Properties

Label 338.10.a.d.1.1
Level $338$
Weight $10$
Character 338.1
Self dual yes
Analytic conductor $174.082$
Analytic rank $0$
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] = [338,10,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(174.082112623\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} +192.000 q^{3} +256.000 q^{4} +1310.00 q^{5} +3072.00 q^{6} +5810.00 q^{7} +4096.00 q^{8} +17181.0 q^{9} +O(q^{10})\) \(q+16.0000 q^{2} +192.000 q^{3} +256.000 q^{4} +1310.00 q^{5} +3072.00 q^{6} +5810.00 q^{7} +4096.00 q^{8} +17181.0 q^{9} +20960.0 q^{10} +4498.00 q^{11} +49152.0 q^{12} +92960.0 q^{14} +251520. q^{15} +65536.0 q^{16} -237498. q^{17} +274896. q^{18} +913014. q^{19} +335360. q^{20} +1.11552e6 q^{21} +71968.0 q^{22} +201544. q^{23} +786432. q^{24} -237025. q^{25} -480384. q^{27} +1.48736e6 q^{28} +1.27683e6 q^{29} +4.02432e6 q^{30} -4.16377e6 q^{31} +1.04858e6 q^{32} +863616. q^{33} -3.79997e6 q^{34} +7.61110e6 q^{35} +4.39834e6 q^{36} +1.84427e7 q^{37} +1.46082e7 q^{38} +5.36576e6 q^{40} +2.26017e7 q^{41} +1.78483e7 q^{42} +1.17263e7 q^{43} +1.15149e6 q^{44} +2.25071e7 q^{45} +3.22470e6 q^{46} -5.92915e7 q^{47} +1.25829e7 q^{48} -6.59751e6 q^{49} -3.79240e6 q^{50} -4.55996e7 q^{51} +1.08159e8 q^{53} -7.68614e6 q^{54} +5.89238e6 q^{55} +2.37978e7 q^{56} +1.75299e8 q^{57} +2.04293e7 q^{58} +1.49202e7 q^{59} +6.43891e7 q^{60} -5.70037e7 q^{61} -6.66203e7 q^{62} +9.98216e7 q^{63} +1.67772e7 q^{64} +1.38179e7 q^{66} -2.20740e7 q^{67} -6.07995e7 q^{68} +3.86964e7 q^{69} +1.21778e8 q^{70} -4.44162e7 q^{71} +7.03734e7 q^{72} -2.65795e8 q^{73} +2.95083e8 q^{74} -4.55088e7 q^{75} +2.33732e8 q^{76} +2.61334e7 q^{77} +4.76755e8 q^{79} +8.58522e7 q^{80} -4.30407e8 q^{81} +3.61627e8 q^{82} +5.05316e8 q^{83} +2.85573e8 q^{84} -3.11122e8 q^{85} +1.87621e8 q^{86} +2.45152e8 q^{87} +1.84238e7 q^{88} -8.90841e8 q^{89} +3.60114e8 q^{90} +5.15953e7 q^{92} -7.99444e8 q^{93} -9.48665e8 q^{94} +1.19605e9 q^{95} +2.01327e8 q^{96} +8.02777e8 q^{97} -1.05560e8 q^{98} +7.72801e7 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) 192.000 1.36853 0.684267 0.729232i \(-0.260122\pi\)
0.684267 + 0.729232i \(0.260122\pi\)
\(4\) 256.000 0.500000
\(5\) 1310.00 0.937360 0.468680 0.883368i \(-0.344730\pi\)
0.468680 + 0.883368i \(0.344730\pi\)
\(6\) 3072.00 0.967700
\(7\) 5810.00 0.914608 0.457304 0.889310i \(-0.348815\pi\)
0.457304 + 0.889310i \(0.348815\pi\)
\(8\) 4096.00 0.353553
\(9\) 17181.0 0.872885
\(10\) 20960.0 0.662813
\(11\) 4498.00 0.0926302 0.0463151 0.998927i \(-0.485252\pi\)
0.0463151 + 0.998927i \(0.485252\pi\)
\(12\) 49152.0 0.684267
\(13\) 0 0
\(14\) 92960.0 0.646725
\(15\) 251520. 1.28281
\(16\) 65536.0 0.250000
\(17\) −237498. −0.689668 −0.344834 0.938664i \(-0.612065\pi\)
−0.344834 + 0.938664i \(0.612065\pi\)
\(18\) 274896. 0.617223
\(19\) 913014. 1.60726 0.803630 0.595129i \(-0.202899\pi\)
0.803630 + 0.595129i \(0.202899\pi\)
\(20\) 335360. 0.468680
\(21\) 1.11552e6 1.25167
\(22\) 71968.0 0.0654994
\(23\) 201544. 0.150174 0.0750870 0.997177i \(-0.476077\pi\)
0.0750870 + 0.997177i \(0.476077\pi\)
\(24\) 786432. 0.483850
\(25\) −237025. −0.121357
\(26\) 0 0
\(27\) −480384. −0.173961
\(28\) 1.48736e6 0.457304
\(29\) 1.27683e6 0.335230 0.167615 0.985852i \(-0.446393\pi\)
0.167615 + 0.985852i \(0.446393\pi\)
\(30\) 4.02432e6 0.907083
\(31\) −4.16377e6 −0.809765 −0.404883 0.914369i \(-0.632688\pi\)
−0.404883 + 0.914369i \(0.632688\pi\)
\(32\) 1.04858e6 0.176777
\(33\) 863616. 0.126768
\(34\) −3.79997e6 −0.487669
\(35\) 7.61110e6 0.857317
\(36\) 4.39834e6 0.436443
\(37\) 1.84427e7 1.61777 0.808883 0.587969i \(-0.200072\pi\)
0.808883 + 0.587969i \(0.200072\pi\)
\(38\) 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 \(0.285273\pi\)
\(42\) 1.78483e7 0.885066
\(43\) 1.17263e7 0.523062 0.261531 0.965195i \(-0.415773\pi\)
0.261531 + 0.965195i \(0.415773\pi\)
\(44\) 1.15149e6 0.0463151
\(45\) 2.25071e7 0.818207
\(46\) 3.22470e6 0.106189
\(47\) −5.92915e7 −1.77236 −0.886181 0.463340i \(-0.846651\pi\)
−0.886181 + 0.463340i \(0.846651\pi\)
\(48\) 1.25829e7 0.342133
\(49\) −6.59751e6 −0.163492
\(50\) −3.79240e6 −0.0858122
\(51\) −4.55996e7 −0.943834
\(52\) 0 0
\(53\) 1.08159e8 1.88287 0.941434 0.337196i \(-0.109479\pi\)
0.941434 + 0.337196i \(0.109479\pi\)
\(54\) −7.68614e6 −0.123009
\(55\) 5.89238e6 0.0868278
\(56\) 2.37978e7 0.323363
\(57\) 1.75299e8 2.19959
\(58\) 2.04293e7 0.237044
\(59\) 1.49202e7 0.160302 0.0801511 0.996783i \(-0.474460\pi\)
0.0801511 + 0.996783i \(0.474460\pi\)
\(60\) 6.43891e7 0.641404
\(61\) −5.70037e7 −0.527132 −0.263566 0.964641i \(-0.584899\pi\)
−0.263566 + 0.964641i \(0.584899\pi\)
\(62\) −6.66203e7 −0.572590
\(63\) 9.98216e7 0.798348
\(64\) 1.67772e7 0.125000
\(65\) 0 0
\(66\) 1.38179e7 0.0896382
\(67\) −2.20740e7 −0.133827 −0.0669136 0.997759i \(-0.521315\pi\)
−0.0669136 + 0.997759i \(0.521315\pi\)
\(68\) −6.07995e7 −0.344834
\(69\) 3.86964e7 0.205518
\(70\) 1.21778e8 0.606214
\(71\) −4.44162e7 −0.207434 −0.103717 0.994607i \(-0.533074\pi\)
−0.103717 + 0.994607i \(0.533074\pi\)
\(72\) 7.03734e7 0.308612
\(73\) −2.65795e8 −1.09545 −0.547726 0.836658i \(-0.684506\pi\)
−0.547726 + 0.836658i \(0.684506\pi\)
\(74\) 2.95083e8 1.14393
\(75\) −4.55088e7 −0.166081
\(76\) 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
\(81\) −4.30407e8 −1.11096
\(82\) 3.61627e8 0.883280
\(83\) 5.05316e8 1.16872 0.584361 0.811494i \(-0.301345\pi\)
0.584361 + 0.811494i \(0.301345\pi\)
\(84\) 2.85573e8 0.625836
\(85\) −3.11122e8 −0.646467
\(86\) 1.87621e8 0.369861
\(87\) 2.45152e8 0.458774
\(88\) 1.84238e7 0.0327497
\(89\) −8.90841e8 −1.50503 −0.752515 0.658575i \(-0.771159\pi\)
−0.752515 + 0.658575i \(0.771159\pi\)
\(90\) 3.60114e8 0.578560
\(91\) 0 0
\(92\) 5.15953e7 0.0750870
\(93\) −7.99444e8 −1.10819
\(94\) −9.48665e8 −1.25325
\(95\) 1.19605e9 1.50658
\(96\) 2.01327e8 0.241925
\(97\) 8.02777e8 0.920708 0.460354 0.887735i \(-0.347722\pi\)
0.460354 + 0.887735i \(0.347722\pi\)
\(98\) −1.05560e8 −0.115607
\(99\) 7.72801e7 0.0808555
\(100\) −6.06784e7 −0.0606784
\(101\) 1.19998e9 1.14743 0.573717 0.819053i \(-0.305501\pi\)
0.573717 + 0.819053i \(0.305501\pi\)
\(102\) −7.29594e8 −0.667391
\(103\) −9.58027e8 −0.838707 −0.419353 0.907823i \(-0.637743\pi\)
−0.419353 + 0.907823i \(0.637743\pi\)
\(104\) 0 0
\(105\) 1.46133e9 1.17327
\(106\) 1.73054e9 1.33139
\(107\) −2.39051e9 −1.76304 −0.881521 0.472145i \(-0.843480\pi\)
−0.881521 + 0.472145i \(0.843480\pi\)
\(108\) −1.22978e8 −0.0869804
\(109\) 1.70171e9 1.15469 0.577346 0.816499i \(-0.304088\pi\)
0.577346 + 0.816499i \(0.304088\pi\)
\(110\) 9.42781e7 0.0613965
\(111\) 3.54099e9 2.21397
\(112\) 3.80764e8 0.228652
\(113\) −1.40793e9 −0.812320 −0.406160 0.913802i \(-0.633132\pi\)
−0.406160 + 0.913802i \(0.633132\pi\)
\(114\) 2.80478e9 1.55535
\(115\) 2.64023e8 0.140767
\(116\) 3.26870e8 0.167615
\(117\) 0 0
\(118\) 2.38722e8 0.113351
\(119\) −1.37986e9 −0.630775
\(120\) 1.03023e9 0.453541
\(121\) −2.33772e9 −0.991420
\(122\) −9.12060e8 −0.372738
\(123\) 4.33952e9 1.70950
\(124\) −1.06593e9 −0.404883
\(125\) −2.86910e9 −1.05111
\(126\) 1.59715e9 0.564517
\(127\) −3.31210e9 −1.12976 −0.564881 0.825172i \(-0.691078\pi\)
−0.564881 + 0.825172i \(0.691078\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 2.25145e9 0.715828
\(130\) 0 0
\(131\) 3.06389e9 0.908977 0.454489 0.890753i \(-0.349822\pi\)
0.454489 + 0.890753i \(0.349822\pi\)
\(132\) 2.21086e8 0.0633838
\(133\) 5.30461e9 1.47001
\(134\) −3.53184e8 −0.0946302
\(135\) −6.29303e8 −0.163064
\(136\) −9.72792e8 −0.243834
\(137\) −5.62781e8 −0.136489 −0.0682444 0.997669i \(-0.521740\pi\)
−0.0682444 + 0.997669i \(0.521740\pi\)
\(138\) 6.19143e8 0.145323
\(139\) −4.60597e8 −0.104654 −0.0523268 0.998630i \(-0.516664\pi\)
−0.0523268 + 0.998630i \(0.516664\pi\)
\(140\) 1.94844e9 0.428658
\(141\) −1.13840e10 −2.42554
\(142\) −7.10660e8 −0.146678
\(143\) 0 0
\(144\) 1.12597e9 0.218221
\(145\) 1.67265e9 0.314232
\(146\) −4.25271e9 −0.774601
\(147\) −1.26672e9 −0.223745
\(148\) 4.72132e9 0.808883
\(149\) 6.01717e8 0.100012 0.0500062 0.998749i \(-0.484076\pi\)
0.0500062 + 0.998749i \(0.484076\pi\)
\(150\) −7.28141e8 −0.117437
\(151\) 1.26695e10 1.98318 0.991589 0.129428i \(-0.0413142\pi\)
0.991589 + 0.129428i \(0.0413142\pi\)
\(152\) 3.73971e9 0.568252
\(153\) −4.08045e9 −0.602001
\(154\) 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
\(159\) 2.07665e10 2.57677
\(160\) 1.37363e9 0.165703
\(161\) 1.17097e9 0.137350
\(162\) −6.88652e9 −0.785565
\(163\) −6.32491e9 −0.701795 −0.350898 0.936414i \(-0.614123\pi\)
−0.350898 + 0.936414i \(0.614123\pi\)
\(164\) 5.78603e9 0.624573
\(165\) 1.13134e9 0.118827
\(166\) 8.08505e9 0.826412
\(167\) −1.51400e10 −1.50627 −0.753134 0.657867i \(-0.771459\pi\)
−0.753134 + 0.657867i \(0.771459\pi\)
\(168\) 4.56917e9 0.442533
\(169\) 0 0
\(170\) −4.97796e9 −0.457121
\(171\) 1.56865e10 1.40295
\(172\) 3.00193e9 0.261531
\(173\) −1.63483e9 −0.138760 −0.0693802 0.997590i \(-0.522102\pi\)
−0.0693802 + 0.997590i \(0.522102\pi\)
\(174\) 3.92243e9 0.324402
\(175\) −1.37712e9 −0.110994
\(176\) 2.94781e8 0.0231575
\(177\) 2.86467e9 0.219379
\(178\) −1.42535e10 −1.06422
\(179\) −4.12980e9 −0.300670 −0.150335 0.988635i \(-0.548035\pi\)
−0.150335 + 0.988635i \(0.548035\pi\)
\(180\) 5.76182e9 0.409104
\(181\) −2.13092e10 −1.47575 −0.737875 0.674937i \(-0.764171\pi\)
−0.737875 + 0.674937i \(0.764171\pi\)
\(182\) 0 0
\(183\) −1.09447e10 −0.721398
\(184\) 8.25524e8 0.0530945
\(185\) 2.41599e10 1.51643
\(186\) −1.27911e10 −0.783610
\(187\) −1.06827e9 −0.0638840
\(188\) −1.51786e10 −0.886181
\(189\) −2.79103e9 −0.159106
\(190\) 1.91368e10 1.06531
\(191\) −3.08641e10 −1.67804 −0.839021 0.544099i \(-0.816872\pi\)
−0.839021 + 0.544099i \(0.816872\pi\)
\(192\) 3.22123e9 0.171067
\(193\) 4.54917e9 0.236007 0.118003 0.993013i \(-0.462351\pi\)
0.118003 + 0.993013i \(0.462351\pi\)
\(194\) 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 \(0.679583\pi\)
\(198\) 1.23648e9 0.0571735
\(199\) 1.05027e10 0.474749 0.237375 0.971418i \(-0.423713\pi\)
0.237375 + 0.971418i \(0.423713\pi\)
\(200\) −9.70854e8 −0.0429061
\(201\) −4.23821e9 −0.183147
\(202\) 1.91997e10 0.811359
\(203\) 7.41841e9 0.306604
\(204\) −1.16735e10 −0.471917
\(205\) 2.96082e10 1.17090
\(206\) −1.53284e10 −0.593055
\(207\) 3.46273e9 0.131085
\(208\) 0 0
\(209\) 4.10674e9 0.148881
\(210\) 2.33813e10 0.829625
\(211\) −5.66420e9 −0.196729 −0.0983643 0.995150i \(-0.531361\pi\)
−0.0983643 + 0.995150i \(0.531361\pi\)
\(212\) 2.76886e10 0.941434
\(213\) −8.52792e9 −0.283880
\(214\) −3.82481e10 −1.24666
\(215\) 1.53615e10 0.490297
\(216\) −1.96765e9 −0.0615045
\(217\) −2.41915e10 −0.740618
\(218\) 2.72274e10 0.816491
\(219\) −5.10326e10 −1.49916
\(220\) 1.50845e9 0.0434139
\(221\) 0 0
\(222\) 5.66559e10 1.56551
\(223\) 3.19607e10 0.865454 0.432727 0.901525i \(-0.357551\pi\)
0.432727 + 0.901525i \(0.357551\pi\)
\(224\) 6.09223e9 0.161681
\(225\) −4.07233e9 −0.105931
\(226\) −2.25268e10 −0.574397
\(227\) 5.07782e10 1.26929 0.634645 0.772804i \(-0.281146\pi\)
0.634645 + 0.772804i \(0.281146\pi\)
\(228\) 4.48765e10 1.09980
\(229\) −5.99836e10 −1.44136 −0.720681 0.693267i \(-0.756171\pi\)
−0.720681 + 0.693267i \(0.756171\pi\)
\(230\) 4.22436e9 0.0995373
\(231\) 5.01761e9 0.115943
\(232\) 5.22991e9 0.118522
\(233\) 4.77228e10 1.06078 0.530389 0.847755i \(-0.322046\pi\)
0.530389 + 0.847755i \(0.322046\pi\)
\(234\) 0 0
\(235\) −7.76719e10 −1.66134
\(236\) 3.81956e9 0.0801511
\(237\) 9.15371e10 1.88464
\(238\) −2.20778e10 −0.446026
\(239\) 8.71569e10 1.72787 0.863936 0.503602i \(-0.167992\pi\)
0.863936 + 0.503602i \(0.167992\pi\)
\(240\) 1.64836e10 0.320702
\(241\) 1.04205e11 1.98981 0.994903 0.100835i \(-0.0321515\pi\)
0.994903 + 0.100835i \(0.0321515\pi\)
\(242\) −3.74035e10 −0.701040
\(243\) −7.31828e10 −1.34642
\(244\) −1.45930e10 −0.263566
\(245\) −8.64273e9 −0.153251
\(246\) 6.94323e10 1.20880
\(247\) 0 0
\(248\) −1.70548e10 −0.286295
\(249\) 9.70206e10 1.59944
\(250\) −4.59055e10 −0.743250
\(251\) 2.82027e9 0.0448496 0.0224248 0.999749i \(-0.492861\pi\)
0.0224248 + 0.999749i \(0.492861\pi\)
\(252\) 2.55543e10 0.399174
\(253\) 9.06545e8 0.0139106
\(254\) −5.29937e10 −0.798863
\(255\) −5.97355e10 −0.884711
\(256\) 4.29497e9 0.0625000
\(257\) −1.41573e10 −0.202433 −0.101216 0.994864i \(-0.532273\pi\)
−0.101216 + 0.994864i \(0.532273\pi\)
\(258\) 3.60232e10 0.506167
\(259\) 1.07152e11 1.47962
\(260\) 0 0
\(261\) 2.19373e10 0.292618
\(262\) 4.90223e10 0.642744
\(263\) −3.58497e10 −0.462045 −0.231023 0.972948i \(-0.574207\pi\)
−0.231023 + 0.972948i \(0.574207\pi\)
\(264\) 3.53737e9 0.0448191
\(265\) 1.41688e11 1.76493
\(266\) 8.48738e10 1.03946
\(267\) −1.71041e11 −2.05968
\(268\) −5.65095e9 −0.0669136
\(269\) −7.14394e10 −0.831864 −0.415932 0.909396i \(-0.636545\pi\)
−0.415932 + 0.909396i \(0.636545\pi\)
\(270\) −1.00688e10 −0.115304
\(271\) 6.79344e9 0.0765117 0.0382558 0.999268i \(-0.487820\pi\)
0.0382558 + 0.999268i \(0.487820\pi\)
\(272\) −1.55647e10 −0.172417
\(273\) 0 0
\(274\) −9.00450e9 −0.0965122
\(275\) −1.06614e9 −0.0112413
\(276\) 9.90629e9 0.102759
\(277\) −6.93103e10 −0.707357 −0.353679 0.935367i \(-0.615069\pi\)
−0.353679 + 0.935367i \(0.615069\pi\)
\(278\) −7.36955e9 −0.0740013
\(279\) −7.15377e10 −0.706832
\(280\) 3.11751e10 0.303107
\(281\) −3.10369e10 −0.296961 −0.148480 0.988915i \(-0.547438\pi\)
−0.148480 + 0.988915i \(0.547438\pi\)
\(282\) −1.82144e11 −1.71511
\(283\) 1.35312e11 1.25400 0.627001 0.779018i \(-0.284282\pi\)
0.627001 + 0.779018i \(0.284282\pi\)
\(284\) −1.13706e10 −0.103717
\(285\) 2.29641e11 2.06181
\(286\) 0 0
\(287\) 1.31316e11 1.14248
\(288\) 1.80156e10 0.154306
\(289\) −6.21826e10 −0.524359
\(290\) 2.67624e10 0.222195
\(291\) 1.54133e11 1.26002
\(292\) −6.80434e10 −0.547726
\(293\) 7.55078e10 0.598532 0.299266 0.954170i \(-0.403258\pi\)
0.299266 + 0.954170i \(0.403258\pi\)
\(294\) −2.02675e10 −0.158212
\(295\) 1.95454e10 0.150261
\(296\) 7.55411e10 0.571967
\(297\) −2.16077e9 −0.0161140
\(298\) 9.62747e9 0.0707195
\(299\) 0 0
\(300\) −1.16503e10 −0.0830405
\(301\) 6.81298e10 0.478397
\(302\) 2.02711e11 1.40232
\(303\) 2.30396e11 1.57030
\(304\) 5.98353e10 0.401815
\(305\) −7.46749e10 −0.494112
\(306\) −6.52873e10 −0.425679
\(307\) −1.42760e10 −0.0917241 −0.0458620 0.998948i \(-0.514603\pi\)
−0.0458620 + 0.998948i \(0.514603\pi\)
\(308\) 6.69015e9 0.0423601
\(309\) −1.83941e11 −1.14780
\(310\) −8.72726e10 −0.536723
\(311\) 3.58426e9 0.0217259 0.0108630 0.999941i \(-0.496542\pi\)
0.0108630 + 0.999941i \(0.496542\pi\)
\(312\) 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
\(315\) 1.30766e11 0.748339
\(316\) 1.22049e11 0.688563
\(317\) −2.40148e10 −0.133571 −0.0667855 0.997767i \(-0.521274\pi\)
−0.0667855 + 0.997767i \(0.521274\pi\)
\(318\) 3.32264e11 1.82205
\(319\) 5.74320e9 0.0310524
\(320\) 2.19782e10 0.117170
\(321\) −4.58977e11 −2.41278
\(322\) 1.87355e10 0.0971213
\(323\) −2.16839e11 −1.10848
\(324\) −1.10184e11 −0.555478
\(325\) 0 0
\(326\) −1.01199e11 −0.496244
\(327\) 3.26728e11 1.58024
\(328\) 9.25764e10 0.441640
\(329\) −3.44484e11 −1.62102
\(330\) 1.81014e10 0.0840232
\(331\) −3.73009e11 −1.70802 −0.854010 0.520257i \(-0.825836\pi\)
−0.854010 + 0.520257i \(0.825836\pi\)
\(332\) 1.29361e11 0.584361
\(333\) 3.16863e11 1.41212
\(334\) −2.42240e11 −1.06509
\(335\) −2.89170e10 −0.125444
\(336\) 7.31067e10 0.312918
\(337\) 1.91157e11 0.807340 0.403670 0.914905i \(-0.367734\pi\)
0.403670 + 0.914905i \(0.367734\pi\)
\(338\) 0 0
\(339\) −2.70322e11 −1.11169
\(340\) −7.96473e10 −0.323233
\(341\) −1.87286e10 −0.0750087
\(342\) 2.50984e11 0.992038
\(343\) −2.72786e11 −1.06414
\(344\) 4.80310e10 0.184930
\(345\) 5.06923e10 0.192644
\(346\) −2.61573e10 −0.0981184
\(347\) 8.60398e10 0.318579 0.159289 0.987232i \(-0.449080\pi\)
0.159289 + 0.987232i \(0.449080\pi\)
\(348\) 6.27589e10 0.229387
\(349\) 1.33612e11 0.482094 0.241047 0.970513i \(-0.422509\pi\)
0.241047 + 0.970513i \(0.422509\pi\)
\(350\) −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 \(0.465904\pi\)
\(354\) 4.58347e10 0.155124
\(355\) −5.81853e10 −0.194440
\(356\) −2.28055e11 −0.752515
\(357\) −2.64934e11 −0.863238
\(358\) −6.60767e10 −0.212606
\(359\) 3.82739e11 1.21612 0.608062 0.793890i \(-0.291947\pi\)
0.608062 + 0.793890i \(0.291947\pi\)
\(360\) 9.21891e10 0.289280
\(361\) 5.10907e11 1.58329
\(362\) −3.40947e11 −1.04351
\(363\) −4.48841e11 −1.35679
\(364\) 0 0
\(365\) −3.48191e11 −1.02683
\(366\) −1.75116e11 −0.510105
\(367\) 2.59802e11 0.747560 0.373780 0.927517i \(-0.378062\pi\)
0.373780 + 0.927517i \(0.378062\pi\)
\(368\) 1.32084e10 0.0375435
\(369\) 3.88319e11 1.09036
\(370\) 3.86558e11 1.07228
\(371\) 6.28402e11 1.72209
\(372\) −2.04658e11 −0.554096
\(373\) 4.70946e11 1.25974 0.629870 0.776700i \(-0.283108\pi\)
0.629870 + 0.776700i \(0.283108\pi\)
\(374\) −1.70923e10 −0.0451728
\(375\) −5.50867e11 −1.43849
\(376\) −2.42858e11 −0.626624
\(377\) 0 0
\(378\) −4.46565e10 −0.112505
\(379\) 3.60046e11 0.896358 0.448179 0.893944i \(-0.352073\pi\)
0.448179 + 0.893944i \(0.352073\pi\)
\(380\) 3.06188e11 0.753291
\(381\) −6.35924e11 −1.54612
\(382\) −4.93825e11 −1.18655
\(383\) 9.59380e10 0.227822 0.113911 0.993491i \(-0.463662\pi\)
0.113911 + 0.993491i \(0.463662\pi\)
\(384\) 5.15396e10 0.120962
\(385\) 3.42347e10 0.0794134
\(386\) 7.27868e10 0.166882
\(387\) 2.01470e11 0.456573
\(388\) 2.05511e11 0.460354
\(389\) −4.60488e11 −1.01964 −0.509818 0.860282i \(-0.670287\pi\)
−0.509818 + 0.860282i \(0.670287\pi\)
\(390\) 0 0
\(391\) −4.78663e10 −0.103570
\(392\) −2.70234e10 −0.0578033
\(393\) 5.88268e11 1.24397
\(394\) −3.61721e11 −0.756208
\(395\) 6.24550e11 1.29086
\(396\) 1.97837e10 0.0404277
\(397\) −2.90299e11 −0.586528 −0.293264 0.956032i \(-0.594741\pi\)
−0.293264 + 0.956032i \(0.594741\pi\)
\(398\) 1.68044e11 0.335698
\(399\) 1.01849e12 2.01176
\(400\) −1.55337e10 −0.0303392
\(401\) −6.85495e11 −1.32390 −0.661949 0.749549i \(-0.730270\pi\)
−0.661949 + 0.749549i \(0.730270\pi\)
\(402\) −6.78114e10 −0.129505
\(403\) 0 0
\(404\) 3.07195e11 0.573717
\(405\) −5.63834e11 −1.04137
\(406\) 1.18694e11 0.216802
\(407\) 8.29551e10 0.149854
\(408\) −1.86776e11 −0.333696
\(409\) −1.00030e12 −1.76756 −0.883779 0.467905i \(-0.845009\pi\)
−0.883779 + 0.467905i \(0.845009\pi\)
\(410\) 4.73731e11 0.827951
\(411\) −1.08054e11 −0.186790
\(412\) −2.45255e11 −0.419353
\(413\) 8.66861e10 0.146614
\(414\) 5.54036e10 0.0926908
\(415\) 6.61964e11 1.09551
\(416\) 0 0
\(417\) −8.84346e10 −0.143222
\(418\) 6.57078e10 0.105275
\(419\) −8.64798e11 −1.37073 −0.685364 0.728200i \(-0.740357\pi\)
−0.685364 + 0.728200i \(0.740357\pi\)
\(420\) 3.74101e11 0.586633
\(421\) 9.57784e10 0.148593 0.0742965 0.997236i \(-0.476329\pi\)
0.0742965 + 0.997236i \(0.476329\pi\)
\(422\) −9.06272e10 −0.139108
\(423\) −1.01869e12 −1.54707
\(424\) 4.43018e11 0.665695
\(425\) 5.62930e10 0.0836959
\(426\) −1.36447e11 −0.200734
\(427\) −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 \(0.471707\pi\)
\(432\) −3.14824e10 −0.0434902
\(433\) −1.55264e11 −0.212264 −0.106132 0.994352i \(-0.533847\pi\)
−0.106132 + 0.994352i \(0.533847\pi\)
\(434\) −3.87064e11 −0.523696
\(435\) 3.21149e11 0.430036
\(436\) 4.35638e11 0.577346
\(437\) 1.84012e11 0.241369
\(438\) −8.16521e11 −1.06007
\(439\) −1.02610e12 −1.31855 −0.659277 0.751901i \(-0.729137\pi\)
−0.659277 + 0.751901i \(0.729137\pi\)
\(440\) 2.41352e10 0.0306983
\(441\) −1.13352e11 −0.142710
\(442\) 0 0
\(443\) 2.52039e11 0.310922 0.155461 0.987842i \(-0.450314\pi\)
0.155461 + 0.987842i \(0.450314\pi\)
\(444\) 9.06494e11 1.10698
\(445\) −1.16700e12 −1.41075
\(446\) 5.11371e11 0.611969
\(447\) 1.15530e11 0.136870
\(448\) 9.74756e10 0.114326
\(449\) −7.66198e11 −0.889678 −0.444839 0.895611i \(-0.646739\pi\)
−0.444839 + 0.895611i \(0.646739\pi\)
\(450\) −6.51572e10 −0.0749042
\(451\) 1.01662e11 0.115709
\(452\) −3.60429e11 −0.406160
\(453\) 2.43254e12 2.71405
\(454\) 8.12451e11 0.897524
\(455\) 0 0
\(456\) 7.18023e11 0.777673
\(457\) −1.75683e12 −1.88411 −0.942057 0.335454i \(-0.891110\pi\)
−0.942057 + 0.335454i \(0.891110\pi\)
\(458\) −9.59738e11 −1.01920
\(459\) 1.14090e11 0.119975
\(460\) 6.75898e10 0.0703835
\(461\) 1.13127e12 1.16657 0.583287 0.812266i \(-0.301766\pi\)
0.583287 + 0.812266i \(0.301766\pi\)
\(462\) 8.02817e10 0.0819838
\(463\) 2.71657e11 0.274730 0.137365 0.990521i \(-0.456137\pi\)
0.137365 + 0.990521i \(0.456137\pi\)
\(464\) 8.36786e10 0.0838076
\(465\) −1.04727e12 −1.03877
\(466\) 7.63565e11 0.750083
\(467\) −9.54617e11 −0.928759 −0.464380 0.885636i \(-0.653723\pi\)
−0.464380 + 0.885636i \(0.653723\pi\)
\(468\) 0 0
\(469\) −1.28250e11 −0.122399
\(470\) −1.24275e12 −1.17474
\(471\) −3.85407e10 −0.0360849
\(472\) 6.11130e10 0.0566754
\(473\) 5.27449e10 0.0484513
\(474\) 1.46459e12 1.33264
\(475\) −2.16407e11 −0.195052
\(476\) −3.53245e11 −0.315388
\(477\) 1.85827e12 1.64353
\(478\) 1.39451e12 1.22179
\(479\) −1.43680e12 −1.24705 −0.623527 0.781802i \(-0.714301\pi\)
−0.623527 + 0.781802i \(0.714301\pi\)
\(480\) 2.63738e11 0.226771
\(481\) 0 0
\(482\) 1.66728e12 1.40701
\(483\) 2.24826e11 0.187969
\(484\) −5.98455e11 −0.495710
\(485\) 1.05164e12 0.863035
\(486\) −1.17093e12 −0.952063
\(487\) 1.22744e12 0.988826 0.494413 0.869227i \(-0.335383\pi\)
0.494413 + 0.869227i \(0.335383\pi\)
\(488\) −2.33487e11 −0.186369
\(489\) −1.21438e12 −0.960431
\(490\) −1.38284e11 −0.108365
\(491\) −1.00389e12 −0.779506 −0.389753 0.920919i \(-0.627440\pi\)
−0.389753 + 0.920919i \(0.627440\pi\)
\(492\) 1.11092e12 0.854749
\(493\) −3.03246e11 −0.231198
\(494\) 0 0
\(495\) 1.01237e11 0.0757907
\(496\) −2.72877e11 −0.202441
\(497\) −2.58058e11 −0.189721
\(498\) 1.55233e12 1.13097
\(499\) 7.58262e11 0.547478 0.273739 0.961804i \(-0.411740\pi\)
0.273739 + 0.961804i \(0.411740\pi\)
\(500\) −7.34489e11 −0.525557
\(501\) −2.90688e12 −2.06138
\(502\) 4.51243e10 0.0317134
\(503\) 1.82032e12 1.26792 0.633959 0.773367i \(-0.281429\pi\)
0.633959 + 0.773367i \(0.281429\pi\)
\(504\) 4.08869e11 0.282259
\(505\) 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 \(0.569603\pi\)
\(510\) −9.55768e11 −0.625585
\(511\) −1.54427e12 −1.00191
\(512\) 6.87195e10 0.0441942
\(513\) −4.38597e11 −0.279600
\(514\) −2.26516e11 −0.143141
\(515\) −1.25502e12 −0.786170
\(516\) 5.76371e11 0.357914
\(517\) −2.66693e11 −0.164174
\(518\) 1.71443e12 1.04625
\(519\) −3.13887e11 −0.189898
\(520\) 0 0
\(521\) 3.17678e11 0.188894 0.0944468 0.995530i \(-0.469892\pi\)
0.0944468 + 0.995530i \(0.469892\pi\)
\(522\) 3.50997e11 0.206912
\(523\) −2.88365e12 −1.68533 −0.842666 0.538436i \(-0.819015\pi\)
−0.842666 + 0.538436i \(0.819015\pi\)
\(524\) 7.84357e11 0.454489
\(525\) −2.64406e11 −0.151899
\(526\) −5.73595e11 −0.326715
\(527\) 9.88887e11 0.558469
\(528\) 5.65979e10 0.0316919
\(529\) −1.76053e12 −0.977448
\(530\) 2.26701e12 1.24799
\(531\) 2.56343e11 0.139925
\(532\) 1.35798e12 0.735007
\(533\) 0 0
\(534\) −2.73666e12 −1.45642
\(535\) −3.13156e12 −1.65260
\(536\) −9.04151e10 −0.0473151
\(537\) −7.92921e11 −0.411477
\(538\) −1.14303e12 −0.588217
\(539\) −2.96756e10 −0.0151443
\(540\) −1.61102e11 −0.0815320
\(541\) −2.16753e12 −1.08787 −0.543937 0.839126i \(-0.683067\pi\)
−0.543937 + 0.839126i \(0.683067\pi\)
\(542\) 1.08695e11 0.0541019
\(543\) −4.09136e12 −2.01961
\(544\) −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
\(549\) −9.79381e11 −0.460125
\(550\) −1.70582e10 −0.00794880
\(551\) 1.16577e12 0.538803
\(552\) 1.58501e11 0.0726616
\(553\) 2.76995e12 1.25953
\(554\) −1.10896e12 −0.500177
\(555\) 4.63870e12 2.07528
\(556\) −1.17913e11 −0.0523268
\(557\) −1.75791e10 −0.00773834 −0.00386917 0.999993i \(-0.501232\pi\)
−0.00386917 + 0.999993i \(0.501232\pi\)
\(558\) −1.14460e12 −0.499806
\(559\) 0 0
\(560\) 4.98801e11 0.214329
\(561\) −2.05107e11 −0.0874274
\(562\) −4.96590e11 −0.209983
\(563\) −3.70644e12 −1.55478 −0.777390 0.629019i \(-0.783457\pi\)
−0.777390 + 0.629019i \(0.783457\pi\)
\(564\) −2.91430e12 −1.21277
\(565\) −1.84438e12 −0.761436
\(566\) 2.16500e12 0.886713
\(567\) −2.50067e12 −1.01609
\(568\) −1.81929e11 −0.0733389
\(569\) −3.40051e12 −1.36000 −0.679999 0.733213i \(-0.738020\pi\)
−0.679999 + 0.733213i \(0.738020\pi\)
\(570\) 3.67426e12 1.45792
\(571\) −4.46270e12 −1.75685 −0.878427 0.477877i \(-0.841406\pi\)
−0.878427 + 0.477877i \(0.841406\pi\)
\(572\) 0 0
\(573\) −5.92590e12 −2.29646
\(574\) 2.10105e12 0.807854
\(575\) −4.77710e10 −0.0182246
\(576\) 2.88249e11 0.109111
\(577\) −3.47193e12 −1.30401 −0.652004 0.758215i \(-0.726071\pi\)
−0.652004 + 0.758215i \(0.726071\pi\)
\(578\) −9.94921e11 −0.370778
\(579\) 8.73441e11 0.322983
\(580\) 4.28199e11 0.157116
\(581\) 2.93588e12 1.06892
\(582\) 2.46613e12 0.890969
\(583\) 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 \(0.665780\pi\)
\(588\) −3.24281e11 −0.111872
\(589\) −3.80158e12 −1.30150
\(590\) 3.12726e11 0.106250
\(591\) −4.34066e12 −1.46356
\(592\) 1.20866e12 0.404442
\(593\) 8.38217e11 0.278362 0.139181 0.990267i \(-0.455553\pi\)
0.139181 + 0.990267i \(0.455553\pi\)
\(594\) −3.45723e10 −0.0113943
\(595\) −1.80762e12 −0.591263
\(596\) 1.54039e11 0.0500062
\(597\) 2.01653e12 0.649710
\(598\) 0 0
\(599\) 3.94489e12 1.25203 0.626013 0.779812i \(-0.284685\pi\)
0.626013 + 0.779812i \(0.284685\pi\)
\(600\) −1.86404e11 −0.0587185
\(601\) 4.99865e12 1.56285 0.781426 0.623998i \(-0.214492\pi\)
0.781426 + 0.623998i \(0.214492\pi\)
\(602\) 1.09008e12 0.338278
\(603\) −3.79254e11 −0.116816
\(604\) 3.24338e12 0.991589
\(605\) −3.06241e12 −0.929317
\(606\) 3.68634e12 1.11037
\(607\) −3.95582e11 −0.118274 −0.0591368 0.998250i \(-0.518835\pi\)
−0.0591368 + 0.998250i \(0.518835\pi\)
\(608\) 9.57365e11 0.284126
\(609\) 1.42433e12 0.419599
\(610\) −1.19480e12 −0.349390
\(611\) 0 0
\(612\) −1.04460e12 −0.301000
\(613\) −5.03617e12 −1.44055 −0.720275 0.693688i \(-0.755984\pi\)
−0.720275 + 0.693688i \(0.755984\pi\)
\(614\) −2.28416e11 −0.0648587
\(615\) 5.68477e12 1.60242
\(616\) 1.07042e11 0.0299531
\(617\) −4.06829e12 −1.13013 −0.565065 0.825046i \(-0.691149\pi\)
−0.565065 + 0.825046i \(0.691149\pi\)
\(618\) −2.94306e12 −0.811616
\(619\) 4.73136e12 1.29532 0.647662 0.761928i \(-0.275747\pi\)
0.647662 + 0.761928i \(0.275747\pi\)
\(620\) −1.39636e12 −0.379521
\(621\) −9.68185e10 −0.0261244
\(622\) 5.73482e10 0.0153626
\(623\) −5.17578e12 −1.37651
\(624\) 0 0
\(625\) −3.29558e12 −0.863916
\(626\) 4.47728e12 1.16528
\(627\) 7.88493e11 0.203748
\(628\) −5.13877e10 −0.0131838
\(629\) −4.38010e12 −1.11572
\(630\) 2.09226e12 0.529156
\(631\) −3.45019e12 −0.866384 −0.433192 0.901302i \(-0.642613\pi\)
−0.433192 + 0.901302i \(0.642613\pi\)
\(632\) 1.95279e12 0.486888
\(633\) −1.08753e12 −0.269230
\(634\) −3.84237e11 −0.0944490
\(635\) −4.33886e12 −1.05899
\(636\) 5.31622e12 1.28838
\(637\) 0 0
\(638\) 9.18912e10 0.0219574
\(639\) −7.63116e11 −0.181066
\(640\) 3.51650e11 0.0828517
\(641\) −3.87461e12 −0.906498 −0.453249 0.891384i \(-0.649735\pi\)
−0.453249 + 0.891384i \(0.649735\pi\)
\(642\) −7.34363e12 −1.70610
\(643\) −2.77752e12 −0.640778 −0.320389 0.947286i \(-0.603814\pi\)
−0.320389 + 0.947286i \(0.603814\pi\)
\(644\) 2.99768e11 0.0686751
\(645\) 2.94940e12 0.670989
\(646\) −3.46942e12 −0.783810
\(647\) 6.12025e12 1.37309 0.686546 0.727086i \(-0.259126\pi\)
0.686546 + 0.727086i \(0.259126\pi\)
\(648\) −1.76295e12 −0.392782
\(649\) 6.71109e10 0.0148488
\(650\) 0 0
\(651\) −4.64477e12 −1.01356
\(652\) −1.61918e12 −0.350898
\(653\) 2.50039e12 0.538143 0.269072 0.963120i \(-0.413283\pi\)
0.269072 + 0.963120i \(0.413283\pi\)
\(654\) 5.22765e12 1.11740
\(655\) 4.01370e12 0.852038
\(656\) 1.48122e12 0.312286
\(657\) −4.56662e12 −0.956204
\(658\) −5.51174e12 −1.14623
\(659\) −6.13676e12 −1.26752 −0.633760 0.773529i \(-0.718489\pi\)
−0.633760 + 0.773529i \(0.718489\pi\)
\(660\) 2.89622e11 0.0594134
\(661\) 6.28369e12 1.28029 0.640145 0.768254i \(-0.278874\pi\)
0.640145 + 0.768254i \(0.278874\pi\)
\(662\) −5.96814e12 −1.20775
\(663\) 0 0
\(664\) 2.06977e12 0.413206
\(665\) 6.94904e12 1.37793
\(666\) 5.06981e12 0.998523
\(667\) 2.57338e11 0.0503429
\(668\) −3.87585e12 −0.753134
\(669\) 6.13645e12 1.18440
\(670\) −4.62671e11 −0.0887025
\(671\) −2.56403e11 −0.0488283
\(672\) 1.16971e12 0.221266
\(673\) −7.98616e12 −1.50062 −0.750309 0.661087i \(-0.770095\pi\)
−0.750309 + 0.661087i \(0.770095\pi\)
\(674\) 3.05852e12 0.570876
\(675\) 1.13863e11 0.0211113
\(676\) 0 0
\(677\) −3.64428e12 −0.666749 −0.333374 0.942794i \(-0.608187\pi\)
−0.333374 + 0.942794i \(0.608187\pi\)
\(678\) −4.32515e12 −0.786082
\(679\) 4.66413e12 0.842087
\(680\) −1.27436e12 −0.228560
\(681\) 9.74942e12 1.73707
\(682\) −2.99658e11 −0.0530391
\(683\) 1.11273e12 0.195658 0.0978291 0.995203i \(-0.468810\pi\)
0.0978291 + 0.995203i \(0.468810\pi\)
\(684\) 4.01574e12 0.701477
\(685\) −7.37244e11 −0.127939
\(686\) −4.36458e12 −0.752460
\(687\) −1.15169e13 −1.97255
\(688\) 7.68495e11 0.130766
\(689\) 0 0
\(690\) 8.11078e11 0.136220
\(691\) 2.75811e12 0.460214 0.230107 0.973165i \(-0.426092\pi\)
0.230107 + 0.973165i \(0.426092\pi\)
\(692\) −4.18517e11 −0.0693802
\(693\) 4.48998e11 0.0739511
\(694\) 1.37664e12 0.225269
\(695\) −6.03382e11 −0.0980982
\(696\) 1.00414e12 0.162201
\(697\) −5.36785e12 −0.861495
\(698\) 2.13779e12 0.340892
\(699\) 9.16278e12 1.45171
\(700\) −3.52542e11 −0.0554969
\(701\) 8.08880e12 1.26518 0.632591 0.774486i \(-0.281991\pi\)
0.632591 + 0.774486i \(0.281991\pi\)
\(702\) 0 0
\(703\) 1.68384e13 2.60017
\(704\) 7.54639e10 0.0115788
\(705\) −1.49130e13 −2.27360
\(706\) 9.98078e11 0.151197
\(707\) 6.97189e12 1.04945
\(708\) 7.33355e11 0.109689
\(709\) −2.19552e11 −0.0326310 −0.0163155 0.999867i \(-0.505194\pi\)
−0.0163155 + 0.999867i \(0.505194\pi\)
\(710\) −9.30965e11 −0.137490
\(711\) 8.19114e12 1.20207
\(712\) −3.64888e12 −0.532108
\(713\) −8.39183e11 −0.121606
\(714\) −4.23894e12 −0.610401
\(715\) 0 0
\(716\) −1.05723e12 −0.150335
\(717\) 1.67341e13 2.36465
\(718\) 6.12382e12 0.859929
\(719\) −8.58532e12 −1.19805 −0.599027 0.800729i \(-0.704446\pi\)
−0.599027 + 0.800729i \(0.704446\pi\)
\(720\) 1.47503e12 0.204552
\(721\) −5.56614e12 −0.767088
\(722\) 8.17451e12 1.11955
\(723\) 2.00073e13 2.72312
\(724\) −5.45515e12 −0.737875
\(725\) −3.02642e11 −0.0406825
\(726\) −7.18146e12 −0.959396
\(727\) 7.59563e11 0.100846 0.0504230 0.998728i \(-0.483943\pi\)
0.0504230 + 0.998728i \(0.483943\pi\)
\(728\) 0 0
\(729\) −5.57939e12 −0.731666
\(730\) −5.57106e12 −0.726080
\(731\) −2.78497e12 −0.360739
\(732\) −2.80185e12 −0.360699
\(733\) 7.83005e12 1.00184 0.500918 0.865495i \(-0.332996\pi\)
0.500918 + 0.865495i \(0.332996\pi\)
\(734\) 4.15684e12 0.528605
\(735\) −1.65940e12 −0.209729
\(736\) 2.11334e11 0.0265473
\(737\) −9.92889e10 −0.0123964
\(738\) 6.21311e12 0.771002
\(739\) −5.41643e12 −0.668056 −0.334028 0.942563i \(-0.608408\pi\)
−0.334028 + 0.942563i \(0.608408\pi\)
\(740\) 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 \(0.551262\pi\)
\(744\) −3.27452e12 −0.391805
\(745\) 7.88249e11 0.0937476
\(746\) 7.53513e12 0.890771
\(747\) 8.68183e12 1.02016
\(748\) −2.73476e11 −0.0319420
\(749\) −1.38888e13 −1.61249
\(750\) −8.81386e12 −1.01716
\(751\) 5.82882e12 0.668653 0.334326 0.942457i \(-0.391491\pi\)
0.334326 + 0.942457i \(0.391491\pi\)
\(752\) −3.88573e12 −0.443090
\(753\) 5.41491e11 0.0613782
\(754\) 0 0
\(755\) 1.65970e13 1.85895
\(756\) −7.14504e11 −0.0795530
\(757\) 5.67869e12 0.628517 0.314258 0.949338i \(-0.398244\pi\)
0.314258 + 0.949338i \(0.398244\pi\)
\(758\) 5.76074e12 0.633821
\(759\) 1.74057e11 0.0190372
\(760\) 4.89901e12 0.532657
\(761\) 1.12490e13 1.21586 0.607931 0.793990i \(-0.292000\pi\)
0.607931 + 0.793990i \(0.292000\pi\)
\(762\) −1.01748e13 −1.09327
\(763\) 9.88694e12 1.05609
\(764\) −7.90120e12 −0.839021
\(765\) −5.34539e12 −0.564291
\(766\) 1.53501e12 0.161095
\(767\) 0 0
\(768\) 8.24634e11 0.0855334
\(769\) −5.02943e12 −0.518621 −0.259311 0.965794i \(-0.583495\pi\)
−0.259311 + 0.965794i \(0.583495\pi\)
\(770\) 5.47756e11 0.0561537
\(771\) −2.71819e12 −0.277036
\(772\) 1.16459e12 0.118003
\(773\) −1.22620e13 −1.23524 −0.617621 0.786476i \(-0.711903\pi\)
−0.617621 + 0.786476i \(0.711903\pi\)
\(774\) 3.22352e12 0.322846
\(775\) 9.86918e11 0.0982705
\(776\) 3.28817e12 0.325520
\(777\) 2.05732e13 2.02491
\(778\) −7.36781e12 −0.720992
\(779\) 2.06356e13 2.00770
\(780\) 0 0
\(781\) −1.99784e11 −0.0192146
\(782\) −7.65861e11 −0.0732351
\(783\) −6.13371e11 −0.0583170
\(784\) −4.32374e11 −0.0408731
\(785\) −2.62960e11 −0.0247159
\(786\) 9.41228e12 0.879617
\(787\) 1.13978e13 1.05909 0.529547 0.848281i \(-0.322362\pi\)
0.529547 + 0.848281i \(0.322362\pi\)
\(788\) −5.78754e12 −0.534720
\(789\) −6.88314e12 −0.632325
\(790\) 9.99279e12 0.912778
\(791\) −8.18005e12 −0.742954
\(792\) 3.16539e11 0.0285867
\(793\) 0 0
\(794\) −4.64479e12 −0.414738
\(795\) 2.72041e13 2.41536
\(796\) 2.68870e12 0.237375
\(797\) 9.66670e12 0.848625 0.424313 0.905516i \(-0.360516\pi\)
0.424313 + 0.905516i \(0.360516\pi\)
\(798\) 1.62958e13 1.42253
\(799\) 1.40816e13 1.22234
\(800\) −2.48539e11 −0.0214531
\(801\) −1.53055e13 −1.31372
\(802\) −1.09679e13 −0.936137
\(803\) −1.19554e12 −0.101472
\(804\) −1.08498e12 −0.0915736
\(805\) 1.53397e12 0.128747
\(806\) 0 0
\(807\) −1.37164e13 −1.13843
\(808\) 4.91512e12 0.405679
\(809\) 4.89988e12 0.402177 0.201088 0.979573i \(-0.435552\pi\)
0.201088 + 0.979573i \(0.435552\pi\)
\(810\) −9.02134e12 −0.736357
\(811\) 9.97393e12 0.809604 0.404802 0.914404i \(-0.367340\pi\)
0.404802 + 0.914404i \(0.367340\pi\)
\(812\) 1.89911e12 0.153302
\(813\) 1.30434e12 0.104709
\(814\) 1.32728e12 0.105963
\(815\) −8.28564e12 −0.657835
\(816\) −2.98842e12 −0.235958
\(817\) 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 \(0.423418\pi\)
\(822\) −1.72886e12 −0.132080
\(823\) 2.05255e13 1.55953 0.779765 0.626072i \(-0.215339\pi\)
0.779765 + 0.626072i \(0.215339\pi\)
\(824\) −3.92408e12 −0.296528
\(825\) −2.04699e11 −0.0153841
\(826\) 1.38698e12 0.103671
\(827\) 1.52447e13 1.13330 0.566649 0.823959i \(-0.308239\pi\)
0.566649 + 0.823959i \(0.308239\pi\)
\(828\) 8.86458e11 0.0655423
\(829\) 1.59195e12 0.117067 0.0585334 0.998285i \(-0.481358\pi\)
0.0585334 + 0.998285i \(0.481358\pi\)
\(830\) 1.05914e13 0.774645
\(831\) −1.33076e13 −0.968042
\(832\) 0 0
\(833\) 1.56689e12 0.112755
\(834\) −1.41495e12 −0.101273
\(835\) −1.98334e13 −1.41192
\(836\) 1.05132e12 0.0744404
\(837\) 2.00021e12 0.140867
\(838\) −1.38368e13 −0.969251
\(839\) 5.56668e11 0.0387853 0.0193927 0.999812i \(-0.493827\pi\)
0.0193927 + 0.999812i \(0.493827\pi\)
\(840\) 5.98561e12 0.414812
\(841\) −1.28768e13 −0.887621
\(842\) 1.53245e12 0.105071
\(843\) −5.95908e12 −0.406401
\(844\) −1.45004e12 −0.0983643
\(845\) 0 0
\(846\) −1.62990e13 −1.09394
\(847\) −1.35821e13 −0.906760
\(848\) 7.08829e12 0.470717
\(849\) 2.59800e13 1.71614
\(850\) 9.00687e11 0.0591819
\(851\) 3.71701e12 0.242946
\(852\) −2.18315e12 −0.141940
\(853\) 1.76959e13 1.14446 0.572231 0.820093i \(-0.306078\pi\)
0.572231 + 0.820093i \(0.306078\pi\)
\(854\) −5.29907e12 −0.340909
\(855\) 2.05493e13 1.31507
\(856\) −9.79151e12 −0.623330
\(857\) 1.34064e13 0.848983 0.424492 0.905432i \(-0.360453\pi\)
0.424492 + 0.905432i \(0.360453\pi\)
\(858\) 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
\(861\) 2.52126e13 1.56352
\(862\) 2.03496e12 0.125537
\(863\) 1.09637e13 0.672838 0.336419 0.941712i \(-0.390784\pi\)
0.336419 + 0.941712i \(0.390784\pi\)
\(864\) −5.03719e11 −0.0307522
\(865\) −2.14163e12 −0.130068
\(866\) −2.48423e12 −0.150093
\(867\) −1.19391e13 −0.717603
\(868\) −6.19302e12 −0.370309
\(869\) 2.14445e12 0.127563
\(870\) 5.13839e12 0.304082
\(871\) 0 0
\(872\) 6.97021e12 0.408245
\(873\) 1.37925e13 0.803673
\(874\) 2.94420e12 0.170673
\(875\) −1.66695e13 −0.961358
\(876\) −1.30643e13 −0.749582
\(877\) −1.20955e13 −0.690440 −0.345220 0.938522i \(-0.612196\pi\)
−0.345220 + 0.938522i \(0.612196\pi\)
\(878\) −1.64175e13 −0.932358
\(879\) 1.44975e13 0.819112
\(880\) 3.86163e11 0.0217069
\(881\) −3.33493e13 −1.86507 −0.932534 0.361083i \(-0.882407\pi\)
−0.932534 + 0.361083i \(0.882407\pi\)
\(882\) −1.81363e12 −0.100911
\(883\) −1.01455e13 −0.561628 −0.280814 0.959762i \(-0.590604\pi\)
−0.280814 + 0.959762i \(0.590604\pi\)
\(884\) 0 0
\(885\) 3.75272e12 0.205637
\(886\) 4.03263e12 0.219855
\(887\) 3.04816e13 1.65341 0.826707 0.562633i \(-0.190211\pi\)
0.826707 + 0.562633i \(0.190211\pi\)
\(888\) 1.45039e13 0.782756
\(889\) −1.92433e13 −1.03329
\(890\) −1.86720e13 −0.997554
\(891\) −1.93597e12 −0.102908
\(892\) 8.18194e12 0.432727
\(893\) −5.41340e13 −2.84865
\(894\) 1.84847e12 0.0967820
\(895\) −5.41003e12 −0.281836
\(896\) 1.55961e12 0.0808407
\(897\) 0 0
\(898\) −1.22592e13 −0.629097
\(899\) −5.31644e12 −0.271458
\(900\) −1.04252e12 −0.0529653
\(901\) −2.56875e13 −1.29855
\(902\) 1.62660e12 0.0818183
\(903\) 1.30809e13 0.654702
\(904\) −5.76687e12 −0.287198
\(905\) −2.79150e13 −1.38331
\(906\) 3.89206e13 1.91912
\(907\) −2.08823e12 −0.102458 −0.0512289 0.998687i \(-0.516314\pi\)
−0.0512289 + 0.998687i \(0.516314\pi\)
\(908\) 1.29992e13 0.634645
\(909\) 2.06169e13 1.00158
\(910\) 0 0
\(911\) −1.70747e13 −0.821336 −0.410668 0.911785i \(-0.634704\pi\)
−0.410668 + 0.911785i \(0.634704\pi\)
\(912\) 1.14884e13 0.549898
\(913\) 2.27291e12 0.108259
\(914\) −2.81093e13 −1.33227
\(915\) −1.43376e13 −0.676209
\(916\) −1.53558e13 −0.720681
\(917\) 1.78012e13 0.831358
\(918\) 1.82544e12 0.0848353
\(919\) −3.86177e12 −0.178594 −0.0892970 0.996005i \(-0.528462\pi\)
−0.0892970 + 0.996005i \(0.528462\pi\)
\(920\) 1.08144e12 0.0497687
\(921\) −2.74099e12 −0.125528
\(922\) 1.81003e13 0.824893
\(923\) 0 0
\(924\) 1.28451e12 0.0579713
\(925\) −4.37137e12 −0.196327
\(926\) 4.34651e12 0.194263
\(927\) −1.64599e13 −0.732095
\(928\) 1.33886e12 0.0592609
\(929\) −2.72392e13 −1.19984 −0.599921 0.800059i \(-0.704801\pi\)
−0.599921 + 0.800059i \(0.704801\pi\)
\(930\) −1.67563e13 −0.734524
\(931\) −6.02362e12 −0.262775
\(932\) 1.22170e13 0.530389
\(933\) 6.88179e11 0.0297327
\(934\) −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
\(939\) 5.37274e13 2.25528
\(940\) −1.98840e13 −0.830670
\(941\) −5.78614e12 −0.240567 −0.120283 0.992740i \(-0.538380\pi\)
−0.120283 + 0.992740i \(0.538380\pi\)
\(942\) −6.16652e11 −0.0255159
\(943\) 4.55523e12 0.187589
\(944\) 9.77807e11 0.0400755
\(945\) −3.65625e12 −0.149140
\(946\) 8.43919e11 0.0342603
\(947\) 4.03191e13 1.62905 0.814527 0.580125i \(-0.196996\pi\)
0.814527 + 0.580125i \(0.196996\pi\)
\(948\) 2.34335e13 0.942322
\(949\) 0 0
\(950\) −3.46251e12 −0.137923
\(951\) −4.61084e12 −0.182797
\(952\) −5.65192e12 −0.223013
\(953\) 1.46625e13 0.575824 0.287912 0.957657i \(-0.407039\pi\)
0.287912 + 0.957657i \(0.407039\pi\)
\(954\) 2.97324e13 1.16215
\(955\) −4.04319e13 −1.57293
\(956\) 2.23122e13 0.863936
\(957\) 1.10269e12 0.0424963
\(958\) −2.29887e13 −0.881801
\(959\) −3.26976e12 −0.124834
\(960\) 4.21981e12 0.160351
\(961\) −9.10264e12 −0.344280
\(962\) 0 0
\(963\) −4.10713e13 −1.53893
\(964\) 2.66764e13 0.994903
\(965\) 5.95942e12 0.221223
\(966\) 3.59722e12 0.132914
\(967\) −1.87662e13 −0.690172 −0.345086 0.938571i \(-0.612150\pi\)
−0.345086 + 0.938571i \(0.612150\pi\)
\(968\) −9.57528e12 −0.350520
\(969\) −4.16331e13 −1.51699
\(970\) 1.68262e13 0.610258
\(971\) 2.66964e13 0.963755 0.481877 0.876239i \(-0.339955\pi\)
0.481877 + 0.876239i \(0.339955\pi\)
\(972\) −1.87348e13 −0.673210
\(973\) −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 \(0.581593\pi\)
\(978\) −1.94301e13 −0.679127
\(979\) −4.00700e12 −0.139411
\(980\) −2.21254e12 −0.0766256
\(981\) 2.92371e13 1.00791
\(982\) −1.60622e13 −0.551194
\(983\) −3.96507e13 −1.35444 −0.677220 0.735781i \(-0.736815\pi\)
−0.677220 + 0.735781i \(0.736815\pi\)
\(984\) 1.77747e13 0.604399
\(985\) −2.96159e13 −1.00245
\(986\) −4.85193e12 −0.163481
\(987\) −6.61409e13 −2.21842
\(988\) 0 0
\(989\) 2.36337e12 0.0785503
\(990\) 1.61979e12 0.0535921
\(991\) −4.04833e13 −1.33335 −0.666675 0.745348i \(-0.732283\pi\)
−0.666675 + 0.745348i \(0.732283\pi\)
\(992\) −4.36603e12 −0.143148
\(993\) −7.16177e13 −2.33748
\(994\) −4.12893e12 −0.134153
\(995\) 1.37586e13 0.445011
\(996\) 2.48373e13 0.799718
\(997\) −2.52148e13 −0.808217 −0.404109 0.914711i \(-0.632418\pi\)
−0.404109 + 0.914711i \(0.632418\pi\)
\(998\) 1.21322e13 0.387125
\(999\) −8.85956e12 −0.281428
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 338.10.a.d.1.1 1
13.12 even 2 26.10.a.b.1.1 1
39.38 odd 2 234.10.a.c.1.1 1
52.51 odd 2 208.10.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.b.1.1 1 13.12 even 2
208.10.a.a.1.1 1 52.51 odd 2
234.10.a.c.1.1 1 39.38 odd 2
338.10.a.d.1.1 1 1.1 even 1 trivial