Properties

Label 304.8.a.d.1.2
Level $304$
Weight $8$
Character 304.1
Self dual yes
Analytic conductor $94.965$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(94.9650477472\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-12.0797\) of defining polynomial
Character \(\chi\) \(=\) 304.1

$q$-expansion

\(f(q)\) \(=\) \(q+72.2392 q^{3} +467.472 q^{5} +1471.23 q^{7} +3031.51 q^{9} +O(q^{10})\) \(q+72.2392 q^{3} +467.472 q^{5} +1471.23 q^{7} +3031.51 q^{9} +3547.04 q^{11} -12361.8 q^{13} +33769.8 q^{15} -13486.3 q^{17} +6859.00 q^{19} +106281. q^{21} +94084.9 q^{23} +140405. q^{25} +61006.6 q^{27} +33541.0 q^{29} -212096. q^{31} +256236. q^{33} +687760. q^{35} +33656.5 q^{37} -893007. q^{39} -480531. q^{41} +561340. q^{43} +1.41715e6 q^{45} -478445. q^{47} +1.34098e6 q^{49} -974237. q^{51} -93127.0 q^{53} +1.65814e6 q^{55} +495489. q^{57} -955034. q^{59} -1.71632e6 q^{61} +4.46005e6 q^{63} -5.77880e6 q^{65} -476255. q^{67} +6.79662e6 q^{69} -1.95518e6 q^{71} +1.21632e6 q^{73} +1.01428e7 q^{75} +5.21852e6 q^{77} -3.94544e6 q^{79} -2.22284e6 q^{81} +3.54252e6 q^{83} -6.30445e6 q^{85} +2.42298e6 q^{87} -7.50202e6 q^{89} -1.81871e7 q^{91} -1.53216e7 q^{93} +3.20639e6 q^{95} +1.27974e7 q^{97} +1.07529e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + O(q^{10}) \) \( 2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + 3295q^{11} - 13427q^{13} + 34782q^{15} - 32256q^{17} + 13718q^{19} + 103797q^{21} + 82525q^{23} + 159919q^{25} + 75141q^{27} - 12749q^{29} - 258944q^{31} + 257052q^{33} + 448167q^{35} - 149260q^{37} - 889557q^{39} + 339130q^{41} + 83869q^{43} + 2097243q^{45} - 1471025q^{47} + 1105372q^{49} - 913437q^{51} - 945643q^{53} + 1736899q^{55} + 473271q^{57} + 969009q^{59} - 1506755q^{61} + 2791179q^{63} - 5445956q^{65} + 1848219q^{67} + 6834063q^{69} + 3417184q^{71} - 2499822q^{73} + 10079553q^{75} + 5025267q^{77} - 2636926q^{79} + 2491398q^{81} + 10059354q^{83} - 439425q^{85} + 2572923q^{87} - 3506160q^{89} - 19003851q^{91} - 15169884q^{93} + 1063145q^{95} + 5893526q^{97} + 11301453q^{99} + O(q^{100}) \)

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
\(3\) 72.2392 1.54472 0.772358 0.635187i \(-0.219077\pi\)
0.772358 + 0.635187i \(0.219077\pi\)
\(4\) 0 0
\(5\) 467.472 1.67248 0.836240 0.548364i \(-0.184749\pi\)
0.836240 + 0.548364i \(0.184749\pi\)
\(6\) 0 0
\(7\) 1471.23 1.62121 0.810603 0.585596i \(-0.199139\pi\)
0.810603 + 0.585596i \(0.199139\pi\)
\(8\) 0 0
\(9\) 3031.51 1.38615
\(10\) 0 0
\(11\) 3547.04 0.803511 0.401756 0.915747i \(-0.368400\pi\)
0.401756 + 0.915747i \(0.368400\pi\)
\(12\) 0 0
\(13\) −12361.8 −1.56056 −0.780279 0.625431i \(-0.784923\pi\)
−0.780279 + 0.625431i \(0.784923\pi\)
\(14\) 0 0
\(15\) 33769.8 2.58351
\(16\) 0 0
\(17\) −13486.3 −0.665764 −0.332882 0.942969i \(-0.608021\pi\)
−0.332882 + 0.942969i \(0.608021\pi\)
\(18\) 0 0
\(19\) 6859.00 0.229416
\(20\) 0 0
\(21\) 106281. 2.50430
\(22\) 0 0
\(23\) 94084.9 1.61240 0.806199 0.591644i \(-0.201521\pi\)
0.806199 + 0.591644i \(0.201521\pi\)
\(24\) 0 0
\(25\) 140405. 1.79719
\(26\) 0 0
\(27\) 61006.6 0.596490
\(28\) 0 0
\(29\) 33541.0 0.255378 0.127689 0.991814i \(-0.459244\pi\)
0.127689 + 0.991814i \(0.459244\pi\)
\(30\) 0 0
\(31\) −212096. −1.27869 −0.639346 0.768919i \(-0.720795\pi\)
−0.639346 + 0.768919i \(0.720795\pi\)
\(32\) 0 0
\(33\) 256236. 1.24120
\(34\) 0 0
\(35\) 687760. 2.71143
\(36\) 0 0
\(37\) 33656.5 0.109235 0.0546176 0.998507i \(-0.482606\pi\)
0.0546176 + 0.998507i \(0.482606\pi\)
\(38\) 0 0
\(39\) −893007. −2.41062
\(40\) 0 0
\(41\) −480531. −1.08887 −0.544437 0.838801i \(-0.683257\pi\)
−0.544437 + 0.838801i \(0.683257\pi\)
\(42\) 0 0
\(43\) 561340. 1.07668 0.538339 0.842728i \(-0.319052\pi\)
0.538339 + 0.842728i \(0.319052\pi\)
\(44\) 0 0
\(45\) 1.41715e6 2.31830
\(46\) 0 0
\(47\) −478445. −0.672187 −0.336093 0.941829i \(-0.609106\pi\)
−0.336093 + 0.941829i \(0.609106\pi\)
\(48\) 0 0
\(49\) 1.34098e6 1.62831
\(50\) 0 0
\(51\) −974237. −1.02842
\(52\) 0 0
\(53\) −93127.0 −0.0859232 −0.0429616 0.999077i \(-0.513679\pi\)
−0.0429616 + 0.999077i \(0.513679\pi\)
\(54\) 0 0
\(55\) 1.65814e6 1.34386
\(56\) 0 0
\(57\) 495489. 0.354382
\(58\) 0 0
\(59\) −955034. −0.605392 −0.302696 0.953087i \(-0.597887\pi\)
−0.302696 + 0.953087i \(0.597887\pi\)
\(60\) 0 0
\(61\) −1.71632e6 −0.968152 −0.484076 0.875026i \(-0.660844\pi\)
−0.484076 + 0.875026i \(0.660844\pi\)
\(62\) 0 0
\(63\) 4.46005e6 2.24723
\(64\) 0 0
\(65\) −5.77880e6 −2.61000
\(66\) 0 0
\(67\) −476255. −0.193454 −0.0967270 0.995311i \(-0.530837\pi\)
−0.0967270 + 0.995311i \(0.530837\pi\)
\(68\) 0 0
\(69\) 6.79662e6 2.49070
\(70\) 0 0
\(71\) −1.95518e6 −0.648311 −0.324155 0.946004i \(-0.605080\pi\)
−0.324155 + 0.946004i \(0.605080\pi\)
\(72\) 0 0
\(73\) 1.21632e6 0.365948 0.182974 0.983118i \(-0.441428\pi\)
0.182974 + 0.983118i \(0.441428\pi\)
\(74\) 0 0
\(75\) 1.01428e7 2.77614
\(76\) 0 0
\(77\) 5.21852e6 1.30266
\(78\) 0 0
\(79\) −3.94544e6 −0.900328 −0.450164 0.892946i \(-0.648634\pi\)
−0.450164 + 0.892946i \(0.648634\pi\)
\(80\) 0 0
\(81\) −2.22284e6 −0.464740
\(82\) 0 0
\(83\) 3.54252e6 0.680048 0.340024 0.940417i \(-0.389565\pi\)
0.340024 + 0.940417i \(0.389565\pi\)
\(84\) 0 0
\(85\) −6.30445e6 −1.11348
\(86\) 0 0
\(87\) 2.42298e6 0.394487
\(88\) 0 0
\(89\) −7.50202e6 −1.12801 −0.564006 0.825771i \(-0.690740\pi\)
−0.564006 + 0.825771i \(0.690740\pi\)
\(90\) 0 0
\(91\) −1.81871e7 −2.52999
\(92\) 0 0
\(93\) −1.53216e7 −1.97522
\(94\) 0 0
\(95\) 3.20639e6 0.383693
\(96\) 0 0
\(97\) 1.27974e7 1.42371 0.711855 0.702326i \(-0.247855\pi\)
0.711855 + 0.702326i \(0.247855\pi\)
\(98\) 0 0
\(99\) 1.07529e7 1.11379
\(100\) 0 0
\(101\) 332855. 0.0321463 0.0160731 0.999871i \(-0.494884\pi\)
0.0160731 + 0.999871i \(0.494884\pi\)
\(102\) 0 0
\(103\) −4.52517e6 −0.408042 −0.204021 0.978967i \(-0.565401\pi\)
−0.204021 + 0.978967i \(0.565401\pi\)
\(104\) 0 0
\(105\) 4.96833e7 4.18840
\(106\) 0 0
\(107\) −7.33885e6 −0.579141 −0.289571 0.957157i \(-0.593513\pi\)
−0.289571 + 0.957157i \(0.593513\pi\)
\(108\) 0 0
\(109\) −1.03920e7 −0.768612 −0.384306 0.923206i \(-0.625559\pi\)
−0.384306 + 0.923206i \(0.625559\pi\)
\(110\) 0 0
\(111\) 2.43132e6 0.168737
\(112\) 0 0
\(113\) 5.33971e6 0.348131 0.174066 0.984734i \(-0.444309\pi\)
0.174066 + 0.984734i \(0.444309\pi\)
\(114\) 0 0
\(115\) 4.39820e7 2.69670
\(116\) 0 0
\(117\) −3.74749e7 −2.16317
\(118\) 0 0
\(119\) −1.98414e7 −1.07934
\(120\) 0 0
\(121\) −6.90567e6 −0.354370
\(122\) 0 0
\(123\) −3.47132e7 −1.68200
\(124\) 0 0
\(125\) 2.91142e7 1.33328
\(126\) 0 0
\(127\) 3.43331e7 1.48730 0.743652 0.668566i \(-0.233092\pi\)
0.743652 + 0.668566i \(0.233092\pi\)
\(128\) 0 0
\(129\) 4.05507e7 1.66316
\(130\) 0 0
\(131\) −250925. −0.00975202 −0.00487601 0.999988i \(-0.501552\pi\)
−0.00487601 + 0.999988i \(0.501552\pi\)
\(132\) 0 0
\(133\) 1.00912e7 0.371930
\(134\) 0 0
\(135\) 2.85189e7 0.997618
\(136\) 0 0
\(137\) −2.54613e7 −0.845978 −0.422989 0.906135i \(-0.639019\pi\)
−0.422989 + 0.906135i \(0.639019\pi\)
\(138\) 0 0
\(139\) −4.33237e7 −1.36828 −0.684138 0.729352i \(-0.739822\pi\)
−0.684138 + 0.729352i \(0.739822\pi\)
\(140\) 0 0
\(141\) −3.45625e7 −1.03834
\(142\) 0 0
\(143\) −4.38478e7 −1.25393
\(144\) 0 0
\(145\) 1.56795e7 0.427114
\(146\) 0 0
\(147\) 9.68716e7 2.51528
\(148\) 0 0
\(149\) −3.98620e7 −0.987204 −0.493602 0.869688i \(-0.664320\pi\)
−0.493602 + 0.869688i \(0.664320\pi\)
\(150\) 0 0
\(151\) 2.13810e6 0.0505369 0.0252684 0.999681i \(-0.491956\pi\)
0.0252684 + 0.999681i \(0.491956\pi\)
\(152\) 0 0
\(153\) −4.08837e7 −0.922847
\(154\) 0 0
\(155\) −9.91489e7 −2.13859
\(156\) 0 0
\(157\) −3.04702e7 −0.628387 −0.314193 0.949359i \(-0.601734\pi\)
−0.314193 + 0.949359i \(0.601734\pi\)
\(158\) 0 0
\(159\) −6.72743e6 −0.132727
\(160\) 0 0
\(161\) 1.38421e8 2.61403
\(162\) 0 0
\(163\) 9.48952e7 1.71628 0.858139 0.513417i \(-0.171621\pi\)
0.858139 + 0.513417i \(0.171621\pi\)
\(164\) 0 0
\(165\) 1.19783e8 2.07588
\(166\) 0 0
\(167\) 1.10332e8 1.83313 0.916563 0.399890i \(-0.130952\pi\)
0.916563 + 0.399890i \(0.130952\pi\)
\(168\) 0 0
\(169\) 9.00657e7 1.43534
\(170\) 0 0
\(171\) 2.07931e7 0.318004
\(172\) 0 0
\(173\) 1.06112e8 1.55813 0.779066 0.626942i \(-0.215694\pi\)
0.779066 + 0.626942i \(0.215694\pi\)
\(174\) 0 0
\(175\) 2.06569e8 2.91361
\(176\) 0 0
\(177\) −6.89909e7 −0.935159
\(178\) 0 0
\(179\) −4.88351e7 −0.636424 −0.318212 0.948020i \(-0.603082\pi\)
−0.318212 + 0.948020i \(0.603082\pi\)
\(180\) 0 0
\(181\) 5.52074e6 0.0692026 0.0346013 0.999401i \(-0.488984\pi\)
0.0346013 + 0.999401i \(0.488984\pi\)
\(182\) 0 0
\(183\) −1.23986e8 −1.49552
\(184\) 0 0
\(185\) 1.57335e7 0.182693
\(186\) 0 0
\(187\) −4.78363e7 −0.534948
\(188\) 0 0
\(189\) 8.97549e7 0.967034
\(190\) 0 0
\(191\) 1.15562e8 1.20005 0.600023 0.799983i \(-0.295158\pi\)
0.600023 + 0.799983i \(0.295158\pi\)
\(192\) 0 0
\(193\) 4.01954e6 0.0402463 0.0201232 0.999798i \(-0.493594\pi\)
0.0201232 + 0.999798i \(0.493594\pi\)
\(194\) 0 0
\(195\) −4.17456e8 −4.03171
\(196\) 0 0
\(197\) 8.92289e7 0.831522 0.415761 0.909474i \(-0.363515\pi\)
0.415761 + 0.909474i \(0.363515\pi\)
\(198\) 0 0
\(199\) 1.73811e8 1.56348 0.781740 0.623604i \(-0.214332\pi\)
0.781740 + 0.623604i \(0.214332\pi\)
\(200\) 0 0
\(201\) −3.44043e7 −0.298832
\(202\) 0 0
\(203\) 4.93467e7 0.414020
\(204\) 0 0
\(205\) −2.24635e8 −1.82112
\(206\) 0 0
\(207\) 2.85219e8 2.23502
\(208\) 0 0
\(209\) 2.43292e7 0.184338
\(210\) 0 0
\(211\) 1.48188e8 1.08598 0.542992 0.839738i \(-0.317291\pi\)
0.542992 + 0.839738i \(0.317291\pi\)
\(212\) 0 0
\(213\) −1.41241e8 −1.00146
\(214\) 0 0
\(215\) 2.62411e8 1.80072
\(216\) 0 0
\(217\) −3.12042e8 −2.07302
\(218\) 0 0
\(219\) 8.78662e7 0.565285
\(220\) 0 0
\(221\) 1.66714e8 1.03896
\(222\) 0 0
\(223\) −1.38594e6 −0.00836907 −0.00418454 0.999991i \(-0.501332\pi\)
−0.00418454 + 0.999991i \(0.501332\pi\)
\(224\) 0 0
\(225\) 4.25639e8 2.49117
\(226\) 0 0
\(227\) −1.35147e7 −0.0766858 −0.0383429 0.999265i \(-0.512208\pi\)
−0.0383429 + 0.999265i \(0.512208\pi\)
\(228\) 0 0
\(229\) 3.09882e7 0.170519 0.0852594 0.996359i \(-0.472828\pi\)
0.0852594 + 0.996359i \(0.472828\pi\)
\(230\) 0 0
\(231\) 3.76982e8 2.01224
\(232\) 0 0
\(233\) −2.04142e8 −1.05727 −0.528635 0.848849i \(-0.677296\pi\)
−0.528635 + 0.848849i \(0.677296\pi\)
\(234\) 0 0
\(235\) −2.23660e8 −1.12422
\(236\) 0 0
\(237\) −2.85016e8 −1.39075
\(238\) 0 0
\(239\) 2.25595e8 1.06890 0.534449 0.845201i \(-0.320519\pi\)
0.534449 + 0.845201i \(0.320519\pi\)
\(240\) 0 0
\(241\) 4.16156e7 0.191512 0.0957562 0.995405i \(-0.469473\pi\)
0.0957562 + 0.995405i \(0.469473\pi\)
\(242\) 0 0
\(243\) −2.93998e8 −1.31438
\(244\) 0 0
\(245\) 6.26872e8 2.72331
\(246\) 0 0
\(247\) −8.47896e7 −0.358017
\(248\) 0 0
\(249\) 2.55909e8 1.05048
\(250\) 0 0
\(251\) −4.55579e8 −1.81847 −0.909233 0.416287i \(-0.863331\pi\)
−0.909233 + 0.416287i \(0.863331\pi\)
\(252\) 0 0
\(253\) 3.33723e8 1.29558
\(254\) 0 0
\(255\) −4.55428e8 −1.72000
\(256\) 0 0
\(257\) −2.46264e8 −0.904972 −0.452486 0.891772i \(-0.649463\pi\)
−0.452486 + 0.891772i \(0.649463\pi\)
\(258\) 0 0
\(259\) 4.95165e7 0.177093
\(260\) 0 0
\(261\) 1.01680e8 0.353992
\(262\) 0 0
\(263\) −4.04810e8 −1.37217 −0.686083 0.727523i \(-0.740671\pi\)
−0.686083 + 0.727523i \(0.740671\pi\)
\(264\) 0 0
\(265\) −4.35343e7 −0.143705
\(266\) 0 0
\(267\) −5.41940e8 −1.74246
\(268\) 0 0
\(269\) −2.44584e8 −0.766118 −0.383059 0.923724i \(-0.625129\pi\)
−0.383059 + 0.923724i \(0.625129\pi\)
\(270\) 0 0
\(271\) −7.88689e7 −0.240721 −0.120360 0.992730i \(-0.538405\pi\)
−0.120360 + 0.992730i \(0.538405\pi\)
\(272\) 0 0
\(273\) −1.31382e9 −3.90811
\(274\) 0 0
\(275\) 4.98023e8 1.44406
\(276\) 0 0
\(277\) −2.30329e8 −0.651133 −0.325567 0.945519i \(-0.605555\pi\)
−0.325567 + 0.945519i \(0.605555\pi\)
\(278\) 0 0
\(279\) −6.42970e8 −1.77246
\(280\) 0 0
\(281\) −5.44589e8 −1.46419 −0.732094 0.681203i \(-0.761457\pi\)
−0.732094 + 0.681203i \(0.761457\pi\)
\(282\) 0 0
\(283\) 3.69887e7 0.0970101 0.0485051 0.998823i \(-0.484554\pi\)
0.0485051 + 0.998823i \(0.484554\pi\)
\(284\) 0 0
\(285\) 2.31627e8 0.592697
\(286\) 0 0
\(287\) −7.06973e8 −1.76529
\(288\) 0 0
\(289\) −2.28460e8 −0.556759
\(290\) 0 0
\(291\) 9.24477e8 2.19923
\(292\) 0 0
\(293\) 1.98050e8 0.459979 0.229990 0.973193i \(-0.426131\pi\)
0.229990 + 0.973193i \(0.426131\pi\)
\(294\) 0 0
\(295\) −4.46452e8 −1.01251
\(296\) 0 0
\(297\) 2.16393e8 0.479287
\(298\) 0 0
\(299\) −1.16306e9 −2.51624
\(300\) 0 0
\(301\) 8.25861e8 1.74552
\(302\) 0 0
\(303\) 2.40452e7 0.0496569
\(304\) 0 0
\(305\) −8.02331e8 −1.61921
\(306\) 0 0
\(307\) −3.28496e8 −0.647957 −0.323979 0.946064i \(-0.605021\pi\)
−0.323979 + 0.946064i \(0.605021\pi\)
\(308\) 0 0
\(309\) −3.26895e8 −0.630309
\(310\) 0 0
\(311\) 5.87246e8 1.10703 0.553514 0.832840i \(-0.313287\pi\)
0.553514 + 0.832840i \(0.313287\pi\)
\(312\) 0 0
\(313\) 6.29976e8 1.16123 0.580616 0.814178i \(-0.302812\pi\)
0.580616 + 0.814178i \(0.302812\pi\)
\(314\) 0 0
\(315\) 2.08495e9 3.75845
\(316\) 0 0
\(317\) 8.00885e8 1.41209 0.706046 0.708166i \(-0.250477\pi\)
0.706046 + 0.708166i \(0.250477\pi\)
\(318\) 0 0
\(319\) 1.18971e8 0.205199
\(320\) 0 0
\(321\) −5.30153e8 −0.894609
\(322\) 0 0
\(323\) −9.25022e7 −0.152737
\(324\) 0 0
\(325\) −1.73566e9 −2.80462
\(326\) 0 0
\(327\) −7.50711e8 −1.18729
\(328\) 0 0
\(329\) −7.03905e8 −1.08975
\(330\) 0 0
\(331\) 4.54262e8 0.688507 0.344253 0.938877i \(-0.388132\pi\)
0.344253 + 0.938877i \(0.388132\pi\)
\(332\) 0 0
\(333\) 1.02030e8 0.151416
\(334\) 0 0
\(335\) −2.22636e8 −0.323548
\(336\) 0 0
\(337\) −9.81183e8 −1.39651 −0.698257 0.715847i \(-0.746041\pi\)
−0.698257 + 0.715847i \(0.746041\pi\)
\(338\) 0 0
\(339\) 3.85737e8 0.537764
\(340\) 0 0
\(341\) −7.52313e8 −1.02744
\(342\) 0 0
\(343\) 7.61275e8 1.01862
\(344\) 0 0
\(345\) 3.17723e9 4.16564
\(346\) 0 0
\(347\) −2.69918e8 −0.346800 −0.173400 0.984852i \(-0.555475\pi\)
−0.173400 + 0.984852i \(0.555475\pi\)
\(348\) 0 0
\(349\) 1.09409e9 1.37773 0.688863 0.724892i \(-0.258110\pi\)
0.688863 + 0.724892i \(0.258110\pi\)
\(350\) 0 0
\(351\) −7.54151e8 −0.930858
\(352\) 0 0
\(353\) 5.02386e8 0.607891 0.303946 0.952689i \(-0.401696\pi\)
0.303946 + 0.952689i \(0.401696\pi\)
\(354\) 0 0
\(355\) −9.13994e8 −1.08429
\(356\) 0 0
\(357\) −1.43333e9 −1.66727
\(358\) 0 0
\(359\) 4.89464e8 0.558329 0.279165 0.960243i \(-0.409942\pi\)
0.279165 + 0.960243i \(0.409942\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) −4.98860e8 −0.547401
\(364\) 0 0
\(365\) 5.68597e8 0.612040
\(366\) 0 0
\(367\) 5.54974e8 0.586059 0.293029 0.956103i \(-0.405337\pi\)
0.293029 + 0.956103i \(0.405337\pi\)
\(368\) 0 0
\(369\) −1.45673e9 −1.50934
\(370\) 0 0
\(371\) −1.37012e8 −0.139299
\(372\) 0 0
\(373\) −1.29835e9 −1.29542 −0.647708 0.761889i \(-0.724272\pi\)
−0.647708 + 0.761889i \(0.724272\pi\)
\(374\) 0 0
\(375\) 2.10319e9 2.05954
\(376\) 0 0
\(377\) −4.14628e8 −0.398532
\(378\) 0 0
\(379\) −7.49227e8 −0.706929 −0.353465 0.935448i \(-0.614997\pi\)
−0.353465 + 0.935448i \(0.614997\pi\)
\(380\) 0 0
\(381\) 2.48020e9 2.29746
\(382\) 0 0
\(383\) −2.12814e8 −0.193555 −0.0967774 0.995306i \(-0.530853\pi\)
−0.0967774 + 0.995306i \(0.530853\pi\)
\(384\) 0 0
\(385\) 2.43951e9 2.17867
\(386\) 0 0
\(387\) 1.70171e9 1.49244
\(388\) 0 0
\(389\) 3.13367e8 0.269917 0.134958 0.990851i \(-0.456910\pi\)
0.134958 + 0.990851i \(0.456910\pi\)
\(390\) 0 0
\(391\) −1.26885e9 −1.07348
\(392\) 0 0
\(393\) −1.81266e7 −0.0150641
\(394\) 0 0
\(395\) −1.84438e9 −1.50578
\(396\) 0 0
\(397\) 1.91013e9 1.53213 0.766066 0.642762i \(-0.222211\pi\)
0.766066 + 0.642762i \(0.222211\pi\)
\(398\) 0 0
\(399\) 7.28980e8 0.574527
\(400\) 0 0
\(401\) −1.27797e9 −0.989726 −0.494863 0.868971i \(-0.664782\pi\)
−0.494863 + 0.868971i \(0.664782\pi\)
\(402\) 0 0
\(403\) 2.62189e9 1.99547
\(404\) 0 0
\(405\) −1.03912e9 −0.777269
\(406\) 0 0
\(407\) 1.19381e8 0.0877716
\(408\) 0 0
\(409\) 2.92650e8 0.211503 0.105752 0.994393i \(-0.466275\pi\)
0.105752 + 0.994393i \(0.466275\pi\)
\(410\) 0 0
\(411\) −1.83931e9 −1.30680
\(412\) 0 0
\(413\) −1.40508e9 −0.981465
\(414\) 0 0
\(415\) 1.65603e9 1.13737
\(416\) 0 0
\(417\) −3.12967e9 −2.11360
\(418\) 0 0
\(419\) −3.81573e8 −0.253413 −0.126706 0.991940i \(-0.540441\pi\)
−0.126706 + 0.991940i \(0.540441\pi\)
\(420\) 0 0
\(421\) 2.64639e9 1.72849 0.864244 0.503073i \(-0.167797\pi\)
0.864244 + 0.503073i \(0.167797\pi\)
\(422\) 0 0
\(423\) −1.45041e9 −0.931751
\(424\) 0 0
\(425\) −1.89354e9 −1.19650
\(426\) 0 0
\(427\) −2.52511e9 −1.56957
\(428\) 0 0
\(429\) −3.16753e9 −1.93696
\(430\) 0 0
\(431\) −1.69877e9 −1.02203 −0.511015 0.859572i \(-0.670730\pi\)
−0.511015 + 0.859572i \(0.670730\pi\)
\(432\) 0 0
\(433\) 2.37930e9 1.40845 0.704226 0.709975i \(-0.251294\pi\)
0.704226 + 0.709975i \(0.251294\pi\)
\(434\) 0 0
\(435\) 1.13267e9 0.659770
\(436\) 0 0
\(437\) 6.45328e8 0.369909
\(438\) 0 0
\(439\) −2.18333e9 −1.23167 −0.615835 0.787875i \(-0.711181\pi\)
−0.615835 + 0.787875i \(0.711181\pi\)
\(440\) 0 0
\(441\) 4.06520e9 2.25708
\(442\) 0 0
\(443\) 2.41144e9 1.31784 0.658922 0.752212i \(-0.271013\pi\)
0.658922 + 0.752212i \(0.271013\pi\)
\(444\) 0 0
\(445\) −3.50699e9 −1.88658
\(446\) 0 0
\(447\) −2.87960e9 −1.52495
\(448\) 0 0
\(449\) −6.02088e8 −0.313905 −0.156952 0.987606i \(-0.550167\pi\)
−0.156952 + 0.987606i \(0.550167\pi\)
\(450\) 0 0
\(451\) −1.70446e9 −0.874923
\(452\) 0 0
\(453\) 1.54455e8 0.0780652
\(454\) 0 0
\(455\) −8.50196e9 −4.23135
\(456\) 0 0
\(457\) −3.28085e9 −1.60798 −0.803989 0.594644i \(-0.797293\pi\)
−0.803989 + 0.594644i \(0.797293\pi\)
\(458\) 0 0
\(459\) −8.22750e8 −0.397122
\(460\) 0 0
\(461\) −3.50631e9 −1.66685 −0.833426 0.552632i \(-0.813624\pi\)
−0.833426 + 0.552632i \(0.813624\pi\)
\(462\) 0 0
\(463\) 3.85170e9 1.80351 0.901757 0.432244i \(-0.142278\pi\)
0.901757 + 0.432244i \(0.142278\pi\)
\(464\) 0 0
\(465\) −7.16244e9 −3.30351
\(466\) 0 0
\(467\) −4.73318e8 −0.215052 −0.107526 0.994202i \(-0.534293\pi\)
−0.107526 + 0.994202i \(0.534293\pi\)
\(468\) 0 0
\(469\) −7.00682e8 −0.313629
\(470\) 0 0
\(471\) −2.20115e9 −0.970680
\(472\) 0 0
\(473\) 1.99109e9 0.865123
\(474\) 0 0
\(475\) 9.63039e8 0.412303
\(476\) 0 0
\(477\) −2.82315e8 −0.119102
\(478\) 0 0
\(479\) −2.43257e9 −1.01132 −0.505662 0.862732i \(-0.668752\pi\)
−0.505662 + 0.862732i \(0.668752\pi\)
\(480\) 0 0
\(481\) −4.16055e8 −0.170468
\(482\) 0 0
\(483\) 9.99941e9 4.03793
\(484\) 0 0
\(485\) 5.98244e9 2.38113
\(486\) 0 0
\(487\) 3.37943e9 1.32584 0.662921 0.748689i \(-0.269317\pi\)
0.662921 + 0.748689i \(0.269317\pi\)
\(488\) 0 0
\(489\) 6.85516e9 2.65116
\(490\) 0 0
\(491\) 1.62343e9 0.618938 0.309469 0.950910i \(-0.399849\pi\)
0.309469 + 0.950910i \(0.399849\pi\)
\(492\) 0 0
\(493\) −4.52343e8 −0.170021
\(494\) 0 0
\(495\) 5.02667e9 1.86278
\(496\) 0 0
\(497\) −2.87653e9 −1.05105
\(498\) 0 0
\(499\) −3.88153e9 −1.39846 −0.699231 0.714896i \(-0.746474\pi\)
−0.699231 + 0.714896i \(0.746474\pi\)
\(500\) 0 0
\(501\) 7.97027e9 2.83166
\(502\) 0 0
\(503\) −2.38583e9 −0.835893 −0.417947 0.908472i \(-0.637250\pi\)
−0.417947 + 0.908472i \(0.637250\pi\)
\(504\) 0 0
\(505\) 1.55600e8 0.0537639
\(506\) 0 0
\(507\) 6.50628e9 2.21720
\(508\) 0 0
\(509\) −3.37295e9 −1.13370 −0.566849 0.823822i \(-0.691838\pi\)
−0.566849 + 0.823822i \(0.691838\pi\)
\(510\) 0 0
\(511\) 1.78949e9 0.593277
\(512\) 0 0
\(513\) 4.18444e8 0.136844
\(514\) 0 0
\(515\) −2.11539e9 −0.682441
\(516\) 0 0
\(517\) −1.69707e9 −0.540109
\(518\) 0 0
\(519\) 7.66547e9 2.40687
\(520\) 0 0
\(521\) 5.32384e9 1.64927 0.824637 0.565662i \(-0.191379\pi\)
0.824637 + 0.565662i \(0.191379\pi\)
\(522\) 0 0
\(523\) −4.07644e9 −1.24602 −0.623010 0.782214i \(-0.714090\pi\)
−0.623010 + 0.782214i \(0.714090\pi\)
\(524\) 0 0
\(525\) 1.49224e10 4.50070
\(526\) 0 0
\(527\) 2.86038e9 0.851307
\(528\) 0 0
\(529\) 5.44713e9 1.59983
\(530\) 0 0
\(531\) −2.89519e9 −0.839163
\(532\) 0 0
\(533\) 5.94023e9 1.69925
\(534\) 0 0
\(535\) −3.43071e9 −0.968602
\(536\) 0 0
\(537\) −3.52781e9 −0.983095
\(538\) 0 0
\(539\) 4.75652e9 1.30837
\(540\) 0 0
\(541\) −3.63310e9 −0.986477 −0.493239 0.869894i \(-0.664187\pi\)
−0.493239 + 0.869894i \(0.664187\pi\)
\(542\) 0 0
\(543\) 3.98814e8 0.106898
\(544\) 0 0
\(545\) −4.85798e9 −1.28549
\(546\) 0 0
\(547\) 5.74113e9 1.49983 0.749914 0.661535i \(-0.230095\pi\)
0.749914 + 0.661535i \(0.230095\pi\)
\(548\) 0 0
\(549\) −5.20303e9 −1.34200
\(550\) 0 0
\(551\) 2.30058e8 0.0585877
\(552\) 0 0
\(553\) −5.80466e9 −1.45962
\(554\) 0 0
\(555\) 1.13657e9 0.282210
\(556\) 0 0
\(557\) 9.95936e8 0.244196 0.122098 0.992518i \(-0.461038\pi\)
0.122098 + 0.992518i \(0.461038\pi\)
\(558\) 0 0
\(559\) −6.93917e9 −1.68022
\(560\) 0 0
\(561\) −3.45566e9 −0.826344
\(562\) 0 0
\(563\) 4.05283e9 0.957147 0.478573 0.878048i \(-0.341154\pi\)
0.478573 + 0.878048i \(0.341154\pi\)
\(564\) 0 0
\(565\) 2.49617e9 0.582242
\(566\) 0 0
\(567\) −3.27031e9 −0.753440
\(568\) 0 0
\(569\) 3.35141e9 0.762666 0.381333 0.924438i \(-0.375465\pi\)
0.381333 + 0.924438i \(0.375465\pi\)
\(570\) 0 0
\(571\) −4.22454e9 −0.949627 −0.474813 0.880087i \(-0.657484\pi\)
−0.474813 + 0.880087i \(0.657484\pi\)
\(572\) 0 0
\(573\) 8.34810e9 1.85373
\(574\) 0 0
\(575\) 1.32100e10 2.89778
\(576\) 0 0
\(577\) −6.43062e9 −1.39360 −0.696799 0.717266i \(-0.745393\pi\)
−0.696799 + 0.717266i \(0.745393\pi\)
\(578\) 0 0
\(579\) 2.90369e8 0.0621691
\(580\) 0 0
\(581\) 5.21188e9 1.10250
\(582\) 0 0
\(583\) −3.30325e8 −0.0690402
\(584\) 0 0
\(585\) −1.75185e10 −3.61785
\(586\) 0 0
\(587\) 2.90754e9 0.593326 0.296663 0.954982i \(-0.404126\pi\)
0.296663 + 0.954982i \(0.404126\pi\)
\(588\) 0 0
\(589\) −1.45476e9 −0.293352
\(590\) 0 0
\(591\) 6.44582e9 1.28447
\(592\) 0 0
\(593\) −5.13346e8 −0.101093 −0.0505463 0.998722i \(-0.516096\pi\)
−0.0505463 + 0.998722i \(0.516096\pi\)
\(594\) 0 0
\(595\) −9.27531e9 −1.80517
\(596\) 0 0
\(597\) 1.25560e10 2.41513
\(598\) 0 0
\(599\) 3.44859e9 0.655613 0.327807 0.944745i \(-0.393691\pi\)
0.327807 + 0.944745i \(0.393691\pi\)
\(600\) 0 0
\(601\) −5.96985e9 −1.12177 −0.560883 0.827895i \(-0.689538\pi\)
−0.560883 + 0.827895i \(0.689538\pi\)
\(602\) 0 0
\(603\) −1.44377e9 −0.268156
\(604\) 0 0
\(605\) −3.22821e9 −0.592676
\(606\) 0 0
\(607\) 1.00863e10 1.83050 0.915249 0.402888i \(-0.131993\pi\)
0.915249 + 0.402888i \(0.131993\pi\)
\(608\) 0 0
\(609\) 3.56477e9 0.639544
\(610\) 0 0
\(611\) 5.91445e9 1.04899
\(612\) 0 0
\(613\) 1.24543e9 0.218378 0.109189 0.994021i \(-0.465175\pi\)
0.109189 + 0.994021i \(0.465175\pi\)
\(614\) 0 0
\(615\) −1.62275e10 −2.81311
\(616\) 0 0
\(617\) −5.96038e8 −0.102159 −0.0510794 0.998695i \(-0.516266\pi\)
−0.0510794 + 0.998695i \(0.516266\pi\)
\(618\) 0 0
\(619\) 3.90038e9 0.660981 0.330491 0.943809i \(-0.392786\pi\)
0.330491 + 0.943809i \(0.392786\pi\)
\(620\) 0 0
\(621\) 5.73979e9 0.961780
\(622\) 0 0
\(623\) −1.10372e10 −1.82874
\(624\) 0 0
\(625\) 2.64094e9 0.432692
\(626\) 0 0
\(627\) 1.75752e9 0.284750
\(628\) 0 0
\(629\) −4.53899e8 −0.0727248
\(630\) 0 0
\(631\) −4.64916e9 −0.736668 −0.368334 0.929694i \(-0.620072\pi\)
−0.368334 + 0.929694i \(0.620072\pi\)
\(632\) 0 0
\(633\) 1.07050e10 1.67754
\(634\) 0 0
\(635\) 1.60498e10 2.48749
\(636\) 0 0
\(637\) −1.65770e10 −2.54107
\(638\) 0 0
\(639\) −5.92715e9 −0.898655
\(640\) 0 0
\(641\) 9.04607e9 1.35662 0.678308 0.734778i \(-0.262714\pi\)
0.678308 + 0.734778i \(0.262714\pi\)
\(642\) 0 0
\(643\) −1.11196e10 −1.64949 −0.824746 0.565503i \(-0.808682\pi\)
−0.824746 + 0.565503i \(0.808682\pi\)
\(644\) 0 0
\(645\) 1.89563e10 2.78161
\(646\) 0 0
\(647\) 3.91564e9 0.568379 0.284189 0.958768i \(-0.408276\pi\)
0.284189 + 0.958768i \(0.408276\pi\)
\(648\) 0 0
\(649\) −3.38754e9 −0.486439
\(650\) 0 0
\(651\) −2.25417e10 −3.20223
\(652\) 0 0
\(653\) −1.07233e10 −1.50707 −0.753536 0.657407i \(-0.771653\pi\)
−0.753536 + 0.657407i \(0.771653\pi\)
\(654\) 0 0
\(655\) −1.17300e8 −0.0163100
\(656\) 0 0
\(657\) 3.68729e9 0.507258
\(658\) 0 0
\(659\) −9.55747e8 −0.130090 −0.0650450 0.997882i \(-0.520719\pi\)
−0.0650450 + 0.997882i \(0.520719\pi\)
\(660\) 0 0
\(661\) −1.17758e10 −1.58594 −0.792971 0.609260i \(-0.791467\pi\)
−0.792971 + 0.609260i \(0.791467\pi\)
\(662\) 0 0
\(663\) 1.20433e10 1.60490
\(664\) 0 0
\(665\) 4.71735e9 0.622046
\(666\) 0 0
\(667\) 3.15570e9 0.411771
\(668\) 0 0
\(669\) −1.00119e8 −0.0129278
\(670\) 0 0
\(671\) −6.08786e9 −0.777921
\(672\) 0 0
\(673\) 5.31058e9 0.671567 0.335783 0.941939i \(-0.390999\pi\)
0.335783 + 0.941939i \(0.390999\pi\)
\(674\) 0 0
\(675\) 8.56564e9 1.07200
\(676\) 0 0
\(677\) 1.06518e10 1.31936 0.659680 0.751547i \(-0.270692\pi\)
0.659680 + 0.751547i \(0.270692\pi\)
\(678\) 0 0
\(679\) 1.88280e10 2.30813
\(680\) 0 0
\(681\) −9.76290e8 −0.118458
\(682\) 0 0
\(683\) 3.14122e8 0.0377247 0.0188624 0.999822i \(-0.493996\pi\)
0.0188624 + 0.999822i \(0.493996\pi\)
\(684\) 0 0
\(685\) −1.19025e10 −1.41488
\(686\) 0 0
\(687\) 2.23856e9 0.263403
\(688\) 0 0
\(689\) 1.15122e9 0.134088
\(690\) 0 0
\(691\) 1.44774e10 1.66923 0.834615 0.550833i \(-0.185690\pi\)
0.834615 + 0.550833i \(0.185690\pi\)
\(692\) 0 0
\(693\) 1.58200e10 1.80568
\(694\) 0 0
\(695\) −2.02526e10 −2.28841
\(696\) 0 0
\(697\) 6.48056e9 0.724933
\(698\) 0 0
\(699\) −1.47470e10 −1.63318
\(700\) 0 0
\(701\) 1.22852e10 1.34701 0.673503 0.739185i \(-0.264789\pi\)
0.673503 + 0.739185i \(0.264789\pi\)
\(702\) 0 0
\(703\) 2.30850e8 0.0250603
\(704\) 0 0
\(705\) −1.61570e10 −1.73660
\(706\) 0 0
\(707\) 4.89707e8 0.0521157
\(708\) 0 0
\(709\) −3.48632e9 −0.367371 −0.183686 0.982985i \(-0.558803\pi\)
−0.183686 + 0.982985i \(0.558803\pi\)
\(710\) 0 0
\(711\) −1.19606e10 −1.24799
\(712\) 0 0
\(713\) −1.99550e10 −2.06176
\(714\) 0 0
\(715\) −2.04976e10 −2.09717
\(716\) 0 0
\(717\) 1.62968e10 1.65115
\(718\) 0 0
\(719\) −6.30005e9 −0.632110 −0.316055 0.948741i \(-0.602358\pi\)
−0.316055 + 0.948741i \(0.602358\pi\)
\(720\) 0 0
\(721\) −6.65758e9 −0.661520
\(722\) 0 0
\(723\) 3.00628e9 0.295832
\(724\) 0 0
\(725\) 4.70933e9 0.458962
\(726\) 0 0
\(727\) 4.41318e9 0.425973 0.212986 0.977055i \(-0.431681\pi\)
0.212986 + 0.977055i \(0.431681\pi\)
\(728\) 0 0
\(729\) −1.63768e10 −1.56561
\(730\) 0 0
\(731\) −7.57037e9 −0.716814
\(732\) 0 0
\(733\) 1.15847e10 1.08648 0.543240 0.839577i \(-0.317197\pi\)
0.543240 + 0.839577i \(0.317197\pi\)
\(734\) 0 0
\(735\) 4.52848e10 4.20675
\(736\) 0 0
\(737\) −1.68930e9 −0.155443
\(738\) 0 0
\(739\) −1.60174e10 −1.45995 −0.729974 0.683475i \(-0.760468\pi\)
−0.729974 + 0.683475i \(0.760468\pi\)
\(740\) 0 0
\(741\) −6.12514e9 −0.553034
\(742\) 0 0
\(743\) −6.54024e9 −0.584969 −0.292484 0.956270i \(-0.594482\pi\)
−0.292484 + 0.956270i \(0.594482\pi\)
\(744\) 0 0
\(745\) −1.86344e10 −1.65108
\(746\) 0 0
\(747\) 1.07392e10 0.942648
\(748\) 0 0
\(749\) −1.07972e10 −0.938908
\(750\) 0 0
\(751\) −1.04818e10 −0.903015 −0.451508 0.892267i \(-0.649114\pi\)
−0.451508 + 0.892267i \(0.649114\pi\)
\(752\) 0 0
\(753\) −3.29107e10 −2.80901
\(754\) 0 0
\(755\) 9.99502e8 0.0845219
\(756\) 0 0
\(757\) 4.11773e9 0.345003 0.172501 0.985009i \(-0.444815\pi\)
0.172501 + 0.985009i \(0.444815\pi\)
\(758\) 0 0
\(759\) 2.41079e10 2.00130
\(760\) 0 0
\(761\) −1.90960e10 −1.57071 −0.785357 0.619043i \(-0.787521\pi\)
−0.785357 + 0.619043i \(0.787521\pi\)
\(762\) 0 0
\(763\) −1.52891e10 −1.24608
\(764\) 0 0
\(765\) −1.91120e10 −1.54344
\(766\) 0 0
\(767\) 1.18059e10 0.944750
\(768\) 0 0
\(769\) −1.20473e10 −0.955315 −0.477658 0.878546i \(-0.658514\pi\)
−0.477658 + 0.878546i \(0.658514\pi\)
\(770\) 0 0
\(771\) −1.77899e10 −1.39792
\(772\) 0 0
\(773\) 4.62608e9 0.360234 0.180117 0.983645i \(-0.442352\pi\)
0.180117 + 0.983645i \(0.442352\pi\)
\(774\) 0 0
\(775\) −2.97793e10 −2.29805
\(776\) 0 0
\(777\) 3.57703e9 0.273558
\(778\) 0 0
\(779\) −3.29596e9 −0.249805
\(780\) 0 0
\(781\) −6.93512e9 −0.520925
\(782\) 0 0
\(783\) 2.04622e9 0.152330
\(784\) 0 0
\(785\) −1.42440e10 −1.05096
\(786\) 0 0
\(787\) −5.71149e9 −0.417675 −0.208837 0.977950i \(-0.566968\pi\)
−0.208837 + 0.977950i \(0.566968\pi\)
\(788\) 0 0
\(789\) −2.92432e10 −2.11961
\(790\) 0 0
\(791\) 7.85596e9 0.564393
\(792\) 0 0
\(793\) 2.12168e10 1.51086
\(794\) 0 0
\(795\) −3.14488e9 −0.221983
\(796\) 0 0
\(797\) −1.15032e10 −0.804848 −0.402424 0.915453i \(-0.631832\pi\)
−0.402424 + 0.915453i \(0.631832\pi\)
\(798\) 0 0
\(799\) 6.45244e9 0.447517
\(800\) 0 0
\(801\) −2.27424e10 −1.56359
\(802\) 0 0
\(803\) 4.31435e9 0.294043
\(804\) 0 0
\(805\) 6.47078e10 4.37191
\(806\) 0 0
\(807\) −1.76686e10 −1.18344
\(808\) 0 0
\(809\) 1.91918e10 1.27437 0.637184 0.770712i \(-0.280099\pi\)
0.637184 + 0.770712i \(0.280099\pi\)
\(810\) 0 0
\(811\) 5.04377e9 0.332034 0.166017 0.986123i \(-0.446909\pi\)
0.166017 + 0.986123i \(0.446909\pi\)
\(812\) 0 0
\(813\) −5.69743e9 −0.371845
\(814\) 0 0
\(815\) 4.43609e10 2.87044
\(816\) 0 0
\(817\) 3.85023e9 0.247007
\(818\) 0 0
\(819\) −5.51343e10 −3.50694
\(820\) 0 0
\(821\) 2.85707e10 1.80185 0.900927 0.433972i \(-0.142888\pi\)
0.900927 + 0.433972i \(0.142888\pi\)
\(822\) 0 0
\(823\) −1.82564e10 −1.14161 −0.570803 0.821087i \(-0.693368\pi\)
−0.570803 + 0.821087i \(0.693368\pi\)
\(824\) 0 0
\(825\) 3.59768e10 2.23066
\(826\) 0 0
\(827\) 1.92073e9 0.118086 0.0590428 0.998255i \(-0.481195\pi\)
0.0590428 + 0.998255i \(0.481195\pi\)
\(828\) 0 0
\(829\) 4.75211e9 0.289698 0.144849 0.989454i \(-0.453730\pi\)
0.144849 + 0.989454i \(0.453730\pi\)
\(830\) 0 0
\(831\) −1.66388e10 −1.00582
\(832\) 0 0
\(833\) −1.80848e10 −1.08407
\(834\) 0 0
\(835\) 5.15770e10 3.06587
\(836\) 0 0
\(837\) −1.29392e10 −0.762728
\(838\) 0 0
\(839\) 1.17511e10 0.686927 0.343463 0.939166i \(-0.388400\pi\)
0.343463 + 0.939166i \(0.388400\pi\)
\(840\) 0 0
\(841\) −1.61249e10 −0.934782
\(842\) 0 0
\(843\) −3.93407e10 −2.26176
\(844\) 0 0
\(845\) 4.21032e10 2.40058
\(846\) 0 0
\(847\) −1.01598e10 −0.574507
\(848\) 0 0
\(849\) 2.67204e9 0.149853
\(850\) 0 0
\(851\) 3.16656e9 0.176130
\(852\) 0 0
\(853\) −1.98045e10 −1.09255 −0.546276 0.837605i \(-0.683955\pi\)
−0.546276 + 0.837605i \(0.683955\pi\)
\(854\) 0 0
\(855\) 9.72020e9 0.531856
\(856\) 0 0
\(857\) 1.83237e10 0.994445 0.497223 0.867623i \(-0.334353\pi\)
0.497223 + 0.867623i \(0.334353\pi\)
\(858\) 0 0
\(859\) −2.65234e10 −1.42775 −0.713876 0.700272i \(-0.753062\pi\)
−0.713876 + 0.700272i \(0.753062\pi\)
\(860\) 0 0
\(861\) −5.10712e10 −2.72687
\(862\) 0 0
\(863\) 1.77522e10 0.940188 0.470094 0.882616i \(-0.344220\pi\)
0.470094 + 0.882616i \(0.344220\pi\)
\(864\) 0 0
\(865\) 4.96045e10 2.60594
\(866\) 0 0
\(867\) −1.65038e10 −0.860034
\(868\) 0 0
\(869\) −1.39946e10 −0.723423
\(870\) 0 0
\(871\) 5.88737e9 0.301897
\(872\) 0 0
\(873\) 3.87955e10 1.97348
\(874\) 0 0
\(875\) 4.28338e10 2.16152
\(876\) 0 0
\(877\) 1.22608e9 0.0613792 0.0306896 0.999529i \(-0.490230\pi\)
0.0306896 + 0.999529i \(0.490230\pi\)
\(878\) 0 0
\(879\) 1.43070e10 0.710537
\(880\) 0 0
\(881\) 6.34486e9 0.312612 0.156306 0.987709i \(-0.450041\pi\)
0.156306 + 0.987709i \(0.450041\pi\)
\(882\) 0 0
\(883\) 2.24216e10 1.09598 0.547992 0.836484i \(-0.315392\pi\)
0.547992 + 0.836484i \(0.315392\pi\)
\(884\) 0 0
\(885\) −3.22513e10 −1.56403
\(886\) 0 0
\(887\) −3.38047e10 −1.62647 −0.813233 0.581938i \(-0.802295\pi\)
−0.813233 + 0.581938i \(0.802295\pi\)
\(888\) 0 0
\(889\) 5.05120e10 2.41123
\(890\) 0 0
\(891\) −7.88450e9 −0.373424
\(892\) 0 0
\(893\) −3.28166e9 −0.154210
\(894\) 0 0
\(895\) −2.28291e10 −1.06441
\(896\) 0 0
\(897\) −8.40185e10 −3.88688
\(898\) 0 0
\(899\) −7.11391e9 −0.326550
\(900\) 0 0
\(901\) 1.25593e9 0.0572045
\(902\) 0 0
\(903\) 5.96596e10 2.69633
\(904\) 0 0
\(905\) 2.58079e9 0.115740
\(906\) 0 0
\(907\) 1.56762e10 0.697616 0.348808 0.937194i \(-0.386587\pi\)
0.348808 + 0.937194i \(0.386587\pi\)
\(908\) 0 0
\(909\) 1.00905e9 0.0445595
\(910\) 0 0
\(911\) −2.35920e10 −1.03383 −0.516916 0.856036i \(-0.672920\pi\)
−0.516916 + 0.856036i \(0.672920\pi\)
\(912\) 0 0
\(913\) 1.25655e10 0.546426
\(914\) 0 0
\(915\) −5.79598e10 −2.50123
\(916\) 0 0
\(917\) −3.69169e8 −0.0158100
\(918\) 0 0
\(919\) 3.39353e10 1.44227 0.721137 0.692793i \(-0.243620\pi\)
0.721137 + 0.692793i \(0.243620\pi\)
\(920\) 0 0
\(921\) −2.37303e10 −1.00091
\(922\) 0 0
\(923\) 2.41696e10 1.01173
\(924\) 0 0
\(925\) 4.72554e9 0.196316
\(926\) 0 0
\(927\) −1.37181e10 −0.565607
\(928\) 0 0
\(929\) 7.75319e9 0.317267 0.158634 0.987338i \(-0.449291\pi\)
0.158634 + 0.987338i \(0.449291\pi\)
\(930\) 0 0
\(931\) 9.19780e9 0.373560
\(932\) 0 0
\(933\) 4.24222e10 1.71004
\(934\) 0 0
\(935\) −2.23621e10 −0.894690
\(936\) 0 0
\(937\) −3.05613e10 −1.21362 −0.606811 0.794846i \(-0.707552\pi\)
−0.606811 + 0.794846i \(0.707552\pi\)
\(938\) 0 0
\(939\) 4.55090e10 1.79377
\(940\) 0 0
\(941\) −4.59802e10 −1.79890 −0.899451 0.437022i \(-0.856033\pi\)
−0.899451 + 0.437022i \(0.856033\pi\)
\(942\) 0 0
\(943\) −4.52107e10 −1.75570
\(944\) 0 0
\(945\) 4.19579e10 1.61734
\(946\) 0 0
\(947\) 3.07383e10 1.17613 0.588065 0.808813i \(-0.299890\pi\)
0.588065 + 0.808813i \(0.299890\pi\)
\(948\) 0 0
\(949\) −1.50359e10 −0.571083
\(950\) 0 0
\(951\) 5.78553e10 2.18128
\(952\) 0 0
\(953\) 1.04326e10 0.390451 0.195226 0.980758i \(-0.437456\pi\)
0.195226 + 0.980758i \(0.437456\pi\)
\(954\) 0 0
\(955\) 5.40220e10 2.00705
\(956\) 0 0
\(957\) 8.59441e9 0.316974
\(958\) 0 0
\(959\) −3.74595e10 −1.37150
\(960\) 0 0
\(961\) 1.74720e10 0.635054
\(962\) 0 0
\(963\) −2.22478e10 −0.802776
\(964\) 0 0
\(965\) 1.87902e9 0.0673111
\(966\) 0 0
\(967\) −1.29860e10 −0.461829 −0.230915 0.972974i \(-0.574172\pi\)
−0.230915 + 0.972974i \(0.574172\pi\)
\(968\) 0 0
\(969\) −6.68229e9 −0.235935
\(970\) 0 0
\(971\) 5.17386e10 1.81362 0.906812 0.421536i \(-0.138509\pi\)
0.906812 + 0.421536i \(0.138509\pi\)
\(972\) 0 0
\(973\) −6.37393e10 −2.21826
\(974\) 0 0
\(975\) −1.25383e11 −4.33234
\(976\) 0 0
\(977\) 3.29063e10 1.12888 0.564440 0.825474i \(-0.309092\pi\)
0.564440 + 0.825474i \(0.309092\pi\)
\(978\) 0 0
\(979\) −2.66100e10 −0.906370
\(980\) 0 0
\(981\) −3.15035e10 −1.06541
\(982\) 0 0
\(983\) 2.58802e9 0.0869021 0.0434510 0.999056i \(-0.486165\pi\)
0.0434510 + 0.999056i \(0.486165\pi\)
\(984\) 0 0
\(985\) 4.17120e10 1.39070
\(986\) 0 0
\(987\) −5.08495e10 −1.68336
\(988\) 0 0
\(989\) 5.28136e10 1.73603
\(990\) 0 0
\(991\) 4.62321e10 1.50899 0.754494 0.656306i \(-0.227882\pi\)
0.754494 + 0.656306i \(0.227882\pi\)
\(992\) 0 0
\(993\) 3.28155e10 1.06355
\(994\) 0 0
\(995\) 8.12520e10 2.61489
\(996\) 0 0
\(997\) 2.54695e10 0.813930 0.406965 0.913444i \(-0.366587\pi\)
0.406965 + 0.913444i \(0.366587\pi\)
\(998\) 0 0
\(999\) 2.05326e9 0.0651577
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 304.8.a.d.1.2 2
4.3 odd 2 38.8.a.d.1.1 2
12.11 even 2 342.8.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.8.a.d.1.1 2 4.3 odd 2
304.8.a.d.1.2 2 1.1 even 1 trivial
342.8.a.g.1.1 2 12.11 even 2