Properties

Label 91.10.a.a.1.5
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-11.0043\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-13.0043 q^{2} -81.8707 q^{3} -342.889 q^{4} -1042.73 q^{5} +1064.67 q^{6} +2401.00 q^{7} +11117.2 q^{8} -12980.2 q^{9} +O(q^{10})\) \(q-13.0043 q^{2} -81.8707 q^{3} -342.889 q^{4} -1042.73 q^{5} +1064.67 q^{6} +2401.00 q^{7} +11117.2 q^{8} -12980.2 q^{9} +13559.9 q^{10} -52645.0 q^{11} +28072.6 q^{12} +28561.0 q^{13} -31223.3 q^{14} +85368.8 q^{15} +30987.9 q^{16} +616500. q^{17} +168798. q^{18} +262035. q^{19} +357539. q^{20} -196572. q^{21} +684609. q^{22} +1.18232e6 q^{23} -910174. q^{24} -865846. q^{25} -371415. q^{26} +2.67416e6 q^{27} -823276. q^{28} -3.03775e6 q^{29} -1.11016e6 q^{30} +5.18698e6 q^{31} -6.09499e6 q^{32} +4.31008e6 q^{33} -8.01713e6 q^{34} -2.50359e6 q^{35} +4.45076e6 q^{36} +8.57803e6 q^{37} -3.40758e6 q^{38} -2.33831e6 q^{39} -1.15922e7 q^{40} +2.51421e7 q^{41} +2.55627e6 q^{42} -2.75282e7 q^{43} +1.80514e7 q^{44} +1.35348e7 q^{45} -1.53752e7 q^{46} -4.94296e6 q^{47} -2.53700e6 q^{48} +5.76480e6 q^{49} +1.12597e7 q^{50} -5.04733e7 q^{51} -9.79325e6 q^{52} -1.01380e8 q^{53} -3.47755e7 q^{54} +5.48943e7 q^{55} +2.66924e7 q^{56} -2.14530e7 q^{57} +3.95038e7 q^{58} -1.47790e8 q^{59} -2.92720e7 q^{60} +8.03304e7 q^{61} -6.74528e7 q^{62} -3.11654e7 q^{63} +6.33950e7 q^{64} -2.97813e7 q^{65} -5.60495e7 q^{66} -5.89802e7 q^{67} -2.11391e8 q^{68} -9.67974e7 q^{69} +3.25573e7 q^{70} +8.65828e7 q^{71} -1.44303e8 q^{72} -3.16816e8 q^{73} -1.11551e8 q^{74} +7.08875e7 q^{75} -8.98490e7 q^{76} -1.26401e8 q^{77} +3.04080e7 q^{78} +3.92511e8 q^{79} -3.23119e7 q^{80} +3.65535e7 q^{81} -3.26954e8 q^{82} -8.65771e6 q^{83} +6.74022e7 q^{84} -6.42841e8 q^{85} +3.57984e8 q^{86} +2.48703e8 q^{87} -5.85265e8 q^{88} +1.81734e8 q^{89} -1.76010e8 q^{90} +6.85750e7 q^{91} -4.05404e8 q^{92} -4.24662e8 q^{93} +6.42797e7 q^{94} -2.73231e8 q^{95} +4.99001e8 q^{96} -1.21586e9 q^{97} -7.49670e7 q^{98} +6.83341e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 21q^{2} - 323q^{3} + 2945q^{4} - 5202q^{5} - 9897q^{6} + 28812q^{7} - 25431q^{8} + 78519q^{9} + O(q^{10}) \) \( 12q - 21q^{2} - 323q^{3} + 2945q^{4} - 5202q^{5} - 9897q^{6} + 28812q^{7} - 25431q^{8} + 78519q^{9} - 65812q^{10} - 80061q^{11} - 184395q^{12} + 342732q^{13} - 50421q^{14} + 160096q^{15} + 385497q^{16} - 1493598q^{17} + 1520858q^{18} - 109038q^{19} - 622260q^{20} - 775523q^{21} + 4636975q^{22} - 3367443q^{23} - 5963895q^{24} - 51480q^{25} - 599781q^{26} - 8158937q^{27} + 7070945q^{28} - 13333098q^{29} + 2915424q^{30} - 3954765q^{31} + 4389297q^{32} - 5790219q^{33} + 14879968q^{34} - 12490002q^{35} + 80697058q^{36} + 580535q^{37} - 19134246q^{38} - 9225203q^{39} + 12365024q^{40} - 27018171q^{41} - 23762697q^{42} + 31237588q^{43} - 125053839q^{44} - 62765470q^{45} - 114008121q^{46} - 21983709q^{47} - 309724207q^{48} + 69177612q^{49} - 131331747q^{50} - 176522692q^{51} + 84112145q^{52} - 196548234q^{53} - 456152547q^{54} - 309055872q^{55} - 61059831q^{56} - 274411494q^{57} - 521980612q^{58} - 215907906q^{59} - 177006648q^{60} - 218340705q^{61} - 673289997q^{62} + 188524119q^{63} - 386667247q^{64} - 148574322q^{65} - 777397365q^{66} + 14544775q^{67} - 1246637448q^{68} - 65252625q^{69} - 158014612q^{70} - 552451776q^{71} + 369379470q^{72} - 349395159q^{73} + 73591023q^{74} + 329300747q^{75} - 1036299002q^{76} - 192226461q^{77} - 282668217q^{78} + 962249727q^{79} - 1494536184q^{80} + 874458108q^{81} - 1417698067q^{82} - 2032575912q^{83} - 442732395q^{84} - 411671064q^{85} - 2139249420q^{86} - 759642172q^{87} + 558651957q^{88} - 280821684q^{89} - 5764700804q^{90} + 822899532q^{91} - 4491569571q^{92} - 1729557923q^{93} - 1591372165q^{94} - 1282463328q^{95} - 2148993055q^{96} - 2115165937q^{97} - 121060821q^{98} - 3595669198q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.0043 −0.574713 −0.287357 0.957824i \(-0.592776\pi\)
−0.287357 + 0.957824i \(0.592776\pi\)
\(3\) −81.8707 −0.583557 −0.291778 0.956486i \(-0.594247\pi\)
−0.291778 + 0.956486i \(0.594247\pi\)
\(4\) −342.889 −0.669705
\(5\) −1042.73 −0.746114 −0.373057 0.927808i \(-0.621690\pi\)
−0.373057 + 0.927808i \(0.621690\pi\)
\(6\) 1064.67 0.335378
\(7\) 2401.00 0.377964
\(8\) 11117.2 0.959601
\(9\) −12980.2 −0.659462
\(10\) 13559.9 0.428802
\(11\) −52645.0 −1.08415 −0.542075 0.840330i \(-0.682361\pi\)
−0.542075 + 0.840330i \(0.682361\pi\)
\(12\) 28072.6 0.390811
\(13\) 28561.0 0.277350
\(14\) −31223.3 −0.217221
\(15\) 85368.8 0.435400
\(16\) 30987.9 0.118210
\(17\) 616500. 1.79025 0.895123 0.445818i \(-0.147087\pi\)
0.895123 + 0.445818i \(0.147087\pi\)
\(18\) 168798. 0.379001
\(19\) 262035. 0.461284 0.230642 0.973039i \(-0.425917\pi\)
0.230642 + 0.973039i \(0.425917\pi\)
\(20\) 357539. 0.499676
\(21\) −196572. −0.220564
\(22\) 684609. 0.623076
\(23\) 1.18232e6 0.880967 0.440483 0.897761i \(-0.354807\pi\)
0.440483 + 0.897761i \(0.354807\pi\)
\(24\) −910174. −0.559982
\(25\) −865846. −0.443313
\(26\) −371415. −0.159397
\(27\) 2.67416e6 0.968390
\(28\) −823276. −0.253125
\(29\) −3.03775e6 −0.797556 −0.398778 0.917047i \(-0.630566\pi\)
−0.398778 + 0.917047i \(0.630566\pi\)
\(30\) −1.11016e6 −0.250230
\(31\) 5.18698e6 1.00876 0.504378 0.863483i \(-0.331722\pi\)
0.504378 + 0.863483i \(0.331722\pi\)
\(32\) −6.09499e6 −1.02754
\(33\) 4.31008e6 0.632663
\(34\) −8.01713e6 −1.02888
\(35\) −2.50359e6 −0.282005
\(36\) 4.45076e6 0.441645
\(37\) 8.57803e6 0.752454 0.376227 0.926528i \(-0.377221\pi\)
0.376227 + 0.926528i \(0.377221\pi\)
\(38\) −3.40758e6 −0.265106
\(39\) −2.33831e6 −0.161850
\(40\) −1.15922e7 −0.715972
\(41\) 2.51421e7 1.38955 0.694774 0.719228i \(-0.255504\pi\)
0.694774 + 0.719228i \(0.255504\pi\)
\(42\) 2.55627e6 0.126761
\(43\) −2.75282e7 −1.22792 −0.613959 0.789338i \(-0.710424\pi\)
−0.613959 + 0.789338i \(0.710424\pi\)
\(44\) 1.80514e7 0.726061
\(45\) 1.35348e7 0.492034
\(46\) −1.53752e7 −0.506303
\(47\) −4.94296e6 −0.147757 −0.0738783 0.997267i \(-0.523538\pi\)
−0.0738783 + 0.997267i \(0.523538\pi\)
\(48\) −2.53700e6 −0.0689820
\(49\) 5.76480e6 0.142857
\(50\) 1.12597e7 0.254778
\(51\) −5.04733e7 −1.04471
\(52\) −9.79325e6 −0.185743
\(53\) −1.01380e8 −1.76486 −0.882430 0.470444i \(-0.844094\pi\)
−0.882430 + 0.470444i \(0.844094\pi\)
\(54\) −3.47755e7 −0.556546
\(55\) 5.48943e7 0.808900
\(56\) 2.66924e7 0.362695
\(57\) −2.14530e7 −0.269186
\(58\) 3.95038e7 0.458366
\(59\) −1.47790e8 −1.58785 −0.793925 0.608016i \(-0.791966\pi\)
−0.793925 + 0.608016i \(0.791966\pi\)
\(60\) −2.92720e7 −0.291590
\(61\) 8.03304e7 0.742841 0.371420 0.928465i \(-0.378871\pi\)
0.371420 + 0.928465i \(0.378871\pi\)
\(62\) −6.74528e7 −0.579746
\(63\) −3.11654e7 −0.249253
\(64\) 6.33950e7 0.472330
\(65\) −2.97813e7 −0.206935
\(66\) −5.60495e7 −0.363600
\(67\) −5.89802e7 −0.357577 −0.178789 0.983888i \(-0.557218\pi\)
−0.178789 + 0.983888i \(0.557218\pi\)
\(68\) −2.11391e8 −1.19894
\(69\) −9.67974e7 −0.514094
\(70\) 3.25573e7 0.162072
\(71\) 8.65828e7 0.404361 0.202180 0.979348i \(-0.435197\pi\)
0.202180 + 0.979348i \(0.435197\pi\)
\(72\) −1.44303e8 −0.632820
\(73\) −3.16816e8 −1.30573 −0.652866 0.757473i \(-0.726434\pi\)
−0.652866 + 0.757473i \(0.726434\pi\)
\(74\) −1.11551e8 −0.432445
\(75\) 7.08875e7 0.258698
\(76\) −8.98490e7 −0.308924
\(77\) −1.26401e8 −0.409770
\(78\) 3.04080e7 0.0930170
\(79\) 3.92511e8 1.13378 0.566891 0.823793i \(-0.308146\pi\)
0.566891 + 0.823793i \(0.308146\pi\)
\(80\) −3.23119e7 −0.0881979
\(81\) 3.65535e7 0.0943511
\(82\) −3.26954e8 −0.798592
\(83\) −8.65771e6 −0.0200240 −0.0100120 0.999950i \(-0.503187\pi\)
−0.0100120 + 0.999950i \(0.503187\pi\)
\(84\) 6.74022e7 0.147713
\(85\) −6.42841e8 −1.33573
\(86\) 3.57984e8 0.705701
\(87\) 2.48703e8 0.465419
\(88\) −5.85265e8 −1.04035
\(89\) 1.81734e8 0.307030 0.153515 0.988146i \(-0.450941\pi\)
0.153515 + 0.988146i \(0.450941\pi\)
\(90\) −1.76010e8 −0.282778
\(91\) 6.85750e7 0.104828
\(92\) −4.05404e8 −0.589988
\(93\) −4.24662e8 −0.588667
\(94\) 6.42797e7 0.0849177
\(95\) −2.73231e8 −0.344171
\(96\) 4.99001e8 0.599627
\(97\) −1.21586e9 −1.39447 −0.697236 0.716841i \(-0.745587\pi\)
−0.697236 + 0.716841i \(0.745587\pi\)
\(98\) −7.49670e7 −0.0821019
\(99\) 6.83341e8 0.714956
\(100\) 2.96889e8 0.296889
\(101\) 1.90976e8 0.182614 0.0913069 0.995823i \(-0.470896\pi\)
0.0913069 + 0.995823i \(0.470896\pi\)
\(102\) 6.56369e8 0.600409
\(103\) 1.22281e9 1.07051 0.535255 0.844690i \(-0.320215\pi\)
0.535255 + 0.844690i \(0.320215\pi\)
\(104\) 3.17519e8 0.266145
\(105\) 2.04970e8 0.164566
\(106\) 1.31837e9 1.01429
\(107\) −2.24053e9 −1.65243 −0.826216 0.563353i \(-0.809511\pi\)
−0.826216 + 0.563353i \(0.809511\pi\)
\(108\) −9.16939e8 −0.648535
\(109\) −8.83074e8 −0.599208 −0.299604 0.954064i \(-0.596855\pi\)
−0.299604 + 0.954064i \(0.596855\pi\)
\(110\) −7.13860e8 −0.464886
\(111\) −7.02290e8 −0.439099
\(112\) 7.44020e7 0.0446790
\(113\) 3.37815e9 1.94907 0.974533 0.224246i \(-0.0719920\pi\)
0.974533 + 0.224246i \(0.0719920\pi\)
\(114\) 2.78981e8 0.154704
\(115\) −1.23284e9 −0.657302
\(116\) 1.04161e9 0.534127
\(117\) −3.70727e8 −0.182902
\(118\) 1.92189e9 0.912558
\(119\) 1.48022e9 0.676650
\(120\) 9.49063e8 0.417810
\(121\) 4.13543e8 0.175383
\(122\) −1.04464e9 −0.426920
\(123\) −2.05840e9 −0.810881
\(124\) −1.77856e9 −0.675569
\(125\) 2.93942e9 1.07688
\(126\) 4.05284e8 0.143249
\(127\) −2.64803e8 −0.0903247 −0.0451623 0.998980i \(-0.514381\pi\)
−0.0451623 + 0.998980i \(0.514381\pi\)
\(128\) 2.29623e9 0.756084
\(129\) 2.25375e9 0.716560
\(130\) 3.87284e8 0.118928
\(131\) −1.88042e9 −0.557872 −0.278936 0.960310i \(-0.589982\pi\)
−0.278936 + 0.960310i \(0.589982\pi\)
\(132\) −1.47788e9 −0.423698
\(133\) 6.29147e8 0.174349
\(134\) 7.66995e8 0.205504
\(135\) −2.78842e9 −0.722530
\(136\) 6.85376e9 1.71792
\(137\) 4.25112e9 1.03101 0.515503 0.856888i \(-0.327605\pi\)
0.515503 + 0.856888i \(0.327605\pi\)
\(138\) 1.25878e9 0.295457
\(139\) 4.96899e9 1.12902 0.564510 0.825426i \(-0.309065\pi\)
0.564510 + 0.825426i \(0.309065\pi\)
\(140\) 8.58452e8 0.188860
\(141\) 4.04684e8 0.0862244
\(142\) −1.12595e9 −0.232391
\(143\) −1.50359e9 −0.300689
\(144\) −4.02229e8 −0.0779547
\(145\) 3.16755e9 0.595068
\(146\) 4.11996e9 0.750421
\(147\) −4.71969e8 −0.0833652
\(148\) −2.94131e9 −0.503922
\(149\) −5.07315e9 −0.843217 −0.421608 0.906778i \(-0.638534\pi\)
−0.421608 + 0.906778i \(0.638534\pi\)
\(150\) −9.21840e8 −0.148677
\(151\) −7.63995e9 −1.19590 −0.597949 0.801534i \(-0.704018\pi\)
−0.597949 + 0.801534i \(0.704018\pi\)
\(152\) 2.91310e9 0.442649
\(153\) −8.00228e9 −1.18060
\(154\) 1.64375e9 0.235500
\(155\) −5.40860e9 −0.752648
\(156\) 8.01781e8 0.108391
\(157\) 3.63784e9 0.477854 0.238927 0.971038i \(-0.423204\pi\)
0.238927 + 0.971038i \(0.423204\pi\)
\(158\) −5.10431e9 −0.651599
\(159\) 8.30004e9 1.02990
\(160\) 6.35540e9 0.766661
\(161\) 2.83875e9 0.332974
\(162\) −4.75352e8 −0.0542248
\(163\) 6.37653e9 0.707523 0.353761 0.935336i \(-0.384902\pi\)
0.353761 + 0.935336i \(0.384902\pi\)
\(164\) −8.62094e9 −0.930588
\(165\) −4.49424e9 −0.472039
\(166\) 1.12587e8 0.0115081
\(167\) 1.82469e9 0.181536 0.0907682 0.995872i \(-0.471068\pi\)
0.0907682 + 0.995872i \(0.471068\pi\)
\(168\) −2.18533e9 −0.211653
\(169\) 8.15731e8 0.0769231
\(170\) 8.35968e9 0.767661
\(171\) −3.40127e9 −0.304199
\(172\) 9.43911e9 0.822343
\(173\) −1.53692e10 −1.30450 −0.652252 0.758003i \(-0.726175\pi\)
−0.652252 + 0.758003i \(0.726175\pi\)
\(174\) −3.23420e9 −0.267483
\(175\) −2.07890e9 −0.167557
\(176\) −1.63136e9 −0.128157
\(177\) 1.20996e10 0.926601
\(178\) −2.36331e9 −0.176454
\(179\) 1.82802e10 1.33089 0.665445 0.746447i \(-0.268242\pi\)
0.665445 + 0.746447i \(0.268242\pi\)
\(180\) −4.64093e9 −0.329517
\(181\) 2.28590e10 1.58308 0.791541 0.611116i \(-0.209279\pi\)
0.791541 + 0.611116i \(0.209279\pi\)
\(182\) −8.91767e8 −0.0602463
\(183\) −6.57671e9 −0.433490
\(184\) 1.31441e10 0.845377
\(185\) −8.94454e9 −0.561417
\(186\) 5.52241e9 0.338315
\(187\) −3.24556e10 −1.94090
\(188\) 1.69489e9 0.0989534
\(189\) 6.42066e9 0.366017
\(190\) 3.55317e9 0.197799
\(191\) −1.64014e10 −0.891724 −0.445862 0.895102i \(-0.647103\pi\)
−0.445862 + 0.895102i \(0.647103\pi\)
\(192\) −5.19020e9 −0.275631
\(193\) −4.24012e9 −0.219973 −0.109987 0.993933i \(-0.535081\pi\)
−0.109987 + 0.993933i \(0.535081\pi\)
\(194\) 1.58113e10 0.801422
\(195\) 2.43822e9 0.120758
\(196\) −1.97669e9 −0.0956721
\(197\) −9.28108e9 −0.439036 −0.219518 0.975608i \(-0.570449\pi\)
−0.219518 + 0.975608i \(0.570449\pi\)
\(198\) −8.88635e9 −0.410894
\(199\) −3.26203e10 −1.47451 −0.737257 0.675613i \(-0.763879\pi\)
−0.737257 + 0.675613i \(0.763879\pi\)
\(200\) −9.62579e9 −0.425404
\(201\) 4.82875e9 0.208667
\(202\) −2.48351e9 −0.104951
\(203\) −7.29364e9 −0.301448
\(204\) 1.73067e10 0.699648
\(205\) −2.62163e10 −1.03676
\(206\) −1.59017e10 −0.615236
\(207\) −1.53467e10 −0.580964
\(208\) 8.85046e8 0.0327854
\(209\) −1.37948e10 −0.500102
\(210\) −2.66549e9 −0.0945781
\(211\) −3.31280e10 −1.15060 −0.575300 0.817943i \(-0.695115\pi\)
−0.575300 + 0.817943i \(0.695115\pi\)
\(212\) 3.47620e10 1.18194
\(213\) −7.08860e9 −0.235967
\(214\) 2.91365e10 0.949674
\(215\) 2.87044e10 0.916168
\(216\) 2.97292e10 0.929268
\(217\) 1.24539e10 0.381274
\(218\) 1.14837e10 0.344373
\(219\) 2.59380e10 0.761969
\(220\) −1.88226e10 −0.541725
\(221\) 1.76079e10 0.496525
\(222\) 9.13277e9 0.252356
\(223\) 4.44395e10 1.20336 0.601682 0.798735i \(-0.294497\pi\)
0.601682 + 0.798735i \(0.294497\pi\)
\(224\) −1.46341e10 −0.388373
\(225\) 1.12388e10 0.292348
\(226\) −4.39304e10 −1.12015
\(227\) −4.04362e10 −1.01077 −0.505387 0.862893i \(-0.668650\pi\)
−0.505387 + 0.862893i \(0.668650\pi\)
\(228\) 7.35601e9 0.180275
\(229\) −2.65160e10 −0.637159 −0.318579 0.947896i \(-0.603206\pi\)
−0.318579 + 0.947896i \(0.603206\pi\)
\(230\) 1.60321e10 0.377760
\(231\) 1.03485e10 0.239124
\(232\) −3.37713e10 −0.765336
\(233\) −5.44565e10 −1.21045 −0.605227 0.796053i \(-0.706918\pi\)
−0.605227 + 0.796053i \(0.706918\pi\)
\(234\) 4.82103e9 0.105116
\(235\) 5.15416e9 0.110243
\(236\) 5.06754e10 1.06339
\(237\) −3.21351e10 −0.661626
\(238\) −1.92491e10 −0.388879
\(239\) −4.72578e10 −0.936878 −0.468439 0.883496i \(-0.655183\pi\)
−0.468439 + 0.883496i \(0.655183\pi\)
\(240\) 2.64540e9 0.0514685
\(241\) −2.80478e10 −0.535577 −0.267789 0.963478i \(-0.586293\pi\)
−0.267789 + 0.963478i \(0.586293\pi\)
\(242\) −5.37783e9 −0.100795
\(243\) −5.56281e10 −1.02345
\(244\) −2.75444e10 −0.497484
\(245\) −6.01111e9 −0.106588
\(246\) 2.67680e10 0.466024
\(247\) 7.48399e9 0.127937
\(248\) 5.76647e10 0.968004
\(249\) 7.08813e8 0.0116852
\(250\) −3.82250e10 −0.618895
\(251\) −7.19345e10 −1.14395 −0.571973 0.820272i \(-0.693822\pi\)
−0.571973 + 0.820272i \(0.693822\pi\)
\(252\) 1.06863e10 0.166926
\(253\) −6.22432e10 −0.955101
\(254\) 3.44357e9 0.0519108
\(255\) 5.26299e10 0.779474
\(256\) −6.23190e10 −0.906861
\(257\) 2.70525e10 0.386819 0.193409 0.981118i \(-0.438045\pi\)
0.193409 + 0.981118i \(0.438045\pi\)
\(258\) −2.93084e10 −0.411816
\(259\) 2.05959e10 0.284401
\(260\) 1.02117e10 0.138585
\(261\) 3.94306e10 0.525958
\(262\) 2.44535e10 0.320617
\(263\) −1.25065e11 −1.61189 −0.805945 0.591990i \(-0.798343\pi\)
−0.805945 + 0.591990i \(0.798343\pi\)
\(264\) 4.79161e10 0.607105
\(265\) 1.05711e11 1.31679
\(266\) −8.18160e9 −0.100201
\(267\) −1.48787e10 −0.179169
\(268\) 2.02237e10 0.239471
\(269\) −2.17977e10 −0.253820 −0.126910 0.991914i \(-0.540506\pi\)
−0.126910 + 0.991914i \(0.540506\pi\)
\(270\) 3.62613e10 0.415247
\(271\) 6.88862e10 0.775837 0.387918 0.921694i \(-0.373194\pi\)
0.387918 + 0.921694i \(0.373194\pi\)
\(272\) 1.91041e10 0.211624
\(273\) −5.61428e9 −0.0611734
\(274\) −5.52827e10 −0.592532
\(275\) 4.55824e10 0.480619
\(276\) 3.31907e10 0.344291
\(277\) 1.37990e10 0.140828 0.0704138 0.997518i \(-0.477568\pi\)
0.0704138 + 0.997518i \(0.477568\pi\)
\(278\) −6.46181e10 −0.648863
\(279\) −6.73279e10 −0.665236
\(280\) −2.78329e10 −0.270612
\(281\) −9.32170e10 −0.891901 −0.445950 0.895058i \(-0.647134\pi\)
−0.445950 + 0.895058i \(0.647134\pi\)
\(282\) −5.26262e9 −0.0495543
\(283\) −2.07181e10 −0.192004 −0.0960021 0.995381i \(-0.530606\pi\)
−0.0960021 + 0.995381i \(0.530606\pi\)
\(284\) −2.96883e10 −0.270802
\(285\) 2.23696e10 0.200843
\(286\) 1.95531e10 0.172810
\(287\) 6.03661e10 0.525200
\(288\) 7.91140e10 0.677622
\(289\) 2.61484e11 2.20498
\(290\) −4.11916e10 −0.341994
\(291\) 9.95432e10 0.813754
\(292\) 1.08633e11 0.874455
\(293\) −5.83186e10 −0.462277 −0.231139 0.972921i \(-0.574245\pi\)
−0.231139 + 0.972921i \(0.574245\pi\)
\(294\) 6.13761e9 0.0479111
\(295\) 1.54104e11 1.18472
\(296\) 9.53638e10 0.722056
\(297\) −1.40781e11 −1.04988
\(298\) 6.59726e10 0.484608
\(299\) 3.37682e10 0.244336
\(300\) −2.43065e10 −0.173252
\(301\) −6.60952e10 −0.464110
\(302\) 9.93520e10 0.687299
\(303\) −1.56354e10 −0.106566
\(304\) 8.11993e9 0.0545282
\(305\) −8.37626e10 −0.554244
\(306\) 1.04064e11 0.678506
\(307\) 1.22866e11 0.789421 0.394710 0.918806i \(-0.370845\pi\)
0.394710 + 0.918806i \(0.370845\pi\)
\(308\) 4.33413e10 0.274425
\(309\) −1.00112e11 −0.624704
\(310\) 7.03349e10 0.432557
\(311\) 1.28181e11 0.776963 0.388482 0.921456i \(-0.373000\pi\)
0.388482 + 0.921456i \(0.373000\pi\)
\(312\) −2.59955e10 −0.155311
\(313\) 1.77718e11 1.04660 0.523301 0.852148i \(-0.324700\pi\)
0.523301 + 0.852148i \(0.324700\pi\)
\(314\) −4.73074e10 −0.274629
\(315\) 3.24970e10 0.185971
\(316\) −1.34588e11 −0.759299
\(317\) 3.20298e11 1.78151 0.890755 0.454485i \(-0.150177\pi\)
0.890755 + 0.454485i \(0.150177\pi\)
\(318\) −1.07936e11 −0.591895
\(319\) 1.59922e11 0.864671
\(320\) −6.61037e10 −0.352412
\(321\) 1.83434e11 0.964288
\(322\) −3.69159e10 −0.191365
\(323\) 1.61545e11 0.825813
\(324\) −1.25338e10 −0.0631874
\(325\) −2.47294e10 −0.122953
\(326\) −8.29222e10 −0.406623
\(327\) 7.22979e10 0.349672
\(328\) 2.79510e11 1.33341
\(329\) −1.18681e10 −0.0558468
\(330\) 5.84443e10 0.271287
\(331\) 2.60895e10 0.119465 0.0597324 0.998214i \(-0.480975\pi\)
0.0597324 + 0.998214i \(0.480975\pi\)
\(332\) 2.96863e9 0.0134102
\(333\) −1.11344e11 −0.496214
\(334\) −2.37287e10 −0.104331
\(335\) 6.15002e10 0.266793
\(336\) −6.09135e9 −0.0260727
\(337\) −1.76487e11 −0.745379 −0.372689 0.927956i \(-0.621564\pi\)
−0.372689 + 0.927956i \(0.621564\pi\)
\(338\) −1.06080e10 −0.0442087
\(339\) −2.76572e11 −1.13739
\(340\) 2.20423e11 0.894544
\(341\) −2.73068e11 −1.09364
\(342\) 4.42310e10 0.174827
\(343\) 1.38413e10 0.0539949
\(344\) −3.06037e11 −1.17831
\(345\) 1.00933e11 0.383573
\(346\) 1.99866e11 0.749715
\(347\) −5.10860e11 −1.89156 −0.945779 0.324812i \(-0.894699\pi\)
−0.945779 + 0.324812i \(0.894699\pi\)
\(348\) −8.52775e10 −0.311694
\(349\) −3.51080e11 −1.26675 −0.633375 0.773845i \(-0.718331\pi\)
−0.633375 + 0.773845i \(0.718331\pi\)
\(350\) 2.70345e10 0.0962970
\(351\) 7.63767e10 0.268583
\(352\) 3.20870e11 1.11401
\(353\) −5.14719e11 −1.76435 −0.882173 0.470926i \(-0.843920\pi\)
−0.882173 + 0.470926i \(0.843920\pi\)
\(354\) −1.57347e11 −0.532529
\(355\) −9.02822e10 −0.301699
\(356\) −6.23145e10 −0.205619
\(357\) −1.21186e11 −0.394864
\(358\) −2.37721e11 −0.764880
\(359\) −1.19334e11 −0.379176 −0.189588 0.981864i \(-0.560715\pi\)
−0.189588 + 0.981864i \(0.560715\pi\)
\(360\) 1.50469e11 0.472156
\(361\) −2.54025e11 −0.787217
\(362\) −2.97265e11 −0.909818
\(363\) −3.38571e10 −0.102346
\(364\) −2.35136e10 −0.0702042
\(365\) 3.30352e11 0.974226
\(366\) 8.55253e10 0.249132
\(367\) 2.44481e11 0.703473 0.351737 0.936099i \(-0.385591\pi\)
0.351737 + 0.936099i \(0.385591\pi\)
\(368\) 3.66376e10 0.104139
\(369\) −3.26349e11 −0.916354
\(370\) 1.16317e11 0.322653
\(371\) −2.43413e11 −0.667054
\(372\) 1.45612e11 0.394233
\(373\) 6.57776e10 0.175949 0.0879747 0.996123i \(-0.471961\pi\)
0.0879747 + 0.996123i \(0.471961\pi\)
\(374\) 4.22062e11 1.11546
\(375\) −2.40652e11 −0.628419
\(376\) −5.49520e10 −0.141787
\(377\) −8.67613e10 −0.221202
\(378\) −8.34959e10 −0.210355
\(379\) 2.61508e10 0.0651042 0.0325521 0.999470i \(-0.489637\pi\)
0.0325521 + 0.999470i \(0.489637\pi\)
\(380\) 9.36880e10 0.230493
\(381\) 2.16796e10 0.0527096
\(382\) 2.13288e11 0.512486
\(383\) −6.46212e11 −1.53455 −0.767274 0.641319i \(-0.778388\pi\)
−0.767274 + 0.641319i \(0.778388\pi\)
\(384\) −1.87994e11 −0.441218
\(385\) 1.31801e11 0.305736
\(386\) 5.51397e10 0.126422
\(387\) 3.57321e11 0.809765
\(388\) 4.16904e11 0.933885
\(389\) −5.23046e11 −1.15815 −0.579077 0.815273i \(-0.696587\pi\)
−0.579077 + 0.815273i \(0.696587\pi\)
\(390\) −3.17073e10 −0.0694013
\(391\) 7.28900e11 1.57715
\(392\) 6.40885e10 0.137086
\(393\) 1.53952e11 0.325550
\(394\) 1.20694e11 0.252320
\(395\) −4.09281e11 −0.845931
\(396\) −2.34310e11 −0.478809
\(397\) −3.11556e10 −0.0629476 −0.0314738 0.999505i \(-0.510020\pi\)
−0.0314738 + 0.999505i \(0.510020\pi\)
\(398\) 4.24203e11 0.847422
\(399\) −5.15087e10 −0.101743
\(400\) −2.68308e10 −0.0524039
\(401\) −1.58937e11 −0.306956 −0.153478 0.988152i \(-0.549047\pi\)
−0.153478 + 0.988152i \(0.549047\pi\)
\(402\) −6.27944e10 −0.119923
\(403\) 1.48145e11 0.279779
\(404\) −6.54837e10 −0.122297
\(405\) −3.81154e10 −0.0703967
\(406\) 9.48485e10 0.173246
\(407\) −4.51590e11 −0.815773
\(408\) −5.61122e11 −1.00251
\(409\) −7.35058e11 −1.29887 −0.649437 0.760416i \(-0.724995\pi\)
−0.649437 + 0.760416i \(0.724995\pi\)
\(410\) 3.40924e11 0.595841
\(411\) −3.48042e11 −0.601650
\(412\) −4.19287e11 −0.716926
\(413\) −3.54843e11 −0.600151
\(414\) 1.99573e11 0.333887
\(415\) 9.02762e9 0.0149402
\(416\) −1.74079e11 −0.284988
\(417\) −4.06815e11 −0.658847
\(418\) 1.79392e11 0.287415
\(419\) 9.05781e11 1.43569 0.717844 0.696204i \(-0.245129\pi\)
0.717844 + 0.696204i \(0.245129\pi\)
\(420\) −7.02821e10 −0.110210
\(421\) 7.38544e11 1.14579 0.572897 0.819627i \(-0.305819\pi\)
0.572897 + 0.819627i \(0.305819\pi\)
\(422\) 4.30806e11 0.661265
\(423\) 6.41606e10 0.0974398
\(424\) −1.12706e12 −1.69356
\(425\) −5.33794e11 −0.793640
\(426\) 9.21821e10 0.135614
\(427\) 1.92873e11 0.280767
\(428\) 7.68253e11 1.10664
\(429\) 1.23100e11 0.175469
\(430\) −3.73279e11 −0.526533
\(431\) 8.85724e11 1.23638 0.618188 0.786030i \(-0.287867\pi\)
0.618188 + 0.786030i \(0.287867\pi\)
\(432\) 8.28667e10 0.114473
\(433\) −4.72594e11 −0.646090 −0.323045 0.946384i \(-0.604706\pi\)
−0.323045 + 0.946384i \(0.604706\pi\)
\(434\) −1.61954e11 −0.219123
\(435\) −2.59329e11 −0.347256
\(436\) 3.02796e11 0.401293
\(437\) 3.09810e11 0.406376
\(438\) −3.37304e11 −0.437913
\(439\) −3.84141e11 −0.493628 −0.246814 0.969063i \(-0.579384\pi\)
−0.246814 + 0.969063i \(0.579384\pi\)
\(440\) 6.10271e11 0.776222
\(441\) −7.48282e10 −0.0942088
\(442\) −2.28977e11 −0.285360
\(443\) −1.13925e12 −1.40541 −0.702706 0.711480i \(-0.748025\pi\)
−0.702706 + 0.711480i \(0.748025\pi\)
\(444\) 2.40807e11 0.294067
\(445\) −1.89499e11 −0.229079
\(446\) −5.77903e11 −0.691590
\(447\) 4.15342e11 0.492065
\(448\) 1.52211e11 0.178524
\(449\) −6.62932e9 −0.00769769 −0.00384884 0.999993i \(-0.501225\pi\)
−0.00384884 + 0.999993i \(0.501225\pi\)
\(450\) −1.46153e11 −0.168016
\(451\) −1.32360e12 −1.50648
\(452\) −1.15833e12 −1.30530
\(453\) 6.25489e11 0.697875
\(454\) 5.25844e11 0.580906
\(455\) −7.15049e10 −0.0782140
\(456\) −2.38498e11 −0.258311
\(457\) 1.43734e12 1.54148 0.770739 0.637150i \(-0.219887\pi\)
0.770739 + 0.637150i \(0.219887\pi\)
\(458\) 3.44821e11 0.366184
\(459\) 1.64862e12 1.73366
\(460\) 4.22726e11 0.440198
\(461\) −1.03478e12 −1.06707 −0.533535 0.845778i \(-0.679137\pi\)
−0.533535 + 0.845778i \(0.679137\pi\)
\(462\) −1.34575e11 −0.137428
\(463\) 4.37463e11 0.442412 0.221206 0.975227i \(-0.429001\pi\)
0.221206 + 0.975227i \(0.429001\pi\)
\(464\) −9.41337e10 −0.0942788
\(465\) 4.42806e11 0.439213
\(466\) 7.08168e11 0.695664
\(467\) −1.37128e12 −1.33414 −0.667070 0.744995i \(-0.732452\pi\)
−0.667070 + 0.744995i \(0.732452\pi\)
\(468\) 1.27118e11 0.122490
\(469\) −1.41612e11 −0.135151
\(470\) −6.70261e10 −0.0633583
\(471\) −2.97833e11 −0.278855
\(472\) −1.64301e12 −1.52370
\(473\) 1.44922e12 1.33125
\(474\) 4.17894e11 0.380245
\(475\) −2.26882e11 −0.204494
\(476\) −5.07550e11 −0.453156
\(477\) 1.31593e12 1.16386
\(478\) 6.14553e11 0.538436
\(479\) 1.16134e12 1.00798 0.503988 0.863711i \(-0.331866\pi\)
0.503988 + 0.863711i \(0.331866\pi\)
\(480\) −5.20322e11 −0.447390
\(481\) 2.44997e11 0.208693
\(482\) 3.64741e11 0.307803
\(483\) −2.32411e11 −0.194309
\(484\) −1.41799e11 −0.117455
\(485\) 1.26781e12 1.04044
\(486\) 7.23403e11 0.588190
\(487\) 1.98588e12 1.59982 0.799911 0.600118i \(-0.204880\pi\)
0.799911 + 0.600118i \(0.204880\pi\)
\(488\) 8.93050e11 0.712831
\(489\) −5.22052e11 −0.412880
\(490\) 7.81701e10 0.0612574
\(491\) 6.43114e11 0.499369 0.249684 0.968327i \(-0.419673\pi\)
0.249684 + 0.968327i \(0.419673\pi\)
\(492\) 7.05803e11 0.543051
\(493\) −1.87277e12 −1.42782
\(494\) −9.73239e10 −0.0735272
\(495\) −7.12538e11 −0.533439
\(496\) 1.60734e11 0.119245
\(497\) 2.07885e11 0.152834
\(498\) −9.21759e9 −0.00671561
\(499\) 7.42912e11 0.536395 0.268198 0.963364i \(-0.413572\pi\)
0.268198 + 0.963364i \(0.413572\pi\)
\(500\) −1.00789e12 −0.721190
\(501\) −1.49388e11 −0.105937
\(502\) 9.35456e11 0.657441
\(503\) 7.15953e11 0.498687 0.249344 0.968415i \(-0.419785\pi\)
0.249344 + 0.968415i \(0.419785\pi\)
\(504\) −3.46472e11 −0.239184
\(505\) −1.99136e11 −0.136251
\(506\) 8.09427e11 0.548909
\(507\) −6.67845e10 −0.0448890
\(508\) 9.07981e10 0.0604909
\(509\) 2.21598e11 0.146331 0.0731656 0.997320i \(-0.476690\pi\)
0.0731656 + 0.997320i \(0.476690\pi\)
\(510\) −6.84413e11 −0.447974
\(511\) −7.60675e11 −0.493520
\(512\) −3.65255e11 −0.234899
\(513\) 7.00724e11 0.446703
\(514\) −3.51797e11 −0.222310
\(515\) −1.27505e12 −0.798723
\(516\) −7.72787e11 −0.479884
\(517\) 2.60222e11 0.160191
\(518\) −2.67834e11 −0.163449
\(519\) 1.25829e12 0.761252
\(520\) −3.31085e11 −0.198575
\(521\) −2.86380e12 −1.70283 −0.851417 0.524489i \(-0.824256\pi\)
−0.851417 + 0.524489i \(0.824256\pi\)
\(522\) −5.12766e11 −0.302275
\(523\) 1.29093e12 0.754474 0.377237 0.926117i \(-0.376874\pi\)
0.377237 + 0.926117i \(0.376874\pi\)
\(524\) 6.44776e11 0.373610
\(525\) 1.70201e11 0.0977788
\(526\) 1.62638e12 0.926375
\(527\) 3.19777e12 1.80592
\(528\) 1.33561e11 0.0747869
\(529\) −4.03273e11 −0.223897
\(530\) −1.37470e12 −0.756775
\(531\) 1.91833e12 1.04713
\(532\) −2.15727e11 −0.116762
\(533\) 7.18083e11 0.385392
\(534\) 1.93486e11 0.102971
\(535\) 2.33626e12 1.23290
\(536\) −6.55695e11 −0.343131
\(537\) −1.49661e12 −0.776650
\(538\) 2.83464e11 0.145874
\(539\) −3.03488e11 −0.154879
\(540\) 9.56117e11 0.483882
\(541\) 3.33288e12 1.67275 0.836376 0.548156i \(-0.184670\pi\)
0.836376 + 0.548156i \(0.184670\pi\)
\(542\) −8.95815e11 −0.445884
\(543\) −1.87148e12 −0.923818
\(544\) −3.75756e12 −1.83955
\(545\) 9.20805e11 0.447078
\(546\) 7.30097e10 0.0351571
\(547\) 9.35481e11 0.446778 0.223389 0.974729i \(-0.428288\pi\)
0.223389 + 0.974729i \(0.428288\pi\)
\(548\) −1.45766e12 −0.690469
\(549\) −1.04270e12 −0.489875
\(550\) −5.92766e11 −0.276218
\(551\) −7.95999e11 −0.367900
\(552\) −1.07612e12 −0.493325
\(553\) 9.42418e11 0.428529
\(554\) −1.79445e11 −0.0809354
\(555\) 7.32296e11 0.327618
\(556\) −1.70381e12 −0.756110
\(557\) 1.23154e12 0.542125 0.271062 0.962562i \(-0.412625\pi\)
0.271062 + 0.962562i \(0.412625\pi\)
\(558\) 8.75550e11 0.382320
\(559\) −7.86232e11 −0.340563
\(560\) −7.75810e10 −0.0333357
\(561\) 2.65717e12 1.13262
\(562\) 1.21222e12 0.512587
\(563\) 1.06358e11 0.0446153 0.0223076 0.999751i \(-0.492899\pi\)
0.0223076 + 0.999751i \(0.492899\pi\)
\(564\) −1.38762e11 −0.0577449
\(565\) −3.52249e12 −1.45423
\(566\) 2.69424e11 0.110347
\(567\) 8.77651e10 0.0356614
\(568\) 9.62559e11 0.388025
\(569\) −3.86394e12 −1.54535 −0.772673 0.634805i \(-0.781081\pi\)
−0.772673 + 0.634805i \(0.781081\pi\)
\(570\) −2.90901e11 −0.115427
\(571\) 5.00767e12 1.97139 0.985697 0.168529i \(-0.0539016\pi\)
0.985697 + 0.168529i \(0.0539016\pi\)
\(572\) 5.15565e11 0.201373
\(573\) 1.34279e12 0.520372
\(574\) −7.85018e11 −0.301839
\(575\) −1.02371e12 −0.390544
\(576\) −8.22879e11 −0.311483
\(577\) −4.06683e12 −1.52744 −0.763722 0.645545i \(-0.776630\pi\)
−0.763722 + 0.645545i \(0.776630\pi\)
\(578\) −3.40041e12 −1.26723
\(579\) 3.47142e11 0.128367
\(580\) −1.08612e12 −0.398520
\(581\) −2.07872e10 −0.00756837
\(582\) −1.29449e12 −0.467675
\(583\) 5.33714e12 1.91337
\(584\) −3.52211e12 −1.25298
\(585\) 3.86567e11 0.136466
\(586\) 7.58390e11 0.265677
\(587\) −2.49711e12 −0.868093 −0.434047 0.900890i \(-0.642915\pi\)
−0.434047 + 0.900890i \(0.642915\pi\)
\(588\) 1.61833e11 0.0558301
\(589\) 1.35917e12 0.465324
\(590\) −2.00401e12 −0.680873
\(591\) 7.59849e11 0.256203
\(592\) 2.65815e11 0.0889472
\(593\) −1.70678e12 −0.566803 −0.283402 0.959001i \(-0.591463\pi\)
−0.283402 + 0.959001i \(0.591463\pi\)
\(594\) 1.83075e12 0.603380
\(595\) −1.54346e12 −0.504858
\(596\) 1.73953e12 0.564706
\(597\) 2.67065e12 0.860462
\(598\) −4.39131e11 −0.140423
\(599\) −2.44672e11 −0.0776540 −0.0388270 0.999246i \(-0.512362\pi\)
−0.0388270 + 0.999246i \(0.512362\pi\)
\(600\) 7.88071e11 0.248247
\(601\) −1.62513e12 −0.508104 −0.254052 0.967191i \(-0.581763\pi\)
−0.254052 + 0.967191i \(0.581763\pi\)
\(602\) 8.59520e11 0.266730
\(603\) 7.65574e11 0.235808
\(604\) 2.61965e12 0.800899
\(605\) −4.31213e11 −0.130856
\(606\) 2.03327e11 0.0612446
\(607\) −5.56025e12 −1.66244 −0.831219 0.555945i \(-0.812356\pi\)
−0.831219 + 0.555945i \(0.812356\pi\)
\(608\) −1.59710e12 −0.473987
\(609\) 5.97136e11 0.175912
\(610\) 1.08927e12 0.318531
\(611\) −1.41176e11 −0.0409803
\(612\) 2.74389e12 0.790653
\(613\) −4.99910e12 −1.42995 −0.714973 0.699152i \(-0.753561\pi\)
−0.714973 + 0.699152i \(0.753561\pi\)
\(614\) −1.59778e12 −0.453691
\(615\) 2.14635e12 0.605010
\(616\) −1.40522e12 −0.393216
\(617\) −2.07654e12 −0.576842 −0.288421 0.957504i \(-0.593130\pi\)
−0.288421 + 0.957504i \(0.593130\pi\)
\(618\) 1.30189e12 0.359025
\(619\) 5.43974e12 1.48926 0.744629 0.667479i \(-0.232626\pi\)
0.744629 + 0.667479i \(0.232626\pi\)
\(620\) 1.85455e12 0.504052
\(621\) 3.16171e12 0.853120
\(622\) −1.66690e12 −0.446531
\(623\) 4.36343e11 0.116046
\(624\) −7.24594e10 −0.0191322
\(625\) −1.37390e12 −0.360160
\(626\) −2.31109e12 −0.601496
\(627\) 1.12939e12 0.291838
\(628\) −1.24737e12 −0.320021
\(629\) 5.28836e12 1.34708
\(630\) −4.22600e11 −0.106880
\(631\) −5.35647e11 −0.134507 −0.0672537 0.997736i \(-0.521424\pi\)
−0.0672537 + 0.997736i \(0.521424\pi\)
\(632\) 4.36362e12 1.08798
\(633\) 2.71221e12 0.671440
\(634\) −4.16525e12 −1.02386
\(635\) 2.76117e11 0.0673926
\(636\) −2.84599e12 −0.689726
\(637\) 1.64648e11 0.0396214
\(638\) −2.07967e12 −0.496938
\(639\) −1.12386e12 −0.266660
\(640\) −2.39434e12 −0.564125
\(641\) −4.66531e12 −1.09149 −0.545744 0.837952i \(-0.683753\pi\)
−0.545744 + 0.837952i \(0.683753\pi\)
\(642\) −2.38542e12 −0.554189
\(643\) −5.57297e12 −1.28569 −0.642846 0.765995i \(-0.722246\pi\)
−0.642846 + 0.765995i \(0.722246\pi\)
\(644\) −9.73376e11 −0.222994
\(645\) −2.35005e12 −0.534636
\(646\) −2.10077e12 −0.474605
\(647\) 3.55930e12 0.798538 0.399269 0.916834i \(-0.369264\pi\)
0.399269 + 0.916834i \(0.369264\pi\)
\(648\) 4.06373e11 0.0905394
\(649\) 7.78037e12 1.72147
\(650\) 3.21588e11 0.0706627
\(651\) −1.01961e12 −0.222495
\(652\) −2.18644e12 −0.473832
\(653\) −1.46933e11 −0.0316235 −0.0158118 0.999875i \(-0.505033\pi\)
−0.0158118 + 0.999875i \(0.505033\pi\)
\(654\) −9.40182e11 −0.200961
\(655\) 1.96077e12 0.416237
\(656\) 7.79101e11 0.164258
\(657\) 4.11233e12 0.861080
\(658\) 1.54335e11 0.0320959
\(659\) −3.34926e12 −0.691774 −0.345887 0.938276i \(-0.612422\pi\)
−0.345887 + 0.938276i \(0.612422\pi\)
\(660\) 1.54102e12 0.316127
\(661\) 2.24643e11 0.0457707 0.0228853 0.999738i \(-0.492715\pi\)
0.0228853 + 0.999738i \(0.492715\pi\)
\(662\) −3.39275e11 −0.0686580
\(663\) −1.44157e12 −0.289751
\(664\) −9.62495e10 −0.0192151
\(665\) −6.56028e11 −0.130084
\(666\) 1.44795e12 0.285181
\(667\) −3.59159e12 −0.702621
\(668\) −6.25664e11 −0.121576
\(669\) −3.63830e12 −0.702232
\(670\) −7.99766e11 −0.153330
\(671\) −4.22899e12 −0.805351
\(672\) 1.19810e12 0.226638
\(673\) −8.26465e12 −1.55295 −0.776474 0.630150i \(-0.782994\pi\)
−0.776474 + 0.630150i \(0.782994\pi\)
\(674\) 2.29508e12 0.428379
\(675\) −2.31541e12 −0.429300
\(676\) −2.79705e11 −0.0515158
\(677\) −2.66007e10 −0.00486680 −0.00243340 0.999997i \(-0.500775\pi\)
−0.00243340 + 0.999997i \(0.500775\pi\)
\(678\) 3.59662e12 0.653673
\(679\) −2.91927e12 −0.527061
\(680\) −7.14660e12 −1.28177
\(681\) 3.31055e12 0.589845
\(682\) 3.55105e12 0.628532
\(683\) −8.70709e12 −1.53102 −0.765508 0.643427i \(-0.777512\pi\)
−0.765508 + 0.643427i \(0.777512\pi\)
\(684\) 1.16626e12 0.203724
\(685\) −4.43276e12 −0.769248
\(686\) −1.79996e11 −0.0310316
\(687\) 2.17088e12 0.371818
\(688\) −8.53041e11 −0.145152
\(689\) −2.89551e12 −0.489484
\(690\) −1.31256e12 −0.220444
\(691\) 4.43513e12 0.740039 0.370020 0.929024i \(-0.379351\pi\)
0.370020 + 0.929024i \(0.379351\pi\)
\(692\) 5.26995e12 0.873632
\(693\) 1.64070e12 0.270228
\(694\) 6.64337e12 1.08710
\(695\) −5.18130e12 −0.842378
\(696\) 2.76488e12 0.446617
\(697\) 1.55001e13 2.48764
\(698\) 4.56554e12 0.728018
\(699\) 4.45840e12 0.706369
\(700\) 7.12831e11 0.112214
\(701\) −3.34710e12 −0.523526 −0.261763 0.965132i \(-0.584304\pi\)
−0.261763 + 0.965132i \(0.584304\pi\)
\(702\) −9.93223e11 −0.154358
\(703\) 2.24775e12 0.347095
\(704\) −3.33743e12 −0.512077
\(705\) −4.21975e11 −0.0643333
\(706\) 6.69354e12 1.01399
\(707\) 4.58535e11 0.0690215
\(708\) −4.14883e12 −0.620549
\(709\) 5.93452e12 0.882018 0.441009 0.897503i \(-0.354621\pi\)
0.441009 + 0.897503i \(0.354621\pi\)
\(710\) 1.17405e12 0.173391
\(711\) −5.09486e12 −0.747685
\(712\) 2.02037e12 0.294626
\(713\) 6.13266e12 0.888682
\(714\) 1.57594e12 0.226933
\(715\) 1.56784e12 0.224349
\(716\) −6.26808e12 −0.891304
\(717\) 3.86903e12 0.546722
\(718\) 1.55186e12 0.217917
\(719\) −9.49911e12 −1.32557 −0.662785 0.748810i \(-0.730626\pi\)
−0.662785 + 0.748810i \(0.730626\pi\)
\(720\) 4.19415e11 0.0581631
\(721\) 2.93596e12 0.404615
\(722\) 3.30341e12 0.452424
\(723\) 2.29629e12 0.312540
\(724\) −7.83810e12 −1.06020
\(725\) 2.63023e12 0.353567
\(726\) 4.40287e11 0.0588195
\(727\) 9.01903e12 1.19744 0.598722 0.800957i \(-0.295675\pi\)
0.598722 + 0.800957i \(0.295675\pi\)
\(728\) 7.62362e11 0.100594
\(729\) 3.83483e12 0.502890
\(730\) −4.29599e12 −0.559900
\(731\) −1.69711e13 −2.19828
\(732\) 2.25508e12 0.290310
\(733\) −8.41189e12 −1.07628 −0.538141 0.842855i \(-0.680873\pi\)
−0.538141 + 0.842855i \(0.680873\pi\)
\(734\) −3.17930e12 −0.404295
\(735\) 4.92134e11 0.0622000
\(736\) −7.20622e12 −0.905227
\(737\) 3.10501e12 0.387668
\(738\) 4.24393e12 0.526641
\(739\) 2.10304e11 0.0259387 0.0129693 0.999916i \(-0.495872\pi\)
0.0129693 + 0.999916i \(0.495872\pi\)
\(740\) 3.06698e12 0.375983
\(741\) −6.12720e11 −0.0746586
\(742\) 3.16541e12 0.383365
\(743\) 1.14246e13 1.37528 0.687638 0.726053i \(-0.258647\pi\)
0.687638 + 0.726053i \(0.258647\pi\)
\(744\) −4.72105e12 −0.564885
\(745\) 5.28990e12 0.629136
\(746\) −8.55389e11 −0.101120
\(747\) 1.12379e11 0.0132051
\(748\) 1.11287e13 1.29983
\(749\) −5.37951e12 −0.624561
\(750\) 3.12951e12 0.361160
\(751\) 4.28464e12 0.491512 0.245756 0.969332i \(-0.420964\pi\)
0.245756 + 0.969332i \(0.420964\pi\)
\(752\) −1.53172e11 −0.0174663
\(753\) 5.88933e12 0.667558
\(754\) 1.12827e12 0.127128
\(755\) 7.96638e12 0.892277
\(756\) −2.20157e12 −0.245123
\(757\) −1.55155e13 −1.71726 −0.858629 0.512597i \(-0.828683\pi\)
−0.858629 + 0.512597i \(0.828683\pi\)
\(758\) −3.40073e11 −0.0374163
\(759\) 5.09589e12 0.557356
\(760\) −3.03757e12 −0.330267
\(761\) 2.21517e12 0.239429 0.119714 0.992808i \(-0.461802\pi\)
0.119714 + 0.992808i \(0.461802\pi\)
\(762\) −2.81928e11 −0.0302929
\(763\) −2.12026e12 −0.226480
\(764\) 5.62386e12 0.597192
\(765\) 8.34419e12 0.880862
\(766\) 8.40352e12 0.881925
\(767\) −4.22102e12 −0.440390
\(768\) 5.10210e12 0.529205
\(769\) 6.11684e12 0.630752 0.315376 0.948967i \(-0.397869\pi\)
0.315376 + 0.948967i \(0.397869\pi\)
\(770\) −1.71398e12 −0.175710
\(771\) −2.21480e12 −0.225731
\(772\) 1.45389e12 0.147317
\(773\) −4.80981e12 −0.484530 −0.242265 0.970210i \(-0.577890\pi\)
−0.242265 + 0.970210i \(0.577890\pi\)
\(774\) −4.64670e12 −0.465382
\(775\) −4.49112e12 −0.447195
\(776\) −1.35169e13 −1.33814
\(777\) −1.68620e12 −0.165964
\(778\) 6.80183e12 0.665606
\(779\) 6.58811e12 0.640977
\(780\) −8.36038e11 −0.0808724
\(781\) −4.55815e12 −0.438388
\(782\) −9.47881e12 −0.906408
\(783\) −8.12343e12 −0.772346
\(784\) 1.78639e11 0.0168871
\(785\) −3.79327e12 −0.356533
\(786\) −2.00203e12 −0.187098
\(787\) 1.82456e13 1.69540 0.847699 0.530478i \(-0.177988\pi\)
0.847699 + 0.530478i \(0.177988\pi\)
\(788\) 3.18238e12 0.294025
\(789\) 1.02392e13 0.940630
\(790\) 5.32240e12 0.486167
\(791\) 8.11095e12 0.736677
\(792\) 7.59684e12 0.686072
\(793\) 2.29432e12 0.206027
\(794\) 4.05156e11 0.0361768
\(795\) −8.65468e12 −0.768420
\(796\) 1.11851e13 0.987489
\(797\) −6.51130e12 −0.571617 −0.285809 0.958287i \(-0.592262\pi\)
−0.285809 + 0.958287i \(0.592262\pi\)
\(798\) 6.69833e11 0.0584728
\(799\) −3.04734e12 −0.264521
\(800\) 5.27732e12 0.455521
\(801\) −2.35894e12 −0.202474
\(802\) 2.06686e12 0.176412
\(803\) 1.66788e13 1.41561
\(804\) −1.65573e12 −0.139745
\(805\) −2.96004e12 −0.248437
\(806\) −1.92652e12 −0.160793
\(807\) 1.78460e12 0.148119
\(808\) 2.12313e12 0.175236
\(809\) −1.91839e13 −1.57459 −0.787296 0.616575i \(-0.788520\pi\)
−0.787296 + 0.616575i \(0.788520\pi\)
\(810\) 4.95662e11 0.0404579
\(811\) −6.92573e10 −0.00562175 −0.00281088 0.999996i \(-0.500895\pi\)
−0.00281088 + 0.999996i \(0.500895\pi\)
\(812\) 2.50091e12 0.201881
\(813\) −5.63976e12 −0.452745
\(814\) 5.87260e12 0.468836
\(815\) −6.64898e12 −0.527893
\(816\) −1.56406e12 −0.123495
\(817\) −7.21336e12 −0.566419
\(818\) 9.55890e12 0.746479
\(819\) −8.90115e11 −0.0691304
\(820\) 8.98928e12 0.694325
\(821\) 4.46303e12 0.342836 0.171418 0.985198i \(-0.445165\pi\)
0.171418 + 0.985198i \(0.445165\pi\)
\(822\) 4.52604e12 0.345776
\(823\) 1.45566e13 1.10602 0.553008 0.833176i \(-0.313480\pi\)
0.553008 + 0.833176i \(0.313480\pi\)
\(824\) 1.35942e13 1.02726
\(825\) −3.73187e12 −0.280468
\(826\) 4.61447e12 0.344915
\(827\) −1.55531e13 −1.15622 −0.578111 0.815958i \(-0.696210\pi\)
−0.578111 + 0.815958i \(0.696210\pi\)
\(828\) 5.26222e12 0.389074
\(829\) −8.28724e12 −0.609417 −0.304708 0.952446i \(-0.598559\pi\)
−0.304708 + 0.952446i \(0.598559\pi\)
\(830\) −1.17398e11 −0.00858634
\(831\) −1.12973e12 −0.0821808
\(832\) 1.81063e12 0.131001
\(833\) 3.55400e12 0.255750
\(834\) 5.29033e12 0.378648
\(835\) −1.90265e12 −0.135447
\(836\) 4.73010e12 0.334921
\(837\) 1.38708e13 0.976870
\(838\) −1.17790e13 −0.825109
\(839\) −1.55211e13 −1.08142 −0.540708 0.841210i \(-0.681844\pi\)
−0.540708 + 0.841210i \(0.681844\pi\)
\(840\) 2.27870e12 0.157917
\(841\) −5.27920e12 −0.363904
\(842\) −9.60422e12 −0.658503
\(843\) 7.63174e12 0.520475
\(844\) 1.13592e13 0.770562
\(845\) −8.50584e11 −0.0573934
\(846\) −8.34362e11 −0.0560000
\(847\) 9.92918e11 0.0662884
\(848\) −3.14155e12 −0.208623
\(849\) 1.69620e12 0.112045
\(850\) 6.94161e12 0.456116
\(851\) 1.01420e13 0.662887
\(852\) 2.43060e12 0.158029
\(853\) −2.47923e13 −1.60341 −0.801706 0.597718i \(-0.796074\pi\)
−0.801706 + 0.597718i \(0.796074\pi\)
\(854\) −2.50818e12 −0.161361
\(855\) 3.54659e12 0.226967
\(856\) −2.49084e13 −1.58568
\(857\) 5.26535e12 0.333437 0.166719 0.986005i \(-0.446683\pi\)
0.166719 + 0.986005i \(0.446683\pi\)
\(858\) −1.60083e12 −0.100844
\(859\) −1.83393e12 −0.114925 −0.0574624 0.998348i \(-0.518301\pi\)
−0.0574624 + 0.998348i \(0.518301\pi\)
\(860\) −9.84241e12 −0.613562
\(861\) −4.94222e12 −0.306484
\(862\) −1.15182e13 −0.710562
\(863\) −2.02830e13 −1.24475 −0.622376 0.782718i \(-0.713833\pi\)
−0.622376 + 0.782718i \(0.713833\pi\)
\(864\) −1.62990e13 −0.995057
\(865\) 1.60259e13 0.973309
\(866\) 6.14574e12 0.371316
\(867\) −2.14079e13 −1.28673
\(868\) −4.27031e12 −0.255341
\(869\) −2.06637e13 −1.22919
\(870\) 3.37239e12 0.199573
\(871\) −1.68453e12 −0.0991741
\(872\) −9.81732e12 −0.575001
\(873\) 1.57821e13 0.919601
\(874\) −4.02885e12 −0.233550
\(875\) 7.05754e12 0.407021
\(876\) −8.89384e12 −0.510294
\(877\) −3.39343e13 −1.93705 −0.968526 0.248913i \(-0.919927\pi\)
−0.968526 + 0.248913i \(0.919927\pi\)
\(878\) 4.99547e12 0.283695
\(879\) 4.77458e12 0.269765
\(880\) 1.70106e12 0.0956198
\(881\) 4.37156e12 0.244481 0.122240 0.992501i \(-0.460992\pi\)
0.122240 + 0.992501i \(0.460992\pi\)
\(882\) 9.73086e11 0.0541430
\(883\) 1.96116e13 1.08565 0.542825 0.839846i \(-0.317355\pi\)
0.542825 + 0.839846i \(0.317355\pi\)
\(884\) −6.03754e12 −0.332525
\(885\) −1.26166e13 −0.691350
\(886\) 1.48152e13 0.807709
\(887\) 1.70930e13 0.927177 0.463589 0.886051i \(-0.346562\pi\)
0.463589 + 0.886051i \(0.346562\pi\)
\(888\) −7.80750e12 −0.421360
\(889\) −6.35793e11 −0.0341395
\(890\) 2.46429e12 0.131655
\(891\) −1.92436e12 −0.102291
\(892\) −1.52378e13 −0.805899
\(893\) −1.29523e12 −0.0681578
\(894\) −5.40122e12 −0.282796
\(895\) −1.90612e13 −0.992996
\(896\) 5.51324e12 0.285773
\(897\) −2.76463e12 −0.142584
\(898\) 8.62095e10 0.00442396
\(899\) −1.57567e13 −0.804541
\(900\) −3.85368e12 −0.195787
\(901\) −6.25007e13 −3.15954
\(902\) 1.72125e13 0.865794
\(903\) 5.41126e12 0.270834
\(904\) 3.75556e13 1.87033
\(905\) −2.38357e13 −1.18116
\(906\) −8.13402e12 −0.401078
\(907\) 4.18652e12 0.205410 0.102705 0.994712i \(-0.467250\pi\)
0.102705 + 0.994712i \(0.467250\pi\)
\(908\) 1.38651e13 0.676921
\(909\) −2.47891e12 −0.120427
\(910\) 9.29870e11 0.0449506
\(911\) −3.75305e12 −0.180531 −0.0902656 0.995918i \(-0.528772\pi\)
−0.0902656 + 0.995918i \(0.528772\pi\)
\(912\) −6.64785e11 −0.0318203
\(913\) 4.55784e11 0.0217091
\(914\) −1.86916e13 −0.885908
\(915\) 6.85771e12 0.323433
\(916\) 9.09203e12 0.426708
\(917\) −4.51490e12 −0.210856
\(918\) −2.14391e13 −0.996355
\(919\) −8.78201e12 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(920\) −1.37057e13 −0.630748
\(921\) −1.00591e13 −0.460672
\(922\) 1.34565e13 0.613260
\(923\) 2.47289e12 0.112150
\(924\) −3.54839e12 −0.160143
\(925\) −7.42726e12 −0.333573
\(926\) −5.68889e12 −0.254260
\(927\) −1.58723e13 −0.705961
\(928\) 1.85151e13 0.819519
\(929\) 2.37759e13 1.04729 0.523643 0.851938i \(-0.324572\pi\)
0.523643 + 0.851938i \(0.324572\pi\)
\(930\) −5.75837e12 −0.252421
\(931\) 1.51058e12 0.0658978
\(932\) 1.86725e13 0.810647
\(933\) −1.04942e13 −0.453402
\(934\) 1.78325e13 0.766748
\(935\) 3.38423e13 1.44813
\(936\) −4.12145e12 −0.175513
\(937\) −1.35698e13 −0.575103 −0.287552 0.957765i \(-0.592841\pi\)
−0.287552 + 0.957765i \(0.592841\pi\)
\(938\) 1.84155e12 0.0776733
\(939\) −1.45499e13 −0.610752
\(940\) −1.76730e12 −0.0738305
\(941\) 1.13296e13 0.471045 0.235522 0.971869i \(-0.424320\pi\)
0.235522 + 0.971869i \(0.424320\pi\)
\(942\) 3.87309e12 0.160261
\(943\) 2.97260e13 1.22415
\(944\) −4.57969e12 −0.187699
\(945\) −6.69499e12 −0.273091
\(946\) −1.88460e13 −0.765086
\(947\) 3.80982e13 1.53932 0.769661 0.638453i \(-0.220425\pi\)
0.769661 + 0.638453i \(0.220425\pi\)
\(948\) 1.10188e13 0.443094
\(949\) −9.04858e12 −0.362145
\(950\) 2.95044e12 0.117525
\(951\) −2.62231e13 −1.03961
\(952\) 1.64559e13 0.649314
\(953\) −3.76777e13 −1.47967 −0.739837 0.672786i \(-0.765098\pi\)
−0.739837 + 0.672786i \(0.765098\pi\)
\(954\) −1.71127e13 −0.668884
\(955\) 1.71022e13 0.665328
\(956\) 1.62042e13 0.627432
\(957\) −1.30930e13 −0.504585
\(958\) −1.51024e13 −0.579297
\(959\) 1.02069e13 0.389683
\(960\) 5.41196e12 0.205652
\(961\) 4.65090e11 0.0175906
\(962\) −3.18601e12 −0.119939
\(963\) 2.90825e13 1.08972
\(964\) 9.61728e12 0.358679
\(965\) 4.42129e12 0.164125
\(966\) 3.02233e12 0.111672
\(967\) 1.67813e13 0.617172 0.308586 0.951196i \(-0.400144\pi\)
0.308586 + 0.951196i \(0.400144\pi\)
\(968\) 4.59745e12 0.168297
\(969\) −1.32258e13 −0.481909
\(970\) −1.64869e13 −0.597952
\(971\) −3.31649e13 −1.19727 −0.598635 0.801022i \(-0.704290\pi\)
−0.598635 + 0.801022i \(0.704290\pi\)
\(972\) 1.90743e13 0.685409
\(973\) 1.19306e13 0.426730
\(974\) −2.58249e13 −0.919439
\(975\) 2.02462e12 0.0717501
\(976\) 2.48927e12 0.0878109
\(977\) 3.27921e13 1.15145 0.575724 0.817644i \(-0.304720\pi\)
0.575724 + 0.817644i \(0.304720\pi\)
\(978\) 6.78890e12 0.237287
\(979\) −9.56736e12 −0.332866
\(980\) 2.06114e12 0.0713823
\(981\) 1.14625e13 0.395155
\(982\) −8.36323e12 −0.286994
\(983\) 2.84584e13 0.972120 0.486060 0.873925i \(-0.338434\pi\)
0.486060 + 0.873925i \(0.338434\pi\)
\(984\) −2.28837e13 −0.778122
\(985\) 9.67763e12 0.327571
\(986\) 2.43541e13 0.820589
\(987\) 9.71647e11 0.0325898
\(988\) −2.56618e12 −0.0856802
\(989\) −3.25471e13 −1.08176
\(990\) 9.26604e12 0.306574
\(991\) 1.68459e13 0.554834 0.277417 0.960750i \(-0.410522\pi\)
0.277417 + 0.960750i \(0.410522\pi\)
\(992\) −3.16145e13 −1.03654
\(993\) −2.13597e12 −0.0697145
\(994\) −2.70340e12 −0.0878357
\(995\) 3.40140e13 1.10016
\(996\) −2.43044e11 −0.00782561
\(997\) 5.13028e13 1.64442 0.822211 0.569183i \(-0.192740\pi\)
0.822211 + 0.569183i \(0.192740\pi\)
\(998\) −9.66103e12 −0.308273
\(999\) 2.29390e13 0.728669
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 91.10.a.a.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.5 12 1.1 even 1 trivial