Properties

Label 387.8.a.d.1.5
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 1.5
Root \(7.08185\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.08185 q^{2} -62.6836 q^{4} -164.718 q^{5} +1314.61 q^{7} +1541.08 q^{8} +O(q^{10})\) \(q-8.08185 q^{2} -62.6836 q^{4} -164.718 q^{5} +1314.61 q^{7} +1541.08 q^{8} +1331.23 q^{10} -8636.39 q^{11} +8425.12 q^{13} -10624.5 q^{14} -4431.25 q^{16} -6309.90 q^{17} +16664.4 q^{19} +10325.1 q^{20} +69798.0 q^{22} -44105.5 q^{23} -50992.9 q^{25} -68090.6 q^{26} -82404.5 q^{28} +23876.9 q^{29} +27160.3 q^{31} -161445. q^{32} +50995.7 q^{34} -216540. q^{35} -114297. q^{37} -134680. q^{38} -253844. q^{40} +792610. q^{41} -79507.0 q^{43} +541361. q^{44} +356454. q^{46} -490927. q^{47} +904653. q^{49} +412117. q^{50} -528117. q^{52} +1.64370e6 q^{53} +1.42257e6 q^{55} +2.02591e6 q^{56} -192970. q^{58} +625984. q^{59} -1.22257e6 q^{61} -219505. q^{62} +1.87198e6 q^{64} -1.38777e6 q^{65} +1.75201e6 q^{67} +395528. q^{68} +1.75005e6 q^{70} -1.15151e6 q^{71} +2.26726e6 q^{73} +923731. q^{74} -1.04459e6 q^{76} -1.13535e7 q^{77} +4.91491e6 q^{79} +729909. q^{80} -6.40576e6 q^{82} +9.34319e6 q^{83} +1.03936e6 q^{85} +642564. q^{86} -1.33093e7 q^{88} -9.44626e6 q^{89} +1.10757e7 q^{91} +2.76469e6 q^{92} +3.96760e6 q^{94} -2.74494e6 q^{95} -6.08401e6 q^{97} -7.31127e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.08185 −0.714342 −0.357171 0.934039i \(-0.616259\pi\)
−0.357171 + 0.934039i \(0.616259\pi\)
\(3\) 0 0
\(4\) −62.6836 −0.489716
\(5\) −164.718 −0.589314 −0.294657 0.955603i \(-0.595205\pi\)
−0.294657 + 0.955603i \(0.595205\pi\)
\(6\) 0 0
\(7\) 1314.61 1.44862 0.724308 0.689476i \(-0.242159\pi\)
0.724308 + 0.689476i \(0.242159\pi\)
\(8\) 1541.08 1.06417
\(9\) 0 0
\(10\) 1331.23 0.420972
\(11\) −8636.39 −1.95640 −0.978201 0.207662i \(-0.933415\pi\)
−0.978201 + 0.207662i \(0.933415\pi\)
\(12\) 0 0
\(13\) 8425.12 1.06359 0.531795 0.846873i \(-0.321518\pi\)
0.531795 + 0.846873i \(0.321518\pi\)
\(14\) −10624.5 −1.03481
\(15\) 0 0
\(16\) −4431.25 −0.270462
\(17\) −6309.90 −0.311495 −0.155748 0.987797i \(-0.549779\pi\)
−0.155748 + 0.987797i \(0.549779\pi\)
\(18\) 0 0
\(19\) 16664.4 0.557382 0.278691 0.960381i \(-0.410099\pi\)
0.278691 + 0.960381i \(0.410099\pi\)
\(20\) 10325.1 0.288597
\(21\) 0 0
\(22\) 69798.0 1.39754
\(23\) −44105.5 −0.755867 −0.377933 0.925833i \(-0.623365\pi\)
−0.377933 + 0.925833i \(0.623365\pi\)
\(24\) 0 0
\(25\) −50992.9 −0.652709
\(26\) −68090.6 −0.759767
\(27\) 0 0
\(28\) −82404.5 −0.709411
\(29\) 23876.9 0.181796 0.0908981 0.995860i \(-0.471026\pi\)
0.0908981 + 0.995860i \(0.471026\pi\)
\(30\) 0 0
\(31\) 27160.3 0.163745 0.0818725 0.996643i \(-0.473910\pi\)
0.0818725 + 0.996643i \(0.473910\pi\)
\(32\) −161445. −0.870964
\(33\) 0 0
\(34\) 50995.7 0.222514
\(35\) −216540. −0.853690
\(36\) 0 0
\(37\) −114297. −0.370961 −0.185481 0.982648i \(-0.559384\pi\)
−0.185481 + 0.982648i \(0.559384\pi\)
\(38\) −134680. −0.398161
\(39\) 0 0
\(40\) −253844. −0.627128
\(41\) 792610. 1.79604 0.898020 0.439954i \(-0.145005\pi\)
0.898020 + 0.439954i \(0.145005\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) 541361. 0.958081
\(45\) 0 0
\(46\) 356454. 0.539947
\(47\) −490927. −0.689722 −0.344861 0.938654i \(-0.612074\pi\)
−0.344861 + 0.938654i \(0.612074\pi\)
\(48\) 0 0
\(49\) 904653. 1.09849
\(50\) 412117. 0.466257
\(51\) 0 0
\(52\) −528117. −0.520857
\(53\) 1.64370e6 1.51655 0.758275 0.651935i \(-0.226042\pi\)
0.758275 + 0.651935i \(0.226042\pi\)
\(54\) 0 0
\(55\) 1.42257e6 1.15293
\(56\) 2.02591e6 1.54157
\(57\) 0 0
\(58\) −192970. −0.129865
\(59\) 625984. 0.396809 0.198404 0.980120i \(-0.436424\pi\)
0.198404 + 0.980120i \(0.436424\pi\)
\(60\) 0 0
\(61\) −1.22257e6 −0.689635 −0.344817 0.938670i \(-0.612059\pi\)
−0.344817 + 0.938670i \(0.612059\pi\)
\(62\) −219505. −0.116970
\(63\) 0 0
\(64\) 1.87198e6 0.892628
\(65\) −1.38777e6 −0.626789
\(66\) 0 0
\(67\) 1.75201e6 0.711665 0.355833 0.934550i \(-0.384197\pi\)
0.355833 + 0.934550i \(0.384197\pi\)
\(68\) 395528. 0.152544
\(69\) 0 0
\(70\) 1.75005e6 0.609826
\(71\) −1.15151e6 −0.381823 −0.190911 0.981607i \(-0.561144\pi\)
−0.190911 + 0.981607i \(0.561144\pi\)
\(72\) 0 0
\(73\) 2.26726e6 0.682137 0.341068 0.940038i \(-0.389211\pi\)
0.341068 + 0.940038i \(0.389211\pi\)
\(74\) 923731. 0.264993
\(75\) 0 0
\(76\) −1.04459e6 −0.272959
\(77\) −1.13535e7 −2.83407
\(78\) 0 0
\(79\) 4.91491e6 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(80\) 729909. 0.159387
\(81\) 0 0
\(82\) −6.40576e6 −1.28299
\(83\) 9.34319e6 1.79358 0.896792 0.442452i \(-0.145891\pi\)
0.896792 + 0.442452i \(0.145891\pi\)
\(84\) 0 0
\(85\) 1.03936e6 0.183569
\(86\) 642564. 0.108936
\(87\) 0 0
\(88\) −1.33093e7 −2.08194
\(89\) −9.44626e6 −1.42035 −0.710174 0.704026i \(-0.751384\pi\)
−0.710174 + 0.704026i \(0.751384\pi\)
\(90\) 0 0
\(91\) 1.10757e7 1.54073
\(92\) 2.76469e6 0.370160
\(93\) 0 0
\(94\) 3.96760e6 0.492697
\(95\) −2.74494e6 −0.328473
\(96\) 0 0
\(97\) −6.08401e6 −0.676845 −0.338422 0.940994i \(-0.609893\pi\)
−0.338422 + 0.940994i \(0.609893\pi\)
\(98\) −7.31127e6 −0.784697
\(99\) 0 0
\(100\) 3.19642e6 0.319642
\(101\) −8.43016e6 −0.814163 −0.407081 0.913392i \(-0.633453\pi\)
−0.407081 + 0.913392i \(0.633453\pi\)
\(102\) 0 0
\(103\) −1.11690e7 −1.00712 −0.503561 0.863959i \(-0.667977\pi\)
−0.503561 + 0.863959i \(0.667977\pi\)
\(104\) 1.29838e7 1.13184
\(105\) 0 0
\(106\) −1.32841e7 −1.08333
\(107\) −9.97447e6 −0.787131 −0.393565 0.919297i \(-0.628758\pi\)
−0.393565 + 0.919297i \(0.628758\pi\)
\(108\) 0 0
\(109\) −1.68276e7 −1.24460 −0.622301 0.782778i \(-0.713802\pi\)
−0.622301 + 0.782778i \(0.713802\pi\)
\(110\) −1.14970e7 −0.823589
\(111\) 0 0
\(112\) −5.82536e6 −0.391796
\(113\) 1.00006e7 0.652006 0.326003 0.945369i \(-0.394298\pi\)
0.326003 + 0.945369i \(0.394298\pi\)
\(114\) 0 0
\(115\) 7.26498e6 0.445443
\(116\) −1.49669e6 −0.0890285
\(117\) 0 0
\(118\) −5.05911e6 −0.283457
\(119\) −8.29505e6 −0.451237
\(120\) 0 0
\(121\) 5.51001e7 2.82751
\(122\) 9.88063e6 0.492635
\(123\) 0 0
\(124\) −1.70250e6 −0.0801886
\(125\) 2.12681e7 0.973965
\(126\) 0 0
\(127\) 1.37203e7 0.594362 0.297181 0.954821i \(-0.403954\pi\)
0.297181 + 0.954821i \(0.403954\pi\)
\(128\) 5.53594e6 0.233322
\(129\) 0 0
\(130\) 1.12158e7 0.447741
\(131\) 2.22841e7 0.866054 0.433027 0.901381i \(-0.357446\pi\)
0.433027 + 0.901381i \(0.357446\pi\)
\(132\) 0 0
\(133\) 2.19072e7 0.807433
\(134\) −1.41595e7 −0.508372
\(135\) 0 0
\(136\) −9.72405e6 −0.331483
\(137\) 1.65210e6 0.0548928 0.0274464 0.999623i \(-0.491262\pi\)
0.0274464 + 0.999623i \(0.491262\pi\)
\(138\) 0 0
\(139\) 167822. 0.00530027 0.00265014 0.999996i \(-0.499156\pi\)
0.00265014 + 0.999996i \(0.499156\pi\)
\(140\) 1.35735e7 0.418066
\(141\) 0 0
\(142\) 9.30630e6 0.272752
\(143\) −7.27626e7 −2.08081
\(144\) 0 0
\(145\) −3.93296e6 −0.107135
\(146\) −1.83237e7 −0.487279
\(147\) 0 0
\(148\) 7.16455e6 0.181666
\(149\) 3.24074e7 0.802587 0.401294 0.915949i \(-0.368561\pi\)
0.401294 + 0.915949i \(0.368561\pi\)
\(150\) 0 0
\(151\) −2.03174e7 −0.480229 −0.240114 0.970745i \(-0.577185\pi\)
−0.240114 + 0.970745i \(0.577185\pi\)
\(152\) 2.56812e7 0.593147
\(153\) 0 0
\(154\) 9.17571e7 2.02450
\(155\) −4.47379e6 −0.0964973
\(156\) 0 0
\(157\) −7.75567e7 −1.59945 −0.799725 0.600366i \(-0.795022\pi\)
−0.799725 + 0.600366i \(0.795022\pi\)
\(158\) −3.97216e7 −0.801174
\(159\) 0 0
\(160\) 2.65930e7 0.513271
\(161\) −5.79814e7 −1.09496
\(162\) 0 0
\(163\) −8.55135e7 −1.54660 −0.773300 0.634040i \(-0.781395\pi\)
−0.773300 + 0.634040i \(0.781395\pi\)
\(164\) −4.96837e7 −0.879550
\(165\) 0 0
\(166\) −7.55103e7 −1.28123
\(167\) 5.30786e7 0.881885 0.440942 0.897535i \(-0.354644\pi\)
0.440942 + 0.897535i \(0.354644\pi\)
\(168\) 0 0
\(169\) 8.23412e6 0.131224
\(170\) −8.39993e6 −0.131131
\(171\) 0 0
\(172\) 4.98379e6 0.0746810
\(173\) −7.43872e7 −1.09229 −0.546144 0.837691i \(-0.683905\pi\)
−0.546144 + 0.837691i \(0.683905\pi\)
\(174\) 0 0
\(175\) −6.70357e7 −0.945525
\(176\) 3.82700e7 0.529133
\(177\) 0 0
\(178\) 7.63433e7 1.01461
\(179\) −1.45355e8 −1.89428 −0.947139 0.320824i \(-0.896040\pi\)
−0.947139 + 0.320824i \(0.896040\pi\)
\(180\) 0 0
\(181\) −4.70208e7 −0.589406 −0.294703 0.955589i \(-0.595221\pi\)
−0.294703 + 0.955589i \(0.595221\pi\)
\(182\) −8.95125e7 −1.10061
\(183\) 0 0
\(184\) −6.79699e7 −0.804368
\(185\) 1.88268e7 0.218613
\(186\) 0 0
\(187\) 5.44948e7 0.609410
\(188\) 3.07731e7 0.337768
\(189\) 0 0
\(190\) 2.21842e7 0.234642
\(191\) 6.43802e7 0.668552 0.334276 0.942475i \(-0.391508\pi\)
0.334276 + 0.942475i \(0.391508\pi\)
\(192\) 0 0
\(193\) −1.29604e8 −1.29768 −0.648842 0.760923i \(-0.724746\pi\)
−0.648842 + 0.760923i \(0.724746\pi\)
\(194\) 4.91701e7 0.483499
\(195\) 0 0
\(196\) −5.67070e7 −0.537948
\(197\) −8.33653e7 −0.776879 −0.388440 0.921474i \(-0.626986\pi\)
−0.388440 + 0.921474i \(0.626986\pi\)
\(198\) 0 0
\(199\) −1.89130e8 −1.70127 −0.850636 0.525755i \(-0.823783\pi\)
−0.850636 + 0.525755i \(0.823783\pi\)
\(200\) −7.85840e7 −0.694591
\(201\) 0 0
\(202\) 6.81313e7 0.581590
\(203\) 3.13888e7 0.263353
\(204\) 0 0
\(205\) −1.30557e8 −1.05843
\(206\) 9.02659e7 0.719430
\(207\) 0 0
\(208\) −3.73338e7 −0.287661
\(209\) −1.43921e8 −1.09046
\(210\) 0 0
\(211\) 1.41316e8 1.03563 0.517814 0.855493i \(-0.326746\pi\)
0.517814 + 0.855493i \(0.326746\pi\)
\(212\) −1.03033e8 −0.742679
\(213\) 0 0
\(214\) 8.06122e7 0.562280
\(215\) 1.30963e7 0.0898696
\(216\) 0 0
\(217\) 3.57051e7 0.237204
\(218\) 1.35998e8 0.889070
\(219\) 0 0
\(220\) −8.91720e7 −0.564611
\(221\) −5.31617e7 −0.331303
\(222\) 0 0
\(223\) −1.10823e8 −0.669211 −0.334605 0.942358i \(-0.608603\pi\)
−0.334605 + 0.942358i \(0.608603\pi\)
\(224\) −2.12237e8 −1.26169
\(225\) 0 0
\(226\) −8.08234e7 −0.465755
\(227\) 1.43855e8 0.816269 0.408134 0.912922i \(-0.366180\pi\)
0.408134 + 0.912922i \(0.366180\pi\)
\(228\) 0 0
\(229\) −3.52277e8 −1.93847 −0.969236 0.246132i \(-0.920840\pi\)
−0.969236 + 0.246132i \(0.920840\pi\)
\(230\) −5.87145e7 −0.318198
\(231\) 0 0
\(232\) 3.67961e7 0.193461
\(233\) −3.78934e6 −0.0196254 −0.00981269 0.999952i \(-0.503124\pi\)
−0.00981269 + 0.999952i \(0.503124\pi\)
\(234\) 0 0
\(235\) 8.08647e7 0.406463
\(236\) −3.92390e7 −0.194324
\(237\) 0 0
\(238\) 6.70394e7 0.322337
\(239\) 9.27614e6 0.0439516 0.0219758 0.999759i \(-0.493004\pi\)
0.0219758 + 0.999759i \(0.493004\pi\)
\(240\) 0 0
\(241\) 3.57012e8 1.64294 0.821472 0.570249i \(-0.193153\pi\)
0.821472 + 0.570249i \(0.193153\pi\)
\(242\) −4.45311e8 −2.01981
\(243\) 0 0
\(244\) 7.66351e7 0.337725
\(245\) −1.49013e8 −0.647355
\(246\) 0 0
\(247\) 1.40400e8 0.592826
\(248\) 4.18561e7 0.174252
\(249\) 0 0
\(250\) −1.71885e8 −0.695744
\(251\) −4.81876e8 −1.92343 −0.961716 0.274047i \(-0.911637\pi\)
−0.961716 + 0.274047i \(0.911637\pi\)
\(252\) 0 0
\(253\) 3.80912e8 1.47878
\(254\) −1.10886e8 −0.424577
\(255\) 0 0
\(256\) −2.84354e8 −1.05930
\(257\) 1.70067e6 0.00624962 0.00312481 0.999995i \(-0.499005\pi\)
0.00312481 + 0.999995i \(0.499005\pi\)
\(258\) 0 0
\(259\) −1.50256e8 −0.537380
\(260\) 8.69906e7 0.306948
\(261\) 0 0
\(262\) −1.80097e8 −0.618658
\(263\) 3.62647e8 1.22925 0.614623 0.788821i \(-0.289308\pi\)
0.614623 + 0.788821i \(0.289308\pi\)
\(264\) 0 0
\(265\) −2.70747e8 −0.893724
\(266\) −1.77051e8 −0.576783
\(267\) 0 0
\(268\) −1.09823e8 −0.348514
\(269\) −3.91885e8 −1.22751 −0.613756 0.789496i \(-0.710342\pi\)
−0.613756 + 0.789496i \(0.710342\pi\)
\(270\) 0 0
\(271\) 5.27272e6 0.0160932 0.00804660 0.999968i \(-0.497439\pi\)
0.00804660 + 0.999968i \(0.497439\pi\)
\(272\) 2.79608e7 0.0842477
\(273\) 0 0
\(274\) −1.33521e7 −0.0392122
\(275\) 4.40394e8 1.27696
\(276\) 0 0
\(277\) 5.31633e8 1.50291 0.751455 0.659784i \(-0.229352\pi\)
0.751455 + 0.659784i \(0.229352\pi\)
\(278\) −1.35632e6 −0.00378620
\(279\) 0 0
\(280\) −3.33705e8 −0.908468
\(281\) −5.83130e8 −1.56781 −0.783905 0.620881i \(-0.786775\pi\)
−0.783905 + 0.620881i \(0.786775\pi\)
\(282\) 0 0
\(283\) −2.16632e8 −0.568159 −0.284079 0.958801i \(-0.591688\pi\)
−0.284079 + 0.958801i \(0.591688\pi\)
\(284\) 7.21806e7 0.186985
\(285\) 0 0
\(286\) 5.88057e8 1.48641
\(287\) 1.04197e9 2.60177
\(288\) 0 0
\(289\) −3.70524e8 −0.902971
\(290\) 3.17856e7 0.0765311
\(291\) 0 0
\(292\) −1.42120e8 −0.334053
\(293\) 9.62588e7 0.223565 0.111782 0.993733i \(-0.464344\pi\)
0.111782 + 0.993733i \(0.464344\pi\)
\(294\) 0 0
\(295\) −1.03111e8 −0.233845
\(296\) −1.76140e8 −0.394764
\(297\) 0 0
\(298\) −2.61912e8 −0.573321
\(299\) −3.71594e8 −0.803932
\(300\) 0 0
\(301\) −1.04521e8 −0.220912
\(302\) 1.64202e8 0.343047
\(303\) 0 0
\(304\) −7.38444e7 −0.150751
\(305\) 2.01380e8 0.406411
\(306\) 0 0
\(307\) 7.18187e8 1.41662 0.708310 0.705902i \(-0.249458\pi\)
0.708310 + 0.705902i \(0.249458\pi\)
\(308\) 7.11677e8 1.38789
\(309\) 0 0
\(310\) 3.61565e7 0.0689320
\(311\) 1.67567e8 0.315883 0.157942 0.987448i \(-0.449514\pi\)
0.157942 + 0.987448i \(0.449514\pi\)
\(312\) 0 0
\(313\) 6.19823e8 1.14252 0.571258 0.820770i \(-0.306455\pi\)
0.571258 + 0.820770i \(0.306455\pi\)
\(314\) 6.26802e8 1.14255
\(315\) 0 0
\(316\) −3.08084e8 −0.549244
\(317\) −2.63030e8 −0.463765 −0.231883 0.972744i \(-0.574489\pi\)
−0.231883 + 0.972744i \(0.574489\pi\)
\(318\) 0 0
\(319\) −2.06210e8 −0.355666
\(320\) −3.08349e8 −0.526038
\(321\) 0 0
\(322\) 4.68598e8 0.782176
\(323\) −1.05151e8 −0.173622
\(324\) 0 0
\(325\) −4.29621e8 −0.694215
\(326\) 6.91108e8 1.10480
\(327\) 0 0
\(328\) 1.22147e9 1.91129
\(329\) −6.45377e8 −0.999143
\(330\) 0 0
\(331\) −9.80269e8 −1.48575 −0.742877 0.669427i \(-0.766540\pi\)
−0.742877 + 0.669427i \(0.766540\pi\)
\(332\) −5.85665e8 −0.878347
\(333\) 0 0
\(334\) −4.28973e8 −0.629967
\(335\) −2.88589e8 −0.419394
\(336\) 0 0
\(337\) −6.28527e8 −0.894580 −0.447290 0.894389i \(-0.647611\pi\)
−0.447290 + 0.894389i \(0.647611\pi\)
\(338\) −6.65470e7 −0.0937389
\(339\) 0 0
\(340\) −6.51507e7 −0.0898965
\(341\) −2.34567e8 −0.320351
\(342\) 0 0
\(343\) 1.06628e8 0.142673
\(344\) −1.22526e8 −0.162284
\(345\) 0 0
\(346\) 6.01187e8 0.780267
\(347\) −1.41484e9 −1.81783 −0.908915 0.416982i \(-0.863088\pi\)
−0.908915 + 0.416982i \(0.863088\pi\)
\(348\) 0 0
\(349\) 1.08261e9 1.36327 0.681637 0.731691i \(-0.261269\pi\)
0.681637 + 0.731691i \(0.261269\pi\)
\(350\) 5.41773e8 0.675428
\(351\) 0 0
\(352\) 1.39430e9 1.70395
\(353\) 1.04347e9 1.26261 0.631306 0.775533i \(-0.282519\pi\)
0.631306 + 0.775533i \(0.282519\pi\)
\(354\) 0 0
\(355\) 1.89674e8 0.225014
\(356\) 5.92126e8 0.695567
\(357\) 0 0
\(358\) 1.17474e9 1.35316
\(359\) −7.36087e8 −0.839650 −0.419825 0.907605i \(-0.637909\pi\)
−0.419825 + 0.907605i \(0.637909\pi\)
\(360\) 0 0
\(361\) −6.16168e8 −0.689325
\(362\) 3.80015e8 0.421037
\(363\) 0 0
\(364\) −6.94267e8 −0.754522
\(365\) −3.73459e8 −0.401993
\(366\) 0 0
\(367\) 5.85636e8 0.618439 0.309219 0.950991i \(-0.399932\pi\)
0.309219 + 0.950991i \(0.399932\pi\)
\(368\) 1.95443e8 0.204433
\(369\) 0 0
\(370\) −1.52155e8 −0.156164
\(371\) 2.16082e9 2.19690
\(372\) 0 0
\(373\) −1.55931e9 −1.55579 −0.777896 0.628393i \(-0.783713\pi\)
−0.777896 + 0.628393i \(0.783713\pi\)
\(374\) −4.40419e8 −0.435327
\(375\) 0 0
\(376\) −7.56556e8 −0.733979
\(377\) 2.01166e8 0.193357
\(378\) 0 0
\(379\) −2.07398e7 −0.0195689 −0.00978446 0.999952i \(-0.503115\pi\)
−0.00978446 + 0.999952i \(0.503115\pi\)
\(380\) 1.72063e8 0.160859
\(381\) 0 0
\(382\) −5.20311e8 −0.477575
\(383\) −1.16713e9 −1.06151 −0.530755 0.847525i \(-0.678092\pi\)
−0.530755 + 0.847525i \(0.678092\pi\)
\(384\) 0 0
\(385\) 1.87013e9 1.67016
\(386\) 1.04744e9 0.926990
\(387\) 0 0
\(388\) 3.81368e8 0.331462
\(389\) 1.12210e9 0.966512 0.483256 0.875479i \(-0.339454\pi\)
0.483256 + 0.875479i \(0.339454\pi\)
\(390\) 0 0
\(391\) 2.78301e8 0.235449
\(392\) 1.39414e9 1.16897
\(393\) 0 0
\(394\) 6.73746e8 0.554957
\(395\) −8.09575e8 −0.660948
\(396\) 0 0
\(397\) 6.25123e8 0.501417 0.250708 0.968063i \(-0.419337\pi\)
0.250708 + 0.968063i \(0.419337\pi\)
\(398\) 1.52852e9 1.21529
\(399\) 0 0
\(400\) 2.25962e8 0.176533
\(401\) −6.80350e8 −0.526899 −0.263449 0.964673i \(-0.584860\pi\)
−0.263449 + 0.964673i \(0.584860\pi\)
\(402\) 0 0
\(403\) 2.28828e8 0.174158
\(404\) 5.28433e8 0.398708
\(405\) 0 0
\(406\) −2.53680e8 −0.188124
\(407\) 9.87113e8 0.725749
\(408\) 0 0
\(409\) 1.45254e9 1.04977 0.524887 0.851172i \(-0.324108\pi\)
0.524887 + 0.851172i \(0.324108\pi\)
\(410\) 1.05515e9 0.756082
\(411\) 0 0
\(412\) 7.00111e8 0.493204
\(413\) 8.22925e8 0.574824
\(414\) 0 0
\(415\) −1.53899e9 −1.05698
\(416\) −1.36019e9 −0.926349
\(417\) 0 0
\(418\) 1.16315e9 0.778963
\(419\) 2.07021e9 1.37488 0.687441 0.726240i \(-0.258734\pi\)
0.687441 + 0.726240i \(0.258734\pi\)
\(420\) 0 0
\(421\) −1.86742e9 −1.21971 −0.609853 0.792514i \(-0.708772\pi\)
−0.609853 + 0.792514i \(0.708772\pi\)
\(422\) −1.14210e9 −0.739792
\(423\) 0 0
\(424\) 2.53307e9 1.61386
\(425\) 3.21760e8 0.203316
\(426\) 0 0
\(427\) −1.60720e9 −0.999016
\(428\) 6.25236e8 0.385470
\(429\) 0 0
\(430\) −1.05842e8 −0.0641976
\(431\) −1.03410e9 −0.622144 −0.311072 0.950386i \(-0.600688\pi\)
−0.311072 + 0.950386i \(0.600688\pi\)
\(432\) 0 0
\(433\) 1.88745e9 1.11730 0.558648 0.829405i \(-0.311320\pi\)
0.558648 + 0.829405i \(0.311320\pi\)
\(434\) −2.88564e8 −0.169444
\(435\) 0 0
\(436\) 1.05482e9 0.609501
\(437\) −7.34993e8 −0.421307
\(438\) 0 0
\(439\) −2.19197e9 −1.23654 −0.618270 0.785965i \(-0.712166\pi\)
−0.618270 + 0.785965i \(0.712166\pi\)
\(440\) 2.19229e9 1.22691
\(441\) 0 0
\(442\) 4.29645e8 0.236664
\(443\) 1.23285e9 0.673745 0.336873 0.941550i \(-0.390631\pi\)
0.336873 + 0.941550i \(0.390631\pi\)
\(444\) 0 0
\(445\) 1.55597e9 0.837031
\(446\) 8.95655e8 0.478045
\(447\) 0 0
\(448\) 2.46092e9 1.29308
\(449\) −2.06378e9 −1.07597 −0.537985 0.842954i \(-0.680814\pi\)
−0.537985 + 0.842954i \(0.680814\pi\)
\(450\) 0 0
\(451\) −6.84529e9 −3.51378
\(452\) −6.26874e8 −0.319298
\(453\) 0 0
\(454\) −1.16261e9 −0.583095
\(455\) −1.82438e9 −0.907976
\(456\) 0 0
\(457\) −3.67119e9 −1.79928 −0.899642 0.436628i \(-0.856173\pi\)
−0.899642 + 0.436628i \(0.856173\pi\)
\(458\) 2.84705e9 1.38473
\(459\) 0 0
\(460\) −4.55395e8 −0.218140
\(461\) −1.50027e9 −0.713206 −0.356603 0.934256i \(-0.616065\pi\)
−0.356603 + 0.934256i \(0.616065\pi\)
\(462\) 0 0
\(463\) 1.00012e9 0.468293 0.234147 0.972201i \(-0.424770\pi\)
0.234147 + 0.972201i \(0.424770\pi\)
\(464\) −1.05805e8 −0.0491690
\(465\) 0 0
\(466\) 3.06249e7 0.0140192
\(467\) 1.13114e9 0.513933 0.256967 0.966420i \(-0.417277\pi\)
0.256967 + 0.966420i \(0.417277\pi\)
\(468\) 0 0
\(469\) 2.30321e9 1.03093
\(470\) −6.53536e8 −0.290354
\(471\) 0 0
\(472\) 9.64690e8 0.422271
\(473\) 6.86654e8 0.298348
\(474\) 0 0
\(475\) −8.49768e8 −0.363808
\(476\) 5.19964e8 0.220978
\(477\) 0 0
\(478\) −7.49684e7 −0.0313964
\(479\) −4.27056e8 −0.177546 −0.0887730 0.996052i \(-0.528295\pi\)
−0.0887730 + 0.996052i \(0.528295\pi\)
\(480\) 0 0
\(481\) −9.62965e8 −0.394551
\(482\) −2.88532e9 −1.17362
\(483\) 0 0
\(484\) −3.45387e9 −1.38467
\(485\) 1.00215e9 0.398874
\(486\) 0 0
\(487\) 1.86591e9 0.732049 0.366025 0.930605i \(-0.380719\pi\)
0.366025 + 0.930605i \(0.380719\pi\)
\(488\) −1.88407e9 −0.733886
\(489\) 0 0
\(490\) 1.20430e9 0.462433
\(491\) 2.67497e9 1.01985 0.509923 0.860220i \(-0.329674\pi\)
0.509923 + 0.860220i \(0.329674\pi\)
\(492\) 0 0
\(493\) −1.50661e8 −0.0566287
\(494\) −1.13469e9 −0.423481
\(495\) 0 0
\(496\) −1.20354e8 −0.0442868
\(497\) −1.51378e9 −0.553115
\(498\) 0 0
\(499\) −2.01727e9 −0.726794 −0.363397 0.931634i \(-0.618383\pi\)
−0.363397 + 0.931634i \(0.618383\pi\)
\(500\) −1.33316e9 −0.476966
\(501\) 0 0
\(502\) 3.89445e9 1.37399
\(503\) −2.78344e9 −0.975201 −0.487601 0.873067i \(-0.662128\pi\)
−0.487601 + 0.873067i \(0.662128\pi\)
\(504\) 0 0
\(505\) 1.38860e9 0.479798
\(506\) −3.07848e9 −1.05635
\(507\) 0 0
\(508\) −8.60039e8 −0.291068
\(509\) 6.34132e8 0.213141 0.106571 0.994305i \(-0.466013\pi\)
0.106571 + 0.994305i \(0.466013\pi\)
\(510\) 0 0
\(511\) 2.98056e9 0.988154
\(512\) 1.58950e9 0.523380
\(513\) 0 0
\(514\) −1.37446e7 −0.00446437
\(515\) 1.83973e9 0.593512
\(516\) 0 0
\(517\) 4.23984e9 1.34937
\(518\) 1.21434e9 0.383873
\(519\) 0 0
\(520\) −2.13866e9 −0.667007
\(521\) 2.55557e9 0.791692 0.395846 0.918317i \(-0.370451\pi\)
0.395846 + 0.918317i \(0.370451\pi\)
\(522\) 0 0
\(523\) 1.86691e9 0.570647 0.285323 0.958431i \(-0.407899\pi\)
0.285323 + 0.958431i \(0.407899\pi\)
\(524\) −1.39685e9 −0.424120
\(525\) 0 0
\(526\) −2.93086e9 −0.878101
\(527\) −1.71379e8 −0.0510058
\(528\) 0 0
\(529\) −1.45953e9 −0.428666
\(530\) 2.18814e9 0.638425
\(531\) 0 0
\(532\) −1.37322e9 −0.395413
\(533\) 6.67783e9 1.91025
\(534\) 0 0
\(535\) 1.64298e9 0.463867
\(536\) 2.69999e9 0.757330
\(537\) 0 0
\(538\) 3.16716e9 0.876863
\(539\) −7.81294e9 −2.14909
\(540\) 0 0
\(541\) −2.51318e8 −0.0682391 −0.0341195 0.999418i \(-0.510863\pi\)
−0.0341195 + 0.999418i \(0.510863\pi\)
\(542\) −4.26134e7 −0.0114960
\(543\) 0 0
\(544\) 1.01870e9 0.271301
\(545\) 2.77182e9 0.733461
\(546\) 0 0
\(547\) 1.95845e9 0.511631 0.255815 0.966726i \(-0.417656\pi\)
0.255815 + 0.966726i \(0.417656\pi\)
\(548\) −1.03560e8 −0.0268819
\(549\) 0 0
\(550\) −3.55920e9 −0.912186
\(551\) 3.97895e8 0.101330
\(552\) 0 0
\(553\) 6.46118e9 1.62470
\(554\) −4.29658e9 −1.07359
\(555\) 0 0
\(556\) −1.05197e7 −0.00259563
\(557\) 9.85537e7 0.0241646 0.0120823 0.999927i \(-0.496154\pi\)
0.0120823 + 0.999927i \(0.496154\pi\)
\(558\) 0 0
\(559\) −6.69856e8 −0.162196
\(560\) 9.59544e8 0.230891
\(561\) 0 0
\(562\) 4.71277e9 1.11995
\(563\) 8.32439e8 0.196595 0.0982976 0.995157i \(-0.468660\pi\)
0.0982976 + 0.995157i \(0.468660\pi\)
\(564\) 0 0
\(565\) −1.64728e9 −0.384236
\(566\) 1.75079e9 0.405859
\(567\) 0 0
\(568\) −1.77456e9 −0.406323
\(569\) 4.82012e8 0.109689 0.0548447 0.998495i \(-0.482534\pi\)
0.0548447 + 0.998495i \(0.482534\pi\)
\(570\) 0 0
\(571\) −3.91868e9 −0.880874 −0.440437 0.897784i \(-0.645176\pi\)
−0.440437 + 0.897784i \(0.645176\pi\)
\(572\) 4.56103e9 1.01901
\(573\) 0 0
\(574\) −8.42107e9 −1.85856
\(575\) 2.24907e9 0.493361
\(576\) 0 0
\(577\) −6.04095e9 −1.30915 −0.654576 0.755996i \(-0.727153\pi\)
−0.654576 + 0.755996i \(0.727153\pi\)
\(578\) 2.99452e9 0.645030
\(579\) 0 0
\(580\) 2.46532e8 0.0524658
\(581\) 1.22826e10 2.59822
\(582\) 0 0
\(583\) −1.41956e10 −2.96698
\(584\) 3.49402e9 0.725907
\(585\) 0 0
\(586\) −7.77949e8 −0.159702
\(587\) −9.41828e9 −1.92193 −0.960967 0.276665i \(-0.910771\pi\)
−0.960967 + 0.276665i \(0.910771\pi\)
\(588\) 0 0
\(589\) 4.52611e8 0.0912686
\(590\) 8.33329e8 0.167045
\(591\) 0 0
\(592\) 5.06479e8 0.100331
\(593\) −2.77653e9 −0.546778 −0.273389 0.961904i \(-0.588145\pi\)
−0.273389 + 0.961904i \(0.588145\pi\)
\(594\) 0 0
\(595\) 1.36635e9 0.265920
\(596\) −2.03141e9 −0.393040
\(597\) 0 0
\(598\) 3.00317e9 0.574282
\(599\) −2.48861e9 −0.473110 −0.236555 0.971618i \(-0.576018\pi\)
−0.236555 + 0.971618i \(0.576018\pi\)
\(600\) 0 0
\(601\) 9.96150e8 0.187182 0.0935910 0.995611i \(-0.470165\pi\)
0.0935910 + 0.995611i \(0.470165\pi\)
\(602\) 8.44720e8 0.157807
\(603\) 0 0
\(604\) 1.27357e9 0.235176
\(605\) −9.07599e9 −1.66629
\(606\) 0 0
\(607\) 4.07704e9 0.739920 0.369960 0.929048i \(-0.379371\pi\)
0.369960 + 0.929048i \(0.379371\pi\)
\(608\) −2.69039e9 −0.485460
\(609\) 0 0
\(610\) −1.62752e9 −0.290317
\(611\) −4.13612e9 −0.733582
\(612\) 0 0
\(613\) 7.37622e9 1.29337 0.646684 0.762758i \(-0.276155\pi\)
0.646684 + 0.762758i \(0.276155\pi\)
\(614\) −5.80428e9 −1.01195
\(615\) 0 0
\(616\) −1.74966e10 −3.01593
\(617\) 3.57832e9 0.613312 0.306656 0.951820i \(-0.400790\pi\)
0.306656 + 0.951820i \(0.400790\pi\)
\(618\) 0 0
\(619\) −2.12273e9 −0.359731 −0.179865 0.983691i \(-0.557566\pi\)
−0.179865 + 0.983691i \(0.557566\pi\)
\(620\) 2.80434e8 0.0472563
\(621\) 0 0
\(622\) −1.35425e9 −0.225649
\(623\) −1.24181e10 −2.05754
\(624\) 0 0
\(625\) 4.80577e8 0.0787377
\(626\) −5.00932e9 −0.816147
\(627\) 0 0
\(628\) 4.86154e9 0.783277
\(629\) 7.21203e8 0.115553
\(630\) 0 0
\(631\) −1.03343e10 −1.63749 −0.818744 0.574159i \(-0.805329\pi\)
−0.818744 + 0.574159i \(0.805329\pi\)
\(632\) 7.57425e9 1.19352
\(633\) 0 0
\(634\) 2.12577e9 0.331287
\(635\) −2.25999e9 −0.350266
\(636\) 0 0
\(637\) 7.62181e9 1.16834
\(638\) 1.66656e9 0.254067
\(639\) 0 0
\(640\) −9.11871e8 −0.137500
\(641\) −4.06584e9 −0.609744 −0.304872 0.952393i \(-0.598614\pi\)
−0.304872 + 0.952393i \(0.598614\pi\)
\(642\) 0 0
\(643\) 1.27525e10 1.89172 0.945860 0.324574i \(-0.105221\pi\)
0.945860 + 0.324574i \(0.105221\pi\)
\(644\) 3.63449e9 0.536220
\(645\) 0 0
\(646\) 8.49815e8 0.124025
\(647\) −6.06984e9 −0.881074 −0.440537 0.897734i \(-0.645212\pi\)
−0.440537 + 0.897734i \(0.645212\pi\)
\(648\) 0 0
\(649\) −5.40625e9 −0.776318
\(650\) 3.47213e9 0.495907
\(651\) 0 0
\(652\) 5.36030e9 0.757395
\(653\) −8.96474e9 −1.25992 −0.629958 0.776629i \(-0.716928\pi\)
−0.629958 + 0.776629i \(0.716928\pi\)
\(654\) 0 0
\(655\) −3.67059e9 −0.510378
\(656\) −3.51226e9 −0.485761
\(657\) 0 0
\(658\) 5.21584e9 0.713730
\(659\) −7.66829e9 −1.04376 −0.521879 0.853020i \(-0.674769\pi\)
−0.521879 + 0.853020i \(0.674769\pi\)
\(660\) 0 0
\(661\) 9.45702e9 1.27365 0.636824 0.771010i \(-0.280248\pi\)
0.636824 + 0.771010i \(0.280248\pi\)
\(662\) 7.92239e9 1.06134
\(663\) 0 0
\(664\) 1.43986e10 1.90867
\(665\) −3.60852e9 −0.475832
\(666\) 0 0
\(667\) −1.05310e9 −0.137414
\(668\) −3.32716e9 −0.431873
\(669\) 0 0
\(670\) 2.33233e9 0.299591
\(671\) 1.05586e10 1.34920
\(672\) 0 0
\(673\) 7.72168e9 0.976470 0.488235 0.872712i \(-0.337641\pi\)
0.488235 + 0.872712i \(0.337641\pi\)
\(674\) 5.07966e9 0.639036
\(675\) 0 0
\(676\) −5.16145e8 −0.0642626
\(677\) 3.61144e9 0.447321 0.223661 0.974667i \(-0.428199\pi\)
0.223661 + 0.974667i \(0.428199\pi\)
\(678\) 0 0
\(679\) −7.99810e9 −0.980489
\(680\) 1.60173e9 0.195347
\(681\) 0 0
\(682\) 1.89573e9 0.228840
\(683\) −1.07988e10 −1.29689 −0.648444 0.761262i \(-0.724580\pi\)
−0.648444 + 0.761262i \(0.724580\pi\)
\(684\) 0 0
\(685\) −2.72132e8 −0.0323491
\(686\) −8.61751e8 −0.101917
\(687\) 0 0
\(688\) 3.52316e8 0.0412451
\(689\) 1.38484e10 1.61299
\(690\) 0 0
\(691\) 3.51258e9 0.404998 0.202499 0.979282i \(-0.435094\pi\)
0.202499 + 0.979282i \(0.435094\pi\)
\(692\) 4.66286e9 0.534911
\(693\) 0 0
\(694\) 1.14345e10 1.29855
\(695\) −2.76434e7 −0.00312353
\(696\) 0 0
\(697\) −5.00129e9 −0.559458
\(698\) −8.74949e9 −0.973843
\(699\) 0 0
\(700\) 4.20204e9 0.463039
\(701\) 2.46256e9 0.270006 0.135003 0.990845i \(-0.456896\pi\)
0.135003 + 0.990845i \(0.456896\pi\)
\(702\) 0 0
\(703\) −1.90469e9 −0.206767
\(704\) −1.61671e10 −1.74634
\(705\) 0 0
\(706\) −8.43320e9 −0.901937
\(707\) −1.10824e10 −1.17941
\(708\) 0 0
\(709\) −1.34920e10 −1.42172 −0.710859 0.703335i \(-0.751693\pi\)
−0.710859 + 0.703335i \(0.751693\pi\)
\(710\) −1.53292e9 −0.160737
\(711\) 0 0
\(712\) −1.45574e10 −1.51149
\(713\) −1.19792e9 −0.123769
\(714\) 0 0
\(715\) 1.19853e10 1.22625
\(716\) 9.11137e9 0.927658
\(717\) 0 0
\(718\) 5.94894e9 0.599797
\(719\) −7.00395e9 −0.702736 −0.351368 0.936237i \(-0.614283\pi\)
−0.351368 + 0.936237i \(0.614283\pi\)
\(720\) 0 0
\(721\) −1.46828e10 −1.45893
\(722\) 4.97978e9 0.492414
\(723\) 0 0
\(724\) 2.94743e9 0.288642
\(725\) −1.21755e9 −0.118660
\(726\) 0 0
\(727\) 8.00714e9 0.772871 0.386436 0.922316i \(-0.373706\pi\)
0.386436 + 0.922316i \(0.373706\pi\)
\(728\) 1.70686e10 1.63960
\(729\) 0 0
\(730\) 3.01824e9 0.287160
\(731\) 5.01681e8 0.0475026
\(732\) 0 0
\(733\) 9.27809e9 0.870151 0.435075 0.900394i \(-0.356722\pi\)
0.435075 + 0.900394i \(0.356722\pi\)
\(734\) −4.73303e9 −0.441777
\(735\) 0 0
\(736\) 7.12062e9 0.658332
\(737\) −1.51311e10 −1.39230
\(738\) 0 0
\(739\) −1.50075e10 −1.36789 −0.683947 0.729532i \(-0.739738\pi\)
−0.683947 + 0.729532i \(0.739738\pi\)
\(740\) −1.18013e9 −0.107058
\(741\) 0 0
\(742\) −1.74634e10 −1.56934
\(743\) −2.52291e9 −0.225653 −0.112827 0.993615i \(-0.535990\pi\)
−0.112827 + 0.993615i \(0.535990\pi\)
\(744\) 0 0
\(745\) −5.33809e9 −0.472976
\(746\) 1.26021e10 1.11137
\(747\) 0 0
\(748\) −3.41593e9 −0.298438
\(749\) −1.31125e10 −1.14025
\(750\) 0 0
\(751\) −9.71269e9 −0.836758 −0.418379 0.908273i \(-0.637402\pi\)
−0.418379 + 0.908273i \(0.637402\pi\)
\(752\) 2.17542e9 0.186544
\(753\) 0 0
\(754\) −1.62579e9 −0.138123
\(755\) 3.34664e9 0.283006
\(756\) 0 0
\(757\) −1.39407e10 −1.16802 −0.584008 0.811748i \(-0.698516\pi\)
−0.584008 + 0.811748i \(0.698516\pi\)
\(758\) 1.67616e8 0.0139789
\(759\) 0 0
\(760\) −4.23016e9 −0.349550
\(761\) −9.33520e9 −0.767852 −0.383926 0.923364i \(-0.625428\pi\)
−0.383926 + 0.923364i \(0.625428\pi\)
\(762\) 0 0
\(763\) −2.21217e10 −1.80295
\(764\) −4.03558e9 −0.327401
\(765\) 0 0
\(766\) 9.43258e9 0.758281
\(767\) 5.27399e9 0.422042
\(768\) 0 0
\(769\) −1.02873e10 −0.815754 −0.407877 0.913037i \(-0.633731\pi\)
−0.407877 + 0.913037i \(0.633731\pi\)
\(770\) −1.51141e10 −1.19307
\(771\) 0 0
\(772\) 8.12408e9 0.635497
\(773\) 6.15947e9 0.479640 0.239820 0.970817i \(-0.422912\pi\)
0.239820 + 0.970817i \(0.422912\pi\)
\(774\) 0 0
\(775\) −1.38498e9 −0.106878
\(776\) −9.37594e9 −0.720275
\(777\) 0 0
\(778\) −9.06863e9 −0.690420
\(779\) 1.32084e10 1.00108
\(780\) 0 0
\(781\) 9.94486e9 0.746999
\(782\) −2.24919e9 −0.168191
\(783\) 0 0
\(784\) −4.00875e9 −0.297100
\(785\) 1.27750e10 0.942579
\(786\) 0 0
\(787\) 6.80084e9 0.497337 0.248669 0.968589i \(-0.420007\pi\)
0.248669 + 0.968589i \(0.420007\pi\)
\(788\) 5.22564e9 0.380450
\(789\) 0 0
\(790\) 6.54287e9 0.472143
\(791\) 1.31469e10 0.944506
\(792\) 0 0
\(793\) −1.03003e10 −0.733489
\(794\) −5.05215e9 −0.358183
\(795\) 0 0
\(796\) 1.18553e10 0.833140
\(797\) 1.73727e10 1.21552 0.607762 0.794119i \(-0.292067\pi\)
0.607762 + 0.794119i \(0.292067\pi\)
\(798\) 0 0
\(799\) 3.09770e9 0.214845
\(800\) 8.23255e9 0.568486
\(801\) 0 0
\(802\) 5.49849e9 0.376386
\(803\) −1.95810e10 −1.33453
\(804\) 0 0
\(805\) 9.55061e9 0.645276
\(806\) −1.84936e9 −0.124408
\(807\) 0 0
\(808\) −1.29915e10 −0.866404
\(809\) −9.63683e9 −0.639903 −0.319952 0.947434i \(-0.603667\pi\)
−0.319952 + 0.947434i \(0.603667\pi\)
\(810\) 0 0
\(811\) 9.66741e9 0.636410 0.318205 0.948022i \(-0.396920\pi\)
0.318205 + 0.948022i \(0.396920\pi\)
\(812\) −1.96756e9 −0.128968
\(813\) 0 0
\(814\) −7.97770e9 −0.518433
\(815\) 1.40856e10 0.911433
\(816\) 0 0
\(817\) −1.32494e9 −0.0850000
\(818\) −1.17392e10 −0.749897
\(819\) 0 0
\(820\) 8.18381e9 0.518331
\(821\) −4.33448e9 −0.273361 −0.136680 0.990615i \(-0.543643\pi\)
−0.136680 + 0.990615i \(0.543643\pi\)
\(822\) 0 0
\(823\) 7.05830e9 0.441368 0.220684 0.975345i \(-0.429171\pi\)
0.220684 + 0.975345i \(0.429171\pi\)
\(824\) −1.72122e10 −1.07175
\(825\) 0 0
\(826\) −6.65076e9 −0.410621
\(827\) −1.97938e10 −1.21691 −0.608457 0.793587i \(-0.708211\pi\)
−0.608457 + 0.793587i \(0.708211\pi\)
\(828\) 0 0
\(829\) 1.55516e10 0.948055 0.474027 0.880510i \(-0.342800\pi\)
0.474027 + 0.880510i \(0.342800\pi\)
\(830\) 1.24379e10 0.755048
\(831\) 0 0
\(832\) 1.57716e10 0.949390
\(833\) −5.70827e9 −0.342174
\(834\) 0 0
\(835\) −8.74302e9 −0.519707
\(836\) 9.02147e9 0.534017
\(837\) 0 0
\(838\) −1.67311e10 −0.982135
\(839\) 4.02189e9 0.235106 0.117553 0.993067i \(-0.462495\pi\)
0.117553 + 0.993067i \(0.462495\pi\)
\(840\) 0 0
\(841\) −1.66798e10 −0.966950
\(842\) 1.50922e10 0.871287
\(843\) 0 0
\(844\) −8.85822e9 −0.507163
\(845\) −1.35631e9 −0.0773323
\(846\) 0 0
\(847\) 7.24351e10 4.09597
\(848\) −7.28365e9 −0.410170
\(849\) 0 0
\(850\) −2.60042e9 −0.145237
\(851\) 5.04112e9 0.280397
\(852\) 0 0
\(853\) −7.65536e9 −0.422322 −0.211161 0.977451i \(-0.567725\pi\)
−0.211161 + 0.977451i \(0.567725\pi\)
\(854\) 1.29892e10 0.713639
\(855\) 0 0
\(856\) −1.53714e10 −0.837638
\(857\) 1.60019e10 0.868437 0.434218 0.900808i \(-0.357025\pi\)
0.434218 + 0.900808i \(0.357025\pi\)
\(858\) 0 0
\(859\) −9.96999e9 −0.536684 −0.268342 0.963324i \(-0.586476\pi\)
−0.268342 + 0.963324i \(0.586476\pi\)
\(860\) −8.20921e8 −0.0440106
\(861\) 0 0
\(862\) 8.35742e9 0.444423
\(863\) −2.21580e10 −1.17353 −0.586763 0.809759i \(-0.699598\pi\)
−0.586763 + 0.809759i \(0.699598\pi\)
\(864\) 0 0
\(865\) 1.22529e10 0.643701
\(866\) −1.52541e10 −0.798131
\(867\) 0 0
\(868\) −2.23813e9 −0.116162
\(869\) −4.24471e10 −2.19421
\(870\) 0 0
\(871\) 1.47609e10 0.756920
\(872\) −2.59327e10 −1.32446
\(873\) 0 0
\(874\) 5.94011e9 0.300957
\(875\) 2.79592e10 1.41090
\(876\) 0 0
\(877\) −8.03090e9 −0.402037 −0.201018 0.979587i \(-0.564425\pi\)
−0.201018 + 0.979587i \(0.564425\pi\)
\(878\) 1.77152e10 0.883313
\(879\) 0 0
\(880\) −6.30378e9 −0.311825
\(881\) −1.58502e10 −0.780944 −0.390472 0.920615i \(-0.627688\pi\)
−0.390472 + 0.920615i \(0.627688\pi\)
\(882\) 0 0
\(883\) −1.53480e10 −0.750222 −0.375111 0.926980i \(-0.622395\pi\)
−0.375111 + 0.926980i \(0.622395\pi\)
\(884\) 3.33237e9 0.162245
\(885\) 0 0
\(886\) −9.96368e9 −0.481284
\(887\) −6.55736e9 −0.315498 −0.157749 0.987479i \(-0.550424\pi\)
−0.157749 + 0.987479i \(0.550424\pi\)
\(888\) 0 0
\(889\) 1.80368e10 0.861002
\(890\) −1.25751e10 −0.597926
\(891\) 0 0
\(892\) 6.94679e9 0.327723
\(893\) −8.18102e9 −0.384439
\(894\) 0 0
\(895\) 2.39426e10 1.11632
\(896\) 7.27760e9 0.337995
\(897\) 0 0
\(898\) 1.66791e10 0.768611
\(899\) 6.48503e8 0.0297682
\(900\) 0 0
\(901\) −1.03716e10 −0.472398
\(902\) 5.53226e10 2.51004
\(903\) 0 0
\(904\) 1.54117e10 0.693842
\(905\) 7.74518e9 0.347345
\(906\) 0 0
\(907\) −3.93411e10 −1.75074 −0.875368 0.483457i \(-0.839381\pi\)
−0.875368 + 0.483457i \(0.839381\pi\)
\(908\) −9.01733e9 −0.399740
\(909\) 0 0
\(910\) 1.47443e10 0.648605
\(911\) −3.45238e10 −1.51288 −0.756440 0.654064i \(-0.773063\pi\)
−0.756440 + 0.654064i \(0.773063\pi\)
\(912\) 0 0
\(913\) −8.06914e10 −3.50897
\(914\) 2.96700e10 1.28530
\(915\) 0 0
\(916\) 2.20820e10 0.949301
\(917\) 2.92948e10 1.25458
\(918\) 0 0
\(919\) −9.91457e9 −0.421376 −0.210688 0.977553i \(-0.567570\pi\)
−0.210688 + 0.977553i \(0.567570\pi\)
\(920\) 1.11959e10 0.474025
\(921\) 0 0
\(922\) 1.21249e10 0.509473
\(923\) −9.70157e9 −0.406103
\(924\) 0 0
\(925\) 5.82833e9 0.242130
\(926\) −8.08281e9 −0.334521
\(927\) 0 0
\(928\) −3.85481e9 −0.158338
\(929\) 4.62667e10 1.89327 0.946636 0.322304i \(-0.104457\pi\)
0.946636 + 0.322304i \(0.104457\pi\)
\(930\) 0 0
\(931\) 1.50755e10 0.612278
\(932\) 2.37530e8 0.00961086
\(933\) 0 0
\(934\) −9.14169e9 −0.367124
\(935\) −8.97629e9 −0.359134
\(936\) 0 0
\(937\) −2.62108e10 −1.04086 −0.520430 0.853904i \(-0.674228\pi\)
−0.520430 + 0.853904i \(0.674228\pi\)
\(938\) −1.86142e10 −0.736436
\(939\) 0 0
\(940\) −5.06889e9 −0.199052
\(941\) −2.90508e10 −1.13657 −0.568283 0.822833i \(-0.692392\pi\)
−0.568283 + 0.822833i \(0.692392\pi\)
\(942\) 0 0
\(943\) −3.49584e10 −1.35757
\(944\) −2.77390e9 −0.107322
\(945\) 0 0
\(946\) −5.54943e9 −0.213123
\(947\) −6.74455e9 −0.258064 −0.129032 0.991640i \(-0.541187\pi\)
−0.129032 + 0.991640i \(0.541187\pi\)
\(948\) 0 0
\(949\) 1.91019e10 0.725514
\(950\) 6.86770e9 0.259883
\(951\) 0 0
\(952\) −1.27833e10 −0.480191
\(953\) −1.83022e10 −0.684981 −0.342490 0.939521i \(-0.611270\pi\)
−0.342490 + 0.939521i \(0.611270\pi\)
\(954\) 0 0
\(955\) −1.06046e10 −0.393987
\(956\) −5.81462e8 −0.0215238
\(957\) 0 0
\(958\) 3.45141e9 0.126828
\(959\) 2.17187e9 0.0795186
\(960\) 0 0
\(961\) −2.67749e10 −0.973188
\(962\) 7.78254e9 0.281844
\(963\) 0 0
\(964\) −2.23788e10 −0.804576
\(965\) 2.13482e10 0.764744
\(966\) 0 0
\(967\) −2.88943e9 −0.102759 −0.0513795 0.998679i \(-0.516362\pi\)
−0.0513795 + 0.998679i \(0.516362\pi\)
\(968\) 8.49135e10 3.00894
\(969\) 0 0
\(970\) −8.09922e9 −0.284933
\(971\) 5.32082e10 1.86514 0.932569 0.360991i \(-0.117562\pi\)
0.932569 + 0.360991i \(0.117562\pi\)
\(972\) 0 0
\(973\) 2.20621e8 0.00767806
\(974\) −1.50800e10 −0.522933
\(975\) 0 0
\(976\) 5.41751e9 0.186520
\(977\) 1.19053e10 0.408423 0.204211 0.978927i \(-0.434537\pi\)
0.204211 + 0.978927i \(0.434537\pi\)
\(978\) 0 0
\(979\) 8.15816e10 2.77877
\(980\) 9.34067e9 0.317020
\(981\) 0 0
\(982\) −2.16187e10 −0.728518
\(983\) 4.93240e10 1.65623 0.828116 0.560557i \(-0.189413\pi\)
0.828116 + 0.560557i \(0.189413\pi\)
\(984\) 0 0
\(985\) 1.37318e10 0.457826
\(986\) 1.21762e9 0.0404522
\(987\) 0 0
\(988\) −8.80078e9 −0.290317
\(989\) 3.50669e9 0.115269
\(990\) 0 0
\(991\) 3.40025e10 1.10982 0.554911 0.831910i \(-0.312752\pi\)
0.554911 + 0.831910i \(0.312752\pi\)
\(992\) −4.38489e9 −0.142616
\(993\) 0 0
\(994\) 1.22341e10 0.395113
\(995\) 3.11531e10 1.00258
\(996\) 0 0
\(997\) −4.30970e10 −1.37725 −0.688627 0.725116i \(-0.741786\pi\)
−0.688627 + 0.725116i \(0.741786\pi\)
\(998\) 1.63033e10 0.519179
\(999\) 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.1.5 13
3.2 odd 2 43.8.a.b.1.9 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.9 13 3.2 odd 2
387.8.a.d.1.5 13 1.1 even 1 trivial