Properties

Label 4600.2.a.be.1.1
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Defining polynomial: \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.30649\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.30649 q^{3} +2.55040 q^{7} +7.93288 q^{9} +O(q^{10})\) \(q-3.30649 q^{3} +2.55040 q^{7} +7.93288 q^{9} -2.72314 q^{11} -7.12637 q^{13} -0.924010 q^{17} +7.51623 q^{19} -8.43286 q^{21} +1.00000 q^{23} -16.3105 q^{27} -2.38248 q^{29} +0.866248 q^{31} +9.00402 q^{33} -0.352855 q^{37} +23.5633 q^{39} +4.34066 q^{41} +13.3239 q^{47} -0.495474 q^{49} +3.05523 q^{51} -3.99262 q^{53} -24.8523 q^{57} -3.84064 q^{59} -9.14262 q^{61} +20.2320 q^{63} +3.15933 q^{67} -3.30649 q^{69} -6.07883 q^{71} +11.3239 q^{73} -6.94508 q^{77} -12.0593 q^{79} +30.1319 q^{81} +6.35285 q^{83} +7.87765 q^{87} -9.71377 q^{89} -18.1751 q^{91} -2.86424 q^{93} -8.76465 q^{97} -21.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} - 4 q^{17} + 7 q^{19} + 6 q^{21} + 5 q^{23} - 3 q^{27} + 4 q^{29} + 19 q^{31} - 17 q^{33} - 15 q^{37} + 19 q^{39} + 25 q^{41} + 11 q^{47} + 25 q^{49} + 19 q^{51} - 3 q^{53} - 48 q^{57} - q^{59} - 5 q^{61} + 41 q^{63} - 9 q^{67} + q^{71} + q^{73} - 18 q^{77} - 2 q^{79} + 57 q^{81} + 45 q^{83} + 9 q^{87} + 6 q^{89} + 11 q^{91} + 39 q^{93} - 25 q^{97} - 65 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −3.30649 −1.90900 −0.954501 0.298206i \(-0.903612\pi\)
−0.954501 + 0.298206i \(0.903612\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.55040 0.963960 0.481980 0.876182i \(-0.339918\pi\)
0.481980 + 0.876182i \(0.339918\pi\)
\(8\) 0 0
\(9\) 7.93288 2.64429
\(10\) 0 0
\(11\) −2.72314 −0.821056 −0.410528 0.911848i \(-0.634656\pi\)
−0.410528 + 0.911848i \(0.634656\pi\)
\(12\) 0 0
\(13\) −7.12637 −1.97650 −0.988250 0.152845i \(-0.951156\pi\)
−0.988250 + 0.152845i \(0.951156\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.924010 −0.224105 −0.112053 0.993702i \(-0.535743\pi\)
−0.112053 + 0.993702i \(0.535743\pi\)
\(18\) 0 0
\(19\) 7.51623 1.72434 0.862171 0.506617i \(-0.169104\pi\)
0.862171 + 0.506617i \(0.169104\pi\)
\(20\) 0 0
\(21\) −8.43286 −1.84020
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −16.3105 −3.13896
\(28\) 0 0
\(29\) −2.38248 −0.442415 −0.221208 0.975227i \(-0.571000\pi\)
−0.221208 + 0.975227i \(0.571000\pi\)
\(30\) 0 0
\(31\) 0.866248 0.155583 0.0777913 0.996970i \(-0.475213\pi\)
0.0777913 + 0.996970i \(0.475213\pi\)
\(32\) 0 0
\(33\) 9.00402 1.56740
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.352855 −0.0580089 −0.0290045 0.999579i \(-0.509234\pi\)
−0.0290045 + 0.999579i \(0.509234\pi\)
\(38\) 0 0
\(39\) 23.5633 3.77315
\(40\) 0 0
\(41\) 4.34066 0.677896 0.338948 0.940805i \(-0.389929\pi\)
0.338948 + 0.940805i \(0.389929\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.3239 1.94349 0.971746 0.236027i \(-0.0758454\pi\)
0.971746 + 0.236027i \(0.0758454\pi\)
\(48\) 0 0
\(49\) −0.495474 −0.0707819
\(50\) 0 0
\(51\) 3.05523 0.427818
\(52\) 0 0
\(53\) −3.99262 −0.548429 −0.274214 0.961669i \(-0.588418\pi\)
−0.274214 + 0.961669i \(0.588418\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −24.8523 −3.29177
\(58\) 0 0
\(59\) −3.84064 −0.500009 −0.250004 0.968245i \(-0.580432\pi\)
−0.250004 + 0.968245i \(0.580432\pi\)
\(60\) 0 0
\(61\) −9.14262 −1.17059 −0.585296 0.810820i \(-0.699022\pi\)
−0.585296 + 0.810820i \(0.699022\pi\)
\(62\) 0 0
\(63\) 20.2320 2.54899
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.15933 0.385974 0.192987 0.981201i \(-0.438183\pi\)
0.192987 + 0.981201i \(0.438183\pi\)
\(68\) 0 0
\(69\) −3.30649 −0.398055
\(70\) 0 0
\(71\) −6.07883 −0.721424 −0.360712 0.932677i \(-0.617466\pi\)
−0.360712 + 0.932677i \(0.617466\pi\)
\(72\) 0 0
\(73\) 11.3239 1.32536 0.662682 0.748901i \(-0.269418\pi\)
0.662682 + 0.748901i \(0.269418\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.94508 −0.791465
\(78\) 0 0
\(79\) −12.0593 −1.35677 −0.678386 0.734706i \(-0.737320\pi\)
−0.678386 + 0.734706i \(0.737320\pi\)
\(80\) 0 0
\(81\) 30.1319 3.34799
\(82\) 0 0
\(83\) 6.35285 0.697316 0.348658 0.937250i \(-0.386637\pi\)
0.348658 + 0.937250i \(0.386637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.87765 0.844572
\(88\) 0 0
\(89\) −9.71377 −1.02966 −0.514829 0.857293i \(-0.672145\pi\)
−0.514829 + 0.857293i \(0.672145\pi\)
\(90\) 0 0
\(91\) −18.1751 −1.90527
\(92\) 0 0
\(93\) −2.86424 −0.297008
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.76465 −0.889916 −0.444958 0.895552i \(-0.646781\pi\)
−0.444958 + 0.895552i \(0.646781\pi\)
\(98\) 0 0
\(99\) −21.6023 −2.17111
\(100\) 0 0
\(101\) 4.74794 0.472438 0.236219 0.971700i \(-0.424092\pi\)
0.236219 + 0.971700i \(0.424092\pi\)
\(102\) 0 0
\(103\) −12.6304 −1.24451 −0.622255 0.782814i \(-0.713783\pi\)
−0.622255 + 0.782814i \(0.713783\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.9658 1.25345 0.626727 0.779239i \(-0.284394\pi\)
0.626727 + 0.779239i \(0.284394\pi\)
\(108\) 0 0
\(109\) −15.0040 −1.43712 −0.718562 0.695463i \(-0.755199\pi\)
−0.718562 + 0.695463i \(0.755199\pi\)
\(110\) 0 0
\(111\) 1.16671 0.110739
\(112\) 0 0
\(113\) −2.13496 −0.200840 −0.100420 0.994945i \(-0.532019\pi\)
−0.100420 + 0.994945i \(0.532019\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −56.5326 −5.22644
\(118\) 0 0
\(119\) −2.35659 −0.216029
\(120\) 0 0
\(121\) −3.58454 −0.325867
\(122\) 0 0
\(123\) −14.3523 −1.29411
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.54181 0.314285 0.157142 0.987576i \(-0.449772\pi\)
0.157142 + 0.987576i \(0.449772\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 19.5499 1.70808 0.854042 0.520205i \(-0.174144\pi\)
0.854042 + 0.520205i \(0.174144\pi\)
\(132\) 0 0
\(133\) 19.1694 1.66220
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.41544 −0.377236 −0.188618 0.982051i \(-0.560401\pi\)
−0.188618 + 0.982051i \(0.560401\pi\)
\(138\) 0 0
\(139\) 21.9954 1.86563 0.932814 0.360358i \(-0.117345\pi\)
0.932814 + 0.360358i \(0.117345\pi\)
\(140\) 0 0
\(141\) −44.0554 −3.71013
\(142\) 0 0
\(143\) 19.4061 1.62282
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.63828 0.135123
\(148\) 0 0
\(149\) 19.0780 1.56293 0.781466 0.623947i \(-0.214472\pi\)
0.781466 + 0.623947i \(0.214472\pi\)
\(150\) 0 0
\(151\) −7.75273 −0.630908 −0.315454 0.948941i \(-0.602157\pi\)
−0.315454 + 0.948941i \(0.602157\pi\)
\(152\) 0 0
\(153\) −7.33006 −0.592600
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −17.5129 −1.39768 −0.698841 0.715277i \(-0.746300\pi\)
−0.698841 + 0.715277i \(0.746300\pi\)
\(158\) 0 0
\(159\) 13.2016 1.04695
\(160\) 0 0
\(161\) 2.55040 0.200999
\(162\) 0 0
\(163\) 2.55491 0.200116 0.100058 0.994982i \(-0.468097\pi\)
0.100058 + 0.994982i \(0.468097\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.61301 0.279583 0.139791 0.990181i \(-0.455357\pi\)
0.139791 + 0.990181i \(0.455357\pi\)
\(168\) 0 0
\(169\) 37.7852 2.90655
\(170\) 0 0
\(171\) 59.6253 4.55966
\(172\) 0 0
\(173\) −12.7824 −0.971827 −0.485913 0.874007i \(-0.661513\pi\)
−0.485913 + 0.874007i \(0.661513\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.6990 0.954519
\(178\) 0 0
\(179\) 10.0149 0.748546 0.374273 0.927318i \(-0.377892\pi\)
0.374273 + 0.927318i \(0.377892\pi\)
\(180\) 0 0
\(181\) 9.96248 0.740505 0.370252 0.928931i \(-0.379271\pi\)
0.370252 + 0.928931i \(0.379271\pi\)
\(182\) 0 0
\(183\) 30.2300 2.23466
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.51621 0.184003
\(188\) 0 0
\(189\) −41.5983 −3.02583
\(190\) 0 0
\(191\) −6.61298 −0.478498 −0.239249 0.970958i \(-0.576901\pi\)
−0.239249 + 0.970958i \(0.576901\pi\)
\(192\) 0 0
\(193\) 19.0540 1.37154 0.685768 0.727820i \(-0.259466\pi\)
0.685768 + 0.727820i \(0.259466\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.0886 0.790030 0.395015 0.918675i \(-0.370739\pi\)
0.395015 + 0.918675i \(0.370739\pi\)
\(198\) 0 0
\(199\) 3.71377 0.263263 0.131631 0.991299i \(-0.457979\pi\)
0.131631 + 0.991299i \(0.457979\pi\)
\(200\) 0 0
\(201\) −10.4463 −0.736825
\(202\) 0 0
\(203\) −6.07627 −0.426471
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.93288 0.551373
\(208\) 0 0
\(209\) −20.4677 −1.41578
\(210\) 0 0
\(211\) −21.7656 −1.49841 −0.749204 0.662339i \(-0.769564\pi\)
−0.749204 + 0.662339i \(0.769564\pi\)
\(212\) 0 0
\(213\) 20.0996 1.37720
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.20928 0.149975
\(218\) 0 0
\(219\) −37.4424 −2.53012
\(220\) 0 0
\(221\) 6.58484 0.442944
\(222\) 0 0
\(223\) −6.90730 −0.462547 −0.231273 0.972889i \(-0.574289\pi\)
−0.231273 + 0.972889i \(0.574289\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.0325 1.52872 0.764359 0.644791i \(-0.223055\pi\)
0.764359 + 0.644791i \(0.223055\pi\)
\(228\) 0 0
\(229\) 12.9665 0.856854 0.428427 0.903576i \(-0.359068\pi\)
0.428427 + 0.903576i \(0.359068\pi\)
\(230\) 0 0
\(231\) 22.9638 1.51091
\(232\) 0 0
\(233\) 4.80934 0.315071 0.157535 0.987513i \(-0.449645\pi\)
0.157535 + 0.987513i \(0.449645\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 39.8738 2.59008
\(238\) 0 0
\(239\) −20.4661 −1.32384 −0.661922 0.749573i \(-0.730259\pi\)
−0.661922 + 0.749573i \(0.730259\pi\)
\(240\) 0 0
\(241\) 26.8657 1.73057 0.865287 0.501277i \(-0.167136\pi\)
0.865287 + 0.501277i \(0.167136\pi\)
\(242\) 0 0
\(243\) −50.6993 −3.25236
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −53.5635 −3.40816
\(248\) 0 0
\(249\) −21.0057 −1.33118
\(250\) 0 0
\(251\) −15.9625 −1.00754 −0.503771 0.863837i \(-0.668055\pi\)
−0.503771 + 0.863837i \(0.668055\pi\)
\(252\) 0 0
\(253\) −2.72314 −0.171202
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.33867 −0.208261 −0.104130 0.994564i \(-0.533206\pi\)
−0.104130 + 0.994564i \(0.533206\pi\)
\(258\) 0 0
\(259\) −0.899919 −0.0559183
\(260\) 0 0
\(261\) −18.8999 −1.16988
\(262\) 0 0
\(263\) 23.3880 1.44217 0.721083 0.692849i \(-0.243645\pi\)
0.721083 + 0.692849i \(0.243645\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 32.1185 1.96562
\(268\) 0 0
\(269\) 14.1378 0.861995 0.430998 0.902353i \(-0.358162\pi\)
0.430998 + 0.902353i \(0.358162\pi\)
\(270\) 0 0
\(271\) 25.9283 1.57503 0.787517 0.616293i \(-0.211366\pi\)
0.787517 + 0.616293i \(0.211366\pi\)
\(272\) 0 0
\(273\) 60.0957 3.63716
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.07117 −0.0643603 −0.0321802 0.999482i \(-0.510245\pi\)
−0.0321802 + 0.999482i \(0.510245\pi\)
\(278\) 0 0
\(279\) 6.87184 0.411406
\(280\) 0 0
\(281\) 12.2260 0.729341 0.364671 0.931137i \(-0.381182\pi\)
0.364671 + 0.931137i \(0.381182\pi\)
\(282\) 0 0
\(283\) 30.5454 1.81573 0.907867 0.419259i \(-0.137710\pi\)
0.907867 + 0.419259i \(0.137710\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.0704 0.653465
\(288\) 0 0
\(289\) −16.1462 −0.949777
\(290\) 0 0
\(291\) 28.9802 1.69885
\(292\) 0 0
\(293\) 8.46838 0.494728 0.247364 0.968923i \(-0.420436\pi\)
0.247364 + 0.968923i \(0.420436\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 44.4157 2.57726
\(298\) 0 0
\(299\) −7.12637 −0.412129
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −15.6990 −0.901885
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.32802 0.0757942 0.0378971 0.999282i \(-0.487934\pi\)
0.0378971 + 0.999282i \(0.487934\pi\)
\(308\) 0 0
\(309\) 41.7623 2.37578
\(310\) 0 0
\(311\) 24.2971 1.37776 0.688882 0.724874i \(-0.258102\pi\)
0.688882 + 0.724874i \(0.258102\pi\)
\(312\) 0 0
\(313\) −20.3478 −1.15013 −0.575063 0.818109i \(-0.695023\pi\)
−0.575063 + 0.818109i \(0.695023\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.57309 0.481513 0.240756 0.970586i \(-0.422604\pi\)
0.240756 + 0.970586i \(0.422604\pi\)
\(318\) 0 0
\(319\) 6.48781 0.363248
\(320\) 0 0
\(321\) −42.8714 −2.39285
\(322\) 0 0
\(323\) −6.94508 −0.386434
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 49.6106 2.74347
\(328\) 0 0
\(329\) 33.9813 1.87345
\(330\) 0 0
\(331\) −20.9767 −1.15299 −0.576493 0.817102i \(-0.695579\pi\)
−0.576493 + 0.817102i \(0.695579\pi\)
\(332\) 0 0
\(333\) −2.79915 −0.153393
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.17314 0.118379 0.0591894 0.998247i \(-0.481148\pi\)
0.0591894 + 0.998247i \(0.481148\pi\)
\(338\) 0 0
\(339\) 7.05922 0.383404
\(340\) 0 0
\(341\) −2.35891 −0.127742
\(342\) 0 0
\(343\) −19.1164 −1.03219
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.1335 1.72502 0.862509 0.506042i \(-0.168892\pi\)
0.862509 + 0.506042i \(0.168892\pi\)
\(348\) 0 0
\(349\) 13.5157 0.723479 0.361740 0.932279i \(-0.382183\pi\)
0.361740 + 0.932279i \(0.382183\pi\)
\(350\) 0 0
\(351\) 116.235 6.20415
\(352\) 0 0
\(353\) 28.0645 1.49372 0.746861 0.664981i \(-0.231560\pi\)
0.746861 + 0.664981i \(0.231560\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.79205 0.412399
\(358\) 0 0
\(359\) 20.2340 1.06791 0.533956 0.845513i \(-0.320705\pi\)
0.533956 + 0.845513i \(0.320705\pi\)
\(360\) 0 0
\(361\) 37.4937 1.97336
\(362\) 0 0
\(363\) 11.8522 0.622081
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.76666 −0.248818 −0.124409 0.992231i \(-0.539703\pi\)
−0.124409 + 0.992231i \(0.539703\pi\)
\(368\) 0 0
\(369\) 34.4339 1.79256
\(370\) 0 0
\(371\) −10.1828 −0.528663
\(372\) 0 0
\(373\) 29.1510 1.50938 0.754690 0.656082i \(-0.227787\pi\)
0.754690 + 0.656082i \(0.227787\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.9784 0.874434
\(378\) 0 0
\(379\) 4.20856 0.216179 0.108090 0.994141i \(-0.465527\pi\)
0.108090 + 0.994141i \(0.465527\pi\)
\(380\) 0 0
\(381\) −11.7110 −0.599971
\(382\) 0 0
\(383\) −3.55206 −0.181502 −0.0907508 0.995874i \(-0.528927\pi\)
−0.0907508 + 0.995874i \(0.528927\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −29.2515 −1.48311 −0.741554 0.670893i \(-0.765911\pi\)
−0.741554 + 0.670893i \(0.765911\pi\)
\(390\) 0 0
\(391\) −0.924010 −0.0467292
\(392\) 0 0
\(393\) −64.6416 −3.26074
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 17.8000 0.893356 0.446678 0.894695i \(-0.352607\pi\)
0.446678 + 0.894695i \(0.352607\pi\)
\(398\) 0 0
\(399\) −63.3834 −3.17314
\(400\) 0 0
\(401\) −11.6210 −0.580327 −0.290164 0.956977i \(-0.593710\pi\)
−0.290164 + 0.956977i \(0.593710\pi\)
\(402\) 0 0
\(403\) −6.17320 −0.307509
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.960871 0.0476286
\(408\) 0 0
\(409\) −13.5710 −0.671041 −0.335521 0.942033i \(-0.608912\pi\)
−0.335521 + 0.942033i \(0.608912\pi\)
\(410\) 0 0
\(411\) 14.5996 0.720145
\(412\) 0 0
\(413\) −9.79516 −0.481988
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −72.7277 −3.56149
\(418\) 0 0
\(419\) 13.4959 0.659317 0.329658 0.944100i \(-0.393066\pi\)
0.329658 + 0.944100i \(0.393066\pi\)
\(420\) 0 0
\(421\) 37.6495 1.83492 0.917462 0.397824i \(-0.130234\pi\)
0.917462 + 0.397824i \(0.130234\pi\)
\(422\) 0 0
\(423\) 105.697 5.13916
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −23.3173 −1.12840
\(428\) 0 0
\(429\) −64.1660 −3.09796
\(430\) 0 0
\(431\) 10.0568 0.484421 0.242210 0.970224i \(-0.422128\pi\)
0.242210 + 0.970224i \(0.422128\pi\)
\(432\) 0 0
\(433\) −2.30001 −0.110531 −0.0552657 0.998472i \(-0.517601\pi\)
−0.0552657 + 0.998472i \(0.517601\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.51623 0.359550
\(438\) 0 0
\(439\) 12.4273 0.593124 0.296562 0.955014i \(-0.404160\pi\)
0.296562 + 0.955014i \(0.404160\pi\)
\(440\) 0 0
\(441\) −3.93053 −0.187168
\(442\) 0 0
\(443\) −2.18268 −0.103702 −0.0518510 0.998655i \(-0.516512\pi\)
−0.0518510 + 0.998655i \(0.516512\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −63.0813 −2.98364
\(448\) 0 0
\(449\) 3.09603 0.146111 0.0730554 0.997328i \(-0.476725\pi\)
0.0730554 + 0.997328i \(0.476725\pi\)
\(450\) 0 0
\(451\) −11.8202 −0.556591
\(452\) 0 0
\(453\) 25.6343 1.20441
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.320363 0.0149860 0.00749298 0.999972i \(-0.497615\pi\)
0.00749298 + 0.999972i \(0.497615\pi\)
\(458\) 0 0
\(459\) 15.0711 0.703458
\(460\) 0 0
\(461\) −15.3239 −0.713706 −0.356853 0.934161i \(-0.616150\pi\)
−0.356853 + 0.934161i \(0.616150\pi\)
\(462\) 0 0
\(463\) −6.81556 −0.316746 −0.158373 0.987379i \(-0.550625\pi\)
−0.158373 + 0.987379i \(0.550625\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −27.2444 −1.26072 −0.630360 0.776303i \(-0.717093\pi\)
−0.630360 + 0.776303i \(0.717093\pi\)
\(468\) 0 0
\(469\) 8.05755 0.372063
\(470\) 0 0
\(471\) 57.9062 2.66818
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −31.6730 −1.45021
\(478\) 0 0
\(479\) 23.5031 1.07388 0.536942 0.843619i \(-0.319579\pi\)
0.536942 + 0.843619i \(0.319579\pi\)
\(480\) 0 0
\(481\) 2.51457 0.114655
\(482\) 0 0
\(483\) −8.43286 −0.383709
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −7.63058 −0.345774 −0.172887 0.984942i \(-0.555310\pi\)
−0.172887 + 0.984942i \(0.555310\pi\)
\(488\) 0 0
\(489\) −8.44779 −0.382022
\(490\) 0 0
\(491\) −19.1206 −0.862902 −0.431451 0.902136i \(-0.641998\pi\)
−0.431451 + 0.902136i \(0.641998\pi\)
\(492\) 0 0
\(493\) 2.20144 0.0991477
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.5034 −0.695424
\(498\) 0 0
\(499\) −25.6581 −1.14861 −0.574306 0.818641i \(-0.694728\pi\)
−0.574306 + 0.818641i \(0.694728\pi\)
\(500\) 0 0
\(501\) −11.9464 −0.533725
\(502\) 0 0
\(503\) 8.42282 0.375555 0.187777 0.982212i \(-0.439872\pi\)
0.187777 + 0.982212i \(0.439872\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −124.936 −5.54862
\(508\) 0 0
\(509\) 18.2328 0.808153 0.404076 0.914725i \(-0.367593\pi\)
0.404076 + 0.914725i \(0.367593\pi\)
\(510\) 0 0
\(511\) 28.8805 1.27760
\(512\) 0 0
\(513\) −122.594 −5.41264
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −36.2828 −1.59572
\(518\) 0 0
\(519\) 42.2648 1.85522
\(520\) 0 0
\(521\) 19.8738 0.870689 0.435345 0.900264i \(-0.356627\pi\)
0.435345 + 0.900264i \(0.356627\pi\)
\(522\) 0 0
\(523\) 19.7228 0.862418 0.431209 0.902252i \(-0.358087\pi\)
0.431209 + 0.902252i \(0.358087\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.800422 −0.0348669
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −30.4673 −1.32217
\(532\) 0 0
\(533\) −30.9331 −1.33986
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −33.1141 −1.42898
\(538\) 0 0
\(539\) 1.34924 0.0581160
\(540\) 0 0
\(541\) −16.4662 −0.707938 −0.353969 0.935257i \(-0.615168\pi\)
−0.353969 + 0.935257i \(0.615168\pi\)
\(542\) 0 0
\(543\) −32.9408 −1.41363
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 38.3286 1.63881 0.819405 0.573215i \(-0.194304\pi\)
0.819405 + 0.573215i \(0.194304\pi\)
\(548\) 0 0
\(549\) −72.5273 −3.09539
\(550\) 0 0
\(551\) −17.9073 −0.762875
\(552\) 0 0
\(553\) −30.7559 −1.30787
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.3973 0.864263 0.432132 0.901811i \(-0.357762\pi\)
0.432132 + 0.901811i \(0.357762\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.31981 −0.351263
\(562\) 0 0
\(563\) 1.72270 0.0726032 0.0363016 0.999341i \(-0.488442\pi\)
0.0363016 + 0.999341i \(0.488442\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 76.8483 3.22733
\(568\) 0 0
\(569\) 25.3283 1.06182 0.530909 0.847429i \(-0.321850\pi\)
0.530909 + 0.847429i \(0.321850\pi\)
\(570\) 0 0
\(571\) −5.68968 −0.238106 −0.119053 0.992888i \(-0.537986\pi\)
−0.119053 + 0.992888i \(0.537986\pi\)
\(572\) 0 0
\(573\) 21.8658 0.913455
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.0377 0.750918 0.375459 0.926839i \(-0.377485\pi\)
0.375459 + 0.926839i \(0.377485\pi\)
\(578\) 0 0
\(579\) −63.0019 −2.61827
\(580\) 0 0
\(581\) 16.2023 0.672185
\(582\) 0 0
\(583\) 10.8724 0.450291
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.1894 −0.544383 −0.272192 0.962243i \(-0.587748\pi\)
−0.272192 + 0.962243i \(0.587748\pi\)
\(588\) 0 0
\(589\) 6.51092 0.268278
\(590\) 0 0
\(591\) −36.6643 −1.50817
\(592\) 0 0
\(593\) 0.326755 0.0134182 0.00670911 0.999977i \(-0.497864\pi\)
0.00670911 + 0.999977i \(0.497864\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.2796 −0.502569
\(598\) 0 0
\(599\) −7.49619 −0.306286 −0.153143 0.988204i \(-0.548939\pi\)
−0.153143 + 0.988204i \(0.548939\pi\)
\(600\) 0 0
\(601\) 0.121724 0.00496523 0.00248262 0.999997i \(-0.499210\pi\)
0.00248262 + 0.999997i \(0.499210\pi\)
\(602\) 0 0
\(603\) 25.0626 1.02063
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.52992 −0.0620975 −0.0310488 0.999518i \(-0.509885\pi\)
−0.0310488 + 0.999518i \(0.509885\pi\)
\(608\) 0 0
\(609\) 20.0911 0.814134
\(610\) 0 0
\(611\) −94.9512 −3.84131
\(612\) 0 0
\(613\) −24.3927 −0.985211 −0.492605 0.870253i \(-0.663955\pi\)
−0.492605 + 0.870253i \(0.663955\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.8906 −1.12283 −0.561416 0.827534i \(-0.689743\pi\)
−0.561416 + 0.827534i \(0.689743\pi\)
\(618\) 0 0
\(619\) 12.8617 0.516957 0.258478 0.966017i \(-0.416779\pi\)
0.258478 + 0.966017i \(0.416779\pi\)
\(620\) 0 0
\(621\) −16.3105 −0.654518
\(622\) 0 0
\(623\) −24.7740 −0.992549
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 67.6763 2.70273
\(628\) 0 0
\(629\) 0.326041 0.0130001
\(630\) 0 0
\(631\) −11.9518 −0.475794 −0.237897 0.971290i \(-0.576458\pi\)
−0.237897 + 0.971290i \(0.576458\pi\)
\(632\) 0 0
\(633\) 71.9679 2.86047
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.53093 0.139901
\(638\) 0 0
\(639\) −48.2226 −1.90766
\(640\) 0 0
\(641\) −23.9139 −0.944544 −0.472272 0.881453i \(-0.656566\pi\)
−0.472272 + 0.881453i \(0.656566\pi\)
\(642\) 0 0
\(643\) −3.87313 −0.152741 −0.0763707 0.997079i \(-0.524333\pi\)
−0.0763707 + 0.997079i \(0.524333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.3330 0.406231 0.203115 0.979155i \(-0.434893\pi\)
0.203115 + 0.979155i \(0.434893\pi\)
\(648\) 0 0
\(649\) 10.4586 0.410535
\(650\) 0 0
\(651\) −7.30495 −0.286303
\(652\) 0 0
\(653\) −21.6319 −0.846522 −0.423261 0.906008i \(-0.639115\pi\)
−0.423261 + 0.906008i \(0.639115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 89.8312 3.50465
\(658\) 0 0
\(659\) 15.7168 0.612238 0.306119 0.951993i \(-0.400970\pi\)
0.306119 + 0.951993i \(0.400970\pi\)
\(660\) 0 0
\(661\) −40.3002 −1.56750 −0.783749 0.621078i \(-0.786695\pi\)
−0.783749 + 0.621078i \(0.786695\pi\)
\(662\) 0 0
\(663\) −21.7727 −0.845582
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.38248 −0.0922500
\(668\) 0 0
\(669\) 22.8389 0.883003
\(670\) 0 0
\(671\) 24.8966 0.961122
\(672\) 0 0
\(673\) 6.69888 0.258223 0.129111 0.991630i \(-0.458788\pi\)
0.129111 + 0.991630i \(0.458788\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.5812 0.867866 0.433933 0.900945i \(-0.357125\pi\)
0.433933 + 0.900945i \(0.357125\pi\)
\(678\) 0 0
\(679\) −22.3533 −0.857843
\(680\) 0 0
\(681\) −76.1566 −2.91833
\(682\) 0 0
\(683\) 1.42797 0.0546398 0.0273199 0.999627i \(-0.491303\pi\)
0.0273199 + 0.999627i \(0.491303\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −42.8738 −1.63574
\(688\) 0 0
\(689\) 28.4529 1.08397
\(690\) 0 0
\(691\) −1.50255 −0.0571595 −0.0285798 0.999592i \(-0.509098\pi\)
−0.0285798 + 0.999592i \(0.509098\pi\)
\(692\) 0 0
\(693\) −55.0944 −2.09286
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.01081 −0.151920
\(698\) 0 0
\(699\) −15.9020 −0.601471
\(700\) 0 0
\(701\) 18.7341 0.707577 0.353789 0.935325i \(-0.384893\pi\)
0.353789 + 0.935325i \(0.384893\pi\)
\(702\) 0 0
\(703\) −2.65214 −0.100027
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.1091 0.455411
\(708\) 0 0
\(709\) −13.5052 −0.507199 −0.253599 0.967309i \(-0.581615\pi\)
−0.253599 + 0.967309i \(0.581615\pi\)
\(710\) 0 0
\(711\) −95.6646 −3.58770
\(712\) 0 0
\(713\) 0.866248 0.0324412
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 67.6711 2.52722
\(718\) 0 0
\(719\) 31.7496 1.18406 0.592030 0.805916i \(-0.298327\pi\)
0.592030 + 0.805916i \(0.298327\pi\)
\(720\) 0 0
\(721\) −32.2126 −1.19966
\(722\) 0 0
\(723\) −88.8313 −3.30367
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.6531 −0.803070 −0.401535 0.915844i \(-0.631523\pi\)
−0.401535 + 0.915844i \(0.631523\pi\)
\(728\) 0 0
\(729\) 77.2411 2.86078
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −0.640507 −0.0236577 −0.0118288 0.999930i \(-0.503765\pi\)
−0.0118288 + 0.999930i \(0.503765\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.60329 −0.316906
\(738\) 0 0
\(739\) 16.8191 0.618700 0.309350 0.950948i \(-0.399889\pi\)
0.309350 + 0.950948i \(0.399889\pi\)
\(740\) 0 0
\(741\) 177.107 6.50619
\(742\) 0 0
\(743\) 15.9691 0.585851 0.292925 0.956135i \(-0.405371\pi\)
0.292925 + 0.956135i \(0.405371\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 50.3964 1.84391
\(748\) 0 0
\(749\) 33.0680 1.20828
\(750\) 0 0
\(751\) −8.09841 −0.295515 −0.147758 0.989024i \(-0.547206\pi\)
−0.147758 + 0.989024i \(0.547206\pi\)
\(752\) 0 0
\(753\) 52.7798 1.92340
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −20.5691 −0.747598 −0.373799 0.927510i \(-0.621945\pi\)
−0.373799 + 0.927510i \(0.621945\pi\)
\(758\) 0 0
\(759\) 9.00402 0.326825
\(760\) 0 0
\(761\) −17.1840 −0.622919 −0.311459 0.950259i \(-0.600818\pi\)
−0.311459 + 0.950259i \(0.600818\pi\)
\(762\) 0 0
\(763\) −38.2662 −1.38533
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.3698 0.988268
\(768\) 0 0
\(769\) 41.8648 1.50968 0.754841 0.655908i \(-0.227714\pi\)
0.754841 + 0.655908i \(0.227714\pi\)
\(770\) 0 0
\(771\) 11.0393 0.397570
\(772\) 0 0
\(773\) 35.7639 1.28634 0.643170 0.765724i \(-0.277619\pi\)
0.643170 + 0.765724i \(0.277619\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.97557 0.106748
\(778\) 0 0
\(779\) 32.6254 1.16893
\(780\) 0 0
\(781\) 16.5535 0.592330
\(782\) 0 0
\(783\) 38.8595 1.38872
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −26.9835 −0.961858 −0.480929 0.876759i \(-0.659701\pi\)
−0.480929 + 0.876759i \(0.659701\pi\)
\(788\) 0 0
\(789\) −77.3321 −2.75310
\(790\) 0 0
\(791\) −5.44500 −0.193602
\(792\) 0 0
\(793\) 65.1537 2.31368
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.7226 −0.946561 −0.473281 0.880912i \(-0.656930\pi\)
−0.473281 + 0.880912i \(0.656930\pi\)
\(798\) 0 0
\(799\) −12.3114 −0.435547
\(800\) 0 0
\(801\) −77.0582 −2.72272
\(802\) 0 0
\(803\) −30.8366 −1.08820
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −46.7464 −1.64555
\(808\) 0 0
\(809\) 50.0319 1.75903 0.879513 0.475874i \(-0.157868\pi\)
0.879513 + 0.475874i \(0.157868\pi\)
\(810\) 0 0
\(811\) 33.5774 1.17906 0.589531 0.807746i \(-0.299313\pi\)
0.589531 + 0.807746i \(0.299313\pi\)
\(812\) 0 0
\(813\) −85.7318 −3.00675
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −144.181 −5.03808
\(820\) 0 0
\(821\) 27.1429 0.947295 0.473647 0.880715i \(-0.342937\pi\)
0.473647 + 0.880715i \(0.342937\pi\)
\(822\) 0 0
\(823\) −32.6392 −1.13773 −0.568866 0.822431i \(-0.692618\pi\)
−0.568866 + 0.822431i \(0.692618\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.0828 1.28949 0.644747 0.764396i \(-0.276963\pi\)
0.644747 + 0.764396i \(0.276963\pi\)
\(828\) 0 0
\(829\) 33.9405 1.17880 0.589401 0.807841i \(-0.299364\pi\)
0.589401 + 0.807841i \(0.299364\pi\)
\(830\) 0 0
\(831\) 3.54181 0.122864
\(832\) 0 0
\(833\) 0.457823 0.0158626
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −14.1289 −0.488368
\(838\) 0 0
\(839\) −1.36929 −0.0472730 −0.0236365 0.999721i \(-0.507524\pi\)
−0.0236365 + 0.999721i \(0.507524\pi\)
\(840\) 0 0
\(841\) −23.3238 −0.804269
\(842\) 0 0
\(843\) −40.4251 −1.39231
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −9.14199 −0.314122
\(848\) 0 0
\(849\) −100.998 −3.46624
\(850\) 0 0
\(851\) −0.352855 −0.0120957
\(852\) 0 0
\(853\) −9.21382 −0.315475 −0.157738 0.987481i \(-0.550420\pi\)
−0.157738 + 0.987481i \(0.550420\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.3564 −0.490405 −0.245202 0.969472i \(-0.578854\pi\)
−0.245202 + 0.969472i \(0.578854\pi\)
\(858\) 0 0
\(859\) 31.3758 1.07053 0.535264 0.844685i \(-0.320212\pi\)
0.535264 + 0.844685i \(0.320212\pi\)
\(860\) 0 0
\(861\) −36.6042 −1.24747
\(862\) 0 0
\(863\) −26.1214 −0.889182 −0.444591 0.895734i \(-0.646651\pi\)
−0.444591 + 0.895734i \(0.646651\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 53.3873 1.81313
\(868\) 0 0
\(869\) 32.8390 1.11399
\(870\) 0 0
\(871\) −22.5146 −0.762877
\(872\) 0 0
\(873\) −69.5289 −2.35320
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.5530 0.390118 0.195059 0.980792i \(-0.437510\pi\)
0.195059 + 0.980792i \(0.437510\pi\)
\(878\) 0 0
\(879\) −28.0006 −0.944437
\(880\) 0 0
\(881\) −34.3559 −1.15748 −0.578740 0.815512i \(-0.696456\pi\)
−0.578740 + 0.815512i \(0.696456\pi\)
\(882\) 0 0
\(883\) −12.8316 −0.431818 −0.215909 0.976414i \(-0.569271\pi\)
−0.215909 + 0.976414i \(0.569271\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.46724 0.0492652 0.0246326 0.999697i \(-0.492158\pi\)
0.0246326 + 0.999697i \(0.492158\pi\)
\(888\) 0 0
\(889\) 9.03303 0.302958
\(890\) 0 0
\(891\) −82.0533 −2.74889
\(892\) 0 0
\(893\) 100.146 3.35125
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 23.5633 0.786755
\(898\) 0 0
\(899\) −2.06382 −0.0688322
\(900\) 0 0
\(901\) 3.68922 0.122906
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.44558 −0.147613 −0.0738066 0.997273i \(-0.523515\pi\)
−0.0738066 + 0.997273i \(0.523515\pi\)
\(908\) 0 0
\(909\) 37.6648 1.24926
\(910\) 0 0
\(911\) 40.1697 1.33088 0.665440 0.746451i \(-0.268244\pi\)
0.665440 + 0.746451i \(0.268244\pi\)
\(912\) 0 0
\(913\) −17.2997 −0.572536
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 49.8600 1.64652
\(918\) 0 0
\(919\) 46.8063 1.54400 0.771999 0.635624i \(-0.219257\pi\)
0.771999 + 0.635624i \(0.219257\pi\)
\(920\) 0 0
\(921\) −4.39109 −0.144691
\(922\) 0 0
\(923\) 43.3200 1.42590
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −100.195 −3.29085
\(928\) 0 0
\(929\) 8.92145 0.292703 0.146352 0.989233i \(-0.453247\pi\)
0.146352 + 0.989233i \(0.453247\pi\)
\(930\) 0 0
\(931\) −3.72409 −0.122052
\(932\) 0 0
\(933\) −80.3382 −2.63016
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −7.70070 −0.251571 −0.125785 0.992057i \(-0.540145\pi\)
−0.125785 + 0.992057i \(0.540145\pi\)
\(938\) 0 0
\(939\) 67.2799 2.19560
\(940\) 0 0
\(941\) −16.8979 −0.550856 −0.275428 0.961322i \(-0.588820\pi\)
−0.275428 + 0.961322i \(0.588820\pi\)
\(942\) 0 0
\(943\) 4.34066 0.141351
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 43.7150 1.42055 0.710273 0.703927i \(-0.248572\pi\)
0.710273 + 0.703927i \(0.248572\pi\)
\(948\) 0 0
\(949\) −80.6985 −2.61958
\(950\) 0 0
\(951\) −28.3469 −0.919210
\(952\) 0 0
\(953\) −30.2353 −0.979418 −0.489709 0.871886i \(-0.662897\pi\)
−0.489709 + 0.871886i \(0.662897\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −21.4519 −0.693441
\(958\) 0 0
\(959\) −11.2611 −0.363641
\(960\) 0 0
\(961\) −30.2496 −0.975794
\(962\) 0 0
\(963\) 102.856 3.31450
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −16.2727 −0.523296 −0.261648 0.965163i \(-0.584266\pi\)
−0.261648 + 0.965163i \(0.584266\pi\)
\(968\) 0 0
\(969\) 22.9638 0.737704
\(970\) 0 0
\(971\) 6.21424 0.199424 0.0997122 0.995016i \(-0.468208\pi\)
0.0997122 + 0.995016i \(0.468208\pi\)
\(972\) 0 0
\(973\) 56.0971 1.79839
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.6705 −0.821271 −0.410636 0.911800i \(-0.634693\pi\)
−0.410636 + 0.911800i \(0.634693\pi\)
\(978\) 0 0
\(979\) 26.4519 0.845407
\(980\) 0 0
\(981\) −119.025 −3.80018
\(982\) 0 0
\(983\) 30.7124 0.979574 0.489787 0.871842i \(-0.337075\pi\)
0.489787 + 0.871842i \(0.337075\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −112.359 −3.57642
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 29.4052 0.934086 0.467043 0.884235i \(-0.345319\pi\)
0.467043 + 0.884235i \(0.345319\pi\)
\(992\) 0 0
\(993\) 69.3594 2.20105
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.59825 −0.177298 −0.0886492 0.996063i \(-0.528255\pi\)
−0.0886492 + 0.996063i \(0.528255\pi\)
\(998\) 0 0
\(999\) 5.75524 0.182088
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 4600.2.a.be.1.1 5
4.3 odd 2 9200.2.a.cu.1.5 5
5.2 odd 4 4600.2.e.u.4049.9 10
5.3 odd 4 4600.2.e.u.4049.2 10
5.4 even 2 920.2.a.j.1.5 5
15.14 odd 2 8280.2.a.bs.1.2 5
20.19 odd 2 1840.2.a.v.1.1 5
40.19 odd 2 7360.2.a.cp.1.5 5
40.29 even 2 7360.2.a.co.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.5 5 5.4 even 2
1840.2.a.v.1.1 5 20.19 odd 2
4600.2.a.be.1.1 5 1.1 even 1 trivial
4600.2.e.u.4049.2 10 5.3 odd 4
4600.2.e.u.4049.9 10 5.2 odd 4
7360.2.a.co.1.1 5 40.29 even 2
7360.2.a.cp.1.5 5 40.19 odd 2
8280.2.a.bs.1.2 5 15.14 odd 2
9200.2.a.cu.1.5 5 4.3 odd 2