Properties

Label 4018.2.a.i.1.1
Level 4018
Weight 2
Character 4018.1
Self dual yes
Analytic conductor 32.084
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0838915322\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 574)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4018.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +4.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} -8.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -4.00000 q^{20} -4.00000 q^{22} -8.00000 q^{23} -2.00000 q^{24} +11.0000 q^{25} +4.00000 q^{26} -4.00000 q^{27} +6.00000 q^{29} +8.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +8.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{38} -8.00000 q^{39} +4.00000 q^{40} +1.00000 q^{41} +4.00000 q^{44} -4.00000 q^{45} +8.00000 q^{46} -4.00000 q^{47} +2.00000 q^{48} -11.0000 q^{50} +4.00000 q^{51} -4.00000 q^{52} +2.00000 q^{53} +4.00000 q^{54} -16.0000 q^{55} +12.0000 q^{57} -6.00000 q^{58} -14.0000 q^{59} -8.00000 q^{60} -4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +16.0000 q^{65} -8.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} -16.0000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} +22.0000 q^{75} +6.00000 q^{76} +8.00000 q^{78} -8.00000 q^{79} -4.00000 q^{80} -11.0000 q^{81} -1.00000 q^{82} -6.00000 q^{83} -8.00000 q^{85} +12.0000 q^{87} -4.00000 q^{88} -10.0000 q^{89} +4.00000 q^{90} -8.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} -24.0000 q^{95} -2.00000 q^{96} -6.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 2.00000 0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −8.00000 −2.06559
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −2.00000 −0.408248
\(25\) 11.0000 2.20000
\(26\) 4.00000 0.784465
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 8.00000 1.46059
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 8.00000 1.39262
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −6.00000 −0.973329
\(39\) −8.00000 −1.28103
\(40\) 4.00000 0.632456
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 4.00000 0.603023
\(45\) −4.00000 −0.596285
\(46\) 8.00000 1.17954
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 2.00000 0.288675
\(49\) 0 0
\(50\) −11.0000 −1.55563
\(51\) 4.00000 0.560112
\(52\) −4.00000 −0.554700
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 4.00000 0.544331
\(55\) −16.0000 −2.15744
\(56\) 0 0
\(57\) 12.0000 1.58944
\(58\) −6.00000 −0.787839
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) −8.00000 −1.03280
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 16.0000 1.98456
\(66\) −8.00000 −0.984732
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) −16.0000 −1.92617
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −2.00000 −0.232495
\(75\) 22.0000 2.54034
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −4.00000 −0.447214
\(81\) −11.0000 −1.22222
\(82\) −1.00000 −0.110432
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) −4.00000 −0.426401
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 4.00000 0.421637
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) −8.00000 −0.829561
\(94\) 4.00000 0.412568
\(95\) −24.0000 −2.46235
\(96\) −2.00000 −0.204124
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 11.0000 1.10000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −4.00000 −0.396059
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −4.00000 −0.384900
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 16.0000 1.52554
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −12.0000 −1.12390
\(115\) 32.0000 2.98402
\(116\) 6.00000 0.557086
\(117\) −4.00000 −0.369800
\(118\) 14.0000 1.28880
\(119\) 0 0
\(120\) 8.00000 0.730297
\(121\) 5.00000 0.454545
\(122\) 4.00000 0.362143
\(123\) 2.00000 0.180334
\(124\) −4.00000 −0.359211
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −16.0000 −1.40329
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 8.00000 0.696311
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 16.0000 1.37706
\(136\) −2.00000 −0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 16.0000 1.36201
\(139\) 22.0000 1.86602 0.933008 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −8.00000 −0.671345
\(143\) −16.0000 −1.33799
\(144\) 1.00000 0.0833333
\(145\) −24.0000 −1.99309
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −22.0000 −1.79629
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −6.00000 −0.486664
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) −8.00000 −0.640513
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 8.00000 0.636446
\(159\) 4.00000 0.317221
\(160\) 4.00000 0.316228
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 1.00000 0.0780869
\(165\) −32.0000 −2.49120
\(166\) 6.00000 0.465690
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 8.00000 0.613572
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −28.0000 −2.10461
\(178\) 10.0000 0.749532
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) −4.00000 −0.298142
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 8.00000 0.589768
\(185\) −8.00000 −0.588172
\(186\) 8.00000 0.586588
\(187\) 8.00000 0.585018
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 2.00000 0.144338
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 6.00000 0.430775
\(195\) 32.0000 2.29157
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −4.00000 −0.284268
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −11.0000 −0.777817
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) −4.00000 −0.279372
\(206\) −12.0000 −0.836080
\(207\) −8.00000 −0.556038
\(208\) −4.00000 −0.277350
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 2.00000 0.137361
\(213\) 16.0000 1.09630
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 28.0000 1.89206
\(220\) −16.0000 −1.07872
\(221\) −8.00000 −0.538138
\(222\) −4.00000 −0.268462
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 10.0000 0.665190
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 12.0000 0.794719
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) −32.0000 −2.11002
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 4.00000 0.261488
\(235\) 16.0000 1.04372
\(236\) −14.0000 −0.911322
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −8.00000 −0.516398
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) −5.00000 −0.321412
\(243\) −10.0000 −0.641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −24.0000 −1.52708
\(248\) 4.00000 0.254000
\(249\) −12.0000 −0.760469
\(250\) 24.0000 1.51789
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) 8.00000 0.501965
\(255\) −16.0000 −1.00196
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 16.0000 0.992278
\(261\) 6.00000 0.371391
\(262\) 6.00000 0.370681
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −8.00000 −0.492366
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) −20.0000 −1.22398
\(268\) −8.00000 −0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −16.0000 −0.973729
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 44.0000 2.65330
\(276\) −16.0000 −0.963087
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −22.0000 −1.31947
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 8.00000 0.476393
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 8.00000 0.474713
\(285\) −48.0000 −2.84327
\(286\) 16.0000 0.946100
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 24.0000 1.40933
\(291\) −12.0000 −0.703452
\(292\) 14.0000 0.819288
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0 0
\(295\) 56.0000 3.26045
\(296\) −2.00000 −0.116248
\(297\) −16.0000 −0.928414
\(298\) 6.00000 0.347571
\(299\) 32.0000 1.85061
\(300\) 22.0000 1.27017
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 16.0000 0.916157
\(306\) −2.00000 −0.114332
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 24.0000 1.36531
\(310\) −16.0000 −0.908739
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 8.00000 0.452911
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) −4.00000 −0.224309
\(319\) 24.0000 1.34374
\(320\) −4.00000 −0.223607
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) −11.0000 −0.611111
\(325\) −44.0000 −2.44068
\(326\) 24.0000 1.32924
\(327\) 12.0000 0.663602
\(328\) −1.00000 −0.0552158
\(329\) 0 0
\(330\) 32.0000 1.76154
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −6.00000 −0.329293
\(333\) 2.00000 0.109599
\(334\) 20.0000 1.09435
\(335\) 32.0000 1.74835
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) −3.00000 −0.163178
\(339\) −20.0000 −1.08625
\(340\) −8.00000 −0.433861
\(341\) −16.0000 −0.866449
\(342\) −6.00000 −0.324443
\(343\) 0 0
\(344\) 0 0
\(345\) 64.0000 3.44564
\(346\) 24.0000 1.29025
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 12.0000 0.643268
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) −4.00000 −0.213201
\(353\) 22.0000 1.17094 0.585471 0.810693i \(-0.300910\pi\)
0.585471 + 0.810693i \(0.300910\pi\)
\(354\) 28.0000 1.48818
\(355\) −32.0000 −1.69838
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 4.00000 0.210819
\(361\) 17.0000 0.894737
\(362\) 20.0000 1.05118
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) −56.0000 −2.93117
\(366\) 8.00000 0.418167
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −8.00000 −0.417029
\(369\) 1.00000 0.0520579
\(370\) 8.00000 0.415900
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) −8.00000 −0.413670
\(375\) −48.0000 −2.47871
\(376\) 4.00000 0.206284
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) −24.0000 −1.23117
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −32.0000 −1.62038
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 10.0000 0.503793
\(395\) 32.0000 1.61009
\(396\) 4.00000 0.201008
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 16.0000 0.798007
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 44.0000 2.18638
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) −4.00000 −0.198030
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 4.00000 0.197546
\(411\) −12.0000 −0.591916
\(412\) 12.0000 0.591198
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 24.0000 1.17811
\(416\) 4.00000 0.196116
\(417\) 44.0000 2.15469
\(418\) −24.0000 −1.17388
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −8.00000 −0.389434
\(423\) −4.00000 −0.194487
\(424\) −2.00000 −0.0971286
\(425\) 22.0000 1.06716
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −32.0000 −1.54497
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −4.00000 −0.192450
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −48.0000 −2.30142
\(436\) 6.00000 0.287348
\(437\) −48.0000 −2.29615
\(438\) −28.0000 −1.33789
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 16.0000 0.762770
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 4.00000 0.189832
\(445\) 40.0000 1.89618
\(446\) 24.0000 1.13643
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −11.0000 −0.518545
\(451\) 4.00000 0.188353
\(452\) −10.0000 −0.470360
\(453\) 32.0000 1.50349
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −20.0000 −0.934539
\(459\) −8.00000 −0.373408
\(460\) 32.0000 1.49201
\(461\) 16.0000 0.745194 0.372597 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 6.00000 0.278543
\(465\) 32.0000 1.48396
\(466\) −10.0000 −0.463241
\(467\) −14.0000 −0.647843 −0.323921 0.946084i \(-0.605001\pi\)
−0.323921 + 0.946084i \(0.605001\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) −16.0000 −0.738025
\(471\) −8.00000 −0.368621
\(472\) 14.0000 0.644402
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 66.0000 3.02829
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 8.00000 0.365148
\(481\) −8.00000 −0.364769
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 24.0000 1.08978
\(486\) 10.0000 0.453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 4.00000 0.181071
\(489\) −48.0000 −2.17064
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) 24.0000 1.07981
\(495\) −16.0000 −0.719147
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −24.0000 −1.07331
\(501\) −40.0000 −1.78707
\(502\) −6.00000 −0.267793
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) 6.00000 0.266469
\(508\) −8.00000 −0.354943
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 16.0000 0.708492
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −24.0000 −1.05963
\(514\) 2.00000 0.0882162
\(515\) −48.0000 −2.11513
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −48.0000 −2.10697
\(520\) −16.0000 −0.701646
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) −6.00000 −0.262613
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −8.00000 −0.348485
\(528\) 8.00000 0.348155
\(529\) 41.0000 1.78261
\(530\) 8.00000 0.347498
\(531\) −14.0000 −0.607548
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 20.0000 0.865485
\(535\) 16.0000 0.691740
\(536\) 8.00000 0.345547
\(537\) −48.0000 −2.07135
\(538\) 0 0
\(539\) 0 0
\(540\) 16.0000 0.688530
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −16.0000 −0.687259
\(543\) −40.0000 −1.71656
\(544\) −2.00000 −0.0857493
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −6.00000 −0.256307
\(549\) −4.00000 −0.170716
\(550\) −44.0000 −1.87617
\(551\) 36.0000 1.53365
\(552\) 16.0000 0.681005
\(553\) 0 0
\(554\) 18.0000 0.764747
\(555\) −16.0000 −0.679162
\(556\) 22.0000 0.933008
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) 0 0
\(561\) 16.0000 0.675521
\(562\) −10.0000 −0.421825
\(563\) 10.0000 0.421450 0.210725 0.977545i \(-0.432418\pi\)
0.210725 + 0.977545i \(0.432418\pi\)
\(564\) −8.00000 −0.336861
\(565\) 40.0000 1.68281
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 48.0000 2.01050
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −16.0000 −0.668994
\(573\) 0 0
\(574\) 0 0
\(575\) −88.0000 −3.66985
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 13.0000 0.540729
\(579\) −36.0000 −1.49611
\(580\) −24.0000 −0.996546
\(581\) 0 0
\(582\) 12.0000 0.497416
\(583\) 8.00000 0.331326
\(584\) −14.0000 −0.579324
\(585\) 16.0000 0.661519
\(586\) −16.0000 −0.660954
\(587\) −38.0000 −1.56843 −0.784214 0.620491i \(-0.786934\pi\)
−0.784214 + 0.620491i \(0.786934\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) −56.0000 −2.30548
\(591\) −20.0000 −0.822690
\(592\) 2.00000 0.0821995
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 16.0000 0.656488
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −40.0000 −1.63709
\(598\) −32.0000 −1.30858
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) −22.0000 −0.898146
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 16.0000 0.651031
\(605\) −20.0000 −0.813116
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) −16.0000 −0.647821
\(611\) 16.0000 0.647291
\(612\) 2.00000 0.0808452
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −26.0000 −1.04927
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −24.0000 −0.965422
\(619\) −30.0000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) 16.0000 0.642575
\(621\) 32.0000 1.28412
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) −8.00000 −0.320256
\(625\) 41.0000 1.64000
\(626\) −34.0000 −1.35891
\(627\) 48.0000 1.91694
\(628\) −4.00000 −0.159617
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000 0.318223
\(633\) 16.0000 0.635943
\(634\) 14.0000 0.556011
\(635\) 32.0000 1.26988
\(636\) 4.00000 0.158610
\(637\) 0 0
\(638\) −24.0000 −0.950169
\(639\) 8.00000 0.316475
\(640\) 4.00000 0.158114
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 8.00000 0.315735
\(643\) −22.0000 −0.867595 −0.433798 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 11.0000 0.432121
\(649\) −56.0000 −2.19819
\(650\) 44.0000 1.72582
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −12.0000 −0.469237
\(655\) 24.0000 0.937758
\(656\) 1.00000 0.0390434
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −32.0000 −1.24560
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 12.0000 0.466393
\(663\) −16.0000 −0.621389
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −48.0000 −1.85857
\(668\) −20.0000 −0.773823
\(669\) −48.0000 −1.85579
\(670\) −32.0000 −1.23627
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −30.0000 −1.15556
\(675\) −44.0000 −1.69356
\(676\) 3.00000 0.115385
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 20.0000 0.768095
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) 12.0000 0.459841
\(682\) 16.0000 0.612672
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 6.00000 0.229416
\(685\) 24.0000 0.916993
\(686\) 0 0
\(687\) 40.0000 1.52610
\(688\) 0 0
\(689\) −8.00000 −0.304776
\(690\) −64.0000 −2.43644
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −88.0000 −3.33803
\(696\) −12.0000 −0.454859
\(697\) 2.00000 0.0757554
\(698\) −16.0000 −0.605609
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −16.0000 −0.603881
\(703\) 12.0000 0.452589
\(704\) 4.00000 0.150756
\(705\) 32.0000 1.20519
\(706\) −22.0000 −0.827981
\(707\) 0 0
\(708\) −28.0000 −1.05230
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 32.0000 1.20094
\(711\) −8.00000 −0.300023
\(712\) 10.0000 0.374766
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 64.0000 2.39346
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) −4.00000 −0.149071
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 52.0000 1.93390
\(724\) −20.0000 −0.743294
\(725\) 66.0000 2.45118
\(726\) −10.0000 −0.371135
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 56.0000 2.07265
\(731\) 0 0
\(732\) −8.00000 −0.295689
\(733\) −20.0000 −0.738717 −0.369358 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −32.0000 −1.17874
\(738\) −1.00000 −0.0368105
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) −8.00000 −0.294086
\(741\) −48.0000 −1.76332
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 8.00000 0.293294
\(745\) 24.0000 0.879292
\(746\) 34.0000 1.24483
\(747\) −6.00000 −0.219529
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 48.0000 1.75271
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −4.00000 −0.145865
\(753\) 12.0000 0.437304
\(754\) 24.0000 0.874028
\(755\) −64.0000 −2.32920
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 32.0000 1.16229
\(759\) −64.0000 −2.32305
\(760\) 24.0000 0.870572
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) −8.00000 −0.289241
\(766\) 20.0000 0.722629
\(767\) 56.0000 2.02204
\(768\) 2.00000 0.0721688
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) −18.0000 −0.647834
\(773\) 48.0000 1.72644 0.863220 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 0 0
\(775\) −44.0000 −1.58053
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 6.00000 0.214972
\(780\) 32.0000 1.14578
\(781\) 32.0000 1.14505
\(782\) 16.0000 0.572159
\(783\) −24.0000 −0.857690
\(784\) 0 0
\(785\) 16.0000 0.571064
\(786\) 12.0000 0.428026
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) −10.0000 −0.356235
\(789\) −32.0000 −1.13923
\(790\) −32.0000 −1.13851
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 16.0000 0.568177
\(794\) −28.0000 −0.993683
\(795\) −16.0000 −0.567462
\(796\) −20.0000 −0.708881
\(797\) 16.0000 0.566749 0.283375 0.959009i \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) −11.0000 −0.388909
\(801\) −10.0000 −0.353333
\(802\) 34.0000 1.20058
\(803\) 56.0000 1.97620
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) 0 0
\(809\) 22.0000 0.773479 0.386739 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) −44.0000 −1.54600
\(811\) 42.0000 1.47482 0.737410 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) −8.00000 −0.280400
\(815\) 96.0000 3.36273
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 12.0000 0.418548
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −12.0000 −0.418040
\(825\) 88.0000 3.06377
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) −8.00000 −0.278019
\(829\) −12.0000 −0.416777 −0.208389 0.978046i \(-0.566822\pi\)
−0.208389 + 0.978046i \(0.566822\pi\)
\(830\) −24.0000 −0.833052
\(831\) −36.0000 −1.24883
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) −44.0000 −1.52360
\(835\) 80.0000 2.76851
\(836\) 24.0000 0.830057
\(837\) 16.0000 0.553041
\(838\) −6.00000 −0.207267
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 20.0000 0.688837
\(844\) 8.00000 0.275371
\(845\) −12.0000 −0.412813
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −28.0000 −0.960958
\(850\) −22.0000 −0.754594
\(851\) −16.0000 −0.548473
\(852\) 16.0000 0.548151
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 4.00000 0.136717
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 32.0000 1.09246
\(859\) 6.00000 0.204717 0.102359 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 56.0000 1.90626 0.953131 0.302558i \(-0.0978405\pi\)
0.953131 + 0.302558i \(0.0978405\pi\)
\(864\) 4.00000 0.136083
\(865\) 96.0000 3.26410
\(866\) 14.0000 0.475739
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 48.0000 1.62735
\(871\) 32.0000 1.08428
\(872\) −6.00000 −0.203186
\(873\) −6.00000 −0.203069
\(874\) 48.0000 1.62362
\(875\) 0 0
\(876\) 28.0000 0.946032
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 8.00000 0.269987
\(879\) 32.0000 1.07933
\(880\) −16.0000 −0.539360
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −8.00000 −0.269069
\(885\) 112.000 3.76484
\(886\) −12.0000 −0.403148
\(887\) 20.0000 0.671534 0.335767 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) −4.00000 −0.134231
\(889\) 0 0
\(890\) −40.0000 −1.34080
\(891\) −44.0000 −1.47406
\(892\) −24.0000 −0.803579
\(893\) −24.0000 −0.803129
\(894\) 12.0000 0.401340
\(895\) 96.0000 3.20893
\(896\) 0 0
\(897\) 64.0000 2.13690
\(898\) 30.0000 1.00111
\(899\) −24.0000 −0.800445
\(900\) 11.0000 0.366667
\(901\) 4.00000 0.133259
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 80.0000 2.65929
\(906\) −32.0000 −1.06313
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 6.00000 0.199117
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 12.0000 0.397360
\(913\) −24.0000 −0.794284
\(914\) −38.0000 −1.25693
\(915\) 32.0000 1.05789
\(916\) 20.0000 0.660819
\(917\) 0 0
\(918\) 8.00000 0.264039
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −32.0000 −1.05501
\(921\) 52.0000 1.71346
\(922\) −16.0000 −0.526932
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 22.0000 0.723356
\(926\) 32.0000 1.05159
\(927\) 12.0000 0.394132
\(928\) −6.00000 −0.196960
\(929\) −38.0000 −1.24674 −0.623370 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) −32.0000 −1.04932
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −16.0000 −0.523816
\(934\) 14.0000 0.458094
\(935\) −32.0000 −1.04651
\(936\) 4.00000 0.130744
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 68.0000 2.21910
\(940\) 16.0000 0.521862
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 8.00000 0.260654
\(943\) −8.00000 −0.260516
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) −16.0000 −0.519656
\(949\) −56.0000 −1.81784
\(950\) −66.0000 −2.14132
\(951\) −28.0000 −0.907962
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 0 0
\(957\) 48.0000 1.55162
\(958\) −36.0000 −1.16311
\(959\) 0 0
\(960\) −8.00000 −0.258199
\(961\) −15.0000 −0.483871
\(962\) 8.00000 0.257930
\(963\) −4.00000 −0.128898
\(964\) 26.0000 0.837404
\(965\) 72.0000 2.31776
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −5.00000 −0.160706
\(969\) 24.0000 0.770991
\(970\) −24.0000 −0.770594
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) −88.0000 −2.81826
\(976\) −4.00000 −0.128037
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 48.0000 1.53487
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 20.0000 0.638226
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 40.0000 1.27451
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 0 0
\(990\) 16.0000 0.508513
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 4.00000 0.127000
\(993\) −24.0000 −0.761617
\(994\) 0 0
\(995\) 80.0000 2.53617
\(996\) −12.0000 −0.380235
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) −32.0000 −1.01294
\(999\) −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 4018.2.a.i.1.1 1
7.6 odd 2 574.2.a.a.1.1 1
21.20 even 2 5166.2.a.u.1.1 1
28.27 even 2 4592.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
574.2.a.a.1.1 1 7.6 odd 2
4018.2.a.i.1.1 1 1.1 even 1 trivial
4592.2.a.k.1.1 1 28.27 even 2
5166.2.a.u.1.1 1 21.20 even 2