Properties

Label 786.2.a.g.1.1
Level 786
Weight 2
Character 786.1
Self dual yes
Analytic conductor 6.276
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 786.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.27624159887\)
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
Root \(0\)
Character \(\chi\) = 786.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -5.00000 q^{13} +2.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} -1.00000 q^{18} -5.00000 q^{19} -2.00000 q^{20} -2.00000 q^{21} -3.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -3.00000 q^{29} +2.00000 q^{30} -7.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} -7.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} +5.00000 q^{38} -5.00000 q^{39} +2.00000 q^{40} -12.0000 q^{41} +2.00000 q^{42} -2.00000 q^{43} +3.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +7.00000 q^{51} -5.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +2.00000 q^{56} -5.00000 q^{57} +3.00000 q^{58} -3.00000 q^{59} -2.00000 q^{60} +13.0000 q^{61} +7.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +10.0000 q^{65} -3.00000 q^{66} +8.00000 q^{67} +7.00000 q^{68} -4.00000 q^{69} -4.00000 q^{70} -16.0000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +8.00000 q^{74} -1.00000 q^{75} -5.00000 q^{76} -6.00000 q^{77} +5.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} +6.00000 q^{83} -2.00000 q^{84} -14.0000 q^{85} +2.00000 q^{86} -3.00000 q^{87} -3.00000 q^{88} -14.0000 q^{89} +2.00000 q^{90} +10.0000 q^{91} -4.00000 q^{92} -7.00000 q^{93} -8.00000 q^{94} +10.0000 q^{95} -1.00000 q^{96} +12.0000 q^{97} +3.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000 0.288675
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 2.00000 0.534522
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −2.00000 −0.447214
\(21\) −2.00000 −0.436436
\(22\) −3.00000 −0.639602
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 5.00000 0.980581
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 2.00000 0.365148
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) −7.00000 −1.20049
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 5.00000 0.811107
\(39\) −5.00000 −0.800641
\(40\) 2.00000 0.316228
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 2.00000 0.308607
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 3.00000 0.452267
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 7.00000 0.980196
\(52\) −5.00000 −0.693375
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.00000 −0.809040
\(56\) 2.00000 0.267261
\(57\) −5.00000 −0.662266
\(58\) 3.00000 0.393919
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) −2.00000 −0.258199
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 7.00000 0.889001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 10.0000 1.24035
\(66\) −3.00000 −0.369274
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 7.00000 0.848875
\(69\) −4.00000 −0.481543
\(70\) −4.00000 −0.478091
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 8.00000 0.929981
\(75\) −1.00000 −0.115470
\(76\) −5.00000 −0.573539
\(77\) −6.00000 −0.683763
\(78\) 5.00000 0.566139
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −2.00000 −0.218218
\(85\) −14.0000 −1.51851
\(86\) 2.00000 0.215666
\(87\) −3.00000 −0.321634
\(88\) −3.00000 −0.319801
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 2.00000 0.210819
\(91\) 10.0000 1.04828
\(92\) −4.00000 −0.417029
\(93\) −7.00000 −0.725866
\(94\) −8.00000 −0.825137
\(95\) 10.0000 1.02598
\(96\) −1.00000 −0.102062
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 3.00000 0.303046
\(99\) 3.00000 0.301511
\(100\) −1.00000 −0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) −7.00000 −0.693103
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 5.00000 0.490290
\(105\) 4.00000 0.390360
\(106\) −4.00000 −0.388514
\(107\) 5.00000 0.483368 0.241684 0.970355i \(-0.422300\pi\)
0.241684 + 0.970355i \(0.422300\pi\)
\(108\) 1.00000 0.0962250
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 6.00000 0.572078
\(111\) −8.00000 −0.759326
\(112\) −2.00000 −0.188982
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 5.00000 0.468293
\(115\) 8.00000 0.746004
\(116\) −3.00000 −0.278543
\(117\) −5.00000 −0.462250
\(118\) 3.00000 0.276172
\(119\) −14.0000 −1.28338
\(120\) 2.00000 0.182574
\(121\) −2.00000 −0.181818
\(122\) −13.0000 −1.17696
\(123\) −12.0000 −1.08200
\(124\) −7.00000 −0.628619
\(125\) 12.0000 1.07331
\(126\) 2.00000 0.178174
\(127\) −19.0000 −1.68598 −0.842989 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) −10.0000 −0.877058
\(131\) 1.00000 0.0873704
\(132\) 3.00000 0.261116
\(133\) 10.0000 0.867110
\(134\) −8.00000 −0.691095
\(135\) −2.00000 −0.172133
\(136\) −7.00000 −0.600245
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 4.00000 0.340503
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 4.00000 0.338062
\(141\) 8.00000 0.673722
\(142\) 16.0000 1.34269
\(143\) −15.0000 −1.25436
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 2.00000 0.165521
\(147\) −3.00000 −0.247436
\(148\) −8.00000 −0.657596
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 1.00000 0.0816497
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 5.00000 0.405554
\(153\) 7.00000 0.565916
\(154\) 6.00000 0.483494
\(155\) 14.0000 1.12451
\(156\) −5.00000 −0.400320
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) 2.00000 0.158114
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) −3.00000 −0.234978 −0.117489 0.993074i \(-0.537485\pi\)
−0.117489 + 0.993074i \(0.537485\pi\)
\(164\) −12.0000 −0.937043
\(165\) −6.00000 −0.467099
\(166\) −6.00000 −0.465690
\(167\) 11.0000 0.851206 0.425603 0.904910i \(-0.360062\pi\)
0.425603 + 0.904910i \(0.360062\pi\)
\(168\) 2.00000 0.154303
\(169\) 12.0000 0.923077
\(170\) 14.0000 1.07375
\(171\) −5.00000 −0.382360
\(172\) −2.00000 −0.152499
\(173\) 1.00000 0.0760286 0.0380143 0.999277i \(-0.487897\pi\)
0.0380143 + 0.999277i \(0.487897\pi\)
\(174\) 3.00000 0.227429
\(175\) 2.00000 0.151186
\(176\) 3.00000 0.226134
\(177\) −3.00000 −0.225494
\(178\) 14.0000 1.04934
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −2.00000 −0.149071
\(181\) −4.00000 −0.297318 −0.148659 0.988889i \(-0.547496\pi\)
−0.148659 + 0.988889i \(0.547496\pi\)
\(182\) −10.0000 −0.741249
\(183\) 13.0000 0.960988
\(184\) 4.00000 0.294884
\(185\) 16.0000 1.17634
\(186\) 7.00000 0.513265
\(187\) 21.0000 1.53567
\(188\) 8.00000 0.583460
\(189\) −2.00000 −0.145479
\(190\) −10.0000 −0.725476
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 1.00000 0.0721688
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) −12.0000 −0.861550
\(195\) 10.0000 0.716115
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −3.00000 −0.213201
\(199\) 13.0000 0.921546 0.460773 0.887518i \(-0.347572\pi\)
0.460773 + 0.887518i \(0.347572\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) 18.0000 1.26648
\(203\) 6.00000 0.421117
\(204\) 7.00000 0.490098
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) −5.00000 −0.346688
\(209\) −15.0000 −1.03757
\(210\) −4.00000 −0.276026
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 4.00000 0.274721
\(213\) −16.0000 −1.09630
\(214\) −5.00000 −0.341793
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 14.0000 0.950382
\(218\) −3.00000 −0.203186
\(219\) −2.00000 −0.135147
\(220\) −6.00000 −0.404520
\(221\) −35.0000 −2.35435
\(222\) 8.00000 0.536925
\(223\) −17.0000 −1.13840 −0.569202 0.822198i \(-0.692748\pi\)
−0.569202 + 0.822198i \(0.692748\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) −12.0000 −0.798228
\(227\) −26.0000 −1.72568 −0.862840 0.505477i \(-0.831317\pi\)
−0.862840 + 0.505477i \(0.831317\pi\)
\(228\) −5.00000 −0.331133
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) −8.00000 −0.527504
\(231\) −6.00000 −0.394771
\(232\) 3.00000 0.196960
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 5.00000 0.326860
\(235\) −16.0000 −1.04372
\(236\) −3.00000 −0.195283
\(237\) 0 0
\(238\) 14.0000 0.907485
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −2.00000 −0.129099
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 13.0000 0.832240
\(245\) 6.00000 0.383326
\(246\) 12.0000 0.765092
\(247\) 25.0000 1.59071
\(248\) 7.00000 0.444500
\(249\) 6.00000 0.380235
\(250\) −12.0000 −0.758947
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) −2.00000 −0.125988
\(253\) −12.0000 −0.754434
\(254\) 19.0000 1.19217
\(255\) −14.0000 −0.876714
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 2.00000 0.124515
\(259\) 16.0000 0.994192
\(260\) 10.0000 0.620174
\(261\) −3.00000 −0.185695
\(262\) −1.00000 −0.0617802
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) −3.00000 −0.184637
\(265\) −8.00000 −0.491436
\(266\) −10.0000 −0.613139
\(267\) −14.0000 −0.856786
\(268\) 8.00000 0.488678
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 2.00000 0.121716
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 7.00000 0.424437
\(273\) 10.0000 0.605228
\(274\) −5.00000 −0.302061
\(275\) −3.00000 −0.180907
\(276\) −4.00000 −0.240772
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 4.00000 0.239904
\(279\) −7.00000 −0.419079
\(280\) −4.00000 −0.239046
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −8.00000 −0.476393
\(283\) −26.0000 −1.54554 −0.772770 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) −16.0000 −0.949425
\(285\) 10.0000 0.592349
\(286\) 15.0000 0.886969
\(287\) 24.0000 1.41668
\(288\) −1.00000 −0.0589256
\(289\) 32.0000 1.88235
\(290\) −6.00000 −0.352332
\(291\) 12.0000 0.703452
\(292\) −2.00000 −0.117041
\(293\) −15.0000 −0.876309 −0.438155 0.898900i \(-0.644368\pi\)
−0.438155 + 0.898900i \(0.644368\pi\)
\(294\) 3.00000 0.174964
\(295\) 6.00000 0.349334
\(296\) 8.00000 0.464991
\(297\) 3.00000 0.174078
\(298\) −21.0000 −1.21650
\(299\) 20.0000 1.15663
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 20.0000 1.15087
\(303\) −18.0000 −1.03407
\(304\) −5.00000 −0.286770
\(305\) −26.0000 −1.48876
\(306\) −7.00000 −0.400163
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) −14.0000 −0.795147
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 5.00000 0.283069
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) −18.0000 −1.01580
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −4.00000 −0.224309
\(319\) −9.00000 −0.503903
\(320\) −2.00000 −0.111803
\(321\) 5.00000 0.279073
\(322\) −8.00000 −0.445823
\(323\) −35.0000 −1.94745
\(324\) 1.00000 0.0555556
\(325\) 5.00000 0.277350
\(326\) 3.00000 0.166155
\(327\) 3.00000 0.165900
\(328\) 12.0000 0.662589
\(329\) −16.0000 −0.882109
\(330\) 6.00000 0.330289
\(331\) −11.0000 −0.604615 −0.302307 0.953211i \(-0.597757\pi\)
−0.302307 + 0.953211i \(0.597757\pi\)
\(332\) 6.00000 0.329293
\(333\) −8.00000 −0.438397
\(334\) −11.0000 −0.601893
\(335\) −16.0000 −0.874173
\(336\) −2.00000 −0.109109
\(337\) −31.0000 −1.68868 −0.844339 0.535810i \(-0.820006\pi\)
−0.844339 + 0.535810i \(0.820006\pi\)
\(338\) −12.0000 −0.652714
\(339\) 12.0000 0.651751
\(340\) −14.0000 −0.759257
\(341\) −21.0000 −1.13721
\(342\) 5.00000 0.270369
\(343\) 20.0000 1.07990
\(344\) 2.00000 0.107833
\(345\) 8.00000 0.430706
\(346\) −1.00000 −0.0537603
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) −3.00000 −0.160817
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −2.00000 −0.106904
\(351\) −5.00000 −0.266880
\(352\) −3.00000 −0.159901
\(353\) −4.00000 −0.212899 −0.106449 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) 3.00000 0.159448
\(355\) 32.0000 1.69838
\(356\) −14.0000 −0.741999
\(357\) −14.0000 −0.740959
\(358\) −20.0000 −1.05703
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 2.00000 0.105409
\(361\) 6.00000 0.315789
\(362\) 4.00000 0.210235
\(363\) −2.00000 −0.104973
\(364\) 10.0000 0.524142
\(365\) 4.00000 0.209370
\(366\) −13.0000 −0.679521
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) −4.00000 −0.208514
\(369\) −12.0000 −0.624695
\(370\) −16.0000 −0.831800
\(371\) −8.00000 −0.415339
\(372\) −7.00000 −0.362933
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −21.0000 −1.08588
\(375\) 12.0000 0.619677
\(376\) −8.00000 −0.412568
\(377\) 15.0000 0.772539
\(378\) 2.00000 0.102869
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 10.0000 0.512989
\(381\) −19.0000 −0.973399
\(382\) −13.0000 −0.665138
\(383\) −23.0000 −1.17525 −0.587623 0.809135i \(-0.699936\pi\)
−0.587623 + 0.809135i \(0.699936\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 12.0000 0.611577
\(386\) −11.0000 −0.559885
\(387\) −2.00000 −0.101666
\(388\) 12.0000 0.609208
\(389\) 31.0000 1.57176 0.785881 0.618378i \(-0.212210\pi\)
0.785881 + 0.618378i \(0.212210\pi\)
\(390\) −10.0000 −0.506370
\(391\) −28.0000 −1.41602
\(392\) 3.00000 0.151523
\(393\) 1.00000 0.0504433
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −13.0000 −0.651631
\(399\) 10.0000 0.500626
\(400\) −1.00000 −0.0500000
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) −8.00000 −0.399004
\(403\) 35.0000 1.74347
\(404\) −18.0000 −0.895533
\(405\) −2.00000 −0.0993808
\(406\) −6.00000 −0.297775
\(407\) −24.0000 −1.18964
\(408\) −7.00000 −0.346552
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −24.0000 −1.18528
\(411\) 5.00000 0.246632
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 4.00000 0.196589
\(415\) −12.0000 −0.589057
\(416\) 5.00000 0.245145
\(417\) −4.00000 −0.195881
\(418\) 15.0000 0.733674
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 4.00000 0.195180
\(421\) 37.0000 1.80327 0.901635 0.432498i \(-0.142368\pi\)
0.901635 + 0.432498i \(0.142368\pi\)
\(422\) 10.0000 0.486792
\(423\) 8.00000 0.388973
\(424\) −4.00000 −0.194257
\(425\) −7.00000 −0.339550
\(426\) 16.0000 0.775203
\(427\) −26.0000 −1.25823
\(428\) 5.00000 0.241684
\(429\) −15.0000 −0.724207
\(430\) −4.00000 −0.192897
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) −14.0000 −0.672022
\(435\) 6.00000 0.287678
\(436\) 3.00000 0.143674
\(437\) 20.0000 0.956730
\(438\) 2.00000 0.0955637
\(439\) −34.0000 −1.62273 −0.811366 0.584539i \(-0.801275\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) 6.00000 0.286039
\(441\) −3.00000 −0.142857
\(442\) 35.0000 1.66478
\(443\) −14.0000 −0.665160 −0.332580 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) −8.00000 −0.379663
\(445\) 28.0000 1.32733
\(446\) 17.0000 0.804973
\(447\) 21.0000 0.993266
\(448\) −2.00000 −0.0944911
\(449\) 11.0000 0.519122 0.259561 0.965727i \(-0.416422\pi\)
0.259561 + 0.965727i \(0.416422\pi\)
\(450\) 1.00000 0.0471405
\(451\) −36.0000 −1.69517
\(452\) 12.0000 0.564433
\(453\) −20.0000 −0.939682
\(454\) 26.0000 1.22024
\(455\) −20.0000 −0.937614
\(456\) 5.00000 0.234146
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −4.00000 −0.186908
\(459\) 7.00000 0.326732
\(460\) 8.00000 0.373002
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 6.00000 0.279145
\(463\) 7.00000 0.325318 0.162659 0.986682i \(-0.447993\pi\)
0.162659 + 0.986682i \(0.447993\pi\)
\(464\) −3.00000 −0.139272
\(465\) 14.0000 0.649234
\(466\) 12.0000 0.555889
\(467\) −17.0000 −0.786666 −0.393333 0.919396i \(-0.628678\pi\)
−0.393333 + 0.919396i \(0.628678\pi\)
\(468\) −5.00000 −0.231125
\(469\) −16.0000 −0.738811
\(470\) 16.0000 0.738025
\(471\) 18.0000 0.829396
\(472\) 3.00000 0.138086
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) −14.0000 −0.641689
\(477\) 4.00000 0.183147
\(478\) −24.0000 −1.09773
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 2.00000 0.0912871
\(481\) 40.0000 1.82384
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) −2.00000 −0.0909091
\(485\) −24.0000 −1.08978
\(486\) −1.00000 −0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −13.0000 −0.588482
\(489\) −3.00000 −0.135665
\(490\) −6.00000 −0.271052
\(491\) −34.0000 −1.53440 −0.767199 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) −12.0000 −0.541002
\(493\) −21.0000 −0.945792
\(494\) −25.0000 −1.12480
\(495\) −6.00000 −0.269680
\(496\) −7.00000 −0.314309
\(497\) 32.0000 1.43540
\(498\) −6.00000 −0.268866
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 12.0000 0.536656
\(501\) 11.0000 0.491444
\(502\) 18.0000 0.803379
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 2.00000 0.0890871
\(505\) 36.0000 1.60198
\(506\) 12.0000 0.533465
\(507\) 12.0000 0.532939
\(508\) −19.0000 −0.842989
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 14.0000 0.619930
\(511\) 4.00000 0.176950
\(512\) −1.00000 −0.0441942
\(513\) −5.00000 −0.220755
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) 24.0000 1.05552
\(518\) −16.0000 −0.703000
\(519\) 1.00000 0.0438951
\(520\) −10.0000 −0.438529
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 3.00000 0.131306
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 1.00000 0.0436852
\(525\) 2.00000 0.0872872
\(526\) −21.0000 −0.915644
\(527\) −49.0000 −2.13447
\(528\) 3.00000 0.130558
\(529\) −7.00000 −0.304348
\(530\) 8.00000 0.347498
\(531\) −3.00000 −0.130189
\(532\) 10.0000 0.433555
\(533\) 60.0000 2.59889
\(534\) 14.0000 0.605839
\(535\) −10.0000 −0.432338
\(536\) −8.00000 −0.345547
\(537\) 20.0000 0.863064
\(538\) −16.0000 −0.689809
\(539\) −9.00000 −0.387657
\(540\) −2.00000 −0.0860663
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 20.0000 0.859074
\(543\) −4.00000 −0.171656
\(544\) −7.00000 −0.300123
\(545\) −6.00000 −0.257012
\(546\) −10.0000 −0.427960
\(547\) −35.0000 −1.49649 −0.748246 0.663421i \(-0.769104\pi\)
−0.748246 + 0.663421i \(0.769104\pi\)
\(548\) 5.00000 0.213589
\(549\) 13.0000 0.554826
\(550\) 3.00000 0.127920
\(551\) 15.0000 0.639021
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) −19.0000 −0.807233
\(555\) 16.0000 0.679162
\(556\) −4.00000 −0.169638
\(557\) 16.0000 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) 7.00000 0.296334
\(559\) 10.0000 0.422955
\(560\) 4.00000 0.169031
\(561\) 21.0000 0.886621
\(562\) −6.00000 −0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 8.00000 0.336861
\(565\) −24.0000 −1.00969
\(566\) 26.0000 1.09286
\(567\) −2.00000 −0.0839921
\(568\) 16.0000 0.671345
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −10.0000 −0.418854
\(571\) −11.0000 −0.460336 −0.230168 0.973151i \(-0.573928\pi\)
−0.230168 + 0.973151i \(0.573928\pi\)
\(572\) −15.0000 −0.627182
\(573\) 13.0000 0.543083
\(574\) −24.0000 −1.00174
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −11.0000 −0.457936 −0.228968 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) −32.0000 −1.33102
\(579\) 11.0000 0.457144
\(580\) 6.00000 0.249136
\(581\) −12.0000 −0.497844
\(582\) −12.0000 −0.497416
\(583\) 12.0000 0.496989
\(584\) 2.00000 0.0827606
\(585\) 10.0000 0.413449
\(586\) 15.0000 0.619644
\(587\) 17.0000 0.701665 0.350833 0.936438i \(-0.385899\pi\)
0.350833 + 0.936438i \(0.385899\pi\)
\(588\) −3.00000 −0.123718
\(589\) 35.0000 1.44215
\(590\) −6.00000 −0.247016
\(591\) 6.00000 0.246807
\(592\) −8.00000 −0.328798
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) −3.00000 −0.123091
\(595\) 28.0000 1.14789
\(596\) 21.0000 0.860194
\(597\) 13.0000 0.532055
\(598\) −20.0000 −0.817861
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 1.00000 0.0408248
\(601\) 45.0000 1.83559 0.917794 0.397057i \(-0.129968\pi\)
0.917794 + 0.397057i \(0.129968\pi\)
\(602\) −4.00000 −0.163028
\(603\) 8.00000 0.325785
\(604\) −20.0000 −0.813788
\(605\) 4.00000 0.162623
\(606\) 18.0000 0.731200
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 5.00000 0.202777
\(609\) 6.00000 0.243132
\(610\) 26.0000 1.05271
\(611\) −40.0000 −1.61823
\(612\) 7.00000 0.282958
\(613\) −11.0000 −0.444286 −0.222143 0.975014i \(-0.571305\pi\)
−0.222143 + 0.975014i \(0.571305\pi\)
\(614\) 26.0000 1.04927
\(615\) 24.0000 0.967773
\(616\) 6.00000 0.241747
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −45.0000 −1.80870 −0.904351 0.426789i \(-0.859645\pi\)
−0.904351 + 0.426789i \(0.859645\pi\)
\(620\) 14.0000 0.562254
\(621\) −4.00000 −0.160514
\(622\) 7.00000 0.280674
\(623\) 28.0000 1.12180
\(624\) −5.00000 −0.200160
\(625\) −19.0000 −0.760000
\(626\) −24.0000 −0.959233
\(627\) −15.0000 −0.599042
\(628\) 18.0000 0.718278
\(629\) −56.0000 −2.23287
\(630\) −4.00000 −0.159364
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) −10.0000 −0.397464
\(634\) −2.00000 −0.0794301
\(635\) 38.0000 1.50798
\(636\) 4.00000 0.158610
\(637\) 15.0000 0.594322
\(638\) 9.00000 0.356313
\(639\) −16.0000 −0.632950
\(640\) 2.00000 0.0790569
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −5.00000 −0.197334
\(643\) −9.00000 −0.354925 −0.177463 0.984128i \(-0.556789\pi\)
−0.177463 + 0.984128i \(0.556789\pi\)
\(644\) 8.00000 0.315244
\(645\) 4.00000 0.157500
\(646\) 35.0000 1.37706
\(647\) −1.00000 −0.0393141 −0.0196570 0.999807i \(-0.506257\pi\)
−0.0196570 + 0.999807i \(0.506257\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −9.00000 −0.353281
\(650\) −5.00000 −0.196116
\(651\) 14.0000 0.548703
\(652\) −3.00000 −0.117489
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −3.00000 −0.117309
\(655\) −2.00000 −0.0781465
\(656\) −12.0000 −0.468521
\(657\) −2.00000 −0.0780274
\(658\) 16.0000 0.623745
\(659\) 19.0000 0.740135 0.370067 0.929005i \(-0.379335\pi\)
0.370067 + 0.929005i \(0.379335\pi\)
\(660\) −6.00000 −0.233550
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 11.0000 0.427527
\(663\) −35.0000 −1.35929
\(664\) −6.00000 −0.232845
\(665\) −20.0000 −0.775567
\(666\) 8.00000 0.309994
\(667\) 12.0000 0.464642
\(668\) 11.0000 0.425603
\(669\) −17.0000 −0.657258
\(670\) 16.0000 0.618134
\(671\) 39.0000 1.50558
\(672\) 2.00000 0.0771517
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 31.0000 1.19408
\(675\) −1.00000 −0.0384900
\(676\) 12.0000 0.461538
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −12.0000 −0.460857
\(679\) −24.0000 −0.921035
\(680\) 14.0000 0.536875
\(681\) −26.0000 −0.996322
\(682\) 21.0000 0.804132
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) −5.00000 −0.191180
\(685\) −10.0000 −0.382080
\(686\) −20.0000 −0.763604
\(687\) 4.00000 0.152610
\(688\) −2.00000 −0.0762493
\(689\) −20.0000 −0.761939
\(690\) −8.00000 −0.304555
\(691\) 48.0000 1.82601 0.913003 0.407953i \(-0.133757\pi\)
0.913003 + 0.407953i \(0.133757\pi\)
\(692\) 1.00000 0.0380143
\(693\) −6.00000 −0.227921
\(694\) −6.00000 −0.227757
\(695\) 8.00000 0.303457
\(696\) 3.00000 0.113715
\(697\) −84.0000 −3.18173
\(698\) −10.0000 −0.378506
\(699\) −12.0000 −0.453882
\(700\) 2.00000 0.0755929
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 5.00000 0.188713
\(703\) 40.0000 1.50863
\(704\) 3.00000 0.113067
\(705\) −16.0000 −0.602595
\(706\) 4.00000 0.150542
\(707\) 36.0000 1.35392
\(708\) −3.00000 −0.112747
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −32.0000 −1.20094
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) 28.0000 1.04861
\(714\) 14.0000 0.523937
\(715\) 30.0000 1.12194
\(716\) 20.0000 0.747435
\(717\) 24.0000 0.896296
\(718\) 18.0000 0.671754
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −6.00000 −0.223297
\(723\) 0 0
\(724\) −4.00000 −0.148659
\(725\) 3.00000 0.111417
\(726\) 2.00000 0.0742270
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) −10.0000 −0.370625
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) −14.0000 −0.517809
\(732\) 13.0000 0.480494
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 20.0000 0.738213
\(735\) 6.00000 0.221313
\(736\) 4.00000 0.147442
\(737\) 24.0000 0.884051
\(738\) 12.0000 0.441726
\(739\) 42.0000 1.54499 0.772497 0.635018i \(-0.219007\pi\)
0.772497 + 0.635018i \(0.219007\pi\)
\(740\) 16.0000 0.588172
\(741\) 25.0000 0.918398
\(742\) 8.00000 0.293689
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 7.00000 0.256632
\(745\) −42.0000 −1.53876
\(746\) 22.0000 0.805477
\(747\) 6.00000 0.219529
\(748\) 21.0000 0.767836
\(749\) −10.0000 −0.365392
\(750\) −12.0000 −0.438178
\(751\) −33.0000 −1.20419 −0.602094 0.798426i \(-0.705667\pi\)
−0.602094 + 0.798426i \(0.705667\pi\)
\(752\) 8.00000 0.291730
\(753\) −18.0000 −0.655956
\(754\) −15.0000 −0.546268
\(755\) 40.0000 1.45575
\(756\) −2.00000 −0.0727393
\(757\) 19.0000 0.690567 0.345283 0.938498i \(-0.387783\pi\)
0.345283 + 0.938498i \(0.387783\pi\)
\(758\) −14.0000 −0.508503
\(759\) −12.0000 −0.435572
\(760\) −10.0000 −0.362738
\(761\) 7.00000 0.253750 0.126875 0.991919i \(-0.459505\pi\)
0.126875 + 0.991919i \(0.459505\pi\)
\(762\) 19.0000 0.688297
\(763\) −6.00000 −0.217215
\(764\) 13.0000 0.470323
\(765\) −14.0000 −0.506171
\(766\) 23.0000 0.831024
\(767\) 15.0000 0.541619
\(768\) 1.00000 0.0360844
\(769\) −41.0000 −1.47850 −0.739249 0.673432i \(-0.764819\pi\)
−0.739249 + 0.673432i \(0.764819\pi\)
\(770\) −12.0000 −0.432450
\(771\) −6.00000 −0.216085
\(772\) 11.0000 0.395899
\(773\) 55.0000 1.97821 0.989106 0.147203i \(-0.0470272\pi\)
0.989106 + 0.147203i \(0.0470272\pi\)
\(774\) 2.00000 0.0718885
\(775\) 7.00000 0.251447
\(776\) −12.0000 −0.430775
\(777\) 16.0000 0.573997
\(778\) −31.0000 −1.11140
\(779\) 60.0000 2.14972
\(780\) 10.0000 0.358057
\(781\) −48.0000 −1.71758
\(782\) 28.0000 1.00128
\(783\) −3.00000 −0.107211
\(784\) −3.00000 −0.107143
\(785\) −36.0000 −1.28490
\(786\) −1.00000 −0.0356688
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) 6.00000 0.213741
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) −3.00000 −0.106600
\(793\) −65.0000 −2.30822
\(794\) −2.00000 −0.0709773
\(795\) −8.00000 −0.283731
\(796\) 13.0000 0.460773
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) −10.0000 −0.353996
\(799\) 56.0000 1.98114
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) −27.0000 −0.953403
\(803\) −6.00000 −0.211735
\(804\) 8.00000 0.282138
\(805\) −16.0000 −0.563926
\(806\) −35.0000 −1.23282
\(807\) 16.0000 0.563227
\(808\) 18.0000 0.633238
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 2.00000 0.0702728
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 6.00000 0.210559
\(813\) −20.0000 −0.701431
\(814\) 24.0000 0.841200
\(815\) 6.00000 0.210171
\(816\) 7.00000 0.245049
\(817\) 10.0000 0.349856
\(818\) 22.0000 0.769212
\(819\) 10.0000 0.349428
\(820\) 24.0000 0.838116
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) −5.00000 −0.174395
\(823\) 37.0000 1.28974 0.644869 0.764293i \(-0.276912\pi\)
0.644869 + 0.764293i \(0.276912\pi\)
\(824\) 0 0
\(825\) −3.00000 −0.104447
\(826\) −6.00000 −0.208767
\(827\) −1.00000 −0.0347734 −0.0173867 0.999849i \(-0.505535\pi\)
−0.0173867 + 0.999849i \(0.505535\pi\)
\(828\) −4.00000 −0.139010
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 12.0000 0.416526
\(831\) 19.0000 0.659103
\(832\) −5.00000 −0.173344
\(833\) −21.0000 −0.727607
\(834\) 4.00000 0.138509
\(835\) −22.0000 −0.761341
\(836\) −15.0000 −0.518786
\(837\) −7.00000 −0.241955
\(838\) −12.0000 −0.414533
\(839\) −3.00000 −0.103572 −0.0517858 0.998658i \(-0.516491\pi\)
−0.0517858 + 0.998658i \(0.516491\pi\)
\(840\) −4.00000 −0.138013
\(841\) −20.0000 −0.689655
\(842\) −37.0000 −1.27510
\(843\) 6.00000 0.206651
\(844\) −10.0000 −0.344214
\(845\) −24.0000 −0.825625
\(846\) −8.00000 −0.275046
\(847\) 4.00000 0.137442
\(848\) 4.00000 0.137361
\(849\) −26.0000 −0.892318
\(850\) 7.00000 0.240098
\(851\) 32.0000 1.09695
\(852\) −16.0000 −0.548151
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 26.0000 0.889702
\(855\) 10.0000 0.341993
\(856\) −5.00000 −0.170896
\(857\) 31.0000 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(858\) 15.0000 0.512092
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 4.00000 0.136399
\(861\) 24.0000 0.817918
\(862\) 12.0000 0.408722
\(863\) 3.00000 0.102121 0.0510606 0.998696i \(-0.483740\pi\)
0.0510606 + 0.998696i \(0.483740\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −2.00000 −0.0680020
\(866\) 10.0000 0.339814
\(867\) 32.0000 1.08678
\(868\) 14.0000 0.475191
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) −40.0000 −1.35535
\(872\) −3.00000 −0.101593
\(873\) 12.0000 0.406138
\(874\) −20.0000 −0.676510
\(875\) −24.0000 −0.811348
\(876\) −2.00000 −0.0675737
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 34.0000 1.14744
\(879\) −15.0000 −0.505937
\(880\) −6.00000 −0.202260
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 3.00000 0.101015
\(883\) 13.0000 0.437485 0.218742 0.975783i \(-0.429805\pi\)
0.218742 + 0.975783i \(0.429805\pi\)
\(884\) −35.0000 −1.17718
\(885\) 6.00000 0.201688
\(886\) 14.0000 0.470339
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 8.00000 0.268462
\(889\) 38.0000 1.27448
\(890\) −28.0000 −0.938562
\(891\) 3.00000 0.100504
\(892\) −17.0000 −0.569202
\(893\) −40.0000 −1.33855
\(894\) −21.0000 −0.702345
\(895\) −40.0000 −1.33705
\(896\) 2.00000 0.0668153
\(897\) 20.0000 0.667781
\(898\) −11.0000 −0.367075
\(899\) 21.0000 0.700389
\(900\) −1.00000 −0.0333333
\(901\) 28.0000 0.932815
\(902\) 36.0000 1.19867
\(903\) 4.00000 0.133112
\(904\) −12.0000 −0.399114
\(905\) 8.00000 0.265929
\(906\) 20.0000 0.664455
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) −26.0000 −0.862840
\(909\) −18.0000 −0.597022
\(910\) 20.0000 0.662994
\(911\) 1.00000 0.0331315 0.0165657 0.999863i \(-0.494727\pi\)
0.0165657 + 0.999863i \(0.494727\pi\)
\(912\) −5.00000 −0.165567
\(913\) 18.0000 0.595713
\(914\) −6.00000 −0.198462
\(915\) −26.0000 −0.859533
\(916\) 4.00000 0.132164
\(917\) −2.00000 −0.0660458
\(918\) −7.00000 −0.231034
\(919\) −13.0000 −0.428830 −0.214415 0.976743i \(-0.568785\pi\)
−0.214415 + 0.976743i \(0.568785\pi\)
\(920\) −8.00000 −0.263752
\(921\) −26.0000 −0.856729
\(922\) 6.00000 0.197599
\(923\) 80.0000 2.63323
\(924\) −6.00000 −0.197386
\(925\) 8.00000 0.263038
\(926\) −7.00000 −0.230034
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) −14.0000 −0.459078
\(931\) 15.0000 0.491605
\(932\) −12.0000 −0.393073
\(933\) −7.00000 −0.229170
\(934\) 17.0000 0.556257
\(935\) −42.0000 −1.37355
\(936\) 5.00000 0.163430
\(937\) 49.0000 1.60076 0.800380 0.599493i \(-0.204631\pi\)
0.800380 + 0.599493i \(0.204631\pi\)
\(938\) 16.0000 0.522419
\(939\) 24.0000 0.783210
\(940\) −16.0000 −0.521862
\(941\) −45.0000 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(942\) −18.0000 −0.586472
\(943\) 48.0000 1.56310
\(944\) −3.00000 −0.0976417
\(945\) 4.00000 0.130120
\(946\) 6.00000 0.195077
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) −5.00000 −0.162221
\(951\) 2.00000 0.0648544
\(952\) 14.0000 0.453743
\(953\) −48.0000 −1.55487 −0.777436 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(954\) −4.00000 −0.129505
\(955\) −26.0000 −0.841340
\(956\) 24.0000 0.776215
\(957\) −9.00000 −0.290929
\(958\) 28.0000 0.904639
\(959\) −10.0000 −0.322917
\(960\) −2.00000 −0.0645497
\(961\) 18.0000 0.580645
\(962\) −40.0000 −1.28965
\(963\) 5.00000 0.161123
\(964\) 0 0
\(965\) −22.0000 −0.708205
\(966\) −8.00000 −0.257396
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) 2.00000 0.0642824
\(969\) −35.0000 −1.12436
\(970\) 24.0000 0.770594
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.00000 0.256468
\(974\) −28.0000 −0.897178
\(975\) 5.00000 0.160128
\(976\) 13.0000 0.416120
\(977\) 36.0000 1.15174 0.575871 0.817541i \(-0.304663\pi\)
0.575871 + 0.817541i \(0.304663\pi\)
\(978\) 3.00000 0.0959294
\(979\) −42.0000 −1.34233
\(980\) 6.00000 0.191663
\(981\) 3.00000 0.0957826
\(982\) 34.0000 1.08498
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 12.0000 0.382546
\(985\) −12.0000 −0.382352
\(986\) 21.0000 0.668776
\(987\) −16.0000 −0.509286
\(988\) 25.0000 0.795356
\(989\) 8.00000 0.254385
\(990\) 6.00000 0.190693
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) 7.00000 0.222250
\(993\) −11.0000 −0.349074
\(994\) −32.0000 −1.01498
\(995\) −26.0000 −0.824255
\(996\) 6.00000 0.190117
\(997\) −13.0000 −0.411714 −0.205857 0.978582i \(-0.565998\pi\)
−0.205857 + 0.978582i \(0.565998\pi\)
\(998\) −16.0000 −0.506471
\(999\) −8.00000 −0.253109
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 786.2.a.g.1.1 1
3.2 odd 2 2358.2.a.v.1.1 1
4.3 odd 2 6288.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
786.2.a.g.1.1 1 1.1 even 1 trivial
2358.2.a.v.1.1 1 3.2 odd 2
6288.2.a.c.1.1 1 4.3 odd 2