Properties

Label 207.8.a.a.1.5
Level $207$
Weight $8$
Character 207.1
Self dual yes
Analytic conductor $64.664$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.6637002752\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-17.8260\) of defining polynomial
Character \(\chi\) \(=\) 207.1

$q$-expansion

\(f(q)\) \(=\) \(q+17.8260 q^{2} +189.767 q^{4} +31.0428 q^{5} -1247.35 q^{7} +1101.05 q^{8} +O(q^{10})\) \(q+17.8260 q^{2} +189.767 q^{4} +31.0428 q^{5} -1247.35 q^{7} +1101.05 q^{8} +553.369 q^{10} +939.114 q^{11} +10151.8 q^{13} -22235.2 q^{14} -4662.78 q^{16} -33965.5 q^{17} +18716.1 q^{19} +5890.88 q^{20} +16740.7 q^{22} -12167.0 q^{23} -77161.3 q^{25} +180965. q^{26} -236704. q^{28} -91722.7 q^{29} -187422. q^{31} -224053. q^{32} -605469. q^{34} -38721.0 q^{35} -112278. q^{37} +333633. q^{38} +34179.6 q^{40} -218882. q^{41} -910269. q^{43} +178212. q^{44} -216889. q^{46} -54041.9 q^{47} +732327. q^{49} -1.37548e6 q^{50} +1.92647e6 q^{52} +135797. q^{53} +29152.7 q^{55} -1.37339e6 q^{56} -1.63505e6 q^{58} +2.26099e6 q^{59} +124600. q^{61} -3.34098e6 q^{62} -3.39714e6 q^{64} +315139. q^{65} +3.30689e6 q^{67} -6.44551e6 q^{68} -690242. q^{70} +740248. q^{71} +2.68349e6 q^{73} -2.00147e6 q^{74} +3.55169e6 q^{76} -1.17140e6 q^{77} -6.55101e6 q^{79} -144746. q^{80} -3.90179e6 q^{82} -3.21765e6 q^{83} -1.05438e6 q^{85} -1.62265e7 q^{86} +1.03401e6 q^{88} +1.14227e7 q^{89} -1.26628e7 q^{91} -2.30889e6 q^{92} -963351. q^{94} +580999. q^{95} -1.41253e7 q^{97} +1.30545e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8} + 1452 q^{10} + 1148 q^{11} - 642 q^{13} - 5756 q^{14} - 22606 q^{16} + 5798 q^{17} - 6036 q^{19} + 27376 q^{20} - 97896 q^{22} - 60835 q^{23} - 262477 q^{25} + 355992 q^{26} - 507124 q^{28} + 169162 q^{29} - 199640 q^{31} + 284794 q^{32} - 1027740 q^{34} + 137680 q^{35} - 202002 q^{37} + 554924 q^{38} - 340904 q^{40} - 541282 q^{41} - 909596 q^{43} + 1236032 q^{44} - 80208 q^{47} + 850589 q^{49} + 941416 q^{50} + 146940 q^{52} + 278138 q^{53} - 933560 q^{55} + 539932 q^{56} - 3522712 q^{58} + 3177380 q^{59} + 147782 q^{61} - 4606456 q^{62} - 4142622 q^{64} - 3877332 q^{65} - 464916 q^{67} - 7513072 q^{68} + 2093200 q^{70} - 1576792 q^{71} - 38190 q^{73} - 12164864 q^{74} + 6889436 q^{76} - 10332384 q^{77} - 3913336 q^{79} - 6334776 q^{80} + 6799360 q^{82} - 15774716 q^{83} - 8520740 q^{85} - 24874084 q^{86} + 53216 q^{88} - 1116482 q^{89} - 27369552 q^{91} - 3285090 q^{92} - 7153744 q^{94} + 6067832 q^{95} - 15738566 q^{97} - 11730488 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.8260 1.57561 0.787806 0.615924i \(-0.211217\pi\)
0.787806 + 0.615924i \(0.211217\pi\)
\(3\) 0 0
\(4\) 189.767 1.48255
\(5\) 31.0428 0.111062 0.0555310 0.998457i \(-0.482315\pi\)
0.0555310 + 0.998457i \(0.482315\pi\)
\(6\) 0 0
\(7\) −1247.35 −1.37450 −0.687248 0.726423i \(-0.741181\pi\)
−0.687248 + 0.726423i \(0.741181\pi\)
\(8\) 1101.05 0.760313
\(9\) 0 0
\(10\) 553.369 0.174991
\(11\) 939.114 0.212737 0.106369 0.994327i \(-0.466078\pi\)
0.106369 + 0.994327i \(0.466078\pi\)
\(12\) 0 0
\(13\) 10151.8 1.28156 0.640781 0.767723i \(-0.278610\pi\)
0.640781 + 0.767723i \(0.278610\pi\)
\(14\) −22235.2 −2.16567
\(15\) 0 0
\(16\) −4662.78 −0.284594
\(17\) −33965.5 −1.67674 −0.838371 0.545100i \(-0.816492\pi\)
−0.838371 + 0.545100i \(0.816492\pi\)
\(18\) 0 0
\(19\) 18716.1 0.626005 0.313002 0.949752i \(-0.398665\pi\)
0.313002 + 0.949752i \(0.398665\pi\)
\(20\) 5890.88 0.164655
\(21\) 0 0
\(22\) 16740.7 0.335192
\(23\) −12167.0 −0.208514
\(24\) 0 0
\(25\) −77161.3 −0.987665
\(26\) 180965. 2.01925
\(27\) 0 0
\(28\) −236704. −2.03776
\(29\) −91722.7 −0.698367 −0.349183 0.937054i \(-0.613541\pi\)
−0.349183 + 0.937054i \(0.613541\pi\)
\(30\) 0 0
\(31\) −187422. −1.12994 −0.564968 0.825113i \(-0.691111\pi\)
−0.564968 + 0.825113i \(0.691111\pi\)
\(32\) −224053. −1.20872
\(33\) 0 0
\(34\) −605469. −2.64189
\(35\) −38721.0 −0.152654
\(36\) 0 0
\(37\) −112278. −0.364408 −0.182204 0.983261i \(-0.558323\pi\)
−0.182204 + 0.983261i \(0.558323\pi\)
\(38\) 333633. 0.986340
\(39\) 0 0
\(40\) 34179.6 0.0844418
\(41\) −218882. −0.495983 −0.247991 0.968762i \(-0.579770\pi\)
−0.247991 + 0.968762i \(0.579770\pi\)
\(42\) 0 0
\(43\) −910269. −1.74594 −0.872971 0.487772i \(-0.837810\pi\)
−0.872971 + 0.487772i \(0.837810\pi\)
\(44\) 178212. 0.315394
\(45\) 0 0
\(46\) −216889. −0.328538
\(47\) −54041.9 −0.0759256 −0.0379628 0.999279i \(-0.512087\pi\)
−0.0379628 + 0.999279i \(0.512087\pi\)
\(48\) 0 0
\(49\) 732327. 0.889239
\(50\) −1.37548e6 −1.55618
\(51\) 0 0
\(52\) 1.92647e6 1.89998
\(53\) 135797. 0.125292 0.0626462 0.998036i \(-0.480046\pi\)
0.0626462 + 0.998036i \(0.480046\pi\)
\(54\) 0 0
\(55\) 29152.7 0.0236270
\(56\) −1.37339e6 −1.04505
\(57\) 0 0
\(58\) −1.63505e6 −1.10035
\(59\) 2.26099e6 1.43323 0.716616 0.697468i \(-0.245690\pi\)
0.716616 + 0.697468i \(0.245690\pi\)
\(60\) 0 0
\(61\) 124600. 0.0702851 0.0351426 0.999382i \(-0.488811\pi\)
0.0351426 + 0.999382i \(0.488811\pi\)
\(62\) −3.34098e6 −1.78034
\(63\) 0 0
\(64\) −3.39714e6 −1.61988
\(65\) 315139. 0.142333
\(66\) 0 0
\(67\) 3.30689e6 1.34325 0.671627 0.740890i \(-0.265596\pi\)
0.671627 + 0.740890i \(0.265596\pi\)
\(68\) −6.44551e6 −2.48586
\(69\) 0 0
\(70\) −690242. −0.240524
\(71\) 740248. 0.245456 0.122728 0.992440i \(-0.460836\pi\)
0.122728 + 0.992440i \(0.460836\pi\)
\(72\) 0 0
\(73\) 2.68349e6 0.807365 0.403683 0.914899i \(-0.367730\pi\)
0.403683 + 0.914899i \(0.367730\pi\)
\(74\) −2.00147e6 −0.574166
\(75\) 0 0
\(76\) 3.55169e6 0.928084
\(77\) −1.17140e6 −0.292407
\(78\) 0 0
\(79\) −6.55101e6 −1.49490 −0.747452 0.664315i \(-0.768723\pi\)
−0.747452 + 0.664315i \(0.768723\pi\)
\(80\) −144746. −0.0316076
\(81\) 0 0
\(82\) −3.90179e6 −0.781476
\(83\) −3.21765e6 −0.617682 −0.308841 0.951114i \(-0.599941\pi\)
−0.308841 + 0.951114i \(0.599941\pi\)
\(84\) 0 0
\(85\) −1.05438e6 −0.186222
\(86\) −1.62265e7 −2.75093
\(87\) 0 0
\(88\) 1.03401e6 0.161747
\(89\) 1.14227e7 1.71753 0.858765 0.512370i \(-0.171232\pi\)
0.858765 + 0.512370i \(0.171232\pi\)
\(90\) 0 0
\(91\) −1.26628e7 −1.76150
\(92\) −2.30889e6 −0.309133
\(93\) 0 0
\(94\) −963351. −0.119629
\(95\) 580999. 0.0695253
\(96\) 0 0
\(97\) −1.41253e7 −1.57143 −0.785717 0.618586i \(-0.787706\pi\)
−0.785717 + 0.618586i \(0.787706\pi\)
\(98\) 1.30545e7 1.40110
\(99\) 0 0
\(100\) −1.46426e7 −1.46426
\(101\) 1.20112e7 1.16001 0.580006 0.814612i \(-0.303050\pi\)
0.580006 + 0.814612i \(0.303050\pi\)
\(102\) 0 0
\(103\) −7.74503e6 −0.698382 −0.349191 0.937052i \(-0.613544\pi\)
−0.349191 + 0.937052i \(0.613544\pi\)
\(104\) 1.11776e7 0.974388
\(105\) 0 0
\(106\) 2.42072e6 0.197412
\(107\) −4.90393e6 −0.386991 −0.193495 0.981101i \(-0.561982\pi\)
−0.193495 + 0.981101i \(0.561982\pi\)
\(108\) 0 0
\(109\) 2.12306e7 1.57025 0.785127 0.619335i \(-0.212598\pi\)
0.785127 + 0.619335i \(0.212598\pi\)
\(110\) 519676. 0.0372270
\(111\) 0 0
\(112\) 5.81610e6 0.391173
\(113\) −7.90376e6 −0.515299 −0.257649 0.966238i \(-0.582948\pi\)
−0.257649 + 0.966238i \(0.582948\pi\)
\(114\) 0 0
\(115\) −377697. −0.0231580
\(116\) −1.74059e7 −1.03536
\(117\) 0 0
\(118\) 4.03044e7 2.25822
\(119\) 4.23667e7 2.30468
\(120\) 0 0
\(121\) −1.86052e7 −0.954743
\(122\) 2.22112e6 0.110742
\(123\) 0 0
\(124\) −3.55663e7 −1.67519
\(125\) −4.82052e6 −0.220754
\(126\) 0 0
\(127\) 3.73010e7 1.61587 0.807936 0.589270i \(-0.200585\pi\)
0.807936 + 0.589270i \(0.200585\pi\)
\(128\) −3.18786e7 −1.34358
\(129\) 0 0
\(130\) 5.61767e6 0.224261
\(131\) 1.67942e6 0.0652694 0.0326347 0.999467i \(-0.489610\pi\)
0.0326347 + 0.999467i \(0.489610\pi\)
\(132\) 0 0
\(133\) −2.33454e7 −0.860441
\(134\) 5.89486e7 2.11644
\(135\) 0 0
\(136\) −3.73977e7 −1.27485
\(137\) −8.56278e6 −0.284507 −0.142253 0.989830i \(-0.545435\pi\)
−0.142253 + 0.989830i \(0.545435\pi\)
\(138\) 0 0
\(139\) −1.00144e7 −0.316281 −0.158140 0.987417i \(-0.550550\pi\)
−0.158140 + 0.987417i \(0.550550\pi\)
\(140\) −7.34796e6 −0.226318
\(141\) 0 0
\(142\) 1.31957e7 0.386743
\(143\) 9.53367e6 0.272636
\(144\) 0 0
\(145\) −2.84733e6 −0.0775620
\(146\) 4.78359e7 1.27209
\(147\) 0 0
\(148\) −2.13066e7 −0.540254
\(149\) −1.78249e7 −0.441443 −0.220722 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(150\) 0 0
\(151\) −1.24711e7 −0.294771 −0.147386 0.989079i \(-0.547086\pi\)
−0.147386 + 0.989079i \(0.547086\pi\)
\(152\) 2.06074e7 0.475959
\(153\) 0 0
\(154\) −2.08814e7 −0.460719
\(155\) −5.81808e6 −0.125493
\(156\) 0 0
\(157\) −2.73688e7 −0.564426 −0.282213 0.959352i \(-0.591069\pi\)
−0.282213 + 0.959352i \(0.591069\pi\)
\(158\) −1.16778e8 −2.35539
\(159\) 0 0
\(160\) −6.95523e6 −0.134243
\(161\) 1.51764e7 0.286602
\(162\) 0 0
\(163\) 8.32721e7 1.50606 0.753031 0.657985i \(-0.228591\pi\)
0.753031 + 0.657985i \(0.228591\pi\)
\(164\) −4.15365e7 −0.735320
\(165\) 0 0
\(166\) −5.73578e7 −0.973227
\(167\) −9.35501e6 −0.155431 −0.0777153 0.996976i \(-0.524763\pi\)
−0.0777153 + 0.996976i \(0.524763\pi\)
\(168\) 0 0
\(169\) 4.03099e7 0.642404
\(170\) −1.87954e7 −0.293414
\(171\) 0 0
\(172\) −1.72738e8 −2.58845
\(173\) 8.62670e7 1.26673 0.633364 0.773854i \(-0.281674\pi\)
0.633364 + 0.773854i \(0.281674\pi\)
\(174\) 0 0
\(175\) 9.62468e7 1.35754
\(176\) −4.37889e6 −0.0605438
\(177\) 0 0
\(178\) 2.03621e8 2.70616
\(179\) 9.64475e7 1.25691 0.628456 0.777845i \(-0.283687\pi\)
0.628456 + 0.777845i \(0.283687\pi\)
\(180\) 0 0
\(181\) −2.62936e7 −0.329591 −0.164796 0.986328i \(-0.552696\pi\)
−0.164796 + 0.986328i \(0.552696\pi\)
\(182\) −2.25726e8 −2.77544
\(183\) 0 0
\(184\) −1.33965e7 −0.158536
\(185\) −3.48542e6 −0.0404719
\(186\) 0 0
\(187\) −3.18974e7 −0.356706
\(188\) −1.02553e7 −0.112564
\(189\) 0 0
\(190\) 1.03569e7 0.109545
\(191\) −5.41983e7 −0.562819 −0.281410 0.959588i \(-0.590802\pi\)
−0.281410 + 0.959588i \(0.590802\pi\)
\(192\) 0 0
\(193\) −1.54438e8 −1.54633 −0.773166 0.634203i \(-0.781328\pi\)
−0.773166 + 0.634203i \(0.781328\pi\)
\(194\) −2.51797e8 −2.47597
\(195\) 0 0
\(196\) 1.38971e8 1.31834
\(197\) −1.17189e8 −1.09208 −0.546041 0.837759i \(-0.683866\pi\)
−0.546041 + 0.837759i \(0.683866\pi\)
\(198\) 0 0
\(199\) −9.17056e7 −0.824917 −0.412458 0.910976i \(-0.635330\pi\)
−0.412458 + 0.910976i \(0.635330\pi\)
\(200\) −8.49585e7 −0.750934
\(201\) 0 0
\(202\) 2.14112e8 1.82773
\(203\) 1.14410e8 0.959903
\(204\) 0 0
\(205\) −6.79470e6 −0.0550848
\(206\) −1.38063e8 −1.10038
\(207\) 0 0
\(208\) −4.73355e7 −0.364725
\(209\) 1.75766e7 0.133175
\(210\) 0 0
\(211\) 3.23106e7 0.236786 0.118393 0.992967i \(-0.462226\pi\)
0.118393 + 0.992967i \(0.462226\pi\)
\(212\) 2.57697e7 0.185752
\(213\) 0 0
\(214\) −8.74174e7 −0.609747
\(215\) −2.82573e7 −0.193908
\(216\) 0 0
\(217\) 2.33779e8 1.55309
\(218\) 3.78457e8 2.47411
\(219\) 0 0
\(220\) 5.53221e6 0.0350283
\(221\) −3.44809e8 −2.14885
\(222\) 0 0
\(223\) 1.25273e6 0.00756468 0.00378234 0.999993i \(-0.498796\pi\)
0.00378234 + 0.999993i \(0.498796\pi\)
\(224\) 2.79472e8 1.66138
\(225\) 0 0
\(226\) −1.40892e8 −0.811911
\(227\) 1.41797e8 0.804594 0.402297 0.915509i \(-0.368212\pi\)
0.402297 + 0.915509i \(0.368212\pi\)
\(228\) 0 0
\(229\) −7.55158e7 −0.415541 −0.207770 0.978178i \(-0.566621\pi\)
−0.207770 + 0.978178i \(0.566621\pi\)
\(230\) −6.73284e6 −0.0364880
\(231\) 0 0
\(232\) −1.00991e8 −0.530977
\(233\) 2.66330e8 1.37935 0.689674 0.724120i \(-0.257754\pi\)
0.689674 + 0.724120i \(0.257754\pi\)
\(234\) 0 0
\(235\) −1.67761e6 −0.00843244
\(236\) 4.29060e8 2.12484
\(237\) 0 0
\(238\) 7.55228e8 3.63127
\(239\) 3.47869e8 1.64825 0.824125 0.566408i \(-0.191667\pi\)
0.824125 + 0.566408i \(0.191667\pi\)
\(240\) 0 0
\(241\) −1.34331e8 −0.618184 −0.309092 0.951032i \(-0.600025\pi\)
−0.309092 + 0.951032i \(0.600025\pi\)
\(242\) −3.31657e8 −1.50430
\(243\) 0 0
\(244\) 2.36449e7 0.104201
\(245\) 2.27335e7 0.0987607
\(246\) 0 0
\(247\) 1.90001e8 0.802265
\(248\) −2.06361e8 −0.859104
\(249\) 0 0
\(250\) −8.59306e7 −0.347823
\(251\) 1.25191e8 0.499707 0.249854 0.968284i \(-0.419618\pi\)
0.249854 + 0.968284i \(0.419618\pi\)
\(252\) 0 0
\(253\) −1.14262e7 −0.0443588
\(254\) 6.64927e8 2.54599
\(255\) 0 0
\(256\) −1.33434e8 −0.497082
\(257\) 3.83946e8 1.41093 0.705463 0.708746i \(-0.250739\pi\)
0.705463 + 0.708746i \(0.250739\pi\)
\(258\) 0 0
\(259\) 1.40049e8 0.500878
\(260\) 5.98028e7 0.211016
\(261\) 0 0
\(262\) 2.99373e7 0.102839
\(263\) 5.17704e8 1.75483 0.877417 0.479729i \(-0.159265\pi\)
0.877417 + 0.479729i \(0.159265\pi\)
\(264\) 0 0
\(265\) 4.21552e6 0.0139152
\(266\) −4.16156e8 −1.35572
\(267\) 0 0
\(268\) 6.27537e8 1.99144
\(269\) 2.04806e8 0.641518 0.320759 0.947161i \(-0.396062\pi\)
0.320759 + 0.947161i \(0.396062\pi\)
\(270\) 0 0
\(271\) −2.30872e8 −0.704659 −0.352329 0.935876i \(-0.614610\pi\)
−0.352329 + 0.935876i \(0.614610\pi\)
\(272\) 1.58374e8 0.477190
\(273\) 0 0
\(274\) −1.52640e8 −0.448272
\(275\) −7.24633e7 −0.210113
\(276\) 0 0
\(277\) −3.59723e8 −1.01693 −0.508463 0.861084i \(-0.669786\pi\)
−0.508463 + 0.861084i \(0.669786\pi\)
\(278\) −1.78517e8 −0.498335
\(279\) 0 0
\(280\) −4.26338e7 −0.116065
\(281\) −5.59198e8 −1.50346 −0.751732 0.659469i \(-0.770781\pi\)
−0.751732 + 0.659469i \(0.770781\pi\)
\(282\) 0 0
\(283\) −2.74952e8 −0.721114 −0.360557 0.932737i \(-0.617413\pi\)
−0.360557 + 0.932737i \(0.617413\pi\)
\(284\) 1.40474e8 0.363900
\(285\) 0 0
\(286\) 1.69947e8 0.429569
\(287\) 2.73021e8 0.681726
\(288\) 0 0
\(289\) 7.43314e8 1.81146
\(290\) −5.07564e7 −0.122208
\(291\) 0 0
\(292\) 5.09237e8 1.19696
\(293\) −2.01642e8 −0.468321 −0.234161 0.972198i \(-0.575234\pi\)
−0.234161 + 0.972198i \(0.575234\pi\)
\(294\) 0 0
\(295\) 7.01873e7 0.159178
\(296\) −1.23624e8 −0.277064
\(297\) 0 0
\(298\) −3.17747e8 −0.695543
\(299\) −1.23517e8 −0.267224
\(300\) 0 0
\(301\) 1.13542e9 2.39979
\(302\) −2.22310e8 −0.464445
\(303\) 0 0
\(304\) −8.72691e7 −0.178157
\(305\) 3.86793e6 0.00780600
\(306\) 0 0
\(307\) −4.38553e8 −0.865043 −0.432521 0.901624i \(-0.642376\pi\)
−0.432521 + 0.901624i \(0.642376\pi\)
\(308\) −2.22292e8 −0.433508
\(309\) 0 0
\(310\) −1.03713e8 −0.197728
\(311\) −6.05423e8 −1.14129 −0.570647 0.821196i \(-0.693307\pi\)
−0.570647 + 0.821196i \(0.693307\pi\)
\(312\) 0 0
\(313\) −1.07052e7 −0.0197329 −0.00986647 0.999951i \(-0.503141\pi\)
−0.00986647 + 0.999951i \(0.503141\pi\)
\(314\) −4.87877e8 −0.889316
\(315\) 0 0
\(316\) −1.24316e9 −2.21627
\(317\) −9.99712e8 −1.76266 −0.881328 0.472505i \(-0.843350\pi\)
−0.881328 + 0.472505i \(0.843350\pi\)
\(318\) 0 0
\(319\) −8.61381e7 −0.148569
\(320\) −1.05457e8 −0.179907
\(321\) 0 0
\(322\) 2.70535e8 0.451574
\(323\) −6.35701e8 −1.04965
\(324\) 0 0
\(325\) −7.83324e8 −1.26576
\(326\) 1.48441e9 2.37297
\(327\) 0 0
\(328\) −2.41000e8 −0.377102
\(329\) 6.74089e7 0.104359
\(330\) 0 0
\(331\) −8.39788e8 −1.27283 −0.636417 0.771345i \(-0.719584\pi\)
−0.636417 + 0.771345i \(0.719584\pi\)
\(332\) −6.10602e8 −0.915745
\(333\) 0 0
\(334\) −1.66762e8 −0.244898
\(335\) 1.02655e8 0.149184
\(336\) 0 0
\(337\) −5.53660e8 −0.788022 −0.394011 0.919106i \(-0.628913\pi\)
−0.394011 + 0.919106i \(0.628913\pi\)
\(338\) 7.18564e8 1.01218
\(339\) 0 0
\(340\) −2.00086e8 −0.276084
\(341\) −1.76010e8 −0.240380
\(342\) 0 0
\(343\) 1.13778e8 0.152240
\(344\) −1.00225e9 −1.32746
\(345\) 0 0
\(346\) 1.53780e9 1.99587
\(347\) −8.05434e8 −1.03485 −0.517424 0.855729i \(-0.673109\pi\)
−0.517424 + 0.855729i \(0.673109\pi\)
\(348\) 0 0
\(349\) 6.32931e8 0.797017 0.398509 0.917165i \(-0.369528\pi\)
0.398509 + 0.917165i \(0.369528\pi\)
\(350\) 1.71570e9 2.13896
\(351\) 0 0
\(352\) −2.10412e8 −0.257140
\(353\) −3.46589e8 −0.419376 −0.209688 0.977768i \(-0.567245\pi\)
−0.209688 + 0.977768i \(0.567245\pi\)
\(354\) 0 0
\(355\) 2.29793e7 0.0272608
\(356\) 2.16765e9 2.54633
\(357\) 0 0
\(358\) 1.71927e9 1.98041
\(359\) 2.14794e8 0.245014 0.122507 0.992468i \(-0.460907\pi\)
0.122507 + 0.992468i \(0.460907\pi\)
\(360\) 0 0
\(361\) −5.43580e8 −0.608118
\(362\) −4.68710e8 −0.519307
\(363\) 0 0
\(364\) −2.40297e9 −2.61152
\(365\) 8.33030e7 0.0896676
\(366\) 0 0
\(367\) 1.71463e9 1.81067 0.905335 0.424699i \(-0.139620\pi\)
0.905335 + 0.424699i \(0.139620\pi\)
\(368\) 5.67321e7 0.0593419
\(369\) 0 0
\(370\) −6.21311e7 −0.0637680
\(371\) −1.69386e8 −0.172214
\(372\) 0 0
\(373\) −8.64659e8 −0.862709 −0.431354 0.902183i \(-0.641964\pi\)
−0.431354 + 0.902183i \(0.641964\pi\)
\(374\) −5.68604e8 −0.562030
\(375\) 0 0
\(376\) −5.95028e7 −0.0577272
\(377\) −9.31147e8 −0.895001
\(378\) 0 0
\(379\) −5.13537e7 −0.0484546 −0.0242273 0.999706i \(-0.507713\pi\)
−0.0242273 + 0.999706i \(0.507713\pi\)
\(380\) 1.10254e8 0.103075
\(381\) 0 0
\(382\) −9.66139e8 −0.886784
\(383\) −2.01952e9 −1.83676 −0.918379 0.395703i \(-0.870501\pi\)
−0.918379 + 0.395703i \(0.870501\pi\)
\(384\) 0 0
\(385\) −3.63635e7 −0.0324753
\(386\) −2.75301e9 −2.43642
\(387\) 0 0
\(388\) −2.68051e9 −2.32973
\(389\) −1.39868e9 −1.20474 −0.602370 0.798217i \(-0.705777\pi\)
−0.602370 + 0.798217i \(0.705777\pi\)
\(390\) 0 0
\(391\) 4.13258e8 0.349625
\(392\) 8.06329e8 0.676100
\(393\) 0 0
\(394\) −2.08901e9 −1.72070
\(395\) −2.03361e8 −0.166027
\(396\) 0 0
\(397\) −1.96246e9 −1.57411 −0.787055 0.616883i \(-0.788395\pi\)
−0.787055 + 0.616883i \(0.788395\pi\)
\(398\) −1.63474e9 −1.29975
\(399\) 0 0
\(400\) 3.59787e8 0.281083
\(401\) 2.26678e9 1.75552 0.877758 0.479104i \(-0.159038\pi\)
0.877758 + 0.479104i \(0.159038\pi\)
\(402\) 0 0
\(403\) −1.90266e9 −1.44808
\(404\) 2.27933e9 1.71978
\(405\) 0 0
\(406\) 2.03947e9 1.51243
\(407\) −1.05442e8 −0.0775233
\(408\) 0 0
\(409\) 2.28154e9 1.64891 0.824453 0.565931i \(-0.191483\pi\)
0.824453 + 0.565931i \(0.191483\pi\)
\(410\) −1.21122e8 −0.0867923
\(411\) 0 0
\(412\) −1.46975e9 −1.03539
\(413\) −2.82023e9 −1.96997
\(414\) 0 0
\(415\) −9.98847e7 −0.0686010
\(416\) −2.27454e9 −1.54905
\(417\) 0 0
\(418\) 3.13320e8 0.209832
\(419\) −6.99241e7 −0.0464384 −0.0232192 0.999730i \(-0.507392\pi\)
−0.0232192 + 0.999730i \(0.507392\pi\)
\(420\) 0 0
\(421\) −1.08544e9 −0.708952 −0.354476 0.935065i \(-0.615341\pi\)
−0.354476 + 0.935065i \(0.615341\pi\)
\(422\) 5.75969e8 0.373083
\(423\) 0 0
\(424\) 1.49519e8 0.0952614
\(425\) 2.62082e9 1.65606
\(426\) 0 0
\(427\) −1.55419e8 −0.0966066
\(428\) −9.30601e8 −0.573734
\(429\) 0 0
\(430\) −5.03714e8 −0.305523
\(431\) 3.16872e9 1.90640 0.953198 0.302348i \(-0.0977705\pi\)
0.953198 + 0.302348i \(0.0977705\pi\)
\(432\) 0 0
\(433\) 3.26047e8 0.193007 0.0965033 0.995333i \(-0.469234\pi\)
0.0965033 + 0.995333i \(0.469234\pi\)
\(434\) 4.16735e9 2.44707
\(435\) 0 0
\(436\) 4.02886e9 2.32798
\(437\) −2.27719e8 −0.130531
\(438\) 0 0
\(439\) −2.63160e9 −1.48455 −0.742273 0.670097i \(-0.766252\pi\)
−0.742273 + 0.670097i \(0.766252\pi\)
\(440\) 3.20986e7 0.0179639
\(441\) 0 0
\(442\) −6.14658e9 −3.38575
\(443\) 3.10154e8 0.169498 0.0847489 0.996402i \(-0.472991\pi\)
0.0847489 + 0.996402i \(0.472991\pi\)
\(444\) 0 0
\(445\) 3.54593e8 0.190752
\(446\) 2.23312e7 0.0119190
\(447\) 0 0
\(448\) 4.23740e9 2.22652
\(449\) 1.98075e9 1.03268 0.516341 0.856383i \(-0.327294\pi\)
0.516341 + 0.856383i \(0.327294\pi\)
\(450\) 0 0
\(451\) −2.05555e8 −0.105514
\(452\) −1.49987e9 −0.763957
\(453\) 0 0
\(454\) 2.52767e9 1.26773
\(455\) −3.93087e8 −0.195636
\(456\) 0 0
\(457\) −1.57211e9 −0.770504 −0.385252 0.922811i \(-0.625885\pi\)
−0.385252 + 0.922811i \(0.625885\pi\)
\(458\) −1.34615e9 −0.654731
\(459\) 0 0
\(460\) −7.16743e7 −0.0343329
\(461\) −2.80297e9 −1.33249 −0.666247 0.745731i \(-0.732101\pi\)
−0.666247 + 0.745731i \(0.732101\pi\)
\(462\) 0 0
\(463\) 3.29626e9 1.54343 0.771717 0.635966i \(-0.219398\pi\)
0.771717 + 0.635966i \(0.219398\pi\)
\(464\) 4.27683e8 0.198751
\(465\) 0 0
\(466\) 4.74759e9 2.17332
\(467\) 2.22099e9 1.00911 0.504554 0.863380i \(-0.331657\pi\)
0.504554 + 0.863380i \(0.331657\pi\)
\(468\) 0 0
\(469\) −4.12483e9 −1.84630
\(470\) −2.99051e7 −0.0132863
\(471\) 0 0
\(472\) 2.48946e9 1.08970
\(473\) −8.54846e8 −0.371427
\(474\) 0 0
\(475\) −1.44416e9 −0.618283
\(476\) 8.03977e9 3.41680
\(477\) 0 0
\(478\) 6.20111e9 2.59700
\(479\) −4.08881e9 −1.69990 −0.849948 0.526867i \(-0.823367\pi\)
−0.849948 + 0.526867i \(0.823367\pi\)
\(480\) 0 0
\(481\) −1.13982e9 −0.467012
\(482\) −2.39459e9 −0.974017
\(483\) 0 0
\(484\) −3.53065e9 −1.41545
\(485\) −4.38488e8 −0.174527
\(486\) 0 0
\(487\) −2.20516e9 −0.865144 −0.432572 0.901599i \(-0.642394\pi\)
−0.432572 + 0.901599i \(0.642394\pi\)
\(488\) 1.37191e8 0.0534387
\(489\) 0 0
\(490\) 4.05247e8 0.155608
\(491\) 2.44960e9 0.933921 0.466961 0.884278i \(-0.345349\pi\)
0.466961 + 0.884278i \(0.345349\pi\)
\(492\) 0 0
\(493\) 3.11540e9 1.17098
\(494\) 3.38697e9 1.26406
\(495\) 0 0
\(496\) 8.73906e8 0.321573
\(497\) −9.23345e8 −0.337378
\(498\) 0 0
\(499\) −4.09114e9 −1.47398 −0.736991 0.675903i \(-0.763754\pi\)
−0.736991 + 0.675903i \(0.763754\pi\)
\(500\) −9.14773e8 −0.327279
\(501\) 0 0
\(502\) 2.23166e9 0.787344
\(503\) −4.53054e8 −0.158731 −0.0793655 0.996846i \(-0.525289\pi\)
−0.0793655 + 0.996846i \(0.525289\pi\)
\(504\) 0 0
\(505\) 3.72862e8 0.128833
\(506\) −2.03684e8 −0.0698923
\(507\) 0 0
\(508\) 7.07847e9 2.39561
\(509\) 2.23184e9 0.750156 0.375078 0.926993i \(-0.377616\pi\)
0.375078 + 0.926993i \(0.377616\pi\)
\(510\) 0 0
\(511\) −3.34724e9 −1.10972
\(512\) 1.70186e9 0.560375
\(513\) 0 0
\(514\) 6.84423e9 2.22307
\(515\) −2.40427e8 −0.0775636
\(516\) 0 0
\(517\) −5.07515e7 −0.0161522
\(518\) 2.49652e9 0.789189
\(519\) 0 0
\(520\) 3.46984e8 0.108218
\(521\) −3.73130e9 −1.15592 −0.577960 0.816065i \(-0.696151\pi\)
−0.577960 + 0.816065i \(0.696151\pi\)
\(522\) 0 0
\(523\) −3.65832e9 −1.11822 −0.559108 0.829095i \(-0.688856\pi\)
−0.559108 + 0.829095i \(0.688856\pi\)
\(524\) 3.18698e8 0.0967652
\(525\) 0 0
\(526\) 9.22859e9 2.76494
\(527\) 6.36586e9 1.89461
\(528\) 0 0
\(529\) 1.48036e8 0.0434783
\(530\) 7.51458e7 0.0219250
\(531\) 0 0
\(532\) −4.43018e9 −1.27565
\(533\) −2.22204e9 −0.635633
\(534\) 0 0
\(535\) −1.52231e8 −0.0429800
\(536\) 3.64105e9 1.02129
\(537\) 0 0
\(538\) 3.65087e9 1.01078
\(539\) 6.87739e8 0.189175
\(540\) 0 0
\(541\) 2.44725e9 0.664490 0.332245 0.943193i \(-0.392194\pi\)
0.332245 + 0.943193i \(0.392194\pi\)
\(542\) −4.11553e9 −1.11027
\(543\) 0 0
\(544\) 7.61007e9 2.02671
\(545\) 6.59057e8 0.174396
\(546\) 0 0
\(547\) 3.76239e9 0.982897 0.491449 0.870907i \(-0.336468\pi\)
0.491449 + 0.870907i \(0.336468\pi\)
\(548\) −1.62493e9 −0.421796
\(549\) 0 0
\(550\) −1.29173e9 −0.331057
\(551\) −1.71669e9 −0.437181
\(552\) 0 0
\(553\) 8.17137e9 2.05474
\(554\) −6.41242e9 −1.60228
\(555\) 0 0
\(556\) −1.90040e9 −0.468902
\(557\) −6.50679e9 −1.59541 −0.797707 0.603045i \(-0.793954\pi\)
−0.797707 + 0.603045i \(0.793954\pi\)
\(558\) 0 0
\(559\) −9.24084e9 −2.23754
\(560\) 1.80548e8 0.0434445
\(561\) 0 0
\(562\) −9.96826e9 −2.36888
\(563\) −2.23762e9 −0.528453 −0.264226 0.964461i \(-0.585117\pi\)
−0.264226 + 0.964461i \(0.585117\pi\)
\(564\) 0 0
\(565\) −2.45355e8 −0.0572301
\(566\) −4.90129e9 −1.13620
\(567\) 0 0
\(568\) 8.15050e8 0.186623
\(569\) −3.55887e9 −0.809877 −0.404939 0.914344i \(-0.632707\pi\)
−0.404939 + 0.914344i \(0.632707\pi\)
\(570\) 0 0
\(571\) −2.69760e9 −0.606389 −0.303194 0.952929i \(-0.598053\pi\)
−0.303194 + 0.952929i \(0.598053\pi\)
\(572\) 1.80917e9 0.404197
\(573\) 0 0
\(574\) 4.86688e9 1.07414
\(575\) 9.38822e8 0.205942
\(576\) 0 0
\(577\) −2.73131e9 −0.591910 −0.295955 0.955202i \(-0.595638\pi\)
−0.295955 + 0.955202i \(0.595638\pi\)
\(578\) 1.32503e10 2.85416
\(579\) 0 0
\(580\) −5.40327e8 −0.114990
\(581\) 4.01352e9 0.849002
\(582\) 0 0
\(583\) 1.27529e8 0.0266544
\(584\) 2.95466e9 0.613850
\(585\) 0 0
\(586\) −3.59447e9 −0.737892
\(587\) 4.25360e9 0.868008 0.434004 0.900911i \(-0.357100\pi\)
0.434004 + 0.900911i \(0.357100\pi\)
\(588\) 0 0
\(589\) −3.50780e9 −0.707345
\(590\) 1.25116e9 0.250802
\(591\) 0 0
\(592\) 5.23528e8 0.103708
\(593\) −8.85467e9 −1.74374 −0.871869 0.489740i \(-0.837092\pi\)
−0.871869 + 0.489740i \(0.837092\pi\)
\(594\) 0 0
\(595\) 1.31518e9 0.255962
\(596\) −3.38257e9 −0.654462
\(597\) 0 0
\(598\) −2.20181e9 −0.421042
\(599\) 8.77964e9 1.66910 0.834551 0.550930i \(-0.185727\pi\)
0.834551 + 0.550930i \(0.185727\pi\)
\(600\) 0 0
\(601\) 4.36354e9 0.819933 0.409967 0.912101i \(-0.365540\pi\)
0.409967 + 0.912101i \(0.365540\pi\)
\(602\) 2.02400e10 3.78114
\(603\) 0 0
\(604\) −2.36659e9 −0.437013
\(605\) −5.77558e8 −0.106036
\(606\) 0 0
\(607\) −1.00264e10 −1.81963 −0.909815 0.415015i \(-0.863776\pi\)
−0.909815 + 0.415015i \(0.863776\pi\)
\(608\) −4.19340e9 −0.756666
\(609\) 0 0
\(610\) 6.89497e7 0.0122992
\(611\) −5.48621e8 −0.0973034
\(612\) 0 0
\(613\) −1.06422e10 −1.86603 −0.933013 0.359842i \(-0.882830\pi\)
−0.933013 + 0.359842i \(0.882830\pi\)
\(614\) −7.81765e9 −1.36297
\(615\) 0 0
\(616\) −1.28977e9 −0.222321
\(617\) 1.31952e9 0.226161 0.113080 0.993586i \(-0.463928\pi\)
0.113080 + 0.993586i \(0.463928\pi\)
\(618\) 0 0
\(619\) 8.51725e9 1.44339 0.721693 0.692214i \(-0.243364\pi\)
0.721693 + 0.692214i \(0.243364\pi\)
\(620\) −1.10408e9 −0.186050
\(621\) 0 0
\(622\) −1.07923e10 −1.79823
\(623\) −1.42481e10 −2.36074
\(624\) 0 0
\(625\) 5.87859e9 0.963148
\(626\) −1.90832e8 −0.0310914
\(627\) 0 0
\(628\) −5.19368e9 −0.836791
\(629\) 3.81357e9 0.611019
\(630\) 0 0
\(631\) 1.32684e9 0.210240 0.105120 0.994460i \(-0.466477\pi\)
0.105120 + 0.994460i \(0.466477\pi\)
\(632\) −7.21299e9 −1.13659
\(633\) 0 0
\(634\) −1.78209e10 −2.77726
\(635\) 1.15792e9 0.179462
\(636\) 0 0
\(637\) 7.43441e9 1.13962
\(638\) −1.53550e9 −0.234087
\(639\) 0 0
\(640\) −9.89600e8 −0.149221
\(641\) −3.00273e9 −0.450312 −0.225156 0.974323i \(-0.572289\pi\)
−0.225156 + 0.974323i \(0.572289\pi\)
\(642\) 0 0
\(643\) −1.08587e10 −1.61080 −0.805398 0.592734i \(-0.798049\pi\)
−0.805398 + 0.592734i \(0.798049\pi\)
\(644\) 2.87998e9 0.424902
\(645\) 0 0
\(646\) −1.13320e10 −1.65384
\(647\) 8.53415e9 1.23878 0.619392 0.785082i \(-0.287379\pi\)
0.619392 + 0.785082i \(0.287379\pi\)
\(648\) 0 0
\(649\) 2.12333e9 0.304902
\(650\) −1.39635e10 −1.99434
\(651\) 0 0
\(652\) 1.58022e10 2.23281
\(653\) 2.44692e9 0.343893 0.171947 0.985106i \(-0.444994\pi\)
0.171947 + 0.985106i \(0.444994\pi\)
\(654\) 0 0
\(655\) 5.21338e7 0.00724895
\(656\) 1.02060e9 0.141154
\(657\) 0 0
\(658\) 1.20163e9 0.164430
\(659\) −7.66941e9 −1.04391 −0.521955 0.852973i \(-0.674797\pi\)
−0.521955 + 0.852973i \(0.674797\pi\)
\(660\) 0 0
\(661\) 1.06623e10 1.43597 0.717987 0.696057i \(-0.245064\pi\)
0.717987 + 0.696057i \(0.245064\pi\)
\(662\) −1.49701e10 −2.00549
\(663\) 0 0
\(664\) −3.54279e9 −0.469632
\(665\) −7.24707e8 −0.0955623
\(666\) 0 0
\(667\) 1.11599e9 0.145620
\(668\) −1.77527e9 −0.230434
\(669\) 0 0
\(670\) 1.82993e9 0.235057
\(671\) 1.17014e8 0.0149523
\(672\) 0 0
\(673\) 1.13404e10 1.43408 0.717042 0.697030i \(-0.245495\pi\)
0.717042 + 0.697030i \(0.245495\pi\)
\(674\) −9.86954e9 −1.24162
\(675\) 0 0
\(676\) 7.64946e9 0.952396
\(677\) 2.59963e9 0.321996 0.160998 0.986955i \(-0.448529\pi\)
0.160998 + 0.986955i \(0.448529\pi\)
\(678\) 0 0
\(679\) 1.76191e10 2.15993
\(680\) −1.16093e9 −0.141587
\(681\) 0 0
\(682\) −3.13756e9 −0.378745
\(683\) −1.25557e10 −1.50789 −0.753943 0.656940i \(-0.771851\pi\)
−0.753943 + 0.656940i \(0.771851\pi\)
\(684\) 0 0
\(685\) −2.65812e8 −0.0315979
\(686\) 2.02821e9 0.239871
\(687\) 0 0
\(688\) 4.24439e9 0.496884
\(689\) 1.37858e9 0.160570
\(690\) 0 0
\(691\) 4.66084e9 0.537392 0.268696 0.963225i \(-0.413407\pi\)
0.268696 + 0.963225i \(0.413407\pi\)
\(692\) 1.63706e10 1.87799
\(693\) 0 0
\(694\) −1.43577e10 −1.63052
\(695\) −3.10874e8 −0.0351268
\(696\) 0 0
\(697\) 7.43443e9 0.831635
\(698\) 1.12826e10 1.25579
\(699\) 0 0
\(700\) 1.82644e10 2.01263
\(701\) 4.37132e9 0.479292 0.239646 0.970860i \(-0.422969\pi\)
0.239646 + 0.970860i \(0.422969\pi\)
\(702\) 0 0
\(703\) −2.10140e9 −0.228121
\(704\) −3.19030e9 −0.344610
\(705\) 0 0
\(706\) −6.17830e9 −0.660774
\(707\) −1.49822e10 −1.59443
\(708\) 0 0
\(709\) −1.06398e10 −1.12117 −0.560585 0.828097i \(-0.689424\pi\)
−0.560585 + 0.828097i \(0.689424\pi\)
\(710\) 4.09630e8 0.0429524
\(711\) 0 0
\(712\) 1.25770e10 1.30586
\(713\) 2.28036e9 0.235608
\(714\) 0 0
\(715\) 2.95951e8 0.0302795
\(716\) 1.83025e10 1.86344
\(717\) 0 0
\(718\) 3.82892e9 0.386048
\(719\) 1.04952e10 1.05302 0.526512 0.850168i \(-0.323499\pi\)
0.526512 + 0.850168i \(0.323499\pi\)
\(720\) 0 0
\(721\) 9.66073e9 0.959923
\(722\) −9.68985e9 −0.958158
\(723\) 0 0
\(724\) −4.98965e9 −0.488635
\(725\) 7.07744e9 0.689753
\(726\) 0 0
\(727\) 1.98822e9 0.191908 0.0959541 0.995386i \(-0.469410\pi\)
0.0959541 + 0.995386i \(0.469410\pi\)
\(728\) −1.39423e10 −1.33929
\(729\) 0 0
\(730\) 1.48496e9 0.141281
\(731\) 3.09177e10 2.92750
\(732\) 0 0
\(733\) 2.85161e9 0.267440 0.133720 0.991019i \(-0.457308\pi\)
0.133720 + 0.991019i \(0.457308\pi\)
\(734\) 3.05650e10 2.85291
\(735\) 0 0
\(736\) 2.72606e9 0.252036
\(737\) 3.10555e9 0.285760
\(738\) 0 0
\(739\) −1.35553e10 −1.23553 −0.617766 0.786362i \(-0.711962\pi\)
−0.617766 + 0.786362i \(0.711962\pi\)
\(740\) −6.61416e8 −0.0600017
\(741\) 0 0
\(742\) −3.01947e9 −0.271342
\(743\) 6.48538e9 0.580062 0.290031 0.957017i \(-0.406334\pi\)
0.290031 + 0.957017i \(0.406334\pi\)
\(744\) 0 0
\(745\) −5.53334e8 −0.0490276
\(746\) −1.54134e10 −1.35929
\(747\) 0 0
\(748\) −6.05307e9 −0.528835
\(749\) 6.11689e9 0.531917
\(750\) 0 0
\(751\) 8.10260e9 0.698047 0.349024 0.937114i \(-0.386513\pi\)
0.349024 + 0.937114i \(0.386513\pi\)
\(752\) 2.51986e8 0.0216079
\(753\) 0 0
\(754\) −1.65986e10 −1.41017
\(755\) −3.87137e8 −0.0327379
\(756\) 0 0
\(757\) 3.72306e9 0.311935 0.155968 0.987762i \(-0.450150\pi\)
0.155968 + 0.987762i \(0.450150\pi\)
\(758\) −9.15432e8 −0.0763456
\(759\) 0 0
\(760\) 6.39709e8 0.0528610
\(761\) −1.48338e10 −1.22013 −0.610065 0.792351i \(-0.708857\pi\)
−0.610065 + 0.792351i \(0.708857\pi\)
\(762\) 0 0
\(763\) −2.64819e10 −2.15831
\(764\) −1.02850e10 −0.834408
\(765\) 0 0
\(766\) −3.59999e10 −2.89402
\(767\) 2.29530e10 1.83678
\(768\) 0 0
\(769\) 9.76344e9 0.774213 0.387107 0.922035i \(-0.373475\pi\)
0.387107 + 0.922035i \(0.373475\pi\)
\(770\) −6.48216e8 −0.0511684
\(771\) 0 0
\(772\) −2.93071e10 −2.29252
\(773\) −1.51981e10 −1.18348 −0.591740 0.806129i \(-0.701559\pi\)
−0.591740 + 0.806129i \(0.701559\pi\)
\(774\) 0 0
\(775\) 1.44617e10 1.11600
\(776\) −1.55526e10 −1.19478
\(777\) 0 0
\(778\) −2.49328e10 −1.89820
\(779\) −4.09662e9 −0.310488
\(780\) 0 0
\(781\) 6.95177e8 0.0522176
\(782\) 7.36674e9 0.550873
\(783\) 0 0
\(784\) −3.41468e9 −0.253072
\(785\) −8.49604e8 −0.0626863
\(786\) 0 0
\(787\) −5.97907e8 −0.0437242 −0.0218621 0.999761i \(-0.506959\pi\)
−0.0218621 + 0.999761i \(0.506959\pi\)
\(788\) −2.22385e10 −1.61907
\(789\) 0 0
\(790\) −3.62512e9 −0.261594
\(791\) 9.85872e9 0.708276
\(792\) 0 0
\(793\) 1.26491e9 0.0900748
\(794\) −3.49829e10 −2.48018
\(795\) 0 0
\(796\) −1.74026e10 −1.22298
\(797\) 6.05571e9 0.423703 0.211851 0.977302i \(-0.432051\pi\)
0.211851 + 0.977302i \(0.432051\pi\)
\(798\) 0 0
\(799\) 1.83556e9 0.127308
\(800\) 1.72883e10 1.19381
\(801\) 0 0
\(802\) 4.04077e10 2.76601
\(803\) 2.52010e9 0.171757
\(804\) 0 0
\(805\) 4.71119e8 0.0318306
\(806\) −3.39168e10 −2.28162
\(807\) 0 0
\(808\) 1.32250e10 0.881972
\(809\) 7.82042e9 0.519290 0.259645 0.965704i \(-0.416394\pi\)
0.259645 + 0.965704i \(0.416394\pi\)
\(810\) 0 0
\(811\) 2.15298e10 1.41731 0.708657 0.705553i \(-0.249301\pi\)
0.708657 + 0.705553i \(0.249301\pi\)
\(812\) 2.17112e10 1.42310
\(813\) 0 0
\(814\) −1.87961e9 −0.122147
\(815\) 2.58500e9 0.167266
\(816\) 0 0
\(817\) −1.70367e10 −1.09297
\(818\) 4.06707e10 2.59803
\(819\) 0 0
\(820\) −1.28941e9 −0.0816661
\(821\) 2.41869e10 1.52539 0.762693 0.646761i \(-0.223877\pi\)
0.762693 + 0.646761i \(0.223877\pi\)
\(822\) 0 0
\(823\) −3.08772e8 −0.0193081 −0.00965403 0.999953i \(-0.503073\pi\)
−0.00965403 + 0.999953i \(0.503073\pi\)
\(824\) −8.52767e9 −0.530988
\(825\) 0 0
\(826\) −5.02735e10 −3.10391
\(827\) 1.13854e10 0.699967 0.349984 0.936756i \(-0.386187\pi\)
0.349984 + 0.936756i \(0.386187\pi\)
\(828\) 0 0
\(829\) −5.67599e9 −0.346019 −0.173010 0.984920i \(-0.555349\pi\)
−0.173010 + 0.984920i \(0.555349\pi\)
\(830\) −1.78054e9 −0.108089
\(831\) 0 0
\(832\) −3.44870e10 −2.07598
\(833\) −2.48738e10 −1.49103
\(834\) 0 0
\(835\) −2.90405e8 −0.0172624
\(836\) 3.33544e9 0.197438
\(837\) 0 0
\(838\) −1.24647e9 −0.0731689
\(839\) −2.26083e10 −1.32160 −0.660801 0.750561i \(-0.729783\pi\)
−0.660801 + 0.750561i \(0.729783\pi\)
\(840\) 0 0
\(841\) −8.83683e9 −0.512284
\(842\) −1.93490e10 −1.11703
\(843\) 0 0
\(844\) 6.13147e9 0.351047
\(845\) 1.25133e9 0.0713466
\(846\) 0 0
\(847\) 2.32072e10 1.31229
\(848\) −6.33193e8 −0.0356575
\(849\) 0 0
\(850\) 4.67188e10 2.60931
\(851\) 1.36609e9 0.0759844
\(852\) 0 0
\(853\) −3.13840e10 −1.73136 −0.865679 0.500599i \(-0.833113\pi\)
−0.865679 + 0.500599i \(0.833113\pi\)
\(854\) −2.77050e9 −0.152214
\(855\) 0 0
\(856\) −5.39947e9 −0.294234
\(857\) −6.44481e9 −0.349766 −0.174883 0.984589i \(-0.555955\pi\)
−0.174883 + 0.984589i \(0.555955\pi\)
\(858\) 0 0
\(859\) −2.36238e10 −1.27167 −0.635834 0.771826i \(-0.719344\pi\)
−0.635834 + 0.771826i \(0.719344\pi\)
\(860\) −5.36228e9 −0.287478
\(861\) 0 0
\(862\) 5.64856e10 3.00374
\(863\) 1.31698e10 0.697496 0.348748 0.937217i \(-0.386607\pi\)
0.348748 + 0.937217i \(0.386607\pi\)
\(864\) 0 0
\(865\) 2.67797e9 0.140685
\(866\) 5.81211e9 0.304103
\(867\) 0 0
\(868\) 4.43635e10 2.30254
\(869\) −6.15215e9 −0.318022
\(870\) 0 0
\(871\) 3.35708e10 1.72146
\(872\) 2.33760e10 1.19388
\(873\) 0 0
\(874\) −4.05932e9 −0.205666
\(875\) 6.01285e9 0.303426
\(876\) 0 0
\(877\) −1.72997e10 −0.866046 −0.433023 0.901383i \(-0.642553\pi\)
−0.433023 + 0.901383i \(0.642553\pi\)
\(878\) −4.69109e10 −2.33907
\(879\) 0 0
\(880\) −1.35933e8 −0.00672411
\(881\) −2.51040e10 −1.23688 −0.618439 0.785832i \(-0.712235\pi\)
−0.618439 + 0.785832i \(0.712235\pi\)
\(882\) 0 0
\(883\) 2.96678e10 1.45018 0.725091 0.688653i \(-0.241797\pi\)
0.725091 + 0.688653i \(0.241797\pi\)
\(884\) −6.54333e10 −3.18578
\(885\) 0 0
\(886\) 5.52881e9 0.267063
\(887\) −1.46350e10 −0.704140 −0.352070 0.935974i \(-0.614522\pi\)
−0.352070 + 0.935974i \(0.614522\pi\)
\(888\) 0 0
\(889\) −4.65272e10 −2.22101
\(890\) 6.32097e9 0.300551
\(891\) 0 0
\(892\) 2.37726e8 0.0112150
\(893\) −1.01145e9 −0.0475298
\(894\) 0 0
\(895\) 2.99400e9 0.139595
\(896\) 3.97636e10 1.84675
\(897\) 0 0
\(898\) 3.53088e10 1.62711
\(899\) 1.71908e10 0.789110
\(900\) 0 0
\(901\) −4.61241e9 −0.210083
\(902\) −3.66423e9 −0.166249
\(903\) 0 0
\(904\) −8.70244e9 −0.391788
\(905\) −8.16227e8 −0.0366050
\(906\) 0 0
\(907\) 1.17379e10 0.522353 0.261177 0.965291i \(-0.415889\pi\)
0.261177 + 0.965291i \(0.415889\pi\)
\(908\) 2.69083e10 1.19285
\(909\) 0 0
\(910\) −7.00717e9 −0.308246
\(911\) 3.54807e9 0.155481 0.0777405 0.996974i \(-0.475229\pi\)
0.0777405 + 0.996974i \(0.475229\pi\)
\(912\) 0 0
\(913\) −3.02174e9 −0.131404
\(914\) −2.80244e10 −1.21401
\(915\) 0 0
\(916\) −1.43304e10 −0.616060
\(917\) −2.09482e9 −0.0897126
\(918\) 0 0
\(919\) 3.70231e10 1.57350 0.786752 0.617269i \(-0.211761\pi\)
0.786752 + 0.617269i \(0.211761\pi\)
\(920\) −4.15864e8 −0.0176073
\(921\) 0 0
\(922\) −4.99658e10 −2.09949
\(923\) 7.51482e9 0.314567
\(924\) 0 0
\(925\) 8.66352e9 0.359914
\(926\) 5.87592e10 2.43185
\(927\) 0 0
\(928\) 2.05508e10 0.844131
\(929\) −2.67577e10 −1.09495 −0.547475 0.836822i \(-0.684411\pi\)
−0.547475 + 0.836822i \(0.684411\pi\)
\(930\) 0 0
\(931\) 1.37063e10 0.556668
\(932\) 5.05405e10 2.04495
\(933\) 0 0
\(934\) 3.95914e10 1.58996
\(935\) −9.90185e8 −0.0396165
\(936\) 0 0
\(937\) −3.93013e9 −0.156069 −0.0780347 0.996951i \(-0.524865\pi\)
−0.0780347 + 0.996951i \(0.524865\pi\)
\(938\) −7.35293e10 −2.90904
\(939\) 0 0
\(940\) −3.18354e8 −0.0125015
\(941\) 2.54150e10 0.994320 0.497160 0.867659i \(-0.334376\pi\)
0.497160 + 0.867659i \(0.334376\pi\)
\(942\) 0 0
\(943\) 2.66314e9 0.103420
\(944\) −1.05425e10 −0.407889
\(945\) 0 0
\(946\) −1.52385e10 −0.585225
\(947\) 2.76072e10 1.05632 0.528161 0.849144i \(-0.322882\pi\)
0.528161 + 0.849144i \(0.322882\pi\)
\(948\) 0 0
\(949\) 2.72422e10 1.03469
\(950\) −2.57436e10 −0.974174
\(951\) 0 0
\(952\) 4.66478e10 1.75227
\(953\) 2.15615e10 0.806962 0.403481 0.914988i \(-0.367800\pi\)
0.403481 + 0.914988i \(0.367800\pi\)
\(954\) 0 0
\(955\) −1.68246e9 −0.0625078
\(956\) 6.60139e10 2.44361
\(957\) 0 0
\(958\) −7.28871e10 −2.67837
\(959\) 1.06807e10 0.391053
\(960\) 0 0
\(961\) 7.61423e9 0.276754
\(962\) −2.03184e10 −0.735830
\(963\) 0 0
\(964\) −2.54916e10 −0.916489
\(965\) −4.79418e9 −0.171739
\(966\) 0 0
\(967\) 8.89528e9 0.316349 0.158175 0.987411i \(-0.449439\pi\)
0.158175 + 0.987411i \(0.449439\pi\)
\(968\) −2.04853e10 −0.725903
\(969\) 0 0
\(970\) −7.81649e9 −0.274986
\(971\) 9.39030e9 0.329164 0.164582 0.986363i \(-0.447372\pi\)
0.164582 + 0.986363i \(0.447372\pi\)
\(972\) 0 0
\(973\) 1.24914e10 0.434727
\(974\) −3.93092e10 −1.36313
\(975\) 0 0
\(976\) −5.80983e8 −0.0200027
\(977\) −4.04777e10 −1.38863 −0.694313 0.719673i \(-0.744292\pi\)
−0.694313 + 0.719673i \(0.744292\pi\)
\(978\) 0 0
\(979\) 1.07272e10 0.365383
\(980\) 4.31405e9 0.146418
\(981\) 0 0
\(982\) 4.36666e10 1.47150
\(983\) −2.17203e10 −0.729336 −0.364668 0.931138i \(-0.618818\pi\)
−0.364668 + 0.931138i \(0.618818\pi\)
\(984\) 0 0
\(985\) −3.63787e9 −0.121289
\(986\) 5.55352e10 1.84501
\(987\) 0 0
\(988\) 3.60559e10 1.18940
\(989\) 1.10752e10 0.364054
\(990\) 0 0
\(991\) 4.02205e10 1.31277 0.656387 0.754425i \(-0.272084\pi\)
0.656387 + 0.754425i \(0.272084\pi\)
\(992\) 4.19924e10 1.36578
\(993\) 0 0
\(994\) −1.64595e10 −0.531576
\(995\) −2.84680e9 −0.0916169
\(996\) 0 0
\(997\) 1.02432e10 0.327342 0.163671 0.986515i \(-0.447666\pi\)
0.163671 + 0.986515i \(0.447666\pi\)
\(998\) −7.29287e10 −2.32242
\(999\) 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 207.8.a.a.1.5 5
3.2 odd 2 69.8.a.a.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.a.1.1 5 3.2 odd 2
207.8.a.a.1.5 5 1.1 even 1 trivial