Properties

Label 10000.2.a.c.1.1
Level $10000$
Weight $2$
Character 10000.1
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.8504020213\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 10000.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.618034 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.618034 q^{7} -2.00000 q^{9} +5.23607 q^{11} +1.85410 q^{13} -5.23607 q^{17} -0.854102 q^{19} +0.618034 q^{21} -3.76393 q^{23} +5.00000 q^{27} -3.61803 q^{29} +3.00000 q^{31} -5.23607 q^{33} -0.236068 q^{37} -1.85410 q^{39} -0.763932 q^{41} +4.85410 q^{43} -0.618034 q^{47} -6.61803 q^{49} +5.23607 q^{51} -3.47214 q^{53} +0.854102 q^{57} +10.8541 q^{59} +8.70820 q^{61} +1.23607 q^{63} -4.76393 q^{67} +3.76393 q^{69} +6.61803 q^{71} -9.00000 q^{73} -3.23607 q^{77} +8.09017 q^{79} +1.00000 q^{81} +6.23607 q^{83} +3.61803 q^{87} -8.94427 q^{89} -1.14590 q^{91} -3.00000 q^{93} -3.85410 q^{97} -10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{7} - 4 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + q^{7} - 4 q^{9} + 6 q^{11} - 3 q^{13} - 6 q^{17} + 5 q^{19} - q^{21} - 12 q^{23} + 10 q^{27} - 5 q^{29} + 6 q^{31} - 6 q^{33} + 4 q^{37} + 3 q^{39} - 6 q^{41} + 3 q^{43} + q^{47} - 11 q^{49} + 6 q^{51} + 2 q^{53} - 5 q^{57} + 15 q^{59} + 4 q^{61} - 2 q^{63} - 14 q^{67} + 12 q^{69} + 11 q^{71} - 18 q^{73} - 2 q^{77} + 5 q^{79} + 2 q^{81} + 8 q^{83} + 5 q^{87} - 9 q^{91} - 6 q^{93} - q^{97} - 12 q^{99} + O(q^{100}) \)

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 0
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.618034 −0.233595 −0.116797 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 5.23607 1.57873 0.789367 0.613922i \(-0.210409\pi\)
0.789367 + 0.613922i \(0.210409\pi\)
\(12\) 0 0
\(13\) 1.85410 0.514235 0.257118 0.966380i \(-0.417227\pi\)
0.257118 + 0.966380i \(0.417227\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.23607 −1.26993 −0.634967 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) 0 0
\(19\) −0.854102 −0.195944 −0.0979722 0.995189i \(-0.531236\pi\)
−0.0979722 + 0.995189i \(0.531236\pi\)
\(20\) 0 0
\(21\) 0.618034 0.134866
\(22\) 0 0
\(23\) −3.76393 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −3.61803 −0.671852 −0.335926 0.941888i \(-0.609049\pi\)
−0.335926 + 0.941888i \(0.609049\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) −5.23607 −0.911482
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.236068 −0.0388093 −0.0194047 0.999812i \(-0.506177\pi\)
−0.0194047 + 0.999812i \(0.506177\pi\)
\(38\) 0 0
\(39\) −1.85410 −0.296894
\(40\) 0 0
\(41\) −0.763932 −0.119306 −0.0596531 0.998219i \(-0.518999\pi\)
−0.0596531 + 0.998219i \(0.518999\pi\)
\(42\) 0 0
\(43\) 4.85410 0.740244 0.370122 0.928983i \(-0.379316\pi\)
0.370122 + 0.928983i \(0.379316\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.618034 −0.0901495 −0.0450748 0.998984i \(-0.514353\pi\)
−0.0450748 + 0.998984i \(0.514353\pi\)
\(48\) 0 0
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) 5.23607 0.733196
\(52\) 0 0
\(53\) −3.47214 −0.476935 −0.238467 0.971151i \(-0.576645\pi\)
−0.238467 + 0.971151i \(0.576645\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.854102 0.113129
\(58\) 0 0
\(59\) 10.8541 1.41308 0.706542 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(60\) 0 0
\(61\) 8.70820 1.11497 0.557486 0.830187i \(-0.311766\pi\)
0.557486 + 0.830187i \(0.311766\pi\)
\(62\) 0 0
\(63\) 1.23607 0.155730
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.76393 −0.582007 −0.291003 0.956722i \(-0.593989\pi\)
−0.291003 + 0.956722i \(0.593989\pi\)
\(68\) 0 0
\(69\) 3.76393 0.453124
\(70\) 0 0
\(71\) 6.61803 0.785416 0.392708 0.919663i \(-0.371538\pi\)
0.392708 + 0.919663i \(0.371538\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.23607 −0.368784
\(78\) 0 0
\(79\) 8.09017 0.910215 0.455108 0.890436i \(-0.349601\pi\)
0.455108 + 0.890436i \(0.349601\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.23607 0.684497 0.342249 0.939609i \(-0.388811\pi\)
0.342249 + 0.939609i \(0.388811\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.61803 0.387894
\(88\) 0 0
\(89\) −8.94427 −0.948091 −0.474045 0.880500i \(-0.657207\pi\)
−0.474045 + 0.880500i \(0.657207\pi\)
\(90\) 0 0
\(91\) −1.14590 −0.120123
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.85410 −0.391325 −0.195662 0.980671i \(-0.562686\pi\)
−0.195662 + 0.980671i \(0.562686\pi\)
\(98\) 0 0
\(99\) −10.4721 −1.05249
\(100\) 0 0
\(101\) 1.47214 0.146483 0.0732415 0.997314i \(-0.476666\pi\)
0.0732415 + 0.997314i \(0.476666\pi\)
\(102\) 0 0
\(103\) −8.56231 −0.843669 −0.421835 0.906673i \(-0.638614\pi\)
−0.421835 + 0.906673i \(0.638614\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4164 1.58703 0.793517 0.608548i \(-0.208248\pi\)
0.793517 + 0.608548i \(0.208248\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0.236068 0.0224066
\(112\) 0 0
\(113\) 16.8541 1.58550 0.792750 0.609547i \(-0.208648\pi\)
0.792750 + 0.609547i \(0.208648\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.70820 −0.342824
\(118\) 0 0
\(119\) 3.23607 0.296650
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) 0 0
\(123\) 0.763932 0.0688814
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −19.8885 −1.76482 −0.882411 0.470479i \(-0.844081\pi\)
−0.882411 + 0.470479i \(0.844081\pi\)
\(128\) 0 0
\(129\) −4.85410 −0.427380
\(130\) 0 0
\(131\) −6.79837 −0.593977 −0.296988 0.954881i \(-0.595982\pi\)
−0.296988 + 0.954881i \(0.595982\pi\)
\(132\) 0 0
\(133\) 0.527864 0.0457716
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.9443 −1.02047 −0.510234 0.860036i \(-0.670441\pi\)
−0.510234 + 0.860036i \(0.670441\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 0.618034 0.0520479
\(142\) 0 0
\(143\) 9.70820 0.811841
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.61803 0.545846
\(148\) 0 0
\(149\) −3.94427 −0.323127 −0.161564 0.986862i \(-0.551654\pi\)
−0.161564 + 0.986862i \(0.551654\pi\)
\(150\) 0 0
\(151\) −14.5623 −1.18506 −0.592532 0.805547i \(-0.701872\pi\)
−0.592532 + 0.805547i \(0.701872\pi\)
\(152\) 0 0
\(153\) 10.4721 0.846622
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.1803 1.05191 0.525953 0.850514i \(-0.323709\pi\)
0.525953 + 0.850514i \(0.323709\pi\)
\(158\) 0 0
\(159\) 3.47214 0.275358
\(160\) 0 0
\(161\) 2.32624 0.183333
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.5623 −1.12687 −0.563433 0.826162i \(-0.690520\pi\)
−0.563433 + 0.826162i \(0.690520\pi\)
\(168\) 0 0
\(169\) −9.56231 −0.735562
\(170\) 0 0
\(171\) 1.70820 0.130630
\(172\) 0 0
\(173\) 18.8885 1.43607 0.718035 0.696007i \(-0.245042\pi\)
0.718035 + 0.696007i \(0.245042\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10.8541 −0.815844
\(178\) 0 0
\(179\) 0.527864 0.0394544 0.0197272 0.999805i \(-0.493720\pi\)
0.0197272 + 0.999805i \(0.493720\pi\)
\(180\) 0 0
\(181\) 0.291796 0.0216890 0.0108445 0.999941i \(-0.496548\pi\)
0.0108445 + 0.999941i \(0.496548\pi\)
\(182\) 0 0
\(183\) −8.70820 −0.643729
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −27.4164 −2.00489
\(188\) 0 0
\(189\) −3.09017 −0.224777
\(190\) 0 0
\(191\) 1.81966 0.131666 0.0658330 0.997831i \(-0.479030\pi\)
0.0658330 + 0.997831i \(0.479030\pi\)
\(192\) 0 0
\(193\) 7.70820 0.554849 0.277424 0.960747i \(-0.410519\pi\)
0.277424 + 0.960747i \(0.410519\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.70820 0.264199 0.132099 0.991236i \(-0.457828\pi\)
0.132099 + 0.991236i \(0.457828\pi\)
\(198\) 0 0
\(199\) 17.5623 1.24496 0.622479 0.782636i \(-0.286125\pi\)
0.622479 + 0.782636i \(0.286125\pi\)
\(200\) 0 0
\(201\) 4.76393 0.336022
\(202\) 0 0
\(203\) 2.23607 0.156941
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.52786 0.523223
\(208\) 0 0
\(209\) −4.47214 −0.309344
\(210\) 0 0
\(211\) 9.18034 0.632001 0.316000 0.948759i \(-0.397660\pi\)
0.316000 + 0.948759i \(0.397660\pi\)
\(212\) 0 0
\(213\) −6.61803 −0.453460
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.85410 −0.125865
\(218\) 0 0
\(219\) 9.00000 0.608164
\(220\) 0 0
\(221\) −9.70820 −0.653044
\(222\) 0 0
\(223\) 0.180340 0.0120765 0.00603823 0.999982i \(-0.498078\pi\)
0.00603823 + 0.999982i \(0.498078\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.7639 −0.979917 −0.489958 0.871746i \(-0.662988\pi\)
−0.489958 + 0.871746i \(0.662988\pi\)
\(228\) 0 0
\(229\) 21.7082 1.43452 0.717259 0.696806i \(-0.245396\pi\)
0.717259 + 0.696806i \(0.245396\pi\)
\(230\) 0 0
\(231\) 3.23607 0.212918
\(232\) 0 0
\(233\) −2.94427 −0.192886 −0.0964428 0.995339i \(-0.530746\pi\)
−0.0964428 + 0.995339i \(0.530746\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.09017 −0.525513
\(238\) 0 0
\(239\) 20.5279 1.32784 0.663919 0.747805i \(-0.268892\pi\)
0.663919 + 0.747805i \(0.268892\pi\)
\(240\) 0 0
\(241\) 2.52786 0.162834 0.0814170 0.996680i \(-0.474055\pi\)
0.0814170 + 0.996680i \(0.474055\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.58359 −0.100762
\(248\) 0 0
\(249\) −6.23607 −0.395195
\(250\) 0 0
\(251\) 29.1803 1.84185 0.920923 0.389744i \(-0.127436\pi\)
0.920923 + 0.389744i \(0.127436\pi\)
\(252\) 0 0
\(253\) −19.7082 −1.23904
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.8541 1.42560 0.712800 0.701367i \(-0.247427\pi\)
0.712800 + 0.701367i \(0.247427\pi\)
\(258\) 0 0
\(259\) 0.145898 0.00906566
\(260\) 0 0
\(261\) 7.23607 0.447901
\(262\) 0 0
\(263\) 10.9098 0.672729 0.336364 0.941732i \(-0.390803\pi\)
0.336364 + 0.941732i \(0.390803\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.94427 0.547381
\(268\) 0 0
\(269\) −12.7639 −0.778231 −0.389115 0.921189i \(-0.627219\pi\)
−0.389115 + 0.921189i \(0.627219\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 1.14590 0.0693529
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −24.7082 −1.48457 −0.742286 0.670083i \(-0.766258\pi\)
−0.742286 + 0.670083i \(0.766258\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 10.0902 0.601929 0.300965 0.953635i \(-0.402691\pi\)
0.300965 + 0.953635i \(0.402691\pi\)
\(282\) 0 0
\(283\) 29.8541 1.77464 0.887321 0.461152i \(-0.152564\pi\)
0.887321 + 0.461152i \(0.152564\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.472136 0.0278693
\(288\) 0 0
\(289\) 10.4164 0.612730
\(290\) 0 0
\(291\) 3.85410 0.225931
\(292\) 0 0
\(293\) −19.5279 −1.14083 −0.570415 0.821357i \(-0.693218\pi\)
−0.570415 + 0.821357i \(0.693218\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 26.1803 1.51914
\(298\) 0 0
\(299\) −6.97871 −0.403589
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) −1.47214 −0.0845720
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.23607 −0.527130 −0.263565 0.964642i \(-0.584898\pi\)
−0.263565 + 0.964642i \(0.584898\pi\)
\(308\) 0 0
\(309\) 8.56231 0.487093
\(310\) 0 0
\(311\) −8.50658 −0.482364 −0.241182 0.970480i \(-0.577535\pi\)
−0.241182 + 0.970480i \(0.577535\pi\)
\(312\) 0 0
\(313\) −16.7639 −0.947553 −0.473777 0.880645i \(-0.657110\pi\)
−0.473777 + 0.880645i \(0.657110\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.65248 0.429806 0.214903 0.976635i \(-0.431056\pi\)
0.214903 + 0.976635i \(0.431056\pi\)
\(318\) 0 0
\(319\) −18.9443 −1.06068
\(320\) 0 0
\(321\) −16.4164 −0.916275
\(322\) 0 0
\(323\) 4.47214 0.248836
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 0.381966 0.0210585
\(330\) 0 0
\(331\) 23.1246 1.27104 0.635522 0.772083i \(-0.280785\pi\)
0.635522 + 0.772083i \(0.280785\pi\)
\(332\) 0 0
\(333\) 0.472136 0.0258729
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.85410 0.427840 0.213920 0.976851i \(-0.431377\pi\)
0.213920 + 0.976851i \(0.431377\pi\)
\(338\) 0 0
\(339\) −16.8541 −0.915389
\(340\) 0 0
\(341\) 15.7082 0.850647
\(342\) 0 0
\(343\) 8.41641 0.454443
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.9098 1.06882 0.534408 0.845227i \(-0.320535\pi\)
0.534408 + 0.845227i \(0.320535\pi\)
\(348\) 0 0
\(349\) 21.7082 1.16201 0.581007 0.813899i \(-0.302659\pi\)
0.581007 + 0.813899i \(0.302659\pi\)
\(350\) 0 0
\(351\) 9.27051 0.494823
\(352\) 0 0
\(353\) 12.9098 0.687121 0.343560 0.939131i \(-0.388367\pi\)
0.343560 + 0.939131i \(0.388367\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3.23607 −0.171271
\(358\) 0 0
\(359\) −13.7426 −0.725309 −0.362655 0.931924i \(-0.618130\pi\)
−0.362655 + 0.931924i \(0.618130\pi\)
\(360\) 0 0
\(361\) −18.2705 −0.961606
\(362\) 0 0
\(363\) −16.4164 −0.861638
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 25.5623 1.33434 0.667171 0.744905i \(-0.267505\pi\)
0.667171 + 0.744905i \(0.267505\pi\)
\(368\) 0 0
\(369\) 1.52786 0.0795374
\(370\) 0 0
\(371\) 2.14590 0.111409
\(372\) 0 0
\(373\) −28.2705 −1.46379 −0.731896 0.681417i \(-0.761364\pi\)
−0.731896 + 0.681417i \(0.761364\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.70820 −0.345490
\(378\) 0 0
\(379\) −14.5967 −0.749785 −0.374892 0.927068i \(-0.622320\pi\)
−0.374892 + 0.927068i \(0.622320\pi\)
\(380\) 0 0
\(381\) 19.8885 1.01892
\(382\) 0 0
\(383\) −33.3607 −1.70465 −0.852326 0.523012i \(-0.824808\pi\)
−0.852326 + 0.523012i \(0.824808\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9.70820 −0.493496
\(388\) 0 0
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 19.7082 0.996687
\(392\) 0 0
\(393\) 6.79837 0.342933
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 29.0344 1.45720 0.728598 0.684941i \(-0.240172\pi\)
0.728598 + 0.684941i \(0.240172\pi\)
\(398\) 0 0
\(399\) −0.527864 −0.0264263
\(400\) 0 0
\(401\) 26.5967 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(402\) 0 0
\(403\) 5.56231 0.277078
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.23607 −0.0612696
\(408\) 0 0
\(409\) 1.58359 0.0783036 0.0391518 0.999233i \(-0.487534\pi\)
0.0391518 + 0.999233i \(0.487534\pi\)
\(410\) 0 0
\(411\) 11.9443 0.589167
\(412\) 0 0
\(413\) −6.70820 −0.330089
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) 9.47214 0.462744 0.231372 0.972865i \(-0.425679\pi\)
0.231372 + 0.972865i \(0.425679\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 0 0
\(423\) 1.23607 0.0600997
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.38197 −0.260452
\(428\) 0 0
\(429\) −9.70820 −0.468717
\(430\) 0 0
\(431\) 29.8328 1.43700 0.718498 0.695529i \(-0.244830\pi\)
0.718498 + 0.695529i \(0.244830\pi\)
\(432\) 0 0
\(433\) 26.8541 1.29053 0.645263 0.763961i \(-0.276748\pi\)
0.645263 + 0.763961i \(0.276748\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.21478 0.153784
\(438\) 0 0
\(439\) 40.9787 1.95581 0.977904 0.209056i \(-0.0670391\pi\)
0.977904 + 0.209056i \(0.0670391\pi\)
\(440\) 0 0
\(441\) 13.2361 0.630289
\(442\) 0 0
\(443\) −29.9443 −1.42270 −0.711348 0.702840i \(-0.751915\pi\)
−0.711348 + 0.702840i \(0.751915\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3.94427 0.186558
\(448\) 0 0
\(449\) 4.67376 0.220568 0.110284 0.993900i \(-0.464824\pi\)
0.110284 + 0.993900i \(0.464824\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) 14.5623 0.684197
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −21.4164 −1.00182 −0.500909 0.865500i \(-0.667001\pi\)
−0.500909 + 0.865500i \(0.667001\pi\)
\(458\) 0 0
\(459\) −26.1803 −1.22199
\(460\) 0 0
\(461\) 0.819660 0.0381754 0.0190877 0.999818i \(-0.493924\pi\)
0.0190877 + 0.999818i \(0.493924\pi\)
\(462\) 0 0
\(463\) 24.1246 1.12117 0.560583 0.828098i \(-0.310577\pi\)
0.560583 + 0.828098i \(0.310577\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −27.4508 −1.27027 −0.635137 0.772400i \(-0.719056\pi\)
−0.635137 + 0.772400i \(0.719056\pi\)
\(468\) 0 0
\(469\) 2.94427 0.135954
\(470\) 0 0
\(471\) −13.1803 −0.607318
\(472\) 0 0
\(473\) 25.4164 1.16865
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.94427 0.317956
\(478\) 0 0
\(479\) 10.8541 0.495937 0.247968 0.968768i \(-0.420237\pi\)
0.247968 + 0.968768i \(0.420237\pi\)
\(480\) 0 0
\(481\) −0.437694 −0.0199571
\(482\) 0 0
\(483\) −2.32624 −0.105847
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 36.4164 1.65018 0.825092 0.564998i \(-0.191123\pi\)
0.825092 + 0.564998i \(0.191123\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) 43.2492 1.95181 0.975905 0.218196i \(-0.0700171\pi\)
0.975905 + 0.218196i \(0.0700171\pi\)
\(492\) 0 0
\(493\) 18.9443 0.853207
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.09017 −0.183469
\(498\) 0 0
\(499\) −7.56231 −0.338535 −0.169268 0.985570i \(-0.554140\pi\)
−0.169268 + 0.985570i \(0.554140\pi\)
\(500\) 0 0
\(501\) 14.5623 0.650596
\(502\) 0 0
\(503\) 37.4164 1.66832 0.834158 0.551526i \(-0.185954\pi\)
0.834158 + 0.551526i \(0.185954\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.56231 0.424677
\(508\) 0 0
\(509\) −20.3262 −0.900945 −0.450472 0.892790i \(-0.648744\pi\)
−0.450472 + 0.892790i \(0.648744\pi\)
\(510\) 0 0
\(511\) 5.56231 0.246062
\(512\) 0 0
\(513\) −4.27051 −0.188548
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.23607 −0.142322
\(518\) 0 0
\(519\) −18.8885 −0.829115
\(520\) 0 0
\(521\) 29.3607 1.28631 0.643157 0.765734i \(-0.277624\pi\)
0.643157 + 0.765734i \(0.277624\pi\)
\(522\) 0 0
\(523\) 13.1459 0.574830 0.287415 0.957806i \(-0.407204\pi\)
0.287415 + 0.957806i \(0.407204\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.7082 −0.684260
\(528\) 0 0
\(529\) −8.83282 −0.384035
\(530\) 0 0
\(531\) −21.7082 −0.942056
\(532\) 0 0
\(533\) −1.41641 −0.0613514
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.527864 −0.0227790
\(538\) 0 0
\(539\) −34.6525 −1.49259
\(540\) 0 0
\(541\) 27.1246 1.16618 0.583089 0.812408i \(-0.301844\pi\)
0.583089 + 0.812408i \(0.301844\pi\)
\(542\) 0 0
\(543\) −0.291796 −0.0125222
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21.2918 0.910371 0.455186 0.890397i \(-0.349573\pi\)
0.455186 + 0.890397i \(0.349573\pi\)
\(548\) 0 0
\(549\) −17.4164 −0.743314
\(550\) 0 0
\(551\) 3.09017 0.131646
\(552\) 0 0
\(553\) −5.00000 −0.212622
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.76393 0.201854 0.100927 0.994894i \(-0.467819\pi\)
0.100927 + 0.994894i \(0.467819\pi\)
\(558\) 0 0
\(559\) 9.00000 0.380659
\(560\) 0 0
\(561\) 27.4164 1.15752
\(562\) 0 0
\(563\) −7.38197 −0.311113 −0.155556 0.987827i \(-0.549717\pi\)
−0.155556 + 0.987827i \(0.549717\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.618034 −0.0259550
\(568\) 0 0
\(569\) 20.5279 0.860573 0.430286 0.902692i \(-0.358413\pi\)
0.430286 + 0.902692i \(0.358413\pi\)
\(570\) 0 0
\(571\) 8.12461 0.340004 0.170002 0.985444i \(-0.445623\pi\)
0.170002 + 0.985444i \(0.445623\pi\)
\(572\) 0 0
\(573\) −1.81966 −0.0760174
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −33.7771 −1.40616 −0.703079 0.711111i \(-0.748192\pi\)
−0.703079 + 0.711111i \(0.748192\pi\)
\(578\) 0 0
\(579\) −7.70820 −0.320342
\(580\) 0 0
\(581\) −3.85410 −0.159895
\(582\) 0 0
\(583\) −18.1803 −0.752953
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.29180 −0.218416 −0.109208 0.994019i \(-0.534831\pi\)
−0.109208 + 0.994019i \(0.534831\pi\)
\(588\) 0 0
\(589\) −2.56231 −0.105578
\(590\) 0 0
\(591\) −3.70820 −0.152535
\(592\) 0 0
\(593\) −10.9098 −0.448013 −0.224007 0.974588i \(-0.571914\pi\)
−0.224007 + 0.974588i \(0.571914\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −17.5623 −0.718777
\(598\) 0 0
\(599\) 9.47214 0.387021 0.193510 0.981098i \(-0.438013\pi\)
0.193510 + 0.981098i \(0.438013\pi\)
\(600\) 0 0
\(601\) 2.72949 0.111338 0.0556691 0.998449i \(-0.482271\pi\)
0.0556691 + 0.998449i \(0.482271\pi\)
\(602\) 0 0
\(603\) 9.52786 0.388005
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 35.5623 1.44343 0.721715 0.692191i \(-0.243354\pi\)
0.721715 + 0.692191i \(0.243354\pi\)
\(608\) 0 0
\(609\) −2.23607 −0.0906100
\(610\) 0 0
\(611\) −1.14590 −0.0463581
\(612\) 0 0
\(613\) −14.9787 −0.604985 −0.302492 0.953152i \(-0.597819\pi\)
−0.302492 + 0.953152i \(0.597819\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.2361 0.573123 0.286561 0.958062i \(-0.407488\pi\)
0.286561 + 0.958062i \(0.407488\pi\)
\(618\) 0 0
\(619\) −30.5279 −1.22702 −0.613509 0.789688i \(-0.710243\pi\)
−0.613509 + 0.789688i \(0.710243\pi\)
\(620\) 0 0
\(621\) −18.8197 −0.755207
\(622\) 0 0
\(623\) 5.52786 0.221469
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4.47214 0.178600
\(628\) 0 0
\(629\) 1.23607 0.0492853
\(630\) 0 0
\(631\) 10.2361 0.407491 0.203746 0.979024i \(-0.434688\pi\)
0.203746 + 0.979024i \(0.434688\pi\)
\(632\) 0 0
\(633\) −9.18034 −0.364886
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −12.2705 −0.486175
\(638\) 0 0
\(639\) −13.2361 −0.523611
\(640\) 0 0
\(641\) −1.09017 −0.0430591 −0.0215296 0.999768i \(-0.506854\pi\)
−0.0215296 + 0.999768i \(0.506854\pi\)
\(642\) 0 0
\(643\) 30.8328 1.21593 0.607964 0.793965i \(-0.291987\pi\)
0.607964 + 0.793965i \(0.291987\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.5410 1.43658 0.718288 0.695746i \(-0.244926\pi\)
0.718288 + 0.695746i \(0.244926\pi\)
\(648\) 0 0
\(649\) 56.8328 2.23088
\(650\) 0 0
\(651\) 1.85410 0.0726680
\(652\) 0 0
\(653\) 19.0902 0.747056 0.373528 0.927619i \(-0.378148\pi\)
0.373528 + 0.927619i \(0.378148\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) −15.5279 −0.604880 −0.302440 0.953168i \(-0.597801\pi\)
−0.302440 + 0.953168i \(0.597801\pi\)
\(660\) 0 0
\(661\) 19.6869 0.765732 0.382866 0.923804i \(-0.374937\pi\)
0.382866 + 0.923804i \(0.374937\pi\)
\(662\) 0 0
\(663\) 9.70820 0.377035
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.6180 0.527292
\(668\) 0 0
\(669\) −0.180340 −0.00697234
\(670\) 0 0
\(671\) 45.5967 1.76024
\(672\) 0 0
\(673\) 12.1803 0.469518 0.234759 0.972054i \(-0.424570\pi\)
0.234759 + 0.972054i \(0.424570\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.6180 0.408084 0.204042 0.978962i \(-0.434592\pi\)
0.204042 + 0.978962i \(0.434592\pi\)
\(678\) 0 0
\(679\) 2.38197 0.0914115
\(680\) 0 0
\(681\) 14.7639 0.565755
\(682\) 0 0
\(683\) 13.4721 0.515497 0.257748 0.966212i \(-0.417019\pi\)
0.257748 + 0.966212i \(0.417019\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −21.7082 −0.828220
\(688\) 0 0
\(689\) −6.43769 −0.245257
\(690\) 0 0
\(691\) −36.2705 −1.37980 −0.689898 0.723907i \(-0.742344\pi\)
−0.689898 + 0.723907i \(0.742344\pi\)
\(692\) 0 0
\(693\) 6.47214 0.245856
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) 2.94427 0.111363
\(700\) 0 0
\(701\) −41.0132 −1.54905 −0.774523 0.632546i \(-0.782010\pi\)
−0.774523 + 0.632546i \(0.782010\pi\)
\(702\) 0 0
\(703\) 0.201626 0.00760447
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.909830 −0.0342177
\(708\) 0 0
\(709\) −33.5410 −1.25966 −0.629830 0.776733i \(-0.716875\pi\)
−0.629830 + 0.776733i \(0.716875\pi\)
\(710\) 0 0
\(711\) −16.1803 −0.606810
\(712\) 0 0
\(713\) −11.2918 −0.422881
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20.5279 −0.766627
\(718\) 0 0
\(719\) 23.2918 0.868637 0.434319 0.900759i \(-0.356989\pi\)
0.434319 + 0.900759i \(0.356989\pi\)
\(720\) 0 0
\(721\) 5.29180 0.197077
\(722\) 0 0
\(723\) −2.52786 −0.0940123
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −24.5623 −0.910966 −0.455483 0.890245i \(-0.650533\pi\)
−0.455483 + 0.890245i \(0.650533\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −25.4164 −0.940060
\(732\) 0 0
\(733\) −19.9787 −0.737931 −0.368965 0.929443i \(-0.620288\pi\)
−0.368965 + 0.929443i \(0.620288\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.9443 −0.918834
\(738\) 0 0
\(739\) −15.9787 −0.587786 −0.293893 0.955838i \(-0.594951\pi\)
−0.293893 + 0.955838i \(0.594951\pi\)
\(740\) 0 0
\(741\) 1.58359 0.0581747
\(742\) 0 0
\(743\) −28.3607 −1.04045 −0.520226 0.854029i \(-0.674152\pi\)
−0.520226 + 0.854029i \(0.674152\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.4721 −0.456332
\(748\) 0 0
\(749\) −10.1459 −0.370723
\(750\) 0 0
\(751\) 5.11146 0.186520 0.0932598 0.995642i \(-0.470271\pi\)
0.0932598 + 0.995642i \(0.470271\pi\)
\(752\) 0 0
\(753\) −29.1803 −1.06339
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30.4164 1.10550 0.552752 0.833346i \(-0.313578\pi\)
0.552752 + 0.833346i \(0.313578\pi\)
\(758\) 0 0
\(759\) 19.7082 0.715362
\(760\) 0 0
\(761\) −18.4508 −0.668843 −0.334421 0.942424i \(-0.608541\pi\)
−0.334421 + 0.942424i \(0.608541\pi\)
\(762\) 0 0
\(763\) −6.18034 −0.223743
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.1246 0.726658
\(768\) 0 0
\(769\) 13.4164 0.483808 0.241904 0.970300i \(-0.422228\pi\)
0.241904 + 0.970300i \(0.422228\pi\)
\(770\) 0 0
\(771\) −22.8541 −0.823070
\(772\) 0 0
\(773\) −36.1591 −1.30055 −0.650275 0.759699i \(-0.725346\pi\)
−0.650275 + 0.759699i \(0.725346\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −0.145898 −0.00523406
\(778\) 0 0
\(779\) 0.652476 0.0233774
\(780\) 0 0
\(781\) 34.6525 1.23996
\(782\) 0 0
\(783\) −18.0902 −0.646490
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 11.8197 0.421325 0.210663 0.977559i \(-0.432438\pi\)
0.210663 + 0.977559i \(0.432438\pi\)
\(788\) 0 0
\(789\) −10.9098 −0.388400
\(790\) 0 0
\(791\) −10.4164 −0.370365
\(792\) 0 0
\(793\) 16.1459 0.573358
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.76393 0.345856 0.172928 0.984934i \(-0.444677\pi\)
0.172928 + 0.984934i \(0.444677\pi\)
\(798\) 0 0
\(799\) 3.23607 0.114484
\(800\) 0 0
\(801\) 17.8885 0.632061
\(802\) 0 0
\(803\) −47.1246 −1.66299
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.7639 0.449312
\(808\) 0 0
\(809\) 30.9787 1.08915 0.544577 0.838711i \(-0.316690\pi\)
0.544577 + 0.838711i \(0.316690\pi\)
\(810\) 0 0
\(811\) 14.7082 0.516475 0.258237 0.966081i \(-0.416858\pi\)
0.258237 + 0.966081i \(0.416858\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.14590 −0.145047
\(818\) 0 0
\(819\) 2.29180 0.0800818
\(820\) 0 0
\(821\) −40.6869 −1.41998 −0.709992 0.704210i \(-0.751301\pi\)
−0.709992 + 0.704210i \(0.751301\pi\)
\(822\) 0 0
\(823\) −47.7082 −1.66300 −0.831502 0.555522i \(-0.812518\pi\)
−0.831502 + 0.555522i \(0.812518\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.965558 0.0335757 0.0167879 0.999859i \(-0.494656\pi\)
0.0167879 + 0.999859i \(0.494656\pi\)
\(828\) 0 0
\(829\) −35.8541 −1.24526 −0.622632 0.782515i \(-0.713937\pi\)
−0.622632 + 0.782515i \(0.713937\pi\)
\(830\) 0 0
\(831\) 24.7082 0.857118
\(832\) 0 0
\(833\) 34.6525 1.20064
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) 0 0
\(839\) −10.8541 −0.374725 −0.187363 0.982291i \(-0.559994\pi\)
−0.187363 + 0.982291i \(0.559994\pi\)
\(840\) 0 0
\(841\) −15.9098 −0.548615
\(842\) 0 0
\(843\) −10.0902 −0.347524
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.1459 −0.348617
\(848\) 0 0
\(849\) −29.8541 −1.02459
\(850\) 0 0
\(851\) 0.888544 0.0304589
\(852\) 0 0
\(853\) −15.3050 −0.524032 −0.262016 0.965064i \(-0.584387\pi\)
−0.262016 + 0.965064i \(0.584387\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.6869 0.672492 0.336246 0.941774i \(-0.390843\pi\)
0.336246 + 0.941774i \(0.390843\pi\)
\(858\) 0 0
\(859\) 1.58359 0.0540315 0.0270157 0.999635i \(-0.491400\pi\)
0.0270157 + 0.999635i \(0.491400\pi\)
\(860\) 0 0
\(861\) −0.472136 −0.0160904
\(862\) 0 0
\(863\) 21.4377 0.729748 0.364874 0.931057i \(-0.381112\pi\)
0.364874 + 0.931057i \(0.381112\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −10.4164 −0.353760
\(868\) 0 0
\(869\) 42.3607 1.43699
\(870\) 0 0
\(871\) −8.83282 −0.299289
\(872\) 0 0
\(873\) 7.70820 0.260883
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36.5410 −1.23390 −0.616951 0.787001i \(-0.711632\pi\)
−0.616951 + 0.787001i \(0.711632\pi\)
\(878\) 0 0
\(879\) 19.5279 0.658659
\(880\) 0 0
\(881\) −40.3607 −1.35979 −0.679893 0.733311i \(-0.737974\pi\)
−0.679893 + 0.733311i \(0.737974\pi\)
\(882\) 0 0
\(883\) 20.5836 0.692693 0.346347 0.938107i \(-0.387422\pi\)
0.346347 + 0.938107i \(0.387422\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −29.8885 −1.00356 −0.501780 0.864996i \(-0.667321\pi\)
−0.501780 + 0.864996i \(0.667321\pi\)
\(888\) 0 0
\(889\) 12.2918 0.412254
\(890\) 0 0
\(891\) 5.23607 0.175415
\(892\) 0 0
\(893\) 0.527864 0.0176643
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.97871 0.233012
\(898\) 0 0
\(899\) −10.8541 −0.362005
\(900\) 0 0
\(901\) 18.1803 0.605675
\(902\) 0 0
\(903\) 3.00000 0.0998337
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 33.2492 1.10402 0.552011 0.833837i \(-0.313861\pi\)
0.552011 + 0.833837i \(0.313861\pi\)
\(908\) 0 0
\(909\) −2.94427 −0.0976553
\(910\) 0 0
\(911\) 40.2361 1.33308 0.666540 0.745469i \(-0.267774\pi\)
0.666540 + 0.745469i \(0.267774\pi\)
\(912\) 0 0
\(913\) 32.6525 1.08064
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.20163 0.138750
\(918\) 0 0
\(919\) 53.2148 1.75539 0.877697 0.479216i \(-0.159079\pi\)
0.877697 + 0.479216i \(0.159079\pi\)
\(920\) 0 0
\(921\) 9.23607 0.304339
\(922\) 0 0
\(923\) 12.2705 0.403889
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 17.1246 0.562446
\(928\) 0 0
\(929\) 41.6312 1.36588 0.682938 0.730477i \(-0.260702\pi\)
0.682938 + 0.730477i \(0.260702\pi\)
\(930\) 0 0
\(931\) 5.65248 0.185252
\(932\) 0 0
\(933\) 8.50658 0.278493
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17.7295 0.579197 0.289599 0.957148i \(-0.406478\pi\)
0.289599 + 0.957148i \(0.406478\pi\)
\(938\) 0 0
\(939\) 16.7639 0.547070
\(940\) 0 0
\(941\) −46.4164 −1.51313 −0.756566 0.653918i \(-0.773124\pi\)
−0.756566 + 0.653918i \(0.773124\pi\)
\(942\) 0 0
\(943\) 2.87539 0.0936355
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.65248 −0.0861939 −0.0430969 0.999071i \(-0.513722\pi\)
−0.0430969 + 0.999071i \(0.513722\pi\)
\(948\) 0 0
\(949\) −16.6869 −0.541680
\(950\) 0 0
\(951\) −7.65248 −0.248149
\(952\) 0 0
\(953\) −7.74265 −0.250809 −0.125404 0.992106i \(-0.540023\pi\)
−0.125404 + 0.992106i \(0.540023\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18.9443 0.612381
\(958\) 0 0
\(959\) 7.38197 0.238376
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −32.8328 −1.05802
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −4.11146 −0.132216 −0.0661078 0.997812i \(-0.521058\pi\)
−0.0661078 + 0.997812i \(0.521058\pi\)
\(968\) 0 0
\(969\) −4.47214 −0.143666
\(970\) 0 0
\(971\) −5.61803 −0.180291 −0.0901456 0.995929i \(-0.528733\pi\)
−0.0901456 + 0.995929i \(0.528733\pi\)
\(972\) 0 0
\(973\) 3.09017 0.0990663
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.34752 −0.0751040 −0.0375520 0.999295i \(-0.511956\pi\)
−0.0375520 + 0.999295i \(0.511956\pi\)
\(978\) 0 0
\(979\) −46.8328 −1.49678
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) 0 0
\(983\) −9.61803 −0.306768 −0.153384 0.988167i \(-0.549017\pi\)
−0.153384 + 0.988167i \(0.549017\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −0.381966 −0.0121581
\(988\) 0 0
\(989\) −18.2705 −0.580968
\(990\) 0 0
\(991\) 15.3607 0.487948 0.243974 0.969782i \(-0.421549\pi\)
0.243974 + 0.969782i \(0.421549\pi\)
\(992\) 0 0
\(993\) −23.1246 −0.733837
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 24.8885 0.788228 0.394114 0.919062i \(-0.371051\pi\)
0.394114 + 0.919062i \(0.371051\pi\)
\(998\) 0 0
\(999\) −1.18034 −0.0373443
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 10000.2.a.c.1.1 2
4.3 odd 2 625.2.a.b.1.1 2
5.4 even 2 10000.2.a.l.1.2 2
12.11 even 2 5625.2.a.f.1.2 2
20.3 even 4 625.2.b.a.624.4 4
20.7 even 4 625.2.b.a.624.1 4
20.19 odd 2 625.2.a.c.1.2 2
25.11 even 5 400.2.u.b.321.1 4
25.16 even 5 400.2.u.b.81.1 4
60.59 even 2 5625.2.a.d.1.1 2
100.3 even 20 625.2.e.c.249.2 8
100.11 odd 10 25.2.d.a.21.1 yes 4
100.19 odd 10 625.2.d.b.251.1 4
100.23 even 20 125.2.e.a.24.2 8
100.27 even 20 125.2.e.a.24.1 8
100.31 odd 10 625.2.d.h.251.1 4
100.39 odd 10 125.2.d.a.101.1 4
100.47 even 20 625.2.e.c.249.1 8
100.59 odd 10 125.2.d.a.26.1 4
100.63 even 20 125.2.e.a.99.1 8
100.67 even 20 625.2.e.c.374.2 8
100.71 odd 10 625.2.d.h.376.1 4
100.79 odd 10 625.2.d.b.376.1 4
100.83 even 20 625.2.e.c.374.1 8
100.87 even 20 125.2.e.a.99.2 8
100.91 odd 10 25.2.d.a.6.1 4
300.11 even 10 225.2.h.b.46.1 4
300.191 even 10 225.2.h.b.181.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.6.1 4 100.91 odd 10
25.2.d.a.21.1 yes 4 100.11 odd 10
125.2.d.a.26.1 4 100.59 odd 10
125.2.d.a.101.1 4 100.39 odd 10
125.2.e.a.24.1 8 100.27 even 20
125.2.e.a.24.2 8 100.23 even 20
125.2.e.a.99.1 8 100.63 even 20
125.2.e.a.99.2 8 100.87 even 20
225.2.h.b.46.1 4 300.11 even 10
225.2.h.b.181.1 4 300.191 even 10
400.2.u.b.81.1 4 25.16 even 5
400.2.u.b.321.1 4 25.11 even 5
625.2.a.b.1.1 2 4.3 odd 2
625.2.a.c.1.2 2 20.19 odd 2
625.2.b.a.624.1 4 20.7 even 4
625.2.b.a.624.4 4 20.3 even 4
625.2.d.b.251.1 4 100.19 odd 10
625.2.d.b.376.1 4 100.79 odd 10
625.2.d.h.251.1 4 100.31 odd 10
625.2.d.h.376.1 4 100.71 odd 10
625.2.e.c.249.1 8 100.47 even 20
625.2.e.c.249.2 8 100.3 even 20
625.2.e.c.374.1 8 100.83 even 20
625.2.e.c.374.2 8 100.67 even 20
5625.2.a.d.1.1 2 60.59 even 2
5625.2.a.f.1.2 2 12.11 even 2
10000.2.a.c.1.1 2 1.1 even 1 trivial
10000.2.a.l.1.2 2 5.4 even 2