Properties

Label 387.8.a.d.1.13
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 1.13
Root \(-21.3781\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q+20.3781 q^{2} +287.267 q^{4} -405.085 q^{5} +52.5965 q^{7} +3245.57 q^{8} +O(q^{10})\) \(q+20.3781 q^{2} +287.267 q^{4} -405.085 q^{5} +52.5965 q^{7} +3245.57 q^{8} -8254.88 q^{10} -2544.97 q^{11} +7138.95 q^{13} +1071.82 q^{14} +29368.3 q^{16} +22507.9 q^{17} -6396.31 q^{19} -116368. q^{20} -51861.7 q^{22} -99842.9 q^{23} +85969.2 q^{25} +145478. q^{26} +15109.2 q^{28} -5608.53 q^{29} -163095. q^{31} +183038. q^{32} +458669. q^{34} -21306.1 q^{35} +198188. q^{37} -130345. q^{38} -1.31473e6 q^{40} -742519. q^{41} -79507.0 q^{43} -731087. q^{44} -2.03461e6 q^{46} -786601. q^{47} -820777. q^{49} +1.75189e6 q^{50} +2.05079e6 q^{52} -2.08070e6 q^{53} +1.03093e6 q^{55} +170705. q^{56} -114291. q^{58} -607486. q^{59} +2.59833e6 q^{61} -3.32357e6 q^{62} -29175.0 q^{64} -2.89188e6 q^{65} -755375. q^{67} +6.46580e6 q^{68} -434177. q^{70} +383737. q^{71} +2.94854e6 q^{73} +4.03869e6 q^{74} -1.83745e6 q^{76} -133856. q^{77} -7.28812e6 q^{79} -1.18967e7 q^{80} -1.51311e7 q^{82} +1.91613e6 q^{83} -9.11764e6 q^{85} -1.62020e6 q^{86} -8.25988e6 q^{88} -1.05744e7 q^{89} +375483. q^{91} -2.86816e7 q^{92} -1.60294e7 q^{94} +2.59105e6 q^{95} +1.18419e7 q^{97} -1.67259e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.3781 1.80119 0.900594 0.434662i \(-0.143132\pi\)
0.900594 + 0.434662i \(0.143132\pi\)
\(3\) 0 0
\(4\) 287.267 2.24428
\(5\) −405.085 −1.44928 −0.724639 0.689129i \(-0.757993\pi\)
−0.724639 + 0.689129i \(0.757993\pi\)
\(6\) 0 0
\(7\) 52.5965 0.0579580 0.0289790 0.999580i \(-0.490774\pi\)
0.0289790 + 0.999580i \(0.490774\pi\)
\(8\) 3245.57 2.24117
\(9\) 0 0
\(10\) −8254.88 −2.61042
\(11\) −2544.97 −0.576512 −0.288256 0.957553i \(-0.593075\pi\)
−0.288256 + 0.957553i \(0.593075\pi\)
\(12\) 0 0
\(13\) 7138.95 0.901224 0.450612 0.892720i \(-0.351206\pi\)
0.450612 + 0.892720i \(0.351206\pi\)
\(14\) 1071.82 0.104393
\(15\) 0 0
\(16\) 29368.3 1.79250
\(17\) 22507.9 1.11113 0.555565 0.831473i \(-0.312502\pi\)
0.555565 + 0.831473i \(0.312502\pi\)
\(18\) 0 0
\(19\) −6396.31 −0.213940 −0.106970 0.994262i \(-0.534115\pi\)
−0.106970 + 0.994262i \(0.534115\pi\)
\(20\) −116368. −3.25258
\(21\) 0 0
\(22\) −51861.7 −1.03841
\(23\) −99842.9 −1.71108 −0.855539 0.517739i \(-0.826774\pi\)
−0.855539 + 0.517739i \(0.826774\pi\)
\(24\) 0 0
\(25\) 85969.2 1.10041
\(26\) 145478. 1.62327
\(27\) 0 0
\(28\) 15109.2 0.130074
\(29\) −5608.53 −0.0427028 −0.0213514 0.999772i \(-0.506797\pi\)
−0.0213514 + 0.999772i \(0.506797\pi\)
\(30\) 0 0
\(31\) −163095. −0.983274 −0.491637 0.870800i \(-0.663601\pi\)
−0.491637 + 0.870800i \(0.663601\pi\)
\(32\) 183038. 0.987453
\(33\) 0 0
\(34\) 458669. 2.00135
\(35\) −21306.1 −0.0839972
\(36\) 0 0
\(37\) 198188. 0.643236 0.321618 0.946869i \(-0.395773\pi\)
0.321618 + 0.946869i \(0.395773\pi\)
\(38\) −130345. −0.385346
\(39\) 0 0
\(40\) −1.31473e6 −3.24808
\(41\) −742519. −1.68254 −0.841268 0.540619i \(-0.818190\pi\)
−0.841268 + 0.540619i \(0.818190\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) −731087. −1.29385
\(45\) 0 0
\(46\) −2.03461e6 −3.08197
\(47\) −786601. −1.10513 −0.552563 0.833471i \(-0.686350\pi\)
−0.552563 + 0.833471i \(0.686350\pi\)
\(48\) 0 0
\(49\) −820777. −0.996641
\(50\) 1.75189e6 1.98204
\(51\) 0 0
\(52\) 2.05079e6 2.02259
\(53\) −2.08070e6 −1.91975 −0.959875 0.280427i \(-0.909524\pi\)
−0.959875 + 0.280427i \(0.909524\pi\)
\(54\) 0 0
\(55\) 1.03093e6 0.835526
\(56\) 170705. 0.129894
\(57\) 0 0
\(58\) −114291. −0.0769157
\(59\) −607486. −0.385083 −0.192542 0.981289i \(-0.561673\pi\)
−0.192542 + 0.981289i \(0.561673\pi\)
\(60\) 0 0
\(61\) 2.59833e6 1.46568 0.732841 0.680400i \(-0.238194\pi\)
0.732841 + 0.680400i \(0.238194\pi\)
\(62\) −3.32357e6 −1.77106
\(63\) 0 0
\(64\) −29175.0 −0.0139117
\(65\) −2.89188e6 −1.30612
\(66\) 0 0
\(67\) −755375. −0.306832 −0.153416 0.988162i \(-0.549027\pi\)
−0.153416 + 0.988162i \(0.549027\pi\)
\(68\) 6.46580e6 2.49368
\(69\) 0 0
\(70\) −434177. −0.151295
\(71\) 383737. 0.127242 0.0636209 0.997974i \(-0.479735\pi\)
0.0636209 + 0.997974i \(0.479735\pi\)
\(72\) 0 0
\(73\) 2.94854e6 0.887109 0.443555 0.896247i \(-0.353717\pi\)
0.443555 + 0.896247i \(0.353717\pi\)
\(74\) 4.03869e6 1.15859
\(75\) 0 0
\(76\) −1.83745e6 −0.480141
\(77\) −133856. −0.0334135
\(78\) 0 0
\(79\) −7.28812e6 −1.66311 −0.831555 0.555443i \(-0.812549\pi\)
−0.831555 + 0.555443i \(0.812549\pi\)
\(80\) −1.18967e7 −2.59783
\(81\) 0 0
\(82\) −1.51311e7 −3.03056
\(83\) 1.91613e6 0.367834 0.183917 0.982942i \(-0.441122\pi\)
0.183917 + 0.982942i \(0.441122\pi\)
\(84\) 0 0
\(85\) −9.11764e6 −1.61034
\(86\) −1.62020e6 −0.274679
\(87\) 0 0
\(88\) −8.25988e6 −1.29206
\(89\) −1.05744e7 −1.58997 −0.794986 0.606627i \(-0.792522\pi\)
−0.794986 + 0.606627i \(0.792522\pi\)
\(90\) 0 0
\(91\) 375483. 0.0522331
\(92\) −2.86816e7 −3.84013
\(93\) 0 0
\(94\) −1.60294e7 −1.99054
\(95\) 2.59105e6 0.310059
\(96\) 0 0
\(97\) 1.18419e7 1.31741 0.658705 0.752402i \(-0.271105\pi\)
0.658705 + 0.752402i \(0.271105\pi\)
\(98\) −1.67259e7 −1.79514
\(99\) 0 0
\(100\) 2.46961e7 2.46961
\(101\) 725420. 0.0700591 0.0350296 0.999386i \(-0.488847\pi\)
0.0350296 + 0.999386i \(0.488847\pi\)
\(102\) 0 0
\(103\) −3.68761e6 −0.332518 −0.166259 0.986082i \(-0.553169\pi\)
−0.166259 + 0.986082i \(0.553169\pi\)
\(104\) 2.31699e7 2.01980
\(105\) 0 0
\(106\) −4.24008e7 −3.45783
\(107\) 2.07350e7 1.63629 0.818145 0.575012i \(-0.195003\pi\)
0.818145 + 0.575012i \(0.195003\pi\)
\(108\) 0 0
\(109\) −3.76731e6 −0.278637 −0.139319 0.990248i \(-0.544491\pi\)
−0.139319 + 0.990248i \(0.544491\pi\)
\(110\) 2.10084e7 1.50494
\(111\) 0 0
\(112\) 1.54467e6 0.103890
\(113\) −2.24379e6 −0.146287 −0.0731437 0.997321i \(-0.523303\pi\)
−0.0731437 + 0.997321i \(0.523303\pi\)
\(114\) 0 0
\(115\) 4.04449e7 2.47983
\(116\) −1.61115e6 −0.0958368
\(117\) 0 0
\(118\) −1.23794e7 −0.693607
\(119\) 1.18384e6 0.0643988
\(120\) 0 0
\(121\) −1.30103e7 −0.667634
\(122\) 5.29491e7 2.63997
\(123\) 0 0
\(124\) −4.68519e7 −2.20674
\(125\) −3.17758e6 −0.145516
\(126\) 0 0
\(127\) 3.84940e7 1.66755 0.833777 0.552101i \(-0.186174\pi\)
0.833777 + 0.552101i \(0.186174\pi\)
\(128\) −2.40234e7 −1.01251
\(129\) 0 0
\(130\) −5.89311e7 −2.35257
\(131\) 1.04514e6 0.0406188 0.0203094 0.999794i \(-0.493535\pi\)
0.0203094 + 0.999794i \(0.493535\pi\)
\(132\) 0 0
\(133\) −336423. −0.0123995
\(134\) −1.53931e7 −0.552662
\(135\) 0 0
\(136\) 7.30511e7 2.49024
\(137\) −3.36576e7 −1.11831 −0.559154 0.829064i \(-0.688874\pi\)
−0.559154 + 0.829064i \(0.688874\pi\)
\(138\) 0 0
\(139\) −3.30283e7 −1.04312 −0.521559 0.853215i \(-0.674650\pi\)
−0.521559 + 0.853215i \(0.674650\pi\)
\(140\) −6.12054e6 −0.188513
\(141\) 0 0
\(142\) 7.81984e6 0.229186
\(143\) −1.81684e7 −0.519566
\(144\) 0 0
\(145\) 2.27193e6 0.0618881
\(146\) 6.00857e7 1.59785
\(147\) 0 0
\(148\) 5.69328e7 1.44360
\(149\) −3.23900e6 −0.0802157 −0.0401079 0.999195i \(-0.512770\pi\)
−0.0401079 + 0.999195i \(0.512770\pi\)
\(150\) 0 0
\(151\) 6.37837e7 1.50761 0.753807 0.657096i \(-0.228215\pi\)
0.753807 + 0.657096i \(0.228215\pi\)
\(152\) −2.07597e7 −0.479477
\(153\) 0 0
\(154\) −2.72774e6 −0.0601840
\(155\) 6.60674e7 1.42504
\(156\) 0 0
\(157\) −5.75867e7 −1.18761 −0.593804 0.804610i \(-0.702375\pi\)
−0.593804 + 0.804610i \(0.702375\pi\)
\(158\) −1.48518e8 −2.99557
\(159\) 0 0
\(160\) −7.41460e7 −1.43109
\(161\) −5.25138e6 −0.0991706
\(162\) 0 0
\(163\) 9.59822e7 1.73594 0.867969 0.496619i \(-0.165425\pi\)
0.867969 + 0.496619i \(0.165425\pi\)
\(164\) −2.13302e8 −3.77607
\(165\) 0 0
\(166\) 3.90471e7 0.662538
\(167\) 5.45343e7 0.906070 0.453035 0.891493i \(-0.350341\pi\)
0.453035 + 0.891493i \(0.350341\pi\)
\(168\) 0 0
\(169\) −1.17839e7 −0.187796
\(170\) −1.85800e8 −2.90052
\(171\) 0 0
\(172\) −2.28398e7 −0.342249
\(173\) 5.48114e6 0.0804841 0.0402420 0.999190i \(-0.487187\pi\)
0.0402420 + 0.999190i \(0.487187\pi\)
\(174\) 0 0
\(175\) 4.52168e6 0.0637773
\(176\) −7.47415e7 −1.03340
\(177\) 0 0
\(178\) −2.15486e8 −2.86384
\(179\) 7.23802e7 0.943266 0.471633 0.881795i \(-0.343665\pi\)
0.471633 + 0.881795i \(0.343665\pi\)
\(180\) 0 0
\(181\) −7.78319e7 −0.975624 −0.487812 0.872949i \(-0.662205\pi\)
−0.487812 + 0.872949i \(0.662205\pi\)
\(182\) 7.65164e6 0.0940816
\(183\) 0 0
\(184\) −3.24047e8 −3.83482
\(185\) −8.02829e7 −0.932228
\(186\) 0 0
\(187\) −5.72821e7 −0.640580
\(188\) −2.25965e8 −2.48021
\(189\) 0 0
\(190\) 5.28008e7 0.558474
\(191\) −9.82338e6 −0.102010 −0.0510052 0.998698i \(-0.516243\pi\)
−0.0510052 + 0.998698i \(0.516243\pi\)
\(192\) 0 0
\(193\) 7.35110e7 0.736040 0.368020 0.929818i \(-0.380036\pi\)
0.368020 + 0.929818i \(0.380036\pi\)
\(194\) 2.41316e8 2.37290
\(195\) 0 0
\(196\) −2.35782e8 −2.23674
\(197\) 3.73612e7 0.348168 0.174084 0.984731i \(-0.444304\pi\)
0.174084 + 0.984731i \(0.444304\pi\)
\(198\) 0 0
\(199\) 2.37151e7 0.213323 0.106662 0.994295i \(-0.465984\pi\)
0.106662 + 0.994295i \(0.465984\pi\)
\(200\) 2.79019e8 2.46620
\(201\) 0 0
\(202\) 1.47827e7 0.126190
\(203\) −294989. −0.00247497
\(204\) 0 0
\(205\) 3.00784e8 2.43846
\(206\) −7.51466e7 −0.598927
\(207\) 0 0
\(208\) 2.09659e8 1.61544
\(209\) 1.62784e7 0.123339
\(210\) 0 0
\(211\) 9.22491e7 0.676042 0.338021 0.941139i \(-0.390242\pi\)
0.338021 + 0.941139i \(0.390242\pi\)
\(212\) −5.97719e8 −4.30845
\(213\) 0 0
\(214\) 4.22540e8 2.94727
\(215\) 3.22071e7 0.221013
\(216\) 0 0
\(217\) −8.57822e6 −0.0569886
\(218\) −7.67707e7 −0.501877
\(219\) 0 0
\(220\) 2.96153e8 1.87515
\(221\) 1.60683e8 1.00138
\(222\) 0 0
\(223\) −2.83890e8 −1.71428 −0.857141 0.515082i \(-0.827762\pi\)
−0.857141 + 0.515082i \(0.827762\pi\)
\(224\) 9.62715e6 0.0572308
\(225\) 0 0
\(226\) −4.57241e7 −0.263491
\(227\) −4.18699e7 −0.237581 −0.118790 0.992919i \(-0.537902\pi\)
−0.118790 + 0.992919i \(0.537902\pi\)
\(228\) 0 0
\(229\) −8.20861e7 −0.451695 −0.225848 0.974163i \(-0.572515\pi\)
−0.225848 + 0.974163i \(0.572515\pi\)
\(230\) 8.24191e8 4.46663
\(231\) 0 0
\(232\) −1.82029e7 −0.0957043
\(233\) 1.98224e8 1.02662 0.513312 0.858202i \(-0.328418\pi\)
0.513312 + 0.858202i \(0.328418\pi\)
\(234\) 0 0
\(235\) 3.18640e8 1.60163
\(236\) −1.74511e8 −0.864233
\(237\) 0 0
\(238\) 2.41244e7 0.115994
\(239\) 4.10134e8 1.94327 0.971635 0.236487i \(-0.0759962\pi\)
0.971635 + 0.236487i \(0.0759962\pi\)
\(240\) 0 0
\(241\) −2.36729e8 −1.08941 −0.544704 0.838628i \(-0.683358\pi\)
−0.544704 + 0.838628i \(0.683358\pi\)
\(242\) −2.65125e8 −1.20253
\(243\) 0 0
\(244\) 7.46415e8 3.28940
\(245\) 3.32485e8 1.44441
\(246\) 0 0
\(247\) −4.56629e7 −0.192808
\(248\) −5.29336e8 −2.20369
\(249\) 0 0
\(250\) −6.47530e7 −0.262102
\(251\) 1.19914e8 0.478644 0.239322 0.970940i \(-0.423075\pi\)
0.239322 + 0.970940i \(0.423075\pi\)
\(252\) 0 0
\(253\) 2.54097e8 0.986457
\(254\) 7.84435e8 3.00358
\(255\) 0 0
\(256\) −4.85817e8 −1.80981
\(257\) −4.20140e8 −1.54393 −0.771965 0.635664i \(-0.780726\pi\)
−0.771965 + 0.635664i \(0.780726\pi\)
\(258\) 0 0
\(259\) 1.04240e7 0.0372807
\(260\) −8.30744e8 −2.93130
\(261\) 0 0
\(262\) 2.12981e7 0.0731620
\(263\) 2.09315e7 0.0709506 0.0354753 0.999371i \(-0.488705\pi\)
0.0354753 + 0.999371i \(0.488705\pi\)
\(264\) 0 0
\(265\) 8.42863e8 2.78225
\(266\) −6.85567e6 −0.0223339
\(267\) 0 0
\(268\) −2.16995e8 −0.688616
\(269\) −4.20833e8 −1.31819 −0.659093 0.752062i \(-0.729060\pi\)
−0.659093 + 0.752062i \(0.729060\pi\)
\(270\) 0 0
\(271\) −1.41459e8 −0.431757 −0.215878 0.976420i \(-0.569261\pi\)
−0.215878 + 0.976420i \(0.569261\pi\)
\(272\) 6.61020e8 1.99170
\(273\) 0 0
\(274\) −6.85879e8 −2.01428
\(275\) −2.18789e8 −0.634398
\(276\) 0 0
\(277\) −4.46880e8 −1.26332 −0.631658 0.775248i \(-0.717625\pi\)
−0.631658 + 0.775248i \(0.717625\pi\)
\(278\) −6.73053e8 −1.87885
\(279\) 0 0
\(280\) −6.91503e7 −0.188252
\(281\) −4.25457e8 −1.14389 −0.571944 0.820293i \(-0.693811\pi\)
−0.571944 + 0.820293i \(0.693811\pi\)
\(282\) 0 0
\(283\) −3.36158e7 −0.0881638 −0.0440819 0.999028i \(-0.514036\pi\)
−0.0440819 + 0.999028i \(0.514036\pi\)
\(284\) 1.10235e8 0.285566
\(285\) 0 0
\(286\) −3.70238e8 −0.935837
\(287\) −3.90539e7 −0.0975164
\(288\) 0 0
\(289\) 9.62690e7 0.234609
\(290\) 4.62977e7 0.111472
\(291\) 0 0
\(292\) 8.47019e8 1.99092
\(293\) 4.64940e8 1.07984 0.539921 0.841715i \(-0.318454\pi\)
0.539921 + 0.841715i \(0.318454\pi\)
\(294\) 0 0
\(295\) 2.46084e8 0.558092
\(296\) 6.43231e8 1.44160
\(297\) 0 0
\(298\) −6.60048e7 −0.144484
\(299\) −7.12773e8 −1.54206
\(300\) 0 0
\(301\) −4.18179e6 −0.00883851
\(302\) 1.29979e9 2.71550
\(303\) 0 0
\(304\) −1.87849e8 −0.383487
\(305\) −1.05255e9 −2.12418
\(306\) 0 0
\(307\) −9.90322e8 −1.95340 −0.976702 0.214600i \(-0.931155\pi\)
−0.976702 + 0.214600i \(0.931155\pi\)
\(308\) −3.84526e7 −0.0749891
\(309\) 0 0
\(310\) 1.34633e9 2.56676
\(311\) 1.57593e8 0.297081 0.148540 0.988906i \(-0.452543\pi\)
0.148540 + 0.988906i \(0.452543\pi\)
\(312\) 0 0
\(313\) 8.72665e8 1.60858 0.804290 0.594237i \(-0.202546\pi\)
0.804290 + 0.594237i \(0.202546\pi\)
\(314\) −1.17351e9 −2.13911
\(315\) 0 0
\(316\) −2.09364e9 −3.73248
\(317\) −1.63595e8 −0.288444 −0.144222 0.989545i \(-0.546068\pi\)
−0.144222 + 0.989545i \(0.546068\pi\)
\(318\) 0 0
\(319\) 1.42735e7 0.0246187
\(320\) 1.18184e7 0.0201620
\(321\) 0 0
\(322\) −1.07013e8 −0.178625
\(323\) −1.43968e8 −0.237715
\(324\) 0 0
\(325\) 6.13730e8 0.991712
\(326\) 1.95594e9 3.12675
\(327\) 0 0
\(328\) −2.40990e9 −3.77086
\(329\) −4.13724e7 −0.0640509
\(330\) 0 0
\(331\) 6.41815e8 0.972773 0.486387 0.873744i \(-0.338315\pi\)
0.486387 + 0.873744i \(0.338315\pi\)
\(332\) 5.50442e8 0.825521
\(333\) 0 0
\(334\) 1.11131e9 1.63200
\(335\) 3.05991e8 0.444685
\(336\) 0 0
\(337\) −2.16017e8 −0.307457 −0.153728 0.988113i \(-0.549128\pi\)
−0.153728 + 0.988113i \(0.549128\pi\)
\(338\) −2.40134e8 −0.338256
\(339\) 0 0
\(340\) −2.61920e9 −3.61404
\(341\) 4.15072e8 0.566870
\(342\) 0 0
\(343\) −8.64854e7 −0.115721
\(344\) −2.58045e8 −0.341776
\(345\) 0 0
\(346\) 1.11695e8 0.144967
\(347\) 1.18322e9 1.52024 0.760118 0.649785i \(-0.225141\pi\)
0.760118 + 0.649785i \(0.225141\pi\)
\(348\) 0 0
\(349\) −4.71635e8 −0.593905 −0.296953 0.954892i \(-0.595970\pi\)
−0.296953 + 0.954892i \(0.595970\pi\)
\(350\) 9.21432e7 0.114875
\(351\) 0 0
\(352\) −4.65827e8 −0.569279
\(353\) 2.14674e8 0.259757 0.129879 0.991530i \(-0.458541\pi\)
0.129879 + 0.991530i \(0.458541\pi\)
\(354\) 0 0
\(355\) −1.55446e8 −0.184409
\(356\) −3.03767e9 −3.56834
\(357\) 0 0
\(358\) 1.47497e9 1.69900
\(359\) −5.07607e8 −0.579024 −0.289512 0.957174i \(-0.593493\pi\)
−0.289512 + 0.957174i \(0.593493\pi\)
\(360\) 0 0
\(361\) −8.52959e8 −0.954230
\(362\) −1.58607e9 −1.75728
\(363\) 0 0
\(364\) 1.07864e8 0.117226
\(365\) −1.19441e9 −1.28567
\(366\) 0 0
\(367\) −7.17068e8 −0.757232 −0.378616 0.925554i \(-0.623600\pi\)
−0.378616 + 0.925554i \(0.623600\pi\)
\(368\) −2.93222e9 −3.06711
\(369\) 0 0
\(370\) −1.63601e9 −1.67912
\(371\) −1.09438e8 −0.111265
\(372\) 0 0
\(373\) 1.47530e9 1.47197 0.735985 0.676998i \(-0.236720\pi\)
0.735985 + 0.676998i \(0.236720\pi\)
\(374\) −1.16730e9 −1.15380
\(375\) 0 0
\(376\) −2.55297e9 −2.47678
\(377\) −4.00390e7 −0.0384847
\(378\) 0 0
\(379\) 1.57868e9 1.48956 0.744778 0.667313i \(-0.232556\pi\)
0.744778 + 0.667313i \(0.232556\pi\)
\(380\) 7.44325e8 0.695857
\(381\) 0 0
\(382\) −2.00182e8 −0.183740
\(383\) −6.74713e8 −0.613653 −0.306827 0.951765i \(-0.599267\pi\)
−0.306827 + 0.951765i \(0.599267\pi\)
\(384\) 0 0
\(385\) 5.42233e7 0.0484254
\(386\) 1.49802e9 1.32575
\(387\) 0 0
\(388\) 3.40179e9 2.95663
\(389\) 5.60683e8 0.482941 0.241470 0.970408i \(-0.422370\pi\)
0.241470 + 0.970408i \(0.422370\pi\)
\(390\) 0 0
\(391\) −2.24726e9 −1.90123
\(392\) −2.66389e9 −2.23365
\(393\) 0 0
\(394\) 7.61350e8 0.627116
\(395\) 2.95231e9 2.41031
\(396\) 0 0
\(397\) 3.50283e8 0.280965 0.140482 0.990083i \(-0.455135\pi\)
0.140482 + 0.990083i \(0.455135\pi\)
\(398\) 4.83268e8 0.384236
\(399\) 0 0
\(400\) 2.52477e9 1.97248
\(401\) −1.19625e9 −0.926440 −0.463220 0.886243i \(-0.653306\pi\)
−0.463220 + 0.886243i \(0.653306\pi\)
\(402\) 0 0
\(403\) −1.16433e9 −0.886150
\(404\) 2.08389e8 0.157232
\(405\) 0 0
\(406\) −6.01131e6 −0.00445788
\(407\) −5.04382e8 −0.370834
\(408\) 0 0
\(409\) 1.57416e9 1.13767 0.568836 0.822451i \(-0.307394\pi\)
0.568836 + 0.822451i \(0.307394\pi\)
\(410\) 6.12940e9 4.39213
\(411\) 0 0
\(412\) −1.05933e9 −0.746262
\(413\) −3.19516e7 −0.0223186
\(414\) 0 0
\(415\) −7.76197e8 −0.533094
\(416\) 1.30670e9 0.889916
\(417\) 0 0
\(418\) 3.31724e8 0.222157
\(419\) −2.03488e9 −1.35142 −0.675710 0.737167i \(-0.736163\pi\)
−0.675710 + 0.737167i \(0.736163\pi\)
\(420\) 0 0
\(421\) −1.11312e9 −0.727032 −0.363516 0.931588i \(-0.618424\pi\)
−0.363516 + 0.931588i \(0.618424\pi\)
\(422\) 1.87986e9 1.21768
\(423\) 0 0
\(424\) −6.75307e9 −4.30250
\(425\) 1.93499e9 1.22269
\(426\) 0 0
\(427\) 1.36663e8 0.0849480
\(428\) 5.95648e9 3.67229
\(429\) 0 0
\(430\) 6.56320e8 0.398085
\(431\) 2.21785e9 1.33432 0.667161 0.744913i \(-0.267509\pi\)
0.667161 + 0.744913i \(0.267509\pi\)
\(432\) 0 0
\(433\) 1.61665e9 0.956992 0.478496 0.878090i \(-0.341182\pi\)
0.478496 + 0.878090i \(0.341182\pi\)
\(434\) −1.74808e8 −0.102647
\(435\) 0 0
\(436\) −1.08223e9 −0.625338
\(437\) 6.38626e8 0.366068
\(438\) 0 0
\(439\) −2.27405e9 −1.28285 −0.641424 0.767187i \(-0.721656\pi\)
−0.641424 + 0.767187i \(0.721656\pi\)
\(440\) 3.34596e9 1.87256
\(441\) 0 0
\(442\) 3.27442e9 1.80367
\(443\) 1.06718e9 0.583209 0.291605 0.956539i \(-0.405811\pi\)
0.291605 + 0.956539i \(0.405811\pi\)
\(444\) 0 0
\(445\) 4.28353e9 2.30431
\(446\) −5.78513e9 −3.08774
\(447\) 0 0
\(448\) −1.53450e6 −0.000806296 0
\(449\) 1.72761e9 0.900707 0.450354 0.892850i \(-0.351298\pi\)
0.450354 + 0.892850i \(0.351298\pi\)
\(450\) 0 0
\(451\) 1.88969e9 0.970002
\(452\) −6.44567e8 −0.328309
\(453\) 0 0
\(454\) −8.53229e8 −0.427927
\(455\) −1.52103e8 −0.0757003
\(456\) 0 0
\(457\) 3.85685e8 0.189028 0.0945139 0.995524i \(-0.469870\pi\)
0.0945139 + 0.995524i \(0.469870\pi\)
\(458\) −1.67276e9 −0.813587
\(459\) 0 0
\(460\) 1.16185e10 5.56542
\(461\) 2.53341e9 1.20435 0.602174 0.798365i \(-0.294301\pi\)
0.602174 + 0.798365i \(0.294301\pi\)
\(462\) 0 0
\(463\) −5.21077e8 −0.243988 −0.121994 0.992531i \(-0.538929\pi\)
−0.121994 + 0.992531i \(0.538929\pi\)
\(464\) −1.64713e8 −0.0765447
\(465\) 0 0
\(466\) 4.03944e9 1.84914
\(467\) 1.97610e9 0.897842 0.448921 0.893571i \(-0.351808\pi\)
0.448921 + 0.893571i \(0.351808\pi\)
\(468\) 0 0
\(469\) −3.97300e7 −0.0177834
\(470\) 6.49329e9 2.88484
\(471\) 0 0
\(472\) −1.97164e9 −0.863039
\(473\) 2.02343e8 0.0879173
\(474\) 0 0
\(475\) −5.49886e8 −0.235421
\(476\) 3.40078e8 0.144529
\(477\) 0 0
\(478\) 8.35775e9 3.50019
\(479\) 1.86520e9 0.775444 0.387722 0.921776i \(-0.373262\pi\)
0.387722 + 0.921776i \(0.373262\pi\)
\(480\) 0 0
\(481\) 1.41485e9 0.579700
\(482\) −4.82408e9 −1.96223
\(483\) 0 0
\(484\) −3.73743e9 −1.49835
\(485\) −4.79699e9 −1.90929
\(486\) 0 0
\(487\) 2.65232e9 1.04058 0.520288 0.853991i \(-0.325824\pi\)
0.520288 + 0.853991i \(0.325824\pi\)
\(488\) 8.43306e9 3.28485
\(489\) 0 0
\(490\) 6.77541e9 2.60165
\(491\) −3.14043e9 −1.19730 −0.598651 0.801010i \(-0.704297\pi\)
−0.598651 + 0.801010i \(0.704297\pi\)
\(492\) 0 0
\(493\) −1.26236e8 −0.0474483
\(494\) −9.30525e8 −0.347283
\(495\) 0 0
\(496\) −4.78982e9 −1.76252
\(497\) 2.01832e7 0.00737468
\(498\) 0 0
\(499\) −1.74081e8 −0.0627191 −0.0313596 0.999508i \(-0.509984\pi\)
−0.0313596 + 0.999508i \(0.509984\pi\)
\(500\) −9.12814e8 −0.326578
\(501\) 0 0
\(502\) 2.44363e9 0.862128
\(503\) −7.08807e8 −0.248336 −0.124168 0.992261i \(-0.539626\pi\)
−0.124168 + 0.992261i \(0.539626\pi\)
\(504\) 0 0
\(505\) −2.93857e8 −0.101535
\(506\) 5.17802e9 1.77679
\(507\) 0 0
\(508\) 1.10581e10 3.74245
\(509\) −5.15128e9 −1.73142 −0.865711 0.500545i \(-0.833133\pi\)
−0.865711 + 0.500545i \(0.833133\pi\)
\(510\) 0 0
\(511\) 1.55083e8 0.0514151
\(512\) −6.82504e9 −2.24730
\(513\) 0 0
\(514\) −8.56165e9 −2.78091
\(515\) 1.49380e9 0.481911
\(516\) 0 0
\(517\) 2.00188e9 0.637119
\(518\) 2.12421e8 0.0671495
\(519\) 0 0
\(520\) −9.38581e9 −2.92725
\(521\) 3.29977e9 1.02224 0.511118 0.859510i \(-0.329231\pi\)
0.511118 + 0.859510i \(0.329231\pi\)
\(522\) 0 0
\(523\) 4.25818e9 1.30157 0.650786 0.759261i \(-0.274440\pi\)
0.650786 + 0.759261i \(0.274440\pi\)
\(524\) 3.00236e8 0.0911598
\(525\) 0 0
\(526\) 4.26545e8 0.127795
\(527\) −3.67093e9 −1.09255
\(528\) 0 0
\(529\) 6.56378e9 1.92779
\(530\) 1.71760e10 5.01136
\(531\) 0 0
\(532\) −9.66435e7 −0.0278280
\(533\) −5.30081e9 −1.51634
\(534\) 0 0
\(535\) −8.39944e9 −2.37144
\(536\) −2.45162e9 −0.687664
\(537\) 0 0
\(538\) −8.57578e9 −2.37430
\(539\) 2.08885e9 0.574576
\(540\) 0 0
\(541\) −7.50896e8 −0.203887 −0.101943 0.994790i \(-0.532506\pi\)
−0.101943 + 0.994790i \(0.532506\pi\)
\(542\) −2.88268e9 −0.777675
\(543\) 0 0
\(544\) 4.11981e9 1.09719
\(545\) 1.52608e9 0.403822
\(546\) 0 0
\(547\) −6.03569e7 −0.0157678 −0.00788390 0.999969i \(-0.502510\pi\)
−0.00788390 + 0.999969i \(0.502510\pi\)
\(548\) −9.66874e9 −2.50979
\(549\) 0 0
\(550\) −4.45851e9 −1.14267
\(551\) 3.58739e7 0.00913583
\(552\) 0 0
\(553\) −3.83329e8 −0.0963905
\(554\) −9.10657e9 −2.27547
\(555\) 0 0
\(556\) −9.48794e9 −2.34105
\(557\) −1.40575e9 −0.344678 −0.172339 0.985038i \(-0.555133\pi\)
−0.172339 + 0.985038i \(0.555133\pi\)
\(558\) 0 0
\(559\) −5.67596e8 −0.137435
\(560\) −6.25723e8 −0.150565
\(561\) 0 0
\(562\) −8.67001e9 −2.06036
\(563\) 5.50487e9 1.30007 0.650036 0.759903i \(-0.274754\pi\)
0.650036 + 0.759903i \(0.274754\pi\)
\(564\) 0 0
\(565\) 9.08926e8 0.212011
\(566\) −6.85026e8 −0.158800
\(567\) 0 0
\(568\) 1.24544e9 0.285171
\(569\) −1.17013e9 −0.266283 −0.133141 0.991097i \(-0.542506\pi\)
−0.133141 + 0.991097i \(0.542506\pi\)
\(570\) 0 0
\(571\) −1.88234e9 −0.423127 −0.211563 0.977364i \(-0.567855\pi\)
−0.211563 + 0.977364i \(0.567855\pi\)
\(572\) −5.21919e9 −1.16605
\(573\) 0 0
\(574\) −7.95844e8 −0.175645
\(575\) −8.58342e9 −1.88288
\(576\) 0 0
\(577\) −6.06026e9 −1.31334 −0.656668 0.754180i \(-0.728035\pi\)
−0.656668 + 0.754180i \(0.728035\pi\)
\(578\) 1.96178e9 0.422574
\(579\) 0 0
\(580\) 6.52652e8 0.138894
\(581\) 1.00782e8 0.0213189
\(582\) 0 0
\(583\) 5.29533e9 1.10676
\(584\) 9.56969e9 1.98817
\(585\) 0 0
\(586\) 9.47460e9 1.94500
\(587\) −7.70428e9 −1.57217 −0.786083 0.618120i \(-0.787894\pi\)
−0.786083 + 0.618120i \(0.787894\pi\)
\(588\) 0 0
\(589\) 1.04321e9 0.210362
\(590\) 5.01472e9 1.00523
\(591\) 0 0
\(592\) 5.82044e9 1.15300
\(593\) 8.40982e9 1.65613 0.828067 0.560629i \(-0.189441\pi\)
0.828067 + 0.560629i \(0.189441\pi\)
\(594\) 0 0
\(595\) −4.79556e8 −0.0933318
\(596\) −9.30460e8 −0.180026
\(597\) 0 0
\(598\) −1.45250e10 −2.77755
\(599\) −2.66302e9 −0.506267 −0.253134 0.967431i \(-0.581461\pi\)
−0.253134 + 0.967431i \(0.581461\pi\)
\(600\) 0 0
\(601\) 5.05119e9 0.949146 0.474573 0.880216i \(-0.342602\pi\)
0.474573 + 0.880216i \(0.342602\pi\)
\(602\) −8.52169e7 −0.0159198
\(603\) 0 0
\(604\) 1.83230e10 3.38350
\(605\) 5.27028e9 0.967587
\(606\) 0 0
\(607\) 1.98881e9 0.360938 0.180469 0.983581i \(-0.442238\pi\)
0.180469 + 0.983581i \(0.442238\pi\)
\(608\) −1.17077e9 −0.211256
\(609\) 0 0
\(610\) −2.14489e10 −3.82605
\(611\) −5.61550e9 −0.995966
\(612\) 0 0
\(613\) 3.81895e9 0.669626 0.334813 0.942285i \(-0.391327\pi\)
0.334813 + 0.942285i \(0.391327\pi\)
\(614\) −2.01809e10 −3.51845
\(615\) 0 0
\(616\) −4.34440e8 −0.0748855
\(617\) 8.24101e9 1.41248 0.706241 0.707972i \(-0.250390\pi\)
0.706241 + 0.707972i \(0.250390\pi\)
\(618\) 0 0
\(619\) 2.93974e9 0.498185 0.249093 0.968480i \(-0.419868\pi\)
0.249093 + 0.968480i \(0.419868\pi\)
\(620\) 1.89790e10 3.19818
\(621\) 0 0
\(622\) 3.21144e9 0.535098
\(623\) −5.56175e8 −0.0921516
\(624\) 0 0
\(625\) −5.42915e9 −0.889513
\(626\) 1.77833e10 2.89735
\(627\) 0 0
\(628\) −1.65428e10 −2.66532
\(629\) 4.46080e9 0.714719
\(630\) 0 0
\(631\) −5.09030e9 −0.806567 −0.403283 0.915075i \(-0.632131\pi\)
−0.403283 + 0.915075i \(0.632131\pi\)
\(632\) −2.36541e10 −3.72732
\(633\) 0 0
\(634\) −3.33375e9 −0.519542
\(635\) −1.55934e10 −2.41675
\(636\) 0 0
\(637\) −5.85948e9 −0.898196
\(638\) 2.90868e8 0.0443428
\(639\) 0 0
\(640\) 9.73153e9 1.46741
\(641\) 1.31257e9 0.196843 0.0984216 0.995145i \(-0.468621\pi\)
0.0984216 + 0.995145i \(0.468621\pi\)
\(642\) 0 0
\(643\) −6.98129e9 −1.03561 −0.517806 0.855498i \(-0.673251\pi\)
−0.517806 + 0.855498i \(0.673251\pi\)
\(644\) −1.50855e9 −0.222566
\(645\) 0 0
\(646\) −2.93379e9 −0.428169
\(647\) 4.67412e9 0.678476 0.339238 0.940700i \(-0.389831\pi\)
0.339238 + 0.940700i \(0.389831\pi\)
\(648\) 0 0
\(649\) 1.54604e9 0.222005
\(650\) 1.25067e10 1.78626
\(651\) 0 0
\(652\) 2.75725e10 3.89592
\(653\) 2.18418e9 0.306968 0.153484 0.988151i \(-0.450951\pi\)
0.153484 + 0.988151i \(0.450951\pi\)
\(654\) 0 0
\(655\) −4.23373e8 −0.0588679
\(656\) −2.18065e10 −3.01594
\(657\) 0 0
\(658\) −8.43091e8 −0.115368
\(659\) −3.50457e9 −0.477019 −0.238510 0.971140i \(-0.576659\pi\)
−0.238510 + 0.971140i \(0.576659\pi\)
\(660\) 0 0
\(661\) −8.46214e9 −1.13966 −0.569830 0.821763i \(-0.692991\pi\)
−0.569830 + 0.821763i \(0.692991\pi\)
\(662\) 1.30790e10 1.75215
\(663\) 0 0
\(664\) 6.21893e9 0.824380
\(665\) 1.36280e8 0.0179704
\(666\) 0 0
\(667\) 5.59972e8 0.0730677
\(668\) 1.56659e10 2.03347
\(669\) 0 0
\(670\) 6.23552e9 0.800961
\(671\) −6.61268e9 −0.844984
\(672\) 0 0
\(673\) 5.78313e9 0.731324 0.365662 0.930748i \(-0.380843\pi\)
0.365662 + 0.930748i \(0.380843\pi\)
\(674\) −4.40203e9 −0.553787
\(675\) 0 0
\(676\) −3.38514e9 −0.421466
\(677\) 2.36319e9 0.292710 0.146355 0.989232i \(-0.453246\pi\)
0.146355 + 0.989232i \(0.453246\pi\)
\(678\) 0 0
\(679\) 6.22843e8 0.0763544
\(680\) −2.95919e10 −3.60904
\(681\) 0 0
\(682\) 8.45839e9 1.02104
\(683\) 5.06343e9 0.608096 0.304048 0.952657i \(-0.401662\pi\)
0.304048 + 0.952657i \(0.401662\pi\)
\(684\) 0 0
\(685\) 1.36342e10 1.62074
\(686\) −1.76241e9 −0.208436
\(687\) 0 0
\(688\) −2.33499e9 −0.273354
\(689\) −1.48540e10 −1.73012
\(690\) 0 0
\(691\) −2.24876e8 −0.0259281 −0.0129640 0.999916i \(-0.504127\pi\)
−0.0129640 + 0.999916i \(0.504127\pi\)
\(692\) 1.57455e9 0.180628
\(693\) 0 0
\(694\) 2.41117e10 2.73823
\(695\) 1.33793e10 1.51177
\(696\) 0 0
\(697\) −1.67126e10 −1.86951
\(698\) −9.61103e9 −1.06973
\(699\) 0 0
\(700\) 1.29893e9 0.143134
\(701\) −7.33460e9 −0.804198 −0.402099 0.915596i \(-0.631719\pi\)
−0.402099 + 0.915596i \(0.631719\pi\)
\(702\) 0 0
\(703\) −1.26767e9 −0.137614
\(704\) 7.42496e7 0.00802028
\(705\) 0 0
\(706\) 4.37465e9 0.467871
\(707\) 3.81545e7 0.00406049
\(708\) 0 0
\(709\) 9.29450e8 0.0979409 0.0489704 0.998800i \(-0.484406\pi\)
0.0489704 + 0.998800i \(0.484406\pi\)
\(710\) −3.16770e9 −0.332155
\(711\) 0 0
\(712\) −3.43199e10 −3.56341
\(713\) 1.62839e10 1.68246
\(714\) 0 0
\(715\) 7.35976e9 0.752996
\(716\) 2.07925e10 2.11695
\(717\) 0 0
\(718\) −1.03441e10 −1.04293
\(719\) −1.46064e10 −1.46552 −0.732759 0.680489i \(-0.761768\pi\)
−0.732759 + 0.680489i \(0.761768\pi\)
\(720\) 0 0
\(721\) −1.93955e8 −0.0192721
\(722\) −1.73817e10 −1.71875
\(723\) 0 0
\(724\) −2.23586e10 −2.18957
\(725\) −4.82161e8 −0.0469904
\(726\) 0 0
\(727\) 6.09535e9 0.588339 0.294170 0.955753i \(-0.404957\pi\)
0.294170 + 0.955753i \(0.404957\pi\)
\(728\) 1.21866e9 0.117064
\(729\) 0 0
\(730\) −2.43398e10 −2.31573
\(731\) −1.78954e9 −0.169446
\(732\) 0 0
\(733\) −5.59722e9 −0.524939 −0.262469 0.964940i \(-0.584537\pi\)
−0.262469 + 0.964940i \(0.584537\pi\)
\(734\) −1.46125e10 −1.36392
\(735\) 0 0
\(736\) −1.82750e10 −1.68961
\(737\) 1.92241e9 0.176892
\(738\) 0 0
\(739\) −8.34763e9 −0.760865 −0.380433 0.924809i \(-0.624225\pi\)
−0.380433 + 0.924809i \(0.624225\pi\)
\(740\) −2.30627e10 −2.09218
\(741\) 0 0
\(742\) −2.23013e9 −0.200409
\(743\) 1.62616e10 1.45446 0.727230 0.686393i \(-0.240807\pi\)
0.727230 + 0.686393i \(0.240807\pi\)
\(744\) 0 0
\(745\) 1.31207e9 0.116255
\(746\) 3.00638e10 2.65129
\(747\) 0 0
\(748\) −1.64553e10 −1.43764
\(749\) 1.09059e9 0.0948361
\(750\) 0 0
\(751\) −1.06354e10 −0.916253 −0.458127 0.888887i \(-0.651479\pi\)
−0.458127 + 0.888887i \(0.651479\pi\)
\(752\) −2.31011e10 −1.98094
\(753\) 0 0
\(754\) −8.15919e8 −0.0693182
\(755\) −2.58378e10 −2.18495
\(756\) 0 0
\(757\) −1.33873e10 −1.12165 −0.560826 0.827934i \(-0.689516\pi\)
−0.560826 + 0.827934i \(0.689516\pi\)
\(758\) 3.21705e10 2.68297
\(759\) 0 0
\(760\) 8.40944e9 0.694895
\(761\) 3.98898e9 0.328107 0.164054 0.986451i \(-0.447543\pi\)
0.164054 + 0.986451i \(0.447543\pi\)
\(762\) 0 0
\(763\) −1.98147e8 −0.0161492
\(764\) −2.82194e9 −0.228939
\(765\) 0 0
\(766\) −1.37494e10 −1.10530
\(767\) −4.33681e9 −0.347046
\(768\) 0 0
\(769\) −1.33758e10 −1.06067 −0.530333 0.847789i \(-0.677933\pi\)
−0.530333 + 0.847789i \(0.677933\pi\)
\(770\) 1.10497e9 0.0872233
\(771\) 0 0
\(772\) 2.11173e10 1.65188
\(773\) −1.88395e10 −1.46704 −0.733518 0.679670i \(-0.762123\pi\)
−0.733518 + 0.679670i \(0.762123\pi\)
\(774\) 0 0
\(775\) −1.40211e10 −1.08200
\(776\) 3.84337e10 2.95254
\(777\) 0 0
\(778\) 1.14257e10 0.869867
\(779\) 4.74938e9 0.359962
\(780\) 0 0
\(781\) −9.76600e8 −0.0733564
\(782\) −4.57949e10 −3.42447
\(783\) 0 0
\(784\) −2.41048e10 −1.78648
\(785\) 2.33275e10 1.72117
\(786\) 0 0
\(787\) 1.86675e8 0.0136513 0.00682566 0.999977i \(-0.497827\pi\)
0.00682566 + 0.999977i \(0.497827\pi\)
\(788\) 1.07326e10 0.781385
\(789\) 0 0
\(790\) 6.01625e10 4.34142
\(791\) −1.18015e8 −0.00847853
\(792\) 0 0
\(793\) 1.85493e10 1.32091
\(794\) 7.13810e9 0.506070
\(795\) 0 0
\(796\) 6.81257e9 0.478757
\(797\) 1.16617e10 0.815938 0.407969 0.912996i \(-0.366237\pi\)
0.407969 + 0.912996i \(0.366237\pi\)
\(798\) 0 0
\(799\) −1.77048e10 −1.22794
\(800\) 1.57356e10 1.08660
\(801\) 0 0
\(802\) −2.43773e10 −1.66869
\(803\) −7.50395e9 −0.511429
\(804\) 0 0
\(805\) 2.12726e9 0.143726
\(806\) −2.37268e10 −1.59612
\(807\) 0 0
\(808\) 2.35440e9 0.157015
\(809\) −2.35019e10 −1.56057 −0.780286 0.625423i \(-0.784927\pi\)
−0.780286 + 0.625423i \(0.784927\pi\)
\(810\) 0 0
\(811\) −2.42645e10 −1.59734 −0.798670 0.601769i \(-0.794463\pi\)
−0.798670 + 0.601769i \(0.794463\pi\)
\(812\) −8.47406e7 −0.00555451
\(813\) 0 0
\(814\) −1.02783e10 −0.667941
\(815\) −3.88810e10 −2.51586
\(816\) 0 0
\(817\) 5.08552e8 0.0326255
\(818\) 3.20784e10 2.04916
\(819\) 0 0
\(820\) 8.64053e10 5.47258
\(821\) −2.94003e10 −1.85417 −0.927087 0.374847i \(-0.877695\pi\)
−0.927087 + 0.374847i \(0.877695\pi\)
\(822\) 0 0
\(823\) 1.83212e10 1.14566 0.572829 0.819675i \(-0.305846\pi\)
0.572829 + 0.819675i \(0.305846\pi\)
\(824\) −1.19684e10 −0.745231
\(825\) 0 0
\(826\) −6.51114e8 −0.0402001
\(827\) 2.71066e10 1.66650 0.833251 0.552896i \(-0.186477\pi\)
0.833251 + 0.552896i \(0.186477\pi\)
\(828\) 0 0
\(829\) −9.94304e9 −0.606147 −0.303074 0.952967i \(-0.598013\pi\)
−0.303074 + 0.952967i \(0.598013\pi\)
\(830\) −1.58174e10 −0.960201
\(831\) 0 0
\(832\) −2.08279e8 −0.0125376
\(833\) −1.84740e10 −1.10740
\(834\) 0 0
\(835\) −2.20910e10 −1.31315
\(836\) 4.67626e9 0.276807
\(837\) 0 0
\(838\) −4.14671e10 −2.43416
\(839\) 2.24029e10 1.30960 0.654798 0.755804i \(-0.272754\pi\)
0.654798 + 0.755804i \(0.272754\pi\)
\(840\) 0 0
\(841\) −1.72184e10 −0.998176
\(842\) −2.26832e10 −1.30952
\(843\) 0 0
\(844\) 2.65002e10 1.51723
\(845\) 4.77350e9 0.272169
\(846\) 0 0
\(847\) −6.84295e8 −0.0386947
\(848\) −6.11068e10 −3.44115
\(849\) 0 0
\(850\) 3.94314e10 2.20230
\(851\) −1.97876e10 −1.10063
\(852\) 0 0
\(853\) 5.08263e9 0.280393 0.140197 0.990124i \(-0.455227\pi\)
0.140197 + 0.990124i \(0.455227\pi\)
\(854\) 2.78493e9 0.153007
\(855\) 0 0
\(856\) 6.72968e10 3.66721
\(857\) −1.75834e10 −0.954267 −0.477134 0.878831i \(-0.658324\pi\)
−0.477134 + 0.878831i \(0.658324\pi\)
\(858\) 0 0
\(859\) 3.92669e7 0.00211374 0.00105687 0.999999i \(-0.499664\pi\)
0.00105687 + 0.999999i \(0.499664\pi\)
\(860\) 9.25206e9 0.496014
\(861\) 0 0
\(862\) 4.51955e10 2.40337
\(863\) −2.83133e10 −1.49952 −0.749761 0.661709i \(-0.769832\pi\)
−0.749761 + 0.661709i \(0.769832\pi\)
\(864\) 0 0
\(865\) −2.22033e9 −0.116644
\(866\) 3.29443e10 1.72372
\(867\) 0 0
\(868\) −2.46424e9 −0.127898
\(869\) 1.85481e10 0.958803
\(870\) 0 0
\(871\) −5.39258e9 −0.276524
\(872\) −1.22271e10 −0.624474
\(873\) 0 0
\(874\) 1.30140e10 0.659357
\(875\) −1.67129e8 −0.00843382
\(876\) 0 0
\(877\) −2.63453e10 −1.31888 −0.659440 0.751757i \(-0.729207\pi\)
−0.659440 + 0.751757i \(0.729207\pi\)
\(878\) −4.63409e10 −2.31065
\(879\) 0 0
\(880\) 3.02767e10 1.49768
\(881\) 7.36199e9 0.362727 0.181363 0.983416i \(-0.441949\pi\)
0.181363 + 0.983416i \(0.441949\pi\)
\(882\) 0 0
\(883\) 2.45713e10 1.20106 0.600531 0.799601i \(-0.294956\pi\)
0.600531 + 0.799601i \(0.294956\pi\)
\(884\) 4.61590e10 2.24736
\(885\) 0 0
\(886\) 2.17471e10 1.05047
\(887\) −3.76588e9 −0.181190 −0.0905950 0.995888i \(-0.528877\pi\)
−0.0905950 + 0.995888i \(0.528877\pi\)
\(888\) 0 0
\(889\) 2.02465e9 0.0966481
\(890\) 8.72902e10 4.15050
\(891\) 0 0
\(892\) −8.15522e10 −3.84732
\(893\) 5.03134e9 0.236431
\(894\) 0 0
\(895\) −2.93202e10 −1.36705
\(896\) −1.26355e9 −0.0586831
\(897\) 0 0
\(898\) 3.52054e10 1.62234
\(899\) 9.14723e8 0.0419885
\(900\) 0 0
\(901\) −4.68324e10 −2.13309
\(902\) 3.85083e10 1.74716
\(903\) 0 0
\(904\) −7.28236e9 −0.327856
\(905\) 3.15286e10 1.41395
\(906\) 0 0
\(907\) −6.47573e9 −0.288180 −0.144090 0.989565i \(-0.546025\pi\)
−0.144090 + 0.989565i \(0.546025\pi\)
\(908\) −1.20279e10 −0.533197
\(909\) 0 0
\(910\) −3.09957e9 −0.136350
\(911\) 1.27336e10 0.558004 0.279002 0.960291i \(-0.409996\pi\)
0.279002 + 0.960291i \(0.409996\pi\)
\(912\) 0 0
\(913\) −4.87650e9 −0.212061
\(914\) 7.85952e9 0.340474
\(915\) 0 0
\(916\) −2.35807e10 −1.01373
\(917\) 5.49709e7 0.00235418
\(918\) 0 0
\(919\) 6.63174e8 0.0281853 0.0140927 0.999901i \(-0.495514\pi\)
0.0140927 + 0.999901i \(0.495514\pi\)
\(920\) 1.31267e11 5.55773
\(921\) 0 0
\(922\) 5.16261e10 2.16926
\(923\) 2.73948e9 0.114673
\(924\) 0 0
\(925\) 1.70380e10 0.707821
\(926\) −1.06186e10 −0.439468
\(927\) 0 0
\(928\) −1.02657e9 −0.0421670
\(929\) −3.43993e10 −1.40765 −0.703825 0.710373i \(-0.748526\pi\)
−0.703825 + 0.710373i \(0.748526\pi\)
\(930\) 0 0
\(931\) 5.24994e9 0.213221
\(932\) 5.69434e10 2.30403
\(933\) 0 0
\(934\) 4.02692e10 1.61718
\(935\) 2.32041e10 0.928378
\(936\) 0 0
\(937\) −9.38341e9 −0.372625 −0.186313 0.982491i \(-0.559654\pi\)
−0.186313 + 0.982491i \(0.559654\pi\)
\(938\) −8.09623e8 −0.0320312
\(939\) 0 0
\(940\) 9.15350e10 3.59451
\(941\) −2.39000e10 −0.935050 −0.467525 0.883980i \(-0.654854\pi\)
−0.467525 + 0.883980i \(0.654854\pi\)
\(942\) 0 0
\(943\) 7.41353e10 2.87895
\(944\) −1.78409e10 −0.690261
\(945\) 0 0
\(946\) 4.12337e9 0.158356
\(947\) −4.09342e10 −1.56625 −0.783126 0.621863i \(-0.786376\pi\)
−0.783126 + 0.621863i \(0.786376\pi\)
\(948\) 0 0
\(949\) 2.10495e10 0.799484
\(950\) −1.12056e10 −0.424037
\(951\) 0 0
\(952\) 3.84223e9 0.144329
\(953\) 2.39304e10 0.895621 0.447811 0.894128i \(-0.352204\pi\)
0.447811 + 0.894128i \(0.352204\pi\)
\(954\) 0 0
\(955\) 3.97931e9 0.147841
\(956\) 1.17818e11 4.36123
\(957\) 0 0
\(958\) 3.80092e10 1.39672
\(959\) −1.77027e9 −0.0648149
\(960\) 0 0
\(961\) −9.12637e8 −0.0331716
\(962\) 2.88320e10 1.04415
\(963\) 0 0
\(964\) −6.80044e10 −2.44493
\(965\) −2.97782e10 −1.06673
\(966\) 0 0
\(967\) −9.99102e9 −0.355318 −0.177659 0.984092i \(-0.556852\pi\)
−0.177659 + 0.984092i \(0.556852\pi\)
\(968\) −4.22258e10 −1.49628
\(969\) 0 0
\(970\) −9.77535e10 −3.43899
\(971\) −2.91846e10 −1.02303 −0.511513 0.859275i \(-0.670915\pi\)
−0.511513 + 0.859275i \(0.670915\pi\)
\(972\) 0 0
\(973\) −1.73717e9 −0.0604571
\(974\) 5.40492e10 1.87427
\(975\) 0 0
\(976\) 7.63086e10 2.62724
\(977\) 1.72546e9 0.0591935 0.0295968 0.999562i \(-0.490578\pi\)
0.0295968 + 0.999562i \(0.490578\pi\)
\(978\) 0 0
\(979\) 2.69115e10 0.916639
\(980\) 9.55120e10 3.24165
\(981\) 0 0
\(982\) −6.39960e10 −2.15657
\(983\) −5.02005e10 −1.68566 −0.842831 0.538178i \(-0.819113\pi\)
−0.842831 + 0.538178i \(0.819113\pi\)
\(984\) 0 0
\(985\) −1.51345e10 −0.504592
\(986\) −2.57246e9 −0.0854633
\(987\) 0 0
\(988\) −1.31175e10 −0.432714
\(989\) 7.93821e9 0.260937
\(990\) 0 0
\(991\) 3.49646e10 1.14122 0.570612 0.821220i \(-0.306706\pi\)
0.570612 + 0.821220i \(0.306706\pi\)
\(992\) −2.98526e10 −0.970937
\(993\) 0 0
\(994\) 4.11296e8 0.0132832
\(995\) −9.60663e9 −0.309165
\(996\) 0 0
\(997\) −2.66912e9 −0.0852972 −0.0426486 0.999090i \(-0.513580\pi\)
−0.0426486 + 0.999090i \(0.513580\pi\)
\(998\) −3.54745e9 −0.112969
\(999\) 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.1.13 13
3.2 odd 2 43.8.a.b.1.1 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.1 13 3.2 odd 2
387.8.a.d.1.13 13 1.1 even 1 trivial