Properties

Label 3450.2.a.z.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 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +8.00000 q^{19} +2.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -10.0000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} -8.00000 q^{37} +8.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} +2.00000 q^{42} -12.0000 q^{43} +2.00000 q^{44} +1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +8.00000 q^{57} -10.0000 q^{58} +4.00000 q^{59} +12.0000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +4.00000 q^{67} +1.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -8.00000 q^{74} +8.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +10.0000 q^{83} +2.00000 q^{84} -12.0000 q^{86} -10.0000 q^{87} +2.00000 q^{88} +4.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 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
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) 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
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 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
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 8.00000 1.29777
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000 0.308607
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 8.00000 1.05963
\(58\) −10.0000 −1.31306
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −10.0000 −1.07211
\(88\) 2.00000 0.213201
\(89\) 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
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −3.00000 −0.303046
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 1.00000 0.0962250
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 2.00000 0.188982
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 12.0000 1.08643
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 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 \(0.613642\pi\)
\(132\) 2.00000 0.174078
\(133\) 16.0000 1.38738
\(134\) 4.00000 0.345547
\(135\) 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
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 16.0000 1.34269
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) −3.00000 −0.247436
\(148\) −8.00000 −0.657596
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 8.00000 0.648886
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 10.0000 0.795557
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −12.0000 −0.914991
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 4.00000 0.296500
\(183\) 12.0000 0.887066
\(184\) 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
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 2.00000 0.142134
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 10.0000 0.703598
\(203\) −20.0000 −1.40372
\(204\) 0 0
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −10.0000 −0.686803
\(213\) 16.0000 1.09630
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −8.00000 −0.541828
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 8.00000 0.529813
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) −10.0000 −0.656532
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 16.0000 1.01806
\(248\) 8.00000 0.508001
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 2.00000 0.125988
\(253\) 2.00000 0.125739
\(254\) 4.00000 0.250982
\(255\) 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
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) −8.00000 −0.494242
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 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
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) −20.0000 −1.19952
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) −8.00000 −0.476393
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 10.0000 0.585206
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 2.00000 0.116052
\(298\) −10.0000 −0.579284
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 20.0000 1.15087
\(303\) 10.0000 0.574485
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 4.00000 0.227921
\(309\) 2.00000 0.113776
\(310\) 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
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −10.0000 −0.560772
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −8.00000 −0.442401
\(328\) −6.00000 −0.331295
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 10.0000 0.548821
\(333\) −8.00000 −0.438397
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 8.00000 0.432590
\(343\) −20.0000 −1.07990
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) −10.0000 −0.536056
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 2.00000 0.106600
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −16.0000 −0.845626
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 16.0000 0.840941
\(363\) −7.00000 −0.367405
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 12.0000 0.627250
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 8.00000 0.414781
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −20.0000 −1.03005
\(378\) 2.00000 0.102869
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) −16.0000 −0.818631
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −12.0000 −0.609994
\(388\) −10.0000 −0.507673
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) −8.00000 −0.403547
\(394\) −26.0000 −1.30986
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) −14.0000 −0.701757
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) −16.0000 −0.793091
\(408\) 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
\(412\) 2.00000 0.0985329
\(413\) 8.00000 0.393654
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −20.0000 −0.979404
\(418\) 16.0000 0.782586
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −36.0000 −1.75453 −0.877266 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 4.00000 0.194717
\(423\) −8.00000 −0.388973
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 24.0000 1.16144
\(428\) 10.0000 0.483368
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 8.00000 0.382692
\(438\) 10.0000 0.477818
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) −10.0000 −0.472984
\(448\) 2.00000 0.0944911
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) −8.00000 −0.376288
\(453\) 20.0000 0.939682
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 4.00000 0.186097
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −10.0000 −0.464238
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 2.00000 0.0924500
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 4.00000 0.184115
\(473\) −24.0000 −1.10352
\(474\) 10.0000 0.459315
\(475\) 0 0
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −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
\(481\) −16.0000 −0.729537
\(482\) 10.0000 0.455488
\(483\) 2.00000 0.0910032
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 12.0000 0.543214
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 32.0000 1.43540
\(498\) 10.0000 0.448111
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −14.0000 −0.624851
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) −9.00000 −0.399704
\(508\) 4.00000 0.177471
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) −16.0000 −0.703679
\(518\) −16.0000 −0.703000
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) −10.0000 −0.437688
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 16.0000 0.693688
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −16.0000 −0.690451
\(538\) 18.0000 0.776035
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 28.0000 1.20270
\(543\) 16.0000 0.686626
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 12.0000 0.512615
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −80.0000 −3.40811
\(552\) 1.00000 0.0425628
\(553\) 20.0000 0.850487
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −16.0000 −0.674919
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 2.00000 0.0839921
\(568\) 16.0000 0.671345
\(569\) −32.0000 −1.34151 −0.670755 0.741679i \(-0.734030\pi\)
−0.670755 + 0.741679i \(0.734030\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 4.00000 0.167248
\(573\) −16.0000 −0.668410
\(574\) −12.0000 −0.500870
\(575\) 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
\(581\) 20.0000 0.829740
\(582\) −10.0000 −0.414513
\(583\) −20.0000 −0.828315
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −3.00000 −0.123718
\(589\) 64.0000 2.63707
\(590\) 0 0
\(591\) −26.0000 −1.06950
\(592\) −8.00000 −0.328798
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −14.0000 −0.572982
\(598\) 2.00000 0.0817861
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 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
\(603\) 4.00000 0.162893
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 8.00000 0.324443
\(609\) −20.0000 −0.810441
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −48.0000 −1.93241 −0.966204 0.257780i \(-0.917009\pi\)
−0.966204 + 0.257780i \(0.917009\pi\)
\(618\) 2.00000 0.0804518
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 16.0000 0.641542
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 16.0000 0.638978
\(628\) 8.00000 0.319235
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 10.0000 0.397779
\(633\) 4.00000 0.158986
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) −6.00000 −0.237729
\(638\) −20.0000 −0.791808
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 10.0000 0.394669
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 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
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −4.00000 −0.156652
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −8.00000 −0.312825
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) −16.0000 −0.623745
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) 10.0000 0.388075
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) −10.0000 −0.387202
\(668\) 8.00000 0.309529
\(669\) 4.00000 0.154649
\(670\) 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 \(0.153071\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) −8.00000 −0.307238
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 16.0000 0.612672
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −16.0000 −0.610438
\(688\) −12.0000 −0.457496
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −52.0000 −1.97817 −0.989087 0.147335i \(-0.952930\pi\)
−0.989087 + 0.147335i \(0.952930\pi\)
\(692\) 2.00000 0.0760286
\(693\) 4.00000 0.151947
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) 0 0
\(698\) −6.00000 −0.227103
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000 0.0754851
\(703\) −64.0000 −2.41381
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 20.0000 0.752177
\(708\) 4.00000 0.150329
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) −16.0000 −0.597531
\(718\) 8.00000 0.298557
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 45.0000 1.67473
\(723\) 10.0000 0.371904
\(724\) 16.0000 0.594635
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 12.0000 0.443533
\(733\) 52.0000 1.92066 0.960332 0.278859i \(-0.0899564\pi\)
0.960332 + 0.278859i \(0.0899564\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 8.00000 0.294684
\(738\) −6.00000 −0.220863
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) −20.0000 −0.734223
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 24.0000 0.878702
\(747\) 10.0000 0.365881
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) −8.00000 −0.291730
\(753\) −14.0000 −0.510188
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 8.00000 0.290573
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 4.00000 0.144905
\(763\) −16.0000 −0.579239
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −14.0000 −0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) −16.0000 −0.573997
\(778\) −2.00000 −0.0717035
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) −10.0000 −0.357371
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −26.0000 −0.926212
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) −16.0000 −0.568895
\(792\) 2.00000 0.0710669
\(793\) 24.0000 0.852265
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 16.0000 0.566394
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) −16.0000 −0.564980
\(803\) 20.0000 0.705785
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 18.0000 0.633630
\(808\) 10.0000 0.351799
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) −20.0000 −0.701862
\(813\) 28.0000 0.982003
\(814\) −16.0000 −0.560800
\(815\) 0 0
\(816\) 0 0
\(817\) −96.0000 −3.35861
\(818\) −10.0000 −0.349642
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 12.0000 0.418548
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 1.00000 0.0347524
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 8.00000 0.276520
\(838\) 30.0000 1.03633
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −36.0000 −1.24064
\(843\) −16.0000 −0.551069
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) −14.0000 −0.481046
\(848\) −10.0000 −0.343401
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 16.0000 0.548151
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 10.0000 0.341793
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 4.00000 0.136558
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) −28.0000 −0.953684
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) −17.0000 −0.577350
\(868\) 16.0000 0.543075
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −8.00000 −0.270914
\(873\) −10.0000 −0.338449
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 16.0000 0.539974
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −3.00000 −0.101015
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −8.00000 −0.268462
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 4.00000 0.133930
\(893\) −64.0000 −2.14168
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 2.00000 0.0667781
\(898\) 18.0000 0.600668
\(899\) −80.0000 −2.66815
\(900\) 0 0
\(901\) 0 0
\(902\) −12.0000 −0.399556
\(903\) −24.0000 −0.798670
\(904\) −8.00000 −0.266076
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −14.0000 −0.464606
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 8.00000 0.264906
\(913\) 20.0000 0.661903
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) −30.0000 −0.987997
\(923\) 32.0000 1.05329
\(924\) 4.00000 0.131590
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 2.00000 0.0656886
\(928\) −10.0000 −0.328266
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) 6.00000 0.196537
\(933\) 16.0000 0.523816
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 8.00000 0.261209
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 8.00000 0.260654
\(943\) −6.00000 −0.195387
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 10.0000 0.324785
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −8.00000 −0.259145 −0.129573 0.991570i \(-0.541361\pi\)
−0.129573 + 0.991570i \(0.541361\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −20.0000 −0.646508
\(958\) 24.0000 0.775405
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −16.0000 −0.515861
\(963\) 10.0000 0.322245
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 1.00000 0.0320750
\(973\) −40.0000 −1.28234
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) 20.0000 0.639857 0.319928 0.947442i \(-0.396341\pi\)
0.319928 + 0.947442i \(0.396341\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) −8.00000 −0.255420
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 16.0000 0.509028
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000 0.254000
\(993\) −12.0000 −0.380808
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) 10.0000 0.316862
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −4.00000 −0.126618
\(999\) −8.00000 −0.253109
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.z.1.1 1
5.2 odd 4 3450.2.d.q.2899.2 2
5.3 odd 4 3450.2.d.q.2899.1 2
5.4 even 2 690.2.a.c.1.1 1
15.14 odd 2 2070.2.a.k.1.1 1
20.19 odd 2 5520.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.c.1.1 1 5.4 even 2
2070.2.a.k.1.1 1 15.14 odd 2
3450.2.a.z.1.1 1 1.1 even 1 trivial
3450.2.d.q.2899.1 2 5.3 odd 4
3450.2.d.q.2899.2 2 5.2 odd 4
5520.2.a.bg.1.1 1 20.19 odd 2