Properties

Label 1850.2.a.f.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} -2.00000 q^{24} +2.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.00000 q^{38} -4.00000 q^{39} -6.00000 q^{41} +4.00000 q^{42} +4.00000 q^{43} +6.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} -12.0000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +2.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} -10.0000 q^{61} +10.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} -6.00000 q^{68} -1.00000 q^{72} -2.00000 q^{73} +1.00000 q^{74} +2.00000 q^{76} +4.00000 q^{78} -10.0000 q^{79} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -4.00000 q^{86} +12.0000 q^{87} -6.00000 q^{89} +4.00000 q^{91} -20.0000 q^{93} -6.00000 q^{94} -2.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −4.00000 −0.769800
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −2.00000 −0.324443
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 2.00000 0.288675
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 4.00000 0.529813
\(58\) −6.00000 −0.787839
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 10.0000 1.27000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −20.0000 −2.07390
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 12.0000 1.18818
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −4.00000 −0.384900
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 10.0000 0.905357
\(123\) −12.0000 −1.08200
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) −6.00000 −0.494872
\(148\) −1.00000 −0.0821995
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −2.00000 −0.162221
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 10.0000 0.795557
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 4.00000 0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 6.00000 0.449719
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −4.00000 −0.296500
\(183\) −20.0000 −1.47844
\(184\) 0 0
\(185\) 0 0
\(186\) 20.0000 1.46647
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 8.00000 0.581914
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 2.00000 0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −18.0000 −1.26648
\(203\) −12.0000 −0.842235
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 20.0000 1.35769
\(218\) −14.0000 −0.948200
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) −20.0000 −1.29914
\(238\) −12.0000 −0.777844
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 11.0000 0.707107
\(243\) −10.0000 −0.641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) −4.00000 −0.254514
\(248\) 10.0000 0.635001
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −8.00000 −0.498058
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 6.00000 0.370681
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) −12.0000 −0.734388
\(268\) −2.00000 −0.122169
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 8.00000 0.484182
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 4.00000 0.239904
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −12.0000 −0.714590
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) −2.00000 −0.117041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −20.0000 −1.15087
\(303\) 36.0000 2.06815
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 4.00000 0.226455
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 12.0000 0.672927
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 28.0000 1.54840
\(328\) 6.00000 0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 6.00000 0.329293
\(333\) −1.00000 −0.0547997
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 9.00000 0.489535
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 12.0000 0.643268
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 24.0000 1.27021
\(358\) −6.00000 −0.317110
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 22.0000 1.15629
\(363\) −22.0000 −1.15470
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 20.0000 1.04542
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −20.0000 −1.03695
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −12.0000 −0.618031
\(378\) −8.00000 −0.411476
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 6.00000 0.306987
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) −12.0000 −0.605320
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 22.0000 1.10276
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 4.00000 0.199502
\(403\) 20.0000 0.996271
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 12.0000 0.594089
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 4.00000 0.197066
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −20.0000 −0.973585
\(423\) 6.00000 0.291730
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −4.00000 −0.192450
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −20.0000 −0.960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −12.0000 −0.570782
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 36.0000 1.70274
\(448\) −2.00000 −0.0944911
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 40.0000 1.87936
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 10.0000 0.467269
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 20.0000 0.918630
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) −6.00000 −0.274721
\(478\) −6.00000 −0.274434
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 10.0000 0.452679
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) −12.0000 −0.541002
\(493\) −36.0000 −1.62136
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) −18.0000 −0.803379
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) −2.00000 −0.0887357
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 36.0000 1.58022
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −6.00000 −0.262613
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 60.0000 2.61364
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 16.0000 0.687259
\(543\) −44.0000 −1.88822
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 10.0000 0.423334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 22.0000 0.923913
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −19.0000 −0.790296
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 4.00000 0.165805
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −6.00000 −0.247436
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 36.0000 1.48084
\(592\) −1.00000 −0.0410997
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −44.0000 −1.80080
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 8.00000 0.326056
\(603\) −2.00000 −0.0814463
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) −36.0000 −1.46240
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) −6.00000 −0.242536
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 26.0000 1.04927
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) −8.00000 −0.321807
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 12.0000 0.480770
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 10.0000 0.397779
\(633\) 40.0000 1.58986
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 12.0000 0.473602
\(643\) −32.0000 −1.26196 −0.630978 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) 40.0000 1.56772
\(652\) 16.0000 0.626608
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −28.0000 −1.09489
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −2.00000 −0.0780274
\(658\) 12.0000 0.467809
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 10.0000 0.388661
\(663\) 24.0000 0.932083
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) −28.0000 −1.08254
\(670\) 0 0
\(671\) 0 0
\(672\) 4.00000 0.154303
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 12.0000 0.460857
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −20.0000 −0.763048
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) 36.0000 1.36360
\(698\) −26.0000 −0.984115
\(699\) −36.0000 −1.36165
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −8.00000 −0.301941
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −36.0000 −1.35392
\(708\) −12.0000 −0.450988
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 12.0000 0.448148
\(718\) −36.0000 −1.34351
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 15.0000 0.558242
\(723\) −44.0000 −1.63638
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) 22.0000 0.816497
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) −4.00000 −0.148250
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) −20.0000 −0.739221
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) −12.0000 −0.440534
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) 20.0000 0.733236
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 6.00000 0.218797
\(753\) 36.0000 1.31191
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 8.00000 0.290957
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 4.00000 0.144905
\(763\) −28.0000 −1.01367
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 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
\(771\) −12.0000 −0.432169
\(772\) −2.00000 −0.0719816
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 4.00000 0.143499
\(778\) −6.00000 −0.215110
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −24.0000 −0.857690
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 10.0000 0.356462 0.178231 0.983989i \(-0.442963\pi\)
0.178231 + 0.983989i \(0.442963\pi\)
\(788\) 18.0000 0.641223
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 8.00000 0.283197
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) −12.0000 −0.422420
\(808\) −18.0000 −0.633238
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −12.0000 −0.421117
\(813\) −32.0000 −1.12229
\(814\) 0 0
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) 8.00000 0.279885
\(818\) −2.00000 −0.0699284
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −12.0000 −0.418548
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) −52.0000 −1.80386
\(832\) −2.00000 −0.0693375
\(833\) 18.0000 0.623663
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 40.0000 1.38260
\(838\) −24.0000 −0.829066
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) 36.0000 1.23991
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 22.0000 0.755929
\(848\) −6.00000 −0.206041
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) −30.0000 −1.02180
\(863\) −54.0000 −1.83818 −0.919091 0.394046i \(-0.871075\pi\)
−0.919091 + 0.394046i \(0.871075\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 38.0000 1.29055
\(868\) 20.0000 0.678844
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) −14.0000 −0.474100
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 22.0000 0.742464
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 3.00000 0.101015
\(883\) −56.0000 −1.88455 −0.942275 0.334840i \(-0.891318\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 2.00000 0.0671156
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) 12.0000 0.401565
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) −60.0000 −2.00111
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −40.0000 −1.32891
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −24.0000 −0.796468
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 12.0000 0.396275
\(918\) −24.0000 −0.792118
\(919\) −46.0000 −1.51740 −0.758700 0.651440i \(-0.774165\pi\)
−0.758700 + 0.651440i \(0.774165\pi\)
\(920\) 0 0
\(921\) −52.0000 −1.71346
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −40.0000 −1.31448
\(927\) 4.00000 0.131377
\(928\) −6.00000 −0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −18.0000 −0.589610
\(933\) −36.0000 −1.17859
\(934\) 0 0
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) −4.00000 −0.130605
\(939\) 44.0000 1.43589
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −20.0000 −0.651635
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −20.0000 −0.649570
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) −12.0000 −0.388922
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 18.0000 0.581554
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −2.00000 −0.0644826
\(963\) −6.00000 −0.193347
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 11.0000 0.353553
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −10.0000 −0.320750
\(973\) 8.00000 0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −32.0000 −1.02325
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) −36.0000 −1.14881
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) −24.0000 −0.763928
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 10.0000 0.317500
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −14.0000 −0.443162
\(999\) 4.00000 0.126554
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 1850.2.a.f.1.1 1
5.2 odd 4 1850.2.b.b.149.1 2
5.3 odd 4 1850.2.b.b.149.2 2
5.4 even 2 370.2.a.d.1.1 1
15.14 odd 2 3330.2.a.d.1.1 1
20.19 odd 2 2960.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.d.1.1 1 5.4 even 2
1850.2.a.f.1.1 1 1.1 even 1 trivial
1850.2.b.b.149.1 2 5.2 odd 4
1850.2.b.b.149.2 2 5.3 odd 4
2960.2.a.m.1.1 1 20.19 odd 2
3330.2.a.d.1.1 1 15.14 odd 2