Properties

Label 546.8.a.r.1.5
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 264981 x^{4} + 17519669 x^{3} + 15113237808 x^{2} - 1787613752904 x - 21984668630064\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-298.282\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +332.282 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +332.282 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +2658.26 q^{10} +4538.16 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -8971.62 q^{15} +4096.00 q^{16} -1683.55 q^{17} +5832.00 q^{18} +43035.4 q^{19} +21266.1 q^{20} +9261.00 q^{21} +36305.3 q^{22} +80014.0 q^{23} -13824.0 q^{24} +32286.5 q^{25} +17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} -175048. q^{29} -71773.0 q^{30} -177686. q^{31} +32768.0 q^{32} -122530. q^{33} -13468.4 q^{34} -113973. q^{35} +46656.0 q^{36} +207013. q^{37} +344283. q^{38} -59319.0 q^{39} +170129. q^{40} +418774. q^{41} +74088.0 q^{42} +511110. q^{43} +290443. q^{44} +242234. q^{45} +640112. q^{46} -154485. q^{47} -110592. q^{48} +117649. q^{49} +258292. q^{50} +45456.0 q^{51} +140608. q^{52} -58699.1 q^{53} -157464. q^{54} +1.50795e6 q^{55} -175616. q^{56} -1.16196e6 q^{57} -1.40038e6 q^{58} +249612. q^{59} -574184. q^{60} -926699. q^{61} -1.42149e6 q^{62} -250047. q^{63} +262144. q^{64} +730024. q^{65} -980243. q^{66} -3.22459e6 q^{67} -107747. q^{68} -2.16038e6 q^{69} -911783. q^{70} -720569. q^{71} +373248. q^{72} +6.11222e6 q^{73} +1.65610e6 q^{74} -871736. q^{75} +2.75427e6 q^{76} -1.55659e6 q^{77} -474552. q^{78} -1.62450e6 q^{79} +1.36103e6 q^{80} +531441. q^{81} +3.35020e6 q^{82} +8.58284e6 q^{83} +592704. q^{84} -559415. q^{85} +4.08888e6 q^{86} +4.72629e6 q^{87} +2.32354e6 q^{88} -7.22166e6 q^{89} +1.93787e6 q^{90} -753571. q^{91} +5.12089e6 q^{92} +4.79753e6 q^{93} -1.23588e6 q^{94} +1.42999e7 q^{95} -884736. q^{96} -6.97736e6 q^{97} +941192. q^{98} +3.30832e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 1624 q^{10} - 2690 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} - 5481 q^{15} + 24576 q^{16} + 2910 q^{17} + 34992 q^{18} - 13055 q^{19} + 12992 q^{20} + 55566 q^{21} - 21520 q^{22} + 11581 q^{23} - 82944 q^{24} + 68081 q^{25} + 105456 q^{26} - 118098 q^{27} - 131712 q^{28} - 92335 q^{29} - 43848 q^{30} - 83081 q^{31} + 196608 q^{32} + 72630 q^{33} + 23280 q^{34} - 69629 q^{35} + 279936 q^{36} - 265114 q^{37} - 104440 q^{38} - 355914 q^{39} + 103936 q^{40} - 367468 q^{41} + 444528 q^{42} + 454955 q^{43} - 172160 q^{44} + 147987 q^{45} + 92648 q^{46} + 733973 q^{47} - 663552 q^{48} + 705894 q^{49} + 544648 q^{50} - 78570 q^{51} + 843648 q^{52} - 1577379 q^{53} - 944784 q^{54} + 2231118 q^{55} - 1053696 q^{56} + 352485 q^{57} - 738680 q^{58} + 2062708 q^{59} - 350784 q^{60} - 271270 q^{61} - 664648 q^{62} - 1500282 q^{63} + 1572864 q^{64} + 445991 q^{65} + 581040 q^{66} - 758674 q^{67} + 186240 q^{68} - 312687 q^{69} - 557032 q^{70} - 6138216 q^{71} + 2239488 q^{72} + 6361979 q^{73} - 2120912 q^{74} - 1838187 q^{75} - 835520 q^{76} + 922670 q^{77} - 2847312 q^{78} - 899781 q^{79} + 831488 q^{80} + 3188646 q^{81} - 2939744 q^{82} + 3313561 q^{83} + 3556224 q^{84} + 5307940 q^{85} + 3639640 q^{86} + 2493045 q^{87} - 1377280 q^{88} + 11210703 q^{89} + 1183896 q^{90} - 4521426 q^{91} + 741184 q^{92} + 2243187 q^{93} + 5871784 q^{94} + 12912395 q^{95} - 5308416 q^{96} + 28682643 q^{97} + 5647152 q^{98} - 1961010 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) 332.282 1.18881 0.594405 0.804166i \(-0.297388\pi\)
0.594405 + 0.804166i \(0.297388\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 2658.26 0.840615
\(11\) 4538.16 1.02803 0.514015 0.857781i \(-0.328158\pi\)
0.514015 + 0.857781i \(0.328158\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −8971.62 −0.686359
\(16\) 4096.00 0.250000
\(17\) −1683.55 −0.0831105 −0.0415552 0.999136i \(-0.513231\pi\)
−0.0415552 + 0.999136i \(0.513231\pi\)
\(18\) 5832.00 0.235702
\(19\) 43035.4 1.43942 0.719712 0.694273i \(-0.244274\pi\)
0.719712 + 0.694273i \(0.244274\pi\)
\(20\) 21266.1 0.594405
\(21\) 9261.00 0.218218
\(22\) 36305.3 0.726927
\(23\) 80014.0 1.37126 0.685628 0.727952i \(-0.259528\pi\)
0.685628 + 0.727952i \(0.259528\pi\)
\(24\) −13824.0 −0.204124
\(25\) 32286.5 0.413267
\(26\) 17576.0 0.196116
\(27\) −19683.0 −0.192450
\(28\) −21952.0 −0.188982
\(29\) −175048. −1.33279 −0.666397 0.745597i \(-0.732165\pi\)
−0.666397 + 0.745597i \(0.732165\pi\)
\(30\) −71773.0 −0.485329
\(31\) −177686. −1.07124 −0.535622 0.844458i \(-0.679923\pi\)
−0.535622 + 0.844458i \(0.679923\pi\)
\(32\) 32768.0 0.176777
\(33\) −122530. −0.593533
\(34\) −13468.4 −0.0587680
\(35\) −113973. −0.449328
\(36\) 46656.0 0.166667
\(37\) 207013. 0.671880 0.335940 0.941883i \(-0.390946\pi\)
0.335940 + 0.941883i \(0.390946\pi\)
\(38\) 344283. 1.01783
\(39\) −59319.0 −0.160128
\(40\) 170129. 0.420308
\(41\) 418774. 0.948935 0.474468 0.880273i \(-0.342641\pi\)
0.474468 + 0.880273i \(0.342641\pi\)
\(42\) 74088.0 0.154303
\(43\) 511110. 0.980336 0.490168 0.871628i \(-0.336935\pi\)
0.490168 + 0.871628i \(0.336935\pi\)
\(44\) 290443. 0.514015
\(45\) 242234. 0.396270
\(46\) 640112. 0.969624
\(47\) −154485. −0.217042 −0.108521 0.994094i \(-0.534612\pi\)
−0.108521 + 0.994094i \(0.534612\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 258292. 0.292224
\(51\) 45456.0 0.0479839
\(52\) 140608. 0.138675
\(53\) −58699.1 −0.0541584 −0.0270792 0.999633i \(-0.508621\pi\)
−0.0270792 + 0.999633i \(0.508621\pi\)
\(54\) −157464. −0.136083
\(55\) 1.50795e6 1.22213
\(56\) −175616. −0.133631
\(57\) −1.16196e6 −0.831051
\(58\) −1.40038e6 −0.942428
\(59\) 249612. 0.158228 0.0791140 0.996866i \(-0.474791\pi\)
0.0791140 + 0.996866i \(0.474791\pi\)
\(60\) −574184. −0.343180
\(61\) −926699. −0.522738 −0.261369 0.965239i \(-0.584174\pi\)
−0.261369 + 0.965239i \(0.584174\pi\)
\(62\) −1.42149e6 −0.757484
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) 730024. 0.329716
\(66\) −980243. −0.419692
\(67\) −3.22459e6 −1.30982 −0.654912 0.755705i \(-0.727294\pi\)
−0.654912 + 0.755705i \(0.727294\pi\)
\(68\) −107747. −0.0415552
\(69\) −2.16038e6 −0.791695
\(70\) −911783. −0.317723
\(71\) −720569. −0.238930 −0.119465 0.992838i \(-0.538118\pi\)
−0.119465 + 0.992838i \(0.538118\pi\)
\(72\) 373248. 0.117851
\(73\) 6.11222e6 1.83895 0.919474 0.393152i \(-0.128615\pi\)
0.919474 + 0.393152i \(0.128615\pi\)
\(74\) 1.65610e6 0.475091
\(75\) −871736. −0.238600
\(76\) 2.75427e6 0.719712
\(77\) −1.55659e6 −0.388559
\(78\) −474552. −0.113228
\(79\) −1.62450e6 −0.370702 −0.185351 0.982672i \(-0.559342\pi\)
−0.185351 + 0.982672i \(0.559342\pi\)
\(80\) 1.36103e6 0.297202
\(81\) 531441. 0.111111
\(82\) 3.35020e6 0.670999
\(83\) 8.58284e6 1.64762 0.823811 0.566864i \(-0.191844\pi\)
0.823811 + 0.566864i \(0.191844\pi\)
\(84\) 592704. 0.109109
\(85\) −559415. −0.0988025
\(86\) 4.08888e6 0.693202
\(87\) 4.72629e6 0.769489
\(88\) 2.32354e6 0.363464
\(89\) −7.22166e6 −1.08586 −0.542928 0.839780i \(-0.682684\pi\)
−0.542928 + 0.839780i \(0.682684\pi\)
\(90\) 1.93787e6 0.280205
\(91\) −753571. −0.104828
\(92\) 5.12089e6 0.685628
\(93\) 4.79753e6 0.618483
\(94\) −1.23588e6 −0.153472
\(95\) 1.42999e7 1.71120
\(96\) −884736. −0.102062
\(97\) −6.97736e6 −0.776230 −0.388115 0.921611i \(-0.626874\pi\)
−0.388115 + 0.921611i \(0.626874\pi\)
\(98\) 941192. 0.101015
\(99\) 3.30832e6 0.342677
\(100\) 2.06634e6 0.206634
\(101\) −430774. −0.0416030 −0.0208015 0.999784i \(-0.506622\pi\)
−0.0208015 + 0.999784i \(0.506622\pi\)
\(102\) 363648. 0.0339297
\(103\) −1.71937e7 −1.55039 −0.775193 0.631724i \(-0.782347\pi\)
−0.775193 + 0.631724i \(0.782347\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 3.07727e6 0.259419
\(106\) −469593. −0.0382958
\(107\) −7.23378e6 −0.570850 −0.285425 0.958401i \(-0.592135\pi\)
−0.285425 + 0.958401i \(0.592135\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 531822. 0.0393345 0.0196673 0.999807i \(-0.493739\pi\)
0.0196673 + 0.999807i \(0.493739\pi\)
\(110\) 1.20636e7 0.864178
\(111\) −5.58935e6 −0.387910
\(112\) −1.40493e6 −0.0944911
\(113\) 1.31514e7 0.857426 0.428713 0.903441i \(-0.358967\pi\)
0.428713 + 0.903441i \(0.358967\pi\)
\(114\) −9.29565e6 −0.587642
\(115\) 2.65872e7 1.63016
\(116\) −1.12030e7 −0.666397
\(117\) 1.60161e6 0.0924500
\(118\) 1.99689e6 0.111884
\(119\) 577459. 0.0314128
\(120\) −4.59347e6 −0.242665
\(121\) 1.10776e6 0.0568458
\(122\) −7.41359e6 −0.369632
\(123\) −1.13069e7 −0.547868
\(124\) −1.13719e7 −0.535622
\(125\) −1.52313e7 −0.697513
\(126\) −2.00038e6 −0.0890871
\(127\) −414009. −0.0179348 −0.00896740 0.999960i \(-0.502854\pi\)
−0.00896740 + 0.999960i \(0.502854\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.38000e7 −0.565997
\(130\) 5.84019e6 0.233145
\(131\) 4.64364e7 1.80472 0.902358 0.430988i \(-0.141835\pi\)
0.902358 + 0.430988i \(0.141835\pi\)
\(132\) −7.84195e6 −0.296767
\(133\) −1.47611e7 −0.544051
\(134\) −2.57967e7 −0.926186
\(135\) −6.54031e6 −0.228786
\(136\) −861980. −0.0293840
\(137\) 8.72739e6 0.289976 0.144988 0.989433i \(-0.453686\pi\)
0.144988 + 0.989433i \(0.453686\pi\)
\(138\) −1.72830e7 −0.559813
\(139\) −1.17087e7 −0.369790 −0.184895 0.982758i \(-0.559195\pi\)
−0.184895 + 0.982758i \(0.559195\pi\)
\(140\) −7.29426e6 −0.224664
\(141\) 4.17110e6 0.125310
\(142\) −5.76455e6 −0.168949
\(143\) 9.97035e6 0.285124
\(144\) 2.98598e6 0.0833333
\(145\) −5.81652e7 −1.58444
\(146\) 4.88978e7 1.30033
\(147\) −3.17652e6 −0.0824786
\(148\) 1.32488e7 0.335940
\(149\) 5.25065e7 1.30035 0.650176 0.759784i \(-0.274695\pi\)
0.650176 + 0.759784i \(0.274695\pi\)
\(150\) −6.97389e6 −0.168716
\(151\) −1.06827e6 −0.0252501 −0.0126251 0.999920i \(-0.504019\pi\)
−0.0126251 + 0.999920i \(0.504019\pi\)
\(152\) 2.20341e7 0.508913
\(153\) −1.22731e6 −0.0277035
\(154\) −1.24527e7 −0.274753
\(155\) −5.90420e7 −1.27350
\(156\) −3.79642e6 −0.0800641
\(157\) 1.85891e7 0.383362 0.191681 0.981457i \(-0.438606\pi\)
0.191681 + 0.981457i \(0.438606\pi\)
\(158\) −1.29960e7 −0.262126
\(159\) 1.58488e6 0.0312684
\(160\) 1.08882e7 0.210154
\(161\) −2.74448e7 −0.518286
\(162\) 4.25153e6 0.0785674
\(163\) 7.81618e7 1.41364 0.706818 0.707395i \(-0.250130\pi\)
0.706818 + 0.707395i \(0.250130\pi\)
\(164\) 2.68016e7 0.474468
\(165\) −4.07147e7 −0.705598
\(166\) 6.86627e7 1.16504
\(167\) −2.97299e7 −0.493954 −0.246977 0.969021i \(-0.579437\pi\)
−0.246977 + 0.969021i \(0.579437\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −4.47532e6 −0.0698639
\(171\) 3.13728e7 0.479808
\(172\) 3.27111e7 0.490168
\(173\) 1.04807e8 1.53897 0.769485 0.638665i \(-0.220513\pi\)
0.769485 + 0.638665i \(0.220513\pi\)
\(174\) 3.78103e7 0.544111
\(175\) −1.10743e7 −0.156200
\(176\) 1.85883e7 0.257008
\(177\) −6.73952e6 −0.0913529
\(178\) −5.77733e7 −0.767816
\(179\) 1.32769e8 1.73026 0.865131 0.501546i \(-0.167235\pi\)
0.865131 + 0.501546i \(0.167235\pi\)
\(180\) 1.55030e7 0.198135
\(181\) −6.86420e7 −0.860428 −0.430214 0.902727i \(-0.641562\pi\)
−0.430214 + 0.902727i \(0.641562\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 2.50209e7 0.301803
\(184\) 4.09671e7 0.484812
\(185\) 6.87868e7 0.798737
\(186\) 3.83803e7 0.437333
\(187\) −7.64024e6 −0.0854401
\(188\) −9.88706e6 −0.108521
\(189\) 6.75127e6 0.0727393
\(190\) 1.14399e8 1.21000
\(191\) −5.98203e7 −0.621200 −0.310600 0.950541i \(-0.600530\pi\)
−0.310600 + 0.950541i \(0.600530\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 1.48627e8 1.48815 0.744076 0.668095i \(-0.232890\pi\)
0.744076 + 0.668095i \(0.232890\pi\)
\(194\) −5.58189e7 −0.548877
\(195\) −1.97107e7 −0.190362
\(196\) 7.52954e6 0.0714286
\(197\) 4.01516e7 0.374172 0.187086 0.982344i \(-0.440096\pi\)
0.187086 + 0.982344i \(0.440096\pi\)
\(198\) 2.64666e7 0.242309
\(199\) 1.22948e8 1.10595 0.552976 0.833197i \(-0.313492\pi\)
0.552976 + 0.833197i \(0.313492\pi\)
\(200\) 1.65307e7 0.146112
\(201\) 8.70640e7 0.756228
\(202\) −3.44619e6 −0.0294178
\(203\) 6.00413e7 0.503749
\(204\) 2.90918e6 0.0239919
\(205\) 1.39151e8 1.12810
\(206\) −1.37550e8 −1.09629
\(207\) 5.83302e7 0.457085
\(208\) 8.99891e6 0.0693375
\(209\) 1.95302e8 1.47977
\(210\) 2.46181e7 0.183437
\(211\) −2.00000e8 −1.46569 −0.732843 0.680397i \(-0.761807\pi\)
−0.732843 + 0.680397i \(0.761807\pi\)
\(212\) −3.75674e6 −0.0270792
\(213\) 1.94554e7 0.137946
\(214\) −5.78702e7 −0.403652
\(215\) 1.69833e8 1.16543
\(216\) −1.00777e7 −0.0680414
\(217\) 6.09464e7 0.404892
\(218\) 4.25458e6 0.0278137
\(219\) −1.65030e8 −1.06172
\(220\) 9.65089e7 0.611066
\(221\) −3.69877e6 −0.0230507
\(222\) −4.47148e7 −0.274294
\(223\) −1.90121e8 −1.14805 −0.574027 0.818836i \(-0.694620\pi\)
−0.574027 + 0.818836i \(0.694620\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 2.35369e7 0.137756
\(226\) 1.05211e8 0.606292
\(227\) 1.97080e8 1.11828 0.559141 0.829073i \(-0.311131\pi\)
0.559141 + 0.829073i \(0.311131\pi\)
\(228\) −7.43652e7 −0.415526
\(229\) 1.92885e8 1.06139 0.530694 0.847564i \(-0.321931\pi\)
0.530694 + 0.847564i \(0.321931\pi\)
\(230\) 2.12698e8 1.15270
\(231\) 4.20279e7 0.224335
\(232\) −8.96244e7 −0.471214
\(233\) 2.15169e8 1.11438 0.557191 0.830385i \(-0.311879\pi\)
0.557191 + 0.830385i \(0.311879\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) −5.13327e7 −0.258022
\(236\) 1.59752e7 0.0791140
\(237\) 4.38615e7 0.214025
\(238\) 4.61967e6 0.0222122
\(239\) −1.36102e8 −0.644868 −0.322434 0.946592i \(-0.604501\pi\)
−0.322434 + 0.946592i \(0.604501\pi\)
\(240\) −3.67478e7 −0.171590
\(241\) 3.48180e8 1.60230 0.801150 0.598464i \(-0.204222\pi\)
0.801150 + 0.598464i \(0.204222\pi\)
\(242\) 8.86211e6 0.0401960
\(243\) −1.43489e7 −0.0641500
\(244\) −5.93087e7 −0.261369
\(245\) 3.90927e7 0.169830
\(246\) −9.04553e7 −0.387401
\(247\) 9.45488e7 0.399224
\(248\) −9.09754e7 −0.378742
\(249\) −2.31737e8 −0.951255
\(250\) −1.21851e8 −0.493216
\(251\) 5.30641e6 0.0211808 0.0105904 0.999944i \(-0.496629\pi\)
0.0105904 + 0.999944i \(0.496629\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) 3.63116e8 1.40969
\(254\) −3.31207e6 −0.0126818
\(255\) 1.51042e7 0.0570437
\(256\) 1.67772e7 0.0625000
\(257\) 1.37965e8 0.506995 0.253498 0.967336i \(-0.418419\pi\)
0.253498 + 0.967336i \(0.418419\pi\)
\(258\) −1.10400e8 −0.400221
\(259\) −7.10055e7 −0.253947
\(260\) 4.67215e7 0.164858
\(261\) −1.27610e8 −0.444265
\(262\) 3.71491e8 1.27613
\(263\) 5.05702e8 1.71415 0.857077 0.515189i \(-0.172278\pi\)
0.857077 + 0.515189i \(0.172278\pi\)
\(264\) −6.27356e7 −0.209846
\(265\) −1.95047e7 −0.0643840
\(266\) −1.18089e8 −0.384702
\(267\) 1.94985e8 0.626919
\(268\) −2.06374e8 −0.654912
\(269\) −4.93505e7 −0.154582 −0.0772909 0.997009i \(-0.524627\pi\)
−0.0772909 + 0.997009i \(0.524627\pi\)
\(270\) −5.23225e7 −0.161776
\(271\) −1.26125e8 −0.384953 −0.192476 0.981302i \(-0.561652\pi\)
−0.192476 + 0.981302i \(0.561652\pi\)
\(272\) −6.89584e6 −0.0207776
\(273\) 2.03464e7 0.0605228
\(274\) 6.98191e7 0.205044
\(275\) 1.46522e8 0.424851
\(276\) −1.38264e8 −0.395847
\(277\) −2.88014e8 −0.814207 −0.407103 0.913382i \(-0.633461\pi\)
−0.407103 + 0.913382i \(0.633461\pi\)
\(278\) −9.36693e7 −0.261481
\(279\) −1.29533e8 −0.357081
\(280\) −5.83541e7 −0.158861
\(281\) −8.91246e7 −0.239621 −0.119811 0.992797i \(-0.538229\pi\)
−0.119811 + 0.992797i \(0.538229\pi\)
\(282\) 3.33688e7 0.0886072
\(283\) 2.68924e7 0.0705304 0.0352652 0.999378i \(-0.488772\pi\)
0.0352652 + 0.999378i \(0.488772\pi\)
\(284\) −4.61164e7 −0.119465
\(285\) −3.86098e8 −0.987961
\(286\) 7.97628e7 0.201613
\(287\) −1.43640e8 −0.358664
\(288\) 2.38879e7 0.0589256
\(289\) −4.07504e8 −0.993093
\(290\) −4.65322e8 −1.12037
\(291\) 1.88389e8 0.448156
\(292\) 3.91182e8 0.919474
\(293\) −1.31694e8 −0.305865 −0.152933 0.988237i \(-0.548872\pi\)
−0.152933 + 0.988237i \(0.548872\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 8.29416e7 0.188103
\(296\) 1.05991e8 0.237545
\(297\) −8.93247e7 −0.197844
\(298\) 4.20052e8 0.919488
\(299\) 1.75791e8 0.380318
\(300\) −5.57911e7 −0.119300
\(301\) −1.75311e8 −0.370532
\(302\) −8.54620e6 −0.0178545
\(303\) 1.16309e7 0.0240195
\(304\) 1.76273e8 0.359856
\(305\) −3.07926e8 −0.621436
\(306\) −9.81849e6 −0.0195893
\(307\) 7.59154e7 0.149743 0.0748714 0.997193i \(-0.476145\pi\)
0.0748714 + 0.997193i \(0.476145\pi\)
\(308\) −9.96218e7 −0.194279
\(309\) 4.64231e8 0.895116
\(310\) −4.72336e8 −0.900504
\(311\) 4.40832e8 0.831021 0.415511 0.909588i \(-0.363603\pi\)
0.415511 + 0.909588i \(0.363603\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) 8.94209e8 1.64829 0.824146 0.566378i \(-0.191656\pi\)
0.824146 + 0.566378i \(0.191656\pi\)
\(314\) 1.48713e8 0.271078
\(315\) −8.30862e7 −0.149776
\(316\) −1.03968e8 −0.185351
\(317\) −9.27330e8 −1.63503 −0.817517 0.575904i \(-0.804650\pi\)
−0.817517 + 0.575904i \(0.804650\pi\)
\(318\) 1.26790e7 0.0221101
\(319\) −7.94395e8 −1.37015
\(320\) 8.71058e7 0.148601
\(321\) 1.95312e8 0.329580
\(322\) −2.19558e8 −0.366483
\(323\) −7.24525e7 −0.119631
\(324\) 3.40122e7 0.0555556
\(325\) 7.09335e7 0.114620
\(326\) 6.25294e8 0.999592
\(327\) −1.43592e7 −0.0227098
\(328\) 2.14413e8 0.335499
\(329\) 5.29885e7 0.0820344
\(330\) −3.25718e8 −0.498933
\(331\) −9.99764e8 −1.51530 −0.757651 0.652660i \(-0.773653\pi\)
−0.757651 + 0.652660i \(0.773653\pi\)
\(332\) 5.49302e8 0.823811
\(333\) 1.50912e8 0.223960
\(334\) −2.37840e8 −0.349278
\(335\) −1.07148e9 −1.55713
\(336\) 3.79331e7 0.0545545
\(337\) 9.36583e8 1.33304 0.666518 0.745489i \(-0.267784\pi\)
0.666518 + 0.745489i \(0.267784\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) −3.55087e8 −0.495035
\(340\) −3.58026e7 −0.0494013
\(341\) −8.06370e8 −1.10127
\(342\) 2.50983e8 0.339275
\(343\) −4.03536e7 −0.0539949
\(344\) 2.61688e8 0.346601
\(345\) −7.17855e8 −0.941174
\(346\) 8.38458e8 1.08822
\(347\) −2.89119e8 −0.371469 −0.185735 0.982600i \(-0.559466\pi\)
−0.185735 + 0.982600i \(0.559466\pi\)
\(348\) 3.02482e8 0.384745
\(349\) 1.05672e9 1.33067 0.665333 0.746546i \(-0.268289\pi\)
0.665333 + 0.746546i \(0.268289\pi\)
\(350\) −8.85942e7 −0.110450
\(351\) −4.32436e7 −0.0533761
\(352\) 1.48707e8 0.181732
\(353\) 2.62139e8 0.317191 0.158595 0.987344i \(-0.449303\pi\)
0.158595 + 0.987344i \(0.449303\pi\)
\(354\) −5.39162e7 −0.0645963
\(355\) −2.39432e8 −0.284043
\(356\) −4.62186e8 −0.542928
\(357\) −1.55914e7 −0.0181362
\(358\) 1.06215e9 1.22348
\(359\) −1.58381e8 −0.180664 −0.0903322 0.995912i \(-0.528793\pi\)
−0.0903322 + 0.995912i \(0.528793\pi\)
\(360\) 1.24024e8 0.140103
\(361\) 9.58176e8 1.07194
\(362\) −5.49136e8 −0.608415
\(363\) −2.99096e7 −0.0328199
\(364\) −4.82285e7 −0.0524142
\(365\) 2.03098e9 2.18616
\(366\) 2.00167e8 0.213407
\(367\) −4.26484e8 −0.450372 −0.225186 0.974316i \(-0.572299\pi\)
−0.225186 + 0.974316i \(0.572299\pi\)
\(368\) 3.27737e8 0.342814
\(369\) 3.05287e8 0.316312
\(370\) 5.50294e8 0.564792
\(371\) 2.01338e7 0.0204700
\(372\) 3.07042e8 0.309241
\(373\) −1.71354e9 −1.70967 −0.854836 0.518898i \(-0.826342\pi\)
−0.854836 + 0.518898i \(0.826342\pi\)
\(374\) −6.11220e7 −0.0604153
\(375\) 4.11246e8 0.402709
\(376\) −7.90965e7 −0.0767361
\(377\) −3.84580e8 −0.369651
\(378\) 5.40102e7 0.0514344
\(379\) −5.98355e8 −0.564575 −0.282288 0.959330i \(-0.591093\pi\)
−0.282288 + 0.959330i \(0.591093\pi\)
\(380\) 9.15194e8 0.855600
\(381\) 1.11782e7 0.0103547
\(382\) −4.78562e8 −0.439255
\(383\) −1.16290e9 −1.05766 −0.528829 0.848728i \(-0.677369\pi\)
−0.528829 + 0.848728i \(0.677369\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −5.17227e8 −0.461922
\(386\) 1.18902e9 1.05228
\(387\) 3.72599e8 0.326779
\(388\) −4.46551e8 −0.388115
\(389\) −1.04953e9 −0.904004 −0.452002 0.892017i \(-0.649290\pi\)
−0.452002 + 0.892017i \(0.649290\pi\)
\(390\) −1.57685e8 −0.134606
\(391\) −1.34708e8 −0.113966
\(392\) 6.02363e7 0.0505076
\(393\) −1.25378e9 −1.04195
\(394\) 3.21213e8 0.264579
\(395\) −5.39793e8 −0.440694
\(396\) 2.11733e8 0.171338
\(397\) −1.09871e9 −0.881284 −0.440642 0.897683i \(-0.645249\pi\)
−0.440642 + 0.897683i \(0.645249\pi\)
\(398\) 9.83586e8 0.782027
\(399\) 3.98551e8 0.314108
\(400\) 1.32246e8 0.103317
\(401\) 1.80907e9 1.40104 0.700520 0.713633i \(-0.252951\pi\)
0.700520 + 0.713633i \(0.252951\pi\)
\(402\) 6.96512e8 0.534734
\(403\) −3.90377e8 −0.297110
\(404\) −2.75695e7 −0.0208015
\(405\) 1.76588e8 0.132090
\(406\) 4.80331e8 0.356204
\(407\) 9.39459e8 0.690713
\(408\) 2.32735e7 0.0169649
\(409\) 8.35953e8 0.604157 0.302079 0.953283i \(-0.402319\pi\)
0.302079 + 0.953283i \(0.402319\pi\)
\(410\) 1.11321e9 0.797689
\(411\) −2.35639e8 −0.167418
\(412\) −1.10040e9 −0.775193
\(413\) −8.56169e7 −0.0598045
\(414\) 4.66641e8 0.323208
\(415\) 2.85193e9 1.95871
\(416\) 7.19913e7 0.0490290
\(417\) 3.16134e8 0.213498
\(418\) 1.56241e9 1.04636
\(419\) −6.07516e8 −0.403467 −0.201734 0.979440i \(-0.564658\pi\)
−0.201734 + 0.979440i \(0.564658\pi\)
\(420\) 1.96945e8 0.129710
\(421\) 2.06243e9 1.34708 0.673539 0.739152i \(-0.264773\pi\)
0.673539 + 0.739152i \(0.264773\pi\)
\(422\) −1.60000e9 −1.03640
\(423\) −1.12620e8 −0.0723475
\(424\) −3.00539e7 −0.0191479
\(425\) −5.43561e7 −0.0343469
\(426\) 1.55643e8 0.0975429
\(427\) 3.17858e8 0.197576
\(428\) −4.62962e8 −0.285425
\(429\) −2.69199e8 −0.164617
\(430\) 1.35866e9 0.824085
\(431\) −1.90276e8 −0.114476 −0.0572378 0.998361i \(-0.518229\pi\)
−0.0572378 + 0.998361i \(0.518229\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) −2.84375e9 −1.68339 −0.841693 0.539957i \(-0.818441\pi\)
−0.841693 + 0.539957i \(0.818441\pi\)
\(434\) 4.87571e8 0.286302
\(435\) 1.57046e9 0.914776
\(436\) 3.40366e7 0.0196673
\(437\) 3.44343e9 1.97382
\(438\) −1.32024e9 −0.750747
\(439\) −2.56732e9 −1.44828 −0.724142 0.689651i \(-0.757764\pi\)
−0.724142 + 0.689651i \(0.757764\pi\)
\(440\) 7.72071e8 0.432089
\(441\) 8.57661e7 0.0476190
\(442\) −2.95901e7 −0.0162993
\(443\) 8.11793e7 0.0443641 0.0221821 0.999754i \(-0.492939\pi\)
0.0221821 + 0.999754i \(0.492939\pi\)
\(444\) −3.57718e8 −0.193955
\(445\) −2.39963e9 −1.29087
\(446\) −1.52097e9 −0.811797
\(447\) −1.41767e9 −0.750759
\(448\) −8.99154e7 −0.0472456
\(449\) 1.41154e9 0.735920 0.367960 0.929842i \(-0.380056\pi\)
0.367960 + 0.929842i \(0.380056\pi\)
\(450\) 1.88295e8 0.0974081
\(451\) 1.90047e9 0.975534
\(452\) 8.41688e8 0.428713
\(453\) 2.88434e7 0.0145782
\(454\) 1.57664e9 0.790745
\(455\) −2.50398e8 −0.124621
\(456\) −5.94922e8 −0.293821
\(457\) −2.45874e8 −0.120505 −0.0602526 0.998183i \(-0.519191\pi\)
−0.0602526 + 0.998183i \(0.519191\pi\)
\(458\) 1.54308e9 0.750515
\(459\) 3.31374e7 0.0159946
\(460\) 1.70158e9 0.815081
\(461\) 7.67759e8 0.364982 0.182491 0.983208i \(-0.441584\pi\)
0.182491 + 0.983208i \(0.441584\pi\)
\(462\) 3.36224e8 0.158628
\(463\) −3.13479e9 −1.46783 −0.733914 0.679242i \(-0.762309\pi\)
−0.733914 + 0.679242i \(0.762309\pi\)
\(464\) −7.16995e8 −0.333199
\(465\) 1.59414e9 0.735258
\(466\) 1.72135e9 0.787986
\(467\) −3.05486e9 −1.38798 −0.693989 0.719985i \(-0.744148\pi\)
−0.693989 + 0.719985i \(0.744148\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) 1.10604e9 0.495067
\(470\) −4.10662e8 −0.182449
\(471\) −5.01905e8 −0.221334
\(472\) 1.27801e8 0.0559420
\(473\) 2.31950e9 1.00782
\(474\) 3.50892e8 0.151339
\(475\) 1.38946e9 0.594867
\(476\) 3.69574e7 0.0157064
\(477\) −4.27917e7 −0.0180528
\(478\) −1.08881e9 −0.455990
\(479\) 8.05606e8 0.334926 0.167463 0.985878i \(-0.446443\pi\)
0.167463 + 0.985878i \(0.446443\pi\)
\(480\) −2.93982e8 −0.121332
\(481\) 4.54808e8 0.186346
\(482\) 2.78544e9 1.13300
\(483\) 7.41009e8 0.299232
\(484\) 7.08969e7 0.0284229
\(485\) −2.31845e9 −0.922789
\(486\) −1.14791e8 −0.0453609
\(487\) −3.88945e9 −1.52594 −0.762969 0.646435i \(-0.776259\pi\)
−0.762969 + 0.646435i \(0.776259\pi\)
\(488\) −4.74470e8 −0.184816
\(489\) −2.11037e9 −0.816164
\(490\) 3.12741e8 0.120088
\(491\) 3.09826e8 0.118122 0.0590612 0.998254i \(-0.481189\pi\)
0.0590612 + 0.998254i \(0.481189\pi\)
\(492\) −7.23642e8 −0.273934
\(493\) 2.94702e8 0.110769
\(494\) 7.56391e8 0.282294
\(495\) 1.09930e9 0.407377
\(496\) −7.27803e8 −0.267811
\(497\) 2.47155e8 0.0903072
\(498\) −1.85389e9 −0.672639
\(499\) −5.94530e8 −0.214201 −0.107101 0.994248i \(-0.534157\pi\)
−0.107101 + 0.994248i \(0.534157\pi\)
\(500\) −9.74804e8 −0.348757
\(501\) 8.02708e8 0.285184
\(502\) 4.24512e7 0.0149771
\(503\) −2.00206e9 −0.701437 −0.350718 0.936481i \(-0.614063\pi\)
−0.350718 + 0.936481i \(0.614063\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) −1.43139e8 −0.0494580
\(506\) 2.90493e9 0.996803
\(507\) −1.30324e8 −0.0444116
\(508\) −2.64966e7 −0.00896740
\(509\) 1.22512e9 0.411782 0.205891 0.978575i \(-0.433991\pi\)
0.205891 + 0.978575i \(0.433991\pi\)
\(510\) 1.20834e8 0.0403360
\(511\) −2.09649e9 −0.695057
\(512\) 1.34218e8 0.0441942
\(513\) −8.47066e8 −0.277017
\(514\) 1.10372e9 0.358500
\(515\) −5.71317e9 −1.84311
\(516\) −8.83199e8 −0.282999
\(517\) −7.01080e8 −0.223126
\(518\) −5.68044e8 −0.179567
\(519\) −2.82980e9 −0.888525
\(520\) 3.73772e8 0.116572
\(521\) 2.07737e9 0.643548 0.321774 0.946816i \(-0.395721\pi\)
0.321774 + 0.946816i \(0.395721\pi\)
\(522\) −1.02088e9 −0.314143
\(523\) 6.24365e8 0.190846 0.0954229 0.995437i \(-0.469580\pi\)
0.0954229 + 0.995437i \(0.469580\pi\)
\(524\) 2.97193e9 0.902358
\(525\) 2.99005e8 0.0901824
\(526\) 4.04562e9 1.21209
\(527\) 2.99145e8 0.0890316
\(528\) −5.01885e8 −0.148383
\(529\) 2.99741e9 0.880341
\(530\) −1.56037e8 −0.0455264
\(531\) 1.81967e8 0.0527426
\(532\) −9.44714e8 −0.272025
\(533\) 9.20047e8 0.263187
\(534\) 1.55988e9 0.443299
\(535\) −2.40366e9 −0.678632
\(536\) −1.65099e9 −0.463093
\(537\) −3.58477e9 −0.998967
\(538\) −3.94804e8 −0.109306
\(539\) 5.33910e8 0.146861
\(540\) −4.18580e8 −0.114393
\(541\) −1.27993e9 −0.347534 −0.173767 0.984787i \(-0.555594\pi\)
−0.173767 + 0.984787i \(0.555594\pi\)
\(542\) −1.00900e9 −0.272203
\(543\) 1.85333e9 0.496769
\(544\) −5.51667e7 −0.0146920
\(545\) 1.76715e8 0.0467612
\(546\) 1.62771e8 0.0427960
\(547\) −1.66165e9 −0.434093 −0.217047 0.976161i \(-0.569642\pi\)
−0.217047 + 0.976161i \(0.569642\pi\)
\(548\) 5.58553e8 0.144988
\(549\) −6.75563e8 −0.174246
\(550\) 1.17217e9 0.300415
\(551\) −7.53325e9 −1.91846
\(552\) −1.10611e9 −0.279906
\(553\) 5.57204e8 0.140112
\(554\) −2.30411e9 −0.575731
\(555\) −1.85724e9 −0.461151
\(556\) −7.49354e8 −0.184895
\(557\) −4.23236e9 −1.03774 −0.518871 0.854852i \(-0.673648\pi\)
−0.518871 + 0.854852i \(0.673648\pi\)
\(558\) −1.03627e9 −0.252495
\(559\) 1.12291e9 0.271896
\(560\) −4.66833e8 −0.112332
\(561\) 2.06287e8 0.0493289
\(562\) −7.12997e8 −0.169438
\(563\) 2.29041e9 0.540922 0.270461 0.962731i \(-0.412824\pi\)
0.270461 + 0.962731i \(0.412824\pi\)
\(564\) 2.66951e8 0.0626548
\(565\) 4.36997e9 1.01932
\(566\) 2.15139e8 0.0498725
\(567\) −1.82284e8 −0.0419961
\(568\) −3.68931e8 −0.0844746
\(569\) 5.70558e9 1.29839 0.649197 0.760620i \(-0.275105\pi\)
0.649197 + 0.760620i \(0.275105\pi\)
\(570\) −3.08878e9 −0.698594
\(571\) −5.78089e8 −0.129948 −0.0649738 0.997887i \(-0.520696\pi\)
−0.0649738 + 0.997887i \(0.520696\pi\)
\(572\) 6.38102e8 0.142562
\(573\) 1.61515e9 0.358650
\(574\) −1.14912e9 −0.253614
\(575\) 2.58337e9 0.566695
\(576\) 1.91103e8 0.0416667
\(577\) 1.85447e9 0.401888 0.200944 0.979603i \(-0.435599\pi\)
0.200944 + 0.979603i \(0.435599\pi\)
\(578\) −3.26003e9 −0.702223
\(579\) −4.01293e9 −0.859185
\(580\) −3.72257e9 −0.792219
\(581\) −2.94391e9 −0.622743
\(582\) 1.50711e9 0.316894
\(583\) −2.66386e8 −0.0556765
\(584\) 3.12946e9 0.650166
\(585\) 5.32188e8 0.109905
\(586\) −1.05355e9 −0.216279
\(587\) 1.59026e7 0.00324516 0.00162258 0.999999i \(-0.499484\pi\)
0.00162258 + 0.999999i \(0.499484\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −7.64681e9 −1.54197
\(590\) 6.63533e8 0.133009
\(591\) −1.08409e9 −0.216028
\(592\) 8.47925e8 0.167970
\(593\) 4.72446e9 0.930380 0.465190 0.885211i \(-0.345986\pi\)
0.465190 + 0.885211i \(0.345986\pi\)
\(594\) −7.14597e8 −0.139897
\(595\) 1.91879e8 0.0373438
\(596\) 3.36041e9 0.650176
\(597\) −3.31960e9 −0.638522
\(598\) 1.40633e9 0.268925
\(599\) 6.54444e9 1.24417 0.622084 0.782951i \(-0.286286\pi\)
0.622084 + 0.782951i \(0.286286\pi\)
\(600\) −4.46329e8 −0.0843579
\(601\) 9.35203e9 1.75730 0.878649 0.477468i \(-0.158445\pi\)
0.878649 + 0.477468i \(0.158445\pi\)
\(602\) −1.40249e9 −0.262006
\(603\) −2.35073e9 −0.436608
\(604\) −6.83696e7 −0.0126251
\(605\) 3.68090e8 0.0675788
\(606\) 9.30472e7 0.0169844
\(607\) 7.23440e9 1.31293 0.656466 0.754355i \(-0.272050\pi\)
0.656466 + 0.754355i \(0.272050\pi\)
\(608\) 1.41018e9 0.254456
\(609\) −1.62112e9 −0.290840
\(610\) −2.46340e9 −0.439422
\(611\) −3.39404e8 −0.0601968
\(612\) −7.85479e7 −0.0138517
\(613\) 1.45381e9 0.254916 0.127458 0.991844i \(-0.459318\pi\)
0.127458 + 0.991844i \(0.459318\pi\)
\(614\) 6.07323e8 0.105884
\(615\) −3.75709e9 −0.651311
\(616\) −7.96974e8 −0.137376
\(617\) −4.53208e9 −0.776782 −0.388391 0.921495i \(-0.626969\pi\)
−0.388391 + 0.921495i \(0.626969\pi\)
\(618\) 3.71385e9 0.632943
\(619\) −1.04972e10 −1.77891 −0.889457 0.457019i \(-0.848917\pi\)
−0.889457 + 0.457019i \(0.848917\pi\)
\(620\) −3.77869e9 −0.636752
\(621\) −1.57491e9 −0.263898
\(622\) 3.52666e9 0.587621
\(623\) 2.47703e9 0.410415
\(624\) −2.42971e8 −0.0400320
\(625\) −7.58348e9 −1.24248
\(626\) 7.15368e9 1.16552
\(627\) −5.27315e9 −0.854346
\(628\) 1.18970e9 0.191681
\(629\) −3.48518e8 −0.0558403
\(630\) −6.64690e8 −0.105908
\(631\) −1.90313e9 −0.301554 −0.150777 0.988568i \(-0.548178\pi\)
−0.150777 + 0.988568i \(0.548178\pi\)
\(632\) −8.31744e8 −0.131063
\(633\) 5.40000e9 0.846215
\(634\) −7.41864e9 −1.15614
\(635\) −1.37568e8 −0.0213210
\(636\) 1.01432e8 0.0156342
\(637\) 2.58475e8 0.0396214
\(638\) −6.35516e9 −0.968845
\(639\) −5.25295e8 −0.0796434
\(640\) 6.96846e8 0.105077
\(641\) −1.28685e10 −1.92986 −0.964930 0.262506i \(-0.915451\pi\)
−0.964930 + 0.262506i \(0.915451\pi\)
\(642\) 1.56250e9 0.233048
\(643\) −1.15273e9 −0.170998 −0.0854989 0.996338i \(-0.527248\pi\)
−0.0854989 + 0.996338i \(0.527248\pi\)
\(644\) −1.75647e9 −0.259143
\(645\) −4.58549e9 −0.672863
\(646\) −5.79620e8 −0.0845920
\(647\) −5.76076e9 −0.836209 −0.418105 0.908399i \(-0.637305\pi\)
−0.418105 + 0.908399i \(0.637305\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 1.13278e9 0.162663
\(650\) 5.67468e8 0.0810484
\(651\) −1.64555e9 −0.233765
\(652\) 5.00235e9 0.706818
\(653\) 3.72855e8 0.0524015 0.0262008 0.999657i \(-0.491659\pi\)
0.0262008 + 0.999657i \(0.491659\pi\)
\(654\) −1.14874e8 −0.0160582
\(655\) 1.54300e10 2.14546
\(656\) 1.71530e9 0.237234
\(657\) 4.45581e9 0.612982
\(658\) 4.23908e8 0.0580070
\(659\) −5.88562e9 −0.801113 −0.400556 0.916272i \(-0.631183\pi\)
−0.400556 + 0.916272i \(0.631183\pi\)
\(660\) −2.60574e9 −0.352799
\(661\) −1.29378e8 −0.0174244 −0.00871218 0.999962i \(-0.502773\pi\)
−0.00871218 + 0.999962i \(0.502773\pi\)
\(662\) −7.99811e9 −1.07148
\(663\) 9.98667e7 0.0133083
\(664\) 4.39441e9 0.582522
\(665\) −4.90487e9 −0.646773
\(666\) 1.20730e9 0.158364
\(667\) −1.40063e10 −1.82760
\(668\) −1.90272e9 −0.246977
\(669\) 5.13326e9 0.662829
\(670\) −8.57180e9 −1.10106
\(671\) −4.20551e9 −0.537390
\(672\) 3.03464e8 0.0385758
\(673\) 7.40929e9 0.936967 0.468483 0.883472i \(-0.344801\pi\)
0.468483 + 0.883472i \(0.344801\pi\)
\(674\) 7.49267e9 0.942599
\(675\) −6.35496e8 −0.0795334
\(676\) 3.08916e8 0.0384615
\(677\) −1.34737e10 −1.66888 −0.834441 0.551098i \(-0.814209\pi\)
−0.834441 + 0.551098i \(0.814209\pi\)
\(678\) −2.84070e9 −0.350043
\(679\) 2.39324e9 0.293387
\(680\) −2.86421e8 −0.0349320
\(681\) −5.32115e9 −0.645640
\(682\) −6.45096e9 −0.778716
\(683\) −1.01931e10 −1.22415 −0.612074 0.790801i \(-0.709664\pi\)
−0.612074 + 0.790801i \(0.709664\pi\)
\(684\) 2.00786e9 0.239904
\(685\) 2.89996e9 0.344726
\(686\) −3.22829e8 −0.0381802
\(687\) −5.20789e9 −0.612793
\(688\) 2.09351e9 0.245084
\(689\) −1.28962e8 −0.0150208
\(690\) −5.74284e9 −0.665510
\(691\) 1.25240e10 1.44401 0.722005 0.691888i \(-0.243221\pi\)
0.722005 + 0.691888i \(0.243221\pi\)
\(692\) 6.70767e9 0.769485
\(693\) −1.13475e9 −0.129520
\(694\) −2.31295e9 −0.262669
\(695\) −3.89058e9 −0.439610
\(696\) 2.41986e9 0.272056
\(697\) −7.05029e8 −0.0788665
\(698\) 8.45373e9 0.940923
\(699\) −5.80956e9 −0.643388
\(700\) −7.08754e8 −0.0781002
\(701\) −2.19973e9 −0.241188 −0.120594 0.992702i \(-0.538480\pi\)
−0.120594 + 0.992702i \(0.538480\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) 8.90889e9 0.967119
\(704\) 1.18965e9 0.128504
\(705\) 1.38598e9 0.148969
\(706\) 2.09711e9 0.224288
\(707\) 1.47756e8 0.0157245
\(708\) −4.31329e8 −0.0456765
\(709\) −9.24653e9 −0.974355 −0.487177 0.873303i \(-0.661974\pi\)
−0.487177 + 0.873303i \(0.661974\pi\)
\(710\) −1.91546e9 −0.200848
\(711\) −1.18426e9 −0.123567
\(712\) −3.69749e9 −0.383908
\(713\) −1.42174e10 −1.46895
\(714\) −1.24731e8 −0.0128242
\(715\) 3.31297e9 0.338958
\(716\) 8.49723e9 0.865131
\(717\) 3.67474e9 0.372314
\(718\) −1.26705e9 −0.127749
\(719\) −5.78088e9 −0.580020 −0.290010 0.957024i \(-0.593659\pi\)
−0.290010 + 0.957024i \(0.593659\pi\)
\(720\) 9.92190e8 0.0990674
\(721\) 5.89745e9 0.585991
\(722\) 7.66541e9 0.757975
\(723\) −9.40085e9 −0.925088
\(724\) −4.39309e9 −0.430214
\(725\) −5.65168e9 −0.550801
\(726\) −2.39277e8 −0.0232072
\(727\) 8.21685e9 0.793113 0.396556 0.918010i \(-0.370205\pi\)
0.396556 + 0.918010i \(0.370205\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 1.62479e10 1.54585
\(731\) −8.60482e8 −0.0814762
\(732\) 1.60134e9 0.150901
\(733\) −2.05550e10 −1.92776 −0.963881 0.266332i \(-0.914188\pi\)
−0.963881 + 0.266332i \(0.914188\pi\)
\(734\) −3.41187e9 −0.318461
\(735\) −1.05550e9 −0.0980513
\(736\) 2.62190e9 0.242406
\(737\) −1.46337e10 −1.34654
\(738\) 2.44229e9 0.223666
\(739\) −1.95410e10 −1.78111 −0.890555 0.454876i \(-0.849683\pi\)
−0.890555 + 0.454876i \(0.849683\pi\)
\(740\) 4.40235e9 0.399368
\(741\) −2.55282e9 −0.230492
\(742\) 1.61070e8 0.0144744
\(743\) −1.92668e10 −1.72325 −0.861625 0.507546i \(-0.830553\pi\)
−0.861625 + 0.507546i \(0.830553\pi\)
\(744\) 2.45634e9 0.218667
\(745\) 1.74470e10 1.54587
\(746\) −1.37083e10 −1.20892
\(747\) 6.25689e9 0.549207
\(748\) −4.88976e8 −0.0427200
\(749\) 2.48119e9 0.215761
\(750\) 3.28996e9 0.284759
\(751\) 1.09052e10 0.939497 0.469749 0.882800i \(-0.344345\pi\)
0.469749 + 0.882800i \(0.344345\pi\)
\(752\) −6.32772e8 −0.0542606
\(753\) −1.43273e8 −0.0122287
\(754\) −3.07664e9 −0.261383
\(755\) −3.54969e8 −0.0300176
\(756\) 4.32081e8 0.0363696
\(757\) 7.82173e7 0.00655341 0.00327670 0.999995i \(-0.498957\pi\)
0.00327670 + 0.999995i \(0.498957\pi\)
\(758\) −4.78684e9 −0.399215
\(759\) −9.80415e9 −0.813886
\(760\) 7.32155e9 0.605000
\(761\) −1.53822e10 −1.26524 −0.632618 0.774464i \(-0.718020\pi\)
−0.632618 + 0.774464i \(0.718020\pi\)
\(762\) 8.94259e7 0.00732185
\(763\) −1.82415e8 −0.0148670
\(764\) −3.82850e9 −0.310600
\(765\) −4.07814e8 −0.0329342
\(766\) −9.30317e9 −0.747877
\(767\) 5.48397e8 0.0438845
\(768\) −4.52985e8 −0.0360844
\(769\) 2.14405e10 1.70017 0.850087 0.526643i \(-0.176549\pi\)
0.850087 + 0.526643i \(0.176549\pi\)
\(770\) −4.13782e9 −0.326628
\(771\) −3.72506e9 −0.292714
\(772\) 9.51213e9 0.744076
\(773\) −4.61629e9 −0.359472 −0.179736 0.983715i \(-0.557524\pi\)
−0.179736 + 0.983715i \(0.557524\pi\)
\(774\) 2.98080e9 0.231067
\(775\) −5.73688e9 −0.442710
\(776\) −3.57241e9 −0.274439
\(777\) 1.91715e9 0.146616
\(778\) −8.39622e9 −0.639227
\(779\) 1.80221e10 1.36592
\(780\) −1.26148e9 −0.0951809
\(781\) −3.27006e9 −0.245628
\(782\) −1.07766e9 −0.0805859
\(783\) 3.44546e9 0.256496
\(784\) 4.81890e8 0.0357143
\(785\) 6.17682e9 0.455744
\(786\) −1.00303e10 −0.736772
\(787\) −6.13485e9 −0.448634 −0.224317 0.974516i \(-0.572015\pi\)
−0.224317 + 0.974516i \(0.572015\pi\)
\(788\) 2.56970e9 0.187086
\(789\) −1.36540e10 −0.989667
\(790\) −4.31834e9 −0.311618
\(791\) −4.51092e9 −0.324077
\(792\) 1.69386e9 0.121155
\(793\) −2.03596e9 −0.144981
\(794\) −8.78967e9 −0.623162
\(795\) 5.26626e8 0.0371721
\(796\) 7.86869e9 0.552976
\(797\) −1.81940e10 −1.27298 −0.636492 0.771283i \(-0.719615\pi\)
−0.636492 + 0.771283i \(0.719615\pi\)
\(798\) 3.18841e9 0.222108
\(799\) 2.60084e8 0.0180385
\(800\) 1.05796e9 0.0730561
\(801\) −5.26459e9 −0.361952
\(802\) 1.44726e10 0.990685
\(803\) 2.77383e10 1.89049
\(804\) 5.57210e9 0.378114
\(805\) −9.11942e9 −0.616143
\(806\) −3.12302e9 −0.210088
\(807\) 1.33246e9 0.0892479
\(808\) −2.20556e8 −0.0147089
\(809\) 1.69125e10 1.12302 0.561511 0.827469i \(-0.310220\pi\)
0.561511 + 0.827469i \(0.310220\pi\)
\(810\) 1.41271e9 0.0934017
\(811\) 1.42003e10 0.934813 0.467407 0.884042i \(-0.345188\pi\)
0.467407 + 0.884042i \(0.345188\pi\)
\(812\) 3.84265e9 0.251875
\(813\) 3.40536e9 0.222253
\(814\) 7.51567e9 0.488408
\(815\) 2.59718e10 1.68054
\(816\) 1.86188e8 0.0119960
\(817\) 2.19958e10 1.41112
\(818\) 6.68762e9 0.427204
\(819\) −5.49353e8 −0.0349428
\(820\) 8.90569e9 0.564052
\(821\) −8.40863e9 −0.530303 −0.265152 0.964207i \(-0.585422\pi\)
−0.265152 + 0.964207i \(0.585422\pi\)
\(822\) −1.88512e9 −0.118382
\(823\) −8.48016e9 −0.530279 −0.265140 0.964210i \(-0.585418\pi\)
−0.265140 + 0.964210i \(0.585418\pi\)
\(824\) −8.80319e9 −0.548144
\(825\) −3.95608e9 −0.245288
\(826\) −6.84935e8 −0.0422882
\(827\) −1.81973e10 −1.11876 −0.559381 0.828911i \(-0.688961\pi\)
−0.559381 + 0.828911i \(0.688961\pi\)
\(828\) 3.73313e9 0.228543
\(829\) −5.46488e9 −0.333150 −0.166575 0.986029i \(-0.553271\pi\)
−0.166575 + 0.986029i \(0.553271\pi\)
\(830\) 2.28154e10 1.38502
\(831\) 7.77638e9 0.470082
\(832\) 5.75930e8 0.0346688
\(833\) −1.98068e8 −0.0118729
\(834\) 2.52907e9 0.150966
\(835\) −9.87873e9 −0.587217
\(836\) 1.24993e10 0.739885
\(837\) 3.49740e9 0.206161
\(838\) −4.86013e9 −0.285294
\(839\) 1.14912e9 0.0671737 0.0335868 0.999436i \(-0.489307\pi\)
0.0335868 + 0.999436i \(0.489307\pi\)
\(840\) 1.57556e9 0.0917186
\(841\) 1.33918e10 0.776342
\(842\) 1.64995e10 0.952528
\(843\) 2.40637e9 0.138346
\(844\) −1.28000e10 −0.732843
\(845\) 1.60386e9 0.0914469
\(846\) −9.00958e8 −0.0511574
\(847\) −3.79963e8 −0.0214857
\(848\) −2.40432e8 −0.0135396
\(849\) −7.26094e8 −0.0407208
\(850\) −4.34849e8 −0.0242869
\(851\) 1.65639e10 0.921319
\(852\) 1.24514e9 0.0689732
\(853\) 1.87086e10 1.03210 0.516048 0.856560i \(-0.327403\pi\)
0.516048 + 0.856560i \(0.327403\pi\)
\(854\) 2.54286e9 0.139708
\(855\) 1.04246e10 0.570400
\(856\) −3.70369e9 −0.201826
\(857\) −1.77412e10 −0.962832 −0.481416 0.876492i \(-0.659877\pi\)
−0.481416 + 0.876492i \(0.659877\pi\)
\(858\) −2.15359e9 −0.116401
\(859\) 1.69542e10 0.912643 0.456322 0.889815i \(-0.349167\pi\)
0.456322 + 0.889815i \(0.349167\pi\)
\(860\) 1.08693e10 0.582716
\(861\) 3.87827e9 0.207075
\(862\) −1.52220e9 −0.0809464
\(863\) −3.76730e10 −1.99523 −0.997613 0.0690540i \(-0.978002\pi\)
−0.997613 + 0.0690540i \(0.978002\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 3.48256e10 1.82954
\(866\) −2.27500e10 −1.19033
\(867\) 1.10026e10 0.573362
\(868\) 3.90057e9 0.202446
\(869\) −7.37225e9 −0.381093
\(870\) 1.25637e10 0.646844
\(871\) −7.08443e9 −0.363280
\(872\) 2.72293e8 0.0139068
\(873\) −5.08650e9 −0.258743
\(874\) 2.75475e10 1.39570
\(875\) 5.22434e9 0.263635
\(876\) −1.05619e10 −0.530858
\(877\) −1.32929e10 −0.665461 −0.332730 0.943022i \(-0.607970\pi\)
−0.332730 + 0.943022i \(0.607970\pi\)
\(878\) −2.05385e10 −1.02409
\(879\) 3.55574e9 0.176591
\(880\) 6.17657e9 0.305533
\(881\) 1.63851e10 0.807299 0.403650 0.914914i \(-0.367742\pi\)
0.403650 + 0.914914i \(0.367742\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 2.98014e10 1.45671 0.728357 0.685198i \(-0.240284\pi\)
0.728357 + 0.685198i \(0.240284\pi\)
\(884\) −2.36721e8 −0.0115254
\(885\) −2.23942e9 −0.108601
\(886\) 6.49434e8 0.0313702
\(887\) −1.54252e10 −0.742163 −0.371081 0.928600i \(-0.621013\pi\)
−0.371081 + 0.928600i \(0.621013\pi\)
\(888\) −2.86175e9 −0.137147
\(889\) 1.42005e8 0.00677872
\(890\) −1.91970e10 −0.912786
\(891\) 2.41177e9 0.114226
\(892\) −1.21677e10 −0.574027
\(893\) −6.64834e9 −0.312416
\(894\) −1.13414e10 −0.530866
\(895\) 4.41169e10 2.05695
\(896\) −7.19323e8 −0.0334077
\(897\) −4.74635e9 −0.219577
\(898\) 1.12923e10 0.520374
\(899\) 3.11036e10 1.42775
\(900\) 1.50636e9 0.0688779
\(901\) 9.88231e7 0.00450113
\(902\) 1.52037e10 0.689807
\(903\) 4.73339e9 0.213927
\(904\) 6.73351e9 0.303146
\(905\) −2.28085e10 −1.02289
\(906\) 2.30747e8 0.0103083
\(907\) −2.20805e10 −0.982614 −0.491307 0.870987i \(-0.663481\pi\)
−0.491307 + 0.870987i \(0.663481\pi\)
\(908\) 1.26131e10 0.559141
\(909\) −3.14034e8 −0.0138677
\(910\) −2.00319e9 −0.0881204
\(911\) −4.36513e10 −1.91286 −0.956428 0.291967i \(-0.905690\pi\)
−0.956428 + 0.291967i \(0.905690\pi\)
\(912\) −4.75937e9 −0.207763
\(913\) 3.89503e10 1.69381
\(914\) −1.96699e9 −0.0852100
\(915\) 8.31399e9 0.358786
\(916\) 1.23446e10 0.530694
\(917\) −1.59277e10 −0.682118
\(918\) 2.65099e8 0.0113099
\(919\) 1.67550e9 0.0712099 0.0356049 0.999366i \(-0.488664\pi\)
0.0356049 + 0.999366i \(0.488664\pi\)
\(920\) 1.36127e10 0.576349
\(921\) −2.04972e9 −0.0864540
\(922\) 6.14207e9 0.258081
\(923\) −1.58309e9 −0.0662673
\(924\) 2.68979e9 0.112167
\(925\) 6.68373e9 0.277666
\(926\) −2.50783e10 −1.03791
\(927\) −1.25342e10 −0.516795
\(928\) −5.73596e9 −0.235607
\(929\) 2.20950e10 0.904149 0.452074 0.891980i \(-0.350684\pi\)
0.452074 + 0.891980i \(0.350684\pi\)
\(930\) 1.27531e10 0.519906
\(931\) 5.06307e9 0.205632
\(932\) 1.37708e10 0.557191
\(933\) −1.19025e10 −0.479790
\(934\) −2.44389e10 −0.981449
\(935\) −2.53872e9 −0.101572
\(936\) 8.20026e8 0.0326860
\(937\) 3.17830e10 1.26214 0.631069 0.775727i \(-0.282617\pi\)
0.631069 + 0.775727i \(0.282617\pi\)
\(938\) 8.84828e9 0.350065
\(939\) −2.41437e10 −0.951642
\(940\) −3.28530e9 −0.129011
\(941\) −2.50884e10 −0.981544 −0.490772 0.871288i \(-0.663285\pi\)
−0.490772 + 0.871288i \(0.663285\pi\)
\(942\) −4.01524e9 −0.156507
\(943\) 3.35078e10 1.30123
\(944\) 1.02241e9 0.0395570
\(945\) 2.24333e9 0.0864731
\(946\) 1.85560e10 0.712633
\(947\) −2.13366e10 −0.816396 −0.408198 0.912893i \(-0.633843\pi\)
−0.408198 + 0.912893i \(0.633843\pi\)
\(948\) 2.80714e9 0.107012
\(949\) 1.34286e10 0.510032
\(950\) 1.11157e10 0.420634
\(951\) 2.50379e10 0.943988
\(952\) 2.95659e8 0.0111061
\(953\) −3.59693e10 −1.34619 −0.673097 0.739555i \(-0.735036\pi\)
−0.673097 + 0.739555i \(0.735036\pi\)
\(954\) −3.42333e8 −0.0127653
\(955\) −1.98772e10 −0.738489
\(956\) −8.71050e9 −0.322434
\(957\) 2.14487e10 0.791058
\(958\) 6.44485e9 0.236828
\(959\) −2.99349e9 −0.109601
\(960\) −2.35186e9 −0.0857949
\(961\) 4.05984e9 0.147563
\(962\) 3.63846e9 0.131766
\(963\) −5.27342e9 −0.190283
\(964\) 2.22835e10 0.801150
\(965\) 4.93861e10 1.76913
\(966\) 5.92807e9 0.211589
\(967\) 3.68747e10 1.31140 0.655702 0.755020i \(-0.272373\pi\)
0.655702 + 0.755020i \(0.272373\pi\)
\(968\) 5.67175e8 0.0200980
\(969\) 1.95622e9 0.0690691
\(970\) −1.85476e10 −0.652510
\(971\) 3.88100e10 1.36043 0.680215 0.733013i \(-0.261886\pi\)
0.680215 + 0.733013i \(0.261886\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 4.01607e9 0.139768
\(974\) −3.11156e10 −1.07900
\(975\) −1.91520e9 −0.0661758
\(976\) −3.79576e9 −0.130685
\(977\) 1.23536e10 0.423801 0.211901 0.977291i \(-0.432035\pi\)
0.211901 + 0.977291i \(0.432035\pi\)
\(978\) −1.68829e10 −0.577115
\(979\) −3.27731e10 −1.11629
\(980\) 2.50193e9 0.0849149
\(981\) 3.87698e8 0.0131115
\(982\) 2.47860e9 0.0835251
\(983\) 1.39366e9 0.0467972 0.0233986 0.999726i \(-0.492551\pi\)
0.0233986 + 0.999726i \(0.492551\pi\)
\(984\) −5.78914e9 −0.193701
\(985\) 1.33417e10 0.444819
\(986\) 2.35762e9 0.0783257
\(987\) −1.43069e9 −0.0473626
\(988\) 6.05112e9 0.199612
\(989\) 4.08960e10 1.34429
\(990\) 8.79437e9 0.288059
\(991\) 3.93376e10 1.28396 0.641978 0.766723i \(-0.278114\pi\)
0.641978 + 0.766723i \(0.278114\pi\)
\(992\) −5.82243e9 −0.189371
\(993\) 2.69936e10 0.874860
\(994\) 1.97724e9 0.0638568
\(995\) 4.08535e10 1.31477
\(996\) −1.48311e10 −0.475628
\(997\) 5.24317e10 1.67556 0.837781 0.546006i \(-0.183853\pi\)
0.837781 + 0.546006i \(0.183853\pi\)
\(998\) −4.75624e9 −0.151463
\(999\) −4.07464e9 −0.129303
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 546.8.a.r.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.r.1.5 6 1.1 even 1 trivial