Properties

Label 8001.2.a.l.1.3
Level 8001
Weight 2
Character 8001.1
Self dual yes
Analytic conductor 63.888
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.118870813.1
Defining polynomial: \(x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.06168\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.692358 q^{2} -1.52064 q^{4} +2.78145 q^{5} +1.00000 q^{7} +2.43754 q^{8} +O(q^{10})\) \(q-0.692358 q^{2} -1.52064 q^{4} +2.78145 q^{5} +1.00000 q^{7} +2.43754 q^{8} -1.92576 q^{10} +0.348849 q^{11} +2.17757 q^{13} -0.692358 q^{14} +1.35363 q^{16} -7.42626 q^{17} -0.255629 q^{19} -4.22958 q^{20} -0.241529 q^{22} +4.98233 q^{23} +2.73644 q^{25} -1.50766 q^{26} -1.52064 q^{28} +6.88124 q^{29} -10.1038 q^{31} -5.81228 q^{32} +5.14163 q^{34} +2.78145 q^{35} -1.64279 q^{37} +0.176987 q^{38} +6.77989 q^{40} -3.56990 q^{41} -9.34628 q^{43} -0.530475 q^{44} -3.44956 q^{46} -9.73885 q^{47} +1.00000 q^{49} -1.89460 q^{50} -3.31130 q^{52} +7.44666 q^{53} +0.970306 q^{55} +2.43754 q^{56} -4.76428 q^{58} +2.01055 q^{59} -7.69549 q^{61} +6.99543 q^{62} +1.31692 q^{64} +6.05678 q^{65} -6.52951 q^{67} +11.2927 q^{68} -1.92576 q^{70} -13.7550 q^{71} -9.37124 q^{73} +1.13740 q^{74} +0.388720 q^{76} +0.348849 q^{77} -5.92206 q^{79} +3.76505 q^{80} +2.47165 q^{82} -1.68660 q^{83} -20.6557 q^{85} +6.47097 q^{86} +0.850335 q^{88} +17.6750 q^{89} +2.17757 q^{91} -7.57634 q^{92} +6.74277 q^{94} -0.711019 q^{95} -0.216772 q^{97} -0.692358 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + O(q^{10}) \) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + 3q^{11} - 23q^{13} + 2q^{14} + 2q^{16} - 3q^{17} - 9q^{19} + 9q^{20} - 19q^{22} - 12q^{23} + 3q^{25} - 18q^{26} + 4q^{28} + 9q^{29} - 33q^{31} - 10q^{32} - 2q^{34} + 8q^{35} - 33q^{37} + 3q^{38} - 9q^{40} + 3q^{41} - 9q^{43} - 2q^{44} - 32q^{46} - 11q^{47} + 7q^{49} - 29q^{50} - 21q^{52} - q^{53} - 16q^{55} + 9q^{56} - 5q^{58} + 30q^{59} - 19q^{61} - 3q^{62} - 21q^{64} - 14q^{65} - 30q^{67} - 24q^{68} - 8q^{71} - 20q^{73} + 9q^{74} - 42q^{76} + 3q^{77} + 8q^{79} - 12q^{80} + 10q^{82} + 34q^{83} - 28q^{85} - 24q^{86} - q^{88} + 12q^{89} - 23q^{91} - 60q^{92} - 3q^{94} - 12q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.692358 −0.489571 −0.244785 0.969577i \(-0.578718\pi\)
−0.244785 + 0.969577i \(0.578718\pi\)
\(3\) 0 0
\(4\) −1.52064 −0.760320
\(5\) 2.78145 1.24390 0.621950 0.783057i \(-0.286341\pi\)
0.621950 + 0.783057i \(0.286341\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.43754 0.861801
\(9\) 0 0
\(10\) −1.92576 −0.608977
\(11\) 0.348849 0.105182 0.0525910 0.998616i \(-0.483252\pi\)
0.0525910 + 0.998616i \(0.483252\pi\)
\(12\) 0 0
\(13\) 2.17757 0.603948 0.301974 0.953316i \(-0.402354\pi\)
0.301974 + 0.953316i \(0.402354\pi\)
\(14\) −0.692358 −0.185040
\(15\) 0 0
\(16\) 1.35363 0.338408
\(17\) −7.42626 −1.80113 −0.900567 0.434718i \(-0.856848\pi\)
−0.900567 + 0.434718i \(0.856848\pi\)
\(18\) 0 0
\(19\) −0.255629 −0.0586454 −0.0293227 0.999570i \(-0.509335\pi\)
−0.0293227 + 0.999570i \(0.509335\pi\)
\(20\) −4.22958 −0.945763
\(21\) 0 0
\(22\) −0.241529 −0.0514941
\(23\) 4.98233 1.03889 0.519444 0.854504i \(-0.326139\pi\)
0.519444 + 0.854504i \(0.326139\pi\)
\(24\) 0 0
\(25\) 2.73644 0.547288
\(26\) −1.50766 −0.295676
\(27\) 0 0
\(28\) −1.52064 −0.287374
\(29\) 6.88124 1.27781 0.638907 0.769284i \(-0.279387\pi\)
0.638907 + 0.769284i \(0.279387\pi\)
\(30\) 0 0
\(31\) −10.1038 −1.81469 −0.907346 0.420385i \(-0.861895\pi\)
−0.907346 + 0.420385i \(0.861895\pi\)
\(32\) −5.81228 −1.02748
\(33\) 0 0
\(34\) 5.14163 0.881782
\(35\) 2.78145 0.470150
\(36\) 0 0
\(37\) −1.64279 −0.270072 −0.135036 0.990841i \(-0.543115\pi\)
−0.135036 + 0.990841i \(0.543115\pi\)
\(38\) 0.176987 0.0287111
\(39\) 0 0
\(40\) 6.77989 1.07200
\(41\) −3.56990 −0.557525 −0.278762 0.960360i \(-0.589924\pi\)
−0.278762 + 0.960360i \(0.589924\pi\)
\(42\) 0 0
\(43\) −9.34628 −1.42529 −0.712647 0.701523i \(-0.752504\pi\)
−0.712647 + 0.701523i \(0.752504\pi\)
\(44\) −0.530475 −0.0799721
\(45\) 0 0
\(46\) −3.44956 −0.508610
\(47\) −9.73885 −1.42056 −0.710279 0.703920i \(-0.751431\pi\)
−0.710279 + 0.703920i \(0.751431\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.89460 −0.267936
\(51\) 0 0
\(52\) −3.31130 −0.459194
\(53\) 7.44666 1.02288 0.511439 0.859320i \(-0.329113\pi\)
0.511439 + 0.859320i \(0.329113\pi\)
\(54\) 0 0
\(55\) 0.970306 0.130836
\(56\) 2.43754 0.325730
\(57\) 0 0
\(58\) −4.76428 −0.625581
\(59\) 2.01055 0.261752 0.130876 0.991399i \(-0.458221\pi\)
0.130876 + 0.991399i \(0.458221\pi\)
\(60\) 0 0
\(61\) −7.69549 −0.985306 −0.492653 0.870226i \(-0.663973\pi\)
−0.492653 + 0.870226i \(0.663973\pi\)
\(62\) 6.99543 0.888420
\(63\) 0 0
\(64\) 1.31692 0.164615
\(65\) 6.05678 0.751252
\(66\) 0 0
\(67\) −6.52951 −0.797706 −0.398853 0.917015i \(-0.630592\pi\)
−0.398853 + 0.917015i \(0.630592\pi\)
\(68\) 11.2927 1.36944
\(69\) 0 0
\(70\) −1.92576 −0.230172
\(71\) −13.7550 −1.63242 −0.816208 0.577758i \(-0.803928\pi\)
−0.816208 + 0.577758i \(0.803928\pi\)
\(72\) 0 0
\(73\) −9.37124 −1.09682 −0.548411 0.836209i \(-0.684767\pi\)
−0.548411 + 0.836209i \(0.684767\pi\)
\(74\) 1.13740 0.132220
\(75\) 0 0
\(76\) 0.388720 0.0445893
\(77\) 0.348849 0.0397551
\(78\) 0 0
\(79\) −5.92206 −0.666283 −0.333142 0.942877i \(-0.608109\pi\)
−0.333142 + 0.942877i \(0.608109\pi\)
\(80\) 3.76505 0.420945
\(81\) 0 0
\(82\) 2.47165 0.272948
\(83\) −1.68660 −0.185128 −0.0925642 0.995707i \(-0.529506\pi\)
−0.0925642 + 0.995707i \(0.529506\pi\)
\(84\) 0 0
\(85\) −20.6557 −2.24043
\(86\) 6.47097 0.697783
\(87\) 0 0
\(88\) 0.850335 0.0906461
\(89\) 17.6750 1.87355 0.936775 0.349932i \(-0.113795\pi\)
0.936775 + 0.349932i \(0.113795\pi\)
\(90\) 0 0
\(91\) 2.17757 0.228271
\(92\) −7.57634 −0.789888
\(93\) 0 0
\(94\) 6.74277 0.695464
\(95\) −0.711019 −0.0729490
\(96\) 0 0
\(97\) −0.216772 −0.0220099 −0.0110049 0.999939i \(-0.503503\pi\)
−0.0110049 + 0.999939i \(0.503503\pi\)
\(98\) −0.692358 −0.0699387
\(99\) 0 0
\(100\) −4.16114 −0.416114
\(101\) 15.6270 1.55494 0.777472 0.628918i \(-0.216502\pi\)
0.777472 + 0.628918i \(0.216502\pi\)
\(102\) 0 0
\(103\) −2.82388 −0.278245 −0.139122 0.990275i \(-0.544428\pi\)
−0.139122 + 0.990275i \(0.544428\pi\)
\(104\) 5.30791 0.520484
\(105\) 0 0
\(106\) −5.15575 −0.500771
\(107\) 13.6505 1.31964 0.659820 0.751424i \(-0.270633\pi\)
0.659820 + 0.751424i \(0.270633\pi\)
\(108\) 0 0
\(109\) −12.4581 −1.19327 −0.596633 0.802514i \(-0.703495\pi\)
−0.596633 + 0.802514i \(0.703495\pi\)
\(110\) −0.671799 −0.0640535
\(111\) 0 0
\(112\) 1.35363 0.127906
\(113\) −17.1295 −1.61140 −0.805702 0.592322i \(-0.798212\pi\)
−0.805702 + 0.592322i \(0.798212\pi\)
\(114\) 0 0
\(115\) 13.8581 1.29227
\(116\) −10.4639 −0.971549
\(117\) 0 0
\(118\) −1.39202 −0.128146
\(119\) −7.42626 −0.680764
\(120\) 0 0
\(121\) −10.8783 −0.988937
\(122\) 5.32803 0.482377
\(123\) 0 0
\(124\) 15.3642 1.37975
\(125\) −6.29597 −0.563128
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 10.7128 0.946885
\(129\) 0 0
\(130\) −4.19346 −0.367791
\(131\) −8.33905 −0.728587 −0.364293 0.931284i \(-0.618689\pi\)
−0.364293 + 0.931284i \(0.618689\pi\)
\(132\) 0 0
\(133\) −0.255629 −0.0221659
\(134\) 4.52075 0.390534
\(135\) 0 0
\(136\) −18.1018 −1.55222
\(137\) −20.9573 −1.79050 −0.895251 0.445563i \(-0.853004\pi\)
−0.895251 + 0.445563i \(0.853004\pi\)
\(138\) 0 0
\(139\) −10.1867 −0.864026 −0.432013 0.901867i \(-0.642197\pi\)
−0.432013 + 0.901867i \(0.642197\pi\)
\(140\) −4.22958 −0.357465
\(141\) 0 0
\(142\) 9.52337 0.799183
\(143\) 0.759643 0.0635245
\(144\) 0 0
\(145\) 19.1398 1.58947
\(146\) 6.48825 0.536972
\(147\) 0 0
\(148\) 2.49809 0.205342
\(149\) 4.86877 0.398865 0.199433 0.979912i \(-0.436090\pi\)
0.199433 + 0.979912i \(0.436090\pi\)
\(150\) 0 0
\(151\) −1.93573 −0.157528 −0.0787638 0.996893i \(-0.525097\pi\)
−0.0787638 + 0.996893i \(0.525097\pi\)
\(152\) −0.623107 −0.0505407
\(153\) 0 0
\(154\) −0.241529 −0.0194629
\(155\) −28.1031 −2.25730
\(156\) 0 0
\(157\) 4.29270 0.342595 0.171297 0.985219i \(-0.445204\pi\)
0.171297 + 0.985219i \(0.445204\pi\)
\(158\) 4.10018 0.326193
\(159\) 0 0
\(160\) −16.1665 −1.27808
\(161\) 4.98233 0.392663
\(162\) 0 0
\(163\) 4.96855 0.389167 0.194583 0.980886i \(-0.437664\pi\)
0.194583 + 0.980886i \(0.437664\pi\)
\(164\) 5.42854 0.423897
\(165\) 0 0
\(166\) 1.16773 0.0906335
\(167\) 14.3188 1.10802 0.554012 0.832508i \(-0.313096\pi\)
0.554012 + 0.832508i \(0.313096\pi\)
\(168\) 0 0
\(169\) −8.25820 −0.635246
\(170\) 14.3012 1.09685
\(171\) 0 0
\(172\) 14.2123 1.08368
\(173\) 22.2870 1.69445 0.847225 0.531235i \(-0.178272\pi\)
0.847225 + 0.531235i \(0.178272\pi\)
\(174\) 0 0
\(175\) 2.73644 0.206855
\(176\) 0.472213 0.0355944
\(177\) 0 0
\(178\) −12.2374 −0.917235
\(179\) −26.5031 −1.98093 −0.990467 0.137748i \(-0.956014\pi\)
−0.990467 + 0.137748i \(0.956014\pi\)
\(180\) 0 0
\(181\) −6.62020 −0.492076 −0.246038 0.969260i \(-0.579129\pi\)
−0.246038 + 0.969260i \(0.579129\pi\)
\(182\) −1.50766 −0.111755
\(183\) 0 0
\(184\) 12.1447 0.895316
\(185\) −4.56932 −0.335943
\(186\) 0 0
\(187\) −2.59065 −0.189447
\(188\) 14.8093 1.08008
\(189\) 0 0
\(190\) 0.492279 0.0357137
\(191\) −9.50979 −0.688104 −0.344052 0.938951i \(-0.611800\pi\)
−0.344052 + 0.938951i \(0.611800\pi\)
\(192\) 0 0
\(193\) 18.2187 1.31141 0.655706 0.755016i \(-0.272371\pi\)
0.655706 + 0.755016i \(0.272371\pi\)
\(194\) 0.150084 0.0107754
\(195\) 0 0
\(196\) −1.52064 −0.108617
\(197\) 6.73369 0.479756 0.239878 0.970803i \(-0.422893\pi\)
0.239878 + 0.970803i \(0.422893\pi\)
\(198\) 0 0
\(199\) 19.1877 1.36018 0.680088 0.733130i \(-0.261941\pi\)
0.680088 + 0.733130i \(0.261941\pi\)
\(200\) 6.67019 0.471654
\(201\) 0 0
\(202\) −10.8195 −0.761255
\(203\) 6.88124 0.482969
\(204\) 0 0
\(205\) −9.92948 −0.693505
\(206\) 1.95513 0.136221
\(207\) 0 0
\(208\) 2.94762 0.204381
\(209\) −0.0891761 −0.00616844
\(210\) 0 0
\(211\) 3.89333 0.268028 0.134014 0.990979i \(-0.457213\pi\)
0.134014 + 0.990979i \(0.457213\pi\)
\(212\) −11.3237 −0.777715
\(213\) 0 0
\(214\) −9.45100 −0.646057
\(215\) −25.9962 −1.77292
\(216\) 0 0
\(217\) −10.1038 −0.685889
\(218\) 8.62544 0.584189
\(219\) 0 0
\(220\) −1.47549 −0.0994773
\(221\) −16.1712 −1.08779
\(222\) 0 0
\(223\) 17.8977 1.19852 0.599258 0.800556i \(-0.295462\pi\)
0.599258 + 0.800556i \(0.295462\pi\)
\(224\) −5.81228 −0.388349
\(225\) 0 0
\(226\) 11.8597 0.788896
\(227\) −8.76418 −0.581699 −0.290850 0.956769i \(-0.593938\pi\)
−0.290850 + 0.956769i \(0.593938\pi\)
\(228\) 0 0
\(229\) −20.1983 −1.33474 −0.667371 0.744726i \(-0.732580\pi\)
−0.667371 + 0.744726i \(0.732580\pi\)
\(230\) −9.59476 −0.632660
\(231\) 0 0
\(232\) 16.7733 1.10122
\(233\) 29.4789 1.93122 0.965612 0.259986i \(-0.0837181\pi\)
0.965612 + 0.259986i \(0.0837181\pi\)
\(234\) 0 0
\(235\) −27.0881 −1.76703
\(236\) −3.05733 −0.199015
\(237\) 0 0
\(238\) 5.14163 0.333282
\(239\) 22.7548 1.47188 0.735941 0.677045i \(-0.236740\pi\)
0.735941 + 0.677045i \(0.236740\pi\)
\(240\) 0 0
\(241\) 12.6179 0.812789 0.406395 0.913698i \(-0.366786\pi\)
0.406395 + 0.913698i \(0.366786\pi\)
\(242\) 7.53168 0.484155
\(243\) 0 0
\(244\) 11.7021 0.749148
\(245\) 2.78145 0.177700
\(246\) 0 0
\(247\) −0.556650 −0.0354188
\(248\) −24.6284 −1.56390
\(249\) 0 0
\(250\) 4.35906 0.275691
\(251\) 17.4819 1.10345 0.551725 0.834026i \(-0.313970\pi\)
0.551725 + 0.834026i \(0.313970\pi\)
\(252\) 0 0
\(253\) 1.73808 0.109272
\(254\) 0.692358 0.0434424
\(255\) 0 0
\(256\) −10.0509 −0.628182
\(257\) −8.26061 −0.515283 −0.257641 0.966241i \(-0.582945\pi\)
−0.257641 + 0.966241i \(0.582945\pi\)
\(258\) 0 0
\(259\) −1.64279 −0.102078
\(260\) −9.21019 −0.571192
\(261\) 0 0
\(262\) 5.77361 0.356695
\(263\) −25.8026 −1.59106 −0.795530 0.605915i \(-0.792807\pi\)
−0.795530 + 0.605915i \(0.792807\pi\)
\(264\) 0 0
\(265\) 20.7125 1.27236
\(266\) 0.176987 0.0108518
\(267\) 0 0
\(268\) 9.92903 0.606512
\(269\) 6.10591 0.372284 0.186142 0.982523i \(-0.440402\pi\)
0.186142 + 0.982523i \(0.440402\pi\)
\(270\) 0 0
\(271\) −16.8740 −1.02503 −0.512513 0.858680i \(-0.671285\pi\)
−0.512513 + 0.858680i \(0.671285\pi\)
\(272\) −10.0524 −0.609517
\(273\) 0 0
\(274\) 14.5099 0.876577
\(275\) 0.954606 0.0575649
\(276\) 0 0
\(277\) 29.0357 1.74459 0.872293 0.488983i \(-0.162632\pi\)
0.872293 + 0.488983i \(0.162632\pi\)
\(278\) 7.05285 0.423002
\(279\) 0 0
\(280\) 6.77989 0.405176
\(281\) −3.71119 −0.221391 −0.110695 0.993854i \(-0.535308\pi\)
−0.110695 + 0.993854i \(0.535308\pi\)
\(282\) 0 0
\(283\) −10.0392 −0.596770 −0.298385 0.954446i \(-0.596448\pi\)
−0.298385 + 0.954446i \(0.596448\pi\)
\(284\) 20.9164 1.24116
\(285\) 0 0
\(286\) −0.525945 −0.0310998
\(287\) −3.56990 −0.210725
\(288\) 0 0
\(289\) 38.1494 2.24408
\(290\) −13.2516 −0.778160
\(291\) 0 0
\(292\) 14.2503 0.833935
\(293\) −30.6862 −1.79271 −0.896355 0.443338i \(-0.853794\pi\)
−0.896355 + 0.443338i \(0.853794\pi\)
\(294\) 0 0
\(295\) 5.59224 0.325593
\(296\) −4.00436 −0.232749
\(297\) 0 0
\(298\) −3.37093 −0.195273
\(299\) 10.8494 0.627435
\(300\) 0 0
\(301\) −9.34628 −0.538711
\(302\) 1.34022 0.0771209
\(303\) 0 0
\(304\) −0.346027 −0.0198460
\(305\) −21.4046 −1.22562
\(306\) 0 0
\(307\) −7.58942 −0.433151 −0.216575 0.976266i \(-0.569489\pi\)
−0.216575 + 0.976266i \(0.569489\pi\)
\(308\) −0.530475 −0.0302266
\(309\) 0 0
\(310\) 19.4574 1.10511
\(311\) 31.5432 1.78865 0.894324 0.447419i \(-0.147657\pi\)
0.894324 + 0.447419i \(0.147657\pi\)
\(312\) 0 0
\(313\) 25.5982 1.44689 0.723447 0.690379i \(-0.242556\pi\)
0.723447 + 0.690379i \(0.242556\pi\)
\(314\) −2.97208 −0.167724
\(315\) 0 0
\(316\) 9.00532 0.506589
\(317\) −20.7732 −1.16674 −0.583370 0.812206i \(-0.698266\pi\)
−0.583370 + 0.812206i \(0.698266\pi\)
\(318\) 0 0
\(319\) 2.40052 0.134403
\(320\) 3.66293 0.204764
\(321\) 0 0
\(322\) −3.44956 −0.192236
\(323\) 1.89837 0.105628
\(324\) 0 0
\(325\) 5.95878 0.330534
\(326\) −3.44001 −0.190525
\(327\) 0 0
\(328\) −8.70178 −0.480476
\(329\) −9.73885 −0.536920
\(330\) 0 0
\(331\) −3.11825 −0.171394 −0.0856971 0.996321i \(-0.527312\pi\)
−0.0856971 + 0.996321i \(0.527312\pi\)
\(332\) 2.56471 0.140757
\(333\) 0 0
\(334\) −9.91376 −0.542457
\(335\) −18.1615 −0.992267
\(336\) 0 0
\(337\) 0.0769121 0.00418967 0.00209483 0.999998i \(-0.499333\pi\)
0.00209483 + 0.999998i \(0.499333\pi\)
\(338\) 5.71763 0.310998
\(339\) 0 0
\(340\) 31.4100 1.70344
\(341\) −3.52470 −0.190873
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −22.7820 −1.22832
\(345\) 0 0
\(346\) −15.4306 −0.829553
\(347\) 7.04794 0.378353 0.189177 0.981943i \(-0.439418\pi\)
0.189177 + 0.981943i \(0.439418\pi\)
\(348\) 0 0
\(349\) 15.4067 0.824703 0.412351 0.911025i \(-0.364708\pi\)
0.412351 + 0.911025i \(0.364708\pi\)
\(350\) −1.89460 −0.101270
\(351\) 0 0
\(352\) −2.02761 −0.108072
\(353\) −15.3530 −0.817157 −0.408578 0.912723i \(-0.633975\pi\)
−0.408578 + 0.912723i \(0.633975\pi\)
\(354\) 0 0
\(355\) −38.2587 −2.03056
\(356\) −26.8774 −1.42450
\(357\) 0 0
\(358\) 18.3496 0.969808
\(359\) 2.58094 0.136217 0.0681084 0.997678i \(-0.478304\pi\)
0.0681084 + 0.997678i \(0.478304\pi\)
\(360\) 0 0
\(361\) −18.9347 −0.996561
\(362\) 4.58355 0.240906
\(363\) 0 0
\(364\) −3.31130 −0.173559
\(365\) −26.0656 −1.36434
\(366\) 0 0
\(367\) 22.1135 1.15431 0.577157 0.816634i \(-0.304162\pi\)
0.577157 + 0.816634i \(0.304162\pi\)
\(368\) 6.74424 0.351568
\(369\) 0 0
\(370\) 3.16360 0.164468
\(371\) 7.44666 0.386612
\(372\) 0 0
\(373\) −16.3558 −0.846873 −0.423436 0.905926i \(-0.639176\pi\)
−0.423436 + 0.905926i \(0.639176\pi\)
\(374\) 1.79365 0.0927477
\(375\) 0 0
\(376\) −23.7389 −1.22424
\(377\) 14.9844 0.771734
\(378\) 0 0
\(379\) 22.3125 1.14612 0.573059 0.819514i \(-0.305757\pi\)
0.573059 + 0.819514i \(0.305757\pi\)
\(380\) 1.08120 0.0554646
\(381\) 0 0
\(382\) 6.58418 0.336876
\(383\) 18.0582 0.922729 0.461365 0.887211i \(-0.347360\pi\)
0.461365 + 0.887211i \(0.347360\pi\)
\(384\) 0 0
\(385\) 0.970306 0.0494514
\(386\) −12.6139 −0.642029
\(387\) 0 0
\(388\) 0.329632 0.0167346
\(389\) 1.96882 0.0998232 0.0499116 0.998754i \(-0.484106\pi\)
0.0499116 + 0.998754i \(0.484106\pi\)
\(390\) 0 0
\(391\) −37.0001 −1.87118
\(392\) 2.43754 0.123114
\(393\) 0 0
\(394\) −4.66212 −0.234874
\(395\) −16.4719 −0.828790
\(396\) 0 0
\(397\) −13.1572 −0.660341 −0.330171 0.943921i \(-0.607106\pi\)
−0.330171 + 0.943921i \(0.607106\pi\)
\(398\) −13.2847 −0.665903
\(399\) 0 0
\(400\) 3.70413 0.185206
\(401\) 27.6771 1.38213 0.691063 0.722794i \(-0.257143\pi\)
0.691063 + 0.722794i \(0.257143\pi\)
\(402\) 0 0
\(403\) −22.0017 −1.09598
\(404\) −23.7630 −1.18226
\(405\) 0 0
\(406\) −4.76428 −0.236447
\(407\) −0.573085 −0.0284068
\(408\) 0 0
\(409\) −23.5157 −1.16278 −0.581388 0.813626i \(-0.697490\pi\)
−0.581388 + 0.813626i \(0.697490\pi\)
\(410\) 6.87475 0.339520
\(411\) 0 0
\(412\) 4.29410 0.211555
\(413\) 2.01055 0.0989328
\(414\) 0 0
\(415\) −4.69119 −0.230281
\(416\) −12.6566 −0.620542
\(417\) 0 0
\(418\) 0.0617418 0.00301989
\(419\) −3.78832 −0.185072 −0.0925359 0.995709i \(-0.529497\pi\)
−0.0925359 + 0.995709i \(0.529497\pi\)
\(420\) 0 0
\(421\) −40.3970 −1.96883 −0.984415 0.175859i \(-0.943730\pi\)
−0.984415 + 0.175859i \(0.943730\pi\)
\(422\) −2.69558 −0.131219
\(423\) 0 0
\(424\) 18.1516 0.881518
\(425\) −20.3215 −0.985739
\(426\) 0 0
\(427\) −7.69549 −0.372411
\(428\) −20.7574 −1.00335
\(429\) 0 0
\(430\) 17.9986 0.867972
\(431\) −26.3339 −1.26846 −0.634230 0.773144i \(-0.718683\pi\)
−0.634230 + 0.773144i \(0.718683\pi\)
\(432\) 0 0
\(433\) −6.47692 −0.311261 −0.155631 0.987815i \(-0.549741\pi\)
−0.155631 + 0.987815i \(0.549741\pi\)
\(434\) 6.99543 0.335791
\(435\) 0 0
\(436\) 18.9443 0.907265
\(437\) −1.27363 −0.0609260
\(438\) 0 0
\(439\) 11.9232 0.569062 0.284531 0.958667i \(-0.408162\pi\)
0.284531 + 0.958667i \(0.408162\pi\)
\(440\) 2.36516 0.112755
\(441\) 0 0
\(442\) 11.1962 0.532551
\(443\) −31.9287 −1.51698 −0.758490 0.651685i \(-0.774062\pi\)
−0.758490 + 0.651685i \(0.774062\pi\)
\(444\) 0 0
\(445\) 49.1622 2.33051
\(446\) −12.3916 −0.586759
\(447\) 0 0
\(448\) 1.31692 0.0622185
\(449\) −31.3379 −1.47893 −0.739463 0.673197i \(-0.764921\pi\)
−0.739463 + 0.673197i \(0.764921\pi\)
\(450\) 0 0
\(451\) −1.24536 −0.0586416
\(452\) 26.0477 1.22518
\(453\) 0 0
\(454\) 6.06795 0.284783
\(455\) 6.05678 0.283946
\(456\) 0 0
\(457\) −17.5797 −0.822344 −0.411172 0.911558i \(-0.634880\pi\)
−0.411172 + 0.911558i \(0.634880\pi\)
\(458\) 13.9844 0.653450
\(459\) 0 0
\(460\) −21.0732 −0.982542
\(461\) 27.4541 1.27867 0.639333 0.768930i \(-0.279211\pi\)
0.639333 + 0.768930i \(0.279211\pi\)
\(462\) 0 0
\(463\) −18.2297 −0.847207 −0.423604 0.905848i \(-0.639235\pi\)
−0.423604 + 0.905848i \(0.639235\pi\)
\(464\) 9.31466 0.432422
\(465\) 0 0
\(466\) −20.4099 −0.945471
\(467\) 6.72456 0.311176 0.155588 0.987822i \(-0.450273\pi\)
0.155588 + 0.987822i \(0.450273\pi\)
\(468\) 0 0
\(469\) −6.52951 −0.301505
\(470\) 18.7546 0.865087
\(471\) 0 0
\(472\) 4.90081 0.225578
\(473\) −3.26044 −0.149915
\(474\) 0 0
\(475\) −0.699514 −0.0320959
\(476\) 11.2927 0.517599
\(477\) 0 0
\(478\) −15.7544 −0.720591
\(479\) −19.3112 −0.882350 −0.441175 0.897421i \(-0.645438\pi\)
−0.441175 + 0.897421i \(0.645438\pi\)
\(480\) 0 0
\(481\) −3.57728 −0.163110
\(482\) −8.73609 −0.397918
\(483\) 0 0
\(484\) 16.5420 0.751909
\(485\) −0.602940 −0.0273781
\(486\) 0 0
\(487\) −38.4870 −1.74401 −0.872007 0.489494i \(-0.837182\pi\)
−0.872007 + 0.489494i \(0.837182\pi\)
\(488\) −18.7581 −0.849138
\(489\) 0 0
\(490\) −1.92576 −0.0869968
\(491\) −19.8484 −0.895746 −0.447873 0.894097i \(-0.647818\pi\)
−0.447873 + 0.894097i \(0.647818\pi\)
\(492\) 0 0
\(493\) −51.1019 −2.30151
\(494\) 0.385401 0.0173400
\(495\) 0 0
\(496\) −13.6768 −0.614105
\(497\) −13.7550 −0.616995
\(498\) 0 0
\(499\) 12.8775 0.576476 0.288238 0.957559i \(-0.406931\pi\)
0.288238 + 0.957559i \(0.406931\pi\)
\(500\) 9.57391 0.428158
\(501\) 0 0
\(502\) −12.1037 −0.540217
\(503\) 35.1228 1.56605 0.783024 0.621992i \(-0.213676\pi\)
0.783024 + 0.621992i \(0.213676\pi\)
\(504\) 0 0
\(505\) 43.4656 1.93419
\(506\) −1.20338 −0.0534966
\(507\) 0 0
\(508\) 1.52064 0.0674675
\(509\) 1.90456 0.0844184 0.0422092 0.999109i \(-0.486560\pi\)
0.0422092 + 0.999109i \(0.486560\pi\)
\(510\) 0 0
\(511\) −9.37124 −0.414559
\(512\) −14.4667 −0.639346
\(513\) 0 0
\(514\) 5.71930 0.252267
\(515\) −7.85446 −0.346109
\(516\) 0 0
\(517\) −3.39739 −0.149417
\(518\) 1.13740 0.0499743
\(519\) 0 0
\(520\) 14.7637 0.647430
\(521\) 31.1698 1.36558 0.682788 0.730617i \(-0.260767\pi\)
0.682788 + 0.730617i \(0.260767\pi\)
\(522\) 0 0
\(523\) −16.1856 −0.707748 −0.353874 0.935293i \(-0.615136\pi\)
−0.353874 + 0.935293i \(0.615136\pi\)
\(524\) 12.6807 0.553959
\(525\) 0 0
\(526\) 17.8647 0.778936
\(527\) 75.0333 3.26850
\(528\) 0 0
\(529\) 1.82366 0.0792896
\(530\) −14.3405 −0.622909
\(531\) 0 0
\(532\) 0.388720 0.0168532
\(533\) −7.77370 −0.336716
\(534\) 0 0
\(535\) 37.9680 1.64150
\(536\) −15.9160 −0.687464
\(537\) 0 0
\(538\) −4.22748 −0.182259
\(539\) 0.348849 0.0150260
\(540\) 0 0
\(541\) −18.5894 −0.799221 −0.399610 0.916685i \(-0.630855\pi\)
−0.399610 + 0.916685i \(0.630855\pi\)
\(542\) 11.6829 0.501822
\(543\) 0 0
\(544\) 43.1635 1.85062
\(545\) −34.6514 −1.48430
\(546\) 0 0
\(547\) 16.3670 0.699802 0.349901 0.936787i \(-0.386215\pi\)
0.349901 + 0.936787i \(0.386215\pi\)
\(548\) 31.8685 1.36135
\(549\) 0 0
\(550\) −0.660929 −0.0281821
\(551\) −1.75905 −0.0749379
\(552\) 0 0
\(553\) −5.92206 −0.251831
\(554\) −20.1031 −0.854099
\(555\) 0 0
\(556\) 15.4903 0.656937
\(557\) −37.7810 −1.60083 −0.800417 0.599444i \(-0.795388\pi\)
−0.800417 + 0.599444i \(0.795388\pi\)
\(558\) 0 0
\(559\) −20.3522 −0.860804
\(560\) 3.76505 0.159102
\(561\) 0 0
\(562\) 2.56947 0.108386
\(563\) −13.6001 −0.573176 −0.286588 0.958054i \(-0.592521\pi\)
−0.286588 + 0.958054i \(0.592521\pi\)
\(564\) 0 0
\(565\) −47.6446 −2.00443
\(566\) 6.95074 0.292161
\(567\) 0 0
\(568\) −33.5284 −1.40682
\(569\) −9.94802 −0.417043 −0.208521 0.978018i \(-0.566865\pi\)
−0.208521 + 0.978018i \(0.566865\pi\)
\(570\) 0 0
\(571\) 8.42532 0.352588 0.176294 0.984338i \(-0.443589\pi\)
0.176294 + 0.984338i \(0.443589\pi\)
\(572\) −1.15514 −0.0482990
\(573\) 0 0
\(574\) 2.47165 0.103165
\(575\) 13.6339 0.568571
\(576\) 0 0
\(577\) −20.5668 −0.856206 −0.428103 0.903730i \(-0.640818\pi\)
−0.428103 + 0.903730i \(0.640818\pi\)
\(578\) −26.4130 −1.09864
\(579\) 0 0
\(580\) −29.1048 −1.20851
\(581\) −1.68660 −0.0699720
\(582\) 0 0
\(583\) 2.59776 0.107588
\(584\) −22.8428 −0.945242
\(585\) 0 0
\(586\) 21.2459 0.877658
\(587\) −23.2118 −0.958053 −0.479027 0.877800i \(-0.659010\pi\)
−0.479027 + 0.877800i \(0.659010\pi\)
\(588\) 0 0
\(589\) 2.58282 0.106423
\(590\) −3.87183 −0.159401
\(591\) 0 0
\(592\) −2.22373 −0.0913945
\(593\) −29.0658 −1.19359 −0.596795 0.802393i \(-0.703560\pi\)
−0.596795 + 0.802393i \(0.703560\pi\)
\(594\) 0 0
\(595\) −20.6557 −0.846803
\(596\) −7.40365 −0.303265
\(597\) 0 0
\(598\) −7.51164 −0.307174
\(599\) 31.7965 1.29917 0.649584 0.760290i \(-0.274943\pi\)
0.649584 + 0.760290i \(0.274943\pi\)
\(600\) 0 0
\(601\) 21.6226 0.882004 0.441002 0.897506i \(-0.354623\pi\)
0.441002 + 0.897506i \(0.354623\pi\)
\(602\) 6.47097 0.263737
\(603\) 0 0
\(604\) 2.94355 0.119771
\(605\) −30.2574 −1.23014
\(606\) 0 0
\(607\) 19.7589 0.801988 0.400994 0.916081i \(-0.368665\pi\)
0.400994 + 0.916081i \(0.368665\pi\)
\(608\) 1.48579 0.0602567
\(609\) 0 0
\(610\) 14.8196 0.600029
\(611\) −21.2070 −0.857944
\(612\) 0 0
\(613\) −7.69528 −0.310809 −0.155405 0.987851i \(-0.549668\pi\)
−0.155405 + 0.987851i \(0.549668\pi\)
\(614\) 5.25459 0.212058
\(615\) 0 0
\(616\) 0.850335 0.0342610
\(617\) 46.4653 1.87062 0.935310 0.353828i \(-0.115120\pi\)
0.935310 + 0.353828i \(0.115120\pi\)
\(618\) 0 0
\(619\) −9.42955 −0.379006 −0.189503 0.981880i \(-0.560688\pi\)
−0.189503 + 0.981880i \(0.560688\pi\)
\(620\) 42.7347 1.71627
\(621\) 0 0
\(622\) −21.8391 −0.875670
\(623\) 17.6750 0.708135
\(624\) 0 0
\(625\) −31.1941 −1.24776
\(626\) −17.7231 −0.708357
\(627\) 0 0
\(628\) −6.52765 −0.260482
\(629\) 12.1998 0.486436
\(630\) 0 0
\(631\) −34.1327 −1.35880 −0.679401 0.733768i \(-0.737760\pi\)
−0.679401 + 0.733768i \(0.737760\pi\)
\(632\) −14.4353 −0.574204
\(633\) 0 0
\(634\) 14.3825 0.571202
\(635\) −2.78145 −0.110378
\(636\) 0 0
\(637\) 2.17757 0.0862783
\(638\) −1.66202 −0.0657999
\(639\) 0 0
\(640\) 29.7970 1.17783
\(641\) −35.7111 −1.41050 −0.705252 0.708956i \(-0.749166\pi\)
−0.705252 + 0.708956i \(0.749166\pi\)
\(642\) 0 0
\(643\) 0.292989 0.0115544 0.00577718 0.999983i \(-0.498161\pi\)
0.00577718 + 0.999983i \(0.498161\pi\)
\(644\) −7.57634 −0.298550
\(645\) 0 0
\(646\) −1.31435 −0.0517124
\(647\) −7.55746 −0.297114 −0.148557 0.988904i \(-0.547463\pi\)
−0.148557 + 0.988904i \(0.547463\pi\)
\(648\) 0 0
\(649\) 0.701380 0.0275316
\(650\) −4.12561 −0.161820
\(651\) 0 0
\(652\) −7.55538 −0.295892
\(653\) 43.1190 1.68738 0.843688 0.536833i \(-0.180380\pi\)
0.843688 + 0.536833i \(0.180380\pi\)
\(654\) 0 0
\(655\) −23.1946 −0.906289
\(656\) −4.83232 −0.188671
\(657\) 0 0
\(658\) 6.74277 0.262861
\(659\) 22.2019 0.864865 0.432433 0.901666i \(-0.357655\pi\)
0.432433 + 0.901666i \(0.357655\pi\)
\(660\) 0 0
\(661\) 42.4146 1.64974 0.824868 0.565326i \(-0.191249\pi\)
0.824868 + 0.565326i \(0.191249\pi\)
\(662\) 2.15894 0.0839096
\(663\) 0 0
\(664\) −4.11116 −0.159544
\(665\) −0.711019 −0.0275721
\(666\) 0 0
\(667\) 34.2847 1.32751
\(668\) −21.7738 −0.842454
\(669\) 0 0
\(670\) 12.5742 0.485785
\(671\) −2.68457 −0.103637
\(672\) 0 0
\(673\) −21.3504 −0.822998 −0.411499 0.911410i \(-0.634995\pi\)
−0.411499 + 0.911410i \(0.634995\pi\)
\(674\) −0.0532507 −0.00205114
\(675\) 0 0
\(676\) 12.5578 0.482991
\(677\) −6.31191 −0.242587 −0.121293 0.992617i \(-0.538704\pi\)
−0.121293 + 0.992617i \(0.538704\pi\)
\(678\) 0 0
\(679\) −0.216772 −0.00831895
\(680\) −50.3493 −1.93081
\(681\) 0 0
\(682\) 2.44035 0.0934459
\(683\) −44.6920 −1.71009 −0.855047 0.518551i \(-0.826472\pi\)
−0.855047 + 0.518551i \(0.826472\pi\)
\(684\) 0 0
\(685\) −58.2915 −2.22721
\(686\) −0.692358 −0.0264343
\(687\) 0 0
\(688\) −12.6514 −0.482330
\(689\) 16.2156 0.617766
\(690\) 0 0
\(691\) −16.3228 −0.620948 −0.310474 0.950582i \(-0.600488\pi\)
−0.310474 + 0.950582i \(0.600488\pi\)
\(692\) −33.8905 −1.28832
\(693\) 0 0
\(694\) −4.87970 −0.185231
\(695\) −28.3338 −1.07476
\(696\) 0 0
\(697\) 26.5110 1.00418
\(698\) −10.6670 −0.403750
\(699\) 0 0
\(700\) −4.16114 −0.157276
\(701\) −16.1072 −0.608360 −0.304180 0.952615i \(-0.598382\pi\)
−0.304180 + 0.952615i \(0.598382\pi\)
\(702\) 0 0
\(703\) 0.419944 0.0158385
\(704\) 0.459406 0.0173145
\(705\) 0 0
\(706\) 10.6298 0.400056
\(707\) 15.6270 0.587713
\(708\) 0 0
\(709\) 18.7118 0.702738 0.351369 0.936237i \(-0.385716\pi\)
0.351369 + 0.936237i \(0.385716\pi\)
\(710\) 26.4887 0.994104
\(711\) 0 0
\(712\) 43.0837 1.61463
\(713\) −50.3404 −1.88526
\(714\) 0 0
\(715\) 2.11291 0.0790182
\(716\) 40.3017 1.50615
\(717\) 0 0
\(718\) −1.78693 −0.0666878
\(719\) −2.91001 −0.108525 −0.0542625 0.998527i \(-0.517281\pi\)
−0.0542625 + 0.998527i \(0.517281\pi\)
\(720\) 0 0
\(721\) −2.82388 −0.105167
\(722\) 13.1096 0.487887
\(723\) 0 0
\(724\) 10.0669 0.374135
\(725\) 18.8301 0.699333
\(726\) 0 0
\(727\) −31.1593 −1.15563 −0.577817 0.816166i \(-0.696095\pi\)
−0.577817 + 0.816166i \(0.696095\pi\)
\(728\) 5.30791 0.196724
\(729\) 0 0
\(730\) 18.0467 0.667939
\(731\) 69.4079 2.56715
\(732\) 0 0
\(733\) −35.0113 −1.29317 −0.646586 0.762841i \(-0.723804\pi\)
−0.646586 + 0.762841i \(0.723804\pi\)
\(734\) −15.3104 −0.565118
\(735\) 0 0
\(736\) −28.9587 −1.06743
\(737\) −2.27782 −0.0839044
\(738\) 0 0
\(739\) −31.9289 −1.17452 −0.587262 0.809397i \(-0.699794\pi\)
−0.587262 + 0.809397i \(0.699794\pi\)
\(740\) 6.94830 0.255424
\(741\) 0 0
\(742\) −5.15575 −0.189274
\(743\) 9.30835 0.341490 0.170745 0.985315i \(-0.445382\pi\)
0.170745 + 0.985315i \(0.445382\pi\)
\(744\) 0 0
\(745\) 13.5422 0.496149
\(746\) 11.3241 0.414604
\(747\) 0 0
\(748\) 3.93944 0.144040
\(749\) 13.6505 0.498777
\(750\) 0 0
\(751\) 51.6363 1.88424 0.942118 0.335282i \(-0.108832\pi\)
0.942118 + 0.335282i \(0.108832\pi\)
\(752\) −13.1828 −0.480728
\(753\) 0 0
\(754\) −10.3745 −0.377819
\(755\) −5.38413 −0.195949
\(756\) 0 0
\(757\) 42.6294 1.54939 0.774695 0.632335i \(-0.217903\pi\)
0.774695 + 0.632335i \(0.217903\pi\)
\(758\) −15.4482 −0.561106
\(759\) 0 0
\(760\) −1.73314 −0.0628675
\(761\) 31.2012 1.13104 0.565520 0.824734i \(-0.308675\pi\)
0.565520 + 0.824734i \(0.308675\pi\)
\(762\) 0 0
\(763\) −12.4581 −0.451012
\(764\) 14.4610 0.523180
\(765\) 0 0
\(766\) −12.5027 −0.451741
\(767\) 4.37811 0.158084
\(768\) 0 0
\(769\) −14.0501 −0.506660 −0.253330 0.967380i \(-0.581526\pi\)
−0.253330 + 0.967380i \(0.581526\pi\)
\(770\) −0.671799 −0.0242099
\(771\) 0 0
\(772\) −27.7041 −0.997094
\(773\) −14.5672 −0.523945 −0.261973 0.965075i \(-0.584373\pi\)
−0.261973 + 0.965075i \(0.584373\pi\)
\(774\) 0 0
\(775\) −27.6484 −0.993159
\(776\) −0.528391 −0.0189681
\(777\) 0 0
\(778\) −1.36313 −0.0488705
\(779\) 0.912570 0.0326962
\(780\) 0 0
\(781\) −4.79842 −0.171701
\(782\) 25.6173 0.916074
\(783\) 0 0
\(784\) 1.35363 0.0483439
\(785\) 11.9399 0.426153
\(786\) 0 0
\(787\) 33.4906 1.19381 0.596905 0.802312i \(-0.296397\pi\)
0.596905 + 0.802312i \(0.296397\pi\)
\(788\) −10.2395 −0.364768
\(789\) 0 0
\(790\) 11.4044 0.405751
\(791\) −17.1295 −0.609053
\(792\) 0 0
\(793\) −16.7574 −0.595074
\(794\) 9.10949 0.323284
\(795\) 0 0
\(796\) −29.1775 −1.03417
\(797\) −18.4316 −0.652880 −0.326440 0.945218i \(-0.605849\pi\)
−0.326440 + 0.945218i \(0.605849\pi\)
\(798\) 0 0
\(799\) 72.3233 2.55861
\(800\) −15.9050 −0.562325
\(801\) 0 0
\(802\) −19.1624 −0.676649
\(803\) −3.26915 −0.115366
\(804\) 0 0
\(805\) 13.8581 0.488434
\(806\) 15.2330 0.536560
\(807\) 0 0
\(808\) 38.0914 1.34005
\(809\) −9.24970 −0.325202 −0.162601 0.986692i \(-0.551988\pi\)
−0.162601 + 0.986692i \(0.551988\pi\)
\(810\) 0 0
\(811\) −43.2167 −1.51755 −0.758773 0.651356i \(-0.774201\pi\)
−0.758773 + 0.651356i \(0.774201\pi\)
\(812\) −10.4639 −0.367211
\(813\) 0 0
\(814\) 0.396780 0.0139071
\(815\) 13.8198 0.484085
\(816\) 0 0
\(817\) 2.38918 0.0835869
\(818\) 16.2813 0.569261
\(819\) 0 0
\(820\) 15.0992 0.527286
\(821\) −29.3683 −1.02496 −0.512481 0.858699i \(-0.671273\pi\)
−0.512481 + 0.858699i \(0.671273\pi\)
\(822\) 0 0
\(823\) 16.4590 0.573726 0.286863 0.957972i \(-0.407388\pi\)
0.286863 + 0.957972i \(0.407388\pi\)
\(824\) −6.88332 −0.239792
\(825\) 0 0
\(826\) −1.39202 −0.0484346
\(827\) −42.9794 −1.49454 −0.747270 0.664521i \(-0.768636\pi\)
−0.747270 + 0.664521i \(0.768636\pi\)
\(828\) 0 0
\(829\) 8.69707 0.302062 0.151031 0.988529i \(-0.451741\pi\)
0.151031 + 0.988529i \(0.451741\pi\)
\(830\) 3.24798 0.112739
\(831\) 0 0
\(832\) 2.86768 0.0994188
\(833\) −7.42626 −0.257305
\(834\) 0 0
\(835\) 39.8271 1.37827
\(836\) 0.135605 0.00468999
\(837\) 0 0
\(838\) 2.62287 0.0906057
\(839\) −14.0856 −0.486289 −0.243144 0.969990i \(-0.578179\pi\)
−0.243144 + 0.969990i \(0.578179\pi\)
\(840\) 0 0
\(841\) 18.3515 0.632811
\(842\) 27.9692 0.963882
\(843\) 0 0
\(844\) −5.92036 −0.203787
\(845\) −22.9697 −0.790183
\(846\) 0 0
\(847\) −10.8783 −0.373783
\(848\) 10.0800 0.346150
\(849\) 0 0
\(850\) 14.0698 0.482589
\(851\) −8.18491 −0.280575
\(852\) 0 0
\(853\) 21.4439 0.734225 0.367113 0.930176i \(-0.380346\pi\)
0.367113 + 0.930176i \(0.380346\pi\)
\(854\) 5.32803 0.182321
\(855\) 0 0
\(856\) 33.2736 1.13727
\(857\) −27.2216 −0.929873 −0.464936 0.885344i \(-0.653923\pi\)
−0.464936 + 0.885344i \(0.653923\pi\)
\(858\) 0 0
\(859\) −47.7733 −1.63000 −0.815001 0.579459i \(-0.803264\pi\)
−0.815001 + 0.579459i \(0.803264\pi\)
\(860\) 39.5308 1.34799
\(861\) 0 0
\(862\) 18.2325 0.621001
\(863\) −8.00836 −0.272608 −0.136304 0.990667i \(-0.543522\pi\)
−0.136304 + 0.990667i \(0.543522\pi\)
\(864\) 0 0
\(865\) 61.9901 2.10773
\(866\) 4.48435 0.152384
\(867\) 0 0
\(868\) 15.3642 0.521495
\(869\) −2.06591 −0.0700811
\(870\) 0 0
\(871\) −14.2184 −0.481773
\(872\) −30.3671 −1.02836
\(873\) 0 0
\(874\) 0.881808 0.0298276
\(875\) −6.29597 −0.212843
\(876\) 0 0
\(877\) −29.0720 −0.981691 −0.490845 0.871247i \(-0.663312\pi\)
−0.490845 + 0.871247i \(0.663312\pi\)
\(878\) −8.25510 −0.278596
\(879\) 0 0
\(880\) 1.31344 0.0442759
\(881\) 34.1165 1.14941 0.574707 0.818360i \(-0.305116\pi\)
0.574707 + 0.818360i \(0.305116\pi\)
\(882\) 0 0
\(883\) 50.0021 1.68271 0.841353 0.540486i \(-0.181760\pi\)
0.841353 + 0.540486i \(0.181760\pi\)
\(884\) 24.5906 0.827070
\(885\) 0 0
\(886\) 22.1061 0.742669
\(887\) −7.91800 −0.265861 −0.132930 0.991125i \(-0.542439\pi\)
−0.132930 + 0.991125i \(0.542439\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) −34.0378 −1.14095
\(891\) 0 0
\(892\) −27.2159 −0.911257
\(893\) 2.48953 0.0833091
\(894\) 0 0
\(895\) −73.7169 −2.46409
\(896\) 10.7128 0.357889
\(897\) 0 0
\(898\) 21.6970 0.724039
\(899\) −69.5266 −2.31884
\(900\) 0 0
\(901\) −55.3009 −1.84234
\(902\) 0.862233 0.0287092
\(903\) 0 0
\(904\) −41.7538 −1.38871
\(905\) −18.4137 −0.612093
\(906\) 0 0
\(907\) 11.4971 0.381756 0.190878 0.981614i \(-0.438867\pi\)
0.190878 + 0.981614i \(0.438867\pi\)
\(908\) 13.3272 0.442278
\(909\) 0 0
\(910\) −4.19346 −0.139012
\(911\) −15.6924 −0.519913 −0.259957 0.965620i \(-0.583708\pi\)
−0.259957 + 0.965620i \(0.583708\pi\)
\(912\) 0 0
\(913\) −0.588370 −0.0194722
\(914\) 12.1714 0.402596
\(915\) 0 0
\(916\) 30.7144 1.01483
\(917\) −8.33905 −0.275380
\(918\) 0 0
\(919\) −17.3175 −0.571252 −0.285626 0.958341i \(-0.592202\pi\)
−0.285626 + 0.958341i \(0.592202\pi\)
\(920\) 33.7797 1.11368
\(921\) 0 0
\(922\) −19.0081 −0.625997
\(923\) −29.9524 −0.985895
\(924\) 0 0
\(925\) −4.49539 −0.147807
\(926\) 12.6215 0.414768
\(927\) 0 0
\(928\) −39.9957 −1.31292
\(929\) −24.5131 −0.804249 −0.402125 0.915585i \(-0.631728\pi\)
−0.402125 + 0.915585i \(0.631728\pi\)
\(930\) 0 0
\(931\) −0.255629 −0.00837791
\(932\) −44.8268 −1.46835
\(933\) 0 0
\(934\) −4.65580 −0.152342
\(935\) −7.20575 −0.235653
\(936\) 0 0
\(937\) 39.1077 1.27759 0.638796 0.769376i \(-0.279433\pi\)
0.638796 + 0.769376i \(0.279433\pi\)
\(938\) 4.52075 0.147608
\(939\) 0 0
\(940\) 41.1913 1.34351
\(941\) 24.8154 0.808958 0.404479 0.914547i \(-0.367453\pi\)
0.404479 + 0.914547i \(0.367453\pi\)
\(942\) 0 0
\(943\) −17.7864 −0.579206
\(944\) 2.72154 0.0885787
\(945\) 0 0
\(946\) 2.25739 0.0733942
\(947\) 7.86086 0.255444 0.127722 0.991810i \(-0.459234\pi\)
0.127722 + 0.991810i \(0.459234\pi\)
\(948\) 0 0
\(949\) −20.4065 −0.662423
\(950\) 0.484314 0.0157132
\(951\) 0 0
\(952\) −18.1018 −0.586684
\(953\) −19.4522 −0.630119 −0.315060 0.949072i \(-0.602024\pi\)
−0.315060 + 0.949072i \(0.602024\pi\)
\(954\) 0 0
\(955\) −26.4510 −0.855933
\(956\) −34.6018 −1.11910
\(957\) 0 0
\(958\) 13.3702 0.431973
\(959\) −20.9573 −0.676746
\(960\) 0 0
\(961\) 71.0863 2.29311
\(962\) 2.47676 0.0798538
\(963\) 0 0
\(964\) −19.1873 −0.617980
\(965\) 50.6744 1.63127
\(966\) 0 0
\(967\) −31.4602 −1.01169 −0.505846 0.862624i \(-0.668819\pi\)
−0.505846 + 0.862624i \(0.668819\pi\)
\(968\) −26.5163 −0.852267
\(969\) 0 0
\(970\) 0.417450 0.0134035
\(971\) −9.05176 −0.290485 −0.145242 0.989396i \(-0.546396\pi\)
−0.145242 + 0.989396i \(0.546396\pi\)
\(972\) 0 0
\(973\) −10.1867 −0.326571
\(974\) 26.6468 0.853818
\(975\) 0 0
\(976\) −10.4168 −0.333435
\(977\) −7.31839 −0.234136 −0.117068 0.993124i \(-0.537350\pi\)
−0.117068 + 0.993124i \(0.537350\pi\)
\(978\) 0 0
\(979\) 6.16593 0.197064
\(980\) −4.22958 −0.135109
\(981\) 0 0
\(982\) 13.7422 0.438531
\(983\) −7.36329 −0.234852 −0.117426 0.993082i \(-0.537464\pi\)
−0.117426 + 0.993082i \(0.537464\pi\)
\(984\) 0 0
\(985\) 18.7294 0.596768
\(986\) 35.3808 1.12675
\(987\) 0 0
\(988\) 0.846464 0.0269296
\(989\) −46.5663 −1.48072
\(990\) 0 0
\(991\) 41.9301 1.33195 0.665976 0.745973i \(-0.268015\pi\)
0.665976 + 0.745973i \(0.268015\pi\)
\(992\) 58.7260 1.86455
\(993\) 0 0
\(994\) 9.52337 0.302063
\(995\) 53.3694 1.69192
\(996\) 0 0
\(997\) 42.0879 1.33294 0.666468 0.745533i \(-0.267805\pi\)
0.666468 + 0.745533i \(0.267805\pi\)
\(998\) −8.91583 −0.282226
\(999\) 0 0
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 8001.2.a.l.1.3 7
3.2 odd 2 2667.2.a.j.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.j.1.5 7 3.2 odd 2
8001.2.a.l.1.3 7 1.1 even 1 trivial