Properties

Label 387.10.a.c.1.11
Level 387
Weight 10
Character 387.1
Self dual yes
Analytic conductor 199.319
Analytic rank 0
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(199.318868595\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} - 11474166224 x^{9} + 47465836576 x^{8} + 5986976782464 x^{7} - 32493903147264 x^{6} - 1516975415483904 x^{5} + 10892588268404224 x^{4} + 139803541742443008 x^{3} - 1349125586394823680 x^{2} + 2103623681144094720 x + 529838441422848000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(16.4876\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q+18.4876 q^{2} -170.210 q^{4} +2363.43 q^{5} +7025.62 q^{7} -12612.4 q^{8} +O(q^{10})\) \(q+18.4876 q^{2} -170.210 q^{4} +2363.43 q^{5} +7025.62 q^{7} -12612.4 q^{8} +43694.1 q^{10} +59098.5 q^{11} -84020.4 q^{13} +129887. q^{14} -146025. q^{16} -199182. q^{17} +293458. q^{19} -402279. q^{20} +1.09259e6 q^{22} -366554. q^{23} +3.63268e6 q^{25} -1.55333e6 q^{26} -1.19583e6 q^{28} +1.87481e6 q^{29} +3.91740e6 q^{31} +3.75790e6 q^{32} -3.68239e6 q^{34} +1.66046e7 q^{35} +2.22050e7 q^{37} +5.42532e6 q^{38} -2.98085e7 q^{40} -3.33190e6 q^{41} -3.41880e6 q^{43} -1.00591e7 q^{44} -6.77669e6 q^{46} +4.54124e7 q^{47} +9.00569e6 q^{49} +6.71594e7 q^{50} +1.43011e7 q^{52} -4.88629e7 q^{53} +1.39675e8 q^{55} -8.86099e7 q^{56} +3.46607e7 q^{58} -1.66035e8 q^{59} -1.65599e8 q^{61} +7.24233e7 q^{62} +1.44239e8 q^{64} -1.98576e8 q^{65} +1.13400e7 q^{67} +3.39027e7 q^{68} +3.06978e8 q^{70} +2.09329e8 q^{71} +7.39810e6 q^{73} +4.10516e8 q^{74} -4.99494e7 q^{76} +4.15204e8 q^{77} +5.09343e8 q^{79} -3.45120e8 q^{80} -6.15988e7 q^{82} +4.59491e8 q^{83} -4.70753e8 q^{85} -6.32053e7 q^{86} -7.45374e8 q^{88} +2.28949e8 q^{89} -5.90295e8 q^{91} +6.23911e7 q^{92} +8.39565e8 q^{94} +6.93567e8 q^{95} -9.54121e8 q^{97} +1.66493e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 32q^{2} + 3242q^{4} + 4717q^{5} - 9680q^{7} + 20394q^{8} + O(q^{10}) \) \( 15q + 32q^{2} + 3242q^{4} + 4717q^{5} - 9680q^{7} + 20394q^{8} - 36237q^{10} + 104484q^{11} - 116174q^{13} - 416064q^{14} + 996762q^{16} + 884265q^{17} - 689535q^{19} + 3077879q^{20} - 7276218q^{22} + 2504077q^{23} + 1315350q^{25} + 13343414q^{26} - 28059568q^{28} + 18406221q^{29} - 12033699q^{31} + 18952630q^{32} - 30383125q^{34} + 27855546q^{35} - 8722847q^{37} + 63941843q^{38} - 39665611q^{40} + 18689389q^{41} - 51282015q^{43} + 68723220q^{44} - 2067521q^{46} + 104960741q^{47} + 92663095q^{49} + 42446347q^{50} + 149226080q^{52} + 215907800q^{53} + 384379852q^{55} - 430441344q^{56} + 295963139q^{58} - 185924544q^{59} + 247538102q^{61} - 139798853q^{62} + 848556290q^{64} - 94294394q^{65} + 467904656q^{67} + 88234341q^{68} + 647526126q^{70} + 8252944q^{71} - 715627902q^{73} - 725122989q^{74} + 346300359q^{76} + 1236779964q^{77} + 560681783q^{79} + 1157214179q^{80} + 941346367q^{82} + 1442854698q^{83} + 699302088q^{85} - 109401632q^{86} - 1464507256q^{88} + 396710008q^{89} - 3278076852q^{91} - 155864647q^{92} + 4666638949q^{94} + 3854114395q^{95} - 3063837815q^{97} + 6161086984q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.4876 0.817043 0.408521 0.912749i \(-0.366045\pi\)
0.408521 + 0.912749i \(0.366045\pi\)
\(3\) 0 0
\(4\) −170.210 −0.332441
\(5\) 2363.43 1.69113 0.845567 0.533870i \(-0.179263\pi\)
0.845567 + 0.533870i \(0.179263\pi\)
\(6\) 0 0
\(7\) 7025.62 1.10597 0.552985 0.833191i \(-0.313489\pi\)
0.552985 + 0.833191i \(0.313489\pi\)
\(8\) −12612.4 −1.08866
\(9\) 0 0
\(10\) 43694.1 1.38173
\(11\) 59098.5 1.21705 0.608526 0.793534i \(-0.291761\pi\)
0.608526 + 0.793534i \(0.291761\pi\)
\(12\) 0 0
\(13\) −84020.4 −0.815905 −0.407952 0.913003i \(-0.633757\pi\)
−0.407952 + 0.913003i \(0.633757\pi\)
\(14\) 129887. 0.903625
\(15\) 0 0
\(16\) −146025. −0.557042
\(17\) −199182. −0.578402 −0.289201 0.957268i \(-0.593390\pi\)
−0.289201 + 0.957268i \(0.593390\pi\)
\(18\) 0 0
\(19\) 293458. 0.516600 0.258300 0.966065i \(-0.416838\pi\)
0.258300 + 0.966065i \(0.416838\pi\)
\(20\) −402279. −0.562202
\(21\) 0 0
\(22\) 1.09259e6 0.994384
\(23\) −366554. −0.273126 −0.136563 0.990631i \(-0.543606\pi\)
−0.136563 + 0.990631i \(0.543606\pi\)
\(24\) 0 0
\(25\) 3.63268e6 1.85993
\(26\) −1.55333e6 −0.666629
\(27\) 0 0
\(28\) −1.19583e6 −0.367670
\(29\) 1.87481e6 0.492228 0.246114 0.969241i \(-0.420846\pi\)
0.246114 + 0.969241i \(0.420846\pi\)
\(30\) 0 0
\(31\) 3.91740e6 0.761852 0.380926 0.924605i \(-0.375605\pi\)
0.380926 + 0.924605i \(0.375605\pi\)
\(32\) 3.75790e6 0.633534
\(33\) 0 0
\(34\) −3.68239e6 −0.472579
\(35\) 1.66046e7 1.87034
\(36\) 0 0
\(37\) 2.22050e7 1.94779 0.973895 0.226997i \(-0.0728907\pi\)
0.973895 + 0.226997i \(0.0728907\pi\)
\(38\) 5.42532e6 0.422084
\(39\) 0 0
\(40\) −2.98085e7 −1.84107
\(41\) −3.33190e6 −0.184147 −0.0920736 0.995752i \(-0.529350\pi\)
−0.0920736 + 0.995752i \(0.529350\pi\)
\(42\) 0 0
\(43\) −3.41880e6 −0.152499
\(44\) −1.00591e7 −0.404598
\(45\) 0 0
\(46\) −6.77669e6 −0.223156
\(47\) 4.54124e7 1.35748 0.678741 0.734378i \(-0.262526\pi\)
0.678741 + 0.734378i \(0.262526\pi\)
\(48\) 0 0
\(49\) 9.00569e6 0.223169
\(50\) 6.71594e7 1.51964
\(51\) 0 0
\(52\) 1.43011e7 0.271240
\(53\) −4.88629e7 −0.850624 −0.425312 0.905047i \(-0.639836\pi\)
−0.425312 + 0.905047i \(0.639836\pi\)
\(54\) 0 0
\(55\) 1.39675e8 2.05820
\(56\) −8.86099e7 −1.20403
\(57\) 0 0
\(58\) 3.46607e7 0.402172
\(59\) −1.66035e8 −1.78387 −0.891937 0.452159i \(-0.850654\pi\)
−0.891937 + 0.452159i \(0.850654\pi\)
\(60\) 0 0
\(61\) −1.65599e8 −1.53135 −0.765673 0.643230i \(-0.777594\pi\)
−0.765673 + 0.643230i \(0.777594\pi\)
\(62\) 7.24233e7 0.622466
\(63\) 0 0
\(64\) 1.44239e8 1.07467
\(65\) −1.98576e8 −1.37980
\(66\) 0 0
\(67\) 1.13400e7 0.0687507 0.0343753 0.999409i \(-0.489056\pi\)
0.0343753 + 0.999409i \(0.489056\pi\)
\(68\) 3.39027e7 0.192285
\(69\) 0 0
\(70\) 3.06978e8 1.52815
\(71\) 2.09329e8 0.977611 0.488806 0.872393i \(-0.337433\pi\)
0.488806 + 0.872393i \(0.337433\pi\)
\(72\) 0 0
\(73\) 7.39810e6 0.0304907 0.0152453 0.999884i \(-0.495147\pi\)
0.0152453 + 0.999884i \(0.495147\pi\)
\(74\) 4.10516e8 1.59143
\(75\) 0 0
\(76\) −4.99494e7 −0.171739
\(77\) 4.15204e8 1.34602
\(78\) 0 0
\(79\) 5.09343e8 1.47126 0.735629 0.677385i \(-0.236887\pi\)
0.735629 + 0.677385i \(0.236887\pi\)
\(80\) −3.45120e8 −0.942032
\(81\) 0 0
\(82\) −6.15988e7 −0.150456
\(83\) 4.59491e8 1.06274 0.531369 0.847141i \(-0.321678\pi\)
0.531369 + 0.847141i \(0.321678\pi\)
\(84\) 0 0
\(85\) −4.70753e8 −0.978155
\(86\) −6.32053e7 −0.124598
\(87\) 0 0
\(88\) −7.45374e8 −1.32496
\(89\) 2.28949e8 0.386798 0.193399 0.981120i \(-0.438049\pi\)
0.193399 + 0.981120i \(0.438049\pi\)
\(90\) 0 0
\(91\) −5.90295e8 −0.902366
\(92\) 6.23911e7 0.0907983
\(93\) 0 0
\(94\) 8.39565e8 1.10912
\(95\) 6.93567e8 0.873639
\(96\) 0 0
\(97\) −9.54121e8 −1.09429 −0.547143 0.837039i \(-0.684285\pi\)
−0.547143 + 0.837039i \(0.684285\pi\)
\(98\) 1.66493e8 0.182339
\(99\) 0 0
\(100\) −6.18318e8 −0.618318
\(101\) −1.29443e9 −1.23775 −0.618876 0.785489i \(-0.712412\pi\)
−0.618876 + 0.785489i \(0.712412\pi\)
\(102\) 0 0
\(103\) 5.67850e8 0.497126 0.248563 0.968616i \(-0.420042\pi\)
0.248563 + 0.968616i \(0.420042\pi\)
\(104\) 1.05970e9 0.888244
\(105\) 0 0
\(106\) −9.03356e8 −0.694996
\(107\) −4.53655e8 −0.334579 −0.167290 0.985908i \(-0.553501\pi\)
−0.167290 + 0.985908i \(0.553501\pi\)
\(108\) 0 0
\(109\) 9.29530e8 0.630731 0.315365 0.948970i \(-0.397873\pi\)
0.315365 + 0.948970i \(0.397873\pi\)
\(110\) 2.58226e9 1.68164
\(111\) 0 0
\(112\) −1.02592e9 −0.616071
\(113\) 3.79267e7 0.0218822 0.0109411 0.999940i \(-0.496517\pi\)
0.0109411 + 0.999940i \(0.496517\pi\)
\(114\) 0 0
\(115\) −8.66325e8 −0.461892
\(116\) −3.19111e8 −0.163637
\(117\) 0 0
\(118\) −3.06957e9 −1.45750
\(119\) −1.39938e9 −0.639696
\(120\) 0 0
\(121\) 1.13469e9 0.481218
\(122\) −3.06152e9 −1.25118
\(123\) 0 0
\(124\) −6.66781e8 −0.253271
\(125\) 3.96951e9 1.45426
\(126\) 0 0
\(127\) −3.97458e9 −1.35573 −0.677867 0.735185i \(-0.737095\pi\)
−0.677867 + 0.735185i \(0.737095\pi\)
\(128\) 7.42589e8 0.244514
\(129\) 0 0
\(130\) −3.67119e9 −1.12736
\(131\) 1.80701e9 0.536093 0.268047 0.963406i \(-0.413622\pi\)
0.268047 + 0.963406i \(0.413622\pi\)
\(132\) 0 0
\(133\) 2.06172e9 0.571344
\(134\) 2.09649e8 0.0561722
\(135\) 0 0
\(136\) 2.51216e9 0.629684
\(137\) 2.82788e9 0.685834 0.342917 0.939366i \(-0.388585\pi\)
0.342917 + 0.939366i \(0.388585\pi\)
\(138\) 0 0
\(139\) 6.82579e9 1.55091 0.775454 0.631404i \(-0.217521\pi\)
0.775454 + 0.631404i \(0.217521\pi\)
\(140\) −2.82626e9 −0.621779
\(141\) 0 0
\(142\) 3.86998e9 0.798750
\(143\) −4.96548e9 −0.992999
\(144\) 0 0
\(145\) 4.43099e9 0.832424
\(146\) 1.36773e8 0.0249122
\(147\) 0 0
\(148\) −3.77950e9 −0.647526
\(149\) 9.22449e9 1.53322 0.766610 0.642113i \(-0.221942\pi\)
0.766610 + 0.642113i \(0.221942\pi\)
\(150\) 0 0
\(151\) 7.71760e9 1.20805 0.604027 0.796964i \(-0.293562\pi\)
0.604027 + 0.796964i \(0.293562\pi\)
\(152\) −3.70121e9 −0.562402
\(153\) 0 0
\(154\) 7.67610e9 1.09976
\(155\) 9.25851e9 1.28839
\(156\) 0 0
\(157\) −1.03691e10 −1.36205 −0.681027 0.732259i \(-0.738466\pi\)
−0.681027 + 0.732259i \(0.738466\pi\)
\(158\) 9.41652e9 1.20208
\(159\) 0 0
\(160\) 8.88153e9 1.07139
\(161\) −2.57527e9 −0.302069
\(162\) 0 0
\(163\) 1.24179e10 1.37786 0.688929 0.724829i \(-0.258081\pi\)
0.688929 + 0.724829i \(0.258081\pi\)
\(164\) 5.67123e8 0.0612181
\(165\) 0 0
\(166\) 8.49488e9 0.868302
\(167\) −8.49813e8 −0.0845472 −0.0422736 0.999106i \(-0.513460\pi\)
−0.0422736 + 0.999106i \(0.513460\pi\)
\(168\) 0 0
\(169\) −3.54508e9 −0.334300
\(170\) −8.70308e9 −0.799195
\(171\) 0 0
\(172\) 5.81914e8 0.0506968
\(173\) 2.05242e10 1.74204 0.871022 0.491244i \(-0.163458\pi\)
0.871022 + 0.491244i \(0.163458\pi\)
\(174\) 0 0
\(175\) 2.55218e10 2.05703
\(176\) −8.62987e9 −0.677949
\(177\) 0 0
\(178\) 4.23271e9 0.316030
\(179\) −2.24623e8 −0.0163537 −0.00817684 0.999967i \(-0.502603\pi\)
−0.00817684 + 0.999967i \(0.502603\pi\)
\(180\) 0 0
\(181\) 1.28374e10 0.889043 0.444522 0.895768i \(-0.353374\pi\)
0.444522 + 0.895768i \(0.353374\pi\)
\(182\) −1.09131e10 −0.737272
\(183\) 0 0
\(184\) 4.62313e9 0.297342
\(185\) 5.24799e10 3.29397
\(186\) 0 0
\(187\) −1.17714e10 −0.703946
\(188\) −7.72964e9 −0.451283
\(189\) 0 0
\(190\) 1.28224e10 0.713801
\(191\) −1.84958e9 −0.100560 −0.0502798 0.998735i \(-0.516011\pi\)
−0.0502798 + 0.998735i \(0.516011\pi\)
\(192\) 0 0
\(193\) 7.73188e9 0.401123 0.200561 0.979681i \(-0.435723\pi\)
0.200561 + 0.979681i \(0.435723\pi\)
\(194\) −1.76394e10 −0.894078
\(195\) 0 0
\(196\) −1.53286e9 −0.0741907
\(197\) −2.41118e10 −1.14060 −0.570298 0.821438i \(-0.693172\pi\)
−0.570298 + 0.821438i \(0.693172\pi\)
\(198\) 0 0
\(199\) 2.46274e10 1.11322 0.556609 0.830775i \(-0.312102\pi\)
0.556609 + 0.830775i \(0.312102\pi\)
\(200\) −4.58168e10 −2.02484
\(201\) 0 0
\(202\) −2.39309e10 −1.01130
\(203\) 1.31717e10 0.544390
\(204\) 0 0
\(205\) −7.87472e9 −0.311417
\(206\) 1.04982e10 0.406173
\(207\) 0 0
\(208\) 1.22691e10 0.454493
\(209\) 1.73429e10 0.628729
\(210\) 0 0
\(211\) −1.54470e10 −0.536504 −0.268252 0.963349i \(-0.586446\pi\)
−0.268252 + 0.963349i \(0.586446\pi\)
\(212\) 8.31695e9 0.282782
\(213\) 0 0
\(214\) −8.38698e9 −0.273365
\(215\) −8.08010e9 −0.257895
\(216\) 0 0
\(217\) 2.75222e10 0.842586
\(218\) 1.71847e10 0.515334
\(219\) 0 0
\(220\) −2.37741e10 −0.684230
\(221\) 1.67353e10 0.471921
\(222\) 0 0
\(223\) −6.69002e10 −1.81157 −0.905786 0.423736i \(-0.860718\pi\)
−0.905786 + 0.423736i \(0.860718\pi\)
\(224\) 2.64016e10 0.700670
\(225\) 0 0
\(226\) 7.01172e8 0.0178787
\(227\) 2.18215e10 0.545466 0.272733 0.962090i \(-0.412072\pi\)
0.272733 + 0.962090i \(0.412072\pi\)
\(228\) 0 0
\(229\) −4.48901e9 −0.107868 −0.0539338 0.998545i \(-0.517176\pi\)
−0.0539338 + 0.998545i \(0.517176\pi\)
\(230\) −1.60162e10 −0.377386
\(231\) 0 0
\(232\) −2.36459e10 −0.535870
\(233\) −2.75398e9 −0.0612151 −0.0306076 0.999531i \(-0.509744\pi\)
−0.0306076 + 0.999531i \(0.509744\pi\)
\(234\) 0 0
\(235\) 1.07329e11 2.29568
\(236\) 2.82607e10 0.593033
\(237\) 0 0
\(238\) −2.58711e10 −0.522659
\(239\) −2.52877e10 −0.501324 −0.250662 0.968075i \(-0.580648\pi\)
−0.250662 + 0.968075i \(0.580648\pi\)
\(240\) 0 0
\(241\) 1.07229e10 0.204755 0.102377 0.994746i \(-0.467355\pi\)
0.102377 + 0.994746i \(0.467355\pi\)
\(242\) 2.09776e10 0.393176
\(243\) 0 0
\(244\) 2.81866e10 0.509082
\(245\) 2.12843e10 0.377409
\(246\) 0 0
\(247\) −2.46564e10 −0.421496
\(248\) −4.94079e10 −0.829399
\(249\) 0 0
\(250\) 7.33866e10 1.18819
\(251\) −4.66245e10 −0.741451 −0.370725 0.928743i \(-0.620891\pi\)
−0.370725 + 0.928743i \(0.620891\pi\)
\(252\) 0 0
\(253\) −2.16628e10 −0.332409
\(254\) −7.34803e10 −1.10769
\(255\) 0 0
\(256\) −6.01218e10 −0.874888
\(257\) 2.77809e10 0.397234 0.198617 0.980077i \(-0.436355\pi\)
0.198617 + 0.980077i \(0.436355\pi\)
\(258\) 0 0
\(259\) 1.56004e11 2.15420
\(260\) 3.37996e10 0.458703
\(261\) 0 0
\(262\) 3.34073e10 0.438011
\(263\) 7.66179e10 0.987482 0.493741 0.869609i \(-0.335629\pi\)
0.493741 + 0.869609i \(0.335629\pi\)
\(264\) 0 0
\(265\) −1.15484e11 −1.43852
\(266\) 3.81162e10 0.466812
\(267\) 0 0
\(268\) −1.93018e9 −0.0228556
\(269\) 5.82715e10 0.678533 0.339266 0.940690i \(-0.389821\pi\)
0.339266 + 0.940690i \(0.389821\pi\)
\(270\) 0 0
\(271\) 5.01640e10 0.564977 0.282488 0.959271i \(-0.408840\pi\)
0.282488 + 0.959271i \(0.408840\pi\)
\(272\) 2.90856e10 0.322194
\(273\) 0 0
\(274\) 5.22807e10 0.560356
\(275\) 2.14686e11 2.26364
\(276\) 0 0
\(277\) −6.47834e10 −0.661157 −0.330579 0.943778i \(-0.607244\pi\)
−0.330579 + 0.943778i \(0.607244\pi\)
\(278\) 1.26192e11 1.26716
\(279\) 0 0
\(280\) −2.09423e11 −2.03617
\(281\) −5.01006e10 −0.479363 −0.239682 0.970852i \(-0.577043\pi\)
−0.239682 + 0.970852i \(0.577043\pi\)
\(282\) 0 0
\(283\) 2.09526e11 1.94177 0.970887 0.239540i \(-0.0769966\pi\)
0.970887 + 0.239540i \(0.0769966\pi\)
\(284\) −3.56298e10 −0.324998
\(285\) 0 0
\(286\) −9.17996e10 −0.811323
\(287\) −2.34087e10 −0.203661
\(288\) 0 0
\(289\) −7.89144e10 −0.665451
\(290\) 8.19182e10 0.680126
\(291\) 0 0
\(292\) −1.25923e9 −0.0101364
\(293\) 2.20081e11 1.74453 0.872264 0.489035i \(-0.162651\pi\)
0.872264 + 0.489035i \(0.162651\pi\)
\(294\) 0 0
\(295\) −3.92411e11 −3.01677
\(296\) −2.80058e11 −2.12048
\(297\) 0 0
\(298\) 1.70538e11 1.25271
\(299\) 3.07980e10 0.222845
\(300\) 0 0
\(301\) −2.40192e10 −0.168659
\(302\) 1.42680e11 0.987031
\(303\) 0 0
\(304\) −4.28522e10 −0.287768
\(305\) −3.91382e11 −2.58971
\(306\) 0 0
\(307\) 4.15422e10 0.266911 0.133455 0.991055i \(-0.457393\pi\)
0.133455 + 0.991055i \(0.457393\pi\)
\(308\) −7.06717e10 −0.447474
\(309\) 0 0
\(310\) 1.71167e11 1.05267
\(311\) −7.16229e10 −0.434140 −0.217070 0.976156i \(-0.569650\pi\)
−0.217070 + 0.976156i \(0.569650\pi\)
\(312\) 0 0
\(313\) −1.61891e11 −0.953398 −0.476699 0.879067i \(-0.658167\pi\)
−0.476699 + 0.879067i \(0.658167\pi\)
\(314\) −1.91700e11 −1.11286
\(315\) 0 0
\(316\) −8.66953e10 −0.489107
\(317\) 7.64203e9 0.0425052 0.0212526 0.999774i \(-0.493235\pi\)
0.0212526 + 0.999774i \(0.493235\pi\)
\(318\) 0 0
\(319\) 1.10799e11 0.599068
\(320\) 3.40900e11 1.81740
\(321\) 0 0
\(322\) −4.76105e10 −0.246803
\(323\) −5.84515e10 −0.298803
\(324\) 0 0
\(325\) −3.05219e11 −1.51753
\(326\) 2.29577e11 1.12577
\(327\) 0 0
\(328\) 4.20233e10 0.200474
\(329\) 3.19050e11 1.50133
\(330\) 0 0
\(331\) 1.41081e11 0.646014 0.323007 0.946396i \(-0.395306\pi\)
0.323007 + 0.946396i \(0.395306\pi\)
\(332\) −7.82100e10 −0.353298
\(333\) 0 0
\(334\) −1.57110e10 −0.0690787
\(335\) 2.68013e10 0.116267
\(336\) 0 0
\(337\) 1.00695e11 0.425279 0.212640 0.977131i \(-0.431794\pi\)
0.212640 + 0.977131i \(0.431794\pi\)
\(338\) −6.55399e10 −0.273137
\(339\) 0 0
\(340\) 8.01268e10 0.325179
\(341\) 2.31513e11 0.927215
\(342\) 0 0
\(343\) −2.20238e11 −0.859151
\(344\) 4.31193e10 0.166019
\(345\) 0 0
\(346\) 3.79443e11 1.42332
\(347\) 2.91880e11 1.08074 0.540370 0.841428i \(-0.318284\pi\)
0.540370 + 0.841428i \(0.318284\pi\)
\(348\) 0 0
\(349\) −7.88123e10 −0.284367 −0.142184 0.989840i \(-0.545412\pi\)
−0.142184 + 0.989840i \(0.545412\pi\)
\(350\) 4.71836e11 1.68068
\(351\) 0 0
\(352\) 2.22086e11 0.771045
\(353\) 4.57862e11 1.56945 0.784726 0.619842i \(-0.212803\pi\)
0.784726 + 0.619842i \(0.212803\pi\)
\(354\) 0 0
\(355\) 4.94734e11 1.65327
\(356\) −3.89694e10 −0.128588
\(357\) 0 0
\(358\) −4.15273e9 −0.0133617
\(359\) 6.39676e10 0.203252 0.101626 0.994823i \(-0.467595\pi\)
0.101626 + 0.994823i \(0.467595\pi\)
\(360\) 0 0
\(361\) −2.36570e11 −0.733125
\(362\) 2.37332e11 0.726387
\(363\) 0 0
\(364\) 1.00474e11 0.299984
\(365\) 1.74849e10 0.0515638
\(366\) 0 0
\(367\) 5.56082e10 0.160008 0.0800039 0.996795i \(-0.474507\pi\)
0.0800039 + 0.996795i \(0.474507\pi\)
\(368\) 5.35261e10 0.152143
\(369\) 0 0
\(370\) 9.70226e11 2.69132
\(371\) −3.43292e11 −0.940765
\(372\) 0 0
\(373\) −2.47592e11 −0.662289 −0.331145 0.943580i \(-0.607435\pi\)
−0.331145 + 0.943580i \(0.607435\pi\)
\(374\) −2.17624e11 −0.575154
\(375\) 0 0
\(376\) −5.72759e11 −1.47784
\(377\) −1.57522e11 −0.401612
\(378\) 0 0
\(379\) 2.02764e11 0.504794 0.252397 0.967624i \(-0.418781\pi\)
0.252397 + 0.967624i \(0.418781\pi\)
\(380\) −1.18052e11 −0.290434
\(381\) 0 0
\(382\) −3.41943e10 −0.0821615
\(383\) 7.84097e10 0.186198 0.0930990 0.995657i \(-0.470323\pi\)
0.0930990 + 0.995657i \(0.470323\pi\)
\(384\) 0 0
\(385\) 9.81305e11 2.27631
\(386\) 1.42944e11 0.327734
\(387\) 0 0
\(388\) 1.62401e11 0.363786
\(389\) −2.70312e11 −0.598539 −0.299270 0.954169i \(-0.596743\pi\)
−0.299270 + 0.954169i \(0.596743\pi\)
\(390\) 0 0
\(391\) 7.30110e10 0.157977
\(392\) −1.13583e11 −0.242956
\(393\) 0 0
\(394\) −4.45769e11 −0.931915
\(395\) 1.20380e12 2.48809
\(396\) 0 0
\(397\) −6.70016e11 −1.35372 −0.676859 0.736113i \(-0.736659\pi\)
−0.676859 + 0.736113i \(0.736659\pi\)
\(398\) 4.55301e11 0.909546
\(399\) 0 0
\(400\) −5.30463e11 −1.03606
\(401\) −3.58493e11 −0.692359 −0.346179 0.938168i \(-0.612521\pi\)
−0.346179 + 0.938168i \(0.612521\pi\)
\(402\) 0 0
\(403\) −3.29142e11 −0.621599
\(404\) 2.20325e11 0.411480
\(405\) 0 0
\(406\) 2.43513e11 0.444790
\(407\) 1.31228e12 2.37056
\(408\) 0 0
\(409\) −2.93226e11 −0.518140 −0.259070 0.965859i \(-0.583416\pi\)
−0.259070 + 0.965859i \(0.583416\pi\)
\(410\) −1.45584e11 −0.254441
\(411\) 0 0
\(412\) −9.66537e10 −0.165265
\(413\) −1.16649e12 −1.97291
\(414\) 0 0
\(415\) 1.08598e12 1.79723
\(416\) −3.15740e11 −0.516904
\(417\) 0 0
\(418\) 3.20628e11 0.513699
\(419\) 2.52617e11 0.400404 0.200202 0.979755i \(-0.435840\pi\)
0.200202 + 0.979755i \(0.435840\pi\)
\(420\) 0 0
\(421\) 1.22137e11 0.189486 0.0947429 0.995502i \(-0.469797\pi\)
0.0947429 + 0.995502i \(0.469797\pi\)
\(422\) −2.85578e11 −0.438347
\(423\) 0 0
\(424\) 6.16278e11 0.926042
\(425\) −7.23564e11 −1.07579
\(426\) 0 0
\(427\) −1.16343e12 −1.69362
\(428\) 7.72166e10 0.111228
\(429\) 0 0
\(430\) −1.49381e11 −0.210712
\(431\) −1.01246e12 −1.41329 −0.706643 0.707571i \(-0.749791\pi\)
−0.706643 + 0.707571i \(0.749791\pi\)
\(432\) 0 0
\(433\) −9.36722e11 −1.28061 −0.640303 0.768123i \(-0.721191\pi\)
−0.640303 + 0.768123i \(0.721191\pi\)
\(434\) 5.08818e11 0.688429
\(435\) 0 0
\(436\) −1.58215e11 −0.209681
\(437\) −1.07568e11 −0.141097
\(438\) 0 0
\(439\) −5.85451e10 −0.0752316 −0.0376158 0.999292i \(-0.511976\pi\)
−0.0376158 + 0.999292i \(0.511976\pi\)
\(440\) −1.76164e12 −2.24068
\(441\) 0 0
\(442\) 3.09396e11 0.385580
\(443\) −1.20731e12 −1.48936 −0.744681 0.667421i \(-0.767398\pi\)
−0.744681 + 0.667421i \(0.767398\pi\)
\(444\) 0 0
\(445\) 5.41105e11 0.654127
\(446\) −1.23682e12 −1.48013
\(447\) 0 0
\(448\) 1.01337e12 1.18855
\(449\) 1.34315e11 0.155961 0.0779807 0.996955i \(-0.475153\pi\)
0.0779807 + 0.996955i \(0.475153\pi\)
\(450\) 0 0
\(451\) −1.96911e11 −0.224117
\(452\) −6.45550e9 −0.00727456
\(453\) 0 0
\(454\) 4.03426e11 0.445669
\(455\) −1.39512e12 −1.52602
\(456\) 0 0
\(457\) 6.04169e11 0.647941 0.323970 0.946067i \(-0.394982\pi\)
0.323970 + 0.946067i \(0.394982\pi\)
\(458\) −8.29909e10 −0.0881325
\(459\) 0 0
\(460\) 1.47457e11 0.153552
\(461\) 7.96508e11 0.821364 0.410682 0.911779i \(-0.365291\pi\)
0.410682 + 0.911779i \(0.365291\pi\)
\(462\) 0 0
\(463\) 1.60119e12 1.61930 0.809652 0.586910i \(-0.199656\pi\)
0.809652 + 0.586910i \(0.199656\pi\)
\(464\) −2.73770e11 −0.274192
\(465\) 0 0
\(466\) −5.09144e10 −0.0500154
\(467\) −1.97526e11 −0.192176 −0.0960879 0.995373i \(-0.530633\pi\)
−0.0960879 + 0.995373i \(0.530633\pi\)
\(468\) 0 0
\(469\) 7.96706e10 0.0760362
\(470\) 1.98425e12 1.87567
\(471\) 0 0
\(472\) 2.09409e12 1.94204
\(473\) −2.02046e11 −0.185599
\(474\) 0 0
\(475\) 1.06604e12 0.960841
\(476\) 2.38188e11 0.212661
\(477\) 0 0
\(478\) −4.67508e11 −0.409604
\(479\) 1.31380e12 1.14030 0.570149 0.821541i \(-0.306885\pi\)
0.570149 + 0.821541i \(0.306885\pi\)
\(480\) 0 0
\(481\) −1.86567e12 −1.58921
\(482\) 1.98240e11 0.167293
\(483\) 0 0
\(484\) −1.93135e11 −0.159977
\(485\) −2.25500e12 −1.85058
\(486\) 0 0
\(487\) −1.13531e12 −0.914604 −0.457302 0.889312i \(-0.651184\pi\)
−0.457302 + 0.889312i \(0.651184\pi\)
\(488\) 2.08860e12 1.66712
\(489\) 0 0
\(490\) 3.93495e11 0.308360
\(491\) 2.16799e11 0.168341 0.0841707 0.996451i \(-0.473176\pi\)
0.0841707 + 0.996451i \(0.473176\pi\)
\(492\) 0 0
\(493\) −3.73429e11 −0.284706
\(494\) −4.55837e11 −0.344380
\(495\) 0 0
\(496\) −5.72040e11 −0.424384
\(497\) 1.47066e12 1.08121
\(498\) 0 0
\(499\) −1.61040e12 −1.16274 −0.581368 0.813641i \(-0.697482\pi\)
−0.581368 + 0.813641i \(0.697482\pi\)
\(500\) −6.75650e11 −0.483456
\(501\) 0 0
\(502\) −8.61973e11 −0.605797
\(503\) 1.85720e12 1.29361 0.646803 0.762657i \(-0.276106\pi\)
0.646803 + 0.762657i \(0.276106\pi\)
\(504\) 0 0
\(505\) −3.05930e12 −2.09320
\(506\) −4.00493e11 −0.271592
\(507\) 0 0
\(508\) 6.76513e11 0.450702
\(509\) −1.23771e11 −0.0817312 −0.0408656 0.999165i \(-0.513012\pi\)
−0.0408656 + 0.999165i \(0.513012\pi\)
\(510\) 0 0
\(511\) 5.19762e10 0.0337218
\(512\) −1.49171e12 −0.959335
\(513\) 0 0
\(514\) 5.13601e11 0.324557
\(515\) 1.34207e12 0.840705
\(516\) 0 0
\(517\) 2.68380e12 1.65213
\(518\) 2.88413e12 1.76007
\(519\) 0 0
\(520\) 2.50452e12 1.50214
\(521\) 3.54135e11 0.210571 0.105286 0.994442i \(-0.466424\pi\)
0.105286 + 0.994442i \(0.466424\pi\)
\(522\) 0 0
\(523\) −9.79180e11 −0.572275 −0.286138 0.958188i \(-0.592371\pi\)
−0.286138 + 0.958188i \(0.592371\pi\)
\(524\) −3.07571e11 −0.178219
\(525\) 0 0
\(526\) 1.41648e12 0.806815
\(527\) −7.80277e11 −0.440657
\(528\) 0 0
\(529\) −1.66679e12 −0.925402
\(530\) −2.13502e12 −1.17533
\(531\) 0 0
\(532\) −3.50925e11 −0.189938
\(533\) 2.79948e11 0.150247
\(534\) 0 0
\(535\) −1.07218e12 −0.565818
\(536\) −1.43025e11 −0.0748462
\(537\) 0 0
\(538\) 1.07730e12 0.554390
\(539\) 5.32223e11 0.271609
\(540\) 0 0
\(541\) −1.88732e12 −0.947234 −0.473617 0.880731i \(-0.657052\pi\)
−0.473617 + 0.880731i \(0.657052\pi\)
\(542\) 9.27411e11 0.461610
\(543\) 0 0
\(544\) −7.48506e11 −0.366438
\(545\) 2.19688e12 1.06665
\(546\) 0 0
\(547\) −9.25517e11 −0.442020 −0.221010 0.975272i \(-0.570935\pi\)
−0.221010 + 0.975272i \(0.570935\pi\)
\(548\) −4.81334e11 −0.227999
\(549\) 0 0
\(550\) 3.96902e12 1.84949
\(551\) 5.50178e11 0.254285
\(552\) 0 0
\(553\) 3.57845e12 1.62717
\(554\) −1.19769e12 −0.540194
\(555\) 0 0
\(556\) −1.16182e12 −0.515586
\(557\) −4.04068e12 −1.77871 −0.889356 0.457215i \(-0.848847\pi\)
−0.889356 + 0.457215i \(0.848847\pi\)
\(558\) 0 0
\(559\) 2.87249e11 0.124424
\(560\) −2.42468e12 −1.04186
\(561\) 0 0
\(562\) −9.26239e11 −0.391660
\(563\) 1.31953e12 0.553517 0.276758 0.960940i \(-0.410740\pi\)
0.276758 + 0.960940i \(0.410740\pi\)
\(564\) 0 0
\(565\) 8.96371e10 0.0370058
\(566\) 3.87362e12 1.58651
\(567\) 0 0
\(568\) −2.64014e12 −1.06429
\(569\) 1.40113e12 0.560366 0.280183 0.959947i \(-0.409605\pi\)
0.280183 + 0.959947i \(0.409605\pi\)
\(570\) 0 0
\(571\) −7.55062e11 −0.297249 −0.148624 0.988894i \(-0.547485\pi\)
−0.148624 + 0.988894i \(0.547485\pi\)
\(572\) 8.45173e11 0.330114
\(573\) 0 0
\(574\) −4.32770e11 −0.166400
\(575\) −1.33157e12 −0.507995
\(576\) 0 0
\(577\) 2.44025e12 0.916521 0.458261 0.888818i \(-0.348473\pi\)
0.458261 + 0.888818i \(0.348473\pi\)
\(578\) −1.45894e12 −0.543702
\(579\) 0 0
\(580\) −7.54198e11 −0.276732
\(581\) 3.22821e12 1.17536
\(582\) 0 0
\(583\) −2.88772e12 −1.03525
\(584\) −9.33078e10 −0.0331940
\(585\) 0 0
\(586\) 4.06876e12 1.42535
\(587\) 5.63814e12 1.96004 0.980019 0.198906i \(-0.0637389\pi\)
0.980019 + 0.198906i \(0.0637389\pi\)
\(588\) 0 0
\(589\) 1.14959e12 0.393573
\(590\) −7.25473e12 −2.46483
\(591\) 0 0
\(592\) −3.24248e12 −1.08500
\(593\) −2.07785e12 −0.690031 −0.345016 0.938597i \(-0.612126\pi\)
−0.345016 + 0.938597i \(0.612126\pi\)
\(594\) 0 0
\(595\) −3.30733e12 −1.08181
\(596\) −1.57010e12 −0.509705
\(597\) 0 0
\(598\) 5.69380e11 0.182074
\(599\) −2.05195e12 −0.651248 −0.325624 0.945499i \(-0.605574\pi\)
−0.325624 + 0.945499i \(0.605574\pi\)
\(600\) 0 0
\(601\) −3.93140e12 −1.22917 −0.614586 0.788850i \(-0.710677\pi\)
−0.614586 + 0.788850i \(0.710677\pi\)
\(602\) −4.44056e11 −0.137801
\(603\) 0 0
\(604\) −1.31361e12 −0.401607
\(605\) 2.68175e12 0.813803
\(606\) 0 0
\(607\) 2.33530e12 0.698221 0.349110 0.937082i \(-0.386484\pi\)
0.349110 + 0.937082i \(0.386484\pi\)
\(608\) 1.10278e12 0.327284
\(609\) 0 0
\(610\) −7.23569e12 −2.11590
\(611\) −3.81557e12 −1.10758
\(612\) 0 0
\(613\) 1.39331e12 0.398544 0.199272 0.979944i \(-0.436142\pi\)
0.199272 + 0.979944i \(0.436142\pi\)
\(614\) 7.68013e11 0.218078
\(615\) 0 0
\(616\) −5.23671e12 −1.46536
\(617\) −6.75919e11 −0.187764 −0.0938818 0.995583i \(-0.529928\pi\)
−0.0938818 + 0.995583i \(0.529928\pi\)
\(618\) 0 0
\(619\) −3.04638e12 −0.834019 −0.417010 0.908902i \(-0.636922\pi\)
−0.417010 + 0.908902i \(0.636922\pi\)
\(620\) −1.57589e12 −0.428315
\(621\) 0 0
\(622\) −1.32413e12 −0.354711
\(623\) 1.60851e12 0.427787
\(624\) 0 0
\(625\) 2.28658e12 0.599414
\(626\) −2.99298e12 −0.778967
\(627\) 0 0
\(628\) 1.76493e12 0.452802
\(629\) −4.42283e12 −1.12661
\(630\) 0 0
\(631\) −5.72611e12 −1.43790 −0.718948 0.695064i \(-0.755376\pi\)
−0.718948 + 0.695064i \(0.755376\pi\)
\(632\) −6.42404e12 −1.60170
\(633\) 0 0
\(634\) 1.41283e11 0.0347286
\(635\) −9.39364e12 −2.29273
\(636\) 0 0
\(637\) −7.56661e11 −0.182085
\(638\) 2.04840e12 0.489464
\(639\) 0 0
\(640\) 1.75506e12 0.413506
\(641\) −7.81185e10 −0.0182765 −0.00913825 0.999958i \(-0.502909\pi\)
−0.00913825 + 0.999958i \(0.502909\pi\)
\(642\) 0 0
\(643\) −1.63383e12 −0.376927 −0.188464 0.982080i \(-0.560351\pi\)
−0.188464 + 0.982080i \(0.560351\pi\)
\(644\) 4.38336e11 0.100420
\(645\) 0 0
\(646\) −1.08063e12 −0.244134
\(647\) −4.02702e12 −0.903471 −0.451735 0.892152i \(-0.649195\pi\)
−0.451735 + 0.892152i \(0.649195\pi\)
\(648\) 0 0
\(649\) −9.81239e12 −2.17107
\(650\) −5.64276e12 −1.23988
\(651\) 0 0
\(652\) −2.11365e12 −0.458057
\(653\) −5.84257e12 −1.25746 −0.628730 0.777623i \(-0.716425\pi\)
−0.628730 + 0.777623i \(0.716425\pi\)
\(654\) 0 0
\(655\) 4.27075e12 0.906605
\(656\) 4.86542e11 0.102578
\(657\) 0 0
\(658\) 5.89846e12 1.22665
\(659\) −4.19256e11 −0.0865955 −0.0432977 0.999062i \(-0.513786\pi\)
−0.0432977 + 0.999062i \(0.513786\pi\)
\(660\) 0 0
\(661\) −4.51791e12 −0.920515 −0.460258 0.887785i \(-0.652243\pi\)
−0.460258 + 0.887785i \(0.652243\pi\)
\(662\) 2.60824e12 0.527821
\(663\) 0 0
\(664\) −5.79529e12 −1.15696
\(665\) 4.87274e12 0.966219
\(666\) 0 0
\(667\) −6.87220e11 −0.134440
\(668\) 1.44647e11 0.0281070
\(669\) 0 0
\(670\) 4.95492e11 0.0949948
\(671\) −9.78665e12 −1.86373
\(672\) 0 0
\(673\) −3.13452e12 −0.588983 −0.294492 0.955654i \(-0.595150\pi\)
−0.294492 + 0.955654i \(0.595150\pi\)
\(674\) 1.86161e12 0.347471
\(675\) 0 0
\(676\) 6.03407e11 0.111135
\(677\) −9.08025e12 −1.66130 −0.830651 0.556793i \(-0.812032\pi\)
−0.830651 + 0.556793i \(0.812032\pi\)
\(678\) 0 0
\(679\) −6.70329e12 −1.21025
\(680\) 5.93732e12 1.06488
\(681\) 0 0
\(682\) 4.28011e12 0.757574
\(683\) −9.92561e12 −1.74528 −0.872638 0.488367i \(-0.837593\pi\)
−0.872638 + 0.488367i \(0.837593\pi\)
\(684\) 0 0
\(685\) 6.68351e12 1.15984
\(686\) −4.07167e12 −0.701963
\(687\) 0 0
\(688\) 4.99231e11 0.0849481
\(689\) 4.10548e12 0.694028
\(690\) 0 0
\(691\) 2.83695e12 0.473370 0.236685 0.971586i \(-0.423939\pi\)
0.236685 + 0.971586i \(0.423939\pi\)
\(692\) −3.49342e12 −0.579127
\(693\) 0 0
\(694\) 5.39614e12 0.883011
\(695\) 1.61323e13 2.62279
\(696\) 0 0
\(697\) 6.63655e11 0.106511
\(698\) −1.45705e12 −0.232340
\(699\) 0 0
\(700\) −4.34406e12 −0.683841
\(701\) 9.36363e11 0.146458 0.0732290 0.997315i \(-0.476670\pi\)
0.0732290 + 0.997315i \(0.476670\pi\)
\(702\) 0 0
\(703\) 6.51622e12 1.00623
\(704\) 8.52433e12 1.30793
\(705\) 0 0
\(706\) 8.46475e12 1.28231
\(707\) −9.09420e12 −1.36892
\(708\) 0 0
\(709\) 5.31394e12 0.789785 0.394892 0.918727i \(-0.370782\pi\)
0.394892 + 0.918727i \(0.370782\pi\)
\(710\) 9.14643e12 1.35079
\(711\) 0 0
\(712\) −2.88760e12 −0.421092
\(713\) −1.43594e12 −0.208082
\(714\) 0 0
\(715\) −1.17356e13 −1.67929
\(716\) 3.82330e10 0.00543663
\(717\) 0 0
\(718\) 1.18261e12 0.166066
\(719\) −6.83793e12 −0.954212 −0.477106 0.878846i \(-0.658314\pi\)
−0.477106 + 0.878846i \(0.658314\pi\)
\(720\) 0 0
\(721\) 3.98950e12 0.549806
\(722\) −4.37361e12 −0.598994
\(723\) 0 0
\(724\) −2.18505e12 −0.295555
\(725\) 6.81059e12 0.915511
\(726\) 0 0
\(727\) 6.36741e12 0.845392 0.422696 0.906272i \(-0.361084\pi\)
0.422696 + 0.906272i \(0.361084\pi\)
\(728\) 7.44504e12 0.982371
\(729\) 0 0
\(730\) 3.23253e11 0.0421299
\(731\) 6.80964e11 0.0882055
\(732\) 0 0
\(733\) 5.47275e12 0.700226 0.350113 0.936708i \(-0.386143\pi\)
0.350113 + 0.936708i \(0.386143\pi\)
\(734\) 1.02806e12 0.130733
\(735\) 0 0
\(736\) −1.37747e12 −0.173035
\(737\) 6.70178e11 0.0836732
\(738\) 0 0
\(739\) −1.09212e13 −1.34701 −0.673503 0.739185i \(-0.735211\pi\)
−0.673503 + 0.739185i \(0.735211\pi\)
\(740\) −8.93260e12 −1.09505
\(741\) 0 0
\(742\) −6.34663e12 −0.768645
\(743\) −1.41066e11 −0.0169813 −0.00849066 0.999964i \(-0.502703\pi\)
−0.00849066 + 0.999964i \(0.502703\pi\)
\(744\) 0 0
\(745\) 2.18014e13 2.59288
\(746\) −4.57738e12 −0.541118
\(747\) 0 0
\(748\) 2.00360e12 0.234021
\(749\) −3.18721e12 −0.370034
\(750\) 0 0
\(751\) 1.07431e13 1.23240 0.616200 0.787590i \(-0.288671\pi\)
0.616200 + 0.787590i \(0.288671\pi\)
\(752\) −6.63135e12 −0.756174
\(753\) 0 0
\(754\) −2.91221e12 −0.328134
\(755\) 1.82400e13 2.04298
\(756\) 0 0
\(757\) −3.92377e12 −0.434282 −0.217141 0.976140i \(-0.569673\pi\)
−0.217141 + 0.976140i \(0.569673\pi\)
\(758\) 3.74861e12 0.412438
\(759\) 0 0
\(760\) −8.74754e12 −0.951097
\(761\) −2.65761e12 −0.287250 −0.143625 0.989632i \(-0.545876\pi\)
−0.143625 + 0.989632i \(0.545876\pi\)
\(762\) 0 0
\(763\) 6.53052e12 0.697569
\(764\) 3.14817e11 0.0334302
\(765\) 0 0
\(766\) 1.44960e12 0.152132
\(767\) 1.39503e13 1.45547
\(768\) 0 0
\(769\) 3.14923e12 0.324740 0.162370 0.986730i \(-0.448086\pi\)
0.162370 + 0.986730i \(0.448086\pi\)
\(770\) 1.81419e13 1.85984
\(771\) 0 0
\(772\) −1.31604e12 −0.133350
\(773\) −3.67725e11 −0.0370438 −0.0185219 0.999828i \(-0.505896\pi\)
−0.0185219 + 0.999828i \(0.505896\pi\)
\(774\) 0 0
\(775\) 1.42307e13 1.41699
\(776\) 1.20338e13 1.19131
\(777\) 0 0
\(778\) −4.99742e12 −0.489032
\(779\) −9.77773e11 −0.0951304
\(780\) 0 0
\(781\) 1.23710e13 1.18980
\(782\) 1.34980e12 0.129074
\(783\) 0 0
\(784\) −1.31506e12 −0.124315
\(785\) −2.45067e13 −2.30341
\(786\) 0 0
\(787\) −1.46553e13 −1.36179 −0.680895 0.732381i \(-0.738409\pi\)
−0.680895 + 0.732381i \(0.738409\pi\)
\(788\) 4.10407e12 0.379181
\(789\) 0 0
\(790\) 2.22553e13 2.03288
\(791\) 2.66458e11 0.0242011
\(792\) 0 0
\(793\) 1.39137e13 1.24943
\(794\) −1.23870e13 −1.10604
\(795\) 0 0
\(796\) −4.19183e12 −0.370079
\(797\) 2.07106e13 1.81815 0.909075 0.416633i \(-0.136790\pi\)
0.909075 + 0.416633i \(0.136790\pi\)
\(798\) 0 0
\(799\) −9.04533e12 −0.785171
\(800\) 1.36512e13 1.17833
\(801\) 0 0
\(802\) −6.62767e12 −0.565687
\(803\) 4.37217e11 0.0371088
\(804\) 0 0
\(805\) −6.08647e12 −0.510839
\(806\) −6.08503e12 −0.507873
\(807\) 0 0
\(808\) 1.63259e13 1.34749
\(809\) −1.94599e12 −0.159725 −0.0798623 0.996806i \(-0.525448\pi\)
−0.0798623 + 0.996806i \(0.525448\pi\)
\(810\) 0 0
\(811\) 1.32679e11 0.0107698 0.00538491 0.999986i \(-0.498286\pi\)
0.00538491 + 0.999986i \(0.498286\pi\)
\(812\) −2.24196e12 −0.180978
\(813\) 0 0
\(814\) 2.42609e13 1.93685
\(815\) 2.93489e13 2.33014
\(816\) 0 0
\(817\) −1.00327e12 −0.0787807
\(818\) −5.42103e12 −0.423343
\(819\) 0 0
\(820\) 1.34036e12 0.103528
\(821\) −1.39889e13 −1.07458 −0.537290 0.843398i \(-0.680552\pi\)
−0.537290 + 0.843398i \(0.680552\pi\)
\(822\) 0 0
\(823\) −2.70521e12 −0.205543 −0.102771 0.994705i \(-0.532771\pi\)
−0.102771 + 0.994705i \(0.532771\pi\)
\(824\) −7.16195e12 −0.541201
\(825\) 0 0
\(826\) −2.15657e13 −1.61195
\(827\) 7.17939e11 0.0533719 0.0266859 0.999644i \(-0.491505\pi\)
0.0266859 + 0.999644i \(0.491505\pi\)
\(828\) 0 0
\(829\) −1.65304e12 −0.121559 −0.0607796 0.998151i \(-0.519359\pi\)
−0.0607796 + 0.998151i \(0.519359\pi\)
\(830\) 2.00771e13 1.46841
\(831\) 0 0
\(832\) −1.21190e13 −0.876825
\(833\) −1.79377e12 −0.129082
\(834\) 0 0
\(835\) −2.00848e12 −0.142981
\(836\) −2.95193e12 −0.209016
\(837\) 0 0
\(838\) 4.67027e12 0.327148
\(839\) −1.51228e13 −1.05366 −0.526832 0.849969i \(-0.676620\pi\)
−0.526832 + 0.849969i \(0.676620\pi\)
\(840\) 0 0
\(841\) −1.09922e13 −0.757711
\(842\) 2.25801e12 0.154818
\(843\) 0 0
\(844\) 2.62923e12 0.178356
\(845\) −8.37855e12 −0.565345
\(846\) 0 0
\(847\) 7.97187e12 0.532212
\(848\) 7.13521e12 0.473833
\(849\) 0 0
\(850\) −1.33769e13 −0.878965
\(851\) −8.13932e12 −0.531992
\(852\) 0 0
\(853\) −1.14817e13 −0.742568 −0.371284 0.928519i \(-0.621082\pi\)
−0.371284 + 0.928519i \(0.621082\pi\)
\(854\) −2.15091e13 −1.38376
\(855\) 0 0
\(856\) 5.72168e12 0.364243
\(857\) −1.99397e13 −1.26271 −0.631356 0.775493i \(-0.717501\pi\)
−0.631356 + 0.775493i \(0.717501\pi\)
\(858\) 0 0
\(859\) 1.38539e13 0.868164 0.434082 0.900873i \(-0.357073\pi\)
0.434082 + 0.900873i \(0.357073\pi\)
\(860\) 1.37531e12 0.0857350
\(861\) 0 0
\(862\) −1.87179e13 −1.15471
\(863\) 2.96018e13 1.81664 0.908321 0.418275i \(-0.137365\pi\)
0.908321 + 0.418275i \(0.137365\pi\)
\(864\) 0 0
\(865\) 4.85076e13 2.94603
\(866\) −1.73177e13 −1.04631
\(867\) 0 0
\(868\) −4.68455e12 −0.280110
\(869\) 3.01014e13 1.79060
\(870\) 0 0
\(871\) −9.52792e11 −0.0560940
\(872\) −1.17236e13 −0.686652
\(873\) 0 0
\(874\) −1.98867e12 −0.115282
\(875\) 2.78883e13 1.60837
\(876\) 0 0
\(877\) −5.98845e12 −0.341835 −0.170918 0.985285i \(-0.554673\pi\)
−0.170918 + 0.985285i \(0.554673\pi\)
\(878\) −1.08236e12 −0.0614674
\(879\) 0 0
\(880\) −2.03961e13 −1.14650
\(881\) −4.47788e10 −0.00250427 −0.00125213 0.999999i \(-0.500399\pi\)
−0.00125213 + 0.999999i \(0.500399\pi\)
\(882\) 0 0
\(883\) 1.82152e13 1.00835 0.504175 0.863601i \(-0.331797\pi\)
0.504175 + 0.863601i \(0.331797\pi\)
\(884\) −2.84852e12 −0.156886
\(885\) 0 0
\(886\) −2.23201e13 −1.21687
\(887\) 1.08674e13 0.589483 0.294741 0.955577i \(-0.404766\pi\)
0.294741 + 0.955577i \(0.404766\pi\)
\(888\) 0 0
\(889\) −2.79239e13 −1.49940
\(890\) 1.00037e13 0.534450
\(891\) 0 0
\(892\) 1.13871e13 0.602241
\(893\) 1.33266e13 0.701275
\(894\) 0 0
\(895\) −5.30881e11 −0.0276562
\(896\) 5.21715e12 0.270425
\(897\) 0 0
\(898\) 2.48316e12 0.127427
\(899\) 7.34440e12 0.375005
\(900\) 0 0
\(901\) 9.73261e12 0.492003
\(902\) −3.64040e12 −0.183113
\(903\) 0 0
\(904\) −4.78347e11 −0.0238224
\(905\) 3.03403e13 1.50349
\(906\) 0 0
\(907\) −2.21516e13 −1.08686 −0.543429 0.839455i \(-0.682874\pi\)
−0.543429 + 0.839455i \(0.682874\pi\)
\(908\) −3.71423e12 −0.181335
\(909\) 0 0
\(910\) −2.57924e13 −1.24682
\(911\) −8.34279e12 −0.401309 −0.200654 0.979662i \(-0.564307\pi\)
−0.200654 + 0.979662i \(0.564307\pi\)
\(912\) 0 0
\(913\) 2.71553e13 1.29341
\(914\) 1.11696e13 0.529395
\(915\) 0 0
\(916\) 7.64074e11 0.0358596
\(917\) 1.26954e13 0.592903
\(918\) 0 0
\(919\) −1.73492e13 −0.802341 −0.401170 0.916003i \(-0.631396\pi\)
−0.401170 + 0.916003i \(0.631396\pi\)
\(920\) 1.09264e13 0.502844
\(921\) 0 0
\(922\) 1.47255e13 0.671090
\(923\) −1.75879e13 −0.797638
\(924\) 0 0
\(925\) 8.06635e13 3.62276
\(926\) 2.96021e13 1.32304
\(927\) 0 0
\(928\) 7.04535e12 0.311844
\(929\) 1.11298e13 0.490247 0.245124 0.969492i \(-0.421171\pi\)
0.245124 + 0.969492i \(0.421171\pi\)
\(930\) 0 0
\(931\) 2.64279e12 0.115289
\(932\) 4.68754e11 0.0203504
\(933\) 0 0
\(934\) −3.65178e12 −0.157016
\(935\) −2.78208e13 −1.19047
\(936\) 0 0
\(937\) −2.31541e13 −0.981297 −0.490648 0.871358i \(-0.663240\pi\)
−0.490648 + 0.871358i \(0.663240\pi\)
\(938\) 1.47292e12 0.0621248
\(939\) 0 0
\(940\) −1.82685e13 −0.763179
\(941\) −6.27432e12 −0.260864 −0.130432 0.991457i \(-0.541636\pi\)
−0.130432 + 0.991457i \(0.541636\pi\)
\(942\) 0 0
\(943\) 1.22132e12 0.0502954
\(944\) 2.42452e13 0.993693
\(945\) 0 0
\(946\) −3.73534e12 −0.151642
\(947\) 1.95848e12 0.0791305 0.0395652 0.999217i \(-0.487403\pi\)
0.0395652 + 0.999217i \(0.487403\pi\)
\(948\) 0 0
\(949\) −6.21591e11 −0.0248775
\(950\) 1.97084e13 0.785048
\(951\) 0 0
\(952\) 1.76495e13 0.696412
\(953\) 3.01343e13 1.18343 0.591716 0.806146i \(-0.298451\pi\)
0.591716 + 0.806146i \(0.298451\pi\)
\(954\) 0 0
\(955\) −4.37136e12 −0.170060
\(956\) 4.30422e12 0.166661
\(957\) 0 0
\(958\) 2.42889e13 0.931672
\(959\) 1.98676e13 0.758512
\(960\) 0 0
\(961\) −1.10936e13 −0.419581
\(962\) −3.44917e13 −1.29845
\(963\) 0 0
\(964\) −1.82514e12 −0.0680689
\(965\) 1.82738e13 0.678352
\(966\) 0 0
\(967\) −4.59576e13 −1.69020 −0.845099 0.534609i \(-0.820459\pi\)
−0.845099 + 0.534609i \(0.820459\pi\)
\(968\) −1.43111e13 −0.523883
\(969\) 0 0
\(970\) −4.16895e13 −1.51201
\(971\) −5.92518e11 −0.0213902 −0.0106951 0.999943i \(-0.503404\pi\)
−0.0106951 + 0.999943i \(0.503404\pi\)
\(972\) 0 0
\(973\) 4.79554e13 1.71526
\(974\) −2.09891e13 −0.747270
\(975\) 0 0
\(976\) 2.41816e13 0.853024
\(977\) 3.04374e13 1.06876 0.534382 0.845243i \(-0.320544\pi\)
0.534382 + 0.845243i \(0.320544\pi\)
\(978\) 0 0
\(979\) 1.35306e13 0.470753
\(980\) −3.62280e12 −0.125466
\(981\) 0 0
\(982\) 4.00809e12 0.137542
\(983\) −2.48900e13 −0.850225 −0.425113 0.905140i \(-0.639765\pi\)
−0.425113 + 0.905140i \(0.639765\pi\)
\(984\) 0 0
\(985\) −5.69866e13 −1.92890
\(986\) −6.90379e12 −0.232617
\(987\) 0 0
\(988\) 4.19677e12 0.140123
\(989\) 1.25318e12 0.0416513
\(990\) 0 0
\(991\) 7.06033e12 0.232538 0.116269 0.993218i \(-0.462907\pi\)
0.116269 + 0.993218i \(0.462907\pi\)
\(992\) 1.47212e13 0.482660
\(993\) 0 0
\(994\) 2.71890e13 0.883394
\(995\) 5.82052e13 1.88260
\(996\) 0 0
\(997\) −1.27122e13 −0.407468 −0.203734 0.979026i \(-0.565308\pi\)
−0.203734 + 0.979026i \(0.565308\pi\)
\(998\) −2.97724e13 −0.950005
\(999\) 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 387.10.a.c.1.11 15
3.2 odd 2 43.10.a.a.1.5 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.5 15 3.2 odd 2
387.10.a.c.1.11 15 1.1 even 1 trivial