Properties

Label 6762.2.a.m.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{30} -6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +4.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} +8.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -1.00000 q^{46} -10.0000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +4.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +2.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -8.00000 q^{65} +4.00000 q^{66} +8.00000 q^{67} +4.00000 q^{68} +1.00000 q^{69} -4.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +6.00000 q^{83} -8.00000 q^{85} -8.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +2.00000 q^{90} +1.00000 q^{92} -6.00000 q^{93} +10.0000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 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
\(8\) −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
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −2.00000 −0.324443
\(39\) 4.00000 0.640513
\(40\) 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
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) −1.00000 −0.147442
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −2.00000 −0.258199
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 4.00000 0.492366
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −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
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) −8.00000 −0.862662
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −6.00000 −0.622171
\(94\) 10.0000 1.03142
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −16.0000 −1.59206 −0.796030 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) −4.00000 −0.396059
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −8.00000 −0.762770
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −2.00000 −0.187317
\(115\) −2.00000 −0.186501
\(116\) −6.00000 −0.557086
\(117\) 4.00000 0.369800
\(118\) −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
\(124\) −6.00000 −0.538816
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 8.00000 0.701646
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) −2.00000 −0.172133
\(136\) −4.00000 −0.342997
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 4.00000 0.335673
\(143\) −16.0000 −1.33799
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) −2.00000 −0.162221
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 4.00000 0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 10.0000 0.780869
\(165\) 8.00000 0.622799
\(166\) −6.00000 −0.465690
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 8.00000 0.613572
\(171\) 2.00000 0.152944
\(172\) 8.00000 0.609994
\(173\) −16.0000 −1.21646 −0.608229 0.793762i \(-0.708120\pi\)
−0.608229 + 0.793762i \(0.708120\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −2.00000 −0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) 4.00000 0.294086
\(186\) 6.00000 0.439941
\(187\) −16.0000 −1.17004
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 1.00000 0.0721688
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −8.00000 −0.574367
\(195\) −8.00000 −0.572892
\(196\) 0 0
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 4.00000 0.284268
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) 16.0000 1.12576
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) −20.0000 −1.39686
\(206\) 12.0000 0.836080
\(207\) 1.00000 0.0695048
\(208\) 4.00000 0.277350
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) −4.00000 −0.274075
\(214\) 20.0000 1.36717
\(215\) −16.0000 −1.09119
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 2.00000 0.135147
\(220\) 8.00000 0.539360
\(221\) 16.0000 1.07628
\(222\) 2.00000 0.134231
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 2.00000 0.132453
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −4.00000 −0.261488
\(235\) 20.0000 1.30466
\(236\) 12.0000 0.781133
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) −2.00000 −0.129099
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) 8.00000 0.509028
\(248\) 6.00000 0.381000
\(249\) 6.00000 0.380235
\(250\) −12.0000 −0.758947
\(251\) −26.0000 −1.64111 −0.820553 0.571571i \(-0.806334\pi\)
−0.820553 + 0.571571i \(0.806334\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −20.0000 −1.25491
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −8.00000 −0.498058
\(259\) 0 0
\(260\) −8.00000 −0.496139
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 4.00000 0.246183
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 2.00000 0.121716
\(271\) 26.0000 1.57939 0.789694 0.613501i \(-0.210239\pi\)
0.789694 + 0.613501i \(0.210239\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 4.00000 0.241209
\(276\) 1.00000 0.0601929
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −16.0000 −0.959616
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 10.0000 0.595491
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) −4.00000 −0.237356
\(285\) −4.00000 −0.236940
\(286\) 16.0000 0.946100
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −12.0000 −0.704664
\(291\) 8.00000 0.468968
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 2.00000 0.116248
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 4.00000 0.231326
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) −16.0000 −0.919176
\(304\) 2.00000 0.114708
\(305\) −20.0000 −1.14520
\(306\) −4.00000 −0.228665
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) −12.0000 −0.681554
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) −4.00000 −0.226455
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 6.00000 0.336463
\(319\) 24.0000 1.34374
\(320\) −2.00000 −0.111803
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −12.0000 −0.664619
\(327\) 2.00000 0.110600
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 6.00000 0.329293
\(333\) −2.00000 −0.109599
\(334\) −18.0000 −0.984916
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −3.00000 −0.163178
\(339\) −6.00000 −0.325875
\(340\) −8.00000 −0.433861
\(341\) 24.0000 1.29967
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) −2.00000 −0.107676
\(346\) 16.0000 0.860165
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −6.00000 −0.321634
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 4.00000 0.213201
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −12.0000 −0.637793
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 2.00000 0.105409
\(361\) −15.0000 −0.789474
\(362\) −2.00000 −0.105118
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) −10.0000 −0.522708
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 1.00000 0.0521286
\(369\) 10.0000 0.520579
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) −6.00000 −0.311086
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 16.0000 0.827340
\(375\) 12.0000 0.619677
\(376\) 10.0000 0.515711
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −4.00000 −0.205196
\(381\) 20.0000 1.02463
\(382\) −16.0000 −0.818631
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 8.00000 0.406663
\(388\) 8.00000 0.406138
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 8.00000 0.405096
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 26.0000 1.30986
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) −28.0000 −1.40528 −0.702640 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) −12.0000 −0.601506
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) −8.00000 −0.399004
\(403\) −24.0000 −1.19553
\(404\) −16.0000 −0.796030
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) −4.00000 −0.198030
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 20.0000 0.987730
\(411\) 6.00000 0.295958
\(412\) −12.0000 −0.591198
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −12.0000 −0.589057
\(416\) −4.00000 −0.196116
\(417\) 16.0000 0.783523
\(418\) 8.00000 0.391293
\(419\) 22.0000 1.07477 0.537385 0.843337i \(-0.319412\pi\)
0.537385 + 0.843337i \(0.319412\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −4.00000 −0.194717
\(423\) −10.0000 −0.486217
\(424\) 6.00000 0.291386
\(425\) −4.00000 −0.194029
\(426\) 4.00000 0.193801
\(427\) 0 0
\(428\) −20.0000 −0.966736
\(429\) −16.0000 −0.772487
\(430\) 16.0000 0.771589
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 2.00000 0.0957826
\(437\) 2.00000 0.0956730
\(438\) −2.00000 −0.0955637
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) −8.00000 −0.381385
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 1.00000 0.0471405
\(451\) −40.0000 −1.88353
\(452\) −6.00000 −0.282216
\(453\) 4.00000 0.187936
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 6.00000 0.280362
\(459\) 4.00000 0.186704
\(460\) −2.00000 −0.0932505
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 12.0000 0.556487
\(466\) −6.00000 −0.277945
\(467\) −22.0000 −1.01804 −0.509019 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) −20.0000 −0.922531
\(471\) −10.0000 −0.460776
\(472\) −12.0000 −0.552345
\(473\) −32.0000 −1.47136
\(474\) 8.00000 0.367452
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −20.0000 −0.914779
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 2.00000 0.0912871
\(481\) −8.00000 −0.364769
\(482\) 12.0000 0.546585
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −16.0000 −0.726523
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −10.0000 −0.452679
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 10.0000 0.450835
\(493\) −24.0000 −1.08091
\(494\) −8.00000 −0.359937
\(495\) 8.00000 0.359573
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 12.0000 0.536656
\(501\) 18.0000 0.804181
\(502\) 26.0000 1.16044
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 32.0000 1.42398
\(506\) 4.00000 0.177822
\(507\) 3.00000 0.133235
\(508\) 20.0000 0.887357
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 8.00000 0.354246
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 2.00000 0.0882162
\(515\) 24.0000 1.05757
\(516\) 8.00000 0.352180
\(517\) 40.0000 1.75920
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 8.00000 0.350823
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 6.00000 0.262613
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −24.0000 −1.04546
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) −12.0000 −0.521247
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) 40.0000 1.72935
\(536\) −8.00000 −0.345547
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −26.0000 −1.11680
\(543\) 2.00000 0.0858282
\(544\) −4.00000 −0.171499
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 6.00000 0.256307
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) −12.0000 −0.511217
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −14.0000 −0.594803
\(555\) 4.00000 0.169791
\(556\) 16.0000 0.678551
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 6.00000 0.254000
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) −14.0000 −0.590554
\(563\) 22.0000 0.927189 0.463595 0.886047i \(-0.346559\pi\)
0.463595 + 0.886047i \(0.346559\pi\)
\(564\) −10.0000 −0.421076
\(565\) 12.0000 0.504844
\(566\) −18.0000 −0.756596
\(567\) 0 0
\(568\) 4.00000 0.167836
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 4.00000 0.167542
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) −16.0000 −0.668994
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 1.00000 0.0415945
\(579\) 6.00000 0.249351
\(580\) 12.0000 0.498273
\(581\) 0 0
\(582\) −8.00000 −0.331611
\(583\) 24.0000 0.993978
\(584\) −2.00000 −0.0827606
\(585\) −8.00000 −0.330759
\(586\) −6.00000 −0.247858
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 24.0000 0.988064
\(591\) −26.0000 −1.06950
\(592\) −2.00000 −0.0821995
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 12.0000 0.491127
\(598\) −4.00000 −0.163572
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000 0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 4.00000 0.162758
\(605\) −10.0000 −0.406558
\(606\) 16.0000 0.649956
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) −40.0000 −1.61823
\(612\) 4.00000 0.161690
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 12.0000 0.482711
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 12.0000 0.481932
\(621\) 1.00000 0.0401286
\(622\) −30.0000 −1.20289
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) −19.0000 −0.760000
\(626\) −20.0000 −0.799361
\(627\) −8.00000 −0.319489
\(628\) −10.0000 −0.399043
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000 0.318223
\(633\) 4.00000 0.158986
\(634\) −30.0000 −1.19145
\(635\) −40.0000 −1.58735
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −24.0000 −0.950169
\(639\) −4.00000 −0.158238
\(640\) 2.00000 0.0790569
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 20.0000 0.789337
\(643\) −10.0000 −0.394362 −0.197181 0.980367i \(-0.563179\pi\)
−0.197181 + 0.980367i \(0.563179\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) −8.00000 −0.314756
\(647\) −14.0000 −0.550397 −0.275198 0.961387i \(-0.588744\pi\)
−0.275198 + 0.961387i \(0.588744\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −48.0000 −1.88416
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −24.0000 −0.937758
\(656\) 10.0000 0.390434
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 8.00000 0.311400
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) 4.00000 0.155464
\(663\) 16.0000 0.621389
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) −6.00000 −0.232321
\(668\) 18.0000 0.696441
\(669\) 14.0000 0.541271
\(670\) 16.0000 0.618134
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 10.0000 0.385186
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) 10.0000 0.383201
\(682\) −24.0000 −0.919007
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 2.00000 0.0764719
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 8.00000 0.304997
\(689\) −24.0000 −0.914327
\(690\) 2.00000 0.0761387
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −16.0000 −0.608229
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) −32.0000 −1.21383
\(696\) 6.00000 0.227429
\(697\) 40.0000 1.51511
\(698\) −16.0000 −0.605609
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −4.00000 −0.150970
\(703\) −4.00000 −0.150863
\(704\) −4.00000 −0.150756
\(705\) 20.0000 0.753244
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) −8.00000 −0.300235
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 32.0000 1.19673
\(716\) 4.00000 0.149487
\(717\) 20.0000 0.746914
\(718\) −16.0000 −0.597115
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) −12.0000 −0.446285
\(724\) 2.00000 0.0743294
\(725\) 6.00000 0.222834
\(726\) −5.00000 −0.185567
\(727\) −20.0000 −0.741759 −0.370879 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) 32.0000 1.18356
\(732\) 10.0000 0.369611
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −32.0000 −1.17874
\(738\) −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 \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 6.00000 0.219971
\(745\) −12.0000 −0.439646
\(746\) −6.00000 −0.219676
\(747\) 6.00000 0.219529
\(748\) −16.0000 −0.585018
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) −10.0000 −0.364662
\(753\) −26.0000 −0.947493
\(754\) 24.0000 0.874028
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 28.0000 1.01701
\(759\) −4.00000 −0.145191
\(760\) 4.00000 0.145095
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) −20.0000 −0.724524
\(763\) 0 0
\(764\) 16.0000 0.578860
\(765\) −8.00000 −0.289241
\(766\) 32.0000 1.15621
\(767\) 48.0000 1.73318
\(768\) 1.00000 0.0360844
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 6.00000 0.215945
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −8.00000 −0.287554
\(775\) 6.00000 0.215526
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 20.0000 0.716574
\(780\) −8.00000 −0.286446
\(781\) 16.0000 0.572525
\(782\) −4.00000 −0.143040
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) −12.0000 −0.428026
\(787\) 6.00000 0.213877 0.106938 0.994266i \(-0.465895\pi\)
0.106938 + 0.994266i \(0.465895\pi\)
\(788\) −26.0000 −0.926212
\(789\) 16.0000 0.569615
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) 40.0000 1.42044
\(794\) 28.0000 0.993683
\(795\) 12.0000 0.425596
\(796\) 12.0000 0.425329
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 2.00000 0.0706225
\(803\) −8.00000 −0.282314
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) 0 0
\(808\) 16.0000 0.562878
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 2.00000 0.0702728
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 26.0000 0.911860
\(814\) −8.00000 −0.280400
\(815\) −24.0000 −0.840683
\(816\) 4.00000 0.140028
\(817\) 16.0000 0.559769
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) −20.0000 −0.698430
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) −6.00000 −0.209274
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 12.0000 0.418040
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 1.00000 0.0347524
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 12.0000 0.416526
\(831\) 14.0000 0.485655
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) −36.0000 −1.24583
\(836\) −8.00000 −0.276686
\(837\) −6.00000 −0.207390
\(838\) −22.0000 −0.759977
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) 14.0000 0.482186
\(844\) 4.00000 0.137686
\(845\) −6.00000 −0.206406
\(846\) 10.0000 0.343807
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 18.0000 0.617758
\(850\) 4.00000 0.137199
\(851\) −2.00000 −0.0685591
\(852\) −4.00000 −0.137038
\(853\) −12.0000 −0.410872 −0.205436 0.978671i \(-0.565861\pi\)
−0.205436 + 0.978671i \(0.565861\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 20.0000 0.683586
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 16.0000 0.546231
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 32.0000 1.08803
\(866\) −16.0000 −0.543702
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) −12.0000 −0.406838
\(871\) 32.0000 1.08428
\(872\) −2.00000 −0.0677285
\(873\) 8.00000 0.270759
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −14.0000 −0.472477
\(879\) 6.00000 0.202375
\(880\) 8.00000 0.269680
\(881\) −20.0000 −0.673817 −0.336909 0.941537i \(-0.609381\pi\)
−0.336909 + 0.941537i \(0.609381\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 16.0000 0.538138
\(885\) −24.0000 −0.806751
\(886\) 4.00000 0.134383
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 14.0000 0.468755
\(893\) −20.0000 −0.669274
\(894\) −6.00000 −0.200670
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) −14.0000 −0.467186
\(899\) 36.0000 1.20067
\(900\) −1.00000 −0.0333333
\(901\) −24.0000 −0.799556
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −4.00000 −0.132964
\(906\) −4.00000 −0.132891
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 10.0000 0.331862
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 2.00000 0.0662266
\(913\) −24.0000 −0.794284
\(914\) −18.0000 −0.595387
\(915\) −20.0000 −0.661180
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) −24.0000 −0.790398
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 32.0000 1.05159
\(927\) −12.0000 −0.394132
\(928\) 6.00000 0.196960
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) −12.0000 −0.393496
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 30.0000 0.982156
\(934\) 22.0000 0.719862
\(935\) 32.0000 1.04651
\(936\) −4.00000 −0.130744
\(937\) −4.00000 −0.130674 −0.0653372 0.997863i \(-0.520812\pi\)
−0.0653372 + 0.997863i \(0.520812\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) 20.0000 0.652328
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 10.0000 0.325818
\(943\) 10.0000 0.325645
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) −8.00000 −0.259828
\(949\) 8.00000 0.259691
\(950\) 2.00000 0.0648886
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 6.00000 0.194257
\(955\) −32.0000 −1.03550
\(956\) 20.0000 0.646846
\(957\) 24.0000 0.775810
\(958\) 4.00000 0.129234
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) 5.00000 0.161290
\(962\) 8.00000 0.257930
\(963\) −20.0000 −0.644491
\(964\) −12.0000 −0.386494
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) −5.00000 −0.160706
\(969\) 8.00000 0.256997
\(970\) 16.0000 0.513729
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) −4.00000 −0.128103
\(976\) 10.0000 0.320092
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) −12.0000 −0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 28.0000 0.893516
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −10.0000 −0.318788
\(985\) 52.0000 1.65686
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 8.00000 0.254385
\(990\) −8.00000 −0.254257
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 6.00000 0.190500
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 6.00000 0.190117
\(997\) 52.0000 1.64686 0.823428 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) −28.0000 −0.886325
\(999\) −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 6762.2.a.m.1.1 1
7.6 odd 2 966.2.a.c.1.1 1
21.20 even 2 2898.2.a.l.1.1 1
28.27 even 2 7728.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.c.1.1 1 7.6 odd 2
2898.2.a.l.1.1 1 21.20 even 2
6762.2.a.m.1.1 1 1.1 even 1 trivial
7728.2.a.t.1.1 1 28.27 even 2