Properties

Label 2850.2.a.a.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +4.00000 q^{21} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +1.00000 q^{38} -2.00000 q^{39} -2.00000 q^{41} -4.00000 q^{42} -12.0000 q^{43} -4.00000 q^{44} -8.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{56} +1.00000 q^{57} -6.00000 q^{58} -10.0000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -10.0000 q^{74} -1.00000 q^{76} +16.0000 q^{77} +2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +8.00000 q^{83} +4.00000 q^{84} +12.0000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +6.00000 q^{89} -8.00000 q^{91} +8.00000 q^{92} -4.00000 q^{93} +1.00000 q^{96} -18.0000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 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
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 1.00000 0.162221
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −4.00000 −0.617213
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 1.00000 0.132453
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 16.0000 1.82337
\(78\) 2.00000 0.226455
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 8.00000 0.834058
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) −9.00000 −0.909137
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 2.00000 0.180334
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000 0.348155
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 8.00000 0.681005
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) −9.00000 −0.742307
\(148\) 10.0000 0.821995
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 1.00000 0.0811107
\(153\) 2.00000 0.161690
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 12.0000 0.954669
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −32.0000 −2.52195
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −12.0000 −0.914991
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 8.00000 0.592999
\(183\) 10.0000 0.739221
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 4.00000 0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 10.0000 0.703598
\(203\) −24.0000 −1.68447
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 8.00000 0.556038
\(208\) 2.00000 0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −6.00000 −0.412082
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) −10.0000 −0.677285
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 10.0000 0.671156
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 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 \(0.369623\pi\)
\(228\) 1.00000 0.0662266
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 8.00000 0.518563
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −2.00000 −0.127257
\(248\) −4.00000 −0.254000
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) −4.00000 −0.251976
\(253\) −32.0000 −2.01182
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −12.0000 −0.747087
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) −6.00000 −0.367194
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 2.00000 0.121268
\(273\) 8.00000 0.484182
\(274\) 14.0000 0.845771
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 12.0000 0.719712
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 8.00000 0.472225
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) −2.00000 −0.117041
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 4.00000 0.232104
\(298\) −22.0000 −1.27443
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 48.0000 2.76667
\(302\) −4.00000 −0.230174
\(303\) 10.0000 0.574485
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 16.0000 0.911685
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −6.00000 −0.336463
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 32.0000 1.78329
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −10.0000 −0.553001
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 8.00000 0.439057
\(333\) 10.0000 0.547997
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 1.00000 0.0540738
\(343\) −8.00000 −0.431959
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −6.00000 −0.321634
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 4.00000 0.213201
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 8.00000 0.423405
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) −5.00000 −0.262432
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 12.0000 0.626395 0.313197 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) 8.00000 0.417029
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) −4.00000 −0.207390
\(373\) −30.0000 −1.55334 −0.776671 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) −12.0000 −0.609994
\(388\) −18.0000 −0.913812
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −9.00000 −0.454569
\(393\) 12.0000 0.605320
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −16.0000 −0.802008
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 4.00000 0.199502
\(403\) 8.00000 0.398508
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) −40.0000 −1.98273
\(408\) 2.00000 0.0990148
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) −4.00000 −0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 40.0000 1.93574
\(428\) 12.0000 0.580042
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −8.00000 −0.382692
\(438\) −2.00000 −0.0955637
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −4.00000 −0.190261
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) −22.0000 −1.04056
\(448\) −4.00000 −0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 6.00000 0.282216
\(453\) −4.00000 −0.187936
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −22.0000 −1.02799
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 16.0000 0.744387
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 2.00000 0.0924500
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) 48.0000 2.20704
\(474\) −12.0000 −0.551178
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) −6.00000 −0.274721
\(478\) 8.00000 0.365911
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) −2.00000 −0.0910975
\(483\) 32.0000 1.45605
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 10.0000 0.452679
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 32.0000 1.43540
\(498\) 8.00000 0.358489
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 28.0000 1.24970
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) 40.0000 1.75750
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −6.00000 −0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 8.00000 0.348485
\(528\) 4.00000 0.174078
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) −4.00000 −0.173259
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 24.0000 1.03089
\(543\) −2.00000 −0.0858282
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −14.0000 −0.598050
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 8.00000 0.340503
\(553\) 48.0000 2.04117
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −4.00000 −0.169334
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −30.0000 −1.26547
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) −4.00000 −0.167984
\(568\) 8.00000 0.335673
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 13.0000 0.540729
\(579\) 18.0000 0.748054
\(580\) 0 0
\(581\) −32.0000 −1.32758
\(582\) −18.0000 −0.746124
\(583\) 24.0000 0.993978
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 40.0000 1.65098 0.825488 0.564419i \(-0.190900\pi\)
0.825488 + 0.564419i \(0.190900\pi\)
\(588\) −9.00000 −0.371154
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 10.0000 0.410997
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) −16.0000 −0.654836
\(598\) −16.0000 −0.654289
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −48.0000 −1.95633
\(603\) 4.00000 0.162893
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 1.00000 0.0405554
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −16.0000 −0.643614
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) −24.0000 −0.962312
\(623\) −24.0000 −0.961540
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) −4.00000 −0.159745
\(628\) 6.00000 0.239426
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 12.0000 0.477334
\(633\) 12.0000 0.476957
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 18.0000 0.713186
\(638\) 24.0000 0.950169
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 12.0000 0.473602
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) −32.0000 −1.26098
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 4.00000 0.156652
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 12.0000 0.466393
\(663\) −4.00000 −0.155347
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 48.0000 1.85857
\(668\) 16.0000 0.619059
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) −4.00000 −0.154303
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 6.00000 0.230429
\(679\) 72.0000 2.76311
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 16.0000 0.612672
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) −22.0000 −0.839352
\(688\) −12.0000 −0.457496
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −14.0000 −0.532200
\(693\) 16.0000 0.607790
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −4.00000 −0.151511
\(698\) 34.0000 1.28692
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 2.00000 0.0754851
\(703\) −10.0000 −0.377157
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 40.0000 1.50435
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −6.00000 −0.224860
\(713\) 32.0000 1.19841
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 0 0
\(717\) 8.00000 0.298765
\(718\) −32.0000 −1.19423
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 64.0000 2.38348
\(722\) −1.00000 −0.0372161
\(723\) −2.00000 −0.0743808
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 10.0000 0.369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −16.0000 −0.589368
\(738\) 2.00000 0.0736210
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) −24.0000 −0.881068
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 30.0000 1.09838
\(747\) 8.00000 0.292705
\(748\) −8.00000 −0.292509
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 0 0
\(753\) 28.0000 1.02038
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 20.0000 0.726433
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −8.00000 −0.289809
\(763\) −40.0000 −1.44810
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −18.0000 −0.647834
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 40.0000 1.43499
\(778\) −6.00000 −0.215110
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) −16.0000 −0.572159
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 18.0000 0.641223
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 4.00000 0.142134
\(793\) −20.0000 −0.710221
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 18.0000 0.635602
\(803\) 8.00000 0.282314
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) −6.00000 −0.211210
\(808\) 10.0000 0.351799
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) −24.0000 −0.842235
\(813\) 24.0000 0.841717
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 12.0000 0.419827
\(818\) 6.00000 0.209785
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) −14.0000 −0.488306
\(823\) −52.0000 −1.81261 −0.906303 0.422628i \(-0.861108\pi\)
−0.906303 + 0.422628i \(0.861108\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 8.00000 0.278019
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 2.00000 0.0693375
\(833\) 18.0000 0.623663
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) −4.00000 −0.138260
\(838\) −12.0000 −0.414533
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 14.0000 0.482472
\(843\) −30.0000 −1.03325
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) −6.00000 −0.206041
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 80.0000 2.74236
\(852\) 8.00000 0.274075
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) −40.0000 −1.36877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) −8.00000 −0.273115
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 13.0000 0.441503
\(868\) −16.0000 −0.543075
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) −18.0000 −0.609208
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 4.00000 0.134993
\(879\) −10.0000 −0.337292
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −9.00000 −0.303046
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 10.0000 0.335578
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 22.0000 0.735790
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) −16.0000 −0.534224
\(898\) −6.00000 −0.200223
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) −8.00000 −0.266371
\(903\) −48.0000 −1.59734
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 4.00000 0.132891
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 12.0000 0.398234
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 1.00000 0.0331133
\(913\) −32.0000 −1.05905
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 48.0000 1.58510
\(918\) 2.00000 0.0660098
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) −14.0000 −0.461065
\(923\) −16.0000 −0.526646
\(924\) −16.0000 −0.526361
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) −16.0000 −0.525509
\(928\) −6.00000 −0.196960
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −9.00000 −0.294963
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 16.0000 0.522419
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 6.00000 0.195491
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 40.0000 1.29983 0.649913 0.760009i \(-0.274805\pi\)
0.649913 + 0.760009i \(0.274805\pi\)
\(948\) 12.0000 0.389742
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 8.00000 0.259281
\(953\) −10.0000 −0.323932 −0.161966 0.986796i \(-0.551783\pi\)
−0.161966 + 0.986796i \(0.551783\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 24.0000 0.775810
\(958\) 16.0000 0.516937
\(959\) 56.0000 1.80833
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −20.0000 −0.644826
\(963\) 12.0000 0.386695
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) −32.0000 −1.02958
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) −5.00000 −0.160706
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 48.0000 1.53881
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −50.0000 −1.59964 −0.799821 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 4.00000 0.127906
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) −28.0000 −0.893516
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) −96.0000 −3.05262
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) −4.00000 −0.127000
\(993\) 12.0000 0.380808
\(994\) −32.0000 −1.01498
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 4.00000 0.126618
\(999\) −10.0000 −0.316386
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 2850.2.a.a.1.1 1
3.2 odd 2 8550.2.a.t.1.1 1
5.2 odd 4 2850.2.d.k.799.1 2
5.3 odd 4 2850.2.d.k.799.2 2
5.4 even 2 570.2.a.m.1.1 1
15.14 odd 2 1710.2.a.f.1.1 1
20.19 odd 2 4560.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.m.1.1 1 5.4 even 2
1710.2.a.f.1.1 1 15.14 odd 2
2850.2.a.a.1.1 1 1.1 even 1 trivial
2850.2.d.k.799.1 2 5.2 odd 4
2850.2.d.k.799.2 2 5.3 odd 4
4560.2.a.k.1.1 1 20.19 odd 2
8550.2.a.t.1.1 1 3.2 odd 2