Properties

Label 342.8.a.g.1.1
Level $342$
Weight $8$
Character 342.1
Self dual yes
Analytic conductor $106.836$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(106.835678716\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.1
Root \(13.0797\) of defining polynomial
Character \(\chi\) \(=\) 342.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +64.0000 q^{4} -467.472 q^{5} -1471.23 q^{7} -512.000 q^{8} +O(q^{10})\) \(q-8.00000 q^{2} +64.0000 q^{4} -467.472 q^{5} -1471.23 q^{7} -512.000 q^{8} +3739.78 q^{10} +3547.04 q^{11} -12361.8 q^{13} +11769.9 q^{14} +4096.00 q^{16} +13486.3 q^{17} -6859.00 q^{19} -29918.2 q^{20} -28376.3 q^{22} +94084.9 q^{23} +140405. q^{25} +98894.4 q^{26} -94158.9 q^{28} -33541.0 q^{29} +212096. q^{31} -32768.0 q^{32} -107890. q^{34} +687760. q^{35} +33656.5 q^{37} +54872.0 q^{38} +239346. q^{40} +480531. q^{41} -561340. q^{43} +227011. q^{44} -752679. q^{46} -478445. q^{47} +1.34098e6 q^{49} -1.12324e6 q^{50} -791156. q^{52} +93127.0 q^{53} -1.65814e6 q^{55} +753271. q^{56} +268328. q^{58} -955034. q^{59} -1.71632e6 q^{61} -1.69677e6 q^{62} +262144. q^{64} +5.77880e6 q^{65} +476255. q^{67} +863120. q^{68} -5.50208e6 q^{70} -1.95518e6 q^{71} +1.21632e6 q^{73} -269252. q^{74} -438976. q^{76} -5.21852e6 q^{77} +3.94544e6 q^{79} -1.91477e6 q^{80} -3.84425e6 q^{82} +3.54252e6 q^{83} -6.30445e6 q^{85} +4.49072e6 q^{86} -1.81609e6 q^{88} +7.50202e6 q^{89} +1.81871e7 q^{91} +6.02143e6 q^{92} +3.82756e6 q^{94} +3.20639e6 q^{95} +1.27974e7 q^{97} -1.07279e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{2} + 128q^{4} - 155q^{5} - 2238q^{7} - 1024q^{8} + O(q^{10}) \) \( 2q - 16q^{2} + 128q^{4} - 155q^{5} - 2238q^{7} - 1024q^{8} + 1240q^{10} + 3295q^{11} - 13427q^{13} + 17904q^{14} + 8192q^{16} + 32256q^{17} - 13718q^{19} - 9920q^{20} - 26360q^{22} + 82525q^{23} + 159919q^{25} + 107416q^{26} - 143232q^{28} + 12749q^{29} + 258944q^{31} - 65536q^{32} - 258048q^{34} + 448167q^{35} - 149260q^{37} + 109744q^{38} + 79360q^{40} - 339130q^{41} - 83869q^{43} + 210880q^{44} - 660200q^{46} - 1471025q^{47} + 1105372q^{49} - 1279352q^{50} - 859328q^{52} + 945643q^{53} - 1736899q^{55} + 1145856q^{56} - 101992q^{58} + 969009q^{59} - 1506755q^{61} - 2071552q^{62} + 524288q^{64} + 5445956q^{65} - 1848219q^{67} + 2064384q^{68} - 3585336q^{70} + 3417184q^{71} - 2499822q^{73} + 1194080q^{74} - 877952q^{76} - 5025267q^{77} + 2636926q^{79} - 634880q^{80} + 2713040q^{82} + 10059354q^{83} - 439425q^{85} + 670952q^{86} - 1687040q^{88} + 3506160q^{89} + 19003851q^{91} + 5281600q^{92} + 11768200q^{94} + 1063145q^{95} + 5893526q^{97} - 8842976q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) −467.472 −1.67248 −0.836240 0.548364i \(-0.815251\pi\)
−0.836240 + 0.548364i \(0.815251\pi\)
\(6\) 0 0
\(7\) −1471.23 −1.62121 −0.810603 0.585596i \(-0.800861\pi\)
−0.810603 + 0.585596i \(0.800861\pi\)
\(8\) −512.000 −0.353553
\(9\) 0 0
\(10\) 3739.78 1.18262
\(11\) 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 \(0.784923\pi\)
\(14\) 11769.9 1.14637
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 13486.3 0.665764 0.332882 0.942969i \(-0.391979\pi\)
0.332882 + 0.942969i \(0.391979\pi\)
\(18\) 0 0
\(19\) −6859.00 −0.229416
\(20\) −29918.2 −0.836240
\(21\) 0 0
\(22\) −28376.3 −0.568168
\(23\) 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
\(26\) 98894.4 1.10348
\(27\) 0 0
\(28\) −94158.9 −0.810603
\(29\) −33541.0 −0.255378 −0.127689 0.991814i \(-0.540756\pi\)
−0.127689 + 0.991814i \(0.540756\pi\)
\(30\) 0 0
\(31\) 212096. 1.27869 0.639346 0.768919i \(-0.279205\pi\)
0.639346 + 0.768919i \(0.279205\pi\)
\(32\) −32768.0 −0.176777
\(33\) 0 0
\(34\) −107890. −0.470766
\(35\) 687760. 2.71143
\(36\) 0 0
\(37\) 33656.5 0.109235 0.0546176 0.998507i \(-0.482606\pi\)
0.0546176 + 0.998507i \(0.482606\pi\)
\(38\) 54872.0 0.162221
\(39\) 0 0
\(40\) 239346. 0.591311
\(41\) 480531. 1.08887 0.544437 0.838801i \(-0.316743\pi\)
0.544437 + 0.838801i \(0.316743\pi\)
\(42\) 0 0
\(43\) −561340. −1.07668 −0.538339 0.842728i \(-0.680948\pi\)
−0.538339 + 0.842728i \(0.680948\pi\)
\(44\) 227011. 0.401756
\(45\) 0 0
\(46\) −752679. −1.14014
\(47\) −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
\(50\) −1.12324e6 −1.27080
\(51\) 0 0
\(52\) −791156. −0.780279
\(53\) 93127.0 0.0859232 0.0429616 0.999077i \(-0.486321\pi\)
0.0429616 + 0.999077i \(0.486321\pi\)
\(54\) 0 0
\(55\) −1.65814e6 −1.34386
\(56\) 753271. 0.573183
\(57\) 0 0
\(58\) 268328. 0.180579
\(59\) −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 \(0.660844\pi\)
\(62\) −1.69677e6 −0.904172
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 5.77880e6 2.61000
\(66\) 0 0
\(67\) 476255. 0.193454 0.0967270 0.995311i \(-0.469163\pi\)
0.0967270 + 0.995311i \(0.469163\pi\)
\(68\) 863120. 0.332882
\(69\) 0 0
\(70\) −5.50208e6 −1.91727
\(71\) −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 \(0.441428\pi\)
\(74\) −269252. −0.0772409
\(75\) 0 0
\(76\) −438976. −0.114708
\(77\) −5.21852e6 −1.30266
\(78\) 0 0
\(79\) 3.94544e6 0.900328 0.450164 0.892946i \(-0.351366\pi\)
0.450164 + 0.892946i \(0.351366\pi\)
\(80\) −1.91477e6 −0.418120
\(81\) 0 0
\(82\) −3.84425e6 −0.769951
\(83\) 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
\(86\) 4.49072e6 0.761327
\(87\) 0 0
\(88\) −1.81609e6 −0.284084
\(89\) 7.50202e6 1.12801 0.564006 0.825771i \(-0.309260\pi\)
0.564006 + 0.825771i \(0.309260\pi\)
\(90\) 0 0
\(91\) 1.81871e7 2.52999
\(92\) 6.02143e6 0.806199
\(93\) 0 0
\(94\) 3.82756e6 0.475308
\(95\) 3.20639e6 0.383693
\(96\) 0 0
\(97\) 1.27974e7 1.42371 0.711855 0.702326i \(-0.247855\pi\)
0.711855 + 0.702326i \(0.247855\pi\)
\(98\) −1.07279e7 −1.15139
\(99\) 0 0
\(100\) 8.98593e6 0.898593
\(101\) −332855. −0.0321463 −0.0160731 0.999871i \(-0.505116\pi\)
−0.0160731 + 0.999871i \(0.505116\pi\)
\(102\) 0 0
\(103\) 4.52517e6 0.408042 0.204021 0.978967i \(-0.434599\pi\)
0.204021 + 0.978967i \(0.434599\pi\)
\(104\) 6.32924e6 0.551741
\(105\) 0 0
\(106\) −745016. −0.0607568
\(107\) −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 \(0.625559\pi\)
\(110\) 1.32651e7 0.950249
\(111\) 0 0
\(112\) −6.02617e6 −0.405302
\(113\) −5.33971e6 −0.348131 −0.174066 0.984734i \(-0.555691\pi\)
−0.174066 + 0.984734i \(0.555691\pi\)
\(114\) 0 0
\(115\) −4.39820e7 −2.69670
\(116\) −2.14663e6 −0.127689
\(117\) 0 0
\(118\) 7.64027e6 0.428077
\(119\) −1.98414e7 −1.07934
\(120\) 0 0
\(121\) −6.90567e6 −0.354370
\(122\) 1.37306e7 0.684587
\(123\) 0 0
\(124\) 1.35741e7 0.639346
\(125\) −2.91142e7 −1.33328
\(126\) 0 0
\(127\) −3.43331e7 −1.48730 −0.743652 0.668566i \(-0.766908\pi\)
−0.743652 + 0.668566i \(0.766908\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 0 0
\(130\) −4.62304e7 −1.84555
\(131\) −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
\(134\) −3.81004e6 −0.136793
\(135\) 0 0
\(136\) −6.90496e6 −0.235383
\(137\) 2.54613e7 0.845978 0.422989 0.906135i \(-0.360981\pi\)
0.422989 + 0.906135i \(0.360981\pi\)
\(138\) 0 0
\(139\) 4.33237e7 1.36828 0.684138 0.729352i \(-0.260178\pi\)
0.684138 + 0.729352i \(0.260178\pi\)
\(140\) 4.40167e7 1.35572
\(141\) 0 0
\(142\) 1.56415e7 0.458425
\(143\) −4.38478e7 −1.25393
\(144\) 0 0
\(145\) 1.56795e7 0.427114
\(146\) −9.73058e6 −0.258764
\(147\) 0 0
\(148\) 2.15401e6 0.0546176
\(149\) 3.98620e7 0.987204 0.493602 0.869688i \(-0.335680\pi\)
0.493602 + 0.869688i \(0.335680\pi\)
\(150\) 0 0
\(151\) −2.13810e6 −0.0505369 −0.0252684 0.999681i \(-0.508044\pi\)
−0.0252684 + 0.999681i \(0.508044\pi\)
\(152\) 3.51181e6 0.0811107
\(153\) 0 0
\(154\) 4.17482e7 0.921118
\(155\) −9.91489e7 −2.13859
\(156\) 0 0
\(157\) −3.04702e7 −0.628387 −0.314193 0.949359i \(-0.601734\pi\)
−0.314193 + 0.949359i \(0.601734\pi\)
\(158\) −3.15635e7 −0.636628
\(159\) 0 0
\(160\) 1.53181e7 0.295655
\(161\) −1.38421e8 −2.61403
\(162\) 0 0
\(163\) −9.48952e7 −1.71628 −0.858139 0.513417i \(-0.828379\pi\)
−0.858139 + 0.513417i \(0.828379\pi\)
\(164\) 3.07540e7 0.544437
\(165\) 0 0
\(166\) −2.83402e7 −0.480867
\(167\) 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
\(170\) 5.04356e7 0.787346
\(171\) 0 0
\(172\) −3.59257e7 −0.538339
\(173\) −1.06112e8 −1.55813 −0.779066 0.626942i \(-0.784306\pi\)
−0.779066 + 0.626942i \(0.784306\pi\)
\(174\) 0 0
\(175\) −2.06569e8 −2.91361
\(176\) 1.45287e7 0.200878
\(177\) 0 0
\(178\) −6.00162e7 −0.797624
\(179\) −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 \(0.488984\pi\)
\(182\) −1.45497e8 −1.78897
\(183\) 0 0
\(184\) −4.81714e7 −0.570069
\(185\) −1.57335e7 −0.182693
\(186\) 0 0
\(187\) 4.78363e7 0.534948
\(188\) −3.06205e7 −0.336093
\(189\) 0 0
\(190\) −2.56511e7 −0.271312
\(191\) 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 \(0.493594\pi\)
\(194\) −1.02379e8 −1.00672
\(195\) 0 0
\(196\) 8.58229e7 0.814155
\(197\) −8.92289e7 −0.831522 −0.415761 0.909474i \(-0.636485\pi\)
−0.415761 + 0.909474i \(0.636485\pi\)
\(198\) 0 0
\(199\) −1.73811e8 −1.56348 −0.781740 0.623604i \(-0.785668\pi\)
−0.781740 + 0.623604i \(0.785668\pi\)
\(200\) −7.18875e7 −0.635401
\(201\) 0 0
\(202\) 2.66284e6 0.0227308
\(203\) 4.93467e7 0.414020
\(204\) 0 0
\(205\) −2.24635e8 −1.82112
\(206\) −3.62014e7 −0.288529
\(207\) 0 0
\(208\) −5.06340e7 −0.390140
\(209\) −2.43292e7 −0.184338
\(210\) 0 0
\(211\) −1.48188e8 −1.08598 −0.542992 0.839738i \(-0.682709\pi\)
−0.542992 + 0.839738i \(0.682709\pi\)
\(212\) 5.96013e6 0.0429616
\(213\) 0 0
\(214\) 5.87108e7 0.409515
\(215\) 2.62411e8 1.80072
\(216\) 0 0
\(217\) −3.12042e8 −2.07302
\(218\) 8.31361e7 0.543491
\(219\) 0 0
\(220\) −1.06121e8 −0.671928
\(221\) −1.66714e8 −1.03896
\(222\) 0 0
\(223\) 1.38594e6 0.00836907 0.00418454 0.999991i \(-0.498668\pi\)
0.00418454 + 0.999991i \(0.498668\pi\)
\(224\) 4.82094e7 0.286591
\(225\) 0 0
\(226\) 4.27177e7 0.246166
\(227\) −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 \(0.472828\pi\)
\(230\) 3.51856e8 1.90686
\(231\) 0 0
\(232\) 1.71730e7 0.0902897
\(233\) 2.04142e8 1.05727 0.528635 0.848849i \(-0.322704\pi\)
0.528635 + 0.848849i \(0.322704\pi\)
\(234\) 0 0
\(235\) 2.23660e8 1.12422
\(236\) −6.11221e7 −0.302696
\(237\) 0 0
\(238\) 1.58731e8 0.763209
\(239\) 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 \(0.469473\pi\)
\(242\) 5.52453e7 0.250577
\(243\) 0 0
\(244\) −1.09844e8 −0.484076
\(245\) −6.26872e8 −2.72331
\(246\) 0 0
\(247\) 8.47896e7 0.358017
\(248\) −1.08593e8 −0.452086
\(249\) 0 0
\(250\) 2.32914e8 0.942769
\(251\) −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
\(254\) 2.74665e8 1.05168
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) 2.46264e8 0.904972 0.452486 0.891772i \(-0.350537\pi\)
0.452486 + 0.891772i \(0.350537\pi\)
\(258\) 0 0
\(259\) −4.95165e7 −0.177093
\(260\) 3.69843e8 1.30500
\(261\) 0 0
\(262\) 2.00740e6 0.00689572
\(263\) −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
\(266\) −8.07295e7 −0.262994
\(267\) 0 0
\(268\) 3.04803e7 0.0967270
\(269\) 2.44584e8 0.766118 0.383059 0.923724i \(-0.374871\pi\)
0.383059 + 0.923724i \(0.374871\pi\)
\(270\) 0 0
\(271\) 7.88689e7 0.240721 0.120360 0.992730i \(-0.461595\pi\)
0.120360 + 0.992730i \(0.461595\pi\)
\(272\) 5.52397e7 0.166441
\(273\) 0 0
\(274\) −2.03691e8 −0.598196
\(275\) 4.98023e8 1.44406
\(276\) 0 0
\(277\) −2.30329e8 −0.651133 −0.325567 0.945519i \(-0.605555\pi\)
−0.325567 + 0.945519i \(0.605555\pi\)
\(278\) −3.46590e8 −0.967518
\(279\) 0 0
\(280\) −3.52133e8 −0.958637
\(281\) 5.44589e8 1.46419 0.732094 0.681203i \(-0.238543\pi\)
0.732094 + 0.681203i \(0.238543\pi\)
\(282\) 0 0
\(283\) −3.69887e7 −0.0970101 −0.0485051 0.998823i \(-0.515446\pi\)
−0.0485051 + 0.998823i \(0.515446\pi\)
\(284\) −1.25132e8 −0.324155
\(285\) 0 0
\(286\) 3.50783e8 0.886660
\(287\) −7.06973e8 −1.76529
\(288\) 0 0
\(289\) −2.28460e8 −0.556759
\(290\) −1.25436e8 −0.302015
\(291\) 0 0
\(292\) 7.78447e7 0.182974
\(293\) −1.98050e8 −0.459979 −0.229990 0.973193i \(-0.573869\pi\)
−0.229990 + 0.973193i \(0.573869\pi\)
\(294\) 0 0
\(295\) 4.46452e8 1.01251
\(296\) −1.72321e7 −0.0386204
\(297\) 0 0
\(298\) −3.18896e8 −0.698058
\(299\) −1.16306e9 −2.51624
\(300\) 0 0
\(301\) 8.25861e8 1.74552
\(302\) 1.71048e7 0.0357350
\(303\) 0 0
\(304\) −2.80945e7 −0.0573539
\(305\) 8.02331e8 1.61921
\(306\) 0 0
\(307\) 3.28496e8 0.647957 0.323979 0.946064i \(-0.394979\pi\)
0.323979 + 0.946064i \(0.394979\pi\)
\(308\) −3.33986e8 −0.651329
\(309\) 0 0
\(310\) 7.93191e8 1.51221
\(311\) 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 \(0.302812\pi\)
\(314\) 2.43762e8 0.444337
\(315\) 0 0
\(316\) 2.52508e8 0.450164
\(317\) −8.00885e8 −1.41209 −0.706046 0.708166i \(-0.749523\pi\)
−0.706046 + 0.708166i \(0.749523\pi\)
\(318\) 0 0
\(319\) −1.18971e8 −0.205199
\(320\) −1.22545e8 −0.209060
\(321\) 0 0
\(322\) 1.10737e9 1.84840
\(323\) −9.25022e7 −0.152737
\(324\) 0 0
\(325\) −1.73566e9 −2.80462
\(326\) 7.59162e8 1.21359
\(327\) 0 0
\(328\) −2.46032e8 −0.384975
\(329\) 7.03905e8 1.08975
\(330\) 0 0
\(331\) −4.54262e8 −0.688507 −0.344253 0.938877i \(-0.611868\pi\)
−0.344253 + 0.938877i \(0.611868\pi\)
\(332\) 2.26722e8 0.340024
\(333\) 0 0
\(334\) −8.82653e8 −1.29622
\(335\) −2.22636e8 −0.323548
\(336\) 0 0
\(337\) −9.81183e8 −1.39651 −0.698257 0.715847i \(-0.746041\pi\)
−0.698257 + 0.715847i \(0.746041\pi\)
\(338\) −7.20526e8 −1.01494
\(339\) 0 0
\(340\) −4.03485e8 −0.556738
\(341\) 7.52313e8 1.02744
\(342\) 0 0
\(343\) −7.61275e8 −1.01862
\(344\) 2.87406e8 0.380663
\(345\) 0 0
\(346\) 8.48898e8 1.10177
\(347\) −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 \(0.258110\pi\)
\(350\) 1.65255e9 2.06023
\(351\) 0 0
\(352\) −1.16229e8 −0.142042
\(353\) −5.02386e8 −0.607891 −0.303946 0.952689i \(-0.598304\pi\)
−0.303946 + 0.952689i \(0.598304\pi\)
\(354\) 0 0
\(355\) 9.13994e8 1.08429
\(356\) 4.80130e8 0.564006
\(357\) 0 0
\(358\) 3.90681e8 0.450020
\(359\) 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
\(362\) −4.41659e7 −0.0489336
\(363\) 0 0
\(364\) 1.16397e9 1.26499
\(365\) −5.68597e8 −0.612040
\(366\) 0 0
\(367\) −5.54974e8 −0.586059 −0.293029 0.956103i \(-0.594663\pi\)
−0.293029 + 0.956103i \(0.594663\pi\)
\(368\) 3.85372e8 0.403100
\(369\) 0 0
\(370\) 1.25868e8 0.129184
\(371\) −1.37012e8 −0.139299
\(372\) 0 0
\(373\) −1.29835e9 −1.29542 −0.647708 0.761889i \(-0.724272\pi\)
−0.647708 + 0.761889i \(0.724272\pi\)
\(374\) −3.82690e8 −0.378266
\(375\) 0 0
\(376\) 2.44964e8 0.237654
\(377\) 4.14628e8 0.398532
\(378\) 0 0
\(379\) 7.49227e8 0.706929 0.353465 0.935448i \(-0.385003\pi\)
0.353465 + 0.935448i \(0.385003\pi\)
\(380\) 2.05209e8 0.191847
\(381\) 0 0
\(382\) −9.24495e8 −0.848561
\(383\) −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
\(386\) −3.21563e7 −0.0284584
\(387\) 0 0
\(388\) 8.19036e8 0.711855
\(389\) −3.13367e8 −0.269917 −0.134958 0.990851i \(-0.543090\pi\)
−0.134958 + 0.990851i \(0.543090\pi\)
\(390\) 0 0
\(391\) 1.26885e9 1.07348
\(392\) −6.86583e8 −0.575694
\(393\) 0 0
\(394\) 7.13831e8 0.587975
\(395\) −1.84438e9 −1.50578
\(396\) 0 0
\(397\) 1.91013e9 1.53213 0.766066 0.642762i \(-0.222211\pi\)
0.766066 + 0.642762i \(0.222211\pi\)
\(398\) 1.39049e9 1.10555
\(399\) 0 0
\(400\) 5.75100e8 0.449297
\(401\) 1.27797e9 0.989726 0.494863 0.868971i \(-0.335218\pi\)
0.494863 + 0.868971i \(0.335218\pi\)
\(402\) 0 0
\(403\) −2.62189e9 −1.99547
\(404\) −2.13027e7 −0.0160731
\(405\) 0 0
\(406\) −3.94773e8 −0.292757
\(407\) 1.19381e8 0.0877716
\(408\) 0 0
\(409\) 2.92650e8 0.211503 0.105752 0.994393i \(-0.466275\pi\)
0.105752 + 0.994393i \(0.466275\pi\)
\(410\) 1.79708e9 1.28773
\(411\) 0 0
\(412\) 2.89611e8 0.204021
\(413\) 1.40508e9 0.981465
\(414\) 0 0
\(415\) −1.65603e9 −1.13737
\(416\) 4.05072e8 0.275870
\(417\) 0 0
\(418\) 1.94633e8 0.130347
\(419\) −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 \(0.167797\pi\)
\(422\) 1.18550e9 0.767906
\(423\) 0 0
\(424\) −4.76810e7 −0.0303784
\(425\) 1.89354e9 1.19650
\(426\) 0 0
\(427\) 2.52511e9 1.56957
\(428\) −4.69686e8 −0.289571
\(429\) 0 0
\(430\) −2.09928e9 −1.27330
\(431\) −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 \(0.251294\pi\)
\(434\) 2.49634e9 1.46585
\(435\) 0 0
\(436\) −6.65089e8 −0.384306
\(437\) −6.45328e8 −0.369909
\(438\) 0 0
\(439\) 2.18333e9 1.23167 0.615835 0.787875i \(-0.288819\pi\)
0.615835 + 0.787875i \(0.288819\pi\)
\(440\) 8.48969e8 0.475125
\(441\) 0 0
\(442\) 1.33372e9 0.734658
\(443\) 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
\(446\) −1.10875e7 −0.00591783
\(447\) 0 0
\(448\) −3.85675e8 −0.202651
\(449\) 6.02088e8 0.313905 0.156952 0.987606i \(-0.449833\pi\)
0.156952 + 0.987606i \(0.449833\pi\)
\(450\) 0 0
\(451\) 1.70446e9 0.874923
\(452\) −3.41741e8 −0.174066
\(453\) 0 0
\(454\) 1.08117e8 0.0542251
\(455\) −8.50196e9 −4.23135
\(456\) 0 0
\(457\) −3.28085e9 −1.60798 −0.803989 0.594644i \(-0.797293\pi\)
−0.803989 + 0.594644i \(0.797293\pi\)
\(458\) −2.47906e8 −0.120575
\(459\) 0 0
\(460\) −2.81485e9 −1.34835
\(461\) 3.50631e9 1.66685 0.833426 0.552632i \(-0.186376\pi\)
0.833426 + 0.552632i \(0.186376\pi\)
\(462\) 0 0
\(463\) −3.85170e9 −1.80351 −0.901757 0.432244i \(-0.857722\pi\)
−0.901757 + 0.432244i \(0.857722\pi\)
\(464\) −1.37384e8 −0.0638445
\(465\) 0 0
\(466\) −1.63313e9 −0.747603
\(467\) −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
\(470\) −1.78928e9 −0.794942
\(471\) 0 0
\(472\) 4.88977e8 0.214038
\(473\) −1.99109e9 −0.865123
\(474\) 0 0
\(475\) −9.63039e8 −0.412303
\(476\) −1.26985e9 −0.539670
\(477\) 0 0
\(478\) −1.80476e9 −0.755826
\(479\) −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
\(482\) −3.32925e8 −0.135420
\(483\) 0 0
\(484\) −4.41963e8 −0.177185
\(485\) −5.98244e9 −2.38113
\(486\) 0 0
\(487\) −3.37943e9 −1.32584 −0.662921 0.748689i \(-0.730683\pi\)
−0.662921 + 0.748689i \(0.730683\pi\)
\(488\) 8.78755e8 0.342294
\(489\) 0 0
\(490\) 5.01498e9 1.92567
\(491\) 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
\(494\) −6.78317e8 −0.253156
\(495\) 0 0
\(496\) 8.68744e8 0.319673
\(497\) 2.87653e9 1.05105
\(498\) 0 0
\(499\) 3.88153e9 1.39846 0.699231 0.714896i \(-0.253526\pi\)
0.699231 + 0.714896i \(0.253526\pi\)
\(500\) −1.86331e9 −0.666639
\(501\) 0 0
\(502\) 3.64463e9 1.28585
\(503\) −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
\(506\) −2.66978e9 −0.916113
\(507\) 0 0
\(508\) −2.19732e9 −0.743652
\(509\) 3.37295e9 1.13370 0.566849 0.823822i \(-0.308162\pi\)
0.566849 + 0.823822i \(0.308162\pi\)
\(510\) 0 0
\(511\) −1.78949e9 −0.593277
\(512\) −1.34218e8 −0.0441942
\(513\) 0 0
\(514\) −1.97011e9 −0.639912
\(515\) −2.11539e9 −0.682441
\(516\) 0 0
\(517\) −1.69707e9 −0.540109
\(518\) 3.96132e8 0.125223
\(519\) 0 0
\(520\) −2.95875e9 −0.922775
\(521\) −5.32384e9 −1.64927 −0.824637 0.565662i \(-0.808621\pi\)
−0.824637 + 0.565662i \(0.808621\pi\)
\(522\) 0 0
\(523\) 4.07644e9 1.24602 0.623010 0.782214i \(-0.285910\pi\)
0.623010 + 0.782214i \(0.285910\pi\)
\(524\) −1.60592e7 −0.00487601
\(525\) 0 0
\(526\) 3.23848e9 0.970268
\(527\) 2.86038e9 0.851307
\(528\) 0 0
\(529\) 5.44713e9 1.59983
\(530\) 3.48274e8 0.101615
\(531\) 0 0
\(532\) 6.45836e8 0.185965
\(533\) −5.94023e9 −1.69925
\(534\) 0 0
\(535\) 3.43071e9 0.968602
\(536\) −2.43843e8 −0.0683964
\(537\) 0 0
\(538\) −1.95668e9 −0.541728
\(539\) 4.75652e9 1.30837
\(540\) 0 0
\(541\) −3.63310e9 −0.986477 −0.493239 0.869894i \(-0.664187\pi\)
−0.493239 + 0.869894i \(0.664187\pi\)
\(542\) −6.30951e8 −0.170215
\(543\) 0 0
\(544\) −4.41918e8 −0.117692
\(545\) 4.85798e9 1.28549
\(546\) 0 0
\(547\) −5.74113e9 −1.49983 −0.749914 0.661535i \(-0.769905\pi\)
−0.749914 + 0.661535i \(0.769905\pi\)
\(548\) 1.62952e9 0.422989
\(549\) 0 0
\(550\) −3.98418e9 −1.02110
\(551\) 2.30058e8 0.0585877
\(552\) 0 0
\(553\) −5.80466e9 −1.45962
\(554\) 1.84263e9 0.460421
\(555\) 0 0
\(556\) 2.77272e9 0.684138
\(557\) −9.95936e8 −0.244196 −0.122098 0.992518i \(-0.538962\pi\)
−0.122098 + 0.992518i \(0.538962\pi\)
\(558\) 0 0
\(559\) 6.93917e9 1.68022
\(560\) 2.81707e9 0.677858
\(561\) 0 0
\(562\) −4.35671e9 −1.03534
\(563\) 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
\(566\) 2.95910e8 0.0685965
\(567\) 0 0
\(568\) 1.00105e9 0.229212
\(569\) −3.35141e9 −0.762666 −0.381333 0.924438i \(-0.624535\pi\)
−0.381333 + 0.924438i \(0.624535\pi\)
\(570\) 0 0
\(571\) 4.22454e9 0.949627 0.474813 0.880087i \(-0.342516\pi\)
0.474813 + 0.880087i \(0.342516\pi\)
\(572\) −2.80626e9 −0.626963
\(573\) 0 0
\(574\) 5.65579e9 1.24825
\(575\) 1.32100e10 2.89778
\(576\) 0 0
\(577\) −6.43062e9 −1.39360 −0.696799 0.717266i \(-0.745393\pi\)
−0.696799 + 0.717266i \(0.745393\pi\)
\(578\) 1.82768e9 0.393688
\(579\) 0 0
\(580\) 1.00349e9 0.213557
\(581\) −5.21188e9 −1.10250
\(582\) 0 0
\(583\) 3.30325e8 0.0690402
\(584\) −6.22757e8 −0.129382
\(585\) 0 0
\(586\) 1.58440e9 0.325254
\(587\) 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
\(590\) −3.57161e9 −0.715949
\(591\) 0 0
\(592\) 1.37857e8 0.0273088
\(593\) 5.13346e8 0.101093 0.0505463 0.998722i \(-0.483904\pi\)
0.0505463 + 0.998722i \(0.483904\pi\)
\(594\) 0 0
\(595\) 9.27531e9 1.80517
\(596\) 2.55117e9 0.493602
\(597\) 0 0
\(598\) 9.30447e9 1.77925
\(599\) 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 \(0.689538\pi\)
\(602\) −6.60689e9 −1.23427
\(603\) 0 0
\(604\) −1.36838e8 −0.0252684
\(605\) 3.22821e9 0.592676
\(606\) 0 0
\(607\) −1.00863e10 −1.83050 −0.915249 0.402888i \(-0.868007\pi\)
−0.915249 + 0.402888i \(0.868007\pi\)
\(608\) 2.24756e8 0.0405554
\(609\) 0 0
\(610\) −6.41865e9 −1.14496
\(611\) 5.91445e9 1.04899
\(612\) 0 0
\(613\) 1.24543e9 0.218378 0.109189 0.994021i \(-0.465175\pi\)
0.109189 + 0.994021i \(0.465175\pi\)
\(614\) −2.62797e9 −0.458175
\(615\) 0 0
\(616\) 2.67188e9 0.460559
\(617\) 5.96038e8 0.102159 0.0510794 0.998695i \(-0.483734\pi\)
0.0510794 + 0.998695i \(0.483734\pi\)
\(618\) 0 0
\(619\) −3.90038e9 −0.660981 −0.330491 0.943809i \(-0.607214\pi\)
−0.330491 + 0.943809i \(0.607214\pi\)
\(620\) −6.34553e9 −1.06929
\(621\) 0 0
\(622\) −4.69797e9 −0.782787
\(623\) −1.10372e10 −1.82874
\(624\) 0 0
\(625\) 2.64094e9 0.432692
\(626\) −5.03981e9 −0.821115
\(627\) 0 0
\(628\) −1.95010e9 −0.314193
\(629\) 4.53899e8 0.0727248
\(630\) 0 0
\(631\) 4.64916e9 0.736668 0.368334 0.929694i \(-0.379928\pi\)
0.368334 + 0.929694i \(0.379928\pi\)
\(632\) −2.02007e9 −0.318314
\(633\) 0 0
\(634\) 6.40708e9 0.998499
\(635\) 1.60498e10 2.48749
\(636\) 0 0
\(637\) −1.65770e10 −2.54107
\(638\) 9.51772e8 0.145098
\(639\) 0 0
\(640\) 9.80360e8 0.147828
\(641\) −9.04607e9 −1.35662 −0.678308 0.734778i \(-0.737286\pi\)
−0.678308 + 0.734778i \(0.737286\pi\)
\(642\) 0 0
\(643\) 1.11196e10 1.64949 0.824746 0.565503i \(-0.191318\pi\)
0.824746 + 0.565503i \(0.191318\pi\)
\(644\) −8.85893e9 −1.30701
\(645\) 0 0
\(646\) 7.40018e8 0.108001
\(647\) 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
\(650\) 1.38853e10 1.98316
\(651\) 0 0
\(652\) −6.07329e9 −0.858139
\(653\) 1.07233e10 1.50707 0.753536 0.657407i \(-0.228347\pi\)
0.753536 + 0.657407i \(0.228347\pi\)
\(654\) 0 0
\(655\) 1.17300e8 0.0163100
\(656\) 1.96826e9 0.272219
\(657\) 0 0
\(658\) −5.63124e9 −0.770572
\(659\) −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 \(0.791467\pi\)
\(662\) 3.63409e9 0.486848
\(663\) 0 0
\(664\) −1.81377e9 −0.240433
\(665\) −4.71735e9 −0.622046
\(666\) 0 0
\(667\) −3.15570e9 −0.411771
\(668\) 7.06122e9 0.916563
\(669\) 0 0
\(670\) 1.78109e9 0.228783
\(671\) −6.08786e9 −0.777921
\(672\) 0 0
\(673\) 5.31058e9 0.671567 0.335783 0.941939i \(-0.390999\pi\)
0.335783 + 0.941939i \(0.390999\pi\)
\(674\) 7.84946e9 0.987485
\(675\) 0 0
\(676\) 5.76421e9 0.717672
\(677\) −1.06518e10 −1.31936 −0.659680 0.751547i \(-0.729308\pi\)
−0.659680 + 0.751547i \(0.729308\pi\)
\(678\) 0 0
\(679\) −1.88280e10 −2.30813
\(680\) 3.22788e9 0.393673
\(681\) 0 0
\(682\) −6.01850e9 −0.726512
\(683\) 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
\(686\) 6.09020e9 0.720273
\(687\) 0 0
\(688\) −2.29925e9 −0.269170
\(689\) −1.15122e9 −0.134088
\(690\) 0 0
\(691\) −1.44774e10 −1.66923 −0.834615 0.550833i \(-0.814310\pi\)
−0.834615 + 0.550833i \(0.814310\pi\)
\(692\) −6.79119e9 −0.779066
\(693\) 0 0
\(694\) 2.15934e9 0.245224
\(695\) −2.02526e10 −2.28841
\(696\) 0 0
\(697\) 6.48056e9 0.724933
\(698\) −8.75269e9 −0.974199
\(699\) 0 0
\(700\) −1.32204e10 −1.45680
\(701\) −1.22852e10 −1.34701 −0.673503 0.739185i \(-0.735211\pi\)
−0.673503 + 0.739185i \(0.735211\pi\)
\(702\) 0 0
\(703\) −2.30850e8 −0.0250603
\(704\) 9.29836e8 0.100439
\(705\) 0 0
\(706\) 4.01909e9 0.429844
\(707\) 4.89707e8 0.0521157
\(708\) 0 0
\(709\) −3.48632e9 −0.367371 −0.183686 0.982985i \(-0.558803\pi\)
−0.183686 + 0.982985i \(0.558803\pi\)
\(710\) −7.31195e9 −0.766706
\(711\) 0 0
\(712\) −3.84104e9 −0.398812
\(713\) 1.99550e10 2.06176
\(714\) 0 0
\(715\) 2.04976e10 2.09717
\(716\) −3.12545e9 −0.318212
\(717\) 0 0
\(718\) −3.91571e9 −0.394798
\(719\) −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
\(722\) −3.76367e8 −0.0372161
\(723\) 0 0
\(724\) 3.53328e8 0.0346013
\(725\) −4.70933e9 −0.458962
\(726\) 0 0
\(727\) −4.41318e9 −0.425973 −0.212986 0.977055i \(-0.568319\pi\)
−0.212986 + 0.977055i \(0.568319\pi\)
\(728\) −9.31179e9 −0.894486
\(729\) 0 0
\(730\) 4.54878e9 0.432777
\(731\) −7.57037e9 −0.716814
\(732\) 0 0
\(733\) 1.15847e10 1.08648 0.543240 0.839577i \(-0.317197\pi\)
0.543240 + 0.839577i \(0.317197\pi\)
\(734\) 4.43979e9 0.414406
\(735\) 0 0
\(736\) −3.08297e9 −0.285034
\(737\) 1.68930e9 0.155443
\(738\) 0 0
\(739\) 1.60174e10 1.45995 0.729974 0.683475i \(-0.239532\pi\)
0.729974 + 0.683475i \(0.239532\pi\)
\(740\) −1.00694e9 −0.0913467
\(741\) 0 0
\(742\) 1.09609e9 0.0984994
\(743\) −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
\(746\) 1.03868e10 0.915997
\(747\) 0 0
\(748\) 3.06152e9 0.267474
\(749\) 1.07972e10 0.938908
\(750\) 0 0
\(751\) 1.04818e10 0.903015 0.451508 0.892267i \(-0.350886\pi\)
0.451508 + 0.892267i \(0.350886\pi\)
\(752\) −1.95971e9 −0.168047
\(753\) 0 0
\(754\) −3.31702e9 −0.281805
\(755\) 9.99502e8 0.0845219
\(756\) 0 0
\(757\) 4.11773e9 0.345003 0.172501 0.985009i \(-0.444815\pi\)
0.172501 + 0.985009i \(0.444815\pi\)
\(758\) −5.99381e9 −0.499874
\(759\) 0 0
\(760\) −1.64167e9 −0.135656
\(761\) 1.90960e10 1.57071 0.785357 0.619043i \(-0.212479\pi\)
0.785357 + 0.619043i \(0.212479\pi\)
\(762\) 0 0
\(763\) 1.52891e10 1.24608
\(764\) 7.39596e9 0.600023
\(765\) 0 0
\(766\) 1.70251e9 0.136864
\(767\) 1.18059e10 0.944750
\(768\) 0 0
\(769\) −1.20473e10 −0.955315 −0.477658 0.878546i \(-0.658514\pi\)
−0.477658 + 0.878546i \(0.658514\pi\)
\(770\) −1.95161e10 −1.54055
\(771\) 0 0
\(772\) 2.57251e8 0.0201232
\(773\) −4.62608e9 −0.360234 −0.180117 0.983645i \(-0.557648\pi\)
−0.180117 + 0.983645i \(0.557648\pi\)
\(774\) 0 0
\(775\) 2.97793e10 2.29805
\(776\) −6.55229e9 −0.503358
\(777\) 0 0
\(778\) 2.50694e9 0.190860
\(779\) −3.29596e9 −0.249805
\(780\) 0 0
\(781\) −6.93512e9 −0.520925
\(782\) −1.01508e10 −0.759062
\(783\) 0 0
\(784\) 5.49267e9 0.407077
\(785\) 1.42440e10 1.05096
\(786\) 0 0
\(787\) 5.71149e9 0.417675 0.208837 0.977950i \(-0.433032\pi\)
0.208837 + 0.977950i \(0.433032\pi\)
\(788\) −5.71065e9 −0.415761
\(789\) 0 0
\(790\) 1.47551e10 1.06475
\(791\) 7.85596e9 0.564393
\(792\) 0 0
\(793\) 2.12168e10 1.51086
\(794\) −1.52810e10 −1.08338
\(795\) 0 0
\(796\) −1.11239e10 −0.781740
\(797\) 1.15032e10 0.804848 0.402424 0.915453i \(-0.368168\pi\)
0.402424 + 0.915453i \(0.368168\pi\)
\(798\) 0 0
\(799\) −6.45244e9 −0.447517
\(800\) −4.60080e9 −0.317701
\(801\) 0 0
\(802\) −1.02237e10 −0.699842
\(803\) 4.31435e9 0.294043
\(804\) 0 0
\(805\) 6.47078e10 4.37191
\(806\) 2.09751e10 1.41101
\(807\) 0 0
\(808\) 1.70422e8 0.0113654
\(809\) −1.91918e10 −1.27437 −0.637184 0.770712i \(-0.719901\pi\)
−0.637184 + 0.770712i \(0.719901\pi\)
\(810\) 0 0
\(811\) −5.04377e9 −0.332034 −0.166017 0.986123i \(-0.553091\pi\)
−0.166017 + 0.986123i \(0.553091\pi\)
\(812\) 3.15819e9 0.207010
\(813\) 0 0
\(814\) −9.55047e8 −0.0620639
\(815\) 4.43609e10 2.87044
\(816\) 0 0
\(817\) 3.85023e9 0.247007
\(818\) −2.34120e9 −0.149556
\(819\) 0 0
\(820\) −1.43766e10 −0.910560
\(821\) −2.85707e10 −1.80185 −0.900927 0.433972i \(-0.857112\pi\)
−0.900927 + 0.433972i \(0.857112\pi\)
\(822\) 0 0
\(823\) 1.82564e10 1.14161 0.570803 0.821087i \(-0.306632\pi\)
0.570803 + 0.821087i \(0.306632\pi\)
\(824\) −2.31689e9 −0.144265
\(825\) 0 0
\(826\) −1.12406e10 −0.694001
\(827\) 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 \(0.453730\pi\)
\(830\) 1.32483e10 0.804239
\(831\) 0 0
\(832\) −3.24057e9 −0.195070
\(833\) 1.80848e10 1.08407
\(834\) 0 0
\(835\) −5.15770e10 −3.06587
\(836\) −1.55707e9 −0.0921690
\(837\) 0 0
\(838\) 3.05258e9 0.179190
\(839\) 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
\(842\) −2.11711e10 −1.22223
\(843\) 0 0
\(844\) −9.48400e9 −0.542992
\(845\) −4.21032e10 −2.40058
\(846\) 0 0
\(847\) 1.01598e10 0.574507
\(848\) 3.81448e8 0.0214808
\(849\) 0 0
\(850\) −1.51483e10 −0.846054
\(851\) 3.16656e9 0.176130
\(852\) 0 0
\(853\) −1.98045e10 −1.09255 −0.546276 0.837605i \(-0.683955\pi\)
−0.546276 + 0.837605i \(0.683955\pi\)
\(854\) −2.02008e10 −1.10986
\(855\) 0 0
\(856\) 3.75749e9 0.204757
\(857\) −1.83237e10 −0.994445 −0.497223 0.867623i \(-0.665647\pi\)
−0.497223 + 0.867623i \(0.665647\pi\)
\(858\) 0 0
\(859\) 2.65234e10 1.42775 0.713876 0.700272i \(-0.246938\pi\)
0.713876 + 0.700272i \(0.246938\pi\)
\(860\) 1.67943e10 0.900361
\(861\) 0 0
\(862\) 1.35901e10 0.722684
\(863\) 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
\(866\) −1.90344e10 −0.995927
\(867\) 0 0
\(868\) −1.99707e10 −1.03651
\(869\) 1.39946e10 0.723423
\(870\) 0 0
\(871\) −5.88737e9 −0.301897
\(872\) 5.32071e9 0.271745
\(873\) 0 0
\(874\) 5.16262e9 0.261566
\(875\) 4.28338e10 2.16152
\(876\) 0 0
\(877\) 1.22608e9 0.0613792 0.0306896 0.999529i \(-0.490230\pi\)
0.0306896 + 0.999529i \(0.490230\pi\)
\(878\) −1.74667e10 −0.870922
\(879\) 0 0
\(880\) −6.79175e9 −0.335964
\(881\) −6.34486e9 −0.312612 −0.156306 0.987709i \(-0.549959\pi\)
−0.156306 + 0.987709i \(0.549959\pi\)
\(882\) 0 0
\(883\) −2.24216e10 −1.09598 −0.547992 0.836484i \(-0.684608\pi\)
−0.547992 + 0.836484i \(0.684608\pi\)
\(884\) −1.06697e10 −0.519482
\(885\) 0 0
\(886\) −1.92915e10 −0.931856
\(887\) −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
\(890\) 2.80559e10 1.33401
\(891\) 0 0
\(892\) 8.87002e7 0.00418454
\(893\) 3.28166e9 0.154210
\(894\) 0 0
\(895\) 2.28291e10 1.06441
\(896\) 3.08540e9 0.143296
\(897\) 0 0
\(898\) −4.81671e9 −0.221964
\(899\) −7.11391e9 −0.326550
\(900\) 0 0
\(901\) 1.25593e9 0.0572045
\(902\) −1.36357e10 −0.618664
\(903\) 0 0
\(904\) 2.73393e9 0.123083
\(905\) −2.58079e9 −0.115740
\(906\) 0 0
\(907\) −1.56762e10 −0.697616 −0.348808 0.937194i \(-0.613413\pi\)
−0.348808 + 0.937194i \(0.613413\pi\)
\(908\) −8.64939e8 −0.0383429
\(909\) 0 0
\(910\) 6.80157e10 2.99202
\(911\) −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
\(914\) 2.62468e10 1.13701
\(915\) 0 0
\(916\) 1.98325e9 0.0852594
\(917\) 3.69169e8 0.0158100
\(918\) 0 0
\(919\) −3.39353e10 −1.44227 −0.721137 0.692793i \(-0.756380\pi\)
−0.721137 + 0.692793i \(0.756380\pi\)
\(920\) 2.25188e10 0.953428
\(921\) 0 0
\(922\) −2.80505e10 −1.17864
\(923\) 2.41696e10 1.01173
\(924\) 0 0
\(925\) 4.72554e9 0.196316
\(926\) 3.08136e10 1.27528
\(927\) 0 0
\(928\) 1.09907e9 0.0451449
\(929\) −7.75319e9 −0.317267 −0.158634 0.987338i \(-0.550709\pi\)
−0.158634 + 0.987338i \(0.550709\pi\)
\(930\) 0 0
\(931\) −9.19780e9 −0.373560
\(932\) 1.30651e10 0.528635
\(933\) 0 0
\(934\) 3.78654e9 0.152065
\(935\) −2.23621e10 −0.894690
\(936\) 0 0
\(937\) −3.05613e10 −1.21362 −0.606811 0.794846i \(-0.707552\pi\)
−0.606811 + 0.794846i \(0.707552\pi\)
\(938\) 5.60546e9 0.221769
\(939\) 0 0
\(940\) 1.43142e10 0.562109
\(941\) 4.59802e10 1.79890 0.899451 0.437022i \(-0.143967\pi\)
0.899451 + 0.437022i \(0.143967\pi\)
\(942\) 0 0
\(943\) 4.52107e10 1.75570
\(944\) −3.91182e9 −0.151348
\(945\) 0 0
\(946\) 1.59288e10 0.611735
\(947\) 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
\(950\) 7.70431e9 0.291542
\(951\) 0 0
\(952\) 1.01588e10 0.381604
\(953\) −1.04326e10 −0.390451 −0.195226 0.980758i \(-0.562544\pi\)
−0.195226 + 0.980758i \(0.562544\pi\)
\(954\) 0 0
\(955\) −5.40220e10 −2.00705
\(956\) 1.44381e10 0.534449
\(957\) 0 0
\(958\) 1.94605e10 0.715114
\(959\) −3.74595e10 −1.37150
\(960\) 0 0
\(961\) 1.74720e10 0.635054
\(962\) 3.32844e9 0.120539
\(963\) 0 0
\(964\) 2.66340e9 0.0957562
\(965\) −1.87902e9 −0.0673111
\(966\) 0 0
\(967\) 1.29860e10 0.461829 0.230915 0.972974i \(-0.425828\pi\)
0.230915 + 0.972974i \(0.425828\pi\)
\(968\) 3.53570e9 0.125289
\(969\) 0 0
\(970\) 4.78595e10 1.68371
\(971\) 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
\(974\) 2.70354e10 0.937512
\(975\) 0 0
\(976\) −7.03004e9 −0.242038
\(977\) −3.29063e10 −1.12888 −0.564440 0.825474i \(-0.690908\pi\)
−0.564440 + 0.825474i \(0.690908\pi\)
\(978\) 0 0
\(979\) 2.66100e10 0.906370
\(980\) −4.01198e10 −1.36166
\(981\) 0 0
\(982\) −1.29874e10 −0.437655
\(983\) 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
\(986\) 3.61874e9 0.120223
\(987\) 0 0
\(988\) 5.42654e9 0.179008
\(989\) −5.28136e10 −1.73603
\(990\) 0 0
\(991\) −4.62321e10 −1.50899 −0.754494 0.656306i \(-0.772118\pi\)
−0.754494 + 0.656306i \(0.772118\pi\)
\(992\) −6.94995e9 −0.226043
\(993\) 0 0
\(994\) −2.30122e10 −0.743201
\(995\) 8.12520e10 2.61489
\(996\) 0 0
\(997\) 2.54695e10 0.813930 0.406965 0.913444i \(-0.366587\pi\)
0.406965 + 0.913444i \(0.366587\pi\)
\(998\) −3.10522e10 −0.988862
\(999\) 0 0
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 342.8.a.g.1.1 2
3.2 odd 2 38.8.a.d.1.1 2
12.11 even 2 304.8.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.8.a.d.1.1 2 3.2 odd 2
304.8.a.d.1.2 2 12.11 even 2
342.8.a.g.1.1 2 1.1 even 1 trivial