Properties

Label 546.8.a.r.1.1
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 1.1
Root \(390.179\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -356.179 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -356.179 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -2849.43 q^{10} -7505.72 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} +9616.83 q^{15} +4096.00 q^{16} -16070.3 q^{17} +5832.00 q^{18} +2775.91 q^{19} -22795.4 q^{20} +9261.00 q^{21} -60045.7 q^{22} -32534.4 q^{23} -13824.0 q^{24} +48738.4 q^{25} +17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} -168180. q^{29} +76934.6 q^{30} -220119. q^{31} +32768.0 q^{32} +202654. q^{33} -128562. q^{34} +122169. q^{35} +46656.0 q^{36} -211079. q^{37} +22207.3 q^{38} -59319.0 q^{39} -182364. q^{40} -824216. q^{41} +74088.0 q^{42} +640034. q^{43} -480366. q^{44} -259654. q^{45} -260275. q^{46} -85878.7 q^{47} -110592. q^{48} +117649. q^{49} +389907. q^{50} +433898. q^{51} +140608. q^{52} -1.72649e6 q^{53} -157464. q^{54} +2.67338e6 q^{55} -175616. q^{56} -74949.7 q^{57} -1.34544e6 q^{58} +2.81760e6 q^{59} +615477. q^{60} -1.08363e6 q^{61} -1.76095e6 q^{62} -250047. q^{63} +262144. q^{64} -782525. q^{65} +1.62123e6 q^{66} +1.27353e6 q^{67} -1.02850e6 q^{68} +878428. q^{69} +977355. q^{70} -2.65081e6 q^{71} +373248. q^{72} +4.18663e6 q^{73} -1.68864e6 q^{74} -1.31594e6 q^{75} +177658. q^{76} +2.57446e6 q^{77} -474552. q^{78} -7.94649e6 q^{79} -1.45891e6 q^{80} +531441. q^{81} -6.59373e6 q^{82} -5.50842e6 q^{83} +592704. q^{84} +5.72390e6 q^{85} +5.12028e6 q^{86} +4.54085e6 q^{87} -3.84293e6 q^{88} +4.44064e6 q^{89} -2.07723e6 q^{90} -753571. q^{91} -2.08220e6 q^{92} +5.94321e6 q^{93} -687030. q^{94} -988722. q^{95} -884736. q^{96} +5.97700e6 q^{97} +941192. q^{98} -5.47167e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 1624 q^{10} - 2690 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} - 5481 q^{15} + 24576 q^{16} + 2910 q^{17} + 34992 q^{18} - 13055 q^{19} + 12992 q^{20} + 55566 q^{21} - 21520 q^{22} + 11581 q^{23} - 82944 q^{24} + 68081 q^{25} + 105456 q^{26} - 118098 q^{27} - 131712 q^{28} - 92335 q^{29} - 43848 q^{30} - 83081 q^{31} + 196608 q^{32} + 72630 q^{33} + 23280 q^{34} - 69629 q^{35} + 279936 q^{36} - 265114 q^{37} - 104440 q^{38} - 355914 q^{39} + 103936 q^{40} - 367468 q^{41} + 444528 q^{42} + 454955 q^{43} - 172160 q^{44} + 147987 q^{45} + 92648 q^{46} + 733973 q^{47} - 663552 q^{48} + 705894 q^{49} + 544648 q^{50} - 78570 q^{51} + 843648 q^{52} - 1577379 q^{53} - 944784 q^{54} + 2231118 q^{55} - 1053696 q^{56} + 352485 q^{57} - 738680 q^{58} + 2062708 q^{59} - 350784 q^{60} - 271270 q^{61} - 664648 q^{62} - 1500282 q^{63} + 1572864 q^{64} + 445991 q^{65} + 581040 q^{66} - 758674 q^{67} + 186240 q^{68} - 312687 q^{69} - 557032 q^{70} - 6138216 q^{71} + 2239488 q^{72} + 6361979 q^{73} - 2120912 q^{74} - 1838187 q^{75} - 835520 q^{76} + 922670 q^{77} - 2847312 q^{78} - 899781 q^{79} + 831488 q^{80} + 3188646 q^{81} - 2939744 q^{82} + 3313561 q^{83} + 3556224 q^{84} + 5307940 q^{85} + 3639640 q^{86} + 2493045 q^{87} - 1377280 q^{88} + 11210703 q^{89} + 1183896 q^{90} - 4521426 q^{91} + 741184 q^{92} + 2243187 q^{93} + 5871784 q^{94} + 12912395 q^{95} - 5308416 q^{96} + 28682643 q^{97} + 5647152 q^{98} - 1961010 q^{99} + O(q^{100}) \)

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
\(5\) −356.179 −1.27430 −0.637152 0.770738i \(-0.719888\pi\)
−0.637152 + 0.770738i \(0.719888\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −2849.43 −0.901069
\(11\) −7505.72 −1.70027 −0.850135 0.526565i \(-0.823480\pi\)
−0.850135 + 0.526565i \(0.823480\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) 9616.83 0.735720
\(16\) 4096.00 0.250000
\(17\) −16070.3 −0.793328 −0.396664 0.917964i \(-0.629832\pi\)
−0.396664 + 0.917964i \(0.629832\pi\)
\(18\) 5832.00 0.235702
\(19\) 2775.91 0.0928471 0.0464236 0.998922i \(-0.485218\pi\)
0.0464236 + 0.998922i \(0.485218\pi\)
\(20\) −22795.4 −0.637152
\(21\) 9261.00 0.218218
\(22\) −60045.7 −1.20227
\(23\) −32534.4 −0.557564 −0.278782 0.960354i \(-0.589931\pi\)
−0.278782 + 0.960354i \(0.589931\pi\)
\(24\) −13824.0 −0.204124
\(25\) 48738.4 0.623851
\(26\) 17576.0 0.196116
\(27\) −19683.0 −0.192450
\(28\) −21952.0 −0.188982
\(29\) −168180. −1.28050 −0.640252 0.768165i \(-0.721170\pi\)
−0.640252 + 0.768165i \(0.721170\pi\)
\(30\) 76934.6 0.520232
\(31\) −220119. −1.32706 −0.663531 0.748148i \(-0.730943\pi\)
−0.663531 + 0.748148i \(0.730943\pi\)
\(32\) 32768.0 0.176777
\(33\) 202654. 0.981651
\(34\) −128562. −0.560968
\(35\) 122169. 0.481642
\(36\) 46656.0 0.166667
\(37\) −211079. −0.685078 −0.342539 0.939504i \(-0.611287\pi\)
−0.342539 + 0.939504i \(0.611287\pi\)
\(38\) 22207.3 0.0656528
\(39\) −59319.0 −0.160128
\(40\) −182364. −0.450535
\(41\) −824216. −1.86766 −0.933830 0.357718i \(-0.883555\pi\)
−0.933830 + 0.357718i \(0.883555\pi\)
\(42\) 74088.0 0.154303
\(43\) 640034. 1.22762 0.613810 0.789454i \(-0.289636\pi\)
0.613810 + 0.789454i \(0.289636\pi\)
\(44\) −480366. −0.850135
\(45\) −259654. −0.424768
\(46\) −260275. −0.394257
\(47\) −85878.7 −0.120654 −0.0603272 0.998179i \(-0.519214\pi\)
−0.0603272 + 0.998179i \(0.519214\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 389907. 0.441129
\(51\) 433898. 0.458028
\(52\) 140608. 0.138675
\(53\) −1.72649e6 −1.59293 −0.796466 0.604683i \(-0.793300\pi\)
−0.796466 + 0.604683i \(0.793300\pi\)
\(54\) −157464. −0.136083
\(55\) 2.67338e6 2.16666
\(56\) −175616. −0.133631
\(57\) −74949.7 −0.0536053
\(58\) −1.34544e6 −0.905452
\(59\) 2.81760e6 1.78606 0.893032 0.449994i \(-0.148574\pi\)
0.893032 + 0.449994i \(0.148574\pi\)
\(60\) 615477. 0.367860
\(61\) −1.08363e6 −0.611261 −0.305630 0.952150i \(-0.598867\pi\)
−0.305630 + 0.952150i \(0.598867\pi\)
\(62\) −1.76095e6 −0.938375
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) −782525. −0.353428
\(66\) 1.62123e6 0.694132
\(67\) 1.27353e6 0.517305 0.258653 0.965970i \(-0.416721\pi\)
0.258653 + 0.965970i \(0.416721\pi\)
\(68\) −1.02850e6 −0.396664
\(69\) 878428. 0.321910
\(70\) 977355. 0.340572
\(71\) −2.65081e6 −0.878970 −0.439485 0.898250i \(-0.644839\pi\)
−0.439485 + 0.898250i \(0.644839\pi\)
\(72\) 373248. 0.117851
\(73\) 4.18663e6 1.25961 0.629803 0.776755i \(-0.283136\pi\)
0.629803 + 0.776755i \(0.283136\pi\)
\(74\) −1.68864e6 −0.484423
\(75\) −1.31594e6 −0.360181
\(76\) 177658. 0.0464236
\(77\) 2.57446e6 0.642641
\(78\) −474552. −0.113228
\(79\) −7.94649e6 −1.81335 −0.906673 0.421835i \(-0.861386\pi\)
−0.906673 + 0.421835i \(0.861386\pi\)
\(80\) −1.45891e6 −0.318576
\(81\) 531441. 0.111111
\(82\) −6.59373e6 −1.32063
\(83\) −5.50842e6 −1.05744 −0.528718 0.848798i \(-0.677327\pi\)
−0.528718 + 0.848798i \(0.677327\pi\)
\(84\) 592704. 0.109109
\(85\) 5.72390e6 1.01094
\(86\) 5.12028e6 0.868058
\(87\) 4.54085e6 0.739299
\(88\) −3.84293e6 −0.601136
\(89\) 4.44064e6 0.667698 0.333849 0.942626i \(-0.391652\pi\)
0.333849 + 0.942626i \(0.391652\pi\)
\(90\) −2.07723e6 −0.300356
\(91\) −753571. −0.104828
\(92\) −2.08220e6 −0.278782
\(93\) 5.94321e6 0.766180
\(94\) −687030. −0.0853155
\(95\) −988722. −0.118315
\(96\) −884736. −0.102062
\(97\) 5.97700e6 0.664940 0.332470 0.943114i \(-0.392118\pi\)
0.332470 + 0.943114i \(0.392118\pi\)
\(98\) 941192. 0.101015
\(99\) −5.47167e6 −0.566756
\(100\) 3.11926e6 0.311926
\(101\) −2.84477e6 −0.274740 −0.137370 0.990520i \(-0.543865\pi\)
−0.137370 + 0.990520i \(0.543865\pi\)
\(102\) 3.47119e6 0.323875
\(103\) −1.45832e7 −1.31499 −0.657494 0.753460i \(-0.728384\pi\)
−0.657494 + 0.753460i \(0.728384\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) −3.29857e6 −0.278076
\(106\) −1.38119e7 −1.12637
\(107\) −1.89107e6 −0.149233 −0.0746163 0.997212i \(-0.523773\pi\)
−0.0746163 + 0.997212i \(0.523773\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 9.23714e6 0.683195 0.341598 0.939846i \(-0.389032\pi\)
0.341598 + 0.939846i \(0.389032\pi\)
\(110\) 2.13870e7 1.53206
\(111\) 5.69915e6 0.395530
\(112\) −1.40493e6 −0.0944911
\(113\) 1.36594e7 0.890547 0.445273 0.895395i \(-0.353107\pi\)
0.445273 + 0.895395i \(0.353107\pi\)
\(114\) −599597. −0.0379047
\(115\) 1.15880e7 0.710506
\(116\) −1.07635e7 −0.640252
\(117\) 1.60161e6 0.0924500
\(118\) 2.25408e7 1.26294
\(119\) 5.51211e6 0.299850
\(120\) 4.92382e6 0.260116
\(121\) 3.68486e7 1.89092
\(122\) −8.66904e6 −0.432227
\(123\) 2.22538e7 1.07829
\(124\) −1.40876e7 −0.663531
\(125\) 1.04669e7 0.479328
\(126\) −2.00038e6 −0.0890871
\(127\) 3.41065e7 1.47749 0.738743 0.673987i \(-0.235420\pi\)
0.738743 + 0.673987i \(0.235420\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.72809e7 −0.708766
\(130\) −6.26020e6 −0.249912
\(131\) 7.43598e6 0.288994 0.144497 0.989505i \(-0.453844\pi\)
0.144497 + 0.989505i \(0.453844\pi\)
\(132\) 1.29699e7 0.490825
\(133\) −952138. −0.0350929
\(134\) 1.01882e7 0.365790
\(135\) 7.01067e6 0.245240
\(136\) −8.22799e6 −0.280484
\(137\) 1.36234e7 0.452652 0.226326 0.974052i \(-0.427328\pi\)
0.226326 + 0.974052i \(0.427328\pi\)
\(138\) 7.02742e6 0.227625
\(139\) 4.13662e7 1.30645 0.653227 0.757162i \(-0.273415\pi\)
0.653227 + 0.757162i \(0.273415\pi\)
\(140\) 7.81884e6 0.240821
\(141\) 2.31872e6 0.0696598
\(142\) −2.12065e7 −0.621526
\(143\) −1.64901e7 −0.471570
\(144\) 2.98598e6 0.0833333
\(145\) 5.99021e7 1.63175
\(146\) 3.34930e7 0.890676
\(147\) −3.17652e6 −0.0824786
\(148\) −1.35091e7 −0.342539
\(149\) −7.21056e7 −1.78574 −0.892868 0.450318i \(-0.851311\pi\)
−0.892868 + 0.450318i \(0.851311\pi\)
\(150\) −1.05275e7 −0.254686
\(151\) 1.68206e6 0.0397577 0.0198788 0.999802i \(-0.493672\pi\)
0.0198788 + 0.999802i \(0.493672\pi\)
\(152\) 1.42127e6 0.0328264
\(153\) −1.17153e7 −0.264443
\(154\) 2.05957e7 0.454416
\(155\) 7.84017e7 1.69108
\(156\) −3.79642e6 −0.0800641
\(157\) −3.18358e7 −0.656548 −0.328274 0.944583i \(-0.606467\pi\)
−0.328274 + 0.944583i \(0.606467\pi\)
\(158\) −6.35719e7 −1.28223
\(159\) 4.66151e7 0.919680
\(160\) −1.16713e7 −0.225267
\(161\) 1.11593e7 0.210739
\(162\) 4.25153e6 0.0785674
\(163\) −9.32110e7 −1.68582 −0.842909 0.538057i \(-0.819159\pi\)
−0.842909 + 0.538057i \(0.819159\pi\)
\(164\) −5.27498e7 −0.933830
\(165\) −7.21812e7 −1.25092
\(166\) −4.40674e7 −0.747720
\(167\) 4.68184e7 0.777873 0.388937 0.921265i \(-0.372843\pi\)
0.388937 + 0.921265i \(0.372843\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 4.57912e7 0.714843
\(171\) 2.02364e6 0.0309490
\(172\) 4.09622e7 0.613810
\(173\) 5.84637e7 0.858469 0.429235 0.903193i \(-0.358783\pi\)
0.429235 + 0.903193i \(0.358783\pi\)
\(174\) 3.63268e7 0.522763
\(175\) −1.67173e7 −0.235794
\(176\) −3.07434e7 −0.425067
\(177\) −7.60751e7 −1.03118
\(178\) 3.55251e7 0.472134
\(179\) −1.04704e8 −1.36451 −0.682255 0.731114i \(-0.739001\pi\)
−0.682255 + 0.731114i \(0.739001\pi\)
\(180\) −1.66179e7 −0.212384
\(181\) 5.59187e7 0.700942 0.350471 0.936574i \(-0.386022\pi\)
0.350471 + 0.936574i \(0.386022\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 2.92580e7 0.352912
\(184\) −1.66576e7 −0.197129
\(185\) 7.51820e7 0.872998
\(186\) 4.75457e7 0.541771
\(187\) 1.20619e8 1.34887
\(188\) −5.49624e6 −0.0603272
\(189\) 6.75127e6 0.0727393
\(190\) −7.90977e6 −0.0836617
\(191\) −5.29705e7 −0.550069 −0.275034 0.961434i \(-0.588689\pi\)
−0.275034 + 0.961434i \(0.588689\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 1.46969e8 1.47155 0.735774 0.677227i \(-0.236818\pi\)
0.735774 + 0.677227i \(0.236818\pi\)
\(194\) 4.78160e7 0.470183
\(195\) 2.11282e7 0.204052
\(196\) 7.52954e6 0.0714286
\(197\) −8.47740e7 −0.790007 −0.395003 0.918680i \(-0.629257\pi\)
−0.395003 + 0.918680i \(0.629257\pi\)
\(198\) −4.37733e7 −0.400757
\(199\) −1.83215e8 −1.64807 −0.824033 0.566541i \(-0.808281\pi\)
−0.824033 + 0.566541i \(0.808281\pi\)
\(200\) 2.49540e7 0.220565
\(201\) −3.43853e7 −0.298666
\(202\) −2.27581e7 −0.194271
\(203\) 5.76856e7 0.483985
\(204\) 2.77695e7 0.229014
\(205\) 2.93568e8 2.37997
\(206\) −1.16665e8 −0.929837
\(207\) −2.37175e7 −0.185855
\(208\) 8.99891e6 0.0693375
\(209\) −2.08352e7 −0.157865
\(210\) −2.63886e7 −0.196629
\(211\) 1.65562e8 1.21331 0.606655 0.794966i \(-0.292511\pi\)
0.606655 + 0.794966i \(0.292511\pi\)
\(212\) −1.10495e8 −0.796466
\(213\) 7.15718e7 0.507474
\(214\) −1.51285e7 −0.105523
\(215\) −2.27967e8 −1.56436
\(216\) −1.00777e7 −0.0680414
\(217\) 7.55008e7 0.501583
\(218\) 7.38971e7 0.483092
\(219\) −1.13039e8 −0.727234
\(220\) 1.71096e8 1.08333
\(221\) −3.53065e7 −0.220030
\(222\) 4.55932e7 0.279682
\(223\) 2.92424e8 1.76582 0.882909 0.469543i \(-0.155581\pi\)
0.882909 + 0.469543i \(0.155581\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 3.55303e7 0.207950
\(226\) 1.09275e8 0.629712
\(227\) 2.88122e8 1.63488 0.817441 0.576012i \(-0.195392\pi\)
0.817441 + 0.576012i \(0.195392\pi\)
\(228\) −4.79678e6 −0.0268027
\(229\) −7.75197e7 −0.426568 −0.213284 0.976990i \(-0.568416\pi\)
−0.213284 + 0.976990i \(0.568416\pi\)
\(230\) 9.27044e7 0.502404
\(231\) −6.95104e7 −0.371029
\(232\) −8.61080e7 −0.452726
\(233\) 2.75907e8 1.42895 0.714475 0.699661i \(-0.246666\pi\)
0.714475 + 0.699661i \(0.246666\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) 3.05882e7 0.153750
\(236\) 1.80326e8 0.893032
\(237\) 2.14555e8 1.04694
\(238\) 4.40969e7 0.212026
\(239\) −2.14321e8 −1.01548 −0.507740 0.861510i \(-0.669519\pi\)
−0.507740 + 0.861510i \(0.669519\pi\)
\(240\) 3.93905e7 0.183930
\(241\) 6.61344e7 0.304346 0.152173 0.988354i \(-0.451373\pi\)
0.152173 + 0.988354i \(0.451373\pi\)
\(242\) 2.94789e8 1.33708
\(243\) −1.43489e7 −0.0641500
\(244\) −6.93523e7 −0.305630
\(245\) −4.19041e7 −0.182043
\(246\) 1.78031e8 0.762469
\(247\) 6.09868e6 0.0257512
\(248\) −1.12701e8 −0.469188
\(249\) 1.48727e8 0.610511
\(250\) 8.37352e7 0.338936
\(251\) 3.68420e8 1.47057 0.735285 0.677758i \(-0.237048\pi\)
0.735285 + 0.677758i \(0.237048\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) 2.44194e8 0.948009
\(254\) 2.72852e8 1.04474
\(255\) −1.54545e8 −0.583667
\(256\) 1.67772e7 0.0625000
\(257\) −2.60448e8 −0.957097 −0.478548 0.878061i \(-0.658837\pi\)
−0.478548 + 0.878061i \(0.658837\pi\)
\(258\) −1.38247e8 −0.501173
\(259\) 7.24003e7 0.258935
\(260\) −5.00816e7 −0.176714
\(261\) −1.22603e8 −0.426834
\(262\) 5.94879e7 0.204350
\(263\) 1.81155e8 0.614052 0.307026 0.951701i \(-0.400666\pi\)
0.307026 + 0.951701i \(0.400666\pi\)
\(264\) 1.03759e8 0.347066
\(265\) 6.14938e8 2.02988
\(266\) −7.61711e6 −0.0248144
\(267\) −1.19897e8 −0.385496
\(268\) 8.15058e7 0.258653
\(269\) −5.00047e8 −1.56631 −0.783155 0.621827i \(-0.786391\pi\)
−0.783155 + 0.621827i \(0.786391\pi\)
\(270\) 5.60853e7 0.173411
\(271\) −8.44196e7 −0.257662 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(272\) −6.58240e7 −0.198332
\(273\) 2.03464e7 0.0605228
\(274\) 1.08988e8 0.320074
\(275\) −3.65816e8 −1.06071
\(276\) 5.62194e7 0.160955
\(277\) −5.05664e8 −1.42950 −0.714748 0.699382i \(-0.753459\pi\)
−0.714748 + 0.699382i \(0.753459\pi\)
\(278\) 3.30930e8 0.923802
\(279\) −1.60467e8 −0.442354
\(280\) 6.25507e7 0.170286
\(281\) −1.25449e8 −0.337283 −0.168641 0.985677i \(-0.553938\pi\)
−0.168641 + 0.985677i \(0.553938\pi\)
\(282\) 1.85498e7 0.0492569
\(283\) −1.51960e8 −0.398544 −0.199272 0.979944i \(-0.563858\pi\)
−0.199272 + 0.979944i \(0.563858\pi\)
\(284\) −1.69652e8 −0.439485
\(285\) 2.66955e7 0.0683095
\(286\) −1.31920e8 −0.333450
\(287\) 2.82706e8 0.705909
\(288\) 2.38879e7 0.0589256
\(289\) −1.52084e8 −0.370631
\(290\) 4.79216e8 1.15382
\(291\) −1.61379e8 −0.383903
\(292\) 2.67944e8 0.629803
\(293\) 4.33667e7 0.100721 0.0503604 0.998731i \(-0.483963\pi\)
0.0503604 + 0.998731i \(0.483963\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −1.00357e9 −2.27599
\(296\) −1.08073e8 −0.242212
\(297\) 1.47735e8 0.327217
\(298\) −5.76845e8 −1.26271
\(299\) −7.14780e7 −0.154640
\(300\) −8.42199e7 −0.180090
\(301\) −2.19532e8 −0.463996
\(302\) 1.34564e7 0.0281129
\(303\) 7.68087e7 0.158621
\(304\) 1.13701e7 0.0232118
\(305\) 3.85966e8 0.778932
\(306\) −9.37220e7 −0.186989
\(307\) −8.17920e8 −1.61334 −0.806671 0.591001i \(-0.798733\pi\)
−0.806671 + 0.591001i \(0.798733\pi\)
\(308\) 1.64765e8 0.321321
\(309\) 3.93746e8 0.759209
\(310\) 6.27214e8 1.19578
\(311\) 8.61446e8 1.62393 0.811964 0.583708i \(-0.198399\pi\)
0.811964 + 0.583708i \(0.198399\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) 2.15722e8 0.397640 0.198820 0.980036i \(-0.436289\pi\)
0.198820 + 0.980036i \(0.436289\pi\)
\(314\) −2.54686e8 −0.464250
\(315\) 8.90614e7 0.160547
\(316\) −5.08575e8 −0.906673
\(317\) 1.12172e8 0.197778 0.0988890 0.995098i \(-0.468471\pi\)
0.0988890 + 0.995098i \(0.468471\pi\)
\(318\) 3.72921e8 0.650312
\(319\) 1.26231e9 2.17720
\(320\) −9.33701e7 −0.159288
\(321\) 5.10588e7 0.0861595
\(322\) 8.92743e7 0.149015
\(323\) −4.46098e7 −0.0736582
\(324\) 3.40122e7 0.0555556
\(325\) 1.07078e8 0.173025
\(326\) −7.45688e8 −1.19205
\(327\) −2.49403e8 −0.394443
\(328\) −4.21999e8 −0.660317
\(329\) 2.94564e7 0.0456031
\(330\) −5.77449e8 −0.884535
\(331\) 5.07956e8 0.769889 0.384945 0.922940i \(-0.374221\pi\)
0.384945 + 0.922940i \(0.374221\pi\)
\(332\) −3.52539e8 −0.528718
\(333\) −1.53877e8 −0.228359
\(334\) 3.74547e8 0.550039
\(335\) −4.53604e8 −0.659204
\(336\) 3.79331e7 0.0545545
\(337\) −8.81685e8 −1.25490 −0.627450 0.778657i \(-0.715901\pi\)
−0.627450 + 0.778657i \(0.715901\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) −3.68803e8 −0.514157
\(340\) 3.66330e8 0.505471
\(341\) 1.65215e9 2.25636
\(342\) 1.61891e7 0.0218843
\(343\) −4.03536e7 −0.0539949
\(344\) 3.27698e8 0.434029
\(345\) −3.12877e8 −0.410211
\(346\) 4.67709e8 0.607029
\(347\) 3.23488e8 0.415629 0.207814 0.978168i \(-0.433365\pi\)
0.207814 + 0.978168i \(0.433365\pi\)
\(348\) 2.90615e8 0.369649
\(349\) 1.46992e9 1.85100 0.925499 0.378750i \(-0.123646\pi\)
0.925499 + 0.378750i \(0.123646\pi\)
\(350\) −1.33738e8 −0.166731
\(351\) −4.32436e7 −0.0533761
\(352\) −2.45947e8 −0.300568
\(353\) 1.28684e9 1.55709 0.778543 0.627591i \(-0.215959\pi\)
0.778543 + 0.627591i \(0.215959\pi\)
\(354\) −6.08601e8 −0.729157
\(355\) 9.44162e8 1.12008
\(356\) 2.84201e8 0.333849
\(357\) −1.48827e8 −0.173118
\(358\) −8.37630e8 −0.964854
\(359\) −1.45809e9 −1.66323 −0.831616 0.555351i \(-0.812584\pi\)
−0.831616 + 0.555351i \(0.812584\pi\)
\(360\) −1.32943e8 −0.150178
\(361\) −8.86166e8 −0.991379
\(362\) 4.47350e8 0.495641
\(363\) −9.94912e8 −1.09172
\(364\) −4.82285e7 −0.0524142
\(365\) −1.49119e9 −1.60512
\(366\) 2.34064e8 0.249546
\(367\) −8.89101e8 −0.938901 −0.469451 0.882959i \(-0.655548\pi\)
−0.469451 + 0.882959i \(0.655548\pi\)
\(368\) −1.33261e8 −0.139391
\(369\) −6.00854e8 −0.622553
\(370\) 6.01456e8 0.617302
\(371\) 5.92184e8 0.602072
\(372\) 3.80365e8 0.383090
\(373\) −2.98628e8 −0.297954 −0.148977 0.988841i \(-0.547598\pi\)
−0.148977 + 0.988841i \(0.547598\pi\)
\(374\) 9.64953e8 0.953796
\(375\) −2.82606e8 −0.276740
\(376\) −4.39699e7 −0.0426578
\(377\) −3.69491e8 −0.355148
\(378\) 5.40102e7 0.0514344
\(379\) −7.55911e8 −0.713236 −0.356618 0.934250i \(-0.616070\pi\)
−0.356618 + 0.934250i \(0.616070\pi\)
\(380\) −6.32782e7 −0.0591577
\(381\) −9.20874e8 −0.853027
\(382\) −4.23764e8 −0.388957
\(383\) 1.16913e9 1.06333 0.531664 0.846955i \(-0.321567\pi\)
0.531664 + 0.846955i \(0.321567\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −9.16968e8 −0.818921
\(386\) 1.17575e9 1.04054
\(387\) 4.66585e8 0.409206
\(388\) 3.82528e8 0.332470
\(389\) 3.70209e8 0.318878 0.159439 0.987208i \(-0.449032\pi\)
0.159439 + 0.987208i \(0.449032\pi\)
\(390\) 1.69025e8 0.144287
\(391\) 5.22837e8 0.442331
\(392\) 6.02363e7 0.0505076
\(393\) −2.00772e8 −0.166851
\(394\) −6.78192e8 −0.558619
\(395\) 2.83037e9 2.31075
\(396\) −3.50187e8 −0.283378
\(397\) −3.90037e8 −0.312852 −0.156426 0.987690i \(-0.549997\pi\)
−0.156426 + 0.987690i \(0.549997\pi\)
\(398\) −1.46572e9 −1.16536
\(399\) 2.57077e7 0.0202609
\(400\) 1.99632e8 0.155963
\(401\) 9.42418e8 0.729858 0.364929 0.931035i \(-0.381093\pi\)
0.364929 + 0.931035i \(0.381093\pi\)
\(402\) −2.75082e8 −0.211189
\(403\) −4.83601e8 −0.368061
\(404\) −1.82065e8 −0.137370
\(405\) −1.89288e8 −0.141589
\(406\) 4.61485e8 0.342229
\(407\) 1.58430e9 1.16482
\(408\) 2.22156e8 0.161937
\(409\) −1.69995e9 −1.22858 −0.614290 0.789080i \(-0.710557\pi\)
−0.614290 + 0.789080i \(0.710557\pi\)
\(410\) 2.34855e9 1.68289
\(411\) −3.67833e8 −0.261339
\(412\) −9.33324e8 −0.657494
\(413\) −9.66436e8 −0.675068
\(414\) −1.89740e8 −0.131419
\(415\) 1.96198e9 1.34749
\(416\) 7.19913e7 0.0490290
\(417\) −1.11689e9 −0.754281
\(418\) −1.66682e8 −0.111627
\(419\) 8.67590e8 0.576190 0.288095 0.957602i \(-0.406978\pi\)
0.288095 + 0.957602i \(0.406978\pi\)
\(420\) −2.11109e8 −0.139038
\(421\) −2.13921e9 −1.39723 −0.698614 0.715499i \(-0.746199\pi\)
−0.698614 + 0.715499i \(0.746199\pi\)
\(422\) 1.32449e9 0.857939
\(423\) −6.26056e7 −0.0402181
\(424\) −8.83960e8 −0.563187
\(425\) −7.83240e8 −0.494919
\(426\) 5.72575e8 0.358838
\(427\) 3.71685e8 0.231035
\(428\) −1.21028e8 −0.0746163
\(429\) 4.45232e8 0.272261
\(430\) −1.82373e9 −1.10617
\(431\) −1.26935e9 −0.763677 −0.381838 0.924229i \(-0.624709\pi\)
−0.381838 + 0.924229i \(0.624709\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 1.77878e9 1.05297 0.526483 0.850186i \(-0.323510\pi\)
0.526483 + 0.850186i \(0.323510\pi\)
\(434\) 6.04006e8 0.354672
\(435\) −1.61736e9 −0.942092
\(436\) 5.91177e8 0.341598
\(437\) −9.03126e7 −0.0517682
\(438\) −9.04312e8 −0.514232
\(439\) −1.14147e9 −0.643932 −0.321966 0.946751i \(-0.604344\pi\)
−0.321966 + 0.946751i \(0.604344\pi\)
\(440\) 1.36877e9 0.766030
\(441\) 8.57661e7 0.0476190
\(442\) −2.82452e8 −0.155584
\(443\) −9.84229e8 −0.537877 −0.268939 0.963157i \(-0.586673\pi\)
−0.268939 + 0.963157i \(0.586673\pi\)
\(444\) 3.64745e8 0.197765
\(445\) −1.58166e9 −0.850851
\(446\) 2.33939e9 1.24862
\(447\) 1.94685e9 1.03100
\(448\) −8.99154e7 −0.0472456
\(449\) 7.14366e8 0.372442 0.186221 0.982508i \(-0.440376\pi\)
0.186221 + 0.982508i \(0.440376\pi\)
\(450\) 2.84242e8 0.147043
\(451\) 6.18633e9 3.17552
\(452\) 8.74201e8 0.445273
\(453\) −4.54155e7 −0.0229541
\(454\) 2.30498e9 1.15604
\(455\) 2.68406e8 0.133583
\(456\) −3.83742e7 −0.0189523
\(457\) −5.28807e8 −0.259173 −0.129587 0.991568i \(-0.541365\pi\)
−0.129587 + 0.991568i \(0.541365\pi\)
\(458\) −6.20158e8 −0.301629
\(459\) 3.16312e8 0.152676
\(460\) 7.41635e8 0.355253
\(461\) 1.12605e9 0.535311 0.267655 0.963515i \(-0.413751\pi\)
0.267655 + 0.963515i \(0.413751\pi\)
\(462\) −5.56083e8 −0.262357
\(463\) −1.93722e9 −0.907078 −0.453539 0.891236i \(-0.649839\pi\)
−0.453539 + 0.891236i \(0.649839\pi\)
\(464\) −6.88864e8 −0.320126
\(465\) −2.11685e9 −0.976346
\(466\) 2.20726e9 1.01042
\(467\) −3.55991e9 −1.61745 −0.808723 0.588190i \(-0.799841\pi\)
−0.808723 + 0.588190i \(0.799841\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) −4.36820e8 −0.195523
\(470\) 2.44705e8 0.108718
\(471\) 8.59566e8 0.379058
\(472\) 1.44261e9 0.631469
\(473\) −4.80392e9 −2.08728
\(474\) 1.71644e9 0.740295
\(475\) 1.35294e8 0.0579228
\(476\) 3.52775e8 0.149925
\(477\) −1.25861e9 −0.530977
\(478\) −1.71457e9 −0.718053
\(479\) 8.33162e8 0.346382 0.173191 0.984888i \(-0.444592\pi\)
0.173191 + 0.984888i \(0.444592\pi\)
\(480\) 3.15124e8 0.130058
\(481\) −4.63742e8 −0.190006
\(482\) 5.29075e8 0.215205
\(483\) −3.01301e8 −0.121670
\(484\) 2.35831e9 0.945458
\(485\) −2.12888e9 −0.847336
\(486\) −1.14791e8 −0.0453609
\(487\) 8.53756e8 0.334952 0.167476 0.985876i \(-0.446438\pi\)
0.167476 + 0.985876i \(0.446438\pi\)
\(488\) −5.54819e8 −0.216113
\(489\) 2.51670e9 0.973307
\(490\) −3.35233e8 −0.128724
\(491\) −4.44043e9 −1.69293 −0.846466 0.532443i \(-0.821274\pi\)
−0.846466 + 0.532443i \(0.821274\pi\)
\(492\) 1.42425e9 0.539147
\(493\) 2.70270e9 1.01586
\(494\) 4.87895e7 0.0182088
\(495\) 1.94889e9 0.722220
\(496\) −9.01607e8 −0.331766
\(497\) 9.09227e8 0.332220
\(498\) 1.18982e9 0.431696
\(499\) −1.71385e8 −0.0617476 −0.0308738 0.999523i \(-0.509829\pi\)
−0.0308738 + 0.999523i \(0.509829\pi\)
\(500\) 6.69881e8 0.239664
\(501\) −1.26410e9 −0.449105
\(502\) 2.94736e9 1.03985
\(503\) −1.62786e9 −0.570332 −0.285166 0.958478i \(-0.592049\pi\)
−0.285166 + 0.958478i \(0.592049\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 1.01325e9 0.350102
\(506\) 1.95355e9 0.670344
\(507\) −1.30324e8 −0.0444116
\(508\) 2.18281e9 0.738743
\(509\) 4.20928e9 1.41480 0.707401 0.706812i \(-0.249867\pi\)
0.707401 + 0.706812i \(0.249867\pi\)
\(510\) −1.23636e9 −0.412715
\(511\) −1.43601e9 −0.476086
\(512\) 1.34218e8 0.0441942
\(513\) −5.46383e7 −0.0178684
\(514\) −2.08359e9 −0.676769
\(515\) 5.19422e9 1.67570
\(516\) −1.10598e9 −0.354383
\(517\) 6.44581e8 0.205145
\(518\) 5.79202e8 0.183095
\(519\) −1.57852e9 −0.495637
\(520\) −4.00653e8 −0.124956
\(521\) −7.80158e8 −0.241685 −0.120843 0.992672i \(-0.538560\pi\)
−0.120843 + 0.992672i \(0.538560\pi\)
\(522\) −9.80824e8 −0.301817
\(523\) 2.32391e9 0.710335 0.355168 0.934803i \(-0.384424\pi\)
0.355168 + 0.934803i \(0.384424\pi\)
\(524\) 4.75903e8 0.144497
\(525\) 4.51366e8 0.136135
\(526\) 1.44924e9 0.434200
\(527\) 3.53738e9 1.05280
\(528\) 8.30072e8 0.245413
\(529\) −2.34634e9 −0.689122
\(530\) 4.91950e9 1.43534
\(531\) 2.05403e9 0.595354
\(532\) −6.09369e7 −0.0175465
\(533\) −1.81080e9 −0.517996
\(534\) −9.59178e8 −0.272587
\(535\) 6.73558e8 0.190168
\(536\) 6.52047e8 0.182895
\(537\) 2.82700e9 0.787800
\(538\) −4.00037e9 −1.10755
\(539\) −8.83040e8 −0.242896
\(540\) 4.48683e8 0.122620
\(541\) 4.54066e9 1.23290 0.616451 0.787393i \(-0.288570\pi\)
0.616451 + 0.787393i \(0.288570\pi\)
\(542\) −6.75357e8 −0.182195
\(543\) −1.50980e9 −0.404689
\(544\) −5.26592e8 −0.140242
\(545\) −3.29007e9 −0.870598
\(546\) 1.62771e8 0.0427960
\(547\) 4.44342e9 1.16081 0.580405 0.814328i \(-0.302894\pi\)
0.580405 + 0.814328i \(0.302894\pi\)
\(548\) 8.71900e8 0.226326
\(549\) −7.89966e8 −0.203754
\(550\) −2.92653e9 −0.750039
\(551\) −4.66852e8 −0.118891
\(552\) 4.49755e8 0.113812
\(553\) 2.72565e9 0.685380
\(554\) −4.04531e9 −1.01081
\(555\) −2.02991e9 −0.504025
\(556\) 2.64744e9 0.653227
\(557\) 6.67789e8 0.163737 0.0818684 0.996643i \(-0.473911\pi\)
0.0818684 + 0.996643i \(0.473911\pi\)
\(558\) −1.28373e9 −0.312792
\(559\) 1.40616e9 0.340480
\(560\) 5.00406e8 0.120410
\(561\) −3.25672e9 −0.778771
\(562\) −1.00359e9 −0.238495
\(563\) −4.39812e9 −1.03870 −0.519348 0.854563i \(-0.673825\pi\)
−0.519348 + 0.854563i \(0.673825\pi\)
\(564\) 1.48398e8 0.0348299
\(565\) −4.86518e9 −1.13483
\(566\) −1.21568e9 −0.281813
\(567\) −1.82284e8 −0.0419961
\(568\) −1.35721e9 −0.310763
\(569\) −7.16581e9 −1.63069 −0.815347 0.578972i \(-0.803454\pi\)
−0.815347 + 0.578972i \(0.803454\pi\)
\(570\) 2.13564e8 0.0483021
\(571\) −1.98106e9 −0.445319 −0.222660 0.974896i \(-0.571474\pi\)
−0.222660 + 0.974896i \(0.571474\pi\)
\(572\) −1.05536e9 −0.235785
\(573\) 1.43020e9 0.317582
\(574\) 2.26165e9 0.499153
\(575\) −1.58567e9 −0.347837
\(576\) 1.91103e8 0.0416667
\(577\) −7.86735e9 −1.70496 −0.852478 0.522763i \(-0.824901\pi\)
−0.852478 + 0.522763i \(0.824901\pi\)
\(578\) −1.21667e9 −0.262075
\(579\) −3.96816e9 −0.849599
\(580\) 3.83373e9 0.815875
\(581\) 1.88939e9 0.399673
\(582\) −1.29103e9 −0.271461
\(583\) 1.29585e10 2.70841
\(584\) 2.14355e9 0.445338
\(585\) −5.70461e8 −0.117809
\(586\) 3.46933e8 0.0712204
\(587\) −1.86121e9 −0.379806 −0.189903 0.981803i \(-0.560817\pi\)
−0.189903 + 0.981803i \(0.560817\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −6.11031e8 −0.123214
\(590\) −8.02855e9 −1.60937
\(591\) 2.28890e9 0.456111
\(592\) −8.64581e8 −0.171269
\(593\) 9.71905e8 0.191396 0.0956979 0.995410i \(-0.469492\pi\)
0.0956979 + 0.995410i \(0.469492\pi\)
\(594\) 1.18188e9 0.231377
\(595\) −1.96330e9 −0.382100
\(596\) −4.61476e9 −0.892868
\(597\) 4.94680e9 0.951512
\(598\) −5.71824e8 −0.109347
\(599\) −7.80541e9 −1.48389 −0.741945 0.670461i \(-0.766096\pi\)
−0.741945 + 0.670461i \(0.766096\pi\)
\(600\) −6.73759e8 −0.127343
\(601\) −6.38951e9 −1.20062 −0.600312 0.799766i \(-0.704957\pi\)
−0.600312 + 0.799766i \(0.704957\pi\)
\(602\) −1.75625e9 −0.328095
\(603\) 9.28403e8 0.172435
\(604\) 1.07652e8 0.0198788
\(605\) −1.31247e10 −2.40960
\(606\) 6.14470e8 0.112162
\(607\) −3.76538e9 −0.683359 −0.341680 0.939817i \(-0.610996\pi\)
−0.341680 + 0.939817i \(0.610996\pi\)
\(608\) 9.09611e7 0.0164132
\(609\) −1.55751e9 −0.279429
\(610\) 3.08773e9 0.550788
\(611\) −1.88675e8 −0.0334635
\(612\) −7.49776e8 −0.132221
\(613\) 4.01054e9 0.703219 0.351610 0.936147i \(-0.385634\pi\)
0.351610 + 0.936147i \(0.385634\pi\)
\(614\) −6.54336e9 −1.14080
\(615\) −7.92635e9 −1.37407
\(616\) 1.31812e9 0.227208
\(617\) 1.41032e9 0.241724 0.120862 0.992669i \(-0.461434\pi\)
0.120862 + 0.992669i \(0.461434\pi\)
\(618\) 3.14997e9 0.536842
\(619\) −9.58969e9 −1.62513 −0.812563 0.582873i \(-0.801929\pi\)
−0.812563 + 0.582873i \(0.801929\pi\)
\(620\) 5.01771e9 0.845541
\(621\) 6.40374e8 0.107303
\(622\) 6.89157e9 1.14829
\(623\) −1.52314e9 −0.252366
\(624\) −2.42971e8 −0.0400320
\(625\) −7.53577e9 −1.23466
\(626\) 1.72578e9 0.281174
\(627\) 5.62551e8 0.0911435
\(628\) −2.03749e9 −0.328274
\(629\) 3.39211e9 0.543492
\(630\) 7.12492e8 0.113524
\(631\) −7.67583e9 −1.21625 −0.608125 0.793842i \(-0.708078\pi\)
−0.608125 + 0.793842i \(0.708078\pi\)
\(632\) −4.06860e9 −0.641114
\(633\) −4.47017e9 −0.700504
\(634\) 8.97378e8 0.139850
\(635\) −1.21480e10 −1.88277
\(636\) 2.98337e9 0.459840
\(637\) 2.58475e8 0.0396214
\(638\) 1.00985e10 1.53951
\(639\) −1.93244e9 −0.292990
\(640\) −7.46961e8 −0.112634
\(641\) −6.28456e9 −0.942480 −0.471240 0.882005i \(-0.656193\pi\)
−0.471240 + 0.882005i \(0.656193\pi\)
\(642\) 4.08471e8 0.0609240
\(643\) 2.44480e9 0.362665 0.181333 0.983422i \(-0.441959\pi\)
0.181333 + 0.983422i \(0.441959\pi\)
\(644\) 7.14194e8 0.105370
\(645\) 6.15510e9 0.903184
\(646\) −3.56878e8 −0.0520842
\(647\) 7.83211e7 0.0113688 0.00568439 0.999984i \(-0.498191\pi\)
0.00568439 + 0.999984i \(0.498191\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −2.11481e10 −3.03679
\(650\) 8.56625e8 0.122347
\(651\) −2.03852e9 −0.289589
\(652\) −5.96550e9 −0.842909
\(653\) −1.02149e10 −1.43562 −0.717811 0.696238i \(-0.754856\pi\)
−0.717811 + 0.696238i \(0.754856\pi\)
\(654\) −1.99522e9 −0.278913
\(655\) −2.64854e9 −0.368266
\(656\) −3.37599e9 −0.466915
\(657\) 3.05205e9 0.419869
\(658\) 2.35651e8 0.0322462
\(659\) −4.57175e9 −0.622276 −0.311138 0.950365i \(-0.600710\pi\)
−0.311138 + 0.950365i \(0.600710\pi\)
\(660\) −4.61960e9 −0.625461
\(661\) 1.10823e10 1.49253 0.746266 0.665648i \(-0.231845\pi\)
0.746266 + 0.665648i \(0.231845\pi\)
\(662\) 4.06365e9 0.544394
\(663\) 9.53274e8 0.127034
\(664\) −2.82031e9 −0.373860
\(665\) 3.39132e8 0.0447190
\(666\) −1.23102e9 −0.161474
\(667\) 5.47162e9 0.713962
\(668\) 2.99638e9 0.388937
\(669\) −7.89545e9 −1.01950
\(670\) −3.62883e9 −0.466128
\(671\) 8.13342e9 1.03931
\(672\) 3.03464e8 0.0385758
\(673\) 1.22126e9 0.154438 0.0772190 0.997014i \(-0.475396\pi\)
0.0772190 + 0.997014i \(0.475396\pi\)
\(674\) −7.05348e9 −0.887348
\(675\) −9.59317e8 −0.120060
\(676\) 3.08916e8 0.0384615
\(677\) 1.31735e9 0.163170 0.0815848 0.996666i \(-0.474002\pi\)
0.0815848 + 0.996666i \(0.474002\pi\)
\(678\) −2.95043e9 −0.363564
\(679\) −2.05011e9 −0.251324
\(680\) 2.93064e9 0.357422
\(681\) −7.77930e9 −0.943899
\(682\) 1.32172e10 1.59549
\(683\) −3.30209e9 −0.396567 −0.198284 0.980145i \(-0.563537\pi\)
−0.198284 + 0.980145i \(0.563537\pi\)
\(684\) 1.29513e8 0.0154745
\(685\) −4.85238e9 −0.576817
\(686\) −3.22829e8 −0.0381802
\(687\) 2.09303e9 0.246279
\(688\) 2.62158e9 0.306905
\(689\) −3.79309e9 −0.441800
\(690\) −2.50302e9 −0.290063
\(691\) −1.58840e10 −1.83141 −0.915706 0.401849i \(-0.868368\pi\)
−0.915706 + 0.401849i \(0.868368\pi\)
\(692\) 3.74168e9 0.429235
\(693\) 1.87678e9 0.214214
\(694\) 2.58791e9 0.293894
\(695\) −1.47338e10 −1.66482
\(696\) 2.32492e9 0.261382
\(697\) 1.32454e10 1.48167
\(698\) 1.17594e10 1.30885
\(699\) −7.44949e9 −0.825005
\(700\) −1.06990e9 −0.117897
\(701\) 1.08169e10 1.18601 0.593005 0.805199i \(-0.297942\pi\)
0.593005 + 0.805199i \(0.297942\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) −5.85938e8 −0.0636075
\(704\) −1.96758e9 −0.212534
\(705\) −8.25881e8 −0.0887678
\(706\) 1.02947e10 1.10103
\(707\) 9.75755e8 0.103842
\(708\) −4.86881e9 −0.515592
\(709\) 1.11238e10 1.17217 0.586085 0.810250i \(-0.300669\pi\)
0.586085 + 0.810250i \(0.300669\pi\)
\(710\) 7.55330e9 0.792013
\(711\) −5.79299e9 −0.604448
\(712\) 2.27361e9 0.236067
\(713\) 7.16143e9 0.739922
\(714\) −1.19062e9 −0.122413
\(715\) 5.87341e9 0.600923
\(716\) −6.70104e9 −0.682255
\(717\) 5.78666e9 0.586288
\(718\) −1.16647e10 −1.17608
\(719\) 5.88745e9 0.590713 0.295356 0.955387i \(-0.404562\pi\)
0.295356 + 0.955387i \(0.404562\pi\)
\(720\) −1.06354e9 −0.106192
\(721\) 5.00203e9 0.497019
\(722\) −7.08933e9 −0.701011
\(723\) −1.78563e9 −0.175714
\(724\) 3.57880e9 0.350471
\(725\) −8.19680e9 −0.798843
\(726\) −7.95930e9 −0.771963
\(727\) 3.66339e9 0.353601 0.176800 0.984247i \(-0.443425\pi\)
0.176800 + 0.984247i \(0.443425\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −1.19295e10 −1.13499
\(731\) −1.02855e10 −0.973905
\(732\) 1.87251e9 0.176456
\(733\) 1.21576e10 1.14020 0.570102 0.821574i \(-0.306904\pi\)
0.570102 + 0.821574i \(0.306904\pi\)
\(734\) −7.11281e9 −0.663904
\(735\) 1.13141e9 0.105103
\(736\) −1.06609e9 −0.0985643
\(737\) −9.55875e9 −0.879558
\(738\) −4.80683e9 −0.440212
\(739\) −1.03201e10 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(740\) 4.81165e9 0.436499
\(741\) −1.64664e8 −0.0148674
\(742\) 4.73748e9 0.425729
\(743\) −9.76148e9 −0.873081 −0.436541 0.899685i \(-0.643796\pi\)
−0.436541 + 0.899685i \(0.643796\pi\)
\(744\) 3.04292e9 0.270886
\(745\) 2.56825e10 2.27557
\(746\) −2.38902e9 −0.210685
\(747\) −4.01564e9 −0.352478
\(748\) 7.71962e9 0.674436
\(749\) 6.48636e8 0.0564046
\(750\) −2.26085e9 −0.195685
\(751\) −3.56446e9 −0.307082 −0.153541 0.988142i \(-0.549068\pi\)
−0.153541 + 0.988142i \(0.549068\pi\)
\(752\) −3.51759e8 −0.0301636
\(753\) −9.94735e9 −0.849034
\(754\) −2.95593e9 −0.251127
\(755\) −5.99112e8 −0.0506634
\(756\) 4.32081e8 0.0363696
\(757\) 6.66060e9 0.558056 0.279028 0.960283i \(-0.409988\pi\)
0.279028 + 0.960283i \(0.409988\pi\)
\(758\) −6.04729e9 −0.504334
\(759\) −6.59323e9 −0.547333
\(760\) −5.06226e8 −0.0418308
\(761\) 1.16411e10 0.957520 0.478760 0.877946i \(-0.341086\pi\)
0.478760 + 0.877946i \(0.341086\pi\)
\(762\) −7.36699e9 −0.603181
\(763\) −3.16834e9 −0.258223
\(764\) −3.39011e9 −0.275034
\(765\) 4.17272e9 0.336980
\(766\) 9.35305e9 0.751887
\(767\) 6.19026e9 0.495365
\(768\) −4.52985e8 −0.0360844
\(769\) 6.06456e9 0.480903 0.240451 0.970661i \(-0.422705\pi\)
0.240451 + 0.970661i \(0.422705\pi\)
\(770\) −7.33575e9 −0.579064
\(771\) 7.03211e9 0.552580
\(772\) 9.40600e9 0.735774
\(773\) −4.87535e9 −0.379645 −0.189822 0.981818i \(-0.560791\pi\)
−0.189822 + 0.981818i \(0.560791\pi\)
\(774\) 3.73268e9 0.289353
\(775\) −1.07282e10 −0.827889
\(776\) 3.06023e9 0.235092
\(777\) −1.95481e9 −0.149496
\(778\) 2.96168e9 0.225480
\(779\) −2.28795e9 −0.173407
\(780\) 1.35220e9 0.102026
\(781\) 1.98962e10 1.49449
\(782\) 4.18270e9 0.312775
\(783\) 3.31028e9 0.246433
\(784\) 4.81890e8 0.0357143
\(785\) 1.13392e10 0.836642
\(786\) −1.60617e9 −0.117981
\(787\) 1.35287e9 0.0989336 0.0494668 0.998776i \(-0.484248\pi\)
0.0494668 + 0.998776i \(0.484248\pi\)
\(788\) −5.42554e9 −0.395003
\(789\) −4.89119e9 −0.354523
\(790\) 2.26430e10 1.63395
\(791\) −4.68517e9 −0.336595
\(792\) −2.80149e9 −0.200379
\(793\) −2.38074e9 −0.169533
\(794\) −3.12030e9 −0.221220
\(795\) −1.66033e10 −1.17195
\(796\) −1.17257e10 −0.824033
\(797\) 1.47844e10 1.03443 0.517214 0.855856i \(-0.326969\pi\)
0.517214 + 0.855856i \(0.326969\pi\)
\(798\) 2.05662e8 0.0143266
\(799\) 1.38010e9 0.0957185
\(800\) 1.59706e9 0.110282
\(801\) 3.23722e9 0.222566
\(802\) 7.53935e9 0.516087
\(803\) −3.14237e10 −2.14167
\(804\) −2.20066e9 −0.149333
\(805\) −3.97470e9 −0.268546
\(806\) −3.86881e9 −0.260258
\(807\) 1.35013e10 0.904309
\(808\) −1.45652e9 −0.0971353
\(809\) −2.00980e8 −0.0133455 −0.00667273 0.999978i \(-0.502124\pi\)
−0.00667273 + 0.999978i \(0.502124\pi\)
\(810\) −1.51430e9 −0.100119
\(811\) 1.02811e10 0.676809 0.338405 0.941001i \(-0.390113\pi\)
0.338405 + 0.941001i \(0.390113\pi\)
\(812\) 3.69188e9 0.241992
\(813\) 2.27933e9 0.148761
\(814\) 1.26744e10 0.823650
\(815\) 3.31998e10 2.14824
\(816\) 1.77725e9 0.114507
\(817\) 1.77668e9 0.113981
\(818\) −1.35996e10 −0.868737
\(819\) −5.49353e8 −0.0349428
\(820\) 1.87884e10 1.18998
\(821\) −1.08015e9 −0.0681213 −0.0340607 0.999420i \(-0.510844\pi\)
−0.0340607 + 0.999420i \(0.510844\pi\)
\(822\) −2.94266e9 −0.184795
\(823\) −8.00736e9 −0.500714 −0.250357 0.968154i \(-0.580548\pi\)
−0.250357 + 0.968154i \(0.580548\pi\)
\(824\) −7.46659e9 −0.464919
\(825\) 9.87704e9 0.612404
\(826\) −7.73149e9 −0.477345
\(827\) 1.60107e10 0.984330 0.492165 0.870502i \(-0.336206\pi\)
0.492165 + 0.870502i \(0.336206\pi\)
\(828\) −1.51792e9 −0.0929273
\(829\) −1.70468e10 −1.03920 −0.519602 0.854408i \(-0.673920\pi\)
−0.519602 + 0.854408i \(0.673920\pi\)
\(830\) 1.56959e10 0.952822
\(831\) 1.36529e10 0.825320
\(832\) 5.75930e8 0.0346688
\(833\) −1.89066e9 −0.113333
\(834\) −8.93510e9 −0.533357
\(835\) −1.66757e10 −0.991247
\(836\) −1.33345e9 −0.0789325
\(837\) 4.33260e9 0.255393
\(838\) 6.94072e9 0.407428
\(839\) 1.28512e10 0.751237 0.375618 0.926774i \(-0.377430\pi\)
0.375618 + 0.926774i \(0.377430\pi\)
\(840\) −1.68887e9 −0.0983147
\(841\) 1.10345e10 0.639688
\(842\) −1.71137e10 −0.987989
\(843\) 3.38712e9 0.194730
\(844\) 1.05960e10 0.606655
\(845\) −1.71921e9 −0.0980234
\(846\) −5.00845e8 −0.0284385
\(847\) −1.26391e10 −0.714699
\(848\) −7.07168e9 −0.398233
\(849\) 4.10292e9 0.230100
\(850\) −6.26592e9 −0.349960
\(851\) 6.86733e9 0.381975
\(852\) 4.58060e9 0.253737
\(853\) −4.40731e9 −0.243137 −0.121569 0.992583i \(-0.538792\pi\)
−0.121569 + 0.992583i \(0.538792\pi\)
\(854\) 2.97348e9 0.163366
\(855\) −7.20778e8 −0.0394385
\(856\) −9.68226e8 −0.0527617
\(857\) −1.04953e10 −0.569587 −0.284794 0.958589i \(-0.591925\pi\)
−0.284794 + 0.958589i \(0.591925\pi\)
\(858\) 3.56185e9 0.192518
\(859\) 3.04414e10 1.63866 0.819328 0.573325i \(-0.194347\pi\)
0.819328 + 0.573325i \(0.194347\pi\)
\(860\) −1.45899e10 −0.782180
\(861\) −7.63307e9 −0.407557
\(862\) −1.01548e10 −0.540001
\(863\) −1.44946e10 −0.767659 −0.383829 0.923404i \(-0.625395\pi\)
−0.383829 + 0.923404i \(0.625395\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −2.08235e10 −1.09395
\(866\) 1.42302e10 0.744559
\(867\) 4.10627e9 0.213984
\(868\) 4.83205e9 0.250791
\(869\) 5.96441e10 3.08318
\(870\) −1.29388e10 −0.666159
\(871\) 2.79794e9 0.143475
\(872\) 4.72942e9 0.241546
\(873\) 4.35723e9 0.221647
\(874\) −7.22501e8 −0.0366057
\(875\) −3.59015e9 −0.181169
\(876\) −7.23450e9 −0.363617
\(877\) 1.93578e10 0.969074 0.484537 0.874771i \(-0.338988\pi\)
0.484537 + 0.874771i \(0.338988\pi\)
\(878\) −9.13179e9 −0.455329
\(879\) −1.17090e9 −0.0581512
\(880\) 1.09502e10 0.541665
\(881\) −7.70488e9 −0.379621 −0.189811 0.981821i \(-0.560787\pi\)
−0.189811 + 0.981821i \(0.560787\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 7.96120e9 0.389149 0.194574 0.980888i \(-0.437667\pi\)
0.194574 + 0.980888i \(0.437667\pi\)
\(884\) −2.25961e9 −0.110015
\(885\) 2.70963e10 1.31404
\(886\) −7.87383e9 −0.380337
\(887\) −3.45970e10 −1.66459 −0.832293 0.554336i \(-0.812972\pi\)
−0.832293 + 0.554336i \(0.812972\pi\)
\(888\) 2.91796e9 0.139841
\(889\) −1.16985e10 −0.558437
\(890\) −1.26533e10 −0.601642
\(891\) −3.98884e9 −0.188919
\(892\) 1.87151e10 0.882909
\(893\) −2.38392e8 −0.0112024
\(894\) 1.55748e10 0.729024
\(895\) 3.72933e10 1.73880
\(896\) −7.19323e8 −0.0334077
\(897\) 1.92991e9 0.0892817
\(898\) 5.71493e9 0.263356
\(899\) 3.70195e10 1.69931
\(900\) 2.27394e9 0.103975
\(901\) 2.77451e10 1.26372
\(902\) 4.94907e10 2.24543
\(903\) 5.92736e9 0.267888
\(904\) 6.99361e9 0.314856
\(905\) −1.99171e10 −0.893213
\(906\) −3.63324e8 −0.0162310
\(907\) −1.75994e9 −0.0783197 −0.0391599 0.999233i \(-0.512468\pi\)
−0.0391599 + 0.999233i \(0.512468\pi\)
\(908\) 1.84398e10 0.817441
\(909\) −2.07384e9 −0.0915800
\(910\) 2.14725e9 0.0944577
\(911\) −2.98095e10 −1.30629 −0.653146 0.757232i \(-0.726551\pi\)
−0.653146 + 0.757232i \(0.726551\pi\)
\(912\) −3.06994e8 −0.0134013
\(913\) 4.13446e10 1.79792
\(914\) −4.23045e9 −0.183263
\(915\) −1.04211e10 −0.449717
\(916\) −4.96126e9 −0.213284
\(917\) −2.55054e9 −0.109229
\(918\) 2.53049e9 0.107958
\(919\) 3.27566e10 1.39218 0.696089 0.717956i \(-0.254922\pi\)
0.696089 + 0.717956i \(0.254922\pi\)
\(920\) 5.93308e9 0.251202
\(921\) 2.20838e10 0.931463
\(922\) 9.00843e9 0.378522
\(923\) −5.82383e9 −0.243782
\(924\) −4.44867e9 −0.185515
\(925\) −1.02877e10 −0.427387
\(926\) −1.54977e10 −0.641401
\(927\) −1.06311e10 −0.438330
\(928\) −5.51091e9 −0.226363
\(929\) −4.44117e10 −1.81737 −0.908683 0.417486i \(-0.862911\pi\)
−0.908683 + 0.417486i \(0.862911\pi\)
\(930\) −1.69348e10 −0.690381
\(931\) 3.26583e8 0.0132639
\(932\) 1.76581e10 0.714475
\(933\) −2.32590e10 −0.937575
\(934\) −2.84793e10 −1.14371
\(935\) −4.29620e10 −1.71887
\(936\) 8.20026e8 0.0326860
\(937\) −2.70253e10 −1.07320 −0.536602 0.843835i \(-0.680292\pi\)
−0.536602 + 0.843835i \(0.680292\pi\)
\(938\) −3.49456e9 −0.138256
\(939\) −5.82451e9 −0.229578
\(940\) 1.95764e9 0.0768752
\(941\) −3.12862e10 −1.22402 −0.612010 0.790850i \(-0.709639\pi\)
−0.612010 + 0.790850i \(0.709639\pi\)
\(942\) 6.87653e9 0.268035
\(943\) 2.68153e10 1.04134
\(944\) 1.15409e10 0.446516
\(945\) −2.40466e9 −0.0926920
\(946\) −3.84313e10 −1.47593
\(947\) 3.80554e10 1.45610 0.728051 0.685523i \(-0.240426\pi\)
0.728051 + 0.685523i \(0.240426\pi\)
\(948\) 1.37315e10 0.523468
\(949\) 9.19803e9 0.349352
\(950\) 1.08235e9 0.0409576
\(951\) −3.02865e9 −0.114187
\(952\) 2.82220e9 0.106013
\(953\) 1.78132e10 0.666681 0.333340 0.942807i \(-0.391824\pi\)
0.333340 + 0.942807i \(0.391824\pi\)
\(954\) −1.00689e10 −0.375458
\(955\) 1.88670e10 0.700955
\(956\) −1.37165e10 −0.507740
\(957\) −3.40823e10 −1.25701
\(958\) 6.66530e9 0.244929
\(959\) −4.67284e9 −0.171087
\(960\) 2.52099e9 0.0919650
\(961\) 2.09397e10 0.761095
\(962\) −3.70993e9 −0.134355
\(963\) −1.37859e9 −0.0497442
\(964\) 4.23260e9 0.152173
\(965\) −5.23472e10 −1.87520
\(966\) −2.41041e9 −0.0860340
\(967\) 1.49059e10 0.530109 0.265054 0.964233i \(-0.414610\pi\)
0.265054 + 0.964233i \(0.414610\pi\)
\(968\) 1.88665e10 0.668540
\(969\) 1.20446e9 0.0425266
\(970\) −1.70311e10 −0.599157
\(971\) 2.94577e10 1.03260 0.516299 0.856408i \(-0.327309\pi\)
0.516299 + 0.856408i \(0.327309\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) −1.41886e10 −0.493793
\(974\) 6.83004e9 0.236847
\(975\) −2.89111e9 −0.0998961
\(976\) −4.43855e9 −0.152815
\(977\) 1.84802e10 0.633980 0.316990 0.948429i \(-0.397328\pi\)
0.316990 + 0.948429i \(0.397328\pi\)
\(978\) 2.01336e10 0.688232
\(979\) −3.33302e10 −1.13527
\(980\) −2.68186e9 −0.0910217
\(981\) 6.73387e9 0.227732
\(982\) −3.55234e10 −1.19708
\(983\) −4.33131e10 −1.45439 −0.727197 0.686429i \(-0.759177\pi\)
−0.727197 + 0.686429i \(0.759177\pi\)
\(984\) 1.13940e10 0.381234
\(985\) 3.01947e10 1.00671
\(986\) 2.16216e10 0.718321
\(987\) −7.95323e8 −0.0263289
\(988\) 3.90316e8 0.0128756
\(989\) −2.08231e10 −0.684476
\(990\) 1.55911e10 0.510687
\(991\) −8.95161e9 −0.292175 −0.146088 0.989272i \(-0.546668\pi\)
−0.146088 + 0.989272i \(0.546668\pi\)
\(992\) −7.21286e9 −0.234594
\(993\) −1.37148e10 −0.444496
\(994\) 7.27382e9 0.234915
\(995\) 6.52572e10 2.10014
\(996\) 9.51855e9 0.305255
\(997\) −4.71755e9 −0.150759 −0.0753795 0.997155i \(-0.524017\pi\)
−0.0753795 + 0.997155i \(0.524017\pi\)
\(998\) −1.37108e9 −0.0436621
\(999\) 4.15468e9 0.131843
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.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.r.1.1 6 1.1 even 1 trivial