Properties

Label 6762.2.a.x.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -3.00000 q^{20} +1.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -7.00000 q^{29} +3.00000 q^{30} +7.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} -10.0000 q^{41} +2.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +11.0000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +2.00000 q^{57} -7.00000 q^{58} +15.0000 q^{59} +3.00000 q^{60} -2.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} -1.00000 q^{66} +2.00000 q^{67} -2.00000 q^{68} +1.00000 q^{69} +1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -2.00000 q^{76} +6.00000 q^{78} -11.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +13.0000 q^{83} +6.00000 q^{85} +2.00000 q^{86} +7.00000 q^{87} +1.00000 q^{88} +8.00000 q^{89} -3.00000 q^{90} -1.00000 q^{92} -7.00000 q^{93} +6.00000 q^{94} +6.00000 q^{95} -1.00000 q^{96} +1.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 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 \(0.573682\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 3.00000 0.547723
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −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
\(39\) 6.00000 0.960769
\(40\) −3.00000 −0.474342
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 1.00000 0.150756
\(45\) −3.00000 −0.447214
\(46\) −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
\(52\) −6.00000 −0.832050
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −7.00000 −0.919145
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 3.00000 0.387298
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 7.00000 0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 18.0000 2.23263
\(66\) −1.00000 −0.123091
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −2.00000 −0.242536
\(69\) 1.00000 0.120386
\(70\) 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 \(-0.698986\pi\)
−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
\(78\) 6.00000 0.679366
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 13.0000 1.42694 0.713468 0.700688i \(-0.247124\pi\)
0.713468 + 0.700688i \(0.247124\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 2.00000 0.215666
\(87\) 7.00000 0.750479
\(88\) 1.00000 0.106600
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −7.00000 −0.725866
\(94\) 6.00000 0.618853
\(95\) 6.00000 0.615587
\(96\) −1.00000 −0.102062
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 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
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) −17.0000 −1.64345 −0.821726 0.569883i \(-0.806989\pi\)
−0.821726 + 0.569883i \(0.806989\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −3.00000 −0.286039
\(111\) 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
\(115\) 3.00000 0.279751
\(116\) −7.00000 −0.649934
\(117\) −6.00000 −0.554700
\(118\) 15.0000 1.38086
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) −10.0000 −0.909091
\(122\) −2.00000 −0.181071
\(123\) 10.0000 0.901670
\(124\) 7.00000 0.628619
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.00000 −0.176090
\(130\) 18.0000 1.57870
\(131\) 7.00000 0.611593 0.305796 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 3.00000 0.258199
\(136\) −2.00000 −0.171499
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 1.00000 0.0851257
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 21.0000 1.74396
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −4.00000 −0.326599
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) −2.00000 −0.162221
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −21.0000 −1.68676
\(156\) 6.00000 0.480384
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −11.0000 −0.875113
\(159\) −11.0000 −0.872357
\(160\) −3.00000 −0.237171
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) −10.0000 −0.780869
\(165\) 3.00000 0.233550
\(166\) 13.0000 1.00900
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 6.00000 0.460179
\(171\) −2.00000 −0.152944
\(172\) 2.00000 0.152499
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 7.00000 0.530669
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −15.0000 −1.12747
\(178\) 8.00000 0.599625
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −3.00000 −0.223607
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −1.00000 −0.0737210
\(185\) 6.00000 0.441129
\(186\) −7.00000 −0.513265
\(187\) −2.00000 −0.146254
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) 26.0000 1.88129 0.940647 0.339387i \(-0.110219\pi\)
0.940647 + 0.339387i \(0.110219\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −23.0000 −1.65558 −0.827788 0.561041i \(-0.810401\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 1.00000 0.0717958
\(195\) −18.0000 −1.28901
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 1.00000 0.0710669
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 4.00000 0.282843
\(201\) −2.00000 −0.141069
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 30.0000 2.09529
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) −6.00000 −0.416025
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 11.0000 0.755483
\(213\) 0 0
\(214\) −17.0000 −1.16210
\(215\) −6.00000 −0.409197
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) 10.0000 0.675737
\(220\) −3.00000 −0.202260
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) 21.0000 1.40626 0.703132 0.711059i \(-0.251784\pi\)
0.703132 + 0.711059i \(0.251784\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −6.00000 −0.399114
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 2.00000 0.132453
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) −7.00000 −0.459573
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −6.00000 −0.392232
\(235\) −18.0000 −1.17419
\(236\) 15.0000 0.976417
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 3.00000 0.193649
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 12.0000 0.763542
\(248\) 7.00000 0.444500
\(249\) −13.0000 −0.823842
\(250\) 3.00000 0.189737
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 17.0000 1.06667
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 18.0000 1.11631
\(261\) −7.00000 −0.433289
\(262\) 7.00000 0.432461
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −33.0000 −2.02717
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 2.00000 0.122169
\(269\) 29.0000 1.76816 0.884081 0.467334i \(-0.154786\pi\)
0.884081 + 0.467334i \(0.154786\pi\)
\(270\) 3.00000 0.182574
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 4.00000 0.241209
\(276\) 1.00000 0.0601929
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −2.00000 −0.119952
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) −6.00000 −0.357295
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) −6.00000 −0.355409
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 21.0000 1.23316
\(291\) −1.00000 −0.0586210
\(292\) −10.0000 −0.585206
\(293\) −1.00000 −0.0584206 −0.0292103 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) −45.0000 −2.62000
\(296\) −2.00000 −0.116248
\(297\) −1.00000 −0.0580259
\(298\) 10.0000 0.579284
\(299\) 6.00000 0.346989
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 19.0000 1.09333
\(303\) 10.0000 0.574485
\(304\) −2.00000 −0.114708
\(305\) 6.00000 0.343559
\(306\) −2.00000 −0.114332
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −21.0000 −1.19272
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 6.00000 0.339683
\(313\) 27.0000 1.52613 0.763065 0.646322i \(-0.223694\pi\)
0.763065 + 0.646322i \(0.223694\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −11.0000 −0.618798
\(317\) −15.0000 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) −11.0000 −0.616849
\(319\) −7.00000 −0.391925
\(320\) −3.00000 −0.167705
\(321\) 17.0000 0.948847
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) −24.0000 −1.33128
\(326\) 10.0000 0.553849
\(327\) 14.0000 0.774202
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 3.00000 0.165145
\(331\) −14.0000 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) 13.0000 0.713468
\(333\) −2.00000 −0.109599
\(334\) 6.00000 0.328305
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 35.0000 1.90657 0.953286 0.302070i \(-0.0976776\pi\)
0.953286 + 0.302070i \(0.0976776\pi\)
\(338\) 23.0000 1.25104
\(339\) 6.00000 0.325875
\(340\) 6.00000 0.325396
\(341\) 7.00000 0.379071
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) −3.00000 −0.161515
\(346\) 10.0000 0.537603
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 7.00000 0.375239
\(349\) −24.0000 −1.28469 −0.642345 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 1.00000 0.0533002
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −15.0000 −0.797241
\(355\) 0 0
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) −3.00000 −0.158114
\(361\) −15.0000 −0.789474
\(362\) −16.0000 −0.840941
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 30.0000 1.57027
\(366\) 2.00000 0.104542
\(367\) 11.0000 0.574195 0.287098 0.957901i \(-0.407310\pi\)
0.287098 + 0.957901i \(0.407310\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −10.0000 −0.520579
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −7.00000 −0.362933
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) −2.00000 −0.103418
\(375\) −3.00000 −0.154919
\(376\) 6.00000 0.309426
\(377\) 42.0000 2.16311
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 6.00000 0.307794
\(381\) −17.0000 −0.870936
\(382\) 26.0000 1.33028
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −23.0000 −1.17067
\(387\) 2.00000 0.101666
\(388\) 1.00000 0.0507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −18.0000 −0.911465
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) −7.00000 −0.353103
\(394\) −2.00000 −0.100759
\(395\) 33.0000 1.66041
\(396\) 1.00000 0.0502519
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −42.0000 −2.09217
\(404\) −10.0000 −0.497519
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 2.00000 0.0990148
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 30.0000 1.48159
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −39.0000 −1.91443
\(416\) −6.00000 −0.294174
\(417\) 2.00000 0.0979404
\(418\) −2.00000 −0.0978232
\(419\) 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 \(0.319894\pi\)
\(422\) 6.00000 0.292075
\(423\) 6.00000 0.291730
\(424\) 11.0000 0.534207
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 0 0
\(428\) −17.0000 −0.821726
\(429\) 6.00000 0.289683
\(430\) −6.00000 −0.289346
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) −21.0000 −1.00687
\(436\) −14.0000 −0.670478
\(437\) 2.00000 0.0956730
\(438\) 10.0000 0.477818
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 11.0000 0.522626 0.261313 0.965254i \(-0.415845\pi\)
0.261313 + 0.965254i \(0.415845\pi\)
\(444\) 2.00000 0.0949158
\(445\) −24.0000 −1.13771
\(446\) 21.0000 0.994379
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 4.00000 0.188562
\(451\) −10.0000 −0.470882
\(452\) −6.00000 −0.282216
\(453\) −19.0000 −0.892698
\(454\) −13.0000 −0.610120
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) −10.0000 −0.467269
\(459\) 2.00000 0.0933520
\(460\) 3.00000 0.139876
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −7.00000 −0.324967
\(465\) 21.0000 0.973852
\(466\) 18.0000 0.833834
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) −18.0000 −0.830278
\(471\) −22.0000 −1.01371
\(472\) 15.0000 0.690431
\(473\) 2.00000 0.0919601
\(474\) 11.0000 0.505247
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 11.0000 0.503655
\(478\) 20.0000 0.914779
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 3.00000 0.136931
\(481\) 12.0000 0.547153
\(482\) −7.00000 −0.318841
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −3.00000 −0.136223
\(486\) −1.00000 −0.0453609
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −29.0000 −1.30875 −0.654376 0.756169i \(-0.727069\pi\)
−0.654376 + 0.756169i \(0.727069\pi\)
\(492\) 10.0000 0.450835
\(493\) 14.0000 0.630528
\(494\) 12.0000 0.539906
\(495\) −3.00000 −0.134840
\(496\) 7.00000 0.314309
\(497\) 0 0
\(498\) −13.0000 −0.582544
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 3.00000 0.134164
\(501\) −6.00000 −0.268060
\(502\) −15.0000 −0.669483
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) 30.0000 1.33498
\(506\) −1.00000 −0.0444554
\(507\) −23.0000 −1.02147
\(508\) 17.0000 0.754253
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) −6.00000 −0.265684
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 18.0000 0.789352
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) −7.00000 −0.306382
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 7.00000 0.305796
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) −14.0000 −0.609850
\(528\) −1.00000 −0.0435194
\(529\) 1.00000 0.0434783
\(530\) −33.0000 −1.43343
\(531\) 15.0000 0.650945
\(532\) 0 0
\(533\) 60.0000 2.59889
\(534\) −8.00000 −0.346194
\(535\) 51.0000 2.20492
\(536\) 2.00000 0.0863868
\(537\) −4.00000 −0.172613
\(538\) 29.0000 1.25028
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) −1.00000 −0.0429537
\(543\) 16.0000 0.686626
\(544\) −2.00000 −0.0857493
\(545\) 42.0000 1.79908
\(546\) 0 0
\(547\) 18.0000 0.769624 0.384812 0.922995i \(-0.374266\pi\)
0.384812 + 0.922995i \(0.374266\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) 14.0000 0.596420
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 8.00000 0.339887
\(555\) −6.00000 −0.254686
\(556\) −2.00000 −0.0848189
\(557\) −27.0000 −1.14403 −0.572013 0.820244i \(-0.693837\pi\)
−0.572013 + 0.820244i \(0.693837\pi\)
\(558\) 7.00000 0.296334
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 20.0000 0.843649
\(563\) 41.0000 1.72794 0.863972 0.503540i \(-0.167969\pi\)
0.863972 + 0.503540i \(0.167969\pi\)
\(564\) −6.00000 −0.252646
\(565\) 18.0000 0.757266
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) −6.00000 −0.251312
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −6.00000 −0.250873
\(573\) −26.0000 −1.08617
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −25.0000 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) −13.0000 −0.540729
\(579\) 23.0000 0.955847
\(580\) 21.0000 0.871978
\(581\) 0 0
\(582\) −1.00000 −0.0414513
\(583\) 11.0000 0.455573
\(584\) −10.0000 −0.413803
\(585\) 18.0000 0.744208
\(586\) −1.00000 −0.0413096
\(587\) −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
\(590\) −45.0000 −1.85262
\(591\) 2.00000 0.0822690
\(592\) −2.00000 −0.0821995
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −20.0000 −0.818546
\(598\) 6.00000 0.245358
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) −4.00000 −0.163299
\(601\) −17.0000 −0.693444 −0.346722 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 19.0000 0.773099
\(605\) 30.0000 1.21967
\(606\) 10.0000 0.406222
\(607\) −19.0000 −0.771186 −0.385593 0.922669i \(-0.626003\pi\)
−0.385593 + 0.922669i \(0.626003\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) −36.0000 −1.45640
\(612\) −2.00000 −0.0808452
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) 6.00000 0.242140
\(615\) −30.0000 −1.20972
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) −21.0000 −0.843380
\(621\) 1.00000 0.0401286
\(622\) 6.00000 0.240578
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) −29.0000 −1.16000
\(626\) 27.0000 1.07914
\(627\) 2.00000 0.0798723
\(628\) 22.0000 0.877896
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −21.0000 −0.835997 −0.417998 0.908448i \(-0.637268\pi\)
−0.417998 + 0.908448i \(0.637268\pi\)
\(632\) −11.0000 −0.437557
\(633\) −6.00000 −0.238479
\(634\) −15.0000 −0.595726
\(635\) −51.0000 −2.02387
\(636\) −11.0000 −0.436178
\(637\) 0 0
\(638\) −7.00000 −0.277133
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −20.0000 −0.789953 −0.394976 0.918691i \(-0.629247\pi\)
−0.394976 + 0.918691i \(0.629247\pi\)
\(642\) 17.0000 0.670936
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 4.00000 0.157378
\(647\) 22.0000 0.864909 0.432455 0.901656i \(-0.357648\pi\)
0.432455 + 0.901656i \(0.357648\pi\)
\(648\) 1.00000 0.0392837
\(649\) 15.0000 0.588802
\(650\) −24.0000 −0.941357
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −37.0000 −1.44792 −0.723961 0.689841i \(-0.757680\pi\)
−0.723961 + 0.689841i \(0.757680\pi\)
\(654\) 14.0000 0.547443
\(655\) −21.0000 −0.820538
\(656\) −10.0000 −0.390434
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 3.00000 0.116775
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) −14.0000 −0.544125
\(663\) −12.0000 −0.466041
\(664\) 13.0000 0.504498
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 7.00000 0.271041
\(668\) 6.00000 0.232147
\(669\) −21.0000 −0.811907
\(670\) −6.00000 −0.231800
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −7.00000 −0.269830 −0.134915 0.990857i \(-0.543076\pi\)
−0.134915 + 0.990857i \(0.543076\pi\)
\(674\) 35.0000 1.34815
\(675\) −4.00000 −0.153960
\(676\) 23.0000 0.884615
\(677\) −23.0000 −0.883962 −0.441981 0.897024i \(-0.645724\pi\)
−0.441981 + 0.897024i \(0.645724\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) 13.0000 0.498161
\(682\) 7.00000 0.268044
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 2.00000 0.0762493
\(689\) −66.0000 −2.51440
\(690\) −3.00000 −0.114208
\(691\) −46.0000 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(692\) 10.0000 0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 6.00000 0.227593
\(696\) 7.00000 0.265334
\(697\) 20.0000 0.757554
\(698\) −24.0000 −0.908413
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −5.00000 −0.188847 −0.0944237 0.995532i \(-0.530101\pi\)
−0.0944237 + 0.995532i \(0.530101\pi\)
\(702\) 6.00000 0.226455
\(703\) 4.00000 0.150863
\(704\) 1.00000 0.0376889
\(705\) 18.0000 0.677919
\(706\) 14.0000 0.526897
\(707\) 0 0
\(708\) −15.0000 −0.563735
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 8.00000 0.299813
\(713\) −7.00000 −0.262152
\(714\) 0 0
\(715\) 18.0000 0.673162
\(716\) 4.00000 0.149487
\(717\) −20.0000 −0.746914
\(718\) 6.00000 0.223918
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) 7.00000 0.260333
\(724\) −16.0000 −0.594635
\(725\) −28.0000 −1.03989
\(726\) 10.0000 0.371135
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 30.0000 1.11035
\(731\) −4.00000 −0.147945
\(732\) 2.00000 0.0739221
\(733\) −20.0000 −0.738717 −0.369358 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) 11.0000 0.406017
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 2.00000 0.0736709
\(738\) −10.0000 −0.368105
\(739\) −14.0000 −0.514998 −0.257499 0.966279i \(-0.582898\pi\)
−0.257499 + 0.966279i \(0.582898\pi\)
\(740\) 6.00000 0.220564
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −7.00000 −0.256632
\(745\) −30.0000 −1.09911
\(746\) 16.0000 0.585802
\(747\) 13.0000 0.475645
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) 47.0000 1.71505 0.857527 0.514439i \(-0.172000\pi\)
0.857527 + 0.514439i \(0.172000\pi\)
\(752\) 6.00000 0.218797
\(753\) 15.0000 0.546630
\(754\) 42.0000 1.52955
\(755\) −57.0000 −2.07444
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) −4.00000 −0.145287
\(759\) 1.00000 0.0362977
\(760\) 6.00000 0.217643
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −17.0000 −0.615845
\(763\) 0 0
\(764\) 26.0000 0.940647
\(765\) 6.00000 0.216930
\(766\) 16.0000 0.578103
\(767\) −90.0000 −3.24971
\(768\) −1.00000 −0.0360844
\(769\) −29.0000 −1.04577 −0.522883 0.852404i \(-0.675144\pi\)
−0.522883 + 0.852404i \(0.675144\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) −23.0000 −0.827788
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 2.00000 0.0718885
\(775\) 28.0000 1.00579
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 20.0000 0.716574
\(780\) −18.0000 −0.644503
\(781\) 0 0
\(782\) 2.00000 0.0715199
\(783\) 7.00000 0.250160
\(784\) 0 0
\(785\) −66.0000 −2.35564
\(786\) −7.00000 −0.249682
\(787\) −54.0000 −1.92489 −0.962446 0.271473i \(-0.912489\pi\)
−0.962446 + 0.271473i \(0.912489\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −6.00000 −0.213606
\(790\) 33.0000 1.17409
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 12.0000 0.426132
\(794\) 2.00000 0.0709773
\(795\) 33.0000 1.17039
\(796\) 20.0000 0.708881
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 4.00000 0.141421
\(801\) 8.00000 0.282666
\(802\) 4.00000 0.141245
\(803\) −10.0000 −0.352892
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) −42.0000 −1.47939
\(807\) −29.0000 −1.02085
\(808\) −10.0000 −0.351799
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) −3.00000 −0.105409
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 1.00000 0.0350715
\(814\) −2.00000 −0.0701000
\(815\) −30.0000 −1.05085
\(816\) 2.00000 0.0700140
\(817\) −4.00000 −0.139942
\(818\) 19.0000 0.664319
\(819\) 0 0
\(820\) 30.0000 1.04765
\(821\) −15.0000 −0.523504 −0.261752 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 12.0000 0.416777 0.208389 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) −39.0000 −1.35371
\(831\) −8.00000 −0.277517
\(832\) −6.00000 −0.208013
\(833\) 0 0
\(834\) 2.00000 0.0692543
\(835\) −18.0000 −0.622916
\(836\) −2.00000 −0.0691714
\(837\) −7.00000 −0.241955
\(838\) 4.00000 0.138178
\(839\) 54.0000 1.86429 0.932144 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 22.0000 0.758170
\(843\) −20.0000 −0.688837
\(844\) 6.00000 0.206529
\(845\) −69.0000 −2.37367
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 11.0000 0.377742
\(849\) −14.0000 −0.480479
\(850\) −8.00000 −0.274398
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) −17.0000 −0.581048
\(857\) 40.0000 1.36637 0.683187 0.730243i \(-0.260593\pi\)
0.683187 + 0.730243i \(0.260593\pi\)
\(858\) 6.00000 0.204837
\(859\) −48.0000 −1.63774 −0.818869 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −30.0000 −1.02003
\(866\) −26.0000 −0.883516
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −11.0000 −0.373149
\(870\) −21.0000 −0.711967
\(871\) −12.0000 −0.406604
\(872\) −14.0000 −0.474100
\(873\) 1.00000 0.0338449
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) −29.0000 −0.978703
\(879\) 1.00000 0.0337292
\(880\) −3.00000 −0.101130
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −54.0000 −1.81724 −0.908622 0.417619i \(-0.862865\pi\)
−0.908622 + 0.417619i \(0.862865\pi\)
\(884\) 12.0000 0.403604
\(885\) 45.0000 1.51266
\(886\) 11.0000 0.369552
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 2.00000 0.0671156
\(889\) 0 0
\(890\) −24.0000 −0.804482
\(891\) 1.00000 0.0335013
\(892\) 21.0000 0.703132
\(893\) −12.0000 −0.401565
\(894\) −10.0000 −0.334450
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) −6.00000 −0.200223
\(899\) −49.0000 −1.63424
\(900\) 4.00000 0.133333
\(901\) −22.0000 −0.732926
\(902\) −10.0000 −0.332964
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 48.0000 1.59557
\(906\) −19.0000 −0.631233
\(907\) 46.0000 1.52740 0.763702 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) −13.0000 −0.431420
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 14.0000 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) 2.00000 0.0662266
\(913\) 13.0000 0.430237
\(914\) −11.0000 −0.363848
\(915\) −6.00000 −0.198354
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 3.00000 0.0989071
\(921\) −6.00000 −0.197707
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) −7.00000 −0.229786
\(929\) −40.0000 −1.31236 −0.656179 0.754606i \(-0.727828\pi\)
−0.656179 + 0.754606i \(0.727828\pi\)
\(930\) 21.0000 0.688617
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −6.00000 −0.196431
\(934\) −20.0000 −0.654420
\(935\) 6.00000 0.196221
\(936\) −6.00000 −0.196116
\(937\) 1.00000 0.0326686 0.0163343 0.999867i \(-0.494800\pi\)
0.0163343 + 0.999867i \(0.494800\pi\)
\(938\) 0 0
\(939\) −27.0000 −0.881112
\(940\) −18.0000 −0.587095
\(941\) 37.0000 1.20617 0.603083 0.797679i \(-0.293939\pi\)
0.603083 + 0.797679i \(0.293939\pi\)
\(942\) −22.0000 −0.716799
\(943\) 10.0000 0.325645
\(944\) 15.0000 0.488208
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 11.0000 0.357263
\(949\) 60.0000 1.94768
\(950\) −8.00000 −0.259554
\(951\) 15.0000 0.486408
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 11.0000 0.356138
\(955\) −78.0000 −2.52402
\(956\) 20.0000 0.646846
\(957\) 7.00000 0.226278
\(958\) 20.0000 0.646171
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) 18.0000 0.580645
\(962\) 12.0000 0.386896
\(963\) −17.0000 −0.547817
\(964\) −7.00000 −0.225455
\(965\) 69.0000 2.22119
\(966\) 0 0
\(967\) 3.00000 0.0964735 0.0482367 0.998836i \(-0.484640\pi\)
0.0482367 + 0.998836i \(0.484640\pi\)
\(968\) −10.0000 −0.321412
\(969\) −4.00000 −0.128499
\(970\) −3.00000 −0.0963242
\(971\) −21.0000 −0.673922 −0.336961 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 1.00000 0.0320421
\(975\) 24.0000 0.768615
\(976\) −2.00000 −0.0640184
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −10.0000 −0.319765
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) −29.0000 −0.925427
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 10.0000 0.318788
\(985\) 6.00000 0.191176
\(986\) 14.0000 0.445851
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −2.00000 −0.0635963
\(990\) −3.00000 −0.0953463
\(991\) −29.0000 −0.921215 −0.460608 0.887604i \(-0.652368\pi\)
−0.460608 + 0.887604i \(0.652368\pi\)
\(992\) 7.00000 0.222250
\(993\) 14.0000 0.444277
\(994\) 0 0
\(995\) −60.0000 −1.90213
\(996\) −13.0000 −0.411921
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −10.0000 −0.316544
\(999\) 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.x.1.1 1
7.2 even 3 966.2.i.d.277.1 2
7.4 even 3 966.2.i.d.415.1 yes 2
7.6 odd 2 6762.2.a.bm.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.d.277.1 2 7.2 even 3
966.2.i.d.415.1 yes 2 7.4 even 3
6762.2.a.x.1.1 1 1.1 even 1 trivial
6762.2.a.bm.1.1 1 7.6 odd 2