Properties

Label 43.8.a.a.1.3
Level $43$
Weight $8$
Character 43.1
Self dual yes
Analytic conductor $13.433$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(13.1261\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-15.1261 q^{2} +62.8737 q^{3} +100.799 q^{4} +12.0619 q^{5} -951.035 q^{6} -1248.65 q^{7} +411.441 q^{8} +1766.10 q^{9} +O(q^{10})\) \(q-15.1261 q^{2} +62.8737 q^{3} +100.799 q^{4} +12.0619 q^{5} -951.035 q^{6} -1248.65 q^{7} +411.441 q^{8} +1766.10 q^{9} -182.450 q^{10} -2043.21 q^{11} +6337.62 q^{12} +9460.36 q^{13} +18887.2 q^{14} +758.376 q^{15} -19125.8 q^{16} -19027.5 q^{17} -26714.2 q^{18} -28326.9 q^{19} +1215.83 q^{20} -78507.2 q^{21} +30905.9 q^{22} +2087.68 q^{23} +25868.8 q^{24} -77979.5 q^{25} -143098. q^{26} -26463.5 q^{27} -125863. q^{28} -111967. q^{29} -11471.3 q^{30} +7978.79 q^{31} +236635. q^{32} -128464. q^{33} +287812. q^{34} -15061.1 q^{35} +178022. q^{36} -185038. q^{37} +428475. q^{38} +594808. q^{39} +4962.76 q^{40} -421826. q^{41} +1.18751e6 q^{42} +79507.0 q^{43} -205954. q^{44} +21302.5 q^{45} -31578.5 q^{46} +197368. q^{47} -1.20251e6 q^{48} +735581. q^{49} +1.17953e6 q^{50} -1.19633e6 q^{51} +953597. q^{52} -2.01317e6 q^{53} +400289. q^{54} -24645.0 q^{55} -513745. q^{56} -1.78101e6 q^{57} +1.69363e6 q^{58} -856041. q^{59} +76443.8 q^{60} +2.61554e6 q^{61} -120688. q^{62} -2.20524e6 q^{63} -1.13126e6 q^{64} +114110. q^{65} +1.94317e6 q^{66} +2.93249e6 q^{67} -1.91796e6 q^{68} +131260. q^{69} +227816. q^{70} -3.20003e6 q^{71} +726646. q^{72} +4.19984e6 q^{73} +2.79891e6 q^{74} -4.90286e6 q^{75} -2.85533e6 q^{76} +2.55125e6 q^{77} -8.99713e6 q^{78} +7.83389e6 q^{79} -230694. q^{80} -5.52632e6 q^{81} +6.38059e6 q^{82} +5.57533e6 q^{83} -7.91347e6 q^{84} -229508. q^{85} -1.20263e6 q^{86} -7.03981e6 q^{87} -840660. q^{88} -889160. q^{89} -322225. q^{90} -1.18127e7 q^{91} +210437. q^{92} +501656. q^{93} -2.98542e6 q^{94} -341676. q^{95} +1.48781e7 q^{96} +1.05709e7 q^{97} -1.11265e7 q^{98} -3.60852e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.1261 −1.33697 −0.668486 0.743725i \(-0.733057\pi\)
−0.668486 + 0.743725i \(0.733057\pi\)
\(3\) 62.8737 1.34445 0.672225 0.740347i \(-0.265339\pi\)
0.672225 + 0.740347i \(0.265339\pi\)
\(4\) 100.799 0.787495
\(5\) 12.0619 0.0431540 0.0215770 0.999767i \(-0.493131\pi\)
0.0215770 + 0.999767i \(0.493131\pi\)
\(6\) −951.035 −1.79749
\(7\) −1248.65 −1.37593 −0.687966 0.725743i \(-0.741496\pi\)
−0.687966 + 0.725743i \(0.741496\pi\)
\(8\) 411.441 0.284114
\(9\) 1766.10 0.807545
\(10\) −182.450 −0.0576956
\(11\) −2043.21 −0.462849 −0.231424 0.972853i \(-0.574339\pi\)
−0.231424 + 0.972853i \(0.574339\pi\)
\(12\) 6337.62 1.05875
\(13\) 9460.36 1.19428 0.597139 0.802137i \(-0.296304\pi\)
0.597139 + 0.802137i \(0.296304\pi\)
\(14\) 18887.2 1.83958
\(15\) 758.376 0.0580183
\(16\) −19125.8 −1.16735
\(17\) −19027.5 −0.939312 −0.469656 0.882849i \(-0.655622\pi\)
−0.469656 + 0.882849i \(0.655622\pi\)
\(18\) −26714.2 −1.07967
\(19\) −28326.9 −0.947460 −0.473730 0.880670i \(-0.657093\pi\)
−0.473730 + 0.880670i \(0.657093\pi\)
\(20\) 1215.83 0.0339835
\(21\) −78507.2 −1.84987
\(22\) 30905.9 0.618816
\(23\) 2087.68 0.0357781 0.0178890 0.999840i \(-0.494305\pi\)
0.0178890 + 0.999840i \(0.494305\pi\)
\(24\) 25868.8 0.381977
\(25\) −77979.5 −0.998138
\(26\) −143098. −1.59672
\(27\) −26463.5 −0.258746
\(28\) −125863. −1.08354
\(29\) −111967. −0.852508 −0.426254 0.904603i \(-0.640167\pi\)
−0.426254 + 0.904603i \(0.640167\pi\)
\(30\) −11471.3 −0.0775689
\(31\) 7978.79 0.0481029 0.0240514 0.999711i \(-0.492343\pi\)
0.0240514 + 0.999711i \(0.492343\pi\)
\(32\) 236635. 1.27660
\(33\) −128464. −0.622277
\(34\) 287812. 1.25583
\(35\) −15061.1 −0.0593769
\(36\) 178022. 0.635937
\(37\) −185038. −0.600559 −0.300279 0.953851i \(-0.597080\pi\)
−0.300279 + 0.953851i \(0.597080\pi\)
\(38\) 428475. 1.26673
\(39\) 594808. 1.60565
\(40\) 4962.76 0.0122606
\(41\) −421826. −0.955851 −0.477925 0.878400i \(-0.658611\pi\)
−0.477925 + 0.878400i \(0.658611\pi\)
\(42\) 1.18751e6 2.47323
\(43\) 79507.0 0.152499
\(44\) −205954. −0.364491
\(45\) 21302.5 0.0348488
\(46\) −31578.5 −0.0478343
\(47\) 197368. 0.277290 0.138645 0.990342i \(-0.455725\pi\)
0.138645 + 0.990342i \(0.455725\pi\)
\(48\) −1.20251e6 −1.56944
\(49\) 735581. 0.893191
\(50\) 1.17953e6 1.33448
\(51\) −1.19633e6 −1.26286
\(52\) 953597. 0.940488
\(53\) −2.01317e6 −1.85744 −0.928722 0.370778i \(-0.879091\pi\)
−0.928722 + 0.370778i \(0.879091\pi\)
\(54\) 400289. 0.345936
\(55\) −24645.0 −0.0199738
\(56\) −513745. −0.390921
\(57\) −1.78101e6 −1.27381
\(58\) 1.69363e6 1.13978
\(59\) −856041. −0.542641 −0.271321 0.962489i \(-0.587460\pi\)
−0.271321 + 0.962489i \(0.587460\pi\)
\(60\) 76443.8 0.0456891
\(61\) 2.61554e6 1.47539 0.737695 0.675135i \(-0.235914\pi\)
0.737695 + 0.675135i \(0.235914\pi\)
\(62\) −120688. −0.0643122
\(63\) −2.20524e6 −1.11113
\(64\) −1.13126e6 −0.539427
\(65\) 114110. 0.0515379
\(66\) 1.94317e6 0.831967
\(67\) 2.93249e6 1.19117 0.595587 0.803291i \(-0.296920\pi\)
0.595587 + 0.803291i \(0.296920\pi\)
\(68\) −1.91796e6 −0.739703
\(69\) 131260. 0.0481018
\(70\) 227816. 0.0793853
\(71\) −3.20003e6 −1.06109 −0.530543 0.847658i \(-0.678012\pi\)
−0.530543 + 0.847658i \(0.678012\pi\)
\(72\) 726646. 0.229435
\(73\) 4.19984e6 1.26358 0.631790 0.775140i \(-0.282321\pi\)
0.631790 + 0.775140i \(0.282321\pi\)
\(74\) 2.79891e6 0.802930
\(75\) −4.90286e6 −1.34195
\(76\) −2.85533e6 −0.746119
\(77\) 2.55125e6 0.636849
\(78\) −8.99713e6 −2.14671
\(79\) 7.83389e6 1.78765 0.893825 0.448416i \(-0.148012\pi\)
0.893825 + 0.448416i \(0.148012\pi\)
\(80\) −230694. −0.0503756
\(81\) −5.52632e6 −1.15542
\(82\) 6.38059e6 1.27795
\(83\) 5.57533e6 1.07028 0.535140 0.844764i \(-0.320259\pi\)
0.535140 + 0.844764i \(0.320259\pi\)
\(84\) −7.91347e6 −1.45676
\(85\) −229508. −0.0405351
\(86\) −1.20263e6 −0.203886
\(87\) −7.03981e6 −1.14615
\(88\) −840660. −0.131502
\(89\) −889160. −0.133695 −0.0668474 0.997763i \(-0.521294\pi\)
−0.0668474 + 0.997763i \(0.521294\pi\)
\(90\) −322225. −0.0465918
\(91\) −1.18127e7 −1.64325
\(92\) 210437. 0.0281750
\(93\) 501656. 0.0646719
\(94\) −2.98542e6 −0.370730
\(95\) −341676. −0.0408866
\(96\) 1.48781e7 1.71632
\(97\) 1.05709e7 1.17601 0.588003 0.808859i \(-0.299914\pi\)
0.588003 + 0.808859i \(0.299914\pi\)
\(98\) −1.11265e7 −1.19417
\(99\) −3.60852e6 −0.373771
\(100\) −7.86028e6 −0.786028
\(101\) 613031. 0.0592049 0.0296024 0.999562i \(-0.490576\pi\)
0.0296024 + 0.999562i \(0.490576\pi\)
\(102\) 1.80958e7 1.68841
\(103\) −1.58867e7 −1.43253 −0.716265 0.697828i \(-0.754150\pi\)
−0.716265 + 0.697828i \(0.754150\pi\)
\(104\) 3.89238e6 0.339311
\(105\) −946945. −0.0798293
\(106\) 3.04515e7 2.48335
\(107\) −2.82221e6 −0.222713 −0.111357 0.993781i \(-0.535520\pi\)
−0.111357 + 0.993781i \(0.535520\pi\)
\(108\) −2.66750e6 −0.203761
\(109\) −2.05758e6 −0.152183 −0.0760913 0.997101i \(-0.524244\pi\)
−0.0760913 + 0.997101i \(0.524244\pi\)
\(110\) 372783. 0.0267044
\(111\) −1.16340e7 −0.807421
\(112\) 2.38814e7 1.60619
\(113\) 1.63742e6 0.106754 0.0533770 0.998574i \(-0.483001\pi\)
0.0533770 + 0.998574i \(0.483001\pi\)
\(114\) 2.69398e7 1.70305
\(115\) 25181.4 0.00154396
\(116\) −1.12862e7 −0.671346
\(117\) 1.67079e7 0.964434
\(118\) 1.29486e7 0.725496
\(119\) 2.37586e7 1.29243
\(120\) 312027. 0.0164838
\(121\) −1.53125e7 −0.785771
\(122\) −3.95629e7 −1.97255
\(123\) −2.65218e7 −1.28509
\(124\) 804256. 0.0378808
\(125\) −1.88292e6 −0.0862276
\(126\) 3.33567e7 1.48555
\(127\) 6.60627e6 0.286183 0.143091 0.989709i \(-0.454296\pi\)
0.143091 + 0.989709i \(0.454296\pi\)
\(128\) −1.31777e7 −0.555397
\(129\) 4.99890e6 0.205027
\(130\) −1.72604e6 −0.0689047
\(131\) −5.79186e6 −0.225096 −0.112548 0.993646i \(-0.535901\pi\)
−0.112548 + 0.993646i \(0.535901\pi\)
\(132\) −1.29491e7 −0.490040
\(133\) 3.53703e7 1.30364
\(134\) −4.43572e7 −1.59257
\(135\) −319200. −0.0111659
\(136\) −7.82868e6 −0.266872
\(137\) 2.00914e7 0.667556 0.333778 0.942652i \(-0.391676\pi\)
0.333778 + 0.942652i \(0.391676\pi\)
\(138\) −1.98546e6 −0.0643108
\(139\) 2.57362e7 0.812817 0.406408 0.913692i \(-0.366781\pi\)
0.406408 + 0.913692i \(0.366781\pi\)
\(140\) −1.51815e6 −0.0467590
\(141\) 1.24093e7 0.372803
\(142\) 4.84041e7 1.41864
\(143\) −1.93295e7 −0.552770
\(144\) −3.37781e7 −0.942685
\(145\) −1.35054e6 −0.0367891
\(146\) −6.35272e7 −1.68937
\(147\) 4.62487e7 1.20085
\(148\) −1.86517e7 −0.472937
\(149\) 2.86333e7 0.709121 0.354560 0.935033i \(-0.384631\pi\)
0.354560 + 0.935033i \(0.384631\pi\)
\(150\) 7.41612e7 1.79414
\(151\) −6.67796e7 −1.57843 −0.789213 0.614119i \(-0.789511\pi\)
−0.789213 + 0.614119i \(0.789511\pi\)
\(152\) −1.16548e7 −0.269186
\(153\) −3.36045e7 −0.758537
\(154\) −3.85906e7 −0.851449
\(155\) 96239.3 0.00207583
\(156\) 5.99562e7 1.26444
\(157\) −8.44900e7 −1.74243 −0.871217 0.490898i \(-0.836669\pi\)
−0.871217 + 0.490898i \(0.836669\pi\)
\(158\) −1.18496e8 −2.39004
\(159\) −1.26576e8 −2.49724
\(160\) 2.85426e6 0.0550902
\(161\) −2.60678e6 −0.0492282
\(162\) 8.35917e7 1.54476
\(163\) 6.39006e7 1.15571 0.577854 0.816140i \(-0.303890\pi\)
0.577854 + 0.816140i \(0.303890\pi\)
\(164\) −4.25198e7 −0.752727
\(165\) −1.54952e6 −0.0268537
\(166\) −8.43330e7 −1.43093
\(167\) −1.62544e7 −0.270061 −0.135031 0.990841i \(-0.543113\pi\)
−0.135031 + 0.990841i \(0.543113\pi\)
\(168\) −3.23010e7 −0.525574
\(169\) 2.67498e7 0.426302
\(170\) 3.47156e6 0.0541942
\(171\) −5.00281e7 −0.765116
\(172\) 8.01425e6 0.120092
\(173\) −1.19428e8 −1.75366 −0.876830 0.480801i \(-0.840346\pi\)
−0.876830 + 0.480801i \(0.840346\pi\)
\(174\) 1.06485e8 1.53238
\(175\) 9.73690e7 1.37337
\(176\) 3.90781e7 0.540305
\(177\) −5.38225e7 −0.729554
\(178\) 1.34495e7 0.178746
\(179\) 6.67618e7 0.870047 0.435023 0.900419i \(-0.356740\pi\)
0.435023 + 0.900419i \(0.356740\pi\)
\(180\) 2.14728e6 0.0274432
\(181\) 1.24721e8 1.56339 0.781693 0.623664i \(-0.214357\pi\)
0.781693 + 0.623664i \(0.214357\pi\)
\(182\) 1.78680e8 2.19698
\(183\) 1.64449e8 1.98359
\(184\) 858957. 0.0101650
\(185\) −2.23191e6 −0.0259165
\(186\) −7.58810e6 −0.0864645
\(187\) 3.88772e7 0.434759
\(188\) 1.98946e7 0.218365
\(189\) 3.30436e7 0.356017
\(190\) 5.16822e6 0.0546643
\(191\) 7.43805e7 0.772400 0.386200 0.922415i \(-0.373787\pi\)
0.386200 + 0.922415i \(0.373787\pi\)
\(192\) −7.11265e7 −0.725233
\(193\) 9.08749e7 0.909900 0.454950 0.890517i \(-0.349657\pi\)
0.454950 + 0.890517i \(0.349657\pi\)
\(194\) −1.59896e8 −1.57229
\(195\) 7.17451e6 0.0692901
\(196\) 7.41460e7 0.703383
\(197\) −1.24750e8 −1.16254 −0.581270 0.813711i \(-0.697444\pi\)
−0.581270 + 0.813711i \(0.697444\pi\)
\(198\) 5.45829e7 0.499722
\(199\) −1.75151e8 −1.57553 −0.787767 0.615973i \(-0.788763\pi\)
−0.787767 + 0.615973i \(0.788763\pi\)
\(200\) −3.20839e7 −0.283585
\(201\) 1.84377e8 1.60147
\(202\) −9.27277e6 −0.0791553
\(203\) 1.39808e8 1.17299
\(204\) −1.20589e8 −0.994494
\(205\) −5.08803e6 −0.0412487
\(206\) 2.40304e8 1.91525
\(207\) 3.68706e6 0.0288924
\(208\) −1.80937e8 −1.39414
\(209\) 5.78778e7 0.438530
\(210\) 1.43236e7 0.106730
\(211\) 8.95521e7 0.656277 0.328138 0.944630i \(-0.393579\pi\)
0.328138 + 0.944630i \(0.393579\pi\)
\(212\) −2.02926e8 −1.46273
\(213\) −2.01198e8 −1.42658
\(214\) 4.26891e7 0.297761
\(215\) 959005. 0.00658092
\(216\) −1.08881e7 −0.0735133
\(217\) −9.96271e6 −0.0661863
\(218\) 3.11233e7 0.203464
\(219\) 2.64059e8 1.69882
\(220\) −2.48420e6 −0.0157292
\(221\) −1.80007e8 −1.12180
\(222\) 1.75978e8 1.07950
\(223\) −1.68028e8 −1.01464 −0.507322 0.861757i \(-0.669364\pi\)
−0.507322 + 0.861757i \(0.669364\pi\)
\(224\) −2.95474e8 −1.75651
\(225\) −1.37720e8 −0.806041
\(226\) −2.47677e7 −0.142727
\(227\) 1.05659e8 0.599538 0.299769 0.954012i \(-0.403090\pi\)
0.299769 + 0.954012i \(0.403090\pi\)
\(228\) −1.79525e8 −1.00312
\(229\) −2.91314e8 −1.60301 −0.801506 0.597987i \(-0.795967\pi\)
−0.801506 + 0.597987i \(0.795967\pi\)
\(230\) −380897. −0.00206424
\(231\) 1.60407e8 0.856211
\(232\) −4.60680e7 −0.242209
\(233\) −2.72762e8 −1.41266 −0.706332 0.707881i \(-0.749651\pi\)
−0.706332 + 0.707881i \(0.749651\pi\)
\(234\) −2.52726e8 −1.28942
\(235\) 2.38064e6 0.0119662
\(236\) −8.62884e7 −0.427327
\(237\) 4.92545e8 2.40340
\(238\) −3.59376e8 −1.72794
\(239\) 2.25857e8 1.07014 0.535069 0.844808i \(-0.320285\pi\)
0.535069 + 0.844808i \(0.320285\pi\)
\(240\) −1.45046e7 −0.0677275
\(241\) 2.02550e8 0.932122 0.466061 0.884753i \(-0.345673\pi\)
0.466061 + 0.884753i \(0.345673\pi\)
\(242\) 2.31618e8 1.05055
\(243\) −2.89584e8 −1.29465
\(244\) 2.63644e8 1.16186
\(245\) 8.87250e6 0.0385447
\(246\) 4.01171e8 1.71813
\(247\) −2.67982e8 −1.13153
\(248\) 3.28280e6 0.0136667
\(249\) 3.50541e8 1.43894
\(250\) 2.84812e7 0.115284
\(251\) 4.56654e8 1.82276 0.911379 0.411567i \(-0.135019\pi\)
0.911379 + 0.411567i \(0.135019\pi\)
\(252\) −2.22287e8 −0.875007
\(253\) −4.26557e6 −0.0165598
\(254\) −9.99272e7 −0.382618
\(255\) −1.44300e7 −0.0544973
\(256\) 3.44128e8 1.28198
\(257\) 6.13602e7 0.225487 0.112743 0.993624i \(-0.464036\pi\)
0.112743 + 0.993624i \(0.464036\pi\)
\(258\) −7.56139e7 −0.274115
\(259\) 2.31048e8 0.826328
\(260\) 1.15022e7 0.0405858
\(261\) −1.97746e8 −0.688439
\(262\) 8.76083e7 0.300947
\(263\) −5.34920e8 −1.81319 −0.906596 0.422000i \(-0.861328\pi\)
−0.906596 + 0.422000i \(0.861328\pi\)
\(264\) −5.28554e7 −0.176797
\(265\) −2.42827e7 −0.0801560
\(266\) −5.35015e8 −1.74293
\(267\) −5.59048e7 −0.179746
\(268\) 2.95593e8 0.938043
\(269\) −1.68231e8 −0.526956 −0.263478 0.964665i \(-0.584870\pi\)
−0.263478 + 0.964665i \(0.584870\pi\)
\(270\) 4.82825e6 0.0149285
\(271\) 3.63723e8 1.11014 0.555071 0.831803i \(-0.312691\pi\)
0.555071 + 0.831803i \(0.312691\pi\)
\(272\) 3.63916e8 1.09650
\(273\) −7.42706e8 −2.20926
\(274\) −3.03904e8 −0.892504
\(275\) 1.59329e8 0.461987
\(276\) 1.32309e7 0.0378799
\(277\) −2.54748e8 −0.720165 −0.360083 0.932920i \(-0.617252\pi\)
−0.360083 + 0.932920i \(0.617252\pi\)
\(278\) −3.89289e8 −1.08671
\(279\) 1.40913e7 0.0388452
\(280\) −6.19674e6 −0.0168698
\(281\) −5.70700e8 −1.53439 −0.767195 0.641414i \(-0.778348\pi\)
−0.767195 + 0.641414i \(0.778348\pi\)
\(282\) −1.87704e8 −0.498427
\(283\) −2.06689e8 −0.542083 −0.271041 0.962568i \(-0.587368\pi\)
−0.271041 + 0.962568i \(0.587368\pi\)
\(284\) −3.22561e8 −0.835599
\(285\) −2.14824e7 −0.0549700
\(286\) 2.92380e8 0.739039
\(287\) 5.26713e8 1.31519
\(288\) 4.17921e8 1.03091
\(289\) −4.82937e7 −0.117692
\(290\) 2.04284e7 0.0491860
\(291\) 6.64629e8 1.58108
\(292\) 4.23341e8 0.995062
\(293\) 8.14731e8 1.89225 0.946123 0.323808i \(-0.104963\pi\)
0.946123 + 0.323808i \(0.104963\pi\)
\(294\) −6.99563e8 −1.60550
\(295\) −1.03255e7 −0.0234171
\(296\) −7.61323e7 −0.170627
\(297\) 5.40705e7 0.119760
\(298\) −4.33111e8 −0.948075
\(299\) 1.97502e7 0.0427290
\(300\) −4.94205e8 −1.05678
\(301\) −9.92763e7 −0.209828
\(302\) 1.01012e9 2.11031
\(303\) 3.85435e7 0.0795980
\(304\) 5.41774e8 1.10601
\(305\) 3.15484e7 0.0636689
\(306\) 5.08305e8 1.01414
\(307\) −1.50628e8 −0.297112 −0.148556 0.988904i \(-0.547463\pi\)
−0.148556 + 0.988904i \(0.547463\pi\)
\(308\) 2.57165e8 0.501515
\(309\) −9.98857e8 −1.92597
\(310\) −1.45573e6 −0.00277533
\(311\) −6.59750e8 −1.24371 −0.621853 0.783134i \(-0.713620\pi\)
−0.621853 + 0.783134i \(0.713620\pi\)
\(312\) 2.44728e8 0.456187
\(313\) −3.46916e8 −0.639469 −0.319735 0.947507i \(-0.603594\pi\)
−0.319735 + 0.947507i \(0.603594\pi\)
\(314\) 1.27801e9 2.32959
\(315\) −2.65994e7 −0.0479496
\(316\) 7.89650e8 1.40776
\(317\) −9.98461e8 −1.76045 −0.880225 0.474557i \(-0.842608\pi\)
−0.880225 + 0.474557i \(0.842608\pi\)
\(318\) 1.91460e9 3.33874
\(319\) 2.28773e8 0.394582
\(320\) −1.36452e7 −0.0232784
\(321\) −1.77443e8 −0.299427
\(322\) 3.94305e7 0.0658167
\(323\) 5.38989e8 0.889961
\(324\) −5.57049e8 −0.909884
\(325\) −7.37714e8 −1.19205
\(326\) −9.66567e8 −1.54515
\(327\) −1.29368e8 −0.204602
\(328\) −1.73556e8 −0.271570
\(329\) −2.46444e8 −0.381533
\(330\) 2.34383e7 0.0359027
\(331\) 7.96597e8 1.20737 0.603686 0.797222i \(-0.293698\pi\)
0.603686 + 0.797222i \(0.293698\pi\)
\(332\) 5.61989e8 0.842839
\(333\) −3.26796e8 −0.484978
\(334\) 2.45865e8 0.361065
\(335\) 3.53714e7 0.0514039
\(336\) 1.50151e9 2.15944
\(337\) −1.10141e9 −1.56764 −0.783818 0.620991i \(-0.786730\pi\)
−0.783818 + 0.620991i \(0.786730\pi\)
\(338\) −4.04621e8 −0.569954
\(339\) 1.02950e8 0.143525
\(340\) −2.31342e7 −0.0319211
\(341\) −1.63024e7 −0.0222643
\(342\) 7.56731e8 1.02294
\(343\) 1.09834e8 0.146963
\(344\) 3.27124e7 0.0433269
\(345\) 1.58325e6 0.00207578
\(346\) 1.80648e9 2.34459
\(347\) −5.91060e8 −0.759414 −0.379707 0.925107i \(-0.623975\pi\)
−0.379707 + 0.925107i \(0.623975\pi\)
\(348\) −7.09607e8 −0.902591
\(349\) 1.16064e9 1.46153 0.730764 0.682631i \(-0.239164\pi\)
0.730764 + 0.682631i \(0.239164\pi\)
\(350\) −1.47281e9 −1.83616
\(351\) −2.50354e8 −0.309015
\(352\) −4.83495e8 −0.590871
\(353\) −2.28005e8 −0.275888 −0.137944 0.990440i \(-0.544049\pi\)
−0.137944 + 0.990440i \(0.544049\pi\)
\(354\) 8.14125e8 0.975393
\(355\) −3.85985e7 −0.0457900
\(356\) −8.96267e7 −0.105284
\(357\) 1.49379e9 1.73761
\(358\) −1.00985e9 −1.16323
\(359\) −1.34720e9 −1.53674 −0.768371 0.640005i \(-0.778932\pi\)
−0.768371 + 0.640005i \(0.778932\pi\)
\(360\) 8.76473e6 0.00990102
\(361\) −9.14611e7 −0.102320
\(362\) −1.88655e9 −2.09020
\(363\) −9.62751e8 −1.05643
\(364\) −1.19071e9 −1.29405
\(365\) 5.06580e7 0.0545285
\(366\) −2.48747e9 −2.65200
\(367\) −9.77318e8 −1.03206 −0.516030 0.856571i \(-0.672591\pi\)
−0.516030 + 0.856571i \(0.672591\pi\)
\(368\) −3.99286e7 −0.0417654
\(369\) −7.44988e8 −0.771893
\(370\) 3.37602e7 0.0346496
\(371\) 2.51375e9 2.55572
\(372\) 5.05666e7 0.0509288
\(373\) 6.10073e8 0.608696 0.304348 0.952561i \(-0.401561\pi\)
0.304348 + 0.952561i \(0.401561\pi\)
\(374\) −5.88060e8 −0.581261
\(375\) −1.18386e8 −0.115929
\(376\) 8.12054e7 0.0787820
\(377\) −1.05925e9 −1.01813
\(378\) −4.99821e8 −0.475985
\(379\) 3.49421e8 0.329695 0.164847 0.986319i \(-0.447287\pi\)
0.164847 + 0.986319i \(0.447287\pi\)
\(380\) −3.44407e7 −0.0321980
\(381\) 4.15361e8 0.384758
\(382\) −1.12509e9 −1.03268
\(383\) 6.97644e8 0.634509 0.317255 0.948340i \(-0.397239\pi\)
0.317255 + 0.948340i \(0.397239\pi\)
\(384\) −8.28529e8 −0.746704
\(385\) 3.07730e7 0.0274825
\(386\) −1.37458e9 −1.21651
\(387\) 1.40417e8 0.123149
\(388\) 1.06554e9 0.926098
\(389\) 1.70449e9 1.46815 0.734077 0.679067i \(-0.237615\pi\)
0.734077 + 0.679067i \(0.237615\pi\)
\(390\) −1.08522e8 −0.0926389
\(391\) −3.97233e7 −0.0336068
\(392\) 3.02648e8 0.253768
\(393\) −3.64155e8 −0.302631
\(394\) 1.88698e9 1.55428
\(395\) 9.44915e7 0.0771442
\(396\) −3.63736e8 −0.294343
\(397\) −1.51386e7 −0.0121428 −0.00607138 0.999982i \(-0.501933\pi\)
−0.00607138 + 0.999982i \(0.501933\pi\)
\(398\) 2.64936e9 2.10645
\(399\) 2.22386e9 1.75268
\(400\) 1.49142e9 1.16517
\(401\) −1.18655e9 −0.918928 −0.459464 0.888196i \(-0.651958\pi\)
−0.459464 + 0.888196i \(0.651958\pi\)
\(402\) −2.78890e9 −2.14112
\(403\) 7.54822e7 0.0574482
\(404\) 6.17931e7 0.0466235
\(405\) −6.66579e7 −0.0498608
\(406\) −2.11475e9 −1.56826
\(407\) 3.78072e8 0.277968
\(408\) −4.92218e8 −0.358795
\(409\) 1.78027e9 1.28663 0.643315 0.765602i \(-0.277559\pi\)
0.643315 + 0.765602i \(0.277559\pi\)
\(410\) 7.69620e7 0.0551484
\(411\) 1.26322e9 0.897495
\(412\) −1.60137e9 −1.12811
\(413\) 1.06890e9 0.746638
\(414\) −5.57708e7 −0.0386283
\(415\) 6.72490e7 0.0461868
\(416\) 2.23865e9 1.52461
\(417\) 1.61813e9 1.09279
\(418\) −8.75466e8 −0.586303
\(419\) 1.26485e9 0.840023 0.420012 0.907519i \(-0.362026\pi\)
0.420012 + 0.907519i \(0.362026\pi\)
\(420\) −9.54515e7 −0.0628652
\(421\) 2.11626e8 0.138224 0.0691118 0.997609i \(-0.477983\pi\)
0.0691118 + 0.997609i \(0.477983\pi\)
\(422\) −1.35457e9 −0.877424
\(423\) 3.48572e8 0.223925
\(424\) −8.28301e8 −0.527725
\(425\) 1.48375e9 0.937563
\(426\) 3.04334e9 1.90729
\(427\) −3.26589e9 −2.03004
\(428\) −2.84477e8 −0.175385
\(429\) −1.21532e9 −0.743172
\(430\) −1.45060e7 −0.00879850
\(431\) 3.00302e8 0.180671 0.0903353 0.995911i \(-0.471206\pi\)
0.0903353 + 0.995911i \(0.471206\pi\)
\(432\) 5.06135e8 0.302046
\(433\) −1.41630e9 −0.838392 −0.419196 0.907896i \(-0.637688\pi\)
−0.419196 + 0.907896i \(0.637688\pi\)
\(434\) 1.50697e8 0.0884893
\(435\) −8.49134e7 −0.0494611
\(436\) −2.07403e8 −0.119843
\(437\) −5.91374e7 −0.0338983
\(438\) −3.99419e9 −2.27127
\(439\) 1.52173e9 0.858445 0.429223 0.903199i \(-0.358788\pi\)
0.429223 + 0.903199i \(0.358788\pi\)
\(440\) −1.01400e7 −0.00567482
\(441\) 1.29911e9 0.721292
\(442\) 2.72280e9 1.49982
\(443\) 2.81273e8 0.153715 0.0768574 0.997042i \(-0.475511\pi\)
0.0768574 + 0.997042i \(0.475511\pi\)
\(444\) −1.17270e9 −0.635840
\(445\) −1.07250e7 −0.00576946
\(446\) 2.54160e9 1.35655
\(447\) 1.80028e9 0.953377
\(448\) 1.41255e9 0.742215
\(449\) 2.07207e9 1.08029 0.540146 0.841571i \(-0.318369\pi\)
0.540146 + 0.841571i \(0.318369\pi\)
\(450\) 2.08316e9 1.07765
\(451\) 8.61880e8 0.442414
\(452\) 1.65050e8 0.0840682
\(453\) −4.19868e9 −2.12212
\(454\) −1.59821e9 −0.801566
\(455\) −1.42483e8 −0.0709126
\(456\) −7.32782e8 −0.361908
\(457\) 6.18336e8 0.303053 0.151526 0.988453i \(-0.451581\pi\)
0.151526 + 0.988453i \(0.451581\pi\)
\(458\) 4.40644e9 2.14318
\(459\) 5.03533e8 0.243043
\(460\) 2.53827e6 0.00121586
\(461\) 1.04519e9 0.496867 0.248433 0.968649i \(-0.420084\pi\)
0.248433 + 0.968649i \(0.420084\pi\)
\(462\) −2.42633e9 −1.14473
\(463\) −6.19870e8 −0.290246 −0.145123 0.989414i \(-0.546358\pi\)
−0.145123 + 0.989414i \(0.546358\pi\)
\(464\) 2.14147e9 0.995173
\(465\) 6.05092e6 0.00279085
\(466\) 4.12584e9 1.88869
\(467\) −4.18640e9 −1.90209 −0.951047 0.309047i \(-0.899990\pi\)
−0.951047 + 0.309047i \(0.899990\pi\)
\(468\) 1.68415e9 0.759487
\(469\) −3.66165e9 −1.63897
\(470\) −3.60098e7 −0.0159985
\(471\) −5.31220e9 −2.34262
\(472\) −3.52210e8 −0.154172
\(473\) −1.62450e8 −0.0705838
\(474\) −7.45030e9 −3.21329
\(475\) 2.20891e9 0.945695
\(476\) 2.39485e9 1.01778
\(477\) −3.55547e9 −1.49997
\(478\) −3.41633e9 −1.43075
\(479\) 2.16526e9 0.900193 0.450097 0.892980i \(-0.351390\pi\)
0.450097 + 0.892980i \(0.351390\pi\)
\(480\) 1.79458e8 0.0740660
\(481\) −1.75053e9 −0.717235
\(482\) −3.06380e9 −1.24622
\(483\) −1.63898e8 −0.0661848
\(484\) −1.54349e9 −0.618791
\(485\) 1.27505e8 0.0507493
\(486\) 4.38029e9 1.73091
\(487\) −4.04525e9 −1.58706 −0.793531 0.608530i \(-0.791760\pi\)
−0.793531 + 0.608530i \(0.791760\pi\)
\(488\) 1.07614e9 0.419178
\(489\) 4.01766e9 1.55379
\(490\) −1.34206e8 −0.0515332
\(491\) 2.63031e9 1.00282 0.501408 0.865211i \(-0.332816\pi\)
0.501408 + 0.865211i \(0.332816\pi\)
\(492\) −2.67338e9 −1.01200
\(493\) 2.13046e9 0.800772
\(494\) 4.05353e9 1.51283
\(495\) −4.35256e7 −0.0161297
\(496\) −1.52601e8 −0.0561527
\(497\) 3.99572e9 1.45998
\(498\) −5.30233e9 −1.92382
\(499\) 3.10184e8 0.111755 0.0558775 0.998438i \(-0.482204\pi\)
0.0558775 + 0.998438i \(0.482204\pi\)
\(500\) −1.89797e8 −0.0679037
\(501\) −1.02197e9 −0.363084
\(502\) −6.90740e9 −2.43698
\(503\) 4.58242e9 1.60549 0.802743 0.596325i \(-0.203373\pi\)
0.802743 + 0.596325i \(0.203373\pi\)
\(504\) −9.07326e8 −0.315687
\(505\) 7.39432e6 0.00255493
\(506\) 6.45216e7 0.0221400
\(507\) 1.68186e9 0.573142
\(508\) 6.65908e8 0.225367
\(509\) −5.65113e8 −0.189943 −0.0949714 0.995480i \(-0.530276\pi\)
−0.0949714 + 0.995480i \(0.530276\pi\)
\(510\) 2.18270e8 0.0728614
\(511\) −5.24412e9 −1.73860
\(512\) −3.51858e9 −1.15857
\(513\) 7.49627e8 0.245151
\(514\) −9.28141e8 −0.301469
\(515\) −1.91624e8 −0.0618194
\(516\) 5.03886e8 0.161457
\(517\) −4.03265e8 −0.128344
\(518\) −3.49486e9 −1.10478
\(519\) −7.50889e9 −2.35771
\(520\) 4.69494e7 0.0146426
\(521\) −5.02091e9 −1.55543 −0.777714 0.628618i \(-0.783621\pi\)
−0.777714 + 0.628618i \(0.783621\pi\)
\(522\) 2.99113e9 0.920424
\(523\) 4.46410e9 1.36452 0.682258 0.731112i \(-0.260998\pi\)
0.682258 + 0.731112i \(0.260998\pi\)
\(524\) −5.83815e8 −0.177262
\(525\) 6.12195e9 1.84643
\(526\) 8.09126e9 2.42419
\(527\) −1.51816e8 −0.0451836
\(528\) 2.45698e9 0.726413
\(529\) −3.40047e9 −0.998720
\(530\) 3.67303e8 0.107166
\(531\) −1.51186e9 −0.438207
\(532\) 3.56530e9 1.02661
\(533\) −3.99063e9 −1.14155
\(534\) 8.45622e8 0.240315
\(535\) −3.40412e7 −0.00961096
\(536\) 1.20655e9 0.338429
\(537\) 4.19756e9 1.16973
\(538\) 2.54469e9 0.704525
\(539\) −1.50295e9 −0.413412
\(540\) −3.21751e7 −0.00879310
\(541\) −4.10317e9 −1.11411 −0.557056 0.830475i \(-0.688069\pi\)
−0.557056 + 0.830475i \(0.688069\pi\)
\(542\) −5.50172e9 −1.48423
\(543\) 7.84170e9 2.10189
\(544\) −4.50256e9 −1.19912
\(545\) −2.48184e7 −0.00656728
\(546\) 1.12343e10 2.95372
\(547\) −1.85474e9 −0.484539 −0.242269 0.970209i \(-0.577892\pi\)
−0.242269 + 0.970209i \(0.577892\pi\)
\(548\) 2.02520e9 0.525697
\(549\) 4.61930e9 1.19144
\(550\) −2.41002e9 −0.617663
\(551\) 3.17168e9 0.807717
\(552\) 5.40058e7 0.0136664
\(553\) −9.78177e9 −2.45969
\(554\) 3.85335e9 0.962841
\(555\) −1.40329e8 −0.0348434
\(556\) 2.59419e9 0.640089
\(557\) −5.61884e9 −1.37770 −0.688848 0.724906i \(-0.741883\pi\)
−0.688848 + 0.724906i \(0.741883\pi\)
\(558\) −2.13147e8 −0.0519350
\(559\) 7.52165e8 0.182126
\(560\) 2.88055e8 0.0693135
\(561\) 2.44435e9 0.584512
\(562\) 8.63247e9 2.05144
\(563\) −4.77175e9 −1.12693 −0.563467 0.826139i \(-0.690533\pi\)
−0.563467 + 0.826139i \(0.690533\pi\)
\(564\) 1.25085e9 0.293580
\(565\) 1.97503e7 0.00460686
\(566\) 3.12641e9 0.724749
\(567\) 6.90043e9 1.58977
\(568\) −1.31662e9 −0.301469
\(569\) −3.47483e9 −0.790753 −0.395377 0.918519i \(-0.629386\pi\)
−0.395377 + 0.918519i \(0.629386\pi\)
\(570\) 3.24945e8 0.0734934
\(571\) 3.59894e9 0.809000 0.404500 0.914538i \(-0.367446\pi\)
0.404500 + 0.914538i \(0.367446\pi\)
\(572\) −1.94840e9 −0.435304
\(573\) 4.67658e9 1.03845
\(574\) −7.96712e9 −1.75837
\(575\) −1.62796e8 −0.0357114
\(576\) −1.99792e9 −0.435612
\(577\) 4.67766e9 1.01371 0.506855 0.862032i \(-0.330808\pi\)
0.506855 + 0.862032i \(0.330808\pi\)
\(578\) 7.30496e8 0.157351
\(579\) 5.71364e9 1.22331
\(580\) −1.36133e8 −0.0289712
\(581\) −6.96163e9 −1.47263
\(582\) −1.00533e10 −2.11386
\(583\) 4.11334e9 0.859715
\(584\) 1.72798e9 0.359000
\(585\) 2.01530e8 0.0416192
\(586\) −1.23237e10 −2.52988
\(587\) 5.27917e9 1.07729 0.538644 0.842533i \(-0.318937\pi\)
0.538644 + 0.842533i \(0.318937\pi\)
\(588\) 4.66184e9 0.945663
\(589\) −2.26014e8 −0.0455755
\(590\) 1.56184e8 0.0313080
\(591\) −7.84348e9 −1.56298
\(592\) 3.53901e9 0.701060
\(593\) −4.65728e9 −0.917151 −0.458575 0.888655i \(-0.651640\pi\)
−0.458575 + 0.888655i \(0.651640\pi\)
\(594\) −8.17876e8 −0.160116
\(595\) 2.86574e8 0.0557735
\(596\) 2.88622e9 0.558429
\(597\) −1.10124e10 −2.11823
\(598\) −2.98744e8 −0.0571274
\(599\) −8.21418e9 −1.56160 −0.780801 0.624780i \(-0.785189\pi\)
−0.780801 + 0.624780i \(0.785189\pi\)
\(600\) −2.01724e9 −0.381265
\(601\) 4.53542e9 0.852229 0.426115 0.904669i \(-0.359882\pi\)
0.426115 + 0.904669i \(0.359882\pi\)
\(602\) 1.50166e9 0.280534
\(603\) 5.17908e9 0.961926
\(604\) −6.73134e9 −1.24300
\(605\) −1.84697e8 −0.0339091
\(606\) −5.83014e8 −0.106420
\(607\) 2.58768e9 0.469625 0.234812 0.972041i \(-0.424552\pi\)
0.234812 + 0.972041i \(0.424552\pi\)
\(608\) −6.70312e9 −1.20952
\(609\) 8.79024e9 1.57703
\(610\) −4.77204e8 −0.0851235
\(611\) 1.86717e9 0.331162
\(612\) −3.38731e9 −0.597344
\(613\) 4.69723e9 0.823626 0.411813 0.911268i \(-0.364896\pi\)
0.411813 + 0.911268i \(0.364896\pi\)
\(614\) 2.27841e9 0.397231
\(615\) −3.19903e8 −0.0554569
\(616\) 1.04969e9 0.180937
\(617\) −6.31709e9 −1.08273 −0.541363 0.840789i \(-0.682092\pi\)
−0.541363 + 0.840789i \(0.682092\pi\)
\(618\) 1.51088e10 2.57496
\(619\) 1.92907e9 0.326912 0.163456 0.986551i \(-0.447736\pi\)
0.163456 + 0.986551i \(0.447736\pi\)
\(620\) 9.70086e6 0.00163470
\(621\) −5.52473e7 −0.00925742
\(622\) 9.97945e9 1.66280
\(623\) 1.11025e9 0.183955
\(624\) −1.13762e10 −1.87435
\(625\) 6.06944e9 0.994417
\(626\) 5.24750e9 0.854953
\(627\) 3.63899e9 0.589582
\(628\) −8.51653e9 −1.37216
\(629\) 3.52081e9 0.564112
\(630\) 4.02345e8 0.0641072
\(631\) −5.13248e9 −0.813251 −0.406626 0.913595i \(-0.633295\pi\)
−0.406626 + 0.913595i \(0.633295\pi\)
\(632\) 3.22318e9 0.507896
\(633\) 5.63047e9 0.882331
\(634\) 1.51028e10 2.35367
\(635\) 7.96842e7 0.0123499
\(636\) −1.27587e10 −1.96656
\(637\) 6.95886e9 1.06672
\(638\) −3.46045e9 −0.527546
\(639\) −5.65158e9 −0.856874
\(640\) −1.58948e8 −0.0239676
\(641\) −1.08820e10 −1.63195 −0.815976 0.578086i \(-0.803800\pi\)
−0.815976 + 0.578086i \(0.803800\pi\)
\(642\) 2.68402e9 0.400325
\(643\) 5.44950e9 0.808385 0.404192 0.914674i \(-0.367553\pi\)
0.404192 + 0.914674i \(0.367553\pi\)
\(644\) −2.62762e8 −0.0387669
\(645\) 6.02962e7 0.00884771
\(646\) −8.15280e9 −1.18985
\(647\) 2.41640e9 0.350754 0.175377 0.984501i \(-0.443886\pi\)
0.175377 + 0.984501i \(0.443886\pi\)
\(648\) −2.27375e9 −0.328270
\(649\) 1.74907e9 0.251161
\(650\) 1.11587e10 1.59374
\(651\) −6.26392e8 −0.0889842
\(652\) 6.44113e9 0.910114
\(653\) 7.41937e9 1.04273 0.521363 0.853335i \(-0.325424\pi\)
0.521363 + 0.853335i \(0.325424\pi\)
\(654\) 1.95683e9 0.273547
\(655\) −6.98608e7 −0.00971380
\(656\) 8.06777e9 1.11581
\(657\) 7.41734e9 1.02040
\(658\) 3.72774e9 0.510099
\(659\) −4.19463e9 −0.570945 −0.285473 0.958387i \(-0.592151\pi\)
−0.285473 + 0.958387i \(0.592151\pi\)
\(660\) −1.56191e8 −0.0211471
\(661\) −5.21211e9 −0.701954 −0.350977 0.936384i \(-0.614151\pi\)
−0.350977 + 0.936384i \(0.614151\pi\)
\(662\) −1.20494e10 −1.61422
\(663\) −1.13177e10 −1.50820
\(664\) 2.29392e9 0.304081
\(665\) 4.26633e8 0.0562573
\(666\) 4.94316e9 0.648402
\(667\) −2.33752e8 −0.0305011
\(668\) −1.63843e9 −0.212672
\(669\) −1.05645e10 −1.36414
\(670\) −5.35032e8 −0.0687255
\(671\) −5.34410e9 −0.682882
\(672\) −1.85775e10 −2.36154
\(673\) −1.45876e10 −1.84473 −0.922363 0.386325i \(-0.873744\pi\)
−0.922363 + 0.386325i \(0.873744\pi\)
\(674\) 1.66601e10 2.09589
\(675\) 2.06361e9 0.258264
\(676\) 2.69636e9 0.335711
\(677\) −5.06173e9 −0.626958 −0.313479 0.949595i \(-0.601495\pi\)
−0.313479 + 0.949595i \(0.601495\pi\)
\(678\) −1.55724e9 −0.191890
\(679\) −1.31993e10 −1.61810
\(680\) −9.44287e7 −0.0115166
\(681\) 6.64318e9 0.806049
\(682\) 2.46591e8 0.0297668
\(683\) 3.88726e9 0.466844 0.233422 0.972376i \(-0.425008\pi\)
0.233422 + 0.972376i \(0.425008\pi\)
\(684\) −5.04280e9 −0.602525
\(685\) 2.42340e8 0.0288077
\(686\) −1.66136e9 −0.196485
\(687\) −1.83160e10 −2.15517
\(688\) −1.52064e9 −0.178019
\(689\) −1.90453e10 −2.21831
\(690\) −2.39484e7 −0.00277526
\(691\) −6.93681e9 −0.799810 −0.399905 0.916557i \(-0.630957\pi\)
−0.399905 + 0.916557i \(0.630957\pi\)
\(692\) −1.20383e10 −1.38100
\(693\) 4.50577e9 0.514284
\(694\) 8.94045e9 1.01532
\(695\) 3.10428e8 0.0350763
\(696\) −2.89646e9 −0.325638
\(697\) 8.02629e9 0.897842
\(698\) −1.75559e10 −1.95402
\(699\) −1.71496e10 −1.89926
\(700\) 9.81473e9 1.08152
\(701\) 3.87694e9 0.425085 0.212543 0.977152i \(-0.431826\pi\)
0.212543 + 0.977152i \(0.431826\pi\)
\(702\) 3.78688e9 0.413144
\(703\) 5.24155e9 0.569005
\(704\) 2.31141e9 0.249673
\(705\) 1.49679e8 0.0160879
\(706\) 3.44883e9 0.368855
\(707\) −7.65460e8 −0.0814619
\(708\) −5.42527e9 −0.574520
\(709\) −1.10614e10 −1.16559 −0.582797 0.812618i \(-0.698042\pi\)
−0.582797 + 0.812618i \(0.698042\pi\)
\(710\) 5.83845e8 0.0612200
\(711\) 1.38354e10 1.44361
\(712\) −3.65836e8 −0.0379846
\(713\) 1.66572e7 0.00172103
\(714\) −2.25953e10 −2.32313
\(715\) −2.33151e8 −0.0238542
\(716\) 6.72955e9 0.685157
\(717\) 1.42004e10 1.43875
\(718\) 2.03779e10 2.05458
\(719\) −1.72872e10 −1.73450 −0.867249 0.497875i \(-0.834114\pi\)
−0.867249 + 0.497875i \(0.834114\pi\)
\(720\) −4.07428e8 −0.0406806
\(721\) 1.98369e10 1.97107
\(722\) 1.38345e9 0.136799
\(723\) 1.27351e10 1.25319
\(724\) 1.25718e10 1.23116
\(725\) 8.73116e9 0.850921
\(726\) 1.45627e10 1.41242
\(727\) 5.92252e9 0.571658 0.285829 0.958281i \(-0.407731\pi\)
0.285829 + 0.958281i \(0.407731\pi\)
\(728\) −4.86021e9 −0.466869
\(729\) −6.12119e9 −0.585180
\(730\) −7.66259e8 −0.0729030
\(731\) −1.51282e9 −0.143244
\(732\) 1.65763e10 1.56206
\(733\) 1.69133e10 1.58623 0.793114 0.609074i \(-0.208459\pi\)
0.793114 + 0.609074i \(0.208459\pi\)
\(734\) 1.47830e10 1.37983
\(735\) 5.57847e8 0.0518214
\(736\) 4.94018e8 0.0456741
\(737\) −5.99170e9 −0.551333
\(738\) 1.12688e10 1.03200
\(739\) −2.73492e9 −0.249281 −0.124640 0.992202i \(-0.539778\pi\)
−0.124640 + 0.992202i \(0.539778\pi\)
\(740\) −2.24975e8 −0.0204091
\(741\) −1.68490e10 −1.52129
\(742\) −3.80232e10 −3.41692
\(743\) −9.30626e9 −0.832366 −0.416183 0.909281i \(-0.636632\pi\)
−0.416183 + 0.909281i \(0.636632\pi\)
\(744\) 2.06402e8 0.0183742
\(745\) 3.45373e8 0.0306014
\(746\) −9.22803e9 −0.813810
\(747\) 9.84659e9 0.864299
\(748\) 3.91879e9 0.342371
\(749\) 3.52395e9 0.306438
\(750\) 1.79072e9 0.154993
\(751\) 2.13212e9 0.183684 0.0918421 0.995774i \(-0.470725\pi\)
0.0918421 + 0.995774i \(0.470725\pi\)
\(752\) −3.77483e9 −0.323694
\(753\) 2.87115e10 2.45061
\(754\) 1.60224e10 1.36122
\(755\) −8.05489e8 −0.0681154
\(756\) 3.33077e9 0.280361
\(757\) 1.54717e10 1.29630 0.648148 0.761515i \(-0.275544\pi\)
0.648148 + 0.761515i \(0.275544\pi\)
\(758\) −5.28539e9 −0.440793
\(759\) −2.68192e8 −0.0222638
\(760\) −1.40579e8 −0.0116165
\(761\) −1.05745e8 −0.00869787 −0.00434894 0.999991i \(-0.501384\pi\)
−0.00434894 + 0.999991i \(0.501384\pi\)
\(762\) −6.28279e9 −0.514411
\(763\) 2.56920e9 0.209393
\(764\) 7.49750e9 0.608261
\(765\) −4.05334e8 −0.0327339
\(766\) −1.05526e10 −0.848321
\(767\) −8.09846e9 −0.648065
\(768\) 2.16366e10 1.72356
\(769\) 1.45940e10 1.15726 0.578632 0.815589i \(-0.303587\pi\)
0.578632 + 0.815589i \(0.303587\pi\)
\(770\) −4.65475e8 −0.0367434
\(771\) 3.85794e9 0.303155
\(772\) 9.16013e9 0.716541
\(773\) 7.67773e9 0.597867 0.298934 0.954274i \(-0.403369\pi\)
0.298934 + 0.954274i \(0.403369\pi\)
\(774\) −2.12397e9 −0.164647
\(775\) −6.22182e8 −0.0480133
\(776\) 4.34928e9 0.334119
\(777\) 1.45268e10 1.11096
\(778\) −2.57823e10 −1.96288
\(779\) 1.19490e10 0.905630
\(780\) 7.23186e8 0.0545656
\(781\) 6.53835e9 0.491122
\(782\) 6.00859e8 0.0449313
\(783\) 2.96305e9 0.220583
\(784\) −1.40686e10 −1.04266
\(785\) −1.01911e9 −0.0751930
\(786\) 5.50826e9 0.404609
\(787\) −9.73198e9 −0.711688 −0.355844 0.934545i \(-0.615807\pi\)
−0.355844 + 0.934545i \(0.615807\pi\)
\(788\) −1.25747e10 −0.915494
\(789\) −3.36324e10 −2.43775
\(790\) −1.42929e9 −0.103140
\(791\) −2.04456e9 −0.146886
\(792\) −1.48469e9 −0.106194
\(793\) 2.47439e10 1.76203
\(794\) 2.28988e8 0.0162345
\(795\) −1.52674e9 −0.107766
\(796\) −1.76551e10 −1.24073
\(797\) −4.83086e9 −0.338003 −0.169001 0.985616i \(-0.554054\pi\)
−0.169001 + 0.985616i \(0.554054\pi\)
\(798\) −3.36384e10 −2.34328
\(799\) −3.75542e9 −0.260462
\(800\) −1.84527e10 −1.27422
\(801\) −1.57035e9 −0.107965
\(802\) 1.79479e10 1.22858
\(803\) −8.58116e9 −0.584846
\(804\) 1.85850e10 1.26115
\(805\) −3.14427e7 −0.00212439
\(806\) −1.14175e9 −0.0768067
\(807\) −1.05773e10 −0.708465
\(808\) 2.52226e8 0.0168209
\(809\) 1.46263e10 0.971216 0.485608 0.874177i \(-0.338598\pi\)
0.485608 + 0.874177i \(0.338598\pi\)
\(810\) 1.00828e9 0.0666625
\(811\) 2.93692e9 0.193339 0.0966694 0.995317i \(-0.469181\pi\)
0.0966694 + 0.995317i \(0.469181\pi\)
\(812\) 1.40925e10 0.923727
\(813\) 2.28686e10 1.49253
\(814\) −5.71876e9 −0.371635
\(815\) 7.70762e8 0.0498734
\(816\) 2.28807e10 1.47419
\(817\) −2.25218e9 −0.144486
\(818\) −2.69285e10 −1.72019
\(819\) −2.08624e10 −1.32700
\(820\) −5.12869e8 −0.0324832
\(821\) 1.70106e10 1.07280 0.536401 0.843964i \(-0.319784\pi\)
0.536401 + 0.843964i \(0.319784\pi\)
\(822\) −1.91076e10 −1.19993
\(823\) 7.47291e9 0.467294 0.233647 0.972321i \(-0.424934\pi\)
0.233647 + 0.972321i \(0.424934\pi\)
\(824\) −6.53644e9 −0.407002
\(825\) 1.00176e10 0.621118
\(826\) −1.61682e10 −0.998234
\(827\) −2.00025e10 −1.22975 −0.614873 0.788626i \(-0.710793\pi\)
−0.614873 + 0.788626i \(0.710793\pi\)
\(828\) 3.71653e8 0.0227526
\(829\) −8.27952e9 −0.504736 −0.252368 0.967631i \(-0.581209\pi\)
−0.252368 + 0.967631i \(0.581209\pi\)
\(830\) −1.01722e9 −0.0617504
\(831\) −1.60170e10 −0.968226
\(832\) −1.07021e10 −0.644226
\(833\) −1.39963e10 −0.838985
\(834\) −2.44760e10 −1.46103
\(835\) −1.96059e8 −0.0116542
\(836\) 5.83404e9 0.345340
\(837\) −2.11146e8 −0.0124464
\(838\) −1.91323e10 −1.12309
\(839\) 1.26666e10 0.740443 0.370222 0.928943i \(-0.379282\pi\)
0.370222 + 0.928943i \(0.379282\pi\)
\(840\) −3.89612e8 −0.0226806
\(841\) −4.71317e9 −0.273229
\(842\) −3.20108e9 −0.184801
\(843\) −3.58820e10 −2.06291
\(844\) 9.02679e9 0.516814
\(845\) 3.22654e8 0.0183966
\(846\) −5.27255e9 −0.299381
\(847\) 1.91199e10 1.08117
\(848\) 3.85036e10 2.16828
\(849\) −1.29953e10 −0.728803
\(850\) −2.24434e10 −1.25350
\(851\) −3.86301e8 −0.0214868
\(852\) −2.02806e10 −1.12342
\(853\) −3.00382e10 −1.65711 −0.828556 0.559906i \(-0.810837\pi\)
−0.828556 + 0.559906i \(0.810837\pi\)
\(854\) 4.94002e10 2.71410
\(855\) −6.03434e8 −0.0330178
\(856\) −1.16117e9 −0.0632759
\(857\) 1.76722e10 0.959088 0.479544 0.877518i \(-0.340802\pi\)
0.479544 + 0.877518i \(0.340802\pi\)
\(858\) 1.83830e10 0.993600
\(859\) 1.84797e9 0.0994763 0.0497382 0.998762i \(-0.484161\pi\)
0.0497382 + 0.998762i \(0.484161\pi\)
\(860\) 9.66671e7 0.00518244
\(861\) 3.31164e10 1.76820
\(862\) −4.54240e9 −0.241551
\(863\) −5.58485e9 −0.295784 −0.147892 0.989004i \(-0.547249\pi\)
−0.147892 + 0.989004i \(0.547249\pi\)
\(864\) −6.26218e9 −0.330314
\(865\) −1.44053e9 −0.0756774
\(866\) 2.14231e10 1.12091
\(867\) −3.03640e9 −0.158231
\(868\) −1.00423e9 −0.0521214
\(869\) −1.60063e10 −0.827411
\(870\) 1.28441e9 0.0661281
\(871\) 2.77424e10 1.42259
\(872\) −8.46574e8 −0.0432372
\(873\) 1.86692e10 0.949677
\(874\) 8.94519e8 0.0453210
\(875\) 2.35110e9 0.118643
\(876\) 2.66170e10 1.33781
\(877\) −2.49280e10 −1.24793 −0.623963 0.781454i \(-0.714478\pi\)
−0.623963 + 0.781454i \(0.714478\pi\)
\(878\) −2.30179e10 −1.14772
\(879\) 5.12251e10 2.54403
\(880\) 4.71356e8 0.0233163
\(881\) −7.08433e9 −0.349047 −0.174523 0.984653i \(-0.555838\pi\)
−0.174523 + 0.984653i \(0.555838\pi\)
\(882\) −1.96505e10 −0.964347
\(883\) −1.79072e9 −0.0875316 −0.0437658 0.999042i \(-0.513936\pi\)
−0.0437658 + 0.999042i \(0.513936\pi\)
\(884\) −1.81446e10 −0.883412
\(885\) −6.49201e8 −0.0314831
\(886\) −4.25457e9 −0.205512
\(887\) 6.97503e9 0.335593 0.167797 0.985822i \(-0.446335\pi\)
0.167797 + 0.985822i \(0.446335\pi\)
\(888\) −4.78672e9 −0.229399
\(889\) −8.24891e9 −0.393768
\(890\) 1.62227e8 0.00771361
\(891\) 1.12914e10 0.534783
\(892\) −1.69371e10 −0.799026
\(893\) −5.59082e9 −0.262722
\(894\) −2.72313e10 −1.27464
\(895\) 8.05275e8 0.0375460
\(896\) 1.64543e10 0.764189
\(897\) 1.24177e9 0.0574470
\(898\) −3.13423e10 −1.44432
\(899\) −8.93364e8 −0.0410081
\(900\) −1.38821e10 −0.634753
\(901\) 3.83056e10 1.74472
\(902\) −1.30369e10 −0.591495
\(903\) −6.24187e9 −0.282103
\(904\) 6.73699e8 0.0303303
\(905\) 1.50438e9 0.0674663
\(906\) 6.35097e10 2.83721
\(907\) −6.43630e9 −0.286425 −0.143213 0.989692i \(-0.545743\pi\)
−0.143213 + 0.989692i \(0.545743\pi\)
\(908\) 1.06504e10 0.472133
\(909\) 1.08267e9 0.0478106
\(910\) 2.15522e9 0.0948082
\(911\) 1.55350e10 0.680766 0.340383 0.940287i \(-0.389443\pi\)
0.340383 + 0.940287i \(0.389443\pi\)
\(912\) 3.40633e10 1.48698
\(913\) −1.13916e10 −0.495377
\(914\) −9.35302e9 −0.405173
\(915\) 1.98356e9 0.0855996
\(916\) −2.93642e10 −1.26236
\(917\) 7.23199e9 0.309717
\(918\) −7.61650e9 −0.324942
\(919\) 3.73555e10 1.58763 0.793816 0.608158i \(-0.208091\pi\)
0.793816 + 0.608158i \(0.208091\pi\)
\(920\) 1.03607e7 0.000438662 0
\(921\) −9.47051e9 −0.399452
\(922\) −1.58096e10 −0.664297
\(923\) −3.02735e10 −1.26723
\(924\) 1.61689e10 0.674261
\(925\) 1.44292e10 0.599440
\(926\) 9.37622e9 0.388051
\(927\) −2.80576e10 −1.15683
\(928\) −2.64954e10 −1.08831
\(929\) 2.51045e10 1.02730 0.513649 0.858001i \(-0.328294\pi\)
0.513649 + 0.858001i \(0.328294\pi\)
\(930\) −9.15270e7 −0.00373129
\(931\) −2.08367e10 −0.846262
\(932\) −2.74943e10 −1.11247
\(933\) −4.14809e10 −1.67210
\(934\) 6.33240e10 2.54305
\(935\) 4.68933e8 0.0187616
\(936\) 6.87433e9 0.274009
\(937\) −3.29027e10 −1.30660 −0.653300 0.757099i \(-0.726616\pi\)
−0.653300 + 0.757099i \(0.726616\pi\)
\(938\) 5.53866e10 2.19126
\(939\) −2.18119e10 −0.859734
\(940\) 2.39967e8 0.00942330
\(941\) 1.27046e10 0.497048 0.248524 0.968626i \(-0.420054\pi\)
0.248524 + 0.968626i \(0.420054\pi\)
\(942\) 8.03529e10 3.13201
\(943\) −8.80638e8 −0.0341985
\(944\) 1.63725e10 0.633450
\(945\) 3.98568e8 0.0153635
\(946\) 2.45723e9 0.0943685
\(947\) −2.04335e9 −0.0781840 −0.0390920 0.999236i \(-0.512447\pi\)
−0.0390920 + 0.999236i \(0.512447\pi\)
\(948\) 4.96482e10 1.89267
\(949\) 3.97320e10 1.50907
\(950\) −3.34123e10 −1.26437
\(951\) −6.27769e10 −2.36684
\(952\) 9.77527e9 0.367197
\(953\) −1.23244e10 −0.461253 −0.230626 0.973042i \(-0.574078\pi\)
−0.230626 + 0.973042i \(0.574078\pi\)
\(954\) 5.37804e10 2.00542
\(955\) 8.97170e8 0.0333321
\(956\) 2.27662e10 0.842729
\(957\) 1.43838e10 0.530496
\(958\) −3.27520e10 −1.20353
\(959\) −2.50871e10 −0.918512
\(960\) −8.57921e8 −0.0312967
\(961\) −2.74490e10 −0.997686
\(962\) 2.64787e10 0.958923
\(963\) −4.98431e9 −0.179851
\(964\) 2.04169e10 0.734041
\(965\) 1.09612e9 0.0392658
\(966\) 2.47914e9 0.0884873
\(967\) 1.28161e10 0.455788 0.227894 0.973686i \(-0.426816\pi\)
0.227894 + 0.973686i \(0.426816\pi\)
\(968\) −6.30017e9 −0.223248
\(969\) 3.38882e10 1.19651
\(970\) −1.92865e9 −0.0678504
\(971\) 2.66467e10 0.934063 0.467032 0.884241i \(-0.345323\pi\)
0.467032 + 0.884241i \(0.345323\pi\)
\(972\) −2.91899e10 −1.01953
\(973\) −3.21355e10 −1.11838
\(974\) 6.11889e10 2.12186
\(975\) −4.63828e10 −1.60266
\(976\) −5.00243e10 −1.72229
\(977\) 8.47518e9 0.290749 0.145374 0.989377i \(-0.453561\pi\)
0.145374 + 0.989377i \(0.453561\pi\)
\(978\) −6.07716e10 −2.07738
\(979\) 1.81674e9 0.0618805
\(980\) 8.94342e8 0.0303538
\(981\) −3.63390e9 −0.122894
\(982\) −3.97863e10 −1.34074
\(983\) −2.76800e9 −0.0929455 −0.0464727 0.998920i \(-0.514798\pi\)
−0.0464727 + 0.998920i \(0.514798\pi\)
\(984\) −1.09121e10 −0.365113
\(985\) −1.50472e9 −0.0501682
\(986\) −3.22255e10 −1.07061
\(987\) −1.54948e10 −0.512952
\(988\) −2.70124e10 −0.891075
\(989\) 1.65985e8 0.00545610
\(990\) 6.58373e8 0.0215650
\(991\) 2.88798e10 0.942621 0.471310 0.881967i \(-0.343781\pi\)
0.471310 + 0.881967i \(0.343781\pi\)
\(992\) 1.88806e9 0.0614080
\(993\) 5.00850e10 1.62325
\(994\) −6.04397e10 −1.95196
\(995\) −2.11266e9 −0.0679906
\(996\) 3.53343e10 1.13315
\(997\) 4.99664e9 0.159678 0.0798390 0.996808i \(-0.474559\pi\)
0.0798390 + 0.996808i \(0.474559\pi\)
\(998\) −4.69187e9 −0.149413
\(999\) 4.89675e9 0.155392
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 43.8.a.a.1.3 11
3.2 odd 2 387.8.a.b.1.9 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.3 11 1.1 even 1 trivial
387.8.a.b.1.9 11 3.2 odd 2