Properties

Label 234.10.a.c.1.1
Level $234$
Weight $10$
Character 234.1
Self dual yes
Analytic conductor $120.518$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [234,10,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 234.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(120.518385662\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} +256.000 q^{4} +1310.00 q^{5} -5810.00 q^{7} +4096.00 q^{8} +O(q^{10})\) \(q+16.0000 q^{2} +256.000 q^{4} +1310.00 q^{5} -5810.00 q^{7} +4096.00 q^{8} +20960.0 q^{10} +4498.00 q^{11} -28561.0 q^{13} -92960.0 q^{14} +65536.0 q^{16} +237498. q^{17} -913014. q^{19} +335360. q^{20} +71968.0 q^{22} -201544. q^{23} -237025. q^{25} -456976. q^{26} -1.48736e6 q^{28} -1.27683e6 q^{29} +4.16377e6 q^{31} +1.04858e6 q^{32} +3.79997e6 q^{34} -7.61110e6 q^{35} -1.84427e7 q^{37} -1.46082e7 q^{38} +5.36576e6 q^{40} +2.26017e7 q^{41} +1.17263e7 q^{43} +1.15149e6 q^{44} -3.22470e6 q^{46} -5.92915e7 q^{47} -6.59751e6 q^{49} -3.79240e6 q^{50} -7.31162e6 q^{52} -1.08159e8 q^{53} +5.89238e6 q^{55} -2.37978e7 q^{56} -2.04293e7 q^{58} +1.49202e7 q^{59} -5.70037e7 q^{61} +6.66203e7 q^{62} +1.67772e7 q^{64} -3.74149e7 q^{65} +2.20740e7 q^{67} +6.07995e7 q^{68} -1.21778e8 q^{70} -4.44162e7 q^{71} +2.65795e8 q^{73} -2.95083e8 q^{74} -2.33732e8 q^{76} -2.61334e7 q^{77} +4.76755e8 q^{79} +8.58522e7 q^{80} +3.61627e8 q^{82} +5.05316e8 q^{83} +3.11122e8 q^{85} +1.87621e8 q^{86} +1.84238e7 q^{88} -8.90841e8 q^{89} +1.65939e8 q^{91} -5.15953e7 q^{92} -9.48665e8 q^{94} -1.19605e9 q^{95} -8.02777e8 q^{97} -1.05560e8 q^{98} +O(q^{100})\)

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
\(5\) 1310.00 0.937360 0.468680 0.883368i \(-0.344730\pi\)
0.468680 + 0.883368i \(0.344730\pi\)
\(6\) 0 0
\(7\) −5810.00 −0.914608 −0.457304 0.889310i \(-0.651185\pi\)
−0.457304 + 0.889310i \(0.651185\pi\)
\(8\) 4096.00 0.353553
\(9\) 0 0
\(10\) 20960.0 0.662813
\(11\) 4498.00 0.0926302 0.0463151 0.998927i \(-0.485252\pi\)
0.0463151 + 0.998927i \(0.485252\pi\)
\(12\) 0 0
\(13\) −28561.0 −0.277350
\(14\) −92960.0 −0.646725
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) 237498. 0.689668 0.344834 0.938664i \(-0.387935\pi\)
0.344834 + 0.938664i \(0.387935\pi\)
\(18\) 0 0
\(19\) −913014. −1.60726 −0.803630 0.595129i \(-0.797101\pi\)
−0.803630 + 0.595129i \(0.797101\pi\)
\(20\) 335360. 0.468680
\(21\) 0 0
\(22\) 71968.0 0.0654994
\(23\) −201544. −0.150174 −0.0750870 0.997177i \(-0.523923\pi\)
−0.0750870 + 0.997177i \(0.523923\pi\)
\(24\) 0 0
\(25\) −237025. −0.121357
\(26\) −456976. −0.196116
\(27\) 0 0
\(28\) −1.48736e6 −0.457304
\(29\) −1.27683e6 −0.335230 −0.167615 0.985852i \(-0.553607\pi\)
−0.167615 + 0.985852i \(0.553607\pi\)
\(30\) 0 0
\(31\) 4.16377e6 0.809765 0.404883 0.914369i \(-0.367312\pi\)
0.404883 + 0.914369i \(0.367312\pi\)
\(32\) 1.04858e6 0.176777
\(33\) 0 0
\(34\) 3.79997e6 0.487669
\(35\) −7.61110e6 −0.857317
\(36\) 0 0
\(37\) −1.84427e7 −1.61777 −0.808883 0.587969i \(-0.799928\pi\)
−0.808883 + 0.587969i \(0.799928\pi\)
\(38\) −1.46082e7 −1.13650
\(39\) 0 0
\(40\) 5.36576e6 0.331407
\(41\) 2.26017e7 1.24915 0.624573 0.780966i \(-0.285273\pi\)
0.624573 + 0.780966i \(0.285273\pi\)
\(42\) 0 0
\(43\) 1.17263e7 0.523062 0.261531 0.965195i \(-0.415773\pi\)
0.261531 + 0.965195i \(0.415773\pi\)
\(44\) 1.15149e6 0.0463151
\(45\) 0 0
\(46\) −3.22470e6 −0.106189
\(47\) −5.92915e7 −1.77236 −0.886181 0.463340i \(-0.846651\pi\)
−0.886181 + 0.463340i \(0.846651\pi\)
\(48\) 0 0
\(49\) −6.59751e6 −0.163492
\(50\) −3.79240e6 −0.0858122
\(51\) 0 0
\(52\) −7.31162e6 −0.138675
\(53\) −1.08159e8 −1.88287 −0.941434 0.337196i \(-0.890521\pi\)
−0.941434 + 0.337196i \(0.890521\pi\)
\(54\) 0 0
\(55\) 5.89238e6 0.0868278
\(56\) −2.37978e7 −0.323363
\(57\) 0 0
\(58\) −2.04293e7 −0.237044
\(59\) 1.49202e7 0.160302 0.0801511 0.996783i \(-0.474460\pi\)
0.0801511 + 0.996783i \(0.474460\pi\)
\(60\) 0 0
\(61\) −5.70037e7 −0.527132 −0.263566 0.964641i \(-0.584899\pi\)
−0.263566 + 0.964641i \(0.584899\pi\)
\(62\) 6.66203e7 0.572590
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) −3.74149e7 −0.259977
\(66\) 0 0
\(67\) 2.20740e7 0.133827 0.0669136 0.997759i \(-0.478685\pi\)
0.0669136 + 0.997759i \(0.478685\pi\)
\(68\) 6.07995e7 0.344834
\(69\) 0 0
\(70\) −1.21778e8 −0.606214
\(71\) −4.44162e7 −0.207434 −0.103717 0.994607i \(-0.533074\pi\)
−0.103717 + 0.994607i \(0.533074\pi\)
\(72\) 0 0
\(73\) 2.65795e8 1.09545 0.547726 0.836658i \(-0.315494\pi\)
0.547726 + 0.836658i \(0.315494\pi\)
\(74\) −2.95083e8 −1.14393
\(75\) 0 0
\(76\) −2.33732e8 −0.803630
\(77\) −2.61334e7 −0.0847203
\(78\) 0 0
\(79\) 4.76755e8 1.37713 0.688563 0.725176i \(-0.258242\pi\)
0.688563 + 0.725176i \(0.258242\pi\)
\(80\) 8.58522e7 0.234340
\(81\) 0 0
\(82\) 3.61627e8 0.883280
\(83\) 5.05316e8 1.16872 0.584361 0.811494i \(-0.301345\pi\)
0.584361 + 0.811494i \(0.301345\pi\)
\(84\) 0 0
\(85\) 3.11122e8 0.646467
\(86\) 1.87621e8 0.369861
\(87\) 0 0
\(88\) 1.84238e7 0.0327497
\(89\) −8.90841e8 −1.50503 −0.752515 0.658575i \(-0.771159\pi\)
−0.752515 + 0.658575i \(0.771159\pi\)
\(90\) 0 0
\(91\) 1.65939e8 0.253667
\(92\) −5.15953e7 −0.0750870
\(93\) 0 0
\(94\) −9.48665e8 −1.25325
\(95\) −1.19605e9 −1.50658
\(96\) 0 0
\(97\) −8.02777e8 −0.920708 −0.460354 0.887735i \(-0.652278\pi\)
−0.460354 + 0.887735i \(0.652278\pi\)
\(98\) −1.05560e8 −0.115607
\(99\) 0 0
\(100\) −6.06784e7 −0.0606784
\(101\) −1.19998e9 −1.14743 −0.573717 0.819053i \(-0.694499\pi\)
−0.573717 + 0.819053i \(0.694499\pi\)
\(102\) 0 0
\(103\) −9.58027e8 −0.838707 −0.419353 0.907823i \(-0.637743\pi\)
−0.419353 + 0.907823i \(0.637743\pi\)
\(104\) −1.16986e8 −0.0980581
\(105\) 0 0
\(106\) −1.73054e9 −1.33139
\(107\) 2.39051e9 1.76304 0.881521 0.472145i \(-0.156520\pi\)
0.881521 + 0.472145i \(0.156520\pi\)
\(108\) 0 0
\(109\) −1.70171e9 −1.15469 −0.577346 0.816499i \(-0.695912\pi\)
−0.577346 + 0.816499i \(0.695912\pi\)
\(110\) 9.42781e7 0.0613965
\(111\) 0 0
\(112\) −3.80764e8 −0.228652
\(113\) 1.40793e9 0.812320 0.406160 0.913802i \(-0.366868\pi\)
0.406160 + 0.913802i \(0.366868\pi\)
\(114\) 0 0
\(115\) −2.64023e8 −0.140767
\(116\) −3.26870e8 −0.167615
\(117\) 0 0
\(118\) 2.38722e8 0.113351
\(119\) −1.37986e9 −0.630775
\(120\) 0 0
\(121\) −2.33772e9 −0.991420
\(122\) −9.12060e8 −0.372738
\(123\) 0 0
\(124\) 1.06593e9 0.404883
\(125\) −2.86910e9 −1.05111
\(126\) 0 0
\(127\) −3.31210e9 −1.12976 −0.564881 0.825172i \(-0.691078\pi\)
−0.564881 + 0.825172i \(0.691078\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 0 0
\(130\) −5.98639e8 −0.183831
\(131\) −3.06389e9 −0.908977 −0.454489 0.890753i \(-0.650178\pi\)
−0.454489 + 0.890753i \(0.650178\pi\)
\(132\) 0 0
\(133\) 5.30461e9 1.47001
\(134\) 3.53184e8 0.0946302
\(135\) 0 0
\(136\) 9.72792e8 0.243834
\(137\) −5.62781e8 −0.136489 −0.0682444 0.997669i \(-0.521740\pi\)
−0.0682444 + 0.997669i \(0.521740\pi\)
\(138\) 0 0
\(139\) −4.60597e8 −0.104654 −0.0523268 0.998630i \(-0.516664\pi\)
−0.0523268 + 0.998630i \(0.516664\pi\)
\(140\) −1.94844e9 −0.428658
\(141\) 0 0
\(142\) −7.10660e8 −0.146678
\(143\) −1.28467e8 −0.0256910
\(144\) 0 0
\(145\) −1.67265e9 −0.314232
\(146\) 4.25271e9 0.774601
\(147\) 0 0
\(148\) −4.72132e9 −0.808883
\(149\) 6.01717e8 0.100012 0.0500062 0.998749i \(-0.484076\pi\)
0.0500062 + 0.998749i \(0.484076\pi\)
\(150\) 0 0
\(151\) −1.26695e10 −1.98318 −0.991589 0.129428i \(-0.958686\pi\)
−0.991589 + 0.129428i \(0.958686\pi\)
\(152\) −3.73971e9 −0.568252
\(153\) 0 0
\(154\) −4.18134e8 −0.0599063
\(155\) 5.45454e9 0.759041
\(156\) 0 0
\(157\) −2.00733e8 −0.0263676 −0.0131838 0.999913i \(-0.504197\pi\)
−0.0131838 + 0.999913i \(0.504197\pi\)
\(158\) 7.62809e9 0.973775
\(159\) 0 0
\(160\) 1.37363e9 0.165703
\(161\) 1.17097e9 0.137350
\(162\) 0 0
\(163\) 6.32491e9 0.701795 0.350898 0.936414i \(-0.385877\pi\)
0.350898 + 0.936414i \(0.385877\pi\)
\(164\) 5.78603e9 0.624573
\(165\) 0 0
\(166\) 8.08505e9 0.826412
\(167\) −1.51400e10 −1.50627 −0.753134 0.657867i \(-0.771459\pi\)
−0.753134 + 0.657867i \(0.771459\pi\)
\(168\) 0 0
\(169\) 8.15731e8 0.0769231
\(170\) 4.97796e9 0.457121
\(171\) 0 0
\(172\) 3.00193e9 0.261531
\(173\) 1.63483e9 0.138760 0.0693802 0.997590i \(-0.477898\pi\)
0.0693802 + 0.997590i \(0.477898\pi\)
\(174\) 0 0
\(175\) 1.37712e9 0.110994
\(176\) 2.94781e8 0.0231575
\(177\) 0 0
\(178\) −1.42535e10 −1.06422
\(179\) 4.12980e9 0.300670 0.150335 0.988635i \(-0.451965\pi\)
0.150335 + 0.988635i \(0.451965\pi\)
\(180\) 0 0
\(181\) −2.13092e10 −1.47575 −0.737875 0.674937i \(-0.764171\pi\)
−0.737875 + 0.674937i \(0.764171\pi\)
\(182\) 2.65503e9 0.179369
\(183\) 0 0
\(184\) −8.25524e8 −0.0530945
\(185\) −2.41599e10 −1.51643
\(186\) 0 0
\(187\) 1.06827e9 0.0638840
\(188\) −1.51786e10 −0.886181
\(189\) 0 0
\(190\) −1.91368e10 −1.06531
\(191\) 3.08641e10 1.67804 0.839021 0.544099i \(-0.183128\pi\)
0.839021 + 0.544099i \(0.183128\pi\)
\(192\) 0 0
\(193\) −4.54917e9 −0.236007 −0.118003 0.993013i \(-0.537649\pi\)
−0.118003 + 0.993013i \(0.537649\pi\)
\(194\) −1.28444e10 −0.651039
\(195\) 0 0
\(196\) −1.68896e9 −0.0817462
\(197\) −2.26076e10 −1.06944 −0.534720 0.845030i \(-0.679583\pi\)
−0.534720 + 0.845030i \(0.679583\pi\)
\(198\) 0 0
\(199\) 1.05027e10 0.474749 0.237375 0.971418i \(-0.423713\pi\)
0.237375 + 0.971418i \(0.423713\pi\)
\(200\) −9.70854e8 −0.0429061
\(201\) 0 0
\(202\) −1.91997e10 −0.811359
\(203\) 7.41841e9 0.306604
\(204\) 0 0
\(205\) 2.96082e10 1.17090
\(206\) −1.53284e10 −0.593055
\(207\) 0 0
\(208\) −1.87177e9 −0.0693375
\(209\) −4.10674e9 −0.148881
\(210\) 0 0
\(211\) −5.66420e9 −0.196729 −0.0983643 0.995150i \(-0.531361\pi\)
−0.0983643 + 0.995150i \(0.531361\pi\)
\(212\) −2.76886e10 −0.941434
\(213\) 0 0
\(214\) 3.82481e10 1.24666
\(215\) 1.53615e10 0.490297
\(216\) 0 0
\(217\) −2.41915e10 −0.740618
\(218\) −2.72274e10 −0.816491
\(219\) 0 0
\(220\) 1.50845e9 0.0434139
\(221\) −6.78318e9 −0.191279
\(222\) 0 0
\(223\) −3.19607e10 −0.865454 −0.432727 0.901525i \(-0.642449\pi\)
−0.432727 + 0.901525i \(0.642449\pi\)
\(224\) −6.09223e9 −0.161681
\(225\) 0 0
\(226\) 2.25268e10 0.574397
\(227\) 5.07782e10 1.26929 0.634645 0.772804i \(-0.281146\pi\)
0.634645 + 0.772804i \(0.281146\pi\)
\(228\) 0 0
\(229\) 5.99836e10 1.44136 0.720681 0.693267i \(-0.243829\pi\)
0.720681 + 0.693267i \(0.243829\pi\)
\(230\) −4.22436e9 −0.0995373
\(231\) 0 0
\(232\) −5.22991e9 −0.118522
\(233\) −4.77228e10 −1.06078 −0.530389 0.847755i \(-0.677954\pi\)
−0.530389 + 0.847755i \(0.677954\pi\)
\(234\) 0 0
\(235\) −7.76719e10 −1.66134
\(236\) 3.81956e9 0.0801511
\(237\) 0 0
\(238\) −2.20778e10 −0.446026
\(239\) 8.71569e10 1.72787 0.863936 0.503602i \(-0.167992\pi\)
0.863936 + 0.503602i \(0.167992\pi\)
\(240\) 0 0
\(241\) −1.04205e11 −1.98981 −0.994903 0.100835i \(-0.967849\pi\)
−0.994903 + 0.100835i \(0.967849\pi\)
\(242\) −3.74035e10 −0.701040
\(243\) 0 0
\(244\) −1.45930e10 −0.263566
\(245\) −8.64273e9 −0.153251
\(246\) 0 0
\(247\) 2.60766e10 0.445774
\(248\) 1.70548e10 0.286295
\(249\) 0 0
\(250\) −4.59055e10 −0.743250
\(251\) −2.82027e9 −0.0448496 −0.0224248 0.999749i \(-0.507139\pi\)
−0.0224248 + 0.999749i \(0.507139\pi\)
\(252\) 0 0
\(253\) −9.06545e8 −0.0139106
\(254\) −5.29937e10 −0.798863
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) 1.41573e10 0.202433 0.101216 0.994864i \(-0.467727\pi\)
0.101216 + 0.994864i \(0.467727\pi\)
\(258\) 0 0
\(259\) 1.07152e11 1.47962
\(260\) −9.57822e9 −0.129988
\(261\) 0 0
\(262\) −4.90223e10 −0.642744
\(263\) 3.58497e10 0.462045 0.231023 0.972948i \(-0.425793\pi\)
0.231023 + 0.972948i \(0.425793\pi\)
\(264\) 0 0
\(265\) −1.41688e11 −1.76493
\(266\) 8.48738e10 1.03946
\(267\) 0 0
\(268\) 5.65095e9 0.0669136
\(269\) 7.14394e10 0.831864 0.415932 0.909396i \(-0.363455\pi\)
0.415932 + 0.909396i \(0.363455\pi\)
\(270\) 0 0
\(271\) −6.79344e9 −0.0765117 −0.0382558 0.999268i \(-0.512180\pi\)
−0.0382558 + 0.999268i \(0.512180\pi\)
\(272\) 1.55647e10 0.172417
\(273\) 0 0
\(274\) −9.00450e9 −0.0965122
\(275\) −1.06614e9 −0.0112413
\(276\) 0 0
\(277\) −6.93103e10 −0.707357 −0.353679 0.935367i \(-0.615069\pi\)
−0.353679 + 0.935367i \(0.615069\pi\)
\(278\) −7.36955e9 −0.0740013
\(279\) 0 0
\(280\) −3.11751e10 −0.303107
\(281\) −3.10369e10 −0.296961 −0.148480 0.988915i \(-0.547438\pi\)
−0.148480 + 0.988915i \(0.547438\pi\)
\(282\) 0 0
\(283\) 1.35312e11 1.25400 0.627001 0.779018i \(-0.284282\pi\)
0.627001 + 0.779018i \(0.284282\pi\)
\(284\) −1.13706e10 −0.103717
\(285\) 0 0
\(286\) −2.05548e9 −0.0181663
\(287\) −1.31316e11 −1.14248
\(288\) 0 0
\(289\) −6.21826e10 −0.524359
\(290\) −2.67624e10 −0.222195
\(291\) 0 0
\(292\) 6.80434e10 0.547726
\(293\) 7.55078e10 0.598532 0.299266 0.954170i \(-0.403258\pi\)
0.299266 + 0.954170i \(0.403258\pi\)
\(294\) 0 0
\(295\) 1.95454e10 0.150261
\(296\) −7.55411e10 −0.571967
\(297\) 0 0
\(298\) 9.62747e9 0.0707195
\(299\) 5.75630e9 0.0416508
\(300\) 0 0
\(301\) −6.81298e10 −0.478397
\(302\) −2.02711e11 −1.40232
\(303\) 0 0
\(304\) −5.98353e10 −0.401815
\(305\) −7.46749e10 −0.494112
\(306\) 0 0
\(307\) 1.42760e10 0.0917241 0.0458620 0.998948i \(-0.485397\pi\)
0.0458620 + 0.998948i \(0.485397\pi\)
\(308\) −6.69015e9 −0.0423601
\(309\) 0 0
\(310\) 8.72726e10 0.536723
\(311\) −3.58426e9 −0.0217259 −0.0108630 0.999941i \(-0.503458\pi\)
−0.0108630 + 0.999941i \(0.503458\pi\)
\(312\) 0 0
\(313\) 2.79830e11 1.64795 0.823977 0.566623i \(-0.191750\pi\)
0.823977 + 0.566623i \(0.191750\pi\)
\(314\) −3.21173e9 −0.0186447
\(315\) 0 0
\(316\) 1.22049e11 0.688563
\(317\) −2.40148e10 −0.133571 −0.0667855 0.997767i \(-0.521274\pi\)
−0.0667855 + 0.997767i \(0.521274\pi\)
\(318\) 0 0
\(319\) −5.74320e9 −0.0310524
\(320\) 2.19782e10 0.117170
\(321\) 0 0
\(322\) 1.87355e10 0.0971213
\(323\) −2.16839e11 −1.10848
\(324\) 0 0
\(325\) 6.76967e9 0.0336583
\(326\) 1.01199e11 0.496244
\(327\) 0 0
\(328\) 9.25764e10 0.441640
\(329\) 3.44484e11 1.62102
\(330\) 0 0
\(331\) 3.73009e11 1.70802 0.854010 0.520257i \(-0.174164\pi\)
0.854010 + 0.520257i \(0.174164\pi\)
\(332\) 1.29361e11 0.584361
\(333\) 0 0
\(334\) −2.42240e11 −1.06509
\(335\) 2.89170e10 0.125444
\(336\) 0 0
\(337\) 1.91157e11 0.807340 0.403670 0.914905i \(-0.367734\pi\)
0.403670 + 0.914905i \(0.367734\pi\)
\(338\) 1.30517e10 0.0543928
\(339\) 0 0
\(340\) 7.96473e10 0.323233
\(341\) 1.87286e10 0.0750087
\(342\) 0 0
\(343\) 2.72786e11 1.06414
\(344\) 4.80310e10 0.184930
\(345\) 0 0
\(346\) 2.61573e10 0.0981184
\(347\) −8.60398e10 −0.318579 −0.159289 0.987232i \(-0.550920\pi\)
−0.159289 + 0.987232i \(0.550920\pi\)
\(348\) 0 0
\(349\) −1.33612e11 −0.482094 −0.241047 0.970513i \(-0.577491\pi\)
−0.241047 + 0.970513i \(0.577491\pi\)
\(350\) 2.20338e10 0.0784845
\(351\) 0 0
\(352\) 4.71649e9 0.0163749
\(353\) 6.23799e10 0.213825 0.106912 0.994268i \(-0.465904\pi\)
0.106912 + 0.994268i \(0.465904\pi\)
\(354\) 0 0
\(355\) −5.81853e10 −0.194440
\(356\) −2.28055e11 −0.752515
\(357\) 0 0
\(358\) 6.60767e10 0.212606
\(359\) 3.82739e11 1.21612 0.608062 0.793890i \(-0.291947\pi\)
0.608062 + 0.793890i \(0.291947\pi\)
\(360\) 0 0
\(361\) 5.10907e11 1.58329
\(362\) −3.40947e11 −1.04351
\(363\) 0 0
\(364\) 4.24805e10 0.126833
\(365\) 3.48191e11 1.02683
\(366\) 0 0
\(367\) 2.59802e11 0.747560 0.373780 0.927517i \(-0.378062\pi\)
0.373780 + 0.927517i \(0.378062\pi\)
\(368\) −1.32084e10 −0.0375435
\(369\) 0 0
\(370\) −3.86558e11 −1.07228
\(371\) 6.28402e11 1.72209
\(372\) 0 0
\(373\) 4.70946e11 1.25974 0.629870 0.776700i \(-0.283108\pi\)
0.629870 + 0.776700i \(0.283108\pi\)
\(374\) 1.70923e10 0.0451728
\(375\) 0 0
\(376\) −2.42858e11 −0.626624
\(377\) 3.64677e10 0.0929762
\(378\) 0 0
\(379\) −3.60046e11 −0.896358 −0.448179 0.893944i \(-0.647927\pi\)
−0.448179 + 0.893944i \(0.647927\pi\)
\(380\) −3.06188e11 −0.753291
\(381\) 0 0
\(382\) 4.93825e11 1.18655
\(383\) 9.59380e10 0.227822 0.113911 0.993491i \(-0.463662\pi\)
0.113911 + 0.993491i \(0.463662\pi\)
\(384\) 0 0
\(385\) −3.42347e10 −0.0794134
\(386\) −7.27868e10 −0.166882
\(387\) 0 0
\(388\) −2.05511e11 −0.460354
\(389\) 4.60488e11 1.01964 0.509818 0.860282i \(-0.329713\pi\)
0.509818 + 0.860282i \(0.329713\pi\)
\(390\) 0 0
\(391\) −4.78663e10 −0.103570
\(392\) −2.70234e10 −0.0578033
\(393\) 0 0
\(394\) −3.61721e11 −0.756208
\(395\) 6.24550e11 1.29086
\(396\) 0 0
\(397\) 2.90299e11 0.586528 0.293264 0.956032i \(-0.405259\pi\)
0.293264 + 0.956032i \(0.405259\pi\)
\(398\) 1.68044e11 0.335698
\(399\) 0 0
\(400\) −1.55337e10 −0.0303392
\(401\) −6.85495e11 −1.32390 −0.661949 0.749549i \(-0.730270\pi\)
−0.661949 + 0.749549i \(0.730270\pi\)
\(402\) 0 0
\(403\) −1.18921e11 −0.224588
\(404\) −3.07195e11 −0.573717
\(405\) 0 0
\(406\) 1.18694e11 0.216802
\(407\) −8.29551e10 −0.149854
\(408\) 0 0
\(409\) 1.00030e12 1.76756 0.883779 0.467905i \(-0.154991\pi\)
0.883779 + 0.467905i \(0.154991\pi\)
\(410\) 4.73731e11 0.827951
\(411\) 0 0
\(412\) −2.45255e11 −0.419353
\(413\) −8.66861e10 −0.146614
\(414\) 0 0
\(415\) 6.61964e11 1.09551
\(416\) −2.99484e10 −0.0490290
\(417\) 0 0
\(418\) −6.57078e10 −0.105275
\(419\) 8.64798e11 1.37073 0.685364 0.728200i \(-0.259643\pi\)
0.685364 + 0.728200i \(0.259643\pi\)
\(420\) 0 0
\(421\) −9.57784e10 −0.148593 −0.0742965 0.997236i \(-0.523671\pi\)
−0.0742965 + 0.997236i \(0.523671\pi\)
\(422\) −9.06272e10 −0.139108
\(423\) 0 0
\(424\) −4.43018e11 −0.665695
\(425\) −5.62930e10 −0.0836959
\(426\) 0 0
\(427\) 3.31192e11 0.482119
\(428\) 6.11969e11 0.881521
\(429\) 0 0
\(430\) 2.45783e11 0.346693
\(431\) 1.27185e11 0.177536 0.0887682 0.996052i \(-0.471707\pi\)
0.0887682 + 0.996052i \(0.471707\pi\)
\(432\) 0 0
\(433\) −1.55264e11 −0.212264 −0.106132 0.994352i \(-0.533847\pi\)
−0.106132 + 0.994352i \(0.533847\pi\)
\(434\) −3.87064e11 −0.523696
\(435\) 0 0
\(436\) −4.35638e11 −0.577346
\(437\) 1.84012e11 0.241369
\(438\) 0 0
\(439\) −1.02610e12 −1.31855 −0.659277 0.751901i \(-0.729137\pi\)
−0.659277 + 0.751901i \(0.729137\pi\)
\(440\) 2.41352e10 0.0306983
\(441\) 0 0
\(442\) −1.08531e11 −0.135255
\(443\) −2.52039e11 −0.310922 −0.155461 0.987842i \(-0.549686\pi\)
−0.155461 + 0.987842i \(0.549686\pi\)
\(444\) 0 0
\(445\) −1.16700e12 −1.41075
\(446\) −5.11371e11 −0.611969
\(447\) 0 0
\(448\) −9.74756e10 −0.114326
\(449\) −7.66198e11 −0.889678 −0.444839 0.895611i \(-0.646739\pi\)
−0.444839 + 0.895611i \(0.646739\pi\)
\(450\) 0 0
\(451\) 1.01662e11 0.115709
\(452\) 3.60429e11 0.406160
\(453\) 0 0
\(454\) 8.12451e11 0.897524
\(455\) 2.17381e11 0.237777
\(456\) 0 0
\(457\) 1.75683e12 1.88411 0.942057 0.335454i \(-0.108890\pi\)
0.942057 + 0.335454i \(0.108890\pi\)
\(458\) 9.59738e11 1.01920
\(459\) 0 0
\(460\) −6.75898e10 −0.0703835
\(461\) 1.13127e12 1.16657 0.583287 0.812266i \(-0.301766\pi\)
0.583287 + 0.812266i \(0.301766\pi\)
\(462\) 0 0
\(463\) −2.71657e11 −0.274730 −0.137365 0.990521i \(-0.543863\pi\)
−0.137365 + 0.990521i \(0.543863\pi\)
\(464\) −8.36786e10 −0.0838076
\(465\) 0 0
\(466\) −7.63565e11 −0.750083
\(467\) 9.54617e11 0.928759 0.464380 0.885636i \(-0.346277\pi\)
0.464380 + 0.885636i \(0.346277\pi\)
\(468\) 0 0
\(469\) −1.28250e11 −0.122399
\(470\) −1.24275e12 −1.17474
\(471\) 0 0
\(472\) 6.11130e10 0.0566754
\(473\) 5.27449e10 0.0484513
\(474\) 0 0
\(475\) 2.16407e11 0.195052
\(476\) −3.53245e11 −0.315388
\(477\) 0 0
\(478\) 1.39451e12 1.22179
\(479\) −1.43680e12 −1.24705 −0.623527 0.781802i \(-0.714301\pi\)
−0.623527 + 0.781802i \(0.714301\pi\)
\(480\) 0 0
\(481\) 5.26741e11 0.448688
\(482\) −1.66728e12 −1.40701
\(483\) 0 0
\(484\) −5.98455e11 −0.495710
\(485\) −1.05164e12 −0.863035
\(486\) 0 0
\(487\) −1.22744e12 −0.988826 −0.494413 0.869227i \(-0.664617\pi\)
−0.494413 + 0.869227i \(0.664617\pi\)
\(488\) −2.33487e11 −0.186369
\(489\) 0 0
\(490\) −1.38284e11 −0.108365
\(491\) 1.00389e12 0.779506 0.389753 0.920919i \(-0.372560\pi\)
0.389753 + 0.920919i \(0.372560\pi\)
\(492\) 0 0
\(493\) −3.03246e11 −0.231198
\(494\) 4.17225e11 0.315210
\(495\) 0 0
\(496\) 2.72877e11 0.202441
\(497\) 2.58058e11 0.189721
\(498\) 0 0
\(499\) −7.58262e11 −0.547478 −0.273739 0.961804i \(-0.588260\pi\)
−0.273739 + 0.961804i \(0.588260\pi\)
\(500\) −7.34489e11 −0.525557
\(501\) 0 0
\(502\) −4.51243e10 −0.0317134
\(503\) −1.82032e12 −1.26792 −0.633959 0.773367i \(-0.718571\pi\)
−0.633959 + 0.773367i \(0.718571\pi\)
\(504\) 0 0
\(505\) −1.57197e12 −1.07556
\(506\) −1.45047e10 −0.00983630
\(507\) 0 0
\(508\) −8.47899e11 −0.564881
\(509\) −6.57012e11 −0.433854 −0.216927 0.976188i \(-0.569603\pi\)
−0.216927 + 0.976188i \(0.569603\pi\)
\(510\) 0 0
\(511\) −1.54427e12 −1.00191
\(512\) 6.87195e10 0.0441942
\(513\) 0 0
\(514\) 2.26516e11 0.143141
\(515\) −1.25502e12 −0.786170
\(516\) 0 0
\(517\) −2.66693e11 −0.164174
\(518\) 1.71443e12 1.04625
\(519\) 0 0
\(520\) −1.53251e11 −0.0919157
\(521\) −3.17678e11 −0.188894 −0.0944468 0.995530i \(-0.530108\pi\)
−0.0944468 + 0.995530i \(0.530108\pi\)
\(522\) 0 0
\(523\) −2.88365e12 −1.68533 −0.842666 0.538436i \(-0.819015\pi\)
−0.842666 + 0.538436i \(0.819015\pi\)
\(524\) −7.84357e11 −0.454489
\(525\) 0 0
\(526\) 5.73595e11 0.326715
\(527\) 9.88887e11 0.558469
\(528\) 0 0
\(529\) −1.76053e12 −0.977448
\(530\) −2.26701e12 −1.24799
\(531\) 0 0
\(532\) 1.35798e12 0.735007
\(533\) −6.45526e11 −0.346451
\(534\) 0 0
\(535\) 3.13156e12 1.65260
\(536\) 9.04151e10 0.0473151
\(537\) 0 0
\(538\) 1.14303e12 0.588217
\(539\) −2.96756e10 −0.0151443
\(540\) 0 0
\(541\) 2.16753e12 1.08787 0.543937 0.839126i \(-0.316933\pi\)
0.543937 + 0.839126i \(0.316933\pi\)
\(542\) −1.08695e11 −0.0541019
\(543\) 0 0
\(544\) 2.49035e11 0.121917
\(545\) −2.22924e12 −1.08236
\(546\) 0 0
\(547\) −9.14427e11 −0.436723 −0.218361 0.975868i \(-0.570071\pi\)
−0.218361 + 0.975868i \(0.570071\pi\)
\(548\) −1.44072e11 −0.0682444
\(549\) 0 0
\(550\) −1.70582e10 −0.00794880
\(551\) 1.16577e12 0.538803
\(552\) 0 0
\(553\) −2.76995e12 −1.25953
\(554\) −1.10896e12 −0.500177
\(555\) 0 0
\(556\) −1.17913e11 −0.0523268
\(557\) −1.75791e10 −0.00773834 −0.00386917 0.999993i \(-0.501232\pi\)
−0.00386917 + 0.999993i \(0.501232\pi\)
\(558\) 0 0
\(559\) −3.34915e11 −0.145071
\(560\) −4.98801e11 −0.214329
\(561\) 0 0
\(562\) −4.96590e11 −0.209983
\(563\) 3.70644e12 1.55478 0.777390 0.629019i \(-0.216543\pi\)
0.777390 + 0.629019i \(0.216543\pi\)
\(564\) 0 0
\(565\) 1.84438e12 0.761436
\(566\) 2.16500e12 0.886713
\(567\) 0 0
\(568\) −1.81929e11 −0.0733389
\(569\) 3.40051e12 1.36000 0.679999 0.733213i \(-0.261980\pi\)
0.679999 + 0.733213i \(0.261980\pi\)
\(570\) 0 0
\(571\) −4.46270e12 −1.75685 −0.878427 0.477877i \(-0.841406\pi\)
−0.878427 + 0.477877i \(0.841406\pi\)
\(572\) −3.28876e10 −0.0128455
\(573\) 0 0
\(574\) −2.10105e12 −0.807854
\(575\) 4.77710e10 0.0182246
\(576\) 0 0
\(577\) 3.47193e12 1.30401 0.652004 0.758215i \(-0.273929\pi\)
0.652004 + 0.758215i \(0.273929\pi\)
\(578\) −9.94921e11 −0.370778
\(579\) 0 0
\(580\) −4.28199e11 −0.157116
\(581\) −2.93588e12 −1.06892
\(582\) 0 0
\(583\) −4.86498e11 −0.174410
\(584\) 1.08869e12 0.387301
\(585\) 0 0
\(586\) 1.20812e12 0.423226
\(587\) −2.86266e12 −0.995171 −0.497586 0.867415i \(-0.665780\pi\)
−0.497586 + 0.867415i \(0.665780\pi\)
\(588\) 0 0
\(589\) −3.80158e12 −1.30150
\(590\) 3.12726e11 0.106250
\(591\) 0 0
\(592\) −1.20866e12 −0.404442
\(593\) 8.38217e11 0.278362 0.139181 0.990267i \(-0.455553\pi\)
0.139181 + 0.990267i \(0.455553\pi\)
\(594\) 0 0
\(595\) −1.80762e12 −0.591263
\(596\) 1.54039e11 0.0500062
\(597\) 0 0
\(598\) 9.21008e10 0.0294515
\(599\) −3.94489e12 −1.25203 −0.626013 0.779812i \(-0.715315\pi\)
−0.626013 + 0.779812i \(0.715315\pi\)
\(600\) 0 0
\(601\) 4.99865e12 1.56285 0.781426 0.623998i \(-0.214492\pi\)
0.781426 + 0.623998i \(0.214492\pi\)
\(602\) −1.09008e12 −0.338278
\(603\) 0 0
\(604\) −3.24338e12 −0.991589
\(605\) −3.06241e12 −0.929317
\(606\) 0 0
\(607\) −3.95582e11 −0.118274 −0.0591368 0.998250i \(-0.518835\pi\)
−0.0591368 + 0.998250i \(0.518835\pi\)
\(608\) −9.57365e11 −0.284126
\(609\) 0 0
\(610\) −1.19480e12 −0.349390
\(611\) 1.69343e12 0.491565
\(612\) 0 0
\(613\) 5.03617e12 1.44055 0.720275 0.693688i \(-0.244016\pi\)
0.720275 + 0.693688i \(0.244016\pi\)
\(614\) 2.28416e11 0.0648587
\(615\) 0 0
\(616\) −1.07042e11 −0.0299531
\(617\) −4.06829e12 −1.13013 −0.565065 0.825046i \(-0.691149\pi\)
−0.565065 + 0.825046i \(0.691149\pi\)
\(618\) 0 0
\(619\) −4.73136e12 −1.29532 −0.647662 0.761928i \(-0.724253\pi\)
−0.647662 + 0.761928i \(0.724253\pi\)
\(620\) 1.39636e12 0.379521
\(621\) 0 0
\(622\) −5.73482e10 −0.0153626
\(623\) 5.17578e12 1.37651
\(624\) 0 0
\(625\) −3.29558e12 −0.863916
\(626\) 4.47728e12 1.16528
\(627\) 0 0
\(628\) −5.13877e10 −0.0131838
\(629\) −4.38010e12 −1.11572
\(630\) 0 0
\(631\) 3.45019e12 0.866384 0.433192 0.901302i \(-0.357387\pi\)
0.433192 + 0.901302i \(0.357387\pi\)
\(632\) 1.95279e12 0.486888
\(633\) 0 0
\(634\) −3.84237e11 −0.0944490
\(635\) −4.33886e12 −1.05899
\(636\) 0 0
\(637\) 1.88431e11 0.0453446
\(638\) −9.18912e10 −0.0219574
\(639\) 0 0
\(640\) 3.51650e11 0.0828517
\(641\) 3.87461e12 0.906498 0.453249 0.891384i \(-0.350265\pi\)
0.453249 + 0.891384i \(0.350265\pi\)
\(642\) 0 0
\(643\) 2.77752e12 0.640778 0.320389 0.947286i \(-0.396186\pi\)
0.320389 + 0.947286i \(0.396186\pi\)
\(644\) 2.99768e11 0.0686751
\(645\) 0 0
\(646\) −3.46942e12 −0.783810
\(647\) −6.12025e12 −1.37309 −0.686546 0.727086i \(-0.740874\pi\)
−0.686546 + 0.727086i \(0.740874\pi\)
\(648\) 0 0
\(649\) 6.71109e10 0.0148488
\(650\) 1.08315e11 0.0238000
\(651\) 0 0
\(652\) 1.61918e12 0.350898
\(653\) −2.50039e12 −0.538143 −0.269072 0.963120i \(-0.586717\pi\)
−0.269072 + 0.963120i \(0.586717\pi\)
\(654\) 0 0
\(655\) −4.01370e12 −0.852038
\(656\) 1.48122e12 0.312286
\(657\) 0 0
\(658\) 5.51174e12 1.14623
\(659\) 6.13676e12 1.26752 0.633760 0.773529i \(-0.281511\pi\)
0.633760 + 0.773529i \(0.281511\pi\)
\(660\) 0 0
\(661\) −6.28369e12 −1.28029 −0.640145 0.768254i \(-0.721126\pi\)
−0.640145 + 0.768254i \(0.721126\pi\)
\(662\) 5.96814e12 1.20775
\(663\) 0 0
\(664\) 2.06977e12 0.413206
\(665\) 6.94904e12 1.37793
\(666\) 0 0
\(667\) 2.57338e11 0.0503429
\(668\) −3.87585e12 −0.753134
\(669\) 0 0
\(670\) 4.62671e11 0.0887025
\(671\) −2.56403e11 −0.0488283
\(672\) 0 0
\(673\) −7.98616e12 −1.50062 −0.750309 0.661087i \(-0.770095\pi\)
−0.750309 + 0.661087i \(0.770095\pi\)
\(674\) 3.05852e12 0.570876
\(675\) 0 0
\(676\) 2.08827e11 0.0384615
\(677\) 3.64428e12 0.666749 0.333374 0.942794i \(-0.391813\pi\)
0.333374 + 0.942794i \(0.391813\pi\)
\(678\) 0 0
\(679\) 4.66413e12 0.842087
\(680\) 1.27436e12 0.228560
\(681\) 0 0
\(682\) 2.99658e11 0.0530391
\(683\) 1.11273e12 0.195658 0.0978291 0.995203i \(-0.468810\pi\)
0.0978291 + 0.995203i \(0.468810\pi\)
\(684\) 0 0
\(685\) −7.37244e11 −0.127939
\(686\) 4.36458e12 0.752460
\(687\) 0 0
\(688\) 7.68495e11 0.130766
\(689\) 3.08912e12 0.522214
\(690\) 0 0
\(691\) −2.75811e12 −0.460214 −0.230107 0.973165i \(-0.573908\pi\)
−0.230107 + 0.973165i \(0.573908\pi\)
\(692\) 4.18517e11 0.0693802
\(693\) 0 0
\(694\) −1.37664e12 −0.225269
\(695\) −6.03382e11 −0.0980982
\(696\) 0 0
\(697\) 5.36785e12 0.861495
\(698\) −2.13779e12 −0.340892
\(699\) 0 0
\(700\) 3.52542e11 0.0554969
\(701\) −8.08880e12 −1.26518 −0.632591 0.774486i \(-0.718009\pi\)
−0.632591 + 0.774486i \(0.718009\pi\)
\(702\) 0 0
\(703\) 1.68384e13 2.60017
\(704\) 7.54639e10 0.0115788
\(705\) 0 0
\(706\) 9.98078e11 0.151197
\(707\) 6.97189e12 1.04945
\(708\) 0 0
\(709\) 2.19552e11 0.0326310 0.0163155 0.999867i \(-0.494806\pi\)
0.0163155 + 0.999867i \(0.494806\pi\)
\(710\) −9.30965e11 −0.137490
\(711\) 0 0
\(712\) −3.64888e12 −0.532108
\(713\) −8.39183e11 −0.121606
\(714\) 0 0
\(715\) −1.68292e11 −0.0240817
\(716\) 1.05723e12 0.150335
\(717\) 0 0
\(718\) 6.12382e12 0.859929
\(719\) 8.58532e12 1.19805 0.599027 0.800729i \(-0.295554\pi\)
0.599027 + 0.800729i \(0.295554\pi\)
\(720\) 0 0
\(721\) 5.56614e12 0.767088
\(722\) 8.17451e12 1.11955
\(723\) 0 0
\(724\) −5.45515e12 −0.737875
\(725\) 3.02642e11 0.0406825
\(726\) 0 0
\(727\) 7.59563e11 0.100846 0.0504230 0.998728i \(-0.483943\pi\)
0.0504230 + 0.998728i \(0.483943\pi\)
\(728\) 6.79688e11 0.0896847
\(729\) 0 0
\(730\) 5.57106e12 0.726080
\(731\) 2.78497e12 0.360739
\(732\) 0 0
\(733\) −7.83005e12 −1.00184 −0.500918 0.865495i \(-0.667004\pi\)
−0.500918 + 0.865495i \(0.667004\pi\)
\(734\) 4.15684e12 0.528605
\(735\) 0 0
\(736\) −2.11334e11 −0.0265473
\(737\) 9.92889e10 0.0123964
\(738\) 0 0
\(739\) 5.41643e12 0.668056 0.334028 0.942563i \(-0.391592\pi\)
0.334028 + 0.942563i \(0.391592\pi\)
\(740\) −6.18493e12 −0.758215
\(741\) 0 0
\(742\) 1.00544e13 1.21770
\(743\) −2.66408e12 −0.320699 −0.160349 0.987060i \(-0.551262\pi\)
−0.160349 + 0.987060i \(0.551262\pi\)
\(744\) 0 0
\(745\) 7.88249e11 0.0937476
\(746\) 7.53513e12 0.890771
\(747\) 0 0
\(748\) 2.73476e11 0.0319420
\(749\) −1.38888e13 −1.61249
\(750\) 0 0
\(751\) 5.82882e12 0.668653 0.334326 0.942457i \(-0.391491\pi\)
0.334326 + 0.942457i \(0.391491\pi\)
\(752\) −3.88573e12 −0.443090
\(753\) 0 0
\(754\) 5.83482e11 0.0657441
\(755\) −1.65970e13 −1.85895
\(756\) 0 0
\(757\) 5.67869e12 0.628517 0.314258 0.949338i \(-0.398244\pi\)
0.314258 + 0.949338i \(0.398244\pi\)
\(758\) −5.76074e12 −0.633821
\(759\) 0 0
\(760\) −4.89901e12 −0.532657
\(761\) 1.12490e13 1.21586 0.607931 0.793990i \(-0.292000\pi\)
0.607931 + 0.793990i \(0.292000\pi\)
\(762\) 0 0
\(763\) 9.88694e12 1.05609
\(764\) 7.90120e12 0.839021
\(765\) 0 0
\(766\) 1.53501e12 0.161095
\(767\) −4.26135e11 −0.0444598
\(768\) 0 0
\(769\) 5.02943e12 0.518621 0.259311 0.965794i \(-0.416505\pi\)
0.259311 + 0.965794i \(0.416505\pi\)
\(770\) −5.47756e11 −0.0561537
\(771\) 0 0
\(772\) −1.16459e12 −0.118003
\(773\) −1.22620e13 −1.23524 −0.617621 0.786476i \(-0.711903\pi\)
−0.617621 + 0.786476i \(0.711903\pi\)
\(774\) 0 0
\(775\) −9.86918e11 −0.0982705
\(776\) −3.28817e12 −0.325520
\(777\) 0 0
\(778\) 7.36781e12 0.720992
\(779\) −2.06356e13 −2.00770
\(780\) 0 0
\(781\) −1.99784e11 −0.0192146
\(782\) −7.65861e11 −0.0732351
\(783\) 0 0
\(784\) −4.32374e11 −0.0408731
\(785\) −2.62960e11 −0.0247159
\(786\) 0 0
\(787\) −1.13978e13 −1.05909 −0.529547 0.848281i \(-0.677638\pi\)
−0.529547 + 0.848281i \(0.677638\pi\)
\(788\) −5.78754e12 −0.534720
\(789\) 0 0
\(790\) 9.99279e12 0.912778
\(791\) −8.18005e12 −0.742954
\(792\) 0 0
\(793\) 1.62808e12 0.146200
\(794\) 4.64479e12 0.414738
\(795\) 0 0
\(796\) 2.68870e12 0.237375
\(797\) −9.66670e12 −0.848625 −0.424313 0.905516i \(-0.639484\pi\)
−0.424313 + 0.905516i \(0.639484\pi\)
\(798\) 0 0
\(799\) −1.40816e13 −1.22234
\(800\) −2.48539e11 −0.0214531
\(801\) 0 0
\(802\) −1.09679e13 −0.936137
\(803\) 1.19554e12 0.101472
\(804\) 0 0
\(805\) 1.53397e12 0.128747
\(806\) −1.90274e12 −0.158808
\(807\) 0 0
\(808\) −4.91512e12 −0.405679
\(809\) −4.89988e12 −0.402177 −0.201088 0.979573i \(-0.564448\pi\)
−0.201088 + 0.979573i \(0.564448\pi\)
\(810\) 0 0
\(811\) −9.97393e12 −0.809604 −0.404802 0.914404i \(-0.632660\pi\)
−0.404802 + 0.914404i \(0.632660\pi\)
\(812\) 1.89911e12 0.153302
\(813\) 0 0
\(814\) −1.32728e12 −0.105963
\(815\) 8.28564e12 0.657835
\(816\) 0 0
\(817\) −1.07063e13 −0.840697
\(818\) 1.60047e13 1.24985
\(819\) 0 0
\(820\) 7.57970e12 0.585450
\(821\) 6.20376e12 0.476552 0.238276 0.971197i \(-0.423418\pi\)
0.238276 + 0.971197i \(0.423418\pi\)
\(822\) 0 0
\(823\) 2.05255e13 1.55953 0.779765 0.626072i \(-0.215339\pi\)
0.779765 + 0.626072i \(0.215339\pi\)
\(824\) −3.92408e12 −0.296528
\(825\) 0 0
\(826\) −1.38698e12 −0.103671
\(827\) 1.52447e13 1.13330 0.566649 0.823959i \(-0.308239\pi\)
0.566649 + 0.823959i \(0.308239\pi\)
\(828\) 0 0
\(829\) 1.59195e12 0.117067 0.0585334 0.998285i \(-0.481358\pi\)
0.0585334 + 0.998285i \(0.481358\pi\)
\(830\) 1.05914e13 0.774645
\(831\) 0 0
\(832\) −4.79174e11 −0.0346688
\(833\) −1.56689e12 −0.112755
\(834\) 0 0
\(835\) −1.98334e13 −1.41192
\(836\) −1.05132e12 −0.0744404
\(837\) 0 0
\(838\) 1.38368e13 0.969251
\(839\) 5.56668e11 0.0387853 0.0193927 0.999812i \(-0.493827\pi\)
0.0193927 + 0.999812i \(0.493827\pi\)
\(840\) 0 0
\(841\) −1.28768e13 −0.887621
\(842\) −1.53245e12 −0.105071
\(843\) 0 0
\(844\) −1.45004e12 −0.0983643
\(845\) 1.06861e12 0.0721046
\(846\) 0 0
\(847\) 1.35821e13 0.906760
\(848\) −7.08829e12 −0.470717
\(849\) 0 0
\(850\) −9.00687e11 −0.0591819
\(851\) 3.71701e12 0.242946
\(852\) 0 0
\(853\) −1.76959e13 −1.14446 −0.572231 0.820093i \(-0.693922\pi\)
−0.572231 + 0.820093i \(0.693922\pi\)
\(854\) 5.29907e12 0.340909
\(855\) 0 0
\(856\) 9.79151e12 0.623330
\(857\) −1.34064e13 −0.848983 −0.424492 0.905432i \(-0.639547\pi\)
−0.424492 + 0.905432i \(0.639547\pi\)
\(858\) 0 0
\(859\) 2.16215e13 1.35493 0.677466 0.735554i \(-0.263078\pi\)
0.677466 + 0.735554i \(0.263078\pi\)
\(860\) 3.93253e12 0.245149
\(861\) 0 0
\(862\) 2.03496e12 0.125537
\(863\) 1.09637e13 0.672838 0.336419 0.941712i \(-0.390784\pi\)
0.336419 + 0.941712i \(0.390784\pi\)
\(864\) 0 0
\(865\) 2.14163e12 0.130068
\(866\) −2.48423e12 −0.150093
\(867\) 0 0
\(868\) −6.19302e12 −0.370309
\(869\) 2.14445e12 0.127563
\(870\) 0 0
\(871\) −6.30456e11 −0.0371170
\(872\) −6.97021e12 −0.408245
\(873\) 0 0
\(874\) 2.94420e12 0.170673
\(875\) 1.66695e13 0.961358
\(876\) 0 0
\(877\) 1.20955e13 0.690440 0.345220 0.938522i \(-0.387804\pi\)
0.345220 + 0.938522i \(0.387804\pi\)
\(878\) −1.64175e13 −0.932358
\(879\) 0 0
\(880\) 3.86163e11 0.0217069
\(881\) 3.33493e13 1.86507 0.932534 0.361083i \(-0.117593\pi\)
0.932534 + 0.361083i \(0.117593\pi\)
\(882\) 0 0
\(883\) −1.01455e13 −0.561628 −0.280814 0.959762i \(-0.590604\pi\)
−0.280814 + 0.959762i \(0.590604\pi\)
\(884\) −1.73649e12 −0.0956397
\(885\) 0 0
\(886\) −4.03263e12 −0.219855
\(887\) −3.04816e13 −1.65341 −0.826707 0.562633i \(-0.809789\pi\)
−0.826707 + 0.562633i \(0.809789\pi\)
\(888\) 0 0
\(889\) 1.92433e13 1.03329
\(890\) −1.86720e13 −0.997554
\(891\) 0 0
\(892\) −8.18194e12 −0.432727
\(893\) 5.41340e13 2.84865
\(894\) 0 0
\(895\) 5.41003e12 0.281836
\(896\) −1.55961e12 −0.0808407
\(897\) 0 0
\(898\) −1.22592e13 −0.629097
\(899\) −5.31644e12 −0.271458
\(900\) 0 0
\(901\) −2.56875e13 −1.29855
\(902\) 1.62660e12 0.0818183
\(903\) 0 0
\(904\) 5.76687e12 0.287198
\(905\) −2.79150e13 −1.38331
\(906\) 0 0
\(907\) −2.08823e12 −0.102458 −0.0512289 0.998687i \(-0.516314\pi\)
−0.0512289 + 0.998687i \(0.516314\pi\)
\(908\) 1.29992e13 0.634645
\(909\) 0 0
\(910\) 3.47809e12 0.168134
\(911\) 1.70747e13 0.821336 0.410668 0.911785i \(-0.365296\pi\)
0.410668 + 0.911785i \(0.365296\pi\)
\(912\) 0 0
\(913\) 2.27291e12 0.108259
\(914\) 2.81093e13 1.33227
\(915\) 0 0
\(916\) 1.53558e13 0.720681
\(917\) 1.78012e13 0.831358
\(918\) 0 0
\(919\) −3.86177e12 −0.178594 −0.0892970 0.996005i \(-0.528462\pi\)
−0.0892970 + 0.996005i \(0.528462\pi\)
\(920\) −1.08144e12 −0.0497687
\(921\) 0 0
\(922\) 1.81003e13 0.824893
\(923\) 1.26857e12 0.0575318
\(924\) 0 0
\(925\) 4.37137e12 0.196327
\(926\) −4.34651e12 −0.194263
\(927\) 0 0
\(928\) −1.33886e12 −0.0592609
\(929\) −2.72392e13 −1.19984 −0.599921 0.800059i \(-0.704801\pi\)
−0.599921 + 0.800059i \(0.704801\pi\)
\(930\) 0 0
\(931\) 6.02362e12 0.262775
\(932\) −1.22170e13 −0.530389
\(933\) 0 0
\(934\) 1.52739e13 0.656732
\(935\) 1.39943e12 0.0598823
\(936\) 0 0
\(937\) −1.33830e13 −0.567184 −0.283592 0.958945i \(-0.591526\pi\)
−0.283592 + 0.958945i \(0.591526\pi\)
\(938\) −2.05200e12 −0.0865495
\(939\) 0 0
\(940\) −1.98840e13 −0.830670
\(941\) −5.78614e12 −0.240567 −0.120283 0.992740i \(-0.538380\pi\)
−0.120283 + 0.992740i \(0.538380\pi\)
\(942\) 0 0
\(943\) −4.55523e12 −0.187589
\(944\) 9.77807e11 0.0400755
\(945\) 0 0
\(946\) 8.43919e11 0.0342603
\(947\) 4.03191e13 1.62905 0.814527 0.580125i \(-0.196996\pi\)
0.814527 + 0.580125i \(0.196996\pi\)
\(948\) 0 0
\(949\) −7.59136e12 −0.303824
\(950\) 3.46251e12 0.137923
\(951\) 0 0
\(952\) −5.65192e12 −0.223013
\(953\) −1.46625e13 −0.575824 −0.287912 0.957657i \(-0.592961\pi\)
−0.287912 + 0.957657i \(0.592961\pi\)
\(954\) 0 0
\(955\) 4.04319e13 1.57293
\(956\) 2.23122e13 0.863936
\(957\) 0 0
\(958\) −2.29887e13 −0.881801
\(959\) 3.26976e12 0.124834
\(960\) 0 0
\(961\) −9.10264e12 −0.344280
\(962\) 8.42785e12 0.317270
\(963\) 0 0
\(964\) −2.66764e13 −0.994903
\(965\) −5.95942e12 −0.221223
\(966\) 0 0
\(967\) 1.87662e13 0.690172 0.345086 0.938571i \(-0.387850\pi\)
0.345086 + 0.938571i \(0.387850\pi\)
\(968\) −9.57528e12 −0.350520
\(969\) 0 0
\(970\) −1.68262e13 −0.610258
\(971\) −2.66964e13 −0.963755 −0.481877 0.876239i \(-0.660045\pi\)
−0.481877 + 0.876239i \(0.660045\pi\)
\(972\) 0 0
\(973\) 2.67607e12 0.0957171
\(974\) −1.96390e13 −0.699206
\(975\) 0 0
\(976\) −3.73580e12 −0.131783
\(977\) −1.44408e13 −0.507068 −0.253534 0.967327i \(-0.581593\pi\)
−0.253534 + 0.967327i \(0.581593\pi\)
\(978\) 0 0
\(979\) −4.00700e12 −0.139411
\(980\) −2.21254e12 −0.0766256
\(981\) 0 0
\(982\) 1.60622e13 0.551194
\(983\) −3.96507e13 −1.35444 −0.677220 0.735781i \(-0.736815\pi\)
−0.677220 + 0.735781i \(0.736815\pi\)
\(984\) 0 0
\(985\) −2.96159e13 −1.00245
\(986\) −4.85193e12 −0.163481
\(987\) 0 0
\(988\) 6.67561e12 0.222887
\(989\) −2.36337e12 −0.0785503
\(990\) 0 0
\(991\) −4.04833e13 −1.33335 −0.666675 0.745348i \(-0.732283\pi\)
−0.666675 + 0.745348i \(0.732283\pi\)
\(992\) 4.36603e12 0.143148
\(993\) 0 0
\(994\) 4.12893e12 0.134153
\(995\) 1.37586e13 0.445011
\(996\) 0 0
\(997\) −2.52148e13 −0.808217 −0.404109 0.914711i \(-0.632418\pi\)
−0.404109 + 0.914711i \(0.632418\pi\)
\(998\) −1.21322e13 −0.387125
\(999\) 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 234.10.a.c.1.1 1
3.2 odd 2 26.10.a.b.1.1 1
12.11 even 2 208.10.a.a.1.1 1
39.38 odd 2 338.10.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.b.1.1 1 3.2 odd 2
208.10.a.a.1.1 1 12.11 even 2
234.10.a.c.1.1 1 1.1 even 1 trivial
338.10.a.d.1.1 1 39.38 odd 2