Properties

Label 112.10.a.b.1.1
Level $112$
Weight $10$
Character 112.1
Self dual yes
Analytic conductor $57.684$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.6840136504\)
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 \(\chi\) \(=\) 112.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.00000 q^{3} +560.000 q^{5} +2401.00 q^{7} -19647.0 q^{9} +O(q^{10})\) \(q+6.00000 q^{3} +560.000 q^{5} +2401.00 q^{7} -19647.0 q^{9} +54152.0 q^{11} -113172. q^{13} +3360.00 q^{15} +6262.00 q^{17} -257078. q^{19} +14406.0 q^{21} +266000. q^{23} -1.63952e6 q^{25} -235980. q^{27} +1.57471e6 q^{29} +4.63748e6 q^{31} +324912. q^{33} +1.34456e6 q^{35} -1.19462e7 q^{37} -679032. q^{39} +2.19091e7 q^{41} -2.75206e7 q^{43} -1.10023e7 q^{45} -5.29278e7 q^{47} +5.76480e6 q^{49} +37572.0 q^{51} +1.62212e7 q^{53} +3.03251e7 q^{55} -1.54247e6 q^{57} +1.40510e8 q^{59} -2.02964e8 q^{61} -4.71724e7 q^{63} -6.33763e7 q^{65} -1.53735e8 q^{67} +1.59600e6 q^{69} -2.79656e8 q^{71} -4.04023e8 q^{73} -9.83715e6 q^{75} +1.30019e8 q^{77} +1.30690e8 q^{79} +3.85296e8 q^{81} -4.20134e8 q^{83} +3.50672e6 q^{85} +9.44828e6 q^{87} -4.69542e8 q^{89} -2.71726e8 q^{91} +2.78249e7 q^{93} -1.43964e8 q^{95} -8.72502e8 q^{97} -1.06392e9 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 6.00000 0.0427667 0.0213833 0.999771i \(-0.493193\pi\)
0.0213833 + 0.999771i \(0.493193\pi\)
\(4\) 0 0
\(5\) 560.000 0.400703 0.200352 0.979724i \(-0.435792\pi\)
0.200352 + 0.979724i \(0.435792\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) −19647.0 −0.998171
\(10\) 0 0
\(11\) 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 \(0.685180\pi\)
\(14\) 0 0
\(15\) 3360.00 0.0171368
\(16\) 0 0
\(17\) 6262.00 0.0181841 0.00909207 0.999959i \(-0.497106\pi\)
0.00909207 + 0.999959i \(0.497106\pi\)
\(18\) 0 0
\(19\) −257078. −0.452557 −0.226279 0.974063i \(-0.572656\pi\)
−0.226279 + 0.974063i \(0.572656\pi\)
\(20\) 0 0
\(21\) 14406.0 0.0161643
\(22\) 0 0
\(23\) 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
\(26\) 0 0
\(27\) −235980. −0.0854552
\(28\) 0 0
\(29\) 1.57471e6 0.413438 0.206719 0.978400i \(-0.433721\pi\)
0.206719 + 0.978400i \(0.433721\pi\)
\(30\) 0 0
\(31\) 4.63748e6 0.901893 0.450946 0.892551i \(-0.351087\pi\)
0.450946 + 0.892551i \(0.351087\pi\)
\(32\) 0 0
\(33\) 324912. 0.0476928
\(34\) 0 0
\(35\) 1.34456e6 0.151452
\(36\) 0 0
\(37\) −1.19462e7 −1.04791 −0.523954 0.851746i \(-0.675544\pi\)
−0.523954 + 0.851746i \(0.675544\pi\)
\(38\) 0 0
\(39\) −679032. −0.0470002
\(40\) 0 0
\(41\) 2.19091e7 1.21087 0.605435 0.795895i \(-0.292999\pi\)
0.605435 + 0.795895i \(0.292999\pi\)
\(42\) 0 0
\(43\) −2.75206e7 −1.22758 −0.613790 0.789469i \(-0.710356\pi\)
−0.613790 + 0.789469i \(0.710356\pi\)
\(44\) 0 0
\(45\) −1.10023e7 −0.399970
\(46\) 0 0
\(47\) −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
\(50\) 0 0
\(51\) 37572.0 0.000777676 0
\(52\) 0 0
\(53\) 1.62212e7 0.282385 0.141193 0.989982i \(-0.454906\pi\)
0.141193 + 0.989982i \(0.454906\pi\)
\(54\) 0 0
\(55\) 3.03251e7 0.446859
\(56\) 0 0
\(57\) −1.54247e6 −0.0193544
\(58\) 0 0
\(59\) 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 \(0.887724\pi\)
\(62\) 0 0
\(63\) −4.71724e7 −0.377273
\(64\) 0 0
\(65\) −6.33763e7 −0.440369
\(66\) 0 0
\(67\) −1.53735e8 −0.932041 −0.466020 0.884774i \(-0.654313\pi\)
−0.466020 + 0.884774i \(0.654313\pi\)
\(68\) 0 0
\(69\) 1.59600e6 0.00847641
\(70\) 0 0
\(71\) −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 \(0.813134\pi\)
\(74\) 0 0
\(75\) −9.83715e6 −0.0358999
\(76\) 0 0
\(77\) 1.30019e8 0.421501
\(78\) 0 0
\(79\) 1.30690e8 0.377503 0.188751 0.982025i \(-0.439556\pi\)
0.188751 + 0.982025i \(0.439556\pi\)
\(80\) 0 0
\(81\) 3.85296e8 0.994516
\(82\) 0 0
\(83\) −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
\(86\) 0 0
\(87\) 9.44828e6 0.0176814
\(88\) 0 0
\(89\) −4.69542e8 −0.793268 −0.396634 0.917977i \(-0.629822\pi\)
−0.396634 + 0.917977i \(0.629822\pi\)
\(90\) 0 0
\(91\) −2.71726e8 −0.415379
\(92\) 0 0
\(93\) 2.78249e7 0.0385710
\(94\) 0 0
\(95\) −1.43964e8 −0.181341
\(96\) 0 0
\(97\) −8.72502e8 −1.00068 −0.500338 0.865830i \(-0.666791\pi\)
−0.500338 + 0.865830i \(0.666791\pi\)
\(98\) 0 0
\(99\) −1.06392e9 −1.11315
\(100\) 0 0
\(101\) −1.20901e9 −1.15607 −0.578037 0.816011i \(-0.696181\pi\)
−0.578037 + 0.816011i \(0.696181\pi\)
\(102\) 0 0
\(103\) −6.90563e8 −0.604555 −0.302277 0.953220i \(-0.597747\pi\)
−0.302277 + 0.953220i \(0.597747\pi\)
\(104\) 0 0
\(105\) 8.06736e6 0.00647708
\(106\) 0 0
\(107\) −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 \(0.683115\pi\)
\(110\) 0 0
\(111\) −7.16774e7 −0.0448156
\(112\) 0 0
\(113\) 1.42785e9 0.823815 0.411908 0.911226i \(-0.364863\pi\)
0.411908 + 0.911226i \(0.364863\pi\)
\(114\) 0 0
\(115\) 1.48960e8 0.0794199
\(116\) 0 0
\(117\) 2.22349e9 1.09698
\(118\) 0 0
\(119\) 1.50351e7 0.00687296
\(120\) 0 0
\(121\) 5.74491e8 0.243640
\(122\) 0 0
\(123\) 1.31455e8 0.0517849
\(124\) 0 0
\(125\) −2.01188e9 −0.737069
\(126\) 0 0
\(127\) 2.35873e9 0.804565 0.402282 0.915516i \(-0.368217\pi\)
0.402282 + 0.915516i \(0.368217\pi\)
\(128\) 0 0
\(129\) −1.65124e8 −0.0524995
\(130\) 0 0
\(131\) −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
\(134\) 0 0
\(135\) −1.32149e8 −0.0342422
\(136\) 0 0
\(137\) −5.16009e9 −1.25145 −0.625726 0.780043i \(-0.715197\pi\)
−0.625726 + 0.780043i \(0.715197\pi\)
\(138\) 0 0
\(139\) 7.14356e9 1.62311 0.811556 0.584275i \(-0.198621\pi\)
0.811556 + 0.584275i \(0.198621\pi\)
\(140\) 0 0
\(141\) −3.17567e8 −0.0676627
\(142\) 0 0
\(143\) −6.12849e9 −1.22558
\(144\) 0 0
\(145\) 8.81840e8 0.165666
\(146\) 0 0
\(147\) 3.45888e7 0.00610953
\(148\) 0 0
\(149\) 9.10424e9 1.51323 0.756616 0.653859i \(-0.226851\pi\)
0.756616 + 0.653859i \(0.226851\pi\)
\(150\) 0 0
\(151\) 2.89432e8 0.0453054 0.0226527 0.999743i \(-0.492789\pi\)
0.0226527 + 0.999743i \(0.492789\pi\)
\(152\) 0 0
\(153\) −1.23030e8 −0.0181509
\(154\) 0 0
\(155\) 2.59699e9 0.361391
\(156\) 0 0
\(157\) 1.39068e10 1.82675 0.913373 0.407124i \(-0.133468\pi\)
0.913373 + 0.407124i \(0.133468\pi\)
\(158\) 0 0
\(159\) 9.73273e7 0.0120767
\(160\) 0 0
\(161\) 6.38666e8 0.0749130
\(162\) 0 0
\(163\) −1.66232e10 −1.84447 −0.922235 0.386629i \(-0.873639\pi\)
−0.922235 + 0.386629i \(0.873639\pi\)
\(164\) 0 0
\(165\) 1.81951e8 0.0191107
\(166\) 0 0
\(167\) 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
\(170\) 0 0
\(171\) 5.05081e9 0.451730
\(172\) 0 0
\(173\) 3.23125e9 0.274260 0.137130 0.990553i \(-0.456212\pi\)
0.137130 + 0.990553i \(0.456212\pi\)
\(174\) 0 0
\(175\) −3.93650e9 −0.317277
\(176\) 0 0
\(177\) 8.43058e8 0.0645621
\(178\) 0 0
\(179\) 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 \(0.543044\pi\)
\(182\) 0 0
\(183\) −1.21778e9 −0.0802674
\(184\) 0 0
\(185\) −6.68989e9 −0.419901
\(186\) 0 0
\(187\) 3.39100e8 0.0202787
\(188\) 0 0
\(189\) −5.66588e8 −0.0322990
\(190\) 0 0
\(191\) 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 \(0.364344\pi\)
\(194\) 0 0
\(195\) −3.80258e8 −0.0188331
\(196\) 0 0
\(197\) −3.34685e9 −0.158321 −0.0791604 0.996862i \(-0.525224\pi\)
−0.0791604 + 0.996862i \(0.525224\pi\)
\(198\) 0 0
\(199\) 1.34261e10 0.606891 0.303445 0.952849i \(-0.401863\pi\)
0.303445 + 0.952849i \(0.401863\pi\)
\(200\) 0 0
\(201\) −9.22407e8 −0.0398603
\(202\) 0 0
\(203\) 3.78089e9 0.156265
\(204\) 0 0
\(205\) 1.22691e10 0.485200
\(206\) 0 0
\(207\) −5.22610e9 −0.197839
\(208\) 0 0
\(209\) −1.39213e10 −0.504686
\(210\) 0 0
\(211\) −3.01702e10 −1.04787 −0.523935 0.851759i \(-0.675536\pi\)
−0.523935 + 0.851759i \(0.675536\pi\)
\(212\) 0 0
\(213\) −1.67794e9 −0.0558556
\(214\) 0 0
\(215\) −1.54115e10 −0.491895
\(216\) 0 0
\(217\) 1.11346e10 0.340883
\(218\) 0 0
\(219\) −2.42414e9 −0.0712129
\(220\) 0 0
\(221\) −7.08683e8 −0.0199842
\(222\) 0 0
\(223\) −5.35030e10 −1.44879 −0.724396 0.689384i \(-0.757881\pi\)
−0.724396 + 0.689384i \(0.757881\pi\)
\(224\) 0 0
\(225\) 3.22117e10 0.837901
\(226\) 0 0
\(227\) 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 \(0.426593\pi\)
\(230\) 0 0
\(231\) 7.80114e8 0.0180262
\(232\) 0 0
\(233\) −3.67748e10 −0.817426 −0.408713 0.912663i \(-0.634022\pi\)
−0.408713 + 0.912663i \(0.634022\pi\)
\(234\) 0 0
\(235\) −2.96396e10 −0.633967
\(236\) 0 0
\(237\) 7.84139e8 0.0161445
\(238\) 0 0
\(239\) −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 \(0.827208\pi\)
\(242\) 0 0
\(243\) 6.95657e9 0.127987
\(244\) 0 0
\(245\) 3.22829e9 0.0572433
\(246\) 0 0
\(247\) 2.90940e10 0.497356
\(248\) 0 0
\(249\) −2.52080e9 −0.0415568
\(250\) 0 0
\(251\) −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
\(254\) 0 0
\(255\) 2.10403e7 0.000311617 0
\(256\) 0 0
\(257\) −8.35575e10 −1.19478 −0.597388 0.801952i \(-0.703795\pi\)
−0.597388 + 0.801952i \(0.703795\pi\)
\(258\) 0 0
\(259\) −2.86829e10 −0.396072
\(260\) 0 0
\(261\) −3.09384e10 −0.412682
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) −2.81725e9 −0.0339254
\(268\) 0 0
\(269\) 1.41401e11 1.64652 0.823258 0.567668i \(-0.192154\pi\)
0.823258 + 0.567668i \(0.192154\pi\)
\(270\) 0 0
\(271\) 9.08353e10 1.02304 0.511520 0.859271i \(-0.329083\pi\)
0.511520 + 0.859271i \(0.329083\pi\)
\(272\) 0 0
\(273\) −1.63036e9 −0.0177644
\(274\) 0 0
\(275\) −8.87836e10 −0.936128
\(276\) 0 0
\(277\) −2.65075e10 −0.270527 −0.135263 0.990810i \(-0.543188\pi\)
−0.135263 + 0.990810i \(0.543188\pi\)
\(278\) 0 0
\(279\) −9.11126e10 −0.900243
\(280\) 0 0
\(281\) −1.86968e11 −1.78891 −0.894455 0.447158i \(-0.852436\pi\)
−0.894455 + 0.447158i \(0.852436\pi\)
\(282\) 0 0
\(283\) 5.33413e9 0.0494338 0.0247169 0.999694i \(-0.492132\pi\)
0.0247169 + 0.999694i \(0.492132\pi\)
\(284\) 0 0
\(285\) −8.63782e8 −0.00775537
\(286\) 0 0
\(287\) 5.26038e10 0.457666
\(288\) 0 0
\(289\) −1.18549e11 −0.999669
\(290\) 0 0
\(291\) −5.23501e9 −0.0427956
\(292\) 0 0
\(293\) 7.65433e10 0.606741 0.303370 0.952873i \(-0.401888\pi\)
0.303370 + 0.952873i \(0.401888\pi\)
\(294\) 0 0
\(295\) 7.86854e10 0.604916
\(296\) 0 0
\(297\) −1.27788e10 −0.0952984
\(298\) 0 0
\(299\) −3.01038e10 −0.217821
\(300\) 0 0
\(301\) −6.60769e10 −0.463982
\(302\) 0 0
\(303\) −7.25409e9 −0.0494414
\(304\) 0 0
\(305\) −1.13660e11 −0.752068
\(306\) 0 0
\(307\) 7.51944e10 0.483128 0.241564 0.970385i \(-0.422340\pi\)
0.241564 + 0.970385i \(0.422340\pi\)
\(308\) 0 0
\(309\) −4.14338e9 −0.0258548
\(310\) 0 0
\(311\) −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 \(0.408868\pi\)
\(314\) 0 0
\(315\) −2.64166e10 −0.151175
\(316\) 0 0
\(317\) 1.70586e11 0.948807 0.474403 0.880308i \(-0.342664\pi\)
0.474403 + 0.880308i \(0.342664\pi\)
\(318\) 0 0
\(319\) 8.52739e10 0.461061
\(320\) 0 0
\(321\) −1.07700e9 −0.00566163
\(322\) 0 0
\(323\) −1.60982e9 −0.00822937
\(324\) 0 0
\(325\) 1.85548e11 0.922533
\(326\) 0 0
\(327\) −9.62165e9 −0.0465355
\(328\) 0 0
\(329\) −1.27080e11 −0.597991
\(330\) 0 0
\(331\) 1.23992e11 0.567762 0.283881 0.958859i \(-0.408378\pi\)
0.283881 + 0.958859i \(0.408378\pi\)
\(332\) 0 0
\(333\) 2.34708e11 1.04599
\(334\) 0 0
\(335\) −8.60914e10 −0.373472
\(336\) 0 0
\(337\) −7.29335e10 −0.308030 −0.154015 0.988069i \(-0.549220\pi\)
−0.154015 + 0.988069i \(0.549220\pi\)
\(338\) 0 0
\(339\) 8.56710e9 0.0352318
\(340\) 0 0
\(341\) 2.51129e11 1.00578
\(342\) 0 0
\(343\) 1.38413e10 0.0539949
\(344\) 0 0
\(345\) 8.93760e8 0.00339653
\(346\) 0 0
\(347\) 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 \(0.437154\pi\)
\(350\) 0 0
\(351\) 2.67063e10 0.0939144
\(352\) 0 0
\(353\) 3.25585e11 1.11604 0.558018 0.829829i \(-0.311562\pi\)
0.558018 + 0.829829i \(0.311562\pi\)
\(354\) 0 0
\(355\) −1.56607e11 −0.523341
\(356\) 0 0
\(357\) 9.02104e7 0.000293934 0
\(358\) 0 0
\(359\) 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
\(362\) 0 0
\(363\) 3.44695e9 0.0104197
\(364\) 0 0
\(365\) −2.26253e11 −0.667231
\(366\) 0 0
\(367\) 4.21993e11 1.21425 0.607125 0.794607i \(-0.292323\pi\)
0.607125 + 0.794607i \(0.292323\pi\)
\(368\) 0 0
\(369\) −4.30449e11 −1.20866
\(370\) 0 0
\(371\) 3.89472e10 0.106732
\(372\) 0 0
\(373\) 3.83283e11 1.02525 0.512625 0.858613i \(-0.328673\pi\)
0.512625 + 0.858613i \(0.328673\pi\)
\(374\) 0 0
\(375\) −1.20713e10 −0.0315220
\(376\) 0 0
\(377\) −1.78214e11 −0.454365
\(378\) 0 0
\(379\) 1.21462e11 0.302386 0.151193 0.988504i \(-0.451688\pi\)
0.151193 + 0.988504i \(0.451688\pi\)
\(380\) 0 0
\(381\) 1.41524e10 0.0344086
\(382\) 0 0
\(383\) 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
\(386\) 0 0
\(387\) 5.40697e11 1.22533
\(388\) 0 0
\(389\) 6.75462e10 0.149564 0.0747821 0.997200i \(-0.476174\pi\)
0.0747821 + 0.997200i \(0.476174\pi\)
\(390\) 0 0
\(391\) 1.66569e9 0.00360412
\(392\) 0 0
\(393\) −3.60999e9 −0.00763378
\(394\) 0 0
\(395\) 7.31863e10 0.151267
\(396\) 0 0
\(397\) 1.24656e11 0.251857 0.125929 0.992039i \(-0.459809\pi\)
0.125929 + 0.992039i \(0.459809\pi\)
\(398\) 0 0
\(399\) −3.70347e9 −0.00731527
\(400\) 0 0
\(401\) 3.51196e11 0.678265 0.339133 0.940739i \(-0.389866\pi\)
0.339133 + 0.940739i \(0.389866\pi\)
\(402\) 0 0
\(403\) −5.24833e11 −0.991171
\(404\) 0 0
\(405\) 2.15766e11 0.398506
\(406\) 0 0
\(407\) −6.46913e11 −1.16861
\(408\) 0 0
\(409\) −3.81956e10 −0.0674930 −0.0337465 0.999430i \(-0.510744\pi\)
−0.0337465 + 0.999430i \(0.510744\pi\)
\(410\) 0 0
\(411\) −3.09605e10 −0.0535205
\(412\) 0 0
\(413\) 3.37364e11 0.570588
\(414\) 0 0
\(415\) −2.35275e11 −0.389367
\(416\) 0 0
\(417\) 4.28614e10 0.0694151
\(418\) 0 0
\(419\) −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 \(0.119519\pi\)
\(422\) 0 0
\(423\) 1.03987e12 1.57924
\(424\) 0 0
\(425\) −1.02667e10 −0.0152644
\(426\) 0 0
\(427\) −4.87316e11 −0.709390
\(428\) 0 0
\(429\) −3.67709e10 −0.0524140
\(430\) 0 0
\(431\) −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 \(0.789489\pi\)
\(434\) 0 0
\(435\) 5.29104e9 0.00708499
\(436\) 0 0
\(437\) −6.83827e10 −0.0896974
\(438\) 0 0
\(439\) 2.12728e11 0.273360 0.136680 0.990615i \(-0.456357\pi\)
0.136680 + 0.990615i \(0.456357\pi\)
\(440\) 0 0
\(441\) −1.13261e11 −0.142596
\(442\) 0 0
\(443\) 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
\(446\) 0 0
\(447\) 5.46255e10 0.0647159
\(448\) 0 0
\(449\) −1.08031e12 −1.25441 −0.627207 0.778853i \(-0.715802\pi\)
−0.627207 + 0.778853i \(0.715802\pi\)
\(450\) 0 0
\(451\) 1.18642e12 1.35035
\(452\) 0 0
\(453\) 1.73659e9 0.00193756
\(454\) 0 0
\(455\) −1.52167e11 −0.166444
\(456\) 0 0
\(457\) −6.46725e10 −0.0693581 −0.0346790 0.999399i \(-0.511041\pi\)
−0.0346790 + 0.999399i \(0.511041\pi\)
\(458\) 0 0
\(459\) −1.47771e9 −0.00155393
\(460\) 0 0
\(461\) 4.29254e11 0.442649 0.221325 0.975200i \(-0.428962\pi\)
0.221325 + 0.975200i \(0.428962\pi\)
\(462\) 0 0
\(463\) −1.61883e12 −1.63715 −0.818574 0.574401i \(-0.805235\pi\)
−0.818574 + 0.574401i \(0.805235\pi\)
\(464\) 0 0
\(465\) 1.55819e10 0.0154555
\(466\) 0 0
\(467\) −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
\(470\) 0 0
\(471\) 8.34407e10 0.0781239
\(472\) 0 0
\(473\) −1.49030e12 −1.36898
\(474\) 0 0
\(475\) 4.21486e11 0.379893
\(476\) 0 0
\(477\) −3.18698e11 −0.281869
\(478\) 0 0
\(479\) −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
\(482\) 0 0
\(483\) 3.83200e9 0.00320378
\(484\) 0 0
\(485\) −4.88601e11 −0.400974
\(486\) 0 0
\(487\) 7.14776e11 0.575824 0.287912 0.957657i \(-0.407039\pi\)
0.287912 + 0.957657i \(0.407039\pi\)
\(488\) 0 0
\(489\) −9.97395e10 −0.0788819
\(490\) 0 0
\(491\) −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
\(494\) 0 0
\(495\) −5.95798e11 −0.446042
\(496\) 0 0
\(497\) −6.71454e11 −0.493642
\(498\) 0 0
\(499\) −1.33412e12 −0.963260 −0.481630 0.876375i \(-0.659955\pi\)
−0.481630 + 0.876375i \(0.659955\pi\)
\(500\) 0 0
\(501\) 9.48116e10 0.0672345
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) 1.32204e10 0.00888606
\(508\) 0 0
\(509\) 3.57173e11 0.235857 0.117928 0.993022i \(-0.462375\pi\)
0.117928 + 0.993022i \(0.462375\pi\)
\(510\) 0 0
\(511\) −9.70059e11 −0.629367
\(512\) 0 0
\(513\) 6.06653e10 0.0386734
\(514\) 0 0
\(515\) −3.86715e11 −0.242247
\(516\) 0 0
\(517\) −2.86615e12 −1.76438
\(518\) 0 0
\(519\) 1.93875e10 0.0117292
\(520\) 0 0
\(521\) 2.17972e12 1.29608 0.648040 0.761606i \(-0.275589\pi\)
0.648040 + 0.761606i \(0.275589\pi\)
\(522\) 0 0
\(523\) 1.57081e12 0.918048 0.459024 0.888424i \(-0.348199\pi\)
0.459024 + 0.888424i \(0.348199\pi\)
\(524\) 0 0
\(525\) −2.36190e10 −0.0135689
\(526\) 0 0
\(527\) 2.90399e10 0.0164001
\(528\) 0 0
\(529\) −1.73040e12 −0.960716
\(530\) 0 0
\(531\) −2.76059e12 −1.50687
\(532\) 0 0
\(533\) −2.47950e12 −1.33074
\(534\) 0 0
\(535\) −1.00520e11 −0.0530468
\(536\) 0 0
\(537\) 1.44845e11 0.0751655
\(538\) 0 0
\(539\) 3.12176e11 0.159312
\(540\) 0 0
\(541\) 2.24544e12 1.12697 0.563486 0.826126i \(-0.309460\pi\)
0.563486 + 0.826126i \(0.309460\pi\)
\(542\) 0 0
\(543\) −2.33599e10 −0.0115311
\(544\) 0 0
\(545\) −8.98021e11 −0.436016
\(546\) 0 0
\(547\) 3.86062e11 0.184380 0.0921899 0.995741i \(-0.470613\pi\)
0.0921899 + 0.995741i \(0.470613\pi\)
\(548\) 0 0
\(549\) 3.98763e12 1.87344
\(550\) 0 0
\(551\) −4.04824e11 −0.187105
\(552\) 0 0
\(553\) 3.13786e11 0.142683
\(554\) 0 0
\(555\) −4.01394e10 −0.0179578
\(556\) 0 0
\(557\) −7.95102e11 −0.350005 −0.175003 0.984568i \(-0.555993\pi\)
−0.175003 + 0.984568i \(0.555993\pi\)
\(558\) 0 0
\(559\) 3.11456e12 1.34910
\(560\) 0 0
\(561\) 2.03460e9 0.000867253 0
\(562\) 0 0
\(563\) 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
\(566\) 0 0
\(567\) 9.25096e11 0.375892
\(568\) 0 0
\(569\) 2.17461e12 0.869714 0.434857 0.900500i \(-0.356799\pi\)
0.434857 + 0.900500i \(0.356799\pi\)
\(570\) 0 0
\(571\) −9.95075e11 −0.391736 −0.195868 0.980630i \(-0.562752\pi\)
−0.195868 + 0.980630i \(0.562752\pi\)
\(572\) 0 0
\(573\) 1.55393e11 0.0602192
\(574\) 0 0
\(575\) −4.36114e11 −0.166377
\(576\) 0 0
\(577\) 4.30588e12 1.61723 0.808614 0.588340i \(-0.200218\pi\)
0.808614 + 0.588340i \(0.200218\pi\)
\(578\) 0 0
\(579\) 9.56204e10 0.0353588
\(580\) 0 0
\(581\) −1.00874e12 −0.367272
\(582\) 0 0
\(583\) 8.78412e11 0.314912
\(584\) 0 0
\(585\) 1.24515e12 0.439564
\(586\) 0 0
\(587\) −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
\(590\) 0 0
\(591\) −2.00811e10 −0.00677085
\(592\) 0 0
\(593\) −2.80300e12 −0.930844 −0.465422 0.885089i \(-0.654097\pi\)
−0.465422 + 0.885089i \(0.654097\pi\)
\(594\) 0 0
\(595\) 8.41963e9 0.00275402
\(596\) 0 0
\(597\) 8.05565e10 0.0259547
\(598\) 0 0
\(599\) −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 \(0.462619\pi\)
\(602\) 0 0
\(603\) 3.02042e12 0.930336
\(604\) 0 0
\(605\) 3.21715e11 0.0976275
\(606\) 0 0
\(607\) 1.74097e12 0.520526 0.260263 0.965538i \(-0.416191\pi\)
0.260263 + 0.965538i \(0.416191\pi\)
\(608\) 0 0
\(609\) 2.26853e10 0.00668294
\(610\) 0 0
\(611\) 5.98995e12 1.73875
\(612\) 0 0
\(613\) −4.03977e12 −1.15554 −0.577770 0.816200i \(-0.696077\pi\)
−0.577770 + 0.816200i \(0.696077\pi\)
\(614\) 0 0
\(615\) 7.36147e10 0.0207504
\(616\) 0 0
\(617\) −2.93367e12 −0.814945 −0.407472 0.913218i \(-0.633590\pi\)
−0.407472 + 0.913218i \(0.633590\pi\)
\(618\) 0 0
\(619\) 5.77691e12 1.58157 0.790784 0.612095i \(-0.209673\pi\)
0.790784 + 0.612095i \(0.209673\pi\)
\(620\) 0 0
\(621\) −6.27707e10 −0.0169373
\(622\) 0 0
\(623\) −1.12737e12 −0.299827
\(624\) 0 0
\(625\) 2.07554e12 0.544091
\(626\) 0 0
\(627\) −8.35277e10 −0.0215837
\(628\) 0 0
\(629\) −7.48073e10 −0.0190553
\(630\) 0 0
\(631\) −3.99985e12 −1.00441 −0.502206 0.864748i \(-0.667478\pi\)
−0.502206 + 0.864748i \(0.667478\pi\)
\(632\) 0 0
\(633\) −1.81021e11 −0.0448139
\(634\) 0 0
\(635\) 1.32089e12 0.322392
\(636\) 0 0
\(637\) −6.52414e11 −0.156999
\(638\) 0 0
\(639\) 5.49440e12 1.30367
\(640\) 0 0
\(641\) 4.68328e12 1.09569 0.547846 0.836579i \(-0.315448\pi\)
0.547846 + 0.836579i \(0.315448\pi\)
\(642\) 0 0
\(643\) 1.54877e12 0.357304 0.178652 0.983912i \(-0.442826\pi\)
0.178652 + 0.983912i \(0.442826\pi\)
\(644\) 0 0
\(645\) −9.24692e10 −0.0210367
\(646\) 0 0
\(647\) 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
\(650\) 0 0
\(651\) 6.68076e10 0.0145785
\(652\) 0 0
\(653\) −2.88925e12 −0.621836 −0.310918 0.950437i \(-0.600636\pi\)
−0.310918 + 0.950437i \(0.600636\pi\)
\(654\) 0 0
\(655\) −3.36933e11 −0.0715249
\(656\) 0 0
\(657\) 7.93784e12 1.66210
\(658\) 0 0
\(659\) −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 \(0.404884\pi\)
\(662\) 0 0
\(663\) −4.25210e9 −0.000854658 0
\(664\) 0 0
\(665\) −3.45657e11 −0.0685406
\(666\) 0 0
\(667\) 4.18874e11 0.0819440
\(668\) 0 0
\(669\) −3.21018e11 −0.0619600
\(670\) 0 0
\(671\) −1.09909e13 −2.09306
\(672\) 0 0
\(673\) 9.46362e12 1.77824 0.889119 0.457677i \(-0.151318\pi\)
0.889119 + 0.457677i \(0.151318\pi\)
\(674\) 0 0
\(675\) 3.86895e11 0.0717342
\(676\) 0 0
\(677\) −6.52268e12 −1.19338 −0.596688 0.802474i \(-0.703517\pi\)
−0.596688 + 0.802474i \(0.703517\pi\)
\(678\) 0 0
\(679\) −2.09488e12 −0.378220
\(680\) 0 0
\(681\) 2.41622e11 0.0430502
\(682\) 0 0
\(683\) −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
\(686\) 0 0
\(687\) 1.14148e11 0.0195508
\(688\) 0 0
\(689\) −1.83579e12 −0.310339
\(690\) 0 0
\(691\) −2.01563e12 −0.336325 −0.168163 0.985759i \(-0.553783\pi\)
−0.168163 + 0.985759i \(0.553783\pi\)
\(692\) 0 0
\(693\) −2.55448e12 −0.420730
\(694\) 0 0
\(695\) 4.00040e12 0.650386
\(696\) 0 0
\(697\) 1.37195e11 0.0220186
\(698\) 0 0
\(699\) −2.20649e11 −0.0349586
\(700\) 0 0
\(701\) −1.06523e13 −1.66614 −0.833068 0.553171i \(-0.813418\pi\)
−0.833068 + 0.553171i \(0.813418\pi\)
\(702\) 0 0
\(703\) 3.07111e12 0.474239
\(704\) 0 0
\(705\) −1.77838e11 −0.0271127
\(706\) 0 0
\(707\) −2.90284e12 −0.436955
\(708\) 0 0
\(709\) 3.46187e12 0.514520 0.257260 0.966342i \(-0.417180\pi\)
0.257260 + 0.966342i \(0.417180\pi\)
\(710\) 0 0
\(711\) −2.56766e12 −0.376812
\(712\) 0 0
\(713\) 1.23357e12 0.178756
\(714\) 0 0
\(715\) −3.43195e12 −0.491094
\(716\) 0 0
\(717\) −3.93666e10 −0.00556277
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) −5.38091e11 −0.0732374
\(724\) 0 0
\(725\) −2.58178e12 −0.347055
\(726\) 0 0
\(727\) 3.13479e12 0.416202 0.208101 0.978107i \(-0.433272\pi\)
0.208101 + 0.978107i \(0.433272\pi\)
\(728\) 0 0
\(729\) −7.54204e12 −0.989043
\(730\) 0 0
\(731\) −1.72334e11 −0.0223225
\(732\) 0 0
\(733\) 6.47775e12 0.828812 0.414406 0.910092i \(-0.363989\pi\)
0.414406 + 0.910092i \(0.363989\pi\)
\(734\) 0 0
\(735\) 1.93697e10 0.00244811
\(736\) 0 0
\(737\) −8.32503e12 −1.03940
\(738\) 0 0
\(739\) 1.12510e12 0.138769 0.0693843 0.997590i \(-0.477897\pi\)
0.0693843 + 0.997590i \(0.477897\pi\)
\(740\) 0 0
\(741\) 1.74564e11 0.0212703
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) 8.25437e12 0.969932
\(748\) 0 0
\(749\) −4.30978e11 −0.0500365
\(750\) 0 0
\(751\) −6.03299e12 −0.692074 −0.346037 0.938221i \(-0.612473\pi\)
−0.346037 + 0.938221i \(0.612473\pi\)
\(752\) 0 0
\(753\) −3.20222e10 −0.00362972
\(754\) 0 0
\(755\) 1.62082e11 0.0181540
\(756\) 0 0
\(757\) −8.02798e12 −0.888535 −0.444268 0.895894i \(-0.646536\pi\)
−0.444268 + 0.895894i \(0.646536\pi\)
\(758\) 0 0
\(759\) 8.64266e10 0.00945278
\(760\) 0 0
\(761\) −6.51923e12 −0.704637 −0.352318 0.935880i \(-0.614607\pi\)
−0.352318 + 0.935880i \(0.614607\pi\)
\(762\) 0 0
\(763\) −3.85026e12 −0.411273
\(764\) 0 0
\(765\) −6.88965e10 −0.00727312
\(766\) 0 0
\(767\) −1.59018e13 −1.65907
\(768\) 0 0
\(769\) −1.34250e13 −1.38435 −0.692175 0.721730i \(-0.743347\pi\)
−0.692175 + 0.721730i \(0.743347\pi\)
\(770\) 0 0
\(771\) −5.01345e11 −0.0510966
\(772\) 0 0
\(773\) 7.85934e12 0.791733 0.395866 0.918308i \(-0.370444\pi\)
0.395866 + 0.918308i \(0.370444\pi\)
\(774\) 0 0
\(775\) −7.60327e12 −0.757082
\(776\) 0 0
\(777\) −1.72098e11 −0.0169387
\(778\) 0 0
\(779\) −5.63235e12 −0.547988
\(780\) 0 0
\(781\) −1.51439e13 −1.45649
\(782\) 0 0
\(783\) −3.71601e11 −0.0353304
\(784\) 0 0
\(785\) 7.78780e12 0.731983
\(786\) 0 0
\(787\) 1.47720e12 0.137263 0.0686316 0.997642i \(-0.478137\pi\)
0.0686316 + 0.997642i \(0.478137\pi\)
\(788\) 0 0
\(789\) −6.51809e11 −0.0598789
\(790\) 0 0
\(791\) 3.42827e12 0.311373
\(792\) 0 0
\(793\) 2.29698e13 2.06266
\(794\) 0 0
\(795\) 5.45033e10 0.00483917
\(796\) 0 0
\(797\) −6.47327e12 −0.568278 −0.284139 0.958783i \(-0.591708\pi\)
−0.284139 + 0.958783i \(0.591708\pi\)
\(798\) 0 0
\(799\) −3.31434e11 −0.0287698
\(800\) 0 0
\(801\) 9.22510e12 0.791817
\(802\) 0 0
\(803\) −2.18786e13 −1.85695
\(804\) 0 0
\(805\) 3.57653e11 0.0300179
\(806\) 0 0
\(807\) 8.48403e11 0.0704160
\(808\) 0 0
\(809\) −1.53631e13 −1.26099 −0.630493 0.776195i \(-0.717147\pi\)
−0.630493 + 0.776195i \(0.717147\pi\)
\(810\) 0 0
\(811\) 1.29056e13 1.04757 0.523787 0.851849i \(-0.324519\pi\)
0.523787 + 0.851849i \(0.324519\pi\)
\(812\) 0 0
\(813\) 5.45012e11 0.0437520
\(814\) 0 0
\(815\) −9.30902e12 −0.739085
\(816\) 0 0
\(817\) 7.07494e12 0.555550
\(818\) 0 0
\(819\) 5.33860e12 0.414620
\(820\) 0 0
\(821\) 8.93771e11 0.0686566 0.0343283 0.999411i \(-0.489071\pi\)
0.0343283 + 0.999411i \(0.489071\pi\)
\(822\) 0 0
\(823\) 2.29844e13 1.74636 0.873182 0.487394i \(-0.162053\pi\)
0.873182 + 0.487394i \(0.162053\pi\)
\(824\) 0 0
\(825\) −5.32701e11 −0.0400351
\(826\) 0 0
\(827\) 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 \(0.537059\pi\)
\(830\) 0 0
\(831\) −1.59045e11 −0.0115695
\(832\) 0 0
\(833\) 3.60992e10 0.00259774
\(834\) 0 0
\(835\) 8.84909e12 0.629955
\(836\) 0 0
\(837\) −1.09435e12 −0.0770714
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) −1.12181e12 −0.0765058
\(844\) 0 0
\(845\) 1.23391e12 0.0832581
\(846\) 0 0
\(847\) 1.37935e12 0.0920874
\(848\) 0 0
\(849\) 3.20048e10 0.00211412
\(850\) 0 0
\(851\) −3.17770e12 −0.207697
\(852\) 0 0
\(853\) −2.60804e13 −1.68672 −0.843362 0.537345i \(-0.819427\pi\)
−0.843362 + 0.537345i \(0.819427\pi\)
\(854\) 0 0
\(855\) 2.82845e12 0.181010
\(856\) 0 0
\(857\) 2.19177e13 1.38797 0.693986 0.719988i \(-0.255853\pi\)
0.693986 + 0.719988i \(0.255853\pi\)
\(858\) 0 0
\(859\) 3.55588e12 0.222832 0.111416 0.993774i \(-0.464461\pi\)
0.111416 + 0.993774i \(0.464461\pi\)
\(860\) 0 0
\(861\) 3.15623e11 0.0195729
\(862\) 0 0
\(863\) −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
\(866\) 0 0
\(867\) −7.11292e11 −0.0427525
\(868\) 0 0
\(869\) 7.07711e12 0.420986
\(870\) 0 0
\(871\) 1.73984e13 1.02430
\(872\) 0 0
\(873\) 1.71420e13 0.998846
\(874\) 0 0
\(875\) −4.83053e12 −0.278586
\(876\) 0 0
\(877\) 3.38004e12 0.192941 0.0964703 0.995336i \(-0.469245\pi\)
0.0964703 + 0.995336i \(0.469245\pi\)
\(878\) 0 0
\(879\) 4.59260e11 0.0259483
\(880\) 0 0
\(881\) 5.25103e11 0.0293665 0.0146833 0.999892i \(-0.495326\pi\)
0.0146833 + 0.999892i \(0.495326\pi\)
\(882\) 0 0
\(883\) −3.33972e13 −1.84879 −0.924393 0.381441i \(-0.875428\pi\)
−0.924393 + 0.381441i \(0.875428\pi\)
\(884\) 0 0
\(885\) 4.72112e11 0.0258702
\(886\) 0 0
\(887\) 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
\(890\) 0 0
\(891\) 2.08646e13 1.10907
\(892\) 0 0
\(893\) 1.36066e13 0.716007
\(894\) 0 0
\(895\) 1.35188e13 0.704264
\(896\) 0 0
\(897\) −1.80623e11 −0.00931549
\(898\) 0 0
\(899\) 7.30271e12 0.372877
\(900\) 0 0
\(901\) 1.01577e11 0.00513494
\(902\) 0 0
\(903\) −3.96462e11 −0.0198430
\(904\) 0 0
\(905\) −2.18026e12 −0.108041
\(906\) 0 0
\(907\) 2.08341e13 1.02221 0.511107 0.859517i \(-0.329236\pi\)
0.511107 + 0.859517i \(0.329236\pi\)
\(908\) 0 0
\(909\) 2.37535e13 1.15396
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) −6.81958e11 −0.0321634
\(916\) 0 0
\(917\) −1.44460e12 −0.0674660
\(918\) 0 0
\(919\) 1.38352e13 0.639830 0.319915 0.947446i \(-0.396346\pi\)
0.319915 + 0.947446i \(0.396346\pi\)
\(920\) 0 0
\(921\) 4.51166e11 0.0206618
\(922\) 0 0
\(923\) 3.16492e13 1.43534
\(924\) 0 0
\(925\) 1.95862e13 0.879653
\(926\) 0 0
\(927\) 1.35675e13 0.603449
\(928\) 0 0
\(929\) −1.74073e13 −0.766761 −0.383380 0.923591i \(-0.625240\pi\)
−0.383380 + 0.923591i \(0.625240\pi\)
\(930\) 0 0
\(931\) −1.48200e12 −0.0646511
\(932\) 0 0
\(933\) −1.29080e12 −0.0557690
\(934\) 0 0
\(935\) 1.89896e11 0.00812575
\(936\) 0 0
\(937\) 1.98361e12 0.0840676 0.0420338 0.999116i \(-0.486616\pi\)
0.0420338 + 0.999116i \(0.486616\pi\)
\(938\) 0 0
\(939\) 5.75445e11 0.0241551
\(940\) 0 0
\(941\) −2.33533e13 −0.970946 −0.485473 0.874252i \(-0.661353\pi\)
−0.485473 + 0.874252i \(0.661353\pi\)
\(942\) 0 0
\(943\) 5.82783e12 0.239996
\(944\) 0 0
\(945\) −3.17289e11 −0.0129423
\(946\) 0 0
\(947\) −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
\(950\) 0 0
\(951\) 1.02352e12 0.0405773
\(952\) 0 0
\(953\) −1.37662e13 −0.540624 −0.270312 0.962773i \(-0.587127\pi\)
−0.270312 + 0.962773i \(0.587127\pi\)
\(954\) 0 0
\(955\) 1.45033e13 0.564225
\(956\) 0 0
\(957\) 5.11643e11 0.0197180
\(958\) 0 0
\(959\) −1.23894e13 −0.473005
\(960\) 0 0
\(961\) −4.93336e12 −0.186590
\(962\) 0 0
\(963\) 3.52662e12 0.132142
\(964\) 0 0
\(965\) 8.92457e12 0.331295
\(966\) 0 0
\(967\) 3.02718e12 0.111332 0.0556659 0.998449i \(-0.482272\pi\)
0.0556659 + 0.998449i \(0.482272\pi\)
\(968\) 0 0
\(969\) −9.65893e9 −0.000351943 0
\(970\) 0 0
\(971\) 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
\(974\) 0 0
\(975\) 1.11329e12 0.0394537
\(976\) 0 0
\(977\) 9.90729e12 0.347880 0.173940 0.984756i \(-0.444350\pi\)
0.173940 + 0.984756i \(0.444350\pi\)
\(978\) 0 0
\(979\) −2.54267e13 −0.884641
\(980\) 0 0
\(981\) 3.15061e13 1.08614
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) −7.62478e11 −0.0255741
\(988\) 0 0
\(989\) −7.32048e12 −0.243308
\(990\) 0 0
\(991\) −3.15782e13 −1.04005 −0.520027 0.854150i \(-0.674078\pi\)
−0.520027 + 0.854150i \(0.674078\pi\)
\(992\) 0 0
\(993\) 7.43950e11 0.0242813
\(994\) 0 0
\(995\) 7.51860e12 0.243183
\(996\) 0 0
\(997\) 9.32241e12 0.298813 0.149407 0.988776i \(-0.452264\pi\)
0.149407 + 0.988776i \(0.452264\pi\)
\(998\) 0 0
\(999\) 2.81907e12 0.0895492
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 112.10.a.b.1.1 1
4.3 odd 2 14.10.a.a.1.1 1
12.11 even 2 126.10.a.e.1.1 1
20.3 even 4 350.10.c.b.99.2 2
20.7 even 4 350.10.c.b.99.1 2
20.19 odd 2 350.10.a.c.1.1 1
28.3 even 6 98.10.c.e.79.1 2
28.11 odd 6 98.10.c.f.79.1 2
28.19 even 6 98.10.c.e.67.1 2
28.23 odd 6 98.10.c.f.67.1 2
28.27 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 4.3 odd 2
98.10.a.a.1.1 1 28.27 even 2
98.10.c.e.67.1 2 28.19 even 6
98.10.c.e.79.1 2 28.3 even 6
98.10.c.f.67.1 2 28.23 odd 6
98.10.c.f.79.1 2 28.11 odd 6
112.10.a.b.1.1 1 1.1 even 1 trivial
126.10.a.e.1.1 1 12.11 even 2
350.10.a.c.1.1 1 20.19 odd 2
350.10.c.b.99.1 2 20.7 even 4
350.10.c.b.99.2 2 20.3 even 4