Properties

Label 644.2.a.d.1.1
Level $644$
Weight $2$
Character 644.1
Self dual yes
Analytic conductor $5.142$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.6963152.1
Defining polynomial: \( x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.11181\) of defining polynomial
Character \(\chi\) \(=\) 644.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.11181 q^{3} +1.77860 q^{5} -1.00000 q^{7} +1.45975 q^{9} +O(q^{10})\) \(q-2.11181 q^{3} +1.77860 q^{5} -1.00000 q^{7} +1.45975 q^{9} +0.612946 q^{11} +1.84681 q^{13} -3.75608 q^{15} +4.26501 q^{17} -4.70034 q^{19} +2.11181 q^{21} -1.00000 q^{23} -1.83657 q^{25} +3.25271 q^{27} +10.2406 q^{29} +10.2181 q^{31} -1.29443 q^{33} -1.77860 q^{35} -2.47671 q^{37} -3.90011 q^{39} +6.37682 q^{41} +10.9240 q^{43} +2.59632 q^{45} +0.544914 q^{47} +1.00000 q^{49} -9.00689 q^{51} -7.61984 q^{53} +1.09019 q^{55} +9.92623 q^{57} -2.74172 q^{59} +15.1454 q^{61} -1.45975 q^{63} +3.28473 q^{65} -5.27937 q^{67} +2.11181 q^{69} -2.21583 q^{71} +5.62764 q^{73} +3.87849 q^{75} -0.612946 q^{77} -2.46755 q^{79} -11.2484 q^{81} +5.84347 q^{83} +7.58575 q^{85} -21.6262 q^{87} +0.00746465 q^{89} -1.84681 q^{91} -21.5786 q^{93} -8.36003 q^{95} -7.58815 q^{97} +0.894750 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100}) \) 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
\(3\) −2.11181 −1.21926 −0.609628 0.792688i \(-0.708681\pi\)
−0.609628 + 0.792688i \(0.708681\pi\)
\(4\) 0 0
\(5\) 1.77860 0.795415 0.397708 0.917512i \(-0.369806\pi\)
0.397708 + 0.917512i \(0.369806\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.45975 0.486584
\(10\) 0 0
\(11\) 0.612946 0.184810 0.0924051 0.995721i \(-0.470545\pi\)
0.0924051 + 0.995721i \(0.470545\pi\)
\(12\) 0 0
\(13\) 1.84681 0.512212 0.256106 0.966649i \(-0.417560\pi\)
0.256106 + 0.966649i \(0.417560\pi\)
\(14\) 0 0
\(15\) −3.75608 −0.969815
\(16\) 0 0
\(17\) 4.26501 1.03442 0.517208 0.855860i \(-0.326971\pi\)
0.517208 + 0.855860i \(0.326971\pi\)
\(18\) 0 0
\(19\) −4.70034 −1.07833 −0.539166 0.842200i \(-0.681260\pi\)
−0.539166 + 0.842200i \(0.681260\pi\)
\(20\) 0 0
\(21\) 2.11181 0.460835
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.83657 −0.367314
\(26\) 0 0
\(27\) 3.25271 0.625985
\(28\) 0 0
\(29\) 10.2406 1.90163 0.950814 0.309761i \(-0.100249\pi\)
0.950814 + 0.309761i \(0.100249\pi\)
\(30\) 0 0
\(31\) 10.2181 1.83522 0.917609 0.397485i \(-0.130117\pi\)
0.917609 + 0.397485i \(0.130117\pi\)
\(32\) 0 0
\(33\) −1.29443 −0.225331
\(34\) 0 0
\(35\) −1.77860 −0.300639
\(36\) 0 0
\(37\) −2.47671 −0.407169 −0.203584 0.979057i \(-0.565259\pi\)
−0.203584 + 0.979057i \(0.565259\pi\)
\(38\) 0 0
\(39\) −3.90011 −0.624517
\(40\) 0 0
\(41\) 6.37682 0.995892 0.497946 0.867208i \(-0.334088\pi\)
0.497946 + 0.867208i \(0.334088\pi\)
\(42\) 0 0
\(43\) 10.9240 1.66589 0.832944 0.553357i \(-0.186653\pi\)
0.832944 + 0.553357i \(0.186653\pi\)
\(44\) 0 0
\(45\) 2.59632 0.387037
\(46\) 0 0
\(47\) 0.544914 0.0794838 0.0397419 0.999210i \(-0.487346\pi\)
0.0397419 + 0.999210i \(0.487346\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.00689 −1.26122
\(52\) 0 0
\(53\) −7.61984 −1.04667 −0.523333 0.852128i \(-0.675312\pi\)
−0.523333 + 0.852128i \(0.675312\pi\)
\(54\) 0 0
\(55\) 1.09019 0.147001
\(56\) 0 0
\(57\) 9.92623 1.31476
\(58\) 0 0
\(59\) −2.74172 −0.356941 −0.178471 0.983945i \(-0.557115\pi\)
−0.178471 + 0.983945i \(0.557115\pi\)
\(60\) 0 0
\(61\) 15.1454 1.93916 0.969582 0.244766i \(-0.0787111\pi\)
0.969582 + 0.244766i \(0.0787111\pi\)
\(62\) 0 0
\(63\) −1.45975 −0.183912
\(64\) 0 0
\(65\) 3.28473 0.407421
\(66\) 0 0
\(67\) −5.27937 −0.644977 −0.322489 0.946573i \(-0.604519\pi\)
−0.322489 + 0.946573i \(0.604519\pi\)
\(68\) 0 0
\(69\) 2.11181 0.254232
\(70\) 0 0
\(71\) −2.21583 −0.262970 −0.131485 0.991318i \(-0.541975\pi\)
−0.131485 + 0.991318i \(0.541975\pi\)
\(72\) 0 0
\(73\) 5.62764 0.658665 0.329333 0.944214i \(-0.393176\pi\)
0.329333 + 0.944214i \(0.393176\pi\)
\(74\) 0 0
\(75\) 3.87849 0.447850
\(76\) 0 0
\(77\) −0.612946 −0.0698517
\(78\) 0 0
\(79\) −2.46755 −0.277621 −0.138810 0.990319i \(-0.544328\pi\)
−0.138810 + 0.990319i \(0.544328\pi\)
\(80\) 0 0
\(81\) −11.2484 −1.24982
\(82\) 0 0
\(83\) 5.84347 0.641404 0.320702 0.947180i \(-0.396081\pi\)
0.320702 + 0.947180i \(0.396081\pi\)
\(84\) 0 0
\(85\) 7.58575 0.822790
\(86\) 0 0
\(87\) −21.6262 −2.31857
\(88\) 0 0
\(89\) 0.00746465 0.000791251 0 0.000395626 1.00000i \(-0.499874\pi\)
0.000395626 1.00000i \(0.499874\pi\)
\(90\) 0 0
\(91\) −1.84681 −0.193598
\(92\) 0 0
\(93\) −21.5786 −2.23760
\(94\) 0 0
\(95\) −8.36003 −0.857721
\(96\) 0 0
\(97\) −7.58815 −0.770460 −0.385230 0.922821i \(-0.625878\pi\)
−0.385230 + 0.922821i \(0.625878\pi\)
\(98\) 0 0
\(99\) 0.894750 0.0899257
\(100\) 0 0
\(101\) 11.5904 1.15329 0.576643 0.816996i \(-0.304362\pi\)
0.576643 + 0.816996i \(0.304362\pi\)
\(102\) 0 0
\(103\) −0.666419 −0.0656642 −0.0328321 0.999461i \(-0.510453\pi\)
−0.0328321 + 0.999461i \(0.510453\pi\)
\(104\) 0 0
\(105\) 3.75608 0.366555
\(106\) 0 0
\(107\) 6.92396 0.669365 0.334682 0.942331i \(-0.391371\pi\)
0.334682 + 0.942331i \(0.391371\pi\)
\(108\) 0 0
\(109\) 4.06263 0.389130 0.194565 0.980890i \(-0.437671\pi\)
0.194565 + 0.980890i \(0.437671\pi\)
\(110\) 0 0
\(111\) 5.23035 0.496443
\(112\) 0 0
\(113\) −12.5926 −1.18462 −0.592308 0.805711i \(-0.701783\pi\)
−0.592308 + 0.805711i \(0.701783\pi\)
\(114\) 0 0
\(115\) −1.77860 −0.165856
\(116\) 0 0
\(117\) 2.69588 0.249234
\(118\) 0 0
\(119\) −4.26501 −0.390973
\(120\) 0 0
\(121\) −10.6243 −0.965845
\(122\) 0 0
\(123\) −13.4666 −1.21425
\(124\) 0 0
\(125\) −12.1595 −1.08758
\(126\) 0 0
\(127\) −12.8535 −1.14057 −0.570283 0.821448i \(-0.693167\pi\)
−0.570283 + 0.821448i \(0.693167\pi\)
\(128\) 0 0
\(129\) −23.0694 −2.03114
\(130\) 0 0
\(131\) −14.6948 −1.28389 −0.641944 0.766752i \(-0.721872\pi\)
−0.641944 + 0.766752i \(0.721872\pi\)
\(132\) 0 0
\(133\) 4.70034 0.407571
\(134\) 0 0
\(135\) 5.78529 0.497918
\(136\) 0 0
\(137\) −19.2303 −1.64296 −0.821480 0.570238i \(-0.806851\pi\)
−0.821480 + 0.570238i \(0.806851\pi\)
\(138\) 0 0
\(139\) −6.14573 −0.521274 −0.260637 0.965437i \(-0.583933\pi\)
−0.260637 + 0.965437i \(0.583933\pi\)
\(140\) 0 0
\(141\) −1.15076 −0.0969111
\(142\) 0 0
\(143\) 1.13199 0.0946620
\(144\) 0 0
\(145\) 18.2139 1.51258
\(146\) 0 0
\(147\) −2.11181 −0.174179
\(148\) 0 0
\(149\) −6.66642 −0.546134 −0.273067 0.961995i \(-0.588038\pi\)
−0.273067 + 0.961995i \(0.588038\pi\)
\(150\) 0 0
\(151\) −1.07270 −0.0872950 −0.0436475 0.999047i \(-0.513898\pi\)
−0.0436475 + 0.999047i \(0.513898\pi\)
\(152\) 0 0
\(153\) 6.22585 0.503330
\(154\) 0 0
\(155\) 18.1739 1.45976
\(156\) 0 0
\(157\) −2.76664 −0.220802 −0.110401 0.993887i \(-0.535214\pi\)
−0.110401 + 0.993887i \(0.535214\pi\)
\(158\) 0 0
\(159\) 16.0917 1.27615
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 5.94336 0.465520 0.232760 0.972534i \(-0.425224\pi\)
0.232760 + 0.972534i \(0.425224\pi\)
\(164\) 0 0
\(165\) −2.30227 −0.179232
\(166\) 0 0
\(167\) 4.52778 0.350371 0.175185 0.984535i \(-0.443948\pi\)
0.175185 + 0.984535i \(0.443948\pi\)
\(168\) 0 0
\(169\) −9.58931 −0.737639
\(170\) 0 0
\(171\) −6.86133 −0.524699
\(172\) 0 0
\(173\) −10.7003 −0.813531 −0.406766 0.913533i \(-0.633343\pi\)
−0.406766 + 0.913533i \(0.633343\pi\)
\(174\) 0 0
\(175\) 1.83657 0.138832
\(176\) 0 0
\(177\) 5.78999 0.435203
\(178\) 0 0
\(179\) 11.3590 0.849008 0.424504 0.905426i \(-0.360448\pi\)
0.424504 + 0.905426i \(0.360448\pi\)
\(180\) 0 0
\(181\) 20.2162 1.50266 0.751328 0.659929i \(-0.229414\pi\)
0.751328 + 0.659929i \(0.229414\pi\)
\(182\) 0 0
\(183\) −31.9842 −2.36434
\(184\) 0 0
\(185\) −4.40509 −0.323868
\(186\) 0 0
\(187\) 2.61422 0.191171
\(188\) 0 0
\(189\) −3.25271 −0.236600
\(190\) 0 0
\(191\) 1.97054 0.142583 0.0712916 0.997456i \(-0.477288\pi\)
0.0712916 + 0.997456i \(0.477288\pi\)
\(192\) 0 0
\(193\) 14.7995 1.06529 0.532645 0.846339i \(-0.321198\pi\)
0.532645 + 0.846339i \(0.321198\pi\)
\(194\) 0 0
\(195\) −6.93674 −0.496751
\(196\) 0 0
\(197\) 15.3496 1.09362 0.546808 0.837258i \(-0.315843\pi\)
0.546808 + 0.837258i \(0.315843\pi\)
\(198\) 0 0
\(199\) 13.7071 0.971668 0.485834 0.874051i \(-0.338516\pi\)
0.485834 + 0.874051i \(0.338516\pi\)
\(200\) 0 0
\(201\) 11.1490 0.786392
\(202\) 0 0
\(203\) −10.2406 −0.718748
\(204\) 0 0
\(205\) 11.3418 0.792148
\(206\) 0 0
\(207\) −1.45975 −0.101460
\(208\) 0 0
\(209\) −2.88105 −0.199287
\(210\) 0 0
\(211\) 22.0376 1.51713 0.758567 0.651596i \(-0.225900\pi\)
0.758567 + 0.651596i \(0.225900\pi\)
\(212\) 0 0
\(213\) 4.67941 0.320628
\(214\) 0 0
\(215\) 19.4294 1.32507
\(216\) 0 0
\(217\) −10.2181 −0.693647
\(218\) 0 0
\(219\) −11.8845 −0.803081
\(220\) 0 0
\(221\) 7.87664 0.529840
\(222\) 0 0
\(223\) −3.01419 −0.201845 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(224\) 0 0
\(225\) −2.68094 −0.178729
\(226\) 0 0
\(227\) −23.3269 −1.54826 −0.774130 0.633026i \(-0.781813\pi\)
−0.774130 + 0.633026i \(0.781813\pi\)
\(228\) 0 0
\(229\) 27.8111 1.83781 0.918904 0.394482i \(-0.129076\pi\)
0.918904 + 0.394482i \(0.129076\pi\)
\(230\) 0 0
\(231\) 1.29443 0.0851671
\(232\) 0 0
\(233\) −22.2000 −1.45437 −0.727185 0.686442i \(-0.759172\pi\)
−0.727185 + 0.686442i \(0.759172\pi\)
\(234\) 0 0
\(235\) 0.969185 0.0632227
\(236\) 0 0
\(237\) 5.21100 0.338491
\(238\) 0 0
\(239\) 13.2136 0.854714 0.427357 0.904083i \(-0.359445\pi\)
0.427357 + 0.904083i \(0.359445\pi\)
\(240\) 0 0
\(241\) 11.9586 0.770322 0.385161 0.922849i \(-0.374146\pi\)
0.385161 + 0.922849i \(0.374146\pi\)
\(242\) 0 0
\(243\) 13.9963 0.897865
\(244\) 0 0
\(245\) 1.77860 0.113631
\(246\) 0 0
\(247\) −8.68061 −0.552334
\(248\) 0 0
\(249\) −12.3403 −0.782035
\(250\) 0 0
\(251\) 15.0805 0.951872 0.475936 0.879480i \(-0.342109\pi\)
0.475936 + 0.879480i \(0.342109\pi\)
\(252\) 0 0
\(253\) −0.612946 −0.0385356
\(254\) 0 0
\(255\) −16.0197 −1.00319
\(256\) 0 0
\(257\) −12.8822 −0.803573 −0.401786 0.915733i \(-0.631611\pi\)
−0.401786 + 0.915733i \(0.631611\pi\)
\(258\) 0 0
\(259\) 2.47671 0.153895
\(260\) 0 0
\(261\) 14.9487 0.925302
\(262\) 0 0
\(263\) −25.4881 −1.57166 −0.785831 0.618442i \(-0.787764\pi\)
−0.785831 + 0.618442i \(0.787764\pi\)
\(264\) 0 0
\(265\) −13.5527 −0.832534
\(266\) 0 0
\(267\) −0.0157639 −0.000964737 0
\(268\) 0 0
\(269\) −15.0772 −0.919270 −0.459635 0.888108i \(-0.652020\pi\)
−0.459635 + 0.888108i \(0.652020\pi\)
\(270\) 0 0
\(271\) 10.4014 0.631841 0.315920 0.948786i \(-0.397687\pi\)
0.315920 + 0.948786i \(0.397687\pi\)
\(272\) 0 0
\(273\) 3.90011 0.236045
\(274\) 0 0
\(275\) −1.12572 −0.0678834
\(276\) 0 0
\(277\) −9.01848 −0.541868 −0.270934 0.962598i \(-0.587333\pi\)
−0.270934 + 0.962598i \(0.587333\pi\)
\(278\) 0 0
\(279\) 14.9158 0.892988
\(280\) 0 0
\(281\) 1.91724 0.114373 0.0571864 0.998364i \(-0.481787\pi\)
0.0571864 + 0.998364i \(0.481787\pi\)
\(282\) 0 0
\(283\) 9.49457 0.564394 0.282197 0.959357i \(-0.408937\pi\)
0.282197 + 0.959357i \(0.408937\pi\)
\(284\) 0 0
\(285\) 17.6548 1.04578
\(286\) 0 0
\(287\) −6.37682 −0.376412
\(288\) 0 0
\(289\) 1.19028 0.0700165
\(290\) 0 0
\(291\) 16.0248 0.939388
\(292\) 0 0
\(293\) −16.6936 −0.975249 −0.487624 0.873053i \(-0.662136\pi\)
−0.487624 + 0.873053i \(0.662136\pi\)
\(294\) 0 0
\(295\) −4.87643 −0.283917
\(296\) 0 0
\(297\) 1.99374 0.115688
\(298\) 0 0
\(299\) −1.84681 −0.106804
\(300\) 0 0
\(301\) −10.9240 −0.629647
\(302\) 0 0
\(303\) −24.4767 −1.40615
\(304\) 0 0
\(305\) 26.9376 1.54244
\(306\) 0 0
\(307\) 25.9285 1.47982 0.739908 0.672709i \(-0.234869\pi\)
0.739908 + 0.672709i \(0.234869\pi\)
\(308\) 0 0
\(309\) 1.40735 0.0800615
\(310\) 0 0
\(311\) −25.4581 −1.44360 −0.721798 0.692104i \(-0.756684\pi\)
−0.721798 + 0.692104i \(0.756684\pi\)
\(312\) 0 0
\(313\) 17.5490 0.991928 0.495964 0.868343i \(-0.334815\pi\)
0.495964 + 0.868343i \(0.334815\pi\)
\(314\) 0 0
\(315\) −2.59632 −0.146286
\(316\) 0 0
\(317\) −4.42250 −0.248392 −0.124196 0.992258i \(-0.539635\pi\)
−0.124196 + 0.992258i \(0.539635\pi\)
\(318\) 0 0
\(319\) 6.27693 0.351440
\(320\) 0 0
\(321\) −14.6221 −0.816126
\(322\) 0 0
\(323\) −20.0470 −1.11544
\(324\) 0 0
\(325\) −3.39179 −0.188143
\(326\) 0 0
\(327\) −8.57952 −0.474449
\(328\) 0 0
\(329\) −0.544914 −0.0300421
\(330\) 0 0
\(331\) 31.9340 1.75525 0.877626 0.479345i \(-0.159126\pi\)
0.877626 + 0.479345i \(0.159126\pi\)
\(332\) 0 0
\(333\) −3.61538 −0.198122
\(334\) 0 0
\(335\) −9.38990 −0.513025
\(336\) 0 0
\(337\) −7.31791 −0.398632 −0.199316 0.979935i \(-0.563872\pi\)
−0.199316 + 0.979935i \(0.563872\pi\)
\(338\) 0 0
\(339\) 26.5933 1.44435
\(340\) 0 0
\(341\) 6.26312 0.339167
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 3.75608 0.202220
\(346\) 0 0
\(347\) 14.0287 0.753101 0.376551 0.926396i \(-0.377110\pi\)
0.376551 + 0.926396i \(0.377110\pi\)
\(348\) 0 0
\(349\) −26.8241 −1.43586 −0.717930 0.696116i \(-0.754910\pi\)
−0.717930 + 0.696116i \(0.754910\pi\)
\(350\) 0 0
\(351\) 6.00713 0.320637
\(352\) 0 0
\(353\) −7.37529 −0.392547 −0.196274 0.980549i \(-0.562884\pi\)
−0.196274 + 0.980549i \(0.562884\pi\)
\(354\) 0 0
\(355\) −3.94108 −0.209171
\(356\) 0 0
\(357\) 9.00689 0.476695
\(358\) 0 0
\(359\) −19.5656 −1.03263 −0.516317 0.856397i \(-0.672697\pi\)
−0.516317 + 0.856397i \(0.672697\pi\)
\(360\) 0 0
\(361\) 3.09316 0.162798
\(362\) 0 0
\(363\) 22.4365 1.17761
\(364\) 0 0
\(365\) 10.0093 0.523912
\(366\) 0 0
\(367\) −8.63696 −0.450846 −0.225423 0.974261i \(-0.572376\pi\)
−0.225423 + 0.974261i \(0.572376\pi\)
\(368\) 0 0
\(369\) 9.30858 0.484585
\(370\) 0 0
\(371\) 7.61984 0.395602
\(372\) 0 0
\(373\) −13.1338 −0.680042 −0.340021 0.940418i \(-0.610434\pi\)
−0.340021 + 0.940418i \(0.610434\pi\)
\(374\) 0 0
\(375\) 25.6787 1.32604
\(376\) 0 0
\(377\) 18.9124 0.974037
\(378\) 0 0
\(379\) 2.07360 0.106514 0.0532569 0.998581i \(-0.483040\pi\)
0.0532569 + 0.998581i \(0.483040\pi\)
\(380\) 0 0
\(381\) 27.1442 1.39064
\(382\) 0 0
\(383\) 7.61051 0.388879 0.194439 0.980915i \(-0.437711\pi\)
0.194439 + 0.980915i \(0.437711\pi\)
\(384\) 0 0
\(385\) −1.09019 −0.0555611
\(386\) 0 0
\(387\) 15.9463 0.810595
\(388\) 0 0
\(389\) −15.6243 −0.792183 −0.396092 0.918211i \(-0.629634\pi\)
−0.396092 + 0.918211i \(0.629634\pi\)
\(390\) 0 0
\(391\) −4.26501 −0.215691
\(392\) 0 0
\(393\) 31.0326 1.56539
\(394\) 0 0
\(395\) −4.38879 −0.220824
\(396\) 0 0
\(397\) 4.53335 0.227522 0.113761 0.993508i \(-0.463710\pi\)
0.113761 + 0.993508i \(0.463710\pi\)
\(398\) 0 0
\(399\) −9.92623 −0.496933
\(400\) 0 0
\(401\) 25.8569 1.29123 0.645616 0.763662i \(-0.276601\pi\)
0.645616 + 0.763662i \(0.276601\pi\)
\(402\) 0 0
\(403\) 18.8708 0.940020
\(404\) 0 0
\(405\) −20.0064 −0.994126
\(406\) 0 0
\(407\) −1.51809 −0.0752490
\(408\) 0 0
\(409\) −14.3124 −0.707702 −0.353851 0.935302i \(-0.615128\pi\)
−0.353851 + 0.935302i \(0.615128\pi\)
\(410\) 0 0
\(411\) 40.6109 2.00319
\(412\) 0 0
\(413\) 2.74172 0.134911
\(414\) 0 0
\(415\) 10.3932 0.510182
\(416\) 0 0
\(417\) 12.9786 0.635566
\(418\) 0 0
\(419\) −23.4145 −1.14387 −0.571936 0.820299i \(-0.693807\pi\)
−0.571936 + 0.820299i \(0.693807\pi\)
\(420\) 0 0
\(421\) 15.4946 0.755159 0.377580 0.925977i \(-0.376756\pi\)
0.377580 + 0.925977i \(0.376756\pi\)
\(422\) 0 0
\(423\) 0.795439 0.0386756
\(424\) 0 0
\(425\) −7.83299 −0.379956
\(426\) 0 0
\(427\) −15.1454 −0.732935
\(428\) 0 0
\(429\) −2.39056 −0.115417
\(430\) 0 0
\(431\) 34.6016 1.66670 0.833349 0.552747i \(-0.186420\pi\)
0.833349 + 0.552747i \(0.186420\pi\)
\(432\) 0 0
\(433\) −18.8051 −0.903713 −0.451857 0.892091i \(-0.649238\pi\)
−0.451857 + 0.892091i \(0.649238\pi\)
\(434\) 0 0
\(435\) −38.4644 −1.84423
\(436\) 0 0
\(437\) 4.70034 0.224848
\(438\) 0 0
\(439\) −37.7000 −1.79933 −0.899663 0.436586i \(-0.856187\pi\)
−0.899663 + 0.436586i \(0.856187\pi\)
\(440\) 0 0
\(441\) 1.45975 0.0695120
\(442\) 0 0
\(443\) −13.6027 −0.646284 −0.323142 0.946350i \(-0.604739\pi\)
−0.323142 + 0.946350i \(0.604739\pi\)
\(444\) 0 0
\(445\) 0.0132766 0.000629373 0
\(446\) 0 0
\(447\) 14.0782 0.665877
\(448\) 0 0
\(449\) 3.95586 0.186689 0.0933443 0.995634i \(-0.470244\pi\)
0.0933443 + 0.995634i \(0.470244\pi\)
\(450\) 0 0
\(451\) 3.90865 0.184051
\(452\) 0 0
\(453\) 2.26534 0.106435
\(454\) 0 0
\(455\) −3.28473 −0.153991
\(456\) 0 0
\(457\) 19.3668 0.905939 0.452969 0.891526i \(-0.350365\pi\)
0.452969 + 0.891526i \(0.350365\pi\)
\(458\) 0 0
\(459\) 13.8728 0.647529
\(460\) 0 0
\(461\) −27.0772 −1.26111 −0.630554 0.776145i \(-0.717172\pi\)
−0.630554 + 0.776145i \(0.717172\pi\)
\(462\) 0 0
\(463\) −20.6172 −0.958164 −0.479082 0.877770i \(-0.659030\pi\)
−0.479082 + 0.877770i \(0.659030\pi\)
\(464\) 0 0
\(465\) −38.3798 −1.77982
\(466\) 0 0
\(467\) 6.54603 0.302914 0.151457 0.988464i \(-0.451603\pi\)
0.151457 + 0.988464i \(0.451603\pi\)
\(468\) 0 0
\(469\) 5.27937 0.243778
\(470\) 0 0
\(471\) 5.84263 0.269214
\(472\) 0 0
\(473\) 6.69580 0.307873
\(474\) 0 0
\(475\) 8.63250 0.396086
\(476\) 0 0
\(477\) −11.1231 −0.509291
\(478\) 0 0
\(479\) −22.8457 −1.04385 −0.521924 0.852992i \(-0.674786\pi\)
−0.521924 + 0.852992i \(0.674786\pi\)
\(480\) 0 0
\(481\) −4.57400 −0.208557
\(482\) 0 0
\(483\) −2.11181 −0.0960908
\(484\) 0 0
\(485\) −13.4963 −0.612836
\(486\) 0 0
\(487\) −29.3458 −1.32979 −0.664893 0.746938i \(-0.731523\pi\)
−0.664893 + 0.746938i \(0.731523\pi\)
\(488\) 0 0
\(489\) −12.5513 −0.567587
\(490\) 0 0
\(491\) 40.7264 1.83796 0.918978 0.394309i \(-0.129016\pi\)
0.918978 + 0.394309i \(0.129016\pi\)
\(492\) 0 0
\(493\) 43.6762 1.96708
\(494\) 0 0
\(495\) 1.59140 0.0715283
\(496\) 0 0
\(497\) 2.21583 0.0993935
\(498\) 0 0
\(499\) −10.9806 −0.491559 −0.245780 0.969326i \(-0.579044\pi\)
−0.245780 + 0.969326i \(0.579044\pi\)
\(500\) 0 0
\(501\) −9.56183 −0.427191
\(502\) 0 0
\(503\) 9.67314 0.431304 0.215652 0.976470i \(-0.430812\pi\)
0.215652 + 0.976470i \(0.430812\pi\)
\(504\) 0 0
\(505\) 20.6147 0.917341
\(506\) 0 0
\(507\) 20.2508 0.899370
\(508\) 0 0
\(509\) 15.7619 0.698634 0.349317 0.937005i \(-0.386414\pi\)
0.349317 + 0.937005i \(0.386414\pi\)
\(510\) 0 0
\(511\) −5.62764 −0.248952
\(512\) 0 0
\(513\) −15.2888 −0.675019
\(514\) 0 0
\(515\) −1.18530 −0.0522303
\(516\) 0 0
\(517\) 0.334003 0.0146894
\(518\) 0 0
\(519\) 22.5971 0.991903
\(520\) 0 0
\(521\) −13.4428 −0.588938 −0.294469 0.955661i \(-0.595143\pi\)
−0.294469 + 0.955661i \(0.595143\pi\)
\(522\) 0 0
\(523\) −33.5415 −1.46667 −0.733334 0.679868i \(-0.762037\pi\)
−0.733334 + 0.679868i \(0.762037\pi\)
\(524\) 0 0
\(525\) −3.87849 −0.169271
\(526\) 0 0
\(527\) 43.5801 1.89838
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −4.00223 −0.173682
\(532\) 0 0
\(533\) 11.7767 0.510108
\(534\) 0 0
\(535\) 12.3150 0.532423
\(536\) 0 0
\(537\) −23.9880 −1.03516
\(538\) 0 0
\(539\) 0.612946 0.0264015
\(540\) 0 0
\(541\) −22.7411 −0.977718 −0.488859 0.872363i \(-0.662587\pi\)
−0.488859 + 0.872363i \(0.662587\pi\)
\(542\) 0 0
\(543\) −42.6927 −1.83212
\(544\) 0 0
\(545\) 7.22581 0.309520
\(546\) 0 0
\(547\) 35.4551 1.51595 0.757975 0.652284i \(-0.226189\pi\)
0.757975 + 0.652284i \(0.226189\pi\)
\(548\) 0 0
\(549\) 22.1085 0.943567
\(550\) 0 0
\(551\) −48.1342 −2.05059
\(552\) 0 0
\(553\) 2.46755 0.104931
\(554\) 0 0
\(555\) 9.30271 0.394878
\(556\) 0 0
\(557\) −37.3269 −1.58159 −0.790796 0.612080i \(-0.790333\pi\)
−0.790796 + 0.612080i \(0.790333\pi\)
\(558\) 0 0
\(559\) 20.1744 0.853288
\(560\) 0 0
\(561\) −5.52074 −0.233086
\(562\) 0 0
\(563\) −7.80022 −0.328740 −0.164370 0.986399i \(-0.552559\pi\)
−0.164370 + 0.986399i \(0.552559\pi\)
\(564\) 0 0
\(565\) −22.3973 −0.942262
\(566\) 0 0
\(567\) 11.2484 0.472388
\(568\) 0 0
\(569\) −13.4450 −0.563643 −0.281822 0.959467i \(-0.590939\pi\)
−0.281822 + 0.959467i \(0.590939\pi\)
\(570\) 0 0
\(571\) −23.6776 −0.990877 −0.495438 0.868643i \(-0.664993\pi\)
−0.495438 + 0.868643i \(0.664993\pi\)
\(572\) 0 0
\(573\) −4.16141 −0.173845
\(574\) 0 0
\(575\) 1.83657 0.0765903
\(576\) 0 0
\(577\) 31.6414 1.31725 0.658625 0.752471i \(-0.271138\pi\)
0.658625 + 0.752471i \(0.271138\pi\)
\(578\) 0 0
\(579\) −31.2537 −1.29886
\(580\) 0 0
\(581\) −5.84347 −0.242428
\(582\) 0 0
\(583\) −4.67055 −0.193435
\(584\) 0 0
\(585\) 4.79490 0.198245
\(586\) 0 0
\(587\) 44.5237 1.83769 0.918844 0.394621i \(-0.129124\pi\)
0.918844 + 0.394621i \(0.129124\pi\)
\(588\) 0 0
\(589\) −48.0283 −1.97897
\(590\) 0 0
\(591\) −32.4155 −1.33340
\(592\) 0 0
\(593\) −0.774107 −0.0317888 −0.0158944 0.999874i \(-0.505060\pi\)
−0.0158944 + 0.999874i \(0.505060\pi\)
\(594\) 0 0
\(595\) −7.58575 −0.310986
\(596\) 0 0
\(597\) −28.9467 −1.18471
\(598\) 0 0
\(599\) 22.1838 0.906404 0.453202 0.891408i \(-0.350282\pi\)
0.453202 + 0.891408i \(0.350282\pi\)
\(600\) 0 0
\(601\) −19.5006 −0.795445 −0.397722 0.917506i \(-0.630199\pi\)
−0.397722 + 0.917506i \(0.630199\pi\)
\(602\) 0 0
\(603\) −7.70657 −0.313836
\(604\) 0 0
\(605\) −18.8964 −0.768248
\(606\) 0 0
\(607\) 8.99336 0.365029 0.182515 0.983203i \(-0.441576\pi\)
0.182515 + 0.983203i \(0.441576\pi\)
\(608\) 0 0
\(609\) 21.6262 0.876338
\(610\) 0 0
\(611\) 1.00635 0.0407126
\(612\) 0 0
\(613\) −5.21656 −0.210695 −0.105347 0.994435i \(-0.533595\pi\)
−0.105347 + 0.994435i \(0.533595\pi\)
\(614\) 0 0
\(615\) −23.9518 −0.965831
\(616\) 0 0
\(617\) −21.1908 −0.853111 −0.426555 0.904461i \(-0.640273\pi\)
−0.426555 + 0.904461i \(0.640273\pi\)
\(618\) 0 0
\(619\) 10.6076 0.426355 0.213177 0.977014i \(-0.431619\pi\)
0.213177 + 0.977014i \(0.431619\pi\)
\(620\) 0 0
\(621\) −3.25271 −0.130527
\(622\) 0 0
\(623\) −0.00746465 −0.000299065 0
\(624\) 0 0
\(625\) −12.4441 −0.497766
\(626\) 0 0
\(627\) 6.08424 0.242981
\(628\) 0 0
\(629\) −10.5632 −0.421182
\(630\) 0 0
\(631\) 0.393777 0.0156760 0.00783802 0.999969i \(-0.497505\pi\)
0.00783802 + 0.999969i \(0.497505\pi\)
\(632\) 0 0
\(633\) −46.5393 −1.84977
\(634\) 0 0
\(635\) −22.8613 −0.907224
\(636\) 0 0
\(637\) 1.84681 0.0731731
\(638\) 0 0
\(639\) −3.23456 −0.127957
\(640\) 0 0
\(641\) −25.8922 −1.02268 −0.511341 0.859378i \(-0.670851\pi\)
−0.511341 + 0.859378i \(0.670851\pi\)
\(642\) 0 0
\(643\) −42.5479 −1.67793 −0.838963 0.544188i \(-0.816838\pi\)
−0.838963 + 0.544188i \(0.816838\pi\)
\(644\) 0 0
\(645\) −41.0312 −1.61560
\(646\) 0 0
\(647\) 48.7238 1.91553 0.957765 0.287551i \(-0.0928411\pi\)
0.957765 + 0.287551i \(0.0928411\pi\)
\(648\) 0 0
\(649\) −1.68053 −0.0659664
\(650\) 0 0
\(651\) 21.5786 0.845733
\(652\) 0 0
\(653\) 19.7238 0.771854 0.385927 0.922529i \(-0.373882\pi\)
0.385927 + 0.922529i \(0.373882\pi\)
\(654\) 0 0
\(655\) −26.1362 −1.02122
\(656\) 0 0
\(657\) 8.21496 0.320496
\(658\) 0 0
\(659\) 6.48604 0.252660 0.126330 0.991988i \(-0.459680\pi\)
0.126330 + 0.991988i \(0.459680\pi\)
\(660\) 0 0
\(661\) 37.1204 1.44382 0.721909 0.691988i \(-0.243265\pi\)
0.721909 + 0.691988i \(0.243265\pi\)
\(662\) 0 0
\(663\) −16.6340 −0.646011
\(664\) 0 0
\(665\) 8.36003 0.324188
\(666\) 0 0
\(667\) −10.2406 −0.396517
\(668\) 0 0
\(669\) 6.36540 0.246101
\(670\) 0 0
\(671\) 9.28329 0.358377
\(672\) 0 0
\(673\) 39.9763 1.54097 0.770486 0.637457i \(-0.220013\pi\)
0.770486 + 0.637457i \(0.220013\pi\)
\(674\) 0 0
\(675\) −5.97384 −0.229933
\(676\) 0 0
\(677\) 12.1898 0.468492 0.234246 0.972177i \(-0.424738\pi\)
0.234246 + 0.972177i \(0.424738\pi\)
\(678\) 0 0
\(679\) 7.58815 0.291207
\(680\) 0 0
\(681\) 49.2620 1.88773
\(682\) 0 0
\(683\) 22.0880 0.845174 0.422587 0.906322i \(-0.361122\pi\)
0.422587 + 0.906322i \(0.361122\pi\)
\(684\) 0 0
\(685\) −34.2032 −1.30684
\(686\) 0 0
\(687\) −58.7318 −2.24076
\(688\) 0 0
\(689\) −14.0724 −0.536115
\(690\) 0 0
\(691\) 29.7055 1.13005 0.565025 0.825074i \(-0.308866\pi\)
0.565025 + 0.825074i \(0.308866\pi\)
\(692\) 0 0
\(693\) −0.894750 −0.0339887
\(694\) 0 0
\(695\) −10.9308 −0.414629
\(696\) 0 0
\(697\) 27.1972 1.03017
\(698\) 0 0
\(699\) 46.8822 1.77325
\(700\) 0 0
\(701\) −37.4972 −1.41625 −0.708125 0.706087i \(-0.750459\pi\)
−0.708125 + 0.706087i \(0.750459\pi\)
\(702\) 0 0
\(703\) 11.6414 0.439063
\(704\) 0 0
\(705\) −2.04674 −0.0770846
\(706\) 0 0
\(707\) −11.5904 −0.435901
\(708\) 0 0
\(709\) 13.4163 0.503861 0.251931 0.967745i \(-0.418935\pi\)
0.251931 + 0.967745i \(0.418935\pi\)
\(710\) 0 0
\(711\) −3.60201 −0.135086
\(712\) 0 0
\(713\) −10.2181 −0.382669
\(714\) 0 0
\(715\) 2.01337 0.0752956
\(716\) 0 0
\(717\) −27.9046 −1.04212
\(718\) 0 0
\(719\) −5.14090 −0.191723 −0.0958616 0.995395i \(-0.530561\pi\)
−0.0958616 + 0.995395i \(0.530561\pi\)
\(720\) 0 0
\(721\) 0.666419 0.0248187
\(722\) 0 0
\(723\) −25.2544 −0.939220
\(724\) 0 0
\(725\) −18.8076 −0.698495
\(726\) 0 0
\(727\) 33.3224 1.23586 0.617929 0.786234i \(-0.287972\pi\)
0.617929 + 0.786234i \(0.287972\pi\)
\(728\) 0 0
\(729\) 4.18752 0.155093
\(730\) 0 0
\(731\) 46.5908 1.72322
\(732\) 0 0
\(733\) −37.2546 −1.37603 −0.688015 0.725696i \(-0.741518\pi\)
−0.688015 + 0.725696i \(0.741518\pi\)
\(734\) 0 0
\(735\) −3.75608 −0.138545
\(736\) 0 0
\(737\) −3.23597 −0.119198
\(738\) 0 0
\(739\) −6.52222 −0.239924 −0.119962 0.992779i \(-0.538277\pi\)
−0.119962 + 0.992779i \(0.538277\pi\)
\(740\) 0 0
\(741\) 18.3318 0.673436
\(742\) 0 0
\(743\) −15.6151 −0.572864 −0.286432 0.958101i \(-0.592469\pi\)
−0.286432 + 0.958101i \(0.592469\pi\)
\(744\) 0 0
\(745\) −11.8569 −0.434404
\(746\) 0 0
\(747\) 8.53001 0.312097
\(748\) 0 0
\(749\) −6.92396 −0.252996
\(750\) 0 0
\(751\) −50.7568 −1.85214 −0.926071 0.377350i \(-0.876835\pi\)
−0.926071 + 0.377350i \(0.876835\pi\)
\(752\) 0 0
\(753\) −31.8472 −1.16058
\(754\) 0 0
\(755\) −1.90790 −0.0694358
\(756\) 0 0
\(757\) −35.5904 −1.29355 −0.646777 0.762679i \(-0.723884\pi\)
−0.646777 + 0.762679i \(0.723884\pi\)
\(758\) 0 0
\(759\) 1.29443 0.0469847
\(760\) 0 0
\(761\) 2.59153 0.0939429 0.0469715 0.998896i \(-0.485043\pi\)
0.0469715 + 0.998896i \(0.485043\pi\)
\(762\) 0 0
\(763\) −4.06263 −0.147077
\(764\) 0 0
\(765\) 11.0733 0.400357
\(766\) 0 0
\(767\) −5.06342 −0.182830
\(768\) 0 0
\(769\) 23.9098 0.862208 0.431104 0.902302i \(-0.358124\pi\)
0.431104 + 0.902302i \(0.358124\pi\)
\(770\) 0 0
\(771\) 27.2049 0.979760
\(772\) 0 0
\(773\) −12.0909 −0.434879 −0.217440 0.976074i \(-0.569771\pi\)
−0.217440 + 0.976074i \(0.569771\pi\)
\(774\) 0 0
\(775\) −18.7662 −0.674101
\(776\) 0 0
\(777\) −5.23035 −0.187638
\(778\) 0 0
\(779\) −29.9732 −1.07390
\(780\) 0 0
\(781\) −1.35818 −0.0485996
\(782\) 0 0
\(783\) 33.3097 1.19039
\(784\) 0 0
\(785\) −4.92076 −0.175629
\(786\) 0 0
\(787\) 10.7979 0.384905 0.192453 0.981306i \(-0.438356\pi\)
0.192453 + 0.981306i \(0.438356\pi\)
\(788\) 0 0
\(789\) 53.8260 1.91626
\(790\) 0 0
\(791\) 12.5926 0.447743
\(792\) 0 0
\(793\) 27.9705 0.993263
\(794\) 0 0
\(795\) 28.6207 1.01507
\(796\) 0 0
\(797\) −20.5076 −0.726417 −0.363209 0.931708i \(-0.618319\pi\)
−0.363209 + 0.931708i \(0.618319\pi\)
\(798\) 0 0
\(799\) 2.32406 0.0822194
\(800\) 0 0
\(801\) 0.0108965 0.000385010 0
\(802\) 0 0
\(803\) 3.44944 0.121728
\(804\) 0 0
\(805\) 1.77860 0.0626875
\(806\) 0 0
\(807\) 31.8401 1.12083
\(808\) 0 0
\(809\) 37.4089 1.31523 0.657614 0.753355i \(-0.271566\pi\)
0.657614 + 0.753355i \(0.271566\pi\)
\(810\) 0 0
\(811\) 19.2121 0.674627 0.337314 0.941392i \(-0.390482\pi\)
0.337314 + 0.941392i \(0.390482\pi\)
\(812\) 0 0
\(813\) −21.9658 −0.770375
\(814\) 0 0
\(815\) 10.5709 0.370282
\(816\) 0 0
\(817\) −51.3463 −1.79638
\(818\) 0 0
\(819\) −2.69588 −0.0942017
\(820\) 0 0
\(821\) 30.3982 1.06091 0.530453 0.847714i \(-0.322022\pi\)
0.530453 + 0.847714i \(0.322022\pi\)
\(822\) 0 0
\(823\) −19.8890 −0.693286 −0.346643 0.937997i \(-0.612678\pi\)
−0.346643 + 0.937997i \(0.612678\pi\)
\(824\) 0 0
\(825\) 2.37731 0.0827673
\(826\) 0 0
\(827\) 22.2952 0.775279 0.387640 0.921811i \(-0.373291\pi\)
0.387640 + 0.921811i \(0.373291\pi\)
\(828\) 0 0
\(829\) 25.1144 0.872260 0.436130 0.899884i \(-0.356349\pi\)
0.436130 + 0.899884i \(0.356349\pi\)
\(830\) 0 0
\(831\) 19.0453 0.660676
\(832\) 0 0
\(833\) 4.26501 0.147774
\(834\) 0 0
\(835\) 8.05313 0.278690
\(836\) 0 0
\(837\) 33.2364 1.14882
\(838\) 0 0
\(839\) 14.8696 0.513355 0.256677 0.966497i \(-0.417372\pi\)
0.256677 + 0.966497i \(0.417372\pi\)
\(840\) 0 0
\(841\) 75.8696 2.61619
\(842\) 0 0
\(843\) −4.04885 −0.139450
\(844\) 0 0
\(845\) −17.0556 −0.586729
\(846\) 0 0
\(847\) 10.6243 0.365055
\(848\) 0 0
\(849\) −20.0508 −0.688140
\(850\) 0 0
\(851\) 2.47671 0.0849005
\(852\) 0 0
\(853\) −46.0320 −1.57611 −0.788053 0.615607i \(-0.788911\pi\)
−0.788053 + 0.615607i \(0.788911\pi\)
\(854\) 0 0
\(855\) −12.2036 −0.417353
\(856\) 0 0
\(857\) −23.0239 −0.786480 −0.393240 0.919436i \(-0.628646\pi\)
−0.393240 + 0.919436i \(0.628646\pi\)
\(858\) 0 0
\(859\) 7.46024 0.254540 0.127270 0.991868i \(-0.459378\pi\)
0.127270 + 0.991868i \(0.459378\pi\)
\(860\) 0 0
\(861\) 13.4666 0.458942
\(862\) 0 0
\(863\) 41.0366 1.39690 0.698450 0.715659i \(-0.253873\pi\)
0.698450 + 0.715659i \(0.253873\pi\)
\(864\) 0 0
\(865\) −19.0316 −0.647095
\(866\) 0 0
\(867\) −2.51365 −0.0853680
\(868\) 0 0
\(869\) −1.51247 −0.0513072
\(870\) 0 0
\(871\) −9.74996 −0.330365
\(872\) 0 0
\(873\) −11.0768 −0.374894
\(874\) 0 0
\(875\) 12.1595 0.411068
\(876\) 0 0
\(877\) 50.4646 1.70407 0.852034 0.523487i \(-0.175369\pi\)
0.852034 + 0.523487i \(0.175369\pi\)
\(878\) 0 0
\(879\) 35.2537 1.18908
\(880\) 0 0
\(881\) 39.7525 1.33930 0.669648 0.742679i \(-0.266445\pi\)
0.669648 + 0.742679i \(0.266445\pi\)
\(882\) 0 0
\(883\) 2.64149 0.0888933 0.0444467 0.999012i \(-0.485848\pi\)
0.0444467 + 0.999012i \(0.485848\pi\)
\(884\) 0 0
\(885\) 10.2981 0.346167
\(886\) 0 0
\(887\) −27.0485 −0.908199 −0.454100 0.890951i \(-0.650039\pi\)
−0.454100 + 0.890951i \(0.650039\pi\)
\(888\) 0 0
\(889\) 12.8535 0.431094
\(890\) 0 0
\(891\) −6.89465 −0.230980
\(892\) 0 0
\(893\) −2.56128 −0.0857099
\(894\) 0 0
\(895\) 20.2031 0.675314
\(896\) 0 0
\(897\) 3.90011 0.130221
\(898\) 0 0
\(899\) 104.639 3.48990
\(900\) 0 0
\(901\) −32.4987 −1.08269
\(902\) 0 0
\(903\) 23.0694 0.767700
\(904\) 0 0
\(905\) 35.9565 1.19524
\(906\) 0 0
\(907\) 21.5751 0.716390 0.358195 0.933647i \(-0.383392\pi\)
0.358195 + 0.933647i \(0.383392\pi\)
\(908\) 0 0
\(909\) 16.9191 0.561171
\(910\) 0 0
\(911\) −47.6282 −1.57799 −0.788996 0.614398i \(-0.789399\pi\)
−0.788996 + 0.614398i \(0.789399\pi\)
\(912\) 0 0
\(913\) 3.58173 0.118538
\(914\) 0 0
\(915\) −56.8871 −1.88063
\(916\) 0 0
\(917\) 14.6948 0.485264
\(918\) 0 0
\(919\) −42.5481 −1.40353 −0.701766 0.712408i \(-0.747605\pi\)
−0.701766 + 0.712408i \(0.747605\pi\)
\(920\) 0 0
\(921\) −54.7560 −1.80427
\(922\) 0 0
\(923\) −4.09221 −0.134697
\(924\) 0 0
\(925\) 4.54866 0.149559
\(926\) 0 0
\(927\) −0.972807 −0.0319512
\(928\) 0 0
\(929\) −48.6564 −1.59636 −0.798182 0.602416i \(-0.794205\pi\)
−0.798182 + 0.602416i \(0.794205\pi\)
\(930\) 0 0
\(931\) −4.70034 −0.154047
\(932\) 0 0
\(933\) 53.7627 1.76011
\(934\) 0 0
\(935\) 4.64966 0.152060
\(936\) 0 0
\(937\) −6.35816 −0.207712 −0.103856 0.994592i \(-0.533118\pi\)
−0.103856 + 0.994592i \(0.533118\pi\)
\(938\) 0 0
\(939\) −37.0602 −1.20941
\(940\) 0 0
\(941\) −22.4618 −0.732234 −0.366117 0.930569i \(-0.619313\pi\)
−0.366117 + 0.930569i \(0.619313\pi\)
\(942\) 0 0
\(943\) −6.37682 −0.207658
\(944\) 0 0
\(945\) −5.78529 −0.188195
\(946\) 0 0
\(947\) −42.2098 −1.37163 −0.685817 0.727774i \(-0.740555\pi\)
−0.685817 + 0.727774i \(0.740555\pi\)
\(948\) 0 0
\(949\) 10.3932 0.337376
\(950\) 0 0
\(951\) 9.33948 0.302853
\(952\) 0 0
\(953\) 60.6146 1.96350 0.981750 0.190177i \(-0.0609061\pi\)
0.981750 + 0.190177i \(0.0609061\pi\)
\(954\) 0 0
\(955\) 3.50481 0.113413
\(956\) 0 0
\(957\) −13.2557 −0.428496
\(958\) 0 0
\(959\) 19.2303 0.620980
\(960\) 0 0
\(961\) 73.4087 2.36802
\(962\) 0 0
\(963\) 10.1073 0.325702
\(964\) 0 0
\(965\) 26.3224 0.847348
\(966\) 0 0
\(967\) −52.1364 −1.67659 −0.838297 0.545213i \(-0.816449\pi\)
−0.838297 + 0.545213i \(0.816449\pi\)
\(968\) 0 0
\(969\) 42.3354 1.36001
\(970\) 0 0
\(971\) −50.8055 −1.63042 −0.815212 0.579162i \(-0.803380\pi\)
−0.815212 + 0.579162i \(0.803380\pi\)
\(972\) 0 0
\(973\) 6.14573 0.197023
\(974\) 0 0
\(975\) 7.16283 0.229394
\(976\) 0 0
\(977\) 8.63804 0.276355 0.138178 0.990407i \(-0.455875\pi\)
0.138178 + 0.990407i \(0.455875\pi\)
\(978\) 0 0
\(979\) 0.00457543 0.000146231 0
\(980\) 0 0
\(981\) 5.93044 0.189344
\(982\) 0 0
\(983\) 25.0179 0.797946 0.398973 0.916963i \(-0.369367\pi\)
0.398973 + 0.916963i \(0.369367\pi\)
\(984\) 0 0
\(985\) 27.3009 0.869879
\(986\) 0 0
\(987\) 1.15076 0.0366290
\(988\) 0 0
\(989\) −10.9240 −0.347362
\(990\) 0 0
\(991\) 11.4716 0.364406 0.182203 0.983261i \(-0.441677\pi\)
0.182203 + 0.983261i \(0.441677\pi\)
\(992\) 0 0
\(993\) −67.4387 −2.14010
\(994\) 0 0
\(995\) 24.3794 0.772879
\(996\) 0 0
\(997\) −14.9322 −0.472908 −0.236454 0.971643i \(-0.575985\pi\)
−0.236454 + 0.971643i \(0.575985\pi\)
\(998\) 0 0
\(999\) −8.05603 −0.254882
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 644.2.a.d.1.1 5
3.2 odd 2 5796.2.a.t.1.3 5
4.3 odd 2 2576.2.a.bb.1.5 5
7.6 odd 2 4508.2.a.f.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
644.2.a.d.1.1 5 1.1 even 1 trivial
2576.2.a.bb.1.5 5 4.3 odd 2
4508.2.a.f.1.5 5 7.6 odd 2
5796.2.a.t.1.3 5 3.2 odd 2