Properties

Label 1386.4.a.m.1.1
Level $1386$
Weight $4$
Character 1386.1
Self dual yes
Analytic conductor $81.777$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(81.7766472680\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +28.0000 q^{10} -11.0000 q^{11} +38.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -54.0000 q^{17} +40.0000 q^{19} +56.0000 q^{20} -22.0000 q^{22} -8.00000 q^{23} +71.0000 q^{25} +76.0000 q^{26} -28.0000 q^{28} +170.000 q^{29} +92.0000 q^{31} +32.0000 q^{32} -108.000 q^{34} -98.0000 q^{35} +294.000 q^{37} +80.0000 q^{38} +112.000 q^{40} +258.000 q^{41} -52.0000 q^{43} -44.0000 q^{44} -16.0000 q^{46} +76.0000 q^{47} +49.0000 q^{49} +142.000 q^{50} +152.000 q^{52} +322.000 q^{53} -154.000 q^{55} -56.0000 q^{56} +340.000 q^{58} -260.000 q^{59} +22.0000 q^{61} +184.000 q^{62} +64.0000 q^{64} +532.000 q^{65} -436.000 q^{67} -216.000 q^{68} -196.000 q^{70} +368.000 q^{71} -2.00000 q^{73} +588.000 q^{74} +160.000 q^{76} +77.0000 q^{77} -200.000 q^{79} +224.000 q^{80} +516.000 q^{82} +952.000 q^{83} -756.000 q^{85} -104.000 q^{86} -88.0000 q^{88} +70.0000 q^{89} -266.000 q^{91} -32.0000 q^{92} +152.000 q^{94} +560.000 q^{95} -1086.00 q^{97} +98.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 14.0000 1.25220 0.626099 0.779744i \(-0.284651\pi\)
0.626099 + 0.779744i \(0.284651\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 28.0000 0.885438
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 38.0000 0.810716 0.405358 0.914158i \(-0.367147\pi\)
0.405358 + 0.914158i \(0.367147\pi\)
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) 0 0
\(19\) 40.0000 0.482980 0.241490 0.970403i \(-0.422364\pi\)
0.241490 + 0.970403i \(0.422364\pi\)
\(20\) 56.0000 0.626099
\(21\) 0 0
\(22\) −22.0000 −0.213201
\(23\) −8.00000 −0.0725268 −0.0362634 0.999342i \(-0.511546\pi\)
−0.0362634 + 0.999342i \(0.511546\pi\)
\(24\) 0 0
\(25\) 71.0000 0.568000
\(26\) 76.0000 0.573263
\(27\) 0 0
\(28\) −28.0000 −0.188982
\(29\) 170.000 1.08856 0.544279 0.838904i \(-0.316803\pi\)
0.544279 + 0.838904i \(0.316803\pi\)
\(30\) 0 0
\(31\) 92.0000 0.533022 0.266511 0.963832i \(-0.414129\pi\)
0.266511 + 0.963832i \(0.414129\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −108.000 −0.544760
\(35\) −98.0000 −0.473286
\(36\) 0 0
\(37\) 294.000 1.30631 0.653153 0.757226i \(-0.273446\pi\)
0.653153 + 0.757226i \(0.273446\pi\)
\(38\) 80.0000 0.341519
\(39\) 0 0
\(40\) 112.000 0.442719
\(41\) 258.000 0.982752 0.491376 0.870948i \(-0.336494\pi\)
0.491376 + 0.870948i \(0.336494\pi\)
\(42\) 0 0
\(43\) −52.0000 −0.184417 −0.0922084 0.995740i \(-0.529393\pi\)
−0.0922084 + 0.995740i \(0.529393\pi\)
\(44\) −44.0000 −0.150756
\(45\) 0 0
\(46\) −16.0000 −0.0512842
\(47\) 76.0000 0.235867 0.117933 0.993022i \(-0.462373\pi\)
0.117933 + 0.993022i \(0.462373\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 142.000 0.401637
\(51\) 0 0
\(52\) 152.000 0.405358
\(53\) 322.000 0.834530 0.417265 0.908785i \(-0.362989\pi\)
0.417265 + 0.908785i \(0.362989\pi\)
\(54\) 0 0
\(55\) −154.000 −0.377552
\(56\) −56.0000 −0.133631
\(57\) 0 0
\(58\) 340.000 0.769727
\(59\) −260.000 −0.573714 −0.286857 0.957973i \(-0.592610\pi\)
−0.286857 + 0.957973i \(0.592610\pi\)
\(60\) 0 0
\(61\) 22.0000 0.0461772 0.0230886 0.999733i \(-0.492650\pi\)
0.0230886 + 0.999733i \(0.492650\pi\)
\(62\) 184.000 0.376904
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 532.000 1.01518
\(66\) 0 0
\(67\) −436.000 −0.795013 −0.397507 0.917599i \(-0.630124\pi\)
−0.397507 + 0.917599i \(0.630124\pi\)
\(68\) −216.000 −0.385204
\(69\) 0 0
\(70\) −196.000 −0.334664
\(71\) 368.000 0.615121 0.307560 0.951529i \(-0.400487\pi\)
0.307560 + 0.951529i \(0.400487\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.00320661 −0.00160330 0.999999i \(-0.500510\pi\)
−0.00160330 + 0.999999i \(0.500510\pi\)
\(74\) 588.000 0.923697
\(75\) 0 0
\(76\) 160.000 0.241490
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) −200.000 −0.284832 −0.142416 0.989807i \(-0.545487\pi\)
−0.142416 + 0.989807i \(0.545487\pi\)
\(80\) 224.000 0.313050
\(81\) 0 0
\(82\) 516.000 0.694911
\(83\) 952.000 1.25898 0.629491 0.777007i \(-0.283263\pi\)
0.629491 + 0.777007i \(0.283263\pi\)
\(84\) 0 0
\(85\) −756.000 −0.964703
\(86\) −104.000 −0.130402
\(87\) 0 0
\(88\) −88.0000 −0.106600
\(89\) 70.0000 0.0833706 0.0416853 0.999131i \(-0.486727\pi\)
0.0416853 + 0.999131i \(0.486727\pi\)
\(90\) 0 0
\(91\) −266.000 −0.306422
\(92\) −32.0000 −0.0362634
\(93\) 0 0
\(94\) 152.000 0.166783
\(95\) 560.000 0.604787
\(96\) 0 0
\(97\) −1086.00 −1.13677 −0.568385 0.822763i \(-0.692431\pi\)
−0.568385 + 0.822763i \(0.692431\pi\)
\(98\) 98.0000 0.101015
\(99\) 0 0
\(100\) 284.000 0.284000
\(101\) −102.000 −0.100489 −0.0502445 0.998737i \(-0.516000\pi\)
−0.0502445 + 0.998737i \(0.516000\pi\)
\(102\) 0 0
\(103\) 188.000 0.179847 0.0899233 0.995949i \(-0.471338\pi\)
0.0899233 + 0.995949i \(0.471338\pi\)
\(104\) 304.000 0.286631
\(105\) 0 0
\(106\) 644.000 0.590102
\(107\) 1716.00 1.55039 0.775196 0.631721i \(-0.217651\pi\)
0.775196 + 0.631721i \(0.217651\pi\)
\(108\) 0 0
\(109\) 310.000 0.272409 0.136205 0.990681i \(-0.456510\pi\)
0.136205 + 0.990681i \(0.456510\pi\)
\(110\) −308.000 −0.266970
\(111\) 0 0
\(112\) −112.000 −0.0944911
\(113\) 1822.00 1.51681 0.758404 0.651785i \(-0.225979\pi\)
0.758404 + 0.651785i \(0.225979\pi\)
\(114\) 0 0
\(115\) −112.000 −0.0908179
\(116\) 680.000 0.544279
\(117\) 0 0
\(118\) −520.000 −0.405677
\(119\) 378.000 0.291187
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 44.0000 0.0326522
\(123\) 0 0
\(124\) 368.000 0.266511
\(125\) −756.000 −0.540950
\(126\) 0 0
\(127\) −1576.00 −1.10116 −0.550580 0.834782i \(-0.685594\pi\)
−0.550580 + 0.834782i \(0.685594\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 1064.00 0.717838
\(131\) −1072.00 −0.714970 −0.357485 0.933919i \(-0.616366\pi\)
−0.357485 + 0.933919i \(0.616366\pi\)
\(132\) 0 0
\(133\) −280.000 −0.182549
\(134\) −872.000 −0.562159
\(135\) 0 0
\(136\) −432.000 −0.272380
\(137\) 246.000 0.153410 0.0767051 0.997054i \(-0.475560\pi\)
0.0767051 + 0.997054i \(0.475560\pi\)
\(138\) 0 0
\(139\) 1640.00 1.00074 0.500370 0.865811i \(-0.333197\pi\)
0.500370 + 0.865811i \(0.333197\pi\)
\(140\) −392.000 −0.236643
\(141\) 0 0
\(142\) 736.000 0.434956
\(143\) −418.000 −0.244440
\(144\) 0 0
\(145\) 2380.00 1.36309
\(146\) −4.00000 −0.00226741
\(147\) 0 0
\(148\) 1176.00 0.653153
\(149\) 850.000 0.467347 0.233674 0.972315i \(-0.424925\pi\)
0.233674 + 0.972315i \(0.424925\pi\)
\(150\) 0 0
\(151\) −1448.00 −0.780375 −0.390187 0.920735i \(-0.627590\pi\)
−0.390187 + 0.920735i \(0.627590\pi\)
\(152\) 320.000 0.170759
\(153\) 0 0
\(154\) 154.000 0.0805823
\(155\) 1288.00 0.667449
\(156\) 0 0
\(157\) −1886.00 −0.958721 −0.479360 0.877618i \(-0.659131\pi\)
−0.479360 + 0.877618i \(0.659131\pi\)
\(158\) −400.000 −0.201407
\(159\) 0 0
\(160\) 448.000 0.221359
\(161\) 56.0000 0.0274125
\(162\) 0 0
\(163\) 228.000 0.109560 0.0547802 0.998498i \(-0.482554\pi\)
0.0547802 + 0.998498i \(0.482554\pi\)
\(164\) 1032.00 0.491376
\(165\) 0 0
\(166\) 1904.00 0.890235
\(167\) −1664.00 −0.771043 −0.385522 0.922699i \(-0.625978\pi\)
−0.385522 + 0.922699i \(0.625978\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) −1512.00 −0.682148
\(171\) 0 0
\(172\) −208.000 −0.0922084
\(173\) −2438.00 −1.07143 −0.535716 0.844398i \(-0.679958\pi\)
−0.535716 + 0.844398i \(0.679958\pi\)
\(174\) 0 0
\(175\) −497.000 −0.214684
\(176\) −176.000 −0.0753778
\(177\) 0 0
\(178\) 140.000 0.0589519
\(179\) −1620.00 −0.676450 −0.338225 0.941065i \(-0.609826\pi\)
−0.338225 + 0.941065i \(0.609826\pi\)
\(180\) 0 0
\(181\) 3602.00 1.47920 0.739598 0.673049i \(-0.235016\pi\)
0.739598 + 0.673049i \(0.235016\pi\)
\(182\) −532.000 −0.216673
\(183\) 0 0
\(184\) −64.0000 −0.0256421
\(185\) 4116.00 1.63575
\(186\) 0 0
\(187\) 594.000 0.232287
\(188\) 304.000 0.117933
\(189\) 0 0
\(190\) 1120.00 0.427649
\(191\) −3472.00 −1.31531 −0.657657 0.753317i \(-0.728453\pi\)
−0.657657 + 0.753317i \(0.728453\pi\)
\(192\) 0 0
\(193\) −222.000 −0.0827975 −0.0413987 0.999143i \(-0.513181\pi\)
−0.0413987 + 0.999143i \(0.513181\pi\)
\(194\) −2172.00 −0.803817
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) 3906.00 1.41264 0.706322 0.707890i \(-0.250353\pi\)
0.706322 + 0.707890i \(0.250353\pi\)
\(198\) 0 0
\(199\) 1900.00 0.676821 0.338411 0.940999i \(-0.390111\pi\)
0.338411 + 0.940999i \(0.390111\pi\)
\(200\) 568.000 0.200818
\(201\) 0 0
\(202\) −204.000 −0.0710564
\(203\) −1190.00 −0.411437
\(204\) 0 0
\(205\) 3612.00 1.23060
\(206\) 376.000 0.127171
\(207\) 0 0
\(208\) 608.000 0.202679
\(209\) −440.000 −0.145624
\(210\) 0 0
\(211\) 1172.00 0.382388 0.191194 0.981552i \(-0.438764\pi\)
0.191194 + 0.981552i \(0.438764\pi\)
\(212\) 1288.00 0.417265
\(213\) 0 0
\(214\) 3432.00 1.09629
\(215\) −728.000 −0.230926
\(216\) 0 0
\(217\) −644.000 −0.201463
\(218\) 620.000 0.192622
\(219\) 0 0
\(220\) −616.000 −0.188776
\(221\) −2052.00 −0.624581
\(222\) 0 0
\(223\) −2372.00 −0.712291 −0.356145 0.934431i \(-0.615909\pi\)
−0.356145 + 0.934431i \(0.615909\pi\)
\(224\) −224.000 −0.0668153
\(225\) 0 0
\(226\) 3644.00 1.07255
\(227\) −1024.00 −0.299406 −0.149703 0.988731i \(-0.547832\pi\)
−0.149703 + 0.988731i \(0.547832\pi\)
\(228\) 0 0
\(229\) 290.000 0.0836845 0.0418422 0.999124i \(-0.486677\pi\)
0.0418422 + 0.999124i \(0.486677\pi\)
\(230\) −224.000 −0.0642179
\(231\) 0 0
\(232\) 1360.00 0.384864
\(233\) 5022.00 1.41203 0.706013 0.708199i \(-0.250492\pi\)
0.706013 + 0.708199i \(0.250492\pi\)
\(234\) 0 0
\(235\) 1064.00 0.295352
\(236\) −1040.00 −0.286857
\(237\) 0 0
\(238\) 756.000 0.205900
\(239\) −200.000 −0.0541294 −0.0270647 0.999634i \(-0.508616\pi\)
−0.0270647 + 0.999634i \(0.508616\pi\)
\(240\) 0 0
\(241\) −1498.00 −0.400393 −0.200196 0.979756i \(-0.564158\pi\)
−0.200196 + 0.979756i \(0.564158\pi\)
\(242\) 242.000 0.0642824
\(243\) 0 0
\(244\) 88.0000 0.0230886
\(245\) 686.000 0.178885
\(246\) 0 0
\(247\) 1520.00 0.391560
\(248\) 736.000 0.188452
\(249\) 0 0
\(250\) −1512.00 −0.382509
\(251\) 7468.00 1.87799 0.938996 0.343928i \(-0.111758\pi\)
0.938996 + 0.343928i \(0.111758\pi\)
\(252\) 0 0
\(253\) 88.0000 0.0218676
\(254\) −3152.00 −0.778638
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2194.00 −0.532521 −0.266261 0.963901i \(-0.585788\pi\)
−0.266261 + 0.963901i \(0.585788\pi\)
\(258\) 0 0
\(259\) −2058.00 −0.493737
\(260\) 2128.00 0.507588
\(261\) 0 0
\(262\) −2144.00 −0.505560
\(263\) −3288.00 −0.770900 −0.385450 0.922729i \(-0.625954\pi\)
−0.385450 + 0.922729i \(0.625954\pi\)
\(264\) 0 0
\(265\) 4508.00 1.04500
\(266\) −560.000 −0.129082
\(267\) 0 0
\(268\) −1744.00 −0.397507
\(269\) 6710.00 1.52088 0.760439 0.649410i \(-0.224984\pi\)
0.760439 + 0.649410i \(0.224984\pi\)
\(270\) 0 0
\(271\) 4952.00 1.11001 0.555005 0.831847i \(-0.312716\pi\)
0.555005 + 0.831847i \(0.312716\pi\)
\(272\) −864.000 −0.192602
\(273\) 0 0
\(274\) 492.000 0.108477
\(275\) −781.000 −0.171258
\(276\) 0 0
\(277\) −3666.00 −0.795193 −0.397597 0.917560i \(-0.630156\pi\)
−0.397597 + 0.917560i \(0.630156\pi\)
\(278\) 3280.00 0.707631
\(279\) 0 0
\(280\) −784.000 −0.167332
\(281\) 6798.00 1.44318 0.721592 0.692319i \(-0.243411\pi\)
0.721592 + 0.692319i \(0.243411\pi\)
\(282\) 0 0
\(283\) −1992.00 −0.418417 −0.209209 0.977871i \(-0.567089\pi\)
−0.209209 + 0.977871i \(0.567089\pi\)
\(284\) 1472.00 0.307560
\(285\) 0 0
\(286\) −836.000 −0.172845
\(287\) −1806.00 −0.371445
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 4760.00 0.963851
\(291\) 0 0
\(292\) −8.00000 −0.00160330
\(293\) 4002.00 0.797950 0.398975 0.916962i \(-0.369366\pi\)
0.398975 + 0.916962i \(0.369366\pi\)
\(294\) 0 0
\(295\) −3640.00 −0.718403
\(296\) 2352.00 0.461849
\(297\) 0 0
\(298\) 1700.00 0.330464
\(299\) −304.000 −0.0587986
\(300\) 0 0
\(301\) 364.000 0.0697030
\(302\) −2896.00 −0.551808
\(303\) 0 0
\(304\) 640.000 0.120745
\(305\) 308.000 0.0578230
\(306\) 0 0
\(307\) 5344.00 0.993479 0.496740 0.867900i \(-0.334530\pi\)
0.496740 + 0.867900i \(0.334530\pi\)
\(308\) 308.000 0.0569803
\(309\) 0 0
\(310\) 2576.00 0.471958
\(311\) 2268.00 0.413526 0.206763 0.978391i \(-0.433707\pi\)
0.206763 + 0.978391i \(0.433707\pi\)
\(312\) 0 0
\(313\) −7422.00 −1.34031 −0.670154 0.742222i \(-0.733772\pi\)
−0.670154 + 0.742222i \(0.733772\pi\)
\(314\) −3772.00 −0.677918
\(315\) 0 0
\(316\) −800.000 −0.142416
\(317\) 7626.00 1.35116 0.675582 0.737285i \(-0.263893\pi\)
0.675582 + 0.737285i \(0.263893\pi\)
\(318\) 0 0
\(319\) −1870.00 −0.328213
\(320\) 896.000 0.156525
\(321\) 0 0
\(322\) 112.000 0.0193836
\(323\) −2160.00 −0.372092
\(324\) 0 0
\(325\) 2698.00 0.460487
\(326\) 456.000 0.0774709
\(327\) 0 0
\(328\) 2064.00 0.347455
\(329\) −532.000 −0.0891493
\(330\) 0 0
\(331\) 1492.00 0.247758 0.123879 0.992297i \(-0.460467\pi\)
0.123879 + 0.992297i \(0.460467\pi\)
\(332\) 3808.00 0.629491
\(333\) 0 0
\(334\) −3328.00 −0.545210
\(335\) −6104.00 −0.995514
\(336\) 0 0
\(337\) 74.0000 0.0119615 0.00598077 0.999982i \(-0.498096\pi\)
0.00598077 + 0.999982i \(0.498096\pi\)
\(338\) −1506.00 −0.242354
\(339\) 0 0
\(340\) −3024.00 −0.482351
\(341\) −1012.00 −0.160712
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −416.000 −0.0652012
\(345\) 0 0
\(346\) −4876.00 −0.757617
\(347\) −11724.0 −1.81377 −0.906884 0.421381i \(-0.861546\pi\)
−0.906884 + 0.421381i \(0.861546\pi\)
\(348\) 0 0
\(349\) 6350.00 0.973948 0.486974 0.873417i \(-0.338101\pi\)
0.486974 + 0.873417i \(0.338101\pi\)
\(350\) −994.000 −0.151804
\(351\) 0 0
\(352\) −352.000 −0.0533002
\(353\) 5822.00 0.877829 0.438915 0.898529i \(-0.355363\pi\)
0.438915 + 0.898529i \(0.355363\pi\)
\(354\) 0 0
\(355\) 5152.00 0.770253
\(356\) 280.000 0.0416853
\(357\) 0 0
\(358\) −3240.00 −0.478322
\(359\) −13320.0 −1.95822 −0.979112 0.203320i \(-0.934827\pi\)
−0.979112 + 0.203320i \(0.934827\pi\)
\(360\) 0 0
\(361\) −5259.00 −0.766730
\(362\) 7204.00 1.04595
\(363\) 0 0
\(364\) −1064.00 −0.153211
\(365\) −28.0000 −0.00401531
\(366\) 0 0
\(367\) −11076.0 −1.57537 −0.787687 0.616075i \(-0.788722\pi\)
−0.787687 + 0.616075i \(0.788722\pi\)
\(368\) −128.000 −0.0181317
\(369\) 0 0
\(370\) 8232.00 1.15665
\(371\) −2254.00 −0.315423
\(372\) 0 0
\(373\) −3922.00 −0.544433 −0.272216 0.962236i \(-0.587757\pi\)
−0.272216 + 0.962236i \(0.587757\pi\)
\(374\) 1188.00 0.164251
\(375\) 0 0
\(376\) 608.000 0.0833915
\(377\) 6460.00 0.882512
\(378\) 0 0
\(379\) −11620.0 −1.57488 −0.787440 0.616392i \(-0.788594\pi\)
−0.787440 + 0.616392i \(0.788594\pi\)
\(380\) 2240.00 0.302394
\(381\) 0 0
\(382\) −6944.00 −0.930068
\(383\) −3628.00 −0.484026 −0.242013 0.970273i \(-0.577808\pi\)
−0.242013 + 0.970273i \(0.577808\pi\)
\(384\) 0 0
\(385\) 1078.00 0.142701
\(386\) −444.000 −0.0585466
\(387\) 0 0
\(388\) −4344.00 −0.568385
\(389\) −9750.00 −1.27081 −0.635404 0.772180i \(-0.719167\pi\)
−0.635404 + 0.772180i \(0.719167\pi\)
\(390\) 0 0
\(391\) 432.000 0.0558751
\(392\) 392.000 0.0505076
\(393\) 0 0
\(394\) 7812.00 0.998891
\(395\) −2800.00 −0.356667
\(396\) 0 0
\(397\) −6606.00 −0.835128 −0.417564 0.908648i \(-0.637116\pi\)
−0.417564 + 0.908648i \(0.637116\pi\)
\(398\) 3800.00 0.478585
\(399\) 0 0
\(400\) 1136.00 0.142000
\(401\) −2.00000 −0.000249065 0 −0.000124533 1.00000i \(-0.500040\pi\)
−0.000124533 1.00000i \(0.500040\pi\)
\(402\) 0 0
\(403\) 3496.00 0.432129
\(404\) −408.000 −0.0502445
\(405\) 0 0
\(406\) −2380.00 −0.290930
\(407\) −3234.00 −0.393866
\(408\) 0 0
\(409\) −13930.0 −1.68409 −0.842047 0.539405i \(-0.818649\pi\)
−0.842047 + 0.539405i \(0.818649\pi\)
\(410\) 7224.00 0.870166
\(411\) 0 0
\(412\) 752.000 0.0899233
\(413\) 1820.00 0.216843
\(414\) 0 0
\(415\) 13328.0 1.57650
\(416\) 1216.00 0.143316
\(417\) 0 0
\(418\) −880.000 −0.102972
\(419\) 7740.00 0.902443 0.451222 0.892412i \(-0.350988\pi\)
0.451222 + 0.892412i \(0.350988\pi\)
\(420\) 0 0
\(421\) −5618.00 −0.650367 −0.325184 0.945651i \(-0.605426\pi\)
−0.325184 + 0.945651i \(0.605426\pi\)
\(422\) 2344.00 0.270389
\(423\) 0 0
\(424\) 2576.00 0.295051
\(425\) −3834.00 −0.437591
\(426\) 0 0
\(427\) −154.000 −0.0174534
\(428\) 6864.00 0.775196
\(429\) 0 0
\(430\) −1456.00 −0.163290
\(431\) 9008.00 1.00673 0.503364 0.864074i \(-0.332095\pi\)
0.503364 + 0.864074i \(0.332095\pi\)
\(432\) 0 0
\(433\) −10702.0 −1.18777 −0.593886 0.804549i \(-0.702407\pi\)
−0.593886 + 0.804549i \(0.702407\pi\)
\(434\) −1288.00 −0.142456
\(435\) 0 0
\(436\) 1240.00 0.136205
\(437\) −320.000 −0.0350290
\(438\) 0 0
\(439\) −4960.00 −0.539243 −0.269622 0.962966i \(-0.586899\pi\)
−0.269622 + 0.962966i \(0.586899\pi\)
\(440\) −1232.00 −0.133485
\(441\) 0 0
\(442\) −4104.00 −0.441646
\(443\) −188.000 −0.0201629 −0.0100814 0.999949i \(-0.503209\pi\)
−0.0100814 + 0.999949i \(0.503209\pi\)
\(444\) 0 0
\(445\) 980.000 0.104397
\(446\) −4744.00 −0.503666
\(447\) 0 0
\(448\) −448.000 −0.0472456
\(449\) 3150.00 0.331086 0.165543 0.986203i \(-0.447062\pi\)
0.165543 + 0.986203i \(0.447062\pi\)
\(450\) 0 0
\(451\) −2838.00 −0.296311
\(452\) 7288.00 0.758404
\(453\) 0 0
\(454\) −2048.00 −0.211712
\(455\) −3724.00 −0.383701
\(456\) 0 0
\(457\) −12806.0 −1.31081 −0.655404 0.755278i \(-0.727502\pi\)
−0.655404 + 0.755278i \(0.727502\pi\)
\(458\) 580.000 0.0591738
\(459\) 0 0
\(460\) −448.000 −0.0454089
\(461\) 1778.00 0.179631 0.0898153 0.995958i \(-0.471372\pi\)
0.0898153 + 0.995958i \(0.471372\pi\)
\(462\) 0 0
\(463\) −15672.0 −1.57309 −0.786544 0.617534i \(-0.788132\pi\)
−0.786544 + 0.617534i \(0.788132\pi\)
\(464\) 2720.00 0.272140
\(465\) 0 0
\(466\) 10044.0 0.998453
\(467\) −4.00000 −0.000396355 0 −0.000198178 1.00000i \(-0.500063\pi\)
−0.000198178 1.00000i \(0.500063\pi\)
\(468\) 0 0
\(469\) 3052.00 0.300487
\(470\) 2128.00 0.208845
\(471\) 0 0
\(472\) −2080.00 −0.202838
\(473\) 572.000 0.0556038
\(474\) 0 0
\(475\) 2840.00 0.274333
\(476\) 1512.00 0.145593
\(477\) 0 0
\(478\) −400.000 −0.0382753
\(479\) −14640.0 −1.39649 −0.698245 0.715859i \(-0.746035\pi\)
−0.698245 + 0.715859i \(0.746035\pi\)
\(480\) 0 0
\(481\) 11172.0 1.05904
\(482\) −2996.00 −0.283120
\(483\) 0 0
\(484\) 484.000 0.0454545
\(485\) −15204.0 −1.42346
\(486\) 0 0
\(487\) −19896.0 −1.85128 −0.925640 0.378404i \(-0.876473\pi\)
−0.925640 + 0.378404i \(0.876473\pi\)
\(488\) 176.000 0.0163261
\(489\) 0 0
\(490\) 1372.00 0.126491
\(491\) −7972.00 −0.732732 −0.366366 0.930471i \(-0.619398\pi\)
−0.366366 + 0.930471i \(0.619398\pi\)
\(492\) 0 0
\(493\) −9180.00 −0.838634
\(494\) 3040.00 0.276875
\(495\) 0 0
\(496\) 1472.00 0.133256
\(497\) −2576.00 −0.232494
\(498\) 0 0
\(499\) −2260.00 −0.202748 −0.101374 0.994848i \(-0.532324\pi\)
−0.101374 + 0.994848i \(0.532324\pi\)
\(500\) −3024.00 −0.270475
\(501\) 0 0
\(502\) 14936.0 1.32794
\(503\) −6288.00 −0.557392 −0.278696 0.960379i \(-0.589902\pi\)
−0.278696 + 0.960379i \(0.589902\pi\)
\(504\) 0 0
\(505\) −1428.00 −0.125832
\(506\) 176.000 0.0154628
\(507\) 0 0
\(508\) −6304.00 −0.550580
\(509\) −12010.0 −1.04584 −0.522921 0.852381i \(-0.675158\pi\)
−0.522921 + 0.852381i \(0.675158\pi\)
\(510\) 0 0
\(511\) 14.0000 0.00121198
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) −4388.00 −0.376549
\(515\) 2632.00 0.225203
\(516\) 0 0
\(517\) −836.000 −0.0711165
\(518\) −4116.00 −0.349125
\(519\) 0 0
\(520\) 4256.00 0.358919
\(521\) −10842.0 −0.911702 −0.455851 0.890056i \(-0.650665\pi\)
−0.455851 + 0.890056i \(0.650665\pi\)
\(522\) 0 0
\(523\) 18808.0 1.57250 0.786249 0.617910i \(-0.212020\pi\)
0.786249 + 0.617910i \(0.212020\pi\)
\(524\) −4288.00 −0.357485
\(525\) 0 0
\(526\) −6576.00 −0.545109
\(527\) −4968.00 −0.410644
\(528\) 0 0
\(529\) −12103.0 −0.994740
\(530\) 9016.00 0.738925
\(531\) 0 0
\(532\) −1120.00 −0.0912747
\(533\) 9804.00 0.796732
\(534\) 0 0
\(535\) 24024.0 1.94140
\(536\) −3488.00 −0.281080
\(537\) 0 0
\(538\) 13420.0 1.07542
\(539\) −539.000 −0.0430730
\(540\) 0 0
\(541\) 15622.0 1.24148 0.620741 0.784015i \(-0.286832\pi\)
0.620741 + 0.784015i \(0.286832\pi\)
\(542\) 9904.00 0.784895
\(543\) 0 0
\(544\) −1728.00 −0.136190
\(545\) 4340.00 0.341110
\(546\) 0 0
\(547\) 2284.00 0.178532 0.0892658 0.996008i \(-0.471548\pi\)
0.0892658 + 0.996008i \(0.471548\pi\)
\(548\) 984.000 0.0767051
\(549\) 0 0
\(550\) −1562.00 −0.121098
\(551\) 6800.00 0.525753
\(552\) 0 0
\(553\) 1400.00 0.107657
\(554\) −7332.00 −0.562287
\(555\) 0 0
\(556\) 6560.00 0.500370
\(557\) −16854.0 −1.28209 −0.641047 0.767501i \(-0.721500\pi\)
−0.641047 + 0.767501i \(0.721500\pi\)
\(558\) 0 0
\(559\) −1976.00 −0.149510
\(560\) −1568.00 −0.118322
\(561\) 0 0
\(562\) 13596.0 1.02049
\(563\) −528.000 −0.0395250 −0.0197625 0.999805i \(-0.506291\pi\)
−0.0197625 + 0.999805i \(0.506291\pi\)
\(564\) 0 0
\(565\) 25508.0 1.89934
\(566\) −3984.00 −0.295866
\(567\) 0 0
\(568\) 2944.00 0.217478
\(569\) −8050.00 −0.593099 −0.296550 0.955017i \(-0.595836\pi\)
−0.296550 + 0.955017i \(0.595836\pi\)
\(570\) 0 0
\(571\) −11308.0 −0.828765 −0.414383 0.910103i \(-0.636002\pi\)
−0.414383 + 0.910103i \(0.636002\pi\)
\(572\) −1672.00 −0.122220
\(573\) 0 0
\(574\) −3612.00 −0.262652
\(575\) −568.000 −0.0411952
\(576\) 0 0
\(577\) 5274.00 0.380519 0.190260 0.981734i \(-0.439067\pi\)
0.190260 + 0.981734i \(0.439067\pi\)
\(578\) −3994.00 −0.287420
\(579\) 0 0
\(580\) 9520.00 0.681546
\(581\) −6664.00 −0.475851
\(582\) 0 0
\(583\) −3542.00 −0.251620
\(584\) −16.0000 −0.00113371
\(585\) 0 0
\(586\) 8004.00 0.564236
\(587\) −11604.0 −0.815926 −0.407963 0.912999i \(-0.633761\pi\)
−0.407963 + 0.912999i \(0.633761\pi\)
\(588\) 0 0
\(589\) 3680.00 0.257439
\(590\) −7280.00 −0.507988
\(591\) 0 0
\(592\) 4704.00 0.326576
\(593\) 13002.0 0.900385 0.450192 0.892932i \(-0.351355\pi\)
0.450192 + 0.892932i \(0.351355\pi\)
\(594\) 0 0
\(595\) 5292.00 0.364623
\(596\) 3400.00 0.233674
\(597\) 0 0
\(598\) −608.000 −0.0415769
\(599\) −16320.0 −1.11322 −0.556609 0.830775i \(-0.687898\pi\)
−0.556609 + 0.830775i \(0.687898\pi\)
\(600\) 0 0
\(601\) −9258.00 −0.628356 −0.314178 0.949364i \(-0.601729\pi\)
−0.314178 + 0.949364i \(0.601729\pi\)
\(602\) 728.000 0.0492875
\(603\) 0 0
\(604\) −5792.00 −0.390187
\(605\) 1694.00 0.113836
\(606\) 0 0
\(607\) 25824.0 1.72679 0.863397 0.504525i \(-0.168332\pi\)
0.863397 + 0.504525i \(0.168332\pi\)
\(608\) 1280.00 0.0853797
\(609\) 0 0
\(610\) 616.000 0.0408871
\(611\) 2888.00 0.191221
\(612\) 0 0
\(613\) 15518.0 1.02246 0.511228 0.859445i \(-0.329191\pi\)
0.511228 + 0.859445i \(0.329191\pi\)
\(614\) 10688.0 0.702496
\(615\) 0 0
\(616\) 616.000 0.0402911
\(617\) 14486.0 0.945194 0.472597 0.881279i \(-0.343317\pi\)
0.472597 + 0.881279i \(0.343317\pi\)
\(618\) 0 0
\(619\) 12460.0 0.809062 0.404531 0.914524i \(-0.367435\pi\)
0.404531 + 0.914524i \(0.367435\pi\)
\(620\) 5152.00 0.333725
\(621\) 0 0
\(622\) 4536.00 0.292407
\(623\) −490.000 −0.0315111
\(624\) 0 0
\(625\) −19459.0 −1.24538
\(626\) −14844.0 −0.947741
\(627\) 0 0
\(628\) −7544.00 −0.479360
\(629\) −15876.0 −1.00639
\(630\) 0 0
\(631\) −16648.0 −1.05031 −0.525156 0.851006i \(-0.675993\pi\)
−0.525156 + 0.851006i \(0.675993\pi\)
\(632\) −1600.00 −0.100703
\(633\) 0 0
\(634\) 15252.0 0.955417
\(635\) −22064.0 −1.37887
\(636\) 0 0
\(637\) 1862.00 0.115817
\(638\) −3740.00 −0.232082
\(639\) 0 0
\(640\) 1792.00 0.110680
\(641\) 10638.0 0.655500 0.327750 0.944764i \(-0.393710\pi\)
0.327750 + 0.944764i \(0.393710\pi\)
\(642\) 0 0
\(643\) 4588.00 0.281389 0.140694 0.990053i \(-0.455066\pi\)
0.140694 + 0.990053i \(0.455066\pi\)
\(644\) 224.000 0.0137063
\(645\) 0 0
\(646\) −4320.00 −0.263109
\(647\) 8796.00 0.534477 0.267238 0.963630i \(-0.413889\pi\)
0.267238 + 0.963630i \(0.413889\pi\)
\(648\) 0 0
\(649\) 2860.00 0.172981
\(650\) 5396.00 0.325613
\(651\) 0 0
\(652\) 912.000 0.0547802
\(653\) −18878.0 −1.13132 −0.565661 0.824638i \(-0.691379\pi\)
−0.565661 + 0.824638i \(0.691379\pi\)
\(654\) 0 0
\(655\) −15008.0 −0.895284
\(656\) 4128.00 0.245688
\(657\) 0 0
\(658\) −1064.00 −0.0630381
\(659\) 20780.0 1.22834 0.614168 0.789175i \(-0.289492\pi\)
0.614168 + 0.789175i \(0.289492\pi\)
\(660\) 0 0
\(661\) 10402.0 0.612089 0.306045 0.952017i \(-0.400994\pi\)
0.306045 + 0.952017i \(0.400994\pi\)
\(662\) 2984.00 0.175191
\(663\) 0 0
\(664\) 7616.00 0.445118
\(665\) −3920.00 −0.228588
\(666\) 0 0
\(667\) −1360.00 −0.0789496
\(668\) −6656.00 −0.385522
\(669\) 0 0
\(670\) −12208.0 −0.703935
\(671\) −242.000 −0.0139230
\(672\) 0 0
\(673\) 23018.0 1.31839 0.659197 0.751971i \(-0.270896\pi\)
0.659197 + 0.751971i \(0.270896\pi\)
\(674\) 148.000 0.00845808
\(675\) 0 0
\(676\) −3012.00 −0.171370
\(677\) 16866.0 0.957479 0.478739 0.877957i \(-0.341094\pi\)
0.478739 + 0.877957i \(0.341094\pi\)
\(678\) 0 0
\(679\) 7602.00 0.429658
\(680\) −6048.00 −0.341074
\(681\) 0 0
\(682\) −2024.00 −0.113641
\(683\) −4668.00 −0.261517 −0.130758 0.991414i \(-0.541741\pi\)
−0.130758 + 0.991414i \(0.541741\pi\)
\(684\) 0 0
\(685\) 3444.00 0.192100
\(686\) −686.000 −0.0381802
\(687\) 0 0
\(688\) −832.000 −0.0461042
\(689\) 12236.0 0.676567
\(690\) 0 0
\(691\) −33108.0 −1.82270 −0.911351 0.411629i \(-0.864960\pi\)
−0.911351 + 0.411629i \(0.864960\pi\)
\(692\) −9752.00 −0.535716
\(693\) 0 0
\(694\) −23448.0 −1.28253
\(695\) 22960.0 1.25313
\(696\) 0 0
\(697\) −13932.0 −0.757119
\(698\) 12700.0 0.688685
\(699\) 0 0
\(700\) −1988.00 −0.107342
\(701\) −7302.00 −0.393428 −0.196714 0.980461i \(-0.563027\pi\)
−0.196714 + 0.980461i \(0.563027\pi\)
\(702\) 0 0
\(703\) 11760.0 0.630920
\(704\) −704.000 −0.0376889
\(705\) 0 0
\(706\) 11644.0 0.620719
\(707\) 714.000 0.0379812
\(708\) 0 0
\(709\) −10570.0 −0.559894 −0.279947 0.960015i \(-0.590317\pi\)
−0.279947 + 0.960015i \(0.590317\pi\)
\(710\) 10304.0 0.544651
\(711\) 0 0
\(712\) 560.000 0.0294760
\(713\) −736.000 −0.0386584
\(714\) 0 0
\(715\) −5852.00 −0.306087
\(716\) −6480.00 −0.338225
\(717\) 0 0
\(718\) −26640.0 −1.38467
\(719\) −20220.0 −1.04879 −0.524394 0.851476i \(-0.675708\pi\)
−0.524394 + 0.851476i \(0.675708\pi\)
\(720\) 0 0
\(721\) −1316.00 −0.0679756
\(722\) −10518.0 −0.542160
\(723\) 0 0
\(724\) 14408.0 0.739598
\(725\) 12070.0 0.618301
\(726\) 0 0
\(727\) −29996.0 −1.53025 −0.765124 0.643883i \(-0.777322\pi\)
−0.765124 + 0.643883i \(0.777322\pi\)
\(728\) −2128.00 −0.108336
\(729\) 0 0
\(730\) −56.0000 −0.00283925
\(731\) 2808.00 0.142076
\(732\) 0 0
\(733\) −2282.00 −0.114990 −0.0574949 0.998346i \(-0.518311\pi\)
−0.0574949 + 0.998346i \(0.518311\pi\)
\(734\) −22152.0 −1.11396
\(735\) 0 0
\(736\) −256.000 −0.0128210
\(737\) 4796.00 0.239705
\(738\) 0 0
\(739\) 3260.00 0.162275 0.0811374 0.996703i \(-0.474145\pi\)
0.0811374 + 0.996703i \(0.474145\pi\)
\(740\) 16464.0 0.817877
\(741\) 0 0
\(742\) −4508.00 −0.223038
\(743\) 14512.0 0.716546 0.358273 0.933617i \(-0.383366\pi\)
0.358273 + 0.933617i \(0.383366\pi\)
\(744\) 0 0
\(745\) 11900.0 0.585211
\(746\) −7844.00 −0.384972
\(747\) 0 0
\(748\) 2376.00 0.116143
\(749\) −12012.0 −0.585993
\(750\) 0 0
\(751\) −20128.0 −0.978004 −0.489002 0.872283i \(-0.662639\pi\)
−0.489002 + 0.872283i \(0.662639\pi\)
\(752\) 1216.00 0.0589667
\(753\) 0 0
\(754\) 12920.0 0.624030
\(755\) −20272.0 −0.977184
\(756\) 0 0
\(757\) −3066.00 −0.147207 −0.0736035 0.997288i \(-0.523450\pi\)
−0.0736035 + 0.997288i \(0.523450\pi\)
\(758\) −23240.0 −1.11361
\(759\) 0 0
\(760\) 4480.00 0.213825
\(761\) −26982.0 −1.28528 −0.642639 0.766169i \(-0.722161\pi\)
−0.642639 + 0.766169i \(0.722161\pi\)
\(762\) 0 0
\(763\) −2170.00 −0.102961
\(764\) −13888.0 −0.657657
\(765\) 0 0
\(766\) −7256.00 −0.342258
\(767\) −9880.00 −0.465119
\(768\) 0 0
\(769\) 3550.00 0.166471 0.0832355 0.996530i \(-0.473475\pi\)
0.0832355 + 0.996530i \(0.473475\pi\)
\(770\) 2156.00 0.100905
\(771\) 0 0
\(772\) −888.000 −0.0413987
\(773\) 33062.0 1.53837 0.769183 0.639028i \(-0.220663\pi\)
0.769183 + 0.639028i \(0.220663\pi\)
\(774\) 0 0
\(775\) 6532.00 0.302757
\(776\) −8688.00 −0.401909
\(777\) 0 0
\(778\) −19500.0 −0.898598
\(779\) 10320.0 0.474650
\(780\) 0 0
\(781\) −4048.00 −0.185466
\(782\) 864.000 0.0395097
\(783\) 0 0
\(784\) 784.000 0.0357143
\(785\) −26404.0 −1.20051
\(786\) 0 0
\(787\) 18824.0 0.852609 0.426304 0.904580i \(-0.359815\pi\)
0.426304 + 0.904580i \(0.359815\pi\)
\(788\) 15624.0 0.706322
\(789\) 0 0
\(790\) −5600.00 −0.252201
\(791\) −12754.0 −0.573300
\(792\) 0 0
\(793\) 836.000 0.0374366
\(794\) −13212.0 −0.590524
\(795\) 0 0
\(796\) 7600.00 0.338411
\(797\) −41314.0 −1.83616 −0.918078 0.396399i \(-0.870260\pi\)
−0.918078 + 0.396399i \(0.870260\pi\)
\(798\) 0 0
\(799\) −4104.00 −0.181713
\(800\) 2272.00 0.100409
\(801\) 0 0
\(802\) −4.00000 −0.000176116 0
\(803\) 22.0000 0.000966828 0
\(804\) 0 0
\(805\) 784.000 0.0343259
\(806\) 6992.00 0.305562
\(807\) 0 0
\(808\) −816.000 −0.0355282
\(809\) −16090.0 −0.699251 −0.349626 0.936889i \(-0.613691\pi\)
−0.349626 + 0.936889i \(0.613691\pi\)
\(810\) 0 0
\(811\) 4472.00 0.193629 0.0968145 0.995302i \(-0.469135\pi\)
0.0968145 + 0.995302i \(0.469135\pi\)
\(812\) −4760.00 −0.205718
\(813\) 0 0
\(814\) −6468.00 −0.278505
\(815\) 3192.00 0.137191
\(816\) 0 0
\(817\) −2080.00 −0.0890698
\(818\) −27860.0 −1.19083
\(819\) 0 0
\(820\) 14448.0 0.615300
\(821\) 44338.0 1.88478 0.942392 0.334512i \(-0.108571\pi\)
0.942392 + 0.334512i \(0.108571\pi\)
\(822\) 0 0
\(823\) −19432.0 −0.823034 −0.411517 0.911402i \(-0.635001\pi\)
−0.411517 + 0.911402i \(0.635001\pi\)
\(824\) 1504.00 0.0635853
\(825\) 0 0
\(826\) 3640.00 0.153331
\(827\) −15764.0 −0.662839 −0.331420 0.943483i \(-0.607528\pi\)
−0.331420 + 0.943483i \(0.607528\pi\)
\(828\) 0 0
\(829\) 16570.0 0.694210 0.347105 0.937826i \(-0.387165\pi\)
0.347105 + 0.937826i \(0.387165\pi\)
\(830\) 26656.0 1.11475
\(831\) 0 0
\(832\) 2432.00 0.101339
\(833\) −2646.00 −0.110058
\(834\) 0 0
\(835\) −23296.0 −0.965499
\(836\) −1760.00 −0.0728120
\(837\) 0 0
\(838\) 15480.0 0.638124
\(839\) 5580.00 0.229610 0.114805 0.993388i \(-0.463376\pi\)
0.114805 + 0.993388i \(0.463376\pi\)
\(840\) 0 0
\(841\) 4511.00 0.184960
\(842\) −11236.0 −0.459879
\(843\) 0 0
\(844\) 4688.00 0.191194
\(845\) −10542.0 −0.429178
\(846\) 0 0
\(847\) −847.000 −0.0343604
\(848\) 5152.00 0.208633
\(849\) 0 0
\(850\) −7668.00 −0.309424
\(851\) −2352.00 −0.0947421
\(852\) 0 0
\(853\) −17522.0 −0.703332 −0.351666 0.936126i \(-0.614385\pi\)
−0.351666 + 0.936126i \(0.614385\pi\)
\(854\) −308.000 −0.0123414
\(855\) 0 0
\(856\) 13728.0 0.548146
\(857\) 37146.0 1.48061 0.740305 0.672271i \(-0.234681\pi\)
0.740305 + 0.672271i \(0.234681\pi\)
\(858\) 0 0
\(859\) 46820.0 1.85969 0.929847 0.367945i \(-0.119939\pi\)
0.929847 + 0.367945i \(0.119939\pi\)
\(860\) −2912.00 −0.115463
\(861\) 0 0
\(862\) 18016.0 0.711865
\(863\) −25808.0 −1.01798 −0.508989 0.860773i \(-0.669981\pi\)
−0.508989 + 0.860773i \(0.669981\pi\)
\(864\) 0 0
\(865\) −34132.0 −1.34164
\(866\) −21404.0 −0.839882
\(867\) 0 0
\(868\) −2576.00 −0.100732
\(869\) 2200.00 0.0858802
\(870\) 0 0
\(871\) −16568.0 −0.644530
\(872\) 2480.00 0.0963112
\(873\) 0 0
\(874\) −640.000 −0.0247692
\(875\) 5292.00 0.204460
\(876\) 0 0
\(877\) −43546.0 −1.67667 −0.838337 0.545152i \(-0.816472\pi\)
−0.838337 + 0.545152i \(0.816472\pi\)
\(878\) −9920.00 −0.381303
\(879\) 0 0
\(880\) −2464.00 −0.0943880
\(881\) 10278.0 0.393047 0.196524 0.980499i \(-0.437035\pi\)
0.196524 + 0.980499i \(0.437035\pi\)
\(882\) 0 0
\(883\) 20708.0 0.789218 0.394609 0.918849i \(-0.370880\pi\)
0.394609 + 0.918849i \(0.370880\pi\)
\(884\) −8208.00 −0.312291
\(885\) 0 0
\(886\) −376.000 −0.0142573
\(887\) 30296.0 1.14683 0.573416 0.819264i \(-0.305618\pi\)
0.573416 + 0.819264i \(0.305618\pi\)
\(888\) 0 0
\(889\) 11032.0 0.416200
\(890\) 1960.00 0.0738195
\(891\) 0 0
\(892\) −9488.00 −0.356145
\(893\) 3040.00 0.113919
\(894\) 0 0
\(895\) −22680.0 −0.847049
\(896\) −896.000 −0.0334077
\(897\) 0 0
\(898\) 6300.00 0.234113
\(899\) 15640.0 0.580226
\(900\) 0 0
\(901\) −17388.0 −0.642928
\(902\) −5676.00 −0.209523
\(903\) 0 0
\(904\) 14576.0 0.536273
\(905\) 50428.0 1.85225
\(906\) 0 0
\(907\) 29604.0 1.08378 0.541888 0.840451i \(-0.317710\pi\)
0.541888 + 0.840451i \(0.317710\pi\)
\(908\) −4096.00 −0.149703
\(909\) 0 0
\(910\) −7448.00 −0.271317
\(911\) −12112.0 −0.440492 −0.220246 0.975444i \(-0.570686\pi\)
−0.220246 + 0.975444i \(0.570686\pi\)
\(912\) 0 0
\(913\) −10472.0 −0.379598
\(914\) −25612.0 −0.926881
\(915\) 0 0
\(916\) 1160.00 0.0418422
\(917\) 7504.00 0.270233
\(918\) 0 0
\(919\) −33320.0 −1.19600 −0.598001 0.801496i \(-0.704038\pi\)
−0.598001 + 0.801496i \(0.704038\pi\)
\(920\) −896.000 −0.0321090
\(921\) 0 0
\(922\) 3556.00 0.127018
\(923\) 13984.0 0.498688
\(924\) 0 0
\(925\) 20874.0 0.741982
\(926\) −31344.0 −1.11234
\(927\) 0 0
\(928\) 5440.00 0.192432
\(929\) 45950.0 1.62279 0.811394 0.584499i \(-0.198709\pi\)
0.811394 + 0.584499i \(0.198709\pi\)
\(930\) 0 0
\(931\) 1960.00 0.0689972
\(932\) 20088.0 0.706013
\(933\) 0 0
\(934\) −8.00000 −0.000280266 0
\(935\) 8316.00 0.290869
\(936\) 0 0
\(937\) 11054.0 0.385399 0.192699 0.981258i \(-0.438276\pi\)
0.192699 + 0.981258i \(0.438276\pi\)
\(938\) 6104.00 0.212476
\(939\) 0 0
\(940\) 4256.00 0.147676
\(941\) 55818.0 1.93370 0.966852 0.255339i \(-0.0821870\pi\)
0.966852 + 0.255339i \(0.0821870\pi\)
\(942\) 0 0
\(943\) −2064.00 −0.0712758
\(944\) −4160.00 −0.143428
\(945\) 0 0
\(946\) 1144.00 0.0393178
\(947\) 33636.0 1.15420 0.577098 0.816675i \(-0.304185\pi\)
0.577098 + 0.816675i \(0.304185\pi\)
\(948\) 0 0
\(949\) −76.0000 −0.00259965
\(950\) 5680.00 0.193983
\(951\) 0 0
\(952\) 3024.00 0.102950
\(953\) −31418.0 −1.06792 −0.533961 0.845509i \(-0.679297\pi\)
−0.533961 + 0.845509i \(0.679297\pi\)
\(954\) 0 0
\(955\) −48608.0 −1.64703
\(956\) −800.000 −0.0270647
\(957\) 0 0
\(958\) −29280.0 −0.987467
\(959\) −1722.00 −0.0579836
\(960\) 0 0
\(961\) −21327.0 −0.715887
\(962\) 22344.0 0.748856
\(963\) 0 0
\(964\) −5992.00 −0.200196
\(965\) −3108.00 −0.103679
\(966\) 0 0
\(967\) −13176.0 −0.438171 −0.219086 0.975706i \(-0.570307\pi\)
−0.219086 + 0.975706i \(0.570307\pi\)
\(968\) 968.000 0.0321412
\(969\) 0 0
\(970\) −30408.0 −1.00654
\(971\) −27132.0 −0.896712 −0.448356 0.893855i \(-0.647990\pi\)
−0.448356 + 0.893855i \(0.647990\pi\)
\(972\) 0 0
\(973\) −11480.0 −0.378245
\(974\) −39792.0 −1.30905
\(975\) 0 0
\(976\) 352.000 0.0115443
\(977\) −27154.0 −0.889185 −0.444592 0.895733i \(-0.646651\pi\)
−0.444592 + 0.895733i \(0.646651\pi\)
\(978\) 0 0
\(979\) −770.000 −0.0251372
\(980\) 2744.00 0.0894427
\(981\) 0 0
\(982\) −15944.0 −0.518120
\(983\) −35068.0 −1.13784 −0.568919 0.822393i \(-0.692638\pi\)
−0.568919 + 0.822393i \(0.692638\pi\)
\(984\) 0 0
\(985\) 54684.0 1.76891
\(986\) −18360.0 −0.593004
\(987\) 0 0
\(988\) 6080.00 0.195780
\(989\) 416.000 0.0133752
\(990\) 0 0
\(991\) 26072.0 0.835726 0.417863 0.908510i \(-0.362779\pi\)
0.417863 + 0.908510i \(0.362779\pi\)
\(992\) 2944.00 0.0942259
\(993\) 0 0
\(994\) −5152.00 −0.164398
\(995\) 26600.0 0.847514
\(996\) 0 0
\(997\) −19866.0 −0.631056 −0.315528 0.948916i \(-0.602182\pi\)
−0.315528 + 0.948916i \(0.602182\pi\)
\(998\) −4520.00 −0.143365
\(999\) 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 1386.4.a.m.1.1 1
3.2 odd 2 462.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.4.a.b.1.1 1 3.2 odd 2
1386.4.a.m.1.1 1 1.1 even 1 trivial