Properties

Label 531.8.a.b.1.11
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 1.11
Root \(7.00808\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.00808 q^{2} -78.8868 q^{4} -449.079 q^{5} -271.717 q^{7} -1449.88 q^{8} +O(q^{10})\) \(q+7.00808 q^{2} -78.8868 q^{4} -449.079 q^{5} -271.717 q^{7} -1449.88 q^{8} -3147.18 q^{10} +3164.33 q^{11} -9448.56 q^{13} -1904.22 q^{14} -63.3641 q^{16} +6793.30 q^{17} +34097.6 q^{19} +35426.4 q^{20} +22175.9 q^{22} +31238.0 q^{23} +123547. q^{25} -66216.3 q^{26} +21434.9 q^{28} -82712.9 q^{29} +283828. q^{31} +185141. q^{32} +47608.0 q^{34} +122023. q^{35} +26823.2 q^{37} +238959. q^{38} +651111. q^{40} +19538.2 q^{41} -417223. q^{43} -249624. q^{44} +218919. q^{46} +27684.6 q^{47} -749713. q^{49} +865829. q^{50} +745366. q^{52} -29017.7 q^{53} -1.42104e6 q^{55} +393957. q^{56} -579659. q^{58} -205379. q^{59} +1.53434e6 q^{61} +1.98909e6 q^{62} +1.30559e6 q^{64} +4.24315e6 q^{65} -2.52949e6 q^{67} -535901. q^{68} +855144. q^{70} +3.35724e6 q^{71} -1.90625e6 q^{73} +187979. q^{74} -2.68985e6 q^{76} -859803. q^{77} +2.26955e6 q^{79} +28455.5 q^{80} +136925. q^{82} +5.96358e6 q^{83} -3.05073e6 q^{85} -2.92393e6 q^{86} -4.58790e6 q^{88} -5.11791e6 q^{89} +2.56733e6 q^{91} -2.46427e6 q^{92} +194016. q^{94} -1.53125e7 q^{95} -145042. q^{97} -5.25405e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.00808 0.619433 0.309716 0.950829i \(-0.399766\pi\)
0.309716 + 0.950829i \(0.399766\pi\)
\(3\) 0 0
\(4\) −78.8868 −0.616303
\(5\) −449.079 −1.60668 −0.803338 0.595524i \(-0.796944\pi\)
−0.803338 + 0.595524i \(0.796944\pi\)
\(6\) 0 0
\(7\) −271.717 −0.299415 −0.149708 0.988730i \(-0.547833\pi\)
−0.149708 + 0.988730i \(0.547833\pi\)
\(8\) −1449.88 −1.00119
\(9\) 0 0
\(10\) −3147.18 −0.995227
\(11\) 3164.33 0.716816 0.358408 0.933565i \(-0.383320\pi\)
0.358408 + 0.933565i \(0.383320\pi\)
\(12\) 0 0
\(13\) −9448.56 −1.19279 −0.596395 0.802691i \(-0.703401\pi\)
−0.596395 + 0.802691i \(0.703401\pi\)
\(14\) −1904.22 −0.185468
\(15\) 0 0
\(16\) −63.3641 −0.00386744
\(17\) 6793.30 0.335358 0.167679 0.985842i \(-0.446373\pi\)
0.167679 + 0.985842i \(0.446373\pi\)
\(18\) 0 0
\(19\) 34097.6 1.14048 0.570239 0.821479i \(-0.306851\pi\)
0.570239 + 0.821479i \(0.306851\pi\)
\(20\) 35426.4 0.990199
\(21\) 0 0
\(22\) 22175.9 0.444019
\(23\) 31238.0 0.535348 0.267674 0.963510i \(-0.413745\pi\)
0.267674 + 0.963510i \(0.413745\pi\)
\(24\) 0 0
\(25\) 123547. 1.58141
\(26\) −66216.3 −0.738853
\(27\) 0 0
\(28\) 21434.9 0.184531
\(29\) −82712.9 −0.629767 −0.314884 0.949130i \(-0.601966\pi\)
−0.314884 + 0.949130i \(0.601966\pi\)
\(30\) 0 0
\(31\) 283828. 1.71116 0.855578 0.517673i \(-0.173202\pi\)
0.855578 + 0.517673i \(0.173202\pi\)
\(32\) 185141. 0.998795
\(33\) 0 0
\(34\) 47608.0 0.207732
\(35\) 122023. 0.481063
\(36\) 0 0
\(37\) 26823.2 0.0870571 0.0435286 0.999052i \(-0.486140\pi\)
0.0435286 + 0.999052i \(0.486140\pi\)
\(38\) 238959. 0.706449
\(39\) 0 0
\(40\) 651111. 1.60859
\(41\) 19538.2 0.0442732 0.0221366 0.999755i \(-0.492953\pi\)
0.0221366 + 0.999755i \(0.492953\pi\)
\(42\) 0 0
\(43\) −417223. −0.800255 −0.400128 0.916459i \(-0.631034\pi\)
−0.400128 + 0.916459i \(0.631034\pi\)
\(44\) −249624. −0.441776
\(45\) 0 0
\(46\) 218919. 0.331612
\(47\) 27684.6 0.0388952 0.0194476 0.999811i \(-0.493809\pi\)
0.0194476 + 0.999811i \(0.493809\pi\)
\(48\) 0 0
\(49\) −749713. −0.910351
\(50\) 865829. 0.979574
\(51\) 0 0
\(52\) 745366. 0.735120
\(53\) −29017.7 −0.0267730 −0.0133865 0.999910i \(-0.504261\pi\)
−0.0133865 + 0.999910i \(0.504261\pi\)
\(54\) 0 0
\(55\) −1.42104e6 −1.15169
\(56\) 393957. 0.299772
\(57\) 0 0
\(58\) −579659. −0.390099
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 1.53434e6 0.865499 0.432749 0.901514i \(-0.357543\pi\)
0.432749 + 0.901514i \(0.357543\pi\)
\(62\) 1.98909e6 1.05995
\(63\) 0 0
\(64\) 1.30559e6 0.622554
\(65\) 4.24315e6 1.91642
\(66\) 0 0
\(67\) −2.52949e6 −1.02748 −0.513738 0.857947i \(-0.671740\pi\)
−0.513738 + 0.857947i \(0.671740\pi\)
\(68\) −535901. −0.206682
\(69\) 0 0
\(70\) 855144. 0.297986
\(71\) 3.35724e6 1.11321 0.556606 0.830776i \(-0.312103\pi\)
0.556606 + 0.830776i \(0.312103\pi\)
\(72\) 0 0
\(73\) −1.90625e6 −0.573521 −0.286761 0.958002i \(-0.592578\pi\)
−0.286761 + 0.958002i \(0.592578\pi\)
\(74\) 187979. 0.0539260
\(75\) 0 0
\(76\) −2.68985e6 −0.702880
\(77\) −859803. −0.214626
\(78\) 0 0
\(79\) 2.26955e6 0.517899 0.258950 0.965891i \(-0.416624\pi\)
0.258950 + 0.965891i \(0.416624\pi\)
\(80\) 28455.5 0.00621372
\(81\) 0 0
\(82\) 136925. 0.0274243
\(83\) 5.96358e6 1.14481 0.572405 0.819971i \(-0.306011\pi\)
0.572405 + 0.819971i \(0.306011\pi\)
\(84\) 0 0
\(85\) −3.05073e6 −0.538812
\(86\) −2.92393e6 −0.495704
\(87\) 0 0
\(88\) −4.58790e6 −0.717670
\(89\) −5.11791e6 −0.769533 −0.384767 0.923014i \(-0.625718\pi\)
−0.384767 + 0.923014i \(0.625718\pi\)
\(90\) 0 0
\(91\) 2.56733e6 0.357139
\(92\) −2.46427e6 −0.329937
\(93\) 0 0
\(94\) 194016. 0.0240930
\(95\) −1.53125e7 −1.83238
\(96\) 0 0
\(97\) −145042. −0.0161359 −0.00806793 0.999967i \(-0.502568\pi\)
−0.00806793 + 0.999967i \(0.502568\pi\)
\(98\) −5.25405e6 −0.563901
\(99\) 0 0
\(100\) −9.74625e6 −0.974625
\(101\) −7.21193e6 −0.696509 −0.348254 0.937400i \(-0.613225\pi\)
−0.348254 + 0.937400i \(0.613225\pi\)
\(102\) 0 0
\(103\) 1.50882e6 0.136052 0.0680262 0.997684i \(-0.478330\pi\)
0.0680262 + 0.997684i \(0.478330\pi\)
\(104\) 1.36993e7 1.19421
\(105\) 0 0
\(106\) −203359. −0.0165841
\(107\) 1.20706e7 0.952545 0.476272 0.879298i \(-0.341988\pi\)
0.476272 + 0.879298i \(0.341988\pi\)
\(108\) 0 0
\(109\) −1.99398e7 −1.47479 −0.737393 0.675464i \(-0.763943\pi\)
−0.737393 + 0.675464i \(0.763943\pi\)
\(110\) −9.95874e6 −0.713395
\(111\) 0 0
\(112\) 17217.1 0.00115797
\(113\) 2.25861e7 1.47254 0.736269 0.676688i \(-0.236586\pi\)
0.736269 + 0.676688i \(0.236586\pi\)
\(114\) 0 0
\(115\) −1.40284e7 −0.860130
\(116\) 6.52495e6 0.388128
\(117\) 0 0
\(118\) −1.43931e6 −0.0806433
\(119\) −1.84585e6 −0.100411
\(120\) 0 0
\(121\) −9.47417e6 −0.486175
\(122\) 1.07528e7 0.536118
\(123\) 0 0
\(124\) −2.23903e7 −1.05459
\(125\) −2.03982e7 −0.934129
\(126\) 0 0
\(127\) −3.83609e7 −1.66179 −0.830895 0.556429i \(-0.812171\pi\)
−0.830895 + 0.556429i \(0.812171\pi\)
\(128\) −1.45483e7 −0.613165
\(129\) 0 0
\(130\) 2.97364e7 1.18710
\(131\) 2.06635e7 0.803070 0.401535 0.915844i \(-0.368477\pi\)
0.401535 + 0.915844i \(0.368477\pi\)
\(132\) 0 0
\(133\) −9.26491e6 −0.341476
\(134\) −1.77269e7 −0.636452
\(135\) 0 0
\(136\) −9.84946e6 −0.335758
\(137\) 2.08600e7 0.693095 0.346547 0.938033i \(-0.387354\pi\)
0.346547 + 0.938033i \(0.387354\pi\)
\(138\) 0 0
\(139\) −4.87015e7 −1.53812 −0.769061 0.639176i \(-0.779276\pi\)
−0.769061 + 0.639176i \(0.779276\pi\)
\(140\) −9.62597e6 −0.296481
\(141\) 0 0
\(142\) 2.35278e7 0.689560
\(143\) −2.98984e7 −0.855011
\(144\) 0 0
\(145\) 3.71447e7 1.01183
\(146\) −1.33591e7 −0.355258
\(147\) 0 0
\(148\) −2.11600e6 −0.0536536
\(149\) −1.01955e7 −0.252497 −0.126249 0.991999i \(-0.540294\pi\)
−0.126249 + 0.991999i \(0.540294\pi\)
\(150\) 0 0
\(151\) 3.94880e7 0.933354 0.466677 0.884428i \(-0.345451\pi\)
0.466677 + 0.884428i \(0.345451\pi\)
\(152\) −4.94375e7 −1.14184
\(153\) 0 0
\(154\) −6.02557e6 −0.132946
\(155\) −1.27461e8 −2.74927
\(156\) 0 0
\(157\) 4.95090e7 1.02102 0.510511 0.859871i \(-0.329456\pi\)
0.510511 + 0.859871i \(0.329456\pi\)
\(158\) 1.59052e7 0.320804
\(159\) 0 0
\(160\) −8.31428e7 −1.60474
\(161\) −8.48791e6 −0.160291
\(162\) 0 0
\(163\) −7.43101e7 −1.34397 −0.671987 0.740563i \(-0.734559\pi\)
−0.671987 + 0.740563i \(0.734559\pi\)
\(164\) −1.54131e6 −0.0272857
\(165\) 0 0
\(166\) 4.17932e7 0.709133
\(167\) −2.90320e7 −0.482358 −0.241179 0.970481i \(-0.577534\pi\)
−0.241179 + 0.970481i \(0.577534\pi\)
\(168\) 0 0
\(169\) 2.65267e7 0.422746
\(170\) −2.13798e7 −0.333758
\(171\) 0 0
\(172\) 3.29134e7 0.493200
\(173\) −5.40154e7 −0.793152 −0.396576 0.918002i \(-0.629802\pi\)
−0.396576 + 0.918002i \(0.629802\pi\)
\(174\) 0 0
\(175\) −3.35699e7 −0.473497
\(176\) −200505. −0.00277224
\(177\) 0 0
\(178\) −3.58667e7 −0.476674
\(179\) 5.61319e7 0.731516 0.365758 0.930710i \(-0.380810\pi\)
0.365758 + 0.930710i \(0.380810\pi\)
\(180\) 0 0
\(181\) 4.15020e7 0.520228 0.260114 0.965578i \(-0.416240\pi\)
0.260114 + 0.965578i \(0.416240\pi\)
\(182\) 1.79921e7 0.221224
\(183\) 0 0
\(184\) −4.52914e7 −0.535986
\(185\) −1.20457e7 −0.139873
\(186\) 0 0
\(187\) 2.14962e7 0.240390
\(188\) −2.18395e6 −0.0239712
\(189\) 0 0
\(190\) −1.07312e8 −1.13503
\(191\) 5.65111e7 0.586837 0.293418 0.955984i \(-0.405207\pi\)
0.293418 + 0.955984i \(0.405207\pi\)
\(192\) 0 0
\(193\) 1.59119e8 1.59321 0.796604 0.604502i \(-0.206628\pi\)
0.796604 + 0.604502i \(0.206628\pi\)
\(194\) −1.01646e6 −0.00999508
\(195\) 0 0
\(196\) 5.91424e7 0.561052
\(197\) −6.46721e7 −0.602678 −0.301339 0.953517i \(-0.597434\pi\)
−0.301339 + 0.953517i \(0.597434\pi\)
\(198\) 0 0
\(199\) −1.88894e8 −1.69915 −0.849576 0.527467i \(-0.823142\pi\)
−0.849576 + 0.527467i \(0.823142\pi\)
\(200\) −1.79129e8 −1.58329
\(201\) 0 0
\(202\) −5.05418e7 −0.431440
\(203\) 2.24745e7 0.188562
\(204\) 0 0
\(205\) −8.77420e6 −0.0711326
\(206\) 1.05739e7 0.0842754
\(207\) 0 0
\(208\) 598700. 0.00461304
\(209\) 1.07896e8 0.817513
\(210\) 0 0
\(211\) 8.23127e7 0.603224 0.301612 0.953431i \(-0.402475\pi\)
0.301612 + 0.953431i \(0.402475\pi\)
\(212\) 2.28912e6 0.0165003
\(213\) 0 0
\(214\) 8.45917e7 0.590037
\(215\) 1.87366e8 1.28575
\(216\) 0 0
\(217\) −7.71210e7 −0.512346
\(218\) −1.39740e8 −0.913531
\(219\) 0 0
\(220\) 1.12101e8 0.709790
\(221\) −6.41868e7 −0.400012
\(222\) 0 0
\(223\) −1.37252e8 −0.828801 −0.414400 0.910095i \(-0.636009\pi\)
−0.414400 + 0.910095i \(0.636009\pi\)
\(224\) −5.03058e7 −0.299055
\(225\) 0 0
\(226\) 1.58285e8 0.912139
\(227\) 1.70002e8 0.964635 0.482317 0.875997i \(-0.339795\pi\)
0.482317 + 0.875997i \(0.339795\pi\)
\(228\) 0 0
\(229\) 2.30825e8 1.27016 0.635082 0.772445i \(-0.280966\pi\)
0.635082 + 0.772445i \(0.280966\pi\)
\(230\) −9.83119e7 −0.532793
\(231\) 0 0
\(232\) 1.19924e8 0.630518
\(233\) 1.24871e8 0.646719 0.323359 0.946276i \(-0.395188\pi\)
0.323359 + 0.946276i \(0.395188\pi\)
\(234\) 0 0
\(235\) −1.24326e7 −0.0624920
\(236\) 1.62017e7 0.0802358
\(237\) 0 0
\(238\) −1.29359e7 −0.0621981
\(239\) 2.73260e8 1.29474 0.647371 0.762175i \(-0.275868\pi\)
0.647371 + 0.762175i \(0.275868\pi\)
\(240\) 0 0
\(241\) −2.70017e8 −1.24260 −0.621301 0.783572i \(-0.713396\pi\)
−0.621301 + 0.783572i \(0.713396\pi\)
\(242\) −6.63957e7 −0.301152
\(243\) 0 0
\(244\) −1.21039e8 −0.533410
\(245\) 3.36681e8 1.46264
\(246\) 0 0
\(247\) −3.22173e8 −1.36035
\(248\) −4.11517e8 −1.71319
\(249\) 0 0
\(250\) −1.42952e8 −0.578630
\(251\) −1.21031e7 −0.0483103 −0.0241551 0.999708i \(-0.507690\pi\)
−0.0241551 + 0.999708i \(0.507690\pi\)
\(252\) 0 0
\(253\) 9.88475e7 0.383746
\(254\) −2.68836e8 −1.02937
\(255\) 0 0
\(256\) −2.69071e8 −1.00237
\(257\) −2.89004e8 −1.06203 −0.531017 0.847361i \(-0.678190\pi\)
−0.531017 + 0.847361i \(0.678190\pi\)
\(258\) 0 0
\(259\) −7.28832e6 −0.0260662
\(260\) −3.34729e8 −1.18110
\(261\) 0 0
\(262\) 1.44811e8 0.497448
\(263\) 7.64989e6 0.0259304 0.0129652 0.999916i \(-0.495873\pi\)
0.0129652 + 0.999916i \(0.495873\pi\)
\(264\) 0 0
\(265\) 1.30313e7 0.0430156
\(266\) −6.49293e7 −0.211522
\(267\) 0 0
\(268\) 1.99544e8 0.633237
\(269\) −2.10692e8 −0.659956 −0.329978 0.943989i \(-0.607041\pi\)
−0.329978 + 0.943989i \(0.607041\pi\)
\(270\) 0 0
\(271\) −3.98910e8 −1.21754 −0.608769 0.793348i \(-0.708336\pi\)
−0.608769 + 0.793348i \(0.708336\pi\)
\(272\) −430451. −0.00129698
\(273\) 0 0
\(274\) 1.46189e8 0.429325
\(275\) 3.90945e8 1.13358
\(276\) 0 0
\(277\) −5.06287e8 −1.43126 −0.715628 0.698481i \(-0.753859\pi\)
−0.715628 + 0.698481i \(0.753859\pi\)
\(278\) −3.41304e8 −0.952763
\(279\) 0 0
\(280\) −1.76918e8 −0.481636
\(281\) 4.35455e8 1.17077 0.585384 0.810756i \(-0.300944\pi\)
0.585384 + 0.810756i \(0.300944\pi\)
\(282\) 0 0
\(283\) −1.36270e7 −0.0357394 −0.0178697 0.999840i \(-0.505688\pi\)
−0.0178697 + 0.999840i \(0.505688\pi\)
\(284\) −2.64842e8 −0.686076
\(285\) 0 0
\(286\) −2.09530e8 −0.529622
\(287\) −5.30886e6 −0.0132561
\(288\) 0 0
\(289\) −3.64190e8 −0.887535
\(290\) 2.60313e8 0.626762
\(291\) 0 0
\(292\) 1.50378e8 0.353463
\(293\) 5.24778e8 1.21882 0.609409 0.792856i \(-0.291407\pi\)
0.609409 + 0.792856i \(0.291407\pi\)
\(294\) 0 0
\(295\) 9.22315e7 0.209171
\(296\) −3.88904e7 −0.0871608
\(297\) 0 0
\(298\) −7.14510e7 −0.156405
\(299\) −2.95154e8 −0.638558
\(300\) 0 0
\(301\) 1.13367e8 0.239609
\(302\) 2.76735e8 0.578150
\(303\) 0 0
\(304\) −2.16057e6 −0.00441073
\(305\) −6.89039e8 −1.39058
\(306\) 0 0
\(307\) −5.40106e8 −1.06536 −0.532678 0.846318i \(-0.678814\pi\)
−0.532678 + 0.846318i \(0.678814\pi\)
\(308\) 6.78271e7 0.132274
\(309\) 0 0
\(310\) −8.93260e8 −1.70299
\(311\) 1.89139e8 0.356549 0.178275 0.983981i \(-0.442948\pi\)
0.178275 + 0.983981i \(0.442948\pi\)
\(312\) 0 0
\(313\) 5.77137e8 1.06383 0.531917 0.846797i \(-0.321472\pi\)
0.531917 + 0.846797i \(0.321472\pi\)
\(314\) 3.46963e8 0.632454
\(315\) 0 0
\(316\) −1.79038e8 −0.319183
\(317\) −1.18644e6 −0.00209189 −0.00104595 0.999999i \(-0.500333\pi\)
−0.00104595 + 0.999999i \(0.500333\pi\)
\(318\) 0 0
\(319\) −2.61731e8 −0.451427
\(320\) −5.86314e8 −1.00024
\(321\) 0 0
\(322\) −5.94840e7 −0.0992897
\(323\) 2.31635e8 0.382469
\(324\) 0 0
\(325\) −1.16734e9 −1.88628
\(326\) −5.20771e8 −0.832502
\(327\) 0 0
\(328\) −2.83280e7 −0.0443259
\(329\) −7.52238e6 −0.0116458
\(330\) 0 0
\(331\) −8.08372e8 −1.22522 −0.612609 0.790386i \(-0.709880\pi\)
−0.612609 + 0.790386i \(0.709880\pi\)
\(332\) −4.70447e8 −0.705550
\(333\) 0 0
\(334\) −2.03459e8 −0.298788
\(335\) 1.13594e9 1.65082
\(336\) 0 0
\(337\) 8.85349e8 1.26011 0.630057 0.776549i \(-0.283032\pi\)
0.630057 + 0.776549i \(0.283032\pi\)
\(338\) 1.85901e8 0.261863
\(339\) 0 0
\(340\) 2.40662e8 0.332072
\(341\) 8.98127e8 1.22658
\(342\) 0 0
\(343\) 4.27481e8 0.571988
\(344\) 6.04923e8 0.801208
\(345\) 0 0
\(346\) −3.78544e8 −0.491304
\(347\) −7.51351e8 −0.965361 −0.482680 0.875797i \(-0.660337\pi\)
−0.482680 + 0.875797i \(0.660337\pi\)
\(348\) 0 0
\(349\) 5.06457e8 0.637755 0.318877 0.947796i \(-0.396694\pi\)
0.318877 + 0.947796i \(0.396694\pi\)
\(350\) −2.35261e8 −0.293299
\(351\) 0 0
\(352\) 5.85846e8 0.715953
\(353\) −7.31006e8 −0.884523 −0.442262 0.896886i \(-0.645824\pi\)
−0.442262 + 0.896886i \(0.645824\pi\)
\(354\) 0 0
\(355\) −1.50767e9 −1.78857
\(356\) 4.03735e8 0.474266
\(357\) 0 0
\(358\) 3.93377e8 0.453125
\(359\) −1.09721e9 −1.25158 −0.625790 0.779992i \(-0.715223\pi\)
−0.625790 + 0.779992i \(0.715223\pi\)
\(360\) 0 0
\(361\) 2.68777e8 0.300688
\(362\) 2.90849e8 0.322246
\(363\) 0 0
\(364\) −2.02529e8 −0.220106
\(365\) 8.56057e8 0.921462
\(366\) 0 0
\(367\) 2.58816e8 0.273313 0.136657 0.990618i \(-0.456364\pi\)
0.136657 + 0.990618i \(0.456364\pi\)
\(368\) −1.97937e6 −0.00207043
\(369\) 0 0
\(370\) −8.44175e7 −0.0866416
\(371\) 7.88461e6 0.00801626
\(372\) 0 0
\(373\) 1.37810e9 1.37499 0.687497 0.726187i \(-0.258709\pi\)
0.687497 + 0.726187i \(0.258709\pi\)
\(374\) 1.50647e8 0.148906
\(375\) 0 0
\(376\) −4.01394e7 −0.0389415
\(377\) 7.81517e8 0.751180
\(378\) 0 0
\(379\) −1.56880e8 −0.148024 −0.0740119 0.997257i \(-0.523580\pi\)
−0.0740119 + 0.997257i \(0.523580\pi\)
\(380\) 1.20796e9 1.12930
\(381\) 0 0
\(382\) 3.96035e8 0.363506
\(383\) 8.83458e8 0.803508 0.401754 0.915748i \(-0.368401\pi\)
0.401754 + 0.915748i \(0.368401\pi\)
\(384\) 0 0
\(385\) 3.86120e8 0.344834
\(386\) 1.11512e9 0.986885
\(387\) 0 0
\(388\) 1.14419e7 0.00994458
\(389\) −9.53014e8 −0.820873 −0.410436 0.911889i \(-0.634624\pi\)
−0.410436 + 0.911889i \(0.634624\pi\)
\(390\) 0 0
\(391\) 2.12209e8 0.179534
\(392\) 1.08699e9 0.911435
\(393\) 0 0
\(394\) −4.53227e8 −0.373318
\(395\) −1.01921e9 −0.832095
\(396\) 0 0
\(397\) −2.30889e9 −1.85198 −0.925992 0.377544i \(-0.876769\pi\)
−0.925992 + 0.377544i \(0.876769\pi\)
\(398\) −1.32378e9 −1.05251
\(399\) 0 0
\(400\) −7.82846e6 −0.00611599
\(401\) −1.15700e9 −0.896039 −0.448019 0.894024i \(-0.647870\pi\)
−0.448019 + 0.894024i \(0.647870\pi\)
\(402\) 0 0
\(403\) −2.68177e9 −2.04105
\(404\) 5.68926e8 0.429260
\(405\) 0 0
\(406\) 1.57503e8 0.116801
\(407\) 8.48775e7 0.0624040
\(408\) 0 0
\(409\) 1.94341e9 1.40454 0.702269 0.711911i \(-0.252170\pi\)
0.702269 + 0.711911i \(0.252170\pi\)
\(410\) −6.14903e7 −0.0440619
\(411\) 0 0
\(412\) −1.19026e8 −0.0838496
\(413\) 5.58050e7 0.0389805
\(414\) 0 0
\(415\) −2.67812e9 −1.83934
\(416\) −1.74931e9 −1.19135
\(417\) 0 0
\(418\) 7.56146e8 0.506394
\(419\) −2.16975e9 −1.44099 −0.720495 0.693460i \(-0.756085\pi\)
−0.720495 + 0.693460i \(0.756085\pi\)
\(420\) 0 0
\(421\) −6.64386e8 −0.433943 −0.216972 0.976178i \(-0.569618\pi\)
−0.216972 + 0.976178i \(0.569618\pi\)
\(422\) 5.76854e8 0.373657
\(423\) 0 0
\(424\) 4.20722e7 0.0268049
\(425\) 8.39293e8 0.530338
\(426\) 0 0
\(427\) −4.16906e8 −0.259144
\(428\) −9.52210e8 −0.587056
\(429\) 0 0
\(430\) 1.31308e9 0.796436
\(431\) 2.35268e9 1.41545 0.707723 0.706490i \(-0.249723\pi\)
0.707723 + 0.706490i \(0.249723\pi\)
\(432\) 0 0
\(433\) 1.41032e8 0.0834852 0.0417426 0.999128i \(-0.486709\pi\)
0.0417426 + 0.999128i \(0.486709\pi\)
\(434\) −5.40470e8 −0.317364
\(435\) 0 0
\(436\) 1.57299e9 0.908915
\(437\) 1.06514e9 0.610552
\(438\) 0 0
\(439\) −2.83909e9 −1.60159 −0.800797 0.598935i \(-0.795591\pi\)
−0.800797 + 0.598935i \(0.795591\pi\)
\(440\) 2.06033e9 1.15306
\(441\) 0 0
\(442\) −4.49827e8 −0.247781
\(443\) 3.44158e9 1.88081 0.940405 0.340058i \(-0.110447\pi\)
0.940405 + 0.340058i \(0.110447\pi\)
\(444\) 0 0
\(445\) 2.29835e9 1.23639
\(446\) −9.61870e8 −0.513386
\(447\) 0 0
\(448\) −3.54751e8 −0.186402
\(449\) −8.59881e8 −0.448308 −0.224154 0.974554i \(-0.571962\pi\)
−0.224154 + 0.974554i \(0.571962\pi\)
\(450\) 0 0
\(451\) 6.18253e7 0.0317357
\(452\) −1.78175e9 −0.907530
\(453\) 0 0
\(454\) 1.19139e9 0.597526
\(455\) −1.15294e9 −0.573807
\(456\) 0 0
\(457\) −1.71007e9 −0.838124 −0.419062 0.907958i \(-0.637641\pi\)
−0.419062 + 0.907958i \(0.637641\pi\)
\(458\) 1.61764e9 0.786781
\(459\) 0 0
\(460\) 1.10665e9 0.530101
\(461\) −2.48943e9 −1.18344 −0.591720 0.806143i \(-0.701551\pi\)
−0.591720 + 0.806143i \(0.701551\pi\)
\(462\) 0 0
\(463\) 2.91198e9 1.36350 0.681749 0.731587i \(-0.261220\pi\)
0.681749 + 0.731587i \(0.261220\pi\)
\(464\) 5.24103e6 0.00243559
\(465\) 0 0
\(466\) 8.75105e8 0.400599
\(467\) −3.22506e9 −1.46531 −0.732654 0.680601i \(-0.761719\pi\)
−0.732654 + 0.680601i \(0.761719\pi\)
\(468\) 0 0
\(469\) 6.87307e8 0.307642
\(470\) −8.71286e7 −0.0387096
\(471\) 0 0
\(472\) 2.97775e8 0.130344
\(473\) −1.32023e9 −0.573636
\(474\) 0 0
\(475\) 4.21267e9 1.80356
\(476\) 1.45614e8 0.0618839
\(477\) 0 0
\(478\) 1.91503e9 0.802006
\(479\) −3.79747e9 −1.57877 −0.789386 0.613897i \(-0.789601\pi\)
−0.789386 + 0.613897i \(0.789601\pi\)
\(480\) 0 0
\(481\) −2.53440e8 −0.103841
\(482\) −1.89230e9 −0.769709
\(483\) 0 0
\(484\) 7.47387e8 0.299631
\(485\) 6.51353e7 0.0259251
\(486\) 0 0
\(487\) 2.70012e9 1.05933 0.529665 0.848207i \(-0.322318\pi\)
0.529665 + 0.848207i \(0.322318\pi\)
\(488\) −2.22460e9 −0.866530
\(489\) 0 0
\(490\) 2.35948e9 0.906006
\(491\) −1.96974e9 −0.750971 −0.375485 0.926828i \(-0.622524\pi\)
−0.375485 + 0.926828i \(0.622524\pi\)
\(492\) 0 0
\(493\) −5.61893e8 −0.211198
\(494\) −2.25782e9 −0.842645
\(495\) 0 0
\(496\) −1.79845e7 −0.00661779
\(497\) −9.12219e8 −0.333313
\(498\) 0 0
\(499\) 6.68650e8 0.240906 0.120453 0.992719i \(-0.461565\pi\)
0.120453 + 0.992719i \(0.461565\pi\)
\(500\) 1.60915e9 0.575707
\(501\) 0 0
\(502\) −8.48197e7 −0.0299250
\(503\) −8.13925e8 −0.285165 −0.142583 0.989783i \(-0.545541\pi\)
−0.142583 + 0.989783i \(0.545541\pi\)
\(504\) 0 0
\(505\) 3.23873e9 1.11906
\(506\) 6.92732e8 0.237705
\(507\) 0 0
\(508\) 3.02617e9 1.02417
\(509\) −3.90482e9 −1.31247 −0.656235 0.754557i \(-0.727852\pi\)
−0.656235 + 0.754557i \(0.727852\pi\)
\(510\) 0 0
\(511\) 5.17960e8 0.171721
\(512\) −2.34907e7 −0.00773483
\(513\) 0 0
\(514\) −2.02537e9 −0.657859
\(515\) −6.77579e8 −0.218592
\(516\) 0 0
\(517\) 8.76033e7 0.0278807
\(518\) −5.10771e7 −0.0161463
\(519\) 0 0
\(520\) −6.15206e9 −1.91871
\(521\) 1.40413e9 0.434984 0.217492 0.976062i \(-0.430212\pi\)
0.217492 + 0.976062i \(0.430212\pi\)
\(522\) 0 0
\(523\) 5.53390e8 0.169151 0.0845756 0.996417i \(-0.473047\pi\)
0.0845756 + 0.996417i \(0.473047\pi\)
\(524\) −1.63007e9 −0.494935
\(525\) 0 0
\(526\) 5.36110e7 0.0160622
\(527\) 1.92813e9 0.573851
\(528\) 0 0
\(529\) −2.42901e9 −0.713402
\(530\) 9.13241e7 0.0266453
\(531\) 0 0
\(532\) 7.30879e8 0.210453
\(533\) −1.84608e8 −0.0528086
\(534\) 0 0
\(535\) −5.42065e9 −1.53043
\(536\) 3.66746e9 1.02870
\(537\) 0 0
\(538\) −1.47655e9 −0.408798
\(539\) −2.37234e9 −0.652554
\(540\) 0 0
\(541\) 2.74177e8 0.0744459 0.0372229 0.999307i \(-0.488149\pi\)
0.0372229 + 0.999307i \(0.488149\pi\)
\(542\) −2.79559e9 −0.754183
\(543\) 0 0
\(544\) 1.25771e9 0.334954
\(545\) 8.95457e9 2.36950
\(546\) 0 0
\(547\) 3.89224e8 0.101682 0.0508410 0.998707i \(-0.483810\pi\)
0.0508410 + 0.998707i \(0.483810\pi\)
\(548\) −1.64558e9 −0.427156
\(549\) 0 0
\(550\) 2.73977e9 0.702174
\(551\) −2.82031e9 −0.718236
\(552\) 0 0
\(553\) −6.16676e8 −0.155067
\(554\) −3.54810e9 −0.886567
\(555\) 0 0
\(556\) 3.84191e9 0.947949
\(557\) 1.89599e9 0.464883 0.232441 0.972610i \(-0.425329\pi\)
0.232441 + 0.972610i \(0.425329\pi\)
\(558\) 0 0
\(559\) 3.94215e9 0.954536
\(560\) −7.73185e6 −0.00186048
\(561\) 0 0
\(562\) 3.05170e9 0.725212
\(563\) 5.24086e9 1.23772 0.618861 0.785501i \(-0.287594\pi\)
0.618861 + 0.785501i \(0.287594\pi\)
\(564\) 0 0
\(565\) −1.01430e10 −2.36589
\(566\) −9.54991e7 −0.0221382
\(567\) 0 0
\(568\) −4.86759e9 −1.11454
\(569\) 6.24592e9 1.42136 0.710678 0.703517i \(-0.248388\pi\)
0.710678 + 0.703517i \(0.248388\pi\)
\(570\) 0 0
\(571\) −4.35962e9 −0.979991 −0.489996 0.871725i \(-0.663002\pi\)
−0.489996 + 0.871725i \(0.663002\pi\)
\(572\) 2.35859e9 0.526946
\(573\) 0 0
\(574\) −3.72049e7 −0.00821124
\(575\) 3.85937e9 0.846602
\(576\) 0 0
\(577\) 1.25822e9 0.272672 0.136336 0.990663i \(-0.456467\pi\)
0.136336 + 0.990663i \(0.456467\pi\)
\(578\) −2.55227e9 −0.549768
\(579\) 0 0
\(580\) −2.93022e9 −0.623595
\(581\) −1.62041e9 −0.342773
\(582\) 0 0
\(583\) −9.18217e7 −0.0191914
\(584\) 2.76383e9 0.574204
\(585\) 0 0
\(586\) 3.67768e9 0.754975
\(587\) 7.19197e9 1.46762 0.733812 0.679353i \(-0.237739\pi\)
0.733812 + 0.679353i \(0.237739\pi\)
\(588\) 0 0
\(589\) 9.67787e9 1.95154
\(590\) 6.46366e8 0.129568
\(591\) 0 0
\(592\) −1.69963e6 −0.000336688 0
\(593\) 8.65682e9 1.70477 0.852387 0.522911i \(-0.175154\pi\)
0.852387 + 0.522911i \(0.175154\pi\)
\(594\) 0 0
\(595\) 8.28935e8 0.161329
\(596\) 8.04291e8 0.155615
\(597\) 0 0
\(598\) −2.06847e9 −0.395543
\(599\) −5.85691e9 −1.11346 −0.556730 0.830693i \(-0.687944\pi\)
−0.556730 + 0.830693i \(0.687944\pi\)
\(600\) 0 0
\(601\) 5.94489e9 1.11708 0.558539 0.829478i \(-0.311362\pi\)
0.558539 + 0.829478i \(0.311362\pi\)
\(602\) 7.94482e8 0.148421
\(603\) 0 0
\(604\) −3.11509e9 −0.575229
\(605\) 4.25465e9 0.781125
\(606\) 0 0
\(607\) 7.31720e9 1.32796 0.663979 0.747751i \(-0.268866\pi\)
0.663979 + 0.747751i \(0.268866\pi\)
\(608\) 6.31285e9 1.13910
\(609\) 0 0
\(610\) −4.82884e9 −0.861368
\(611\) −2.61580e8 −0.0463938
\(612\) 0 0
\(613\) 8.17131e8 0.143278 0.0716391 0.997431i \(-0.477177\pi\)
0.0716391 + 0.997431i \(0.477177\pi\)
\(614\) −3.78511e9 −0.659917
\(615\) 0 0
\(616\) 1.24661e9 0.214881
\(617\) 4.81332e9 0.824985 0.412493 0.910961i \(-0.364658\pi\)
0.412493 + 0.910961i \(0.364658\pi\)
\(618\) 0 0
\(619\) −4.14223e8 −0.0701968 −0.0350984 0.999384i \(-0.511174\pi\)
−0.0350984 + 0.999384i \(0.511174\pi\)
\(620\) 1.00550e10 1.69439
\(621\) 0 0
\(622\) 1.32550e9 0.220858
\(623\) 1.39062e9 0.230410
\(624\) 0 0
\(625\) −4.91719e8 −0.0805632
\(626\) 4.04462e9 0.658973
\(627\) 0 0
\(628\) −3.90560e9 −0.629259
\(629\) 1.82218e8 0.0291953
\(630\) 0 0
\(631\) −2.20305e9 −0.349078 −0.174539 0.984650i \(-0.555843\pi\)
−0.174539 + 0.984650i \(0.555843\pi\)
\(632\) −3.29058e9 −0.518516
\(633\) 0 0
\(634\) −8.31469e6 −0.00129579
\(635\) 1.72271e10 2.66996
\(636\) 0 0
\(637\) 7.08370e9 1.08586
\(638\) −1.83423e9 −0.279629
\(639\) 0 0
\(640\) 6.53334e9 0.985157
\(641\) −4.40972e9 −0.661315 −0.330657 0.943751i \(-0.607270\pi\)
−0.330657 + 0.943751i \(0.607270\pi\)
\(642\) 0 0
\(643\) 8.75195e9 1.29827 0.649137 0.760671i \(-0.275130\pi\)
0.649137 + 0.760671i \(0.275130\pi\)
\(644\) 6.69584e8 0.0987881
\(645\) 0 0
\(646\) 1.62332e9 0.236914
\(647\) −4.86345e9 −0.705959 −0.352979 0.935631i \(-0.614831\pi\)
−0.352979 + 0.935631i \(0.614831\pi\)
\(648\) 0 0
\(649\) −6.49888e8 −0.0933215
\(650\) −8.18084e9 −1.16843
\(651\) 0 0
\(652\) 5.86208e9 0.828296
\(653\) 9.35709e9 1.31506 0.657529 0.753429i \(-0.271602\pi\)
0.657529 + 0.753429i \(0.271602\pi\)
\(654\) 0 0
\(655\) −9.27953e9 −1.29027
\(656\) −1.23802e6 −0.000171224 0
\(657\) 0 0
\(658\) −5.27175e7 −0.00721380
\(659\) −5.87424e9 −0.799562 −0.399781 0.916611i \(-0.630914\pi\)
−0.399781 + 0.916611i \(0.630914\pi\)
\(660\) 0 0
\(661\) 4.45812e8 0.0600408 0.0300204 0.999549i \(-0.490443\pi\)
0.0300204 + 0.999549i \(0.490443\pi\)
\(662\) −5.66514e9 −0.758940
\(663\) 0 0
\(664\) −8.64647e9 −1.14617
\(665\) 4.16068e9 0.548641
\(666\) 0 0
\(667\) −2.58379e9 −0.337145
\(668\) 2.29024e9 0.297279
\(669\) 0 0
\(670\) 7.96078e9 1.02257
\(671\) 4.85515e9 0.620404
\(672\) 0 0
\(673\) 2.90168e9 0.366942 0.183471 0.983025i \(-0.441267\pi\)
0.183471 + 0.983025i \(0.441267\pi\)
\(674\) 6.20460e9 0.780556
\(675\) 0 0
\(676\) −2.09261e9 −0.260540
\(677\) −8.77680e9 −1.08712 −0.543558 0.839372i \(-0.682923\pi\)
−0.543558 + 0.839372i \(0.682923\pi\)
\(678\) 0 0
\(679\) 3.94103e7 0.00483132
\(680\) 4.42319e9 0.539454
\(681\) 0 0
\(682\) 6.29415e9 0.759787
\(683\) −6.96062e9 −0.835940 −0.417970 0.908461i \(-0.637258\pi\)
−0.417970 + 0.908461i \(0.637258\pi\)
\(684\) 0 0
\(685\) −9.36780e9 −1.11358
\(686\) 2.99582e9 0.354308
\(687\) 0 0
\(688\) 2.64370e7 0.00309494
\(689\) 2.74176e8 0.0319346
\(690\) 0 0
\(691\) −5.57378e8 −0.0642653 −0.0321326 0.999484i \(-0.510230\pi\)
−0.0321326 + 0.999484i \(0.510230\pi\)
\(692\) 4.26110e9 0.488822
\(693\) 0 0
\(694\) −5.26553e9 −0.597976
\(695\) 2.18708e10 2.47126
\(696\) 0 0
\(697\) 1.32729e8 0.0148474
\(698\) 3.54929e9 0.395046
\(699\) 0 0
\(700\) 2.64822e9 0.291817
\(701\) −1.27904e10 −1.40240 −0.701201 0.712963i \(-0.747353\pi\)
−0.701201 + 0.712963i \(0.747353\pi\)
\(702\) 0 0
\(703\) 9.14607e8 0.0992867
\(704\) 4.13132e9 0.446257
\(705\) 0 0
\(706\) −5.12295e9 −0.547903
\(707\) 1.95960e9 0.208545
\(708\) 0 0
\(709\) −1.68769e10 −1.77841 −0.889205 0.457508i \(-0.848742\pi\)
−0.889205 + 0.457508i \(0.848742\pi\)
\(710\) −1.05659e10 −1.10790
\(711\) 0 0
\(712\) 7.42035e9 0.770450
\(713\) 8.86624e9 0.916065
\(714\) 0 0
\(715\) 1.34267e10 1.37372
\(716\) −4.42806e9 −0.450836
\(717\) 0 0
\(718\) −7.68933e9 −0.775270
\(719\) −1.65145e10 −1.65697 −0.828484 0.560013i \(-0.810796\pi\)
−0.828484 + 0.560013i \(0.810796\pi\)
\(720\) 0 0
\(721\) −4.09972e8 −0.0407362
\(722\) 1.88361e9 0.186256
\(723\) 0 0
\(724\) −3.27396e9 −0.320618
\(725\) −1.02190e10 −0.995917
\(726\) 0 0
\(727\) 1.21987e10 1.17745 0.588727 0.808332i \(-0.299629\pi\)
0.588727 + 0.808332i \(0.299629\pi\)
\(728\) −3.72233e9 −0.357565
\(729\) 0 0
\(730\) 5.99932e9 0.570784
\(731\) −2.83432e9 −0.268372
\(732\) 0 0
\(733\) −9.19756e9 −0.862598 −0.431299 0.902209i \(-0.641945\pi\)
−0.431299 + 0.902209i \(0.641945\pi\)
\(734\) 1.81381e9 0.169299
\(735\) 0 0
\(736\) 5.78343e9 0.534703
\(737\) −8.00416e9 −0.736511
\(738\) 0 0
\(739\) −1.94615e10 −1.77386 −0.886932 0.461900i \(-0.847168\pi\)
−0.886932 + 0.461900i \(0.847168\pi\)
\(740\) 9.50250e8 0.0862039
\(741\) 0 0
\(742\) 5.52560e7 0.00496553
\(743\) −1.39586e10 −1.24848 −0.624238 0.781235i \(-0.714590\pi\)
−0.624238 + 0.781235i \(0.714590\pi\)
\(744\) 0 0
\(745\) 4.57859e9 0.405681
\(746\) 9.65786e9 0.851717
\(747\) 0 0
\(748\) −1.69577e9 −0.148153
\(749\) −3.27979e9 −0.285206
\(750\) 0 0
\(751\) −9.03865e9 −0.778688 −0.389344 0.921092i \(-0.627298\pi\)
−0.389344 + 0.921092i \(0.627298\pi\)
\(752\) −1.75421e6 −0.000150425 0
\(753\) 0 0
\(754\) 5.47694e9 0.465305
\(755\) −1.77333e10 −1.49960
\(756\) 0 0
\(757\) 4.80250e9 0.402376 0.201188 0.979553i \(-0.435520\pi\)
0.201188 + 0.979553i \(0.435520\pi\)
\(758\) −1.09943e9 −0.0916908
\(759\) 0 0
\(760\) 2.22013e10 1.83456
\(761\) 1.41275e10 1.16203 0.581015 0.813893i \(-0.302656\pi\)
0.581015 + 0.813893i \(0.302656\pi\)
\(762\) 0 0
\(763\) 5.41800e9 0.441573
\(764\) −4.45798e9 −0.361669
\(765\) 0 0
\(766\) 6.19135e9 0.497719
\(767\) 1.94054e9 0.155288
\(768\) 0 0
\(769\) 1.40261e10 1.11223 0.556113 0.831106i \(-0.312292\pi\)
0.556113 + 0.831106i \(0.312292\pi\)
\(770\) 2.70596e9 0.213601
\(771\) 0 0
\(772\) −1.25524e10 −0.981899
\(773\) −2.02962e8 −0.0158047 −0.00790236 0.999969i \(-0.502515\pi\)
−0.00790236 + 0.999969i \(0.502515\pi\)
\(774\) 0 0
\(775\) 3.50662e10 2.70603
\(776\) 2.10293e8 0.0161551
\(777\) 0 0
\(778\) −6.67880e9 −0.508475
\(779\) 6.66206e8 0.0504926
\(780\) 0 0
\(781\) 1.06234e10 0.797969
\(782\) 1.48718e9 0.111209
\(783\) 0 0
\(784\) 4.75049e7 0.00352073
\(785\) −2.22335e10 −1.64045
\(786\) 0 0
\(787\) 8.88769e9 0.649946 0.324973 0.945723i \(-0.394645\pi\)
0.324973 + 0.945723i \(0.394645\pi\)
\(788\) 5.10178e9 0.371432
\(789\) 0 0
\(790\) −7.14270e9 −0.515427
\(791\) −6.13703e9 −0.440901
\(792\) 0 0
\(793\) −1.44973e10 −1.03236
\(794\) −1.61809e10 −1.14718
\(795\) 0 0
\(796\) 1.49012e10 1.04719
\(797\) 9.59859e9 0.671589 0.335794 0.941935i \(-0.390995\pi\)
0.335794 + 0.941935i \(0.390995\pi\)
\(798\) 0 0
\(799\) 1.88070e8 0.0130438
\(800\) 2.28736e10 1.57950
\(801\) 0 0
\(802\) −8.10833e9 −0.555036
\(803\) −6.03200e9 −0.411109
\(804\) 0 0
\(805\) 3.81174e9 0.257536
\(806\) −1.87940e10 −1.26429
\(807\) 0 0
\(808\) 1.04564e10 0.697338
\(809\) −1.06933e10 −0.710054 −0.355027 0.934856i \(-0.615528\pi\)
−0.355027 + 0.934856i \(0.615528\pi\)
\(810\) 0 0
\(811\) −1.87262e10 −1.23275 −0.616376 0.787452i \(-0.711400\pi\)
−0.616376 + 0.787452i \(0.711400\pi\)
\(812\) −1.77294e9 −0.116211
\(813\) 0 0
\(814\) 5.94828e8 0.0386551
\(815\) 3.33711e10 2.15933
\(816\) 0 0
\(817\) −1.42263e10 −0.912673
\(818\) 1.36196e10 0.870017
\(819\) 0 0
\(820\) 6.92168e8 0.0438393
\(821\) 1.83953e10 1.16013 0.580065 0.814570i \(-0.303027\pi\)
0.580065 + 0.814570i \(0.303027\pi\)
\(822\) 0 0
\(823\) −6.29971e9 −0.393932 −0.196966 0.980410i \(-0.563109\pi\)
−0.196966 + 0.980410i \(0.563109\pi\)
\(824\) −2.18760e9 −0.136215
\(825\) 0 0
\(826\) 3.91086e8 0.0241458
\(827\) 1.00387e10 0.617175 0.308587 0.951196i \(-0.400144\pi\)
0.308587 + 0.951196i \(0.400144\pi\)
\(828\) 0 0
\(829\) 2.76027e8 0.0168271 0.00841357 0.999965i \(-0.497322\pi\)
0.00841357 + 0.999965i \(0.497322\pi\)
\(830\) −1.87685e10 −1.13935
\(831\) 0 0
\(832\) −1.23359e10 −0.742576
\(833\) −5.09302e9 −0.305294
\(834\) 0 0
\(835\) 1.30377e10 0.774992
\(836\) −8.51159e9 −0.503835
\(837\) 0 0
\(838\) −1.52058e10 −0.892597
\(839\) −2.00764e10 −1.17360 −0.586799 0.809733i \(-0.699612\pi\)
−0.586799 + 0.809733i \(0.699612\pi\)
\(840\) 0 0
\(841\) −1.04085e10 −0.603393
\(842\) −4.65607e9 −0.268799
\(843\) 0 0
\(844\) −6.49339e9 −0.371769
\(845\) −1.19126e10 −0.679216
\(846\) 0 0
\(847\) 2.57429e9 0.145568
\(848\) 1.83868e6 0.000103543 0
\(849\) 0 0
\(850\) 5.88183e9 0.328508
\(851\) 8.37904e8 0.0466059
\(852\) 0 0
\(853\) 7.43953e9 0.410415 0.205208 0.978718i \(-0.434213\pi\)
0.205208 + 0.978718i \(0.434213\pi\)
\(854\) −2.92171e9 −0.160522
\(855\) 0 0
\(856\) −1.75009e10 −0.953679
\(857\) 2.02954e10 1.10145 0.550726 0.834686i \(-0.314351\pi\)
0.550726 + 0.834686i \(0.314351\pi\)
\(858\) 0 0
\(859\) 1.59155e10 0.856728 0.428364 0.903606i \(-0.359090\pi\)
0.428364 + 0.903606i \(0.359090\pi\)
\(860\) −1.47807e10 −0.792412
\(861\) 0 0
\(862\) 1.64878e10 0.876774
\(863\) −8.64458e9 −0.457832 −0.228916 0.973446i \(-0.573518\pi\)
−0.228916 + 0.973446i \(0.573518\pi\)
\(864\) 0 0
\(865\) 2.42572e10 1.27434
\(866\) 9.88363e8 0.0517135
\(867\) 0 0
\(868\) 6.08383e9 0.315761
\(869\) 7.18161e9 0.371238
\(870\) 0 0
\(871\) 2.39001e10 1.22556
\(872\) 2.89104e10 1.47654
\(873\) 0 0
\(874\) 7.46461e9 0.378196
\(875\) 5.54254e9 0.279692
\(876\) 0 0
\(877\) −3.61467e10 −1.80955 −0.904775 0.425890i \(-0.859961\pi\)
−0.904775 + 0.425890i \(0.859961\pi\)
\(878\) −1.98965e10 −0.992080
\(879\) 0 0
\(880\) 9.00427e7 0.00445409
\(881\) 2.55814e10 1.26040 0.630199 0.776433i \(-0.282973\pi\)
0.630199 + 0.776433i \(0.282973\pi\)
\(882\) 0 0
\(883\) −3.64985e10 −1.78407 −0.892036 0.451964i \(-0.850724\pi\)
−0.892036 + 0.451964i \(0.850724\pi\)
\(884\) 5.06349e9 0.246529
\(885\) 0 0
\(886\) 2.41189e10 1.16503
\(887\) −1.11780e10 −0.537812 −0.268906 0.963166i \(-0.586662\pi\)
−0.268906 + 0.963166i \(0.586662\pi\)
\(888\) 0 0
\(889\) 1.04233e10 0.497565
\(890\) 1.61070e10 0.765860
\(891\) 0 0
\(892\) 1.08273e10 0.510792
\(893\) 9.43980e8 0.0443591
\(894\) 0 0
\(895\) −2.52077e10 −1.17531
\(896\) 3.95302e9 0.183591
\(897\) 0 0
\(898\) −6.02612e9 −0.277697
\(899\) −2.34763e10 −1.07763
\(900\) 0 0
\(901\) −1.97126e8 −0.00897857
\(902\) 4.33277e8 0.0196582
\(903\) 0 0
\(904\) −3.27471e10 −1.47429
\(905\) −1.86377e10 −0.835837
\(906\) 0 0
\(907\) −3.45269e10 −1.53650 −0.768250 0.640150i \(-0.778872\pi\)
−0.768250 + 0.640150i \(0.778872\pi\)
\(908\) −1.34109e10 −0.594507
\(909\) 0 0
\(910\) −8.07988e9 −0.355435
\(911\) −1.85678e10 −0.813665 −0.406832 0.913503i \(-0.633367\pi\)
−0.406832 + 0.913503i \(0.633367\pi\)
\(912\) 0 0
\(913\) 1.88707e10 0.820618
\(914\) −1.19843e10 −0.519161
\(915\) 0 0
\(916\) −1.82091e10 −0.782806
\(917\) −5.61462e9 −0.240451
\(918\) 0 0
\(919\) −3.89221e10 −1.65422 −0.827108 0.562044i \(-0.810015\pi\)
−0.827108 + 0.562044i \(0.810015\pi\)
\(920\) 2.03394e10 0.861155
\(921\) 0 0
\(922\) −1.74461e10 −0.733062
\(923\) −3.17211e10 −1.32783
\(924\) 0 0
\(925\) 3.31393e9 0.137673
\(926\) 2.04074e10 0.844595
\(927\) 0 0
\(928\) −1.53135e10 −0.629009
\(929\) 7.85450e9 0.321413 0.160707 0.987002i \(-0.448623\pi\)
0.160707 + 0.987002i \(0.448623\pi\)
\(930\) 0 0
\(931\) −2.55634e10 −1.03823
\(932\) −9.85066e9 −0.398575
\(933\) 0 0
\(934\) −2.26015e10 −0.907660
\(935\) −9.65352e9 −0.386229
\(936\) 0 0
\(937\) 1.64761e10 0.654283 0.327142 0.944975i \(-0.393915\pi\)
0.327142 + 0.944975i \(0.393915\pi\)
\(938\) 4.81670e9 0.190563
\(939\) 0 0
\(940\) 9.80767e8 0.0385140
\(941\) −3.54317e10 −1.38621 −0.693105 0.720837i \(-0.743758\pi\)
−0.693105 + 0.720837i \(0.743758\pi\)
\(942\) 0 0
\(943\) 6.10335e8 0.0237016
\(944\) 1.30137e7 0.000503498 0
\(945\) 0 0
\(946\) −9.25229e9 −0.355329
\(947\) 1.07477e10 0.411237 0.205618 0.978632i \(-0.434079\pi\)
0.205618 + 0.978632i \(0.434079\pi\)
\(948\) 0 0
\(949\) 1.80113e10 0.684090
\(950\) 2.95227e10 1.11718
\(951\) 0 0
\(952\) 2.67627e9 0.100531
\(953\) 4.52936e10 1.69516 0.847581 0.530666i \(-0.178058\pi\)
0.847581 + 0.530666i \(0.178058\pi\)
\(954\) 0 0
\(955\) −2.53780e10 −0.942856
\(956\) −2.15566e10 −0.797953
\(957\) 0 0
\(958\) −2.66129e10 −0.977943
\(959\) −5.66802e9 −0.207523
\(960\) 0 0
\(961\) 5.30459e10 1.92806
\(962\) −1.77613e9 −0.0643224
\(963\) 0 0
\(964\) 2.13008e10 0.765820
\(965\) −7.14572e10 −2.55977
\(966\) 0 0
\(967\) −4.08323e10 −1.45215 −0.726074 0.687617i \(-0.758657\pi\)
−0.726074 + 0.687617i \(0.758657\pi\)
\(968\) 1.37364e10 0.486754
\(969\) 0 0
\(970\) 4.56473e8 0.0160588
\(971\) −3.91677e10 −1.37297 −0.686484 0.727145i \(-0.740847\pi\)
−0.686484 + 0.727145i \(0.740847\pi\)
\(972\) 0 0
\(973\) 1.32330e10 0.460537
\(974\) 1.89226e10 0.656183
\(975\) 0 0
\(976\) −9.72220e7 −0.00334726
\(977\) −4.38079e10 −1.50287 −0.751435 0.659808i \(-0.770638\pi\)
−0.751435 + 0.659808i \(0.770638\pi\)
\(978\) 0 0
\(979\) −1.61948e10 −0.551614
\(980\) −2.65596e10 −0.901428
\(981\) 0 0
\(982\) −1.38041e10 −0.465176
\(983\) −3.95635e10 −1.32849 −0.664244 0.747516i \(-0.731246\pi\)
−0.664244 + 0.747516i \(0.731246\pi\)
\(984\) 0 0
\(985\) 2.90429e10 0.968308
\(986\) −3.93779e9 −0.130823
\(987\) 0 0
\(988\) 2.54152e10 0.838387
\(989\) −1.30332e10 −0.428415
\(990\) 0 0
\(991\) −6.12206e10 −1.99820 −0.999102 0.0423753i \(-0.986507\pi\)
−0.999102 + 0.0423753i \(0.986507\pi\)
\(992\) 5.25481e10 1.70910
\(993\) 0 0
\(994\) −6.39291e9 −0.206465
\(995\) 8.48283e10 2.72998
\(996\) 0 0
\(997\) −2.27542e10 −0.727156 −0.363578 0.931564i \(-0.618445\pi\)
−0.363578 + 0.931564i \(0.618445\pi\)
\(998\) 4.68595e9 0.149225
\(999\) 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.1.11 16
3.2 odd 2 177.8.a.a.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.6 16 3.2 odd 2
531.8.a.b.1.11 16 1.1 even 1 trivial