Properties

Label 387.8.a.d.1.1
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 1.1
Root \(21.1822\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q-22.1822 q^{2} +364.050 q^{4} -122.553 q^{5} -90.8173 q^{7} -5236.10 q^{8} +O(q^{10})\) \(q-22.1822 q^{2} +364.050 q^{4} -122.553 q^{5} -90.8173 q^{7} -5236.10 q^{8} +2718.48 q^{10} -3410.31 q^{11} -11275.0 q^{13} +2014.53 q^{14} +69549.8 q^{16} -9720.37 q^{17} +25351.7 q^{19} -44615.2 q^{20} +75648.2 q^{22} +52795.4 q^{23} -63105.9 q^{25} +250104. q^{26} -33062.0 q^{28} +216763. q^{29} +142113. q^{31} -872545. q^{32} +215619. q^{34} +11129.9 q^{35} -74290.8 q^{37} -562357. q^{38} +641697. q^{40} +273026. q^{41} -79507.0 q^{43} -1.24152e6 q^{44} -1.17112e6 q^{46} +112553. q^{47} -815295. q^{49} +1.39983e6 q^{50} -4.10466e6 q^{52} -1.41102e6 q^{53} +417942. q^{55} +475528. q^{56} -4.80827e6 q^{58} +1.44713e6 q^{59} +3.19474e6 q^{61} -3.15239e6 q^{62} +1.04526e7 q^{64} +1.38178e6 q^{65} -3.52359e6 q^{67} -3.53870e6 q^{68} -246885. q^{70} +327344. q^{71} +2.80064e6 q^{73} +1.64793e6 q^{74} +9.22928e6 q^{76} +309715. q^{77} +3.35723e6 q^{79} -8.52350e6 q^{80} -6.05633e6 q^{82} -7.10311e6 q^{83} +1.19126e6 q^{85} +1.76364e6 q^{86} +1.78567e7 q^{88} +1.32585e6 q^{89} +1.02396e6 q^{91} +1.92201e7 q^{92} -2.49668e6 q^{94} -3.10692e6 q^{95} +1.29568e7 q^{97} +1.80850e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −22.1822 −1.96065 −0.980324 0.197397i \(-0.936751\pi\)
−0.980324 + 0.197397i \(0.936751\pi\)
\(3\) 0 0
\(4\) 364.050 2.84414
\(5\) −122.553 −0.438457 −0.219229 0.975674i \(-0.570354\pi\)
−0.219229 + 0.975674i \(0.570354\pi\)
\(6\) 0 0
\(7\) −90.8173 −0.100075 −0.0500375 0.998747i \(-0.515934\pi\)
−0.0500375 + 0.998747i \(0.515934\pi\)
\(8\) −5236.10 −3.61570
\(9\) 0 0
\(10\) 2718.48 0.859660
\(11\) −3410.31 −0.772538 −0.386269 0.922386i \(-0.626236\pi\)
−0.386269 + 0.922386i \(0.626236\pi\)
\(12\) 0 0
\(13\) −11275.0 −1.42336 −0.711680 0.702504i \(-0.752065\pi\)
−0.711680 + 0.702504i \(0.752065\pi\)
\(14\) 2014.53 0.196212
\(15\) 0 0
\(16\) 69549.8 4.24498
\(17\) −9720.37 −0.479857 −0.239928 0.970791i \(-0.577124\pi\)
−0.239928 + 0.970791i \(0.577124\pi\)
\(18\) 0 0
\(19\) 25351.7 0.847949 0.423975 0.905674i \(-0.360635\pi\)
0.423975 + 0.905674i \(0.360635\pi\)
\(20\) −44615.2 −1.24703
\(21\) 0 0
\(22\) 75648.2 1.51467
\(23\) 52795.4 0.904792 0.452396 0.891817i \(-0.350569\pi\)
0.452396 + 0.891817i \(0.350569\pi\)
\(24\) 0 0
\(25\) −63105.9 −0.807755
\(26\) 250104. 2.79071
\(27\) 0 0
\(28\) −33062.0 −0.284627
\(29\) 216763. 1.65041 0.825204 0.564835i \(-0.191060\pi\)
0.825204 + 0.564835i \(0.191060\pi\)
\(30\) 0 0
\(31\) 142113. 0.856780 0.428390 0.903594i \(-0.359081\pi\)
0.428390 + 0.903594i \(0.359081\pi\)
\(32\) −872545. −4.70721
\(33\) 0 0
\(34\) 215619. 0.940830
\(35\) 11129.9 0.0438786
\(36\) 0 0
\(37\) −74290.8 −0.241118 −0.120559 0.992706i \(-0.538469\pi\)
−0.120559 + 0.992706i \(0.538469\pi\)
\(38\) −562357. −1.66253
\(39\) 0 0
\(40\) 641697. 1.58533
\(41\) 273026. 0.618673 0.309337 0.950953i \(-0.399893\pi\)
0.309337 + 0.950953i \(0.399893\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) −1.24152e6 −2.19720
\(45\) 0 0
\(46\) −1.17112e6 −1.77398
\(47\) 112553. 0.158130 0.0790652 0.996869i \(-0.474806\pi\)
0.0790652 + 0.996869i \(0.474806\pi\)
\(48\) 0 0
\(49\) −815295. −0.989985
\(50\) 1.39983e6 1.58372
\(51\) 0 0
\(52\) −4.10466e6 −4.04823
\(53\) −1.41102e6 −1.30187 −0.650936 0.759132i \(-0.725624\pi\)
−0.650936 + 0.759132i \(0.725624\pi\)
\(54\) 0 0
\(55\) 417942. 0.338725
\(56\) 475528. 0.361841
\(57\) 0 0
\(58\) −4.80827e6 −3.23587
\(59\) 1.44713e6 0.917329 0.458665 0.888609i \(-0.348328\pi\)
0.458665 + 0.888609i \(0.348328\pi\)
\(60\) 0 0
\(61\) 3.19474e6 1.80211 0.901054 0.433708i \(-0.142795\pi\)
0.901054 + 0.433708i \(0.142795\pi\)
\(62\) −3.15239e6 −1.67984
\(63\) 0 0
\(64\) 1.04526e7 4.98419
\(65\) 1.38178e6 0.624083
\(66\) 0 0
\(67\) −3.52359e6 −1.43128 −0.715638 0.698472i \(-0.753864\pi\)
−0.715638 + 0.698472i \(0.753864\pi\)
\(68\) −3.53870e6 −1.36478
\(69\) 0 0
\(70\) −246885. −0.0860304
\(71\) 327344. 0.108542 0.0542712 0.998526i \(-0.482716\pi\)
0.0542712 + 0.998526i \(0.482716\pi\)
\(72\) 0 0
\(73\) 2.80064e6 0.842613 0.421306 0.906918i \(-0.361572\pi\)
0.421306 + 0.906918i \(0.361572\pi\)
\(74\) 1.64793e6 0.472746
\(75\) 0 0
\(76\) 9.22928e6 2.41168
\(77\) 309715. 0.0773117
\(78\) 0 0
\(79\) 3.35723e6 0.766101 0.383051 0.923727i \(-0.374874\pi\)
0.383051 + 0.923727i \(0.374874\pi\)
\(80\) −8.52350e6 −1.86124
\(81\) 0 0
\(82\) −6.05633e6 −1.21300
\(83\) −7.10311e6 −1.36356 −0.681781 0.731556i \(-0.738794\pi\)
−0.681781 + 0.731556i \(0.738794\pi\)
\(84\) 0 0
\(85\) 1.19126e6 0.210397
\(86\) 1.76364e6 0.298996
\(87\) 0 0
\(88\) 1.78567e7 2.79327
\(89\) 1.32585e6 0.199356 0.0996778 0.995020i \(-0.468219\pi\)
0.0996778 + 0.995020i \(0.468219\pi\)
\(90\) 0 0
\(91\) 1.02396e6 0.142443
\(92\) 1.92201e7 2.57335
\(93\) 0 0
\(94\) −2.49668e6 −0.310038
\(95\) −3.10692e6 −0.371789
\(96\) 0 0
\(97\) 1.29568e7 1.44144 0.720718 0.693228i \(-0.243812\pi\)
0.720718 + 0.693228i \(0.243812\pi\)
\(98\) 1.80850e7 1.94101
\(99\) 0 0
\(100\) −2.29737e7 −2.29737
\(101\) −309915. −0.0299308 −0.0149654 0.999888i \(-0.504764\pi\)
−0.0149654 + 0.999888i \(0.504764\pi\)
\(102\) 0 0
\(103\) 1.99810e7 1.80172 0.900858 0.434113i \(-0.142938\pi\)
0.900858 + 0.434113i \(0.142938\pi\)
\(104\) 5.90370e7 5.14645
\(105\) 0 0
\(106\) 3.12996e7 2.55251
\(107\) −1.77323e6 −0.139934 −0.0699669 0.997549i \(-0.522289\pi\)
−0.0699669 + 0.997549i \(0.522289\pi\)
\(108\) 0 0
\(109\) 7.12031e6 0.526631 0.263315 0.964710i \(-0.415184\pi\)
0.263315 + 0.964710i \(0.415184\pi\)
\(110\) −9.27088e6 −0.664120
\(111\) 0 0
\(112\) −6.31632e6 −0.424816
\(113\) 9.47406e6 0.617677 0.308839 0.951114i \(-0.400060\pi\)
0.308839 + 0.951114i \(0.400060\pi\)
\(114\) 0 0
\(115\) −6.47021e6 −0.396713
\(116\) 7.89123e7 4.69399
\(117\) 0 0
\(118\) −3.21005e7 −1.79856
\(119\) 882778. 0.0480216
\(120\) 0 0
\(121\) −7.85694e6 −0.403185
\(122\) −7.08663e7 −3.53330
\(123\) 0 0
\(124\) 5.17363e7 2.43680
\(125\) 1.73082e7 0.792623
\(126\) 0 0
\(127\) 2.94089e7 1.27399 0.636994 0.770868i \(-0.280177\pi\)
0.636994 + 0.770868i \(0.280177\pi\)
\(128\) −1.20176e8 −5.06503
\(129\) 0 0
\(130\) −3.06509e7 −1.22361
\(131\) −4.80770e7 −1.86848 −0.934238 0.356650i \(-0.883919\pi\)
−0.934238 + 0.356650i \(0.883919\pi\)
\(132\) 0 0
\(133\) −2.30237e6 −0.0848585
\(134\) 7.81609e7 2.80623
\(135\) 0 0
\(136\) 5.08968e7 1.73502
\(137\) −5.87744e6 −0.195284 −0.0976419 0.995222i \(-0.531130\pi\)
−0.0976419 + 0.995222i \(0.531130\pi\)
\(138\) 0 0
\(139\) −5.12513e7 −1.61865 −0.809326 0.587360i \(-0.800167\pi\)
−0.809326 + 0.587360i \(0.800167\pi\)
\(140\) 4.05183e6 0.124797
\(141\) 0 0
\(142\) −7.26120e6 −0.212813
\(143\) 3.84513e7 1.09960
\(144\) 0 0
\(145\) −2.65648e7 −0.723633
\(146\) −6.21244e7 −1.65207
\(147\) 0 0
\(148\) −2.70455e7 −0.685771
\(149\) 6.91179e6 0.171174 0.0855871 0.996331i \(-0.472723\pi\)
0.0855871 + 0.996331i \(0.472723\pi\)
\(150\) 0 0
\(151\) −6.97335e7 −1.64825 −0.824124 0.566410i \(-0.808332\pi\)
−0.824124 + 0.566410i \(0.808332\pi\)
\(152\) −1.32744e8 −3.06593
\(153\) 0 0
\(154\) −6.87016e6 −0.151581
\(155\) −1.74164e7 −0.375661
\(156\) 0 0
\(157\) −6.10206e7 −1.25843 −0.629213 0.777233i \(-0.716623\pi\)
−0.629213 + 0.777233i \(0.716623\pi\)
\(158\) −7.44707e7 −1.50205
\(159\) 0 0
\(160\) 1.06933e8 2.06391
\(161\) −4.79474e6 −0.0905470
\(162\) 0 0
\(163\) −1.28985e6 −0.0233282 −0.0116641 0.999932i \(-0.503713\pi\)
−0.0116641 + 0.999932i \(0.503713\pi\)
\(164\) 9.93952e7 1.75959
\(165\) 0 0
\(166\) 1.57563e8 2.67347
\(167\) 8.25906e7 1.37222 0.686109 0.727499i \(-0.259317\pi\)
0.686109 + 0.727499i \(0.259317\pi\)
\(168\) 0 0
\(169\) 6.43771e7 1.02595
\(170\) −2.64247e7 −0.412514
\(171\) 0 0
\(172\) −2.89445e7 −0.433727
\(173\) 6.06382e6 0.0890400 0.0445200 0.999008i \(-0.485824\pi\)
0.0445200 + 0.999008i \(0.485824\pi\)
\(174\) 0 0
\(175\) 5.73110e6 0.0808361
\(176\) −2.37186e8 −3.27941
\(177\) 0 0
\(178\) −2.94102e7 −0.390866
\(179\) 5.06661e7 0.660286 0.330143 0.943931i \(-0.392903\pi\)
0.330143 + 0.943931i \(0.392903\pi\)
\(180\) 0 0
\(181\) −2.08989e6 −0.0261968 −0.0130984 0.999914i \(-0.504169\pi\)
−0.0130984 + 0.999914i \(0.504169\pi\)
\(182\) −2.27138e7 −0.279280
\(183\) 0 0
\(184\) −2.76442e8 −3.27146
\(185\) 9.10452e6 0.105720
\(186\) 0 0
\(187\) 3.31495e7 0.370707
\(188\) 4.09749e7 0.449744
\(189\) 0 0
\(190\) 6.89182e7 0.728948
\(191\) −1.78113e8 −1.84960 −0.924800 0.380454i \(-0.875768\pi\)
−0.924800 + 0.380454i \(0.875768\pi\)
\(192\) 0 0
\(193\) 1.26757e8 1.26917 0.634586 0.772852i \(-0.281170\pi\)
0.634586 + 0.772852i \(0.281170\pi\)
\(194\) −2.87410e8 −2.82615
\(195\) 0 0
\(196\) −2.96808e8 −2.81565
\(197\) 1.27815e8 1.19110 0.595550 0.803318i \(-0.296934\pi\)
0.595550 + 0.803318i \(0.296934\pi\)
\(198\) 0 0
\(199\) −1.29981e8 −1.16921 −0.584606 0.811318i \(-0.698751\pi\)
−0.584606 + 0.811318i \(0.698751\pi\)
\(200\) 3.30429e8 2.92060
\(201\) 0 0
\(202\) 6.87459e6 0.0586837
\(203\) −1.96858e7 −0.165164
\(204\) 0 0
\(205\) −3.34601e7 −0.271262
\(206\) −4.43222e8 −3.53253
\(207\) 0 0
\(208\) −7.84173e8 −6.04214
\(209\) −8.64573e7 −0.655073
\(210\) 0 0
\(211\) −1.85689e8 −1.36081 −0.680406 0.732836i \(-0.738196\pi\)
−0.680406 + 0.732836i \(0.738196\pi\)
\(212\) −5.13682e8 −3.70270
\(213\) 0 0
\(214\) 3.93342e7 0.274361
\(215\) 9.74378e6 0.0668641
\(216\) 0 0
\(217\) −1.29064e7 −0.0857422
\(218\) −1.57944e8 −1.03254
\(219\) 0 0
\(220\) 1.52152e8 0.963380
\(221\) 1.09597e8 0.683009
\(222\) 0 0
\(223\) 1.57910e7 0.0953545 0.0476773 0.998863i \(-0.484818\pi\)
0.0476773 + 0.998863i \(0.484818\pi\)
\(224\) 7.92422e7 0.471073
\(225\) 0 0
\(226\) −2.10155e8 −1.21105
\(227\) −3.37705e7 −0.191623 −0.0958113 0.995400i \(-0.530545\pi\)
−0.0958113 + 0.995400i \(0.530545\pi\)
\(228\) 0 0
\(229\) 2.60087e8 1.43118 0.715590 0.698520i \(-0.246158\pi\)
0.715590 + 0.698520i \(0.246158\pi\)
\(230\) 1.43523e8 0.777813
\(231\) 0 0
\(232\) −1.13499e9 −5.96738
\(233\) −1.45083e8 −0.751398 −0.375699 0.926742i \(-0.622597\pi\)
−0.375699 + 0.926742i \(0.622597\pi\)
\(234\) 0 0
\(235\) −1.37937e7 −0.0693334
\(236\) 5.26827e8 2.60901
\(237\) 0 0
\(238\) −1.95819e7 −0.0941535
\(239\) 1.55295e8 0.735808 0.367904 0.929864i \(-0.380076\pi\)
0.367904 + 0.929864i \(0.380076\pi\)
\(240\) 0 0
\(241\) 3.22999e8 1.48642 0.743211 0.669057i \(-0.233302\pi\)
0.743211 + 0.669057i \(0.233302\pi\)
\(242\) 1.74284e8 0.790504
\(243\) 0 0
\(244\) 1.16304e9 5.12544
\(245\) 9.99165e7 0.434066
\(246\) 0 0
\(247\) −2.85841e8 −1.20694
\(248\) −7.44119e8 −3.09786
\(249\) 0 0
\(250\) −3.83934e8 −1.55405
\(251\) 2.35414e8 0.939668 0.469834 0.882755i \(-0.344314\pi\)
0.469834 + 0.882755i \(0.344314\pi\)
\(252\) 0 0
\(253\) −1.80049e8 −0.698986
\(254\) −6.52354e8 −2.49784
\(255\) 0 0
\(256\) 1.32783e9 4.94655
\(257\) 4.25190e7 0.156249 0.0781245 0.996944i \(-0.475107\pi\)
0.0781245 + 0.996944i \(0.475107\pi\)
\(258\) 0 0
\(259\) 6.74689e6 0.0241298
\(260\) 5.03036e8 1.77498
\(261\) 0 0
\(262\) 1.06645e9 3.66342
\(263\) −1.04522e8 −0.354292 −0.177146 0.984185i \(-0.556686\pi\)
−0.177146 + 0.984185i \(0.556686\pi\)
\(264\) 0 0
\(265\) 1.72924e8 0.570815
\(266\) 5.10717e7 0.166378
\(267\) 0 0
\(268\) −1.28276e9 −4.07074
\(269\) 1.86374e8 0.583784 0.291892 0.956451i \(-0.405715\pi\)
0.291892 + 0.956451i \(0.405715\pi\)
\(270\) 0 0
\(271\) 3.37243e8 1.02932 0.514660 0.857394i \(-0.327918\pi\)
0.514660 + 0.857394i \(0.327918\pi\)
\(272\) −6.76049e8 −2.03698
\(273\) 0 0
\(274\) 1.30375e8 0.382883
\(275\) 2.15211e8 0.624022
\(276\) 0 0
\(277\) −4.22893e8 −1.19550 −0.597752 0.801681i \(-0.703939\pi\)
−0.597752 + 0.801681i \(0.703939\pi\)
\(278\) 1.13687e9 3.17360
\(279\) 0 0
\(280\) −5.82772e7 −0.158652
\(281\) −3.41588e8 −0.918396 −0.459198 0.888334i \(-0.651863\pi\)
−0.459198 + 0.888334i \(0.651863\pi\)
\(282\) 0 0
\(283\) −1.93785e8 −0.508238 −0.254119 0.967173i \(-0.581785\pi\)
−0.254119 + 0.967173i \(0.581785\pi\)
\(284\) 1.19169e8 0.308710
\(285\) 0 0
\(286\) −8.52934e8 −2.15593
\(287\) −2.47955e7 −0.0619137
\(288\) 0 0
\(289\) −3.15853e8 −0.769738
\(290\) 5.89265e8 1.41879
\(291\) 0 0
\(292\) 1.01957e9 2.39651
\(293\) −4.63942e8 −1.07752 −0.538762 0.842458i \(-0.681108\pi\)
−0.538762 + 0.842458i \(0.681108\pi\)
\(294\) 0 0
\(295\) −1.77349e8 −0.402210
\(296\) 3.88994e8 0.871809
\(297\) 0 0
\(298\) −1.53319e8 −0.335612
\(299\) −5.95268e8 −1.28785
\(300\) 0 0
\(301\) 7.22061e6 0.0152613
\(302\) 1.54684e9 3.23163
\(303\) 0 0
\(304\) 1.76321e9 3.59953
\(305\) −3.91523e8 −0.790147
\(306\) 0 0
\(307\) 1.10340e8 0.217644 0.108822 0.994061i \(-0.465292\pi\)
0.108822 + 0.994061i \(0.465292\pi\)
\(308\) 1.12752e8 0.219885
\(309\) 0 0
\(310\) 3.86333e8 0.736539
\(311\) 1.65769e8 0.312494 0.156247 0.987718i \(-0.450060\pi\)
0.156247 + 0.987718i \(0.450060\pi\)
\(312\) 0 0
\(313\) −1.02978e9 −1.89818 −0.949091 0.315003i \(-0.897994\pi\)
−0.949091 + 0.315003i \(0.897994\pi\)
\(314\) 1.35357e9 2.46733
\(315\) 0 0
\(316\) 1.22220e9 2.17890
\(317\) −6.40081e8 −1.12857 −0.564284 0.825581i \(-0.690848\pi\)
−0.564284 + 0.825581i \(0.690848\pi\)
\(318\) 0 0
\(319\) −7.39228e8 −1.27500
\(320\) −1.28099e9 −2.18535
\(321\) 0 0
\(322\) 1.06358e8 0.177531
\(323\) −2.46428e8 −0.406894
\(324\) 0 0
\(325\) 7.11519e8 1.14973
\(326\) 2.86116e7 0.0457383
\(327\) 0 0
\(328\) −1.42959e9 −2.23694
\(329\) −1.02218e7 −0.0158249
\(330\) 0 0
\(331\) −4.29838e8 −0.651489 −0.325744 0.945458i \(-0.605615\pi\)
−0.325744 + 0.945458i \(0.605615\pi\)
\(332\) −2.58588e9 −3.87816
\(333\) 0 0
\(334\) −1.83204e9 −2.69044
\(335\) 4.31824e8 0.627553
\(336\) 0 0
\(337\) −3.78144e8 −0.538211 −0.269105 0.963111i \(-0.586728\pi\)
−0.269105 + 0.963111i \(0.586728\pi\)
\(338\) −1.42803e9 −2.01153
\(339\) 0 0
\(340\) 4.33676e8 0.598397
\(341\) −4.84651e8 −0.661895
\(342\) 0 0
\(343\) 1.48835e8 0.199148
\(344\) 4.16306e8 0.551389
\(345\) 0 0
\(346\) −1.34509e8 −0.174576
\(347\) 3.18524e8 0.409250 0.204625 0.978840i \(-0.434403\pi\)
0.204625 + 0.978840i \(0.434403\pi\)
\(348\) 0 0
\(349\) 9.98844e8 1.25779 0.628896 0.777489i \(-0.283507\pi\)
0.628896 + 0.777489i \(0.283507\pi\)
\(350\) −1.27128e8 −0.158491
\(351\) 0 0
\(352\) 2.97565e9 3.63649
\(353\) −1.22295e8 −0.147978 −0.0739890 0.997259i \(-0.523573\pi\)
−0.0739890 + 0.997259i \(0.523573\pi\)
\(354\) 0 0
\(355\) −4.01168e7 −0.0475912
\(356\) 4.82674e8 0.566994
\(357\) 0 0
\(358\) −1.12389e9 −1.29459
\(359\) 1.04011e9 1.18645 0.593225 0.805037i \(-0.297854\pi\)
0.593225 + 0.805037i \(0.297854\pi\)
\(360\) 0 0
\(361\) −2.51162e8 −0.280982
\(362\) 4.63583e7 0.0513627
\(363\) 0 0
\(364\) 3.72774e8 0.405127
\(365\) −3.43226e8 −0.369450
\(366\) 0 0
\(367\) 3.08148e8 0.325408 0.162704 0.986675i \(-0.447978\pi\)
0.162704 + 0.986675i \(0.447978\pi\)
\(368\) 3.67191e9 3.84082
\(369\) 0 0
\(370\) −2.01958e8 −0.207279
\(371\) 1.28145e8 0.130285
\(372\) 0 0
\(373\) −1.52740e9 −1.52395 −0.761977 0.647604i \(-0.775771\pi\)
−0.761977 + 0.647604i \(0.775771\pi\)
\(374\) −7.35329e8 −0.726827
\(375\) 0 0
\(376\) −5.89339e8 −0.571752
\(377\) −2.44400e9 −2.34912
\(378\) 0 0
\(379\) −6.94059e8 −0.654876 −0.327438 0.944873i \(-0.606185\pi\)
−0.327438 + 0.944873i \(0.606185\pi\)
\(380\) −1.13107e9 −1.05742
\(381\) 0 0
\(382\) 3.95093e9 3.62641
\(383\) −4.76633e8 −0.433500 −0.216750 0.976227i \(-0.569546\pi\)
−0.216750 + 0.976227i \(0.569546\pi\)
\(384\) 0 0
\(385\) −3.79564e7 −0.0338979
\(386\) −2.81174e9 −2.48840
\(387\) 0 0
\(388\) 4.71691e9 4.09964
\(389\) −7.93506e8 −0.683482 −0.341741 0.939794i \(-0.611016\pi\)
−0.341741 + 0.939794i \(0.611016\pi\)
\(390\) 0 0
\(391\) −5.13191e8 −0.434171
\(392\) 4.26896e9 3.57949
\(393\) 0 0
\(394\) −2.83521e9 −2.33533
\(395\) −4.11437e8 −0.335903
\(396\) 0 0
\(397\) 1.68006e9 1.34759 0.673795 0.738918i \(-0.264663\pi\)
0.673795 + 0.738918i \(0.264663\pi\)
\(398\) 2.88326e9 2.29241
\(399\) 0 0
\(400\) −4.38900e9 −3.42891
\(401\) 7.52355e8 0.582663 0.291331 0.956622i \(-0.405902\pi\)
0.291331 + 0.956622i \(0.405902\pi\)
\(402\) 0 0
\(403\) −1.60233e9 −1.21951
\(404\) −1.12824e8 −0.0851272
\(405\) 0 0
\(406\) 4.36674e8 0.323829
\(407\) 2.53355e8 0.186272
\(408\) 0 0
\(409\) −4.62613e8 −0.334338 −0.167169 0.985928i \(-0.553463\pi\)
−0.167169 + 0.985928i \(0.553463\pi\)
\(410\) 7.42218e8 0.531848
\(411\) 0 0
\(412\) 7.27407e9 5.12433
\(413\) −1.31424e8 −0.0918017
\(414\) 0 0
\(415\) 8.70504e8 0.597864
\(416\) 9.83795e9 6.70005
\(417\) 0 0
\(418\) 1.91781e9 1.28437
\(419\) −1.30611e9 −0.867420 −0.433710 0.901053i \(-0.642796\pi\)
−0.433710 + 0.901053i \(0.642796\pi\)
\(420\) 0 0
\(421\) −2.16516e8 −0.141417 −0.0707087 0.997497i \(-0.522526\pi\)
−0.0707087 + 0.997497i \(0.522526\pi\)
\(422\) 4.11899e9 2.66807
\(423\) 0 0
\(424\) 7.38825e9 4.70718
\(425\) 6.13413e8 0.387607
\(426\) 0 0
\(427\) −2.90137e8 −0.180346
\(428\) −6.45545e8 −0.397991
\(429\) 0 0
\(430\) −2.16138e8 −0.131097
\(431\) −2.46067e9 −1.48041 −0.740207 0.672379i \(-0.765273\pi\)
−0.740207 + 0.672379i \(0.765273\pi\)
\(432\) 0 0
\(433\) −1.42408e9 −0.843001 −0.421500 0.906828i \(-0.638496\pi\)
−0.421500 + 0.906828i \(0.638496\pi\)
\(434\) 2.86291e8 0.168110
\(435\) 0 0
\(436\) 2.59215e9 1.49781
\(437\) 1.33845e9 0.767218
\(438\) 0 0
\(439\) −1.86431e9 −1.05170 −0.525851 0.850577i \(-0.676253\pi\)
−0.525851 + 0.850577i \(0.676253\pi\)
\(440\) −2.18839e9 −1.22473
\(441\) 0 0
\(442\) −2.43111e9 −1.33914
\(443\) 4.59781e8 0.251269 0.125634 0.992077i \(-0.459903\pi\)
0.125634 + 0.992077i \(0.459903\pi\)
\(444\) 0 0
\(445\) −1.62486e8 −0.0874089
\(446\) −3.50278e8 −0.186957
\(447\) 0 0
\(448\) −9.49277e8 −0.498792
\(449\) −1.81677e9 −0.947192 −0.473596 0.880742i \(-0.657044\pi\)
−0.473596 + 0.880742i \(0.657044\pi\)
\(450\) 0 0
\(451\) −9.31106e8 −0.477948
\(452\) 3.44903e9 1.75676
\(453\) 0 0
\(454\) 7.49103e8 0.375704
\(455\) −1.25489e8 −0.0624550
\(456\) 0 0
\(457\) −1.28773e9 −0.631131 −0.315566 0.948904i \(-0.602194\pi\)
−0.315566 + 0.948904i \(0.602194\pi\)
\(458\) −5.76930e9 −2.80604
\(459\) 0 0
\(460\) −2.35548e9 −1.12831
\(461\) 1.09265e8 0.0519430 0.0259715 0.999663i \(-0.491732\pi\)
0.0259715 + 0.999663i \(0.491732\pi\)
\(462\) 0 0
\(463\) −1.73669e9 −0.813185 −0.406592 0.913610i \(-0.633283\pi\)
−0.406592 + 0.913610i \(0.633283\pi\)
\(464\) 1.50758e10 7.00595
\(465\) 0 0
\(466\) 3.21825e9 1.47323
\(467\) 2.48860e9 1.13070 0.565349 0.824852i \(-0.308741\pi\)
0.565349 + 0.824852i \(0.308741\pi\)
\(468\) 0 0
\(469\) 3.20003e8 0.143235
\(470\) 3.05974e8 0.135938
\(471\) 0 0
\(472\) −7.57731e9 −3.31679
\(473\) 2.71144e8 0.117811
\(474\) 0 0
\(475\) −1.59984e9 −0.684935
\(476\) 3.21375e8 0.136580
\(477\) 0 0
\(478\) −3.44478e9 −1.44266
\(479\) −8.40928e8 −0.349611 −0.174805 0.984603i \(-0.555930\pi\)
−0.174805 + 0.984603i \(0.555930\pi\)
\(480\) 0 0
\(481\) 8.37628e8 0.343197
\(482\) −7.16483e9 −2.91435
\(483\) 0 0
\(484\) −2.86032e9 −1.14671
\(485\) −1.58788e9 −0.632008
\(486\) 0 0
\(487\) −1.41392e9 −0.554718 −0.277359 0.960766i \(-0.589459\pi\)
−0.277359 + 0.960766i \(0.589459\pi\)
\(488\) −1.67279e10 −6.51588
\(489\) 0 0
\(490\) −2.21637e9 −0.851050
\(491\) 2.99438e9 1.14162 0.570810 0.821082i \(-0.306629\pi\)
0.570810 + 0.821082i \(0.306629\pi\)
\(492\) 0 0
\(493\) −2.10701e9 −0.791959
\(494\) 6.34057e9 2.36638
\(495\) 0 0
\(496\) 9.88395e9 3.63701
\(497\) −2.97285e7 −0.0108624
\(498\) 0 0
\(499\) −3.88719e9 −1.40050 −0.700251 0.713897i \(-0.746929\pi\)
−0.700251 + 0.713897i \(0.746929\pi\)
\(500\) 6.30104e9 2.25433
\(501\) 0 0
\(502\) −5.22200e9 −1.84236
\(503\) 2.25318e9 0.789420 0.394710 0.918806i \(-0.370845\pi\)
0.394710 + 0.918806i \(0.370845\pi\)
\(504\) 0 0
\(505\) 3.79809e7 0.0131234
\(506\) 3.99388e9 1.37047
\(507\) 0 0
\(508\) 1.07063e10 3.62340
\(509\) −2.23850e9 −0.752393 −0.376196 0.926540i \(-0.622768\pi\)
−0.376196 + 0.926540i \(0.622768\pi\)
\(510\) 0 0
\(511\) −2.54347e8 −0.0843244
\(512\) −1.40717e10 −4.63341
\(513\) 0 0
\(514\) −9.43165e8 −0.306349
\(515\) −2.44872e9 −0.789976
\(516\) 0 0
\(517\) −3.83841e8 −0.122162
\(518\) −1.49661e8 −0.0473101
\(519\) 0 0
\(520\) −7.23513e9 −2.25650
\(521\) −3.53193e7 −0.0109416 −0.00547079 0.999985i \(-0.501741\pi\)
−0.00547079 + 0.999985i \(0.501741\pi\)
\(522\) 0 0
\(523\) 1.93242e9 0.590671 0.295335 0.955394i \(-0.404569\pi\)
0.295335 + 0.955394i \(0.404569\pi\)
\(524\) −1.75024e10 −5.31420
\(525\) 0 0
\(526\) 2.31852e9 0.694642
\(527\) −1.38139e9 −0.411131
\(528\) 0 0
\(529\) −6.17469e8 −0.181351
\(530\) −3.83584e9 −1.11917
\(531\) 0 0
\(532\) −8.38178e8 −0.241349
\(533\) −3.07837e9 −0.880595
\(534\) 0 0
\(535\) 2.17314e8 0.0613550
\(536\) 1.84498e10 5.17507
\(537\) 0 0
\(538\) −4.13419e9 −1.14460
\(539\) 2.78041e9 0.764801
\(540\) 0 0
\(541\) 4.16649e9 1.13130 0.565652 0.824644i \(-0.308625\pi\)
0.565652 + 0.824644i \(0.308625\pi\)
\(542\) −7.48079e9 −2.01813
\(543\) 0 0
\(544\) 8.48147e9 2.25878
\(545\) −8.72612e8 −0.230905
\(546\) 0 0
\(547\) −2.80094e9 −0.731725 −0.365862 0.930669i \(-0.619226\pi\)
−0.365862 + 0.930669i \(0.619226\pi\)
\(548\) −2.13968e9 −0.555414
\(549\) 0 0
\(550\) −4.77385e9 −1.22349
\(551\) 5.49530e9 1.39946
\(552\) 0 0
\(553\) −3.04894e8 −0.0766675
\(554\) 9.38068e9 2.34396
\(555\) 0 0
\(556\) −1.86580e10 −4.60367
\(557\) −4.87182e9 −1.19453 −0.597266 0.802043i \(-0.703746\pi\)
−0.597266 + 0.802043i \(0.703746\pi\)
\(558\) 0 0
\(559\) 8.96441e8 0.217060
\(560\) 7.74081e8 0.186264
\(561\) 0 0
\(562\) 7.57716e9 1.80065
\(563\) −6.20948e9 −1.46648 −0.733240 0.679970i \(-0.761993\pi\)
−0.733240 + 0.679970i \(0.761993\pi\)
\(564\) 0 0
\(565\) −1.16107e9 −0.270825
\(566\) 4.29857e9 0.996475
\(567\) 0 0
\(568\) −1.71400e9 −0.392457
\(569\) 5.71470e9 1.30047 0.650235 0.759733i \(-0.274670\pi\)
0.650235 + 0.759733i \(0.274670\pi\)
\(570\) 0 0
\(571\) 2.96607e9 0.666737 0.333368 0.942797i \(-0.391815\pi\)
0.333368 + 0.942797i \(0.391815\pi\)
\(572\) 1.39982e10 3.12741
\(573\) 0 0
\(574\) 5.50019e8 0.121391
\(575\) −3.33170e9 −0.730851
\(576\) 0 0
\(577\) 5.24837e9 1.13739 0.568694 0.822549i \(-0.307449\pi\)
0.568694 + 0.822549i \(0.307449\pi\)
\(578\) 7.00631e9 1.50918
\(579\) 0 0
\(580\) −9.67090e9 −2.05811
\(581\) 6.45085e8 0.136458
\(582\) 0 0
\(583\) 4.81203e9 1.00575
\(584\) −1.46644e10 −3.04664
\(585\) 0 0
\(586\) 1.02912e10 2.11264
\(587\) −1.22772e9 −0.250534 −0.125267 0.992123i \(-0.539979\pi\)
−0.125267 + 0.992123i \(0.539979\pi\)
\(588\) 0 0
\(589\) 3.60282e9 0.726505
\(590\) 3.93400e9 0.788591
\(591\) 0 0
\(592\) −5.16690e9 −1.02354
\(593\) −6.05113e9 −1.19164 −0.595820 0.803118i \(-0.703173\pi\)
−0.595820 + 0.803118i \(0.703173\pi\)
\(594\) 0 0
\(595\) −1.08187e8 −0.0210554
\(596\) 2.51623e9 0.486843
\(597\) 0 0
\(598\) 1.32044e10 2.52501
\(599\) −5.84902e9 −1.11196 −0.555980 0.831196i \(-0.687657\pi\)
−0.555980 + 0.831196i \(0.687657\pi\)
\(600\) 0 0
\(601\) −8.45352e9 −1.58846 −0.794231 0.607616i \(-0.792126\pi\)
−0.794231 + 0.607616i \(0.792126\pi\)
\(602\) −1.60169e8 −0.0299220
\(603\) 0 0
\(604\) −2.53865e10 −4.68784
\(605\) 9.62888e8 0.176779
\(606\) 0 0
\(607\) −9.03739e8 −0.164015 −0.0820074 0.996632i \(-0.526133\pi\)
−0.0820074 + 0.996632i \(0.526133\pi\)
\(608\) −2.21205e10 −3.99147
\(609\) 0 0
\(610\) 8.68484e9 1.54920
\(611\) −1.26904e9 −0.225076
\(612\) 0 0
\(613\) −3.64313e9 −0.638797 −0.319399 0.947620i \(-0.603481\pi\)
−0.319399 + 0.947620i \(0.603481\pi\)
\(614\) −2.44757e9 −0.426723
\(615\) 0 0
\(616\) −1.62170e9 −0.279536
\(617\) −8.10529e9 −1.38922 −0.694609 0.719387i \(-0.744423\pi\)
−0.694609 + 0.719387i \(0.744423\pi\)
\(618\) 0 0
\(619\) −7.22020e9 −1.22358 −0.611789 0.791021i \(-0.709550\pi\)
−0.611789 + 0.791021i \(0.709550\pi\)
\(620\) −6.34042e9 −1.06843
\(621\) 0 0
\(622\) −3.67712e9 −0.612691
\(623\) −1.20410e8 −0.0199505
\(624\) 0 0
\(625\) 2.80898e9 0.460224
\(626\) 2.28427e10 3.72166
\(627\) 0 0
\(628\) −2.22145e10 −3.57914
\(629\) 7.22134e8 0.115702
\(630\) 0 0
\(631\) 4.23517e8 0.0671070 0.0335535 0.999437i \(-0.489318\pi\)
0.0335535 + 0.999437i \(0.489318\pi\)
\(632\) −1.75788e10 −2.76999
\(633\) 0 0
\(634\) 1.41984e10 2.21272
\(635\) −3.60413e9 −0.558590
\(636\) 0 0
\(637\) 9.19245e9 1.40911
\(638\) 1.63977e10 2.49983
\(639\) 0 0
\(640\) 1.47278e10 2.22080
\(641\) −3.03264e9 −0.454797 −0.227398 0.973802i \(-0.573022\pi\)
−0.227398 + 0.973802i \(0.573022\pi\)
\(642\) 0 0
\(643\) −2.78734e9 −0.413478 −0.206739 0.978396i \(-0.566285\pi\)
−0.206739 + 0.978396i \(0.566285\pi\)
\(644\) −1.74552e9 −0.257528
\(645\) 0 0
\(646\) 5.46631e9 0.797776
\(647\) 1.73353e9 0.251632 0.125816 0.992054i \(-0.459845\pi\)
0.125816 + 0.992054i \(0.459845\pi\)
\(648\) 0 0
\(649\) −4.93516e9 −0.708672
\(650\) −1.57830e10 −2.25421
\(651\) 0 0
\(652\) −4.69568e8 −0.0663486
\(653\) 9.51546e9 1.33731 0.668657 0.743571i \(-0.266869\pi\)
0.668657 + 0.743571i \(0.266869\pi\)
\(654\) 0 0
\(655\) 5.89195e9 0.819247
\(656\) 1.89889e10 2.62626
\(657\) 0 0
\(658\) 2.26741e8 0.0310270
\(659\) −7.55803e9 −1.02875 −0.514375 0.857565i \(-0.671976\pi\)
−0.514375 + 0.857565i \(0.671976\pi\)
\(660\) 0 0
\(661\) −1.39999e9 −0.188547 −0.0942733 0.995546i \(-0.530053\pi\)
−0.0942733 + 0.995546i \(0.530053\pi\)
\(662\) 9.53475e9 1.27734
\(663\) 0 0
\(664\) 3.71926e10 4.93024
\(665\) 2.82162e8 0.0372068
\(666\) 0 0
\(667\) 1.14441e10 1.49328
\(668\) 3.00671e10 3.90278
\(669\) 0 0
\(670\) −9.57881e9 −1.23041
\(671\) −1.08950e10 −1.39220
\(672\) 0 0
\(673\) −5.04985e8 −0.0638595 −0.0319298 0.999490i \(-0.510165\pi\)
−0.0319298 + 0.999490i \(0.510165\pi\)
\(674\) 8.38806e9 1.05524
\(675\) 0 0
\(676\) 2.34365e10 2.91796
\(677\) 8.77580e9 1.08699 0.543496 0.839412i \(-0.317100\pi\)
0.543496 + 0.839412i \(0.317100\pi\)
\(678\) 0 0
\(679\) −1.17670e9 −0.144252
\(680\) −6.23753e9 −0.760732
\(681\) 0 0
\(682\) 1.07506e10 1.29774
\(683\) −5.66666e9 −0.680541 −0.340271 0.940328i \(-0.610519\pi\)
−0.340271 + 0.940328i \(0.610519\pi\)
\(684\) 0 0
\(685\) 7.20295e8 0.0856236
\(686\) −3.30148e9 −0.390458
\(687\) 0 0
\(688\) −5.52969e9 −0.647353
\(689\) 1.59093e10 1.85303
\(690\) 0 0
\(691\) −7.72356e9 −0.890522 −0.445261 0.895401i \(-0.646889\pi\)
−0.445261 + 0.895401i \(0.646889\pi\)
\(692\) 2.20753e9 0.253242
\(693\) 0 0
\(694\) −7.06555e9 −0.802394
\(695\) 6.28098e9 0.709709
\(696\) 0 0
\(697\) −2.65392e9 −0.296874
\(698\) −2.21566e10 −2.46609
\(699\) 0 0
\(700\) 2.08641e9 0.229909
\(701\) 9.02472e9 0.989511 0.494755 0.869032i \(-0.335258\pi\)
0.494755 + 0.869032i \(0.335258\pi\)
\(702\) 0 0
\(703\) −1.88340e9 −0.204455
\(704\) −3.56466e10 −3.85047
\(705\) 0 0
\(706\) 2.71277e9 0.290133
\(707\) 2.81456e7 0.00299532
\(708\) 0 0
\(709\) 1.01828e10 1.07301 0.536505 0.843897i \(-0.319744\pi\)
0.536505 + 0.843897i \(0.319744\pi\)
\(710\) 8.89878e8 0.0933096
\(711\) 0 0
\(712\) −6.94226e9 −0.720810
\(713\) 7.50294e9 0.775207
\(714\) 0 0
\(715\) −4.71230e9 −0.482127
\(716\) 1.84450e10 1.87794
\(717\) 0 0
\(718\) −2.30720e10 −2.32621
\(719\) 3.88344e9 0.389642 0.194821 0.980839i \(-0.437587\pi\)
0.194821 + 0.980839i \(0.437587\pi\)
\(720\) 0 0
\(721\) −1.81462e9 −0.180307
\(722\) 5.57133e9 0.550907
\(723\) 0 0
\(724\) −7.60823e8 −0.0745073
\(725\) −1.36790e10 −1.33313
\(726\) 0 0
\(727\) 3.21178e9 0.310009 0.155005 0.987914i \(-0.450461\pi\)
0.155005 + 0.987914i \(0.450461\pi\)
\(728\) −5.36158e9 −0.515030
\(729\) 0 0
\(730\) 7.61351e9 0.724360
\(731\) 7.72837e8 0.0731775
\(732\) 0 0
\(733\) 2.53300e9 0.237559 0.118779 0.992921i \(-0.462102\pi\)
0.118779 + 0.992921i \(0.462102\pi\)
\(734\) −6.83540e9 −0.638010
\(735\) 0 0
\(736\) −4.60664e10 −4.25904
\(737\) 1.20165e10 1.10571
\(738\) 0 0
\(739\) 8.64031e9 0.787542 0.393771 0.919209i \(-0.371170\pi\)
0.393771 + 0.919209i \(0.371170\pi\)
\(740\) 3.31450e9 0.300681
\(741\) 0 0
\(742\) −2.84254e9 −0.255443
\(743\) 7.96383e9 0.712297 0.356148 0.934429i \(-0.384090\pi\)
0.356148 + 0.934429i \(0.384090\pi\)
\(744\) 0 0
\(745\) −8.47057e8 −0.0750526
\(746\) 3.38811e10 2.98794
\(747\) 0 0
\(748\) 1.20681e10 1.05434
\(749\) 1.61040e8 0.0140039
\(750\) 0 0
\(751\) 3.65497e9 0.314879 0.157439 0.987529i \(-0.449676\pi\)
0.157439 + 0.987529i \(0.449676\pi\)
\(752\) 7.82804e9 0.671260
\(753\) 0 0
\(754\) 5.42132e10 4.60580
\(755\) 8.54602e9 0.722686
\(756\) 0 0
\(757\) 1.20006e9 0.100546 0.0502732 0.998736i \(-0.483991\pi\)
0.0502732 + 0.998736i \(0.483991\pi\)
\(758\) 1.53958e10 1.28398
\(759\) 0 0
\(760\) 1.62681e10 1.34428
\(761\) −2.32910e9 −0.191576 −0.0957880 0.995402i \(-0.530537\pi\)
−0.0957880 + 0.995402i \(0.530537\pi\)
\(762\) 0 0
\(763\) −6.46647e8 −0.0527025
\(764\) −6.48418e10 −5.26052
\(765\) 0 0
\(766\) 1.05728e10 0.849940
\(767\) −1.63164e10 −1.30569
\(768\) 0 0
\(769\) −4.56990e9 −0.362381 −0.181190 0.983448i \(-0.557995\pi\)
−0.181190 + 0.983448i \(0.557995\pi\)
\(770\) 8.41956e8 0.0664618
\(771\) 0 0
\(772\) 4.61458e10 3.60970
\(773\) −1.34361e10 −1.04627 −0.523135 0.852250i \(-0.675238\pi\)
−0.523135 + 0.852250i \(0.675238\pi\)
\(774\) 0 0
\(775\) −8.96819e9 −0.692068
\(776\) −6.78429e10 −5.21181
\(777\) 0 0
\(778\) 1.76017e10 1.34007
\(779\) 6.92169e9 0.524603
\(780\) 0 0
\(781\) −1.11634e9 −0.0838532
\(782\) 1.13837e10 0.851255
\(783\) 0 0
\(784\) −5.67036e10 −4.20247
\(785\) 7.47823e9 0.551766
\(786\) 0 0
\(787\) 2.05668e10 1.50403 0.752014 0.659147i \(-0.229082\pi\)
0.752014 + 0.659147i \(0.229082\pi\)
\(788\) 4.65308e10 3.38765
\(789\) 0 0
\(790\) 9.12657e9 0.658586
\(791\) −8.60408e8 −0.0618140
\(792\) 0 0
\(793\) −3.60207e10 −2.56505
\(794\) −3.72674e10 −2.64215
\(795\) 0 0
\(796\) −4.73194e10 −3.32540
\(797\) −1.49135e10 −1.04346 −0.521729 0.853111i \(-0.674713\pi\)
−0.521729 + 0.853111i \(0.674713\pi\)
\(798\) 0 0
\(799\) −1.09406e9 −0.0758799
\(800\) 5.50628e10 3.80227
\(801\) 0 0
\(802\) −1.66889e10 −1.14240
\(803\) −9.55107e9 −0.650950
\(804\) 0 0
\(805\) 5.87607e8 0.0397010
\(806\) 3.55432e10 2.39102
\(807\) 0 0
\(808\) 1.62275e9 0.108221
\(809\) 2.29446e10 1.52356 0.761781 0.647835i \(-0.224325\pi\)
0.761781 + 0.647835i \(0.224325\pi\)
\(810\) 0 0
\(811\) 4.15707e9 0.273662 0.136831 0.990594i \(-0.456308\pi\)
0.136831 + 0.990594i \(0.456308\pi\)
\(812\) −7.16660e9 −0.469750
\(813\) 0 0
\(814\) −5.61996e9 −0.365215
\(815\) 1.58074e8 0.0102284
\(816\) 0 0
\(817\) −2.01564e9 −0.129311
\(818\) 1.02618e10 0.655520
\(819\) 0 0
\(820\) −1.21811e10 −0.771506
\(821\) −1.06838e10 −0.673790 −0.336895 0.941542i \(-0.609377\pi\)
−0.336895 + 0.941542i \(0.609377\pi\)
\(822\) 0 0
\(823\) −1.14304e10 −0.714766 −0.357383 0.933958i \(-0.616331\pi\)
−0.357383 + 0.933958i \(0.616331\pi\)
\(824\) −1.04622e11 −6.51447
\(825\) 0 0
\(826\) 2.91528e9 0.179991
\(827\) 1.17022e10 0.719448 0.359724 0.933059i \(-0.382871\pi\)
0.359724 + 0.933059i \(0.382871\pi\)
\(828\) 0 0
\(829\) −1.53161e10 −0.933699 −0.466850 0.884337i \(-0.654611\pi\)
−0.466850 + 0.884337i \(0.654611\pi\)
\(830\) −1.93097e10 −1.17220
\(831\) 0 0
\(832\) −1.17853e11 −7.09430
\(833\) 7.92497e9 0.475051
\(834\) 0 0
\(835\) −1.01217e10 −0.601659
\(836\) −3.14747e10 −1.86312
\(837\) 0 0
\(838\) 2.89723e10 1.70070
\(839\) −1.05214e10 −0.615043 −0.307521 0.951541i \(-0.599500\pi\)
−0.307521 + 0.951541i \(0.599500\pi\)
\(840\) 0 0
\(841\) 2.97361e10 1.72385
\(842\) 4.80280e9 0.277270
\(843\) 0 0
\(844\) −6.76001e10 −3.87034
\(845\) −7.88958e9 −0.449837
\(846\) 0 0
\(847\) 7.13546e8 0.0403487
\(848\) −9.81363e10 −5.52642
\(849\) 0 0
\(850\) −1.36068e10 −0.759960
\(851\) −3.92221e9 −0.218161
\(852\) 0 0
\(853\) −2.95826e10 −1.63198 −0.815989 0.578068i \(-0.803807\pi\)
−0.815989 + 0.578068i \(0.803807\pi\)
\(854\) 6.43588e9 0.353594
\(855\) 0 0
\(856\) 9.28482e9 0.505959
\(857\) −6.19614e9 −0.336270 −0.168135 0.985764i \(-0.553775\pi\)
−0.168135 + 0.985764i \(0.553775\pi\)
\(858\) 0 0
\(859\) 2.78077e10 1.49689 0.748443 0.663199i \(-0.230802\pi\)
0.748443 + 0.663199i \(0.230802\pi\)
\(860\) 3.54722e9 0.190171
\(861\) 0 0
\(862\) 5.45831e10 2.90257
\(863\) −3.15999e10 −1.67359 −0.836793 0.547519i \(-0.815572\pi\)
−0.836793 + 0.547519i \(0.815572\pi\)
\(864\) 0 0
\(865\) −7.43137e8 −0.0390402
\(866\) 3.15893e10 1.65283
\(867\) 0 0
\(868\) −4.69855e9 −0.243863
\(869\) −1.14492e10 −0.591842
\(870\) 0 0
\(871\) 3.97284e10 2.03722
\(872\) −3.72826e10 −1.90414
\(873\) 0 0
\(874\) −2.96899e10 −1.50424
\(875\) −1.57188e9 −0.0793217
\(876\) 0 0
\(877\) 2.48099e10 1.24201 0.621006 0.783806i \(-0.286724\pi\)
0.621006 + 0.783806i \(0.286724\pi\)
\(878\) 4.13545e10 2.06201
\(879\) 0 0
\(880\) 2.90678e10 1.43788
\(881\) −2.37579e9 −0.117056 −0.0585279 0.998286i \(-0.518641\pi\)
−0.0585279 + 0.998286i \(0.518641\pi\)
\(882\) 0 0
\(883\) −1.16100e10 −0.567505 −0.283752 0.958898i \(-0.591579\pi\)
−0.283752 + 0.958898i \(0.591579\pi\)
\(884\) 3.98988e10 1.94257
\(885\) 0 0
\(886\) −1.01990e10 −0.492649
\(887\) −1.53556e10 −0.738813 −0.369407 0.929268i \(-0.620439\pi\)
−0.369407 + 0.929268i \(0.620439\pi\)
\(888\) 0 0
\(889\) −2.67084e9 −0.127494
\(890\) 3.60429e9 0.171378
\(891\) 0 0
\(892\) 5.74869e9 0.271201
\(893\) 2.85342e9 0.134086
\(894\) 0 0
\(895\) −6.20926e9 −0.289507
\(896\) 1.09140e10 0.506883
\(897\) 0 0
\(898\) 4.03000e10 1.85711
\(899\) 3.08049e10 1.41404
\(900\) 0 0
\(901\) 1.37157e10 0.624712
\(902\) 2.06540e10 0.937088
\(903\) 0 0
\(904\) −4.96071e10 −2.23334
\(905\) 2.56121e8 0.0114862
\(906\) 0 0
\(907\) 3.31807e9 0.147659 0.0738295 0.997271i \(-0.476478\pi\)
0.0738295 + 0.997271i \(0.476478\pi\)
\(908\) −1.22941e10 −0.545001
\(909\) 0 0
\(910\) 2.78363e9 0.122452
\(911\) 3.21961e9 0.141087 0.0705437 0.997509i \(-0.477527\pi\)
0.0705437 + 0.997509i \(0.477527\pi\)
\(912\) 0 0
\(913\) 2.42238e10 1.05340
\(914\) 2.85648e10 1.23743
\(915\) 0 0
\(916\) 9.46846e10 4.07047
\(917\) 4.36622e9 0.186988
\(918\) 0 0
\(919\) 7.57125e9 0.321783 0.160892 0.986972i \(-0.448563\pi\)
0.160892 + 0.986972i \(0.448563\pi\)
\(920\) 3.38787e10 1.43439
\(921\) 0 0
\(922\) −2.42373e9 −0.101842
\(923\) −3.69080e9 −0.154495
\(924\) 0 0
\(925\) 4.68818e9 0.194764
\(926\) 3.85236e10 1.59437
\(927\) 0 0
\(928\) −1.89135e11 −7.76881
\(929\) 2.89017e10 1.18268 0.591342 0.806421i \(-0.298598\pi\)
0.591342 + 0.806421i \(0.298598\pi\)
\(930\) 0 0
\(931\) −2.06691e10 −0.839457
\(932\) −5.28173e10 −2.13708
\(933\) 0 0
\(934\) −5.52027e10 −2.21690
\(935\) −4.06255e9 −0.162539
\(936\) 0 0
\(937\) 3.91577e10 1.55499 0.777496 0.628888i \(-0.216489\pi\)
0.777496 + 0.628888i \(0.216489\pi\)
\(938\) −7.09836e9 −0.280833
\(939\) 0 0
\(940\) −5.02158e9 −0.197194
\(941\) 2.41576e10 0.945127 0.472564 0.881297i \(-0.343329\pi\)
0.472564 + 0.881297i \(0.343329\pi\)
\(942\) 0 0
\(943\) 1.44145e10 0.559771
\(944\) 1.00647e11 3.89404
\(945\) 0 0
\(946\) −6.01456e9 −0.230986
\(947\) −3.20511e10 −1.22636 −0.613179 0.789944i \(-0.710110\pi\)
−0.613179 + 0.789944i \(0.710110\pi\)
\(948\) 0 0
\(949\) −3.15773e10 −1.19934
\(950\) 3.54880e10 1.34292
\(951\) 0 0
\(952\) −4.62231e9 −0.173632
\(953\) −4.93663e9 −0.184759 −0.0923794 0.995724i \(-0.529447\pi\)
−0.0923794 + 0.995724i \(0.529447\pi\)
\(954\) 0 0
\(955\) 2.18281e10 0.810970
\(956\) 5.65350e10 2.09274
\(957\) 0 0
\(958\) 1.86536e10 0.685463
\(959\) 5.33773e8 0.0195430
\(960\) 0 0
\(961\) −7.31640e9 −0.265929
\(962\) −1.85804e10 −0.672888
\(963\) 0 0
\(964\) 1.17588e11 4.22759
\(965\) −1.55344e10 −0.556478
\(966\) 0 0
\(967\) 3.10919e10 1.10574 0.552872 0.833266i \(-0.313532\pi\)
0.552872 + 0.833266i \(0.313532\pi\)
\(968\) 4.11397e10 1.45780
\(969\) 0 0
\(970\) 3.52228e10 1.23915
\(971\) −3.37297e10 −1.18235 −0.591174 0.806544i \(-0.701335\pi\)
−0.591174 + 0.806544i \(0.701335\pi\)
\(972\) 0 0
\(973\) 4.65451e9 0.161986
\(974\) 3.13637e10 1.08761
\(975\) 0 0
\(976\) 2.22193e11 7.64991
\(977\) 2.60034e10 0.892072 0.446036 0.895015i \(-0.352835\pi\)
0.446036 + 0.895015i \(0.352835\pi\)
\(978\) 0 0
\(979\) −4.52155e9 −0.154010
\(980\) 3.63746e10 1.23454
\(981\) 0 0
\(982\) −6.64219e10 −2.23831
\(983\) 4.35623e9 0.146276 0.0731381 0.997322i \(-0.476699\pi\)
0.0731381 + 0.997322i \(0.476699\pi\)
\(984\) 0 0
\(985\) −1.56640e10 −0.522247
\(986\) 4.67381e10 1.55275
\(987\) 0 0
\(988\) −1.04060e11 −3.43269
\(989\) −4.19761e9 −0.137980
\(990\) 0 0
\(991\) −3.99586e10 −1.30423 −0.652113 0.758122i \(-0.726117\pi\)
−0.652113 + 0.758122i \(0.726117\pi\)
\(992\) −1.24000e11 −4.03304
\(993\) 0 0
\(994\) 6.59442e8 0.0212973
\(995\) 1.59295e10 0.512649
\(996\) 0 0
\(997\) 1.79270e10 0.572894 0.286447 0.958096i \(-0.407526\pi\)
0.286447 + 0.958096i \(0.407526\pi\)
\(998\) 8.62264e10 2.74589
\(999\) 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.1.1 13
3.2 odd 2 43.8.a.b.1.13 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.13 13 3.2 odd 2
387.8.a.d.1.1 13 1.1 even 1 trivial