Properties

Label 177.8.a.a.1.6
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(7.00808\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.00808 q^{2} -27.0000 q^{3} -78.8868 q^{4} +449.079 q^{5} +189.218 q^{6} -271.717 q^{7} +1449.88 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-7.00808 q^{2} -27.0000 q^{3} -78.8868 q^{4} +449.079 q^{5} +189.218 q^{6} -271.717 q^{7} +1449.88 q^{8} +729.000 q^{9} -3147.18 q^{10} -3164.33 q^{11} +2129.94 q^{12} -9448.56 q^{13} +1904.22 q^{14} -12125.1 q^{15} -63.3641 q^{16} -6793.30 q^{17} -5108.89 q^{18} +34097.6 q^{19} -35426.4 q^{20} +7336.36 q^{21} +22175.9 q^{22} -31238.0 q^{23} -39146.7 q^{24} +123547. q^{25} +66216.3 q^{26} -19683.0 q^{27} +21434.9 q^{28} +82712.9 q^{29} +84974.0 q^{30} +283828. q^{31} -185141. q^{32} +85437.0 q^{33} +47608.0 q^{34} -122023. q^{35} -57508.5 q^{36} +26823.2 q^{37} -238959. q^{38} +255111. q^{39} +651111. q^{40} -19538.2 q^{41} -51413.8 q^{42} -417223. q^{43} +249624. q^{44} +327379. q^{45} +218919. q^{46} -27684.6 q^{47} +1710.83 q^{48} -749713. q^{49} -865829. q^{50} +183419. q^{51} +745366. q^{52} +29017.7 q^{53} +137940. q^{54} -1.42104e6 q^{55} -393957. q^{56} -920636. q^{57} -579659. q^{58} +205379. q^{59} +956514. q^{60} +1.53434e6 q^{61} -1.98909e6 q^{62} -198082. q^{63} +1.30559e6 q^{64} -4.24315e6 q^{65} -598749. q^{66} -2.52949e6 q^{67} +535901. q^{68} +843427. q^{69} +855144. q^{70} -3.35724e6 q^{71} +1.05696e6 q^{72} -1.90625e6 q^{73} -187979. q^{74} -3.33578e6 q^{75} -2.68985e6 q^{76} +859803. q^{77} -1.78784e6 q^{78} +2.26955e6 q^{79} -28455.5 q^{80} +531441. q^{81} +136925. q^{82} -5.96358e6 q^{83} -578742. q^{84} -3.05073e6 q^{85} +2.92393e6 q^{86} -2.23325e6 q^{87} -4.58790e6 q^{88} +5.11791e6 q^{89} -2.29430e6 q^{90} +2.56733e6 q^{91} +2.46427e6 q^{92} -7.66336e6 q^{93} +194016. q^{94} +1.53125e7 q^{95} +4.99879e6 q^{96} -145042. q^{97} +5.25405e6 q^{98} -2.30680e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.00808 −0.619433 −0.309716 0.950829i \(-0.600234\pi\)
−0.309716 + 0.950829i \(0.600234\pi\)
\(3\) −27.0000 −0.577350
\(4\) −78.8868 −0.616303
\(5\) 449.079 1.60668 0.803338 0.595524i \(-0.203056\pi\)
0.803338 + 0.595524i \(0.203056\pi\)
\(6\) 189.218 0.357630
\(7\) −271.717 −0.299415 −0.149708 0.988730i \(-0.547833\pi\)
−0.149708 + 0.988730i \(0.547833\pi\)
\(8\) 1449.88 1.00119
\(9\) 729.000 0.333333
\(10\) −3147.18 −0.995227
\(11\) −3164.33 −0.716816 −0.358408 0.933565i \(-0.616680\pi\)
−0.358408 + 0.933565i \(0.616680\pi\)
\(12\) 2129.94 0.355823
\(13\) −9448.56 −1.19279 −0.596395 0.802691i \(-0.703401\pi\)
−0.596395 + 0.802691i \(0.703401\pi\)
\(14\) 1904.22 0.185468
\(15\) −12125.1 −0.927614
\(16\) −63.3641 −0.00386744
\(17\) −6793.30 −0.335358 −0.167679 0.985842i \(-0.553627\pi\)
−0.167679 + 0.985842i \(0.553627\pi\)
\(18\) −5108.89 −0.206478
\(19\) 34097.6 1.14048 0.570239 0.821479i \(-0.306851\pi\)
0.570239 + 0.821479i \(0.306851\pi\)
\(20\) −35426.4 −0.990199
\(21\) 7336.36 0.172867
\(22\) 22175.9 0.444019
\(23\) −31238.0 −0.535348 −0.267674 0.963510i \(-0.586255\pi\)
−0.267674 + 0.963510i \(0.586255\pi\)
\(24\) −39146.7 −0.578038
\(25\) 123547. 1.58141
\(26\) 66216.3 0.738853
\(27\) −19683.0 −0.192450
\(28\) 21434.9 0.184531
\(29\) 82712.9 0.629767 0.314884 0.949130i \(-0.398034\pi\)
0.314884 + 0.949130i \(0.398034\pi\)
\(30\) 84974.0 0.574595
\(31\) 283828. 1.71116 0.855578 0.517673i \(-0.173202\pi\)
0.855578 + 0.517673i \(0.173202\pi\)
\(32\) −185141. −0.998795
\(33\) 85437.0 0.413854
\(34\) 47608.0 0.207732
\(35\) −122023. −0.481063
\(36\) −57508.5 −0.205434
\(37\) 26823.2 0.0870571 0.0435286 0.999052i \(-0.486140\pi\)
0.0435286 + 0.999052i \(0.486140\pi\)
\(38\) −238959. −0.706449
\(39\) 255111. 0.688657
\(40\) 651111. 1.60859
\(41\) −19538.2 −0.0442732 −0.0221366 0.999755i \(-0.507047\pi\)
−0.0221366 + 0.999755i \(0.507047\pi\)
\(42\) −51413.8 −0.107080
\(43\) −417223. −0.800255 −0.400128 0.916459i \(-0.631034\pi\)
−0.400128 + 0.916459i \(0.631034\pi\)
\(44\) 249624. 0.441776
\(45\) 327379. 0.535558
\(46\) 218919. 0.331612
\(47\) −27684.6 −0.0388952 −0.0194476 0.999811i \(-0.506191\pi\)
−0.0194476 + 0.999811i \(0.506191\pi\)
\(48\) 1710.83 0.00223287
\(49\) −749713. −0.910351
\(50\) −865829. −0.979574
\(51\) 183419. 0.193619
\(52\) 745366. 0.735120
\(53\) 29017.7 0.0267730 0.0133865 0.999910i \(-0.495739\pi\)
0.0133865 + 0.999910i \(0.495739\pi\)
\(54\) 137940. 0.119210
\(55\) −1.42104e6 −1.15169
\(56\) −393957. −0.299772
\(57\) −920636. −0.658455
\(58\) −579659. −0.390099
\(59\) 205379. 0.130189
\(60\) 956514. 0.571692
\(61\) 1.53434e6 0.865499 0.432749 0.901514i \(-0.357543\pi\)
0.432749 + 0.901514i \(0.357543\pi\)
\(62\) −1.98909e6 −1.05995
\(63\) −198082. −0.0998051
\(64\) 1.30559e6 0.622554
\(65\) −4.24315e6 −1.91642
\(66\) −598749. −0.256355
\(67\) −2.52949e6 −1.02748 −0.513738 0.857947i \(-0.671740\pi\)
−0.513738 + 0.857947i \(0.671740\pi\)
\(68\) 535901. 0.206682
\(69\) 843427. 0.309083
\(70\) 855144. 0.297986
\(71\) −3.35724e6 −1.11321 −0.556606 0.830776i \(-0.687897\pi\)
−0.556606 + 0.830776i \(0.687897\pi\)
\(72\) 1.05696e6 0.333730
\(73\) −1.90625e6 −0.573521 −0.286761 0.958002i \(-0.592578\pi\)
−0.286761 + 0.958002i \(0.592578\pi\)
\(74\) −187979. −0.0539260
\(75\) −3.33578e6 −0.913025
\(76\) −2.68985e6 −0.702880
\(77\) 859803. 0.214626
\(78\) −1.78784e6 −0.426577
\(79\) 2.26955e6 0.517899 0.258950 0.965891i \(-0.416624\pi\)
0.258950 + 0.965891i \(0.416624\pi\)
\(80\) −28455.5 −0.00621372
\(81\) 531441. 0.111111
\(82\) 136925. 0.0274243
\(83\) −5.96358e6 −1.14481 −0.572405 0.819971i \(-0.693989\pi\)
−0.572405 + 0.819971i \(0.693989\pi\)
\(84\) −578742. −0.106539
\(85\) −3.05073e6 −0.538812
\(86\) 2.92393e6 0.495704
\(87\) −2.23325e6 −0.363596
\(88\) −4.58790e6 −0.717670
\(89\) 5.11791e6 0.769533 0.384767 0.923014i \(-0.374282\pi\)
0.384767 + 0.923014i \(0.374282\pi\)
\(90\) −2.29430e6 −0.331742
\(91\) 2.56733e6 0.357139
\(92\) 2.46427e6 0.329937
\(93\) −7.66336e6 −0.987937
\(94\) 194016. 0.0240930
\(95\) 1.53125e7 1.83238
\(96\) 4.99879e6 0.576655
\(97\) −145042. −0.0161359 −0.00806793 0.999967i \(-0.502568\pi\)
−0.00806793 + 0.999967i \(0.502568\pi\)
\(98\) 5.25405e6 0.563901
\(99\) −2.30680e6 −0.238939
\(100\) −9.74625e6 −0.974625
\(101\) 7.21193e6 0.696509 0.348254 0.937400i \(-0.386775\pi\)
0.348254 + 0.937400i \(0.386775\pi\)
\(102\) −1.28542e6 −0.119934
\(103\) 1.50882e6 0.136052 0.0680262 0.997684i \(-0.478330\pi\)
0.0680262 + 0.997684i \(0.478330\pi\)
\(104\) −1.36993e7 −1.19421
\(105\) 3.29461e6 0.277742
\(106\) −203359. −0.0165841
\(107\) −1.20706e7 −0.952545 −0.476272 0.879298i \(-0.658012\pi\)
−0.476272 + 0.879298i \(0.658012\pi\)
\(108\) 1.55273e6 0.118608
\(109\) −1.99398e7 −1.47479 −0.737393 0.675464i \(-0.763943\pi\)
−0.737393 + 0.675464i \(0.763943\pi\)
\(110\) 9.95874e6 0.713395
\(111\) −724226. −0.0502625
\(112\) 17217.1 0.00115797
\(113\) −2.25861e7 −1.47254 −0.736269 0.676688i \(-0.763414\pi\)
−0.736269 + 0.676688i \(0.763414\pi\)
\(114\) 6.45189e6 0.407869
\(115\) −1.40284e7 −0.860130
\(116\) −6.52495e6 −0.388128
\(117\) −6.88800e6 −0.397596
\(118\) −1.43931e6 −0.0806433
\(119\) 1.84585e6 0.100411
\(120\) −1.75800e7 −0.928719
\(121\) −9.47417e6 −0.486175
\(122\) −1.07528e7 −0.536118
\(123\) 527531. 0.0255611
\(124\) −2.23903e7 −1.05459
\(125\) 2.03982e7 0.934129
\(126\) 1.38817e6 0.0618225
\(127\) −3.83609e7 −1.66179 −0.830895 0.556429i \(-0.812171\pi\)
−0.830895 + 0.556429i \(0.812171\pi\)
\(128\) 1.45483e7 0.613165
\(129\) 1.12650e7 0.462027
\(130\) 2.97364e7 1.18710
\(131\) −2.06635e7 −0.803070 −0.401535 0.915844i \(-0.631523\pi\)
−0.401535 + 0.915844i \(0.631523\pi\)
\(132\) −6.73985e6 −0.255059
\(133\) −9.26491e6 −0.341476
\(134\) 1.77269e7 0.636452
\(135\) −8.83923e6 −0.309205
\(136\) −9.84946e6 −0.335758
\(137\) −2.08600e7 −0.693095 −0.346547 0.938033i \(-0.612646\pi\)
−0.346547 + 0.938033i \(0.612646\pi\)
\(138\) −5.91080e6 −0.191456
\(139\) −4.87015e7 −1.53812 −0.769061 0.639176i \(-0.779276\pi\)
−0.769061 + 0.639176i \(0.779276\pi\)
\(140\) 9.62597e6 0.296481
\(141\) 747485. 0.0224562
\(142\) 2.35278e7 0.689560
\(143\) 2.98984e7 0.855011
\(144\) −46192.4 −0.00128915
\(145\) 3.71447e7 1.01183
\(146\) 1.33591e7 0.355258
\(147\) 2.02422e7 0.525591
\(148\) −2.11600e6 −0.0536536
\(149\) 1.01955e7 0.252497 0.126249 0.991999i \(-0.459706\pi\)
0.126249 + 0.991999i \(0.459706\pi\)
\(150\) 2.33774e7 0.565557
\(151\) 3.94880e7 0.933354 0.466677 0.884428i \(-0.345451\pi\)
0.466677 + 0.884428i \(0.345451\pi\)
\(152\) 4.94375e7 1.14184
\(153\) −4.95231e6 −0.111786
\(154\) −6.02557e6 −0.132946
\(155\) 1.27461e8 2.74927
\(156\) −2.01249e7 −0.424422
\(157\) 4.95090e7 1.02102 0.510511 0.859871i \(-0.329456\pi\)
0.510511 + 0.859871i \(0.329456\pi\)
\(158\) −1.59052e7 −0.320804
\(159\) −783479. −0.0154574
\(160\) −8.31428e7 −1.60474
\(161\) 8.48791e6 0.160291
\(162\) −3.72438e6 −0.0688259
\(163\) −7.43101e7 −1.34397 −0.671987 0.740563i \(-0.734559\pi\)
−0.671987 + 0.740563i \(0.734559\pi\)
\(164\) 1.54131e6 0.0272857
\(165\) 3.83680e7 0.664929
\(166\) 4.17932e7 0.709133
\(167\) 2.90320e7 0.482358 0.241179 0.970481i \(-0.422466\pi\)
0.241179 + 0.970481i \(0.422466\pi\)
\(168\) 1.06368e7 0.173073
\(169\) 2.65267e7 0.422746
\(170\) 2.13798e7 0.333758
\(171\) 2.48572e7 0.380159
\(172\) 3.29134e7 0.493200
\(173\) 5.40154e7 0.793152 0.396576 0.918002i \(-0.370198\pi\)
0.396576 + 0.918002i \(0.370198\pi\)
\(174\) 1.56508e7 0.225224
\(175\) −3.35699e7 −0.473497
\(176\) 200505. 0.00277224
\(177\) −5.54523e6 −0.0751646
\(178\) −3.58667e7 −0.476674
\(179\) −5.61319e7 −0.731516 −0.365758 0.930710i \(-0.619190\pi\)
−0.365758 + 0.930710i \(0.619190\pi\)
\(180\) −2.58259e7 −0.330066
\(181\) 4.15020e7 0.520228 0.260114 0.965578i \(-0.416240\pi\)
0.260114 + 0.965578i \(0.416240\pi\)
\(182\) −1.79921e7 −0.221224
\(183\) −4.14271e7 −0.499696
\(184\) −4.52914e7 −0.535986
\(185\) 1.20457e7 0.139873
\(186\) 5.37055e7 0.611960
\(187\) 2.14962e7 0.240390
\(188\) 2.18395e6 0.0239712
\(189\) 5.34821e6 0.0576225
\(190\) −1.07312e8 −1.13503
\(191\) −5.65111e7 −0.586837 −0.293418 0.955984i \(-0.594793\pi\)
−0.293418 + 0.955984i \(0.594793\pi\)
\(192\) −3.52509e7 −0.359432
\(193\) 1.59119e8 1.59321 0.796604 0.604502i \(-0.206628\pi\)
0.796604 + 0.604502i \(0.206628\pi\)
\(194\) 1.01646e6 0.00999508
\(195\) 1.14565e8 1.10645
\(196\) 5.91424e7 0.561052
\(197\) 6.46721e7 0.602678 0.301339 0.953517i \(-0.402566\pi\)
0.301339 + 0.953517i \(0.402566\pi\)
\(198\) 1.61662e7 0.148006
\(199\) −1.88894e8 −1.69915 −0.849576 0.527467i \(-0.823142\pi\)
−0.849576 + 0.527467i \(0.823142\pi\)
\(200\) 1.79129e8 1.58329
\(201\) 6.82963e7 0.593214
\(202\) −5.05418e7 −0.431440
\(203\) −2.24745e7 −0.188562
\(204\) −1.44693e7 −0.119328
\(205\) −8.77420e6 −0.0711326
\(206\) −1.05739e7 −0.0842754
\(207\) −2.27725e7 −0.178449
\(208\) 598700. 0.00461304
\(209\) −1.07896e8 −0.817513
\(210\) −2.30889e7 −0.172042
\(211\) 8.23127e7 0.603224 0.301612 0.953431i \(-0.402475\pi\)
0.301612 + 0.953431i \(0.402475\pi\)
\(212\) −2.28912e6 −0.0165003
\(213\) 9.06455e7 0.642714
\(214\) 8.45917e7 0.590037
\(215\) −1.87366e8 −1.28575
\(216\) −2.85380e7 −0.192679
\(217\) −7.71210e7 −0.512346
\(218\) 1.39740e8 0.913531
\(219\) 5.14687e7 0.331123
\(220\) 1.12101e8 0.709790
\(221\) 6.41868e7 0.400012
\(222\) 5.07544e6 0.0311342
\(223\) −1.37252e8 −0.828801 −0.414400 0.910095i \(-0.636009\pi\)
−0.414400 + 0.910095i \(0.636009\pi\)
\(224\) 5.03058e7 0.299055
\(225\) 9.00660e7 0.527135
\(226\) 1.58285e8 0.912139
\(227\) −1.70002e8 −0.964635 −0.482317 0.875997i \(-0.660205\pi\)
−0.482317 + 0.875997i \(0.660205\pi\)
\(228\) 7.26260e7 0.405808
\(229\) 2.30825e8 1.27016 0.635082 0.772445i \(-0.280966\pi\)
0.635082 + 0.772445i \(0.280966\pi\)
\(230\) 9.83119e7 0.532793
\(231\) −2.32147e7 −0.123914
\(232\) 1.19924e8 0.630518
\(233\) −1.24871e8 −0.646719 −0.323359 0.946276i \(-0.604812\pi\)
−0.323359 + 0.946276i \(0.604812\pi\)
\(234\) 4.82717e7 0.246284
\(235\) −1.24326e7 −0.0624920
\(236\) −1.62017e7 −0.0802358
\(237\) −6.12779e7 −0.299009
\(238\) −1.29359e7 −0.0621981
\(239\) −2.73260e8 −1.29474 −0.647371 0.762175i \(-0.724132\pi\)
−0.647371 + 0.762175i \(0.724132\pi\)
\(240\) 768299. 0.00358749
\(241\) −2.70017e8 −1.24260 −0.621301 0.783572i \(-0.713396\pi\)
−0.621301 + 0.783572i \(0.713396\pi\)
\(242\) 6.63957e7 0.301152
\(243\) −1.43489e7 −0.0641500
\(244\) −1.21039e8 −0.533410
\(245\) −3.36681e8 −1.46264
\(246\) −3.69698e6 −0.0158334
\(247\) −3.22173e8 −1.36035
\(248\) 4.11517e8 1.71319
\(249\) 1.61017e8 0.660956
\(250\) −1.42952e8 −0.578630
\(251\) 1.21031e7 0.0483103 0.0241551 0.999708i \(-0.492310\pi\)
0.0241551 + 0.999708i \(0.492310\pi\)
\(252\) 1.56260e7 0.0615102
\(253\) 9.88475e7 0.383746
\(254\) 2.68836e8 1.02937
\(255\) 8.23697e7 0.311083
\(256\) −2.69071e8 −1.00237
\(257\) 2.89004e8 1.06203 0.531017 0.847361i \(-0.321810\pi\)
0.531017 + 0.847361i \(0.321810\pi\)
\(258\) −7.89461e7 −0.286195
\(259\) −7.28832e6 −0.0260662
\(260\) 3.34729e8 1.18110
\(261\) 6.02977e7 0.209922
\(262\) 1.44811e8 0.497448
\(263\) −7.64989e6 −0.0259304 −0.0129652 0.999916i \(-0.504127\pi\)
−0.0129652 + 0.999916i \(0.504127\pi\)
\(264\) 1.23873e8 0.414347
\(265\) 1.30313e7 0.0430156
\(266\) 6.49293e7 0.211522
\(267\) −1.38183e8 −0.444290
\(268\) 1.99544e8 0.633237
\(269\) 2.10692e8 0.659956 0.329978 0.943989i \(-0.392959\pi\)
0.329978 + 0.943989i \(0.392959\pi\)
\(270\) 6.19460e7 0.191532
\(271\) −3.98910e8 −1.21754 −0.608769 0.793348i \(-0.708336\pi\)
−0.608769 + 0.793348i \(0.708336\pi\)
\(272\) 430451. 0.00129698
\(273\) −6.93180e7 −0.206194
\(274\) 1.46189e8 0.429325
\(275\) −3.90945e8 −1.13358
\(276\) −6.65352e7 −0.190489
\(277\) −5.06287e8 −1.43126 −0.715628 0.698481i \(-0.753859\pi\)
−0.715628 + 0.698481i \(0.753859\pi\)
\(278\) 3.41304e8 0.952763
\(279\) 2.06911e8 0.570386
\(280\) −1.76918e8 −0.481636
\(281\) −4.35455e8 −1.17077 −0.585384 0.810756i \(-0.699056\pi\)
−0.585384 + 0.810756i \(0.699056\pi\)
\(282\) −5.23843e6 −0.0139101
\(283\) −1.36270e7 −0.0357394 −0.0178697 0.999840i \(-0.505688\pi\)
−0.0178697 + 0.999840i \(0.505688\pi\)
\(284\) 2.64842e8 0.686076
\(285\) −4.13439e8 −1.05792
\(286\) −2.09530e8 −0.529622
\(287\) 5.30886e6 0.0132561
\(288\) −1.34967e8 −0.332932
\(289\) −3.64190e8 −0.887535
\(290\) −2.60313e8 −0.626762
\(291\) 3.91613e6 0.00931604
\(292\) 1.50378e8 0.353463
\(293\) −5.24778e8 −1.21882 −0.609409 0.792856i \(-0.708593\pi\)
−0.609409 + 0.792856i \(0.708593\pi\)
\(294\) −1.41859e8 −0.325568
\(295\) 9.22315e7 0.209171
\(296\) 3.88904e7 0.0871608
\(297\) 6.22836e7 0.137951
\(298\) −7.14510e7 −0.156405
\(299\) 2.95154e8 0.638558
\(300\) 2.63149e8 0.562700
\(301\) 1.13367e8 0.239609
\(302\) −2.76735e8 −0.578150
\(303\) −1.94722e8 −0.402129
\(304\) −2.16057e6 −0.00441073
\(305\) 6.89039e8 1.39058
\(306\) 3.47062e7 0.0692440
\(307\) −5.40106e8 −1.06536 −0.532678 0.846318i \(-0.678814\pi\)
−0.532678 + 0.846318i \(0.678814\pi\)
\(308\) −6.78271e7 −0.132274
\(309\) −4.07381e7 −0.0785499
\(310\) −8.93260e8 −1.70299
\(311\) −1.89139e8 −0.356549 −0.178275 0.983981i \(-0.557052\pi\)
−0.178275 + 0.983981i \(0.557052\pi\)
\(312\) 3.69880e8 0.689477
\(313\) 5.77137e8 1.06383 0.531917 0.846797i \(-0.321472\pi\)
0.531917 + 0.846797i \(0.321472\pi\)
\(314\) −3.46963e8 −0.632454
\(315\) −8.89544e7 −0.160354
\(316\) −1.79038e8 −0.319183
\(317\) 1.18644e6 0.00209189 0.00104595 0.999999i \(-0.499667\pi\)
0.00104595 + 0.999999i \(0.499667\pi\)
\(318\) 5.49068e6 0.00957484
\(319\) −2.61731e8 −0.451427
\(320\) 5.86314e8 1.00024
\(321\) 3.25906e8 0.549952
\(322\) −5.94840e7 −0.0992897
\(323\) −2.31635e8 −0.382469
\(324\) −4.19237e7 −0.0684781
\(325\) −1.16734e9 −1.88628
\(326\) 5.20771e8 0.832502
\(327\) 5.38376e8 0.851468
\(328\) −2.83280e7 −0.0443259
\(329\) 7.52238e6 0.0116458
\(330\) −2.68886e8 −0.411879
\(331\) −8.08372e8 −1.22522 −0.612609 0.790386i \(-0.709880\pi\)
−0.612609 + 0.790386i \(0.709880\pi\)
\(332\) 4.70447e8 0.705550
\(333\) 1.95541e7 0.0290190
\(334\) −2.03459e8 −0.298788
\(335\) −1.13594e9 −1.65082
\(336\) −464862. −0.000668554 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
\(339\) 6.09825e8 0.850171
\(340\) 2.40662e8 0.332072
\(341\) −8.98127e8 −1.22658
\(342\) −1.74201e8 −0.235483
\(343\) 4.27481e8 0.571988
\(344\) −6.04923e8 −0.801208
\(345\) 3.78766e8 0.496597
\(346\) −3.78544e8 −0.491304
\(347\) 7.51351e8 0.965361 0.482680 0.875797i \(-0.339663\pi\)
0.482680 + 0.875797i \(0.339663\pi\)
\(348\) 1.76174e8 0.224086
\(349\) 5.06457e8 0.637755 0.318877 0.947796i \(-0.396694\pi\)
0.318877 + 0.947796i \(0.396694\pi\)
\(350\) 2.35261e8 0.293299
\(351\) 1.85976e8 0.229552
\(352\) 5.85846e8 0.715953
\(353\) 7.31006e8 0.884523 0.442262 0.896886i \(-0.354176\pi\)
0.442262 + 0.896886i \(0.354176\pi\)
\(354\) 3.88614e7 0.0465594
\(355\) −1.50767e9 −1.78857
\(356\) −4.03735e8 −0.474266
\(357\) −4.98381e7 −0.0579726
\(358\) 3.93377e8 0.453125
\(359\) 1.09721e9 1.25158 0.625790 0.779992i \(-0.284777\pi\)
0.625790 + 0.779992i \(0.284777\pi\)
\(360\) 4.74660e8 0.536196
\(361\) 2.68777e8 0.300688
\(362\) −2.90849e8 −0.322246
\(363\) 2.55803e8 0.280693
\(364\) −2.02529e8 −0.220106
\(365\) −8.56057e8 −0.921462
\(366\) 2.90325e8 0.309528
\(367\) 2.58816e8 0.273313 0.136657 0.990618i \(-0.456364\pi\)
0.136657 + 0.990618i \(0.456364\pi\)
\(368\) 1.97937e6 0.00207043
\(369\) −1.42433e7 −0.0147577
\(370\) −8.44175e7 −0.0866416
\(371\) −7.88461e6 −0.00801626
\(372\) 6.04538e8 0.608868
\(373\) 1.37810e9 1.37499 0.687497 0.726187i \(-0.258709\pi\)
0.687497 + 0.726187i \(0.258709\pi\)
\(374\) −1.50647e8 −0.148906
\(375\) −5.50751e8 −0.539320
\(376\) −4.01394e7 −0.0389415
\(377\) −7.81517e8 −0.751180
\(378\) −3.74807e7 −0.0356933
\(379\) −1.56880e8 −0.148024 −0.0740119 0.997257i \(-0.523580\pi\)
−0.0740119 + 0.997257i \(0.523580\pi\)
\(380\) −1.20796e9 −1.12930
\(381\) 1.03575e9 0.959435
\(382\) 3.96035e8 0.363506
\(383\) −8.83458e8 −0.803508 −0.401754 0.915748i \(-0.631599\pi\)
−0.401754 + 0.915748i \(0.631599\pi\)
\(384\) −3.92804e8 −0.354011
\(385\) 3.86120e8 0.344834
\(386\) −1.11512e9 −0.986885
\(387\) −3.04155e8 −0.266752
\(388\) 1.14419e7 0.00994458
\(389\) 9.53014e8 0.820873 0.410436 0.911889i \(-0.365376\pi\)
0.410436 + 0.911889i \(0.365376\pi\)
\(390\) −8.02882e8 −0.685370
\(391\) 2.12209e8 0.179534
\(392\) −1.08699e9 −0.911435
\(393\) 5.57913e8 0.463653
\(394\) −4.53227e8 −0.373318
\(395\) 1.01921e9 0.832095
\(396\) 1.81976e8 0.147259
\(397\) −2.30889e9 −1.85198 −0.925992 0.377544i \(-0.876769\pi\)
−0.925992 + 0.377544i \(0.876769\pi\)
\(398\) 1.32378e9 1.05251
\(399\) 2.50153e8 0.197151
\(400\) −7.82846e6 −0.00611599
\(401\) 1.15700e9 0.896039 0.448019 0.894024i \(-0.352130\pi\)
0.448019 + 0.894024i \(0.352130\pi\)
\(402\) −4.78626e8 −0.367456
\(403\) −2.68177e9 −2.04105
\(404\) −5.68926e8 −0.429260
\(405\) 2.38659e8 0.178519
\(406\) 1.57503e8 0.116801
\(407\) −8.48775e7 −0.0624040
\(408\) 2.65935e8 0.193850
\(409\) 1.94341e9 1.40454 0.702269 0.711911i \(-0.252170\pi\)
0.702269 + 0.711911i \(0.252170\pi\)
\(410\) 6.14903e7 0.0440619
\(411\) 5.63220e8 0.400158
\(412\) −1.19026e8 −0.0838496
\(413\) −5.58050e7 −0.0389805
\(414\) 1.59592e8 0.110537
\(415\) −2.67812e9 −1.83934
\(416\) 1.74931e9 1.19135
\(417\) 1.31494e9 0.888035
\(418\) 7.56146e8 0.506394
\(419\) 2.16975e9 1.44099 0.720495 0.693460i \(-0.243915\pi\)
0.720495 + 0.693460i \(0.243915\pi\)
\(420\) −2.59901e8 −0.171173
\(421\) −6.64386e8 −0.433943 −0.216972 0.976178i \(-0.569618\pi\)
−0.216972 + 0.976178i \(0.569618\pi\)
\(422\) −5.76854e8 −0.373657
\(423\) −2.01821e7 −0.0129651
\(424\) 4.20722e7 0.0268049
\(425\) −8.39293e8 −0.530338
\(426\) −6.35251e8 −0.398118
\(427\) −4.16906e8 −0.259144
\(428\) 9.52210e8 0.587056
\(429\) −8.07256e8 −0.493641
\(430\) 1.31308e9 0.796436
\(431\) −2.35268e9 −1.41545 −0.707723 0.706490i \(-0.750277\pi\)
−0.707723 + 0.706490i \(0.750277\pi\)
\(432\) 1.24720e6 0.000744289 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
\(435\) −1.00291e9 −0.584181
\(436\) 1.57299e9 0.908915
\(437\) −1.06514e9 −0.610552
\(438\) −3.60697e8 −0.205108
\(439\) −2.83909e9 −1.60159 −0.800797 0.598935i \(-0.795591\pi\)
−0.800797 + 0.598935i \(0.795591\pi\)
\(440\) −2.06033e9 −1.15306
\(441\) −5.46541e8 −0.303450
\(442\) −4.49827e8 −0.247781
\(443\) −3.44158e9 −1.88081 −0.940405 0.340058i \(-0.889553\pi\)
−0.940405 + 0.340058i \(0.889553\pi\)
\(444\) 5.71319e7 0.0309769
\(445\) 2.29835e9 1.23639
\(446\) 9.61870e8 0.513386
\(447\) −2.75279e8 −0.145779
\(448\) −3.54751e8 −0.186402
\(449\) 8.59881e8 0.448308 0.224154 0.974554i \(-0.428038\pi\)
0.224154 + 0.974554i \(0.428038\pi\)
\(450\) −6.31190e8 −0.326525
\(451\) 6.18253e7 0.0317357
\(452\) 1.78175e9 0.907530
\(453\) −1.06618e9 −0.538872
\(454\) 1.19139e9 0.597526
\(455\) 1.15294e9 0.573807
\(456\) −1.33481e9 −0.659239
\(457\) −1.71007e9 −0.838124 −0.419062 0.907958i \(-0.637641\pi\)
−0.419062 + 0.907958i \(0.637641\pi\)
\(458\) −1.61764e9 −0.786781
\(459\) 1.33712e8 0.0645398
\(460\) 1.10665e9 0.530101
\(461\) 2.48943e9 1.18344 0.591720 0.806143i \(-0.298449\pi\)
0.591720 + 0.806143i \(0.298449\pi\)
\(462\) 1.62690e8 0.0767565
\(463\) 2.91198e9 1.36350 0.681749 0.731587i \(-0.261220\pi\)
0.681749 + 0.731587i \(0.261220\pi\)
\(464\) −5.24103e6 −0.00243559
\(465\) −3.44146e9 −1.58729
\(466\) 8.75105e8 0.400599
\(467\) 3.22506e9 1.46531 0.732654 0.680601i \(-0.238281\pi\)
0.732654 + 0.680601i \(0.238281\pi\)
\(468\) 5.43372e8 0.245040
\(469\) 6.87307e8 0.307642
\(470\) 8.71286e7 0.0387096
\(471\) −1.33674e9 −0.589487
\(472\) 2.97775e8 0.130344
\(473\) 1.32023e9 0.573636
\(474\) 4.29440e8 0.185216
\(475\) 4.21267e9 1.80356
\(476\) −1.45614e8 −0.0618839
\(477\) 2.11539e7 0.00892435
\(478\) 1.91503e9 0.802006
\(479\) 3.79747e9 1.57877 0.789386 0.613897i \(-0.210399\pi\)
0.789386 + 0.613897i \(0.210399\pi\)
\(480\) 2.24486e9 0.926497
\(481\) −2.53440e8 −0.103841
\(482\) 1.89230e9 0.769709
\(483\) −2.29174e8 −0.0925443
\(484\) 7.47387e8 0.299631
\(485\) −6.51353e7 −0.0259251
\(486\) 1.00558e8 0.0397366
\(487\) 2.70012e9 1.05933 0.529665 0.848207i \(-0.322318\pi\)
0.529665 + 0.848207i \(0.322318\pi\)
\(488\) 2.22460e9 0.866530
\(489\) 2.00637e9 0.775944
\(490\) 2.35948e9 0.906006
\(491\) 1.96974e9 0.750971 0.375485 0.926828i \(-0.377476\pi\)
0.375485 + 0.926828i \(0.377476\pi\)
\(492\) −4.16152e7 −0.0157534
\(493\) −5.61893e8 −0.211198
\(494\) 2.25782e9 0.842645
\(495\) −1.03594e9 −0.383897
\(496\) −1.79845e7 −0.00661779
\(497\) 9.12219e8 0.333313
\(498\) −1.12842e9 −0.409418
\(499\) 6.68650e8 0.240906 0.120453 0.992719i \(-0.461565\pi\)
0.120453 + 0.992719i \(0.461565\pi\)
\(500\) −1.60915e9 −0.575707
\(501\) −7.83864e8 −0.278489
\(502\) −8.48197e7 −0.0299250
\(503\) 8.13925e8 0.285165 0.142583 0.989783i \(-0.454459\pi\)
0.142583 + 0.989783i \(0.454459\pi\)
\(504\) −2.87195e8 −0.0999239
\(505\) 3.23873e9 1.11906
\(506\) −6.92732e8 −0.237705
\(507\) −7.16221e8 −0.244073
\(508\) 3.02617e9 1.02417
\(509\) 3.90482e9 1.31247 0.656235 0.754557i \(-0.272148\pi\)
0.656235 + 0.754557i \(0.272148\pi\)
\(510\) −5.77253e8 −0.192695
\(511\) 5.17960e8 0.171721
\(512\) 2.34907e7 0.00773483
\(513\) −6.71144e8 −0.219485
\(514\) −2.02537e9 −0.657859
\(515\) 6.77579e8 0.218592
\(516\) −8.88661e8 −0.284749
\(517\) 8.76033e7 0.0278807
\(518\) 5.10771e7 0.0161463
\(519\) −1.45842e9 −0.457926
\(520\) −6.15206e9 −1.91871
\(521\) −1.40413e9 −0.434984 −0.217492 0.976062i \(-0.569788\pi\)
−0.217492 + 0.976062i \(0.569788\pi\)
\(522\) −4.22571e8 −0.130033
\(523\) 5.53390e8 0.169151 0.0845756 0.996417i \(-0.473047\pi\)
0.0845756 + 0.996417i \(0.473047\pi\)
\(524\) 1.63007e9 0.494935
\(525\) 9.06387e8 0.273373
\(526\) 5.36110e7 0.0160622
\(527\) −1.92813e9 −0.573851
\(528\) −5.41364e6 −0.00160056
\(529\) −2.42901e9 −0.713402
\(530\) −9.13241e7 −0.0266453
\(531\) 1.49721e8 0.0433963
\(532\) 7.30879e8 0.210453
\(533\) 1.84608e8 0.0528086
\(534\) 9.68401e8 0.275208
\(535\) −5.42065e9 −1.53043
\(536\) −3.66746e9 −1.02870
\(537\) 1.51556e9 0.422341
\(538\) −1.47655e9 −0.408798
\(539\) 2.37234e9 0.652554
\(540\) 6.97298e8 0.190564
\(541\) 2.74177e8 0.0744459 0.0372229 0.999307i \(-0.488149\pi\)
0.0372229 + 0.999307i \(0.488149\pi\)
\(542\) 2.79559e9 0.754183
\(543\) −1.12055e9 −0.300354
\(544\) 1.25771e9 0.334954
\(545\) −8.95457e9 −2.36950
\(546\) 4.85786e8 0.127724
\(547\) 3.89224e8 0.101682 0.0508410 0.998707i \(-0.483810\pi\)
0.0508410 + 0.998707i \(0.483810\pi\)
\(548\) 1.64558e9 0.427156
\(549\) 1.11853e9 0.288500
\(550\) 2.73977e9 0.702174
\(551\) 2.82031e9 0.718236
\(552\) 1.22287e9 0.309452
\(553\) −6.16676e8 −0.155067
\(554\) 3.54810e9 0.886567
\(555\) −3.25235e8 −0.0807554
\(556\) 3.84191e9 0.947949
\(557\) −1.89599e9 −0.464883 −0.232441 0.972610i \(-0.574671\pi\)
−0.232441 + 0.972610i \(0.574671\pi\)
\(558\) −1.45005e9 −0.353316
\(559\) 3.94215e9 0.954536
\(560\) 7.73185e6 0.00186048
\(561\) −5.80399e8 −0.138789
\(562\) 3.05170e9 0.725212
\(563\) −5.24086e9 −1.23772 −0.618861 0.785501i \(-0.712406\pi\)
−0.618861 + 0.785501i \(0.712406\pi\)
\(564\) −5.89667e7 −0.0138398
\(565\) −1.01430e10 −2.36589
\(566\) 9.54991e7 0.0221382
\(567\) −1.44402e8 −0.0332684
\(568\) −4.86759e9 −1.11454
\(569\) −6.24592e9 −1.42136 −0.710678 0.703517i \(-0.751612\pi\)
−0.710678 + 0.703517i \(0.751612\pi\)
\(570\) 2.89741e9 0.655312
\(571\) −4.35962e9 −0.979991 −0.489996 0.871725i \(-0.663002\pi\)
−0.489996 + 0.871725i \(0.663002\pi\)
\(572\) −2.35859e9 −0.526946
\(573\) 1.52580e9 0.338810
\(574\) −3.72049e7 −0.00821124
\(575\) −3.85937e9 −0.846602
\(576\) 9.51775e8 0.207518
\(577\) 1.25822e9 0.272672 0.136336 0.990663i \(-0.456467\pi\)
0.136336 + 0.990663i \(0.456467\pi\)
\(578\) 2.55227e9 0.549768
\(579\) −4.29622e9 −0.919839
\(580\) −2.93022e9 −0.623595
\(581\) 1.62041e9 0.342773
\(582\) −2.74446e7 −0.00577066
\(583\) −9.18217e7 −0.0191914
\(584\) −2.76383e9 −0.574204
\(585\) −3.09326e9 −0.638808
\(586\) 3.67768e9 0.754975
\(587\) −7.19197e9 −1.46762 −0.733812 0.679353i \(-0.762261\pi\)
−0.733812 + 0.679353i \(0.762261\pi\)
\(588\) −1.59685e9 −0.323923
\(589\) 9.67787e9 1.95154
\(590\) −6.46366e8 −0.129568
\(591\) −1.74615e9 −0.347956
\(592\) −1.69963e6 −0.000336688 0
\(593\) −8.65682e9 −1.70477 −0.852387 0.522911i \(-0.824846\pi\)
−0.852387 + 0.522911i \(0.824846\pi\)
\(594\) −4.36488e8 −0.0854516
\(595\) 8.28935e8 0.161329
\(596\) −8.04291e8 −0.155615
\(597\) 5.10013e9 0.981005
\(598\) −2.06847e9 −0.395543
\(599\) 5.85691e9 1.11346 0.556730 0.830693i \(-0.312056\pi\)
0.556730 + 0.830693i \(0.312056\pi\)
\(600\) −4.83647e9 −0.914112
\(601\) 5.94489e9 1.11708 0.558539 0.829478i \(-0.311362\pi\)
0.558539 + 0.829478i \(0.311362\pi\)
\(602\) −7.94482e8 −0.148421
\(603\) −1.84400e9 −0.342492
\(604\) −3.11509e9 −0.575229
\(605\) −4.25465e9 −0.781125
\(606\) 1.36463e9 0.249092
\(607\) 7.31720e9 1.32796 0.663979 0.747751i \(-0.268866\pi\)
0.663979 + 0.747751i \(0.268866\pi\)
\(608\) −6.31285e9 −1.13910
\(609\) 6.06812e8 0.108866
\(610\) −4.82884e9 −0.861368
\(611\) 2.61580e8 0.0463938
\(612\) 3.90672e8 0.0688942
\(613\) 8.17131e8 0.143278 0.0716391 0.997431i \(-0.477177\pi\)
0.0716391 + 0.997431i \(0.477177\pi\)
\(614\) 3.78511e9 0.659917
\(615\) 2.36903e8 0.0410685
\(616\) 1.24661e9 0.214881
\(617\) −4.81332e9 −0.824985 −0.412493 0.910961i \(-0.635342\pi\)
−0.412493 + 0.910961i \(0.635342\pi\)
\(618\) 2.85496e8 0.0486564
\(619\) −4.14223e8 −0.0701968 −0.0350984 0.999384i \(-0.511174\pi\)
−0.0350984 + 0.999384i \(0.511174\pi\)
\(620\) −1.00550e10 −1.69439
\(621\) 6.14858e8 0.103028
\(622\) 1.32550e9 0.220858
\(623\) −1.39062e9 −0.230410
\(624\) −1.61649e7 −0.00266334
\(625\) −4.91719e8 −0.0805632
\(626\) −4.04462e9 −0.658973
\(627\) 2.91320e9 0.471991
\(628\) −3.90560e9 −0.629259
\(629\) −1.82218e8 −0.0291953
\(630\) 6.23400e8 0.0993287
\(631\) −2.20305e9 −0.349078 −0.174539 0.984650i \(-0.555843\pi\)
−0.174539 + 0.984650i \(0.555843\pi\)
\(632\) 3.29058e9 0.518516
\(633\) −2.22244e9 −0.348271
\(634\) −8.31469e6 −0.00129579
\(635\) −1.72271e10 −2.66996
\(636\) 6.18061e7 0.00952646
\(637\) 7.08370e9 1.08586
\(638\) 1.83423e9 0.279629
\(639\) −2.44743e9 −0.371071
\(640\) 6.53334e9 0.985157
\(641\) 4.40972e9 0.661315 0.330657 0.943751i \(-0.392730\pi\)
0.330657 + 0.943751i \(0.392730\pi\)
\(642\) −2.28398e9 −0.340658
\(643\) 8.75195e9 1.29827 0.649137 0.760671i \(-0.275130\pi\)
0.649137 + 0.760671i \(0.275130\pi\)
\(644\) −6.69584e8 −0.0987881
\(645\) 5.05889e9 0.742328
\(646\) 1.62332e9 0.236914
\(647\) 4.86345e9 0.705959 0.352979 0.935631i \(-0.385169\pi\)
0.352979 + 0.935631i \(0.385169\pi\)
\(648\) 7.70525e8 0.111243
\(649\) −6.49888e8 −0.0933215
\(650\) 8.18084e9 1.16843
\(651\) 2.08227e9 0.295803
\(652\) 5.86208e9 0.828296
\(653\) −9.35709e9 −1.31506 −0.657529 0.753429i \(-0.728398\pi\)
−0.657529 + 0.753429i \(0.728398\pi\)
\(654\) −3.77298e9 −0.527427
\(655\) −9.27953e9 −1.29027
\(656\) 1.23802e6 0.000171224 0
\(657\) −1.38966e9 −0.191174
\(658\) −5.27175e7 −0.00721380
\(659\) 5.87424e9 0.799562 0.399781 0.916611i \(-0.369086\pi\)
0.399781 + 0.916611i \(0.369086\pi\)
\(660\) −3.02673e9 −0.409798
\(661\) 4.45812e8 0.0600408 0.0300204 0.999549i \(-0.490443\pi\)
0.0300204 + 0.999549i \(0.490443\pi\)
\(662\) 5.66514e9 0.758940
\(663\) −1.73304e9 −0.230947
\(664\) −8.64647e9 −1.14617
\(665\) −4.16068e9 −0.548641
\(666\) −1.37037e8 −0.0179753
\(667\) −2.58379e9 −0.337145
\(668\) −2.29024e9 −0.297279
\(669\) 3.70579e9 0.478508
\(670\) 7.96078e9 1.02257
\(671\) −4.85515e9 −0.620404
\(672\) −1.35826e9 −0.172659
\(673\) 2.90168e9 0.366942 0.183471 0.983025i \(-0.441267\pi\)
0.183471 + 0.983025i \(0.441267\pi\)
\(674\) −6.20460e9 −0.780556
\(675\) −2.43178e9 −0.304342
\(676\) −2.09261e9 −0.260540
\(677\) 8.77680e9 1.08712 0.543558 0.839372i \(-0.317077\pi\)
0.543558 + 0.839372i \(0.317077\pi\)
\(678\) −4.27370e9 −0.526624
\(679\) 3.94103e7 0.00483132
\(680\) −4.42319e9 −0.539454
\(681\) 4.59005e9 0.556932
\(682\) 6.29415e9 0.759787
\(683\) 6.96062e9 0.835940 0.417970 0.908461i \(-0.362742\pi\)
0.417970 + 0.908461i \(0.362742\pi\)
\(684\) −1.96090e9 −0.234293
\(685\) −9.36780e9 −1.11358
\(686\) −2.99582e9 −0.354308
\(687\) −6.23229e9 −0.733329
\(688\) 2.64370e7 0.00309494
\(689\) −2.74176e8 −0.0319346
\(690\) −2.65442e9 −0.307608
\(691\) −5.57378e8 −0.0642653 −0.0321326 0.999484i \(-0.510230\pi\)
−0.0321326 + 0.999484i \(0.510230\pi\)
\(692\) −4.26110e9 −0.488822
\(693\) 6.26797e8 0.0715419
\(694\) −5.26553e9 −0.597976
\(695\) −2.18708e10 −2.47126
\(696\) −3.23794e9 −0.364029
\(697\) 1.32729e8 0.0148474
\(698\) −3.54929e9 −0.395046
\(699\) 3.37151e9 0.373383
\(700\) 2.64822e9 0.291817
\(701\) 1.27904e10 1.40240 0.701201 0.712963i \(-0.252647\pi\)
0.701201 + 0.712963i \(0.252647\pi\)
\(702\) −1.30333e9 −0.142192
\(703\) 9.14607e8 0.0992867
\(704\) −4.13132e9 −0.446257
\(705\) 3.35680e8 0.0360797
\(706\) −5.12295e9 −0.547903
\(707\) −1.95960e9 −0.208545
\(708\) 4.37446e8 0.0463242
\(709\) −1.68769e10 −1.77841 −0.889205 0.457508i \(-0.848742\pi\)
−0.889205 + 0.457508i \(0.848742\pi\)
\(710\) 1.05659e10 1.10790
\(711\) 1.65450e9 0.172633
\(712\) 7.42035e9 0.770450
\(713\) −8.86624e9 −0.916065
\(714\) 3.49269e8 0.0359101
\(715\) 1.34267e10 1.37372
\(716\) 4.42806e9 0.450836
\(717\) 7.37802e9 0.747520
\(718\) −7.68933e9 −0.775270
\(719\) 1.65145e10 1.65697 0.828484 0.560013i \(-0.189204\pi\)
0.828484 + 0.560013i \(0.189204\pi\)
\(720\) −2.07441e7 −0.00207124
\(721\) −4.09972e8 −0.0407362
\(722\) −1.88361e9 −0.186256
\(723\) 7.29047e9 0.717417
\(724\) −3.27396e9 −0.320618
\(725\) 1.02190e10 0.995917
\(726\) −1.79269e9 −0.173870
\(727\) 1.21987e10 1.17745 0.588727 0.808332i \(-0.299629\pi\)
0.588727 + 0.808332i \(0.299629\pi\)
\(728\) 3.72233e9 0.357565
\(729\) 3.87420e8 0.0370370
\(730\) 5.99932e9 0.570784
\(731\) 2.83432e9 0.268372
\(732\) 3.26805e9 0.307964
\(733\) −9.19756e9 −0.862598 −0.431299 0.902209i \(-0.641945\pi\)
−0.431299 + 0.902209i \(0.641945\pi\)
\(734\) −1.81381e9 −0.169299
\(735\) 9.09037e9 0.844454
\(736\) 5.78343e9 0.534703
\(737\) 8.00416e9 0.736511
\(738\) 9.98185e7 0.00914142
\(739\) −1.94615e10 −1.77386 −0.886932 0.461900i \(-0.847168\pi\)
−0.886932 + 0.461900i \(0.847168\pi\)
\(740\) −9.50250e8 −0.0862039
\(741\) 8.69868e9 0.785398
\(742\) 5.52560e7 0.00496553
\(743\) 1.39586e10 1.24848 0.624238 0.781235i \(-0.285410\pi\)
0.624238 + 0.781235i \(0.285410\pi\)
\(744\) −1.11110e10 −0.989113
\(745\) 4.57859e9 0.405681
\(746\) −9.65786e9 −0.851717
\(747\) −4.34745e9 −0.381603
\(748\) −1.69577e9 −0.148153
\(749\) 3.27979e9 0.285206
\(750\) 3.85971e9 0.334072
\(751\) −9.03865e9 −0.778688 −0.389344 0.921092i \(-0.627298\pi\)
−0.389344 + 0.921092i \(0.627298\pi\)
\(752\) 1.75421e6 0.000150425 0
\(753\) −3.26784e8 −0.0278919
\(754\) 5.47694e9 0.465305
\(755\) 1.77333e10 1.49960
\(756\) −4.21903e8 −0.0355129
\(757\) 4.80250e9 0.402376 0.201188 0.979553i \(-0.435520\pi\)
0.201188 + 0.979553i \(0.435520\pi\)
\(758\) 1.09943e9 0.0916908
\(759\) −2.66888e9 −0.221556
\(760\) 2.22013e10 1.83456
\(761\) −1.41275e10 −1.16203 −0.581015 0.813893i \(-0.697344\pi\)
−0.581015 + 0.813893i \(0.697344\pi\)
\(762\) −7.25859e9 −0.594305
\(763\) 5.41800e9 0.441573
\(764\) 4.45798e9 0.361669
\(765\) −2.22398e9 −0.179604
\(766\) 6.19135e9 0.497719
\(767\) −1.94054e9 −0.155288
\(768\) 7.26492e9 0.578718
\(769\) 1.40261e10 1.11223 0.556113 0.831106i \(-0.312292\pi\)
0.556113 + 0.831106i \(0.312292\pi\)
\(770\) −2.70596e9 −0.213601
\(771\) −7.80312e9 −0.613166
\(772\) −1.25524e10 −0.981899
\(773\) 2.02962e8 0.0158047 0.00790236 0.999969i \(-0.497485\pi\)
0.00790236 + 0.999969i \(0.497485\pi\)
\(774\) 2.13155e9 0.165235
\(775\) 3.50662e10 2.70603
\(776\) −2.10293e8 −0.0161551
\(777\) 1.96785e8 0.0150493
\(778\) −6.67880e9 −0.508475
\(779\) −6.66206e8 −0.0504926
\(780\) −9.03767e9 −0.681908
\(781\) 1.06234e10 0.797969
\(782\) −1.48718e9 −0.111209
\(783\) −1.62804e9 −0.121199
\(784\) 4.75049e7 0.00352073
\(785\) 2.22335e10 1.64045
\(786\) −3.90990e9 −0.287202
\(787\) 8.88769e9 0.649946 0.324973 0.945723i \(-0.394645\pi\)
0.324973 + 0.945723i \(0.394645\pi\)
\(788\) −5.10178e9 −0.371432
\(789\) 2.06547e8 0.0149709
\(790\) −7.14270e9 −0.515427
\(791\) 6.13703e9 0.440901
\(792\) −3.34458e9 −0.239223
\(793\) −1.44973e10 −1.03236
\(794\) 1.61809e10 1.14718
\(795\) −3.51844e8 −0.0248351
\(796\) 1.49012e10 1.04719
\(797\) −9.59859e9 −0.671589 −0.335794 0.941935i \(-0.609005\pi\)
−0.335794 + 0.941935i \(0.609005\pi\)
\(798\) −1.75309e9 −0.122122
\(799\) 1.88070e8 0.0130438
\(800\) −2.28736e10 −1.57950
\(801\) 3.73095e9 0.256511
\(802\) −8.10833e9 −0.555036
\(803\) 6.03200e9 0.411109
\(804\) −5.38768e9 −0.365599
\(805\) 3.81174e9 0.257536
\(806\) 1.87940e10 1.26429
\(807\) −5.68868e9 −0.381026
\(808\) 1.04564e10 0.697338
\(809\) 1.06933e10 0.710054 0.355027 0.934856i \(-0.384472\pi\)
0.355027 + 0.934856i \(0.384472\pi\)
\(810\) −1.67254e9 −0.110581
\(811\) −1.87262e10 −1.23275 −0.616376 0.787452i \(-0.711400\pi\)
−0.616376 + 0.787452i \(0.711400\pi\)
\(812\) 1.77294e9 0.116211
\(813\) 1.07706e10 0.702946
\(814\) 5.94828e8 0.0386551
\(815\) −3.33711e10 −2.15933
\(816\) −1.16222e7 −0.000748811 0
\(817\) −1.42263e10 −0.912673
\(818\) −1.36196e10 −0.870017
\(819\) 1.87159e9 0.119046
\(820\) 6.92168e8 0.0438393
\(821\) −1.83953e10 −1.16013 −0.580065 0.814570i \(-0.696973\pi\)
−0.580065 + 0.814570i \(0.696973\pi\)
\(822\) −3.94709e9 −0.247871
\(823\) −6.29971e9 −0.393932 −0.196966 0.980410i \(-0.563109\pi\)
−0.196966 + 0.980410i \(0.563109\pi\)
\(824\) 2.18760e9 0.136215
\(825\) 1.05555e10 0.654471
\(826\) 3.91086e8 0.0241458
\(827\) −1.00387e10 −0.617175 −0.308587 0.951196i \(-0.599856\pi\)
−0.308587 + 0.951196i \(0.599856\pi\)
\(828\) 1.79645e9 0.109979
\(829\) 2.76027e8 0.0168271 0.00841357 0.999965i \(-0.497322\pi\)
0.00841357 + 0.999965i \(0.497322\pi\)
\(830\) 1.87685e10 1.13935
\(831\) 1.36697e10 0.826336
\(832\) −1.23359e10 −0.742576
\(833\) 5.09302e9 0.305294
\(834\) −9.21521e9 −0.550078
\(835\) 1.30377e10 0.774992
\(836\) 8.51159e9 0.503835
\(837\) −5.58659e9 −0.329312
\(838\) −1.52058e10 −0.892597
\(839\) 2.00764e10 1.17360 0.586799 0.809733i \(-0.300388\pi\)
0.586799 + 0.809733i \(0.300388\pi\)
\(840\) 4.77679e9 0.278073
\(841\) −1.04085e10 −0.603393
\(842\) 4.65607e9 0.268799
\(843\) 1.17573e10 0.675944
\(844\) −6.49339e9 −0.371769
\(845\) 1.19126e10 0.679216
\(846\) 1.41438e8 0.00803099
\(847\) 2.57429e9 0.145568
\(848\) −1.83868e6 −0.000103543 0
\(849\) 3.67929e8 0.0206342
\(850\) 5.88183e9 0.328508
\(851\) −8.37904e8 −0.0466059
\(852\) −7.15073e9 −0.396106
\(853\) 7.43953e9 0.410415 0.205208 0.978718i \(-0.434213\pi\)
0.205208 + 0.978718i \(0.434213\pi\)
\(854\) 2.92171e9 0.160522
\(855\) 1.11628e10 0.610792
\(856\) −1.75009e10 −0.953679
\(857\) −2.02954e10 −1.10145 −0.550726 0.834686i \(-0.685649\pi\)
−0.550726 + 0.834686i \(0.685649\pi\)
\(858\) 5.65732e9 0.305777
\(859\) 1.59155e10 0.856728 0.428364 0.903606i \(-0.359090\pi\)
0.428364 + 0.903606i \(0.359090\pi\)
\(860\) 1.47807e10 0.792412
\(861\) −1.43339e8 −0.00765339
\(862\) 1.64878e10 0.876774
\(863\) 8.64458e9 0.457832 0.228916 0.973446i \(-0.426482\pi\)
0.228916 + 0.973446i \(0.426482\pi\)
\(864\) 3.64412e9 0.192218
\(865\) 2.42572e10 1.27434
\(866\) −9.88363e8 −0.0517135
\(867\) 9.83312e9 0.512418
\(868\) 6.08383e9 0.315761
\(869\) −7.18161e9 −0.371238
\(870\) 7.02844e9 0.361861
\(871\) 2.39001e10 1.22556
\(872\) −2.89104e10 −1.47654
\(873\) −1.05735e8 −0.00537862
\(874\) 7.46461e9 0.378196
\(875\) −5.54254e9 −0.279692
\(876\) −4.06020e9 −0.204072
\(877\) −3.61467e10 −1.80955 −0.904775 0.425890i \(-0.859961\pi\)
−0.904775 + 0.425890i \(0.859961\pi\)
\(878\) 1.98965e10 0.992080
\(879\) 1.41690e10 0.703684
\(880\) 9.00427e7 0.00445409
\(881\) −2.55814e10 −1.26040 −0.630199 0.776433i \(-0.717027\pi\)
−0.630199 + 0.776433i \(0.717027\pi\)
\(882\) 3.83020e9 0.187967
\(883\) −3.64985e10 −1.78407 −0.892036 0.451964i \(-0.850724\pi\)
−0.892036 + 0.451964i \(0.850724\pi\)
\(884\) −5.06349e9 −0.246529
\(885\) −2.49025e9 −0.120765
\(886\) 2.41189e10 1.16503
\(887\) 1.11780e10 0.537812 0.268906 0.963166i \(-0.413338\pi\)
0.268906 + 0.963166i \(0.413338\pi\)
\(888\) −1.05004e9 −0.0503223
\(889\) 1.04233e10 0.497565
\(890\) −1.61070e10 −0.765860
\(891\) −1.68166e9 −0.0796462
\(892\) 1.08273e10 0.510792
\(893\) −9.43980e8 −0.0443591
\(894\) 1.92918e9 0.0903006
\(895\) −2.52077e10 −1.17531
\(896\) −3.95302e9 −0.183591
\(897\) −7.96917e9 −0.368671
\(898\) −6.02612e9 −0.277697
\(899\) 2.34763e10 1.07763
\(900\) −7.10501e9 −0.324875
\(901\) −1.97126e8 −0.00897857
\(902\) −4.33277e8 −0.0196582
\(903\) −3.06090e9 −0.138338
\(904\) −3.27471e10 −1.47429
\(905\) 1.86377e10 0.835837
\(906\) 7.47186e9 0.333795
\(907\) −3.45269e10 −1.53650 −0.768250 0.640150i \(-0.778872\pi\)
−0.768250 + 0.640150i \(0.778872\pi\)
\(908\) 1.34109e10 0.594507
\(909\) 5.25749e9 0.232170
\(910\) −8.07988e9 −0.355435
\(911\) 1.85678e10 0.813665 0.406832 0.913503i \(-0.366633\pi\)
0.406832 + 0.913503i \(0.366633\pi\)
\(912\) 5.83353e7 0.00254653
\(913\) 1.88707e10 0.820618
\(914\) 1.19843e10 0.519161
\(915\) −1.86041e10 −0.802849
\(916\) −1.82091e10 −0.782806
\(917\) 5.61462e9 0.240451
\(918\) −9.37068e8 −0.0399780
\(919\) −3.89221e10 −1.65422 −0.827108 0.562044i \(-0.810015\pi\)
−0.827108 + 0.562044i \(0.810015\pi\)
\(920\) −2.03394e10 −0.861155
\(921\) 1.45829e10 0.615084
\(922\) −1.74461e10 −0.733062
\(923\) 3.17211e10 1.32783
\(924\) 1.83133e9 0.0763687
\(925\) 3.31393e9 0.137673
\(926\) −2.04074e10 −0.844595
\(927\) 1.09993e9 0.0453508
\(928\) −1.53135e10 −0.629009
\(929\) −7.85450e9 −0.321413 −0.160707 0.987002i \(-0.551377\pi\)
−0.160707 + 0.987002i \(0.551377\pi\)
\(930\) 2.41180e10 0.983222
\(931\) −2.55634e10 −1.03823
\(932\) 9.85066e9 0.398575
\(933\) 5.10675e9 0.205854
\(934\) −2.26015e10 −0.907660
\(935\) 9.65352e9 0.386229
\(936\) −9.98677e9 −0.398070
\(937\) 1.64761e10 0.654283 0.327142 0.944975i \(-0.393915\pi\)
0.327142 + 0.944975i \(0.393915\pi\)
\(938\) −4.81670e9 −0.190563
\(939\) −1.55827e10 −0.614205
\(940\) 9.80767e8 0.0385140
\(941\) 3.54317e10 1.38621 0.693105 0.720837i \(-0.256242\pi\)
0.693105 + 0.720837i \(0.256242\pi\)
\(942\) 9.36800e9 0.365148
\(943\) 6.10335e8 0.0237016
\(944\) −1.30137e7 −0.000503498 0
\(945\) 2.40177e9 0.0925806
\(946\) −9.25229e9 −0.355329
\(947\) −1.07477e10 −0.411237 −0.205618 0.978632i \(-0.565921\pi\)
−0.205618 + 0.978632i \(0.565921\pi\)
\(948\) 4.83401e9 0.184280
\(949\) 1.80113e10 0.684090
\(950\) −2.95227e10 −1.11718
\(951\) −3.20339e7 −0.00120775
\(952\) 2.67627e9 0.100531
\(953\) −4.52936e10 −1.69516 −0.847581 0.530666i \(-0.821942\pi\)
−0.847581 + 0.530666i \(0.821942\pi\)
\(954\) −1.48248e8 −0.00552803
\(955\) −2.53780e10 −0.942856
\(956\) 2.15566e10 0.797953
\(957\) 7.06674e9 0.260632
\(958\) −2.66129e10 −0.977943
\(959\) 5.66802e9 0.207523
\(960\) −1.58305e10 −0.577490
\(961\) 5.30459e10 1.92806
\(962\) 1.77613e9 0.0643224
\(963\) −8.79946e9 −0.317515
\(964\) 2.13008e10 0.765820
\(965\) 7.14572e10 2.55977
\(966\) 1.60607e9 0.0573249
\(967\) −4.08323e10 −1.45215 −0.726074 0.687617i \(-0.758657\pi\)
−0.726074 + 0.687617i \(0.758657\pi\)
\(968\) −1.37364e10 −0.486754
\(969\) 6.25415e9 0.220818
\(970\) 4.56473e8 0.0160588
\(971\) 3.91677e10 1.37297 0.686484 0.727145i \(-0.259153\pi\)
0.686484 + 0.727145i \(0.259153\pi\)
\(972\) 1.13194e9 0.0395359
\(973\) 1.32330e10 0.460537
\(974\) −1.89226e10 −0.656183
\(975\) 3.15183e10 1.08905
\(976\) −9.72220e7 −0.00334726
\(977\) 4.38079e10 1.50287 0.751435 0.659808i \(-0.229362\pi\)
0.751435 + 0.659808i \(0.229362\pi\)
\(978\) −1.40608e10 −0.480645
\(979\) −1.61948e10 −0.551614
\(980\) 2.65596e10 0.901428
\(981\) −1.45361e10 −0.491595
\(982\) −1.38041e10 −0.465176
\(983\) 3.95635e10 1.32849 0.664244 0.747516i \(-0.268754\pi\)
0.664244 + 0.747516i \(0.268754\pi\)
\(984\) 7.64857e8 0.0255916
\(985\) 2.90429e10 0.968308
\(986\) 3.93779e9 0.130823
\(987\) −2.03104e8 −0.00672371
\(988\) 2.54152e10 0.838387
\(989\) 1.30332e10 0.428415
\(990\) 7.25992e9 0.237798
\(991\) −6.12206e10 −1.99820 −0.999102 0.0423753i \(-0.986507\pi\)
−0.999102 + 0.0423753i \(0.986507\pi\)
\(992\) −5.25481e10 −1.70910
\(993\) 2.18261e10 0.707380
\(994\) −6.39291e9 −0.206465
\(995\) −8.48283e10 −2.72998
\(996\) −1.27021e10 −0.407349
\(997\) −2.27542e10 −0.727156 −0.363578 0.931564i \(-0.618445\pi\)
−0.363578 + 0.931564i \(0.618445\pi\)
\(998\) −4.68595e9 −0.149225
\(999\) −5.27961e8 −0.0167542
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 177.8.a.a.1.6 16
3.2 odd 2 531.8.a.b.1.11 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.6 16 1.1 even 1 trivial
531.8.a.b.1.11 16 3.2 odd 2