Properties

Label 8001.2.a.l.1.2
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 1.2
Root \(2.69855\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.840819 q^{2} -1.29302 q^{4} -2.74724 q^{5} +1.00000 q^{7} +2.76884 q^{8} +O(q^{10})\) \(q-0.840819 q^{2} -1.29302 q^{4} -2.74724 q^{5} +1.00000 q^{7} +2.76884 q^{8} +2.30993 q^{10} +5.62069 q^{11} -3.65275 q^{13} -0.840819 q^{14} +0.257957 q^{16} +5.86462 q^{17} -2.05868 q^{19} +3.55225 q^{20} -4.72598 q^{22} -2.78500 q^{23} +2.54733 q^{25} +3.07130 q^{26} -1.29302 q^{28} +4.83208 q^{29} -6.87419 q^{31} -5.75457 q^{32} -4.93108 q^{34} -2.74724 q^{35} -8.71097 q^{37} +1.73097 q^{38} -7.60666 q^{40} +4.09940 q^{41} -2.48024 q^{43} -7.26768 q^{44} +2.34168 q^{46} -12.6576 q^{47} +1.00000 q^{49} -2.14185 q^{50} +4.72310 q^{52} -4.92291 q^{53} -15.4414 q^{55} +2.76884 q^{56} -4.06291 q^{58} +9.41559 q^{59} +4.96498 q^{61} +5.77995 q^{62} +4.32264 q^{64} +10.0350 q^{65} +4.57422 q^{67} -7.58309 q^{68} +2.30993 q^{70} -2.13466 q^{71} +9.59565 q^{73} +7.32435 q^{74} +2.66192 q^{76} +5.62069 q^{77} -10.0298 q^{79} -0.708669 q^{80} -3.44685 q^{82} +5.06691 q^{83} -16.1115 q^{85} +2.08543 q^{86} +15.5628 q^{88} -2.73622 q^{89} -3.65275 q^{91} +3.60107 q^{92} +10.6428 q^{94} +5.65568 q^{95} +14.0813 q^{97} -0.840819 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + O(q^{10}) \) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + 3q^{11} - 23q^{13} + 2q^{14} + 2q^{16} - 3q^{17} - 9q^{19} + 9q^{20} - 19q^{22} - 12q^{23} + 3q^{25} - 18q^{26} + 4q^{28} + 9q^{29} - 33q^{31} - 10q^{32} - 2q^{34} + 8q^{35} - 33q^{37} + 3q^{38} - 9q^{40} + 3q^{41} - 9q^{43} - 2q^{44} - 32q^{46} - 11q^{47} + 7q^{49} - 29q^{50} - 21q^{52} - q^{53} - 16q^{55} + 9q^{56} - 5q^{58} + 30q^{59} - 19q^{61} - 3q^{62} - 21q^{64} - 14q^{65} - 30q^{67} - 24q^{68} - 8q^{71} - 20q^{73} + 9q^{74} - 42q^{76} + 3q^{77} + 8q^{79} - 12q^{80} + 10q^{82} + 34q^{83} - 28q^{85} - 24q^{86} - q^{88} + 12q^{89} - 23q^{91} - 60q^{92} - 3q^{94} - 12q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.840819 −0.594549 −0.297274 0.954792i \(-0.596078\pi\)
−0.297274 + 0.954792i \(0.596078\pi\)
\(3\) 0 0
\(4\) −1.29302 −0.646512
\(5\) −2.74724 −1.22860 −0.614302 0.789071i \(-0.710562\pi\)
−0.614302 + 0.789071i \(0.710562\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.76884 0.978932
\(9\) 0 0
\(10\) 2.30993 0.730465
\(11\) 5.62069 1.69470 0.847351 0.531034i \(-0.178196\pi\)
0.847351 + 0.531034i \(0.178196\pi\)
\(12\) 0 0
\(13\) −3.65275 −1.01309 −0.506546 0.862213i \(-0.669078\pi\)
−0.506546 + 0.862213i \(0.669078\pi\)
\(14\) −0.840819 −0.224718
\(15\) 0 0
\(16\) 0.257957 0.0644892
\(17\) 5.86462 1.42238 0.711189 0.703000i \(-0.248157\pi\)
0.711189 + 0.703000i \(0.248157\pi\)
\(18\) 0 0
\(19\) −2.05868 −0.472293 −0.236146 0.971718i \(-0.575884\pi\)
−0.236146 + 0.971718i \(0.575884\pi\)
\(20\) 3.55225 0.794307
\(21\) 0 0
\(22\) −4.72598 −1.00758
\(23\) −2.78500 −0.580713 −0.290356 0.956919i \(-0.593774\pi\)
−0.290356 + 0.956919i \(0.593774\pi\)
\(24\) 0 0
\(25\) 2.54733 0.509467
\(26\) 3.07130 0.602332
\(27\) 0 0
\(28\) −1.29302 −0.244358
\(29\) 4.83208 0.897295 0.448648 0.893709i \(-0.351906\pi\)
0.448648 + 0.893709i \(0.351906\pi\)
\(30\) 0 0
\(31\) −6.87419 −1.23464 −0.617321 0.786712i \(-0.711782\pi\)
−0.617321 + 0.786712i \(0.711782\pi\)
\(32\) −5.75457 −1.01727
\(33\) 0 0
\(34\) −4.93108 −0.845674
\(35\) −2.74724 −0.464369
\(36\) 0 0
\(37\) −8.71097 −1.43207 −0.716037 0.698062i \(-0.754046\pi\)
−0.716037 + 0.698062i \(0.754046\pi\)
\(38\) 1.73097 0.280801
\(39\) 0 0
\(40\) −7.60666 −1.20272
\(41\) 4.09940 0.640219 0.320109 0.947381i \(-0.396280\pi\)
0.320109 + 0.947381i \(0.396280\pi\)
\(42\) 0 0
\(43\) −2.48024 −0.378233 −0.189116 0.981955i \(-0.560562\pi\)
−0.189116 + 0.981955i \(0.560562\pi\)
\(44\) −7.26768 −1.09564
\(45\) 0 0
\(46\) 2.34168 0.345262
\(47\) −12.6576 −1.84630 −0.923152 0.384434i \(-0.874397\pi\)
−0.923152 + 0.384434i \(0.874397\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.14185 −0.302903
\(51\) 0 0
\(52\) 4.72310 0.654975
\(53\) −4.92291 −0.676214 −0.338107 0.941108i \(-0.609787\pi\)
−0.338107 + 0.941108i \(0.609787\pi\)
\(54\) 0 0
\(55\) −15.4414 −2.08212
\(56\) 2.76884 0.370001
\(57\) 0 0
\(58\) −4.06291 −0.533486
\(59\) 9.41559 1.22581 0.612903 0.790158i \(-0.290002\pi\)
0.612903 + 0.790158i \(0.290002\pi\)
\(60\) 0 0
\(61\) 4.96498 0.635701 0.317850 0.948141i \(-0.397039\pi\)
0.317850 + 0.948141i \(0.397039\pi\)
\(62\) 5.77995 0.734055
\(63\) 0 0
\(64\) 4.32264 0.540330
\(65\) 10.0350 1.24469
\(66\) 0 0
\(67\) 4.57422 0.558829 0.279415 0.960171i \(-0.409860\pi\)
0.279415 + 0.960171i \(0.409860\pi\)
\(68\) −7.58309 −0.919585
\(69\) 0 0
\(70\) 2.30993 0.276090
\(71\) −2.13466 −0.253338 −0.126669 0.991945i \(-0.540429\pi\)
−0.126669 + 0.991945i \(0.540429\pi\)
\(72\) 0 0
\(73\) 9.59565 1.12309 0.561543 0.827448i \(-0.310208\pi\)
0.561543 + 0.827448i \(0.310208\pi\)
\(74\) 7.32435 0.851438
\(75\) 0 0
\(76\) 2.66192 0.305343
\(77\) 5.62069 0.640537
\(78\) 0 0
\(79\) −10.0298 −1.12844 −0.564220 0.825624i \(-0.690823\pi\)
−0.564220 + 0.825624i \(0.690823\pi\)
\(80\) −0.708669 −0.0792316
\(81\) 0 0
\(82\) −3.44685 −0.380641
\(83\) 5.06691 0.556166 0.278083 0.960557i \(-0.410301\pi\)
0.278083 + 0.960557i \(0.410301\pi\)
\(84\) 0 0
\(85\) −16.1115 −1.74754
\(86\) 2.08543 0.224878
\(87\) 0 0
\(88\) 15.5628 1.65900
\(89\) −2.73622 −0.290039 −0.145019 0.989429i \(-0.546324\pi\)
−0.145019 + 0.989429i \(0.546324\pi\)
\(90\) 0 0
\(91\) −3.65275 −0.382913
\(92\) 3.60107 0.375437
\(93\) 0 0
\(94\) 10.6428 1.09772
\(95\) 5.65568 0.580260
\(96\) 0 0
\(97\) 14.0813 1.42974 0.714872 0.699255i \(-0.246485\pi\)
0.714872 + 0.699255i \(0.246485\pi\)
\(98\) −0.840819 −0.0849355
\(99\) 0 0
\(100\) −3.29376 −0.329376
\(101\) 2.91727 0.290279 0.145139 0.989411i \(-0.453637\pi\)
0.145139 + 0.989411i \(0.453637\pi\)
\(102\) 0 0
\(103\) 17.2096 1.69571 0.847855 0.530228i \(-0.177894\pi\)
0.847855 + 0.530228i \(0.177894\pi\)
\(104\) −10.1139 −0.991747
\(105\) 0 0
\(106\) 4.13928 0.402042
\(107\) −3.25655 −0.314823 −0.157411 0.987533i \(-0.550315\pi\)
−0.157411 + 0.987533i \(0.550315\pi\)
\(108\) 0 0
\(109\) 9.69847 0.928945 0.464472 0.885588i \(-0.346244\pi\)
0.464472 + 0.885588i \(0.346244\pi\)
\(110\) 12.9834 1.23792
\(111\) 0 0
\(112\) 0.257957 0.0243746
\(113\) −1.89566 −0.178329 −0.0891645 0.996017i \(-0.528420\pi\)
−0.0891645 + 0.996017i \(0.528420\pi\)
\(114\) 0 0
\(115\) 7.65107 0.713466
\(116\) −6.24800 −0.580112
\(117\) 0 0
\(118\) −7.91681 −0.728801
\(119\) 5.86462 0.537609
\(120\) 0 0
\(121\) 20.5921 1.87201
\(122\) −4.17465 −0.377955
\(123\) 0 0
\(124\) 8.88849 0.798210
\(125\) 6.73807 0.602671
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 7.87458 0.696021
\(129\) 0 0
\(130\) −8.43761 −0.740028
\(131\) −5.84500 −0.510680 −0.255340 0.966851i \(-0.582187\pi\)
−0.255340 + 0.966851i \(0.582187\pi\)
\(132\) 0 0
\(133\) −2.05868 −0.178510
\(134\) −3.84609 −0.332251
\(135\) 0 0
\(136\) 16.2382 1.39241
\(137\) 10.6774 0.912233 0.456117 0.889920i \(-0.349240\pi\)
0.456117 + 0.889920i \(0.349240\pi\)
\(138\) 0 0
\(139\) −21.7313 −1.84323 −0.921613 0.388111i \(-0.873128\pi\)
−0.921613 + 0.388111i \(0.873128\pi\)
\(140\) 3.55225 0.300220
\(141\) 0 0
\(142\) 1.79486 0.150622
\(143\) −20.5310 −1.71689
\(144\) 0 0
\(145\) −13.2749 −1.10242
\(146\) −8.06821 −0.667729
\(147\) 0 0
\(148\) 11.2635 0.925853
\(149\) 16.2091 1.32790 0.663952 0.747775i \(-0.268878\pi\)
0.663952 + 0.747775i \(0.268878\pi\)
\(150\) 0 0
\(151\) −22.1101 −1.79929 −0.899646 0.436621i \(-0.856175\pi\)
−0.899646 + 0.436621i \(0.856175\pi\)
\(152\) −5.70014 −0.462342
\(153\) 0 0
\(154\) −4.72598 −0.380830
\(155\) 18.8851 1.51688
\(156\) 0 0
\(157\) 23.2986 1.85943 0.929714 0.368282i \(-0.120054\pi\)
0.929714 + 0.368282i \(0.120054\pi\)
\(158\) 8.43325 0.670913
\(159\) 0 0
\(160\) 15.8092 1.24983
\(161\) −2.78500 −0.219489
\(162\) 0 0
\(163\) −21.5951 −1.69146 −0.845731 0.533609i \(-0.820835\pi\)
−0.845731 + 0.533609i \(0.820835\pi\)
\(164\) −5.30062 −0.413909
\(165\) 0 0
\(166\) −4.26036 −0.330668
\(167\) 3.26635 0.252758 0.126379 0.991982i \(-0.459664\pi\)
0.126379 + 0.991982i \(0.459664\pi\)
\(168\) 0 0
\(169\) 0.342604 0.0263542
\(170\) 13.5469 1.03900
\(171\) 0 0
\(172\) 3.20701 0.244532
\(173\) −9.97561 −0.758431 −0.379216 0.925308i \(-0.623806\pi\)
−0.379216 + 0.925308i \(0.623806\pi\)
\(174\) 0 0
\(175\) 2.54733 0.192560
\(176\) 1.44989 0.109290
\(177\) 0 0
\(178\) 2.30066 0.172442
\(179\) −21.3721 −1.59743 −0.798714 0.601711i \(-0.794486\pi\)
−0.798714 + 0.601711i \(0.794486\pi\)
\(180\) 0 0
\(181\) −23.7938 −1.76858 −0.884288 0.466942i \(-0.845356\pi\)
−0.884288 + 0.466942i \(0.845356\pi\)
\(182\) 3.07130 0.227660
\(183\) 0 0
\(184\) −7.71121 −0.568478
\(185\) 23.9311 1.75945
\(186\) 0 0
\(187\) 32.9632 2.41051
\(188\) 16.3666 1.19366
\(189\) 0 0
\(190\) −4.75540 −0.344993
\(191\) −17.1115 −1.23814 −0.619072 0.785335i \(-0.712491\pi\)
−0.619072 + 0.785335i \(0.712491\pi\)
\(192\) 0 0
\(193\) −21.1983 −1.52589 −0.762945 0.646464i \(-0.776247\pi\)
−0.762945 + 0.646464i \(0.776247\pi\)
\(194\) −11.8399 −0.850053
\(195\) 0 0
\(196\) −1.29302 −0.0923588
\(197\) −9.64407 −0.687111 −0.343556 0.939132i \(-0.611631\pi\)
−0.343556 + 0.939132i \(0.611631\pi\)
\(198\) 0 0
\(199\) −8.11930 −0.575562 −0.287781 0.957696i \(-0.592918\pi\)
−0.287781 + 0.957696i \(0.592918\pi\)
\(200\) 7.05315 0.498733
\(201\) 0 0
\(202\) −2.45289 −0.172585
\(203\) 4.83208 0.339146
\(204\) 0 0
\(205\) −11.2620 −0.786575
\(206\) −14.4701 −1.00818
\(207\) 0 0
\(208\) −0.942252 −0.0653334
\(209\) −11.5712 −0.800395
\(210\) 0 0
\(211\) −10.1539 −0.699023 −0.349511 0.936932i \(-0.613652\pi\)
−0.349511 + 0.936932i \(0.613652\pi\)
\(212\) 6.36544 0.437180
\(213\) 0 0
\(214\) 2.73817 0.187178
\(215\) 6.81382 0.464698
\(216\) 0 0
\(217\) −6.87419 −0.466651
\(218\) −8.15466 −0.552303
\(219\) 0 0
\(220\) 19.9661 1.34611
\(221\) −21.4220 −1.44100
\(222\) 0 0
\(223\) 22.3276 1.49517 0.747583 0.664168i \(-0.231214\pi\)
0.747583 + 0.664168i \(0.231214\pi\)
\(224\) −5.75457 −0.384493
\(225\) 0 0
\(226\) 1.59391 0.106025
\(227\) 23.3340 1.54873 0.774365 0.632739i \(-0.218069\pi\)
0.774365 + 0.632739i \(0.218069\pi\)
\(228\) 0 0
\(229\) −8.57841 −0.566877 −0.283439 0.958990i \(-0.591475\pi\)
−0.283439 + 0.958990i \(0.591475\pi\)
\(230\) −6.43316 −0.424190
\(231\) 0 0
\(232\) 13.3793 0.878391
\(233\) −14.6183 −0.957677 −0.478839 0.877903i \(-0.658942\pi\)
−0.478839 + 0.877903i \(0.658942\pi\)
\(234\) 0 0
\(235\) 34.7736 2.26838
\(236\) −12.1746 −0.792498
\(237\) 0 0
\(238\) −4.93108 −0.319635
\(239\) −5.05219 −0.326799 −0.163399 0.986560i \(-0.552246\pi\)
−0.163399 + 0.986560i \(0.552246\pi\)
\(240\) 0 0
\(241\) −15.9324 −1.02629 −0.513147 0.858301i \(-0.671520\pi\)
−0.513147 + 0.858301i \(0.671520\pi\)
\(242\) −17.3143 −1.11300
\(243\) 0 0
\(244\) −6.41984 −0.410988
\(245\) −2.74724 −0.175515
\(246\) 0 0
\(247\) 7.51983 0.478475
\(248\) −19.0335 −1.20863
\(249\) 0 0
\(250\) −5.66549 −0.358317
\(251\) 11.8771 0.749678 0.374839 0.927090i \(-0.377698\pi\)
0.374839 + 0.927090i \(0.377698\pi\)
\(252\) 0 0
\(253\) −15.6536 −0.984134
\(254\) 0.840819 0.0527577
\(255\) 0 0
\(256\) −15.2664 −0.954148
\(257\) 19.3728 1.20844 0.604221 0.796817i \(-0.293484\pi\)
0.604221 + 0.796817i \(0.293484\pi\)
\(258\) 0 0
\(259\) −8.71097 −0.541273
\(260\) −12.9755 −0.804705
\(261\) 0 0
\(262\) 4.91459 0.303624
\(263\) 25.0402 1.54405 0.772024 0.635594i \(-0.219245\pi\)
0.772024 + 0.635594i \(0.219245\pi\)
\(264\) 0 0
\(265\) 13.5244 0.830799
\(266\) 1.73097 0.106133
\(267\) 0 0
\(268\) −5.91457 −0.361290
\(269\) 20.8253 1.26974 0.634870 0.772619i \(-0.281054\pi\)
0.634870 + 0.772619i \(0.281054\pi\)
\(270\) 0 0
\(271\) 14.7479 0.895871 0.447936 0.894066i \(-0.352159\pi\)
0.447936 + 0.894066i \(0.352159\pi\)
\(272\) 1.51282 0.0917280
\(273\) 0 0
\(274\) −8.97778 −0.542367
\(275\) 14.3178 0.863394
\(276\) 0 0
\(277\) −12.6554 −0.760389 −0.380195 0.924907i \(-0.624143\pi\)
−0.380195 + 0.924907i \(0.624143\pi\)
\(278\) 18.2721 1.09589
\(279\) 0 0
\(280\) −7.60666 −0.454585
\(281\) 30.2858 1.80670 0.903350 0.428903i \(-0.141100\pi\)
0.903350 + 0.428903i \(0.141100\pi\)
\(282\) 0 0
\(283\) −0.554307 −0.0329501 −0.0164751 0.999864i \(-0.505244\pi\)
−0.0164751 + 0.999864i \(0.505244\pi\)
\(284\) 2.76017 0.163786
\(285\) 0 0
\(286\) 17.2628 1.02077
\(287\) 4.09940 0.241980
\(288\) 0 0
\(289\) 17.3937 1.02316
\(290\) 11.1618 0.655443
\(291\) 0 0
\(292\) −12.4074 −0.726088
\(293\) −10.7329 −0.627023 −0.313512 0.949584i \(-0.601505\pi\)
−0.313512 + 0.949584i \(0.601505\pi\)
\(294\) 0 0
\(295\) −25.8669 −1.50603
\(296\) −24.1192 −1.40190
\(297\) 0 0
\(298\) −13.6289 −0.789504
\(299\) 10.1729 0.588315
\(300\) 0 0
\(301\) −2.48024 −0.142959
\(302\) 18.5906 1.06977
\(303\) 0 0
\(304\) −0.531049 −0.0304578
\(305\) −13.6400 −0.781024
\(306\) 0 0
\(307\) 5.96740 0.340577 0.170289 0.985394i \(-0.445530\pi\)
0.170289 + 0.985394i \(0.445530\pi\)
\(308\) −7.26768 −0.414115
\(309\) 0 0
\(310\) −15.8789 −0.901862
\(311\) −26.7130 −1.51475 −0.757377 0.652978i \(-0.773519\pi\)
−0.757377 + 0.652978i \(0.773519\pi\)
\(312\) 0 0
\(313\) −25.9107 −1.46456 −0.732279 0.681005i \(-0.761543\pi\)
−0.732279 + 0.681005i \(0.761543\pi\)
\(314\) −19.5899 −1.10552
\(315\) 0 0
\(316\) 12.9688 0.729550
\(317\) 6.29184 0.353385 0.176693 0.984266i \(-0.443460\pi\)
0.176693 + 0.984266i \(0.443460\pi\)
\(318\) 0 0
\(319\) 27.1596 1.52065
\(320\) −11.8753 −0.663851
\(321\) 0 0
\(322\) 2.34168 0.130497
\(323\) −12.0733 −0.671779
\(324\) 0 0
\(325\) −9.30478 −0.516136
\(326\) 18.1576 1.00566
\(327\) 0 0
\(328\) 11.3506 0.626731
\(329\) −12.6576 −0.697838
\(330\) 0 0
\(331\) 6.26075 0.344122 0.172061 0.985086i \(-0.444957\pi\)
0.172061 + 0.985086i \(0.444957\pi\)
\(332\) −6.55164 −0.359568
\(333\) 0 0
\(334\) −2.74641 −0.150277
\(335\) −12.5665 −0.686580
\(336\) 0 0
\(337\) 10.6291 0.579003 0.289502 0.957178i \(-0.406510\pi\)
0.289502 + 0.957178i \(0.406510\pi\)
\(338\) −0.288068 −0.0156688
\(339\) 0 0
\(340\) 20.8326 1.12980
\(341\) −38.6377 −2.09235
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −6.86738 −0.370264
\(345\) 0 0
\(346\) 8.38768 0.450925
\(347\) −24.2445 −1.30151 −0.650757 0.759286i \(-0.725548\pi\)
−0.650757 + 0.759286i \(0.725548\pi\)
\(348\) 0 0
\(349\) 19.2656 1.03127 0.515633 0.856810i \(-0.327557\pi\)
0.515633 + 0.856810i \(0.327557\pi\)
\(350\) −2.14185 −0.114487
\(351\) 0 0
\(352\) −32.3446 −1.72397
\(353\) −2.47593 −0.131780 −0.0658902 0.997827i \(-0.520989\pi\)
−0.0658902 + 0.997827i \(0.520989\pi\)
\(354\) 0 0
\(355\) 5.86443 0.311251
\(356\) 3.53800 0.187513
\(357\) 0 0
\(358\) 17.9701 0.949749
\(359\) −31.8474 −1.68084 −0.840422 0.541933i \(-0.817693\pi\)
−0.840422 + 0.541933i \(0.817693\pi\)
\(360\) 0 0
\(361\) −14.7619 −0.776940
\(362\) 20.0062 1.05150
\(363\) 0 0
\(364\) 4.72310 0.247557
\(365\) −26.3616 −1.37983
\(366\) 0 0
\(367\) 9.45164 0.493372 0.246686 0.969095i \(-0.420658\pi\)
0.246686 + 0.969095i \(0.420658\pi\)
\(368\) −0.718409 −0.0374497
\(369\) 0 0
\(370\) −20.1217 −1.04608
\(371\) −4.92291 −0.255585
\(372\) 0 0
\(373\) −8.74070 −0.452576 −0.226288 0.974060i \(-0.572659\pi\)
−0.226288 + 0.974060i \(0.572659\pi\)
\(374\) −27.7161 −1.43316
\(375\) 0 0
\(376\) −35.0469 −1.80741
\(377\) −17.6504 −0.909042
\(378\) 0 0
\(379\) −15.3774 −0.789882 −0.394941 0.918707i \(-0.629235\pi\)
−0.394941 + 0.918707i \(0.629235\pi\)
\(380\) −7.31292 −0.375145
\(381\) 0 0
\(382\) 14.3877 0.736137
\(383\) −19.4375 −0.993210 −0.496605 0.867977i \(-0.665420\pi\)
−0.496605 + 0.867977i \(0.665420\pi\)
\(384\) 0 0
\(385\) −15.4414 −0.786966
\(386\) 17.8240 0.907216
\(387\) 0 0
\(388\) −18.2075 −0.924347
\(389\) −20.2992 −1.02921 −0.514604 0.857428i \(-0.672061\pi\)
−0.514604 + 0.857428i \(0.672061\pi\)
\(390\) 0 0
\(391\) −16.3330 −0.825993
\(392\) 2.76884 0.139847
\(393\) 0 0
\(394\) 8.10891 0.408521
\(395\) 27.5543 1.38641
\(396\) 0 0
\(397\) −5.04735 −0.253319 −0.126660 0.991946i \(-0.540426\pi\)
−0.126660 + 0.991946i \(0.540426\pi\)
\(398\) 6.82686 0.342200
\(399\) 0 0
\(400\) 0.657102 0.0328551
\(401\) −0.828202 −0.0413584 −0.0206792 0.999786i \(-0.506583\pi\)
−0.0206792 + 0.999786i \(0.506583\pi\)
\(402\) 0 0
\(403\) 25.1097 1.25080
\(404\) −3.77209 −0.187669
\(405\) 0 0
\(406\) −4.06291 −0.201639
\(407\) −48.9616 −2.42694
\(408\) 0 0
\(409\) 7.63539 0.377546 0.188773 0.982021i \(-0.439549\pi\)
0.188773 + 0.982021i \(0.439549\pi\)
\(410\) 9.46934 0.467657
\(411\) 0 0
\(412\) −22.2524 −1.09630
\(413\) 9.41559 0.463311
\(414\) 0 0
\(415\) −13.9200 −0.683308
\(416\) 21.0200 1.03059
\(417\) 0 0
\(418\) 9.72926 0.475874
\(419\) 0.606948 0.0296514 0.0148257 0.999890i \(-0.495281\pi\)
0.0148257 + 0.999890i \(0.495281\pi\)
\(420\) 0 0
\(421\) 12.2074 0.594951 0.297476 0.954729i \(-0.403855\pi\)
0.297476 + 0.954729i \(0.403855\pi\)
\(422\) 8.53759 0.415603
\(423\) 0 0
\(424\) −13.6307 −0.661967
\(425\) 14.9391 0.724655
\(426\) 0 0
\(427\) 4.96498 0.240272
\(428\) 4.21080 0.203537
\(429\) 0 0
\(430\) −5.72919 −0.276286
\(431\) −27.6025 −1.32957 −0.664783 0.747037i \(-0.731476\pi\)
−0.664783 + 0.747037i \(0.731476\pi\)
\(432\) 0 0
\(433\) −21.0747 −1.01278 −0.506392 0.862304i \(-0.669021\pi\)
−0.506392 + 0.862304i \(0.669021\pi\)
\(434\) 5.77995 0.277447
\(435\) 0 0
\(436\) −12.5403 −0.600574
\(437\) 5.73341 0.274266
\(438\) 0 0
\(439\) −11.8630 −0.566189 −0.283094 0.959092i \(-0.591361\pi\)
−0.283094 + 0.959092i \(0.591361\pi\)
\(440\) −42.7547 −2.03825
\(441\) 0 0
\(442\) 18.0120 0.856745
\(443\) −11.0804 −0.526446 −0.263223 0.964735i \(-0.584786\pi\)
−0.263223 + 0.964735i \(0.584786\pi\)
\(444\) 0 0
\(445\) 7.51705 0.356342
\(446\) −18.7735 −0.888949
\(447\) 0 0
\(448\) 4.32264 0.204225
\(449\) −21.1462 −0.997949 −0.498974 0.866617i \(-0.666290\pi\)
−0.498974 + 0.866617i \(0.666290\pi\)
\(450\) 0 0
\(451\) 23.0415 1.08498
\(452\) 2.45114 0.115292
\(453\) 0 0
\(454\) −19.6196 −0.920796
\(455\) 10.0350 0.470448
\(456\) 0 0
\(457\) −0.600191 −0.0280758 −0.0140379 0.999901i \(-0.504469\pi\)
−0.0140379 + 0.999901i \(0.504469\pi\)
\(458\) 7.21289 0.337036
\(459\) 0 0
\(460\) −9.89301 −0.461264
\(461\) 31.0470 1.44600 0.723001 0.690847i \(-0.242762\pi\)
0.723001 + 0.690847i \(0.242762\pi\)
\(462\) 0 0
\(463\) 9.93441 0.461691 0.230846 0.972990i \(-0.425851\pi\)
0.230846 + 0.972990i \(0.425851\pi\)
\(464\) 1.24647 0.0578658
\(465\) 0 0
\(466\) 12.2913 0.569386
\(467\) 6.15650 0.284889 0.142444 0.989803i \(-0.454504\pi\)
0.142444 + 0.989803i \(0.454504\pi\)
\(468\) 0 0
\(469\) 4.57422 0.211218
\(470\) −29.2383 −1.34866
\(471\) 0 0
\(472\) 26.0702 1.19998
\(473\) −13.9407 −0.640992
\(474\) 0 0
\(475\) −5.24413 −0.240617
\(476\) −7.58309 −0.347570
\(477\) 0 0
\(478\) 4.24798 0.194298
\(479\) −15.7671 −0.720417 −0.360209 0.932872i \(-0.617294\pi\)
−0.360209 + 0.932872i \(0.617294\pi\)
\(480\) 0 0
\(481\) 31.8190 1.45082
\(482\) 13.3962 0.610182
\(483\) 0 0
\(484\) −26.6261 −1.21028
\(485\) −38.6849 −1.75659
\(486\) 0 0
\(487\) −6.30114 −0.285532 −0.142766 0.989756i \(-0.545600\pi\)
−0.142766 + 0.989756i \(0.545600\pi\)
\(488\) 13.7472 0.622308
\(489\) 0 0
\(490\) 2.30993 0.104352
\(491\) 2.75670 0.124408 0.0622040 0.998063i \(-0.480187\pi\)
0.0622040 + 0.998063i \(0.480187\pi\)
\(492\) 0 0
\(493\) 28.3383 1.27629
\(494\) −6.32282 −0.284477
\(495\) 0 0
\(496\) −1.77324 −0.0796210
\(497\) −2.13466 −0.0957526
\(498\) 0 0
\(499\) 1.20293 0.0538506 0.0269253 0.999637i \(-0.491428\pi\)
0.0269253 + 0.999637i \(0.491428\pi\)
\(500\) −8.71248 −0.389634
\(501\) 0 0
\(502\) −9.98652 −0.445720
\(503\) −29.0110 −1.29354 −0.646769 0.762686i \(-0.723880\pi\)
−0.646769 + 0.762686i \(0.723880\pi\)
\(504\) 0 0
\(505\) −8.01443 −0.356637
\(506\) 13.1619 0.585116
\(507\) 0 0
\(508\) 1.29302 0.0573686
\(509\) 9.23054 0.409136 0.204568 0.978852i \(-0.434421\pi\)
0.204568 + 0.978852i \(0.434421\pi\)
\(510\) 0 0
\(511\) 9.59565 0.424487
\(512\) −2.91291 −0.128734
\(513\) 0 0
\(514\) −16.2890 −0.718478
\(515\) −47.2789 −2.08336
\(516\) 0 0
\(517\) −71.1446 −3.12894
\(518\) 7.32435 0.321813
\(519\) 0 0
\(520\) 27.7853 1.21846
\(521\) 29.3043 1.28385 0.641923 0.766769i \(-0.278137\pi\)
0.641923 + 0.766769i \(0.278137\pi\)
\(522\) 0 0
\(523\) 5.55943 0.243097 0.121549 0.992585i \(-0.461214\pi\)
0.121549 + 0.992585i \(0.461214\pi\)
\(524\) 7.55772 0.330161
\(525\) 0 0
\(526\) −21.0543 −0.918012
\(527\) −40.3145 −1.75613
\(528\) 0 0
\(529\) −15.2438 −0.662773
\(530\) −11.3716 −0.493951
\(531\) 0 0
\(532\) 2.66192 0.115409
\(533\) −14.9741 −0.648600
\(534\) 0 0
\(535\) 8.94654 0.386793
\(536\) 12.6653 0.547056
\(537\) 0 0
\(538\) −17.5103 −0.754922
\(539\) 5.62069 0.242100
\(540\) 0 0
\(541\) −12.1035 −0.520371 −0.260186 0.965559i \(-0.583784\pi\)
−0.260186 + 0.965559i \(0.583784\pi\)
\(542\) −12.4003 −0.532639
\(543\) 0 0
\(544\) −33.7483 −1.44695
\(545\) −26.6440 −1.14130
\(546\) 0 0
\(547\) 27.7176 1.18512 0.592561 0.805526i \(-0.298117\pi\)
0.592561 + 0.805526i \(0.298117\pi\)
\(548\) −13.8062 −0.589770
\(549\) 0 0
\(550\) −12.0387 −0.513330
\(551\) −9.94769 −0.423786
\(552\) 0 0
\(553\) −10.0298 −0.426510
\(554\) 10.6409 0.452088
\(555\) 0 0
\(556\) 28.0991 1.19167
\(557\) −2.98048 −0.126287 −0.0631435 0.998004i \(-0.520113\pi\)
−0.0631435 + 0.998004i \(0.520113\pi\)
\(558\) 0 0
\(559\) 9.05970 0.383185
\(560\) −0.708669 −0.0299467
\(561\) 0 0
\(562\) −25.4649 −1.07417
\(563\) 36.8092 1.55132 0.775662 0.631149i \(-0.217416\pi\)
0.775662 + 0.631149i \(0.217416\pi\)
\(564\) 0 0
\(565\) 5.20784 0.219096
\(566\) 0.466072 0.0195904
\(567\) 0 0
\(568\) −5.91052 −0.248000
\(569\) 30.9224 1.29633 0.648167 0.761499i \(-0.275536\pi\)
0.648167 + 0.761499i \(0.275536\pi\)
\(570\) 0 0
\(571\) 13.7621 0.575925 0.287963 0.957642i \(-0.407022\pi\)
0.287963 + 0.957642i \(0.407022\pi\)
\(572\) 26.5470 1.10999
\(573\) 0 0
\(574\) −3.44685 −0.143869
\(575\) −7.09432 −0.295854
\(576\) 0 0
\(577\) 35.2200 1.46623 0.733114 0.680106i \(-0.238066\pi\)
0.733114 + 0.680106i \(0.238066\pi\)
\(578\) −14.6250 −0.608320
\(579\) 0 0
\(580\) 17.1648 0.712728
\(581\) 5.06691 0.210211
\(582\) 0 0
\(583\) −27.6702 −1.14598
\(584\) 26.5688 1.09942
\(585\) 0 0
\(586\) 9.02443 0.372796
\(587\) −17.1613 −0.708322 −0.354161 0.935184i \(-0.615234\pi\)
−0.354161 + 0.935184i \(0.615234\pi\)
\(588\) 0 0
\(589\) 14.1517 0.583112
\(590\) 21.7494 0.895408
\(591\) 0 0
\(592\) −2.24705 −0.0923533
\(593\) 41.1163 1.68845 0.844223 0.535992i \(-0.180062\pi\)
0.844223 + 0.535992i \(0.180062\pi\)
\(594\) 0 0
\(595\) −16.1115 −0.660508
\(596\) −20.9588 −0.858506
\(597\) 0 0
\(598\) −8.55358 −0.349782
\(599\) −15.4682 −0.632013 −0.316007 0.948757i \(-0.602342\pi\)
−0.316007 + 0.948757i \(0.602342\pi\)
\(600\) 0 0
\(601\) −14.5724 −0.594421 −0.297210 0.954812i \(-0.596056\pi\)
−0.297210 + 0.954812i \(0.596056\pi\)
\(602\) 2.08543 0.0849959
\(603\) 0 0
\(604\) 28.5888 1.16326
\(605\) −56.5716 −2.29996
\(606\) 0 0
\(607\) 41.7362 1.69402 0.847009 0.531578i \(-0.178401\pi\)
0.847009 + 0.531578i \(0.178401\pi\)
\(608\) 11.8468 0.480451
\(609\) 0 0
\(610\) 11.4688 0.464357
\(611\) 46.2352 1.87048
\(612\) 0 0
\(613\) −43.5236 −1.75790 −0.878951 0.476913i \(-0.841756\pi\)
−0.878951 + 0.476913i \(0.841756\pi\)
\(614\) −5.01750 −0.202490
\(615\) 0 0
\(616\) 15.5628 0.627042
\(617\) −18.8239 −0.757823 −0.378912 0.925433i \(-0.623702\pi\)
−0.378912 + 0.925433i \(0.623702\pi\)
\(618\) 0 0
\(619\) −29.5177 −1.18642 −0.593209 0.805049i \(-0.702139\pi\)
−0.593209 + 0.805049i \(0.702139\pi\)
\(620\) −24.4188 −0.980684
\(621\) 0 0
\(622\) 22.4608 0.900595
\(623\) −2.73622 −0.109624
\(624\) 0 0
\(625\) −31.2478 −1.24991
\(626\) 21.7862 0.870751
\(627\) 0 0
\(628\) −30.1256 −1.20214
\(629\) −51.0865 −2.03695
\(630\) 0 0
\(631\) −24.2239 −0.964337 −0.482169 0.876078i \(-0.660151\pi\)
−0.482169 + 0.876078i \(0.660151\pi\)
\(632\) −27.7709 −1.10467
\(633\) 0 0
\(634\) −5.29030 −0.210105
\(635\) 2.74724 0.109021
\(636\) 0 0
\(637\) −3.65275 −0.144727
\(638\) −22.8363 −0.904099
\(639\) 0 0
\(640\) −21.6334 −0.855134
\(641\) −28.0303 −1.10713 −0.553564 0.832806i \(-0.686733\pi\)
−0.553564 + 0.832806i \(0.686733\pi\)
\(642\) 0 0
\(643\) −42.3142 −1.66871 −0.834354 0.551229i \(-0.814159\pi\)
−0.834354 + 0.551229i \(0.814159\pi\)
\(644\) 3.60107 0.141902
\(645\) 0 0
\(646\) 10.1515 0.399405
\(647\) 28.5548 1.12260 0.561302 0.827611i \(-0.310301\pi\)
0.561302 + 0.827611i \(0.310301\pi\)
\(648\) 0 0
\(649\) 52.9221 2.07737
\(650\) 7.82364 0.306868
\(651\) 0 0
\(652\) 27.9230 1.09355
\(653\) 0.00473242 0.000185194 0 9.25970e−5 1.00000i \(-0.499971\pi\)
9.25970e−5 1.00000i \(0.499971\pi\)
\(654\) 0 0
\(655\) 16.0576 0.627423
\(656\) 1.05747 0.0412872
\(657\) 0 0
\(658\) 10.6428 0.414899
\(659\) −1.08891 −0.0424181 −0.0212090 0.999775i \(-0.506752\pi\)
−0.0212090 + 0.999775i \(0.506752\pi\)
\(660\) 0 0
\(661\) −34.1649 −1.32886 −0.664431 0.747350i \(-0.731326\pi\)
−0.664431 + 0.747350i \(0.731326\pi\)
\(662\) −5.26415 −0.204597
\(663\) 0 0
\(664\) 14.0295 0.544449
\(665\) 5.65568 0.219318
\(666\) 0 0
\(667\) −13.4574 −0.521071
\(668\) −4.22347 −0.163411
\(669\) 0 0
\(670\) 10.5661 0.408205
\(671\) 27.9066 1.07732
\(672\) 0 0
\(673\) −5.87608 −0.226506 −0.113253 0.993566i \(-0.536127\pi\)
−0.113253 + 0.993566i \(0.536127\pi\)
\(674\) −8.93714 −0.344246
\(675\) 0 0
\(676\) −0.442995 −0.0170383
\(677\) 3.17149 0.121890 0.0609452 0.998141i \(-0.480589\pi\)
0.0609452 + 0.998141i \(0.480589\pi\)
\(678\) 0 0
\(679\) 14.0813 0.540393
\(680\) −44.6102 −1.71072
\(681\) 0 0
\(682\) 32.4873 1.24400
\(683\) −16.1790 −0.619074 −0.309537 0.950887i \(-0.600174\pi\)
−0.309537 + 0.950887i \(0.600174\pi\)
\(684\) 0 0
\(685\) −29.3334 −1.12077
\(686\) −0.840819 −0.0321026
\(687\) 0 0
\(688\) −0.639794 −0.0243919
\(689\) 17.9822 0.685067
\(690\) 0 0
\(691\) −20.6251 −0.784617 −0.392309 0.919834i \(-0.628323\pi\)
−0.392309 + 0.919834i \(0.628323\pi\)
\(692\) 12.8987 0.490335
\(693\) 0 0
\(694\) 20.3852 0.773813
\(695\) 59.7011 2.26459
\(696\) 0 0
\(697\) 24.0414 0.910634
\(698\) −16.1989 −0.613138
\(699\) 0 0
\(700\) −3.29376 −0.124493
\(701\) 4.12137 0.155662 0.0778311 0.996967i \(-0.475201\pi\)
0.0778311 + 0.996967i \(0.475201\pi\)
\(702\) 0 0
\(703\) 17.9331 0.676358
\(704\) 24.2962 0.915697
\(705\) 0 0
\(706\) 2.08181 0.0783498
\(707\) 2.91727 0.109715
\(708\) 0 0
\(709\) −0.871563 −0.0327322 −0.0163661 0.999866i \(-0.505210\pi\)
−0.0163661 + 0.999866i \(0.505210\pi\)
\(710\) −4.93092 −0.185054
\(711\) 0 0
\(712\) −7.57614 −0.283928
\(713\) 19.1446 0.716972
\(714\) 0 0
\(715\) 56.4036 2.10937
\(716\) 27.6347 1.03276
\(717\) 0 0
\(718\) 26.7779 0.999343
\(719\) 40.4977 1.51031 0.755154 0.655547i \(-0.227562\pi\)
0.755154 + 0.655547i \(0.227562\pi\)
\(720\) 0 0
\(721\) 17.2096 0.640918
\(722\) 12.4120 0.461929
\(723\) 0 0
\(724\) 30.7659 1.14340
\(725\) 12.3089 0.457142
\(726\) 0 0
\(727\) 20.3846 0.756022 0.378011 0.925801i \(-0.376608\pi\)
0.378011 + 0.925801i \(0.376608\pi\)
\(728\) −10.1139 −0.374845
\(729\) 0 0
\(730\) 22.1653 0.820375
\(731\) −14.5457 −0.537991
\(732\) 0 0
\(733\) −45.0929 −1.66554 −0.832772 0.553617i \(-0.813247\pi\)
−0.832772 + 0.553617i \(0.813247\pi\)
\(734\) −7.94712 −0.293334
\(735\) 0 0
\(736\) 16.0265 0.590743
\(737\) 25.7102 0.947049
\(738\) 0 0
\(739\) −18.1927 −0.669230 −0.334615 0.942355i \(-0.608606\pi\)
−0.334615 + 0.942355i \(0.608606\pi\)
\(740\) −30.9435 −1.13751
\(741\) 0 0
\(742\) 4.13928 0.151958
\(743\) −33.9156 −1.24424 −0.622122 0.782921i \(-0.713729\pi\)
−0.622122 + 0.782921i \(0.713729\pi\)
\(744\) 0 0
\(745\) −44.5304 −1.63147
\(746\) 7.34935 0.269079
\(747\) 0 0
\(748\) −42.6222 −1.55842
\(749\) −3.25655 −0.118992
\(750\) 0 0
\(751\) 26.1611 0.954631 0.477316 0.878732i \(-0.341610\pi\)
0.477316 + 0.878732i \(0.341610\pi\)
\(752\) −3.26512 −0.119067
\(753\) 0 0
\(754\) 14.8408 0.540470
\(755\) 60.7417 2.21062
\(756\) 0 0
\(757\) −8.54392 −0.310534 −0.155267 0.987873i \(-0.549624\pi\)
−0.155267 + 0.987873i \(0.549624\pi\)
\(758\) 12.9296 0.469623
\(759\) 0 0
\(760\) 15.6596 0.568035
\(761\) −6.45289 −0.233917 −0.116959 0.993137i \(-0.537314\pi\)
−0.116959 + 0.993137i \(0.537314\pi\)
\(762\) 0 0
\(763\) 9.69847 0.351108
\(764\) 22.1255 0.800474
\(765\) 0 0
\(766\) 16.3434 0.590512
\(767\) −34.3928 −1.24185
\(768\) 0 0
\(769\) −12.6368 −0.455695 −0.227847 0.973697i \(-0.573169\pi\)
−0.227847 + 0.973697i \(0.573169\pi\)
\(770\) 12.9834 0.467890
\(771\) 0 0
\(772\) 27.4099 0.986505
\(773\) −1.63697 −0.0588776 −0.0294388 0.999567i \(-0.509372\pi\)
−0.0294388 + 0.999567i \(0.509372\pi\)
\(774\) 0 0
\(775\) −17.5109 −0.629009
\(776\) 38.9890 1.39962
\(777\) 0 0
\(778\) 17.0679 0.611914
\(779\) −8.43934 −0.302371
\(780\) 0 0
\(781\) −11.9983 −0.429331
\(782\) 13.7331 0.491093
\(783\) 0 0
\(784\) 0.257957 0.00921274
\(785\) −64.0068 −2.28450
\(786\) 0 0
\(787\) 32.4089 1.15525 0.577626 0.816301i \(-0.303979\pi\)
0.577626 + 0.816301i \(0.303979\pi\)
\(788\) 12.4700 0.444225
\(789\) 0 0
\(790\) −23.1682 −0.824286
\(791\) −1.89566 −0.0674020
\(792\) 0 0
\(793\) −18.1359 −0.644023
\(794\) 4.24391 0.150611
\(795\) 0 0
\(796\) 10.4985 0.372108
\(797\) −28.3883 −1.00557 −0.502783 0.864413i \(-0.667690\pi\)
−0.502783 + 0.864413i \(0.667690\pi\)
\(798\) 0 0
\(799\) −74.2322 −2.62614
\(800\) −14.6588 −0.518267
\(801\) 0 0
\(802\) 0.696368 0.0245896
\(803\) 53.9342 1.90330
\(804\) 0 0
\(805\) 7.65107 0.269665
\(806\) −21.1127 −0.743664
\(807\) 0 0
\(808\) 8.07743 0.284163
\(809\) −0.0199237 −0.000700480 0 −0.000350240 1.00000i \(-0.500111\pi\)
−0.000350240 1.00000i \(0.500111\pi\)
\(810\) 0 0
\(811\) 8.52421 0.299326 0.149663 0.988737i \(-0.452181\pi\)
0.149663 + 0.988737i \(0.452181\pi\)
\(812\) −6.24800 −0.219262
\(813\) 0 0
\(814\) 41.1679 1.44293
\(815\) 59.3271 2.07814
\(816\) 0 0
\(817\) 5.10601 0.178637
\(818\) −6.41998 −0.224469
\(819\) 0 0
\(820\) 14.5621 0.508530
\(821\) 21.9384 0.765655 0.382828 0.923820i \(-0.374950\pi\)
0.382828 + 0.923820i \(0.374950\pi\)
\(822\) 0 0
\(823\) 16.5378 0.576471 0.288236 0.957559i \(-0.406931\pi\)
0.288236 + 0.957559i \(0.406931\pi\)
\(824\) 47.6505 1.65998
\(825\) 0 0
\(826\) −7.91681 −0.275461
\(827\) 0.791654 0.0275285 0.0137643 0.999905i \(-0.495619\pi\)
0.0137643 + 0.999905i \(0.495619\pi\)
\(828\) 0 0
\(829\) −41.5550 −1.44326 −0.721632 0.692277i \(-0.756608\pi\)
−0.721632 + 0.692277i \(0.756608\pi\)
\(830\) 11.7042 0.406260
\(831\) 0 0
\(832\) −15.7895 −0.547403
\(833\) 5.86462 0.203197
\(834\) 0 0
\(835\) −8.97345 −0.310539
\(836\) 14.9618 0.517465
\(837\) 0 0
\(838\) −0.510334 −0.0176292
\(839\) −44.5008 −1.53634 −0.768169 0.640247i \(-0.778832\pi\)
−0.768169 + 0.640247i \(0.778832\pi\)
\(840\) 0 0
\(841\) −5.65097 −0.194861
\(842\) −10.2642 −0.353728
\(843\) 0 0
\(844\) 13.1292 0.451926
\(845\) −0.941216 −0.0323788
\(846\) 0 0
\(847\) 20.5921 0.707554
\(848\) −1.26990 −0.0436085
\(849\) 0 0
\(850\) −12.5611 −0.430843
\(851\) 24.2600 0.831623
\(852\) 0 0
\(853\) −18.5412 −0.634838 −0.317419 0.948285i \(-0.602816\pi\)
−0.317419 + 0.948285i \(0.602816\pi\)
\(854\) −4.17465 −0.142854
\(855\) 0 0
\(856\) −9.01686 −0.308190
\(857\) −37.2675 −1.27304 −0.636518 0.771262i \(-0.719626\pi\)
−0.636518 + 0.771262i \(0.719626\pi\)
\(858\) 0 0
\(859\) −16.3924 −0.559303 −0.279651 0.960102i \(-0.590219\pi\)
−0.279651 + 0.960102i \(0.590219\pi\)
\(860\) −8.81042 −0.300433
\(861\) 0 0
\(862\) 23.2087 0.790491
\(863\) 47.5599 1.61896 0.809479 0.587149i \(-0.199750\pi\)
0.809479 + 0.587149i \(0.199750\pi\)
\(864\) 0 0
\(865\) 27.4054 0.931812
\(866\) 17.7200 0.602149
\(867\) 0 0
\(868\) 8.88849 0.301695
\(869\) −56.3744 −1.91237
\(870\) 0 0
\(871\) −16.7085 −0.566145
\(872\) 26.8535 0.909373
\(873\) 0 0
\(874\) −4.82076 −0.163065
\(875\) 6.73807 0.227788
\(876\) 0 0
\(877\) −22.3076 −0.753275 −0.376637 0.926361i \(-0.622920\pi\)
−0.376637 + 0.926361i \(0.622920\pi\)
\(878\) 9.97462 0.336627
\(879\) 0 0
\(880\) −3.98321 −0.134274
\(881\) −20.9990 −0.707474 −0.353737 0.935345i \(-0.615089\pi\)
−0.353737 + 0.935345i \(0.615089\pi\)
\(882\) 0 0
\(883\) 24.4512 0.822849 0.411425 0.911444i \(-0.365031\pi\)
0.411425 + 0.911444i \(0.365031\pi\)
\(884\) 27.6992 0.931623
\(885\) 0 0
\(886\) 9.31662 0.312998
\(887\) −17.1615 −0.576227 −0.288113 0.957596i \(-0.593028\pi\)
−0.288113 + 0.957596i \(0.593028\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) −6.32048 −0.211863
\(891\) 0 0
\(892\) −28.8701 −0.966643
\(893\) 26.0579 0.871996
\(894\) 0 0
\(895\) 58.7144 1.96261
\(896\) 7.87458 0.263071
\(897\) 0 0
\(898\) 17.7801 0.593329
\(899\) −33.2167 −1.10784
\(900\) 0 0
\(901\) −28.8710 −0.961833
\(902\) −19.3737 −0.645073
\(903\) 0 0
\(904\) −5.24878 −0.174572
\(905\) 65.3672 2.17288
\(906\) 0 0
\(907\) 31.9309 1.06025 0.530124 0.847920i \(-0.322145\pi\)
0.530124 + 0.847920i \(0.322145\pi\)
\(908\) −30.1714 −1.00127
\(909\) 0 0
\(910\) −8.43761 −0.279704
\(911\) −37.7485 −1.25067 −0.625333 0.780358i \(-0.715037\pi\)
−0.625333 + 0.780358i \(0.715037\pi\)
\(912\) 0 0
\(913\) 28.4795 0.942535
\(914\) 0.504652 0.0166924
\(915\) 0 0
\(916\) 11.0921 0.366493
\(917\) −5.84500 −0.193019
\(918\) 0 0
\(919\) 48.2941 1.59308 0.796538 0.604588i \(-0.206662\pi\)
0.796538 + 0.604588i \(0.206662\pi\)
\(920\) 21.1845 0.698434
\(921\) 0 0
\(922\) −26.1049 −0.859718
\(923\) 7.79739 0.256654
\(924\) 0 0
\(925\) −22.1897 −0.729594
\(926\) −8.35304 −0.274498
\(927\) 0 0
\(928\) −27.8066 −0.912795
\(929\) −59.7282 −1.95962 −0.979810 0.199932i \(-0.935928\pi\)
−0.979810 + 0.199932i \(0.935928\pi\)
\(930\) 0 0
\(931\) −2.05868 −0.0674704
\(932\) 18.9018 0.619149
\(933\) 0 0
\(934\) −5.17650 −0.169380
\(935\) −90.5578 −2.96156
\(936\) 0 0
\(937\) −27.0530 −0.883782 −0.441891 0.897069i \(-0.645692\pi\)
−0.441891 + 0.897069i \(0.645692\pi\)
\(938\) −3.84609 −0.125579
\(939\) 0 0
\(940\) −44.9630 −1.46653
\(941\) −32.2760 −1.05217 −0.526084 0.850433i \(-0.676340\pi\)
−0.526084 + 0.850433i \(0.676340\pi\)
\(942\) 0 0
\(943\) −11.4168 −0.371783
\(944\) 2.42882 0.0790512
\(945\) 0 0
\(946\) 11.7216 0.381101
\(947\) −4.40313 −0.143083 −0.0715413 0.997438i \(-0.522792\pi\)
−0.0715413 + 0.997438i \(0.522792\pi\)
\(948\) 0 0
\(949\) −35.0505 −1.13779
\(950\) 4.40937 0.143059
\(951\) 0 0
\(952\) 16.2382 0.526282
\(953\) −17.0635 −0.552741 −0.276370 0.961051i \(-0.589132\pi\)
−0.276370 + 0.961051i \(0.589132\pi\)
\(954\) 0 0
\(955\) 47.0094 1.52119
\(956\) 6.53260 0.211279
\(957\) 0 0
\(958\) 13.2573 0.428323
\(959\) 10.6774 0.344792
\(960\) 0 0
\(961\) 16.2545 0.524340
\(962\) −26.7540 −0.862584
\(963\) 0 0
\(964\) 20.6009 0.663511
\(965\) 58.2369 1.87471
\(966\) 0 0
\(967\) −26.9109 −0.865398 −0.432699 0.901539i \(-0.642439\pi\)
−0.432699 + 0.901539i \(0.642439\pi\)
\(968\) 57.0163 1.83257
\(969\) 0 0
\(970\) 32.5270 1.04438
\(971\) 28.9843 0.930151 0.465076 0.885271i \(-0.346027\pi\)
0.465076 + 0.885271i \(0.346027\pi\)
\(972\) 0 0
\(973\) −21.7313 −0.696674
\(974\) 5.29812 0.169763
\(975\) 0 0
\(976\) 1.28075 0.0409958
\(977\) 50.2194 1.60666 0.803330 0.595534i \(-0.203059\pi\)
0.803330 + 0.595534i \(0.203059\pi\)
\(978\) 0 0
\(979\) −15.3794 −0.491529
\(980\) 3.55225 0.113472
\(981\) 0 0
\(982\) −2.31788 −0.0739667
\(983\) 1.32859 0.0423756 0.0211878 0.999776i \(-0.493255\pi\)
0.0211878 + 0.999776i \(0.493255\pi\)
\(984\) 0 0
\(985\) 26.4946 0.844187
\(986\) −23.8274 −0.758819
\(987\) 0 0
\(988\) −9.72332 −0.309340
\(989\) 6.90747 0.219645
\(990\) 0 0
\(991\) −38.1866 −1.21304 −0.606518 0.795070i \(-0.707434\pi\)
−0.606518 + 0.795070i \(0.707434\pi\)
\(992\) 39.5580 1.25597
\(993\) 0 0
\(994\) 1.79486 0.0569296
\(995\) 22.3057 0.707138
\(996\) 0 0
\(997\) 34.2759 1.08553 0.542764 0.839885i \(-0.317378\pi\)
0.542764 + 0.839885i \(0.317378\pi\)
\(998\) −1.01145 −0.0320168
\(999\) 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.1.2 7
3.2 odd 2 2667.2.a.j.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.j.1.6 7 3.2 odd 2
8001.2.a.l.1.2 7 1.1 even 1 trivial