Properties

Label 8007.2.a.f.1.31
Level 8007
Weight 2
Character 8007.1
Self dual yes
Analytic conductor 63.936
Analytic rank 1
Dimension 48
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8007.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9362168984\)
Analytic rank: \(1\)
Dimension: \(48\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.31
Character \(\chi\) = 8007.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.901623 q^{2} -1.00000 q^{3} -1.18708 q^{4} -2.82523 q^{5} -0.901623 q^{6} -0.878690 q^{7} -2.87354 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.901623 q^{2} -1.00000 q^{3} -1.18708 q^{4} -2.82523 q^{5} -0.901623 q^{6} -0.878690 q^{7} -2.87354 q^{8} +1.00000 q^{9} -2.54729 q^{10} +4.81398 q^{11} +1.18708 q^{12} -5.14267 q^{13} -0.792247 q^{14} +2.82523 q^{15} -0.216697 q^{16} -1.00000 q^{17} +0.901623 q^{18} -0.823997 q^{19} +3.35376 q^{20} +0.878690 q^{21} +4.34039 q^{22} +2.70529 q^{23} +2.87354 q^{24} +2.98191 q^{25} -4.63675 q^{26} -1.00000 q^{27} +1.04307 q^{28} -7.58704 q^{29} +2.54729 q^{30} +6.60216 q^{31} +5.55170 q^{32} -4.81398 q^{33} -0.901623 q^{34} +2.48250 q^{35} -1.18708 q^{36} +7.06009 q^{37} -0.742935 q^{38} +5.14267 q^{39} +8.11840 q^{40} +2.65942 q^{41} +0.792247 q^{42} -2.47244 q^{43} -5.71456 q^{44} -2.82523 q^{45} +2.43915 q^{46} +6.04421 q^{47} +0.216697 q^{48} -6.22790 q^{49} +2.68855 q^{50} +1.00000 q^{51} +6.10474 q^{52} -2.88461 q^{53} -0.901623 q^{54} -13.6006 q^{55} +2.52495 q^{56} +0.823997 q^{57} -6.84065 q^{58} +4.30346 q^{59} -3.35376 q^{60} +12.7081 q^{61} +5.95266 q^{62} -0.878690 q^{63} +5.43894 q^{64} +14.5292 q^{65} -4.34039 q^{66} +5.27025 q^{67} +1.18708 q^{68} -2.70529 q^{69} +2.23828 q^{70} +0.559293 q^{71} -2.87354 q^{72} +8.67777 q^{73} +6.36554 q^{74} -2.98191 q^{75} +0.978147 q^{76} -4.22999 q^{77} +4.63675 q^{78} -11.9113 q^{79} +0.612219 q^{80} +1.00000 q^{81} +2.39780 q^{82} +6.37939 q^{83} -1.04307 q^{84} +2.82523 q^{85} -2.22921 q^{86} +7.58704 q^{87} -13.8332 q^{88} -0.915887 q^{89} -2.54729 q^{90} +4.51881 q^{91} -3.21139 q^{92} -6.60216 q^{93} +5.44960 q^{94} +2.32798 q^{95} -5.55170 q^{96} -0.734920 q^{97} -5.61522 q^{98} +4.81398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} + O(q^{10}) \) \( 48q - q^{2} - 48q^{3} + 45q^{4} + q^{5} + q^{6} - 13q^{7} - 6q^{8} + 48q^{9} - 20q^{10} + 5q^{11} - 45q^{12} - 8q^{13} + 4q^{14} - q^{15} + 39q^{16} - 48q^{17} - q^{18} - 6q^{19} + 6q^{20} + 13q^{21} - 35q^{22} - 8q^{23} + 6q^{24} + 13q^{25} + 17q^{26} - 48q^{27} - 38q^{28} + q^{29} + 20q^{30} - 21q^{31} - 3q^{32} - 5q^{33} + q^{34} + 19q^{35} + 45q^{36} - 58q^{37} - 14q^{38} + 8q^{39} - 54q^{40} - 3q^{41} - 4q^{42} - 33q^{43} + 2q^{44} + q^{45} - 26q^{46} + 9q^{47} - 39q^{48} + 11q^{49} + 4q^{50} + 48q^{51} - 31q^{52} - 33q^{53} + q^{54} - 21q^{55} + 6q^{57} - 55q^{58} + 77q^{59} - 6q^{60} - 29q^{61} - 46q^{62} - 13q^{63} + 24q^{64} - 49q^{65} + 35q^{66} - 44q^{67} - 45q^{68} + 8q^{69} + 4q^{70} + 22q^{71} - 6q^{72} - 63q^{73} - 16q^{74} - 13q^{75} - 46q^{76} - 30q^{77} - 17q^{78} - 46q^{79} - 14q^{80} + 48q^{81} - 75q^{82} + 11q^{83} + 38q^{84} - q^{85} + 8q^{86} - q^{87} - 116q^{88} + 10q^{89} - 20q^{90} - 67q^{91} - 64q^{92} + 21q^{93} - 16q^{94} - 8q^{95} + 3q^{96} - 96q^{97} - 46q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.901623 0.637544 0.318772 0.947831i \(-0.396730\pi\)
0.318772 + 0.947831i \(0.396730\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.18708 −0.593538
\(5\) −2.82523 −1.26348 −0.631740 0.775180i \(-0.717659\pi\)
−0.631740 + 0.775180i \(0.717659\pi\)
\(6\) −0.901623 −0.368086
\(7\) −0.878690 −0.332113 −0.166057 0.986116i \(-0.553103\pi\)
−0.166057 + 0.986116i \(0.553103\pi\)
\(8\) −2.87354 −1.01595
\(9\) 1.00000 0.333333
\(10\) −2.54729 −0.805523
\(11\) 4.81398 1.45147 0.725734 0.687975i \(-0.241500\pi\)
0.725734 + 0.687975i \(0.241500\pi\)
\(12\) 1.18708 0.342679
\(13\) −5.14267 −1.42632 −0.713160 0.701001i \(-0.752737\pi\)
−0.713160 + 0.701001i \(0.752737\pi\)
\(14\) −0.792247 −0.211737
\(15\) 2.82523 0.729470
\(16\) −0.216697 −0.0541743
\(17\) −1.00000 −0.242536
\(18\) 0.901623 0.212515
\(19\) −0.823997 −0.189038 −0.0945190 0.995523i \(-0.530131\pi\)
−0.0945190 + 0.995523i \(0.530131\pi\)
\(20\) 3.35376 0.749923
\(21\) 0.878690 0.191746
\(22\) 4.34039 0.925374
\(23\) 2.70529 0.564092 0.282046 0.959401i \(-0.408987\pi\)
0.282046 + 0.959401i \(0.408987\pi\)
\(24\) 2.87354 0.586559
\(25\) 2.98191 0.596381
\(26\) −4.63675 −0.909342
\(27\) −1.00000 −0.192450
\(28\) 1.04307 0.197122
\(29\) −7.58704 −1.40888 −0.704439 0.709765i \(-0.748801\pi\)
−0.704439 + 0.709765i \(0.748801\pi\)
\(30\) 2.54729 0.465069
\(31\) 6.60216 1.18578 0.592892 0.805282i \(-0.297986\pi\)
0.592892 + 0.805282i \(0.297986\pi\)
\(32\) 5.55170 0.981412
\(33\) −4.81398 −0.838006
\(34\) −0.901623 −0.154627
\(35\) 2.48250 0.419619
\(36\) −1.18708 −0.197846
\(37\) 7.06009 1.16067 0.580336 0.814377i \(-0.302921\pi\)
0.580336 + 0.814377i \(0.302921\pi\)
\(38\) −0.742935 −0.120520
\(39\) 5.14267 0.823487
\(40\) 8.11840 1.28363
\(41\) 2.65942 0.415332 0.207666 0.978200i \(-0.433413\pi\)
0.207666 + 0.978200i \(0.433413\pi\)
\(42\) 0.792247 0.122246
\(43\) −2.47244 −0.377044 −0.188522 0.982069i \(-0.560370\pi\)
−0.188522 + 0.982069i \(0.560370\pi\)
\(44\) −5.71456 −0.861502
\(45\) −2.82523 −0.421160
\(46\) 2.43915 0.359633
\(47\) 6.04421 0.881639 0.440819 0.897596i \(-0.354688\pi\)
0.440819 + 0.897596i \(0.354688\pi\)
\(48\) 0.216697 0.0312776
\(49\) −6.22790 −0.889701
\(50\) 2.68855 0.380219
\(51\) 1.00000 0.140028
\(52\) 6.10474 0.846576
\(53\) −2.88461 −0.396231 −0.198116 0.980179i \(-0.563482\pi\)
−0.198116 + 0.980179i \(0.563482\pi\)
\(54\) −0.901623 −0.122695
\(55\) −13.6006 −1.83390
\(56\) 2.52495 0.337411
\(57\) 0.823997 0.109141
\(58\) −6.84065 −0.898221
\(59\) 4.30346 0.560263 0.280132 0.959962i \(-0.409622\pi\)
0.280132 + 0.959962i \(0.409622\pi\)
\(60\) −3.35376 −0.432968
\(61\) 12.7081 1.62711 0.813555 0.581489i \(-0.197529\pi\)
0.813555 + 0.581489i \(0.197529\pi\)
\(62\) 5.95266 0.755989
\(63\) −0.878690 −0.110704
\(64\) 5.43894 0.679867
\(65\) 14.5292 1.80213
\(66\) −4.34039 −0.534265
\(67\) 5.27025 0.643864 0.321932 0.946763i \(-0.395668\pi\)
0.321932 + 0.946763i \(0.395668\pi\)
\(68\) 1.18708 0.143954
\(69\) −2.70529 −0.325679
\(70\) 2.23828 0.267525
\(71\) 0.559293 0.0663759 0.0331879 0.999449i \(-0.489434\pi\)
0.0331879 + 0.999449i \(0.489434\pi\)
\(72\) −2.87354 −0.338650
\(73\) 8.67777 1.01566 0.507828 0.861458i \(-0.330449\pi\)
0.507828 + 0.861458i \(0.330449\pi\)
\(74\) 6.36554 0.739979
\(75\) −2.98191 −0.344321
\(76\) 0.978147 0.112201
\(77\) −4.22999 −0.482052
\(78\) 4.63675 0.525009
\(79\) −11.9113 −1.34012 −0.670062 0.742305i \(-0.733732\pi\)
−0.670062 + 0.742305i \(0.733732\pi\)
\(80\) 0.612219 0.0684482
\(81\) 1.00000 0.111111
\(82\) 2.39780 0.264792
\(83\) 6.37939 0.700229 0.350114 0.936707i \(-0.386143\pi\)
0.350114 + 0.936707i \(0.386143\pi\)
\(84\) −1.04307 −0.113808
\(85\) 2.82523 0.306439
\(86\) −2.22921 −0.240382
\(87\) 7.58704 0.813416
\(88\) −13.8332 −1.47462
\(89\) −0.915887 −0.0970838 −0.0485419 0.998821i \(-0.515457\pi\)
−0.0485419 + 0.998821i \(0.515457\pi\)
\(90\) −2.54729 −0.268508
\(91\) 4.51881 0.473700
\(92\) −3.21139 −0.334810
\(93\) −6.60216 −0.684612
\(94\) 5.44960 0.562083
\(95\) 2.32798 0.238846
\(96\) −5.55170 −0.566618
\(97\) −0.734920 −0.0746199 −0.0373099 0.999304i \(-0.511879\pi\)
−0.0373099 + 0.999304i \(0.511879\pi\)
\(98\) −5.61522 −0.567223
\(99\) 4.81398 0.483823
\(100\) −3.53975 −0.353975
\(101\) −6.61199 −0.657917 −0.328959 0.944344i \(-0.606698\pi\)
−0.328959 + 0.944344i \(0.606698\pi\)
\(102\) 0.901623 0.0892740
\(103\) −0.797810 −0.0786106 −0.0393053 0.999227i \(-0.512514\pi\)
−0.0393053 + 0.999227i \(0.512514\pi\)
\(104\) 14.7777 1.44907
\(105\) −2.48250 −0.242267
\(106\) −2.60083 −0.252615
\(107\) −9.88807 −0.955916 −0.477958 0.878383i \(-0.658623\pi\)
−0.477958 + 0.878383i \(0.658623\pi\)
\(108\) 1.18708 0.114226
\(109\) 7.86949 0.753761 0.376880 0.926262i \(-0.376997\pi\)
0.376880 + 0.926262i \(0.376997\pi\)
\(110\) −12.2626 −1.16919
\(111\) −7.06009 −0.670114
\(112\) 0.190410 0.0179920
\(113\) 7.60312 0.715241 0.357621 0.933867i \(-0.383588\pi\)
0.357621 + 0.933867i \(0.383588\pi\)
\(114\) 0.742935 0.0695822
\(115\) −7.64306 −0.712719
\(116\) 9.00639 0.836223
\(117\) −5.14267 −0.475440
\(118\) 3.88010 0.357192
\(119\) 0.878690 0.0805493
\(120\) −8.11840 −0.741105
\(121\) 12.1744 1.10676
\(122\) 11.4579 1.03735
\(123\) −2.65942 −0.239792
\(124\) −7.83727 −0.703808
\(125\) 5.70157 0.509964
\(126\) −0.792247 −0.0705789
\(127\) 13.5975 1.20659 0.603293 0.797519i \(-0.293855\pi\)
0.603293 + 0.797519i \(0.293855\pi\)
\(128\) −6.19954 −0.547967
\(129\) 2.47244 0.217687
\(130\) 13.0999 1.14893
\(131\) −6.85124 −0.598596 −0.299298 0.954160i \(-0.596752\pi\)
−0.299298 + 0.954160i \(0.596752\pi\)
\(132\) 5.71456 0.497388
\(133\) 0.724038 0.0627820
\(134\) 4.75178 0.410491
\(135\) 2.82523 0.243157
\(136\) 2.87354 0.246404
\(137\) −13.6270 −1.16424 −0.582119 0.813104i \(-0.697776\pi\)
−0.582119 + 0.813104i \(0.697776\pi\)
\(138\) −2.43915 −0.207634
\(139\) 2.30778 0.195744 0.0978718 0.995199i \(-0.468796\pi\)
0.0978718 + 0.995199i \(0.468796\pi\)
\(140\) −2.94691 −0.249060
\(141\) −6.04421 −0.509014
\(142\) 0.504271 0.0423175
\(143\) −24.7567 −2.07026
\(144\) −0.216697 −0.0180581
\(145\) 21.4351 1.78009
\(146\) 7.82408 0.647525
\(147\) 6.22790 0.513669
\(148\) −8.38087 −0.688903
\(149\) 5.05450 0.414081 0.207040 0.978332i \(-0.433617\pi\)
0.207040 + 0.978332i \(0.433617\pi\)
\(150\) −2.68855 −0.219520
\(151\) −14.5908 −1.18739 −0.593693 0.804692i \(-0.702330\pi\)
−0.593693 + 0.804692i \(0.702330\pi\)
\(152\) 2.36779 0.192053
\(153\) −1.00000 −0.0808452
\(154\) −3.81386 −0.307329
\(155\) −18.6526 −1.49821
\(156\) −6.10474 −0.488771
\(157\) −1.00000 −0.0798087
\(158\) −10.7395 −0.854387
\(159\) 2.88461 0.228764
\(160\) −15.6848 −1.23999
\(161\) −2.37711 −0.187343
\(162\) 0.901623 0.0708382
\(163\) −20.9990 −1.64477 −0.822386 0.568931i \(-0.807357\pi\)
−0.822386 + 0.568931i \(0.807357\pi\)
\(164\) −3.15694 −0.246515
\(165\) 13.6006 1.05880
\(166\) 5.75180 0.446426
\(167\) −15.0163 −1.16200 −0.580998 0.813905i \(-0.697337\pi\)
−0.580998 + 0.813905i \(0.697337\pi\)
\(168\) −2.52495 −0.194804
\(169\) 13.4471 1.03439
\(170\) 2.54729 0.195368
\(171\) −0.823997 −0.0630126
\(172\) 2.93498 0.223790
\(173\) −3.09532 −0.235333 −0.117666 0.993053i \(-0.537541\pi\)
−0.117666 + 0.993053i \(0.537541\pi\)
\(174\) 6.84065 0.518588
\(175\) −2.62017 −0.198066
\(176\) −1.04318 −0.0786324
\(177\) −4.30346 −0.323468
\(178\) −0.825784 −0.0618952
\(179\) −18.7948 −1.40479 −0.702394 0.711789i \(-0.747885\pi\)
−0.702394 + 0.711789i \(0.747885\pi\)
\(180\) 3.35376 0.249974
\(181\) 21.5460 1.60150 0.800751 0.598998i \(-0.204434\pi\)
0.800751 + 0.598998i \(0.204434\pi\)
\(182\) 4.07426 0.302005
\(183\) −12.7081 −0.939412
\(184\) −7.77376 −0.573089
\(185\) −19.9464 −1.46649
\(186\) −5.95266 −0.436470
\(187\) −4.81398 −0.352033
\(188\) −7.17494 −0.523286
\(189\) 0.878690 0.0639153
\(190\) 2.09896 0.152274
\(191\) −14.7815 −1.06955 −0.534775 0.844995i \(-0.679604\pi\)
−0.534775 + 0.844995i \(0.679604\pi\)
\(192\) −5.43894 −0.392521
\(193\) −8.44007 −0.607529 −0.303765 0.952747i \(-0.598244\pi\)
−0.303765 + 0.952747i \(0.598244\pi\)
\(194\) −0.662621 −0.0475734
\(195\) −14.5292 −1.04046
\(196\) 7.39300 0.528071
\(197\) 24.0682 1.71479 0.857396 0.514658i \(-0.172081\pi\)
0.857396 + 0.514658i \(0.172081\pi\)
\(198\) 4.34039 0.308458
\(199\) −3.80884 −0.270002 −0.135001 0.990845i \(-0.543104\pi\)
−0.135001 + 0.990845i \(0.543104\pi\)
\(200\) −8.56863 −0.605893
\(201\) −5.27025 −0.371735
\(202\) −5.96152 −0.419451
\(203\) 6.66665 0.467907
\(204\) −1.18708 −0.0831120
\(205\) −7.51347 −0.524764
\(206\) −0.719324 −0.0501177
\(207\) 2.70529 0.188031
\(208\) 1.11440 0.0772700
\(209\) −3.96670 −0.274383
\(210\) −2.23828 −0.154456
\(211\) −12.9833 −0.893809 −0.446905 0.894582i \(-0.647474\pi\)
−0.446905 + 0.894582i \(0.647474\pi\)
\(212\) 3.42425 0.235178
\(213\) −0.559293 −0.0383221
\(214\) −8.91531 −0.609438
\(215\) 6.98522 0.476388
\(216\) 2.87354 0.195520
\(217\) −5.80125 −0.393815
\(218\) 7.09532 0.480555
\(219\) −8.67777 −0.586389
\(220\) 16.1449 1.08849
\(221\) 5.14267 0.345934
\(222\) −6.36554 −0.427227
\(223\) −12.2485 −0.820219 −0.410110 0.912036i \(-0.634510\pi\)
−0.410110 + 0.912036i \(0.634510\pi\)
\(224\) −4.87822 −0.325940
\(225\) 2.98191 0.198794
\(226\) 6.85514 0.455997
\(227\) 7.45130 0.494560 0.247280 0.968944i \(-0.420463\pi\)
0.247280 + 0.968944i \(0.420463\pi\)
\(228\) −0.978147 −0.0647794
\(229\) −4.65306 −0.307483 −0.153741 0.988111i \(-0.549132\pi\)
−0.153741 + 0.988111i \(0.549132\pi\)
\(230\) −6.89116 −0.454389
\(231\) 4.22999 0.278313
\(232\) 21.8017 1.43135
\(233\) −2.11476 −0.138542 −0.0692712 0.997598i \(-0.522067\pi\)
−0.0692712 + 0.997598i \(0.522067\pi\)
\(234\) −4.63675 −0.303114
\(235\) −17.0763 −1.11393
\(236\) −5.10854 −0.332538
\(237\) 11.9113 0.773721
\(238\) 0.792247 0.0513537
\(239\) −5.97051 −0.386200 −0.193100 0.981179i \(-0.561854\pi\)
−0.193100 + 0.981179i \(0.561854\pi\)
\(240\) −0.612219 −0.0395186
\(241\) 0.110401 0.00711155 0.00355578 0.999994i \(-0.498868\pi\)
0.00355578 + 0.999994i \(0.498868\pi\)
\(242\) 10.9767 0.705608
\(243\) −1.00000 −0.0641500
\(244\) −15.0855 −0.965751
\(245\) 17.5952 1.12412
\(246\) −2.39780 −0.152878
\(247\) 4.23755 0.269629
\(248\) −18.9716 −1.20470
\(249\) −6.37939 −0.404277
\(250\) 5.14067 0.325125
\(251\) −1.45020 −0.0915355 −0.0457678 0.998952i \(-0.514573\pi\)
−0.0457678 + 0.998952i \(0.514573\pi\)
\(252\) 1.04307 0.0657073
\(253\) 13.0232 0.818762
\(254\) 12.2599 0.769252
\(255\) −2.82523 −0.176923
\(256\) −16.4675 −1.02922
\(257\) −10.1713 −0.634467 −0.317233 0.948347i \(-0.602754\pi\)
−0.317233 + 0.948347i \(0.602754\pi\)
\(258\) 2.22921 0.138785
\(259\) −6.20363 −0.385475
\(260\) −17.2473 −1.06963
\(261\) −7.58704 −0.469626
\(262\) −6.17724 −0.381631
\(263\) 2.42569 0.149575 0.0747873 0.997200i \(-0.476172\pi\)
0.0747873 + 0.997200i \(0.476172\pi\)
\(264\) 13.8332 0.851372
\(265\) 8.14967 0.500630
\(266\) 0.652809 0.0400263
\(267\) 0.915887 0.0560514
\(268\) −6.25619 −0.382158
\(269\) 17.9118 1.09210 0.546049 0.837753i \(-0.316131\pi\)
0.546049 + 0.837753i \(0.316131\pi\)
\(270\) 2.54729 0.155023
\(271\) −14.4200 −0.875952 −0.437976 0.898987i \(-0.644304\pi\)
−0.437976 + 0.898987i \(0.644304\pi\)
\(272\) 0.216697 0.0131392
\(273\) −4.51881 −0.273491
\(274\) −12.2865 −0.742252
\(275\) 14.3548 0.865628
\(276\) 3.21139 0.193303
\(277\) −7.40497 −0.444921 −0.222461 0.974942i \(-0.571409\pi\)
−0.222461 + 0.974942i \(0.571409\pi\)
\(278\) 2.08075 0.124795
\(279\) 6.60216 0.395261
\(280\) −7.13356 −0.426312
\(281\) −0.470128 −0.0280455 −0.0140227 0.999902i \(-0.504464\pi\)
−0.0140227 + 0.999902i \(0.504464\pi\)
\(282\) −5.44960 −0.324519
\(283\) 2.75781 0.163935 0.0819673 0.996635i \(-0.473880\pi\)
0.0819673 + 0.996635i \(0.473880\pi\)
\(284\) −0.663924 −0.0393966
\(285\) −2.32798 −0.137898
\(286\) −22.3212 −1.31988
\(287\) −2.33681 −0.137937
\(288\) 5.55170 0.327137
\(289\) 1.00000 0.0588235
\(290\) 19.3264 1.13488
\(291\) 0.734920 0.0430818
\(292\) −10.3012 −0.602831
\(293\) 6.64041 0.387937 0.193968 0.981008i \(-0.437864\pi\)
0.193968 + 0.981008i \(0.437864\pi\)
\(294\) 5.61522 0.327486
\(295\) −12.1583 −0.707881
\(296\) −20.2875 −1.17918
\(297\) −4.81398 −0.279335
\(298\) 4.55725 0.263994
\(299\) −13.9124 −0.804576
\(300\) 3.53975 0.204368
\(301\) 2.17251 0.125221
\(302\) −13.1554 −0.757010
\(303\) 6.61199 0.379849
\(304\) 0.178558 0.0102410
\(305\) −35.9033 −2.05582
\(306\) −0.901623 −0.0515423
\(307\) 28.8938 1.64906 0.824528 0.565822i \(-0.191441\pi\)
0.824528 + 0.565822i \(0.191441\pi\)
\(308\) 5.02132 0.286116
\(309\) 0.797810 0.0453858
\(310\) −16.8176 −0.955176
\(311\) −4.99039 −0.282979 −0.141490 0.989940i \(-0.545189\pi\)
−0.141490 + 0.989940i \(0.545189\pi\)
\(312\) −14.7777 −0.836621
\(313\) −19.2970 −1.09073 −0.545365 0.838199i \(-0.683609\pi\)
−0.545365 + 0.838199i \(0.683609\pi\)
\(314\) −0.901623 −0.0508815
\(315\) 2.48250 0.139873
\(316\) 14.1396 0.795415
\(317\) −29.5183 −1.65791 −0.828957 0.559312i \(-0.811065\pi\)
−0.828957 + 0.559312i \(0.811065\pi\)
\(318\) 2.60083 0.145847
\(319\) −36.5238 −2.04494
\(320\) −15.3662 −0.858998
\(321\) 9.88807 0.551898
\(322\) −2.14326 −0.119439
\(323\) 0.823997 0.0458484
\(324\) −1.18708 −0.0659487
\(325\) −15.3350 −0.850631
\(326\) −18.9332 −1.04861
\(327\) −7.86949 −0.435184
\(328\) −7.64196 −0.421957
\(329\) −5.31098 −0.292804
\(330\) 12.2626 0.675033
\(331\) −26.2865 −1.44484 −0.722419 0.691456i \(-0.756970\pi\)
−0.722419 + 0.691456i \(0.756970\pi\)
\(332\) −7.57282 −0.415613
\(333\) 7.06009 0.386891
\(334\) −13.5390 −0.740823
\(335\) −14.8897 −0.813509
\(336\) −0.190410 −0.0103877
\(337\) −4.32529 −0.235614 −0.117807 0.993037i \(-0.537586\pi\)
−0.117807 + 0.993037i \(0.537586\pi\)
\(338\) 12.1242 0.659469
\(339\) −7.60312 −0.412945
\(340\) −3.35376 −0.181883
\(341\) 31.7827 1.72113
\(342\) −0.742935 −0.0401733
\(343\) 11.6232 0.627595
\(344\) 7.10467 0.383058
\(345\) 7.64306 0.411488
\(346\) −2.79081 −0.150035
\(347\) 22.9267 1.23077 0.615384 0.788227i \(-0.289001\pi\)
0.615384 + 0.788227i \(0.289001\pi\)
\(348\) −9.00639 −0.482793
\(349\) 0.973027 0.0520850 0.0260425 0.999661i \(-0.491709\pi\)
0.0260425 + 0.999661i \(0.491709\pi\)
\(350\) −2.36240 −0.126276
\(351\) 5.14267 0.274496
\(352\) 26.7258 1.42449
\(353\) −25.9428 −1.38080 −0.690398 0.723430i \(-0.742565\pi\)
−0.690398 + 0.723430i \(0.742565\pi\)
\(354\) −3.88010 −0.206225
\(355\) −1.58013 −0.0838646
\(356\) 1.08723 0.0576229
\(357\) −0.878690 −0.0465052
\(358\) −16.9458 −0.895613
\(359\) 33.1794 1.75114 0.875570 0.483091i \(-0.160486\pi\)
0.875570 + 0.483091i \(0.160486\pi\)
\(360\) 8.11840 0.427877
\(361\) −18.3210 −0.964265
\(362\) 19.4264 1.02103
\(363\) −12.1744 −0.638989
\(364\) −5.36418 −0.281159
\(365\) −24.5167 −1.28326
\(366\) −11.4579 −0.598916
\(367\) 17.3166 0.903917 0.451958 0.892039i \(-0.350726\pi\)
0.451958 + 0.892039i \(0.350726\pi\)
\(368\) −0.586229 −0.0305593
\(369\) 2.65942 0.138444
\(370\) −17.9841 −0.934948
\(371\) 2.53467 0.131594
\(372\) 7.83727 0.406344
\(373\) −24.7665 −1.28236 −0.641181 0.767390i \(-0.721555\pi\)
−0.641181 + 0.767390i \(0.721555\pi\)
\(374\) −4.34039 −0.224436
\(375\) −5.70157 −0.294428
\(376\) −17.3683 −0.895701
\(377\) 39.0177 2.00951
\(378\) 0.792247 0.0407488
\(379\) −21.3604 −1.09721 −0.548605 0.836082i \(-0.684841\pi\)
−0.548605 + 0.836082i \(0.684841\pi\)
\(380\) −2.76349 −0.141764
\(381\) −13.5975 −0.696623
\(382\) −13.3273 −0.681885
\(383\) −7.39582 −0.377909 −0.188954 0.981986i \(-0.560510\pi\)
−0.188954 + 0.981986i \(0.560510\pi\)
\(384\) 6.19954 0.316369
\(385\) 11.9507 0.609063
\(386\) −7.60976 −0.387326
\(387\) −2.47244 −0.125681
\(388\) 0.872406 0.0442897
\(389\) 20.9675 1.06309 0.531547 0.847029i \(-0.321611\pi\)
0.531547 + 0.847029i \(0.321611\pi\)
\(390\) −13.0999 −0.663338
\(391\) −2.70529 −0.136812
\(392\) 17.8961 0.903891
\(393\) 6.85124 0.345600
\(394\) 21.7005 1.09325
\(395\) 33.6521 1.69322
\(396\) −5.71456 −0.287167
\(397\) −9.72098 −0.487882 −0.243941 0.969790i \(-0.578440\pi\)
−0.243941 + 0.969790i \(0.578440\pi\)
\(398\) −3.43414 −0.172138
\(399\) −0.724038 −0.0362472
\(400\) −0.646171 −0.0323086
\(401\) −27.8355 −1.39004 −0.695019 0.718992i \(-0.744604\pi\)
−0.695019 + 0.718992i \(0.744604\pi\)
\(402\) −4.75178 −0.236997
\(403\) −33.9528 −1.69131
\(404\) 7.84893 0.390499
\(405\) −2.82523 −0.140387
\(406\) 6.01080 0.298311
\(407\) 33.9871 1.68468
\(408\) −2.87354 −0.142261
\(409\) −10.3677 −0.512651 −0.256325 0.966591i \(-0.582512\pi\)
−0.256325 + 0.966591i \(0.582512\pi\)
\(410\) −6.77432 −0.334560
\(411\) 13.6270 0.672173
\(412\) 0.947061 0.0466584
\(413\) −3.78141 −0.186071
\(414\) 2.43915 0.119878
\(415\) −18.0232 −0.884725
\(416\) −28.5506 −1.39981
\(417\) −2.30778 −0.113013
\(418\) −3.57647 −0.174931
\(419\) −7.30960 −0.357098 −0.178549 0.983931i \(-0.557140\pi\)
−0.178549 + 0.983931i \(0.557140\pi\)
\(420\) 2.94691 0.143795
\(421\) 10.1206 0.493249 0.246624 0.969111i \(-0.420679\pi\)
0.246624 + 0.969111i \(0.420679\pi\)
\(422\) −11.7061 −0.569843
\(423\) 6.04421 0.293880
\(424\) 8.28903 0.402551
\(425\) −2.98191 −0.144644
\(426\) −0.504271 −0.0244320
\(427\) −11.1665 −0.540385
\(428\) 11.7379 0.567373
\(429\) 24.7567 1.19526
\(430\) 6.29803 0.303718
\(431\) 19.7501 0.951331 0.475665 0.879626i \(-0.342207\pi\)
0.475665 + 0.879626i \(0.342207\pi\)
\(432\) 0.216697 0.0104259
\(433\) 24.0596 1.15623 0.578115 0.815955i \(-0.303788\pi\)
0.578115 + 0.815955i \(0.303788\pi\)
\(434\) −5.23054 −0.251074
\(435\) −21.4351 −1.02773
\(436\) −9.34169 −0.447386
\(437\) −2.22915 −0.106635
\(438\) −7.82408 −0.373849
\(439\) 6.74427 0.321886 0.160943 0.986964i \(-0.448546\pi\)
0.160943 + 0.986964i \(0.448546\pi\)
\(440\) 39.0818 1.86315
\(441\) −6.22790 −0.296567
\(442\) 4.63675 0.220548
\(443\) 38.7949 1.84320 0.921601 0.388139i \(-0.126882\pi\)
0.921601 + 0.388139i \(0.126882\pi\)
\(444\) 8.38087 0.397738
\(445\) 2.58759 0.122663
\(446\) −11.0435 −0.522926
\(447\) −5.05450 −0.239070
\(448\) −4.77914 −0.225793
\(449\) −23.2150 −1.09558 −0.547791 0.836615i \(-0.684531\pi\)
−0.547791 + 0.836615i \(0.684531\pi\)
\(450\) 2.68855 0.126740
\(451\) 12.8024 0.602842
\(452\) −9.02548 −0.424523
\(453\) 14.5908 0.685537
\(454\) 6.71826 0.315304
\(455\) −12.7667 −0.598511
\(456\) −2.36779 −0.110882
\(457\) −24.2862 −1.13606 −0.568030 0.823008i \(-0.692294\pi\)
−0.568030 + 0.823008i \(0.692294\pi\)
\(458\) −4.19530 −0.196034
\(459\) 1.00000 0.0466760
\(460\) 9.07289 0.423026
\(461\) 13.7110 0.638586 0.319293 0.947656i \(-0.396555\pi\)
0.319293 + 0.947656i \(0.396555\pi\)
\(462\) 3.81386 0.177437
\(463\) −19.4030 −0.901732 −0.450866 0.892592i \(-0.648885\pi\)
−0.450866 + 0.892592i \(0.648885\pi\)
\(464\) 1.64409 0.0763250
\(465\) 18.6526 0.864994
\(466\) −1.90671 −0.0883268
\(467\) 28.3586 1.31228 0.656139 0.754640i \(-0.272188\pi\)
0.656139 + 0.754640i \(0.272188\pi\)
\(468\) 6.10474 0.282192
\(469\) −4.63092 −0.213836
\(470\) −15.3963 −0.710181
\(471\) 1.00000 0.0460776
\(472\) −12.3662 −0.569200
\(473\) −11.9023 −0.547268
\(474\) 10.7395 0.493281
\(475\) −2.45708 −0.112739
\(476\) −1.04307 −0.0478091
\(477\) −2.88461 −0.132077
\(478\) −5.38315 −0.246219
\(479\) −26.1238 −1.19363 −0.596813 0.802380i \(-0.703567\pi\)
−0.596813 + 0.802380i \(0.703567\pi\)
\(480\) 15.6848 0.715911
\(481\) −36.3077 −1.65549
\(482\) 0.0995401 0.00453393
\(483\) 2.37711 0.108162
\(484\) −14.4519 −0.656905
\(485\) 2.07632 0.0942807
\(486\) −0.901623 −0.0408984
\(487\) −11.7214 −0.531147 −0.265574 0.964091i \(-0.585561\pi\)
−0.265574 + 0.964091i \(0.585561\pi\)
\(488\) −36.5173 −1.65306
\(489\) 20.9990 0.949609
\(490\) 15.8643 0.716675
\(491\) 25.1444 1.13475 0.567375 0.823459i \(-0.307959\pi\)
0.567375 + 0.823459i \(0.307959\pi\)
\(492\) 3.15694 0.142326
\(493\) 7.58704 0.341703
\(494\) 3.82067 0.171900
\(495\) −13.6006 −0.611300
\(496\) −1.43067 −0.0642390
\(497\) −0.491445 −0.0220443
\(498\) −5.75180 −0.257744
\(499\) −20.6991 −0.926620 −0.463310 0.886196i \(-0.653338\pi\)
−0.463310 + 0.886196i \(0.653338\pi\)
\(500\) −6.76820 −0.302683
\(501\) 15.0163 0.670878
\(502\) −1.30753 −0.0583579
\(503\) −26.8753 −1.19831 −0.599156 0.800632i \(-0.704497\pi\)
−0.599156 + 0.800632i \(0.704497\pi\)
\(504\) 2.52495 0.112470
\(505\) 18.6804 0.831265
\(506\) 11.7420 0.521996
\(507\) −13.4471 −0.597206
\(508\) −16.1413 −0.716155
\(509\) 43.0113 1.90644 0.953221 0.302273i \(-0.0977454\pi\)
0.953221 + 0.302273i \(0.0977454\pi\)
\(510\) −2.54729 −0.112796
\(511\) −7.62507 −0.337313
\(512\) −2.44842 −0.108206
\(513\) 0.823997 0.0363804
\(514\) −9.17066 −0.404500
\(515\) 2.25399 0.0993228
\(516\) −2.93498 −0.129205
\(517\) 29.0967 1.27967
\(518\) −5.59333 −0.245757
\(519\) 3.09532 0.135869
\(520\) −41.7503 −1.83087
\(521\) 9.88187 0.432933 0.216466 0.976290i \(-0.430547\pi\)
0.216466 + 0.976290i \(0.430547\pi\)
\(522\) −6.84065 −0.299407
\(523\) −35.0848 −1.53415 −0.767075 0.641557i \(-0.778289\pi\)
−0.767075 + 0.641557i \(0.778289\pi\)
\(524\) 8.13295 0.355290
\(525\) 2.62017 0.114354
\(526\) 2.18706 0.0953603
\(527\) −6.60216 −0.287595
\(528\) 1.04318 0.0453984
\(529\) −15.6814 −0.681800
\(530\) 7.34792 0.319173
\(531\) 4.30346 0.186754
\(532\) −0.859488 −0.0372635
\(533\) −13.6765 −0.592397
\(534\) 0.825784 0.0357352
\(535\) 27.9360 1.20778
\(536\) −15.1443 −0.654134
\(537\) 18.7948 0.811055
\(538\) 16.1496 0.696261
\(539\) −29.9810 −1.29137
\(540\) −3.35376 −0.144323
\(541\) 26.8771 1.15554 0.577768 0.816201i \(-0.303924\pi\)
0.577768 + 0.816201i \(0.303924\pi\)
\(542\) −13.0014 −0.558458
\(543\) −21.5460 −0.924627
\(544\) −5.55170 −0.238027
\(545\) −22.2331 −0.952361
\(546\) −4.07426 −0.174362
\(547\) 13.7078 0.586103 0.293052 0.956097i \(-0.405329\pi\)
0.293052 + 0.956097i \(0.405329\pi\)
\(548\) 16.1763 0.691019
\(549\) 12.7081 0.542370
\(550\) 12.9426 0.551876
\(551\) 6.25170 0.266331
\(552\) 7.77376 0.330873
\(553\) 10.4663 0.445073
\(554\) −6.67649 −0.283657
\(555\) 19.9464 0.846676
\(556\) −2.73952 −0.116181
\(557\) −15.1966 −0.643901 −0.321950 0.946757i \(-0.604338\pi\)
−0.321950 + 0.946757i \(0.604338\pi\)
\(558\) 5.95266 0.251996
\(559\) 12.7150 0.537786
\(560\) −0.537951 −0.0227326
\(561\) 4.81398 0.203246
\(562\) −0.423878 −0.0178802
\(563\) 8.62789 0.363622 0.181811 0.983333i \(-0.441804\pi\)
0.181811 + 0.983333i \(0.441804\pi\)
\(564\) 7.17494 0.302119
\(565\) −21.4805 −0.903693
\(566\) 2.48650 0.104515
\(567\) −0.878690 −0.0369015
\(568\) −1.60715 −0.0674346
\(569\) 15.7209 0.659054 0.329527 0.944146i \(-0.393111\pi\)
0.329527 + 0.944146i \(0.393111\pi\)
\(570\) −2.09896 −0.0879157
\(571\) −6.26602 −0.262225 −0.131112 0.991368i \(-0.541855\pi\)
−0.131112 + 0.991368i \(0.541855\pi\)
\(572\) 29.3881 1.22878
\(573\) 14.7815 0.617505
\(574\) −2.10692 −0.0879411
\(575\) 8.06692 0.336414
\(576\) 5.43894 0.226622
\(577\) −29.7294 −1.23765 −0.618825 0.785529i \(-0.712391\pi\)
−0.618825 + 0.785529i \(0.712391\pi\)
\(578\) 0.901623 0.0375026
\(579\) 8.44007 0.350757
\(580\) −25.4451 −1.05655
\(581\) −5.60550 −0.232555
\(582\) 0.662621 0.0274665
\(583\) −13.8864 −0.575117
\(584\) −24.9359 −1.03186
\(585\) 14.5292 0.600709
\(586\) 5.98714 0.247327
\(587\) 13.0419 0.538297 0.269148 0.963099i \(-0.413258\pi\)
0.269148 + 0.963099i \(0.413258\pi\)
\(588\) −7.39300 −0.304882
\(589\) −5.44016 −0.224158
\(590\) −10.9622 −0.451305
\(591\) −24.0682 −0.990035
\(592\) −1.52990 −0.0628786
\(593\) −33.7402 −1.38554 −0.692772 0.721156i \(-0.743611\pi\)
−0.692772 + 0.721156i \(0.743611\pi\)
\(594\) −4.34039 −0.178088
\(595\) −2.48250 −0.101772
\(596\) −6.00007 −0.245773
\(597\) 3.80884 0.155885
\(598\) −12.5438 −0.512952
\(599\) 37.4511 1.53021 0.765104 0.643906i \(-0.222687\pi\)
0.765104 + 0.643906i \(0.222687\pi\)
\(600\) 8.56863 0.349813
\(601\) −0.0910357 −0.00371342 −0.00185671 0.999998i \(-0.500591\pi\)
−0.00185671 + 0.999998i \(0.500591\pi\)
\(602\) 1.95879 0.0798341
\(603\) 5.27025 0.214621
\(604\) 17.3204 0.704758
\(605\) −34.3954 −1.39837
\(606\) 5.96152 0.242170
\(607\) −0.823536 −0.0334263 −0.0167132 0.999860i \(-0.505320\pi\)
−0.0167132 + 0.999860i \(0.505320\pi\)
\(608\) −4.57459 −0.185524
\(609\) −6.66665 −0.270146
\(610\) −32.3713 −1.31067
\(611\) −31.0834 −1.25750
\(612\) 1.18708 0.0479847
\(613\) −5.87403 −0.237250 −0.118625 0.992939i \(-0.537849\pi\)
−0.118625 + 0.992939i \(0.537849\pi\)
\(614\) 26.0513 1.05134
\(615\) 7.51347 0.302973
\(616\) 12.1551 0.489741
\(617\) −33.7166 −1.35738 −0.678690 0.734425i \(-0.737452\pi\)
−0.678690 + 0.734425i \(0.737452\pi\)
\(618\) 0.719324 0.0289354
\(619\) −27.6436 −1.11109 −0.555546 0.831486i \(-0.687491\pi\)
−0.555546 + 0.831486i \(0.687491\pi\)
\(620\) 22.1421 0.889247
\(621\) −2.70529 −0.108560
\(622\) −4.49945 −0.180412
\(623\) 0.804780 0.0322428
\(624\) −1.11440 −0.0446118
\(625\) −31.0178 −1.24071
\(626\) −17.3986 −0.695388
\(627\) 3.96670 0.158415
\(628\) 1.18708 0.0473695
\(629\) −7.06009 −0.281504
\(630\) 2.23828 0.0891751
\(631\) 19.8445 0.789998 0.394999 0.918682i \(-0.370745\pi\)
0.394999 + 0.918682i \(0.370745\pi\)
\(632\) 34.2275 1.36150
\(633\) 12.9833 0.516041
\(634\) −26.6144 −1.05699
\(635\) −38.4161 −1.52450
\(636\) −3.42425 −0.135780
\(637\) 32.0281 1.26900
\(638\) −32.9307 −1.30374
\(639\) 0.559293 0.0221253
\(640\) 17.5151 0.692345
\(641\) 23.1925 0.916047 0.458023 0.888940i \(-0.348558\pi\)
0.458023 + 0.888940i \(0.348558\pi\)
\(642\) 8.91531 0.351859
\(643\) −30.5209 −1.20363 −0.601814 0.798636i \(-0.705555\pi\)
−0.601814 + 0.798636i \(0.705555\pi\)
\(644\) 2.82181 0.111195
\(645\) −6.98522 −0.275043
\(646\) 0.742935 0.0292304
\(647\) −17.1517 −0.674305 −0.337152 0.941450i \(-0.609464\pi\)
−0.337152 + 0.941450i \(0.609464\pi\)
\(648\) −2.87354 −0.112883
\(649\) 20.7168 0.813205
\(650\) −13.8264 −0.542314
\(651\) 5.80125 0.227369
\(652\) 24.9275 0.976234
\(653\) −27.9655 −1.09437 −0.547187 0.837011i \(-0.684301\pi\)
−0.547187 + 0.837011i \(0.684301\pi\)
\(654\) −7.09532 −0.277449
\(655\) 19.3563 0.756314
\(656\) −0.576290 −0.0225003
\(657\) 8.67777 0.338552
\(658\) −4.78850 −0.186675
\(659\) −50.6218 −1.97195 −0.985973 0.166903i \(-0.946623\pi\)
−0.985973 + 0.166903i \(0.946623\pi\)
\(660\) −16.1449 −0.628440
\(661\) 45.8747 1.78432 0.892160 0.451719i \(-0.149189\pi\)
0.892160 + 0.451719i \(0.149189\pi\)
\(662\) −23.7005 −0.921147
\(663\) −5.14267 −0.199725
\(664\) −18.3314 −0.711398
\(665\) −2.04557 −0.0793238
\(666\) 6.36554 0.246660
\(667\) −20.5251 −0.794737
\(668\) 17.8255 0.689689
\(669\) 12.2485 0.473554
\(670\) −13.4249 −0.518647
\(671\) 61.1766 2.36170
\(672\) 4.87822 0.188182
\(673\) 16.8397 0.649124 0.324562 0.945864i \(-0.394783\pi\)
0.324562 + 0.945864i \(0.394783\pi\)
\(674\) −3.89978 −0.150214
\(675\) −2.98191 −0.114774
\(676\) −15.9627 −0.613950
\(677\) −26.7675 −1.02876 −0.514379 0.857563i \(-0.671977\pi\)
−0.514379 + 0.857563i \(0.671977\pi\)
\(678\) −6.85514 −0.263270
\(679\) 0.645767 0.0247823
\(680\) −8.11840 −0.311327
\(681\) −7.45130 −0.285534
\(682\) 28.6560 1.09729
\(683\) −3.37662 −0.129203 −0.0646014 0.997911i \(-0.520578\pi\)
−0.0646014 + 0.997911i \(0.520578\pi\)
\(684\) 0.978147 0.0374004
\(685\) 38.4995 1.47099
\(686\) 10.4798 0.400119
\(687\) 4.65306 0.177525
\(688\) 0.535772 0.0204261
\(689\) 14.8346 0.565153
\(690\) 6.89116 0.262342
\(691\) 23.2760 0.885459 0.442729 0.896655i \(-0.354010\pi\)
0.442729 + 0.896655i \(0.354010\pi\)
\(692\) 3.67438 0.139679
\(693\) −4.22999 −0.160684
\(694\) 20.6712 0.784668
\(695\) −6.52001 −0.247318
\(696\) −21.8017 −0.826390
\(697\) −2.65942 −0.100733
\(698\) 0.877304 0.0332064
\(699\) 2.11476 0.0799875
\(700\) 3.11034 0.117560
\(701\) 29.3464 1.10840 0.554200 0.832384i \(-0.313024\pi\)
0.554200 + 0.832384i \(0.313024\pi\)
\(702\) 4.63675 0.175003
\(703\) −5.81750 −0.219411
\(704\) 26.1829 0.986806
\(705\) 17.0763 0.643129
\(706\) −23.3906 −0.880318
\(707\) 5.80988 0.218503
\(708\) 5.10854 0.191991
\(709\) 33.8224 1.27023 0.635113 0.772419i \(-0.280953\pi\)
0.635113 + 0.772419i \(0.280953\pi\)
\(710\) −1.42468 −0.0534673
\(711\) −11.9113 −0.446708
\(712\) 2.63184 0.0986323
\(713\) 17.8608 0.668891
\(714\) −0.792247 −0.0296491
\(715\) 69.9433 2.61573
\(716\) 22.3108 0.833795
\(717\) 5.97051 0.222973
\(718\) 29.9153 1.11643
\(719\) 3.63404 0.135527 0.0677634 0.997701i \(-0.478414\pi\)
0.0677634 + 0.997701i \(0.478414\pi\)
\(720\) 0.612219 0.0228161
\(721\) 0.701027 0.0261076
\(722\) −16.5187 −0.614761
\(723\) −0.110401 −0.00410586
\(724\) −25.5767 −0.950552
\(725\) −22.6238 −0.840228
\(726\) −10.9767 −0.407383
\(727\) −20.9761 −0.777960 −0.388980 0.921246i \(-0.627172\pi\)
−0.388980 + 0.921246i \(0.627172\pi\)
\(728\) −12.9850 −0.481256
\(729\) 1.00000 0.0370370
\(730\) −22.1048 −0.818135
\(731\) 2.47244 0.0914467
\(732\) 15.0855 0.557577
\(733\) 13.4772 0.497791 0.248895 0.968530i \(-0.419932\pi\)
0.248895 + 0.968530i \(0.419932\pi\)
\(734\) 15.6130 0.576286
\(735\) −17.5952 −0.649010
\(736\) 15.0190 0.553606
\(737\) 25.3709 0.934548
\(738\) 2.39780 0.0882641
\(739\) 27.2751 1.00333 0.501665 0.865062i \(-0.332721\pi\)
0.501665 + 0.865062i \(0.332721\pi\)
\(740\) 23.6778 0.870415
\(741\) −4.23755 −0.155670
\(742\) 2.28532 0.0838967
\(743\) −31.9774 −1.17314 −0.586568 0.809900i \(-0.699521\pi\)
−0.586568 + 0.809900i \(0.699521\pi\)
\(744\) 18.9716 0.695532
\(745\) −14.2801 −0.523182
\(746\) −22.3300 −0.817561
\(747\) 6.37939 0.233410
\(748\) 5.71456 0.208945
\(749\) 8.68855 0.317473
\(750\) −5.14067 −0.187711
\(751\) 27.8372 1.01579 0.507897 0.861418i \(-0.330423\pi\)
0.507897 + 0.861418i \(0.330423\pi\)
\(752\) −1.30976 −0.0477622
\(753\) 1.45020 0.0528481
\(754\) 35.1792 1.28115
\(755\) 41.2224 1.50024
\(756\) −1.04307 −0.0379361
\(757\) −1.36797 −0.0497198 −0.0248599 0.999691i \(-0.507914\pi\)
−0.0248599 + 0.999691i \(0.507914\pi\)
\(758\) −19.2590 −0.699519
\(759\) −13.0232 −0.472712
\(760\) −6.68954 −0.242655
\(761\) 46.1362 1.67244 0.836218 0.548397i \(-0.184762\pi\)
0.836218 + 0.548397i \(0.184762\pi\)
\(762\) −12.2599 −0.444128
\(763\) −6.91484 −0.250334
\(764\) 17.5467 0.634819
\(765\) 2.82523 0.102146
\(766\) −6.66824 −0.240933
\(767\) −22.1313 −0.799115
\(768\) 16.4675 0.594220
\(769\) −23.2948 −0.840031 −0.420016 0.907517i \(-0.637975\pi\)
−0.420016 + 0.907517i \(0.637975\pi\)
\(770\) 10.7750 0.388304
\(771\) 10.1713 0.366310
\(772\) 10.0190 0.360592
\(773\) 26.7362 0.961635 0.480818 0.876821i \(-0.340340\pi\)
0.480818 + 0.876821i \(0.340340\pi\)
\(774\) −2.22921 −0.0801274
\(775\) 19.6870 0.707179
\(776\) 2.11182 0.0758100
\(777\) 6.20363 0.222554
\(778\) 18.9048 0.677768
\(779\) −2.19136 −0.0785135
\(780\) 17.2473 0.617552
\(781\) 2.69242 0.0963425
\(782\) −2.43915 −0.0872239
\(783\) 7.58704 0.271139
\(784\) 1.34957 0.0481989
\(785\) 2.82523 0.100837
\(786\) 6.17724 0.220335
\(787\) 45.5050 1.62208 0.811040 0.584991i \(-0.198902\pi\)
0.811040 + 0.584991i \(0.198902\pi\)
\(788\) −28.5708 −1.01779
\(789\) −2.42569 −0.0863569
\(790\) 30.3415 1.07950
\(791\) −6.68078 −0.237541
\(792\) −13.8332 −0.491540
\(793\) −65.3537 −2.32078
\(794\) −8.76466 −0.311046
\(795\) −8.14967 −0.289039
\(796\) 4.52138 0.160256
\(797\) 5.23937 0.185588 0.0927941 0.995685i \(-0.470420\pi\)
0.0927941 + 0.995685i \(0.470420\pi\)
\(798\) −0.652809 −0.0231092
\(799\) −6.04421 −0.213829
\(800\) 16.5547 0.585295
\(801\) −0.915887 −0.0323613
\(802\) −25.0971 −0.886209
\(803\) 41.7746 1.47419
\(804\) 6.25619 0.220639
\(805\) 6.71588 0.236704
\(806\) −30.6126 −1.07828
\(807\) −17.9118 −0.630524
\(808\) 18.9998 0.668411
\(809\) 40.1169 1.41044 0.705218 0.708990i \(-0.250849\pi\)
0.705218 + 0.708990i \(0.250849\pi\)
\(810\) −2.54729 −0.0895026
\(811\) 2.53994 0.0891894 0.0445947 0.999005i \(-0.485800\pi\)
0.0445947 + 0.999005i \(0.485800\pi\)
\(812\) −7.91382 −0.277721
\(813\) 14.4200 0.505731
\(814\) 30.6436 1.07406
\(815\) 59.3270 2.07813
\(816\) −0.216697 −0.00758593
\(817\) 2.03729 0.0712757
\(818\) −9.34777 −0.326837
\(819\) 4.51881 0.157900
\(820\) 8.91907 0.311467
\(821\) 24.8007 0.865551 0.432776 0.901502i \(-0.357534\pi\)
0.432776 + 0.901502i \(0.357534\pi\)
\(822\) 12.2865 0.428539
\(823\) −25.7022 −0.895921 −0.447961 0.894053i \(-0.647850\pi\)
−0.447961 + 0.894053i \(0.647850\pi\)
\(824\) 2.29254 0.0798644
\(825\) −14.3548 −0.499771
\(826\) −3.40941 −0.118628
\(827\) −30.5023 −1.06067 −0.530335 0.847788i \(-0.677934\pi\)
−0.530335 + 0.847788i \(0.677934\pi\)
\(828\) −3.21139 −0.111603
\(829\) 2.29307 0.0796417 0.0398209 0.999207i \(-0.487321\pi\)
0.0398209 + 0.999207i \(0.487321\pi\)
\(830\) −16.2501 −0.564051
\(831\) 7.40497 0.256875
\(832\) −27.9707 −0.969708
\(833\) 6.22790 0.215784
\(834\) −2.08075 −0.0720505
\(835\) 42.4244 1.46816
\(836\) 4.70878 0.162857
\(837\) −6.60216 −0.228204
\(838\) −6.59051 −0.227665
\(839\) 31.9689 1.10369 0.551844 0.833947i \(-0.313924\pi\)
0.551844 + 0.833947i \(0.313924\pi\)
\(840\) 7.13356 0.246131
\(841\) 28.5631 0.984936
\(842\) 9.12498 0.314468
\(843\) 0.470128 0.0161921
\(844\) 15.4122 0.530510
\(845\) −37.9910 −1.30693
\(846\) 5.44960 0.187361
\(847\) −10.6975 −0.367570
\(848\) 0.625087 0.0214656
\(849\) −2.75781 −0.0946477
\(850\) −2.68855 −0.0922166
\(851\) 19.0996 0.654726
\(852\) 0.663924 0.0227456
\(853\) 22.4462 0.768544 0.384272 0.923220i \(-0.374453\pi\)
0.384272 + 0.923220i \(0.374453\pi\)
\(854\) −10.0680 −0.344519
\(855\) 2.32798 0.0796152
\(856\) 28.4138 0.971163
\(857\) −56.9347 −1.94485 −0.972425 0.233214i \(-0.925076\pi\)
−0.972425 + 0.233214i \(0.925076\pi\)
\(858\) 22.3212 0.762034
\(859\) −18.8210 −0.642166 −0.321083 0.947051i \(-0.604047\pi\)
−0.321083 + 0.947051i \(0.604047\pi\)
\(860\) −8.29198 −0.282754
\(861\) 2.33681 0.0796382
\(862\) 17.8072 0.606515
\(863\) 8.07558 0.274896 0.137448 0.990509i \(-0.456110\pi\)
0.137448 + 0.990509i \(0.456110\pi\)
\(864\) −5.55170 −0.188873
\(865\) 8.74498 0.297338
\(866\) 21.6927 0.737147
\(867\) −1.00000 −0.0339618
\(868\) 6.88653 0.233744
\(869\) −57.3406 −1.94515
\(870\) −19.3264 −0.655226
\(871\) −27.1032 −0.918356
\(872\) −22.6133 −0.765783
\(873\) −0.734920 −0.0248733
\(874\) −2.00985 −0.0679843
\(875\) −5.00991 −0.169366
\(876\) 10.3012 0.348044
\(877\) −32.7860 −1.10711 −0.553553 0.832814i \(-0.686728\pi\)
−0.553553 + 0.832814i \(0.686728\pi\)
\(878\) 6.08079 0.205217
\(879\) −6.64041 −0.223975
\(880\) 2.94721 0.0993504
\(881\) 24.3734 0.821162 0.410581 0.911824i \(-0.365326\pi\)
0.410581 + 0.911824i \(0.365326\pi\)
\(882\) −5.61522 −0.189074
\(883\) 20.9762 0.705904 0.352952 0.935641i \(-0.385178\pi\)
0.352952 + 0.935641i \(0.385178\pi\)
\(884\) −6.10474 −0.205325
\(885\) 12.1583 0.408696
\(886\) 34.9784 1.17512
\(887\) 37.1600 1.24771 0.623855 0.781540i \(-0.285566\pi\)
0.623855 + 0.781540i \(0.285566\pi\)
\(888\) 20.2875 0.680803
\(889\) −11.9480 −0.400724
\(890\) 2.33303 0.0782033
\(891\) 4.81398 0.161274
\(892\) 14.5399 0.486831
\(893\) −4.98041 −0.166663
\(894\) −4.55725 −0.152417
\(895\) 53.0995 1.77492
\(896\) 5.44747 0.181987
\(897\) 13.9124 0.464522
\(898\) −20.9312 −0.698482
\(899\) −50.0909 −1.67062
\(900\) −3.53975 −0.117992
\(901\) 2.88461 0.0961002
\(902\) 11.5429 0.384338
\(903\) −2.17251 −0.0722966
\(904\) −21.8479 −0.726649
\(905\) −60.8723 −2.02346
\(906\) 13.1554 0.437060
\(907\) −49.6774 −1.64951 −0.824755 0.565491i \(-0.808687\pi\)
−0.824755 + 0.565491i \(0.808687\pi\)
\(908\) −8.84526 −0.293540
\(909\) −6.61199 −0.219306
\(910\) −11.5107 −0.381577
\(911\) 23.1227 0.766088 0.383044 0.923730i \(-0.374876\pi\)
0.383044 + 0.923730i \(0.374876\pi\)
\(912\) −0.178558 −0.00591265
\(913\) 30.7102 1.01636
\(914\) −21.8970 −0.724287
\(915\) 35.9033 1.18693
\(916\) 5.52354 0.182503
\(917\) 6.02012 0.198802
\(918\) 0.901623 0.0297580
\(919\) 8.04319 0.265320 0.132660 0.991162i \(-0.457648\pi\)
0.132660 + 0.991162i \(0.457648\pi\)
\(920\) 21.9626 0.724087
\(921\) −28.8938 −0.952083
\(922\) 12.3622 0.407126
\(923\) −2.87626 −0.0946733
\(924\) −5.02132 −0.165189
\(925\) 21.0525 0.692203
\(926\) −17.4942 −0.574894
\(927\) −0.797810 −0.0262035
\(928\) −42.1210 −1.38269
\(929\) −0.00389609 −0.000127827 0 −6.39133e−5 1.00000i \(-0.500020\pi\)
−6.39133e−5 1.00000i \(0.500020\pi\)
\(930\) 16.8176 0.551471
\(931\) 5.13178 0.168187
\(932\) 2.51038 0.0822302
\(933\) 4.99039 0.163378
\(934\) 25.5687 0.836635
\(935\) 13.6006 0.444786
\(936\) 14.7777 0.483024
\(937\) −21.2904 −0.695525 −0.347763 0.937583i \(-0.613059\pi\)
−0.347763 + 0.937583i \(0.613059\pi\)
\(938\) −4.17534 −0.136330
\(939\) 19.2970 0.629733
\(940\) 20.2708 0.661162
\(941\) 2.02523 0.0660206 0.0330103 0.999455i \(-0.489491\pi\)
0.0330103 + 0.999455i \(0.489491\pi\)
\(942\) 0.901623 0.0293765
\(943\) 7.19451 0.234286
\(944\) −0.932550 −0.0303519
\(945\) −2.48250 −0.0807556
\(946\) −10.7314 −0.348907
\(947\) −28.6393 −0.930653 −0.465326 0.885139i \(-0.654063\pi\)
−0.465326 + 0.885139i \(0.654063\pi\)
\(948\) −14.1396 −0.459233
\(949\) −44.6269 −1.44865
\(950\) −2.21536 −0.0718758
\(951\) 29.5183 0.957197
\(952\) −2.52495 −0.0818341
\(953\) 3.55625 0.115198 0.0575991 0.998340i \(-0.481655\pi\)
0.0575991 + 0.998340i \(0.481655\pi\)
\(954\) −2.60083 −0.0842049
\(955\) 41.7610 1.35135
\(956\) 7.08745 0.229225
\(957\) 36.5238 1.18065
\(958\) −23.5538 −0.760988
\(959\) 11.9739 0.386659
\(960\) 15.3662 0.495943
\(961\) 12.5885 0.406082
\(962\) −32.7359 −1.05545
\(963\) −9.88807 −0.318639
\(964\) −0.131054 −0.00422098
\(965\) 23.8451 0.767601
\(966\) 2.14326 0.0689582
\(967\) −43.3529 −1.39413 −0.697067 0.717006i \(-0.745512\pi\)
−0.697067 + 0.717006i \(0.745512\pi\)
\(968\) −34.9835 −1.12441
\(969\) −0.823997 −0.0264706
\(970\) 1.87205 0.0601080
\(971\) −26.9132 −0.863685 −0.431842 0.901949i \(-0.642136\pi\)
−0.431842 + 0.901949i \(0.642136\pi\)
\(972\) 1.18708 0.0380755
\(973\) −2.02783 −0.0650091
\(974\) −10.5683 −0.338629
\(975\) 15.3350 0.491112
\(976\) −2.75382 −0.0881476
\(977\) 56.1951 1.79784 0.898920 0.438112i \(-0.144353\pi\)
0.898920 + 0.438112i \(0.144353\pi\)
\(978\) 18.9332 0.605417
\(979\) −4.40906 −0.140914
\(980\) −20.8869 −0.667207
\(981\) 7.86949 0.251254
\(982\) 22.6708 0.723453
\(983\) 29.6813 0.946686 0.473343 0.880878i \(-0.343047\pi\)
0.473343 + 0.880878i \(0.343047\pi\)
\(984\) 7.64196 0.243617
\(985\) −67.9982 −2.16660
\(986\) 6.84065 0.217851
\(987\) 5.31098 0.169051
\(988\) −5.03029 −0.160035
\(989\) −6.68868 −0.212688
\(990\) −12.2626 −0.389731
\(991\) −19.2055 −0.610083 −0.305042 0.952339i \(-0.598670\pi\)
−0.305042 + 0.952339i \(0.598670\pi\)
\(992\) 36.6532 1.16374
\(993\) 26.2865 0.834177
\(994\) −0.443098 −0.0140542
\(995\) 10.7608 0.341141
\(996\) 7.57282 0.239954
\(997\) −29.7547 −0.942340 −0.471170 0.882042i \(-0.656168\pi\)
−0.471170 + 0.882042i \(0.656168\pi\)
\(998\) −18.6628 −0.590761
\(999\) −7.06009 −0.223371
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 8007.2.a.f.1.31 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8007.2.a.f.1.31 48 1.1 even 1 trivial