Properties

Label 546.8.a.i.1.4
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 86504 x^{3} - 9117228 x^{2} + 89606664 x + 21810067776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5\cdot 13 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(44.7350\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -12.1505 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -12.1505 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +97.2036 q^{10} +2351.17 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -328.062 q^{15} +4096.00 q^{16} -23753.6 q^{17} -5832.00 q^{18} +40958.6 q^{19} -777.629 q^{20} +9261.00 q^{21} -18809.3 q^{22} -36727.0 q^{23} -13824.0 q^{24} -77977.4 q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -145789. q^{29} +2624.50 q^{30} -91087.9 q^{31} -32768.0 q^{32} +63481.5 q^{33} +190028. q^{34} -4167.60 q^{35} +46656.0 q^{36} +389729. q^{37} -327669. q^{38} +59319.0 q^{39} +6221.03 q^{40} +592344. q^{41} -74088.0 q^{42} -408257. q^{43} +150475. q^{44} -8857.68 q^{45} +293816. q^{46} -1.19508e6 q^{47} +110592. q^{48} +117649. q^{49} +623819. q^{50} -641346. q^{51} +140608. q^{52} -1.59585e6 q^{53} -157464. q^{54} -28567.7 q^{55} -175616. q^{56} +1.10588e6 q^{57} +1.16631e6 q^{58} +697099. q^{59} -20996.0 q^{60} -2.63653e6 q^{61} +728703. q^{62} +250047. q^{63} +262144. q^{64} -26694.5 q^{65} -507852. q^{66} +2.94033e6 q^{67} -1.52023e6 q^{68} -991628. q^{69} +33340.8 q^{70} +2.79525e6 q^{71} -373248. q^{72} +3.60050e6 q^{73} -3.11783e6 q^{74} -2.10539e6 q^{75} +2.62135e6 q^{76} +806450. q^{77} -474552. q^{78} -5.23592e6 q^{79} -49768.2 q^{80} +531441. q^{81} -4.73875e6 q^{82} -3.05612e6 q^{83} +592704. q^{84} +288616. q^{85} +3.26606e6 q^{86} -3.93631e6 q^{87} -1.20380e6 q^{88} -2.20102e6 q^{89} +70861.4 q^{90} +753571. q^{91} -2.35053e6 q^{92} -2.45937e6 q^{93} +9.56063e6 q^{94} -497666. q^{95} -884736. q^{96} -1.71655e7 q^{97} -941192. q^{98} +1.71400e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99} + O(q^{100}) \)

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
\(5\) −12.1505 −0.0434708 −0.0217354 0.999764i \(-0.506919\pi\)
−0.0217354 + 0.999764i \(0.506919\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 97.2036 0.0307385
\(11\) 2351.17 0.532609 0.266305 0.963889i \(-0.414197\pi\)
0.266305 + 0.963889i \(0.414197\pi\)
\(12\) 1728.00 0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −328.062 −0.0250979
\(16\) 4096.00 0.250000
\(17\) −23753.6 −1.17262 −0.586310 0.810087i \(-0.699420\pi\)
−0.586310 + 0.810087i \(0.699420\pi\)
\(18\) −5832.00 −0.235702
\(19\) 40958.6 1.36996 0.684980 0.728562i \(-0.259811\pi\)
0.684980 + 0.728562i \(0.259811\pi\)
\(20\) −777.629 −0.0217354
\(21\) 9261.00 0.218218
\(22\) −18809.3 −0.376612
\(23\) −36727.0 −0.629416 −0.314708 0.949189i \(-0.601907\pi\)
−0.314708 + 0.949189i \(0.601907\pi\)
\(24\) −13824.0 −0.204124
\(25\) −77977.4 −0.998110
\(26\) −17576.0 −0.196116
\(27\) 19683.0 0.192450
\(28\) 21952.0 0.188982
\(29\) −145789. −1.11002 −0.555012 0.831842i \(-0.687286\pi\)
−0.555012 + 0.831842i \(0.687286\pi\)
\(30\) 2624.50 0.0177469
\(31\) −91087.9 −0.549155 −0.274577 0.961565i \(-0.588538\pi\)
−0.274577 + 0.961565i \(0.588538\pi\)
\(32\) −32768.0 −0.176777
\(33\) 63481.5 0.307502
\(34\) 190028. 0.829168
\(35\) −4167.60 −0.0164304
\(36\) 46656.0 0.166667
\(37\) 389729. 1.26490 0.632451 0.774601i \(-0.282049\pi\)
0.632451 + 0.774601i \(0.282049\pi\)
\(38\) −327669. −0.968708
\(39\) 59319.0 0.160128
\(40\) 6221.03 0.0153692
\(41\) 592344. 1.34224 0.671120 0.741349i \(-0.265813\pi\)
0.671120 + 0.741349i \(0.265813\pi\)
\(42\) −74088.0 −0.154303
\(43\) −408257. −0.783059 −0.391530 0.920166i \(-0.628054\pi\)
−0.391530 + 0.920166i \(0.628054\pi\)
\(44\) 150475. 0.266305
\(45\) −8857.68 −0.0144903
\(46\) 293816. 0.445064
\(47\) −1.19508e6 −1.67901 −0.839507 0.543349i \(-0.817156\pi\)
−0.839507 + 0.543349i \(0.817156\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) 623819. 0.705771
\(51\) −641346. −0.677013
\(52\) 140608. 0.138675
\(53\) −1.59585e6 −1.47240 −0.736201 0.676763i \(-0.763382\pi\)
−0.736201 + 0.676763i \(0.763382\pi\)
\(54\) −157464. −0.136083
\(55\) −28567.7 −0.0231529
\(56\) −175616. −0.133631
\(57\) 1.10588e6 0.790946
\(58\) 1.16631e6 0.784906
\(59\) 697099. 0.441888 0.220944 0.975286i \(-0.429086\pi\)
0.220944 + 0.975286i \(0.429086\pi\)
\(60\) −20996.0 −0.0125489
\(61\) −2.63653e6 −1.48723 −0.743615 0.668609i \(-0.766890\pi\)
−0.743615 + 0.668609i \(0.766890\pi\)
\(62\) 728703. 0.388311
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −26694.5 −0.0120566
\(66\) −507852. −0.217437
\(67\) 2.94033e6 1.19436 0.597179 0.802108i \(-0.296288\pi\)
0.597179 + 0.802108i \(0.296288\pi\)
\(68\) −1.52023e6 −0.586310
\(69\) −991628. −0.363393
\(70\) 33340.8 0.0116181
\(71\) 2.79525e6 0.926865 0.463432 0.886132i \(-0.346618\pi\)
0.463432 + 0.886132i \(0.346618\pi\)
\(72\) −373248. −0.117851
\(73\) 3.60050e6 1.08326 0.541630 0.840617i \(-0.317808\pi\)
0.541630 + 0.840617i \(0.317808\pi\)
\(74\) −3.11783e6 −0.894420
\(75\) −2.10539e6 −0.576259
\(76\) 2.62135e6 0.684980
\(77\) 806450. 0.201307
\(78\) −474552. −0.113228
\(79\) −5.23592e6 −1.19481 −0.597404 0.801940i \(-0.703801\pi\)
−0.597404 + 0.801940i \(0.703801\pi\)
\(80\) −49768.2 −0.0108677
\(81\) 531441. 0.111111
\(82\) −4.73875e6 −0.949107
\(83\) −3.05612e6 −0.586675 −0.293337 0.956009i \(-0.594766\pi\)
−0.293337 + 0.956009i \(0.594766\pi\)
\(84\) 592704. 0.109109
\(85\) 288616. 0.0509747
\(86\) 3.26606e6 0.553706
\(87\) −3.93631e6 −0.640873
\(88\) −1.20380e6 −0.188306
\(89\) −2.20102e6 −0.330948 −0.165474 0.986214i \(-0.552915\pi\)
−0.165474 + 0.986214i \(0.552915\pi\)
\(90\) 70861.4 0.0102462
\(91\) 753571. 0.104828
\(92\) −2.35053e6 −0.314708
\(93\) −2.45937e6 −0.317055
\(94\) 9.56063e6 1.18724
\(95\) −497666. −0.0595532
\(96\) −884736. −0.102062
\(97\) −1.71655e7 −1.90966 −0.954830 0.297153i \(-0.903963\pi\)
−0.954830 + 0.297153i \(0.903963\pi\)
\(98\) −941192. −0.101015
\(99\) 1.71400e6 0.177536
\(100\) −4.99055e6 −0.499055
\(101\) −1.25438e7 −1.21145 −0.605723 0.795676i \(-0.707116\pi\)
−0.605723 + 0.795676i \(0.707116\pi\)
\(102\) 5.13077e6 0.478720
\(103\) 34707.7 0.00312965 0.00156482 0.999999i \(-0.499502\pi\)
0.00156482 + 0.999999i \(0.499502\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −112525. −0.00948610
\(106\) 1.27668e7 1.04114
\(107\) 1.42610e7 1.12540 0.562700 0.826662i \(-0.309763\pi\)
0.562700 + 0.826662i \(0.309763\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −9.09132e6 −0.672410 −0.336205 0.941789i \(-0.609143\pi\)
−0.336205 + 0.941789i \(0.609143\pi\)
\(110\) 228542. 0.0163716
\(111\) 1.05227e7 0.730291
\(112\) 1.40493e6 0.0944911
\(113\) 6.11807e6 0.398878 0.199439 0.979910i \(-0.436088\pi\)
0.199439 + 0.979910i \(0.436088\pi\)
\(114\) −8.84706e6 −0.559284
\(115\) 446249. 0.0273612
\(116\) −9.33051e6 −0.555012
\(117\) 1.60161e6 0.0924500
\(118\) −5.57679e6 −0.312462
\(119\) −8.14747e6 −0.443209
\(120\) 167968. 0.00887343
\(121\) −1.39592e7 −0.716327
\(122\) 2.10922e7 1.05163
\(123\) 1.59933e7 0.774943
\(124\) −5.82963e6 −0.274577
\(125\) 1.89671e6 0.0868594
\(126\) −2.00038e6 −0.0890871
\(127\) 2.62050e7 1.13520 0.567598 0.823306i \(-0.307873\pi\)
0.567598 + 0.823306i \(0.307873\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) −1.10230e7 −0.452099
\(130\) 213556. 0.00852532
\(131\) 3.22701e7 1.25415 0.627077 0.778957i \(-0.284251\pi\)
0.627077 + 0.778957i \(0.284251\pi\)
\(132\) 4.06281e6 0.153751
\(133\) 1.40488e7 0.517796
\(134\) −2.35227e7 −0.844539
\(135\) −239157. −0.00836595
\(136\) 1.21618e7 0.414584
\(137\) 4.94777e6 0.164394 0.0821972 0.996616i \(-0.473806\pi\)
0.0821972 + 0.996616i \(0.473806\pi\)
\(138\) 7.93303e6 0.256958
\(139\) 4.73278e7 1.49474 0.747368 0.664410i \(-0.231317\pi\)
0.747368 + 0.664410i \(0.231317\pi\)
\(140\) −266727. −0.00821520
\(141\) −3.22671e7 −0.969379
\(142\) −2.23620e7 −0.655392
\(143\) 5.16551e6 0.147719
\(144\) 2.98598e6 0.0833333
\(145\) 1.77140e6 0.0482536
\(146\) −2.88040e7 −0.765980
\(147\) 3.17652e6 0.0824786
\(148\) 2.49426e7 0.632451
\(149\) −8.61144e6 −0.213267 −0.106634 0.994298i \(-0.534007\pi\)
−0.106634 + 0.994298i \(0.534007\pi\)
\(150\) 1.68431e7 0.407477
\(151\) −4.00833e7 −0.947424 −0.473712 0.880680i \(-0.657086\pi\)
−0.473712 + 0.880680i \(0.657086\pi\)
\(152\) −2.09708e7 −0.484354
\(153\) −1.73163e7 −0.390873
\(154\) −6.45160e6 −0.142346
\(155\) 1.10676e6 0.0238722
\(156\) 3.79642e6 0.0800641
\(157\) 2.87214e6 0.0592320 0.0296160 0.999561i \(-0.490572\pi\)
0.0296160 + 0.999561i \(0.490572\pi\)
\(158\) 4.18874e7 0.844857
\(159\) −4.30879e7 −0.850091
\(160\) 398146. 0.00768462
\(161\) −1.25974e7 −0.237897
\(162\) −4.25153e6 −0.0785674
\(163\) 9.22930e7 1.66921 0.834607 0.550846i \(-0.185695\pi\)
0.834607 + 0.550846i \(0.185695\pi\)
\(164\) 3.79100e7 0.671120
\(165\) −771328. −0.0133674
\(166\) 2.44490e7 0.414842
\(167\) −4.13487e7 −0.686995 −0.343498 0.939154i \(-0.611612\pi\)
−0.343498 + 0.939154i \(0.611612\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −2.30893e6 −0.0360446
\(171\) 2.98588e7 0.456653
\(172\) −2.61285e7 −0.391530
\(173\) −7.56258e7 −1.11047 −0.555237 0.831692i \(-0.687372\pi\)
−0.555237 + 0.831692i \(0.687372\pi\)
\(174\) 3.14905e7 0.453165
\(175\) −2.67462e7 −0.377250
\(176\) 9.63037e6 0.133152
\(177\) 1.88217e7 0.255124
\(178\) 1.76082e7 0.234015
\(179\) −1.37316e8 −1.78952 −0.894759 0.446549i \(-0.852653\pi\)
−0.894759 + 0.446549i \(0.852653\pi\)
\(180\) −566891. −0.00724513
\(181\) 1.35670e8 1.70063 0.850314 0.526276i \(-0.176412\pi\)
0.850314 + 0.526276i \(0.176412\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −7.11862e7 −0.858652
\(184\) 1.88042e7 0.222532
\(185\) −4.73538e6 −0.0549862
\(186\) 1.96750e7 0.224192
\(187\) −5.58485e7 −0.624549
\(188\) −7.64851e7 −0.839507
\(189\) 6.75127e6 0.0727393
\(190\) 3.98133e6 0.0421105
\(191\) −1.52695e6 −0.0158565 −0.00792825 0.999969i \(-0.502524\pi\)
−0.00792825 + 0.999969i \(0.502524\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 1.63010e8 1.63216 0.816080 0.577939i \(-0.196143\pi\)
0.816080 + 0.577939i \(0.196143\pi\)
\(194\) 1.37324e8 1.35033
\(195\) −720753. −0.00696090
\(196\) 7.52954e6 0.0714286
\(197\) −4.38681e7 −0.408806 −0.204403 0.978887i \(-0.565525\pi\)
−0.204403 + 0.978887i \(0.565525\pi\)
\(198\) −1.37120e7 −0.125537
\(199\) 4.42403e6 0.0397953 0.0198977 0.999802i \(-0.493666\pi\)
0.0198977 + 0.999802i \(0.493666\pi\)
\(200\) 3.99244e7 0.352885
\(201\) 7.93890e7 0.689563
\(202\) 1.00350e8 0.856622
\(203\) −5.00057e7 −0.419550
\(204\) −4.10461e7 −0.338506
\(205\) −7.19724e6 −0.0583482
\(206\) −277662. −0.00221300
\(207\) −2.67740e7 −0.209805
\(208\) 8.99891e6 0.0693375
\(209\) 9.63005e7 0.729653
\(210\) 900203. 0.00670769
\(211\) −1.40630e8 −1.03060 −0.515298 0.857011i \(-0.672319\pi\)
−0.515298 + 0.857011i \(0.672319\pi\)
\(212\) −1.02134e8 −0.736201
\(213\) 7.54717e7 0.535126
\(214\) −1.14088e8 −0.795777
\(215\) 4.96051e6 0.0340402
\(216\) −1.00777e7 −0.0680414
\(217\) −3.12432e7 −0.207561
\(218\) 7.27305e7 0.475465
\(219\) 9.72134e7 0.625420
\(220\) −1.82833e6 −0.0115765
\(221\) −5.21866e7 −0.325226
\(222\) −8.41814e7 −0.516394
\(223\) −1.68816e8 −1.01940 −0.509701 0.860352i \(-0.670244\pi\)
−0.509701 + 0.860352i \(0.670244\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −5.68455e7 −0.332703
\(226\) −4.89446e7 −0.282049
\(227\) 1.76137e8 0.999445 0.499723 0.866185i \(-0.333435\pi\)
0.499723 + 0.866185i \(0.333435\pi\)
\(228\) 7.07765e7 0.395473
\(229\) −3.15877e8 −1.73818 −0.869089 0.494655i \(-0.835294\pi\)
−0.869089 + 0.494655i \(0.835294\pi\)
\(230\) −3.56999e6 −0.0193473
\(231\) 2.17741e7 0.116225
\(232\) 7.46441e7 0.392453
\(233\) −1.57810e8 −0.817311 −0.408656 0.912689i \(-0.634002\pi\)
−0.408656 + 0.912689i \(0.634002\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 1.45208e7 0.0729880
\(236\) 4.46144e7 0.220944
\(237\) −1.41370e8 −0.689823
\(238\) 6.51798e7 0.313396
\(239\) −3.00642e8 −1.42448 −0.712240 0.701936i \(-0.752319\pi\)
−0.712240 + 0.701936i \(0.752319\pi\)
\(240\) −1.34374e6 −0.00627447
\(241\) −3.91798e8 −1.80303 −0.901514 0.432750i \(-0.857543\pi\)
−0.901514 + 0.432750i \(0.857543\pi\)
\(242\) 1.11674e8 0.506520
\(243\) 1.43489e7 0.0641500
\(244\) −1.68738e8 −0.743615
\(245\) −1.42949e6 −0.00621011
\(246\) −1.27946e8 −0.547967
\(247\) 8.99861e7 0.379958
\(248\) 4.66370e7 0.194156
\(249\) −8.25153e7 −0.338717
\(250\) −1.51737e7 −0.0614189
\(251\) −3.11086e8 −1.24172 −0.620858 0.783923i \(-0.713215\pi\)
−0.620858 + 0.783923i \(0.713215\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) −8.63512e7 −0.335233
\(254\) −2.09640e8 −0.802704
\(255\) 7.79264e6 0.0294303
\(256\) 1.67772e7 0.0625000
\(257\) 2.78740e8 1.02431 0.512157 0.858892i \(-0.328846\pi\)
0.512157 + 0.858892i \(0.328846\pi\)
\(258\) 8.81836e7 0.319683
\(259\) 1.33677e8 0.478088
\(260\) −1.70845e6 −0.00602831
\(261\) −1.06280e8 −0.370008
\(262\) −2.58161e8 −0.886821
\(263\) 2.01753e8 0.683871 0.341936 0.939723i \(-0.388918\pi\)
0.341936 + 0.939723i \(0.388918\pi\)
\(264\) −3.25025e7 −0.108718
\(265\) 1.93903e7 0.0640064
\(266\) −1.12390e8 −0.366137
\(267\) −5.94276e7 −0.191073
\(268\) 1.88181e8 0.597179
\(269\) −4.48126e8 −1.40368 −0.701838 0.712337i \(-0.747637\pi\)
−0.701838 + 0.712337i \(0.747637\pi\)
\(270\) 1.91326e6 0.00591562
\(271\) 2.78349e8 0.849566 0.424783 0.905295i \(-0.360350\pi\)
0.424783 + 0.905295i \(0.360350\pi\)
\(272\) −9.72946e7 −0.293155
\(273\) 2.03464e7 0.0605228
\(274\) −3.95821e7 −0.116244
\(275\) −1.83338e8 −0.531603
\(276\) −6.34642e7 −0.181697
\(277\) −1.41961e7 −0.0401320 −0.0200660 0.999799i \(-0.506388\pi\)
−0.0200660 + 0.999799i \(0.506388\pi\)
\(278\) −3.78623e8 −1.05694
\(279\) −6.64031e7 −0.183052
\(280\) 2.13381e6 0.00580903
\(281\) −3.72754e8 −1.00219 −0.501095 0.865392i \(-0.667069\pi\)
−0.501095 + 0.865392i \(0.667069\pi\)
\(282\) 2.58137e8 0.685454
\(283\) −1.77407e8 −0.465284 −0.232642 0.972562i \(-0.574737\pi\)
−0.232642 + 0.972562i \(0.574737\pi\)
\(284\) 1.78896e8 0.463432
\(285\) −1.34370e7 −0.0343831
\(286\) −4.13241e7 −0.104453
\(287\) 2.03174e8 0.507319
\(288\) −2.38879e7 −0.0589256
\(289\) 1.53893e8 0.375039
\(290\) −1.41712e7 −0.0341205
\(291\) −4.63469e8 −1.10254
\(292\) 2.30432e8 0.541630
\(293\) −4.77202e8 −1.10832 −0.554160 0.832410i \(-0.686961\pi\)
−0.554160 + 0.832410i \(0.686961\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −8.47007e6 −0.0192092
\(296\) −1.99541e8 −0.447210
\(297\) 4.62780e7 0.102501
\(298\) 6.88915e7 0.150803
\(299\) −8.06892e7 −0.174569
\(300\) −1.34745e8 −0.288130
\(301\) −1.40032e8 −0.295968
\(302\) 3.20667e8 0.669930
\(303\) −3.38682e8 −0.699429
\(304\) 1.67767e8 0.342490
\(305\) 3.20350e7 0.0646510
\(306\) 1.38531e8 0.276389
\(307\) −8.98202e8 −1.77170 −0.885849 0.463973i \(-0.846423\pi\)
−0.885849 + 0.463973i \(0.846423\pi\)
\(308\) 5.16128e7 0.100654
\(309\) 937108. 0.00180690
\(310\) −8.85407e6 −0.0168802
\(311\) −2.44076e8 −0.460112 −0.230056 0.973177i \(-0.573891\pi\)
−0.230056 + 0.973177i \(0.573891\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) −6.87239e8 −1.26678 −0.633392 0.773831i \(-0.718338\pi\)
−0.633392 + 0.773831i \(0.718338\pi\)
\(314\) −2.29771e7 −0.0418834
\(315\) −3.03818e6 −0.00547680
\(316\) −3.35099e8 −0.597404
\(317\) −1.13427e8 −0.199991 −0.0999954 0.994988i \(-0.531883\pi\)
−0.0999954 + 0.994988i \(0.531883\pi\)
\(318\) 3.44703e8 0.601105
\(319\) −3.42775e8 −0.591209
\(320\) −3.18517e6 −0.00543385
\(321\) 3.85047e8 0.649749
\(322\) 1.00779e8 0.168218
\(323\) −9.72913e8 −1.60644
\(324\) 3.40122e7 0.0555556
\(325\) −1.71316e8 −0.276826
\(326\) −7.38344e8 −1.18031
\(327\) −2.45466e8 −0.388216
\(328\) −3.03280e8 −0.474554
\(329\) −4.09912e8 −0.634607
\(330\) 6.17063e6 0.00945215
\(331\) −2.73163e8 −0.414022 −0.207011 0.978339i \(-0.566374\pi\)
−0.207011 + 0.978339i \(0.566374\pi\)
\(332\) −1.95592e8 −0.293337
\(333\) 2.84112e8 0.421634
\(334\) 3.30789e8 0.485779
\(335\) −3.57264e7 −0.0519197
\(336\) 3.79331e7 0.0545545
\(337\) −4.38234e8 −0.623736 −0.311868 0.950125i \(-0.600955\pi\)
−0.311868 + 0.950125i \(0.600955\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) 1.65188e8 0.230292
\(340\) 1.84715e7 0.0254874
\(341\) −2.14163e8 −0.292485
\(342\) −2.38871e8 −0.322903
\(343\) 4.03536e7 0.0539949
\(344\) 2.09028e8 0.276853
\(345\) 1.20487e7 0.0157970
\(346\) 6.05006e8 0.785224
\(347\) 1.99731e8 0.256622 0.128311 0.991734i \(-0.459044\pi\)
0.128311 + 0.991734i \(0.459044\pi\)
\(348\) −2.51924e8 −0.320436
\(349\) 8.27137e8 1.04157 0.520785 0.853688i \(-0.325639\pi\)
0.520785 + 0.853688i \(0.325639\pi\)
\(350\) 2.13970e8 0.266756
\(351\) 4.32436e7 0.0533761
\(352\) −7.70430e7 −0.0941529
\(353\) 6.66700e7 0.0806713 0.0403356 0.999186i \(-0.487157\pi\)
0.0403356 + 0.999186i \(0.487157\pi\)
\(354\) −1.50573e8 −0.180400
\(355\) −3.39635e7 −0.0402915
\(356\) −1.40865e8 −0.165474
\(357\) −2.19982e8 −0.255887
\(358\) 1.09853e9 1.26538
\(359\) −3.63566e8 −0.414718 −0.207359 0.978265i \(-0.566487\pi\)
−0.207359 + 0.978265i \(0.566487\pi\)
\(360\) 4.53513e6 0.00512308
\(361\) 7.83737e8 0.876789
\(362\) −1.08536e9 −1.20253
\(363\) −3.76898e8 −0.413572
\(364\) 4.82285e7 0.0524142
\(365\) −4.37477e7 −0.0470901
\(366\) 5.69490e8 0.607159
\(367\) 1.04729e9 1.10595 0.552973 0.833199i \(-0.313493\pi\)
0.552973 + 0.833199i \(0.313493\pi\)
\(368\) −1.50434e8 −0.157354
\(369\) 4.31819e8 0.447413
\(370\) 3.78831e7 0.0388811
\(371\) −5.47376e8 −0.556515
\(372\) −1.57400e8 −0.158527
\(373\) −1.49314e9 −1.48977 −0.744884 0.667194i \(-0.767495\pi\)
−0.744884 + 0.667194i \(0.767495\pi\)
\(374\) 4.46788e8 0.441623
\(375\) 5.12113e7 0.0501483
\(376\) 6.11881e8 0.593621
\(377\) −3.20299e8 −0.307865
\(378\) −5.40102e7 −0.0514344
\(379\) 4.44467e8 0.419375 0.209688 0.977768i \(-0.432755\pi\)
0.209688 + 0.977768i \(0.432755\pi\)
\(380\) −3.18506e7 −0.0297766
\(381\) 7.07534e8 0.655405
\(382\) 1.22156e7 0.0112122
\(383\) 1.71656e9 1.56122 0.780609 0.625020i \(-0.214909\pi\)
0.780609 + 0.625020i \(0.214909\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −9.79873e6 −0.00875099
\(386\) −1.30408e9 −1.15411
\(387\) −2.97620e8 −0.261020
\(388\) −1.09859e9 −0.954830
\(389\) −1.89671e9 −1.63372 −0.816861 0.576834i \(-0.804288\pi\)
−0.816861 + 0.576834i \(0.804288\pi\)
\(390\) 5.76602e6 0.00492210
\(391\) 8.72396e8 0.738066
\(392\) −6.02363e7 −0.0505076
\(393\) 8.71293e8 0.724086
\(394\) 3.50945e8 0.289069
\(395\) 6.36188e7 0.0519392
\(396\) 1.09696e8 0.0887682
\(397\) −2.05358e9 −1.64720 −0.823599 0.567173i \(-0.808037\pi\)
−0.823599 + 0.567173i \(0.808037\pi\)
\(398\) −3.53922e7 −0.0281395
\(399\) 3.79318e8 0.298950
\(400\) −3.19395e8 −0.249528
\(401\) −1.61799e9 −1.25306 −0.626530 0.779398i \(-0.715525\pi\)
−0.626530 + 0.779398i \(0.715525\pi\)
\(402\) −6.35112e8 −0.487595
\(403\) −2.00120e8 −0.152308
\(404\) −8.02803e8 −0.605723
\(405\) −6.45725e6 −0.00483009
\(406\) 4.00046e8 0.296666
\(407\) 9.16317e8 0.673698
\(408\) 3.28369e8 0.239360
\(409\) −8.98794e8 −0.649574 −0.324787 0.945787i \(-0.605293\pi\)
−0.324787 + 0.945787i \(0.605293\pi\)
\(410\) 5.75780e7 0.0412584
\(411\) 1.33590e8 0.0949132
\(412\) 2.22129e6 0.00156482
\(413\) 2.39105e8 0.167018
\(414\) 2.14192e8 0.148355
\(415\) 3.71333e7 0.0255032
\(416\) −7.19913e7 −0.0490290
\(417\) 1.27785e9 0.862987
\(418\) −7.70404e8 −0.515943
\(419\) −8.08723e8 −0.537095 −0.268547 0.963266i \(-0.586544\pi\)
−0.268547 + 0.963266i \(0.586544\pi\)
\(420\) −7.20162e6 −0.00474305
\(421\) 1.27096e9 0.830129 0.415064 0.909792i \(-0.363759\pi\)
0.415064 + 0.909792i \(0.363759\pi\)
\(422\) 1.12504e9 0.728741
\(423\) −8.71213e8 −0.559671
\(424\) 8.17075e8 0.520572
\(425\) 1.85224e9 1.17040
\(426\) −6.03774e8 −0.378391
\(427\) −9.04329e8 −0.562120
\(428\) 9.12704e8 0.562700
\(429\) 1.39469e8 0.0852857
\(430\) −3.96841e7 −0.0240700
\(431\) −1.09537e9 −0.659008 −0.329504 0.944154i \(-0.606882\pi\)
−0.329504 + 0.944154i \(0.606882\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 2.16541e9 1.28183 0.640917 0.767610i \(-0.278554\pi\)
0.640917 + 0.767610i \(0.278554\pi\)
\(434\) 2.49945e8 0.146768
\(435\) 4.78279e7 0.0278592
\(436\) −5.81844e8 −0.336205
\(437\) −1.50429e9 −0.862274
\(438\) −7.77707e8 −0.442239
\(439\) 7.97498e8 0.449887 0.224944 0.974372i \(-0.427780\pi\)
0.224944 + 0.974372i \(0.427780\pi\)
\(440\) 1.46267e7 0.00818580
\(441\) 8.57661e7 0.0476190
\(442\) 4.17493e8 0.229970
\(443\) −1.06435e8 −0.0581665 −0.0290832 0.999577i \(-0.509259\pi\)
−0.0290832 + 0.999577i \(0.509259\pi\)
\(444\) 6.73452e8 0.365145
\(445\) 2.67434e7 0.0143866
\(446\) 1.35052e9 0.720826
\(447\) −2.32509e8 −0.123130
\(448\) 8.99154e7 0.0472456
\(449\) −9.25513e8 −0.482526 −0.241263 0.970460i \(-0.577562\pi\)
−0.241263 + 0.970460i \(0.577562\pi\)
\(450\) 4.54764e8 0.235257
\(451\) 1.39270e9 0.714890
\(452\) 3.91557e8 0.199439
\(453\) −1.08225e9 −0.546995
\(454\) −1.40909e9 −0.706715
\(455\) −9.15623e6 −0.00455698
\(456\) −5.66212e8 −0.279642
\(457\) −2.23261e8 −0.109423 −0.0547113 0.998502i \(-0.517424\pi\)
−0.0547113 + 0.998502i \(0.517424\pi\)
\(458\) 2.52702e9 1.22908
\(459\) −4.67541e8 −0.225671
\(460\) 2.85600e7 0.0136806
\(461\) −3.98913e9 −1.89638 −0.948190 0.317705i \(-0.897088\pi\)
−0.948190 + 0.317705i \(0.897088\pi\)
\(462\) −1.74193e8 −0.0821834
\(463\) −4.66671e8 −0.218513 −0.109256 0.994014i \(-0.534847\pi\)
−0.109256 + 0.994014i \(0.534847\pi\)
\(464\) −5.97153e8 −0.277506
\(465\) 2.98825e7 0.0137826
\(466\) 1.26248e9 0.577926
\(467\) 1.70197e9 0.773291 0.386645 0.922228i \(-0.373634\pi\)
0.386645 + 0.922228i \(0.373634\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) 1.00853e9 0.451425
\(470\) −1.16166e8 −0.0516103
\(471\) 7.75477e7 0.0341976
\(472\) −3.56915e8 −0.156231
\(473\) −9.59881e8 −0.417065
\(474\) 1.13096e9 0.487778
\(475\) −3.19385e9 −1.36737
\(476\) −5.21438e8 −0.221604
\(477\) −1.16337e9 −0.490800
\(478\) 2.40513e9 1.00726
\(479\) 1.31402e9 0.546297 0.273149 0.961972i \(-0.411935\pi\)
0.273149 + 0.961972i \(0.411935\pi\)
\(480\) 1.07499e7 0.00443672
\(481\) 8.56234e8 0.350820
\(482\) 3.13438e9 1.27493
\(483\) −3.40129e8 −0.137350
\(484\) −8.93388e8 −0.358164
\(485\) 2.08569e8 0.0830144
\(486\) −1.14791e8 −0.0453609
\(487\) 2.58937e9 1.01588 0.507941 0.861392i \(-0.330407\pi\)
0.507941 + 0.861392i \(0.330407\pi\)
\(488\) 1.34990e9 0.525815
\(489\) 2.49191e9 0.963721
\(490\) 1.14359e7 0.00439121
\(491\) −2.73335e9 −1.04210 −0.521051 0.853525i \(-0.674460\pi\)
−0.521051 + 0.853525i \(0.674460\pi\)
\(492\) 1.02357e9 0.387471
\(493\) 3.46301e9 1.30164
\(494\) −7.19889e8 −0.268671
\(495\) −2.08259e7 −0.00771765
\(496\) −3.73096e8 −0.137289
\(497\) 9.58771e8 0.350322
\(498\) 6.60122e8 0.239509
\(499\) 4.97895e9 1.79385 0.896925 0.442182i \(-0.145796\pi\)
0.896925 + 0.442182i \(0.145796\pi\)
\(500\) 1.21390e8 0.0434297
\(501\) −1.11641e9 −0.396637
\(502\) 2.48869e9 0.878025
\(503\) 4.03599e8 0.141404 0.0707020 0.997497i \(-0.477476\pi\)
0.0707020 + 0.997497i \(0.477476\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 1.52413e8 0.0526625
\(506\) 6.90810e8 0.237045
\(507\) 1.30324e8 0.0444116
\(508\) 1.67712e9 0.567598
\(509\) 1.42963e9 0.480521 0.240261 0.970708i \(-0.422767\pi\)
0.240261 + 0.970708i \(0.422767\pi\)
\(510\) −6.23412e7 −0.0208103
\(511\) 1.23497e9 0.409434
\(512\) −1.34218e8 −0.0441942
\(513\) 8.06189e8 0.263649
\(514\) −2.22992e9 −0.724300
\(515\) −421714. −0.000136048 0
\(516\) −7.05469e8 −0.226050
\(517\) −2.80983e9 −0.894258
\(518\) −1.06942e9 −0.338059
\(519\) −2.04190e9 −0.641132
\(520\) 1.36676e7 0.00426266
\(521\) −3.69792e9 −1.14558 −0.572791 0.819702i \(-0.694139\pi\)
−0.572791 + 0.819702i \(0.694139\pi\)
\(522\) 8.50243e8 0.261635
\(523\) 4.45878e9 1.36289 0.681444 0.731870i \(-0.261352\pi\)
0.681444 + 0.731870i \(0.261352\pi\)
\(524\) 2.06529e9 0.627077
\(525\) −7.22148e8 −0.217806
\(526\) −1.61402e9 −0.483570
\(527\) 2.16366e9 0.643950
\(528\) 2.60020e8 0.0768755
\(529\) −2.05595e9 −0.603836
\(530\) −1.55122e8 −0.0452594
\(531\) 5.08185e8 0.147296
\(532\) 8.99124e8 0.258898
\(533\) 1.30138e9 0.372271
\(534\) 4.75421e8 0.135109
\(535\) −1.73277e8 −0.0489220
\(536\) −1.50545e9 −0.422269
\(537\) −3.70754e9 −1.03318
\(538\) 3.58501e9 0.992549
\(539\) 2.76612e8 0.0760870
\(540\) −1.53061e7 −0.00418298
\(541\) 2.07517e9 0.563459 0.281730 0.959494i \(-0.409092\pi\)
0.281730 + 0.959494i \(0.409092\pi\)
\(542\) −2.22679e9 −0.600734
\(543\) 3.66309e9 0.981858
\(544\) 7.78357e8 0.207292
\(545\) 1.10464e8 0.0292302
\(546\) −1.62771e8 −0.0427960
\(547\) 5.81898e9 1.52017 0.760084 0.649825i \(-0.225158\pi\)
0.760084 + 0.649825i \(0.225158\pi\)
\(548\) 3.16657e8 0.0821972
\(549\) −1.92203e9 −0.495743
\(550\) 1.46670e9 0.375900
\(551\) −5.97133e9 −1.52069
\(552\) 5.07714e8 0.128479
\(553\) −1.79592e9 −0.451595
\(554\) 1.13569e8 0.0283776
\(555\) −1.27855e8 −0.0317463
\(556\) 3.02898e9 0.747368
\(557\) 5.32423e9 1.30546 0.652730 0.757591i \(-0.273624\pi\)
0.652730 + 0.757591i \(0.273624\pi\)
\(558\) 5.31225e8 0.129437
\(559\) −8.96942e8 −0.217181
\(560\) −1.70705e7 −0.00410760
\(561\) −1.50791e9 −0.360583
\(562\) 2.98203e9 0.708655
\(563\) 1.91319e9 0.451834 0.225917 0.974147i \(-0.427462\pi\)
0.225917 + 0.974147i \(0.427462\pi\)
\(564\) −2.06510e9 −0.484689
\(565\) −7.43373e7 −0.0173395
\(566\) 1.41926e9 0.329006
\(567\) 1.82284e8 0.0419961
\(568\) −1.43117e9 −0.327696
\(569\) 2.81457e9 0.640499 0.320250 0.947333i \(-0.396233\pi\)
0.320250 + 0.947333i \(0.396233\pi\)
\(570\) 1.07496e8 0.0243125
\(571\) −4.87197e9 −1.09516 −0.547581 0.836753i \(-0.684451\pi\)
−0.547581 + 0.836753i \(0.684451\pi\)
\(572\) 3.30593e8 0.0738596
\(573\) −4.12276e7 −0.00915476
\(574\) −1.62539e9 −0.358729
\(575\) 2.86387e9 0.628227
\(576\) 1.91103e8 0.0416667
\(577\) 5.36429e9 1.16251 0.581255 0.813722i \(-0.302562\pi\)
0.581255 + 0.813722i \(0.302562\pi\)
\(578\) −1.23114e9 −0.265192
\(579\) 4.40126e9 0.942328
\(580\) 1.13370e8 0.0241268
\(581\) −1.04825e9 −0.221742
\(582\) 3.70775e9 0.779615
\(583\) −3.75210e9 −0.784215
\(584\) −1.84345e9 −0.382990
\(585\) −1.94603e7 −0.00401887
\(586\) 3.81762e9 0.783701
\(587\) −6.31539e9 −1.28874 −0.644372 0.764712i \(-0.722881\pi\)
−0.644372 + 0.764712i \(0.722881\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) −3.73084e9 −0.752320
\(590\) 6.77606e7 0.0135830
\(591\) −1.18444e9 −0.236024
\(592\) 1.59633e9 0.316225
\(593\) 7.01786e9 1.38202 0.691008 0.722847i \(-0.257167\pi\)
0.691008 + 0.722847i \(0.257167\pi\)
\(594\) −3.70224e8 −0.0724789
\(595\) 9.89954e7 0.0192666
\(596\) −5.51132e8 −0.106634
\(597\) 1.19449e8 0.0229758
\(598\) 6.45513e8 0.123439
\(599\) 9.45941e8 0.179833 0.0899166 0.995949i \(-0.471340\pi\)
0.0899166 + 0.995949i \(0.471340\pi\)
\(600\) 1.07796e9 0.203738
\(601\) −1.51203e9 −0.284119 −0.142060 0.989858i \(-0.545372\pi\)
−0.142060 + 0.989858i \(0.545372\pi\)
\(602\) 1.12026e9 0.209281
\(603\) 2.14350e9 0.398119
\(604\) −2.56533e9 −0.473712
\(605\) 1.69610e8 0.0311393
\(606\) 2.70946e9 0.494571
\(607\) 6.06641e9 1.10096 0.550479 0.834849i \(-0.314445\pi\)
0.550479 + 0.834849i \(0.314445\pi\)
\(608\) −1.34213e9 −0.242177
\(609\) −1.35015e9 −0.242227
\(610\) −2.56280e8 −0.0457152
\(611\) −2.62559e9 −0.465675
\(612\) −1.10825e9 −0.195437
\(613\) −1.53297e8 −0.0268796 −0.0134398 0.999910i \(-0.504278\pi\)
−0.0134398 + 0.999910i \(0.504278\pi\)
\(614\) 7.18562e9 1.25278
\(615\) −1.94326e8 −0.0336874
\(616\) −4.12902e8 −0.0711729
\(617\) 5.78295e9 0.991177 0.495589 0.868557i \(-0.334952\pi\)
0.495589 + 0.868557i \(0.334952\pi\)
\(618\) −7.49687e6 −0.00127767
\(619\) −1.12405e10 −1.90488 −0.952442 0.304720i \(-0.901437\pi\)
−0.952442 + 0.304720i \(0.901437\pi\)
\(620\) 7.08326e7 0.0119361
\(621\) −7.22897e8 −0.121131
\(622\) 1.95261e9 0.325348
\(623\) −7.54951e8 −0.125086
\(624\) 2.42971e8 0.0400320
\(625\) 6.06894e9 0.994334
\(626\) 5.49791e9 0.895751
\(627\) 2.60011e9 0.421265
\(628\) 1.83817e8 0.0296160
\(629\) −9.25745e9 −1.48325
\(630\) 2.43055e7 0.00387268
\(631\) 8.68673e9 1.37643 0.688214 0.725508i \(-0.258395\pi\)
0.688214 + 0.725508i \(0.258395\pi\)
\(632\) 2.68079e9 0.422429
\(633\) −3.79700e9 −0.595015
\(634\) 9.07418e8 0.141415
\(635\) −3.18402e8 −0.0493478
\(636\) −2.75763e9 −0.425046
\(637\) 2.58475e8 0.0396214
\(638\) 2.74220e9 0.418048
\(639\) 2.03774e9 0.308955
\(640\) 2.54813e7 0.00384231
\(641\) 7.98647e9 1.19771 0.598855 0.800857i \(-0.295623\pi\)
0.598855 + 0.800857i \(0.295623\pi\)
\(642\) −3.08037e9 −0.459442
\(643\) −5.00901e9 −0.743043 −0.371521 0.928424i \(-0.621164\pi\)
−0.371521 + 0.928424i \(0.621164\pi\)
\(644\) −8.06231e8 −0.118948
\(645\) 1.33934e8 0.0196531
\(646\) 7.78330e9 1.13593
\(647\) −3.56209e9 −0.517059 −0.258530 0.966003i \(-0.583238\pi\)
−0.258530 + 0.966003i \(0.583238\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) 1.63900e9 0.235354
\(650\) 1.37053e9 0.195746
\(651\) −8.43565e8 −0.119835
\(652\) 5.90675e9 0.834607
\(653\) −1.25486e10 −1.76360 −0.881799 0.471625i \(-0.843667\pi\)
−0.881799 + 0.471625i \(0.843667\pi\)
\(654\) 1.96372e9 0.274510
\(655\) −3.92096e8 −0.0545191
\(656\) 2.42624e9 0.335560
\(657\) 2.62476e9 0.361086
\(658\) 3.27930e9 0.448735
\(659\) −1.66119e9 −0.226110 −0.113055 0.993589i \(-0.536064\pi\)
−0.113055 + 0.993589i \(0.536064\pi\)
\(660\) −4.93650e7 −0.00668368
\(661\) 1.05642e10 1.42276 0.711380 0.702807i \(-0.248070\pi\)
0.711380 + 0.702807i \(0.248070\pi\)
\(662\) 2.18530e9 0.292758
\(663\) −1.40904e9 −0.187770
\(664\) 1.56473e9 0.207421
\(665\) −1.70699e8 −0.0225090
\(666\) −2.27290e9 −0.298140
\(667\) 5.35440e9 0.698667
\(668\) −2.64631e9 −0.343498
\(669\) −4.55802e9 −0.588552
\(670\) 2.85811e8 0.0367128
\(671\) −6.19891e9 −0.792112
\(672\) −3.03464e8 −0.0385758
\(673\) −1.47467e10 −1.86484 −0.932421 0.361374i \(-0.882308\pi\)
−0.932421 + 0.361374i \(0.882308\pi\)
\(674\) 3.50587e9 0.441048
\(675\) −1.53483e9 −0.192086
\(676\) 3.08916e8 0.0384615
\(677\) −1.14704e10 −1.42076 −0.710378 0.703820i \(-0.751476\pi\)
−0.710378 + 0.703820i \(0.751476\pi\)
\(678\) −1.32150e9 −0.162841
\(679\) −5.88777e9 −0.721783
\(680\) −1.47772e8 −0.0180223
\(681\) 4.75569e9 0.577030
\(682\) 1.71330e9 0.206818
\(683\) −1.28166e10 −1.53922 −0.769610 0.638514i \(-0.779549\pi\)
−0.769610 + 0.638514i \(0.779549\pi\)
\(684\) 1.91097e9 0.228327
\(685\) −6.01176e7 −0.00714636
\(686\) −3.22829e8 −0.0381802
\(687\) −8.52869e9 −1.00354
\(688\) −1.67222e9 −0.195765
\(689\) −3.50608e9 −0.408371
\(690\) −9.63899e7 −0.0111702
\(691\) 9.69675e9 1.11803 0.559015 0.829158i \(-0.311180\pi\)
0.559015 + 0.829158i \(0.311180\pi\)
\(692\) −4.84005e9 −0.555237
\(693\) 5.87902e8 0.0671025
\(694\) −1.59785e9 −0.181459
\(695\) −5.75054e8 −0.0649774
\(696\) 2.01539e9 0.226583
\(697\) −1.40703e10 −1.57394
\(698\) −6.61710e9 −0.736502
\(699\) −4.26086e9 −0.471875
\(700\) −1.71176e9 −0.188625
\(701\) 6.43479e9 0.705539 0.352770 0.935710i \(-0.385240\pi\)
0.352770 + 0.935710i \(0.385240\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) 1.59628e10 1.73286
\(704\) 6.16344e8 0.0665762
\(705\) 3.92060e8 0.0421397
\(706\) −5.33360e8 −0.0570432
\(707\) −4.30252e9 −0.457884
\(708\) 1.20459e9 0.127562
\(709\) 1.04957e10 1.10598 0.552992 0.833187i \(-0.313486\pi\)
0.552992 + 0.833187i \(0.313486\pi\)
\(710\) 2.71708e8 0.0284904
\(711\) −3.81699e9 −0.398269
\(712\) 1.12692e9 0.117008
\(713\) 3.34538e9 0.345647
\(714\) 1.75985e9 0.180939
\(715\) −6.27633e7 −0.00642147
\(716\) −8.78823e9 −0.894759
\(717\) −8.11732e9 −0.822424
\(718\) 2.90853e9 0.293250
\(719\) −1.67170e10 −1.67728 −0.838641 0.544685i \(-0.816649\pi\)
−0.838641 + 0.544685i \(0.816649\pi\)
\(720\) −3.62811e7 −0.00362256
\(721\) 1.19047e7 0.00118290
\(722\) −6.26990e9 −0.619983
\(723\) −1.05785e10 −1.04098
\(724\) 8.68289e9 0.850314
\(725\) 1.13683e10 1.10793
\(726\) 3.01519e9 0.292439
\(727\) 3.00200e9 0.289762 0.144881 0.989449i \(-0.453720\pi\)
0.144881 + 0.989449i \(0.453720\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 3.49981e8 0.0332977
\(731\) 9.69757e9 0.918231
\(732\) −4.55592e9 −0.429326
\(733\) 1.68142e10 1.57693 0.788465 0.615080i \(-0.210876\pi\)
0.788465 + 0.615080i \(0.210876\pi\)
\(734\) −8.37829e9 −0.782022
\(735\) −3.85962e7 −0.00358541
\(736\) 1.20347e9 0.111266
\(737\) 6.91321e9 0.636126
\(738\) −3.45455e9 −0.316369
\(739\) −1.78287e9 −0.162504 −0.0812518 0.996694i \(-0.525892\pi\)
−0.0812518 + 0.996694i \(0.525892\pi\)
\(740\) −3.03064e8 −0.0274931
\(741\) 2.42962e9 0.219369
\(742\) 4.37901e9 0.393516
\(743\) 6.48101e9 0.579671 0.289836 0.957076i \(-0.406399\pi\)
0.289836 + 0.957076i \(0.406399\pi\)
\(744\) 1.25920e9 0.112096
\(745\) 1.04633e8 0.00927089
\(746\) 1.19451e10 1.05343
\(747\) −2.22791e9 −0.195558
\(748\) −3.57431e9 −0.312274
\(749\) 4.89152e9 0.425361
\(750\) −4.09690e8 −0.0354602
\(751\) 7.54558e9 0.650059 0.325030 0.945704i \(-0.394626\pi\)
0.325030 + 0.945704i \(0.394626\pi\)
\(752\) −4.89504e9 −0.419753
\(753\) −8.39931e9 −0.716905
\(754\) 2.56239e9 0.217694
\(755\) 4.87030e8 0.0411853
\(756\) 4.32081e8 0.0363696
\(757\) −2.12009e10 −1.77631 −0.888156 0.459542i \(-0.848013\pi\)
−0.888156 + 0.459542i \(0.848013\pi\)
\(758\) −3.55574e9 −0.296543
\(759\) −2.33148e9 −0.193547
\(760\) 2.54805e8 0.0210552
\(761\) −1.43518e10 −1.18048 −0.590241 0.807227i \(-0.700967\pi\)
−0.590241 + 0.807227i \(0.700967\pi\)
\(762\) −5.66027e9 −0.463442
\(763\) −3.11832e9 −0.254147
\(764\) −9.77247e7 −0.00792825
\(765\) 2.10401e8 0.0169916
\(766\) −1.37325e10 −1.10395
\(767\) 1.53153e9 0.122558
\(768\) 4.52985e8 0.0360844
\(769\) 4.75972e9 0.377433 0.188716 0.982032i \(-0.439567\pi\)
0.188716 + 0.982032i \(0.439567\pi\)
\(770\) 7.83898e7 0.00618788
\(771\) 7.52598e9 0.591388
\(772\) 1.04326e10 0.816080
\(773\) −3.10511e9 −0.241796 −0.120898 0.992665i \(-0.538577\pi\)
−0.120898 + 0.992665i \(0.538577\pi\)
\(774\) 2.38096e9 0.184569
\(775\) 7.10280e9 0.548117
\(776\) 8.78875e9 0.675167
\(777\) 3.60928e9 0.276024
\(778\) 1.51737e10 1.15522
\(779\) 2.42616e10 1.83882
\(780\) −4.61282e7 −0.00348045
\(781\) 6.57209e9 0.493657
\(782\) −6.97917e9 −0.521891
\(783\) −2.86957e9 −0.213624
\(784\) 4.81890e8 0.0357143
\(785\) −3.48978e7 −0.00257486
\(786\) −6.97035e9 −0.512006
\(787\) −1.45473e10 −1.06382 −0.531912 0.846800i \(-0.678526\pi\)
−0.531912 + 0.846800i \(0.678526\pi\)
\(788\) −2.80756e9 −0.204403
\(789\) 5.44732e9 0.394833
\(790\) −5.08950e8 −0.0367266
\(791\) 2.09850e9 0.150762
\(792\) −8.77568e8 −0.0627686
\(793\) −5.79245e9 −0.412483
\(794\) 1.64287e10 1.16474
\(795\) 5.23538e8 0.0369541
\(796\) 2.83138e8 0.0198977
\(797\) −1.42033e10 −0.993766 −0.496883 0.867818i \(-0.665522\pi\)
−0.496883 + 0.867818i \(0.665522\pi\)
\(798\) −3.03454e9 −0.211389
\(799\) 2.83874e10 1.96885
\(800\) 2.55516e9 0.176443
\(801\) −1.60454e9 −0.110316
\(802\) 1.29439e10 0.886047
\(803\) 8.46536e9 0.576954
\(804\) 5.08089e9 0.344782
\(805\) 1.53064e8 0.0103416
\(806\) 1.60096e9 0.107698
\(807\) −1.20994e10 −0.810413
\(808\) 6.42242e9 0.428311
\(809\) 1.14889e10 0.762883 0.381442 0.924393i \(-0.375428\pi\)
0.381442 + 0.924393i \(0.375428\pi\)
\(810\) 5.16580e7 0.00341539
\(811\) −1.27152e10 −0.837048 −0.418524 0.908206i \(-0.637452\pi\)
−0.418524 + 0.908206i \(0.637452\pi\)
\(812\) −3.20037e9 −0.209775
\(813\) 7.51542e9 0.490497
\(814\) −7.33054e9 −0.476376
\(815\) −1.12140e9 −0.0725620
\(816\) −2.62695e9 −0.169253
\(817\) −1.67217e10 −1.07276
\(818\) 7.19035e9 0.459318
\(819\) 5.49353e8 0.0349428
\(820\) −4.60624e8 −0.0291741
\(821\) 2.47879e9 0.156329 0.0781644 0.996940i \(-0.475094\pi\)
0.0781644 + 0.996940i \(0.475094\pi\)
\(822\) −1.06872e9 −0.0671138
\(823\) 1.25365e10 0.783927 0.391964 0.919981i \(-0.371796\pi\)
0.391964 + 0.919981i \(0.371796\pi\)
\(824\) −1.77703e7 −0.00110650
\(825\) −4.95012e9 −0.306921
\(826\) −1.91284e9 −0.118100
\(827\) −1.01541e10 −0.624269 −0.312135 0.950038i \(-0.601044\pi\)
−0.312135 + 0.950038i \(0.601044\pi\)
\(828\) −1.71353e9 −0.104903
\(829\) −1.14635e10 −0.698837 −0.349419 0.936967i \(-0.613621\pi\)
−0.349419 + 0.936967i \(0.613621\pi\)
\(830\) −2.97066e8 −0.0180335
\(831\) −3.83295e8 −0.0231702
\(832\) 5.75930e8 0.0346688
\(833\) −2.79458e9 −0.167517
\(834\) −1.02228e10 −0.610224
\(835\) 5.02405e8 0.0298642
\(836\) 6.16323e9 0.364827
\(837\) −1.79288e9 −0.105685
\(838\) 6.46979e9 0.379783
\(839\) −1.33243e10 −0.778892 −0.389446 0.921049i \(-0.627333\pi\)
−0.389446 + 0.921049i \(0.627333\pi\)
\(840\) 5.76130e7 0.00335384
\(841\) 4.00462e9 0.232154
\(842\) −1.01677e10 −0.586990
\(843\) −1.00644e10 −0.578614
\(844\) −9.00030e9 −0.515298
\(845\) −5.86479e7 −0.00334391
\(846\) 6.96970e9 0.395747
\(847\) −4.78800e9 −0.270746
\(848\) −6.53660e9 −0.368100
\(849\) −4.78999e9 −0.268632
\(850\) −1.48179e10 −0.827601
\(851\) −1.43136e10 −0.796149
\(852\) 4.83019e9 0.267563
\(853\) 3.41609e10 1.88455 0.942274 0.334843i \(-0.108683\pi\)
0.942274 + 0.334843i \(0.108683\pi\)
\(854\) 7.23463e9 0.397479
\(855\) −3.62798e8 −0.0198511
\(856\) −7.30163e9 −0.397889
\(857\) −3.06420e10 −1.66297 −0.831485 0.555548i \(-0.812509\pi\)
−0.831485 + 0.555548i \(0.812509\pi\)
\(858\) −1.11575e9 −0.0603061
\(859\) −8.93738e9 −0.481099 −0.240549 0.970637i \(-0.577328\pi\)
−0.240549 + 0.970637i \(0.577328\pi\)
\(860\) 3.17473e8 0.0170201
\(861\) 5.48570e9 0.292901
\(862\) 8.76296e9 0.465989
\(863\) −9.35681e9 −0.495553 −0.247777 0.968817i \(-0.579700\pi\)
−0.247777 + 0.968817i \(0.579700\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 9.18887e8 0.0482732
\(866\) −1.73233e10 −0.906394
\(867\) 4.15511e9 0.216529
\(868\) −1.99956e9 −0.103781
\(869\) −1.23105e10 −0.636366
\(870\) −3.82623e8 −0.0196995
\(871\) 6.45991e9 0.331255
\(872\) 4.65475e9 0.237733
\(873\) −1.25137e10 −0.636553
\(874\) 1.20343e10 0.609720
\(875\) 6.50573e8 0.0328298
\(876\) 6.22166e9 0.312710
\(877\) −4.31179e9 −0.215853 −0.107927 0.994159i \(-0.534421\pi\)
−0.107927 + 0.994159i \(0.534421\pi\)
\(878\) −6.37998e9 −0.318118
\(879\) −1.28845e10 −0.639889
\(880\) −1.17013e8 −0.00578824
\(881\) 2.17775e10 1.07298 0.536491 0.843906i \(-0.319750\pi\)
0.536491 + 0.843906i \(0.319750\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) 3.20221e10 1.56526 0.782631 0.622486i \(-0.213877\pi\)
0.782631 + 0.622486i \(0.213877\pi\)
\(884\) −3.33994e9 −0.162613
\(885\) −2.28692e8 −0.0110905
\(886\) 8.51483e8 0.0411299
\(887\) 1.71630e10 0.825774 0.412887 0.910782i \(-0.364520\pi\)
0.412887 + 0.910782i \(0.364520\pi\)
\(888\) −5.38761e9 −0.258197
\(889\) 8.98831e9 0.429064
\(890\) −2.13947e8 −0.0101728
\(891\) 1.24951e9 0.0591788
\(892\) −1.08042e10 −0.509701
\(893\) −4.89488e10 −2.30018
\(894\) 1.86007e9 0.0870660
\(895\) 1.66845e9 0.0777917
\(896\) −7.19323e8 −0.0334077
\(897\) −2.17861e9 −0.100787
\(898\) 7.40411e9 0.341197
\(899\) 1.32796e10 0.609575
\(900\) −3.63811e9 −0.166352
\(901\) 3.79071e10 1.72657
\(902\) −1.11416e10 −0.505503
\(903\) −3.78087e9 −0.170877
\(904\) −3.13245e9 −0.141025
\(905\) −1.64845e9 −0.0739276
\(906\) 8.65800e9 0.386784
\(907\) 1.20055e10 0.534263 0.267132 0.963660i \(-0.413924\pi\)
0.267132 + 0.963660i \(0.413924\pi\)
\(908\) 1.12727e10 0.499723
\(909\) −9.14443e9 −0.403815
\(910\) 7.32498e7 0.00322227
\(911\) 1.59171e10 0.697509 0.348755 0.937214i \(-0.386605\pi\)
0.348755 + 0.937214i \(0.386605\pi\)
\(912\) 4.52970e9 0.197737
\(913\) −7.18545e9 −0.312468
\(914\) 1.78609e9 0.0773735
\(915\) 8.64945e8 0.0373263
\(916\) −2.02162e10 −0.869089
\(917\) 1.10687e10 0.474026
\(918\) 3.74033e9 0.159573
\(919\) −5.58329e9 −0.237293 −0.118647 0.992937i \(-0.537856\pi\)
−0.118647 + 0.992937i \(0.537856\pi\)
\(920\) −2.28480e8 −0.00967365
\(921\) −2.42515e10 −1.02289
\(922\) 3.19131e10 1.34094
\(923\) 6.14116e9 0.257066
\(924\) 1.39355e9 0.0581124
\(925\) −3.03900e10 −1.26251
\(926\) 3.73337e9 0.154512
\(927\) 2.53019e7 0.00104322
\(928\) 4.77722e9 0.196226
\(929\) 3.96979e10 1.62447 0.812237 0.583328i \(-0.198250\pi\)
0.812237 + 0.583328i \(0.198250\pi\)
\(930\) −2.39060e8 −0.00974578
\(931\) 4.81874e9 0.195708
\(932\) −1.00998e10 −0.408656
\(933\) −6.59004e9 −0.265646
\(934\) −1.36158e10 −0.546799
\(935\) 6.78585e8 0.0271496
\(936\) −8.20026e8 −0.0326860
\(937\) −3.87603e10 −1.53921 −0.769606 0.638519i \(-0.779547\pi\)
−0.769606 + 0.638519i \(0.779547\pi\)
\(938\) −8.06827e9 −0.319206
\(939\) −1.85554e10 −0.731378
\(940\) 9.29328e8 0.0364940
\(941\) −9.23500e9 −0.361304 −0.180652 0.983547i \(-0.557821\pi\)
−0.180652 + 0.983547i \(0.557821\pi\)
\(942\) −6.20382e8 −0.0241814
\(943\) −2.17550e10 −0.844828
\(944\) 2.85532e9 0.110472
\(945\) −8.20310e7 −0.00316203
\(946\) 7.67905e9 0.294909
\(947\) 3.97440e10 1.52071 0.760355 0.649508i \(-0.225025\pi\)
0.760355 + 0.649508i \(0.225025\pi\)
\(948\) −9.04767e9 −0.344911
\(949\) 7.91029e9 0.300442
\(950\) 2.55508e10 0.966877
\(951\) −3.06254e9 −0.115465
\(952\) 4.17150e9 0.156698
\(953\) 2.76232e10 1.03383 0.516914 0.856037i \(-0.327081\pi\)
0.516914 + 0.856037i \(0.327081\pi\)
\(954\) 9.30699e9 0.347048
\(955\) 1.85531e7 0.000689295 0
\(956\) −1.92411e10 −0.712240
\(957\) −9.25491e9 −0.341335
\(958\) −1.05122e10 −0.386291
\(959\) 1.69708e9 0.0621353
\(960\) −8.59995e7 −0.00313723
\(961\) −1.92156e10 −0.698429
\(962\) −6.84987e9 −0.248068
\(963\) 1.03963e10 0.375133
\(964\) −2.50751e10 −0.901514
\(965\) −1.98064e9 −0.0709512
\(966\) 2.72103e9 0.0971210
\(967\) −4.48219e10 −1.59403 −0.797017 0.603957i \(-0.793590\pi\)
−0.797017 + 0.603957i \(0.793590\pi\)
\(968\) 7.14711e9 0.253260
\(969\) −2.62687e10 −0.927480
\(970\) −1.66855e9 −0.0587000
\(971\) 3.63237e10 1.27328 0.636638 0.771163i \(-0.280325\pi\)
0.636638 + 0.771163i \(0.280325\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 1.62334e10 0.564957
\(974\) −2.07150e10 −0.718336
\(975\) −4.62554e9 −0.159826
\(976\) −1.07992e10 −0.371807
\(977\) 3.98690e10 1.36774 0.683871 0.729603i \(-0.260295\pi\)
0.683871 + 0.729603i \(0.260295\pi\)
\(978\) −1.99353e10 −0.681454
\(979\) −5.17497e9 −0.176266
\(980\) −9.14873e7 −0.00310506
\(981\) −6.62757e9 −0.224137
\(982\) 2.18668e10 0.736878
\(983\) 6.50063e9 0.218282 0.109141 0.994026i \(-0.465190\pi\)
0.109141 + 0.994026i \(0.465190\pi\)
\(984\) −8.18856e9 −0.273984
\(985\) 5.33017e8 0.0177711
\(986\) −2.77041e10 −0.920396
\(987\) −1.10676e10 −0.366391
\(988\) 5.75911e9 0.189979
\(989\) 1.49941e10 0.492870
\(990\) 1.66607e8 0.00545720
\(991\) 2.84761e10 0.929443 0.464721 0.885457i \(-0.346154\pi\)
0.464721 + 0.885457i \(0.346154\pi\)
\(992\) 2.98477e9 0.0970778
\(993\) −7.37539e9 −0.239036
\(994\) −7.67017e9 −0.247715
\(995\) −5.37539e7 −0.00172993
\(996\) −5.28098e9 −0.169358
\(997\) 1.31123e10 0.419030 0.209515 0.977805i \(-0.432811\pi\)
0.209515 + 0.977805i \(0.432811\pi\)
\(998\) −3.98316e10 −1.26844
\(999\) 7.67103e9 0.243430
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.i.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.i.1.4 5 1.1 even 1 trivial