506.0 −0.853781
\(534\) 3080.00 0.249597
\(535\) 0 0
\(536\) 848.000 0.0683359
\(537\) −364.000 −0.0292509
\(538\) −108.000 −0.00865467
\(539\) 0 0
\(540\) 0 0
\(541\) −8537.00 −0.678437 −0.339218 0.940708i \(-0.610163\pi\)
−0.339218 + 0.940708i \(0.610163\pi\)
\(542\) 5864.00 0.464724
\(543\) 17234.0 1.36203
\(544\) −1312.00 −0.103404
\(545\) 0 0
\(546\) 0 0
\(547\) 13060.0 1.02085 0.510425 0.859922i \(-0.329488\pi\)
0.510425 + 0.859922i \(0.329488\pi\)
\(548\) −768.000 −0.0598674
\(549\) 20680.0 1.60765
\(550\) 0 0
\(551\) −26892.0 −2.07920
\(552\) 3920.00 0.302258
\(553\) 0 0
\(554\) 6508.00 0.499095
\(555\) 0 0
\(556\) −5608.00 −0.427756
\(557\) −21372.0 −1.62578 −0.812891 0.582416i \(-0.802108\pi\)
−0.812891 + 0.582416i \(0.802108\pi\)
\(558\) −5896.00 −0.447307
\(559\) 19176.0 1.45091
\(560\) 0 0
\(561\) 10619.0 0.799170
\(562\) −6654.00 −0.499434
\(563\) −12704.0 −0.950994 −0.475497 0.879717i \(-0.657732\pi\)
−0.475497 + 0.879717i \(0.657732\pi\)
\(564\) 8036.00 0.599959
\(565\) 0 0
\(566\) −9254.00 −0.687234
\(567\) 0 0
\(568\) −3648.00 −0.269484
\(569\) 8762.00 0.645557 0.322779 0.946474i \(-0.395383\pi\)
0.322779 + 0.946474i \(0.395383\pi\)
\(570\) 0 0
\(571\) −24764.0 −1.81496 −0.907479 0.420097i \(-0.861996\pi\)
−0.907479 + 0.420097i \(0.861996\pi\)
\(572\) −7548.00 −0.551744
\(573\) −22113.0 −1.61219
\(574\) 0 0
\(575\) 0 0
\(576\) 1408.00 0.101852
\(577\) 1811.00 0.130664 0.0653318 0.997864i \(-0.479189\pi\)
0.0653318 + 0.997864i \(0.479189\pi\)
\(578\) 6464.00 0.465168
\(579\) 14420.0 1.03502
\(580\) 0 0
\(581\) 0 0
\(582\) −14770.0 −1.05195
\(583\) −222.000 −0.0157707
\(584\) −5200.00 −0.368455
\(585\) 0 0
\(586\) 8166.00 0.575656
\(587\) 10548.0 0.741674 0.370837 0.928698i \(-0.379071\pi\)
0.370837 + 0.928698i \(0.379071\pi\)
\(588\) 0 0
\(589\) 14472.0 1.01241
\(590\) 0 0
\(591\) 12166.0 0.846772
\(592\) 5344.00 0.371009
\(593\) 17439.0 1.20765 0.603823 0.797119i \(-0.293643\pi\)
0.603823 + 0.797119i \(0.293643\pi\)
\(594\) 2590.00 0.178904
\(595\) 0 0
\(596\) −1208.00 −0.0830228
\(597\) −6258.00 −0.429017
\(598\) −7140.00 −0.488255
\(599\) 2451.00 0.167187 0.0835936 0.996500i \(-0.473360\pi\)
0.0835936 + 0.996500i \(0.473360\pi\)
\(600\) 0 0
\(601\) 7792.00 0.528856 0.264428 0.964405i \(-0.414817\pi\)
0.264428 + 0.964405i \(0.414817\pi\)
\(602\) 0 0
\(603\) −2332.00 −0.157490
\(604\) −12668.0 −0.853400
\(605\) 0 0
\(606\) −27440.0 −1.83940
\(607\) −1937.00 −0.129523 −0.0647615 0.997901i \(-0.520629\pi\)
−0.0647615 + 0.997901i \(0.520629\pi\)
\(608\) −3456.00 −0.230525
\(609\) 0 0
\(610\) 0 0
\(611\) −14637.0 −0.969148
\(612\) 3608.00 0.238308
\(613\) −5036.00 −0.331814 −0.165907 0.986141i \(-0.553055\pi\)
−0.165907 + 0.986141i \(0.553055\pi\)
\(614\) 8178.00 0.537520
\(615\) 0 0
\(616\) 0 0
\(617\) 27286.0 1.78038 0.890189 0.455592i \(-0.150572\pi\)
0.890189 + 0.455592i \(0.150572\pi\)
\(618\) 25550.0 1.66306
\(619\) −28538.0 −1.85305 −0.926526 0.376231i \(-0.877220\pi\)
−0.926526 + 0.376231i \(0.877220\pi\)
\(620\) 0 0
\(621\) 2450.00 0.158317
\(622\) −8016.00 −0.516740
\(623\) 0 0
\(624\) −5712.00 −0.366447
\(625\) 0 0
\(626\) −14710.0 −0.939185
\(627\) 27972.0 1.78165
\(628\) −1880.00 −0.119459
\(629\) 13694.0 0.868069
\(630\) 0 0
\(631\) 25007.0 1.57768 0.788838 0.614602i \(-0.210683\pi\)
0.788838 + 0.614602i \(0.210683\pi\)
\(632\) 9912.00 0.623858
\(633\) 28581.0 1.79462
\(634\) −3368.00 −0.210978
\(635\) 0 0
\(636\) −168.000 −0.0104743
\(637\) 0 0
\(638\) −18426.0 −1.14340
\(639\) 10032.0 0.621064
\(640\) 0 0
\(641\) −12130.0 −0.747436 −0.373718 0.927542i \(-0.621917\pi\)
−0.373718 + 0.927542i \(0.621917\pi\)
\(642\) 2016.00 0.123933
\(643\) 14385.0 0.882254 0.441127 0.897445i \(-0.354579\pi\)
0.441127 + 0.897445i \(0.354579\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8856.00 −0.539373
\(647\) −2208.00 −0.134166 −0.0670830 0.997747i \(-0.521369\pi\)
−0.0670830 + 0.997747i \(0.521369\pi\)
\(648\) 6712.00 0.406902
\(649\) −74.0000 −0.00447574
\(650\) 0 0
\(651\) 0 0
\(652\) 9560.00 0.574231
\(653\) −22448.0 −1.34527 −0.672633 0.739977i \(-0.734837\pi\)
−0.672633 + 0.739977i \(0.734837\pi\)
\(654\) 23534.0 1.40711
\(655\) 0 0
\(656\) −3296.00 −0.196169
\(657\) 14300.0 0.849157
\(658\) 0 0
\(659\) 8791.00 0.519649 0.259825 0.965656i \(-0.416335\pi\)
0.259825 + 0.965656i \(0.416335\pi\)
\(660\) 0 0
\(661\) 13180.0 0.775556 0.387778 0.921753i \(-0.373243\pi\)
0.387778 + 0.921753i \(0.373243\pi\)
\(662\) 2920.00 0.171434
\(663\) −14637.0 −0.857397
\(664\) −3424.00 −0.200116
\(665\) 0 0
\(666\) −14696.0 −0.855043
\(667\) −17430.0 −1.01183
\(668\) −10524.0 −0.609560
\(669\) −2639.00 −0.152511
\(670\) 0 0
\(671\) −34780.0 −2.00099
\(672\) 0 0
\(673\) 7164.00 0.410330 0.205165 0.978727i \(-0.434227\pi\)
0.205165 + 0.978727i \(0.434227\pi\)
\(674\) 15028.0 0.858838
\(675\) 0 0
\(676\) 1616.00 0.0919436
\(677\) 12335.0 0.700255 0.350127 0.936702i \(-0.386138\pi\)
0.350127 + 0.936702i \(0.386138\pi\)
\(678\) −11172.0 −0.632829
\(679\) 0 0
\(680\) 0 0
\(681\) 17857.0 1.00482
\(682\) 9916.00 0.556750
\(683\) 15436.0 0.864776 0.432388 0.901688i \(-0.357671\pi\)
0.432388 + 0.901688i \(0.357671\pi\)
\(684\) 9504.00 0.531279
\(685\) 0 0
\(686\) 0 0
\(687\) 518.000 0.0287670
\(688\) 6016.00 0.333369
\(689\) 306.000 0.0169197
\(690\) 0 0
\(691\) 19184.0 1.05614 0.528071 0.849200i \(-0.322916\pi\)
0.528071 + 0.849200i \(0.322916\pi\)
\(692\) −8972.00 −0.492867
\(693\) 0 0
\(694\) −5724.00 −0.313084
\(695\) 0 0
\(696\) −13944.0 −0.759405
\(697\) −8446.00 −0.458989
\(698\) −12736.0 −0.690637
\(699\) 13216.0 0.715129
\(700\) 0 0
\(701\) 32975.0 1.77667 0.888337 0.459192i \(-0.151861\pi\)
0.888337 + 0.459192i \(0.151861\pi\)
\(702\) −3570.00 −0.191939
\(703\) 36072.0 1.93525
\(704\) −2368.00 −0.126772
\(705\) 0 0
\(706\) 7270.00 0.387550
\(707\) 0 0
\(708\) −56.0000 −0.00297261
\(709\) −31497.0 −1.66840 −0.834199 0.551463i \(-0.814070\pi\)
−0.834199 + 0.551463i \(0.814070\pi\)
\(710\) 0 0
\(711\) −27258.0 −1.43777
\(712\) −1760.00 −0.0926387
\(713\) 9380.00 0.492684
\(714\) 0 0
\(715\) 0 0
\(716\) 208.000 0.0108566
\(717\) −34979.0 −1.82192
\(718\) −14232.0 −0.739740
\(719\) 18610.0 0.965279 0.482640 0.875819i \(-0.339678\pi\)
0.482640 + 0.875819i \(0.339678\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −9610.00 −0.495356
\(723\) −26810.0 −1.37908
\(724\) −9848.00 −0.505522
\(725\) 0 0
\(726\) 532.000 0.0271961
\(727\) −17508.0 −0.893172 −0.446586 0.894741i \(-0.647360\pi\)
−0.446586 + 0.894741i \(0.647360\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) 15416.0 0.780002
\(732\) −26320.0 −1.32898
\(733\) −4685.00 −0.236077 −0.118038 0.993009i \(-0.537661\pi\)
−0.118038 + 0.993009i \(0.537661\pi\)
\(734\) 638.000 0.0320831
\(735\) 0 0
\(736\) −2240.00 −0.112184
\(737\) 3922.00 0.196023
\(738\) 9064.00 0.452101
\(739\) −25925.0 −1.29048 −0.645241 0.763979i \(-0.723243\pi\)
−0.645241 + 0.763979i \(0.723243\pi\)
\(740\) 0 0
\(741\) −38556.0 −1.91146
\(742\) 0 0
\(743\) −25578.0 −1.26294 −0.631471 0.775400i \(-0.717548\pi\)
−0.631471 + 0.775400i \(0.717548\pi\)
\(744\) 7504.00 0.369771
\(745\) 0 0
\(746\) −23304.0 −1.14373
\(747\) 9416.00 0.461196
\(748\) −6068.00 −0.296615
\(749\) 0 0
\(750\) 0 0
\(751\) −4291.00 −0.208496 −0.104248 0.994551i \(-0.533244\pi\)
−0.104248 + 0.994551i \(0.533244\pi\)
\(752\) −4592.00 −0.222677
\(753\) −23730.0 −1.14843
\(754\) 25398.0 1.22671
\(755\) 0 0
\(756\) 0 0
\(757\) 31528.0 1.51374 0.756872 0.653563i \(-0.226726\pi\)
0.756872 + 0.653563i \(0.226726\pi\)
\(758\) −15496.0 −0.742533
\(759\) 18130.0 0.867032
\(760\) 0 0
\(761\) 23154.0 1.10293 0.551466 0.834197i \(-0.314068\pi\)
0.551466 + 0.834197i \(0.314068\pi\)
\(762\) −6076.00 −0.288859
\(763\) 0 0
\(764\) 12636.0 0.598370
\(765\) 0 0
\(766\) −17360.0 −0.818854
\(767\) 102.000 0.00480183
\(768\) −1792.00 −0.0841969
\(769\) −13992.0 −0.656131 −0.328065 0.944655i \(-0.606397\pi\)
−0.328065 + 0.944655i \(0.606397\pi\)
\(770\) 0 0
\(771\) −50190.0 −2.34442
\(772\) −8240.00 −0.384150
\(773\) −21681.0 −1.00881 −0.504406 0.863467i \(-0.668288\pi\)
−0.504406 + 0.863467i \(0.668288\pi\)
\(774\) −16544.0 −0.768297
\(775\) 0 0
\(776\) 8440.00 0.390436
\(777\) 0 0
\(778\) 3422.00 0.157692
\(779\) −22248.0 −1.02326
\(780\) 0 0
\(781\) −16872.0 −0.773019
\(782\) −5740.00 −0.262483
\(783\) −8715.00 −0.397763
\(784\) 0 0
\(785\) 0 0
\(786\) 18060.0 0.819566
\(787\) −16903.0 −0.765600 −0.382800 0.923831i \(-0.625040\pi\)
−0.382800 + 0.923831i \(0.625040\pi\)
\(788\) −6952.00 −0.314283
\(789\) −53704.0 −2.42321
\(790\) 0 0
\(791\) 0 0
\(792\) 6512.00 0.292164
\(793\) 47940.0 2.14678
\(794\) −3178.00 −0.142044
\(795\) 0 0
\(796\) 3576.00 0.159231
\(797\) 18905.0 0.840213 0.420106 0.907475i \(-0.361993\pi\)
0.420106 + 0.907475i \(0.361993\pi\)
\(798\) 0 0
\(799\) −11767.0 −0.521009
\(800\) 0 0
\(801\) 4840.00 0.213499
\(802\) 10294.0 0.453234
\(803\) −24050.0 −1.05692
\(804\) 2968.00 0.130191
\(805\) 0 0
\(806\) −13668.0 −0.597314
\(807\) −378.000 −0.0164885
\(808\) 15680.0 0.682699
\(809\) −5571.00 −0.242109 −0.121054 0.992646i \(-0.538628\pi\)
−0.121054 + 0.992646i \(0.538628\pi\)
\(810\) 0 0
\(811\) −10894.0 −0.471689 −0.235845 0.971791i \(-0.575786\pi\)
−0.235845 + 0.971791i \(0.575786\pi\)
\(812\) 0 0
\(813\) 20524.0 0.885373
\(814\) 24716.0 1.06424
\(815\) 0 0
\(816\) −4592.00 −0.197000
\(817\) 40608.0 1.73892
\(818\) −18200.0 −0.777932
\(819\) 0 0
\(820\) 0 0
\(821\) −30731.0 −1.30636 −0.653179 0.757204i \(-0.726565\pi\)
−0.653179 + 0.757204i \(0.726565\pi\)
\(822\) −2688.00 −0.114057
\(823\) −1038.00 −0.0439640 −0.0219820 0.999758i \(-0.506998\pi\)
−0.0219820 + 0.999758i \(0.506998\pi\)
\(824\) −14600.0 −0.617251
\(825\) 0 0
\(826\) 0 0
\(827\) −7958.00 −0.334615 −0.167308 0.985905i \(-0.553507\pi\)
−0.167308 + 0.985905i \(0.553507\pi\)
\(828\) 6160.00 0.258544
\(829\) 30666.0 1.28477 0.642385 0.766382i \(-0.277945\pi\)
0.642385 + 0.766382i \(0.277945\pi\)
\(830\) 0 0
\(831\) 22778.0 0.950854
\(832\) 3264.00 0.136008
\(833\) 0 0
\(834\) −19628.0 −0.814943
\(835\) 0 0
\(836\) −15984.0 −0.661266
\(837\) 4690.00 0.193680
\(838\) 5236.00 0.215841
\(839\) 5354.00 0.220311 0.110155 0.993914i \(-0.464865\pi\)
0.110155 + 0.993914i \(0.464865\pi\)
\(840\) 0 0
\(841\) 37612.0 1.54217
\(842\) 7390.00 0.302466
\(843\) −23289.0 −0.951502
\(844\) −16332.0 −0.666079
\(845\) 0 0
\(846\) 12628.0 0.513191
\(847\) 0 0
\(848\) 96.0000 0.00388756
\(849\) −32389.0 −1.30929
\(850\) 0 0
\(851\) 23380.0 0.941782
\(852\) −12768.0 −0.513409
\(853\) −42890.0 −1.72160 −0.860800 0.508943i \(-0.830037\pi\)
−0.860800 + 0.508943i \(0.830037\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1152.00 −0.0459983
\(857\) 22950.0 0.914769 0.457385 0.889269i \(-0.348786\pi\)
0.457385 + 0.889269i \(0.348786\pi\)
\(858\) −26418.0 −1.05116
\(859\) 2824.00 0.112170 0.0560848 0.998426i \(-0.482138\pi\)
0.0560848 + 0.998426i \(0.482138\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −31558.0 −1.24695
\(863\) 4866.00 0.191936 0.0959679 0.995384i \(-0.469405\pi\)
0.0959679 + 0.995384i \(0.469405\pi\)
\(864\) −1120.00 −0.0441009
\(865\) 0 0
\(866\) −14476.0 −0.568031
\(867\) 22624.0 0.886218
\(868\) 0 0
\(869\) 45843.0 1.78955
\(870\) 0 0
\(871\) −5406.00 −0.210305
\(872\) −13448.0 −0.522255
\(873\) −23210.0 −0.899816
\(874\) −15120.0 −0.585173
\(875\) 0 0
\(876\) −18200.0 −0.701965
\(877\) 10676.0 0.411064 0.205532 0.978650i \(-0.434108\pi\)
0.205532 + 0.978650i \(0.434108\pi\)
\(878\) −5292.00 −0.203413
\(879\) 28581.0 1.09672
\(880\) 0 0
\(881\) 29856.0 1.14174 0.570871 0.821040i \(-0.306606\pi\)
0.570871 + 0.821040i \(0.306606\pi\)
\(882\) 0 0
\(883\) 1944.00 0.0740893 0.0370446 0.999314i \(-0.488206\pi\)
0.0370446 + 0.999314i \(0.488206\pi\)
\(884\) 8364.00 0.318226
\(885\) 0 0
\(886\) 11376.0 0.431359
\(887\) −14628.0 −0.553732 −0.276866 0.960909i \(-0.589296\pi\)
−0.276866 + 0.960909i \(0.589296\pi\)
\(888\) 18704.0 0.706830
\(889\) 0 0
\(890\) 0 0
\(891\) 31043.0 1.16720
\(892\) 1508.00 0.0566049
\(893\) −30996.0 −1.16152
\(894\) −4228.00 −0.158172
\(895\) 0 0
\(896\) 0 0
\(897\) −24990.0 −0.930203
\(898\) 6570.00 0.244147
\(899\) −33366.0 −1.23784
\(900\) 0 0
\(901\) 246.000 0.00909595
\(902\) −15244.0 −0.562716
\(903\) 0 0
\(904\) 6384.00 0.234877
\(905\) 0 0
\(906\) −44338.0 −1.62586
\(907\) 12858.0 0.470720 0.235360 0.971908i \(-0.424373\pi\)
0.235360 + 0.971908i \(0.424373\pi\)
\(908\) −10204.0 −0.372942
\(909\) −43120.0 −1.57338
\(910\) 0 0
\(911\) −18324.0 −0.666412 −0.333206 0.942854i \(-0.608130\pi\)
−0.333206 + 0.942854i \(0.608130\pi\)
\(912\) −12096.0 −0.439187
\(913\) −15836.0 −0.574036
\(914\) 29668.0 1.07367
\(915\) 0 0
\(916\) −296.000 −0.0106770
\(917\) 0 0
\(918\) −2870.00 −0.103185
\(919\) −14751.0 −0.529478 −0.264739 0.964320i \(-0.585286\pi\)
−0.264739 + 0.964320i \(0.585286\pi\)
\(920\) 0 0
\(921\) 28623.0 1.02406
\(922\) −19944.0 −0.712387
\(923\) 23256.0 0.829340
\(924\) 0 0
\(925\) 0 0
\(926\) −18192.0 −0.645601
\(927\) 40150.0 1.42254
\(928\) 7968.00 0.281856
\(929\) 47922.0 1.69243 0.846216 0.532840i \(-0.178875\pi\)
0.846216 + 0.532840i \(0.178875\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −7552.00 −0.265423
\(933\) −28056.0 −0.984472
\(934\) −31734.0 −1.11174
\(935\) 0 0
\(936\) −8976.00 −0.313451
\(937\) −44987.0 −1.56848 −0.784238 0.620461i \(-0.786946\pi\)
−0.784238 + 0.620461i \(0.786946\pi\)
\(938\) 0 0
\(939\) −51485.0 −1.78930
\(940\) 0 0
\(941\) 20356.0 0.705193 0.352597 0.935775i \(-0.385299\pi\)
0.352597 + 0.935775i \(0.385299\pi\)
\(942\) −6580.00 −0.227588
\(943\) −14420.0 −0.497964
\(944\) 32.0000 0.00110330
\(945\) 0 0
\(946\) 27824.0 0.956275
\(947\) 27786.0 0.953457 0.476728 0.879051i \(-0.341823\pi\)
0.476728 + 0.879051i \(0.341823\pi\)
\(948\) 34692.0 1.18855
\(949\) 33150.0 1.13392
\(950\) 0 0
\(951\) −11788.0 −0.401948
\(952\) 0 0
\(953\) 48674.0 1.65447 0.827233 0.561859i \(-0.189914\pi\)
0.827233 + 0.561859i \(0.189914\pi\)
\(954\) −264.000 −0.00895945
\(955\) 0 0
\(956\) 19988.0 0.676211
\(957\) −64491.0 −2.17837
\(958\) 484.000 0.0163229
\(959\) 0 0
\(960\) 0 0
\(961\) −11835.0 −0.397268
\(962\) −34068.0 −1.14178
\(963\) 3168.00 0.106010
\(964\) 15320.0 0.511851
\(965\) 0 0
\(966\) 0 0
\(967\) 11168.0 0.371395 0.185697 0.982607i \(-0.440546\pi\)
0.185697 + 0.982607i \(0.440546\pi\)
\(968\) −304.000 −0.0100939
\(969\) −30996.0 −1.02759
\(970\) 0 0
\(971\) −20094.0 −0.664106 −0.332053 0.943261i \(-0.607741\pi\)
−0.332053 + 0.943261i \(0.607741\pi\)
\(972\) 19712.0 0.650476
\(973\) 0 0
\(974\) −7116.00 −0.234098
\(975\) 0 0
\(976\) 15040.0 0.493257
\(977\) −49104.0 −1.60796 −0.803980 0.594657i \(-0.797288\pi\)
−0.803980 + 0.594657i \(0.797288\pi\)
\(978\) 33460.0 1.09400
\(979\) −8140.00 −0.265736
\(980\) 0 0
\(981\) 36982.0 1.20361
\(982\) −2946.00 −0.0957338
\(983\) 27751.0 0.900427 0.450213 0.892921i \(-0.351348\pi\)
0.450213 + 0.892921i \(0.351348\pi\)
\(984\) −11536.0 −0.373734
\(985\) 0 0
\(986\) 20418.0 0.659474
\(987\) 0 0
\(988\) 22032.0 0.709445
\(989\) 26320.0 0.846236
\(990\) 0 0
\(991\) 37600.0 1.20525 0.602625 0.798024i \(-0.294121\pi\)
0.602625 + 0.798024i \(0.294121\pi\)
\(992\) −4288.00 −0.137242
\(993\) 10220.0 0.326608
\(994\) 0 0
\(995\) 0 0
\(996\) −11984.0 −0.381253
\(997\) −10911.0 −0.346595 −0.173297 0.984870i \(-0.555442\pi\)
−0.173297 + 0.984870i \(0.555442\pi\)
\(998\) −1206.00 −0.0382518
\(999\) 11690.0 0.370225
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 2450.4.a.c.1.1 1
5.2 odd 4 490.4.c.a.99.1 2
5.3 odd 4 490.4.c.a.99.2 2
5.4 even 2 2450.4.a.bn.1.1 1
7.6 odd 2 350.4.a.i.1.1 1
35.13 even 4 70.4.c.a.29.2 yes 2
35.27 even 4 70.4.c.a.29.1 2
35.34 odd 2 350.4.a.m.1.1 1
105.62 odd 4 630.4.g.a.379.2 2
105.83 odd 4 630.4.g.a.379.1 2
140.27 odd 4 560.4.g.c.449.2 2
140.83 odd 4 560.4.g.c.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.c.a.29.1 2 35.27 even 4
70.4.c.a.29.2 yes 2 35.13 even 4
350.4.a.i.1.1 1 7.6 odd 2
350.4.a.m.1.1 1 35.34 odd 2
490.4.c.a.99.1 2 5.2 odd 4
490.4.c.a.99.2 2 5.3 odd 4
560.4.g.c.449.1 2 140.83 odd 4
560.4.g.c.449.2 2 140.27 odd 4
630.4.g.a.379.1 2 105.83 odd 4
630.4.g.a.379.2 2 105.62 odd 4
2450.4.a.c.1.1 1 1.1 even 1 trivial
2450.4.a.bn.1.1 1 5.4 even 2