Properties

Label 177.8.a.a.1.8
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.8
Root \(1.97136\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.97136 q^{2} -27.0000 q^{3} -124.114 q^{4} -339.775 q^{5} +53.2268 q^{6} +364.700 q^{7} +497.007 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-1.97136 q^{2} -27.0000 q^{3} -124.114 q^{4} -339.775 q^{5} +53.2268 q^{6} +364.700 q^{7} +497.007 q^{8} +729.000 q^{9} +669.820 q^{10} -3588.22 q^{11} +3351.07 q^{12} +857.316 q^{13} -718.956 q^{14} +9173.94 q^{15} +14906.8 q^{16} +3649.35 q^{17} -1437.12 q^{18} -9650.25 q^{19} +42170.8 q^{20} -9846.91 q^{21} +7073.68 q^{22} +116133. q^{23} -13419.2 q^{24} +37322.4 q^{25} -1690.08 q^{26} -19683.0 q^{27} -45264.3 q^{28} +247675. q^{29} -18085.1 q^{30} +114054. q^{31} -93003.6 q^{32} +96882.0 q^{33} -7194.19 q^{34} -123916. q^{35} -90478.9 q^{36} -194589. q^{37} +19024.1 q^{38} -23147.5 q^{39} -168871. q^{40} -594925. q^{41} +19411.8 q^{42} +247788. q^{43} +445348. q^{44} -247696. q^{45} -228939. q^{46} -1.30718e6 q^{47} -402483. q^{48} -690537. q^{49} -73575.9 q^{50} -98532.5 q^{51} -106405. q^{52} +1.20394e6 q^{53} +38802.3 q^{54} +1.21919e6 q^{55} +181259. q^{56} +260557. q^{57} -488257. q^{58} +205379. q^{59} -1.13861e6 q^{60} +428862. q^{61} -224843. q^{62} +265866. q^{63} -1.72472e6 q^{64} -291295. q^{65} -190989. q^{66} -2.01026e6 q^{67} -452934. q^{68} -3.13558e6 q^{69} +244284. q^{70} +1.87858e6 q^{71} +362318. q^{72} +2.35049e6 q^{73} +383605. q^{74} -1.00770e6 q^{75} +1.19773e6 q^{76} -1.30863e6 q^{77} +45632.2 q^{78} -3.74874e6 q^{79} -5.06496e6 q^{80} +531441. q^{81} +1.17281e6 q^{82} +3.21289e6 q^{83} +1.22214e6 q^{84} -1.23996e6 q^{85} -488480. q^{86} -6.68722e6 q^{87} -1.78337e6 q^{88} -544547. q^{89} +488299. q^{90} +312663. q^{91} -1.44136e7 q^{92} -3.07947e6 q^{93} +2.57692e6 q^{94} +3.27892e6 q^{95} +2.51110e6 q^{96} -2.91764e6 q^{97} +1.36130e6 q^{98} -2.61581e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.97136 −0.174245 −0.0871227 0.996198i \(-0.527767\pi\)
−0.0871227 + 0.996198i \(0.527767\pi\)
\(3\) −27.0000 −0.577350
\(4\) −124.114 −0.969639
\(5\) −339.775 −1.21562 −0.607809 0.794083i \(-0.707951\pi\)
−0.607809 + 0.794083i \(0.707951\pi\)
\(6\) 53.2268 0.100601
\(7\) 364.700 0.401877 0.200938 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(8\) 497.007 0.343200
\(9\) 729.000 0.333333
\(10\) 669.820 0.211816
\(11\) −3588.22 −0.812840 −0.406420 0.913686i \(-0.633223\pi\)
−0.406420 + 0.913686i \(0.633223\pi\)
\(12\) 3351.07 0.559821
\(13\) 857.316 0.108228 0.0541140 0.998535i \(-0.482767\pi\)
0.0541140 + 0.998535i \(0.482767\pi\)
\(14\) −718.956 −0.0700252
\(15\) 9173.94 0.701837
\(16\) 14906.8 0.909837
\(17\) 3649.35 0.180154 0.0900771 0.995935i \(-0.471289\pi\)
0.0900771 + 0.995935i \(0.471289\pi\)
\(18\) −1437.12 −0.0580818
\(19\) −9650.25 −0.322776 −0.161388 0.986891i \(-0.551597\pi\)
−0.161388 + 0.986891i \(0.551597\pi\)
\(20\) 42170.8 1.17871
\(21\) −9846.91 −0.232024
\(22\) 7073.68 0.141634
\(23\) 116133. 1.99024 0.995122 0.0986484i \(-0.0314519\pi\)
0.995122 + 0.0986484i \(0.0314519\pi\)
\(24\) −13419.2 −0.198147
\(25\) 37322.4 0.477727
\(26\) −1690.08 −0.0188582
\(27\) −19683.0 −0.192450
\(28\) −45264.3 −0.389675
\(29\) 247675. 1.88577 0.942886 0.333116i \(-0.108100\pi\)
0.942886 + 0.333116i \(0.108100\pi\)
\(30\) −18085.1 −0.122292
\(31\) 114054. 0.687617 0.343808 0.939040i \(-0.388283\pi\)
0.343808 + 0.939040i \(0.388283\pi\)
\(32\) −93003.6 −0.501735
\(33\) 96882.0 0.469293
\(34\) −7194.19 −0.0313910
\(35\) −123916. −0.488529
\(36\) −90478.9 −0.323213
\(37\) −194589. −0.631556 −0.315778 0.948833i \(-0.602266\pi\)
−0.315778 + 0.948833i \(0.602266\pi\)
\(38\) 19024.1 0.0562422
\(39\) −23147.5 −0.0624854
\(40\) −168871. −0.417201
\(41\) −594925. −1.34809 −0.674045 0.738691i \(-0.735444\pi\)
−0.674045 + 0.738691i \(0.735444\pi\)
\(42\) 19411.8 0.0404290
\(43\) 247788. 0.475270 0.237635 0.971354i \(-0.423628\pi\)
0.237635 + 0.971354i \(0.423628\pi\)
\(44\) 445348. 0.788161
\(45\) −247696. −0.405206
\(46\) −228939. −0.346791
\(47\) −1.30718e6 −1.83651 −0.918254 0.395993i \(-0.870400\pi\)
−0.918254 + 0.395993i \(0.870400\pi\)
\(48\) −402483. −0.525295
\(49\) −690537. −0.838495
\(50\) −73575.9 −0.0832416
\(51\) −98532.5 −0.104012
\(52\) −106405. −0.104942
\(53\) 1.20394e6 1.11081 0.555403 0.831581i \(-0.312564\pi\)
0.555403 + 0.831581i \(0.312564\pi\)
\(54\) 38802.3 0.0335335
\(55\) 1.21919e6 0.988102
\(56\) 181259. 0.137924
\(57\) 260557. 0.186355
\(58\) −488257. −0.328587
\(59\) 205379. 0.130189
\(60\) −1.13861e6 −0.680528
\(61\) 428862. 0.241915 0.120958 0.992658i \(-0.461404\pi\)
0.120958 + 0.992658i \(0.461404\pi\)
\(62\) −224843. −0.119814
\(63\) 265866. 0.133959
\(64\) −1.72472e6 −0.822412
\(65\) −291295. −0.131564
\(66\) −190989. −0.0817722
\(67\) −2.01026e6 −0.816565 −0.408283 0.912856i \(-0.633872\pi\)
−0.408283 + 0.912856i \(0.633872\pi\)
\(68\) −452934. −0.174684
\(69\) −3.13558e6 −1.14907
\(70\) 244284. 0.0851238
\(71\) 1.87858e6 0.622909 0.311454 0.950261i \(-0.399184\pi\)
0.311454 + 0.950261i \(0.399184\pi\)
\(72\) 362318. 0.114400
\(73\) 2.35049e6 0.707178 0.353589 0.935401i \(-0.384961\pi\)
0.353589 + 0.935401i \(0.384961\pi\)
\(74\) 383605. 0.110046
\(75\) −1.00770e6 −0.275816
\(76\) 1.19773e6 0.312976
\(77\) −1.30863e6 −0.326661
\(78\) 45632.2 0.0108878
\(79\) −3.74874e6 −0.855442 −0.427721 0.903911i \(-0.640683\pi\)
−0.427721 + 0.903911i \(0.640683\pi\)
\(80\) −5.06496e6 −1.10601
\(81\) 531441. 0.111111
\(82\) 1.17281e6 0.234898
\(83\) 3.21289e6 0.616768 0.308384 0.951262i \(-0.400212\pi\)
0.308384 + 0.951262i \(0.400212\pi\)
\(84\) 1.22214e6 0.224979
\(85\) −1.23996e6 −0.218999
\(86\) −488480. −0.0828137
\(87\) −6.68722e6 −1.08875
\(88\) −1.78337e6 −0.278967
\(89\) −544547. −0.0818786 −0.0409393 0.999162i \(-0.513035\pi\)
−0.0409393 + 0.999162i \(0.513035\pi\)
\(90\) 488299. 0.0706053
\(91\) 312663. 0.0434943
\(92\) −1.44136e7 −1.92982
\(93\) −3.07947e6 −0.396996
\(94\) 2.57692e6 0.320003
\(95\) 3.27892e6 0.392372
\(96\) 2.51110e6 0.289677
\(97\) −2.91764e6 −0.324586 −0.162293 0.986743i \(-0.551889\pi\)
−0.162293 + 0.986743i \(0.551889\pi\)
\(98\) 1.36130e6 0.146104
\(99\) −2.61581e6 −0.270947
\(100\) −4.63222e6 −0.463222
\(101\) −1.99790e7 −1.92952 −0.964761 0.263129i \(-0.915246\pi\)
−0.964761 + 0.263129i \(0.915246\pi\)
\(102\) 194243. 0.0181236
\(103\) −1.14245e7 −1.03016 −0.515082 0.857141i \(-0.672239\pi\)
−0.515082 + 0.857141i \(0.672239\pi\)
\(104\) 426092. 0.0371439
\(105\) 3.34574e6 0.282052
\(106\) −2.37340e6 −0.193553
\(107\) 2.29986e7 1.81492 0.907460 0.420138i \(-0.138018\pi\)
0.907460 + 0.420138i \(0.138018\pi\)
\(108\) 2.44293e6 0.186607
\(109\) 2.02074e7 1.49457 0.747287 0.664502i \(-0.231356\pi\)
0.747287 + 0.664502i \(0.231356\pi\)
\(110\) −2.40346e6 −0.172172
\(111\) 5.25390e6 0.364629
\(112\) 5.43650e6 0.365643
\(113\) −1.26444e7 −0.824374 −0.412187 0.911099i \(-0.635235\pi\)
−0.412187 + 0.911099i \(0.635235\pi\)
\(114\) −513651. −0.0324714
\(115\) −3.94590e7 −2.41938
\(116\) −3.07399e7 −1.82852
\(117\) 624983. 0.0360760
\(118\) −404876. −0.0226848
\(119\) 1.33092e6 0.0723998
\(120\) 4.55951e6 0.240871
\(121\) −6.61184e6 −0.339292
\(122\) −845441. −0.0421526
\(123\) 1.60630e7 0.778320
\(124\) −1.41557e7 −0.666740
\(125\) 1.38637e7 0.634885
\(126\) −524119. −0.0233417
\(127\) 3.20643e7 1.38902 0.694509 0.719484i \(-0.255621\pi\)
0.694509 + 0.719484i \(0.255621\pi\)
\(128\) 1.53045e7 0.645037
\(129\) −6.69028e6 −0.274398
\(130\) 574248. 0.0229244
\(131\) −2.38103e7 −0.925369 −0.462684 0.886523i \(-0.653114\pi\)
−0.462684 + 0.886523i \(0.653114\pi\)
\(132\) −1.20244e7 −0.455045
\(133\) −3.51945e6 −0.129716
\(134\) 3.96295e6 0.142283
\(135\) 6.68780e6 0.233946
\(136\) 1.81375e6 0.0618290
\(137\) 2.13799e7 0.710370 0.355185 0.934796i \(-0.384418\pi\)
0.355185 + 0.934796i \(0.384418\pi\)
\(138\) 6.18136e6 0.200220
\(139\) −3.64257e7 −1.15042 −0.575209 0.818006i \(-0.695079\pi\)
−0.575209 + 0.818006i \(0.695079\pi\)
\(140\) 1.53797e7 0.473696
\(141\) 3.52938e7 1.06031
\(142\) −3.70335e6 −0.108539
\(143\) −3.07624e6 −0.0879719
\(144\) 1.08670e7 0.303279
\(145\) −8.41539e7 −2.29238
\(146\) −4.63367e6 −0.123222
\(147\) 1.86445e7 0.484105
\(148\) 2.41512e7 0.612381
\(149\) 95229.9 0.00235842 0.00117921 0.999999i \(-0.499625\pi\)
0.00117921 + 0.999999i \(0.499625\pi\)
\(150\) 1.98655e6 0.0480596
\(151\) 4.71653e7 1.11482 0.557408 0.830238i \(-0.311796\pi\)
0.557408 + 0.830238i \(0.311796\pi\)
\(152\) −4.79624e6 −0.110777
\(153\) 2.66038e6 0.0600514
\(154\) 2.57977e6 0.0569192
\(155\) −3.87529e7 −0.835879
\(156\) 2.87293e6 0.0605883
\(157\) 4.44571e7 0.916838 0.458419 0.888736i \(-0.348416\pi\)
0.458419 + 0.888736i \(0.348416\pi\)
\(158\) 7.39012e6 0.149057
\(159\) −3.25063e7 −0.641324
\(160\) 3.16003e7 0.609918
\(161\) 4.23536e7 0.799833
\(162\) −1.04766e6 −0.0193606
\(163\) −9.85994e7 −1.78327 −0.891636 0.452753i \(-0.850442\pi\)
−0.891636 + 0.452753i \(0.850442\pi\)
\(164\) 7.38384e7 1.30716
\(165\) −3.29181e7 −0.570481
\(166\) −6.33376e6 −0.107469
\(167\) 1.60639e7 0.266897 0.133449 0.991056i \(-0.457395\pi\)
0.133449 + 0.991056i \(0.457395\pi\)
\(168\) −4.89398e6 −0.0796306
\(169\) −6.20135e7 −0.988287
\(170\) 2.44441e6 0.0381595
\(171\) −7.03503e6 −0.107592
\(172\) −3.07539e7 −0.460841
\(173\) 9.63596e7 1.41493 0.707463 0.706751i \(-0.249840\pi\)
0.707463 + 0.706751i \(0.249840\pi\)
\(174\) 1.31829e7 0.189710
\(175\) 1.36115e7 0.191987
\(176\) −5.34888e7 −0.739552
\(177\) −5.54523e6 −0.0751646
\(178\) 1.07350e6 0.0142670
\(179\) 6.96623e7 0.907846 0.453923 0.891041i \(-0.350024\pi\)
0.453923 + 0.891041i \(0.350024\pi\)
\(180\) 3.07425e7 0.392903
\(181\) −1.04761e8 −1.31319 −0.656593 0.754245i \(-0.728003\pi\)
−0.656593 + 0.754245i \(0.728003\pi\)
\(182\) −616372. −0.00757868
\(183\) −1.15793e7 −0.139670
\(184\) 5.77187e7 0.683053
\(185\) 6.61165e7 0.767731
\(186\) 6.07075e6 0.0691747
\(187\) −1.30947e7 −0.146436
\(188\) 1.62239e8 1.78075
\(189\) −7.17839e6 −0.0773412
\(190\) −6.46393e6 −0.0683690
\(191\) 3.38716e7 0.351738 0.175869 0.984414i \(-0.443727\pi\)
0.175869 + 0.984414i \(0.443727\pi\)
\(192\) 4.65675e7 0.474820
\(193\) −5.74536e7 −0.575264 −0.287632 0.957741i \(-0.592868\pi\)
−0.287632 + 0.957741i \(0.592868\pi\)
\(194\) 5.75172e6 0.0565576
\(195\) 7.86496e6 0.0759584
\(196\) 8.57051e7 0.813037
\(197\) 7.82632e7 0.729333 0.364667 0.931138i \(-0.381183\pi\)
0.364667 + 0.931138i \(0.381183\pi\)
\(198\) 5.15671e6 0.0472112
\(199\) −1.18386e8 −1.06491 −0.532455 0.846458i \(-0.678731\pi\)
−0.532455 + 0.846458i \(0.678731\pi\)
\(200\) 1.85495e7 0.163956
\(201\) 5.42771e7 0.471444
\(202\) 3.93859e7 0.336210
\(203\) 9.03271e7 0.757848
\(204\) 1.22292e7 0.100854
\(205\) 2.02141e8 1.63876
\(206\) 2.25218e7 0.179501
\(207\) 8.46606e7 0.663415
\(208\) 1.27798e7 0.0984698
\(209\) 3.46272e7 0.262365
\(210\) −6.59566e6 −0.0491463
\(211\) 4.78501e7 0.350666 0.175333 0.984509i \(-0.443900\pi\)
0.175333 + 0.984509i \(0.443900\pi\)
\(212\) −1.49425e8 −1.07708
\(213\) −5.07215e7 −0.359637
\(214\) −4.53385e7 −0.316241
\(215\) −8.41923e7 −0.577747
\(216\) −9.78259e6 −0.0660489
\(217\) 4.15957e7 0.276337
\(218\) −3.98361e7 −0.260423
\(219\) −6.34632e7 −0.408289
\(220\) −1.51318e8 −0.958102
\(221\) 3.12865e6 0.0194977
\(222\) −1.03573e7 −0.0635349
\(223\) 1.24666e8 0.752800 0.376400 0.926457i \(-0.377162\pi\)
0.376400 + 0.926457i \(0.377162\pi\)
\(224\) −3.39184e7 −0.201636
\(225\) 2.72080e7 0.159242
\(226\) 2.49267e7 0.143643
\(227\) 7.13606e7 0.404919 0.202460 0.979291i \(-0.435107\pi\)
0.202460 + 0.979291i \(0.435107\pi\)
\(228\) −3.23387e7 −0.180697
\(229\) −3.39682e8 −1.86917 −0.934584 0.355743i \(-0.884228\pi\)
−0.934584 + 0.355743i \(0.884228\pi\)
\(230\) 7.77879e7 0.421565
\(231\) 3.53329e7 0.188598
\(232\) 1.23096e8 0.647198
\(233\) −3.19085e8 −1.65257 −0.826286 0.563250i \(-0.809551\pi\)
−0.826286 + 0.563250i \(0.809551\pi\)
\(234\) −1.23207e6 −0.00628607
\(235\) 4.44147e8 2.23249
\(236\) −2.54904e7 −0.126236
\(237\) 1.01216e8 0.493890
\(238\) −2.62372e6 −0.0126153
\(239\) 2.48784e8 1.17877 0.589387 0.807851i \(-0.299369\pi\)
0.589387 + 0.807851i \(0.299369\pi\)
\(240\) 1.36754e8 0.638558
\(241\) −1.02085e8 −0.469788 −0.234894 0.972021i \(-0.575474\pi\)
−0.234894 + 0.972021i \(0.575474\pi\)
\(242\) 1.30343e7 0.0591200
\(243\) −1.43489e7 −0.0641500
\(244\) −5.32276e7 −0.234570
\(245\) 2.34627e8 1.01929
\(246\) −3.16659e7 −0.135619
\(247\) −8.27331e6 −0.0349333
\(248\) 5.66859e7 0.235990
\(249\) −8.67479e7 −0.356091
\(250\) −2.73304e7 −0.110626
\(251\) 1.69390e8 0.676131 0.338066 0.941123i \(-0.390227\pi\)
0.338066 + 0.941123i \(0.390227\pi\)
\(252\) −3.29977e7 −0.129892
\(253\) −4.16709e8 −1.61775
\(254\) −6.32102e7 −0.242030
\(255\) 3.34789e7 0.126439
\(256\) 1.90594e8 0.710018
\(257\) −4.15160e8 −1.52563 −0.762815 0.646617i \(-0.776183\pi\)
−0.762815 + 0.646617i \(0.776183\pi\)
\(258\) 1.31890e7 0.0478125
\(259\) −7.09666e7 −0.253808
\(260\) 3.61537e7 0.127569
\(261\) 1.80555e8 0.628591
\(262\) 4.69387e7 0.161241
\(263\) −2.46530e8 −0.835650 −0.417825 0.908527i \(-0.637208\pi\)
−0.417825 + 0.908527i \(0.637208\pi\)
\(264\) 4.81511e7 0.161062
\(265\) −4.09068e8 −1.35032
\(266\) 6.93810e6 0.0226024
\(267\) 1.47028e7 0.0472726
\(268\) 2.49501e8 0.791773
\(269\) 3.18928e8 0.998987 0.499494 0.866318i \(-0.333519\pi\)
0.499494 + 0.866318i \(0.333519\pi\)
\(270\) −1.31841e7 −0.0407640
\(271\) −3.87904e8 −1.18394 −0.591972 0.805958i \(-0.701651\pi\)
−0.591972 + 0.805958i \(0.701651\pi\)
\(272\) 5.44000e7 0.163911
\(273\) −8.44191e6 −0.0251114
\(274\) −4.21476e7 −0.123779
\(275\) −1.33921e8 −0.388315
\(276\) 3.89168e8 1.11418
\(277\) −7.42290e7 −0.209843 −0.104921 0.994481i \(-0.533459\pi\)
−0.104921 + 0.994481i \(0.533459\pi\)
\(278\) 7.18082e7 0.200455
\(279\) 8.31457e7 0.229206
\(280\) −6.15873e7 −0.167663
\(281\) −6.16298e8 −1.65698 −0.828492 0.560001i \(-0.810801\pi\)
−0.828492 + 0.560001i \(0.810801\pi\)
\(282\) −6.95769e7 −0.184754
\(283\) 7.96612e7 0.208927 0.104463 0.994529i \(-0.466687\pi\)
0.104463 + 0.994529i \(0.466687\pi\)
\(284\) −2.33157e8 −0.603996
\(285\) −8.85308e7 −0.226536
\(286\) 6.06438e6 0.0153287
\(287\) −2.16969e8 −0.541766
\(288\) −6.77996e7 −0.167245
\(289\) −3.97021e8 −0.967544
\(290\) 1.65898e8 0.399436
\(291\) 7.87762e7 0.187400
\(292\) −2.91728e8 −0.685707
\(293\) 2.50390e8 0.581542 0.290771 0.956793i \(-0.406088\pi\)
0.290771 + 0.956793i \(0.406088\pi\)
\(294\) −3.67550e7 −0.0843531
\(295\) −6.97828e7 −0.158260
\(296\) −9.67121e7 −0.216750
\(297\) 7.06270e7 0.156431
\(298\) −187732. −0.000410944 0
\(299\) 9.95623e7 0.215400
\(300\) 1.25070e8 0.267441
\(301\) 9.03684e7 0.191000
\(302\) −9.29799e7 −0.194252
\(303\) 5.39434e8 1.11401
\(304\) −1.43854e8 −0.293673
\(305\) −1.45717e8 −0.294076
\(306\) −5.24456e6 −0.0104637
\(307\) −3.31396e8 −0.653677 −0.326839 0.945080i \(-0.605983\pi\)
−0.326839 + 0.945080i \(0.605983\pi\)
\(308\) 1.62418e8 0.316743
\(309\) 3.08461e8 0.594766
\(310\) 7.63960e7 0.145648
\(311\) −9.41919e7 −0.177563 −0.0887815 0.996051i \(-0.528297\pi\)
−0.0887815 + 0.996051i \(0.528297\pi\)
\(312\) −1.15045e7 −0.0214450
\(313\) −5.39491e8 −0.994441 −0.497220 0.867624i \(-0.665646\pi\)
−0.497220 + 0.867624i \(0.665646\pi\)
\(314\) −8.76411e7 −0.159755
\(315\) −9.03349e7 −0.162843
\(316\) 4.65270e8 0.829470
\(317\) 6.91838e8 1.21982 0.609912 0.792469i \(-0.291205\pi\)
0.609912 + 0.792469i \(0.291205\pi\)
\(318\) 6.40817e7 0.111748
\(319\) −8.88713e8 −1.53283
\(320\) 5.86019e8 0.999739
\(321\) −6.20961e8 −1.04784
\(322\) −8.34942e7 −0.139367
\(323\) −3.52171e7 −0.0581494
\(324\) −6.59591e7 −0.107738
\(325\) 3.19971e7 0.0517033
\(326\) 1.94375e8 0.310727
\(327\) −5.45599e8 −0.862893
\(328\) −2.95682e8 −0.462665
\(329\) −4.76729e8 −0.738050
\(330\) 6.48935e7 0.0994037
\(331\) −8.54742e8 −1.29550 −0.647750 0.761853i \(-0.724290\pi\)
−0.647750 + 0.761853i \(0.724290\pi\)
\(332\) −3.98763e8 −0.598042
\(333\) −1.41855e8 −0.210519
\(334\) −3.16678e7 −0.0465056
\(335\) 6.83038e8 0.992631
\(336\) −1.46786e8 −0.211104
\(337\) 1.27131e8 0.180945 0.0904723 0.995899i \(-0.471162\pi\)
0.0904723 + 0.995899i \(0.471162\pi\)
\(338\) 1.22251e8 0.172204
\(339\) 3.41399e8 0.475952
\(340\) 1.53896e8 0.212349
\(341\) −4.09253e8 −0.558922
\(342\) 1.38686e7 0.0187474
\(343\) −5.52185e8 −0.738848
\(344\) 1.23152e8 0.163113
\(345\) 1.06539e9 1.39683
\(346\) −1.89960e8 −0.246544
\(347\) −6.17139e7 −0.0792920 −0.0396460 0.999214i \(-0.512623\pi\)
−0.0396460 + 0.999214i \(0.512623\pi\)
\(348\) 8.29976e8 1.05569
\(349\) 3.92987e8 0.494868 0.247434 0.968905i \(-0.420413\pi\)
0.247434 + 0.968905i \(0.420413\pi\)
\(350\) −2.68332e7 −0.0334529
\(351\) −1.68746e7 −0.0208285
\(352\) 3.33717e8 0.407830
\(353\) −1.45872e9 −1.76506 −0.882531 0.470254i \(-0.844162\pi\)
−0.882531 + 0.470254i \(0.844162\pi\)
\(354\) 1.09317e7 0.0130971
\(355\) −6.38294e8 −0.757219
\(356\) 6.75857e7 0.0793926
\(357\) −3.59348e7 −0.0418000
\(358\) −1.37330e8 −0.158188
\(359\) −1.00957e9 −1.15162 −0.575808 0.817585i \(-0.695313\pi\)
−0.575808 + 0.817585i \(0.695313\pi\)
\(360\) −1.23107e8 −0.139067
\(361\) −8.00744e8 −0.895816
\(362\) 2.06523e8 0.228817
\(363\) 1.78520e8 0.195890
\(364\) −3.88058e7 −0.0421737
\(365\) −7.98639e8 −0.859658
\(366\) 2.28269e7 0.0243368
\(367\) 6.89693e8 0.728324 0.364162 0.931336i \(-0.381355\pi\)
0.364162 + 0.931336i \(0.381355\pi\)
\(368\) 1.73116e9 1.81080
\(369\) −4.33700e8 −0.449363
\(370\) −1.30340e8 −0.133774
\(371\) 4.39076e8 0.446407
\(372\) 3.82205e8 0.384942
\(373\) 1.52953e7 0.0152608 0.00763041 0.999971i \(-0.497571\pi\)
0.00763041 + 0.999971i \(0.497571\pi\)
\(374\) 2.58143e7 0.0255159
\(375\) −3.74321e8 −0.366551
\(376\) −6.49678e8 −0.630290
\(377\) 2.12336e8 0.204093
\(378\) 1.41512e7 0.0134763
\(379\) −1.31066e9 −1.23667 −0.618335 0.785915i \(-0.712192\pi\)
−0.618335 + 0.785915i \(0.712192\pi\)
\(380\) −4.06959e8 −0.380459
\(381\) −8.65735e8 −0.801950
\(382\) −6.67731e7 −0.0612886
\(383\) 6.67359e8 0.606966 0.303483 0.952837i \(-0.401851\pi\)
0.303483 + 0.952837i \(0.401851\pi\)
\(384\) −4.13222e8 −0.372412
\(385\) 4.44639e8 0.397095
\(386\) 1.13262e8 0.100237
\(387\) 1.80638e8 0.158423
\(388\) 3.62119e8 0.314731
\(389\) 2.25715e9 1.94418 0.972089 0.234611i \(-0.0753816\pi\)
0.972089 + 0.234611i \(0.0753816\pi\)
\(390\) −1.55047e7 −0.0132354
\(391\) 4.23808e8 0.358551
\(392\) −3.43202e8 −0.287772
\(393\) 6.42877e8 0.534262
\(394\) −1.54285e8 −0.127083
\(395\) 1.27373e9 1.03989
\(396\) 3.24658e8 0.262720
\(397\) −9.81086e8 −0.786938 −0.393469 0.919338i \(-0.628725\pi\)
−0.393469 + 0.919338i \(0.628725\pi\)
\(398\) 2.33381e8 0.185556
\(399\) 9.50251e7 0.0748916
\(400\) 5.56357e8 0.434654
\(401\) −5.90695e7 −0.0457465 −0.0228732 0.999738i \(-0.507281\pi\)
−0.0228732 + 0.999738i \(0.507281\pi\)
\(402\) −1.07000e8 −0.0821470
\(403\) 9.77807e7 0.0744193
\(404\) 2.47967e9 1.87094
\(405\) −1.80571e8 −0.135069
\(406\) −1.78067e8 −0.132051
\(407\) 6.98228e8 0.513354
\(408\) −4.89713e7 −0.0356970
\(409\) −7.74847e8 −0.559995 −0.279998 0.960001i \(-0.590334\pi\)
−0.279998 + 0.960001i \(0.590334\pi\)
\(410\) −3.98493e8 −0.285547
\(411\) −5.77258e8 −0.410132
\(412\) 1.41794e9 0.998887
\(413\) 7.49018e7 0.0523199
\(414\) −1.66897e8 −0.115597
\(415\) −1.09166e9 −0.749754
\(416\) −7.97335e7 −0.0543018
\(417\) 9.83493e8 0.664194
\(418\) −6.82628e7 −0.0457159
\(419\) 2.71966e8 0.180620 0.0903098 0.995914i \(-0.471214\pi\)
0.0903098 + 0.995914i \(0.471214\pi\)
\(420\) −4.15252e8 −0.273489
\(421\) −8.59819e8 −0.561591 −0.280795 0.959768i \(-0.590598\pi\)
−0.280795 + 0.959768i \(0.590598\pi\)
\(422\) −9.43298e7 −0.0611020
\(423\) −9.52934e8 −0.612169
\(424\) 5.98366e8 0.381229
\(425\) 1.36202e8 0.0860644
\(426\) 9.99905e7 0.0626650
\(427\) 1.56406e8 0.0972200
\(428\) −2.85444e9 −1.75982
\(429\) 8.30585e7 0.0507906
\(430\) 1.65973e8 0.100670
\(431\) −6.62428e8 −0.398536 −0.199268 0.979945i \(-0.563856\pi\)
−0.199268 + 0.979945i \(0.563856\pi\)
\(432\) −2.93410e8 −0.175098
\(433\) 2.01470e9 1.19262 0.596312 0.802753i \(-0.296632\pi\)
0.596312 + 0.802753i \(0.296632\pi\)
\(434\) −8.20001e7 −0.0481505
\(435\) 2.27215e9 1.32350
\(436\) −2.50801e9 −1.44920
\(437\) −1.12071e9 −0.642403
\(438\) 1.25109e8 0.0711425
\(439\) 2.41587e9 1.36285 0.681425 0.731888i \(-0.261360\pi\)
0.681425 + 0.731888i \(0.261360\pi\)
\(440\) 6.05946e8 0.339117
\(441\) −5.03401e8 −0.279498
\(442\) −6.16769e6 −0.00339739
\(443\) −2.74104e9 −1.49796 −0.748982 0.662590i \(-0.769457\pi\)
−0.748982 + 0.662590i \(0.769457\pi\)
\(444\) −6.52081e8 −0.353558
\(445\) 1.85024e8 0.0995330
\(446\) −2.45761e8 −0.131172
\(447\) −2.57121e6 −0.00136163
\(448\) −6.29007e8 −0.330508
\(449\) −1.37537e9 −0.717063 −0.358532 0.933518i \(-0.616723\pi\)
−0.358532 + 0.933518i \(0.616723\pi\)
\(450\) −5.36368e7 −0.0277472
\(451\) 2.13472e9 1.09578
\(452\) 1.56935e9 0.799344
\(453\) −1.27346e9 −0.643640
\(454\) −1.40678e8 −0.0705553
\(455\) −1.06235e8 −0.0528724
\(456\) 1.29499e8 0.0639570
\(457\) 1.14427e9 0.560816 0.280408 0.959881i \(-0.409530\pi\)
0.280408 + 0.959881i \(0.409530\pi\)
\(458\) 6.69636e8 0.325694
\(459\) −7.18302e7 −0.0346707
\(460\) 4.89740e9 2.34592
\(461\) 7.69708e8 0.365909 0.182954 0.983121i \(-0.441434\pi\)
0.182954 + 0.983121i \(0.441434\pi\)
\(462\) −6.96539e7 −0.0328623
\(463\) −2.23996e8 −0.104884 −0.0524418 0.998624i \(-0.516700\pi\)
−0.0524418 + 0.998624i \(0.516700\pi\)
\(464\) 3.69204e9 1.71575
\(465\) 1.04633e9 0.482595
\(466\) 6.29032e8 0.287953
\(467\) 3.61971e8 0.164462 0.0822308 0.996613i \(-0.473796\pi\)
0.0822308 + 0.996613i \(0.473796\pi\)
\(468\) −7.75690e7 −0.0349806
\(469\) −7.33143e8 −0.328159
\(470\) −8.75575e8 −0.389001
\(471\) −1.20034e9 −0.529337
\(472\) 1.02075e8 0.0446809
\(473\) −8.89119e8 −0.386319
\(474\) −1.99533e8 −0.0860580
\(475\) −3.60170e8 −0.154199
\(476\) −1.65185e8 −0.0702016
\(477\) 8.77670e8 0.370269
\(478\) −4.90444e8 −0.205396
\(479\) −1.79221e9 −0.745100 −0.372550 0.928012i \(-0.621516\pi\)
−0.372550 + 0.928012i \(0.621516\pi\)
\(480\) −8.53209e8 −0.352137
\(481\) −1.66824e8 −0.0683520
\(482\) 2.01246e8 0.0818583
\(483\) −1.14355e9 −0.461784
\(484\) 8.20620e8 0.328990
\(485\) 9.91341e8 0.394573
\(486\) 2.82869e7 0.0111778
\(487\) 1.52055e9 0.596554 0.298277 0.954479i \(-0.403588\pi\)
0.298277 + 0.954479i \(0.403588\pi\)
\(488\) 2.13147e8 0.0830253
\(489\) 2.66218e9 1.02957
\(490\) −4.62536e8 −0.177606
\(491\) −3.45739e9 −1.31815 −0.659073 0.752079i \(-0.729051\pi\)
−0.659073 + 0.752079i \(0.729051\pi\)
\(492\) −1.99364e9 −0.754689
\(493\) 9.03853e8 0.339730
\(494\) 1.63097e7 0.00608697
\(495\) 8.88789e8 0.329367
\(496\) 1.70018e9 0.625619
\(497\) 6.85117e8 0.250333
\(498\) 1.71011e8 0.0620472
\(499\) −5.24675e8 −0.189033 −0.0945167 0.995523i \(-0.530131\pi\)
−0.0945167 + 0.995523i \(0.530131\pi\)
\(500\) −1.72068e9 −0.615609
\(501\) −4.33726e8 −0.154093
\(502\) −3.33930e8 −0.117813
\(503\) −2.08499e9 −0.730492 −0.365246 0.930911i \(-0.619015\pi\)
−0.365246 + 0.930911i \(0.619015\pi\)
\(504\) 1.32138e8 0.0459748
\(505\) 6.78839e9 2.34556
\(506\) 8.21485e8 0.281885
\(507\) 1.67437e9 0.570588
\(508\) −3.97961e9 −1.34685
\(509\) 3.13440e9 1.05352 0.526760 0.850014i \(-0.323407\pi\)
0.526760 + 0.850014i \(0.323407\pi\)
\(510\) −6.59990e7 −0.0220314
\(511\) 8.57224e8 0.284198
\(512\) −2.33471e9 −0.768754
\(513\) 1.89946e8 0.0621182
\(514\) 8.18430e8 0.265834
\(515\) 3.88176e9 1.25229
\(516\) 8.30355e8 0.266066
\(517\) 4.69045e9 1.49279
\(518\) 1.39901e8 0.0442248
\(519\) −2.60171e9 −0.816908
\(520\) −1.44776e8 −0.0451527
\(521\) 9.76015e8 0.302360 0.151180 0.988506i \(-0.451693\pi\)
0.151180 + 0.988506i \(0.451693\pi\)
\(522\) −3.55939e8 −0.109529
\(523\) −1.99309e9 −0.609216 −0.304608 0.952478i \(-0.598526\pi\)
−0.304608 + 0.952478i \(0.598526\pi\)
\(524\) 2.95518e9 0.897273
\(525\) −3.67510e8 −0.110844
\(526\) 4.86000e8 0.145608
\(527\) 4.16225e8 0.123877
\(528\) 1.44420e9 0.426981
\(529\) 1.00819e10 2.96107
\(530\) 8.06422e8 0.235286
\(531\) 1.49721e8 0.0433963
\(532\) 4.36812e8 0.125778
\(533\) −5.10039e8 −0.145901
\(534\) −2.89845e7 −0.00823703
\(535\) −7.81435e9 −2.20625
\(536\) −9.99115e8 −0.280246
\(537\) −1.88088e9 −0.524145
\(538\) −6.28723e8 −0.174069
\(539\) 2.47780e9 0.681562
\(540\) −8.30048e8 −0.226843
\(541\) 3.17125e9 0.861072 0.430536 0.902573i \(-0.358325\pi\)
0.430536 + 0.902573i \(0.358325\pi\)
\(542\) 7.64698e8 0.206297
\(543\) 2.82856e9 0.758168
\(544\) −3.39403e8 −0.0903897
\(545\) −6.86597e9 −1.81683
\(546\) 1.66421e7 0.00437555
\(547\) 2.77827e9 0.725803 0.362901 0.931828i \(-0.381786\pi\)
0.362901 + 0.931828i \(0.381786\pi\)
\(548\) −2.65354e9 −0.688802
\(549\) 3.12640e8 0.0806383
\(550\) 2.64007e8 0.0676621
\(551\) −2.39012e9 −0.608681
\(552\) −1.55841e9 −0.394361
\(553\) −1.36717e9 −0.343782
\(554\) 1.46332e8 0.0365641
\(555\) −1.78515e9 −0.443250
\(556\) 4.52093e9 1.11549
\(557\) −7.99807e8 −0.196107 −0.0980533 0.995181i \(-0.531262\pi\)
−0.0980533 + 0.995181i \(0.531262\pi\)
\(558\) −1.63910e8 −0.0399380
\(559\) 2.12433e8 0.0514375
\(560\) −1.84719e9 −0.444482
\(561\) 3.53556e8 0.0845451
\(562\) 1.21495e9 0.288722
\(563\) −1.37529e9 −0.324800 −0.162400 0.986725i \(-0.551923\pi\)
−0.162400 + 0.986725i \(0.551923\pi\)
\(564\) −4.38045e9 −1.02812
\(565\) 4.29626e9 1.00212
\(566\) −1.57041e8 −0.0364045
\(567\) 1.93817e8 0.0446530
\(568\) 9.33666e8 0.213783
\(569\) 4.45989e8 0.101492 0.0507459 0.998712i \(-0.483840\pi\)
0.0507459 + 0.998712i \(0.483840\pi\)
\(570\) 1.74526e8 0.0394728
\(571\) −7.16360e9 −1.61029 −0.805147 0.593076i \(-0.797914\pi\)
−0.805147 + 0.593076i \(0.797914\pi\)
\(572\) 3.81804e8 0.0853010
\(573\) −9.14533e8 −0.203076
\(574\) 4.27725e8 0.0944002
\(575\) 4.33434e9 0.950793
\(576\) −1.25732e9 −0.274137
\(577\) −5.75941e9 −1.24814 −0.624069 0.781369i \(-0.714522\pi\)
−0.624069 + 0.781369i \(0.714522\pi\)
\(578\) 7.82672e8 0.168590
\(579\) 1.55125e9 0.332129
\(580\) 1.04447e10 2.22278
\(581\) 1.17174e9 0.247865
\(582\) −1.55296e8 −0.0326536
\(583\) −4.31999e9 −0.902908
\(584\) 1.16821e9 0.242704
\(585\) −2.12354e8 −0.0438546
\(586\) −4.93610e8 −0.101331
\(587\) −2.89213e9 −0.590180 −0.295090 0.955469i \(-0.595350\pi\)
−0.295090 + 0.955469i \(0.595350\pi\)
\(588\) −2.31404e9 −0.469407
\(589\) −1.10065e9 −0.221946
\(590\) 1.37567e8 0.0275761
\(591\) −2.11311e9 −0.421081
\(592\) −2.90069e9 −0.574613
\(593\) −2.52669e9 −0.497577 −0.248789 0.968558i \(-0.580032\pi\)
−0.248789 + 0.968558i \(0.580032\pi\)
\(594\) −1.39231e8 −0.0272574
\(595\) −4.52214e8 −0.0880104
\(596\) −1.18193e7 −0.00228682
\(597\) 3.19641e9 0.614826
\(598\) −1.96273e8 −0.0375325
\(599\) −9.87541e9 −1.87742 −0.938709 0.344709i \(-0.887977\pi\)
−0.938709 + 0.344709i \(0.887977\pi\)
\(600\) −5.00836e8 −0.0946600
\(601\) −1.69668e9 −0.318815 −0.159408 0.987213i \(-0.550958\pi\)
−0.159408 + 0.987213i \(0.550958\pi\)
\(602\) −1.78149e8 −0.0332809
\(603\) −1.46548e9 −0.272188
\(604\) −5.85386e9 −1.08097
\(605\) 2.24654e9 0.412449
\(606\) −1.06342e9 −0.194111
\(607\) −9.97951e9 −1.81113 −0.905564 0.424211i \(-0.860552\pi\)
−0.905564 + 0.424211i \(0.860552\pi\)
\(608\) 8.97507e8 0.161948
\(609\) −2.43883e9 −0.437544
\(610\) 2.87260e8 0.0512414
\(611\) −1.12067e9 −0.198761
\(612\) −3.30189e8 −0.0582281
\(613\) −6.06527e9 −1.06350 −0.531751 0.846901i \(-0.678466\pi\)
−0.531751 + 0.846901i \(0.678466\pi\)
\(614\) 6.53302e8 0.113900
\(615\) −5.45780e9 −0.946139
\(616\) −6.50396e8 −0.112110
\(617\) −1.40806e9 −0.241336 −0.120668 0.992693i \(-0.538504\pi\)
−0.120668 + 0.992693i \(0.538504\pi\)
\(618\) −6.08089e8 −0.103635
\(619\) −2.17547e9 −0.368668 −0.184334 0.982864i \(-0.559013\pi\)
−0.184334 + 0.982864i \(0.559013\pi\)
\(620\) 4.80977e9 0.810501
\(621\) −2.28584e9 −0.383023
\(622\) 1.85686e8 0.0309395
\(623\) −1.98596e8 −0.0329051
\(624\) −3.45055e8 −0.0568516
\(625\) −7.62637e9 −1.24950
\(626\) 1.06353e9 0.173277
\(627\) −9.34935e8 −0.151476
\(628\) −5.51774e9 −0.889001
\(629\) −7.10123e8 −0.113777
\(630\) 1.78083e8 0.0283746
\(631\) −3.80261e9 −0.602531 −0.301266 0.953540i \(-0.597409\pi\)
−0.301266 + 0.953540i \(0.597409\pi\)
\(632\) −1.86315e9 −0.293588
\(633\) −1.29195e9 −0.202457
\(634\) −1.36386e9 −0.212549
\(635\) −1.08946e10 −1.68852
\(636\) 4.03448e9 0.621853
\(637\) −5.92008e8 −0.0907486
\(638\) 1.75197e9 0.267089
\(639\) 1.36948e9 0.207636
\(640\) −5.20010e9 −0.784118
\(641\) 3.59687e9 0.539413 0.269706 0.962943i \(-0.413073\pi\)
0.269706 + 0.962943i \(0.413073\pi\)
\(642\) 1.22414e9 0.182582
\(643\) 4.76649e9 0.707066 0.353533 0.935422i \(-0.384980\pi\)
0.353533 + 0.935422i \(0.384980\pi\)
\(644\) −5.25666e9 −0.775549
\(645\) 2.27319e9 0.333562
\(646\) 6.94257e7 0.0101323
\(647\) 1.17944e9 0.171203 0.0856016 0.996329i \(-0.472719\pi\)
0.0856016 + 0.996329i \(0.472719\pi\)
\(648\) 2.64130e8 0.0381334
\(649\) −7.36945e8 −0.105823
\(650\) −6.30778e7 −0.00900907
\(651\) −1.12308e9 −0.159543
\(652\) 1.22375e10 1.72913
\(653\) 8.76325e9 1.23160 0.615799 0.787903i \(-0.288833\pi\)
0.615799 + 0.787903i \(0.288833\pi\)
\(654\) 1.07557e9 0.150355
\(655\) 8.09015e9 1.12489
\(656\) −8.86841e9 −1.22654
\(657\) 1.71351e9 0.235726
\(658\) 9.39804e8 0.128602
\(659\) −1.34224e10 −1.82697 −0.913487 0.406868i \(-0.866621\pi\)
−0.913487 + 0.406868i \(0.866621\pi\)
\(660\) 4.08559e9 0.553161
\(661\) −5.67841e9 −0.764754 −0.382377 0.924006i \(-0.624894\pi\)
−0.382377 + 0.924006i \(0.624894\pi\)
\(662\) 1.68501e9 0.225735
\(663\) −8.44735e7 −0.0112570
\(664\) 1.59683e9 0.211675
\(665\) 1.19582e9 0.157685
\(666\) 2.79648e8 0.0366819
\(667\) 2.87631e10 3.75315
\(668\) −1.99375e9 −0.258794
\(669\) −3.36597e9 −0.434630
\(670\) −1.34651e9 −0.172961
\(671\) −1.53885e9 −0.196638
\(672\) 9.15797e8 0.116414
\(673\) 2.49548e9 0.315574 0.157787 0.987473i \(-0.449564\pi\)
0.157787 + 0.987473i \(0.449564\pi\)
\(674\) −2.50621e8 −0.0315288
\(675\) −7.34617e8 −0.0919385
\(676\) 7.69673e9 0.958281
\(677\) 5.54895e9 0.687307 0.343654 0.939097i \(-0.388335\pi\)
0.343654 + 0.939097i \(0.388335\pi\)
\(678\) −6.73021e8 −0.0829325
\(679\) −1.06406e9 −0.130444
\(680\) −6.16269e8 −0.0751604
\(681\) −1.92674e9 −0.233780
\(682\) 8.06785e8 0.0973896
\(683\) 4.93572e9 0.592759 0.296379 0.955070i \(-0.404221\pi\)
0.296379 + 0.955070i \(0.404221\pi\)
\(684\) 8.73144e8 0.104325
\(685\) −7.26438e9 −0.863538
\(686\) 1.08856e9 0.128741
\(687\) 9.17141e9 1.07916
\(688\) 3.69372e9 0.432419
\(689\) 1.03215e9 0.120220
\(690\) −2.10027e9 −0.243391
\(691\) −8.63015e9 −0.995050 −0.497525 0.867450i \(-0.665758\pi\)
−0.497525 + 0.867450i \(0.665758\pi\)
\(692\) −1.19595e10 −1.37197
\(693\) −9.53988e8 −0.108887
\(694\) 1.21660e8 0.0138163
\(695\) 1.23766e10 1.39847
\(696\) −3.32360e9 −0.373660
\(697\) −2.17109e9 −0.242864
\(698\) −7.74720e8 −0.0862285
\(699\) 8.61529e9 0.954113
\(700\) −1.68937e9 −0.186158
\(701\) 8.06267e9 0.884028 0.442014 0.897008i \(-0.354264\pi\)
0.442014 + 0.897008i \(0.354264\pi\)
\(702\) 3.32658e7 0.00362926
\(703\) 1.87783e9 0.203851
\(704\) 6.18869e9 0.668489
\(705\) −1.19920e10 −1.28893
\(706\) 2.87566e9 0.307554
\(707\) −7.28636e9 −0.775430
\(708\) 6.88240e8 0.0728825
\(709\) 1.13085e10 1.19163 0.595817 0.803120i \(-0.296828\pi\)
0.595817 + 0.803120i \(0.296828\pi\)
\(710\) 1.25831e9 0.131942
\(711\) −2.73283e9 −0.285147
\(712\) −2.70644e8 −0.0281008
\(713\) 1.32454e10 1.36853
\(714\) 7.08405e7 0.00728346
\(715\) 1.04523e9 0.106940
\(716\) −8.64605e9 −0.880283
\(717\) −6.71718e9 −0.680565
\(718\) 1.99024e9 0.200664
\(719\) −1.36390e10 −1.36846 −0.684231 0.729266i \(-0.739862\pi\)
−0.684231 + 0.729266i \(0.739862\pi\)
\(720\) −3.69235e9 −0.368672
\(721\) −4.16652e9 −0.413999
\(722\) 1.57856e9 0.156092
\(723\) 2.75629e9 0.271232
\(724\) 1.30023e10 1.27332
\(725\) 9.24382e9 0.900883
\(726\) −3.51927e8 −0.0341330
\(727\) 3.69976e9 0.357111 0.178556 0.983930i \(-0.442858\pi\)
0.178556 + 0.983930i \(0.442858\pi\)
\(728\) 1.55396e8 0.0149273
\(729\) 3.87420e8 0.0370370
\(730\) 1.57441e9 0.149791
\(731\) 9.04265e8 0.0856219
\(732\) 1.43715e9 0.135429
\(733\) 1.64513e10 1.54289 0.771447 0.636294i \(-0.219533\pi\)
0.771447 + 0.636294i \(0.219533\pi\)
\(734\) −1.35963e9 −0.126907
\(735\) −6.33494e9 −0.588487
\(736\) −1.08007e10 −0.998576
\(737\) 7.21327e9 0.663737
\(738\) 8.54980e8 0.0782994
\(739\) −4.40695e9 −0.401682 −0.200841 0.979624i \(-0.564367\pi\)
−0.200841 + 0.979624i \(0.564367\pi\)
\(740\) −8.20597e9 −0.744421
\(741\) 2.23379e8 0.0201688
\(742\) −8.65578e8 −0.0777844
\(743\) 2.97820e9 0.266374 0.133187 0.991091i \(-0.457479\pi\)
0.133187 + 0.991091i \(0.457479\pi\)
\(744\) −1.53052e9 −0.136249
\(745\) −3.23568e7 −0.00286694
\(746\) −3.01526e7 −0.00265913
\(747\) 2.34219e9 0.205589
\(748\) 1.62523e9 0.141990
\(749\) 8.38758e9 0.729374
\(750\) 7.37921e8 0.0638698
\(751\) 6.84373e9 0.589594 0.294797 0.955560i \(-0.404748\pi\)
0.294797 + 0.955560i \(0.404748\pi\)
\(752\) −1.94858e10 −1.67092
\(753\) −4.57354e9 −0.390364
\(754\) −4.18590e8 −0.0355623
\(755\) −1.60256e10 −1.35519
\(756\) 8.90937e8 0.0749930
\(757\) −1.28064e10 −1.07298 −0.536491 0.843906i \(-0.680250\pi\)
−0.536491 + 0.843906i \(0.680250\pi\)
\(758\) 2.58379e9 0.215484
\(759\) 1.12512e10 0.934008
\(760\) 1.62965e9 0.134662
\(761\) −1.33708e9 −0.109979 −0.0549897 0.998487i \(-0.517513\pi\)
−0.0549897 + 0.998487i \(0.517513\pi\)
\(762\) 1.70668e9 0.139736
\(763\) 7.36964e9 0.600634
\(764\) −4.20393e9 −0.341058
\(765\) −9.03931e8 −0.0729995
\(766\) −1.31561e9 −0.105761
\(767\) 1.76075e8 0.0140901
\(768\) −5.14604e9 −0.409929
\(769\) 1.10898e10 0.879390 0.439695 0.898147i \(-0.355086\pi\)
0.439695 + 0.898147i \(0.355086\pi\)
\(770\) −8.76544e8 −0.0691920
\(771\) 1.12093e10 0.880823
\(772\) 7.13078e9 0.557798
\(773\) −1.78044e10 −1.38644 −0.693218 0.720728i \(-0.743808\pi\)
−0.693218 + 0.720728i \(0.743808\pi\)
\(774\) −3.56102e8 −0.0276046
\(775\) 4.25679e9 0.328493
\(776\) −1.45009e9 −0.111398
\(777\) 1.91610e9 0.146536
\(778\) −4.44965e9 −0.338764
\(779\) 5.74117e9 0.435130
\(780\) −9.76150e8 −0.0736522
\(781\) −6.74075e9 −0.506325
\(782\) −8.35479e8 −0.0624758
\(783\) −4.87499e9 −0.362917
\(784\) −1.02937e10 −0.762894
\(785\) −1.51054e10 −1.11452
\(786\) −1.26734e9 −0.0930927
\(787\) 1.97486e10 1.44419 0.722095 0.691794i \(-0.243179\pi\)
0.722095 + 0.691794i \(0.243179\pi\)
\(788\) −9.71354e9 −0.707190
\(789\) 6.65631e9 0.482463
\(790\) −2.51098e9 −0.181196
\(791\) −4.61142e9 −0.331297
\(792\) −1.30008e9 −0.0929890
\(793\) 3.67670e8 0.0261820
\(794\) 1.93408e9 0.137120
\(795\) 1.10448e10 0.779605
\(796\) 1.46933e10 1.03258
\(797\) 1.84228e10 1.28900 0.644499 0.764605i \(-0.277066\pi\)
0.644499 + 0.764605i \(0.277066\pi\)
\(798\) −1.87329e8 −0.0130495
\(799\) −4.77035e9 −0.330854
\(800\) −3.47112e9 −0.239692
\(801\) −3.96975e8 −0.0272929
\(802\) 1.16447e8 0.00797111
\(803\) −8.43408e9 −0.574822
\(804\) −6.73653e9 −0.457130
\(805\) −1.43907e10 −0.972291
\(806\) −1.92761e8 −0.0129672
\(807\) −8.61106e9 −0.576766
\(808\) −9.92972e9 −0.662213
\(809\) 1.12171e10 0.744838 0.372419 0.928065i \(-0.378528\pi\)
0.372419 + 0.928065i \(0.378528\pi\)
\(810\) 3.55970e8 0.0235351
\(811\) 1.91381e10 1.25987 0.629934 0.776649i \(-0.283082\pi\)
0.629934 + 0.776649i \(0.283082\pi\)
\(812\) −1.12108e10 −0.734838
\(813\) 1.04734e10 0.683551
\(814\) −1.37646e9 −0.0894495
\(815\) 3.35017e10 2.16778
\(816\) −1.46880e9 −0.0946341
\(817\) −2.39122e9 −0.153406
\(818\) 1.52750e9 0.0975766
\(819\) 2.27932e8 0.0144981
\(820\) −2.50885e10 −1.58901
\(821\) 1.26324e10 0.796685 0.398342 0.917237i \(-0.369586\pi\)
0.398342 + 0.917237i \(0.369586\pi\)
\(822\) 1.13799e9 0.0714637
\(823\) −5.60518e9 −0.350502 −0.175251 0.984524i \(-0.556074\pi\)
−0.175251 + 0.984524i \(0.556074\pi\)
\(824\) −5.67806e9 −0.353553
\(825\) 3.61587e9 0.224194
\(826\) −1.47658e8 −0.00911650
\(827\) −9.05725e9 −0.556836 −0.278418 0.960460i \(-0.589810\pi\)
−0.278418 + 0.960460i \(0.589810\pi\)
\(828\) −1.05075e10 −0.643273
\(829\) 1.80271e10 1.09897 0.549483 0.835505i \(-0.314825\pi\)
0.549483 + 0.835505i \(0.314825\pi\)
\(830\) 2.15206e9 0.130641
\(831\) 2.00418e9 0.121153
\(832\) −1.47863e9 −0.0890080
\(833\) −2.52001e9 −0.151058
\(834\) −1.93882e9 −0.115733
\(835\) −5.45813e9 −0.324445
\(836\) −4.29771e9 −0.254399
\(837\) −2.24493e9 −0.132332
\(838\) −5.36143e8 −0.0314721
\(839\) 2.52101e10 1.47370 0.736848 0.676059i \(-0.236313\pi\)
0.736848 + 0.676059i \(0.236313\pi\)
\(840\) 1.66286e9 0.0968004
\(841\) 4.40930e10 2.55613
\(842\) 1.69501e9 0.0978546
\(843\) 1.66400e10 0.956660
\(844\) −5.93885e9 −0.340020
\(845\) 2.10707e10 1.20138
\(846\) 1.87858e9 0.106668
\(847\) −2.41134e9 −0.136353
\(848\) 1.79468e10 1.01065
\(849\) −2.15085e9 −0.120624
\(850\) −2.68504e8 −0.0149963
\(851\) −2.25981e10 −1.25695
\(852\) 6.29524e9 0.348717
\(853\) −4.60485e9 −0.254035 −0.127017 0.991900i \(-0.540540\pi\)
−0.127017 + 0.991900i \(0.540540\pi\)
\(854\) −3.08333e8 −0.0169401
\(855\) 2.39033e9 0.130791
\(856\) 1.14305e10 0.622881
\(857\) 2.64820e10 1.43720 0.718600 0.695423i \(-0.244783\pi\)
0.718600 + 0.695423i \(0.244783\pi\)
\(858\) −1.63738e8 −0.00885003
\(859\) −3.30189e10 −1.77740 −0.888702 0.458485i \(-0.848392\pi\)
−0.888702 + 0.458485i \(0.848392\pi\)
\(860\) 1.04494e10 0.560206
\(861\) 5.85817e9 0.312789
\(862\) 1.30588e9 0.0694431
\(863\) −1.44548e10 −0.765552 −0.382776 0.923841i \(-0.625032\pi\)
−0.382776 + 0.923841i \(0.625032\pi\)
\(864\) 1.83059e9 0.0965590
\(865\) −3.27406e10 −1.72001
\(866\) −3.97171e9 −0.207809
\(867\) 1.07196e10 0.558612
\(868\) −5.16260e9 −0.267947
\(869\) 1.34513e10 0.695337
\(870\) −4.47924e9 −0.230615
\(871\) −1.72343e9 −0.0883751
\(872\) 1.00432e10 0.512938
\(873\) −2.12696e9 −0.108195
\(874\) 2.20932e9 0.111936
\(875\) 5.05610e9 0.255145
\(876\) 7.87666e9 0.395893
\(877\) 1.61500e9 0.0808491 0.0404245 0.999183i \(-0.487129\pi\)
0.0404245 + 0.999183i \(0.487129\pi\)
\(878\) −4.76256e9 −0.237470
\(879\) −6.76054e9 −0.335753
\(880\) 1.81742e10 0.899013
\(881\) 3.62460e10 1.78585 0.892925 0.450206i \(-0.148649\pi\)
0.892925 + 0.450206i \(0.148649\pi\)
\(882\) 9.92386e8 0.0487013
\(883\) 1.90627e10 0.931796 0.465898 0.884838i \(-0.345731\pi\)
0.465898 + 0.884838i \(0.345731\pi\)
\(884\) −3.88308e8 −0.0189057
\(885\) 1.88413e9 0.0913714
\(886\) 5.40357e9 0.261013
\(887\) −1.75867e9 −0.0846157 −0.0423079 0.999105i \(-0.513471\pi\)
−0.0423079 + 0.999105i \(0.513471\pi\)
\(888\) 2.61123e9 0.125141
\(889\) 1.16938e10 0.558214
\(890\) −3.64748e8 −0.0173432
\(891\) −1.90693e9 −0.0903155
\(892\) −1.54727e10 −0.729944
\(893\) 1.26146e10 0.592780
\(894\) 5.06878e6 0.000237259 0
\(895\) −2.36695e10 −1.10359
\(896\) 5.58156e9 0.259225
\(897\) −2.68818e9 −0.124361
\(898\) 2.71135e9 0.124945
\(899\) 2.82484e10 1.29669
\(900\) −3.37689e9 −0.154407
\(901\) 4.39359e9 0.200116
\(902\) −4.20831e9 −0.190935
\(903\) −2.43995e9 −0.110274
\(904\) −6.28436e9 −0.282925
\(905\) 3.55954e10 1.59633
\(906\) 2.51046e9 0.112151
\(907\) −3.74603e10 −1.66704 −0.833520 0.552489i \(-0.813678\pi\)
−0.833520 + 0.552489i \(0.813678\pi\)
\(908\) −8.85684e9 −0.392625
\(909\) −1.45647e10 −0.643174
\(910\) 2.09428e8 0.00921277
\(911\) 8.20611e9 0.359603 0.179801 0.983703i \(-0.442455\pi\)
0.179801 + 0.983703i \(0.442455\pi\)
\(912\) 3.88406e9 0.169552
\(913\) −1.15285e10 −0.501334
\(914\) −2.25576e9 −0.0977196
\(915\) 3.93435e9 0.169785
\(916\) 4.21592e10 1.81242
\(917\) −8.68361e9 −0.371884
\(918\) 1.41603e8 0.00604121
\(919\) −1.35638e9 −0.0576472 −0.0288236 0.999585i \(-0.509176\pi\)
−0.0288236 + 0.999585i \(0.509176\pi\)
\(920\) −1.96114e10 −0.830331
\(921\) 8.94770e9 0.377401
\(922\) −1.51737e9 −0.0637579
\(923\) 1.61053e9 0.0674161
\(924\) −4.38530e9 −0.182872
\(925\) −7.26252e9 −0.301711
\(926\) 4.41578e8 0.0182755
\(927\) −8.32846e9 −0.343388
\(928\) −2.30347e10 −0.946158
\(929\) 6.23174e9 0.255008 0.127504 0.991838i \(-0.459303\pi\)
0.127504 + 0.991838i \(0.459303\pi\)
\(930\) −2.06269e9 −0.0840899
\(931\) 6.66385e9 0.270646
\(932\) 3.96028e10 1.60240
\(933\) 2.54318e9 0.102516
\(934\) −7.13575e8 −0.0286567
\(935\) 4.44925e9 0.178011
\(936\) 3.10621e8 0.0123813
\(937\) −3.18495e10 −1.26478 −0.632389 0.774651i \(-0.717926\pi\)
−0.632389 + 0.774651i \(0.717926\pi\)
\(938\) 1.44529e9 0.0571801
\(939\) 1.45663e10 0.574141
\(940\) −5.51248e10 −2.16471
\(941\) −4.56004e10 −1.78404 −0.892021 0.451995i \(-0.850713\pi\)
−0.892021 + 0.451995i \(0.850713\pi\)
\(942\) 2.36631e9 0.0922345
\(943\) −6.90901e10 −2.68303
\(944\) 3.06154e9 0.118451
\(945\) 2.43904e9 0.0940174
\(946\) 1.75277e9 0.0673142
\(947\) 1.39598e10 0.534138 0.267069 0.963677i \(-0.413945\pi\)
0.267069 + 0.963677i \(0.413945\pi\)
\(948\) −1.25623e10 −0.478894
\(949\) 2.01511e9 0.0765363
\(950\) 7.10026e8 0.0268684
\(951\) −1.86796e10 −0.704266
\(952\) 6.61476e8 0.0248476
\(953\) −1.88757e10 −0.706444 −0.353222 0.935540i \(-0.614914\pi\)
−0.353222 + 0.935540i \(0.614914\pi\)
\(954\) −1.73021e9 −0.0645176
\(955\) −1.15087e10 −0.427578
\(956\) −3.08776e10 −1.14298
\(957\) 2.39952e10 0.884980
\(958\) 3.53309e9 0.129830
\(959\) 7.79727e9 0.285481
\(960\) −1.58225e10 −0.577200
\(961\) −1.45042e10 −0.527183
\(962\) 3.28871e8 0.0119100
\(963\) 1.67660e10 0.604973
\(964\) 1.26701e10 0.455524
\(965\) 1.95213e10 0.699301
\(966\) 2.25434e9 0.0804637
\(967\) −4.38222e10 −1.55848 −0.779241 0.626724i \(-0.784395\pi\)
−0.779241 + 0.626724i \(0.784395\pi\)
\(968\) −3.28613e9 −0.116445
\(969\) 9.50862e8 0.0335726
\(970\) −1.95429e9 −0.0687525
\(971\) 7.68907e8 0.0269529 0.0134765 0.999909i \(-0.495710\pi\)
0.0134765 + 0.999909i \(0.495710\pi\)
\(972\) 1.78090e9 0.0622023
\(973\) −1.32845e10 −0.462326
\(974\) −2.99756e9 −0.103947
\(975\) −8.63921e8 −0.0298509
\(976\) 6.39295e9 0.220103
\(977\) −3.25858e10 −1.11788 −0.558942 0.829206i \(-0.688793\pi\)
−0.558942 + 0.829206i \(0.688793\pi\)
\(978\) −5.24813e9 −0.179398
\(979\) 1.95395e9 0.0665541
\(980\) −2.91205e10 −0.988342
\(981\) 1.47312e10 0.498191
\(982\) 6.81577e9 0.229681
\(983\) −3.29965e10 −1.10798 −0.553988 0.832525i \(-0.686894\pi\)
−0.553988 + 0.832525i \(0.686894\pi\)
\(984\) 7.98341e9 0.267120
\(985\) −2.65919e10 −0.886590
\(986\) −1.78182e9 −0.0591963
\(987\) 1.28717e10 0.426113
\(988\) 1.02683e9 0.0338727
\(989\) 2.87763e10 0.945904
\(990\) −1.75213e9 −0.0573908
\(991\) 2.11125e10 0.689100 0.344550 0.938768i \(-0.388031\pi\)
0.344550 + 0.938768i \(0.388031\pi\)
\(992\) −1.06075e10 −0.345002
\(993\) 2.30780e10 0.747957
\(994\) −1.35061e9 −0.0436193
\(995\) 4.02245e10 1.29452
\(996\) 1.07666e10 0.345280
\(997\) −2.81284e10 −0.898902 −0.449451 0.893305i \(-0.648380\pi\)
−0.449451 + 0.893305i \(0.648380\pi\)
\(998\) 1.03432e9 0.0329382
\(999\) 3.83009e9 0.121543
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.1.8 16
3.2 odd 2 531.8.a.b.1.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.8 16 1.1 even 1 trivial
531.8.a.b.1.9 16 3.2 odd 2