Properties

Label 76.8.a.a.1.5
Level $76$
Weight $8$
Character 76.1
Self dual yes
Analytic conductor $23.741$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.7412619368\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 5014 x^{3} + 113222 x^{2} - 625803 x + 567036\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.12286\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q+73.9019 q^{3} -358.964 q^{5} -109.800 q^{7} +3274.48 q^{9} +O(q^{10})\) \(q+73.9019 q^{3} -358.964 q^{5} -109.800 q^{7} +3274.48 q^{9} -7153.73 q^{11} -3123.17 q^{13} -26528.1 q^{15} -23898.6 q^{17} -6859.00 q^{19} -8114.45 q^{21} +50622.1 q^{23} +50730.0 q^{25} +80367.1 q^{27} -134788. q^{29} +199472. q^{31} -528674. q^{33} +39414.3 q^{35} -95421.5 q^{37} -230808. q^{39} -488804. q^{41} +162905. q^{43} -1.17542e6 q^{45} -832599. q^{47} -811487. q^{49} -1.76615e6 q^{51} +953158. q^{53} +2.56793e6 q^{55} -506893. q^{57} +2.46032e6 q^{59} +1.15629e6 q^{61} -359539. q^{63} +1.12110e6 q^{65} -1.59366e6 q^{67} +3.74107e6 q^{69} -5.90057e6 q^{71} -1.17056e6 q^{73} +3.74904e6 q^{75} +785482. q^{77} +7.58482e6 q^{79} -1.22202e6 q^{81} +7.26944e6 q^{83} +8.57871e6 q^{85} -9.96106e6 q^{87} -7.80137e6 q^{89} +342925. q^{91} +1.47414e7 q^{93} +2.46213e6 q^{95} -6.56007e6 q^{97} -2.34248e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} + O(q^{10}) \) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} - 2662q^{11} - 602q^{13} - 20800q^{15} - 27366q^{17} - 34295q^{19} - 59964q^{21} - 67096q^{23} - 109115q^{25} - 178778q^{27} - 372398q^{29} - 271372q^{31} - 792700q^{33} - 608250q^{35} - 562630q^{37} - 963904q^{39} - 956714q^{41} - 827362q^{43} - 1165100q^{45} - 1812982q^{47} - 862031q^{49} - 2458254q^{51} + 486998q^{53} + 467930q^{55} + 96026q^{57} - 367182q^{59} + 1879732q^{61} - 1007274q^{63} + 1790920q^{65} - 1046394q^{67} + 7261712q^{69} - 4664572q^{71} + 4224942q^{73} + 8194850q^{75} + 8611110q^{77} + 9574024q^{79} + 11351813q^{81} + 11754804q^{83} + 18711750q^{85} + 3801472q^{87} + 2782542q^{89} + 7385214q^{91} + 29535004q^{93} + 1920520q^{95} + 1291574q^{97} - 9760310q^{99} + O(q^{100}) \)

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
\(3\) 73.9019 1.58027 0.790134 0.612934i \(-0.210011\pi\)
0.790134 + 0.612934i \(0.210011\pi\)
\(4\) 0 0
\(5\) −358.964 −1.28427 −0.642134 0.766592i \(-0.721951\pi\)
−0.642134 + 0.766592i \(0.721951\pi\)
\(6\) 0 0
\(7\) −109.800 −0.120993 −0.0604965 0.998168i \(-0.519268\pi\)
−0.0604965 + 0.998168i \(0.519268\pi\)
\(8\) 0 0
\(9\) 3274.48 1.49725
\(10\) 0 0
\(11\) −7153.73 −1.62053 −0.810267 0.586061i \(-0.800678\pi\)
−0.810267 + 0.586061i \(0.800678\pi\)
\(12\) 0 0
\(13\) −3123.17 −0.394270 −0.197135 0.980376i \(-0.563164\pi\)
−0.197135 + 0.980376i \(0.563164\pi\)
\(14\) 0 0
\(15\) −26528.1 −2.02949
\(16\) 0 0
\(17\) −23898.6 −1.17978 −0.589889 0.807484i \(-0.700829\pi\)
−0.589889 + 0.807484i \(0.700829\pi\)
\(18\) 0 0
\(19\) −6859.00 −0.229416
\(20\) 0 0
\(21\) −8114.45 −0.191202
\(22\) 0 0
\(23\) 50622.1 0.867547 0.433773 0.901022i \(-0.357182\pi\)
0.433773 + 0.901022i \(0.357182\pi\)
\(24\) 0 0
\(25\) 50730.0 0.649344
\(26\) 0 0
\(27\) 80367.1 0.785788
\(28\) 0 0
\(29\) −134788. −1.02626 −0.513130 0.858311i \(-0.671514\pi\)
−0.513130 + 0.858311i \(0.671514\pi\)
\(30\) 0 0
\(31\) 199472. 1.20259 0.601293 0.799029i \(-0.294652\pi\)
0.601293 + 0.799029i \(0.294652\pi\)
\(32\) 0 0
\(33\) −528674. −2.56088
\(34\) 0 0
\(35\) 39414.3 0.155388
\(36\) 0 0
\(37\) −95421.5 −0.309699 −0.154850 0.987938i \(-0.549489\pi\)
−0.154850 + 0.987938i \(0.549489\pi\)
\(38\) 0 0
\(39\) −230808. −0.623052
\(40\) 0 0
\(41\) −488804. −1.10762 −0.553811 0.832642i \(-0.686827\pi\)
−0.553811 + 0.832642i \(0.686827\pi\)
\(42\) 0 0
\(43\) 162905. 0.312460 0.156230 0.987721i \(-0.450066\pi\)
0.156230 + 0.987721i \(0.450066\pi\)
\(44\) 0 0
\(45\) −1.17542e6 −1.92287
\(46\) 0 0
\(47\) −832599. −1.16975 −0.584875 0.811123i \(-0.698857\pi\)
−0.584875 + 0.811123i \(0.698857\pi\)
\(48\) 0 0
\(49\) −811487. −0.985361
\(50\) 0 0
\(51\) −1.76615e6 −1.86437
\(52\) 0 0
\(53\) 953158. 0.879426 0.439713 0.898138i \(-0.355080\pi\)
0.439713 + 0.898138i \(0.355080\pi\)
\(54\) 0 0
\(55\) 2.56793e6 2.08120
\(56\) 0 0
\(57\) −506893. −0.362539
\(58\) 0 0
\(59\) 2.46032e6 1.55958 0.779792 0.626039i \(-0.215325\pi\)
0.779792 + 0.626039i \(0.215325\pi\)
\(60\) 0 0
\(61\) 1.15629e6 0.652245 0.326122 0.945328i \(-0.394258\pi\)
0.326122 + 0.945328i \(0.394258\pi\)
\(62\) 0 0
\(63\) −359539. −0.181157
\(64\) 0 0
\(65\) 1.12110e6 0.506348
\(66\) 0 0
\(67\) −1.59366e6 −0.647344 −0.323672 0.946169i \(-0.604917\pi\)
−0.323672 + 0.946169i \(0.604917\pi\)
\(68\) 0 0
\(69\) 3.74107e6 1.37096
\(70\) 0 0
\(71\) −5.90057e6 −1.95654 −0.978272 0.207326i \(-0.933524\pi\)
−0.978272 + 0.207326i \(0.933524\pi\)
\(72\) 0 0
\(73\) −1.17056e6 −0.352178 −0.176089 0.984374i \(-0.556345\pi\)
−0.176089 + 0.984374i \(0.556345\pi\)
\(74\) 0 0
\(75\) 3.74904e6 1.02614
\(76\) 0 0
\(77\) 785482. 0.196073
\(78\) 0 0
\(79\) 7.58482e6 1.73082 0.865408 0.501069i \(-0.167060\pi\)
0.865408 + 0.501069i \(0.167060\pi\)
\(80\) 0 0
\(81\) −1.22202e6 −0.255494
\(82\) 0 0
\(83\) 7.26944e6 1.39549 0.697747 0.716345i \(-0.254186\pi\)
0.697747 + 0.716345i \(0.254186\pi\)
\(84\) 0 0
\(85\) 8.57871e6 1.51515
\(86\) 0 0
\(87\) −9.96106e6 −1.62177
\(88\) 0 0
\(89\) −7.80137e6 −1.17302 −0.586511 0.809941i \(-0.699499\pi\)
−0.586511 + 0.809941i \(0.699499\pi\)
\(90\) 0 0
\(91\) 342925. 0.0477039
\(92\) 0 0
\(93\) 1.47414e7 1.90041
\(94\) 0 0
\(95\) 2.46213e6 0.294631
\(96\) 0 0
\(97\) −6.56007e6 −0.729806 −0.364903 0.931046i \(-0.618898\pi\)
−0.364903 + 0.931046i \(0.618898\pi\)
\(98\) 0 0
\(99\) −2.34248e7 −2.42634
\(100\) 0 0
\(101\) −149684. −0.0144561 −0.00722805 0.999974i \(-0.502301\pi\)
−0.00722805 + 0.999974i \(0.502301\pi\)
\(102\) 0 0
\(103\) 9.12098e6 0.822453 0.411226 0.911533i \(-0.365101\pi\)
0.411226 + 0.911533i \(0.365101\pi\)
\(104\) 0 0
\(105\) 2.91279e6 0.245554
\(106\) 0 0
\(107\) 8.96589e6 0.707538 0.353769 0.935333i \(-0.384900\pi\)
0.353769 + 0.935333i \(0.384900\pi\)
\(108\) 0 0
\(109\) 1.55911e7 1.15314 0.576572 0.817047i \(-0.304390\pi\)
0.576572 + 0.817047i \(0.304390\pi\)
\(110\) 0 0
\(111\) −7.05182e6 −0.489408
\(112\) 0 0
\(113\) 1.68951e7 1.10150 0.550752 0.834669i \(-0.314341\pi\)
0.550752 + 0.834669i \(0.314341\pi\)
\(114\) 0 0
\(115\) −1.81715e7 −1.11416
\(116\) 0 0
\(117\) −1.02268e7 −0.590320
\(118\) 0 0
\(119\) 2.62407e6 0.142745
\(120\) 0 0
\(121\) 3.16887e7 1.62613
\(122\) 0 0
\(123\) −3.61235e7 −1.75034
\(124\) 0 0
\(125\) 9.83381e6 0.450336
\(126\) 0 0
\(127\) 1.19961e7 0.519668 0.259834 0.965653i \(-0.416332\pi\)
0.259834 + 0.965653i \(0.416332\pi\)
\(128\) 0 0
\(129\) 1.20390e7 0.493771
\(130\) 0 0
\(131\) −1.52350e7 −0.592096 −0.296048 0.955173i \(-0.595669\pi\)
−0.296048 + 0.955173i \(0.595669\pi\)
\(132\) 0 0
\(133\) 753120. 0.0277577
\(134\) 0 0
\(135\) −2.88489e7 −1.00916
\(136\) 0 0
\(137\) −4.60519e7 −1.53012 −0.765060 0.643959i \(-0.777291\pi\)
−0.765060 + 0.643959i \(0.777291\pi\)
\(138\) 0 0
\(139\) −2.26208e7 −0.714424 −0.357212 0.934023i \(-0.616273\pi\)
−0.357212 + 0.934023i \(0.616273\pi\)
\(140\) 0 0
\(141\) −6.15306e7 −1.84852
\(142\) 0 0
\(143\) 2.23423e7 0.638927
\(144\) 0 0
\(145\) 4.83839e7 1.31799
\(146\) 0 0
\(147\) −5.99704e7 −1.55713
\(148\) 0 0
\(149\) 4.97987e7 1.23329 0.616646 0.787240i \(-0.288491\pi\)
0.616646 + 0.787240i \(0.288491\pi\)
\(150\) 0 0
\(151\) −3.51439e7 −0.830675 −0.415337 0.909667i \(-0.636337\pi\)
−0.415337 + 0.909667i \(0.636337\pi\)
\(152\) 0 0
\(153\) −7.82554e7 −1.76642
\(154\) 0 0
\(155\) −7.16032e7 −1.54444
\(156\) 0 0
\(157\) −7.17788e7 −1.48029 −0.740146 0.672447i \(-0.765243\pi\)
−0.740146 + 0.672447i \(0.765243\pi\)
\(158\) 0 0
\(159\) 7.04401e7 1.38973
\(160\) 0 0
\(161\) −5.55833e6 −0.104967
\(162\) 0 0
\(163\) −9.56289e7 −1.72955 −0.864773 0.502162i \(-0.832538\pi\)
−0.864773 + 0.502162i \(0.832538\pi\)
\(164\) 0 0
\(165\) 1.89775e8 3.28885
\(166\) 0 0
\(167\) 7.29897e7 1.21270 0.606351 0.795197i \(-0.292633\pi\)
0.606351 + 0.795197i \(0.292633\pi\)
\(168\) 0 0
\(169\) −5.29944e7 −0.844551
\(170\) 0 0
\(171\) −2.24597e7 −0.343493
\(172\) 0 0
\(173\) 5.12303e7 0.752257 0.376128 0.926568i \(-0.377255\pi\)
0.376128 + 0.926568i \(0.377255\pi\)
\(174\) 0 0
\(175\) −5.57017e6 −0.0785661
\(176\) 0 0
\(177\) 1.81822e8 2.46456
\(178\) 0 0
\(179\) 4.58207e7 0.597140 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(180\) 0 0
\(181\) 7.39469e7 0.926925 0.463463 0.886116i \(-0.346607\pi\)
0.463463 + 0.886116i \(0.346607\pi\)
\(182\) 0 0
\(183\) 8.54516e7 1.03072
\(184\) 0 0
\(185\) 3.42529e7 0.397737
\(186\) 0 0
\(187\) 1.70964e8 1.91187
\(188\) 0 0
\(189\) −8.82433e6 −0.0950748
\(190\) 0 0
\(191\) −1.01526e8 −1.05429 −0.527144 0.849776i \(-0.676737\pi\)
−0.527144 + 0.849776i \(0.676737\pi\)
\(192\) 0 0
\(193\) −4.98679e7 −0.499310 −0.249655 0.968335i \(-0.580317\pi\)
−0.249655 + 0.968335i \(0.580317\pi\)
\(194\) 0 0
\(195\) 8.28516e7 0.800166
\(196\) 0 0
\(197\) −1.39368e8 −1.29876 −0.649382 0.760463i \(-0.724972\pi\)
−0.649382 + 0.760463i \(0.724972\pi\)
\(198\) 0 0
\(199\) −1.01709e8 −0.914898 −0.457449 0.889236i \(-0.651237\pi\)
−0.457449 + 0.889236i \(0.651237\pi\)
\(200\) 0 0
\(201\) −1.17775e8 −1.02298
\(202\) 0 0
\(203\) 1.47997e7 0.124170
\(204\) 0 0
\(205\) 1.75463e8 1.42248
\(206\) 0 0
\(207\) 1.65761e8 1.29893
\(208\) 0 0
\(209\) 4.90674e7 0.371776
\(210\) 0 0
\(211\) 9.48482e7 0.695089 0.347545 0.937663i \(-0.387015\pi\)
0.347545 + 0.937663i \(0.387015\pi\)
\(212\) 0 0
\(213\) −4.36063e8 −3.09187
\(214\) 0 0
\(215\) −5.84769e7 −0.401282
\(216\) 0 0
\(217\) −2.19021e7 −0.145505
\(218\) 0 0
\(219\) −8.65063e7 −0.556536
\(220\) 0 0
\(221\) 7.46391e7 0.465151
\(222\) 0 0
\(223\) 1.13197e8 0.683548 0.341774 0.939782i \(-0.388972\pi\)
0.341774 + 0.939782i \(0.388972\pi\)
\(224\) 0 0
\(225\) 1.66115e8 0.972230
\(226\) 0 0
\(227\) −3.23618e8 −1.83629 −0.918147 0.396241i \(-0.870314\pi\)
−0.918147 + 0.396241i \(0.870314\pi\)
\(228\) 0 0
\(229\) 2.63829e7 0.145177 0.0725887 0.997362i \(-0.476874\pi\)
0.0725887 + 0.997362i \(0.476874\pi\)
\(230\) 0 0
\(231\) 5.80486e7 0.309849
\(232\) 0 0
\(233\) −2.91009e8 −1.50716 −0.753581 0.657355i \(-0.771675\pi\)
−0.753581 + 0.657355i \(0.771675\pi\)
\(234\) 0 0
\(235\) 2.98873e8 1.50227
\(236\) 0 0
\(237\) 5.60533e8 2.73515
\(238\) 0 0
\(239\) −3.00022e8 −1.42155 −0.710773 0.703422i \(-0.751655\pi\)
−0.710773 + 0.703422i \(0.751655\pi\)
\(240\) 0 0
\(241\) 1.30849e8 0.602157 0.301078 0.953599i \(-0.402653\pi\)
0.301078 + 0.953599i \(0.402653\pi\)
\(242\) 0 0
\(243\) −2.66072e8 −1.18954
\(244\) 0 0
\(245\) 2.91294e8 1.26547
\(246\) 0 0
\(247\) 2.14218e7 0.0904517
\(248\) 0 0
\(249\) 5.37225e8 2.20525
\(250\) 0 0
\(251\) −4.29176e8 −1.71308 −0.856539 0.516083i \(-0.827390\pi\)
−0.856539 + 0.516083i \(0.827390\pi\)
\(252\) 0 0
\(253\) −3.62137e8 −1.40589
\(254\) 0 0
\(255\) 6.33983e8 2.39435
\(256\) 0 0
\(257\) −3.14319e8 −1.15506 −0.577530 0.816370i \(-0.695983\pi\)
−0.577530 + 0.816370i \(0.695983\pi\)
\(258\) 0 0
\(259\) 1.04773e7 0.0374715
\(260\) 0 0
\(261\) −4.41360e8 −1.53657
\(262\) 0 0
\(263\) −4.31366e8 −1.46218 −0.731090 0.682281i \(-0.760988\pi\)
−0.731090 + 0.682281i \(0.760988\pi\)
\(264\) 0 0
\(265\) −3.42149e8 −1.12942
\(266\) 0 0
\(267\) −5.76536e8 −1.85369
\(268\) 0 0
\(269\) 1.12548e8 0.352537 0.176269 0.984342i \(-0.443597\pi\)
0.176269 + 0.984342i \(0.443597\pi\)
\(270\) 0 0
\(271\) 4.25465e8 1.29859 0.649294 0.760537i \(-0.275064\pi\)
0.649294 + 0.760537i \(0.275064\pi\)
\(272\) 0 0
\(273\) 2.53428e7 0.0753850
\(274\) 0 0
\(275\) −3.62909e8 −1.05228
\(276\) 0 0
\(277\) −3.22706e8 −0.912280 −0.456140 0.889908i \(-0.650768\pi\)
−0.456140 + 0.889908i \(0.650768\pi\)
\(278\) 0 0
\(279\) 6.53168e8 1.80057
\(280\) 0 0
\(281\) 3.95029e8 1.06208 0.531039 0.847347i \(-0.321802\pi\)
0.531039 + 0.847347i \(0.321802\pi\)
\(282\) 0 0
\(283\) 2.22370e7 0.0583209 0.0291605 0.999575i \(-0.490717\pi\)
0.0291605 + 0.999575i \(0.490717\pi\)
\(284\) 0 0
\(285\) 1.81956e8 0.465597
\(286\) 0 0
\(287\) 5.36709e7 0.134015
\(288\) 0 0
\(289\) 1.60802e8 0.391877
\(290\) 0 0
\(291\) −4.84801e8 −1.15329
\(292\) 0 0
\(293\) −2.90759e8 −0.675299 −0.337649 0.941272i \(-0.609632\pi\)
−0.337649 + 0.941272i \(0.609632\pi\)
\(294\) 0 0
\(295\) −8.83164e8 −2.00292
\(296\) 0 0
\(297\) −5.74925e8 −1.27340
\(298\) 0 0
\(299\) −1.58101e8 −0.342047
\(300\) 0 0
\(301\) −1.78870e7 −0.0378055
\(302\) 0 0
\(303\) −1.10619e7 −0.0228445
\(304\) 0 0
\(305\) −4.15064e8 −0.837657
\(306\) 0 0
\(307\) −3.45260e8 −0.681023 −0.340512 0.940240i \(-0.610600\pi\)
−0.340512 + 0.940240i \(0.610600\pi\)
\(308\) 0 0
\(309\) 6.74057e8 1.29970
\(310\) 0 0
\(311\) −1.53537e8 −0.289435 −0.144718 0.989473i \(-0.546227\pi\)
−0.144718 + 0.989473i \(0.546227\pi\)
\(312\) 0 0
\(313\) 3.62886e8 0.668906 0.334453 0.942412i \(-0.391449\pi\)
0.334453 + 0.942412i \(0.391449\pi\)
\(314\) 0 0
\(315\) 1.29062e8 0.232654
\(316\) 0 0
\(317\) −2.06047e8 −0.363294 −0.181647 0.983364i \(-0.558143\pi\)
−0.181647 + 0.983364i \(0.558143\pi\)
\(318\) 0 0
\(319\) 9.64234e8 1.66309
\(320\) 0 0
\(321\) 6.62596e8 1.11810
\(322\) 0 0
\(323\) 1.63920e8 0.270660
\(324\) 0 0
\(325\) −1.58438e8 −0.256017
\(326\) 0 0
\(327\) 1.15221e9 1.82228
\(328\) 0 0
\(329\) 9.14196e7 0.141532
\(330\) 0 0
\(331\) 5.59579e8 0.848131 0.424066 0.905631i \(-0.360603\pi\)
0.424066 + 0.905631i \(0.360603\pi\)
\(332\) 0 0
\(333\) −3.12456e8 −0.463697
\(334\) 0 0
\(335\) 5.72068e8 0.831363
\(336\) 0 0
\(337\) −4.56120e8 −0.649195 −0.324597 0.945852i \(-0.605229\pi\)
−0.324597 + 0.945852i \(0.605229\pi\)
\(338\) 0 0
\(339\) 1.24858e9 1.74067
\(340\) 0 0
\(341\) −1.42697e9 −1.94883
\(342\) 0 0
\(343\) 1.79527e8 0.240215
\(344\) 0 0
\(345\) −1.34291e9 −1.76068
\(346\) 0 0
\(347\) 9.84759e8 1.26525 0.632626 0.774458i \(-0.281977\pi\)
0.632626 + 0.774458i \(0.281977\pi\)
\(348\) 0 0
\(349\) 6.02914e8 0.759218 0.379609 0.925147i \(-0.376059\pi\)
0.379609 + 0.925147i \(0.376059\pi\)
\(350\) 0 0
\(351\) −2.51000e8 −0.309812
\(352\) 0 0
\(353\) 2.23097e7 0.0269949 0.0134975 0.999909i \(-0.495703\pi\)
0.0134975 + 0.999909i \(0.495703\pi\)
\(354\) 0 0
\(355\) 2.11809e9 2.51273
\(356\) 0 0
\(357\) 1.93924e8 0.225575
\(358\) 0 0
\(359\) −9.61552e8 −1.09684 −0.548418 0.836204i \(-0.684770\pi\)
−0.548418 + 0.836204i \(0.684770\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) 2.34185e9 2.56972
\(364\) 0 0
\(365\) 4.20188e8 0.452291
\(366\) 0 0
\(367\) 6.58001e8 0.694856 0.347428 0.937707i \(-0.387055\pi\)
0.347428 + 0.937707i \(0.387055\pi\)
\(368\) 0 0
\(369\) −1.60058e9 −1.65839
\(370\) 0 0
\(371\) −1.04657e8 −0.106404
\(372\) 0 0
\(373\) −1.20948e9 −1.20675 −0.603373 0.797459i \(-0.706177\pi\)
−0.603373 + 0.797459i \(0.706177\pi\)
\(374\) 0 0
\(375\) 7.26737e8 0.711652
\(376\) 0 0
\(377\) 4.20964e8 0.404623
\(378\) 0 0
\(379\) 1.22771e9 1.15840 0.579198 0.815187i \(-0.303366\pi\)
0.579198 + 0.815187i \(0.303366\pi\)
\(380\) 0 0
\(381\) 8.86531e8 0.821214
\(382\) 0 0
\(383\) −1.21666e9 −1.10655 −0.553276 0.832998i \(-0.686623\pi\)
−0.553276 + 0.832998i \(0.686623\pi\)
\(384\) 0 0
\(385\) −2.81960e8 −0.251811
\(386\) 0 0
\(387\) 5.33429e8 0.467830
\(388\) 0 0
\(389\) 1.34163e9 1.15561 0.577803 0.816176i \(-0.303910\pi\)
0.577803 + 0.816176i \(0.303910\pi\)
\(390\) 0 0
\(391\) −1.20980e9 −1.02351
\(392\) 0 0
\(393\) −1.12589e9 −0.935670
\(394\) 0 0
\(395\) −2.72268e9 −2.22283
\(396\) 0 0
\(397\) −1.91153e9 −1.53325 −0.766626 0.642094i \(-0.778066\pi\)
−0.766626 + 0.642094i \(0.778066\pi\)
\(398\) 0 0
\(399\) 5.56570e7 0.0438647
\(400\) 0 0
\(401\) −1.14237e9 −0.884708 −0.442354 0.896841i \(-0.645856\pi\)
−0.442354 + 0.896841i \(0.645856\pi\)
\(402\) 0 0
\(403\) −6.22984e8 −0.474143
\(404\) 0 0
\(405\) 4.38661e8 0.328123
\(406\) 0 0
\(407\) 6.82619e8 0.501878
\(408\) 0 0
\(409\) 2.00435e9 1.44858 0.724290 0.689496i \(-0.242168\pi\)
0.724290 + 0.689496i \(0.242168\pi\)
\(410\) 0 0
\(411\) −3.40332e9 −2.41800
\(412\) 0 0
\(413\) −2.70143e8 −0.188699
\(414\) 0 0
\(415\) −2.60947e9 −1.79219
\(416\) 0 0
\(417\) −1.67172e9 −1.12898
\(418\) 0 0
\(419\) 1.02610e9 0.681461 0.340730 0.940161i \(-0.389326\pi\)
0.340730 + 0.940161i \(0.389326\pi\)
\(420\) 0 0
\(421\) 2.29081e9 1.49624 0.748122 0.663562i \(-0.230956\pi\)
0.748122 + 0.663562i \(0.230956\pi\)
\(422\) 0 0
\(423\) −2.72633e9 −1.75141
\(424\) 0 0
\(425\) −1.21237e9 −0.766082
\(426\) 0 0
\(427\) −1.26960e8 −0.0789171
\(428\) 0 0
\(429\) 1.65114e9 1.00968
\(430\) 0 0
\(431\) −4.13824e8 −0.248969 −0.124484 0.992222i \(-0.539728\pi\)
−0.124484 + 0.992222i \(0.539728\pi\)
\(432\) 0 0
\(433\) 2.83448e9 1.67790 0.838950 0.544209i \(-0.183170\pi\)
0.838950 + 0.544209i \(0.183170\pi\)
\(434\) 0 0
\(435\) 3.57566e9 2.08278
\(436\) 0 0
\(437\) −3.47217e8 −0.199029
\(438\) 0 0
\(439\) −2.83393e9 −1.59869 −0.799344 0.600874i \(-0.794819\pi\)
−0.799344 + 0.600874i \(0.794819\pi\)
\(440\) 0 0
\(441\) −2.65720e9 −1.47533
\(442\) 0 0
\(443\) 2.13580e9 1.16720 0.583602 0.812040i \(-0.301643\pi\)
0.583602 + 0.812040i \(0.301643\pi\)
\(444\) 0 0
\(445\) 2.80041e9 1.50647
\(446\) 0 0
\(447\) 3.68022e9 1.94893
\(448\) 0 0
\(449\) 8.43253e8 0.439638 0.219819 0.975541i \(-0.429453\pi\)
0.219819 + 0.975541i \(0.429453\pi\)
\(450\) 0 0
\(451\) 3.49677e9 1.79494
\(452\) 0 0
\(453\) −2.59720e9 −1.31269
\(454\) 0 0
\(455\) −1.23098e8 −0.0612646
\(456\) 0 0
\(457\) 2.59270e9 1.27071 0.635354 0.772221i \(-0.280854\pi\)
0.635354 + 0.772221i \(0.280854\pi\)
\(458\) 0 0
\(459\) −1.92066e9 −0.927055
\(460\) 0 0
\(461\) 1.25804e9 0.598053 0.299027 0.954245i \(-0.403338\pi\)
0.299027 + 0.954245i \(0.403338\pi\)
\(462\) 0 0
\(463\) −2.70904e9 −1.26847 −0.634236 0.773139i \(-0.718685\pi\)
−0.634236 + 0.773139i \(0.718685\pi\)
\(464\) 0 0
\(465\) −5.29161e9 −2.44063
\(466\) 0 0
\(467\) −1.94781e9 −0.884987 −0.442494 0.896772i \(-0.645906\pi\)
−0.442494 + 0.896772i \(0.645906\pi\)
\(468\) 0 0
\(469\) 1.74985e8 0.0783241
\(470\) 0 0
\(471\) −5.30458e9 −2.33926
\(472\) 0 0
\(473\) −1.16538e9 −0.506352
\(474\) 0 0
\(475\) −3.47957e8 −0.148970
\(476\) 0 0
\(477\) 3.12110e9 1.31672
\(478\) 0 0
\(479\) −1.46097e9 −0.607390 −0.303695 0.952769i \(-0.598220\pi\)
−0.303695 + 0.952769i \(0.598220\pi\)
\(480\) 0 0
\(481\) 2.98017e8 0.122105
\(482\) 0 0
\(483\) −4.10771e8 −0.165876
\(484\) 0 0
\(485\) 2.35483e9 0.937266
\(486\) 0 0
\(487\) 1.26613e9 0.496736 0.248368 0.968666i \(-0.420106\pi\)
0.248368 + 0.968666i \(0.420106\pi\)
\(488\) 0 0
\(489\) −7.06715e9 −2.73315
\(490\) 0 0
\(491\) −3.83771e9 −1.46314 −0.731572 0.681765i \(-0.761213\pi\)
−0.731572 + 0.681765i \(0.761213\pi\)
\(492\) 0 0
\(493\) 3.22123e9 1.21076
\(494\) 0 0
\(495\) 8.40865e9 3.11607
\(496\) 0 0
\(497\) 6.47884e8 0.236728
\(498\) 0 0
\(499\) −3.49904e9 −1.26066 −0.630328 0.776329i \(-0.717080\pi\)
−0.630328 + 0.776329i \(0.717080\pi\)
\(500\) 0 0
\(501\) 5.39408e9 1.91640
\(502\) 0 0
\(503\) −2.11713e9 −0.741754 −0.370877 0.928682i \(-0.620943\pi\)
−0.370877 + 0.928682i \(0.620943\pi\)
\(504\) 0 0
\(505\) 5.37312e7 0.0185655
\(506\) 0 0
\(507\) −3.91638e9 −1.33462
\(508\) 0 0
\(509\) −1.01744e9 −0.341977 −0.170988 0.985273i \(-0.554696\pi\)
−0.170988 + 0.985273i \(0.554696\pi\)
\(510\) 0 0
\(511\) 1.28528e8 0.0426111
\(512\) 0 0
\(513\) −5.51238e8 −0.180272
\(514\) 0 0
\(515\) −3.27410e9 −1.05625
\(516\) 0 0
\(517\) 5.95619e9 1.89562
\(518\) 0 0
\(519\) 3.78602e9 1.18877
\(520\) 0 0
\(521\) −2.02605e9 −0.627650 −0.313825 0.949481i \(-0.601611\pi\)
−0.313825 + 0.949481i \(0.601611\pi\)
\(522\) 0 0
\(523\) −1.04115e8 −0.0318241 −0.0159120 0.999873i \(-0.505065\pi\)
−0.0159120 + 0.999873i \(0.505065\pi\)
\(524\) 0 0
\(525\) −4.11646e8 −0.124156
\(526\) 0 0
\(527\) −4.76709e9 −1.41878
\(528\) 0 0
\(529\) −8.42227e8 −0.247363
\(530\) 0 0
\(531\) 8.05626e9 2.33509
\(532\) 0 0
\(533\) 1.52662e9 0.436702
\(534\) 0 0
\(535\) −3.21843e9 −0.908669
\(536\) 0 0
\(537\) 3.38623e9 0.943641
\(538\) 0 0
\(539\) 5.80516e9 1.59681
\(540\) 0 0
\(541\) 2.86709e9 0.778487 0.389243 0.921135i \(-0.372737\pi\)
0.389243 + 0.921135i \(0.372737\pi\)
\(542\) 0 0
\(543\) 5.46481e9 1.46479
\(544\) 0 0
\(545\) −5.59663e9 −1.48095
\(546\) 0 0
\(547\) 4.20539e9 1.09863 0.549313 0.835616i \(-0.314889\pi\)
0.549313 + 0.835616i \(0.314889\pi\)
\(548\) 0 0
\(549\) 3.78624e9 0.976573
\(550\) 0 0
\(551\) 9.24508e8 0.235440
\(552\) 0 0
\(553\) −8.32816e8 −0.209417
\(554\) 0 0
\(555\) 2.53135e9 0.628531
\(556\) 0 0
\(557\) 6.90397e9 1.69280 0.846400 0.532548i \(-0.178765\pi\)
0.846400 + 0.532548i \(0.178765\pi\)
\(558\) 0 0
\(559\) −5.08779e8 −0.123193
\(560\) 0 0
\(561\) 1.26345e10 3.02127
\(562\) 0 0
\(563\) −2.07530e9 −0.490119 −0.245060 0.969508i \(-0.578808\pi\)
−0.245060 + 0.969508i \(0.578808\pi\)
\(564\) 0 0
\(565\) −6.06473e9 −1.41463
\(566\) 0 0
\(567\) 1.34178e8 0.0309130
\(568\) 0 0
\(569\) 3.50410e9 0.797414 0.398707 0.917078i \(-0.369459\pi\)
0.398707 + 0.917078i \(0.369459\pi\)
\(570\) 0 0
\(571\) 5.15559e9 1.15892 0.579458 0.815002i \(-0.303264\pi\)
0.579458 + 0.815002i \(0.303264\pi\)
\(572\) 0 0
\(573\) −7.50294e9 −1.66606
\(574\) 0 0
\(575\) 2.56806e9 0.563336
\(576\) 0 0
\(577\) −6.08613e9 −1.31894 −0.659471 0.751730i \(-0.729220\pi\)
−0.659471 + 0.751730i \(0.729220\pi\)
\(578\) 0 0
\(579\) −3.68533e9 −0.789045
\(580\) 0 0
\(581\) −7.98187e8 −0.168845
\(582\) 0 0
\(583\) −6.81863e9 −1.42514
\(584\) 0 0
\(585\) 3.67104e9 0.758129
\(586\) 0 0
\(587\) 3.10318e9 0.633247 0.316623 0.948551i \(-0.397451\pi\)
0.316623 + 0.948551i \(0.397451\pi\)
\(588\) 0 0
\(589\) −1.36818e9 −0.275892
\(590\) 0 0
\(591\) −1.02995e10 −2.05240
\(592\) 0 0
\(593\) 8.15698e9 1.60634 0.803170 0.595749i \(-0.203145\pi\)
0.803170 + 0.595749i \(0.203145\pi\)
\(594\) 0 0
\(595\) −9.41946e8 −0.183323
\(596\) 0 0
\(597\) −7.51647e9 −1.44578
\(598\) 0 0
\(599\) −3.83681e9 −0.729418 −0.364709 0.931122i \(-0.618831\pi\)
−0.364709 + 0.931122i \(0.618831\pi\)
\(600\) 0 0
\(601\) −5.74255e9 −1.07906 −0.539529 0.841967i \(-0.681398\pi\)
−0.539529 + 0.841967i \(0.681398\pi\)
\(602\) 0 0
\(603\) −5.21843e9 −0.969235
\(604\) 0 0
\(605\) −1.13751e10 −2.08839
\(606\) 0 0
\(607\) 2.66716e9 0.484048 0.242024 0.970270i \(-0.422189\pi\)
0.242024 + 0.970270i \(0.422189\pi\)
\(608\) 0 0
\(609\) 1.09373e9 0.196222
\(610\) 0 0
\(611\) 2.60034e9 0.461197
\(612\) 0 0
\(613\) 2.97013e9 0.520791 0.260396 0.965502i \(-0.416147\pi\)
0.260396 + 0.965502i \(0.416147\pi\)
\(614\) 0 0
\(615\) 1.29670e10 2.24791
\(616\) 0 0
\(617\) −6.26837e9 −1.07438 −0.537189 0.843462i \(-0.680514\pi\)
−0.537189 + 0.843462i \(0.680514\pi\)
\(618\) 0 0
\(619\) −6.46559e9 −1.09570 −0.547849 0.836577i \(-0.684553\pi\)
−0.547849 + 0.836577i \(0.684553\pi\)
\(620\) 0 0
\(621\) 4.06835e9 0.681707
\(622\) 0 0
\(623\) 8.56594e8 0.141928
\(624\) 0 0
\(625\) −7.49326e9 −1.22770
\(626\) 0 0
\(627\) 3.62617e9 0.587506
\(628\) 0 0
\(629\) 2.28043e9 0.365376
\(630\) 0 0
\(631\) −9.86287e9 −1.56279 −0.781395 0.624037i \(-0.785491\pi\)
−0.781395 + 0.624037i \(0.785491\pi\)
\(632\) 0 0
\(633\) 7.00946e9 1.09843
\(634\) 0 0
\(635\) −4.30615e9 −0.667392
\(636\) 0 0
\(637\) 2.53441e9 0.388498
\(638\) 0 0
\(639\) −1.93213e10 −2.92943
\(640\) 0 0
\(641\) 1.04849e10 1.57240 0.786198 0.617974i \(-0.212046\pi\)
0.786198 + 0.617974i \(0.212046\pi\)
\(642\) 0 0
\(643\) −9.27539e8 −0.137592 −0.0687961 0.997631i \(-0.521916\pi\)
−0.0687961 + 0.997631i \(0.521916\pi\)
\(644\) 0 0
\(645\) −4.32155e9 −0.634134
\(646\) 0 0
\(647\) 1.07688e10 1.56316 0.781578 0.623808i \(-0.214415\pi\)
0.781578 + 0.623808i \(0.214415\pi\)
\(648\) 0 0
\(649\) −1.76004e10 −2.52736
\(650\) 0 0
\(651\) −1.61861e9 −0.229936
\(652\) 0 0
\(653\) 7.05077e9 0.990925 0.495462 0.868629i \(-0.334999\pi\)
0.495462 + 0.868629i \(0.334999\pi\)
\(654\) 0 0
\(655\) 5.46880e9 0.760409
\(656\) 0 0
\(657\) −3.83297e9 −0.527299
\(658\) 0 0
\(659\) 9.71506e8 0.132235 0.0661175 0.997812i \(-0.478939\pi\)
0.0661175 + 0.997812i \(0.478939\pi\)
\(660\) 0 0
\(661\) −7.03635e9 −0.947637 −0.473819 0.880622i \(-0.657125\pi\)
−0.473819 + 0.880622i \(0.657125\pi\)
\(662\) 0 0
\(663\) 5.51597e9 0.735063
\(664\) 0 0
\(665\) −2.70343e8 −0.0356483
\(666\) 0 0
\(667\) −6.82324e9 −0.890328
\(668\) 0 0
\(669\) 8.36548e9 1.08019
\(670\) 0 0
\(671\) −8.27175e9 −1.05698
\(672\) 0 0
\(673\) 9.35466e9 1.18297 0.591487 0.806314i \(-0.298541\pi\)
0.591487 + 0.806314i \(0.298541\pi\)
\(674\) 0 0
\(675\) 4.07702e9 0.510246
\(676\) 0 0
\(677\) 5.64449e9 0.699140 0.349570 0.936910i \(-0.386328\pi\)
0.349570 + 0.936910i \(0.386328\pi\)
\(678\) 0 0
\(679\) 7.20298e8 0.0883015
\(680\) 0 0
\(681\) −2.39160e10 −2.90184
\(682\) 0 0
\(683\) −1.50975e10 −1.81314 −0.906571 0.422054i \(-0.861309\pi\)
−0.906571 + 0.422054i \(0.861309\pi\)
\(684\) 0 0
\(685\) 1.65310e10 1.96508
\(686\) 0 0
\(687\) 1.94975e9 0.229419
\(688\) 0 0
\(689\) −2.97687e9 −0.346731
\(690\) 0 0
\(691\) −1.25008e9 −0.144133 −0.0720666 0.997400i \(-0.522959\pi\)
−0.0720666 + 0.997400i \(0.522959\pi\)
\(692\) 0 0
\(693\) 2.57205e9 0.293571
\(694\) 0 0
\(695\) 8.12004e9 0.917511
\(696\) 0 0
\(697\) 1.16817e10 1.30675
\(698\) 0 0
\(699\) −2.15061e10 −2.38172
\(700\) 0 0
\(701\) 4.65921e9 0.510857 0.255428 0.966828i \(-0.417784\pi\)
0.255428 + 0.966828i \(0.417784\pi\)
\(702\) 0 0
\(703\) 6.54496e8 0.0710499
\(704\) 0 0
\(705\) 2.20873e10 2.37399
\(706\) 0 0
\(707\) 1.64354e7 0.00174909
\(708\) 0 0
\(709\) −5.70761e9 −0.601441 −0.300720 0.953712i \(-0.597227\pi\)
−0.300720 + 0.953712i \(0.597227\pi\)
\(710\) 0 0
\(711\) 2.48364e10 2.59146
\(712\) 0 0
\(713\) 1.00977e10 1.04330
\(714\) 0 0
\(715\) −8.02007e9 −0.820554
\(716\) 0 0
\(717\) −2.21722e10 −2.24642
\(718\) 0 0
\(719\) 1.08664e10 1.09027 0.545136 0.838347i \(-0.316478\pi\)
0.545136 + 0.838347i \(0.316478\pi\)
\(720\) 0 0
\(721\) −1.00149e9 −0.0995111
\(722\) 0 0
\(723\) 9.66996e9 0.951569
\(724\) 0 0
\(725\) −6.83778e9 −0.666395
\(726\) 0 0
\(727\) −3.49074e9 −0.336936 −0.168468 0.985707i \(-0.553882\pi\)
−0.168468 + 0.985707i \(0.553882\pi\)
\(728\) 0 0
\(729\) −1.69907e10 −1.62429
\(730\) 0 0
\(731\) −3.89319e9 −0.368633
\(732\) 0 0
\(733\) −1.94782e9 −0.182677 −0.0913386 0.995820i \(-0.529115\pi\)
−0.0913386 + 0.995820i \(0.529115\pi\)
\(734\) 0 0
\(735\) 2.15272e10 1.99978
\(736\) 0 0
\(737\) 1.14006e10 1.04904
\(738\) 0 0
\(739\) −5.89076e9 −0.536928 −0.268464 0.963290i \(-0.586516\pi\)
−0.268464 + 0.963290i \(0.586516\pi\)
\(740\) 0 0
\(741\) 1.58311e9 0.142938
\(742\) 0 0
\(743\) 1.25115e10 1.11905 0.559525 0.828813i \(-0.310984\pi\)
0.559525 + 0.828813i \(0.310984\pi\)
\(744\) 0 0
\(745\) −1.78759e10 −1.58388
\(746\) 0 0
\(747\) 2.38037e10 2.08940
\(748\) 0 0
\(749\) −9.84457e8 −0.0856072
\(750\) 0 0
\(751\) 1.24647e10 1.07385 0.536925 0.843630i \(-0.319586\pi\)
0.536925 + 0.843630i \(0.319586\pi\)
\(752\) 0 0
\(753\) −3.17169e10 −2.70712
\(754\) 0 0
\(755\) 1.26154e10 1.06681
\(756\) 0 0
\(757\) 1.27300e10 1.06658 0.533290 0.845933i \(-0.320956\pi\)
0.533290 + 0.845933i \(0.320956\pi\)
\(758\) 0 0
\(759\) −2.67626e10 −2.22168
\(760\) 0 0
\(761\) −2.03284e10 −1.67208 −0.836041 0.548666i \(-0.815136\pi\)
−0.836041 + 0.548666i \(0.815136\pi\)
\(762\) 0 0
\(763\) −1.71191e9 −0.139522
\(764\) 0 0
\(765\) 2.80909e10 2.26856
\(766\) 0 0
\(767\) −7.68397e9 −0.614897
\(768\) 0 0
\(769\) −9.40790e9 −0.746020 −0.373010 0.927827i \(-0.621674\pi\)
−0.373010 + 0.927827i \(0.621674\pi\)
\(770\) 0 0
\(771\) −2.32287e10 −1.82530
\(772\) 0 0
\(773\) 3.86805e9 0.301206 0.150603 0.988594i \(-0.451878\pi\)
0.150603 + 0.988594i \(0.451878\pi\)
\(774\) 0 0
\(775\) 1.01192e10 0.780892
\(776\) 0 0
\(777\) 7.74293e8 0.0592150
\(778\) 0 0
\(779\) 3.35271e9 0.254106
\(780\) 0 0
\(781\) 4.22111e10 3.17065
\(782\) 0 0
\(783\) −1.08325e10 −0.806422
\(784\) 0 0
\(785\) 2.57660e10 1.90109
\(786\) 0 0
\(787\) 2.08450e10 1.52437 0.762185 0.647359i \(-0.224127\pi\)
0.762185 + 0.647359i \(0.224127\pi\)
\(788\) 0 0
\(789\) −3.18787e10 −2.31064
\(790\) 0 0
\(791\) −1.85509e9 −0.133274
\(792\) 0 0
\(793\) −3.61127e9 −0.257160
\(794\) 0 0
\(795\) −2.52854e10 −1.78478
\(796\) 0 0
\(797\) 2.08677e9 0.146006 0.0730028 0.997332i \(-0.476742\pi\)
0.0730028 + 0.997332i \(0.476742\pi\)
\(798\) 0 0
\(799\) 1.98979e10 1.38005
\(800\) 0 0
\(801\) −2.55455e10 −1.75631
\(802\) 0 0
\(803\) 8.37385e9 0.570717
\(804\) 0 0
\(805\) 1.99524e9 0.134806
\(806\) 0 0
\(807\) 8.31751e9 0.557104
\(808\) 0 0
\(809\) −1.85609e10 −1.23248 −0.616238 0.787560i \(-0.711344\pi\)
−0.616238 + 0.787560i \(0.711344\pi\)
\(810\) 0 0
\(811\) 1.76811e10 1.16396 0.581979 0.813204i \(-0.302279\pi\)
0.581979 + 0.813204i \(0.302279\pi\)
\(812\) 0 0
\(813\) 3.14427e10 2.05212
\(814\) 0 0
\(815\) 3.43273e10 2.22120
\(816\) 0 0
\(817\) −1.11736e9 −0.0716832
\(818\) 0 0
\(819\) 1.12290e9 0.0714246
\(820\) 0 0
\(821\) 2.32701e10 1.46757 0.733784 0.679383i \(-0.237753\pi\)
0.733784 + 0.679383i \(0.237753\pi\)
\(822\) 0 0
\(823\) −1.10963e10 −0.693874 −0.346937 0.937888i \(-0.612778\pi\)
−0.346937 + 0.937888i \(0.612778\pi\)
\(824\) 0 0
\(825\) −2.68196e10 −1.66289
\(826\) 0 0
\(827\) 6.85391e9 0.421375 0.210688 0.977553i \(-0.432430\pi\)
0.210688 + 0.977553i \(0.432430\pi\)
\(828\) 0 0
\(829\) −1.93058e9 −0.117692 −0.0588459 0.998267i \(-0.518742\pi\)
−0.0588459 + 0.998267i \(0.518742\pi\)
\(830\) 0 0
\(831\) −2.38486e10 −1.44165
\(832\) 0 0
\(833\) 1.93934e10 1.16251
\(834\) 0 0
\(835\) −2.62007e10 −1.55743
\(836\) 0 0
\(837\) 1.60310e10 0.944977
\(838\) 0 0
\(839\) −1.22571e10 −0.716508 −0.358254 0.933624i \(-0.616628\pi\)
−0.358254 + 0.933624i \(0.616628\pi\)
\(840\) 0 0
\(841\) 9.17827e8 0.0532077
\(842\) 0 0
\(843\) 2.91934e10 1.67837
\(844\) 0 0
\(845\) 1.90231e10 1.08463
\(846\) 0 0
\(847\) −3.47943e9 −0.196751
\(848\) 0 0
\(849\) 1.64336e9 0.0921627
\(850\) 0 0
\(851\) −4.83044e9 −0.268679
\(852\) 0 0
\(853\) −2.04007e10 −1.12545 −0.562723 0.826646i \(-0.690246\pi\)
−0.562723 + 0.826646i \(0.690246\pi\)
\(854\) 0 0
\(855\) 8.06221e9 0.441136
\(856\) 0 0
\(857\) 1.22548e10 0.665080 0.332540 0.943089i \(-0.392094\pi\)
0.332540 + 0.943089i \(0.392094\pi\)
\(858\) 0 0
\(859\) −9.32872e9 −0.502164 −0.251082 0.967966i \(-0.580786\pi\)
−0.251082 + 0.967966i \(0.580786\pi\)
\(860\) 0 0
\(861\) 3.96638e9 0.211779
\(862\) 0 0
\(863\) 5.79054e9 0.306677 0.153339 0.988174i \(-0.450998\pi\)
0.153339 + 0.988174i \(0.450998\pi\)
\(864\) 0 0
\(865\) −1.83898e10 −0.966099
\(866\) 0 0
\(867\) 1.18836e10 0.619270
\(868\) 0 0
\(869\) −5.42598e10 −2.80484
\(870\) 0 0
\(871\) 4.97728e9 0.255228
\(872\) 0 0
\(873\) −2.14808e10 −1.09270
\(874\) 0 0
\(875\) −1.07976e9 −0.0544876
\(876\) 0 0
\(877\) −1.87189e10 −0.937090 −0.468545 0.883440i \(-0.655222\pi\)
−0.468545 + 0.883440i \(0.655222\pi\)
\(878\) 0 0
\(879\) −2.14876e10 −1.06715
\(880\) 0 0
\(881\) 1.71691e10 0.845927 0.422964 0.906147i \(-0.360990\pi\)
0.422964 + 0.906147i \(0.360990\pi\)
\(882\) 0 0
\(883\) −5.07048e9 −0.247848 −0.123924 0.992292i \(-0.539548\pi\)
−0.123924 + 0.992292i \(0.539548\pi\)
\(884\) 0 0
\(885\) −6.52675e10 −3.16516
\(886\) 0 0
\(887\) 2.31390e10 1.11330 0.556650 0.830747i \(-0.312086\pi\)
0.556650 + 0.830747i \(0.312086\pi\)
\(888\) 0 0
\(889\) −1.31717e9 −0.0628762
\(890\) 0 0
\(891\) 8.74199e9 0.414036
\(892\) 0 0
\(893\) 5.71079e9 0.268359
\(894\) 0 0
\(895\) −1.64480e10 −0.766887
\(896\) 0 0
\(897\) −1.16840e10 −0.540527
\(898\) 0 0
\(899\) −2.68864e10 −1.23416
\(900\) 0 0
\(901\) −2.27791e10 −1.03753
\(902\) 0 0
\(903\) −1.32188e9 −0.0597428
\(904\) 0 0
\(905\) −2.65443e10 −1.19042
\(906\) 0 0
\(907\) 1.75358e10 0.780369 0.390184 0.920737i \(-0.372411\pi\)
0.390184 + 0.920737i \(0.372411\pi\)
\(908\) 0 0
\(909\) −4.90138e8 −0.0216444
\(910\) 0 0
\(911\) 2.89796e10 1.26992 0.634962 0.772543i \(-0.281016\pi\)
0.634962 + 0.772543i \(0.281016\pi\)
\(912\) 0 0
\(913\) −5.20036e10 −2.26144
\(914\) 0 0
\(915\) −3.06740e10 −1.32372
\(916\) 0 0
\(917\) 1.67280e9 0.0716395
\(918\) 0 0
\(919\) 2.54457e10 1.08146 0.540729 0.841197i \(-0.318148\pi\)
0.540729 + 0.841197i \(0.318148\pi\)
\(920\) 0 0
\(921\) −2.55154e10 −1.07620
\(922\) 0 0
\(923\) 1.84285e10 0.771406
\(924\) 0 0
\(925\) −4.84073e9 −0.201101
\(926\) 0 0
\(927\) 2.98665e10 1.23142
\(928\) 0 0
\(929\) 3.45187e10 1.41254 0.706268 0.707945i \(-0.250378\pi\)
0.706268 + 0.707945i \(0.250378\pi\)
\(930\) 0 0
\(931\) 5.56599e9 0.226057
\(932\) 0 0
\(933\) −1.13467e10 −0.457385
\(934\) 0 0
\(935\) −6.13698e10 −2.45535
\(936\) 0 0
\(937\) 1.01764e10 0.404116 0.202058 0.979374i \(-0.435237\pi\)
0.202058 + 0.979374i \(0.435237\pi\)
\(938\) 0 0
\(939\) 2.68179e10 1.05705
\(940\) 0 0
\(941\) 5.96144e9 0.233232 0.116616 0.993177i \(-0.462795\pi\)
0.116616 + 0.993177i \(0.462795\pi\)
\(942\) 0 0
\(943\) −2.47443e10 −0.960914
\(944\) 0 0
\(945\) 3.16762e9 0.122102
\(946\) 0 0
\(947\) −4.69268e10 −1.79554 −0.897772 0.440461i \(-0.854815\pi\)
−0.897772 + 0.440461i \(0.854815\pi\)
\(948\) 0 0
\(949\) 3.65584e9 0.138853
\(950\) 0 0
\(951\) −1.52273e10 −0.574103
\(952\) 0 0
\(953\) −5.55628e9 −0.207950 −0.103975 0.994580i \(-0.533156\pi\)
−0.103975 + 0.994580i \(0.533156\pi\)
\(954\) 0 0
\(955\) 3.64440e10 1.35399
\(956\) 0 0
\(957\) 7.12587e10 2.62813
\(958\) 0 0
\(959\) 5.05652e9 0.185134
\(960\) 0 0
\(961\) 1.22765e10 0.446213
\(962\) 0 0
\(963\) 2.93587e10 1.05936
\(964\) 0 0
\(965\) 1.79008e10 0.641248
\(966\) 0 0
\(967\) 2.14955e10 0.764461 0.382231 0.924067i \(-0.375156\pi\)
0.382231 + 0.924067i \(0.375156\pi\)
\(968\) 0 0
\(969\) 1.21140e10 0.427715
\(970\) 0 0
\(971\) −1.64141e10 −0.575375 −0.287688 0.957724i \(-0.592886\pi\)
−0.287688 + 0.957724i \(0.592886\pi\)
\(972\) 0 0
\(973\) 2.48377e9 0.0864403
\(974\) 0 0
\(975\) −1.17089e10 −0.404575
\(976\) 0 0
\(977\) 3.51550e10 1.20602 0.603012 0.797732i \(-0.293967\pi\)
0.603012 + 0.797732i \(0.293967\pi\)
\(978\) 0 0
\(979\) 5.58089e10 1.90092
\(980\) 0 0
\(981\) 5.10527e10 1.72654
\(982\) 0 0
\(983\) −1.34333e10 −0.451073 −0.225536 0.974235i \(-0.572413\pi\)
−0.225536 + 0.974235i \(0.572413\pi\)
\(984\) 0 0
\(985\) 5.00279e10 1.66796
\(986\) 0 0
\(987\) 6.75608e9 0.223658
\(988\) 0 0
\(989\) 8.24659e9 0.271074
\(990\) 0 0
\(991\) −4.25370e10 −1.38838 −0.694191 0.719791i \(-0.744238\pi\)
−0.694191 + 0.719791i \(0.744238\pi\)
\(992\) 0 0
\(993\) 4.13539e10 1.34028
\(994\) 0 0
\(995\) 3.65098e10 1.17497
\(996\) 0 0
\(997\) 2.44581e10 0.781609 0.390805 0.920474i \(-0.372197\pi\)
0.390805 + 0.920474i \(0.372197\pi\)
\(998\) 0 0
\(999\) −7.66875e9 −0.243358
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 76.8.a.a.1.5 5
4.3 odd 2 304.8.a.g.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.8.a.a.1.5 5 1.1 even 1 trivial
304.8.a.g.1.1 5 4.3 odd 2