Properties

Label 8001.2.a.z.1.23
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 1.23
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.18013 q^{2} -0.607291 q^{4} -4.43826 q^{5} -1.00000 q^{7} -3.07694 q^{8} +O(q^{10})\) \(q+1.18013 q^{2} -0.607291 q^{4} -4.43826 q^{5} -1.00000 q^{7} -3.07694 q^{8} -5.23773 q^{10} +5.41915 q^{11} -2.38471 q^{13} -1.18013 q^{14} -2.41662 q^{16} -2.36152 q^{17} -2.75497 q^{19} +2.69531 q^{20} +6.39530 q^{22} +5.73685 q^{23} +14.6981 q^{25} -2.81427 q^{26} +0.607291 q^{28} +3.90830 q^{29} +2.83738 q^{31} +3.30196 q^{32} -2.78691 q^{34} +4.43826 q^{35} -3.58729 q^{37} -3.25123 q^{38} +13.6563 q^{40} +0.707118 q^{41} +9.07783 q^{43} -3.29100 q^{44} +6.77024 q^{46} -4.92884 q^{47} +1.00000 q^{49} +17.3457 q^{50} +1.44821 q^{52} -10.8388 q^{53} -24.0516 q^{55} +3.07694 q^{56} +4.61231 q^{58} -8.70245 q^{59} +15.4770 q^{61} +3.34848 q^{62} +8.72998 q^{64} +10.5839 q^{65} +5.37268 q^{67} +1.43413 q^{68} +5.23773 q^{70} +8.79079 q^{71} -14.5845 q^{73} -4.23348 q^{74} +1.67307 q^{76} -5.41915 q^{77} -13.5647 q^{79} +10.7256 q^{80} +0.834492 q^{82} +5.85553 q^{83} +10.4810 q^{85} +10.7130 q^{86} -16.6744 q^{88} -5.77576 q^{89} +2.38471 q^{91} -3.48394 q^{92} -5.81668 q^{94} +12.2273 q^{95} +13.8577 q^{97} +1.18013 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.18013 0.834479 0.417239 0.908797i \(-0.362998\pi\)
0.417239 + 0.908797i \(0.362998\pi\)
\(3\) 0 0
\(4\) −0.607291 −0.303645
\(5\) −4.43826 −1.98485 −0.992425 0.122856i \(-0.960795\pi\)
−0.992425 + 0.122856i \(0.960795\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −3.07694 −1.08786
\(9\) 0 0
\(10\) −5.23773 −1.65631
\(11\) 5.41915 1.63393 0.816967 0.576685i \(-0.195654\pi\)
0.816967 + 0.576685i \(0.195654\pi\)
\(12\) 0 0
\(13\) −2.38471 −0.661398 −0.330699 0.943736i \(-0.607285\pi\)
−0.330699 + 0.943736i \(0.607285\pi\)
\(14\) −1.18013 −0.315403
\(15\) 0 0
\(16\) −2.41662 −0.604154
\(17\) −2.36152 −0.572753 −0.286377 0.958117i \(-0.592451\pi\)
−0.286377 + 0.958117i \(0.592451\pi\)
\(18\) 0 0
\(19\) −2.75497 −0.632034 −0.316017 0.948753i \(-0.602346\pi\)
−0.316017 + 0.948753i \(0.602346\pi\)
\(20\) 2.69531 0.602690
\(21\) 0 0
\(22\) 6.39530 1.36348
\(23\) 5.73685 1.19622 0.598108 0.801416i \(-0.295919\pi\)
0.598108 + 0.801416i \(0.295919\pi\)
\(24\) 0 0
\(25\) 14.6981 2.93963
\(26\) −2.81427 −0.551923
\(27\) 0 0
\(28\) 0.607291 0.114767
\(29\) 3.90830 0.725754 0.362877 0.931837i \(-0.381795\pi\)
0.362877 + 0.931837i \(0.381795\pi\)
\(30\) 0 0
\(31\) 2.83738 0.509609 0.254804 0.966993i \(-0.417989\pi\)
0.254804 + 0.966993i \(0.417989\pi\)
\(32\) 3.30196 0.583710
\(33\) 0 0
\(34\) −2.78691 −0.477950
\(35\) 4.43826 0.750202
\(36\) 0 0
\(37\) −3.58729 −0.589747 −0.294874 0.955536i \(-0.595278\pi\)
−0.294874 + 0.955536i \(0.595278\pi\)
\(38\) −3.25123 −0.527419
\(39\) 0 0
\(40\) 13.6563 2.15925
\(41\) 0.707118 0.110433 0.0552166 0.998474i \(-0.482415\pi\)
0.0552166 + 0.998474i \(0.482415\pi\)
\(42\) 0 0
\(43\) 9.07783 1.38436 0.692178 0.721727i \(-0.256651\pi\)
0.692178 + 0.721727i \(0.256651\pi\)
\(44\) −3.29100 −0.496136
\(45\) 0 0
\(46\) 6.77024 0.998217
\(47\) −4.92884 −0.718945 −0.359473 0.933156i \(-0.617043\pi\)
−0.359473 + 0.933156i \(0.617043\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 17.3457 2.45306
\(51\) 0 0
\(52\) 1.44821 0.200830
\(53\) −10.8388 −1.48883 −0.744414 0.667719i \(-0.767271\pi\)
−0.744414 + 0.667719i \(0.767271\pi\)
\(54\) 0 0
\(55\) −24.0516 −3.24311
\(56\) 3.07694 0.411174
\(57\) 0 0
\(58\) 4.61231 0.605626
\(59\) −8.70245 −1.13296 −0.566481 0.824075i \(-0.691696\pi\)
−0.566481 + 0.824075i \(0.691696\pi\)
\(60\) 0 0
\(61\) 15.4770 1.98163 0.990816 0.135214i \(-0.0431723\pi\)
0.990816 + 0.135214i \(0.0431723\pi\)
\(62\) 3.34848 0.425258
\(63\) 0 0
\(64\) 8.72998 1.09125
\(65\) 10.5839 1.31278
\(66\) 0 0
\(67\) 5.37268 0.656377 0.328189 0.944612i \(-0.393562\pi\)
0.328189 + 0.944612i \(0.393562\pi\)
\(68\) 1.43413 0.173914
\(69\) 0 0
\(70\) 5.23773 0.626028
\(71\) 8.79079 1.04327 0.521637 0.853167i \(-0.325321\pi\)
0.521637 + 0.853167i \(0.325321\pi\)
\(72\) 0 0
\(73\) −14.5845 −1.70698 −0.853492 0.521107i \(-0.825519\pi\)
−0.853492 + 0.521107i \(0.825519\pi\)
\(74\) −4.23348 −0.492132
\(75\) 0 0
\(76\) 1.67307 0.191914
\(77\) −5.41915 −0.617569
\(78\) 0 0
\(79\) −13.5647 −1.52615 −0.763077 0.646308i \(-0.776312\pi\)
−0.763077 + 0.646308i \(0.776312\pi\)
\(80\) 10.7256 1.19915
\(81\) 0 0
\(82\) 0.834492 0.0921542
\(83\) 5.85553 0.642728 0.321364 0.946956i \(-0.395859\pi\)
0.321364 + 0.946956i \(0.395859\pi\)
\(84\) 0 0
\(85\) 10.4810 1.13683
\(86\) 10.7130 1.15522
\(87\) 0 0
\(88\) −16.6744 −1.77750
\(89\) −5.77576 −0.612229 −0.306114 0.951995i \(-0.599029\pi\)
−0.306114 + 0.951995i \(0.599029\pi\)
\(90\) 0 0
\(91\) 2.38471 0.249985
\(92\) −3.48394 −0.363225
\(93\) 0 0
\(94\) −5.81668 −0.599944
\(95\) 12.2273 1.25449
\(96\) 0 0
\(97\) 13.8577 1.40703 0.703517 0.710678i \(-0.251612\pi\)
0.703517 + 0.710678i \(0.251612\pi\)
\(98\) 1.18013 0.119211
\(99\) 0 0
\(100\) −8.92604 −0.892604
\(101\) 1.80702 0.179806 0.0899028 0.995951i \(-0.471344\pi\)
0.0899028 + 0.995951i \(0.471344\pi\)
\(102\) 0 0
\(103\) −10.8246 −1.06658 −0.533288 0.845934i \(-0.679044\pi\)
−0.533288 + 0.845934i \(0.679044\pi\)
\(104\) 7.33761 0.719512
\(105\) 0 0
\(106\) −12.7912 −1.24239
\(107\) 12.8202 1.23937 0.619686 0.784849i \(-0.287260\pi\)
0.619686 + 0.784849i \(0.287260\pi\)
\(108\) 0 0
\(109\) −9.99649 −0.957490 −0.478745 0.877954i \(-0.658908\pi\)
−0.478745 + 0.877954i \(0.658908\pi\)
\(110\) −28.3840 −2.70631
\(111\) 0 0
\(112\) 2.41662 0.228349
\(113\) 8.33535 0.784124 0.392062 0.919939i \(-0.371762\pi\)
0.392062 + 0.919939i \(0.371762\pi\)
\(114\) 0 0
\(115\) −25.4616 −2.37431
\(116\) −2.37348 −0.220372
\(117\) 0 0
\(118\) −10.2700 −0.945433
\(119\) 2.36152 0.216480
\(120\) 0 0
\(121\) 18.3671 1.66974
\(122\) 18.2649 1.65363
\(123\) 0 0
\(124\) −1.72311 −0.154740
\(125\) −43.0428 −3.84986
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 3.69860 0.326913
\(129\) 0 0
\(130\) 12.4904 1.09548
\(131\) 9.14569 0.799062 0.399531 0.916720i \(-0.369173\pi\)
0.399531 + 0.916720i \(0.369173\pi\)
\(132\) 0 0
\(133\) 2.75497 0.238886
\(134\) 6.34047 0.547733
\(135\) 0 0
\(136\) 7.26627 0.623078
\(137\) −13.9083 −1.18826 −0.594132 0.804368i \(-0.702504\pi\)
−0.594132 + 0.804368i \(0.702504\pi\)
\(138\) 0 0
\(139\) −17.0334 −1.44476 −0.722379 0.691497i \(-0.756951\pi\)
−0.722379 + 0.691497i \(0.756951\pi\)
\(140\) −2.69531 −0.227795
\(141\) 0 0
\(142\) 10.3743 0.870590
\(143\) −12.9231 −1.08068
\(144\) 0 0
\(145\) −17.3461 −1.44051
\(146\) −17.2116 −1.42444
\(147\) 0 0
\(148\) 2.17853 0.179074
\(149\) −12.3798 −1.01419 −0.507097 0.861889i \(-0.669281\pi\)
−0.507097 + 0.861889i \(0.669281\pi\)
\(150\) 0 0
\(151\) −12.4328 −1.01176 −0.505882 0.862603i \(-0.668833\pi\)
−0.505882 + 0.862603i \(0.668833\pi\)
\(152\) 8.47690 0.687567
\(153\) 0 0
\(154\) −6.39530 −0.515348
\(155\) −12.5930 −1.01150
\(156\) 0 0
\(157\) −19.3845 −1.54705 −0.773524 0.633767i \(-0.781508\pi\)
−0.773524 + 0.633767i \(0.781508\pi\)
\(158\) −16.0082 −1.27354
\(159\) 0 0
\(160\) −14.6550 −1.15858
\(161\) −5.73685 −0.452127
\(162\) 0 0
\(163\) −3.46010 −0.271016 −0.135508 0.990776i \(-0.543267\pi\)
−0.135508 + 0.990776i \(0.543267\pi\)
\(164\) −0.429426 −0.0335325
\(165\) 0 0
\(166\) 6.91029 0.536343
\(167\) 19.5272 1.51106 0.755531 0.655113i \(-0.227379\pi\)
0.755531 + 0.655113i \(0.227379\pi\)
\(168\) 0 0
\(169\) −7.31318 −0.562552
\(170\) 12.3690 0.948659
\(171\) 0 0
\(172\) −5.51288 −0.420353
\(173\) 19.1759 1.45792 0.728959 0.684558i \(-0.240005\pi\)
0.728959 + 0.684558i \(0.240005\pi\)
\(174\) 0 0
\(175\) −14.6981 −1.11107
\(176\) −13.0960 −0.987148
\(177\) 0 0
\(178\) −6.81615 −0.510892
\(179\) 7.02739 0.525252 0.262626 0.964898i \(-0.415412\pi\)
0.262626 + 0.964898i \(0.415412\pi\)
\(180\) 0 0
\(181\) −15.3645 −1.14203 −0.571017 0.820938i \(-0.693451\pi\)
−0.571017 + 0.820938i \(0.693451\pi\)
\(182\) 2.81427 0.208607
\(183\) 0 0
\(184\) −17.6520 −1.30132
\(185\) 15.9213 1.17056
\(186\) 0 0
\(187\) −12.7974 −0.935841
\(188\) 2.99324 0.218304
\(189\) 0 0
\(190\) 14.4298 1.04685
\(191\) 7.76445 0.561816 0.280908 0.959735i \(-0.409364\pi\)
0.280908 + 0.959735i \(0.409364\pi\)
\(192\) 0 0
\(193\) −24.7433 −1.78106 −0.890529 0.454926i \(-0.849666\pi\)
−0.890529 + 0.454926i \(0.849666\pi\)
\(194\) 16.3539 1.17414
\(195\) 0 0
\(196\) −0.607291 −0.0433779
\(197\) 10.5794 0.753754 0.376877 0.926263i \(-0.376998\pi\)
0.376877 + 0.926263i \(0.376998\pi\)
\(198\) 0 0
\(199\) 0.813046 0.0576353 0.0288177 0.999585i \(-0.490826\pi\)
0.0288177 + 0.999585i \(0.490826\pi\)
\(200\) −45.2253 −3.19791
\(201\) 0 0
\(202\) 2.13253 0.150044
\(203\) −3.90830 −0.274309
\(204\) 0 0
\(205\) −3.13837 −0.219193
\(206\) −12.7744 −0.890035
\(207\) 0 0
\(208\) 5.76292 0.399587
\(209\) −14.9296 −1.03270
\(210\) 0 0
\(211\) −2.74396 −0.188902 −0.0944509 0.995530i \(-0.530110\pi\)
−0.0944509 + 0.995530i \(0.530110\pi\)
\(212\) 6.58232 0.452075
\(213\) 0 0
\(214\) 15.1295 1.03423
\(215\) −40.2897 −2.74774
\(216\) 0 0
\(217\) −2.83738 −0.192614
\(218\) −11.7972 −0.799005
\(219\) 0 0
\(220\) 14.6063 0.984756
\(221\) 5.63153 0.378818
\(222\) 0 0
\(223\) 19.4867 1.30493 0.652464 0.757820i \(-0.273735\pi\)
0.652464 + 0.757820i \(0.273735\pi\)
\(224\) −3.30196 −0.220622
\(225\) 0 0
\(226\) 9.83680 0.654334
\(227\) −12.5024 −0.829813 −0.414906 0.909864i \(-0.636186\pi\)
−0.414906 + 0.909864i \(0.636186\pi\)
\(228\) 0 0
\(229\) −25.3292 −1.67380 −0.836900 0.547356i \(-0.815634\pi\)
−0.836900 + 0.547356i \(0.815634\pi\)
\(230\) −30.0480 −1.98131
\(231\) 0 0
\(232\) −12.0256 −0.789521
\(233\) 9.91538 0.649578 0.324789 0.945786i \(-0.394707\pi\)
0.324789 + 0.945786i \(0.394707\pi\)
\(234\) 0 0
\(235\) 21.8755 1.42700
\(236\) 5.28491 0.344019
\(237\) 0 0
\(238\) 2.78691 0.180648
\(239\) −18.4248 −1.19180 −0.595900 0.803058i \(-0.703205\pi\)
−0.595900 + 0.803058i \(0.703205\pi\)
\(240\) 0 0
\(241\) 8.87131 0.571451 0.285725 0.958312i \(-0.407765\pi\)
0.285725 + 0.958312i \(0.407765\pi\)
\(242\) 21.6756 1.39336
\(243\) 0 0
\(244\) −9.39906 −0.601713
\(245\) −4.43826 −0.283550
\(246\) 0 0
\(247\) 6.56980 0.418026
\(248\) −8.73046 −0.554385
\(249\) 0 0
\(250\) −50.7961 −3.21263
\(251\) −0.561631 −0.0354498 −0.0177249 0.999843i \(-0.505642\pi\)
−0.0177249 + 0.999843i \(0.505642\pi\)
\(252\) 0 0
\(253\) 31.0888 1.95454
\(254\) −1.18013 −0.0740480
\(255\) 0 0
\(256\) −13.0951 −0.818446
\(257\) 3.53641 0.220595 0.110298 0.993899i \(-0.464820\pi\)
0.110298 + 0.993899i \(0.464820\pi\)
\(258\) 0 0
\(259\) 3.58729 0.222904
\(260\) −6.42753 −0.398618
\(261\) 0 0
\(262\) 10.7931 0.666801
\(263\) 10.8675 0.670118 0.335059 0.942197i \(-0.391244\pi\)
0.335059 + 0.942197i \(0.391244\pi\)
\(264\) 0 0
\(265\) 48.1055 2.95510
\(266\) 3.25123 0.199346
\(267\) 0 0
\(268\) −3.26278 −0.199306
\(269\) 15.7001 0.957254 0.478627 0.878018i \(-0.341135\pi\)
0.478627 + 0.878018i \(0.341135\pi\)
\(270\) 0 0
\(271\) −27.0181 −1.64123 −0.820617 0.571478i \(-0.806370\pi\)
−0.820617 + 0.571478i \(0.806370\pi\)
\(272\) 5.70689 0.346031
\(273\) 0 0
\(274\) −16.4136 −0.991581
\(275\) 79.6513 4.80315
\(276\) 0 0
\(277\) −20.7983 −1.24965 −0.624823 0.780766i \(-0.714829\pi\)
−0.624823 + 0.780766i \(0.714829\pi\)
\(278\) −20.1017 −1.20562
\(279\) 0 0
\(280\) −13.6563 −0.816118
\(281\) −11.3243 −0.675554 −0.337777 0.941226i \(-0.609675\pi\)
−0.337777 + 0.941226i \(0.609675\pi\)
\(282\) 0 0
\(283\) 7.09116 0.421526 0.210763 0.977537i \(-0.432405\pi\)
0.210763 + 0.977537i \(0.432405\pi\)
\(284\) −5.33856 −0.316785
\(285\) 0 0
\(286\) −15.2509 −0.901805
\(287\) −0.707118 −0.0417398
\(288\) 0 0
\(289\) −11.4232 −0.671954
\(290\) −20.4706 −1.20208
\(291\) 0 0
\(292\) 8.85701 0.518317
\(293\) −1.07254 −0.0626582 −0.0313291 0.999509i \(-0.509974\pi\)
−0.0313291 + 0.999509i \(0.509974\pi\)
\(294\) 0 0
\(295\) 38.6237 2.24876
\(296\) 11.0379 0.641565
\(297\) 0 0
\(298\) −14.6098 −0.846324
\(299\) −13.6807 −0.791175
\(300\) 0 0
\(301\) −9.07783 −0.523237
\(302\) −14.6723 −0.844296
\(303\) 0 0
\(304\) 6.65772 0.381846
\(305\) −68.6911 −3.93324
\(306\) 0 0
\(307\) 18.1259 1.03450 0.517249 0.855835i \(-0.326956\pi\)
0.517249 + 0.855835i \(0.326956\pi\)
\(308\) 3.29100 0.187522
\(309\) 0 0
\(310\) −14.8614 −0.844072
\(311\) −6.03589 −0.342264 −0.171132 0.985248i \(-0.554742\pi\)
−0.171132 + 0.985248i \(0.554742\pi\)
\(312\) 0 0
\(313\) 16.9041 0.955475 0.477737 0.878503i \(-0.341457\pi\)
0.477737 + 0.878503i \(0.341457\pi\)
\(314\) −22.8762 −1.29098
\(315\) 0 0
\(316\) 8.23774 0.463409
\(317\) 18.1372 1.01868 0.509342 0.860564i \(-0.329889\pi\)
0.509342 + 0.860564i \(0.329889\pi\)
\(318\) 0 0
\(319\) 21.1797 1.18583
\(320\) −38.7459 −2.16596
\(321\) 0 0
\(322\) −6.77024 −0.377290
\(323\) 6.50593 0.362000
\(324\) 0 0
\(325\) −35.0507 −1.94426
\(326\) −4.08337 −0.226157
\(327\) 0 0
\(328\) −2.17576 −0.120136
\(329\) 4.92884 0.271736
\(330\) 0 0
\(331\) −8.15730 −0.448366 −0.224183 0.974547i \(-0.571971\pi\)
−0.224183 + 0.974547i \(0.571971\pi\)
\(332\) −3.55601 −0.195161
\(333\) 0 0
\(334\) 23.0447 1.26095
\(335\) −23.8453 −1.30281
\(336\) 0 0
\(337\) −9.35375 −0.509531 −0.254765 0.967003i \(-0.581998\pi\)
−0.254765 + 0.967003i \(0.581998\pi\)
\(338\) −8.63051 −0.469438
\(339\) 0 0
\(340\) −6.36504 −0.345193
\(341\) 15.3762 0.832667
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −27.9320 −1.50599
\(345\) 0 0
\(346\) 22.6301 1.21660
\(347\) 2.98500 0.160243 0.0801216 0.996785i \(-0.474469\pi\)
0.0801216 + 0.996785i \(0.474469\pi\)
\(348\) 0 0
\(349\) 30.6058 1.63829 0.819146 0.573585i \(-0.194448\pi\)
0.819146 + 0.573585i \(0.194448\pi\)
\(350\) −17.3457 −0.927168
\(351\) 0 0
\(352\) 17.8938 0.953744
\(353\) −6.64509 −0.353682 −0.176841 0.984239i \(-0.556588\pi\)
−0.176841 + 0.984239i \(0.556588\pi\)
\(354\) 0 0
\(355\) −39.0158 −2.07074
\(356\) 3.50756 0.185900
\(357\) 0 0
\(358\) 8.29324 0.438311
\(359\) 32.2341 1.70125 0.850626 0.525772i \(-0.176223\pi\)
0.850626 + 0.525772i \(0.176223\pi\)
\(360\) 0 0
\(361\) −11.4101 −0.600533
\(362\) −18.1321 −0.953003
\(363\) 0 0
\(364\) −1.44821 −0.0759068
\(365\) 64.7296 3.38810
\(366\) 0 0
\(367\) 12.1394 0.633670 0.316835 0.948481i \(-0.397380\pi\)
0.316835 + 0.948481i \(0.397380\pi\)
\(368\) −13.8638 −0.722699
\(369\) 0 0
\(370\) 18.7893 0.976807
\(371\) 10.8388 0.562724
\(372\) 0 0
\(373\) −32.5972 −1.68782 −0.843910 0.536485i \(-0.819752\pi\)
−0.843910 + 0.536485i \(0.819752\pi\)
\(374\) −15.1026 −0.780939
\(375\) 0 0
\(376\) 15.1658 0.782115
\(377\) −9.32015 −0.480012
\(378\) 0 0
\(379\) 21.7712 1.11831 0.559155 0.829063i \(-0.311126\pi\)
0.559155 + 0.829063i \(0.311126\pi\)
\(380\) −7.42551 −0.380921
\(381\) 0 0
\(382\) 9.16307 0.468824
\(383\) −3.86635 −0.197561 −0.0987806 0.995109i \(-0.531494\pi\)
−0.0987806 + 0.995109i \(0.531494\pi\)
\(384\) 0 0
\(385\) 24.0516 1.22578
\(386\) −29.2003 −1.48626
\(387\) 0 0
\(388\) −8.41564 −0.427239
\(389\) −17.3609 −0.880230 −0.440115 0.897941i \(-0.645062\pi\)
−0.440115 + 0.897941i \(0.645062\pi\)
\(390\) 0 0
\(391\) −13.5477 −0.685137
\(392\) −3.07694 −0.155409
\(393\) 0 0
\(394\) 12.4851 0.628991
\(395\) 60.2038 3.02918
\(396\) 0 0
\(397\) −36.5315 −1.83346 −0.916732 0.399502i \(-0.869183\pi\)
−0.916732 + 0.399502i \(0.869183\pi\)
\(398\) 0.959501 0.0480955
\(399\) 0 0
\(400\) −35.5197 −1.77599
\(401\) −15.3087 −0.764482 −0.382241 0.924063i \(-0.624848\pi\)
−0.382241 + 0.924063i \(0.624848\pi\)
\(402\) 0 0
\(403\) −6.76632 −0.337054
\(404\) −1.09739 −0.0545971
\(405\) 0 0
\(406\) −4.61231 −0.228905
\(407\) −19.4401 −0.963608
\(408\) 0 0
\(409\) 27.7504 1.37217 0.686083 0.727523i \(-0.259328\pi\)
0.686083 + 0.727523i \(0.259328\pi\)
\(410\) −3.70369 −0.182912
\(411\) 0 0
\(412\) 6.57366 0.323861
\(413\) 8.70245 0.428219
\(414\) 0 0
\(415\) −25.9883 −1.27572
\(416\) −7.87421 −0.386065
\(417\) 0 0
\(418\) −17.6189 −0.861768
\(419\) 7.51179 0.366975 0.183488 0.983022i \(-0.441261\pi\)
0.183488 + 0.983022i \(0.441261\pi\)
\(420\) 0 0
\(421\) 6.05425 0.295066 0.147533 0.989057i \(-0.452867\pi\)
0.147533 + 0.989057i \(0.452867\pi\)
\(422\) −3.23823 −0.157634
\(423\) 0 0
\(424\) 33.3505 1.61964
\(425\) −34.7100 −1.68368
\(426\) 0 0
\(427\) −15.4770 −0.748987
\(428\) −7.78557 −0.376330
\(429\) 0 0
\(430\) −47.5472 −2.29293
\(431\) −33.5629 −1.61667 −0.808334 0.588724i \(-0.799631\pi\)
−0.808334 + 0.588724i \(0.799631\pi\)
\(432\) 0 0
\(433\) 14.8488 0.713586 0.356793 0.934183i \(-0.383870\pi\)
0.356793 + 0.934183i \(0.383870\pi\)
\(434\) −3.34848 −0.160732
\(435\) 0 0
\(436\) 6.07078 0.290737
\(437\) −15.8049 −0.756049
\(438\) 0 0
\(439\) 18.5869 0.887103 0.443552 0.896249i \(-0.353718\pi\)
0.443552 + 0.896249i \(0.353718\pi\)
\(440\) 74.0053 3.52807
\(441\) 0 0
\(442\) 6.64595 0.316116
\(443\) −2.21728 −0.105346 −0.0526730 0.998612i \(-0.516774\pi\)
−0.0526730 + 0.998612i \(0.516774\pi\)
\(444\) 0 0
\(445\) 25.6343 1.21518
\(446\) 22.9969 1.08893
\(447\) 0 0
\(448\) −8.72998 −0.412453
\(449\) −12.9587 −0.611558 −0.305779 0.952102i \(-0.598917\pi\)
−0.305779 + 0.952102i \(0.598917\pi\)
\(450\) 0 0
\(451\) 3.83197 0.180441
\(452\) −5.06198 −0.238095
\(453\) 0 0
\(454\) −14.7545 −0.692461
\(455\) −10.5839 −0.496183
\(456\) 0 0
\(457\) −36.1883 −1.69282 −0.846410 0.532532i \(-0.821240\pi\)
−0.846410 + 0.532532i \(0.821240\pi\)
\(458\) −29.8918 −1.39675
\(459\) 0 0
\(460\) 15.4626 0.720948
\(461\) −41.2133 −1.91949 −0.959747 0.280866i \(-0.909378\pi\)
−0.959747 + 0.280866i \(0.909378\pi\)
\(462\) 0 0
\(463\) −39.1721 −1.82048 −0.910240 0.414080i \(-0.864103\pi\)
−0.910240 + 0.414080i \(0.864103\pi\)
\(464\) −9.44487 −0.438467
\(465\) 0 0
\(466\) 11.7014 0.542059
\(467\) 27.5007 1.27258 0.636291 0.771449i \(-0.280468\pi\)
0.636291 + 0.771449i \(0.280468\pi\)
\(468\) 0 0
\(469\) −5.37268 −0.248087
\(470\) 25.8159 1.19080
\(471\) 0 0
\(472\) 26.7769 1.23251
\(473\) 49.1941 2.26195
\(474\) 0 0
\(475\) −40.4930 −1.85794
\(476\) −1.43413 −0.0657332
\(477\) 0 0
\(478\) −21.7437 −0.994532
\(479\) 2.25308 0.102946 0.0514728 0.998674i \(-0.483608\pi\)
0.0514728 + 0.998674i \(0.483608\pi\)
\(480\) 0 0
\(481\) 8.55464 0.390058
\(482\) 10.4693 0.476864
\(483\) 0 0
\(484\) −11.1542 −0.507009
\(485\) −61.5039 −2.79275
\(486\) 0 0
\(487\) 4.33373 0.196380 0.0981901 0.995168i \(-0.468695\pi\)
0.0981901 + 0.995168i \(0.468695\pi\)
\(488\) −47.6220 −2.15575
\(489\) 0 0
\(490\) −5.23773 −0.236616
\(491\) 3.53012 0.159312 0.0796560 0.996822i \(-0.474618\pi\)
0.0796560 + 0.996822i \(0.474618\pi\)
\(492\) 0 0
\(493\) −9.22954 −0.415678
\(494\) 7.75323 0.348834
\(495\) 0 0
\(496\) −6.85686 −0.307882
\(497\) −8.79079 −0.394321
\(498\) 0 0
\(499\) −19.5211 −0.873884 −0.436942 0.899490i \(-0.643939\pi\)
−0.436942 + 0.899490i \(0.643939\pi\)
\(500\) 26.1395 1.16899
\(501\) 0 0
\(502\) −0.662798 −0.0295821
\(503\) 25.9921 1.15893 0.579465 0.814997i \(-0.303261\pi\)
0.579465 + 0.814997i \(0.303261\pi\)
\(504\) 0 0
\(505\) −8.02004 −0.356887
\(506\) 36.6889 1.63102
\(507\) 0 0
\(508\) 0.607291 0.0269442
\(509\) −39.4799 −1.74992 −0.874959 0.484198i \(-0.839112\pi\)
−0.874959 + 0.484198i \(0.839112\pi\)
\(510\) 0 0
\(511\) 14.5845 0.645179
\(512\) −22.8512 −1.00989
\(513\) 0 0
\(514\) 4.17343 0.184082
\(515\) 48.0422 2.11699
\(516\) 0 0
\(517\) −26.7101 −1.17471
\(518\) 4.23348 0.186008
\(519\) 0 0
\(520\) −32.5662 −1.42812
\(521\) 10.6512 0.466639 0.233320 0.972400i \(-0.425041\pi\)
0.233320 + 0.972400i \(0.425041\pi\)
\(522\) 0 0
\(523\) 25.7574 1.12629 0.563147 0.826357i \(-0.309591\pi\)
0.563147 + 0.826357i \(0.309591\pi\)
\(524\) −5.55409 −0.242632
\(525\) 0 0
\(526\) 12.8251 0.559199
\(527\) −6.70054 −0.291880
\(528\) 0 0
\(529\) 9.91145 0.430933
\(530\) 56.7708 2.46597
\(531\) 0 0
\(532\) −1.67307 −0.0725368
\(533\) −1.68627 −0.0730404
\(534\) 0 0
\(535\) −56.8992 −2.45997
\(536\) −16.5314 −0.714049
\(537\) 0 0
\(538\) 18.5282 0.798808
\(539\) 5.41915 0.233419
\(540\) 0 0
\(541\) −27.6386 −1.18828 −0.594138 0.804363i \(-0.702507\pi\)
−0.594138 + 0.804363i \(0.702507\pi\)
\(542\) −31.8849 −1.36958
\(543\) 0 0
\(544\) −7.79766 −0.334322
\(545\) 44.3670 1.90047
\(546\) 0 0
\(547\) 19.2255 0.822024 0.411012 0.911630i \(-0.365175\pi\)
0.411012 + 0.911630i \(0.365175\pi\)
\(548\) 8.44636 0.360811
\(549\) 0 0
\(550\) 93.9990 4.00813
\(551\) −10.7673 −0.458701
\(552\) 0 0
\(553\) 13.5647 0.576832
\(554\) −24.5447 −1.04280
\(555\) 0 0
\(556\) 10.3442 0.438694
\(557\) 29.8537 1.26494 0.632471 0.774584i \(-0.282041\pi\)
0.632471 + 0.774584i \(0.282041\pi\)
\(558\) 0 0
\(559\) −21.6480 −0.915611
\(560\) −10.7256 −0.453238
\(561\) 0 0
\(562\) −13.3642 −0.563735
\(563\) 32.2423 1.35885 0.679425 0.733745i \(-0.262229\pi\)
0.679425 + 0.733745i \(0.262229\pi\)
\(564\) 0 0
\(565\) −36.9944 −1.55637
\(566\) 8.36850 0.351754
\(567\) 0 0
\(568\) −27.0488 −1.13494
\(569\) −27.2451 −1.14217 −0.571086 0.820890i \(-0.693478\pi\)
−0.571086 + 0.820890i \(0.693478\pi\)
\(570\) 0 0
\(571\) −23.3646 −0.977779 −0.488890 0.872346i \(-0.662598\pi\)
−0.488890 + 0.872346i \(0.662598\pi\)
\(572\) 7.84806 0.328144
\(573\) 0 0
\(574\) −0.834492 −0.0348310
\(575\) 84.3210 3.51643
\(576\) 0 0
\(577\) −14.4610 −0.602018 −0.301009 0.953621i \(-0.597323\pi\)
−0.301009 + 0.953621i \(0.597323\pi\)
\(578\) −13.4809 −0.560731
\(579\) 0 0
\(580\) 10.5341 0.437404
\(581\) −5.85553 −0.242928
\(582\) 0 0
\(583\) −58.7372 −2.43265
\(584\) 44.8756 1.85697
\(585\) 0 0
\(586\) −1.26573 −0.0522870
\(587\) −15.5314 −0.641049 −0.320524 0.947240i \(-0.603859\pi\)
−0.320524 + 0.947240i \(0.603859\pi\)
\(588\) 0 0
\(589\) −7.81691 −0.322090
\(590\) 45.5810 1.87654
\(591\) 0 0
\(592\) 8.66911 0.356298
\(593\) −15.6281 −0.641769 −0.320884 0.947118i \(-0.603980\pi\)
−0.320884 + 0.947118i \(0.603980\pi\)
\(594\) 0 0
\(595\) −10.4810 −0.429681
\(596\) 7.51815 0.307955
\(597\) 0 0
\(598\) −16.1450 −0.660219
\(599\) −8.04131 −0.328559 −0.164280 0.986414i \(-0.552530\pi\)
−0.164280 + 0.986414i \(0.552530\pi\)
\(600\) 0 0
\(601\) −46.7643 −1.90755 −0.953777 0.300514i \(-0.902842\pi\)
−0.953777 + 0.300514i \(0.902842\pi\)
\(602\) −10.7130 −0.436630
\(603\) 0 0
\(604\) 7.55030 0.307217
\(605\) −81.5181 −3.31418
\(606\) 0 0
\(607\) 4.05313 0.164511 0.0822557 0.996611i \(-0.473788\pi\)
0.0822557 + 0.996611i \(0.473788\pi\)
\(608\) −9.09682 −0.368925
\(609\) 0 0
\(610\) −81.0645 −3.28221
\(611\) 11.7538 0.475509
\(612\) 0 0
\(613\) 7.60595 0.307201 0.153601 0.988133i \(-0.450913\pi\)
0.153601 + 0.988133i \(0.450913\pi\)
\(614\) 21.3909 0.863267
\(615\) 0 0
\(616\) 16.6744 0.671831
\(617\) −23.6786 −0.953265 −0.476632 0.879103i \(-0.658143\pi\)
−0.476632 + 0.879103i \(0.658143\pi\)
\(618\) 0 0
\(619\) 30.7464 1.23580 0.617901 0.786256i \(-0.287983\pi\)
0.617901 + 0.786256i \(0.287983\pi\)
\(620\) 7.64763 0.307136
\(621\) 0 0
\(622\) −7.12314 −0.285612
\(623\) 5.77576 0.231401
\(624\) 0 0
\(625\) 117.544 4.70177
\(626\) 19.9490 0.797323
\(627\) 0 0
\(628\) 11.7720 0.469754
\(629\) 8.47147 0.337780
\(630\) 0 0
\(631\) −39.5487 −1.57441 −0.787205 0.616691i \(-0.788473\pi\)
−0.787205 + 0.616691i \(0.788473\pi\)
\(632\) 41.7380 1.66025
\(633\) 0 0
\(634\) 21.4042 0.850071
\(635\) 4.43826 0.176127
\(636\) 0 0
\(637\) −2.38471 −0.0944855
\(638\) 24.9948 0.989553
\(639\) 0 0
\(640\) −16.4153 −0.648873
\(641\) 21.3454 0.843091 0.421546 0.906807i \(-0.361488\pi\)
0.421546 + 0.906807i \(0.361488\pi\)
\(642\) 0 0
\(643\) 9.79675 0.386346 0.193173 0.981165i \(-0.438122\pi\)
0.193173 + 0.981165i \(0.438122\pi\)
\(644\) 3.48394 0.137286
\(645\) 0 0
\(646\) 7.67785 0.302081
\(647\) −9.33818 −0.367122 −0.183561 0.983008i \(-0.558762\pi\)
−0.183561 + 0.983008i \(0.558762\pi\)
\(648\) 0 0
\(649\) −47.1598 −1.85118
\(650\) −41.3644 −1.62245
\(651\) 0 0
\(652\) 2.10129 0.0822927
\(653\) −1.02438 −0.0400872 −0.0200436 0.999799i \(-0.506380\pi\)
−0.0200436 + 0.999799i \(0.506380\pi\)
\(654\) 0 0
\(655\) −40.5909 −1.58602
\(656\) −1.70883 −0.0667187
\(657\) 0 0
\(658\) 5.81668 0.226758
\(659\) 18.9099 0.736623 0.368312 0.929702i \(-0.379936\pi\)
0.368312 + 0.929702i \(0.379936\pi\)
\(660\) 0 0
\(661\) −27.3932 −1.06547 −0.532736 0.846282i \(-0.678836\pi\)
−0.532736 + 0.846282i \(0.678836\pi\)
\(662\) −9.62668 −0.374152
\(663\) 0 0
\(664\) −18.0171 −0.699200
\(665\) −12.2273 −0.474154
\(666\) 0 0
\(667\) 22.4213 0.868158
\(668\) −11.8587 −0.458827
\(669\) 0 0
\(670\) −28.1406 −1.08717
\(671\) 83.8724 3.23786
\(672\) 0 0
\(673\) 11.6979 0.450920 0.225460 0.974252i \(-0.427611\pi\)
0.225460 + 0.974252i \(0.427611\pi\)
\(674\) −11.0386 −0.425193
\(675\) 0 0
\(676\) 4.44122 0.170816
\(677\) −12.2230 −0.469768 −0.234884 0.972023i \(-0.575471\pi\)
−0.234884 + 0.972023i \(0.575471\pi\)
\(678\) 0 0
\(679\) −13.8577 −0.531809
\(680\) −32.2496 −1.23672
\(681\) 0 0
\(682\) 18.1459 0.694843
\(683\) 45.0284 1.72296 0.861482 0.507788i \(-0.169537\pi\)
0.861482 + 0.507788i \(0.169537\pi\)
\(684\) 0 0
\(685\) 61.7285 2.35852
\(686\) −1.18013 −0.0450576
\(687\) 0 0
\(688\) −21.9376 −0.836365
\(689\) 25.8474 0.984708
\(690\) 0 0
\(691\) 18.8479 0.717009 0.358504 0.933528i \(-0.383287\pi\)
0.358504 + 0.933528i \(0.383287\pi\)
\(692\) −11.6453 −0.442690
\(693\) 0 0
\(694\) 3.52269 0.133719
\(695\) 75.5988 2.86763
\(696\) 0 0
\(697\) −1.66987 −0.0632510
\(698\) 36.1189 1.36712
\(699\) 0 0
\(700\) 8.92604 0.337372
\(701\) 6.02478 0.227553 0.113776 0.993506i \(-0.463705\pi\)
0.113776 + 0.993506i \(0.463705\pi\)
\(702\) 0 0
\(703\) 9.88290 0.372740
\(704\) 47.3091 1.78303
\(705\) 0 0
\(706\) −7.84208 −0.295140
\(707\) −1.80702 −0.0679601
\(708\) 0 0
\(709\) 2.99684 0.112549 0.0562744 0.998415i \(-0.482078\pi\)
0.0562744 + 0.998415i \(0.482078\pi\)
\(710\) −46.0437 −1.72799
\(711\) 0 0
\(712\) 17.7717 0.666022
\(713\) 16.2776 0.609602
\(714\) 0 0
\(715\) 57.3559 2.14499
\(716\) −4.26767 −0.159490
\(717\) 0 0
\(718\) 38.0405 1.41966
\(719\) 36.4922 1.36093 0.680465 0.732781i \(-0.261778\pi\)
0.680465 + 0.732781i \(0.261778\pi\)
\(720\) 0 0
\(721\) 10.8246 0.403128
\(722\) −13.4654 −0.501132
\(723\) 0 0
\(724\) 9.33071 0.346773
\(725\) 57.4447 2.13344
\(726\) 0 0
\(727\) 8.36457 0.310225 0.155112 0.987897i \(-0.450426\pi\)
0.155112 + 0.987897i \(0.450426\pi\)
\(728\) −7.33761 −0.271950
\(729\) 0 0
\(730\) 76.3895 2.82730
\(731\) −21.4375 −0.792894
\(732\) 0 0
\(733\) 28.0240 1.03509 0.517545 0.855656i \(-0.326846\pi\)
0.517545 + 0.855656i \(0.326846\pi\)
\(734\) 14.3260 0.528784
\(735\) 0 0
\(736\) 18.9429 0.698244
\(737\) 29.1153 1.07248
\(738\) 0 0
\(739\) −44.3664 −1.63204 −0.816022 0.578021i \(-0.803825\pi\)
−0.816022 + 0.578021i \(0.803825\pi\)
\(740\) −9.66887 −0.355435
\(741\) 0 0
\(742\) 12.7912 0.469581
\(743\) 27.2520 0.999778 0.499889 0.866089i \(-0.333374\pi\)
0.499889 + 0.866089i \(0.333374\pi\)
\(744\) 0 0
\(745\) 54.9448 2.01302
\(746\) −38.4690 −1.40845
\(747\) 0 0
\(748\) 7.77176 0.284164
\(749\) −12.8202 −0.468439
\(750\) 0 0
\(751\) 20.4756 0.747167 0.373583 0.927597i \(-0.378129\pi\)
0.373583 + 0.927597i \(0.378129\pi\)
\(752\) 11.9111 0.434354
\(753\) 0 0
\(754\) −10.9990 −0.400560
\(755\) 55.1798 2.00820
\(756\) 0 0
\(757\) 3.88978 0.141376 0.0706882 0.997498i \(-0.477480\pi\)
0.0706882 + 0.997498i \(0.477480\pi\)
\(758\) 25.6928 0.933206
\(759\) 0 0
\(760\) −37.6227 −1.36472
\(761\) −2.39284 −0.0867405 −0.0433702 0.999059i \(-0.513810\pi\)
−0.0433702 + 0.999059i \(0.513810\pi\)
\(762\) 0 0
\(763\) 9.99649 0.361897
\(764\) −4.71528 −0.170593
\(765\) 0 0
\(766\) −4.56280 −0.164861
\(767\) 20.7528 0.749339
\(768\) 0 0
\(769\) 17.0826 0.616014 0.308007 0.951384i \(-0.400338\pi\)
0.308007 + 0.951384i \(0.400338\pi\)
\(770\) 28.3840 1.02289
\(771\) 0 0
\(772\) 15.0263 0.540810
\(773\) −38.7362 −1.39324 −0.696621 0.717439i \(-0.745314\pi\)
−0.696621 + 0.717439i \(0.745314\pi\)
\(774\) 0 0
\(775\) 41.7042 1.49806
\(776\) −42.6393 −1.53066
\(777\) 0 0
\(778\) −20.4881 −0.734533
\(779\) −1.94809 −0.0697976
\(780\) 0 0
\(781\) 47.6385 1.70464
\(782\) −15.9881 −0.571732
\(783\) 0 0
\(784\) −2.41662 −0.0863077
\(785\) 86.0332 3.07066
\(786\) 0 0
\(787\) 13.3375 0.475430 0.237715 0.971335i \(-0.423602\pi\)
0.237715 + 0.971335i \(0.423602\pi\)
\(788\) −6.42479 −0.228874
\(789\) 0 0
\(790\) 71.0484 2.52779
\(791\) −8.33535 −0.296371
\(792\) 0 0
\(793\) −36.9082 −1.31065
\(794\) −43.1120 −1.52999
\(795\) 0 0
\(796\) −0.493755 −0.0175007
\(797\) 5.73847 0.203267 0.101633 0.994822i \(-0.467593\pi\)
0.101633 + 0.994822i \(0.467593\pi\)
\(798\) 0 0
\(799\) 11.6396 0.411778
\(800\) 48.5327 1.71589
\(801\) 0 0
\(802\) −18.0663 −0.637944
\(803\) −79.0354 −2.78910
\(804\) 0 0
\(805\) 25.4616 0.897404
\(806\) −7.98514 −0.281265
\(807\) 0 0
\(808\) −5.56011 −0.195604
\(809\) −32.9630 −1.15892 −0.579458 0.815002i \(-0.696736\pi\)
−0.579458 + 0.815002i \(0.696736\pi\)
\(810\) 0 0
\(811\) 35.8058 1.25731 0.628656 0.777684i \(-0.283605\pi\)
0.628656 + 0.777684i \(0.283605\pi\)
\(812\) 2.37348 0.0832927
\(813\) 0 0
\(814\) −22.9418 −0.804110
\(815\) 15.3568 0.537926
\(816\) 0 0
\(817\) −25.0092 −0.874960
\(818\) 32.7491 1.14504
\(819\) 0 0
\(820\) 1.90590 0.0665570
\(821\) 45.5266 1.58889 0.794445 0.607336i \(-0.207762\pi\)
0.794445 + 0.607336i \(0.207762\pi\)
\(822\) 0 0
\(823\) 4.84441 0.168866 0.0844328 0.996429i \(-0.473092\pi\)
0.0844328 + 0.996429i \(0.473092\pi\)
\(824\) 33.3066 1.16029
\(825\) 0 0
\(826\) 10.2700 0.357340
\(827\) −29.6775 −1.03199 −0.515993 0.856593i \(-0.672577\pi\)
−0.515993 + 0.856593i \(0.672577\pi\)
\(828\) 0 0
\(829\) −38.9206 −1.35177 −0.675884 0.737008i \(-0.736238\pi\)
−0.675884 + 0.737008i \(0.736238\pi\)
\(830\) −30.6696 −1.06456
\(831\) 0 0
\(832\) −20.8184 −0.721750
\(833\) −2.36152 −0.0818219
\(834\) 0 0
\(835\) −86.6668 −2.99923
\(836\) 9.06661 0.313575
\(837\) 0 0
\(838\) 8.86490 0.306233
\(839\) −11.2390 −0.388015 −0.194007 0.981000i \(-0.562149\pi\)
−0.194007 + 0.981000i \(0.562149\pi\)
\(840\) 0 0
\(841\) −13.7252 −0.473282
\(842\) 7.14480 0.246226
\(843\) 0 0
\(844\) 1.66638 0.0573591
\(845\) 32.4578 1.11658
\(846\) 0 0
\(847\) −18.3671 −0.631102
\(848\) 26.1933 0.899481
\(849\) 0 0
\(850\) −40.9623 −1.40500
\(851\) −20.5798 −0.705465
\(852\) 0 0
\(853\) −3.28040 −0.112319 −0.0561594 0.998422i \(-0.517885\pi\)
−0.0561594 + 0.998422i \(0.517885\pi\)
\(854\) −18.2649 −0.625014
\(855\) 0 0
\(856\) −39.4470 −1.34827
\(857\) 18.3698 0.627499 0.313750 0.949506i \(-0.398415\pi\)
0.313750 + 0.949506i \(0.398415\pi\)
\(858\) 0 0
\(859\) −39.9179 −1.36198 −0.680991 0.732292i \(-0.738451\pi\)
−0.680991 + 0.732292i \(0.738451\pi\)
\(860\) 24.4676 0.834338
\(861\) 0 0
\(862\) −39.6086 −1.34908
\(863\) −2.21429 −0.0753753 −0.0376876 0.999290i \(-0.511999\pi\)
−0.0376876 + 0.999290i \(0.511999\pi\)
\(864\) 0 0
\(865\) −85.1076 −2.89375
\(866\) 17.5235 0.595472
\(867\) 0 0
\(868\) 1.72311 0.0584863
\(869\) −73.5093 −2.49363
\(870\) 0 0
\(871\) −12.8123 −0.434127
\(872\) 30.7586 1.04162
\(873\) 0 0
\(874\) −18.6518 −0.630907
\(875\) 43.0428 1.45511
\(876\) 0 0
\(877\) −34.7091 −1.17204 −0.586021 0.810296i \(-0.699306\pi\)
−0.586021 + 0.810296i \(0.699306\pi\)
\(878\) 21.9349 0.740269
\(879\) 0 0
\(880\) 58.1234 1.95934
\(881\) −0.174959 −0.00589451 −0.00294725 0.999996i \(-0.500938\pi\)
−0.00294725 + 0.999996i \(0.500938\pi\)
\(882\) 0 0
\(883\) 47.3602 1.59380 0.796899 0.604113i \(-0.206473\pi\)
0.796899 + 0.604113i \(0.206473\pi\)
\(884\) −3.41998 −0.115026
\(885\) 0 0
\(886\) −2.61668 −0.0879090
\(887\) −40.2851 −1.35264 −0.676321 0.736607i \(-0.736427\pi\)
−0.676321 + 0.736607i \(0.736427\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 30.2518 1.01404
\(891\) 0 0
\(892\) −11.8341 −0.396235
\(893\) 13.5788 0.454398
\(894\) 0 0
\(895\) −31.1894 −1.04255
\(896\) −3.69860 −0.123561
\(897\) 0 0
\(898\) −15.2929 −0.510332
\(899\) 11.0893 0.369850
\(900\) 0 0
\(901\) 25.5961 0.852731
\(902\) 4.52223 0.150574
\(903\) 0 0
\(904\) −25.6474 −0.853020
\(905\) 68.1916 2.26677
\(906\) 0 0
\(907\) −16.1755 −0.537100 −0.268550 0.963266i \(-0.586544\pi\)
−0.268550 + 0.963266i \(0.586544\pi\)
\(908\) 7.59259 0.251969
\(909\) 0 0
\(910\) −12.4904 −0.414054
\(911\) 5.34893 0.177218 0.0886090 0.996066i \(-0.471758\pi\)
0.0886090 + 0.996066i \(0.471758\pi\)
\(912\) 0 0
\(913\) 31.7320 1.05017
\(914\) −42.7070 −1.41262
\(915\) 0 0
\(916\) 15.3822 0.508242
\(917\) −9.14569 −0.302017
\(918\) 0 0
\(919\) −26.5607 −0.876156 −0.438078 0.898937i \(-0.644341\pi\)
−0.438078 + 0.898937i \(0.644341\pi\)
\(920\) 78.3440 2.58292
\(921\) 0 0
\(922\) −48.6371 −1.60178
\(923\) −20.9634 −0.690020
\(924\) 0 0
\(925\) −52.7265 −1.73364
\(926\) −46.2282 −1.51915
\(927\) 0 0
\(928\) 12.9051 0.423630
\(929\) −42.5323 −1.39544 −0.697719 0.716371i \(-0.745802\pi\)
−0.697719 + 0.716371i \(0.745802\pi\)
\(930\) 0 0
\(931\) −2.75497 −0.0902906
\(932\) −6.02152 −0.197241
\(933\) 0 0
\(934\) 32.4545 1.06194
\(935\) 56.7983 1.85750
\(936\) 0 0
\(937\) 8.57209 0.280038 0.140019 0.990149i \(-0.455284\pi\)
0.140019 + 0.990149i \(0.455284\pi\)
\(938\) −6.34047 −0.207024
\(939\) 0 0
\(940\) −13.2848 −0.433301
\(941\) −30.0271 −0.978857 −0.489428 0.872044i \(-0.662795\pi\)
−0.489428 + 0.872044i \(0.662795\pi\)
\(942\) 0 0
\(943\) 4.05663 0.132102
\(944\) 21.0305 0.684484
\(945\) 0 0
\(946\) 58.0555 1.88755
\(947\) −27.6070 −0.897108 −0.448554 0.893756i \(-0.648061\pi\)
−0.448554 + 0.893756i \(0.648061\pi\)
\(948\) 0 0
\(949\) 34.7797 1.12900
\(950\) −47.7870 −1.55041
\(951\) 0 0
\(952\) −7.26627 −0.235501
\(953\) −56.3567 −1.82557 −0.912786 0.408437i \(-0.866074\pi\)
−0.912786 + 0.408437i \(0.866074\pi\)
\(954\) 0 0
\(955\) −34.4606 −1.11512
\(956\) 11.1892 0.361885
\(957\) 0 0
\(958\) 2.65893 0.0859060
\(959\) 13.9083 0.449121
\(960\) 0 0
\(961\) −22.9493 −0.740299
\(962\) 10.0956 0.325495
\(963\) 0 0
\(964\) −5.38746 −0.173518
\(965\) 109.817 3.53513
\(966\) 0 0
\(967\) −10.0649 −0.323664 −0.161832 0.986818i \(-0.551740\pi\)
−0.161832 + 0.986818i \(0.551740\pi\)
\(968\) −56.5147 −1.81645
\(969\) 0 0
\(970\) −72.5827 −2.33049
\(971\) −47.6227 −1.52829 −0.764143 0.645047i \(-0.776838\pi\)
−0.764143 + 0.645047i \(0.776838\pi\)
\(972\) 0 0
\(973\) 17.0334 0.546067
\(974\) 5.11437 0.163875
\(975\) 0 0
\(976\) −37.4021 −1.19721
\(977\) 2.84827 0.0911243 0.0455621 0.998962i \(-0.485492\pi\)
0.0455621 + 0.998962i \(0.485492\pi\)
\(978\) 0 0
\(979\) −31.2997 −1.00034
\(980\) 2.69531 0.0860986
\(981\) 0 0
\(982\) 4.16600 0.132942
\(983\) 37.5274 1.19694 0.598470 0.801146i \(-0.295776\pi\)
0.598470 + 0.801146i \(0.295776\pi\)
\(984\) 0 0
\(985\) −46.9543 −1.49609
\(986\) −10.8921 −0.346874
\(987\) 0 0
\(988\) −3.98978 −0.126932
\(989\) 52.0782 1.65599
\(990\) 0 0
\(991\) −42.7222 −1.35711 −0.678557 0.734547i \(-0.737395\pi\)
−0.678557 + 0.734547i \(0.737395\pi\)
\(992\) 9.36893 0.297464
\(993\) 0 0
\(994\) −10.3743 −0.329052
\(995\) −3.60851 −0.114397
\(996\) 0 0
\(997\) 3.76493 0.119237 0.0596183 0.998221i \(-0.481012\pi\)
0.0596183 + 0.998221i \(0.481012\pi\)
\(998\) −23.0374 −0.729238
\(999\) 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.1.23 yes 32
3.2 odd 2 inner 8001.2.a.z.1.10 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.10 32 3.2 odd 2 inner
8001.2.a.z.1.23 yes 32 1.1 even 1 trivial