Properties

Label 722.2.a.a.1.1
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} -3.00000 q^{12} +3.00000 q^{13} +3.00000 q^{14} -6.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -6.00000 q^{18} +2.00000 q^{20} +9.00000 q^{21} +2.00000 q^{22} +5.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -3.00000 q^{26} -9.00000 q^{27} -3.00000 q^{28} +3.00000 q^{29} +6.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{34} -6.00000 q^{35} +6.00000 q^{36} -6.00000 q^{37} -9.00000 q^{39} -2.00000 q^{40} -12.0000 q^{41} -9.00000 q^{42} -10.0000 q^{43} -2.00000 q^{44} +12.0000 q^{45} -5.00000 q^{46} -8.00000 q^{47} -3.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +3.00000 q^{51} +3.00000 q^{52} +3.00000 q^{53} +9.00000 q^{54} -4.00000 q^{55} +3.00000 q^{56} -3.00000 q^{58} -3.00000 q^{59} -6.00000 q^{60} -6.00000 q^{62} -18.0000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -6.00000 q^{66} -15.0000 q^{67} -1.00000 q^{68} -15.0000 q^{69} +6.00000 q^{70} -6.00000 q^{72} -11.0000 q^{73} +6.00000 q^{74} +3.00000 q^{75} +6.00000 q^{77} +9.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +12.0000 q^{82} +2.00000 q^{83} +9.00000 q^{84} -2.00000 q^{85} +10.0000 q^{86} -9.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} -12.0000 q^{90} -9.00000 q^{91} +5.00000 q^{92} -18.0000 q^{93} +8.00000 q^{94} +3.00000 q^{96} -12.0000 q^{97} -2.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 3.00000 1.22474
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) −2.00000 −0.632456
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −3.00000 −0.866025
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 3.00000 0.801784
\(15\) −6.00000 −1.54919
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) −6.00000 −1.41421
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) 9.00000 1.96396
\(22\) 2.00000 0.426401
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 3.00000 0.612372
\(25\) −1.00000 −0.200000
\(26\) −3.00000 −0.588348
\(27\) −9.00000 −1.73205
\(28\) −3.00000 −0.566947
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 6.00000 1.09545
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) 1.00000 0.171499
\(35\) −6.00000 −1.01419
\(36\) 6.00000 1.00000
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −9.00000 −1.44115
\(40\) −2.00000 −0.316228
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) −9.00000 −1.38873
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) −2.00000 −0.301511
\(45\) 12.0000 1.78885
\(46\) −5.00000 −0.737210
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −3.00000 −0.433013
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 3.00000 0.420084
\(52\) 3.00000 0.416025
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 9.00000 1.22474
\(55\) −4.00000 −0.539360
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) −6.00000 −0.774597
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −6.00000 −0.762001
\(63\) −18.0000 −2.26779
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) −6.00000 −0.738549
\(67\) −15.0000 −1.83254 −0.916271 0.400559i \(-0.868816\pi\)
−0.916271 + 0.400559i \(0.868816\pi\)
\(68\) −1.00000 −0.121268
\(69\) −15.0000 −1.80579
\(70\) 6.00000 0.717137
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −6.00000 −0.707107
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 6.00000 0.697486
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 9.00000 1.01905
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 2.00000 0.223607
\(81\) 9.00000 1.00000
\(82\) 12.0000 1.32518
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 9.00000 0.981981
\(85\) −2.00000 −0.216930
\(86\) 10.0000 1.07833
\(87\) −9.00000 −0.964901
\(88\) 2.00000 0.213201
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −12.0000 −1.26491
\(91\) −9.00000 −0.943456
\(92\) 5.00000 0.521286
\(93\) −18.0000 −1.86651
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 3.00000 0.306186
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −2.00000 −0.202031
\(99\) −12.0000 −1.20605
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −3.00000 −0.297044
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −3.00000 −0.294174
\(105\) 18.0000 1.75662
\(106\) −3.00000 −0.291386
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −9.00000 −0.866025
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 4.00000 0.381385
\(111\) 18.0000 1.70848
\(112\) −3.00000 −0.283473
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 10.0000 0.932505
\(116\) 3.00000 0.278543
\(117\) 18.0000 1.66410
\(118\) 3.00000 0.276172
\(119\) 3.00000 0.275010
\(120\) 6.00000 0.547723
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 36.0000 3.24601
\(124\) 6.00000 0.538816
\(125\) −12.0000 −1.07331
\(126\) 18.0000 1.60357
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 30.0000 2.64135
\(130\) −6.00000 −0.526235
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) 15.0000 1.29580
\(135\) −18.0000 −1.54919
\(136\) 1.00000 0.0857493
\(137\) 19.0000 1.62328 0.811640 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) 15.0000 1.27688
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) −6.00000 −0.507093
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 6.00000 0.500000
\(145\) 6.00000 0.498273
\(146\) 11.0000 0.910366
\(147\) −6.00000 −0.494872
\(148\) −6.00000 −0.493197
\(149\) −8.00000 −0.655386 −0.327693 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) −3.00000 −0.244949
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) −6.00000 −0.483494
\(155\) 12.0000 0.963863
\(156\) −9.00000 −0.720577
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −12.0000 −0.954669
\(159\) −9.00000 −0.713746
\(160\) −2.00000 −0.158114
\(161\) −15.0000 −1.18217
\(162\) −9.00000 −0.707107
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −12.0000 −0.937043
\(165\) 12.0000 0.934199
\(166\) −2.00000 −0.155230
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −9.00000 −0.694365
\(169\) −4.00000 −0.307692
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 9.00000 0.682288
\(175\) 3.00000 0.226779
\(176\) −2.00000 −0.150756
\(177\) 9.00000 0.676481
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 12.0000 0.894427
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 9.00000 0.667124
\(183\) 0 0
\(184\) −5.00000 −0.368605
\(185\) −12.0000 −0.882258
\(186\) 18.0000 1.31982
\(187\) 2.00000 0.146254
\(188\) −8.00000 −0.583460
\(189\) 27.0000 1.96396
\(190\) 0 0
\(191\) −11.0000 −0.795932 −0.397966 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) −3.00000 −0.216506
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 12.0000 0.861550
\(195\) −18.0000 −1.28901
\(196\) 2.00000 0.142857
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 12.0000 0.852803
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 1.00000 0.0707107
\(201\) 45.0000 3.17406
\(202\) −10.0000 −0.703598
\(203\) −9.00000 −0.631676
\(204\) 3.00000 0.210042
\(205\) −24.0000 −1.67623
\(206\) 6.00000 0.418040
\(207\) 30.0000 2.08514
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) −18.0000 −1.24212
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) −20.0000 −1.36399
\(216\) 9.00000 0.612372
\(217\) −18.0000 −1.22192
\(218\) 3.00000 0.203186
\(219\) 33.0000 2.22993
\(220\) −4.00000 −0.269680
\(221\) −3.00000 −0.201802
\(222\) −18.0000 −1.20808
\(223\) −18.0000 −1.20537 −0.602685 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 3.00000 0.200446
\(225\) −6.00000 −0.400000
\(226\) 12.0000 0.798228
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −10.0000 −0.659380
\(231\) −18.0000 −1.18431
\(232\) −3.00000 −0.196960
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) −18.0000 −1.17670
\(235\) −16.0000 −1.04372
\(236\) −3.00000 −0.195283
\(237\) −36.0000 −2.33845
\(238\) −3.00000 −0.194461
\(239\) 1.00000 0.0646846 0.0323423 0.999477i \(-0.489703\pi\)
0.0323423 + 0.999477i \(0.489703\pi\)
\(240\) −6.00000 −0.387298
\(241\) −24.0000 −1.54598 −0.772988 0.634421i \(-0.781239\pi\)
−0.772988 + 0.634421i \(0.781239\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) −36.0000 −2.29528
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) −6.00000 −0.380235
\(250\) 12.0000 0.758947
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −18.0000 −1.13389
\(253\) −10.0000 −0.628695
\(254\) 12.0000 0.752947
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −30.0000 −1.86772
\(259\) 18.0000 1.11847
\(260\) 6.00000 0.372104
\(261\) 18.0000 1.11417
\(262\) −14.0000 −0.864923
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) −6.00000 −0.369274
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) −15.0000 −0.916271
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 18.0000 1.09545
\(271\) −11.0000 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 27.0000 1.63411
\(274\) −19.0000 −1.14783
\(275\) 2.00000 0.120605
\(276\) −15.0000 −0.902894
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) −6.00000 −0.359856
\(279\) 36.0000 2.15526
\(280\) 6.00000 0.358569
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) −24.0000 −1.42918
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 36.0000 2.12501
\(288\) −6.00000 −0.353553
\(289\) −16.0000 −0.941176
\(290\) −6.00000 −0.352332
\(291\) 36.0000 2.11036
\(292\) −11.0000 −0.643726
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 6.00000 0.349927
\(295\) −6.00000 −0.349334
\(296\) 6.00000 0.348743
\(297\) 18.0000 1.04447
\(298\) 8.00000 0.463428
\(299\) 15.0000 0.867472
\(300\) 3.00000 0.173205
\(301\) 30.0000 1.72917
\(302\) −18.0000 −1.03578
\(303\) −30.0000 −1.72345
\(304\) 0 0
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 6.00000 0.341882
\(309\) 18.0000 1.02398
\(310\) −12.0000 −0.681554
\(311\) −11.0000 −0.623753 −0.311876 0.950123i \(-0.600957\pi\)
−0.311876 + 0.950123i \(0.600957\pi\)
\(312\) 9.00000 0.509525
\(313\) 21.0000 1.18699 0.593495 0.804838i \(-0.297748\pi\)
0.593495 + 0.804838i \(0.297748\pi\)
\(314\) 0 0
\(315\) −36.0000 −2.02837
\(316\) 12.0000 0.675053
\(317\) 33.0000 1.85346 0.926732 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 9.00000 0.504695
\(319\) −6.00000 −0.335936
\(320\) 2.00000 0.111803
\(321\) −9.00000 −0.502331
\(322\) 15.0000 0.835917
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) −3.00000 −0.166410
\(326\) 6.00000 0.332309
\(327\) 9.00000 0.497701
\(328\) 12.0000 0.662589
\(329\) 24.0000 1.32316
\(330\) −12.0000 −0.660578
\(331\) −9.00000 −0.494685 −0.247342 0.968928i \(-0.579557\pi\)
−0.247342 + 0.968928i \(0.579557\pi\)
\(332\) 2.00000 0.109764
\(333\) −36.0000 −1.97279
\(334\) −12.0000 −0.656611
\(335\) −30.0000 −1.63908
\(336\) 9.00000 0.490990
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 4.00000 0.217571
\(339\) 36.0000 1.95525
\(340\) −2.00000 −0.108465
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 10.0000 0.539164
\(345\) −30.0000 −1.61515
\(346\) −18.0000 −0.967686
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) −9.00000 −0.482451
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) −3.00000 −0.160357
\(351\) −27.0000 −1.44115
\(352\) 2.00000 0.106600
\(353\) −31.0000 −1.64996 −0.824982 0.565159i \(-0.808815\pi\)
−0.824982 + 0.565159i \(0.808815\pi\)
\(354\) −9.00000 −0.478345
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −9.00000 −0.476331
\(358\) −12.0000 −0.634220
\(359\) 19.0000 1.00278 0.501391 0.865221i \(-0.332822\pi\)
0.501391 + 0.865221i \(0.332822\pi\)
\(360\) −12.0000 −0.632456
\(361\) 0 0
\(362\) 18.0000 0.946059
\(363\) 21.0000 1.10221
\(364\) −9.00000 −0.471728
\(365\) −22.0000 −1.15153
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 5.00000 0.260643
\(369\) −72.0000 −3.74817
\(370\) 12.0000 0.623850
\(371\) −9.00000 −0.467257
\(372\) −18.0000 −0.933257
\(373\) 21.0000 1.08734 0.543669 0.839299i \(-0.317035\pi\)
0.543669 + 0.839299i \(0.317035\pi\)
\(374\) −2.00000 −0.103418
\(375\) 36.0000 1.85903
\(376\) 8.00000 0.412568
\(377\) 9.00000 0.463524
\(378\) −27.0000 −1.38873
\(379\) −3.00000 −0.154100 −0.0770498 0.997027i \(-0.524550\pi\)
−0.0770498 + 0.997027i \(0.524550\pi\)
\(380\) 0 0
\(381\) 36.0000 1.84434
\(382\) 11.0000 0.562809
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 3.00000 0.153093
\(385\) 12.0000 0.611577
\(386\) 6.00000 0.305392
\(387\) −60.0000 −3.04997
\(388\) −12.0000 −0.609208
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 18.0000 0.911465
\(391\) −5.00000 −0.252861
\(392\) −2.00000 −0.101015
\(393\) −42.0000 −2.11862
\(394\) −4.00000 −0.201517
\(395\) 24.0000 1.20757
\(396\) −12.0000 −0.603023
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) −45.0000 −2.24440
\(403\) 18.0000 0.896644
\(404\) 10.0000 0.497519
\(405\) 18.0000 0.894427
\(406\) 9.00000 0.446663
\(407\) 12.0000 0.594818
\(408\) −3.00000 −0.148522
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 24.0000 1.18528
\(411\) −57.0000 −2.81160
\(412\) −6.00000 −0.295599
\(413\) 9.00000 0.442861
\(414\) −30.0000 −1.47442
\(415\) 4.00000 0.196352
\(416\) −3.00000 −0.147087
\(417\) −18.0000 −0.881464
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 18.0000 0.878310
\(421\) 27.0000 1.31590 0.657950 0.753062i \(-0.271424\pi\)
0.657950 + 0.753062i \(0.271424\pi\)
\(422\) 3.00000 0.146038
\(423\) −48.0000 −2.33384
\(424\) −3.00000 −0.145693
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) 18.0000 0.869048
\(430\) 20.0000 0.964486
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −9.00000 −0.433013
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 18.0000 0.864028
\(435\) −18.0000 −0.863034
\(436\) −3.00000 −0.143674
\(437\) 0 0
\(438\) −33.0000 −1.57680
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 4.00000 0.190693
\(441\) 12.0000 0.571429
\(442\) 3.00000 0.142695
\(443\) −22.0000 −1.04525 −0.522626 0.852562i \(-0.675047\pi\)
−0.522626 + 0.852562i \(0.675047\pi\)
\(444\) 18.0000 0.854242
\(445\) −12.0000 −0.568855
\(446\) 18.0000 0.852325
\(447\) 24.0000 1.13516
\(448\) −3.00000 −0.141737
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 6.00000 0.282843
\(451\) 24.0000 1.13012
\(452\) −12.0000 −0.564433
\(453\) −54.0000 −2.53714
\(454\) 3.00000 0.140797
\(455\) −18.0000 −0.843853
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 12.0000 0.560723
\(459\) 9.00000 0.420084
\(460\) 10.0000 0.466252
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 18.0000 0.837436
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 3.00000 0.139272
\(465\) −36.0000 −1.66946
\(466\) −14.0000 −0.648537
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 18.0000 0.832050
\(469\) 45.0000 2.07791
\(470\) 16.0000 0.738025
\(471\) 0 0
\(472\) 3.00000 0.138086
\(473\) 20.0000 0.919601
\(474\) 36.0000 1.65353
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 18.0000 0.824163
\(478\) −1.00000 −0.0457389
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 6.00000 0.273861
\(481\) −18.0000 −0.820729
\(482\) 24.0000 1.09317
\(483\) 45.0000 2.04757
\(484\) −7.00000 −0.318182
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 0 0
\(489\) 18.0000 0.813988
\(490\) −4.00000 −0.180702
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 36.0000 1.62301
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) −24.0000 −1.07872
\(496\) 6.00000 0.269408
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) 18.0000 0.805791 0.402895 0.915246i \(-0.368004\pi\)
0.402895 + 0.915246i \(0.368004\pi\)
\(500\) −12.0000 −0.536656
\(501\) −36.0000 −1.60836
\(502\) 20.0000 0.892644
\(503\) 1.00000 0.0445878 0.0222939 0.999751i \(-0.492903\pi\)
0.0222939 + 0.999751i \(0.492903\pi\)
\(504\) 18.0000 0.801784
\(505\) 20.0000 0.889988
\(506\) 10.0000 0.444554
\(507\) 12.0000 0.532939
\(508\) −12.0000 −0.532414
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −6.00000 −0.265684
\(511\) 33.0000 1.45983
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −12.0000 −0.528783
\(516\) 30.0000 1.32068
\(517\) 16.0000 0.703679
\(518\) −18.0000 −0.790875
\(519\) −54.0000 −2.37034
\(520\) −6.00000 −0.263117
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) −18.0000 −0.787839
\(523\) −9.00000 −0.393543 −0.196771 0.980449i \(-0.563046\pi\)
−0.196771 + 0.980449i \(0.563046\pi\)
\(524\) 14.0000 0.611593
\(525\) −9.00000 −0.392792
\(526\) 8.00000 0.348817
\(527\) −6.00000 −0.261364
\(528\) 6.00000 0.261116
\(529\) 2.00000 0.0869565
\(530\) −6.00000 −0.260623
\(531\) −18.0000 −0.781133
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) −18.0000 −0.778936
\(535\) 6.00000 0.259403
\(536\) 15.0000 0.647901
\(537\) −36.0000 −1.55351
\(538\) −6.00000 −0.258678
\(539\) −4.00000 −0.172292
\(540\) −18.0000 −0.774597
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 11.0000 0.472490
\(543\) 54.0000 2.31736
\(544\) 1.00000 0.0428746
\(545\) −6.00000 −0.257012
\(546\) −27.0000 −1.15549
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 19.0000 0.811640
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 0 0
\(552\) 15.0000 0.638442
\(553\) −36.0000 −1.53088
\(554\) −30.0000 −1.27458
\(555\) 36.0000 1.52811
\(556\) 6.00000 0.254457
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) −36.0000 −1.52400
\(559\) −30.0000 −1.26886
\(560\) −6.00000 −0.253546
\(561\) −6.00000 −0.253320
\(562\) 12.0000 0.506189
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 24.0000 1.01058
\(565\) −24.0000 −1.00969
\(566\) 14.0000 0.588464
\(567\) −27.0000 −1.13389
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −6.00000 −0.250873
\(573\) 33.0000 1.37859
\(574\) −36.0000 −1.50261
\(575\) −5.00000 −0.208514
\(576\) 6.00000 0.250000
\(577\) 15.0000 0.624458 0.312229 0.950007i \(-0.398924\pi\)
0.312229 + 0.950007i \(0.398924\pi\)
\(578\) 16.0000 0.665512
\(579\) 18.0000 0.748054
\(580\) 6.00000 0.249136
\(581\) −6.00000 −0.248922
\(582\) −36.0000 −1.49225
\(583\) −6.00000 −0.248495
\(584\) 11.0000 0.455183
\(585\) 36.0000 1.48842
\(586\) −9.00000 −0.371787
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) −6.00000 −0.247436
\(589\) 0 0
\(590\) 6.00000 0.247016
\(591\) −12.0000 −0.493614
\(592\) −6.00000 −0.246598
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) −18.0000 −0.738549
\(595\) 6.00000 0.245976
\(596\) −8.00000 −0.327693
\(597\) 21.0000 0.859473
\(598\) −15.0000 −0.613396
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) −3.00000 −0.122474
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) −30.0000 −1.22271
\(603\) −90.0000 −3.66508
\(604\) 18.0000 0.732410
\(605\) −14.0000 −0.569181
\(606\) 30.0000 1.21867
\(607\) 12.0000 0.487065 0.243532 0.969893i \(-0.421694\pi\)
0.243532 + 0.969893i \(0.421694\pi\)
\(608\) 0 0
\(609\) 27.0000 1.09410
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) −6.00000 −0.242536
\(613\) 18.0000 0.727013 0.363507 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) 12.0000 0.484281
\(615\) 72.0000 2.90332
\(616\) −6.00000 −0.241747
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) −18.0000 −0.724066
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 12.0000 0.481932
\(621\) −45.0000 −1.80579
\(622\) 11.0000 0.441060
\(623\) 18.0000 0.721155
\(624\) −9.00000 −0.360288
\(625\) −19.0000 −0.760000
\(626\) −21.0000 −0.839329
\(627\) 0 0
\(628\) 0 0
\(629\) 6.00000 0.239236
\(630\) 36.0000 1.43427
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −12.0000 −0.477334
\(633\) 9.00000 0.357718
\(634\) −33.0000 −1.31060
\(635\) −24.0000 −0.952411
\(636\) −9.00000 −0.356873
\(637\) 6.00000 0.237729
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 9.00000 0.355202
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) −15.0000 −0.591083
\(645\) 60.0000 2.36250
\(646\) 0 0
\(647\) −23.0000 −0.904223 −0.452112 0.891961i \(-0.649329\pi\)
−0.452112 + 0.891961i \(0.649329\pi\)
\(648\) −9.00000 −0.353553
\(649\) 6.00000 0.235521
\(650\) 3.00000 0.117670
\(651\) 54.0000 2.11643
\(652\) −6.00000 −0.234978
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) −9.00000 −0.351928
\(655\) 28.0000 1.09405
\(656\) −12.0000 −0.468521
\(657\) −66.0000 −2.57491
\(658\) −24.0000 −0.935617
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 12.0000 0.467099
\(661\) −15.0000 −0.583432 −0.291716 0.956505i \(-0.594226\pi\)
−0.291716 + 0.956505i \(0.594226\pi\)
\(662\) 9.00000 0.349795
\(663\) 9.00000 0.349531
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 36.0000 1.39497
\(667\) 15.0000 0.580802
\(668\) 12.0000 0.464294
\(669\) 54.0000 2.08776
\(670\) 30.0000 1.15900
\(671\) 0 0
\(672\) −9.00000 −0.347183
\(673\) 48.0000 1.85026 0.925132 0.379646i \(-0.123954\pi\)
0.925132 + 0.379646i \(0.123954\pi\)
\(674\) 18.0000 0.693334
\(675\) 9.00000 0.346410
\(676\) −4.00000 −0.153846
\(677\) −3.00000 −0.115299 −0.0576497 0.998337i \(-0.518361\pi\)
−0.0576497 + 0.998337i \(0.518361\pi\)
\(678\) −36.0000 −1.38257
\(679\) 36.0000 1.38155
\(680\) 2.00000 0.0766965
\(681\) 9.00000 0.344881
\(682\) 12.0000 0.459504
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 38.0000 1.45191
\(686\) −15.0000 −0.572703
\(687\) 36.0000 1.37349
\(688\) −10.0000 −0.381246
\(689\) 9.00000 0.342873
\(690\) 30.0000 1.14208
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 18.0000 0.684257
\(693\) 36.0000 1.36753
\(694\) −16.0000 −0.607352
\(695\) 12.0000 0.455186
\(696\) 9.00000 0.341144
\(697\) 12.0000 0.454532
\(698\) −28.0000 −1.05982
\(699\) −42.0000 −1.58859
\(700\) 3.00000 0.113389
\(701\) −40.0000 −1.51078 −0.755390 0.655276i \(-0.772552\pi\)
−0.755390 + 0.655276i \(0.772552\pi\)
\(702\) 27.0000 1.01905
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 48.0000 1.80778
\(706\) 31.0000 1.16670
\(707\) −30.0000 −1.12827
\(708\) 9.00000 0.338241
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) 72.0000 2.70021
\(712\) 6.00000 0.224860
\(713\) 30.0000 1.12351
\(714\) 9.00000 0.336817
\(715\) −12.0000 −0.448775
\(716\) 12.0000 0.448461
\(717\) −3.00000 −0.112037
\(718\) −19.0000 −0.709074
\(719\) −43.0000 −1.60363 −0.801815 0.597573i \(-0.796132\pi\)
−0.801815 + 0.597573i \(0.796132\pi\)
\(720\) 12.0000 0.447214
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 72.0000 2.67771
\(724\) −18.0000 −0.668965
\(725\) −3.00000 −0.111417
\(726\) −21.0000 −0.779383
\(727\) −35.0000 −1.29808 −0.649039 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) 9.00000 0.333562
\(729\) −27.0000 −1.00000
\(730\) 22.0000 0.814257
\(731\) 10.0000 0.369863
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −8.00000 −0.295285
\(735\) −12.0000 −0.442627
\(736\) −5.00000 −0.184302
\(737\) 30.0000 1.10506
\(738\) 72.0000 2.65036
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 9.00000 0.330400
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 18.0000 0.659912
\(745\) −16.0000 −0.586195
\(746\) −21.0000 −0.768865
\(747\) 12.0000 0.439057
\(748\) 2.00000 0.0731272
\(749\) −9.00000 −0.328853
\(750\) −36.0000 −1.31453
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −8.00000 −0.291730
\(753\) 60.0000 2.18652
\(754\) −9.00000 −0.327761
\(755\) 36.0000 1.31017
\(756\) 27.0000 0.981981
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) 3.00000 0.108965
\(759\) 30.0000 1.08893
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) −36.0000 −1.30414
\(763\) 9.00000 0.325822
\(764\) −11.0000 −0.397966
\(765\) −12.0000 −0.433861
\(766\) 18.0000 0.650366
\(767\) −9.00000 −0.324971
\(768\) −3.00000 −0.108253
\(769\) −15.0000 −0.540914 −0.270457 0.962732i \(-0.587175\pi\)
−0.270457 + 0.962732i \(0.587175\pi\)
\(770\) −12.0000 −0.432450
\(771\) −54.0000 −1.94476
\(772\) −6.00000 −0.215945
\(773\) −15.0000 −0.539513 −0.269756 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 60.0000 2.15666
\(775\) −6.00000 −0.215526
\(776\) 12.0000 0.430775
\(777\) −54.0000 −1.93724
\(778\) −26.0000 −0.932145
\(779\) 0 0
\(780\) −18.0000 −0.644503
\(781\) 0 0
\(782\) 5.00000 0.178800
\(783\) −27.0000 −0.964901
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 42.0000 1.49809
\(787\) 9.00000 0.320815 0.160408 0.987051i \(-0.448719\pi\)
0.160408 + 0.987051i \(0.448719\pi\)
\(788\) 4.00000 0.142494
\(789\) 24.0000 0.854423
\(790\) −24.0000 −0.853882
\(791\) 36.0000 1.28001
\(792\) 12.0000 0.426401
\(793\) 0 0
\(794\) −2.00000 −0.0709773
\(795\) −18.0000 −0.638394
\(796\) −7.00000 −0.248108
\(797\) −33.0000 −1.16892 −0.584460 0.811423i \(-0.698694\pi\)
−0.584460 + 0.811423i \(0.698694\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 1.00000 0.0353553
\(801\) −36.0000 −1.27200
\(802\) 36.0000 1.27120
\(803\) 22.0000 0.776363
\(804\) 45.0000 1.58703
\(805\) −30.0000 −1.05736
\(806\) −18.0000 −0.634023
\(807\) −18.0000 −0.633630
\(808\) −10.0000 −0.351799
\(809\) −11.0000 −0.386739 −0.193370 0.981126i \(-0.561942\pi\)
−0.193370 + 0.981126i \(0.561942\pi\)
\(810\) −18.0000 −0.632456
\(811\) −33.0000 −1.15879 −0.579393 0.815048i \(-0.696710\pi\)
−0.579393 + 0.815048i \(0.696710\pi\)
\(812\) −9.00000 −0.315838
\(813\) 33.0000 1.15736
\(814\) −12.0000 −0.420600
\(815\) −12.0000 −0.420342
\(816\) 3.00000 0.105021
\(817\) 0 0
\(818\) 6.00000 0.209785
\(819\) −54.0000 −1.88691
\(820\) −24.0000 −0.838116
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 57.0000 1.98810
\(823\) 3.00000 0.104573 0.0522867 0.998632i \(-0.483349\pi\)
0.0522867 + 0.998632i \(0.483349\pi\)
\(824\) 6.00000 0.209020
\(825\) −6.00000 −0.208893
\(826\) −9.00000 −0.313150
\(827\) −15.0000 −0.521601 −0.260801 0.965393i \(-0.583986\pi\)
−0.260801 + 0.965393i \(0.583986\pi\)
\(828\) 30.0000 1.04257
\(829\) −51.0000 −1.77130 −0.885652 0.464350i \(-0.846288\pi\)
−0.885652 + 0.464350i \(0.846288\pi\)
\(830\) −4.00000 −0.138842
\(831\) −90.0000 −3.12207
\(832\) 3.00000 0.104006
\(833\) −2.00000 −0.0692959
\(834\) 18.0000 0.623289
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) −54.0000 −1.86651
\(838\) 14.0000 0.483622
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) −18.0000 −0.621059
\(841\) −20.0000 −0.689655
\(842\) −27.0000 −0.930481
\(843\) 36.0000 1.23991
\(844\) −3.00000 −0.103264
\(845\) −8.00000 −0.275208
\(846\) 48.0000 1.65027
\(847\) 21.0000 0.721569
\(848\) 3.00000 0.103020
\(849\) 42.0000 1.44144
\(850\) −1.00000 −0.0342997
\(851\) −30.0000 −1.02839
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.00000 −0.102538
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) −18.0000 −0.614510
\(859\) 54.0000 1.84246 0.921228 0.389023i \(-0.127187\pi\)
0.921228 + 0.389023i \(0.127187\pi\)
\(860\) −20.0000 −0.681994
\(861\) −108.000 −3.68063
\(862\) 24.0000 0.817443
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 9.00000 0.306186
\(865\) 36.0000 1.22404
\(866\) 30.0000 1.01944
\(867\) 48.0000 1.63017
\(868\) −18.0000 −0.610960
\(869\) −24.0000 −0.814144
\(870\) 18.0000 0.610257
\(871\) −45.0000 −1.52477
\(872\) 3.00000 0.101593
\(873\) −72.0000 −2.43683
\(874\) 0 0
\(875\) 36.0000 1.21702
\(876\) 33.0000 1.11497
\(877\) −27.0000 −0.911725 −0.455863 0.890050i \(-0.650669\pi\)
−0.455863 + 0.890050i \(0.650669\pi\)
\(878\) 0 0
\(879\) −27.0000 −0.910687
\(880\) −4.00000 −0.134840
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) −12.0000 −0.404061
\(883\) −24.0000 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) −3.00000 −0.100901
\(885\) 18.0000 0.605063
\(886\) 22.0000 0.739104
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −18.0000 −0.604040
\(889\) 36.0000 1.20740
\(890\) 12.0000 0.402241
\(891\) −18.0000 −0.603023
\(892\) −18.0000 −0.602685
\(893\) 0 0
\(894\) −24.0000 −0.802680
\(895\) 24.0000 0.802232
\(896\) 3.00000 0.100223
\(897\) −45.0000 −1.50251
\(898\) 6.00000 0.200223
\(899\) 18.0000 0.600334
\(900\) −6.00000 −0.200000
\(901\) −3.00000 −0.0999445
\(902\) −24.0000 −0.799113
\(903\) −90.0000 −2.99501
\(904\) 12.0000 0.399114
\(905\) −36.0000 −1.19668
\(906\) 54.0000 1.79403
\(907\) 15.0000 0.498067 0.249033 0.968495i \(-0.419887\pi\)
0.249033 + 0.968495i \(0.419887\pi\)
\(908\) −3.00000 −0.0995585
\(909\) 60.0000 1.99007
\(910\) 18.0000 0.596694
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) −1.00000 −0.0330771
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) −42.0000 −1.38696
\(918\) −9.00000 −0.297044
\(919\) 15.0000 0.494804 0.247402 0.968913i \(-0.420423\pi\)
0.247402 + 0.968913i \(0.420423\pi\)
\(920\) −10.0000 −0.329690
\(921\) 36.0000 1.18624
\(922\) −4.00000 −0.131733
\(923\) 0 0
\(924\) −18.0000 −0.592157
\(925\) 6.00000 0.197279
\(926\) −32.0000 −1.05159
\(927\) −36.0000 −1.18240
\(928\) −3.00000 −0.0984798
\(929\) 41.0000 1.34517 0.672583 0.740022i \(-0.265185\pi\)
0.672583 + 0.740022i \(0.265185\pi\)
\(930\) 36.0000 1.18049
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 33.0000 1.08037
\(934\) −8.00000 −0.261768
\(935\) 4.00000 0.130814
\(936\) −18.0000 −0.588348
\(937\) −47.0000 −1.53542 −0.767712 0.640796i \(-0.778605\pi\)
−0.767712 + 0.640796i \(0.778605\pi\)
\(938\) −45.0000 −1.46930
\(939\) −63.0000 −2.05593
\(940\) −16.0000 −0.521862
\(941\) 27.0000 0.880175 0.440087 0.897955i \(-0.354947\pi\)
0.440087 + 0.897955i \(0.354947\pi\)
\(942\) 0 0
\(943\) −60.0000 −1.95387
\(944\) −3.00000 −0.0976417
\(945\) 54.0000 1.75662
\(946\) −20.0000 −0.650256
\(947\) −46.0000 −1.49480 −0.747400 0.664375i \(-0.768698\pi\)
−0.747400 + 0.664375i \(0.768698\pi\)
\(948\) −36.0000 −1.16923
\(949\) −33.0000 −1.07123
\(950\) 0 0
\(951\) −99.0000 −3.21029
\(952\) −3.00000 −0.0972306
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) −18.0000 −0.582772
\(955\) −22.0000 −0.711903
\(956\) 1.00000 0.0323423
\(957\) 18.0000 0.581857
\(958\) 40.0000 1.29234
\(959\) −57.0000 −1.84063
\(960\) −6.00000 −0.193649
\(961\) 5.00000 0.161290
\(962\) 18.0000 0.580343
\(963\) 18.0000 0.580042
\(964\) −24.0000 −0.772988
\(965\) −12.0000 −0.386294
\(966\) −45.0000 −1.44785
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) −18.0000 −0.577054
\(974\) −18.0000 −0.576757
\(975\) 9.00000 0.288231
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −18.0000 −0.575577
\(979\) 12.0000 0.383522
\(980\) 4.00000 0.127775
\(981\) −18.0000 −0.574696
\(982\) −8.00000 −0.255290
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −36.0000 −1.14764
\(985\) 8.00000 0.254901
\(986\) 3.00000 0.0955395
\(987\) −72.0000 −2.29179
\(988\) 0 0
\(989\) −50.0000 −1.58991
\(990\) 24.0000 0.762770
\(991\) −6.00000 −0.190596 −0.0952981 0.995449i \(-0.530380\pi\)
−0.0952981 + 0.995449i \(0.530380\pi\)
\(992\) −6.00000 −0.190500
\(993\) 27.0000 0.856819
\(994\) 0 0
\(995\) −14.0000 −0.443830
\(996\) −6.00000 −0.190117
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) −18.0000 −0.569780
\(999\) 54.0000 1.70848
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 722.2.a.a.1.1 1
3.2 odd 2 6498.2.a.m.1.1 1
4.3 odd 2 5776.2.a.q.1.1 1
19.2 odd 18 722.2.e.g.99.1 6
19.3 odd 18 722.2.e.g.389.1 6
19.4 even 9 722.2.e.h.415.1 6
19.5 even 9 722.2.e.h.595.1 6
19.6 even 9 722.2.e.h.245.1 6
19.7 even 3 722.2.c.g.429.1 2
19.8 odd 6 722.2.c.a.653.1 2
19.9 even 9 722.2.e.h.423.1 6
19.10 odd 18 722.2.e.g.423.1 6
19.11 even 3 722.2.c.g.653.1 2
19.12 odd 6 722.2.c.a.429.1 2
19.13 odd 18 722.2.e.g.245.1 6
19.14 odd 18 722.2.e.g.595.1 6
19.15 odd 18 722.2.e.g.415.1 6
19.16 even 9 722.2.e.h.389.1 6
19.17 even 9 722.2.e.h.99.1 6
19.18 odd 2 722.2.a.f.1.1 yes 1
57.56 even 2 6498.2.a.a.1.1 1
76.75 even 2 5776.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.a.1.1 1 1.1 even 1 trivial
722.2.a.f.1.1 yes 1 19.18 odd 2
722.2.c.a.429.1 2 19.12 odd 6
722.2.c.a.653.1 2 19.8 odd 6
722.2.c.g.429.1 2 19.7 even 3
722.2.c.g.653.1 2 19.11 even 3
722.2.e.g.99.1 6 19.2 odd 18
722.2.e.g.245.1 6 19.13 odd 18
722.2.e.g.389.1 6 19.3 odd 18
722.2.e.g.415.1 6 19.15 odd 18
722.2.e.g.423.1 6 19.10 odd 18
722.2.e.g.595.1 6 19.14 odd 18
722.2.e.h.99.1 6 19.17 even 9
722.2.e.h.245.1 6 19.6 even 9
722.2.e.h.389.1 6 19.16 even 9
722.2.e.h.415.1 6 19.4 even 9
722.2.e.h.423.1 6 19.9 even 9
722.2.e.h.595.1 6 19.5 even 9
5776.2.a.a.1.1 1 76.75 even 2
5776.2.a.q.1.1 1 4.3 odd 2
6498.2.a.a.1.1 1 57.56 even 2
6498.2.a.m.1.1 1 3.2 odd 2