Properties

Label 387.8.a.b.1.6
Level $387$
Weight $8$
Character 387.1
Self dual yes
Analytic conductor $120.893$
Analytic rank $0$
Dimension $11$
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(3.52766\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.52766 q^{2} -97.4450 q^{4} -304.619 q^{5} +1421.71 q^{7} -1246.18 q^{8} +O(q^{10})\) \(q+5.52766 q^{2} -97.4450 q^{4} -304.619 q^{5} +1421.71 q^{7} -1246.18 q^{8} -1683.83 q^{10} +7143.27 q^{11} -1897.38 q^{13} +7858.73 q^{14} +5584.49 q^{16} -1140.96 q^{17} -16176.5 q^{19} +29683.6 q^{20} +39485.6 q^{22} -91608.2 q^{23} +14667.6 q^{25} -10488.0 q^{26} -138539. q^{28} +229144. q^{29} -212418. q^{31} +190381. q^{32} -6306.85 q^{34} -433079. q^{35} -482724. q^{37} -89418.0 q^{38} +379611. q^{40} +384844. q^{41} +79507.0 q^{43} -696076. q^{44} -506379. q^{46} -380758. q^{47} +1.19772e6 q^{49} +81077.5 q^{50} +184890. q^{52} +361687. q^{53} -2.17598e6 q^{55} -1.77171e6 q^{56} +1.26663e6 q^{58} +1.95582e6 q^{59} -479312. q^{61} -1.17417e6 q^{62} +337544. q^{64} +577976. q^{65} -2.46363e6 q^{67} +111181. q^{68} -2.39392e6 q^{70} +1.56017e6 q^{71} +2.98243e6 q^{73} -2.66833e6 q^{74} +1.57632e6 q^{76} +1.01557e7 q^{77} +2.64376e6 q^{79} -1.70114e6 q^{80} +2.12728e6 q^{82} -5.54958e6 q^{83} +347559. q^{85} +439488. q^{86} -8.90183e6 q^{88} +9.15035e6 q^{89} -2.69752e6 q^{91} +8.92676e6 q^{92} -2.10470e6 q^{94} +4.92766e6 q^{95} +4.33147e6 q^{97} +6.62056e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 24q^{2} + 602q^{4} + 752q^{5} - 12q^{7} + 3810q^{8} + O(q^{10}) \) \( 11q + 24q^{2} + 602q^{4} + 752q^{5} - 12q^{7} + 3810q^{8} - 1333q^{10} - 1333q^{11} - 17967q^{13} + 22352q^{14} - 34406q^{16} + 63095q^{17} - 54524q^{19} + 280995q^{20} - 289358q^{22} + 138139q^{23} + 3455q^{25} + 132946q^{26} - 12704q^{28} + 308658q^{29} - 209523q^{31} + 644934q^{32} + 762435q^{34} + 578892q^{35} - 298472q^{37} + 369707q^{38} + 2633173q^{40} + 1346735q^{41} + 874577q^{43} - 3134292q^{44} + 3588111q^{46} - 499284q^{47} + 2544563q^{49} - 3049745q^{50} + 983088q^{52} + 2210495q^{53} - 1855072q^{55} + 469976q^{56} + 4397067q^{58} + 5824216q^{59} - 4453034q^{61} - 1002789q^{62} + 4757538q^{64} + 2162872q^{65} - 6859513q^{67} + 9397005q^{68} + 845078q^{70} + 10726554q^{71} - 4456898q^{73} - 1046637q^{74} + 5861267q^{76} + 17019816q^{77} - 15541320q^{79} + 15680911q^{80} + 20233655q^{82} + 11146767q^{83} - 12471976q^{85} + 1908168q^{86} - 24463544q^{88} + 13531356q^{89} - 19746448q^{91} + 26023161q^{92} + 20288857q^{94} + 12291624q^{95} - 10999901q^{97} - 29909168q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.52766 0.488581 0.244290 0.969702i \(-0.421445\pi\)
0.244290 + 0.969702i \(0.421445\pi\)
\(3\) 0 0
\(4\) −97.4450 −0.761289
\(5\) −304.619 −1.08984 −0.544919 0.838489i \(-0.683439\pi\)
−0.544919 + 0.838489i \(0.683439\pi\)
\(6\) 0 0
\(7\) 1421.71 1.56663 0.783317 0.621622i \(-0.213526\pi\)
0.783317 + 0.621622i \(0.213526\pi\)
\(8\) −1246.18 −0.860532
\(9\) 0 0
\(10\) −1683.83 −0.532473
\(11\) 7143.27 1.61817 0.809083 0.587695i \(-0.199964\pi\)
0.809083 + 0.587695i \(0.199964\pi\)
\(12\) 0 0
\(13\) −1897.38 −0.239525 −0.119763 0.992803i \(-0.538213\pi\)
−0.119763 + 0.992803i \(0.538213\pi\)
\(14\) 7858.73 0.765427
\(15\) 0 0
\(16\) 5584.49 0.340850
\(17\) −1140.96 −0.0563249 −0.0281624 0.999603i \(-0.508966\pi\)
−0.0281624 + 0.999603i \(0.508966\pi\)
\(18\) 0 0
\(19\) −16176.5 −0.541061 −0.270531 0.962711i \(-0.587199\pi\)
−0.270531 + 0.962711i \(0.587199\pi\)
\(20\) 29683.6 0.829681
\(21\) 0 0
\(22\) 39485.6 0.790604
\(23\) −91608.2 −1.56995 −0.784977 0.619525i \(-0.787325\pi\)
−0.784977 + 0.619525i \(0.787325\pi\)
\(24\) 0 0
\(25\) 14667.6 0.187745
\(26\) −10488.0 −0.117027
\(27\) 0 0
\(28\) −138539. −1.19266
\(29\) 229144. 1.74468 0.872341 0.488898i \(-0.162601\pi\)
0.872341 + 0.488898i \(0.162601\pi\)
\(30\) 0 0
\(31\) −212418. −1.28064 −0.640318 0.768110i \(-0.721197\pi\)
−0.640318 + 0.768110i \(0.721197\pi\)
\(32\) 190381. 1.02706
\(33\) 0 0
\(34\) −6306.85 −0.0275192
\(35\) −433079. −1.70738
\(36\) 0 0
\(37\) −482724. −1.56672 −0.783362 0.621566i \(-0.786497\pi\)
−0.783362 + 0.621566i \(0.786497\pi\)
\(38\) −89418.0 −0.264352
\(39\) 0 0
\(40\) 379611. 0.937839
\(41\) 384844. 0.872049 0.436025 0.899935i \(-0.356386\pi\)
0.436025 + 0.899935i \(0.356386\pi\)
\(42\) 0 0
\(43\) 79507.0 0.152499
\(44\) −696076. −1.23189
\(45\) 0 0
\(46\) −506379. −0.767049
\(47\) −380758. −0.534941 −0.267471 0.963566i \(-0.586188\pi\)
−0.267471 + 0.963566i \(0.586188\pi\)
\(48\) 0 0
\(49\) 1.19772e6 1.45435
\(50\) 81077.5 0.0917287
\(51\) 0 0
\(52\) 184890. 0.182348
\(53\) 361687. 0.333709 0.166854 0.985982i \(-0.446639\pi\)
0.166854 + 0.985982i \(0.446639\pi\)
\(54\) 0 0
\(55\) −2.17598e6 −1.76354
\(56\) −1.77171e6 −1.34814
\(57\) 0 0
\(58\) 1.26663e6 0.852418
\(59\) 1.95582e6 1.23979 0.619894 0.784685i \(-0.287176\pi\)
0.619894 + 0.784685i \(0.287176\pi\)
\(60\) 0 0
\(61\) −479312. −0.270374 −0.135187 0.990820i \(-0.543163\pi\)
−0.135187 + 0.990820i \(0.543163\pi\)
\(62\) −1.17417e6 −0.625694
\(63\) 0 0
\(64\) 337544. 0.160954
\(65\) 577976. 0.261044
\(66\) 0 0
\(67\) −2.46363e6 −1.00072 −0.500362 0.865816i \(-0.666800\pi\)
−0.500362 + 0.865816i \(0.666800\pi\)
\(68\) 111181. 0.0428795
\(69\) 0 0
\(70\) −2.39392e6 −0.834191
\(71\) 1.56017e6 0.517332 0.258666 0.965967i \(-0.416717\pi\)
0.258666 + 0.965967i \(0.416717\pi\)
\(72\) 0 0
\(73\) 2.98243e6 0.897305 0.448652 0.893706i \(-0.351904\pi\)
0.448652 + 0.893706i \(0.351904\pi\)
\(74\) −2.66833e6 −0.765471
\(75\) 0 0
\(76\) 1.57632e6 0.411904
\(77\) 1.01557e7 2.53507
\(78\) 0 0
\(79\) 2.64376e6 0.603292 0.301646 0.953420i \(-0.402464\pi\)
0.301646 + 0.953420i \(0.402464\pi\)
\(80\) −1.70114e6 −0.371471
\(81\) 0 0
\(82\) 2.12728e6 0.426066
\(83\) −5.54958e6 −1.06534 −0.532668 0.846324i \(-0.678810\pi\)
−0.532668 + 0.846324i \(0.678810\pi\)
\(84\) 0 0
\(85\) 347559. 0.0613850
\(86\) 439488. 0.0745078
\(87\) 0 0
\(88\) −8.90183e6 −1.39248
\(89\) 9.15035e6 1.37585 0.687927 0.725779i \(-0.258521\pi\)
0.687927 + 0.725779i \(0.258521\pi\)
\(90\) 0 0
\(91\) −2.69752e6 −0.375249
\(92\) 8.92676e6 1.19519
\(93\) 0 0
\(94\) −2.10470e6 −0.261362
\(95\) 4.92766e6 0.589669
\(96\) 0 0
\(97\) 4.33147e6 0.481875 0.240937 0.970541i \(-0.422545\pi\)
0.240937 + 0.970541i \(0.422545\pi\)
\(98\) 6.62056e6 0.710565
\(99\) 0 0
\(100\) −1.42928e6 −0.142928
\(101\) 5.03114e6 0.485894 0.242947 0.970040i \(-0.421886\pi\)
0.242947 + 0.970040i \(0.421886\pi\)
\(102\) 0 0
\(103\) 1.58973e7 1.43348 0.716741 0.697339i \(-0.245633\pi\)
0.716741 + 0.697339i \(0.245633\pi\)
\(104\) 2.36448e6 0.206119
\(105\) 0 0
\(106\) 1.99928e6 0.163044
\(107\) 1.56322e7 1.23361 0.616805 0.787116i \(-0.288427\pi\)
0.616805 + 0.787116i \(0.288427\pi\)
\(108\) 0 0
\(109\) 1.27583e7 0.943624 0.471812 0.881699i \(-0.343600\pi\)
0.471812 + 0.881699i \(0.343600\pi\)
\(110\) −1.20280e7 −0.861630
\(111\) 0 0
\(112\) 7.93952e6 0.533987
\(113\) 1.58622e7 1.03416 0.517081 0.855937i \(-0.327019\pi\)
0.517081 + 0.855937i \(0.327019\pi\)
\(114\) 0 0
\(115\) 2.79056e7 1.71099
\(116\) −2.23290e7 −1.32821
\(117\) 0 0
\(118\) 1.08111e7 0.605737
\(119\) −1.62212e6 −0.0882405
\(120\) 0 0
\(121\) 3.15392e7 1.61846
\(122\) −2.64948e6 −0.132099
\(123\) 0 0
\(124\) 2.06991e7 0.974934
\(125\) 1.93303e7 0.885225
\(126\) 0 0
\(127\) −2.44940e7 −1.06108 −0.530539 0.847660i \(-0.678011\pi\)
−0.530539 + 0.847660i \(0.678011\pi\)
\(128\) −2.25029e7 −0.948425
\(129\) 0 0
\(130\) 3.19486e6 0.127541
\(131\) −3.59703e7 −1.39796 −0.698979 0.715142i \(-0.746362\pi\)
−0.698979 + 0.715142i \(0.746362\pi\)
\(132\) 0 0
\(133\) −2.29983e7 −0.847645
\(134\) −1.36181e7 −0.488934
\(135\) 0 0
\(136\) 1.42185e6 0.0484693
\(137\) 1.88193e7 0.625289 0.312644 0.949870i \(-0.398785\pi\)
0.312644 + 0.949870i \(0.398785\pi\)
\(138\) 0 0
\(139\) −3.72014e7 −1.17492 −0.587459 0.809254i \(-0.699872\pi\)
−0.587459 + 0.809254i \(0.699872\pi\)
\(140\) 4.22014e7 1.29981
\(141\) 0 0
\(142\) 8.62411e6 0.252758
\(143\) −1.35535e7 −0.387592
\(144\) 0 0
\(145\) −6.98017e7 −1.90142
\(146\) 1.64858e7 0.438406
\(147\) 0 0
\(148\) 4.70390e7 1.19273
\(149\) 4.39692e7 1.08892 0.544461 0.838786i \(-0.316734\pi\)
0.544461 + 0.838786i \(0.316734\pi\)
\(150\) 0 0
\(151\) −3.09651e7 −0.731902 −0.365951 0.930634i \(-0.619256\pi\)
−0.365951 + 0.930634i \(0.619256\pi\)
\(152\) 2.01588e7 0.465600
\(153\) 0 0
\(154\) 5.61370e7 1.23859
\(155\) 6.47065e7 1.39568
\(156\) 0 0
\(157\) −1.67885e6 −0.0346228 −0.0173114 0.999850i \(-0.505511\pi\)
−0.0173114 + 0.999850i \(0.505511\pi\)
\(158\) 1.46138e7 0.294757
\(159\) 0 0
\(160\) −5.79935e7 −1.11933
\(161\) −1.30240e8 −2.45954
\(162\) 0 0
\(163\) 8.30485e6 0.150202 0.0751009 0.997176i \(-0.476072\pi\)
0.0751009 + 0.997176i \(0.476072\pi\)
\(164\) −3.75011e7 −0.663881
\(165\) 0 0
\(166\) −3.06762e7 −0.520502
\(167\) −4.62254e7 −0.768021 −0.384010 0.923329i \(-0.625457\pi\)
−0.384010 + 0.923329i \(0.625457\pi\)
\(168\) 0 0
\(169\) −5.91485e7 −0.942628
\(170\) 1.92119e6 0.0299915
\(171\) 0 0
\(172\) −7.74756e6 −0.116095
\(173\) 4.56672e7 0.670568 0.335284 0.942117i \(-0.391168\pi\)
0.335284 + 0.942117i \(0.391168\pi\)
\(174\) 0 0
\(175\) 2.08531e7 0.294128
\(176\) 3.98915e7 0.551552
\(177\) 0 0
\(178\) 5.05800e7 0.672216
\(179\) −7.68660e7 −1.00173 −0.500863 0.865527i \(-0.666984\pi\)
−0.500863 + 0.865527i \(0.666984\pi\)
\(180\) 0 0
\(181\) 7.38362e7 0.925538 0.462769 0.886479i \(-0.346856\pi\)
0.462769 + 0.886479i \(0.346856\pi\)
\(182\) −1.49110e7 −0.183339
\(183\) 0 0
\(184\) 1.14161e8 1.35100
\(185\) 1.47047e8 1.70747
\(186\) 0 0
\(187\) −8.15021e6 −0.0911430
\(188\) 3.71029e7 0.407245
\(189\) 0 0
\(190\) 2.72384e7 0.288101
\(191\) 4.61537e7 0.479280 0.239640 0.970862i \(-0.422971\pi\)
0.239640 + 0.970862i \(0.422971\pi\)
\(192\) 0 0
\(193\) 1.04470e8 1.04603 0.523013 0.852325i \(-0.324808\pi\)
0.523013 + 0.852325i \(0.324808\pi\)
\(194\) 2.39429e7 0.235435
\(195\) 0 0
\(196\) −1.16711e8 −1.10718
\(197\) −3.24556e7 −0.302453 −0.151226 0.988499i \(-0.548322\pi\)
−0.151226 + 0.988499i \(0.548322\pi\)
\(198\) 0 0
\(199\) 1.90706e8 1.71545 0.857725 0.514109i \(-0.171877\pi\)
0.857725 + 0.514109i \(0.171877\pi\)
\(200\) −1.82785e7 −0.161561
\(201\) 0 0
\(202\) 2.78104e7 0.237398
\(203\) 3.25777e8 2.73328
\(204\) 0 0
\(205\) −1.17231e8 −0.950392
\(206\) 8.78747e7 0.700372
\(207\) 0 0
\(208\) −1.05959e7 −0.0816422
\(209\) −1.15553e8 −0.875526
\(210\) 0 0
\(211\) −4.93604e6 −0.0361735 −0.0180867 0.999836i \(-0.505758\pi\)
−0.0180867 + 0.999836i \(0.505758\pi\)
\(212\) −3.52446e7 −0.254049
\(213\) 0 0
\(214\) 8.64097e7 0.602718
\(215\) −2.42193e7 −0.166199
\(216\) 0 0
\(217\) −3.01997e8 −2.00629
\(218\) 7.05233e7 0.461036
\(219\) 0 0
\(220\) 2.12038e8 1.34256
\(221\) 2.16484e6 0.0134912
\(222\) 0 0
\(223\) 2.77562e8 1.67608 0.838038 0.545612i \(-0.183703\pi\)
0.838038 + 0.545612i \(0.183703\pi\)
\(224\) 2.70666e8 1.60903
\(225\) 0 0
\(226\) 8.76807e7 0.505271
\(227\) 1.27335e8 0.722535 0.361267 0.932462i \(-0.382344\pi\)
0.361267 + 0.932462i \(0.382344\pi\)
\(228\) 0 0
\(229\) 1.66075e8 0.913861 0.456931 0.889502i \(-0.348949\pi\)
0.456931 + 0.889502i \(0.348949\pi\)
\(230\) 1.54252e8 0.835959
\(231\) 0 0
\(232\) −2.85556e8 −1.50135
\(233\) 1.97183e8 1.02123 0.510616 0.859809i \(-0.329417\pi\)
0.510616 + 0.859809i \(0.329417\pi\)
\(234\) 0 0
\(235\) 1.15986e8 0.582999
\(236\) −1.90585e8 −0.943837
\(237\) 0 0
\(238\) −8.96652e6 −0.0431126
\(239\) −7.22966e7 −0.342551 −0.171275 0.985223i \(-0.554789\pi\)
−0.171275 + 0.985223i \(0.554789\pi\)
\(240\) 0 0
\(241\) −3.09906e8 −1.42617 −0.713084 0.701079i \(-0.752702\pi\)
−0.713084 + 0.701079i \(0.752702\pi\)
\(242\) 1.74338e8 0.790748
\(243\) 0 0
\(244\) 4.67066e7 0.205832
\(245\) −3.64847e8 −1.58500
\(246\) 0 0
\(247\) 3.06929e7 0.129598
\(248\) 2.64712e8 1.10203
\(249\) 0 0
\(250\) 1.06851e8 0.432504
\(251\) −1.11482e8 −0.444988 −0.222494 0.974934i \(-0.571420\pi\)
−0.222494 + 0.974934i \(0.571420\pi\)
\(252\) 0 0
\(253\) −6.54383e8 −2.54045
\(254\) −1.35395e8 −0.518422
\(255\) 0 0
\(256\) −1.67594e8 −0.624336
\(257\) −8.26622e7 −0.303767 −0.151884 0.988398i \(-0.548534\pi\)
−0.151884 + 0.988398i \(0.548534\pi\)
\(258\) 0 0
\(259\) −6.86293e8 −2.45448
\(260\) −5.63209e7 −0.198730
\(261\) 0 0
\(262\) −1.98831e8 −0.683015
\(263\) 2.74508e8 0.930486 0.465243 0.885183i \(-0.345967\pi\)
0.465243 + 0.885183i \(0.345967\pi\)
\(264\) 0 0
\(265\) −1.10177e8 −0.363688
\(266\) −1.27126e8 −0.414143
\(267\) 0 0
\(268\) 2.40069e8 0.761840
\(269\) 9.06481e7 0.283940 0.141970 0.989871i \(-0.454656\pi\)
0.141970 + 0.989871i \(0.454656\pi\)
\(270\) 0 0
\(271\) 2.46059e8 0.751011 0.375505 0.926820i \(-0.377469\pi\)
0.375505 + 0.926820i \(0.377469\pi\)
\(272\) −6.37169e6 −0.0191983
\(273\) 0 0
\(274\) 1.04026e8 0.305504
\(275\) 1.04775e8 0.303803
\(276\) 0 0
\(277\) 6.60621e8 1.86755 0.933777 0.357857i \(-0.116492\pi\)
0.933777 + 0.357857i \(0.116492\pi\)
\(278\) −2.05637e8 −0.574042
\(279\) 0 0
\(280\) 5.39696e8 1.46925
\(281\) −2.01108e7 −0.0540700 −0.0270350 0.999634i \(-0.508607\pi\)
−0.0270350 + 0.999634i \(0.508607\pi\)
\(282\) 0 0
\(283\) 2.51117e8 0.658602 0.329301 0.944225i \(-0.393187\pi\)
0.329301 + 0.944225i \(0.393187\pi\)
\(284\) −1.52031e8 −0.393839
\(285\) 0 0
\(286\) −7.49190e7 −0.189370
\(287\) 5.47136e8 1.36618
\(288\) 0 0
\(289\) −4.09037e8 −0.996828
\(290\) −3.85840e8 −0.928997
\(291\) 0 0
\(292\) −2.90623e8 −0.683108
\(293\) 3.74733e8 0.870333 0.435166 0.900350i \(-0.356690\pi\)
0.435166 + 0.900350i \(0.356690\pi\)
\(294\) 0 0
\(295\) −5.95780e8 −1.35117
\(296\) 6.01562e8 1.34822
\(297\) 0 0
\(298\) 2.43047e8 0.532027
\(299\) 1.73815e8 0.376044
\(300\) 0 0
\(301\) 1.13036e8 0.238910
\(302\) −1.71164e8 −0.357593
\(303\) 0 0
\(304\) −9.03373e7 −0.184421
\(305\) 1.46008e8 0.294663
\(306\) 0 0
\(307\) 3.69155e8 0.728156 0.364078 0.931369i \(-0.381384\pi\)
0.364078 + 0.931369i \(0.381384\pi\)
\(308\) −9.89619e8 −1.92992
\(309\) 0 0
\(310\) 3.57676e8 0.681904
\(311\) 7.67471e8 1.44677 0.723387 0.690443i \(-0.242584\pi\)
0.723387 + 0.690443i \(0.242584\pi\)
\(312\) 0 0
\(313\) 3.94815e8 0.727761 0.363881 0.931446i \(-0.381452\pi\)
0.363881 + 0.931446i \(0.381452\pi\)
\(314\) −9.28008e6 −0.0169160
\(315\) 0 0
\(316\) −2.57621e8 −0.459279
\(317\) −5.25976e8 −0.927381 −0.463691 0.885997i \(-0.653475\pi\)
−0.463691 + 0.885997i \(0.653475\pi\)
\(318\) 0 0
\(319\) 1.63684e9 2.82318
\(320\) −1.02822e8 −0.175413
\(321\) 0 0
\(322\) −7.19924e8 −1.20169
\(323\) 1.84568e7 0.0304752
\(324\) 0 0
\(325\) −2.78299e7 −0.0449698
\(326\) 4.59064e7 0.0733857
\(327\) 0 0
\(328\) −4.79586e8 −0.750426
\(329\) −5.41327e8 −0.838058
\(330\) 0 0
\(331\) 1.17755e9 1.78477 0.892386 0.451273i \(-0.149030\pi\)
0.892386 + 0.451273i \(0.149030\pi\)
\(332\) 5.40778e8 0.811028
\(333\) 0 0
\(334\) −2.55518e8 −0.375240
\(335\) 7.50469e8 1.09063
\(336\) 0 0
\(337\) −4.76656e8 −0.678423 −0.339212 0.940710i \(-0.610160\pi\)
−0.339212 + 0.940710i \(0.610160\pi\)
\(338\) −3.26953e8 −0.460550
\(339\) 0 0
\(340\) −3.38679e7 −0.0467317
\(341\) −1.51736e9 −2.07228
\(342\) 0 0
\(343\) 5.31965e8 0.711793
\(344\) −9.90803e7 −0.131230
\(345\) 0 0
\(346\) 2.52433e8 0.327626
\(347\) −1.08505e8 −0.139411 −0.0697057 0.997568i \(-0.522206\pi\)
−0.0697057 + 0.997568i \(0.522206\pi\)
\(348\) 0 0
\(349\) −7.46559e8 −0.940102 −0.470051 0.882639i \(-0.655765\pi\)
−0.470051 + 0.882639i \(0.655765\pi\)
\(350\) 1.15269e8 0.143705
\(351\) 0 0
\(352\) 1.35994e9 1.66196
\(353\) 4.13749e8 0.500640 0.250320 0.968163i \(-0.419464\pi\)
0.250320 + 0.968163i \(0.419464\pi\)
\(354\) 0 0
\(355\) −4.75259e8 −0.563807
\(356\) −8.91656e8 −1.04742
\(357\) 0 0
\(358\) −4.24889e8 −0.489424
\(359\) 1.58431e8 0.180721 0.0903607 0.995909i \(-0.471198\pi\)
0.0903607 + 0.995909i \(0.471198\pi\)
\(360\) 0 0
\(361\) −6.32193e8 −0.707253
\(362\) 4.08141e8 0.452200
\(363\) 0 0
\(364\) 2.62860e8 0.285673
\(365\) −9.08504e8 −0.977916
\(366\) 0 0
\(367\) 5.61609e7 0.0593065 0.0296533 0.999560i \(-0.490560\pi\)
0.0296533 + 0.999560i \(0.490560\pi\)
\(368\) −5.11585e8 −0.535119
\(369\) 0 0
\(370\) 8.12824e8 0.834239
\(371\) 5.14214e8 0.522800
\(372\) 0 0
\(373\) 1.21691e9 1.21416 0.607080 0.794641i \(-0.292341\pi\)
0.607080 + 0.794641i \(0.292341\pi\)
\(374\) −4.50516e7 −0.0445307
\(375\) 0 0
\(376\) 4.74494e8 0.460334
\(377\) −4.34773e8 −0.417896
\(378\) 0 0
\(379\) −1.99936e9 −1.88649 −0.943245 0.332098i \(-0.892244\pi\)
−0.943245 + 0.332098i \(0.892244\pi\)
\(380\) −4.80176e8 −0.448908
\(381\) 0 0
\(382\) 2.55122e8 0.234167
\(383\) 7.65586e8 0.696303 0.348151 0.937438i \(-0.386809\pi\)
0.348151 + 0.937438i \(0.386809\pi\)
\(384\) 0 0
\(385\) −3.09361e9 −2.76282
\(386\) 5.77476e8 0.511068
\(387\) 0 0
\(388\) −4.22080e8 −0.366846
\(389\) 6.56361e8 0.565352 0.282676 0.959215i \(-0.408778\pi\)
0.282676 + 0.959215i \(0.408778\pi\)
\(390\) 0 0
\(391\) 1.04522e8 0.0884275
\(392\) −1.49257e9 −1.25151
\(393\) 0 0
\(394\) −1.79403e8 −0.147773
\(395\) −8.05339e8 −0.657490
\(396\) 0 0
\(397\) 9.12150e8 0.731644 0.365822 0.930685i \(-0.380788\pi\)
0.365822 + 0.930685i \(0.380788\pi\)
\(398\) 1.05416e9 0.838136
\(399\) 0 0
\(400\) 8.19110e7 0.0639930
\(401\) 1.15178e8 0.0891997 0.0445999 0.999005i \(-0.485799\pi\)
0.0445999 + 0.999005i \(0.485799\pi\)
\(402\) 0 0
\(403\) 4.03037e8 0.306745
\(404\) −4.90259e8 −0.369906
\(405\) 0 0
\(406\) 1.80078e9 1.33543
\(407\) −3.44823e9 −2.53522
\(408\) 0 0
\(409\) −8.39721e8 −0.606881 −0.303440 0.952850i \(-0.598135\pi\)
−0.303440 + 0.952850i \(0.598135\pi\)
\(410\) −6.48011e8 −0.464343
\(411\) 0 0
\(412\) −1.54911e9 −1.09129
\(413\) 2.78061e9 1.94230
\(414\) 0 0
\(415\) 1.69051e9 1.16104
\(416\) −3.61223e8 −0.246008
\(417\) 0 0
\(418\) −6.38738e8 −0.427765
\(419\) −9.18009e8 −0.609674 −0.304837 0.952404i \(-0.598602\pi\)
−0.304837 + 0.952404i \(0.598602\pi\)
\(420\) 0 0
\(421\) 9.86735e7 0.0644486 0.0322243 0.999481i \(-0.489741\pi\)
0.0322243 + 0.999481i \(0.489741\pi\)
\(422\) −2.72847e7 −0.0176737
\(423\) 0 0
\(424\) −4.50728e8 −0.287167
\(425\) −1.67352e7 −0.0105747
\(426\) 0 0
\(427\) −6.81443e8 −0.423577
\(428\) −1.52328e9 −0.939134
\(429\) 0 0
\(430\) −1.33876e8 −0.0812014
\(431\) −2.42488e9 −1.45888 −0.729440 0.684045i \(-0.760219\pi\)
−0.729440 + 0.684045i \(0.760219\pi\)
\(432\) 0 0
\(433\) 1.63675e9 0.968888 0.484444 0.874822i \(-0.339022\pi\)
0.484444 + 0.874822i \(0.339022\pi\)
\(434\) −1.66934e9 −0.980233
\(435\) 0 0
\(436\) −1.24323e9 −0.718370
\(437\) 1.48190e9 0.849441
\(438\) 0 0
\(439\) −1.40300e9 −0.791465 −0.395733 0.918366i \(-0.629509\pi\)
−0.395733 + 0.918366i \(0.629509\pi\)
\(440\) 2.71166e9 1.51758
\(441\) 0 0
\(442\) 1.19665e7 0.00659156
\(443\) −4.39909e8 −0.240409 −0.120204 0.992749i \(-0.538355\pi\)
−0.120204 + 0.992749i \(0.538355\pi\)
\(444\) 0 0
\(445\) −2.78737e9 −1.49946
\(446\) 1.53427e9 0.818898
\(447\) 0 0
\(448\) 4.79890e8 0.252156
\(449\) −2.11015e9 −1.10015 −0.550073 0.835116i \(-0.685400\pi\)
−0.550073 + 0.835116i \(0.685400\pi\)
\(450\) 0 0
\(451\) 2.74904e9 1.41112
\(452\) −1.54569e9 −0.787295
\(453\) 0 0
\(454\) 7.03866e8 0.353016
\(455\) 8.21715e8 0.408960
\(456\) 0 0
\(457\) −2.13547e9 −1.04661 −0.523307 0.852144i \(-0.675302\pi\)
−0.523307 + 0.852144i \(0.675302\pi\)
\(458\) 9.18006e8 0.446495
\(459\) 0 0
\(460\) −2.71926e9 −1.30256
\(461\) −7.44367e8 −0.353862 −0.176931 0.984223i \(-0.556617\pi\)
−0.176931 + 0.984223i \(0.556617\pi\)
\(462\) 0 0
\(463\) 1.70021e9 0.796105 0.398052 0.917363i \(-0.369686\pi\)
0.398052 + 0.917363i \(0.369686\pi\)
\(464\) 1.27965e9 0.594675
\(465\) 0 0
\(466\) 1.08996e9 0.498954
\(467\) 3.93170e9 1.78637 0.893185 0.449689i \(-0.148465\pi\)
0.893185 + 0.449689i \(0.148465\pi\)
\(468\) 0 0
\(469\) −3.50257e9 −1.56777
\(470\) 6.41131e8 0.284842
\(471\) 0 0
\(472\) −2.43731e9 −1.06688
\(473\) 5.67940e8 0.246768
\(474\) 0 0
\(475\) −2.37270e8 −0.101582
\(476\) 1.58067e8 0.0671766
\(477\) 0 0
\(478\) −3.99631e8 −0.167364
\(479\) 1.45058e9 0.603071 0.301536 0.953455i \(-0.402501\pi\)
0.301536 + 0.953455i \(0.402501\pi\)
\(480\) 0 0
\(481\) 9.15908e8 0.375270
\(482\) −1.71305e9 −0.696798
\(483\) 0 0
\(484\) −3.07334e9 −1.23212
\(485\) −1.31945e9 −0.525165
\(486\) 0 0
\(487\) 4.29131e9 1.68360 0.841800 0.539790i \(-0.181496\pi\)
0.841800 + 0.539790i \(0.181496\pi\)
\(488\) 5.97311e8 0.232665
\(489\) 0 0
\(490\) −2.01675e9 −0.774400
\(491\) 3.50889e8 0.133778 0.0668889 0.997760i \(-0.478693\pi\)
0.0668889 + 0.997760i \(0.478693\pi\)
\(492\) 0 0
\(493\) −2.61445e8 −0.0982690
\(494\) 1.69660e8 0.0633190
\(495\) 0 0
\(496\) −1.18625e9 −0.436505
\(497\) 2.21812e9 0.810470
\(498\) 0 0
\(499\) −3.72276e9 −1.34126 −0.670630 0.741792i \(-0.733976\pi\)
−0.670630 + 0.741792i \(0.733976\pi\)
\(500\) −1.88364e9 −0.673912
\(501\) 0 0
\(502\) −6.16236e8 −0.217412
\(503\) −5.48803e9 −1.92277 −0.961387 0.275199i \(-0.911256\pi\)
−0.961387 + 0.275199i \(0.911256\pi\)
\(504\) 0 0
\(505\) −1.53258e9 −0.529545
\(506\) −3.61720e9 −1.24121
\(507\) 0 0
\(508\) 2.38682e9 0.807788
\(509\) −3.85451e8 −0.129556 −0.0647778 0.997900i \(-0.520634\pi\)
−0.0647778 + 0.997900i \(0.520634\pi\)
\(510\) 0 0
\(511\) 4.24015e9 1.40575
\(512\) 1.95397e9 0.643387
\(513\) 0 0
\(514\) −4.56928e8 −0.148415
\(515\) −4.84261e9 −1.56226
\(516\) 0 0
\(517\) −2.71986e9 −0.865623
\(518\) −3.79359e9 −1.19921
\(519\) 0 0
\(520\) −7.20264e8 −0.224636
\(521\) 3.74846e9 1.16124 0.580618 0.814176i \(-0.302811\pi\)
0.580618 + 0.814176i \(0.302811\pi\)
\(522\) 0 0
\(523\) 2.64980e9 0.809949 0.404974 0.914328i \(-0.367280\pi\)
0.404974 + 0.914328i \(0.367280\pi\)
\(524\) 3.50512e9 1.06425
\(525\) 0 0
\(526\) 1.51739e9 0.454617
\(527\) 2.42361e8 0.0721316
\(528\) 0 0
\(529\) 4.98724e9 1.46476
\(530\) −6.09019e8 −0.177691
\(531\) 0 0
\(532\) 2.24106e9 0.645303
\(533\) −7.30193e8 −0.208878
\(534\) 0 0
\(535\) −4.76187e9 −1.34443
\(536\) 3.07014e9 0.861155
\(537\) 0 0
\(538\) 5.01072e8 0.138727
\(539\) 8.55561e9 2.35337
\(540\) 0 0
\(541\) −5.07052e9 −1.37677 −0.688386 0.725344i \(-0.741681\pi\)
−0.688386 + 0.725344i \(0.741681\pi\)
\(542\) 1.36013e9 0.366929
\(543\) 0 0
\(544\) −2.17217e8 −0.0578493
\(545\) −3.88641e9 −1.02840
\(546\) 0 0
\(547\) 3.00582e9 0.785249 0.392624 0.919699i \(-0.371567\pi\)
0.392624 + 0.919699i \(0.371567\pi\)
\(548\) −1.83384e9 −0.476026
\(549\) 0 0
\(550\) 5.79159e8 0.148432
\(551\) −3.70675e9 −0.943980
\(552\) 0 0
\(553\) 3.75866e9 0.945138
\(554\) 3.65169e9 0.912450
\(555\) 0 0
\(556\) 3.62509e9 0.894452
\(557\) −5.43732e9 −1.33319 −0.666594 0.745421i \(-0.732249\pi\)
−0.666594 + 0.745421i \(0.732249\pi\)
\(558\) 0 0
\(559\) −1.50855e8 −0.0365273
\(560\) −2.41853e9 −0.581959
\(561\) 0 0
\(562\) −1.11166e8 −0.0264176
\(563\) 4.80656e9 1.13516 0.567578 0.823320i \(-0.307881\pi\)
0.567578 + 0.823320i \(0.307881\pi\)
\(564\) 0 0
\(565\) −4.83192e9 −1.12707
\(566\) 1.38809e9 0.321780
\(567\) 0 0
\(568\) −1.94426e9 −0.445180
\(569\) −7.07656e9 −1.61038 −0.805191 0.593015i \(-0.797937\pi\)
−0.805191 + 0.593015i \(0.797937\pi\)
\(570\) 0 0
\(571\) −3.80378e9 −0.855045 −0.427523 0.904005i \(-0.640614\pi\)
−0.427523 + 0.904005i \(0.640614\pi\)
\(572\) 1.32072e9 0.295069
\(573\) 0 0
\(574\) 3.02438e9 0.667490
\(575\) −1.34367e9 −0.294751
\(576\) 0 0
\(577\) −2.81170e8 −0.0609332 −0.0304666 0.999536i \(-0.509699\pi\)
−0.0304666 + 0.999536i \(0.509699\pi\)
\(578\) −2.26102e9 −0.487031
\(579\) 0 0
\(580\) 6.80182e9 1.44753
\(581\) −7.88989e9 −1.66899
\(582\) 0 0
\(583\) 2.58363e9 0.539996
\(584\) −3.71665e9 −0.772159
\(585\) 0 0
\(586\) 2.07140e9 0.425228
\(587\) −8.79495e8 −0.179473 −0.0897367 0.995966i \(-0.528603\pi\)
−0.0897367 + 0.995966i \(0.528603\pi\)
\(588\) 0 0
\(589\) 3.43618e9 0.692902
\(590\) −3.29327e9 −0.660154
\(591\) 0 0
\(592\) −2.69576e9 −0.534018
\(593\) 6.78081e8 0.133534 0.0667668 0.997769i \(-0.478732\pi\)
0.0667668 + 0.997769i \(0.478732\pi\)
\(594\) 0 0
\(595\) 4.94128e8 0.0961678
\(596\) −4.28458e9 −0.828985
\(597\) 0 0
\(598\) 9.60791e8 0.183728
\(599\) 6.13578e9 1.16648 0.583238 0.812301i \(-0.301785\pi\)
0.583238 + 0.812301i \(0.301785\pi\)
\(600\) 0 0
\(601\) −4.78315e8 −0.0898780 −0.0449390 0.998990i \(-0.514309\pi\)
−0.0449390 + 0.998990i \(0.514309\pi\)
\(602\) 6.24824e8 0.116727
\(603\) 0 0
\(604\) 3.01739e9 0.557189
\(605\) −9.60743e9 −1.76386
\(606\) 0 0
\(607\) −8.97283e9 −1.62843 −0.814215 0.580563i \(-0.802832\pi\)
−0.814215 + 0.580563i \(0.802832\pi\)
\(608\) −3.07969e9 −0.555705
\(609\) 0 0
\(610\) 8.07080e8 0.143967
\(611\) 7.22440e8 0.128132
\(612\) 0 0
\(613\) 7.77928e9 1.36404 0.682021 0.731333i \(-0.261101\pi\)
0.682021 + 0.731333i \(0.261101\pi\)
\(614\) 2.04056e9 0.355763
\(615\) 0 0
\(616\) −1.26558e10 −2.18151
\(617\) 8.81774e9 1.51133 0.755665 0.654958i \(-0.227314\pi\)
0.755665 + 0.654958i \(0.227314\pi\)
\(618\) 0 0
\(619\) −4.50857e9 −0.764050 −0.382025 0.924152i \(-0.624773\pi\)
−0.382025 + 0.924152i \(0.624773\pi\)
\(620\) −6.30533e9 −1.06252
\(621\) 0 0
\(622\) 4.24232e9 0.706865
\(623\) 1.30091e10 2.15546
\(624\) 0 0
\(625\) −7.03428e9 −1.15250
\(626\) 2.18240e9 0.355570
\(627\) 0 0
\(628\) 1.63595e8 0.0263579
\(629\) 5.50770e8 0.0882456
\(630\) 0 0
\(631\) −1.47926e9 −0.234391 −0.117195 0.993109i \(-0.537390\pi\)
−0.117195 + 0.993109i \(0.537390\pi\)
\(632\) −3.29461e9 −0.519152
\(633\) 0 0
\(634\) −2.90741e9 −0.453101
\(635\) 7.46135e9 1.15640
\(636\) 0 0
\(637\) −2.27252e9 −0.348353
\(638\) 9.04790e9 1.37935
\(639\) 0 0
\(640\) 6.85480e9 1.03363
\(641\) −6.56349e9 −0.984309 −0.492155 0.870508i \(-0.663791\pi\)
−0.492155 + 0.870508i \(0.663791\pi\)
\(642\) 0 0
\(643\) 6.43261e9 0.954221 0.477110 0.878843i \(-0.341684\pi\)
0.477110 + 0.878843i \(0.341684\pi\)
\(644\) 1.26913e10 1.87242
\(645\) 0 0
\(646\) 1.02023e8 0.0148896
\(647\) −1.74903e8 −0.0253882 −0.0126941 0.999919i \(-0.504041\pi\)
−0.0126941 + 0.999919i \(0.504041\pi\)
\(648\) 0 0
\(649\) 1.39710e10 2.00618
\(650\) −1.53834e8 −0.0219714
\(651\) 0 0
\(652\) −8.09266e8 −0.114347
\(653\) 1.10036e10 1.54646 0.773230 0.634126i \(-0.218640\pi\)
0.773230 + 0.634126i \(0.218640\pi\)
\(654\) 0 0
\(655\) 1.09572e10 1.52355
\(656\) 2.14915e9 0.297238
\(657\) 0 0
\(658\) −2.99227e9 −0.409459
\(659\) 1.09969e10 1.49683 0.748416 0.663230i \(-0.230815\pi\)
0.748416 + 0.663230i \(0.230815\pi\)
\(660\) 0 0
\(661\) 1.20713e10 1.62574 0.812869 0.582447i \(-0.197905\pi\)
0.812869 + 0.582447i \(0.197905\pi\)
\(662\) 6.50911e9 0.872005
\(663\) 0 0
\(664\) 6.91579e9 0.916755
\(665\) 7.00570e9 0.923795
\(666\) 0 0
\(667\) −2.09915e10 −2.73907
\(668\) 4.50443e9 0.584686
\(669\) 0 0
\(670\) 4.14834e9 0.532859
\(671\) −3.42386e9 −0.437509
\(672\) 0 0
\(673\) 4.08129e9 0.516113 0.258057 0.966130i \(-0.416918\pi\)
0.258057 + 0.966130i \(0.416918\pi\)
\(674\) −2.63479e9 −0.331464
\(675\) 0 0
\(676\) 5.76372e9 0.717612
\(677\) 7.05725e9 0.874128 0.437064 0.899430i \(-0.356018\pi\)
0.437064 + 0.899430i \(0.356018\pi\)
\(678\) 0 0
\(679\) 6.15809e9 0.754922
\(680\) −4.33122e8 −0.0528237
\(681\) 0 0
\(682\) −8.38745e9 −1.01248
\(683\) −7.13819e9 −0.857266 −0.428633 0.903479i \(-0.641005\pi\)
−0.428633 + 0.903479i \(0.641005\pi\)
\(684\) 0 0
\(685\) −5.73270e9 −0.681463
\(686\) 2.94052e9 0.347768
\(687\) 0 0
\(688\) 4.44006e8 0.0519791
\(689\) −6.86257e8 −0.0799317
\(690\) 0 0
\(691\) −1.09452e10 −1.26197 −0.630987 0.775793i \(-0.717350\pi\)
−0.630987 + 0.775793i \(0.717350\pi\)
\(692\) −4.45004e9 −0.510496
\(693\) 0 0
\(694\) −5.99781e8 −0.0681137
\(695\) 1.13322e10 1.28047
\(696\) 0 0
\(697\) −4.39093e8 −0.0491181
\(698\) −4.12672e9 −0.459316
\(699\) 0 0
\(700\) −2.03203e9 −0.223917
\(701\) −1.00558e10 −1.10256 −0.551282 0.834319i \(-0.685861\pi\)
−0.551282 + 0.834319i \(0.685861\pi\)
\(702\) 0 0
\(703\) 7.80877e9 0.847694
\(704\) 2.41117e9 0.260450
\(705\) 0 0
\(706\) 2.28706e9 0.244603
\(707\) 7.15282e9 0.761218
\(708\) 0 0
\(709\) 1.17573e10 1.23892 0.619462 0.785027i \(-0.287351\pi\)
0.619462 + 0.785027i \(0.287351\pi\)
\(710\) −2.62707e9 −0.275465
\(711\) 0 0
\(712\) −1.14030e10 −1.18397
\(713\) 1.94592e10 2.01054
\(714\) 0 0
\(715\) 4.12864e9 0.422412
\(716\) 7.49021e9 0.762603
\(717\) 0 0
\(718\) 8.75753e8 0.0882970
\(719\) 1.29980e9 0.130414 0.0652072 0.997872i \(-0.479229\pi\)
0.0652072 + 0.997872i \(0.479229\pi\)
\(720\) 0 0
\(721\) 2.26013e10 2.24574
\(722\) −3.49455e9 −0.345550
\(723\) 0 0
\(724\) −7.19497e9 −0.704602
\(725\) 3.36100e9 0.327556
\(726\) 0 0
\(727\) 1.55479e10 1.50073 0.750363 0.661026i \(-0.229879\pi\)
0.750363 + 0.661026i \(0.229879\pi\)
\(728\) 3.36160e9 0.322914
\(729\) 0 0
\(730\) −5.02190e9 −0.477791
\(731\) −9.07145e7 −0.00858946
\(732\) 0 0
\(733\) 2.52604e9 0.236906 0.118453 0.992960i \(-0.462206\pi\)
0.118453 + 0.992960i \(0.462206\pi\)
\(734\) 3.10438e8 0.0289760
\(735\) 0 0
\(736\) −1.74404e10 −1.61244
\(737\) −1.75984e10 −1.61934
\(738\) 0 0
\(739\) −8.96231e9 −0.816892 −0.408446 0.912783i \(-0.633929\pi\)
−0.408446 + 0.912783i \(0.633929\pi\)
\(740\) −1.43290e10 −1.29988
\(741\) 0 0
\(742\) 2.84240e9 0.255430
\(743\) −9.22267e9 −0.824890 −0.412445 0.910983i \(-0.635325\pi\)
−0.412445 + 0.910983i \(0.635325\pi\)
\(744\) 0 0
\(745\) −1.33939e10 −1.18675
\(746\) 6.72664e9 0.593215
\(747\) 0 0
\(748\) 7.94197e8 0.0693862
\(749\) 2.22245e10 1.93262
\(750\) 0 0
\(751\) 1.49684e10 1.28954 0.644770 0.764377i \(-0.276953\pi\)
0.644770 + 0.764377i \(0.276953\pi\)
\(752\) −2.12634e9 −0.182335
\(753\) 0 0
\(754\) −2.40328e9 −0.204176
\(755\) 9.43255e9 0.797654
\(756\) 0 0
\(757\) −1.31799e10 −1.10427 −0.552136 0.833754i \(-0.686187\pi\)
−0.552136 + 0.833754i \(0.686187\pi\)
\(758\) −1.10518e10 −0.921702
\(759\) 0 0
\(760\) −6.14076e9 −0.507428
\(761\) 5.65319e9 0.464994 0.232497 0.972597i \(-0.425310\pi\)
0.232497 + 0.972597i \(0.425310\pi\)
\(762\) 0 0
\(763\) 1.81385e10 1.47831
\(764\) −4.49745e9 −0.364871
\(765\) 0 0
\(766\) 4.23190e9 0.340200
\(767\) −3.71093e9 −0.296961
\(768\) 0 0
\(769\) 2.55751e9 0.202803 0.101402 0.994846i \(-0.467667\pi\)
0.101402 + 0.994846i \(0.467667\pi\)
\(770\) −1.71004e10 −1.34986
\(771\) 0 0
\(772\) −1.01801e10 −0.796328
\(773\) −6.24612e9 −0.486387 −0.243194 0.969978i \(-0.578195\pi\)
−0.243194 + 0.969978i \(0.578195\pi\)
\(774\) 0 0
\(775\) −3.11566e9 −0.240433
\(776\) −5.39780e9 −0.414668
\(777\) 0 0
\(778\) 3.62814e9 0.276220
\(779\) −6.22542e9 −0.471832
\(780\) 0 0
\(781\) 1.11448e10 0.837128
\(782\) 5.77760e8 0.0432040
\(783\) 0 0
\(784\) 6.68863e9 0.495713
\(785\) 5.11408e8 0.0377332
\(786\) 0 0
\(787\) −2.24074e10 −1.63863 −0.819314 0.573345i \(-0.805646\pi\)
−0.819314 + 0.573345i \(0.805646\pi\)
\(788\) 3.16263e9 0.230254
\(789\) 0 0
\(790\) −4.45164e9 −0.321237
\(791\) 2.25514e10 1.62015
\(792\) 0 0
\(793\) 9.09436e8 0.0647613
\(794\) 5.04206e9 0.357467
\(795\) 0 0
\(796\) −1.85833e10 −1.30595
\(797\) −7.81380e9 −0.546712 −0.273356 0.961913i \(-0.588134\pi\)
−0.273356 + 0.961913i \(0.588134\pi\)
\(798\) 0 0
\(799\) 4.34430e8 0.0301305
\(800\) 2.79243e9 0.192826
\(801\) 0 0
\(802\) 6.36664e8 0.0435813
\(803\) 2.13043e10 1.45199
\(804\) 0 0
\(805\) 3.96736e10 2.68050
\(806\) 2.22785e9 0.149870
\(807\) 0 0
\(808\) −6.26972e9 −0.418127
\(809\) −1.22193e9 −0.0811386 −0.0405693 0.999177i \(-0.512917\pi\)
−0.0405693 + 0.999177i \(0.512917\pi\)
\(810\) 0 0
\(811\) −1.05088e10 −0.691797 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(812\) −3.17453e10 −2.08082
\(813\) 0 0
\(814\) −1.90606e10 −1.23866
\(815\) −2.52981e9 −0.163695
\(816\) 0 0
\(817\) −1.28614e9 −0.0825111
\(818\) −4.64169e9 −0.296510
\(819\) 0 0
\(820\) 1.14235e10 0.723523
\(821\) 1.87258e10 1.18097 0.590486 0.807048i \(-0.298936\pi\)
0.590486 + 0.807048i \(0.298936\pi\)
\(822\) 0 0
\(823\) −3.81070e9 −0.238290 −0.119145 0.992877i \(-0.538015\pi\)
−0.119145 + 0.992877i \(0.538015\pi\)
\(824\) −1.98109e10 −1.23356
\(825\) 0 0
\(826\) 1.53703e10 0.948968
\(827\) −1.63714e10 −1.00651 −0.503254 0.864139i \(-0.667864\pi\)
−0.503254 + 0.864139i \(0.667864\pi\)
\(828\) 0 0
\(829\) −6.71162e9 −0.409153 −0.204577 0.978851i \(-0.565582\pi\)
−0.204577 + 0.978851i \(0.565582\pi\)
\(830\) 9.34453e9 0.567263
\(831\) 0 0
\(832\) −6.40448e8 −0.0385525
\(833\) −1.36655e9 −0.0819158
\(834\) 0 0
\(835\) 1.40811e10 0.837018
\(836\) 1.12601e10 0.666529
\(837\) 0 0
\(838\) −5.07444e9 −0.297875
\(839\) 1.35926e10 0.794577 0.397289 0.917694i \(-0.369951\pi\)
0.397289 + 0.917694i \(0.369951\pi\)
\(840\) 0 0
\(841\) 3.52573e10 2.04391
\(842\) 5.45434e8 0.0314883
\(843\) 0 0
\(844\) 4.80993e8 0.0275385
\(845\) 1.80177e10 1.02731
\(846\) 0 0
\(847\) 4.48396e10 2.53554
\(848\) 2.01984e9 0.113745
\(849\) 0 0
\(850\) −9.25064e7 −0.00516661
\(851\) 4.42214e10 2.45968
\(852\) 0 0
\(853\) −8.31755e9 −0.458853 −0.229427 0.973326i \(-0.573685\pi\)
−0.229427 + 0.973326i \(0.573685\pi\)
\(854\) −3.76679e9 −0.206951
\(855\) 0 0
\(856\) −1.94806e10 −1.06156
\(857\) 5.32901e8 0.0289210 0.0144605 0.999895i \(-0.495397\pi\)
0.0144605 + 0.999895i \(0.495397\pi\)
\(858\) 0 0
\(859\) −3.23176e10 −1.73965 −0.869827 0.493357i \(-0.835770\pi\)
−0.869827 + 0.493357i \(0.835770\pi\)
\(860\) 2.36005e9 0.126525
\(861\) 0 0
\(862\) −1.34039e10 −0.712780
\(863\) −5.97266e9 −0.316323 −0.158161 0.987413i \(-0.550557\pi\)
−0.158161 + 0.987413i \(0.550557\pi\)
\(864\) 0 0
\(865\) −1.39111e10 −0.730810
\(866\) 9.04737e9 0.473380
\(867\) 0 0
\(868\) 2.94281e10 1.52737
\(869\) 1.88851e10 0.976226
\(870\) 0 0
\(871\) 4.67444e9 0.239699
\(872\) −1.58991e10 −0.812018
\(873\) 0 0
\(874\) 8.19143e9 0.415020
\(875\) 2.74821e10 1.38683
\(876\) 0 0
\(877\) −7.05767e9 −0.353315 −0.176658 0.984272i \(-0.556529\pi\)
−0.176658 + 0.984272i \(0.556529\pi\)
\(878\) −7.75530e9 −0.386695
\(879\) 0 0
\(880\) −1.21517e10 −0.601102
\(881\) −2.62054e10 −1.29114 −0.645572 0.763700i \(-0.723381\pi\)
−0.645572 + 0.763700i \(0.723381\pi\)
\(882\) 0 0
\(883\) −5.36619e8 −0.0262303 −0.0131152 0.999914i \(-0.504175\pi\)
−0.0131152 + 0.999914i \(0.504175\pi\)
\(884\) −2.10952e8 −0.0102707
\(885\) 0 0
\(886\) −2.43167e9 −0.117459
\(887\) 2.81727e10 1.35549 0.677744 0.735298i \(-0.262958\pi\)
0.677744 + 0.735298i \(0.262958\pi\)
\(888\) 0 0
\(889\) −3.48234e10 −1.66232
\(890\) −1.54076e10 −0.732606
\(891\) 0 0
\(892\) −2.70471e10 −1.27598
\(893\) 6.15932e9 0.289436
\(894\) 0 0
\(895\) 2.34148e10 1.09172
\(896\) −3.19926e10 −1.48584
\(897\) 0 0
\(898\) −1.16642e10 −0.537510
\(899\) −4.86744e10 −2.23430
\(900\) 0 0
\(901\) −4.12672e8 −0.0187961
\(902\) 1.51958e10 0.689446
\(903\) 0 0
\(904\) −1.97672e10 −0.889928
\(905\) −2.24919e10 −1.00869
\(906\) 0 0
\(907\) 3.54442e9 0.157732 0.0788659 0.996885i \(-0.474870\pi\)
0.0788659 + 0.996885i \(0.474870\pi\)
\(908\) −1.24082e10 −0.550058
\(909\) 0 0
\(910\) 4.54216e9 0.199810
\(911\) −9.71330e9 −0.425650 −0.212825 0.977090i \(-0.568266\pi\)
−0.212825 + 0.977090i \(0.568266\pi\)
\(912\) 0 0
\(913\) −3.96421e10 −1.72389
\(914\) −1.18041e10 −0.511355
\(915\) 0 0
\(916\) −1.61832e10 −0.695713
\(917\) −5.11393e10 −2.19009
\(918\) 0 0
\(919\) 1.30593e10 0.555029 0.277515 0.960721i \(-0.410489\pi\)
0.277515 + 0.960721i \(0.410489\pi\)
\(920\) −3.47755e10 −1.47236
\(921\) 0 0
\(922\) −4.11461e9 −0.172890
\(923\) −2.96024e9 −0.123914
\(924\) 0 0
\(925\) −7.08040e9 −0.294145
\(926\) 9.39820e9 0.388961
\(927\) 0 0
\(928\) 4.36246e10 1.79190
\(929\) 1.76183e10 0.720955 0.360478 0.932768i \(-0.382614\pi\)
0.360478 + 0.932768i \(0.382614\pi\)
\(930\) 0 0
\(931\) −1.93748e10 −0.786890
\(932\) −1.92145e10 −0.777452
\(933\) 0 0
\(934\) 2.17331e10 0.872786
\(935\) 2.48271e9 0.0993310
\(936\) 0 0
\(937\) 2.93955e10 1.16733 0.583664 0.811996i \(-0.301619\pi\)
0.583664 + 0.811996i \(0.301619\pi\)
\(938\) −1.93610e10 −0.765982
\(939\) 0 0
\(940\) −1.13022e10 −0.443831
\(941\) −5.66655e9 −0.221694 −0.110847 0.993837i \(-0.535356\pi\)
−0.110847 + 0.993837i \(0.535356\pi\)
\(942\) 0 0
\(943\) −3.52548e10 −1.36908
\(944\) 1.09223e10 0.422582
\(945\) 0 0
\(946\) 3.13938e9 0.120566
\(947\) −2.82326e10 −1.08025 −0.540126 0.841584i \(-0.681624\pi\)
−0.540126 + 0.841584i \(0.681624\pi\)
\(948\) 0 0
\(949\) −5.65879e9 −0.214927
\(950\) −1.31155e9 −0.0496308
\(951\) 0 0
\(952\) 2.02146e9 0.0759338
\(953\) 2.48053e10 0.928366 0.464183 0.885739i \(-0.346348\pi\)
0.464183 + 0.885739i \(0.346348\pi\)
\(954\) 0 0
\(955\) −1.40593e10 −0.522338
\(956\) 7.04494e9 0.260780
\(957\) 0 0
\(958\) 8.01833e9 0.294649
\(959\) 2.67555e10 0.979599
\(960\) 0 0
\(961\) 1.76088e10 0.640027
\(962\) 5.06283e9 0.183350
\(963\) 0 0
\(964\) 3.01988e10 1.08573
\(965\) −3.18236e10 −1.14000
\(966\) 0 0
\(967\) −3.08573e10 −1.09740 −0.548700 0.836019i \(-0.684877\pi\)
−0.548700 + 0.836019i \(0.684877\pi\)
\(968\) −3.93036e10 −1.39274
\(969\) 0 0
\(970\) −7.29345e9 −0.256585
\(971\) 3.26691e9 0.114517 0.0572585 0.998359i \(-0.481764\pi\)
0.0572585 + 0.998359i \(0.481764\pi\)
\(972\) 0 0
\(973\) −5.28896e10 −1.84067
\(974\) 2.37209e10 0.822574
\(975\) 0 0
\(976\) −2.67671e9 −0.0921568
\(977\) −1.92954e10 −0.661948 −0.330974 0.943640i \(-0.607377\pi\)
−0.330974 + 0.943640i \(0.607377\pi\)
\(978\) 0 0
\(979\) 6.53635e10 2.22636
\(980\) 3.55525e10 1.20664
\(981\) 0 0
\(982\) 1.93959e9 0.0653612
\(983\) −4.50199e10 −1.51170 −0.755852 0.654742i \(-0.772777\pi\)
−0.755852 + 0.654742i \(0.772777\pi\)
\(984\) 0 0
\(985\) 9.88658e9 0.329624
\(986\) −1.44518e9 −0.0480123
\(987\) 0 0
\(988\) −2.99087e9 −0.0986615
\(989\) −7.28349e9 −0.239416
\(990\) 0 0
\(991\) −4.47544e10 −1.46076 −0.730379 0.683042i \(-0.760657\pi\)
−0.730379 + 0.683042i \(0.760657\pi\)
\(992\) −4.04403e10 −1.31529
\(993\) 0 0
\(994\) 1.22610e10 0.395980
\(995\) −5.80926e10 −1.86956
\(996\) 0 0
\(997\) 4.11948e10 1.31647 0.658233 0.752815i \(-0.271304\pi\)
0.658233 + 0.752815i \(0.271304\pi\)
\(998\) −2.05781e10 −0.655314
\(999\) 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.b.1.6 11
3.2 odd 2 43.8.a.a.1.6 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.6 11 3.2 odd 2
387.8.a.b.1.6 11 1.1 even 1 trivial