Properties

Label 3330.2.a.m.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -6.00000 q^{11} +2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} -1.00000 q^{20} -6.00000 q^{22} +1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} -6.00000 q^{44} -6.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +6.00000 q^{55} -4.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{70} +14.0000 q^{73} +1.00000 q^{74} +2.00000 q^{76} +24.0000 q^{77} -16.0000 q^{79} -1.00000 q^{80} +6.00000 q^{82} +12.0000 q^{83} -6.00000 q^{85} +8.00000 q^{86} -6.00000 q^{88} -6.00000 q^{89} -8.00000 q^{91} -6.00000 q^{94} -2.00000 q^{95} +8.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 24.0000 2.73505
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 6.00000 0.572078
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) 24.0000 1.93398
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −16.0000 −1.27289
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −6.00000 −0.452267
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −36.0000 −2.63258
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 0 0
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −32.0000 −2.17230
\(218\) 20.0000 1.35457
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) −24.0000 −1.55569
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 25.0000 1.60706
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 12.0000 0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 24.0000 1.36753
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 36.0000 2.01561
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −48.0000 −2.59935
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) −16.0000 −0.814379
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) −48.0000 −2.36193
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −16.0000 −0.778868
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −32.0000 −1.54859
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 38.0000 1.82616 0.913082 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 6.00000 0.286039
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 26.0000 1.21490
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) −34.0000 −1.58011 −0.790057 0.613033i \(-0.789949\pi\)
−0.790057 + 0.613033i \(0.789949\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −24.0000 −1.10004
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −56.0000 −2.47729
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −30.0000 −1.32324
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) 36.0000 1.58328
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 32.0000 1.39926 0.699631 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 24.0000 1.03471
\(539\) −54.0000 −2.32594
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) −6.00000 −0.255841
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) 64.0000 2.72156
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 12.0000 0.491539
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −32.0000 −1.30422
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) −25.0000 −1.01639
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 24.0000 0.966988
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −16.0000 −0.639489
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −16.0000 −0.636446
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) −72.0000 −2.82625
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 14.0000 0.544125
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 0 0
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −32.0000 −1.22805
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) −48.0000 −1.83801
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 0 0
\(709\) 44.0000 1.65245 0.826227 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −56.0000 −2.08555
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 38.0000 1.40934 0.704671 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) −8.00000 −0.296500
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) −36.0000 −1.31629
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −80.0000 −2.89619
\(764\) 0 0
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −24.0000 −0.864900
\(771\) 0 0
\(772\) −16.0000 −0.575853
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 16.0000 0.569254
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) −84.0000 −2.96430
\(804\) 0 0
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 24.0000 0.842235
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −28.0000 −0.972480 −0.486240 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) −6.00000 −0.207267
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 8.00000 0.275698
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −100.000 −3.43604
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −32.0000 −1.09502
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 0 0
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 38.0000 1.29129
\(867\) 0 0
\(868\) −32.0000 −1.08615
\(869\) 96.0000 3.25658
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 20.0000 0.677285
\(873\) 0 0
\(874\) 0 0
\(875\) 4.00000 0.135225
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 6.00000 0.202260
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −36.0000 −1.19867
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −72.0000 −2.38285
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.00000 0.197599
\(923\) 0 0
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −34.0000 −1.11731
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 36.0000 1.17733
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) −24.0000 −0.777844
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) −46.0000 −1.47926 −0.739630 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) 25.0000 0.803530
\(969\) 0 0
\(970\) −8.00000 −0.256865
\(971\) 54.0000 1.73294 0.866471 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) 6.00000 0.191468
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 14.0000 0.443162
\(999\) 0 0
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 3330.2.a.m.1.1 1
3.2 odd 2 1110.2.a.h.1.1 1
12.11 even 2 8880.2.a.n.1.1 1
15.14 odd 2 5550.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.h.1.1 1 3.2 odd 2
3330.2.a.m.1.1 1 1.1 even 1 trivial
5550.2.a.bd.1.1 1 15.14 odd 2
8880.2.a.n.1.1 1 12.11 even 2