Properties

Label 272.10.a.f.1.1
Level $272$
Weight $10$
Character 272.1
Self dual yes
Analytic conductor $140.090$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 272.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(140.089747437\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.77274\) of defining polynomial
Character \(\chi\) \(=\) 272.1

$q$-expansion

\(f(q)\) \(=\) \(q-177.437 q^{3} -1620.18 q^{5} +1834.42 q^{7} +11801.0 q^{9} +O(q^{10})\) \(q-177.437 q^{3} -1620.18 q^{5} +1834.42 q^{7} +11801.0 q^{9} +31779.3 q^{11} -132363. q^{13} +287481. q^{15} -83521.0 q^{17} +1603.79 q^{19} -325495. q^{21} -23254.0 q^{23} +671873. q^{25} +1.39856e6 q^{27} +3.73267e6 q^{29} -8.91479e6 q^{31} -5.63884e6 q^{33} -2.97211e6 q^{35} -1.20475e7 q^{37} +2.34861e7 q^{39} -1.26779e7 q^{41} -2.86352e7 q^{43} -1.91198e7 q^{45} +7.17761e6 q^{47} -3.69885e7 q^{49} +1.48197e7 q^{51} -5.96065e7 q^{53} -5.14884e7 q^{55} -284573. q^{57} -1.85990e8 q^{59} -2.00037e8 q^{61} +2.16480e7 q^{63} +2.14453e8 q^{65} +1.27030e8 q^{67} +4.12613e6 q^{69} +3.27860e8 q^{71} -1.48678e8 q^{73} -1.19215e8 q^{75} +5.82968e7 q^{77} +2.58778e8 q^{79} -4.80436e8 q^{81} -3.45060e8 q^{83} +1.35319e8 q^{85} -6.62314e8 q^{87} +4.03936e8 q^{89} -2.42810e8 q^{91} +1.58182e9 q^{93} -2.59844e6 q^{95} -9.89973e8 q^{97} +3.75028e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + O(q^{10}) \) \( 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + 68036q^{11} - 158862q^{13} + 687324q^{15} - 417605q^{17} + 370992q^{19} + 1783880q^{21} - 1645870q^{23} + 3270239q^{25} + 2998268q^{27} + 3668616q^{29} + 7262362q^{31} - 11334900q^{33} + 26503988q^{35} - 31420708q^{37} + 42449884q^{39} - 7996938q^{41} + 56908268q^{43} + 12799536q^{45} + 16903336q^{47} - 11784059q^{49} - 19710956q^{51} - 83362982q^{53} - 6363364q^{55} + 136615904q^{57} + 37946604q^{59} - 77685452q^{61} + 191945278q^{63} - 40321288q^{65} + 304503600q^{67} - 333409272q^{69} + 476602922q^{71} - 289980486q^{73} + 153685772q^{75} - 143385648q^{77} + 828240610q^{79} + 891328609q^{81} - 194681148q^{83} - 123611080q^{85} - 158149884q^{87} + 376848106q^{89} - 194543664q^{91} + 3494835920q^{93} - 1498679864q^{95} + 692035246q^{97} - 2027106408q^{99} + O(q^{100}) \)

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
\(3\) −177.437 −1.26473 −0.632367 0.774669i \(-0.717917\pi\)
−0.632367 + 0.774669i \(0.717917\pi\)
\(4\) 0 0
\(5\) −1620.18 −1.15931 −0.579655 0.814862i \(-0.696813\pi\)
−0.579655 + 0.814862i \(0.696813\pi\)
\(6\) 0 0
\(7\) 1834.42 0.288774 0.144387 0.989521i \(-0.453879\pi\)
0.144387 + 0.989521i \(0.453879\pi\)
\(8\) 0 0
\(9\) 11801.0 0.599553
\(10\) 0 0
\(11\) 31779.3 0.654452 0.327226 0.944946i \(-0.393886\pi\)
0.327226 + 0.944946i \(0.393886\pi\)
\(12\) 0 0
\(13\) −132363. −1.28535 −0.642675 0.766139i \(-0.722176\pi\)
−0.642675 + 0.766139i \(0.722176\pi\)
\(14\) 0 0
\(15\) 287481. 1.46622
\(16\) 0 0
\(17\) −83521.0 −0.242536
\(18\) 0 0
\(19\) 1603.79 0.00282330 0.00141165 0.999999i \(-0.499551\pi\)
0.00141165 + 0.999999i \(0.499551\pi\)
\(20\) 0 0
\(21\) −325495. −0.365223
\(22\) 0 0
\(23\) −23254.0 −0.0173270 −0.00866348 0.999962i \(-0.502758\pi\)
−0.00866348 + 0.999962i \(0.502758\pi\)
\(24\) 0 0
\(25\) 671873. 0.343999
\(26\) 0 0
\(27\) 1.39856e6 0.506460
\(28\) 0 0
\(29\) 3.73267e6 0.980005 0.490002 0.871721i \(-0.336996\pi\)
0.490002 + 0.871721i \(0.336996\pi\)
\(30\) 0 0
\(31\) −8.91479e6 −1.73374 −0.866869 0.498536i \(-0.833871\pi\)
−0.866869 + 0.498536i \(0.833871\pi\)
\(32\) 0 0
\(33\) −5.63884e6 −0.827707
\(34\) 0 0
\(35\) −2.97211e6 −0.334779
\(36\) 0 0
\(37\) −1.20475e7 −1.05679 −0.528395 0.848999i \(-0.677206\pi\)
−0.528395 + 0.848999i \(0.677206\pi\)
\(38\) 0 0
\(39\) 2.34861e7 1.62563
\(40\) 0 0
\(41\) −1.26779e7 −0.700680 −0.350340 0.936623i \(-0.613934\pi\)
−0.350340 + 0.936623i \(0.613934\pi\)
\(42\) 0 0
\(43\) −2.86352e7 −1.27730 −0.638650 0.769498i \(-0.720507\pi\)
−0.638650 + 0.769498i \(0.720507\pi\)
\(44\) 0 0
\(45\) −1.91198e7 −0.695067
\(46\) 0 0
\(47\) 7.17761e6 0.214555 0.107278 0.994229i \(-0.465787\pi\)
0.107278 + 0.994229i \(0.465787\pi\)
\(48\) 0 0
\(49\) −3.69885e7 −0.916609
\(50\) 0 0
\(51\) 1.48197e7 0.306743
\(52\) 0 0
\(53\) −5.96065e7 −1.03765 −0.518826 0.854880i \(-0.673631\pi\)
−0.518826 + 0.854880i \(0.673631\pi\)
\(54\) 0 0
\(55\) −5.14884e7 −0.758712
\(56\) 0 0
\(57\) −284573. −0.00357072
\(58\) 0 0
\(59\) −1.85990e8 −1.99828 −0.999139 0.0414925i \(-0.986789\pi\)
−0.999139 + 0.0414925i \(0.986789\pi\)
\(60\) 0 0
\(61\) −2.00037e8 −1.84981 −0.924905 0.380198i \(-0.875856\pi\)
−0.924905 + 0.380198i \(0.875856\pi\)
\(62\) 0 0
\(63\) 2.16480e7 0.173135
\(64\) 0 0
\(65\) 2.14453e8 1.49012
\(66\) 0 0
\(67\) 1.27030e8 0.770137 0.385069 0.922888i \(-0.374178\pi\)
0.385069 + 0.922888i \(0.374178\pi\)
\(68\) 0 0
\(69\) 4.12613e6 0.0219140
\(70\) 0 0
\(71\) 3.27860e8 1.53118 0.765590 0.643329i \(-0.222447\pi\)
0.765590 + 0.643329i \(0.222447\pi\)
\(72\) 0 0
\(73\) −1.48678e8 −0.612766 −0.306383 0.951908i \(-0.599119\pi\)
−0.306383 + 0.951908i \(0.599119\pi\)
\(74\) 0 0
\(75\) −1.19215e8 −0.435067
\(76\) 0 0
\(77\) 5.82968e7 0.188989
\(78\) 0 0
\(79\) 2.58778e8 0.747491 0.373746 0.927531i \(-0.378073\pi\)
0.373746 + 0.927531i \(0.378073\pi\)
\(80\) 0 0
\(81\) −4.80436e8 −1.24009
\(82\) 0 0
\(83\) −3.45060e8 −0.798073 −0.399037 0.916935i \(-0.630655\pi\)
−0.399037 + 0.916935i \(0.630655\pi\)
\(84\) 0 0
\(85\) 1.35319e8 0.281174
\(86\) 0 0
\(87\) −6.62314e8 −1.23945
\(88\) 0 0
\(89\) 4.03936e8 0.682428 0.341214 0.939986i \(-0.389162\pi\)
0.341214 + 0.939986i \(0.389162\pi\)
\(90\) 0 0
\(91\) −2.42810e8 −0.371176
\(92\) 0 0
\(93\) 1.58182e9 2.19272
\(94\) 0 0
\(95\) −2.59844e6 −0.00327308
\(96\) 0 0
\(97\) −9.89973e8 −1.13540 −0.567702 0.823234i \(-0.692167\pi\)
−0.567702 + 0.823234i \(0.692167\pi\)
\(98\) 0 0
\(99\) 3.75028e8 0.392378
\(100\) 0 0
\(101\) −1.11917e9 −1.07017 −0.535084 0.844799i \(-0.679720\pi\)
−0.535084 + 0.844799i \(0.679720\pi\)
\(102\) 0 0
\(103\) −6.58796e6 −0.00576744 −0.00288372 0.999996i \(-0.500918\pi\)
−0.00288372 + 0.999996i \(0.500918\pi\)
\(104\) 0 0
\(105\) 5.27363e8 0.423407
\(106\) 0 0
\(107\) −1.55590e9 −1.14751 −0.573754 0.819028i \(-0.694513\pi\)
−0.573754 + 0.819028i \(0.694513\pi\)
\(108\) 0 0
\(109\) 6.94766e8 0.471432 0.235716 0.971822i \(-0.424256\pi\)
0.235716 + 0.971822i \(0.424256\pi\)
\(110\) 0 0
\(111\) 2.13767e9 1.33656
\(112\) 0 0
\(113\) 2.09735e9 1.21009 0.605047 0.796190i \(-0.293154\pi\)
0.605047 + 0.796190i \(0.293154\pi\)
\(114\) 0 0
\(115\) 3.76758e7 0.0200873
\(116\) 0 0
\(117\) −1.56202e9 −0.770635
\(118\) 0 0
\(119\) −1.53213e8 −0.0700381
\(120\) 0 0
\(121\) −1.34802e9 −0.571693
\(122\) 0 0
\(123\) 2.24953e9 0.886173
\(124\) 0 0
\(125\) 2.07586e9 0.760508
\(126\) 0 0
\(127\) 4.92229e9 1.67900 0.839499 0.543361i \(-0.182849\pi\)
0.839499 + 0.543361i \(0.182849\pi\)
\(128\) 0 0
\(129\) 5.08096e9 1.61544
\(130\) 0 0
\(131\) −4.54604e9 −1.34869 −0.674345 0.738416i \(-0.735574\pi\)
−0.674345 + 0.738416i \(0.735574\pi\)
\(132\) 0 0
\(133\) 2.94204e6 0.000815297 0
\(134\) 0 0
\(135\) −2.26593e9 −0.587143
\(136\) 0 0
\(137\) 5.25313e9 1.27402 0.637010 0.770856i \(-0.280171\pi\)
0.637010 + 0.770856i \(0.280171\pi\)
\(138\) 0 0
\(139\) −1.55602e9 −0.353549 −0.176774 0.984251i \(-0.556566\pi\)
−0.176774 + 0.984251i \(0.556566\pi\)
\(140\) 0 0
\(141\) −1.27358e9 −0.271356
\(142\) 0 0
\(143\) −4.20641e9 −0.841200
\(144\) 0 0
\(145\) −6.04761e9 −1.13613
\(146\) 0 0
\(147\) 6.56314e9 1.15927
\(148\) 0 0
\(149\) 2.13936e9 0.355587 0.177794 0.984068i \(-0.443104\pi\)
0.177794 + 0.984068i \(0.443104\pi\)
\(150\) 0 0
\(151\) 1.65162e9 0.258531 0.129266 0.991610i \(-0.458738\pi\)
0.129266 + 0.991610i \(0.458738\pi\)
\(152\) 0 0
\(153\) −9.85631e8 −0.145413
\(154\) 0 0
\(155\) 1.44436e10 2.00994
\(156\) 0 0
\(157\) −4.44203e9 −0.583490 −0.291745 0.956496i \(-0.594236\pi\)
−0.291745 + 0.956496i \(0.594236\pi\)
\(158\) 0 0
\(159\) 1.05764e10 1.31235
\(160\) 0 0
\(161\) −4.26577e7 −0.00500358
\(162\) 0 0
\(163\) −2.66196e9 −0.295363 −0.147682 0.989035i \(-0.547181\pi\)
−0.147682 + 0.989035i \(0.547181\pi\)
\(164\) 0 0
\(165\) 9.13596e9 0.959569
\(166\) 0 0
\(167\) 1.58326e10 1.57518 0.787588 0.616203i \(-0.211330\pi\)
0.787588 + 0.616203i \(0.211330\pi\)
\(168\) 0 0
\(169\) 6.91547e9 0.652126
\(170\) 0 0
\(171\) 1.89264e7 0.00169272
\(172\) 0 0
\(173\) −1.27988e10 −1.08633 −0.543165 0.839626i \(-0.682774\pi\)
−0.543165 + 0.839626i \(0.682774\pi\)
\(174\) 0 0
\(175\) 1.23250e9 0.0993382
\(176\) 0 0
\(177\) 3.30016e10 2.52729
\(178\) 0 0
\(179\) 8.33273e9 0.606665 0.303332 0.952885i \(-0.401901\pi\)
0.303332 + 0.952885i \(0.401901\pi\)
\(180\) 0 0
\(181\) −8.70236e9 −0.602676 −0.301338 0.953517i \(-0.597433\pi\)
−0.301338 + 0.953517i \(0.597433\pi\)
\(182\) 0 0
\(183\) 3.54941e10 2.33952
\(184\) 0 0
\(185\) 1.95192e10 1.22515
\(186\) 0 0
\(187\) −2.65424e9 −0.158728
\(188\) 0 0
\(189\) 2.56556e9 0.146253
\(190\) 0 0
\(191\) −2.29558e10 −1.24808 −0.624040 0.781392i \(-0.714510\pi\)
−0.624040 + 0.781392i \(0.714510\pi\)
\(192\) 0 0
\(193\) 2.04685e10 1.06189 0.530943 0.847407i \(-0.321838\pi\)
0.530943 + 0.847407i \(0.321838\pi\)
\(194\) 0 0
\(195\) −3.80519e10 −1.88460
\(196\) 0 0
\(197\) 5.81009e9 0.274843 0.137422 0.990513i \(-0.456118\pi\)
0.137422 + 0.990513i \(0.456118\pi\)
\(198\) 0 0
\(199\) −4.49526e9 −0.203196 −0.101598 0.994826i \(-0.532396\pi\)
−0.101598 + 0.994826i \(0.532396\pi\)
\(200\) 0 0
\(201\) −2.25398e10 −0.974019
\(202\) 0 0
\(203\) 6.84730e9 0.283000
\(204\) 0 0
\(205\) 2.05405e10 0.812305
\(206\) 0 0
\(207\) −2.74420e8 −0.0103884
\(208\) 0 0
\(209\) 5.09674e7 0.00184771
\(210\) 0 0
\(211\) −3.47865e10 −1.20820 −0.604101 0.796908i \(-0.706467\pi\)
−0.604101 + 0.796908i \(0.706467\pi\)
\(212\) 0 0
\(213\) −5.81746e10 −1.93654
\(214\) 0 0
\(215\) 4.63944e10 1.48079
\(216\) 0 0
\(217\) −1.63535e10 −0.500659
\(218\) 0 0
\(219\) 2.63811e10 0.774986
\(220\) 0 0
\(221\) 1.10551e10 0.311743
\(222\) 0 0
\(223\) 2.46348e10 0.667078 0.333539 0.942736i \(-0.391757\pi\)
0.333539 + 0.942736i \(0.391757\pi\)
\(224\) 0 0
\(225\) 7.92877e9 0.206246
\(226\) 0 0
\(227\) −4.45657e10 −1.11400 −0.556999 0.830513i \(-0.688047\pi\)
−0.556999 + 0.830513i \(0.688047\pi\)
\(228\) 0 0
\(229\) 6.61817e10 1.59030 0.795148 0.606415i \(-0.207393\pi\)
0.795148 + 0.606415i \(0.207393\pi\)
\(230\) 0 0
\(231\) −1.03440e10 −0.239021
\(232\) 0 0
\(233\) −4.35721e10 −0.968516 −0.484258 0.874925i \(-0.660910\pi\)
−0.484258 + 0.874925i \(0.660910\pi\)
\(234\) 0 0
\(235\) −1.16291e10 −0.248736
\(236\) 0 0
\(237\) −4.59169e10 −0.945378
\(238\) 0 0
\(239\) −3.67128e10 −0.727825 −0.363912 0.931433i \(-0.618559\pi\)
−0.363912 + 0.931433i \(0.618559\pi\)
\(240\) 0 0
\(241\) 3.62693e10 0.692568 0.346284 0.938130i \(-0.387443\pi\)
0.346284 + 0.938130i \(0.387443\pi\)
\(242\) 0 0
\(243\) 5.77194e10 1.06192
\(244\) 0 0
\(245\) 5.99282e10 1.06263
\(246\) 0 0
\(247\) −2.12283e8 −0.00362893
\(248\) 0 0
\(249\) 6.12264e10 1.00935
\(250\) 0 0
\(251\) 9.34890e10 1.48672 0.743359 0.668892i \(-0.233231\pi\)
0.743359 + 0.668892i \(0.233231\pi\)
\(252\) 0 0
\(253\) −7.38996e8 −0.0113397
\(254\) 0 0
\(255\) −2.40107e10 −0.355610
\(256\) 0 0
\(257\) 1.91619e10 0.273993 0.136996 0.990572i \(-0.456255\pi\)
0.136996 + 0.990572i \(0.456255\pi\)
\(258\) 0 0
\(259\) −2.21002e10 −0.305174
\(260\) 0 0
\(261\) 4.40492e10 0.587564
\(262\) 0 0
\(263\) −3.75028e10 −0.483352 −0.241676 0.970357i \(-0.577697\pi\)
−0.241676 + 0.970357i \(0.577697\pi\)
\(264\) 0 0
\(265\) 9.65735e10 1.20296
\(266\) 0 0
\(267\) −7.16732e10 −0.863090
\(268\) 0 0
\(269\) −9.36804e10 −1.09085 −0.545423 0.838161i \(-0.683631\pi\)
−0.545423 + 0.838161i \(0.683631\pi\)
\(270\) 0 0
\(271\) 9.81985e10 1.10597 0.552985 0.833191i \(-0.313489\pi\)
0.552985 + 0.833191i \(0.313489\pi\)
\(272\) 0 0
\(273\) 4.30836e10 0.469439
\(274\) 0 0
\(275\) 2.13517e10 0.225131
\(276\) 0 0
\(277\) 9.27606e9 0.0946683 0.0473341 0.998879i \(-0.484927\pi\)
0.0473341 + 0.998879i \(0.484927\pi\)
\(278\) 0 0
\(279\) −1.05203e11 −1.03947
\(280\) 0 0
\(281\) −1.04165e11 −0.996649 −0.498325 0.866991i \(-0.666051\pi\)
−0.498325 + 0.866991i \(0.666051\pi\)
\(282\) 0 0
\(283\) 9.57066e10 0.886958 0.443479 0.896285i \(-0.353744\pi\)
0.443479 + 0.896285i \(0.353744\pi\)
\(284\) 0 0
\(285\) 4.61060e8 0.00413958
\(286\) 0 0
\(287\) −2.32566e10 −0.202338
\(288\) 0 0
\(289\) 6.97576e9 0.0588235
\(290\) 0 0
\(291\) 1.75658e11 1.43598
\(292\) 0 0
\(293\) −1.62664e11 −1.28940 −0.644698 0.764437i \(-0.723017\pi\)
−0.644698 + 0.764437i \(0.723017\pi\)
\(294\) 0 0
\(295\) 3.01338e11 2.31662
\(296\) 0 0
\(297\) 4.44453e10 0.331453
\(298\) 0 0
\(299\) 3.07797e9 0.0222712
\(300\) 0 0
\(301\) −5.25292e10 −0.368851
\(302\) 0 0
\(303\) 1.98583e11 1.35348
\(304\) 0 0
\(305\) 3.24098e11 2.14450
\(306\) 0 0
\(307\) 2.26286e11 1.45390 0.726952 0.686688i \(-0.240936\pi\)
0.726952 + 0.686688i \(0.240936\pi\)
\(308\) 0 0
\(309\) 1.16895e9 0.00729428
\(310\) 0 0
\(311\) −1.17225e11 −0.710556 −0.355278 0.934761i \(-0.615614\pi\)
−0.355278 + 0.934761i \(0.615614\pi\)
\(312\) 0 0
\(313\) 2.85060e11 1.67876 0.839378 0.543549i \(-0.182920\pi\)
0.839378 + 0.543549i \(0.182920\pi\)
\(314\) 0 0
\(315\) −3.50738e10 −0.200718
\(316\) 0 0
\(317\) −9.98016e9 −0.0555099 −0.0277550 0.999615i \(-0.508836\pi\)
−0.0277550 + 0.999615i \(0.508836\pi\)
\(318\) 0 0
\(319\) 1.18622e11 0.641365
\(320\) 0 0
\(321\) 2.76075e11 1.45129
\(322\) 0 0
\(323\) −1.33950e8 −0.000684751 0
\(324\) 0 0
\(325\) −8.89312e10 −0.442159
\(326\) 0 0
\(327\) −1.23277e11 −0.596236
\(328\) 0 0
\(329\) 1.31668e10 0.0619581
\(330\) 0 0
\(331\) 2.14841e11 0.983766 0.491883 0.870661i \(-0.336309\pi\)
0.491883 + 0.870661i \(0.336309\pi\)
\(332\) 0 0
\(333\) −1.42172e11 −0.633601
\(334\) 0 0
\(335\) −2.05811e11 −0.892828
\(336\) 0 0
\(337\) −3.33228e11 −1.40737 −0.703683 0.710514i \(-0.748463\pi\)
−0.703683 + 0.710514i \(0.748463\pi\)
\(338\) 0 0
\(339\) −3.72149e11 −1.53045
\(340\) 0 0
\(341\) −2.83306e11 −1.13465
\(342\) 0 0
\(343\) −1.41878e11 −0.553468
\(344\) 0 0
\(345\) −6.68509e9 −0.0254051
\(346\) 0 0
\(347\) 1.63241e10 0.0604432 0.0302216 0.999543i \(-0.490379\pi\)
0.0302216 + 0.999543i \(0.490379\pi\)
\(348\) 0 0
\(349\) 1.61528e11 0.582819 0.291410 0.956598i \(-0.405876\pi\)
0.291410 + 0.956598i \(0.405876\pi\)
\(350\) 0 0
\(351\) −1.85118e11 −0.650978
\(352\) 0 0
\(353\) 2.89170e11 0.991212 0.495606 0.868547i \(-0.334946\pi\)
0.495606 + 0.868547i \(0.334946\pi\)
\(354\) 0 0
\(355\) −5.31194e11 −1.77511
\(356\) 0 0
\(357\) 2.71857e10 0.0885796
\(358\) 0 0
\(359\) −2.34120e11 −0.743898 −0.371949 0.928253i \(-0.621310\pi\)
−0.371949 + 0.928253i \(0.621310\pi\)
\(360\) 0 0
\(361\) −3.22685e11 −0.999992
\(362\) 0 0
\(363\) 2.39190e11 0.723040
\(364\) 0 0
\(365\) 2.40886e11 0.710385
\(366\) 0 0
\(367\) 1.08932e11 0.313442 0.156721 0.987643i \(-0.449908\pi\)
0.156721 + 0.987643i \(0.449908\pi\)
\(368\) 0 0
\(369\) −1.49612e11 −0.420094
\(370\) 0 0
\(371\) −1.09344e11 −0.299648
\(372\) 0 0
\(373\) −6.82746e11 −1.82629 −0.913145 0.407635i \(-0.866353\pi\)
−0.913145 + 0.407635i \(0.866353\pi\)
\(374\) 0 0
\(375\) −3.68336e11 −0.961841
\(376\) 0 0
\(377\) −4.94067e11 −1.25965
\(378\) 0 0
\(379\) −1.87645e11 −0.467156 −0.233578 0.972338i \(-0.575043\pi\)
−0.233578 + 0.972338i \(0.575043\pi\)
\(380\) 0 0
\(381\) −8.73397e11 −2.12349
\(382\) 0 0
\(383\) −1.01456e11 −0.240925 −0.120462 0.992718i \(-0.538438\pi\)
−0.120462 + 0.992718i \(0.538438\pi\)
\(384\) 0 0
\(385\) −9.44515e10 −0.219097
\(386\) 0 0
\(387\) −3.37924e11 −0.765808
\(388\) 0 0
\(389\) −4.62050e11 −1.02310 −0.511548 0.859255i \(-0.670928\pi\)
−0.511548 + 0.859255i \(0.670928\pi\)
\(390\) 0 0
\(391\) 1.94220e9 0.00420240
\(392\) 0 0
\(393\) 8.06637e11 1.70573
\(394\) 0 0
\(395\) −4.19269e11 −0.866574
\(396\) 0 0
\(397\) 8.72806e11 1.76344 0.881720 0.471774i \(-0.156386\pi\)
0.881720 + 0.471774i \(0.156386\pi\)
\(398\) 0 0
\(399\) −5.22027e8 −0.00103113
\(400\) 0 0
\(401\) −1.68101e11 −0.324654 −0.162327 0.986737i \(-0.551900\pi\)
−0.162327 + 0.986737i \(0.551900\pi\)
\(402\) 0 0
\(403\) 1.17999e12 2.22846
\(404\) 0 0
\(405\) 7.78395e11 1.43765
\(406\) 0 0
\(407\) −3.82861e11 −0.691618
\(408\) 0 0
\(409\) −6.99127e11 −1.23538 −0.617691 0.786421i \(-0.711932\pi\)
−0.617691 + 0.786421i \(0.711932\pi\)
\(410\) 0 0
\(411\) −9.32102e11 −1.61130
\(412\) 0 0
\(413\) −3.41185e11 −0.577052
\(414\) 0 0
\(415\) 5.59060e11 0.925214
\(416\) 0 0
\(417\) 2.76097e11 0.447145
\(418\) 0 0
\(419\) −9.68451e11 −1.53502 −0.767511 0.641036i \(-0.778505\pi\)
−0.767511 + 0.641036i \(0.778505\pi\)
\(420\) 0 0
\(421\) −5.47606e11 −0.849569 −0.424784 0.905295i \(-0.639650\pi\)
−0.424784 + 0.905295i \(0.639650\pi\)
\(422\) 0 0
\(423\) 8.47029e10 0.128637
\(424\) 0 0
\(425\) −5.61155e10 −0.0834320
\(426\) 0 0
\(427\) −3.66954e11 −0.534178
\(428\) 0 0
\(429\) 7.46373e11 1.06389
\(430\) 0 0
\(431\) −4.80670e11 −0.670964 −0.335482 0.942047i \(-0.608899\pi\)
−0.335482 + 0.942047i \(0.608899\pi\)
\(432\) 0 0
\(433\) −7.09547e11 −0.970031 −0.485016 0.874506i \(-0.661186\pi\)
−0.485016 + 0.874506i \(0.661186\pi\)
\(434\) 0 0
\(435\) 1.07307e12 1.43690
\(436\) 0 0
\(437\) −3.72946e7 −4.89192e−5 0
\(438\) 0 0
\(439\) −1.17101e12 −1.50477 −0.752387 0.658721i \(-0.771098\pi\)
−0.752387 + 0.658721i \(0.771098\pi\)
\(440\) 0 0
\(441\) −4.36501e11 −0.549555
\(442\) 0 0
\(443\) 4.75055e11 0.586040 0.293020 0.956106i \(-0.405340\pi\)
0.293020 + 0.956106i \(0.405340\pi\)
\(444\) 0 0
\(445\) −6.54450e11 −0.791146
\(446\) 0 0
\(447\) −3.79603e11 −0.449723
\(448\) 0 0
\(449\) −9.33820e11 −1.08431 −0.542156 0.840278i \(-0.682392\pi\)
−0.542156 + 0.840278i \(0.682392\pi\)
\(450\) 0 0
\(451\) −4.02895e11 −0.458561
\(452\) 0 0
\(453\) −2.93058e11 −0.326973
\(454\) 0 0
\(455\) 3.93397e11 0.430308
\(456\) 0 0
\(457\) 1.28584e12 1.37900 0.689498 0.724288i \(-0.257831\pi\)
0.689498 + 0.724288i \(0.257831\pi\)
\(458\) 0 0
\(459\) −1.16809e11 −0.122834
\(460\) 0 0
\(461\) 5.18444e11 0.534623 0.267311 0.963610i \(-0.413865\pi\)
0.267311 + 0.963610i \(0.413865\pi\)
\(462\) 0 0
\(463\) 7.76808e11 0.785596 0.392798 0.919625i \(-0.371507\pi\)
0.392798 + 0.919625i \(0.371507\pi\)
\(464\) 0 0
\(465\) −2.56283e12 −2.54204
\(466\) 0 0
\(467\) −8.28768e10 −0.0806319 −0.0403159 0.999187i \(-0.512836\pi\)
−0.0403159 + 0.999187i \(0.512836\pi\)
\(468\) 0 0
\(469\) 2.33026e11 0.222396
\(470\) 0 0
\(471\) 7.88182e11 0.737959
\(472\) 0 0
\(473\) −9.10008e11 −0.835930
\(474\) 0 0
\(475\) 1.07755e9 0.000971213 0
\(476\) 0 0
\(477\) −7.03416e11 −0.622127
\(478\) 0 0
\(479\) 2.08230e12 1.80731 0.903657 0.428257i \(-0.140872\pi\)
0.903657 + 0.428257i \(0.140872\pi\)
\(480\) 0 0
\(481\) 1.59464e12 1.35835
\(482\) 0 0
\(483\) 7.56907e9 0.00632820
\(484\) 0 0
\(485\) 1.60394e12 1.31629
\(486\) 0 0
\(487\) 2.69427e11 0.217051 0.108525 0.994094i \(-0.465387\pi\)
0.108525 + 0.994094i \(0.465387\pi\)
\(488\) 0 0
\(489\) 4.72330e11 0.373556
\(490\) 0 0
\(491\) 5.87772e10 0.0456396 0.0228198 0.999740i \(-0.492736\pi\)
0.0228198 + 0.999740i \(0.492736\pi\)
\(492\) 0 0
\(493\) −3.11756e11 −0.237686
\(494\) 0 0
\(495\) −6.07614e11 −0.454888
\(496\) 0 0
\(497\) 6.01435e11 0.442166
\(498\) 0 0
\(499\) 3.13142e11 0.226094 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(500\) 0 0
\(501\) −2.80930e12 −1.99218
\(502\) 0 0
\(503\) −7.31761e11 −0.509699 −0.254849 0.966981i \(-0.582026\pi\)
−0.254849 + 0.966981i \(0.582026\pi\)
\(504\) 0 0
\(505\) 1.81327e12 1.24066
\(506\) 0 0
\(507\) −1.22706e12 −0.824766
\(508\) 0 0
\(509\) 2.84601e12 1.87935 0.939673 0.342073i \(-0.111129\pi\)
0.939673 + 0.342073i \(0.111129\pi\)
\(510\) 0 0
\(511\) −2.72739e11 −0.176951
\(512\) 0 0
\(513\) 2.24300e9 0.00142989
\(514\) 0 0
\(515\) 1.06737e10 0.00668625
\(516\) 0 0
\(517\) 2.28100e11 0.140416
\(518\) 0 0
\(519\) 2.27099e12 1.37392
\(520\) 0 0
\(521\) −1.42270e10 −0.00845949 −0.00422974 0.999991i \(-0.501346\pi\)
−0.00422974 + 0.999991i \(0.501346\pi\)
\(522\) 0 0
\(523\) −1.20853e12 −0.706316 −0.353158 0.935564i \(-0.614892\pi\)
−0.353158 + 0.935564i \(0.614892\pi\)
\(524\) 0 0
\(525\) −2.18692e11 −0.125636
\(526\) 0 0
\(527\) 7.44572e11 0.420493
\(528\) 0 0
\(529\) −1.80061e12 −0.999700
\(530\) 0 0
\(531\) −2.19487e12 −1.19807
\(532\) 0 0
\(533\) 1.67808e12 0.900619
\(534\) 0 0
\(535\) 2.52085e12 1.33032
\(536\) 0 0
\(537\) −1.47854e12 −0.767270
\(538\) 0 0
\(539\) −1.17547e12 −0.599876
\(540\) 0 0
\(541\) 2.35337e12 1.18115 0.590573 0.806985i \(-0.298902\pi\)
0.590573 + 0.806985i \(0.298902\pi\)
\(542\) 0 0
\(543\) 1.54412e12 0.762224
\(544\) 0 0
\(545\) −1.12565e12 −0.546536
\(546\) 0 0
\(547\) −1.30236e12 −0.621996 −0.310998 0.950411i \(-0.600663\pi\)
−0.310998 + 0.950411i \(0.600663\pi\)
\(548\) 0 0
\(549\) −2.36064e12 −1.10906
\(550\) 0 0
\(551\) 5.98642e9 0.00276685
\(552\) 0 0
\(553\) 4.74710e11 0.215856
\(554\) 0 0
\(555\) −3.46343e12 −1.54949
\(556\) 0 0
\(557\) 3.33519e12 1.46816 0.734078 0.679065i \(-0.237615\pi\)
0.734078 + 0.679065i \(0.237615\pi\)
\(558\) 0 0
\(559\) 3.79025e12 1.64178
\(560\) 0 0
\(561\) 4.70961e11 0.200748
\(562\) 0 0
\(563\) 9.51343e11 0.399070 0.199535 0.979891i \(-0.436057\pi\)
0.199535 + 0.979891i \(0.436057\pi\)
\(564\) 0 0
\(565\) −3.39810e12 −1.40287
\(566\) 0 0
\(567\) −8.81324e11 −0.358106
\(568\) 0 0
\(569\) −8.14899e10 −0.0325911 −0.0162955 0.999867i \(-0.505187\pi\)
−0.0162955 + 0.999867i \(0.505187\pi\)
\(570\) 0 0
\(571\) −1.06368e12 −0.418744 −0.209372 0.977836i \(-0.567142\pi\)
−0.209372 + 0.977836i \(0.567142\pi\)
\(572\) 0 0
\(573\) 4.07322e12 1.57849
\(574\) 0 0
\(575\) −1.56237e10 −0.00596046
\(576\) 0 0
\(577\) −3.44925e12 −1.29549 −0.647743 0.761859i \(-0.724287\pi\)
−0.647743 + 0.761859i \(0.724287\pi\)
\(578\) 0 0
\(579\) −3.63187e12 −1.34300
\(580\) 0 0
\(581\) −6.32986e11 −0.230463
\(582\) 0 0
\(583\) −1.89425e12 −0.679093
\(584\) 0 0
\(585\) 2.53075e12 0.893405
\(586\) 0 0
\(587\) 3.31180e12 1.15131 0.575655 0.817693i \(-0.304747\pi\)
0.575655 + 0.817693i \(0.304747\pi\)
\(588\) 0 0
\(589\) −1.42975e10 −0.00489486
\(590\) 0 0
\(591\) −1.03093e12 −0.347604
\(592\) 0 0
\(593\) 9.22763e10 0.0306439 0.0153219 0.999883i \(-0.495123\pi\)
0.0153219 + 0.999883i \(0.495123\pi\)
\(594\) 0 0
\(595\) 2.48233e11 0.0811958
\(596\) 0 0
\(597\) 7.97627e11 0.256989
\(598\) 0 0
\(599\) 4.28034e11 0.135849 0.0679246 0.997690i \(-0.478362\pi\)
0.0679246 + 0.997690i \(0.478362\pi\)
\(600\) 0 0
\(601\) 2.92420e12 0.914266 0.457133 0.889398i \(-0.348876\pi\)
0.457133 + 0.889398i \(0.348876\pi\)
\(602\) 0 0
\(603\) 1.49908e12 0.461738
\(604\) 0 0
\(605\) 2.18405e12 0.662770
\(606\) 0 0
\(607\) 1.45237e12 0.434237 0.217118 0.976145i \(-0.430334\pi\)
0.217118 + 0.976145i \(0.430334\pi\)
\(608\) 0 0
\(609\) −1.21497e12 −0.357920
\(610\) 0 0
\(611\) −9.50050e11 −0.275779
\(612\) 0 0
\(613\) 5.13743e12 1.46951 0.734757 0.678331i \(-0.237296\pi\)
0.734757 + 0.678331i \(0.237296\pi\)
\(614\) 0 0
\(615\) −3.64465e12 −1.02735
\(616\) 0 0
\(617\) −5.31905e12 −1.47758 −0.738790 0.673936i \(-0.764603\pi\)
−0.738790 + 0.673936i \(0.764603\pi\)
\(618\) 0 0
\(619\) 3.23470e12 0.885577 0.442789 0.896626i \(-0.353989\pi\)
0.442789 + 0.896626i \(0.353989\pi\)
\(620\) 0 0
\(621\) −3.25222e10 −0.00877540
\(622\) 0 0
\(623\) 7.40989e11 0.197068
\(624\) 0 0
\(625\) −4.67554e12 −1.22566
\(626\) 0 0
\(627\) −9.04353e9 −0.00233687
\(628\) 0 0
\(629\) 1.00622e12 0.256309
\(630\) 0 0
\(631\) 7.91021e12 1.98635 0.993175 0.116630i \(-0.0372091\pi\)
0.993175 + 0.116630i \(0.0372091\pi\)
\(632\) 0 0
\(633\) 6.17242e12 1.52805
\(634\) 0 0
\(635\) −7.97501e12 −1.94648
\(636\) 0 0
\(637\) 4.89591e12 1.17816
\(638\) 0 0
\(639\) 3.86908e12 0.918023
\(640\) 0 0
\(641\) −2.28438e12 −0.534451 −0.267225 0.963634i \(-0.586107\pi\)
−0.267225 + 0.963634i \(0.586107\pi\)
\(642\) 0 0
\(643\) −7.52203e11 −0.173534 −0.0867672 0.996229i \(-0.527654\pi\)
−0.0867672 + 0.996229i \(0.527654\pi\)
\(644\) 0 0
\(645\) −8.23209e12 −1.87280
\(646\) 0 0
\(647\) 4.59159e12 1.03013 0.515067 0.857150i \(-0.327767\pi\)
0.515067 + 0.857150i \(0.327767\pi\)
\(648\) 0 0
\(649\) −5.91064e12 −1.30778
\(650\) 0 0
\(651\) 2.90172e12 0.633201
\(652\) 0 0
\(653\) 4.27664e12 0.920436 0.460218 0.887806i \(-0.347771\pi\)
0.460218 + 0.887806i \(0.347771\pi\)
\(654\) 0 0
\(655\) 7.36542e12 1.56355
\(656\) 0 0
\(657\) −1.75455e12 −0.367385
\(658\) 0 0
\(659\) 8.26860e12 1.70784 0.853920 0.520404i \(-0.174218\pi\)
0.853920 + 0.520404i \(0.174218\pi\)
\(660\) 0 0
\(661\) 4.45784e12 0.908276 0.454138 0.890931i \(-0.349947\pi\)
0.454138 + 0.890931i \(0.349947\pi\)
\(662\) 0 0
\(663\) −1.96159e12 −0.394272
\(664\) 0 0
\(665\) −4.76664e9 −0.000945182 0
\(666\) 0 0
\(667\) −8.67994e10 −0.0169805
\(668\) 0 0
\(669\) −4.37112e12 −0.843676
\(670\) 0 0
\(671\) −6.35705e12 −1.21061
\(672\) 0 0
\(673\) 3.90397e12 0.733566 0.366783 0.930307i \(-0.380459\pi\)
0.366783 + 0.930307i \(0.380459\pi\)
\(674\) 0 0
\(675\) 9.39656e11 0.174222
\(676\) 0 0
\(677\) 4.84414e12 0.886273 0.443137 0.896454i \(-0.353866\pi\)
0.443137 + 0.896454i \(0.353866\pi\)
\(678\) 0 0
\(679\) −1.81603e12 −0.327876
\(680\) 0 0
\(681\) 7.90761e12 1.40891
\(682\) 0 0
\(683\) 3.67160e12 0.645598 0.322799 0.946468i \(-0.395376\pi\)
0.322799 + 0.946468i \(0.395376\pi\)
\(684\) 0 0
\(685\) −8.51105e12 −1.47698
\(686\) 0 0
\(687\) −1.17431e13 −2.01130
\(688\) 0 0
\(689\) 7.88969e12 1.33375
\(690\) 0 0
\(691\) −3.94359e12 −0.658023 −0.329011 0.944326i \(-0.606715\pi\)
−0.329011 + 0.944326i \(0.606715\pi\)
\(692\) 0 0
\(693\) 6.87960e11 0.113309
\(694\) 0 0
\(695\) 2.52104e12 0.409873
\(696\) 0 0
\(697\) 1.05887e12 0.169940
\(698\) 0 0
\(699\) 7.73131e12 1.22492
\(700\) 0 0
\(701\) −9.80834e12 −1.53414 −0.767069 0.641564i \(-0.778286\pi\)
−0.767069 + 0.641564i \(0.778286\pi\)
\(702\) 0 0
\(703\) −1.93217e10 −0.00298364
\(704\) 0 0
\(705\) 2.06343e12 0.314585
\(706\) 0 0
\(707\) −2.05304e12 −0.309037
\(708\) 0 0
\(709\) 4.04313e12 0.600910 0.300455 0.953796i \(-0.402861\pi\)
0.300455 + 0.953796i \(0.402861\pi\)
\(710\) 0 0
\(711\) 3.05384e12 0.448160
\(712\) 0 0
\(713\) 2.07305e11 0.0300404
\(714\) 0 0
\(715\) 6.81515e12 0.975211
\(716\) 0 0
\(717\) 6.51422e12 0.920505
\(718\) 0 0
\(719\) 4.48823e12 0.626319 0.313159 0.949701i \(-0.398613\pi\)
0.313159 + 0.949701i \(0.398613\pi\)
\(720\) 0 0
\(721\) −1.20851e10 −0.00166549
\(722\) 0 0
\(723\) −6.43553e12 −0.875915
\(724\) 0 0
\(725\) 2.50788e12 0.337121
\(726\) 0 0
\(727\) 1.02503e13 1.36092 0.680459 0.732786i \(-0.261780\pi\)
0.680459 + 0.732786i \(0.261780\pi\)
\(728\) 0 0
\(729\) −7.85147e11 −0.102962
\(730\) 0 0
\(731\) 2.39164e12 0.309791
\(732\) 0 0
\(733\) −1.77332e12 −0.226892 −0.113446 0.993544i \(-0.536189\pi\)
−0.113446 + 0.993544i \(0.536189\pi\)
\(734\) 0 0
\(735\) −1.06335e13 −1.34395
\(736\) 0 0
\(737\) 4.03691e12 0.504018
\(738\) 0 0
\(739\) −2.07204e12 −0.255563 −0.127781 0.991802i \(-0.540786\pi\)
−0.127781 + 0.991802i \(0.540786\pi\)
\(740\) 0 0
\(741\) 3.76669e10 0.00458963
\(742\) 0 0
\(743\) −7.60966e12 −0.916042 −0.458021 0.888941i \(-0.651442\pi\)
−0.458021 + 0.888941i \(0.651442\pi\)
\(744\) 0 0
\(745\) −3.46616e12 −0.412236
\(746\) 0 0
\(747\) −4.07205e12 −0.478487
\(748\) 0 0
\(749\) −2.85419e12 −0.331371
\(750\) 0 0
\(751\) 7.56877e12 0.868252 0.434126 0.900852i \(-0.357057\pi\)
0.434126 + 0.900852i \(0.357057\pi\)
\(752\) 0 0
\(753\) −1.65884e13 −1.88030
\(754\) 0 0
\(755\) −2.67592e12 −0.299718
\(756\) 0 0
\(757\) −6.86633e12 −0.759965 −0.379982 0.924994i \(-0.624070\pi\)
−0.379982 + 0.924994i \(0.624070\pi\)
\(758\) 0 0
\(759\) 1.31125e11 0.0143416
\(760\) 0 0
\(761\) 9.51816e12 1.02878 0.514389 0.857557i \(-0.328019\pi\)
0.514389 + 0.857557i \(0.328019\pi\)
\(762\) 0 0
\(763\) 1.27450e12 0.136138
\(764\) 0 0
\(765\) 1.59690e12 0.168579
\(766\) 0 0
\(767\) 2.46182e13 2.56849
\(768\) 0 0
\(769\) −6.30061e12 −0.649702 −0.324851 0.945765i \(-0.605314\pi\)
−0.324851 + 0.945765i \(0.605314\pi\)
\(770\) 0 0
\(771\) −3.40003e12 −0.346528
\(772\) 0 0
\(773\) −1.40318e13 −1.41354 −0.706768 0.707446i \(-0.749847\pi\)
−0.706768 + 0.707446i \(0.749847\pi\)
\(774\) 0 0
\(775\) −5.98961e12 −0.596404
\(776\) 0 0
\(777\) 3.92140e12 0.385964
\(778\) 0 0
\(779\) −2.03327e10 −0.00197823
\(780\) 0 0
\(781\) 1.04192e13 1.00208
\(782\) 0 0
\(783\) 5.22036e12 0.496333
\(784\) 0 0
\(785\) 7.19691e12 0.676445
\(786\) 0 0
\(787\) −1.03038e13 −0.957440 −0.478720 0.877968i \(-0.658899\pi\)
−0.478720 + 0.877968i \(0.658899\pi\)
\(788\) 0 0
\(789\) 6.65440e12 0.611312
\(790\) 0 0
\(791\) 3.84744e12 0.349444
\(792\) 0 0
\(793\) 2.64776e13 2.37765
\(794\) 0 0
\(795\) −1.71357e13 −1.52143
\(796\) 0 0
\(797\) 1.57641e13 1.38390 0.691952 0.721944i \(-0.256751\pi\)
0.691952 + 0.721944i \(0.256751\pi\)
\(798\) 0 0
\(799\) −5.99481e11 −0.0520373
\(800\) 0 0
\(801\) 4.76684e12 0.409152
\(802\) 0 0
\(803\) −4.72489e12 −0.401026
\(804\) 0 0
\(805\) 6.91134e10 0.00580070
\(806\) 0 0
\(807\) 1.66224e13 1.37963
\(808\) 0 0
\(809\) 1.37214e13 1.12623 0.563117 0.826377i \(-0.309602\pi\)
0.563117 + 0.826377i \(0.309602\pi\)
\(810\) 0 0
\(811\) −1.55198e13 −1.25977 −0.629887 0.776687i \(-0.716899\pi\)
−0.629887 + 0.776687i \(0.716899\pi\)
\(812\) 0 0
\(813\) −1.74241e13 −1.39876
\(814\) 0 0
\(815\) 4.31286e12 0.342418
\(816\) 0 0
\(817\) −4.59250e10 −0.00360620
\(818\) 0 0
\(819\) −2.86540e12 −0.222540
\(820\) 0 0
\(821\) 3.40134e12 0.261280 0.130640 0.991430i \(-0.458297\pi\)
0.130640 + 0.991430i \(0.458297\pi\)
\(822\) 0 0
\(823\) −4.01848e12 −0.305325 −0.152663 0.988278i \(-0.548785\pi\)
−0.152663 + 0.988278i \(0.548785\pi\)
\(824\) 0 0
\(825\) −3.78858e12 −0.284731
\(826\) 0 0
\(827\) 4.31147e12 0.320517 0.160258 0.987075i \(-0.448767\pi\)
0.160258 + 0.987075i \(0.448767\pi\)
\(828\) 0 0
\(829\) 4.42248e12 0.325215 0.162608 0.986691i \(-0.448010\pi\)
0.162608 + 0.986691i \(0.448010\pi\)
\(830\) 0 0
\(831\) −1.64592e12 −0.119730
\(832\) 0 0
\(833\) 3.08932e12 0.222310
\(834\) 0 0
\(835\) −2.56518e13 −1.82612
\(836\) 0 0
\(837\) −1.24679e13 −0.878068
\(838\) 0 0
\(839\) −1.12787e13 −0.785835 −0.392917 0.919574i \(-0.628534\pi\)
−0.392917 + 0.919574i \(0.628534\pi\)
\(840\) 0 0
\(841\) −5.74353e11 −0.0395910
\(842\) 0 0
\(843\) 1.84827e13 1.26050
\(844\) 0 0
\(845\) −1.12043e13 −0.756016
\(846\) 0 0
\(847\) −2.47285e12 −0.165090
\(848\) 0 0
\(849\) −1.69819e13 −1.12177
\(850\) 0 0
\(851\) 2.80152e11 0.0183110
\(852\) 0 0
\(853\) −3.07683e12 −0.198991 −0.0994954 0.995038i \(-0.531723\pi\)
−0.0994954 + 0.995038i \(0.531723\pi\)
\(854\) 0 0
\(855\) −3.06642e10 −0.00196238
\(856\) 0 0
\(857\) −3.10840e13 −1.96845 −0.984223 0.176930i \(-0.943383\pi\)
−0.984223 + 0.176930i \(0.943383\pi\)
\(858\) 0 0
\(859\) 7.61893e12 0.477446 0.238723 0.971088i \(-0.423271\pi\)
0.238723 + 0.971088i \(0.423271\pi\)
\(860\) 0 0
\(861\) 4.12659e12 0.255904
\(862\) 0 0
\(863\) 2.61427e13 1.60436 0.802179 0.597083i \(-0.203674\pi\)
0.802179 + 0.597083i \(0.203674\pi\)
\(864\) 0 0
\(865\) 2.07364e13 1.25939
\(866\) 0 0
\(867\) −1.23776e12 −0.0743961
\(868\) 0 0
\(869\) 8.22380e12 0.489197
\(870\) 0 0
\(871\) −1.68140e13 −0.989896
\(872\) 0 0
\(873\) −1.16827e13 −0.680735
\(874\) 0 0
\(875\) 3.80802e12 0.219615
\(876\) 0 0
\(877\) −3.00641e13 −1.71613 −0.858065 0.513541i \(-0.828333\pi\)
−0.858065 + 0.513541i \(0.828333\pi\)
\(878\) 0 0
\(879\) 2.88626e13 1.63074
\(880\) 0 0
\(881\) −2.06409e13 −1.15435 −0.577175 0.816620i \(-0.695845\pi\)
−0.577175 + 0.816620i \(0.695845\pi\)
\(882\) 0 0
\(883\) −1.85711e13 −1.02805 −0.514026 0.857774i \(-0.671847\pi\)
−0.514026 + 0.857774i \(0.671847\pi\)
\(884\) 0 0
\(885\) −5.34687e13 −2.92991
\(886\) 0 0
\(887\) −1.35590e11 −0.00735482 −0.00367741 0.999993i \(-0.501171\pi\)
−0.00367741 + 0.999993i \(0.501171\pi\)
\(888\) 0 0
\(889\) 9.02957e12 0.484852
\(890\) 0 0
\(891\) −1.52679e13 −0.811578
\(892\) 0 0
\(893\) 1.15114e10 0.000605754 0
\(894\) 0 0
\(895\) −1.35006e13 −0.703313
\(896\) 0 0
\(897\) −5.46146e11 −0.0281672
\(898\) 0 0
\(899\) −3.32759e13 −1.69907
\(900\) 0 0
\(901\) 4.97839e12 0.251668
\(902\) 0 0
\(903\) 9.32064e12 0.466499
\(904\) 0 0
\(905\) 1.40994e13 0.698688
\(906\) 0 0
\(907\) 1.36972e13 0.672045 0.336022 0.941854i \(-0.390918\pi\)
0.336022 + 0.941854i \(0.390918\pi\)
\(908\) 0 0
\(909\) −1.32074e13 −0.641622
\(910\) 0 0
\(911\) 2.24844e13 1.08156 0.540778 0.841165i \(-0.318130\pi\)
0.540778 + 0.841165i \(0.318130\pi\)
\(912\) 0 0
\(913\) −1.09658e13 −0.522300
\(914\) 0 0
\(915\) −5.75070e13 −2.71223
\(916\) 0 0
\(917\) −8.33936e12 −0.389467
\(918\) 0 0
\(919\) −9.66637e12 −0.447037 −0.223519 0.974700i \(-0.571754\pi\)
−0.223519 + 0.974700i \(0.571754\pi\)
\(920\) 0 0
\(921\) −4.01516e13 −1.83880
\(922\) 0 0
\(923\) −4.33966e13 −1.96810
\(924\) 0 0
\(925\) −8.09439e12 −0.363535
\(926\) 0 0
\(927\) −7.77444e10 −0.00345788
\(928\) 0 0
\(929\) 8.19216e12 0.360851 0.180425 0.983589i \(-0.442253\pi\)
0.180425 + 0.983589i \(0.442253\pi\)
\(930\) 0 0
\(931\) −5.93219e10 −0.00258786
\(932\) 0 0
\(933\) 2.08001e13 0.898665
\(934\) 0 0
\(935\) 4.30036e12 0.184015
\(936\) 0 0
\(937\) 3.49872e12 0.148279 0.0741396 0.997248i \(-0.476379\pi\)
0.0741396 + 0.997248i \(0.476379\pi\)
\(938\) 0 0
\(939\) −5.05803e13 −2.12318
\(940\) 0 0
\(941\) −1.51926e12 −0.0631655 −0.0315827 0.999501i \(-0.510055\pi\)
−0.0315827 + 0.999501i \(0.510055\pi\)
\(942\) 0 0
\(943\) 2.94811e11 0.0121406
\(944\) 0 0
\(945\) −4.15668e12 −0.169552
\(946\) 0 0
\(947\) 2.12785e13 0.859738 0.429869 0.902891i \(-0.358560\pi\)
0.429869 + 0.902891i \(0.358560\pi\)
\(948\) 0 0
\(949\) 1.96795e13 0.787619
\(950\) 0 0
\(951\) 1.77085e12 0.0702053
\(952\) 0 0
\(953\) 9.61359e12 0.377544 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(954\) 0 0
\(955\) 3.71927e13 1.44691
\(956\) 0 0
\(957\) −2.10479e13 −0.811157
\(958\) 0 0
\(959\) 9.63648e12 0.367904
\(960\) 0 0
\(961\) 5.30339e13 2.00585
\(962\) 0 0
\(963\) −1.83612e13 −0.687992
\(964\) 0 0
\(965\) −3.31627e13 −1.23106
\(966\) 0 0
\(967\) 1.08700e13 0.399772 0.199886 0.979819i \(-0.435943\pi\)
0.199886 + 0.979819i \(0.435943\pi\)
\(968\) 0 0
\(969\) 2.37678e10 0.000866028 0
\(970\) 0 0
\(971\) −2.79438e13 −1.00879 −0.504393 0.863474i \(-0.668284\pi\)
−0.504393 + 0.863474i \(0.668284\pi\)
\(972\) 0 0
\(973\) −2.85441e12 −0.102096
\(974\) 0 0
\(975\) 1.57797e13 0.559214
\(976\) 0 0
\(977\) −1.49620e13 −0.525369 −0.262685 0.964882i \(-0.584608\pi\)
−0.262685 + 0.964882i \(0.584608\pi\)
\(978\) 0 0
\(979\) 1.28368e13 0.446616
\(980\) 0 0
\(981\) 8.19893e12 0.282648
\(982\) 0 0
\(983\) −2.54966e13 −0.870946 −0.435473 0.900202i \(-0.643419\pi\)
−0.435473 + 0.900202i \(0.643419\pi\)
\(984\) 0 0
\(985\) −9.41343e12 −0.318629
\(986\) 0 0
\(987\) −2.33628e12 −0.0783605
\(988\) 0 0
\(989\) 6.65884e11 0.0221317
\(990\) 0 0
\(991\) 9.59893e12 0.316149 0.158074 0.987427i \(-0.449471\pi\)
0.158074 + 0.987427i \(0.449471\pi\)
\(992\) 0 0
\(993\) −3.81209e13 −1.24420
\(994\) 0 0
\(995\) 7.28315e12 0.235568
\(996\) 0 0
\(997\) −3.17533e13 −1.01780 −0.508898 0.860827i \(-0.669947\pi\)
−0.508898 + 0.860827i \(0.669947\pi\)
\(998\) 0 0
\(999\) −1.68492e13 −0.535222
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 272.10.a.f.1.1 5
4.3 odd 2 17.10.a.a.1.3 5
12.11 even 2 153.10.a.c.1.3 5
68.67 odd 2 289.10.a.a.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.10.a.a.1.3 5 4.3 odd 2
153.10.a.c.1.3 5 12.11 even 2
272.10.a.f.1.1 5 1.1 even 1 trivial
289.10.a.a.1.3 5 68.67 odd 2