Properties

Label 102.2.a.c.1.1
Level 102
Weight 2
Character 102.1
Self dual yes
Analytic conductor 0.814
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 102.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.814474100617\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 102.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{41} +12.0000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} +4.00000 q^{57} -10.0000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +1.00000 q^{68} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} -2.00000 q^{85} +12.0000 q^{86} -10.0000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} +8.00000 q^{93} -8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) −2.00000 −0.316228
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −10.0000 −1.31306
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −2.00000 −0.258199
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 1.00000 0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 12.0000 1.29399
\(87\) −10.0000 −1.07211
\(88\) −4.00000 −0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −7.00000 −0.707107
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 1.00000 0.0990148
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 8.00000 0.762770
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) −2.00000 −0.184900
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 10.0000 0.901670
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) −2.00000 −0.172133
\(136\) 1.00000 0.0857493
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 20.0000 1.66091
\(146\) 10.0000 0.827606
\(147\) −7.00000 −0.577350
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 24.0000 1.95309 0.976546 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) 4.00000 0.324443
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) −2.00000 −0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 10.0000 0.780869
\(165\) 8.00000 0.622799
\(166\) 4.00000 0.310460
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 4.00000 0.305888
\(172\) 12.0000 0.914991
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 12.0000 0.901975
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 8.00000 0.586588
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −14.0000 −1.00514
\(195\) 4.00000 0.286446
\(196\) −7.00000 −0.500000
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) −4.00000 −0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.0000 −0.846415
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 1.00000 0.0700140
\(205\) −20.0000 −1.39686
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −24.0000 −1.63679
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 10.0000 0.675737
\(220\) 8.00000 0.539360
\(221\) −2.00000 −0.134535
\(222\) −2.00000 −0.134231
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 2.00000 0.133038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 4.00000 0.264906
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −2.00000 −0.129099
\(241\) 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
\(245\) 14.0000 0.894427
\(246\) 10.0000 0.637577
\(247\) −8.00000 −0.509028
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 12.0000 0.758947
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) −10.0000 −0.618984
\(262\) −12.0000 −0.741362
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) −4.00000 −0.246183
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −12.0000 −0.733017
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −2.00000 −0.121716
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) −4.00000 −0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 20.0000 1.17444
\(291\) −14.0000 −0.820695
\(292\) 10.0000 0.585206
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) −7.00000 −0.408248
\(295\) −24.0000 −1.39733
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 24.0000 1.38104
\(303\) −10.0000 −0.574485
\(304\) 4.00000 0.229416
\(305\) 20.0000 1.14520
\(306\) 1.00000 0.0571662
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) −16.0000 −0.908739
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −2.00000 −0.113228
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 6.00000 0.336463
\(319\) 40.0000 2.23957
\(320\) −2.00000 −0.111803
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) −10.0000 −0.553001
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 8.00000 0.440386
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) −2.00000 −0.109599
\(334\) 16.0000 0.875481
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 2.00000 0.108625
\(340\) −2.00000 −0.108465
\(341\) −32.0000 −1.73290
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −10.0000 −0.536056
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −4.00000 −0.213201
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) −10.0000 −0.522708
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −4.00000 −0.206835
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 12.0000 0.609994
\(388\) −14.0000 −0.710742
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) −7.00000 −0.353553
\(393\) −12.0000 −0.605320
\(394\) 14.0000 0.705310
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) −12.0000 −0.598506
\(403\) −16.0000 −0.797017
\(404\) −10.0000 −0.497519
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 1.00000 0.0495074
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −20.0000 −0.987730
\(411\) 10.0000 0.493264
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) −2.00000 −0.0980581
\(417\) −4.00000 −0.195881
\(418\) −16.0000 −0.782586
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −28.0000 −1.36302
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 8.00000 0.386244
\(430\) −24.0000 −1.15738
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 20.0000 0.958927
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 8.00000 0.381385
\(441\) −7.00000 −0.333333
\(442\) −2.00000 −0.0951303
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 12.0000 0.568855
\(446\) 16.0000 0.757622
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −40.0000 −1.88353
\(452\) 2.00000 0.0940721
\(453\) 24.0000 1.12762
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −26.0000 −1.21490
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −10.0000 −0.464238
\(465\) −16.0000 −0.741982
\(466\) 26.0000 1.20443
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 12.0000 0.552345
\(473\) −48.0000 −2.20704
\(474\) −8.00000 −0.367452
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 4.00000 0.182384
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 28.0000 1.27141
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −10.0000 −0.452679
\(489\) 4.00000 0.180886
\(490\) 14.0000 0.632456
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 10.0000 0.450835
\(493\) −10.0000 −0.450377
\(494\) −8.00000 −0.359937
\(495\) 8.00000 0.359573
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 12.0000 0.536656
\(501\) 16.0000 0.714827
\(502\) 28.0000 1.24970
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 2.00000 0.0882162
\(515\) 16.0000 0.705044
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 4.00000 0.175412
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) −10.0000 −0.437688
\(523\) −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
\(526\) −8.00000 −0.348817
\(527\) 8.00000 0.348485
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) −12.0000 −0.521247
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) −6.00000 −0.259645
\(535\) 8.00000 0.345870
\(536\) −12.0000 −0.518321
\(537\) −12.0000 −0.517838
\(538\) 6.00000 0.258678
\(539\) 28.0000 1.20605
\(540\) −2.00000 −0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 16.0000 0.687259
\(543\) 14.0000 0.600798
\(544\) 1.00000 0.0428746
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 10.0000 0.427179
\(549\) −10.0000 −0.426790
\(550\) 4.00000 0.170561
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) 0 0
\(554\) 30.0000 1.27458
\(555\) 4.00000 0.169791
\(556\) −4.00000 −0.169638
\(557\) −34.0000 −1.44063 −0.720313 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −6.00000 −0.253095
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −8.00000 −0.335083
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 8.00000 0.334497
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 1.00000 0.0415945
\(579\) 18.0000 0.748054
\(580\) 20.0000 0.830455
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) −24.0000 −0.993978
\(584\) 10.0000 0.413803
\(585\) 4.00000 0.165380
\(586\) −26.0000 −1.07405
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) −7.00000 −0.288675
\(589\) 32.0000 1.31854
\(590\) −24.0000 −0.988064
\(591\) 14.0000 0.575883
\(592\) −2.00000 −0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 24.0000 0.976546
\(605\) −10.0000 −0.406558
\(606\) −10.0000 −0.406222
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) 0 0
\(612\) 1.00000 0.0404226
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −12.0000 −0.484281
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −8.00000 −0.321807
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) 10.0000 0.399680
\(627\) −16.0000 −0.638978
\(628\) −2.00000 −0.0798087
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) −8.00000 −0.318223
\(633\) −28.0000 −1.11290
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 14.0000 0.554700
\(638\) 40.0000 1.58362
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −4.00000 −0.157867
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 4.00000 0.157378
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) 1.00000 0.0392837
\(649\) −48.0000 −1.88416
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −10.0000 −0.391031
\(655\) 24.0000 0.937758
\(656\) 10.0000 0.390434
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 8.00000 0.311400
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) −20.0000 −0.777322
\(663\) −2.00000 −0.0776736
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 16.0000 0.619059
\(669\) 16.0000 0.618596
\(670\) 24.0000 0.927201
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −14.0000 −0.539260
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 2.00000 0.0768095
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 4.00000 0.153280
\(682\) −32.0000 −1.22534
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 4.00000 0.152944
\(685\) −20.0000 −0.764161
\(686\) 0 0
\(687\) −26.0000 −0.991962
\(688\) 12.0000 0.457496
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 8.00000 0.303457
\(696\) −10.0000 −0.379049
\(697\) 10.0000 0.378777
\(698\) 14.0000 0.529908
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) 46.0000 1.73740 0.868698 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −8.00000 −0.301726
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 2.00000 0.0743808
\(724\) 14.0000 0.520306
\(725\) 10.0000 0.371391
\(726\) 5.00000 0.185567
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −20.0000 −0.740233
\(731\) 12.0000 0.443836
\(732\) −10.0000 −0.369611
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 24.0000 0.885856
\(735\) 14.0000 0.516398
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 10.0000 0.368105
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 4.00000 0.147043
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 8.00000 0.293294
\(745\) 20.0000 0.732743
\(746\) 6.00000 0.219676
\(747\) 4.00000 0.146352
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 28.0000 1.02038
\(754\) 20.0000 0.728357
\(755\) −48.0000 −1.74690
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) −2.00000 −0.0723102
\(766\) −16.0000 −0.578103
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 18.0000 0.647834
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 12.0000 0.431331
\(775\) −8.00000 −0.287368
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) 40.0000 1.43315
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) −10.0000 −0.357371
\(784\) −7.00000 −0.250000
\(785\) 4.00000 0.142766
\(786\) −12.0000 −0.428026
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 14.0000 0.498729
\(789\) −8.00000 −0.284808
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) −4.00000 −0.142134
\(793\) 20.0000 0.710221
\(794\) −26.0000 −0.922705
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −14.0000 −0.494357
\(803\) −40.0000 −1.41157
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 6.00000 0.211210
\(808\) −10.0000 −0.351799
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 8.00000 0.280400
\(815\) −8.00000 −0.280228
\(816\) 1.00000 0.0350070
\(817\) 48.0000 1.67931
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) −20.0000 −0.698430
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) 10.0000 0.348790
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −8.00000 −0.278693
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −8.00000 −0.277684
\(831\) 30.0000 1.04069
\(832\) −2.00000 −0.0693375
\(833\) −7.00000 −0.242536
\(834\) −4.00000 −0.138509
\(835\) −32.0000 −1.10741
\(836\) −16.0000 −0.553372
\(837\) 8.00000 0.276520
\(838\) 4.00000 0.138178
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 22.0000 0.758170
\(843\) −6.00000 −0.206651
\(844\) −28.0000 −0.963800
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 12.0000 0.411839
\(850\) −1.00000 −0.0342997
\(851\) 0 0
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) −4.00000 −0.136717
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 8.00000 0.273115
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 1.00000 0.0340207
\(865\) −12.0000 −0.408012
\(866\) −14.0000 −0.475739
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 20.0000 0.678064
\(871\) 24.0000 0.813209
\(872\) −10.0000 −0.338643
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 16.0000 0.539974
\(879\) −26.0000 −0.876958
\(880\) 8.00000 0.269680
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) −7.00000 −0.235702
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −24.0000 −0.806751
\(886\) −4.00000 −0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) −4.00000 −0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) −80.0000 −2.66815
\(900\) −1.00000 −0.0333333
\(901\) 6.00000 0.199889
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −28.0000 −0.930751
\(906\) 24.0000 0.797347
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 4.00000 0.132745
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000 0.132453
\(913\) −16.0000 −0.529523
\(914\) 10.0000 0.330771
\(915\) 20.0000 0.661180
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 32.0000 1.05159
\(927\) −8.00000 −0.262754
\(928\) −10.0000 −0.328266
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) −16.0000 −0.524661
\(931\) −28.0000 −0.917663
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 8.00000 0.261628
\(936\) −2.00000 −0.0653720
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) −2.00000 −0.0651635
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) −8.00000 −0.259828
\(949\) −20.0000 −0.649227
\(950\) −4.00000 −0.129777
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 6.00000 0.194257
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) 40.0000 1.29302
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) −4.00000 −0.128898
\(964\) 2.00000 0.0644157
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 5.00000 0.160706
\(969\) 4.00000 0.128499
\(970\) 28.0000 0.899026
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 2.00000 0.0640513
\(976\) −10.0000 −0.320092
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 4.00000 0.127906
\(979\) 24.0000 0.767043
\(980\) 14.0000 0.447214
\(981\) −10.0000 −0.319275
\(982\) 12.0000 0.382935
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 10.0000 0.318788
\(985\) −28.0000 −0.892154
\(986\) −10.0000 −0.318465
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 8.00000 0.254257
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 8.00000 0.254000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 4.00000 0.126618
\(999\) −2.00000 −0.0632772
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 102.2.a.c.1.1 1
3.2 odd 2 306.2.a.b.1.1 1
4.3 odd 2 816.2.a.b.1.1 1
5.2 odd 4 2550.2.d.m.2449.2 2
5.3 odd 4 2550.2.d.m.2449.1 2
5.4 even 2 2550.2.a.c.1.1 1
7.6 odd 2 4998.2.a.be.1.1 1
8.3 odd 2 3264.2.a.bc.1.1 1
8.5 even 2 3264.2.a.m.1.1 1
12.11 even 2 2448.2.a.p.1.1 1
15.14 odd 2 7650.2.a.ca.1.1 1
17.2 even 8 1734.2.f.e.1483.2 4
17.4 even 4 1734.2.b.b.577.1 2
17.8 even 8 1734.2.f.e.829.1 4
17.9 even 8 1734.2.f.e.829.2 4
17.13 even 4 1734.2.b.b.577.2 2
17.15 even 8 1734.2.f.e.1483.1 4
17.16 even 2 1734.2.a.j.1.1 1
24.5 odd 2 9792.2.a.k.1.1 1
24.11 even 2 9792.2.a.l.1.1 1
51.50 odd 2 5202.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.c.1.1 1 1.1 even 1 trivial
306.2.a.b.1.1 1 3.2 odd 2
816.2.a.b.1.1 1 4.3 odd 2
1734.2.a.j.1.1 1 17.16 even 2
1734.2.b.b.577.1 2 17.4 even 4
1734.2.b.b.577.2 2 17.13 even 4
1734.2.f.e.829.1 4 17.8 even 8
1734.2.f.e.829.2 4 17.9 even 8
1734.2.f.e.1483.1 4 17.15 even 8
1734.2.f.e.1483.2 4 17.2 even 8
2448.2.a.p.1.1 1 12.11 even 2
2550.2.a.c.1.1 1 5.4 even 2
2550.2.d.m.2449.1 2 5.3 odd 4
2550.2.d.m.2449.2 2 5.2 odd 4
3264.2.a.m.1.1 1 8.5 even 2
3264.2.a.bc.1.1 1 8.3 odd 2
4998.2.a.be.1.1 1 7.6 odd 2
5202.2.a.c.1.1 1 51.50 odd 2
7650.2.a.ca.1.1 1 15.14 odd 2
9792.2.a.k.1.1 1 24.5 odd 2
9792.2.a.l.1.1 1 24.11 even 2