Properties

Label 531.8.a.b.1.1
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 1.1
Root \(-19.6562\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-19.6562 q^{2} +258.365 q^{4} +436.761 q^{5} +956.841 q^{7} -2562.48 q^{8} +O(q^{10})\) \(q-19.6562 q^{2} +258.365 q^{4} +436.761 q^{5} +956.841 q^{7} -2562.48 q^{8} -8585.06 q^{10} -1972.61 q^{11} -5308.65 q^{13} -18807.8 q^{14} +17297.8 q^{16} +19184.9 q^{17} -21442.5 q^{19} +112844. q^{20} +38773.9 q^{22} +34320.3 q^{23} +112635. q^{25} +104348. q^{26} +247214. q^{28} -40010.4 q^{29} -95465.0 q^{31} -12010.9 q^{32} -377103. q^{34} +417911. q^{35} -269178. q^{37} +421478. q^{38} -1.11919e6 q^{40} +360504. q^{41} -929097. q^{43} -509653. q^{44} -674606. q^{46} -54052.8 q^{47} +92001.6 q^{49} -2.21398e6 q^{50} -1.37157e6 q^{52} -680499. q^{53} -861559. q^{55} -2.45188e6 q^{56} +786452. q^{58} -205379. q^{59} -1.27358e6 q^{61} +1.87648e6 q^{62} -1.97803e6 q^{64} -2.31861e6 q^{65} +630075. q^{67} +4.95672e6 q^{68} -8.21453e6 q^{70} -2.05155e6 q^{71} -5.36743e6 q^{73} +5.29101e6 q^{74} -5.54000e6 q^{76} -1.88747e6 q^{77} +4.57098e6 q^{79} +7.55500e6 q^{80} -7.08613e6 q^{82} -9.49376e6 q^{83} +8.37924e6 q^{85} +1.82625e7 q^{86} +5.05477e6 q^{88} +2.45830e6 q^{89} -5.07953e6 q^{91} +8.86717e6 q^{92} +1.06247e6 q^{94} -9.36526e6 q^{95} -1.41914e7 q^{97} -1.80840e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.6562 −1.73738 −0.868688 0.495359i \(-0.835036\pi\)
−0.868688 + 0.495359i \(0.835036\pi\)
\(3\) 0 0
\(4\) 258.365 2.01848
\(5\) 436.761 1.56260 0.781302 0.624153i \(-0.214556\pi\)
0.781302 + 0.624153i \(0.214556\pi\)
\(6\) 0 0
\(7\) 956.841 1.05438 0.527189 0.849748i \(-0.323246\pi\)
0.527189 + 0.849748i \(0.323246\pi\)
\(8\) −2562.48 −1.76948
\(9\) 0 0
\(10\) −8585.06 −2.71483
\(11\) −1972.61 −0.446855 −0.223427 0.974721i \(-0.571725\pi\)
−0.223427 + 0.974721i \(0.571725\pi\)
\(12\) 0 0
\(13\) −5308.65 −0.670166 −0.335083 0.942189i \(-0.608764\pi\)
−0.335083 + 0.942189i \(0.608764\pi\)
\(14\) −18807.8 −1.83185
\(15\) 0 0
\(16\) 17297.8 1.05577
\(17\) 19184.9 0.947086 0.473543 0.880771i \(-0.342975\pi\)
0.473543 + 0.880771i \(0.342975\pi\)
\(18\) 0 0
\(19\) −21442.5 −0.717196 −0.358598 0.933492i \(-0.616745\pi\)
−0.358598 + 0.933492i \(0.616745\pi\)
\(20\) 112844. 3.15408
\(21\) 0 0
\(22\) 38773.9 0.776355
\(23\) 34320.3 0.588171 0.294086 0.955779i \(-0.404985\pi\)
0.294086 + 0.955779i \(0.404985\pi\)
\(24\) 0 0
\(25\) 112635. 1.44173
\(26\) 104348. 1.16433
\(27\) 0 0
\(28\) 247214. 2.12824
\(29\) −40010.4 −0.304635 −0.152318 0.988332i \(-0.548674\pi\)
−0.152318 + 0.988332i \(0.548674\pi\)
\(30\) 0 0
\(31\) −95465.0 −0.575544 −0.287772 0.957699i \(-0.592915\pi\)
−0.287772 + 0.957699i \(0.592915\pi\)
\(32\) −12010.9 −0.0647961
\(33\) 0 0
\(34\) −377103. −1.64544
\(35\) 417911. 1.64758
\(36\) 0 0
\(37\) −269178. −0.873642 −0.436821 0.899549i \(-0.643896\pi\)
−0.436821 + 0.899549i \(0.643896\pi\)
\(38\) 421478. 1.24604
\(39\) 0 0
\(40\) −1.11919e6 −2.76499
\(41\) 360504. 0.816896 0.408448 0.912782i \(-0.366070\pi\)
0.408448 + 0.912782i \(0.366070\pi\)
\(42\) 0 0
\(43\) −929097. −1.78206 −0.891028 0.453949i \(-0.850015\pi\)
−0.891028 + 0.453949i \(0.850015\pi\)
\(44\) −509653. −0.901967
\(45\) 0 0
\(46\) −674606. −1.02187
\(47\) −54052.8 −0.0759408 −0.0379704 0.999279i \(-0.512089\pi\)
−0.0379704 + 0.999279i \(0.512089\pi\)
\(48\) 0 0
\(49\) 92001.6 0.111714
\(50\) −2.21398e6 −2.50483
\(51\) 0 0
\(52\) −1.37157e6 −1.35271
\(53\) −680499. −0.627859 −0.313929 0.949446i \(-0.601646\pi\)
−0.313929 + 0.949446i \(0.601646\pi\)
\(54\) 0 0
\(55\) −861559. −0.698258
\(56\) −2.45188e6 −1.86570
\(57\) 0 0
\(58\) 786452. 0.529266
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −1.27358e6 −0.718408 −0.359204 0.933259i \(-0.616952\pi\)
−0.359204 + 0.933259i \(0.616952\pi\)
\(62\) 1.87648e6 0.999936
\(63\) 0 0
\(64\) −1.97803e6 −0.943197
\(65\) −2.31861e6 −1.04720
\(66\) 0 0
\(67\) 630075. 0.255936 0.127968 0.991778i \(-0.459155\pi\)
0.127968 + 0.991778i \(0.459155\pi\)
\(68\) 4.95672e6 1.91167
\(69\) 0 0
\(70\) −8.21453e6 −2.86246
\(71\) −2.05155e6 −0.680265 −0.340133 0.940377i \(-0.610472\pi\)
−0.340133 + 0.940377i \(0.610472\pi\)
\(72\) 0 0
\(73\) −5.36743e6 −1.61487 −0.807433 0.589959i \(-0.799144\pi\)
−0.807433 + 0.589959i \(0.799144\pi\)
\(74\) 5.29101e6 1.51784
\(75\) 0 0
\(76\) −5.54000e6 −1.44764
\(77\) −1.88747e6 −0.471154
\(78\) 0 0
\(79\) 4.57098e6 1.04307 0.521536 0.853229i \(-0.325359\pi\)
0.521536 + 0.853229i \(0.325359\pi\)
\(80\) 7.55500e6 1.64975
\(81\) 0 0
\(82\) −7.08613e6 −1.41926
\(83\) −9.49376e6 −1.82249 −0.911245 0.411865i \(-0.864878\pi\)
−0.911245 + 0.411865i \(0.864878\pi\)
\(84\) 0 0
\(85\) 8.37924e6 1.47992
\(86\) 1.82625e7 3.09610
\(87\) 0 0
\(88\) 5.05477e6 0.790700
\(89\) 2.45830e6 0.369633 0.184816 0.982773i \(-0.440831\pi\)
0.184816 + 0.982773i \(0.440831\pi\)
\(90\) 0 0
\(91\) −5.07953e6 −0.706609
\(92\) 8.86717e6 1.18721
\(93\) 0 0
\(94\) 1.06247e6 0.131938
\(95\) −9.36526e6 −1.12069
\(96\) 0 0
\(97\) −1.41914e7 −1.57879 −0.789397 0.613883i \(-0.789607\pi\)
−0.789397 + 0.613883i \(0.789607\pi\)
\(98\) −1.80840e6 −0.194090
\(99\) 0 0
\(100\) 2.91011e7 2.91011
\(101\) −3.81799e6 −0.368731 −0.184366 0.982858i \(-0.559023\pi\)
−0.184366 + 0.982858i \(0.559023\pi\)
\(102\) 0 0
\(103\) 1.77672e7 1.60210 0.801049 0.598599i \(-0.204276\pi\)
0.801049 + 0.598599i \(0.204276\pi\)
\(104\) 1.36033e7 1.18584
\(105\) 0 0
\(106\) 1.33760e7 1.09083
\(107\) −2.35965e7 −1.86210 −0.931051 0.364888i \(-0.881107\pi\)
−0.931051 + 0.364888i \(0.881107\pi\)
\(108\) 0 0
\(109\) 3.99327e6 0.295349 0.147675 0.989036i \(-0.452821\pi\)
0.147675 + 0.989036i \(0.452821\pi\)
\(110\) 1.69350e7 1.21314
\(111\) 0 0
\(112\) 1.65512e7 1.11318
\(113\) 2.69156e6 0.175481 0.0877405 0.996143i \(-0.472035\pi\)
0.0877405 + 0.996143i \(0.472035\pi\)
\(114\) 0 0
\(115\) 1.49898e7 0.919079
\(116\) −1.03373e7 −0.614900
\(117\) 0 0
\(118\) 4.03696e6 0.226187
\(119\) 1.83569e7 0.998587
\(120\) 0 0
\(121\) −1.55960e7 −0.800321
\(122\) 2.50337e7 1.24815
\(123\) 0 0
\(124\) −2.46648e7 −1.16172
\(125\) 1.50728e7 0.690255
\(126\) 0 0
\(127\) 1.72960e6 0.0749261 0.0374631 0.999298i \(-0.488072\pi\)
0.0374631 + 0.999298i \(0.488072\pi\)
\(128\) 4.04178e7 1.70348
\(129\) 0 0
\(130\) 4.55750e7 1.81939
\(131\) −1.21153e7 −0.470854 −0.235427 0.971892i \(-0.575649\pi\)
−0.235427 + 0.971892i \(0.575649\pi\)
\(132\) 0 0
\(133\) −2.05171e7 −0.756196
\(134\) −1.23849e7 −0.444657
\(135\) 0 0
\(136\) −4.91610e7 −1.67585
\(137\) 3.48264e7 1.15714 0.578571 0.815632i \(-0.303611\pi\)
0.578571 + 0.815632i \(0.303611\pi\)
\(138\) 0 0
\(139\) 3.29388e7 1.04029 0.520147 0.854077i \(-0.325877\pi\)
0.520147 + 0.854077i \(0.325877\pi\)
\(140\) 1.07974e8 3.32560
\(141\) 0 0
\(142\) 4.03257e7 1.18188
\(143\) 1.04719e7 0.299467
\(144\) 0 0
\(145\) −1.74750e7 −0.476025
\(146\) 1.05503e8 2.80563
\(147\) 0 0
\(148\) −6.95461e7 −1.76343
\(149\) −3.14675e7 −0.779311 −0.389656 0.920961i \(-0.627406\pi\)
−0.389656 + 0.920961i \(0.627406\pi\)
\(150\) 0 0
\(151\) −2.53600e6 −0.0599419 −0.0299710 0.999551i \(-0.509541\pi\)
−0.0299710 + 0.999551i \(0.509541\pi\)
\(152\) 5.49460e7 1.26906
\(153\) 0 0
\(154\) 3.71005e7 0.818572
\(155\) −4.16954e7 −0.899348
\(156\) 0 0
\(157\) −5.81479e7 −1.19918 −0.599591 0.800306i \(-0.704670\pi\)
−0.599591 + 0.800306i \(0.704670\pi\)
\(158\) −8.98479e7 −1.81221
\(159\) 0 0
\(160\) −5.24588e6 −0.101251
\(161\) 3.28391e7 0.620155
\(162\) 0 0
\(163\) −4.39268e7 −0.794461 −0.397230 0.917719i \(-0.630029\pi\)
−0.397230 + 0.917719i \(0.630029\pi\)
\(164\) 9.31417e7 1.64889
\(165\) 0 0
\(166\) 1.86611e8 3.16635
\(167\) 9.02250e7 1.49906 0.749530 0.661970i \(-0.230280\pi\)
0.749530 + 0.661970i \(0.230280\pi\)
\(168\) 0 0
\(169\) −3.45668e7 −0.550878
\(170\) −1.64704e8 −2.57118
\(171\) 0 0
\(172\) −2.40046e8 −3.59704
\(173\) −2.25125e7 −0.330569 −0.165285 0.986246i \(-0.552854\pi\)
−0.165285 + 0.986246i \(0.552854\pi\)
\(174\) 0 0
\(175\) 1.07774e8 1.52013
\(176\) −3.41217e7 −0.471777
\(177\) 0 0
\(178\) −4.83208e7 −0.642191
\(179\) −1.26881e8 −1.65353 −0.826766 0.562546i \(-0.809822\pi\)
−0.826766 + 0.562546i \(0.809822\pi\)
\(180\) 0 0
\(181\) 9.29702e7 1.16538 0.582691 0.812694i \(-0.302000\pi\)
0.582691 + 0.812694i \(0.302000\pi\)
\(182\) 9.98442e7 1.22765
\(183\) 0 0
\(184\) −8.79450e7 −1.04076
\(185\) −1.17566e8 −1.36516
\(186\) 0 0
\(187\) −3.78444e7 −0.423210
\(188\) −1.39653e7 −0.153285
\(189\) 0 0
\(190\) 1.84085e8 1.94707
\(191\) 1.55664e8 1.61649 0.808243 0.588850i \(-0.200419\pi\)
0.808243 + 0.588850i \(0.200419\pi\)
\(192\) 0 0
\(193\) 1.85473e7 0.185708 0.0928539 0.995680i \(-0.470401\pi\)
0.0928539 + 0.995680i \(0.470401\pi\)
\(194\) 2.78949e8 2.74296
\(195\) 0 0
\(196\) 2.37700e7 0.225493
\(197\) −3.60337e7 −0.335798 −0.167899 0.985804i \(-0.553698\pi\)
−0.167899 + 0.985804i \(0.553698\pi\)
\(198\) 0 0
\(199\) 3.41912e7 0.307559 0.153780 0.988105i \(-0.450855\pi\)
0.153780 + 0.988105i \(0.450855\pi\)
\(200\) −2.88626e8 −2.55112
\(201\) 0 0
\(202\) 7.50471e7 0.640625
\(203\) −3.82836e7 −0.321201
\(204\) 0 0
\(205\) 1.57454e8 1.27649
\(206\) −3.49235e8 −2.78345
\(207\) 0 0
\(208\) −9.18278e7 −0.707543
\(209\) 4.22977e7 0.320483
\(210\) 0 0
\(211\) 1.66203e8 1.21801 0.609005 0.793166i \(-0.291569\pi\)
0.609005 + 0.793166i \(0.291569\pi\)
\(212\) −1.75817e8 −1.26732
\(213\) 0 0
\(214\) 4.63816e8 3.23517
\(215\) −4.05793e8 −2.78465
\(216\) 0 0
\(217\) −9.13449e7 −0.606841
\(218\) −7.84923e7 −0.513133
\(219\) 0 0
\(220\) −2.22597e8 −1.40942
\(221\) −1.01846e8 −0.634705
\(222\) 0 0
\(223\) −2.57838e8 −1.55697 −0.778485 0.627663i \(-0.784012\pi\)
−0.778485 + 0.627663i \(0.784012\pi\)
\(224\) −1.14925e7 −0.0683197
\(225\) 0 0
\(226\) −5.29058e7 −0.304877
\(227\) 2.27009e8 1.28811 0.644055 0.764979i \(-0.277251\pi\)
0.644055 + 0.764979i \(0.277251\pi\)
\(228\) 0 0
\(229\) −7.40017e7 −0.407209 −0.203605 0.979053i \(-0.565266\pi\)
−0.203605 + 0.979053i \(0.565266\pi\)
\(230\) −2.94642e8 −1.59679
\(231\) 0 0
\(232\) 1.02526e8 0.539046
\(233\) −6.77414e7 −0.350839 −0.175420 0.984494i \(-0.556128\pi\)
−0.175420 + 0.984494i \(0.556128\pi\)
\(234\) 0 0
\(235\) −2.36082e7 −0.118665
\(236\) −5.30628e7 −0.262783
\(237\) 0 0
\(238\) −3.60827e8 −1.73492
\(239\) −1.28770e8 −0.610128 −0.305064 0.952332i \(-0.598678\pi\)
−0.305064 + 0.952332i \(0.598678\pi\)
\(240\) 0 0
\(241\) −2.97076e8 −1.36712 −0.683561 0.729893i \(-0.739570\pi\)
−0.683561 + 0.729893i \(0.739570\pi\)
\(242\) 3.06557e8 1.39046
\(243\) 0 0
\(244\) −3.29048e8 −1.45009
\(245\) 4.01827e7 0.174565
\(246\) 0 0
\(247\) 1.13831e8 0.480640
\(248\) 2.44627e8 1.01841
\(249\) 0 0
\(250\) −2.96274e8 −1.19923
\(251\) 6.81145e7 0.271883 0.135941 0.990717i \(-0.456594\pi\)
0.135941 + 0.990717i \(0.456594\pi\)
\(252\) 0 0
\(253\) −6.77005e7 −0.262827
\(254\) −3.39974e7 −0.130175
\(255\) 0 0
\(256\) −5.41272e8 −2.01640
\(257\) 1.12934e8 0.415010 0.207505 0.978234i \(-0.433466\pi\)
0.207505 + 0.978234i \(0.433466\pi\)
\(258\) 0 0
\(259\) −2.57560e8 −0.921149
\(260\) −5.99049e8 −2.11376
\(261\) 0 0
\(262\) 2.38141e8 0.818050
\(263\) −5.13397e8 −1.74024 −0.870119 0.492842i \(-0.835958\pi\)
−0.870119 + 0.492842i \(0.835958\pi\)
\(264\) 0 0
\(265\) −2.97216e8 −0.981095
\(266\) 4.03287e8 1.31380
\(267\) 0 0
\(268\) 1.62789e8 0.516600
\(269\) 7.41447e7 0.232245 0.116123 0.993235i \(-0.462953\pi\)
0.116123 + 0.993235i \(0.462953\pi\)
\(270\) 0 0
\(271\) 1.72102e8 0.525285 0.262642 0.964893i \(-0.415406\pi\)
0.262642 + 0.964893i \(0.415406\pi\)
\(272\) 3.31857e8 0.999907
\(273\) 0 0
\(274\) −6.84554e8 −2.01039
\(275\) −2.22186e8 −0.644246
\(276\) 0 0
\(277\) 4.25097e8 1.20174 0.600868 0.799348i \(-0.294822\pi\)
0.600868 + 0.799348i \(0.294822\pi\)
\(278\) −6.47451e8 −1.80738
\(279\) 0 0
\(280\) −1.07089e9 −2.91535
\(281\) −3.70445e7 −0.0995983 −0.0497991 0.998759i \(-0.515858\pi\)
−0.0497991 + 0.998759i \(0.515858\pi\)
\(282\) 0 0
\(283\) −2.60297e7 −0.0682679 −0.0341339 0.999417i \(-0.510867\pi\)
−0.0341339 + 0.999417i \(0.510867\pi\)
\(284\) −5.30049e8 −1.37310
\(285\) 0 0
\(286\) −2.05837e8 −0.520287
\(287\) 3.44945e8 0.861318
\(288\) 0 0
\(289\) −4.22765e7 −0.103028
\(290\) 3.43492e8 0.827034
\(291\) 0 0
\(292\) −1.38676e9 −3.25957
\(293\) −6.51952e8 −1.51418 −0.757092 0.653308i \(-0.773381\pi\)
−0.757092 + 0.653308i \(0.773381\pi\)
\(294\) 0 0
\(295\) −8.97016e7 −0.203434
\(296\) 6.89762e8 1.54589
\(297\) 0 0
\(298\) 6.18532e8 1.35396
\(299\) −1.82194e8 −0.394172
\(300\) 0 0
\(301\) −8.88998e8 −1.87896
\(302\) 4.98481e7 0.104142
\(303\) 0 0
\(304\) −3.70908e8 −0.757196
\(305\) −5.56250e8 −1.12259
\(306\) 0 0
\(307\) 8.07706e8 1.59320 0.796598 0.604510i \(-0.206631\pi\)
0.796598 + 0.604510i \(0.206631\pi\)
\(308\) −4.87657e8 −0.951014
\(309\) 0 0
\(310\) 8.19573e8 1.56251
\(311\) 1.60231e8 0.302055 0.151027 0.988530i \(-0.451742\pi\)
0.151027 + 0.988530i \(0.451742\pi\)
\(312\) 0 0
\(313\) 2.15368e8 0.396987 0.198494 0.980102i \(-0.436395\pi\)
0.198494 + 0.980102i \(0.436395\pi\)
\(314\) 1.14296e9 2.08343
\(315\) 0 0
\(316\) 1.18098e9 2.10542
\(317\) 8.64021e8 1.52341 0.761705 0.647924i \(-0.224362\pi\)
0.761705 + 0.647924i \(0.224362\pi\)
\(318\) 0 0
\(319\) 7.89250e7 0.136128
\(320\) −8.63926e8 −1.47384
\(321\) 0 0
\(322\) −6.45490e8 −1.07744
\(323\) −4.11373e8 −0.679246
\(324\) 0 0
\(325\) −5.97942e8 −0.966201
\(326\) 8.63432e8 1.38028
\(327\) 0 0
\(328\) −9.23784e8 −1.44548
\(329\) −5.17199e7 −0.0800704
\(330\) 0 0
\(331\) 9.12406e8 1.38290 0.691449 0.722425i \(-0.256973\pi\)
0.691449 + 0.722425i \(0.256973\pi\)
\(332\) −2.45286e9 −3.67865
\(333\) 0 0
\(334\) −1.77348e9 −2.60443
\(335\) 2.75193e8 0.399926
\(336\) 0 0
\(337\) 5.45796e8 0.776829 0.388414 0.921485i \(-0.373023\pi\)
0.388414 + 0.921485i \(0.373023\pi\)
\(338\) 6.79450e8 0.957082
\(339\) 0 0
\(340\) 2.16490e9 2.98719
\(341\) 1.88315e8 0.257185
\(342\) 0 0
\(343\) −6.99969e8 −0.936589
\(344\) 2.38079e9 3.15331
\(345\) 0 0
\(346\) 4.42510e8 0.574323
\(347\) 5.84036e8 0.750389 0.375194 0.926946i \(-0.377576\pi\)
0.375194 + 0.926946i \(0.377576\pi\)
\(348\) 0 0
\(349\) −7.64386e8 −0.962551 −0.481276 0.876569i \(-0.659826\pi\)
−0.481276 + 0.876569i \(0.659826\pi\)
\(350\) −2.11843e9 −2.64104
\(351\) 0 0
\(352\) 2.36927e7 0.0289545
\(353\) 1.06373e9 1.28712 0.643559 0.765396i \(-0.277457\pi\)
0.643559 + 0.765396i \(0.277457\pi\)
\(354\) 0 0
\(355\) −8.96039e8 −1.06299
\(356\) 6.35140e8 0.746095
\(357\) 0 0
\(358\) 2.49400e9 2.87281
\(359\) 1.09179e9 1.24540 0.622700 0.782461i \(-0.286036\pi\)
0.622700 + 0.782461i \(0.286036\pi\)
\(360\) 0 0
\(361\) −4.34091e8 −0.485630
\(362\) −1.82744e9 −2.02471
\(363\) 0 0
\(364\) −1.31237e9 −1.42627
\(365\) −2.34429e9 −2.52340
\(366\) 0 0
\(367\) −1.92123e8 −0.202884 −0.101442 0.994841i \(-0.532346\pi\)
−0.101442 + 0.994841i \(0.532346\pi\)
\(368\) 5.93665e8 0.620975
\(369\) 0 0
\(370\) 2.31091e9 2.37179
\(371\) −6.51129e8 −0.662001
\(372\) 0 0
\(373\) −1.24365e9 −1.24085 −0.620423 0.784267i \(-0.713039\pi\)
−0.620423 + 0.784267i \(0.713039\pi\)
\(374\) 7.43876e8 0.735275
\(375\) 0 0
\(376\) 1.38509e8 0.134376
\(377\) 2.12401e8 0.204156
\(378\) 0 0
\(379\) −2.03376e9 −1.91894 −0.959472 0.281805i \(-0.909067\pi\)
−0.959472 + 0.281805i \(0.909067\pi\)
\(380\) −2.41966e9 −2.26210
\(381\) 0 0
\(382\) −3.05976e9 −2.80844
\(383\) 8.58688e8 0.780979 0.390490 0.920607i \(-0.372306\pi\)
0.390490 + 0.920607i \(0.372306\pi\)
\(384\) 0 0
\(385\) −8.24375e8 −0.736228
\(386\) −3.64569e8 −0.322645
\(387\) 0 0
\(388\) −3.66657e9 −3.18676
\(389\) 1.79662e9 1.54751 0.773755 0.633485i \(-0.218376\pi\)
0.773755 + 0.633485i \(0.218376\pi\)
\(390\) 0 0
\(391\) 6.58433e8 0.557049
\(392\) −2.35752e8 −0.197676
\(393\) 0 0
\(394\) 7.08285e8 0.583407
\(395\) 1.99643e9 1.62991
\(396\) 0 0
\(397\) −1.32366e9 −1.06172 −0.530860 0.847460i \(-0.678131\pi\)
−0.530860 + 0.847460i \(0.678131\pi\)
\(398\) −6.72069e8 −0.534347
\(399\) 0 0
\(400\) 1.94834e9 1.52214
\(401\) 1.25074e9 0.968635 0.484317 0.874892i \(-0.339068\pi\)
0.484317 + 0.874892i \(0.339068\pi\)
\(402\) 0 0
\(403\) 5.06790e8 0.385710
\(404\) −9.86435e8 −0.744276
\(405\) 0 0
\(406\) 7.52510e8 0.558047
\(407\) 5.30983e8 0.390391
\(408\) 0 0
\(409\) 1.84026e9 1.32999 0.664993 0.746850i \(-0.268435\pi\)
0.664993 + 0.746850i \(0.268435\pi\)
\(410\) −3.09495e9 −2.21774
\(411\) 0 0
\(412\) 4.59043e9 3.23380
\(413\) −1.96515e8 −0.137268
\(414\) 0 0
\(415\) −4.14651e9 −2.84783
\(416\) 6.37614e7 0.0434242
\(417\) 0 0
\(418\) −8.31411e8 −0.556799
\(419\) −1.97514e9 −1.31174 −0.655871 0.754873i \(-0.727699\pi\)
−0.655871 + 0.754873i \(0.727699\pi\)
\(420\) 0 0
\(421\) 1.04914e9 0.685247 0.342624 0.939473i \(-0.388684\pi\)
0.342624 + 0.939473i \(0.388684\pi\)
\(422\) −3.26692e9 −2.11614
\(423\) 0 0
\(424\) 1.74376e9 1.11098
\(425\) 2.16091e9 1.36545
\(426\) 0 0
\(427\) −1.21861e9 −0.757474
\(428\) −6.09650e9 −3.75861
\(429\) 0 0
\(430\) 7.97635e9 4.83798
\(431\) −1.15205e9 −0.693109 −0.346554 0.938030i \(-0.612648\pi\)
−0.346554 + 0.938030i \(0.612648\pi\)
\(432\) 0 0
\(433\) 2.37623e9 1.40663 0.703317 0.710877i \(-0.251702\pi\)
0.703317 + 0.710877i \(0.251702\pi\)
\(434\) 1.79549e9 1.05431
\(435\) 0 0
\(436\) 1.03172e9 0.596155
\(437\) −7.35913e8 −0.421834
\(438\) 0 0
\(439\) −6.70393e8 −0.378185 −0.189092 0.981959i \(-0.560555\pi\)
−0.189092 + 0.981959i \(0.560555\pi\)
\(440\) 2.20773e9 1.23555
\(441\) 0 0
\(442\) 2.00191e9 1.10272
\(443\) 2.24468e9 1.22671 0.613354 0.789808i \(-0.289820\pi\)
0.613354 + 0.789808i \(0.289820\pi\)
\(444\) 0 0
\(445\) 1.07369e9 0.577590
\(446\) 5.06811e9 2.70504
\(447\) 0 0
\(448\) −1.89266e9 −0.994487
\(449\) 5.76894e8 0.300769 0.150385 0.988628i \(-0.451949\pi\)
0.150385 + 0.988628i \(0.451949\pi\)
\(450\) 0 0
\(451\) −7.11134e8 −0.365034
\(452\) 6.95406e8 0.354204
\(453\) 0 0
\(454\) −4.46213e9 −2.23793
\(455\) −2.21854e9 −1.10415
\(456\) 0 0
\(457\) 9.02311e8 0.442231 0.221116 0.975248i \(-0.429030\pi\)
0.221116 + 0.975248i \(0.429030\pi\)
\(458\) 1.45459e9 0.707476
\(459\) 0 0
\(460\) 3.87284e9 1.85514
\(461\) −1.56907e9 −0.745913 −0.372956 0.927849i \(-0.621656\pi\)
−0.372956 + 0.927849i \(0.621656\pi\)
\(462\) 0 0
\(463\) 6.01283e8 0.281543 0.140772 0.990042i \(-0.455042\pi\)
0.140772 + 0.990042i \(0.455042\pi\)
\(464\) −6.92092e8 −0.321626
\(465\) 0 0
\(466\) 1.33154e9 0.609540
\(467\) 2.18611e9 0.993260 0.496630 0.867962i \(-0.334571\pi\)
0.496630 + 0.867962i \(0.334571\pi\)
\(468\) 0 0
\(469\) 6.02882e8 0.269853
\(470\) 4.64046e8 0.206167
\(471\) 0 0
\(472\) 5.26279e8 0.230366
\(473\) 1.83274e9 0.796320
\(474\) 0 0
\(475\) −2.41519e9 −1.03401
\(476\) 4.74279e9 2.01563
\(477\) 0 0
\(478\) 2.53112e9 1.06002
\(479\) −1.17364e9 −0.487936 −0.243968 0.969783i \(-0.578449\pi\)
−0.243968 + 0.969783i \(0.578449\pi\)
\(480\) 0 0
\(481\) 1.42897e9 0.585485
\(482\) 5.83937e9 2.37521
\(483\) 0 0
\(484\) −4.02946e9 −1.61543
\(485\) −6.19827e9 −2.46703
\(486\) 0 0
\(487\) 1.37532e9 0.539574 0.269787 0.962920i \(-0.413047\pi\)
0.269787 + 0.962920i \(0.413047\pi\)
\(488\) 3.26352e9 1.27121
\(489\) 0 0
\(490\) −7.89838e8 −0.303286
\(491\) −4.93551e9 −1.88168 −0.940842 0.338845i \(-0.889964\pi\)
−0.940842 + 0.338845i \(0.889964\pi\)
\(492\) 0 0
\(493\) −7.67598e8 −0.288516
\(494\) −2.23748e9 −0.835053
\(495\) 0 0
\(496\) −1.65133e9 −0.607643
\(497\) −1.96301e9 −0.717257
\(498\) 0 0
\(499\) −4.54580e9 −1.63779 −0.818895 0.573943i \(-0.805413\pi\)
−0.818895 + 0.573943i \(0.805413\pi\)
\(500\) 3.89429e9 1.39326
\(501\) 0 0
\(502\) −1.33887e9 −0.472362
\(503\) −1.13170e9 −0.396500 −0.198250 0.980151i \(-0.563526\pi\)
−0.198250 + 0.980151i \(0.563526\pi\)
\(504\) 0 0
\(505\) −1.66755e9 −0.576181
\(506\) 1.33073e9 0.456630
\(507\) 0 0
\(508\) 4.46869e8 0.151237
\(509\) −3.02720e9 −1.01749 −0.508743 0.860918i \(-0.669890\pi\)
−0.508743 + 0.860918i \(0.669890\pi\)
\(510\) 0 0
\(511\) −5.13578e9 −1.70268
\(512\) 5.46586e9 1.79976
\(513\) 0 0
\(514\) −2.21985e9 −0.721028
\(515\) 7.76003e9 2.50345
\(516\) 0 0
\(517\) 1.06625e8 0.0339345
\(518\) 5.06265e9 1.60038
\(519\) 0 0
\(520\) 5.94139e9 1.85301
\(521\) −9.37018e8 −0.290279 −0.145140 0.989411i \(-0.546363\pi\)
−0.145140 + 0.989411i \(0.546363\pi\)
\(522\) 0 0
\(523\) 2.52142e9 0.770708 0.385354 0.922769i \(-0.374079\pi\)
0.385354 + 0.922769i \(0.374079\pi\)
\(524\) −3.13018e9 −0.950407
\(525\) 0 0
\(526\) 1.00914e10 3.02345
\(527\) −1.83149e9 −0.545089
\(528\) 0 0
\(529\) −2.22694e9 −0.654055
\(530\) 5.84212e9 1.70453
\(531\) 0 0
\(532\) −5.30089e9 −1.52637
\(533\) −1.91379e9 −0.547456
\(534\) 0 0
\(535\) −1.03060e10 −2.90973
\(536\) −1.61455e9 −0.452872
\(537\) 0 0
\(538\) −1.45740e9 −0.403497
\(539\) −1.81483e8 −0.0499201
\(540\) 0 0
\(541\) 5.53185e8 0.150204 0.0751018 0.997176i \(-0.476072\pi\)
0.0751018 + 0.997176i \(0.476072\pi\)
\(542\) −3.38288e9 −0.912617
\(543\) 0 0
\(544\) −2.30428e8 −0.0613675
\(545\) 1.74410e9 0.461514
\(546\) 0 0
\(547\) 4.50977e8 0.117814 0.0589072 0.998263i \(-0.481238\pi\)
0.0589072 + 0.998263i \(0.481238\pi\)
\(548\) 8.99792e9 2.33566
\(549\) 0 0
\(550\) 4.36732e9 1.11930
\(551\) 8.57924e8 0.218483
\(552\) 0 0
\(553\) 4.37370e9 1.09979
\(554\) −8.35578e9 −2.08787
\(555\) 0 0
\(556\) 8.51024e9 2.09981
\(557\) 3.86605e9 0.947926 0.473963 0.880545i \(-0.342823\pi\)
0.473963 + 0.880545i \(0.342823\pi\)
\(558\) 0 0
\(559\) 4.93225e9 1.19427
\(560\) 7.22893e9 1.73947
\(561\) 0 0
\(562\) 7.28153e8 0.173040
\(563\) −2.10838e9 −0.497930 −0.248965 0.968512i \(-0.580090\pi\)
−0.248965 + 0.968512i \(0.580090\pi\)
\(564\) 0 0
\(565\) 1.17557e9 0.274207
\(566\) 5.11644e8 0.118607
\(567\) 0 0
\(568\) 5.25706e9 1.20371
\(569\) 5.98974e9 1.36306 0.681530 0.731790i \(-0.261315\pi\)
0.681530 + 0.731790i \(0.261315\pi\)
\(570\) 0 0
\(571\) 5.77991e9 1.29925 0.649627 0.760253i \(-0.274925\pi\)
0.649627 + 0.760253i \(0.274925\pi\)
\(572\) 2.70557e9 0.604467
\(573\) 0 0
\(574\) −6.78030e9 −1.49643
\(575\) 3.86568e9 0.847986
\(576\) 0 0
\(577\) 8.69442e8 0.188419 0.0942096 0.995552i \(-0.469968\pi\)
0.0942096 + 0.995552i \(0.469968\pi\)
\(578\) 8.30994e8 0.178999
\(579\) 0 0
\(580\) −4.51493e9 −0.960845
\(581\) −9.08402e9 −1.92159
\(582\) 0 0
\(583\) 1.34236e9 0.280562
\(584\) 1.37539e10 2.85747
\(585\) 0 0
\(586\) 1.28149e10 2.63071
\(587\) 1.19117e9 0.243074 0.121537 0.992587i \(-0.461218\pi\)
0.121537 + 0.992587i \(0.461218\pi\)
\(588\) 0 0
\(589\) 2.04701e9 0.412778
\(590\) 1.76319e9 0.353441
\(591\) 0 0
\(592\) −4.65618e9 −0.922367
\(593\) −5.37577e9 −1.05864 −0.529321 0.848421i \(-0.677553\pi\)
−0.529321 + 0.848421i \(0.677553\pi\)
\(594\) 0 0
\(595\) 8.01760e9 1.56040
\(596\) −8.13012e9 −1.57302
\(597\) 0 0
\(598\) 3.58125e9 0.684826
\(599\) 7.00918e9 1.33252 0.666260 0.745720i \(-0.267894\pi\)
0.666260 + 0.745720i \(0.267894\pi\)
\(600\) 0 0
\(601\) 5.18595e9 0.974469 0.487235 0.873271i \(-0.338006\pi\)
0.487235 + 0.873271i \(0.338006\pi\)
\(602\) 1.74743e10 3.26446
\(603\) 0 0
\(604\) −6.55215e8 −0.120991
\(605\) −6.81172e9 −1.25058
\(606\) 0 0
\(607\) 5.60023e9 1.01636 0.508178 0.861252i \(-0.330319\pi\)
0.508178 + 0.861252i \(0.330319\pi\)
\(608\) 2.57543e8 0.0464715
\(609\) 0 0
\(610\) 1.09337e10 1.95036
\(611\) 2.86947e8 0.0508929
\(612\) 0 0
\(613\) 3.23965e9 0.568051 0.284025 0.958817i \(-0.408330\pi\)
0.284025 + 0.958817i \(0.408330\pi\)
\(614\) −1.58764e10 −2.76798
\(615\) 0 0
\(616\) 4.83661e9 0.833697
\(617\) −1.21036e9 −0.207451 −0.103726 0.994606i \(-0.533076\pi\)
−0.103726 + 0.994606i \(0.533076\pi\)
\(618\) 0 0
\(619\) −9.96098e9 −1.68805 −0.844024 0.536305i \(-0.819820\pi\)
−0.844024 + 0.536305i \(0.819820\pi\)
\(620\) −1.07726e10 −1.81531
\(621\) 0 0
\(622\) −3.14953e9 −0.524783
\(623\) 2.35221e9 0.389733
\(624\) 0 0
\(625\) −2.21642e9 −0.363138
\(626\) −4.23332e9 −0.689717
\(627\) 0 0
\(628\) −1.50234e10 −2.42052
\(629\) −5.16416e9 −0.827414
\(630\) 0 0
\(631\) 8.86210e9 1.40422 0.702108 0.712071i \(-0.252243\pi\)
0.702108 + 0.712071i \(0.252243\pi\)
\(632\) −1.17130e10 −1.84569
\(633\) 0 0
\(634\) −1.69833e10 −2.64674
\(635\) 7.55424e8 0.117080
\(636\) 0 0
\(637\) −4.88404e8 −0.0748671
\(638\) −1.55136e9 −0.236505
\(639\) 0 0
\(640\) 1.76529e10 2.66187
\(641\) −1.08624e10 −1.62901 −0.814507 0.580154i \(-0.802992\pi\)
−0.814507 + 0.580154i \(0.802992\pi\)
\(642\) 0 0
\(643\) −5.64420e9 −0.837267 −0.418634 0.908155i \(-0.637491\pi\)
−0.418634 + 0.908155i \(0.637491\pi\)
\(644\) 8.48447e9 1.25177
\(645\) 0 0
\(646\) 8.08603e9 1.18011
\(647\) −1.50140e7 −0.00217937 −0.00108968 0.999999i \(-0.500347\pi\)
−0.00108968 + 0.999999i \(0.500347\pi\)
\(648\) 0 0
\(649\) 4.05132e8 0.0581756
\(650\) 1.17533e10 1.67865
\(651\) 0 0
\(652\) −1.13491e10 −1.60360
\(653\) 3.25704e9 0.457748 0.228874 0.973456i \(-0.426496\pi\)
0.228874 + 0.973456i \(0.426496\pi\)
\(654\) 0 0
\(655\) −5.29151e9 −0.735758
\(656\) 6.23592e9 0.862457
\(657\) 0 0
\(658\) 1.01661e9 0.139112
\(659\) −9.09574e9 −1.23805 −0.619026 0.785370i \(-0.712473\pi\)
−0.619026 + 0.785370i \(0.712473\pi\)
\(660\) 0 0
\(661\) −8.48609e9 −1.14289 −0.571443 0.820642i \(-0.693616\pi\)
−0.571443 + 0.820642i \(0.693616\pi\)
\(662\) −1.79344e10 −2.40261
\(663\) 0 0
\(664\) 2.43276e10 3.22486
\(665\) −8.96106e9 −1.18164
\(666\) 0 0
\(667\) −1.37317e9 −0.179178
\(668\) 2.33110e10 3.02582
\(669\) 0 0
\(670\) −5.40923e9 −0.694822
\(671\) 2.51227e9 0.321024
\(672\) 0 0
\(673\) −1.43232e10 −1.81129 −0.905646 0.424035i \(-0.860613\pi\)
−0.905646 + 0.424035i \(0.860613\pi\)
\(674\) −1.07283e10 −1.34964
\(675\) 0 0
\(676\) −8.93084e9 −1.11193
\(677\) 2.38817e9 0.295805 0.147902 0.989002i \(-0.452748\pi\)
0.147902 + 0.989002i \(0.452748\pi\)
\(678\) 0 0
\(679\) −1.35789e10 −1.66465
\(680\) −2.14716e10 −2.61869
\(681\) 0 0
\(682\) −3.70156e9 −0.446827
\(683\) −1.52714e10 −1.83403 −0.917013 0.398857i \(-0.869407\pi\)
−0.917013 + 0.398857i \(0.869407\pi\)
\(684\) 0 0
\(685\) 1.52108e10 1.80816
\(686\) 1.37587e10 1.62721
\(687\) 0 0
\(688\) −1.60713e10 −1.88145
\(689\) 3.61253e9 0.420770
\(690\) 0 0
\(691\) −1.01080e10 −1.16544 −0.582721 0.812672i \(-0.698012\pi\)
−0.582721 + 0.812672i \(0.698012\pi\)
\(692\) −5.81645e9 −0.667247
\(693\) 0 0
\(694\) −1.14799e10 −1.30371
\(695\) 1.43864e10 1.62557
\(696\) 0 0
\(697\) 6.91626e9 0.773671
\(698\) 1.50249e10 1.67231
\(699\) 0 0
\(700\) 2.78451e10 3.06835
\(701\) −1.18838e10 −1.30299 −0.651494 0.758654i \(-0.725857\pi\)
−0.651494 + 0.758654i \(0.725857\pi\)
\(702\) 0 0
\(703\) 5.77185e9 0.626572
\(704\) 3.90188e9 0.421472
\(705\) 0 0
\(706\) −2.09088e10 −2.23621
\(707\) −3.65321e9 −0.388782
\(708\) 0 0
\(709\) −1.60939e10 −1.69589 −0.847946 0.530082i \(-0.822161\pi\)
−0.847946 + 0.530082i \(0.822161\pi\)
\(710\) 1.76127e10 1.84681
\(711\) 0 0
\(712\) −6.29935e9 −0.654057
\(713\) −3.27639e9 −0.338518
\(714\) 0 0
\(715\) 4.57372e9 0.467949
\(716\) −3.27817e10 −3.33762
\(717\) 0 0
\(718\) −2.14604e10 −2.16373
\(719\) −6.07025e9 −0.609053 −0.304527 0.952504i \(-0.598498\pi\)
−0.304527 + 0.952504i \(0.598498\pi\)
\(720\) 0 0
\(721\) 1.70004e10 1.68922
\(722\) 8.53256e9 0.843721
\(723\) 0 0
\(724\) 2.40202e10 2.35230
\(725\) −4.50659e9 −0.439203
\(726\) 0 0
\(727\) −1.05256e10 −1.01596 −0.507979 0.861369i \(-0.669607\pi\)
−0.507979 + 0.861369i \(0.669607\pi\)
\(728\) 1.30162e10 1.25033
\(729\) 0 0
\(730\) 4.60797e10 4.38409
\(731\) −1.78247e10 −1.68776
\(732\) 0 0
\(733\) −8.12696e9 −0.762192 −0.381096 0.924536i \(-0.624453\pi\)
−0.381096 + 0.924536i \(0.624453\pi\)
\(734\) 3.77639e9 0.352485
\(735\) 0 0
\(736\) −4.12216e8 −0.0381112
\(737\) −1.24289e9 −0.114366
\(738\) 0 0
\(739\) −5.50602e9 −0.501860 −0.250930 0.968005i \(-0.580736\pi\)
−0.250930 + 0.968005i \(0.580736\pi\)
\(740\) −3.03751e10 −2.75554
\(741\) 0 0
\(742\) 1.27987e10 1.15014
\(743\) 4.28129e9 0.382925 0.191462 0.981500i \(-0.438677\pi\)
0.191462 + 0.981500i \(0.438677\pi\)
\(744\) 0 0
\(745\) −1.37438e10 −1.21776
\(746\) 2.44455e10 2.15582
\(747\) 0 0
\(748\) −9.77767e9 −0.854240
\(749\) −2.25781e10 −1.96336
\(750\) 0 0
\(751\) 7.38153e9 0.635926 0.317963 0.948103i \(-0.397001\pi\)
0.317963 + 0.948103i \(0.397001\pi\)
\(752\) −9.34992e8 −0.0801762
\(753\) 0 0
\(754\) −4.17500e9 −0.354696
\(755\) −1.10763e9 −0.0936655
\(756\) 0 0
\(757\) −2.14892e10 −1.80046 −0.900232 0.435410i \(-0.856604\pi\)
−0.900232 + 0.435410i \(0.856604\pi\)
\(758\) 3.99759e10 3.33393
\(759\) 0 0
\(760\) 2.39983e10 1.98304
\(761\) −9.48902e9 −0.780504 −0.390252 0.920708i \(-0.627612\pi\)
−0.390252 + 0.920708i \(0.627612\pi\)
\(762\) 0 0
\(763\) 3.82092e9 0.311410
\(764\) 4.02182e10 3.26284
\(765\) 0 0
\(766\) −1.68785e10 −1.35685
\(767\) 1.09029e9 0.0872482
\(768\) 0 0
\(769\) −2.02433e9 −0.160524 −0.0802620 0.996774i \(-0.525576\pi\)
−0.0802620 + 0.996774i \(0.525576\pi\)
\(770\) 1.62041e10 1.27911
\(771\) 0 0
\(772\) 4.79198e9 0.374847
\(773\) 5.39051e9 0.419761 0.209880 0.977727i \(-0.432693\pi\)
0.209880 + 0.977727i \(0.432693\pi\)
\(774\) 0 0
\(775\) −1.07527e10 −0.829781
\(776\) 3.63652e10 2.79364
\(777\) 0 0
\(778\) −3.53147e10 −2.68861
\(779\) −7.73012e9 −0.585875
\(780\) 0 0
\(781\) 4.04691e9 0.303980
\(782\) −1.29423e10 −0.967803
\(783\) 0 0
\(784\) 1.59142e9 0.117945
\(785\) −2.53968e10 −1.87385
\(786\) 0 0
\(787\) −3.44573e8 −0.0251982 −0.0125991 0.999921i \(-0.504011\pi\)
−0.0125991 + 0.999921i \(0.504011\pi\)
\(788\) −9.30986e9 −0.677800
\(789\) 0 0
\(790\) −3.92421e10 −2.83177
\(791\) 2.57540e9 0.185023
\(792\) 0 0
\(793\) 6.76098e9 0.481453
\(794\) 2.60181e10 1.84461
\(795\) 0 0
\(796\) 8.83382e9 0.620802
\(797\) −7.94031e9 −0.555563 −0.277781 0.960644i \(-0.589599\pi\)
−0.277781 + 0.960644i \(0.589599\pi\)
\(798\) 0 0
\(799\) −1.03700e9 −0.0719225
\(800\) −1.35285e9 −0.0934188
\(801\) 0 0
\(802\) −2.45847e10 −1.68288
\(803\) 1.05878e10 0.721611
\(804\) 0 0
\(805\) 1.43428e10 0.969057
\(806\) −9.96156e9 −0.670123
\(807\) 0 0
\(808\) 9.78352e9 0.652462
\(809\) −1.83573e10 −1.21896 −0.609478 0.792803i \(-0.708621\pi\)
−0.609478 + 0.792803i \(0.708621\pi\)
\(810\) 0 0
\(811\) −2.86473e10 −1.88587 −0.942934 0.332979i \(-0.891946\pi\)
−0.942934 + 0.332979i \(0.891946\pi\)
\(812\) −9.89115e9 −0.648337
\(813\) 0 0
\(814\) −1.04371e10 −0.678256
\(815\) −1.91855e10 −1.24143
\(816\) 0 0
\(817\) 1.99222e10 1.27808
\(818\) −3.61724e10 −2.31069
\(819\) 0 0
\(820\) 4.06807e10 2.57656
\(821\) 1.87449e10 1.18218 0.591088 0.806607i \(-0.298698\pi\)
0.591088 + 0.806607i \(0.298698\pi\)
\(822\) 0 0
\(823\) 2.20970e10 1.38176 0.690881 0.722969i \(-0.257223\pi\)
0.690881 + 0.722969i \(0.257223\pi\)
\(824\) −4.55281e10 −2.83488
\(825\) 0 0
\(826\) 3.86273e9 0.238487
\(827\) −7.00279e9 −0.430529 −0.215264 0.976556i \(-0.569061\pi\)
−0.215264 + 0.976556i \(0.569061\pi\)
\(828\) 0 0
\(829\) −1.55766e10 −0.949581 −0.474791 0.880099i \(-0.657476\pi\)
−0.474791 + 0.880099i \(0.657476\pi\)
\(830\) 8.15045e10 4.94776
\(831\) 0 0
\(832\) 1.05007e10 0.632099
\(833\) 1.76505e9 0.105803
\(834\) 0 0
\(835\) 3.94068e10 2.34244
\(836\) 1.09282e10 0.646887
\(837\) 0 0
\(838\) 3.88237e10 2.27899
\(839\) 2.48824e10 1.45454 0.727268 0.686353i \(-0.240790\pi\)
0.727268 + 0.686353i \(0.240790\pi\)
\(840\) 0 0
\(841\) −1.56490e10 −0.907197
\(842\) −2.06221e10 −1.19053
\(843\) 0 0
\(844\) 4.29411e10 2.45853
\(845\) −1.50974e10 −0.860804
\(846\) 0 0
\(847\) −1.49229e10 −0.843841
\(848\) −1.17711e10 −0.662876
\(849\) 0 0
\(850\) −4.24751e10 −2.37229
\(851\) −9.23827e9 −0.513851
\(852\) 0 0
\(853\) −1.67988e10 −0.926736 −0.463368 0.886166i \(-0.653359\pi\)
−0.463368 + 0.886166i \(0.653359\pi\)
\(854\) 2.39533e10 1.31602
\(855\) 0 0
\(856\) 6.04654e10 3.29495
\(857\) 2.93017e10 1.59023 0.795115 0.606458i \(-0.207410\pi\)
0.795115 + 0.606458i \(0.207410\pi\)
\(858\) 0 0
\(859\) 2.65222e10 1.42769 0.713844 0.700305i \(-0.246952\pi\)
0.713844 + 0.700305i \(0.246952\pi\)
\(860\) −1.04843e11 −5.62075
\(861\) 0 0
\(862\) 2.26449e10 1.20419
\(863\) −2.67744e10 −1.41802 −0.709009 0.705199i \(-0.750858\pi\)
−0.709009 + 0.705199i \(0.750858\pi\)
\(864\) 0 0
\(865\) −9.83259e9 −0.516549
\(866\) −4.67076e10 −2.44385
\(867\) 0 0
\(868\) −2.36003e10 −1.22489
\(869\) −9.01675e9 −0.466102
\(870\) 0 0
\(871\) −3.34485e9 −0.171519
\(872\) −1.02327e10 −0.522614
\(873\) 0 0
\(874\) 1.44652e10 0.732885
\(875\) 1.44223e10 0.727790
\(876\) 0 0
\(877\) 2.17802e10 1.09034 0.545171 0.838325i \(-0.316465\pi\)
0.545171 + 0.838325i \(0.316465\pi\)
\(878\) 1.31774e10 0.657049
\(879\) 0 0
\(880\) −1.49031e10 −0.737201
\(881\) 2.03738e10 1.00382 0.501910 0.864920i \(-0.332631\pi\)
0.501910 + 0.864920i \(0.332631\pi\)
\(882\) 0 0
\(883\) 2.67217e10 1.30618 0.653088 0.757282i \(-0.273473\pi\)
0.653088 + 0.757282i \(0.273473\pi\)
\(884\) −2.63135e10 −1.28114
\(885\) 0 0
\(886\) −4.41218e10 −2.13125
\(887\) −6.91217e9 −0.332569 −0.166285 0.986078i \(-0.553177\pi\)
−0.166285 + 0.986078i \(0.553177\pi\)
\(888\) 0 0
\(889\) 1.65495e9 0.0790005
\(890\) −2.11047e10 −1.00349
\(891\) 0 0
\(892\) −6.66164e10 −3.14271
\(893\) 1.15903e9 0.0544645
\(894\) 0 0
\(895\) −5.54169e10 −2.58382
\(896\) 3.86734e10 1.79612
\(897\) 0 0
\(898\) −1.13395e10 −0.522550
\(899\) 3.81960e9 0.175331
\(900\) 0 0
\(901\) −1.30553e10 −0.594636
\(902\) 1.39782e10 0.634202
\(903\) 0 0
\(904\) −6.89707e9 −0.310510
\(905\) 4.06058e10 1.82103
\(906\) 0 0
\(907\) −1.77215e10 −0.788632 −0.394316 0.918975i \(-0.629018\pi\)
−0.394316 + 0.918975i \(0.629018\pi\)
\(908\) 5.86512e10 2.60002
\(909\) 0 0
\(910\) 4.36081e10 1.91832
\(911\) 2.97591e10 1.30408 0.652042 0.758183i \(-0.273913\pi\)
0.652042 + 0.758183i \(0.273913\pi\)
\(912\) 0 0
\(913\) 1.87275e10 0.814389
\(914\) −1.77360e10 −0.768323
\(915\) 0 0
\(916\) −1.91195e10 −0.821942
\(917\) −1.15925e10 −0.496458
\(918\) 0 0
\(919\) −7.49318e8 −0.0318465 −0.0159233 0.999873i \(-0.505069\pi\)
−0.0159233 + 0.999873i \(0.505069\pi\)
\(920\) −3.84110e10 −1.62629
\(921\) 0 0
\(922\) 3.08418e10 1.29593
\(923\) 1.08910e10 0.455891
\(924\) 0 0
\(925\) −3.03190e10 −1.25956
\(926\) −1.18189e10 −0.489147
\(927\) 0 0
\(928\) 4.80560e8 0.0197392
\(929\) −3.70776e10 −1.51725 −0.758623 0.651529i \(-0.774128\pi\)
−0.758623 + 0.651529i \(0.774128\pi\)
\(930\) 0 0
\(931\) −1.97274e9 −0.0801211
\(932\) −1.75020e10 −0.708161
\(933\) 0 0
\(934\) −4.29706e10 −1.72567
\(935\) −1.65290e10 −0.661310
\(936\) 0 0
\(937\) 2.52127e10 1.00122 0.500611 0.865672i \(-0.333109\pi\)
0.500611 + 0.865672i \(0.333109\pi\)
\(938\) −1.18504e10 −0.468836
\(939\) 0 0
\(940\) −6.09952e9 −0.239524
\(941\) −2.49972e10 −0.977975 −0.488988 0.872291i \(-0.662634\pi\)
−0.488988 + 0.872291i \(0.662634\pi\)
\(942\) 0 0
\(943\) 1.23726e10 0.480475
\(944\) −3.55260e9 −0.137450
\(945\) 0 0
\(946\) −3.60247e10 −1.38351
\(947\) 1.83651e10 0.702696 0.351348 0.936245i \(-0.385723\pi\)
0.351348 + 0.936245i \(0.385723\pi\)
\(948\) 0 0
\(949\) 2.84938e10 1.08223
\(950\) 4.74733e10 1.79646
\(951\) 0 0
\(952\) −4.70393e10 −1.76698
\(953\) 1.85113e10 0.692805 0.346403 0.938086i \(-0.387403\pi\)
0.346403 + 0.938086i \(0.387403\pi\)
\(954\) 0 0
\(955\) 6.79881e10 2.52593
\(956\) −3.32696e10 −1.23153
\(957\) 0 0
\(958\) 2.30694e10 0.847728
\(959\) 3.33233e10 1.22007
\(960\) 0 0
\(961\) −1.83990e10 −0.668749
\(962\) −2.80881e10 −1.01721
\(963\) 0 0
\(964\) −7.67540e10 −2.75950
\(965\) 8.10075e9 0.290188
\(966\) 0 0
\(967\) −4.59158e10 −1.63294 −0.816469 0.577389i \(-0.804072\pi\)
−0.816469 + 0.577389i \(0.804072\pi\)
\(968\) 3.99644e10 1.41615
\(969\) 0 0
\(970\) 1.21834e11 4.28616
\(971\) −9.02722e9 −0.316437 −0.158218 0.987404i \(-0.550575\pi\)
−0.158218 + 0.987404i \(0.550575\pi\)
\(972\) 0 0
\(973\) 3.15172e10 1.09686
\(974\) −2.70335e10 −0.937444
\(975\) 0 0
\(976\) −2.20301e10 −0.758476
\(977\) −2.33504e10 −0.801058 −0.400529 0.916284i \(-0.631174\pi\)
−0.400529 + 0.916284i \(0.631174\pi\)
\(978\) 0 0
\(979\) −4.84927e9 −0.165172
\(980\) 1.03818e10 0.352356
\(981\) 0 0
\(982\) 9.70132e10 3.26919
\(983\) 4.78814e10 1.60779 0.803895 0.594771i \(-0.202757\pi\)
0.803895 + 0.594771i \(0.202757\pi\)
\(984\) 0 0
\(985\) −1.57381e10 −0.524719
\(986\) 1.50880e10 0.501261
\(987\) 0 0
\(988\) 2.94099e10 0.970162
\(989\) −3.18869e10 −1.04815
\(990\) 0 0
\(991\) −1.35826e10 −0.443329 −0.221664 0.975123i \(-0.571149\pi\)
−0.221664 + 0.975123i \(0.571149\pi\)
\(992\) 1.14662e9 0.0372930
\(993\) 0 0
\(994\) 3.85852e10 1.24615
\(995\) 1.49334e10 0.480594
\(996\) 0 0
\(997\) 4.46503e10 1.42689 0.713446 0.700710i \(-0.247133\pi\)
0.713446 + 0.700710i \(0.247133\pi\)
\(998\) 8.93530e10 2.84546
\(999\) 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.1.1 16
3.2 odd 2 177.8.a.a.1.16 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.16 16 3.2 odd 2
531.8.a.b.1.1 16 1.1 even 1 trivial