Properties

Label 6762.2.a.bb.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -7.00000 q^{29} -1.00000 q^{30} +7.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} -6.00000 q^{43} -3.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +2.00000 q^{57} -7.00000 q^{58} -9.00000 q^{59} -1.00000 q^{60} +6.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +3.00000 q^{66} +10.0000 q^{67} -2.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} +4.00000 q^{75} -2.00000 q^{76} -2.00000 q^{78} -15.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +1.00000 q^{83} -2.00000 q^{85} -6.00000 q^{86} +7.00000 q^{87} -3.00000 q^{88} +1.00000 q^{90} +1.00000 q^{92} -7.00000 q^{93} +6.00000 q^{94} -2.00000 q^{95} -1.00000 q^{96} +5.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) −1.00000 −0.182574
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) −2.00000 −0.342997
\(35\) 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
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −3.00000 −0.452267
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 2.00000 0.280056
\(52\) 2.00000 0.277350
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −7.00000 −0.919145
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 3.00000 0.369274
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) −2.00000 −0.242536
\(69\) −1.00000 −0.120386
\(70\) 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
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 4.00000 0.461880
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −6.00000 −0.646997
\(87\) 7.00000 0.750479
\(88\) −3.00000 −0.319801
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −7.00000 −0.725866
\(94\) 6.00000 0.618853
\(95\) −2.00000 −0.205196
\(96\) −1.00000 −0.102062
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) −4.00000 −0.400000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) −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
\(106\) −9.00000 −0.874157
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −3.00000 −0.286039
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 2.00000 0.187317
\(115\) 1.00000 0.0932505
\(116\) −7.00000 −0.649934
\(117\) 2.00000 0.184900
\(118\) −9.00000 −0.828517
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) 6.00000 0.543214
\(123\) 2.00000 0.180334
\(124\) 7.00000 0.628619
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.00000 0.528271
\(130\) 2.00000 0.175412
\(131\) 7.00000 0.611593 0.305796 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) −1.00000 −0.0860663
\(136\) −2.00000 −0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −8.00000 −0.671345
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) −7.00000 −0.581318
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 4.00000 0.326599
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) −2.00000 −0.162221
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 7.00000 0.562254
\(156\) −2.00000 −0.160128
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −15.0000 −1.19334
\(159\) 9.00000 0.713746
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −2.00000 −0.156174
\(165\) 3.00000 0.233550
\(166\) 1.00000 0.0776151
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) −2.00000 −0.152944
\(172\) −6.00000 −0.457496
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 7.00000 0.530669
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 9.00000 0.676481
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 1.00000 0.0737210
\(185\) −2.00000 −0.147043
\(186\) −7.00000 −0.513265
\(187\) 6.00000 0.438763
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −23.0000 −1.65558 −0.827788 0.561041i \(-0.810401\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 5.00000 0.358979
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) −3.00000 −0.213201
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −4.00000 −0.282843
\(201\) −10.0000 −0.705346
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) −2.00000 −0.139686
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) −9.00000 −0.618123
\(213\) 8.00000 0.548151
\(214\) 3.00000 0.205076
\(215\) −6.00000 −0.409197
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 10.0000 0.675737
\(220\) −3.00000 −0.202260
\(221\) −4.00000 −0.269069
\(222\) 2.00000 0.134231
\(223\) 13.0000 0.870544 0.435272 0.900299i \(-0.356652\pi\)
0.435272 + 0.900299i \(0.356652\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 10.0000 0.665190
\(227\) −9.00000 −0.597351 −0.298675 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 2.00000 0.132453
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) −7.00000 −0.459573
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) 6.00000 0.391397
\(236\) −9.00000 −0.585850
\(237\) 15.0000 0.974355
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −27.0000 −1.73922 −0.869611 0.493737i \(-0.835631\pi\)
−0.869611 + 0.493737i \(0.835631\pi\)
\(242\) −2.00000 −0.128565
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) −4.00000 −0.254514
\(248\) 7.00000 0.444500
\(249\) −1.00000 −0.0633724
\(250\) −9.00000 −0.569210
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) −7.00000 −0.439219
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −4.00000 −0.249513 −0.124757 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(258\) 6.00000 0.373544
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) −7.00000 −0.433289
\(262\) 7.00000 0.432461
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 3.00000 0.184637
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) −27.0000 −1.64622 −0.823110 0.567883i \(-0.807763\pi\)
−0.823110 + 0.567883i \(0.807763\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −9.00000 −0.546711 −0.273356 0.961913i \(-0.588134\pi\)
−0.273356 + 0.961913i \(0.588134\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 12.0000 0.723627
\(276\) −1.00000 −0.0601929
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 14.0000 0.839664
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) −6.00000 −0.357295
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) −8.00000 −0.474713
\(285\) 2.00000 0.118470
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −7.00000 −0.411054
\(291\) −5.00000 −0.293105
\(292\) −10.0000 −0.585206
\(293\) 19.0000 1.10999 0.554996 0.831853i \(-0.312720\pi\)
0.554996 + 0.831853i \(0.312720\pi\)
\(294\) 0 0
\(295\) −9.00000 −0.524000
\(296\) −2.00000 −0.116248
\(297\) 3.00000 0.174078
\(298\) −22.0000 −1.27443
\(299\) 2.00000 0.115663
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) −5.00000 −0.287718
\(303\) 10.0000 0.574485
\(304\) −2.00000 −0.114708
\(305\) 6.00000 0.343559
\(306\) −2.00000 −0.114332
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 7.00000 0.397573
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) −2.00000 −0.113228
\(313\) −25.0000 −1.41308 −0.706542 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −15.0000 −0.843816
\(317\) −31.0000 −1.74113 −0.870567 0.492050i \(-0.836248\pi\)
−0.870567 + 0.492050i \(0.836248\pi\)
\(318\) 9.00000 0.504695
\(319\) 21.0000 1.17577
\(320\) 1.00000 0.0559017
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) −8.00000 −0.443760
\(326\) 2.00000 0.110770
\(327\) −10.0000 −0.553001
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 3.00000 0.165145
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 1.00000 0.0548821
\(333\) −2.00000 −0.109599
\(334\) 6.00000 0.328305
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) −9.00000 −0.489535
\(339\) −10.0000 −0.543125
\(340\) −2.00000 −0.108465
\(341\) −21.0000 −1.13721
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) −1.00000 −0.0538382
\(346\) −6.00000 −0.322562
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 7.00000 0.375239
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −3.00000 −0.159901
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 9.00000 0.478345
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 1.00000 0.0527046
\(361\) −15.0000 −0.789474
\(362\) 16.0000 0.840941
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) −6.00000 −0.313625
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) 1.00000 0.0521286
\(369\) −2.00000 −0.104116
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) −7.00000 −0.362933
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 6.00000 0.310253
\(375\) 9.00000 0.464758
\(376\) 6.00000 0.309426
\(377\) −14.0000 −0.721037
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −2.00000 −0.102598
\(381\) 7.00000 0.358621
\(382\) 10.0000 0.511645
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −23.0000 −1.17067
\(387\) −6.00000 −0.304997
\(388\) 5.00000 0.253837
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) −2.00000 −0.101274
\(391\) −2.00000 −0.101144
\(392\) 0 0
\(393\) −7.00000 −0.353103
\(394\) 14.0000 0.705310
\(395\) −15.0000 −0.754732
\(396\) −3.00000 −0.150756
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) −10.0000 −0.498755
\(403\) 14.0000 0.697390
\(404\) −10.0000 −0.497519
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 2.00000 0.0990148
\(409\) −29.0000 −1.43396 −0.716979 0.697095i \(-0.754476\pi\)
−0.716979 + 0.697095i \(0.754476\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 6.00000 0.295958
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 1.00000 0.0490881
\(416\) 2.00000 0.0980581
\(417\) −14.0000 −0.685583
\(418\) 6.00000 0.293470
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 6.00000 0.292075
\(423\) 6.00000 0.291730
\(424\) −9.00000 −0.437079
\(425\) 8.00000 0.388057
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) 6.00000 0.289683
\(430\) −6.00000 −0.289346
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 38.0000 1.82616 0.913082 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) 0 0
\(435\) 7.00000 0.335624
\(436\) 10.0000 0.478913
\(437\) −2.00000 −0.0956730
\(438\) 10.0000 0.477818
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −29.0000 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 13.0000 0.615568
\(447\) 22.0000 1.04056
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −4.00000 −0.188562
\(451\) 6.00000 0.282529
\(452\) 10.0000 0.470360
\(453\) 5.00000 0.234920
\(454\) −9.00000 −0.422391
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 22.0000 1.02799
\(459\) 2.00000 0.0933520
\(460\) 1.00000 0.0466252
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −7.00000 −0.324967
\(465\) −7.00000 −0.324617
\(466\) −6.00000 −0.277945
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 6.00000 0.276759
\(471\) −6.00000 −0.276465
\(472\) −9.00000 −0.414259
\(473\) 18.0000 0.827641
\(474\) 15.0000 0.688973
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) −12.0000 −0.548867
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −4.00000 −0.182384
\(482\) −27.0000 −1.22982
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 5.00000 0.227038
\(486\) −1.00000 −0.0453609
\(487\) −15.0000 −0.679715 −0.339857 0.940477i \(-0.610379\pi\)
−0.339857 + 0.940477i \(0.610379\pi\)
\(488\) 6.00000 0.271607
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) 19.0000 0.857458 0.428729 0.903433i \(-0.358962\pi\)
0.428729 + 0.903433i \(0.358962\pi\)
\(492\) 2.00000 0.0901670
\(493\) 14.0000 0.630528
\(494\) −4.00000 −0.179969
\(495\) −3.00000 −0.134840
\(496\) 7.00000 0.314309
\(497\) 0 0
\(498\) −1.00000 −0.0448111
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) −9.00000 −0.402492
\(501\) −6.00000 −0.268060
\(502\) −3.00000 −0.133897
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) −3.00000 −0.133366
\(507\) 9.00000 0.399704
\(508\) −7.00000 −0.310575
\(509\) 17.0000 0.753512 0.376756 0.926313i \(-0.377040\pi\)
0.376756 + 0.926313i \(0.377040\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −4.00000 −0.176432
\(515\) −8.00000 −0.352522
\(516\) 6.00000 0.264135
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 2.00000 0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −7.00000 −0.306382
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 7.00000 0.305796
\(525\) 0 0
\(526\) −2.00000 −0.0872041
\(527\) −14.0000 −0.609850
\(528\) 3.00000 0.130558
\(529\) 1.00000 0.0434783
\(530\) −9.00000 −0.390935
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 3.00000 0.129701
\(536\) 10.0000 0.431934
\(537\) 12.0000 0.517838
\(538\) −27.0000 −1.16405
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −9.00000 −0.386583
\(543\) −16.0000 −0.686626
\(544\) −2.00000 −0.0857493
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −38.0000 −1.62476 −0.812381 0.583127i \(-0.801829\pi\)
−0.812381 + 0.583127i \(0.801829\pi\)
\(548\) −6.00000 −0.256307
\(549\) 6.00000 0.256074
\(550\) 12.0000 0.511682
\(551\) 14.0000 0.596420
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) 2.00000 0.0848953
\(556\) 14.0000 0.593732
\(557\) 33.0000 1.39825 0.699127 0.714997i \(-0.253572\pi\)
0.699127 + 0.714997i \(0.253572\pi\)
\(558\) 7.00000 0.296334
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) −4.00000 −0.168730
\(563\) −3.00000 −0.126435 −0.0632175 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(564\) −6.00000 −0.252646
\(565\) 10.0000 0.420703
\(566\) 6.00000 0.252199
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 2.00000 0.0837708
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −6.00000 −0.250873
\(573\) −10.0000 −0.417756
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −33.0000 −1.37381 −0.686904 0.726748i \(-0.741031\pi\)
−0.686904 + 0.726748i \(0.741031\pi\)
\(578\) −13.0000 −0.540729
\(579\) 23.0000 0.955847
\(580\) −7.00000 −0.290659
\(581\) 0 0
\(582\) −5.00000 −0.207257
\(583\) 27.0000 1.11823
\(584\) −10.0000 −0.413803
\(585\) 2.00000 0.0826898
\(586\) 19.0000 0.784883
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) −9.00000 −0.370524
\(591\) −14.0000 −0.575883
\(592\) −2.00000 −0.0821995
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) 4.00000 0.163709
\(598\) 2.00000 0.0817861
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 4.00000 0.163299
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) −5.00000 −0.203447
\(605\) −2.00000 −0.0813116
\(606\) 10.0000 0.406222
\(607\) 21.0000 0.852364 0.426182 0.904638i \(-0.359858\pi\)
0.426182 + 0.904638i \(0.359858\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) 12.0000 0.485468
\(612\) −2.00000 −0.0808452
\(613\) 36.0000 1.45403 0.727013 0.686624i \(-0.240908\pi\)
0.727013 + 0.686624i \(0.240908\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 2.00000 0.0806478
\(616\) 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
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 7.00000 0.281127
\(621\) −1.00000 −0.0401286
\(622\) −2.00000 −0.0801927
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 11.0000 0.440000
\(626\) −25.0000 −0.999201
\(627\) −6.00000 −0.239617
\(628\) 6.00000 0.239426
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 31.0000 1.23409 0.617045 0.786928i \(-0.288330\pi\)
0.617045 + 0.786928i \(0.288330\pi\)
\(632\) −15.0000 −0.596668
\(633\) −6.00000 −0.238479
\(634\) −31.0000 −1.23117
\(635\) −7.00000 −0.277787
\(636\) 9.00000 0.356873
\(637\) 0 0
\(638\) 21.0000 0.831398
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) −3.00000 −0.118401
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 4.00000 0.157378
\(647\) 38.0000 1.49393 0.746967 0.664861i \(-0.231509\pi\)
0.746967 + 0.664861i \(0.231509\pi\)
\(648\) 1.00000 0.0392837
\(649\) 27.0000 1.05984
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) 11.0000 0.430463 0.215232 0.976563i \(-0.430949\pi\)
0.215232 + 0.976563i \(0.430949\pi\)
\(654\) −10.0000 −0.391031
\(655\) 7.00000 0.273513
\(656\) −2.00000 −0.0780869
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 3.00000 0.116775
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 10.0000 0.388661
\(663\) 4.00000 0.155347
\(664\) 1.00000 0.0388075
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −7.00000 −0.271041
\(668\) 6.00000 0.232147
\(669\) −13.0000 −0.502609
\(670\) 10.0000 0.386334
\(671\) −18.0000 −0.694882
\(672\) 0 0
\(673\) 25.0000 0.963679 0.481840 0.876259i \(-0.339969\pi\)
0.481840 + 0.876259i \(0.339969\pi\)
\(674\) −9.00000 −0.346667
\(675\) 4.00000 0.153960
\(676\) −9.00000 −0.346154
\(677\) −35.0000 −1.34516 −0.672580 0.740025i \(-0.734814\pi\)
−0.672580 + 0.740025i \(0.734814\pi\)
\(678\) −10.0000 −0.384048
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 9.00000 0.344881
\(682\) −21.0000 −0.804132
\(683\) −23.0000 −0.880071 −0.440035 0.897980i \(-0.645034\pi\)
−0.440035 + 0.897980i \(0.645034\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) −6.00000 −0.228748
\(689\) −18.0000 −0.685745
\(690\) −1.00000 −0.0380693
\(691\) −22.0000 −0.836919 −0.418460 0.908235i \(-0.637430\pi\)
−0.418460 + 0.908235i \(0.637430\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 36.0000 1.36654
\(695\) 14.0000 0.531050
\(696\) 7.00000 0.265334
\(697\) 4.00000 0.151511
\(698\) 16.0000 0.605609
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −1.00000 −0.0377695 −0.0188847 0.999822i \(-0.506012\pi\)
−0.0188847 + 0.999822i \(0.506012\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 4.00000 0.150863
\(704\) −3.00000 −0.113067
\(705\) −6.00000 −0.225973
\(706\) −26.0000 −0.978523
\(707\) 0 0
\(708\) 9.00000 0.338241
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) −8.00000 −0.300235
\(711\) −15.0000 −0.562544
\(712\) 0 0
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) 30.0000 1.11959
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) 27.0000 1.00414
\(724\) 16.0000 0.594635
\(725\) 28.0000 1.03989
\(726\) 2.00000 0.0742270
\(727\) 5.00000 0.185440 0.0927199 0.995692i \(-0.470444\pi\)
0.0927199 + 0.995692i \(0.470444\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) 12.0000 0.443836
\(732\) −6.00000 −0.221766
\(733\) −28.0000 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) 7.00000 0.258375
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −30.0000 −1.10506
\(738\) −2.00000 −0.0736210
\(739\) −14.0000 −0.514998 −0.257499 0.966279i \(-0.582898\pi\)
−0.257499 + 0.966279i \(0.582898\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) −7.00000 −0.256632
\(745\) −22.0000 −0.806018
\(746\) 32.0000 1.17160
\(747\) 1.00000 0.0365881
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 9.00000 0.328634
\(751\) 11.0000 0.401396 0.200698 0.979653i \(-0.435679\pi\)
0.200698 + 0.979653i \(0.435679\pi\)
\(752\) 6.00000 0.218797
\(753\) 3.00000 0.109326
\(754\) −14.0000 −0.509850
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −4.00000 −0.145287
\(759\) 3.00000 0.108893
\(760\) −2.00000 −0.0725476
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 7.00000 0.253583
\(763\) 0 0
\(764\) 10.0000 0.361787
\(765\) −2.00000 −0.0723102
\(766\) 8.00000 0.289052
\(767\) −18.0000 −0.649942
\(768\) −1.00000 −0.0360844
\(769\) 15.0000 0.540914 0.270457 0.962732i \(-0.412825\pi\)
0.270457 + 0.962732i \(0.412825\pi\)
\(770\) 0 0
\(771\) 4.00000 0.144056
\(772\) −23.0000 −0.827788
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) −6.00000 −0.215666
\(775\) −28.0000 −1.00579
\(776\) 5.00000 0.179490
\(777\) 0 0
\(778\) 38.0000 1.36237
\(779\) 4.00000 0.143315
\(780\) −2.00000 −0.0716115
\(781\) 24.0000 0.858788
\(782\) −2.00000 −0.0715199
\(783\) 7.00000 0.250160
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) −7.00000 −0.249682
\(787\) −14.0000 −0.499046 −0.249523 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 14.0000 0.498729
\(789\) 2.00000 0.0712019
\(790\) −15.0000 −0.533676
\(791\) 0 0
\(792\) −3.00000 −0.106600
\(793\) 12.0000 0.426132
\(794\) −38.0000 −1.34857
\(795\) 9.00000 0.319197
\(796\) −4.00000 −0.141776
\(797\) −15.0000 −0.531327 −0.265664 0.964066i \(-0.585591\pi\)
−0.265664 + 0.964066i \(0.585591\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) 4.00000 0.141245
\(803\) 30.0000 1.05868
\(804\) −10.0000 −0.352673
\(805\) 0 0
\(806\) 14.0000 0.493129
\(807\) 27.0000 0.950445
\(808\) −10.0000 −0.351799
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 1.00000 0.0351364
\(811\) 46.0000 1.61528 0.807639 0.589677i \(-0.200745\pi\)
0.807639 + 0.589677i \(0.200745\pi\)
\(812\) 0 0
\(813\) 9.00000 0.315644
\(814\) 6.00000 0.210300
\(815\) 2.00000 0.0700569
\(816\) 2.00000 0.0700140
\(817\) 12.0000 0.419827
\(818\) −29.0000 −1.01396
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) 17.0000 0.593304 0.296652 0.954986i \(-0.404130\pi\)
0.296652 + 0.954986i \(0.404130\pi\)
\(822\) 6.00000 0.209274
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −8.00000 −0.278693
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 5.00000 0.173867 0.0869335 0.996214i \(-0.472293\pi\)
0.0869335 + 0.996214i \(0.472293\pi\)
\(828\) 1.00000 0.0347524
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 1.00000 0.0347105
\(831\) 8.00000 0.277517
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) −14.0000 −0.484780
\(835\) 6.00000 0.207639
\(836\) 6.00000 0.207514
\(837\) −7.00000 −0.241955
\(838\) −36.0000 −1.24360
\(839\) −26.0000 −0.897620 −0.448810 0.893627i \(-0.648152\pi\)
−0.448810 + 0.893627i \(0.648152\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) −18.0000 −0.620321
\(843\) 4.00000 0.137767
\(844\) 6.00000 0.206529
\(845\) −9.00000 −0.309609
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) −6.00000 −0.205919
\(850\) 8.00000 0.274398
\(851\) −2.00000 −0.0685591
\(852\) 8.00000 0.274075
\(853\) 54.0000 1.84892 0.924462 0.381273i \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 3.00000 0.102538
\(857\) 16.0000 0.546550 0.273275 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 6.00000 0.204837
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −6.00000 −0.204006
\(866\) 38.0000 1.29129
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 45.0000 1.52652
\(870\) 7.00000 0.237322
\(871\) 20.0000 0.677674
\(872\) 10.0000 0.338643
\(873\) 5.00000 0.169224
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 35.0000 1.18119
\(879\) −19.0000 −0.640854
\(880\) −3.00000 −0.101130
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) −4.00000 −0.134535
\(885\) 9.00000 0.302532
\(886\) −29.0000 −0.974274
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 13.0000 0.435272
\(893\) −12.0000 −0.401565
\(894\) 22.0000 0.735790
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) −30.0000 −1.00111
\(899\) −49.0000 −1.63424
\(900\) −4.00000 −0.133333
\(901\) 18.0000 0.599667
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 16.0000 0.531858
\(906\) 5.00000 0.166114
\(907\) 46.0000 1.52740 0.763702 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) −9.00000 −0.298675
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 22.0000 0.728893 0.364446 0.931224i \(-0.381258\pi\)
0.364446 + 0.931224i \(0.381258\pi\)
\(912\) 2.00000 0.0662266
\(913\) −3.00000 −0.0992855
\(914\) 1.00000 0.0330771
\(915\) −6.00000 −0.198354
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 1.00000 0.0329690
\(921\) 2.00000 0.0659022
\(922\) 2.00000 0.0658665
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −8.00000 −0.262896
\(927\) −8.00000 −0.262754
\(928\) −7.00000 −0.229786
\(929\) 32.0000 1.04989 0.524943 0.851137i \(-0.324087\pi\)
0.524943 + 0.851137i \(0.324087\pi\)
\(930\) −7.00000 −0.229539
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 2.00000 0.0654771
\(934\) 4.00000 0.130884
\(935\) 6.00000 0.196221
\(936\) 2.00000 0.0653720
\(937\) 29.0000 0.947389 0.473694 0.880689i \(-0.342920\pi\)
0.473694 + 0.880689i \(0.342920\pi\)
\(938\) 0 0
\(939\) 25.0000 0.815844
\(940\) 6.00000 0.195698
\(941\) −15.0000 −0.488986 −0.244493 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) −6.00000 −0.195491
\(943\) −2.00000 −0.0651290
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 18.0000 0.585230
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 15.0000 0.487177
\(949\) −20.0000 −0.649227
\(950\) 8.00000 0.259554
\(951\) 31.0000 1.00524
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) −9.00000 −0.291386
\(955\) 10.0000 0.323592
\(956\) −12.0000 −0.388108
\(957\) −21.0000 −0.678834
\(958\) 28.0000 0.904639
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 18.0000 0.580645
\(962\) −4.00000 −0.128965
\(963\) 3.00000 0.0966736
\(964\) −27.0000 −0.869611
\(965\) −23.0000 −0.740396
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −4.00000 −0.128499
\(970\) 5.00000 0.160540
\(971\) −17.0000 −0.545556 −0.272778 0.962077i \(-0.587942\pi\)
−0.272778 + 0.962077i \(0.587942\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −15.0000 −0.480631
\(975\) 8.00000 0.256205
\(976\) 6.00000 0.192055
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 19.0000 0.606314
\(983\) −50.0000 −1.59475 −0.797376 0.603483i \(-0.793779\pi\)
−0.797376 + 0.603483i \(0.793779\pi\)
\(984\) 2.00000 0.0637577
\(985\) 14.0000 0.446077
\(986\) 14.0000 0.445851
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −6.00000 −0.190789
\(990\) −3.00000 −0.0953463
\(991\) 3.00000 0.0952981 0.0476491 0.998864i \(-0.484827\pi\)
0.0476491 + 0.998864i \(0.484827\pi\)
\(992\) 7.00000 0.222250
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) −1.00000 −0.0316862
\(997\) 36.0000 1.14013 0.570066 0.821599i \(-0.306918\pi\)
0.570066 + 0.821599i \(0.306918\pi\)
\(998\) 6.00000 0.189927
\(999\) 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.bb.1.1 1
7.2 even 3 966.2.i.c.277.1 2
7.4 even 3 966.2.i.c.415.1 yes 2
7.6 odd 2 6762.2.a.bh.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.c.277.1 2 7.2 even 3
966.2.i.c.415.1 yes 2 7.4 even 3
6762.2.a.bb.1.1 1 1.1 even 1 trivial
6762.2.a.bh.1.1 1 7.6 odd 2