Properties

Label 69.8.a.a.1.1
Level $69$
Weight $8$
Character 69.1
Self dual yes
Analytic conductor $21.555$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5545667584\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-17.8260\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q-17.8260 q^{2} -27.0000 q^{3} +189.767 q^{4} -31.0428 q^{5} +481.302 q^{6} -1247.35 q^{7} -1101.05 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-17.8260 q^{2} -27.0000 q^{3} +189.767 q^{4} -31.0428 q^{5} +481.302 q^{6} -1247.35 q^{7} -1101.05 q^{8} +729.000 q^{9} +553.369 q^{10} -939.114 q^{11} -5123.70 q^{12} +10151.8 q^{13} +22235.2 q^{14} +838.155 q^{15} -4662.78 q^{16} +33965.5 q^{17} -12995.2 q^{18} +18716.1 q^{19} -5890.88 q^{20} +33678.3 q^{21} +16740.7 q^{22} +12167.0 q^{23} +29728.4 q^{24} -77161.3 q^{25} -180965. q^{26} -19683.0 q^{27} -236704. q^{28} +91722.7 q^{29} -14941.0 q^{30} -187422. q^{31} +224053. q^{32} +25356.1 q^{33} -605469. q^{34} +38721.0 q^{35} +138340. q^{36} -112278. q^{37} -333633. q^{38} -274098. q^{39} +34179.6 q^{40} +218882. q^{41} -600350. q^{42} -910269. q^{43} -178212. q^{44} -22630.2 q^{45} -216889. q^{46} +54041.9 q^{47} +125895. q^{48} +732327. q^{49} +1.37548e6 q^{50} -917067. q^{51} +1.92647e6 q^{52} -135797. q^{53} +350869. q^{54} +29152.7 q^{55} +1.37339e6 q^{56} -505335. q^{57} -1.63505e6 q^{58} -2.26099e6 q^{59} +159054. q^{60} +124600. q^{61} +3.34098e6 q^{62} -909315. q^{63} -3.39714e6 q^{64} -315139. q^{65} -451998. q^{66} +3.30689e6 q^{67} +6.44551e6 q^{68} -328509. q^{69} -690242. q^{70} -740248. q^{71} -802666. q^{72} +2.68349e6 q^{73} +2.00147e6 q^{74} +2.08336e6 q^{75} +3.55169e6 q^{76} +1.17140e6 q^{77} +4.88607e6 q^{78} -6.55101e6 q^{79} +144746. q^{80} +531441. q^{81} -3.90179e6 q^{82} +3.21765e6 q^{83} +6.39102e6 q^{84} -1.05438e6 q^{85} +1.62265e7 q^{86} -2.47651e6 q^{87} +1.03401e6 q^{88} -1.14227e7 q^{89} +403406. q^{90} -1.26628e7 q^{91} +2.30889e6 q^{92} +5.06038e6 q^{93} -963351. q^{94} -580999. q^{95} -6.04944e6 q^{96} -1.41253e7 q^{97} -1.30545e7 q^{98} -684614. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −17.8260 −1.57561 −0.787806 0.615924i \(-0.788783\pi\)
−0.787806 + 0.615924i \(0.788783\pi\)
\(3\) −27.0000 −0.577350
\(4\) 189.767 1.48255
\(5\) −31.0428 −0.111062 −0.0555310 0.998457i \(-0.517685\pi\)
−0.0555310 + 0.998457i \(0.517685\pi\)
\(6\) 481.302 0.909680
\(7\) −1247.35 −1.37450 −0.687248 0.726423i \(-0.741181\pi\)
−0.687248 + 0.726423i \(0.741181\pi\)
\(8\) −1101.05 −0.760313
\(9\) 729.000 0.333333
\(10\) 553.369 0.174991
\(11\) −939.114 −0.212737 −0.106369 0.994327i \(-0.533922\pi\)
−0.106369 + 0.994327i \(0.533922\pi\)
\(12\) −5123.70 −0.855951
\(13\) 10151.8 1.28156 0.640781 0.767723i \(-0.278610\pi\)
0.640781 + 0.767723i \(0.278610\pi\)
\(14\) 22235.2 2.16567
\(15\) 838.155 0.0641217
\(16\) −4662.78 −0.284594
\(17\) 33965.5 1.67674 0.838371 0.545100i \(-0.183508\pi\)
0.838371 + 0.545100i \(0.183508\pi\)
\(18\) −12995.2 −0.525204
\(19\) 18716.1 0.626005 0.313002 0.949752i \(-0.398665\pi\)
0.313002 + 0.949752i \(0.398665\pi\)
\(20\) −5890.88 −0.164655
\(21\) 33678.3 0.793566
\(22\) 16740.7 0.335192
\(23\) 12167.0 0.208514
\(24\) 29728.4 0.438967
\(25\) −77161.3 −0.987665
\(26\) −180965. −2.01925
\(27\) −19683.0 −0.192450
\(28\) −236704. −2.03776
\(29\) 91722.7 0.698367 0.349183 0.937054i \(-0.386459\pi\)
0.349183 + 0.937054i \(0.386459\pi\)
\(30\) −14941.0 −0.101031
\(31\) −187422. −1.12994 −0.564968 0.825113i \(-0.691111\pi\)
−0.564968 + 0.825113i \(0.691111\pi\)
\(32\) 224053. 1.20872
\(33\) 25356.1 0.122824
\(34\) −605469. −2.64189
\(35\) 38721.0 0.152654
\(36\) 138340. 0.494184
\(37\) −112278. −0.364408 −0.182204 0.983261i \(-0.558323\pi\)
−0.182204 + 0.983261i \(0.558323\pi\)
\(38\) −333633. −0.986340
\(39\) −274098. −0.739911
\(40\) 34179.6 0.0844418
\(41\) 218882. 0.495983 0.247991 0.968762i \(-0.420230\pi\)
0.247991 + 0.968762i \(0.420230\pi\)
\(42\) −600350. −1.25035
\(43\) −910269. −1.74594 −0.872971 0.487772i \(-0.837810\pi\)
−0.872971 + 0.487772i \(0.837810\pi\)
\(44\) −178212. −0.315394
\(45\) −22630.2 −0.0370207
\(46\) −216889. −0.328538
\(47\) 54041.9 0.0759256 0.0379628 0.999279i \(-0.487913\pi\)
0.0379628 + 0.999279i \(0.487913\pi\)
\(48\) 125895. 0.164310
\(49\) 732327. 0.889239
\(50\) 1.37548e6 1.55618
\(51\) −917067. −0.968068
\(52\) 1.92647e6 1.89998
\(53\) −135797. −0.125292 −0.0626462 0.998036i \(-0.519954\pi\)
−0.0626462 + 0.998036i \(0.519954\pi\)
\(54\) 350869. 0.303227
\(55\) 29152.7 0.0236270
\(56\) 1.37339e6 1.04505
\(57\) −505335. −0.361424
\(58\) −1.63505e6 −1.10035
\(59\) −2.26099e6 −1.43323 −0.716616 0.697468i \(-0.754310\pi\)
−0.716616 + 0.697468i \(0.754310\pi\)
\(60\) 159054. 0.0950636
\(61\) 124600. 0.0702851 0.0351426 0.999382i \(-0.488811\pi\)
0.0351426 + 0.999382i \(0.488811\pi\)
\(62\) 3.34098e6 1.78034
\(63\) −909315. −0.458165
\(64\) −3.39714e6 −1.61988
\(65\) −315139. −0.142333
\(66\) −451998. −0.193523
\(67\) 3.30689e6 1.34325 0.671627 0.740890i \(-0.265596\pi\)
0.671627 + 0.740890i \(0.265596\pi\)
\(68\) 6.44551e6 2.48586
\(69\) −328509. −0.120386
\(70\) −690242. −0.240524
\(71\) −740248. −0.245456 −0.122728 0.992440i \(-0.539164\pi\)
−0.122728 + 0.992440i \(0.539164\pi\)
\(72\) −802666. −0.253438
\(73\) 2.68349e6 0.807365 0.403683 0.914899i \(-0.367730\pi\)
0.403683 + 0.914899i \(0.367730\pi\)
\(74\) 2.00147e6 0.574166
\(75\) 2.08336e6 0.570229
\(76\) 3.55169e6 0.928084
\(77\) 1.17140e6 0.292407
\(78\) 4.88607e6 1.16581
\(79\) −6.55101e6 −1.49490 −0.747452 0.664315i \(-0.768723\pi\)
−0.747452 + 0.664315i \(0.768723\pi\)
\(80\) 144746. 0.0316076
\(81\) 531441. 0.111111
\(82\) −3.90179e6 −0.781476
\(83\) 3.21765e6 0.617682 0.308841 0.951114i \(-0.400059\pi\)
0.308841 + 0.951114i \(0.400059\pi\)
\(84\) 6.39102e6 1.17650
\(85\) −1.05438e6 −0.186222
\(86\) 1.62265e7 2.75093
\(87\) −2.47651e6 −0.403202
\(88\) 1.03401e6 0.161747
\(89\) −1.14227e7 −1.71753 −0.858765 0.512370i \(-0.828768\pi\)
−0.858765 + 0.512370i \(0.828768\pi\)
\(90\) 403406. 0.0583302
\(91\) −1.26628e7 −1.76150
\(92\) 2.30889e6 0.309133
\(93\) 5.06038e6 0.652369
\(94\) −963351. −0.119629
\(95\) −580999. −0.0695253
\(96\) −6.04944e6 −0.697856
\(97\) −1.41253e7 −1.57143 −0.785717 0.618586i \(-0.787706\pi\)
−0.785717 + 0.618586i \(0.787706\pi\)
\(98\) −1.30545e7 −1.40110
\(99\) −684614. −0.0709125
\(100\) −1.46426e7 −1.46426
\(101\) −1.20112e7 −1.16001 −0.580006 0.814612i \(-0.696950\pi\)
−0.580006 + 0.814612i \(0.696950\pi\)
\(102\) 1.63476e7 1.52530
\(103\) −7.74503e6 −0.698382 −0.349191 0.937052i \(-0.613544\pi\)
−0.349191 + 0.937052i \(0.613544\pi\)
\(104\) −1.11776e7 −0.974388
\(105\) −1.04547e6 −0.0881350
\(106\) 2.42072e6 0.197412
\(107\) 4.90393e6 0.386991 0.193495 0.981101i \(-0.438018\pi\)
0.193495 + 0.981101i \(0.438018\pi\)
\(108\) −3.73517e6 −0.285317
\(109\) 2.12306e7 1.57025 0.785127 0.619335i \(-0.212598\pi\)
0.785127 + 0.619335i \(0.212598\pi\)
\(110\) −519676. −0.0372270
\(111\) 3.03150e6 0.210391
\(112\) 5.81610e6 0.391173
\(113\) 7.90376e6 0.515299 0.257649 0.966238i \(-0.417052\pi\)
0.257649 + 0.966238i \(0.417052\pi\)
\(114\) 9.00810e6 0.569464
\(115\) −377697. −0.0231580
\(116\) 1.74059e7 1.03536
\(117\) 7.40064e6 0.427188
\(118\) 4.03044e7 2.25822
\(119\) −4.23667e7 −2.30468
\(120\) −922850. −0.0487525
\(121\) −1.86052e7 −0.954743
\(122\) −2.22112e6 −0.110742
\(123\) −5.90982e6 −0.286356
\(124\) −3.55663e7 −1.67519
\(125\) 4.82052e6 0.220754
\(126\) 1.62094e7 0.721890
\(127\) 3.73010e7 1.61587 0.807936 0.589270i \(-0.200585\pi\)
0.807936 + 0.589270i \(0.200585\pi\)
\(128\) 3.18786e7 1.34358
\(129\) 2.45773e7 1.00802
\(130\) 5.61767e6 0.224261
\(131\) −1.67942e6 −0.0652694 −0.0326347 0.999467i \(-0.510390\pi\)
−0.0326347 + 0.999467i \(0.510390\pi\)
\(132\) 4.81174e6 0.182093
\(133\) −2.33454e7 −0.860441
\(134\) −5.89486e7 −2.11644
\(135\) 611015. 0.0213739
\(136\) −3.73977e7 −1.27485
\(137\) 8.56278e6 0.284507 0.142253 0.989830i \(-0.454565\pi\)
0.142253 + 0.989830i \(0.454565\pi\)
\(138\) 5.85600e6 0.189681
\(139\) −1.00144e7 −0.316281 −0.158140 0.987417i \(-0.550550\pi\)
−0.158140 + 0.987417i \(0.550550\pi\)
\(140\) 7.34796e6 0.226318
\(141\) −1.45913e6 −0.0438357
\(142\) 1.31957e7 0.386743
\(143\) −9.53367e6 −0.272636
\(144\) −3.39917e6 −0.0948646
\(145\) −2.84733e6 −0.0775620
\(146\) −4.78359e7 −1.27209
\(147\) −1.97728e7 −0.513403
\(148\) −2.13066e7 −0.540254
\(149\) 1.78249e7 0.441443 0.220722 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(150\) −3.71379e7 −0.898459
\(151\) −1.24711e7 −0.294771 −0.147386 0.989079i \(-0.547086\pi\)
−0.147386 + 0.989079i \(0.547086\pi\)
\(152\) −2.06074e7 −0.475959
\(153\) 2.47608e7 0.558914
\(154\) −2.08814e7 −0.460719
\(155\) 5.81808e6 0.125493
\(156\) −5.20146e7 −1.09696
\(157\) −2.73688e7 −0.564426 −0.282213 0.959352i \(-0.591069\pi\)
−0.282213 + 0.959352i \(0.591069\pi\)
\(158\) 1.16778e8 2.35539
\(159\) 3.66652e6 0.0723376
\(160\) −6.95523e6 −0.134243
\(161\) −1.51764e7 −0.286602
\(162\) −9.47347e6 −0.175068
\(163\) 8.32721e7 1.50606 0.753031 0.657985i \(-0.228591\pi\)
0.753031 + 0.657985i \(0.228591\pi\)
\(164\) 4.15365e7 0.735320
\(165\) −787123. −0.0136411
\(166\) −5.73578e7 −0.973227
\(167\) 9.35501e6 0.155431 0.0777153 0.996976i \(-0.475237\pi\)
0.0777153 + 0.996976i \(0.475237\pi\)
\(168\) −3.70815e7 −0.603358
\(169\) 4.03099e7 0.642404
\(170\) 1.87954e7 0.293414
\(171\) 1.36440e7 0.208668
\(172\) −1.72738e8 −2.58845
\(173\) −8.62670e7 −1.26673 −0.633364 0.773854i \(-0.718326\pi\)
−0.633364 + 0.773854i \(0.718326\pi\)
\(174\) 4.41463e7 0.635290
\(175\) 9.62468e7 1.35754
\(176\) 4.37889e6 0.0605438
\(177\) 6.10467e7 0.827476
\(178\) 2.03621e8 2.70616
\(179\) −9.64475e7 −1.25691 −0.628456 0.777845i \(-0.716313\pi\)
−0.628456 + 0.777845i \(0.716313\pi\)
\(180\) −4.29445e6 −0.0548850
\(181\) −2.62936e7 −0.329591 −0.164796 0.986328i \(-0.552696\pi\)
−0.164796 + 0.986328i \(0.552696\pi\)
\(182\) 2.25726e8 2.77544
\(183\) −3.36420e6 −0.0405791
\(184\) −1.33965e7 −0.158536
\(185\) 3.48542e6 0.0404719
\(186\) −9.02064e7 −1.02788
\(187\) −3.18974e7 −0.356706
\(188\) 1.02553e7 0.112564
\(189\) 2.45515e7 0.264522
\(190\) 1.03569e7 0.109545
\(191\) 5.41983e7 0.562819 0.281410 0.959588i \(-0.409198\pi\)
0.281410 + 0.959588i \(0.409198\pi\)
\(192\) 9.17227e7 0.935239
\(193\) −1.54438e8 −1.54633 −0.773166 0.634203i \(-0.781328\pi\)
−0.773166 + 0.634203i \(0.781328\pi\)
\(194\) 2.51797e8 2.47597
\(195\) 8.50875e6 0.0821759
\(196\) 1.38971e8 1.31834
\(197\) 1.17189e8 1.09208 0.546041 0.837759i \(-0.316134\pi\)
0.546041 + 0.837759i \(0.316134\pi\)
\(198\) 1.22039e7 0.111731
\(199\) −9.17056e7 −0.824917 −0.412458 0.910976i \(-0.635330\pi\)
−0.412458 + 0.910976i \(0.635330\pi\)
\(200\) 8.49585e7 0.750934
\(201\) −8.92860e7 −0.775528
\(202\) 2.14112e8 1.82773
\(203\) −1.14410e8 −0.959903
\(204\) −1.74029e8 −1.43521
\(205\) −6.79470e6 −0.0550848
\(206\) 1.38063e8 1.10038
\(207\) 8.86974e6 0.0695048
\(208\) −4.73355e7 −0.364725
\(209\) −1.75766e7 −0.133175
\(210\) 1.86365e7 0.138866
\(211\) 3.23106e7 0.236786 0.118393 0.992967i \(-0.462226\pi\)
0.118393 + 0.992967i \(0.462226\pi\)
\(212\) −2.57697e7 −0.185752
\(213\) 1.99867e7 0.141714
\(214\) −8.74174e7 −0.609747
\(215\) 2.82573e7 0.193908
\(216\) 2.16720e7 0.146322
\(217\) 2.33779e8 1.55309
\(218\) −3.78457e8 −2.47411
\(219\) −7.24543e7 −0.466133
\(220\) 5.53221e6 0.0350283
\(221\) 3.44809e8 2.14885
\(222\) −5.40396e7 −0.331495
\(223\) 1.25273e6 0.00756468 0.00378234 0.999993i \(-0.498796\pi\)
0.00378234 + 0.999993i \(0.498796\pi\)
\(224\) −2.79472e8 −1.66138
\(225\) −5.62506e7 −0.329222
\(226\) −1.40892e8 −0.811911
\(227\) −1.41797e8 −0.804594 −0.402297 0.915509i \(-0.631788\pi\)
−0.402297 + 0.915509i \(0.631788\pi\)
\(228\) −9.58956e7 −0.535829
\(229\) −7.55158e7 −0.415541 −0.207770 0.978178i \(-0.566621\pi\)
−0.207770 + 0.978178i \(0.566621\pi\)
\(230\) 6.73284e6 0.0364880
\(231\) −3.16278e7 −0.168821
\(232\) −1.00991e8 −0.530977
\(233\) −2.66330e8 −1.37935 −0.689674 0.724120i \(-0.742246\pi\)
−0.689674 + 0.724120i \(0.742246\pi\)
\(234\) −1.31924e8 −0.673082
\(235\) −1.67761e6 −0.00843244
\(236\) −4.29060e8 −2.12484
\(237\) 1.76877e8 0.863084
\(238\) 7.55228e8 3.63127
\(239\) −3.47869e8 −1.64825 −0.824125 0.566408i \(-0.808333\pi\)
−0.824125 + 0.566408i \(0.808333\pi\)
\(240\) −3.90814e6 −0.0182486
\(241\) −1.34331e8 −0.618184 −0.309092 0.951032i \(-0.600025\pi\)
−0.309092 + 0.951032i \(0.600025\pi\)
\(242\) 3.31657e8 1.50430
\(243\) −1.43489e7 −0.0641500
\(244\) 2.36449e7 0.104201
\(245\) −2.27335e7 −0.0987607
\(246\) 1.05348e8 0.451185
\(247\) 1.90001e8 0.802265
\(248\) 2.06361e8 0.859104
\(249\) −8.68765e7 −0.356619
\(250\) −8.59306e7 −0.347823
\(251\) −1.25191e8 −0.499707 −0.249854 0.968284i \(-0.580382\pi\)
−0.249854 + 0.968284i \(0.580382\pi\)
\(252\) −1.72557e8 −0.679253
\(253\) −1.14262e7 −0.0443588
\(254\) −6.64927e8 −2.54599
\(255\) 2.84683e7 0.107515
\(256\) −1.33434e8 −0.497082
\(257\) −3.83946e8 −1.41093 −0.705463 0.708746i \(-0.749261\pi\)
−0.705463 + 0.708746i \(0.749261\pi\)
\(258\) −4.38114e8 −1.58825
\(259\) 1.40049e8 0.500878
\(260\) −5.98028e7 −0.211016
\(261\) 6.68658e7 0.232789
\(262\) 2.99373e7 0.102839
\(263\) −5.17704e8 −1.75483 −0.877417 0.479729i \(-0.840735\pi\)
−0.877417 + 0.479729i \(0.840735\pi\)
\(264\) −2.79183e7 −0.0933847
\(265\) 4.21552e6 0.0139152
\(266\) 4.16156e8 1.35572
\(267\) 3.08413e8 0.991616
\(268\) 6.27537e8 1.99144
\(269\) −2.04806e8 −0.641518 −0.320759 0.947161i \(-0.603938\pi\)
−0.320759 + 0.947161i \(0.603938\pi\)
\(270\) −1.08920e7 −0.0336769
\(271\) −2.30872e8 −0.704659 −0.352329 0.935876i \(-0.614610\pi\)
−0.352329 + 0.935876i \(0.614610\pi\)
\(272\) −1.58374e8 −0.477190
\(273\) 3.41894e8 1.01700
\(274\) −1.52640e8 −0.448272
\(275\) 7.24633e7 0.210113
\(276\) −6.23400e7 −0.178478
\(277\) −3.59723e8 −1.01693 −0.508463 0.861084i \(-0.669786\pi\)
−0.508463 + 0.861084i \(0.669786\pi\)
\(278\) 1.78517e8 0.498335
\(279\) −1.36630e8 −0.376645
\(280\) −4.26338e7 −0.116065
\(281\) 5.59198e8 1.50346 0.751732 0.659469i \(-0.229219\pi\)
0.751732 + 0.659469i \(0.229219\pi\)
\(282\) 2.60105e7 0.0690679
\(283\) −2.74952e8 −0.721114 −0.360557 0.932737i \(-0.617413\pi\)
−0.360557 + 0.932737i \(0.617413\pi\)
\(284\) −1.40474e8 −0.363900
\(285\) 1.56870e7 0.0401405
\(286\) 1.69947e8 0.429569
\(287\) −2.73021e8 −0.681726
\(288\) 1.63335e8 0.402907
\(289\) 7.43314e8 1.81146
\(290\) 5.07564e7 0.122208
\(291\) 3.81383e8 0.907268
\(292\) 5.09237e8 1.19696
\(293\) 2.01642e8 0.468321 0.234161 0.972198i \(-0.424766\pi\)
0.234161 + 0.972198i \(0.424766\pi\)
\(294\) 3.52471e8 0.808923
\(295\) 7.01873e7 0.159178
\(296\) 1.23624e8 0.277064
\(297\) 1.84846e7 0.0409413
\(298\) −3.17747e8 −0.695543
\(299\) 1.23517e8 0.267224
\(300\) 3.95351e8 0.845393
\(301\) 1.13542e9 2.39979
\(302\) 2.22310e8 0.464445
\(303\) 3.24303e8 0.669734
\(304\) −8.72691e7 −0.178157
\(305\) −3.86793e6 −0.00780600
\(306\) −4.41387e8 −0.880631
\(307\) −4.38553e8 −0.865043 −0.432521 0.901624i \(-0.642376\pi\)
−0.432521 + 0.901624i \(0.642376\pi\)
\(308\) 2.22292e8 0.433508
\(309\) 2.09116e8 0.403211
\(310\) −1.03713e8 −0.197728
\(311\) 6.05423e8 1.14129 0.570647 0.821196i \(-0.306693\pi\)
0.570647 + 0.821196i \(0.306693\pi\)
\(312\) 3.01795e8 0.562563
\(313\) −1.07052e7 −0.0197329 −0.00986647 0.999951i \(-0.503141\pi\)
−0.00986647 + 0.999951i \(0.503141\pi\)
\(314\) 4.87877e8 0.889316
\(315\) 2.82276e7 0.0508848
\(316\) −1.24316e9 −2.21627
\(317\) 9.99712e8 1.76266 0.881328 0.472505i \(-0.156650\pi\)
0.881328 + 0.472505i \(0.156650\pi\)
\(318\) −6.53594e7 −0.113976
\(319\) −8.61381e7 −0.148569
\(320\) 1.05457e8 0.179907
\(321\) −1.32406e8 −0.223429
\(322\) 2.70535e8 0.451574
\(323\) 6.35701e8 1.04965
\(324\) 1.00850e8 0.164728
\(325\) −7.83324e8 −1.26576
\(326\) −1.48441e9 −2.37297
\(327\) −5.73227e8 −0.906587
\(328\) −2.41000e8 −0.377102
\(329\) −6.74089e7 −0.104359
\(330\) 1.40313e7 0.0214930
\(331\) −8.39788e8 −1.27283 −0.636417 0.771345i \(-0.719584\pi\)
−0.636417 + 0.771345i \(0.719584\pi\)
\(332\) 6.10602e8 0.915745
\(333\) −8.18506e7 −0.121469
\(334\) −1.66762e8 −0.244898
\(335\) −1.02655e8 −0.149184
\(336\) −1.57035e8 −0.225844
\(337\) −5.53660e8 −0.788022 −0.394011 0.919106i \(-0.628913\pi\)
−0.394011 + 0.919106i \(0.628913\pi\)
\(338\) −7.18564e8 −1.01218
\(339\) −2.13402e8 −0.297508
\(340\) −2.00086e8 −0.276084
\(341\) 1.76010e8 0.240380
\(342\) −2.43219e8 −0.328780
\(343\) 1.13778e8 0.152240
\(344\) 1.00225e9 1.32746
\(345\) 1.01978e7 0.0133703
\(346\) 1.53780e9 1.99587
\(347\) 8.05434e8 1.03485 0.517424 0.855729i \(-0.326891\pi\)
0.517424 + 0.855729i \(0.326891\pi\)
\(348\) −4.69959e8 −0.597768
\(349\) 6.32931e8 0.797017 0.398509 0.917165i \(-0.369528\pi\)
0.398509 + 0.917165i \(0.369528\pi\)
\(350\) −1.71570e9 −2.13896
\(351\) −1.99817e8 −0.246637
\(352\) −2.10412e8 −0.257140
\(353\) 3.46589e8 0.419376 0.209688 0.977768i \(-0.432755\pi\)
0.209688 + 0.977768i \(0.432755\pi\)
\(354\) −1.08822e9 −1.30378
\(355\) 2.29793e7 0.0272608
\(356\) −2.16765e9 −2.54633
\(357\) 1.14390e9 1.33060
\(358\) 1.71927e9 1.98041
\(359\) −2.14794e8 −0.245014 −0.122507 0.992468i \(-0.539093\pi\)
−0.122507 + 0.992468i \(0.539093\pi\)
\(360\) 2.49170e7 0.0281473
\(361\) −5.43580e8 −0.608118
\(362\) 4.68710e8 0.519307
\(363\) 5.02341e8 0.551221
\(364\) −2.40297e9 −2.61152
\(365\) −8.33030e7 −0.0896676
\(366\) 5.99702e7 0.0639369
\(367\) 1.71463e9 1.81067 0.905335 0.424699i \(-0.139620\pi\)
0.905335 + 0.424699i \(0.139620\pi\)
\(368\) −5.67321e7 −0.0593419
\(369\) 1.59565e8 0.165328
\(370\) −6.21311e7 −0.0637680
\(371\) 1.69386e8 0.172214
\(372\) 9.60291e8 0.967170
\(373\) −8.64659e8 −0.862709 −0.431354 0.902183i \(-0.641964\pi\)
−0.431354 + 0.902183i \(0.641964\pi\)
\(374\) 5.68604e8 0.562030
\(375\) −1.30154e8 −0.127452
\(376\) −5.95028e7 −0.0577272
\(377\) 9.31147e8 0.895001
\(378\) −4.37655e8 −0.416784
\(379\) −5.13537e7 −0.0484546 −0.0242273 0.999706i \(-0.507713\pi\)
−0.0242273 + 0.999706i \(0.507713\pi\)
\(380\) −1.10254e8 −0.103075
\(381\) −1.00713e9 −0.932924
\(382\) −9.66139e8 −0.886784
\(383\) 2.01952e9 1.83676 0.918379 0.395703i \(-0.129499\pi\)
0.918379 + 0.395703i \(0.129499\pi\)
\(384\) −8.60722e8 −0.775718
\(385\) −3.63635e7 −0.0324753
\(386\) 2.75301e9 2.43642
\(387\) −6.63586e8 −0.581981
\(388\) −2.68051e9 −2.32973
\(389\) 1.39868e9 1.20474 0.602370 0.798217i \(-0.294223\pi\)
0.602370 + 0.798217i \(0.294223\pi\)
\(390\) −1.51677e8 −0.129477
\(391\) 4.13258e8 0.349625
\(392\) −8.06329e8 −0.676100
\(393\) 4.53443e7 0.0376833
\(394\) −2.08901e9 −1.72070
\(395\) 2.03361e8 0.166027
\(396\) −1.29917e8 −0.105131
\(397\) −1.96246e9 −1.57411 −0.787055 0.616883i \(-0.788395\pi\)
−0.787055 + 0.616883i \(0.788395\pi\)
\(398\) 1.63474e9 1.29975
\(399\) 6.30327e8 0.496776
\(400\) 3.59787e8 0.281083
\(401\) −2.26678e9 −1.75552 −0.877758 0.479104i \(-0.840962\pi\)
−0.877758 + 0.479104i \(0.840962\pi\)
\(402\) 1.59161e9 1.22193
\(403\) −1.90266e9 −1.44808
\(404\) −2.27933e9 −1.71978
\(405\) −1.64974e7 −0.0123402
\(406\) 2.03947e9 1.51243
\(407\) 1.05442e8 0.0775233
\(408\) 1.00974e9 0.736034
\(409\) 2.28154e9 1.64891 0.824453 0.565931i \(-0.191483\pi\)
0.824453 + 0.565931i \(0.191483\pi\)
\(410\) 1.21122e8 0.0867923
\(411\) −2.31195e8 −0.164260
\(412\) −1.46975e9 −1.03539
\(413\) 2.82023e9 1.96997
\(414\) −1.58112e8 −0.109513
\(415\) −9.98847e7 −0.0686010
\(416\) 2.27454e9 1.54905
\(417\) 2.70389e8 0.182605
\(418\) 3.13320e8 0.209832
\(419\) 6.99241e7 0.0464384 0.0232192 0.999730i \(-0.492608\pi\)
0.0232192 + 0.999730i \(0.492608\pi\)
\(420\) −1.98395e8 −0.130665
\(421\) −1.08544e9 −0.708952 −0.354476 0.935065i \(-0.615341\pi\)
−0.354476 + 0.935065i \(0.615341\pi\)
\(422\) −5.75969e8 −0.373083
\(423\) 3.93965e7 0.0253085
\(424\) 1.49519e8 0.0952614
\(425\) −2.62082e9 −1.65606
\(426\) −3.56283e8 −0.223286
\(427\) −1.55419e8 −0.0966066
\(428\) 9.30601e8 0.573734
\(429\) 2.57409e8 0.157407
\(430\) −5.03714e8 −0.305523
\(431\) −3.16872e9 −1.90640 −0.953198 0.302348i \(-0.902229\pi\)
−0.953198 + 0.302348i \(0.902229\pi\)
\(432\) 9.17776e7 0.0547701
\(433\) 3.26047e8 0.193007 0.0965033 0.995333i \(-0.469234\pi\)
0.0965033 + 0.995333i \(0.469234\pi\)
\(434\) −4.16735e9 −2.44707
\(435\) 7.68778e7 0.0447805
\(436\) 4.02886e9 2.32798
\(437\) 2.27719e8 0.130531
\(438\) 1.29157e9 0.734444
\(439\) −2.63160e9 −1.48455 −0.742273 0.670097i \(-0.766252\pi\)
−0.742273 + 0.670097i \(0.766252\pi\)
\(440\) −3.20986e7 −0.0179639
\(441\) 5.33866e8 0.296413
\(442\) −6.14658e9 −3.38575
\(443\) −3.10154e8 −0.169498 −0.0847489 0.996402i \(-0.527009\pi\)
−0.0847489 + 0.996402i \(0.527009\pi\)
\(444\) 5.75278e8 0.311916
\(445\) 3.54593e8 0.190752
\(446\) −2.23312e7 −0.0119190
\(447\) −4.81272e8 −0.254868
\(448\) 4.23740e9 2.22652
\(449\) −1.98075e9 −1.03268 −0.516341 0.856383i \(-0.672706\pi\)
−0.516341 + 0.856383i \(0.672706\pi\)
\(450\) 1.00272e9 0.518725
\(451\) −2.05555e8 −0.105514
\(452\) 1.49987e9 0.763957
\(453\) 3.36719e8 0.170186
\(454\) 2.52767e9 1.26773
\(455\) 3.93087e8 0.195636
\(456\) 5.56399e8 0.274795
\(457\) −1.57211e9 −0.770504 −0.385252 0.922811i \(-0.625885\pi\)
−0.385252 + 0.922811i \(0.625885\pi\)
\(458\) 1.34615e9 0.654731
\(459\) −6.68542e8 −0.322689
\(460\) −7.16743e7 −0.0343329
\(461\) 2.80297e9 1.33249 0.666247 0.745731i \(-0.267899\pi\)
0.666247 + 0.745731i \(0.267899\pi\)
\(462\) 5.63797e8 0.265997
\(463\) 3.29626e9 1.54343 0.771717 0.635966i \(-0.219398\pi\)
0.771717 + 0.635966i \(0.219398\pi\)
\(464\) −4.27683e8 −0.198751
\(465\) −1.57088e8 −0.0724533
\(466\) 4.74759e9 2.17332
\(467\) −2.22099e9 −1.00911 −0.504554 0.863380i \(-0.668343\pi\)
−0.504554 + 0.863380i \(0.668343\pi\)
\(468\) 1.40439e9 0.633327
\(469\) −4.12483e9 −1.84630
\(470\) 2.99051e7 0.0132863
\(471\) 7.38958e8 0.325872
\(472\) 2.48946e9 1.08970
\(473\) 8.54846e8 0.371427
\(474\) −3.15302e9 −1.35988
\(475\) −1.44416e9 −0.618283
\(476\) −8.03977e9 −3.41680
\(477\) −9.89961e7 −0.0417642
\(478\) 6.20111e9 2.59700
\(479\) 4.08881e9 1.69990 0.849948 0.526867i \(-0.176633\pi\)
0.849948 + 0.526867i \(0.176633\pi\)
\(480\) 1.87791e8 0.0775053
\(481\) −1.13982e9 −0.467012
\(482\) 2.39459e9 0.974017
\(483\) 4.09764e8 0.165470
\(484\) −3.53065e9 −1.41545
\(485\) 4.38488e8 0.174527
\(486\) 2.55784e8 0.101076
\(487\) −2.20516e9 −0.865144 −0.432572 0.901599i \(-0.642394\pi\)
−0.432572 + 0.901599i \(0.642394\pi\)
\(488\) −1.37191e8 −0.0534387
\(489\) −2.24835e9 −0.869525
\(490\) 4.05247e8 0.155608
\(491\) −2.44960e9 −0.933921 −0.466961 0.884278i \(-0.654651\pi\)
−0.466961 + 0.884278i \(0.654651\pi\)
\(492\) −1.12148e9 −0.424537
\(493\) 3.11540e9 1.17098
\(494\) −3.38697e9 −1.26406
\(495\) 2.12523e7 0.00787568
\(496\) 8.73906e8 0.321573
\(497\) 9.23345e8 0.337378
\(498\) 1.54866e9 0.561893
\(499\) −4.09114e9 −1.47398 −0.736991 0.675903i \(-0.763754\pi\)
−0.736991 + 0.675903i \(0.763754\pi\)
\(500\) 9.14773e8 0.327279
\(501\) −2.52585e8 −0.0897379
\(502\) 2.23166e9 0.787344
\(503\) 4.53054e8 0.158731 0.0793655 0.996846i \(-0.474711\pi\)
0.0793655 + 0.996846i \(0.474711\pi\)
\(504\) 1.00120e9 0.348349
\(505\) 3.72862e8 0.128833
\(506\) 2.03684e8 0.0698923
\(507\) −1.08837e9 −0.370892
\(508\) 7.07847e9 2.39561
\(509\) −2.23184e9 −0.750156 −0.375078 0.926993i \(-0.622384\pi\)
−0.375078 + 0.926993i \(0.622384\pi\)
\(510\) −5.07476e8 −0.169403
\(511\) −3.34724e9 −1.10972
\(512\) −1.70186e9 −0.560375
\(513\) −3.68389e8 −0.120475
\(514\) 6.84423e9 2.22307
\(515\) 2.40427e8 0.0775636
\(516\) 4.66394e9 1.49444
\(517\) −5.07515e7 −0.0161522
\(518\) −2.49652e9 −0.789189
\(519\) 2.32921e9 0.731346
\(520\) 3.46984e8 0.108218
\(521\) 3.73130e9 1.15592 0.577960 0.816065i \(-0.303849\pi\)
0.577960 + 0.816065i \(0.303849\pi\)
\(522\) −1.19195e9 −0.366785
\(523\) −3.65832e9 −1.11822 −0.559108 0.829095i \(-0.688856\pi\)
−0.559108 + 0.829095i \(0.688856\pi\)
\(524\) −3.18698e8 −0.0967652
\(525\) −2.59866e9 −0.783777
\(526\) 9.22859e9 2.76494
\(527\) −6.36586e9 −1.89461
\(528\) −1.18230e8 −0.0349550
\(529\) 1.48036e8 0.0434783
\(530\) −7.51458e7 −0.0219250
\(531\) −1.64826e9 −0.477744
\(532\) −4.43018e9 −1.27565
\(533\) 2.22204e9 0.635633
\(534\) −5.49778e9 −1.56240
\(535\) −1.52231e8 −0.0429800
\(536\) −3.64105e9 −1.02129
\(537\) 2.60408e9 0.725679
\(538\) 3.65087e9 1.01078
\(539\) −6.87739e8 −0.189175
\(540\) 1.15950e8 0.0316879
\(541\) 2.44725e9 0.664490 0.332245 0.943193i \(-0.392194\pi\)
0.332245 + 0.943193i \(0.392194\pi\)
\(542\) 4.11553e9 1.11027
\(543\) 7.09928e8 0.190289
\(544\) 7.61007e9 2.02671
\(545\) −6.59057e8 −0.174396
\(546\) −6.09461e9 −1.60240
\(547\) 3.76239e9 0.982897 0.491449 0.870907i \(-0.336468\pi\)
0.491449 + 0.870907i \(0.336468\pi\)
\(548\) 1.62493e9 0.421796
\(549\) 9.08333e7 0.0234284
\(550\) −1.29173e9 −0.331057
\(551\) 1.71669e9 0.437181
\(552\) 3.61705e8 0.0915309
\(553\) 8.17137e9 2.05474
\(554\) 6.41242e9 1.60228
\(555\) −9.41063e7 −0.0233665
\(556\) −1.90040e9 −0.468902
\(557\) 6.50679e9 1.59541 0.797707 0.603045i \(-0.206046\pi\)
0.797707 + 0.603045i \(0.206046\pi\)
\(558\) 2.43557e9 0.593446
\(559\) −9.24084e9 −2.23754
\(560\) −1.80548e8 −0.0434445
\(561\) 8.61231e8 0.205944
\(562\) −9.96826e9 −2.36888
\(563\) 2.23762e9 0.528453 0.264226 0.964461i \(-0.414883\pi\)
0.264226 + 0.964461i \(0.414883\pi\)
\(564\) −2.76894e8 −0.0649886
\(565\) −2.45355e8 −0.0572301
\(566\) 4.90129e9 1.13620
\(567\) −6.62890e8 −0.152722
\(568\) 8.15050e8 0.186623
\(569\) 3.55887e9 0.809877 0.404939 0.914344i \(-0.367293\pi\)
0.404939 + 0.914344i \(0.367293\pi\)
\(570\) −2.79636e8 −0.0632458
\(571\) −2.69760e9 −0.606389 −0.303194 0.952929i \(-0.598053\pi\)
−0.303194 + 0.952929i \(0.598053\pi\)
\(572\) −1.80917e9 −0.404197
\(573\) −1.46335e9 −0.324944
\(574\) 4.86688e9 1.07414
\(575\) −9.38822e8 −0.205942
\(576\) −2.47651e9 −0.539961
\(577\) −2.73131e9 −0.591910 −0.295955 0.955202i \(-0.595638\pi\)
−0.295955 + 0.955202i \(0.595638\pi\)
\(578\) −1.32503e10 −2.85416
\(579\) 4.16982e9 0.892776
\(580\) −5.40327e8 −0.114990
\(581\) −4.01352e9 −0.849002
\(582\) −6.79853e9 −1.42950
\(583\) 1.27529e8 0.0266544
\(584\) −2.95466e9 −0.613850
\(585\) −2.29736e8 −0.0474443
\(586\) −3.59447e9 −0.737892
\(587\) −4.25360e9 −0.868008 −0.434004 0.900911i \(-0.642900\pi\)
−0.434004 + 0.900911i \(0.642900\pi\)
\(588\) −3.75222e9 −0.761145
\(589\) −3.50780e9 −0.707345
\(590\) −1.25116e9 −0.250802
\(591\) −3.16410e9 −0.630513
\(592\) 5.23528e8 0.103708
\(593\) 8.85467e9 1.74374 0.871869 0.489740i \(-0.162908\pi\)
0.871869 + 0.489740i \(0.162908\pi\)
\(594\) −3.29506e8 −0.0645076
\(595\) 1.31518e9 0.255962
\(596\) 3.38257e9 0.654462
\(597\) 2.47605e9 0.476266
\(598\) −2.20181e9 −0.421042
\(599\) −8.77964e9 −1.66910 −0.834551 0.550930i \(-0.814273\pi\)
−0.834551 + 0.550930i \(0.814273\pi\)
\(600\) −2.29388e9 −0.433552
\(601\) 4.36354e9 0.819933 0.409967 0.912101i \(-0.365540\pi\)
0.409967 + 0.912101i \(0.365540\pi\)
\(602\) −2.02400e10 −3.78114
\(603\) 2.41072e9 0.447751
\(604\) −2.36659e9 −0.437013
\(605\) 5.77558e8 0.106036
\(606\) −5.78103e9 −1.05524
\(607\) −1.00264e10 −1.81963 −0.909815 0.415015i \(-0.863776\pi\)
−0.909815 + 0.415015i \(0.863776\pi\)
\(608\) 4.19340e9 0.756666
\(609\) 3.08907e9 0.554200
\(610\) 6.89497e7 0.0122992
\(611\) 5.48621e8 0.0973034
\(612\) 4.69877e9 0.828618
\(613\) −1.06422e10 −1.86603 −0.933013 0.359842i \(-0.882830\pi\)
−0.933013 + 0.359842i \(0.882830\pi\)
\(614\) 7.81765e9 1.36297
\(615\) 1.83457e8 0.0318032
\(616\) −1.28977e9 −0.222321
\(617\) −1.31952e9 −0.226161 −0.113080 0.993586i \(-0.536072\pi\)
−0.113080 + 0.993586i \(0.536072\pi\)
\(618\) −3.72770e9 −0.635304
\(619\) 8.51725e9 1.44339 0.721693 0.692214i \(-0.243364\pi\)
0.721693 + 0.692214i \(0.243364\pi\)
\(620\) 1.10408e9 0.186050
\(621\) −2.39483e8 −0.0401286
\(622\) −1.07923e10 −1.79823
\(623\) 1.42481e10 2.36074
\(624\) 1.27806e9 0.210574
\(625\) 5.87859e9 0.963148
\(626\) 1.90832e8 0.0310914
\(627\) 4.74567e8 0.0768884
\(628\) −5.19368e9 −0.836791
\(629\) −3.81357e9 −0.611019
\(630\) −5.03186e8 −0.0801746
\(631\) 1.32684e9 0.210240 0.105120 0.994460i \(-0.466477\pi\)
0.105120 + 0.994460i \(0.466477\pi\)
\(632\) 7.21299e9 1.13659
\(633\) −8.72386e8 −0.136708
\(634\) −1.78209e10 −2.77726
\(635\) −1.15792e9 −0.179462
\(636\) 6.95783e8 0.107244
\(637\) 7.43441e9 1.13962
\(638\) 1.53550e9 0.234087
\(639\) −5.39641e8 −0.0818185
\(640\) −9.89600e8 −0.149221
\(641\) 3.00273e9 0.450312 0.225156 0.974323i \(-0.427711\pi\)
0.225156 + 0.974323i \(0.427711\pi\)
\(642\) 2.36027e9 0.352038
\(643\) −1.08587e10 −1.61080 −0.805398 0.592734i \(-0.798049\pi\)
−0.805398 + 0.592734i \(0.798049\pi\)
\(644\) −2.87998e9 −0.424902
\(645\) −7.62946e8 −0.111953
\(646\) −1.13320e10 −1.65384
\(647\) −8.53415e9 −1.23878 −0.619392 0.785082i \(-0.712621\pi\)
−0.619392 + 0.785082i \(0.712621\pi\)
\(648\) −5.85143e8 −0.0844792
\(649\) 2.12333e9 0.304902
\(650\) 1.39635e10 1.99434
\(651\) −6.31204e9 −0.896678
\(652\) 1.58022e10 2.23281
\(653\) −2.44692e9 −0.343893 −0.171947 0.985106i \(-0.555006\pi\)
−0.171947 + 0.985106i \(0.555006\pi\)
\(654\) 1.02183e10 1.42843
\(655\) 5.21338e7 0.00724895
\(656\) −1.02060e9 −0.141154
\(657\) 1.95627e9 0.269122
\(658\) 1.20163e9 0.164430
\(659\) 7.66941e9 1.04391 0.521955 0.852973i \(-0.325203\pi\)
0.521955 + 0.852973i \(0.325203\pi\)
\(660\) −1.49370e8 −0.0202236
\(661\) 1.06623e10 1.43597 0.717987 0.696057i \(-0.245064\pi\)
0.717987 + 0.696057i \(0.245064\pi\)
\(662\) 1.49701e10 2.00549
\(663\) −9.30986e9 −1.24064
\(664\) −3.54279e9 −0.469632
\(665\) 7.24707e8 0.0955623
\(666\) 1.45907e9 0.191389
\(667\) 1.11599e9 0.145620
\(668\) 1.77527e9 0.230434
\(669\) −3.38237e7 −0.00436747
\(670\) 1.82993e9 0.235057
\(671\) −1.17014e8 −0.0149523
\(672\) 7.54574e9 0.959200
\(673\) 1.13404e10 1.43408 0.717042 0.697030i \(-0.245495\pi\)
0.717042 + 0.697030i \(0.245495\pi\)
\(674\) 9.86954e9 1.24162
\(675\) 1.51877e9 0.190076
\(676\) 7.64946e9 0.952396
\(677\) −2.59963e9 −0.321996 −0.160998 0.986955i \(-0.551471\pi\)
−0.160998 + 0.986955i \(0.551471\pi\)
\(678\) 3.80410e9 0.468757
\(679\) 1.76191e10 2.15993
\(680\) 1.16093e9 0.141587
\(681\) 3.82852e9 0.464532
\(682\) −3.13756e9 −0.378745
\(683\) 1.25557e10 1.50789 0.753943 0.656940i \(-0.228149\pi\)
0.753943 + 0.656940i \(0.228149\pi\)
\(684\) 2.58918e9 0.309361
\(685\) −2.65812e8 −0.0315979
\(686\) −2.02821e9 −0.239871
\(687\) 2.03893e9 0.239913
\(688\) 4.24439e9 0.496884
\(689\) −1.37858e9 −0.160570
\(690\) −1.81787e8 −0.0210664
\(691\) 4.66084e9 0.537392 0.268696 0.963225i \(-0.413407\pi\)
0.268696 + 0.963225i \(0.413407\pi\)
\(692\) −1.63706e10 −1.87799
\(693\) 8.53950e8 0.0974689
\(694\) −1.43577e10 −1.63052
\(695\) 3.10874e8 0.0351268
\(696\) 2.72676e9 0.306560
\(697\) 7.43443e9 0.831635
\(698\) −1.12826e10 −1.25579
\(699\) 7.19090e9 0.796367
\(700\) 1.82644e10 2.01263
\(701\) −4.37132e9 −0.479292 −0.239646 0.970860i \(-0.577031\pi\)
−0.239646 + 0.970860i \(0.577031\pi\)
\(702\) 3.56194e9 0.388604
\(703\) −2.10140e9 −0.228121
\(704\) 3.19030e9 0.344610
\(705\) 4.52955e7 0.00486847
\(706\) −6.17830e9 −0.660774
\(707\) 1.49822e10 1.59443
\(708\) 1.15846e10 1.22678
\(709\) −1.06398e10 −1.12117 −0.560585 0.828097i \(-0.689424\pi\)
−0.560585 + 0.828097i \(0.689424\pi\)
\(710\) −4.09630e8 −0.0429524
\(711\) −4.77569e9 −0.498302
\(712\) 1.25770e10 1.30586
\(713\) −2.28036e9 −0.235608
\(714\) −2.03912e10 −2.09652
\(715\) 2.95951e8 0.0302795
\(716\) −1.83025e10 −1.86344
\(717\) 9.39246e9 0.951617
\(718\) 3.82892e9 0.386048
\(719\) −1.04952e10 −1.05302 −0.526512 0.850168i \(-0.676501\pi\)
−0.526512 + 0.850168i \(0.676501\pi\)
\(720\) 1.05520e8 0.0105359
\(721\) 9.66073e9 0.959923
\(722\) 9.68985e9 0.958158
\(723\) 3.62694e9 0.356908
\(724\) −4.98965e9 −0.488635
\(725\) −7.07744e9 −0.689753
\(726\) −8.95474e9 −0.868510
\(727\) 1.98822e9 0.191908 0.0959541 0.995386i \(-0.469410\pi\)
0.0959541 + 0.995386i \(0.469410\pi\)
\(728\) 1.39423e10 1.33929
\(729\) 3.87420e8 0.0370370
\(730\) 1.48496e9 0.141281
\(731\) −3.09177e10 −2.92750
\(732\) −6.38412e8 −0.0601606
\(733\) 2.85161e9 0.267440 0.133720 0.991019i \(-0.457308\pi\)
0.133720 + 0.991019i \(0.457308\pi\)
\(734\) −3.05650e10 −2.85291
\(735\) 6.13803e8 0.0570195
\(736\) 2.72606e9 0.252036
\(737\) −3.10555e9 −0.285760
\(738\) −2.84441e9 −0.260492
\(739\) −1.35553e10 −1.23553 −0.617766 0.786362i \(-0.711962\pi\)
−0.617766 + 0.786362i \(0.711962\pi\)
\(740\) 6.61416e8 0.0600017
\(741\) −5.13004e9 −0.463188
\(742\) −3.01947e9 −0.271342
\(743\) −6.48538e9 −0.580062 −0.290031 0.957017i \(-0.593666\pi\)
−0.290031 + 0.957017i \(0.593666\pi\)
\(744\) −5.57173e9 −0.496004
\(745\) −5.53334e8 −0.0490276
\(746\) 1.54134e10 1.35929
\(747\) 2.34567e9 0.205894
\(748\) −6.05307e9 −0.528835
\(749\) −6.11689e9 −0.531917
\(750\) 2.32013e9 0.200815
\(751\) 8.10260e9 0.698047 0.349024 0.937114i \(-0.386513\pi\)
0.349024 + 0.937114i \(0.386513\pi\)
\(752\) −2.51986e8 −0.0216079
\(753\) 3.38016e9 0.288506
\(754\) −1.65986e10 −1.41017
\(755\) 3.87137e8 0.0327379
\(756\) 4.65905e9 0.392167
\(757\) 3.72306e9 0.311935 0.155968 0.987762i \(-0.450150\pi\)
0.155968 + 0.987762i \(0.450150\pi\)
\(758\) 9.15432e8 0.0763456
\(759\) 3.08507e8 0.0256106
\(760\) 6.39709e8 0.0528610
\(761\) 1.48338e10 1.22013 0.610065 0.792351i \(-0.291143\pi\)
0.610065 + 0.792351i \(0.291143\pi\)
\(762\) 1.79530e10 1.46993
\(763\) −2.64819e10 −2.15831
\(764\) 1.02850e10 0.834408
\(765\) −7.68644e8 −0.0620741
\(766\) −3.59999e10 −2.89402
\(767\) −2.29530e10 −1.83678
\(768\) 3.60273e9 0.286990
\(769\) 9.76344e9 0.774213 0.387107 0.922035i \(-0.373475\pi\)
0.387107 + 0.922035i \(0.373475\pi\)
\(770\) 6.48216e8 0.0511684
\(771\) 1.03665e10 0.814599
\(772\) −2.93071e10 −2.29252
\(773\) 1.51981e10 1.18348 0.591740 0.806129i \(-0.298441\pi\)
0.591740 + 0.806129i \(0.298441\pi\)
\(774\) 1.18291e10 0.916976
\(775\) 1.44617e10 1.11600
\(776\) 1.55526e10 1.19478
\(777\) −3.78133e9 −0.289182
\(778\) −2.49328e10 −1.89820
\(779\) 4.09662e9 0.310488
\(780\) 1.61468e9 0.121830
\(781\) 6.95177e8 0.0522176
\(782\) −7.36674e9 −0.550873
\(783\) −1.80538e9 −0.134401
\(784\) −3.41468e9 −0.253072
\(785\) 8.49604e8 0.0626863
\(786\) −8.08308e8 −0.0593743
\(787\) −5.97907e8 −0.0437242 −0.0218621 0.999761i \(-0.506959\pi\)
−0.0218621 + 0.999761i \(0.506959\pi\)
\(788\) 2.22385e10 1.61907
\(789\) 1.39780e10 1.01315
\(790\) −3.62512e9 −0.261594
\(791\) −9.85872e9 −0.708276
\(792\) 7.53795e8 0.0539157
\(793\) 1.26491e9 0.0900748
\(794\) 3.49829e10 2.48018
\(795\) −1.13819e8 −0.00803396
\(796\) −1.74026e10 −1.22298
\(797\) −6.05571e9 −0.423703 −0.211851 0.977302i \(-0.567949\pi\)
−0.211851 + 0.977302i \(0.567949\pi\)
\(798\) −1.12362e10 −0.782726
\(799\) 1.83556e9 0.127308
\(800\) −1.72883e10 −1.19381
\(801\) −8.32716e9 −0.572510
\(802\) 4.04077e10 2.76601
\(803\) −2.52010e9 −0.171757
\(804\) −1.69435e10 −1.14976
\(805\) 4.71119e8 0.0318306
\(806\) 3.39168e10 2.28162
\(807\) 5.52975e9 0.370381
\(808\) 1.32250e10 0.881972
\(809\) −7.82042e9 −0.519290 −0.259645 0.965704i \(-0.583606\pi\)
−0.259645 + 0.965704i \(0.583606\pi\)
\(810\) 2.94083e8 0.0194434
\(811\) 2.15298e10 1.41731 0.708657 0.705553i \(-0.249301\pi\)
0.708657 + 0.705553i \(0.249301\pi\)
\(812\) −2.17112e10 −1.42310
\(813\) 6.23355e9 0.406835
\(814\) −1.87961e9 −0.122147
\(815\) −2.58500e9 −0.167266
\(816\) 4.27609e9 0.275506
\(817\) −1.70367e10 −1.09297
\(818\) −4.06707e10 −2.59803
\(819\) −9.23115e9 −0.587168
\(820\) −1.28941e9 −0.0816661
\(821\) −2.41869e10 −1.52539 −0.762693 0.646761i \(-0.776123\pi\)
−0.762693 + 0.646761i \(0.776123\pi\)
\(822\) 4.12128e9 0.258810
\(823\) −3.08772e8 −0.0193081 −0.00965403 0.999953i \(-0.503073\pi\)
−0.00965403 + 0.999953i \(0.503073\pi\)
\(824\) 8.52767e9 0.530988
\(825\) −1.95651e9 −0.121309
\(826\) −5.02735e10 −3.10391
\(827\) −1.13854e10 −0.699967 −0.349984 0.936756i \(-0.613813\pi\)
−0.349984 + 0.936756i \(0.613813\pi\)
\(828\) 1.68318e9 0.103044
\(829\) −5.67599e9 −0.346019 −0.173010 0.984920i \(-0.555349\pi\)
−0.173010 + 0.984920i \(0.555349\pi\)
\(830\) 1.78054e9 0.108089
\(831\) 9.71252e9 0.587122
\(832\) −3.44870e10 −2.07598
\(833\) 2.48738e10 1.49103
\(834\) −4.81995e9 −0.287714
\(835\) −2.90405e8 −0.0172624
\(836\) −3.33544e9 −0.197438
\(837\) 3.68902e9 0.217456
\(838\) −1.24647e9 −0.0731689
\(839\) 2.26083e10 1.32160 0.660801 0.750561i \(-0.270217\pi\)
0.660801 + 0.750561i \(0.270217\pi\)
\(840\) 1.15111e9 0.0670101
\(841\) −8.83683e9 −0.512284
\(842\) 1.93490e10 1.11703
\(843\) −1.50983e10 −0.868026
\(844\) 6.13147e9 0.351047
\(845\) −1.25133e9 −0.0713466
\(846\) −7.02283e8 −0.0398764
\(847\) 2.32072e10 1.31229
\(848\) 6.33193e8 0.0356575
\(849\) 7.42370e9 0.416336
\(850\) 4.67188e10 2.60931
\(851\) −1.36609e9 −0.0759844
\(852\) 3.79280e9 0.210098
\(853\) −3.13840e10 −1.73136 −0.865679 0.500599i \(-0.833113\pi\)
−0.865679 + 0.500599i \(0.833113\pi\)
\(854\) 2.77050e9 0.152214
\(855\) −4.23549e8 −0.0231751
\(856\) −5.39947e9 −0.294234
\(857\) 6.44481e9 0.349766 0.174883 0.984589i \(-0.444045\pi\)
0.174883 + 0.984589i \(0.444045\pi\)
\(858\) −4.58858e9 −0.248012
\(859\) −2.36238e10 −1.27167 −0.635834 0.771826i \(-0.719344\pi\)
−0.635834 + 0.771826i \(0.719344\pi\)
\(860\) 5.36228e9 0.287478
\(861\) 7.37158e9 0.393595
\(862\) 5.64856e10 3.00374
\(863\) −1.31698e10 −0.697496 −0.348748 0.937217i \(-0.613393\pi\)
−0.348748 + 0.937217i \(0.613393\pi\)
\(864\) −4.41004e9 −0.232619
\(865\) 2.67797e9 0.140685
\(866\) −5.81211e9 −0.304103
\(867\) −2.00695e10 −1.04585
\(868\) 4.43635e10 2.30254
\(869\) 6.15215e9 0.318022
\(870\) −1.37042e9 −0.0705566
\(871\) 3.35708e10 1.72146
\(872\) −2.33760e10 −1.19388
\(873\) −1.02973e10 −0.523811
\(874\) −4.05932e9 −0.205666
\(875\) −6.01285e9 −0.303426
\(876\) −1.37494e10 −0.691065
\(877\) −1.72997e10 −0.866046 −0.433023 0.901383i \(-0.642553\pi\)
−0.433023 + 0.901383i \(0.642553\pi\)
\(878\) 4.69109e10 2.33907
\(879\) −5.44433e9 −0.270385
\(880\) −1.35933e8 −0.00672411
\(881\) 2.51040e10 1.23688 0.618439 0.785832i \(-0.287765\pi\)
0.618439 + 0.785832i \(0.287765\pi\)
\(882\) −9.51670e9 −0.467032
\(883\) 2.96678e10 1.45018 0.725091 0.688653i \(-0.241797\pi\)
0.725091 + 0.688653i \(0.241797\pi\)
\(884\) 6.54333e10 3.18578
\(885\) −1.89506e9 −0.0919012
\(886\) 5.52881e9 0.267063
\(887\) 1.46350e10 0.704140 0.352070 0.935974i \(-0.385478\pi\)
0.352070 + 0.935974i \(0.385478\pi\)
\(888\) −3.33784e9 −0.159963
\(889\) −4.65272e10 −2.22101
\(890\) −6.32097e9 −0.300551
\(891\) −4.99084e8 −0.0236375
\(892\) 2.37726e8 0.0112150
\(893\) 1.01145e9 0.0475298
\(894\) 8.57916e9 0.401572
\(895\) 2.99400e9 0.139595
\(896\) −3.97636e10 −1.84675
\(897\) −3.33495e9 −0.154282
\(898\) 3.53088e10 1.62711
\(899\) −1.71908e10 −0.789110
\(900\) −1.06745e10 −0.488088
\(901\) −4.61241e9 −0.210083
\(902\) 3.66423e9 0.166249
\(903\) −3.06563e10 −1.38552
\(904\) −8.70244e9 −0.391788
\(905\) 8.16227e8 0.0366050
\(906\) −6.00236e9 −0.268147
\(907\) 1.17379e10 0.522353 0.261177 0.965291i \(-0.415889\pi\)
0.261177 + 0.965291i \(0.415889\pi\)
\(908\) −2.69083e10 −1.19285
\(909\) −8.75619e9 −0.386671
\(910\) −7.00717e9 −0.308246
\(911\) −3.54807e9 −0.155481 −0.0777405 0.996974i \(-0.524771\pi\)
−0.0777405 + 0.996974i \(0.524771\pi\)
\(912\) 2.35627e9 0.102859
\(913\) −3.02174e9 −0.131404
\(914\) 2.80244e10 1.21401
\(915\) 1.04434e8 0.00450680
\(916\) −1.43304e10 −0.616060
\(917\) 2.09482e9 0.0897126
\(918\) 1.19174e10 0.508433
\(919\) 3.70231e10 1.57350 0.786752 0.617269i \(-0.211761\pi\)
0.786752 + 0.617269i \(0.211761\pi\)
\(920\) 4.15864e8 0.0176073
\(921\) 1.18409e10 0.499433
\(922\) −4.99658e10 −2.09949
\(923\) −7.51482e9 −0.314567
\(924\) −6.00189e9 −0.250286
\(925\) 8.66352e9 0.359914
\(926\) −5.87592e10 −2.43185
\(927\) −5.64613e9 −0.232794
\(928\) 2.05508e10 0.844131
\(929\) 2.67577e10 1.09495 0.547475 0.836822i \(-0.315589\pi\)
0.547475 + 0.836822i \(0.315589\pi\)
\(930\) 2.80026e9 0.114158
\(931\) 1.37063e10 0.556668
\(932\) −5.05405e10 −2.04495
\(933\) −1.63464e10 −0.658926
\(934\) 3.95914e10 1.58996
\(935\) 9.90185e8 0.0396165
\(936\) −8.14847e9 −0.324796
\(937\) −3.93013e9 −0.156069 −0.0780347 0.996951i \(-0.524865\pi\)
−0.0780347 + 0.996951i \(0.524865\pi\)
\(938\) 7.35293e10 2.90904
\(939\) 2.89042e8 0.0113928
\(940\) −3.18354e8 −0.0125015
\(941\) −2.54150e10 −0.994320 −0.497160 0.867659i \(-0.665624\pi\)
−0.497160 + 0.867659i \(0.665624\pi\)
\(942\) −1.31727e10 −0.513447
\(943\) 2.66314e9 0.103420
\(944\) 1.05425e10 0.407889
\(945\) −7.62146e8 −0.0293783
\(946\) −1.52385e10 −0.585225
\(947\) −2.76072e10 −1.05632 −0.528161 0.849144i \(-0.677118\pi\)
−0.528161 + 0.849144i \(0.677118\pi\)
\(948\) 3.35654e10 1.27957
\(949\) 2.72422e10 1.03469
\(950\) 2.57436e10 0.974174
\(951\) −2.69922e10 −1.01767
\(952\) 4.66478e10 1.75227
\(953\) −2.15615e10 −0.806962 −0.403481 0.914988i \(-0.632200\pi\)
−0.403481 + 0.914988i \(0.632200\pi\)
\(954\) 1.76470e9 0.0658041
\(955\) −1.68246e9 −0.0625078
\(956\) −6.60139e10 −2.44361
\(957\) 2.32573e9 0.0857763
\(958\) −7.28871e10 −2.67837
\(959\) −1.06807e10 −0.391053
\(960\) −2.84733e9 −0.103870
\(961\) 7.61423e9 0.276754
\(962\) 2.03184e10 0.735830
\(963\) 3.57496e9 0.128997
\(964\) −2.54916e10 −0.916489
\(965\) 4.79418e9 0.171739
\(966\) −7.30446e9 −0.260716
\(967\) 8.89528e9 0.316349 0.158175 0.987411i \(-0.449439\pi\)
0.158175 + 0.987411i \(0.449439\pi\)
\(968\) 2.04853e10 0.725903
\(969\) −1.71639e10 −0.606015
\(970\) −7.81649e9 −0.274986
\(971\) −9.39030e9 −0.329164 −0.164582 0.986363i \(-0.552628\pi\)
−0.164582 + 0.986363i \(0.552628\pi\)
\(972\) −2.72294e9 −0.0951057
\(973\) 1.24914e10 0.434727
\(974\) 3.93092e10 1.36313
\(975\) 2.11498e10 0.730784
\(976\) −5.80983e8 −0.0200027
\(977\) 4.04777e10 1.38863 0.694313 0.719673i \(-0.255708\pi\)
0.694313 + 0.719673i \(0.255708\pi\)
\(978\) 4.00790e10 1.37003
\(979\) 1.07272e10 0.365383
\(980\) −4.31405e9 −0.146418
\(981\) 1.54771e10 0.523418
\(982\) 4.36666e10 1.47150
\(983\) 2.17203e10 0.729336 0.364668 0.931138i \(-0.381182\pi\)
0.364668 + 0.931138i \(0.381182\pi\)
\(984\) 6.50700e9 0.217720
\(985\) −3.63787e9 −0.121289
\(986\) −5.55352e10 −1.84501
\(987\) 1.82004e9 0.0602519
\(988\) 3.60559e10 1.18940
\(989\) −1.10752e10 −0.364054
\(990\) −3.78844e8 −0.0124090
\(991\) 4.02205e10 1.31277 0.656387 0.754425i \(-0.272084\pi\)
0.656387 + 0.754425i \(0.272084\pi\)
\(992\) −4.19924e10 −1.36578
\(993\) 2.26743e10 0.734871
\(994\) −1.64595e10 −0.531576
\(995\) 2.84680e9 0.0916169
\(996\) −1.64862e10 −0.528706
\(997\) 1.02432e10 0.327342 0.163671 0.986515i \(-0.447666\pi\)
0.163671 + 0.986515i \(0.447666\pi\)
\(998\) 7.29287e10 2.32242
\(999\) 2.20997e9 0.0701304
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 69.8.a.a.1.1 5
3.2 odd 2 207.8.a.a.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.a.1.1 5 1.1 even 1 trivial
207.8.a.a.1.5 5 3.2 odd 2