Properties

Label 8001.2.a.l.1.7
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 1.7
Root \(1.13462\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.48001 q^{2} +4.15043 q^{4} +0.783950 q^{5} +1.00000 q^{7} +5.33307 q^{8} +O(q^{10})\) \(q+2.48001 q^{2} +4.15043 q^{4} +0.783950 q^{5} +1.00000 q^{7} +5.33307 q^{8} +1.94420 q^{10} -4.88599 q^{11} -6.65543 q^{13} +2.48001 q^{14} +4.92520 q^{16} -1.25567 q^{17} -2.07830 q^{19} +3.25373 q^{20} -12.1173 q^{22} -6.86976 q^{23} -4.38542 q^{25} -16.5055 q^{26} +4.15043 q^{28} +9.79804 q^{29} -3.45591 q^{31} +1.54837 q^{32} -3.11406 q^{34} +0.783950 q^{35} -2.67789 q^{37} -5.15419 q^{38} +4.18086 q^{40} +4.19977 q^{41} -4.11683 q^{43} -20.2789 q^{44} -17.0370 q^{46} +6.51041 q^{47} +1.00000 q^{49} -10.8759 q^{50} -27.6229 q^{52} -3.48731 q^{53} -3.83037 q^{55} +5.33307 q^{56} +24.2992 q^{58} -12.2268 q^{59} +6.13863 q^{61} -8.57067 q^{62} -6.01043 q^{64} -5.21753 q^{65} -15.0576 q^{67} -5.21155 q^{68} +1.94420 q^{70} +7.18924 q^{71} +2.24951 q^{73} -6.64118 q^{74} -8.62583 q^{76} -4.88599 q^{77} -7.38716 q^{79} +3.86111 q^{80} +10.4154 q^{82} +3.82920 q^{83} -0.984380 q^{85} -10.2098 q^{86} -26.0573 q^{88} +16.4372 q^{89} -6.65543 q^{91} -28.5124 q^{92} +16.1458 q^{94} -1.62928 q^{95} +16.8967 q^{97} +2.48001 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + O(q^{10}) \) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + 3q^{11} - 23q^{13} + 2q^{14} + 2q^{16} - 3q^{17} - 9q^{19} + 9q^{20} - 19q^{22} - 12q^{23} + 3q^{25} - 18q^{26} + 4q^{28} + 9q^{29} - 33q^{31} - 10q^{32} - 2q^{34} + 8q^{35} - 33q^{37} + 3q^{38} - 9q^{40} + 3q^{41} - 9q^{43} - 2q^{44} - 32q^{46} - 11q^{47} + 7q^{49} - 29q^{50} - 21q^{52} - q^{53} - 16q^{55} + 9q^{56} - 5q^{58} + 30q^{59} - 19q^{61} - 3q^{62} - 21q^{64} - 14q^{65} - 30q^{67} - 24q^{68} - 8q^{71} - 20q^{73} + 9q^{74} - 42q^{76} + 3q^{77} + 8q^{79} - 12q^{80} + 10q^{82} + 34q^{83} - 28q^{85} - 24q^{86} - q^{88} + 12q^{89} - 23q^{91} - 60q^{92} - 3q^{94} - 12q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.48001 1.75363 0.876814 0.480829i \(-0.159664\pi\)
0.876814 + 0.480829i \(0.159664\pi\)
\(3\) 0 0
\(4\) 4.15043 2.07521
\(5\) 0.783950 0.350593 0.175297 0.984516i \(-0.443912\pi\)
0.175297 + 0.984516i \(0.443912\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 5.33307 1.88553
\(9\) 0 0
\(10\) 1.94420 0.614810
\(11\) −4.88599 −1.47318 −0.736590 0.676340i \(-0.763565\pi\)
−0.736590 + 0.676340i \(0.763565\pi\)
\(12\) 0 0
\(13\) −6.65543 −1.84589 −0.922943 0.384937i \(-0.874223\pi\)
−0.922943 + 0.384937i \(0.874223\pi\)
\(14\) 2.48001 0.662809
\(15\) 0 0
\(16\) 4.92520 1.23130
\(17\) −1.25567 −0.304544 −0.152272 0.988339i \(-0.548659\pi\)
−0.152272 + 0.988339i \(0.548659\pi\)
\(18\) 0 0
\(19\) −2.07830 −0.476795 −0.238397 0.971168i \(-0.576622\pi\)
−0.238397 + 0.971168i \(0.576622\pi\)
\(20\) 3.25373 0.727556
\(21\) 0 0
\(22\) −12.1173 −2.58341
\(23\) −6.86976 −1.43244 −0.716222 0.697872i \(-0.754130\pi\)
−0.716222 + 0.697872i \(0.754130\pi\)
\(24\) 0 0
\(25\) −4.38542 −0.877084
\(26\) −16.5055 −3.23700
\(27\) 0 0
\(28\) 4.15043 0.784357
\(29\) 9.79804 1.81945 0.909726 0.415210i \(-0.136292\pi\)
0.909726 + 0.415210i \(0.136292\pi\)
\(30\) 0 0
\(31\) −3.45591 −0.620699 −0.310349 0.950623i \(-0.600446\pi\)
−0.310349 + 0.950623i \(0.600446\pi\)
\(32\) 1.54837 0.273715
\(33\) 0 0
\(34\) −3.11406 −0.534057
\(35\) 0.783950 0.132512
\(36\) 0 0
\(37\) −2.67789 −0.440242 −0.220121 0.975473i \(-0.570645\pi\)
−0.220121 + 0.975473i \(0.570645\pi\)
\(38\) −5.15419 −0.836121
\(39\) 0 0
\(40\) 4.18086 0.661052
\(41\) 4.19977 0.655893 0.327947 0.944696i \(-0.393643\pi\)
0.327947 + 0.944696i \(0.393643\pi\)
\(42\) 0 0
\(43\) −4.11683 −0.627811 −0.313906 0.949454i \(-0.601638\pi\)
−0.313906 + 0.949454i \(0.601638\pi\)
\(44\) −20.2789 −3.05716
\(45\) 0 0
\(46\) −17.0370 −2.51198
\(47\) 6.51041 0.949640 0.474820 0.880083i \(-0.342513\pi\)
0.474820 + 0.880083i \(0.342513\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −10.8759 −1.53808
\(51\) 0 0
\(52\) −27.6229 −3.83061
\(53\) −3.48731 −0.479019 −0.239510 0.970894i \(-0.576987\pi\)
−0.239510 + 0.970894i \(0.576987\pi\)
\(54\) 0 0
\(55\) −3.83037 −0.516487
\(56\) 5.33307 0.712662
\(57\) 0 0
\(58\) 24.2992 3.19064
\(59\) −12.2268 −1.59180 −0.795899 0.605430i \(-0.793001\pi\)
−0.795899 + 0.605430i \(0.793001\pi\)
\(60\) 0 0
\(61\) 6.13863 0.785971 0.392986 0.919545i \(-0.371442\pi\)
0.392986 + 0.919545i \(0.371442\pi\)
\(62\) −8.57067 −1.08848
\(63\) 0 0
\(64\) −6.01043 −0.751304
\(65\) −5.21753 −0.647155
\(66\) 0 0
\(67\) −15.0576 −1.83958 −0.919790 0.392412i \(-0.871641\pi\)
−0.919790 + 0.392412i \(0.871641\pi\)
\(68\) −5.21155 −0.631994
\(69\) 0 0
\(70\) 1.94420 0.232376
\(71\) 7.18924 0.853206 0.426603 0.904439i \(-0.359710\pi\)
0.426603 + 0.904439i \(0.359710\pi\)
\(72\) 0 0
\(73\) 2.24951 0.263285 0.131643 0.991297i \(-0.457975\pi\)
0.131643 + 0.991297i \(0.457975\pi\)
\(74\) −6.64118 −0.772021
\(75\) 0 0
\(76\) −8.62583 −0.989451
\(77\) −4.88599 −0.556810
\(78\) 0 0
\(79\) −7.38716 −0.831120 −0.415560 0.909566i \(-0.636414\pi\)
−0.415560 + 0.909566i \(0.636414\pi\)
\(80\) 3.86111 0.431685
\(81\) 0 0
\(82\) 10.4154 1.15019
\(83\) 3.82920 0.420310 0.210155 0.977668i \(-0.432603\pi\)
0.210155 + 0.977668i \(0.432603\pi\)
\(84\) 0 0
\(85\) −0.984380 −0.106771
\(86\) −10.2098 −1.10095
\(87\) 0 0
\(88\) −26.0573 −2.77772
\(89\) 16.4372 1.74234 0.871171 0.490980i \(-0.163361\pi\)
0.871171 + 0.490980i \(0.163361\pi\)
\(90\) 0 0
\(91\) −6.65543 −0.697679
\(92\) −28.5124 −2.97263
\(93\) 0 0
\(94\) 16.1458 1.66532
\(95\) −1.62928 −0.167161
\(96\) 0 0
\(97\) 16.8967 1.71560 0.857798 0.513987i \(-0.171832\pi\)
0.857798 + 0.513987i \(0.171832\pi\)
\(98\) 2.48001 0.250518
\(99\) 0 0
\(100\) −18.2014 −1.82014
\(101\) −0.691702 −0.0688269 −0.0344134 0.999408i \(-0.510956\pi\)
−0.0344134 + 0.999408i \(0.510956\pi\)
\(102\) 0 0
\(103\) −16.5225 −1.62801 −0.814003 0.580861i \(-0.802716\pi\)
−0.814003 + 0.580861i \(0.802716\pi\)
\(104\) −35.4939 −3.48047
\(105\) 0 0
\(106\) −8.64855 −0.840022
\(107\) 19.1211 1.84851 0.924255 0.381776i \(-0.124687\pi\)
0.924255 + 0.381776i \(0.124687\pi\)
\(108\) 0 0
\(109\) 14.7383 1.41167 0.705837 0.708374i \(-0.250571\pi\)
0.705837 + 0.708374i \(0.250571\pi\)
\(110\) −9.49933 −0.905726
\(111\) 0 0
\(112\) 4.92520 0.465387
\(113\) −12.5353 −1.17922 −0.589612 0.807686i \(-0.700719\pi\)
−0.589612 + 0.807686i \(0.700719\pi\)
\(114\) 0 0
\(115\) −5.38555 −0.502205
\(116\) 40.6661 3.77575
\(117\) 0 0
\(118\) −30.3226 −2.79142
\(119\) −1.25567 −0.115107
\(120\) 0 0
\(121\) 12.8729 1.17026
\(122\) 15.2238 1.37830
\(123\) 0 0
\(124\) −14.3435 −1.28808
\(125\) −7.35770 −0.658093
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −18.0026 −1.59122
\(129\) 0 0
\(130\) −12.9395 −1.13487
\(131\) 0.687639 0.0600793 0.0300397 0.999549i \(-0.490437\pi\)
0.0300397 + 0.999549i \(0.490437\pi\)
\(132\) 0 0
\(133\) −2.07830 −0.180211
\(134\) −37.3429 −3.22594
\(135\) 0 0
\(136\) −6.69656 −0.574225
\(137\) 7.96226 0.680262 0.340131 0.940378i \(-0.389529\pi\)
0.340131 + 0.940378i \(0.389529\pi\)
\(138\) 0 0
\(139\) −0.410908 −0.0348528 −0.0174264 0.999848i \(-0.505547\pi\)
−0.0174264 + 0.999848i \(0.505547\pi\)
\(140\) 3.25373 0.274990
\(141\) 0 0
\(142\) 17.8294 1.49621
\(143\) 32.5184 2.71932
\(144\) 0 0
\(145\) 7.68118 0.637887
\(146\) 5.57880 0.461705
\(147\) 0 0
\(148\) −11.1144 −0.913597
\(149\) −23.6085 −1.93408 −0.967041 0.254620i \(-0.918050\pi\)
−0.967041 + 0.254620i \(0.918050\pi\)
\(150\) 0 0
\(151\) −6.43593 −0.523748 −0.261874 0.965102i \(-0.584341\pi\)
−0.261874 + 0.965102i \(0.584341\pi\)
\(152\) −11.0837 −0.899009
\(153\) 0 0
\(154\) −12.1173 −0.976438
\(155\) −2.70926 −0.217613
\(156\) 0 0
\(157\) −14.4987 −1.15712 −0.578561 0.815639i \(-0.696385\pi\)
−0.578561 + 0.815639i \(0.696385\pi\)
\(158\) −18.3202 −1.45748
\(159\) 0 0
\(160\) 1.21384 0.0959627
\(161\) −6.86976 −0.541413
\(162\) 0 0
\(163\) 8.06383 0.631608 0.315804 0.948824i \(-0.397726\pi\)
0.315804 + 0.948824i \(0.397726\pi\)
\(164\) 17.4308 1.36112
\(165\) 0 0
\(166\) 9.49645 0.737067
\(167\) −4.60533 −0.356371 −0.178186 0.983997i \(-0.557023\pi\)
−0.178186 + 0.983997i \(0.557023\pi\)
\(168\) 0 0
\(169\) 31.2948 2.40729
\(170\) −2.44127 −0.187237
\(171\) 0 0
\(172\) −17.0866 −1.30284
\(173\) 11.5019 0.874475 0.437237 0.899346i \(-0.355957\pi\)
0.437237 + 0.899346i \(0.355957\pi\)
\(174\) 0 0
\(175\) −4.38542 −0.331507
\(176\) −24.0644 −1.81393
\(177\) 0 0
\(178\) 40.7644 3.05542
\(179\) −22.8385 −1.70703 −0.853515 0.521068i \(-0.825534\pi\)
−0.853515 + 0.521068i \(0.825534\pi\)
\(180\) 0 0
\(181\) 4.40434 0.327372 0.163686 0.986512i \(-0.447662\pi\)
0.163686 + 0.986512i \(0.447662\pi\)
\(182\) −16.5055 −1.22347
\(183\) 0 0
\(184\) −36.6369 −2.70091
\(185\) −2.09933 −0.154346
\(186\) 0 0
\(187\) 6.13517 0.448648
\(188\) 27.0210 1.97071
\(189\) 0 0
\(190\) −4.04063 −0.293138
\(191\) −20.5561 −1.48739 −0.743695 0.668520i \(-0.766928\pi\)
−0.743695 + 0.668520i \(0.766928\pi\)
\(192\) 0 0
\(193\) −18.0715 −1.30082 −0.650409 0.759584i \(-0.725402\pi\)
−0.650409 + 0.759584i \(0.725402\pi\)
\(194\) 41.9038 3.00852
\(195\) 0 0
\(196\) 4.15043 0.296459
\(197\) 24.7109 1.76058 0.880291 0.474434i \(-0.157347\pi\)
0.880291 + 0.474434i \(0.157347\pi\)
\(198\) 0 0
\(199\) 16.6604 1.18102 0.590511 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(200\) −23.3878 −1.65377
\(201\) 0 0
\(202\) −1.71542 −0.120697
\(203\) 9.79804 0.687688
\(204\) 0 0
\(205\) 3.29241 0.229952
\(206\) −40.9758 −2.85492
\(207\) 0 0
\(208\) −32.7793 −2.27284
\(209\) 10.1545 0.702404
\(210\) 0 0
\(211\) 14.5672 1.00284 0.501422 0.865203i \(-0.332810\pi\)
0.501422 + 0.865203i \(0.332810\pi\)
\(212\) −14.4738 −0.994067
\(213\) 0 0
\(214\) 47.4205 3.24160
\(215\) −3.22739 −0.220106
\(216\) 0 0
\(217\) −3.45591 −0.234602
\(218\) 36.5511 2.47555
\(219\) 0 0
\(220\) −15.8977 −1.07182
\(221\) 8.35701 0.562153
\(222\) 0 0
\(223\) −9.76267 −0.653757 −0.326878 0.945066i \(-0.605997\pi\)
−0.326878 + 0.945066i \(0.605997\pi\)
\(224\) 1.54837 0.103455
\(225\) 0 0
\(226\) −31.0877 −2.06792
\(227\) 26.0747 1.73064 0.865321 0.501219i \(-0.167115\pi\)
0.865321 + 0.501219i \(0.167115\pi\)
\(228\) 0 0
\(229\) 6.24716 0.412824 0.206412 0.978465i \(-0.433821\pi\)
0.206412 + 0.978465i \(0.433821\pi\)
\(230\) −13.3562 −0.880681
\(231\) 0 0
\(232\) 52.2537 3.43062
\(233\) −6.77809 −0.444047 −0.222024 0.975041i \(-0.571266\pi\)
−0.222024 + 0.975041i \(0.571266\pi\)
\(234\) 0 0
\(235\) 5.10383 0.332937
\(236\) −50.7466 −3.30332
\(237\) 0 0
\(238\) −3.11406 −0.201855
\(239\) 23.7065 1.53345 0.766724 0.641977i \(-0.221885\pi\)
0.766724 + 0.641977i \(0.221885\pi\)
\(240\) 0 0
\(241\) −9.28434 −0.598057 −0.299028 0.954244i \(-0.596663\pi\)
−0.299028 + 0.954244i \(0.596663\pi\)
\(242\) 31.9247 2.05220
\(243\) 0 0
\(244\) 25.4779 1.63106
\(245\) 0.783950 0.0500847
\(246\) 0 0
\(247\) 13.8320 0.880108
\(248\) −18.4306 −1.17034
\(249\) 0 0
\(250\) −18.2471 −1.15405
\(251\) −0.539366 −0.0340445 −0.0170222 0.999855i \(-0.505419\pi\)
−0.0170222 + 0.999855i \(0.505419\pi\)
\(252\) 0 0
\(253\) 33.5656 2.11025
\(254\) −2.48001 −0.155609
\(255\) 0 0
\(256\) −32.6258 −2.03911
\(257\) 4.16847 0.260022 0.130011 0.991513i \(-0.458499\pi\)
0.130011 + 0.991513i \(0.458499\pi\)
\(258\) 0 0
\(259\) −2.67789 −0.166396
\(260\) −21.6550 −1.34298
\(261\) 0 0
\(262\) 1.70535 0.105357
\(263\) 23.9281 1.47547 0.737734 0.675092i \(-0.235896\pi\)
0.737734 + 0.675092i \(0.235896\pi\)
\(264\) 0 0
\(265\) −2.73388 −0.167941
\(266\) −5.15419 −0.316024
\(267\) 0 0
\(268\) −62.4955 −3.81752
\(269\) 30.9708 1.88832 0.944160 0.329487i \(-0.106876\pi\)
0.944160 + 0.329487i \(0.106876\pi\)
\(270\) 0 0
\(271\) −23.8214 −1.44705 −0.723523 0.690300i \(-0.757478\pi\)
−0.723523 + 0.690300i \(0.757478\pi\)
\(272\) −6.18440 −0.374985
\(273\) 0 0
\(274\) 19.7465 1.19293
\(275\) 21.4271 1.29210
\(276\) 0 0
\(277\) −14.3357 −0.861351 −0.430675 0.902507i \(-0.641725\pi\)
−0.430675 + 0.902507i \(0.641725\pi\)
\(278\) −1.01905 −0.0611188
\(279\) 0 0
\(280\) 4.18086 0.249854
\(281\) −20.6212 −1.23016 −0.615079 0.788466i \(-0.710876\pi\)
−0.615079 + 0.788466i \(0.710876\pi\)
\(282\) 0 0
\(283\) 3.77609 0.224465 0.112233 0.993682i \(-0.464200\pi\)
0.112233 + 0.993682i \(0.464200\pi\)
\(284\) 29.8384 1.77058
\(285\) 0 0
\(286\) 80.6457 4.76868
\(287\) 4.19977 0.247904
\(288\) 0 0
\(289\) −15.4233 −0.907253
\(290\) 19.0494 1.11862
\(291\) 0 0
\(292\) 9.33644 0.546374
\(293\) 24.2919 1.41915 0.709573 0.704632i \(-0.248888\pi\)
0.709573 + 0.704632i \(0.248888\pi\)
\(294\) 0 0
\(295\) −9.58522 −0.558073
\(296\) −14.2814 −0.830088
\(297\) 0 0
\(298\) −58.5492 −3.39166
\(299\) 45.7212 2.64413
\(300\) 0 0
\(301\) −4.11683 −0.237290
\(302\) −15.9611 −0.918460
\(303\) 0 0
\(304\) −10.2360 −0.587077
\(305\) 4.81238 0.275556
\(306\) 0 0
\(307\) −18.6482 −1.06431 −0.532153 0.846648i \(-0.678617\pi\)
−0.532153 + 0.846648i \(0.678617\pi\)
\(308\) −20.2789 −1.15550
\(309\) 0 0
\(310\) −6.71897 −0.381612
\(311\) −20.8747 −1.18370 −0.591848 0.806050i \(-0.701601\pi\)
−0.591848 + 0.806050i \(0.701601\pi\)
\(312\) 0 0
\(313\) −7.93347 −0.448426 −0.224213 0.974540i \(-0.571981\pi\)
−0.224213 + 0.974540i \(0.571981\pi\)
\(314\) −35.9568 −2.02916
\(315\) 0 0
\(316\) −30.6599 −1.72475
\(317\) −11.3644 −0.638290 −0.319145 0.947706i \(-0.603396\pi\)
−0.319145 + 0.947706i \(0.603396\pi\)
\(318\) 0 0
\(319\) −47.8731 −2.68038
\(320\) −4.71188 −0.263402
\(321\) 0 0
\(322\) −17.0370 −0.949437
\(323\) 2.60965 0.145205
\(324\) 0 0
\(325\) 29.1869 1.61900
\(326\) 19.9983 1.10761
\(327\) 0 0
\(328\) 22.3977 1.23670
\(329\) 6.51041 0.358930
\(330\) 0 0
\(331\) −19.1983 −1.05523 −0.527616 0.849483i \(-0.676914\pi\)
−0.527616 + 0.849483i \(0.676914\pi\)
\(332\) 15.8928 0.872233
\(333\) 0 0
\(334\) −11.4213 −0.624943
\(335\) −11.8044 −0.644944
\(336\) 0 0
\(337\) 8.90770 0.485234 0.242617 0.970122i \(-0.421994\pi\)
0.242617 + 0.970122i \(0.421994\pi\)
\(338\) 77.6113 4.22150
\(339\) 0 0
\(340\) −4.08560 −0.221573
\(341\) 16.8855 0.914401
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −21.9554 −1.18375
\(345\) 0 0
\(346\) 28.5248 1.53350
\(347\) 1.17952 0.0633201 0.0316601 0.999499i \(-0.489921\pi\)
0.0316601 + 0.999499i \(0.489921\pi\)
\(348\) 0 0
\(349\) −29.5422 −1.58136 −0.790679 0.612231i \(-0.790272\pi\)
−0.790679 + 0.612231i \(0.790272\pi\)
\(350\) −10.8759 −0.581340
\(351\) 0 0
\(352\) −7.56530 −0.403232
\(353\) −10.1955 −0.542653 −0.271327 0.962487i \(-0.587462\pi\)
−0.271327 + 0.962487i \(0.587462\pi\)
\(354\) 0 0
\(355\) 5.63601 0.299128
\(356\) 68.2215 3.61573
\(357\) 0 0
\(358\) −56.6396 −2.99350
\(359\) −30.6151 −1.61580 −0.807900 0.589319i \(-0.799396\pi\)
−0.807900 + 0.589319i \(0.799396\pi\)
\(360\) 0 0
\(361\) −14.6807 −0.772667
\(362\) 10.9228 0.574089
\(363\) 0 0
\(364\) −27.6229 −1.44783
\(365\) 1.76350 0.0923061
\(366\) 0 0
\(367\) −10.8181 −0.564702 −0.282351 0.959311i \(-0.591114\pi\)
−0.282351 + 0.959311i \(0.591114\pi\)
\(368\) −33.8349 −1.76377
\(369\) 0 0
\(370\) −5.20635 −0.270665
\(371\) −3.48731 −0.181052
\(372\) 0 0
\(373\) 10.2006 0.528169 0.264085 0.964499i \(-0.414930\pi\)
0.264085 + 0.964499i \(0.414930\pi\)
\(374\) 15.2153 0.786762
\(375\) 0 0
\(376\) 34.7205 1.79057
\(377\) −65.2102 −3.35850
\(378\) 0 0
\(379\) 6.58231 0.338110 0.169055 0.985607i \(-0.445928\pi\)
0.169055 + 0.985607i \(0.445928\pi\)
\(380\) −6.76222 −0.346895
\(381\) 0 0
\(382\) −50.9793 −2.60833
\(383\) 20.2431 1.03438 0.517188 0.855872i \(-0.326979\pi\)
0.517188 + 0.855872i \(0.326979\pi\)
\(384\) 0 0
\(385\) −3.83037 −0.195214
\(386\) −44.8175 −2.28115
\(387\) 0 0
\(388\) 70.1283 3.56023
\(389\) −8.10612 −0.410997 −0.205498 0.978657i \(-0.565882\pi\)
−0.205498 + 0.978657i \(0.565882\pi\)
\(390\) 0 0
\(391\) 8.62613 0.436242
\(392\) 5.33307 0.269361
\(393\) 0 0
\(394\) 61.2833 3.08741
\(395\) −5.79116 −0.291385
\(396\) 0 0
\(397\) −15.9348 −0.799745 −0.399872 0.916571i \(-0.630946\pi\)
−0.399872 + 0.916571i \(0.630946\pi\)
\(398\) 41.3178 2.07107
\(399\) 0 0
\(400\) −21.5991 −1.07995
\(401\) −35.9972 −1.79761 −0.898807 0.438344i \(-0.855565\pi\)
−0.898807 + 0.438344i \(0.855565\pi\)
\(402\) 0 0
\(403\) 23.0006 1.14574
\(404\) −2.87086 −0.142831
\(405\) 0 0
\(406\) 24.2992 1.20595
\(407\) 13.0841 0.648556
\(408\) 0 0
\(409\) −21.3091 −1.05367 −0.526833 0.849969i \(-0.676621\pi\)
−0.526833 + 0.849969i \(0.676621\pi\)
\(410\) 8.16519 0.403250
\(411\) 0 0
\(412\) −68.5752 −3.37846
\(413\) −12.2268 −0.601643
\(414\) 0 0
\(415\) 3.00190 0.147358
\(416\) −10.3051 −0.505247
\(417\) 0 0
\(418\) 25.1833 1.23176
\(419\) 26.8399 1.31122 0.655608 0.755101i \(-0.272412\pi\)
0.655608 + 0.755101i \(0.272412\pi\)
\(420\) 0 0
\(421\) 1.71312 0.0834923 0.0417462 0.999128i \(-0.486708\pi\)
0.0417462 + 0.999128i \(0.486708\pi\)
\(422\) 36.1266 1.75862
\(423\) 0 0
\(424\) −18.5981 −0.903203
\(425\) 5.50663 0.267111
\(426\) 0 0
\(427\) 6.13863 0.297069
\(428\) 79.3609 3.83605
\(429\) 0 0
\(430\) −8.00395 −0.385985
\(431\) −29.8518 −1.43791 −0.718954 0.695057i \(-0.755379\pi\)
−0.718954 + 0.695057i \(0.755379\pi\)
\(432\) 0 0
\(433\) −6.53077 −0.313849 −0.156924 0.987611i \(-0.550158\pi\)
−0.156924 + 0.987611i \(0.550158\pi\)
\(434\) −8.57067 −0.411405
\(435\) 0 0
\(436\) 61.1703 2.92953
\(437\) 14.2774 0.682982
\(438\) 0 0
\(439\) −14.9276 −0.712457 −0.356228 0.934399i \(-0.615938\pi\)
−0.356228 + 0.934399i \(0.615938\pi\)
\(440\) −20.4276 −0.973849
\(441\) 0 0
\(442\) 20.7254 0.985808
\(443\) 4.63421 0.220178 0.110089 0.993922i \(-0.464886\pi\)
0.110089 + 0.993922i \(0.464886\pi\)
\(444\) 0 0
\(445\) 12.8860 0.610853
\(446\) −24.2115 −1.14645
\(447\) 0 0
\(448\) −6.01043 −0.283966
\(449\) 6.63579 0.313162 0.156581 0.987665i \(-0.449953\pi\)
0.156581 + 0.987665i \(0.449953\pi\)
\(450\) 0 0
\(451\) −20.5200 −0.966249
\(452\) −52.0270 −2.44714
\(453\) 0 0
\(454\) 64.6655 3.03490
\(455\) −5.21753 −0.244601
\(456\) 0 0
\(457\) −17.6826 −0.827157 −0.413578 0.910468i \(-0.635721\pi\)
−0.413578 + 0.910468i \(0.635721\pi\)
\(458\) 15.4930 0.723940
\(459\) 0 0
\(460\) −22.3523 −1.04218
\(461\) 40.2668 1.87541 0.937706 0.347431i \(-0.112946\pi\)
0.937706 + 0.347431i \(0.112946\pi\)
\(462\) 0 0
\(463\) −3.88536 −0.180568 −0.0902841 0.995916i \(-0.528777\pi\)
−0.0902841 + 0.995916i \(0.528777\pi\)
\(464\) 48.2573 2.24029
\(465\) 0 0
\(466\) −16.8097 −0.778694
\(467\) 1.32929 0.0615123 0.0307561 0.999527i \(-0.490208\pi\)
0.0307561 + 0.999527i \(0.490208\pi\)
\(468\) 0 0
\(469\) −15.0576 −0.695296
\(470\) 12.6575 0.583849
\(471\) 0 0
\(472\) −65.2066 −3.00138
\(473\) 20.1148 0.924879
\(474\) 0 0
\(475\) 9.11422 0.418189
\(476\) −5.21155 −0.238871
\(477\) 0 0
\(478\) 58.7923 2.68910
\(479\) 10.9654 0.501022 0.250511 0.968114i \(-0.419401\pi\)
0.250511 + 0.968114i \(0.419401\pi\)
\(480\) 0 0
\(481\) 17.8225 0.812637
\(482\) −23.0252 −1.04877
\(483\) 0 0
\(484\) 53.4278 2.42854
\(485\) 13.2461 0.601476
\(486\) 0 0
\(487\) 7.42363 0.336397 0.168198 0.985753i \(-0.446205\pi\)
0.168198 + 0.985753i \(0.446205\pi\)
\(488\) 32.7378 1.48197
\(489\) 0 0
\(490\) 1.94420 0.0878300
\(491\) −10.4742 −0.472694 −0.236347 0.971669i \(-0.575950\pi\)
−0.236347 + 0.971669i \(0.575950\pi\)
\(492\) 0 0
\(493\) −12.3031 −0.554103
\(494\) 34.3034 1.54338
\(495\) 0 0
\(496\) −17.0210 −0.764266
\(497\) 7.18924 0.322482
\(498\) 0 0
\(499\) 26.6794 1.19433 0.597167 0.802117i \(-0.296293\pi\)
0.597167 + 0.802117i \(0.296293\pi\)
\(500\) −30.5376 −1.36568
\(501\) 0 0
\(502\) −1.33763 −0.0597013
\(503\) −27.1665 −1.21129 −0.605646 0.795734i \(-0.707085\pi\)
−0.605646 + 0.795734i \(0.707085\pi\)
\(504\) 0 0
\(505\) −0.542259 −0.0241302
\(506\) 83.2428 3.70059
\(507\) 0 0
\(508\) −4.15043 −0.184145
\(509\) 5.67100 0.251363 0.125681 0.992071i \(-0.459888\pi\)
0.125681 + 0.992071i \(0.459888\pi\)
\(510\) 0 0
\(511\) 2.24951 0.0995126
\(512\) −44.9069 −1.98462
\(513\) 0 0
\(514\) 10.3378 0.455982
\(515\) −12.9528 −0.570767
\(516\) 0 0
\(517\) −31.8098 −1.39899
\(518\) −6.64118 −0.291797
\(519\) 0 0
\(520\) −27.8255 −1.22023
\(521\) −32.7163 −1.43333 −0.716664 0.697419i \(-0.754332\pi\)
−0.716664 + 0.697419i \(0.754332\pi\)
\(522\) 0 0
\(523\) 20.8450 0.911486 0.455743 0.890111i \(-0.349374\pi\)
0.455743 + 0.890111i \(0.349374\pi\)
\(524\) 2.85400 0.124677
\(525\) 0 0
\(526\) 59.3417 2.58742
\(527\) 4.33946 0.189030
\(528\) 0 0
\(529\) 24.1936 1.05190
\(530\) −6.78003 −0.294506
\(531\) 0 0
\(532\) −8.62583 −0.373977
\(533\) −27.9513 −1.21070
\(534\) 0 0
\(535\) 14.9900 0.648075
\(536\) −80.3033 −3.46857
\(537\) 0 0
\(538\) 76.8077 3.31141
\(539\) −4.88599 −0.210454
\(540\) 0 0
\(541\) −21.1270 −0.908319 −0.454159 0.890920i \(-0.650060\pi\)
−0.454159 + 0.890920i \(0.650060\pi\)
\(542\) −59.0772 −2.53758
\(543\) 0 0
\(544\) −1.94423 −0.0833583
\(545\) 11.5541 0.494923
\(546\) 0 0
\(547\) 10.1713 0.434895 0.217448 0.976072i \(-0.430227\pi\)
0.217448 + 0.976072i \(0.430227\pi\)
\(548\) 33.0468 1.41169
\(549\) 0 0
\(550\) 53.1394 2.26587
\(551\) −20.3633 −0.867504
\(552\) 0 0
\(553\) −7.38716 −0.314134
\(554\) −35.5527 −1.51049
\(555\) 0 0
\(556\) −1.70544 −0.0723270
\(557\) 29.5187 1.25075 0.625373 0.780326i \(-0.284947\pi\)
0.625373 + 0.780326i \(0.284947\pi\)
\(558\) 0 0
\(559\) 27.3993 1.15887
\(560\) 3.86111 0.163162
\(561\) 0 0
\(562\) −51.1407 −2.15724
\(563\) −29.4720 −1.24210 −0.621049 0.783772i \(-0.713293\pi\)
−0.621049 + 0.783772i \(0.713293\pi\)
\(564\) 0 0
\(565\) −9.82707 −0.413428
\(566\) 9.36473 0.393629
\(567\) 0 0
\(568\) 38.3408 1.60874
\(569\) −18.1639 −0.761471 −0.380735 0.924684i \(-0.624329\pi\)
−0.380735 + 0.924684i \(0.624329\pi\)
\(570\) 0 0
\(571\) −5.20524 −0.217832 −0.108916 0.994051i \(-0.534738\pi\)
−0.108916 + 0.994051i \(0.534738\pi\)
\(572\) 134.965 5.64317
\(573\) 0 0
\(574\) 10.4154 0.434732
\(575\) 30.1268 1.25637
\(576\) 0 0
\(577\) 22.5036 0.936835 0.468418 0.883507i \(-0.344824\pi\)
0.468418 + 0.883507i \(0.344824\pi\)
\(578\) −38.2499 −1.59099
\(579\) 0 0
\(580\) 31.8802 1.32375
\(581\) 3.82920 0.158862
\(582\) 0 0
\(583\) 17.0390 0.705681
\(584\) 11.9968 0.496432
\(585\) 0 0
\(586\) 60.2440 2.48866
\(587\) 11.1742 0.461208 0.230604 0.973048i \(-0.425930\pi\)
0.230604 + 0.973048i \(0.425930\pi\)
\(588\) 0 0
\(589\) 7.18241 0.295946
\(590\) −23.7714 −0.978653
\(591\) 0 0
\(592\) −13.1891 −0.542070
\(593\) −10.7553 −0.441669 −0.220835 0.975311i \(-0.570878\pi\)
−0.220835 + 0.975311i \(0.570878\pi\)
\(594\) 0 0
\(595\) −0.984380 −0.0403556
\(596\) −97.9853 −4.01364
\(597\) 0 0
\(598\) 113.389 4.63682
\(599\) −16.4493 −0.672099 −0.336049 0.941844i \(-0.609091\pi\)
−0.336049 + 0.941844i \(0.609091\pi\)
\(600\) 0 0
\(601\) 24.6614 1.00596 0.502979 0.864298i \(-0.332237\pi\)
0.502979 + 0.864298i \(0.332237\pi\)
\(602\) −10.2098 −0.416119
\(603\) 0 0
\(604\) −26.7118 −1.08689
\(605\) 10.0917 0.410285
\(606\) 0 0
\(607\) −6.40510 −0.259975 −0.129988 0.991516i \(-0.541494\pi\)
−0.129988 + 0.991516i \(0.541494\pi\)
\(608\) −3.21797 −0.130506
\(609\) 0 0
\(610\) 11.9347 0.483223
\(611\) −43.3296 −1.75293
\(612\) 0 0
\(613\) 9.77647 0.394868 0.197434 0.980316i \(-0.436739\pi\)
0.197434 + 0.980316i \(0.436739\pi\)
\(614\) −46.2475 −1.86640
\(615\) 0 0
\(616\) −26.0573 −1.04988
\(617\) −36.0074 −1.44960 −0.724802 0.688958i \(-0.758069\pi\)
−0.724802 + 0.688958i \(0.758069\pi\)
\(618\) 0 0
\(619\) −18.4220 −0.740441 −0.370220 0.928944i \(-0.620718\pi\)
−0.370220 + 0.928944i \(0.620718\pi\)
\(620\) −11.2446 −0.451593
\(621\) 0 0
\(622\) −51.7694 −2.07576
\(623\) 16.4372 0.658543
\(624\) 0 0
\(625\) 16.1590 0.646362
\(626\) −19.6750 −0.786373
\(627\) 0 0
\(628\) −60.1758 −2.40127
\(629\) 3.36254 0.134073
\(630\) 0 0
\(631\) −21.8713 −0.870682 −0.435341 0.900266i \(-0.643372\pi\)
−0.435341 + 0.900266i \(0.643372\pi\)
\(632\) −39.3962 −1.56710
\(633\) 0 0
\(634\) −28.1838 −1.11932
\(635\) −0.783950 −0.0311101
\(636\) 0 0
\(637\) −6.65543 −0.263698
\(638\) −118.726 −4.70039
\(639\) 0 0
\(640\) −14.1132 −0.557872
\(641\) 5.35545 0.211527 0.105764 0.994391i \(-0.466271\pi\)
0.105764 + 0.994391i \(0.466271\pi\)
\(642\) 0 0
\(643\) 12.1586 0.479488 0.239744 0.970836i \(-0.422936\pi\)
0.239744 + 0.970836i \(0.422936\pi\)
\(644\) −28.5124 −1.12355
\(645\) 0 0
\(646\) 6.47195 0.254635
\(647\) 3.92615 0.154353 0.0771764 0.997017i \(-0.475410\pi\)
0.0771764 + 0.997017i \(0.475410\pi\)
\(648\) 0 0
\(649\) 59.7401 2.34500
\(650\) 72.3837 2.83912
\(651\) 0 0
\(652\) 33.4683 1.31072
\(653\) −3.49414 −0.136736 −0.0683681 0.997660i \(-0.521779\pi\)
−0.0683681 + 0.997660i \(0.521779\pi\)
\(654\) 0 0
\(655\) 0.539075 0.0210634
\(656\) 20.6847 0.807601
\(657\) 0 0
\(658\) 16.1458 0.629431
\(659\) −26.1167 −1.01736 −0.508682 0.860955i \(-0.669867\pi\)
−0.508682 + 0.860955i \(0.669867\pi\)
\(660\) 0 0
\(661\) −34.1023 −1.32643 −0.663213 0.748431i \(-0.730808\pi\)
−0.663213 + 0.748431i \(0.730808\pi\)
\(662\) −47.6118 −1.85049
\(663\) 0 0
\(664\) 20.4214 0.792505
\(665\) −1.62928 −0.0631809
\(666\) 0 0
\(667\) −67.3102 −2.60626
\(668\) −19.1141 −0.739547
\(669\) 0 0
\(670\) −29.2750 −1.13099
\(671\) −29.9933 −1.15788
\(672\) 0 0
\(673\) −11.3642 −0.438058 −0.219029 0.975718i \(-0.570289\pi\)
−0.219029 + 0.975718i \(0.570289\pi\)
\(674\) 22.0912 0.850920
\(675\) 0 0
\(676\) 129.887 4.99565
\(677\) 8.47890 0.325870 0.162935 0.986637i \(-0.447904\pi\)
0.162935 + 0.986637i \(0.447904\pi\)
\(678\) 0 0
\(679\) 16.8967 0.648434
\(680\) −5.24977 −0.201319
\(681\) 0 0
\(682\) 41.8761 1.60352
\(683\) −13.5516 −0.518537 −0.259269 0.965805i \(-0.583482\pi\)
−0.259269 + 0.965805i \(0.583482\pi\)
\(684\) 0 0
\(685\) 6.24201 0.238495
\(686\) 2.48001 0.0946871
\(687\) 0 0
\(688\) −20.2762 −0.773023
\(689\) 23.2096 0.884214
\(690\) 0 0
\(691\) −43.4677 −1.65359 −0.826794 0.562505i \(-0.809838\pi\)
−0.826794 + 0.562505i \(0.809838\pi\)
\(692\) 47.7379 1.81472
\(693\) 0 0
\(694\) 2.92522 0.111040
\(695\) −0.322131 −0.0122191
\(696\) 0 0
\(697\) −5.27351 −0.199748
\(698\) −73.2649 −2.77312
\(699\) 0 0
\(700\) −18.2014 −0.687947
\(701\) −28.8144 −1.08831 −0.544153 0.838986i \(-0.683149\pi\)
−0.544153 + 0.838986i \(0.683149\pi\)
\(702\) 0 0
\(703\) 5.56545 0.209905
\(704\) 29.3669 1.10681
\(705\) 0 0
\(706\) −25.2850 −0.951612
\(707\) −0.691702 −0.0260141
\(708\) 0 0
\(709\) −20.9862 −0.788154 −0.394077 0.919077i \(-0.628936\pi\)
−0.394077 + 0.919077i \(0.628936\pi\)
\(710\) 13.9773 0.524560
\(711\) 0 0
\(712\) 87.6609 3.28523
\(713\) 23.7412 0.889117
\(714\) 0 0
\(715\) 25.4928 0.953375
\(716\) −94.7896 −3.54245
\(717\) 0 0
\(718\) −75.9255 −2.83351
\(719\) 4.40085 0.164124 0.0820620 0.996627i \(-0.473849\pi\)
0.0820620 + 0.996627i \(0.473849\pi\)
\(720\) 0 0
\(721\) −16.5225 −0.615328
\(722\) −36.4081 −1.35497
\(723\) 0 0
\(724\) 18.2799 0.679368
\(725\) −42.9686 −1.59581
\(726\) 0 0
\(727\) 21.6396 0.802568 0.401284 0.915954i \(-0.368564\pi\)
0.401284 + 0.915954i \(0.368564\pi\)
\(728\) −35.4939 −1.31549
\(729\) 0 0
\(730\) 4.37350 0.161871
\(731\) 5.16937 0.191196
\(732\) 0 0
\(733\) 30.9010 1.14136 0.570678 0.821174i \(-0.306680\pi\)
0.570678 + 0.821174i \(0.306680\pi\)
\(734\) −26.8291 −0.990279
\(735\) 0 0
\(736\) −10.6369 −0.392082
\(737\) 73.5712 2.71003
\(738\) 0 0
\(739\) −32.2746 −1.18724 −0.593620 0.804745i \(-0.702302\pi\)
−0.593620 + 0.804745i \(0.702302\pi\)
\(740\) −8.71312 −0.320301
\(741\) 0 0
\(742\) −8.64855 −0.317498
\(743\) −28.5122 −1.04601 −0.523006 0.852329i \(-0.675190\pi\)
−0.523006 + 0.852329i \(0.675190\pi\)
\(744\) 0 0
\(745\) −18.5079 −0.678076
\(746\) 25.2977 0.926213
\(747\) 0 0
\(748\) 25.4636 0.931040
\(749\) 19.1211 0.698671
\(750\) 0 0
\(751\) −24.0517 −0.877659 −0.438829 0.898570i \(-0.644607\pi\)
−0.438829 + 0.898570i \(0.644607\pi\)
\(752\) 32.0650 1.16929
\(753\) 0 0
\(754\) −161.722 −5.88956
\(755\) −5.04544 −0.183623
\(756\) 0 0
\(757\) −33.1952 −1.20650 −0.603249 0.797553i \(-0.706128\pi\)
−0.603249 + 0.797553i \(0.706128\pi\)
\(758\) 16.3242 0.592920
\(759\) 0 0
\(760\) −8.68908 −0.315186
\(761\) −27.5176 −0.997511 −0.498755 0.866743i \(-0.666209\pi\)
−0.498755 + 0.866743i \(0.666209\pi\)
\(762\) 0 0
\(763\) 14.7383 0.533563
\(764\) −85.3167 −3.08665
\(765\) 0 0
\(766\) 50.2031 1.81391
\(767\) 81.3748 2.93828
\(768\) 0 0
\(769\) −14.6074 −0.526757 −0.263378 0.964693i \(-0.584837\pi\)
−0.263378 + 0.964693i \(0.584837\pi\)
\(770\) −9.49933 −0.342332
\(771\) 0 0
\(772\) −75.0046 −2.69948
\(773\) −4.97622 −0.178982 −0.0894912 0.995988i \(-0.528524\pi\)
−0.0894912 + 0.995988i \(0.528524\pi\)
\(774\) 0 0
\(775\) 15.1556 0.544405
\(776\) 90.1111 3.23480
\(777\) 0 0
\(778\) −20.1032 −0.720736
\(779\) −8.72837 −0.312726
\(780\) 0 0
\(781\) −35.1265 −1.25693
\(782\) 21.3928 0.765007
\(783\) 0 0
\(784\) 4.92520 0.175900
\(785\) −11.3662 −0.405679
\(786\) 0 0
\(787\) 20.1321 0.717633 0.358816 0.933408i \(-0.383180\pi\)
0.358816 + 0.933408i \(0.383180\pi\)
\(788\) 102.561 3.65358
\(789\) 0 0
\(790\) −14.3621 −0.510981
\(791\) −12.5353 −0.445705
\(792\) 0 0
\(793\) −40.8553 −1.45081
\(794\) −39.5184 −1.40246
\(795\) 0 0
\(796\) 69.1476 2.45087
\(797\) 35.3654 1.25271 0.626354 0.779539i \(-0.284546\pi\)
0.626354 + 0.779539i \(0.284546\pi\)
\(798\) 0 0
\(799\) −8.17490 −0.289207
\(800\) −6.79025 −0.240071
\(801\) 0 0
\(802\) −89.2733 −3.15235
\(803\) −10.9911 −0.387867
\(804\) 0 0
\(805\) −5.38555 −0.189816
\(806\) 57.0415 2.00920
\(807\) 0 0
\(808\) −3.68890 −0.129775
\(809\) −9.88279 −0.347460 −0.173730 0.984793i \(-0.555582\pi\)
−0.173730 + 0.984793i \(0.555582\pi\)
\(810\) 0 0
\(811\) 29.3646 1.03113 0.515566 0.856850i \(-0.327582\pi\)
0.515566 + 0.856850i \(0.327582\pi\)
\(812\) 40.6661 1.42710
\(813\) 0 0
\(814\) 32.4487 1.13733
\(815\) 6.32164 0.221437
\(816\) 0 0
\(817\) 8.55601 0.299337
\(818\) −52.8466 −1.84774
\(819\) 0 0
\(820\) 13.6649 0.477199
\(821\) 11.5855 0.404338 0.202169 0.979351i \(-0.435201\pi\)
0.202169 + 0.979351i \(0.435201\pi\)
\(822\) 0 0
\(823\) 32.4979 1.13281 0.566404 0.824128i \(-0.308334\pi\)
0.566404 + 0.824128i \(0.308334\pi\)
\(824\) −88.1154 −3.06965
\(825\) 0 0
\(826\) −30.3226 −1.05506
\(827\) 34.8327 1.21125 0.605626 0.795749i \(-0.292923\pi\)
0.605626 + 0.795749i \(0.292923\pi\)
\(828\) 0 0
\(829\) 25.0563 0.870241 0.435120 0.900372i \(-0.356706\pi\)
0.435120 + 0.900372i \(0.356706\pi\)
\(830\) 7.44474 0.258411
\(831\) 0 0
\(832\) 40.0020 1.38682
\(833\) −1.25567 −0.0435063
\(834\) 0 0
\(835\) −3.61035 −0.124941
\(836\) 42.1457 1.45764
\(837\) 0 0
\(838\) 66.5632 2.29939
\(839\) 16.0782 0.555082 0.277541 0.960714i \(-0.410481\pi\)
0.277541 + 0.960714i \(0.410481\pi\)
\(840\) 0 0
\(841\) 67.0017 2.31040
\(842\) 4.24855 0.146415
\(843\) 0 0
\(844\) 60.4599 2.08112
\(845\) 24.5336 0.843980
\(846\) 0 0
\(847\) 12.8729 0.442316
\(848\) −17.1757 −0.589816
\(849\) 0 0
\(850\) 13.6565 0.468413
\(851\) 18.3965 0.630622
\(852\) 0 0
\(853\) 9.71995 0.332805 0.166402 0.986058i \(-0.446785\pi\)
0.166402 + 0.986058i \(0.446785\pi\)
\(854\) 15.2238 0.520949
\(855\) 0 0
\(856\) 101.974 3.48541
\(857\) −41.2846 −1.41025 −0.705127 0.709081i \(-0.749110\pi\)
−0.705127 + 0.709081i \(0.749110\pi\)
\(858\) 0 0
\(859\) −1.05236 −0.0359061 −0.0179531 0.999839i \(-0.505715\pi\)
−0.0179531 + 0.999839i \(0.505715\pi\)
\(860\) −13.3951 −0.456768
\(861\) 0 0
\(862\) −74.0325 −2.52156
\(863\) −34.4783 −1.17365 −0.586827 0.809712i \(-0.699623\pi\)
−0.586827 + 0.809712i \(0.699623\pi\)
\(864\) 0 0
\(865\) 9.01693 0.306585
\(866\) −16.1963 −0.550374
\(867\) 0 0
\(868\) −14.3435 −0.486850
\(869\) 36.0935 1.22439
\(870\) 0 0
\(871\) 100.215 3.39565
\(872\) 78.6005 2.66175
\(873\) 0 0
\(874\) 35.4081 1.19770
\(875\) −7.35770 −0.248736
\(876\) 0 0
\(877\) 41.2483 1.39285 0.696427 0.717627i \(-0.254772\pi\)
0.696427 + 0.717627i \(0.254772\pi\)
\(878\) −37.0206 −1.24939
\(879\) 0 0
\(880\) −18.8653 −0.635950
\(881\) −16.3701 −0.551523 −0.275761 0.961226i \(-0.588930\pi\)
−0.275761 + 0.961226i \(0.588930\pi\)
\(882\) 0 0
\(883\) 39.4843 1.32875 0.664376 0.747398i \(-0.268697\pi\)
0.664376 + 0.747398i \(0.268697\pi\)
\(884\) 34.6851 1.16659
\(885\) 0 0
\(886\) 11.4929 0.386110
\(887\) 14.0226 0.470831 0.235416 0.971895i \(-0.424355\pi\)
0.235416 + 0.971895i \(0.424355\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 31.9572 1.07121
\(891\) 0 0
\(892\) −40.5193 −1.35669
\(893\) −13.5306 −0.452783
\(894\) 0 0
\(895\) −17.9042 −0.598473
\(896\) −18.0026 −0.601426
\(897\) 0 0
\(898\) 16.4568 0.549171
\(899\) −33.8611 −1.12933
\(900\) 0 0
\(901\) 4.37890 0.145882
\(902\) −50.8897 −1.69444
\(903\) 0 0
\(904\) −66.8518 −2.22346
\(905\) 3.45278 0.114774
\(906\) 0 0
\(907\) −35.6241 −1.18288 −0.591440 0.806349i \(-0.701440\pi\)
−0.591440 + 0.806349i \(0.701440\pi\)
\(908\) 108.221 3.59145
\(909\) 0 0
\(910\) −12.9395 −0.428940
\(911\) −4.45318 −0.147541 −0.0737703 0.997275i \(-0.523503\pi\)
−0.0737703 + 0.997275i \(0.523503\pi\)
\(912\) 0 0
\(913\) −18.7094 −0.619192
\(914\) −43.8529 −1.45053
\(915\) 0 0
\(916\) 25.9284 0.856698
\(917\) 0.687639 0.0227078
\(918\) 0 0
\(919\) 38.4633 1.26879 0.634394 0.773010i \(-0.281250\pi\)
0.634394 + 0.773010i \(0.281250\pi\)
\(920\) −28.7215 −0.946921
\(921\) 0 0
\(922\) 99.8618 3.28877
\(923\) −47.8475 −1.57492
\(924\) 0 0
\(925\) 11.7437 0.386130
\(926\) −9.63572 −0.316649
\(927\) 0 0
\(928\) 15.1710 0.498012
\(929\) −41.8359 −1.37259 −0.686295 0.727323i \(-0.740764\pi\)
−0.686295 + 0.727323i \(0.740764\pi\)
\(930\) 0 0
\(931\) −2.07830 −0.0681135
\(932\) −28.1320 −0.921493
\(933\) 0 0
\(934\) 3.29665 0.107870
\(935\) 4.80966 0.157293
\(936\) 0 0
\(937\) 27.1190 0.885938 0.442969 0.896537i \(-0.353925\pi\)
0.442969 + 0.896537i \(0.353925\pi\)
\(938\) −37.3429 −1.21929
\(939\) 0 0
\(940\) 21.1831 0.690916
\(941\) 28.1483 0.917608 0.458804 0.888538i \(-0.348278\pi\)
0.458804 + 0.888538i \(0.348278\pi\)
\(942\) 0 0
\(943\) −28.8514 −0.939531
\(944\) −60.2195 −1.95998
\(945\) 0 0
\(946\) 49.8848 1.62189
\(947\) −26.7498 −0.869251 −0.434625 0.900611i \(-0.643119\pi\)
−0.434625 + 0.900611i \(0.643119\pi\)
\(948\) 0 0
\(949\) −14.9715 −0.485995
\(950\) 22.6033 0.733349
\(951\) 0 0
\(952\) −6.69656 −0.217037
\(953\) 59.4329 1.92522 0.962611 0.270888i \(-0.0873174\pi\)
0.962611 + 0.270888i \(0.0873174\pi\)
\(954\) 0 0
\(955\) −16.1150 −0.521468
\(956\) 98.3923 3.18223
\(957\) 0 0
\(958\) 27.1943 0.878607
\(959\) 7.96226 0.257115
\(960\) 0 0
\(961\) −19.0567 −0.614733
\(962\) 44.1999 1.42506
\(963\) 0 0
\(964\) −38.5340 −1.24110
\(965\) −14.1672 −0.456058
\(966\) 0 0
\(967\) −4.26909 −0.137285 −0.0686424 0.997641i \(-0.521867\pi\)
−0.0686424 + 0.997641i \(0.521867\pi\)
\(968\) 68.6519 2.20655
\(969\) 0 0
\(970\) 32.8505 1.05477
\(971\) −53.2969 −1.71038 −0.855190 0.518315i \(-0.826559\pi\)
−0.855190 + 0.518315i \(0.826559\pi\)
\(972\) 0 0
\(973\) −0.410908 −0.0131731
\(974\) 18.4106 0.589915
\(975\) 0 0
\(976\) 30.2340 0.967766
\(977\) −19.3371 −0.618647 −0.309324 0.950957i \(-0.600103\pi\)
−0.309324 + 0.950957i \(0.600103\pi\)
\(978\) 0 0
\(979\) −80.3120 −2.56678
\(980\) 3.25373 0.103937
\(981\) 0 0
\(982\) −25.9761 −0.828930
\(983\) 38.4570 1.22659 0.613294 0.789855i \(-0.289844\pi\)
0.613294 + 0.789855i \(0.289844\pi\)
\(984\) 0 0
\(985\) 19.3721 0.617248
\(986\) −30.5117 −0.971690
\(987\) 0 0
\(988\) 57.4087 1.82641
\(989\) 28.2817 0.899305
\(990\) 0 0
\(991\) 25.2038 0.800626 0.400313 0.916379i \(-0.368901\pi\)
0.400313 + 0.916379i \(0.368901\pi\)
\(992\) −5.35101 −0.169895
\(993\) 0 0
\(994\) 17.8294 0.565513
\(995\) 13.0609 0.414058
\(996\) 0 0
\(997\) 42.0562 1.33193 0.665966 0.745982i \(-0.268019\pi\)
0.665966 + 0.745982i \(0.268019\pi\)
\(998\) 66.1651 2.09442
\(999\) 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.1.7 7
3.2 odd 2 2667.2.a.j.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.j.1.1 7 3.2 odd 2
8001.2.a.l.1.7 7 1.1 even 1 trivial