Properties

Label 76.8.a.a.1.1
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 1.1
Root \(57.9656\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q-84.8970 q^{3} +72.4872 q^{5} -210.006 q^{7} +5020.49 q^{9} +O(q^{10})\) \(q-84.8970 q^{3} +72.4872 q^{5} -210.006 q^{7} +5020.49 q^{9} +2350.13 q^{11} +2219.57 q^{13} -6153.94 q^{15} +22406.1 q^{17} -6859.00 q^{19} +17828.8 q^{21} -33836.7 q^{23} -72870.6 q^{25} -240555. q^{27} -93654.9 q^{29} -222646. q^{31} -199519. q^{33} -15222.7 q^{35} +228916. q^{37} -188435. q^{39} +67353.8 q^{41} +162691. q^{43} +363922. q^{45} +81632.8 q^{47} -779441. q^{49} -1.90221e6 q^{51} +905874. q^{53} +170354. q^{55} +582308. q^{57} -1.42600e6 q^{59} +2.18670e6 q^{61} -1.05433e6 q^{63} +160891. q^{65} -4.73974e6 q^{67} +2.87263e6 q^{69} -3.58523e6 q^{71} -3.22283e6 q^{73} +6.18649e6 q^{75} -493541. q^{77} +334481. q^{79} +9.44257e6 q^{81} +9.68747e6 q^{83} +1.62416e6 q^{85} +7.95102e6 q^{87} -6.74280e6 q^{89} -466123. q^{91} +1.89020e7 q^{93} -497190. q^{95} -483068. q^{97} +1.17988e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} + O(q^{10}) \) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} - 2662q^{11} - 602q^{13} - 20800q^{15} - 27366q^{17} - 34295q^{19} - 59964q^{21} - 67096q^{23} - 109115q^{25} - 178778q^{27} - 372398q^{29} - 271372q^{31} - 792700q^{33} - 608250q^{35} - 562630q^{37} - 963904q^{39} - 956714q^{41} - 827362q^{43} - 1165100q^{45} - 1812982q^{47} - 862031q^{49} - 2458254q^{51} + 486998q^{53} + 467930q^{55} + 96026q^{57} - 367182q^{59} + 1879732q^{61} - 1007274q^{63} + 1790920q^{65} - 1046394q^{67} + 7261712q^{69} - 4664572q^{71} + 4224942q^{73} + 8194850q^{75} + 8611110q^{77} + 9574024q^{79} + 11351813q^{81} + 11754804q^{83} + 18711750q^{85} + 3801472q^{87} + 2782542q^{89} + 7385214q^{91} + 29535004q^{93} + 1920520q^{95} + 1291574q^{97} - 9760310q^{99} + O(q^{100}) \)

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
\(3\) −84.8970 −1.81538 −0.907690 0.419640i \(-0.862156\pi\)
−0.907690 + 0.419640i \(0.862156\pi\)
\(4\) 0 0
\(5\) 72.4872 0.259338 0.129669 0.991557i \(-0.458608\pi\)
0.129669 + 0.991557i \(0.458608\pi\)
\(6\) 0 0
\(7\) −210.006 −0.231413 −0.115707 0.993283i \(-0.536913\pi\)
−0.115707 + 0.993283i \(0.536913\pi\)
\(8\) 0 0
\(9\) 5020.49 2.29561
\(10\) 0 0
\(11\) 2350.13 0.532375 0.266188 0.963921i \(-0.414236\pi\)
0.266188 + 0.963921i \(0.414236\pi\)
\(12\) 0 0
\(13\) 2219.57 0.280200 0.140100 0.990137i \(-0.455258\pi\)
0.140100 + 0.990137i \(0.455258\pi\)
\(14\) 0 0
\(15\) −6153.94 −0.470798
\(16\) 0 0
\(17\) 22406.1 1.10610 0.553051 0.833147i \(-0.313464\pi\)
0.553051 + 0.833147i \(0.313464\pi\)
\(18\) 0 0
\(19\) −6859.00 −0.229416
\(20\) 0 0
\(21\) 17828.8 0.420103
\(22\) 0 0
\(23\) −33836.7 −0.579883 −0.289941 0.957044i \(-0.593636\pi\)
−0.289941 + 0.957044i \(0.593636\pi\)
\(24\) 0 0
\(25\) −72870.6 −0.932744
\(26\) 0 0
\(27\) −240555. −2.35202
\(28\) 0 0
\(29\) −93654.9 −0.713079 −0.356539 0.934280i \(-0.616043\pi\)
−0.356539 + 0.934280i \(0.616043\pi\)
\(30\) 0 0
\(31\) −222646. −1.34230 −0.671150 0.741322i \(-0.734199\pi\)
−0.671150 + 0.741322i \(0.734199\pi\)
\(32\) 0 0
\(33\) −199519. −0.966464
\(34\) 0 0
\(35\) −15222.7 −0.0600142
\(36\) 0 0
\(37\) 228916. 0.742968 0.371484 0.928439i \(-0.378849\pi\)
0.371484 + 0.928439i \(0.378849\pi\)
\(38\) 0 0
\(39\) −188435. −0.508669
\(40\) 0 0
\(41\) 67353.8 0.152623 0.0763113 0.997084i \(-0.475686\pi\)
0.0763113 + 0.997084i \(0.475686\pi\)
\(42\) 0 0
\(43\) 162691. 0.312049 0.156025 0.987753i \(-0.450132\pi\)
0.156025 + 0.987753i \(0.450132\pi\)
\(44\) 0 0
\(45\) 363922. 0.595339
\(46\) 0 0
\(47\) 81632.8 0.114689 0.0573445 0.998354i \(-0.481737\pi\)
0.0573445 + 0.998354i \(0.481737\pi\)
\(48\) 0 0
\(49\) −779441. −0.946448
\(50\) 0 0
\(51\) −1.90221e6 −2.00800
\(52\) 0 0
\(53\) 905874. 0.835800 0.417900 0.908493i \(-0.362766\pi\)
0.417900 + 0.908493i \(0.362766\pi\)
\(54\) 0 0
\(55\) 170354. 0.138065
\(56\) 0 0
\(57\) 582308. 0.416477
\(58\) 0 0
\(59\) −1.42600e6 −0.903934 −0.451967 0.892035i \(-0.649277\pi\)
−0.451967 + 0.892035i \(0.649277\pi\)
\(60\) 0 0
\(61\) 2.18670e6 1.23349 0.616745 0.787163i \(-0.288451\pi\)
0.616745 + 0.787163i \(0.288451\pi\)
\(62\) 0 0
\(63\) −1.05433e6 −0.531234
\(64\) 0 0
\(65\) 160891. 0.0726665
\(66\) 0 0
\(67\) −4.73974e6 −1.92527 −0.962636 0.270797i \(-0.912713\pi\)
−0.962636 + 0.270797i \(0.912713\pi\)
\(68\) 0 0
\(69\) 2.87263e6 1.05271
\(70\) 0 0
\(71\) −3.58523e6 −1.18881 −0.594406 0.804165i \(-0.702613\pi\)
−0.594406 + 0.804165i \(0.702613\pi\)
\(72\) 0 0
\(73\) −3.22283e6 −0.969632 −0.484816 0.874616i \(-0.661113\pi\)
−0.484816 + 0.874616i \(0.661113\pi\)
\(74\) 0 0
\(75\) 6.18649e6 1.69329
\(76\) 0 0
\(77\) −493541. −0.123199
\(78\) 0 0
\(79\) 334481. 0.0763268 0.0381634 0.999272i \(-0.487849\pi\)
0.0381634 + 0.999272i \(0.487849\pi\)
\(80\) 0 0
\(81\) 9.44257e6 1.97421
\(82\) 0 0
\(83\) 9.68747e6 1.85967 0.929837 0.367972i \(-0.119948\pi\)
0.929837 + 0.367972i \(0.119948\pi\)
\(84\) 0 0
\(85\) 1.62416e6 0.286854
\(86\) 0 0
\(87\) 7.95102e6 1.29451
\(88\) 0 0
\(89\) −6.74280e6 −1.01385 −0.506927 0.861989i \(-0.669219\pi\)
−0.506927 + 0.861989i \(0.669219\pi\)
\(90\) 0 0
\(91\) −466123. −0.0648419
\(92\) 0 0
\(93\) 1.89020e7 2.43679
\(94\) 0 0
\(95\) −497190. −0.0594963
\(96\) 0 0
\(97\) −483068. −0.0537412 −0.0268706 0.999639i \(-0.508554\pi\)
−0.0268706 + 0.999639i \(0.508554\pi\)
\(98\) 0 0
\(99\) 1.17988e7 1.22212
\(100\) 0 0
\(101\) −1.87119e7 −1.80714 −0.903571 0.428439i \(-0.859064\pi\)
−0.903571 + 0.428439i \(0.859064\pi\)
\(102\) 0 0
\(103\) 4.79581e6 0.432446 0.216223 0.976344i \(-0.430626\pi\)
0.216223 + 0.976344i \(0.430626\pi\)
\(104\) 0 0
\(105\) 1.29236e6 0.108949
\(106\) 0 0
\(107\) −9.12089e6 −0.719770 −0.359885 0.932997i \(-0.617184\pi\)
−0.359885 + 0.932997i \(0.617184\pi\)
\(108\) 0 0
\(109\) −1.07583e7 −0.795706 −0.397853 0.917449i \(-0.630245\pi\)
−0.397853 + 0.917449i \(0.630245\pi\)
\(110\) 0 0
\(111\) −1.94343e7 −1.34877
\(112\) 0 0
\(113\) 1.08959e7 0.710378 0.355189 0.934795i \(-0.384417\pi\)
0.355189 + 0.934795i \(0.384417\pi\)
\(114\) 0 0
\(115\) −2.45273e6 −0.150386
\(116\) 0 0
\(117\) 1.11434e7 0.643229
\(118\) 0 0
\(119\) −4.70541e6 −0.255966
\(120\) 0 0
\(121\) −1.39641e7 −0.716577
\(122\) 0 0
\(123\) −5.71814e6 −0.277068
\(124\) 0 0
\(125\) −1.09453e7 −0.501234
\(126\) 0 0
\(127\) −3.16808e7 −1.37241 −0.686204 0.727409i \(-0.740724\pi\)
−0.686204 + 0.727409i \(0.740724\pi\)
\(128\) 0 0
\(129\) −1.38119e7 −0.566488
\(130\) 0 0
\(131\) −4.08202e7 −1.58645 −0.793223 0.608931i \(-0.791598\pi\)
−0.793223 + 0.608931i \(0.791598\pi\)
\(132\) 0 0
\(133\) 1.44043e6 0.0530898
\(134\) 0 0
\(135\) −1.74372e7 −0.609969
\(136\) 0 0
\(137\) 2.17104e7 0.721351 0.360676 0.932691i \(-0.382546\pi\)
0.360676 + 0.932691i \(0.382546\pi\)
\(138\) 0 0
\(139\) 3.68005e7 1.16226 0.581128 0.813812i \(-0.302612\pi\)
0.581128 + 0.813812i \(0.302612\pi\)
\(140\) 0 0
\(141\) −6.93037e6 −0.208204
\(142\) 0 0
\(143\) 5.21629e6 0.149171
\(144\) 0 0
\(145\) −6.78878e6 −0.184929
\(146\) 0 0
\(147\) 6.61721e7 1.71816
\(148\) 0 0
\(149\) −7.59158e6 −0.188010 −0.0940049 0.995572i \(-0.529967\pi\)
−0.0940049 + 0.995572i \(0.529967\pi\)
\(150\) 0 0
\(151\) −1.51149e7 −0.357260 −0.178630 0.983916i \(-0.557167\pi\)
−0.178630 + 0.983916i \(0.557167\pi\)
\(152\) 0 0
\(153\) 1.12490e8 2.53918
\(154\) 0 0
\(155\) −1.61390e7 −0.348110
\(156\) 0 0
\(157\) −3.20106e7 −0.660154 −0.330077 0.943954i \(-0.607075\pi\)
−0.330077 + 0.943954i \(0.607075\pi\)
\(158\) 0 0
\(159\) −7.69060e7 −1.51730
\(160\) 0 0
\(161\) 7.10589e6 0.134192
\(162\) 0 0
\(163\) 2.86614e7 0.518372 0.259186 0.965827i \(-0.416546\pi\)
0.259186 + 0.965827i \(0.416546\pi\)
\(164\) 0 0
\(165\) −1.44626e7 −0.250641
\(166\) 0 0
\(167\) −1.00827e8 −1.67522 −0.837608 0.546271i \(-0.816047\pi\)
−0.837608 + 0.546271i \(0.816047\pi\)
\(168\) 0 0
\(169\) −5.78220e7 −0.921488
\(170\) 0 0
\(171\) −3.44356e7 −0.526648
\(172\) 0 0
\(173\) −3.62991e6 −0.0533009 −0.0266505 0.999645i \(-0.508484\pi\)
−0.0266505 + 0.999645i \(0.508484\pi\)
\(174\) 0 0
\(175\) 1.53032e7 0.215849
\(176\) 0 0
\(177\) 1.21063e8 1.64098
\(178\) 0 0
\(179\) −6.65481e7 −0.867261 −0.433630 0.901091i \(-0.642768\pi\)
−0.433630 + 0.901091i \(0.642768\pi\)
\(180\) 0 0
\(181\) 1.41404e8 1.77250 0.886250 0.463207i \(-0.153301\pi\)
0.886250 + 0.463207i \(0.153301\pi\)
\(182\) 0 0
\(183\) −1.85645e8 −2.23925
\(184\) 0 0
\(185\) 1.65935e7 0.192680
\(186\) 0 0
\(187\) 5.26573e7 0.588861
\(188\) 0 0
\(189\) 5.05179e7 0.544288
\(190\) 0 0
\(191\) 1.12407e8 1.16728 0.583641 0.812012i \(-0.301627\pi\)
0.583641 + 0.812012i \(0.301627\pi\)
\(192\) 0 0
\(193\) 5.05226e7 0.505866 0.252933 0.967484i \(-0.418605\pi\)
0.252933 + 0.967484i \(0.418605\pi\)
\(194\) 0 0
\(195\) −1.36591e7 −0.131917
\(196\) 0 0
\(197\) −8.40182e7 −0.782964 −0.391482 0.920186i \(-0.628037\pi\)
−0.391482 + 0.920186i \(0.628037\pi\)
\(198\) 0 0
\(199\) −1.15328e8 −1.03740 −0.518701 0.854956i \(-0.673584\pi\)
−0.518701 + 0.854956i \(0.673584\pi\)
\(200\) 0 0
\(201\) 4.02389e8 3.49510
\(202\) 0 0
\(203\) 1.96681e7 0.165016
\(204\) 0 0
\(205\) 4.88229e6 0.0395809
\(206\) 0 0
\(207\) −1.69877e8 −1.33118
\(208\) 0 0
\(209\) −1.61196e7 −0.122135
\(210\) 0 0
\(211\) 2.20711e6 0.0161747 0.00808733 0.999967i \(-0.497426\pi\)
0.00808733 + 0.999967i \(0.497426\pi\)
\(212\) 0 0
\(213\) 3.04375e8 2.15815
\(214\) 0 0
\(215\) 1.17930e7 0.0809262
\(216\) 0 0
\(217\) 4.67570e7 0.310626
\(218\) 0 0
\(219\) 2.73608e8 1.76025
\(220\) 0 0
\(221\) 4.97320e7 0.309929
\(222\) 0 0
\(223\) 1.35844e8 0.820300 0.410150 0.912018i \(-0.365476\pi\)
0.410150 + 0.912018i \(0.365476\pi\)
\(224\) 0 0
\(225\) −3.65846e8 −2.14121
\(226\) 0 0
\(227\) 2.40869e8 1.36675 0.683377 0.730066i \(-0.260511\pi\)
0.683377 + 0.730066i \(0.260511\pi\)
\(228\) 0 0
\(229\) 2.98261e8 1.64124 0.820621 0.571472i \(-0.193628\pi\)
0.820621 + 0.571472i \(0.193628\pi\)
\(230\) 0 0
\(231\) 4.19001e7 0.223652
\(232\) 0 0
\(233\) −1.54139e8 −0.798302 −0.399151 0.916885i \(-0.630695\pi\)
−0.399151 + 0.916885i \(0.630695\pi\)
\(234\) 0 0
\(235\) 5.91733e6 0.0297433
\(236\) 0 0
\(237\) −2.83965e7 −0.138562
\(238\) 0 0
\(239\) −1.87849e8 −0.890054 −0.445027 0.895517i \(-0.646806\pi\)
−0.445027 + 0.895517i \(0.646806\pi\)
\(240\) 0 0
\(241\) −3.93989e8 −1.81311 −0.906555 0.422089i \(-0.861297\pi\)
−0.906555 + 0.422089i \(0.861297\pi\)
\(242\) 0 0
\(243\) −2.75552e8 −1.23192
\(244\) 0 0
\(245\) −5.64995e7 −0.245450
\(246\) 0 0
\(247\) −1.52240e7 −0.0642822
\(248\) 0 0
\(249\) −8.22436e8 −3.37602
\(250\) 0 0
\(251\) 3.64401e8 1.45453 0.727263 0.686359i \(-0.240792\pi\)
0.727263 + 0.686359i \(0.240792\pi\)
\(252\) 0 0
\(253\) −7.95206e7 −0.308715
\(254\) 0 0
\(255\) −1.37886e8 −0.520750
\(256\) 0 0
\(257\) −1.31627e8 −0.483702 −0.241851 0.970313i \(-0.577755\pi\)
−0.241851 + 0.970313i \(0.577755\pi\)
\(258\) 0 0
\(259\) −4.80737e7 −0.171933
\(260\) 0 0
\(261\) −4.70194e8 −1.63695
\(262\) 0 0
\(263\) −5.06319e8 −1.71625 −0.858123 0.513445i \(-0.828369\pi\)
−0.858123 + 0.513445i \(0.828369\pi\)
\(264\) 0 0
\(265\) 6.56643e7 0.216755
\(266\) 0 0
\(267\) 5.72443e8 1.84053
\(268\) 0 0
\(269\) 3.36604e7 0.105435 0.0527176 0.998609i \(-0.483212\pi\)
0.0527176 + 0.998609i \(0.483212\pi\)
\(270\) 0 0
\(271\) 5.12274e8 1.56354 0.781772 0.623564i \(-0.214316\pi\)
0.781772 + 0.623564i \(0.214316\pi\)
\(272\) 0 0
\(273\) 3.95724e7 0.117713
\(274\) 0 0
\(275\) −1.71255e8 −0.496570
\(276\) 0 0
\(277\) 6.54393e8 1.84995 0.924974 0.380030i \(-0.124086\pi\)
0.924974 + 0.380030i \(0.124086\pi\)
\(278\) 0 0
\(279\) −1.11779e9 −3.08139
\(280\) 0 0
\(281\) −1.27528e8 −0.342874 −0.171437 0.985195i \(-0.554841\pi\)
−0.171437 + 0.985195i \(0.554841\pi\)
\(282\) 0 0
\(283\) 4.21241e8 1.10479 0.552393 0.833584i \(-0.313715\pi\)
0.552393 + 0.833584i \(0.313715\pi\)
\(284\) 0 0
\(285\) 4.22099e7 0.108008
\(286\) 0 0
\(287\) −1.41447e7 −0.0353189
\(288\) 0 0
\(289\) 9.16947e7 0.223461
\(290\) 0 0
\(291\) 4.10110e7 0.0975608
\(292\) 0 0
\(293\) −3.75311e8 −0.871675 −0.435837 0.900025i \(-0.643548\pi\)
−0.435837 + 0.900025i \(0.643548\pi\)
\(294\) 0 0
\(295\) −1.03367e8 −0.234425
\(296\) 0 0
\(297\) −5.65336e8 −1.25216
\(298\) 0 0
\(299\) −7.51030e7 −0.162483
\(300\) 0 0
\(301\) −3.41659e7 −0.0722122
\(302\) 0 0
\(303\) 1.58858e9 3.28065
\(304\) 0 0
\(305\) 1.58508e8 0.319891
\(306\) 0 0
\(307\) 2.15576e7 0.0425223 0.0212611 0.999774i \(-0.493232\pi\)
0.0212611 + 0.999774i \(0.493232\pi\)
\(308\) 0 0
\(309\) −4.07150e8 −0.785054
\(310\) 0 0
\(311\) 5.96289e8 1.12408 0.562038 0.827111i \(-0.310017\pi\)
0.562038 + 0.827111i \(0.310017\pi\)
\(312\) 0 0
\(313\) 5.04393e8 0.929746 0.464873 0.885377i \(-0.346100\pi\)
0.464873 + 0.885377i \(0.346100\pi\)
\(314\) 0 0
\(315\) −7.64256e7 −0.137769
\(316\) 0 0
\(317\) −7.71147e8 −1.35966 −0.679829 0.733371i \(-0.737946\pi\)
−0.679829 + 0.733371i \(0.737946\pi\)
\(318\) 0 0
\(319\) −2.20101e8 −0.379625
\(320\) 0 0
\(321\) 7.74336e8 1.30666
\(322\) 0 0
\(323\) −1.53683e8 −0.253757
\(324\) 0 0
\(325\) −1.61742e8 −0.261354
\(326\) 0 0
\(327\) 9.13351e8 1.44451
\(328\) 0 0
\(329\) −1.71433e7 −0.0265405
\(330\) 0 0
\(331\) 3.04843e8 0.462039 0.231020 0.972949i \(-0.425794\pi\)
0.231020 + 0.972949i \(0.425794\pi\)
\(332\) 0 0
\(333\) 1.14927e9 1.70556
\(334\) 0 0
\(335\) −3.43570e8 −0.499297
\(336\) 0 0
\(337\) −7.56300e8 −1.07644 −0.538219 0.842805i \(-0.680903\pi\)
−0.538219 + 0.842805i \(0.680903\pi\)
\(338\) 0 0
\(339\) −9.25031e8 −1.28961
\(340\) 0 0
\(341\) −5.23248e8 −0.714607
\(342\) 0 0
\(343\) 3.36636e8 0.450433
\(344\) 0 0
\(345\) 2.08229e8 0.273007
\(346\) 0 0
\(347\) 1.93595e8 0.248737 0.124368 0.992236i \(-0.460310\pi\)
0.124368 + 0.992236i \(0.460310\pi\)
\(348\) 0 0
\(349\) −7.55673e8 −0.951579 −0.475790 0.879559i \(-0.657838\pi\)
−0.475790 + 0.879559i \(0.657838\pi\)
\(350\) 0 0
\(351\) −5.33929e8 −0.659036
\(352\) 0 0
\(353\) −3.97258e7 −0.0480686 −0.0240343 0.999711i \(-0.507651\pi\)
−0.0240343 + 0.999711i \(0.507651\pi\)
\(354\) 0 0
\(355\) −2.59884e8 −0.308304
\(356\) 0 0
\(357\) 3.99475e8 0.464676
\(358\) 0 0
\(359\) −1.08901e9 −1.24223 −0.621113 0.783721i \(-0.713319\pi\)
−0.621113 + 0.783721i \(0.713319\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) 1.18551e9 1.30086
\(364\) 0 0
\(365\) −2.33614e8 −0.251463
\(366\) 0 0
\(367\) 1.47931e9 1.56216 0.781082 0.624428i \(-0.214668\pi\)
0.781082 + 0.624428i \(0.214668\pi\)
\(368\) 0 0
\(369\) 3.38149e8 0.350362
\(370\) 0 0
\(371\) −1.90239e8 −0.193415
\(372\) 0 0
\(373\) 1.36968e9 1.36659 0.683294 0.730144i \(-0.260547\pi\)
0.683294 + 0.730144i \(0.260547\pi\)
\(374\) 0 0
\(375\) 9.29219e8 0.909931
\(376\) 0 0
\(377\) −2.07874e8 −0.199804
\(378\) 0 0
\(379\) −3.11167e7 −0.0293600 −0.0146800 0.999892i \(-0.504673\pi\)
−0.0146800 + 0.999892i \(0.504673\pi\)
\(380\) 0 0
\(381\) 2.68961e9 2.49144
\(382\) 0 0
\(383\) −8.84422e8 −0.804384 −0.402192 0.915555i \(-0.631752\pi\)
−0.402192 + 0.915555i \(0.631752\pi\)
\(384\) 0 0
\(385\) −3.57754e7 −0.0319501
\(386\) 0 0
\(387\) 8.16787e8 0.716342
\(388\) 0 0
\(389\) −1.48902e9 −1.28256 −0.641280 0.767307i \(-0.721596\pi\)
−0.641280 + 0.767307i \(0.721596\pi\)
\(390\) 0 0
\(391\) −7.58148e8 −0.641409
\(392\) 0 0
\(393\) 3.46551e9 2.88000
\(394\) 0 0
\(395\) 2.42456e7 0.0197945
\(396\) 0 0
\(397\) −1.12415e9 −0.901692 −0.450846 0.892602i \(-0.648878\pi\)
−0.450846 + 0.892602i \(0.648878\pi\)
\(398\) 0 0
\(399\) −1.22288e8 −0.0963782
\(400\) 0 0
\(401\) −1.49136e9 −1.15499 −0.577494 0.816395i \(-0.695969\pi\)
−0.577494 + 0.816395i \(0.695969\pi\)
\(402\) 0 0
\(403\) −4.94180e8 −0.376112
\(404\) 0 0
\(405\) 6.84466e8 0.511987
\(406\) 0 0
\(407\) 5.37983e8 0.395538
\(408\) 0 0
\(409\) 1.49909e9 1.08342 0.541710 0.840566i \(-0.317777\pi\)
0.541710 + 0.840566i \(0.317777\pi\)
\(410\) 0 0
\(411\) −1.84315e9 −1.30953
\(412\) 0 0
\(413\) 2.99467e8 0.209182
\(414\) 0 0
\(415\) 7.02217e8 0.482284
\(416\) 0 0
\(417\) −3.12425e9 −2.10994
\(418\) 0 0
\(419\) −2.26383e9 −1.50347 −0.751735 0.659465i \(-0.770783\pi\)
−0.751735 + 0.659465i \(0.770783\pi\)
\(420\) 0 0
\(421\) −7.52298e8 −0.491363 −0.245682 0.969351i \(-0.579012\pi\)
−0.245682 + 0.969351i \(0.579012\pi\)
\(422\) 0 0
\(423\) 4.09837e8 0.263281
\(424\) 0 0
\(425\) −1.63275e9 −1.03171
\(426\) 0 0
\(427\) −4.59220e8 −0.285446
\(428\) 0 0
\(429\) −4.42847e8 −0.270803
\(430\) 0 0
\(431\) 2.07720e9 1.24970 0.624852 0.780743i \(-0.285159\pi\)
0.624852 + 0.780743i \(0.285159\pi\)
\(432\) 0 0
\(433\) −2.52531e9 −1.49488 −0.747440 0.664329i \(-0.768717\pi\)
−0.747440 + 0.664329i \(0.768717\pi\)
\(434\) 0 0
\(435\) 5.76347e8 0.335716
\(436\) 0 0
\(437\) 2.32086e8 0.133034
\(438\) 0 0
\(439\) 1.91118e9 1.07814 0.539071 0.842260i \(-0.318775\pi\)
0.539071 + 0.842260i \(0.318775\pi\)
\(440\) 0 0
\(441\) −3.91318e9 −2.17267
\(442\) 0 0
\(443\) 3.04554e7 0.0166438 0.00832188 0.999965i \(-0.497351\pi\)
0.00832188 + 0.999965i \(0.497351\pi\)
\(444\) 0 0
\(445\) −4.88767e8 −0.262931
\(446\) 0 0
\(447\) 6.44502e8 0.341309
\(448\) 0 0
\(449\) −3.70483e9 −1.93155 −0.965775 0.259381i \(-0.916481\pi\)
−0.965775 + 0.259381i \(0.916481\pi\)
\(450\) 0 0
\(451\) 1.58290e8 0.0812525
\(452\) 0 0
\(453\) 1.28321e9 0.648564
\(454\) 0 0
\(455\) −3.37879e7 −0.0168160
\(456\) 0 0
\(457\) −1.34064e9 −0.657059 −0.328529 0.944494i \(-0.606553\pi\)
−0.328529 + 0.944494i \(0.606553\pi\)
\(458\) 0 0
\(459\) −5.38990e9 −2.60157
\(460\) 0 0
\(461\) −2.91246e9 −1.38454 −0.692272 0.721637i \(-0.743390\pi\)
−0.692272 + 0.721637i \(0.743390\pi\)
\(462\) 0 0
\(463\) 7.56134e8 0.354051 0.177025 0.984206i \(-0.443353\pi\)
0.177025 + 0.984206i \(0.443353\pi\)
\(464\) 0 0
\(465\) 1.37015e9 0.631951
\(466\) 0 0
\(467\) −2.01789e9 −0.916831 −0.458415 0.888738i \(-0.651583\pi\)
−0.458415 + 0.888738i \(0.651583\pi\)
\(468\) 0 0
\(469\) 9.95371e8 0.445533
\(470\) 0 0
\(471\) 2.71761e9 1.19843
\(472\) 0 0
\(473\) 3.82344e8 0.166127
\(474\) 0 0
\(475\) 4.99819e8 0.213986
\(476\) 0 0
\(477\) 4.54794e9 1.91867
\(478\) 0 0
\(479\) −1.39778e9 −0.581118 −0.290559 0.956857i \(-0.593841\pi\)
−0.290559 + 0.956857i \(0.593841\pi\)
\(480\) 0 0
\(481\) 5.08096e8 0.208179
\(482\) 0 0
\(483\) −6.03269e8 −0.243610
\(484\) 0 0
\(485\) −3.50163e7 −0.0139371
\(486\) 0 0
\(487\) 1.34378e9 0.527203 0.263602 0.964632i \(-0.415090\pi\)
0.263602 + 0.964632i \(0.415090\pi\)
\(488\) 0 0
\(489\) −2.43327e9 −0.941042
\(490\) 0 0
\(491\) 9.27643e8 0.353668 0.176834 0.984241i \(-0.443414\pi\)
0.176834 + 0.984241i \(0.443414\pi\)
\(492\) 0 0
\(493\) −2.09844e9 −0.788738
\(494\) 0 0
\(495\) 8.55264e8 0.316943
\(496\) 0 0
\(497\) 7.52919e8 0.275107
\(498\) 0 0
\(499\) 2.84992e9 1.02679 0.513394 0.858153i \(-0.328388\pi\)
0.513394 + 0.858153i \(0.328388\pi\)
\(500\) 0 0
\(501\) 8.55994e9 3.04116
\(502\) 0 0
\(503\) 2.38596e9 0.835940 0.417970 0.908461i \(-0.362742\pi\)
0.417970 + 0.908461i \(0.362742\pi\)
\(504\) 0 0
\(505\) −1.35637e9 −0.468661
\(506\) 0 0
\(507\) 4.90891e9 1.67285
\(508\) 0 0
\(509\) 5.57715e9 1.87456 0.937282 0.348573i \(-0.113333\pi\)
0.937282 + 0.348573i \(0.113333\pi\)
\(510\) 0 0
\(511\) 6.76812e8 0.224385
\(512\) 0 0
\(513\) 1.64997e9 0.539591
\(514\) 0 0
\(515\) 3.47635e8 0.112150
\(516\) 0 0
\(517\) 1.91848e8 0.0610576
\(518\) 0 0
\(519\) 3.08168e8 0.0967615
\(520\) 0 0
\(521\) 3.55947e9 1.10269 0.551344 0.834278i \(-0.314115\pi\)
0.551344 + 0.834278i \(0.314115\pi\)
\(522\) 0 0
\(523\) −3.38191e9 −1.03373 −0.516864 0.856067i \(-0.672901\pi\)
−0.516864 + 0.856067i \(0.672901\pi\)
\(524\) 0 0
\(525\) −1.29920e9 −0.391848
\(526\) 0 0
\(527\) −4.98864e9 −1.48472
\(528\) 0 0
\(529\) −2.25990e9 −0.663736
\(530\) 0 0
\(531\) −7.15921e9 −2.07508
\(532\) 0 0
\(533\) 1.49497e8 0.0427648
\(534\) 0 0
\(535\) −6.61148e8 −0.186664
\(536\) 0 0
\(537\) 5.64973e9 1.57441
\(538\) 0 0
\(539\) −1.83179e9 −0.503865
\(540\) 0 0
\(541\) 2.48160e9 0.673817 0.336908 0.941537i \(-0.390619\pi\)
0.336908 + 0.941537i \(0.390619\pi\)
\(542\) 0 0
\(543\) −1.20048e10 −3.21776
\(544\) 0 0
\(545\) −7.79843e8 −0.206357
\(546\) 0 0
\(547\) 4.17148e9 1.08977 0.544884 0.838511i \(-0.316573\pi\)
0.544884 + 0.838511i \(0.316573\pi\)
\(548\) 0 0
\(549\) 1.09783e10 2.83161
\(550\) 0 0
\(551\) 6.42379e8 0.163591
\(552\) 0 0
\(553\) −7.02430e7 −0.0176630
\(554\) 0 0
\(555\) −1.40874e9 −0.349788
\(556\) 0 0
\(557\) −1.77249e9 −0.434600 −0.217300 0.976105i \(-0.569725\pi\)
−0.217300 + 0.976105i \(0.569725\pi\)
\(558\) 0 0
\(559\) 3.61104e8 0.0874360
\(560\) 0 0
\(561\) −4.47044e9 −1.06901
\(562\) 0 0
\(563\) 5.96074e9 1.40774 0.703868 0.710331i \(-0.251455\pi\)
0.703868 + 0.710331i \(0.251455\pi\)
\(564\) 0 0
\(565\) 7.89815e8 0.184228
\(566\) 0 0
\(567\) −1.98299e9 −0.456857
\(568\) 0 0
\(569\) 2.59626e9 0.590819 0.295410 0.955371i \(-0.404544\pi\)
0.295410 + 0.955371i \(0.404544\pi\)
\(570\) 0 0
\(571\) −2.53949e9 −0.570848 −0.285424 0.958401i \(-0.592134\pi\)
−0.285424 + 0.958401i \(0.592134\pi\)
\(572\) 0 0
\(573\) −9.54300e9 −2.11906
\(574\) 0 0
\(575\) 2.46570e9 0.540882
\(576\) 0 0
\(577\) 5.51763e9 1.19574 0.597871 0.801592i \(-0.296014\pi\)
0.597871 + 0.801592i \(0.296014\pi\)
\(578\) 0 0
\(579\) −4.28922e9 −0.918339
\(580\) 0 0
\(581\) −2.03442e9 −0.430353
\(582\) 0 0
\(583\) 2.12892e9 0.444959
\(584\) 0 0
\(585\) 8.07751e8 0.166814
\(586\) 0 0
\(587\) 6.29754e9 1.28510 0.642550 0.766243i \(-0.277876\pi\)
0.642550 + 0.766243i \(0.277876\pi\)
\(588\) 0 0
\(589\) 1.52713e9 0.307945
\(590\) 0 0
\(591\) 7.13289e9 1.42138
\(592\) 0 0
\(593\) −4.02283e9 −0.792209 −0.396105 0.918205i \(-0.629638\pi\)
−0.396105 + 0.918205i \(0.629638\pi\)
\(594\) 0 0
\(595\) −3.41082e8 −0.0663819
\(596\) 0 0
\(597\) 9.79096e9 1.88328
\(598\) 0 0
\(599\) 1.25980e9 0.239502 0.119751 0.992804i \(-0.461790\pi\)
0.119751 + 0.992804i \(0.461790\pi\)
\(600\) 0 0
\(601\) −2.33022e9 −0.437860 −0.218930 0.975741i \(-0.570257\pi\)
−0.218930 + 0.975741i \(0.570257\pi\)
\(602\) 0 0
\(603\) −2.37958e10 −4.41967
\(604\) 0 0
\(605\) −1.01222e9 −0.185836
\(606\) 0 0
\(607\) −7.72032e8 −0.140112 −0.0700559 0.997543i \(-0.522318\pi\)
−0.0700559 + 0.997543i \(0.522318\pi\)
\(608\) 0 0
\(609\) −1.66976e9 −0.299566
\(610\) 0 0
\(611\) 1.81190e8 0.0321358
\(612\) 0 0
\(613\) 1.24050e9 0.217514 0.108757 0.994068i \(-0.465313\pi\)
0.108757 + 0.994068i \(0.465313\pi\)
\(614\) 0 0
\(615\) −4.14492e8 −0.0718543
\(616\) 0 0
\(617\) −3.30230e8 −0.0566003 −0.0283002 0.999599i \(-0.509009\pi\)
−0.0283002 + 0.999599i \(0.509009\pi\)
\(618\) 0 0
\(619\) 5.95644e9 1.00941 0.504707 0.863291i \(-0.331600\pi\)
0.504707 + 0.863291i \(0.331600\pi\)
\(620\) 0 0
\(621\) 8.13958e9 1.36390
\(622\) 0 0
\(623\) 1.41603e9 0.234619
\(624\) 0 0
\(625\) 4.89962e9 0.802755
\(626\) 0 0
\(627\) 1.36850e9 0.221722
\(628\) 0 0
\(629\) 5.12912e9 0.821798
\(630\) 0 0
\(631\) −2.68160e9 −0.424905 −0.212452 0.977171i \(-0.568145\pi\)
−0.212452 + 0.977171i \(0.568145\pi\)
\(632\) 0 0
\(633\) −1.87377e8 −0.0293632
\(634\) 0 0
\(635\) −2.29646e9 −0.355918
\(636\) 0 0
\(637\) −1.73003e9 −0.265194
\(638\) 0 0
\(639\) −1.79996e10 −2.72905
\(640\) 0 0
\(641\) 2.85995e9 0.428899 0.214449 0.976735i \(-0.431204\pi\)
0.214449 + 0.976735i \(0.431204\pi\)
\(642\) 0 0
\(643\) 1.77154e9 0.262792 0.131396 0.991330i \(-0.458054\pi\)
0.131396 + 0.991330i \(0.458054\pi\)
\(644\) 0 0
\(645\) −1.00119e9 −0.146912
\(646\) 0 0
\(647\) 6.32710e9 0.918416 0.459208 0.888329i \(-0.348133\pi\)
0.459208 + 0.888329i \(0.348133\pi\)
\(648\) 0 0
\(649\) −3.35128e9 −0.481232
\(650\) 0 0
\(651\) −3.96953e9 −0.563904
\(652\) 0 0
\(653\) 1.04854e10 1.47363 0.736817 0.676093i \(-0.236328\pi\)
0.736817 + 0.676093i \(0.236328\pi\)
\(654\) 0 0
\(655\) −2.95894e9 −0.411426
\(656\) 0 0
\(657\) −1.61802e10 −2.22589
\(658\) 0 0
\(659\) 5.46117e8 0.0743339 0.0371669 0.999309i \(-0.488167\pi\)
0.0371669 + 0.999309i \(0.488167\pi\)
\(660\) 0 0
\(661\) −9.98464e9 −1.34471 −0.672353 0.740231i \(-0.734716\pi\)
−0.672353 + 0.740231i \(0.734716\pi\)
\(662\) 0 0
\(663\) −4.22209e9 −0.562640
\(664\) 0 0
\(665\) 1.04413e8 0.0137682
\(666\) 0 0
\(667\) 3.16897e9 0.413502
\(668\) 0 0
\(669\) −1.15327e10 −1.48916
\(670\) 0 0
\(671\) 5.13904e9 0.656679
\(672\) 0 0
\(673\) −3.92194e9 −0.495962 −0.247981 0.968765i \(-0.579767\pi\)
−0.247981 + 0.968765i \(0.579767\pi\)
\(674\) 0 0
\(675\) 1.75294e10 2.19383
\(676\) 0 0
\(677\) 1.14925e9 0.142349 0.0711745 0.997464i \(-0.477325\pi\)
0.0711745 + 0.997464i \(0.477325\pi\)
\(678\) 0 0
\(679\) 1.01447e8 0.0124364
\(680\) 0 0
\(681\) −2.04490e10 −2.48118
\(682\) 0 0
\(683\) −1.45436e10 −1.74662 −0.873311 0.487164i \(-0.838031\pi\)
−0.873311 + 0.487164i \(0.838031\pi\)
\(684\) 0 0
\(685\) 1.57373e9 0.187074
\(686\) 0 0
\(687\) −2.53215e10 −2.97948
\(688\) 0 0
\(689\) 2.01065e9 0.234191
\(690\) 0 0
\(691\) −8.07157e8 −0.0930647 −0.0465323 0.998917i \(-0.514817\pi\)
−0.0465323 + 0.998917i \(0.514817\pi\)
\(692\) 0 0
\(693\) −2.47782e9 −0.282816
\(694\) 0 0
\(695\) 2.66756e9 0.301417
\(696\) 0 0
\(697\) 1.50914e9 0.168816
\(698\) 0 0
\(699\) 1.30859e10 1.44922
\(700\) 0 0
\(701\) −1.29139e10 −1.41594 −0.707970 0.706242i \(-0.750389\pi\)
−0.707970 + 0.706242i \(0.750389\pi\)
\(702\) 0 0
\(703\) −1.57014e9 −0.170449
\(704\) 0 0
\(705\) −5.02364e8 −0.0539953
\(706\) 0 0
\(707\) 3.92960e9 0.418196
\(708\) 0 0
\(709\) 6.99785e9 0.737399 0.368700 0.929549i \(-0.379803\pi\)
0.368700 + 0.929549i \(0.379803\pi\)
\(710\) 0 0
\(711\) 1.67926e9 0.175216
\(712\) 0 0
\(713\) 7.53361e9 0.778377
\(714\) 0 0
\(715\) 3.78114e8 0.0386858
\(716\) 0 0
\(717\) 1.59478e10 1.61579
\(718\) 0 0
\(719\) 1.02312e10 1.02654 0.513271 0.858227i \(-0.328434\pi\)
0.513271 + 0.858227i \(0.328434\pi\)
\(720\) 0 0
\(721\) −1.00715e9 −0.100074
\(722\) 0 0
\(723\) 3.34484e10 3.29148
\(724\) 0 0
\(725\) 6.82469e9 0.665120
\(726\) 0 0
\(727\) −5.55225e9 −0.535918 −0.267959 0.963430i \(-0.586349\pi\)
−0.267959 + 0.963430i \(0.586349\pi\)
\(728\) 0 0
\(729\) 2.74259e9 0.262189
\(730\) 0 0
\(731\) 3.64526e9 0.345158
\(732\) 0 0
\(733\) −1.22155e10 −1.14564 −0.572818 0.819683i \(-0.694150\pi\)
−0.572818 + 0.819683i \(0.694150\pi\)
\(734\) 0 0
\(735\) 4.79663e9 0.445585
\(736\) 0 0
\(737\) −1.11390e10 −1.02497
\(738\) 0 0
\(739\) −4.14234e9 −0.377564 −0.188782 0.982019i \(-0.560454\pi\)
−0.188782 + 0.982019i \(0.560454\pi\)
\(740\) 0 0
\(741\) 1.29248e9 0.116697
\(742\) 0 0
\(743\) −4.49275e9 −0.401839 −0.200919 0.979608i \(-0.564393\pi\)
−0.200919 + 0.979608i \(0.564393\pi\)
\(744\) 0 0
\(745\) −5.50293e8 −0.0487581
\(746\) 0 0
\(747\) 4.86359e10 4.26908
\(748\) 0 0
\(749\) 1.91544e9 0.166564
\(750\) 0 0
\(751\) 6.17152e9 0.531683 0.265841 0.964017i \(-0.414350\pi\)
0.265841 + 0.964017i \(0.414350\pi\)
\(752\) 0 0
\(753\) −3.09365e10 −2.64052
\(754\) 0 0
\(755\) −1.09563e9 −0.0926513
\(756\) 0 0
\(757\) −1.23563e10 −1.03527 −0.517634 0.855602i \(-0.673187\pi\)
−0.517634 + 0.855602i \(0.673187\pi\)
\(758\) 0 0
\(759\) 6.75106e9 0.560436
\(760\) 0 0
\(761\) 1.46973e10 1.20891 0.604453 0.796641i \(-0.293392\pi\)
0.604453 + 0.796641i \(0.293392\pi\)
\(762\) 0 0
\(763\) 2.25931e9 0.184137
\(764\) 0 0
\(765\) 8.15407e9 0.658505
\(766\) 0 0
\(767\) −3.16510e9 −0.253282
\(768\) 0 0
\(769\) −1.99912e10 −1.58525 −0.792625 0.609710i \(-0.791286\pi\)
−0.792625 + 0.609710i \(0.791286\pi\)
\(770\) 0 0
\(771\) 1.11747e10 0.878104
\(772\) 0 0
\(773\) 1.87659e10 1.46130 0.730651 0.682751i \(-0.239217\pi\)
0.730651 + 0.682751i \(0.239217\pi\)
\(774\) 0 0
\(775\) 1.62244e10 1.25202
\(776\) 0 0
\(777\) 4.08131e9 0.312123
\(778\) 0 0
\(779\) −4.61980e8 −0.0350140
\(780\) 0 0
\(781\) −8.42577e9 −0.632894
\(782\) 0 0
\(783\) 2.25292e10 1.67718
\(784\) 0 0
\(785\) −2.32036e9 −0.171203
\(786\) 0 0
\(787\) −1.33857e10 −0.978877 −0.489439 0.872038i \(-0.662798\pi\)
−0.489439 + 0.872038i \(0.662798\pi\)
\(788\) 0 0
\(789\) 4.29850e10 3.11564
\(790\) 0 0
\(791\) −2.28821e9 −0.164391
\(792\) 0 0
\(793\) 4.85355e9 0.345624
\(794\) 0 0
\(795\) −5.57470e9 −0.393493
\(796\) 0 0
\(797\) 2.57977e10 1.80500 0.902498 0.430693i \(-0.141731\pi\)
0.902498 + 0.430693i \(0.141731\pi\)
\(798\) 0 0
\(799\) 1.82907e9 0.126858
\(800\) 0 0
\(801\) −3.38522e10 −2.32741
\(802\) 0 0
\(803\) −7.57406e9 −0.516208
\(804\) 0 0
\(805\) 5.15087e8 0.0348012
\(806\) 0 0
\(807\) −2.85766e9 −0.191405
\(808\) 0 0
\(809\) 1.24018e10 0.823504 0.411752 0.911296i \(-0.364917\pi\)
0.411752 + 0.911296i \(0.364917\pi\)
\(810\) 0 0
\(811\) −1.28908e10 −0.848607 −0.424303 0.905520i \(-0.639481\pi\)
−0.424303 + 0.905520i \(0.639481\pi\)
\(812\) 0 0
\(813\) −4.34905e10 −2.83843
\(814\) 0 0
\(815\) 2.07759e9 0.134434
\(816\) 0 0
\(817\) −1.11589e9 −0.0715890
\(818\) 0 0
\(819\) −2.34017e9 −0.148851
\(820\) 0 0
\(821\) −1.73179e9 −0.109218 −0.0546089 0.998508i \(-0.517391\pi\)
−0.0546089 + 0.998508i \(0.517391\pi\)
\(822\) 0 0
\(823\) −1.33148e10 −0.832599 −0.416299 0.909228i \(-0.636673\pi\)
−0.416299 + 0.909228i \(0.636673\pi\)
\(824\) 0 0
\(825\) 1.45391e10 0.901463
\(826\) 0 0
\(827\) 7.23050e9 0.444528 0.222264 0.974987i \(-0.428655\pi\)
0.222264 + 0.974987i \(0.428655\pi\)
\(828\) 0 0
\(829\) −2.85814e10 −1.74238 −0.871189 0.490947i \(-0.836651\pi\)
−0.871189 + 0.490947i \(0.836651\pi\)
\(830\) 0 0
\(831\) −5.55560e10 −3.35836
\(832\) 0 0
\(833\) −1.74642e10 −1.04687
\(834\) 0 0
\(835\) −7.30870e9 −0.434448
\(836\) 0 0
\(837\) 5.35587e10 3.15712
\(838\) 0 0
\(839\) −2.31016e10 −1.35044 −0.675219 0.737618i \(-0.735951\pi\)
−0.675219 + 0.737618i \(0.735951\pi\)
\(840\) 0 0
\(841\) −8.47864e9 −0.491519
\(842\) 0 0
\(843\) 1.08268e10 0.622447
\(844\) 0 0
\(845\) −4.19136e9 −0.238977
\(846\) 0 0
\(847\) 2.93253e9 0.165825
\(848\) 0 0
\(849\) −3.57621e10 −2.00561
\(850\) 0 0
\(851\) −7.74576e9 −0.430834
\(852\) 0 0
\(853\) 3.69001e9 0.203566 0.101783 0.994807i \(-0.467545\pi\)
0.101783 + 0.994807i \(0.467545\pi\)
\(854\) 0 0
\(855\) −2.49614e9 −0.136580
\(856\) 0 0
\(857\) 8.27437e9 0.449057 0.224529 0.974467i \(-0.427916\pi\)
0.224529 + 0.974467i \(0.427916\pi\)
\(858\) 0 0
\(859\) 3.37678e10 1.81772 0.908859 0.417104i \(-0.136955\pi\)
0.908859 + 0.417104i \(0.136955\pi\)
\(860\) 0 0
\(861\) 1.20084e9 0.0641172
\(862\) 0 0
\(863\) −7.31538e9 −0.387435 −0.193718 0.981057i \(-0.562055\pi\)
−0.193718 + 0.981057i \(0.562055\pi\)
\(864\) 0 0
\(865\) −2.63122e8 −0.0138230
\(866\) 0 0
\(867\) −7.78460e9 −0.405667
\(868\) 0 0
\(869\) 7.86075e8 0.0406345
\(870\) 0 0
\(871\) −1.05202e10 −0.539461
\(872\) 0 0
\(873\) −2.42524e9 −0.123369
\(874\) 0 0
\(875\) 2.29856e9 0.115992
\(876\) 0 0
\(877\) 2.90785e10 1.45570 0.727852 0.685734i \(-0.240519\pi\)
0.727852 + 0.685734i \(0.240519\pi\)
\(878\) 0 0
\(879\) 3.18628e10 1.58242
\(880\) 0 0
\(881\) 1.37853e10 0.679205 0.339602 0.940569i \(-0.389708\pi\)
0.339602 + 0.940569i \(0.389708\pi\)
\(882\) 0 0
\(883\) −1.41202e10 −0.690208 −0.345104 0.938564i \(-0.612156\pi\)
−0.345104 + 0.938564i \(0.612156\pi\)
\(884\) 0 0
\(885\) 8.77551e9 0.425570
\(886\) 0 0
\(887\) −3.17818e10 −1.52914 −0.764568 0.644543i \(-0.777048\pi\)
−0.764568 + 0.644543i \(0.777048\pi\)
\(888\) 0 0
\(889\) 6.65315e9 0.317593
\(890\) 0 0
\(891\) 2.21913e10 1.05102
\(892\) 0 0
\(893\) −5.59919e8 −0.0263115
\(894\) 0 0
\(895\) −4.82388e9 −0.224914
\(896\) 0 0
\(897\) 6.37601e9 0.294969
\(898\) 0 0
\(899\) 2.08519e10 0.957166
\(900\) 0 0
\(901\) 2.02971e10 0.924480
\(902\) 0 0
\(903\) 2.90058e9 0.131093
\(904\) 0 0
\(905\) 1.02500e10 0.459677
\(906\) 0 0
\(907\) −3.87339e10 −1.72372 −0.861858 0.507149i \(-0.830699\pi\)
−0.861858 + 0.507149i \(0.830699\pi\)
\(908\) 0 0
\(909\) −9.39428e10 −4.14849
\(910\) 0 0
\(911\) 4.43662e10 1.94419 0.972093 0.234597i \(-0.0753772\pi\)
0.972093 + 0.234597i \(0.0753772\pi\)
\(912\) 0 0
\(913\) 2.27668e10 0.990044
\(914\) 0 0
\(915\) −1.34569e10 −0.580724
\(916\) 0 0
\(917\) 8.57247e9 0.367124
\(918\) 0 0
\(919\) −1.49991e10 −0.637474 −0.318737 0.947843i \(-0.603259\pi\)
−0.318737 + 0.947843i \(0.603259\pi\)
\(920\) 0 0
\(921\) −1.83018e9 −0.0771941
\(922\) 0 0
\(923\) −7.95768e9 −0.333105
\(924\) 0 0
\(925\) −1.66812e10 −0.692999
\(926\) 0 0
\(927\) 2.40773e10 0.992726
\(928\) 0 0
\(929\) −2.26273e9 −0.0925931 −0.0462965 0.998928i \(-0.514742\pi\)
−0.0462965 + 0.998928i \(0.514742\pi\)
\(930\) 0 0
\(931\) 5.34618e9 0.217130
\(932\) 0 0
\(933\) −5.06232e10 −2.04063
\(934\) 0 0
\(935\) 3.81698e9 0.152714
\(936\) 0 0
\(937\) −1.38459e10 −0.549834 −0.274917 0.961468i \(-0.588650\pi\)
−0.274917 + 0.961468i \(0.588650\pi\)
\(938\) 0 0
\(939\) −4.28215e10 −1.68784
\(940\) 0 0
\(941\) −1.00343e10 −0.392574 −0.196287 0.980547i \(-0.562888\pi\)
−0.196287 + 0.980547i \(0.562888\pi\)
\(942\) 0 0
\(943\) −2.27903e9 −0.0885032
\(944\) 0 0
\(945\) 3.66190e9 0.141155
\(946\) 0 0
\(947\) −1.99331e10 −0.762693 −0.381347 0.924432i \(-0.624540\pi\)
−0.381347 + 0.924432i \(0.624540\pi\)
\(948\) 0 0
\(949\) −7.15330e9 −0.271691
\(950\) 0 0
\(951\) 6.54680e10 2.46830
\(952\) 0 0
\(953\) 5.95607e9 0.222913 0.111456 0.993769i \(-0.464448\pi\)
0.111456 + 0.993769i \(0.464448\pi\)
\(954\) 0 0
\(955\) 8.14806e9 0.302721
\(956\) 0 0
\(957\) 1.86859e10 0.689165
\(958\) 0 0
\(959\) −4.55932e9 −0.166930
\(960\) 0 0
\(961\) 2.20588e10 0.801769
\(962\) 0 0
\(963\) −4.57914e10 −1.65231
\(964\) 0 0
\(965\) 3.66225e9 0.131190
\(966\) 0 0
\(967\) 1.33084e10 0.473295 0.236648 0.971596i \(-0.423951\pi\)
0.236648 + 0.971596i \(0.423951\pi\)
\(968\) 0 0
\(969\) 1.30473e10 0.460666
\(970\) 0 0
\(971\) 7.00846e9 0.245672 0.122836 0.992427i \(-0.460801\pi\)
0.122836 + 0.992427i \(0.460801\pi\)
\(972\) 0 0
\(973\) −7.72830e9 −0.268961
\(974\) 0 0
\(975\) 1.37314e10 0.474458
\(976\) 0 0
\(977\) 4.45829e10 1.52946 0.764729 0.644352i \(-0.222873\pi\)
0.764729 + 0.644352i \(0.222873\pi\)
\(978\) 0 0
\(979\) −1.58465e10 −0.539750
\(980\) 0 0
\(981\) −5.40122e10 −1.82663
\(982\) 0 0
\(983\) −1.18845e10 −0.399066 −0.199533 0.979891i \(-0.563943\pi\)
−0.199533 + 0.979891i \(0.563943\pi\)
\(984\) 0 0
\(985\) −6.09025e9 −0.203052
\(986\) 0 0
\(987\) 1.45542e9 0.0481812
\(988\) 0 0
\(989\) −5.50491e9 −0.180952
\(990\) 0 0
\(991\) 8.83738e9 0.288447 0.144223 0.989545i \(-0.453932\pi\)
0.144223 + 0.989545i \(0.453932\pi\)
\(992\) 0 0
\(993\) −2.58803e10 −0.838777
\(994\) 0 0
\(995\) −8.35977e9 −0.269038
\(996\) 0 0
\(997\) 4.97053e10 1.58844 0.794219 0.607632i \(-0.207880\pi\)
0.794219 + 0.607632i \(0.207880\pi\)
\(998\) 0 0
\(999\) −5.50669e10 −1.74748
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.1.1 5
4.3 odd 2 304.8.a.g.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.8.a.a.1.1 5 1.1 even 1 trivial
304.8.a.g.1.5 5 4.3 odd 2