Properties

Label 8001.2.a.z.1.21
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $32$
CM no
Inner twists $2$

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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: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.21
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.942656 q^{2} -1.11140 q^{4} +2.02754 q^{5} -1.00000 q^{7} -2.93298 q^{8} +O(q^{10})\) \(q+0.942656 q^{2} -1.11140 q^{4} +2.02754 q^{5} -1.00000 q^{7} -2.93298 q^{8} +1.91127 q^{10} +0.393608 q^{11} -1.52460 q^{13} -0.942656 q^{14} -0.541991 q^{16} -7.22498 q^{17} +5.33043 q^{19} -2.25341 q^{20} +0.371037 q^{22} +6.14596 q^{23} -0.889072 q^{25} -1.43717 q^{26} +1.11140 q^{28} +10.3543 q^{29} -5.51081 q^{31} +5.35505 q^{32} -6.81067 q^{34} -2.02754 q^{35} +1.11637 q^{37} +5.02476 q^{38} -5.94674 q^{40} -12.0128 q^{41} -0.215108 q^{43} -0.437456 q^{44} +5.79352 q^{46} -4.05988 q^{47} +1.00000 q^{49} -0.838089 q^{50} +1.69443 q^{52} +6.05109 q^{53} +0.798057 q^{55} +2.93298 q^{56} +9.76057 q^{58} +3.92402 q^{59} -3.48704 q^{61} -5.19480 q^{62} +6.13195 q^{64} -3.09118 q^{65} +9.31796 q^{67} +8.02984 q^{68} -1.91127 q^{70} -6.78717 q^{71} -11.2119 q^{73} +1.05235 q^{74} -5.92424 q^{76} -0.393608 q^{77} -5.16136 q^{79} -1.09891 q^{80} -11.3239 q^{82} -6.24052 q^{83} -14.6490 q^{85} -0.202772 q^{86} -1.15444 q^{88} -13.9448 q^{89} +1.52460 q^{91} -6.83062 q^{92} -3.82707 q^{94} +10.8077 q^{95} +10.5474 q^{97} +0.942656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.942656 0.666558 0.333279 0.942828i \(-0.391845\pi\)
0.333279 + 0.942828i \(0.391845\pi\)
\(3\) 0 0
\(4\) −1.11140 −0.555700
\(5\) 2.02754 0.906745 0.453372 0.891321i \(-0.350221\pi\)
0.453372 + 0.891321i \(0.350221\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.93298 −1.03696
\(9\) 0 0
\(10\) 1.91127 0.604398
\(11\) 0.393608 0.118677 0.0593387 0.998238i \(-0.481101\pi\)
0.0593387 + 0.998238i \(0.481101\pi\)
\(12\) 0 0
\(13\) −1.52460 −0.422847 −0.211423 0.977395i \(-0.567810\pi\)
−0.211423 + 0.977395i \(0.567810\pi\)
\(14\) −0.942656 −0.251935
\(15\) 0 0
\(16\) −0.541991 −0.135498
\(17\) −7.22498 −1.75232 −0.876158 0.482025i \(-0.839902\pi\)
−0.876158 + 0.482025i \(0.839902\pi\)
\(18\) 0 0
\(19\) 5.33043 1.22288 0.611442 0.791289i \(-0.290590\pi\)
0.611442 + 0.791289i \(0.290590\pi\)
\(20\) −2.25341 −0.503878
\(21\) 0 0
\(22\) 0.371037 0.0791054
\(23\) 6.14596 1.28152 0.640760 0.767741i \(-0.278619\pi\)
0.640760 + 0.767741i \(0.278619\pi\)
\(24\) 0 0
\(25\) −0.889072 −0.177814
\(26\) −1.43717 −0.281852
\(27\) 0 0
\(28\) 1.11140 0.210035
\(29\) 10.3543 1.92275 0.961376 0.275239i \(-0.0887570\pi\)
0.961376 + 0.275239i \(0.0887570\pi\)
\(30\) 0 0
\(31\) −5.51081 −0.989771 −0.494885 0.868958i \(-0.664790\pi\)
−0.494885 + 0.868958i \(0.664790\pi\)
\(32\) 5.35505 0.946648
\(33\) 0 0
\(34\) −6.81067 −1.16802
\(35\) −2.02754 −0.342717
\(36\) 0 0
\(37\) 1.11637 0.183529 0.0917647 0.995781i \(-0.470749\pi\)
0.0917647 + 0.995781i \(0.470749\pi\)
\(38\) 5.02476 0.815124
\(39\) 0 0
\(40\) −5.94674 −0.940262
\(41\) −12.0128 −1.87609 −0.938043 0.346519i \(-0.887364\pi\)
−0.938043 + 0.346519i \(0.887364\pi\)
\(42\) 0 0
\(43\) −0.215108 −0.0328036 −0.0164018 0.999865i \(-0.505221\pi\)
−0.0164018 + 0.999865i \(0.505221\pi\)
\(44\) −0.437456 −0.0659490
\(45\) 0 0
\(46\) 5.79352 0.854208
\(47\) −4.05988 −0.592194 −0.296097 0.955158i \(-0.595685\pi\)
−0.296097 + 0.955158i \(0.595685\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −0.838089 −0.118524
\(51\) 0 0
\(52\) 1.69443 0.234976
\(53\) 6.05109 0.831181 0.415591 0.909552i \(-0.363575\pi\)
0.415591 + 0.909552i \(0.363575\pi\)
\(54\) 0 0
\(55\) 0.798057 0.107610
\(56\) 2.93298 0.391936
\(57\) 0 0
\(58\) 9.76057 1.28163
\(59\) 3.92402 0.510864 0.255432 0.966827i \(-0.417782\pi\)
0.255432 + 0.966827i \(0.417782\pi\)
\(60\) 0 0
\(61\) −3.48704 −0.446469 −0.223235 0.974765i \(-0.571662\pi\)
−0.223235 + 0.974765i \(0.571662\pi\)
\(62\) −5.19480 −0.659740
\(63\) 0 0
\(64\) 6.13195 0.766494
\(65\) −3.09118 −0.383414
\(66\) 0 0
\(67\) 9.31796 1.13837 0.569185 0.822209i \(-0.307259\pi\)
0.569185 + 0.822209i \(0.307259\pi\)
\(68\) 8.02984 0.973761
\(69\) 0 0
\(70\) −1.91127 −0.228441
\(71\) −6.78717 −0.805489 −0.402744 0.915313i \(-0.631944\pi\)
−0.402744 + 0.915313i \(0.631944\pi\)
\(72\) 0 0
\(73\) −11.2119 −1.31225 −0.656125 0.754652i \(-0.727806\pi\)
−0.656125 + 0.754652i \(0.727806\pi\)
\(74\) 1.05235 0.122333
\(75\) 0 0
\(76\) −5.92424 −0.679557
\(77\) −0.393608 −0.0448558
\(78\) 0 0
\(79\) −5.16136 −0.580698 −0.290349 0.956921i \(-0.593771\pi\)
−0.290349 + 0.956921i \(0.593771\pi\)
\(80\) −1.09891 −0.122862
\(81\) 0 0
\(82\) −11.3239 −1.25052
\(83\) −6.24052 −0.684986 −0.342493 0.939520i \(-0.611271\pi\)
−0.342493 + 0.939520i \(0.611271\pi\)
\(84\) 0 0
\(85\) −14.6490 −1.58890
\(86\) −0.202772 −0.0218655
\(87\) 0 0
\(88\) −1.15444 −0.123064
\(89\) −13.9448 −1.47815 −0.739074 0.673624i \(-0.764737\pi\)
−0.739074 + 0.673624i \(0.764737\pi\)
\(90\) 0 0
\(91\) 1.52460 0.159821
\(92\) −6.83062 −0.712141
\(93\) 0 0
\(94\) −3.82707 −0.394732
\(95\) 10.8077 1.10884
\(96\) 0 0
\(97\) 10.5474 1.07093 0.535465 0.844557i \(-0.320136\pi\)
0.535465 + 0.844557i \(0.320136\pi\)
\(98\) 0.942656 0.0952226
\(99\) 0 0
\(100\) 0.988114 0.0988114
\(101\) −15.7128 −1.56348 −0.781739 0.623605i \(-0.785667\pi\)
−0.781739 + 0.623605i \(0.785667\pi\)
\(102\) 0 0
\(103\) 4.27528 0.421256 0.210628 0.977566i \(-0.432449\pi\)
0.210628 + 0.977566i \(0.432449\pi\)
\(104\) 4.47161 0.438477
\(105\) 0 0
\(106\) 5.70410 0.554031
\(107\) −8.30037 −0.802427 −0.401214 0.915985i \(-0.631412\pi\)
−0.401214 + 0.915985i \(0.631412\pi\)
\(108\) 0 0
\(109\) 5.24166 0.502060 0.251030 0.967979i \(-0.419231\pi\)
0.251030 + 0.967979i \(0.419231\pi\)
\(110\) 0.752293 0.0717284
\(111\) 0 0
\(112\) 0.541991 0.0512133
\(113\) −1.08075 −0.101669 −0.0508344 0.998707i \(-0.516188\pi\)
−0.0508344 + 0.998707i \(0.516188\pi\)
\(114\) 0 0
\(115\) 12.4612 1.16201
\(116\) −11.5078 −1.06847
\(117\) 0 0
\(118\) 3.69900 0.340521
\(119\) 7.22498 0.662313
\(120\) 0 0
\(121\) −10.8451 −0.985916
\(122\) −3.28708 −0.297598
\(123\) 0 0
\(124\) 6.12472 0.550016
\(125\) −11.9403 −1.06798
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −4.92978 −0.435735
\(129\) 0 0
\(130\) −2.91392 −0.255568
\(131\) −7.36931 −0.643860 −0.321930 0.946764i \(-0.604332\pi\)
−0.321930 + 0.946764i \(0.604332\pi\)
\(132\) 0 0
\(133\) −5.33043 −0.462207
\(134\) 8.78363 0.758790
\(135\) 0 0
\(136\) 21.1907 1.81709
\(137\) −2.37680 −0.203063 −0.101532 0.994832i \(-0.532374\pi\)
−0.101532 + 0.994832i \(0.532374\pi\)
\(138\) 0 0
\(139\) −14.1222 −1.19783 −0.598916 0.800812i \(-0.704402\pi\)
−0.598916 + 0.800812i \(0.704402\pi\)
\(140\) 2.25341 0.190448
\(141\) 0 0
\(142\) −6.39796 −0.536905
\(143\) −0.600093 −0.0501823
\(144\) 0 0
\(145\) 20.9939 1.74344
\(146\) −10.5689 −0.874691
\(147\) 0 0
\(148\) −1.24073 −0.101987
\(149\) 13.5063 1.10648 0.553240 0.833022i \(-0.313391\pi\)
0.553240 + 0.833022i \(0.313391\pi\)
\(150\) 0 0
\(151\) 21.6796 1.76426 0.882129 0.471007i \(-0.156109\pi\)
0.882129 + 0.471007i \(0.156109\pi\)
\(152\) −15.6340 −1.26809
\(153\) 0 0
\(154\) −0.371037 −0.0298990
\(155\) −11.1734 −0.897469
\(156\) 0 0
\(157\) −12.2097 −0.974438 −0.487219 0.873280i \(-0.661989\pi\)
−0.487219 + 0.873280i \(0.661989\pi\)
\(158\) −4.86538 −0.387069
\(159\) 0 0
\(160\) 10.8576 0.858368
\(161\) −6.14596 −0.484369
\(162\) 0 0
\(163\) −11.4746 −0.898759 −0.449379 0.893341i \(-0.648355\pi\)
−0.449379 + 0.893341i \(0.648355\pi\)
\(164\) 13.3510 1.04254
\(165\) 0 0
\(166\) −5.88266 −0.456583
\(167\) −9.48483 −0.733958 −0.366979 0.930229i \(-0.619608\pi\)
−0.366979 + 0.930229i \(0.619608\pi\)
\(168\) 0 0
\(169\) −10.6756 −0.821201
\(170\) −13.8089 −1.05910
\(171\) 0 0
\(172\) 0.239071 0.0182290
\(173\) −21.7904 −1.65669 −0.828346 0.560216i \(-0.810718\pi\)
−0.828346 + 0.560216i \(0.810718\pi\)
\(174\) 0 0
\(175\) 0.889072 0.0672075
\(176\) −0.213332 −0.0160805
\(177\) 0 0
\(178\) −13.1452 −0.985272
\(179\) 4.32122 0.322983 0.161492 0.986874i \(-0.448370\pi\)
0.161492 + 0.986874i \(0.448370\pi\)
\(180\) 0 0
\(181\) 9.17405 0.681902 0.340951 0.940081i \(-0.389251\pi\)
0.340951 + 0.940081i \(0.389251\pi\)
\(182\) 1.43717 0.106530
\(183\) 0 0
\(184\) −18.0260 −1.32889
\(185\) 2.26348 0.166414
\(186\) 0 0
\(187\) −2.84381 −0.207960
\(188\) 4.51215 0.329082
\(189\) 0 0
\(190\) 10.1879 0.739109
\(191\) −14.7115 −1.06449 −0.532244 0.846591i \(-0.678651\pi\)
−0.532244 + 0.846591i \(0.678651\pi\)
\(192\) 0 0
\(193\) −12.2579 −0.882345 −0.441173 0.897422i \(-0.645437\pi\)
−0.441173 + 0.897422i \(0.645437\pi\)
\(194\) 9.94261 0.713838
\(195\) 0 0
\(196\) −1.11140 −0.0793857
\(197\) −3.36875 −0.240013 −0.120007 0.992773i \(-0.538292\pi\)
−0.120007 + 0.992773i \(0.538292\pi\)
\(198\) 0 0
\(199\) −20.0623 −1.42218 −0.711089 0.703102i \(-0.751798\pi\)
−0.711089 + 0.703102i \(0.751798\pi\)
\(200\) 2.60763 0.184387
\(201\) 0 0
\(202\) −14.8117 −1.04215
\(203\) −10.3543 −0.726732
\(204\) 0 0
\(205\) −24.3565 −1.70113
\(206\) 4.03012 0.280792
\(207\) 0 0
\(208\) 0.826316 0.0572947
\(209\) 2.09810 0.145129
\(210\) 0 0
\(211\) −26.1999 −1.80368 −0.901839 0.432071i \(-0.857783\pi\)
−0.901839 + 0.432071i \(0.857783\pi\)
\(212\) −6.72518 −0.461887
\(213\) 0 0
\(214\) −7.82440 −0.534865
\(215\) −0.436140 −0.0297445
\(216\) 0 0
\(217\) 5.51081 0.374098
\(218\) 4.94108 0.334652
\(219\) 0 0
\(220\) −0.886961 −0.0597989
\(221\) 11.0152 0.740961
\(222\) 0 0
\(223\) −7.28885 −0.488098 −0.244049 0.969763i \(-0.578476\pi\)
−0.244049 + 0.969763i \(0.578476\pi\)
\(224\) −5.35505 −0.357799
\(225\) 0 0
\(226\) −1.01878 −0.0677681
\(227\) 25.9000 1.71904 0.859522 0.511098i \(-0.170761\pi\)
0.859522 + 0.511098i \(0.170761\pi\)
\(228\) 0 0
\(229\) 29.9555 1.97951 0.989757 0.142764i \(-0.0455990\pi\)
0.989757 + 0.142764i \(0.0455990\pi\)
\(230\) 11.7466 0.774549
\(231\) 0 0
\(232\) −30.3691 −1.99383
\(233\) 12.2443 0.802152 0.401076 0.916045i \(-0.368636\pi\)
0.401076 + 0.916045i \(0.368636\pi\)
\(234\) 0 0
\(235\) −8.23157 −0.536969
\(236\) −4.36116 −0.283887
\(237\) 0 0
\(238\) 6.81067 0.441470
\(239\) −0.293187 −0.0189647 −0.00948234 0.999955i \(-0.503018\pi\)
−0.00948234 + 0.999955i \(0.503018\pi\)
\(240\) 0 0
\(241\) −17.4415 −1.12351 −0.561753 0.827305i \(-0.689873\pi\)
−0.561753 + 0.827305i \(0.689873\pi\)
\(242\) −10.2232 −0.657170
\(243\) 0 0
\(244\) 3.87549 0.248103
\(245\) 2.02754 0.129535
\(246\) 0 0
\(247\) −8.12675 −0.517093
\(248\) 16.1631 1.02636
\(249\) 0 0
\(250\) −11.2556 −0.711869
\(251\) −0.315392 −0.0199074 −0.00995368 0.999950i \(-0.503168\pi\)
−0.00995368 + 0.999950i \(0.503168\pi\)
\(252\) 0 0
\(253\) 2.41910 0.152087
\(254\) −0.942656 −0.0591475
\(255\) 0 0
\(256\) −16.9110 −1.05694
\(257\) 6.88363 0.429389 0.214695 0.976681i \(-0.431124\pi\)
0.214695 + 0.976681i \(0.431124\pi\)
\(258\) 0 0
\(259\) −1.11637 −0.0693676
\(260\) 3.43554 0.213063
\(261\) 0 0
\(262\) −6.94672 −0.429170
\(263\) 17.9942 1.10957 0.554784 0.831994i \(-0.312801\pi\)
0.554784 + 0.831994i \(0.312801\pi\)
\(264\) 0 0
\(265\) 12.2688 0.753669
\(266\) −5.02476 −0.308088
\(267\) 0 0
\(268\) −10.3560 −0.632592
\(269\) 6.25866 0.381597 0.190799 0.981629i \(-0.438892\pi\)
0.190799 + 0.981629i \(0.438892\pi\)
\(270\) 0 0
\(271\) −15.0536 −0.914443 −0.457222 0.889353i \(-0.651155\pi\)
−0.457222 + 0.889353i \(0.651155\pi\)
\(272\) 3.91587 0.237435
\(273\) 0 0
\(274\) −2.24050 −0.135354
\(275\) −0.349946 −0.0211025
\(276\) 0 0
\(277\) −14.9805 −0.900090 −0.450045 0.893006i \(-0.648592\pi\)
−0.450045 + 0.893006i \(0.648592\pi\)
\(278\) −13.3124 −0.798425
\(279\) 0 0
\(280\) 5.94674 0.355386
\(281\) 28.4947 1.69985 0.849925 0.526903i \(-0.176647\pi\)
0.849925 + 0.526903i \(0.176647\pi\)
\(282\) 0 0
\(283\) 31.7506 1.88737 0.943687 0.330839i \(-0.107332\pi\)
0.943687 + 0.330839i \(0.107332\pi\)
\(284\) 7.54326 0.447610
\(285\) 0 0
\(286\) −0.565681 −0.0334494
\(287\) 12.0128 0.709094
\(288\) 0 0
\(289\) 35.2003 2.07061
\(290\) 19.7900 1.16211
\(291\) 0 0
\(292\) 12.4609 0.729217
\(293\) −26.7569 −1.56316 −0.781578 0.623807i \(-0.785585\pi\)
−0.781578 + 0.623807i \(0.785585\pi\)
\(294\) 0 0
\(295\) 7.95613 0.463224
\(296\) −3.27428 −0.190314
\(297\) 0 0
\(298\) 12.7318 0.737533
\(299\) −9.37010 −0.541887
\(300\) 0 0
\(301\) 0.215108 0.0123986
\(302\) 20.4364 1.17598
\(303\) 0 0
\(304\) −2.88904 −0.165698
\(305\) −7.07011 −0.404834
\(306\) 0 0
\(307\) 12.9334 0.738147 0.369073 0.929400i \(-0.379675\pi\)
0.369073 + 0.929400i \(0.379675\pi\)
\(308\) 0.437456 0.0249264
\(309\) 0 0
\(310\) −10.5327 −0.598216
\(311\) 33.8815 1.92124 0.960620 0.277864i \(-0.0896264\pi\)
0.960620 + 0.277864i \(0.0896264\pi\)
\(312\) 0 0
\(313\) −29.8784 −1.68883 −0.844413 0.535692i \(-0.820051\pi\)
−0.844413 + 0.535692i \(0.820051\pi\)
\(314\) −11.5095 −0.649520
\(315\) 0 0
\(316\) 5.73633 0.322694
\(317\) 12.1570 0.682804 0.341402 0.939917i \(-0.389098\pi\)
0.341402 + 0.939917i \(0.389098\pi\)
\(318\) 0 0
\(319\) 4.07555 0.228187
\(320\) 12.4328 0.695014
\(321\) 0 0
\(322\) −5.79352 −0.322860
\(323\) −38.5123 −2.14288
\(324\) 0 0
\(325\) 1.35547 0.0751882
\(326\) −10.8166 −0.599075
\(327\) 0 0
\(328\) 35.2333 1.94544
\(329\) 4.05988 0.223828
\(330\) 0 0
\(331\) −3.00611 −0.165231 −0.0826155 0.996581i \(-0.526327\pi\)
−0.0826155 + 0.996581i \(0.526327\pi\)
\(332\) 6.93571 0.380647
\(333\) 0 0
\(334\) −8.94093 −0.489226
\(335\) 18.8926 1.03221
\(336\) 0 0
\(337\) −16.6283 −0.905799 −0.452899 0.891562i \(-0.649610\pi\)
−0.452899 + 0.891562i \(0.649610\pi\)
\(338\) −10.0634 −0.547378
\(339\) 0 0
\(340\) 16.2808 0.882953
\(341\) −2.16910 −0.117463
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0.630906 0.0340162
\(345\) 0 0
\(346\) −20.5408 −1.10428
\(347\) −22.0246 −1.18234 −0.591172 0.806546i \(-0.701334\pi\)
−0.591172 + 0.806546i \(0.701334\pi\)
\(348\) 0 0
\(349\) 5.78704 0.309773 0.154887 0.987932i \(-0.450499\pi\)
0.154887 + 0.987932i \(0.450499\pi\)
\(350\) 0.838089 0.0447977
\(351\) 0 0
\(352\) 2.10779 0.112346
\(353\) 19.2731 1.02580 0.512902 0.858447i \(-0.328571\pi\)
0.512902 + 0.858447i \(0.328571\pi\)
\(354\) 0 0
\(355\) −13.7613 −0.730372
\(356\) 15.4983 0.821407
\(357\) 0 0
\(358\) 4.07343 0.215287
\(359\) −6.30280 −0.332649 −0.166325 0.986071i \(-0.553190\pi\)
−0.166325 + 0.986071i \(0.553190\pi\)
\(360\) 0 0
\(361\) 9.41349 0.495447
\(362\) 8.64798 0.454528
\(363\) 0 0
\(364\) −1.69443 −0.0888125
\(365\) −22.7325 −1.18988
\(366\) 0 0
\(367\) −15.8154 −0.825559 −0.412780 0.910831i \(-0.635442\pi\)
−0.412780 + 0.910831i \(0.635442\pi\)
\(368\) −3.33105 −0.173643
\(369\) 0 0
\(370\) 2.13368 0.110925
\(371\) −6.05109 −0.314157
\(372\) 0 0
\(373\) −28.5380 −1.47764 −0.738821 0.673902i \(-0.764617\pi\)
−0.738821 + 0.673902i \(0.764617\pi\)
\(374\) −2.68074 −0.138618
\(375\) 0 0
\(376\) 11.9075 0.614084
\(377\) −15.7862 −0.813029
\(378\) 0 0
\(379\) −33.4675 −1.71911 −0.859554 0.511045i \(-0.829258\pi\)
−0.859554 + 0.511045i \(0.829258\pi\)
\(380\) −12.0116 −0.616185
\(381\) 0 0
\(382\) −13.8679 −0.709543
\(383\) 22.2341 1.13611 0.568055 0.822990i \(-0.307696\pi\)
0.568055 + 0.822990i \(0.307696\pi\)
\(384\) 0 0
\(385\) −0.798057 −0.0406728
\(386\) −11.5550 −0.588135
\(387\) 0 0
\(388\) −11.7224 −0.595116
\(389\) −19.0317 −0.964943 −0.482472 0.875912i \(-0.660261\pi\)
−0.482472 + 0.875912i \(0.660261\pi\)
\(390\) 0 0
\(391\) −44.4044 −2.24563
\(392\) −2.93298 −0.148138
\(393\) 0 0
\(394\) −3.17557 −0.159983
\(395\) −10.4649 −0.526545
\(396\) 0 0
\(397\) −6.12683 −0.307497 −0.153748 0.988110i \(-0.549135\pi\)
−0.153748 + 0.988110i \(0.549135\pi\)
\(398\) −18.9118 −0.947964
\(399\) 0 0
\(400\) 0.481868 0.0240934
\(401\) 7.88878 0.393947 0.196974 0.980409i \(-0.436889\pi\)
0.196974 + 0.980409i \(0.436889\pi\)
\(402\) 0 0
\(403\) 8.40176 0.418521
\(404\) 17.4632 0.868825
\(405\) 0 0
\(406\) −9.76057 −0.484409
\(407\) 0.439411 0.0217808
\(408\) 0 0
\(409\) 34.4698 1.70442 0.852211 0.523198i \(-0.175261\pi\)
0.852211 + 0.523198i \(0.175261\pi\)
\(410\) −22.9598 −1.13390
\(411\) 0 0
\(412\) −4.75155 −0.234092
\(413\) −3.92402 −0.193089
\(414\) 0 0
\(415\) −12.6529 −0.621107
\(416\) −8.16428 −0.400287
\(417\) 0 0
\(418\) 1.97779 0.0967367
\(419\) −24.4461 −1.19427 −0.597135 0.802141i \(-0.703694\pi\)
−0.597135 + 0.802141i \(0.703694\pi\)
\(420\) 0 0
\(421\) 29.1674 1.42153 0.710767 0.703428i \(-0.248348\pi\)
0.710767 + 0.703428i \(0.248348\pi\)
\(422\) −24.6975 −1.20226
\(423\) 0 0
\(424\) −17.7477 −0.861906
\(425\) 6.42353 0.311587
\(426\) 0 0
\(427\) 3.48704 0.168750
\(428\) 9.22503 0.445909
\(429\) 0 0
\(430\) −0.411130 −0.0198264
\(431\) 8.12744 0.391485 0.195742 0.980655i \(-0.437288\pi\)
0.195742 + 0.980655i \(0.437288\pi\)
\(432\) 0 0
\(433\) −12.8060 −0.615419 −0.307709 0.951480i \(-0.599562\pi\)
−0.307709 + 0.951480i \(0.599562\pi\)
\(434\) 5.19480 0.249358
\(435\) 0 0
\(436\) −5.82558 −0.278995
\(437\) 32.7606 1.56715
\(438\) 0 0
\(439\) −18.2637 −0.871679 −0.435840 0.900024i \(-0.643549\pi\)
−0.435840 + 0.900024i \(0.643549\pi\)
\(440\) −2.34069 −0.111588
\(441\) 0 0
\(442\) 10.3835 0.493893
\(443\) −6.35839 −0.302096 −0.151048 0.988526i \(-0.548265\pi\)
−0.151048 + 0.988526i \(0.548265\pi\)
\(444\) 0 0
\(445\) −28.2737 −1.34030
\(446\) −6.87088 −0.325346
\(447\) 0 0
\(448\) −6.13195 −0.289707
\(449\) −37.4797 −1.76878 −0.884388 0.466753i \(-0.845424\pi\)
−0.884388 + 0.466753i \(0.845424\pi\)
\(450\) 0 0
\(451\) −4.72834 −0.222649
\(452\) 1.20115 0.0564973
\(453\) 0 0
\(454\) 24.4148 1.14584
\(455\) 3.09118 0.144917
\(456\) 0 0
\(457\) 38.1237 1.78335 0.891677 0.452673i \(-0.149529\pi\)
0.891677 + 0.452673i \(0.149529\pi\)
\(458\) 28.2377 1.31946
\(459\) 0 0
\(460\) −13.8494 −0.645730
\(461\) 4.34534 0.202382 0.101191 0.994867i \(-0.467735\pi\)
0.101191 + 0.994867i \(0.467735\pi\)
\(462\) 0 0
\(463\) 32.5836 1.51429 0.757144 0.653248i \(-0.226594\pi\)
0.757144 + 0.653248i \(0.226594\pi\)
\(464\) −5.61195 −0.260528
\(465\) 0 0
\(466\) 11.5422 0.534681
\(467\) −7.91842 −0.366421 −0.183210 0.983074i \(-0.558649\pi\)
−0.183210 + 0.983074i \(0.558649\pi\)
\(468\) 0 0
\(469\) −9.31796 −0.430263
\(470\) −7.75954 −0.357921
\(471\) 0 0
\(472\) −11.5091 −0.529748
\(473\) −0.0846681 −0.00389304
\(474\) 0 0
\(475\) −4.73914 −0.217446
\(476\) −8.02984 −0.368047
\(477\) 0 0
\(478\) −0.276374 −0.0126411
\(479\) 28.9770 1.32399 0.661996 0.749508i \(-0.269710\pi\)
0.661996 + 0.749508i \(0.269710\pi\)
\(480\) 0 0
\(481\) −1.70201 −0.0776048
\(482\) −16.4413 −0.748883
\(483\) 0 0
\(484\) 12.0532 0.547873
\(485\) 21.3854 0.971061
\(486\) 0 0
\(487\) −9.20818 −0.417262 −0.208631 0.977994i \(-0.566901\pi\)
−0.208631 + 0.977994i \(0.566901\pi\)
\(488\) 10.2274 0.462973
\(489\) 0 0
\(490\) 1.91127 0.0863426
\(491\) −36.3971 −1.64258 −0.821288 0.570513i \(-0.806744\pi\)
−0.821288 + 0.570513i \(0.806744\pi\)
\(492\) 0 0
\(493\) −74.8099 −3.36927
\(494\) −7.66073 −0.344672
\(495\) 0 0
\(496\) 2.98681 0.134112
\(497\) 6.78717 0.304446
\(498\) 0 0
\(499\) 8.80459 0.394148 0.197074 0.980389i \(-0.436856\pi\)
0.197074 + 0.980389i \(0.436856\pi\)
\(500\) 13.2705 0.593475
\(501\) 0 0
\(502\) −0.297306 −0.0132694
\(503\) 30.0843 1.34139 0.670696 0.741732i \(-0.265996\pi\)
0.670696 + 0.741732i \(0.265996\pi\)
\(504\) 0 0
\(505\) −31.8583 −1.41768
\(506\) 2.28038 0.101375
\(507\) 0 0
\(508\) 1.11140 0.0493104
\(509\) 14.6613 0.649849 0.324924 0.945740i \(-0.394661\pi\)
0.324924 + 0.945740i \(0.394661\pi\)
\(510\) 0 0
\(511\) 11.2119 0.495984
\(512\) −6.08168 −0.268775
\(513\) 0 0
\(514\) 6.48890 0.286213
\(515\) 8.66831 0.381972
\(516\) 0 0
\(517\) −1.59800 −0.0702800
\(518\) −1.05235 −0.0462376
\(519\) 0 0
\(520\) 9.06637 0.397587
\(521\) −23.4040 −1.02535 −0.512674 0.858583i \(-0.671345\pi\)
−0.512674 + 0.858583i \(0.671345\pi\)
\(522\) 0 0
\(523\) −1.58294 −0.0692173 −0.0346087 0.999401i \(-0.511018\pi\)
−0.0346087 + 0.999401i \(0.511018\pi\)
\(524\) 8.19025 0.357793
\(525\) 0 0
\(526\) 16.9623 0.739592
\(527\) 39.8155 1.73439
\(528\) 0 0
\(529\) 14.7728 0.642296
\(530\) 11.5653 0.502364
\(531\) 0 0
\(532\) 5.92424 0.256848
\(533\) 18.3147 0.793297
\(534\) 0 0
\(535\) −16.8294 −0.727597
\(536\) −27.3294 −1.18045
\(537\) 0 0
\(538\) 5.89976 0.254357
\(539\) 0.393608 0.0169539
\(540\) 0 0
\(541\) −16.8950 −0.726372 −0.363186 0.931717i \(-0.618311\pi\)
−0.363186 + 0.931717i \(0.618311\pi\)
\(542\) −14.1904 −0.609530
\(543\) 0 0
\(544\) −38.6901 −1.65883
\(545\) 10.6277 0.455240
\(546\) 0 0
\(547\) −28.0686 −1.20013 −0.600064 0.799952i \(-0.704858\pi\)
−0.600064 + 0.799952i \(0.704858\pi\)
\(548\) 2.64157 0.112842
\(549\) 0 0
\(550\) −0.329879 −0.0140661
\(551\) 55.1931 2.35130
\(552\) 0 0
\(553\) 5.16136 0.219483
\(554\) −14.1214 −0.599963
\(555\) 0 0
\(556\) 15.6954 0.665635
\(557\) 30.4192 1.28890 0.644452 0.764645i \(-0.277086\pi\)
0.644452 + 0.764645i \(0.277086\pi\)
\(558\) 0 0
\(559\) 0.327952 0.0138709
\(560\) 1.09891 0.0464374
\(561\) 0 0
\(562\) 26.8607 1.13305
\(563\) −38.0568 −1.60390 −0.801952 0.597388i \(-0.796205\pi\)
−0.801952 + 0.597388i \(0.796205\pi\)
\(564\) 0 0
\(565\) −2.19127 −0.0921876
\(566\) 29.9298 1.25805
\(567\) 0 0
\(568\) 19.9066 0.835263
\(569\) −8.39927 −0.352116 −0.176058 0.984380i \(-0.556335\pi\)
−0.176058 + 0.984380i \(0.556335\pi\)
\(570\) 0 0
\(571\) −33.2531 −1.39160 −0.695799 0.718236i \(-0.744950\pi\)
−0.695799 + 0.718236i \(0.744950\pi\)
\(572\) 0.666943 0.0278863
\(573\) 0 0
\(574\) 11.3239 0.472652
\(575\) −5.46420 −0.227873
\(576\) 0 0
\(577\) −18.8513 −0.784790 −0.392395 0.919797i \(-0.628353\pi\)
−0.392395 + 0.919797i \(0.628353\pi\)
\(578\) 33.1818 1.38018
\(579\) 0 0
\(580\) −23.3326 −0.968832
\(581\) 6.24052 0.258900
\(582\) 0 0
\(583\) 2.38176 0.0986423
\(584\) 32.8842 1.36076
\(585\) 0 0
\(586\) −25.2226 −1.04194
\(587\) −28.6206 −1.18130 −0.590648 0.806929i \(-0.701128\pi\)
−0.590648 + 0.806929i \(0.701128\pi\)
\(588\) 0 0
\(589\) −29.3750 −1.21038
\(590\) 7.49989 0.308766
\(591\) 0 0
\(592\) −0.605060 −0.0248678
\(593\) −13.7191 −0.563377 −0.281689 0.959506i \(-0.590895\pi\)
−0.281689 + 0.959506i \(0.590895\pi\)
\(594\) 0 0
\(595\) 14.6490 0.600549
\(596\) −15.0109 −0.614870
\(597\) 0 0
\(598\) −8.83278 −0.361199
\(599\) 40.0016 1.63442 0.817211 0.576338i \(-0.195519\pi\)
0.817211 + 0.576338i \(0.195519\pi\)
\(600\) 0 0
\(601\) −10.8800 −0.443806 −0.221903 0.975069i \(-0.571227\pi\)
−0.221903 + 0.975069i \(0.571227\pi\)
\(602\) 0.202772 0.00826439
\(603\) 0 0
\(604\) −24.0947 −0.980398
\(605\) −21.9888 −0.893974
\(606\) 0 0
\(607\) −20.1563 −0.818119 −0.409060 0.912508i \(-0.634143\pi\)
−0.409060 + 0.912508i \(0.634143\pi\)
\(608\) 28.5447 1.15764
\(609\) 0 0
\(610\) −6.66468 −0.269845
\(611\) 6.18967 0.250407
\(612\) 0 0
\(613\) 24.4960 0.989382 0.494691 0.869069i \(-0.335281\pi\)
0.494691 + 0.869069i \(0.335281\pi\)
\(614\) 12.1917 0.492018
\(615\) 0 0
\(616\) 1.15444 0.0465139
\(617\) −33.8571 −1.36304 −0.681518 0.731801i \(-0.738680\pi\)
−0.681518 + 0.731801i \(0.738680\pi\)
\(618\) 0 0
\(619\) −47.1828 −1.89644 −0.948219 0.317619i \(-0.897117\pi\)
−0.948219 + 0.317619i \(0.897117\pi\)
\(620\) 12.4181 0.498724
\(621\) 0 0
\(622\) 31.9386 1.28062
\(623\) 13.9448 0.558688
\(624\) 0 0
\(625\) −19.7642 −0.790568
\(626\) −28.1650 −1.12570
\(627\) 0 0
\(628\) 13.5698 0.541495
\(629\) −8.06572 −0.321601
\(630\) 0 0
\(631\) −6.13033 −0.244045 −0.122022 0.992527i \(-0.538938\pi\)
−0.122022 + 0.992527i \(0.538938\pi\)
\(632\) 15.1381 0.602163
\(633\) 0 0
\(634\) 11.4599 0.455129
\(635\) −2.02754 −0.0804606
\(636\) 0 0
\(637\) −1.52460 −0.0604067
\(638\) 3.84184 0.152100
\(639\) 0 0
\(640\) −9.99534 −0.395100
\(641\) −15.4511 −0.610281 −0.305140 0.952307i \(-0.598703\pi\)
−0.305140 + 0.952307i \(0.598703\pi\)
\(642\) 0 0
\(643\) 28.9559 1.14191 0.570955 0.820982i \(-0.306573\pi\)
0.570955 + 0.820982i \(0.306573\pi\)
\(644\) 6.83062 0.269164
\(645\) 0 0
\(646\) −36.3038 −1.42835
\(647\) 17.7543 0.697993 0.348996 0.937124i \(-0.386523\pi\)
0.348996 + 0.937124i \(0.386523\pi\)
\(648\) 0 0
\(649\) 1.54453 0.0606280
\(650\) 1.27775 0.0501173
\(651\) 0 0
\(652\) 12.7529 0.499440
\(653\) 47.4601 1.85726 0.928628 0.371013i \(-0.120989\pi\)
0.928628 + 0.371013i \(0.120989\pi\)
\(654\) 0 0
\(655\) −14.9416 −0.583816
\(656\) 6.51083 0.254205
\(657\) 0 0
\(658\) 3.82707 0.149195
\(659\) −4.97190 −0.193678 −0.0968388 0.995300i \(-0.530873\pi\)
−0.0968388 + 0.995300i \(0.530873\pi\)
\(660\) 0 0
\(661\) −5.29780 −0.206060 −0.103030 0.994678i \(-0.532854\pi\)
−0.103030 + 0.994678i \(0.532854\pi\)
\(662\) −2.83373 −0.110136
\(663\) 0 0
\(664\) 18.3033 0.710306
\(665\) −10.8077 −0.419104
\(666\) 0 0
\(667\) 63.6373 2.46405
\(668\) 10.5414 0.407860
\(669\) 0 0
\(670\) 17.8092 0.688029
\(671\) −1.37253 −0.0529858
\(672\) 0 0
\(673\) −30.4834 −1.17505 −0.587524 0.809206i \(-0.699897\pi\)
−0.587524 + 0.809206i \(0.699897\pi\)
\(674\) −15.6747 −0.603768
\(675\) 0 0
\(676\) 11.8649 0.456341
\(677\) −1.37014 −0.0526588 −0.0263294 0.999653i \(-0.508382\pi\)
−0.0263294 + 0.999653i \(0.508382\pi\)
\(678\) 0 0
\(679\) −10.5474 −0.404774
\(680\) 42.9651 1.64764
\(681\) 0 0
\(682\) −2.04472 −0.0782962
\(683\) 39.0704 1.49499 0.747494 0.664268i \(-0.231257\pi\)
0.747494 + 0.664268i \(0.231257\pi\)
\(684\) 0 0
\(685\) −4.81905 −0.184127
\(686\) −0.942656 −0.0359908
\(687\) 0 0
\(688\) 0.116586 0.00444481
\(689\) −9.22546 −0.351462
\(690\) 0 0
\(691\) 39.0221 1.48447 0.742235 0.670140i \(-0.233766\pi\)
0.742235 + 0.670140i \(0.233766\pi\)
\(692\) 24.2178 0.920624
\(693\) 0 0
\(694\) −20.7616 −0.788101
\(695\) −28.6334 −1.08613
\(696\) 0 0
\(697\) 86.7923 3.28749
\(698\) 5.45519 0.206482
\(699\) 0 0
\(700\) −0.988114 −0.0373472
\(701\) −5.33026 −0.201321 −0.100661 0.994921i \(-0.532096\pi\)
−0.100661 + 0.994921i \(0.532096\pi\)
\(702\) 0 0
\(703\) 5.95071 0.224435
\(704\) 2.41359 0.0909654
\(705\) 0 0
\(706\) 18.1679 0.683758
\(707\) 15.7128 0.590939
\(708\) 0 0
\(709\) 8.31243 0.312180 0.156090 0.987743i \(-0.450111\pi\)
0.156090 + 0.987743i \(0.450111\pi\)
\(710\) −12.9721 −0.486836
\(711\) 0 0
\(712\) 40.8999 1.53279
\(713\) −33.8692 −1.26841
\(714\) 0 0
\(715\) −1.21671 −0.0455025
\(716\) −4.80261 −0.179482
\(717\) 0 0
\(718\) −5.94137 −0.221730
\(719\) −15.6948 −0.585319 −0.292659 0.956217i \(-0.594540\pi\)
−0.292659 + 0.956217i \(0.594540\pi\)
\(720\) 0 0
\(721\) −4.27528 −0.159220
\(722\) 8.87368 0.330244
\(723\) 0 0
\(724\) −10.1960 −0.378933
\(725\) −9.20575 −0.341893
\(726\) 0 0
\(727\) 24.8055 0.919985 0.459993 0.887923i \(-0.347852\pi\)
0.459993 + 0.887923i \(0.347852\pi\)
\(728\) −4.47161 −0.165729
\(729\) 0 0
\(730\) −21.4290 −0.793121
\(731\) 1.55415 0.0574822
\(732\) 0 0
\(733\) −40.1964 −1.48469 −0.742344 0.670018i \(-0.766286\pi\)
−0.742344 + 0.670018i \(0.766286\pi\)
\(734\) −14.9085 −0.550283
\(735\) 0 0
\(736\) 32.9119 1.21315
\(737\) 3.66763 0.135099
\(738\) 0 0
\(739\) −8.54978 −0.314509 −0.157254 0.987558i \(-0.550264\pi\)
−0.157254 + 0.987558i \(0.550264\pi\)
\(740\) −2.51563 −0.0924764
\(741\) 0 0
\(742\) −5.70410 −0.209404
\(743\) −16.6201 −0.609732 −0.304866 0.952395i \(-0.598612\pi\)
−0.304866 + 0.952395i \(0.598612\pi\)
\(744\) 0 0
\(745\) 27.3846 1.00329
\(746\) −26.9015 −0.984934
\(747\) 0 0
\(748\) 3.16061 0.115563
\(749\) 8.30037 0.303289
\(750\) 0 0
\(751\) −24.5495 −0.895825 −0.447912 0.894077i \(-0.647832\pi\)
−0.447912 + 0.894077i \(0.647832\pi\)
\(752\) 2.20041 0.0802409
\(753\) 0 0
\(754\) −14.8809 −0.541931
\(755\) 43.9563 1.59973
\(756\) 0 0
\(757\) −11.8571 −0.430952 −0.215476 0.976509i \(-0.569130\pi\)
−0.215476 + 0.976509i \(0.569130\pi\)
\(758\) −31.5483 −1.14589
\(759\) 0 0
\(760\) −31.6987 −1.14983
\(761\) 48.1062 1.74385 0.871924 0.489641i \(-0.162872\pi\)
0.871924 + 0.489641i \(0.162872\pi\)
\(762\) 0 0
\(763\) −5.24166 −0.189761
\(764\) 16.3504 0.591536
\(765\) 0 0
\(766\) 20.9591 0.757284
\(767\) −5.98255 −0.216017
\(768\) 0 0
\(769\) −33.7773 −1.21804 −0.609021 0.793154i \(-0.708437\pi\)
−0.609021 + 0.793154i \(0.708437\pi\)
\(770\) −0.752293 −0.0271108
\(771\) 0 0
\(772\) 13.6235 0.490319
\(773\) 22.2790 0.801321 0.400661 0.916227i \(-0.368781\pi\)
0.400661 + 0.916227i \(0.368781\pi\)
\(774\) 0 0
\(775\) 4.89951 0.175995
\(776\) −30.9354 −1.11052
\(777\) 0 0
\(778\) −17.9403 −0.643191
\(779\) −64.0335 −2.29424
\(780\) 0 0
\(781\) −2.67148 −0.0955932
\(782\) −41.8581 −1.49684
\(783\) 0 0
\(784\) −0.541991 −0.0193568
\(785\) −24.7556 −0.883566
\(786\) 0 0
\(787\) −22.1520 −0.789634 −0.394817 0.918760i \(-0.629192\pi\)
−0.394817 + 0.918760i \(0.629192\pi\)
\(788\) 3.74403 0.133375
\(789\) 0 0
\(790\) −9.86477 −0.350973
\(791\) 1.08075 0.0384272
\(792\) 0 0
\(793\) 5.31632 0.188788
\(794\) −5.77549 −0.204965
\(795\) 0 0
\(796\) 22.2972 0.790304
\(797\) −6.03663 −0.213828 −0.106914 0.994268i \(-0.534097\pi\)
−0.106914 + 0.994268i \(0.534097\pi\)
\(798\) 0 0
\(799\) 29.3325 1.03771
\(800\) −4.76102 −0.168328
\(801\) 0 0
\(802\) 7.43641 0.262589
\(803\) −4.41308 −0.155734
\(804\) 0 0
\(805\) −12.4612 −0.439199
\(806\) 7.91997 0.278969
\(807\) 0 0
\(808\) 46.0852 1.62127
\(809\) 1.64082 0.0576880 0.0288440 0.999584i \(-0.490817\pi\)
0.0288440 + 0.999584i \(0.490817\pi\)
\(810\) 0 0
\(811\) 24.1361 0.847532 0.423766 0.905772i \(-0.360708\pi\)
0.423766 + 0.905772i \(0.360708\pi\)
\(812\) 11.5078 0.403845
\(813\) 0 0
\(814\) 0.414213 0.0145182
\(815\) −23.2652 −0.814945
\(816\) 0 0
\(817\) −1.14662 −0.0401150
\(818\) 32.4932 1.13610
\(819\) 0 0
\(820\) 27.0698 0.945318
\(821\) 8.59622 0.300010 0.150005 0.988685i \(-0.452071\pi\)
0.150005 + 0.988685i \(0.452071\pi\)
\(822\) 0 0
\(823\) 11.4797 0.400156 0.200078 0.979780i \(-0.435880\pi\)
0.200078 + 0.979780i \(0.435880\pi\)
\(824\) −12.5393 −0.436828
\(825\) 0 0
\(826\) −3.69900 −0.128705
\(827\) −12.1662 −0.423061 −0.211531 0.977371i \(-0.567845\pi\)
−0.211531 + 0.977371i \(0.567845\pi\)
\(828\) 0 0
\(829\) −18.9606 −0.658530 −0.329265 0.944237i \(-0.606801\pi\)
−0.329265 + 0.944237i \(0.606801\pi\)
\(830\) −11.9273 −0.414004
\(831\) 0 0
\(832\) −9.34874 −0.324109
\(833\) −7.22498 −0.250331
\(834\) 0 0
\(835\) −19.2309 −0.665512
\(836\) −2.33183 −0.0806480
\(837\) 0 0
\(838\) −23.0443 −0.796051
\(839\) −37.6366 −1.29936 −0.649680 0.760208i \(-0.725097\pi\)
−0.649680 + 0.760208i \(0.725097\pi\)
\(840\) 0 0
\(841\) 78.2122 2.69697
\(842\) 27.4949 0.947535
\(843\) 0 0
\(844\) 29.1186 1.00230
\(845\) −21.6453 −0.744619
\(846\) 0 0
\(847\) 10.8451 0.372641
\(848\) −3.27963 −0.112623
\(849\) 0 0
\(850\) 6.05517 0.207691
\(851\) 6.86114 0.235197
\(852\) 0 0
\(853\) 23.5742 0.807165 0.403583 0.914943i \(-0.367765\pi\)
0.403583 + 0.914943i \(0.367765\pi\)
\(854\) 3.28708 0.112481
\(855\) 0 0
\(856\) 24.3448 0.832089
\(857\) 46.2533 1.57998 0.789991 0.613118i \(-0.210085\pi\)
0.789991 + 0.613118i \(0.210085\pi\)
\(858\) 0 0
\(859\) −38.4044 −1.31034 −0.655171 0.755480i \(-0.727404\pi\)
−0.655171 + 0.755480i \(0.727404\pi\)
\(860\) 0.484726 0.0165290
\(861\) 0 0
\(862\) 7.66138 0.260948
\(863\) 40.9841 1.39511 0.697557 0.716529i \(-0.254270\pi\)
0.697557 + 0.716529i \(0.254270\pi\)
\(864\) 0 0
\(865\) −44.1809 −1.50220
\(866\) −12.0717 −0.410212
\(867\) 0 0
\(868\) −6.12472 −0.207886
\(869\) −2.03155 −0.0689157
\(870\) 0 0
\(871\) −14.2061 −0.481356
\(872\) −15.3737 −0.520618
\(873\) 0 0
\(874\) 30.8820 1.04460
\(875\) 11.9403 0.403657
\(876\) 0 0
\(877\) 11.1446 0.376325 0.188163 0.982138i \(-0.439747\pi\)
0.188163 + 0.982138i \(0.439747\pi\)
\(878\) −17.2164 −0.581025
\(879\) 0 0
\(880\) −0.432539 −0.0145809
\(881\) 50.5138 1.70185 0.850926 0.525285i \(-0.176041\pi\)
0.850926 + 0.525285i \(0.176041\pi\)
\(882\) 0 0
\(883\) 26.0412 0.876357 0.438179 0.898888i \(-0.355624\pi\)
0.438179 + 0.898888i \(0.355624\pi\)
\(884\) −12.2423 −0.411752
\(885\) 0 0
\(886\) −5.99378 −0.201365
\(887\) 31.8404 1.06910 0.534548 0.845138i \(-0.320482\pi\)
0.534548 + 0.845138i \(0.320482\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −26.6524 −0.893390
\(891\) 0 0
\(892\) 8.10083 0.271236
\(893\) −21.6409 −0.724185
\(894\) 0 0
\(895\) 8.76146 0.292863
\(896\) 4.92978 0.164692
\(897\) 0 0
\(898\) −35.3304 −1.17899
\(899\) −57.0608 −1.90308
\(900\) 0 0
\(901\) −43.7190 −1.45649
\(902\) −4.45720 −0.148408
\(903\) 0 0
\(904\) 3.16983 0.105427
\(905\) 18.6008 0.618311
\(906\) 0 0
\(907\) −32.2737 −1.07163 −0.535815 0.844335i \(-0.679996\pi\)
−0.535815 + 0.844335i \(0.679996\pi\)
\(908\) −28.7853 −0.955273
\(909\) 0 0
\(910\) 2.91392 0.0965955
\(911\) 34.4345 1.14087 0.570433 0.821344i \(-0.306776\pi\)
0.570433 + 0.821344i \(0.306776\pi\)
\(912\) 0 0
\(913\) −2.45632 −0.0812923
\(914\) 35.9376 1.18871
\(915\) 0 0
\(916\) −33.2925 −1.10002
\(917\) 7.36931 0.243356
\(918\) 0 0
\(919\) 5.27192 0.173904 0.0869522 0.996212i \(-0.472287\pi\)
0.0869522 + 0.996212i \(0.472287\pi\)
\(920\) −36.5484 −1.20497
\(921\) 0 0
\(922\) 4.09616 0.134900
\(923\) 10.3477 0.340598
\(924\) 0 0
\(925\) −0.992530 −0.0326342
\(926\) 30.7151 1.00936
\(927\) 0 0
\(928\) 55.4480 1.82017
\(929\) −7.44226 −0.244173 −0.122086 0.992519i \(-0.538958\pi\)
−0.122086 + 0.992519i \(0.538958\pi\)
\(930\) 0 0
\(931\) 5.33043 0.174698
\(932\) −13.6083 −0.445756
\(933\) 0 0
\(934\) −7.46435 −0.244241
\(935\) −5.76595 −0.188567
\(936\) 0 0
\(937\) 47.8635 1.56363 0.781816 0.623510i \(-0.214294\pi\)
0.781816 + 0.623510i \(0.214294\pi\)
\(938\) −8.78363 −0.286796
\(939\) 0 0
\(940\) 9.14857 0.298393
\(941\) −34.0814 −1.11102 −0.555511 0.831509i \(-0.687477\pi\)
−0.555511 + 0.831509i \(0.687477\pi\)
\(942\) 0 0
\(943\) −73.8302 −2.40424
\(944\) −2.12678 −0.0692209
\(945\) 0 0
\(946\) −0.0798129 −0.00259494
\(947\) 2.35915 0.0766620 0.0383310 0.999265i \(-0.487796\pi\)
0.0383310 + 0.999265i \(0.487796\pi\)
\(948\) 0 0
\(949\) 17.0936 0.554880
\(950\) −4.46737 −0.144941
\(951\) 0 0
\(952\) −21.1907 −0.686795
\(953\) 38.2987 1.24062 0.620308 0.784358i \(-0.287007\pi\)
0.620308 + 0.784358i \(0.287007\pi\)
\(954\) 0 0
\(955\) −29.8282 −0.965218
\(956\) 0.325848 0.0105387
\(957\) 0 0
\(958\) 27.3153 0.882517
\(959\) 2.37680 0.0767507
\(960\) 0 0
\(961\) −0.630958 −0.0203535
\(962\) −1.60441 −0.0517281
\(963\) 0 0
\(964\) 19.3845 0.624333
\(965\) −24.8535 −0.800062
\(966\) 0 0
\(967\) 26.3209 0.846423 0.423211 0.906031i \(-0.360903\pi\)
0.423211 + 0.906031i \(0.360903\pi\)
\(968\) 31.8084 1.02236
\(969\) 0 0
\(970\) 20.1591 0.647269
\(971\) 28.2912 0.907908 0.453954 0.891025i \(-0.350013\pi\)
0.453954 + 0.891025i \(0.350013\pi\)
\(972\) 0 0
\(973\) 14.1222 0.452738
\(974\) −8.68014 −0.278130
\(975\) 0 0
\(976\) 1.88994 0.0604955
\(977\) 12.7823 0.408943 0.204471 0.978873i \(-0.434452\pi\)
0.204471 + 0.978873i \(0.434452\pi\)
\(978\) 0 0
\(979\) −5.48880 −0.175423
\(980\) −2.25341 −0.0719826
\(981\) 0 0
\(982\) −34.3099 −1.09487
\(983\) −23.7874 −0.758701 −0.379350 0.925253i \(-0.623852\pi\)
−0.379350 + 0.925253i \(0.623852\pi\)
\(984\) 0 0
\(985\) −6.83028 −0.217631
\(986\) −70.5200 −2.24581
\(987\) 0 0
\(988\) 9.03207 0.287348
\(989\) −1.32204 −0.0420385
\(990\) 0 0
\(991\) −13.9078 −0.441794 −0.220897 0.975297i \(-0.570899\pi\)
−0.220897 + 0.975297i \(0.570899\pi\)
\(992\) −29.5107 −0.936964
\(993\) 0 0
\(994\) 6.39796 0.202931
\(995\) −40.6771 −1.28955
\(996\) 0 0
\(997\) −29.1698 −0.923816 −0.461908 0.886928i \(-0.652835\pi\)
−0.461908 + 0.886928i \(0.652835\pi\)
\(998\) 8.29970 0.262722
\(999\) 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.z.1.21 yes 32
3.2 odd 2 inner 8001.2.a.z.1.12 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.12 32 3.2 odd 2 inner
8001.2.a.z.1.21 yes 32 1.1 even 1 trivial