Properties

Label 2450.2.a.bf.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +3.00000 q^{22} -9.00000 q^{23} +2.00000 q^{24} +1.00000 q^{26} -4.00000 q^{27} +6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +7.00000 q^{37} -1.00000 q^{38} +2.00000 q^{39} +3.00000 q^{41} -2.00000 q^{43} +3.00000 q^{44} -9.00000 q^{46} -9.00000 q^{47} +2.00000 q^{48} +12.0000 q^{51} +1.00000 q^{52} -9.00000 q^{53} -4.00000 q^{54} -2.00000 q^{57} +6.00000 q^{58} +8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} -18.0000 q^{69} +1.00000 q^{72} +4.00000 q^{73} +7.00000 q^{74} -1.00000 q^{76} +2.00000 q^{78} -10.0000 q^{79} -11.0000 q^{81} +3.00000 q^{82} -2.00000 q^{86} +12.0000 q^{87} +3.00000 q^{88} +6.00000 q^{89} -9.00000 q^{92} +16.0000 q^{93} -9.00000 q^{94} +2.00000 q^{96} +10.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.00000 0.577350
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 2.00000 0.288675
\(49\) 0 0
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 1.00000 0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 6.00000 0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) −18.0000 −2.16695
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 3.00000 0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 12.0000 1.28654
\(88\) 3.00000 0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −9.00000 −0.938315
\(93\) 16.0000 1.65912
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 12.0000 1.18818
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −4.00000 −0.384900
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 14.0000 1.32882
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 8.00000 0.724286
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −18.0000 −1.53226
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −18.0000 −1.51587
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 7.00000 0.575396
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −23.0000 −1.83560 −0.917800 0.397043i \(-0.870036\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(158\) −10.0000 −0.795557
\(159\) −18.0000 −1.42749
\(160\) 0 0
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −2.00000 −0.152499
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 16.0000 1.18275
\(184\) −9.00000 −0.663489
\(185\) 0 0
\(186\) 16.0000 1.17318
\(187\) 18.0000 1.31629
\(188\) −9.00000 −0.656392
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 2.00000 0.144338
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 3.00000 0.213201
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −9.00000 −0.625543
\(208\) 1.00000 0.0693375
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 14.0000 0.939618
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −2.00000 −0.132453
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 0 0
\(237\) −20.0000 −1.29914
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) −2.00000 −0.128565
\(243\) −10.0000 −0.641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) −1.00000 −0.0636285
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) −27.0000 −1.69748
\(254\) 1.00000 0.0627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 3.00000 0.185341
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) −8.00000 −0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −18.0000 −1.08347
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −4.00000 −0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −27.0000 −1.61068 −0.805342 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) −18.0000 −1.07188
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) 4.00000 0.234082
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) −12.0000 −0.696311
\(298\) −6.00000 −0.347571
\(299\) −9.00000 −0.520483
\(300\) 0 0
\(301\) 0 0
\(302\) −10.0000 −0.575435
\(303\) 24.0000 1.37876
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 2.00000 0.113228
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) −23.0000 −1.29797
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −18.0000 −1.00939
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) −32.0000 −1.76960
\(328\) 3.00000 0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 7.00000 0.383598
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 12.0000 0.643268
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 3.00000 0.159901
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −3.00000 −0.158555
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 2.00000 0.105118
\(363\) −4.00000 −0.209946
\(364\) 0 0
\(365\) 0 0
\(366\) 16.0000 0.836333
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) −9.00000 −0.469157
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 0 0
\(372\) 16.0000 0.829561
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 12.0000 0.613973
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) −2.00000 −0.101666
\(388\) 10.0000 0.507673
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −54.0000 −2.73090
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) −16.0000 −0.798007
\(403\) 8.00000 0.398508
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 21.0000 1.04093
\(408\) 12.0000 0.594089
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) −9.00000 −0.442326
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −8.00000 −0.391762
\(418\) −3.00000 −0.146735
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 23.0000 1.11962
\(423\) −9.00000 −0.437595
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −4.00000 −0.192450
\(433\) 40.0000 1.92228 0.961139 0.276066i \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 9.00000 0.430528
\(438\) 8.00000 0.382255
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 14.0000 0.664411
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) −20.0000 −0.939682
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) −4.00000 −0.186908
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 1.00000 0.0464739 0.0232370 0.999730i \(-0.492603\pi\)
0.0232370 + 0.999730i \(0.492603\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −46.0000 −2.11957
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) −20.0000 −0.918630
\(475\) 0 0
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) −6.00000 −0.274434
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 7.00000 0.319173
\(482\) −1.00000 −0.0455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 8.00000 0.362143
\(489\) −40.0000 −1.80886
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 6.00000 0.270501
\(493\) 36.0000 1.62136
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) −15.0000 −0.669483
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −27.0000 −1.20030
\(507\) −24.0000 −1.06588
\(508\) 1.00000 0.0443678
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) −27.0000 −1.18746
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 6.00000 0.262613
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 6.00000 0.261116
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.00000 0.129944
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) −6.00000 −0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −16.0000 −0.687259
\(543\) 4.00000 0.171656
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −12.0000 −0.512615
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) −18.0000 −0.766131
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 9.00000 0.381342 0.190671 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) 8.00000 0.338667
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) −27.0000 −1.13893
\(563\) −42.0000 −1.77009 −0.885044 0.465506i \(-0.845872\pi\)
−0.885044 + 0.465506i \(0.845872\pi\)
\(564\) −18.0000 −0.757937
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 3.00000 0.125436
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −44.0000 −1.83174 −0.915872 0.401470i \(-0.868499\pi\)
−0.915872 + 0.401470i \(0.868499\pi\)
\(578\) 19.0000 0.790296
\(579\) 32.0000 1.32987
\(580\) 0 0
\(581\) 0 0
\(582\) 20.0000 0.829027
\(583\) −27.0000 −1.11823
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −30.0000 −1.23404
\(592\) 7.00000 0.287698
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −32.0000 −1.30967
\(598\) −9.00000 −0.368037
\(599\) −42.0000 −1.71607 −0.858037 0.513588i \(-0.828316\pi\)
−0.858037 + 0.513588i \(0.828316\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 6.00000 0.242536
\(613\) −29.0000 −1.17130 −0.585649 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) −14.0000 −0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 8.00000 0.321807
\(619\) 23.0000 0.924448 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 0 0
\(621\) 36.0000 1.44463
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 28.0000 1.11911
\(627\) −6.00000 −0.239617
\(628\) −23.0000 −0.917800
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −10.0000 −0.397779
\(633\) 46.0000 1.82834
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −18.0000 −0.713746
\(637\) 0 0
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) 0 0
\(641\) 27.0000 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(642\) 24.0000 0.947204
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) −33.0000 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −20.0000 −0.783260
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) −32.0000 −1.25130
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) −7.00000 −0.272063
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) −54.0000 −2.09089
\(668\) −3.00000 −0.116073
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 24.0000 0.919007
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) −2.00000 −0.0762493
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) −9.00000 −0.342129
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 18.0000 0.681799
\(698\) 26.0000 0.984115
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −4.00000 −0.150970
\(703\) −7.00000 −0.264010
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 6.00000 0.224860
\(713\) −72.0000 −2.69642
\(714\) 0 0
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) −12.0000 −0.448148
\(718\) 18.0000 0.671754
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.0000 −0.669891
\(723\) −2.00000 −0.0743808
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 16.0000 0.591377
\(733\) 43.0000 1.58824 0.794121 0.607760i \(-0.207932\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) 19.0000 0.701303
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) −24.0000 −0.884051
\(738\) 3.00000 0.110432
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 45.0000 1.65089 0.825445 0.564483i \(-0.190924\pi\)
0.825445 + 0.564483i \(0.190924\pi\)
\(744\) 16.0000 0.586588
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) 0 0
\(748\) 18.0000 0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) −9.00000 −0.328196
\(753\) −30.0000 −1.09326
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 23.0000 0.835398
\(759\) −54.0000 −1.96008
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 2.00000 0.0724524
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −21.0000 −0.758761
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.0000 0.575853
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) 0 0
\(782\) −54.0000 −1.93104
\(783\) −24.0000 −0.857690
\(784\) 0 0
\(785\) 0 0
\(786\) 6.00000 0.214013
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −15.0000 −0.534353
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) 8.00000 0.284088
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) −54.0000 −1.91038
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −27.0000 −0.953403
\(803\) 12.0000 0.423471
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 0 0
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 0 0
\(813\) −32.0000 −1.12229
\(814\) 21.0000 0.736050
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 2.00000 0.0699711
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −24.0000 −0.837096
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) −9.00000 −0.312772
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −3.00000 −0.103757
\(837\) −32.0000 −1.10608
\(838\) −9.00000 −0.310900
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 2.00000 0.0689246
\(843\) −54.0000 −1.85986
\(844\) 23.0000 0.791693
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −63.0000 −2.15961
\(852\) 0 0
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 6.00000 0.204837
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 3.00000 0.102121 0.0510606 0.998696i \(-0.483740\pi\)
0.0510606 + 0.998696i \(0.483740\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 40.0000 1.35926
\(867\) 38.0000 1.29055
\(868\) 0 0
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −16.0000 −0.541828
\(873\) 10.0000 0.338449
\(874\) 9.00000 0.304430
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 26.0000 0.877457
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 14.0000 0.469809
\(889\) 0 0
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) −8.00000 −0.267860
\(893\) 9.00000 0.301174
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) −18.0000 −0.601003
\(898\) 21.0000 0.700779
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −54.0000 −1.79900
\(902\) 9.00000 0.299667
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 12.0000 0.398234
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 0 0
\(914\) −14.0000 −0.463079
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) −24.0000 −0.792118
\(919\) −22.0000 −0.725713 −0.362857 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 1.00000 0.0328620
\(927\) 4.00000 0.131377
\(928\) 6.00000 0.196960
\(929\) 57.0000 1.87011 0.935055 0.354504i \(-0.115350\pi\)
0.935055 + 0.354504i \(0.115350\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) −48.0000 −1.57145
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 56.0000 1.82749
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) −46.0000 −1.49876
\(943\) −27.0000 −0.879241
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −6.00000 −0.194974 −0.0974869 0.995237i \(-0.531080\pi\)
−0.0974869 + 0.995237i \(0.531080\pi\)
\(948\) −20.0000 −0.649570
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 36.0000 1.16371
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 7.00000 0.225689
\(963\) 12.0000 0.386695
\(964\) −1.00000 −0.0322078
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −40.0000 −1.27906
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 36.0000 1.14881
\(983\) −3.00000 −0.0956851 −0.0478426 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 8.00000 0.254000
\(993\) −14.0000 −0.444277
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −4.00000 −0.126618
\(999\) −28.0000 −0.885881
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 2450.2.a.bf.1.1 1
5.2 odd 4 2450.2.c.q.99.2 2
5.3 odd 4 2450.2.c.q.99.1 2
5.4 even 2 490.2.a.a.1.1 1
7.2 even 3 350.2.e.b.151.1 2
7.4 even 3 350.2.e.b.51.1 2
7.6 odd 2 2450.2.a.v.1.1 1
15.14 odd 2 4410.2.a.x.1.1 1
20.19 odd 2 3920.2.a.bh.1.1 1
35.2 odd 12 350.2.j.d.249.2 4
35.4 even 6 70.2.e.d.51.1 yes 2
35.9 even 6 70.2.e.d.11.1 2
35.13 even 4 2450.2.c.e.99.1 2
35.18 odd 12 350.2.j.d.149.2 4
35.19 odd 6 490.2.e.g.361.1 2
35.23 odd 12 350.2.j.d.249.1 4
35.24 odd 6 490.2.e.g.471.1 2
35.27 even 4 2450.2.c.e.99.2 2
35.32 odd 12 350.2.j.d.149.1 4
35.34 odd 2 490.2.a.d.1.1 1
105.44 odd 6 630.2.k.d.361.1 2
105.74 odd 6 630.2.k.d.541.1 2
105.104 even 2 4410.2.a.bg.1.1 1
140.39 odd 6 560.2.q.b.401.1 2
140.79 odd 6 560.2.q.b.81.1 2
140.139 even 2 3920.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 35.9 even 6
70.2.e.d.51.1 yes 2 35.4 even 6
350.2.e.b.51.1 2 7.4 even 3
350.2.e.b.151.1 2 7.2 even 3
350.2.j.d.149.1 4 35.32 odd 12
350.2.j.d.149.2 4 35.18 odd 12
350.2.j.d.249.1 4 35.23 odd 12
350.2.j.d.249.2 4 35.2 odd 12
490.2.a.a.1.1 1 5.4 even 2
490.2.a.d.1.1 1 35.34 odd 2
490.2.e.g.361.1 2 35.19 odd 6
490.2.e.g.471.1 2 35.24 odd 6
560.2.q.b.81.1 2 140.79 odd 6
560.2.q.b.401.1 2 140.39 odd 6
630.2.k.d.361.1 2 105.44 odd 6
630.2.k.d.541.1 2 105.74 odd 6
2450.2.a.v.1.1 1 7.6 odd 2
2450.2.a.bf.1.1 1 1.1 even 1 trivial
2450.2.c.e.99.1 2 35.13 even 4
2450.2.c.e.99.2 2 35.27 even 4
2450.2.c.q.99.1 2 5.3 odd 4
2450.2.c.q.99.2 2 5.2 odd 4
3920.2.a.e.1.1 1 140.139 even 2
3920.2.a.bh.1.1 1 20.19 odd 2
4410.2.a.x.1.1 1 15.14 odd 2
4410.2.a.bg.1.1 1 105.104 even 2