Properties

Label 387.8.a.d.1.11
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 1.11
Root \(-16.1540\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q+15.1540 q^{2} +101.645 q^{4} -210.950 q^{5} +1100.87 q^{7} -399.390 q^{8} +O(q^{10})\) \(q+15.1540 q^{2} +101.645 q^{4} -210.950 q^{5} +1100.87 q^{7} -399.390 q^{8} -3196.74 q^{10} +3136.09 q^{11} -5743.36 q^{13} +16682.6 q^{14} -19062.9 q^{16} -27231.5 q^{17} +54685.6 q^{19} -21441.9 q^{20} +47524.4 q^{22} -12024.9 q^{23} -33625.2 q^{25} -87035.1 q^{26} +111897. q^{28} -132127. q^{29} +281517. q^{31} -237758. q^{32} -412666. q^{34} -232228. q^{35} -214511. q^{37} +828707. q^{38} +84251.4 q^{40} -302212. q^{41} -79507.0 q^{43} +318767. q^{44} -182226. q^{46} -235041. q^{47} +388365. q^{49} -509557. q^{50} -583782. q^{52} -55032.5 q^{53} -661558. q^{55} -439676. q^{56} -2.00226e6 q^{58} -150144. q^{59} -1.65598e6 q^{61} +4.26612e6 q^{62} -1.16294e6 q^{64} +1.21156e6 q^{65} +1.76796e6 q^{67} -2.76793e6 q^{68} -3.51919e6 q^{70} -3.57159e6 q^{71} -430044. q^{73} -3.25070e6 q^{74} +5.55849e6 q^{76} +3.45242e6 q^{77} -7.44014e6 q^{79} +4.02131e6 q^{80} -4.57974e6 q^{82} -5.50388e6 q^{83} +5.74447e6 q^{85} -1.20485e6 q^{86} -1.25252e6 q^{88} +3.53700e6 q^{89} -6.32268e6 q^{91} -1.22227e6 q^{92} -3.56182e6 q^{94} -1.15359e7 q^{95} -2.08749e6 q^{97} +5.88530e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 15.1540 1.33944 0.669720 0.742614i \(-0.266414\pi\)
0.669720 + 0.742614i \(0.266414\pi\)
\(3\) 0 0
\(4\) 101.645 0.794098
\(5\) −210.950 −0.754717 −0.377359 0.926067i \(-0.623168\pi\)
−0.377359 + 0.926067i \(0.623168\pi\)
\(6\) 0 0
\(7\) 1100.87 1.21309 0.606543 0.795051i \(-0.292556\pi\)
0.606543 + 0.795051i \(0.292556\pi\)
\(8\) −399.390 −0.275793
\(9\) 0 0
\(10\) −3196.74 −1.01090
\(11\) 3136.09 0.710419 0.355209 0.934787i \(-0.384410\pi\)
0.355209 + 0.934787i \(0.384410\pi\)
\(12\) 0 0
\(13\) −5743.36 −0.725044 −0.362522 0.931975i \(-0.618084\pi\)
−0.362522 + 0.931975i \(0.618084\pi\)
\(14\) 16682.6 1.62486
\(15\) 0 0
\(16\) −19062.9 −1.16351
\(17\) −27231.5 −1.34431 −0.672155 0.740410i \(-0.734631\pi\)
−0.672155 + 0.740410i \(0.734631\pi\)
\(18\) 0 0
\(19\) 54685.6 1.82909 0.914545 0.404484i \(-0.132549\pi\)
0.914545 + 0.404484i \(0.132549\pi\)
\(20\) −21441.9 −0.599320
\(21\) 0 0
\(22\) 47524.4 0.951563
\(23\) −12024.9 −0.206079 −0.103040 0.994677i \(-0.532857\pi\)
−0.103040 + 0.994677i \(0.532857\pi\)
\(24\) 0 0
\(25\) −33625.2 −0.430402
\(26\) −87035.1 −0.971153
\(27\) 0 0
\(28\) 111897. 0.963310
\(29\) −132127. −1.00600 −0.503001 0.864286i \(-0.667771\pi\)
−0.503001 + 0.864286i \(0.667771\pi\)
\(30\) 0 0
\(31\) 281517. 1.69722 0.848612 0.529016i \(-0.177439\pi\)
0.848612 + 0.529016i \(0.177439\pi\)
\(32\) −237758. −1.28265
\(33\) 0 0
\(34\) −412666. −1.80062
\(35\) −232228. −0.915537
\(36\) 0 0
\(37\) −214511. −0.696215 −0.348107 0.937455i \(-0.613176\pi\)
−0.348107 + 0.937455i \(0.613176\pi\)
\(38\) 828707. 2.44996
\(39\) 0 0
\(40\) 84251.4 0.208145
\(41\) −302212. −0.684808 −0.342404 0.939553i \(-0.611241\pi\)
−0.342404 + 0.939553i \(0.611241\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) 318767. 0.564142
\(45\) 0 0
\(46\) −182226. −0.276030
\(47\) −235041. −0.330218 −0.165109 0.986275i \(-0.552798\pi\)
−0.165109 + 0.986275i \(0.552798\pi\)
\(48\) 0 0
\(49\) 388365. 0.471578
\(50\) −509557. −0.576498
\(51\) 0 0
\(52\) −583782. −0.575757
\(53\) −55032.5 −0.0507754 −0.0253877 0.999678i \(-0.508082\pi\)
−0.0253877 + 0.999678i \(0.508082\pi\)
\(54\) 0 0
\(55\) −661558. −0.536165
\(56\) −439676. −0.334560
\(57\) 0 0
\(58\) −2.00226e6 −1.34748
\(59\) −150144. −0.0951754 −0.0475877 0.998867i \(-0.515153\pi\)
−0.0475877 + 0.998867i \(0.515153\pi\)
\(60\) 0 0
\(61\) −1.65598e6 −0.934113 −0.467057 0.884227i \(-0.654686\pi\)
−0.467057 + 0.884227i \(0.654686\pi\)
\(62\) 4.26612e6 2.27333
\(63\) 0 0
\(64\) −1.16294e6 −0.554531
\(65\) 1.21156e6 0.547203
\(66\) 0 0
\(67\) 1.76796e6 0.718144 0.359072 0.933310i \(-0.383093\pi\)
0.359072 + 0.933310i \(0.383093\pi\)
\(68\) −2.76793e6 −1.06752
\(69\) 0 0
\(70\) −3.51919e6 −1.22631
\(71\) −3.57159e6 −1.18429 −0.592144 0.805832i \(-0.701718\pi\)
−0.592144 + 0.805832i \(0.701718\pi\)
\(72\) 0 0
\(73\) −430044. −0.129385 −0.0646923 0.997905i \(-0.520607\pi\)
−0.0646923 + 0.997905i \(0.520607\pi\)
\(74\) −3.25070e6 −0.932538
\(75\) 0 0
\(76\) 5.55849e6 1.45248
\(77\) 3.45242e6 0.861799
\(78\) 0 0
\(79\) −7.44014e6 −1.69780 −0.848900 0.528554i \(-0.822734\pi\)
−0.848900 + 0.528554i \(0.822734\pi\)
\(80\) 4.02131e6 0.878118
\(81\) 0 0
\(82\) −4.57974e6 −0.917259
\(83\) −5.50388e6 −1.05656 −0.528282 0.849069i \(-0.677163\pi\)
−0.528282 + 0.849069i \(0.677163\pi\)
\(84\) 0 0
\(85\) 5.74447e6 1.01457
\(86\) −1.20485e6 −0.204263
\(87\) 0 0
\(88\) −1.25252e6 −0.195928
\(89\) 3.53700e6 0.531826 0.265913 0.963997i \(-0.414327\pi\)
0.265913 + 0.963997i \(0.414327\pi\)
\(90\) 0 0
\(91\) −6.32268e6 −0.879541
\(92\) −1.22227e6 −0.163647
\(93\) 0 0
\(94\) −3.56182e6 −0.442308
\(95\) −1.15359e7 −1.38045
\(96\) 0 0
\(97\) −2.08749e6 −0.232233 −0.116117 0.993236i \(-0.537045\pi\)
−0.116117 + 0.993236i \(0.537045\pi\)
\(98\) 5.88530e6 0.631651
\(99\) 0 0
\(100\) −3.41782e6 −0.341782
\(101\) −1.59947e7 −1.54473 −0.772364 0.635180i \(-0.780926\pi\)
−0.772364 + 0.635180i \(0.780926\pi\)
\(102\) 0 0
\(103\) −1.14391e7 −1.03148 −0.515739 0.856746i \(-0.672483\pi\)
−0.515739 + 0.856746i \(0.672483\pi\)
\(104\) 2.29384e6 0.199962
\(105\) 0 0
\(106\) −833964. −0.0680106
\(107\) −5.13105e6 −0.404914 −0.202457 0.979291i \(-0.564893\pi\)
−0.202457 + 0.979291i \(0.564893\pi\)
\(108\) 0 0
\(109\) −2.10496e7 −1.55687 −0.778433 0.627727i \(-0.783985\pi\)
−0.778433 + 0.627727i \(0.783985\pi\)
\(110\) −1.00253e7 −0.718161
\(111\) 0 0
\(112\) −2.09857e7 −1.41143
\(113\) 1.44050e7 0.939156 0.469578 0.882891i \(-0.344406\pi\)
0.469578 + 0.882891i \(0.344406\pi\)
\(114\) 0 0
\(115\) 2.53665e6 0.155531
\(116\) −1.34300e7 −0.798865
\(117\) 0 0
\(118\) −2.27528e6 −0.127482
\(119\) −2.99782e7 −1.63077
\(120\) 0 0
\(121\) −9.65210e6 −0.495306
\(122\) −2.50947e7 −1.25119
\(123\) 0 0
\(124\) 2.86147e7 1.34776
\(125\) 2.35737e7 1.07955
\(126\) 0 0
\(127\) −2.66397e7 −1.15403 −0.577014 0.816734i \(-0.695782\pi\)
−0.577014 + 0.816734i \(0.695782\pi\)
\(128\) 1.28098e7 0.539893
\(129\) 0 0
\(130\) 1.83600e7 0.732946
\(131\) 1.53052e7 0.594825 0.297413 0.954749i \(-0.403876\pi\)
0.297413 + 0.954749i \(0.403876\pi\)
\(132\) 0 0
\(133\) 6.02015e7 2.21884
\(134\) 2.67918e7 0.961911
\(135\) 0 0
\(136\) 1.08760e7 0.370751
\(137\) 2.88018e7 0.956968 0.478484 0.878096i \(-0.341186\pi\)
0.478484 + 0.878096i \(0.341186\pi\)
\(138\) 0 0
\(139\) 2.41973e7 0.764213 0.382106 0.924118i \(-0.375199\pi\)
0.382106 + 0.924118i \(0.375199\pi\)
\(140\) −2.36047e7 −0.727026
\(141\) 0 0
\(142\) −5.41240e7 −1.58628
\(143\) −1.80117e7 −0.515085
\(144\) 0 0
\(145\) 2.78722e7 0.759247
\(146\) −6.51690e6 −0.173303
\(147\) 0 0
\(148\) −2.18039e7 −0.552863
\(149\) −1.88108e7 −0.465859 −0.232929 0.972494i \(-0.574831\pi\)
−0.232929 + 0.972494i \(0.574831\pi\)
\(150\) 0 0
\(151\) −5.28673e7 −1.24959 −0.624796 0.780788i \(-0.714818\pi\)
−0.624796 + 0.780788i \(0.714818\pi\)
\(152\) −2.18409e7 −0.504450
\(153\) 0 0
\(154\) 5.23181e7 1.15433
\(155\) −5.93860e7 −1.28092
\(156\) 0 0
\(157\) −5.80087e7 −1.19631 −0.598156 0.801380i \(-0.704100\pi\)
−0.598156 + 0.801380i \(0.704100\pi\)
\(158\) −1.12748e8 −2.27410
\(159\) 0 0
\(160\) 5.01549e7 0.968040
\(161\) −1.32378e7 −0.249992
\(162\) 0 0
\(163\) 9.85826e7 1.78297 0.891484 0.453051i \(-0.149665\pi\)
0.891484 + 0.453051i \(0.149665\pi\)
\(164\) −3.07183e7 −0.543805
\(165\) 0 0
\(166\) −8.34059e7 −1.41520
\(167\) −3.30309e7 −0.548798 −0.274399 0.961616i \(-0.588479\pi\)
−0.274399 + 0.961616i \(0.588479\pi\)
\(168\) 0 0
\(169\) −2.97623e7 −0.474311
\(170\) 8.70519e7 1.35896
\(171\) 0 0
\(172\) −8.08146e6 −0.121099
\(173\) 9.21797e7 1.35355 0.676774 0.736191i \(-0.263377\pi\)
0.676774 + 0.736191i \(0.263377\pi\)
\(174\) 0 0
\(175\) −3.70168e7 −0.522115
\(176\) −5.97829e7 −0.826576
\(177\) 0 0
\(178\) 5.35998e7 0.712349
\(179\) −8.22086e7 −1.07135 −0.535675 0.844424i \(-0.679943\pi\)
−0.535675 + 0.844424i \(0.679943\pi\)
\(180\) 0 0
\(181\) −4.53246e7 −0.568144 −0.284072 0.958803i \(-0.591686\pi\)
−0.284072 + 0.958803i \(0.591686\pi\)
\(182\) −9.58141e7 −1.17809
\(183\) 0 0
\(184\) 4.80263e6 0.0568351
\(185\) 4.52510e7 0.525445
\(186\) 0 0
\(187\) −8.54003e7 −0.955023
\(188\) −2.38907e7 −0.262226
\(189\) 0 0
\(190\) −1.74816e8 −1.84902
\(191\) 1.21596e8 1.26270 0.631352 0.775496i \(-0.282500\pi\)
0.631352 + 0.775496i \(0.282500\pi\)
\(192\) 0 0
\(193\) 1.40361e8 1.40538 0.702691 0.711495i \(-0.251981\pi\)
0.702691 + 0.711495i \(0.251981\pi\)
\(194\) −3.16339e7 −0.311062
\(195\) 0 0
\(196\) 3.94752e7 0.374480
\(197\) 5.68563e7 0.529842 0.264921 0.964270i \(-0.414654\pi\)
0.264921 + 0.964270i \(0.414654\pi\)
\(198\) 0 0
\(199\) 3.29464e7 0.296361 0.148181 0.988960i \(-0.452658\pi\)
0.148181 + 0.988960i \(0.452658\pi\)
\(200\) 1.34296e7 0.118702
\(201\) 0 0
\(202\) −2.42385e8 −2.06907
\(203\) −1.45454e8 −1.22037
\(204\) 0 0
\(205\) 6.37517e7 0.516836
\(206\) −1.73348e8 −1.38160
\(207\) 0 0
\(208\) 1.09485e8 0.843593
\(209\) 1.71499e8 1.29942
\(210\) 0 0
\(211\) −2.35087e8 −1.72282 −0.861411 0.507908i \(-0.830419\pi\)
−0.861411 + 0.507908i \(0.830419\pi\)
\(212\) −5.59375e6 −0.0403207
\(213\) 0 0
\(214\) −7.77561e7 −0.542358
\(215\) 1.67720e7 0.115093
\(216\) 0 0
\(217\) 3.09913e8 2.05888
\(218\) −3.18986e8 −2.08533
\(219\) 0 0
\(220\) −6.72438e7 −0.425768
\(221\) 1.56400e8 0.974685
\(222\) 0 0
\(223\) 6.86931e7 0.414807 0.207404 0.978255i \(-0.433499\pi\)
0.207404 + 0.978255i \(0.433499\pi\)
\(224\) −2.61739e8 −1.55597
\(225\) 0 0
\(226\) 2.18293e8 1.25794
\(227\) 3.33942e8 1.89487 0.947436 0.319945i \(-0.103664\pi\)
0.947436 + 0.319945i \(0.103664\pi\)
\(228\) 0 0
\(229\) 3.08822e8 1.69935 0.849677 0.527303i \(-0.176797\pi\)
0.849677 + 0.527303i \(0.176797\pi\)
\(230\) 3.84405e7 0.208325
\(231\) 0 0
\(232\) 5.27703e7 0.277448
\(233\) −1.57739e8 −0.816947 −0.408473 0.912770i \(-0.633939\pi\)
−0.408473 + 0.912770i \(0.633939\pi\)
\(234\) 0 0
\(235\) 4.95819e7 0.249221
\(236\) −1.52613e7 −0.0755787
\(237\) 0 0
\(238\) −4.54291e8 −2.18431
\(239\) 1.84694e8 0.875105 0.437553 0.899193i \(-0.355845\pi\)
0.437553 + 0.899193i \(0.355845\pi\)
\(240\) 0 0
\(241\) 4.79565e7 0.220692 0.110346 0.993893i \(-0.464804\pi\)
0.110346 + 0.993893i \(0.464804\pi\)
\(242\) −1.46268e8 −0.663432
\(243\) 0 0
\(244\) −1.68321e8 −0.741778
\(245\) −8.19256e7 −0.355908
\(246\) 0 0
\(247\) −3.14079e8 −1.32617
\(248\) −1.12435e8 −0.468082
\(249\) 0 0
\(250\) 3.57236e8 1.44599
\(251\) 4.64988e8 1.85602 0.928012 0.372550i \(-0.121517\pi\)
0.928012 + 0.372550i \(0.121517\pi\)
\(252\) 0 0
\(253\) −3.77112e7 −0.146402
\(254\) −4.03699e8 −1.54575
\(255\) 0 0
\(256\) 3.42976e8 1.27768
\(257\) −2.59186e8 −0.952456 −0.476228 0.879322i \(-0.657996\pi\)
−0.476228 + 0.879322i \(0.657996\pi\)
\(258\) 0 0
\(259\) −2.36148e8 −0.844569
\(260\) 1.23149e8 0.434533
\(261\) 0 0
\(262\) 2.31935e8 0.796732
\(263\) 2.19827e8 0.745137 0.372569 0.928005i \(-0.378477\pi\)
0.372569 + 0.928005i \(0.378477\pi\)
\(264\) 0 0
\(265\) 1.16091e7 0.0383211
\(266\) 9.12296e8 2.97201
\(267\) 0 0
\(268\) 1.79704e8 0.570277
\(269\) −5.19950e8 −1.62865 −0.814327 0.580407i \(-0.802893\pi\)
−0.814327 + 0.580407i \(0.802893\pi\)
\(270\) 0 0
\(271\) 2.80890e8 0.857323 0.428661 0.903465i \(-0.358985\pi\)
0.428661 + 0.903465i \(0.358985\pi\)
\(272\) 5.19110e8 1.56411
\(273\) 0 0
\(274\) 4.36463e8 1.28180
\(275\) −1.05452e8 −0.305766
\(276\) 0 0
\(277\) 2.46843e8 0.697818 0.348909 0.937157i \(-0.386552\pi\)
0.348909 + 0.937157i \(0.386552\pi\)
\(278\) 3.66686e8 1.02362
\(279\) 0 0
\(280\) 9.27495e7 0.252498
\(281\) −1.74389e8 −0.468863 −0.234432 0.972133i \(-0.575323\pi\)
−0.234432 + 0.972133i \(0.575323\pi\)
\(282\) 0 0
\(283\) −5.30249e8 −1.39068 −0.695341 0.718680i \(-0.744746\pi\)
−0.695341 + 0.718680i \(0.744746\pi\)
\(284\) −3.63033e8 −0.940441
\(285\) 0 0
\(286\) −2.72950e8 −0.689925
\(287\) −3.32696e8 −0.830731
\(288\) 0 0
\(289\) 3.31214e8 0.807172
\(290\) 4.22376e8 1.01697
\(291\) 0 0
\(292\) −4.37116e7 −0.102744
\(293\) −2.11533e8 −0.491294 −0.245647 0.969359i \(-0.579000\pi\)
−0.245647 + 0.969359i \(0.579000\pi\)
\(294\) 0 0
\(295\) 3.16728e7 0.0718305
\(296\) 8.56736e7 0.192011
\(297\) 0 0
\(298\) −2.85059e8 −0.623990
\(299\) 6.90633e7 0.149416
\(300\) 0 0
\(301\) −8.75266e7 −0.184994
\(302\) −8.01153e8 −1.67375
\(303\) 0 0
\(304\) −1.04246e9 −2.12816
\(305\) 3.49328e8 0.704991
\(306\) 0 0
\(307\) −3.83496e8 −0.756444 −0.378222 0.925715i \(-0.623464\pi\)
−0.378222 + 0.925715i \(0.623464\pi\)
\(308\) 3.50920e8 0.684353
\(309\) 0 0
\(310\) −8.99937e8 −1.71572
\(311\) −2.71805e8 −0.512386 −0.256193 0.966626i \(-0.582468\pi\)
−0.256193 + 0.966626i \(0.582468\pi\)
\(312\) 0 0
\(313\) 2.82814e8 0.521309 0.260655 0.965432i \(-0.416062\pi\)
0.260655 + 0.965432i \(0.416062\pi\)
\(314\) −8.79065e8 −1.60239
\(315\) 0 0
\(316\) −7.56250e8 −1.34822
\(317\) −1.50905e8 −0.266070 −0.133035 0.991111i \(-0.542472\pi\)
−0.133035 + 0.991111i \(0.542472\pi\)
\(318\) 0 0
\(319\) −4.14363e8 −0.714683
\(320\) 2.45321e8 0.418514
\(321\) 0 0
\(322\) −2.00606e8 −0.334849
\(323\) −1.48917e9 −2.45887
\(324\) 0 0
\(325\) 1.93122e8 0.312061
\(326\) 1.49392e9 2.38818
\(327\) 0 0
\(328\) 1.20701e8 0.188865
\(329\) −2.58749e8 −0.400583
\(330\) 0 0
\(331\) 4.82795e8 0.731754 0.365877 0.930663i \(-0.380769\pi\)
0.365877 + 0.930663i \(0.380769\pi\)
\(332\) −5.59439e8 −0.839015
\(333\) 0 0
\(334\) −5.00551e8 −0.735082
\(335\) −3.72952e8 −0.541996
\(336\) 0 0
\(337\) −5.19140e8 −0.738891 −0.369445 0.929252i \(-0.620452\pi\)
−0.369445 + 0.929252i \(0.620452\pi\)
\(338\) −4.51019e8 −0.635311
\(339\) 0 0
\(340\) 5.83895e8 0.805672
\(341\) 8.82863e8 1.20574
\(342\) 0 0
\(343\) −4.79073e8 −0.641021
\(344\) 3.17543e7 0.0420580
\(345\) 0 0
\(346\) 1.39689e9 1.81300
\(347\) −2.17439e8 −0.279373 −0.139686 0.990196i \(-0.544609\pi\)
−0.139686 + 0.990196i \(0.544609\pi\)
\(348\) 0 0
\(349\) −1.23345e8 −0.155322 −0.0776609 0.996980i \(-0.524745\pi\)
−0.0776609 + 0.996980i \(0.524745\pi\)
\(350\) −5.60954e8 −0.699341
\(351\) 0 0
\(352\) −7.45629e8 −0.911221
\(353\) −3.57356e8 −0.432404 −0.216202 0.976349i \(-0.569367\pi\)
−0.216202 + 0.976349i \(0.569367\pi\)
\(354\) 0 0
\(355\) 7.53426e8 0.893802
\(356\) 3.59517e8 0.422322
\(357\) 0 0
\(358\) −1.24579e9 −1.43501
\(359\) 2.89591e8 0.330335 0.165168 0.986266i \(-0.447184\pi\)
0.165168 + 0.986266i \(0.447184\pi\)
\(360\) 0 0
\(361\) 2.09664e9 2.34557
\(362\) −6.86850e8 −0.760995
\(363\) 0 0
\(364\) −6.42666e8 −0.698442
\(365\) 9.07177e7 0.0976488
\(366\) 0 0
\(367\) −8.92000e8 −0.941963 −0.470981 0.882143i \(-0.656100\pi\)
−0.470981 + 0.882143i \(0.656100\pi\)
\(368\) 2.29229e8 0.239774
\(369\) 0 0
\(370\) 6.85735e8 0.703802
\(371\) −6.05834e7 −0.0615950
\(372\) 0 0
\(373\) 1.71138e9 1.70752 0.853761 0.520665i \(-0.174316\pi\)
0.853761 + 0.520665i \(0.174316\pi\)
\(374\) −1.29416e9 −1.27920
\(375\) 0 0
\(376\) 9.38732e7 0.0910718
\(377\) 7.58854e8 0.729396
\(378\) 0 0
\(379\) 4.74180e7 0.0447410 0.0223705 0.999750i \(-0.492879\pi\)
0.0223705 + 0.999750i \(0.492879\pi\)
\(380\) −1.17256e9 −1.09621
\(381\) 0 0
\(382\) 1.84267e9 1.69132
\(383\) 3.02938e8 0.275523 0.137761 0.990465i \(-0.456009\pi\)
0.137761 + 0.990465i \(0.456009\pi\)
\(384\) 0 0
\(385\) −7.28287e8 −0.650414
\(386\) 2.12703e9 1.88243
\(387\) 0 0
\(388\) −2.12182e8 −0.184416
\(389\) −2.48110e8 −0.213708 −0.106854 0.994275i \(-0.534078\pi\)
−0.106854 + 0.994275i \(0.534078\pi\)
\(390\) 0 0
\(391\) 3.27455e8 0.277034
\(392\) −1.55109e8 −0.130058
\(393\) 0 0
\(394\) 8.61601e8 0.709691
\(395\) 1.56950e9 1.28136
\(396\) 0 0
\(397\) −1.17389e9 −0.941589 −0.470794 0.882243i \(-0.656033\pi\)
−0.470794 + 0.882243i \(0.656033\pi\)
\(398\) 4.99270e8 0.396958
\(399\) 0 0
\(400\) 6.40993e8 0.500776
\(401\) 4.69179e8 0.363357 0.181679 0.983358i \(-0.441847\pi\)
0.181679 + 0.983358i \(0.441847\pi\)
\(402\) 0 0
\(403\) −1.61686e9 −1.23056
\(404\) −1.62578e9 −1.22667
\(405\) 0 0
\(406\) −2.20422e9 −1.63461
\(407\) −6.72726e8 −0.494604
\(408\) 0 0
\(409\) −2.36052e9 −1.70599 −0.852995 0.521918i \(-0.825217\pi\)
−0.852995 + 0.521918i \(0.825217\pi\)
\(410\) 9.66094e8 0.692271
\(411\) 0 0
\(412\) −1.16272e9 −0.819096
\(413\) −1.65288e8 −0.115456
\(414\) 0 0
\(415\) 1.16104e9 0.797406
\(416\) 1.36553e9 0.929981
\(417\) 0 0
\(418\) 2.59890e9 1.74049
\(419\) 2.55941e9 1.69977 0.849886 0.526967i \(-0.176671\pi\)
0.849886 + 0.526967i \(0.176671\pi\)
\(420\) 0 0
\(421\) −3.64111e7 −0.0237819 −0.0118909 0.999929i \(-0.503785\pi\)
−0.0118909 + 0.999929i \(0.503785\pi\)
\(422\) −3.56252e9 −2.30762
\(423\) 0 0
\(424\) 2.19794e7 0.0140035
\(425\) 9.15662e8 0.578594
\(426\) 0 0
\(427\) −1.82301e9 −1.13316
\(428\) −5.21543e8 −0.321542
\(429\) 0 0
\(430\) 2.54163e8 0.154160
\(431\) 1.88575e9 1.13452 0.567262 0.823538i \(-0.308003\pi\)
0.567262 + 0.823538i \(0.308003\pi\)
\(432\) 0 0
\(433\) −4.74195e8 −0.280705 −0.140352 0.990102i \(-0.544824\pi\)
−0.140352 + 0.990102i \(0.544824\pi\)
\(434\) 4.69643e9 2.75774
\(435\) 0 0
\(436\) −2.13958e9 −1.23631
\(437\) −6.57588e8 −0.376937
\(438\) 0 0
\(439\) 1.83494e9 1.03513 0.517565 0.855644i \(-0.326838\pi\)
0.517565 + 0.855644i \(0.326838\pi\)
\(440\) 2.64220e8 0.147870
\(441\) 0 0
\(442\) 2.37009e9 1.30553
\(443\) −2.78161e9 −1.52014 −0.760070 0.649841i \(-0.774836\pi\)
−0.760070 + 0.649841i \(0.774836\pi\)
\(444\) 0 0
\(445\) −7.46129e8 −0.401378
\(446\) 1.04098e9 0.555609
\(447\) 0 0
\(448\) −1.28024e9 −0.672694
\(449\) 2.43689e9 1.27050 0.635250 0.772307i \(-0.280897\pi\)
0.635250 + 0.772307i \(0.280897\pi\)
\(450\) 0 0
\(451\) −9.47766e8 −0.486500
\(452\) 1.46419e9 0.745782
\(453\) 0 0
\(454\) 5.06056e9 2.53807
\(455\) 1.33377e9 0.663805
\(456\) 0 0
\(457\) 3.65375e9 1.79074 0.895369 0.445324i \(-0.146912\pi\)
0.895369 + 0.445324i \(0.146912\pi\)
\(458\) 4.67990e9 2.27618
\(459\) 0 0
\(460\) 2.57837e8 0.123507
\(461\) 1.07049e9 0.508897 0.254449 0.967086i \(-0.418106\pi\)
0.254449 + 0.967086i \(0.418106\pi\)
\(462\) 0 0
\(463\) 3.98983e9 1.86819 0.934095 0.357025i \(-0.116209\pi\)
0.934095 + 0.357025i \(0.116209\pi\)
\(464\) 2.51872e9 1.17049
\(465\) 0 0
\(466\) −2.39038e9 −1.09425
\(467\) 3.19032e9 1.44952 0.724762 0.688999i \(-0.241950\pi\)
0.724762 + 0.688999i \(0.241950\pi\)
\(468\) 0 0
\(469\) 1.94629e9 0.871171
\(470\) 7.51365e8 0.333817
\(471\) 0 0
\(472\) 5.99659e7 0.0262487
\(473\) −2.49341e8 −0.108338
\(474\) 0 0
\(475\) −1.83881e9 −0.787244
\(476\) −3.04712e9 −1.29499
\(477\) 0 0
\(478\) 2.79886e9 1.17215
\(479\) 2.57106e9 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(480\) 0 0
\(481\) 1.23201e9 0.504787
\(482\) 7.26734e8 0.295604
\(483\) 0 0
\(484\) −9.81084e8 −0.393321
\(485\) 4.40356e8 0.175270
\(486\) 0 0
\(487\) 2.60295e9 1.02121 0.510604 0.859816i \(-0.329422\pi\)
0.510604 + 0.859816i \(0.329422\pi\)
\(488\) 6.61381e8 0.257622
\(489\) 0 0
\(490\) −1.24150e9 −0.476718
\(491\) −1.12697e9 −0.429662 −0.214831 0.976651i \(-0.568920\pi\)
−0.214831 + 0.976651i \(0.568920\pi\)
\(492\) 0 0
\(493\) 3.59801e9 1.35238
\(494\) −4.75956e9 −1.77633
\(495\) 0 0
\(496\) −5.36653e9 −1.97473
\(497\) −3.93184e9 −1.43664
\(498\) 0 0
\(499\) −4.16790e9 −1.50164 −0.750820 0.660507i \(-0.770341\pi\)
−0.750820 + 0.660507i \(0.770341\pi\)
\(500\) 2.39614e9 0.857268
\(501\) 0 0
\(502\) 7.04644e9 2.48603
\(503\) −4.24704e9 −1.48799 −0.743993 0.668187i \(-0.767070\pi\)
−0.743993 + 0.668187i \(0.767070\pi\)
\(504\) 0 0
\(505\) 3.37409e9 1.16583
\(506\) −5.71476e8 −0.196097
\(507\) 0 0
\(508\) −2.70778e9 −0.916412
\(509\) −4.51429e9 −1.51732 −0.758661 0.651486i \(-0.774146\pi\)
−0.758661 + 0.651486i \(0.774146\pi\)
\(510\) 0 0
\(511\) −4.73421e8 −0.156955
\(512\) 3.55781e9 1.17149
\(513\) 0 0
\(514\) −3.92771e9 −1.27576
\(515\) 2.41307e9 0.778475
\(516\) 0 0
\(517\) −7.37110e8 −0.234593
\(518\) −3.57859e9 −1.13125
\(519\) 0 0
\(520\) −4.83886e8 −0.150915
\(521\) 1.82834e9 0.566402 0.283201 0.959061i \(-0.408604\pi\)
0.283201 + 0.959061i \(0.408604\pi\)
\(522\) 0 0
\(523\) −8.74437e8 −0.267284 −0.133642 0.991030i \(-0.542667\pi\)
−0.133642 + 0.991030i \(0.542667\pi\)
\(524\) 1.55569e9 0.472350
\(525\) 0 0
\(526\) 3.33127e9 0.998066
\(527\) −7.66612e9 −2.28160
\(528\) 0 0
\(529\) −3.26023e9 −0.957531
\(530\) 1.75924e8 0.0513288
\(531\) 0 0
\(532\) 6.11916e9 1.76198
\(533\) 1.73572e9 0.496516
\(534\) 0 0
\(535\) 1.08239e9 0.305596
\(536\) −7.06108e8 −0.198059
\(537\) 0 0
\(538\) −7.87934e9 −2.18148
\(539\) 1.21795e9 0.335018
\(540\) 0 0
\(541\) 4.52769e9 1.22938 0.614690 0.788769i \(-0.289281\pi\)
0.614690 + 0.788769i \(0.289281\pi\)
\(542\) 4.25662e9 1.14833
\(543\) 0 0
\(544\) 6.47448e9 1.72429
\(545\) 4.44041e9 1.17499
\(546\) 0 0
\(547\) 2.38403e9 0.622811 0.311406 0.950277i \(-0.399200\pi\)
0.311406 + 0.950277i \(0.399200\pi\)
\(548\) 2.92755e9 0.759927
\(549\) 0 0
\(550\) −1.59802e9 −0.409555
\(551\) −7.22545e9 −1.84007
\(552\) 0 0
\(553\) −8.19061e9 −2.05958
\(554\) 3.74067e9 0.934685
\(555\) 0 0
\(556\) 2.45952e9 0.606860
\(557\) 2.04626e9 0.501727 0.250864 0.968022i \(-0.419285\pi\)
0.250864 + 0.968022i \(0.419285\pi\)
\(558\) 0 0
\(559\) 4.56638e8 0.110568
\(560\) 4.42693e9 1.06523
\(561\) 0 0
\(562\) −2.64269e9 −0.628014
\(563\) −3.85093e9 −0.909465 −0.454733 0.890628i \(-0.650265\pi\)
−0.454733 + 0.890628i \(0.650265\pi\)
\(564\) 0 0
\(565\) −3.03873e9 −0.708797
\(566\) −8.03541e9 −1.86273
\(567\) 0 0
\(568\) 1.42646e9 0.326618
\(569\) 1.30991e9 0.298091 0.149045 0.988830i \(-0.452380\pi\)
0.149045 + 0.988830i \(0.452380\pi\)
\(570\) 0 0
\(571\) 3.44368e9 0.774098 0.387049 0.922059i \(-0.373494\pi\)
0.387049 + 0.922059i \(0.373494\pi\)
\(572\) −1.83079e9 −0.409028
\(573\) 0 0
\(574\) −5.04168e9 −1.11271
\(575\) 4.04339e8 0.0886969
\(576\) 0 0
\(577\) 2.53520e9 0.549411 0.274706 0.961528i \(-0.411420\pi\)
0.274706 + 0.961528i \(0.411420\pi\)
\(578\) 5.01922e9 1.08116
\(579\) 0 0
\(580\) 2.83306e9 0.602917
\(581\) −6.05904e9 −1.28170
\(582\) 0 0
\(583\) −1.72587e8 −0.0360718
\(584\) 1.71755e8 0.0356833
\(585\) 0 0
\(586\) −3.20558e9 −0.658059
\(587\) 7.98017e9 1.62847 0.814233 0.580539i \(-0.197158\pi\)
0.814233 + 0.580539i \(0.197158\pi\)
\(588\) 0 0
\(589\) 1.53949e10 3.10437
\(590\) 4.79970e8 0.0962127
\(591\) 0 0
\(592\) 4.08920e9 0.810050
\(593\) −4.65675e8 −0.0917047 −0.0458523 0.998948i \(-0.514600\pi\)
−0.0458523 + 0.998948i \(0.514600\pi\)
\(594\) 0 0
\(595\) 6.32390e9 1.23077
\(596\) −1.91201e9 −0.369938
\(597\) 0 0
\(598\) 1.04659e9 0.200134
\(599\) −6.41918e9 −1.22035 −0.610177 0.792265i \(-0.708902\pi\)
−0.610177 + 0.792265i \(0.708902\pi\)
\(600\) 0 0
\(601\) −5.93318e9 −1.11488 −0.557438 0.830219i \(-0.688216\pi\)
−0.557438 + 0.830219i \(0.688216\pi\)
\(602\) −1.32638e9 −0.247788
\(603\) 0 0
\(604\) −5.37368e9 −0.992299
\(605\) 2.03611e9 0.373816
\(606\) 0 0
\(607\) −8.51996e9 −1.54624 −0.773121 0.634259i \(-0.781305\pi\)
−0.773121 + 0.634259i \(0.781305\pi\)
\(608\) −1.30019e10 −2.34609
\(609\) 0 0
\(610\) 5.29373e9 0.944293
\(611\) 1.34993e9 0.239423
\(612\) 0 0
\(613\) 5.16503e9 0.905652 0.452826 0.891599i \(-0.350416\pi\)
0.452826 + 0.891599i \(0.350416\pi\)
\(614\) −5.81152e9 −1.01321
\(615\) 0 0
\(616\) −1.37886e9 −0.237678
\(617\) 2.47374e9 0.423991 0.211995 0.977271i \(-0.432004\pi\)
0.211995 + 0.977271i \(0.432004\pi\)
\(618\) 0 0
\(619\) −5.19636e9 −0.880607 −0.440304 0.897849i \(-0.645129\pi\)
−0.440304 + 0.897849i \(0.645129\pi\)
\(620\) −6.03627e9 −1.01718
\(621\) 0 0
\(622\) −4.11895e9 −0.686309
\(623\) 3.89376e9 0.645151
\(624\) 0 0
\(625\) −2.34590e9 −0.384352
\(626\) 4.28577e9 0.698263
\(627\) 0 0
\(628\) −5.89627e9 −0.949989
\(629\) 5.84144e9 0.935929
\(630\) 0 0
\(631\) 4.50866e9 0.714405 0.357203 0.934027i \(-0.383731\pi\)
0.357203 + 0.934027i \(0.383731\pi\)
\(632\) 2.97152e9 0.468241
\(633\) 0 0
\(634\) −2.28681e9 −0.356384
\(635\) 5.61964e9 0.870965
\(636\) 0 0
\(637\) −2.23052e9 −0.341915
\(638\) −6.27926e9 −0.957274
\(639\) 0 0
\(640\) −2.70223e9 −0.407466
\(641\) 7.93187e8 0.118952 0.0594761 0.998230i \(-0.481057\pi\)
0.0594761 + 0.998230i \(0.481057\pi\)
\(642\) 0 0
\(643\) −1.11969e10 −1.66095 −0.830477 0.557053i \(-0.811932\pi\)
−0.830477 + 0.557053i \(0.811932\pi\)
\(644\) −1.34555e9 −0.198518
\(645\) 0 0
\(646\) −2.25669e10 −3.29350
\(647\) −1.19259e10 −1.73111 −0.865554 0.500815i \(-0.833034\pi\)
−0.865554 + 0.500815i \(0.833034\pi\)
\(648\) 0 0
\(649\) −4.70864e8 −0.0676144
\(650\) 2.92657e9 0.417986
\(651\) 0 0
\(652\) 1.00204e10 1.41585
\(653\) 2.22608e9 0.312857 0.156428 0.987689i \(-0.450002\pi\)
0.156428 + 0.987689i \(0.450002\pi\)
\(654\) 0 0
\(655\) −3.22863e9 −0.448925
\(656\) 5.76104e9 0.796778
\(657\) 0 0
\(658\) −3.92109e9 −0.536557
\(659\) 8.32960e7 0.0113377 0.00566885 0.999984i \(-0.498196\pi\)
0.00566885 + 0.999984i \(0.498196\pi\)
\(660\) 0 0
\(661\) 6.64097e9 0.894389 0.447194 0.894437i \(-0.352423\pi\)
0.447194 + 0.894437i \(0.352423\pi\)
\(662\) 7.31629e9 0.980140
\(663\) 0 0
\(664\) 2.19820e9 0.291392
\(665\) −1.26995e10 −1.67460
\(666\) 0 0
\(667\) 1.58881e9 0.207316
\(668\) −3.35741e9 −0.435800
\(669\) 0 0
\(670\) −5.65172e9 −0.725970
\(671\) −5.19329e9 −0.663612
\(672\) 0 0
\(673\) 3.72281e9 0.470780 0.235390 0.971901i \(-0.424363\pi\)
0.235390 + 0.971901i \(0.424363\pi\)
\(674\) −7.86707e9 −0.989699
\(675\) 0 0
\(676\) −3.02518e9 −0.376649
\(677\) −1.49598e10 −1.85296 −0.926480 0.376343i \(-0.877181\pi\)
−0.926480 + 0.376343i \(0.877181\pi\)
\(678\) 0 0
\(679\) −2.29805e9 −0.281719
\(680\) −2.29429e9 −0.279812
\(681\) 0 0
\(682\) 1.33789e10 1.61501
\(683\) −6.36282e9 −0.764148 −0.382074 0.924132i \(-0.624790\pi\)
−0.382074 + 0.924132i \(0.624790\pi\)
\(684\) 0 0
\(685\) −6.07573e9 −0.722240
\(686\) −7.25989e9 −0.858609
\(687\) 0 0
\(688\) 1.51563e9 0.177433
\(689\) 3.16071e8 0.0368144
\(690\) 0 0
\(691\) 1.18625e10 1.36774 0.683870 0.729604i \(-0.260296\pi\)
0.683870 + 0.729604i \(0.260296\pi\)
\(692\) 9.36957e9 1.07485
\(693\) 0 0
\(694\) −3.29507e9 −0.374203
\(695\) −5.10441e9 −0.576764
\(696\) 0 0
\(697\) 8.22969e9 0.920595
\(698\) −1.86917e9 −0.208044
\(699\) 0 0
\(700\) −3.76256e9 −0.414611
\(701\) −8.69584e9 −0.953451 −0.476725 0.879052i \(-0.658176\pi\)
−0.476725 + 0.879052i \(0.658176\pi\)
\(702\) 0 0
\(703\) −1.17306e10 −1.27344
\(704\) −3.64707e9 −0.393949
\(705\) 0 0
\(706\) −5.41539e9 −0.579179
\(707\) −1.76081e10 −1.87389
\(708\) 0 0
\(709\) −6.50844e9 −0.685828 −0.342914 0.939367i \(-0.611414\pi\)
−0.342914 + 0.939367i \(0.611414\pi\)
\(710\) 1.14174e10 1.19719
\(711\) 0 0
\(712\) −1.41264e9 −0.146674
\(713\) −3.38521e9 −0.349762
\(714\) 0 0
\(715\) 3.79957e9 0.388743
\(716\) −8.35606e9 −0.850758
\(717\) 0 0
\(718\) 4.38847e9 0.442464
\(719\) 3.90453e9 0.391757 0.195879 0.980628i \(-0.437244\pi\)
0.195879 + 0.980628i \(0.437244\pi\)
\(720\) 0 0
\(721\) −1.25929e10 −1.25127
\(722\) 3.17725e10 3.14175
\(723\) 0 0
\(724\) −4.60700e9 −0.451163
\(725\) 4.44280e9 0.432986
\(726\) 0 0
\(727\) −1.72837e10 −1.66827 −0.834133 0.551563i \(-0.814032\pi\)
−0.834133 + 0.551563i \(0.814032\pi\)
\(728\) 2.52522e9 0.242571
\(729\) 0 0
\(730\) 1.37474e9 0.130795
\(731\) 2.16509e9 0.205005
\(732\) 0 0
\(733\) 8.26113e9 0.774775 0.387387 0.921917i \(-0.373378\pi\)
0.387387 + 0.921917i \(0.373378\pi\)
\(734\) −1.35174e10 −1.26170
\(735\) 0 0
\(736\) 2.85901e9 0.264328
\(737\) 5.54450e9 0.510183
\(738\) 0 0
\(739\) 6.43474e9 0.586510 0.293255 0.956034i \(-0.405261\pi\)
0.293255 + 0.956034i \(0.405261\pi\)
\(740\) 4.59952e9 0.417255
\(741\) 0 0
\(742\) −9.18083e8 −0.0825027
\(743\) −9.72354e9 −0.869689 −0.434844 0.900506i \(-0.643197\pi\)
−0.434844 + 0.900506i \(0.643197\pi\)
\(744\) 0 0
\(745\) 3.96813e9 0.351592
\(746\) 2.59344e10 2.28712
\(747\) 0 0
\(748\) −8.68048e9 −0.758383
\(749\) −5.64860e9 −0.491196
\(750\) 0 0
\(751\) −1.14363e10 −0.985252 −0.492626 0.870241i \(-0.663963\pi\)
−0.492626 + 0.870241i \(0.663963\pi\)
\(752\) 4.48056e9 0.384211
\(753\) 0 0
\(754\) 1.14997e10 0.976982
\(755\) 1.11524e10 0.943088
\(756\) 0 0
\(757\) −7.15016e9 −0.599073 −0.299537 0.954085i \(-0.596832\pi\)
−0.299537 + 0.954085i \(0.596832\pi\)
\(758\) 7.18573e8 0.0599278
\(759\) 0 0
\(760\) 4.60733e9 0.380717
\(761\) −3.88824e9 −0.319821 −0.159910 0.987132i \(-0.551121\pi\)
−0.159910 + 0.987132i \(0.551121\pi\)
\(762\) 0 0
\(763\) −2.31728e10 −1.88861
\(764\) 1.23596e10 1.00271
\(765\) 0 0
\(766\) 4.59073e9 0.369046
\(767\) 8.62330e8 0.0690064
\(768\) 0 0
\(769\) −1.58580e10 −1.25750 −0.628748 0.777609i \(-0.716432\pi\)
−0.628748 + 0.777609i \(0.716432\pi\)
\(770\) −1.10365e10 −0.871191
\(771\) 0 0
\(772\) 1.42669e10 1.11601
\(773\) 6.91144e9 0.538196 0.269098 0.963113i \(-0.413274\pi\)
0.269098 + 0.963113i \(0.413274\pi\)
\(774\) 0 0
\(775\) −9.46606e9 −0.730488
\(776\) 8.33725e8 0.0640482
\(777\) 0 0
\(778\) −3.75987e9 −0.286249
\(779\) −1.65267e10 −1.25258
\(780\) 0 0
\(781\) −1.12008e10 −0.841340
\(782\) 4.96227e9 0.371071
\(783\) 0 0
\(784\) −7.40336e9 −0.548684
\(785\) 1.22369e10 0.902877
\(786\) 0 0
\(787\) −8.31218e9 −0.607860 −0.303930 0.952694i \(-0.598299\pi\)
−0.303930 + 0.952694i \(0.598299\pi\)
\(788\) 5.77913e9 0.420747
\(789\) 0 0
\(790\) 2.37842e10 1.71630
\(791\) 1.58580e10 1.13928
\(792\) 0 0
\(793\) 9.51087e9 0.677274
\(794\) −1.77892e10 −1.26120
\(795\) 0 0
\(796\) 3.34882e9 0.235340
\(797\) −2.22972e10 −1.56008 −0.780038 0.625733i \(-0.784800\pi\)
−0.780038 + 0.625733i \(0.784800\pi\)
\(798\) 0 0
\(799\) 6.40051e9 0.443916
\(800\) 7.99464e9 0.552057
\(801\) 0 0
\(802\) 7.10996e9 0.486695
\(803\) −1.34866e9 −0.0919172
\(804\) 0 0
\(805\) 2.79251e9 0.188673
\(806\) −2.45019e10 −1.64826
\(807\) 0 0
\(808\) 6.38814e9 0.426025
\(809\) −1.32718e10 −0.881275 −0.440638 0.897685i \(-0.645248\pi\)
−0.440638 + 0.897685i \(0.645248\pi\)
\(810\) 0 0
\(811\) 4.23818e9 0.279002 0.139501 0.990222i \(-0.455450\pi\)
0.139501 + 0.990222i \(0.455450\pi\)
\(812\) −1.47847e10 −0.969092
\(813\) 0 0
\(814\) −1.01945e10 −0.662492
\(815\) −2.07960e10 −1.34564
\(816\) 0 0
\(817\) −4.34789e9 −0.278934
\(818\) −3.57714e10 −2.28507
\(819\) 0 0
\(820\) 6.48001e9 0.410419
\(821\) 1.61244e10 1.01691 0.508455 0.861089i \(-0.330217\pi\)
0.508455 + 0.861089i \(0.330217\pi\)
\(822\) 0 0
\(823\) −1.67540e9 −0.104766 −0.0523829 0.998627i \(-0.516682\pi\)
−0.0523829 + 0.998627i \(0.516682\pi\)
\(824\) 4.56865e9 0.284474
\(825\) 0 0
\(826\) −2.50478e9 −0.154646
\(827\) 2.26903e10 1.39499 0.697496 0.716588i \(-0.254297\pi\)
0.697496 + 0.716588i \(0.254297\pi\)
\(828\) 0 0
\(829\) −2.14017e10 −1.30469 −0.652345 0.757922i \(-0.726215\pi\)
−0.652345 + 0.757922i \(0.726215\pi\)
\(830\) 1.75945e10 1.06808
\(831\) 0 0
\(832\) 6.67916e9 0.402059
\(833\) −1.05758e10 −0.633948
\(834\) 0 0
\(835\) 6.96786e9 0.414187
\(836\) 1.74319e10 1.03187
\(837\) 0 0
\(838\) 3.87854e10 2.27674
\(839\) −3.21757e10 −1.88088 −0.940439 0.339962i \(-0.889586\pi\)
−0.940439 + 0.339962i \(0.889586\pi\)
\(840\) 0 0
\(841\) 2.07698e8 0.0120406
\(842\) −5.51774e8 −0.0318544
\(843\) 0 0
\(844\) −2.38953e10 −1.36809
\(845\) 6.27835e9 0.357970
\(846\) 0 0
\(847\) −1.06257e10 −0.600848
\(848\) 1.04908e9 0.0590775
\(849\) 0 0
\(850\) 1.38760e10 0.774992
\(851\) 2.57947e9 0.143475
\(852\) 0 0
\(853\) −8.66660e9 −0.478109 −0.239055 0.971006i \(-0.576838\pi\)
−0.239055 + 0.971006i \(0.576838\pi\)
\(854\) −2.76259e10 −1.51780
\(855\) 0 0
\(856\) 2.04929e9 0.111672
\(857\) −4.64158e9 −0.251903 −0.125951 0.992036i \(-0.540198\pi\)
−0.125951 + 0.992036i \(0.540198\pi\)
\(858\) 0 0
\(859\) 1.96063e10 1.05541 0.527703 0.849429i \(-0.323053\pi\)
0.527703 + 0.849429i \(0.323053\pi\)
\(860\) 1.70478e9 0.0913954
\(861\) 0 0
\(862\) 2.85767e10 1.51963
\(863\) −2.94733e10 −1.56096 −0.780478 0.625183i \(-0.785024\pi\)
−0.780478 + 0.625183i \(0.785024\pi\)
\(864\) 0 0
\(865\) −1.94453e10 −1.02155
\(866\) −7.18597e9 −0.375987
\(867\) 0 0
\(868\) 3.15010e10 1.63495
\(869\) −2.33330e10 −1.20615
\(870\) 0 0
\(871\) −1.01541e10 −0.520686
\(872\) 8.40702e9 0.429372
\(873\) 0 0
\(874\) −9.96511e9 −0.504885
\(875\) 2.59515e10 1.30959
\(876\) 0 0
\(877\) −2.55789e9 −0.128051 −0.0640256 0.997948i \(-0.520394\pi\)
−0.0640256 + 0.997948i \(0.520394\pi\)
\(878\) 2.78067e10 1.38650
\(879\) 0 0
\(880\) 1.26112e10 0.623831
\(881\) 2.66718e10 1.31412 0.657062 0.753837i \(-0.271799\pi\)
0.657062 + 0.753837i \(0.271799\pi\)
\(882\) 0 0
\(883\) −4.99020e9 −0.243924 −0.121962 0.992535i \(-0.538919\pi\)
−0.121962 + 0.992535i \(0.538919\pi\)
\(884\) 1.58972e10 0.773996
\(885\) 0 0
\(886\) −4.21527e10 −2.03614
\(887\) 2.25283e10 1.08392 0.541959 0.840405i \(-0.317683\pi\)
0.541959 + 0.840405i \(0.317683\pi\)
\(888\) 0 0
\(889\) −2.93268e10 −1.39994
\(890\) −1.13069e10 −0.537622
\(891\) 0 0
\(892\) 6.98228e9 0.329398
\(893\) −1.28534e10 −0.603999
\(894\) 0 0
\(895\) 1.73419e10 0.808567
\(896\) 1.41019e10 0.654937
\(897\) 0 0
\(898\) 3.69288e10 1.70176
\(899\) −3.71960e10 −1.70741
\(900\) 0 0
\(901\) 1.49861e9 0.0682579
\(902\) −1.43625e10 −0.651638
\(903\) 0 0
\(904\) −5.75321e9 −0.259012
\(905\) 9.56121e9 0.428788
\(906\) 0 0
\(907\) 2.20475e10 0.981145 0.490573 0.871400i \(-0.336788\pi\)
0.490573 + 0.871400i \(0.336788\pi\)
\(908\) 3.39434e10 1.50472
\(909\) 0 0
\(910\) 2.02120e10 0.889127
\(911\) −4.20990e9 −0.184484 −0.0922418 0.995737i \(-0.529403\pi\)
−0.0922418 + 0.995737i \(0.529403\pi\)
\(912\) 0 0
\(913\) −1.72607e10 −0.750602
\(914\) 5.53690e10 2.39859
\(915\) 0 0
\(916\) 3.13901e10 1.34945
\(917\) 1.68490e10 0.721574
\(918\) 0 0
\(919\) −1.55285e10 −0.659972 −0.329986 0.943986i \(-0.607044\pi\)
−0.329986 + 0.943986i \(0.607044\pi\)
\(920\) −1.01311e9 −0.0428944
\(921\) 0 0
\(922\) 1.62223e10 0.681637
\(923\) 2.05129e10 0.858661
\(924\) 0 0
\(925\) 7.21296e9 0.299652
\(926\) 6.04620e10 2.50233
\(927\) 0 0
\(928\) 3.14142e10 1.29035
\(929\) 2.69496e10 1.10280 0.551401 0.834240i \(-0.314093\pi\)
0.551401 + 0.834240i \(0.314093\pi\)
\(930\) 0 0
\(931\) 2.12380e10 0.862560
\(932\) −1.60333e10 −0.648736
\(933\) 0 0
\(934\) 4.83462e10 1.94155
\(935\) 1.80152e10 0.720772
\(936\) 0 0
\(937\) −1.04584e10 −0.415315 −0.207657 0.978202i \(-0.566584\pi\)
−0.207657 + 0.978202i \(0.566584\pi\)
\(938\) 2.94942e10 1.16688
\(939\) 0 0
\(940\) 5.03973e9 0.197906
\(941\) −2.29124e10 −0.896412 −0.448206 0.893930i \(-0.647937\pi\)
−0.448206 + 0.893930i \(0.647937\pi\)
\(942\) 0 0
\(943\) 3.63407e9 0.141125
\(944\) 2.86217e9 0.110737
\(945\) 0 0
\(946\) −3.77852e9 −0.145112
\(947\) 2.22008e10 0.849460 0.424730 0.905320i \(-0.360369\pi\)
0.424730 + 0.905320i \(0.360369\pi\)
\(948\) 0 0
\(949\) 2.46990e9 0.0938096
\(950\) −2.78654e10 −1.05447
\(951\) 0 0
\(952\) 1.19730e10 0.449753
\(953\) −1.45448e10 −0.544357 −0.272178 0.962247i \(-0.587744\pi\)
−0.272178 + 0.962247i \(0.587744\pi\)
\(954\) 0 0
\(955\) −2.56506e10 −0.952985
\(956\) 1.87732e10 0.694920
\(957\) 0 0
\(958\) 3.89619e10 1.43173
\(959\) 3.17069e10 1.16088
\(960\) 0 0
\(961\) 5.17393e10 1.88057
\(962\) 1.86700e10 0.676131
\(963\) 0 0
\(964\) 4.87452e9 0.175252
\(965\) −2.96090e10 −1.06067
\(966\) 0 0
\(967\) −4.60488e10 −1.63767 −0.818833 0.574032i \(-0.805379\pi\)
−0.818833 + 0.574032i \(0.805379\pi\)
\(968\) 3.85496e9 0.136602
\(969\) 0 0
\(970\) 6.67317e9 0.234764
\(971\) −2.30696e10 −0.808671 −0.404335 0.914611i \(-0.632497\pi\)
−0.404335 + 0.914611i \(0.632497\pi\)
\(972\) 0 0
\(973\) 2.66380e10 0.927056
\(974\) 3.94451e10 1.36785
\(975\) 0 0
\(976\) 3.15677e10 1.08685
\(977\) −5.54001e10 −1.90055 −0.950276 0.311408i \(-0.899199\pi\)
−0.950276 + 0.311408i \(0.899199\pi\)
\(978\) 0 0
\(979\) 1.10923e10 0.377819
\(980\) −8.32729e9 −0.282626
\(981\) 0 0
\(982\) −1.70781e10 −0.575506
\(983\) 3.63783e10 1.22153 0.610766 0.791811i \(-0.290861\pi\)
0.610766 + 0.791811i \(0.290861\pi\)
\(984\) 0 0
\(985\) −1.19938e10 −0.399881
\(986\) 5.45244e10 1.81143
\(987\) 0 0
\(988\) −3.19244e10 −1.05311
\(989\) 9.56063e8 0.0314268
\(990\) 0 0
\(991\) 1.33718e9 0.0436446 0.0218223 0.999762i \(-0.493053\pi\)
0.0218223 + 0.999762i \(0.493053\pi\)
\(992\) −6.69328e10 −2.17695
\(993\) 0 0
\(994\) −5.95833e10 −1.92430
\(995\) −6.95003e9 −0.223669
\(996\) 0 0
\(997\) 2.40234e10 0.767718 0.383859 0.923392i \(-0.374595\pi\)
0.383859 + 0.923392i \(0.374595\pi\)
\(998\) −6.31605e10 −2.01135
\(999\) 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.1.11 13
3.2 odd 2 43.8.a.b.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.3 13 3.2 odd 2
387.8.a.d.1.11 13 1.1 even 1 trivial