Properties

Label 2450.2.c.q.99.2
Level $2450$
Weight $2$
Character 2450.99
Analytic conductor $19.563$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2450.99
Dual form 2450.2.c.q.99.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +3.00000i q^{22} +9.00000i q^{23} -2.00000 q^{24} +1.00000 q^{26} -4.00000i q^{27} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} +3.00000 q^{41} +2.00000i q^{43} -3.00000 q^{44} -9.00000 q^{46} -9.00000i q^{47} -2.00000i q^{48} +12.0000 q^{51} +1.00000i q^{52} +9.00000i q^{53} +4.00000 q^{54} -2.00000i q^{57} -6.00000i q^{58} +8.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +18.0000 q^{69} +1.00000i q^{72} -4.00000i q^{73} -7.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +3.00000i q^{82} -2.00000 q^{86} +12.0000i q^{87} -3.00000i q^{88} -6.00000 q^{89} -9.00000i q^{92} -16.0000i q^{93} +9.00000 q^{94} +2.00000 q^{96} +10.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + 6q^{11} + 2q^{16} + 2q^{19} - 4q^{24} + 2q^{26} - 12q^{29} + 16q^{31} - 12q^{34} + 2q^{36} - 4q^{39} + 6q^{41} - 6q^{44} - 18q^{46} + 24q^{51} + 8q^{54} + 16q^{61} - 2q^{64} + 12q^{66} + 36q^{69} - 14q^{74} - 2q^{76} + 20q^{79} - 22q^{81} - 4q^{86} - 12q^{89} + 18q^{94} + 4q^{96} - 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.00000i 0.577350i
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000i 0.639602i
\(23\) 9.00000i 1.87663i 0.345782 + 0.938315i \(0.387614\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 6.00000i − 1.04447i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 1.00000i 0.162221i
\(39\) −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
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) − 9.00000i − 1.31278i −0.754420 0.656392i \(-0.772082\pi\)
0.754420 0.656392i \(-0.227918\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) 0 0
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 1.00000i 0.138675i
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) − 6.00000i − 0.787839i
\(59\) 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 \(0.328850\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 18.0000 2.16695
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 3.00000i 0.331295i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 12.0000i 1.28654i
\(88\) − 3.00000i − 0.319801i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 9.00000i − 0.938315i
\(93\) − 16.0000i − 1.65912i
\(94\) 9.00000 0.928279
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 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 \(0.296350\pi\)
\(102\) 12.0000i 1.18818i
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 14.0000 1.32882
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 8.00000i 0.724286i
\(123\) − 6.00000i − 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000i 0.0887357i 0.999015 + 0.0443678i \(0.0141274\pi\)
−0.999015 + 0.0443678i \(0.985873\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 18.0000i 1.53226i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −18.0000 −1.51587
\(142\) 0 0
\(143\) − 3.00000i − 0.250873i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) − 7.00000i − 0.575396i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 23.0000i − 1.83560i −0.397043 0.917800i \(-0.629964\pi\)
0.397043 0.917800i \(-0.370036\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) 0 0
\(162\) − 11.0000i − 0.864242i
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 0 0
\(167\) − 3.00000i − 0.232147i −0.993241 0.116073i \(-0.962969\pi\)
0.993241 0.116073i \(-0.0370308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) − 2.00000i − 0.152499i
\(173\) 9.00000i 0.684257i 0.939653 + 0.342129i \(0.111148\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) − 6.00000i − 0.449719i
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) − 16.0000i − 1.18275i
\(184\) 9.00000 0.663489
\(185\) 0 0
\(186\) 16.0000 1.17318
\(187\) 18.0000i 1.31629i
\(188\) 9.00000i 0.656392i
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 2.00000i 0.144338i
\(193\) − 16.0000i − 1.15171i −0.817554 0.575853i \(-0.804670\pi\)
0.817554 0.575853i \(-0.195330\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.0000i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) − 3.00000i − 0.213201i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 12.0000i 0.844317i
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) − 9.00000i − 0.625543i
\(208\) − 1.00000i − 0.0693375i
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) − 9.00000i − 0.618123i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) 16.0000i 1.08366i
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 14.0000i 0.939618i
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 2.00000i 0.132453i
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 0 0
\(237\) − 20.0000i − 1.29914i
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 10.0000i 0.641500i
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) − 1.00000i − 0.0636285i
\(248\) − 8.00000i − 0.508001i
\(249\) 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
\(253\) 27.0000i 1.69748i
\(254\) −1.00000 −0.0627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 3.00000i 0.185341i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 8.00000i 0.488678i
\(269\) 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 \(0.661532\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −18.0000 −1.08347
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) 4.00000i 0.239904i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −27.0000 −1.61068 −0.805342 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) − 18.0000i − 1.07188i
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) 4.00000i 0.234082i
\(293\) − 9.00000i − 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) − 12.0000i − 0.696311i
\(298\) 6.00000i 0.347571i
\(299\) 9.00000 0.520483
\(300\) 0 0
\(301\) 0 0
\(302\) − 10.0000i − 0.575435i
\(303\) − 24.0000i − 1.37876i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 14.0000i − 0.799022i −0.916728 0.399511i \(-0.869180\pi\)
0.916728 0.399511i \(-0.130820\pi\)
\(308\) 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 \(0.738219\pi\)
\(312\) 2.00000i 0.113228i
\(313\) − 28.0000i − 1.58265i −0.611393 0.791327i \(-0.709391\pi\)
0.611393 0.791327i \(-0.290609\pi\)
\(314\) 23.0000 1.29797
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 18.0000i 1.00939i
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 6.00000i 0.333849i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) − 32.0000i − 1.76960i
\(328\) − 3.00000i − 0.165647i
\(329\) 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
\(333\) − 7.00000i − 0.383598i
\(334\) 3.00000 0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) − 1.00000i − 0.0540738i
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) − 12.0000i − 0.643268i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 3.00000i 0.159901i
\(353\) 12.0000i 0.638696i 0.947638 + 0.319348i \(0.103464\pi\)
−0.947638 + 0.319348i \(0.896536\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 3.00000i 0.158555i
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 2.00000i 0.105118i
\(363\) 4.00000i 0.209946i
\(364\) 0 0
\(365\) 0 0
\(366\) 16.0000 0.836333
\(367\) 19.0000i 0.991792i 0.868382 + 0.495896i \(0.165160\pi\)
−0.868382 + 0.495896i \(0.834840\pi\)
\(368\) 9.00000i 0.469157i
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) 0 0
\(372\) 16.0000i 0.829561i
\(373\) 2.00000i 0.103556i 0.998659 + 0.0517780i \(0.0164888\pi\)
−0.998659 + 0.0517780i \(0.983511\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) −23.0000 −1.18143 −0.590715 0.806880i \(-0.701154\pi\)
−0.590715 + 0.806880i \(0.701154\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 12.0000i 0.613973i
\(383\) 21.0000i 1.07305i 0.843884 + 0.536525i \(0.180263\pi\)
−0.843884 + 0.536525i \(0.819737\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) − 2.00000i − 0.101666i
\(388\) − 10.0000i − 0.507673i
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) −54.0000 −2.73090
\(392\) 0 0
\(393\) − 6.00000i − 0.302660i
\(394\) 15.0000 0.755689
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) − 16.0000i − 0.798007i
\(403\) − 8.00000i − 0.398508i
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 21.0000i 1.04093i
\(408\) − 12.0000i − 0.594089i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 4.00000i 0.197066i
\(413\) 0 0
\(414\) 9.00000 0.442326
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) − 8.00000i − 0.391762i
\(418\) 3.00000i 0.146735i
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 23.0000i 1.11962i
\(423\) 9.00000i 0.437595i
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) − 40.0000i − 1.92228i −0.276066 0.961139i \(-0.589031\pi\)
0.276066 0.961139i \(-0.410969\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 9.00000i 0.430528i
\(438\) − 8.00000i − 0.382255i
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.00000i 0.285391i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) −14.0000 −0.664411
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) − 12.0000i − 0.567581i
\(448\) 0 0
\(449\) −21.0000 −0.991051 −0.495526 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) 20.0000i 0.939682i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) − 14.0000i − 0.654892i −0.944870 0.327446i \(-0.893812\pi\)
0.944870 0.327446i \(-0.106188\pi\)
\(458\) 4.00000i 0.186908i
\(459\) 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
\(463\) − 1.00000i − 0.0464739i −0.999730 0.0232370i \(-0.992603\pi\)
0.999730 0.0232370i \(-0.00739722\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) 0 0
\(470\) 0 0
\(471\) −46.0000 −2.11957
\(472\) 0 0
\(473\) 6.00000i 0.275880i
\(474\) 20.0000 0.918630
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.00000i − 0.412082i
\(478\) 6.00000i 0.274434i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 7.00000 0.319173
\(482\) − 1.00000i − 0.0455488i
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) − 8.00000i − 0.362143i
\(489\) 40.0000 1.80886
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 6.00000i 0.270501i
\(493\) − 36.0000i − 1.62136i
\(494\) 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 \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) − 15.0000i − 0.669483i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −27.0000 −1.20030
\(507\) − 24.0000i − 1.06588i
\(508\) − 1.00000i − 0.0443678i
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) − 27.0000i − 1.18746i
\(518\) 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 \(0.606570\pi\)
\(522\) 6.00000i 0.262613i
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) −3.00000 −0.131056
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000i 2.09091i
\(528\) − 6.00000i − 0.261116i
\(529\) −58.0000 −2.52174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 3.00000i − 0.129944i
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) − 6.00000i − 0.258919i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) − 4.00000i − 0.171656i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 12.0000i 0.512615i
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) − 18.0000i − 0.766131i
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 9.00000i 0.381342i 0.981654 + 0.190671i \(0.0610664\pi\)
−0.981654 + 0.190671i \(0.938934\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) − 27.0000i − 1.13893i
\(563\) 42.0000i 1.77009i 0.465506 + 0.885044i \(0.345872\pi\)
−0.465506 + 0.885044i \(0.654128\pi\)
\(564\) 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 \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 3.00000i 0.125436i
\(573\) − 24.0000i − 1.00261i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 44.0000i − 1.83174i −0.401470 0.915872i \(-0.631501\pi\)
0.401470 0.915872i \(-0.368499\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) −32.0000 −1.32987
\(580\) 0 0
\(581\) 0 0
\(582\) 20.0000i 0.829027i
\(583\) 27.0000i 1.11823i
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 24.0000i 0.990586i 0.868726 + 0.495293i \(0.164939\pi\)
−0.868726 + 0.495293i \(0.835061\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −30.0000 −1.23404
\(592\) 7.00000i 0.287698i
\(593\) 24.0000i 0.985562i 0.870153 + 0.492781i \(0.164020\pi\)
−0.870153 + 0.492781i \(0.835980\pi\)
\(594\) 12.0000 0.492366
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) − 32.0000i − 1.30967i
\(598\) 9.00000i 0.368037i
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) 1.00000i 0.0405887i 0.999794 + 0.0202944i \(0.00646034\pi\)
−0.999794 + 0.0202944i \(0.993540\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 6.00000i 0.242536i
\(613\) 29.0000i 1.17130i 0.810564 + 0.585649i \(0.199160\pi\)
−0.810564 + 0.585649i \(0.800840\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) −23.0000 −0.924448 −0.462224 0.886763i \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) 0 0
\(621\) 36.0000 1.44463
\(622\) − 24.0000i − 0.962312i
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 28.0000 1.11911
\(627\) − 6.00000i − 0.239617i
\(628\) 23.0000i 0.917800i
\(629\) −42.0000 −1.67465
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) − 10.0000i − 0.397779i
\(633\) − 46.0000i − 1.82834i
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −18.0000 −0.713746
\(637\) 0 0
\(638\) − 18.0000i − 0.712627i
\(639\) 0 0
\(640\) 0 0
\(641\) 27.0000 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(642\) 24.0000i 0.947204i
\(643\) 2.00000i 0.0788723i 0.999222 + 0.0394362i \(0.0125562\pi\)
−0.999222 + 0.0394362i \(0.987444\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) − 33.0000i − 1.29736i −0.761060 0.648682i \(-0.775321\pi\)
0.761060 0.648682i \(-0.224679\pi\)
\(648\) 11.0000i 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) − 20.0000i − 0.783260i
\(653\) − 9.00000i − 0.352197i −0.984373 0.176099i \(-0.943652\pi\)
0.984373 0.176099i \(-0.0563478\pi\)
\(654\) 32.0000 1.25130
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) − 7.00000i − 0.272063i
\(663\) − 12.0000i − 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) − 54.0000i − 2.09089i
\(668\) 3.00000i 0.116073i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) − 34.0000i − 1.31060i −0.755367 0.655302i \(-0.772541\pi\)
0.755367 0.655302i \(-0.227459\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 9.00000i 0.345898i 0.984931 + 0.172949i \(0.0553296\pi\)
−0.984931 + 0.172949i \(0.944670\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 24.0000i 0.919007i
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) − 8.00000i − 0.305219i
\(688\) 2.00000i 0.0762493i
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) − 9.00000i − 0.342129i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 18.0000i 0.681799i
\(698\) − 26.0000i − 0.984115i
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 7.00000i 0.264010i
\(704\) −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 \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 6.00000i 0.224860i
\(713\) 72.0000i 2.69642i
\(714\) 0 0
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) − 12.0000i − 0.448148i
\(718\) − 18.0000i − 0.671754i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 18.0000i − 0.669891i
\(723\) 2.00000i 0.0743808i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) 1.00000i 0.0370879i 0.999828 + 0.0185440i \(0.00590307\pi\)
−0.999828 + 0.0185440i \(0.994097\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 16.0000i 0.591377i
\(733\) − 43.0000i − 1.58824i −0.607760 0.794121i \(-0.707932\pi\)
0.607760 0.794121i \(-0.292068\pi\)
\(734\) −19.0000 −0.701303
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) − 24.0000i − 0.884051i
\(738\) − 3.00000i − 0.110432i
\(739\) −35.0000 −1.28750 −0.643748 0.765238i \(-0.722621\pi\)
−0.643748 + 0.765238i \(0.722621\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) − 45.0000i − 1.65089i −0.564483 0.825445i \(-0.690924\pi\)
0.564483 0.825445i \(-0.309076\pi\)
\(744\) −16.0000 −0.586588
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) 0 0
\(748\) − 18.0000i − 0.658145i
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) − 9.00000i − 0.328196i
\(753\) 30.0000i 1.09326i
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) − 38.0000i − 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) − 23.0000i − 0.835398i
\(759\) 54.0000 1.96008
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 2.00000i 0.0724524i
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −21.0000 −0.758761
\(767\) 0 0
\(768\) − 2.00000i − 0.0721688i
\(769\) −23.0000 −0.829401 −0.414701 0.909958i \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.0000i 0.575853i
\(773\) − 51.0000i − 1.83434i −0.398493 0.917171i \(-0.630467\pi\)
0.398493 0.917171i \(-0.369533\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 12.0000i 0.430221i
\(779\) 3.00000 0.107486
\(780\) 0 0
\(781\) 0 0
\(782\) − 54.0000i − 1.93104i
\(783\) 24.0000i 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 6.00000 0.214013
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) 15.0000i 0.534353i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 3.00000i 0.106600i
\(793\) − 8.00000i − 0.284088i
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) 54.0000 1.91038
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) − 27.0000i − 0.953403i
\(803\) − 12.0000i − 0.423471i
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) − 12.0000i − 0.422159i
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 0 0
\(813\) 32.0000i 1.12229i
\(814\) −21.0000 −0.736050
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 2.00000i 0.0699711i
\(818\) − 26.0000i − 0.909069i
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) − 24.0000i − 0.837096i
\(823\) − 4.00000i − 0.139431i −0.997567 0.0697156i \(-0.977791\pi\)
0.997567 0.0697156i \(-0.0222092\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) − 6.00000i − 0.208640i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) 9.00000i 0.312772i
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 1.00000i 0.0346688i
\(833\) 0 0
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) −3.00000 −0.103757
\(837\) − 32.0000i − 1.10608i
\(838\) 9.00000i 0.310900i
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 2.00000i 0.0689246i
\(843\) 54.0000i 1.85986i
\(844\) −23.0000 −0.791693
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) 0 0
\(848\) 9.00000i 0.309061i
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) −63.0000 −2.15961
\(852\) 0 0
\(853\) − 19.0000i − 0.650548i −0.945620 0.325274i \(-0.894544\pi\)
0.945620 0.325274i \(-0.105456\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) − 6.00000i − 0.204837i
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000i 0.408722i
\(863\) − 3.00000i − 0.102121i −0.998696 0.0510606i \(-0.983740\pi\)
0.998696 0.0510606i \(-0.0162602\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 40.0000 1.35926
\(867\) 38.0000i 1.29055i
\(868\) 0 0
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 16.0000i − 0.541828i
\(873\) − 10.0000i − 0.338449i
\(874\) −9.00000 −0.304430
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 13.0000i 0.438979i 0.975615 + 0.219489i \(0.0704391\pi\)
−0.975615 + 0.219489i \(0.929561\pi\)
\(878\) − 26.0000i − 0.877457i
\(879\) −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
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) − 14.0000i − 0.469809i
\(889\) 0 0
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) − 8.00000i − 0.267860i
\(893\) − 9.00000i − 0.301174i
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) − 18.0000i − 0.601003i
\(898\) − 21.0000i − 0.700779i
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −54.0000 −1.79900
\(902\) 9.00000i 0.299667i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) 10.0000i 0.332045i 0.986122 + 0.166022i \(0.0530924\pi\)
−0.986122 + 0.166022i \(0.946908\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) − 2.00000i − 0.0662266i
\(913\) 0 0
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) 24.0000i 0.792118i
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) − 30.0000i − 0.987997i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 1.00000 0.0328620
\(927\) 4.00000i 0.131377i
\(928\) − 6.00000i − 0.196960i
\(929\) −57.0000 −1.87011 −0.935055 0.354504i \(-0.884650\pi\)
−0.935055 + 0.354504i \(0.884650\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000i 0.196537i
\(933\) 48.0000i 1.57145i
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 10.0000i 0.326686i 0.986569 + 0.163343i \(0.0522277\pi\)
−0.986569 + 0.163343i \(0.947772\pi\)
\(938\) 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 \(0.214007\pi\)
\(942\) − 46.0000i − 1.49876i
\(943\) 27.0000i 0.879241i
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) − 6.00000i − 0.194974i −0.995237 0.0974869i \(-0.968920\pi\)
0.995237 0.0974869i \(-0.0310804\pi\)
\(948\) 20.0000i 0.649570i
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 36.0000i 1.16615i 0.812417 + 0.583077i \(0.198151\pi\)
−0.812417 + 0.583077i \(0.801849\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 36.0000i 1.16371i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 7.00000i 0.225689i
\(963\) − 12.0000i − 0.386695i
\(964\) 1.00000 0.0322078
\(965\) 0 0
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 40.0000i 1.27906i
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 36.0000i 1.14881i
\(983\) 3.00000i 0.0956851i 0.998855 + 0.0478426i \(0.0152346\pi\)
−0.998855 + 0.0478426i \(0.984765\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 1.00000i 0.0318142i
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 14.0000i 0.444277i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.0000i 0.696747i 0.937356 + 0.348373i \(0.113266\pi\)
−0.937356 + 0.348373i \(0.886734\pi\)
\(998\) 4.00000i 0.126618i
\(999\) 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.c.q.99.2 2
5.2 odd 4 490.2.a.a.1.1 1
5.3 odd 4 2450.2.a.bf.1.1 1
5.4 even 2 inner 2450.2.c.q.99.1 2
7.2 even 3 350.2.j.d.249.2 4
7.4 even 3 350.2.j.d.149.1 4
7.6 odd 2 2450.2.c.e.99.2 2
15.2 even 4 4410.2.a.x.1.1 1
20.7 even 4 3920.2.a.bh.1.1 1
35.2 odd 12 70.2.e.d.11.1 2
35.4 even 6 350.2.j.d.149.2 4
35.9 even 6 350.2.j.d.249.1 4
35.12 even 12 490.2.e.g.361.1 2
35.13 even 4 2450.2.a.v.1.1 1
35.17 even 12 490.2.e.g.471.1 2
35.18 odd 12 350.2.e.b.51.1 2
35.23 odd 12 350.2.e.b.151.1 2
35.27 even 4 490.2.a.d.1.1 1
35.32 odd 12 70.2.e.d.51.1 yes 2
35.34 odd 2 2450.2.c.e.99.1 2
105.2 even 12 630.2.k.d.361.1 2
105.32 even 12 630.2.k.d.541.1 2
105.62 odd 4 4410.2.a.bg.1.1 1
140.27 odd 4 3920.2.a.e.1.1 1
140.67 even 12 560.2.q.b.401.1 2
140.107 even 12 560.2.q.b.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 35.2 odd 12
70.2.e.d.51.1 yes 2 35.32 odd 12
350.2.e.b.51.1 2 35.18 odd 12
350.2.e.b.151.1 2 35.23 odd 12
350.2.j.d.149.1 4 7.4 even 3
350.2.j.d.149.2 4 35.4 even 6
350.2.j.d.249.1 4 35.9 even 6
350.2.j.d.249.2 4 7.2 even 3
490.2.a.a.1.1 1 5.2 odd 4
490.2.a.d.1.1 1 35.27 even 4
490.2.e.g.361.1 2 35.12 even 12
490.2.e.g.471.1 2 35.17 even 12
560.2.q.b.81.1 2 140.107 even 12
560.2.q.b.401.1 2 140.67 even 12
630.2.k.d.361.1 2 105.2 even 12
630.2.k.d.541.1 2 105.32 even 12
2450.2.a.v.1.1 1 35.13 even 4
2450.2.a.bf.1.1 1 5.3 odd 4
2450.2.c.e.99.1 2 35.34 odd 2
2450.2.c.e.99.2 2 7.6 odd 2
2450.2.c.q.99.1 2 5.4 even 2 inner
2450.2.c.q.99.2 2 1.1 even 1 trivial
3920.2.a.e.1.1 1 140.27 odd 4
3920.2.a.bh.1.1 1 20.7 even 4
4410.2.a.x.1.1 1 15.2 even 4
4410.2.a.bg.1.1 1 105.62 odd 4