Properties

Label 6498.2.a.u.1.1
Level $6498$
Weight $2$
Character 6498.1
Self dual yes
Analytic conductor $51.887$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6498.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.8867912334\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6498.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +2.00000 q^{11} -3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +2.00000 q^{22} -4.00000 q^{23} -5.00000 q^{25} -3.00000 q^{26} +1.00000 q^{28} -3.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -5.00000 q^{37} -4.00000 q^{41} -9.00000 q^{43} +2.00000 q^{44} -4.00000 q^{46} -10.0000 q^{47} -6.00000 q^{49} -5.00000 q^{50} -3.00000 q^{52} +4.00000 q^{53} +1.00000 q^{56} +14.0000 q^{59} +11.0000 q^{61} -3.00000 q^{62} +1.00000 q^{64} +3.00000 q^{67} -4.00000 q^{68} -14.0000 q^{71} -11.0000 q^{73} -5.00000 q^{74} +2.00000 q^{77} +1.00000 q^{79} -4.00000 q^{82} -8.00000 q^{83} -9.00000 q^{86} +2.00000 q^{88} +14.0000 q^{89} -3.00000 q^{91} -4.00000 q^{92} -10.0000 q^{94} +2.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −3.00000 −0.588348
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) −3.00000 −0.416025
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 0 0
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) −3.00000 −0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −5.00000 −0.581238
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −4.00000 −0.441726
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −9.00000 −0.970495
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −3.00000 −0.295599 −0.147799 0.989017i \(-0.547219\pi\)
−0.147799 + 0.989017i \(0.547219\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 14.0000 1.28880
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 11.0000 0.995893
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −14.0000 −1.17485
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) −5.00000 −0.410997
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) −21.0000 −1.67598 −0.837991 0.545684i \(-0.816270\pi\)
−0.837991 + 0.545684i \(0.816270\pi\)
\(158\) 1.00000 0.0795557
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) −9.00000 −0.686244
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −3.00000 −0.222375
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) −5.00000 −0.353553
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −3.00000 −0.209020
\(207\) 0 0
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) 0 0
\(211\) 25.0000 1.72107 0.860535 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.0000 0.911322
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 11.0000 0.704203
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 3.00000 0.183254
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −10.0000 −0.603023
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −19.0000 −1.13954
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) −11.0000 −0.643726
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.00000 −0.290619
\(297\) 0 0
\(298\) 22.0000 1.27443
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −21.0000 −1.18510
\(315\) 0 0
\(316\) 1.00000 0.0562544
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) 0 0
\(324\) 0 0
\(325\) 15.0000 0.832050
\(326\) 11.0000 0.609234
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) 15.0000 0.824475 0.412237 0.911077i \(-0.364747\pi\)
0.412237 + 0.911077i \(0.364747\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) 0 0
\(337\) −19.0000 −1.03500 −0.517498 0.855684i \(-0.673136\pi\)
−0.517498 + 0.855684i \(0.673136\pi\)
\(338\) −4.00000 −0.217571
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −9.00000 −0.485247
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) −29.0000 −1.55233 −0.776167 0.630527i \(-0.782839\pi\)
−0.776167 + 0.630527i \(0.782839\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −18.0000 −0.946059
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) 0 0
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 0 0
\(378\) 0 0
\(379\) −13.0000 −0.667765 −0.333883 0.942615i \(-0.608359\pi\)
−0.333883 + 0.942615i \(0.608359\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −13.0000 −0.661683
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 15.0000 0.752828 0.376414 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(398\) 5.00000 0.250627
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 9.00000 0.448322
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.00000 −0.147799
\(413\) 14.0000 0.688895
\(414\) 0 0
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) 0 0
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 25.0000 1.21698
\(423\) 0 0
\(424\) 4.00000 0.194257
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) 11.0000 0.532327
\(428\) 10.0000 0.483368
\(429\) 0 0
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 0 0
\(433\) −9.00000 −0.432512 −0.216256 0.976337i \(-0.569385\pi\)
−0.216256 + 0.976337i \(0.569385\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 0 0
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) −11.0000 −0.513996
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −9.00000 −0.418265 −0.209133 0.977887i \(-0.567064\pi\)
−0.209133 + 0.977887i \(0.567064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) 0 0
\(472\) 14.0000 0.644402
\(473\) −18.0000 −0.827641
\(474\) 0 0
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) 15.0000 0.683941
\(482\) 25.0000 1.13872
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 11.0000 0.497947
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −14.0000 −0.627986
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) −20.0000 −0.879599
\(518\) −5.00000 −0.219687
\(519\) 0 0
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) 37.0000 1.61790 0.808949 0.587879i \(-0.200037\pi\)
0.808949 + 0.587879i \(0.200037\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 3.00000 0.129580
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −31.0000 −1.33279 −0.666397 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) −10.0000 −0.426401
\(551\) 0 0
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −19.0000 −0.805779
\(557\) −34.0000 −1.44063 −0.720313 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 0 0
\(559\) 27.0000 1.14198
\(560\) 0 0
\(561\) 0 0
\(562\) 8.00000 0.337460
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) −14.0000 −0.587427
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 20.0000 0.834058
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −5.00000 −0.205499
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 12.0000 0.490716
\(599\) 46.0000 1.87951 0.939755 0.341850i \(-0.111053\pi\)
0.939755 + 0.341850i \(0.111053\pi\)
\(600\) 0 0
\(601\) 27.0000 1.10135 0.550676 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) −9.00000 −0.366813
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 5.00000 0.202944 0.101472 0.994838i \(-0.467645\pi\)
0.101472 + 0.994838i \(0.467645\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 30.0000 1.21367
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −10.0000 −0.400963
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −21.0000 −0.837991
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 1.00000 0.0397779
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 37.0000 1.45914 0.729569 0.683907i \(-0.239721\pi\)
0.729569 + 0.683907i \(0.239721\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 28.0000 1.09910
\(650\) 15.0000 0.588348
\(651\) 0 0
\(652\) 11.0000 0.430793
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) −10.0000 −0.389841
\(659\) 14.0000 0.545363 0.272681 0.962104i \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 15.0000 0.582992
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −6.00000 −0.232147
\(669\) 0 0
\(670\) 0 0
\(671\) 22.0000 0.849301
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −19.0000 −0.731853
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) −6.00000 −0.229752
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) −9.00000 −0.343122
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) −29.0000 −1.09767
\(699\) 0 0
\(700\) −5.00000 −0.188982
\(701\) 44.0000 1.66186 0.830929 0.556379i \(-0.187810\pi\)
0.830929 + 0.556379i \(0.187810\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 0 0
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) −3.00000 −0.111187
\(729\) 0 0
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 23.0000 0.848945
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) −10.0000 −0.364662
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.00000 0.181728 0.0908640 0.995863i \(-0.471037\pi\)
0.0908640 + 0.995863i \(0.471037\pi\)
\(758\) −13.0000 −0.472181
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −14.0000 −0.505841
\(767\) −42.0000 −1.51653
\(768\) 0 0
\(769\) 27.0000 0.973645 0.486822 0.873501i \(-0.338156\pi\)
0.486822 + 0.873501i \(0.338156\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13.0000 −0.467880
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 15.0000 0.538816
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 16.0000 0.573628
\(779\) 0 0
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) −47.0000 −1.67537 −0.837685 0.546154i \(-0.816091\pi\)
−0.837685 + 0.546154i \(0.816091\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) −33.0000 −1.17186
\(794\) 15.0000 0.532330
\(795\) 0 0
\(796\) 5.00000 0.177220
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) −22.0000 −0.776363
\(804\) 0 0
\(805\) 0 0
\(806\) 9.00000 0.317011
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −10.0000 −0.350500
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) 0 0
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −3.00000 −0.104510
\(825\) 0 0
\(826\) 14.0000 0.487122
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −31.0000 −1.07667 −0.538337 0.842729i \(-0.680947\pi\)
−0.538337 + 0.842729i \(0.680947\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.00000 −0.104006
\(833\) 24.0000 0.831551
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −18.0000 −0.621800
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 25.0000 0.860535
\(845\) 0 0
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 4.00000 0.137361
\(849\) 0 0
\(850\) 20.0000 0.685994
\(851\) 20.0000 0.685591
\(852\) 0 0
\(853\) 3.00000 0.102718 0.0513590 0.998680i \(-0.483645\pi\)
0.0513590 + 0.998680i \(0.483645\pi\)
\(854\) 11.0000 0.376412
\(855\) 0 0
\(856\) 10.0000 0.341793
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 7.00000 0.238837 0.119418 0.992844i \(-0.461897\pi\)
0.119418 + 0.992844i \(0.461897\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 10.0000 0.340601
\(863\) 22.0000 0.748889 0.374444 0.927249i \(-0.377833\pi\)
0.374444 + 0.927249i \(0.377833\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −9.00000 −0.305832
\(867\) 0 0
\(868\) −3.00000 −0.101827
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) −9.00000 −0.304953
\(872\) 14.0000 0.474100
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −25.0000 −0.844190 −0.422095 0.906552i \(-0.638705\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) 19.0000 0.641219
\(879\) 0 0
\(880\) 0 0
\(881\) −36.0000 −1.21287 −0.606435 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) −9.00000 −0.302874 −0.151437 0.988467i \(-0.548390\pi\)
−0.151437 + 0.988467i \(0.548390\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 36.0000 1.20134
\(899\) 0 0
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 60.0000 1.98789 0.993944 0.109885i \(-0.0350482\pi\)
0.993944 + 0.109885i \(0.0350482\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 5.00000 0.165385
\(915\) 0 0
\(916\) −11.0000 −0.363450
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 55.0000 1.81428 0.907141 0.420826i \(-0.138260\pi\)
0.907141 + 0.420826i \(0.138260\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 42.0000 1.38245
\(924\) 0 0
\(925\) 25.0000 0.821995
\(926\) −9.00000 −0.295758
\(927\) 0 0
\(928\) 0 0
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) 3.00000 0.0979535
\(939\) 0 0
\(940\) 0 0
\(941\) 32.0000 1.04317 0.521585 0.853199i \(-0.325341\pi\)
0.521585 + 0.853199i \(0.325341\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) −18.0000 −0.585230
\(947\) −10.0000 −0.324956 −0.162478 0.986712i \(-0.551949\pi\)
−0.162478 + 0.986712i \(0.551949\pi\)
\(948\) 0 0
\(949\) 33.0000 1.07123
\(950\) 0 0
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 15.0000 0.483619
\(963\) 0 0
\(964\) 25.0000 0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) 51.0000 1.64005 0.820025 0.572328i \(-0.193960\pi\)
0.820025 + 0.572328i \(0.193960\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) −19.0000 −0.609112
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) 11.0000 0.352101
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 28.0000 0.894884
\(980\) 0 0
\(981\) 0 0
\(982\) −30.0000 −0.957338
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 0 0
\(994\) −14.0000 −0.444053
\(995\) 0 0
\(996\) 0 0
\(997\) −49.0000 −1.55185 −0.775923 0.630828i \(-0.782715\pi\)
−0.775923 + 0.630828i \(0.782715\pi\)
\(998\) 5.00000 0.158272
\(999\) 0 0
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 6498.2.a.u.1.1 1
3.2 odd 2 2166.2.a.b.1.1 1
19.7 even 3 342.2.g.c.163.1 2
19.11 even 3 342.2.g.c.235.1 2
19.18 odd 2 6498.2.a.g.1.1 1
57.11 odd 6 114.2.e.b.7.1 2
57.26 odd 6 114.2.e.b.49.1 yes 2
57.56 even 2 2166.2.a.h.1.1 1
76.7 odd 6 2736.2.s.k.1873.1 2
76.11 odd 6 2736.2.s.k.577.1 2
228.11 even 6 912.2.q.b.577.1 2
228.83 even 6 912.2.q.b.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.e.b.7.1 2 57.11 odd 6
114.2.e.b.49.1 yes 2 57.26 odd 6
342.2.g.c.163.1 2 19.7 even 3
342.2.g.c.235.1 2 19.11 even 3
912.2.q.b.49.1 2 228.83 even 6
912.2.q.b.577.1 2 228.11 even 6
2166.2.a.b.1.1 1 3.2 odd 2
2166.2.a.h.1.1 1 57.56 even 2
2736.2.s.k.577.1 2 76.11 odd 6
2736.2.s.k.1873.1 2 76.7 odd 6
6498.2.a.g.1.1 1 19.18 odd 2
6498.2.a.u.1.1 1 1.1 even 1 trivial