Properties

Label 5550.2.a.k.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +5.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -5.00000 q^{14} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -5.00000 q^{21} +5.00000 q^{22} -3.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +5.00000 q^{28} +6.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} -5.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{39} +5.00000 q^{42} -4.00000 q^{43} -5.00000 q^{44} +3.00000 q^{46} -1.00000 q^{48} +18.0000 q^{49} -5.00000 q^{51} +1.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} -5.00000 q^{56} +3.00000 q^{57} -6.00000 q^{58} -10.0000 q^{59} +10.0000 q^{61} +6.00000 q^{62} +5.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +14.0000 q^{67} +5.00000 q^{68} +3.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +9.00000 q^{73} -1.00000 q^{74} -3.00000 q^{76} -25.0000 q^{77} +1.00000 q^{78} +1.00000 q^{81} -5.00000 q^{83} -5.00000 q^{84} +4.00000 q^{86} -6.00000 q^{87} +5.00000 q^{88} +13.0000 q^{89} +5.00000 q^{91} -3.00000 q^{92} +6.00000 q^{93} +1.00000 q^{96} +4.00000 q^{97} -18.0000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(7\) 5.00000 1.88982 0.944911 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −5.00000 −1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) −5.00000 −1.09109
\(22\) 5.00000 1.06600
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 5.00000 0.944911
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.00000 0.870388
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 3.00000 0.486664
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 5.00000 0.771517
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 1.00000 0.138675
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −5.00000 −0.668153
\(57\) 3.00000 0.397360
\(58\) −6.00000 −0.787839
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) 5.00000 0.629941
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 5.00000 0.606339
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) −25.0000 −2.84901
\(78\) 1.00000 0.113228
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.00000 −0.548821 −0.274411 0.961613i \(-0.588483\pi\)
−0.274411 + 0.961613i \(0.588483\pi\)
\(84\) −5.00000 −0.545545
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) 5.00000 0.533002
\(89\) 13.0000 1.37800 0.688999 0.724763i \(-0.258051\pi\)
0.688999 + 0.724763i \(0.258051\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) −3.00000 −0.312772
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) −18.0000 −1.81827
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 5.00000 0.495074
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −15.0000 −1.45010 −0.725052 0.688694i \(-0.758184\pi\)
−0.725052 + 0.688694i \(0.758184\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 5.00000 0.472456
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 1.00000 0.0924500
\(118\) 10.0000 0.920575
\(119\) 25.0000 2.29175
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) −5.00000 −0.445435
\(127\) 3.00000 0.266207 0.133103 0.991102i \(-0.457506\pi\)
0.133103 + 0.991102i \(0.457506\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 5.00000 0.435194
\(133\) −15.0000 −1.30066
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) −5.00000 −0.428746
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −3.00000 −0.255377
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −5.00000 −0.418121
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −9.00000 −0.744845
\(147\) −18.0000 −1.48461
\(148\) 1.00000 0.0821995
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 15.0000 1.22068 0.610341 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) 3.00000 0.243332
\(153\) 5.00000 0.404226
\(154\) 25.0000 2.01456
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 20.0000 1.59617 0.798087 0.602542i \(-0.205846\pi\)
0.798087 + 0.602542i \(0.205846\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −15.0000 −1.18217
\(162\) −1.00000 −0.0785674
\(163\) 23.0000 1.80150 0.900750 0.434339i \(-0.143018\pi\)
0.900750 + 0.434339i \(0.143018\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 5.00000 0.388075
\(167\) −5.00000 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 5.00000 0.385758
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) −4.00000 −0.304997
\(173\) −11.0000 −0.836315 −0.418157 0.908375i \(-0.637324\pi\)
−0.418157 + 0.908375i \(0.637324\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 10.0000 0.751646
\(178\) −13.0000 −0.974391
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −5.00000 −0.370625
\(183\) −10.0000 −0.739221
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) −25.0000 −1.82818
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) −23.0000 −1.63868 −0.819341 0.573306i \(-0.805660\pi\)
−0.819341 + 0.573306i \(0.805660\pi\)
\(198\) 5.00000 0.355335
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) 6.00000 0.422159
\(203\) 30.0000 2.10559
\(204\) −5.00000 −0.350070
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) −3.00000 −0.208514
\(208\) 1.00000 0.0693375
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 3.00000 0.206041
\(213\) −6.00000 −0.411113
\(214\) 15.0000 1.02538
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −30.0000 −2.03653
\(218\) 5.00000 0.338643
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) 5.00000 0.336336
\(222\) 1.00000 0.0671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 3.00000 0.198680
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 25.0000 1.64488
\(232\) −6.00000 −0.393919
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) −25.0000 −1.62051
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −14.0000 −0.899954
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) −3.00000 −0.190885
\(248\) 6.00000 0.381000
\(249\) 5.00000 0.316862
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 5.00000 0.314970
\(253\) 15.0000 0.943042
\(254\) −3.00000 −0.188237
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) −4.00000 −0.249029
\(259\) 5.00000 0.310685
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 20.0000 1.23560
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −5.00000 −0.307729
\(265\) 0 0
\(266\) 15.0000 0.919709
\(267\) −13.0000 −0.795587
\(268\) 14.0000 0.855186
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 5.00000 0.303170
\(273\) −5.00000 −0.302614
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) −25.0000 −1.50210 −0.751052 0.660243i \(-0.770453\pi\)
−0.751052 + 0.660243i \(0.770453\pi\)
\(278\) −2.00000 −0.119952
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 1.00000 0.0596550 0.0298275 0.999555i \(-0.490504\pi\)
0.0298275 + 0.999555i \(0.490504\pi\)
\(282\) 0 0
\(283\) −5.00000 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 9.00000 0.526685
\(293\) 11.0000 0.642627 0.321313 0.946973i \(-0.395876\pi\)
0.321313 + 0.946973i \(0.395876\pi\)
\(294\) 18.0000 1.04978
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 5.00000 0.290129
\(298\) −2.00000 −0.115857
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −15.0000 −0.863153
\(303\) 6.00000 0.344691
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −5.00000 −0.285831
\(307\) 30.0000 1.71219 0.856095 0.516818i \(-0.172884\pi\)
0.856095 + 0.516818i \(0.172884\pi\)
\(308\) −25.0000 −1.42451
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 1.00000 0.0566139
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −20.0000 −1.12867
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 3.00000 0.168232
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 15.0000 0.835917
\(323\) −15.0000 −0.834622
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −23.0000 −1.27385
\(327\) 5.00000 0.276501
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −5.00000 −0.274411
\(333\) 1.00000 0.0547997
\(334\) 5.00000 0.273588
\(335\) 0 0
\(336\) −5.00000 −0.272772
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 12.0000 0.652714
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 3.00000 0.162221
\(343\) 55.0000 2.96972
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 11.0000 0.591364
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) −6.00000 −0.321634
\(349\) 24.0000 1.28469 0.642345 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 5.00000 0.266501
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) 13.0000 0.688999
\(357\) −25.0000 −1.32314
\(358\) −8.00000 −0.422813
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 10.0000 0.525588
\(363\) −14.0000 −0.734809
\(364\) 5.00000 0.262071
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 5.00000 0.260998 0.130499 0.991448i \(-0.458342\pi\)
0.130499 + 0.991448i \(0.458342\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 0 0
\(371\) 15.0000 0.778761
\(372\) 6.00000 0.311086
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 25.0000 1.29272
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 5.00000 0.257172
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) −3.00000 −0.153695
\(382\) 7.00000 0.358151
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) 4.00000 0.203069
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) −15.0000 −0.758583
\(392\) −18.0000 −0.909137
\(393\) 20.0000 1.00887
\(394\) 23.0000 1.15872
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 36.0000 1.80679 0.903394 0.428811i \(-0.141067\pi\)
0.903394 + 0.428811i \(0.141067\pi\)
\(398\) 24.0000 1.20301
\(399\) 15.0000 0.750939
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 14.0000 0.698257
\(403\) −6.00000 −0.298881
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −30.0000 −1.48888
\(407\) −5.00000 −0.247841
\(408\) 5.00000 0.247537
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 16.0000 0.788263
\(413\) −50.0000 −2.46034
\(414\) 3.00000 0.147442
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) −2.00000 −0.0979404
\(418\) −15.0000 −0.733674
\(419\) −19.0000 −0.928211 −0.464105 0.885780i \(-0.653624\pi\)
−0.464105 + 0.885780i \(0.653624\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 2.00000 0.0973585
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) 50.0000 2.41967
\(428\) −15.0000 −0.725052
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −21.0000 −1.00920 −0.504598 0.863355i \(-0.668359\pi\)
−0.504598 + 0.863355i \(0.668359\pi\)
\(434\) 30.0000 1.44005
\(435\) 0 0
\(436\) −5.00000 −0.239457
\(437\) 9.00000 0.430528
\(438\) 9.00000 0.430037
\(439\) 36.0000 1.71819 0.859093 0.511819i \(-0.171028\pi\)
0.859093 + 0.511819i \(0.171028\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) −5.00000 −0.237826
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −2.00000 −0.0945968
\(448\) 5.00000 0.236228
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.0000 0.470360
\(453\) −15.0000 −0.704761
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) −6.00000 −0.280362
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) −25.0000 −1.16311
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 1.00000 0.0462250
\(469\) 70.0000 3.23230
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 10.0000 0.460287
\(473\) 20.0000 0.919601
\(474\) 0 0
\(475\) 0 0
\(476\) 25.0000 1.14587
\(477\) 3.00000 0.137361
\(478\) −12.0000 −0.548867
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) 1.00000 0.0455961
\(482\) 18.0000 0.819878
\(483\) 15.0000 0.682524
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) −10.0000 −0.452679
\(489\) −23.0000 −1.04010
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 30.0000 1.35113
\(494\) 3.00000 0.134976
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 30.0000 1.34568
\(498\) −5.00000 −0.224055
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 5.00000 0.223384
\(502\) −30.0000 −1.33897
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) −5.00000 −0.222718
\(505\) 0 0
\(506\) −15.0000 −0.666831
\(507\) 12.0000 0.532939
\(508\) 3.00000 0.133103
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 45.0000 1.99068
\(512\) −1.00000 −0.0441942
\(513\) 3.00000 0.132453
\(514\) 15.0000 0.661622
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −5.00000 −0.219687
\(519\) 11.0000 0.482846
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) −6.00000 −0.262613
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −30.0000 −1.30682
\(528\) 5.00000 0.217597
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) −15.0000 −0.650332
\(533\) 0 0
\(534\) 13.0000 0.562565
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) −8.00000 −0.345225
\(538\) −9.00000 −0.388018
\(539\) −90.0000 −3.87657
\(540\) 0 0
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) −20.0000 −0.859074
\(543\) 10.0000 0.429141
\(544\) −5.00000 −0.214373
\(545\) 0 0
\(546\) 5.00000 0.213980
\(547\) 43.0000 1.83855 0.919274 0.393619i \(-0.128777\pi\)
0.919274 + 0.393619i \(0.128777\pi\)
\(548\) 2.00000 0.0854358
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) −3.00000 −0.127688
\(553\) 0 0
\(554\) 25.0000 1.06215
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 6.00000 0.254000
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 25.0000 1.05550
\(562\) −1.00000 −0.0421825
\(563\) −22.0000 −0.927189 −0.463595 0.886047i \(-0.653441\pi\)
−0.463595 + 0.886047i \(0.653441\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 5.00000 0.210166
\(567\) 5.00000 0.209980
\(568\) −6.00000 −0.251754
\(569\) −13.0000 −0.544988 −0.272494 0.962157i \(-0.587849\pi\)
−0.272494 + 0.962157i \(0.587849\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) −5.00000 −0.209061
\(573\) 7.00000 0.292429
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) −8.00000 −0.332756
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −25.0000 −1.03717
\(582\) 4.00000 0.165805
\(583\) −15.0000 −0.621237
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) −11.0000 −0.454406
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) −18.0000 −0.742307
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) 23.0000 0.946094
\(592\) 1.00000 0.0410997
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 24.0000 0.982255
\(598\) 3.00000 0.122679
\(599\) 44.0000 1.79779 0.898896 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 20.0000 0.815139
\(603\) 14.0000 0.570124
\(604\) 15.0000 0.610341
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 3.00000 0.121666
\(609\) −30.0000 −1.21566
\(610\) 0 0
\(611\) 0 0
\(612\) 5.00000 0.202113
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −30.0000 −1.21070
\(615\) 0 0
\(616\) 25.0000 1.00728
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 16.0000 0.643614
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) −8.00000 −0.320771
\(623\) 65.0000 2.60417
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 24.0000 0.959233
\(627\) −15.0000 −0.599042
\(628\) 20.0000 0.798087
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 2.00000 0.0794929
\(634\) −10.0000 −0.397151
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 18.0000 0.713186
\(638\) 30.0000 1.18771
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) −15.0000 −0.592003
\(643\) −23.0000 −0.907031 −0.453516 0.891248i \(-0.649830\pi\)
−0.453516 + 0.891248i \(0.649830\pi\)
\(644\) −15.0000 −0.591083
\(645\) 0 0
\(646\) 15.0000 0.590167
\(647\) −3.00000 −0.117942 −0.0589711 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 50.0000 1.96267
\(650\) 0 0
\(651\) 30.0000 1.17579
\(652\) 23.0000 0.900750
\(653\) 48.0000 1.87839 0.939193 0.343391i \(-0.111576\pi\)
0.939193 + 0.343391i \(0.111576\pi\)
\(654\) −5.00000 −0.195515
\(655\) 0 0
\(656\) 0 0
\(657\) 9.00000 0.351123
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −47.0000 −1.82809 −0.914044 0.405615i \(-0.867057\pi\)
−0.914044 + 0.405615i \(0.867057\pi\)
\(662\) −4.00000 −0.155464
\(663\) −5.00000 −0.194184
\(664\) 5.00000 0.194038
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) −18.0000 −0.696963
\(668\) −5.00000 −0.193456
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −50.0000 −1.93023
\(672\) 5.00000 0.192879
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) −9.00000 −0.346667
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) 10.0000 0.384048
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) −30.0000 −1.14876
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) −3.00000 −0.114708
\(685\) 0 0
\(686\) −55.0000 −2.09991
\(687\) −6.00000 −0.228914
\(688\) −4.00000 −0.152499
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) −22.0000 −0.836919 −0.418460 0.908235i \(-0.637430\pi\)
−0.418460 + 0.908235i \(0.637430\pi\)
\(692\) −11.0000 −0.418157
\(693\) −25.0000 −0.949671
\(694\) −36.0000 −1.36654
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −24.0000 −0.908413
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 1.00000 0.0377426
\(703\) −3.00000 −0.113147
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) −30.0000 −1.12827
\(708\) 10.0000 0.375823
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −13.0000 −0.487196
\(713\) 18.0000 0.674105
\(714\) 25.0000 0.935601
\(715\) 0 0
\(716\) 8.00000 0.298974
\(717\) −12.0000 −0.448148
\(718\) −36.0000 −1.34351
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) 80.0000 2.97936
\(722\) 10.0000 0.372161
\(723\) 18.0000 0.669427
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 30.0000 1.11264 0.556319 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) −5.00000 −0.185312
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) −10.0000 −0.369611
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) −5.00000 −0.184553
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) −70.0000 −2.57848
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 0 0
\(741\) 3.00000 0.110208
\(742\) −15.0000 −0.550667
\(743\) −10.0000 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) −5.00000 −0.182940
\(748\) −25.0000 −0.914091
\(749\) −75.0000 −2.74044
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 0 0
\(753\) −30.0000 −1.09326
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) −5.00000 −0.181848
\(757\) −37.0000 −1.34479 −0.672394 0.740193i \(-0.734734\pi\)
−0.672394 + 0.740193i \(0.734734\pi\)
\(758\) 26.0000 0.944363
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 3.00000 0.108679
\(763\) −25.0000 −0.905061
\(764\) −7.00000 −0.253251
\(765\) 0 0
\(766\) 9.00000 0.325183
\(767\) −10.0000 −0.361079
\(768\) −1.00000 −0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 15.0000 0.540212
\(772\) 2.00000 0.0719816
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −4.00000 −0.143592
\(777\) −5.00000 −0.179374
\(778\) −34.0000 −1.21896
\(779\) 0 0
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 15.0000 0.536399
\(783\) −6.00000 −0.214423
\(784\) 18.0000 0.642857
\(785\) 0 0
\(786\) −20.0000 −0.713376
\(787\) −42.0000 −1.49714 −0.748569 0.663057i \(-0.769259\pi\)
−0.748569 + 0.663057i \(0.769259\pi\)
\(788\) −23.0000 −0.819341
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 50.0000 1.77780
\(792\) 5.00000 0.177667
\(793\) 10.0000 0.355110
\(794\) −36.0000 −1.27759
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) −15.0000 −0.530994
\(799\) 0 0
\(800\) 0 0
\(801\) 13.0000 0.459332
\(802\) 27.0000 0.953403
\(803\) −45.0000 −1.58802
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) −9.00000 −0.316815
\(808\) 6.00000 0.211079
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 30.0000 1.05279
\(813\) −20.0000 −0.701431
\(814\) 5.00000 0.175250
\(815\) 0 0
\(816\) −5.00000 −0.175035
\(817\) 12.0000 0.419827
\(818\) −24.0000 −0.839140
\(819\) 5.00000 0.174714
\(820\) 0 0
\(821\) 39.0000 1.36111 0.680555 0.732697i \(-0.261739\pi\)
0.680555 + 0.732697i \(0.261739\pi\)
\(822\) 2.00000 0.0697580
\(823\) 35.0000 1.22002 0.610012 0.792392i \(-0.291165\pi\)
0.610012 + 0.792392i \(0.291165\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 50.0000 1.73972
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −3.00000 −0.104257
\(829\) 7.00000 0.243120 0.121560 0.992584i \(-0.461210\pi\)
0.121560 + 0.992584i \(0.461210\pi\)
\(830\) 0 0
\(831\) 25.0000 0.867240
\(832\) 1.00000 0.0346688
\(833\) 90.0000 3.11832
\(834\) 2.00000 0.0692543
\(835\) 0 0
\(836\) 15.0000 0.518786
\(837\) 6.00000 0.207390
\(838\) 19.0000 0.656344
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) −1.00000 −0.0344418
\(844\) −2.00000 −0.0688428
\(845\) 0 0
\(846\) 0 0
\(847\) 70.0000 2.40523
\(848\) 3.00000 0.103020
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) −6.00000 −0.205557
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) −50.0000 −1.71096
\(855\) 0 0
\(856\) 15.0000 0.512689
\(857\) −9.00000 −0.307434 −0.153717 0.988115i \(-0.549124\pi\)
−0.153717 + 0.988115i \(0.549124\pi\)
\(858\) −5.00000 −0.170697
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 33.0000 1.12398
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 21.0000 0.713609
\(867\) −8.00000 −0.271694
\(868\) −30.0000 −1.01827
\(869\) 0 0
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 5.00000 0.169321
\(873\) 4.00000 0.135379
\(874\) −9.00000 −0.304430
\(875\) 0 0
\(876\) −9.00000 −0.304082
\(877\) 52.0000 1.75592 0.877958 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) −36.0000 −1.21494
\(879\) −11.0000 −0.371021
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) −18.0000 −0.606092
\(883\) −21.0000 −0.706706 −0.353353 0.935490i \(-0.614959\pi\)
−0.353353 + 0.935490i \(0.614959\pi\)
\(884\) 5.00000 0.168168
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) 38.0000 1.27592 0.637958 0.770072i \(-0.279780\pi\)
0.637958 + 0.770072i \(0.279780\pi\)
\(888\) 1.00000 0.0335578
\(889\) 15.0000 0.503084
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 2.00000 0.0668900
\(895\) 0 0
\(896\) −5.00000 −0.167038
\(897\) 3.00000 0.100167
\(898\) −30.0000 −1.00111
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) 0 0
\(903\) 20.0000 0.665558
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) 15.0000 0.498342
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −8.00000 −0.265489
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 3.00000 0.0993399
\(913\) 25.0000 0.827379
\(914\) 4.00000 0.132308
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −100.000 −3.30229
\(918\) 5.00000 0.165025
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −30.0000 −0.988534
\(922\) −26.0000 −0.856264
\(923\) 6.00000 0.197492
\(924\) 25.0000 0.822440
\(925\) 0 0
\(926\) −28.0000 −0.920137
\(927\) 16.0000 0.525509
\(928\) −6.00000 −0.196960
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 26.0000 0.851658
\(933\) −8.00000 −0.261908
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) −70.0000 −2.28558
\(939\) 24.0000 0.783210
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) −10.0000 −0.324272
\(952\) −25.0000 −0.810255
\(953\) −14.0000 −0.453504 −0.226752 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 30.0000 0.969762
\(958\) 21.0000 0.678479
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −1.00000 −0.0322413
\(963\) −15.0000 −0.483368
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) −15.0000 −0.482617
\(967\) 18.0000 0.578841 0.289420 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) −14.0000 −0.449977
\(969\) 15.0000 0.481869
\(970\) 0 0
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 10.0000 0.320585
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −39.0000 −1.24772 −0.623860 0.781536i \(-0.714437\pi\)
−0.623860 + 0.781536i \(0.714437\pi\)
\(978\) 23.0000 0.735459
\(979\) −65.0000 −2.07741
\(980\) 0 0
\(981\) −5.00000 −0.159638
\(982\) 9.00000 0.287202
\(983\) 58.0000 1.84991 0.924956 0.380073i \(-0.124101\pi\)
0.924956 + 0.380073i \(0.124101\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −30.0000 −0.955395
\(987\) 0 0
\(988\) −3.00000 −0.0954427
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 6.00000 0.190500
\(993\) −4.00000 −0.126936
\(994\) −30.0000 −0.951542
\(995\) 0 0
\(996\) 5.00000 0.158431
\(997\) 39.0000 1.23514 0.617571 0.786515i \(-0.288117\pi\)
0.617571 + 0.786515i \(0.288117\pi\)
\(998\) 5.00000 0.158272
\(999\) −1.00000 −0.0316386
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 5550.2.a.k.1.1 1
5.4 even 2 1110.2.a.m.1.1 1
15.14 odd 2 3330.2.a.f.1.1 1
20.19 odd 2 8880.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.m.1.1 1 5.4 even 2
3330.2.a.f.1.1 1 15.14 odd 2
5550.2.a.k.1.1 1 1.1 even 1 trivial
8880.2.a.i.1.1 1 20.19 odd 2