Properties

Label 273.12.a.c.1.2
Level $273$
Weight $12$
Character 273.1
Self dual yes
Analytic conductor $209.758$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 273.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(209.757688293\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} - 1473489774048 x^{10} - 6951601887328 x^{9} + 4072481535854976 x^{8} + 21355761647276288 x^{7} - 5703452025696599040 x^{6} - 25041700255043457024 x^{5} + 3737994671267475603456 x^{4} + 6711995200063092490240 x^{3} - 963518272786595267739648 x^{2} + 1287697466669064032092160 x + 45555354651594891689197568\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12}\cdot 5\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-81.4415\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q-85.4415 q^{2} -243.000 q^{3} +5252.25 q^{4} +8053.09 q^{5} +20762.3 q^{6} +16807.0 q^{7} -273776. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-85.4415 q^{2} -243.000 q^{3} +5252.25 q^{4} +8053.09 q^{5} +20762.3 q^{6} +16807.0 q^{7} -273776. q^{8} +59049.0 q^{9} -688068. q^{10} -763843. q^{11} -1.27630e6 q^{12} -371293. q^{13} -1.43602e6 q^{14} -1.95690e6 q^{15} +1.26353e7 q^{16} -1.13101e7 q^{17} -5.04524e6 q^{18} +1.19332e7 q^{19} +4.22969e7 q^{20} -4.08410e6 q^{21} +6.52639e7 q^{22} -2.31448e7 q^{23} +6.65277e7 q^{24} +1.60241e7 q^{25} +3.17238e7 q^{26} -1.43489e7 q^{27} +8.82746e7 q^{28} +6.17965e7 q^{29} +1.67201e8 q^{30} +2.82040e8 q^{31} -5.18881e8 q^{32} +1.85614e8 q^{33} +9.66351e8 q^{34} +1.35348e8 q^{35} +3.10140e8 q^{36} +1.13149e8 q^{37} -1.01959e9 q^{38} +9.02242e7 q^{39} -2.20474e9 q^{40} +8.17246e8 q^{41} +3.48952e8 q^{42} -1.03346e9 q^{43} -4.01190e9 q^{44} +4.75527e8 q^{45} +1.97753e9 q^{46} -9.12860e8 q^{47} -3.07037e9 q^{48} +2.82475e8 q^{49} -1.36912e9 q^{50} +2.74835e9 q^{51} -1.95013e9 q^{52} +3.83883e8 q^{53} +1.22599e9 q^{54} -6.15129e9 q^{55} -4.60136e9 q^{56} -2.89977e9 q^{57} -5.27998e9 q^{58} +7.57765e9 q^{59} -1.02781e10 q^{60} -4.32479e9 q^{61} -2.40980e10 q^{62} +9.92437e8 q^{63} +1.84570e10 q^{64} -2.99005e9 q^{65} -1.58591e10 q^{66} +5.01329e9 q^{67} -5.94034e10 q^{68} +5.62420e9 q^{69} -1.15644e10 q^{70} +2.74383e10 q^{71} -1.61662e10 q^{72} +1.00245e10 q^{73} -9.66760e9 q^{74} -3.89385e9 q^{75} +6.26762e10 q^{76} -1.28379e10 q^{77} -7.70889e9 q^{78} -1.20185e10 q^{79} +1.01753e11 q^{80} +3.48678e9 q^{81} -6.98268e10 q^{82} -5.80640e10 q^{83} -2.14507e10 q^{84} -9.10811e10 q^{85} +8.83003e10 q^{86} -1.50165e10 q^{87} +2.09122e11 q^{88} -2.43139e10 q^{89} -4.06297e10 q^{90} -6.24032e9 q^{91} -1.21563e11 q^{92} -6.85358e10 q^{93} +7.79962e10 q^{94} +9.60991e10 q^{95} +1.26088e11 q^{96} +1.17628e11 q^{97} -2.41351e10 q^{98} -4.51041e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −85.4415 −1.88801 −0.944004 0.329933i \(-0.892974\pi\)
−0.944004 + 0.329933i \(0.892974\pi\)
\(3\) −243.000 −0.577350
\(4\) 5252.25 2.56458
\(5\) 8053.09 1.15246 0.576232 0.817286i \(-0.304522\pi\)
0.576232 + 0.817286i \(0.304522\pi\)
\(6\) 20762.3 1.09004
\(7\) 16807.0 0.377964
\(8\) −273776. −2.95394
\(9\) 59049.0 0.333333
\(10\) −688068. −2.17586
\(11\) −763843. −1.43003 −0.715013 0.699111i \(-0.753579\pi\)
−0.715013 + 0.699111i \(0.753579\pi\)
\(12\) −1.27630e6 −1.48066
\(13\) −371293. −0.277350
\(14\) −1.43602e6 −0.713600
\(15\) −1.95690e6 −0.665375
\(16\) 1.26353e7 3.01248
\(17\) −1.13101e7 −1.93195 −0.965977 0.258628i \(-0.916729\pi\)
−0.965977 + 0.258628i \(0.916729\pi\)
\(18\) −5.04524e6 −0.629336
\(19\) 1.19332e7 1.10564 0.552818 0.833302i \(-0.313553\pi\)
0.552818 + 0.833302i \(0.313553\pi\)
\(20\) 4.22969e7 2.95558
\(21\) −4.08410e6 −0.218218
\(22\) 6.52639e7 2.69990
\(23\) −2.31448e7 −0.749810 −0.374905 0.927063i \(-0.622325\pi\)
−0.374905 + 0.927063i \(0.622325\pi\)
\(24\) 6.65277e7 1.70546
\(25\) 1.60241e7 0.328173
\(26\) 3.17238e7 0.523639
\(27\) −1.43489e7 −0.192450
\(28\) 8.82746e7 0.969319
\(29\) 6.17965e7 0.559467 0.279734 0.960078i \(-0.409754\pi\)
0.279734 + 0.960078i \(0.409754\pi\)
\(30\) 1.67201e8 1.25623
\(31\) 2.82040e8 1.76938 0.884692 0.466176i \(-0.154369\pi\)
0.884692 + 0.466176i \(0.154369\pi\)
\(32\) −5.18881e8 −2.73365
\(33\) 1.85614e8 0.825626
\(34\) 9.66351e8 3.64755
\(35\) 1.35348e8 0.435590
\(36\) 3.10140e8 0.854859
\(37\) 1.13149e8 0.268250 0.134125 0.990964i \(-0.457178\pi\)
0.134125 + 0.990964i \(0.457178\pi\)
\(38\) −1.01959e9 −2.08745
\(39\) 9.02242e7 0.160128
\(40\) −2.20474e9 −3.40430
\(41\) 8.17246e8 1.10165 0.550823 0.834622i \(-0.314314\pi\)
0.550823 + 0.834622i \(0.314314\pi\)
\(42\) 3.48952e8 0.411997
\(43\) −1.03346e9 −1.07205 −0.536027 0.844201i \(-0.680076\pi\)
−0.536027 + 0.844201i \(0.680076\pi\)
\(44\) −4.01190e9 −3.66741
\(45\) 4.75527e8 0.384155
\(46\) 1.97753e9 1.41565
\(47\) −9.12860e8 −0.580585 −0.290293 0.956938i \(-0.593753\pi\)
−0.290293 + 0.956938i \(0.593753\pi\)
\(48\) −3.07037e9 −1.73926
\(49\) 2.82475e8 0.142857
\(50\) −1.36912e9 −0.619594
\(51\) 2.74835e9 1.11541
\(52\) −1.95013e9 −0.711286
\(53\) 3.83883e8 0.126090 0.0630451 0.998011i \(-0.479919\pi\)
0.0630451 + 0.998011i \(0.479919\pi\)
\(54\) 1.22599e9 0.363347
\(55\) −6.15129e9 −1.64805
\(56\) −4.60136e9 −1.11648
\(57\) −2.89977e9 −0.638339
\(58\) −5.27998e9 −1.05628
\(59\) 7.57765e9 1.37990 0.689951 0.723856i \(-0.257632\pi\)
0.689951 + 0.723856i \(0.257632\pi\)
\(60\) −1.02781e10 −1.70641
\(61\) −4.32479e9 −0.655618 −0.327809 0.944744i \(-0.606310\pi\)
−0.327809 + 0.944744i \(0.606310\pi\)
\(62\) −2.40980e10 −3.34061
\(63\) 9.92437e8 0.125988
\(64\) 1.84570e10 2.14868
\(65\) −2.99005e9 −0.319636
\(66\) −1.58591e10 −1.55879
\(67\) 5.01329e9 0.453640 0.226820 0.973937i \(-0.427167\pi\)
0.226820 + 0.973937i \(0.427167\pi\)
\(68\) −5.94034e10 −4.95464
\(69\) 5.62420e9 0.432903
\(70\) −1.15644e10 −0.822399
\(71\) 2.74383e10 1.80483 0.902416 0.430867i \(-0.141792\pi\)
0.902416 + 0.430867i \(0.141792\pi\)
\(72\) −1.61662e10 −0.984645
\(73\) 1.00245e10 0.565964 0.282982 0.959125i \(-0.408676\pi\)
0.282982 + 0.959125i \(0.408676\pi\)
\(74\) −9.66760e9 −0.506459
\(75\) −3.89385e9 −0.189471
\(76\) 6.26762e10 2.83549
\(77\) −1.28379e10 −0.540499
\(78\) −7.70889e9 −0.302323
\(79\) −1.20185e10 −0.439441 −0.219721 0.975563i \(-0.570515\pi\)
−0.219721 + 0.975563i \(0.570515\pi\)
\(80\) 1.01753e11 3.47177
\(81\) 3.48678e9 0.111111
\(82\) −6.98268e10 −2.07992
\(83\) −5.80640e10 −1.61800 −0.808998 0.587812i \(-0.799990\pi\)
−0.808998 + 0.587812i \(0.799990\pi\)
\(84\) −2.14507e10 −0.559637
\(85\) −9.10811e10 −2.22651
\(86\) 8.83003e10 2.02405
\(87\) −1.50165e10 −0.323009
\(88\) 2.09122e11 4.22421
\(89\) −2.43139e10 −0.461540 −0.230770 0.973008i \(-0.574125\pi\)
−0.230770 + 0.973008i \(0.574125\pi\)
\(90\) −4.06297e10 −0.725287
\(91\) −6.24032e9 −0.104828
\(92\) −1.21563e11 −1.92295
\(93\) −6.85358e10 −1.02155
\(94\) 7.79962e10 1.09615
\(95\) 9.60991e10 1.27421
\(96\) 1.26088e11 1.57827
\(97\) 1.17628e11 1.39081 0.695405 0.718618i \(-0.255225\pi\)
0.695405 + 0.718618i \(0.255225\pi\)
\(98\) −2.41351e10 −0.269716
\(99\) −4.51041e10 −0.476676
\(100\) 8.41626e10 0.841626
\(101\) 1.26568e11 1.19828 0.599138 0.800646i \(-0.295510\pi\)
0.599138 + 0.800646i \(0.295510\pi\)
\(102\) −2.34823e11 −2.10591
\(103\) −9.18850e10 −0.780980 −0.390490 0.920607i \(-0.627694\pi\)
−0.390490 + 0.920607i \(0.627694\pi\)
\(104\) 1.01651e11 0.819274
\(105\) −3.28896e10 −0.251488
\(106\) −3.27995e10 −0.238059
\(107\) 6.94702e10 0.478837 0.239419 0.970916i \(-0.423043\pi\)
0.239419 + 0.970916i \(0.423043\pi\)
\(108\) −7.53641e10 −0.493553
\(109\) 3.47768e10 0.216493 0.108247 0.994124i \(-0.465476\pi\)
0.108247 + 0.994124i \(0.465476\pi\)
\(110\) 5.25576e11 3.11154
\(111\) −2.74951e10 −0.154874
\(112\) 2.12361e11 1.13861
\(113\) 2.16735e11 1.10662 0.553309 0.832976i \(-0.313365\pi\)
0.553309 + 0.832976i \(0.313365\pi\)
\(114\) 2.47761e11 1.20519
\(115\) −1.86387e11 −0.864129
\(116\) 3.24571e11 1.43480
\(117\) −2.19245e10 −0.0924500
\(118\) −6.47446e11 −2.60527
\(119\) −1.90089e11 −0.730210
\(120\) 5.35753e11 1.96548
\(121\) 2.98144e11 1.04498
\(122\) 3.69517e11 1.23781
\(123\) −1.98591e11 −0.636035
\(124\) 1.48135e12 4.53772
\(125\) −2.64174e11 −0.774256
\(126\) −8.47953e10 −0.237867
\(127\) 2.57449e11 0.691467 0.345734 0.938333i \(-0.387630\pi\)
0.345734 + 0.938333i \(0.387630\pi\)
\(128\) −5.14326e11 −1.32307
\(129\) 2.51131e11 0.618951
\(130\) 2.55475e11 0.603476
\(131\) −5.15177e11 −1.16671 −0.583357 0.812216i \(-0.698261\pi\)
−0.583357 + 0.812216i \(0.698261\pi\)
\(132\) 9.74891e11 2.11738
\(133\) 2.00561e11 0.417891
\(134\) −4.28343e11 −0.856476
\(135\) −1.15553e11 −0.221792
\(136\) 3.09643e12 5.70687
\(137\) −7.73092e11 −1.36857 −0.684287 0.729213i \(-0.739886\pi\)
−0.684287 + 0.729213i \(0.739886\pi\)
\(138\) −4.80540e11 −0.817325
\(139\) 1.41717e11 0.231655 0.115827 0.993269i \(-0.463048\pi\)
0.115827 + 0.993269i \(0.463048\pi\)
\(140\) 7.10883e11 1.11711
\(141\) 2.21825e11 0.335201
\(142\) −2.34437e12 −3.40754
\(143\) 2.83609e11 0.396618
\(144\) 7.46099e11 1.00416
\(145\) 4.97652e11 0.644766
\(146\) −8.56512e11 −1.06854
\(147\) −6.86415e10 −0.0824786
\(148\) 5.94286e11 0.687949
\(149\) −1.07931e12 −1.20399 −0.601994 0.798501i \(-0.705627\pi\)
−0.601994 + 0.798501i \(0.705627\pi\)
\(150\) 3.32697e11 0.357723
\(151\) −7.61202e11 −0.789090 −0.394545 0.918877i \(-0.629098\pi\)
−0.394545 + 0.918877i \(0.629098\pi\)
\(152\) −3.26703e12 −3.26598
\(153\) −6.67849e11 −0.643985
\(154\) 1.09689e12 1.02047
\(155\) 2.27130e12 2.03915
\(156\) 4.73880e11 0.410661
\(157\) 2.01629e12 1.68696 0.843480 0.537160i \(-0.180503\pi\)
0.843480 + 0.537160i \(0.180503\pi\)
\(158\) 1.02688e12 0.829669
\(159\) −9.32835e10 −0.0727982
\(160\) −4.17860e12 −3.15043
\(161\) −3.88995e11 −0.283402
\(162\) −2.97916e11 −0.209779
\(163\) 7.39508e11 0.503398 0.251699 0.967806i \(-0.419011\pi\)
0.251699 + 0.967806i \(0.419011\pi\)
\(164\) 4.29238e12 2.82525
\(165\) 1.49476e12 0.951505
\(166\) 4.96108e12 3.05479
\(167\) −1.50716e12 −0.897879 −0.448940 0.893562i \(-0.648198\pi\)
−0.448940 + 0.893562i \(0.648198\pi\)
\(168\) 1.11813e12 0.644602
\(169\) 1.37858e11 0.0769231
\(170\) 7.78211e12 4.20367
\(171\) 7.04643e11 0.368545
\(172\) −5.42799e12 −2.74937
\(173\) 2.41520e9 0.00118495 0.000592475 1.00000i \(-0.499811\pi\)
0.000592475 1.00000i \(0.499811\pi\)
\(174\) 1.28304e12 0.609843
\(175\) 2.69317e11 0.124038
\(176\) −9.65134e12 −4.30793
\(177\) −1.84137e12 −0.796687
\(178\) 2.07742e12 0.871392
\(179\) −1.14518e12 −0.465782 −0.232891 0.972503i \(-0.574819\pi\)
−0.232891 + 0.972503i \(0.574819\pi\)
\(180\) 2.49759e12 0.985194
\(181\) −2.11231e12 −0.808214 −0.404107 0.914712i \(-0.632418\pi\)
−0.404107 + 0.914712i \(0.632418\pi\)
\(182\) 5.33183e11 0.197917
\(183\) 1.05092e12 0.378521
\(184\) 6.33651e12 2.21489
\(185\) 9.11197e11 0.309149
\(186\) 5.85581e12 1.92870
\(187\) 8.63912e12 2.76275
\(188\) −4.79457e12 −1.48896
\(189\) −2.41162e11 −0.0727393
\(190\) −8.21085e12 −2.40571
\(191\) −3.12702e12 −0.890116 −0.445058 0.895502i \(-0.646817\pi\)
−0.445058 + 0.895502i \(0.646817\pi\)
\(192\) −4.48505e12 −1.24054
\(193\) −4.45408e12 −1.19727 −0.598636 0.801021i \(-0.704290\pi\)
−0.598636 + 0.801021i \(0.704290\pi\)
\(194\) −1.00504e13 −2.62586
\(195\) 7.26583e11 0.184542
\(196\) 1.48363e12 0.366368
\(197\) 1.68054e12 0.403538 0.201769 0.979433i \(-0.435331\pi\)
0.201769 + 0.979433i \(0.435331\pi\)
\(198\) 3.85377e12 0.899968
\(199\) −4.33759e12 −0.985272 −0.492636 0.870235i \(-0.663967\pi\)
−0.492636 + 0.870235i \(0.663967\pi\)
\(200\) −4.38702e12 −0.969403
\(201\) −1.21823e12 −0.261909
\(202\) −1.08142e13 −2.26236
\(203\) 1.03861e12 0.211459
\(204\) 1.44350e13 2.86057
\(205\) 6.58136e12 1.26961
\(206\) 7.85079e12 1.47450
\(207\) −1.36668e12 −0.249937
\(208\) −4.69138e12 −0.835511
\(209\) −9.11508e12 −1.58109
\(210\) 2.81014e12 0.474812
\(211\) 3.61505e12 0.595061 0.297530 0.954712i \(-0.403837\pi\)
0.297530 + 0.954712i \(0.403837\pi\)
\(212\) 2.01625e12 0.323368
\(213\) −6.66751e12 −1.04202
\(214\) −5.93564e12 −0.904049
\(215\) −8.32254e12 −1.23550
\(216\) 3.92839e12 0.568485
\(217\) 4.74025e12 0.668764
\(218\) −2.97139e12 −0.408741
\(219\) −2.43596e12 −0.326759
\(220\) −3.23081e13 −4.22656
\(221\) 4.19935e12 0.535828
\(222\) 2.34923e12 0.292404
\(223\) 9.05337e12 1.09934 0.549672 0.835381i \(-0.314753\pi\)
0.549672 + 0.835381i \(0.314753\pi\)
\(224\) −8.72084e12 −1.03322
\(225\) 9.46206e11 0.109391
\(226\) −1.85182e13 −2.08930
\(227\) 7.05927e12 0.777351 0.388676 0.921375i \(-0.372933\pi\)
0.388676 + 0.921375i \(0.372933\pi\)
\(228\) −1.52303e13 −1.63707
\(229\) −8.49506e12 −0.891397 −0.445699 0.895183i \(-0.647045\pi\)
−0.445699 + 0.895183i \(0.647045\pi\)
\(230\) 1.59252e13 1.63148
\(231\) 3.11961e12 0.312057
\(232\) −1.69184e13 −1.65263
\(233\) −1.34373e13 −1.28190 −0.640950 0.767583i \(-0.721459\pi\)
−0.640950 + 0.767583i \(0.721459\pi\)
\(234\) 1.87326e12 0.174546
\(235\) −7.35134e12 −0.669104
\(236\) 3.97997e13 3.53887
\(237\) 2.92049e12 0.253712
\(238\) 1.62415e13 1.37864
\(239\) 1.23059e13 1.02076 0.510381 0.859949i \(-0.329504\pi\)
0.510381 + 0.859949i \(0.329504\pi\)
\(240\) −2.47259e13 −2.00443
\(241\) −6.61712e12 −0.524294 −0.262147 0.965028i \(-0.584431\pi\)
−0.262147 + 0.965028i \(0.584431\pi\)
\(242\) −2.54739e13 −1.97292
\(243\) −8.47289e11 −0.0641500
\(244\) −2.27149e13 −1.68138
\(245\) 2.27480e12 0.164638
\(246\) 1.69679e13 1.20084
\(247\) −4.43071e12 −0.306648
\(248\) −7.72160e13 −5.22665
\(249\) 1.41096e13 0.934150
\(250\) 2.25714e13 1.46180
\(251\) 2.08534e13 1.32121 0.660604 0.750734i \(-0.270300\pi\)
0.660604 + 0.750734i \(0.270300\pi\)
\(252\) 5.21253e12 0.323106
\(253\) 1.76790e13 1.07225
\(254\) −2.19969e13 −1.30550
\(255\) 2.21327e13 1.28547
\(256\) 6.14486e12 0.349295
\(257\) −7.72597e12 −0.429854 −0.214927 0.976630i \(-0.568951\pi\)
−0.214927 + 0.976630i \(0.568951\pi\)
\(258\) −2.14570e13 −1.16858
\(259\) 1.90169e12 0.101389
\(260\) −1.57045e13 −0.819731
\(261\) 3.64902e12 0.186489
\(262\) 4.40175e13 2.20276
\(263\) −1.44121e13 −0.706270 −0.353135 0.935572i \(-0.614884\pi\)
−0.353135 + 0.935572i \(0.614884\pi\)
\(264\) −5.08167e13 −2.43885
\(265\) 3.09144e12 0.145314
\(266\) −1.71363e13 −0.788982
\(267\) 5.90828e12 0.266471
\(268\) 2.63311e13 1.16340
\(269\) −4.21199e13 −1.82327 −0.911633 0.411005i \(-0.865178\pi\)
−0.911633 + 0.411005i \(0.865178\pi\)
\(270\) 9.87302e12 0.418745
\(271\) −3.75772e13 −1.56169 −0.780843 0.624728i \(-0.785210\pi\)
−0.780843 + 0.624728i \(0.785210\pi\)
\(272\) −1.42906e14 −5.81997
\(273\) 1.51640e12 0.0605228
\(274\) 6.60542e13 2.58388
\(275\) −1.22399e13 −0.469296
\(276\) 2.95397e13 1.11021
\(277\) 1.26653e13 0.466633 0.233316 0.972401i \(-0.425042\pi\)
0.233316 + 0.972401i \(0.425042\pi\)
\(278\) −1.21085e13 −0.437366
\(279\) 1.66542e13 0.589795
\(280\) −3.70551e13 −1.28671
\(281\) 4.15864e13 1.41601 0.708005 0.706208i \(-0.249596\pi\)
0.708005 + 0.706208i \(0.249596\pi\)
\(282\) −1.89531e13 −0.632863
\(283\) 4.39981e13 1.44082 0.720408 0.693551i \(-0.243955\pi\)
0.720408 + 0.693551i \(0.243955\pi\)
\(284\) 1.44113e14 4.62863
\(285\) −2.33521e13 −0.735663
\(286\) −2.42320e13 −0.748818
\(287\) 1.37355e13 0.416383
\(288\) −3.06394e13 −0.911217
\(289\) 9.36461e13 2.73245
\(290\) −4.25202e13 −1.21732
\(291\) −2.85837e13 −0.802985
\(292\) 5.26514e13 1.45146
\(293\) −2.77921e13 −0.751880 −0.375940 0.926644i \(-0.622680\pi\)
−0.375940 + 0.926644i \(0.622680\pi\)
\(294\) 5.86483e12 0.155720
\(295\) 6.10235e13 1.59029
\(296\) −3.09775e13 −0.792394
\(297\) 1.09603e13 0.275209
\(298\) 9.22180e13 2.27314
\(299\) 8.59352e12 0.207960
\(300\) −2.04515e13 −0.485913
\(301\) −1.73694e13 −0.405199
\(302\) 6.50383e13 1.48981
\(303\) −3.07561e13 −0.691825
\(304\) 1.50779e14 3.33070
\(305\) −3.48279e13 −0.755576
\(306\) 5.70620e13 1.21585
\(307\) −8.88361e13 −1.85921 −0.929605 0.368558i \(-0.879852\pi\)
−0.929605 + 0.368558i \(0.879852\pi\)
\(308\) −6.74279e13 −1.38615
\(309\) 2.23280e13 0.450899
\(310\) −1.94063e14 −3.84994
\(311\) −8.37420e13 −1.63215 −0.816077 0.577943i \(-0.803856\pi\)
−0.816077 + 0.577943i \(0.803856\pi\)
\(312\) −2.47013e13 −0.473008
\(313\) −2.78398e13 −0.523809 −0.261904 0.965094i \(-0.584351\pi\)
−0.261904 + 0.965094i \(0.584351\pi\)
\(314\) −1.72275e14 −3.18500
\(315\) 7.99218e12 0.145197
\(316\) −6.31242e13 −1.12698
\(317\) 2.04725e12 0.0359207 0.0179604 0.999839i \(-0.494283\pi\)
0.0179604 + 0.999839i \(0.494283\pi\)
\(318\) 7.97028e12 0.137444
\(319\) −4.72028e13 −0.800053
\(320\) 1.48636e14 2.47627
\(321\) −1.68813e13 −0.276457
\(322\) 3.32364e13 0.535065
\(323\) −1.34965e14 −2.13604
\(324\) 1.83135e13 0.284953
\(325\) −5.94963e12 −0.0910189
\(326\) −6.31847e13 −0.950419
\(327\) −8.45077e12 −0.124992
\(328\) −2.23743e14 −3.25419
\(329\) −1.53424e13 −0.219441
\(330\) −1.27715e14 −1.79645
\(331\) 5.51332e13 0.762710 0.381355 0.924429i \(-0.375458\pi\)
0.381355 + 0.924429i \(0.375458\pi\)
\(332\) −3.04967e14 −4.14948
\(333\) 6.68132e12 0.0894168
\(334\) 1.28774e14 1.69520
\(335\) 4.03725e13 0.522804
\(336\) −5.16036e13 −0.657377
\(337\) −1.01528e14 −1.27240 −0.636199 0.771525i \(-0.719494\pi\)
−0.636199 + 0.771525i \(0.719494\pi\)
\(338\) −1.17788e13 −0.145231
\(339\) −5.26666e13 −0.638906
\(340\) −4.78381e14 −5.71005
\(341\) −2.15435e14 −2.53027
\(342\) −6.02058e13 −0.695817
\(343\) 4.74756e12 0.0539949
\(344\) 2.82937e14 3.16678
\(345\) 4.52922e13 0.498905
\(346\) −2.06359e11 −0.00223720
\(347\) 9.72058e13 1.03724 0.518621 0.855004i \(-0.326446\pi\)
0.518621 + 0.855004i \(0.326446\pi\)
\(348\) −7.88707e13 −0.828380
\(349\) 5.50463e13 0.569100 0.284550 0.958661i \(-0.408156\pi\)
0.284550 + 0.958661i \(0.408156\pi\)
\(350\) −2.30108e13 −0.234185
\(351\) 5.32765e12 0.0533761
\(352\) 3.96344e14 3.90919
\(353\) 9.47241e13 0.919813 0.459906 0.887967i \(-0.347883\pi\)
0.459906 + 0.887967i \(0.347883\pi\)
\(354\) 1.57329e14 1.50415
\(355\) 2.20963e14 2.08000
\(356\) −1.27703e14 −1.18366
\(357\) 4.61915e13 0.421587
\(358\) 9.78461e13 0.879401
\(359\) −6.11442e13 −0.541173 −0.270586 0.962696i \(-0.587218\pi\)
−0.270586 + 0.962696i \(0.587218\pi\)
\(360\) −1.30188e14 −1.13477
\(361\) 2.59109e13 0.222430
\(362\) 1.80479e14 1.52592
\(363\) −7.24490e13 −0.603317
\(364\) −3.27758e13 −0.268841
\(365\) 8.07285e13 0.652253
\(366\) −8.97926e13 −0.714652
\(367\) 1.88415e14 1.47724 0.738622 0.674119i \(-0.235477\pi\)
0.738622 + 0.674119i \(0.235477\pi\)
\(368\) −2.92441e14 −2.25879
\(369\) 4.82576e13 0.367215
\(370\) −7.78540e13 −0.583676
\(371\) 6.45191e12 0.0476576
\(372\) −3.59968e14 −2.61985
\(373\) −2.59905e14 −1.86388 −0.931938 0.362619i \(-0.881883\pi\)
−0.931938 + 0.362619i \(0.881883\pi\)
\(374\) −7.38140e14 −5.21609
\(375\) 6.41942e13 0.447017
\(376\) 2.49919e14 1.71501
\(377\) −2.29446e13 −0.155168
\(378\) 2.06053e13 0.137332
\(379\) −4.36538e13 −0.286752 −0.143376 0.989668i \(-0.545796\pi\)
−0.143376 + 0.989668i \(0.545796\pi\)
\(380\) 5.04737e14 3.26780
\(381\) −6.25602e13 −0.399219
\(382\) 2.67177e14 1.68055
\(383\) −5.00699e13 −0.310444 −0.155222 0.987880i \(-0.549609\pi\)
−0.155222 + 0.987880i \(0.549609\pi\)
\(384\) 1.24981e14 0.763877
\(385\) −1.03385e14 −0.622906
\(386\) 3.80563e14 2.26046
\(387\) −6.10247e13 −0.357352
\(388\) 6.17815e14 3.56684
\(389\) −9.34160e12 −0.0531739 −0.0265870 0.999647i \(-0.508464\pi\)
−0.0265870 + 0.999647i \(0.508464\pi\)
\(390\) −6.20804e13 −0.348417
\(391\) 2.61770e14 1.44860
\(392\) −7.73350e13 −0.421991
\(393\) 1.25188e14 0.673602
\(394\) −1.43588e14 −0.761883
\(395\) −9.67859e13 −0.506440
\(396\) −2.36898e14 −1.22247
\(397\) 1.73516e14 0.883065 0.441533 0.897245i \(-0.354435\pi\)
0.441533 + 0.897245i \(0.354435\pi\)
\(398\) 3.70610e14 1.86020
\(399\) −4.87364e13 −0.241269
\(400\) 2.02468e14 0.988615
\(401\) 3.80819e14 1.83411 0.917054 0.398763i \(-0.130560\pi\)
0.917054 + 0.398763i \(0.130560\pi\)
\(402\) 1.04087e14 0.494487
\(403\) −1.04720e14 −0.490739
\(404\) 6.64768e14 3.07307
\(405\) 2.80794e13 0.128052
\(406\) −8.87407e13 −0.399236
\(407\) −8.64278e13 −0.383605
\(408\) −7.52433e14 −3.29486
\(409\) −4.09780e14 −1.77041 −0.885203 0.465205i \(-0.845981\pi\)
−0.885203 + 0.465205i \(0.845981\pi\)
\(410\) −5.62321e14 −2.39703
\(411\) 1.87861e14 0.790147
\(412\) −4.82603e14 −2.00288
\(413\) 1.27358e14 0.521554
\(414\) 1.16771e14 0.471883
\(415\) −4.67594e14 −1.86468
\(416\) 1.92657e14 0.758178
\(417\) −3.44373e13 −0.133746
\(418\) 7.78807e14 2.98511
\(419\) −1.19915e14 −0.453625 −0.226813 0.973938i \(-0.572830\pi\)
−0.226813 + 0.973938i \(0.572830\pi\)
\(420\) −1.72745e14 −0.644961
\(421\) −3.01588e13 −0.111138 −0.0555690 0.998455i \(-0.517697\pi\)
−0.0555690 + 0.998455i \(0.517697\pi\)
\(422\) −3.08876e14 −1.12348
\(423\) −5.39035e13 −0.193528
\(424\) −1.05098e14 −0.372462
\(425\) −1.81234e14 −0.634016
\(426\) 5.69682e14 1.96734
\(427\) −7.26868e13 −0.247800
\(428\) 3.64875e14 1.22801
\(429\) −6.89171e13 −0.228988
\(430\) 7.11090e14 2.33264
\(431\) 9.13897e13 0.295987 0.147993 0.988988i \(-0.452719\pi\)
0.147993 + 0.988988i \(0.452719\pi\)
\(432\) −1.81302e14 −0.579752
\(433\) 6.01116e14 1.89791 0.948953 0.315418i \(-0.102145\pi\)
0.948953 + 0.315418i \(0.102145\pi\)
\(434\) −4.05015e14 −1.26263
\(435\) −1.20930e14 −0.372256
\(436\) 1.82657e14 0.555214
\(437\) −2.76192e14 −0.829017
\(438\) 2.08132e14 0.616925
\(439\) 2.75739e14 0.807131 0.403565 0.914951i \(-0.367771\pi\)
0.403565 + 0.914951i \(0.367771\pi\)
\(440\) 1.68408e15 4.86825
\(441\) 1.66799e13 0.0476190
\(442\) −3.58799e14 −1.01165
\(443\) 4.86155e13 0.135380 0.0676899 0.997706i \(-0.478437\pi\)
0.0676899 + 0.997706i \(0.478437\pi\)
\(444\) −1.44411e14 −0.397187
\(445\) −1.95802e14 −0.531909
\(446\) −7.73534e14 −2.07557
\(447\) 2.62273e14 0.695123
\(448\) 3.10207e14 0.812124
\(449\) −2.40909e14 −0.623015 −0.311507 0.950244i \(-0.600834\pi\)
−0.311507 + 0.950244i \(0.600834\pi\)
\(450\) −8.08453e13 −0.206531
\(451\) −6.24248e14 −1.57538
\(452\) 1.13835e15 2.83801
\(453\) 1.84972e14 0.455581
\(454\) −6.03155e14 −1.46765
\(455\) −5.02539e13 −0.120811
\(456\) 7.93888e14 1.88561
\(457\) 5.45002e14 1.27897 0.639483 0.768805i \(-0.279149\pi\)
0.639483 + 0.768805i \(0.279149\pi\)
\(458\) 7.25831e14 1.68297
\(459\) 1.62287e14 0.371805
\(460\) −9.78954e14 −2.21613
\(461\) 7.41577e14 1.65883 0.829414 0.558635i \(-0.188675\pi\)
0.829414 + 0.558635i \(0.188675\pi\)
\(462\) −2.66544e14 −0.589167
\(463\) 2.31282e14 0.505181 0.252590 0.967573i \(-0.418717\pi\)
0.252590 + 0.967573i \(0.418717\pi\)
\(464\) 7.80814e14 1.68538
\(465\) −5.51925e14 −1.17730
\(466\) 1.14810e15 2.42024
\(467\) −2.84119e14 −0.591913 −0.295956 0.955201i \(-0.595638\pi\)
−0.295956 + 0.955201i \(0.595638\pi\)
\(468\) −1.15153e14 −0.237095
\(469\) 8.42584e13 0.171460
\(470\) 6.28110e14 1.26327
\(471\) −4.89958e14 −0.973967
\(472\) −2.07458e15 −4.07614
\(473\) 7.89400e14 1.53307
\(474\) −2.49531e14 −0.479010
\(475\) 1.91219e14 0.362840
\(476\) −9.98393e14 −1.87268
\(477\) 2.26679e13 0.0420300
\(478\) −1.05143e15 −1.92721
\(479\) −1.73156e14 −0.313757 −0.156878 0.987618i \(-0.550143\pi\)
−0.156878 + 0.987618i \(0.550143\pi\)
\(480\) 1.01540e15 1.81890
\(481\) −4.20113e13 −0.0743993
\(482\) 5.65377e14 0.989872
\(483\) 9.45259e13 0.163622
\(484\) 1.56593e15 2.67992
\(485\) 9.47272e14 1.60286
\(486\) 7.23936e13 0.121116
\(487\) −9.71549e14 −1.60715 −0.803574 0.595205i \(-0.797071\pi\)
−0.803574 + 0.595205i \(0.797071\pi\)
\(488\) 1.18403e15 1.93665
\(489\) −1.79701e14 −0.290637
\(490\) −1.94362e14 −0.310837
\(491\) 4.23365e14 0.669525 0.334763 0.942303i \(-0.391344\pi\)
0.334763 + 0.942303i \(0.391344\pi\)
\(492\) −1.04305e15 −1.63116
\(493\) −6.98923e14 −1.08086
\(494\) 3.78567e14 0.578954
\(495\) −3.63228e14 −0.549351
\(496\) 3.56365e15 5.33023
\(497\) 4.61156e14 0.682162
\(498\) −1.20554e15 −1.76368
\(499\) −6.23180e14 −0.901697 −0.450849 0.892600i \(-0.648879\pi\)
−0.450849 + 0.892600i \(0.648879\pi\)
\(500\) −1.38751e15 −1.98564
\(501\) 3.66239e14 0.518391
\(502\) −1.78175e15 −2.49445
\(503\) −9.28331e14 −1.28552 −0.642760 0.766068i \(-0.722211\pi\)
−0.642760 + 0.766068i \(0.722211\pi\)
\(504\) −2.71706e14 −0.372161
\(505\) 1.01926e15 1.38097
\(506\) −1.51052e15 −2.02441
\(507\) −3.34996e13 −0.0444116
\(508\) 1.35219e15 1.77332
\(509\) −4.82159e14 −0.625521 −0.312761 0.949832i \(-0.601254\pi\)
−0.312761 + 0.949832i \(0.601254\pi\)
\(510\) −1.89105e15 −2.42699
\(511\) 1.68482e14 0.213914
\(512\) 5.28314e14 0.663601
\(513\) −1.71228e14 −0.212780
\(514\) 6.60119e14 0.811568
\(515\) −7.39958e14 −0.900051
\(516\) 1.31900e15 1.58735
\(517\) 6.97281e14 0.830252
\(518\) −1.62483e14 −0.191424
\(519\) −5.86894e11 −0.000684131 0
\(520\) 8.18606e14 0.944184
\(521\) 8.45383e14 0.964820 0.482410 0.875946i \(-0.339762\pi\)
0.482410 + 0.875946i \(0.339762\pi\)
\(522\) −3.11778e14 −0.352093
\(523\) 1.01965e15 1.13944 0.569721 0.821838i \(-0.307051\pi\)
0.569721 + 0.821838i \(0.307051\pi\)
\(524\) −2.70584e15 −2.99213
\(525\) −6.54440e13 −0.0716133
\(526\) 1.23139e15 1.33344
\(527\) −3.18990e15 −3.41837
\(528\) 2.34528e15 2.48718
\(529\) −4.17126e14 −0.437785
\(530\) −2.64137e14 −0.274355
\(531\) 4.47452e14 0.459967
\(532\) 1.05340e15 1.07171
\(533\) −3.03438e14 −0.305541
\(534\) −5.04812e14 −0.503099
\(535\) 5.59450e14 0.551843
\(536\) −1.37252e15 −1.34002
\(537\) 2.78279e14 0.268920
\(538\) 3.59879e15 3.44234
\(539\) −2.15767e14 −0.204290
\(540\) −6.06914e14 −0.568802
\(541\) −3.73294e14 −0.346311 −0.173155 0.984895i \(-0.555396\pi\)
−0.173155 + 0.984895i \(0.555396\pi\)
\(542\) 3.21065e15 2.94848
\(543\) 5.13292e14 0.466623
\(544\) 5.86859e15 5.28129
\(545\) 2.80061e14 0.249501
\(546\) −1.29563e14 −0.114267
\(547\) −1.08627e15 −0.948436 −0.474218 0.880407i \(-0.657269\pi\)
−0.474218 + 0.880407i \(0.657269\pi\)
\(548\) −4.06048e15 −3.50981
\(549\) −2.55375e14 −0.218539
\(550\) 1.04579e15 0.886036
\(551\) 7.37429e14 0.618567
\(552\) −1.53977e15 −1.27877
\(553\) −2.01995e14 −0.166093
\(554\) −1.08214e15 −0.881007
\(555\) −2.21421e14 −0.178487
\(556\) 7.44335e14 0.594096
\(557\) −1.81624e15 −1.43539 −0.717696 0.696356i \(-0.754803\pi\)
−0.717696 + 0.696356i \(0.754803\pi\)
\(558\) −1.42296e15 −1.11354
\(559\) 3.83716e14 0.297334
\(560\) 1.71016e15 1.31221
\(561\) −2.09931e15 −1.59507
\(562\) −3.55320e15 −2.67344
\(563\) −2.36106e15 −1.75918 −0.879592 0.475728i \(-0.842184\pi\)
−0.879592 + 0.475728i \(0.842184\pi\)
\(564\) 1.16508e15 0.859649
\(565\) 1.74539e15 1.27534
\(566\) −3.75926e15 −2.72027
\(567\) 5.86024e13 0.0419961
\(568\) −7.51196e15 −5.33136
\(569\) 2.14496e15 1.50766 0.753829 0.657071i \(-0.228205\pi\)
0.753829 + 0.657071i \(0.228205\pi\)
\(570\) 1.99524e15 1.38894
\(571\) −1.64159e14 −0.113179 −0.0565897 0.998398i \(-0.518023\pi\)
−0.0565897 + 0.998398i \(0.518023\pi\)
\(572\) 1.48959e15 1.01716
\(573\) 7.59865e14 0.513909
\(574\) −1.17358e15 −0.786134
\(575\) −3.70875e14 −0.246068
\(576\) 1.08987e15 0.716226
\(577\) −2.00527e15 −1.30529 −0.652645 0.757664i \(-0.726340\pi\)
−0.652645 + 0.757664i \(0.726340\pi\)
\(578\) −8.00126e15 −5.15888
\(579\) 1.08234e15 0.691245
\(580\) 2.61380e15 1.65355
\(581\) −9.75882e14 −0.611545
\(582\) 2.44224e15 1.51604
\(583\) −2.93226e14 −0.180312
\(584\) −2.74448e15 −1.67182
\(585\) −1.76560e14 −0.106545
\(586\) 2.37460e15 1.41956
\(587\) −2.73941e15 −1.62236 −0.811179 0.584798i \(-0.801174\pi\)
−0.811179 + 0.584798i \(0.801174\pi\)
\(588\) −3.60523e14 −0.211523
\(589\) 3.36564e15 1.95629
\(590\) −5.21394e15 −3.00248
\(591\) −4.08371e14 −0.232983
\(592\) 1.42966e15 0.808099
\(593\) −2.62471e15 −1.46988 −0.734939 0.678133i \(-0.762789\pi\)
−0.734939 + 0.678133i \(0.762789\pi\)
\(594\) −9.36465e14 −0.519597
\(595\) −1.53080e15 −0.841541
\(596\) −5.66882e15 −3.08772
\(597\) 1.05403e15 0.568847
\(598\) −7.34243e14 −0.392630
\(599\) −7.10912e14 −0.376676 −0.188338 0.982104i \(-0.560310\pi\)
−0.188338 + 0.982104i \(0.560310\pi\)
\(600\) 1.06604e15 0.559685
\(601\) −2.15854e15 −1.12292 −0.561461 0.827503i \(-0.689761\pi\)
−0.561461 + 0.827503i \(0.689761\pi\)
\(602\) 1.48406e15 0.765018
\(603\) 2.96030e14 0.151213
\(604\) −3.99803e15 −2.02368
\(605\) 2.40098e15 1.20430
\(606\) 2.62785e15 1.30617
\(607\) −2.00799e14 −0.0989061 −0.0494531 0.998776i \(-0.515748\pi\)
−0.0494531 + 0.998776i \(0.515748\pi\)
\(608\) −6.19191e15 −3.02242
\(609\) −2.52383e14 −0.122086
\(610\) 2.97575e15 1.42653
\(611\) 3.38939e14 0.161025
\(612\) −3.50771e15 −1.65155
\(613\) −2.09302e15 −0.976654 −0.488327 0.872661i \(-0.662393\pi\)
−0.488327 + 0.872661i \(0.662393\pi\)
\(614\) 7.59029e15 3.51020
\(615\) −1.59927e15 −0.733008
\(616\) 3.51471e15 1.59660
\(617\) 1.25578e15 0.565384 0.282692 0.959211i \(-0.408773\pi\)
0.282692 + 0.959211i \(0.408773\pi\)
\(618\) −1.90774e15 −0.851301
\(619\) −1.82948e15 −0.809149 −0.404575 0.914505i \(-0.632580\pi\)
−0.404575 + 0.914505i \(0.632580\pi\)
\(620\) 1.19294e16 5.22956
\(621\) 3.32103e14 0.144301
\(622\) 7.15505e15 3.08152
\(623\) −4.08644e14 −0.174446
\(624\) 1.14001e15 0.482383
\(625\) −2.90984e15 −1.22048
\(626\) 2.37868e15 0.988956
\(627\) 2.21497e15 0.912842
\(628\) 1.05901e16 4.32634
\(629\) −1.27972e15 −0.518247
\(630\) −6.82864e14 −0.274133
\(631\) −3.62896e15 −1.44418 −0.722090 0.691800i \(-0.756818\pi\)
−0.722090 + 0.691800i \(0.756818\pi\)
\(632\) 3.29038e15 1.29808
\(633\) −8.78458e14 −0.343558
\(634\) −1.74920e14 −0.0678186
\(635\) 2.07326e15 0.796891
\(636\) −4.89948e14 −0.186697
\(637\) −1.04881e14 −0.0396214
\(638\) 4.03308e15 1.51051
\(639\) 1.62021e15 0.601610
\(640\) −4.14191e15 −1.52479
\(641\) −2.58370e15 −0.943023 −0.471512 0.881860i \(-0.656291\pi\)
−0.471512 + 0.881860i \(0.656291\pi\)
\(642\) 1.44236e15 0.521953
\(643\) −2.12651e15 −0.762969 −0.381485 0.924375i \(-0.624587\pi\)
−0.381485 + 0.924375i \(0.624587\pi\)
\(644\) −2.04310e15 −0.726805
\(645\) 2.02238e15 0.713319
\(646\) 1.15317e16 4.03286
\(647\) −3.79093e15 −1.31454 −0.657268 0.753657i \(-0.728288\pi\)
−0.657268 + 0.753657i \(0.728288\pi\)
\(648\) −9.54599e14 −0.328215
\(649\) −5.78813e15 −1.97330
\(650\) 5.08345e14 0.171844
\(651\) −1.15188e15 −0.386111
\(652\) 3.88409e15 1.29100
\(653\) −2.56060e15 −0.843955 −0.421977 0.906606i \(-0.638664\pi\)
−0.421977 + 0.906606i \(0.638664\pi\)
\(654\) 7.22047e14 0.235987
\(655\) −4.14876e15 −1.34459
\(656\) 1.03261e16 3.31868
\(657\) 5.91939e14 0.188655
\(658\) 1.31088e15 0.414306
\(659\) −6.67060e14 −0.209072 −0.104536 0.994521i \(-0.533336\pi\)
−0.104536 + 0.994521i \(0.533336\pi\)
\(660\) 7.85088e15 2.44021
\(661\) 3.63231e15 1.11963 0.559816 0.828617i \(-0.310872\pi\)
0.559816 + 0.828617i \(0.310872\pi\)
\(662\) −4.71067e15 −1.44000
\(663\) −1.02044e15 −0.309360
\(664\) 1.58965e16 4.77946
\(665\) 1.61514e15 0.481604
\(666\) −5.70862e14 −0.168820
\(667\) −1.43027e15 −0.419494
\(668\) −7.91597e15 −2.30268
\(669\) −2.19997e15 −0.634706
\(670\) −3.44948e15 −0.987058
\(671\) 3.30346e15 0.937551
\(672\) 2.11916e15 0.596532
\(673\) 1.60730e15 0.448760 0.224380 0.974502i \(-0.427964\pi\)
0.224380 + 0.974502i \(0.427964\pi\)
\(674\) 8.67474e15 2.40230
\(675\) −2.29928e14 −0.0631570
\(676\) 7.24068e14 0.197275
\(677\) −5.15043e14 −0.139189 −0.0695947 0.997575i \(-0.522171\pi\)
−0.0695947 + 0.997575i \(0.522171\pi\)
\(678\) 4.49991e15 1.20626
\(679\) 1.97698e15 0.525677
\(680\) 2.49358e16 6.57696
\(681\) −1.71540e15 −0.448804
\(682\) 1.84071e16 4.77716
\(683\) 6.98363e15 1.79791 0.898953 0.438045i \(-0.144329\pi\)
0.898953 + 0.438045i \(0.144329\pi\)
\(684\) 3.70097e15 0.945163
\(685\) −6.22578e15 −1.57723
\(686\) −4.05639e14 −0.101943
\(687\) 2.06430e15 0.514648
\(688\) −1.30580e16 −3.22954
\(689\) −1.42533e14 −0.0349711
\(690\) −3.86983e15 −0.941937
\(691\) −6.84269e15 −1.65233 −0.826167 0.563426i \(-0.809483\pi\)
−0.826167 + 0.563426i \(0.809483\pi\)
\(692\) 1.26853e13 0.00303890
\(693\) −7.58065e14 −0.180166
\(694\) −8.30541e15 −1.95832
\(695\) 1.14126e15 0.266974
\(696\) 4.11117e15 0.954146
\(697\) −9.24312e15 −2.12833
\(698\) −4.70324e15 −1.07446
\(699\) 3.26526e15 0.740105
\(700\) 1.41452e15 0.318105
\(701\) −3.02022e15 −0.673892 −0.336946 0.941524i \(-0.609394\pi\)
−0.336946 + 0.941524i \(0.609394\pi\)
\(702\) −4.55202e14 −0.100774
\(703\) 1.35023e15 0.296587
\(704\) −1.40983e16 −3.07267
\(705\) 1.78638e15 0.386307
\(706\) −8.09337e15 −1.73661
\(707\) 2.12723e15 0.452906
\(708\) −9.67133e15 −2.04317
\(709\) 4.15720e15 0.871457 0.435729 0.900078i \(-0.356491\pi\)
0.435729 + 0.900078i \(0.356491\pi\)
\(710\) −1.88794e16 −3.92706
\(711\) −7.09680e14 −0.146480
\(712\) 6.65657e15 1.36336
\(713\) −6.52778e15 −1.32670
\(714\) −3.94667e15 −0.795960
\(715\) 2.28393e15 0.457088
\(716\) −6.01479e15 −1.19453
\(717\) −2.99033e15 −0.589337
\(718\) 5.22426e15 1.02174
\(719\) 2.21409e15 0.429722 0.214861 0.976645i \(-0.431070\pi\)
0.214861 + 0.976645i \(0.431070\pi\)
\(720\) 6.00840e15 1.15726
\(721\) −1.54431e15 −0.295183
\(722\) −2.21387e15 −0.419950
\(723\) 1.60796e15 0.302701
\(724\) −1.10944e16 −2.07273
\(725\) 9.90232e14 0.183602
\(726\) 6.19015e15 1.13907
\(727\) −5.01733e15 −0.916290 −0.458145 0.888877i \(-0.651486\pi\)
−0.458145 + 0.888877i \(0.651486\pi\)
\(728\) 1.70845e15 0.309657
\(729\) 2.05891e14 0.0370370
\(730\) −6.89756e15 −1.23146
\(731\) 1.16885e16 2.07116
\(732\) 5.51972e15 0.970747
\(733\) −3.99723e14 −0.0697730 −0.0348865 0.999391i \(-0.511107\pi\)
−0.0348865 + 0.999391i \(0.511107\pi\)
\(734\) −1.60985e16 −2.78905
\(735\) −5.52776e14 −0.0950536
\(736\) 1.20094e16 2.04972
\(737\) −3.82936e15 −0.648717
\(738\) −4.12320e15 −0.693305
\(739\) −5.53787e15 −0.924269 −0.462135 0.886810i \(-0.652916\pi\)
−0.462135 + 0.886810i \(0.652916\pi\)
\(740\) 4.78584e15 0.792836
\(741\) 1.07666e15 0.177043
\(742\) −5.51261e14 −0.0899780
\(743\) 3.67629e15 0.595622 0.297811 0.954625i \(-0.403743\pi\)
0.297811 + 0.954625i \(0.403743\pi\)
\(744\) 1.87635e16 3.01761
\(745\) −8.69179e15 −1.38755
\(746\) 2.22067e16 3.51901
\(747\) −3.42862e15 −0.539332
\(748\) 4.53749e16 7.08527
\(749\) 1.16759e15 0.180983
\(750\) −5.48485e15 −0.843972
\(751\) 7.63847e14 0.116677 0.0583387 0.998297i \(-0.481420\pi\)
0.0583387 + 0.998297i \(0.481420\pi\)
\(752\) −1.15342e16 −1.74900
\(753\) −5.06738e15 −0.762800
\(754\) 1.96042e15 0.292959
\(755\) −6.13003e15 −0.909398
\(756\) −1.26664e15 −0.186546
\(757\) 8.23903e15 1.20462 0.602308 0.798264i \(-0.294248\pi\)
0.602308 + 0.798264i \(0.294248\pi\)
\(758\) 3.72985e15 0.541390
\(759\) −4.29600e15 −0.619063
\(760\) −2.63097e16 −3.76392
\(761\) 2.54839e15 0.361951 0.180976 0.983488i \(-0.442075\pi\)
0.180976 + 0.983488i \(0.442075\pi\)
\(762\) 5.34524e15 0.753728
\(763\) 5.84494e14 0.0818267
\(764\) −1.64239e16 −2.28277
\(765\) −5.37825e15 −0.742169
\(766\) 4.27805e15 0.586121
\(767\) −2.81353e15 −0.382716
\(768\) −1.49320e15 −0.201665
\(769\) 6.77648e15 0.908676 0.454338 0.890829i \(-0.349876\pi\)
0.454338 + 0.890829i \(0.349876\pi\)
\(770\) 8.83335e15 1.17605
\(771\) 1.87741e15 0.248176
\(772\) −2.33939e16 −3.07049
\(773\) 1.08648e16 1.41591 0.707955 0.706257i \(-0.249618\pi\)
0.707955 + 0.706257i \(0.249618\pi\)
\(774\) 5.21405e15 0.674683
\(775\) 4.51944e15 0.580664
\(776\) −3.22039e16 −4.10836
\(777\) −4.62111e14 −0.0585370
\(778\) 7.98161e14 0.100393
\(779\) 9.75236e15 1.21802
\(780\) 3.81620e15 0.473272
\(781\) −2.09586e16 −2.58096
\(782\) −2.23660e16 −2.73497
\(783\) −8.86712e14 −0.107670
\(784\) 3.56915e15 0.430354
\(785\) 1.62374e16 1.94416
\(786\) −1.06962e16 −1.27177
\(787\) 6.41445e15 0.757354 0.378677 0.925529i \(-0.376379\pi\)
0.378677 + 0.925529i \(0.376379\pi\)
\(788\) 8.82662e15 1.03490
\(789\) 3.50214e15 0.407765
\(790\) 8.26954e15 0.956164
\(791\) 3.64266e15 0.418262
\(792\) 1.23484e16 1.40807
\(793\) 1.60576e15 0.181836
\(794\) −1.48255e16 −1.66723
\(795\) −7.51220e14 −0.0838973
\(796\) −2.27821e16 −2.52681
\(797\) −6.94319e15 −0.764783 −0.382392 0.924000i \(-0.624899\pi\)
−0.382392 + 0.924000i \(0.624899\pi\)
\(798\) 4.16411e15 0.455519
\(799\) 1.03245e16 1.12166
\(800\) −8.31460e15 −0.897111
\(801\) −1.43571e15 −0.153847
\(802\) −3.25378e16 −3.46281
\(803\) −7.65717e15 −0.809343
\(804\) −6.39845e15 −0.671686
\(805\) −3.13261e15 −0.326610
\(806\) 8.94741e15 0.926519
\(807\) 1.02351e16 1.05266
\(808\) −3.46514e16 −3.53963
\(809\) 1.64346e16 1.66741 0.833707 0.552207i \(-0.186214\pi\)
0.833707 + 0.552207i \(0.186214\pi\)
\(810\) −2.39914e15 −0.241762
\(811\) −1.08054e16 −1.08150 −0.540749 0.841184i \(-0.681859\pi\)
−0.540749 + 0.841184i \(0.681859\pi\)
\(812\) 5.45506e15 0.542302
\(813\) 9.13126e15 0.901639
\(814\) 7.38453e15 0.724250
\(815\) 5.95532e15 0.580148
\(816\) 3.47261e16 3.36016
\(817\) −1.23325e16 −1.18530
\(818\) 3.50122e16 3.34254
\(819\) −3.68485e14 −0.0349428
\(820\) 3.45669e16 3.25600
\(821\) −1.43731e16 −1.34482 −0.672410 0.740179i \(-0.734741\pi\)
−0.672410 + 0.740179i \(0.734741\pi\)
\(822\) −1.60512e16 −1.49180
\(823\) −5.47150e14 −0.0505135 −0.0252568 0.999681i \(-0.508040\pi\)
−0.0252568 + 0.999681i \(0.508040\pi\)
\(824\) 2.51559e16 2.30696
\(825\) 2.97429e15 0.270948
\(826\) −1.08816e16 −0.984699
\(827\) −1.17334e16 −1.05473 −0.527367 0.849637i \(-0.676821\pi\)
−0.527367 + 0.849637i \(0.676821\pi\)
\(828\) −7.17815e15 −0.640982
\(829\) −1.76415e16 −1.56490 −0.782450 0.622714i \(-0.786030\pi\)
−0.782450 + 0.622714i \(0.786030\pi\)
\(830\) 3.99520e16 3.52054
\(831\) −3.07766e15 −0.269411
\(832\) −6.85296e15 −0.595936
\(833\) −3.19482e15 −0.275993
\(834\) 2.94237e15 0.252513
\(835\) −1.21373e16 −1.03477
\(836\) −4.78747e16 −4.05482
\(837\) −4.04697e15 −0.340518
\(838\) 1.02457e16 0.856448
\(839\) −1.43593e16 −1.19246 −0.596229 0.802814i \(-0.703335\pi\)
−0.596229 + 0.802814i \(0.703335\pi\)
\(840\) 9.00440e15 0.742880
\(841\) −8.38171e15 −0.686996
\(842\) 2.57681e15 0.209829
\(843\) −1.01055e16 −0.817533
\(844\) 1.89872e16 1.52608
\(845\) 1.11019e15 0.0886511
\(846\) 4.60559e15 0.365383
\(847\) 5.01090e15 0.394964
\(848\) 4.85045e15 0.379844
\(849\) −1.06915e16 −0.831855
\(850\) 1.54849e16 1.19703
\(851\) −2.61881e15 −0.201137
\(852\) −3.50195e16 −2.67234
\(853\) 4.70434e15 0.356680 0.178340 0.983969i \(-0.442927\pi\)
0.178340 + 0.983969i \(0.442927\pi\)
\(854\) 6.21047e15 0.467849
\(855\) 5.67455e15 0.424735
\(856\) −1.90193e16 −1.41445
\(857\) 2.41359e16 1.78348 0.891740 0.452547i \(-0.149485\pi\)
0.891740 + 0.452547i \(0.149485\pi\)
\(858\) 5.88838e15 0.432330
\(859\) −2.29146e16 −1.67167 −0.835835 0.548980i \(-0.815016\pi\)
−0.835835 + 0.548980i \(0.815016\pi\)
\(860\) −4.37121e16 −3.16855
\(861\) −3.33772e15 −0.240399
\(862\) −7.80848e15 −0.558826
\(863\) −5.87430e15 −0.417731 −0.208866 0.977944i \(-0.566977\pi\)
−0.208866 + 0.977944i \(0.566977\pi\)
\(864\) 7.44538e15 0.526091
\(865\) 1.94498e13 0.00136561
\(866\) −5.13602e16 −3.58326
\(867\) −2.27560e16 −1.57758
\(868\) 2.48970e16 1.71510
\(869\) 9.18023e15 0.628413
\(870\) 1.03324e16 0.702822
\(871\) −1.86140e15 −0.125817
\(872\) −9.52107e15 −0.639507
\(873\) 6.94584e15 0.463604
\(874\) 2.35983e16 1.56519
\(875\) −4.43997e15 −0.292641
\(876\) −1.27943e16 −0.838000
\(877\) 9.65356e15 0.628333 0.314166 0.949368i \(-0.398275\pi\)
0.314166 + 0.949368i \(0.398275\pi\)
\(878\) −2.35596e16 −1.52387
\(879\) 6.75347e15 0.434098
\(880\) −7.77231e16 −4.96473
\(881\) −4.62485e15 −0.293583 −0.146791 0.989167i \(-0.546895\pi\)
−0.146791 + 0.989167i \(0.546895\pi\)
\(882\) −1.42515e15 −0.0899052
\(883\) −8.06211e15 −0.505434 −0.252717 0.967540i \(-0.581324\pi\)
−0.252717 + 0.967540i \(0.581324\pi\)
\(884\) 2.20561e16 1.37417
\(885\) −1.48287e16 −0.918153
\(886\) −4.15378e15 −0.255598
\(887\) −1.13706e16 −0.695352 −0.347676 0.937615i \(-0.613029\pi\)
−0.347676 + 0.937615i \(0.613029\pi\)
\(888\) 7.52752e15 0.457489
\(889\) 4.32695e15 0.261350
\(890\) 1.67296e16 1.00425
\(891\) −2.66335e15 −0.158892
\(892\) 4.75506e16 2.81935
\(893\) −1.08933e16 −0.641916
\(894\) −2.24090e16 −1.31240
\(895\) −9.22225e15 −0.536797
\(896\) −8.64428e15 −0.500075
\(897\) −2.08823e15 −0.120066
\(898\) 2.05836e16 1.17626
\(899\) 1.74291e16 0.989912
\(900\) 4.96972e15 0.280542
\(901\) −4.34174e15 −0.243600
\(902\) 5.33367e16 2.97433
\(903\) 4.22075e15 0.233942
\(904\) −5.93369e16 −3.26888
\(905\) −1.70107e16 −0.931438
\(906\) −1.58043e16 −0.860142
\(907\) 1.85203e16 1.00186 0.500931 0.865487i \(-0.332991\pi\)
0.500931 + 0.865487i \(0.332991\pi\)
\(908\) 3.70771e16 1.99358
\(909\) 7.47372e15 0.399425
\(910\) 4.29377e15 0.228092
\(911\) −3.26336e16 −1.72311 −0.861557 0.507660i \(-0.830511\pi\)
−0.861557 + 0.507660i \(0.830511\pi\)
\(912\) −3.66393e16 −1.92298
\(913\) 4.43518e16 2.31378
\(914\) −4.65658e16 −2.41470
\(915\) 8.46318e15 0.436232
\(916\) −4.46182e16 −2.28606
\(917\) −8.65857e15 −0.440976
\(918\) −1.38661e16 −0.701971
\(919\) 1.85187e16 0.931915 0.465957 0.884807i \(-0.345710\pi\)
0.465957 + 0.884807i \(0.345710\pi\)
\(920\) 5.10285e16 2.55258
\(921\) 2.15872e16 1.07342
\(922\) −6.33615e16 −3.13188
\(923\) −1.01877e16 −0.500570
\(924\) 1.63850e16 0.800295
\(925\) 1.81311e15 0.0880326
\(926\) −1.97611e16 −0.953786
\(927\) −5.42571e15 −0.260327
\(928\) −3.20650e16 −1.52939
\(929\) 1.99057e16 0.943826 0.471913 0.881645i \(-0.343564\pi\)
0.471913 + 0.881645i \(0.343564\pi\)
\(930\) 4.71573e16 2.22276
\(931\) 3.37083e15 0.157948
\(932\) −7.05761e16 −3.28753
\(933\) 2.03493e16 0.942325
\(934\) 2.42756e16 1.11754
\(935\) 6.95716e16 3.18396
\(936\) 6.00240e15 0.273091
\(937\) −1.64080e15 −0.0742142 −0.0371071 0.999311i \(-0.511814\pi\)
−0.0371071 + 0.999311i \(0.511814\pi\)
\(938\) −7.19916e15 −0.323718
\(939\) 6.76508e15 0.302421
\(940\) −3.86111e16 −1.71597
\(941\) −3.81330e16 −1.68484 −0.842419 0.538823i \(-0.818869\pi\)
−0.842419 + 0.538823i \(0.818869\pi\)
\(942\) 4.18628e16 1.83886
\(943\) −1.89150e16 −0.826025
\(944\) 9.57455e16 4.15693
\(945\) −1.94210e15 −0.0838294
\(946\) −6.74476e16 −2.89444
\(947\) 6.18471e15 0.263873 0.131936 0.991258i \(-0.457881\pi\)
0.131936 + 0.991258i \(0.457881\pi\)
\(948\) 1.53392e16 0.650663
\(949\) −3.72204e15 −0.156970
\(950\) −1.63380e16 −0.685045
\(951\) −4.97482e14 −0.0207388
\(952\) 5.20418e16 2.15699
\(953\) 3.57430e16 1.47292 0.736461 0.676480i \(-0.236496\pi\)
0.736461 + 0.676480i \(0.236496\pi\)
\(954\) −1.93678e15 −0.0793531
\(955\) −2.51821e16 −1.02583
\(956\) 6.46336e16 2.61782
\(957\) 1.14703e16 0.461911
\(958\) 1.47947e16 0.592375
\(959\) −1.29934e16 −0.517272
\(960\) −3.61185e16 −1.42968
\(961\) 5.41383e16 2.13072
\(962\) 3.58951e15 0.140466
\(963\) 4.10215e15 0.159612
\(964\) −3.47548e16 −1.34459
\(965\) −3.58691e16 −1.37981
\(966\) −8.07644e15 −0.308920
\(967\) 1.56698e16 0.595960 0.297980 0.954572i \(-0.403687\pi\)
0.297980 + 0.954572i \(0.403687\pi\)
\(968\) −8.16248e16 −3.08679
\(969\) 3.27966e16 1.23324
\(970\) −8.09364e16 −3.02621
\(971\) −1.05307e16 −0.391519 −0.195759 0.980652i \(-0.562717\pi\)
−0.195759 + 0.980652i \(0.562717\pi\)
\(972\) −4.45018e15 −0.164518
\(973\) 2.38184e15 0.0875572
\(974\) 8.30107e16 3.03431
\(975\) 1.44576e15 0.0525498
\(976\) −5.46448e16 −1.97504
\(977\) 1.49941e15 0.0538891 0.0269446 0.999637i \(-0.491422\pi\)
0.0269446 + 0.999637i \(0.491422\pi\)
\(978\) 1.53539e16 0.548725
\(979\) 1.85720e16 0.660015
\(980\) 1.19478e16 0.422226
\(981\) 2.05354e15 0.0721644
\(982\) −3.61730e16 −1.26407
\(983\) 2.30235e15 0.0800068 0.0400034 0.999200i \(-0.487263\pi\)
0.0400034 + 0.999200i \(0.487263\pi\)
\(984\) 5.43695e16 1.87881
\(985\) 1.35335e16 0.465063
\(986\) 5.97171e16 2.04068
\(987\) 3.72821e15 0.126694
\(988\) −2.32712e16 −0.786423
\(989\) 2.39193e16 0.803837
\(990\) 3.10347e16 1.03718
\(991\) 2.32084e15 0.0771329 0.0385665 0.999256i \(-0.487721\pi\)
0.0385665 + 0.999256i \(0.487721\pi\)
\(992\) −1.46346e17 −4.83688
\(993\) −1.33974e16 −0.440351
\(994\) −3.94019e16 −1.28793
\(995\) −3.49310e16 −1.13549
\(996\) 7.41069e16 2.39570
\(997\) −5.73575e16 −1.84403 −0.922013 0.387158i \(-0.873457\pi\)
−0.922013 + 0.387158i \(0.873457\pi\)
\(998\) 5.32455e16 1.70241
\(999\) −1.62356e15 −0.0516248
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 273.12.a.c.1.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.2 16 1.1 even 1 trivial