Properties

Label 387.8.a.d.1.7
Level $387$
Weight $8$
Character 387.1
Self dual yes
Analytic conductor $120.893$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(120.893004862\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.684341\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.315659 q^{2} -127.900 q^{4} -531.642 q^{5} -349.841 q^{7} +80.7773 q^{8} +O(q^{10})\) \(q-0.315659 q^{2} -127.900 q^{4} -531.642 q^{5} -349.841 q^{7} +80.7773 q^{8} +167.818 q^{10} -3331.87 q^{11} -8584.81 q^{13} +110.430 q^{14} +16345.7 q^{16} +25786.3 q^{17} +16993.6 q^{19} +67997.2 q^{20} +1051.73 q^{22} -80723.2 q^{23} +204518. q^{25} +2709.87 q^{26} +44744.7 q^{28} +67762.5 q^{29} +18887.5 q^{31} -15499.2 q^{32} -8139.67 q^{34} +185990. q^{35} -179478. q^{37} -5364.19 q^{38} -42944.6 q^{40} +611192. q^{41} -79507.0 q^{43} +426147. q^{44} +25481.0 q^{46} +1.18047e6 q^{47} -701155. q^{49} -64557.9 q^{50} +1.09800e6 q^{52} -538854. q^{53} +1.77136e6 q^{55} -28259.2 q^{56} -21389.9 q^{58} +1.96283e6 q^{59} +2.10797e6 q^{61} -5962.00 q^{62} -2.08736e6 q^{64} +4.56404e6 q^{65} -813166. q^{67} -3.29807e6 q^{68} -58709.4 q^{70} -1.76280e6 q^{71} -3.47624e6 q^{73} +56653.7 q^{74} -2.17349e6 q^{76} +1.16562e6 q^{77} -3.68544e6 q^{79} -8.69008e6 q^{80} -192928. q^{82} -3.77929e6 q^{83} -1.37091e7 q^{85} +25097.1 q^{86} -269139. q^{88} +2.83826e6 q^{89} +3.00331e6 q^{91} +1.03245e7 q^{92} -372627. q^{94} -9.03452e6 q^{95} +4.40344e6 q^{97} +221326. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.315659 −0.0279006 −0.0139503 0.999903i \(-0.504441\pi\)
−0.0139503 + 0.999903i \(0.504441\pi\)
\(3\) 0 0
\(4\) −127.900 −0.999222
\(5\) −531.642 −1.90206 −0.951029 0.309100i \(-0.899972\pi\)
−0.951029 + 0.309100i \(0.899972\pi\)
\(6\) 0 0
\(7\) −349.841 −0.385502 −0.192751 0.981248i \(-0.561741\pi\)
−0.192751 + 0.981248i \(0.561741\pi\)
\(8\) 80.7773 0.0557795
\(9\) 0 0
\(10\) 167.818 0.0530686
\(11\) −3331.87 −0.754767 −0.377384 0.926057i \(-0.623176\pi\)
−0.377384 + 0.926057i \(0.623176\pi\)
\(12\) 0 0
\(13\) −8584.81 −1.08375 −0.541875 0.840459i \(-0.682285\pi\)
−0.541875 + 0.840459i \(0.682285\pi\)
\(14\) 110.430 0.0107557
\(15\) 0 0
\(16\) 16345.7 0.997665
\(17\) 25786.3 1.27297 0.636484 0.771290i \(-0.280388\pi\)
0.636484 + 0.771290i \(0.280388\pi\)
\(18\) 0 0
\(19\) 16993.6 0.568392 0.284196 0.958766i \(-0.408273\pi\)
0.284196 + 0.958766i \(0.408273\pi\)
\(20\) 67997.2 1.90058
\(21\) 0 0
\(22\) 1051.73 0.0210585
\(23\) −80723.2 −1.38341 −0.691705 0.722180i \(-0.743140\pi\)
−0.691705 + 0.722180i \(0.743140\pi\)
\(24\) 0 0
\(25\) 204518. 2.61783
\(26\) 2709.87 0.0302373
\(27\) 0 0
\(28\) 44744.7 0.385202
\(29\) 67762.5 0.515937 0.257968 0.966153i \(-0.416947\pi\)
0.257968 + 0.966153i \(0.416947\pi\)
\(30\) 0 0
\(31\) 18887.5 0.113870 0.0569348 0.998378i \(-0.481867\pi\)
0.0569348 + 0.998378i \(0.481867\pi\)
\(32\) −15499.2 −0.0836150
\(33\) 0 0
\(34\) −8139.67 −0.0355166
\(35\) 185990. 0.733248
\(36\) 0 0
\(37\) −179478. −0.582511 −0.291255 0.956645i \(-0.594073\pi\)
−0.291255 + 0.956645i \(0.594073\pi\)
\(38\) −5364.19 −0.0158585
\(39\) 0 0
\(40\) −42944.6 −0.106096
\(41\) 611192. 1.38495 0.692475 0.721442i \(-0.256520\pi\)
0.692475 + 0.721442i \(0.256520\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) 426147. 0.754180
\(45\) 0 0
\(46\) 25481.0 0.0385980
\(47\) 1.18047e6 1.65849 0.829245 0.558885i \(-0.188771\pi\)
0.829245 + 0.558885i \(0.188771\pi\)
\(48\) 0 0
\(49\) −701155. −0.851388
\(50\) −64557.9 −0.0730390
\(51\) 0 0
\(52\) 1.09800e6 1.08291
\(53\) −538854. −0.497170 −0.248585 0.968610i \(-0.579966\pi\)
−0.248585 + 0.968610i \(0.579966\pi\)
\(54\) 0 0
\(55\) 1.77136e6 1.43561
\(56\) −28259.2 −0.0215031
\(57\) 0 0
\(58\) −21389.9 −0.0143949
\(59\) 1.96283e6 1.24423 0.622113 0.782927i \(-0.286274\pi\)
0.622113 + 0.782927i \(0.286274\pi\)
\(60\) 0 0
\(61\) 2.10797e6 1.18907 0.594537 0.804068i \(-0.297335\pi\)
0.594537 + 0.804068i \(0.297335\pi\)
\(62\) −5962.00 −0.00317703
\(63\) 0 0
\(64\) −2.08736e6 −0.995332
\(65\) 4.56404e6 2.06136
\(66\) 0 0
\(67\) −813166. −0.330307 −0.165153 0.986268i \(-0.552812\pi\)
−0.165153 + 0.986268i \(0.552812\pi\)
\(68\) −3.29807e6 −1.27198
\(69\) 0 0
\(70\) −58709.4 −0.0204581
\(71\) −1.76280e6 −0.584521 −0.292260 0.956339i \(-0.594407\pi\)
−0.292260 + 0.956339i \(0.594407\pi\)
\(72\) 0 0
\(73\) −3.47624e6 −1.04588 −0.522938 0.852371i \(-0.675164\pi\)
−0.522938 + 0.852371i \(0.675164\pi\)
\(74\) 56653.7 0.0162524
\(75\) 0 0
\(76\) −2.17349e6 −0.567950
\(77\) 1.16562e6 0.290965
\(78\) 0 0
\(79\) −3.68544e6 −0.840997 −0.420498 0.907293i \(-0.638145\pi\)
−0.420498 + 0.907293i \(0.638145\pi\)
\(80\) −8.69008e6 −1.89762
\(81\) 0 0
\(82\) −192928. −0.0386409
\(83\) −3.77929e6 −0.725500 −0.362750 0.931887i \(-0.618162\pi\)
−0.362750 + 0.931887i \(0.618162\pi\)
\(84\) 0 0
\(85\) −1.37091e7 −2.42126
\(86\) 25097.1 0.00425480
\(87\) 0 0
\(88\) −269139. −0.0421005
\(89\) 2.83826e6 0.426764 0.213382 0.976969i \(-0.431552\pi\)
0.213382 + 0.976969i \(0.431552\pi\)
\(90\) 0 0
\(91\) 3.00331e6 0.417788
\(92\) 1.03245e7 1.38233
\(93\) 0 0
\(94\) −372627. −0.0462729
\(95\) −9.03452e6 −1.08112
\(96\) 0 0
\(97\) 4.40344e6 0.489882 0.244941 0.969538i \(-0.421231\pi\)
0.244941 + 0.969538i \(0.421231\pi\)
\(98\) 221326. 0.0237542
\(99\) 0 0
\(100\) −2.61579e7 −2.61579
\(101\) 5.88409e6 0.568269 0.284135 0.958784i \(-0.408294\pi\)
0.284135 + 0.958784i \(0.408294\pi\)
\(102\) 0 0
\(103\) 698841. 0.0630156 0.0315078 0.999504i \(-0.489969\pi\)
0.0315078 + 0.999504i \(0.489969\pi\)
\(104\) −693458. −0.0604510
\(105\) 0 0
\(106\) 170094. 0.0138714
\(107\) 2.58664e6 0.204123 0.102062 0.994778i \(-0.467456\pi\)
0.102062 + 0.994778i \(0.467456\pi\)
\(108\) 0 0
\(109\) 1.24939e7 0.924070 0.462035 0.886862i \(-0.347119\pi\)
0.462035 + 0.886862i \(0.347119\pi\)
\(110\) −559146. −0.0400544
\(111\) 0 0
\(112\) −5.71841e6 −0.384602
\(113\) −1.92969e7 −1.25810 −0.629048 0.777367i \(-0.716555\pi\)
−0.629048 + 0.777367i \(0.716555\pi\)
\(114\) 0 0
\(115\) 4.29158e7 2.63133
\(116\) −8.66685e6 −0.515535
\(117\) 0 0
\(118\) −619584. −0.0347147
\(119\) −9.02108e6 −0.490732
\(120\) 0 0
\(121\) −8.38585e6 −0.430326
\(122\) −665399. −0.0331759
\(123\) 0 0
\(124\) −2.41571e6 −0.113781
\(125\) −6.71957e7 −3.07720
\(126\) 0 0
\(127\) −5.00899e6 −0.216989 −0.108494 0.994097i \(-0.534603\pi\)
−0.108494 + 0.994097i \(0.534603\pi\)
\(128\) 2.64279e6 0.111385
\(129\) 0 0
\(130\) −1.44068e6 −0.0575131
\(131\) 4.18902e7 1.62803 0.814016 0.580843i \(-0.197277\pi\)
0.814016 + 0.580843i \(0.197277\pi\)
\(132\) 0 0
\(133\) −5.94506e6 −0.219117
\(134\) 256683. 0.00921575
\(135\) 0 0
\(136\) 2.08295e6 0.0710055
\(137\) −3.89920e7 −1.29555 −0.647774 0.761833i \(-0.724300\pi\)
−0.647774 + 0.761833i \(0.724300\pi\)
\(138\) 0 0
\(139\) 6.03467e7 1.90591 0.952954 0.303115i \(-0.0980267\pi\)
0.952954 + 0.303115i \(0.0980267\pi\)
\(140\) −2.37882e7 −0.732677
\(141\) 0 0
\(142\) 556446. 0.0163085
\(143\) 2.86034e7 0.817978
\(144\) 0 0
\(145\) −3.60254e7 −0.981342
\(146\) 1.09731e6 0.0291806
\(147\) 0 0
\(148\) 2.29552e7 0.582057
\(149\) 1.86557e7 0.462018 0.231009 0.972952i \(-0.425797\pi\)
0.231009 + 0.972952i \(0.425797\pi\)
\(150\) 0 0
\(151\) −1.15952e7 −0.274070 −0.137035 0.990566i \(-0.543757\pi\)
−0.137035 + 0.990566i \(0.543757\pi\)
\(152\) 1.37270e6 0.0317046
\(153\) 0 0
\(154\) −367939. −0.00811809
\(155\) −1.00414e7 −0.216587
\(156\) 0 0
\(157\) 2.89477e7 0.596988 0.298494 0.954412i \(-0.403516\pi\)
0.298494 + 0.954412i \(0.403516\pi\)
\(158\) 1.16334e6 0.0234643
\(159\) 0 0
\(160\) 8.24001e6 0.159041
\(161\) 2.82403e7 0.533308
\(162\) 0 0
\(163\) 779647. 0.0141007 0.00705036 0.999975i \(-0.497756\pi\)
0.00705036 + 0.999975i \(0.497756\pi\)
\(164\) −7.81716e7 −1.38387
\(165\) 0 0
\(166\) 1.19297e6 0.0202419
\(167\) 4.18156e7 0.694754 0.347377 0.937726i \(-0.387072\pi\)
0.347377 + 0.937726i \(0.387072\pi\)
\(168\) 0 0
\(169\) 1.09504e7 0.174513
\(170\) 4.32739e6 0.0675546
\(171\) 0 0
\(172\) 1.01690e7 0.152380
\(173\) −9.94492e7 −1.46029 −0.730147 0.683291i \(-0.760548\pi\)
−0.730147 + 0.683291i \(0.760548\pi\)
\(174\) 0 0
\(175\) −7.15486e7 −1.00918
\(176\) −5.44618e7 −0.753005
\(177\) 0 0
\(178\) −895924. −0.0119070
\(179\) −7.81297e7 −1.01819 −0.509097 0.860709i \(-0.670021\pi\)
−0.509097 + 0.860709i \(0.670021\pi\)
\(180\) 0 0
\(181\) 1.22327e7 0.153337 0.0766684 0.997057i \(-0.475572\pi\)
0.0766684 + 0.997057i \(0.475572\pi\)
\(182\) −948024. −0.0116565
\(183\) 0 0
\(184\) −6.52061e6 −0.0771659
\(185\) 9.54177e7 1.10797
\(186\) 0 0
\(187\) −8.59164e7 −0.960794
\(188\) −1.50983e8 −1.65720
\(189\) 0 0
\(190\) 2.85183e6 0.0301638
\(191\) 9.63352e7 1.00039 0.500194 0.865914i \(-0.333262\pi\)
0.500194 + 0.865914i \(0.333262\pi\)
\(192\) 0 0
\(193\) 2.89017e7 0.289383 0.144692 0.989477i \(-0.453781\pi\)
0.144692 + 0.989477i \(0.453781\pi\)
\(194\) −1.38999e6 −0.0136680
\(195\) 0 0
\(196\) 8.96779e7 0.850725
\(197\) −4.17794e7 −0.389342 −0.194671 0.980869i \(-0.562364\pi\)
−0.194671 + 0.980869i \(0.562364\pi\)
\(198\) 0 0
\(199\) 5.54068e7 0.498399 0.249199 0.968452i \(-0.419833\pi\)
0.249199 + 0.968452i \(0.419833\pi\)
\(200\) 1.65204e7 0.146021
\(201\) 0 0
\(202\) −1.85737e6 −0.0158551
\(203\) −2.37061e7 −0.198895
\(204\) 0 0
\(205\) −3.24935e8 −2.63426
\(206\) −220596. −0.00175817
\(207\) 0 0
\(208\) −1.40325e8 −1.08122
\(209\) −5.66205e7 −0.429004
\(210\) 0 0
\(211\) −9.44194e7 −0.691947 −0.345973 0.938244i \(-0.612451\pi\)
−0.345973 + 0.938244i \(0.612451\pi\)
\(212\) 6.89196e7 0.496783
\(213\) 0 0
\(214\) −816497. −0.00569517
\(215\) 4.22692e7 0.290061
\(216\) 0 0
\(217\) −6.60760e6 −0.0438970
\(218\) −3.94381e6 −0.0257821
\(219\) 0 0
\(220\) −2.26557e8 −1.43449
\(221\) −2.21370e8 −1.37958
\(222\) 0 0
\(223\) 1.04942e7 0.0633697 0.0316848 0.999498i \(-0.489913\pi\)
0.0316848 + 0.999498i \(0.489913\pi\)
\(224\) 5.42224e6 0.0322338
\(225\) 0 0
\(226\) 6.09126e6 0.0351016
\(227\) −2.42554e7 −0.137632 −0.0688158 0.997629i \(-0.521922\pi\)
−0.0688158 + 0.997629i \(0.521922\pi\)
\(228\) 0 0
\(229\) 7.20589e7 0.396519 0.198259 0.980150i \(-0.436471\pi\)
0.198259 + 0.980150i \(0.436471\pi\)
\(230\) −1.35468e7 −0.0734157
\(231\) 0 0
\(232\) 5.47367e6 0.0287787
\(233\) −6.20516e7 −0.321372 −0.160686 0.987006i \(-0.551371\pi\)
−0.160686 + 0.987006i \(0.551371\pi\)
\(234\) 0 0
\(235\) −6.27588e8 −3.15455
\(236\) −2.51046e8 −1.24326
\(237\) 0 0
\(238\) 2.84759e6 0.0136917
\(239\) −1.05074e8 −0.497853 −0.248926 0.968522i \(-0.580078\pi\)
−0.248926 + 0.968522i \(0.580078\pi\)
\(240\) 0 0
\(241\) 3.02273e8 1.39104 0.695520 0.718507i \(-0.255174\pi\)
0.695520 + 0.718507i \(0.255174\pi\)
\(242\) 2.64707e6 0.0120064
\(243\) 0 0
\(244\) −2.69610e8 −1.18815
\(245\) 3.72763e8 1.61939
\(246\) 0 0
\(247\) −1.45887e8 −0.615995
\(248\) 1.52568e6 0.00635158
\(249\) 0 0
\(250\) 2.12109e7 0.0858558
\(251\) 1.60365e8 0.640105 0.320052 0.947400i \(-0.396299\pi\)
0.320052 + 0.947400i \(0.396299\pi\)
\(252\) 0 0
\(253\) 2.68959e8 1.04415
\(254\) 1.58114e6 0.00605412
\(255\) 0 0
\(256\) 2.66348e8 0.992225
\(257\) −9.94729e7 −0.365543 −0.182772 0.983155i \(-0.558507\pi\)
−0.182772 + 0.983155i \(0.558507\pi\)
\(258\) 0 0
\(259\) 6.27885e7 0.224559
\(260\) −5.83743e8 −2.05975
\(261\) 0 0
\(262\) −1.32230e7 −0.0454231
\(263\) −1.41037e7 −0.0478066 −0.0239033 0.999714i \(-0.507609\pi\)
−0.0239033 + 0.999714i \(0.507609\pi\)
\(264\) 0 0
\(265\) 2.86477e8 0.945647
\(266\) 1.87661e6 0.00611349
\(267\) 0 0
\(268\) 1.04004e8 0.330049
\(269\) −9.31759e7 −0.291857 −0.145929 0.989295i \(-0.546617\pi\)
−0.145929 + 0.989295i \(0.546617\pi\)
\(270\) 0 0
\(271\) 4.68939e8 1.43128 0.715639 0.698470i \(-0.246135\pi\)
0.715639 + 0.698470i \(0.246135\pi\)
\(272\) 4.21496e8 1.27000
\(273\) 0 0
\(274\) 1.23082e7 0.0361466
\(275\) −6.81426e8 −1.97585
\(276\) 0 0
\(277\) −6.06565e7 −0.171474 −0.0857370 0.996318i \(-0.527324\pi\)
−0.0857370 + 0.996318i \(0.527324\pi\)
\(278\) −1.90490e7 −0.0531760
\(279\) 0 0
\(280\) 1.50238e7 0.0409002
\(281\) 6.12917e8 1.64789 0.823947 0.566667i \(-0.191767\pi\)
0.823947 + 0.566667i \(0.191767\pi\)
\(282\) 0 0
\(283\) −6.99501e8 −1.83458 −0.917288 0.398224i \(-0.869627\pi\)
−0.917288 + 0.398224i \(0.869627\pi\)
\(284\) 2.25463e8 0.584066
\(285\) 0 0
\(286\) −9.02894e6 −0.0228221
\(287\) −2.13820e8 −0.533901
\(288\) 0 0
\(289\) 2.54593e8 0.620446
\(290\) 1.13717e7 0.0273800
\(291\) 0 0
\(292\) 4.44613e8 1.04506
\(293\) −7.00364e8 −1.62662 −0.813312 0.581828i \(-0.802338\pi\)
−0.813312 + 0.581828i \(0.802338\pi\)
\(294\) 0 0
\(295\) −1.04352e9 −2.36659
\(296\) −1.44977e7 −0.0324922
\(297\) 0 0
\(298\) −5.88884e6 −0.0128906
\(299\) 6.92994e8 1.49927
\(300\) 0 0
\(301\) 2.78148e7 0.0587886
\(302\) 3.66015e6 0.00764671
\(303\) 0 0
\(304\) 2.77773e8 0.567065
\(305\) −1.12068e9 −2.26169
\(306\) 0 0
\(307\) 5.88458e8 1.16073 0.580365 0.814357i \(-0.302910\pi\)
0.580365 + 0.814357i \(0.302910\pi\)
\(308\) −1.49083e8 −0.290738
\(309\) 0 0
\(310\) 3.16965e6 0.00604290
\(311\) −7.48619e7 −0.141123 −0.0705617 0.997507i \(-0.522479\pi\)
−0.0705617 + 0.997507i \(0.522479\pi\)
\(312\) 0 0
\(313\) 2.61454e8 0.481937 0.240969 0.970533i \(-0.422535\pi\)
0.240969 + 0.970533i \(0.422535\pi\)
\(314\) −9.13762e6 −0.0166563
\(315\) 0 0
\(316\) 4.71369e8 0.840342
\(317\) 1.08258e9 1.90877 0.954384 0.298582i \(-0.0965138\pi\)
0.954384 + 0.298582i \(0.0965138\pi\)
\(318\) 0 0
\(319\) −2.25775e8 −0.389412
\(320\) 1.10973e9 1.89318
\(321\) 0 0
\(322\) −8.91430e6 −0.0148796
\(323\) 4.38202e8 0.723545
\(324\) 0 0
\(325\) −1.75575e9 −2.83707
\(326\) −246103. −0.000393419 0
\(327\) 0 0
\(328\) 4.93704e7 0.0772518
\(329\) −4.12977e8 −0.639352
\(330\) 0 0
\(331\) −4.20559e8 −0.637425 −0.318713 0.947851i \(-0.603250\pi\)
−0.318713 + 0.947851i \(0.603250\pi\)
\(332\) 4.83373e8 0.724935
\(333\) 0 0
\(334\) −1.31995e7 −0.0193841
\(335\) 4.32313e8 0.628263
\(336\) 0 0
\(337\) 1.18120e9 1.68119 0.840597 0.541662i \(-0.182205\pi\)
0.840597 + 0.541662i \(0.182205\pi\)
\(338\) −3.45660e6 −0.00486901
\(339\) 0 0
\(340\) 1.75339e9 2.41937
\(341\) −6.29305e7 −0.0859450
\(342\) 0 0
\(343\) 5.33401e8 0.713714
\(344\) −6.42236e6 −0.00850629
\(345\) 0 0
\(346\) 3.13921e7 0.0407431
\(347\) −1.04092e9 −1.33741 −0.668707 0.743526i \(-0.733152\pi\)
−0.668707 + 0.743526i \(0.733152\pi\)
\(348\) 0 0
\(349\) −6.58581e8 −0.829316 −0.414658 0.909977i \(-0.636099\pi\)
−0.414658 + 0.909977i \(0.636099\pi\)
\(350\) 2.25850e7 0.0281567
\(351\) 0 0
\(352\) 5.16412e7 0.0631098
\(353\) −3.03399e8 −0.367115 −0.183557 0.983009i \(-0.558761\pi\)
−0.183557 + 0.983009i \(0.558761\pi\)
\(354\) 0 0
\(355\) 9.37180e8 1.11179
\(356\) −3.63015e8 −0.426431
\(357\) 0 0
\(358\) 2.46624e7 0.0284082
\(359\) −1.42234e9 −1.62245 −0.811226 0.584733i \(-0.801199\pi\)
−0.811226 + 0.584733i \(0.801199\pi\)
\(360\) 0 0
\(361\) −6.05089e8 −0.676930
\(362\) −3.86136e6 −0.00427819
\(363\) 0 0
\(364\) −3.84125e8 −0.417463
\(365\) 1.84812e9 1.98932
\(366\) 0 0
\(367\) 5.93187e8 0.626413 0.313206 0.949685i \(-0.398597\pi\)
0.313206 + 0.949685i \(0.398597\pi\)
\(368\) −1.31948e9 −1.38018
\(369\) 0 0
\(370\) −3.01195e7 −0.0309130
\(371\) 1.88513e8 0.191660
\(372\) 0 0
\(373\) −3.41490e8 −0.340719 −0.170360 0.985382i \(-0.554493\pi\)
−0.170360 + 0.985382i \(0.554493\pi\)
\(374\) 2.71203e7 0.0268067
\(375\) 0 0
\(376\) 9.53553e7 0.0925098
\(377\) −5.81728e8 −0.559146
\(378\) 0 0
\(379\) −1.21788e9 −1.14912 −0.574561 0.818462i \(-0.694827\pi\)
−0.574561 + 0.818462i \(0.694827\pi\)
\(380\) 1.15552e9 1.08027
\(381\) 0 0
\(382\) −3.04091e7 −0.0279114
\(383\) −1.21649e9 −1.10640 −0.553200 0.833048i \(-0.686594\pi\)
−0.553200 + 0.833048i \(0.686594\pi\)
\(384\) 0 0
\(385\) −6.19693e8 −0.553432
\(386\) −9.12310e6 −0.00807397
\(387\) 0 0
\(388\) −5.63202e8 −0.489500
\(389\) 2.23009e7 0.0192087 0.00960436 0.999954i \(-0.496943\pi\)
0.00960436 + 0.999954i \(0.496943\pi\)
\(390\) 0 0
\(391\) −2.08155e9 −1.76104
\(392\) −5.66374e7 −0.0474900
\(393\) 0 0
\(394\) 1.31881e7 0.0108629
\(395\) 1.95933e9 1.59962
\(396\) 0 0
\(397\) −8.33419e8 −0.668493 −0.334246 0.942486i \(-0.608482\pi\)
−0.334246 + 0.942486i \(0.608482\pi\)
\(398\) −1.74897e7 −0.0139056
\(399\) 0 0
\(400\) 3.34300e9 2.61172
\(401\) 1.28321e9 0.993786 0.496893 0.867812i \(-0.334474\pi\)
0.496893 + 0.867812i \(0.334474\pi\)
\(402\) 0 0
\(403\) −1.62145e8 −0.123406
\(404\) −7.52577e8 −0.567827
\(405\) 0 0
\(406\) 7.48304e6 0.00554928
\(407\) 5.97995e8 0.439660
\(408\) 0 0
\(409\) 3.97834e8 0.287522 0.143761 0.989612i \(-0.454080\pi\)
0.143761 + 0.989612i \(0.454080\pi\)
\(410\) 1.02569e8 0.0734973
\(411\) 0 0
\(412\) −8.93820e7 −0.0629665
\(413\) −6.86676e8 −0.479652
\(414\) 0 0
\(415\) 2.00923e9 1.37994
\(416\) 1.33058e8 0.0906177
\(417\) 0 0
\(418\) 1.78728e7 0.0119695
\(419\) 1.83847e9 1.22097 0.610487 0.792026i \(-0.290974\pi\)
0.610487 + 0.792026i \(0.290974\pi\)
\(420\) 0 0
\(421\) 2.99208e9 1.95428 0.977139 0.212604i \(-0.0681943\pi\)
0.977139 + 0.212604i \(0.0681943\pi\)
\(422\) 2.98044e7 0.0193057
\(423\) 0 0
\(424\) −4.35272e7 −0.0277319
\(425\) 5.27375e9 3.33241
\(426\) 0 0
\(427\) −7.37452e8 −0.458391
\(428\) −3.30832e8 −0.203965
\(429\) 0 0
\(430\) −1.33427e7 −0.00809288
\(431\) −7.52649e7 −0.0452816 −0.0226408 0.999744i \(-0.507207\pi\)
−0.0226408 + 0.999744i \(0.507207\pi\)
\(432\) 0 0
\(433\) 2.36792e9 1.40172 0.700858 0.713301i \(-0.252801\pi\)
0.700858 + 0.713301i \(0.252801\pi\)
\(434\) 2.08575e6 0.00122475
\(435\) 0 0
\(436\) −1.59797e9 −0.923351
\(437\) −1.37178e9 −0.786320
\(438\) 0 0
\(439\) 1.98503e9 1.11980 0.559901 0.828559i \(-0.310839\pi\)
0.559901 + 0.828559i \(0.310839\pi\)
\(440\) 1.43086e8 0.0800777
\(441\) 0 0
\(442\) 6.98775e7 0.0384910
\(443\) 2.46053e9 1.34467 0.672335 0.740247i \(-0.265291\pi\)
0.672335 + 0.740247i \(0.265291\pi\)
\(444\) 0 0
\(445\) −1.50894e9 −0.811730
\(446\) −3.31259e6 −0.00176805
\(447\) 0 0
\(448\) 7.30244e8 0.383703
\(449\) 1.53414e9 0.799839 0.399920 0.916550i \(-0.369038\pi\)
0.399920 + 0.916550i \(0.369038\pi\)
\(450\) 0 0
\(451\) −2.03641e9 −1.04531
\(452\) 2.46808e9 1.25712
\(453\) 0 0
\(454\) 7.65645e6 0.00384000
\(455\) −1.59669e9 −0.794657
\(456\) 0 0
\(457\) 1.63707e9 0.802345 0.401173 0.916002i \(-0.368603\pi\)
0.401173 + 0.916002i \(0.368603\pi\)
\(458\) −2.27461e7 −0.0110631
\(459\) 0 0
\(460\) −5.48895e9 −2.62928
\(461\) −3.59530e9 −1.70916 −0.854579 0.519321i \(-0.826185\pi\)
−0.854579 + 0.519321i \(0.826185\pi\)
\(462\) 0 0
\(463\) −3.31932e9 −1.55423 −0.777115 0.629359i \(-0.783318\pi\)
−0.777115 + 0.629359i \(0.783318\pi\)
\(464\) 1.10763e9 0.514732
\(465\) 0 0
\(466\) 1.95872e7 0.00896646
\(467\) −2.20717e9 −1.00283 −0.501415 0.865207i \(-0.667187\pi\)
−0.501415 + 0.865207i \(0.667187\pi\)
\(468\) 0 0
\(469\) 2.84478e8 0.127334
\(470\) 1.98104e8 0.0880138
\(471\) 0 0
\(472\) 1.58552e8 0.0694023
\(473\) 2.64907e8 0.115101
\(474\) 0 0
\(475\) 3.47550e9 1.48795
\(476\) 1.15380e9 0.490350
\(477\) 0 0
\(478\) 3.31675e7 0.0138904
\(479\) −1.10499e9 −0.459394 −0.229697 0.973262i \(-0.573774\pi\)
−0.229697 + 0.973262i \(0.573774\pi\)
\(480\) 0 0
\(481\) 1.54078e9 0.631296
\(482\) −9.54152e7 −0.0388108
\(483\) 0 0
\(484\) 1.07255e9 0.429991
\(485\) −2.34105e9 −0.931784
\(486\) 0 0
\(487\) 1.39608e9 0.547719 0.273859 0.961770i \(-0.411700\pi\)
0.273859 + 0.961770i \(0.411700\pi\)
\(488\) 1.70276e8 0.0663260
\(489\) 0 0
\(490\) −1.17666e8 −0.0451820
\(491\) 4.32305e9 1.64818 0.824091 0.566458i \(-0.191687\pi\)
0.824091 + 0.566458i \(0.191687\pi\)
\(492\) 0 0
\(493\) 1.74734e9 0.656770
\(494\) 4.60506e7 0.0171866
\(495\) 0 0
\(496\) 3.08730e8 0.113604
\(497\) 6.16700e8 0.225334
\(498\) 0 0
\(499\) 5.09888e8 0.183706 0.0918528 0.995773i \(-0.470721\pi\)
0.0918528 + 0.995773i \(0.470721\pi\)
\(500\) 8.59435e9 3.07481
\(501\) 0 0
\(502\) −5.06207e7 −0.0178593
\(503\) 3.07862e9 1.07862 0.539310 0.842108i \(-0.318685\pi\)
0.539310 + 0.842108i \(0.318685\pi\)
\(504\) 0 0
\(505\) −3.12823e9 −1.08088
\(506\) −8.48994e7 −0.0291325
\(507\) 0 0
\(508\) 6.40652e8 0.216820
\(509\) −3.54159e9 −1.19038 −0.595191 0.803584i \(-0.702924\pi\)
−0.595191 + 0.803584i \(0.702924\pi\)
\(510\) 0 0
\(511\) 1.21613e9 0.403188
\(512\) −4.22353e8 −0.139069
\(513\) 0 0
\(514\) 3.13996e7 0.0101989
\(515\) −3.71533e8 −0.119859
\(516\) 0 0
\(517\) −3.93317e9 −1.25177
\(518\) −1.98198e7 −0.00626534
\(519\) 0 0
\(520\) 3.68671e8 0.114981
\(521\) 2.36861e9 0.733773 0.366886 0.930266i \(-0.380424\pi\)
0.366886 + 0.930266i \(0.380424\pi\)
\(522\) 0 0
\(523\) 2.62233e9 0.801551 0.400776 0.916176i \(-0.368741\pi\)
0.400776 + 0.916176i \(0.368741\pi\)
\(524\) −5.35777e9 −1.62676
\(525\) 0 0
\(526\) 4.45196e6 0.00133383
\(527\) 4.87037e8 0.144952
\(528\) 0 0
\(529\) 3.11142e9 0.913825
\(530\) −9.04291e7 −0.0263841
\(531\) 0 0
\(532\) 7.60375e8 0.218946
\(533\) −5.24696e9 −1.50094
\(534\) 0 0
\(535\) −1.37517e9 −0.388255
\(536\) −6.56854e7 −0.0184243
\(537\) 0 0
\(538\) 2.94118e7 0.00814300
\(539\) 2.33615e9 0.642600
\(540\) 0 0
\(541\) −3.96326e9 −1.07612 −0.538062 0.842905i \(-0.680843\pi\)
−0.538062 + 0.842905i \(0.680843\pi\)
\(542\) −1.48025e8 −0.0399335
\(543\) 0 0
\(544\) −3.99666e8 −0.106439
\(545\) −6.64227e9 −1.75764
\(546\) 0 0
\(547\) −5.89175e9 −1.53918 −0.769588 0.638540i \(-0.779538\pi\)
−0.769588 + 0.638540i \(0.779538\pi\)
\(548\) 4.98709e9 1.29454
\(549\) 0 0
\(550\) 2.15098e8 0.0551274
\(551\) 1.15153e9 0.293254
\(552\) 0 0
\(553\) 1.28932e9 0.324206
\(554\) 1.91468e7 0.00478423
\(555\) 0 0
\(556\) −7.71837e9 −1.90442
\(557\) −9.78407e8 −0.239898 −0.119949 0.992780i \(-0.538273\pi\)
−0.119949 + 0.992780i \(0.538273\pi\)
\(558\) 0 0
\(559\) 6.82552e8 0.165270
\(560\) 3.04014e9 0.731536
\(561\) 0 0
\(562\) −1.93473e8 −0.0459772
\(563\) −2.19945e9 −0.519439 −0.259719 0.965684i \(-0.583630\pi\)
−0.259719 + 0.965684i \(0.583630\pi\)
\(564\) 0 0
\(565\) 1.02591e10 2.39297
\(566\) 2.20804e8 0.0511858
\(567\) 0 0
\(568\) −1.42395e8 −0.0326043
\(569\) 5.53125e9 1.25872 0.629362 0.777112i \(-0.283316\pi\)
0.629362 + 0.777112i \(0.283316\pi\)
\(570\) 0 0
\(571\) −2.15374e9 −0.484135 −0.242067 0.970259i \(-0.577825\pi\)
−0.242067 + 0.970259i \(0.577825\pi\)
\(572\) −3.65839e9 −0.817342
\(573\) 0 0
\(574\) 6.74942e7 0.0148962
\(575\) −1.65093e10 −3.62153
\(576\) 0 0
\(577\) −8.51167e9 −1.84459 −0.922294 0.386490i \(-0.873688\pi\)
−0.922294 + 0.386490i \(0.873688\pi\)
\(578\) −8.03646e7 −0.0173108
\(579\) 0 0
\(580\) 4.60766e9 0.980578
\(581\) 1.32215e9 0.279682
\(582\) 0 0
\(583\) 1.79539e9 0.375248
\(584\) −2.80802e8 −0.0583384
\(585\) 0 0
\(586\) 2.21076e8 0.0453838
\(587\) 6.09853e9 1.24449 0.622246 0.782822i \(-0.286221\pi\)
0.622246 + 0.782822i \(0.286221\pi\)
\(588\) 0 0
\(589\) 3.20966e8 0.0647226
\(590\) 3.29397e8 0.0660294
\(591\) 0 0
\(592\) −2.93369e9 −0.581151
\(593\) 6.67019e9 1.31355 0.656775 0.754087i \(-0.271920\pi\)
0.656775 + 0.754087i \(0.271920\pi\)
\(594\) 0 0
\(595\) 4.79598e9 0.933401
\(596\) −2.38607e9 −0.461659
\(597\) 0 0
\(598\) −2.18750e8 −0.0418306
\(599\) −2.30726e8 −0.0438635 −0.0219318 0.999759i \(-0.506982\pi\)
−0.0219318 + 0.999759i \(0.506982\pi\)
\(600\) 0 0
\(601\) 6.63912e9 1.24753 0.623763 0.781613i \(-0.285603\pi\)
0.623763 + 0.781613i \(0.285603\pi\)
\(602\) −8.77999e6 −0.00164024
\(603\) 0 0
\(604\) 1.48304e9 0.273856
\(605\) 4.45826e9 0.818506
\(606\) 0 0
\(607\) −4.67413e9 −0.848282 −0.424141 0.905596i \(-0.639424\pi\)
−0.424141 + 0.905596i \(0.639424\pi\)
\(608\) −2.63387e8 −0.0475261
\(609\) 0 0
\(610\) 3.53754e8 0.0631025
\(611\) −1.01341e10 −1.79739
\(612\) 0 0
\(613\) −1.56186e9 −0.273861 −0.136931 0.990581i \(-0.543724\pi\)
−0.136931 + 0.990581i \(0.543724\pi\)
\(614\) −1.85752e8 −0.0323850
\(615\) 0 0
\(616\) 9.41558e7 0.0162299
\(617\) −3.07047e9 −0.526268 −0.263134 0.964759i \(-0.584756\pi\)
−0.263134 + 0.964759i \(0.584756\pi\)
\(618\) 0 0
\(619\) −1.00411e9 −0.170162 −0.0850811 0.996374i \(-0.527115\pi\)
−0.0850811 + 0.996374i \(0.527115\pi\)
\(620\) 1.28429e9 0.216418
\(621\) 0 0
\(622\) 2.36308e7 0.00393743
\(623\) −9.92939e8 −0.164518
\(624\) 0 0
\(625\) 1.97461e10 3.23519
\(626\) −8.25305e7 −0.0134463
\(627\) 0 0
\(628\) −3.70242e9 −0.596523
\(629\) −4.62805e9 −0.741517
\(630\) 0 0
\(631\) −1.07127e9 −0.169745 −0.0848725 0.996392i \(-0.527048\pi\)
−0.0848725 + 0.996392i \(0.527048\pi\)
\(632\) −2.97700e8 −0.0469104
\(633\) 0 0
\(634\) −3.41727e8 −0.0532558
\(635\) 2.66299e9 0.412726
\(636\) 0 0
\(637\) 6.01928e9 0.922691
\(638\) 7.12681e7 0.0108648
\(639\) 0 0
\(640\) −1.40502e9 −0.211861
\(641\) −3.42898e9 −0.514236 −0.257118 0.966380i \(-0.582773\pi\)
−0.257118 + 0.966380i \(0.582773\pi\)
\(642\) 0 0
\(643\) 4.41718e9 0.655249 0.327624 0.944808i \(-0.393752\pi\)
0.327624 + 0.944808i \(0.393752\pi\)
\(644\) −3.61194e9 −0.532893
\(645\) 0 0
\(646\) −1.38323e8 −0.0201873
\(647\) 6.14010e9 0.891272 0.445636 0.895214i \(-0.352978\pi\)
0.445636 + 0.895214i \(0.352978\pi\)
\(648\) 0 0
\(649\) −6.53987e9 −0.939102
\(650\) 5.54218e8 0.0791559
\(651\) 0 0
\(652\) −9.97171e7 −0.0140897
\(653\) −7.83011e9 −1.10045 −0.550227 0.835015i \(-0.685459\pi\)
−0.550227 + 0.835015i \(0.685459\pi\)
\(654\) 0 0
\(655\) −2.22706e10 −3.09661
\(656\) 9.99039e9 1.38172
\(657\) 0 0
\(658\) 1.30360e8 0.0178383
\(659\) −9.87256e9 −1.34379 −0.671894 0.740647i \(-0.734519\pi\)
−0.671894 + 0.740647i \(0.734519\pi\)
\(660\) 0 0
\(661\) 1.73018e9 0.233017 0.116508 0.993190i \(-0.462830\pi\)
0.116508 + 0.993190i \(0.462830\pi\)
\(662\) 1.32753e8 0.0177845
\(663\) 0 0
\(664\) −3.05281e8 −0.0404680
\(665\) 3.16064e9 0.416773
\(666\) 0 0
\(667\) −5.47001e9 −0.713752
\(668\) −5.34823e9 −0.694213
\(669\) 0 0
\(670\) −1.36464e8 −0.0175289
\(671\) −7.02346e9 −0.897474
\(672\) 0 0
\(673\) 1.06458e10 1.34625 0.673127 0.739527i \(-0.264951\pi\)
0.673127 + 0.739527i \(0.264951\pi\)
\(674\) −3.72856e8 −0.0469063
\(675\) 0 0
\(676\) −1.40056e9 −0.174377
\(677\) −8.45984e9 −1.04786 −0.523928 0.851763i \(-0.675534\pi\)
−0.523928 + 0.851763i \(0.675534\pi\)
\(678\) 0 0
\(679\) −1.54050e9 −0.188851
\(680\) −1.10738e9 −0.135057
\(681\) 0 0
\(682\) 1.98646e7 0.00239792
\(683\) 7.82807e9 0.940117 0.470059 0.882635i \(-0.344233\pi\)
0.470059 + 0.882635i \(0.344233\pi\)
\(684\) 0 0
\(685\) 2.07298e10 2.46421
\(686\) −1.68373e8 −0.0199131
\(687\) 0 0
\(688\) −1.29960e9 −0.152143
\(689\) 4.62595e9 0.538808
\(690\) 0 0
\(691\) −8.92999e9 −1.02962 −0.514811 0.857304i \(-0.672138\pi\)
−0.514811 + 0.857304i \(0.672138\pi\)
\(692\) 1.27196e10 1.45916
\(693\) 0 0
\(694\) 3.28577e8 0.0373147
\(695\) −3.20828e10 −3.62515
\(696\) 0 0
\(697\) 1.57604e10 1.76300
\(698\) 2.07887e8 0.0231384
\(699\) 0 0
\(700\) 9.15110e9 1.00839
\(701\) −1.21758e10 −1.33501 −0.667504 0.744606i \(-0.732637\pi\)
−0.667504 + 0.744606i \(0.732637\pi\)
\(702\) 0 0
\(703\) −3.04997e9 −0.331095
\(704\) 6.95481e9 0.751244
\(705\) 0 0
\(706\) 9.57706e7 0.0102427
\(707\) −2.05849e9 −0.219069
\(708\) 0 0
\(709\) −1.53335e10 −1.61577 −0.807885 0.589340i \(-0.799388\pi\)
−0.807885 + 0.589340i \(0.799388\pi\)
\(710\) −2.95830e8 −0.0310197
\(711\) 0 0
\(712\) 2.29267e8 0.0238047
\(713\) −1.52466e9 −0.157528
\(714\) 0 0
\(715\) −1.52068e10 −1.55584
\(716\) 9.99282e9 1.01740
\(717\) 0 0
\(718\) 4.48974e8 0.0452674
\(719\) −3.82428e8 −0.0383706 −0.0191853 0.999816i \(-0.506107\pi\)
−0.0191853 + 0.999816i \(0.506107\pi\)
\(720\) 0 0
\(721\) −2.44483e8 −0.0242926
\(722\) 1.91002e8 0.0188868
\(723\) 0 0
\(724\) −1.56456e9 −0.153217
\(725\) 1.38586e10 1.35063
\(726\) 0 0
\(727\) −3.79482e9 −0.366286 −0.183143 0.983086i \(-0.558627\pi\)
−0.183143 + 0.983086i \(0.558627\pi\)
\(728\) 2.42600e8 0.0233040
\(729\) 0 0
\(730\) −5.83375e8 −0.0555032
\(731\) −2.05019e9 −0.194126
\(732\) 0 0
\(733\) −1.11789e10 −1.04842 −0.524211 0.851588i \(-0.675640\pi\)
−0.524211 + 0.851588i \(0.675640\pi\)
\(734\) −1.87245e8 −0.0174773
\(735\) 0 0
\(736\) 1.25114e9 0.115674
\(737\) 2.70936e9 0.249305
\(738\) 0 0
\(739\) −1.77973e10 −1.62218 −0.811091 0.584921i \(-0.801126\pi\)
−0.811091 + 0.584921i \(0.801126\pi\)
\(740\) −1.22040e10 −1.10711
\(741\) 0 0
\(742\) −5.95058e7 −0.00534744
\(743\) −2.77646e7 −0.00248331 −0.00124165 0.999999i \(-0.500395\pi\)
−0.00124165 + 0.999999i \(0.500395\pi\)
\(744\) 0 0
\(745\) −9.91814e9 −0.878786
\(746\) 1.07794e8 0.00950627
\(747\) 0 0
\(748\) 1.09887e10 0.960046
\(749\) −9.04912e8 −0.0786901
\(750\) 0 0
\(751\) −1.10248e10 −0.949798 −0.474899 0.880040i \(-0.657515\pi\)
−0.474899 + 0.880040i \(0.657515\pi\)
\(752\) 1.92957e10 1.65462
\(753\) 0 0
\(754\) 1.83628e8 0.0156005
\(755\) 6.16452e9 0.521296
\(756\) 0 0
\(757\) −2.07834e10 −1.74133 −0.870664 0.491879i \(-0.836310\pi\)
−0.870664 + 0.491879i \(0.836310\pi\)
\(758\) 3.84434e8 0.0320612
\(759\) 0 0
\(760\) −7.29784e8 −0.0603041
\(761\) −8.64340e9 −0.710949 −0.355474 0.934686i \(-0.615681\pi\)
−0.355474 + 0.934686i \(0.615681\pi\)
\(762\) 0 0
\(763\) −4.37087e9 −0.356231
\(764\) −1.23213e10 −0.999609
\(765\) 0 0
\(766\) 3.83996e8 0.0308692
\(767\) −1.68505e10 −1.34843
\(768\) 0 0
\(769\) 2.17566e9 0.172523 0.0862617 0.996273i \(-0.472508\pi\)
0.0862617 + 0.996273i \(0.472508\pi\)
\(770\) 1.95612e8 0.0154411
\(771\) 0 0
\(772\) −3.69654e9 −0.289158
\(773\) −5.58417e9 −0.434841 −0.217421 0.976078i \(-0.569764\pi\)
−0.217421 + 0.976078i \(0.569764\pi\)
\(774\) 0 0
\(775\) 3.86282e9 0.298091
\(776\) 3.55698e8 0.0273253
\(777\) 0 0
\(778\) −7.03948e6 −0.000535935 0
\(779\) 1.03864e10 0.787195
\(780\) 0 0
\(781\) 5.87343e9 0.441177
\(782\) 6.57061e8 0.0491340
\(783\) 0 0
\(784\) −1.14609e10 −0.849400
\(785\) −1.53898e10 −1.13551
\(786\) 0 0
\(787\) −7.26404e9 −0.531211 −0.265605 0.964082i \(-0.585572\pi\)
−0.265605 + 0.964082i \(0.585572\pi\)
\(788\) 5.34360e9 0.389038
\(789\) 0 0
\(790\) −6.18481e8 −0.0446305
\(791\) 6.75085e9 0.484999
\(792\) 0 0
\(793\) −1.80965e10 −1.28866
\(794\) 2.63076e8 0.0186514
\(795\) 0 0
\(796\) −7.08654e9 −0.498011
\(797\) −1.81556e10 −1.27030 −0.635150 0.772389i \(-0.719062\pi\)
−0.635150 + 0.772389i \(0.719062\pi\)
\(798\) 0 0
\(799\) 3.04400e10 2.11120
\(800\) −3.16986e9 −0.218890
\(801\) 0 0
\(802\) −4.05058e8 −0.0277272
\(803\) 1.15824e10 0.789393
\(804\) 0 0
\(805\) −1.50137e10 −1.01438
\(806\) 5.11826e7 0.00344310
\(807\) 0 0
\(808\) 4.75301e8 0.0316978
\(809\) 2.32465e10 1.54361 0.771806 0.635858i \(-0.219353\pi\)
0.771806 + 0.635858i \(0.219353\pi\)
\(810\) 0 0
\(811\) −1.13477e10 −0.747024 −0.373512 0.927625i \(-0.621847\pi\)
−0.373512 + 0.927625i \(0.621847\pi\)
\(812\) 3.03201e9 0.198740
\(813\) 0 0
\(814\) −1.88763e8 −0.0122668
\(815\) −4.14493e8 −0.0268204
\(816\) 0 0
\(817\) −1.35111e9 −0.0866790
\(818\) −1.25580e8 −0.00802202
\(819\) 0 0
\(820\) 4.15593e10 2.63221
\(821\) 6.60382e9 0.416480 0.208240 0.978078i \(-0.433226\pi\)
0.208240 + 0.978078i \(0.433226\pi\)
\(822\) 0 0
\(823\) −3.82454e9 −0.239155 −0.119577 0.992825i \(-0.538154\pi\)
−0.119577 + 0.992825i \(0.538154\pi\)
\(824\) 5.64505e7 0.00351498
\(825\) 0 0
\(826\) 2.16756e8 0.0133826
\(827\) −1.37532e10 −0.845541 −0.422770 0.906237i \(-0.638942\pi\)
−0.422770 + 0.906237i \(0.638942\pi\)
\(828\) 0 0
\(829\) 1.83250e10 1.11713 0.558563 0.829462i \(-0.311353\pi\)
0.558563 + 0.829462i \(0.311353\pi\)
\(830\) −6.34232e8 −0.0385012
\(831\) 0 0
\(832\) 1.79196e10 1.07869
\(833\) −1.80802e10 −1.08379
\(834\) 0 0
\(835\) −2.22309e10 −1.32146
\(836\) 7.24178e9 0.428670
\(837\) 0 0
\(838\) −5.80329e8 −0.0340659
\(839\) −1.70662e10 −0.997630 −0.498815 0.866708i \(-0.666231\pi\)
−0.498815 + 0.866708i \(0.666231\pi\)
\(840\) 0 0
\(841\) −1.26581e10 −0.733809
\(842\) −9.44478e8 −0.0545255
\(843\) 0 0
\(844\) 1.20763e10 0.691408
\(845\) −5.82169e9 −0.331933
\(846\) 0 0
\(847\) 2.93371e9 0.165892
\(848\) −8.80796e9 −0.496010
\(849\) 0 0
\(850\) −1.66471e9 −0.0929762
\(851\) 1.44880e10 0.805852
\(852\) 0 0
\(853\) −6.70515e9 −0.369902 −0.184951 0.982748i \(-0.559213\pi\)
−0.184951 + 0.982748i \(0.559213\pi\)
\(854\) 2.32784e8 0.0127894
\(855\) 0 0
\(856\) 2.08942e8 0.0113859
\(857\) −2.45826e9 −0.133412 −0.0667061 0.997773i \(-0.521249\pi\)
−0.0667061 + 0.997773i \(0.521249\pi\)
\(858\) 0 0
\(859\) −2.04474e10 −1.10068 −0.550340 0.834941i \(-0.685502\pi\)
−0.550340 + 0.834941i \(0.685502\pi\)
\(860\) −5.40625e9 −0.289835
\(861\) 0 0
\(862\) 2.37581e7 0.00126338
\(863\) 2.86480e10 1.51725 0.758624 0.651529i \(-0.225872\pi\)
0.758624 + 0.651529i \(0.225872\pi\)
\(864\) 0 0
\(865\) 5.28713e10 2.77756
\(866\) −7.47457e8 −0.0391087
\(867\) 0 0
\(868\) 8.45114e8 0.0438628
\(869\) 1.22794e10 0.634757
\(870\) 0 0
\(871\) 6.98087e9 0.357970
\(872\) 1.00922e9 0.0515442
\(873\) 0 0
\(874\) 4.33015e8 0.0219388
\(875\) 2.35078e10 1.18627
\(876\) 0 0
\(877\) 2.41157e10 1.20726 0.603631 0.797264i \(-0.293720\pi\)
0.603631 + 0.797264i \(0.293720\pi\)
\(878\) −6.26593e8 −0.0312432
\(879\) 0 0
\(880\) 2.89542e10 1.43226
\(881\) −8.73417e9 −0.430334 −0.215167 0.976577i \(-0.569030\pi\)
−0.215167 + 0.976577i \(0.569030\pi\)
\(882\) 0 0
\(883\) −1.22297e10 −0.597795 −0.298898 0.954285i \(-0.596619\pi\)
−0.298898 + 0.954285i \(0.596619\pi\)
\(884\) 2.83133e10 1.37850
\(885\) 0 0
\(886\) −7.76690e8 −0.0375171
\(887\) 3.43901e9 0.165463 0.0827315 0.996572i \(-0.473636\pi\)
0.0827315 + 0.996572i \(0.473636\pi\)
\(888\) 0 0
\(889\) 1.75235e9 0.0836497
\(890\) 4.76310e8 0.0226477
\(891\) 0 0
\(892\) −1.34221e9 −0.0633204
\(893\) 2.00605e10 0.942674
\(894\) 0 0
\(895\) 4.15370e10 1.93667
\(896\) −9.24556e8 −0.0429393
\(897\) 0 0
\(898\) −4.84266e8 −0.0223160
\(899\) 1.27986e9 0.0587495
\(900\) 0 0
\(901\) −1.38950e10 −0.632882
\(902\) 6.42811e8 0.0291649
\(903\) 0 0
\(904\) −1.55875e9 −0.0701759
\(905\) −6.50340e9 −0.291655
\(906\) 0 0
\(907\) 2.11330e10 0.940451 0.470226 0.882546i \(-0.344173\pi\)
0.470226 + 0.882546i \(0.344173\pi\)
\(908\) 3.10228e9 0.137524
\(909\) 0 0
\(910\) 5.04009e8 0.0221714
\(911\) −3.25882e10 −1.42806 −0.714029 0.700116i \(-0.753131\pi\)
−0.714029 + 0.700116i \(0.753131\pi\)
\(912\) 0 0
\(913\) 1.25921e10 0.547583
\(914\) −5.16757e8 −0.0223859
\(915\) 0 0
\(916\) −9.21636e9 −0.396210
\(917\) −1.46549e10 −0.627610
\(918\) 0 0
\(919\) 2.90516e10 1.23471 0.617356 0.786684i \(-0.288204\pi\)
0.617356 + 0.786684i \(0.288204\pi\)
\(920\) 3.46663e9 0.146774
\(921\) 0 0
\(922\) 1.13489e9 0.0476865
\(923\) 1.51333e10 0.633474
\(924\) 0 0
\(925\) −3.67063e10 −1.52491
\(926\) 1.04777e9 0.0433639
\(927\) 0 0
\(928\) −1.05026e9 −0.0431400
\(929\) −5.17888e9 −0.211924 −0.105962 0.994370i \(-0.533792\pi\)
−0.105962 + 0.994370i \(0.533792\pi\)
\(930\) 0 0
\(931\) −1.19152e10 −0.483922
\(932\) 7.93642e9 0.321121
\(933\) 0 0
\(934\) 6.96715e8 0.0279796
\(935\) 4.56767e10 1.82749
\(936\) 0 0
\(937\) 5.65702e9 0.224646 0.112323 0.993672i \(-0.464171\pi\)
0.112323 + 0.993672i \(0.464171\pi\)
\(938\) −8.97982e7 −0.00355269
\(939\) 0 0
\(940\) 8.02687e10 3.15209
\(941\) −1.93533e10 −0.757168 −0.378584 0.925567i \(-0.623589\pi\)
−0.378584 + 0.925567i \(0.623589\pi\)
\(942\) 0 0
\(943\) −4.93374e10 −1.91595
\(944\) 3.20838e10 1.24132
\(945\) 0 0
\(946\) −8.36202e7 −0.00321139
\(947\) −2.30104e10 −0.880438 −0.440219 0.897890i \(-0.645099\pi\)
−0.440219 + 0.897890i \(0.645099\pi\)
\(948\) 0 0
\(949\) 2.98429e10 1.13347
\(950\) −1.09707e9 −0.0415148
\(951\) 0 0
\(952\) −7.28699e8 −0.0273728
\(953\) 1.01231e10 0.378867 0.189434 0.981894i \(-0.439335\pi\)
0.189434 + 0.981894i \(0.439335\pi\)
\(954\) 0 0
\(955\) −5.12158e10 −1.90280
\(956\) 1.34390e10 0.497465
\(957\) 0 0
\(958\) 3.48801e8 0.0128174
\(959\) 1.36410e10 0.499437
\(960\) 0 0
\(961\) −2.71559e10 −0.987034
\(962\) −4.86362e8 −0.0176135
\(963\) 0 0
\(964\) −3.86608e10 −1.38996
\(965\) −1.53654e10 −0.550424
\(966\) 0 0
\(967\) 6.28463e8 0.0223505 0.0111752 0.999938i \(-0.496443\pi\)
0.0111752 + 0.999938i \(0.496443\pi\)
\(968\) −6.77386e8 −0.0240034
\(969\) 0 0
\(970\) 7.38975e8 0.0259973
\(971\) −2.05584e10 −0.720645 −0.360322 0.932828i \(-0.617333\pi\)
−0.360322 + 0.932828i \(0.617333\pi\)
\(972\) 0 0
\(973\) −2.11117e10 −0.734732
\(974\) −4.40684e8 −0.0152817
\(975\) 0 0
\(976\) 3.44563e10 1.18630
\(977\) −4.09891e10 −1.40617 −0.703084 0.711107i \(-0.748194\pi\)
−0.703084 + 0.711107i \(0.748194\pi\)
\(978\) 0 0
\(979\) −9.45670e9 −0.322107
\(980\) −4.76765e10 −1.61813
\(981\) 0 0
\(982\) −1.36461e9 −0.0459853
\(983\) −3.77468e10 −1.26748 −0.633742 0.773544i \(-0.718482\pi\)
−0.633742 + 0.773544i \(0.718482\pi\)
\(984\) 0 0
\(985\) 2.22117e10 0.740551
\(986\) −5.51565e8 −0.0183243
\(987\) 0 0
\(988\) 1.86590e10 0.615515
\(989\) 6.41806e9 0.210968
\(990\) 0 0
\(991\) −1.40838e10 −0.459688 −0.229844 0.973227i \(-0.573822\pi\)
−0.229844 + 0.973227i \(0.573822\pi\)
\(992\) −2.92740e8 −0.00952119
\(993\) 0 0
\(994\) −1.94667e8 −0.00628696
\(995\) −2.94565e10 −0.947984
\(996\) 0 0
\(997\) 8.84518e9 0.282666 0.141333 0.989962i \(-0.454861\pi\)
0.141333 + 0.989962i \(0.454861\pi\)
\(998\) −1.60951e8 −0.00512550
\(999\) 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 387.8.a.d.1.7 13
3.2 odd 2 43.8.a.b.1.7 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.7 13 3.2 odd 2
387.8.a.d.1.7 13 1.1 even 1 trivial