Properties

Label 1521.4.a.e.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -7.00000 q^{4} +7.00000 q^{5} +10.0000 q^{7} +15.0000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -7.00000 q^{4} +7.00000 q^{5} +10.0000 q^{7} +15.0000 q^{8} -7.00000 q^{10} -22.0000 q^{11} -10.0000 q^{14} +41.0000 q^{16} -37.0000 q^{17} -30.0000 q^{19} -49.0000 q^{20} +22.0000 q^{22} +162.000 q^{23} -76.0000 q^{25} -70.0000 q^{28} +113.000 q^{29} -196.000 q^{31} -161.000 q^{32} +37.0000 q^{34} +70.0000 q^{35} -13.0000 q^{37} +30.0000 q^{38} +105.000 q^{40} +285.000 q^{41} -246.000 q^{43} +154.000 q^{44} -162.000 q^{46} -462.000 q^{47} -243.000 q^{49} +76.0000 q^{50} +537.000 q^{53} -154.000 q^{55} +150.000 q^{56} -113.000 q^{58} +576.000 q^{59} -635.000 q^{61} +196.000 q^{62} -167.000 q^{64} -202.000 q^{67} +259.000 q^{68} -70.0000 q^{70} -1086.00 q^{71} +805.000 q^{73} +13.0000 q^{74} +210.000 q^{76} -220.000 q^{77} +884.000 q^{79} +287.000 q^{80} -285.000 q^{82} +518.000 q^{83} -259.000 q^{85} +246.000 q^{86} -330.000 q^{88} +194.000 q^{89} -1134.00 q^{92} +462.000 q^{94} -210.000 q^{95} +1202.00 q^{97} +243.000 q^{98} +O(q^{100})\)

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
\(5\) 7.00000 0.626099 0.313050 0.949737i \(-0.398649\pi\)
0.313050 + 0.949737i \(0.398649\pi\)
\(6\) 0 0
\(7\) 10.0000 0.539949 0.269975 0.962867i \(-0.412985\pi\)
0.269975 + 0.962867i \(0.412985\pi\)
\(8\) 15.0000 0.662913
\(9\) 0 0
\(10\) −7.00000 −0.221359
\(11\) −22.0000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −10.0000 −0.190901
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −37.0000 −0.527872 −0.263936 0.964540i \(-0.585021\pi\)
−0.263936 + 0.964540i \(0.585021\pi\)
\(18\) 0 0
\(19\) −30.0000 −0.362235 −0.181118 0.983461i \(-0.557971\pi\)
−0.181118 + 0.983461i \(0.557971\pi\)
\(20\) −49.0000 −0.547837
\(21\) 0 0
\(22\) 22.0000 0.213201
\(23\) 162.000 1.46867 0.734333 0.678789i \(-0.237495\pi\)
0.734333 + 0.678789i \(0.237495\pi\)
\(24\) 0 0
\(25\) −76.0000 −0.608000
\(26\) 0 0
\(27\) 0 0
\(28\) −70.0000 −0.472456
\(29\) 113.000 0.723571 0.361786 0.932261i \(-0.382167\pi\)
0.361786 + 0.932261i \(0.382167\pi\)
\(30\) 0 0
\(31\) −196.000 −1.13557 −0.567785 0.823177i \(-0.692199\pi\)
−0.567785 + 0.823177i \(0.692199\pi\)
\(32\) −161.000 −0.889408
\(33\) 0 0
\(34\) 37.0000 0.186631
\(35\) 70.0000 0.338062
\(36\) 0 0
\(37\) −13.0000 −0.0577618 −0.0288809 0.999583i \(-0.509194\pi\)
−0.0288809 + 0.999583i \(0.509194\pi\)
\(38\) 30.0000 0.128070
\(39\) 0 0
\(40\) 105.000 0.415049
\(41\) 285.000 1.08560 0.542799 0.839863i \(-0.317365\pi\)
0.542799 + 0.839863i \(0.317365\pi\)
\(42\) 0 0
\(43\) −246.000 −0.872434 −0.436217 0.899842i \(-0.643682\pi\)
−0.436217 + 0.899842i \(0.643682\pi\)
\(44\) 154.000 0.527645
\(45\) 0 0
\(46\) −162.000 −0.519252
\(47\) −462.000 −1.43382 −0.716911 0.697165i \(-0.754445\pi\)
−0.716911 + 0.697165i \(0.754445\pi\)
\(48\) 0 0
\(49\) −243.000 −0.708455
\(50\) 76.0000 0.214960
\(51\) 0 0
\(52\) 0 0
\(53\) 537.000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(54\) 0 0
\(55\) −154.000 −0.377552
\(56\) 150.000 0.357939
\(57\) 0 0
\(58\) −113.000 −0.255821
\(59\) 576.000 1.27100 0.635498 0.772102i \(-0.280795\pi\)
0.635498 + 0.772102i \(0.280795\pi\)
\(60\) 0 0
\(61\) −635.000 −1.33284 −0.666421 0.745575i \(-0.732175\pi\)
−0.666421 + 0.745575i \(0.732175\pi\)
\(62\) 196.000 0.401484
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) −202.000 −0.368332 −0.184166 0.982895i \(-0.558958\pi\)
−0.184166 + 0.982895i \(0.558958\pi\)
\(68\) 259.000 0.461888
\(69\) 0 0
\(70\) −70.0000 −0.119523
\(71\) −1086.00 −1.81527 −0.907637 0.419755i \(-0.862116\pi\)
−0.907637 + 0.419755i \(0.862116\pi\)
\(72\) 0 0
\(73\) 805.000 1.29066 0.645330 0.763904i \(-0.276720\pi\)
0.645330 + 0.763904i \(0.276720\pi\)
\(74\) 13.0000 0.0204219
\(75\) 0 0
\(76\) 210.000 0.316956
\(77\) −220.000 −0.325602
\(78\) 0 0
\(79\) 884.000 1.25896 0.629480 0.777017i \(-0.283268\pi\)
0.629480 + 0.777017i \(0.283268\pi\)
\(80\) 287.000 0.401095
\(81\) 0 0
\(82\) −285.000 −0.383817
\(83\) 518.000 0.685035 0.342517 0.939511i \(-0.388720\pi\)
0.342517 + 0.939511i \(0.388720\pi\)
\(84\) 0 0
\(85\) −259.000 −0.330500
\(86\) 246.000 0.308452
\(87\) 0 0
\(88\) −330.000 −0.399751
\(89\) 194.000 0.231056 0.115528 0.993304i \(-0.463144\pi\)
0.115528 + 0.993304i \(0.463144\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1134.00 −1.28508
\(93\) 0 0
\(94\) 462.000 0.506933
\(95\) −210.000 −0.226795
\(96\) 0 0
\(97\) 1202.00 1.25819 0.629096 0.777328i \(-0.283425\pi\)
0.629096 + 0.777328i \(0.283425\pi\)
\(98\) 243.000 0.250477
\(99\) 0 0
\(100\) 532.000 0.532000
\(101\) 429.000 0.422645 0.211322 0.977416i \(-0.432223\pi\)
0.211322 + 0.977416i \(0.432223\pi\)
\(102\) 0 0
\(103\) −1302.00 −1.24553 −0.622766 0.782408i \(-0.713991\pi\)
−0.622766 + 0.782408i \(0.713991\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −537.000 −0.492057
\(107\) 1338.00 1.20887 0.604436 0.796654i \(-0.293398\pi\)
0.604436 + 0.796654i \(0.293398\pi\)
\(108\) 0 0
\(109\) 1034.00 0.908617 0.454308 0.890844i \(-0.349886\pi\)
0.454308 + 0.890844i \(0.349886\pi\)
\(110\) 154.000 0.133485
\(111\) 0 0
\(112\) 410.000 0.345905
\(113\) −1077.00 −0.896599 −0.448299 0.893884i \(-0.647970\pi\)
−0.448299 + 0.893884i \(0.647970\pi\)
\(114\) 0 0
\(115\) 1134.00 0.919531
\(116\) −791.000 −0.633125
\(117\) 0 0
\(118\) −576.000 −0.449365
\(119\) −370.000 −0.285024
\(120\) 0 0
\(121\) −847.000 −0.636364
\(122\) 635.000 0.471231
\(123\) 0 0
\(124\) 1372.00 0.993623
\(125\) −1407.00 −1.00677
\(126\) 0 0
\(127\) −988.000 −0.690321 −0.345161 0.938544i \(-0.612176\pi\)
−0.345161 + 0.938544i \(0.612176\pi\)
\(128\) 1455.00 1.00473
\(129\) 0 0
\(130\) 0 0
\(131\) −560.000 −0.373492 −0.186746 0.982408i \(-0.559794\pi\)
−0.186746 + 0.982408i \(0.559794\pi\)
\(132\) 0 0
\(133\) −300.000 −0.195589
\(134\) 202.000 0.130225
\(135\) 0 0
\(136\) −555.000 −0.349933
\(137\) −519.000 −0.323658 −0.161829 0.986819i \(-0.551739\pi\)
−0.161829 + 0.986819i \(0.551739\pi\)
\(138\) 0 0
\(139\) −348.000 −0.212352 −0.106176 0.994347i \(-0.533861\pi\)
−0.106176 + 0.994347i \(0.533861\pi\)
\(140\) −490.000 −0.295804
\(141\) 0 0
\(142\) 1086.00 0.641796
\(143\) 0 0
\(144\) 0 0
\(145\) 791.000 0.453027
\(146\) −805.000 −0.456317
\(147\) 0 0
\(148\) 91.0000 0.0505416
\(149\) −645.000 −0.354634 −0.177317 0.984154i \(-0.556742\pi\)
−0.177317 + 0.984154i \(0.556742\pi\)
\(150\) 0 0
\(151\) −2914.00 −1.57045 −0.785225 0.619211i \(-0.787453\pi\)
−0.785225 + 0.619211i \(0.787453\pi\)
\(152\) −450.000 −0.240130
\(153\) 0 0
\(154\) 220.000 0.115118
\(155\) −1372.00 −0.710979
\(156\) 0 0
\(157\) −2079.00 −1.05683 −0.528415 0.848986i \(-0.677213\pi\)
−0.528415 + 0.848986i \(0.677213\pi\)
\(158\) −884.000 −0.445109
\(159\) 0 0
\(160\) −1127.00 −0.556857
\(161\) 1620.00 0.793006
\(162\) 0 0
\(163\) −1700.00 −0.816897 −0.408449 0.912781i \(-0.633930\pi\)
−0.408449 + 0.912781i \(0.633930\pi\)
\(164\) −1995.00 −0.949898
\(165\) 0 0
\(166\) −518.000 −0.242196
\(167\) 3680.00 1.70519 0.852596 0.522571i \(-0.175027\pi\)
0.852596 + 0.522571i \(0.175027\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 259.000 0.116849
\(171\) 0 0
\(172\) 1722.00 0.763379
\(173\) −4146.00 −1.82205 −0.911025 0.412352i \(-0.864707\pi\)
−0.911025 + 0.412352i \(0.864707\pi\)
\(174\) 0 0
\(175\) −760.000 −0.328289
\(176\) −902.000 −0.386311
\(177\) 0 0
\(178\) −194.000 −0.0816905
\(179\) −3674.00 −1.53412 −0.767060 0.641575i \(-0.778281\pi\)
−0.767060 + 0.641575i \(0.778281\pi\)
\(180\) 0 0
\(181\) −3283.00 −1.34820 −0.674098 0.738642i \(-0.735467\pi\)
−0.674098 + 0.738642i \(0.735467\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2430.00 0.973598
\(185\) −91.0000 −0.0361646
\(186\) 0 0
\(187\) 814.000 0.318319
\(188\) 3234.00 1.25459
\(189\) 0 0
\(190\) 210.000 0.0801842
\(191\) 596.000 0.225786 0.112893 0.993607i \(-0.463988\pi\)
0.112893 + 0.993607i \(0.463988\pi\)
\(192\) 0 0
\(193\) 393.000 0.146574 0.0732869 0.997311i \(-0.476651\pi\)
0.0732869 + 0.997311i \(0.476651\pi\)
\(194\) −1202.00 −0.444838
\(195\) 0 0
\(196\) 1701.00 0.619898
\(197\) −3522.00 −1.27377 −0.636884 0.770960i \(-0.719777\pi\)
−0.636884 + 0.770960i \(0.719777\pi\)
\(198\) 0 0
\(199\) 2018.00 0.718855 0.359428 0.933173i \(-0.382972\pi\)
0.359428 + 0.933173i \(0.382972\pi\)
\(200\) −1140.00 −0.403051
\(201\) 0 0
\(202\) −429.000 −0.149427
\(203\) 1130.00 0.390692
\(204\) 0 0
\(205\) 1995.00 0.679692
\(206\) 1302.00 0.440362
\(207\) 0 0
\(208\) 0 0
\(209\) 660.000 0.218436
\(210\) 0 0
\(211\) 160.000 0.0522031 0.0261016 0.999659i \(-0.491691\pi\)
0.0261016 + 0.999659i \(0.491691\pi\)
\(212\) −3759.00 −1.21778
\(213\) 0 0
\(214\) −1338.00 −0.427401
\(215\) −1722.00 −0.546230
\(216\) 0 0
\(217\) −1960.00 −0.613150
\(218\) −1034.00 −0.321245
\(219\) 0 0
\(220\) 1078.00 0.330358
\(221\) 0 0
\(222\) 0 0
\(223\) −4072.00 −1.22279 −0.611393 0.791327i \(-0.709391\pi\)
−0.611393 + 0.791327i \(0.709391\pi\)
\(224\) −1610.00 −0.480235
\(225\) 0 0
\(226\) 1077.00 0.316995
\(227\) −5794.00 −1.69410 −0.847051 0.531511i \(-0.821624\pi\)
−0.847051 + 0.531511i \(0.821624\pi\)
\(228\) 0 0
\(229\) −6482.00 −1.87049 −0.935246 0.353999i \(-0.884822\pi\)
−0.935246 + 0.353999i \(0.884822\pi\)
\(230\) −1134.00 −0.325103
\(231\) 0 0
\(232\) 1695.00 0.479665
\(233\) −6890.00 −1.93725 −0.968624 0.248530i \(-0.920053\pi\)
−0.968624 + 0.248530i \(0.920053\pi\)
\(234\) 0 0
\(235\) −3234.00 −0.897714
\(236\) −4032.00 −1.11212
\(237\) 0 0
\(238\) 370.000 0.100771
\(239\) 2466.00 0.667415 0.333708 0.942677i \(-0.391700\pi\)
0.333708 + 0.942677i \(0.391700\pi\)
\(240\) 0 0
\(241\) 3617.00 0.966770 0.483385 0.875408i \(-0.339407\pi\)
0.483385 + 0.875408i \(0.339407\pi\)
\(242\) 847.000 0.224989
\(243\) 0 0
\(244\) 4445.00 1.16624
\(245\) −1701.00 −0.443563
\(246\) 0 0
\(247\) 0 0
\(248\) −2940.00 −0.752783
\(249\) 0 0
\(250\) 1407.00 0.355946
\(251\) −4860.00 −1.22215 −0.611077 0.791571i \(-0.709263\pi\)
−0.611077 + 0.791571i \(0.709263\pi\)
\(252\) 0 0
\(253\) −3564.00 −0.885639
\(254\) 988.000 0.244065
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) −565.000 −0.137135 −0.0685676 0.997646i \(-0.521843\pi\)
−0.0685676 + 0.997646i \(0.521843\pi\)
\(258\) 0 0
\(259\) −130.000 −0.0311884
\(260\) 0 0
\(261\) 0 0
\(262\) 560.000 0.132049
\(263\) 498.000 0.116760 0.0583802 0.998294i \(-0.481406\pi\)
0.0583802 + 0.998294i \(0.481406\pi\)
\(264\) 0 0
\(265\) 3759.00 0.871372
\(266\) 300.000 0.0691511
\(267\) 0 0
\(268\) 1414.00 0.322290
\(269\) −5546.00 −1.25705 −0.628523 0.777791i \(-0.716340\pi\)
−0.628523 + 0.777791i \(0.716340\pi\)
\(270\) 0 0
\(271\) 2256.00 0.505691 0.252845 0.967507i \(-0.418634\pi\)
0.252845 + 0.967507i \(0.418634\pi\)
\(272\) −1517.00 −0.338168
\(273\) 0 0
\(274\) 519.000 0.114430
\(275\) 1672.00 0.366638
\(276\) 0 0
\(277\) 2309.00 0.500846 0.250423 0.968137i \(-0.419430\pi\)
0.250423 + 0.968137i \(0.419430\pi\)
\(278\) 348.000 0.0750779
\(279\) 0 0
\(280\) 1050.00 0.224105
\(281\) 5833.00 1.23832 0.619159 0.785265i \(-0.287473\pi\)
0.619159 + 0.785265i \(0.287473\pi\)
\(282\) 0 0
\(283\) 1650.00 0.346581 0.173290 0.984871i \(-0.444560\pi\)
0.173290 + 0.984871i \(0.444560\pi\)
\(284\) 7602.00 1.58837
\(285\) 0 0
\(286\) 0 0
\(287\) 2850.00 0.586168
\(288\) 0 0
\(289\) −3544.00 −0.721352
\(290\) −791.000 −0.160169
\(291\) 0 0
\(292\) −5635.00 −1.12933
\(293\) 2991.00 0.596369 0.298184 0.954508i \(-0.403619\pi\)
0.298184 + 0.954508i \(0.403619\pi\)
\(294\) 0 0
\(295\) 4032.00 0.795770
\(296\) −195.000 −0.0382910
\(297\) 0 0
\(298\) 645.000 0.125382
\(299\) 0 0
\(300\) 0 0
\(301\) −2460.00 −0.471070
\(302\) 2914.00 0.555238
\(303\) 0 0
\(304\) −1230.00 −0.232057
\(305\) −4445.00 −0.834492
\(306\) 0 0
\(307\) 2422.00 0.450263 0.225132 0.974328i \(-0.427719\pi\)
0.225132 + 0.974328i \(0.427719\pi\)
\(308\) 1540.00 0.284901
\(309\) 0 0
\(310\) 1372.00 0.251369
\(311\) 3402.00 0.620288 0.310144 0.950690i \(-0.399623\pi\)
0.310144 + 0.950690i \(0.399623\pi\)
\(312\) 0 0
\(313\) 2310.00 0.417153 0.208577 0.978006i \(-0.433117\pi\)
0.208577 + 0.978006i \(0.433117\pi\)
\(314\) 2079.00 0.373646
\(315\) 0 0
\(316\) −6188.00 −1.10159
\(317\) −257.000 −0.0455349 −0.0227674 0.999741i \(-0.507248\pi\)
−0.0227674 + 0.999741i \(0.507248\pi\)
\(318\) 0 0
\(319\) −2486.00 −0.436330
\(320\) −1169.00 −0.204216
\(321\) 0 0
\(322\) −1620.00 −0.280370
\(323\) 1110.00 0.191214
\(324\) 0 0
\(325\) 0 0
\(326\) 1700.00 0.288817
\(327\) 0 0
\(328\) 4275.00 0.719657
\(329\) −4620.00 −0.774191
\(330\) 0 0
\(331\) −1028.00 −0.170707 −0.0853535 0.996351i \(-0.527202\pi\)
−0.0853535 + 0.996351i \(0.527202\pi\)
\(332\) −3626.00 −0.599405
\(333\) 0 0
\(334\) −3680.00 −0.602876
\(335\) −1414.00 −0.230612
\(336\) 0 0
\(337\) 2487.00 0.402005 0.201002 0.979591i \(-0.435580\pi\)
0.201002 + 0.979591i \(0.435580\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1813.00 0.289187
\(341\) 4312.00 0.684774
\(342\) 0 0
\(343\) −5860.00 −0.922479
\(344\) −3690.00 −0.578347
\(345\) 0 0
\(346\) 4146.00 0.644192
\(347\) 2850.00 0.440911 0.220455 0.975397i \(-0.429246\pi\)
0.220455 + 0.975397i \(0.429246\pi\)
\(348\) 0 0
\(349\) 2018.00 0.309516 0.154758 0.987952i \(-0.450540\pi\)
0.154758 + 0.987952i \(0.450540\pi\)
\(350\) 760.000 0.116068
\(351\) 0 0
\(352\) 3542.00 0.536333
\(353\) −5287.00 −0.797163 −0.398582 0.917133i \(-0.630497\pi\)
−0.398582 + 0.917133i \(0.630497\pi\)
\(354\) 0 0
\(355\) −7602.00 −1.13654
\(356\) −1358.00 −0.202174
\(357\) 0 0
\(358\) 3674.00 0.542394
\(359\) −7278.00 −1.06997 −0.534983 0.844863i \(-0.679682\pi\)
−0.534983 + 0.844863i \(0.679682\pi\)
\(360\) 0 0
\(361\) −5959.00 −0.868786
\(362\) 3283.00 0.476659
\(363\) 0 0
\(364\) 0 0
\(365\) 5635.00 0.808080
\(366\) 0 0
\(367\) −4202.00 −0.597664 −0.298832 0.954306i \(-0.596597\pi\)
−0.298832 + 0.954306i \(0.596597\pi\)
\(368\) 6642.00 0.940865
\(369\) 0 0
\(370\) 91.0000 0.0127861
\(371\) 5370.00 0.751473
\(372\) 0 0
\(373\) −1583.00 −0.219744 −0.109872 0.993946i \(-0.535044\pi\)
−0.109872 + 0.993946i \(0.535044\pi\)
\(374\) −814.000 −0.112543
\(375\) 0 0
\(376\) −6930.00 −0.950499
\(377\) 0 0
\(378\) 0 0
\(379\) 2052.00 0.278111 0.139056 0.990285i \(-0.455593\pi\)
0.139056 + 0.990285i \(0.455593\pi\)
\(380\) 1470.00 0.198446
\(381\) 0 0
\(382\) −596.000 −0.0798273
\(383\) −6872.00 −0.916822 −0.458411 0.888740i \(-0.651581\pi\)
−0.458411 + 0.888740i \(0.651581\pi\)
\(384\) 0 0
\(385\) −1540.00 −0.203859
\(386\) −393.000 −0.0518217
\(387\) 0 0
\(388\) −8414.00 −1.10092
\(389\) 11653.0 1.51884 0.759422 0.650598i \(-0.225482\pi\)
0.759422 + 0.650598i \(0.225482\pi\)
\(390\) 0 0
\(391\) −5994.00 −0.775268
\(392\) −3645.00 −0.469644
\(393\) 0 0
\(394\) 3522.00 0.450345
\(395\) 6188.00 0.788233
\(396\) 0 0
\(397\) −6134.00 −0.775458 −0.387729 0.921774i \(-0.626740\pi\)
−0.387729 + 0.921774i \(0.626740\pi\)
\(398\) −2018.00 −0.254154
\(399\) 0 0
\(400\) −3116.00 −0.389500
\(401\) −10795.0 −1.34433 −0.672165 0.740401i \(-0.734636\pi\)
−0.672165 + 0.740401i \(0.734636\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3003.00 −0.369814
\(405\) 0 0
\(406\) −1130.00 −0.138130
\(407\) 286.000 0.0348317
\(408\) 0 0
\(409\) 8489.00 1.02629 0.513147 0.858301i \(-0.328480\pi\)
0.513147 + 0.858301i \(0.328480\pi\)
\(410\) −1995.00 −0.240307
\(411\) 0 0
\(412\) 9114.00 1.08984
\(413\) 5760.00 0.686274
\(414\) 0 0
\(415\) 3626.00 0.428900
\(416\) 0 0
\(417\) 0 0
\(418\) −660.000 −0.0772288
\(419\) −1496.00 −0.174426 −0.0872129 0.996190i \(-0.527796\pi\)
−0.0872129 + 0.996190i \(0.527796\pi\)
\(420\) 0 0
\(421\) 11695.0 1.35387 0.676935 0.736043i \(-0.263308\pi\)
0.676935 + 0.736043i \(0.263308\pi\)
\(422\) −160.000 −0.0184566
\(423\) 0 0
\(424\) 8055.00 0.922607
\(425\) 2812.00 0.320946
\(426\) 0 0
\(427\) −6350.00 −0.719668
\(428\) −9366.00 −1.05776
\(429\) 0 0
\(430\) 1722.00 0.193121
\(431\) 10590.0 1.18353 0.591766 0.806110i \(-0.298431\pi\)
0.591766 + 0.806110i \(0.298431\pi\)
\(432\) 0 0
\(433\) −13949.0 −1.54814 −0.774072 0.633098i \(-0.781783\pi\)
−0.774072 + 0.633098i \(0.781783\pi\)
\(434\) 1960.00 0.216781
\(435\) 0 0
\(436\) −7238.00 −0.795040
\(437\) −4860.00 −0.532003
\(438\) 0 0
\(439\) −10726.0 −1.16611 −0.583057 0.812431i \(-0.698144\pi\)
−0.583057 + 0.812431i \(0.698144\pi\)
\(440\) −2310.00 −0.250284
\(441\) 0 0
\(442\) 0 0
\(443\) −16228.0 −1.74044 −0.870221 0.492662i \(-0.836024\pi\)
−0.870221 + 0.492662i \(0.836024\pi\)
\(444\) 0 0
\(445\) 1358.00 0.144664
\(446\) 4072.00 0.432320
\(447\) 0 0
\(448\) −1670.00 −0.176116
\(449\) 7538.00 0.792294 0.396147 0.918187i \(-0.370347\pi\)
0.396147 + 0.918187i \(0.370347\pi\)
\(450\) 0 0
\(451\) −6270.00 −0.654640
\(452\) 7539.00 0.784524
\(453\) 0 0
\(454\) 5794.00 0.598956
\(455\) 0 0
\(456\) 0 0
\(457\) −15539.0 −1.59056 −0.795278 0.606245i \(-0.792675\pi\)
−0.795278 + 0.606245i \(0.792675\pi\)
\(458\) 6482.00 0.661319
\(459\) 0 0
\(460\) −7938.00 −0.804589
\(461\) 4811.00 0.486053 0.243027 0.970020i \(-0.421860\pi\)
0.243027 + 0.970020i \(0.421860\pi\)
\(462\) 0 0
\(463\) −562.000 −0.0564111 −0.0282056 0.999602i \(-0.508979\pi\)
−0.0282056 + 0.999602i \(0.508979\pi\)
\(464\) 4633.00 0.463538
\(465\) 0 0
\(466\) 6890.00 0.684921
\(467\) −4914.00 −0.486922 −0.243461 0.969911i \(-0.578283\pi\)
−0.243461 + 0.969911i \(0.578283\pi\)
\(468\) 0 0
\(469\) −2020.00 −0.198880
\(470\) 3234.00 0.317390
\(471\) 0 0
\(472\) 8640.00 0.842560
\(473\) 5412.00 0.526097
\(474\) 0 0
\(475\) 2280.00 0.220239
\(476\) 2590.00 0.249396
\(477\) 0 0
\(478\) −2466.00 −0.235967
\(479\) −3600.00 −0.343399 −0.171700 0.985149i \(-0.554926\pi\)
−0.171700 + 0.985149i \(0.554926\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −3617.00 −0.341805
\(483\) 0 0
\(484\) 5929.00 0.556818
\(485\) 8414.00 0.787753
\(486\) 0 0
\(487\) 17130.0 1.59391 0.796955 0.604038i \(-0.206443\pi\)
0.796955 + 0.604038i \(0.206443\pi\)
\(488\) −9525.00 −0.883558
\(489\) 0 0
\(490\) 1701.00 0.156823
\(491\) 11838.0 1.08807 0.544034 0.839063i \(-0.316896\pi\)
0.544034 + 0.839063i \(0.316896\pi\)
\(492\) 0 0
\(493\) −4181.00 −0.381953
\(494\) 0 0
\(495\) 0 0
\(496\) −8036.00 −0.727474
\(497\) −10860.0 −0.980156
\(498\) 0 0
\(499\) −8976.00 −0.805252 −0.402626 0.915364i \(-0.631903\pi\)
−0.402626 + 0.915364i \(0.631903\pi\)
\(500\) 9849.00 0.880921
\(501\) 0 0
\(502\) 4860.00 0.432096
\(503\) −1682.00 −0.149099 −0.0745494 0.997217i \(-0.523752\pi\)
−0.0745494 + 0.997217i \(0.523752\pi\)
\(504\) 0 0
\(505\) 3003.00 0.264617
\(506\) 3564.00 0.313121
\(507\) 0 0
\(508\) 6916.00 0.604031
\(509\) 15167.0 1.32076 0.660379 0.750933i \(-0.270396\pi\)
0.660379 + 0.750933i \(0.270396\pi\)
\(510\) 0 0
\(511\) 8050.00 0.696890
\(512\) −11521.0 −0.994455
\(513\) 0 0
\(514\) 565.000 0.0484846
\(515\) −9114.00 −0.779827
\(516\) 0 0
\(517\) 10164.0 0.864627
\(518\) 130.000 0.0110268
\(519\) 0 0
\(520\) 0 0
\(521\) 6783.00 0.570381 0.285191 0.958471i \(-0.407943\pi\)
0.285191 + 0.958471i \(0.407943\pi\)
\(522\) 0 0
\(523\) −13918.0 −1.16366 −0.581828 0.813312i \(-0.697662\pi\)
−0.581828 + 0.813312i \(0.697662\pi\)
\(524\) 3920.00 0.326805
\(525\) 0 0
\(526\) −498.000 −0.0412810
\(527\) 7252.00 0.599435
\(528\) 0 0
\(529\) 14077.0 1.15698
\(530\) −3759.00 −0.308076
\(531\) 0 0
\(532\) 2100.00 0.171140
\(533\) 0 0
\(534\) 0 0
\(535\) 9366.00 0.756874
\(536\) −3030.00 −0.244172
\(537\) 0 0
\(538\) 5546.00 0.444433
\(539\) 5346.00 0.427214
\(540\) 0 0
\(541\) 1335.00 0.106093 0.0530463 0.998592i \(-0.483107\pi\)
0.0530463 + 0.998592i \(0.483107\pi\)
\(542\) −2256.00 −0.178789
\(543\) 0 0
\(544\) 5957.00 0.469493
\(545\) 7238.00 0.568884
\(546\) 0 0
\(547\) −3806.00 −0.297501 −0.148750 0.988875i \(-0.547525\pi\)
−0.148750 + 0.988875i \(0.547525\pi\)
\(548\) 3633.00 0.283201
\(549\) 0 0
\(550\) −1672.00 −0.129626
\(551\) −3390.00 −0.262103
\(552\) 0 0
\(553\) 8840.00 0.679774
\(554\) −2309.00 −0.177076
\(555\) 0 0
\(556\) 2436.00 0.185808
\(557\) −1905.00 −0.144915 −0.0724573 0.997372i \(-0.523084\pi\)
−0.0724573 + 0.997372i \(0.523084\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 2870.00 0.216571
\(561\) 0 0
\(562\) −5833.00 −0.437812
\(563\) 4800.00 0.359318 0.179659 0.983729i \(-0.442501\pi\)
0.179659 + 0.983729i \(0.442501\pi\)
\(564\) 0 0
\(565\) −7539.00 −0.561359
\(566\) −1650.00 −0.122535
\(567\) 0 0
\(568\) −16290.0 −1.20337
\(569\) −14678.0 −1.08143 −0.540715 0.841206i \(-0.681846\pi\)
−0.540715 + 0.841206i \(0.681846\pi\)
\(570\) 0 0
\(571\) −586.000 −0.0429481 −0.0214740 0.999769i \(-0.506836\pi\)
−0.0214740 + 0.999769i \(0.506836\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2850.00 −0.207242
\(575\) −12312.0 −0.892949
\(576\) 0 0
\(577\) −8939.00 −0.644949 −0.322474 0.946578i \(-0.604515\pi\)
−0.322474 + 0.946578i \(0.604515\pi\)
\(578\) 3544.00 0.255036
\(579\) 0 0
\(580\) −5537.00 −0.396399
\(581\) 5180.00 0.369884
\(582\) 0 0
\(583\) −11814.0 −0.839255
\(584\) 12075.0 0.855594
\(585\) 0 0
\(586\) −2991.00 −0.210848
\(587\) −13792.0 −0.969773 −0.484887 0.874577i \(-0.661139\pi\)
−0.484887 + 0.874577i \(0.661139\pi\)
\(588\) 0 0
\(589\) 5880.00 0.411343
\(590\) −4032.00 −0.281347
\(591\) 0 0
\(592\) −533.000 −0.0370037
\(593\) 9569.00 0.662650 0.331325 0.943517i \(-0.392504\pi\)
0.331325 + 0.943517i \(0.392504\pi\)
\(594\) 0 0
\(595\) −2590.00 −0.178453
\(596\) 4515.00 0.310305
\(597\) 0 0
\(598\) 0 0
\(599\) 5192.00 0.354156 0.177078 0.984197i \(-0.443336\pi\)
0.177078 + 0.984197i \(0.443336\pi\)
\(600\) 0 0
\(601\) −3677.00 −0.249564 −0.124782 0.992184i \(-0.539823\pi\)
−0.124782 + 0.992184i \(0.539823\pi\)
\(602\) 2460.00 0.166548
\(603\) 0 0
\(604\) 20398.0 1.37414
\(605\) −5929.00 −0.398427
\(606\) 0 0
\(607\) −10960.0 −0.732871 −0.366435 0.930443i \(-0.619422\pi\)
−0.366435 + 0.930443i \(0.619422\pi\)
\(608\) 4830.00 0.322175
\(609\) 0 0
\(610\) 4445.00 0.295037
\(611\) 0 0
\(612\) 0 0
\(613\) 26027.0 1.71488 0.857439 0.514585i \(-0.172054\pi\)
0.857439 + 0.514585i \(0.172054\pi\)
\(614\) −2422.00 −0.159192
\(615\) 0 0
\(616\) −3300.00 −0.215845
\(617\) 17681.0 1.15366 0.576832 0.816863i \(-0.304289\pi\)
0.576832 + 0.816863i \(0.304289\pi\)
\(618\) 0 0
\(619\) −3192.00 −0.207265 −0.103633 0.994616i \(-0.533047\pi\)
−0.103633 + 0.994616i \(0.533047\pi\)
\(620\) 9604.00 0.622106
\(621\) 0 0
\(622\) −3402.00 −0.219305
\(623\) 1940.00 0.124758
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) −2310.00 −0.147486
\(627\) 0 0
\(628\) 14553.0 0.924726
\(629\) 481.000 0.0304908
\(630\) 0 0
\(631\) −7580.00 −0.478217 −0.239109 0.970993i \(-0.576855\pi\)
−0.239109 + 0.970993i \(0.576855\pi\)
\(632\) 13260.0 0.834580
\(633\) 0 0
\(634\) 257.000 0.0160990
\(635\) −6916.00 −0.432210
\(636\) 0 0
\(637\) 0 0
\(638\) 2486.00 0.154266
\(639\) 0 0
\(640\) 10185.0 0.629059
\(641\) 27707.0 1.70727 0.853635 0.520871i \(-0.174393\pi\)
0.853635 + 0.520871i \(0.174393\pi\)
\(642\) 0 0
\(643\) −11216.0 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(644\) −11340.0 −0.693880
\(645\) 0 0
\(646\) −1110.00 −0.0676043
\(647\) 2536.00 0.154097 0.0770483 0.997027i \(-0.475450\pi\)
0.0770483 + 0.997027i \(0.475450\pi\)
\(648\) 0 0
\(649\) −12672.0 −0.766440
\(650\) 0 0
\(651\) 0 0
\(652\) 11900.0 0.714785
\(653\) −17730.0 −1.06252 −0.531262 0.847207i \(-0.678282\pi\)
−0.531262 + 0.847207i \(0.678282\pi\)
\(654\) 0 0
\(655\) −3920.00 −0.233843
\(656\) 11685.0 0.695461
\(657\) 0 0
\(658\) 4620.00 0.273718
\(659\) −18920.0 −1.11839 −0.559195 0.829036i \(-0.688890\pi\)
−0.559195 + 0.829036i \(0.688890\pi\)
\(660\) 0 0
\(661\) −5241.00 −0.308398 −0.154199 0.988040i \(-0.549280\pi\)
−0.154199 + 0.988040i \(0.549280\pi\)
\(662\) 1028.00 0.0603540
\(663\) 0 0
\(664\) 7770.00 0.454118
\(665\) −2100.00 −0.122458
\(666\) 0 0
\(667\) 18306.0 1.06269
\(668\) −25760.0 −1.49204
\(669\) 0 0
\(670\) 1414.00 0.0815337
\(671\) 13970.0 0.803735
\(672\) 0 0
\(673\) 20467.0 1.17228 0.586140 0.810210i \(-0.300647\pi\)
0.586140 + 0.810210i \(0.300647\pi\)
\(674\) −2487.00 −0.142130
\(675\) 0 0
\(676\) 0 0
\(677\) 70.0000 0.00397388 0.00198694 0.999998i \(-0.499368\pi\)
0.00198694 + 0.999998i \(0.499368\pi\)
\(678\) 0 0
\(679\) 12020.0 0.679360
\(680\) −3885.00 −0.219093
\(681\) 0 0
\(682\) −4312.00 −0.242104
\(683\) 6432.00 0.360342 0.180171 0.983635i \(-0.442335\pi\)
0.180171 + 0.983635i \(0.442335\pi\)
\(684\) 0 0
\(685\) −3633.00 −0.202642
\(686\) 5860.00 0.326146
\(687\) 0 0
\(688\) −10086.0 −0.558903
\(689\) 0 0
\(690\) 0 0
\(691\) 6666.00 0.366985 0.183492 0.983021i \(-0.441260\pi\)
0.183492 + 0.983021i \(0.441260\pi\)
\(692\) 29022.0 1.59429
\(693\) 0 0
\(694\) −2850.00 −0.155885
\(695\) −2436.00 −0.132954
\(696\) 0 0
\(697\) −10545.0 −0.573056
\(698\) −2018.00 −0.109430
\(699\) 0 0
\(700\) 5320.00 0.287253
\(701\) 14054.0 0.757221 0.378611 0.925556i \(-0.376402\pi\)
0.378611 + 0.925556i \(0.376402\pi\)
\(702\) 0 0
\(703\) 390.000 0.0209234
\(704\) 3674.00 0.196689
\(705\) 0 0
\(706\) 5287.00 0.281840
\(707\) 4290.00 0.228207
\(708\) 0 0
\(709\) 71.0000 0.00376088 0.00188044 0.999998i \(-0.499401\pi\)
0.00188044 + 0.999998i \(0.499401\pi\)
\(710\) 7602.00 0.401828
\(711\) 0 0
\(712\) 2910.00 0.153170
\(713\) −31752.0 −1.66777
\(714\) 0 0
\(715\) 0 0
\(716\) 25718.0 1.34236
\(717\) 0 0
\(718\) 7278.00 0.378290
\(719\) −3936.00 −0.204156 −0.102078 0.994776i \(-0.532549\pi\)
−0.102078 + 0.994776i \(0.532549\pi\)
\(720\) 0 0
\(721\) −13020.0 −0.672524
\(722\) 5959.00 0.307162
\(723\) 0 0
\(724\) 22981.0 1.17967
\(725\) −8588.00 −0.439931
\(726\) 0 0
\(727\) 34202.0 1.74482 0.872409 0.488777i \(-0.162557\pi\)
0.872409 + 0.488777i \(0.162557\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −5635.00 −0.285700
\(731\) 9102.00 0.460533
\(732\) 0 0
\(733\) 27363.0 1.37882 0.689410 0.724371i \(-0.257870\pi\)
0.689410 + 0.724371i \(0.257870\pi\)
\(734\) 4202.00 0.211306
\(735\) 0 0
\(736\) −26082.0 −1.30624
\(737\) 4444.00 0.222112
\(738\) 0 0
\(739\) −21776.0 −1.08396 −0.541978 0.840393i \(-0.682324\pi\)
−0.541978 + 0.840393i \(0.682324\pi\)
\(740\) 637.000 0.0316440
\(741\) 0 0
\(742\) −5370.00 −0.265686
\(743\) 2484.00 0.122650 0.0613251 0.998118i \(-0.480467\pi\)
0.0613251 + 0.998118i \(0.480467\pi\)
\(744\) 0 0
\(745\) −4515.00 −0.222036
\(746\) 1583.00 0.0776914
\(747\) 0 0
\(748\) −5698.00 −0.278529
\(749\) 13380.0 0.652730
\(750\) 0 0
\(751\) 32906.0 1.59888 0.799439 0.600748i \(-0.205130\pi\)
0.799439 + 0.600748i \(0.205130\pi\)
\(752\) −18942.0 −0.918542
\(753\) 0 0
\(754\) 0 0
\(755\) −20398.0 −0.983257
\(756\) 0 0
\(757\) −3914.00 −0.187922 −0.0939609 0.995576i \(-0.529953\pi\)
−0.0939609 + 0.995576i \(0.529953\pi\)
\(758\) −2052.00 −0.0983272
\(759\) 0 0
\(760\) −3150.00 −0.150345
\(761\) −33038.0 −1.57375 −0.786877 0.617110i \(-0.788303\pi\)
−0.786877 + 0.617110i \(0.788303\pi\)
\(762\) 0 0
\(763\) 10340.0 0.490607
\(764\) −4172.00 −0.197562
\(765\) 0 0
\(766\) 6872.00 0.324145
\(767\) 0 0
\(768\) 0 0
\(769\) 17586.0 0.824665 0.412332 0.911033i \(-0.364714\pi\)
0.412332 + 0.911033i \(0.364714\pi\)
\(770\) 1540.00 0.0720750
\(771\) 0 0
\(772\) −2751.00 −0.128252
\(773\) 18314.0 0.852146 0.426073 0.904689i \(-0.359897\pi\)
0.426073 + 0.904689i \(0.359897\pi\)
\(774\) 0 0
\(775\) 14896.0 0.690426
\(776\) 18030.0 0.834071
\(777\) 0 0
\(778\) −11653.0 −0.536993
\(779\) −8550.00 −0.393242
\(780\) 0 0
\(781\) 23892.0 1.09465
\(782\) 5994.00 0.274098
\(783\) 0 0
\(784\) −9963.00 −0.453854
\(785\) −14553.0 −0.661680
\(786\) 0 0
\(787\) 42068.0 1.90542 0.952708 0.303888i \(-0.0982847\pi\)
0.952708 + 0.303888i \(0.0982847\pi\)
\(788\) 24654.0 1.11455
\(789\) 0 0
\(790\) −6188.00 −0.278682
\(791\) −10770.0 −0.484118
\(792\) 0 0
\(793\) 0 0
\(794\) 6134.00 0.274166
\(795\) 0 0
\(796\) −14126.0 −0.628998
\(797\) 4282.00 0.190309 0.0951545 0.995463i \(-0.469665\pi\)
0.0951545 + 0.995463i \(0.469665\pi\)
\(798\) 0 0
\(799\) 17094.0 0.756874
\(800\) 12236.0 0.540760
\(801\) 0 0
\(802\) 10795.0 0.475293
\(803\) −17710.0 −0.778297
\(804\) 0 0
\(805\) 11340.0 0.496500
\(806\) 0 0
\(807\) 0 0
\(808\) 6435.00 0.280176
\(809\) −40221.0 −1.74795 −0.873977 0.485967i \(-0.838468\pi\)
−0.873977 + 0.485967i \(0.838468\pi\)
\(810\) 0 0
\(811\) 7084.00 0.306724 0.153362 0.988170i \(-0.450990\pi\)
0.153362 + 0.988170i \(0.450990\pi\)
\(812\) −7910.00 −0.341855
\(813\) 0 0
\(814\) −286.000 −0.0123149
\(815\) −11900.0 −0.511459
\(816\) 0 0
\(817\) 7380.00 0.316026
\(818\) −8489.00 −0.362850
\(819\) 0 0
\(820\) −13965.0 −0.594730
\(821\) 17338.0 0.737028 0.368514 0.929622i \(-0.379867\pi\)
0.368514 + 0.929622i \(0.379867\pi\)
\(822\) 0 0
\(823\) 35496.0 1.50342 0.751709 0.659495i \(-0.229230\pi\)
0.751709 + 0.659495i \(0.229230\pi\)
\(824\) −19530.0 −0.825679
\(825\) 0 0
\(826\) −5760.00 −0.242634
\(827\) −14992.0 −0.630378 −0.315189 0.949029i \(-0.602068\pi\)
−0.315189 + 0.949029i \(0.602068\pi\)
\(828\) 0 0
\(829\) −20659.0 −0.865521 −0.432760 0.901509i \(-0.642460\pi\)
−0.432760 + 0.901509i \(0.642460\pi\)
\(830\) −3626.00 −0.151639
\(831\) 0 0
\(832\) 0 0
\(833\) 8991.00 0.373973
\(834\) 0 0
\(835\) 25760.0 1.06762
\(836\) −4620.00 −0.191132
\(837\) 0 0
\(838\) 1496.00 0.0616688
\(839\) 28716.0 1.18163 0.590814 0.806808i \(-0.298807\pi\)
0.590814 + 0.806808i \(0.298807\pi\)
\(840\) 0 0
\(841\) −11620.0 −0.476444
\(842\) −11695.0 −0.478665
\(843\) 0 0
\(844\) −1120.00 −0.0456777
\(845\) 0 0
\(846\) 0 0
\(847\) −8470.00 −0.343604
\(848\) 22017.0 0.891588
\(849\) 0 0
\(850\) −2812.00 −0.113472
\(851\) −2106.00 −0.0848328
\(852\) 0 0
\(853\) −13377.0 −0.536952 −0.268476 0.963286i \(-0.586520\pi\)
−0.268476 + 0.963286i \(0.586520\pi\)
\(854\) 6350.00 0.254441
\(855\) 0 0
\(856\) 20070.0 0.801377
\(857\) 27419.0 1.09290 0.546450 0.837492i \(-0.315979\pi\)
0.546450 + 0.837492i \(0.315979\pi\)
\(858\) 0 0
\(859\) 2422.00 0.0962021 0.0481010 0.998842i \(-0.484683\pi\)
0.0481010 + 0.998842i \(0.484683\pi\)
\(860\) 12054.0 0.477951
\(861\) 0 0
\(862\) −10590.0 −0.418442
\(863\) −34522.0 −1.36169 −0.680847 0.732425i \(-0.738388\pi\)
−0.680847 + 0.732425i \(0.738388\pi\)
\(864\) 0 0
\(865\) −29022.0 −1.14078
\(866\) 13949.0 0.547351
\(867\) 0 0
\(868\) 13720.0 0.536506
\(869\) −19448.0 −0.759181
\(870\) 0 0
\(871\) 0 0
\(872\) 15510.0 0.602334
\(873\) 0 0
\(874\) 4860.00 0.188091
\(875\) −14070.0 −0.543603
\(876\) 0 0
\(877\) −13733.0 −0.528769 −0.264385 0.964417i \(-0.585169\pi\)
−0.264385 + 0.964417i \(0.585169\pi\)
\(878\) 10726.0 0.412284
\(879\) 0 0
\(880\) −6314.00 −0.241869
\(881\) 22759.0 0.870341 0.435170 0.900348i \(-0.356688\pi\)
0.435170 + 0.900348i \(0.356688\pi\)
\(882\) 0 0
\(883\) −2168.00 −0.0826263 −0.0413131 0.999146i \(-0.513154\pi\)
−0.0413131 + 0.999146i \(0.513154\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 16228.0 0.615339
\(887\) −15888.0 −0.601428 −0.300714 0.953714i \(-0.597225\pi\)
−0.300714 + 0.953714i \(0.597225\pi\)
\(888\) 0 0
\(889\) −9880.00 −0.372739
\(890\) −1358.00 −0.0511464
\(891\) 0 0
\(892\) 28504.0 1.06994
\(893\) 13860.0 0.519381
\(894\) 0 0
\(895\) −25718.0 −0.960512
\(896\) 14550.0 0.542502
\(897\) 0 0
\(898\) −7538.00 −0.280118
\(899\) −22148.0 −0.821665
\(900\) 0 0
\(901\) −19869.0 −0.734664
\(902\) 6270.00 0.231450
\(903\) 0 0
\(904\) −16155.0 −0.594366
\(905\) −22981.0 −0.844104
\(906\) 0 0
\(907\) −11628.0 −0.425691 −0.212845 0.977086i \(-0.568273\pi\)
−0.212845 + 0.977086i \(0.568273\pi\)
\(908\) 40558.0 1.48234
\(909\) 0 0
\(910\) 0 0
\(911\) 12584.0 0.457658 0.228829 0.973467i \(-0.426510\pi\)
0.228829 + 0.973467i \(0.426510\pi\)
\(912\) 0 0
\(913\) −11396.0 −0.413092
\(914\) 15539.0 0.562346
\(915\) 0 0
\(916\) 45374.0 1.63668
\(917\) −5600.00 −0.201667
\(918\) 0 0
\(919\) 17184.0 0.616809 0.308405 0.951255i \(-0.400205\pi\)
0.308405 + 0.951255i \(0.400205\pi\)
\(920\) 17010.0 0.609569
\(921\) 0 0
\(922\) −4811.00 −0.171846
\(923\) 0 0
\(924\) 0 0
\(925\) 988.000 0.0351192
\(926\) 562.000 0.0199443
\(927\) 0 0
\(928\) −18193.0 −0.643550
\(929\) 12777.0 0.451238 0.225619 0.974216i \(-0.427560\pi\)
0.225619 + 0.974216i \(0.427560\pi\)
\(930\) 0 0
\(931\) 7290.00 0.256627
\(932\) 48230.0 1.69509
\(933\) 0 0
\(934\) 4914.00 0.172153
\(935\) 5698.00 0.199299
\(936\) 0 0
\(937\) 9191.00 0.320445 0.160222 0.987081i \(-0.448779\pi\)
0.160222 + 0.987081i \(0.448779\pi\)
\(938\) 2020.00 0.0703149
\(939\) 0 0
\(940\) 22638.0 0.785500
\(941\) 50498.0 1.74940 0.874701 0.484662i \(-0.161058\pi\)
0.874701 + 0.484662i \(0.161058\pi\)
\(942\) 0 0
\(943\) 46170.0 1.59438
\(944\) 23616.0 0.814232
\(945\) 0 0
\(946\) −5412.00 −0.186003
\(947\) 1560.00 0.0535303 0.0267651 0.999642i \(-0.491479\pi\)
0.0267651 + 0.999642i \(0.491479\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2280.00 −0.0778663
\(951\) 0 0
\(952\) −5550.00 −0.188946
\(953\) 21498.0 0.730733 0.365366 0.930864i \(-0.380944\pi\)
0.365366 + 0.930864i \(0.380944\pi\)
\(954\) 0 0
\(955\) 4172.00 0.141364
\(956\) −17262.0 −0.583988
\(957\) 0 0
\(958\) 3600.00 0.121410
\(959\) −5190.00 −0.174759
\(960\) 0 0
\(961\) 8625.00 0.289517
\(962\) 0 0
\(963\) 0 0
\(964\) −25319.0 −0.845923
\(965\) 2751.00 0.0917698
\(966\) 0 0
\(967\) 418.000 0.0139007 0.00695035 0.999976i \(-0.497788\pi\)
0.00695035 + 0.999976i \(0.497788\pi\)
\(968\) −12705.0 −0.421853
\(969\) 0 0
\(970\) −8414.00 −0.278513
\(971\) −18132.0 −0.599262 −0.299631 0.954055i \(-0.596864\pi\)
−0.299631 + 0.954055i \(0.596864\pi\)
\(972\) 0 0
\(973\) −3480.00 −0.114659
\(974\) −17130.0 −0.563532
\(975\) 0 0
\(976\) −26035.0 −0.853853
\(977\) 12501.0 0.409358 0.204679 0.978829i \(-0.434385\pi\)
0.204679 + 0.978829i \(0.434385\pi\)
\(978\) 0 0
\(979\) −4268.00 −0.139332
\(980\) 11907.0 0.388118
\(981\) 0 0
\(982\) −11838.0 −0.384690
\(983\) −43708.0 −1.41818 −0.709089 0.705119i \(-0.750894\pi\)
−0.709089 + 0.705119i \(0.750894\pi\)
\(984\) 0 0
\(985\) −24654.0 −0.797504
\(986\) 4181.00 0.135041
\(987\) 0 0
\(988\) 0 0
\(989\) −39852.0 −1.28131
\(990\) 0 0
\(991\) −39614.0 −1.26981 −0.634904 0.772591i \(-0.718960\pi\)
−0.634904 + 0.772591i \(0.718960\pi\)
\(992\) 31556.0 1.00998
\(993\) 0 0
\(994\) 10860.0 0.346538
\(995\) 14126.0 0.450075
\(996\) 0 0
\(997\) −36503.0 −1.15954 −0.579770 0.814780i \(-0.696858\pi\)
−0.579770 + 0.814780i \(0.696858\pi\)
\(998\) 8976.00 0.284700
\(999\) 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 1521.4.a.e.1.1 1
3.2 odd 2 507.4.a.d.1.1 1
13.4 even 6 117.4.g.a.55.1 2
13.10 even 6 117.4.g.a.100.1 2
13.12 even 2 1521.4.a.h.1.1 1
39.5 even 4 507.4.b.d.337.1 2
39.8 even 4 507.4.b.d.337.2 2
39.17 odd 6 39.4.e.b.16.1 2
39.23 odd 6 39.4.e.b.22.1 yes 2
39.38 odd 2 507.4.a.b.1.1 1
156.23 even 6 624.4.q.c.529.1 2
156.95 even 6 624.4.q.c.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.b.16.1 2 39.17 odd 6
39.4.e.b.22.1 yes 2 39.23 odd 6
117.4.g.a.55.1 2 13.4 even 6
117.4.g.a.100.1 2 13.10 even 6
507.4.a.b.1.1 1 39.38 odd 2
507.4.a.d.1.1 1 3.2 odd 2
507.4.b.d.337.1 2 39.5 even 4
507.4.b.d.337.2 2 39.8 even 4
624.4.q.c.289.1 2 156.95 even 6
624.4.q.c.529.1 2 156.23 even 6
1521.4.a.e.1.1 1 1.1 even 1 trivial
1521.4.a.h.1.1 1 13.12 even 2