Properties

Label 77.2.a.c.1.1
Level 77
Weight 2
Character 77.1
Self dual yes
Analytic conductor 0.615
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 77.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +4.00000 q^{13} -1.00000 q^{14} -4.00000 q^{15} -1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -6.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{30} +10.0000 q^{31} +5.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} +8.00000 q^{39} +6.00000 q^{40} +4.00000 q^{41} -2.00000 q^{42} +12.0000 q^{43} -1.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} -10.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +8.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} -2.00000 q^{55} +3.00000 q^{56} -6.00000 q^{58} +2.00000 q^{59} +4.00000 q^{60} +10.0000 q^{62} -1.00000 q^{63} +7.00000 q^{64} -8.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} -4.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} -12.0000 q^{71} -3.00000 q^{72} -8.00000 q^{73} -6.00000 q^{74} -2.00000 q^{75} -1.00000 q^{77} +8.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} -11.0000 q^{81} +4.00000 q^{82} +2.00000 q^{84} -8.00000 q^{85} +12.0000 q^{86} -12.0000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -4.00000 q^{91} +4.00000 q^{92} +20.0000 q^{93} -10.0000 q^{94} +10.0000 q^{96} -10.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) −2.00000 −0.577350
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) −4.00000 −1.03280
\(16\) −1.00000 −0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −2.00000 −0.436436
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −6.00000 −1.22474
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −4.00000 −0.730297
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 5.00000 0.883883
\(33\) 2.00000 0.348155
\(34\) 4.00000 0.685994
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 6.00000 0.948683
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) −2.00000 −0.308607
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.00000 −0.298142
\(46\) −4.00000 −0.589768
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 8.00000 1.12022
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −4.00000 −0.544331
\(55\) −2.00000 −0.269680
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 4.00000 0.516398
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 10.0000 1.27000
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) −8.00000 −0.992278
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −4.00000 −0.485071
\(69\) −8.00000 −0.963087
\(70\) 2.00000 0.239046
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −3.00000 −0.353553
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) −6.00000 −0.697486
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 8.00000 0.905822
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) −11.0000 −1.22222
\(82\) 4.00000 0.441726
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) −8.00000 −0.867722
\(86\) 12.0000 1.29399
\(87\) −12.0000 −1.28654
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −2.00000 −0.210819
\(91\) −4.00000 −0.419314
\(92\) 4.00000 0.417029
\(93\) 20.0000 2.07390
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) 10.0000 1.02062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 8.00000 0.792118
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −12.0000 −1.17670
\(105\) 4.00000 0.390360
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 4.00000 0.384900
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −2.00000 −0.190693
\(111\) −12.0000 −1.13899
\(112\) 1.00000 0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 6.00000 0.557086
\(117\) 4.00000 0.369800
\(118\) 2.00000 0.184115
\(119\) −4.00000 −0.366679
\(120\) 12.0000 1.09545
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) −10.0000 −0.898027
\(125\) 12.0000 1.07331
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −3.00000 −0.265165
\(129\) 24.0000 2.11308
\(130\) −8.00000 −0.701646
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 8.00000 0.688530
\(136\) −12.0000 −1.02899
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −8.00000 −0.681005
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −2.00000 −0.169031
\(141\) −20.0000 −1.68430
\(142\) −12.0000 −1.00702
\(143\) 4.00000 0.334497
\(144\) −1.00000 −0.0833333
\(145\) 12.0000 0.996546
\(146\) −8.00000 −0.662085
\(147\) 2.00000 0.164957
\(148\) 6.00000 0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −2.00000 −0.163299
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) −1.00000 −0.0805823
\(155\) −20.0000 −1.60644
\(156\) −8.00000 −0.640513
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) −12.0000 −0.951662
\(160\) −10.0000 −0.790569
\(161\) 4.00000 0.315244
\(162\) −11.0000 −0.864242
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −4.00000 −0.312348
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 6.00000 0.462910
\(169\) 3.00000 0.230769
\(170\) −8.00000 −0.613572
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) −12.0000 −0.909718
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 12.0000 0.882258
\(186\) 20.0000 1.46647
\(187\) 4.00000 0.292509
\(188\) 10.0000 0.729325
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 14.0000 1.01036
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) −16.0000 −1.14578
\(196\) −1.00000 −0.0714286
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 1.00000 0.0710669
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 3.00000 0.212132
\(201\) 16.0000 1.12855
\(202\) −4.00000 −0.281439
\(203\) 6.00000 0.421117
\(204\) −8.00000 −0.560112
\(205\) −8.00000 −0.558744
\(206\) 14.0000 0.975426
\(207\) −4.00000 −0.278019
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 4.00000 0.276026
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000 0.412082
\(213\) −24.0000 −1.64445
\(214\) 12.0000 0.820303
\(215\) −24.0000 −1.63679
\(216\) 12.0000 0.816497
\(217\) −10.0000 −0.678844
\(218\) −14.0000 −0.948200
\(219\) −16.0000 −1.08118
\(220\) 2.00000 0.134840
\(221\) 16.0000 1.07628
\(222\) −12.0000 −0.805387
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) −5.00000 −0.334077
\(225\) −1.00000 −0.0666667
\(226\) 18.0000 1.19734
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 8.00000 0.527504
\(231\) −2.00000 −0.131590
\(232\) 18.0000 1.18176
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 4.00000 0.261488
\(235\) 20.0000 1.30466
\(236\) −2.00000 −0.130189
\(237\) 16.0000 1.03931
\(238\) −4.00000 −0.259281
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 4.00000 0.258199
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) −2.00000 −0.127775
\(246\) 8.00000 0.510061
\(247\) 0 0
\(248\) −30.0000 −1.90500
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) −16.0000 −1.00196
\(256\) −17.0000 −1.06250
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 24.0000 1.49417
\(259\) 6.00000 0.372822
\(260\) 8.00000 0.496139
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −6.00000 −0.369274
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) −8.00000 −0.488678
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 8.00000 0.486864
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) −4.00000 −0.242536
\(273\) −8.00000 −0.484182
\(274\) −10.0000 −0.604122
\(275\) −1.00000 −0.0603023
\(276\) 8.00000 0.481543
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −8.00000 −0.479808
\(279\) 10.0000 0.598684
\(280\) −6.00000 −0.358569
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −20.0000 −1.19098
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −4.00000 −0.236113
\(288\) 5.00000 0.294628
\(289\) −1.00000 −0.0588235
\(290\) 12.0000 0.704664
\(291\) −20.0000 −1.17242
\(292\) 8.00000 0.468165
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 2.00000 0.116642
\(295\) −4.00000 −0.232889
\(296\) 18.0000 1.04623
\(297\) −4.00000 −0.232104
\(298\) −10.0000 −0.579284
\(299\) −16.0000 −0.925304
\(300\) 2.00000 0.115470
\(301\) −12.0000 −0.691669
\(302\) −16.0000 −0.920697
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 1.00000 0.0569803
\(309\) 28.0000 1.59286
\(310\) −20.0000 −1.13592
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −24.0000 −1.35873
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 14.0000 0.790066
\(315\) 2.00000 0.112687
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −12.0000 −0.672927
\(319\) −6.00000 −0.335936
\(320\) −14.0000 −0.782624
\(321\) 24.0000 1.33955
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) 11.0000 0.611111
\(325\) −4.00000 −0.221880
\(326\) −8.00000 −0.443079
\(327\) −28.0000 −1.54840
\(328\) −12.0000 −0.662589
\(329\) 10.0000 0.551318
\(330\) −4.00000 −0.220193
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 2.00000 0.109109
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 3.00000 0.163178
\(339\) 36.0000 1.95525
\(340\) 8.00000 0.433861
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −36.0000 −1.94099
\(345\) 16.0000 0.861411
\(346\) 12.0000 0.645124
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 12.0000 0.643268
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 1.00000 0.0534522
\(351\) −16.0000 −0.854017
\(352\) 5.00000 0.266501
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 4.00000 0.212598
\(355\) 24.0000 1.27379
\(356\) 6.00000 0.317999
\(357\) −8.00000 −0.423405
\(358\) 12.0000 0.634220
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 6.00000 0.316228
\(361\) −19.0000 −1.00000
\(362\) 10.0000 0.525588
\(363\) 2.00000 0.104973
\(364\) 4.00000 0.209657
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 4.00000 0.208514
\(369\) 4.00000 0.208232
\(370\) 12.0000 0.623850
\(371\) 6.00000 0.311504
\(372\) −20.0000 −1.03695
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 4.00000 0.206835
\(375\) 24.0000 1.23935
\(376\) 30.0000 1.54713
\(377\) −24.0000 −1.23606
\(378\) 4.00000 0.205738
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 8.00000 0.409316
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) −6.00000 −0.306186
\(385\) 2.00000 0.101929
\(386\) −14.0000 −0.712581
\(387\) 12.0000 0.609994
\(388\) 10.0000 0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −16.0000 −0.810191
\(391\) −16.0000 −0.809155
\(392\) −3.00000 −0.151523
\(393\) 24.0000 1.21064
\(394\) 22.0000 1.10834
\(395\) −16.0000 −0.805047
\(396\) −1.00000 −0.0502519
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −18.0000 −0.902258
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 16.0000 0.798007
\(403\) 40.0000 1.99254
\(404\) 4.00000 0.199007
\(405\) 22.0000 1.09319
\(406\) 6.00000 0.297775
\(407\) −6.00000 −0.297409
\(408\) −24.0000 −1.18818
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) −8.00000 −0.395092
\(411\) −20.0000 −0.986527
\(412\) −14.0000 −0.689730
\(413\) −2.00000 −0.0984136
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 20.0000 0.980581
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) 2.00000 0.0977064 0.0488532 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) −4.00000 −0.195180
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) −12.0000 −0.584151
\(423\) −10.0000 −0.486217
\(424\) 18.0000 0.874157
\(425\) −4.00000 −0.194029
\(426\) −24.0000 −1.16280
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 8.00000 0.386244
\(430\) −24.0000 −1.15738
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 4.00000 0.192450
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −10.0000 −0.480015
\(435\) 24.0000 1.15071
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) −16.0000 −0.764510
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 6.00000 0.286039
\(441\) 1.00000 0.0476190
\(442\) 16.0000 0.761042
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 12.0000 0.569495
\(445\) 12.0000 0.568855
\(446\) 22.0000 1.04173
\(447\) −20.0000 −0.945968
\(448\) −7.00000 −0.330719
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 4.00000 0.188353
\(452\) −18.0000 −0.846649
\(453\) −32.0000 −1.50349
\(454\) 12.0000 0.563188
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 18.0000 0.841085
\(459\) −16.0000 −0.746816
\(460\) −8.00000 −0.373002
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 6.00000 0.278543
\(465\) −40.0000 −1.85496
\(466\) −18.0000 −0.833834
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) −4.00000 −0.184900
\(469\) −8.00000 −0.369406
\(470\) 20.0000 0.922531
\(471\) 28.0000 1.29017
\(472\) −6.00000 −0.276172
\(473\) 12.0000 0.551761
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) −20.0000 −0.912871
\(481\) −24.0000 −1.09431
\(482\) −20.0000 −0.910975
\(483\) 8.00000 0.364013
\(484\) −1.00000 −0.0454545
\(485\) 20.0000 0.908153
\(486\) −10.0000 −0.453609
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) −2.00000 −0.0903508
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −8.00000 −0.360668
\(493\) −24.0000 −1.08091
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) −10.0000 −0.449013
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −2.00000 −0.0892644
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 3.00000 0.133631
\(505\) 8.00000 0.355995
\(506\) −4.00000 −0.177822
\(507\) 6.00000 0.266469
\(508\) −8.00000 −0.354943
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) −16.0000 −0.708492
\(511\) 8.00000 0.353899
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −28.0000 −1.23383
\(516\) −24.0000 −1.05654
\(517\) −10.0000 −0.439799
\(518\) 6.00000 0.263625
\(519\) 24.0000 1.05348
\(520\) 24.0000 1.05247
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −12.0000 −0.524222
\(525\) 2.00000 0.0872872
\(526\) 8.00000 0.348817
\(527\) 40.0000 1.74243
\(528\) −2.00000 −0.0870388
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) −12.0000 −0.519291
\(535\) −24.0000 −1.03761
\(536\) −24.0000 −1.03664
\(537\) 24.0000 1.03568
\(538\) 10.0000 0.431131
\(539\) 1.00000 0.0430730
\(540\) −8.00000 −0.344265
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) −4.00000 −0.171815
\(543\) 20.0000 0.858282
\(544\) 20.0000 0.857493
\(545\) 28.0000 1.19939
\(546\) −8.00000 −0.342368
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 10.0000 0.427179
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 24.0000 1.02151
\(553\) −8.00000 −0.340195
\(554\) 22.0000 0.934690
\(555\) 24.0000 1.01874
\(556\) 8.00000 0.339276
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 10.0000 0.423334
\(559\) 48.0000 2.03018
\(560\) −2.00000 −0.0845154
\(561\) 8.00000 0.337760
\(562\) 6.00000 0.253095
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 20.0000 0.842152
\(565\) −36.0000 −1.51453
\(566\) 4.00000 0.168133
\(567\) 11.0000 0.461957
\(568\) 36.0000 1.51053
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −4.00000 −0.167248
\(573\) 16.0000 0.668410
\(574\) −4.00000 −0.166957
\(575\) 4.00000 0.166812
\(576\) 7.00000 0.291667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −28.0000 −1.16364
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) −20.0000 −0.829027
\(583\) −6.00000 −0.248495
\(584\) 24.0000 0.993127
\(585\) −8.00000 −0.330759
\(586\) −24.0000 −0.991431
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 44.0000 1.80992
\(592\) 6.00000 0.246598
\(593\) 32.0000 1.31408 0.657041 0.753855i \(-0.271808\pi\)
0.657041 + 0.753855i \(0.271808\pi\)
\(594\) −4.00000 −0.164122
\(595\) 8.00000 0.327968
\(596\) 10.0000 0.409616
\(597\) −36.0000 −1.47338
\(598\) −16.0000 −0.654289
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 6.00000 0.244949
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) −12.0000 −0.489083
\(603\) 8.00000 0.325785
\(604\) 16.0000 0.651031
\(605\) −2.00000 −0.0813116
\(606\) −8.00000 −0.324978
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −40.0000 −1.61823
\(612\) −4.00000 −0.161690
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 20.0000 0.807134
\(615\) −16.0000 −0.645182
\(616\) 3.00000 0.120873
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 28.0000 1.12633
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 20.0000 0.803219
\(621\) 16.0000 0.642058
\(622\) −18.0000 −0.721734
\(623\) 6.00000 0.240385
\(624\) −8.00000 −0.320256
\(625\) −19.0000 −0.760000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) −24.0000 −0.956943
\(630\) 2.00000 0.0796819
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −24.0000 −0.954669
\(633\) −24.0000 −0.953914
\(634\) −2.00000 −0.0794301
\(635\) −16.0000 −0.634941
\(636\) 12.0000 0.475831
\(637\) 4.00000 0.158486
\(638\) −6.00000 −0.237542
\(639\) −12.0000 −0.474713
\(640\) 6.00000 0.237171
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 24.0000 0.947204
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) −4.00000 −0.157622
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) −22.0000 −0.864909 −0.432455 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(648\) 33.0000 1.29636
\(649\) 2.00000 0.0785069
\(650\) −4.00000 −0.156893
\(651\) −20.0000 −0.783862
\(652\) 8.00000 0.313304
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) −28.0000 −1.09489
\(655\) −24.0000 −0.937758
\(656\) −4.00000 −0.156174
\(657\) −8.00000 −0.312110
\(658\) 10.0000 0.389841
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 4.00000 0.155700
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −20.0000 −0.777322
\(663\) 32.0000 1.24278
\(664\) 0 0
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 44.0000 1.70114
\(670\) −16.0000 −0.618134
\(671\) 0 0
\(672\) −10.0000 −0.385758
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 14.0000 0.539260
\(675\) 4.00000 0.153960
\(676\) −3.00000 −0.115385
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 36.0000 1.38257
\(679\) 10.0000 0.383765
\(680\) 24.0000 0.920358
\(681\) 24.0000 0.919682
\(682\) 10.0000 0.382920
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 20.0000 0.764161
\(686\) −1.00000 −0.0381802
\(687\) 36.0000 1.37349
\(688\) −12.0000 −0.457496
\(689\) −24.0000 −0.914327
\(690\) 16.0000 0.609110
\(691\) −46.0000 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(692\) −12.0000 −0.456172
\(693\) −1.00000 −0.0379869
\(694\) 4.00000 0.151838
\(695\) 16.0000 0.606915
\(696\) 36.0000 1.36458
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) −36.0000 −1.36165
\(700\) −1.00000 −0.0377964
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) −16.0000 −0.603881
\(703\) 0 0
\(704\) 7.00000 0.263822
\(705\) 40.0000 1.50649
\(706\) −30.0000 −1.12906
\(707\) 4.00000 0.150435
\(708\) −4.00000 −0.150329
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 24.0000 0.900704
\(711\) 8.00000 0.300023
\(712\) 18.0000 0.674579
\(713\) −40.0000 −1.49801
\(714\) −8.00000 −0.299392
\(715\) −8.00000 −0.299183
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 2.00000 0.0745356
\(721\) −14.0000 −0.521387
\(722\) −19.0000 −0.707107
\(723\) −40.0000 −1.48762
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) 2.00000 0.0742270
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 12.0000 0.444750
\(729\) 13.0000 0.481481
\(730\) 16.0000 0.592187
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 22.0000 0.812035
\(735\) −4.00000 −0.147542
\(736\) −20.0000 −0.737210
\(737\) 8.00000 0.294684
\(738\) 4.00000 0.147242
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −12.0000 −0.441129
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −60.0000 −2.19971
\(745\) 20.0000 0.732743
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) −12.0000 −0.438470
\(750\) 24.0000 0.876356
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 10.0000 0.364662
\(753\) −4.00000 −0.145768
\(754\) −24.0000 −0.874028
\(755\) 32.0000 1.16460
\(756\) −4.00000 −0.145479
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −8.00000 −0.290573
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 16.0000 0.579619
\(763\) 14.0000 0.506834
\(764\) −8.00000 −0.289430
\(765\) −8.00000 −0.289241
\(766\) 2.00000 0.0722629
\(767\) 8.00000 0.288863
\(768\) −34.0000 −1.22687
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 2.00000 0.0720750
\(771\) −28.0000 −1.00840
\(772\) 14.0000 0.503871
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 12.0000 0.431331
\(775\) −10.0000 −0.359211
\(776\) 30.0000 1.07694
\(777\) 12.0000 0.430498
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 16.0000 0.572892
\(781\) −12.0000 −0.429394
\(782\) −16.0000 −0.572159
\(783\) 24.0000 0.857690
\(784\) −1.00000 −0.0357143
\(785\) −28.0000 −0.999363
\(786\) 24.0000 0.856052
\(787\) 16.0000 0.570338 0.285169 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) −22.0000 −0.783718
\(789\) 16.0000 0.569615
\(790\) −16.0000 −0.569254
\(791\) −18.0000 −0.640006
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) 22.0000 0.780751
\(795\) 24.0000 0.851192
\(796\) 18.0000 0.637993
\(797\) 14.0000 0.495905 0.247953 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) −5.00000 −0.176777
\(801\) −6.00000 −0.212000
\(802\) −22.0000 −0.776847
\(803\) −8.00000 −0.282314
\(804\) −16.0000 −0.564276
\(805\) −8.00000 −0.281963
\(806\) 40.0000 1.40894
\(807\) 20.0000 0.704033
\(808\) 12.0000 0.422159
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 22.0000 0.773001
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −6.00000 −0.210559
\(813\) −8.00000 −0.280572
\(814\) −6.00000 −0.210300
\(815\) 16.0000 0.560456
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) 24.0000 0.839140
\(819\) −4.00000 −0.139771
\(820\) 8.00000 0.279372
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) −20.0000 −0.697580
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −42.0000 −1.46314
\(825\) −2.00000 −0.0696311
\(826\) −2.00000 −0.0695889
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 4.00000 0.139010
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 28.0000 0.970725
\(833\) 4.00000 0.138592
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) 2.00000 0.0690889
\(839\) 34.0000 1.17381 0.586905 0.809656i \(-0.300346\pi\)
0.586905 + 0.809656i \(0.300346\pi\)
\(840\) −12.0000 −0.414039
\(841\) 7.00000 0.241379
\(842\) −14.0000 −0.482472
\(843\) 12.0000 0.413302
\(844\) 12.0000 0.413057
\(845\) −6.00000 −0.206406
\(846\) −10.0000 −0.343807
\(847\) −1.00000 −0.0343604
\(848\) 6.00000 0.206041
\(849\) 8.00000 0.274559
\(850\) −4.00000 −0.137199
\(851\) 24.0000 0.822709
\(852\) 24.0000 0.822226
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) 56.0000 1.91292 0.956462 0.291858i \(-0.0942733\pi\)
0.956462 + 0.291858i \(0.0942733\pi\)
\(858\) 8.00000 0.273115
\(859\) 6.00000 0.204717 0.102359 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) 24.0000 0.818393
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −20.0000 −0.680414
\(865\) −24.0000 −0.816024
\(866\) −26.0000 −0.883516
\(867\) −2.00000 −0.0679236
\(868\) 10.0000 0.339422
\(869\) 8.00000 0.271381
\(870\) 24.0000 0.813676
\(871\) 32.0000 1.08428
\(872\) 42.0000 1.42230
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 16.0000 0.540590
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) −20.0000 −0.674967
\(879\) −48.0000 −1.61900
\(880\) 2.00000 0.0674200
\(881\) −34.0000 −1.14549 −0.572745 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) 1.00000 0.0336718
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −16.0000 −0.538138
\(885\) −8.00000 −0.268917
\(886\) 4.00000 0.134383
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 36.0000 1.20808
\(889\) −8.00000 −0.268311
\(890\) 12.0000 0.402241
\(891\) −11.0000 −0.368514
\(892\) −22.0000 −0.736614
\(893\) 0 0
\(894\) −20.0000 −0.668900
\(895\) −24.0000 −0.802232
\(896\) 3.00000 0.100223
\(897\) −32.0000 −1.06845
\(898\) −10.0000 −0.333704
\(899\) −60.0000 −2.00111
\(900\) 1.00000 0.0333333
\(901\) −24.0000 −0.799556
\(902\) 4.00000 0.133185
\(903\) −24.0000 −0.798670
\(904\) −54.0000 −1.79601
\(905\) −20.0000 −0.664822
\(906\) −32.0000 −1.06313
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) −12.0000 −0.398234
\(909\) −4.00000 −0.132672
\(910\) 8.00000 0.265197
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) −18.0000 −0.594737
\(917\) −12.0000 −0.396275
\(918\) −16.0000 −0.528079
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −24.0000 −0.791257
\(921\) 40.0000 1.31804
\(922\) 0 0
\(923\) −48.0000 −1.57994
\(924\) 2.00000 0.0657952
\(925\) 6.00000 0.197279
\(926\) 4.00000 0.131448
\(927\) 14.0000 0.459820
\(928\) −30.0000 −0.984798
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) −40.0000 −1.31165
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −36.0000 −1.17859
\(934\) 30.0000 0.981630
\(935\) −8.00000 −0.261628
\(936\) −12.0000 −0.392232
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) −8.00000 −0.261209
\(939\) 4.00000 0.130535
\(940\) −20.0000 −0.652328
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 28.0000 0.912289
\(943\) −16.0000 −0.521032
\(944\) −2.00000 −0.0650945
\(945\) −8.00000 −0.260240
\(946\) 12.0000 0.390154
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −16.0000 −0.519656
\(949\) −32.0000 −1.03876
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 12.0000 0.388922
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −6.00000 −0.194257
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) −4.00000 −0.129234
\(959\) 10.0000 0.322917
\(960\) −28.0000 −0.903696
\(961\) 69.0000 2.22581
\(962\) −24.0000 −0.773791
\(963\) 12.0000 0.386695
\(964\) 20.0000 0.644157
\(965\) 28.0000 0.901352
\(966\) 8.00000 0.257396
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) 14.0000 0.449281 0.224641 0.974442i \(-0.427879\pi\)
0.224641 + 0.974442i \(0.427879\pi\)
\(972\) 10.0000 0.320750
\(973\) 8.00000 0.256468
\(974\) −28.0000 −0.897178
\(975\) −8.00000 −0.256205
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −16.0000 −0.511624
\(979\) −6.00000 −0.191761
\(980\) 2.00000 0.0638877
\(981\) −14.0000 −0.446986
\(982\) 28.0000 0.893516
\(983\) 54.0000 1.72233 0.861166 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) −24.0000 −0.765092
\(985\) −44.0000 −1.40196
\(986\) −24.0000 −0.764316
\(987\) 20.0000 0.636607
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) −2.00000 −0.0635642
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 50.0000 1.58750
\(993\) −40.0000 −1.26936
\(994\) 12.0000 0.380617
\(995\) 36.0000 1.14128
\(996\) 0 0
\(997\) 20.0000 0.633406 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) −16.0000 −0.506471
\(999\) 24.0000 0.759326
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 77.2.a.c.1.1 1
3.2 odd 2 693.2.a.a.1.1 1
4.3 odd 2 1232.2.a.a.1.1 1
5.2 odd 4 1925.2.b.d.1849.2 2
5.3 odd 4 1925.2.b.d.1849.1 2
5.4 even 2 1925.2.a.c.1.1 1
7.2 even 3 539.2.e.a.67.1 2
7.3 odd 6 539.2.e.b.177.1 2
7.4 even 3 539.2.e.a.177.1 2
7.5 odd 6 539.2.e.b.67.1 2
7.6 odd 2 539.2.a.d.1.1 1
8.3 odd 2 4928.2.a.bi.1.1 1
8.5 even 2 4928.2.a.g.1.1 1
11.2 odd 10 847.2.f.k.323.1 4
11.3 even 5 847.2.f.e.372.1 4
11.4 even 5 847.2.f.e.148.1 4
11.5 even 5 847.2.f.e.729.1 4
11.6 odd 10 847.2.f.k.729.1 4
11.7 odd 10 847.2.f.k.148.1 4
11.8 odd 10 847.2.f.k.372.1 4
11.9 even 5 847.2.f.e.323.1 4
11.10 odd 2 847.2.a.a.1.1 1
21.20 even 2 4851.2.a.a.1.1 1
28.27 even 2 8624.2.a.bc.1.1 1
33.32 even 2 7623.2.a.n.1.1 1
77.76 even 2 5929.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.c.1.1 1 1.1 even 1 trivial
539.2.a.d.1.1 1 7.6 odd 2
539.2.e.a.67.1 2 7.2 even 3
539.2.e.a.177.1 2 7.4 even 3
539.2.e.b.67.1 2 7.5 odd 6
539.2.e.b.177.1 2 7.3 odd 6
693.2.a.a.1.1 1 3.2 odd 2
847.2.a.a.1.1 1 11.10 odd 2
847.2.f.e.148.1 4 11.4 even 5
847.2.f.e.323.1 4 11.9 even 5
847.2.f.e.372.1 4 11.3 even 5
847.2.f.e.729.1 4 11.5 even 5
847.2.f.k.148.1 4 11.7 odd 10
847.2.f.k.323.1 4 11.2 odd 10
847.2.f.k.372.1 4 11.8 odd 10
847.2.f.k.729.1 4 11.6 odd 10
1232.2.a.a.1.1 1 4.3 odd 2
1925.2.a.c.1.1 1 5.4 even 2
1925.2.b.d.1849.1 2 5.3 odd 4
1925.2.b.d.1849.2 2 5.2 odd 4
4851.2.a.a.1.1 1 21.20 even 2
4928.2.a.g.1.1 1 8.5 even 2
4928.2.a.bi.1.1 1 8.3 odd 2
5929.2.a.b.1.1 1 77.76 even 2
7623.2.a.n.1.1 1 33.32 even 2
8624.2.a.bc.1.1 1 28.27 even 2