Properties

Label 72.10.a.e.1.1
Level $72$
Weight $10$
Character 72.1
Self dual yes
Analytic conductor $37.083$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 72.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.0825802038\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 72.1

$q$-expansion

\(f(q)\) \(=\) \(q+2074.00 q^{5} -4344.00 q^{7} +O(q^{10})\) \(q+2074.00 q^{5} -4344.00 q^{7} -93644.0 q^{11} -12242.0 q^{13} +319598. q^{17} -553516. q^{19} +712936. q^{23} +2.34835e6 q^{25} -2.07584e6 q^{29} -6.42045e6 q^{31} -9.00946e6 q^{35} -1.81978e7 q^{37} -9.03383e6 q^{41} +1.95947e7 q^{43} +1.84842e7 q^{47} -2.14833e7 q^{49} -1.02558e7 q^{53} -1.94218e8 q^{55} -1.21667e8 q^{59} -4.59490e7 q^{61} -2.53899e7 q^{65} +5.05354e7 q^{67} -2.67045e8 q^{71} -1.76213e8 q^{73} +4.06790e8 q^{77} -2.69686e8 q^{79} +2.27033e8 q^{83} +6.62846e8 q^{85} -7.21416e7 q^{89} +5.31792e7 q^{91} -1.14799e9 q^{95} +2.28777e8 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2074.00 1.48403 0.742017 0.670381i \(-0.233869\pi\)
0.742017 + 0.670381i \(0.233869\pi\)
\(6\) 0 0
\(7\) −4344.00 −0.683831 −0.341915 0.939731i \(-0.611076\pi\)
−0.341915 + 0.939731i \(0.611076\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −93644.0 −1.92847 −0.964235 0.265049i \(-0.914612\pi\)
−0.964235 + 0.265049i \(0.914612\pi\)
\(12\) 0 0
\(13\) −12242.0 −0.118880 −0.0594398 0.998232i \(-0.518931\pi\)
−0.0594398 + 0.998232i \(0.518931\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 319598. 0.928077 0.464038 0.885815i \(-0.346400\pi\)
0.464038 + 0.885815i \(0.346400\pi\)
\(18\) 0 0
\(19\) −553516. −0.974404 −0.487202 0.873289i \(-0.661982\pi\)
−0.487202 + 0.873289i \(0.661982\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 712936. 0.531221 0.265611 0.964080i \(-0.414426\pi\)
0.265611 + 0.964080i \(0.414426\pi\)
\(24\) 0 0
\(25\) 2.34835e6 1.20236
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.07584e6 −0.545007 −0.272504 0.962155i \(-0.587852\pi\)
−0.272504 + 0.962155i \(0.587852\pi\)
\(30\) 0 0
\(31\) −6.42045e6 −1.24864 −0.624321 0.781168i \(-0.714624\pi\)
−0.624321 + 0.781168i \(0.714624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.00946e6 −1.01483
\(36\) 0 0
\(37\) −1.81978e7 −1.59628 −0.798142 0.602470i \(-0.794183\pi\)
−0.798142 + 0.602470i \(0.794183\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.03383e6 −0.499281 −0.249640 0.968339i \(-0.580312\pi\)
−0.249640 + 0.968339i \(0.580312\pi\)
\(42\) 0 0
\(43\) 1.95947e7 0.874040 0.437020 0.899452i \(-0.356034\pi\)
0.437020 + 0.899452i \(0.356034\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.84842e7 0.552535 0.276267 0.961081i \(-0.410902\pi\)
0.276267 + 0.961081i \(0.410902\pi\)
\(48\) 0 0
\(49\) −2.14833e7 −0.532375
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.02558e7 −0.178536 −0.0892682 0.996008i \(-0.528453\pi\)
−0.0892682 + 0.996008i \(0.528453\pi\)
\(54\) 0 0
\(55\) −1.94218e8 −2.86191
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.21667e8 −1.30719 −0.653593 0.756847i \(-0.726739\pi\)
−0.653593 + 0.756847i \(0.726739\pi\)
\(60\) 0 0
\(61\) −4.59490e7 −0.424905 −0.212452 0.977171i \(-0.568145\pi\)
−0.212452 + 0.977171i \(0.568145\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.53899e7 −0.176421
\(66\) 0 0
\(67\) 5.05354e7 0.306379 0.153190 0.988197i \(-0.451045\pi\)
0.153190 + 0.988197i \(0.451045\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.67045e8 −1.24716 −0.623579 0.781760i \(-0.714322\pi\)
−0.623579 + 0.781760i \(0.714322\pi\)
\(72\) 0 0
\(73\) −1.76213e8 −0.726250 −0.363125 0.931740i \(-0.618290\pi\)
−0.363125 + 0.931740i \(0.618290\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.06790e8 1.31875
\(78\) 0 0
\(79\) −2.69686e8 −0.778997 −0.389499 0.921027i \(-0.627352\pi\)
−0.389499 + 0.921027i \(0.627352\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.27033e8 0.525094 0.262547 0.964919i \(-0.415438\pi\)
0.262547 + 0.964919i \(0.415438\pi\)
\(84\) 0 0
\(85\) 6.62846e8 1.37730
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.21416e7 −0.121880 −0.0609398 0.998141i \(-0.519410\pi\)
−0.0609398 + 0.998141i \(0.519410\pi\)
\(90\) 0 0
\(91\) 5.31792e7 0.0812935
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.14799e9 −1.44605
\(96\) 0 0
\(97\) 2.28777e8 0.262385 0.131192 0.991357i \(-0.458119\pi\)
0.131192 + 0.991357i \(0.458119\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.03256e8 0.768082 0.384041 0.923316i \(-0.374532\pi\)
0.384041 + 0.923316i \(0.374532\pi\)
\(102\) 0 0
\(103\) 7.81726e8 0.684363 0.342182 0.939634i \(-0.388834\pi\)
0.342182 + 0.939634i \(0.388834\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.00756e9 0.743093 0.371546 0.928414i \(-0.378828\pi\)
0.371546 + 0.928414i \(0.378828\pi\)
\(108\) 0 0
\(109\) −4.80692e8 −0.326173 −0.163086 0.986612i \(-0.552145\pi\)
−0.163086 + 0.986612i \(0.552145\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.89781e9 1.67193 0.835963 0.548786i \(-0.184910\pi\)
0.835963 + 0.548786i \(0.184910\pi\)
\(114\) 0 0
\(115\) 1.47863e9 0.788350
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.38833e9 −0.634647
\(120\) 0 0
\(121\) 6.41125e9 2.71900
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.19699e8 0.300303
\(126\) 0 0
\(127\) −4.24330e9 −1.44740 −0.723698 0.690117i \(-0.757559\pi\)
−0.723698 + 0.690117i \(0.757559\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.89728e9 −0.859546 −0.429773 0.902937i \(-0.641406\pi\)
−0.429773 + 0.902937i \(0.641406\pi\)
\(132\) 0 0
\(133\) 2.40447e9 0.666327
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.35617e9 0.571432 0.285716 0.958314i \(-0.407769\pi\)
0.285716 + 0.958314i \(0.407769\pi\)
\(138\) 0 0
\(139\) −2.71527e9 −0.616946 −0.308473 0.951233i \(-0.599818\pi\)
−0.308473 + 0.951233i \(0.599818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.14639e9 0.229256
\(144\) 0 0
\(145\) −4.30529e9 −0.808809
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.67402e9 −0.278242 −0.139121 0.990275i \(-0.544428\pi\)
−0.139121 + 0.990275i \(0.544428\pi\)
\(150\) 0 0
\(151\) 5.32709e9 0.833860 0.416930 0.908938i \(-0.363106\pi\)
0.416930 + 0.908938i \(0.363106\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.33160e10 −1.85303
\(156\) 0 0
\(157\) −1.15835e10 −1.52156 −0.760782 0.649008i \(-0.775184\pi\)
−0.760782 + 0.649008i \(0.775184\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.09699e9 −0.363265
\(162\) 0 0
\(163\) 9.48418e8 0.105234 0.0526169 0.998615i \(-0.483244\pi\)
0.0526169 + 0.998615i \(0.483244\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.44718e10 1.43978 0.719891 0.694087i \(-0.244192\pi\)
0.719891 + 0.694087i \(0.244192\pi\)
\(168\) 0 0
\(169\) −1.04546e10 −0.985868
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.39886e10 1.18732 0.593658 0.804717i \(-0.297683\pi\)
0.593658 + 0.804717i \(0.297683\pi\)
\(174\) 0 0
\(175\) −1.02012e10 −0.822208
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.54924e9 0.331207 0.165604 0.986192i \(-0.447043\pi\)
0.165604 + 0.986192i \(0.447043\pi\)
\(180\) 0 0
\(181\) 1.56484e10 1.08372 0.541859 0.840469i \(-0.317721\pi\)
0.541859 + 0.840469i \(0.317721\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.77421e10 −2.36894
\(186\) 0 0
\(187\) −2.99284e10 −1.78977
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.02052e10 −1.09853 −0.549267 0.835647i \(-0.685093\pi\)
−0.549267 + 0.835647i \(0.685093\pi\)
\(192\) 0 0
\(193\) −7.10827e9 −0.368770 −0.184385 0.982854i \(-0.559029\pi\)
−0.184385 + 0.982854i \(0.559029\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.25924e10 −1.06872 −0.534359 0.845257i \(-0.679447\pi\)
−0.534359 + 0.845257i \(0.679447\pi\)
\(198\) 0 0
\(199\) 3.55506e10 1.60697 0.803485 0.595325i \(-0.202977\pi\)
0.803485 + 0.595325i \(0.202977\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.01744e9 0.372693
\(204\) 0 0
\(205\) −1.87362e10 −0.740949
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.18335e10 1.87911
\(210\) 0 0
\(211\) −5.58480e9 −0.193971 −0.0969854 0.995286i \(-0.530920\pi\)
−0.0969854 + 0.995286i \(0.530920\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.06395e10 1.29710
\(216\) 0 0
\(217\) 2.78904e10 0.853859
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.91252e9 −0.110329
\(222\) 0 0
\(223\) 4.74713e10 1.28546 0.642731 0.766092i \(-0.277801\pi\)
0.642731 + 0.766092i \(0.277801\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.37702e10 0.844146 0.422073 0.906562i \(-0.361303\pi\)
0.422073 + 0.906562i \(0.361303\pi\)
\(228\) 0 0
\(229\) 7.28989e9 0.175171 0.0875854 0.996157i \(-0.472085\pi\)
0.0875854 + 0.996157i \(0.472085\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.79739e10 −1.51092 −0.755458 0.655197i \(-0.772586\pi\)
−0.755458 + 0.655197i \(0.772586\pi\)
\(234\) 0 0
\(235\) 3.83362e10 0.819980
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.11283e10 0.617114 0.308557 0.951206i \(-0.400154\pi\)
0.308557 + 0.951206i \(0.400154\pi\)
\(240\) 0 0
\(241\) 1.42372e10 0.271861 0.135931 0.990718i \(-0.456598\pi\)
0.135931 + 0.990718i \(0.456598\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.45563e10 −0.790063
\(246\) 0 0
\(247\) 6.77614e9 0.115837
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.78389e10 0.919789 0.459894 0.887974i \(-0.347887\pi\)
0.459894 + 0.887974i \(0.347887\pi\)
\(252\) 0 0
\(253\) −6.67622e10 −1.02444
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.87176e10 0.267641 0.133820 0.991006i \(-0.457276\pi\)
0.133820 + 0.991006i \(0.457276\pi\)
\(258\) 0 0
\(259\) 7.90510e10 1.09159
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.80437e10 0.361439 0.180719 0.983535i \(-0.442157\pi\)
0.180719 + 0.983535i \(0.442157\pi\)
\(264\) 0 0
\(265\) −2.12705e10 −0.264954
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.46600e10 0.520036 0.260018 0.965604i \(-0.416271\pi\)
0.260018 + 0.965604i \(0.416271\pi\)
\(270\) 0 0
\(271\) −1.03375e11 −1.16427 −0.582137 0.813090i \(-0.697783\pi\)
−0.582137 + 0.813090i \(0.697783\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.19909e11 −2.31871
\(276\) 0 0
\(277\) 1.81403e11 1.85133 0.925666 0.378341i \(-0.123505\pi\)
0.925666 + 0.378341i \(0.123505\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.25487e11 1.20066 0.600332 0.799751i \(-0.295035\pi\)
0.600332 + 0.799751i \(0.295035\pi\)
\(282\) 0 0
\(283\) 1.33561e11 1.23777 0.618886 0.785481i \(-0.287584\pi\)
0.618886 + 0.785481i \(0.287584\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.92430e10 0.341423
\(288\) 0 0
\(289\) −1.64450e10 −0.138673
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.50635e9 0.0277940 0.0138970 0.999903i \(-0.495576\pi\)
0.0138970 + 0.999903i \(0.495576\pi\)
\(294\) 0 0
\(295\) −2.52336e11 −1.93991
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.72776e9 −0.0631513
\(300\) 0 0
\(301\) −8.51195e10 −0.597695
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.52981e10 −0.630573
\(306\) 0 0
\(307\) −2.94357e11 −1.89126 −0.945629 0.325246i \(-0.894553\pi\)
−0.945629 + 0.325246i \(0.894553\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.40305e10 −0.145660 −0.0728301 0.997344i \(-0.523203\pi\)
−0.0728301 + 0.997344i \(0.523203\pi\)
\(312\) 0 0
\(313\) −2.55229e11 −1.50308 −0.751539 0.659689i \(-0.770688\pi\)
−0.751539 + 0.659689i \(0.770688\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.30255e11 −1.28069 −0.640343 0.768089i \(-0.721208\pi\)
−0.640343 + 0.768089i \(0.721208\pi\)
\(318\) 0 0
\(319\) 1.94390e11 1.05103
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.76903e11 −0.904322
\(324\) 0 0
\(325\) −2.87485e10 −0.142936
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.02953e10 −0.377840
\(330\) 0 0
\(331\) −1.21212e11 −0.555035 −0.277518 0.960721i \(-0.589512\pi\)
−0.277518 + 0.960721i \(0.589512\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.04810e11 0.454677
\(336\) 0 0
\(337\) 2.52249e11 1.06536 0.532678 0.846318i \(-0.321186\pi\)
0.532678 + 0.846318i \(0.321186\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.01236e11 2.40797
\(342\) 0 0
\(343\) 2.68619e11 1.04789
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.99996e11 −1.11079 −0.555397 0.831585i \(-0.687434\pi\)
−0.555397 + 0.831585i \(0.687434\pi\)
\(348\) 0 0
\(349\) −1.25625e11 −0.453275 −0.226638 0.973979i \(-0.572773\pi\)
−0.226638 + 0.973979i \(0.572773\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.31672e11 1.47968 0.739841 0.672782i \(-0.234901\pi\)
0.739841 + 0.672782i \(0.234901\pi\)
\(354\) 0 0
\(355\) −5.53851e11 −1.85082
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.83615e11 −0.583421 −0.291711 0.956507i \(-0.594224\pi\)
−0.291711 + 0.956507i \(0.594224\pi\)
\(360\) 0 0
\(361\) −1.63077e10 −0.0505372
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.65467e11 −1.07778
\(366\) 0 0
\(367\) 3.77185e11 1.08532 0.542659 0.839953i \(-0.317418\pi\)
0.542659 + 0.839953i \(0.317418\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.45510e10 0.122089
\(372\) 0 0
\(373\) 2.69400e11 0.720623 0.360312 0.932832i \(-0.382670\pi\)
0.360312 + 0.932832i \(0.382670\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.54124e10 0.0647903
\(378\) 0 0
\(379\) −2.04102e11 −0.508124 −0.254062 0.967188i \(-0.581767\pi\)
−0.254062 + 0.967188i \(0.581767\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.10631e10 0.0975117 0.0487559 0.998811i \(-0.484474\pi\)
0.0487559 + 0.998811i \(0.484474\pi\)
\(384\) 0 0
\(385\) 8.43681e11 1.95706
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.86342e11 0.634032 0.317016 0.948420i \(-0.397319\pi\)
0.317016 + 0.948420i \(0.397319\pi\)
\(390\) 0 0
\(391\) 2.27853e11 0.493014
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.59328e11 −1.15606
\(396\) 0 0
\(397\) 3.73016e11 0.753651 0.376826 0.926284i \(-0.377016\pi\)
0.376826 + 0.926284i \(0.377016\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.70676e11 −0.909018 −0.454509 0.890742i \(-0.650185\pi\)
−0.454509 + 0.890742i \(0.650185\pi\)
\(402\) 0 0
\(403\) 7.85991e10 0.148438
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.70411e12 3.07838
\(408\) 0 0
\(409\) −8.60520e11 −1.52057 −0.760284 0.649590i \(-0.774940\pi\)
−0.760284 + 0.649590i \(0.774940\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.28520e11 0.893894
\(414\) 0 0
\(415\) 4.70866e11 0.779257
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8.46565e11 −1.34183 −0.670914 0.741535i \(-0.734098\pi\)
−0.670914 + 0.741535i \(0.734098\pi\)
\(420\) 0 0
\(421\) −2.27835e11 −0.353468 −0.176734 0.984259i \(-0.556553\pi\)
−0.176734 + 0.984259i \(0.556553\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.50528e11 1.11588
\(426\) 0 0
\(427\) 1.99602e11 0.290563
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.47351e11 0.903633 0.451817 0.892111i \(-0.350776\pi\)
0.451817 + 0.892111i \(0.350776\pi\)
\(432\) 0 0
\(433\) 5.69898e11 0.779114 0.389557 0.921002i \(-0.372628\pi\)
0.389557 + 0.921002i \(0.372628\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.94621e11 −0.517624
\(438\) 0 0
\(439\) 5.98042e11 0.768496 0.384248 0.923230i \(-0.374461\pi\)
0.384248 + 0.923230i \(0.374461\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.10867e11 −0.383494 −0.191747 0.981444i \(-0.561415\pi\)
−0.191747 + 0.981444i \(0.561415\pi\)
\(444\) 0 0
\(445\) −1.49622e11 −0.180873
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.47114e11 −0.867517 −0.433759 0.901029i \(-0.642813\pi\)
−0.433759 + 0.901029i \(0.642813\pi\)
\(450\) 0 0
\(451\) 8.45964e11 0.962848
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.10294e11 0.120642
\(456\) 0 0
\(457\) −1.54275e12 −1.65452 −0.827260 0.561819i \(-0.810102\pi\)
−0.827260 + 0.561819i \(0.810102\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.62766e12 −1.67846 −0.839230 0.543777i \(-0.816994\pi\)
−0.839230 + 0.543777i \(0.816994\pi\)
\(462\) 0 0
\(463\) −1.11591e12 −1.12854 −0.564268 0.825592i \(-0.690842\pi\)
−0.564268 + 0.825592i \(0.690842\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.30194e11 0.515832 0.257916 0.966167i \(-0.416964\pi\)
0.257916 + 0.966167i \(0.416964\pi\)
\(468\) 0 0
\(469\) −2.19526e11 −0.209512
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.83493e12 −1.68556
\(474\) 0 0
\(475\) −1.29985e12 −1.17158
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.10019e12 1.82284 0.911422 0.411473i \(-0.134986\pi\)
0.911422 + 0.411473i \(0.134986\pi\)
\(480\) 0 0
\(481\) 2.22777e11 0.189766
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.74483e11 0.389388
\(486\) 0 0
\(487\) −1.05307e12 −0.848351 −0.424176 0.905580i \(-0.639436\pi\)
−0.424176 + 0.905580i \(0.639436\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.10556e12 −1.63494 −0.817470 0.575971i \(-0.804624\pi\)
−0.817470 + 0.575971i \(0.804624\pi\)
\(492\) 0 0
\(493\) −6.63434e11 −0.505809
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.16004e12 0.852845
\(498\) 0 0
\(499\) 2.88807e11 0.208523 0.104262 0.994550i \(-0.466752\pi\)
0.104262 + 0.994550i \(0.466752\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.17681e11 −0.360584 −0.180292 0.983613i \(-0.557704\pi\)
−0.180292 + 0.983613i \(0.557704\pi\)
\(504\) 0 0
\(505\) 1.66595e12 1.13986
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.01747e11 −0.397360 −0.198680 0.980064i \(-0.563665\pi\)
−0.198680 + 0.980064i \(0.563665\pi\)
\(510\) 0 0
\(511\) 7.65471e11 0.496632
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.62130e12 1.01562
\(516\) 0 0
\(517\) −1.73093e12 −1.06555
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.67285e11 0.0994691 0.0497345 0.998762i \(-0.484162\pi\)
0.0497345 + 0.998762i \(0.484162\pi\)
\(522\) 0 0
\(523\) −1.57966e12 −0.923222 −0.461611 0.887083i \(-0.652728\pi\)
−0.461611 + 0.887083i \(0.652728\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.05196e12 −1.15884
\(528\) 0 0
\(529\) −1.29287e12 −0.717804
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.10592e11 0.0593543
\(534\) 0 0
\(535\) 2.08968e12 1.10277
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.01178e12 1.02667
\(540\) 0 0
\(541\) −3.08736e12 −1.54953 −0.774765 0.632250i \(-0.782132\pi\)
−0.774765 + 0.632250i \(0.782132\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.96955e11 −0.484051
\(546\) 0 0
\(547\) 2.62136e11 0.125194 0.0625969 0.998039i \(-0.480062\pi\)
0.0625969 + 0.998039i \(0.480062\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.14901e12 0.531057
\(552\) 0 0
\(553\) 1.17151e12 0.532702
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.64238e11 −0.160338 −0.0801691 0.996781i \(-0.525546\pi\)
−0.0801691 + 0.996781i \(0.525546\pi\)
\(558\) 0 0
\(559\) −2.39879e11 −0.103906
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.04052e12 −1.27544 −0.637721 0.770267i \(-0.720123\pi\)
−0.637721 + 0.770267i \(0.720123\pi\)
\(564\) 0 0
\(565\) 6.01006e12 2.48119
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.35845e11 0.294294 0.147147 0.989115i \(-0.452991\pi\)
0.147147 + 0.989115i \(0.452991\pi\)
\(570\) 0 0
\(571\) 1.44618e12 0.569324 0.284662 0.958628i \(-0.408119\pi\)
0.284662 + 0.958628i \(0.408119\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.67422e12 0.638717
\(576\) 0 0
\(577\) −2.26945e12 −0.852371 −0.426186 0.904636i \(-0.640143\pi\)
−0.426186 + 0.904636i \(0.640143\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.86229e11 −0.359075
\(582\) 0 0
\(583\) 9.60391e11 0.344302
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.41977e12 −1.18885 −0.594423 0.804153i \(-0.702619\pi\)
−0.594423 + 0.804153i \(0.702619\pi\)
\(588\) 0 0
\(589\) 3.55382e12 1.21668
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.32482e12 0.439959 0.219979 0.975505i \(-0.429401\pi\)
0.219979 + 0.975505i \(0.429401\pi\)
\(594\) 0 0
\(595\) −2.87940e12 −0.941838
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.19936e12 1.33279 0.666395 0.745599i \(-0.267836\pi\)
0.666395 + 0.745599i \(0.267836\pi\)
\(600\) 0 0
\(601\) 1.05682e12 0.330418 0.165209 0.986259i \(-0.447170\pi\)
0.165209 + 0.986259i \(0.447170\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.32969e13 4.03508
\(606\) 0 0
\(607\) 5.97096e12 1.78523 0.892617 0.450816i \(-0.148867\pi\)
0.892617 + 0.450816i \(0.148867\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.26283e11 −0.0656851
\(612\) 0 0
\(613\) 2.80650e12 0.802774 0.401387 0.915909i \(-0.368528\pi\)
0.401387 + 0.915909i \(0.368528\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.48302e12 −0.411968 −0.205984 0.978555i \(-0.566039\pi\)
−0.205984 + 0.978555i \(0.566039\pi\)
\(618\) 0 0
\(619\) 1.53469e12 0.420158 0.210079 0.977684i \(-0.432628\pi\)
0.210079 + 0.977684i \(0.432628\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.13383e11 0.0833450
\(624\) 0 0
\(625\) −2.88657e12 −0.756696
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.81597e12 −1.48147
\(630\) 0 0
\(631\) −4.43498e12 −1.11368 −0.556839 0.830620i \(-0.687986\pi\)
−0.556839 + 0.830620i \(0.687986\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.80061e12 −2.14798
\(636\) 0 0
\(637\) 2.62998e11 0.0632886
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.56257e12 1.06745 0.533725 0.845658i \(-0.320792\pi\)
0.533725 + 0.845658i \(0.320792\pi\)
\(642\) 0 0
\(643\) −3.32818e12 −0.767818 −0.383909 0.923371i \(-0.625422\pi\)
−0.383909 + 0.923371i \(0.625422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.31374e12 −0.519093 −0.259547 0.965731i \(-0.583573\pi\)
−0.259547 + 0.965731i \(0.583573\pi\)
\(648\) 0 0
\(649\) 1.13933e13 2.52087
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.03697e12 −1.51453 −0.757263 0.653110i \(-0.773464\pi\)
−0.757263 + 0.653110i \(0.773464\pi\)
\(654\) 0 0
\(655\) −6.00895e12 −1.27559
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.20320e12 −0.455060 −0.227530 0.973771i \(-0.573065\pi\)
−0.227530 + 0.973771i \(0.573065\pi\)
\(660\) 0 0
\(661\) −7.29570e12 −1.48648 −0.743242 0.669022i \(-0.766713\pi\)
−0.743242 + 0.669022i \(0.766713\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.98688e12 0.988852
\(666\) 0 0
\(667\) −1.47994e12 −0.289519
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.30284e12 0.819416
\(672\) 0 0
\(673\) −4.47079e12 −0.840073 −0.420036 0.907507i \(-0.637983\pi\)
−0.420036 + 0.907507i \(0.637983\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.42095e12 0.625890 0.312945 0.949771i \(-0.398684\pi\)
0.312945 + 0.949771i \(0.398684\pi\)
\(678\) 0 0
\(679\) −9.93805e11 −0.179427
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.18730e12 1.61546 0.807728 0.589556i \(-0.200697\pi\)
0.807728 + 0.589556i \(0.200697\pi\)
\(684\) 0 0
\(685\) 4.88670e12 0.848024
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.25551e11 0.0212243
\(690\) 0 0
\(691\) −1.88811e12 −0.315047 −0.157524 0.987515i \(-0.550351\pi\)
−0.157524 + 0.987515i \(0.550351\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.63148e12 −0.915568
\(696\) 0 0
\(697\) −2.88720e12 −0.463371
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.61907e12 −0.253242 −0.126621 0.991951i \(-0.540413\pi\)
−0.126621 + 0.991951i \(0.540413\pi\)
\(702\) 0 0
\(703\) 1.00727e13 1.55542
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.48934e12 −0.525238
\(708\) 0 0
\(709\) −1.06375e13 −1.58099 −0.790497 0.612466i \(-0.790178\pi\)
−0.790497 + 0.612466i \(0.790178\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.57737e12 −0.663305
\(714\) 0 0
\(715\) 2.37761e12 0.340223
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.32770e13 −1.85276 −0.926380 0.376589i \(-0.877097\pi\)
−0.926380 + 0.376589i \(0.877097\pi\)
\(720\) 0 0
\(721\) −3.39582e12 −0.467989
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.87480e12 −0.655293
\(726\) 0 0
\(727\) 2.60017e12 0.345221 0.172611 0.984990i \(-0.444780\pi\)
0.172611 + 0.984990i \(0.444780\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.26244e12 0.811176
\(732\) 0 0
\(733\) −1.14818e13 −1.46906 −0.734532 0.678574i \(-0.762598\pi\)
−0.734532 + 0.678574i \(0.762598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.73234e12 −0.590843
\(738\) 0 0
\(739\) −7.75984e12 −0.957090 −0.478545 0.878063i \(-0.658836\pi\)
−0.478545 + 0.878063i \(0.658836\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.58115e12 0.310717 0.155358 0.987858i \(-0.450347\pi\)
0.155358 + 0.987858i \(0.450347\pi\)
\(744\) 0 0
\(745\) −3.47191e12 −0.412920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.37683e12 −0.508150
\(750\) 0 0
\(751\) −8.39208e12 −0.962697 −0.481349 0.876529i \(-0.659853\pi\)
−0.481349 + 0.876529i \(0.659853\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.10484e13 1.23748
\(756\) 0 0
\(757\) 8.15875e12 0.903009 0.451505 0.892269i \(-0.350887\pi\)
0.451505 + 0.892269i \(0.350887\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.27433e12 0.678167 0.339083 0.940756i \(-0.389883\pi\)
0.339083 + 0.940756i \(0.389883\pi\)
\(762\) 0 0
\(763\) 2.08813e12 0.223047
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.48944e12 0.155398
\(768\) 0 0
\(769\) 6.12027e12 0.631106 0.315553 0.948908i \(-0.397810\pi\)
0.315553 + 0.948908i \(0.397810\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.62875e12 0.667765 0.333883 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(774\) 0 0
\(775\) −1.50775e13 −1.50131
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.00037e12 0.486501
\(780\) 0 0
\(781\) 2.50071e13 2.40511
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.40241e13 −2.25805
\(786\) 0 0
\(787\) 1.19503e13 1.11043 0.555216 0.831706i \(-0.312636\pi\)
0.555216 + 0.831706i \(0.312636\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.25881e13 −1.14331
\(792\) 0 0
\(793\) 5.62507e11 0.0505125
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.18887e13 1.04369 0.521844 0.853041i \(-0.325244\pi\)
0.521844 + 0.853041i \(0.325244\pi\)
\(798\) 0 0
\(799\) 5.90751e12 0.512795
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.65013e13 1.40055
\(804\) 0 0
\(805\) −6.42317e12 −0.539098
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.68063e13 1.37944 0.689722 0.724074i \(-0.257733\pi\)
0.689722 + 0.724074i \(0.257733\pi\)
\(810\) 0 0
\(811\) 1.98473e13 1.61104 0.805521 0.592567i \(-0.201886\pi\)
0.805521 + 0.592567i \(0.201886\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.96702e12 0.156171
\(816\) 0 0
\(817\) −1.08460e13 −0.851668
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.83992e12 −0.371787 −0.185893 0.982570i \(-0.559518\pi\)
−0.185893 + 0.982570i \(0.559518\pi\)
\(822\) 0 0
\(823\) 8.41664e12 0.639499 0.319749 0.947502i \(-0.396401\pi\)
0.319749 + 0.947502i \(0.396401\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.16658e13 1.61064 0.805321 0.592839i \(-0.201993\pi\)
0.805321 + 0.592839i \(0.201993\pi\)
\(828\) 0 0
\(829\) −3.67734e12 −0.270420 −0.135210 0.990817i \(-0.543171\pi\)
−0.135210 + 0.990817i \(0.543171\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.86601e12 −0.494085
\(834\) 0 0
\(835\) 3.00144e13 2.13669
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.46942e13 1.02381 0.511903 0.859043i \(-0.328941\pi\)
0.511903 + 0.859043i \(0.328941\pi\)
\(840\) 0 0
\(841\) −1.01980e13 −0.702967
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.16829e13 −1.46306
\(846\) 0 0
\(847\) −2.78505e13 −1.85933
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.29738e13 −0.847979
\(852\) 0 0
\(853\) −1.99845e13 −1.29248 −0.646238 0.763136i \(-0.723659\pi\)
−0.646238 + 0.763136i \(0.723659\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.11989e13 −1.34245 −0.671226 0.741252i \(-0.734232\pi\)
−0.671226 + 0.741252i \(0.734232\pi\)
\(858\) 0 0
\(859\) 2.51809e13 1.57798 0.788992 0.614404i \(-0.210603\pi\)
0.788992 + 0.614404i \(0.210603\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.15905e13 −1.32500 −0.662498 0.749064i \(-0.730504\pi\)
−0.662498 + 0.749064i \(0.730504\pi\)
\(864\) 0 0
\(865\) 2.90123e13 1.76202
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.52544e13 1.50227
\(870\) 0 0
\(871\) −6.18655e11 −0.0364222
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.56077e12 −0.205356
\(876\) 0 0
\(877\) 2.63182e13 1.50230 0.751152 0.660129i \(-0.229498\pi\)
0.751152 + 0.660129i \(0.229498\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.23513e12 −0.236851 −0.118425 0.992963i \(-0.537785\pi\)
−0.118425 + 0.992963i \(0.537785\pi\)
\(882\) 0 0
\(883\) −2.56557e13 −1.42024 −0.710119 0.704081i \(-0.751359\pi\)
−0.710119 + 0.704081i \(0.751359\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.14044e13 1.70347 0.851735 0.523973i \(-0.175551\pi\)
0.851735 + 0.523973i \(0.175551\pi\)
\(888\) 0 0
\(889\) 1.84329e13 0.989774
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.02313e13 −0.538392
\(894\) 0 0
\(895\) 9.43512e12 0.491523
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.33278e13 0.680519
\(900\) 0 0
\(901\) −3.27772e12 −0.165695
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.24548e13 1.60827
\(906\) 0 0
\(907\) −3.87001e13 −1.89880 −0.949400 0.314070i \(-0.898307\pi\)
−0.949400 + 0.314070i \(0.898307\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.73436e13 −0.834269 −0.417135 0.908845i \(-0.636966\pi\)
−0.417135 + 0.908845i \(0.636966\pi\)
\(912\) 0 0
\(913\) −2.12602e13 −1.01263
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.25858e13 0.587784
\(918\) 0 0
\(919\) −1.75232e12 −0.0810387 −0.0405194 0.999179i \(-0.512901\pi\)
−0.0405194 + 0.999179i \(0.512901\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.26916e12 0.148262
\(924\) 0 0
\(925\) −4.27347e13 −1.91930
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.99977e13 0.880865 0.440432 0.897786i \(-0.354825\pi\)
0.440432 + 0.897786i \(0.354825\pi\)
\(930\) 0 0
\(931\) 1.18913e13 0.518749
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.20716e13 −2.65608
\(936\) 0 0
\(937\) 4.62137e12 0.195859 0.0979293 0.995193i \(-0.468778\pi\)
0.0979293 + 0.995193i \(0.468778\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.64959e13 −0.685838 −0.342919 0.939365i \(-0.611416\pi\)
−0.342919 + 0.939365i \(0.611416\pi\)
\(942\) 0 0
\(943\) −6.44055e12 −0.265228
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.99606e13 −0.806490 −0.403245 0.915092i \(-0.632118\pi\)
−0.403245 + 0.915092i \(0.632118\pi\)
\(948\) 0 0
\(949\) 2.15720e12 0.0863363
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.83087e12 0.346805 0.173402 0.984851i \(-0.444524\pi\)
0.173402 + 0.984851i \(0.444524\pi\)
\(954\) 0 0
\(955\) −4.19056e13 −1.63026
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.02352e13 −0.390763
\(960\) 0 0
\(961\) 1.47825e13 0.559105
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.47426e13 −0.547268
\(966\) 0 0
\(967\) 2.03562e13 0.748647 0.374323 0.927298i \(-0.377875\pi\)
0.374323 + 0.927298i \(0.377875\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.66888e12 −0.276851 −0.138425 0.990373i \(-0.544204\pi\)
−0.138425 + 0.990373i \(0.544204\pi\)
\(972\) 0 0
\(973\) 1.17951e13 0.421886
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −6.23017e12 −0.218763 −0.109382 0.994000i \(-0.534887\pi\)
−0.109382 + 0.994000i \(0.534887\pi\)
\(978\) 0 0
\(979\) 6.75563e12 0.235041
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.22800e13 −1.78585 −0.892924 0.450207i \(-0.851350\pi\)
−0.892924 + 0.450207i \(0.851350\pi\)
\(984\) 0 0
\(985\) −4.68565e13 −1.58601
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.39698e13 0.464308
\(990\) 0 0
\(991\) −3.73672e13 −1.23072 −0.615359 0.788247i \(-0.710989\pi\)
−0.615359 + 0.788247i \(0.710989\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.37319e13 2.38480
\(996\) 0 0
\(997\) 3.66230e13 1.17389 0.586943 0.809628i \(-0.300331\pi\)
0.586943 + 0.809628i \(0.300331\pi\)
\(998\) 0 0
\(999\) 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 72.10.a.e.1.1 1
3.2 odd 2 8.10.a.a.1.1 1
4.3 odd 2 144.10.a.n.1.1 1
12.11 even 2 16.10.a.c.1.1 1
15.2 even 4 200.10.c.b.49.2 2
15.8 even 4 200.10.c.b.49.1 2
15.14 odd 2 200.10.a.b.1.1 1
21.20 even 2 392.10.a.b.1.1 1
24.5 odd 2 64.10.a.f.1.1 1
24.11 even 2 64.10.a.d.1.1 1
48.5 odd 4 256.10.b.i.129.2 2
48.11 even 4 256.10.b.c.129.1 2
48.29 odd 4 256.10.b.i.129.1 2
48.35 even 4 256.10.b.c.129.2 2
60.23 odd 4 400.10.c.g.49.2 2
60.47 odd 4 400.10.c.g.49.1 2
60.59 even 2 400.10.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.10.a.a.1.1 1 3.2 odd 2
16.10.a.c.1.1 1 12.11 even 2
64.10.a.d.1.1 1 24.11 even 2
64.10.a.f.1.1 1 24.5 odd 2
72.10.a.e.1.1 1 1.1 even 1 trivial
144.10.a.n.1.1 1 4.3 odd 2
200.10.a.b.1.1 1 15.14 odd 2
200.10.c.b.49.1 2 15.8 even 4
200.10.c.b.49.2 2 15.2 even 4
256.10.b.c.129.1 2 48.11 even 4
256.10.b.c.129.2 2 48.35 even 4
256.10.b.i.129.1 2 48.29 odd 4
256.10.b.i.129.2 2 48.5 odd 4
392.10.a.b.1.1 1 21.20 even 2
400.10.a.d.1.1 1 60.59 even 2
400.10.c.g.49.1 2 60.47 odd 4
400.10.c.g.49.2 2 60.23 odd 4