Properties

Label 234.8.a.a.1.1
Level $234$
Weight $8$
Character 234.1
Self dual yes
Analytic conductor $73.098$
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] = [234,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(73.0980959633\)
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 \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +64.0000 q^{4} -321.000 q^{5} -181.000 q^{7} -512.000 q^{8} +O(q^{10})\) \(q-8.00000 q^{2} +64.0000 q^{4} -321.000 q^{5} -181.000 q^{7} -512.000 q^{8} +2568.00 q^{10} -7782.00 q^{11} +2197.00 q^{13} +1448.00 q^{14} +4096.00 q^{16} -9069.00 q^{17} -37150.0 q^{19} -20544.0 q^{20} +62256.0 q^{22} -19008.0 q^{23} +24916.0 q^{25} -17576.0 q^{26} -11584.0 q^{28} -174750. q^{29} +29012.0 q^{31} -32768.0 q^{32} +72552.0 q^{34} +58101.0 q^{35} +323669. q^{37} +297200. q^{38} +164352. q^{40} -795312. q^{41} -314137. q^{43} -498048. q^{44} +152064. q^{46} +447441. q^{47} -790782. q^{49} -199328. q^{50} +140608. q^{52} +1.46923e6 q^{53} +2.49802e6 q^{55} +92672.0 q^{56} +1.39800e6 q^{58} -1.62777e6 q^{59} -2.39961e6 q^{61} -232096. q^{62} +262144. q^{64} -705237. q^{65} -64066.0 q^{67} -580416. q^{68} -464808. q^{70} +322383. q^{71} -4.45478e6 q^{73} -2.58935e6 q^{74} -2.37760e6 q^{76} +1.40854e6 q^{77} +753560. q^{79} -1.31482e6 q^{80} +6.36250e6 q^{82} +1.21909e6 q^{83} +2.91115e6 q^{85} +2.51310e6 q^{86} +3.98438e6 q^{88} -3.39033e6 q^{89} -397657. q^{91} -1.21651e6 q^{92} -3.57953e6 q^{94} +1.19252e7 q^{95} +1.62877e6 q^{97} +6.32626e6 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) −321.000 −1.14844 −0.574222 0.818699i \(-0.694695\pi\)
−0.574222 + 0.818699i \(0.694695\pi\)
\(6\) 0 0
\(7\) −181.000 −0.199451 −0.0997253 0.995015i \(-0.531796\pi\)
−0.0997253 + 0.995015i \(0.531796\pi\)
\(8\) −512.000 −0.353553
\(9\) 0 0
\(10\) 2568.00 0.812073
\(11\) −7782.00 −1.76286 −0.881428 0.472318i \(-0.843417\pi\)
−0.881428 + 0.472318i \(0.843417\pi\)
\(12\) 0 0
\(13\) 2197.00 0.277350
\(14\) 1448.00 0.141033
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) −9069.00 −0.447701 −0.223851 0.974623i \(-0.571863\pi\)
−0.223851 + 0.974623i \(0.571863\pi\)
\(18\) 0 0
\(19\) −37150.0 −1.24257 −0.621286 0.783584i \(-0.713389\pi\)
−0.621286 + 0.783584i \(0.713389\pi\)
\(20\) −20544.0 −0.574222
\(21\) 0 0
\(22\) 62256.0 1.24653
\(23\) −19008.0 −0.325753 −0.162877 0.986646i \(-0.552077\pi\)
−0.162877 + 0.986646i \(0.552077\pi\)
\(24\) 0 0
\(25\) 24916.0 0.318925
\(26\) −17576.0 −0.196116
\(27\) 0 0
\(28\) −11584.0 −0.0997253
\(29\) −174750. −1.33053 −0.665264 0.746608i \(-0.731681\pi\)
−0.665264 + 0.746608i \(0.731681\pi\)
\(30\) 0 0
\(31\) 29012.0 0.174909 0.0874544 0.996169i \(-0.472127\pi\)
0.0874544 + 0.996169i \(0.472127\pi\)
\(32\) −32768.0 −0.176777
\(33\) 0 0
\(34\) 72552.0 0.316572
\(35\) 58101.0 0.229058
\(36\) 0 0
\(37\) 323669. 1.05050 0.525249 0.850949i \(-0.323972\pi\)
0.525249 + 0.850949i \(0.323972\pi\)
\(38\) 297200. 0.878630
\(39\) 0 0
\(40\) 164352. 0.406036
\(41\) −795312. −1.80216 −0.901081 0.433650i \(-0.857225\pi\)
−0.901081 + 0.433650i \(0.857225\pi\)
\(42\) 0 0
\(43\) −314137. −0.602531 −0.301266 0.953540i \(-0.597409\pi\)
−0.301266 + 0.953540i \(0.597409\pi\)
\(44\) −498048. −0.881428
\(45\) 0 0
\(46\) 152064. 0.230342
\(47\) 447441. 0.628627 0.314314 0.949319i \(-0.398226\pi\)
0.314314 + 0.949319i \(0.398226\pi\)
\(48\) 0 0
\(49\) −790782. −0.960219
\(50\) −199328. −0.225514
\(51\) 0 0
\(52\) 140608. 0.138675
\(53\) 1.46923e6 1.35558 0.677790 0.735256i \(-0.262938\pi\)
0.677790 + 0.735256i \(0.262938\pi\)
\(54\) 0 0
\(55\) 2.49802e6 2.02454
\(56\) 92672.0 0.0705165
\(57\) 0 0
\(58\) 1.39800e6 0.940826
\(59\) −1.62777e6 −1.03184 −0.515918 0.856638i \(-0.672549\pi\)
−0.515918 + 0.856638i \(0.672549\pi\)
\(60\) 0 0
\(61\) −2.39961e6 −1.35359 −0.676793 0.736173i \(-0.736631\pi\)
−0.676793 + 0.736173i \(0.736631\pi\)
\(62\) −232096. −0.123679
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −705237. −0.318521
\(66\) 0 0
\(67\) −64066.0 −0.0260235 −0.0130118 0.999915i \(-0.504142\pi\)
−0.0130118 + 0.999915i \(0.504142\pi\)
\(68\) −580416. −0.223851
\(69\) 0 0
\(70\) −464808. −0.161968
\(71\) 322383. 0.106898 0.0534488 0.998571i \(-0.482979\pi\)
0.0534488 + 0.998571i \(0.482979\pi\)
\(72\) 0 0
\(73\) −4.45478e6 −1.34028 −0.670141 0.742233i \(-0.733767\pi\)
−0.670141 + 0.742233i \(0.733767\pi\)
\(74\) −2.58935e6 −0.742814
\(75\) 0 0
\(76\) −2.37760e6 −0.621286
\(77\) 1.40854e6 0.351603
\(78\) 0 0
\(79\) 753560. 0.171958 0.0859791 0.996297i \(-0.472598\pi\)
0.0859791 + 0.996297i \(0.472598\pi\)
\(80\) −1.31482e6 −0.287111
\(81\) 0 0
\(82\) 6.36250e6 1.27432
\(83\) 1.21909e6 0.234025 0.117013 0.993130i \(-0.462668\pi\)
0.117013 + 0.993130i \(0.462668\pi\)
\(84\) 0 0
\(85\) 2.91115e6 0.514160
\(86\) 2.51310e6 0.426054
\(87\) 0 0
\(88\) 3.98438e6 0.623264
\(89\) −3.39033e6 −0.509773 −0.254887 0.966971i \(-0.582038\pi\)
−0.254887 + 0.966971i \(0.582038\pi\)
\(90\) 0 0
\(91\) −397657. −0.0553177
\(92\) −1.21651e6 −0.162877
\(93\) 0 0
\(94\) −3.57953e6 −0.444507
\(95\) 1.19252e7 1.42702
\(96\) 0 0
\(97\) 1.62877e6 0.181201 0.0906003 0.995887i \(-0.471121\pi\)
0.0906003 + 0.995887i \(0.471121\pi\)
\(98\) 6.32626e6 0.678978
\(99\) 0 0
\(100\) 1.59462e6 0.159462
\(101\) 1.53503e7 1.48249 0.741244 0.671236i \(-0.234236\pi\)
0.741244 + 0.671236i \(0.234236\pi\)
\(102\) 0 0
\(103\) 6.87643e6 0.620058 0.310029 0.950727i \(-0.399661\pi\)
0.310029 + 0.950727i \(0.399661\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) 0 0
\(106\) −1.17539e7 −0.958539
\(107\) 1.52027e7 1.19971 0.599857 0.800107i \(-0.295224\pi\)
0.599857 + 0.800107i \(0.295224\pi\)
\(108\) 0 0
\(109\) 6.73260e6 0.497955 0.248978 0.968509i \(-0.419905\pi\)
0.248978 + 0.968509i \(0.419905\pi\)
\(110\) −1.99842e7 −1.43157
\(111\) 0 0
\(112\) −741376. −0.0498627
\(113\) 1.15292e7 0.751667 0.375833 0.926687i \(-0.377356\pi\)
0.375833 + 0.926687i \(0.377356\pi\)
\(114\) 0 0
\(115\) 6.10157e6 0.374110
\(116\) −1.11840e7 −0.665264
\(117\) 0 0
\(118\) 1.30222e7 0.729619
\(119\) 1.64149e6 0.0892943
\(120\) 0 0
\(121\) 4.10724e7 2.10766
\(122\) 1.91969e7 0.957130
\(123\) 0 0
\(124\) 1.85677e6 0.0874544
\(125\) 1.70801e7 0.782177
\(126\) 0 0
\(127\) 2.06699e7 0.895418 0.447709 0.894179i \(-0.352240\pi\)
0.447709 + 0.894179i \(0.352240\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 0 0
\(130\) 5.64190e6 0.225228
\(131\) 1.90949e7 0.742107 0.371054 0.928611i \(-0.378997\pi\)
0.371054 + 0.928611i \(0.378997\pi\)
\(132\) 0 0
\(133\) 6.72415e6 0.247832
\(134\) 512528. 0.0184014
\(135\) 0 0
\(136\) 4.64333e6 0.158286
\(137\) −2.96901e7 −0.986482 −0.493241 0.869893i \(-0.664188\pi\)
−0.493241 + 0.869893i \(0.664188\pi\)
\(138\) 0 0
\(139\) 1.55652e7 0.491591 0.245795 0.969322i \(-0.420951\pi\)
0.245795 + 0.969322i \(0.420951\pi\)
\(140\) 3.71846e6 0.114529
\(141\) 0 0
\(142\) −2.57906e6 −0.0755880
\(143\) −1.70971e7 −0.488928
\(144\) 0 0
\(145\) 5.60948e7 1.52804
\(146\) 3.56383e7 0.947723
\(147\) 0 0
\(148\) 2.07148e7 0.525249
\(149\) 2.49675e6 0.0618334 0.0309167 0.999522i \(-0.490157\pi\)
0.0309167 + 0.999522i \(0.490157\pi\)
\(150\) 0 0
\(151\) 2.39802e7 0.566804 0.283402 0.959001i \(-0.408537\pi\)
0.283402 + 0.959001i \(0.408537\pi\)
\(152\) 1.90208e7 0.439315
\(153\) 0 0
\(154\) −1.12683e7 −0.248621
\(155\) −9.31285e6 −0.200873
\(156\) 0 0
\(157\) 1.70550e7 0.351725 0.175863 0.984415i \(-0.443729\pi\)
0.175863 + 0.984415i \(0.443729\pi\)
\(158\) −6.02848e6 −0.121593
\(159\) 0 0
\(160\) 1.05185e7 0.203018
\(161\) 3.44045e6 0.0649717
\(162\) 0 0
\(163\) −7.34586e7 −1.32857 −0.664287 0.747477i \(-0.731265\pi\)
−0.664287 + 0.747477i \(0.731265\pi\)
\(164\) −5.09000e7 −0.901081
\(165\) 0 0
\(166\) −9.75274e6 −0.165481
\(167\) −4.66860e7 −0.775674 −0.387837 0.921728i \(-0.626778\pi\)
−0.387837 + 0.921728i \(0.626778\pi\)
\(168\) 0 0
\(169\) 4.82681e6 0.0769231
\(170\) −2.32892e7 −0.363566
\(171\) 0 0
\(172\) −2.01048e7 −0.301266
\(173\) 7.80931e7 1.14670 0.573352 0.819309i \(-0.305643\pi\)
0.573352 + 0.819309i \(0.305643\pi\)
\(174\) 0 0
\(175\) −4.50980e6 −0.0636098
\(176\) −3.18751e7 −0.440714
\(177\) 0 0
\(178\) 2.71226e7 0.360464
\(179\) 5.56163e7 0.724797 0.362399 0.932023i \(-0.381958\pi\)
0.362399 + 0.932023i \(0.381958\pi\)
\(180\) 0 0
\(181\) −1.19435e8 −1.49711 −0.748557 0.663070i \(-0.769253\pi\)
−0.748557 + 0.663070i \(0.769253\pi\)
\(182\) 3.18126e6 0.0391155
\(183\) 0 0
\(184\) 9.73210e6 0.115171
\(185\) −1.03898e8 −1.20644
\(186\) 0 0
\(187\) 7.05750e7 0.789233
\(188\) 2.86362e7 0.314314
\(189\) 0 0
\(190\) −9.54012e7 −1.00906
\(191\) −1.05485e8 −1.09540 −0.547700 0.836675i \(-0.684496\pi\)
−0.547700 + 0.836675i \(0.684496\pi\)
\(192\) 0 0
\(193\) 2.12059e7 0.212327 0.106164 0.994349i \(-0.466143\pi\)
0.106164 + 0.994349i \(0.466143\pi\)
\(194\) −1.30302e7 −0.128128
\(195\) 0 0
\(196\) −5.06100e7 −0.480110
\(197\) 1.66535e8 1.55194 0.775969 0.630771i \(-0.217261\pi\)
0.775969 + 0.630771i \(0.217261\pi\)
\(198\) 0 0
\(199\) −1.26351e8 −1.13656 −0.568279 0.822836i \(-0.692391\pi\)
−0.568279 + 0.822836i \(0.692391\pi\)
\(200\) −1.27570e7 −0.112757
\(201\) 0 0
\(202\) −1.22802e8 −1.04828
\(203\) 3.16298e7 0.265375
\(204\) 0 0
\(205\) 2.55295e8 2.06968
\(206\) −5.50114e7 −0.438448
\(207\) 0 0
\(208\) 8.99891e6 0.0693375
\(209\) 2.89101e8 2.19047
\(210\) 0 0
\(211\) 1.08571e8 0.795655 0.397828 0.917460i \(-0.369764\pi\)
0.397828 + 0.917460i \(0.369764\pi\)
\(212\) 9.40308e7 0.677790
\(213\) 0 0
\(214\) −1.21622e8 −0.848326
\(215\) 1.00838e8 0.691974
\(216\) 0 0
\(217\) −5.25117e6 −0.0348857
\(218\) −5.38608e7 −0.352108
\(219\) 0 0
\(220\) 1.59873e8 1.01227
\(221\) −1.99246e7 −0.124170
\(222\) 0 0
\(223\) −1.25603e8 −0.758459 −0.379229 0.925303i \(-0.623811\pi\)
−0.379229 + 0.925303i \(0.623811\pi\)
\(224\) 5.93101e6 0.0352582
\(225\) 0 0
\(226\) −9.22338e7 −0.531509
\(227\) −1.90774e8 −1.08250 −0.541252 0.840861i \(-0.682049\pi\)
−0.541252 + 0.840861i \(0.682049\pi\)
\(228\) 0 0
\(229\) −5.28911e7 −0.291044 −0.145522 0.989355i \(-0.546486\pi\)
−0.145522 + 0.989355i \(0.546486\pi\)
\(230\) −4.88125e7 −0.264536
\(231\) 0 0
\(232\) 8.94720e7 0.470413
\(233\) −1.51254e8 −0.783359 −0.391680 0.920102i \(-0.628106\pi\)
−0.391680 + 0.920102i \(0.628106\pi\)
\(234\) 0 0
\(235\) −1.43629e8 −0.721944
\(236\) −1.04177e8 −0.515918
\(237\) 0 0
\(238\) −1.31319e7 −0.0631406
\(239\) −2.61917e8 −1.24100 −0.620498 0.784208i \(-0.713070\pi\)
−0.620498 + 0.784208i \(0.713070\pi\)
\(240\) 0 0
\(241\) −1.31752e8 −0.606312 −0.303156 0.952941i \(-0.598040\pi\)
−0.303156 + 0.952941i \(0.598040\pi\)
\(242\) −3.28579e8 −1.49034
\(243\) 0 0
\(244\) −1.53575e8 −0.676793
\(245\) 2.53841e8 1.10276
\(246\) 0 0
\(247\) −8.16186e7 −0.344627
\(248\) −1.48541e7 −0.0618396
\(249\) 0 0
\(250\) −1.36641e8 −0.553083
\(251\) −2.47061e8 −0.986159 −0.493080 0.869984i \(-0.664129\pi\)
−0.493080 + 0.869984i \(0.664129\pi\)
\(252\) 0 0
\(253\) 1.47920e8 0.574256
\(254\) −1.65359e8 −0.633156
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) −2.27286e8 −0.835231 −0.417616 0.908624i \(-0.637134\pi\)
−0.417616 + 0.908624i \(0.637134\pi\)
\(258\) 0 0
\(259\) −5.85841e7 −0.209522
\(260\) −4.51352e7 −0.159261
\(261\) 0 0
\(262\) −1.52759e8 −0.524749
\(263\) 4.25872e8 1.44356 0.721779 0.692124i \(-0.243325\pi\)
0.721779 + 0.692124i \(0.243325\pi\)
\(264\) 0 0
\(265\) −4.71623e8 −1.55681
\(266\) −5.37932e7 −0.175243
\(267\) 0 0
\(268\) −4.10022e6 −0.0130118
\(269\) 5.14154e8 1.61050 0.805250 0.592936i \(-0.202031\pi\)
0.805250 + 0.592936i \(0.202031\pi\)
\(270\) 0 0
\(271\) 4.57096e7 0.139513 0.0697565 0.997564i \(-0.477778\pi\)
0.0697565 + 0.997564i \(0.477778\pi\)
\(272\) −3.71466e7 −0.111925
\(273\) 0 0
\(274\) 2.37521e8 0.697548
\(275\) −1.93896e8 −0.562218
\(276\) 0 0
\(277\) −2.73964e8 −0.774487 −0.387244 0.921977i \(-0.626573\pi\)
−0.387244 + 0.921977i \(0.626573\pi\)
\(278\) −1.24522e8 −0.347607
\(279\) 0 0
\(280\) −2.97477e7 −0.0809842
\(281\) −4.21707e8 −1.13381 −0.566903 0.823784i \(-0.691859\pi\)
−0.566903 + 0.823784i \(0.691859\pi\)
\(282\) 0 0
\(283\) 3.81957e8 1.00176 0.500878 0.865518i \(-0.333010\pi\)
0.500878 + 0.865518i \(0.333010\pi\)
\(284\) 2.06325e7 0.0534488
\(285\) 0 0
\(286\) 1.36776e8 0.345724
\(287\) 1.43951e8 0.359443
\(288\) 0 0
\(289\) −3.28092e8 −0.799564
\(290\) −4.48758e8 −1.08049
\(291\) 0 0
\(292\) −2.85106e8 −0.670141
\(293\) 4.04833e8 0.940240 0.470120 0.882602i \(-0.344211\pi\)
0.470120 + 0.882602i \(0.344211\pi\)
\(294\) 0 0
\(295\) 5.22514e8 1.18501
\(296\) −1.65719e8 −0.371407
\(297\) 0 0
\(298\) −1.99740e7 −0.0437228
\(299\) −4.17606e7 −0.0903477
\(300\) 0 0
\(301\) 5.68588e7 0.120175
\(302\) −1.91841e8 −0.400791
\(303\) 0 0
\(304\) −1.52166e8 −0.310643
\(305\) 7.70274e8 1.55452
\(306\) 0 0
\(307\) −4.75520e7 −0.0937960 −0.0468980 0.998900i \(-0.514934\pi\)
−0.0468980 + 0.998900i \(0.514934\pi\)
\(308\) 9.01467e7 0.175801
\(309\) 0 0
\(310\) 7.45028e7 0.142039
\(311\) 3.02841e8 0.570892 0.285446 0.958395i \(-0.407858\pi\)
0.285446 + 0.958395i \(0.407858\pi\)
\(312\) 0 0
\(313\) −6.31685e8 −1.16438 −0.582191 0.813052i \(-0.697804\pi\)
−0.582191 + 0.813052i \(0.697804\pi\)
\(314\) −1.36440e8 −0.248707
\(315\) 0 0
\(316\) 4.82278e7 0.0859791
\(317\) −7.93332e8 −1.39877 −0.699387 0.714743i \(-0.746544\pi\)
−0.699387 + 0.714743i \(0.746544\pi\)
\(318\) 0 0
\(319\) 1.35990e9 2.34553
\(320\) −8.41482e7 −0.143556
\(321\) 0 0
\(322\) −2.75236e7 −0.0459420
\(323\) 3.36913e8 0.556300
\(324\) 0 0
\(325\) 5.47405e7 0.0884538
\(326\) 5.87669e8 0.939444
\(327\) 0 0
\(328\) 4.07200e8 0.637161
\(329\) −8.09868e7 −0.125380
\(330\) 0 0
\(331\) −1.21628e9 −1.84346 −0.921731 0.387829i \(-0.873225\pi\)
−0.921731 + 0.387829i \(0.873225\pi\)
\(332\) 7.80219e7 0.117013
\(333\) 0 0
\(334\) 3.73488e8 0.548484
\(335\) 2.05652e7 0.0298866
\(336\) 0 0
\(337\) −1.51221e8 −0.215232 −0.107616 0.994193i \(-0.534322\pi\)
−0.107616 + 0.994193i \(0.534322\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) 0 0
\(340\) 1.86314e8 0.257080
\(341\) −2.25771e8 −0.308339
\(342\) 0 0
\(343\) 2.92193e8 0.390967
\(344\) 1.60838e8 0.213027
\(345\) 0 0
\(346\) −6.24745e8 −0.810842
\(347\) 5.97234e8 0.767347 0.383673 0.923469i \(-0.374659\pi\)
0.383673 + 0.923469i \(0.374659\pi\)
\(348\) 0 0
\(349\) 1.19600e8 0.150606 0.0753029 0.997161i \(-0.476008\pi\)
0.0753029 + 0.997161i \(0.476008\pi\)
\(350\) 3.60784e7 0.0449789
\(351\) 0 0
\(352\) 2.55001e8 0.311632
\(353\) 4.66414e8 0.564366 0.282183 0.959361i \(-0.408942\pi\)
0.282183 + 0.959361i \(0.408942\pi\)
\(354\) 0 0
\(355\) −1.03485e8 −0.122766
\(356\) −2.16981e8 −0.254887
\(357\) 0 0
\(358\) −4.44931e8 −0.512509
\(359\) 7.70102e8 0.878451 0.439225 0.898377i \(-0.355253\pi\)
0.439225 + 0.898377i \(0.355253\pi\)
\(360\) 0 0
\(361\) 4.86251e8 0.543983
\(362\) 9.55477e8 1.05862
\(363\) 0 0
\(364\) −2.54500e7 −0.0276588
\(365\) 1.42999e9 1.53924
\(366\) 0 0
\(367\) −8.55319e8 −0.903227 −0.451613 0.892214i \(-0.649151\pi\)
−0.451613 + 0.892214i \(0.649151\pi\)
\(368\) −7.78568e7 −0.0814384
\(369\) 0 0
\(370\) 8.31182e8 0.853081
\(371\) −2.65931e8 −0.270371
\(372\) 0 0
\(373\) −5.29609e8 −0.528414 −0.264207 0.964466i \(-0.585110\pi\)
−0.264207 + 0.964466i \(0.585110\pi\)
\(374\) −5.64600e8 −0.558072
\(375\) 0 0
\(376\) −2.29090e8 −0.222253
\(377\) −3.83926e8 −0.369022
\(378\) 0 0
\(379\) 1.98358e9 1.87159 0.935797 0.352540i \(-0.114682\pi\)
0.935797 + 0.352540i \(0.114682\pi\)
\(380\) 7.63210e8 0.713512
\(381\) 0 0
\(382\) 8.43877e8 0.774564
\(383\) −8.98756e8 −0.817422 −0.408711 0.912664i \(-0.634022\pi\)
−0.408711 + 0.912664i \(0.634022\pi\)
\(384\) 0 0
\(385\) −4.52142e8 −0.403796
\(386\) −1.69647e8 −0.150138
\(387\) 0 0
\(388\) 1.04242e8 0.0906003
\(389\) −1.82475e9 −1.57174 −0.785868 0.618395i \(-0.787783\pi\)
−0.785868 + 0.618395i \(0.787783\pi\)
\(390\) 0 0
\(391\) 1.72384e8 0.145840
\(392\) 4.04880e8 0.339489
\(393\) 0 0
\(394\) −1.33228e9 −1.09739
\(395\) −2.41893e8 −0.197485
\(396\) 0 0
\(397\) 4.93083e8 0.395506 0.197753 0.980252i \(-0.436636\pi\)
0.197753 + 0.980252i \(0.436636\pi\)
\(398\) 1.01081e9 0.803668
\(399\) 0 0
\(400\) 1.02056e8 0.0797312
\(401\) 5.68280e8 0.440105 0.220053 0.975488i \(-0.429377\pi\)
0.220053 + 0.975488i \(0.429377\pi\)
\(402\) 0 0
\(403\) 6.37394e7 0.0485110
\(404\) 9.82417e8 0.741244
\(405\) 0 0
\(406\) −2.53038e8 −0.187648
\(407\) −2.51879e9 −1.85188
\(408\) 0 0
\(409\) −1.28472e9 −0.928489 −0.464245 0.885707i \(-0.653674\pi\)
−0.464245 + 0.885707i \(0.653674\pi\)
\(410\) −2.04236e9 −1.46349
\(411\) 0 0
\(412\) 4.40091e8 0.310029
\(413\) 2.94626e8 0.205801
\(414\) 0 0
\(415\) −3.91329e8 −0.268765
\(416\) −7.19913e7 −0.0490290
\(417\) 0 0
\(418\) −2.31281e9 −1.54890
\(419\) −2.74847e8 −0.182533 −0.0912667 0.995826i \(-0.529092\pi\)
−0.0912667 + 0.995826i \(0.529092\pi\)
\(420\) 0 0
\(421\) 7.51368e8 0.490756 0.245378 0.969428i \(-0.421088\pi\)
0.245378 + 0.969428i \(0.421088\pi\)
\(422\) −8.68568e8 −0.562613
\(423\) 0 0
\(424\) −7.52247e8 −0.479270
\(425\) −2.25963e8 −0.142783
\(426\) 0 0
\(427\) 4.34329e8 0.269974
\(428\) 9.72974e8 0.599857
\(429\) 0 0
\(430\) −8.06704e8 −0.489299
\(431\) 1.30756e8 0.0786668 0.0393334 0.999226i \(-0.487477\pi\)
0.0393334 + 0.999226i \(0.487477\pi\)
\(432\) 0 0
\(433\) 1.66736e9 0.987010 0.493505 0.869743i \(-0.335716\pi\)
0.493505 + 0.869743i \(0.335716\pi\)
\(434\) 4.20094e7 0.0246679
\(435\) 0 0
\(436\) 4.30887e8 0.248978
\(437\) 7.06147e8 0.404772
\(438\) 0 0
\(439\) 2.31478e9 1.30582 0.652910 0.757436i \(-0.273548\pi\)
0.652910 + 0.757436i \(0.273548\pi\)
\(440\) −1.27899e9 −0.715784
\(441\) 0 0
\(442\) 1.59397e8 0.0878014
\(443\) 6.90047e8 0.377108 0.188554 0.982063i \(-0.439620\pi\)
0.188554 + 0.982063i \(0.439620\pi\)
\(444\) 0 0
\(445\) 1.08830e9 0.585446
\(446\) 1.00482e9 0.536311
\(447\) 0 0
\(448\) −4.74481e7 −0.0249313
\(449\) −2.63806e9 −1.37538 −0.687690 0.726004i \(-0.741375\pi\)
−0.687690 + 0.726004i \(0.741375\pi\)
\(450\) 0 0
\(451\) 6.18912e9 3.17695
\(452\) 7.37870e8 0.375833
\(453\) 0 0
\(454\) 1.52619e9 0.765445
\(455\) 1.27648e8 0.0635293
\(456\) 0 0
\(457\) −6.16222e8 −0.302016 −0.151008 0.988533i \(-0.548252\pi\)
−0.151008 + 0.988533i \(0.548252\pi\)
\(458\) 4.23129e8 0.205799
\(459\) 0 0
\(460\) 3.90500e8 0.187055
\(461\) −1.23621e9 −0.587679 −0.293839 0.955855i \(-0.594933\pi\)
−0.293839 + 0.955855i \(0.594933\pi\)
\(462\) 0 0
\(463\) 6.78469e7 0.0317685 0.0158843 0.999874i \(-0.494944\pi\)
0.0158843 + 0.999874i \(0.494944\pi\)
\(464\) −7.15776e8 −0.332632
\(465\) 0 0
\(466\) 1.21003e9 0.553919
\(467\) 1.17502e9 0.533869 0.266934 0.963715i \(-0.413989\pi\)
0.266934 + 0.963715i \(0.413989\pi\)
\(468\) 0 0
\(469\) 1.15959e7 0.00519040
\(470\) 1.14903e9 0.510491
\(471\) 0 0
\(472\) 8.33418e8 0.364809
\(473\) 2.44461e9 1.06218
\(474\) 0 0
\(475\) −9.25629e8 −0.396287
\(476\) 1.05055e8 0.0446471
\(477\) 0 0
\(478\) 2.09533e9 0.877517
\(479\) −3.96154e8 −0.164699 −0.0823494 0.996604i \(-0.526242\pi\)
−0.0823494 + 0.996604i \(0.526242\pi\)
\(480\) 0 0
\(481\) 7.11101e8 0.291356
\(482\) 1.05401e9 0.428728
\(483\) 0 0
\(484\) 2.62863e9 1.05383
\(485\) −5.22836e8 −0.208099
\(486\) 0 0
\(487\) −3.03665e9 −1.19136 −0.595680 0.803222i \(-0.703117\pi\)
−0.595680 + 0.803222i \(0.703117\pi\)
\(488\) 1.22860e9 0.478565
\(489\) 0 0
\(490\) −2.03073e9 −0.779768
\(491\) 2.91974e9 1.11316 0.556582 0.830793i \(-0.312113\pi\)
0.556582 + 0.830793i \(0.312113\pi\)
\(492\) 0 0
\(493\) 1.58481e9 0.595679
\(494\) 6.52948e8 0.243688
\(495\) 0 0
\(496\) 1.18833e8 0.0437272
\(497\) −5.83513e7 −0.0213208
\(498\) 0 0
\(499\) −1.62343e9 −0.584898 −0.292449 0.956281i \(-0.594470\pi\)
−0.292449 + 0.956281i \(0.594470\pi\)
\(500\) 1.09313e9 0.391089
\(501\) 0 0
\(502\) 1.97649e9 0.697320
\(503\) −4.75888e9 −1.66731 −0.833655 0.552285i \(-0.813756\pi\)
−0.833655 + 0.552285i \(0.813756\pi\)
\(504\) 0 0
\(505\) −4.92744e9 −1.70256
\(506\) −1.18336e9 −0.406061
\(507\) 0 0
\(508\) 1.32288e9 0.447709
\(509\) 9.19375e8 0.309016 0.154508 0.987992i \(-0.450621\pi\)
0.154508 + 0.987992i \(0.450621\pi\)
\(510\) 0 0
\(511\) 8.06316e8 0.267320
\(512\) −1.34218e8 −0.0441942
\(513\) 0 0
\(514\) 1.81829e9 0.590598
\(515\) −2.20733e9 −0.712103
\(516\) 0 0
\(517\) −3.48199e9 −1.10818
\(518\) 4.68673e8 0.148155
\(519\) 0 0
\(520\) 3.61081e8 0.112614
\(521\) 1.46089e9 0.452569 0.226284 0.974061i \(-0.427342\pi\)
0.226284 + 0.974061i \(0.427342\pi\)
\(522\) 0 0
\(523\) 2.12856e9 0.650624 0.325312 0.945607i \(-0.394531\pi\)
0.325312 + 0.945607i \(0.394531\pi\)
\(524\) 1.22207e9 0.371054
\(525\) 0 0
\(526\) −3.40698e9 −1.02075
\(527\) −2.63110e8 −0.0783069
\(528\) 0 0
\(529\) −3.04352e9 −0.893885
\(530\) 3.77299e9 1.10083
\(531\) 0 0
\(532\) 4.30346e8 0.123916
\(533\) −1.74730e9 −0.499830
\(534\) 0 0
\(535\) −4.88007e9 −1.37781
\(536\) 3.28018e7 0.00920070
\(537\) 0 0
\(538\) −4.11323e9 −1.13879
\(539\) 6.15387e9 1.69273
\(540\) 0 0
\(541\) −1.72479e7 −0.00468324 −0.00234162 0.999997i \(-0.500745\pi\)
−0.00234162 + 0.999997i \(0.500745\pi\)
\(542\) −3.65677e8 −0.0986507
\(543\) 0 0
\(544\) 2.97173e8 0.0791431
\(545\) −2.16117e9 −0.571874
\(546\) 0 0
\(547\) 7.51154e8 0.196234 0.0981168 0.995175i \(-0.468718\pi\)
0.0981168 + 0.995175i \(0.468718\pi\)
\(548\) −1.90016e9 −0.493241
\(549\) 0 0
\(550\) 1.55117e9 0.397549
\(551\) 6.49196e9 1.65328
\(552\) 0 0
\(553\) −1.36394e8 −0.0342972
\(554\) 2.19171e9 0.547645
\(555\) 0 0
\(556\) 9.96175e8 0.245795
\(557\) −3.00701e9 −0.737295 −0.368647 0.929569i \(-0.620179\pi\)
−0.368647 + 0.929569i \(0.620179\pi\)
\(558\) 0 0
\(559\) −6.90159e8 −0.167112
\(560\) 2.37982e8 0.0572645
\(561\) 0 0
\(562\) 3.37366e9 0.801722
\(563\) −2.82880e9 −0.668070 −0.334035 0.942561i \(-0.608410\pi\)
−0.334035 + 0.942561i \(0.608410\pi\)
\(564\) 0 0
\(565\) −3.70088e9 −0.863248
\(566\) −3.05566e9 −0.708349
\(567\) 0 0
\(568\) −1.65060e8 −0.0377940
\(569\) −7.67290e9 −1.74609 −0.873045 0.487639i \(-0.837858\pi\)
−0.873045 + 0.487639i \(0.837858\pi\)
\(570\) 0 0
\(571\) −3.09363e9 −0.695411 −0.347706 0.937604i \(-0.613039\pi\)
−0.347706 + 0.937604i \(0.613039\pi\)
\(572\) −1.09421e9 −0.244464
\(573\) 0 0
\(574\) −1.15161e9 −0.254164
\(575\) −4.73603e8 −0.103891
\(576\) 0 0
\(577\) 3.71815e9 0.805770 0.402885 0.915251i \(-0.368007\pi\)
0.402885 + 0.915251i \(0.368007\pi\)
\(578\) 2.62474e9 0.565377
\(579\) 0 0
\(580\) 3.59006e9 0.764019
\(581\) −2.20656e8 −0.0466765
\(582\) 0 0
\(583\) −1.14336e10 −2.38969
\(584\) 2.28085e9 0.473862
\(585\) 0 0
\(586\) −3.23866e9 −0.664850
\(587\) 2.74853e9 0.560876 0.280438 0.959872i \(-0.409520\pi\)
0.280438 + 0.959872i \(0.409520\pi\)
\(588\) 0 0
\(589\) −1.07780e9 −0.217337
\(590\) −4.18011e9 −0.837927
\(591\) 0 0
\(592\) 1.32575e9 0.262624
\(593\) 9.11262e9 1.79453 0.897267 0.441488i \(-0.145549\pi\)
0.897267 + 0.441488i \(0.145549\pi\)
\(594\) 0 0
\(595\) −5.26918e8 −0.102550
\(596\) 1.59792e8 0.0309167
\(597\) 0 0
\(598\) 3.34085e8 0.0638855
\(599\) 5.52493e9 1.05035 0.525174 0.850995i \(-0.324000\pi\)
0.525174 + 0.850995i \(0.324000\pi\)
\(600\) 0 0
\(601\) −1.78219e9 −0.334883 −0.167441 0.985882i \(-0.553551\pi\)
−0.167441 + 0.985882i \(0.553551\pi\)
\(602\) −4.54870e8 −0.0849767
\(603\) 0 0
\(604\) 1.53473e9 0.283402
\(605\) −1.31842e10 −2.42053
\(606\) 0 0
\(607\) 9.53705e9 1.73083 0.865414 0.501058i \(-0.167056\pi\)
0.865414 + 0.501058i \(0.167056\pi\)
\(608\) 1.21733e9 0.219658
\(609\) 0 0
\(610\) −6.16219e9 −1.09921
\(611\) 9.83028e8 0.174350
\(612\) 0 0
\(613\) −1.18627e9 −0.208004 −0.104002 0.994577i \(-0.533165\pi\)
−0.104002 + 0.994577i \(0.533165\pi\)
\(614\) 3.80416e8 0.0663238
\(615\) 0 0
\(616\) −7.21174e8 −0.124310
\(617\) −1.32256e9 −0.226682 −0.113341 0.993556i \(-0.536155\pi\)
−0.113341 + 0.993556i \(0.536155\pi\)
\(618\) 0 0
\(619\) −3.59450e9 −0.609147 −0.304573 0.952489i \(-0.598514\pi\)
−0.304573 + 0.952489i \(0.598514\pi\)
\(620\) −5.96023e8 −0.100437
\(621\) 0 0
\(622\) −2.42273e9 −0.403681
\(623\) 6.13650e8 0.101675
\(624\) 0 0
\(625\) −7.42927e9 −1.21721
\(626\) 5.05348e9 0.823343
\(627\) 0 0
\(628\) 1.09152e9 0.175863
\(629\) −2.93535e9 −0.470309
\(630\) 0 0
\(631\) 7.49102e6 0.00118697 0.000593483 1.00000i \(-0.499811\pi\)
0.000593483 1.00000i \(0.499811\pi\)
\(632\) −3.85823e8 −0.0607964
\(633\) 0 0
\(634\) 6.34666e9 0.989083
\(635\) −6.63505e9 −1.02834
\(636\) 0 0
\(637\) −1.73735e9 −0.266317
\(638\) −1.08792e10 −1.65854
\(639\) 0 0
\(640\) 6.73186e8 0.101509
\(641\) −4.06396e9 −0.609462 −0.304731 0.952438i \(-0.598567\pi\)
−0.304731 + 0.952438i \(0.598567\pi\)
\(642\) 0 0
\(643\) 1.56544e6 0.000232219 0 0.000116109 1.00000i \(-0.499963\pi\)
0.000116109 1.00000i \(0.499963\pi\)
\(644\) 2.20189e8 0.0324859
\(645\) 0 0
\(646\) −2.69531e9 −0.393364
\(647\) −1.31025e10 −1.90191 −0.950956 0.309325i \(-0.899897\pi\)
−0.950956 + 0.309325i \(0.899897\pi\)
\(648\) 0 0
\(649\) 1.26673e10 1.81898
\(650\) −4.37924e8 −0.0625463
\(651\) 0 0
\(652\) −4.70135e9 −0.664287
\(653\) −7.63326e9 −1.07279 −0.536394 0.843968i \(-0.680214\pi\)
−0.536394 + 0.843968i \(0.680214\pi\)
\(654\) 0 0
\(655\) −6.12945e9 −0.852269
\(656\) −3.25760e9 −0.450541
\(657\) 0 0
\(658\) 6.47895e8 0.0886571
\(659\) 9.25900e9 1.26027 0.630137 0.776484i \(-0.282999\pi\)
0.630137 + 0.776484i \(0.282999\pi\)
\(660\) 0 0
\(661\) 4.79962e9 0.646401 0.323201 0.946330i \(-0.395241\pi\)
0.323201 + 0.946330i \(0.395241\pi\)
\(662\) 9.73021e9 1.30352
\(663\) 0 0
\(664\) −6.24175e8 −0.0827405
\(665\) −2.15845e9 −0.284621
\(666\) 0 0
\(667\) 3.32165e9 0.433424
\(668\) −2.98791e9 −0.387837
\(669\) 0 0
\(670\) −1.64521e8 −0.0211330
\(671\) 1.86737e10 2.38618
\(672\) 0 0
\(673\) −1.08997e10 −1.37836 −0.689182 0.724589i \(-0.742030\pi\)
−0.689182 + 0.724589i \(0.742030\pi\)
\(674\) 1.20977e9 0.152192
\(675\) 0 0
\(676\) 3.08916e8 0.0384615
\(677\) −3.44099e9 −0.426210 −0.213105 0.977029i \(-0.568358\pi\)
−0.213105 + 0.977029i \(0.568358\pi\)
\(678\) 0 0
\(679\) −2.94808e8 −0.0361406
\(680\) −1.49051e9 −0.181783
\(681\) 0 0
\(682\) 1.80617e9 0.218029
\(683\) −5.53553e9 −0.664794 −0.332397 0.943140i \(-0.607857\pi\)
−0.332397 + 0.943140i \(0.607857\pi\)
\(684\) 0 0
\(685\) 9.53051e9 1.13292
\(686\) −2.33754e9 −0.276455
\(687\) 0 0
\(688\) −1.28671e9 −0.150633
\(689\) 3.22790e9 0.375970
\(690\) 0 0
\(691\) 4.21595e8 0.0486097 0.0243048 0.999705i \(-0.492263\pi\)
0.0243048 + 0.999705i \(0.492263\pi\)
\(692\) 4.99796e9 0.573352
\(693\) 0 0
\(694\) −4.77788e9 −0.542596
\(695\) −4.99644e9 −0.564565
\(696\) 0 0
\(697\) 7.21268e9 0.806830
\(698\) −9.56799e8 −0.106494
\(699\) 0 0
\(700\) −2.88627e8 −0.0318049
\(701\) 5.14995e9 0.564663 0.282332 0.959317i \(-0.408892\pi\)
0.282332 + 0.959317i \(0.408892\pi\)
\(702\) 0 0
\(703\) −1.20243e10 −1.30532
\(704\) −2.04000e9 −0.220357
\(705\) 0 0
\(706\) −3.73132e9 −0.399067
\(707\) −2.77840e9 −0.295683
\(708\) 0 0
\(709\) 1.05683e10 1.11363 0.556817 0.830635i \(-0.312022\pi\)
0.556817 + 0.830635i \(0.312022\pi\)
\(710\) 8.27880e8 0.0868086
\(711\) 0 0
\(712\) 1.73585e9 0.180232
\(713\) −5.51460e8 −0.0569772
\(714\) 0 0
\(715\) 5.48815e9 0.561507
\(716\) 3.55944e9 0.362399
\(717\) 0 0
\(718\) −6.16081e9 −0.621159
\(719\) −1.53690e10 −1.54204 −0.771020 0.636811i \(-0.780253\pi\)
−0.771020 + 0.636811i \(0.780253\pi\)
\(720\) 0 0
\(721\) −1.24463e9 −0.123671
\(722\) −3.89001e9 −0.384654
\(723\) 0 0
\(724\) −7.64381e9 −0.748557
\(725\) −4.35407e9 −0.424339
\(726\) 0 0
\(727\) 4.88599e9 0.471609 0.235804 0.971801i \(-0.424228\pi\)
0.235804 + 0.971801i \(0.424228\pi\)
\(728\) 2.03600e8 0.0195577
\(729\) 0 0
\(730\) −1.14399e10 −1.08841
\(731\) 2.84891e9 0.269754
\(732\) 0 0
\(733\) 3.59889e9 0.337524 0.168762 0.985657i \(-0.446023\pi\)
0.168762 + 0.985657i \(0.446023\pi\)
\(734\) 6.84255e9 0.638678
\(735\) 0 0
\(736\) 6.22854e8 0.0575856
\(737\) 4.98562e8 0.0458757
\(738\) 0 0
\(739\) 2.78886e9 0.254198 0.127099 0.991890i \(-0.459433\pi\)
0.127099 + 0.991890i \(0.459433\pi\)
\(740\) −6.64946e9 −0.603219
\(741\) 0 0
\(742\) 2.12745e9 0.191181
\(743\) 3.08130e9 0.275597 0.137798 0.990460i \(-0.455997\pi\)
0.137798 + 0.990460i \(0.455997\pi\)
\(744\) 0 0
\(745\) −8.01457e8 −0.0710122
\(746\) 4.23687e9 0.373645
\(747\) 0 0
\(748\) 4.51680e9 0.394616
\(749\) −2.75169e9 −0.239284
\(750\) 0 0
\(751\) −6.41281e8 −0.0552470 −0.0276235 0.999618i \(-0.508794\pi\)
−0.0276235 + 0.999618i \(0.508794\pi\)
\(752\) 1.83272e9 0.157157
\(753\) 0 0
\(754\) 3.07141e9 0.260938
\(755\) −7.69763e9 −0.650943
\(756\) 0 0
\(757\) −1.60219e10 −1.34239 −0.671195 0.741280i \(-0.734219\pi\)
−0.671195 + 0.741280i \(0.734219\pi\)
\(758\) −1.58686e10 −1.32342
\(759\) 0 0
\(760\) −6.10568e9 −0.504529
\(761\) 5.73623e9 0.471824 0.235912 0.971774i \(-0.424192\pi\)
0.235912 + 0.971774i \(0.424192\pi\)
\(762\) 0 0
\(763\) −1.21860e9 −0.0993175
\(764\) −6.75102e9 −0.547700
\(765\) 0 0
\(766\) 7.19005e9 0.578005
\(767\) −3.57621e9 −0.286180
\(768\) 0 0
\(769\) 2.45874e10 1.94971 0.974857 0.222832i \(-0.0715300\pi\)
0.974857 + 0.222832i \(0.0715300\pi\)
\(770\) 3.61714e9 0.285527
\(771\) 0 0
\(772\) 1.35718e9 0.106164
\(773\) 1.31517e10 1.02413 0.512065 0.858947i \(-0.328881\pi\)
0.512065 + 0.858947i \(0.328881\pi\)
\(774\) 0 0
\(775\) 7.22863e8 0.0557828
\(776\) −8.33932e8 −0.0640641
\(777\) 0 0
\(778\) 1.45980e10 1.11138
\(779\) 2.95458e10 2.23932
\(780\) 0 0
\(781\) −2.50878e9 −0.188445
\(782\) −1.37907e9 −0.103125
\(783\) 0 0
\(784\) −3.23904e9 −0.240055
\(785\) −5.47466e9 −0.403937
\(786\) 0 0
\(787\) −7.38863e9 −0.540322 −0.270161 0.962815i \(-0.587077\pi\)
−0.270161 + 0.962815i \(0.587077\pi\)
\(788\) 1.06583e10 0.775969
\(789\) 0 0
\(790\) 1.93514e9 0.139643
\(791\) −2.08679e9 −0.149920
\(792\) 0 0
\(793\) −5.27194e9 −0.375417
\(794\) −3.94467e9 −0.279665
\(795\) 0 0
\(796\) −8.08645e9 −0.568279
\(797\) 5.22399e9 0.365509 0.182754 0.983159i \(-0.441499\pi\)
0.182754 + 0.983159i \(0.441499\pi\)
\(798\) 0 0
\(799\) −4.05784e9 −0.281437
\(800\) −8.16447e8 −0.0563785
\(801\) 0 0
\(802\) −4.54624e9 −0.311202
\(803\) 3.46671e10 2.36273
\(804\) 0 0
\(805\) −1.10438e9 −0.0746164
\(806\) −5.09915e8 −0.0343024
\(807\) 0 0
\(808\) −7.85934e9 −0.524139
\(809\) 7.92102e9 0.525970 0.262985 0.964800i \(-0.415293\pi\)
0.262985 + 0.964800i \(0.415293\pi\)
\(810\) 0 0
\(811\) −8.16607e9 −0.537576 −0.268788 0.963199i \(-0.586623\pi\)
−0.268788 + 0.963199i \(0.586623\pi\)
\(812\) 2.02430e9 0.132687
\(813\) 0 0
\(814\) 2.01503e10 1.30947
\(815\) 2.35802e10 1.52579
\(816\) 0 0
\(817\) 1.16702e10 0.748688
\(818\) 1.02778e10 0.656541
\(819\) 0 0
\(820\) 1.63389e10 1.03484
\(821\) −2.63749e10 −1.66338 −0.831688 0.555244i \(-0.812625\pi\)
−0.831688 + 0.555244i \(0.812625\pi\)
\(822\) 0 0
\(823\) 2.04085e10 1.27618 0.638090 0.769962i \(-0.279725\pi\)
0.638090 + 0.769962i \(0.279725\pi\)
\(824\) −3.52073e9 −0.219224
\(825\) 0 0
\(826\) −2.35701e9 −0.145523
\(827\) 2.55307e10 1.56962 0.784809 0.619738i \(-0.212761\pi\)
0.784809 + 0.619738i \(0.212761\pi\)
\(828\) 0 0
\(829\) 8.48208e9 0.517085 0.258542 0.966000i \(-0.416758\pi\)
0.258542 + 0.966000i \(0.416758\pi\)
\(830\) 3.13063e9 0.190046
\(831\) 0 0
\(832\) 5.75930e8 0.0346688
\(833\) 7.17160e9 0.429891
\(834\) 0 0
\(835\) 1.49862e10 0.890819
\(836\) 1.85025e10 1.09524
\(837\) 0 0
\(838\) 2.19878e9 0.129071
\(839\) 2.29323e10 1.34055 0.670273 0.742115i \(-0.266177\pi\)
0.670273 + 0.742115i \(0.266177\pi\)
\(840\) 0 0
\(841\) 1.32877e10 0.770306
\(842\) −6.01094e9 −0.347017
\(843\) 0 0
\(844\) 6.94854e9 0.397828
\(845\) −1.54941e9 −0.0883419
\(846\) 0 0
\(847\) −7.43410e9 −0.420374
\(848\) 6.01797e9 0.338895
\(849\) 0 0
\(850\) 1.80771e9 0.100963
\(851\) −6.15230e9 −0.342203
\(852\) 0 0
\(853\) 2.47175e10 1.36358 0.681792 0.731546i \(-0.261201\pi\)
0.681792 + 0.731546i \(0.261201\pi\)
\(854\) −3.47463e9 −0.190900
\(855\) 0 0
\(856\) −7.78379e9 −0.424163
\(857\) −1.19081e10 −0.646265 −0.323133 0.946354i \(-0.604736\pi\)
−0.323133 + 0.946354i \(0.604736\pi\)
\(858\) 0 0
\(859\) −4.94214e9 −0.266035 −0.133018 0.991114i \(-0.542467\pi\)
−0.133018 + 0.991114i \(0.542467\pi\)
\(860\) 6.45363e9 0.345987
\(861\) 0 0
\(862\) −1.04605e9 −0.0556259
\(863\) 2.05387e10 1.08776 0.543881 0.839162i \(-0.316954\pi\)
0.543881 + 0.839162i \(0.316954\pi\)
\(864\) 0 0
\(865\) −2.50679e10 −1.31693
\(866\) −1.33389e10 −0.697921
\(867\) 0 0
\(868\) −3.36075e8 −0.0174428
\(869\) −5.86420e9 −0.303138
\(870\) 0 0
\(871\) −1.40753e8 −0.00721762
\(872\) −3.44709e9 −0.176054
\(873\) 0 0
\(874\) −5.64918e9 −0.286217
\(875\) −3.09150e9 −0.156006
\(876\) 0 0
\(877\) −1.42584e10 −0.713791 −0.356895 0.934144i \(-0.616165\pi\)
−0.356895 + 0.934144i \(0.616165\pi\)
\(878\) −1.85182e10 −0.923354
\(879\) 0 0
\(880\) 1.02319e10 0.506136
\(881\) −1.78398e10 −0.878971 −0.439486 0.898250i \(-0.644839\pi\)
−0.439486 + 0.898250i \(0.644839\pi\)
\(882\) 0 0
\(883\) 3.79954e10 1.85724 0.928622 0.371028i \(-0.120995\pi\)
0.928622 + 0.371028i \(0.120995\pi\)
\(884\) −1.27517e9 −0.0620850
\(885\) 0 0
\(886\) −5.52038e9 −0.266656
\(887\) −8.45195e7 −0.00406653 −0.00203327 0.999998i \(-0.500647\pi\)
−0.00203327 + 0.999998i \(0.500647\pi\)
\(888\) 0 0
\(889\) −3.74126e9 −0.178592
\(890\) −8.70637e9 −0.413973
\(891\) 0 0
\(892\) −8.03857e9 −0.379229
\(893\) −1.66224e10 −0.781114
\(894\) 0 0
\(895\) −1.78528e10 −0.832390
\(896\) 3.79585e8 0.0176291
\(897\) 0 0
\(898\) 2.11045e10 0.972541
\(899\) −5.06985e9 −0.232721
\(900\) 0 0
\(901\) −1.33245e10 −0.606894
\(902\) −4.95129e10 −2.24645
\(903\) 0 0
\(904\) −5.90296e9 −0.265754
\(905\) 3.83385e10 1.71935
\(906\) 0 0
\(907\) 1.82024e10 0.810033 0.405017 0.914309i \(-0.367266\pi\)
0.405017 + 0.914309i \(0.367266\pi\)
\(908\) −1.22096e10 −0.541252
\(909\) 0 0
\(910\) −1.02118e9 −0.0449220
\(911\) 3.66963e10 1.60808 0.804040 0.594575i \(-0.202680\pi\)
0.804040 + 0.594575i \(0.202680\pi\)
\(912\) 0 0
\(913\) −9.48697e9 −0.412553
\(914\) 4.92978e9 0.213558
\(915\) 0 0
\(916\) −3.38503e9 −0.145522
\(917\) −3.45617e9 −0.148014
\(918\) 0 0
\(919\) 1.33474e10 0.567275 0.283638 0.958932i \(-0.408459\pi\)
0.283638 + 0.958932i \(0.408459\pi\)
\(920\) −3.12400e9 −0.132268
\(921\) 0 0
\(922\) 9.88970e9 0.415552
\(923\) 7.08275e8 0.0296481
\(924\) 0 0
\(925\) 8.06454e9 0.335030
\(926\) −5.42775e8 −0.0224637
\(927\) 0 0
\(928\) 5.72621e9 0.235206
\(929\) −2.71771e10 −1.11211 −0.556055 0.831146i \(-0.687686\pi\)
−0.556055 + 0.831146i \(0.687686\pi\)
\(930\) 0 0
\(931\) 2.93776e10 1.19314
\(932\) −9.68025e9 −0.391680
\(933\) 0 0
\(934\) −9.40012e9 −0.377502
\(935\) −2.26546e10 −0.906390
\(936\) 0 0
\(937\) 4.04333e10 1.60565 0.802825 0.596214i \(-0.203329\pi\)
0.802825 + 0.596214i \(0.203329\pi\)
\(938\) −9.27676e7 −0.00367017
\(939\) 0 0
\(940\) −9.19223e9 −0.360972
\(941\) 8.49843e9 0.332487 0.166244 0.986085i \(-0.446836\pi\)
0.166244 + 0.986085i \(0.446836\pi\)
\(942\) 0 0
\(943\) 1.51173e10 0.587061
\(944\) −6.66735e9 −0.257959
\(945\) 0 0
\(946\) −1.95569e10 −0.751072
\(947\) 4.40082e9 0.168387 0.0841935 0.996449i \(-0.473169\pi\)
0.0841935 + 0.996449i \(0.473169\pi\)
\(948\) 0 0
\(949\) −9.78716e9 −0.371728
\(950\) 7.40504e9 0.280217
\(951\) 0 0
\(952\) −8.40442e8 −0.0315703
\(953\) 1.73133e10 0.647970 0.323985 0.946062i \(-0.394977\pi\)
0.323985 + 0.946062i \(0.394977\pi\)
\(954\) 0 0
\(955\) 3.38606e10 1.25801
\(956\) −1.67627e10 −0.620498
\(957\) 0 0
\(958\) 3.16924e9 0.116460
\(959\) 5.37390e9 0.196754
\(960\) 0 0
\(961\) −2.66709e10 −0.969407
\(962\) −5.68881e9 −0.206020
\(963\) 0 0
\(964\) −8.43211e9 −0.303156
\(965\) −6.80708e9 −0.243846
\(966\) 0 0
\(967\) 1.40918e10 0.501158 0.250579 0.968096i \(-0.419379\pi\)
0.250579 + 0.968096i \(0.419379\pi\)
\(968\) −2.10290e10 −0.745171
\(969\) 0 0
\(970\) 4.18269e9 0.147148
\(971\) 7.27843e9 0.255135 0.127568 0.991830i \(-0.459283\pi\)
0.127568 + 0.991830i \(0.459283\pi\)
\(972\) 0 0
\(973\) −2.81731e9 −0.0980481
\(974\) 2.42932e10 0.842419
\(975\) 0 0
\(976\) −9.82879e9 −0.338397
\(977\) −2.43791e10 −0.836348 −0.418174 0.908367i \(-0.637330\pi\)
−0.418174 + 0.908367i \(0.637330\pi\)
\(978\) 0 0
\(979\) 2.63835e10 0.898657
\(980\) 1.62458e10 0.551379
\(981\) 0 0
\(982\) −2.33579e10 −0.787126
\(983\) −4.06556e10 −1.36516 −0.682579 0.730811i \(-0.739142\pi\)
−0.682579 + 0.730811i \(0.739142\pi\)
\(984\) 0 0
\(985\) −5.34578e10 −1.78231
\(986\) −1.26785e10 −0.421209
\(987\) 0 0
\(988\) −5.22359e9 −0.172314
\(989\) 5.97112e9 0.196277
\(990\) 0 0
\(991\) −4.86636e10 −1.58835 −0.794175 0.607689i \(-0.792097\pi\)
−0.794175 + 0.607689i \(0.792097\pi\)
\(992\) −9.50665e8 −0.0309198
\(993\) 0 0
\(994\) 4.66811e8 0.0150761
\(995\) 4.05586e10 1.30527
\(996\) 0 0
\(997\) −1.76682e10 −0.564622 −0.282311 0.959323i \(-0.591101\pi\)
−0.282311 + 0.959323i \(0.591101\pi\)
\(998\) 1.29874e10 0.413586
\(999\) 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 234.8.a.a.1.1 1
3.2 odd 2 26.8.a.b.1.1 1
12.11 even 2 208.8.a.e.1.1 1
39.5 even 4 338.8.b.a.337.1 2
39.8 even 4 338.8.b.a.337.2 2
39.38 odd 2 338.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.b.1.1 1 3.2 odd 2
208.8.a.e.1.1 1 12.11 even 2
234.8.a.a.1.1 1 1.1 even 1 trivial
338.8.a.a.1.1 1 39.38 odd 2
338.8.b.a.337.1 2 39.5 even 4
338.8.b.a.337.2 2 39.8 even 4