Properties

Label 387.8.a.d.1.10
Level $387$
Weight $8$
Character 387.1
Self dual yes
Analytic conductor $120.893$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(120.893004862\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-8.80627\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.80627 q^{2} -67.0621 q^{4} +402.005 q^{5} -356.766 q^{7} -1522.71 q^{8} +O(q^{10})\) \(q+7.80627 q^{2} -67.0621 q^{4} +402.005 q^{5} -356.766 q^{7} -1522.71 q^{8} +3138.16 q^{10} -338.626 q^{11} -1901.99 q^{13} -2785.01 q^{14} -3302.71 q^{16} +13710.6 q^{17} +32375.1 q^{19} -26959.3 q^{20} -2643.41 q^{22} -103235. q^{23} +83482.7 q^{25} -14847.4 q^{26} +23925.5 q^{28} -97.3595 q^{29} -2919.43 q^{31} +169125. q^{32} +107029. q^{34} -143421. q^{35} -280235. q^{37} +252729. q^{38} -612136. q^{40} +68080.9 q^{41} -79507.0 q^{43} +22709.0 q^{44} -805883. q^{46} -145265. q^{47} -696261. q^{49} +651689. q^{50} +127551. q^{52} +450286. q^{53} -136129. q^{55} +543250. q^{56} -760.015 q^{58} -2.16940e6 q^{59} -240945. q^{61} -22789.9 q^{62} +1.74298e6 q^{64} -764608. q^{65} +763129. q^{67} -919465. q^{68} -1.11959e6 q^{70} -256406. q^{71} -4.25607e6 q^{73} -2.18759e6 q^{74} -2.17114e6 q^{76} +120810. q^{77} +6.60338e6 q^{79} -1.32771e6 q^{80} +531458. q^{82} -1.57834e6 q^{83} +5.51174e6 q^{85} -620653. q^{86} +515629. q^{88} -1.01079e7 q^{89} +678564. q^{91} +6.92319e6 q^{92} -1.13398e6 q^{94} +1.30149e7 q^{95} -1.90764e6 q^{97} -5.43520e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.80627 0.689983 0.344992 0.938606i \(-0.387882\pi\)
0.344992 + 0.938606i \(0.387882\pi\)
\(3\) 0 0
\(4\) −67.0621 −0.523923
\(5\) 402.005 1.43826 0.719128 0.694878i \(-0.244542\pi\)
0.719128 + 0.694878i \(0.244542\pi\)
\(6\) 0 0
\(7\) −356.766 −0.393133 −0.196567 0.980490i \(-0.562979\pi\)
−0.196567 + 0.980490i \(0.562979\pi\)
\(8\) −1522.71 −1.05148
\(9\) 0 0
\(10\) 3138.16 0.992372
\(11\) −338.626 −0.0767090 −0.0383545 0.999264i \(-0.512212\pi\)
−0.0383545 + 0.999264i \(0.512212\pi\)
\(12\) 0 0
\(13\) −1901.99 −0.240108 −0.120054 0.992767i \(-0.538307\pi\)
−0.120054 + 0.992767i \(0.538307\pi\)
\(14\) −2785.01 −0.271255
\(15\) 0 0
\(16\) −3302.71 −0.201582
\(17\) 13710.6 0.676841 0.338421 0.940995i \(-0.390107\pi\)
0.338421 + 0.940995i \(0.390107\pi\)
\(18\) 0 0
\(19\) 32375.1 1.08286 0.541431 0.840745i \(-0.317883\pi\)
0.541431 + 0.840745i \(0.317883\pi\)
\(20\) −26959.3 −0.753535
\(21\) 0 0
\(22\) −2643.41 −0.0529279
\(23\) −103235. −1.76922 −0.884609 0.466334i \(-0.845574\pi\)
−0.884609 + 0.466334i \(0.845574\pi\)
\(24\) 0 0
\(25\) 83482.7 1.06858
\(26\) −14847.4 −0.165670
\(27\) 0 0
\(28\) 23925.5 0.205972
\(29\) −97.3595 −0.000741285 0 −0.000370643 1.00000i \(-0.500118\pi\)
−0.000370643 1.00000i \(0.500118\pi\)
\(30\) 0 0
\(31\) −2919.43 −0.0176008 −0.00880039 0.999961i \(-0.502801\pi\)
−0.00880039 + 0.999961i \(0.502801\pi\)
\(32\) 169125. 0.912394
\(33\) 0 0
\(34\) 107029. 0.467009
\(35\) −143421. −0.565426
\(36\) 0 0
\(37\) −280235. −0.909529 −0.454764 0.890612i \(-0.650277\pi\)
−0.454764 + 0.890612i \(0.650277\pi\)
\(38\) 252729. 0.747157
\(39\) 0 0
\(40\) −612136. −1.51230
\(41\) 68080.9 0.154270 0.0771350 0.997021i \(-0.475423\pi\)
0.0771350 + 0.997021i \(0.475423\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) 22709.0 0.0401896
\(45\) 0 0
\(46\) −805883. −1.22073
\(47\) −145265. −0.204089 −0.102044 0.994780i \(-0.532538\pi\)
−0.102044 + 0.994780i \(0.532538\pi\)
\(48\) 0 0
\(49\) −696261. −0.845446
\(50\) 651689. 0.737301
\(51\) 0 0
\(52\) 127551. 0.125798
\(53\) 450286. 0.415454 0.207727 0.978187i \(-0.433393\pi\)
0.207727 + 0.978187i \(0.433393\pi\)
\(54\) 0 0
\(55\) −136129. −0.110327
\(56\) 543250. 0.413372
\(57\) 0 0
\(58\) −760.015 −0.000511474 0
\(59\) −2.16940e6 −1.37518 −0.687588 0.726101i \(-0.741330\pi\)
−0.687588 + 0.726101i \(0.741330\pi\)
\(60\) 0 0
\(61\) −240945. −0.135914 −0.0679570 0.997688i \(-0.521648\pi\)
−0.0679570 + 0.997688i \(0.521648\pi\)
\(62\) −22789.9 −0.0121442
\(63\) 0 0
\(64\) 1.74298e6 0.831118
\(65\) −764608. −0.345336
\(66\) 0 0
\(67\) 763129. 0.309982 0.154991 0.987916i \(-0.450465\pi\)
0.154991 + 0.987916i \(0.450465\pi\)
\(68\) −919465. −0.354613
\(69\) 0 0
\(70\) −1.11959e6 −0.390135
\(71\) −256406. −0.0850206 −0.0425103 0.999096i \(-0.513536\pi\)
−0.0425103 + 0.999096i \(0.513536\pi\)
\(72\) 0 0
\(73\) −4.25607e6 −1.28050 −0.640249 0.768168i \(-0.721169\pi\)
−0.640249 + 0.768168i \(0.721169\pi\)
\(74\) −2.18759e6 −0.627560
\(75\) 0 0
\(76\) −2.17114e6 −0.567337
\(77\) 120810. 0.0301568
\(78\) 0 0
\(79\) 6.60338e6 1.50686 0.753428 0.657531i \(-0.228399\pi\)
0.753428 + 0.657531i \(0.228399\pi\)
\(80\) −1.32771e6 −0.289926
\(81\) 0 0
\(82\) 531458. 0.106444
\(83\) −1.57834e6 −0.302989 −0.151495 0.988458i \(-0.548409\pi\)
−0.151495 + 0.988458i \(0.548409\pi\)
\(84\) 0 0
\(85\) 5.51174e6 0.973470
\(86\) −620653. −0.105221
\(87\) 0 0
\(88\) 515629. 0.0806580
\(89\) −1.01079e7 −1.51983 −0.759914 0.650024i \(-0.774759\pi\)
−0.759914 + 0.650024i \(0.774759\pi\)
\(90\) 0 0
\(91\) 678564. 0.0943944
\(92\) 6.92319e6 0.926934
\(93\) 0 0
\(94\) −1.13398e6 −0.140818
\(95\) 1.30149e7 1.55743
\(96\) 0 0
\(97\) −1.90764e6 −0.212224 −0.106112 0.994354i \(-0.533840\pi\)
−0.106112 + 0.994354i \(0.533840\pi\)
\(98\) −5.43520e6 −0.583344
\(99\) 0 0
\(100\) −5.59853e6 −0.559853
\(101\) 8.79838e6 0.849724 0.424862 0.905258i \(-0.360323\pi\)
0.424862 + 0.905258i \(0.360323\pi\)
\(102\) 0 0
\(103\) 1.98046e7 1.78581 0.892907 0.450242i \(-0.148662\pi\)
0.892907 + 0.450242i \(0.148662\pi\)
\(104\) 2.89617e6 0.252469
\(105\) 0 0
\(106\) 3.51506e6 0.286657
\(107\) −1.19023e7 −0.939264 −0.469632 0.882862i \(-0.655613\pi\)
−0.469632 + 0.882862i \(0.655613\pi\)
\(108\) 0 0
\(109\) −1.61101e7 −1.19153 −0.595766 0.803158i \(-0.703151\pi\)
−0.595766 + 0.803158i \(0.703151\pi\)
\(110\) −1.06266e6 −0.0761238
\(111\) 0 0
\(112\) 1.17829e6 0.0792484
\(113\) −7.15228e6 −0.466305 −0.233153 0.972440i \(-0.574904\pi\)
−0.233153 + 0.972440i \(0.574904\pi\)
\(114\) 0 0
\(115\) −4.15011e7 −2.54459
\(116\) 6529.14 0.000388376 0
\(117\) 0 0
\(118\) −1.69349e7 −0.948849
\(119\) −4.89149e6 −0.266089
\(120\) 0 0
\(121\) −1.93725e7 −0.994116
\(122\) −1.88088e6 −0.0937784
\(123\) 0 0
\(124\) 195783. 0.00922146
\(125\) 2.15382e6 0.0986336
\(126\) 0 0
\(127\) −1.95885e7 −0.848569 −0.424284 0.905529i \(-0.639474\pi\)
−0.424284 + 0.905529i \(0.639474\pi\)
\(128\) −8.04179e6 −0.338936
\(129\) 0 0
\(130\) −5.96874e6 −0.238276
\(131\) −3.84474e7 −1.49423 −0.747115 0.664695i \(-0.768562\pi\)
−0.747115 + 0.664695i \(0.768562\pi\)
\(132\) 0 0
\(133\) −1.15503e7 −0.425709
\(134\) 5.95719e6 0.213882
\(135\) 0 0
\(136\) −2.08773e7 −0.711686
\(137\) −5.35857e7 −1.78044 −0.890219 0.455533i \(-0.849449\pi\)
−0.890219 + 0.455533i \(0.849449\pi\)
\(138\) 0 0
\(139\) −2.44467e7 −0.772091 −0.386046 0.922480i \(-0.626159\pi\)
−0.386046 + 0.922480i \(0.626159\pi\)
\(140\) 9.61815e6 0.296240
\(141\) 0 0
\(142\) −2.00158e6 −0.0586628
\(143\) 644063. 0.0184184
\(144\) 0 0
\(145\) −39139.0 −0.00106616
\(146\) −3.32240e7 −0.883522
\(147\) 0 0
\(148\) 1.87932e7 0.476523
\(149\) −6.83645e7 −1.69308 −0.846542 0.532321i \(-0.821320\pi\)
−0.846542 + 0.532321i \(0.821320\pi\)
\(150\) 0 0
\(151\) 4.48140e7 1.05924 0.529620 0.848235i \(-0.322335\pi\)
0.529620 + 0.848235i \(0.322335\pi\)
\(152\) −4.92978e7 −1.13861
\(153\) 0 0
\(154\) 943077. 0.0208077
\(155\) −1.17362e6 −0.0253144
\(156\) 0 0
\(157\) 4.45418e7 0.918585 0.459293 0.888285i \(-0.348103\pi\)
0.459293 + 0.888285i \(0.348103\pi\)
\(158\) 5.15478e7 1.03970
\(159\) 0 0
\(160\) 6.79889e7 1.31225
\(161\) 3.68308e7 0.695538
\(162\) 0 0
\(163\) −2.04717e7 −0.370251 −0.185126 0.982715i \(-0.559269\pi\)
−0.185126 + 0.982715i \(0.559269\pi\)
\(164\) −4.56565e6 −0.0808256
\(165\) 0 0
\(166\) −1.23209e7 −0.209057
\(167\) −9.99656e6 −0.166090 −0.0830449 0.996546i \(-0.526464\pi\)
−0.0830449 + 0.996546i \(0.526464\pi\)
\(168\) 0 0
\(169\) −5.91310e7 −0.942348
\(170\) 4.30262e7 0.671678
\(171\) 0 0
\(172\) 5.33191e6 0.0798975
\(173\) −9.15039e7 −1.34363 −0.671813 0.740721i \(-0.734484\pi\)
−0.671813 + 0.740721i \(0.734484\pi\)
\(174\) 0 0
\(175\) −2.97838e7 −0.420094
\(176\) 1.11838e6 0.0154631
\(177\) 0 0
\(178\) −7.89047e7 −1.04866
\(179\) 7.48516e7 0.975474 0.487737 0.872991i \(-0.337823\pi\)
0.487737 + 0.872991i \(0.337823\pi\)
\(180\) 0 0
\(181\) −1.08411e7 −0.135894 −0.0679470 0.997689i \(-0.521645\pi\)
−0.0679470 + 0.997689i \(0.521645\pi\)
\(182\) 5.29706e6 0.0651305
\(183\) 0 0
\(184\) 1.57197e8 1.86030
\(185\) −1.12656e8 −1.30813
\(186\) 0 0
\(187\) −4.64278e6 −0.0519198
\(188\) 9.74179e6 0.106927
\(189\) 0 0
\(190\) 1.01598e8 1.07460
\(191\) 1.37400e8 1.42682 0.713410 0.700747i \(-0.247150\pi\)
0.713410 + 0.700747i \(0.247150\pi\)
\(192\) 0 0
\(193\) −1.16548e8 −1.16695 −0.583477 0.812129i \(-0.698308\pi\)
−0.583477 + 0.812129i \(0.698308\pi\)
\(194\) −1.48915e7 −0.146431
\(195\) 0 0
\(196\) 4.66928e7 0.442949
\(197\) 1.79080e8 1.66884 0.834422 0.551126i \(-0.185802\pi\)
0.834422 + 0.551126i \(0.185802\pi\)
\(198\) 0 0
\(199\) −1.30708e8 −1.17575 −0.587875 0.808952i \(-0.700035\pi\)
−0.587875 + 0.808952i \(0.700035\pi\)
\(200\) −1.27120e8 −1.12359
\(201\) 0 0
\(202\) 6.86825e7 0.586295
\(203\) 34734.5 0.000291424 0
\(204\) 0 0
\(205\) 2.73688e7 0.221880
\(206\) 1.54600e8 1.23218
\(207\) 0 0
\(208\) 6.28172e6 0.0484013
\(209\) −1.09631e7 −0.0830653
\(210\) 0 0
\(211\) 3.03753e7 0.222604 0.111302 0.993787i \(-0.464498\pi\)
0.111302 + 0.993787i \(0.464498\pi\)
\(212\) −3.01972e7 −0.217666
\(213\) 0 0
\(214\) −9.29126e7 −0.648076
\(215\) −3.19622e7 −0.219332
\(216\) 0 0
\(217\) 1.04155e6 0.00691945
\(218\) −1.25760e8 −0.822137
\(219\) 0 0
\(220\) 9.12912e6 0.0578029
\(221\) −2.60775e7 −0.162515
\(222\) 0 0
\(223\) 8.68403e7 0.524390 0.262195 0.965015i \(-0.415554\pi\)
0.262195 + 0.965015i \(0.415554\pi\)
\(224\) −6.03379e7 −0.358692
\(225\) 0 0
\(226\) −5.58326e7 −0.321743
\(227\) −1.67604e8 −0.951028 −0.475514 0.879708i \(-0.657738\pi\)
−0.475514 + 0.879708i \(0.657738\pi\)
\(228\) 0 0
\(229\) 1.15619e8 0.636218 0.318109 0.948054i \(-0.396952\pi\)
0.318109 + 0.948054i \(0.396952\pi\)
\(230\) −3.23969e8 −1.75572
\(231\) 0 0
\(232\) 148250. 0.000779448 0
\(233\) −1.02835e8 −0.532592 −0.266296 0.963891i \(-0.585800\pi\)
−0.266296 + 0.963891i \(0.585800\pi\)
\(234\) 0 0
\(235\) −5.83973e7 −0.293532
\(236\) 1.45485e8 0.720486
\(237\) 0 0
\(238\) −3.81843e7 −0.183597
\(239\) −2.55890e8 −1.21244 −0.606221 0.795296i \(-0.707315\pi\)
−0.606221 + 0.795296i \(0.707315\pi\)
\(240\) 0 0
\(241\) −1.95565e6 −0.00899976 −0.00449988 0.999990i \(-0.501432\pi\)
−0.00449988 + 0.999990i \(0.501432\pi\)
\(242\) −1.51227e8 −0.685923
\(243\) 0 0
\(244\) 1.61583e7 0.0712085
\(245\) −2.79900e8 −1.21597
\(246\) 0 0
\(247\) −6.15771e7 −0.260004
\(248\) 4.44544e6 0.0185069
\(249\) 0 0
\(250\) 1.68133e7 0.0680556
\(251\) 2.99606e8 1.19589 0.597946 0.801536i \(-0.295984\pi\)
0.597946 + 0.801536i \(0.295984\pi\)
\(252\) 0 0
\(253\) 3.49582e7 0.135715
\(254\) −1.52913e8 −0.585498
\(255\) 0 0
\(256\) −2.85878e8 −1.06498
\(257\) 3.65657e8 1.34372 0.671859 0.740679i \(-0.265496\pi\)
0.671859 + 0.740679i \(0.265496\pi\)
\(258\) 0 0
\(259\) 9.99782e7 0.357566
\(260\) 5.12763e7 0.180930
\(261\) 0 0
\(262\) −3.00131e8 −1.03099
\(263\) −4.38062e8 −1.48488 −0.742438 0.669914i \(-0.766331\pi\)
−0.742438 + 0.669914i \(0.766331\pi\)
\(264\) 0 0
\(265\) 1.81017e8 0.597529
\(266\) −9.01649e7 −0.293732
\(267\) 0 0
\(268\) −5.11771e7 −0.162407
\(269\) 6.37770e7 0.199771 0.0998853 0.994999i \(-0.468152\pi\)
0.0998853 + 0.994999i \(0.468152\pi\)
\(270\) 0 0
\(271\) 3.49200e8 1.06582 0.532908 0.846173i \(-0.321099\pi\)
0.532908 + 0.846173i \(0.321099\pi\)
\(272\) −4.52823e7 −0.136439
\(273\) 0 0
\(274\) −4.18304e8 −1.22847
\(275\) −2.82694e7 −0.0819695
\(276\) 0 0
\(277\) 2.62910e8 0.743238 0.371619 0.928385i \(-0.378803\pi\)
0.371619 + 0.928385i \(0.378803\pi\)
\(278\) −1.90838e8 −0.532730
\(279\) 0 0
\(280\) 2.18389e8 0.594535
\(281\) 3.13673e8 0.843345 0.421673 0.906748i \(-0.361443\pi\)
0.421673 + 0.906748i \(0.361443\pi\)
\(282\) 0 0
\(283\) 3.05065e7 0.0800093 0.0400046 0.999199i \(-0.487263\pi\)
0.0400046 + 0.999199i \(0.487263\pi\)
\(284\) 1.71951e7 0.0445442
\(285\) 0 0
\(286\) 5.02773e6 0.0127084
\(287\) −2.42889e7 −0.0606487
\(288\) 0 0
\(289\) −2.22357e8 −0.541886
\(290\) −305529. −0.000735631 0
\(291\) 0 0
\(292\) 2.85421e8 0.670882
\(293\) −6.50794e8 −1.51149 −0.755747 0.654863i \(-0.772726\pi\)
−0.755747 + 0.654863i \(0.772726\pi\)
\(294\) 0 0
\(295\) −8.72110e8 −1.97785
\(296\) 4.26716e8 0.956353
\(297\) 0 0
\(298\) −5.33672e8 −1.16820
\(299\) 1.96353e8 0.424803
\(300\) 0 0
\(301\) 2.83654e7 0.0599523
\(302\) 3.49830e8 0.730858
\(303\) 0 0
\(304\) −1.06926e8 −0.218285
\(305\) −9.68611e7 −0.195479
\(306\) 0 0
\(307\) −1.64397e8 −0.324273 −0.162136 0.986768i \(-0.551838\pi\)
−0.162136 + 0.986768i \(0.551838\pi\)
\(308\) −8.10179e6 −0.0157999
\(309\) 0 0
\(310\) −9.16163e6 −0.0174665
\(311\) −6.67878e7 −0.125903 −0.0629515 0.998017i \(-0.520051\pi\)
−0.0629515 + 0.998017i \(0.520051\pi\)
\(312\) 0 0
\(313\) 3.66247e8 0.675101 0.337550 0.941307i \(-0.390402\pi\)
0.337550 + 0.941307i \(0.390402\pi\)
\(314\) 3.47706e8 0.633808
\(315\) 0 0
\(316\) −4.42837e8 −0.789476
\(317\) −8.45942e8 −1.49153 −0.745767 0.666206i \(-0.767917\pi\)
−0.745767 + 0.666206i \(0.767917\pi\)
\(318\) 0 0
\(319\) 32968.5 5.68632e−5 0
\(320\) 7.00686e8 1.19536
\(321\) 0 0
\(322\) 2.87511e8 0.479910
\(323\) 4.43883e8 0.732926
\(324\) 0 0
\(325\) −1.58783e8 −0.256574
\(326\) −1.59807e8 −0.255467
\(327\) 0 0
\(328\) −1.03667e8 −0.162212
\(329\) 5.18256e7 0.0802341
\(330\) 0 0
\(331\) −1.56474e8 −0.237162 −0.118581 0.992944i \(-0.537835\pi\)
−0.118581 + 0.992944i \(0.537835\pi\)
\(332\) 1.05847e8 0.158743
\(333\) 0 0
\(334\) −7.80358e7 −0.114599
\(335\) 3.06781e8 0.445833
\(336\) 0 0
\(337\) 1.10367e9 1.57085 0.785424 0.618958i \(-0.212445\pi\)
0.785424 + 0.618958i \(0.212445\pi\)
\(338\) −4.61592e8 −0.650205
\(339\) 0 0
\(340\) −3.69629e8 −0.510024
\(341\) 988595. 0.00135014
\(342\) 0 0
\(343\) 5.42214e8 0.725506
\(344\) 1.21066e8 0.160349
\(345\) 0 0
\(346\) −7.14304e8 −0.927079
\(347\) 7.51448e7 0.0965485 0.0482743 0.998834i \(-0.484628\pi\)
0.0482743 + 0.998834i \(0.484628\pi\)
\(348\) 0 0
\(349\) 4.40370e8 0.554534 0.277267 0.960793i \(-0.410571\pi\)
0.277267 + 0.960793i \(0.410571\pi\)
\(350\) −2.32500e8 −0.289858
\(351\) 0 0
\(352\) −5.72701e7 −0.0699888
\(353\) 8.89528e7 0.107634 0.0538168 0.998551i \(-0.482861\pi\)
0.0538168 + 0.998551i \(0.482861\pi\)
\(354\) 0 0
\(355\) −1.03076e8 −0.122281
\(356\) 6.77855e8 0.796273
\(357\) 0 0
\(358\) 5.84312e8 0.673061
\(359\) 1.09248e9 1.24619 0.623095 0.782146i \(-0.285875\pi\)
0.623095 + 0.782146i \(0.285875\pi\)
\(360\) 0 0
\(361\) 1.54275e8 0.172591
\(362\) −8.46289e7 −0.0937645
\(363\) 0 0
\(364\) −4.55060e7 −0.0494554
\(365\) −1.71096e9 −1.84168
\(366\) 0 0
\(367\) −5.91559e8 −0.624693 −0.312346 0.949968i \(-0.601115\pi\)
−0.312346 + 0.949968i \(0.601115\pi\)
\(368\) 3.40957e8 0.356642
\(369\) 0 0
\(370\) −8.79422e8 −0.902591
\(371\) −1.60647e8 −0.163329
\(372\) 0 0
\(373\) 1.17638e9 1.17373 0.586863 0.809686i \(-0.300363\pi\)
0.586863 + 0.809686i \(0.300363\pi\)
\(374\) −3.62428e7 −0.0358238
\(375\) 0 0
\(376\) 2.21196e8 0.214595
\(377\) 185177. 0.000177988 0
\(378\) 0 0
\(379\) 1.58878e9 1.49908 0.749541 0.661958i \(-0.230274\pi\)
0.749541 + 0.661958i \(0.230274\pi\)
\(380\) −8.72810e8 −0.815975
\(381\) 0 0
\(382\) 1.07258e9 0.984482
\(383\) 1.52083e9 1.38320 0.691599 0.722281i \(-0.256906\pi\)
0.691599 + 0.722281i \(0.256906\pi\)
\(384\) 0 0
\(385\) 4.85662e7 0.0433732
\(386\) −9.09805e8 −0.805179
\(387\) 0 0
\(388\) 1.27930e8 0.111189
\(389\) −1.81957e9 −1.56728 −0.783639 0.621217i \(-0.786639\pi\)
−0.783639 + 0.621217i \(0.786639\pi\)
\(390\) 0 0
\(391\) −1.41542e9 −1.19748
\(392\) 1.06020e9 0.888971
\(393\) 0 0
\(394\) 1.39795e9 1.15147
\(395\) 2.65459e9 2.16724
\(396\) 0 0
\(397\) 2.11509e9 1.69653 0.848265 0.529571i \(-0.177647\pi\)
0.848265 + 0.529571i \(0.177647\pi\)
\(398\) −1.02034e9 −0.811248
\(399\) 0 0
\(400\) −2.75719e8 −0.215406
\(401\) 2.18744e9 1.69407 0.847034 0.531539i \(-0.178386\pi\)
0.847034 + 0.531539i \(0.178386\pi\)
\(402\) 0 0
\(403\) 5.55272e6 0.00422609
\(404\) −5.90038e8 −0.445190
\(405\) 0 0
\(406\) 271147. 0.000201078 0
\(407\) 9.48949e7 0.0697690
\(408\) 0 0
\(409\) −1.16511e9 −0.842045 −0.421022 0.907050i \(-0.638329\pi\)
−0.421022 + 0.907050i \(0.638329\pi\)
\(410\) 2.13648e8 0.153093
\(411\) 0 0
\(412\) −1.32814e9 −0.935629
\(413\) 7.73968e8 0.540627
\(414\) 0 0
\(415\) −6.34499e8 −0.435776
\(416\) −3.21673e8 −0.219073
\(417\) 0 0
\(418\) −8.55805e7 −0.0573136
\(419\) 9.59686e8 0.637353 0.318677 0.947864i \(-0.396762\pi\)
0.318677 + 0.947864i \(0.396762\pi\)
\(420\) 0 0
\(421\) −1.02361e7 −0.00668572 −0.00334286 0.999994i \(-0.501064\pi\)
−0.00334286 + 0.999994i \(0.501064\pi\)
\(422\) 2.37118e8 0.153593
\(423\) 0 0
\(424\) −6.85655e8 −0.436843
\(425\) 1.14460e9 0.723258
\(426\) 0 0
\(427\) 8.59610e7 0.0534323
\(428\) 7.98194e8 0.492102
\(429\) 0 0
\(430\) −2.49505e8 −0.151335
\(431\) 3.14206e9 1.89036 0.945179 0.326552i \(-0.105887\pi\)
0.945179 + 0.326552i \(0.105887\pi\)
\(432\) 0 0
\(433\) −2.31364e9 −1.36958 −0.684792 0.728739i \(-0.740107\pi\)
−0.684792 + 0.728739i \(0.740107\pi\)
\(434\) 8.13063e6 0.00477431
\(435\) 0 0
\(436\) 1.08038e9 0.624271
\(437\) −3.34225e9 −1.91582
\(438\) 0 0
\(439\) 1.64208e9 0.926337 0.463168 0.886270i \(-0.346713\pi\)
0.463168 + 0.886270i \(0.346713\pi\)
\(440\) 2.07285e8 0.116007
\(441\) 0 0
\(442\) −2.03568e8 −0.112133
\(443\) −7.12250e8 −0.389242 −0.194621 0.980879i \(-0.562348\pi\)
−0.194621 + 0.980879i \(0.562348\pi\)
\(444\) 0 0
\(445\) −4.06341e9 −2.18590
\(446\) 6.77899e8 0.361820
\(447\) 0 0
\(448\) −6.21835e8 −0.326740
\(449\) −2.15899e9 −1.12561 −0.562805 0.826590i \(-0.690278\pi\)
−0.562805 + 0.826590i \(0.690278\pi\)
\(450\) 0 0
\(451\) −2.30540e7 −0.0118339
\(452\) 4.79647e8 0.244308
\(453\) 0 0
\(454\) −1.30836e9 −0.656193
\(455\) 2.72786e8 0.135763
\(456\) 0 0
\(457\) −1.05753e9 −0.518308 −0.259154 0.965836i \(-0.583444\pi\)
−0.259154 + 0.965836i \(0.583444\pi\)
\(458\) 9.02555e8 0.438980
\(459\) 0 0
\(460\) 2.78315e9 1.33317
\(461\) 2.93548e9 1.39549 0.697743 0.716348i \(-0.254188\pi\)
0.697743 + 0.716348i \(0.254188\pi\)
\(462\) 0 0
\(463\) 2.80303e9 1.31248 0.656242 0.754550i \(-0.272145\pi\)
0.656242 + 0.754550i \(0.272145\pi\)
\(464\) 321551. 0.000149429 0
\(465\) 0 0
\(466\) −8.02756e8 −0.367479
\(467\) −1.35420e9 −0.615281 −0.307640 0.951503i \(-0.599539\pi\)
−0.307640 + 0.951503i \(0.599539\pi\)
\(468\) 0 0
\(469\) −2.72258e8 −0.121864
\(470\) −4.55865e8 −0.202532
\(471\) 0 0
\(472\) 3.30337e9 1.44597
\(473\) 2.69231e7 0.0116980
\(474\) 0 0
\(475\) 2.70276e9 1.15712
\(476\) 3.28034e8 0.139410
\(477\) 0 0
\(478\) −1.99755e9 −0.836565
\(479\) −1.17660e9 −0.489165 −0.244583 0.969628i \(-0.578651\pi\)
−0.244583 + 0.969628i \(0.578651\pi\)
\(480\) 0 0
\(481\) 5.33004e8 0.218385
\(482\) −1.52663e7 −0.00620969
\(483\) 0 0
\(484\) 1.29916e9 0.520840
\(485\) −7.66879e8 −0.305233
\(486\) 0 0
\(487\) −1.73453e9 −0.680504 −0.340252 0.940334i \(-0.610512\pi\)
−0.340252 + 0.940334i \(0.610512\pi\)
\(488\) 3.66889e8 0.142911
\(489\) 0 0
\(490\) −2.18498e9 −0.838997
\(491\) −7.94788e8 −0.303016 −0.151508 0.988456i \(-0.548413\pi\)
−0.151508 + 0.988456i \(0.548413\pi\)
\(492\) 0 0
\(493\) −1.33486e6 −0.000501732 0
\(494\) −4.80687e8 −0.179398
\(495\) 0 0
\(496\) 9.64204e6 0.00354799
\(497\) 9.14769e7 0.0334244
\(498\) 0 0
\(499\) −2.73492e8 −0.0985354 −0.0492677 0.998786i \(-0.515689\pi\)
−0.0492677 + 0.998786i \(0.515689\pi\)
\(500\) −1.44440e8 −0.0516764
\(501\) 0 0
\(502\) 2.33880e9 0.825146
\(503\) −3.20927e9 −1.12439 −0.562197 0.827004i \(-0.690044\pi\)
−0.562197 + 0.827004i \(0.690044\pi\)
\(504\) 0 0
\(505\) 3.53699e9 1.22212
\(506\) 2.72893e8 0.0936409
\(507\) 0 0
\(508\) 1.31364e9 0.444585
\(509\) 2.06801e9 0.695090 0.347545 0.937663i \(-0.387015\pi\)
0.347545 + 0.937663i \(0.387015\pi\)
\(510\) 0 0
\(511\) 1.51842e9 0.503406
\(512\) −1.20229e9 −0.395881
\(513\) 0 0
\(514\) 2.85442e9 0.927143
\(515\) 7.96155e9 2.56846
\(516\) 0 0
\(517\) 4.91906e7 0.0156554
\(518\) 7.80457e8 0.246715
\(519\) 0 0
\(520\) 1.16428e9 0.363115
\(521\) 3.73056e8 0.115569 0.0577846 0.998329i \(-0.481596\pi\)
0.0577846 + 0.998329i \(0.481596\pi\)
\(522\) 0 0
\(523\) 3.23366e9 0.988414 0.494207 0.869344i \(-0.335459\pi\)
0.494207 + 0.869344i \(0.335459\pi\)
\(524\) 2.57837e9 0.782862
\(525\) 0 0
\(526\) −3.41963e9 −1.02454
\(527\) −4.00273e7 −0.0119129
\(528\) 0 0
\(529\) 7.25272e9 2.13013
\(530\) 1.41307e9 0.412285
\(531\) 0 0
\(532\) 7.74589e8 0.223039
\(533\) −1.29489e8 −0.0370414
\(534\) 0 0
\(535\) −4.78478e9 −1.35090
\(536\) −1.16202e9 −0.325940
\(537\) 0 0
\(538\) 4.97861e8 0.137838
\(539\) 2.35772e8 0.0648533
\(540\) 0 0
\(541\) −6.33940e9 −1.72131 −0.860653 0.509192i \(-0.829944\pi\)
−0.860653 + 0.509192i \(0.829944\pi\)
\(542\) 2.72595e9 0.735395
\(543\) 0 0
\(544\) 2.31881e9 0.617545
\(545\) −6.47633e9 −1.71373
\(546\) 0 0
\(547\) 2.18850e9 0.571731 0.285865 0.958270i \(-0.407719\pi\)
0.285865 + 0.958270i \(0.407719\pi\)
\(548\) 3.59357e9 0.932812
\(549\) 0 0
\(550\) −2.20679e8 −0.0565576
\(551\) −3.15202e6 −0.000802710 0
\(552\) 0 0
\(553\) −2.35586e9 −0.592395
\(554\) 2.05235e9 0.512822
\(555\) 0 0
\(556\) 1.63945e9 0.404516
\(557\) 2.65106e9 0.650019 0.325009 0.945711i \(-0.394633\pi\)
0.325009 + 0.945711i \(0.394633\pi\)
\(558\) 0 0
\(559\) 1.51221e8 0.0366161
\(560\) 4.73680e8 0.113979
\(561\) 0 0
\(562\) 2.44862e9 0.581894
\(563\) 4.33945e9 1.02484 0.512419 0.858736i \(-0.328749\pi\)
0.512419 + 0.858736i \(0.328749\pi\)
\(564\) 0 0
\(565\) −2.87525e9 −0.670666
\(566\) 2.38142e8 0.0552051
\(567\) 0 0
\(568\) 3.90432e8 0.0893976
\(569\) −6.16980e8 −0.140404 −0.0702018 0.997533i \(-0.522364\pi\)
−0.0702018 + 0.997533i \(0.522364\pi\)
\(570\) 0 0
\(571\) 5.73913e9 1.29009 0.645045 0.764145i \(-0.276839\pi\)
0.645045 + 0.764145i \(0.276839\pi\)
\(572\) −4.31923e7 −0.00964983
\(573\) 0 0
\(574\) −1.89606e8 −0.0418466
\(575\) −8.61837e9 −1.89055
\(576\) 0 0
\(577\) 5.29584e9 1.14768 0.573838 0.818969i \(-0.305454\pi\)
0.573838 + 0.818969i \(0.305454\pi\)
\(578\) −1.73578e9 −0.373892
\(579\) 0 0
\(580\) 2.62474e6 0.000558585 0
\(581\) 5.63097e8 0.119115
\(582\) 0 0
\(583\) −1.52479e8 −0.0318691
\(584\) 6.48075e9 1.34642
\(585\) 0 0
\(586\) −5.08027e9 −1.04291
\(587\) −4.99464e8 −0.101923 −0.0509613 0.998701i \(-0.516229\pi\)
−0.0509613 + 0.998701i \(0.516229\pi\)
\(588\) 0 0
\(589\) −9.45168e7 −0.0190592
\(590\) −6.80793e9 −1.36469
\(591\) 0 0
\(592\) 9.25536e8 0.183344
\(593\) 1.20268e9 0.236842 0.118421 0.992963i \(-0.462217\pi\)
0.118421 + 0.992963i \(0.462217\pi\)
\(594\) 0 0
\(595\) −1.96640e9 −0.382704
\(596\) 4.58467e9 0.887046
\(597\) 0 0
\(598\) 1.53278e9 0.293107
\(599\) 5.47286e9 1.04045 0.520224 0.854030i \(-0.325849\pi\)
0.520224 + 0.854030i \(0.325849\pi\)
\(600\) 0 0
\(601\) −4.60823e9 −0.865912 −0.432956 0.901415i \(-0.642529\pi\)
−0.432956 + 0.901415i \(0.642529\pi\)
\(602\) 2.21428e8 0.0413661
\(603\) 0 0
\(604\) −3.00532e9 −0.554960
\(605\) −7.78784e9 −1.42979
\(606\) 0 0
\(607\) 7.85259e9 1.42512 0.712562 0.701610i \(-0.247535\pi\)
0.712562 + 0.701610i \(0.247535\pi\)
\(608\) 5.47543e9 0.987997
\(609\) 0 0
\(610\) −7.56124e8 −0.134877
\(611\) 2.76293e8 0.0490033
\(612\) 0 0
\(613\) −1.04035e10 −1.82419 −0.912095 0.409980i \(-0.865536\pi\)
−0.912095 + 0.409980i \(0.865536\pi\)
\(614\) −1.28333e9 −0.223743
\(615\) 0 0
\(616\) −1.83959e8 −0.0317094
\(617\) −2.89619e9 −0.496397 −0.248198 0.968709i \(-0.579838\pi\)
−0.248198 + 0.968709i \(0.579838\pi\)
\(618\) 0 0
\(619\) −1.69036e9 −0.286458 −0.143229 0.989690i \(-0.545749\pi\)
−0.143229 + 0.989690i \(0.545749\pi\)
\(620\) 7.87057e7 0.0132628
\(621\) 0 0
\(622\) −5.21364e8 −0.0868710
\(623\) 3.60614e9 0.597495
\(624\) 0 0
\(625\) −5.65624e9 −0.926718
\(626\) 2.85902e9 0.465808
\(627\) 0 0
\(628\) −2.98707e9 −0.481268
\(629\) −3.84220e9 −0.615606
\(630\) 0 0
\(631\) −3.25816e9 −0.516261 −0.258131 0.966110i \(-0.583106\pi\)
−0.258131 + 0.966110i \(0.583106\pi\)
\(632\) −1.00550e10 −1.58443
\(633\) 0 0
\(634\) −6.60366e9 −1.02913
\(635\) −7.87465e9 −1.22046
\(636\) 0 0
\(637\) 1.32428e9 0.202998
\(638\) 257361. 3.92347e−5 0
\(639\) 0 0
\(640\) −3.23284e9 −0.487477
\(641\) −1.62115e9 −0.243119 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(642\) 0 0
\(643\) −4.87271e8 −0.0722823 −0.0361412 0.999347i \(-0.511507\pi\)
−0.0361412 + 0.999347i \(0.511507\pi\)
\(644\) −2.46995e9 −0.364408
\(645\) 0 0
\(646\) 3.46507e9 0.505707
\(647\) −1.32422e10 −1.92219 −0.961095 0.276219i \(-0.910918\pi\)
−0.961095 + 0.276219i \(0.910918\pi\)
\(648\) 0 0
\(649\) 7.34617e8 0.105488
\(650\) −1.23950e9 −0.177032
\(651\) 0 0
\(652\) 1.37287e9 0.193983
\(653\) −1.05133e10 −1.47755 −0.738774 0.673953i \(-0.764595\pi\)
−0.738774 + 0.673953i \(0.764595\pi\)
\(654\) 0 0
\(655\) −1.54560e10 −2.14908
\(656\) −2.24852e8 −0.0310980
\(657\) 0 0
\(658\) 4.04565e8 0.0553602
\(659\) −5.82980e9 −0.793514 −0.396757 0.917924i \(-0.629864\pi\)
−0.396757 + 0.917924i \(0.629864\pi\)
\(660\) 0 0
\(661\) 4.98936e9 0.671954 0.335977 0.941870i \(-0.390934\pi\)
0.335977 + 0.941870i \(0.390934\pi\)
\(662\) −1.22148e9 −0.163638
\(663\) 0 0
\(664\) 2.40335e9 0.318587
\(665\) −4.64328e9 −0.612279
\(666\) 0 0
\(667\) 1.00509e7 0.00131149
\(668\) 6.70391e8 0.0870183
\(669\) 0 0
\(670\) 2.39482e9 0.307617
\(671\) 8.15904e7 0.0104258
\(672\) 0 0
\(673\) −9.45755e9 −1.19599 −0.597993 0.801502i \(-0.704035\pi\)
−0.597993 + 0.801502i \(0.704035\pi\)
\(674\) 8.61554e9 1.08386
\(675\) 0 0
\(676\) 3.96545e9 0.493718
\(677\) 1.17413e10 1.45431 0.727155 0.686474i \(-0.240842\pi\)
0.727155 + 0.686474i \(0.240842\pi\)
\(678\) 0 0
\(679\) 6.80579e8 0.0834324
\(680\) −8.39277e9 −1.02359
\(681\) 0 0
\(682\) 7.71724e6 0.000931572 0
\(683\) −8.01180e9 −0.962183 −0.481091 0.876670i \(-0.659759\pi\)
−0.481091 + 0.876670i \(0.659759\pi\)
\(684\) 0 0
\(685\) −2.15417e10 −2.56072
\(686\) 4.23267e9 0.500587
\(687\) 0 0
\(688\) 2.62589e8 0.0307409
\(689\) −8.56440e8 −0.0997538
\(690\) 0 0
\(691\) 9.40691e9 1.08461 0.542305 0.840181i \(-0.317552\pi\)
0.542305 + 0.840181i \(0.317552\pi\)
\(692\) 6.13645e9 0.703956
\(693\) 0 0
\(694\) 5.86600e8 0.0666169
\(695\) −9.82769e9 −1.11046
\(696\) 0 0
\(697\) 9.33433e8 0.104416
\(698\) 3.43765e9 0.382619
\(699\) 0 0
\(700\) 1.99736e9 0.220097
\(701\) −5.77903e9 −0.633639 −0.316819 0.948486i \(-0.602615\pi\)
−0.316819 + 0.948486i \(0.602615\pi\)
\(702\) 0 0
\(703\) −9.07264e9 −0.984895
\(704\) −5.90219e8 −0.0637542
\(705\) 0 0
\(706\) 6.94389e8 0.0742654
\(707\) −3.13896e9 −0.334055
\(708\) 0 0
\(709\) 1.04068e10 1.09662 0.548311 0.836274i \(-0.315271\pi\)
0.548311 + 0.836274i \(0.315271\pi\)
\(710\) −8.04642e8 −0.0843721
\(711\) 0 0
\(712\) 1.53913e10 1.59807
\(713\) 3.01388e8 0.0311396
\(714\) 0 0
\(715\) 2.58916e8 0.0264904
\(716\) −5.01971e9 −0.511073
\(717\) 0 0
\(718\) 8.52822e9 0.859851
\(719\) −1.83550e10 −1.84164 −0.920819 0.389991i \(-0.872478\pi\)
−0.920819 + 0.389991i \(0.872478\pi\)
\(720\) 0 0
\(721\) −7.06560e9 −0.702063
\(722\) 1.20431e9 0.119085
\(723\) 0 0
\(724\) 7.27030e8 0.0711980
\(725\) −8.12784e6 −0.000792122 0
\(726\) 0 0
\(727\) 1.38023e9 0.133223 0.0666116 0.997779i \(-0.478781\pi\)
0.0666116 + 0.997779i \(0.478781\pi\)
\(728\) −1.03325e9 −0.0992539
\(729\) 0 0
\(730\) −1.33562e10 −1.27073
\(731\) −1.09009e9 −0.103217
\(732\) 0 0
\(733\) 2.24467e9 0.210518 0.105259 0.994445i \(-0.466433\pi\)
0.105259 + 0.994445i \(0.466433\pi\)
\(734\) −4.61787e9 −0.431028
\(735\) 0 0
\(736\) −1.74597e10 −1.61422
\(737\) −2.58415e8 −0.0237784
\(738\) 0 0
\(739\) 4.41123e9 0.402072 0.201036 0.979584i \(-0.435569\pi\)
0.201036 + 0.979584i \(0.435569\pi\)
\(740\) 7.55494e9 0.685362
\(741\) 0 0
\(742\) −1.25405e9 −0.112694
\(743\) −1.31411e10 −1.17536 −0.587679 0.809094i \(-0.699958\pi\)
−0.587679 + 0.809094i \(0.699958\pi\)
\(744\) 0 0
\(745\) −2.74828e10 −2.43509
\(746\) 9.18315e9 0.809852
\(747\) 0 0
\(748\) 3.11355e8 0.0272020
\(749\) 4.24633e9 0.369256
\(750\) 0 0
\(751\) 8.90398e9 0.767087 0.383543 0.923523i \(-0.374704\pi\)
0.383543 + 0.923523i \(0.374704\pi\)
\(752\) 4.79769e8 0.0411405
\(753\) 0 0
\(754\) 1.44554e6 0.000122809 0
\(755\) 1.80154e10 1.52346
\(756\) 0 0
\(757\) −1.98575e10 −1.66375 −0.831877 0.554960i \(-0.812734\pi\)
−0.831877 + 0.554960i \(0.812734\pi\)
\(758\) 1.24024e10 1.03434
\(759\) 0 0
\(760\) −1.98179e10 −1.63761
\(761\) −4.10456e9 −0.337614 −0.168807 0.985649i \(-0.553991\pi\)
−0.168807 + 0.985649i \(0.553991\pi\)
\(762\) 0 0
\(763\) 5.74753e9 0.468431
\(764\) −9.21432e9 −0.747544
\(765\) 0 0
\(766\) 1.18720e10 0.954384
\(767\) 4.12618e9 0.330190
\(768\) 0 0
\(769\) 1.69512e10 1.34418 0.672091 0.740468i \(-0.265396\pi\)
0.672091 + 0.740468i \(0.265396\pi\)
\(770\) 3.79121e8 0.0299268
\(771\) 0 0
\(772\) 7.81595e9 0.611394
\(773\) −7.71790e8 −0.0600995 −0.0300498 0.999548i \(-0.509567\pi\)
−0.0300498 + 0.999548i \(0.509567\pi\)
\(774\) 0 0
\(775\) −2.43722e8 −0.0188078
\(776\) 2.90478e9 0.223150
\(777\) 0 0
\(778\) −1.42041e10 −1.08140
\(779\) 2.20412e9 0.167053
\(780\) 0 0
\(781\) 8.68258e7 0.00652184
\(782\) −1.10492e10 −0.826240
\(783\) 0 0
\(784\) 2.29955e9 0.170426
\(785\) 1.79060e10 1.32116
\(786\) 0 0
\(787\) −1.55107e10 −1.13428 −0.567139 0.823622i \(-0.691950\pi\)
−0.567139 + 0.823622i \(0.691950\pi\)
\(788\) −1.20095e10 −0.874346
\(789\) 0 0
\(790\) 2.07224e10 1.49536
\(791\) 2.55169e9 0.183320
\(792\) 0 0
\(793\) 4.58275e8 0.0326340
\(794\) 1.65110e10 1.17058
\(795\) 0 0
\(796\) 8.76553e9 0.616003
\(797\) −1.44069e10 −1.00801 −0.504006 0.863700i \(-0.668141\pi\)
−0.504006 + 0.863700i \(0.668141\pi\)
\(798\) 0 0
\(799\) −1.99168e9 −0.138136
\(800\) 1.41190e10 0.974964
\(801\) 0 0
\(802\) 1.70757e10 1.16888
\(803\) 1.44122e9 0.0982256
\(804\) 0 0
\(805\) 1.48062e10 1.00036
\(806\) 4.33460e7 0.00291593
\(807\) 0 0
\(808\) −1.33974e10 −0.893469
\(809\) 1.32032e10 0.876714 0.438357 0.898801i \(-0.355560\pi\)
0.438357 + 0.898801i \(0.355560\pi\)
\(810\) 0 0
\(811\) 2.59038e10 1.70526 0.852629 0.522517i \(-0.175007\pi\)
0.852629 + 0.522517i \(0.175007\pi\)
\(812\) −2.32937e6 −0.000152684 0
\(813\) 0 0
\(814\) 7.40775e8 0.0481394
\(815\) −8.22971e9 −0.532516
\(816\) 0 0
\(817\) −2.57405e9 −0.165135
\(818\) −9.09516e9 −0.580997
\(819\) 0 0
\(820\) −1.83541e9 −0.116248
\(821\) 1.80408e9 0.113777 0.0568885 0.998381i \(-0.481882\pi\)
0.0568885 + 0.998381i \(0.481882\pi\)
\(822\) 0 0
\(823\) 2.15004e10 1.34445 0.672227 0.740345i \(-0.265338\pi\)
0.672227 + 0.740345i \(0.265338\pi\)
\(824\) −3.01566e10 −1.87775
\(825\) 0 0
\(826\) 6.04181e9 0.373024
\(827\) −1.95017e10 −1.19895 −0.599477 0.800392i \(-0.704625\pi\)
−0.599477 + 0.800392i \(0.704625\pi\)
\(828\) 0 0
\(829\) 1.50758e10 0.919049 0.459524 0.888165i \(-0.348020\pi\)
0.459524 + 0.888165i \(0.348020\pi\)
\(830\) −4.95307e9 −0.300678
\(831\) 0 0
\(832\) −3.31513e9 −0.199558
\(833\) −9.54619e9 −0.572233
\(834\) 0 0
\(835\) −4.01866e9 −0.238880
\(836\) 7.35206e8 0.0435198
\(837\) 0 0
\(838\) 7.49157e9 0.439763
\(839\) 3.01700e10 1.76363 0.881816 0.471594i \(-0.156321\pi\)
0.881816 + 0.471594i \(0.156321\pi\)
\(840\) 0 0
\(841\) −1.72499e10 −0.999999
\(842\) −7.99060e7 −0.00461304
\(843\) 0 0
\(844\) −2.03703e9 −0.116627
\(845\) −2.37709e10 −1.35534
\(846\) 0 0
\(847\) 6.91144e9 0.390820
\(848\) −1.48717e9 −0.0837480
\(849\) 0 0
\(850\) 8.93507e9 0.499036
\(851\) 2.89302e10 1.60915
\(852\) 0 0
\(853\) −1.64145e9 −0.0905534 −0.0452767 0.998974i \(-0.514417\pi\)
−0.0452767 + 0.998974i \(0.514417\pi\)
\(854\) 6.71035e8 0.0368674
\(855\) 0 0
\(856\) 1.81237e10 0.987619
\(857\) 2.57589e10 1.39796 0.698981 0.715141i \(-0.253637\pi\)
0.698981 + 0.715141i \(0.253637\pi\)
\(858\) 0 0
\(859\) −2.14742e10 −1.15595 −0.577977 0.816053i \(-0.696158\pi\)
−0.577977 + 0.816053i \(0.696158\pi\)
\(860\) 2.14345e9 0.114913
\(861\) 0 0
\(862\) 2.45278e10 1.30432
\(863\) 3.38483e9 0.179266 0.0896332 0.995975i \(-0.471431\pi\)
0.0896332 + 0.995975i \(0.471431\pi\)
\(864\) 0 0
\(865\) −3.67850e10 −1.93248
\(866\) −1.80609e10 −0.944990
\(867\) 0 0
\(868\) −6.98487e7 −0.00362526
\(869\) −2.23608e9 −0.115589
\(870\) 0 0
\(871\) −1.45146e9 −0.0744290
\(872\) 2.45310e10 1.25287
\(873\) 0 0
\(874\) −2.60905e10 −1.32188
\(875\) −7.68410e8 −0.0387762
\(876\) 0 0
\(877\) −2.37382e10 −1.18836 −0.594182 0.804331i \(-0.702524\pi\)
−0.594182 + 0.804331i \(0.702524\pi\)
\(878\) 1.28185e10 0.639157
\(879\) 0 0
\(880\) 4.49596e8 0.0222399
\(881\) 1.49518e10 0.736678 0.368339 0.929692i \(-0.379927\pi\)
0.368339 + 0.929692i \(0.379927\pi\)
\(882\) 0 0
\(883\) 2.73527e10 1.33702 0.668510 0.743703i \(-0.266932\pi\)
0.668510 + 0.743703i \(0.266932\pi\)
\(884\) 1.74881e9 0.0851453
\(885\) 0 0
\(886\) −5.56002e9 −0.268570
\(887\) −3.74606e10 −1.80236 −0.901181 0.433443i \(-0.857299\pi\)
−0.901181 + 0.433443i \(0.857299\pi\)
\(888\) 0 0
\(889\) 6.98849e9 0.333601
\(890\) −3.17201e10 −1.50823
\(891\) 0 0
\(892\) −5.82370e9 −0.274740
\(893\) −4.70297e9 −0.221000
\(894\) 0 0
\(895\) 3.00907e10 1.40298
\(896\) 2.86903e9 0.133247
\(897\) 0 0
\(898\) −1.68536e10 −0.776652
\(899\) 284234. 1.30472e−5 0
\(900\) 0 0
\(901\) 6.17372e9 0.281197
\(902\) −1.79965e8 −0.00816519
\(903\) 0 0
\(904\) 1.08908e10 0.490311
\(905\) −4.35819e9 −0.195450
\(906\) 0 0
\(907\) −4.76936e9 −0.212243 −0.106122 0.994353i \(-0.533843\pi\)
−0.106122 + 0.994353i \(0.533843\pi\)
\(908\) 1.12399e10 0.498265
\(909\) 0 0
\(910\) 2.12944e9 0.0936743
\(911\) 4.59972e9 0.201566 0.100783 0.994908i \(-0.467865\pi\)
0.100783 + 0.994908i \(0.467865\pi\)
\(912\) 0 0
\(913\) 5.34467e8 0.0232420
\(914\) −8.25540e9 −0.357624
\(915\) 0 0
\(916\) −7.75368e9 −0.333329
\(917\) 1.37167e10 0.587432
\(918\) 0 0
\(919\) 1.59138e10 0.676348 0.338174 0.941084i \(-0.390191\pi\)
0.338174 + 0.941084i \(0.390191\pi\)
\(920\) 6.31941e10 2.67559
\(921\) 0 0
\(922\) 2.29151e10 0.962862
\(923\) 4.87682e8 0.0204141
\(924\) 0 0
\(925\) −2.33948e10 −0.971903
\(926\) 2.18812e10 0.905592
\(927\) 0 0
\(928\) −1.64659e7 −0.000676344 0
\(929\) 2.22650e10 0.911102 0.455551 0.890210i \(-0.349442\pi\)
0.455551 + 0.890210i \(0.349442\pi\)
\(930\) 0 0
\(931\) −2.25415e10 −0.915502
\(932\) 6.89632e9 0.279037
\(933\) 0 0
\(934\) −1.05712e10 −0.424533
\(935\) −1.86642e9 −0.0746739
\(936\) 0 0
\(937\) −2.34525e10 −0.931322 −0.465661 0.884963i \(-0.654183\pi\)
−0.465661 + 0.884963i \(0.654183\pi\)
\(938\) −2.12532e9 −0.0840842
\(939\) 0 0
\(940\) 3.91625e9 0.153788
\(941\) −2.99250e10 −1.17077 −0.585383 0.810757i \(-0.699056\pi\)
−0.585383 + 0.810757i \(0.699056\pi\)
\(942\) 0 0
\(943\) −7.02836e9 −0.272937
\(944\) 7.16492e9 0.277210
\(945\) 0 0
\(946\) 2.10169e8 0.00807143
\(947\) 2.94339e10 1.12622 0.563109 0.826383i \(-0.309605\pi\)
0.563109 + 0.826383i \(0.309605\pi\)
\(948\) 0 0
\(949\) 8.09500e9 0.307457
\(950\) 2.10985e10 0.798396
\(951\) 0 0
\(952\) 7.44830e9 0.279787
\(953\) −3.72084e10 −1.39257 −0.696283 0.717767i \(-0.745164\pi\)
−0.696283 + 0.717767i \(0.745164\pi\)
\(954\) 0 0
\(955\) 5.52353e10 2.05213
\(956\) 1.71605e10 0.635226
\(957\) 0 0
\(958\) −9.18488e9 −0.337516
\(959\) 1.91175e10 0.699949
\(960\) 0 0
\(961\) −2.75041e10 −0.999690
\(962\) 4.16077e9 0.150682
\(963\) 0 0
\(964\) 1.31150e8 0.00471518
\(965\) −4.68528e10 −1.67838
\(966\) 0 0
\(967\) −1.49635e10 −0.532159 −0.266080 0.963951i \(-0.585728\pi\)
−0.266080 + 0.963951i \(0.585728\pi\)
\(968\) 2.94987e10 1.04529
\(969\) 0 0
\(970\) −5.98647e9 −0.210605
\(971\) −3.15367e10 −1.10548 −0.552738 0.833355i \(-0.686417\pi\)
−0.552738 + 0.833355i \(0.686417\pi\)
\(972\) 0 0
\(973\) 8.72174e9 0.303535
\(974\) −1.35402e10 −0.469536
\(975\) 0 0
\(976\) 7.95773e8 0.0273978
\(977\) −1.36797e10 −0.469295 −0.234648 0.972080i \(-0.575394\pi\)
−0.234648 + 0.972080i \(0.575394\pi\)
\(978\) 0 0
\(979\) 3.42279e9 0.116584
\(980\) 1.87707e10 0.637073
\(981\) 0 0
\(982\) −6.20433e9 −0.209076
\(983\) −1.28389e10 −0.431112 −0.215556 0.976491i \(-0.569156\pi\)
−0.215556 + 0.976491i \(0.569156\pi\)
\(984\) 0 0
\(985\) 7.19910e10 2.40022
\(986\) −1.04203e7 −0.000346187 0
\(987\) 0 0
\(988\) 4.12949e9 0.136222
\(989\) 8.20794e9 0.269803
\(990\) 0 0
\(991\) −5.51139e10 −1.79889 −0.899443 0.437038i \(-0.856028\pi\)
−0.899443 + 0.437038i \(0.856028\pi\)
\(992\) −4.93748e8 −0.0160588
\(993\) 0 0
\(994\) 7.14093e8 0.0230623
\(995\) −5.25451e10 −1.69103
\(996\) 0 0
\(997\) 5.87820e10 1.87850 0.939250 0.343234i \(-0.111522\pi\)
0.939250 + 0.343234i \(0.111522\pi\)
\(998\) −2.13495e9 −0.0679878
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 387.8.a.d.1.10 13
3.2 odd 2 43.8.a.b.1.4 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.4 13 3.2 odd 2
387.8.a.d.1.10 13 1.1 even 1 trivial