Properties

Label 546.8.a.a.1.1
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -135.000 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -135.000 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -1080.00 q^{10} +4197.00 q^{11} -1728.00 q^{12} +2197.00 q^{13} +2744.00 q^{14} +3645.00 q^{15} +4096.00 q^{16} -12735.0 q^{17} +5832.00 q^{18} -28213.0 q^{19} -8640.00 q^{20} -9261.00 q^{21} +33576.0 q^{22} -57039.0 q^{23} -13824.0 q^{24} -59900.0 q^{25} +17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} -10269.0 q^{29} +29160.0 q^{30} +98276.0 q^{31} +32768.0 q^{32} -113319. q^{33} -101880. q^{34} -46305.0 q^{35} +46656.0 q^{36} -352033. q^{37} -225704. q^{38} -59319.0 q^{39} -69120.0 q^{40} +473172. q^{41} -74088.0 q^{42} +891395. q^{43} +268608. q^{44} -98415.0 q^{45} -456312. q^{46} +684984. q^{47} -110592. q^{48} +117649. q^{49} -479200. q^{50} +343845. q^{51} +140608. q^{52} +271002. q^{53} -157464. q^{54} -566595. q^{55} +175616. q^{56} +761751. q^{57} -82152.0 q^{58} -954024. q^{59} +233280. q^{60} +3.19716e6 q^{61} +786208. q^{62} +250047. q^{63} +262144. q^{64} -296595. q^{65} -906552. q^{66} -2.90202e6 q^{67} -815040. q^{68} +1.54005e6 q^{69} -370440. q^{70} -4.59977e6 q^{71} +373248. q^{72} +1.27711e6 q^{73} -2.81626e6 q^{74} +1.61730e6 q^{75} -1.80563e6 q^{76} +1.43957e6 q^{77} -474552. q^{78} -1.92849e6 q^{79} -552960. q^{80} +531441. q^{81} +3.78538e6 q^{82} -355674. q^{83} -592704. q^{84} +1.71922e6 q^{85} +7.13116e6 q^{86} +277263. q^{87} +2.14886e6 q^{88} -217854. q^{89} -787320. q^{90} +753571. q^{91} -3.65050e6 q^{92} -2.65345e6 q^{93} +5.47987e6 q^{94} +3.80876e6 q^{95} -884736. q^{96} -4.40076e6 q^{97} +941192. q^{98} +3.05961e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) −135.000 −0.482991 −0.241495 0.970402i \(-0.577638\pi\)
−0.241495 + 0.970402i \(0.577638\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −1080.00 −0.341526
\(11\) 4197.00 0.950746 0.475373 0.879784i \(-0.342313\pi\)
0.475373 + 0.879784i \(0.342313\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) 2744.00 0.267261
\(15\) 3645.00 0.278855
\(16\) 4096.00 0.250000
\(17\) −12735.0 −0.628677 −0.314339 0.949311i \(-0.601783\pi\)
−0.314339 + 0.949311i \(0.601783\pi\)
\(18\) 5832.00 0.235702
\(19\) −28213.0 −0.943652 −0.471826 0.881692i \(-0.656405\pi\)
−0.471826 + 0.881692i \(0.656405\pi\)
\(20\) −8640.00 −0.241495
\(21\) −9261.00 −0.218218
\(22\) 33576.0 0.672279
\(23\) −57039.0 −0.977517 −0.488759 0.872419i \(-0.662550\pi\)
−0.488759 + 0.872419i \(0.662550\pi\)
\(24\) −13824.0 −0.204124
\(25\) −59900.0 −0.766720
\(26\) 17576.0 0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) −10269.0 −0.0781871 −0.0390936 0.999236i \(-0.512447\pi\)
−0.0390936 + 0.999236i \(0.512447\pi\)
\(30\) 29160.0 0.197180
\(31\) 98276.0 0.592491 0.296245 0.955112i \(-0.404265\pi\)
0.296245 + 0.955112i \(0.404265\pi\)
\(32\) 32768.0 0.176777
\(33\) −113319. −0.548914
\(34\) −101880. −0.444542
\(35\) −46305.0 −0.182553
\(36\) 46656.0 0.166667
\(37\) −352033. −1.14256 −0.571278 0.820757i \(-0.693552\pi\)
−0.571278 + 0.820757i \(0.693552\pi\)
\(38\) −225704. −0.667262
\(39\) −59319.0 −0.160128
\(40\) −69120.0 −0.170763
\(41\) 473172. 1.07220 0.536100 0.844155i \(-0.319897\pi\)
0.536100 + 0.844155i \(0.319897\pi\)
\(42\) −74088.0 −0.154303
\(43\) 891395. 1.70974 0.854871 0.518841i \(-0.173636\pi\)
0.854871 + 0.518841i \(0.173636\pi\)
\(44\) 268608. 0.475373
\(45\) −98415.0 −0.160997
\(46\) −456312. −0.691209
\(47\) 684984. 0.962361 0.481180 0.876622i \(-0.340208\pi\)
0.481180 + 0.876622i \(0.340208\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) −479200. −0.542153
\(51\) 343845. 0.362967
\(52\) 140608. 0.138675
\(53\) 271002. 0.250039 0.125019 0.992154i \(-0.460101\pi\)
0.125019 + 0.992154i \(0.460101\pi\)
\(54\) −157464. −0.136083
\(55\) −566595. −0.459202
\(56\) 175616. 0.133631
\(57\) 761751. 0.544817
\(58\) −82152.0 −0.0552866
\(59\) −954024. −0.604752 −0.302376 0.953189i \(-0.597780\pi\)
−0.302376 + 0.953189i \(0.597780\pi\)
\(60\) 233280. 0.139427
\(61\) 3.19716e6 1.80347 0.901737 0.432285i \(-0.142293\pi\)
0.901737 + 0.432285i \(0.142293\pi\)
\(62\) 786208. 0.418954
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −296595. −0.133958
\(66\) −906552. −0.388140
\(67\) −2.90202e6 −1.17880 −0.589398 0.807843i \(-0.700635\pi\)
−0.589398 + 0.807843i \(0.700635\pi\)
\(68\) −815040. −0.314339
\(69\) 1.54005e6 0.564370
\(70\) −370440. −0.129085
\(71\) −4.59977e6 −1.52522 −0.762609 0.646860i \(-0.776082\pi\)
−0.762609 + 0.646860i \(0.776082\pi\)
\(72\) 373248. 0.117851
\(73\) 1.27711e6 0.384237 0.192118 0.981372i \(-0.438464\pi\)
0.192118 + 0.981372i \(0.438464\pi\)
\(74\) −2.81626e6 −0.807909
\(75\) 1.61730e6 0.442666
\(76\) −1.80563e6 −0.471826
\(77\) 1.43957e6 0.359348
\(78\) −474552. −0.113228
\(79\) −1.92849e6 −0.440072 −0.220036 0.975492i \(-0.570617\pi\)
−0.220036 + 0.975492i \(0.570617\pi\)
\(80\) −552960. −0.120748
\(81\) 531441. 0.111111
\(82\) 3.78538e6 0.758159
\(83\) −355674. −0.0682777 −0.0341388 0.999417i \(-0.510869\pi\)
−0.0341388 + 0.999417i \(0.510869\pi\)
\(84\) −592704. −0.109109
\(85\) 1.71922e6 0.303645
\(86\) 7.13116e6 1.20897
\(87\) 277263. 0.0451413
\(88\) 2.14886e6 0.336140
\(89\) −217854. −0.0327567 −0.0163784 0.999866i \(-0.505214\pi\)
−0.0163784 + 0.999866i \(0.505214\pi\)
\(90\) −787320. −0.113842
\(91\) 753571. 0.104828
\(92\) −3.65050e6 −0.488759
\(93\) −2.65345e6 −0.342075
\(94\) 5.47987e6 0.680492
\(95\) 3.80876e6 0.455775
\(96\) −884736. −0.102062
\(97\) −4.40076e6 −0.489583 −0.244792 0.969576i \(-0.578720\pi\)
−0.244792 + 0.969576i \(0.578720\pi\)
\(98\) 941192. 0.101015
\(99\) 3.05961e6 0.316915
\(100\) −3.83360e6 −0.383360
\(101\) −1.30268e7 −1.25810 −0.629049 0.777366i \(-0.716555\pi\)
−0.629049 + 0.777366i \(0.716555\pi\)
\(102\) 2.75076e6 0.256656
\(103\) −1.20237e7 −1.08420 −0.542098 0.840316i \(-0.682370\pi\)
−0.542098 + 0.840316i \(0.682370\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 1.25024e6 0.105397
\(106\) 2.16802e6 0.176804
\(107\) −1.08829e7 −0.858821 −0.429410 0.903110i \(-0.641279\pi\)
−0.429410 + 0.903110i \(0.641279\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) −2.15309e7 −1.59246 −0.796230 0.604994i \(-0.793175\pi\)
−0.796230 + 0.604994i \(0.793175\pi\)
\(110\) −4.53276e6 −0.324705
\(111\) 9.50489e6 0.659655
\(112\) 1.40493e6 0.0944911
\(113\) −1.51964e7 −0.990755 −0.495378 0.868678i \(-0.664970\pi\)
−0.495378 + 0.868678i \(0.664970\pi\)
\(114\) 6.09401e6 0.385244
\(115\) 7.70026e6 0.472132
\(116\) −657216. −0.0390936
\(117\) 1.60161e6 0.0924500
\(118\) −7.63219e6 −0.427624
\(119\) −4.36810e6 −0.237618
\(120\) 1.86624e6 0.0985901
\(121\) −1.87236e6 −0.0960818
\(122\) 2.55773e7 1.27525
\(123\) −1.27756e7 −0.619035
\(124\) 6.28966e6 0.296245
\(125\) 1.86334e7 0.853309
\(126\) 2.00038e6 0.0890871
\(127\) 8.10709e6 0.351198 0.175599 0.984462i \(-0.443814\pi\)
0.175599 + 0.984462i \(0.443814\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −2.40677e7 −0.987120
\(130\) −2.37276e6 −0.0947223
\(131\) 8.57031e6 0.333079 0.166539 0.986035i \(-0.446741\pi\)
0.166539 + 0.986035i \(0.446741\pi\)
\(132\) −7.25242e6 −0.274457
\(133\) −9.67706e6 −0.356667
\(134\) −2.32161e7 −0.833534
\(135\) 2.65720e6 0.0929516
\(136\) −6.52032e6 −0.222271
\(137\) −2.62757e7 −0.873036 −0.436518 0.899695i \(-0.643789\pi\)
−0.436518 + 0.899695i \(0.643789\pi\)
\(138\) 1.23204e7 0.399070
\(139\) 725828. 0.0229236 0.0114618 0.999934i \(-0.496352\pi\)
0.0114618 + 0.999934i \(0.496352\pi\)
\(140\) −2.96352e6 −0.0912767
\(141\) −1.84946e7 −0.555619
\(142\) −3.67981e7 −1.07849
\(143\) 9.22081e6 0.263690
\(144\) 2.98598e6 0.0833333
\(145\) 1.38632e6 0.0377636
\(146\) 1.02169e7 0.271696
\(147\) −3.17652e6 −0.0824786
\(148\) −2.25301e7 −0.571278
\(149\) 1.90708e7 0.472299 0.236150 0.971717i \(-0.424114\pi\)
0.236150 + 0.971717i \(0.424114\pi\)
\(150\) 1.29384e7 0.313012
\(151\) −7.41798e7 −1.75334 −0.876671 0.481091i \(-0.840241\pi\)
−0.876671 + 0.481091i \(0.840241\pi\)
\(152\) −1.44451e7 −0.333631
\(153\) −9.28382e6 −0.209559
\(154\) 1.15166e7 0.254098
\(155\) −1.32673e7 −0.286167
\(156\) −3.79642e6 −0.0800641
\(157\) 2.24399e7 0.462777 0.231388 0.972861i \(-0.425673\pi\)
0.231388 + 0.972861i \(0.425673\pi\)
\(158\) −1.54280e7 −0.311178
\(159\) −7.31705e6 −0.144360
\(160\) −4.42368e6 −0.0853815
\(161\) −1.95644e7 −0.369467
\(162\) 4.25153e6 0.0785674
\(163\) −5.02668e7 −0.909127 −0.454564 0.890714i \(-0.650205\pi\)
−0.454564 + 0.890714i \(0.650205\pi\)
\(164\) 3.02830e7 0.536100
\(165\) 1.52981e7 0.265120
\(166\) −2.84539e6 −0.0482796
\(167\) 5.15138e7 0.855887 0.427943 0.903806i \(-0.359238\pi\)
0.427943 + 0.903806i \(0.359238\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 1.37538e7 0.214710
\(171\) −2.05673e7 −0.314551
\(172\) 5.70493e7 0.854871
\(173\) 1.31669e7 0.193340 0.0966699 0.995316i \(-0.469181\pi\)
0.0966699 + 0.995316i \(0.469181\pi\)
\(174\) 2.21810e6 0.0319198
\(175\) −2.05457e7 −0.289793
\(176\) 1.71909e7 0.237687
\(177\) 2.57586e7 0.349154
\(178\) −1.74283e6 −0.0231625
\(179\) −6.29758e7 −0.820707 −0.410354 0.911926i \(-0.634595\pi\)
−0.410354 + 0.911926i \(0.634595\pi\)
\(180\) −6.29856e6 −0.0804984
\(181\) 8.61178e6 0.107949 0.0539744 0.998542i \(-0.482811\pi\)
0.0539744 + 0.998542i \(0.482811\pi\)
\(182\) 6.02857e6 0.0741249
\(183\) −8.63233e7 −1.04124
\(184\) −2.92040e7 −0.345605
\(185\) 4.75245e7 0.551844
\(186\) −2.12276e7 −0.241883
\(187\) −5.34488e7 −0.597712
\(188\) 4.38390e7 0.481180
\(189\) −6.75127e6 −0.0727393
\(190\) 3.04700e7 0.322282
\(191\) 1.15446e8 1.19884 0.599422 0.800433i \(-0.295397\pi\)
0.599422 + 0.800433i \(0.295397\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 2.02053e6 0.0202309 0.0101154 0.999949i \(-0.496780\pi\)
0.0101154 + 0.999949i \(0.496780\pi\)
\(194\) −3.52061e7 −0.346188
\(195\) 8.00806e6 0.0773404
\(196\) 7.52954e6 0.0714286
\(197\) −1.02161e8 −0.952032 −0.476016 0.879437i \(-0.657919\pi\)
−0.476016 + 0.879437i \(0.657919\pi\)
\(198\) 2.44769e7 0.224093
\(199\) −1.76061e8 −1.58372 −0.791858 0.610705i \(-0.790886\pi\)
−0.791858 + 0.610705i \(0.790886\pi\)
\(200\) −3.06688e7 −0.271076
\(201\) 7.83545e7 0.680578
\(202\) −1.04215e8 −0.889609
\(203\) −3.52227e6 −0.0295519
\(204\) 2.20061e7 0.181483
\(205\) −6.38782e7 −0.517862
\(206\) −9.61895e7 −0.766642
\(207\) −4.15814e7 −0.325839
\(208\) 8.99891e6 0.0693375
\(209\) −1.18410e8 −0.897173
\(210\) 1.00019e7 0.0745271
\(211\) −7.29003e6 −0.0534245 −0.0267123 0.999643i \(-0.508504\pi\)
−0.0267123 + 0.999643i \(0.508504\pi\)
\(212\) 1.73441e7 0.125019
\(213\) 1.24194e8 0.880585
\(214\) −8.70634e7 −0.607278
\(215\) −1.20338e8 −0.825789
\(216\) −1.00777e7 −0.0680414
\(217\) 3.37087e7 0.223940
\(218\) −1.72247e8 −1.12604
\(219\) −3.44820e7 −0.221839
\(220\) −3.62621e7 −0.229601
\(221\) −2.79788e7 −0.174364
\(222\) 7.60391e7 0.466446
\(223\) −8.42454e7 −0.508721 −0.254360 0.967110i \(-0.581865\pi\)
−0.254360 + 0.967110i \(0.581865\pi\)
\(224\) 1.12394e7 0.0668153
\(225\) −4.36671e7 −0.255573
\(226\) −1.21571e8 −0.700570
\(227\) 2.74120e8 1.55543 0.777714 0.628618i \(-0.216379\pi\)
0.777714 + 0.628618i \(0.216379\pi\)
\(228\) 4.87521e7 0.272409
\(229\) −1.36185e8 −0.749386 −0.374693 0.927149i \(-0.622252\pi\)
−0.374693 + 0.927149i \(0.622252\pi\)
\(230\) 6.16021e7 0.333848
\(231\) −3.88684e7 −0.207470
\(232\) −5.25773e6 −0.0276433
\(233\) −1.92333e8 −0.996113 −0.498057 0.867145i \(-0.665953\pi\)
−0.498057 + 0.867145i \(0.665953\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) −9.24728e7 −0.464811
\(236\) −6.10575e7 −0.302376
\(237\) 5.20693e7 0.254076
\(238\) −3.49448e7 −0.168021
\(239\) 2.61286e7 0.123801 0.0619003 0.998082i \(-0.480284\pi\)
0.0619003 + 0.998082i \(0.480284\pi\)
\(240\) 1.49299e7 0.0697137
\(241\) 8.10060e7 0.372784 0.186392 0.982475i \(-0.440321\pi\)
0.186392 + 0.982475i \(0.440321\pi\)
\(242\) −1.49789e7 −0.0679401
\(243\) −1.43489e7 −0.0641500
\(244\) 2.04618e8 0.901737
\(245\) −1.58826e7 −0.0689987
\(246\) −1.02205e8 −0.437724
\(247\) −6.19840e7 −0.261722
\(248\) 5.03173e7 0.209477
\(249\) 9.60320e6 0.0394201
\(250\) 1.49067e8 0.603381
\(251\) −4.92098e8 −1.96423 −0.982117 0.188270i \(-0.939712\pi\)
−0.982117 + 0.188270i \(0.939712\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) −2.39393e8 −0.929371
\(254\) 6.48568e7 0.248335
\(255\) −4.64191e7 −0.175310
\(256\) 1.67772e7 0.0625000
\(257\) −4.01406e7 −0.147509 −0.0737544 0.997276i \(-0.523498\pi\)
−0.0737544 + 0.997276i \(0.523498\pi\)
\(258\) −1.92541e8 −0.697999
\(259\) −1.20747e8 −0.431845
\(260\) −1.89821e7 −0.0669788
\(261\) −7.48610e6 −0.0260624
\(262\) 6.85625e7 0.235522
\(263\) 7.44163e7 0.252245 0.126123 0.992015i \(-0.459747\pi\)
0.126123 + 0.992015i \(0.459747\pi\)
\(264\) −5.80193e7 −0.194070
\(265\) −3.65853e7 −0.120766
\(266\) −7.74165e7 −0.252201
\(267\) 5.88206e6 0.0189121
\(268\) −1.85729e8 −0.589398
\(269\) −2.94494e8 −0.922453 −0.461226 0.887283i \(-0.652590\pi\)
−0.461226 + 0.887283i \(0.652590\pi\)
\(270\) 2.12576e7 0.0657267
\(271\) 1.50668e8 0.459863 0.229932 0.973207i \(-0.426150\pi\)
0.229932 + 0.973207i \(0.426150\pi\)
\(272\) −5.21626e7 −0.157169
\(273\) −2.03464e7 −0.0605228
\(274\) −2.10206e8 −0.617330
\(275\) −2.51400e8 −0.728956
\(276\) 9.85634e7 0.282185
\(277\) 3.94010e8 1.11385 0.556926 0.830562i \(-0.311981\pi\)
0.556926 + 0.830562i \(0.311981\pi\)
\(278\) 5.80662e6 0.0162094
\(279\) 7.16432e7 0.197497
\(280\) −2.37082e7 −0.0645423
\(281\) 4.42694e8 1.19023 0.595116 0.803640i \(-0.297106\pi\)
0.595116 + 0.803640i \(0.297106\pi\)
\(282\) −1.47957e8 −0.392882
\(283\) −6.68393e7 −0.175299 −0.0876495 0.996151i \(-0.527936\pi\)
−0.0876495 + 0.996151i \(0.527936\pi\)
\(284\) −2.94385e8 −0.762609
\(285\) −1.02836e8 −0.263142
\(286\) 7.37665e7 0.186457
\(287\) 1.62298e8 0.405253
\(288\) 2.38879e7 0.0589256
\(289\) −2.48158e8 −0.604765
\(290\) 1.10905e7 0.0267029
\(291\) 1.18820e8 0.282661
\(292\) 8.17351e7 0.192118
\(293\) 2.91003e8 0.675867 0.337934 0.941170i \(-0.390272\pi\)
0.337934 + 0.941170i \(0.390272\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 1.28793e8 0.292090
\(296\) −1.80241e8 −0.403954
\(297\) −8.26096e7 −0.182971
\(298\) 1.52566e8 0.333966
\(299\) −1.25315e8 −0.271115
\(300\) 1.03507e8 0.221333
\(301\) 3.05748e8 0.646222
\(302\) −5.93439e8 −1.23980
\(303\) 3.51725e8 0.726363
\(304\) −1.15560e8 −0.235913
\(305\) −4.31616e8 −0.871061
\(306\) −7.42705e7 −0.148181
\(307\) −4.66264e8 −0.919702 −0.459851 0.887996i \(-0.652097\pi\)
−0.459851 + 0.887996i \(0.652097\pi\)
\(308\) 9.21325e7 0.179674
\(309\) 3.24640e8 0.625960
\(310\) −1.06138e8 −0.202351
\(311\) −8.01986e8 −1.51184 −0.755919 0.654665i \(-0.772810\pi\)
−0.755919 + 0.654665i \(0.772810\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) −2.28927e8 −0.421981 −0.210990 0.977488i \(-0.567669\pi\)
−0.210990 + 0.977488i \(0.567669\pi\)
\(314\) 1.79519e8 0.327233
\(315\) −3.37563e7 −0.0608511
\(316\) −1.23424e8 −0.220036
\(317\) −4.87643e7 −0.0859795 −0.0429897 0.999076i \(-0.513688\pi\)
−0.0429897 + 0.999076i \(0.513688\pi\)
\(318\) −5.85364e7 −0.102078
\(319\) −4.30990e7 −0.0743361
\(320\) −3.53894e7 −0.0603738
\(321\) 2.93839e8 0.495840
\(322\) −1.56515e8 −0.261253
\(323\) 3.59293e8 0.593252
\(324\) 3.40122e7 0.0555556
\(325\) −1.31600e8 −0.212650
\(326\) −4.02134e8 −0.642850
\(327\) 5.81333e8 0.919407
\(328\) 2.42264e8 0.379080
\(329\) 2.34950e8 0.363738
\(330\) 1.22385e8 0.187468
\(331\) 7.97529e8 1.20878 0.604391 0.796688i \(-0.293416\pi\)
0.604391 + 0.796688i \(0.293416\pi\)
\(332\) −2.27631e7 −0.0341388
\(333\) −2.56632e8 −0.380852
\(334\) 4.12111e8 0.605203
\(335\) 3.91772e8 0.569347
\(336\) −3.79331e7 −0.0545545
\(337\) 4.47244e8 0.636560 0.318280 0.947997i \(-0.396895\pi\)
0.318280 + 0.947997i \(0.396895\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) 4.10303e8 0.572013
\(340\) 1.10030e8 0.151823
\(341\) 4.12464e8 0.563308
\(342\) −1.64538e8 −0.222421
\(343\) 4.03536e7 0.0539949
\(344\) 4.56394e8 0.604485
\(345\) −2.07907e8 −0.272585
\(346\) 1.05335e8 0.136712
\(347\) −1.11624e9 −1.43419 −0.717094 0.696977i \(-0.754528\pi\)
−0.717094 + 0.696977i \(0.754528\pi\)
\(348\) 1.77448e7 0.0225707
\(349\) −5.83656e7 −0.0734967 −0.0367483 0.999325i \(-0.511700\pi\)
−0.0367483 + 0.999325i \(0.511700\pi\)
\(350\) −1.64366e8 −0.204915
\(351\) −4.32436e7 −0.0533761
\(352\) 1.37527e8 0.168070
\(353\) 1.17201e9 1.41814 0.709068 0.705140i \(-0.249116\pi\)
0.709068 + 0.705140i \(0.249116\pi\)
\(354\) 2.06069e8 0.246889
\(355\) 6.20969e8 0.736666
\(356\) −1.39427e7 −0.0163784
\(357\) 1.17939e8 0.137189
\(358\) −5.03807e8 −0.580328
\(359\) 2.76337e8 0.315216 0.157608 0.987502i \(-0.449622\pi\)
0.157608 + 0.987502i \(0.449622\pi\)
\(360\) −5.03885e7 −0.0569210
\(361\) −9.78984e7 −0.109522
\(362\) 6.88942e7 0.0763313
\(363\) 5.05538e7 0.0554728
\(364\) 4.82285e7 0.0524142
\(365\) −1.72410e8 −0.185583
\(366\) −6.90586e8 −0.736265
\(367\) 6.82458e8 0.720683 0.360342 0.932820i \(-0.382660\pi\)
0.360342 + 0.932820i \(0.382660\pi\)
\(368\) −2.33632e8 −0.244379
\(369\) 3.44942e8 0.357400
\(370\) 3.80196e8 0.390212
\(371\) 9.29537e7 0.0945057
\(372\) −1.69821e8 −0.171037
\(373\) −1.10862e9 −1.10612 −0.553062 0.833140i \(-0.686541\pi\)
−0.553062 + 0.833140i \(0.686541\pi\)
\(374\) −4.27590e8 −0.422647
\(375\) −5.03101e8 −0.492658
\(376\) 3.50712e8 0.340246
\(377\) −2.25610e7 −0.0216852
\(378\) −5.40102e7 −0.0514344
\(379\) −8.56782e8 −0.808413 −0.404206 0.914668i \(-0.632452\pi\)
−0.404206 + 0.914668i \(0.632452\pi\)
\(380\) 2.43760e8 0.227887
\(381\) −2.18892e8 −0.202764
\(382\) 9.23569e8 0.847711
\(383\) −1.04811e9 −0.953259 −0.476630 0.879104i \(-0.658142\pi\)
−0.476630 + 0.879104i \(0.658142\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −1.94342e8 −0.173562
\(386\) 1.61643e7 0.0143054
\(387\) 6.49827e8 0.569914
\(388\) −2.81649e8 −0.244792
\(389\) −3.25143e8 −0.280060 −0.140030 0.990147i \(-0.544720\pi\)
−0.140030 + 0.990147i \(0.544720\pi\)
\(390\) 6.40645e7 0.0546879
\(391\) 7.26392e8 0.614543
\(392\) 6.02363e7 0.0505076
\(393\) −2.31398e8 −0.192303
\(394\) −8.17284e8 −0.673188
\(395\) 2.60347e8 0.212551
\(396\) 1.95815e8 0.158458
\(397\) 1.12836e9 0.905065 0.452532 0.891748i \(-0.350521\pi\)
0.452532 + 0.891748i \(0.350521\pi\)
\(398\) −1.40849e9 −1.11986
\(399\) 2.61281e8 0.205922
\(400\) −2.45350e8 −0.191680
\(401\) 3.93234e8 0.304541 0.152271 0.988339i \(-0.451341\pi\)
0.152271 + 0.988339i \(0.451341\pi\)
\(402\) 6.26836e8 0.481241
\(403\) 2.15912e8 0.164327
\(404\) −8.33718e8 −0.629049
\(405\) −7.17445e7 −0.0536656
\(406\) −2.81781e7 −0.0208964
\(407\) −1.47748e9 −1.08628
\(408\) 1.76049e8 0.128328
\(409\) −8.76131e8 −0.633195 −0.316597 0.948560i \(-0.602540\pi\)
−0.316597 + 0.948560i \(0.602540\pi\)
\(410\) −5.11026e8 −0.366184
\(411\) 7.09444e8 0.504048
\(412\) −7.69516e8 −0.542098
\(413\) −3.27230e8 −0.228575
\(414\) −3.32651e8 −0.230403
\(415\) 4.80160e7 0.0329775
\(416\) 7.19913e7 0.0490290
\(417\) −1.95974e7 −0.0132349
\(418\) −9.47280e8 −0.634397
\(419\) 1.50539e9 0.999772 0.499886 0.866091i \(-0.333375\pi\)
0.499886 + 0.866091i \(0.333375\pi\)
\(420\) 8.00150e7 0.0526986
\(421\) −1.45302e9 −0.949042 −0.474521 0.880244i \(-0.657379\pi\)
−0.474521 + 0.880244i \(0.657379\pi\)
\(422\) −5.83202e7 −0.0377769
\(423\) 4.99353e8 0.320787
\(424\) 1.38753e8 0.0884020
\(425\) 7.62826e8 0.482019
\(426\) 9.93550e8 0.622667
\(427\) 1.09663e9 0.681649
\(428\) −6.96507e8 −0.429410
\(429\) −2.48962e8 −0.152241
\(430\) −9.62707e8 −0.583921
\(431\) 7.01355e8 0.421956 0.210978 0.977491i \(-0.432335\pi\)
0.210978 + 0.977491i \(0.432335\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) −2.01359e9 −1.19196 −0.595981 0.802998i \(-0.703237\pi\)
−0.595981 + 0.802998i \(0.703237\pi\)
\(434\) 2.69669e8 0.158350
\(435\) −3.74305e7 −0.0218029
\(436\) −1.37797e9 −0.796230
\(437\) 1.60924e9 0.922436
\(438\) −2.75856e8 −0.156864
\(439\) −3.23642e9 −1.82574 −0.912871 0.408248i \(-0.866140\pi\)
−0.912871 + 0.408248i \(0.866140\pi\)
\(440\) −2.90097e8 −0.162352
\(441\) 8.57661e7 0.0476190
\(442\) −2.23830e8 −0.123294
\(443\) 4.09585e8 0.223837 0.111918 0.993717i \(-0.464300\pi\)
0.111918 + 0.993717i \(0.464300\pi\)
\(444\) 6.08313e8 0.329827
\(445\) 2.94103e7 0.0158212
\(446\) −6.73964e8 −0.359720
\(447\) −5.14912e8 −0.272682
\(448\) 8.99154e7 0.0472456
\(449\) −2.07365e9 −1.08112 −0.540558 0.841307i \(-0.681787\pi\)
−0.540558 + 0.841307i \(0.681787\pi\)
\(450\) −3.49337e8 −0.180718
\(451\) 1.98590e9 1.01939
\(452\) −9.72570e8 −0.495378
\(453\) 2.00286e9 1.01229
\(454\) 2.19296e9 1.09985
\(455\) −1.01732e8 −0.0506312
\(456\) 3.90017e8 0.192622
\(457\) −1.08948e9 −0.533966 −0.266983 0.963701i \(-0.586027\pi\)
−0.266983 + 0.963701i \(0.586027\pi\)
\(458\) −1.08948e9 −0.529896
\(459\) 2.50663e8 0.120989
\(460\) 4.92817e8 0.236066
\(461\) −1.46678e9 −0.697286 −0.348643 0.937256i \(-0.613358\pi\)
−0.348643 + 0.937256i \(0.613358\pi\)
\(462\) −3.10947e8 −0.146703
\(463\) 1.59802e7 0.00748252 0.00374126 0.999993i \(-0.498809\pi\)
0.00374126 + 0.999993i \(0.498809\pi\)
\(464\) −4.20618e7 −0.0195468
\(465\) 3.58216e8 0.165219
\(466\) −1.53867e9 −0.704358
\(467\) −2.86456e9 −1.30152 −0.650758 0.759285i \(-0.725549\pi\)
−0.650758 + 0.759285i \(0.725549\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) −9.95392e8 −0.445543
\(470\) −7.39783e8 −0.328671
\(471\) −6.05876e8 −0.267184
\(472\) −4.88460e8 −0.213812
\(473\) 3.74118e9 1.62553
\(474\) 4.16555e8 0.179659
\(475\) 1.68996e9 0.723517
\(476\) −2.79559e8 −0.118809
\(477\) 1.97560e8 0.0833462
\(478\) 2.09028e8 0.0875403
\(479\) −4.11927e9 −1.71256 −0.856281 0.516510i \(-0.827231\pi\)
−0.856281 + 0.516510i \(0.827231\pi\)
\(480\) 1.19439e8 0.0492950
\(481\) −7.73417e8 −0.316888
\(482\) 6.48048e8 0.263598
\(483\) 5.28238e8 0.213312
\(484\) −1.19831e8 −0.0480409
\(485\) 5.94102e8 0.236464
\(486\) −1.14791e8 −0.0453609
\(487\) −1.12488e8 −0.0441321 −0.0220661 0.999757i \(-0.507024\pi\)
−0.0220661 + 0.999757i \(0.507024\pi\)
\(488\) 1.63695e9 0.637624
\(489\) 1.35720e9 0.524885
\(490\) −1.27061e8 −0.0487894
\(491\) −1.52267e8 −0.0580525 −0.0290263 0.999579i \(-0.509241\pi\)
−0.0290263 + 0.999579i \(0.509241\pi\)
\(492\) −8.17641e8 −0.309517
\(493\) 1.30776e8 0.0491545
\(494\) −4.95872e8 −0.185065
\(495\) −4.13048e8 −0.153067
\(496\) 4.02538e8 0.148123
\(497\) −1.57772e9 −0.576478
\(498\) 7.68256e7 0.0278742
\(499\) 3.33132e9 1.20023 0.600116 0.799913i \(-0.295121\pi\)
0.600116 + 0.799913i \(0.295121\pi\)
\(500\) 1.19254e9 0.426655
\(501\) −1.39087e9 −0.494146
\(502\) −3.93678e9 −1.38892
\(503\) −1.50448e9 −0.527108 −0.263554 0.964645i \(-0.584895\pi\)
−0.263554 + 0.964645i \(0.584895\pi\)
\(504\) 1.28024e8 0.0445435
\(505\) 1.75862e9 0.607649
\(506\) −1.91514e9 −0.657164
\(507\) −1.30324e8 −0.0444116
\(508\) 5.18854e8 0.175599
\(509\) 1.43003e9 0.480655 0.240327 0.970692i \(-0.422745\pi\)
0.240327 + 0.970692i \(0.422745\pi\)
\(510\) −3.71353e8 −0.123963
\(511\) 4.38049e8 0.145228
\(512\) 1.34218e8 0.0441942
\(513\) 5.55316e8 0.181606
\(514\) −3.21125e8 −0.104305
\(515\) 1.62320e9 0.523656
\(516\) −1.54033e9 −0.493560
\(517\) 2.87488e9 0.914961
\(518\) −9.65979e8 −0.305361
\(519\) −3.55506e8 −0.111625
\(520\) −1.51857e8 −0.0473611
\(521\) −1.26240e8 −0.0391080 −0.0195540 0.999809i \(-0.506225\pi\)
−0.0195540 + 0.999809i \(0.506225\pi\)
\(522\) −5.98888e7 −0.0184289
\(523\) −5.95449e8 −0.182007 −0.0910037 0.995851i \(-0.529008\pi\)
−0.0910037 + 0.995851i \(0.529008\pi\)
\(524\) 5.48500e8 0.166539
\(525\) 5.54734e8 0.167312
\(526\) 5.95331e8 0.178364
\(527\) −1.25154e9 −0.372485
\(528\) −4.64155e8 −0.137228
\(529\) −1.51378e8 −0.0444598
\(530\) −2.92682e8 −0.0853947
\(531\) −6.95483e8 −0.201584
\(532\) −6.19332e8 −0.178333
\(533\) 1.03956e9 0.297375
\(534\) 4.70565e7 0.0133729
\(535\) 1.46920e9 0.414802
\(536\) −1.48583e9 −0.416767
\(537\) 1.70035e9 0.473836
\(538\) −2.35596e9 −0.652272
\(539\) 4.93773e8 0.135821
\(540\) 1.70061e8 0.0464758
\(541\) −1.19809e8 −0.0325311 −0.0162655 0.999868i \(-0.505178\pi\)
−0.0162655 + 0.999868i \(0.505178\pi\)
\(542\) 1.20535e9 0.325173
\(543\) −2.32518e8 −0.0623243
\(544\) −4.17300e8 −0.111135
\(545\) 2.90666e9 0.769143
\(546\) −1.62771e8 −0.0427960
\(547\) 6.05138e9 1.58088 0.790440 0.612540i \(-0.209852\pi\)
0.790440 + 0.612540i \(0.209852\pi\)
\(548\) −1.68164e9 −0.436518
\(549\) 2.33073e9 0.601158
\(550\) −2.01120e9 −0.515450
\(551\) 2.89719e8 0.0737814
\(552\) 7.88507e8 0.199535
\(553\) −6.61473e8 −0.166331
\(554\) 3.15208e9 0.787613
\(555\) −1.28316e9 −0.318607
\(556\) 4.64530e7 0.0114618
\(557\) −7.71199e9 −1.89092 −0.945461 0.325736i \(-0.894388\pi\)
−0.945461 + 0.325736i \(0.894388\pi\)
\(558\) 5.73146e8 0.139651
\(559\) 1.95839e9 0.474197
\(560\) −1.89665e8 −0.0456383
\(561\) 1.44312e9 0.345089
\(562\) 3.54155e9 0.841621
\(563\) −1.03973e8 −0.0245552 −0.0122776 0.999925i \(-0.503908\pi\)
−0.0122776 + 0.999925i \(0.503908\pi\)
\(564\) −1.18365e9 −0.277810
\(565\) 2.05152e9 0.478526
\(566\) −5.34714e8 −0.123955
\(567\) 1.82284e8 0.0419961
\(568\) −2.35508e9 −0.539246
\(569\) 6.29152e9 1.43174 0.715868 0.698236i \(-0.246031\pi\)
0.715868 + 0.698236i \(0.246031\pi\)
\(570\) −8.22691e8 −0.186069
\(571\) −8.55133e9 −1.92224 −0.961119 0.276133i \(-0.910947\pi\)
−0.961119 + 0.276133i \(0.910947\pi\)
\(572\) 5.90132e8 0.131845
\(573\) −3.11705e9 −0.692153
\(574\) 1.29838e9 0.286557
\(575\) 3.41664e9 0.749482
\(576\) 1.91103e8 0.0416667
\(577\) 6.59657e9 1.42956 0.714781 0.699348i \(-0.246526\pi\)
0.714781 + 0.699348i \(0.246526\pi\)
\(578\) −1.98527e9 −0.427633
\(579\) −5.45544e7 −0.0116803
\(580\) 8.87242e7 0.0188818
\(581\) −1.21996e8 −0.0258065
\(582\) 9.50564e8 0.199871
\(583\) 1.13740e9 0.237723
\(584\) 6.53881e8 0.135848
\(585\) −2.16218e8 −0.0446525
\(586\) 2.32803e9 0.477910
\(587\) 5.25537e9 1.07243 0.536216 0.844081i \(-0.319853\pi\)
0.536216 + 0.844081i \(0.319853\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −2.77266e9 −0.559105
\(590\) 1.03035e9 0.206538
\(591\) 2.75833e9 0.549656
\(592\) −1.44193e9 −0.285639
\(593\) 2.40969e9 0.474537 0.237269 0.971444i \(-0.423748\pi\)
0.237269 + 0.971444i \(0.423748\pi\)
\(594\) −6.60876e8 −0.129380
\(595\) 5.89694e8 0.114767
\(596\) 1.22053e9 0.236150
\(597\) 4.75365e9 0.914359
\(598\) −1.00252e9 −0.191707
\(599\) 1.02623e10 1.95096 0.975482 0.220080i \(-0.0706319\pi\)
0.975482 + 0.220080i \(0.0706319\pi\)
\(600\) 8.28058e8 0.156506
\(601\) −2.12909e9 −0.400067 −0.200034 0.979789i \(-0.564105\pi\)
−0.200034 + 0.979789i \(0.564105\pi\)
\(602\) 2.44599e9 0.456948
\(603\) −2.11557e9 −0.392932
\(604\) −4.74751e9 −0.876671
\(605\) 2.52769e8 0.0464066
\(606\) 2.81380e9 0.513616
\(607\) 4.97253e9 0.902438 0.451219 0.892413i \(-0.350989\pi\)
0.451219 + 0.892413i \(0.350989\pi\)
\(608\) −9.24484e8 −0.166816
\(609\) 9.51012e7 0.0170618
\(610\) −3.45293e9 −0.615933
\(611\) 1.50491e9 0.266911
\(612\) −5.94164e8 −0.104780
\(613\) 2.23562e8 0.0392000 0.0196000 0.999808i \(-0.493761\pi\)
0.0196000 + 0.999808i \(0.493761\pi\)
\(614\) −3.73011e9 −0.650328
\(615\) 1.72471e9 0.298988
\(616\) 7.37060e8 0.127049
\(617\) −4.15176e9 −0.711597 −0.355798 0.934563i \(-0.615791\pi\)
−0.355798 + 0.934563i \(0.615791\pi\)
\(618\) 2.59712e9 0.442621
\(619\) 4.08459e9 0.692200 0.346100 0.938198i \(-0.387506\pi\)
0.346100 + 0.938198i \(0.387506\pi\)
\(620\) −8.49105e8 −0.143084
\(621\) 1.12270e9 0.188123
\(622\) −6.41588e9 −1.06903
\(623\) −7.47239e7 −0.0123809
\(624\) −2.42971e8 −0.0400320
\(625\) 2.16418e9 0.354580
\(626\) −1.83142e9 −0.298386
\(627\) 3.19707e9 0.517983
\(628\) 1.43615e9 0.231388
\(629\) 4.48314e9 0.718299
\(630\) −2.70051e8 −0.0430282
\(631\) −1.01736e10 −1.61202 −0.806009 0.591903i \(-0.798377\pi\)
−0.806009 + 0.591903i \(0.798377\pi\)
\(632\) −9.87389e8 −0.155589
\(633\) 1.96831e8 0.0308447
\(634\) −3.90115e8 −0.0607967
\(635\) −1.09446e9 −0.169625
\(636\) −4.68291e8 −0.0721799
\(637\) 2.58475e8 0.0396214
\(638\) −3.44792e8 −0.0525636
\(639\) −3.35323e9 −0.508406
\(640\) −2.83116e8 −0.0426907
\(641\) 3.92635e9 0.588824 0.294412 0.955679i \(-0.404876\pi\)
0.294412 + 0.955679i \(0.404876\pi\)
\(642\) 2.35071e9 0.350612
\(643\) 5.24722e9 0.778379 0.389189 0.921158i \(-0.372755\pi\)
0.389189 + 0.921158i \(0.372755\pi\)
\(644\) −1.25212e9 −0.184733
\(645\) 3.24913e9 0.476770
\(646\) 2.87434e9 0.419493
\(647\) 8.70304e9 1.26330 0.631649 0.775254i \(-0.282378\pi\)
0.631649 + 0.775254i \(0.282378\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −4.00404e9 −0.574966
\(650\) −1.05280e9 −0.150366
\(651\) −9.10134e8 −0.129292
\(652\) −3.21708e9 −0.454564
\(653\) 1.32545e10 1.86281 0.931403 0.363990i \(-0.118586\pi\)
0.931403 + 0.363990i \(0.118586\pi\)
\(654\) 4.65066e9 0.650119
\(655\) −1.15699e9 −0.160874
\(656\) 1.93811e9 0.268050
\(657\) 9.31014e8 0.128079
\(658\) 1.87960e9 0.257202
\(659\) 1.32355e10 1.80153 0.900764 0.434308i \(-0.143007\pi\)
0.900764 + 0.434308i \(0.143007\pi\)
\(660\) 9.79076e8 0.132560
\(661\) 4.15598e9 0.559716 0.279858 0.960041i \(-0.409713\pi\)
0.279858 + 0.960041i \(0.409713\pi\)
\(662\) 6.38023e9 0.854739
\(663\) 7.55427e8 0.100669
\(664\) −1.82105e8 −0.0241398
\(665\) 1.30640e9 0.172267
\(666\) −2.05306e9 −0.269303
\(667\) 5.85733e8 0.0764293
\(668\) 3.29689e9 0.427943
\(669\) 2.27463e9 0.293710
\(670\) 3.13418e9 0.402589
\(671\) 1.34185e10 1.71465
\(672\) −3.03464e8 −0.0385758
\(673\) −2.50664e9 −0.316986 −0.158493 0.987360i \(-0.550664\pi\)
−0.158493 + 0.987360i \(0.550664\pi\)
\(674\) 3.57795e9 0.450116
\(675\) 1.17901e9 0.147555
\(676\) 3.08916e8 0.0384615
\(677\) −3.73190e9 −0.462243 −0.231121 0.972925i \(-0.574239\pi\)
−0.231121 + 0.972925i \(0.574239\pi\)
\(678\) 3.28242e9 0.404474
\(679\) −1.50946e9 −0.185045
\(680\) 8.80243e8 0.107355
\(681\) −7.40124e9 −0.898027
\(682\) 3.29971e9 0.398319
\(683\) −8.40851e8 −0.100983 −0.0504913 0.998725i \(-0.516079\pi\)
−0.0504913 + 0.998725i \(0.516079\pi\)
\(684\) −1.31631e9 −0.157275
\(685\) 3.54722e9 0.421668
\(686\) 3.22829e8 0.0381802
\(687\) 3.67700e9 0.432658
\(688\) 3.65115e9 0.427436
\(689\) 5.95391e8 0.0693482
\(690\) −1.66326e9 −0.192747
\(691\) 7.01310e9 0.808606 0.404303 0.914625i \(-0.367514\pi\)
0.404303 + 0.914625i \(0.367514\pi\)
\(692\) 8.42680e8 0.0966699
\(693\) 1.04945e9 0.119783
\(694\) −8.92995e9 −1.01412
\(695\) −9.79868e7 −0.0110719
\(696\) 1.41959e8 0.0159599
\(697\) −6.02585e9 −0.674067
\(698\) −4.66924e8 −0.0519700
\(699\) 5.19300e9 0.575106
\(700\) −1.31492e9 −0.144896
\(701\) 4.56598e9 0.500635 0.250317 0.968164i \(-0.419465\pi\)
0.250317 + 0.968164i \(0.419465\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) 9.93191e9 1.07817
\(704\) 1.10022e9 0.118843
\(705\) 2.49677e9 0.268359
\(706\) 9.37604e9 1.00277
\(707\) −4.46821e9 −0.475516
\(708\) 1.64855e9 0.174577
\(709\) −3.91082e9 −0.412103 −0.206051 0.978541i \(-0.566061\pi\)
−0.206051 + 0.978541i \(0.566061\pi\)
\(710\) 4.96775e9 0.520901
\(711\) −1.40587e9 −0.146691
\(712\) −1.11541e8 −0.0115813
\(713\) −5.60556e9 −0.579170
\(714\) 9.43511e8 0.0970070
\(715\) −1.24481e9 −0.127360
\(716\) −4.03045e9 −0.410354
\(717\) −7.05471e8 −0.0714763
\(718\) 2.21069e9 0.222891
\(719\) −5.06745e9 −0.508438 −0.254219 0.967147i \(-0.581818\pi\)
−0.254219 + 0.967147i \(0.581818\pi\)
\(720\) −4.03108e8 −0.0402492
\(721\) −4.12413e9 −0.409787
\(722\) −7.83187e8 −0.0774436
\(723\) −2.18716e9 −0.215227
\(724\) 5.51154e8 0.0539744
\(725\) 6.15113e8 0.0599476
\(726\) 4.04430e8 0.0392252
\(727\) 1.52509e10 1.47206 0.736031 0.676948i \(-0.236698\pi\)
0.736031 + 0.676948i \(0.236698\pi\)
\(728\) 3.85828e8 0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −1.37928e9 −0.131227
\(731\) −1.13519e10 −1.07488
\(732\) −5.52469e9 −0.520618
\(733\) −1.09604e10 −1.02793 −0.513963 0.857812i \(-0.671823\pi\)
−0.513963 + 0.857812i \(0.671823\pi\)
\(734\) 5.45966e9 0.509600
\(735\) 4.28831e8 0.0398364
\(736\) −1.86905e9 −0.172802
\(737\) −1.21798e10 −1.12073
\(738\) 2.75954e9 0.252720
\(739\) −1.25222e10 −1.14136 −0.570682 0.821171i \(-0.693321\pi\)
−0.570682 + 0.821171i \(0.693321\pi\)
\(740\) 3.04157e9 0.275922
\(741\) 1.67357e9 0.151105
\(742\) 7.43629e8 0.0668256
\(743\) 7.60040e9 0.679791 0.339896 0.940463i \(-0.389608\pi\)
0.339896 + 0.940463i \(0.389608\pi\)
\(744\) −1.35857e9 −0.120942
\(745\) −2.57456e9 −0.228116
\(746\) −8.86900e9 −0.782147
\(747\) −2.59286e8 −0.0227592
\(748\) −3.42072e9 −0.298856
\(749\) −3.73284e9 −0.324604
\(750\) −4.02481e9 −0.348362
\(751\) −5.51279e9 −0.474932 −0.237466 0.971396i \(-0.576317\pi\)
−0.237466 + 0.971396i \(0.576317\pi\)
\(752\) 2.80569e9 0.240590
\(753\) 1.32866e10 1.13405
\(754\) −1.80488e8 −0.0153338
\(755\) 1.00143e10 0.846848
\(756\) −4.32081e8 −0.0363696
\(757\) 3.18779e9 0.267088 0.133544 0.991043i \(-0.457364\pi\)
0.133544 + 0.991043i \(0.457364\pi\)
\(758\) −6.85426e9 −0.571634
\(759\) 6.46360e9 0.536573
\(760\) 1.95008e9 0.161141
\(761\) 1.15651e10 0.951268 0.475634 0.879643i \(-0.342219\pi\)
0.475634 + 0.879643i \(0.342219\pi\)
\(762\) −1.75113e9 −0.143376
\(763\) −7.38508e9 −0.601893
\(764\) 7.38856e9 0.599422
\(765\) 1.25332e9 0.101215
\(766\) −8.38488e9 −0.674056
\(767\) −2.09599e9 −0.167728
\(768\) −4.52985e8 −0.0360844
\(769\) −4.91430e9 −0.389690 −0.194845 0.980834i \(-0.562420\pi\)
−0.194845 + 0.980834i \(0.562420\pi\)
\(770\) −1.55474e9 −0.122727
\(771\) 1.08380e9 0.0851643
\(772\) 1.29314e8 0.0101154
\(773\) 8.20751e9 0.639121 0.319561 0.947566i \(-0.396465\pi\)
0.319561 + 0.947566i \(0.396465\pi\)
\(774\) 5.19862e9 0.402990
\(775\) −5.88673e9 −0.454274
\(776\) −2.25319e9 −0.173094
\(777\) 3.26018e9 0.249326
\(778\) −2.60114e9 −0.198032
\(779\) −1.33496e10 −1.01178
\(780\) 5.12516e8 0.0386702
\(781\) −1.93052e10 −1.45009
\(782\) 5.81113e9 0.434547
\(783\) 2.02125e8 0.0150471
\(784\) 4.81890e8 0.0357143
\(785\) −3.02938e9 −0.223517
\(786\) −1.85119e9 −0.135979
\(787\) 2.95314e9 0.215960 0.107980 0.994153i \(-0.465562\pi\)
0.107980 + 0.994153i \(0.465562\pi\)
\(788\) −6.53827e9 −0.476016
\(789\) −2.00924e9 −0.145634
\(790\) 2.08277e9 0.150296
\(791\) −5.21237e9 −0.374470
\(792\) 1.56652e9 0.112047
\(793\) 7.02416e9 0.500194
\(794\) 9.02685e9 0.639977
\(795\) 9.87802e8 0.0697244
\(796\) −1.12679e10 −0.791858
\(797\) 2.51364e10 1.75873 0.879364 0.476151i \(-0.157968\pi\)
0.879364 + 0.476151i \(0.157968\pi\)
\(798\) 2.09024e9 0.145609
\(799\) −8.72327e9 −0.605014
\(800\) −1.96280e9 −0.135538
\(801\) −1.58816e8 −0.0109189
\(802\) 3.14587e9 0.215343
\(803\) 5.36003e9 0.365311
\(804\) 5.01469e9 0.340289
\(805\) 2.64119e9 0.178449
\(806\) 1.72730e9 0.116197
\(807\) 7.95135e9 0.532578
\(808\) −6.66974e9 −0.444805
\(809\) −3.78645e9 −0.251428 −0.125714 0.992067i \(-0.540122\pi\)
−0.125714 + 0.992067i \(0.540122\pi\)
\(810\) −5.73956e8 −0.0379473
\(811\) −1.42483e10 −0.937974 −0.468987 0.883205i \(-0.655381\pi\)
−0.468987 + 0.883205i \(0.655381\pi\)
\(812\) −2.25425e8 −0.0147760
\(813\) −4.06804e9 −0.265502
\(814\) −1.18199e10 −0.768116
\(815\) 6.78602e9 0.439100
\(816\) 1.40839e9 0.0907417
\(817\) −2.51489e10 −1.61340
\(818\) −7.00905e9 −0.447736
\(819\) 5.49353e8 0.0349428
\(820\) −4.08821e9 −0.258931
\(821\) 1.47716e10 0.931596 0.465798 0.884891i \(-0.345767\pi\)
0.465798 + 0.884891i \(0.345767\pi\)
\(822\) 5.67555e9 0.356416
\(823\) 1.69875e10 1.06226 0.531129 0.847291i \(-0.321768\pi\)
0.531129 + 0.847291i \(0.321768\pi\)
\(824\) −6.15613e9 −0.383321
\(825\) 6.78781e9 0.420863
\(826\) −2.61784e9 −0.161627
\(827\) 1.71925e10 1.05699 0.528493 0.848938i \(-0.322757\pi\)
0.528493 + 0.848938i \(0.322757\pi\)
\(828\) −2.66121e9 −0.162920
\(829\) −1.31925e10 −0.804239 −0.402120 0.915587i \(-0.631726\pi\)
−0.402120 + 0.915587i \(0.631726\pi\)
\(830\) 3.84128e8 0.0233186
\(831\) −1.06383e10 −0.643083
\(832\) 5.75930e8 0.0346688
\(833\) −1.49826e9 −0.0898110
\(834\) −1.56779e8 −0.00935850
\(835\) −6.95437e9 −0.413385
\(836\) −7.57824e9 −0.448587
\(837\) −1.93437e9 −0.114025
\(838\) 1.20431e10 0.706945
\(839\) 2.23722e10 1.30780 0.653901 0.756581i \(-0.273131\pi\)
0.653901 + 0.756581i \(0.273131\pi\)
\(840\) 6.40120e8 0.0372635
\(841\) −1.71444e10 −0.993887
\(842\) −1.16242e10 −0.671074
\(843\) −1.19527e10 −0.687181
\(844\) −4.66562e8 −0.0267123
\(845\) −6.51619e8 −0.0371531
\(846\) 3.99483e9 0.226831
\(847\) −6.42220e8 −0.0363155
\(848\) 1.11002e9 0.0625096
\(849\) 1.80466e9 0.101209
\(850\) 6.10261e9 0.340839
\(851\) 2.00796e10 1.11687
\(852\) 7.94840e9 0.440292
\(853\) −9.13574e9 −0.503990 −0.251995 0.967728i \(-0.581087\pi\)
−0.251995 + 0.967728i \(0.581087\pi\)
\(854\) 8.77300e9 0.481999
\(855\) 2.77658e9 0.151925
\(856\) −5.57206e9 −0.303639
\(857\) −9.30815e9 −0.505162 −0.252581 0.967576i \(-0.581279\pi\)
−0.252581 + 0.967576i \(0.581279\pi\)
\(858\) −1.99169e9 −0.107651
\(859\) −2.40324e10 −1.29366 −0.646832 0.762632i \(-0.723907\pi\)
−0.646832 + 0.762632i \(0.723907\pi\)
\(860\) −7.70165e9 −0.412895
\(861\) −4.38205e9 −0.233973
\(862\) 5.61084e9 0.298368
\(863\) 2.08803e9 0.110586 0.0552929 0.998470i \(-0.482391\pi\)
0.0552929 + 0.998470i \(0.482391\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −1.77753e9 −0.0933813
\(866\) −1.61087e10 −0.842845
\(867\) 6.70028e9 0.349161
\(868\) 2.15735e9 0.111970
\(869\) −8.09389e9 −0.418397
\(870\) −2.99444e8 −0.0154169
\(871\) −6.37573e9 −0.326939
\(872\) −1.10238e10 −0.563020
\(873\) −3.20815e9 −0.163194
\(874\) 1.28739e10 0.652261
\(875\) 6.39125e9 0.322521
\(876\) −2.20685e9 −0.110920
\(877\) −3.28101e10 −1.64251 −0.821256 0.570560i \(-0.806726\pi\)
−0.821256 + 0.570560i \(0.806726\pi\)
\(878\) −2.58914e10 −1.29099
\(879\) −7.85709e9 −0.390212
\(880\) −2.32077e9 −0.114800
\(881\) 4.25886e9 0.209835 0.104917 0.994481i \(-0.466542\pi\)
0.104917 + 0.994481i \(0.466542\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 2.84443e10 1.39038 0.695188 0.718828i \(-0.255321\pi\)
0.695188 + 0.718828i \(0.255321\pi\)
\(884\) −1.79064e9 −0.0871818
\(885\) −3.47742e9 −0.168638
\(886\) 3.27668e9 0.158276
\(887\) −9.78539e9 −0.470810 −0.235405 0.971897i \(-0.575642\pi\)
−0.235405 + 0.971897i \(0.575642\pi\)
\(888\) 4.86650e9 0.233223
\(889\) 2.78073e9 0.132740
\(890\) 2.35282e8 0.0111873
\(891\) 2.23046e9 0.105638
\(892\) −5.39171e9 −0.254360
\(893\) −1.93255e10 −0.908133
\(894\) −4.11929e9 −0.192815
\(895\) 8.50174e9 0.396394
\(896\) 7.19323e8 0.0334077
\(897\) 3.38350e9 0.156528
\(898\) −1.65892e10 −0.764465
\(899\) −1.00920e9 −0.0463251
\(900\) −2.79469e9 −0.127787
\(901\) −3.45121e9 −0.157194
\(902\) 1.58872e10 0.720817
\(903\) −8.25521e9 −0.373096
\(904\) −7.78056e9 −0.350285
\(905\) −1.16259e9 −0.0521383
\(906\) 1.60228e10 0.715799
\(907\) −1.40669e10 −0.625998 −0.312999 0.949753i \(-0.601334\pi\)
−0.312999 + 0.949753i \(0.601334\pi\)
\(908\) 1.75437e10 0.777714
\(909\) −9.49657e9 −0.419366
\(910\) −8.13857e8 −0.0358017
\(911\) 4.24394e10 1.85975 0.929875 0.367876i \(-0.119915\pi\)
0.929875 + 0.367876i \(0.119915\pi\)
\(912\) 3.12013e9 0.136204
\(913\) −1.49276e9 −0.0649147
\(914\) −8.71586e9 −0.377571
\(915\) 1.16536e10 0.502907
\(916\) −8.71584e9 −0.374693
\(917\) 2.93962e9 0.125892
\(918\) 2.00530e9 0.0855521
\(919\) −4.22209e10 −1.79442 −0.897209 0.441606i \(-0.854409\pi\)
−0.897209 + 0.441606i \(0.854409\pi\)
\(920\) 3.94254e9 0.166924
\(921\) 1.25891e10 0.530990
\(922\) −1.17342e10 −0.493056
\(923\) −1.01057e10 −0.423019
\(924\) −2.48758e9 −0.103735
\(925\) 2.10868e10 0.876020
\(926\) 1.27841e8 0.00529094
\(927\) −8.76527e9 −0.361398
\(928\) −3.36495e8 −0.0138217
\(929\) −5.03682e9 −0.206111 −0.103055 0.994676i \(-0.532862\pi\)
−0.103055 + 0.994676i \(0.532862\pi\)
\(930\) 2.86573e9 0.116827
\(931\) −3.31923e9 −0.134807
\(932\) −1.23093e10 −0.498057
\(933\) 2.16536e10 0.872860
\(934\) −2.29165e10 −0.920310
\(935\) 7.21559e9 0.288690
\(936\) 8.20026e8 0.0326860
\(937\) −2.72484e10 −1.08206 −0.541032 0.841002i \(-0.681966\pi\)
−0.541032 + 0.841002i \(0.681966\pi\)
\(938\) −7.96314e9 −0.315046
\(939\) 6.18104e9 0.243631
\(940\) −5.91826e9 −0.232406
\(941\) 2.08440e10 0.815487 0.407743 0.913097i \(-0.366316\pi\)
0.407743 + 0.913097i \(0.366316\pi\)
\(942\) −4.84701e9 −0.188928
\(943\) −2.69893e10 −1.04809
\(944\) −3.90768e9 −0.151188
\(945\) 9.11421e8 0.0351324
\(946\) 2.99295e10 1.14942
\(947\) −4.02745e10 −1.54101 −0.770505 0.637434i \(-0.779996\pi\)
−0.770505 + 0.637434i \(0.779996\pi\)
\(948\) 3.33244e9 0.127038
\(949\) 2.80581e9 0.106568
\(950\) 1.35197e10 0.511603
\(951\) 1.31664e9 0.0496403
\(952\) −2.23647e9 −0.0840105
\(953\) −1.77964e10 −0.666052 −0.333026 0.942918i \(-0.608070\pi\)
−0.333026 + 0.942918i \(0.608070\pi\)
\(954\) 1.58048e9 0.0589346
\(955\) −1.55852e10 −0.579031
\(956\) 1.67223e9 0.0619003
\(957\) 1.16367e9 0.0429180
\(958\) −3.29542e10 −1.21096
\(959\) −9.01257e9 −0.329977
\(960\) 9.55515e8 0.0348569
\(961\) −1.78544e10 −0.648955
\(962\) −6.18733e9 −0.224074
\(963\) −7.93365e9 −0.286274
\(964\) 5.18438e9 0.186392
\(965\) −2.72772e8 −0.00977134
\(966\) 4.22591e9 0.150834
\(967\) −1.37616e8 −0.00489414 −0.00244707 0.999997i \(-0.500779\pi\)
−0.00244707 + 0.999997i \(0.500779\pi\)
\(968\) −9.58649e8 −0.0339700
\(969\) −9.70090e9 −0.342514
\(970\) 4.75282e9 0.167205
\(971\) 1.07329e10 0.376226 0.188113 0.982147i \(-0.439763\pi\)
0.188113 + 0.982147i \(0.439763\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 2.48959e8 0.00866429
\(974\) −8.99904e8 −0.0312061
\(975\) 3.55321e9 0.122773
\(976\) 1.30956e10 0.450868
\(977\) −1.34908e10 −0.462814 −0.231407 0.972857i \(-0.574333\pi\)
−0.231407 + 0.972857i \(0.574333\pi\)
\(978\) 1.08576e10 0.371150
\(979\) −9.14333e8 −0.0311433
\(980\) −1.01649e9 −0.0344993
\(981\) −1.56960e10 −0.530820
\(982\) −1.21814e9 −0.0410493
\(983\) 3.52787e10 1.18461 0.592305 0.805714i \(-0.298218\pi\)
0.592305 + 0.805714i \(0.298218\pi\)
\(984\) −6.54113e9 −0.218862
\(985\) 1.37917e10 0.459822
\(986\) 1.04621e9 0.0347574
\(987\) −6.34364e9 −0.210004
\(988\) −3.96697e9 −0.130861
\(989\) −5.08443e10 −1.67130
\(990\) −3.30438e9 −0.108235
\(991\) 4.46894e10 1.45864 0.729318 0.684175i \(-0.239838\pi\)
0.729318 + 0.684175i \(0.239838\pi\)
\(992\) 3.22031e9 0.104739
\(993\) −2.15333e10 −0.697891
\(994\) −1.26218e10 −0.407631
\(995\) 2.37682e10 0.764920
\(996\) 6.14605e8 0.0197101
\(997\) −5.95582e10 −1.90330 −0.951652 0.307177i \(-0.900616\pi\)
−0.951652 + 0.307177i \(0.900616\pi\)
\(998\) 2.66506e10 0.848692
\(999\) 6.92907e9 0.219885
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 546.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.a.1.1 1 1.1 even 1 trivial