Properties

Label 5520.2.a.bm.1.2
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} +5.41421 q^{11} +0.585786 q^{13} -1.00000 q^{15} +8.07107 q^{17} -2.24264 q^{19} +3.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.41421 q^{29} +9.82843 q^{31} +5.41421 q^{33} -3.82843 q^{35} +3.00000 q^{37} +0.585786 q^{39} -7.58579 q^{41} -6.00000 q^{43} -1.00000 q^{45} +11.4142 q^{47} +7.65685 q^{49} +8.07107 q^{51} -1.24264 q^{53} -5.41421 q^{55} -2.24264 q^{57} -12.8995 q^{59} +2.58579 q^{61} +3.82843 q^{63} -0.585786 q^{65} -5.48528 q^{67} +1.00000 q^{69} +10.8995 q^{71} -1.75736 q^{73} +1.00000 q^{75} +20.7279 q^{77} -7.31371 q^{79} +1.00000 q^{81} +0.0710678 q^{83} -8.07107 q^{85} -6.41421 q^{87} -18.1421 q^{89} +2.24264 q^{91} +9.82843 q^{93} +2.24264 q^{95} -6.82843 q^{97} +5.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 8 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{17} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 14 q^{31} + 8 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39} - 18 q^{41} - 12 q^{43} - 2 q^{45} + 20 q^{47} + 4 q^{49} + 2 q^{51} + 6 q^{53} - 8 q^{55} + 4 q^{57} - 6 q^{59} + 8 q^{61} + 2 q^{63} - 4 q^{65} + 6 q^{67} + 2 q^{69} + 2 q^{71} - 12 q^{73} + 2 q^{75} + 16 q^{77} + 8 q^{79} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 8 q^{89} - 4 q^{91} + 14 q^{93} - 4 q^{95} - 8 q^{97} + 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.82843 1.44701 0.723505 0.690319i \(-0.242530\pi\)
0.723505 + 0.690319i \(0.242530\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.41421 1.63245 0.816223 0.577736i \(-0.196064\pi\)
0.816223 + 0.577736i \(0.196064\pi\)
\(12\) 0 0
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 8.07107 1.95752 0.978761 0.205006i \(-0.0657214\pi\)
0.978761 + 0.205006i \(0.0657214\pi\)
\(18\) 0 0
\(19\) −2.24264 −0.514497 −0.257249 0.966345i \(-0.582816\pi\)
−0.257249 + 0.966345i \(0.582816\pi\)
\(20\) 0 0
\(21\) 3.82843 0.835431
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.41421 −1.19109 −0.595545 0.803322i \(-0.703064\pi\)
−0.595545 + 0.803322i \(0.703064\pi\)
\(30\) 0 0
\(31\) 9.82843 1.76524 0.882619 0.470089i \(-0.155778\pi\)
0.882619 + 0.470089i \(0.155778\pi\)
\(32\) 0 0
\(33\) 5.41421 0.942494
\(34\) 0 0
\(35\) −3.82843 −0.647122
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0.585786 0.0938009
\(40\) 0 0
\(41\) −7.58579 −1.18470 −0.592350 0.805680i \(-0.701800\pi\)
−0.592350 + 0.805680i \(0.701800\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 11.4142 1.66493 0.832467 0.554075i \(-0.186928\pi\)
0.832467 + 0.554075i \(0.186928\pi\)
\(48\) 0 0
\(49\) 7.65685 1.09384
\(50\) 0 0
\(51\) 8.07107 1.13018
\(52\) 0 0
\(53\) −1.24264 −0.170690 −0.0853449 0.996351i \(-0.527199\pi\)
−0.0853449 + 0.996351i \(0.527199\pi\)
\(54\) 0 0
\(55\) −5.41421 −0.730052
\(56\) 0 0
\(57\) −2.24264 −0.297045
\(58\) 0 0
\(59\) −12.8995 −1.67937 −0.839686 0.543073i \(-0.817261\pi\)
−0.839686 + 0.543073i \(0.817261\pi\)
\(60\) 0 0
\(61\) 2.58579 0.331076 0.165538 0.986203i \(-0.447064\pi\)
0.165538 + 0.986203i \(0.447064\pi\)
\(62\) 0 0
\(63\) 3.82843 0.482336
\(64\) 0 0
\(65\) −0.585786 −0.0726579
\(66\) 0 0
\(67\) −5.48528 −0.670134 −0.335067 0.942194i \(-0.608759\pi\)
−0.335067 + 0.942194i \(0.608759\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.8995 1.29353 0.646766 0.762688i \(-0.276121\pi\)
0.646766 + 0.762688i \(0.276121\pi\)
\(72\) 0 0
\(73\) −1.75736 −0.205683 −0.102842 0.994698i \(-0.532794\pi\)
−0.102842 + 0.994698i \(0.532794\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 20.7279 2.36217
\(78\) 0 0
\(79\) −7.31371 −0.822856 −0.411428 0.911442i \(-0.634970\pi\)
−0.411428 + 0.911442i \(0.634970\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.0710678 0.00780071 0.00390035 0.999992i \(-0.498758\pi\)
0.00390035 + 0.999992i \(0.498758\pi\)
\(84\) 0 0
\(85\) −8.07107 −0.875430
\(86\) 0 0
\(87\) −6.41421 −0.687676
\(88\) 0 0
\(89\) −18.1421 −1.92306 −0.961531 0.274696i \(-0.911423\pi\)
−0.961531 + 0.274696i \(0.911423\pi\)
\(90\) 0 0
\(91\) 2.24264 0.235093
\(92\) 0 0
\(93\) 9.82843 1.01916
\(94\) 0 0
\(95\) 2.24264 0.230090
\(96\) 0 0
\(97\) −6.82843 −0.693322 −0.346661 0.937991i \(-0.612685\pi\)
−0.346661 + 0.937991i \(0.612685\pi\)
\(98\) 0 0
\(99\) 5.41421 0.544149
\(100\) 0 0
\(101\) −18.5563 −1.84643 −0.923213 0.384289i \(-0.874447\pi\)
−0.923213 + 0.384289i \(0.874447\pi\)
\(102\) 0 0
\(103\) 4.82843 0.475759 0.237880 0.971295i \(-0.423548\pi\)
0.237880 + 0.971295i \(0.423548\pi\)
\(104\) 0 0
\(105\) −3.82843 −0.373616
\(106\) 0 0
\(107\) 1.58579 0.153304 0.0766519 0.997058i \(-0.475577\pi\)
0.0766519 + 0.997058i \(0.475577\pi\)
\(108\) 0 0
\(109\) 5.75736 0.551455 0.275728 0.961236i \(-0.411081\pi\)
0.275728 + 0.961236i \(0.411081\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) −7.58579 −0.713611 −0.356805 0.934179i \(-0.616134\pi\)
−0.356805 + 0.934179i \(0.616134\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0.585786 0.0541560
\(118\) 0 0
\(119\) 30.8995 2.83255
\(120\) 0 0
\(121\) 18.3137 1.66488
\(122\) 0 0
\(123\) −7.58579 −0.683987
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.24264 −0.731416 −0.365708 0.930730i \(-0.619173\pi\)
−0.365708 + 0.930730i \(0.619173\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 2.34315 0.204722 0.102361 0.994747i \(-0.467360\pi\)
0.102361 + 0.994747i \(0.467360\pi\)
\(132\) 0 0
\(133\) −8.58579 −0.744482
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −2.82843 −0.241649 −0.120824 0.992674i \(-0.538554\pi\)
−0.120824 + 0.992674i \(0.538554\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 11.4142 0.961250
\(142\) 0 0
\(143\) 3.17157 0.265220
\(144\) 0 0
\(145\) 6.41421 0.532671
\(146\) 0 0
\(147\) 7.65685 0.631527
\(148\) 0 0
\(149\) −5.07107 −0.415438 −0.207719 0.978189i \(-0.566604\pi\)
−0.207719 + 0.978189i \(0.566604\pi\)
\(150\) 0 0
\(151\) −13.3137 −1.08345 −0.541727 0.840554i \(-0.682229\pi\)
−0.541727 + 0.840554i \(0.682229\pi\)
\(152\) 0 0
\(153\) 8.07107 0.652507
\(154\) 0 0
\(155\) −9.82843 −0.789438
\(156\) 0 0
\(157\) 15.4853 1.23586 0.617930 0.786233i \(-0.287972\pi\)
0.617930 + 0.786233i \(0.287972\pi\)
\(158\) 0 0
\(159\) −1.24264 −0.0985478
\(160\) 0 0
\(161\) 3.82843 0.301722
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) −5.41421 −0.421496
\(166\) 0 0
\(167\) −18.7279 −1.44921 −0.724605 0.689164i \(-0.757978\pi\)
−0.724605 + 0.689164i \(0.757978\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) −2.24264 −0.171499
\(172\) 0 0
\(173\) 2.34315 0.178146 0.0890730 0.996025i \(-0.471610\pi\)
0.0890730 + 0.996025i \(0.471610\pi\)
\(174\) 0 0
\(175\) 3.82843 0.289402
\(176\) 0 0
\(177\) −12.8995 −0.969585
\(178\) 0 0
\(179\) −3.65685 −0.273326 −0.136663 0.990618i \(-0.543638\pi\)
−0.136663 + 0.990618i \(0.543638\pi\)
\(180\) 0 0
\(181\) −1.17157 −0.0870823 −0.0435412 0.999052i \(-0.513864\pi\)
−0.0435412 + 0.999052i \(0.513864\pi\)
\(182\) 0 0
\(183\) 2.58579 0.191147
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 43.6985 3.19555
\(188\) 0 0
\(189\) 3.82843 0.278477
\(190\) 0 0
\(191\) −1.75736 −0.127158 −0.0635790 0.997977i \(-0.520251\pi\)
−0.0635790 + 0.997977i \(0.520251\pi\)
\(192\) 0 0
\(193\) 8.48528 0.610784 0.305392 0.952227i \(-0.401213\pi\)
0.305392 + 0.952227i \(0.401213\pi\)
\(194\) 0 0
\(195\) −0.585786 −0.0419490
\(196\) 0 0
\(197\) 11.1716 0.795942 0.397971 0.917398i \(-0.369715\pi\)
0.397971 + 0.917398i \(0.369715\pi\)
\(198\) 0 0
\(199\) 6.48528 0.459729 0.229865 0.973223i \(-0.426172\pi\)
0.229865 + 0.973223i \(0.426172\pi\)
\(200\) 0 0
\(201\) −5.48528 −0.386902
\(202\) 0 0
\(203\) −24.5563 −1.72352
\(204\) 0 0
\(205\) 7.58579 0.529814
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −12.1421 −0.839889
\(210\) 0 0
\(211\) 10.6569 0.733648 0.366824 0.930290i \(-0.380445\pi\)
0.366824 + 0.930290i \(0.380445\pi\)
\(212\) 0 0
\(213\) 10.8995 0.746821
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 37.6274 2.55432
\(218\) 0 0
\(219\) −1.75736 −0.118751
\(220\) 0 0
\(221\) 4.72792 0.318034
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.34315 −0.421009 −0.210505 0.977593i \(-0.567511\pi\)
−0.210505 + 0.977593i \(0.567511\pi\)
\(228\) 0 0
\(229\) −15.7990 −1.04403 −0.522013 0.852937i \(-0.674819\pi\)
−0.522013 + 0.852937i \(0.674819\pi\)
\(230\) 0 0
\(231\) 20.7279 1.36380
\(232\) 0 0
\(233\) −1.65685 −0.108544 −0.0542721 0.998526i \(-0.517284\pi\)
−0.0542721 + 0.998526i \(0.517284\pi\)
\(234\) 0 0
\(235\) −11.4142 −0.744581
\(236\) 0 0
\(237\) −7.31371 −0.475076
\(238\) 0 0
\(239\) 4.41421 0.285532 0.142766 0.989756i \(-0.454400\pi\)
0.142766 + 0.989756i \(0.454400\pi\)
\(240\) 0 0
\(241\) 20.5858 1.32605 0.663024 0.748599i \(-0.269273\pi\)
0.663024 + 0.748599i \(0.269273\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −7.65685 −0.489178
\(246\) 0 0
\(247\) −1.31371 −0.0835893
\(248\) 0 0
\(249\) 0.0710678 0.00450374
\(250\) 0 0
\(251\) −24.6274 −1.55447 −0.777234 0.629211i \(-0.783378\pi\)
−0.777234 + 0.629211i \(0.783378\pi\)
\(252\) 0 0
\(253\) 5.41421 0.340389
\(254\) 0 0
\(255\) −8.07107 −0.505430
\(256\) 0 0
\(257\) −18.0416 −1.12541 −0.562703 0.826659i \(-0.690239\pi\)
−0.562703 + 0.826659i \(0.690239\pi\)
\(258\) 0 0
\(259\) 11.4853 0.713661
\(260\) 0 0
\(261\) −6.41421 −0.397030
\(262\) 0 0
\(263\) 21.7279 1.33980 0.669901 0.742451i \(-0.266337\pi\)
0.669901 + 0.742451i \(0.266337\pi\)
\(264\) 0 0
\(265\) 1.24264 0.0763348
\(266\) 0 0
\(267\) −18.1421 −1.11028
\(268\) 0 0
\(269\) 4.55635 0.277806 0.138903 0.990306i \(-0.455642\pi\)
0.138903 + 0.990306i \(0.455642\pi\)
\(270\) 0 0
\(271\) 23.4853 1.42663 0.713315 0.700844i \(-0.247193\pi\)
0.713315 + 0.700844i \(0.247193\pi\)
\(272\) 0 0
\(273\) 2.24264 0.135731
\(274\) 0 0
\(275\) 5.41421 0.326489
\(276\) 0 0
\(277\) 2.68629 0.161404 0.0807018 0.996738i \(-0.474284\pi\)
0.0807018 + 0.996738i \(0.474284\pi\)
\(278\) 0 0
\(279\) 9.82843 0.588413
\(280\) 0 0
\(281\) 14.5858 0.870115 0.435058 0.900403i \(-0.356728\pi\)
0.435058 + 0.900403i \(0.356728\pi\)
\(282\) 0 0
\(283\) 17.3431 1.03094 0.515472 0.856907i \(-0.327617\pi\)
0.515472 + 0.856907i \(0.327617\pi\)
\(284\) 0 0
\(285\) 2.24264 0.132843
\(286\) 0 0
\(287\) −29.0416 −1.71427
\(288\) 0 0
\(289\) 48.1421 2.83189
\(290\) 0 0
\(291\) −6.82843 −0.400289
\(292\) 0 0
\(293\) −22.8995 −1.33780 −0.668901 0.743351i \(-0.733235\pi\)
−0.668901 + 0.743351i \(0.733235\pi\)
\(294\) 0 0
\(295\) 12.8995 0.751038
\(296\) 0 0
\(297\) 5.41421 0.314165
\(298\) 0 0
\(299\) 0.585786 0.0338769
\(300\) 0 0
\(301\) −22.9706 −1.32400
\(302\) 0 0
\(303\) −18.5563 −1.06603
\(304\) 0 0
\(305\) −2.58579 −0.148062
\(306\) 0 0
\(307\) 15.2132 0.868263 0.434132 0.900849i \(-0.357055\pi\)
0.434132 + 0.900849i \(0.357055\pi\)
\(308\) 0 0
\(309\) 4.82843 0.274680
\(310\) 0 0
\(311\) −6.34315 −0.359687 −0.179843 0.983695i \(-0.557559\pi\)
−0.179843 + 0.983695i \(0.557559\pi\)
\(312\) 0 0
\(313\) −7.97056 −0.450523 −0.225261 0.974298i \(-0.572324\pi\)
−0.225261 + 0.974298i \(0.572324\pi\)
\(314\) 0 0
\(315\) −3.82843 −0.215707
\(316\) 0 0
\(317\) 1.55635 0.0874133 0.0437066 0.999044i \(-0.486083\pi\)
0.0437066 + 0.999044i \(0.486083\pi\)
\(318\) 0 0
\(319\) −34.7279 −1.94439
\(320\) 0 0
\(321\) 1.58579 0.0885100
\(322\) 0 0
\(323\) −18.1005 −1.00714
\(324\) 0 0
\(325\) 0.585786 0.0324936
\(326\) 0 0
\(327\) 5.75736 0.318383
\(328\) 0 0
\(329\) 43.6985 2.40918
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 3.00000 0.164399
\(334\) 0 0
\(335\) 5.48528 0.299693
\(336\) 0 0
\(337\) −2.97056 −0.161817 −0.0809084 0.996722i \(-0.525782\pi\)
−0.0809084 + 0.996722i \(0.525782\pi\)
\(338\) 0 0
\(339\) −7.58579 −0.412003
\(340\) 0 0
\(341\) 53.2132 2.88166
\(342\) 0 0
\(343\) 2.51472 0.135782
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −18.8284 −1.01076 −0.505381 0.862896i \(-0.668648\pi\)
−0.505381 + 0.862896i \(0.668648\pi\)
\(348\) 0 0
\(349\) 30.1127 1.61190 0.805948 0.591986i \(-0.201656\pi\)
0.805948 + 0.591986i \(0.201656\pi\)
\(350\) 0 0
\(351\) 0.585786 0.0312670
\(352\) 0 0
\(353\) 30.5269 1.62478 0.812392 0.583112i \(-0.198165\pi\)
0.812392 + 0.583112i \(0.198165\pi\)
\(354\) 0 0
\(355\) −10.8995 −0.578485
\(356\) 0 0
\(357\) 30.8995 1.63537
\(358\) 0 0
\(359\) 25.0711 1.32320 0.661600 0.749857i \(-0.269878\pi\)
0.661600 + 0.749857i \(0.269878\pi\)
\(360\) 0 0
\(361\) −13.9706 −0.735293
\(362\) 0 0
\(363\) 18.3137 0.961220
\(364\) 0 0
\(365\) 1.75736 0.0919844
\(366\) 0 0
\(367\) −19.9706 −1.04245 −0.521227 0.853418i \(-0.674526\pi\)
−0.521227 + 0.853418i \(0.674526\pi\)
\(368\) 0 0
\(369\) −7.58579 −0.394900
\(370\) 0 0
\(371\) −4.75736 −0.246990
\(372\) 0 0
\(373\) 20.9706 1.08581 0.542907 0.839793i \(-0.317323\pi\)
0.542907 + 0.839793i \(0.317323\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −3.75736 −0.193514
\(378\) 0 0
\(379\) 4.97056 0.255321 0.127660 0.991818i \(-0.459253\pi\)
0.127660 + 0.991818i \(0.459253\pi\)
\(380\) 0 0
\(381\) −8.24264 −0.422283
\(382\) 0 0
\(383\) −11.5858 −0.592006 −0.296003 0.955187i \(-0.595654\pi\)
−0.296003 + 0.955187i \(0.595654\pi\)
\(384\) 0 0
\(385\) −20.7279 −1.05639
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) −2.48528 −0.126009 −0.0630044 0.998013i \(-0.520068\pi\)
−0.0630044 + 0.998013i \(0.520068\pi\)
\(390\) 0 0
\(391\) 8.07107 0.408171
\(392\) 0 0
\(393\) 2.34315 0.118196
\(394\) 0 0
\(395\) 7.31371 0.367993
\(396\) 0 0
\(397\) 7.31371 0.367065 0.183532 0.983014i \(-0.441247\pi\)
0.183532 + 0.983014i \(0.441247\pi\)
\(398\) 0 0
\(399\) −8.58579 −0.429827
\(400\) 0 0
\(401\) −3.85786 −0.192653 −0.0963263 0.995350i \(-0.530709\pi\)
−0.0963263 + 0.995350i \(0.530709\pi\)
\(402\) 0 0
\(403\) 5.75736 0.286794
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 16.2426 0.805118
\(408\) 0 0
\(409\) −15.4853 −0.765698 −0.382849 0.923811i \(-0.625057\pi\)
−0.382849 + 0.923811i \(0.625057\pi\)
\(410\) 0 0
\(411\) −2.82843 −0.139516
\(412\) 0 0
\(413\) −49.3848 −2.43007
\(414\) 0 0
\(415\) −0.0710678 −0.00348858
\(416\) 0 0
\(417\) 11.0000 0.538672
\(418\) 0 0
\(419\) −24.0416 −1.17451 −0.587255 0.809402i \(-0.699792\pi\)
−0.587255 + 0.809402i \(0.699792\pi\)
\(420\) 0 0
\(421\) −10.2426 −0.499196 −0.249598 0.968350i \(-0.580298\pi\)
−0.249598 + 0.968350i \(0.580298\pi\)
\(422\) 0 0
\(423\) 11.4142 0.554978
\(424\) 0 0
\(425\) 8.07107 0.391504
\(426\) 0 0
\(427\) 9.89949 0.479070
\(428\) 0 0
\(429\) 3.17157 0.153125
\(430\) 0 0
\(431\) −11.7990 −0.568337 −0.284169 0.958774i \(-0.591718\pi\)
−0.284169 + 0.958774i \(0.591718\pi\)
\(432\) 0 0
\(433\) 3.82843 0.183982 0.0919912 0.995760i \(-0.470677\pi\)
0.0919912 + 0.995760i \(0.470677\pi\)
\(434\) 0 0
\(435\) 6.41421 0.307538
\(436\) 0 0
\(437\) −2.24264 −0.107280
\(438\) 0 0
\(439\) 1.51472 0.0722936 0.0361468 0.999346i \(-0.488492\pi\)
0.0361468 + 0.999346i \(0.488492\pi\)
\(440\) 0 0
\(441\) 7.65685 0.364612
\(442\) 0 0
\(443\) 34.5269 1.64042 0.820212 0.572060i \(-0.193856\pi\)
0.820212 + 0.572060i \(0.193856\pi\)
\(444\) 0 0
\(445\) 18.1421 0.860020
\(446\) 0 0
\(447\) −5.07107 −0.239853
\(448\) 0 0
\(449\) −4.75736 −0.224514 −0.112257 0.993679i \(-0.535808\pi\)
−0.112257 + 0.993679i \(0.535808\pi\)
\(450\) 0 0
\(451\) −41.0711 −1.93396
\(452\) 0 0
\(453\) −13.3137 −0.625533
\(454\) 0 0
\(455\) −2.24264 −0.105137
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) 8.07107 0.376725
\(460\) 0 0
\(461\) −26.4853 −1.23354 −0.616771 0.787142i \(-0.711560\pi\)
−0.616771 + 0.787142i \(0.711560\pi\)
\(462\) 0 0
\(463\) −2.92893 −0.136119 −0.0680595 0.997681i \(-0.521681\pi\)
−0.0680595 + 0.997681i \(0.521681\pi\)
\(464\) 0 0
\(465\) −9.82843 −0.455782
\(466\) 0 0
\(467\) −6.41421 −0.296814 −0.148407 0.988926i \(-0.547415\pi\)
−0.148407 + 0.988926i \(0.547415\pi\)
\(468\) 0 0
\(469\) −21.0000 −0.969690
\(470\) 0 0
\(471\) 15.4853 0.713524
\(472\) 0 0
\(473\) −32.4853 −1.49367
\(474\) 0 0
\(475\) −2.24264 −0.102899
\(476\) 0 0
\(477\) −1.24264 −0.0568966
\(478\) 0 0
\(479\) −6.72792 −0.307407 −0.153703 0.988117i \(-0.549120\pi\)
−0.153703 + 0.988117i \(0.549120\pi\)
\(480\) 0 0
\(481\) 1.75736 0.0801287
\(482\) 0 0
\(483\) 3.82843 0.174199
\(484\) 0 0
\(485\) 6.82843 0.310063
\(486\) 0 0
\(487\) 26.3848 1.19561 0.597804 0.801642i \(-0.296040\pi\)
0.597804 + 0.801642i \(0.296040\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) 12.0711 0.544760 0.272380 0.962190i \(-0.412189\pi\)
0.272380 + 0.962190i \(0.412189\pi\)
\(492\) 0 0
\(493\) −51.7696 −2.33158
\(494\) 0 0
\(495\) −5.41421 −0.243351
\(496\) 0 0
\(497\) 41.7279 1.87175
\(498\) 0 0
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 0 0
\(501\) −18.7279 −0.836702
\(502\) 0 0
\(503\) 39.7279 1.77138 0.885690 0.464277i \(-0.153686\pi\)
0.885690 + 0.464277i \(0.153686\pi\)
\(504\) 0 0
\(505\) 18.5563 0.825747
\(506\) 0 0
\(507\) −12.6569 −0.562111
\(508\) 0 0
\(509\) 7.79899 0.345684 0.172842 0.984950i \(-0.444705\pi\)
0.172842 + 0.984950i \(0.444705\pi\)
\(510\) 0 0
\(511\) −6.72792 −0.297626
\(512\) 0 0
\(513\) −2.24264 −0.0990150
\(514\) 0 0
\(515\) −4.82843 −0.212766
\(516\) 0 0
\(517\) 61.7990 2.71792
\(518\) 0 0
\(519\) 2.34315 0.102853
\(520\) 0 0
\(521\) −31.5563 −1.38251 −0.691254 0.722612i \(-0.742942\pi\)
−0.691254 + 0.722612i \(0.742942\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) 3.82843 0.167086
\(526\) 0 0
\(527\) 79.3259 3.45549
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −12.8995 −0.559790
\(532\) 0 0
\(533\) −4.44365 −0.192476
\(534\) 0 0
\(535\) −1.58579 −0.0685595
\(536\) 0 0
\(537\) −3.65685 −0.157805
\(538\) 0 0
\(539\) 41.4558 1.78563
\(540\) 0 0
\(541\) 5.85786 0.251849 0.125925 0.992040i \(-0.459810\pi\)
0.125925 + 0.992040i \(0.459810\pi\)
\(542\) 0 0
\(543\) −1.17157 −0.0502770
\(544\) 0 0
\(545\) −5.75736 −0.246618
\(546\) 0 0
\(547\) −12.9706 −0.554581 −0.277291 0.960786i \(-0.589436\pi\)
−0.277291 + 0.960786i \(0.589436\pi\)
\(548\) 0 0
\(549\) 2.58579 0.110359
\(550\) 0 0
\(551\) 14.3848 0.612812
\(552\) 0 0
\(553\) −28.0000 −1.19068
\(554\) 0 0
\(555\) −3.00000 −0.127343
\(556\) 0 0
\(557\) −6.89949 −0.292341 −0.146170 0.989259i \(-0.546695\pi\)
−0.146170 + 0.989259i \(0.546695\pi\)
\(558\) 0 0
\(559\) −3.51472 −0.148657
\(560\) 0 0
\(561\) 43.6985 1.84495
\(562\) 0 0
\(563\) −20.8995 −0.880809 −0.440404 0.897800i \(-0.645165\pi\)
−0.440404 + 0.897800i \(0.645165\pi\)
\(564\) 0 0
\(565\) 7.58579 0.319136
\(566\) 0 0
\(567\) 3.82843 0.160779
\(568\) 0 0
\(569\) 12.1421 0.509025 0.254512 0.967070i \(-0.418085\pi\)
0.254512 + 0.967070i \(0.418085\pi\)
\(570\) 0 0
\(571\) −28.7279 −1.20223 −0.601113 0.799164i \(-0.705276\pi\)
−0.601113 + 0.799164i \(0.705276\pi\)
\(572\) 0 0
\(573\) −1.75736 −0.0734147
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −3.02944 −0.126117 −0.0630586 0.998010i \(-0.520085\pi\)
−0.0630586 + 0.998010i \(0.520085\pi\)
\(578\) 0 0
\(579\) 8.48528 0.352636
\(580\) 0 0
\(581\) 0.272078 0.0112877
\(582\) 0 0
\(583\) −6.72792 −0.278642
\(584\) 0 0
\(585\) −0.585786 −0.0242193
\(586\) 0 0
\(587\) 4.62742 0.190994 0.0954970 0.995430i \(-0.469556\pi\)
0.0954970 + 0.995430i \(0.469556\pi\)
\(588\) 0 0
\(589\) −22.0416 −0.908210
\(590\) 0 0
\(591\) 11.1716 0.459537
\(592\) 0 0
\(593\) −33.5563 −1.37799 −0.688997 0.724764i \(-0.741949\pi\)
−0.688997 + 0.724764i \(0.741949\pi\)
\(594\) 0 0
\(595\) −30.8995 −1.26676
\(596\) 0 0
\(597\) 6.48528 0.265425
\(598\) 0 0
\(599\) −38.8284 −1.58649 −0.793243 0.608905i \(-0.791609\pi\)
−0.793243 + 0.608905i \(0.791609\pi\)
\(600\) 0 0
\(601\) 4.31371 0.175960 0.0879799 0.996122i \(-0.471959\pi\)
0.0879799 + 0.996122i \(0.471959\pi\)
\(602\) 0 0
\(603\) −5.48528 −0.223378
\(604\) 0 0
\(605\) −18.3137 −0.744558
\(606\) 0 0
\(607\) 1.61522 0.0655599 0.0327800 0.999463i \(-0.489564\pi\)
0.0327800 + 0.999463i \(0.489564\pi\)
\(608\) 0 0
\(609\) −24.5563 −0.995073
\(610\) 0 0
\(611\) 6.68629 0.270498
\(612\) 0 0
\(613\) 12.6274 0.510017 0.255008 0.966939i \(-0.417922\pi\)
0.255008 + 0.966939i \(0.417922\pi\)
\(614\) 0 0
\(615\) 7.58579 0.305888
\(616\) 0 0
\(617\) −14.8995 −0.599831 −0.299916 0.953966i \(-0.596959\pi\)
−0.299916 + 0.953966i \(0.596959\pi\)
\(618\) 0 0
\(619\) −8.28427 −0.332973 −0.166486 0.986044i \(-0.553242\pi\)
−0.166486 + 0.986044i \(0.553242\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −69.4558 −2.78269
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −12.1421 −0.484910
\(628\) 0 0
\(629\) 24.2132 0.965444
\(630\) 0 0
\(631\) 9.21320 0.366772 0.183386 0.983041i \(-0.441294\pi\)
0.183386 + 0.983041i \(0.441294\pi\)
\(632\) 0 0
\(633\) 10.6569 0.423572
\(634\) 0 0
\(635\) 8.24264 0.327099
\(636\) 0 0
\(637\) 4.48528 0.177713
\(638\) 0 0
\(639\) 10.8995 0.431177
\(640\) 0 0
\(641\) 19.0711 0.753262 0.376631 0.926363i \(-0.377082\pi\)
0.376631 + 0.926363i \(0.377082\pi\)
\(642\) 0 0
\(643\) −33.0000 −1.30139 −0.650696 0.759338i \(-0.725523\pi\)
−0.650696 + 0.759338i \(0.725523\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 21.2132 0.833977 0.416989 0.908912i \(-0.363086\pi\)
0.416989 + 0.908912i \(0.363086\pi\)
\(648\) 0 0
\(649\) −69.8406 −2.74148
\(650\) 0 0
\(651\) 37.6274 1.47473
\(652\) 0 0
\(653\) −5.41421 −0.211875 −0.105937 0.994373i \(-0.533784\pi\)
−0.105937 + 0.994373i \(0.533784\pi\)
\(654\) 0 0
\(655\) −2.34315 −0.0915543
\(656\) 0 0
\(657\) −1.75736 −0.0685611
\(658\) 0 0
\(659\) 34.3848 1.33944 0.669720 0.742613i \(-0.266414\pi\)
0.669720 + 0.742613i \(0.266414\pi\)
\(660\) 0 0
\(661\) −39.4558 −1.53465 −0.767327 0.641256i \(-0.778414\pi\)
−0.767327 + 0.641256i \(0.778414\pi\)
\(662\) 0 0
\(663\) 4.72792 0.183617
\(664\) 0 0
\(665\) 8.58579 0.332943
\(666\) 0 0
\(667\) −6.41421 −0.248359
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) −17.6985 −0.682226 −0.341113 0.940022i \(-0.610804\pi\)
−0.341113 + 0.940022i \(0.610804\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −10.7574 −0.413439 −0.206719 0.978400i \(-0.566279\pi\)
−0.206719 + 0.978400i \(0.566279\pi\)
\(678\) 0 0
\(679\) −26.1421 −1.00324
\(680\) 0 0
\(681\) −6.34315 −0.243070
\(682\) 0 0
\(683\) −44.1838 −1.69064 −0.845322 0.534257i \(-0.820592\pi\)
−0.845322 + 0.534257i \(0.820592\pi\)
\(684\) 0 0
\(685\) 2.82843 0.108069
\(686\) 0 0
\(687\) −15.7990 −0.602769
\(688\) 0 0
\(689\) −0.727922 −0.0277316
\(690\) 0 0
\(691\) 4.34315 0.165221 0.0826105 0.996582i \(-0.473674\pi\)
0.0826105 + 0.996582i \(0.473674\pi\)
\(692\) 0 0
\(693\) 20.7279 0.787389
\(694\) 0 0
\(695\) −11.0000 −0.417254
\(696\) 0 0
\(697\) −61.2254 −2.31908
\(698\) 0 0
\(699\) −1.65685 −0.0626680
\(700\) 0 0
\(701\) 0.384776 0.0145328 0.00726640 0.999974i \(-0.497687\pi\)
0.00726640 + 0.999974i \(0.497687\pi\)
\(702\) 0 0
\(703\) −6.72792 −0.253748
\(704\) 0 0
\(705\) −11.4142 −0.429884
\(706\) 0 0
\(707\) −71.0416 −2.67180
\(708\) 0 0
\(709\) −8.72792 −0.327784 −0.163892 0.986478i \(-0.552405\pi\)
−0.163892 + 0.986478i \(0.552405\pi\)
\(710\) 0 0
\(711\) −7.31371 −0.274285
\(712\) 0 0
\(713\) 9.82843 0.368077
\(714\) 0 0
\(715\) −3.17157 −0.118610
\(716\) 0 0
\(717\) 4.41421 0.164852
\(718\) 0 0
\(719\) 6.21320 0.231713 0.115857 0.993266i \(-0.463039\pi\)
0.115857 + 0.993266i \(0.463039\pi\)
\(720\) 0 0
\(721\) 18.4853 0.688428
\(722\) 0 0
\(723\) 20.5858 0.765594
\(724\) 0 0
\(725\) −6.41421 −0.238218
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.4264 −1.79112
\(732\) 0 0
\(733\) −0.514719 −0.0190116 −0.00950578 0.999955i \(-0.503026\pi\)
−0.00950578 + 0.999955i \(0.503026\pi\)
\(734\) 0 0
\(735\) −7.65685 −0.282427
\(736\) 0 0
\(737\) −29.6985 −1.09396
\(738\) 0 0
\(739\) 8.37258 0.307990 0.153995 0.988072i \(-0.450786\pi\)
0.153995 + 0.988072i \(0.450786\pi\)
\(740\) 0 0
\(741\) −1.31371 −0.0482603
\(742\) 0 0
\(743\) 14.3431 0.526199 0.263099 0.964769i \(-0.415255\pi\)
0.263099 + 0.964769i \(0.415255\pi\)
\(744\) 0 0
\(745\) 5.07107 0.185790
\(746\) 0 0
\(747\) 0.0710678 0.00260024
\(748\) 0 0
\(749\) 6.07107 0.221832
\(750\) 0 0
\(751\) 34.7279 1.26724 0.633620 0.773644i \(-0.281568\pi\)
0.633620 + 0.773644i \(0.281568\pi\)
\(752\) 0 0
\(753\) −24.6274 −0.897473
\(754\) 0 0
\(755\) 13.3137 0.484535
\(756\) 0 0
\(757\) 27.0000 0.981332 0.490666 0.871348i \(-0.336754\pi\)
0.490666 + 0.871348i \(0.336754\pi\)
\(758\) 0 0
\(759\) 5.41421 0.196524
\(760\) 0 0
\(761\) −34.0711 −1.23508 −0.617538 0.786541i \(-0.711870\pi\)
−0.617538 + 0.786541i \(0.711870\pi\)
\(762\) 0 0
\(763\) 22.0416 0.797961
\(764\) 0 0
\(765\) −8.07107 −0.291810
\(766\) 0 0
\(767\) −7.55635 −0.272844
\(768\) 0 0
\(769\) −28.0416 −1.01121 −0.505604 0.862766i \(-0.668730\pi\)
−0.505604 + 0.862766i \(0.668730\pi\)
\(770\) 0 0
\(771\) −18.0416 −0.649753
\(772\) 0 0
\(773\) −8.82843 −0.317536 −0.158768 0.987316i \(-0.550752\pi\)
−0.158768 + 0.987316i \(0.550752\pi\)
\(774\) 0 0
\(775\) 9.82843 0.353048
\(776\) 0 0
\(777\) 11.4853 0.412032
\(778\) 0 0
\(779\) 17.0122 0.609525
\(780\) 0 0
\(781\) 59.0122 2.11162
\(782\) 0 0
\(783\) −6.41421 −0.229225
\(784\) 0 0
\(785\) −15.4853 −0.552693
\(786\) 0 0
\(787\) 41.6274 1.48386 0.741929 0.670479i \(-0.233911\pi\)
0.741929 + 0.670479i \(0.233911\pi\)
\(788\) 0 0
\(789\) 21.7279 0.773535
\(790\) 0 0
\(791\) −29.0416 −1.03260
\(792\) 0 0
\(793\) 1.51472 0.0537892
\(794\) 0 0
\(795\) 1.24264 0.0440719
\(796\) 0 0
\(797\) 25.2426 0.894140 0.447070 0.894499i \(-0.352467\pi\)
0.447070 + 0.894499i \(0.352467\pi\)
\(798\) 0 0
\(799\) 92.1249 3.25914
\(800\) 0 0
\(801\) −18.1421 −0.641021
\(802\) 0 0
\(803\) −9.51472 −0.335767
\(804\) 0 0
\(805\) −3.82843 −0.134934
\(806\) 0 0
\(807\) 4.55635 0.160391
\(808\) 0 0
\(809\) 47.8701 1.68302 0.841511 0.540240i \(-0.181667\pi\)
0.841511 + 0.540240i \(0.181667\pi\)
\(810\) 0 0
\(811\) 17.2843 0.606933 0.303466 0.952842i \(-0.401856\pi\)
0.303466 + 0.952842i \(0.401856\pi\)
\(812\) 0 0
\(813\) 23.4853 0.823665
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) 13.4558 0.470760
\(818\) 0 0
\(819\) 2.24264 0.0783642
\(820\) 0 0
\(821\) −4.20101 −0.146616 −0.0733081 0.997309i \(-0.523356\pi\)
−0.0733081 + 0.997309i \(0.523356\pi\)
\(822\) 0 0
\(823\) −34.3431 −1.19713 −0.598563 0.801075i \(-0.704262\pi\)
−0.598563 + 0.801075i \(0.704262\pi\)
\(824\) 0 0
\(825\) 5.41421 0.188499
\(826\) 0 0
\(827\) −32.0122 −1.11317 −0.556587 0.830790i \(-0.687889\pi\)
−0.556587 + 0.830790i \(0.687889\pi\)
\(828\) 0 0
\(829\) 6.51472 0.226266 0.113133 0.993580i \(-0.463911\pi\)
0.113133 + 0.993580i \(0.463911\pi\)
\(830\) 0 0
\(831\) 2.68629 0.0931864
\(832\) 0 0
\(833\) 61.7990 2.14121
\(834\) 0 0
\(835\) 18.7279 0.648106
\(836\) 0 0
\(837\) 9.82843 0.339720
\(838\) 0 0
\(839\) −40.6274 −1.40261 −0.701307 0.712859i \(-0.747400\pi\)
−0.701307 + 0.712859i \(0.747400\pi\)
\(840\) 0 0
\(841\) 12.1421 0.418694
\(842\) 0 0
\(843\) 14.5858 0.502361
\(844\) 0 0
\(845\) 12.6569 0.435409
\(846\) 0 0
\(847\) 70.1127 2.40910
\(848\) 0 0
\(849\) 17.3431 0.595215
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) 19.4558 0.666155 0.333078 0.942899i \(-0.391913\pi\)
0.333078 + 0.942899i \(0.391913\pi\)
\(854\) 0 0
\(855\) 2.24264 0.0766967
\(856\) 0 0
\(857\) −57.5980 −1.96751 −0.983755 0.179518i \(-0.942546\pi\)
−0.983755 + 0.179518i \(0.942546\pi\)
\(858\) 0 0
\(859\) 57.0000 1.94481 0.972407 0.233289i \(-0.0749488\pi\)
0.972407 + 0.233289i \(0.0749488\pi\)
\(860\) 0 0
\(861\) −29.0416 −0.989736
\(862\) 0 0
\(863\) −16.8284 −0.572846 −0.286423 0.958103i \(-0.592466\pi\)
−0.286423 + 0.958103i \(0.592466\pi\)
\(864\) 0 0
\(865\) −2.34315 −0.0796693
\(866\) 0 0
\(867\) 48.1421 1.63499
\(868\) 0 0
\(869\) −39.5980 −1.34327
\(870\) 0 0
\(871\) −3.21320 −0.108875
\(872\) 0 0
\(873\) −6.82843 −0.231107
\(874\) 0 0
\(875\) −3.82843 −0.129424
\(876\) 0 0
\(877\) −18.4853 −0.624204 −0.312102 0.950049i \(-0.601033\pi\)
−0.312102 + 0.950049i \(0.601033\pi\)
\(878\) 0 0
\(879\) −22.8995 −0.772381
\(880\) 0 0
\(881\) 16.5858 0.558789 0.279395 0.960176i \(-0.409866\pi\)
0.279395 + 0.960176i \(0.409866\pi\)
\(882\) 0 0
\(883\) −17.8995 −0.602366 −0.301183 0.953566i \(-0.597381\pi\)
−0.301183 + 0.953566i \(0.597381\pi\)
\(884\) 0 0
\(885\) 12.8995 0.433612
\(886\) 0 0
\(887\) 17.7990 0.597632 0.298816 0.954311i \(-0.403408\pi\)
0.298816 + 0.954311i \(0.403408\pi\)
\(888\) 0 0
\(889\) −31.5563 −1.05837
\(890\) 0 0
\(891\) 5.41421 0.181383
\(892\) 0 0
\(893\) −25.5980 −0.856604
\(894\) 0 0
\(895\) 3.65685 0.122235
\(896\) 0 0
\(897\) 0.585786 0.0195588
\(898\) 0 0
\(899\) −63.0416 −2.10256
\(900\) 0 0
\(901\) −10.0294 −0.334129
\(902\) 0 0
\(903\) −22.9706 −0.764412
\(904\) 0 0
\(905\) 1.17157 0.0389444
\(906\) 0 0
\(907\) −2.85786 −0.0948938 −0.0474469 0.998874i \(-0.515108\pi\)
−0.0474469 + 0.998874i \(0.515108\pi\)
\(908\) 0 0
\(909\) −18.5563 −0.615475
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) 0.384776 0.0127342
\(914\) 0 0
\(915\) −2.58579 −0.0854835
\(916\) 0 0
\(917\) 8.97056 0.296234
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 15.2132 0.501292
\(922\) 0 0
\(923\) 6.38478 0.210157
\(924\) 0 0
\(925\) 3.00000 0.0986394
\(926\) 0 0
\(927\) 4.82843 0.158586
\(928\) 0 0
\(929\) −11.8701 −0.389444 −0.194722 0.980858i \(-0.562380\pi\)
−0.194722 + 0.980858i \(0.562380\pi\)
\(930\) 0 0
\(931\) −17.1716 −0.562776
\(932\) 0 0
\(933\) −6.34315 −0.207665
\(934\) 0 0
\(935\) −43.6985 −1.42909
\(936\) 0 0
\(937\) −20.2010 −0.659938 −0.329969 0.943992i \(-0.607038\pi\)
−0.329969 + 0.943992i \(0.607038\pi\)
\(938\) 0 0
\(939\) −7.97056 −0.260109
\(940\) 0 0
\(941\) −4.58579 −0.149492 −0.0747462 0.997203i \(-0.523815\pi\)
−0.0747462 + 0.997203i \(0.523815\pi\)
\(942\) 0 0
\(943\) −7.58579 −0.247027
\(944\) 0 0
\(945\) −3.82843 −0.124539
\(946\) 0 0
\(947\) −35.7990 −1.16331 −0.581655 0.813435i \(-0.697595\pi\)
−0.581655 + 0.813435i \(0.697595\pi\)
\(948\) 0 0
\(949\) −1.02944 −0.0334169
\(950\) 0 0
\(951\) 1.55635 0.0504681
\(952\) 0 0
\(953\) 52.6274 1.70477 0.852385 0.522915i \(-0.175156\pi\)
0.852385 + 0.522915i \(0.175156\pi\)
\(954\) 0 0
\(955\) 1.75736 0.0568668
\(956\) 0 0
\(957\) −34.7279 −1.12259
\(958\) 0 0
\(959\) −10.8284 −0.349668
\(960\) 0 0
\(961\) 65.5980 2.11606
\(962\) 0 0
\(963\) 1.58579 0.0511013
\(964\) 0 0
\(965\) −8.48528 −0.273151
\(966\) 0 0
\(967\) −31.5563 −1.01478 −0.507392 0.861715i \(-0.669390\pi\)
−0.507392 + 0.861715i \(0.669390\pi\)
\(968\) 0 0
\(969\) −18.1005 −0.581472
\(970\) 0 0
\(971\) 54.4264 1.74663 0.873313 0.487159i \(-0.161967\pi\)
0.873313 + 0.487159i \(0.161967\pi\)
\(972\) 0 0
\(973\) 42.1127 1.35007
\(974\) 0 0
\(975\) 0.585786 0.0187602
\(976\) 0 0
\(977\) −18.4142 −0.589123 −0.294561 0.955633i \(-0.595174\pi\)
−0.294561 + 0.955633i \(0.595174\pi\)
\(978\) 0 0
\(979\) −98.2254 −3.13930
\(980\) 0 0
\(981\) 5.75736 0.183818
\(982\) 0 0
\(983\) 58.0122 1.85030 0.925151 0.379600i \(-0.123938\pi\)
0.925151 + 0.379600i \(0.123938\pi\)
\(984\) 0 0
\(985\) −11.1716 −0.355956
\(986\) 0 0
\(987\) 43.6985 1.39094
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −12.6569 −0.402058 −0.201029 0.979585i \(-0.564429\pi\)
−0.201029 + 0.979585i \(0.564429\pi\)
\(992\) 0 0
\(993\) −1.00000 −0.0317340
\(994\) 0 0
\(995\) −6.48528 −0.205597
\(996\) 0 0
\(997\) −5.85786 −0.185520 −0.0927602 0.995688i \(-0.529569\pi\)
−0.0927602 + 0.995688i \(0.529569\pi\)
\(998\) 0 0
\(999\) 3.00000 0.0949158
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 5520.2.a.bm.1.2 2
4.3 odd 2 345.2.a.h.1.2 2
12.11 even 2 1035.2.a.j.1.1 2
20.3 even 4 1725.2.b.s.1174.1 4
20.7 even 4 1725.2.b.s.1174.4 4
20.19 odd 2 1725.2.a.z.1.1 2
60.59 even 2 5175.2.a.bj.1.2 2
92.91 even 2 7935.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.2.a.h.1.2 2 4.3 odd 2
1035.2.a.j.1.1 2 12.11 even 2
1725.2.a.z.1.1 2 20.19 odd 2
1725.2.b.s.1174.1 4 20.3 even 4
1725.2.b.s.1174.4 4 20.7 even 4
5175.2.a.bj.1.2 2 60.59 even 2
5520.2.a.bm.1.2 2 1.1 even 1 trivial
7935.2.a.q.1.2 2 92.91 even 2