Properties

Label 2450.2.c.e.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.e.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 \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) −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 \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000i 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 \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 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 \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 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 \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 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 \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 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 \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −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 \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 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 \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 12.0000i 1.18818i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 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 \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) − 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 \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 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 \(0.370036\pi\)
−0.397043 + 0.917800i \(0.629964\pi\)
\(158\) 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 \(0.0370308\pi\)
−0.993241 + 0.116073i \(0.962969\pi\)
\(168\) 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 \(-0.888852\pi\)
0.939653 0.342129i \(-0.111148\pi\)
\(174\) 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 \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 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 \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 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 \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 2.00000i 0.132453i
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 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 \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) − 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 \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 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 \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) − 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 \(-0.863394\pi\)
0.909316 0.416107i \(-0.136606\pi\)
\(284\) 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 \(0.0846766\pi\)
−0.964825 + 0.262893i \(0.915323\pi\)
\(294\) 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 \(0.130820\pi\)
−0.916728 + 0.399511i \(0.869180\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000i 0.113228i
\(313\) 28.0000i 1.58265i 0.611393 + 0.791327i \(0.290609\pi\)
−0.611393 + 0.791327i \(0.709391\pi\)
\(314\) −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 \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 3.00000i 0.159901i
\(353\) − 12.0000i − 0.638696i −0.947638 0.319348i \(-0.896536\pi\)
0.947638 0.319348i \(-0.103464\pi\)
\(354\) 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 \(-0.834840\pi\)
0.868382 0.495896i \(-0.165160\pi\)
\(368\) 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 \(-0.819737\pi\)
0.843884 0.536525i \(-0.180263\pi\)
\(384\) 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 \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) − 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 \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 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 \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 23.0000i 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 \(0.410969\pi\)
−0.276066 + 0.961139i \(0.589031\pi\)
\(434\) 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 \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 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 \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 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 \(-0.955668\pi\)
0.990317 0.138823i \(-0.0443321\pi\)
\(468\) 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 \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 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 \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 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 \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 6.00000i 0.262613i
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 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 \(-0.654128\pi\)
0.465506 0.885044i \(-0.345872\pi\)
\(564\) 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 \(0.368499\pi\)
−0.401470 + 0.915872i \(0.631501\pi\)
\(578\) − 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 \(-0.835061\pi\)
0.868726 0.495293i \(-0.164939\pi\)
\(588\) 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 \(-0.835980\pi\)
0.870153 0.492781i \(-0.164020\pi\)
\(594\) −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 \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 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 \(-0.993540\pi\)
0.999794 0.0202944i \(-0.00646034\pi\)
\(608\) − 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 \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 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 \(-0.987444\pi\)
0.999222 0.0394362i \(-0.0125562\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 33.0000i 1.29736i 0.761060 + 0.648682i \(0.224679\pi\)
−0.761060 + 0.648682i \(0.775321\pi\)
\(648\) 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 \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) − 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 \(-0.944670\pi\)
0.984931 0.172949i \(-0.0553296\pi\)
\(678\) 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 \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 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 \(-0.994097\pi\)
0.999828 0.0185440i \(-0.00590307\pi\)
\(728\) 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 \(0.292068\pi\)
−0.607760 + 0.794121i \(0.707932\pi\)
\(734\) 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 \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) − 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 \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.0000i 0.575853i
\(773\) 51.0000i 1.83434i 0.398493 + 0.917171i \(0.369533\pi\)
−0.398493 + 0.917171i \(0.630467\pi\)
\(774\) 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 \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) 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 \(0.0338893\pi\)
−0.994338 + 0.106265i \(0.966111\pi\)
\(798\) 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 \(-0.355356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(812\) 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 \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 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 \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 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 \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) − 6.00000i − 0.204837i
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000i 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 \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 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 \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) 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 \(-0.115350\pi\)
0.935055 + 0.354504i \(0.115350\pi\)
\(930\) 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 \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 0 0
\(939\) −56.0000 −1.82749
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) − 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 \(-0.243196\pi\)
0.722059 + 0.691831i \(0.243196\pi\)
\(972\) 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 \(-0.984765\pi\)
0.998855 0.0478426i \(-0.0152346\pi\)
\(984\) −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 \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) 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.e.99.2 2
5.2 odd 4 490.2.a.d.1.1 1
5.3 odd 4 2450.2.a.v.1.1 1
5.4 even 2 inner 2450.2.c.e.99.1 2
7.3 odd 6 350.2.j.d.149.1 4
7.5 odd 6 350.2.j.d.249.2 4
7.6 odd 2 2450.2.c.q.99.2 2
15.2 even 4 4410.2.a.bg.1.1 1
20.7 even 4 3920.2.a.e.1.1 1
35.2 odd 12 490.2.e.g.361.1 2
35.3 even 12 350.2.e.b.51.1 2
35.12 even 12 70.2.e.d.11.1 2
35.13 even 4 2450.2.a.bf.1.1 1
35.17 even 12 70.2.e.d.51.1 yes 2
35.19 odd 6 350.2.j.d.249.1 4
35.24 odd 6 350.2.j.d.149.2 4
35.27 even 4 490.2.a.a.1.1 1
35.32 odd 12 490.2.e.g.471.1 2
35.33 even 12 350.2.e.b.151.1 2
35.34 odd 2 2450.2.c.q.99.1 2
105.17 odd 12 630.2.k.d.541.1 2
105.47 odd 12 630.2.k.d.361.1 2
105.62 odd 4 4410.2.a.x.1.1 1
140.27 odd 4 3920.2.a.bh.1.1 1
140.47 odd 12 560.2.q.b.81.1 2
140.87 odd 12 560.2.q.b.401.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 35.12 even 12
70.2.e.d.51.1 yes 2 35.17 even 12
350.2.e.b.51.1 2 35.3 even 12
350.2.e.b.151.1 2 35.33 even 12
350.2.j.d.149.1 4 7.3 odd 6
350.2.j.d.149.2 4 35.24 odd 6
350.2.j.d.249.1 4 35.19 odd 6
350.2.j.d.249.2 4 7.5 odd 6
490.2.a.a.1.1 1 35.27 even 4
490.2.a.d.1.1 1 5.2 odd 4
490.2.e.g.361.1 2 35.2 odd 12
490.2.e.g.471.1 2 35.32 odd 12
560.2.q.b.81.1 2 140.47 odd 12
560.2.q.b.401.1 2 140.87 odd 12
630.2.k.d.361.1 2 105.47 odd 12
630.2.k.d.541.1 2 105.17 odd 12
2450.2.a.v.1.1 1 5.3 odd 4
2450.2.a.bf.1.1 1 35.13 even 4
2450.2.c.e.99.1 2 5.4 even 2 inner
2450.2.c.e.99.2 2 1.1 even 1 trivial
2450.2.c.q.99.1 2 35.34 odd 2
2450.2.c.q.99.2 2 7.6 odd 2
3920.2.a.e.1.1 1 20.7 even 4
3920.2.a.bh.1.1 1 140.27 odd 4
4410.2.a.x.1.1 1 105.62 odd 4
4410.2.a.bg.1.1 1 15.2 even 4