Properties

Label 8.8.a.a.1.1
Level $8$
Weight $8$
Character 8.1
Self dual yes
Analytic conductor $2.499$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.49908020387\)
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 \(\chi\) \(=\) 8.1

$q$-expansion

\(f(q)\) \(=\) \(q-84.0000 q^{3} -82.0000 q^{5} -456.000 q^{7} +4869.00 q^{9} +O(q^{10})\) \(q-84.0000 q^{3} -82.0000 q^{5} -456.000 q^{7} +4869.00 q^{9} -2524.00 q^{11} -10778.0 q^{13} +6888.00 q^{15} -11150.0 q^{17} +4124.00 q^{19} +38304.0 q^{21} +81704.0 q^{23} -71401.0 q^{25} -225288. q^{27} +99798.0 q^{29} -40480.0 q^{31} +212016. q^{33} +37392.0 q^{35} -419442. q^{37} +905352. q^{39} +141402. q^{41} -690428. q^{43} -399258. q^{45} -682032. q^{47} -615607. q^{49} +936600. q^{51} +1.81312e6 q^{53} +206968. q^{55} -346416. q^{57} -966028. q^{59} +1.88767e6 q^{61} -2.22026e6 q^{63} +883796. q^{65} +2.96587e6 q^{67} -6.86314e6 q^{69} -2.54823e6 q^{71} -1.68033e6 q^{73} +5.99768e6 q^{75} +1.15094e6 q^{77} +4.03806e6 q^{79} +8.27569e6 q^{81} -5.38576e6 q^{83} +914300. q^{85} -8.38303e6 q^{87} -6.47305e6 q^{89} +4.91477e6 q^{91} +3.40032e6 q^{93} -338168. q^{95} -6.06576e6 q^{97} -1.22894e7 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −84.0000 −1.79620 −0.898100 0.439790i \(-0.855053\pi\)
−0.898100 + 0.439790i \(0.855053\pi\)
\(4\) 0 0
\(5\) −82.0000 −0.293372 −0.146686 0.989183i \(-0.546861\pi\)
−0.146686 + 0.989183i \(0.546861\pi\)
\(6\) 0 0
\(7\) −456.000 −0.502483 −0.251242 0.967924i \(-0.580839\pi\)
−0.251242 + 0.967924i \(0.580839\pi\)
\(8\) 0 0
\(9\) 4869.00 2.22634
\(10\) 0 0
\(11\) −2524.00 −0.571762 −0.285881 0.958265i \(-0.592286\pi\)
−0.285881 + 0.958265i \(0.592286\pi\)
\(12\) 0 0
\(13\) −10778.0 −1.36062 −0.680309 0.732925i \(-0.738155\pi\)
−0.680309 + 0.732925i \(0.738155\pi\)
\(14\) 0 0
\(15\) 6888.00 0.526955
\(16\) 0 0
\(17\) −11150.0 −0.550432 −0.275216 0.961382i \(-0.588749\pi\)
−0.275216 + 0.961382i \(0.588749\pi\)
\(18\) 0 0
\(19\) 4124.00 0.137937 0.0689685 0.997619i \(-0.478029\pi\)
0.0689685 + 0.997619i \(0.478029\pi\)
\(20\) 0 0
\(21\) 38304.0 0.902561
\(22\) 0 0
\(23\) 81704.0 1.40022 0.700109 0.714036i \(-0.253135\pi\)
0.700109 + 0.714036i \(0.253135\pi\)
\(24\) 0 0
\(25\) −71401.0 −0.913933
\(26\) 0 0
\(27\) −225288. −2.20275
\(28\) 0 0
\(29\) 99798.0 0.759852 0.379926 0.925017i \(-0.375949\pi\)
0.379926 + 0.925017i \(0.375949\pi\)
\(30\) 0 0
\(31\) −40480.0 −0.244048 −0.122024 0.992527i \(-0.538938\pi\)
−0.122024 + 0.992527i \(0.538938\pi\)
\(32\) 0 0
\(33\) 212016. 1.02700
\(34\) 0 0
\(35\) 37392.0 0.147415
\(36\) 0 0
\(37\) −419442. −1.36134 −0.680669 0.732591i \(-0.738311\pi\)
−0.680669 + 0.732591i \(0.738311\pi\)
\(38\) 0 0
\(39\) 905352. 2.44394
\(40\) 0 0
\(41\) 141402. 0.320414 0.160207 0.987083i \(-0.448784\pi\)
0.160207 + 0.987083i \(0.448784\pi\)
\(42\) 0 0
\(43\) −690428. −1.32428 −0.662138 0.749382i \(-0.730351\pi\)
−0.662138 + 0.749382i \(0.730351\pi\)
\(44\) 0 0
\(45\) −399258. −0.653145
\(46\) 0 0
\(47\) −682032. −0.958213 −0.479107 0.877757i \(-0.659039\pi\)
−0.479107 + 0.877757i \(0.659039\pi\)
\(48\) 0 0
\(49\) −615607. −0.747510
\(50\) 0 0
\(51\) 936600. 0.988686
\(52\) 0 0
\(53\) 1.81312e6 1.67286 0.836432 0.548071i \(-0.184638\pi\)
0.836432 + 0.548071i \(0.184638\pi\)
\(54\) 0 0
\(55\) 206968. 0.167739
\(56\) 0 0
\(57\) −346416. −0.247763
\(58\) 0 0
\(59\) −966028. −0.612361 −0.306181 0.951973i \(-0.599051\pi\)
−0.306181 + 0.951973i \(0.599051\pi\)
\(60\) 0 0
\(61\) 1.88767e6 1.06481 0.532404 0.846490i \(-0.321289\pi\)
0.532404 + 0.846490i \(0.321289\pi\)
\(62\) 0 0
\(63\) −2.22026e6 −1.11870
\(64\) 0 0
\(65\) 883796. 0.399168
\(66\) 0 0
\(67\) 2.96587e6 1.20473 0.602365 0.798220i \(-0.294225\pi\)
0.602365 + 0.798220i \(0.294225\pi\)
\(68\) 0 0
\(69\) −6.86314e6 −2.51507
\(70\) 0 0
\(71\) −2.54823e6 −0.844957 −0.422479 0.906373i \(-0.638840\pi\)
−0.422479 + 0.906373i \(0.638840\pi\)
\(72\) 0 0
\(73\) −1.68033e6 −0.505549 −0.252775 0.967525i \(-0.581343\pi\)
−0.252775 + 0.967525i \(0.581343\pi\)
\(74\) 0 0
\(75\) 5.99768e6 1.64161
\(76\) 0 0
\(77\) 1.15094e6 0.287301
\(78\) 0 0
\(79\) 4.03806e6 0.921464 0.460732 0.887539i \(-0.347587\pi\)
0.460732 + 0.887539i \(0.347587\pi\)
\(80\) 0 0
\(81\) 8.27569e6 1.73024
\(82\) 0 0
\(83\) −5.38576e6 −1.03389 −0.516945 0.856019i \(-0.672931\pi\)
−0.516945 + 0.856019i \(0.672931\pi\)
\(84\) 0 0
\(85\) 914300. 0.161481
\(86\) 0 0
\(87\) −8.38303e6 −1.36485
\(88\) 0 0
\(89\) −6.47305e6 −0.973293 −0.486647 0.873599i \(-0.661780\pi\)
−0.486647 + 0.873599i \(0.661780\pi\)
\(90\) 0 0
\(91\) 4.91477e6 0.683688
\(92\) 0 0
\(93\) 3.40032e6 0.438359
\(94\) 0 0
\(95\) −338168. −0.0404669
\(96\) 0 0
\(97\) −6.06576e6 −0.674814 −0.337407 0.941359i \(-0.609550\pi\)
−0.337407 + 0.941359i \(0.609550\pi\)
\(98\) 0 0
\(99\) −1.22894e7 −1.27293
\(100\) 0 0
\(101\) 9.70069e6 0.936866 0.468433 0.883499i \(-0.344819\pi\)
0.468433 + 0.883499i \(0.344819\pi\)
\(102\) 0 0
\(103\) 4.10159e6 0.369847 0.184924 0.982753i \(-0.440796\pi\)
0.184924 + 0.982753i \(0.440796\pi\)
\(104\) 0 0
\(105\) −3.14093e6 −0.264786
\(106\) 0 0
\(107\) 72900.0 0.00575287 0.00287643 0.999996i \(-0.499084\pi\)
0.00287643 + 0.999996i \(0.499084\pi\)
\(108\) 0 0
\(109\) 9.55841e6 0.706957 0.353478 0.935443i \(-0.384999\pi\)
0.353478 + 0.935443i \(0.384999\pi\)
\(110\) 0 0
\(111\) 3.52331e7 2.44524
\(112\) 0 0
\(113\) 9.33890e6 0.608865 0.304433 0.952534i \(-0.401533\pi\)
0.304433 + 0.952534i \(0.401533\pi\)
\(114\) 0 0
\(115\) −6.69973e6 −0.410785
\(116\) 0 0
\(117\) −5.24781e7 −3.02920
\(118\) 0 0
\(119\) 5.08440e6 0.276583
\(120\) 0 0
\(121\) −1.31166e7 −0.673089
\(122\) 0 0
\(123\) −1.18778e7 −0.575529
\(124\) 0 0
\(125\) 1.22611e7 0.561495
\(126\) 0 0
\(127\) −3.59794e7 −1.55862 −0.779311 0.626637i \(-0.784431\pi\)
−0.779311 + 0.626637i \(0.784431\pi\)
\(128\) 0 0
\(129\) 5.79960e7 2.37867
\(130\) 0 0
\(131\) −676052. −0.0262743 −0.0131371 0.999914i \(-0.504182\pi\)
−0.0131371 + 0.999914i \(0.504182\pi\)
\(132\) 0 0
\(133\) −1.88054e6 −0.0693111
\(134\) 0 0
\(135\) 1.84736e7 0.646225
\(136\) 0 0
\(137\) −2.95841e7 −0.982962 −0.491481 0.870888i \(-0.663544\pi\)
−0.491481 + 0.870888i \(0.663544\pi\)
\(138\) 0 0
\(139\) −3.19084e7 −1.00775 −0.503876 0.863776i \(-0.668093\pi\)
−0.503876 + 0.863776i \(0.668093\pi\)
\(140\) 0 0
\(141\) 5.72907e7 1.72114
\(142\) 0 0
\(143\) 2.72037e7 0.777949
\(144\) 0 0
\(145\) −8.18344e6 −0.222919
\(146\) 0 0
\(147\) 5.17110e7 1.34268
\(148\) 0 0
\(149\) −1.16603e7 −0.288773 −0.144386 0.989521i \(-0.546121\pi\)
−0.144386 + 0.989521i \(0.546121\pi\)
\(150\) 0 0
\(151\) −1.76295e7 −0.416698 −0.208349 0.978055i \(-0.566809\pi\)
−0.208349 + 0.978055i \(0.566809\pi\)
\(152\) 0 0
\(153\) −5.42894e7 −1.22545
\(154\) 0 0
\(155\) 3.31936e6 0.0715968
\(156\) 0 0
\(157\) 6.34658e6 0.130885 0.0654427 0.997856i \(-0.479154\pi\)
0.0654427 + 0.997856i \(0.479154\pi\)
\(158\) 0 0
\(159\) −1.52302e8 −3.00480
\(160\) 0 0
\(161\) −3.72570e7 −0.703587
\(162\) 0 0
\(163\) 8.04234e7 1.45454 0.727271 0.686351i \(-0.240789\pi\)
0.727271 + 0.686351i \(0.240789\pi\)
\(164\) 0 0
\(165\) −1.73853e7 −0.301293
\(166\) 0 0
\(167\) 1.14767e8 1.90682 0.953411 0.301673i \(-0.0975451\pi\)
0.953411 + 0.301673i \(0.0975451\pi\)
\(168\) 0 0
\(169\) 5.34168e7 0.851283
\(170\) 0 0
\(171\) 2.00798e7 0.307095
\(172\) 0 0
\(173\) −6.33755e7 −0.930594 −0.465297 0.885155i \(-0.654053\pi\)
−0.465297 + 0.885155i \(0.654053\pi\)
\(174\) 0 0
\(175\) 3.25589e7 0.459236
\(176\) 0 0
\(177\) 8.11464e7 1.09992
\(178\) 0 0
\(179\) −1.13228e7 −0.147559 −0.0737796 0.997275i \(-0.523506\pi\)
−0.0737796 + 0.997275i \(0.523506\pi\)
\(180\) 0 0
\(181\) −5.22650e6 −0.0655143 −0.0327571 0.999463i \(-0.510429\pi\)
−0.0327571 + 0.999463i \(0.510429\pi\)
\(182\) 0 0
\(183\) −1.58564e8 −1.91261
\(184\) 0 0
\(185\) 3.43942e7 0.399379
\(186\) 0 0
\(187\) 2.81426e7 0.314716
\(188\) 0 0
\(189\) 1.02731e8 1.10684
\(190\) 0 0
\(191\) −8.50301e7 −0.882990 −0.441495 0.897264i \(-0.645552\pi\)
−0.441495 + 0.897264i \(0.645552\pi\)
\(192\) 0 0
\(193\) 1.15092e8 1.15237 0.576186 0.817319i \(-0.304540\pi\)
0.576186 + 0.817319i \(0.304540\pi\)
\(194\) 0 0
\(195\) −7.42389e7 −0.716985
\(196\) 0 0
\(197\) −1.38522e8 −1.29088 −0.645441 0.763810i \(-0.723326\pi\)
−0.645441 + 0.763810i \(0.723326\pi\)
\(198\) 0 0
\(199\) −2.19614e7 −0.197548 −0.0987742 0.995110i \(-0.531492\pi\)
−0.0987742 + 0.995110i \(0.531492\pi\)
\(200\) 0 0
\(201\) −2.49133e8 −2.16394
\(202\) 0 0
\(203\) −4.55079e7 −0.381813
\(204\) 0 0
\(205\) −1.15950e7 −0.0940007
\(206\) 0 0
\(207\) 3.97817e8 3.11736
\(208\) 0 0
\(209\) −1.04090e7 −0.0788671
\(210\) 0 0
\(211\) −6.10208e7 −0.447187 −0.223594 0.974682i \(-0.571779\pi\)
−0.223594 + 0.974682i \(0.571779\pi\)
\(212\) 0 0
\(213\) 2.14051e8 1.51771
\(214\) 0 0
\(215\) 5.66151e7 0.388506
\(216\) 0 0
\(217\) 1.84589e7 0.122630
\(218\) 0 0
\(219\) 1.41147e8 0.908068
\(220\) 0 0
\(221\) 1.20175e8 0.748928
\(222\) 0 0
\(223\) −4.22448e7 −0.255098 −0.127549 0.991832i \(-0.540711\pi\)
−0.127549 + 0.991832i \(0.540711\pi\)
\(224\) 0 0
\(225\) −3.47651e8 −2.03472
\(226\) 0 0
\(227\) −2.39102e8 −1.35673 −0.678364 0.734726i \(-0.737311\pi\)
−0.678364 + 0.734726i \(0.737311\pi\)
\(228\) 0 0
\(229\) −4.67889e7 −0.257465 −0.128733 0.991679i \(-0.541091\pi\)
−0.128733 + 0.991679i \(0.541091\pi\)
\(230\) 0 0
\(231\) −9.66793e7 −0.516050
\(232\) 0 0
\(233\) 3.45225e8 1.78795 0.893977 0.448113i \(-0.147904\pi\)
0.893977 + 0.448113i \(0.147904\pi\)
\(234\) 0 0
\(235\) 5.59266e7 0.281113
\(236\) 0 0
\(237\) −3.39197e8 −1.65513
\(238\) 0 0
\(239\) 2.34413e8 1.11068 0.555340 0.831624i \(-0.312588\pi\)
0.555340 + 0.831624i \(0.312588\pi\)
\(240\) 0 0
\(241\) −1.09557e8 −0.504175 −0.252087 0.967705i \(-0.581117\pi\)
−0.252087 + 0.967705i \(0.581117\pi\)
\(242\) 0 0
\(243\) −2.02453e8 −0.905112
\(244\) 0 0
\(245\) 5.04798e7 0.219299
\(246\) 0 0
\(247\) −4.44485e7 −0.187680
\(248\) 0 0
\(249\) 4.52404e8 1.85707
\(250\) 0 0
\(251\) −3.94031e8 −1.57280 −0.786398 0.617720i \(-0.788057\pi\)
−0.786398 + 0.617720i \(0.788057\pi\)
\(252\) 0 0
\(253\) −2.06221e8 −0.800591
\(254\) 0 0
\(255\) −7.68012e7 −0.290053
\(256\) 0 0
\(257\) 3.19064e8 1.17250 0.586248 0.810131i \(-0.300604\pi\)
0.586248 + 0.810131i \(0.300604\pi\)
\(258\) 0 0
\(259\) 1.91266e8 0.684050
\(260\) 0 0
\(261\) 4.85916e8 1.69169
\(262\) 0 0
\(263\) 2.19359e8 0.743549 0.371774 0.928323i \(-0.378750\pi\)
0.371774 + 0.928323i \(0.378750\pi\)
\(264\) 0 0
\(265\) −1.48676e8 −0.490772
\(266\) 0 0
\(267\) 5.43736e8 1.74823
\(268\) 0 0
\(269\) −1.48033e8 −0.463687 −0.231844 0.972753i \(-0.574476\pi\)
−0.231844 + 0.972753i \(0.574476\pi\)
\(270\) 0 0
\(271\) −3.69934e8 −1.12910 −0.564549 0.825399i \(-0.690950\pi\)
−0.564549 + 0.825399i \(0.690950\pi\)
\(272\) 0 0
\(273\) −4.12841e8 −1.22804
\(274\) 0 0
\(275\) 1.80216e8 0.522552
\(276\) 0 0
\(277\) −3.95860e8 −1.11908 −0.559541 0.828803i \(-0.689023\pi\)
−0.559541 + 0.828803i \(0.689023\pi\)
\(278\) 0 0
\(279\) −1.97097e8 −0.543332
\(280\) 0 0
\(281\) −5.97760e8 −1.60714 −0.803572 0.595208i \(-0.797070\pi\)
−0.803572 + 0.595208i \(0.797070\pi\)
\(282\) 0 0
\(283\) 8.05797e7 0.211336 0.105668 0.994401i \(-0.466302\pi\)
0.105668 + 0.994401i \(0.466302\pi\)
\(284\) 0 0
\(285\) 2.84061e7 0.0726867
\(286\) 0 0
\(287\) −6.44793e7 −0.161003
\(288\) 0 0
\(289\) −2.86016e8 −0.697025
\(290\) 0 0
\(291\) 5.09524e8 1.21210
\(292\) 0 0
\(293\) 7.54530e8 1.75243 0.876213 0.481924i \(-0.160062\pi\)
0.876213 + 0.481924i \(0.160062\pi\)
\(294\) 0 0
\(295\) 7.92143e7 0.179650
\(296\) 0 0
\(297\) 5.68627e8 1.25945
\(298\) 0 0
\(299\) −8.80606e8 −1.90516
\(300\) 0 0
\(301\) 3.14835e8 0.665427
\(302\) 0 0
\(303\) −8.14858e8 −1.68280
\(304\) 0 0
\(305\) −1.54789e8 −0.312385
\(306\) 0 0
\(307\) 8.20472e8 1.61838 0.809188 0.587549i \(-0.199907\pi\)
0.809188 + 0.587549i \(0.199907\pi\)
\(308\) 0 0
\(309\) −3.44534e8 −0.664320
\(310\) 0 0
\(311\) 6.53503e8 1.23193 0.615965 0.787773i \(-0.288766\pi\)
0.615965 + 0.787773i \(0.288766\pi\)
\(312\) 0 0
\(313\) 6.63587e8 1.22319 0.611594 0.791172i \(-0.290529\pi\)
0.611594 + 0.791172i \(0.290529\pi\)
\(314\) 0 0
\(315\) 1.82062e8 0.328195
\(316\) 0 0
\(317\) −3.54718e8 −0.625426 −0.312713 0.949848i \(-0.601238\pi\)
−0.312713 + 0.949848i \(0.601238\pi\)
\(318\) 0 0
\(319\) −2.51890e8 −0.434454
\(320\) 0 0
\(321\) −6.12360e6 −0.0103333
\(322\) 0 0
\(323\) −4.59826e7 −0.0759250
\(324\) 0 0
\(325\) 7.69560e8 1.24351
\(326\) 0 0
\(327\) −8.02906e8 −1.26984
\(328\) 0 0
\(329\) 3.11007e8 0.481486
\(330\) 0 0
\(331\) −3.05543e8 −0.463100 −0.231550 0.972823i \(-0.574380\pi\)
−0.231550 + 0.972823i \(0.574380\pi\)
\(332\) 0 0
\(333\) −2.04226e9 −3.03080
\(334\) 0 0
\(335\) −2.43201e8 −0.353434
\(336\) 0 0
\(337\) 3.54965e7 0.0505220 0.0252610 0.999681i \(-0.491958\pi\)
0.0252610 + 0.999681i \(0.491958\pi\)
\(338\) 0 0
\(339\) −7.84467e8 −1.09364
\(340\) 0 0
\(341\) 1.02172e8 0.139537
\(342\) 0 0
\(343\) 6.56252e8 0.878095
\(344\) 0 0
\(345\) 5.62777e8 0.737853
\(346\) 0 0
\(347\) −1.90594e8 −0.244882 −0.122441 0.992476i \(-0.539072\pi\)
−0.122441 + 0.992476i \(0.539072\pi\)
\(348\) 0 0
\(349\) 8.60864e8 1.08404 0.542020 0.840366i \(-0.317660\pi\)
0.542020 + 0.840366i \(0.317660\pi\)
\(350\) 0 0
\(351\) 2.42815e9 2.99710
\(352\) 0 0
\(353\) −1.04544e9 −1.26500 −0.632498 0.774562i \(-0.717970\pi\)
−0.632498 + 0.774562i \(0.717970\pi\)
\(354\) 0 0
\(355\) 2.08955e8 0.247887
\(356\) 0 0
\(357\) −4.27090e8 −0.496798
\(358\) 0 0
\(359\) 7.63303e8 0.870696 0.435348 0.900262i \(-0.356625\pi\)
0.435348 + 0.900262i \(0.356625\pi\)
\(360\) 0 0
\(361\) −8.76864e8 −0.980973
\(362\) 0 0
\(363\) 1.10179e9 1.20900
\(364\) 0 0
\(365\) 1.37787e8 0.148314
\(366\) 0 0
\(367\) −1.38692e9 −1.46460 −0.732302 0.680980i \(-0.761554\pi\)
−0.732302 + 0.680980i \(0.761554\pi\)
\(368\) 0 0
\(369\) 6.88486e8 0.713351
\(370\) 0 0
\(371\) −8.26782e8 −0.840586
\(372\) 0 0
\(373\) 4.77105e8 0.476029 0.238015 0.971262i \(-0.423503\pi\)
0.238015 + 0.971262i \(0.423503\pi\)
\(374\) 0 0
\(375\) −1.02994e9 −1.00856
\(376\) 0 0
\(377\) −1.07562e9 −1.03387
\(378\) 0 0
\(379\) −3.92468e8 −0.370311 −0.185156 0.982709i \(-0.559279\pi\)
−0.185156 + 0.982709i \(0.559279\pi\)
\(380\) 0 0
\(381\) 3.02227e9 2.79960
\(382\) 0 0
\(383\) 2.10409e9 1.91368 0.956839 0.290617i \(-0.0938605\pi\)
0.956839 + 0.290617i \(0.0938605\pi\)
\(384\) 0 0
\(385\) −9.43774e7 −0.0842860
\(386\) 0 0
\(387\) −3.36169e9 −2.94829
\(388\) 0 0
\(389\) −1.26019e9 −1.08546 −0.542730 0.839907i \(-0.682609\pi\)
−0.542730 + 0.839907i \(0.682609\pi\)
\(390\) 0 0
\(391\) −9.11000e8 −0.770725
\(392\) 0 0
\(393\) 5.67884e7 0.0471939
\(394\) 0 0
\(395\) −3.31121e8 −0.270332
\(396\) 0 0
\(397\) −9.81298e8 −0.787107 −0.393554 0.919302i \(-0.628754\pi\)
−0.393554 + 0.919302i \(0.628754\pi\)
\(398\) 0 0
\(399\) 1.57966e8 0.124497
\(400\) 0 0
\(401\) 9.09981e8 0.704737 0.352369 0.935861i \(-0.385376\pi\)
0.352369 + 0.935861i \(0.385376\pi\)
\(402\) 0 0
\(403\) 4.36293e8 0.332056
\(404\) 0 0
\(405\) −6.78606e8 −0.507604
\(406\) 0 0
\(407\) 1.05867e9 0.778361
\(408\) 0 0
\(409\) −3.55609e7 −0.0257004 −0.0128502 0.999917i \(-0.504090\pi\)
−0.0128502 + 0.999917i \(0.504090\pi\)
\(410\) 0 0
\(411\) 2.48507e9 1.76560
\(412\) 0 0
\(413\) 4.40509e8 0.307701
\(414\) 0 0
\(415\) 4.41633e8 0.303314
\(416\) 0 0
\(417\) 2.68031e9 1.81013
\(418\) 0 0
\(419\) 2.65360e9 1.76233 0.881163 0.472813i \(-0.156761\pi\)
0.881163 + 0.472813i \(0.156761\pi\)
\(420\) 0 0
\(421\) −1.12113e9 −0.732264 −0.366132 0.930563i \(-0.619318\pi\)
−0.366132 + 0.930563i \(0.619318\pi\)
\(422\) 0 0
\(423\) −3.32081e9 −2.13331
\(424\) 0 0
\(425\) 7.96121e8 0.503058
\(426\) 0 0
\(427\) −8.60778e8 −0.535049
\(428\) 0 0
\(429\) −2.28511e9 −1.39735
\(430\) 0 0
\(431\) −1.06344e9 −0.639799 −0.319900 0.947451i \(-0.603649\pi\)
−0.319900 + 0.947451i \(0.603649\pi\)
\(432\) 0 0
\(433\) −7.05962e8 −0.417901 −0.208951 0.977926i \(-0.567005\pi\)
−0.208951 + 0.977926i \(0.567005\pi\)
\(434\) 0 0
\(435\) 6.87409e8 0.400408
\(436\) 0 0
\(437\) 3.36947e8 0.193142
\(438\) 0 0
\(439\) 1.48506e9 0.837760 0.418880 0.908042i \(-0.362423\pi\)
0.418880 + 0.908042i \(0.362423\pi\)
\(440\) 0 0
\(441\) −2.99739e9 −1.66421
\(442\) 0 0
\(443\) −7.22153e8 −0.394654 −0.197327 0.980338i \(-0.563226\pi\)
−0.197327 + 0.980338i \(0.563226\pi\)
\(444\) 0 0
\(445\) 5.30790e8 0.285537
\(446\) 0 0
\(447\) 9.79462e8 0.518694
\(448\) 0 0
\(449\) −1.22968e9 −0.641109 −0.320554 0.947230i \(-0.603869\pi\)
−0.320554 + 0.947230i \(0.603869\pi\)
\(450\) 0 0
\(451\) −3.56899e8 −0.183201
\(452\) 0 0
\(453\) 1.48088e9 0.748473
\(454\) 0 0
\(455\) −4.03011e8 −0.200575
\(456\) 0 0
\(457\) −8.85551e7 −0.0434017 −0.0217009 0.999765i \(-0.506908\pi\)
−0.0217009 + 0.999765i \(0.506908\pi\)
\(458\) 0 0
\(459\) 2.51196e9 1.21246
\(460\) 0 0
\(461\) 2.10937e8 0.100277 0.0501384 0.998742i \(-0.484034\pi\)
0.0501384 + 0.998742i \(0.484034\pi\)
\(462\) 0 0
\(463\) −3.29775e9 −1.54413 −0.772066 0.635543i \(-0.780776\pi\)
−0.772066 + 0.635543i \(0.780776\pi\)
\(464\) 0 0
\(465\) −2.78826e8 −0.128602
\(466\) 0 0
\(467\) 8.82873e7 0.0401134 0.0200567 0.999799i \(-0.493615\pi\)
0.0200567 + 0.999799i \(0.493615\pi\)
\(468\) 0 0
\(469\) −1.35244e9 −0.605357
\(470\) 0 0
\(471\) −5.33113e8 −0.235096
\(472\) 0 0
\(473\) 1.74264e9 0.757171
\(474\) 0 0
\(475\) −2.94458e8 −0.126065
\(476\) 0 0
\(477\) 8.82807e9 3.72436
\(478\) 0 0
\(479\) −4.51507e9 −1.87711 −0.938557 0.345125i \(-0.887836\pi\)
−0.938557 + 0.345125i \(0.887836\pi\)
\(480\) 0 0
\(481\) 4.52075e9 1.85226
\(482\) 0 0
\(483\) 3.12959e9 1.26378
\(484\) 0 0
\(485\) 4.97392e8 0.197972
\(486\) 0 0
\(487\) 3.31338e9 1.29993 0.649964 0.759965i \(-0.274784\pi\)
0.649964 + 0.759965i \(0.274784\pi\)
\(488\) 0 0
\(489\) −6.75557e9 −2.61265
\(490\) 0 0
\(491\) −4.01694e9 −1.53147 −0.765737 0.643154i \(-0.777626\pi\)
−0.765737 + 0.643154i \(0.777626\pi\)
\(492\) 0 0
\(493\) −1.11275e9 −0.418247
\(494\) 0 0
\(495\) 1.00773e9 0.373443
\(496\) 0 0
\(497\) 1.16199e9 0.424577
\(498\) 0 0
\(499\) 2.70976e9 0.976290 0.488145 0.872763i \(-0.337674\pi\)
0.488145 + 0.872763i \(0.337674\pi\)
\(500\) 0 0
\(501\) −9.64045e9 −3.42504
\(502\) 0 0
\(503\) 3.04579e8 0.106712 0.0533558 0.998576i \(-0.483008\pi\)
0.0533558 + 0.998576i \(0.483008\pi\)
\(504\) 0 0
\(505\) −7.95456e8 −0.274850
\(506\) 0 0
\(507\) −4.48701e9 −1.52908
\(508\) 0 0
\(509\) −1.88202e8 −0.0632575 −0.0316287 0.999500i \(-0.510069\pi\)
−0.0316287 + 0.999500i \(0.510069\pi\)
\(510\) 0 0
\(511\) 7.66229e8 0.254030
\(512\) 0 0
\(513\) −9.29088e8 −0.303841
\(514\) 0 0
\(515\) −3.36331e8 −0.108503
\(516\) 0 0
\(517\) 1.72145e9 0.547870
\(518\) 0 0
\(519\) 5.32355e9 1.67153
\(520\) 0 0
\(521\) 4.14963e9 1.28552 0.642758 0.766069i \(-0.277790\pi\)
0.642758 + 0.766069i \(0.277790\pi\)
\(522\) 0 0
\(523\) 2.51360e9 0.768318 0.384159 0.923267i \(-0.374491\pi\)
0.384159 + 0.923267i \(0.374491\pi\)
\(524\) 0 0
\(525\) −2.73494e9 −0.824880
\(526\) 0 0
\(527\) 4.51352e8 0.134332
\(528\) 0 0
\(529\) 3.27072e9 0.960613
\(530\) 0 0
\(531\) −4.70359e9 −1.36332
\(532\) 0 0
\(533\) −1.52403e9 −0.435962
\(534\) 0 0
\(535\) −5.97780e6 −0.00168773
\(536\) 0 0
\(537\) 9.51112e8 0.265046
\(538\) 0 0
\(539\) 1.55379e9 0.427398
\(540\) 0 0
\(541\) −1.32416e9 −0.359543 −0.179772 0.983708i \(-0.557536\pi\)
−0.179772 + 0.983708i \(0.557536\pi\)
\(542\) 0 0
\(543\) 4.39026e8 0.117677
\(544\) 0 0
\(545\) −7.83789e8 −0.207401
\(546\) 0 0
\(547\) 5.58047e8 0.145786 0.0728929 0.997340i \(-0.476777\pi\)
0.0728929 + 0.997340i \(0.476777\pi\)
\(548\) 0 0
\(549\) 9.19107e9 2.37062
\(550\) 0 0
\(551\) 4.11567e8 0.104812
\(552\) 0 0
\(553\) −1.84136e9 −0.463020
\(554\) 0 0
\(555\) −2.88912e9 −0.717364
\(556\) 0 0
\(557\) −3.30331e9 −0.809946 −0.404973 0.914329i \(-0.632719\pi\)
−0.404973 + 0.914329i \(0.632719\pi\)
\(558\) 0 0
\(559\) 7.44143e9 1.80184
\(560\) 0 0
\(561\) −2.36398e9 −0.565293
\(562\) 0 0
\(563\) −1.22011e8 −0.0288152 −0.0144076 0.999896i \(-0.504586\pi\)
−0.0144076 + 0.999896i \(0.504586\pi\)
\(564\) 0 0
\(565\) −7.65790e8 −0.178624
\(566\) 0 0
\(567\) −3.77371e9 −0.869417
\(568\) 0 0
\(569\) 5.00925e8 0.113993 0.0569967 0.998374i \(-0.481848\pi\)
0.0569967 + 0.998374i \(0.481848\pi\)
\(570\) 0 0
\(571\) −6.98702e9 −1.57060 −0.785300 0.619116i \(-0.787491\pi\)
−0.785300 + 0.619116i \(0.787491\pi\)
\(572\) 0 0
\(573\) 7.14253e9 1.58603
\(574\) 0 0
\(575\) −5.83375e9 −1.27971
\(576\) 0 0
\(577\) −8.16573e9 −1.76962 −0.884809 0.465954i \(-0.845711\pi\)
−0.884809 + 0.465954i \(0.845711\pi\)
\(578\) 0 0
\(579\) −9.66769e9 −2.06989
\(580\) 0 0
\(581\) 2.45591e9 0.519512
\(582\) 0 0
\(583\) −4.57631e9 −0.956479
\(584\) 0 0
\(585\) 4.30320e9 0.888682
\(586\) 0 0
\(587\) 8.53182e9 1.74104 0.870519 0.492135i \(-0.163783\pi\)
0.870519 + 0.492135i \(0.163783\pi\)
\(588\) 0 0
\(589\) −1.66940e8 −0.0336632
\(590\) 0 0
\(591\) 1.16358e10 2.31868
\(592\) 0 0
\(593\) −1.71175e9 −0.337092 −0.168546 0.985694i \(-0.553907\pi\)
−0.168546 + 0.985694i \(0.553907\pi\)
\(594\) 0 0
\(595\) −4.16921e8 −0.0811417
\(596\) 0 0
\(597\) 1.84475e9 0.354836
\(598\) 0 0
\(599\) −4.77362e9 −0.907516 −0.453758 0.891125i \(-0.649917\pi\)
−0.453758 + 0.891125i \(0.649917\pi\)
\(600\) 0 0
\(601\) 7.89998e8 0.148445 0.0742224 0.997242i \(-0.476353\pi\)
0.0742224 + 0.997242i \(0.476353\pi\)
\(602\) 0 0
\(603\) 1.44408e10 2.68214
\(604\) 0 0
\(605\) 1.07556e9 0.197465
\(606\) 0 0
\(607\) −1.82652e9 −0.331485 −0.165743 0.986169i \(-0.553002\pi\)
−0.165743 + 0.986169i \(0.553002\pi\)
\(608\) 0 0
\(609\) 3.82266e9 0.685813
\(610\) 0 0
\(611\) 7.35094e9 1.30376
\(612\) 0 0
\(613\) −6.90339e9 −1.21046 −0.605231 0.796050i \(-0.706919\pi\)
−0.605231 + 0.796050i \(0.706919\pi\)
\(614\) 0 0
\(615\) 9.73977e8 0.168844
\(616\) 0 0
\(617\) −5.69235e9 −0.975649 −0.487825 0.872942i \(-0.662209\pi\)
−0.487825 + 0.872942i \(0.662209\pi\)
\(618\) 0 0
\(619\) −4.28594e9 −0.726321 −0.363161 0.931727i \(-0.618302\pi\)
−0.363161 + 0.931727i \(0.618302\pi\)
\(620\) 0 0
\(621\) −1.84069e10 −3.08433
\(622\) 0 0
\(623\) 2.95171e9 0.489064
\(624\) 0 0
\(625\) 4.57279e9 0.749206
\(626\) 0 0
\(627\) 8.74354e8 0.141661
\(628\) 0 0
\(629\) 4.67678e9 0.749324
\(630\) 0 0
\(631\) −5.61602e8 −0.0889869 −0.0444935 0.999010i \(-0.514167\pi\)
−0.0444935 + 0.999010i \(0.514167\pi\)
\(632\) 0 0
\(633\) 5.12575e9 0.803238
\(634\) 0 0
\(635\) 2.95031e9 0.457256
\(636\) 0 0
\(637\) 6.63501e9 1.01708
\(638\) 0 0
\(639\) −1.24073e10 −1.88116
\(640\) 0 0
\(641\) 5.17445e9 0.775998 0.387999 0.921660i \(-0.373166\pi\)
0.387999 + 0.921660i \(0.373166\pi\)
\(642\) 0 0
\(643\) −1.04374e10 −1.54830 −0.774148 0.633004i \(-0.781822\pi\)
−0.774148 + 0.633004i \(0.781822\pi\)
\(644\) 0 0
\(645\) −4.75567e9 −0.697835
\(646\) 0 0
\(647\) −9.71623e8 −0.141037 −0.0705185 0.997510i \(-0.522465\pi\)
−0.0705185 + 0.997510i \(0.522465\pi\)
\(648\) 0 0
\(649\) 2.43825e9 0.350125
\(650\) 0 0
\(651\) −1.55055e9 −0.220268
\(652\) 0 0
\(653\) −7.25223e9 −1.01924 −0.509619 0.860400i \(-0.670214\pi\)
−0.509619 + 0.860400i \(0.670214\pi\)
\(654\) 0 0
\(655\) 5.54363e7 0.00770814
\(656\) 0 0
\(657\) −8.18151e9 −1.12552
\(658\) 0 0
\(659\) 3.81924e9 0.519851 0.259925 0.965629i \(-0.416302\pi\)
0.259925 + 0.965629i \(0.416302\pi\)
\(660\) 0 0
\(661\) 1.07881e10 1.45292 0.726459 0.687210i \(-0.241165\pi\)
0.726459 + 0.687210i \(0.241165\pi\)
\(662\) 0 0
\(663\) −1.00947e10 −1.34523
\(664\) 0 0
\(665\) 1.54205e8 0.0203339
\(666\) 0 0
\(667\) 8.15390e9 1.06396
\(668\) 0 0
\(669\) 3.54857e9 0.458207
\(670\) 0 0
\(671\) −4.76448e9 −0.608817
\(672\) 0 0
\(673\) −6.34833e9 −0.802798 −0.401399 0.915903i \(-0.631476\pi\)
−0.401399 + 0.915903i \(0.631476\pi\)
\(674\) 0 0
\(675\) 1.60858e10 2.01316
\(676\) 0 0
\(677\) 8.82566e9 1.09317 0.546584 0.837404i \(-0.315928\pi\)
0.546584 + 0.837404i \(0.315928\pi\)
\(678\) 0 0
\(679\) 2.76599e9 0.339083
\(680\) 0 0
\(681\) 2.00846e10 2.43696
\(682\) 0 0
\(683\) 4.92331e9 0.591268 0.295634 0.955301i \(-0.404469\pi\)
0.295634 + 0.955301i \(0.404469\pi\)
\(684\) 0 0
\(685\) 2.42590e9 0.288374
\(686\) 0 0
\(687\) 3.93027e9 0.462459
\(688\) 0 0
\(689\) −1.95418e10 −2.27613
\(690\) 0 0
\(691\) −5.68449e9 −0.655418 −0.327709 0.944779i \(-0.606277\pi\)
−0.327709 + 0.944779i \(0.606277\pi\)
\(692\) 0 0
\(693\) 5.60395e9 0.639628
\(694\) 0 0
\(695\) 2.61649e9 0.295646
\(696\) 0 0
\(697\) −1.57663e9 −0.176366
\(698\) 0 0
\(699\) −2.89989e10 −3.21152
\(700\) 0 0
\(701\) −1.70567e9 −0.187017 −0.0935085 0.995618i \(-0.529808\pi\)
−0.0935085 + 0.995618i \(0.529808\pi\)
\(702\) 0 0
\(703\) −1.72978e9 −0.187779
\(704\) 0 0
\(705\) −4.69784e9 −0.504936
\(706\) 0 0
\(707\) −4.42351e9 −0.470760
\(708\) 0 0
\(709\) 4.52189e9 0.476495 0.238248 0.971204i \(-0.423427\pi\)
0.238248 + 0.971204i \(0.423427\pi\)
\(710\) 0 0
\(711\) 1.96613e10 2.05149
\(712\) 0 0
\(713\) −3.30738e9 −0.341720
\(714\) 0 0
\(715\) −2.23070e9 −0.228229
\(716\) 0 0
\(717\) −1.96907e10 −1.99500
\(718\) 0 0
\(719\) −3.09206e9 −0.310239 −0.155120 0.987896i \(-0.549576\pi\)
−0.155120 + 0.987896i \(0.549576\pi\)
\(720\) 0 0
\(721\) −1.87033e9 −0.185842
\(722\) 0 0
\(723\) 9.20280e9 0.905599
\(724\) 0 0
\(725\) −7.12568e9 −0.694453
\(726\) 0 0
\(727\) 1.44622e10 1.39593 0.697965 0.716132i \(-0.254089\pi\)
0.697965 + 0.716132i \(0.254089\pi\)
\(728\) 0 0
\(729\) −1.09288e9 −0.104478
\(730\) 0 0
\(731\) 7.69827e9 0.728924
\(732\) 0 0
\(733\) 3.15415e9 0.295814 0.147907 0.989001i \(-0.452746\pi\)
0.147907 + 0.989001i \(0.452746\pi\)
\(734\) 0 0
\(735\) −4.24030e9 −0.393905
\(736\) 0 0
\(737\) −7.48585e9 −0.688819
\(738\) 0 0
\(739\) 1.54236e10 1.40582 0.702912 0.711277i \(-0.251883\pi\)
0.702912 + 0.711277i \(0.251883\pi\)
\(740\) 0 0
\(741\) 3.73367e9 0.337111
\(742\) 0 0
\(743\) −1.59520e10 −1.42677 −0.713385 0.700772i \(-0.752839\pi\)
−0.713385 + 0.700772i \(0.752839\pi\)
\(744\) 0 0
\(745\) 9.56141e8 0.0847179
\(746\) 0 0
\(747\) −2.62233e10 −2.30179
\(748\) 0 0
\(749\) −3.32424e7 −0.00289072
\(750\) 0 0
\(751\) 6.13964e9 0.528936 0.264468 0.964395i \(-0.414804\pi\)
0.264468 + 0.964395i \(0.414804\pi\)
\(752\) 0 0
\(753\) 3.30986e10 2.82506
\(754\) 0 0
\(755\) 1.44562e9 0.122248
\(756\) 0 0
\(757\) −1.42818e10 −1.19660 −0.598299 0.801273i \(-0.704157\pi\)
−0.598299 + 0.801273i \(0.704157\pi\)
\(758\) 0 0
\(759\) 1.73226e10 1.43802
\(760\) 0 0
\(761\) 1.47536e10 1.21353 0.606767 0.794880i \(-0.292466\pi\)
0.606767 + 0.794880i \(0.292466\pi\)
\(762\) 0 0
\(763\) −4.35863e9 −0.355234
\(764\) 0 0
\(765\) 4.45173e9 0.359512
\(766\) 0 0
\(767\) 1.04118e10 0.833190
\(768\) 0 0
\(769\) 1.97592e10 1.56685 0.783424 0.621487i \(-0.213471\pi\)
0.783424 + 0.621487i \(0.213471\pi\)
\(770\) 0 0
\(771\) −2.68014e10 −2.10604
\(772\) 0 0
\(773\) 1.01370e10 0.789374 0.394687 0.918816i \(-0.370853\pi\)
0.394687 + 0.918816i \(0.370853\pi\)
\(774\) 0 0
\(775\) 2.89031e9 0.223043
\(776\) 0 0
\(777\) −1.60663e10 −1.22869
\(778\) 0 0
\(779\) 5.83142e8 0.0441970
\(780\) 0 0
\(781\) 6.43174e9 0.483114
\(782\) 0 0
\(783\) −2.24833e10 −1.67376
\(784\) 0 0
\(785\) −5.20420e8 −0.0383981
\(786\) 0 0
\(787\) −1.27882e10 −0.935188 −0.467594 0.883943i \(-0.654879\pi\)
−0.467594 + 0.883943i \(0.654879\pi\)
\(788\) 0 0
\(789\) −1.84261e10 −1.33556
\(790\) 0 0
\(791\) −4.25854e9 −0.305945
\(792\) 0 0
\(793\) −2.03453e10 −1.44880
\(794\) 0 0
\(795\) 1.24888e10 0.881524
\(796\) 0 0
\(797\) −7.38617e9 −0.516791 −0.258396 0.966039i \(-0.583194\pi\)
−0.258396 + 0.966039i \(0.583194\pi\)
\(798\) 0 0
\(799\) 7.60466e9 0.527431
\(800\) 0 0
\(801\) −3.15173e10 −2.16688
\(802\) 0 0
\(803\) 4.24114e9 0.289054
\(804\) 0 0
\(805\) 3.05508e9 0.206413
\(806\) 0 0
\(807\) 1.24348e10 0.832875
\(808\) 0 0
\(809\) 1.53742e10 1.02087 0.510437 0.859915i \(-0.329484\pi\)
0.510437 + 0.859915i \(0.329484\pi\)
\(810\) 0 0
\(811\) 9.77882e9 0.643744 0.321872 0.946783i \(-0.395688\pi\)
0.321872 + 0.946783i \(0.395688\pi\)
\(812\) 0 0
\(813\) 3.10745e10 2.02809
\(814\) 0 0
\(815\) −6.59472e9 −0.426722
\(816\) 0 0
\(817\) −2.84733e9 −0.182667
\(818\) 0 0
\(819\) 2.39300e10 1.52212
\(820\) 0 0
\(821\) 1.83470e10 1.15708 0.578540 0.815654i \(-0.303623\pi\)
0.578540 + 0.815654i \(0.303623\pi\)
\(822\) 0 0
\(823\) −3.16960e10 −1.98201 −0.991004 0.133829i \(-0.957273\pi\)
−0.991004 + 0.133829i \(0.957273\pi\)
\(824\) 0 0
\(825\) −1.51382e10 −0.938608
\(826\) 0 0
\(827\) −6.12845e9 −0.376774 −0.188387 0.982095i \(-0.560326\pi\)
−0.188387 + 0.982095i \(0.560326\pi\)
\(828\) 0 0
\(829\) −1.24652e10 −0.759904 −0.379952 0.925006i \(-0.624060\pi\)
−0.379952 + 0.925006i \(0.624060\pi\)
\(830\) 0 0
\(831\) 3.32522e10 2.01010
\(832\) 0 0
\(833\) 6.86402e9 0.411454
\(834\) 0 0
\(835\) −9.41091e9 −0.559409
\(836\) 0 0
\(837\) 9.11966e9 0.537575
\(838\) 0 0
\(839\) −1.82237e10 −1.06530 −0.532648 0.846337i \(-0.678803\pi\)
−0.532648 + 0.846337i \(0.678803\pi\)
\(840\) 0 0
\(841\) −7.29024e9 −0.422625
\(842\) 0 0
\(843\) 5.02118e10 2.88675
\(844\) 0 0
\(845\) −4.38017e9 −0.249743
\(846\) 0 0
\(847\) 5.98117e9 0.338216
\(848\) 0 0
\(849\) −6.76870e9 −0.379602
\(850\) 0 0
\(851\) −3.42701e10 −1.90617
\(852\) 0 0
\(853\) −2.48619e10 −1.37155 −0.685777 0.727812i \(-0.740537\pi\)
−0.685777 + 0.727812i \(0.740537\pi\)
\(854\) 0 0
\(855\) −1.64654e9 −0.0900930
\(856\) 0 0
\(857\) 2.96761e10 1.61055 0.805275 0.592902i \(-0.202018\pi\)
0.805275 + 0.592902i \(0.202018\pi\)
\(858\) 0 0
\(859\) 1.14772e10 0.617819 0.308910 0.951091i \(-0.400036\pi\)
0.308910 + 0.951091i \(0.400036\pi\)
\(860\) 0 0
\(861\) 5.41626e9 0.289194
\(862\) 0 0
\(863\) 2.13485e10 1.13066 0.565328 0.824866i \(-0.308750\pi\)
0.565328 + 0.824866i \(0.308750\pi\)
\(864\) 0 0
\(865\) 5.19679e9 0.273010
\(866\) 0 0
\(867\) 2.40254e10 1.25200
\(868\) 0 0
\(869\) −1.01921e10 −0.526858
\(870\) 0 0
\(871\) −3.19661e10 −1.63918
\(872\) 0 0
\(873\) −2.95342e10 −1.50236
\(874\) 0 0
\(875\) −5.59108e9 −0.282142
\(876\) 0 0
\(877\) −7.92753e9 −0.396862 −0.198431 0.980115i \(-0.563585\pi\)
−0.198431 + 0.980115i \(0.563585\pi\)
\(878\) 0 0
\(879\) −6.33805e10 −3.14771
\(880\) 0 0
\(881\) 7.32045e9 0.360680 0.180340 0.983604i \(-0.442280\pi\)
0.180340 + 0.983604i \(0.442280\pi\)
\(882\) 0 0
\(883\) −3.54988e9 −0.173521 −0.0867604 0.996229i \(-0.527651\pi\)
−0.0867604 + 0.996229i \(0.527651\pi\)
\(884\) 0 0
\(885\) −6.65400e9 −0.322687
\(886\) 0 0
\(887\) 5.80634e9 0.279364 0.139682 0.990196i \(-0.455392\pi\)
0.139682 + 0.990196i \(0.455392\pi\)
\(888\) 0 0
\(889\) 1.64066e10 0.783182
\(890\) 0 0
\(891\) −2.08878e10 −0.989285
\(892\) 0 0
\(893\) −2.81270e9 −0.132173
\(894\) 0 0
\(895\) 9.28466e8 0.0432898
\(896\) 0 0
\(897\) 7.39709e10 3.42206
\(898\) 0 0
\(899\) −4.03982e9 −0.185440
\(900\) 0 0
\(901\) −2.02163e10 −0.920798
\(902\) 0 0
\(903\) −2.64462e10 −1.19524
\(904\) 0 0
\(905\) 4.28573e8 0.0192201
\(906\) 0 0
\(907\) 1.78240e10 0.793196 0.396598 0.917992i \(-0.370191\pi\)
0.396598 + 0.917992i \(0.370191\pi\)
\(908\) 0 0
\(909\) 4.72326e10 2.08578
\(910\) 0 0
\(911\) 1.87703e10 0.822538 0.411269 0.911514i \(-0.365086\pi\)
0.411269 + 0.911514i \(0.365086\pi\)
\(912\) 0 0
\(913\) 1.35937e10 0.591138
\(914\) 0 0
\(915\) 1.30023e10 0.561107
\(916\) 0 0
\(917\) 3.08280e8 0.0132024
\(918\) 0 0
\(919\) 3.75844e10 1.59736 0.798681 0.601754i \(-0.205531\pi\)
0.798681 + 0.601754i \(0.205531\pi\)
\(920\) 0 0
\(921\) −6.89197e10 −2.90693
\(922\) 0 0
\(923\) 2.74648e10 1.14966
\(924\) 0 0
\(925\) 2.99486e10 1.24417
\(926\) 0 0
\(927\) 1.99707e10 0.823404
\(928\) 0 0
\(929\) −1.92372e10 −0.787205 −0.393602 0.919281i \(-0.628771\pi\)
−0.393602 + 0.919281i \(0.628771\pi\)
\(930\) 0 0
\(931\) −2.53876e9 −0.103109
\(932\) 0 0
\(933\) −5.48943e10 −2.21279
\(934\) 0 0
\(935\) −2.30769e9 −0.0923289
\(936\) 0 0
\(937\) 1.04732e9 0.0415900 0.0207950 0.999784i \(-0.493380\pi\)
0.0207950 + 0.999784i \(0.493380\pi\)
\(938\) 0 0
\(939\) −5.57413e10 −2.19709
\(940\) 0 0
\(941\) −7.97861e9 −0.312150 −0.156075 0.987745i \(-0.549884\pi\)
−0.156075 + 0.987745i \(0.549884\pi\)
\(942\) 0 0
\(943\) 1.15531e10 0.448650
\(944\) 0 0
\(945\) −8.42397e9 −0.324717
\(946\) 0 0
\(947\) −4.26943e9 −0.163360 −0.0816799 0.996659i \(-0.526029\pi\)
−0.0816799 + 0.996659i \(0.526029\pi\)
\(948\) 0 0
\(949\) 1.81106e10 0.687860
\(950\) 0 0
\(951\) 2.97963e10 1.12339
\(952\) 0 0
\(953\) −1.06048e10 −0.396897 −0.198449 0.980111i \(-0.563590\pi\)
−0.198449 + 0.980111i \(0.563590\pi\)
\(954\) 0 0
\(955\) 6.97247e9 0.259045
\(956\) 0 0
\(957\) 2.11588e10 0.780367
\(958\) 0 0
\(959\) 1.34904e10 0.493922
\(960\) 0 0
\(961\) −2.58740e10 −0.940441
\(962\) 0 0
\(963\) 3.54950e8 0.0128078
\(964\) 0 0
\(965\) −9.43750e9 −0.338074
\(966\) 0 0
\(967\) −1.65090e10 −0.587123 −0.293562 0.955940i \(-0.594841\pi\)
−0.293562 + 0.955940i \(0.594841\pi\)
\(968\) 0 0
\(969\) 3.86254e9 0.136377
\(970\) 0 0
\(971\) −2.46094e10 −0.862649 −0.431324 0.902197i \(-0.641954\pi\)
−0.431324 + 0.902197i \(0.641954\pi\)
\(972\) 0 0
\(973\) 1.45503e10 0.506379
\(974\) 0 0
\(975\) −6.46430e10 −2.23360
\(976\) 0 0
\(977\) −2.53886e10 −0.870979 −0.435489 0.900194i \(-0.643425\pi\)
−0.435489 + 0.900194i \(0.643425\pi\)
\(978\) 0 0
\(979\) 1.63380e10 0.556492
\(980\) 0 0
\(981\) 4.65399e10 1.57392
\(982\) 0 0
\(983\) 1.87585e10 0.629884 0.314942 0.949111i \(-0.398015\pi\)
0.314942 + 0.949111i \(0.398015\pi\)
\(984\) 0 0
\(985\) 1.13588e10 0.378709
\(986\) 0 0
\(987\) −2.61246e10 −0.864846
\(988\) 0 0
\(989\) −5.64107e10 −1.85428
\(990\) 0 0
\(991\) −3.59792e9 −0.117434 −0.0587170 0.998275i \(-0.518701\pi\)
−0.0587170 + 0.998275i \(0.518701\pi\)
\(992\) 0 0
\(993\) 2.56656e10 0.831821
\(994\) 0 0
\(995\) 1.80083e9 0.0579552
\(996\) 0 0
\(997\) −1.34287e10 −0.429143 −0.214571 0.976708i \(-0.568835\pi\)
−0.214571 + 0.976708i \(0.568835\pi\)
\(998\) 0 0
\(999\) 9.44952e10 2.99868
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 8.8.a.a.1.1 1
3.2 odd 2 72.8.a.d.1.1 1
4.3 odd 2 16.8.a.c.1.1 1
5.2 odd 4 200.8.c.a.49.2 2
5.3 odd 4 200.8.c.a.49.1 2
5.4 even 2 200.8.a.i.1.1 1
7.6 odd 2 392.8.a.d.1.1 1
8.3 odd 2 64.8.a.a.1.1 1
8.5 even 2 64.8.a.g.1.1 1
12.11 even 2 144.8.a.g.1.1 1
16.3 odd 4 256.8.b.c.129.2 2
16.5 even 4 256.8.b.e.129.2 2
16.11 odd 4 256.8.b.c.129.1 2
16.13 even 4 256.8.b.e.129.1 2
20.3 even 4 400.8.c.b.49.2 2
20.7 even 4 400.8.c.b.49.1 2
20.19 odd 2 400.8.a.b.1.1 1
24.5 odd 2 576.8.a.j.1.1 1
24.11 even 2 576.8.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 1.1 even 1 trivial
16.8.a.c.1.1 1 4.3 odd 2
64.8.a.a.1.1 1 8.3 odd 2
64.8.a.g.1.1 1 8.5 even 2
72.8.a.d.1.1 1 3.2 odd 2
144.8.a.g.1.1 1 12.11 even 2
200.8.a.i.1.1 1 5.4 even 2
200.8.c.a.49.1 2 5.3 odd 4
200.8.c.a.49.2 2 5.2 odd 4
256.8.b.c.129.1 2 16.11 odd 4
256.8.b.c.129.2 2 16.3 odd 4
256.8.b.e.129.1 2 16.13 even 4
256.8.b.e.129.2 2 16.5 even 4
392.8.a.d.1.1 1 7.6 odd 2
400.8.a.b.1.1 1 20.19 odd 2
400.8.c.b.49.1 2 20.7 even 4
400.8.c.b.49.2 2 20.3 even 4
576.8.a.j.1.1 1 24.5 odd 2
576.8.a.k.1.1 1 24.11 even 2