Properties

Label 3450.2.a.y.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +2.00000 q^{21} -6.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} +1.00000 q^{36} +2.00000 q^{39} +10.0000 q^{41} +2.00000 q^{42} +12.0000 q^{43} -6.00000 q^{44} +1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +6.00000 q^{58} -12.0000 q^{59} +4.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +12.0000 q^{67} +1.00000 q^{69} +1.00000 q^{72} +10.0000 q^{73} -12.0000 q^{77} +2.00000 q^{78} -6.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} -14.0000 q^{83} +2.00000 q^{84} +12.0000 q^{86} +6.00000 q^{87} -6.00000 q^{88} +4.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +6.00000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −6.00000 −1.27920
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 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
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 2.00000 0.308607
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 2.00000 0.226455
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 6.00000 0.643268
\(88\) −6.00000 −0.639602
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −3.00000 −0.303046
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 4.00000 0.362143
\(123\) 10.0000 0.901670
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 1.00000 0.0851257
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −16.0000 −1.27694 −0.638470 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) −6.00000 −0.477334
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −14.0000 −1.08661
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 4.00000 0.296500
\(183\) 4.00000 0.295689
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −6.00000 −0.426401
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) −6.00000 −0.422159
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) −14.0000 −0.957020
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −16.0000 −1.08366
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 8.00000 0.532152
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 0 0
\(229\) −24.0000 −1.58596 −0.792982 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 6.00000 0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 2.00000 0.125988
\(253\) −6.00000 −0.377217
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −8.00000 −0.494242
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 12.0000 0.719712
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 8.00000 0.476393
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 20.0000 1.18056
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 10.0000 0.585206
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 0 0
\(297\) −6.00000 −0.348155
\(298\) −18.0000 −1.04271
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) −12.0000 −0.690522
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −12.0000 −0.683763
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 2.00000 0.113228
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −16.0000 −0.902932
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −2.00000 −0.112154
\(319\) −36.0000 −2.01561
\(320\) 0 0
\(321\) −14.0000 −0.781404
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) −16.0000 −0.884802
\(328\) 10.0000 0.552158
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) −9.00000 −0.489535
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 6.00000 0.321634
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −6.00000 −0.319801
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −16.0000 −0.845626
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 8.00000 0.420471
\(363\) 25.0000 1.31216
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 1.00000 0.0521286
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 8.00000 0.414781
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 12.0000 0.618031
\(378\) 2.00000 0.102869
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 12.0000 0.609994
\(388\) 6.00000 0.304604
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) −8.00000 −0.403547
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) 0 0
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 12.0000 0.598506
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) −14.0000 −0.689730
\(413\) −24.0000 −1.18096
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 20.0000 0.973585
\(423\) 8.00000 0.388973
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) −14.0000 −0.676716
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) −18.0000 −0.851371
\(448\) 2.00000 0.0944911
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −60.0000 −2.82529
\(452\) 8.00000 0.376288
\(453\) −12.0000 −0.563809
\(454\) 10.0000 0.469323
\(455\) 0 0
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −24.0000 −1.12145
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) −12.0000 −0.558291
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 2.00000 0.0924500
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −16.0000 −0.737241
\(472\) −12.0000 −0.552345
\(473\) −72.0000 −3.31056
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −16.0000 −0.731823
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −6.00000 −0.273293
\(483\) 2.00000 0.0910032
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 4.00000 0.181071
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −14.0000 −0.627355
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) −6.00000 −0.267793
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −9.00000 −0.399704
\(508\) −12.0000 −0.532414
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) −48.0000 −2.11104
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 6.00000 0.262613
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) −6.00000 −0.261116
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) −16.0000 −0.690451
\(538\) 18.0000 0.776035
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 28.0000 1.20270
\(543\) 8.00000 0.343313
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 12.0000 0.512615
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) −12.0000 −0.510292
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 8.00000 0.338667
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) −14.0000 −0.590030 −0.295015 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −24.0000 −1.00880
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 16.0000 0.670755 0.335377 0.942084i \(-0.391136\pi\)
0.335377 + 0.942084i \(0.391136\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) 20.0000 0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −17.0000 −0.707107
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −28.0000 −1.16164
\(582\) 6.00000 0.248708
\(583\) 12.0000 0.496989
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 2.00000 0.0818546
\(598\) 2.00000 0.0817861
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 24.0000 0.978167
\(603\) 12.0000 0.488678
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 32.0000 1.28827 0.644136 0.764911i \(-0.277217\pi\)
0.644136 + 0.764911i \(0.277217\pi\)
\(618\) −14.0000 −0.563163
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 16.0000 0.641542
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) −6.00000 −0.238667
\(633\) 20.0000 0.794929
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) −6.00000 −0.237729
\(638\) −36.0000 −1.42525
\(639\) 0 0
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) −14.0000 −0.552536
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) 1.00000 0.0392837
\(649\) 72.0000 2.82625
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 12.0000 0.469956
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) −16.0000 −0.625650
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 10.0000 0.390137
\(658\) 16.0000 0.623745
\(659\) 42.0000 1.63609 0.818044 0.575156i \(-0.195059\pi\)
0.818044 + 0.575156i \(0.195059\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000 0.232321
\(668\) −8.00000 −0.309529
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 2.00000 0.0771517
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 8.00000 0.307238
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) −48.0000 −1.83801
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −24.0000 −0.915657
\(688\) 12.0000 0.457496
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 18.0000 0.684257
\(693\) −12.0000 −0.455842
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) −12.0000 −0.451306
\(708\) −12.0000 −0.450988
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) −16.0000 −0.597531
\(718\) 24.0000 0.895672
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) −19.0000 −0.707107
\(723\) −6.00000 −0.223142
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 4.00000 0.147844
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −72.0000 −2.65215
\(738\) 10.0000 0.368105
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) −14.0000 −0.512233
\(748\) 0 0
\(749\) −28.0000 −1.02310
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 8.00000 0.291730
\(753\) −6.00000 −0.218652
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −12.0000 −0.434714
\(763\) −32.0000 −1.15848
\(764\) 0 0
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −14.0000 −0.503871
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 6.00000 0.213741
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) −6.00000 −0.213201
\(793\) 8.00000 0.284088
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) −32.0000 −1.12996
\(803\) −60.0000 −2.11735
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 18.0000 0.633630
\(808\) −6.00000 −0.211079
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 12.0000 0.421117
\(813\) 28.0000 0.982003
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 12.0000 0.418548
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 1.00000 0.0347524
\(829\) 42.0000 1.45872 0.729360 0.684130i \(-0.239818\pi\)
0.729360 + 0.684130i \(0.239818\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −26.0000 −0.898155
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 4.00000 0.137849
\(843\) 16.0000 0.551069
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 50.0000 1.71802
\(848\) −2.00000 −0.0686803
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) −2.00000 −0.0683187 −0.0341593 0.999416i \(-0.510875\pi\)
−0.0341593 + 0.999416i \(0.510875\pi\)
\(858\) −12.0000 −0.409673
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) −12.0000 −0.408722
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) −17.0000 −0.577350
\(868\) 16.0000 0.543075
\(869\) 36.0000 1.22122
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) −16.0000 −0.541828
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −16.0000 −0.539974
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) −3.00000 −0.101015
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) −12.0000 −0.401790
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 2.00000 0.0667781
\(898\) 2.00000 0.0667409
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 0 0
\(902\) −60.0000 −1.99778
\(903\) 24.0000 0.798670
\(904\) 8.00000 0.266076
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 10.0000 0.331862
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 84.0000 2.77999
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −24.0000 −0.792982
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) −14.0000 −0.459820
\(928\) 6.00000 0.196960
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −10.0000 −0.327561
\(933\) 16.0000 0.523816
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 24.0000 0.783628
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) −16.0000 −0.521308
\(943\) 10.0000 0.325645
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −72.0000 −2.34092
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) −6.00000 −0.194871
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −36.0000 −1.16371
\(958\) 24.0000 0.775405
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −14.0000 −0.451144
\(964\) −6.00000 −0.193247
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 25.0000 0.803530
\(969\) 0 0
\(970\) 0 0
\(971\) −10.0000 −0.320915 −0.160458 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) 1.00000 0.0320750
\(973\) 24.0000 0.769405
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −32.0000 −1.02116
\(983\) −52.0000 −1.65854 −0.829271 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000 0.254000
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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 3450.2.a.y.1.1 1
5.2 odd 4 3450.2.d.j.2899.2 2
5.3 odd 4 3450.2.d.j.2899.1 2
5.4 even 2 138.2.a.a.1.1 1
15.14 odd 2 414.2.a.d.1.1 1
20.19 odd 2 1104.2.a.e.1.1 1
35.34 odd 2 6762.2.a.q.1.1 1
40.19 odd 2 4416.2.a.m.1.1 1
40.29 even 2 4416.2.a.z.1.1 1
60.59 even 2 3312.2.a.n.1.1 1
115.114 odd 2 3174.2.a.b.1.1 1
345.344 even 2 9522.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 1 5.4 even 2
414.2.a.d.1.1 1 15.14 odd 2
1104.2.a.e.1.1 1 20.19 odd 2
3174.2.a.b.1.1 1 115.114 odd 2
3312.2.a.n.1.1 1 60.59 even 2
3450.2.a.y.1.1 1 1.1 even 1 trivial
3450.2.d.j.2899.1 2 5.3 odd 4
3450.2.d.j.2899.2 2 5.2 odd 4
4416.2.a.m.1.1 1 40.19 odd 2
4416.2.a.z.1.1 1 40.29 even 2
6762.2.a.q.1.1 1 35.34 odd 2
9522.2.a.i.1.1 1 345.344 even 2