Properties

Label 26.10.a.b.1.1
Level $26$
Weight $10$
Character 26.1
Self dual yes
Analytic conductor $13.391$
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] = [26,10,Mod(1,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, 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("26.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 26.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: \(13.3909317403\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 26.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.0000 q^{2} +192.000 q^{3} +256.000 q^{4} -1310.00 q^{5} -3072.00 q^{6} -5810.00 q^{7} -4096.00 q^{8} +17181.0 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} +192.000 q^{3} +256.000 q^{4} -1310.00 q^{5} -3072.00 q^{6} -5810.00 q^{7} -4096.00 q^{8} +17181.0 q^{9} +20960.0 q^{10} -4498.00 q^{11} +49152.0 q^{12} -28561.0 q^{13} +92960.0 q^{14} -251520. q^{15} +65536.0 q^{16} -237498. q^{17} -274896. q^{18} -913014. q^{19} -335360. q^{20} -1.11552e6 q^{21} +71968.0 q^{22} +201544. q^{23} -786432. q^{24} -237025. q^{25} +456976. q^{26} -480384. q^{27} -1.48736e6 q^{28} +1.27683e6 q^{29} +4.02432e6 q^{30} +4.16377e6 q^{31} -1.04858e6 q^{32} -863616. q^{33} +3.79997e6 q^{34} +7.61110e6 q^{35} +4.39834e6 q^{36} -1.84427e7 q^{37} +1.46082e7 q^{38} -5.48371e6 q^{39} +5.36576e6 q^{40} -2.26017e7 q^{41} +1.78483e7 q^{42} +1.17263e7 q^{43} -1.15149e6 q^{44} -2.25071e7 q^{45} -3.22470e6 q^{46} +5.92915e7 q^{47} +1.25829e7 q^{48} -6.59751e6 q^{49} +3.79240e6 q^{50} -4.55996e7 q^{51} -7.31162e6 q^{52} +1.08159e8 q^{53} +7.68614e6 q^{54} +5.89238e6 q^{55} +2.37978e7 q^{56} -1.75299e8 q^{57} -2.04293e7 q^{58} -1.49202e7 q^{59} -6.43891e7 q^{60} -5.70037e7 q^{61} -6.66203e7 q^{62} -9.98216e7 q^{63} +1.67772e7 q^{64} +3.74149e7 q^{65} +1.38179e7 q^{66} +2.20740e7 q^{67} -6.07995e7 q^{68} +3.86964e7 q^{69} -1.21778e8 q^{70} +4.44162e7 q^{71} -7.03734e7 q^{72} +2.65795e8 q^{73} +2.95083e8 q^{74} -4.55088e7 q^{75} -2.33732e8 q^{76} +2.61334e7 q^{77} +8.77394e7 q^{78} +4.76755e8 q^{79} -8.58522e7 q^{80} -4.30407e8 q^{81} +3.61627e8 q^{82} -5.05316e8 q^{83} -2.85573e8 q^{84} +3.11122e8 q^{85} -1.87621e8 q^{86} +2.45152e8 q^{87} +1.84238e7 q^{88} +8.90841e8 q^{89} +3.60114e8 q^{90} +1.65939e8 q^{91} +5.15953e7 q^{92} +7.99444e8 q^{93} -9.48665e8 q^{94} +1.19605e9 q^{95} -2.01327e8 q^{96} -8.02777e8 q^{97} +1.05560e8 q^{98} -7.72801e7 q^{99} +O(q^{100})\)

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 \(-0.655270\pi\)
−0.468680 + 0.883368i \(0.655270\pi\)
\(6\) −3072.00 −0.967700
\(7\) −5810.00 −0.914608 −0.457304 0.889310i \(-0.651185\pi\)
−0.457304 + 0.889310i \(0.651185\pi\)
\(8\) −4096.00 −0.353553
\(9\) 17181.0 0.872885
\(10\) 20960.0 0.662813
\(11\) −4498.00 −0.0926302 −0.0463151 0.998927i \(-0.514748\pi\)
−0.0463151 + 0.998927i \(0.514748\pi\)
\(12\) 49152.0 0.684267
\(13\) −28561.0 −0.277350
\(14\) 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 \(-0.797101\pi\)
−0.803630 + 0.595129i \(0.797101\pi\)
\(20\) −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
\(26\) 456976. 0.196116
\(27\) −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 \(-0.367312\pi\)
0.404883 + 0.914369i \(0.367312\pi\)
\(32\) −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 \(-0.799928\pi\)
−0.808883 + 0.587969i \(0.799928\pi\)
\(38\) 1.46082e7 1.13650
\(39\) −5.48371e6 −0.379563
\(40\) 5.36576e6 0.331407
\(41\) −2.26017e7 −1.24915 −0.624573 0.780966i \(-0.714727\pi\)
−0.624573 + 0.780966i \(0.714727\pi\)
\(42\) 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 \(-0.153349\pi\)
0.886181 + 0.463340i \(0.153349\pi\)
\(48\) 1.25829e7 0.342133
\(49\) −6.59751e6 −0.163492
\(50\) 3.79240e6 0.0858122
\(51\) −4.55996e7 −0.943834
\(52\) −7.31162e6 −0.138675
\(53\) 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 \(-0.525540\pi\)
−0.0801511 + 0.996783i \(0.525540\pi\)
\(60\) −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
\(65\) 3.74149e7 0.259977
\(66\) 1.38179e7 0.0896382
\(67\) 2.20740e7 0.133827 0.0669136 0.997759i \(-0.478685\pi\)
0.0669136 + 0.997759i \(0.478685\pi\)
\(68\) −6.07995e7 −0.344834
\(69\) 3.86964e7 0.205518
\(70\) −1.21778e8 −0.606214
\(71\) 4.44162e7 0.207434 0.103717 0.994607i \(-0.466926\pi\)
0.103717 + 0.994607i \(0.466926\pi\)
\(72\) −7.03734e7 −0.308612
\(73\) 2.65795e8 1.09545 0.547726 0.836658i \(-0.315494\pi\)
0.547726 + 0.836658i \(0.315494\pi\)
\(74\) 2.95083e8 1.14393
\(75\) −4.55088e7 −0.166081
\(76\) −2.33732e8 −0.803630
\(77\) 2.61334e7 0.0847203
\(78\) 8.77394e7 0.268392
\(79\) 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 \(-0.698655\pi\)
−0.584361 + 0.811494i \(0.698655\pi\)
\(84\) −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 \(-0.228841\pi\)
0.752515 + 0.658575i \(0.228841\pi\)
\(90\) 3.60114e8 0.578560
\(91\) 1.65939e8 0.253667
\(92\) 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 \(-0.652278\pi\)
−0.460354 + 0.887735i \(0.652278\pi\)
\(98\) 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 \(0.637743\pi\)
\(104\) 1.16986e8 0.0980581
\(105\) 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 \(-0.695912\pi\)
−0.577346 + 0.816499i \(0.695912\pi\)
\(110\) −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
\(117\) −4.90707e8 −0.242095
\(118\) 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
\(130\) −5.98639e8 −0.183831
\(131\) 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 \(-0.478260\pi\)
0.0682444 + 0.997669i \(0.478260\pi\)
\(138\) −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
\(143\) 1.28467e8 0.0256910
\(144\) 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 \(-0.515924\pi\)
−0.0500062 + 0.998749i \(0.515924\pi\)
\(150\) 7.28141e8 0.117437
\(151\) −1.26695e10 −1.98318 −0.991589 0.129428i \(-0.958686\pi\)
−0.991589 + 0.129428i \(0.958686\pi\)
\(152\) 3.73971e9 0.568252
\(153\) −4.08045e9 −0.602001
\(154\) −4.18134e8 −0.0599063
\(155\) −5.45454e9 −0.759041
\(156\) −1.40383e9 −0.189782
\(157\) −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 \(-0.385877\pi\)
0.350898 + 0.936414i \(0.385877\pi\)
\(164\) −5.78603e9 −0.624573
\(165\) 1.13134e9 0.118827
\(166\) 8.08505e9 0.826412
\(167\) 1.51400e10 1.50627 0.753134 0.657867i \(-0.228541\pi\)
0.753134 + 0.657867i \(0.228541\pi\)
\(168\) 4.56917e9 0.442533
\(169\) 8.15731e8 0.0769231
\(170\) −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 \(0.764171\pi\)
\(182\) −2.65503e9 −0.179369
\(183\) −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 \(-0.537649\pi\)
−0.118003 + 0.993013i \(0.537649\pi\)
\(194\) 1.28444e10 0.651039
\(195\) 7.18366e9 0.355787
\(196\) −1.68896e9 −0.0817462
\(197\) 2.26076e10 1.06944 0.534720 0.845030i \(-0.320417\pi\)
0.534720 + 0.845030i \(0.320417\pi\)
\(198\) 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
\(208\) −1.87177e9 −0.0693375
\(209\) 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
\(221\) 6.78318e9 0.191279
\(222\) 5.66559e10 1.56551
\(223\) −3.19607e10 −0.865454 −0.432727 0.901525i \(-0.642449\pi\)
−0.432727 + 0.901525i \(0.642449\pi\)
\(224\) 6.09223e9 0.161681
\(225\) −4.07233e9 −0.105931
\(226\) 2.25268e10 0.574397
\(227\) −5.07782e10 −1.26929 −0.634645 0.772804i \(-0.718854\pi\)
−0.634645 + 0.772804i \(0.718854\pi\)
\(228\) −4.48765e10 −1.09980
\(229\) 5.99836e10 1.44136 0.720681 0.693267i \(-0.243829\pi\)
0.720681 + 0.693267i \(0.243829\pi\)
\(230\) 4.22436e9 0.0995373
\(231\) 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 \(0.322046\pi\)
\(234\) 7.85130e9 0.171187
\(235\) −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 \(-0.832008\pi\)
−0.863936 + 0.503602i \(0.832008\pi\)
\(240\) −1.64836e10 −0.320702
\(241\) −1.04205e11 −1.98981 −0.994903 0.100835i \(-0.967849\pi\)
−0.994903 + 0.100835i \(0.967849\pi\)
\(242\) 3.74035e10 0.701040
\(243\) −7.31828e10 −1.34642
\(244\) −1.45930e10 −0.263566
\(245\) 8.64273e9 0.153251
\(246\) 6.94323e10 1.20880
\(247\) 2.60766e10 0.445774
\(248\) −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
\(260\) 9.57822e9 0.129988
\(261\) 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 \(-0.512180\pi\)
−0.0382558 + 0.999268i \(0.512180\pi\)
\(272\) −1.55647e10 −0.172417
\(273\) 3.18604e10 0.347151
\(274\) −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 \(-0.452562\pi\)
0.148480 + 0.988915i \(0.452562\pi\)
\(282\) −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
\(286\) −2.05548e9 −0.0181663
\(287\) 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 \(-0.596742\pi\)
−0.299266 + 0.954170i \(0.596742\pi\)
\(294\) 2.02675e10 0.158212
\(295\) 1.95454e10 0.150261
\(296\) 7.55411e10 0.571967
\(297\) 2.16077e9 0.0161140
\(298\) 9.62747e9 0.0707195
\(299\) −5.75630e9 −0.0416508
\(300\) −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 \(-0.485397\pi\)
0.0458620 + 0.998948i \(0.485397\pi\)
\(308\) 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 \(0.496542\pi\)
\(312\) 2.24613e10 0.134196
\(313\) 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 \(-0.478726\pi\)
0.0667855 + 0.997767i \(0.478726\pi\)
\(318\) −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
\(325\) 6.76967e9 0.0336583
\(326\) −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 \(-0.174164\pi\)
0.854010 + 0.520257i \(0.174164\pi\)
\(332\) −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 \(0.367734\pi\)
\(338\) −1.30517e10 −0.0543928
\(339\) −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 \(-0.577491\pi\)
−0.241047 + 0.970513i \(0.577491\pi\)
\(350\) −2.20338e10 −0.0784845
\(351\) 1.37202e10 0.0482481
\(352\) 4.71649e9 0.0163749
\(353\) −6.23799e10 −0.213825 −0.106912 0.994268i \(-0.534096\pi\)
−0.106912 + 0.994268i \(0.534096\pi\)
\(354\) 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 \(-0.708053\pi\)
−0.608062 + 0.793890i \(0.708053\pi\)
\(360\) 9.21891e10 0.289280
\(361\) 5.10907e11 1.58329
\(362\) 3.40947e11 1.04351
\(363\) −4.48841e11 −1.35679
\(364\) 4.24805e10 0.126833
\(365\) −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
\(377\) −3.64677e10 −0.0929762
\(378\) −4.46565e10 −0.112505
\(379\) −3.60046e11 −0.896358 −0.448179 0.893944i \(-0.647927\pi\)
−0.448179 + 0.893944i \(0.647927\pi\)
\(380\) 3.06188e11 0.753291
\(381\) −6.35924e11 −1.54612
\(382\) 4.93825e11 1.18655
\(383\) −9.59380e10 −0.227822 −0.113911 0.993491i \(-0.536338\pi\)
−0.113911 + 0.993491i \(0.536338\pi\)
\(384\) −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 \(0.670287\pi\)
\(390\) −1.14939e11 −0.251579
\(391\) −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 \(-0.405259\pi\)
0.293264 + 0.956032i \(0.405259\pi\)
\(398\) −1.68044e11 −0.335698
\(399\) 1.01849e12 2.01176
\(400\) −1.55337e10 −0.0303392
\(401\) 6.85495e11 1.32390 0.661949 0.749549i \(-0.269730\pi\)
0.661949 + 0.749549i \(0.269730\pi\)
\(402\) −6.78114e10 −0.129505
\(403\) −1.18921e11 −0.224588
\(404\) 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 \(-0.154991\pi\)
0.883779 + 0.467905i \(0.154991\pi\)
\(410\) −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
\(416\) 2.99484e10 0.0490290
\(417\) −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 \(-0.523671\pi\)
−0.0742965 + 0.997236i \(0.523671\pi\)
\(422\) 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
\(429\) 2.46657e10 0.0351590
\(430\) 2.45783e11 0.346693
\(431\) −1.27185e11 −0.177536 −0.0887682 0.996052i \(-0.528293\pi\)
−0.0887682 + 0.996052i \(0.528293\pi\)
\(432\) −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
\(442\) −1.08531e11 −0.135255
\(443\) 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 \(-0.353261\pi\)
0.444839 + 0.895611i \(0.353261\pi\)
\(450\) 6.51572e10 0.0749042
\(451\) 1.01662e11 0.115709
\(452\) −3.60429e11 −0.406160
\(453\) −2.43254e12 −2.71405
\(454\) 8.12451e11 0.897524
\(455\) −2.17381e11 −0.237777
\(456\) 7.18023e11 0.777673
\(457\) 1.75683e12 1.88411 0.942057 0.335454i \(-0.108890\pi\)
0.942057 + 0.335454i \(0.108890\pi\)
\(458\) −9.59738e11 −1.01920
\(459\) 1.14090e11 0.119975
\(460\) −6.75898e10 −0.0703835
\(461\) −1.13127e12 −1.16657 −0.583287 0.812266i \(-0.698234\pi\)
−0.583287 + 0.812266i \(0.698234\pi\)
\(462\) −8.02817e10 −0.0819838
\(463\) −2.71657e11 −0.274730 −0.137365 0.990521i \(-0.543863\pi\)
−0.137365 + 0.990521i \(0.543863\pi\)
\(464\) 8.36786e10 0.0838076
\(465\) −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 \(0.653723\pi\)
\(468\) −1.25621e11 −0.121047
\(469\) −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 \(-0.285699\pi\)
0.623527 + 0.781802i \(0.285699\pi\)
\(480\) 2.63738e11 0.226771
\(481\) 5.26741e11 0.448688
\(482\) 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 \(-0.664617\pi\)
−0.494413 + 0.869227i \(0.664617\pi\)
\(488\) 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
\(494\) −4.17225e11 −0.315210
\(495\) 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 \(-0.588260\pi\)
−0.273739 + 0.961804i \(0.588260\pi\)
\(500\) 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
\(507\) 1.56620e11 0.105272
\(508\) −8.47899e11 −0.564881
\(509\) 6.57012e11 0.433854 0.216927 0.976188i \(-0.430397\pi\)
0.216927 + 0.976188i \(0.430397\pi\)
\(510\) −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
\(520\) −1.53251e11 −0.0919157
\(521\) 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
\(533\) 6.45526e11 0.346451
\(534\) −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 \(-0.316933\pi\)
0.543937 + 0.839126i \(0.316933\pi\)
\(542\) 1.08695e11 0.0541019
\(543\) −4.09136e12 −2.01961
\(544\) 2.49035e11 0.121917
\(545\) 2.22924e12 1.08236
\(546\) −5.09766e11 −0.245473
\(547\) −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 \(-0.498768\pi\)
0.00386917 + 0.999993i \(0.498768\pi\)
\(558\) −1.14460e12 −0.499806
\(559\) −3.34915e11 −0.145071
\(560\) 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 \(0.841406\pi\)
\(572\) 3.28876e10 0.0128455
\(573\) −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 \(-0.273929\pi\)
0.652004 + 0.758215i \(0.273929\pi\)
\(578\) 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
\(585\) 6.42826e11 0.226930
\(586\) 1.20812e12 0.423226
\(587\) 2.86266e12 0.995171 0.497586 0.867415i \(-0.334220\pi\)
0.497586 + 0.867415i \(0.334220\pi\)
\(588\) −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 \(-0.544447\pi\)
−0.139181 + 0.990267i \(0.544447\pi\)
\(594\) −3.45723e10 −0.0113943
\(595\) −1.80762e12 −0.591263
\(596\) −1.54039e11 −0.0500062
\(597\) 2.01653e12 0.649710
\(598\) 9.21008e10 0.0294515
\(599\) 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
\(611\) −1.69343e12 −0.491565
\(612\) −1.04460e12 −0.301000
\(613\) 5.03617e12 1.44055 0.720275 0.693688i \(-0.244016\pi\)
0.720275 + 0.693688i \(0.244016\pi\)
\(614\) −2.28416e11 −0.0648587
\(615\) 5.68477e12 1.60242
\(616\) −1.07042e11 −0.0299531
\(617\) 4.06829e12 1.13013 0.565065 0.825046i \(-0.308851\pi\)
0.565065 + 0.825046i \(0.308851\pi\)
\(618\) 2.94306e12 0.811616
\(619\) −4.73136e12 −1.29532 −0.647662 0.761928i \(-0.724253\pi\)
−0.647662 + 0.761928i \(0.724253\pi\)
\(620\) −1.39636e12 −0.379521
\(621\) −9.68185e10 −0.0261244
\(622\) −5.73482e10 −0.0153626
\(623\) −5.17578e12 −1.37651
\(624\) −3.59381e11 −0.0948908
\(625\) −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 \(-0.357387\pi\)
0.433192 + 0.901302i \(0.357387\pi\)
\(632\) −1.95279e12 −0.486888
\(633\) −1.08753e12 −0.269230
\(634\) −3.84237e11 −0.0944490
\(635\) 4.33886e12 1.05899
\(636\) 5.31622e12 1.28838
\(637\) 1.88431e11 0.0453446
\(638\) 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 \(-0.396186\pi\)
0.320389 + 0.947286i \(0.396186\pi\)
\(644\) −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
\(650\) −1.08315e11 −0.0238000
\(651\) −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 \(-0.721126\pi\)
−0.640145 + 0.768254i \(0.721126\pi\)
\(662\) −5.96814e12 −1.20775
\(663\) 1.30237e12 0.261772
\(664\) 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
\(676\) 2.08827e11 0.0384615
\(677\) −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 \(-0.531190\pi\)
−0.0978291 + 0.995203i \(0.531190\pi\)
\(684\) −4.01574e12 −0.701477
\(685\) −7.37244e11 −0.127939
\(686\) −4.36458e12 −0.752460
\(687\) 1.15169e13 1.97255
\(688\) 7.68495e11 0.130766
\(689\) −3.08912e12 −0.522214
\(690\) 8.11078e11 0.136220
\(691\) −2.75811e12 −0.460214 −0.230107 0.973165i \(-0.573908\pi\)
−0.230107 + 0.973165i \(0.573908\pi\)
\(692\) −4.18517e11 −0.0693802
\(693\) 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 \(0.281991\pi\)
\(702\) −2.19524e11 −0.0341165
\(703\) 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 \(-0.494806\pi\)
0.0163155 + 0.999867i \(0.494806\pi\)
\(710\) 9.30965e11 0.137490
\(711\) 8.19114e12 1.20207
\(712\) −3.64888e12 −0.532108
\(713\) 8.39183e11 0.121606
\(714\) −4.23894e12 −0.610401
\(715\) −1.68292e11 −0.0240817
\(716\) −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 \(0.483943\pi\)
\(728\) −6.79688e11 −0.0896847
\(729\) −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 \(-0.667004\pi\)
−0.500918 + 0.865495i \(0.667004\pi\)
\(734\) −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 \(-0.391592\pi\)
0.334028 + 0.942563i \(0.391592\pi\)
\(740\) 6.18493e12 0.758215
\(741\) 5.00671e12 0.610057
\(742\) 1.00544e13 1.21770
\(743\) 2.66408e12 0.320699 0.160349 0.987060i \(-0.448738\pi\)
0.160349 + 0.987060i \(0.448738\pi\)
\(744\) −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
\(754\) 5.83482e11 0.0657441
\(755\) 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 \(-0.708000\pi\)
−0.607931 + 0.793990i \(0.708000\pi\)
\(762\) 1.01748e13 1.09327
\(763\) 9.88694e12 1.05609
\(764\) −7.90120e12 −0.839021
\(765\) 5.34539e12 0.564291
\(766\) 1.53501e12 0.161095
\(767\) 4.26135e11 0.0444598
\(768\) 8.24634e11 0.0855334
\(769\) 5.02943e12 0.518621 0.259311 0.965794i \(-0.416505\pi\)
0.259311 + 0.965794i \(0.416505\pi\)
\(770\) 5.47756e11 0.0561537
\(771\) −2.71819e12 −0.277036
\(772\) −1.16459e12 −0.118003
\(773\) 1.22620e13 1.23524 0.617621 0.786476i \(-0.288097\pi\)
0.617621 + 0.786476i \(0.288097\pi\)
\(774\) −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
\(780\) 1.83902e12 0.177894
\(781\) −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 \(-0.677638\pi\)
−0.529547 + 0.848281i \(0.677638\pi\)
\(788\) 5.78754e12 0.534720
\(789\) −6.88314e12 −0.632325
\(790\) 9.99279e12 0.912778
\(791\) 8.18005e12 0.742954
\(792\) 3.16539e11 0.0285867
\(793\) 1.62808e12 0.146200
\(794\) −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
\(806\) 1.90274e12 0.158808
\(807\) −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 \(-0.632660\pi\)
−0.404802 + 0.914404i \(0.632660\pi\)
\(812\) −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
\(819\) 2.85101e12 0.221422
\(820\) 7.57970e12 0.585450
\(821\) −6.20376e12 −0.476552 −0.238276 0.971197i \(-0.576582\pi\)
−0.238276 + 0.971197i \(0.576582\pi\)
\(822\) −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 \(-0.691761\pi\)
−0.566649 + 0.823959i \(0.691761\pi\)
\(828\) 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
\(832\) −4.79174e11 −0.0346688
\(833\) 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 \(-0.506173\pi\)
−0.0193927 + 0.999812i \(0.506173\pi\)
\(840\) −5.98561e12 −0.414812
\(841\) −1.28768e13 −0.887621
\(842\) 1.53245e12 0.105071
\(843\) 5.95908e12 0.406401
\(844\) −1.45004e12 −0.0983643
\(845\) −1.06861e12 −0.0721046
\(846\) −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 \(-0.693922\pi\)
−0.572231 + 0.820093i \(0.693922\pi\)
\(854\) −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 \(0.360453\pi\)
\(858\) −3.94652e11 −0.0248612
\(859\) 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 \(-0.609216\pi\)
−0.336419 + 0.941712i \(0.609216\pi\)
\(864\) 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
\(871\) −6.30456e11 −0.0371170
\(872\) 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 \(-0.387804\pi\)
0.345220 + 0.938522i \(0.387804\pi\)
\(878\) 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 \(0.590604\pi\)
\(884\) 1.73649e12 0.0956397
\(885\) 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
\(897\) −1.10521e12 −0.0570005
\(898\) −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
\(910\) 3.47809e12 0.168134
\(911\) −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
\(923\) −1.26857e12 −0.0575318
\(924\) 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 \(-0.295199\pi\)
0.599921 + 0.800059i \(0.295199\pi\)
\(930\) 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
\(936\) 2.00993e12 0.0855934
\(937\) −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 \(-0.461620\pi\)
0.120283 + 0.992740i \(0.461620\pi\)
\(942\) 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 \(-0.803004\pi\)
−0.814527 + 0.580125i \(0.803004\pi\)
\(948\) 2.34335e13 0.942322
\(949\) −7.59136e12 −0.303824
\(950\) −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
\(962\) −8.42785e12 −0.317270
\(963\) −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 \(-0.387850\pi\)
0.345086 + 0.938571i \(0.387850\pi\)
\(968\) 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
\(975\) 1.29978e12 0.0460626
\(976\) −3.73580e12 −0.131783
\(977\) 1.44408e13 0.507068 0.253534 0.967327i \(-0.418407\pi\)
0.253534 + 0.967327i \(0.418407\pi\)
\(978\) −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 \(-0.263185\pi\)
0.677220 + 0.735781i \(0.263185\pi\)
\(984\) 1.77747e13 0.604399
\(985\) −2.96159e13 −1.00245
\(986\) 4.85193e12 0.163481
\(987\) −6.61409e13 −2.21842
\(988\) 6.67561e12 0.222887
\(989\) 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 26.10.a.b.1.1 1
3.2 odd 2 234.10.a.c.1.1 1
4.3 odd 2 208.10.a.a.1.1 1
13.12 even 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 1.1 even 1 trivial
208.10.a.a.1.1 1 4.3 odd 2
234.10.a.c.1.1 1 3.2 odd 2
338.10.a.d.1.1 1 13.12 even 2