Properties

Label 531.8.a.b.1.8
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.05882\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.05882 q^{2} -126.879 q^{4} -151.597 q^{5} -1574.54 q^{7} -269.870 q^{8} +O(q^{10})\) \(q+1.05882 q^{2} -126.879 q^{4} -151.597 q^{5} -1574.54 q^{7} -269.870 q^{8} -160.513 q^{10} -1597.23 q^{11} +8756.36 q^{13} -1667.15 q^{14} +15954.8 q^{16} -19841.7 q^{17} -4839.26 q^{19} +19234.5 q^{20} -1691.17 q^{22} +30741.2 q^{23} -55143.3 q^{25} +9271.37 q^{26} +199776. q^{28} -14740.2 q^{29} +72269.3 q^{31} +51436.5 q^{32} -21008.7 q^{34} +238696. q^{35} +357556. q^{37} -5123.88 q^{38} +40911.4 q^{40} -562020. q^{41} +682037. q^{43} +202654. q^{44} +32549.3 q^{46} +1.24121e6 q^{47} +1.65563e6 q^{49} -58386.6 q^{50} -1.11100e6 q^{52} +1.94069e6 q^{53} +242135. q^{55} +424921. q^{56} -15607.1 q^{58} -205379. q^{59} -2.48399e6 q^{61} +76519.8 q^{62} -1.98775e6 q^{64} -1.32744e6 q^{65} -1.19534e6 q^{67} +2.51749e6 q^{68} +252735. q^{70} +5.50733e6 q^{71} -1.03549e6 q^{73} +378585. q^{74} +614000. q^{76} +2.51490e6 q^{77} -1.05062e6 q^{79} -2.41869e6 q^{80} -595075. q^{82} -8.29947e6 q^{83} +3.00794e6 q^{85} +722151. q^{86} +431043. q^{88} -1.51623e6 q^{89} -1.37872e7 q^{91} -3.90041e6 q^{92} +1.31421e6 q^{94} +733618. q^{95} +1.02696e7 q^{97} +1.75301e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.05882 0.0935869 0.0467935 0.998905i \(-0.485100\pi\)
0.0467935 + 0.998905i \(0.485100\pi\)
\(3\) 0 0
\(4\) −126.879 −0.991241
\(5\) −151.597 −0.542370 −0.271185 0.962527i \(-0.587416\pi\)
−0.271185 + 0.962527i \(0.587416\pi\)
\(6\) 0 0
\(7\) −1574.54 −1.73504 −0.867522 0.497398i \(-0.834289\pi\)
−0.867522 + 0.497398i \(0.834289\pi\)
\(8\) −269.870 −0.186354
\(9\) 0 0
\(10\) −160.513 −0.0507587
\(11\) −1597.23 −0.361819 −0.180910 0.983500i \(-0.557904\pi\)
−0.180910 + 0.983500i \(0.557904\pi\)
\(12\) 0 0
\(13\) 8756.36 1.10541 0.552703 0.833378i \(-0.313596\pi\)
0.552703 + 0.833378i \(0.313596\pi\)
\(14\) −1667.15 −0.162378
\(15\) 0 0
\(16\) 15954.8 0.973801
\(17\) −19841.7 −0.979507 −0.489754 0.871861i \(-0.662913\pi\)
−0.489754 + 0.871861i \(0.662913\pi\)
\(18\) 0 0
\(19\) −4839.26 −0.161861 −0.0809304 0.996720i \(-0.525789\pi\)
−0.0809304 + 0.996720i \(0.525789\pi\)
\(20\) 19234.5 0.537620
\(21\) 0 0
\(22\) −1691.17 −0.0338616
\(23\) 30741.2 0.526834 0.263417 0.964682i \(-0.415151\pi\)
0.263417 + 0.964682i \(0.415151\pi\)
\(24\) 0 0
\(25\) −55143.3 −0.705835
\(26\) 9271.37 0.103452
\(27\) 0 0
\(28\) 199776. 1.71985
\(29\) −14740.2 −0.112230 −0.0561152 0.998424i \(-0.517871\pi\)
−0.0561152 + 0.998424i \(0.517871\pi\)
\(30\) 0 0
\(31\) 72269.3 0.435700 0.217850 0.975982i \(-0.430096\pi\)
0.217850 + 0.975982i \(0.430096\pi\)
\(32\) 51436.5 0.277489
\(33\) 0 0
\(34\) −21008.7 −0.0916691
\(35\) 238696. 0.941036
\(36\) 0 0
\(37\) 357556. 1.16048 0.580240 0.814445i \(-0.302959\pi\)
0.580240 + 0.814445i \(0.302959\pi\)
\(38\) −5123.88 −0.0151480
\(39\) 0 0
\(40\) 40911.4 0.101073
\(41\) −562020. −1.27353 −0.636764 0.771059i \(-0.719727\pi\)
−0.636764 + 0.771059i \(0.719727\pi\)
\(42\) 0 0
\(43\) 682037. 1.30818 0.654092 0.756415i \(-0.273051\pi\)
0.654092 + 0.756415i \(0.273051\pi\)
\(44\) 202654. 0.358650
\(45\) 0 0
\(46\) 32549.3 0.0493047
\(47\) 1.24121e6 1.74382 0.871911 0.489665i \(-0.162881\pi\)
0.871911 + 0.489665i \(0.162881\pi\)
\(48\) 0 0
\(49\) 1.65563e6 2.01038
\(50\) −58386.6 −0.0660569
\(51\) 0 0
\(52\) −1.11100e6 −1.09572
\(53\) 1.94069e6 1.79057 0.895286 0.445493i \(-0.146972\pi\)
0.895286 + 0.445493i \(0.146972\pi\)
\(54\) 0 0
\(55\) 242135. 0.196240
\(56\) 424921. 0.323333
\(57\) 0 0
\(58\) −15607.1 −0.0105033
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −2.48399e6 −1.40119 −0.700593 0.713561i \(-0.747081\pi\)
−0.700593 + 0.713561i \(0.747081\pi\)
\(62\) 76519.8 0.0407758
\(63\) 0 0
\(64\) −1.98775e6 −0.947832
\(65\) −1.32744e6 −0.599539
\(66\) 0 0
\(67\) −1.19534e6 −0.485546 −0.242773 0.970083i \(-0.578057\pi\)
−0.242773 + 0.970083i \(0.578057\pi\)
\(68\) 2.51749e6 0.970928
\(69\) 0 0
\(70\) 252735. 0.0880687
\(71\) 5.50733e6 1.82615 0.913076 0.407789i \(-0.133700\pi\)
0.913076 + 0.407789i \(0.133700\pi\)
\(72\) 0 0
\(73\) −1.03549e6 −0.311542 −0.155771 0.987793i \(-0.549786\pi\)
−0.155771 + 0.987793i \(0.549786\pi\)
\(74\) 378585. 0.108606
\(75\) 0 0
\(76\) 614000. 0.160443
\(77\) 2.51490e6 0.627773
\(78\) 0 0
\(79\) −1.05062e6 −0.239745 −0.119873 0.992789i \(-0.538249\pi\)
−0.119873 + 0.992789i \(0.538249\pi\)
\(80\) −2.41869e6 −0.528161
\(81\) 0 0
\(82\) −595075. −0.119185
\(83\) −8.29947e6 −1.59323 −0.796613 0.604490i \(-0.793377\pi\)
−0.796613 + 0.604490i \(0.793377\pi\)
\(84\) 0 0
\(85\) 3.00794e6 0.531256
\(86\) 722151. 0.122429
\(87\) 0 0
\(88\) 431043. 0.0674266
\(89\) −1.51623e6 −0.227982 −0.113991 0.993482i \(-0.536363\pi\)
−0.113991 + 0.993482i \(0.536363\pi\)
\(90\) 0 0
\(91\) −1.37872e7 −1.91793
\(92\) −3.90041e6 −0.522219
\(93\) 0 0
\(94\) 1.31421e6 0.163199
\(95\) 733618. 0.0877884
\(96\) 0 0
\(97\) 1.02696e7 1.14249 0.571247 0.820778i \(-0.306460\pi\)
0.571247 + 0.820778i \(0.306460\pi\)
\(98\) 1.75301e6 0.188145
\(99\) 0 0
\(100\) 6.99653e6 0.699653
\(101\) 1.21422e7 1.17267 0.586333 0.810070i \(-0.300571\pi\)
0.586333 + 0.810070i \(0.300571\pi\)
\(102\) 0 0
\(103\) −2.11994e7 −1.91159 −0.955793 0.294041i \(-0.905000\pi\)
−0.955793 + 0.294041i \(0.905000\pi\)
\(104\) −2.36308e6 −0.205997
\(105\) 0 0
\(106\) 2.05484e6 0.167574
\(107\) −58058.2 −0.00458163 −0.00229082 0.999997i \(-0.500729\pi\)
−0.00229082 + 0.999997i \(0.500729\pi\)
\(108\) 0 0
\(109\) 1.06543e7 0.788011 0.394005 0.919108i \(-0.371089\pi\)
0.394005 + 0.919108i \(0.371089\pi\)
\(110\) 256376. 0.0183655
\(111\) 0 0
\(112\) −2.51214e7 −1.68959
\(113\) 4.86803e6 0.317379 0.158690 0.987329i \(-0.449273\pi\)
0.158690 + 0.987329i \(0.449273\pi\)
\(114\) 0 0
\(115\) −4.66028e6 −0.285739
\(116\) 1.87022e6 0.111247
\(117\) 0 0
\(118\) −217458. −0.0121840
\(119\) 3.12416e7 1.69949
\(120\) 0 0
\(121\) −1.69360e7 −0.869087
\(122\) −2.63009e6 −0.131133
\(123\) 0 0
\(124\) −9.16945e6 −0.431884
\(125\) 2.02031e7 0.925194
\(126\) 0 0
\(127\) −3.32597e7 −1.44081 −0.720403 0.693556i \(-0.756043\pi\)
−0.720403 + 0.693556i \(0.756043\pi\)
\(128\) −8.68852e6 −0.366194
\(129\) 0 0
\(130\) −1.40551e6 −0.0561090
\(131\) −3.81285e6 −0.148184 −0.0740918 0.997251i \(-0.523606\pi\)
−0.0740918 + 0.997251i \(0.523606\pi\)
\(132\) 0 0
\(133\) 7.61961e6 0.280836
\(134\) −1.26565e6 −0.0454407
\(135\) 0 0
\(136\) 5.35467e6 0.182535
\(137\) −4.60178e7 −1.52899 −0.764494 0.644630i \(-0.777011\pi\)
−0.764494 + 0.644630i \(0.777011\pi\)
\(138\) 0 0
\(139\) 4.52736e7 1.42986 0.714930 0.699196i \(-0.246458\pi\)
0.714930 + 0.699196i \(0.246458\pi\)
\(140\) −3.02855e7 −0.932794
\(141\) 0 0
\(142\) 5.83125e6 0.170904
\(143\) −1.39859e7 −0.399957
\(144\) 0 0
\(145\) 2.23457e6 0.0608704
\(146\) −1.09639e6 −0.0291562
\(147\) 0 0
\(148\) −4.53663e7 −1.15032
\(149\) −2.55791e7 −0.633481 −0.316740 0.948512i \(-0.602588\pi\)
−0.316740 + 0.948512i \(0.602588\pi\)
\(150\) 0 0
\(151\) 2.20103e7 0.520243 0.260121 0.965576i \(-0.416237\pi\)
0.260121 + 0.965576i \(0.416237\pi\)
\(152\) 1.30597e6 0.0301634
\(153\) 0 0
\(154\) 2.66281e6 0.0587513
\(155\) −1.09558e7 −0.236311
\(156\) 0 0
\(157\) 2.45776e7 0.506862 0.253431 0.967353i \(-0.418441\pi\)
0.253431 + 0.967353i \(0.418441\pi\)
\(158\) −1.11241e6 −0.0224370
\(159\) 0 0
\(160\) −7.79762e6 −0.150502
\(161\) −4.84033e7 −0.914080
\(162\) 0 0
\(163\) −7.12800e7 −1.28917 −0.644586 0.764532i \(-0.722970\pi\)
−0.644586 + 0.764532i \(0.722970\pi\)
\(164\) 7.13085e7 1.26237
\(165\) 0 0
\(166\) −8.78761e6 −0.149105
\(167\) −1.72745e7 −0.287010 −0.143505 0.989650i \(-0.545837\pi\)
−0.143505 + 0.989650i \(0.545837\pi\)
\(168\) 0 0
\(169\) 1.39253e7 0.221923
\(170\) 3.18486e6 0.0497186
\(171\) 0 0
\(172\) −8.65361e7 −1.29673
\(173\) 4.27240e7 0.627351 0.313676 0.949530i \(-0.398440\pi\)
0.313676 + 0.949530i \(0.398440\pi\)
\(174\) 0 0
\(175\) 8.68254e7 1.22465
\(176\) −2.54833e7 −0.352340
\(177\) 0 0
\(178\) −1.60541e6 −0.0213361
\(179\) −5.80252e7 −0.756191 −0.378095 0.925767i \(-0.623421\pi\)
−0.378095 + 0.925767i \(0.623421\pi\)
\(180\) 0 0
\(181\) −4.33616e7 −0.543539 −0.271769 0.962362i \(-0.587609\pi\)
−0.271769 + 0.962362i \(0.587609\pi\)
\(182\) −1.45981e7 −0.179493
\(183\) 0 0
\(184\) −8.29612e6 −0.0981777
\(185\) −5.42044e7 −0.629410
\(186\) 0 0
\(187\) 3.16917e7 0.354405
\(188\) −1.57483e8 −1.72855
\(189\) 0 0
\(190\) 776766. 0.00821585
\(191\) 6.93621e7 0.720287 0.360143 0.932897i \(-0.382728\pi\)
0.360143 + 0.932897i \(0.382728\pi\)
\(192\) 0 0
\(193\) 3.17140e7 0.317541 0.158771 0.987315i \(-0.449247\pi\)
0.158771 + 0.987315i \(0.449247\pi\)
\(194\) 1.08736e7 0.106922
\(195\) 0 0
\(196\) −2.10065e8 −1.99277
\(197\) 1.33959e8 1.24836 0.624178 0.781282i \(-0.285434\pi\)
0.624178 + 0.781282i \(0.285434\pi\)
\(198\) 0 0
\(199\) 7.57957e7 0.681803 0.340902 0.940099i \(-0.389268\pi\)
0.340902 + 0.940099i \(0.389268\pi\)
\(200\) 1.48815e7 0.131535
\(201\) 0 0
\(202\) 1.28564e7 0.109746
\(203\) 2.32090e7 0.194725
\(204\) 0 0
\(205\) 8.52006e7 0.690723
\(206\) −2.24463e7 −0.178899
\(207\) 0 0
\(208\) 1.39706e8 1.07645
\(209\) 7.72939e6 0.0585644
\(210\) 0 0
\(211\) −2.11972e8 −1.55343 −0.776713 0.629854i \(-0.783115\pi\)
−0.776713 + 0.629854i \(0.783115\pi\)
\(212\) −2.46233e8 −1.77489
\(213\) 0 0
\(214\) −61472.9 −0.000428781 0
\(215\) −1.03395e8 −0.709519
\(216\) 0 0
\(217\) −1.13791e8 −0.755959
\(218\) 1.12809e7 0.0737475
\(219\) 0 0
\(220\) −3.07218e7 −0.194521
\(221\) −1.73741e8 −1.08275
\(222\) 0 0
\(223\) −1.80114e8 −1.08763 −0.543814 0.839206i \(-0.683020\pi\)
−0.543814 + 0.839206i \(0.683020\pi\)
\(224\) −8.09888e7 −0.481456
\(225\) 0 0
\(226\) 5.15434e6 0.0297025
\(227\) −7.73940e7 −0.439154 −0.219577 0.975595i \(-0.570468\pi\)
−0.219577 + 0.975595i \(0.570468\pi\)
\(228\) 0 0
\(229\) 1.39410e8 0.767134 0.383567 0.923513i \(-0.374696\pi\)
0.383567 + 0.923513i \(0.374696\pi\)
\(230\) −4.93437e6 −0.0267414
\(231\) 0 0
\(232\) 3.97793e6 0.0209146
\(233\) −1.24112e8 −0.642787 −0.321393 0.946946i \(-0.604151\pi\)
−0.321393 + 0.946946i \(0.604151\pi\)
\(234\) 0 0
\(235\) −1.88163e8 −0.945796
\(236\) 2.60583e7 0.129049
\(237\) 0 0
\(238\) 3.30790e7 0.159050
\(239\) 1.55356e8 0.736096 0.368048 0.929807i \(-0.380026\pi\)
0.368048 + 0.929807i \(0.380026\pi\)
\(240\) 0 0
\(241\) 3.44462e8 1.58519 0.792597 0.609746i \(-0.208729\pi\)
0.792597 + 0.609746i \(0.208729\pi\)
\(242\) −1.79321e7 −0.0813352
\(243\) 0 0
\(244\) 3.15166e8 1.38891
\(245\) −2.50989e8 −1.09037
\(246\) 0 0
\(247\) −4.23743e7 −0.178922
\(248\) −1.95033e7 −0.0811945
\(249\) 0 0
\(250\) 2.13913e7 0.0865860
\(251\) 4.67431e8 1.86578 0.932888 0.360166i \(-0.117280\pi\)
0.932888 + 0.360166i \(0.117280\pi\)
\(252\) 0 0
\(253\) −4.91007e7 −0.190619
\(254\) −3.52159e7 −0.134841
\(255\) 0 0
\(256\) 2.45232e8 0.913561
\(257\) 4.09247e8 1.50390 0.751952 0.659218i \(-0.229113\pi\)
0.751952 + 0.659218i \(0.229113\pi\)
\(258\) 0 0
\(259\) −5.62986e8 −2.01349
\(260\) 1.68424e8 0.594288
\(261\) 0 0
\(262\) −4.03710e6 −0.0138680
\(263\) −1.04382e8 −0.353818 −0.176909 0.984227i \(-0.556610\pi\)
−0.176909 + 0.984227i \(0.556610\pi\)
\(264\) 0 0
\(265\) −2.94204e8 −0.971152
\(266\) 8.06776e6 0.0262825
\(267\) 0 0
\(268\) 1.51664e8 0.481293
\(269\) 1.95159e8 0.611302 0.305651 0.952144i \(-0.401126\pi\)
0.305651 + 0.952144i \(0.401126\pi\)
\(270\) 0 0
\(271\) 3.63467e8 1.10936 0.554681 0.832063i \(-0.312840\pi\)
0.554681 + 0.832063i \(0.312840\pi\)
\(272\) −3.16570e8 −0.953845
\(273\) 0 0
\(274\) −4.87244e7 −0.143093
\(275\) 8.80763e7 0.255385
\(276\) 0 0
\(277\) −3.76920e8 −1.06554 −0.532771 0.846260i \(-0.678849\pi\)
−0.532771 + 0.846260i \(0.678849\pi\)
\(278\) 4.79364e7 0.133816
\(279\) 0 0
\(280\) −6.44167e7 −0.175366
\(281\) −3.38642e8 −0.910475 −0.455238 0.890370i \(-0.650446\pi\)
−0.455238 + 0.890370i \(0.650446\pi\)
\(282\) 0 0
\(283\) −2.79439e8 −0.732882 −0.366441 0.930441i \(-0.619424\pi\)
−0.366441 + 0.930441i \(0.619424\pi\)
\(284\) −6.98764e8 −1.81016
\(285\) 0 0
\(286\) −1.48085e7 −0.0374308
\(287\) 8.84923e8 2.20963
\(288\) 0 0
\(289\) −1.66455e7 −0.0405653
\(290\) 2.36600e6 0.00569667
\(291\) 0 0
\(292\) 1.31382e8 0.308813
\(293\) −6.25978e8 −1.45386 −0.726930 0.686712i \(-0.759054\pi\)
−0.726930 + 0.686712i \(0.759054\pi\)
\(294\) 0 0
\(295\) 3.11349e7 0.0706106
\(296\) −9.64935e7 −0.216260
\(297\) 0 0
\(298\) −2.70836e7 −0.0592855
\(299\) 2.69181e8 0.582365
\(300\) 0 0
\(301\) −1.07390e9 −2.26976
\(302\) 2.33048e7 0.0486879
\(303\) 0 0
\(304\) −7.72092e7 −0.157620
\(305\) 3.76566e8 0.759962
\(306\) 0 0
\(307\) −2.80506e8 −0.553297 −0.276649 0.960971i \(-0.589224\pi\)
−0.276649 + 0.960971i \(0.589224\pi\)
\(308\) −3.19087e8 −0.622275
\(309\) 0 0
\(310\) −1.16002e7 −0.0221156
\(311\) −3.02914e8 −0.571029 −0.285515 0.958374i \(-0.592165\pi\)
−0.285515 + 0.958374i \(0.592165\pi\)
\(312\) 0 0
\(313\) −7.65931e8 −1.41184 −0.705918 0.708294i \(-0.749465\pi\)
−0.705918 + 0.708294i \(0.749465\pi\)
\(314\) 2.60231e7 0.0474357
\(315\) 0 0
\(316\) 1.33301e8 0.237645
\(317\) 1.06128e9 1.87120 0.935601 0.353058i \(-0.114858\pi\)
0.935601 + 0.353058i \(0.114858\pi\)
\(318\) 0 0
\(319\) 2.35434e7 0.0406071
\(320\) 3.01337e8 0.514076
\(321\) 0 0
\(322\) −5.12501e7 −0.0855459
\(323\) 9.60192e7 0.158544
\(324\) 0 0
\(325\) −4.82855e8 −0.780234
\(326\) −7.54724e7 −0.120650
\(327\) 0 0
\(328\) 1.51672e8 0.237327
\(329\) −1.95433e9 −3.02561
\(330\) 0 0
\(331\) −3.08627e8 −0.467774 −0.233887 0.972264i \(-0.575145\pi\)
−0.233887 + 0.972264i \(0.575145\pi\)
\(332\) 1.05303e9 1.57927
\(333\) 0 0
\(334\) −1.82905e7 −0.0268604
\(335\) 1.81210e8 0.263346
\(336\) 0 0
\(337\) −7.55380e8 −1.07513 −0.537565 0.843222i \(-0.680656\pi\)
−0.537565 + 0.843222i \(0.680656\pi\)
\(338\) 1.47443e7 0.0207691
\(339\) 0 0
\(340\) −3.81645e8 −0.526603
\(341\) −1.15430e8 −0.157645
\(342\) 0 0
\(343\) −1.31016e9 −1.75306
\(344\) −1.84061e8 −0.243785
\(345\) 0 0
\(346\) 4.52368e7 0.0587119
\(347\) 2.59655e8 0.333614 0.166807 0.985990i \(-0.446654\pi\)
0.166807 + 0.985990i \(0.446654\pi\)
\(348\) 0 0
\(349\) −4.22817e8 −0.532431 −0.266215 0.963914i \(-0.585773\pi\)
−0.266215 + 0.963914i \(0.585773\pi\)
\(350\) 9.19321e7 0.114612
\(351\) 0 0
\(352\) −8.21556e7 −0.100401
\(353\) −1.55553e9 −1.88221 −0.941105 0.338116i \(-0.890210\pi\)
−0.941105 + 0.338116i \(0.890210\pi\)
\(354\) 0 0
\(355\) −8.34895e8 −0.990450
\(356\) 1.92378e8 0.225985
\(357\) 0 0
\(358\) −6.14380e7 −0.0707696
\(359\) 7.61688e8 0.868853 0.434427 0.900707i \(-0.356951\pi\)
0.434427 + 0.900707i \(0.356951\pi\)
\(360\) 0 0
\(361\) −8.70453e8 −0.973801
\(362\) −4.59119e7 −0.0508681
\(363\) 0 0
\(364\) 1.74931e9 1.90113
\(365\) 1.56977e8 0.168971
\(366\) 0 0
\(367\) 1.37332e8 0.145024 0.0725122 0.997368i \(-0.476898\pi\)
0.0725122 + 0.997368i \(0.476898\pi\)
\(368\) 4.90469e8 0.513031
\(369\) 0 0
\(370\) −5.73924e7 −0.0589045
\(371\) −3.05570e9 −3.10672
\(372\) 0 0
\(373\) −2.94107e8 −0.293443 −0.146722 0.989178i \(-0.546872\pi\)
−0.146722 + 0.989178i \(0.546872\pi\)
\(374\) 3.35556e7 0.0331677
\(375\) 0 0
\(376\) −3.34964e8 −0.324968
\(377\) −1.29070e8 −0.124060
\(378\) 0 0
\(379\) 6.95112e8 0.655870 0.327935 0.944700i \(-0.393647\pi\)
0.327935 + 0.944700i \(0.393647\pi\)
\(380\) −9.30806e7 −0.0870195
\(381\) 0 0
\(382\) 7.34417e7 0.0674094
\(383\) 5.22431e8 0.475153 0.237576 0.971369i \(-0.423647\pi\)
0.237576 + 0.971369i \(0.423647\pi\)
\(384\) 0 0
\(385\) −3.81251e8 −0.340485
\(386\) 3.35792e7 0.0297177
\(387\) 0 0
\(388\) −1.30300e9 −1.13249
\(389\) 1.26347e9 1.08829 0.544143 0.838993i \(-0.316855\pi\)
0.544143 + 0.838993i \(0.316855\pi\)
\(390\) 0 0
\(391\) −6.09958e8 −0.516038
\(392\) −4.46806e8 −0.374643
\(393\) 0 0
\(394\) 1.41837e8 0.116830
\(395\) 1.59271e8 0.130031
\(396\) 0 0
\(397\) 7.32040e8 0.587175 0.293588 0.955932i \(-0.405151\pi\)
0.293588 + 0.955932i \(0.405151\pi\)
\(398\) 8.02537e7 0.0638079
\(399\) 0 0
\(400\) −8.79799e8 −0.687343
\(401\) 1.29683e9 1.00434 0.502168 0.864770i \(-0.332536\pi\)
0.502168 + 0.864770i \(0.332536\pi\)
\(402\) 0 0
\(403\) 6.32816e8 0.481626
\(404\) −1.54060e9 −1.16240
\(405\) 0 0
\(406\) 2.45741e7 0.0182237
\(407\) −5.71097e8 −0.419884
\(408\) 0 0
\(409\) 2.56134e9 1.85113 0.925563 0.378594i \(-0.123592\pi\)
0.925563 + 0.378594i \(0.123592\pi\)
\(410\) 9.02116e7 0.0646426
\(411\) 0 0
\(412\) 2.68976e9 1.89484
\(413\) 3.23378e8 0.225884
\(414\) 0 0
\(415\) 1.25818e9 0.864118
\(416\) 4.50396e8 0.306738
\(417\) 0 0
\(418\) 8.18400e6 0.00548086
\(419\) 1.65931e9 1.10199 0.550995 0.834509i \(-0.314248\pi\)
0.550995 + 0.834509i \(0.314248\pi\)
\(420\) 0 0
\(421\) −2.44008e9 −1.59374 −0.796870 0.604151i \(-0.793512\pi\)
−0.796870 + 0.604151i \(0.793512\pi\)
\(422\) −2.24440e8 −0.145380
\(423\) 0 0
\(424\) −5.23734e8 −0.333680
\(425\) 1.09414e9 0.691370
\(426\) 0 0
\(427\) 3.91115e9 2.43112
\(428\) 7.36636e6 0.00454150
\(429\) 0 0
\(430\) −1.09476e8 −0.0664017
\(431\) −5.66963e8 −0.341102 −0.170551 0.985349i \(-0.554555\pi\)
−0.170551 + 0.985349i \(0.554555\pi\)
\(432\) 0 0
\(433\) −1.17068e9 −0.692996 −0.346498 0.938051i \(-0.612629\pi\)
−0.346498 + 0.938051i \(0.612629\pi\)
\(434\) −1.20484e8 −0.0707479
\(435\) 0 0
\(436\) −1.35181e9 −0.781109
\(437\) −1.48765e8 −0.0852737
\(438\) 0 0
\(439\) −1.09400e9 −0.617151 −0.308575 0.951200i \(-0.599852\pi\)
−0.308575 + 0.951200i \(0.599852\pi\)
\(440\) −6.53448e7 −0.0365701
\(441\) 0 0
\(442\) −1.83960e8 −0.101332
\(443\) −1.26000e8 −0.0688584 −0.0344292 0.999407i \(-0.510961\pi\)
−0.0344292 + 0.999407i \(0.510961\pi\)
\(444\) 0 0
\(445\) 2.29856e8 0.123650
\(446\) −1.90707e8 −0.101788
\(447\) 0 0
\(448\) 3.12979e9 1.64453
\(449\) 2.96820e8 0.154750 0.0773750 0.997002i \(-0.475346\pi\)
0.0773750 + 0.997002i \(0.475346\pi\)
\(450\) 0 0
\(451\) 8.97673e8 0.460787
\(452\) −6.17650e8 −0.314599
\(453\) 0 0
\(454\) −8.19459e7 −0.0410990
\(455\) 2.09011e9 1.04023
\(456\) 0 0
\(457\) 2.03128e8 0.0995552 0.0497776 0.998760i \(-0.484149\pi\)
0.0497776 + 0.998760i \(0.484149\pi\)
\(458\) 1.47610e8 0.0717937
\(459\) 0 0
\(460\) 5.91291e8 0.283236
\(461\) 1.66300e9 0.790567 0.395283 0.918559i \(-0.370646\pi\)
0.395283 + 0.918559i \(0.370646\pi\)
\(462\) 0 0
\(463\) −8.36357e8 −0.391614 −0.195807 0.980642i \(-0.562733\pi\)
−0.195807 + 0.980642i \(0.562733\pi\)
\(464\) −2.35176e8 −0.109290
\(465\) 0 0
\(466\) −1.31411e8 −0.0601564
\(467\) 3.67614e9 1.67026 0.835129 0.550054i \(-0.185393\pi\)
0.835129 + 0.550054i \(0.185393\pi\)
\(468\) 0 0
\(469\) 1.88211e9 0.842444
\(470\) −1.99230e8 −0.0885142
\(471\) 0 0
\(472\) 5.54256e7 0.0242612
\(473\) −1.08937e9 −0.473326
\(474\) 0 0
\(475\) 2.66853e8 0.114247
\(476\) −3.96390e9 −1.68460
\(477\) 0 0
\(478\) 1.64493e8 0.0688890
\(479\) −4.47354e9 −1.85985 −0.929924 0.367752i \(-0.880127\pi\)
−0.929924 + 0.367752i \(0.880127\pi\)
\(480\) 0 0
\(481\) 3.13089e9 1.28280
\(482\) 3.64722e8 0.148353
\(483\) 0 0
\(484\) 2.14883e9 0.861475
\(485\) −1.55685e9 −0.619654
\(486\) 0 0
\(487\) 5.90180e8 0.231544 0.115772 0.993276i \(-0.463066\pi\)
0.115772 + 0.993276i \(0.463066\pi\)
\(488\) 6.70354e8 0.261117
\(489\) 0 0
\(490\) −2.65751e8 −0.102044
\(491\) −1.22115e9 −0.465568 −0.232784 0.972528i \(-0.574784\pi\)
−0.232784 + 0.972528i \(0.574784\pi\)
\(492\) 0 0
\(493\) 2.92471e8 0.109930
\(494\) −4.48666e7 −0.0167447
\(495\) 0 0
\(496\) 1.15304e9 0.424285
\(497\) −8.67151e9 −3.16846
\(498\) 0 0
\(499\) 1.48542e9 0.535176 0.267588 0.963533i \(-0.413773\pi\)
0.267588 + 0.963533i \(0.413773\pi\)
\(500\) −2.56335e9 −0.917090
\(501\) 0 0
\(502\) 4.94923e8 0.174612
\(503\) 2.28973e9 0.802224 0.401112 0.916029i \(-0.368624\pi\)
0.401112 + 0.916029i \(0.368624\pi\)
\(504\) 0 0
\(505\) −1.84073e9 −0.636019
\(506\) −5.19885e7 −0.0178394
\(507\) 0 0
\(508\) 4.21996e9 1.42819
\(509\) 4.18910e9 1.40802 0.704009 0.710191i \(-0.251392\pi\)
0.704009 + 0.710191i \(0.251392\pi\)
\(510\) 0 0
\(511\) 1.63042e9 0.540539
\(512\) 1.37179e9 0.451691
\(513\) 0 0
\(514\) 4.33317e8 0.140746
\(515\) 3.21377e9 1.03679
\(516\) 0 0
\(517\) −1.98249e9 −0.630948
\(518\) −5.96098e8 −0.188436
\(519\) 0 0
\(520\) 3.58235e8 0.111727
\(521\) −1.43063e9 −0.443196 −0.221598 0.975138i \(-0.571127\pi\)
−0.221598 + 0.975138i \(0.571127\pi\)
\(522\) 0 0
\(523\) 1.34628e9 0.411509 0.205755 0.978604i \(-0.434035\pi\)
0.205755 + 0.978604i \(0.434035\pi\)
\(524\) 4.83770e8 0.146886
\(525\) 0 0
\(526\) −1.10521e8 −0.0331127
\(527\) −1.43395e9 −0.426772
\(528\) 0 0
\(529\) −2.45980e9 −0.722446
\(530\) −3.11507e8 −0.0908871
\(531\) 0 0
\(532\) −9.66768e8 −0.278376
\(533\) −4.92125e9 −1.40776
\(534\) 0 0
\(535\) 8.80145e6 0.00248494
\(536\) 3.22586e8 0.0904835
\(537\) 0 0
\(538\) 2.06637e8 0.0572099
\(539\) −2.64442e9 −0.727395
\(540\) 0 0
\(541\) −3.77248e9 −1.02432 −0.512162 0.858889i \(-0.671155\pi\)
−0.512162 + 0.858889i \(0.671155\pi\)
\(542\) 3.84845e8 0.103822
\(543\) 0 0
\(544\) −1.02059e9 −0.271803
\(545\) −1.61516e9 −0.427393
\(546\) 0 0
\(547\) 1.81833e9 0.475026 0.237513 0.971384i \(-0.423668\pi\)
0.237513 + 0.971384i \(0.423668\pi\)
\(548\) 5.83869e9 1.51560
\(549\) 0 0
\(550\) 9.32566e7 0.0239007
\(551\) 7.13317e7 0.0181657
\(552\) 0 0
\(553\) 1.65424e9 0.415969
\(554\) −3.99089e8 −0.0997208
\(555\) 0 0
\(556\) −5.74427e9 −1.41734
\(557\) 5.99186e7 0.0146916 0.00734579 0.999973i \(-0.497662\pi\)
0.00734579 + 0.999973i \(0.497662\pi\)
\(558\) 0 0
\(559\) 5.97216e9 1.44607
\(560\) 3.80833e9 0.916382
\(561\) 0 0
\(562\) −3.58559e8 −0.0852086
\(563\) 5.52477e9 1.30477 0.652386 0.757887i \(-0.273768\pi\)
0.652386 + 0.757887i \(0.273768\pi\)
\(564\) 0 0
\(565\) −7.37978e8 −0.172137
\(566\) −2.95874e8 −0.0685882
\(567\) 0 0
\(568\) −1.48626e9 −0.340311
\(569\) 5.19237e9 1.18161 0.590803 0.806816i \(-0.298811\pi\)
0.590803 + 0.806816i \(0.298811\pi\)
\(570\) 0 0
\(571\) 1.98094e9 0.445293 0.222647 0.974899i \(-0.428530\pi\)
0.222647 + 0.974899i \(0.428530\pi\)
\(572\) 1.77451e9 0.396454
\(573\) 0 0
\(574\) 9.36970e8 0.206792
\(575\) −1.69517e9 −0.371858
\(576\) 0 0
\(577\) −3.25873e9 −0.706209 −0.353104 0.935584i \(-0.614874\pi\)
−0.353104 + 0.935584i \(0.614874\pi\)
\(578\) −1.76245e7 −0.00379638
\(579\) 0 0
\(580\) −2.83520e8 −0.0603372
\(581\) 1.30679e10 2.76432
\(582\) 0 0
\(583\) −3.09973e9 −0.647863
\(584\) 2.79448e8 0.0580571
\(585\) 0 0
\(586\) −6.62795e8 −0.136062
\(587\) 7.32754e8 0.149529 0.0747643 0.997201i \(-0.476180\pi\)
0.0747643 + 0.997201i \(0.476180\pi\)
\(588\) 0 0
\(589\) −3.49730e8 −0.0705227
\(590\) 3.29661e7 0.00660823
\(591\) 0 0
\(592\) 5.70472e9 1.13008
\(593\) −7.42096e9 −1.46140 −0.730699 0.682700i \(-0.760806\pi\)
−0.730699 + 0.682700i \(0.760806\pi\)
\(594\) 0 0
\(595\) −4.73613e9 −0.921752
\(596\) 3.24545e9 0.627933
\(597\) 0 0
\(598\) 2.85013e8 0.0545018
\(599\) 1.05376e9 0.200331 0.100165 0.994971i \(-0.468063\pi\)
0.100165 + 0.994971i \(0.468063\pi\)
\(600\) 0 0
\(601\) −7.07267e9 −1.32899 −0.664496 0.747292i \(-0.731354\pi\)
−0.664496 + 0.747292i \(0.731354\pi\)
\(602\) −1.13706e9 −0.212420
\(603\) 0 0
\(604\) −2.79264e9 −0.515686
\(605\) 2.56745e9 0.471367
\(606\) 0 0
\(607\) 4.55310e9 0.826317 0.413159 0.910659i \(-0.364426\pi\)
0.413159 + 0.910659i \(0.364426\pi\)
\(608\) −2.48914e8 −0.0449146
\(609\) 0 0
\(610\) 3.98714e8 0.0711225
\(611\) 1.08685e10 1.92763
\(612\) 0 0
\(613\) 6.47628e9 1.13557 0.567785 0.823177i \(-0.307800\pi\)
0.567785 + 0.823177i \(0.307800\pi\)
\(614\) −2.97004e8 −0.0517814
\(615\) 0 0
\(616\) −6.78694e8 −0.116988
\(617\) −4.44035e9 −0.761061 −0.380531 0.924768i \(-0.624259\pi\)
−0.380531 + 0.924768i \(0.624259\pi\)
\(618\) 0 0
\(619\) 2.96260e9 0.502060 0.251030 0.967979i \(-0.419231\pi\)
0.251030 + 0.967979i \(0.419231\pi\)
\(620\) 1.39006e9 0.234241
\(621\) 0 0
\(622\) −3.20730e8 −0.0534409
\(623\) 2.38737e9 0.395559
\(624\) 0 0
\(625\) 1.24534e9 0.204037
\(626\) −8.10979e8 −0.132129
\(627\) 0 0
\(628\) −3.11837e9 −0.502423
\(629\) −7.09452e9 −1.13670
\(630\) 0 0
\(631\) 1.60390e9 0.254142 0.127071 0.991894i \(-0.459442\pi\)
0.127071 + 0.991894i \(0.459442\pi\)
\(632\) 2.83530e8 0.0446775
\(633\) 0 0
\(634\) 1.12369e9 0.175120
\(635\) 5.04207e9 0.781450
\(636\) 0 0
\(637\) 1.44973e10 2.22229
\(638\) 2.49281e7 0.00380030
\(639\) 0 0
\(640\) 1.31715e9 0.198613
\(641\) 6.32307e8 0.0948254 0.0474127 0.998875i \(-0.484902\pi\)
0.0474127 + 0.998875i \(0.484902\pi\)
\(642\) 0 0
\(643\) −4.72453e9 −0.700842 −0.350421 0.936592i \(-0.613961\pi\)
−0.350421 + 0.936592i \(0.613961\pi\)
\(644\) 6.14136e9 0.906074
\(645\) 0 0
\(646\) 1.01667e8 0.0148376
\(647\) 6.52515e9 0.947165 0.473582 0.880750i \(-0.342961\pi\)
0.473582 + 0.880750i \(0.342961\pi\)
\(648\) 0 0
\(649\) 3.28037e8 0.0471049
\(650\) −5.11254e8 −0.0730197
\(651\) 0 0
\(652\) 9.04393e9 1.27788
\(653\) 7.88114e9 1.10762 0.553812 0.832641i \(-0.313172\pi\)
0.553812 + 0.832641i \(0.313172\pi\)
\(654\) 0 0
\(655\) 5.78017e8 0.0803703
\(656\) −8.96689e9 −1.24016
\(657\) 0 0
\(658\) −2.06928e9 −0.283157
\(659\) −7.14701e9 −0.972805 −0.486402 0.873735i \(-0.661691\pi\)
−0.486402 + 0.873735i \(0.661691\pi\)
\(660\) 0 0
\(661\) −5.05842e9 −0.681255 −0.340627 0.940198i \(-0.610639\pi\)
−0.340627 + 0.940198i \(0.610639\pi\)
\(662\) −3.26779e8 −0.0437776
\(663\) 0 0
\(664\) 2.23978e9 0.296904
\(665\) −1.15511e9 −0.152317
\(666\) 0 0
\(667\) −4.53132e8 −0.0591267
\(668\) 2.19177e9 0.284496
\(669\) 0 0
\(670\) 1.91868e8 0.0246457
\(671\) 3.96750e9 0.506977
\(672\) 0 0
\(673\) −1.46795e10 −1.85634 −0.928169 0.372158i \(-0.878618\pi\)
−0.928169 + 0.372158i \(0.878618\pi\)
\(674\) −7.99808e8 −0.100618
\(675\) 0 0
\(676\) −1.76683e9 −0.219979
\(677\) −8.90911e9 −1.10350 −0.551752 0.834008i \(-0.686041\pi\)
−0.551752 + 0.834008i \(0.686041\pi\)
\(678\) 0 0
\(679\) −1.61700e10 −1.98228
\(680\) −8.11753e8 −0.0990017
\(681\) 0 0
\(682\) −1.22219e8 −0.0147535
\(683\) 1.04640e10 1.25668 0.628338 0.777940i \(-0.283735\pi\)
0.628338 + 0.777940i \(0.283735\pi\)
\(684\) 0 0
\(685\) 6.97617e9 0.829278
\(686\) −1.38722e9 −0.164063
\(687\) 0 0
\(688\) 1.08817e10 1.27391
\(689\) 1.69934e10 1.97931
\(690\) 0 0
\(691\) −9.56029e9 −1.10230 −0.551148 0.834408i \(-0.685810\pi\)
−0.551148 + 0.834408i \(0.685810\pi\)
\(692\) −5.42078e9 −0.621857
\(693\) 0 0
\(694\) 2.74927e8 0.0312219
\(695\) −6.86335e9 −0.775513
\(696\) 0 0
\(697\) 1.11514e10 1.24743
\(698\) −4.47685e8 −0.0498285
\(699\) 0 0
\(700\) −1.10163e10 −1.21393
\(701\) −4.24136e9 −0.465041 −0.232521 0.972591i \(-0.574697\pi\)
−0.232521 + 0.972591i \(0.574697\pi\)
\(702\) 0 0
\(703\) −1.73031e9 −0.187836
\(704\) 3.17488e9 0.342944
\(705\) 0 0
\(706\) −1.64702e9 −0.176150
\(707\) −1.91185e10 −2.03463
\(708\) 0 0
\(709\) 1.04133e10 1.09730 0.548652 0.836051i \(-0.315141\pi\)
0.548652 + 0.836051i \(0.315141\pi\)
\(710\) −8.84000e8 −0.0926932
\(711\) 0 0
\(712\) 4.09184e8 0.0424853
\(713\) 2.22164e9 0.229542
\(714\) 0 0
\(715\) 2.12022e9 0.216925
\(716\) 7.36218e9 0.749567
\(717\) 0 0
\(718\) 8.06487e8 0.0813133
\(719\) 6.99544e9 0.701882 0.350941 0.936398i \(-0.385862\pi\)
0.350941 + 0.936398i \(0.385862\pi\)
\(720\) 0 0
\(721\) 3.33794e10 3.31669
\(722\) −9.21649e8 −0.0911351
\(723\) 0 0
\(724\) 5.50168e9 0.538778
\(725\) 8.12824e8 0.0792161
\(726\) 0 0
\(727\) 1.85412e10 1.78965 0.894824 0.446418i \(-0.147301\pi\)
0.894824 + 0.446418i \(0.147301\pi\)
\(728\) 3.72076e9 0.357414
\(729\) 0 0
\(730\) 1.66210e8 0.0158135
\(731\) −1.35328e10 −1.28137
\(732\) 0 0
\(733\) −7.68407e9 −0.720655 −0.360327 0.932826i \(-0.617335\pi\)
−0.360327 + 0.932826i \(0.617335\pi\)
\(734\) 1.45409e8 0.0135724
\(735\) 0 0
\(736\) 1.58122e9 0.146191
\(737\) 1.90923e9 0.175680
\(738\) 0 0
\(739\) −1.62766e9 −0.148357 −0.0741785 0.997245i \(-0.523633\pi\)
−0.0741785 + 0.997245i \(0.523633\pi\)
\(740\) 6.87740e9 0.623897
\(741\) 0 0
\(742\) −3.23542e9 −0.290748
\(743\) 1.23451e9 0.110417 0.0552084 0.998475i \(-0.482418\pi\)
0.0552084 + 0.998475i \(0.482418\pi\)
\(744\) 0 0
\(745\) 3.87772e9 0.343581
\(746\) −3.11405e8 −0.0274625
\(747\) 0 0
\(748\) −4.02101e9 −0.351301
\(749\) 9.14149e7 0.00794934
\(750\) 0 0
\(751\) 1.87234e10 1.61304 0.806520 0.591206i \(-0.201348\pi\)
0.806520 + 0.591206i \(0.201348\pi\)
\(752\) 1.98032e10 1.69814
\(753\) 0 0
\(754\) −1.36662e8 −0.0116104
\(755\) −3.33669e9 −0.282164
\(756\) 0 0
\(757\) −3.90822e9 −0.327449 −0.163724 0.986506i \(-0.552351\pi\)
−0.163724 + 0.986506i \(0.552351\pi\)
\(758\) 7.35995e8 0.0613809
\(759\) 0 0
\(760\) −1.97981e8 −0.0163597
\(761\) −8.54296e9 −0.702687 −0.351344 0.936247i \(-0.614275\pi\)
−0.351344 + 0.936247i \(0.614275\pi\)
\(762\) 0 0
\(763\) −1.67756e10 −1.36723
\(764\) −8.80059e9 −0.713978
\(765\) 0 0
\(766\) 5.53158e8 0.0444681
\(767\) −1.79837e9 −0.143912
\(768\) 0 0
\(769\) −8.98742e9 −0.712677 −0.356338 0.934357i \(-0.615975\pi\)
−0.356338 + 0.934357i \(0.615975\pi\)
\(770\) −4.03674e8 −0.0318650
\(771\) 0 0
\(772\) −4.02384e9 −0.314760
\(773\) −7.14394e9 −0.556301 −0.278150 0.960538i \(-0.589721\pi\)
−0.278150 + 0.960538i \(0.589721\pi\)
\(774\) 0 0
\(775\) −3.98517e9 −0.307532
\(776\) −2.77146e9 −0.212908
\(777\) 0 0
\(778\) 1.33779e9 0.101849
\(779\) 2.71976e9 0.206134
\(780\) 0 0
\(781\) −8.79645e9 −0.660737
\(782\) −6.45833e8 −0.0482944
\(783\) 0 0
\(784\) 2.64153e10 1.95771
\(785\) −3.72588e9 −0.274907
\(786\) 0 0
\(787\) 2.12694e9 0.155541 0.0777705 0.996971i \(-0.475220\pi\)
0.0777705 + 0.996971i \(0.475220\pi\)
\(788\) −1.69965e10 −1.23742
\(789\) 0 0
\(790\) 1.68638e8 0.0121692
\(791\) −7.66490e9 −0.550667
\(792\) 0 0
\(793\) −2.17507e10 −1.54888
\(794\) 7.75095e8 0.0549519
\(795\) 0 0
\(796\) −9.61688e9 −0.675832
\(797\) 7.00833e9 0.490355 0.245177 0.969478i \(-0.421154\pi\)
0.245177 + 0.969478i \(0.421154\pi\)
\(798\) 0 0
\(799\) −2.46277e10 −1.70809
\(800\) −2.83638e9 −0.195862
\(801\) 0 0
\(802\) 1.37311e9 0.0939926
\(803\) 1.65391e9 0.112722
\(804\) 0 0
\(805\) 7.33780e9 0.495770
\(806\) 6.70035e8 0.0450739
\(807\) 0 0
\(808\) −3.27682e9 −0.218531
\(809\) −2.24886e10 −1.49329 −0.746643 0.665225i \(-0.768335\pi\)
−0.746643 + 0.665225i \(0.768335\pi\)
\(810\) 0 0
\(811\) −7.32750e9 −0.482373 −0.241186 0.970479i \(-0.577536\pi\)
−0.241186 + 0.970479i \(0.577536\pi\)
\(812\) −2.94474e9 −0.193019
\(813\) 0 0
\(814\) −6.04686e8 −0.0392957
\(815\) 1.08058e10 0.699209
\(816\) 0 0
\(817\) −3.30056e9 −0.211743
\(818\) 2.71199e9 0.173241
\(819\) 0 0
\(820\) −1.08102e10 −0.684673
\(821\) 1.86005e10 1.17307 0.586534 0.809924i \(-0.300492\pi\)
0.586534 + 0.809924i \(0.300492\pi\)
\(822\) 0 0
\(823\) −3.44677e9 −0.215532 −0.107766 0.994176i \(-0.534370\pi\)
−0.107766 + 0.994176i \(0.534370\pi\)
\(824\) 5.72108e9 0.356232
\(825\) 0 0
\(826\) 3.42397e8 0.0211398
\(827\) −5.34418e9 −0.328558 −0.164279 0.986414i \(-0.552530\pi\)
−0.164279 + 0.986414i \(0.552530\pi\)
\(828\) 0 0
\(829\) 1.24427e10 0.758532 0.379266 0.925288i \(-0.376176\pi\)
0.379266 + 0.925288i \(0.376176\pi\)
\(830\) 1.33218e9 0.0808701
\(831\) 0 0
\(832\) −1.74054e10 −1.04774
\(833\) −3.28506e10 −1.96918
\(834\) 0 0
\(835\) 2.61876e9 0.155666
\(836\) −9.80697e8 −0.0580514
\(837\) 0 0
\(838\) 1.75690e9 0.103132
\(839\) −2.08037e10 −1.21611 −0.608056 0.793894i \(-0.708050\pi\)
−0.608056 + 0.793894i \(0.708050\pi\)
\(840\) 0 0
\(841\) −1.70326e10 −0.987404
\(842\) −2.58360e9 −0.149153
\(843\) 0 0
\(844\) 2.68948e10 1.53982
\(845\) −2.11104e9 −0.120364
\(846\) 0 0
\(847\) 2.66665e10 1.50790
\(848\) 3.09633e10 1.74366
\(849\) 0 0
\(850\) 1.15849e9 0.0647032
\(851\) 1.09917e10 0.611380
\(852\) 0 0
\(853\) 3.30134e10 1.82125 0.910623 0.413239i \(-0.135603\pi\)
0.910623 + 0.413239i \(0.135603\pi\)
\(854\) 4.14118e9 0.227521
\(855\) 0 0
\(856\) 1.56681e7 0.000853806 0
\(857\) −7.37991e9 −0.400515 −0.200257 0.979743i \(-0.564178\pi\)
−0.200257 + 0.979743i \(0.564178\pi\)
\(858\) 0 0
\(859\) 8.66866e9 0.466633 0.233317 0.972401i \(-0.425042\pi\)
0.233317 + 0.972401i \(0.425042\pi\)
\(860\) 1.31186e10 0.703305
\(861\) 0 0
\(862\) −6.00309e8 −0.0319227
\(863\) 9.44417e8 0.0500180 0.0250090 0.999687i \(-0.492039\pi\)
0.0250090 + 0.999687i \(0.492039\pi\)
\(864\) 0 0
\(865\) −6.47684e9 −0.340257
\(866\) −1.23953e9 −0.0648554
\(867\) 0 0
\(868\) 1.44377e10 0.749338
\(869\) 1.67807e9 0.0867445
\(870\) 0 0
\(871\) −1.04668e10 −0.536725
\(872\) −2.87527e9 −0.146849
\(873\) 0 0
\(874\) −1.57514e8 −0.00798050
\(875\) −3.18106e10 −1.60525
\(876\) 0 0
\(877\) −1.94376e10 −0.973071 −0.486535 0.873661i \(-0.661740\pi\)
−0.486535 + 0.873661i \(0.661740\pi\)
\(878\) −1.15834e9 −0.0577573
\(879\) 0 0
\(880\) 3.86320e9 0.191099
\(881\) −3.61733e10 −1.78227 −0.891134 0.453741i \(-0.850089\pi\)
−0.891134 + 0.453741i \(0.850089\pi\)
\(882\) 0 0
\(883\) −2.10672e10 −1.02978 −0.514890 0.857256i \(-0.672167\pi\)
−0.514890 + 0.857256i \(0.672167\pi\)
\(884\) 2.20441e10 1.07327
\(885\) 0 0
\(886\) −1.33410e8 −0.00644424
\(887\) −3.08019e10 −1.48199 −0.740993 0.671512i \(-0.765645\pi\)
−0.740993 + 0.671512i \(0.765645\pi\)
\(888\) 0 0
\(889\) 5.23688e10 2.49986
\(890\) 2.43375e8 0.0115721
\(891\) 0 0
\(892\) 2.28527e10 1.07810
\(893\) −6.00653e9 −0.282256
\(894\) 0 0
\(895\) 8.79645e9 0.410135
\(896\) 1.36804e10 0.635363
\(897\) 0 0
\(898\) 3.14277e8 0.0144826
\(899\) −1.06526e9 −0.0488988
\(900\) 0 0
\(901\) −3.85067e10 −1.75388
\(902\) 9.50469e8 0.0431236
\(903\) 0 0
\(904\) −1.31373e9 −0.0591449
\(905\) 6.57349e9 0.294799
\(906\) 0 0
\(907\) 1.57628e10 0.701466 0.350733 0.936475i \(-0.385932\pi\)
0.350733 + 0.936475i \(0.385932\pi\)
\(908\) 9.81966e9 0.435307
\(909\) 0 0
\(910\) 2.21304e9 0.0973517
\(911\) −8.64623e9 −0.378890 −0.189445 0.981891i \(-0.560669\pi\)
−0.189445 + 0.981891i \(0.560669\pi\)
\(912\) 0 0
\(913\) 1.32561e10 0.576460
\(914\) 2.15075e8 0.00931706
\(915\) 0 0
\(916\) −1.76883e10 −0.760415
\(917\) 6.00348e9 0.257105
\(918\) 0 0
\(919\) −1.72145e10 −0.731628 −0.365814 0.930688i \(-0.619209\pi\)
−0.365814 + 0.930688i \(0.619209\pi\)
\(920\) 1.25767e9 0.0532486
\(921\) 0 0
\(922\) 1.76081e9 0.0739867
\(923\) 4.82242e10 2.01864
\(924\) 0 0
\(925\) −1.97168e10 −0.819107
\(926\) −8.85547e8 −0.0366499
\(927\) 0 0
\(928\) −7.58184e8 −0.0311427
\(929\) −2.13454e10 −0.873471 −0.436735 0.899590i \(-0.643865\pi\)
−0.436735 + 0.899590i \(0.643865\pi\)
\(930\) 0 0
\(931\) −8.01205e9 −0.325402
\(932\) 1.57472e10 0.637157
\(933\) 0 0
\(934\) 3.89236e9 0.156314
\(935\) −4.80436e9 −0.192219
\(936\) 0 0
\(937\) 2.94917e9 0.117115 0.0585573 0.998284i \(-0.481350\pi\)
0.0585573 + 0.998284i \(0.481350\pi\)
\(938\) 1.99281e9 0.0788417
\(939\) 0 0
\(940\) 2.38740e10 0.937513
\(941\) −8.66911e9 −0.339165 −0.169582 0.985516i \(-0.554242\pi\)
−0.169582 + 0.985516i \(0.554242\pi\)
\(942\) 0 0
\(943\) −1.72772e10 −0.670937
\(944\) −3.27677e9 −0.126778
\(945\) 0 0
\(946\) −1.15344e9 −0.0442971
\(947\) −1.86313e10 −0.712884 −0.356442 0.934317i \(-0.616010\pi\)
−0.356442 + 0.934317i \(0.616010\pi\)
\(948\) 0 0
\(949\) −9.06713e9 −0.344380
\(950\) 2.82548e8 0.0106920
\(951\) 0 0
\(952\) −8.43115e9 −0.316707
\(953\) −3.78151e10 −1.41527 −0.707637 0.706576i \(-0.750239\pi\)
−0.707637 + 0.706576i \(0.750239\pi\)
\(954\) 0 0
\(955\) −1.05151e10 −0.390662
\(956\) −1.97114e10 −0.729649
\(957\) 0 0
\(958\) −4.73666e9 −0.174057
\(959\) 7.24570e10 2.65286
\(960\) 0 0
\(961\) −2.22898e10 −0.810165
\(962\) 3.31503e9 0.120054
\(963\) 0 0
\(964\) −4.37050e10 −1.57131
\(965\) −4.80775e9 −0.172225
\(966\) 0 0
\(967\) −3.55682e10 −1.26494 −0.632470 0.774585i \(-0.717959\pi\)
−0.632470 + 0.774585i \(0.717959\pi\)
\(968\) 4.57052e9 0.161958
\(969\) 0 0
\(970\) −1.64841e9 −0.0579915
\(971\) −2.79517e10 −0.979806 −0.489903 0.871777i \(-0.662968\pi\)
−0.489903 + 0.871777i \(0.662968\pi\)
\(972\) 0 0
\(973\) −7.12852e10 −2.48087
\(974\) 6.24891e8 0.0216695
\(975\) 0 0
\(976\) −3.96315e10 −1.36448
\(977\) 6.61358e9 0.226885 0.113443 0.993545i \(-0.463812\pi\)
0.113443 + 0.993545i \(0.463812\pi\)
\(978\) 0 0
\(979\) 2.42176e9 0.0824882
\(980\) 3.18453e10 1.08082
\(981\) 0 0
\(982\) −1.29297e9 −0.0435711
\(983\) −1.77476e10 −0.595939 −0.297970 0.954575i \(-0.596309\pi\)
−0.297970 + 0.954575i \(0.596309\pi\)
\(984\) 0 0
\(985\) −2.03077e10 −0.677071
\(986\) 3.09672e8 0.0102881
\(987\) 0 0
\(988\) 5.37641e9 0.177355
\(989\) 2.09667e10 0.689195
\(990\) 0 0
\(991\) −4.22116e9 −0.137776 −0.0688880 0.997624i \(-0.521945\pi\)
−0.0688880 + 0.997624i \(0.521945\pi\)
\(992\) 3.71728e9 0.120902
\(993\) 0 0
\(994\) −9.18153e9 −0.296526
\(995\) −1.14904e10 −0.369790
\(996\) 0 0
\(997\) −2.02379e10 −0.646742 −0.323371 0.946272i \(-0.604816\pi\)
−0.323371 + 0.946272i \(0.604816\pi\)
\(998\) 1.57278e9 0.0500854
\(999\) 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 531.8.a.b.1.8 16
3.2 odd 2 177.8.a.a.1.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.9 16 3.2 odd 2
531.8.a.b.1.8 16 1.1 even 1 trivial