Properties

Label 126.10.a.e.1.1
Level $126$
Weight $10$
Character 126.1
Self dual yes
Analytic conductor $64.895$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.8945153566\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 126.1

$q$-expansion

\(f(q)\) \(=\) \(q+16.0000 q^{2} +256.000 q^{4} -560.000 q^{5} -2401.00 q^{7} +4096.00 q^{8} +O(q^{10})\) \(q+16.0000 q^{2} +256.000 q^{4} -560.000 q^{5} -2401.00 q^{7} +4096.00 q^{8} -8960.00 q^{10} +54152.0 q^{11} -113172. q^{13} -38416.0 q^{14} +65536.0 q^{16} -6262.00 q^{17} +257078. q^{19} -143360. q^{20} +866432. q^{22} +266000. q^{23} -1.63952e6 q^{25} -1.81075e6 q^{26} -614656. q^{28} -1.57471e6 q^{29} -4.63748e6 q^{31} +1.04858e6 q^{32} -100192. q^{34} +1.34456e6 q^{35} -1.19462e7 q^{37} +4.11325e6 q^{38} -2.29376e6 q^{40} -2.19091e7 q^{41} +2.75206e7 q^{43} +1.38629e7 q^{44} +4.25600e6 q^{46} -5.29278e7 q^{47} +5.76480e6 q^{49} -2.62324e7 q^{50} -2.89720e7 q^{52} -1.62212e7 q^{53} -3.03251e7 q^{55} -9.83450e6 q^{56} -2.51954e7 q^{58} +1.40510e8 q^{59} -2.02964e8 q^{61} -7.41997e7 q^{62} +1.67772e7 q^{64} +6.33763e7 q^{65} +1.53735e8 q^{67} -1.60307e6 q^{68} +2.15130e7 q^{70} -2.79656e8 q^{71} -4.04023e8 q^{73} -1.91140e8 q^{74} +6.58120e7 q^{76} -1.30019e8 q^{77} -1.30690e8 q^{79} -3.67002e7 q^{80} -3.50546e8 q^{82} -4.20134e8 q^{83} +3.50672e6 q^{85} +4.40329e8 q^{86} +2.21807e8 q^{88} +4.69542e8 q^{89} +2.71726e8 q^{91} +6.80960e7 q^{92} -8.46845e8 q^{94} -1.43964e8 q^{95} -8.72502e8 q^{97} +9.22368e7 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) 0 0
\(4\) 256.000 0.500000
\(5\) −560.000 −0.400703 −0.200352 0.979724i \(-0.564208\pi\)
−0.200352 + 0.979724i \(0.564208\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 4096.00 0.353553
\(9\) 0 0
\(10\) −8960.00 −0.283340
\(11\) 54152.0 1.11519 0.557593 0.830114i \(-0.311725\pi\)
0.557593 + 0.830114i \(0.311725\pi\)
\(12\) 0 0
\(13\) −113172. −1.09899 −0.549495 0.835497i \(-0.685180\pi\)
−0.549495 + 0.835497i \(0.685180\pi\)
\(14\) −38416.0 −0.267261
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) −6262.00 −0.0181841 −0.00909207 0.999959i \(-0.502894\pi\)
−0.00909207 + 0.999959i \(0.502894\pi\)
\(18\) 0 0
\(19\) 257078. 0.452557 0.226279 0.974063i \(-0.427344\pi\)
0.226279 + 0.974063i \(0.427344\pi\)
\(20\) −143360. −0.200352
\(21\) 0 0
\(22\) 866432. 0.788556
\(23\) 266000. 0.198201 0.0991006 0.995077i \(-0.468403\pi\)
0.0991006 + 0.995077i \(0.468403\pi\)
\(24\) 0 0
\(25\) −1.63952e6 −0.839437
\(26\) −1.81075e6 −0.777104
\(27\) 0 0
\(28\) −614656. −0.188982
\(29\) −1.57471e6 −0.413438 −0.206719 0.978400i \(-0.566279\pi\)
−0.206719 + 0.978400i \(0.566279\pi\)
\(30\) 0 0
\(31\) −4.63748e6 −0.901893 −0.450946 0.892551i \(-0.648913\pi\)
−0.450946 + 0.892551i \(0.648913\pi\)
\(32\) 1.04858e6 0.176777
\(33\) 0 0
\(34\) −100192. −0.0128581
\(35\) 1.34456e6 0.151452
\(36\) 0 0
\(37\) −1.19462e7 −1.04791 −0.523954 0.851746i \(-0.675544\pi\)
−0.523954 + 0.851746i \(0.675544\pi\)
\(38\) 4.11325e6 0.320006
\(39\) 0 0
\(40\) −2.29376e6 −0.141670
\(41\) −2.19091e7 −1.21087 −0.605435 0.795895i \(-0.707001\pi\)
−0.605435 + 0.795895i \(0.707001\pi\)
\(42\) 0 0
\(43\) 2.75206e7 1.22758 0.613790 0.789469i \(-0.289644\pi\)
0.613790 + 0.789469i \(0.289644\pi\)
\(44\) 1.38629e7 0.557593
\(45\) 0 0
\(46\) 4.25600e6 0.140149
\(47\) −5.29278e7 −1.58214 −0.791068 0.611728i \(-0.790475\pi\)
−0.791068 + 0.611728i \(0.790475\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) −2.62324e7 −0.593571
\(51\) 0 0
\(52\) −2.89720e7 −0.549495
\(53\) −1.62212e7 −0.282385 −0.141193 0.989982i \(-0.545094\pi\)
−0.141193 + 0.989982i \(0.545094\pi\)
\(54\) 0 0
\(55\) −3.03251e7 −0.446859
\(56\) −9.83450e6 −0.133631
\(57\) 0 0
\(58\) −2.51954e7 −0.292345
\(59\) 1.40510e8 1.50964 0.754818 0.655935i \(-0.227725\pi\)
0.754818 + 0.655935i \(0.227725\pi\)
\(60\) 0 0
\(61\) −2.02964e8 −1.87687 −0.938434 0.345458i \(-0.887724\pi\)
−0.938434 + 0.345458i \(0.887724\pi\)
\(62\) −7.41997e7 −0.637734
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) 6.33763e7 0.440369
\(66\) 0 0
\(67\) 1.53735e8 0.932041 0.466020 0.884774i \(-0.345687\pi\)
0.466020 + 0.884774i \(0.345687\pi\)
\(68\) −1.60307e6 −0.00909207
\(69\) 0 0
\(70\) 2.15130e7 0.107092
\(71\) −2.79656e8 −1.30606 −0.653028 0.757334i \(-0.726501\pi\)
−0.653028 + 0.757334i \(0.726501\pi\)
\(72\) 0 0
\(73\) −4.04023e8 −1.66515 −0.832574 0.553913i \(-0.813134\pi\)
−0.832574 + 0.553913i \(0.813134\pi\)
\(74\) −1.91140e8 −0.740983
\(75\) 0 0
\(76\) 6.58120e7 0.226279
\(77\) −1.30019e8 −0.421501
\(78\) 0 0
\(79\) −1.30690e8 −0.377503 −0.188751 0.982025i \(-0.560444\pi\)
−0.188751 + 0.982025i \(0.560444\pi\)
\(80\) −3.67002e7 −0.100176
\(81\) 0 0
\(82\) −3.50546e8 −0.856215
\(83\) −4.20134e8 −0.971709 −0.485855 0.874040i \(-0.661492\pi\)
−0.485855 + 0.874040i \(0.661492\pi\)
\(84\) 0 0
\(85\) 3.50672e6 0.00728645
\(86\) 4.40329e8 0.868030
\(87\) 0 0
\(88\) 2.21807e8 0.394278
\(89\) 4.69542e8 0.793268 0.396634 0.917977i \(-0.370178\pi\)
0.396634 + 0.917977i \(0.370178\pi\)
\(90\) 0 0
\(91\) 2.71726e8 0.415379
\(92\) 6.80960e7 0.0991006
\(93\) 0 0
\(94\) −8.46845e8 −1.11874
\(95\) −1.43964e8 −0.181341
\(96\) 0 0
\(97\) −8.72502e8 −1.00068 −0.500338 0.865830i \(-0.666791\pi\)
−0.500338 + 0.865830i \(0.666791\pi\)
\(98\) 9.22368e7 0.101015
\(99\) 0 0
\(100\) −4.19718e8 −0.419718
\(101\) 1.20901e9 1.15607 0.578037 0.816011i \(-0.303819\pi\)
0.578037 + 0.816011i \(0.303819\pi\)
\(102\) 0 0
\(103\) 6.90563e8 0.604555 0.302277 0.953220i \(-0.402253\pi\)
0.302277 + 0.953220i \(0.402253\pi\)
\(104\) −4.63553e8 −0.388552
\(105\) 0 0
\(106\) −2.59540e8 −0.199677
\(107\) −1.79499e8 −0.132384 −0.0661921 0.997807i \(-0.521085\pi\)
−0.0661921 + 0.997807i \(0.521085\pi\)
\(108\) 0 0
\(109\) −1.60361e9 −1.08813 −0.544063 0.839044i \(-0.683115\pi\)
−0.544063 + 0.839044i \(0.683115\pi\)
\(110\) −4.85202e8 −0.315977
\(111\) 0 0
\(112\) −1.57352e8 −0.0944911
\(113\) −1.42785e9 −0.823815 −0.411908 0.911226i \(-0.635137\pi\)
−0.411908 + 0.911226i \(0.635137\pi\)
\(114\) 0 0
\(115\) −1.48960e8 −0.0794199
\(116\) −4.03127e8 −0.206719
\(117\) 0 0
\(118\) 2.24815e9 1.06747
\(119\) 1.50351e7 0.00687296
\(120\) 0 0
\(121\) 5.74491e8 0.243640
\(122\) −3.24742e9 −1.32715
\(123\) 0 0
\(124\) −1.18720e9 −0.450946
\(125\) 2.01188e9 0.737069
\(126\) 0 0
\(127\) −2.35873e9 −0.804565 −0.402282 0.915516i \(-0.631783\pi\)
−0.402282 + 0.915516i \(0.631783\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 0 0
\(130\) 1.01402e9 0.311388
\(131\) −6.01665e8 −0.178498 −0.0892492 0.996009i \(-0.528447\pi\)
−0.0892492 + 0.996009i \(0.528447\pi\)
\(132\) 0 0
\(133\) −6.17244e8 −0.171051
\(134\) 2.45975e9 0.659052
\(135\) 0 0
\(136\) −2.56492e7 −0.00642907
\(137\) 5.16009e9 1.25145 0.625726 0.780043i \(-0.284803\pi\)
0.625726 + 0.780043i \(0.284803\pi\)
\(138\) 0 0
\(139\) −7.14356e9 −1.62311 −0.811556 0.584275i \(-0.801379\pi\)
−0.811556 + 0.584275i \(0.801379\pi\)
\(140\) 3.44207e8 0.0757258
\(141\) 0 0
\(142\) −4.47449e9 −0.923520
\(143\) −6.12849e9 −1.22558
\(144\) 0 0
\(145\) 8.81840e8 0.165666
\(146\) −6.46437e9 −1.17744
\(147\) 0 0
\(148\) −3.05824e9 −0.523954
\(149\) −9.10424e9 −1.51323 −0.756616 0.653859i \(-0.773149\pi\)
−0.756616 + 0.653859i \(0.773149\pi\)
\(150\) 0 0
\(151\) −2.89432e8 −0.0453054 −0.0226527 0.999743i \(-0.507211\pi\)
−0.0226527 + 0.999743i \(0.507211\pi\)
\(152\) 1.05299e9 0.160003
\(153\) 0 0
\(154\) −2.08030e9 −0.298046
\(155\) 2.59699e9 0.361391
\(156\) 0 0
\(157\) 1.39068e10 1.82675 0.913373 0.407124i \(-0.133468\pi\)
0.913373 + 0.407124i \(0.133468\pi\)
\(158\) −2.09104e9 −0.266935
\(159\) 0 0
\(160\) −5.87203e8 −0.0708350
\(161\) −6.38666e8 −0.0749130
\(162\) 0 0
\(163\) 1.66232e10 1.84447 0.922235 0.386629i \(-0.126361\pi\)
0.922235 + 0.386629i \(0.126361\pi\)
\(164\) −5.60874e9 −0.605435
\(165\) 0 0
\(166\) −6.72214e9 −0.687102
\(167\) 1.58019e10 1.57212 0.786061 0.618149i \(-0.212117\pi\)
0.786061 + 0.618149i \(0.212117\pi\)
\(168\) 0 0
\(169\) 2.20340e9 0.207780
\(170\) 5.61075e7 0.00515230
\(171\) 0 0
\(172\) 7.04527e9 0.613790
\(173\) −3.23125e9 −0.274260 −0.137130 0.990553i \(-0.543788\pi\)
−0.137130 + 0.990553i \(0.543788\pi\)
\(174\) 0 0
\(175\) 3.93650e9 0.317277
\(176\) 3.54891e9 0.278797
\(177\) 0 0
\(178\) 7.51268e9 0.560925
\(179\) 2.41408e10 1.75757 0.878785 0.477218i \(-0.158355\pi\)
0.878785 + 0.477218i \(0.158355\pi\)
\(180\) 0 0
\(181\) −3.89332e9 −0.269629 −0.134814 0.990871i \(-0.543044\pi\)
−0.134814 + 0.990871i \(0.543044\pi\)
\(182\) 4.34762e9 0.293718
\(183\) 0 0
\(184\) 1.08954e9 0.0700747
\(185\) 6.68989e9 0.419901
\(186\) 0 0
\(187\) −3.39100e8 −0.0202787
\(188\) −1.35495e10 −0.791068
\(189\) 0 0
\(190\) −2.30342e9 −0.128228
\(191\) 2.58988e10 1.40809 0.704043 0.710157i \(-0.251376\pi\)
0.704043 + 0.710157i \(0.251376\pi\)
\(192\) 0 0
\(193\) 1.59367e10 0.826783 0.413391 0.910553i \(-0.364344\pi\)
0.413391 + 0.910553i \(0.364344\pi\)
\(194\) −1.39600e10 −0.707585
\(195\) 0 0
\(196\) 1.47579e9 0.0714286
\(197\) 3.34685e9 0.158321 0.0791604 0.996862i \(-0.474776\pi\)
0.0791604 + 0.996862i \(0.474776\pi\)
\(198\) 0 0
\(199\) −1.34261e10 −0.606891 −0.303445 0.952849i \(-0.598137\pi\)
−0.303445 + 0.952849i \(0.598137\pi\)
\(200\) −6.71549e9 −0.296786
\(201\) 0 0
\(202\) 1.93442e10 0.817467
\(203\) 3.78089e9 0.156265
\(204\) 0 0
\(205\) 1.22691e10 0.485200
\(206\) 1.10490e10 0.427485
\(207\) 0 0
\(208\) −7.41684e9 −0.274748
\(209\) 1.39213e10 0.504686
\(210\) 0 0
\(211\) 3.01702e10 1.04787 0.523935 0.851759i \(-0.324464\pi\)
0.523935 + 0.851759i \(0.324464\pi\)
\(212\) −4.15263e9 −0.141193
\(213\) 0 0
\(214\) −2.87199e9 −0.0936097
\(215\) −1.54115e10 −0.491895
\(216\) 0 0
\(217\) 1.11346e10 0.340883
\(218\) −2.56577e10 −0.769421
\(219\) 0 0
\(220\) −7.76323e9 −0.223429
\(221\) 7.08683e8 0.0199842
\(222\) 0 0
\(223\) 5.35030e10 1.44879 0.724396 0.689384i \(-0.242119\pi\)
0.724396 + 0.689384i \(0.242119\pi\)
\(224\) −2.51763e9 −0.0668153
\(225\) 0 0
\(226\) −2.28456e10 −0.582525
\(227\) 4.02704e10 1.00663 0.503315 0.864103i \(-0.332114\pi\)
0.503315 + 0.864103i \(0.332114\pi\)
\(228\) 0 0
\(229\) 1.90247e10 0.457150 0.228575 0.973526i \(-0.426593\pi\)
0.228575 + 0.973526i \(0.426593\pi\)
\(230\) −2.38336e9 −0.0561584
\(231\) 0 0
\(232\) −6.45003e9 −0.146173
\(233\) 3.67748e10 0.817426 0.408713 0.912663i \(-0.365978\pi\)
0.408713 + 0.912663i \(0.365978\pi\)
\(234\) 0 0
\(235\) 2.96396e10 0.633967
\(236\) 3.59705e10 0.754818
\(237\) 0 0
\(238\) 2.40561e8 0.00485992
\(239\) −6.56110e9 −0.130073 −0.0650363 0.997883i \(-0.520716\pi\)
−0.0650363 + 0.997883i \(0.520716\pi\)
\(240\) 0 0
\(241\) −8.96818e10 −1.71249 −0.856244 0.516572i \(-0.827208\pi\)
−0.856244 + 0.516572i \(0.827208\pi\)
\(242\) 9.19186e9 0.172280
\(243\) 0 0
\(244\) −5.19587e10 −0.938434
\(245\) −3.22829e9 −0.0572433
\(246\) 0 0
\(247\) −2.90940e10 −0.497356
\(248\) −1.89951e10 −0.318867
\(249\) 0 0
\(250\) 3.21901e10 0.521186
\(251\) −5.33703e9 −0.0848727 −0.0424363 0.999099i \(-0.513512\pi\)
−0.0424363 + 0.999099i \(0.513512\pi\)
\(252\) 0 0
\(253\) 1.44044e10 0.221031
\(254\) −3.77396e10 −0.568913
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) 8.35575e10 1.19478 0.597388 0.801952i \(-0.296205\pi\)
0.597388 + 0.801952i \(0.296205\pi\)
\(258\) 0 0
\(259\) 2.86829e10 0.396072
\(260\) 1.62243e10 0.220185
\(261\) 0 0
\(262\) −9.62665e9 −0.126217
\(263\) −1.08635e11 −1.40013 −0.700065 0.714079i \(-0.746846\pi\)
−0.700065 + 0.714079i \(0.746846\pi\)
\(264\) 0 0
\(265\) 9.08388e9 0.113153
\(266\) −9.87591e9 −0.120951
\(267\) 0 0
\(268\) 3.93561e10 0.466020
\(269\) −1.41401e11 −1.64652 −0.823258 0.567668i \(-0.807846\pi\)
−0.823258 + 0.567668i \(0.807846\pi\)
\(270\) 0 0
\(271\) −9.08353e10 −1.02304 −0.511520 0.859271i \(-0.670917\pi\)
−0.511520 + 0.859271i \(0.670917\pi\)
\(272\) −4.10386e8 −0.00454604
\(273\) 0 0
\(274\) 8.25614e10 0.884911
\(275\) −8.87836e10 −0.936128
\(276\) 0 0
\(277\) −2.65075e10 −0.270527 −0.135263 0.990810i \(-0.543188\pi\)
−0.135263 + 0.990810i \(0.543188\pi\)
\(278\) −1.14297e11 −1.14771
\(279\) 0 0
\(280\) 5.50732e9 0.0535462
\(281\) 1.86968e11 1.78891 0.894455 0.447158i \(-0.147564\pi\)
0.894455 + 0.447158i \(0.147564\pi\)
\(282\) 0 0
\(283\) −5.33413e9 −0.0494338 −0.0247169 0.999694i \(-0.507868\pi\)
−0.0247169 + 0.999694i \(0.507868\pi\)
\(284\) −7.15919e10 −0.653028
\(285\) 0 0
\(286\) −9.80558e10 −0.866615
\(287\) 5.26038e10 0.457666
\(288\) 0 0
\(289\) −1.18549e11 −0.999669
\(290\) 1.41094e10 0.117144
\(291\) 0 0
\(292\) −1.03430e11 −0.832574
\(293\) −7.65433e10 −0.606741 −0.303370 0.952873i \(-0.598112\pi\)
−0.303370 + 0.952873i \(0.598112\pi\)
\(294\) 0 0
\(295\) −7.86854e10 −0.604916
\(296\) −4.89318e10 −0.370492
\(297\) 0 0
\(298\) −1.45668e11 −1.07002
\(299\) −3.01038e10 −0.217821
\(300\) 0 0
\(301\) −6.60769e10 −0.463982
\(302\) −4.63091e9 −0.0320358
\(303\) 0 0
\(304\) 1.68479e10 0.113139
\(305\) 1.13660e11 0.752068
\(306\) 0 0
\(307\) −7.51944e10 −0.483128 −0.241564 0.970385i \(-0.577660\pi\)
−0.241564 + 0.970385i \(0.577660\pi\)
\(308\) −3.32849e10 −0.210750
\(309\) 0 0
\(310\) 4.15519e10 0.255542
\(311\) −2.15134e11 −1.30403 −0.652014 0.758207i \(-0.726076\pi\)
−0.652014 + 0.758207i \(0.726076\pi\)
\(312\) 0 0
\(313\) 9.59075e10 0.564811 0.282405 0.959295i \(-0.408868\pi\)
0.282405 + 0.959295i \(0.408868\pi\)
\(314\) 2.22508e11 1.29170
\(315\) 0 0
\(316\) −3.34566e10 −0.188751
\(317\) −1.70586e11 −0.948807 −0.474403 0.880308i \(-0.657336\pi\)
−0.474403 + 0.880308i \(0.657336\pi\)
\(318\) 0 0
\(319\) −8.52739e10 −0.461061
\(320\) −9.39524e9 −0.0500879
\(321\) 0 0
\(322\) −1.02187e10 −0.0529715
\(323\) −1.60982e9 −0.00822937
\(324\) 0 0
\(325\) 1.85548e11 0.922533
\(326\) 2.65972e11 1.30424
\(327\) 0 0
\(328\) −8.97398e10 −0.428107
\(329\) 1.27080e11 0.597991
\(330\) 0 0
\(331\) −1.23992e11 −0.567762 −0.283881 0.958859i \(-0.591622\pi\)
−0.283881 + 0.958859i \(0.591622\pi\)
\(332\) −1.07554e11 −0.485855
\(333\) 0 0
\(334\) 2.52831e11 1.11166
\(335\) −8.60914e10 −0.373472
\(336\) 0 0
\(337\) −7.29335e10 −0.308030 −0.154015 0.988069i \(-0.549220\pi\)
−0.154015 + 0.988069i \(0.549220\pi\)
\(338\) 3.52544e10 0.146923
\(339\) 0 0
\(340\) 8.97720e8 0.00364322
\(341\) −2.51129e11 −1.00578
\(342\) 0 0
\(343\) −1.38413e10 −0.0539949
\(344\) 1.12724e11 0.434015
\(345\) 0 0
\(346\) −5.17000e10 −0.193931
\(347\) 1.55720e11 0.576584 0.288292 0.957542i \(-0.406913\pi\)
0.288292 + 0.957542i \(0.406913\pi\)
\(348\) 0 0
\(349\) 1.08728e11 0.392310 0.196155 0.980573i \(-0.437154\pi\)
0.196155 + 0.980573i \(0.437154\pi\)
\(350\) 6.29840e10 0.224349
\(351\) 0 0
\(352\) 5.67825e10 0.197139
\(353\) −3.25585e11 −1.11604 −0.558018 0.829829i \(-0.688438\pi\)
−0.558018 + 0.829829i \(0.688438\pi\)
\(354\) 0 0
\(355\) 1.56607e11 0.523341
\(356\) 1.20203e11 0.396634
\(357\) 0 0
\(358\) 3.86252e11 1.24279
\(359\) 2.27550e11 0.723022 0.361511 0.932368i \(-0.382261\pi\)
0.361511 + 0.932368i \(0.382261\pi\)
\(360\) 0 0
\(361\) −2.56599e11 −0.795192
\(362\) −6.22931e10 −0.190656
\(363\) 0 0
\(364\) 6.95618e10 0.207690
\(365\) 2.26253e11 0.667231
\(366\) 0 0
\(367\) −4.21993e11 −1.21425 −0.607125 0.794607i \(-0.707677\pi\)
−0.607125 + 0.794607i \(0.707677\pi\)
\(368\) 1.74326e10 0.0495503
\(369\) 0 0
\(370\) 1.07038e11 0.296915
\(371\) 3.89472e10 0.106732
\(372\) 0 0
\(373\) 3.83283e11 1.02525 0.512625 0.858613i \(-0.328673\pi\)
0.512625 + 0.858613i \(0.328673\pi\)
\(374\) −5.42560e9 −0.0143392
\(375\) 0 0
\(376\) −2.16792e11 −0.559370
\(377\) 1.78214e11 0.454365
\(378\) 0 0
\(379\) −1.21462e11 −0.302386 −0.151193 0.988504i \(-0.548312\pi\)
−0.151193 + 0.988504i \(0.548312\pi\)
\(380\) −3.68547e10 −0.0906706
\(381\) 0 0
\(382\) 4.14381e11 0.995667
\(383\) 3.97721e11 0.944461 0.472230 0.881475i \(-0.343449\pi\)
0.472230 + 0.881475i \(0.343449\pi\)
\(384\) 0 0
\(385\) 7.28106e10 0.168897
\(386\) 2.54988e11 0.584624
\(387\) 0 0
\(388\) −2.23360e11 −0.500338
\(389\) −6.75462e10 −0.149564 −0.0747821 0.997200i \(-0.523826\pi\)
−0.0747821 + 0.997200i \(0.523826\pi\)
\(390\) 0 0
\(391\) −1.66569e9 −0.00360412
\(392\) 2.36126e10 0.0505076
\(393\) 0 0
\(394\) 5.35495e10 0.111950
\(395\) 7.31863e10 0.151267
\(396\) 0 0
\(397\) 1.24656e11 0.251857 0.125929 0.992039i \(-0.459809\pi\)
0.125929 + 0.992039i \(0.459809\pi\)
\(398\) −2.14817e11 −0.429136
\(399\) 0 0
\(400\) −1.07448e11 −0.209859
\(401\) −3.51196e11 −0.678265 −0.339133 0.940739i \(-0.610134\pi\)
−0.339133 + 0.940739i \(0.610134\pi\)
\(402\) 0 0
\(403\) 5.24833e11 0.991171
\(404\) 3.09508e11 0.578037
\(405\) 0 0
\(406\) 6.04942e10 0.110496
\(407\) −6.46913e11 −1.16861
\(408\) 0 0
\(409\) −3.81956e10 −0.0674930 −0.0337465 0.999430i \(-0.510744\pi\)
−0.0337465 + 0.999430i \(0.510744\pi\)
\(410\) 1.96306e11 0.343088
\(411\) 0 0
\(412\) 1.76784e11 0.302277
\(413\) −3.37364e11 −0.570588
\(414\) 0 0
\(415\) 2.35275e11 0.389367
\(416\) −1.18669e11 −0.194276
\(417\) 0 0
\(418\) 2.22741e11 0.356867
\(419\) −2.15268e11 −0.341205 −0.170603 0.985340i \(-0.554571\pi\)
−0.170603 + 0.985340i \(0.554571\pi\)
\(420\) 0 0
\(421\) 1.19933e12 1.86066 0.930332 0.366718i \(-0.119519\pi\)
0.930332 + 0.366718i \(0.119519\pi\)
\(422\) 4.82723e11 0.740955
\(423\) 0 0
\(424\) −6.64421e10 −0.0998383
\(425\) 1.02667e10 0.0152644
\(426\) 0 0
\(427\) 4.87316e11 0.709390
\(428\) −4.59518e10 −0.0661921
\(429\) 0 0
\(430\) −2.46585e11 −0.347823
\(431\) −7.91117e11 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(432\) 0 0
\(433\) −1.15451e12 −1.57834 −0.789170 0.614174i \(-0.789489\pi\)
−0.789170 + 0.614174i \(0.789489\pi\)
\(434\) 1.78154e11 0.241041
\(435\) 0 0
\(436\) −4.10524e11 −0.544063
\(437\) 6.83827e10 0.0896974
\(438\) 0 0
\(439\) −2.12728e11 −0.273360 −0.136680 0.990615i \(-0.543643\pi\)
−0.136680 + 0.990615i \(0.543643\pi\)
\(440\) −1.24212e11 −0.157988
\(441\) 0 0
\(442\) 1.13389e10 0.0141310
\(443\) 6.48300e10 0.0799759 0.0399880 0.999200i \(-0.487268\pi\)
0.0399880 + 0.999200i \(0.487268\pi\)
\(444\) 0 0
\(445\) −2.62944e11 −0.317865
\(446\) 8.56047e11 1.02445
\(447\) 0 0
\(448\) −4.02821e10 −0.0472456
\(449\) 1.08031e12 1.25441 0.627207 0.778853i \(-0.284198\pi\)
0.627207 + 0.778853i \(0.284198\pi\)
\(450\) 0 0
\(451\) −1.18642e12 −1.35035
\(452\) −3.65530e11 −0.411908
\(453\) 0 0
\(454\) 6.44326e11 0.711794
\(455\) −1.52167e11 −0.166444
\(456\) 0 0
\(457\) −6.46725e10 −0.0693581 −0.0346790 0.999399i \(-0.511041\pi\)
−0.0346790 + 0.999399i \(0.511041\pi\)
\(458\) 3.04395e11 0.323254
\(459\) 0 0
\(460\) −3.81338e10 −0.0397100
\(461\) −4.29254e11 −0.442649 −0.221325 0.975200i \(-0.571038\pi\)
−0.221325 + 0.975200i \(0.571038\pi\)
\(462\) 0 0
\(463\) 1.61883e12 1.63715 0.818574 0.574401i \(-0.194765\pi\)
0.818574 + 0.574401i \(0.194765\pi\)
\(464\) −1.03200e11 −0.103360
\(465\) 0 0
\(466\) 5.88396e11 0.578007
\(467\) −3.27321e11 −0.318455 −0.159228 0.987242i \(-0.550900\pi\)
−0.159228 + 0.987242i \(0.550900\pi\)
\(468\) 0 0
\(469\) −3.69117e11 −0.352278
\(470\) 4.74233e11 0.448283
\(471\) 0 0
\(472\) 5.75527e11 0.533737
\(473\) 1.49030e12 1.36898
\(474\) 0 0
\(475\) −4.21486e11 −0.379893
\(476\) 3.84898e9 0.00343648
\(477\) 0 0
\(478\) −1.04978e11 −0.0919752
\(479\) −2.84811e11 −0.247199 −0.123600 0.992332i \(-0.539444\pi\)
−0.123600 + 0.992332i \(0.539444\pi\)
\(480\) 0 0
\(481\) 1.35198e12 1.15164
\(482\) −1.43491e12 −1.21091
\(483\) 0 0
\(484\) 1.47070e11 0.121820
\(485\) 4.88601e11 0.400974
\(486\) 0 0
\(487\) −7.14776e11 −0.575824 −0.287912 0.957657i \(-0.592961\pi\)
−0.287912 + 0.957657i \(0.592961\pi\)
\(488\) −8.31339e11 −0.663573
\(489\) 0 0
\(490\) −5.16526e10 −0.0404772
\(491\) −1.01506e12 −0.788176 −0.394088 0.919073i \(-0.628939\pi\)
−0.394088 + 0.919073i \(0.628939\pi\)
\(492\) 0 0
\(493\) 9.86086e9 0.00751802
\(494\) −4.65505e11 −0.351684
\(495\) 0 0
\(496\) −3.03922e11 −0.225473
\(497\) 6.71454e11 0.493642
\(498\) 0 0
\(499\) 1.33412e12 0.963260 0.481630 0.876375i \(-0.340045\pi\)
0.481630 + 0.876375i \(0.340045\pi\)
\(500\) 5.15042e11 0.368534
\(501\) 0 0
\(502\) −8.53925e10 −0.0600140
\(503\) 5.68445e11 0.395943 0.197971 0.980208i \(-0.436565\pi\)
0.197971 + 0.980208i \(0.436565\pi\)
\(504\) 0 0
\(505\) −6.77048e11 −0.463243
\(506\) 2.30471e11 0.156293
\(507\) 0 0
\(508\) −6.03834e11 −0.402282
\(509\) −3.57173e11 −0.235857 −0.117928 0.993022i \(-0.537625\pi\)
−0.117928 + 0.993022i \(0.537625\pi\)
\(510\) 0 0
\(511\) 9.70059e11 0.629367
\(512\) 6.87195e10 0.0441942
\(513\) 0 0
\(514\) 1.33692e12 0.844834
\(515\) −3.86715e11 −0.242247
\(516\) 0 0
\(517\) −2.86615e12 −1.76438
\(518\) 4.58927e11 0.280065
\(519\) 0 0
\(520\) 2.59589e11 0.155694
\(521\) −2.17972e12 −1.29608 −0.648040 0.761606i \(-0.724411\pi\)
−0.648040 + 0.761606i \(0.724411\pi\)
\(522\) 0 0
\(523\) −1.57081e12 −0.918048 −0.459024 0.888424i \(-0.651801\pi\)
−0.459024 + 0.888424i \(0.651801\pi\)
\(524\) −1.54026e11 −0.0892492
\(525\) 0 0
\(526\) −1.73816e12 −0.990042
\(527\) 2.90399e10 0.0164001
\(528\) 0 0
\(529\) −1.73040e12 −0.960716
\(530\) 1.45342e11 0.0800111
\(531\) 0 0
\(532\) −1.58015e11 −0.0855253
\(533\) 2.47950e12 1.33074
\(534\) 0 0
\(535\) 1.00520e11 0.0530468
\(536\) 6.29697e11 0.329526
\(537\) 0 0
\(538\) −2.26241e12 −1.16426
\(539\) 3.12176e11 0.159312
\(540\) 0 0
\(541\) 2.24544e12 1.12697 0.563486 0.826126i \(-0.309460\pi\)
0.563486 + 0.826126i \(0.309460\pi\)
\(542\) −1.45336e12 −0.723399
\(543\) 0 0
\(544\) −6.56618e9 −0.00321453
\(545\) 8.98021e11 0.436016
\(546\) 0 0
\(547\) −3.86062e11 −0.184380 −0.0921899 0.995741i \(-0.529387\pi\)
−0.0921899 + 0.995741i \(0.529387\pi\)
\(548\) 1.32098e12 0.625726
\(549\) 0 0
\(550\) −1.42054e12 −0.661943
\(551\) −4.04824e11 −0.187105
\(552\) 0 0
\(553\) 3.13786e11 0.142683
\(554\) −4.24120e11 −0.191291
\(555\) 0 0
\(556\) −1.82875e12 −0.811556
\(557\) 7.95102e11 0.350005 0.175003 0.984568i \(-0.444007\pi\)
0.175003 + 0.984568i \(0.444007\pi\)
\(558\) 0 0
\(559\) −3.11456e12 −1.34910
\(560\) 8.81171e10 0.0378629
\(561\) 0 0
\(562\) 2.99149e12 1.26495
\(563\) 2.13667e12 0.896292 0.448146 0.893960i \(-0.352084\pi\)
0.448146 + 0.893960i \(0.352084\pi\)
\(564\) 0 0
\(565\) 7.99596e11 0.330105
\(566\) −8.53460e10 −0.0349550
\(567\) 0 0
\(568\) −1.14547e12 −0.461760
\(569\) −2.17461e12 −0.869714 −0.434857 0.900500i \(-0.643201\pi\)
−0.434857 + 0.900500i \(0.643201\pi\)
\(570\) 0 0
\(571\) 9.95075e11 0.391736 0.195868 0.980630i \(-0.437248\pi\)
0.195868 + 0.980630i \(0.437248\pi\)
\(572\) −1.56889e12 −0.612790
\(573\) 0 0
\(574\) 8.41661e11 0.323619
\(575\) −4.36114e11 −0.166377
\(576\) 0 0
\(577\) 4.30588e12 1.61723 0.808614 0.588340i \(-0.200218\pi\)
0.808614 + 0.588340i \(0.200218\pi\)
\(578\) −1.89678e12 −0.706873
\(579\) 0 0
\(580\) 2.25751e11 0.0828331
\(581\) 1.00874e12 0.367272
\(582\) 0 0
\(583\) −8.78412e11 −0.314912
\(584\) −1.65488e12 −0.588719
\(585\) 0 0
\(586\) −1.22469e12 −0.429030
\(587\) −5.05762e12 −1.75822 −0.879112 0.476615i \(-0.841864\pi\)
−0.879112 + 0.476615i \(0.841864\pi\)
\(588\) 0 0
\(589\) −1.19220e12 −0.408158
\(590\) −1.25897e12 −0.427740
\(591\) 0 0
\(592\) −7.82909e11 −0.261977
\(593\) 2.80300e12 0.930844 0.465422 0.885089i \(-0.345903\pi\)
0.465422 + 0.885089i \(0.345903\pi\)
\(594\) 0 0
\(595\) −8.41963e9 −0.00275402
\(596\) −2.33069e12 −0.756616
\(597\) 0 0
\(598\) −4.81660e11 −0.154023
\(599\) −3.20907e12 −1.01849 −0.509247 0.860620i \(-0.670076\pi\)
−0.509247 + 0.860620i \(0.670076\pi\)
\(600\) 0 0
\(601\) 7.49502e11 0.234335 0.117168 0.993112i \(-0.462619\pi\)
0.117168 + 0.993112i \(0.462619\pi\)
\(602\) −1.05723e12 −0.328084
\(603\) 0 0
\(604\) −7.40946e10 −0.0226527
\(605\) −3.21715e11 −0.0976275
\(606\) 0 0
\(607\) −1.74097e12 −0.520526 −0.260263 0.965538i \(-0.583809\pi\)
−0.260263 + 0.965538i \(0.583809\pi\)
\(608\) 2.69566e11 0.0800016
\(609\) 0 0
\(610\) 1.81855e12 0.531792
\(611\) 5.98995e12 1.73875
\(612\) 0 0
\(613\) −4.03977e12 −1.15554 −0.577770 0.816200i \(-0.696077\pi\)
−0.577770 + 0.816200i \(0.696077\pi\)
\(614\) −1.20311e12 −0.341623
\(615\) 0 0
\(616\) −5.32558e11 −0.149023
\(617\) 2.93367e12 0.814945 0.407472 0.913218i \(-0.366410\pi\)
0.407472 + 0.913218i \(0.366410\pi\)
\(618\) 0 0
\(619\) −5.77691e12 −1.58157 −0.790784 0.612095i \(-0.790327\pi\)
−0.790784 + 0.612095i \(0.790327\pi\)
\(620\) 6.64830e11 0.180696
\(621\) 0 0
\(622\) −3.44214e12 −0.922088
\(623\) −1.12737e12 −0.299827
\(624\) 0 0
\(625\) 2.07554e12 0.544091
\(626\) 1.53452e12 0.399381
\(627\) 0 0
\(628\) 3.56014e12 0.913373
\(629\) 7.48073e10 0.0190553
\(630\) 0 0
\(631\) 3.99985e12 1.00441 0.502206 0.864748i \(-0.332522\pi\)
0.502206 + 0.864748i \(0.332522\pi\)
\(632\) −5.35305e11 −0.133467
\(633\) 0 0
\(634\) −2.72938e12 −0.670908
\(635\) 1.32089e12 0.322392
\(636\) 0 0
\(637\) −6.52414e11 −0.156999
\(638\) −1.36438e12 −0.326019
\(639\) 0 0
\(640\) −1.50324e11 −0.0354175
\(641\) −4.68328e12 −1.09569 −0.547846 0.836579i \(-0.684552\pi\)
−0.547846 + 0.836579i \(0.684552\pi\)
\(642\) 0 0
\(643\) −1.54877e12 −0.357304 −0.178652 0.983912i \(-0.557174\pi\)
−0.178652 + 0.983912i \(0.557174\pi\)
\(644\) −1.63498e11 −0.0374565
\(645\) 0 0
\(646\) −2.57572e10 −0.00581904
\(647\) 8.14493e12 1.82733 0.913667 0.406463i \(-0.133238\pi\)
0.913667 + 0.406463i \(0.133238\pi\)
\(648\) 0 0
\(649\) 7.60888e12 1.68352
\(650\) 2.96877e12 0.652329
\(651\) 0 0
\(652\) 4.25555e12 0.922235
\(653\) 2.88925e12 0.621836 0.310918 0.950437i \(-0.399364\pi\)
0.310918 + 0.950437i \(0.399364\pi\)
\(654\) 0 0
\(655\) 3.36933e11 0.0715249
\(656\) −1.43584e12 −0.302718
\(657\) 0 0
\(658\) 2.03328e12 0.422844
\(659\) −5.20255e12 −1.07456 −0.537281 0.843403i \(-0.680549\pi\)
−0.537281 + 0.843403i \(0.680549\pi\)
\(660\) 0 0
\(661\) 2.88973e12 0.588777 0.294388 0.955686i \(-0.404884\pi\)
0.294388 + 0.955686i \(0.404884\pi\)
\(662\) −1.98387e12 −0.401469
\(663\) 0 0
\(664\) −1.72087e12 −0.343551
\(665\) 3.45657e11 0.0685406
\(666\) 0 0
\(667\) −4.18874e11 −0.0819440
\(668\) 4.04530e12 0.786061
\(669\) 0 0
\(670\) −1.37746e12 −0.264085
\(671\) −1.09909e13 −2.09306
\(672\) 0 0
\(673\) 9.46362e12 1.77824 0.889119 0.457677i \(-0.151318\pi\)
0.889119 + 0.457677i \(0.151318\pi\)
\(674\) −1.16694e12 −0.217810
\(675\) 0 0
\(676\) 5.64071e11 0.103890
\(677\) 6.52268e12 1.19338 0.596688 0.802474i \(-0.296483\pi\)
0.596688 + 0.802474i \(0.296483\pi\)
\(678\) 0 0
\(679\) 2.09488e12 0.378220
\(680\) 1.43635e10 0.00257615
\(681\) 0 0
\(682\) −4.01806e12 −0.711193
\(683\) −5.37240e12 −0.944660 −0.472330 0.881422i \(-0.656587\pi\)
−0.472330 + 0.881422i \(0.656587\pi\)
\(684\) 0 0
\(685\) −2.88965e12 −0.501461
\(686\) −2.21461e11 −0.0381802
\(687\) 0 0
\(688\) 1.80359e12 0.306895
\(689\) 1.83579e12 0.310339
\(690\) 0 0
\(691\) 2.01563e12 0.336325 0.168163 0.985759i \(-0.446217\pi\)
0.168163 + 0.985759i \(0.446217\pi\)
\(692\) −8.27200e11 −0.137130
\(693\) 0 0
\(694\) 2.49153e12 0.407707
\(695\) 4.00040e12 0.650386
\(696\) 0 0
\(697\) 1.37195e11 0.0220186
\(698\) 1.73966e12 0.277405
\(699\) 0 0
\(700\) 1.00774e12 0.158639
\(701\) 1.06523e13 1.66614 0.833068 0.553171i \(-0.186582\pi\)
0.833068 + 0.553171i \(0.186582\pi\)
\(702\) 0 0
\(703\) −3.07111e12 −0.474239
\(704\) 9.08520e11 0.139398
\(705\) 0 0
\(706\) −5.20936e12 −0.789157
\(707\) −2.90284e12 −0.436955
\(708\) 0 0
\(709\) 3.46187e12 0.514520 0.257260 0.966342i \(-0.417180\pi\)
0.257260 + 0.966342i \(0.417180\pi\)
\(710\) 2.50572e12 0.370058
\(711\) 0 0
\(712\) 1.92325e12 0.280462
\(713\) −1.23357e12 −0.178756
\(714\) 0 0
\(715\) 3.43195e12 0.491094
\(716\) 6.18004e12 0.878785
\(717\) 0 0
\(718\) 3.64080e12 0.511253
\(719\) 9.62025e12 1.34248 0.671238 0.741242i \(-0.265763\pi\)
0.671238 + 0.741242i \(0.265763\pi\)
\(720\) 0 0
\(721\) −1.65804e12 −0.228500
\(722\) −4.10558e12 −0.562285
\(723\) 0 0
\(724\) −9.96689e11 −0.134814
\(725\) 2.58178e12 0.347055
\(726\) 0 0
\(727\) −3.13479e12 −0.416202 −0.208101 0.978107i \(-0.566728\pi\)
−0.208101 + 0.978107i \(0.566728\pi\)
\(728\) 1.11299e12 0.146859
\(729\) 0 0
\(730\) 3.62004e12 0.471803
\(731\) −1.72334e11 −0.0223225
\(732\) 0 0
\(733\) 6.47775e12 0.828812 0.414406 0.910092i \(-0.363989\pi\)
0.414406 + 0.910092i \(0.363989\pi\)
\(734\) −6.75189e12 −0.858604
\(735\) 0 0
\(736\) 2.78921e11 0.0350374
\(737\) 8.32503e12 1.03940
\(738\) 0 0
\(739\) −1.12510e12 −0.138769 −0.0693843 0.997590i \(-0.522103\pi\)
−0.0693843 + 0.997590i \(0.522103\pi\)
\(740\) 1.71261e12 0.209950
\(741\) 0 0
\(742\) 6.23154e11 0.0754707
\(743\) −1.65266e12 −0.198945 −0.0994725 0.995040i \(-0.531716\pi\)
−0.0994725 + 0.995040i \(0.531716\pi\)
\(744\) 0 0
\(745\) 5.09838e12 0.606357
\(746\) 6.13253e12 0.724961
\(747\) 0 0
\(748\) −8.68096e10 −0.0101394
\(749\) 4.30978e11 0.0500365
\(750\) 0 0
\(751\) 6.03299e12 0.692074 0.346037 0.938221i \(-0.387527\pi\)
0.346037 + 0.938221i \(0.387527\pi\)
\(752\) −3.46868e12 −0.395534
\(753\) 0 0
\(754\) 2.85142e12 0.321284
\(755\) 1.62082e11 0.0181540
\(756\) 0 0
\(757\) −8.02798e12 −0.888535 −0.444268 0.895894i \(-0.646536\pi\)
−0.444268 + 0.895894i \(0.646536\pi\)
\(758\) −1.94338e12 −0.213820
\(759\) 0 0
\(760\) −5.89675e11 −0.0641138
\(761\) 6.51923e12 0.704637 0.352318 0.935880i \(-0.385393\pi\)
0.352318 + 0.935880i \(0.385393\pi\)
\(762\) 0 0
\(763\) 3.85026e12 0.411273
\(764\) 6.63009e12 0.704043
\(765\) 0 0
\(766\) 6.36354e12 0.667835
\(767\) −1.59018e13 −1.65907
\(768\) 0 0
\(769\) −1.34250e13 −1.38435 −0.692175 0.721730i \(-0.743347\pi\)
−0.692175 + 0.721730i \(0.743347\pi\)
\(770\) 1.16497e12 0.119428
\(771\) 0 0
\(772\) 4.07980e12 0.413391
\(773\) −7.85934e12 −0.791733 −0.395866 0.918308i \(-0.629556\pi\)
−0.395866 + 0.918308i \(0.629556\pi\)
\(774\) 0 0
\(775\) 7.60327e12 0.757082
\(776\) −3.57377e12 −0.353792
\(777\) 0 0
\(778\) −1.08074e12 −0.105758
\(779\) −5.63235e12 −0.547988
\(780\) 0 0
\(781\) −1.51439e13 −1.45649
\(782\) −2.66511e10 −0.00254850
\(783\) 0 0
\(784\) 3.77802e11 0.0357143
\(785\) −7.78780e12 −0.731983
\(786\) 0 0
\(787\) −1.47720e12 −0.137263 −0.0686316 0.997642i \(-0.521863\pi\)
−0.0686316 + 0.997642i \(0.521863\pi\)
\(788\) 8.56793e11 0.0791604
\(789\) 0 0
\(790\) 1.17098e12 0.106962
\(791\) 3.42827e12 0.311373
\(792\) 0 0
\(793\) 2.29698e13 2.06266
\(794\) 1.99449e12 0.178090
\(795\) 0 0
\(796\) −3.43708e12 −0.303445
\(797\) 6.47327e12 0.568278 0.284139 0.958783i \(-0.408292\pi\)
0.284139 + 0.958783i \(0.408292\pi\)
\(798\) 0 0
\(799\) 3.31434e11 0.0287698
\(800\) −1.71917e12 −0.148393
\(801\) 0 0
\(802\) −5.61913e12 −0.479606
\(803\) −2.18786e13 −1.85695
\(804\) 0 0
\(805\) 3.57653e11 0.0300179
\(806\) 8.39733e12 0.700864
\(807\) 0 0
\(808\) 4.95212e12 0.408734
\(809\) 1.53631e13 1.26099 0.630493 0.776195i \(-0.282853\pi\)
0.630493 + 0.776195i \(0.282853\pi\)
\(810\) 0 0
\(811\) −1.29056e13 −1.04757 −0.523787 0.851849i \(-0.675481\pi\)
−0.523787 + 0.851849i \(0.675481\pi\)
\(812\) 9.67907e11 0.0781325
\(813\) 0 0
\(814\) −1.03506e13 −0.826334
\(815\) −9.30902e12 −0.739085
\(816\) 0 0
\(817\) 7.07494e12 0.555550
\(818\) −6.11130e11 −0.0477247
\(819\) 0 0
\(820\) 3.14089e12 0.242600
\(821\) −8.93771e11 −0.0686566 −0.0343283 0.999411i \(-0.510929\pi\)
−0.0343283 + 0.999411i \(0.510929\pi\)
\(822\) 0 0
\(823\) −2.29844e13 −1.74636 −0.873182 0.487394i \(-0.837947\pi\)
−0.873182 + 0.487394i \(0.837947\pi\)
\(824\) 2.82854e12 0.213742
\(825\) 0 0
\(826\) −5.39782e12 −0.403467
\(827\) 2.75618e12 0.204896 0.102448 0.994738i \(-0.467333\pi\)
0.102448 + 0.994738i \(0.467333\pi\)
\(828\) 0 0
\(829\) −3.15925e12 −0.232321 −0.116160 0.993230i \(-0.537059\pi\)
−0.116160 + 0.993230i \(0.537059\pi\)
\(830\) 3.76440e12 0.275324
\(831\) 0 0
\(832\) −1.89871e12 −0.137374
\(833\) −3.60992e10 −0.00259774
\(834\) 0 0
\(835\) −8.84909e12 −0.629955
\(836\) 3.56385e12 0.252343
\(837\) 0 0
\(838\) −3.44428e12 −0.241269
\(839\) −1.92403e13 −1.34055 −0.670277 0.742111i \(-0.733825\pi\)
−0.670277 + 0.742111i \(0.733825\pi\)
\(840\) 0 0
\(841\) −1.20274e13 −0.829069
\(842\) 1.91892e13 1.31569
\(843\) 0 0
\(844\) 7.72357e12 0.523935
\(845\) −1.23391e12 −0.0832581
\(846\) 0 0
\(847\) −1.37935e12 −0.0920874
\(848\) −1.06307e12 −0.0705963
\(849\) 0 0
\(850\) 1.64267e11 0.0107936
\(851\) −3.17770e12 −0.207697
\(852\) 0 0
\(853\) −2.60804e13 −1.68672 −0.843362 0.537345i \(-0.819427\pi\)
−0.843362 + 0.537345i \(0.819427\pi\)
\(854\) 7.79705e12 0.501614
\(855\) 0 0
\(856\) −7.35229e11 −0.0468049
\(857\) −2.19177e13 −1.38797 −0.693986 0.719988i \(-0.744147\pi\)
−0.693986 + 0.719988i \(0.744147\pi\)
\(858\) 0 0
\(859\) −3.55588e12 −0.222832 −0.111416 0.993774i \(-0.535539\pi\)
−0.111416 + 0.993774i \(0.535539\pi\)
\(860\) −3.94535e12 −0.245948
\(861\) 0 0
\(862\) −1.26579e13 −0.780869
\(863\) −2.22084e13 −1.36292 −0.681458 0.731858i \(-0.738654\pi\)
−0.681458 + 0.731858i \(0.738654\pi\)
\(864\) 0 0
\(865\) 1.80950e12 0.109897
\(866\) −1.84721e13 −1.11606
\(867\) 0 0
\(868\) 2.85046e12 0.170442
\(869\) −7.07711e12 −0.420986
\(870\) 0 0
\(871\) −1.73984e13 −1.02430
\(872\) −6.56838e12 −0.384711
\(873\) 0 0
\(874\) 1.09412e12 0.0634257
\(875\) −4.83053e12 −0.278586
\(876\) 0 0
\(877\) 3.38004e12 0.192941 0.0964703 0.995336i \(-0.469245\pi\)
0.0964703 + 0.995336i \(0.469245\pi\)
\(878\) −3.40365e12 −0.193295
\(879\) 0 0
\(880\) −1.98739e12 −0.111715
\(881\) −5.25103e11 −0.0293665 −0.0146833 0.999892i \(-0.504674\pi\)
−0.0146833 + 0.999892i \(0.504674\pi\)
\(882\) 0 0
\(883\) 3.33972e13 1.84879 0.924393 0.381441i \(-0.124572\pi\)
0.924393 + 0.381441i \(0.124572\pi\)
\(884\) 1.81423e11 0.00999210
\(885\) 0 0
\(886\) 1.03728e12 0.0565515
\(887\) 9.61964e12 0.521798 0.260899 0.965366i \(-0.415981\pi\)
0.260899 + 0.965366i \(0.415981\pi\)
\(888\) 0 0
\(889\) 5.66331e12 0.304097
\(890\) −4.20710e12 −0.224765
\(891\) 0 0
\(892\) 1.36968e13 0.724396
\(893\) −1.36066e13 −0.716007
\(894\) 0 0
\(895\) −1.35188e13 −0.704264
\(896\) −6.44514e11 −0.0334077
\(897\) 0 0
\(898\) 1.72850e13 0.887004
\(899\) 7.30271e12 0.372877
\(900\) 0 0
\(901\) 1.01577e11 0.00513494
\(902\) −1.89828e13 −0.954839
\(903\) 0 0
\(904\) −5.84848e12 −0.291263
\(905\) 2.18026e12 0.108041
\(906\) 0 0
\(907\) −2.08341e13 −1.02221 −0.511107 0.859517i \(-0.670764\pi\)
−0.511107 + 0.859517i \(0.670764\pi\)
\(908\) 1.03092e13 0.503315
\(909\) 0 0
\(910\) −2.43466e12 −0.117694
\(911\) 1.33789e13 0.643560 0.321780 0.946814i \(-0.395719\pi\)
0.321780 + 0.946814i \(0.395719\pi\)
\(912\) 0 0
\(913\) −2.27511e13 −1.08364
\(914\) −1.03476e12 −0.0490436
\(915\) 0 0
\(916\) 4.87033e12 0.228575
\(917\) 1.44460e12 0.0674660
\(918\) 0 0
\(919\) −1.38352e13 −0.639830 −0.319915 0.947446i \(-0.603654\pi\)
−0.319915 + 0.947446i \(0.603654\pi\)
\(920\) −6.10140e11 −0.0280792
\(921\) 0 0
\(922\) −6.86806e12 −0.313000
\(923\) 3.16492e13 1.43534
\(924\) 0 0
\(925\) 1.95862e13 0.879653
\(926\) 2.59013e13 1.15764
\(927\) 0 0
\(928\) −1.65121e12 −0.0730863
\(929\) 1.74073e13 0.766761 0.383380 0.923591i \(-0.374760\pi\)
0.383380 + 0.923591i \(0.374760\pi\)
\(930\) 0 0
\(931\) 1.48200e12 0.0646511
\(932\) 9.41434e12 0.408713
\(933\) 0 0
\(934\) −5.23714e12 −0.225182
\(935\) 1.89896e11 0.00812575
\(936\) 0 0
\(937\) 1.98361e12 0.0840676 0.0420338 0.999116i \(-0.486616\pi\)
0.0420338 + 0.999116i \(0.486616\pi\)
\(938\) −5.90587e12 −0.249098
\(939\) 0 0
\(940\) 7.58773e12 0.316984
\(941\) 2.33533e13 0.970946 0.485473 0.874252i \(-0.338647\pi\)
0.485473 + 0.874252i \(0.338647\pi\)
\(942\) 0 0
\(943\) −5.82783e12 −0.239996
\(944\) 9.20844e12 0.377409
\(945\) 0 0
\(946\) 2.38447e13 0.968015
\(947\) −3.59882e13 −1.45407 −0.727035 0.686600i \(-0.759102\pi\)
−0.727035 + 0.686600i \(0.759102\pi\)
\(948\) 0 0
\(949\) 4.57241e13 1.82998
\(950\) −6.74377e12 −0.268625
\(951\) 0 0
\(952\) 6.15836e10 0.00242996
\(953\) 1.37662e13 0.540624 0.270312 0.962773i \(-0.412873\pi\)
0.270312 + 0.962773i \(0.412873\pi\)
\(954\) 0 0
\(955\) −1.45033e13 −0.564225
\(956\) −1.67964e12 −0.0650363
\(957\) 0 0
\(958\) −4.55698e12 −0.174796
\(959\) −1.23894e13 −0.473005
\(960\) 0 0
\(961\) −4.93336e12 −0.186590
\(962\) 2.16317e13 0.814334
\(963\) 0 0
\(964\) −2.29585e13 −0.856244
\(965\) −8.92457e12 −0.331295
\(966\) 0 0
\(967\) −3.02718e12 −0.111332 −0.0556659 0.998449i \(-0.517728\pi\)
−0.0556659 + 0.998449i \(0.517728\pi\)
\(968\) 2.35312e12 0.0861399
\(969\) 0 0
\(970\) 7.81762e12 0.283532
\(971\) 2.53183e13 0.914003 0.457002 0.889466i \(-0.348923\pi\)
0.457002 + 0.889466i \(0.348923\pi\)
\(972\) 0 0
\(973\) 1.71517e13 0.613478
\(974\) −1.14364e13 −0.407169
\(975\) 0 0
\(976\) −1.33014e13 −0.469217
\(977\) −9.90729e12 −0.347880 −0.173940 0.984756i \(-0.555650\pi\)
−0.173940 + 0.984756i \(0.555650\pi\)
\(978\) 0 0
\(979\) 2.54267e13 0.884641
\(980\) −8.26442e11 −0.0286217
\(981\) 0 0
\(982\) −1.62409e13 −0.557325
\(983\) −2.40523e13 −0.821610 −0.410805 0.911723i \(-0.634752\pi\)
−0.410805 + 0.911723i \(0.634752\pi\)
\(984\) 0 0
\(985\) −1.87423e12 −0.0634397
\(986\) 1.57774e11 0.00531604
\(987\) 0 0
\(988\) −7.44807e12 −0.248678
\(989\) 7.32048e12 0.243308
\(990\) 0 0
\(991\) 3.15782e13 1.04005 0.520027 0.854150i \(-0.325922\pi\)
0.520027 + 0.854150i \(0.325922\pi\)
\(992\) −4.86275e12 −0.159434
\(993\) 0 0
\(994\) 1.07433e13 0.349058
\(995\) 7.51860e12 0.243183
\(996\) 0 0
\(997\) 9.32241e12 0.298813 0.149407 0.988776i \(-0.452264\pi\)
0.149407 + 0.988776i \(0.452264\pi\)
\(998\) 2.13460e13 0.681128
\(999\) 0 0
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 126.10.a.e.1.1 1
3.2 odd 2 14.10.a.a.1.1 1
12.11 even 2 112.10.a.b.1.1 1
15.2 even 4 350.10.c.b.99.1 2
15.8 even 4 350.10.c.b.99.2 2
15.14 odd 2 350.10.a.c.1.1 1
21.2 odd 6 98.10.c.f.67.1 2
21.5 even 6 98.10.c.e.67.1 2
21.11 odd 6 98.10.c.f.79.1 2
21.17 even 6 98.10.c.e.79.1 2
21.20 even 2 98.10.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.10.a.a.1.1 1 3.2 odd 2
98.10.a.a.1.1 1 21.20 even 2
98.10.c.e.67.1 2 21.5 even 6
98.10.c.e.79.1 2 21.17 even 6
98.10.c.f.67.1 2 21.2 odd 6
98.10.c.f.79.1 2 21.11 odd 6
112.10.a.b.1.1 1 12.11 even 2
126.10.a.e.1.1 1 1.1 even 1 trivial
350.10.a.c.1.1 1 15.14 odd 2
350.10.c.b.99.1 2 15.2 even 4
350.10.c.b.99.2 2 15.8 even 4