Properties

Label 153.10.a.c.1.3
Level $153$
Weight $10$
Character 153.1
Self dual yes
Analytic conductor $78.800$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 153.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.8004829331\)
Analytic rank: \(1\)
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: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.77274\) of defining polynomial
Character \(\chi\) \(=\) 153.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.22726 q^{2} -510.494 q^{4} +1620.18 q^{5} -1834.42 q^{7} -1254.87 q^{8} +O(q^{10})\) \(q+1.22726 q^{2} -510.494 q^{4} +1620.18 q^{5} -1834.42 q^{7} -1254.87 q^{8} +1988.39 q^{10} +31779.3 q^{11} -132363. q^{13} -2251.32 q^{14} +259833. q^{16} +83521.0 q^{17} -1603.79 q^{19} -827094. q^{20} +39001.6 q^{22} -23254.0 q^{23} +671873. q^{25} -162444. q^{26} +936463. q^{28} -3.73267e6 q^{29} +8.91479e6 q^{31} +961376. q^{32} +102502. q^{34} -2.97211e6 q^{35} -1.20475e7 q^{37} -1968.28 q^{38} -2.03312e6 q^{40} +1.26779e7 q^{41} +2.86352e7 q^{43} -1.62231e7 q^{44} -28538.8 q^{46} +7.17761e6 q^{47} -3.69885e7 q^{49} +824565. q^{50} +6.75705e7 q^{52} +5.96065e7 q^{53} +5.14884e7 q^{55} +2.30196e6 q^{56} -4.58096e6 q^{58} -1.85990e8 q^{59} -2.00037e8 q^{61} +1.09408e7 q^{62} -1.31855e8 q^{64} -2.14453e8 q^{65} -1.27030e8 q^{67} -4.26370e7 q^{68} -3.64756e6 q^{70} +3.27860e8 q^{71} -1.48678e8 q^{73} -1.47854e7 q^{74} +818726. q^{76} -5.82968e7 q^{77} -2.58778e8 q^{79} +4.20977e8 q^{80} +1.55591e7 q^{82} -3.45060e8 q^{83} +1.35319e8 q^{85} +3.51430e7 q^{86} -3.98789e7 q^{88} -4.03936e8 q^{89} +2.42810e8 q^{91} +1.18710e7 q^{92} +8.80881e6 q^{94} -2.59844e6 q^{95} -9.89973e8 q^{97} -4.53946e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 33q^{2} + 853q^{4} - 1480q^{5} - 13202q^{7} + 42423q^{8} + O(q^{10}) \) \( 5q + 33q^{2} + 853q^{4} - 1480q^{5} - 13202q^{7} + 42423q^{8} - 89328q^{10} + 68036q^{11} - 158862q^{13} + 84700q^{14} + 350225q^{16} + 417605q^{17} - 370992q^{19} - 1632640q^{20} + 122290q^{22} - 1645870q^{23} + 3270239q^{25} - 734846q^{26} + 183372q^{28} - 3668616q^{29} - 7262362q^{31} + 5605919q^{32} + 2756193q^{34} + 26503988q^{35} - 31420708q^{37} - 18513700q^{38} - 53930464q^{40} + 7996938q^{41} - 56908268q^{43} - 43323054q^{44} - 32063472q^{46} + 16903336q^{47} - 11784059q^{49} - 85921093q^{50} + 173619082q^{52} + 83362982q^{53} + 6363364q^{55} - 317409372q^{56} + 64577488q^{58} + 37946604q^{59} - 77685452q^{61} - 324855300q^{62} + 131623105q^{64} + 40321288q^{65} - 304503600q^{67} + 71243413q^{68} - 122787392q^{70} + 476602922q^{71} - 289980486q^{73} - 262289012q^{74} - 1031276084q^{76} + 143385648q^{77} - 828240610q^{79} - 912750944q^{80} - 1109615654q^{82} - 194681148q^{83} - 123611080q^{85} - 1164707144q^{86} - 1017979978q^{88} - 376848106q^{89} + 194543664q^{91} - 2506713088q^{92} - 2244811104q^{94} - 1498679864q^{95} + 692035246q^{97} - 871744055q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22726 0.0542379 0.0271189 0.999632i \(-0.491367\pi\)
0.0271189 + 0.999632i \(0.491367\pi\)
\(3\) 0 0
\(4\) −510.494 −0.997058
\(5\) 1620.18 1.15931 0.579655 0.814862i \(-0.303187\pi\)
0.579655 + 0.814862i \(0.303187\pi\)
\(6\) 0 0
\(7\) −1834.42 −0.288774 −0.144387 0.989521i \(-0.546121\pi\)
−0.144387 + 0.989521i \(0.546121\pi\)
\(8\) −1254.87 −0.108316
\(9\) 0 0
\(10\) 1988.39 0.0628785
\(11\) 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 \(0.722176\pi\)
\(14\) −2251.32 −0.0156625
\(15\) 0 0
\(16\) 259833. 0.991183
\(17\) 83521.0 0.242536
\(18\) 0 0
\(19\) −1603.79 −0.00282330 −0.00141165 0.999999i \(-0.500449\pi\)
−0.00141165 + 0.999999i \(0.500449\pi\)
\(20\) −827094. −1.15590
\(21\) 0 0
\(22\) 39001.6 0.0354961
\(23\) −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
\(26\) −162444. −0.0697147
\(27\) 0 0
\(28\) 936463. 0.287925
\(29\) −3.73267e6 −0.980005 −0.490002 0.871721i \(-0.663004\pi\)
−0.490002 + 0.871721i \(0.663004\pi\)
\(30\) 0 0
\(31\) 8.91479e6 1.73374 0.866869 0.498536i \(-0.166129\pi\)
0.866869 + 0.498536i \(0.166129\pi\)
\(32\) 961376. 0.162076
\(33\) 0 0
\(34\) 102502. 0.0131546
\(35\) −2.97211e6 −0.334779
\(36\) 0 0
\(37\) −1.20475e7 −1.05679 −0.528395 0.848999i \(-0.677206\pi\)
−0.528395 + 0.848999i \(0.677206\pi\)
\(38\) −1968.28 −0.000153130 0
\(39\) 0 0
\(40\) −2.03312e6 −0.125572
\(41\) 1.26779e7 0.700680 0.350340 0.936623i \(-0.386066\pi\)
0.350340 + 0.936623i \(0.386066\pi\)
\(42\) 0 0
\(43\) 2.86352e7 1.27730 0.638650 0.769498i \(-0.279493\pi\)
0.638650 + 0.769498i \(0.279493\pi\)
\(44\) −1.62231e7 −0.652526
\(45\) 0 0
\(46\) −28538.8 −0.000939777 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
\(50\) 824565. 0.0186578
\(51\) 0 0
\(52\) 6.75705e7 1.28157
\(53\) 5.96065e7 1.03765 0.518826 0.854880i \(-0.326369\pi\)
0.518826 + 0.854880i \(0.326369\pi\)
\(54\) 0 0
\(55\) 5.14884e7 0.758712
\(56\) 2.30196e6 0.0312790
\(57\) 0 0
\(58\) −4.58096e6 −0.0531534
\(59\) −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 \(0.875856\pi\)
\(62\) 1.09408e7 0.0940343
\(63\) 0 0
\(64\) −1.31855e8 −0.982393
\(65\) −2.14453e8 −1.49012
\(66\) 0 0
\(67\) −1.27030e8 −0.770137 −0.385069 0.922888i \(-0.625822\pi\)
−0.385069 + 0.922888i \(0.625822\pi\)
\(68\) −4.26370e7 −0.241822
\(69\) 0 0
\(70\) −3.64756e6 −0.0181577
\(71\) 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 \(0.599119\pi\)
\(74\) −1.47854e7 −0.0573181
\(75\) 0 0
\(76\) 818726. 0.00281500
\(77\) −5.82968e7 −0.188989
\(78\) 0 0
\(79\) −2.58778e8 −0.747491 −0.373746 0.927531i \(-0.621927\pi\)
−0.373746 + 0.927531i \(0.621927\pi\)
\(80\) 4.20977e8 1.14909
\(81\) 0 0
\(82\) 1.55591e7 0.0380034
\(83\) −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
\(86\) 3.51430e7 0.0692780
\(87\) 0 0
\(88\) −3.98789e7 −0.0708877
\(89\) −4.03936e8 −0.682428 −0.341214 0.939986i \(-0.610838\pi\)
−0.341214 + 0.939986i \(0.610838\pi\)
\(90\) 0 0
\(91\) 2.42810e8 0.371176
\(92\) 1.18710e7 0.0172760
\(93\) 0 0
\(94\) 8.80881e6 0.0116370
\(95\) −2.59844e6 −0.00327308
\(96\) 0 0
\(97\) −9.89973e8 −1.13540 −0.567702 0.823234i \(-0.692167\pi\)
−0.567702 + 0.823234i \(0.692167\pi\)
\(98\) −4.53946e7 −0.0497149
\(99\) 0 0
\(100\) −3.42987e8 −0.342987
\(101\) 1.11917e9 1.07017 0.535084 0.844799i \(-0.320280\pi\)
0.535084 + 0.844799i \(0.320280\pi\)
\(102\) 0 0
\(103\) 6.58796e6 0.00576744 0.00288372 0.999996i \(-0.499082\pi\)
0.00288372 + 0.999996i \(0.499082\pi\)
\(104\) 1.66098e8 0.139224
\(105\) 0 0
\(106\) 7.31528e7 0.0562801
\(107\) −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 \(0.424256\pi\)
\(110\) 6.31898e7 0.0411509
\(111\) 0 0
\(112\) −4.76644e8 −0.286228
\(113\) −2.09735e9 −1.21009 −0.605047 0.796190i \(-0.706846\pi\)
−0.605047 + 0.796190i \(0.706846\pi\)
\(114\) 0 0
\(115\) −3.76758e7 −0.0200873
\(116\) 1.90550e9 0.977122
\(117\) 0 0
\(118\) −2.28259e8 −0.108382
\(119\) −1.53213e8 −0.0700381
\(120\) 0 0
\(121\) −1.34802e9 −0.571693
\(122\) −2.45499e8 −0.100330
\(123\) 0 0
\(124\) −4.55095e9 −1.72864
\(125\) −2.07586e9 −0.760508
\(126\) 0 0
\(127\) −4.92229e9 −1.67900 −0.839499 0.543361i \(-0.817151\pi\)
−0.839499 + 0.543361i \(0.817151\pi\)
\(128\) −6.54045e8 −0.215359
\(129\) 0 0
\(130\) −2.63190e8 −0.0808209
\(131\) −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
\(134\) −1.55899e8 −0.0417706
\(135\) 0 0
\(136\) −1.04808e8 −0.0262705
\(137\) −5.25313e9 −1.27402 −0.637010 0.770856i \(-0.719829\pi\)
−0.637010 + 0.770856i \(0.719829\pi\)
\(138\) 0 0
\(139\) 1.55602e9 0.353549 0.176774 0.984251i \(-0.443434\pi\)
0.176774 + 0.984251i \(0.443434\pi\)
\(140\) 1.51724e9 0.333794
\(141\) 0 0
\(142\) 4.02371e8 0.0830479
\(143\) −4.20641e9 −0.841200
\(144\) 0 0
\(145\) −6.04761e9 −1.13613
\(146\) −1.82467e8 −0.0332351
\(147\) 0 0
\(148\) 6.15017e9 1.05368
\(149\) −2.13936e9 −0.355587 −0.177794 0.984068i \(-0.556896\pi\)
−0.177794 + 0.984068i \(0.556896\pi\)
\(150\) 0 0
\(151\) −1.65162e9 −0.258531 −0.129266 0.991610i \(-0.541262\pi\)
−0.129266 + 0.991610i \(0.541262\pi\)
\(152\) 2.01255e6 0.000305809 0
\(153\) 0 0
\(154\) −7.15455e7 −0.0102504
\(155\) 1.44436e10 2.00994
\(156\) 0 0
\(157\) −4.44203e9 −0.583490 −0.291745 0.956496i \(-0.594236\pi\)
−0.291745 + 0.956496i \(0.594236\pi\)
\(158\) −3.17589e8 −0.0405423
\(159\) 0 0
\(160\) 1.55761e9 0.187896
\(161\) 4.26577e7 0.00500358
\(162\) 0 0
\(163\) 2.66196e9 0.295363 0.147682 0.989035i \(-0.452819\pi\)
0.147682 + 0.989035i \(0.452819\pi\)
\(164\) −6.47198e9 −0.698618
\(165\) 0 0
\(166\) −4.23479e8 −0.0432858
\(167\) 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
\(170\) 1.66073e8 0.0152503
\(171\) 0 0
\(172\) −1.46181e10 −1.27354
\(173\) 1.27988e10 1.08633 0.543165 0.839626i \(-0.317226\pi\)
0.543165 + 0.839626i \(0.317226\pi\)
\(174\) 0 0
\(175\) −1.23250e9 −0.0993382
\(176\) 8.25731e9 0.648681
\(177\) 0 0
\(178\) −4.95735e8 −0.0370135
\(179\) 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 \(0.597433\pi\)
\(182\) 2.97992e8 0.0201318
\(183\) 0 0
\(184\) 2.91807e7 0.00187679
\(185\) −1.95192e10 −1.22515
\(186\) 0 0
\(187\) 2.65424e9 0.158728
\(188\) −3.66413e9 −0.213924
\(189\) 0 0
\(190\) −3.18897e6 −0.000177525 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 \(0.321838\pi\)
\(194\) −1.21496e9 −0.0615819
\(195\) 0 0
\(196\) 1.88824e10 0.913913
\(197\) −5.81009e9 −0.274843 −0.137422 0.990513i \(-0.543882\pi\)
−0.137422 + 0.990513i \(0.543882\pi\)
\(198\) 0 0
\(199\) 4.49526e9 0.203196 0.101598 0.994826i \(-0.467604\pi\)
0.101598 + 0.994826i \(0.467604\pi\)
\(200\) −8.43113e8 −0.0372607
\(201\) 0 0
\(202\) 1.37352e9 0.0580436
\(203\) 6.84730e9 0.283000
\(204\) 0 0
\(205\) 2.05405e10 0.812305
\(206\) 8.08516e6 0.000312814 0
\(207\) 0 0
\(208\) −3.43922e10 −1.27402
\(209\) −5.09674e7 −0.00184771
\(210\) 0 0
\(211\) 3.47865e10 1.20820 0.604101 0.796908i \(-0.293533\pi\)
0.604101 + 0.796908i \(0.293533\pi\)
\(212\) −3.04287e10 −1.03460
\(213\) 0 0
\(214\) −1.90950e9 −0.0622384
\(215\) 4.63944e10 1.48079
\(216\) 0 0
\(217\) −1.63535e10 −0.500659
\(218\) 8.52661e8 0.0255695
\(219\) 0 0
\(220\) −2.62845e10 −0.756480
\(221\) −1.10551e10 −0.311743
\(222\) 0 0
\(223\) −2.46348e10 −0.667078 −0.333539 0.942736i \(-0.608243\pi\)
−0.333539 + 0.942736i \(0.608243\pi\)
\(224\) −1.76357e9 −0.0468034
\(225\) 0 0
\(226\) −2.57401e9 −0.0656329
\(227\) −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 \(0.207393\pi\)
\(230\) −4.62381e7 −0.00108949
\(231\) 0 0
\(232\) 4.68401e9 0.106150
\(233\) 4.35721e10 0.968516 0.484258 0.874925i \(-0.339090\pi\)
0.484258 + 0.874925i \(0.339090\pi\)
\(234\) 0 0
\(235\) 1.16291e10 0.248736
\(236\) 9.49468e10 1.99240
\(237\) 0 0
\(238\) −1.88033e8 −0.00379872
\(239\) −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 \(0.387443\pi\)
\(242\) −1.65438e9 −0.0310074
\(243\) 0 0
\(244\) 1.02118e11 1.84437
\(245\) −5.99282e10 −1.06263
\(246\) 0 0
\(247\) 2.12283e8 0.00362893
\(248\) −1.11869e10 −0.187792
\(249\) 0 0
\(250\) −2.54763e9 −0.0412484
\(251\) 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
\(254\) −6.04094e9 −0.0910653
\(255\) 0 0
\(256\) 6.67068e10 0.970712
\(257\) −1.91619e10 −0.273993 −0.136996 0.990572i \(-0.543745\pi\)
−0.136996 + 0.990572i \(0.543745\pi\)
\(258\) 0 0
\(259\) 2.21002e10 0.305174
\(260\) 1.09477e11 1.48574
\(261\) 0 0
\(262\) −5.57918e9 −0.0731501
\(263\) −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
\(266\) 3.61065e6 4.42200e−5 0
\(267\) 0 0
\(268\) 6.48478e10 0.767872
\(269\) 9.36804e10 1.09085 0.545423 0.838161i \(-0.316369\pi\)
0.545423 + 0.838161i \(0.316369\pi\)
\(270\) 0 0
\(271\) −9.81985e10 −1.10597 −0.552985 0.833191i \(-0.686511\pi\)
−0.552985 + 0.833191i \(0.686511\pi\)
\(272\) 2.17015e10 0.240397
\(273\) 0 0
\(274\) −6.44698e9 −0.0691001
\(275\) 2.13517e10 0.225131
\(276\) 0 0
\(277\) 9.27606e9 0.0946683 0.0473341 0.998879i \(-0.484927\pi\)
0.0473341 + 0.998879i \(0.484927\pi\)
\(278\) 1.90965e9 0.0191757
\(279\) 0 0
\(280\) 3.72960e9 0.0362620
\(281\) 1.04165e11 0.996649 0.498325 0.866991i \(-0.333949\pi\)
0.498325 + 0.866991i \(0.333949\pi\)
\(282\) 0 0
\(283\) −9.57066e10 −0.886958 −0.443479 0.896285i \(-0.646256\pi\)
−0.443479 + 0.896285i \(0.646256\pi\)
\(284\) −1.67371e11 −1.52668
\(285\) 0 0
\(286\) −5.16237e9 −0.0456249
\(287\) −2.32566e10 −0.202338
\(288\) 0 0
\(289\) 6.97576e9 0.0588235
\(290\) −7.42201e9 −0.0616212
\(291\) 0 0
\(292\) 7.58993e10 0.610963
\(293\) 1.62664e11 1.28940 0.644698 0.764437i \(-0.276983\pi\)
0.644698 + 0.764437i \(0.276983\pi\)
\(294\) 0 0
\(295\) −3.01338e11 −2.31662
\(296\) 1.51180e10 0.114468
\(297\) 0 0
\(298\) −2.62556e9 −0.0192863
\(299\) 3.07797e9 0.0222712
\(300\) 0 0
\(301\) −5.25292e10 −0.368851
\(302\) −2.02697e9 −0.0140222
\(303\) 0 0
\(304\) −4.16718e8 −0.00279841
\(305\) −3.24098e11 −2.14450
\(306\) 0 0
\(307\) −2.26286e11 −1.45390 −0.726952 0.686688i \(-0.759064\pi\)
−0.726952 + 0.686688i \(0.759064\pi\)
\(308\) 2.97601e10 0.188433
\(309\) 0 0
\(310\) 1.77261e10 0.109015
\(311\) −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 \(0.182920\pi\)
\(314\) −5.45154e9 −0.0316472
\(315\) 0 0
\(316\) 1.32105e11 0.745292
\(317\) 9.98016e9 0.0555099 0.0277550 0.999615i \(-0.491164\pi\)
0.0277550 + 0.999615i \(0.491164\pi\)
\(318\) 0 0
\(319\) −1.18622e11 −0.641365
\(320\) −2.13629e11 −1.13890
\(321\) 0 0
\(322\) 5.23522e7 0.000271384 0
\(323\) −1.33950e8 −0.000684751 0
\(324\) 0 0
\(325\) −8.89312e10 −0.442159
\(326\) 3.26692e9 0.0160199
\(327\) 0 0
\(328\) −1.59091e10 −0.0758949
\(329\) −1.31668e10 −0.0619581
\(330\) 0 0
\(331\) −2.14841e11 −0.983766 −0.491883 0.870661i \(-0.663691\pi\)
−0.491883 + 0.870661i \(0.663691\pi\)
\(332\) 1.76151e11 0.795725
\(333\) 0 0
\(334\) 1.94308e10 0.0854342
\(335\) −2.05811e11 −0.892828
\(336\) 0 0
\(337\) −3.33228e11 −1.40737 −0.703683 0.710514i \(-0.748463\pi\)
−0.703683 + 0.710514i \(0.748463\pi\)
\(338\) 8.48710e9 0.0353699
\(339\) 0 0
\(340\) −6.90797e10 −0.280347
\(341\) 2.83306e11 1.13465
\(342\) 0 0
\(343\) 1.41878e11 0.553468
\(344\) −3.59335e10 −0.138352
\(345\) 0 0
\(346\) 1.57075e10 0.0589203
\(347\) 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 \(0.405876\pi\)
\(350\) −1.51260e9 −0.00538789
\(351\) 0 0
\(352\) 3.05519e10 0.106071
\(353\) −2.89170e11 −0.991212 −0.495606 0.868547i \(-0.665054\pi\)
−0.495606 + 0.868547i \(0.665054\pi\)
\(354\) 0 0
\(355\) 5.31194e11 1.77511
\(356\) 2.06207e11 0.680421
\(357\) 0 0
\(358\) 1.02265e10 0.0329042
\(359\) −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
\(362\) −1.06801e10 −0.0326878
\(363\) 0 0
\(364\) −1.23953e11 −0.370084
\(365\) −2.40886e11 −0.710385
\(366\) 0 0
\(367\) −1.08932e11 −0.313442 −0.156721 0.987643i \(-0.550092\pi\)
−0.156721 + 0.987643i \(0.550092\pi\)
\(368\) −6.04215e9 −0.0171742
\(369\) 0 0
\(370\) −2.39551e10 −0.0664494
\(371\) −1.09344e11 −0.299648
\(372\) 0 0
\(373\) −6.82746e11 −1.82629 −0.913145 0.407635i \(-0.866353\pi\)
−0.913145 + 0.407635i \(0.866353\pi\)
\(374\) 3.25745e9 0.00860906
\(375\) 0 0
\(376\) −9.00696e9 −0.0232398
\(377\) 4.94067e11 1.25965
\(378\) 0 0
\(379\) 1.87645e11 0.467156 0.233578 0.972338i \(-0.424957\pi\)
0.233578 + 0.972338i \(0.424957\pi\)
\(380\) 1.32649e9 0.00326345
\(381\) 0 0
\(382\) −2.81728e10 −0.0676932
\(383\) −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
\(386\) 2.51202e10 0.0575945
\(387\) 0 0
\(388\) 5.05375e11 1.13206
\(389\) 4.62050e11 1.02310 0.511548 0.859255i \(-0.329072\pi\)
0.511548 + 0.859255i \(0.329072\pi\)
\(390\) 0 0
\(391\) −1.94220e9 −0.00420240
\(392\) 4.64157e10 0.0992836
\(393\) 0 0
\(394\) −7.13051e9 −0.0149069
\(395\) −4.19269e11 −0.866574
\(396\) 0 0
\(397\) 8.72806e11 1.76344 0.881720 0.471774i \(-0.156386\pi\)
0.881720 + 0.471774i \(0.156386\pi\)
\(398\) 5.51687e9 0.0110209
\(399\) 0 0
\(400\) 1.74575e11 0.340966
\(401\) 1.68101e11 0.324654 0.162327 0.986737i \(-0.448100\pi\)
0.162327 + 0.986737i \(0.448100\pi\)
\(402\) 0 0
\(403\) −1.17999e12 −2.22846
\(404\) −5.71332e11 −1.06702
\(405\) 0 0
\(406\) 8.40343e9 0.0153493
\(407\) −3.82861e11 −0.691618
\(408\) 0 0
\(409\) −6.99127e11 −1.23538 −0.617691 0.786421i \(-0.711932\pi\)
−0.617691 + 0.786421i \(0.711932\pi\)
\(410\) 2.52086e10 0.0440577
\(411\) 0 0
\(412\) −3.36311e9 −0.00575047
\(413\) 3.41185e11 0.577052
\(414\) 0 0
\(415\) −5.59060e11 −0.925214
\(416\) −1.27251e11 −0.208324
\(417\) 0 0
\(418\) −6.25505e7 −0.000100216 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 \(0.639650\pi\)
\(422\) 4.26921e10 0.0655303
\(423\) 0 0
\(424\) −7.47983e10 −0.112395
\(425\) 5.61155e10 0.0834320
\(426\) 0 0
\(427\) 3.66954e11 0.534178
\(428\) 7.94279e11 1.14413
\(429\) 0 0
\(430\) 5.69381e10 0.0803147
\(431\) −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 \(0.661186\pi\)
\(434\) −2.00701e10 −0.0271547
\(435\) 0 0
\(436\) −3.54674e11 −0.470045
\(437\) 3.72946e7 4.89192e−5 0
\(438\) 0 0
\(439\) 1.17101e12 1.50477 0.752387 0.658721i \(-0.228902\pi\)
0.752387 + 0.658721i \(0.228902\pi\)
\(440\) −6.46111e10 −0.0821808
\(441\) 0 0
\(442\) −1.35675e10 −0.0169083
\(443\) 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
\(446\) −3.02333e10 −0.0361809
\(447\) 0 0
\(448\) 2.41877e11 0.283690
\(449\) 9.33820e11 1.08431 0.542156 0.840278i \(-0.317608\pi\)
0.542156 + 0.840278i \(0.317608\pi\)
\(450\) 0 0
\(451\) 4.02895e11 0.458561
\(452\) 1.07069e12 1.20653
\(453\) 0 0
\(454\) −5.46938e10 −0.0604209
\(455\) 3.93397e11 0.430308
\(456\) 0 0
\(457\) 1.28584e12 1.37900 0.689498 0.724288i \(-0.257831\pi\)
0.689498 + 0.724288i \(0.257831\pi\)
\(458\) 8.12223e10 0.0862543
\(459\) 0 0
\(460\) 1.92332e10 0.0200282
\(461\) −5.18444e11 −0.534623 −0.267311 0.963610i \(-0.586135\pi\)
−0.267311 + 0.963610i \(0.586135\pi\)
\(462\) 0 0
\(463\) −7.76808e11 −0.785596 −0.392798 0.919625i \(-0.628493\pi\)
−0.392798 + 0.919625i \(0.628493\pi\)
\(464\) −9.69869e11 −0.971364
\(465\) 0 0
\(466\) 5.34744e10 0.0525302
\(467\) −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
\(470\) 1.42719e10 0.0134909
\(471\) 0 0
\(472\) 2.33393e11 0.216446
\(473\) 9.10008e11 0.835930
\(474\) 0 0
\(475\) −1.07755e9 −0.000971213 0
\(476\) 7.82143e10 0.0698321
\(477\) 0 0
\(478\) −4.50562e10 −0.0394757
\(479\) 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
\(482\) 4.45120e10 0.0375634
\(483\) 0 0
\(484\) 6.88157e11 0.570011
\(485\) −1.60394e12 −1.31629
\(486\) 0 0
\(487\) −2.69427e11 −0.217051 −0.108525 0.994094i \(-0.534613\pi\)
−0.108525 + 0.994094i \(0.534613\pi\)
\(488\) 2.51021e11 0.200364
\(489\) 0 0
\(490\) −7.35477e10 −0.0576350
\(491\) 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
\(494\) 2.60527e8 0.000196826 0
\(495\) 0 0
\(496\) 2.31635e12 1.71845
\(497\) −6.01435e11 −0.442166
\(498\) 0 0
\(499\) −3.13142e11 −0.226094 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(500\) 1.05972e12 0.758271
\(501\) 0 0
\(502\) 1.14736e11 0.0806365
\(503\) −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
\(506\) −9.06943e8 −0.000615039 0
\(507\) 0 0
\(508\) 2.51280e12 1.67406
\(509\) −2.84601e12 −1.87935 −0.939673 0.342073i \(-0.888871\pi\)
−0.939673 + 0.342073i \(0.888871\pi\)
\(510\) 0 0
\(511\) 2.72739e11 0.176951
\(512\) 4.16738e11 0.268008
\(513\) 0 0
\(514\) −2.35167e10 −0.0148608
\(515\) 1.06737e10 0.00668625
\(516\) 0 0
\(517\) 2.28100e11 0.140416
\(518\) 2.71228e10 0.0165520
\(519\) 0 0
\(520\) 2.69110e11 0.161404
\(521\) 1.42270e10 0.00845949 0.00422974 0.999991i \(-0.498654\pi\)
0.00422974 + 0.999991i \(0.498654\pi\)
\(522\) 0 0
\(523\) 1.20853e12 0.706316 0.353158 0.935564i \(-0.385108\pi\)
0.353158 + 0.935564i \(0.385108\pi\)
\(524\) 2.32072e12 1.34472
\(525\) 0 0
\(526\) −4.60259e10 −0.0262160
\(527\) 7.44572e11 0.420493
\(528\) 0 0
\(529\) −1.80061e12 −0.999700
\(530\) 1.18521e11 0.0652460
\(531\) 0 0
\(532\) −1.50189e9 −0.000812899 0
\(533\) −1.67808e12 −0.900619
\(534\) 0 0
\(535\) −2.52085e12 −1.33032
\(536\) 1.59405e11 0.0834184
\(537\) 0 0
\(538\) 1.14971e11 0.0591652
\(539\) −1.17547e12 −0.599876
\(540\) 0 0
\(541\) 2.35337e12 1.18115 0.590573 0.806985i \(-0.298902\pi\)
0.590573 + 0.806985i \(0.298902\pi\)
\(542\) −1.20515e11 −0.0599854
\(543\) 0 0
\(544\) 8.02951e10 0.0393092
\(545\) 1.12565e12 0.546536
\(546\) 0 0
\(547\) 1.30236e12 0.621996 0.310998 0.950411i \(-0.399337\pi\)
0.310998 + 0.950411i \(0.399337\pi\)
\(548\) 2.68169e12 1.27027
\(549\) 0 0
\(550\) 2.62041e10 0.0122106
\(551\) 5.98642e9 0.00276685
\(552\) 0 0
\(553\) 4.74710e11 0.215856
\(554\) 1.13842e10 0.00513461
\(555\) 0 0
\(556\) −7.94340e11 −0.352509
\(557\) −3.33519e12 −1.46816 −0.734078 0.679065i \(-0.762385\pi\)
−0.734078 + 0.679065i \(0.762385\pi\)
\(558\) 0 0
\(559\) −3.79025e12 −1.64178
\(560\) −7.72251e11 −0.331827
\(561\) 0 0
\(562\) 1.27838e11 0.0540561
\(563\) 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
\(566\) −1.17457e11 −0.0481067
\(567\) 0 0
\(568\) −4.11422e11 −0.165852
\(569\) 8.14899e10 0.0325911 0.0162955 0.999867i \(-0.494813\pi\)
0.0162955 + 0.999867i \(0.494813\pi\)
\(570\) 0 0
\(571\) 1.06368e12 0.418744 0.209372 0.977836i \(-0.432858\pi\)
0.209372 + 0.977836i \(0.432858\pi\)
\(572\) 2.14734e12 0.838725
\(573\) 0 0
\(574\) −2.85420e10 −0.0109744
\(575\) −1.56237e10 −0.00596046
\(576\) 0 0
\(577\) −3.44925e12 −1.29549 −0.647743 0.761859i \(-0.724287\pi\)
−0.647743 + 0.761859i \(0.724287\pi\)
\(578\) 8.56109e9 0.00319046
\(579\) 0 0
\(580\) 3.08727e12 1.13279
\(581\) 6.32986e11 0.230463
\(582\) 0 0
\(583\) 1.89425e12 0.679093
\(584\) 1.86572e11 0.0663725
\(585\) 0 0
\(586\) 1.99631e11 0.0699341
\(587\) 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
\(590\) −3.69821e11 −0.125649
\(591\) 0 0
\(592\) −3.13033e12 −1.04747
\(593\) −9.22763e10 −0.0306439 −0.0153219 0.999883i \(-0.504877\pi\)
−0.0153219 + 0.999883i \(0.504877\pi\)
\(594\) 0 0
\(595\) −2.48233e11 −0.0811958
\(596\) 1.09213e12 0.354541
\(597\) 0 0
\(598\) 3.77748e9 0.00120794
\(599\) 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 \(0.348876\pi\)
\(602\) −6.44671e10 −0.0200057
\(603\) 0 0
\(604\) 8.43140e11 0.257771
\(605\) −2.18405e12 −0.662770
\(606\) 0 0
\(607\) −1.45237e12 −0.434237 −0.217118 0.976145i \(-0.569666\pi\)
−0.217118 + 0.976145i \(0.569666\pi\)
\(608\) −1.54185e9 −0.000457589 0
\(609\) 0 0
\(610\) −3.97753e11 −0.116313
\(611\) −9.50050e11 −0.275779
\(612\) 0 0
\(613\) 5.13743e12 1.46951 0.734757 0.678331i \(-0.237296\pi\)
0.734757 + 0.678331i \(0.237296\pi\)
\(614\) −2.77713e11 −0.0788567
\(615\) 0 0
\(616\) 7.31548e10 0.0204706
\(617\) 5.31905e12 1.47758 0.738790 0.673936i \(-0.235397\pi\)
0.738790 + 0.673936i \(0.235397\pi\)
\(618\) 0 0
\(619\) −3.23470e12 −0.885577 −0.442789 0.896626i \(-0.646011\pi\)
−0.442789 + 0.896626i \(0.646011\pi\)
\(620\) −7.37337e12 −2.00403
\(621\) 0 0
\(622\) −1.43866e11 −0.0385391
\(623\) 7.40989e11 0.197068
\(624\) 0 0
\(625\) −4.67554e12 −1.22566
\(626\) 3.49844e11 0.0910521
\(627\) 0 0
\(628\) 2.26763e12 0.581773
\(629\) −1.00622e12 −0.256309
\(630\) 0 0
\(631\) −7.91021e12 −1.98635 −0.993175 0.116630i \(-0.962791\pi\)
−0.993175 + 0.116630i \(0.962791\pi\)
\(632\) 3.24733e11 0.0809654
\(633\) 0 0
\(634\) 1.22483e10 0.00301074
\(635\) −7.97501e12 −1.94648
\(636\) 0 0
\(637\) 4.89591e12 1.17816
\(638\) −1.45580e11 −0.0347863
\(639\) 0 0
\(640\) −1.05967e12 −0.249668
\(641\) 2.28438e12 0.534451 0.267225 0.963634i \(-0.413893\pi\)
0.267225 + 0.963634i \(0.413893\pi\)
\(642\) 0 0
\(643\) 7.52203e11 0.173534 0.0867672 0.996229i \(-0.472346\pi\)
0.0867672 + 0.996229i \(0.472346\pi\)
\(644\) −2.17765e10 −0.00498886
\(645\) 0 0
\(646\) −1.64392e8 −3.71394e−5 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
\(650\) −1.09142e11 −0.0239818
\(651\) 0 0
\(652\) −1.35891e12 −0.294495
\(653\) −4.27664e12 −0.920436 −0.460218 0.887806i \(-0.652229\pi\)
−0.460218 + 0.887806i \(0.652229\pi\)
\(654\) 0 0
\(655\) −7.36542e12 −1.56355
\(656\) 3.29413e12 0.694502
\(657\) 0 0
\(658\) −1.61591e10 −0.00336048
\(659\) 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 \(0.349947\pi\)
\(662\) −2.63667e11 −0.0533574
\(663\) 0 0
\(664\) 4.33005e11 0.0864442
\(665\) 4.76664e9 0.000945182 0
\(666\) 0 0
\(667\) 8.67994e10 0.0169805
\(668\) −8.08246e12 −1.57054
\(669\) 0 0
\(670\) −2.52585e11 −0.0484251
\(671\) −6.35705e12 −1.21061
\(672\) 0 0
\(673\) 3.90397e12 0.733566 0.366783 0.930307i \(-0.380459\pi\)
0.366783 + 0.930307i \(0.380459\pi\)
\(674\) −4.08959e11 −0.0763326
\(675\) 0 0
\(676\) −3.53030e12 −0.650207
\(677\) −4.84414e12 −0.886273 −0.443137 0.896454i \(-0.646134\pi\)
−0.443137 + 0.896454i \(0.646134\pi\)
\(678\) 0 0
\(679\) 1.81603e12 0.327876
\(680\) −1.69808e11 −0.0304557
\(681\) 0 0
\(682\) 3.47691e11 0.0615409
\(683\) 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
\(686\) 1.74122e11 0.0300189
\(687\) 0 0
\(688\) 7.44037e12 1.26604
\(689\) −7.88969e12 −1.33375
\(690\) 0 0
\(691\) 3.94359e12 0.658023 0.329011 0.944326i \(-0.393285\pi\)
0.329011 + 0.944326i \(0.393285\pi\)
\(692\) −6.53371e12 −1.08314
\(693\) 0 0
\(694\) 2.00340e10 0.00327831
\(695\) 2.52104e12 0.409873
\(696\) 0 0
\(697\) 1.05887e12 0.169940
\(698\) 1.98238e11 0.0316109
\(699\) 0 0
\(700\) 6.29184e11 0.0990459
\(701\) 9.80834e12 1.53414 0.767069 0.641564i \(-0.221714\pi\)
0.767069 + 0.641564i \(0.221714\pi\)
\(702\) 0 0
\(703\) 1.93217e10 0.00298364
\(704\) −4.19025e12 −0.642928
\(705\) 0 0
\(706\) −3.54887e11 −0.0537613
\(707\) −2.05304e12 −0.309037
\(708\) 0 0
\(709\) 4.04313e12 0.600910 0.300455 0.953796i \(-0.402861\pi\)
0.300455 + 0.953796i \(0.402861\pi\)
\(710\) 6.51915e11 0.0962783
\(711\) 0 0
\(712\) 5.06886e11 0.0739180
\(713\) −2.07305e11 −0.0300404
\(714\) 0 0
\(715\) −6.81515e12 −0.975211
\(716\) −4.25381e12 −0.604880
\(717\) 0 0
\(718\) −2.87327e11 −0.0403475
\(719\) 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
\(722\) −3.96020e11 −0.0542374
\(723\) 0 0
\(724\) 4.44250e12 0.600903
\(725\) −2.50788e12 −0.337121
\(726\) 0 0
\(727\) −1.02503e13 −1.36092 −0.680459 0.732786i \(-0.738220\pi\)
−0.680459 + 0.732786i \(0.738220\pi\)
\(728\) −3.04695e11 −0.0402044
\(729\) 0 0
\(730\) −2.95631e11 −0.0385298
\(731\) 2.39164e12 0.309791
\(732\) 0 0
\(733\) −1.77332e12 −0.226892 −0.113446 0.993544i \(-0.536189\pi\)
−0.113446 + 0.993544i \(0.536189\pi\)
\(734\) −1.33688e11 −0.0170004
\(735\) 0 0
\(736\) −2.23558e10 −0.00280828
\(737\) −4.03691e12 −0.504018
\(738\) 0 0
\(739\) 2.07204e12 0.255563 0.127781 0.991802i \(-0.459214\pi\)
0.127781 + 0.991802i \(0.459214\pi\)
\(740\) 9.96441e12 1.22154
\(741\) 0 0
\(742\) −1.34193e11 −0.0162523
\(743\) −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
\(746\) −8.37909e11 −0.0990541
\(747\) 0 0
\(748\) −1.35497e12 −0.158261
\(749\) 2.85419e12 0.331371
\(750\) 0 0
\(751\) −7.56877e12 −0.868252 −0.434126 0.900852i \(-0.642943\pi\)
−0.434126 + 0.900852i \(0.642943\pi\)
\(752\) 1.86498e12 0.212664
\(753\) 0 0
\(754\) 6.06350e11 0.0683207
\(755\) −2.67592e12 −0.299718
\(756\) 0 0
\(757\) −6.86633e12 −0.759965 −0.379982 0.924994i \(-0.624070\pi\)
−0.379982 + 0.924994i \(0.624070\pi\)
\(758\) 2.30290e11 0.0253375
\(759\) 0 0
\(760\) 3.26070e9 0.000354528 0
\(761\) −9.51816e12 −1.02878 −0.514389 0.857557i \(-0.671981\pi\)
−0.514389 + 0.857557i \(0.671981\pi\)
\(762\) 0 0
\(763\) −1.27450e12 −0.136138
\(764\) 1.17188e13 1.24441
\(765\) 0 0
\(766\) −1.24513e11 −0.0130672
\(767\) 2.46182e13 2.56849
\(768\) 0 0
\(769\) −6.30061e12 −0.649702 −0.324851 0.945765i \(-0.605314\pi\)
−0.324851 + 0.945765i \(0.605314\pi\)
\(770\) −1.15917e11 −0.0118833
\(771\) 0 0
\(772\) −1.04490e13 −1.05876
\(773\) 1.40318e13 1.41354 0.706768 0.707446i \(-0.250153\pi\)
0.706768 + 0.707446i \(0.250153\pi\)
\(774\) 0 0
\(775\) 5.98961e12 0.596404
\(776\) 1.24229e12 0.122983
\(777\) 0 0
\(778\) 5.67057e11 0.0554905
\(779\) −2.03327e10 −0.00197823
\(780\) 0 0
\(781\) 1.04192e13 1.00208
\(782\) −2.38359e9 −0.000227930 0
\(783\) 0 0
\(784\) −9.61082e12 −0.908528
\(785\) −7.19691e12 −0.676445
\(786\) 0 0
\(787\) 1.03038e13 0.957440 0.478720 0.877968i \(-0.341101\pi\)
0.478720 + 0.877968i \(0.341101\pi\)
\(788\) 2.96602e12 0.274035
\(789\) 0 0
\(790\) −5.14553e11 −0.0470011
\(791\) 3.84744e12 0.349444
\(792\) 0 0
\(793\) 2.64776e13 2.37765
\(794\) 1.07116e12 0.0956452
\(795\) 0 0
\(796\) −2.29480e12 −0.202599
\(797\) −1.57641e13 −1.38390 −0.691952 0.721944i \(-0.743249\pi\)
−0.691952 + 0.721944i \(0.743249\pi\)
\(798\) 0 0
\(799\) 5.99481e11 0.0520373
\(800\) 6.45923e11 0.0557540
\(801\) 0 0
\(802\) 2.06304e11 0.0176086
\(803\) −4.72489e12 −0.401026
\(804\) 0 0
\(805\) 6.91134e10 0.00580070
\(806\) −1.44816e12 −0.120867
\(807\) 0 0
\(808\) −1.40442e12 −0.115916
\(809\) −1.37214e13 −1.12623 −0.563117 0.826377i \(-0.690398\pi\)
−0.563117 + 0.826377i \(0.690398\pi\)
\(810\) 0 0
\(811\) 1.55198e13 1.25977 0.629887 0.776687i \(-0.283101\pi\)
0.629887 + 0.776687i \(0.283101\pi\)
\(812\) −3.49550e12 −0.282168
\(813\) 0 0
\(814\) −4.69871e11 −0.0375119
\(815\) 4.31286e12 0.342418
\(816\) 0 0
\(817\) −4.59250e10 −0.00360620
\(818\) −8.58013e11 −0.0670045
\(819\) 0 0
\(820\) −1.04858e13 −0.809915
\(821\) −3.40134e12 −0.261280 −0.130640 0.991430i \(-0.541703\pi\)
−0.130640 + 0.991430i \(0.541703\pi\)
\(822\) 0 0
\(823\) 4.01848e12 0.305325 0.152663 0.988278i \(-0.451215\pi\)
0.152663 + 0.988278i \(0.451215\pi\)
\(824\) −8.26702e9 −0.000624707 0
\(825\) 0 0
\(826\) 4.18724e11 0.0312981
\(827\) 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 \(0.448010\pi\)
\(830\) −6.86114e11 −0.0501816
\(831\) 0 0
\(832\) 1.74527e13 1.26272
\(833\) −3.08932e12 −0.222310
\(834\) 0 0
\(835\) 2.56518e13 1.82612
\(836\) 2.60186e10 0.00184228
\(837\) 0 0
\(838\) −1.18854e12 −0.0832563
\(839\) −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
\(842\) −6.72056e11 −0.0460788
\(843\) 0 0
\(844\) −1.77583e13 −1.20465
\(845\) 1.12043e13 0.756016
\(846\) 0 0
\(847\) 2.47285e12 0.165090
\(848\) 1.54877e13 1.02850
\(849\) 0 0
\(850\) 6.88685e10 0.00452518
\(851\) 2.80152e11 0.0183110
\(852\) 0 0
\(853\) −3.07683e12 −0.198991 −0.0994954 0.995038i \(-0.531723\pi\)
−0.0994954 + 0.995038i \(0.531723\pi\)
\(854\) 4.50349e11 0.0289727
\(855\) 0 0
\(856\) 1.95246e12 0.124294
\(857\) 3.10840e13 1.96845 0.984223 0.176930i \(-0.0566166\pi\)
0.984223 + 0.176930i \(0.0566166\pi\)
\(858\) 0 0
\(859\) −7.61893e12 −0.477446 −0.238723 0.971088i \(-0.576729\pi\)
−0.238723 + 0.971088i \(0.576729\pi\)
\(860\) −2.36840e13 −1.47643
\(861\) 0 0
\(862\) −5.89908e11 −0.0363917
\(863\) 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
\(866\) −8.70801e11 −0.0526124
\(867\) 0 0
\(868\) 8.34837e12 0.499186
\(869\) −8.22380e12 −0.489197
\(870\) 0 0
\(871\) 1.68140e13 0.989896
\(872\) −8.71840e11 −0.0510638
\(873\) 0 0
\(874\) 4.57703e7 2.65327e−6 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 \(0.828333\pi\)
\(878\) 1.43714e12 0.0816158
\(879\) 0 0
\(880\) 1.33784e13 0.752023
\(881\) 2.06409e13 1.15435 0.577175 0.816620i \(-0.304155\pi\)
0.577175 + 0.816620i \(0.304155\pi\)
\(882\) 0 0
\(883\) 1.85711e13 1.02805 0.514026 0.857774i \(-0.328153\pi\)
0.514026 + 0.857774i \(0.328153\pi\)
\(884\) 5.64356e12 0.310826
\(885\) 0 0
\(886\) 5.83017e11 0.0317855
\(887\) −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
\(890\) −8.03182e11 −0.0429101
\(891\) 0 0
\(892\) 1.25759e13 0.665115
\(893\) −1.15114e10 −0.000605754 0
\(894\) 0 0
\(895\) 1.35006e13 0.703313
\(896\) 1.19980e12 0.0621901
\(897\) 0 0
\(898\) 1.14604e12 0.0588108
\(899\) −3.32759e13 −1.69907
\(900\) 0 0
\(901\) 4.97839e12 0.251668
\(902\) 4.94458e11 0.0248714
\(903\) 0 0
\(904\) 2.63190e12 0.131073
\(905\) −1.40994e13 −0.698688
\(906\) 0 0
\(907\) −1.36972e13 −0.672045 −0.336022 0.941854i \(-0.609082\pi\)
−0.336022 + 0.941854i \(0.609082\pi\)
\(908\) 2.27505e13 1.11072
\(909\) 0 0
\(910\) 4.82802e11 0.0233390
\(911\) 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
\(914\) 1.57806e12 0.0747938
\(915\) 0 0
\(916\) −3.37853e13 −1.58562
\(917\) 8.33936e12 0.389467
\(918\) 0 0
\(919\) 9.66637e12 0.447037 0.223519 0.974700i \(-0.428246\pi\)
0.223519 + 0.974700i \(0.428246\pi\)
\(920\) 4.72781e10 0.00217578
\(921\) 0 0
\(922\) −6.36267e11 −0.0289968
\(923\) −4.33966e13 −1.96810
\(924\) 0 0
\(925\) −8.09439e12 −0.363535
\(926\) −9.53348e11 −0.0426091
\(927\) 0 0
\(928\) −3.58850e12 −0.158835
\(929\) −8.19216e12 −0.360851 −0.180425 0.983589i \(-0.557747\pi\)
−0.180425 + 0.983589i \(0.557747\pi\)
\(930\) 0 0
\(931\) 5.93219e10 0.00258786
\(932\) −2.22433e13 −0.965667
\(933\) 0 0
\(934\) −1.01712e11 −0.00437330
\(935\) 4.30036e12 0.184015
\(936\) 0 0
\(937\) 3.49872e12 0.148279 0.0741396 0.997248i \(-0.476379\pi\)
0.0741396 + 0.997248i \(0.476379\pi\)
\(938\) 2.85984e11 0.0120623
\(939\) 0 0
\(940\) −5.93656e12 −0.248004
\(941\) 1.51926e12 0.0631655 0.0315827 0.999501i \(-0.489945\pi\)
0.0315827 + 0.999501i \(0.489945\pi\)
\(942\) 0 0
\(943\) −2.94811e11 −0.0121406
\(944\) −4.83263e13 −1.98066
\(945\) 0 0
\(946\) 1.11682e12 0.0453391
\(947\) 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
\(950\) −1.32243e9 −5.26765e−5 0
\(951\) 0 0
\(952\) 1.92262e11 0.00758626
\(953\) −9.61359e12 −0.377544 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(954\) 0 0
\(955\) −3.71927e13 −1.44691
\(956\) 1.87417e13 0.725684
\(957\) 0 0
\(958\) 2.55553e12 0.0980249
\(959\) 9.63648e12 0.367904
\(960\) 0 0
\(961\) 5.30339e13 2.00585
\(962\) 1.95705e12 0.0736738
\(963\) 0 0
\(964\) −1.85153e13 −0.690531
\(965\) 3.31627e13 1.23106
\(966\) 0 0
\(967\) −1.08700e13 −0.399772 −0.199886 0.979819i \(-0.564057\pi\)
−0.199886 + 0.979819i \(0.564057\pi\)
\(968\) 1.69159e12 0.0619236
\(969\) 0 0
\(970\) −1.96846e12 −0.0713925
\(971\) −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
\(974\) −3.30658e11 −0.0117724
\(975\) 0 0
\(976\) −5.19763e13 −1.83350
\(977\) 1.49620e13 0.525369 0.262685 0.964882i \(-0.415392\pi\)
0.262685 + 0.964882i \(0.415392\pi\)
\(978\) 0 0
\(979\) −1.28368e13 −0.446616
\(980\) 3.05930e13 1.05951
\(981\) 0 0
\(982\) 7.21350e10 0.00247540
\(983\) −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
\(986\) −3.82607e11 −0.0128916
\(987\) 0 0
\(988\) −1.08369e11 −0.00361826
\(989\) −6.65884e11 −0.0221317
\(990\) 0 0
\(991\) −9.59893e12 −0.316149 −0.158074 0.987427i \(-0.550529\pi\)
−0.158074 + 0.987427i \(0.550529\pi\)
\(992\) 8.57047e12 0.280997
\(993\) 0 0
\(994\) −7.38119e11 −0.0239821
\(995\) 7.28315e12 0.235568
\(996\) 0 0
\(997\) −3.17533e13 −1.01780 −0.508898 0.860827i \(-0.669947\pi\)
−0.508898 + 0.860827i \(0.669947\pi\)
\(998\) −3.84308e11 −0.0122629
\(999\) 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 153.10.a.c.1.3 5
3.2 odd 2 17.10.a.a.1.3 5
12.11 even 2 272.10.a.f.1.1 5
51.50 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 3.2 odd 2
153.10.a.c.1.3 5 1.1 even 1 trivial
272.10.a.f.1.1 5 12.11 even 2
289.10.a.a.1.3 5 51.50 odd 2