Properties

Label 76.8.a.a.1.4
Level $76$
Weight $8$
Character 76.1
Self dual yes
Analytic conductor $23.741$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.7412619368\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 5014 x^{3} + 113222 x^{2} - 625803 x + 567036\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(6.88490\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q+25.4201 q^{3} -3.35619 q^{5} -173.262 q^{7} -1540.82 q^{9} +O(q^{10})\) \(q+25.4201 q^{3} -3.35619 q^{5} -173.262 q^{7} -1540.82 q^{9} +3804.09 q^{11} -14439.0 q^{13} -85.3148 q^{15} +3292.85 q^{17} -6859.00 q^{19} -4404.35 q^{21} +643.599 q^{23} -78113.7 q^{25} -94761.6 q^{27} +93767.3 q^{29} -103884. q^{31} +96700.4 q^{33} +581.501 q^{35} -322176. q^{37} -367042. q^{39} -843931. q^{41} -289803. q^{43} +5171.27 q^{45} +772392. q^{47} -793523. q^{49} +83704.8 q^{51} -596172. q^{53} -12767.2 q^{55} -174357. q^{57} -2.76656e6 q^{59} +2.39723e6 q^{61} +266965. q^{63} +48460.1 q^{65} +2.71365e6 q^{67} +16360.4 q^{69} +5.03316e6 q^{71} +3.65348e6 q^{73} -1.98566e6 q^{75} -659104. q^{77} -4.72959e6 q^{79} +960912. q^{81} +2.28839e6 q^{83} -11051.4 q^{85} +2.38358e6 q^{87} +8.80023e6 q^{89} +2.50174e6 q^{91} -2.64074e6 q^{93} +23020.1 q^{95} +1.21813e7 q^{97} -5.86140e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} + O(q^{10}) \) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} - 2662q^{11} - 602q^{13} - 20800q^{15} - 27366q^{17} - 34295q^{19} - 59964q^{21} - 67096q^{23} - 109115q^{25} - 178778q^{27} - 372398q^{29} - 271372q^{31} - 792700q^{33} - 608250q^{35} - 562630q^{37} - 963904q^{39} - 956714q^{41} - 827362q^{43} - 1165100q^{45} - 1812982q^{47} - 862031q^{49} - 2458254q^{51} + 486998q^{53} + 467930q^{55} + 96026q^{57} - 367182q^{59} + 1879732q^{61} - 1007274q^{63} + 1790920q^{65} - 1046394q^{67} + 7261712q^{69} - 4664572q^{71} + 4224942q^{73} + 8194850q^{75} + 8611110q^{77} + 9574024q^{79} + 11351813q^{81} + 11754804q^{83} + 18711750q^{85} + 3801472q^{87} + 2782542q^{89} + 7385214q^{91} + 29535004q^{93} + 1920520q^{95} + 1291574q^{97} - 9760310q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 25.4201 0.543568 0.271784 0.962358i \(-0.412386\pi\)
0.271784 + 0.962358i \(0.412386\pi\)
\(4\) 0 0
\(5\) −3.35619 −0.0120075 −0.00600374 0.999982i \(-0.501911\pi\)
−0.00600374 + 0.999982i \(0.501911\pi\)
\(6\) 0 0
\(7\) −173.262 −0.190924 −0.0954620 0.995433i \(-0.530433\pi\)
−0.0954620 + 0.995433i \(0.530433\pi\)
\(8\) 0 0
\(9\) −1540.82 −0.704534
\(10\) 0 0
\(11\) 3804.09 0.861740 0.430870 0.902414i \(-0.358207\pi\)
0.430870 + 0.902414i \(0.358207\pi\)
\(12\) 0 0
\(13\) −14439.0 −1.82279 −0.911394 0.411536i \(-0.864993\pi\)
−0.911394 + 0.411536i \(0.864993\pi\)
\(14\) 0 0
\(15\) −85.3148 −0.00652687
\(16\) 0 0
\(17\) 3292.85 0.162555 0.0812776 0.996691i \(-0.474100\pi\)
0.0812776 + 0.996691i \(0.474100\pi\)
\(18\) 0 0
\(19\) −6859.00 −0.229416
\(20\) 0 0
\(21\) −4404.35 −0.103780
\(22\) 0 0
\(23\) 643.599 0.0110298 0.00551491 0.999985i \(-0.498245\pi\)
0.00551491 + 0.999985i \(0.498245\pi\)
\(24\) 0 0
\(25\) −78113.7 −0.999856
\(26\) 0 0
\(27\) −94761.6 −0.926530
\(28\) 0 0
\(29\) 93767.3 0.713935 0.356967 0.934117i \(-0.383811\pi\)
0.356967 + 0.934117i \(0.383811\pi\)
\(30\) 0 0
\(31\) −103884. −0.626299 −0.313150 0.949704i \(-0.601384\pi\)
−0.313150 + 0.949704i \(0.601384\pi\)
\(32\) 0 0
\(33\) 96700.4 0.468414
\(34\) 0 0
\(35\) 581.501 0.00229251
\(36\) 0 0
\(37\) −322176. −1.04565 −0.522825 0.852440i \(-0.675122\pi\)
−0.522825 + 0.852440i \(0.675122\pi\)
\(38\) 0 0
\(39\) −367042. −0.990808
\(40\) 0 0
\(41\) −843931. −1.91233 −0.956166 0.292824i \(-0.905405\pi\)
−0.956166 + 0.292824i \(0.905405\pi\)
\(42\) 0 0
\(43\) −289803. −0.555858 −0.277929 0.960602i \(-0.589648\pi\)
−0.277929 + 0.960602i \(0.589648\pi\)
\(44\) 0 0
\(45\) 5171.27 0.00845968
\(46\) 0 0
\(47\) 772392. 1.08516 0.542582 0.840003i \(-0.317447\pi\)
0.542582 + 0.840003i \(0.317447\pi\)
\(48\) 0 0
\(49\) −793523. −0.963548
\(50\) 0 0
\(51\) 83704.8 0.0883598
\(52\) 0 0
\(53\) −596172. −0.550055 −0.275028 0.961436i \(-0.588687\pi\)
−0.275028 + 0.961436i \(0.588687\pi\)
\(54\) 0 0
\(55\) −12767.2 −0.0103473
\(56\) 0 0
\(57\) −174357. −0.124703
\(58\) 0 0
\(59\) −2.76656e6 −1.75371 −0.876856 0.480753i \(-0.840363\pi\)
−0.876856 + 0.480753i \(0.840363\pi\)
\(60\) 0 0
\(61\) 2.39723e6 1.35224 0.676121 0.736791i \(-0.263660\pi\)
0.676121 + 0.736791i \(0.263660\pi\)
\(62\) 0 0
\(63\) 266965. 0.134513
\(64\) 0 0
\(65\) 48460.1 0.0218871
\(66\) 0 0
\(67\) 2.71365e6 1.10228 0.551140 0.834413i \(-0.314193\pi\)
0.551140 + 0.834413i \(0.314193\pi\)
\(68\) 0 0
\(69\) 16360.4 0.00599545
\(70\) 0 0
\(71\) 5.03316e6 1.66893 0.834463 0.551065i \(-0.185778\pi\)
0.834463 + 0.551065i \(0.185778\pi\)
\(72\) 0 0
\(73\) 3.65348e6 1.09920 0.549600 0.835428i \(-0.314780\pi\)
0.549600 + 0.835428i \(0.314780\pi\)
\(74\) 0 0
\(75\) −1.98566e6 −0.543489
\(76\) 0 0
\(77\) −659104. −0.164527
\(78\) 0 0
\(79\) −4.72959e6 −1.07927 −0.539633 0.841900i \(-0.681437\pi\)
−0.539633 + 0.841900i \(0.681437\pi\)
\(80\) 0 0
\(81\) 960912. 0.200903
\(82\) 0 0
\(83\) 2.28839e6 0.439296 0.219648 0.975579i \(-0.429509\pi\)
0.219648 + 0.975579i \(0.429509\pi\)
\(84\) 0 0
\(85\) −11051.4 −0.00195188
\(86\) 0 0
\(87\) 2.38358e6 0.388072
\(88\) 0 0
\(89\) 8.80023e6 1.32321 0.661605 0.749852i \(-0.269876\pi\)
0.661605 + 0.749852i \(0.269876\pi\)
\(90\) 0 0
\(91\) 2.50174e6 0.348014
\(92\) 0 0
\(93\) −2.64074e6 −0.340436
\(94\) 0 0
\(95\) 23020.1 0.00275470
\(96\) 0 0
\(97\) 1.21813e7 1.35516 0.677581 0.735448i \(-0.263028\pi\)
0.677581 + 0.735448i \(0.263028\pi\)
\(98\) 0 0
\(99\) −5.86140e6 −0.607125
\(100\) 0 0
\(101\) 2.27068e6 0.219296 0.109648 0.993970i \(-0.465028\pi\)
0.109648 + 0.993970i \(0.465028\pi\)
\(102\) 0 0
\(103\) −4.11736e6 −0.371269 −0.185634 0.982619i \(-0.559434\pi\)
−0.185634 + 0.982619i \(0.559434\pi\)
\(104\) 0 0
\(105\) 14781.8 0.00124614
\(106\) 0 0
\(107\) −2.18710e6 −0.172594 −0.0862970 0.996269i \(-0.527503\pi\)
−0.0862970 + 0.996269i \(0.527503\pi\)
\(108\) 0 0
\(109\) −6.58258e6 −0.486860 −0.243430 0.969919i \(-0.578273\pi\)
−0.243430 + 0.969919i \(0.578273\pi\)
\(110\) 0 0
\(111\) −8.18975e6 −0.568382
\(112\) 0 0
\(113\) 1.01191e7 0.659732 0.329866 0.944028i \(-0.392996\pi\)
0.329866 + 0.944028i \(0.392996\pi\)
\(114\) 0 0
\(115\) −2160.04 −0.000132440 0
\(116\) 0 0
\(117\) 2.22479e7 1.28422
\(118\) 0 0
\(119\) −570527. −0.0310357
\(120\) 0 0
\(121\) −5.01609e6 −0.257405
\(122\) 0 0
\(123\) −2.14528e7 −1.03948
\(124\) 0 0
\(125\) 524367. 0.0240132
\(126\) 0 0
\(127\) −5.27049e6 −0.228317 −0.114158 0.993463i \(-0.536417\pi\)
−0.114158 + 0.993463i \(0.536417\pi\)
\(128\) 0 0
\(129\) −7.36684e6 −0.302146
\(130\) 0 0
\(131\) −3.23909e7 −1.25885 −0.629425 0.777061i \(-0.716710\pi\)
−0.629425 + 0.777061i \(0.716710\pi\)
\(132\) 0 0
\(133\) 1.18840e6 0.0438010
\(134\) 0 0
\(135\) 318038. 0.0111253
\(136\) 0 0
\(137\) −3.71836e7 −1.23546 −0.617730 0.786390i \(-0.711948\pi\)
−0.617730 + 0.786390i \(0.711948\pi\)
\(138\) 0 0
\(139\) 1.43249e7 0.452418 0.226209 0.974079i \(-0.427367\pi\)
0.226209 + 0.974079i \(0.427367\pi\)
\(140\) 0 0
\(141\) 1.96343e7 0.589860
\(142\) 0 0
\(143\) −5.49273e7 −1.57077
\(144\) 0 0
\(145\) −314701. −0.00857255
\(146\) 0 0
\(147\) −2.01715e7 −0.523753
\(148\) 0 0
\(149\) 4.39404e6 0.108821 0.0544104 0.998519i \(-0.482672\pi\)
0.0544104 + 0.998519i \(0.482672\pi\)
\(150\) 0 0
\(151\) −2.48731e7 −0.587908 −0.293954 0.955819i \(-0.594971\pi\)
−0.293954 + 0.955819i \(0.594971\pi\)
\(152\) 0 0
\(153\) −5.07368e6 −0.114526
\(154\) 0 0
\(155\) 348654. 0.00752027
\(156\) 0 0
\(157\) 4.07527e6 0.0840441 0.0420221 0.999117i \(-0.486620\pi\)
0.0420221 + 0.999117i \(0.486620\pi\)
\(158\) 0 0
\(159\) −1.51548e7 −0.298992
\(160\) 0 0
\(161\) −111511. −0.00210586
\(162\) 0 0
\(163\) −7.57086e7 −1.36927 −0.684635 0.728887i \(-0.740038\pi\)
−0.684635 + 0.728887i \(0.740038\pi\)
\(164\) 0 0
\(165\) −324545. −0.00562446
\(166\) 0 0
\(167\) 3.19156e7 0.530268 0.265134 0.964212i \(-0.414584\pi\)
0.265134 + 0.964212i \(0.414584\pi\)
\(168\) 0 0
\(169\) 1.45737e8 2.32255
\(170\) 0 0
\(171\) 1.05685e7 0.161631
\(172\) 0 0
\(173\) 6.07975e7 0.892738 0.446369 0.894849i \(-0.352717\pi\)
0.446369 + 0.894849i \(0.352717\pi\)
\(174\) 0 0
\(175\) 1.35342e7 0.190896
\(176\) 0 0
\(177\) −7.03264e7 −0.953261
\(178\) 0 0
\(179\) 1.12224e7 0.146252 0.0731259 0.997323i \(-0.476703\pi\)
0.0731259 + 0.997323i \(0.476703\pi\)
\(180\) 0 0
\(181\) 5.70223e7 0.714775 0.357388 0.933956i \(-0.383668\pi\)
0.357388 + 0.933956i \(0.383668\pi\)
\(182\) 0 0
\(183\) 6.09378e7 0.735035
\(184\) 0 0
\(185\) 1.08128e6 0.0125556
\(186\) 0 0
\(187\) 1.25263e7 0.140080
\(188\) 0 0
\(189\) 1.64186e7 0.176897
\(190\) 0 0
\(191\) −3.66566e7 −0.380658 −0.190329 0.981720i \(-0.560956\pi\)
−0.190329 + 0.981720i \(0.560956\pi\)
\(192\) 0 0
\(193\) −1.06650e8 −1.06785 −0.533924 0.845533i \(-0.679283\pi\)
−0.533924 + 0.845533i \(0.679283\pi\)
\(194\) 0 0
\(195\) 1.23186e6 0.0118971
\(196\) 0 0
\(197\) 1.20480e8 1.12275 0.561375 0.827561i \(-0.310272\pi\)
0.561375 + 0.827561i \(0.310272\pi\)
\(198\) 0 0
\(199\) 2.12531e8 1.91177 0.955885 0.293741i \(-0.0949002\pi\)
0.955885 + 0.293741i \(0.0949002\pi\)
\(200\) 0 0
\(201\) 6.89813e7 0.599164
\(202\) 0 0
\(203\) −1.62463e7 −0.136307
\(204\) 0 0
\(205\) 2.83239e6 0.0229623
\(206\) 0 0
\(207\) −991668. −0.00777088
\(208\) 0 0
\(209\) −2.60922e7 −0.197697
\(210\) 0 0
\(211\) −1.23530e8 −0.905279 −0.452640 0.891693i \(-0.649518\pi\)
−0.452640 + 0.891693i \(0.649518\pi\)
\(212\) 0 0
\(213\) 1.27944e8 0.907174
\(214\) 0 0
\(215\) 972635. 0.00667445
\(216\) 0 0
\(217\) 1.79991e7 0.119576
\(218\) 0 0
\(219\) 9.28719e7 0.597489
\(220\) 0 0
\(221\) −4.75456e7 −0.296304
\(222\) 0 0
\(223\) 1.75568e7 0.106018 0.0530088 0.998594i \(-0.483119\pi\)
0.0530088 + 0.998594i \(0.483119\pi\)
\(224\) 0 0
\(225\) 1.20359e8 0.704433
\(226\) 0 0
\(227\) −2.25659e8 −1.28045 −0.640225 0.768187i \(-0.721159\pi\)
−0.640225 + 0.768187i \(0.721159\pi\)
\(228\) 0 0
\(229\) −2.22000e8 −1.22160 −0.610800 0.791785i \(-0.709152\pi\)
−0.610800 + 0.791785i \(0.709152\pi\)
\(230\) 0 0
\(231\) −1.67545e7 −0.0894314
\(232\) 0 0
\(233\) 1.03946e8 0.538347 0.269174 0.963092i \(-0.413249\pi\)
0.269174 + 0.963092i \(0.413249\pi\)
\(234\) 0 0
\(235\) −2.59229e6 −0.0130301
\(236\) 0 0
\(237\) −1.20227e8 −0.586654
\(238\) 0 0
\(239\) −2.75718e8 −1.30639 −0.653195 0.757189i \(-0.726572\pi\)
−0.653195 + 0.757189i \(0.726572\pi\)
\(240\) 0 0
\(241\) 3.22608e8 1.48462 0.742311 0.670055i \(-0.233730\pi\)
0.742311 + 0.670055i \(0.233730\pi\)
\(242\) 0 0
\(243\) 2.31670e8 1.03573
\(244\) 0 0
\(245\) 2.66322e6 0.0115698
\(246\) 0 0
\(247\) 9.90372e7 0.418176
\(248\) 0 0
\(249\) 5.81713e7 0.238787
\(250\) 0 0
\(251\) −3.37531e8 −1.34727 −0.673637 0.739062i \(-0.735269\pi\)
−0.673637 + 0.739062i \(0.735269\pi\)
\(252\) 0 0
\(253\) 2.44831e6 0.00950482
\(254\) 0 0
\(255\) −280929. −0.00106098
\(256\) 0 0
\(257\) −3.32504e8 −1.22189 −0.610944 0.791674i \(-0.709210\pi\)
−0.610944 + 0.791674i \(0.709210\pi\)
\(258\) 0 0
\(259\) 5.58208e7 0.199640
\(260\) 0 0
\(261\) −1.44478e8 −0.502992
\(262\) 0 0
\(263\) −7.63765e7 −0.258890 −0.129445 0.991587i \(-0.541320\pi\)
−0.129445 + 0.991587i \(0.541320\pi\)
\(264\) 0 0
\(265\) 2.00087e6 0.00660477
\(266\) 0 0
\(267\) 2.23703e8 0.719254
\(268\) 0 0
\(269\) −3.10285e8 −0.971914 −0.485957 0.873983i \(-0.661529\pi\)
−0.485957 + 0.873983i \(0.661529\pi\)
\(270\) 0 0
\(271\) −1.79448e8 −0.547704 −0.273852 0.961772i \(-0.588298\pi\)
−0.273852 + 0.961772i \(0.588298\pi\)
\(272\) 0 0
\(273\) 6.35945e7 0.189169
\(274\) 0 0
\(275\) −2.97151e8 −0.861615
\(276\) 0 0
\(277\) 2.27108e8 0.642027 0.321014 0.947075i \(-0.395976\pi\)
0.321014 + 0.947075i \(0.395976\pi\)
\(278\) 0 0
\(279\) 1.60066e8 0.441249
\(280\) 0 0
\(281\) −2.49532e8 −0.670895 −0.335448 0.942059i \(-0.608888\pi\)
−0.335448 + 0.942059i \(0.608888\pi\)
\(282\) 0 0
\(283\) −4.15917e8 −1.09082 −0.545412 0.838168i \(-0.683627\pi\)
−0.545412 + 0.838168i \(0.683627\pi\)
\(284\) 0 0
\(285\) 585174. 0.00149737
\(286\) 0 0
\(287\) 1.46221e8 0.365110
\(288\) 0 0
\(289\) −3.99496e8 −0.973576
\(290\) 0 0
\(291\) 3.09649e8 0.736622
\(292\) 0 0
\(293\) −3.24502e8 −0.753670 −0.376835 0.926281i \(-0.622988\pi\)
−0.376835 + 0.926281i \(0.622988\pi\)
\(294\) 0 0
\(295\) 9.28511e6 0.0210577
\(296\) 0 0
\(297\) −3.60481e8 −0.798427
\(298\) 0 0
\(299\) −9.29294e6 −0.0201050
\(300\) 0 0
\(301\) 5.02119e7 0.106127
\(302\) 0 0
\(303\) 5.77211e7 0.119202
\(304\) 0 0
\(305\) −8.04555e6 −0.0162370
\(306\) 0 0
\(307\) 5.69052e8 1.12245 0.561226 0.827663i \(-0.310330\pi\)
0.561226 + 0.827663i \(0.310330\pi\)
\(308\) 0 0
\(309\) −1.04664e8 −0.201810
\(310\) 0 0
\(311\) −1.05324e8 −0.198549 −0.0992744 0.995060i \(-0.531652\pi\)
−0.0992744 + 0.995060i \(0.531652\pi\)
\(312\) 0 0
\(313\) −4.13697e8 −0.762566 −0.381283 0.924458i \(-0.624518\pi\)
−0.381283 + 0.924458i \(0.624518\pi\)
\(314\) 0 0
\(315\) −895986. −0.00161516
\(316\) 0 0
\(317\) 3.97209e8 0.700344 0.350172 0.936685i \(-0.386123\pi\)
0.350172 + 0.936685i \(0.386123\pi\)
\(318\) 0 0
\(319\) 3.56699e8 0.615226
\(320\) 0 0
\(321\) −5.55964e7 −0.0938165
\(322\) 0 0
\(323\) −2.25857e7 −0.0372927
\(324\) 0 0
\(325\) 1.12789e9 1.82252
\(326\) 0 0
\(327\) −1.67330e8 −0.264641
\(328\) 0 0
\(329\) −1.33826e8 −0.207184
\(330\) 0 0
\(331\) 5.36235e8 0.812750 0.406375 0.913706i \(-0.366793\pi\)
0.406375 + 0.913706i \(0.366793\pi\)
\(332\) 0 0
\(333\) 4.96413e8 0.736697
\(334\) 0 0
\(335\) −9.10752e6 −0.0132356
\(336\) 0 0
\(337\) −3.86284e8 −0.549796 −0.274898 0.961473i \(-0.588644\pi\)
−0.274898 + 0.961473i \(0.588644\pi\)
\(338\) 0 0
\(339\) 2.57229e8 0.358609
\(340\) 0 0
\(341\) −3.95183e8 −0.539707
\(342\) 0 0
\(343\) 2.80176e8 0.374888
\(344\) 0 0
\(345\) −54908.6 −7.19902e−5 0
\(346\) 0 0
\(347\) 7.18308e8 0.922906 0.461453 0.887165i \(-0.347328\pi\)
0.461453 + 0.887165i \(0.347328\pi\)
\(348\) 0 0
\(349\) 2.29357e8 0.288817 0.144409 0.989518i \(-0.453872\pi\)
0.144409 + 0.989518i \(0.453872\pi\)
\(350\) 0 0
\(351\) 1.36826e9 1.68887
\(352\) 0 0
\(353\) −1.26980e9 −1.53647 −0.768237 0.640166i \(-0.778866\pi\)
−0.768237 + 0.640166i \(0.778866\pi\)
\(354\) 0 0
\(355\) −1.68923e7 −0.0200396
\(356\) 0 0
\(357\) −1.45029e7 −0.0168700
\(358\) 0 0
\(359\) 1.43972e9 1.64228 0.821142 0.570724i \(-0.193337\pi\)
0.821142 + 0.570724i \(0.193337\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) −1.27510e8 −0.139917
\(364\) 0 0
\(365\) −1.22618e7 −0.0131986
\(366\) 0 0
\(367\) 8.07332e8 0.852552 0.426276 0.904593i \(-0.359825\pi\)
0.426276 + 0.904593i \(0.359825\pi\)
\(368\) 0 0
\(369\) 1.30034e9 1.34730
\(370\) 0 0
\(371\) 1.03294e8 0.105019
\(372\) 0 0
\(373\) 2.62122e8 0.261530 0.130765 0.991413i \(-0.458257\pi\)
0.130765 + 0.991413i \(0.458257\pi\)
\(374\) 0 0
\(375\) 1.33295e7 0.0130528
\(376\) 0 0
\(377\) −1.35391e9 −1.30135
\(378\) 0 0
\(379\) −3.34919e8 −0.316011 −0.158006 0.987438i \(-0.550506\pi\)
−0.158006 + 0.987438i \(0.550506\pi\)
\(380\) 0 0
\(381\) −1.33977e8 −0.124106
\(382\) 0 0
\(383\) −2.17105e9 −1.97457 −0.987287 0.158951i \(-0.949189\pi\)
−0.987287 + 0.158951i \(0.949189\pi\)
\(384\) 0 0
\(385\) 2.21208e6 0.00197555
\(386\) 0 0
\(387\) 4.46534e8 0.391621
\(388\) 0 0
\(389\) −5.23430e8 −0.450853 −0.225426 0.974260i \(-0.572378\pi\)
−0.225426 + 0.974260i \(0.572378\pi\)
\(390\) 0 0
\(391\) 2.11928e6 0.00179295
\(392\) 0 0
\(393\) −8.23382e8 −0.684270
\(394\) 0 0
\(395\) 1.58734e7 0.0129593
\(396\) 0 0
\(397\) 8.46818e8 0.679240 0.339620 0.940563i \(-0.389702\pi\)
0.339620 + 0.940563i \(0.389702\pi\)
\(398\) 0 0
\(399\) 3.02094e7 0.0238088
\(400\) 0 0
\(401\) −1.56110e9 −1.20900 −0.604499 0.796606i \(-0.706627\pi\)
−0.604499 + 0.796606i \(0.706627\pi\)
\(402\) 0 0
\(403\) 1.49998e9 1.14161
\(404\) 0 0
\(405\) −3.22501e6 −0.00241234
\(406\) 0 0
\(407\) −1.22558e9 −0.901078
\(408\) 0 0
\(409\) 7.63724e8 0.551957 0.275978 0.961164i \(-0.410998\pi\)
0.275978 + 0.961164i \(0.410998\pi\)
\(410\) 0 0
\(411\) −9.45211e8 −0.671556
\(412\) 0 0
\(413\) 4.79340e8 0.334826
\(414\) 0 0
\(415\) −7.68029e6 −0.00527484
\(416\) 0 0
\(417\) 3.64141e8 0.245920
\(418\) 0 0
\(419\) 1.80707e9 1.20013 0.600063 0.799953i \(-0.295142\pi\)
0.600063 + 0.799953i \(0.295142\pi\)
\(420\) 0 0
\(421\) −1.17005e9 −0.764215 −0.382108 0.924118i \(-0.624802\pi\)
−0.382108 + 0.924118i \(0.624802\pi\)
\(422\) 0 0
\(423\) −1.19011e9 −0.764535
\(424\) 0 0
\(425\) −2.57217e8 −0.162532
\(426\) 0 0
\(427\) −4.15348e8 −0.258175
\(428\) 0 0
\(429\) −1.39626e9 −0.853818
\(430\) 0 0
\(431\) −3.33701e8 −0.200764 −0.100382 0.994949i \(-0.532007\pi\)
−0.100382 + 0.994949i \(0.532007\pi\)
\(432\) 0 0
\(433\) −1.18240e9 −0.699934 −0.349967 0.936762i \(-0.613807\pi\)
−0.349967 + 0.936762i \(0.613807\pi\)
\(434\) 0 0
\(435\) −7.99974e6 −0.00465976
\(436\) 0 0
\(437\) −4.41445e6 −0.00253041
\(438\) 0 0
\(439\) 1.36067e8 0.0767585 0.0383792 0.999263i \(-0.487781\pi\)
0.0383792 + 0.999263i \(0.487781\pi\)
\(440\) 0 0
\(441\) 1.22267e9 0.678853
\(442\) 0 0
\(443\) −2.42005e9 −1.32255 −0.661274 0.750144i \(-0.729984\pi\)
−0.661274 + 0.750144i \(0.729984\pi\)
\(444\) 0 0
\(445\) −2.95352e7 −0.0158884
\(446\) 0 0
\(447\) 1.11697e8 0.0591515
\(448\) 0 0
\(449\) −1.03162e9 −0.537845 −0.268923 0.963162i \(-0.586668\pi\)
−0.268923 + 0.963162i \(0.586668\pi\)
\(450\) 0 0
\(451\) −3.21039e9 −1.64793
\(452\) 0 0
\(453\) −6.32276e8 −0.319568
\(454\) 0 0
\(455\) −8.39630e6 −0.00417877
\(456\) 0 0
\(457\) 2.98845e8 0.146467 0.0732335 0.997315i \(-0.476668\pi\)
0.0732335 + 0.997315i \(0.476668\pi\)
\(458\) 0 0
\(459\) −3.12036e8 −0.150612
\(460\) 0 0
\(461\) 2.56152e9 1.21771 0.608857 0.793280i \(-0.291628\pi\)
0.608857 + 0.793280i \(0.291628\pi\)
\(462\) 0 0
\(463\) 3.45788e9 1.61911 0.809556 0.587042i \(-0.199708\pi\)
0.809556 + 0.587042i \(0.199708\pi\)
\(464\) 0 0
\(465\) 8.86283e6 0.00408778
\(466\) 0 0
\(467\) 2.11651e9 0.961638 0.480819 0.876820i \(-0.340339\pi\)
0.480819 + 0.876820i \(0.340339\pi\)
\(468\) 0 0
\(469\) −4.70173e8 −0.210452
\(470\) 0 0
\(471\) 1.03594e8 0.0456837
\(472\) 0 0
\(473\) −1.10244e9 −0.479004
\(474\) 0 0
\(475\) 5.35782e8 0.229383
\(476\) 0 0
\(477\) 9.18592e8 0.387533
\(478\) 0 0
\(479\) −3.94044e9 −1.63822 −0.819108 0.573640i \(-0.805531\pi\)
−0.819108 + 0.573640i \(0.805531\pi\)
\(480\) 0 0
\(481\) 4.65190e9 1.90600
\(482\) 0 0
\(483\) −2.83463e6 −0.00114467
\(484\) 0 0
\(485\) −4.08827e7 −0.0162721
\(486\) 0 0
\(487\) −4.70134e9 −1.84446 −0.922231 0.386639i \(-0.873636\pi\)
−0.922231 + 0.386639i \(0.873636\pi\)
\(488\) 0 0
\(489\) −1.92452e9 −0.744290
\(490\) 0 0
\(491\) −3.24351e9 −1.23660 −0.618302 0.785941i \(-0.712179\pi\)
−0.618302 + 0.785941i \(0.712179\pi\)
\(492\) 0 0
\(493\) 3.08762e8 0.116054
\(494\) 0 0
\(495\) 1.96720e7 0.00729004
\(496\) 0 0
\(497\) −8.72057e8 −0.318638
\(498\) 0 0
\(499\) 3.31230e8 0.119338 0.0596689 0.998218i \(-0.480996\pi\)
0.0596689 + 0.998218i \(0.480996\pi\)
\(500\) 0 0
\(501\) 8.11298e8 0.288236
\(502\) 0 0
\(503\) −3.42162e9 −1.19879 −0.599396 0.800453i \(-0.704593\pi\)
−0.599396 + 0.800453i \(0.704593\pi\)
\(504\) 0 0
\(505\) −7.62084e6 −0.00263320
\(506\) 0 0
\(507\) 3.70465e9 1.26246
\(508\) 0 0
\(509\) 4.79856e9 1.61287 0.806435 0.591323i \(-0.201394\pi\)
0.806435 + 0.591323i \(0.201394\pi\)
\(510\) 0 0
\(511\) −6.33009e8 −0.209864
\(512\) 0 0
\(513\) 6.49970e8 0.212560
\(514\) 0 0
\(515\) 1.38186e7 0.00445800
\(516\) 0 0
\(517\) 2.93825e9 0.935128
\(518\) 0 0
\(519\) 1.54548e9 0.485263
\(520\) 0 0
\(521\) −3.01575e9 −0.934250 −0.467125 0.884191i \(-0.654710\pi\)
−0.467125 + 0.884191i \(0.654710\pi\)
\(522\) 0 0
\(523\) 5.76903e9 1.76338 0.881692 0.471826i \(-0.156405\pi\)
0.881692 + 0.471826i \(0.156405\pi\)
\(524\) 0 0
\(525\) 3.44040e8 0.103765
\(526\) 0 0
\(527\) −3.42074e8 −0.101808
\(528\) 0 0
\(529\) −3.40441e9 −0.999878
\(530\) 0 0
\(531\) 4.26276e9 1.23555
\(532\) 0 0
\(533\) 1.21855e10 3.48578
\(534\) 0 0
\(535\) 7.34033e6 0.00207242
\(536\) 0 0
\(537\) 2.85276e8 0.0794978
\(538\) 0 0
\(539\) −3.01863e9 −0.830327
\(540\) 0 0
\(541\) −3.72538e8 −0.101153 −0.0505767 0.998720i \(-0.516106\pi\)
−0.0505767 + 0.998720i \(0.516106\pi\)
\(542\) 0 0
\(543\) 1.44951e9 0.388529
\(544\) 0 0
\(545\) 2.20924e7 0.00584595
\(546\) 0 0
\(547\) −5.36797e9 −1.40234 −0.701172 0.712992i \(-0.747339\pi\)
−0.701172 + 0.712992i \(0.747339\pi\)
\(548\) 0 0
\(549\) −3.69368e9 −0.952701
\(550\) 0 0
\(551\) −6.43150e8 −0.163788
\(552\) 0 0
\(553\) 8.19459e8 0.206058
\(554\) 0 0
\(555\) 2.74864e7 0.00682483
\(556\) 0 0
\(557\) 6.93092e9 1.69941 0.849705 0.527259i \(-0.176780\pi\)
0.849705 + 0.527259i \(0.176780\pi\)
\(558\) 0 0
\(559\) 4.18447e9 1.01321
\(560\) 0 0
\(561\) 3.18420e8 0.0761431
\(562\) 0 0
\(563\) −1.24931e9 −0.295047 −0.147523 0.989059i \(-0.547130\pi\)
−0.147523 + 0.989059i \(0.547130\pi\)
\(564\) 0 0
\(565\) −3.39617e7 −0.00792172
\(566\) 0 0
\(567\) −1.66490e8 −0.0383572
\(568\) 0 0
\(569\) −3.91417e9 −0.890732 −0.445366 0.895349i \(-0.646926\pi\)
−0.445366 + 0.895349i \(0.646926\pi\)
\(570\) 0 0
\(571\) 4.12496e9 0.927242 0.463621 0.886034i \(-0.346550\pi\)
0.463621 + 0.886034i \(0.346550\pi\)
\(572\) 0 0
\(573\) −9.31816e8 −0.206914
\(574\) 0 0
\(575\) −5.02739e7 −0.0110282
\(576\) 0 0
\(577\) 2.84776e9 0.617147 0.308574 0.951200i \(-0.400148\pi\)
0.308574 + 0.951200i \(0.400148\pi\)
\(578\) 0 0
\(579\) −2.71105e9 −0.580447
\(580\) 0 0
\(581\) −3.96492e8 −0.0838722
\(582\) 0 0
\(583\) −2.26789e9 −0.474004
\(584\) 0 0
\(585\) −7.46681e7 −0.0154202
\(586\) 0 0
\(587\) 7.85805e9 1.60355 0.801773 0.597629i \(-0.203890\pi\)
0.801773 + 0.597629i \(0.203890\pi\)
\(588\) 0 0
\(589\) 7.12539e8 0.143683
\(590\) 0 0
\(591\) 3.06262e9 0.610291
\(592\) 0 0
\(593\) −2.49384e9 −0.491109 −0.245554 0.969383i \(-0.578970\pi\)
−0.245554 + 0.969383i \(0.578970\pi\)
\(594\) 0 0
\(595\) 1.91480e6 0.000372660 0
\(596\) 0 0
\(597\) 5.40256e9 1.03918
\(598\) 0 0
\(599\) 1.77144e9 0.336769 0.168385 0.985721i \(-0.446145\pi\)
0.168385 + 0.985721i \(0.446145\pi\)
\(600\) 0 0
\(601\) −6.53447e9 −1.22786 −0.613931 0.789360i \(-0.710413\pi\)
−0.613931 + 0.789360i \(0.710413\pi\)
\(602\) 0 0
\(603\) −4.18124e9 −0.776594
\(604\) 0 0
\(605\) 1.68350e7 0.00309078
\(606\) 0 0
\(607\) −6.98740e9 −1.26811 −0.634053 0.773290i \(-0.718610\pi\)
−0.634053 + 0.773290i \(0.718610\pi\)
\(608\) 0 0
\(609\) −4.12984e8 −0.0740922
\(610\) 0 0
\(611\) −1.11526e10 −1.97802
\(612\) 0 0
\(613\) −4.48052e9 −0.785627 −0.392814 0.919618i \(-0.628498\pi\)
−0.392814 + 0.919618i \(0.628498\pi\)
\(614\) 0 0
\(615\) 7.19998e7 0.0124816
\(616\) 0 0
\(617\) 3.28136e9 0.562413 0.281207 0.959647i \(-0.409265\pi\)
0.281207 + 0.959647i \(0.409265\pi\)
\(618\) 0 0
\(619\) −1.03952e10 −1.76163 −0.880817 0.473457i \(-0.843006\pi\)
−0.880817 + 0.473457i \(0.843006\pi\)
\(620\) 0 0
\(621\) −6.09885e7 −0.0102194
\(622\) 0 0
\(623\) −1.52475e9 −0.252633
\(624\) 0 0
\(625\) 6.10088e9 0.999567
\(626\) 0 0
\(627\) −6.63268e8 −0.107461
\(628\) 0 0
\(629\) −1.06088e9 −0.169976
\(630\) 0 0
\(631\) 3.86959e9 0.613143 0.306572 0.951848i \(-0.400818\pi\)
0.306572 + 0.951848i \(0.400818\pi\)
\(632\) 0 0
\(633\) −3.14014e9 −0.492081
\(634\) 0 0
\(635\) 1.76888e7 0.00274151
\(636\) 0 0
\(637\) 1.14577e10 1.75634
\(638\) 0 0
\(639\) −7.75518e9 −1.17581
\(640\) 0 0
\(641\) −7.14267e9 −1.07117 −0.535584 0.844482i \(-0.679908\pi\)
−0.535584 + 0.844482i \(0.679908\pi\)
\(642\) 0 0
\(643\) −9.59951e9 −1.42400 −0.712001 0.702178i \(-0.752211\pi\)
−0.712001 + 0.702178i \(0.752211\pi\)
\(644\) 0 0
\(645\) 2.47245e7 0.00362801
\(646\) 0 0
\(647\) 9.05075e9 1.31377 0.656885 0.753991i \(-0.271874\pi\)
0.656885 + 0.753991i \(0.271874\pi\)
\(648\) 0 0
\(649\) −1.05242e10 −1.51124
\(650\) 0 0
\(651\) 4.57541e8 0.0649974
\(652\) 0 0
\(653\) 2.68998e9 0.378053 0.189026 0.981972i \(-0.439467\pi\)
0.189026 + 0.981972i \(0.439467\pi\)
\(654\) 0 0
\(655\) 1.08710e8 0.0151156
\(656\) 0 0
\(657\) −5.62934e9 −0.774424
\(658\) 0 0
\(659\) 2.21462e9 0.301439 0.150720 0.988577i \(-0.451841\pi\)
0.150720 + 0.988577i \(0.451841\pi\)
\(660\) 0 0
\(661\) −2.65391e9 −0.357422 −0.178711 0.983902i \(-0.557193\pi\)
−0.178711 + 0.983902i \(0.557193\pi\)
\(662\) 0 0
\(663\) −1.20861e9 −0.161061
\(664\) 0 0
\(665\) −3.98851e6 −0.000525939 0
\(666\) 0 0
\(667\) 6.03486e7 0.00787457
\(668\) 0 0
\(669\) 4.46296e8 0.0576278
\(670\) 0 0
\(671\) 9.11925e9 1.16528
\(672\) 0 0
\(673\) −5.88089e9 −0.743688 −0.371844 0.928295i \(-0.621274\pi\)
−0.371844 + 0.928295i \(0.621274\pi\)
\(674\) 0 0
\(675\) 7.40218e9 0.926396
\(676\) 0 0
\(677\) −7.08195e9 −0.877188 −0.438594 0.898685i \(-0.644523\pi\)
−0.438594 + 0.898685i \(0.644523\pi\)
\(678\) 0 0
\(679\) −2.11055e9 −0.258733
\(680\) 0 0
\(681\) −5.73629e9 −0.696011
\(682\) 0 0
\(683\) 3.66860e8 0.0440583 0.0220292 0.999757i \(-0.492987\pi\)
0.0220292 + 0.999757i \(0.492987\pi\)
\(684\) 0 0
\(685\) 1.24795e8 0.0148348
\(686\) 0 0
\(687\) −5.64327e9 −0.664022
\(688\) 0 0
\(689\) 8.60814e9 1.00263
\(690\) 0 0
\(691\) 3.15284e9 0.363520 0.181760 0.983343i \(-0.441821\pi\)
0.181760 + 0.983343i \(0.441821\pi\)
\(692\) 0 0
\(693\) 1.01556e9 0.115915
\(694\) 0 0
\(695\) −4.80771e7 −0.00543239
\(696\) 0 0
\(697\) −2.77894e9 −0.310860
\(698\) 0 0
\(699\) 2.64232e9 0.292628
\(700\) 0 0
\(701\) 8.59305e9 0.942181 0.471091 0.882085i \(-0.343860\pi\)
0.471091 + 0.882085i \(0.343860\pi\)
\(702\) 0 0
\(703\) 2.20980e9 0.239889
\(704\) 0 0
\(705\) −6.58965e7 −0.00708272
\(706\) 0 0
\(707\) −3.93423e8 −0.0418690
\(708\) 0 0
\(709\) 7.95777e9 0.838551 0.419275 0.907859i \(-0.362284\pi\)
0.419275 + 0.907859i \(0.362284\pi\)
\(710\) 0 0
\(711\) 7.28743e9 0.760380
\(712\) 0 0
\(713\) −6.68596e7 −0.00690797
\(714\) 0 0
\(715\) 1.84346e8 0.0188610
\(716\) 0 0
\(717\) −7.00880e9 −0.710112
\(718\) 0 0
\(719\) −1.47897e10 −1.48391 −0.741955 0.670450i \(-0.766101\pi\)
−0.741955 + 0.670450i \(0.766101\pi\)
\(720\) 0 0
\(721\) 7.13382e8 0.0708841
\(722\) 0 0
\(723\) 8.20075e9 0.806993
\(724\) 0 0
\(725\) −7.32452e9 −0.713832
\(726\) 0 0
\(727\) 8.97168e9 0.865971 0.432986 0.901401i \(-0.357460\pi\)
0.432986 + 0.901401i \(0.357460\pi\)
\(728\) 0 0
\(729\) 3.78757e9 0.362088
\(730\) 0 0
\(731\) −9.54279e8 −0.0903576
\(732\) 0 0
\(733\) −1.73669e9 −0.162876 −0.0814382 0.996678i \(-0.525951\pi\)
−0.0814382 + 0.996678i \(0.525951\pi\)
\(734\) 0 0
\(735\) 6.76993e7 0.00628896
\(736\) 0 0
\(737\) 1.03230e10 0.949878
\(738\) 0 0
\(739\) −2.95911e9 −0.269716 −0.134858 0.990865i \(-0.543058\pi\)
−0.134858 + 0.990865i \(0.543058\pi\)
\(740\) 0 0
\(741\) 2.51754e9 0.227307
\(742\) 0 0
\(743\) 2.67191e9 0.238979 0.119490 0.992835i \(-0.461874\pi\)
0.119490 + 0.992835i \(0.461874\pi\)
\(744\) 0 0
\(745\) −1.47472e7 −0.00130666
\(746\) 0 0
\(747\) −3.52600e9 −0.309499
\(748\) 0 0
\(749\) 3.78942e8 0.0329523
\(750\) 0 0
\(751\) −1.12390e10 −0.968248 −0.484124 0.874999i \(-0.660862\pi\)
−0.484124 + 0.874999i \(0.660862\pi\)
\(752\) 0 0
\(753\) −8.58009e9 −0.732335
\(754\) 0 0
\(755\) 8.34787e7 0.00705930
\(756\) 0 0
\(757\) 6.77994e9 0.568055 0.284027 0.958816i \(-0.408329\pi\)
0.284027 + 0.958816i \(0.408329\pi\)
\(758\) 0 0
\(759\) 6.22363e7 0.00516651
\(760\) 0 0
\(761\) 1.53202e10 1.26014 0.630068 0.776540i \(-0.283027\pi\)
0.630068 + 0.776540i \(0.283027\pi\)
\(762\) 0 0
\(763\) 1.14051e9 0.0929532
\(764\) 0 0
\(765\) 1.70282e7 0.00137517
\(766\) 0 0
\(767\) 3.99464e10 3.19664
\(768\) 0 0
\(769\) −1.94253e10 −1.54037 −0.770187 0.637818i \(-0.779837\pi\)
−0.770187 + 0.637818i \(0.779837\pi\)
\(770\) 0 0
\(771\) −8.45230e9 −0.664178
\(772\) 0 0
\(773\) 2.09447e10 1.63097 0.815487 0.578776i \(-0.196469\pi\)
0.815487 + 0.578776i \(0.196469\pi\)
\(774\) 0 0
\(775\) 8.11475e9 0.626209
\(776\) 0 0
\(777\) 1.41897e9 0.108518
\(778\) 0 0
\(779\) 5.78852e9 0.438719
\(780\) 0 0
\(781\) 1.91466e10 1.43818
\(782\) 0 0
\(783\) −8.88554e9 −0.661482
\(784\) 0 0
\(785\) −1.36774e7 −0.00100916
\(786\) 0 0
\(787\) 3.02490e9 0.221207 0.110604 0.993865i \(-0.464722\pi\)
0.110604 + 0.993865i \(0.464722\pi\)
\(788\) 0 0
\(789\) −1.94150e9 −0.140724
\(790\) 0 0
\(791\) −1.75326e9 −0.125959
\(792\) 0 0
\(793\) −3.46136e10 −2.46485
\(794\) 0 0
\(795\) 5.08623e7 0.00359014
\(796\) 0 0
\(797\) 4.28296e9 0.299667 0.149834 0.988711i \(-0.452126\pi\)
0.149834 + 0.988711i \(0.452126\pi\)
\(798\) 0 0
\(799\) 2.54337e9 0.176399
\(800\) 0 0
\(801\) −1.35595e10 −0.932247
\(802\) 0 0
\(803\) 1.38981e10 0.947224
\(804\) 0 0
\(805\) 374253. 2.52860e−5 0
\(806\) 0 0
\(807\) −7.88749e9 −0.528301
\(808\) 0 0
\(809\) −4.60253e9 −0.305617 −0.152808 0.988256i \(-0.548832\pi\)
−0.152808 + 0.988256i \(0.548832\pi\)
\(810\) 0 0
\(811\) 2.38895e10 1.57266 0.786328 0.617810i \(-0.211980\pi\)
0.786328 + 0.617810i \(0.211980\pi\)
\(812\) 0 0
\(813\) −4.56159e9 −0.297714
\(814\) 0 0
\(815\) 2.54093e8 0.0164415
\(816\) 0 0
\(817\) 1.98776e9 0.127522
\(818\) 0 0
\(819\) −3.85472e9 −0.245188
\(820\) 0 0
\(821\) −4.67646e9 −0.294928 −0.147464 0.989067i \(-0.547111\pi\)
−0.147464 + 0.989067i \(0.547111\pi\)
\(822\) 0 0
\(823\) 3.03215e10 1.89605 0.948027 0.318189i \(-0.103075\pi\)
0.948027 + 0.318189i \(0.103075\pi\)
\(824\) 0 0
\(825\) −7.55363e9 −0.468346
\(826\) 0 0
\(827\) 2.53758e10 1.56009 0.780047 0.625721i \(-0.215195\pi\)
0.780047 + 0.625721i \(0.215195\pi\)
\(828\) 0 0
\(829\) −2.75451e10 −1.67920 −0.839601 0.543204i \(-0.817211\pi\)
−0.839601 + 0.543204i \(0.817211\pi\)
\(830\) 0 0
\(831\) 5.77312e9 0.348985
\(832\) 0 0
\(833\) −2.61296e9 −0.156630
\(834\) 0 0
\(835\) −1.07115e8 −0.00636717
\(836\) 0 0
\(837\) 9.84420e9 0.580285
\(838\) 0 0
\(839\) −1.46274e10 −0.855065 −0.427533 0.904000i \(-0.640617\pi\)
−0.427533 + 0.904000i \(0.640617\pi\)
\(840\) 0 0
\(841\) −8.45756e9 −0.490297
\(842\) 0 0
\(843\) −6.34314e9 −0.364677
\(844\) 0 0
\(845\) −4.89120e8 −0.0278880
\(846\) 0 0
\(847\) 8.69099e8 0.0491448
\(848\) 0 0
\(849\) −1.05727e10 −0.592936
\(850\) 0 0
\(851\) −2.07352e8 −0.0115333
\(852\) 0 0
\(853\) 1.16171e10 0.640880 0.320440 0.947269i \(-0.396169\pi\)
0.320440 + 0.947269i \(0.396169\pi\)
\(854\) 0 0
\(855\) −3.54698e7 −0.00194078
\(856\) 0 0
\(857\) 1.76278e10 0.956678 0.478339 0.878175i \(-0.341239\pi\)
0.478339 + 0.878175i \(0.341239\pi\)
\(858\) 0 0
\(859\) −1.92669e10 −1.03714 −0.518569 0.855036i \(-0.673535\pi\)
−0.518569 + 0.855036i \(0.673535\pi\)
\(860\) 0 0
\(861\) 3.71697e9 0.198462
\(862\) 0 0
\(863\) 2.06979e10 1.09620 0.548098 0.836414i \(-0.315352\pi\)
0.548098 + 0.836414i \(0.315352\pi\)
\(864\) 0 0
\(865\) −2.04048e8 −0.0107195
\(866\) 0 0
\(867\) −1.01552e10 −0.529204
\(868\) 0 0
\(869\) −1.79918e10 −0.930046
\(870\) 0 0
\(871\) −3.91824e10 −2.00922
\(872\) 0 0
\(873\) −1.87691e10 −0.954758
\(874\) 0 0
\(875\) −9.08529e7 −0.00458470
\(876\) 0 0
\(877\) −2.31739e10 −1.16011 −0.580056 0.814576i \(-0.696969\pi\)
−0.580056 + 0.814576i \(0.696969\pi\)
\(878\) 0 0
\(879\) −8.24889e9 −0.409670
\(880\) 0 0
\(881\) −1.29374e10 −0.637426 −0.318713 0.947851i \(-0.603251\pi\)
−0.318713 + 0.947851i \(0.603251\pi\)
\(882\) 0 0
\(883\) 8.34205e9 0.407765 0.203883 0.978995i \(-0.434644\pi\)
0.203883 + 0.978995i \(0.434644\pi\)
\(884\) 0 0
\(885\) 2.36029e8 0.0114463
\(886\) 0 0
\(887\) −5.97628e9 −0.287540 −0.143770 0.989611i \(-0.545923\pi\)
−0.143770 + 0.989611i \(0.545923\pi\)
\(888\) 0 0
\(889\) 9.13176e8 0.0435912
\(890\) 0 0
\(891\) 3.65539e9 0.173126
\(892\) 0 0
\(893\) −5.29783e9 −0.248954
\(894\) 0 0
\(895\) −3.76646e7 −0.00175612
\(896\) 0 0
\(897\) −2.36228e8 −0.0109284
\(898\) 0 0
\(899\) −9.74091e9 −0.447137
\(900\) 0 0
\(901\) −1.96311e9 −0.0894144
\(902\) 0 0
\(903\) 1.27639e9 0.0576870
\(904\) 0 0
\(905\) −1.91378e8 −0.00858264
\(906\) 0 0
\(907\) 5.54811e8 0.0246899 0.0123450 0.999924i \(-0.496070\pi\)
0.0123450 + 0.999924i \(0.496070\pi\)
\(908\) 0 0
\(909\) −3.49871e9 −0.154502
\(910\) 0 0
\(911\) −2.42935e10 −1.06458 −0.532288 0.846564i \(-0.678668\pi\)
−0.532288 + 0.846564i \(0.678668\pi\)
\(912\) 0 0
\(913\) 8.70525e9 0.378559
\(914\) 0 0
\(915\) −2.04519e8 −0.00882591
\(916\) 0 0
\(917\) 5.61212e9 0.240345
\(918\) 0 0
\(919\) 6.29454e9 0.267522 0.133761 0.991014i \(-0.457295\pi\)
0.133761 + 0.991014i \(0.457295\pi\)
\(920\) 0 0
\(921\) 1.44654e10 0.610128
\(922\) 0 0
\(923\) −7.26739e10 −3.04210
\(924\) 0 0
\(925\) 2.51663e10 1.04550
\(926\) 0 0
\(927\) 6.34409e9 0.261572
\(928\) 0 0
\(929\) 8.78717e9 0.359579 0.179789 0.983705i \(-0.442458\pi\)
0.179789 + 0.983705i \(0.442458\pi\)
\(930\) 0 0
\(931\) 5.44278e9 0.221053
\(932\) 0 0
\(933\) −2.67736e9 −0.107925
\(934\) 0 0
\(935\) −4.20407e7 −0.00168201
\(936\) 0 0
\(937\) −4.18401e9 −0.166152 −0.0830758 0.996543i \(-0.526474\pi\)
−0.0830758 + 0.996543i \(0.526474\pi\)
\(938\) 0 0
\(939\) −1.05162e10 −0.414506
\(940\) 0 0
\(941\) 2.92628e10 1.14486 0.572429 0.819954i \(-0.306001\pi\)
0.572429 + 0.819954i \(0.306001\pi\)
\(942\) 0 0
\(943\) −5.43153e8 −0.0210927
\(944\) 0 0
\(945\) −5.51040e7 −0.00212408
\(946\) 0 0
\(947\) 3.35848e10 1.28504 0.642521 0.766268i \(-0.277889\pi\)
0.642521 + 0.766268i \(0.277889\pi\)
\(948\) 0 0
\(949\) −5.27526e10 −2.00361
\(950\) 0 0
\(951\) 1.00971e10 0.380684
\(952\) 0 0
\(953\) −3.10142e10 −1.16074 −0.580370 0.814353i \(-0.697092\pi\)
−0.580370 + 0.814353i \(0.697092\pi\)
\(954\) 0 0
\(955\) 1.23027e8 0.00457075
\(956\) 0 0
\(957\) 9.06734e9 0.334417
\(958\) 0 0
\(959\) 6.44250e9 0.235879
\(960\) 0 0
\(961\) −1.67208e10 −0.607749
\(962\) 0 0
\(963\) 3.36992e9 0.121598
\(964\) 0 0
\(965\) 3.57937e8 0.0128222
\(966\) 0 0
\(967\) 3.04256e10 1.08205 0.541025 0.841007i \(-0.318037\pi\)
0.541025 + 0.841007i \(0.318037\pi\)
\(968\) 0 0
\(969\) −5.74131e8 −0.0202711
\(970\) 0 0
\(971\) 1.46525e10 0.513623 0.256811 0.966462i \(-0.417328\pi\)
0.256811 + 0.966462i \(0.417328\pi\)
\(972\) 0 0
\(973\) −2.48196e9 −0.0863774
\(974\) 0 0
\(975\) 2.86710e10 0.990665
\(976\) 0 0
\(977\) 5.10251e10 1.75046 0.875232 0.483704i \(-0.160709\pi\)
0.875232 + 0.483704i \(0.160709\pi\)
\(978\) 0 0
\(979\) 3.34768e10 1.14026
\(980\) 0 0
\(981\) 1.01426e10 0.343009
\(982\) 0 0
\(983\) −1.29089e10 −0.433463 −0.216732 0.976231i \(-0.569540\pi\)
−0.216732 + 0.976231i \(0.569540\pi\)
\(984\) 0 0
\(985\) −4.04354e8 −0.0134814
\(986\) 0 0
\(987\) −3.40188e9 −0.112618
\(988\) 0 0
\(989\) −1.86517e8 −0.00613100
\(990\) 0 0
\(991\) −4.14437e10 −1.35270 −0.676348 0.736582i \(-0.736439\pi\)
−0.676348 + 0.736582i \(0.736439\pi\)
\(992\) 0 0
\(993\) 1.36312e10 0.441785
\(994\) 0 0
\(995\) −7.13293e8 −0.0229555
\(996\) 0 0
\(997\) −5.44529e9 −0.174015 −0.0870077 0.996208i \(-0.527730\pi\)
−0.0870077 + 0.996208i \(0.527730\pi\)
\(998\) 0 0
\(999\) 3.05299e10 0.968826
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 76.8.a.a.1.4 5
4.3 odd 2 304.8.a.g.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.8.a.a.1.4 5 1.1 even 1 trivial
304.8.a.g.1.2 5 4.3 odd 2