Properties

Label 3.12.a.a.1.1
Level $3$
Weight $12$
Character 3.1
Self dual yes
Analytic conductor $2.305$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(2.30502954168\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q+78.0000 q^{2} -243.000 q^{3} +4036.00 q^{4} -5370.00 q^{5} -18954.0 q^{6} -27760.0 q^{7} +155064. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+78.0000 q^{2} -243.000 q^{3} +4036.00 q^{4} -5370.00 q^{5} -18954.0 q^{6} -27760.0 q^{7} +155064. q^{8} +59049.0 q^{9} -418860. q^{10} +637836. q^{11} -980748. q^{12} +766214. q^{13} -2.16528e6 q^{14} +1.30491e6 q^{15} +3.82926e6 q^{16} +3.08435e6 q^{17} +4.60582e6 q^{18} -1.95114e7 q^{19} -2.16733e7 q^{20} +6.74568e6 q^{21} +4.97512e7 q^{22} +1.53124e7 q^{23} -3.76806e7 q^{24} -1.99912e7 q^{25} +5.97647e7 q^{26} -1.43489e7 q^{27} -1.12039e8 q^{28} +1.07513e7 q^{29} +1.01783e8 q^{30} -5.09374e7 q^{31} -1.88885e7 q^{32} -1.54994e8 q^{33} +2.40580e8 q^{34} +1.49071e8 q^{35} +2.38322e8 q^{36} +6.64741e8 q^{37} -1.52189e9 q^{38} -1.86190e8 q^{39} -8.32694e8 q^{40} +8.98833e8 q^{41} +5.26163e8 q^{42} -9.57947e8 q^{43} +2.57431e9 q^{44} -3.17093e8 q^{45} +1.19436e9 q^{46} -1.55574e9 q^{47} -9.30511e8 q^{48} -1.20671e9 q^{49} -1.55932e9 q^{50} -7.49498e8 q^{51} +3.09244e9 q^{52} +3.79242e9 q^{53} -1.11921e9 q^{54} -3.42518e9 q^{55} -4.30458e9 q^{56} +4.74127e9 q^{57} +8.38598e8 q^{58} +5.55307e8 q^{59} +5.26662e9 q^{60} +4.95042e9 q^{61} -3.97312e9 q^{62} -1.63920e9 q^{63} -9.31563e9 q^{64} -4.11457e9 q^{65} -1.20895e10 q^{66} +5.29240e9 q^{67} +1.24485e10 q^{68} -3.72090e9 q^{69} +1.16276e10 q^{70} -1.48311e10 q^{71} +9.15637e9 q^{72} +1.39710e10 q^{73} +5.18498e10 q^{74} +4.85787e9 q^{75} -7.87480e10 q^{76} -1.77063e10 q^{77} -1.45228e10 q^{78} +3.72054e9 q^{79} -2.05631e10 q^{80} +3.48678e9 q^{81} +7.01090e10 q^{82} +8.76845e9 q^{83} +2.72256e10 q^{84} -1.65630e10 q^{85} -7.47199e10 q^{86} -2.61256e9 q^{87} +9.89054e10 q^{88} -2.54728e10 q^{89} -2.47333e10 q^{90} -2.12701e10 q^{91} +6.18007e10 q^{92} +1.23778e10 q^{93} -1.21348e11 q^{94} +1.04776e11 q^{95} +4.58990e9 q^{96} -3.90925e10 q^{97} -9.41233e10 q^{98} +3.76636e10 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 78.0000 1.72357 0.861786 0.507271i \(-0.169346\pi\)
0.861786 + 0.507271i \(0.169346\pi\)
\(3\) −243.000 −0.577350
\(4\) 4036.00 1.97070
\(5\) −5370.00 −0.768492 −0.384246 0.923231i \(-0.625539\pi\)
−0.384246 + 0.923231i \(0.625539\pi\)
\(6\) −18954.0 −0.995105
\(7\) −27760.0 −0.624281 −0.312141 0.950036i \(-0.601046\pi\)
−0.312141 + 0.950036i \(0.601046\pi\)
\(8\) 155064. 1.67308
\(9\) 59049.0 0.333333
\(10\) −418860. −1.32455
\(11\) 637836. 1.19412 0.597062 0.802195i \(-0.296335\pi\)
0.597062 + 0.802195i \(0.296335\pi\)
\(12\) −980748. −1.13779
\(13\) 766214. 0.572350 0.286175 0.958177i \(-0.407616\pi\)
0.286175 + 0.958177i \(0.407616\pi\)
\(14\) −2.16528e6 −1.07599
\(15\) 1.30491e6 0.443689
\(16\) 3.82926e6 0.912968
\(17\) 3.08435e6 0.526860 0.263430 0.964679i \(-0.415146\pi\)
0.263430 + 0.964679i \(0.415146\pi\)
\(18\) 4.60582e6 0.574524
\(19\) −1.95114e7 −1.80777 −0.903886 0.427773i \(-0.859298\pi\)
−0.903886 + 0.427773i \(0.859298\pi\)
\(20\) −2.16733e7 −1.51447
\(21\) 6.74568e6 0.360429
\(22\) 4.97512e7 2.05816
\(23\) 1.53124e7 0.496066 0.248033 0.968752i \(-0.420216\pi\)
0.248033 + 0.968752i \(0.420216\pi\)
\(24\) −3.76806e7 −0.965952
\(25\) −1.99912e7 −0.409420
\(26\) 5.97647e7 0.986487
\(27\) −1.43489e7 −0.192450
\(28\) −1.12039e8 −1.23027
\(29\) 1.07513e7 0.0973353 0.0486677 0.998815i \(-0.484502\pi\)
0.0486677 + 0.998815i \(0.484502\pi\)
\(30\) 1.01783e8 0.764730
\(31\) −5.09374e7 −0.319556 −0.159778 0.987153i \(-0.551078\pi\)
−0.159778 + 0.987153i \(0.551078\pi\)
\(32\) −1.88885e7 −0.0995112
\(33\) −1.54994e8 −0.689428
\(34\) 2.40580e8 0.908081
\(35\) 1.49071e8 0.479755
\(36\) 2.38322e8 0.656901
\(37\) 6.64741e8 1.57595 0.787976 0.615706i \(-0.211129\pi\)
0.787976 + 0.615706i \(0.211129\pi\)
\(38\) −1.52189e9 −3.11583
\(39\) −1.86190e8 −0.330446
\(40\) −8.32694e8 −1.28575
\(41\) 8.98833e8 1.21162 0.605812 0.795608i \(-0.292848\pi\)
0.605812 + 0.795608i \(0.292848\pi\)
\(42\) 5.26163e8 0.621225
\(43\) −9.57947e8 −0.993722 −0.496861 0.867830i \(-0.665514\pi\)
−0.496861 + 0.867830i \(0.665514\pi\)
\(44\) 2.57431e9 2.35326
\(45\) −3.17093e8 −0.256164
\(46\) 1.19436e9 0.855005
\(47\) −1.55574e9 −0.989462 −0.494731 0.869046i \(-0.664733\pi\)
−0.494731 + 0.869046i \(0.664733\pi\)
\(48\) −9.30511e8 −0.527102
\(49\) −1.20671e9 −0.610273
\(50\) −1.55932e9 −0.705666
\(51\) −7.49498e8 −0.304183
\(52\) 3.09244e9 1.12793
\(53\) 3.79242e9 1.24566 0.622829 0.782358i \(-0.285983\pi\)
0.622829 + 0.782358i \(0.285983\pi\)
\(54\) −1.11921e9 −0.331702
\(55\) −3.42518e9 −0.917674
\(56\) −4.30458e9 −1.04447
\(57\) 4.74127e9 1.04372
\(58\) 8.38598e8 0.167765
\(59\) 5.55307e8 0.101122 0.0505612 0.998721i \(-0.483899\pi\)
0.0505612 + 0.998721i \(0.483899\pi\)
\(60\) 5.26662e9 0.874379
\(61\) 4.95042e9 0.750461 0.375230 0.926932i \(-0.377564\pi\)
0.375230 + 0.926932i \(0.377564\pi\)
\(62\) −3.97312e9 −0.550779
\(63\) −1.63920e9 −0.208094
\(64\) −9.31563e9 −1.08448
\(65\) −4.11457e9 −0.439846
\(66\) −1.20895e10 −1.18828
\(67\) 5.29240e9 0.478896 0.239448 0.970909i \(-0.423034\pi\)
0.239448 + 0.970909i \(0.423034\pi\)
\(68\) 1.24485e10 1.03828
\(69\) −3.72090e9 −0.286404
\(70\) 1.16276e10 0.826893
\(71\) −1.48311e10 −0.975556 −0.487778 0.872968i \(-0.662192\pi\)
−0.487778 + 0.872968i \(0.662192\pi\)
\(72\) 9.15637e9 0.557692
\(73\) 1.39710e10 0.788773 0.394386 0.918945i \(-0.370957\pi\)
0.394386 + 0.918945i \(0.370957\pi\)
\(74\) 5.18498e10 2.71627
\(75\) 4.85787e9 0.236379
\(76\) −7.87480e10 −3.56258
\(77\) −1.77063e10 −0.745469
\(78\) −1.45228e10 −0.569548
\(79\) 3.72054e9 0.136037 0.0680185 0.997684i \(-0.478332\pi\)
0.0680185 + 0.997684i \(0.478332\pi\)
\(80\) −2.05631e10 −0.701608
\(81\) 3.48678e9 0.111111
\(82\) 7.01090e10 2.08832
\(83\) 8.76845e9 0.244339 0.122170 0.992509i \(-0.461015\pi\)
0.122170 + 0.992509i \(0.461015\pi\)
\(84\) 2.72256e10 0.710298
\(85\) −1.65630e10 −0.404888
\(86\) −7.47199e10 −1.71275
\(87\) −2.61256e9 −0.0561966
\(88\) 9.89054e10 1.99786
\(89\) −2.54728e10 −0.483539 −0.241769 0.970334i \(-0.577728\pi\)
−0.241769 + 0.970334i \(0.577728\pi\)
\(90\) −2.47333e10 −0.441517
\(91\) −2.12701e10 −0.357307
\(92\) 6.18007e10 0.977598
\(93\) 1.23778e10 0.184496
\(94\) −1.21348e11 −1.70541
\(95\) 1.04776e11 1.38926
\(96\) 4.58990e9 0.0574528
\(97\) −3.90925e10 −0.462220 −0.231110 0.972928i \(-0.574236\pi\)
−0.231110 + 0.972928i \(0.574236\pi\)
\(98\) −9.41233e10 −1.05185
\(99\) 3.76636e10 0.398041
\(100\) −8.06846e10 −0.806846
\(101\) 9.31078e9 0.0881492 0.0440746 0.999028i \(-0.485966\pi\)
0.0440746 + 0.999028i \(0.485966\pi\)
\(102\) −5.84608e10 −0.524281
\(103\) 4.85751e10 0.412865 0.206433 0.978461i \(-0.433815\pi\)
0.206433 + 0.978461i \(0.433815\pi\)
\(104\) 1.18812e11 0.957586
\(105\) −3.62243e10 −0.276987
\(106\) 2.95809e11 2.14698
\(107\) 2.25596e11 1.55496 0.777482 0.628906i \(-0.216497\pi\)
0.777482 + 0.628906i \(0.216497\pi\)
\(108\) −5.79122e10 −0.379262
\(109\) −6.94512e10 −0.432348 −0.216174 0.976355i \(-0.569358\pi\)
−0.216174 + 0.976355i \(0.569358\pi\)
\(110\) −2.67164e11 −1.58168
\(111\) −1.61532e11 −0.909876
\(112\) −1.06300e11 −0.569949
\(113\) −3.59665e11 −1.83640 −0.918198 0.396122i \(-0.870356\pi\)
−0.918198 + 0.396122i \(0.870356\pi\)
\(114\) 3.69819e11 1.79892
\(115\) −8.22274e10 −0.381222
\(116\) 4.33921e10 0.191819
\(117\) 4.52442e10 0.190783
\(118\) 4.33139e10 0.174292
\(119\) −8.56217e10 −0.328909
\(120\) 2.02345e11 0.742326
\(121\) 1.21523e11 0.425931
\(122\) 3.86133e11 1.29347
\(123\) −2.18417e11 −0.699532
\(124\) −2.05583e11 −0.629751
\(125\) 3.69560e11 1.08313
\(126\) −1.27858e11 −0.358665
\(127\) 2.50273e11 0.672192 0.336096 0.941828i \(-0.390893\pi\)
0.336096 + 0.941828i \(0.390893\pi\)
\(128\) −6.87936e11 −1.76967
\(129\) 2.32781e11 0.573726
\(130\) −3.20936e11 −0.758107
\(131\) −9.78918e10 −0.221694 −0.110847 0.993837i \(-0.535356\pi\)
−0.110847 + 0.993837i \(0.535356\pi\)
\(132\) −6.25556e11 −1.35866
\(133\) 5.41637e11 1.12856
\(134\) 4.12807e11 0.825412
\(135\) 7.70536e10 0.147896
\(136\) 4.78272e11 0.881477
\(137\) −1.55015e11 −0.274416 −0.137208 0.990542i \(-0.543813\pi\)
−0.137208 + 0.990542i \(0.543813\pi\)
\(138\) −2.90230e11 −0.493637
\(139\) −1.12627e12 −1.84102 −0.920512 0.390715i \(-0.872228\pi\)
−0.920512 + 0.390715i \(0.872228\pi\)
\(140\) 6.01651e11 0.945455
\(141\) 3.78045e11 0.571266
\(142\) −1.15682e12 −1.68144
\(143\) 4.88719e11 0.683457
\(144\) 2.26114e11 0.304323
\(145\) −5.77343e10 −0.0748014
\(146\) 1.08974e12 1.35951
\(147\) 2.93230e11 0.352341
\(148\) 2.68289e12 3.10573
\(149\) −1.38458e12 −1.54452 −0.772261 0.635306i \(-0.780874\pi\)
−0.772261 + 0.635306i \(0.780874\pi\)
\(150\) 3.78914e11 0.407416
\(151\) −6.98601e11 −0.724196 −0.362098 0.932140i \(-0.617939\pi\)
−0.362098 + 0.932140i \(0.617939\pi\)
\(152\) −3.02552e12 −3.02454
\(153\) 1.82128e11 0.175620
\(154\) −1.38109e12 −1.28487
\(155\) 2.73534e11 0.245576
\(156\) −7.51463e11 −0.651212
\(157\) 2.13127e12 1.78316 0.891580 0.452863i \(-0.149597\pi\)
0.891580 + 0.452863i \(0.149597\pi\)
\(158\) 2.90202e11 0.234470
\(159\) −9.21557e11 −0.719181
\(160\) 1.01431e11 0.0764736
\(161\) −4.25071e11 −0.309684
\(162\) 2.71969e11 0.191508
\(163\) −1.63564e11 −0.111342 −0.0556708 0.998449i \(-0.517730\pi\)
−0.0556708 + 0.998449i \(0.517730\pi\)
\(164\) 3.62769e12 2.38775
\(165\) 8.32319e11 0.529819
\(166\) 6.83939e11 0.421137
\(167\) −8.80943e9 −0.00524816 −0.00262408 0.999997i \(-0.500835\pi\)
−0.00262408 + 0.999997i \(0.500835\pi\)
\(168\) 1.04601e12 0.603025
\(169\) −1.20508e12 −0.672416
\(170\) −1.29191e12 −0.697853
\(171\) −1.15213e12 −0.602591
\(172\) −3.86627e12 −1.95833
\(173\) −7.30852e11 −0.358571 −0.179286 0.983797i \(-0.557379\pi\)
−0.179286 + 0.983797i \(0.557379\pi\)
\(174\) −2.03779e11 −0.0968589
\(175\) 5.54956e11 0.255593
\(176\) 2.44244e12 1.09020
\(177\) −1.34940e11 −0.0583830
\(178\) −1.98688e12 −0.833414
\(179\) 3.92371e12 1.59590 0.797950 0.602724i \(-0.205918\pi\)
0.797950 + 0.602724i \(0.205918\pi\)
\(180\) −1.27979e12 −0.504823
\(181\) 2.27931e12 0.872110 0.436055 0.899920i \(-0.356375\pi\)
0.436055 + 0.899920i \(0.356375\pi\)
\(182\) −1.65907e12 −0.615845
\(183\) −1.20295e12 −0.433279
\(184\) 2.37440e12 0.829956
\(185\) −3.56966e12 −1.21111
\(186\) 9.65467e11 0.317992
\(187\) 1.96731e12 0.629136
\(188\) −6.27897e12 −1.94994
\(189\) 3.98326e11 0.120143
\(190\) 8.17255e12 2.39449
\(191\) 3.38709e12 0.964147 0.482073 0.876131i \(-0.339884\pi\)
0.482073 + 0.876131i \(0.339884\pi\)
\(192\) 2.26370e12 0.626126
\(193\) −4.92921e12 −1.32499 −0.662494 0.749067i \(-0.730502\pi\)
−0.662494 + 0.749067i \(0.730502\pi\)
\(194\) −3.04921e12 −0.796670
\(195\) 9.99840e11 0.253945
\(196\) −4.87028e12 −1.20267
\(197\) 6.35785e12 1.52667 0.763337 0.646001i \(-0.223560\pi\)
0.763337 + 0.646001i \(0.223560\pi\)
\(198\) 2.93776e12 0.686053
\(199\) −3.78554e12 −0.859875 −0.429938 0.902859i \(-0.641464\pi\)
−0.429938 + 0.902859i \(0.641464\pi\)
\(200\) −3.09992e12 −0.684992
\(201\) −1.28605e12 −0.276491
\(202\) 7.26241e11 0.151932
\(203\) −2.98455e11 −0.0607646
\(204\) −3.02497e12 −0.599454
\(205\) −4.82674e12 −0.931123
\(206\) 3.78885e12 0.711604
\(207\) 9.04180e11 0.165355
\(208\) 2.93404e12 0.522537
\(209\) −1.24451e13 −2.15870
\(210\) −2.82550e12 −0.477407
\(211\) 1.79494e11 0.0295458 0.0147729 0.999891i \(-0.495297\pi\)
0.0147729 + 0.999891i \(0.495297\pi\)
\(212\) 1.53062e13 2.45482
\(213\) 3.60395e12 0.563237
\(214\) 1.75965e13 2.68009
\(215\) 5.14418e12 0.763668
\(216\) −2.22500e12 −0.321984
\(217\) 1.41402e12 0.199493
\(218\) −5.41719e12 −0.745184
\(219\) −3.39495e12 −0.455398
\(220\) −1.38240e13 −1.80846
\(221\) 2.36328e12 0.301548
\(222\) −1.25995e13 −1.56824
\(223\) 2.22568e12 0.270263 0.135132 0.990828i \(-0.456854\pi\)
0.135132 + 0.990828i \(0.456854\pi\)
\(224\) 5.24344e11 0.0621230
\(225\) −1.18046e12 −0.136473
\(226\) −2.80538e13 −3.16516
\(227\) −4.15848e11 −0.0457923 −0.0228961 0.999738i \(-0.507289\pi\)
−0.0228961 + 0.999738i \(0.507289\pi\)
\(228\) 1.91358e13 2.05686
\(229\) 1.81248e13 1.90186 0.950928 0.309414i \(-0.100133\pi\)
0.950928 + 0.309414i \(0.100133\pi\)
\(230\) −6.41374e12 −0.657064
\(231\) 4.30264e12 0.430397
\(232\) 1.66713e12 0.162850
\(233\) −1.87641e10 −0.00179007 −0.000895033 1.00000i \(-0.500285\pi\)
−0.000895033 1.00000i \(0.500285\pi\)
\(234\) 3.52905e12 0.328829
\(235\) 8.35433e12 0.760394
\(236\) 2.24122e12 0.199282
\(237\) −9.04092e11 −0.0785410
\(238\) −6.67849e12 −0.566898
\(239\) −1.76252e13 −1.46200 −0.730999 0.682379i \(-0.760946\pi\)
−0.730999 + 0.682379i \(0.760946\pi\)
\(240\) 4.99684e12 0.405074
\(241\) −8.90117e11 −0.0705267 −0.0352633 0.999378i \(-0.511227\pi\)
−0.0352633 + 0.999378i \(0.511227\pi\)
\(242\) 9.47880e12 0.734123
\(243\) −8.47289e11 −0.0641500
\(244\) 1.99799e13 1.47894
\(245\) 6.48003e12 0.468990
\(246\) −1.70365e13 −1.20569
\(247\) −1.49499e13 −1.03468
\(248\) −7.89856e12 −0.534643
\(249\) −2.13073e12 −0.141069
\(250\) 2.88257e13 1.86685
\(251\) −2.42280e13 −1.53502 −0.767508 0.641040i \(-0.778503\pi\)
−0.767508 + 0.641040i \(0.778503\pi\)
\(252\) −6.61581e12 −0.410091
\(253\) 9.76677e12 0.592364
\(254\) 1.95213e13 1.15857
\(255\) 4.02480e12 0.233762
\(256\) −3.45806e13 −1.96568
\(257\) 7.80492e12 0.434246 0.217123 0.976144i \(-0.430333\pi\)
0.217123 + 0.976144i \(0.430333\pi\)
\(258\) 1.81569e13 0.988858
\(259\) −1.84532e13 −0.983837
\(260\) −1.66064e13 −0.866806
\(261\) 6.34851e11 0.0324451
\(262\) −7.63556e12 −0.382106
\(263\) 1.65956e13 0.813272 0.406636 0.913590i \(-0.366702\pi\)
0.406636 + 0.913590i \(0.366702\pi\)
\(264\) −2.40340e13 −1.15347
\(265\) −2.03653e13 −0.957278
\(266\) 4.22477e13 1.94515
\(267\) 6.18988e12 0.279171
\(268\) 2.13601e13 0.943762
\(269\) 2.85236e13 1.23471 0.617357 0.786683i \(-0.288203\pi\)
0.617357 + 0.786683i \(0.288203\pi\)
\(270\) 6.01018e12 0.254910
\(271\) 2.33800e13 0.971658 0.485829 0.874054i \(-0.338518\pi\)
0.485829 + 0.874054i \(0.338518\pi\)
\(272\) 1.18108e13 0.481006
\(273\) 5.16863e12 0.206291
\(274\) −1.20911e13 −0.472976
\(275\) −1.27511e13 −0.488898
\(276\) −1.50176e13 −0.564416
\(277\) −3.03641e13 −1.11872 −0.559361 0.828924i \(-0.688953\pi\)
−0.559361 + 0.828924i \(0.688953\pi\)
\(278\) −8.78487e13 −3.17314
\(279\) −3.00780e12 −0.106519
\(280\) 2.31156e13 0.802667
\(281\) 1.59749e13 0.543942 0.271971 0.962306i \(-0.412325\pi\)
0.271971 + 0.962306i \(0.412325\pi\)
\(282\) 2.94875e13 0.984619
\(283\) 2.87045e13 0.939993 0.469997 0.882668i \(-0.344255\pi\)
0.469997 + 0.882668i \(0.344255\pi\)
\(284\) −5.98583e13 −1.92253
\(285\) −2.54606e13 −0.802089
\(286\) 3.81201e13 1.17799
\(287\) −2.49516e13 −0.756394
\(288\) −1.11535e12 −0.0331704
\(289\) −2.47587e13 −0.722419
\(290\) −4.50327e12 −0.128926
\(291\) 9.49948e12 0.266863
\(292\) 5.63870e13 1.55444
\(293\) 4.81754e13 1.30333 0.651663 0.758508i \(-0.274071\pi\)
0.651663 + 0.758508i \(0.274071\pi\)
\(294\) 2.28720e13 0.607286
\(295\) −2.98200e12 −0.0777117
\(296\) 1.03077e14 2.63669
\(297\) −9.15225e12 −0.229809
\(298\) −1.07997e14 −2.66209
\(299\) 1.17325e13 0.283923
\(300\) 1.96064e13 0.465833
\(301\) 2.65926e13 0.620362
\(302\) −5.44909e13 −1.24820
\(303\) −2.26252e12 −0.0508930
\(304\) −7.47143e13 −1.65044
\(305\) −2.65838e13 −0.576723
\(306\) 1.42060e13 0.302694
\(307\) 2.57350e13 0.538597 0.269298 0.963057i \(-0.413208\pi\)
0.269298 + 0.963057i \(0.413208\pi\)
\(308\) −7.14627e13 −1.46910
\(309\) −1.18037e13 −0.238368
\(310\) 2.13356e13 0.423269
\(311\) 3.46043e13 0.674446 0.337223 0.941425i \(-0.390512\pi\)
0.337223 + 0.941425i \(0.390512\pi\)
\(312\) −2.88714e13 −0.552862
\(313\) 1.26066e13 0.237194 0.118597 0.992942i \(-0.462160\pi\)
0.118597 + 0.992942i \(0.462160\pi\)
\(314\) 1.66239e14 3.07341
\(315\) 8.80251e12 0.159918
\(316\) 1.50161e13 0.268089
\(317\) −8.18243e13 −1.43568 −0.717838 0.696210i \(-0.754868\pi\)
−0.717838 + 0.696210i \(0.754868\pi\)
\(318\) −7.18815e13 −1.23956
\(319\) 6.85754e12 0.116230
\(320\) 5.00250e13 0.833416
\(321\) −5.48198e13 −0.897758
\(322\) −3.31555e13 −0.533764
\(323\) −6.01801e13 −0.952443
\(324\) 1.40727e13 0.218967
\(325\) −1.53176e13 −0.234332
\(326\) −1.27580e13 −0.191905
\(327\) 1.68766e13 0.249616
\(328\) 1.39377e14 2.02714
\(329\) 4.31874e13 0.617703
\(330\) 6.49208e13 0.913182
\(331\) −3.40115e13 −0.470513 −0.235256 0.971933i \(-0.575593\pi\)
−0.235256 + 0.971933i \(0.575593\pi\)
\(332\) 3.53895e13 0.481520
\(333\) 3.92523e13 0.525317
\(334\) −6.87135e11 −0.00904558
\(335\) −2.84202e13 −0.368028
\(336\) 2.58310e13 0.329060
\(337\) −5.99439e13 −0.751244 −0.375622 0.926773i \(-0.622571\pi\)
−0.375622 + 0.926773i \(0.622571\pi\)
\(338\) −9.39960e13 −1.15896
\(339\) 8.73985e13 1.06024
\(340\) −6.68482e13 −0.797913
\(341\) −3.24897e13 −0.381590
\(342\) −8.98661e13 −1.03861
\(343\) 8.83888e13 1.00526
\(344\) −1.48543e14 −1.66257
\(345\) 1.99813e13 0.220099
\(346\) −5.70064e13 −0.618024
\(347\) 9.78685e13 1.04431 0.522157 0.852850i \(-0.325128\pi\)
0.522157 + 0.852850i \(0.325128\pi\)
\(348\) −1.05443e13 −0.110747
\(349\) −1.42790e14 −1.47624 −0.738120 0.674670i \(-0.764286\pi\)
−0.738120 + 0.674670i \(0.764286\pi\)
\(350\) 4.32866e13 0.440534
\(351\) −1.09943e13 −0.110149
\(352\) −1.20478e13 −0.118829
\(353\) 1.44246e14 1.40069 0.700346 0.713804i \(-0.253029\pi\)
0.700346 + 0.713804i \(0.253029\pi\)
\(354\) −1.05253e13 −0.100627
\(355\) 7.96429e13 0.749707
\(356\) −1.02808e14 −0.952911
\(357\) 2.08061e13 0.189896
\(358\) 3.06050e14 2.75065
\(359\) −1.24349e14 −1.10059 −0.550293 0.834972i \(-0.685484\pi\)
−0.550293 + 0.834972i \(0.685484\pi\)
\(360\) −4.91697e13 −0.428582
\(361\) 2.64205e14 2.26804
\(362\) 1.77786e14 1.50314
\(363\) −2.95301e13 −0.245911
\(364\) −8.58461e13 −0.704147
\(365\) −7.50243e13 −0.606165
\(366\) −9.38303e13 −0.746788
\(367\) −1.60110e14 −1.25532 −0.627659 0.778488i \(-0.715987\pi\)
−0.627659 + 0.778488i \(0.715987\pi\)
\(368\) 5.86351e13 0.452892
\(369\) 5.30752e13 0.403875
\(370\) −2.78433e14 −2.08743
\(371\) −1.05277e14 −0.777641
\(372\) 4.99568e13 0.363587
\(373\) −3.47258e13 −0.249031 −0.124516 0.992218i \(-0.539738\pi\)
−0.124516 + 0.992218i \(0.539738\pi\)
\(374\) 1.53450e14 1.08436
\(375\) −8.98031e13 −0.625344
\(376\) −2.41239e14 −1.65545
\(377\) 8.23777e12 0.0557099
\(378\) 3.10694e13 0.207075
\(379\) −1.46500e14 −0.962327 −0.481163 0.876631i \(-0.659786\pi\)
−0.481163 + 0.876631i \(0.659786\pi\)
\(380\) 4.22877e14 2.73782
\(381\) −6.08164e13 −0.388090
\(382\) 2.64193e14 1.66178
\(383\) 6.43419e13 0.398933 0.199467 0.979905i \(-0.436079\pi\)
0.199467 + 0.979905i \(0.436079\pi\)
\(384\) 1.67168e14 1.02172
\(385\) 9.50830e13 0.572887
\(386\) −3.84478e14 −2.28371
\(387\) −5.65658e13 −0.331241
\(388\) −1.57777e14 −0.910899
\(389\) −3.98900e13 −0.227061 −0.113530 0.993535i \(-0.536216\pi\)
−0.113530 + 0.993535i \(0.536216\pi\)
\(390\) 7.79875e13 0.437693
\(391\) 4.72287e13 0.261357
\(392\) −1.87117e14 −1.02103
\(393\) 2.37877e13 0.127995
\(394\) 4.95912e14 2.63133
\(395\) −1.99793e13 −0.104543
\(396\) 1.52010e14 0.784421
\(397\) 1.06552e14 0.542268 0.271134 0.962542i \(-0.412601\pi\)
0.271134 + 0.962542i \(0.412601\pi\)
\(398\) −2.95272e14 −1.48206
\(399\) −1.31618e14 −0.651573
\(400\) −7.65517e13 −0.373787
\(401\) 3.41445e13 0.164447 0.0822236 0.996614i \(-0.473798\pi\)
0.0822236 + 0.996614i \(0.473798\pi\)
\(402\) −1.00312e14 −0.476552
\(403\) −3.90289e13 −0.182898
\(404\) 3.75783e13 0.173716
\(405\) −1.87240e13 −0.0853880
\(406\) −2.32795e13 −0.104732
\(407\) 4.23996e14 1.88188
\(408\) −1.16220e14 −0.508921
\(409\) 5.33349e13 0.230427 0.115213 0.993341i \(-0.463245\pi\)
0.115213 + 0.993341i \(0.463245\pi\)
\(410\) −3.76485e14 −1.60486
\(411\) 3.76686e13 0.158434
\(412\) 1.96049e14 0.813635
\(413\) −1.54153e13 −0.0631288
\(414\) 7.05260e13 0.285002
\(415\) −4.70866e13 −0.187773
\(416\) −1.44726e13 −0.0569553
\(417\) 2.73682e14 1.06292
\(418\) −9.70716e14 −3.72068
\(419\) −1.01288e14 −0.383159 −0.191580 0.981477i \(-0.561361\pi\)
−0.191580 + 0.981477i \(0.561361\pi\)
\(420\) −1.46201e14 −0.545859
\(421\) −1.57928e14 −0.581981 −0.290991 0.956726i \(-0.593985\pi\)
−0.290991 + 0.956726i \(0.593985\pi\)
\(422\) 1.40005e13 0.0509244
\(423\) −9.18650e13 −0.329821
\(424\) 5.88067e14 2.08408
\(425\) −6.16600e13 −0.215707
\(426\) 2.81108e14 0.970780
\(427\) −1.37424e14 −0.468499
\(428\) 9.10504e14 3.06437
\(429\) −1.18759e14 −0.394594
\(430\) 4.01246e14 1.31624
\(431\) 5.13171e14 1.66202 0.831012 0.556254i \(-0.187762\pi\)
0.831012 + 0.556254i \(0.187762\pi\)
\(432\) −5.49458e13 −0.175701
\(433\) −7.49248e13 −0.236560 −0.118280 0.992980i \(-0.537738\pi\)
−0.118280 + 0.992980i \(0.537738\pi\)
\(434\) 1.10294e14 0.343841
\(435\) 1.40294e13 0.0431866
\(436\) −2.80305e14 −0.852030
\(437\) −2.98766e14 −0.896773
\(438\) −2.64806e14 −0.784912
\(439\) −3.68335e14 −1.07817 −0.539086 0.842250i \(-0.681230\pi\)
−0.539086 + 0.842250i \(0.681230\pi\)
\(440\) −5.31122e14 −1.53534
\(441\) −7.12550e13 −0.203424
\(442\) 1.84335e14 0.519740
\(443\) −1.11248e14 −0.309793 −0.154896 0.987931i \(-0.549504\pi\)
−0.154896 + 0.987931i \(0.549504\pi\)
\(444\) −6.51943e14 −1.79310
\(445\) 1.36789e14 0.371595
\(446\) 1.73603e14 0.465818
\(447\) 3.36453e14 0.891730
\(448\) 2.58602e14 0.677022
\(449\) −8.83314e13 −0.228434 −0.114217 0.993456i \(-0.536436\pi\)
−0.114217 + 0.993456i \(0.536436\pi\)
\(450\) −9.20760e13 −0.235222
\(451\) 5.73308e14 1.44683
\(452\) −1.45161e15 −3.61899
\(453\) 1.69760e14 0.418115
\(454\) −3.24361e13 −0.0789263
\(455\) 1.14220e14 0.274588
\(456\) 7.35200e14 1.74622
\(457\) −9.42094e12 −0.0221083 −0.0110541 0.999939i \(-0.503519\pi\)
−0.0110541 + 0.999939i \(0.503519\pi\)
\(458\) 1.41373e15 3.27799
\(459\) −4.42571e13 −0.101394
\(460\) −3.31870e14 −0.751276
\(461\) −6.97134e14 −1.55941 −0.779706 0.626146i \(-0.784631\pi\)
−0.779706 + 0.626146i \(0.784631\pi\)
\(462\) 3.35606e14 0.741820
\(463\) −1.87941e14 −0.410513 −0.205256 0.978708i \(-0.565803\pi\)
−0.205256 + 0.978708i \(0.565803\pi\)
\(464\) 4.11694e13 0.0888640
\(465\) −6.64687e13 −0.141784
\(466\) −1.46360e12 −0.00308531
\(467\) 3.20007e14 0.666678 0.333339 0.942807i \(-0.391825\pi\)
0.333339 + 0.942807i \(0.391825\pi\)
\(468\) 1.82605e14 0.375977
\(469\) −1.46917e14 −0.298966
\(470\) 6.51638e14 1.31059
\(471\) −5.17899e14 −1.02951
\(472\) 8.61081e13 0.169185
\(473\) −6.11013e14 −1.18663
\(474\) −7.05192e13 −0.135371
\(475\) 3.90057e14 0.740139
\(476\) −3.45569e14 −0.648181
\(477\) 2.23938e14 0.415219
\(478\) −1.37477e15 −2.51986
\(479\) −1.50382e14 −0.272491 −0.136245 0.990675i \(-0.543504\pi\)
−0.136245 + 0.990675i \(0.543504\pi\)
\(480\) −2.46478e13 −0.0441520
\(481\) 5.09334e14 0.901996
\(482\) −6.94291e13 −0.121558
\(483\) 1.03292e14 0.178796
\(484\) 4.90467e14 0.839384
\(485\) 2.09927e14 0.355212
\(486\) −6.60885e13 −0.110567
\(487\) 1.76546e14 0.292045 0.146022 0.989281i \(-0.453353\pi\)
0.146022 + 0.989281i \(0.453353\pi\)
\(488\) 7.67632e14 1.25558
\(489\) 3.97462e13 0.0642831
\(490\) 5.05442e14 0.808338
\(491\) −8.60958e14 −1.36155 −0.680775 0.732492i \(-0.738357\pi\)
−0.680775 + 0.732492i \(0.738357\pi\)
\(492\) −8.81529e14 −1.37857
\(493\) 3.31607e13 0.0512821
\(494\) −1.16609e15 −1.78334
\(495\) −2.02253e14 −0.305891
\(496\) −1.95053e14 −0.291745
\(497\) 4.11711e14 0.609021
\(498\) −1.66197e14 −0.243143
\(499\) 6.01209e14 0.869907 0.434953 0.900453i \(-0.356765\pi\)
0.434953 + 0.900453i \(0.356765\pi\)
\(500\) 1.49154e15 2.13452
\(501\) 2.14069e12 0.00303003
\(502\) −1.88979e15 −2.64571
\(503\) 1.09203e15 1.51221 0.756103 0.654453i \(-0.227101\pi\)
0.756103 + 0.654453i \(0.227101\pi\)
\(504\) −2.54181e14 −0.348157
\(505\) −4.99989e13 −0.0677420
\(506\) 7.61808e14 1.02098
\(507\) 2.92834e14 0.388219
\(508\) 1.01010e15 1.32469
\(509\) 8.76371e14 1.13695 0.568474 0.822702i \(-0.307534\pi\)
0.568474 + 0.822702i \(0.307534\pi\)
\(510\) 3.13935e14 0.402906
\(511\) −3.87835e14 −0.492416
\(512\) −1.28839e15 −1.61832
\(513\) 2.79967e14 0.347906
\(514\) 6.08784e14 0.748455
\(515\) −2.60848e14 −0.317284
\(516\) 9.39505e14 1.13064
\(517\) −9.92308e14 −1.18154
\(518\) −1.43935e15 −1.69571
\(519\) 1.77597e14 0.207021
\(520\) −6.38022e14 −0.735897
\(521\) −3.71989e14 −0.424544 −0.212272 0.977211i \(-0.568086\pi\)
−0.212272 + 0.977211i \(0.568086\pi\)
\(522\) 4.95184e13 0.0559215
\(523\) −9.73507e14 −1.08788 −0.543939 0.839125i \(-0.683068\pi\)
−0.543939 + 0.839125i \(0.683068\pi\)
\(524\) −3.95091e14 −0.436893
\(525\) −1.34854e14 −0.147567
\(526\) 1.29445e15 1.40173
\(527\) −1.57109e14 −0.168361
\(528\) −5.93514e14 −0.629425
\(529\) −7.18341e14 −0.753919
\(530\) −1.58849e15 −1.64994
\(531\) 3.27903e13 0.0337074
\(532\) 2.18605e15 2.22405
\(533\) 6.88699e14 0.693473
\(534\) 4.82811e14 0.481172
\(535\) −1.21145e15 −1.19498
\(536\) 8.20661e14 0.801230
\(537\) −9.53463e14 −0.921393
\(538\) 2.22484e15 2.12812
\(539\) −7.69683e14 −0.728741
\(540\) 3.10988e14 0.291460
\(541\) 1.74606e15 1.61985 0.809925 0.586533i \(-0.199508\pi\)
0.809925 + 0.586533i \(0.199508\pi\)
\(542\) 1.82364e15 1.67472
\(543\) −5.53872e14 −0.503513
\(544\) −5.82588e13 −0.0524285
\(545\) 3.72953e14 0.332256
\(546\) 4.03153e14 0.355558
\(547\) 1.74624e14 0.152467 0.0762333 0.997090i \(-0.475711\pi\)
0.0762333 + 0.997090i \(0.475711\pi\)
\(548\) −6.25639e14 −0.540793
\(549\) 2.92317e14 0.250154
\(550\) −9.94588e14 −0.842652
\(551\) −2.09772e14 −0.175960
\(552\) −5.76978e14 −0.479175
\(553\) −1.03282e14 −0.0849254
\(554\) −2.36840e15 −1.92820
\(555\) 8.67427e14 0.699232
\(556\) −4.54561e15 −3.62811
\(557\) 1.58365e15 1.25157 0.625785 0.779995i \(-0.284779\pi\)
0.625785 + 0.779995i \(0.284779\pi\)
\(558\) −2.34609e14 −0.183593
\(559\) −7.33993e14 −0.568757
\(560\) 5.70833e14 0.438001
\(561\) −4.78057e14 −0.363232
\(562\) 1.24604e15 0.937523
\(563\) −9.75798e14 −0.727049 −0.363525 0.931585i \(-0.618427\pi\)
−0.363525 + 0.931585i \(0.618427\pi\)
\(564\) 1.52579e15 1.12580
\(565\) 1.93140e15 1.41125
\(566\) 2.23895e15 1.62015
\(567\) −9.67931e13 −0.0693646
\(568\) −2.29977e15 −1.63218
\(569\) 1.92427e15 1.35254 0.676269 0.736655i \(-0.263596\pi\)
0.676269 + 0.736655i \(0.263596\pi\)
\(570\) −1.98593e15 −1.38246
\(571\) 1.61132e15 1.11092 0.555461 0.831543i \(-0.312542\pi\)
0.555461 + 0.831543i \(0.312542\pi\)
\(572\) 1.97247e15 1.34689
\(573\) −8.23063e14 −0.556650
\(574\) −1.94623e15 −1.30370
\(575\) −3.06113e14 −0.203099
\(576\) −5.50079e14 −0.361494
\(577\) −2.53250e15 −1.64848 −0.824239 0.566242i \(-0.808397\pi\)
−0.824239 + 0.566242i \(0.808397\pi\)
\(578\) −1.93118e15 −1.24514
\(579\) 1.19780e15 0.764982
\(580\) −2.33016e14 −0.147411
\(581\) −2.43412e14 −0.152536
\(582\) 7.40959e14 0.459958
\(583\) 2.41894e15 1.48747
\(584\) 2.16640e15 1.31968
\(585\) −2.42961e14 −0.146615
\(586\) 3.75768e15 2.24638
\(587\) −3.11508e15 −1.84484 −0.922421 0.386186i \(-0.873792\pi\)
−0.922421 + 0.386186i \(0.873792\pi\)
\(588\) 1.18348e15 0.694360
\(589\) 9.93860e14 0.577685
\(590\) −2.32596e14 −0.133942
\(591\) −1.54496e15 −0.881425
\(592\) 2.54547e15 1.43879
\(593\) −2.20723e15 −1.23608 −0.618040 0.786146i \(-0.712073\pi\)
−0.618040 + 0.786146i \(0.712073\pi\)
\(594\) −7.13875e14 −0.396093
\(595\) 4.59788e14 0.252764
\(596\) −5.58817e15 −3.04379
\(597\) 9.19885e14 0.496449
\(598\) 9.15138e14 0.489362
\(599\) 3.41159e15 1.80763 0.903813 0.427927i \(-0.140756\pi\)
0.903813 + 0.427927i \(0.140756\pi\)
\(600\) 7.53280e14 0.395480
\(601\) −1.57545e15 −0.819588 −0.409794 0.912178i \(-0.634399\pi\)
−0.409794 + 0.912178i \(0.634399\pi\)
\(602\) 2.07422e15 1.06924
\(603\) 3.12511e14 0.159632
\(604\) −2.81956e15 −1.42718
\(605\) −6.52579e14 −0.327325
\(606\) −1.76477e14 −0.0877178
\(607\) −5.05592e14 −0.249036 −0.124518 0.992217i \(-0.539738\pi\)
−0.124518 + 0.992217i \(0.539738\pi\)
\(608\) 3.68541e14 0.179894
\(609\) 7.25246e13 0.0350825
\(610\) −2.07353e15 −0.994024
\(611\) −1.19203e15 −0.566319
\(612\) 7.35069e14 0.346095
\(613\) −1.98375e15 −0.925665 −0.462832 0.886446i \(-0.653167\pi\)
−0.462832 + 0.886446i \(0.653167\pi\)
\(614\) 2.00733e15 0.928310
\(615\) 1.17290e15 0.537584
\(616\) −2.74561e15 −1.24723
\(617\) −1.93099e15 −0.869382 −0.434691 0.900580i \(-0.643142\pi\)
−0.434691 + 0.900580i \(0.643142\pi\)
\(618\) −9.20692e14 −0.410845
\(619\) 9.47689e14 0.419148 0.209574 0.977793i \(-0.432792\pi\)
0.209574 + 0.977793i \(0.432792\pi\)
\(620\) 1.10398e15 0.483958
\(621\) −2.19716e14 −0.0954679
\(622\) 2.69913e15 1.16246
\(623\) 7.07124e14 0.301864
\(624\) −7.12971e14 −0.301687
\(625\) −1.00840e15 −0.422955
\(626\) 9.83314e14 0.408821
\(627\) 3.02415e15 1.24633
\(628\) 8.60181e15 3.51408
\(629\) 2.05030e15 0.830306
\(630\) 6.86595e14 0.275631
\(631\) 1.00593e15 0.400318 0.200159 0.979763i \(-0.435854\pi\)
0.200159 + 0.979763i \(0.435854\pi\)
\(632\) 5.76922e14 0.227601
\(633\) −4.36170e13 −0.0170583
\(634\) −6.38230e15 −2.47449
\(635\) −1.34397e15 −0.516574
\(636\) −3.71941e15 −1.41729
\(637\) −9.24597e14 −0.349290
\(638\) 5.34888e14 0.200332
\(639\) −8.75761e14 −0.325185
\(640\) 3.69422e15 1.35998
\(641\) 1.13144e15 0.412966 0.206483 0.978450i \(-0.433798\pi\)
0.206483 + 0.978450i \(0.433798\pi\)
\(642\) −4.27594e15 −1.54735
\(643\) −8.68306e14 −0.311539 −0.155769 0.987793i \(-0.549786\pi\)
−0.155769 + 0.987793i \(0.549786\pi\)
\(644\) −1.71559e15 −0.610296
\(645\) −1.25003e15 −0.440904
\(646\) −4.69405e15 −1.64160
\(647\) −1.37636e15 −0.477265 −0.238633 0.971110i \(-0.576699\pi\)
−0.238633 + 0.971110i \(0.576699\pi\)
\(648\) 5.40675e14 0.185897
\(649\) 3.54195e14 0.120753
\(650\) −1.19477e15 −0.403888
\(651\) −3.43607e14 −0.115177
\(652\) −6.60146e14 −0.219421
\(653\) 2.55276e15 0.841370 0.420685 0.907207i \(-0.361790\pi\)
0.420685 + 0.907207i \(0.361790\pi\)
\(654\) 1.31638e15 0.430232
\(655\) 5.25679e14 0.170370
\(656\) 3.44187e15 1.10617
\(657\) 8.24974e14 0.262924
\(658\) 3.36862e15 1.06466
\(659\) 9.26792e14 0.290478 0.145239 0.989397i \(-0.453605\pi\)
0.145239 + 0.989397i \(0.453605\pi\)
\(660\) 3.35924e15 1.04412
\(661\) −1.90332e15 −0.586683 −0.293342 0.956008i \(-0.594767\pi\)
−0.293342 + 0.956008i \(0.594767\pi\)
\(662\) −2.65289e15 −0.810963
\(663\) −5.74276e14 −0.174099
\(664\) 1.35967e15 0.408799
\(665\) −2.90859e15 −0.867288
\(666\) 3.06168e15 0.905422
\(667\) 1.64627e14 0.0482847
\(668\) −3.55548e13 −0.0103426
\(669\) −5.40841e14 −0.156037
\(670\) −2.21677e15 −0.634322
\(671\) 3.15756e15 0.896143
\(672\) −1.27416e14 −0.0358667
\(673\) 4.92990e15 1.37643 0.688217 0.725505i \(-0.258394\pi\)
0.688217 + 0.725505i \(0.258394\pi\)
\(674\) −4.67563e15 −1.29482
\(675\) 2.86852e14 0.0787930
\(676\) −4.86369e15 −1.32513
\(677\) 4.30293e14 0.116286 0.0581429 0.998308i \(-0.481482\pi\)
0.0581429 + 0.998308i \(0.481482\pi\)
\(678\) 6.81708e15 1.82741
\(679\) 1.08521e15 0.288555
\(680\) −2.56832e15 −0.677408
\(681\) 1.01051e14 0.0264382
\(682\) −2.53420e15 −0.657698
\(683\) −3.81244e15 −0.981498 −0.490749 0.871301i \(-0.663277\pi\)
−0.490749 + 0.871301i \(0.663277\pi\)
\(684\) −4.64999e15 −1.18753
\(685\) 8.32429e14 0.210887
\(686\) 6.89433e15 1.73264
\(687\) −4.40432e15 −1.09804
\(688\) −3.66823e15 −0.907236
\(689\) 2.90580e15 0.712952
\(690\) 1.55854e15 0.379356
\(691\) 4.03729e15 0.974901 0.487450 0.873151i \(-0.337927\pi\)
0.487450 + 0.873151i \(0.337927\pi\)
\(692\) −2.94972e15 −0.706638
\(693\) −1.04554e15 −0.248490
\(694\) 7.63374e15 1.79995
\(695\) 6.04805e15 1.41481
\(696\) −4.05113e14 −0.0940212
\(697\) 2.77232e15 0.638356
\(698\) −1.11376e16 −2.54441
\(699\) 4.55967e12 0.00103350
\(700\) 2.23980e15 0.503699
\(701\) −4.71267e15 −1.05152 −0.525761 0.850632i \(-0.676219\pi\)
−0.525761 + 0.850632i \(0.676219\pi\)
\(702\) −8.57558e14 −0.189849
\(703\) −1.29700e16 −2.84896
\(704\) −5.94185e15 −1.29501
\(705\) −2.03010e15 −0.439013
\(706\) 1.12512e16 2.41419
\(707\) −2.58467e14 −0.0550299
\(708\) −5.44616e14 −0.115056
\(709\) 8.66706e14 0.181684 0.0908422 0.995865i \(-0.471044\pi\)
0.0908422 + 0.995865i \(0.471044\pi\)
\(710\) 6.21215e15 1.29217
\(711\) 2.19694e14 0.0453457
\(712\) −3.94991e15 −0.808997
\(713\) −7.79972e14 −0.158521
\(714\) 1.62287e15 0.327299
\(715\) −2.62442e15 −0.525231
\(716\) 1.58361e16 3.14505
\(717\) 4.28293e15 0.844085
\(718\) −9.69924e15 −1.89694
\(719\) 2.62666e13 0.00509795 0.00254897 0.999997i \(-0.499189\pi\)
0.00254897 + 0.999997i \(0.499189\pi\)
\(720\) −1.21423e15 −0.233869
\(721\) −1.34844e15 −0.257744
\(722\) 2.06080e16 3.90913
\(723\) 2.16298e14 0.0407186
\(724\) 9.19929e15 1.71867
\(725\) −2.14931e14 −0.0398511
\(726\) −2.30335e15 −0.423846
\(727\) −5.68018e15 −1.03734 −0.518672 0.854973i \(-0.673573\pi\)
−0.518672 + 0.854973i \(0.673573\pi\)
\(728\) −3.29823e15 −0.597803
\(729\) 2.05891e14 0.0370370
\(730\) −5.85190e15 −1.04477
\(731\) −2.95465e15 −0.523552
\(732\) −4.85512e15 −0.853864
\(733\) −1.62312e15 −0.283320 −0.141660 0.989915i \(-0.545244\pi\)
−0.141660 + 0.989915i \(0.545244\pi\)
\(734\) −1.24885e16 −2.16363
\(735\) −1.57465e15 −0.270771
\(736\) −2.89227e14 −0.0493641
\(737\) 3.37568e15 0.571861
\(738\) 4.13987e15 0.696108
\(739\) 3.64794e15 0.608840 0.304420 0.952538i \(-0.401537\pi\)
0.304420 + 0.952538i \(0.401537\pi\)
\(740\) −1.44071e16 −2.38673
\(741\) 3.63283e15 0.597372
\(742\) −8.21164e15 −1.34032
\(743\) 6.76045e15 1.09531 0.547655 0.836704i \(-0.315521\pi\)
0.547655 + 0.836704i \(0.315521\pi\)
\(744\) 1.91935e15 0.308676
\(745\) 7.43520e15 1.18695
\(746\) −2.70861e15 −0.429224
\(747\) 5.17768e14 0.0814465
\(748\) 7.94007e15 1.23984
\(749\) −6.26254e15 −0.970734
\(750\) −7.00464e15 −1.07783
\(751\) 2.64833e13 0.00404532 0.00202266 0.999998i \(-0.499356\pi\)
0.00202266 + 0.999998i \(0.499356\pi\)
\(752\) −5.95734e15 −0.903347
\(753\) 5.88741e15 0.886242
\(754\) 6.42546e14 0.0960200
\(755\) 3.75149e15 0.556539
\(756\) 1.60764e15 0.236766
\(757\) 7.37364e15 1.07809 0.539045 0.842277i \(-0.318785\pi\)
0.539045 + 0.842277i \(0.318785\pi\)
\(758\) −1.14270e16 −1.65864
\(759\) −2.37333e15 −0.342001
\(760\) 1.62470e16 2.32434
\(761\) 6.20983e15 0.881991 0.440996 0.897509i \(-0.354625\pi\)
0.440996 + 0.897509i \(0.354625\pi\)
\(762\) −4.74368e15 −0.668902
\(763\) 1.92796e15 0.269907
\(764\) 1.36703e16 1.90005
\(765\) −9.78027e14 −0.134963
\(766\) 5.01866e15 0.687591
\(767\) 4.25484e14 0.0578774
\(768\) 8.40308e15 1.13488
\(769\) −8.86195e15 −1.18832 −0.594162 0.804346i \(-0.702516\pi\)
−0.594162 + 0.804346i \(0.702516\pi\)
\(770\) 7.41647e15 0.987412
\(771\) −1.89659e15 −0.250712
\(772\) −1.98943e16 −2.61116
\(773\) −2.95047e15 −0.384507 −0.192254 0.981345i \(-0.561580\pi\)
−0.192254 + 0.981345i \(0.561580\pi\)
\(774\) −4.41213e15 −0.570918
\(775\) 1.01830e15 0.130833
\(776\) −6.06184e15 −0.773330
\(777\) 4.48413e15 0.568019
\(778\) −3.11142e15 −0.391355
\(779\) −1.75375e16 −2.19034
\(780\) 4.03536e15 0.500451
\(781\) −9.45980e15 −1.16493
\(782\) 3.68384e15 0.450468
\(783\) −1.54269e14 −0.0187322
\(784\) −4.62081e15 −0.557160
\(785\) −1.14449e16 −1.37034
\(786\) 1.85544e15 0.220609
\(787\) 1.35545e16 1.60038 0.800190 0.599746i \(-0.204732\pi\)
0.800190 + 0.599746i \(0.204732\pi\)
\(788\) 2.56603e16 3.00862
\(789\) −4.03272e15 −0.469543
\(790\) −1.55839e15 −0.180188
\(791\) 9.98429e15 1.14643
\(792\) 5.84027e15 0.665954
\(793\) 3.79308e15 0.429526
\(794\) 8.31105e15 0.934638
\(795\) 4.94876e15 0.552685
\(796\) −1.52784e16 −1.69456
\(797\) −1.59922e16 −1.76152 −0.880762 0.473558i \(-0.842969\pi\)
−0.880762 + 0.473558i \(0.842969\pi\)
\(798\) −1.02662e16 −1.12303
\(799\) −4.79846e15 −0.521308
\(800\) 3.77604e14 0.0407419
\(801\) −1.50414e15 −0.161180
\(802\) 2.66327e15 0.283437
\(803\) 8.91121e15 0.941892
\(804\) −5.19051e15 −0.544881
\(805\) 2.28263e15 0.237990
\(806\) −3.04426e15 −0.315238
\(807\) −6.93123e15 −0.712863
\(808\) 1.44377e15 0.147481
\(809\) 9.45859e15 0.959643 0.479821 0.877366i \(-0.340701\pi\)
0.479821 + 0.877366i \(0.340701\pi\)
\(810\) −1.46047e15 −0.147172
\(811\) −1.07996e16 −1.08092 −0.540461 0.841369i \(-0.681750\pi\)
−0.540461 + 0.841369i \(0.681750\pi\)
\(812\) −1.20456e15 −0.119749
\(813\) −5.68134e15 −0.560987
\(814\) 3.30717e16 3.24356
\(815\) 8.78341e14 0.0855651
\(816\) −2.87003e15 −0.277709
\(817\) 1.86909e16 1.79642
\(818\) 4.16012e15 0.397158
\(819\) −1.25598e15 −0.119102
\(820\) −1.94807e16 −1.83497
\(821\) −1.10411e16 −1.03306 −0.516529 0.856270i \(-0.672776\pi\)
−0.516529 + 0.856270i \(0.672776\pi\)
\(822\) 2.93815e15 0.273073
\(823\) 1.26104e16 1.16420 0.582101 0.813117i \(-0.302231\pi\)
0.582101 + 0.813117i \(0.302231\pi\)
\(824\) 7.53224e15 0.690756
\(825\) 3.09852e15 0.282266
\(826\) −1.20239e15 −0.108807
\(827\) −7.86172e14 −0.0706704 −0.0353352 0.999376i \(-0.511250\pi\)
−0.0353352 + 0.999376i \(0.511250\pi\)
\(828\) 3.64927e15 0.325866
\(829\) −5.92589e15 −0.525659 −0.262829 0.964842i \(-0.584656\pi\)
−0.262829 + 0.964842i \(0.584656\pi\)
\(830\) −3.67275e15 −0.323640
\(831\) 7.37848e15 0.645894
\(832\) −7.13777e15 −0.620704
\(833\) −3.72192e15 −0.321528
\(834\) 2.13472e16 1.83201
\(835\) 4.73066e13 0.00403317
\(836\) −5.02283e16 −4.25416
\(837\) 7.30896e14 0.0614986
\(838\) −7.90044e15 −0.660403
\(839\) 2.10209e16 1.74566 0.872831 0.488022i \(-0.162282\pi\)
0.872831 + 0.488022i \(0.162282\pi\)
\(840\) −5.61709e15 −0.463420
\(841\) −1.20849e16 −0.990526
\(842\) −1.23184e16 −1.00309
\(843\) −3.88189e15 −0.314045
\(844\) 7.24437e14 0.0582260
\(845\) 6.47126e15 0.516746
\(846\) −7.16547e15 −0.568470
\(847\) −3.37348e15 −0.265901
\(848\) 1.45222e16 1.13725
\(849\) −6.97520e15 −0.542705
\(850\) −4.80948e15 −0.371787
\(851\) 1.01788e16 0.781775
\(852\) 1.45456e16 1.10997
\(853\) −1.98378e16 −1.50409 −0.752046 0.659110i \(-0.770933\pi\)
−0.752046 + 0.659110i \(0.770933\pi\)
\(854\) −1.07190e16 −0.807491
\(855\) 6.18693e15 0.463086
\(856\) 3.49818e16 2.60157
\(857\) 7.96639e15 0.588663 0.294332 0.955703i \(-0.404903\pi\)
0.294332 + 0.955703i \(0.404903\pi\)
\(858\) −9.26318e15 −0.680111
\(859\) −1.59749e16 −1.16540 −0.582701 0.812687i \(-0.698004\pi\)
−0.582701 + 0.812687i \(0.698004\pi\)
\(860\) 2.07619e16 1.50496
\(861\) 6.06324e15 0.436704
\(862\) 4.00274e16 2.86462
\(863\) −4.02926e15 −0.286527 −0.143264 0.989685i \(-0.545760\pi\)
−0.143264 + 0.989685i \(0.545760\pi\)
\(864\) 2.71029e14 0.0191509
\(865\) 3.92467e15 0.275559
\(866\) −5.84413e15 −0.407729
\(867\) 6.01635e15 0.417089
\(868\) 5.70699e15 0.393142
\(869\) 2.37310e15 0.162445
\(870\) 1.09430e15 0.0744353
\(871\) 4.05511e15 0.274096
\(872\) −1.07694e16 −0.723352
\(873\) −2.30837e15 −0.154073
\(874\) −2.33037e16 −1.54565
\(875\) −1.02590e16 −0.676176
\(876\) −1.37020e16 −0.897455
\(877\) 9.76958e15 0.635884 0.317942 0.948110i \(-0.397008\pi\)
0.317942 + 0.948110i \(0.397008\pi\)
\(878\) −2.87301e16 −1.85831
\(879\) −1.17066e16 −0.752476
\(880\) −1.31159e16 −0.837807
\(881\) 1.62583e16 1.03207 0.516033 0.856569i \(-0.327408\pi\)
0.516033 + 0.856569i \(0.327408\pi\)
\(882\) −5.55789e15 −0.350617
\(883\) 1.66061e16 1.04108 0.520540 0.853837i \(-0.325731\pi\)
0.520540 + 0.853837i \(0.325731\pi\)
\(884\) 9.53818e15 0.594262
\(885\) 7.24626e14 0.0448669
\(886\) −8.67733e15 −0.533950
\(887\) −1.91314e16 −1.16995 −0.584975 0.811051i \(-0.698896\pi\)
−0.584975 + 0.811051i \(0.698896\pi\)
\(888\) −2.50478e16 −1.52229
\(889\) −6.94758e15 −0.419637
\(890\) 1.06695e16 0.640472
\(891\) 2.22400e15 0.132680
\(892\) 8.98286e15 0.532608
\(893\) 3.03547e16 1.78872
\(894\) 2.62433e16 1.53696
\(895\) −2.10703e16 −1.22644
\(896\) 1.90971e16 1.10477
\(897\) −2.85101e15 −0.163923
\(898\) −6.88985e15 −0.393722
\(899\) −5.47641e14 −0.0311041
\(900\) −4.76434e15 −0.268949
\(901\) 1.16972e16 0.656287
\(902\) 4.47180e16 2.49372
\(903\) −6.46201e15 −0.358166
\(904\) −5.57710e16 −3.07243
\(905\) −1.22399e16 −0.670209
\(906\) 1.32413e16 0.720651
\(907\) −1.07252e16 −0.580182 −0.290091 0.956999i \(-0.593686\pi\)
−0.290091 + 0.956999i \(0.593686\pi\)
\(908\) −1.67836e15 −0.0902430
\(909\) 5.49792e14 0.0293831
\(910\) 8.90919e15 0.473272
\(911\) −2.82249e16 −1.49032 −0.745162 0.666883i \(-0.767628\pi\)
−0.745162 + 0.666883i \(0.767628\pi\)
\(912\) 1.81556e16 0.952881
\(913\) 5.59284e15 0.291771
\(914\) −7.34833e14 −0.0381052
\(915\) 6.45985e15 0.332971
\(916\) 7.31516e16 3.74799
\(917\) 2.71748e15 0.138399
\(918\) −3.45205e15 −0.174760
\(919\) 1.83551e16 0.923679 0.461839 0.886964i \(-0.347190\pi\)
0.461839 + 0.886964i \(0.347190\pi\)
\(920\) −1.27505e16 −0.637814
\(921\) −6.25361e15 −0.310959
\(922\) −5.43764e16 −2.68776
\(923\) −1.13638e16 −0.558359
\(924\) 1.73654e16 0.848184
\(925\) −1.32890e16 −0.645227
\(926\) −1.46594e16 −0.707549
\(927\) 2.86831e15 0.137622
\(928\) −2.03075e14 −0.00968596
\(929\) −1.10125e16 −0.522155 −0.261078 0.965318i \(-0.584078\pi\)
−0.261078 + 0.965318i \(0.584078\pi\)
\(930\) −5.18456e15 −0.244374
\(931\) 2.35446e16 1.10323
\(932\) −7.57317e13 −0.00352769
\(933\) −8.40884e15 −0.389392
\(934\) 2.49605e16 1.14907
\(935\) −1.05645e16 −0.483486
\(936\) 7.01574e15 0.319195
\(937\) 8.36357e15 0.378289 0.189145 0.981949i \(-0.439429\pi\)
0.189145 + 0.981949i \(0.439429\pi\)
\(938\) −1.14595e16 −0.515289
\(939\) −3.06340e15 −0.136944
\(940\) 3.37181e16 1.49851
\(941\) 2.22942e16 0.985028 0.492514 0.870305i \(-0.336078\pi\)
0.492514 + 0.870305i \(0.336078\pi\)
\(942\) −4.03961e16 −1.77443
\(943\) 1.37633e16 0.601045
\(944\) 2.12642e15 0.0923214
\(945\) −2.13901e15 −0.0923289
\(946\) −4.76590e16 −2.04524
\(947\) −3.78038e16 −1.61291 −0.806457 0.591293i \(-0.798618\pi\)
−0.806457 + 0.591293i \(0.798618\pi\)
\(948\) −3.64891e15 −0.154781
\(949\) 1.07048e16 0.451454
\(950\) 3.04244e16 1.27568
\(951\) 1.98833e16 0.828888
\(952\) −1.32768e16 −0.550290
\(953\) 4.79568e15 0.197624 0.0988119 0.995106i \(-0.468496\pi\)
0.0988119 + 0.995106i \(0.468496\pi\)
\(954\) 1.74672e16 0.715661
\(955\) −1.81887e16 −0.740939
\(956\) −7.11355e16 −2.88116
\(957\) −1.66638e15 −0.0671057
\(958\) −1.17298e16 −0.469657
\(959\) 4.30321e15 0.171313
\(960\) −1.21561e16 −0.481173
\(961\) −2.28139e16 −0.897884
\(962\) 3.97280e16 1.55466
\(963\) 1.33212e16 0.518321
\(964\) −3.59251e15 −0.138987
\(965\) 2.64698e16 1.01824
\(966\) 8.05680e15 0.308169
\(967\) 3.37420e16 1.28329 0.641645 0.767002i \(-0.278252\pi\)
0.641645 + 0.767002i \(0.278252\pi\)
\(968\) 1.88439e16 0.712616
\(969\) 1.46238e16 0.549893
\(970\) 1.63743e16 0.612234
\(971\) 3.58587e16 1.33318 0.666590 0.745425i \(-0.267753\pi\)
0.666590 + 0.745425i \(0.267753\pi\)
\(972\) −3.41966e15 −0.126421
\(973\) 3.12651e16 1.14932
\(974\) 1.37706e16 0.503360
\(975\) 3.72217e15 0.135291
\(976\) 1.89565e16 0.685147
\(977\) −1.20023e16 −0.431366 −0.215683 0.976463i \(-0.569198\pi\)
−0.215683 + 0.976463i \(0.569198\pi\)
\(978\) 3.10020e15 0.110797
\(979\) −1.62474e16 −0.577405
\(980\) 2.61534e16 0.924240
\(981\) −4.10102e15 −0.144116
\(982\) −6.71547e16 −2.34673
\(983\) −4.42687e15 −0.153834 −0.0769170 0.997038i \(-0.524508\pi\)
−0.0769170 + 0.997038i \(0.524508\pi\)
\(984\) −3.38685e16 −1.17037
\(985\) −3.41417e16 −1.17324
\(986\) 2.58653e15 0.0883884
\(987\) −1.04945e16 −0.356631
\(988\) −6.03378e16 −2.03904
\(989\) −1.46684e16 −0.492951
\(990\) −1.57758e16 −0.527226
\(991\) −3.79167e16 −1.26016 −0.630080 0.776530i \(-0.716978\pi\)
−0.630080 + 0.776530i \(0.716978\pi\)
\(992\) 9.62130e14 0.0317994
\(993\) 8.26479e15 0.271651
\(994\) 3.21135e16 1.04969
\(995\) 2.03283e16 0.660807
\(996\) −8.59964e15 −0.278006
\(997\) 5.76003e13 0.00185183 0.000925915 1.00000i \(-0.499705\pi\)
0.000925915 1.00000i \(0.499705\pi\)
\(998\) 4.68943e16 1.49935
\(999\) −9.53830e15 −0.303292
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 3.12.a.a.1.1 1
3.2 odd 2 9.12.a.a.1.1 1
4.3 odd 2 48.12.a.f.1.1 1
5.2 odd 4 75.12.b.a.49.2 2
5.3 odd 4 75.12.b.a.49.1 2
5.4 even 2 75.12.a.a.1.1 1
7.6 odd 2 147.12.a.c.1.1 1
8.3 odd 2 192.12.a.g.1.1 1
8.5 even 2 192.12.a.q.1.1 1
9.2 odd 6 81.12.c.e.28.1 2
9.4 even 3 81.12.c.a.55.1 2
9.5 odd 6 81.12.c.e.55.1 2
9.7 even 3 81.12.c.a.28.1 2
12.11 even 2 144.12.a.l.1.1 1
15.2 even 4 225.12.b.a.199.1 2
15.8 even 4 225.12.b.a.199.2 2
15.14 odd 2 225.12.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.12.a.a.1.1 1 1.1 even 1 trivial
9.12.a.a.1.1 1 3.2 odd 2
48.12.a.f.1.1 1 4.3 odd 2
75.12.a.a.1.1 1 5.4 even 2
75.12.b.a.49.1 2 5.3 odd 4
75.12.b.a.49.2 2 5.2 odd 4
81.12.c.a.28.1 2 9.7 even 3
81.12.c.a.55.1 2 9.4 even 3
81.12.c.e.28.1 2 9.2 odd 6
81.12.c.e.55.1 2 9.5 odd 6
144.12.a.l.1.1 1 12.11 even 2
147.12.a.c.1.1 1 7.6 odd 2
192.12.a.g.1.1 1 8.3 odd 2
192.12.a.q.1.1 1 8.5 even 2
225.12.a.f.1.1 1 15.14 odd 2
225.12.b.a.199.1 2 15.2 even 4
225.12.b.a.199.2 2 15.8 even 4