Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -3.00000 q^{21} +3.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} -3.00000 q^{28} +3.00000 q^{29} -10.0000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +3.00000 q^{38} +3.00000 q^{39} -9.00000 q^{41} +3.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{51} +3.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000 q^{57} -3.00000 q^{58} +8.00000 q^{59} +12.0000 q^{61} +10.0000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -8.00000 q^{67} +4.00000 q^{68} +1.00000 q^{69} -14.0000 q^{71} -1.00000 q^{72} -13.0000 q^{73} -8.00000 q^{74} -3.00000 q^{76} +9.00000 q^{77} -3.00000 q^{78} -17.0000 q^{79} +1.00000 q^{81} +9.00000 q^{82} -1.00000 q^{83} -3.00000 q^{84} +1.00000 q^{86} +3.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} -9.00000 q^{91} +1.00000 q^{92} -10.0000 q^{93} +2.00000 q^{94} -1.00000 q^{96} -6.00000 q^{97} -2.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −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
\(21\) −3.00000 −0.654654
\(22\) 3.00000 0.639602
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 1.00000 0.192450
\(28\) −3.00000 −0.566947
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 3.00000 0.486664
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 3.00000 0.462910
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 3.00000 0.416025
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −3.00000 −0.397360
\(58\) −3.00000 −0.393919
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 10.0000 1.27000
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) −1.00000 −0.117851
\(73\) −13.0000 −1.52153 −0.760767 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 9.00000 1.02565
\(78\) −3.00000 −0.339683
\(79\) −17.0000 −1.91265 −0.956325 0.292306i \(-0.905577\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.00000 0.993884
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 3.00000 0.321634
\(88\) 3.00000 0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) 1.00000 0.104257
\(93\) −10.0000 −1.03695
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −2.00000 −0.202031
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −4.00000 −0.396059
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) −3.00000 −0.283473
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 3.00000 0.277350
\(118\) −8.00000 −0.736460
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −12.0000 −1.08643
\(123\) −9.00000 −0.811503
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −3.00000 −0.261116
\(133\) 9.00000 0.780399
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 14.0000 1.17485
\(143\) −9.00000 −0.752618
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 13.0000 1.07589
\(147\) 2.00000 0.164957
\(148\) 8.00000 0.657596
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) 3.00000 0.243332
\(153\) 4.00000 0.323381
\(154\) −9.00000 −0.725241
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 17.0000 1.35245
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) −1.00000 −0.0785674
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 1.00000 0.0776151
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 3.00000 0.231455
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) −1.00000 −0.0762493
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 8.00000 0.601317
\(178\) 6.00000 0.449719
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 9.00000 0.667124
\(183\) 12.0000 0.887066
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) −12.0000 −0.877527
\(188\) −2.00000 −0.145865
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 17.0000 1.21120 0.605600 0.795769i \(-0.292933\pi\)
0.605600 + 0.795769i \(0.292933\pi\)
\(198\) 3.00000 0.213201
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 6.00000 0.422159
\(203\) −9.00000 −0.631676
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) 1.00000 0.0695048
\(208\) 3.00000 0.208013
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 6.00000 0.412082
\(213\) −14.0000 −0.959264
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 30.0000 2.03653
\(218\) 0 0
\(219\) −13.0000 −0.878459
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −8.00000 −0.536925
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −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
\(231\) 9.00000 0.592157
\(232\) −3.00000 −0.196960
\(233\) 1.00000 0.0655122 0.0327561 0.999463i \(-0.489572\pi\)
0.0327561 + 0.999463i \(0.489572\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) −17.0000 −1.10427
\(238\) 12.0000 0.777844
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) −9.00000 −0.572656
\(248\) 10.0000 0.635001
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) −3.00000 −0.188982
\(253\) −3.00000 −0.188608
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 1.00000 0.0622573
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) −10.0000 −0.617802
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −9.00000 −0.551825
\(267\) −6.00000 −0.367194
\(268\) −8.00000 −0.488678
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 4.00000 0.242536
\(273\) −9.00000 −0.544705
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) −8.00000 −0.479808
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 2.00000 0.119098
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) 9.00000 0.532181
\(287\) 27.0000 1.59376
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) −13.0000 −0.760767
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) −3.00000 −0.174078
\(298\) 6.00000 0.347571
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 22.0000 1.26596
\(303\) −6.00000 −0.344691
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 9.00000 0.512823
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −3.00000 −0.169842
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −17.0000 −0.956325
\(317\) −31.0000 −1.74113 −0.870567 0.492050i \(-0.836248\pi\)
−0.870567 + 0.492050i \(0.836248\pi\)
\(318\) −6.00000 −0.336463
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 3.00000 0.167183
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) −1.00000 −0.0548821
\(333\) 8.00000 0.438397
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) −3.00000 −0.163663
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 4.00000 0.217571
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 3.00000 0.162221
\(343\) 15.0000 0.809924
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 15.0000 0.806405
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 3.00000 0.160817
\(349\) −5.00000 −0.267644 −0.133822 0.991005i \(-0.542725\pi\)
−0.133822 + 0.991005i \(0.542725\pi\)
\(350\) 0 0
\(351\) 3.00000 0.160128
\(352\) 3.00000 0.159901
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −12.0000 −0.635107
\(358\) 10.0000 0.528516
\(359\) 31.0000 1.63612 0.818059 0.575135i \(-0.195050\pi\)
0.818059 + 0.575135i \(0.195050\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 16.0000 0.840941
\(363\) −2.00000 −0.104973
\(364\) −9.00000 −0.471728
\(365\) 0 0
\(366\) −12.0000 −0.627250
\(367\) −27.0000 −1.40939 −0.704694 0.709511i \(-0.748916\pi\)
−0.704694 + 0.709511i \(0.748916\pi\)
\(368\) 1.00000 0.0521286
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) −10.0000 −0.518476
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 9.00000 0.463524
\(378\) 3.00000 0.154303
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 15.0000 0.767467
\(383\) 29.0000 1.48183 0.740915 0.671598i \(-0.234392\pi\)
0.740915 + 0.671598i \(0.234392\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −1.00000 −0.0508329
\(388\) −6.00000 −0.304604
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) −2.00000 −0.101015
\(393\) 10.0000 0.504433
\(394\) −17.0000 −0.856448
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −11.0000 −0.551380
\(399\) 9.00000 0.450564
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 8.00000 0.399004
\(403\) −30.0000 −1.49441
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) −24.0000 −1.18964
\(408\) −4.00000 −0.198030
\(409\) −23.0000 −1.13728 −0.568638 0.822588i \(-0.692530\pi\)
−0.568638 + 0.822588i \(0.692530\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) −13.0000 −0.640464
\(413\) −24.0000 −1.18096
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) 8.00000 0.391762
\(418\) −9.00000 −0.440204
\(419\) 19.0000 0.928211 0.464105 0.885780i \(-0.346376\pi\)
0.464105 + 0.885780i \(0.346376\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −22.0000 −1.07094
\(423\) −2.00000 −0.0972433
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 14.0000 0.678302
\(427\) −36.0000 −1.74216
\(428\) 12.0000 0.580042
\(429\) −9.00000 −0.434524
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) −30.0000 −1.44005
\(435\) 0 0
\(436\) 0 0
\(437\) −3.00000 −0.143509
\(438\) 13.0000 0.621164
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) −12.0000 −0.570782
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) −6.00000 −0.283790
\(448\) −3.00000 −0.141737
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) −6.00000 −0.282216
\(453\) −22.0000 −1.03365
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −6.00000 −0.280362
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) −9.00000 −0.418718
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −1.00000 −0.0463241
\(467\) −11.0000 −0.509019 −0.254510 0.967070i \(-0.581914\pi\)
−0.254510 + 0.967070i \(0.581914\pi\)
\(468\) 3.00000 0.138675
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) −8.00000 −0.368230
\(473\) 3.00000 0.137940
\(474\) 17.0000 0.780836
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) 6.00000 0.274721
\(478\) 10.0000 0.457389
\(479\) 33.0000 1.50781 0.753904 0.656984i \(-0.228168\pi\)
0.753904 + 0.656984i \(0.228168\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 10.0000 0.455488
\(483\) −3.00000 −0.136505
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −12.0000 −0.543214
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −9.00000 −0.405751
\(493\) 12.0000 0.540453
\(494\) 9.00000 0.404929
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 42.0000 1.88396
\(498\) 1.00000 0.0448111
\(499\) 34.0000 1.52205 0.761025 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 16.0000 0.714115
\(503\) 35.0000 1.56057 0.780286 0.625422i \(-0.215073\pi\)
0.780286 + 0.625422i \(0.215073\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 3.00000 0.133366
\(507\) −4.00000 −0.177646
\(508\) −16.0000 −0.709885
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 39.0000 1.72526
\(512\) −1.00000 −0.0441942
\(513\) −3.00000 −0.132453
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) 6.00000 0.263880
\(518\) 24.0000 1.05450
\(519\) −15.0000 −0.658427
\(520\) 0 0
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) −3.00000 −0.131306
\(523\) −11.0000 −0.480996 −0.240498 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) −40.0000 −1.74243
\(528\) −3.00000 −0.130558
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 9.00000 0.390199
\(533\) −27.0000 −1.16950
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) −10.0000 −0.431532
\(538\) 5.00000 0.215565
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) −22.0000 −0.944981
\(543\) −16.0000 −0.686626
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 9.00000 0.385164
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 18.0000 0.768922
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) −9.00000 −0.383413
\(552\) −1.00000 −0.0425628
\(553\) 51.0000 2.16874
\(554\) −13.0000 −0.552317
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 32.0000 1.35588 0.677942 0.735116i \(-0.262872\pi\)
0.677942 + 0.735116i \(0.262872\pi\)
\(558\) 10.0000 0.423334
\(559\) −3.00000 −0.126886
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 10.0000 0.421825
\(563\) −5.00000 −0.210725 −0.105362 0.994434i \(-0.533600\pi\)
−0.105362 + 0.994434i \(0.533600\pi\)
\(564\) −2.00000 −0.0842152
\(565\) 0 0
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 14.0000 0.587427
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −9.00000 −0.376309
\(573\) −15.0000 −0.626634
\(574\) −27.0000 −1.12696
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 7.00000 0.291414 0.145707 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 1.00000 0.0415945
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 6.00000 0.248708
\(583\) −18.0000 −0.745484
\(584\) 13.0000 0.537944
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 2.00000 0.0824786
\(589\) 30.0000 1.23613
\(590\) 0 0
\(591\) 17.0000 0.699287
\(592\) 8.00000 0.328798
\(593\) 21.0000 0.862367 0.431183 0.902264i \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 11.0000 0.450200
\(598\) −3.00000 −0.122679
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) −3.00000 −0.122271
\(603\) −8.00000 −0.325785
\(604\) −22.0000 −0.895167
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 3.00000 0.121666
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 4.00000 0.161690
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) 16.0000 0.644136 0.322068 0.946717i \(-0.395622\pi\)
0.322068 + 0.946717i \(0.395622\pi\)
\(618\) 13.0000 0.522937
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −24.0000 −0.962312
\(623\) 18.0000 0.721155
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) 24.0000 0.959233
\(627\) 9.00000 0.359425
\(628\) −6.00000 −0.239426
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 17.0000 0.676224
\(633\) 22.0000 0.874421
\(634\) 31.0000 1.23117
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 6.00000 0.237729
\(638\) 9.00000 0.356313
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) −12.0000 −0.473602
\(643\) 31.0000 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 30.0000 1.17579
\(652\) −6.00000 −0.234978
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) −13.0000 −0.507178
\(658\) −6.00000 −0.233904
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 26.0000 1.01052
\(663\) 12.0000 0.466041
\(664\) 1.00000 0.0388075
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 3.00000 0.116160
\(668\) −2.00000 −0.0773823
\(669\) 6.00000 0.231973
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 3.00000 0.115728
\(673\) −3.00000 −0.115642 −0.0578208 0.998327i \(-0.518415\pi\)
−0.0578208 + 0.998327i \(0.518415\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 6.00000 0.230429
\(679\) 18.0000 0.690777
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) −30.0000 −1.14876
\(683\) −38.0000 −1.45403 −0.727015 0.686622i \(-0.759093\pi\)
−0.727015 + 0.686622i \(0.759093\pi\)
\(684\) −3.00000 −0.114708
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 6.00000 0.228914
\(688\) −1.00000 −0.0381246
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) −15.0000 −0.570214
\(693\) 9.00000 0.341882
\(694\) 0 0
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −36.0000 −1.36360
\(698\) 5.00000 0.189253
\(699\) 1.00000 0.0378235
\(700\) 0 0
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) −3.00000 −0.113228
\(703\) −24.0000 −0.905177
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 9.00000 0.338719
\(707\) 18.0000 0.676960
\(708\) 8.00000 0.300658
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −17.0000 −0.637550
\(712\) 6.00000 0.224860
\(713\) −10.0000 −0.374503
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) −10.0000 −0.373457
\(718\) −31.0000 −1.15691
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 39.0000 1.45244
\(722\) 10.0000 0.372161
\(723\) −10.0000 −0.371904
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 9.00000 0.333562
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 12.0000 0.443533
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 27.0000 0.996588
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 24.0000 0.884051
\(738\) 9.00000 0.331295
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) 0 0
\(741\) −9.00000 −0.330623
\(742\) 18.0000 0.660801
\(743\) 29.0000 1.06391 0.531953 0.846774i \(-0.321458\pi\)
0.531953 + 0.846774i \(0.321458\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) −16.0000 −0.585802
\(747\) −1.00000 −0.0365881
\(748\) −12.0000 −0.438763
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) 45.0000 1.64207 0.821037 0.570875i \(-0.193396\pi\)
0.821037 + 0.570875i \(0.193396\pi\)
\(752\) −2.00000 −0.0729325
\(753\) −16.0000 −0.583072
\(754\) −9.00000 −0.327761
\(755\) 0 0
\(756\) −3.00000 −0.109109
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) −12.0000 −0.435860
\(759\) −3.00000 −0.108893
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) −15.0000 −0.542681
\(765\) 0 0
\(766\) −29.0000 −1.04781
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 36.0000 1.29819 0.649097 0.760706i \(-0.275147\pi\)
0.649097 + 0.760706i \(0.275147\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 6.00000 0.215945
\(773\) 40.0000 1.43870 0.719350 0.694648i \(-0.244440\pi\)
0.719350 + 0.694648i \(0.244440\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) −24.0000 −0.860995
\(778\) −38.0000 −1.36237
\(779\) 27.0000 0.967375
\(780\) 0 0
\(781\) 42.0000 1.50288
\(782\) −4.00000 −0.143040
\(783\) 3.00000 0.107211
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) 5.00000 0.178231 0.0891154 0.996021i \(-0.471596\pi\)
0.0891154 + 0.996021i \(0.471596\pi\)
\(788\) 17.0000 0.605600
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 3.00000 0.106600
\(793\) 36.0000 1.27840
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 11.0000 0.389885
\(797\) −24.0000 −0.850124 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(798\) −9.00000 −0.318597
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 28.0000 0.988714
\(803\) 39.0000 1.37628
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 30.0000 1.05670
\(807\) −5.00000 −0.176008
\(808\) 6.00000 0.211079
\(809\) 21.0000 0.738321 0.369160 0.929366i \(-0.379645\pi\)
0.369160 + 0.929366i \(0.379645\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) −9.00000 −0.315838
\(813\) 22.0000 0.771574
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 3.00000 0.104957
\(818\) 23.0000 0.804176
\(819\) −9.00000 −0.314485
\(820\) 0 0
\(821\) −55.0000 −1.91951 −0.959757 0.280833i \(-0.909389\pi\)
−0.959757 + 0.280833i \(0.909389\pi\)
\(822\) −18.0000 −0.627822
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 1.00000 0.0347524
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) 13.0000 0.450965
\(832\) 3.00000 0.104006
\(833\) 8.00000 0.277184
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 9.00000 0.311272
\(837\) −10.0000 −0.345651
\(838\) −19.0000 −0.656344
\(839\) 33.0000 1.13929 0.569643 0.821892i \(-0.307081\pi\)
0.569643 + 0.821892i \(0.307081\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 10.0000 0.344623
\(843\) −10.0000 −0.344418
\(844\) 22.0000 0.757271
\(845\) 0 0
\(846\) 2.00000 0.0687614
\(847\) 6.00000 0.206162
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) −14.0000 −0.479632
\(853\) 23.0000 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(854\) 36.0000 1.23189
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 9.00000 0.307255
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 27.0000 0.920158
\(862\) −16.0000 −0.544962
\(863\) −54.0000 −1.83818 −0.919091 0.394046i \(-0.871075\pi\)
−0.919091 + 0.394046i \(0.871075\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) −1.00000 −0.0339618
\(868\) 30.0000 1.01827
\(869\) 51.0000 1.73006
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 3.00000 0.101477
\(875\) 0 0
\(876\) −13.0000 −0.439229
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 4.00000 0.134993
\(879\) 0 0
\(880\) 0 0
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 26.0000 0.873487
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −8.00000 −0.268462
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 6.00000 0.200895
\(893\) 6.00000 0.200782
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 3.00000 0.100167
\(898\) 26.0000 0.867631
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) −27.0000 −0.899002
\(903\) 3.00000 0.0998337
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 1.00000 0.0332045 0.0166022 0.999862i \(-0.494715\pi\)
0.0166022 + 0.999862i \(0.494715\pi\)
\(908\) 12.0000 0.398234
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 55.0000 1.82223 0.911116 0.412151i \(-0.135222\pi\)
0.911116 + 0.412151i \(0.135222\pi\)
\(912\) −3.00000 −0.0993399
\(913\) 3.00000 0.0992855
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −30.0000 −0.990687
\(918\) −4.00000 −0.132020
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 9.00000 0.296399
\(923\) −42.0000 −1.38245
\(924\) 9.00000 0.296078
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) −13.0000 −0.426976
\(928\) −3.00000 −0.0984798
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 1.00000 0.0327561
\(933\) 24.0000 0.785725
\(934\) 11.0000 0.359931
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −24.0000 −0.783628
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 6.00000 0.195491
\(943\) −9.00000 −0.293080
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) −2.00000 −0.0649913 −0.0324956 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) −17.0000 −0.552134
\(949\) −39.0000 −1.26599
\(950\) 0 0
\(951\) −31.0000 −1.00524
\(952\) 12.0000 0.388922
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −10.0000 −0.323423
\(957\) −9.00000 −0.290929
\(958\) −33.0000 −1.06618
\(959\) −54.0000 −1.74375
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −24.0000 −0.773791
\(963\) 12.0000 0.386695
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) 2.00000 0.0642824
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) 43.0000 1.37994 0.689968 0.723840i \(-0.257625\pi\)
0.689968 + 0.723840i \(0.257625\pi\)
\(972\) 1.00000 0.0320750
\(973\) −24.0000 −0.769405
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 6.00000 0.191859
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 6.00000 0.190982
\(988\) −9.00000 −0.286328
\(989\) −1.00000 −0.0317982
\(990\) 0 0
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) 10.0000 0.317500
\(993\) −26.0000 −0.825085
\(994\) −42.0000 −1.33216
\(995\) 0 0
\(996\) −1.00000 −0.0316862
\(997\) 49.0000 1.55185 0.775923 0.630828i \(-0.217285\pi\)
0.775923 + 0.630828i \(0.217285\pi\)
\(998\) −34.0000 −1.07625
\(999\) 8.00000 0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3450.2.a.h.1.1 1
5.2 odd 4 3450.2.d.b.2899.1 2
5.3 odd 4 3450.2.d.b.2899.2 2
5.4 even 2 3450.2.a.s.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.h.1.1 1 1.1 even 1 trivial
3450.2.a.s.1.1 yes 1 5.4 even 2
3450.2.d.b.2899.1 2 5.2 odd 4
3450.2.d.b.2899.2 2 5.3 odd 4