Properties

Label 3696.2.a.bl.1.2
Level $3696$
Weight $2$
Character 3696.1
Self dual yes
Analytic conductor $29.513$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 3696.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} +7.58258 q^{17} +6.58258 q^{19} -1.00000 q^{21} +5.58258 q^{23} +4.00000 q^{25} +1.00000 q^{27} -8.16515 q^{29} -3.58258 q^{31} +1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +1.00000 q^{39} -11.1652 q^{41} -1.58258 q^{43} +3.00000 q^{45} -1.41742 q^{47} +1.00000 q^{49} +7.58258 q^{51} -9.58258 q^{53} +3.00000 q^{55} +6.58258 q^{57} -4.58258 q^{59} +10.0000 q^{61} -1.00000 q^{63} +3.00000 q^{65} -8.58258 q^{67} +5.58258 q^{69} -11.1652 q^{71} +7.00000 q^{73} +4.00000 q^{75} -1.00000 q^{77} -7.16515 q^{79} +1.00000 q^{81} +11.5826 q^{83} +22.7477 q^{85} -8.16515 q^{87} +9.16515 q^{89} -1.00000 q^{91} -3.58258 q^{93} +19.7477 q^{95} -2.41742 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 7.58258 1.83904 0.919522 0.393038i \(-0.128576\pi\)
0.919522 + 0.393038i \(0.128576\pi\)
\(18\) 0 0
\(19\) 6.58258 1.51015 0.755073 0.655640i \(-0.227601\pi\)
0.755073 + 0.655640i \(0.227601\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 5.58258 1.16405 0.582024 0.813172i \(-0.302261\pi\)
0.582024 + 0.813172i \(0.302261\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.16515 −1.51623 −0.758115 0.652121i \(-0.773880\pi\)
−0.758115 + 0.652121i \(0.773880\pi\)
\(30\) 0 0
\(31\) −3.58258 −0.643450 −0.321725 0.946833i \(-0.604263\pi\)
−0.321725 + 0.946833i \(0.604263\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −11.1652 −1.74370 −0.871852 0.489770i \(-0.837081\pi\)
−0.871852 + 0.489770i \(0.837081\pi\)
\(42\) 0 0
\(43\) −1.58258 −0.241341 −0.120670 0.992693i \(-0.538504\pi\)
−0.120670 + 0.992693i \(0.538504\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −1.41742 −0.206753 −0.103376 0.994642i \(-0.532965\pi\)
−0.103376 + 0.994642i \(0.532965\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 7.58258 1.06177
\(52\) 0 0
\(53\) −9.58258 −1.31627 −0.658134 0.752901i \(-0.728654\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 6.58258 0.871883
\(58\) 0 0
\(59\) −4.58258 −0.596601 −0.298300 0.954472i \(-0.596420\pi\)
−0.298300 + 0.954472i \(0.596420\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −8.58258 −1.04853 −0.524264 0.851556i \(-0.675660\pi\)
−0.524264 + 0.851556i \(0.675660\pi\)
\(68\) 0 0
\(69\) 5.58258 0.672063
\(70\) 0 0
\(71\) −11.1652 −1.32506 −0.662530 0.749036i \(-0.730517\pi\)
−0.662530 + 0.749036i \(0.730517\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −7.16515 −0.806143 −0.403071 0.915169i \(-0.632057\pi\)
−0.403071 + 0.915169i \(0.632057\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.5826 1.27135 0.635676 0.771956i \(-0.280721\pi\)
0.635676 + 0.771956i \(0.280721\pi\)
\(84\) 0 0
\(85\) 22.7477 2.46734
\(86\) 0 0
\(87\) −8.16515 −0.875396
\(88\) 0 0
\(89\) 9.16515 0.971504 0.485752 0.874097i \(-0.338546\pi\)
0.485752 + 0.874097i \(0.338546\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −3.58258 −0.371496
\(94\) 0 0
\(95\) 19.7477 2.02607
\(96\) 0 0
\(97\) −2.41742 −0.245452 −0.122726 0.992441i \(-0.539164\pi\)
−0.122726 + 0.992441i \(0.539164\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 11.5826 1.15251 0.576255 0.817270i \(-0.304514\pi\)
0.576255 + 0.817270i \(0.304514\pi\)
\(102\) 0 0
\(103\) 1.16515 0.114806 0.0574029 0.998351i \(-0.481718\pi\)
0.0574029 + 0.998351i \(0.481718\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 12.5826 1.21640 0.608202 0.793782i \(-0.291891\pi\)
0.608202 + 0.793782i \(0.291891\pi\)
\(108\) 0 0
\(109\) 3.58258 0.343149 0.171574 0.985171i \(-0.445115\pi\)
0.171574 + 0.985171i \(0.445115\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) 9.16515 0.862185 0.431092 0.902308i \(-0.358128\pi\)
0.431092 + 0.902308i \(0.358128\pi\)
\(114\) 0 0
\(115\) 16.7477 1.56173
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −7.58258 −0.695094
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −11.1652 −1.00673
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 11.5826 1.02779 0.513894 0.857854i \(-0.328203\pi\)
0.513894 + 0.857854i \(0.328203\pi\)
\(128\) 0 0
\(129\) −1.58258 −0.139338
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) −6.58258 −0.570782
\(134\) 0 0
\(135\) 3.00000 0.258199
\(136\) 0 0
\(137\) −11.5826 −0.989566 −0.494783 0.869016i \(-0.664752\pi\)
−0.494783 + 0.869016i \(0.664752\pi\)
\(138\) 0 0
\(139\) −11.1652 −0.947016 −0.473508 0.880790i \(-0.657012\pi\)
−0.473508 + 0.880790i \(0.657012\pi\)
\(140\) 0 0
\(141\) −1.41742 −0.119369
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −24.4955 −2.03424
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 6.16515 0.505069 0.252534 0.967588i \(-0.418736\pi\)
0.252534 + 0.967588i \(0.418736\pi\)
\(150\) 0 0
\(151\) −3.58258 −0.291546 −0.145773 0.989318i \(-0.546567\pi\)
−0.145773 + 0.989318i \(0.546567\pi\)
\(152\) 0 0
\(153\) 7.58258 0.613015
\(154\) 0 0
\(155\) −10.7477 −0.863278
\(156\) 0 0
\(157\) 19.1652 1.52955 0.764773 0.644300i \(-0.222851\pi\)
0.764773 + 0.644300i \(0.222851\pi\)
\(158\) 0 0
\(159\) −9.58258 −0.759948
\(160\) 0 0
\(161\) −5.58258 −0.439969
\(162\) 0 0
\(163\) −8.58258 −0.672239 −0.336120 0.941819i \(-0.609115\pi\)
−0.336120 + 0.941819i \(0.609115\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) 4.74773 0.367390 0.183695 0.982983i \(-0.441194\pi\)
0.183695 + 0.982983i \(0.441194\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 6.58258 0.503382
\(172\) 0 0
\(173\) −7.16515 −0.544756 −0.272378 0.962190i \(-0.587810\pi\)
−0.272378 + 0.962190i \(0.587810\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −4.58258 −0.344447
\(178\) 0 0
\(179\) −14.3303 −1.07110 −0.535549 0.844504i \(-0.679895\pi\)
−0.535549 + 0.844504i \(0.679895\pi\)
\(180\) 0 0
\(181\) −5.58258 −0.414950 −0.207475 0.978240i \(-0.566524\pi\)
−0.207475 + 0.978240i \(0.566524\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 7.58258 0.554493
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −11.5826 −0.838086 −0.419043 0.907966i \(-0.637634\pi\)
−0.419043 + 0.907966i \(0.637634\pi\)
\(192\) 0 0
\(193\) −2.41742 −0.174010 −0.0870050 0.996208i \(-0.527730\pi\)
−0.0870050 + 0.996208i \(0.527730\pi\)
\(194\) 0 0
\(195\) 3.00000 0.214834
\(196\) 0 0
\(197\) 5.16515 0.368002 0.184001 0.982926i \(-0.441095\pi\)
0.184001 + 0.982926i \(0.441095\pi\)
\(198\) 0 0
\(199\) 9.58258 0.679291 0.339645 0.940554i \(-0.389693\pi\)
0.339645 + 0.940554i \(0.389693\pi\)
\(200\) 0 0
\(201\) −8.58258 −0.605368
\(202\) 0 0
\(203\) 8.16515 0.573081
\(204\) 0 0
\(205\) −33.4955 −2.33942
\(206\) 0 0
\(207\) 5.58258 0.388016
\(208\) 0 0
\(209\) 6.58258 0.455326
\(210\) 0 0
\(211\) 13.1652 0.906326 0.453163 0.891428i \(-0.350295\pi\)
0.453163 + 0.891428i \(0.350295\pi\)
\(212\) 0 0
\(213\) −11.1652 −0.765024
\(214\) 0 0
\(215\) −4.74773 −0.323792
\(216\) 0 0
\(217\) 3.58258 0.243201
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 7.58258 0.510059
\(222\) 0 0
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) 0.747727 0.0494112 0.0247056 0.999695i \(-0.492135\pi\)
0.0247056 + 0.999695i \(0.492135\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) −4.25227 −0.277388
\(236\) 0 0
\(237\) −7.16515 −0.465427
\(238\) 0 0
\(239\) −16.5826 −1.07264 −0.536319 0.844015i \(-0.680186\pi\)
−0.536319 + 0.844015i \(0.680186\pi\)
\(240\) 0 0
\(241\) −10.1652 −0.654795 −0.327397 0.944887i \(-0.606172\pi\)
−0.327397 + 0.944887i \(0.606172\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 6.58258 0.418839
\(248\) 0 0
\(249\) 11.5826 0.734016
\(250\) 0 0
\(251\) −7.41742 −0.468184 −0.234092 0.972214i \(-0.575212\pi\)
−0.234092 + 0.972214i \(0.575212\pi\)
\(252\) 0 0
\(253\) 5.58258 0.350974
\(254\) 0 0
\(255\) 22.7477 1.42452
\(256\) 0 0
\(257\) 19.0000 1.18519 0.592594 0.805502i \(-0.298104\pi\)
0.592594 + 0.805502i \(0.298104\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −8.16515 −0.505410
\(262\) 0 0
\(263\) 22.9129 1.41287 0.706434 0.707779i \(-0.250303\pi\)
0.706434 + 0.707779i \(0.250303\pi\)
\(264\) 0 0
\(265\) −28.7477 −1.76596
\(266\) 0 0
\(267\) 9.16515 0.560898
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −5.41742 −0.329085 −0.164543 0.986370i \(-0.552615\pi\)
−0.164543 + 0.986370i \(0.552615\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −19.1652 −1.15152 −0.575761 0.817618i \(-0.695294\pi\)
−0.575761 + 0.817618i \(0.695294\pi\)
\(278\) 0 0
\(279\) −3.58258 −0.214483
\(280\) 0 0
\(281\) −27.3303 −1.63039 −0.815195 0.579187i \(-0.803370\pi\)
−0.815195 + 0.579187i \(0.803370\pi\)
\(282\) 0 0
\(283\) −27.7477 −1.64943 −0.824716 0.565548i \(-0.808665\pi\)
−0.824716 + 0.565548i \(0.808665\pi\)
\(284\) 0 0
\(285\) 19.7477 1.16975
\(286\) 0 0
\(287\) 11.1652 0.659058
\(288\) 0 0
\(289\) 40.4955 2.38209
\(290\) 0 0
\(291\) −2.41742 −0.141712
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −13.7477 −0.800424
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 5.58258 0.322849
\(300\) 0 0
\(301\) 1.58258 0.0912181
\(302\) 0 0
\(303\) 11.5826 0.665402
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 1.16515 0.0662831
\(310\) 0 0
\(311\) −14.3303 −0.812597 −0.406298 0.913740i \(-0.633181\pi\)
−0.406298 + 0.913740i \(0.633181\pi\)
\(312\) 0 0
\(313\) 19.5826 1.10687 0.553436 0.832891i \(-0.313316\pi\)
0.553436 + 0.832891i \(0.313316\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −22.4174 −1.25909 −0.629544 0.776965i \(-0.716758\pi\)
−0.629544 + 0.776965i \(0.716758\pi\)
\(318\) 0 0
\(319\) −8.16515 −0.457161
\(320\) 0 0
\(321\) 12.5826 0.702291
\(322\) 0 0
\(323\) 49.9129 2.77723
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 3.58258 0.198117
\(328\) 0 0
\(329\) 1.41742 0.0781451
\(330\) 0 0
\(331\) 3.16515 0.173972 0.0869862 0.996210i \(-0.472276\pi\)
0.0869862 + 0.996210i \(0.472276\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) −25.7477 −1.40675
\(336\) 0 0
\(337\) 17.5826 0.957784 0.478892 0.877874i \(-0.341039\pi\)
0.478892 + 0.877874i \(0.341039\pi\)
\(338\) 0 0
\(339\) 9.16515 0.497783
\(340\) 0 0
\(341\) −3.58258 −0.194007
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 16.7477 0.901667
\(346\) 0 0
\(347\) −26.3303 −1.41348 −0.706742 0.707471i \(-0.749836\pi\)
−0.706742 + 0.707471i \(0.749836\pi\)
\(348\) 0 0
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 24.1652 1.28618 0.643091 0.765790i \(-0.277652\pi\)
0.643091 + 0.765790i \(0.277652\pi\)
\(354\) 0 0
\(355\) −33.4955 −1.77775
\(356\) 0 0
\(357\) −7.58258 −0.401312
\(358\) 0 0
\(359\) 8.83485 0.466285 0.233143 0.972443i \(-0.425099\pi\)
0.233143 + 0.972443i \(0.425099\pi\)
\(360\) 0 0
\(361\) 24.3303 1.28054
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 21.0000 1.09919
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) −11.1652 −0.581235
\(370\) 0 0
\(371\) 9.58258 0.497503
\(372\) 0 0
\(373\) −34.7477 −1.79917 −0.899585 0.436747i \(-0.856131\pi\)
−0.899585 + 0.436747i \(0.856131\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) −8.16515 −0.420527
\(378\) 0 0
\(379\) 12.5826 0.646323 0.323162 0.946344i \(-0.395254\pi\)
0.323162 + 0.946344i \(0.395254\pi\)
\(380\) 0 0
\(381\) 11.5826 0.593393
\(382\) 0 0
\(383\) −10.3303 −0.527854 −0.263927 0.964543i \(-0.585018\pi\)
−0.263927 + 0.964543i \(0.585018\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) −1.58258 −0.0804468
\(388\) 0 0
\(389\) −26.3303 −1.33500 −0.667500 0.744610i \(-0.732635\pi\)
−0.667500 + 0.744610i \(0.732635\pi\)
\(390\) 0 0
\(391\) 42.3303 2.14074
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) −21.4955 −1.08155
\(396\) 0 0
\(397\) −31.5826 −1.58508 −0.792542 0.609817i \(-0.791243\pi\)
−0.792542 + 0.609817i \(0.791243\pi\)
\(398\) 0 0
\(399\) −6.58258 −0.329541
\(400\) 0 0
\(401\) 31.9129 1.59365 0.796827 0.604208i \(-0.206510\pi\)
0.796827 + 0.604208i \(0.206510\pi\)
\(402\) 0 0
\(403\) −3.58258 −0.178461
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 1.00000 0.0495682
\(408\) 0 0
\(409\) 8.33030 0.411907 0.205953 0.978562i \(-0.433970\pi\)
0.205953 + 0.978562i \(0.433970\pi\)
\(410\) 0 0
\(411\) −11.5826 −0.571326
\(412\) 0 0
\(413\) 4.58258 0.225494
\(414\) 0 0
\(415\) 34.7477 1.70570
\(416\) 0 0
\(417\) −11.1652 −0.546760
\(418\) 0 0
\(419\) −2.58258 −0.126167 −0.0630835 0.998008i \(-0.520093\pi\)
−0.0630835 + 0.998008i \(0.520093\pi\)
\(420\) 0 0
\(421\) −33.6606 −1.64052 −0.820259 0.571993i \(-0.806171\pi\)
−0.820259 + 0.571993i \(0.806171\pi\)
\(422\) 0 0
\(423\) −1.41742 −0.0689175
\(424\) 0 0
\(425\) 30.3303 1.47124
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −17.7477 −0.854878 −0.427439 0.904044i \(-0.640584\pi\)
−0.427439 + 0.904044i \(0.640584\pi\)
\(432\) 0 0
\(433\) −11.1652 −0.536563 −0.268281 0.963341i \(-0.586456\pi\)
−0.268281 + 0.963341i \(0.586456\pi\)
\(434\) 0 0
\(435\) −24.4955 −1.17447
\(436\) 0 0
\(437\) 36.7477 1.75788
\(438\) 0 0
\(439\) −17.4174 −0.831288 −0.415644 0.909527i \(-0.636444\pi\)
−0.415644 + 0.909527i \(0.636444\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 23.1652 1.10061 0.550305 0.834964i \(-0.314512\pi\)
0.550305 + 0.834964i \(0.314512\pi\)
\(444\) 0 0
\(445\) 27.4955 1.30341
\(446\) 0 0
\(447\) 6.16515 0.291602
\(448\) 0 0
\(449\) −18.3303 −0.865060 −0.432530 0.901619i \(-0.642379\pi\)
−0.432530 + 0.901619i \(0.642379\pi\)
\(450\) 0 0
\(451\) −11.1652 −0.525746
\(452\) 0 0
\(453\) −3.58258 −0.168324
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −19.9129 −0.931485 −0.465743 0.884920i \(-0.654213\pi\)
−0.465743 + 0.884920i \(0.654213\pi\)
\(458\) 0 0
\(459\) 7.58258 0.353924
\(460\) 0 0
\(461\) 18.3303 0.853727 0.426864 0.904316i \(-0.359618\pi\)
0.426864 + 0.904316i \(0.359618\pi\)
\(462\) 0 0
\(463\) −8.58258 −0.398866 −0.199433 0.979911i \(-0.563910\pi\)
−0.199433 + 0.979911i \(0.563910\pi\)
\(464\) 0 0
\(465\) −10.7477 −0.498414
\(466\) 0 0
\(467\) 38.5826 1.78539 0.892694 0.450663i \(-0.148812\pi\)
0.892694 + 0.450663i \(0.148812\pi\)
\(468\) 0 0
\(469\) 8.58258 0.396307
\(470\) 0 0
\(471\) 19.1652 0.883084
\(472\) 0 0
\(473\) −1.58258 −0.0727669
\(474\) 0 0
\(475\) 26.3303 1.20812
\(476\) 0 0
\(477\) −9.58258 −0.438756
\(478\) 0 0
\(479\) 15.5826 0.711986 0.355993 0.934489i \(-0.384143\pi\)
0.355993 + 0.934489i \(0.384143\pi\)
\(480\) 0 0
\(481\) 1.00000 0.0455961
\(482\) 0 0
\(483\) −5.58258 −0.254016
\(484\) 0 0
\(485\) −7.25227 −0.329309
\(486\) 0 0
\(487\) −10.3303 −0.468111 −0.234055 0.972223i \(-0.575200\pi\)
−0.234055 + 0.972223i \(0.575200\pi\)
\(488\) 0 0
\(489\) −8.58258 −0.388117
\(490\) 0 0
\(491\) 22.9129 1.03404 0.517022 0.855972i \(-0.327041\pi\)
0.517022 + 0.855972i \(0.327041\pi\)
\(492\) 0 0
\(493\) −61.9129 −2.78842
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 0 0
\(497\) 11.1652 0.500825
\(498\) 0 0
\(499\) −41.7477 −1.86888 −0.934442 0.356114i \(-0.884101\pi\)
−0.934442 + 0.356114i \(0.884101\pi\)
\(500\) 0 0
\(501\) 4.74773 0.212113
\(502\) 0 0
\(503\) 0.747727 0.0333395 0.0166698 0.999861i \(-0.494694\pi\)
0.0166698 + 0.999861i \(0.494694\pi\)
\(504\) 0 0
\(505\) 34.7477 1.54625
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.309662
\(512\) 0 0
\(513\) 6.58258 0.290628
\(514\) 0 0
\(515\) 3.49545 0.154028
\(516\) 0 0
\(517\) −1.41742 −0.0623382
\(518\) 0 0
\(519\) −7.16515 −0.314515
\(520\) 0 0
\(521\) 15.8348 0.693737 0.346869 0.937914i \(-0.387245\pi\)
0.346869 + 0.937914i \(0.387245\pi\)
\(522\) 0 0
\(523\) −15.4174 −0.674157 −0.337078 0.941477i \(-0.609439\pi\)
−0.337078 + 0.941477i \(0.609439\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) −27.1652 −1.18333
\(528\) 0 0
\(529\) 8.16515 0.355007
\(530\) 0 0
\(531\) −4.58258 −0.198867
\(532\) 0 0
\(533\) −11.1652 −0.483616
\(534\) 0 0
\(535\) 37.7477 1.63198
\(536\) 0 0
\(537\) −14.3303 −0.618398
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 18.3303 0.788081 0.394041 0.919093i \(-0.371077\pi\)
0.394041 + 0.919093i \(0.371077\pi\)
\(542\) 0 0
\(543\) −5.58258 −0.239571
\(544\) 0 0
\(545\) 10.7477 0.460382
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −53.7477 −2.28973
\(552\) 0 0
\(553\) 7.16515 0.304693
\(554\) 0 0
\(555\) 3.00000 0.127343
\(556\) 0 0
\(557\) 9.33030 0.395338 0.197669 0.980269i \(-0.436663\pi\)
0.197669 + 0.980269i \(0.436663\pi\)
\(558\) 0 0
\(559\) −1.58258 −0.0669358
\(560\) 0 0
\(561\) 7.58258 0.320137
\(562\) 0 0
\(563\) 37.5826 1.58392 0.791958 0.610575i \(-0.209062\pi\)
0.791958 + 0.610575i \(0.209062\pi\)
\(564\) 0 0
\(565\) 27.4955 1.15674
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −26.6606 −1.11767 −0.558835 0.829279i \(-0.688752\pi\)
−0.558835 + 0.829279i \(0.688752\pi\)
\(570\) 0 0
\(571\) −28.8348 −1.20670 −0.603350 0.797476i \(-0.706168\pi\)
−0.603350 + 0.797476i \(0.706168\pi\)
\(572\) 0 0
\(573\) −11.5826 −0.483869
\(574\) 0 0
\(575\) 22.3303 0.931238
\(576\) 0 0
\(577\) −21.9129 −0.912245 −0.456123 0.889917i \(-0.650762\pi\)
−0.456123 + 0.889917i \(0.650762\pi\)
\(578\) 0 0
\(579\) −2.41742 −0.100465
\(580\) 0 0
\(581\) −11.5826 −0.480526
\(582\) 0 0
\(583\) −9.58258 −0.396870
\(584\) 0 0
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) −37.7477 −1.55802 −0.779008 0.627014i \(-0.784277\pi\)
−0.779008 + 0.627014i \(0.784277\pi\)
\(588\) 0 0
\(589\) −23.5826 −0.971703
\(590\) 0 0
\(591\) 5.16515 0.212466
\(592\) 0 0
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) −22.7477 −0.932566
\(596\) 0 0
\(597\) 9.58258 0.392189
\(598\) 0 0
\(599\) −7.16515 −0.292760 −0.146380 0.989228i \(-0.546762\pi\)
−0.146380 + 0.989228i \(0.546762\pi\)
\(600\) 0 0
\(601\) 24.4955 0.999190 0.499595 0.866259i \(-0.333482\pi\)
0.499595 + 0.866259i \(0.333482\pi\)
\(602\) 0 0
\(603\) −8.58258 −0.349510
\(604\) 0 0
\(605\) 3.00000 0.121967
\(606\) 0 0
\(607\) 21.7477 0.882713 0.441357 0.897332i \(-0.354497\pi\)
0.441357 + 0.897332i \(0.354497\pi\)
\(608\) 0 0
\(609\) 8.16515 0.330869
\(610\) 0 0
\(611\) −1.41742 −0.0573428
\(612\) 0 0
\(613\) −26.7477 −1.08033 −0.540165 0.841559i \(-0.681638\pi\)
−0.540165 + 0.841559i \(0.681638\pi\)
\(614\) 0 0
\(615\) −33.4955 −1.35067
\(616\) 0 0
\(617\) 2.83485 0.114127 0.0570634 0.998371i \(-0.481826\pi\)
0.0570634 + 0.998371i \(0.481826\pi\)
\(618\) 0 0
\(619\) 29.0780 1.16874 0.584372 0.811486i \(-0.301341\pi\)
0.584372 + 0.811486i \(0.301341\pi\)
\(620\) 0 0
\(621\) 5.58258 0.224021
\(622\) 0 0
\(623\) −9.16515 −0.367194
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 6.58258 0.262883
\(628\) 0 0
\(629\) 7.58258 0.302337
\(630\) 0 0
\(631\) −23.1652 −0.922190 −0.461095 0.887351i \(-0.652543\pi\)
−0.461095 + 0.887351i \(0.652543\pi\)
\(632\) 0 0
\(633\) 13.1652 0.523268
\(634\) 0 0
\(635\) 34.7477 1.37892
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −11.1652 −0.441687
\(640\) 0 0
\(641\) 43.5826 1.72141 0.860704 0.509106i \(-0.170024\pi\)
0.860704 + 0.509106i \(0.170024\pi\)
\(642\) 0 0
\(643\) −38.2432 −1.50816 −0.754082 0.656780i \(-0.771918\pi\)
−0.754082 + 0.656780i \(0.771918\pi\)
\(644\) 0 0
\(645\) −4.74773 −0.186942
\(646\) 0 0
\(647\) 10.9129 0.429030 0.214515 0.976721i \(-0.431183\pi\)
0.214515 + 0.976721i \(0.431183\pi\)
\(648\) 0 0
\(649\) −4.58258 −0.179882
\(650\) 0 0
\(651\) 3.58258 0.140412
\(652\) 0 0
\(653\) −30.3303 −1.18692 −0.593458 0.804865i \(-0.702238\pi\)
−0.593458 + 0.804865i \(0.702238\pi\)
\(654\) 0 0
\(655\) 48.0000 1.87552
\(656\) 0 0
\(657\) 7.00000 0.273096
\(658\) 0 0
\(659\) 28.5826 1.11342 0.556710 0.830707i \(-0.312064\pi\)
0.556710 + 0.830707i \(0.312064\pi\)
\(660\) 0 0
\(661\) −39.0780 −1.51996 −0.759980 0.649947i \(-0.774791\pi\)
−0.759980 + 0.649947i \(0.774791\pi\)
\(662\) 0 0
\(663\) 7.58258 0.294483
\(664\) 0 0
\(665\) −19.7477 −0.765784
\(666\) 0 0
\(667\) −45.5826 −1.76496
\(668\) 0 0
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 11.2523 0.433743 0.216872 0.976200i \(-0.430415\pi\)
0.216872 + 0.976200i \(0.430415\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) −45.1652 −1.73584 −0.867919 0.496706i \(-0.834543\pi\)
−0.867919 + 0.496706i \(0.834543\pi\)
\(678\) 0 0
\(679\) 2.41742 0.0927722
\(680\) 0 0
\(681\) 22.0000 0.843042
\(682\) 0 0
\(683\) 33.0780 1.26570 0.632848 0.774276i \(-0.281886\pi\)
0.632848 + 0.774276i \(0.281886\pi\)
\(684\) 0 0
\(685\) −34.7477 −1.32764
\(686\) 0 0
\(687\) 0.747727 0.0285276
\(688\) 0 0
\(689\) −9.58258 −0.365067
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −33.4955 −1.27055
\(696\) 0 0
\(697\) −84.6606 −3.20675
\(698\) 0 0
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) 6.58258 0.248267
\(704\) 0 0
\(705\) −4.25227 −0.160150
\(706\) 0 0
\(707\) −11.5826 −0.435608
\(708\) 0 0
\(709\) 27.6606 1.03882 0.519408 0.854526i \(-0.326153\pi\)
0.519408 + 0.854526i \(0.326153\pi\)
\(710\) 0 0
\(711\) −7.16515 −0.268714
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) −16.5826 −0.619288
\(718\) 0 0
\(719\) 14.0780 0.525022 0.262511 0.964929i \(-0.415449\pi\)
0.262511 + 0.964929i \(0.415449\pi\)
\(720\) 0 0
\(721\) −1.16515 −0.0433925
\(722\) 0 0
\(723\) −10.1652 −0.378046
\(724\) 0 0
\(725\) −32.6606 −1.21298
\(726\) 0 0
\(727\) 15.9129 0.590176 0.295088 0.955470i \(-0.404651\pi\)
0.295088 + 0.955470i \(0.404651\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) −8.58258 −0.316143
\(738\) 0 0
\(739\) 31.9129 1.17393 0.586967 0.809611i \(-0.300322\pi\)
0.586967 + 0.809611i \(0.300322\pi\)
\(740\) 0 0
\(741\) 6.58258 0.241817
\(742\) 0 0
\(743\) 53.2432 1.95330 0.976651 0.214830i \(-0.0689198\pi\)
0.976651 + 0.214830i \(0.0689198\pi\)
\(744\) 0 0
\(745\) 18.4955 0.677621
\(746\) 0 0
\(747\) 11.5826 0.423784
\(748\) 0 0
\(749\) −12.5826 −0.459757
\(750\) 0 0
\(751\) 8.91288 0.325236 0.162618 0.986689i \(-0.448006\pi\)
0.162618 + 0.986689i \(0.448006\pi\)
\(752\) 0 0
\(753\) −7.41742 −0.270306
\(754\) 0 0
\(755\) −10.7477 −0.391150
\(756\) 0 0
\(757\) 27.3303 0.993337 0.496668 0.867940i \(-0.334557\pi\)
0.496668 + 0.867940i \(0.334557\pi\)
\(758\) 0 0
\(759\) 5.58258 0.202635
\(760\) 0 0
\(761\) 42.3303 1.53447 0.767236 0.641365i \(-0.221631\pi\)
0.767236 + 0.641365i \(0.221631\pi\)
\(762\) 0 0
\(763\) −3.58258 −0.129698
\(764\) 0 0
\(765\) 22.7477 0.822446
\(766\) 0 0
\(767\) −4.58258 −0.165467
\(768\) 0 0
\(769\) 6.49545 0.234232 0.117116 0.993118i \(-0.462635\pi\)
0.117116 + 0.993118i \(0.462635\pi\)
\(770\) 0 0
\(771\) 19.0000 0.684268
\(772\) 0 0
\(773\) 6.16515 0.221745 0.110873 0.993835i \(-0.464635\pi\)
0.110873 + 0.993835i \(0.464635\pi\)
\(774\) 0 0
\(775\) −14.3303 −0.514760
\(776\) 0 0
\(777\) −1.00000 −0.0358748
\(778\) 0 0
\(779\) −73.4955 −2.63325
\(780\) 0 0
\(781\) −11.1652 −0.399521
\(782\) 0 0
\(783\) −8.16515 −0.291799
\(784\) 0 0
\(785\) 57.4955 2.05210
\(786\) 0 0
\(787\) −38.5826 −1.37532 −0.687660 0.726033i \(-0.741362\pi\)
−0.687660 + 0.726033i \(0.741362\pi\)
\(788\) 0 0
\(789\) 22.9129 0.815720
\(790\) 0 0
\(791\) −9.16515 −0.325875
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) −28.7477 −1.01958
\(796\) 0 0
\(797\) −52.4955 −1.85948 −0.929742 0.368211i \(-0.879970\pi\)
−0.929742 + 0.368211i \(0.879970\pi\)
\(798\) 0 0
\(799\) −10.7477 −0.380227
\(800\) 0 0
\(801\) 9.16515 0.323835
\(802\) 0 0
\(803\) 7.00000 0.247025
\(804\) 0 0
\(805\) −16.7477 −0.590280
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) 9.33030 0.328036 0.164018 0.986457i \(-0.447554\pi\)
0.164018 + 0.986457i \(0.447554\pi\)
\(810\) 0 0
\(811\) 2.25227 0.0790880 0.0395440 0.999218i \(-0.487409\pi\)
0.0395440 + 0.999218i \(0.487409\pi\)
\(812\) 0 0
\(813\) −5.41742 −0.189997
\(814\) 0 0
\(815\) −25.7477 −0.901904
\(816\) 0 0
\(817\) −10.4174 −0.364460
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −47.0000 −1.64031 −0.820156 0.572140i \(-0.806113\pi\)
−0.820156 + 0.572140i \(0.806113\pi\)
\(822\) 0 0
\(823\) 30.5826 1.06604 0.533021 0.846102i \(-0.321057\pi\)
0.533021 + 0.846102i \(0.321057\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −8.91288 −0.309931 −0.154966 0.987920i \(-0.549527\pi\)
−0.154966 + 0.987920i \(0.549527\pi\)
\(828\) 0 0
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 0 0
\(831\) −19.1652 −0.664832
\(832\) 0 0
\(833\) 7.58258 0.262721
\(834\) 0 0
\(835\) 14.2432 0.492906
\(836\) 0 0
\(837\) −3.58258 −0.123832
\(838\) 0 0
\(839\) −7.08712 −0.244675 −0.122337 0.992489i \(-0.539039\pi\)
−0.122337 + 0.992489i \(0.539039\pi\)
\(840\) 0 0
\(841\) 37.6697 1.29896
\(842\) 0 0
\(843\) −27.3303 −0.941306
\(844\) 0 0
\(845\) −36.0000 −1.23844
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −27.7477 −0.952300
\(850\) 0 0
\(851\) 5.58258 0.191368
\(852\) 0 0
\(853\) 17.1652 0.587724 0.293862 0.955848i \(-0.405059\pi\)
0.293862 + 0.955848i \(0.405059\pi\)
\(854\) 0 0
\(855\) 19.7477 0.675358
\(856\) 0 0
\(857\) −7.66970 −0.261992 −0.130996 0.991383i \(-0.541817\pi\)
−0.130996 + 0.991383i \(0.541817\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 0 0
\(861\) 11.1652 0.380507
\(862\) 0 0
\(863\) −14.4174 −0.490775 −0.245387 0.969425i \(-0.578915\pi\)
−0.245387 + 0.969425i \(0.578915\pi\)
\(864\) 0 0
\(865\) −21.4955 −0.730867
\(866\) 0 0
\(867\) 40.4955 1.37530
\(868\) 0 0
\(869\) −7.16515 −0.243061
\(870\) 0 0
\(871\) −8.58258 −0.290809
\(872\) 0 0
\(873\) −2.41742 −0.0818174
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 9.49545 0.320639 0.160319 0.987065i \(-0.448748\pi\)
0.160319 + 0.987065i \(0.448748\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −37.6606 −1.26882 −0.634409 0.772998i \(-0.718756\pi\)
−0.634409 + 0.772998i \(0.718756\pi\)
\(882\) 0 0
\(883\) 55.7477 1.87606 0.938030 0.346554i \(-0.112648\pi\)
0.938030 + 0.346554i \(0.112648\pi\)
\(884\) 0 0
\(885\) −13.7477 −0.462125
\(886\) 0 0
\(887\) −38.7477 −1.30102 −0.650511 0.759497i \(-0.725445\pi\)
−0.650511 + 0.759497i \(0.725445\pi\)
\(888\) 0 0
\(889\) −11.5826 −0.388467
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −9.33030 −0.312227
\(894\) 0 0
\(895\) −42.9909 −1.43703
\(896\) 0 0
\(897\) 5.58258 0.186397
\(898\) 0 0
\(899\) 29.2523 0.975618
\(900\) 0 0
\(901\) −72.6606 −2.42068
\(902\) 0 0
\(903\) 1.58258 0.0526648
\(904\) 0 0
\(905\) −16.7477 −0.556713
\(906\) 0 0
\(907\) −42.3303 −1.40555 −0.702777 0.711410i \(-0.748057\pi\)
−0.702777 + 0.711410i \(0.748057\pi\)
\(908\) 0 0
\(909\) 11.5826 0.384170
\(910\) 0 0
\(911\) 3.49545 0.115810 0.0579048 0.998322i \(-0.481558\pi\)
0.0579048 + 0.998322i \(0.481558\pi\)
\(912\) 0 0
\(913\) 11.5826 0.383327
\(914\) 0 0
\(915\) 30.0000 0.991769
\(916\) 0 0
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) 8.08712 0.266770 0.133385 0.991064i \(-0.457415\pi\)
0.133385 + 0.991064i \(0.457415\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −11.1652 −0.367505
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 0 0
\(927\) 1.16515 0.0382686
\(928\) 0 0
\(929\) −21.3303 −0.699825 −0.349912 0.936782i \(-0.613789\pi\)
−0.349912 + 0.936782i \(0.613789\pi\)
\(930\) 0 0
\(931\) 6.58258 0.215735
\(932\) 0 0
\(933\) −14.3303 −0.469153
\(934\) 0 0
\(935\) 22.7477 0.743930
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 19.5826 0.639053
\(940\) 0 0
\(941\) 35.1652 1.14635 0.573176 0.819433i \(-0.305711\pi\)
0.573176 + 0.819433i \(0.305711\pi\)
\(942\) 0 0
\(943\) −62.3303 −2.02975
\(944\) 0 0
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) −45.1652 −1.46767 −0.733835 0.679328i \(-0.762272\pi\)
−0.733835 + 0.679328i \(0.762272\pi\)
\(948\) 0 0
\(949\) 7.00000 0.227230
\(950\) 0 0
\(951\) −22.4174 −0.726935
\(952\) 0 0
\(953\) −28.1652 −0.912359 −0.456179 0.889888i \(-0.650782\pi\)
−0.456179 + 0.889888i \(0.650782\pi\)
\(954\) 0 0
\(955\) −34.7477 −1.12441
\(956\) 0 0
\(957\) −8.16515 −0.263942
\(958\) 0 0
\(959\) 11.5826 0.374021
\(960\) 0 0
\(961\) −18.1652 −0.585973
\(962\) 0 0
\(963\) 12.5826 0.405468
\(964\) 0 0
\(965\) −7.25227 −0.233459
\(966\) 0 0
\(967\) 13.1652 0.423363 0.211681 0.977339i \(-0.432106\pi\)
0.211681 + 0.977339i \(0.432106\pi\)
\(968\) 0 0
\(969\) 49.9129 1.60343
\(970\) 0 0
\(971\) 12.5826 0.403794 0.201897 0.979407i \(-0.435289\pi\)
0.201897 + 0.979407i \(0.435289\pi\)
\(972\) 0 0
\(973\) 11.1652 0.357938
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) 23.2523 0.743906 0.371953 0.928252i \(-0.378688\pi\)
0.371953 + 0.928252i \(0.378688\pi\)
\(978\) 0 0
\(979\) 9.16515 0.292920
\(980\) 0 0
\(981\) 3.58258 0.114383
\(982\) 0 0
\(983\) 4.83485 0.154208 0.0771039 0.997023i \(-0.475433\pi\)
0.0771039 + 0.997023i \(0.475433\pi\)
\(984\) 0 0
\(985\) 15.4955 0.493726
\(986\) 0 0
\(987\) 1.41742 0.0451171
\(988\) 0 0
\(989\) −8.83485 −0.280932
\(990\) 0 0
\(991\) 20.2523 0.643335 0.321667 0.946853i \(-0.395757\pi\)
0.321667 + 0.946853i \(0.395757\pi\)
\(992\) 0 0
\(993\) 3.16515 0.100443
\(994\) 0 0
\(995\) 28.7477 0.911364
\(996\) 0 0
\(997\) 10.6606 0.337625 0.168812 0.985648i \(-0.446007\pi\)
0.168812 + 0.985648i \(0.446007\pi\)
\(998\) 0 0
\(999\) 1.00000 0.0316386
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 3696.2.a.bl.1.2 2
4.3 odd 2 231.2.a.b.1.2 2
12.11 even 2 693.2.a.j.1.1 2
20.19 odd 2 5775.2.a.bn.1.1 2
28.27 even 2 1617.2.a.o.1.2 2
44.43 even 2 2541.2.a.z.1.1 2
84.83 odd 2 4851.2.a.ba.1.1 2
132.131 odd 2 7623.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.b.1.2 2 4.3 odd 2
693.2.a.j.1.1 2 12.11 even 2
1617.2.a.o.1.2 2 28.27 even 2
2541.2.a.z.1.1 2 44.43 even 2
3696.2.a.bl.1.2 2 1.1 even 1 trivial
4851.2.a.ba.1.1 2 84.83 odd 2
5775.2.a.bn.1.1 2 20.19 odd 2
7623.2.a.bf.1.2 2 132.131 odd 2