Properties

Label 546.8.a.i.1.1
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 1.1
Root \(-65.0947\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -428.929 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -428.929 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3431.43 q^{10} -5641.82 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -11581.1 q^{15} +4096.00 q^{16} +5030.39 q^{17} -5832.00 q^{18} -7398.89 q^{19} -27451.5 q^{20} +9261.00 q^{21} +45134.5 q^{22} +72906.4 q^{23} -13824.0 q^{24} +105855. q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -59340.1 q^{29} +92648.7 q^{30} -113966. q^{31} -32768.0 q^{32} -152329. q^{33} -40243.1 q^{34} -147123. q^{35} +46656.0 q^{36} +112781. q^{37} +59191.2 q^{38} +59319.0 q^{39} +219612. q^{40} +224080. q^{41} -74088.0 q^{42} +509419. q^{43} -361076. q^{44} -312689. q^{45} -583251. q^{46} -454823. q^{47} +110592. q^{48} +117649. q^{49} -846841. q^{50} +135820. q^{51} +140608. q^{52} +869626. q^{53} -157464. q^{54} +2.41994e6 q^{55} -175616. q^{56} -199770. q^{57} +474721. q^{58} +1.63913e6 q^{59} -741189. q^{60} +2.14907e6 q^{61} +911729. q^{62} +250047. q^{63} +262144. q^{64} -942357. q^{65} +1.21863e6 q^{66} +55040.3 q^{67} +321945. q^{68} +1.96847e6 q^{69} +1.17698e6 q^{70} -4.22051e6 q^{71} -373248. q^{72} +1.72232e6 q^{73} -902247. q^{74} +2.85809e6 q^{75} -473529. q^{76} -1.93514e6 q^{77} -474552. q^{78} -3.59169e6 q^{79} -1.75689e6 q^{80} +531441. q^{81} -1.79264e6 q^{82} -8.13749e6 q^{83} +592704. q^{84} -2.15768e6 q^{85} -4.07536e6 q^{86} -1.60218e6 q^{87} +2.88861e6 q^{88} -4.86198e6 q^{89} +2.50151e6 q^{90} +753571. q^{91} +4.66601e6 q^{92} -3.07708e6 q^{93} +3.63859e6 q^{94} +3.17360e6 q^{95} -884736. q^{96} +6.88065e6 q^{97} -941192. q^{98} -4.11288e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99} + O(q^{100}) \)

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
\(5\) −428.929 −1.53458 −0.767292 0.641298i \(-0.778396\pi\)
−0.767292 + 0.641298i \(0.778396\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 3431.43 1.08511
\(11\) −5641.82 −1.27804 −0.639020 0.769190i \(-0.720660\pi\)
−0.639020 + 0.769190i \(0.720660\pi\)
\(12\) 1728.00 0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −11581.1 −0.885992
\(16\) 4096.00 0.250000
\(17\) 5030.39 0.248331 0.124165 0.992262i \(-0.460375\pi\)
0.124165 + 0.992262i \(0.460375\pi\)
\(18\) −5832.00 −0.235702
\(19\) −7398.89 −0.247474 −0.123737 0.992315i \(-0.539488\pi\)
−0.123737 + 0.992315i \(0.539488\pi\)
\(20\) −27451.5 −0.767292
\(21\) 9261.00 0.218218
\(22\) 45134.5 0.903711
\(23\) 72906.4 1.24945 0.624724 0.780846i \(-0.285211\pi\)
0.624724 + 0.780846i \(0.285211\pi\)
\(24\) −13824.0 −0.204124
\(25\) 105855. 1.35495
\(26\) −17576.0 −0.196116
\(27\) 19683.0 0.192450
\(28\) 21952.0 0.188982
\(29\) −59340.1 −0.451809 −0.225905 0.974149i \(-0.572534\pi\)
−0.225905 + 0.974149i \(0.572534\pi\)
\(30\) 92648.7 0.626491
\(31\) −113966. −0.687084 −0.343542 0.939137i \(-0.611627\pi\)
−0.343542 + 0.939137i \(0.611627\pi\)
\(32\) −32768.0 −0.176777
\(33\) −152329. −0.737877
\(34\) −40243.1 −0.175596
\(35\) −147123. −0.580018
\(36\) 46656.0 0.166667
\(37\) 112781. 0.366041 0.183020 0.983109i \(-0.441413\pi\)
0.183020 + 0.983109i \(0.441413\pi\)
\(38\) 59191.2 0.174990
\(39\) 59319.0 0.160128
\(40\) 219612. 0.542557
\(41\) 224080. 0.507762 0.253881 0.967235i \(-0.418293\pi\)
0.253881 + 0.967235i \(0.418293\pi\)
\(42\) −74088.0 −0.154303
\(43\) 509419. 0.977093 0.488546 0.872538i \(-0.337527\pi\)
0.488546 + 0.872538i \(0.337527\pi\)
\(44\) −361076. −0.639020
\(45\) −312689. −0.511528
\(46\) −583251. −0.883493
\(47\) −454823. −0.638999 −0.319500 0.947586i \(-0.603515\pi\)
−0.319500 + 0.947586i \(0.603515\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) −846841. −0.958091
\(51\) 135820. 0.143374
\(52\) 140608. 0.138675
\(53\) 869626. 0.802356 0.401178 0.916000i \(-0.368601\pi\)
0.401178 + 0.916000i \(0.368601\pi\)
\(54\) −157464. −0.136083
\(55\) 2.41994e6 1.96126
\(56\) −175616. −0.133631
\(57\) −199770. −0.142879
\(58\) 474721. 0.319477
\(59\) 1.63913e6 1.03904 0.519520 0.854458i \(-0.326111\pi\)
0.519520 + 0.854458i \(0.326111\pi\)
\(60\) −741189. −0.442996
\(61\) 2.14907e6 1.21226 0.606130 0.795366i \(-0.292721\pi\)
0.606130 + 0.795366i \(0.292721\pi\)
\(62\) 911729. 0.485842
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −942357. −0.425617
\(66\) 1.21863e6 0.521758
\(67\) 55040.3 0.0223573 0.0111786 0.999938i \(-0.496442\pi\)
0.0111786 + 0.999938i \(0.496442\pi\)
\(68\) 321945. 0.124165
\(69\) 1.96847e6 0.721369
\(70\) 1.17698e6 0.410135
\(71\) −4.22051e6 −1.39946 −0.699730 0.714408i \(-0.746696\pi\)
−0.699730 + 0.714408i \(0.746696\pi\)
\(72\) −373248. −0.117851
\(73\) 1.72232e6 0.518182 0.259091 0.965853i \(-0.416577\pi\)
0.259091 + 0.965853i \(0.416577\pi\)
\(74\) −902247. −0.258830
\(75\) 2.85809e6 0.782278
\(76\) −473529. −0.123737
\(77\) −1.93514e6 −0.483054
\(78\) −474552. −0.113228
\(79\) −3.59169e6 −0.819603 −0.409802 0.912175i \(-0.634402\pi\)
−0.409802 + 0.912175i \(0.634402\pi\)
\(80\) −1.75689e6 −0.383646
\(81\) 531441. 0.111111
\(82\) −1.79264e6 −0.359042
\(83\) −8.13749e6 −1.56213 −0.781065 0.624450i \(-0.785323\pi\)
−0.781065 + 0.624450i \(0.785323\pi\)
\(84\) 592704. 0.109109
\(85\) −2.15768e6 −0.381084
\(86\) −4.07536e6 −0.690909
\(87\) −1.60218e6 −0.260852
\(88\) 2.88861e6 0.451856
\(89\) −4.86198e6 −0.731051 −0.365526 0.930801i \(-0.619111\pi\)
−0.365526 + 0.930801i \(0.619111\pi\)
\(90\) 2.50151e6 0.361705
\(91\) 753571. 0.104828
\(92\) 4.66601e6 0.624724
\(93\) −3.07708e6 −0.396688
\(94\) 3.63859e6 0.451841
\(95\) 3.17360e6 0.379769
\(96\) −884736. −0.102062
\(97\) 6.88065e6 0.765470 0.382735 0.923858i \(-0.374982\pi\)
0.382735 + 0.923858i \(0.374982\pi\)
\(98\) −941192. −0.101015
\(99\) −4.11288e6 −0.426014
\(100\) 6.77473e6 0.677473
\(101\) 7.61335e6 0.735277 0.367638 0.929969i \(-0.380166\pi\)
0.367638 + 0.929969i \(0.380166\pi\)
\(102\) −1.08656e6 −0.101381
\(103\) 6.69446e6 0.603651 0.301825 0.953363i \(-0.402404\pi\)
0.301825 + 0.953363i \(0.402404\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −3.97231e6 −0.334874
\(106\) −6.95701e6 −0.567351
\(107\) −1.40813e7 −1.11122 −0.555608 0.831445i \(-0.687514\pi\)
−0.555608 + 0.831445i \(0.687514\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 1.67215e7 1.23675 0.618376 0.785882i \(-0.287791\pi\)
0.618376 + 0.785882i \(0.287791\pi\)
\(110\) −1.93595e7 −1.38682
\(111\) 3.04508e6 0.211334
\(112\) 1.40493e6 0.0944911
\(113\) −1.98555e7 −1.29451 −0.647255 0.762274i \(-0.724083\pi\)
−0.647255 + 0.762274i \(0.724083\pi\)
\(114\) 1.59816e6 0.101031
\(115\) −3.12717e7 −1.91738
\(116\) −3.79776e6 −0.225905
\(117\) 1.60161e6 0.0924500
\(118\) −1.31131e7 −0.734712
\(119\) 1.72542e6 0.0938601
\(120\) 5.92952e6 0.313245
\(121\) 1.23429e7 0.633388
\(122\) −1.71925e7 −0.857197
\(123\) 6.05017e6 0.293156
\(124\) −7.29383e6 −0.343542
\(125\) −1.18943e7 −0.544694
\(126\) −2.00038e6 −0.0890871
\(127\) 1.12422e7 0.487009 0.243504 0.969900i \(-0.421703\pi\)
0.243504 + 0.969900i \(0.421703\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 1.37543e7 0.564125
\(130\) 7.53886e6 0.300957
\(131\) −1.49026e7 −0.579178 −0.289589 0.957151i \(-0.593519\pi\)
−0.289589 + 0.957151i \(0.593519\pi\)
\(132\) −9.74906e6 −0.368939
\(133\) −2.53782e6 −0.0935363
\(134\) −440322. −0.0158090
\(135\) −8.44261e6 −0.295331
\(136\) −2.57556e6 −0.0877981
\(137\) 2.84703e7 0.945953 0.472976 0.881075i \(-0.343180\pi\)
0.472976 + 0.881075i \(0.343180\pi\)
\(138\) −1.57478e7 −0.510085
\(139\) 3.57651e7 1.12956 0.564778 0.825243i \(-0.308962\pi\)
0.564778 + 0.825243i \(0.308962\pi\)
\(140\) −9.41585e6 −0.290009
\(141\) −1.22802e7 −0.368926
\(142\) 3.37640e7 0.989567
\(143\) −1.23951e7 −0.354465
\(144\) 2.98598e6 0.0833333
\(145\) 2.54527e7 0.693339
\(146\) −1.37785e7 −0.366410
\(147\) 3.17652e6 0.0824786
\(148\) 7.21797e6 0.183020
\(149\) 869193. 0.0215261 0.0107630 0.999942i \(-0.496574\pi\)
0.0107630 + 0.999942i \(0.496574\pi\)
\(150\) −2.28647e7 −0.553154
\(151\) −1.96266e7 −0.463901 −0.231950 0.972728i \(-0.574511\pi\)
−0.231950 + 0.972728i \(0.574511\pi\)
\(152\) 3.78823e6 0.0874952
\(153\) 3.66715e6 0.0827768
\(154\) 1.54811e7 0.341571
\(155\) 4.88834e7 1.05439
\(156\) 3.79642e6 0.0800641
\(157\) −6.86346e7 −1.41545 −0.707725 0.706488i \(-0.750278\pi\)
−0.707725 + 0.706488i \(0.750278\pi\)
\(158\) 2.87335e7 0.579547
\(159\) 2.34799e7 0.463241
\(160\) 1.40551e7 0.271279
\(161\) 2.50069e7 0.472247
\(162\) −4.25153e6 −0.0785674
\(163\) −6.13655e7 −1.10986 −0.554929 0.831898i \(-0.687255\pi\)
−0.554929 + 0.831898i \(0.687255\pi\)
\(164\) 1.43411e7 0.253881
\(165\) 6.53384e7 1.13233
\(166\) 6.50999e7 1.10459
\(167\) −5.21552e7 −0.866543 −0.433272 0.901263i \(-0.642641\pi\)
−0.433272 + 0.901263i \(0.642641\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 1.72614e7 0.269467
\(171\) −5.39379e6 −0.0824913
\(172\) 3.26028e7 0.488546
\(173\) −1.06934e7 −0.157020 −0.0785098 0.996913i \(-0.525016\pi\)
−0.0785098 + 0.996913i \(0.525016\pi\)
\(174\) 1.28175e7 0.184450
\(175\) 3.63083e7 0.512121
\(176\) −2.31089e7 −0.319510
\(177\) 4.42566e7 0.599890
\(178\) 3.88958e7 0.516931
\(179\) 2.83238e7 0.369119 0.184559 0.982821i \(-0.440914\pi\)
0.184559 + 0.982821i \(0.440914\pi\)
\(180\) −2.00121e7 −0.255764
\(181\) 5.39796e7 0.676635 0.338318 0.941032i \(-0.390142\pi\)
0.338318 + 0.941032i \(0.390142\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 5.80248e7 0.699899
\(184\) −3.73281e7 −0.441747
\(185\) −4.83750e7 −0.561720
\(186\) 2.46167e7 0.280501
\(187\) −2.83805e7 −0.317376
\(188\) −2.91087e7 −0.319500
\(189\) 6.75127e6 0.0727393
\(190\) −2.53888e7 −0.268537
\(191\) 3.80423e7 0.395048 0.197524 0.980298i \(-0.436710\pi\)
0.197524 + 0.980298i \(0.436710\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 8.81233e7 0.882348 0.441174 0.897422i \(-0.354562\pi\)
0.441174 + 0.897422i \(0.354562\pi\)
\(194\) −5.50452e7 −0.541269
\(195\) −2.54436e7 −0.245730
\(196\) 7.52954e6 0.0714286
\(197\) 8.53734e7 0.795593 0.397797 0.917474i \(-0.369775\pi\)
0.397797 + 0.917474i \(0.369775\pi\)
\(198\) 3.29031e7 0.301237
\(199\) −4.12660e7 −0.371199 −0.185600 0.982625i \(-0.559423\pi\)
−0.185600 + 0.982625i \(0.559423\pi\)
\(200\) −5.41978e7 −0.479046
\(201\) 1.48609e6 0.0129080
\(202\) −6.09068e7 −0.519919
\(203\) −2.03536e7 −0.170768
\(204\) 8.69251e6 0.0716868
\(205\) −9.61145e7 −0.779203
\(206\) −5.35557e7 −0.426845
\(207\) 5.31488e7 0.416483
\(208\) 8.99891e6 0.0693375
\(209\) 4.17432e7 0.316282
\(210\) 3.17785e7 0.236791
\(211\) −9.56908e7 −0.701264 −0.350632 0.936513i \(-0.614033\pi\)
−0.350632 + 0.936513i \(0.614033\pi\)
\(212\) 5.56561e7 0.401178
\(213\) −1.13954e8 −0.807978
\(214\) 1.12650e8 0.785748
\(215\) −2.18505e8 −1.49943
\(216\) −1.00777e7 −0.0680414
\(217\) −3.90904e7 −0.259693
\(218\) −1.33772e8 −0.874516
\(219\) 4.65025e7 0.299173
\(220\) 1.54876e8 0.980630
\(221\) 1.10518e7 0.0688745
\(222\) −2.43607e7 −0.149435
\(223\) −1.26066e8 −0.761254 −0.380627 0.924729i \(-0.624292\pi\)
−0.380627 + 0.924729i \(0.624292\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 7.71684e7 0.451649
\(226\) 1.58844e8 0.915356
\(227\) −2.67097e8 −1.51558 −0.757790 0.652499i \(-0.773721\pi\)
−0.757790 + 0.652499i \(0.773721\pi\)
\(228\) −1.27853e7 −0.0714395
\(229\) 1.13200e8 0.622903 0.311452 0.950262i \(-0.399185\pi\)
0.311452 + 0.950262i \(0.399185\pi\)
\(230\) 2.50173e8 1.35579
\(231\) −5.22489e7 −0.278891
\(232\) 3.03821e7 0.159739
\(233\) 4.56825e7 0.236594 0.118297 0.992978i \(-0.462256\pi\)
0.118297 + 0.992978i \(0.462256\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 1.95087e8 0.980597
\(236\) 1.04905e8 0.519520
\(237\) −9.69755e7 −0.473198
\(238\) −1.38034e7 −0.0663691
\(239\) −3.73712e8 −1.77070 −0.885349 0.464927i \(-0.846081\pi\)
−0.885349 + 0.464927i \(0.846081\pi\)
\(240\) −4.74361e7 −0.221498
\(241\) −1.57552e8 −0.725045 −0.362522 0.931975i \(-0.618084\pi\)
−0.362522 + 0.931975i \(0.618084\pi\)
\(242\) −9.87435e7 −0.447873
\(243\) 1.43489e7 0.0641500
\(244\) 1.37540e8 0.606130
\(245\) −5.04631e7 −0.219226
\(246\) −4.84013e7 −0.207293
\(247\) −1.62554e7 −0.0686369
\(248\) 5.83506e7 0.242921
\(249\) −2.19712e8 −0.901896
\(250\) 9.51541e7 0.385157
\(251\) 8.41091e7 0.335726 0.167863 0.985810i \(-0.446313\pi\)
0.167863 + 0.985810i \(0.446313\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) −4.11325e8 −1.59685
\(254\) −8.99373e7 −0.344367
\(255\) −5.82573e7 −0.220019
\(256\) 1.67772e7 0.0625000
\(257\) −2.77967e8 −1.02147 −0.510737 0.859737i \(-0.670627\pi\)
−0.510737 + 0.859737i \(0.670627\pi\)
\(258\) −1.10035e8 −0.398897
\(259\) 3.86838e7 0.138350
\(260\) −6.03109e7 −0.212808
\(261\) −4.32589e7 −0.150603
\(262\) 1.19221e8 0.409541
\(263\) −3.24918e8 −1.10136 −0.550679 0.834717i \(-0.685631\pi\)
−0.550679 + 0.834717i \(0.685631\pi\)
\(264\) 7.79925e7 0.260879
\(265\) −3.73008e8 −1.23128
\(266\) 2.03026e7 0.0661402
\(267\) −1.31273e8 −0.422073
\(268\) 3.52258e6 0.0111786
\(269\) −7.17429e7 −0.224722 −0.112361 0.993667i \(-0.535841\pi\)
−0.112361 + 0.993667i \(0.535841\pi\)
\(270\) 6.75409e7 0.208830
\(271\) 2.13653e8 0.652102 0.326051 0.945352i \(-0.394282\pi\)
0.326051 + 0.945352i \(0.394282\pi\)
\(272\) 2.06045e7 0.0620826
\(273\) 2.03464e7 0.0605228
\(274\) −2.27762e8 −0.668889
\(275\) −5.97215e8 −1.73168
\(276\) 1.25982e8 0.360685
\(277\) −5.22223e8 −1.47631 −0.738153 0.674633i \(-0.764302\pi\)
−0.738153 + 0.674633i \(0.764302\pi\)
\(278\) −2.86121e8 −0.798717
\(279\) −8.30813e7 −0.229028
\(280\) 7.53268e7 0.205067
\(281\) −3.80140e8 −1.02205 −0.511025 0.859566i \(-0.670734\pi\)
−0.511025 + 0.859566i \(0.670734\pi\)
\(282\) 9.82418e7 0.260870
\(283\) 5.95810e7 0.156263 0.0781314 0.996943i \(-0.475105\pi\)
0.0781314 + 0.996943i \(0.475105\pi\)
\(284\) −2.70112e8 −0.699730
\(285\) 8.56872e7 0.219260
\(286\) 9.91606e7 0.250644
\(287\) 7.68595e7 0.191916
\(288\) −2.38879e7 −0.0589256
\(289\) −3.85034e8 −0.938332
\(290\) −2.03621e8 −0.490265
\(291\) 1.85777e8 0.441944
\(292\) 1.10228e8 0.259091
\(293\) −3.72478e8 −0.865095 −0.432548 0.901611i \(-0.642385\pi\)
−0.432548 + 0.901611i \(0.642385\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −7.03072e8 −1.59449
\(296\) −5.77438e7 −0.129415
\(297\) −1.11048e8 −0.245959
\(298\) −6.95355e6 −0.0152212
\(299\) 1.60175e8 0.346535
\(300\) 1.82918e8 0.391139
\(301\) 1.74731e8 0.369306
\(302\) 1.57013e8 0.328027
\(303\) 2.05560e8 0.424512
\(304\) −3.03059e7 −0.0618685
\(305\) −9.21798e8 −1.86031
\(306\) −2.93372e7 −0.0585321
\(307\) −4.41815e8 −0.871477 −0.435739 0.900073i \(-0.643513\pi\)
−0.435739 + 0.900073i \(0.643513\pi\)
\(308\) −1.23849e8 −0.241527
\(309\) 1.80751e8 0.348518
\(310\) −3.91067e8 −0.745565
\(311\) −1.19881e8 −0.225989 −0.112995 0.993596i \(-0.536044\pi\)
−0.112995 + 0.993596i \(0.536044\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) −1.18663e8 −0.218732 −0.109366 0.994002i \(-0.534882\pi\)
−0.109366 + 0.994002i \(0.534882\pi\)
\(314\) 5.49077e8 1.00087
\(315\) −1.07252e8 −0.193339
\(316\) −2.29868e8 −0.409802
\(317\) 8.52854e7 0.150372 0.0751860 0.997170i \(-0.476045\pi\)
0.0751860 + 0.997170i \(0.476045\pi\)
\(318\) −1.87839e8 −0.327561
\(319\) 3.34786e8 0.577430
\(320\) −1.12441e8 −0.191823
\(321\) −3.80194e8 −0.641561
\(322\) −2.00055e8 −0.333929
\(323\) −3.72193e7 −0.0614553
\(324\) 3.40122e7 0.0555556
\(325\) 2.32564e8 0.375794
\(326\) 4.90924e8 0.784788
\(327\) 4.51481e8 0.714040
\(328\) −1.14729e8 −0.179521
\(329\) −1.56004e8 −0.241519
\(330\) −5.22707e8 −0.800681
\(331\) 2.94895e8 0.446961 0.223481 0.974708i \(-0.428258\pi\)
0.223481 + 0.974708i \(0.428258\pi\)
\(332\) −5.20799e8 −0.781065
\(333\) 8.22172e7 0.122014
\(334\) 4.17242e8 0.612739
\(335\) −2.36084e7 −0.0343091
\(336\) 3.79331e7 0.0545545
\(337\) −7.07623e8 −1.00716 −0.503579 0.863949i \(-0.667984\pi\)
−0.503579 + 0.863949i \(0.667984\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) −5.36097e8 −0.747385
\(340\) −1.38091e8 −0.190542
\(341\) 6.42976e8 0.878121
\(342\) 4.31504e7 0.0583301
\(343\) 4.03536e7 0.0539949
\(344\) −2.60823e8 −0.345455
\(345\) −8.44335e8 −1.10700
\(346\) 8.55471e7 0.111030
\(347\) −4.77862e8 −0.613974 −0.306987 0.951714i \(-0.599321\pi\)
−0.306987 + 0.951714i \(0.599321\pi\)
\(348\) −1.02540e8 −0.130426
\(349\) −7.46777e8 −0.940377 −0.470189 0.882566i \(-0.655814\pi\)
−0.470189 + 0.882566i \(0.655814\pi\)
\(350\) −2.90467e8 −0.362124
\(351\) 4.32436e7 0.0533761
\(352\) 1.84871e8 0.225928
\(353\) 7.56646e8 0.915548 0.457774 0.889069i \(-0.348647\pi\)
0.457774 + 0.889069i \(0.348647\pi\)
\(354\) −3.54053e8 −0.424186
\(355\) 1.81030e9 2.14759
\(356\) −3.11166e8 −0.365526
\(357\) 4.65864e7 0.0541902
\(358\) −2.26590e8 −0.261006
\(359\) −1.01218e8 −0.115458 −0.0577292 0.998332i \(-0.518386\pi\)
−0.0577292 + 0.998332i \(0.518386\pi\)
\(360\) 1.60097e8 0.180852
\(361\) −8.39128e8 −0.938757
\(362\) −4.31837e8 −0.478453
\(363\) 3.33259e8 0.365687
\(364\) 4.82285e7 0.0524142
\(365\) −7.38751e8 −0.795194
\(366\) −4.64199e8 −0.494903
\(367\) −1.22591e9 −1.29457 −0.647286 0.762248i \(-0.724096\pi\)
−0.647286 + 0.762248i \(0.724096\pi\)
\(368\) 2.98625e8 0.312362
\(369\) 1.63354e8 0.169254
\(370\) 3.87000e8 0.397196
\(371\) 2.98282e8 0.303262
\(372\) −1.96933e8 −0.198344
\(373\) −5.25342e8 −0.524157 −0.262079 0.965047i \(-0.584408\pi\)
−0.262079 + 0.965047i \(0.584408\pi\)
\(374\) 2.27044e8 0.224419
\(375\) −3.21145e8 −0.314479
\(376\) 2.32870e8 0.225920
\(377\) −1.30370e8 −0.125309
\(378\) −5.40102e7 −0.0514344
\(379\) 5.47041e8 0.516158 0.258079 0.966124i \(-0.416911\pi\)
0.258079 + 0.966124i \(0.416911\pi\)
\(380\) 2.03110e8 0.189885
\(381\) 3.03538e8 0.281175
\(382\) −3.04338e8 −0.279341
\(383\) 4.51036e8 0.410219 0.205110 0.978739i \(-0.434245\pi\)
0.205110 + 0.978739i \(0.434245\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 8.30039e8 0.741286
\(386\) −7.04986e8 −0.623914
\(387\) 3.71367e8 0.325698
\(388\) 4.40361e8 0.382735
\(389\) −3.75167e8 −0.323148 −0.161574 0.986861i \(-0.551657\pi\)
−0.161574 + 0.986861i \(0.551657\pi\)
\(390\) 2.03549e8 0.173757
\(391\) 3.66747e8 0.310276
\(392\) −6.02363e7 −0.0505076
\(393\) −4.02370e8 −0.334389
\(394\) −6.82987e8 −0.562569
\(395\) 1.54058e9 1.25775
\(396\) −2.63225e8 −0.213007
\(397\) −8.62998e8 −0.692218 −0.346109 0.938194i \(-0.612497\pi\)
−0.346109 + 0.938194i \(0.612497\pi\)
\(398\) 3.30128e8 0.262478
\(399\) −6.85212e7 −0.0540032
\(400\) 4.33583e8 0.338736
\(401\) 1.21875e9 0.943862 0.471931 0.881635i \(-0.343557\pi\)
0.471931 + 0.881635i \(0.343557\pi\)
\(402\) −1.18887e7 −0.00912732
\(403\) −2.50384e8 −0.190563
\(404\) 4.87254e8 0.367638
\(405\) −2.27950e8 −0.170509
\(406\) 1.62829e8 0.120751
\(407\) −6.36289e8 −0.467815
\(408\) −6.95401e7 −0.0506903
\(409\) 6.28426e8 0.454174 0.227087 0.973874i \(-0.427080\pi\)
0.227087 + 0.973874i \(0.427080\pi\)
\(410\) 7.68916e8 0.550980
\(411\) 7.68697e8 0.546146
\(412\) 4.28446e8 0.301825
\(413\) 5.62223e8 0.392720
\(414\) −4.25190e8 −0.294498
\(415\) 3.49041e9 2.39722
\(416\) −7.19913e7 −0.0490290
\(417\) 9.65658e8 0.652150
\(418\) −3.33946e8 −0.223645
\(419\) 2.28168e9 1.51532 0.757661 0.652648i \(-0.226342\pi\)
0.757661 + 0.652648i \(0.226342\pi\)
\(420\) −2.54228e8 −0.167437
\(421\) −2.96806e9 −1.93859 −0.969294 0.245906i \(-0.920915\pi\)
−0.969294 + 0.245906i \(0.920915\pi\)
\(422\) 7.65526e8 0.495869
\(423\) −3.31566e8 −0.213000
\(424\) −4.45249e8 −0.283676
\(425\) 5.32492e8 0.336474
\(426\) 9.11629e8 0.571327
\(427\) 7.37130e8 0.458191
\(428\) −9.01201e8 −0.555608
\(429\) −3.34667e8 −0.204650
\(430\) 1.74804e9 1.06026
\(431\) 9.96218e8 0.599355 0.299677 0.954041i \(-0.403121\pi\)
0.299677 + 0.954041i \(0.403121\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −2.75006e9 −1.62793 −0.813963 0.580917i \(-0.802694\pi\)
−0.813963 + 0.580917i \(0.802694\pi\)
\(434\) 3.12723e8 0.183631
\(435\) 6.87222e8 0.400299
\(436\) 1.07018e9 0.618376
\(437\) −5.39427e8 −0.309206
\(438\) −3.72020e8 −0.211547
\(439\) −4.24947e8 −0.239723 −0.119861 0.992791i \(-0.538245\pi\)
−0.119861 + 0.992791i \(0.538245\pi\)
\(440\) −1.23901e9 −0.693410
\(441\) 8.57661e7 0.0476190
\(442\) −8.84141e7 −0.0487016
\(443\) 1.92764e9 1.05345 0.526724 0.850036i \(-0.323420\pi\)
0.526724 + 0.850036i \(0.323420\pi\)
\(444\) 1.94885e8 0.105667
\(445\) 2.08544e9 1.12186
\(446\) 1.00853e9 0.538288
\(447\) 2.34682e7 0.0124281
\(448\) 8.99154e7 0.0472456
\(449\) −1.97858e9 −1.03155 −0.515776 0.856724i \(-0.672496\pi\)
−0.515776 + 0.856724i \(0.672496\pi\)
\(450\) −6.17347e8 −0.319364
\(451\) −1.26422e9 −0.648940
\(452\) −1.27075e9 −0.647255
\(453\) −5.29917e8 −0.267833
\(454\) 2.13678e9 1.07168
\(455\) −3.23229e8 −0.160868
\(456\) 1.02282e8 0.0505154
\(457\) 2.66925e9 1.30822 0.654112 0.756398i \(-0.273043\pi\)
0.654112 + 0.756398i \(0.273043\pi\)
\(458\) −9.05596e8 −0.440459
\(459\) 9.90131e7 0.0477912
\(460\) −2.00139e9 −0.958691
\(461\) −2.19410e9 −1.04304 −0.521522 0.853238i \(-0.674635\pi\)
−0.521522 + 0.853238i \(0.674635\pi\)
\(462\) 4.17991e8 0.197206
\(463\) 4.35755e8 0.204037 0.102019 0.994782i \(-0.467470\pi\)
0.102019 + 0.994782i \(0.467470\pi\)
\(464\) −2.43057e8 −0.112952
\(465\) 1.31985e9 0.608751
\(466\) −3.65460e8 −0.167298
\(467\) 3.76838e8 0.171216 0.0856082 0.996329i \(-0.472717\pi\)
0.0856082 + 0.996329i \(0.472717\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) 1.88788e7 0.00845025
\(470\) −1.56070e9 −0.693387
\(471\) −1.85313e9 −0.817210
\(472\) −8.39236e8 −0.367356
\(473\) −2.87405e9 −1.24876
\(474\) 7.75804e8 0.334602
\(475\) −7.83211e8 −0.335314
\(476\) 1.10427e8 0.0469301
\(477\) 6.33958e8 0.267452
\(478\) 2.98970e9 1.25207
\(479\) −1.75733e8 −0.0730600 −0.0365300 0.999333i \(-0.511630\pi\)
−0.0365300 + 0.999333i \(0.511630\pi\)
\(480\) 3.79489e8 0.156623
\(481\) 2.47779e8 0.101521
\(482\) 1.26042e9 0.512684
\(483\) 6.75186e8 0.272652
\(484\) 7.89948e8 0.316694
\(485\) −2.95131e9 −1.17468
\(486\) −1.14791e8 −0.0453609
\(487\) −3.91286e9 −1.53512 −0.767560 0.640977i \(-0.778529\pi\)
−0.767560 + 0.640977i \(0.778529\pi\)
\(488\) −1.10032e9 −0.428599
\(489\) −1.65687e9 −0.640777
\(490\) 4.03705e8 0.155016
\(491\) −3.92505e8 −0.149644 −0.0748221 0.997197i \(-0.523839\pi\)
−0.0748221 + 0.997197i \(0.523839\pi\)
\(492\) 3.87211e8 0.146578
\(493\) −2.98503e8 −0.112198
\(494\) 1.30043e8 0.0485336
\(495\) 1.76414e9 0.653753
\(496\) −4.66805e8 −0.171771
\(497\) −1.44763e9 −0.528946
\(498\) 1.75770e9 0.637737
\(499\) −2.32523e9 −0.837748 −0.418874 0.908044i \(-0.637575\pi\)
−0.418874 + 0.908044i \(0.637575\pi\)
\(500\) −7.61233e8 −0.272347
\(501\) −1.40819e9 −0.500299
\(502\) −6.72873e8 −0.237394
\(503\) 1.41640e9 0.496246 0.248123 0.968729i \(-0.420186\pi\)
0.248123 + 0.968729i \(0.420186\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) −3.26559e9 −1.12834
\(506\) 3.29060e9 1.12914
\(507\) 1.30324e8 0.0444116
\(508\) 7.19498e8 0.243504
\(509\) 1.15453e9 0.388053 0.194027 0.980996i \(-0.437845\pi\)
0.194027 + 0.980996i \(0.437845\pi\)
\(510\) 4.66059e8 0.155577
\(511\) 5.90754e8 0.195855
\(512\) −1.34218e8 −0.0441942
\(513\) −1.45632e8 −0.0476264
\(514\) 2.22374e9 0.722291
\(515\) −2.87145e9 −0.926352
\(516\) 8.80277e8 0.282062
\(517\) 2.56603e9 0.816667
\(518\) −3.09471e8 −0.0978285
\(519\) −2.88721e8 −0.0906553
\(520\) 4.82487e8 0.150478
\(521\) −4.96724e9 −1.53880 −0.769401 0.638766i \(-0.779445\pi\)
−0.769401 + 0.638766i \(0.779445\pi\)
\(522\) 3.46071e8 0.106492
\(523\) −4.20708e8 −0.128595 −0.0642976 0.997931i \(-0.520481\pi\)
−0.0642976 + 0.997931i \(0.520481\pi\)
\(524\) −9.53766e8 −0.289589
\(525\) 9.80324e8 0.295673
\(526\) 2.59934e9 0.778778
\(527\) −5.73294e8 −0.170624
\(528\) −6.23940e8 −0.184469
\(529\) 1.91052e9 0.561121
\(530\) 2.98406e9 0.870648
\(531\) 1.19493e9 0.346347
\(532\) −1.62421e8 −0.0467682
\(533\) 4.92304e8 0.140828
\(534\) 1.05019e9 0.298450
\(535\) 6.03986e9 1.70525
\(536\) −2.81806e7 −0.00790449
\(537\) 7.64743e8 0.213111
\(538\) 5.73943e8 0.158903
\(539\) −6.63754e8 −0.182577
\(540\) −5.40327e8 −0.147665
\(541\) 5.73067e9 1.55602 0.778010 0.628252i \(-0.216229\pi\)
0.778010 + 0.628252i \(0.216229\pi\)
\(542\) −1.70922e9 −0.461106
\(543\) 1.45745e9 0.390656
\(544\) −1.64836e8 −0.0438990
\(545\) −7.17234e9 −1.89790
\(546\) −1.62771e8 −0.0427960
\(547\) −1.35334e9 −0.353549 −0.176775 0.984251i \(-0.556566\pi\)
−0.176775 + 0.984251i \(0.556566\pi\)
\(548\) 1.82210e9 0.472976
\(549\) 1.56667e9 0.404087
\(550\) 4.77772e9 1.22448
\(551\) 4.39051e8 0.111811
\(552\) −1.00786e9 −0.255043
\(553\) −1.23195e9 −0.309781
\(554\) 4.17778e9 1.04391
\(555\) −1.30612e9 −0.324309
\(556\) 2.28897e9 0.564778
\(557\) −3.92720e9 −0.962919 −0.481459 0.876468i \(-0.659893\pi\)
−0.481459 + 0.876468i \(0.659893\pi\)
\(558\) 6.64650e8 0.161947
\(559\) 1.11919e9 0.270997
\(560\) −6.02614e8 −0.145004
\(561\) −7.66274e8 −0.183237
\(562\) 3.04112e9 0.722698
\(563\) 3.44486e9 0.813564 0.406782 0.913525i \(-0.366651\pi\)
0.406782 + 0.913525i \(0.366651\pi\)
\(564\) −7.85935e8 −0.184463
\(565\) 8.51658e9 1.98653
\(566\) −4.76648e8 −0.110494
\(567\) 1.82284e8 0.0419961
\(568\) 2.16090e9 0.494784
\(569\) 5.20185e9 1.18376 0.591882 0.806024i \(-0.298385\pi\)
0.591882 + 0.806024i \(0.298385\pi\)
\(570\) −6.85498e8 −0.155040
\(571\) 4.94153e9 1.11080 0.555398 0.831584i \(-0.312566\pi\)
0.555398 + 0.831584i \(0.312566\pi\)
\(572\) −7.93285e8 −0.177232
\(573\) 1.02714e9 0.228081
\(574\) −6.14876e8 −0.135705
\(575\) 7.71752e9 1.69293
\(576\) 1.91103e8 0.0416667
\(577\) −1.20233e8 −0.0260561 −0.0130280 0.999915i \(-0.504147\pi\)
−0.0130280 + 0.999915i \(0.504147\pi\)
\(578\) 3.08027e9 0.663501
\(579\) 2.37933e9 0.509424
\(580\) 1.62897e9 0.346669
\(581\) −2.79116e9 −0.590429
\(582\) −1.48622e9 −0.312502
\(583\) −4.90627e9 −1.02544
\(584\) −8.81825e8 −0.183205
\(585\) −6.86978e8 −0.141872
\(586\) 2.97982e9 0.611715
\(587\) −1.35701e9 −0.276917 −0.138459 0.990368i \(-0.544215\pi\)
−0.138459 + 0.990368i \(0.544215\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) 8.43223e8 0.170035
\(590\) 5.62458e9 1.12748
\(591\) 2.30508e9 0.459336
\(592\) 4.61950e8 0.0915101
\(593\) 8.65668e8 0.170475 0.0852373 0.996361i \(-0.472835\pi\)
0.0852373 + 0.996361i \(0.472835\pi\)
\(594\) 8.88383e8 0.173919
\(595\) −7.40084e8 −0.144036
\(596\) 5.56284e7 0.0107630
\(597\) −1.11418e9 −0.214312
\(598\) −1.28140e9 −0.245037
\(599\) 2.71350e9 0.515865 0.257932 0.966163i \(-0.416959\pi\)
0.257932 + 0.966163i \(0.416959\pi\)
\(600\) −1.46334e9 −0.276577
\(601\) 7.29865e9 1.37146 0.685728 0.727858i \(-0.259484\pi\)
0.685728 + 0.727858i \(0.259484\pi\)
\(602\) −1.39785e9 −0.261139
\(603\) 4.01244e7 0.00745242
\(604\) −1.25610e9 −0.231950
\(605\) −5.29424e9 −0.971986
\(606\) −1.64448e9 −0.300176
\(607\) 7.24336e9 1.31456 0.657279 0.753647i \(-0.271707\pi\)
0.657279 + 0.753647i \(0.271707\pi\)
\(608\) 2.42447e8 0.0437476
\(609\) −5.49548e8 −0.0985928
\(610\) 7.37438e9 1.31544
\(611\) −9.99247e8 −0.177226
\(612\) 2.34698e8 0.0413884
\(613\) 5.90927e9 1.03615 0.518075 0.855335i \(-0.326649\pi\)
0.518075 + 0.855335i \(0.326649\pi\)
\(614\) 3.53452e9 0.616227
\(615\) −2.59509e9 −0.449873
\(616\) 9.90793e8 0.170785
\(617\) −6.28936e9 −1.07797 −0.538987 0.842314i \(-0.681193\pi\)
−0.538987 + 0.842314i \(0.681193\pi\)
\(618\) −1.44600e9 −0.246439
\(619\) −3.11689e9 −0.528207 −0.264103 0.964494i \(-0.585076\pi\)
−0.264103 + 0.964494i \(0.585076\pi\)
\(620\) 3.12854e9 0.527194
\(621\) 1.43502e9 0.240456
\(622\) 9.59045e8 0.159799
\(623\) −1.66766e9 −0.276311
\(624\) 2.42971e8 0.0400320
\(625\) −3.16814e9 −0.519068
\(626\) 9.49308e8 0.154667
\(627\) 1.12707e9 0.182605
\(628\) −4.39262e9 −0.707725
\(629\) 5.67331e8 0.0908990
\(630\) 8.58019e8 0.136712
\(631\) 1.46856e9 0.232695 0.116348 0.993209i \(-0.462881\pi\)
0.116348 + 0.993209i \(0.462881\pi\)
\(632\) 1.83894e9 0.289773
\(633\) −2.58365e9 −0.404875
\(634\) −6.82283e8 −0.106329
\(635\) −4.82209e9 −0.747355
\(636\) 1.50271e9 0.231620
\(637\) 2.58475e8 0.0396214
\(638\) −2.67829e9 −0.408305
\(639\) −3.07675e9 −0.466486
\(640\) 8.99529e8 0.135639
\(641\) −6.63645e9 −0.995250 −0.497625 0.867392i \(-0.665795\pi\)
−0.497625 + 0.867392i \(0.665795\pi\)
\(642\) 3.04155e9 0.453652
\(643\) −3.68719e8 −0.0546961 −0.0273481 0.999626i \(-0.508706\pi\)
−0.0273481 + 0.999626i \(0.508706\pi\)
\(644\) 1.60044e9 0.236124
\(645\) −5.89963e9 −0.865697
\(646\) 2.97754e8 0.0434555
\(647\) −4.62229e9 −0.670953 −0.335476 0.942049i \(-0.608897\pi\)
−0.335476 + 0.942049i \(0.608897\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) −9.24769e9 −1.32794
\(650\) −1.86051e9 −0.265727
\(651\) −1.05544e9 −0.149934
\(652\) −3.92739e9 −0.554929
\(653\) 6.70504e9 0.942335 0.471168 0.882044i \(-0.343833\pi\)
0.471168 + 0.882044i \(0.343833\pi\)
\(654\) −3.61185e9 −0.504902
\(655\) 6.39216e9 0.888797
\(656\) 9.17833e8 0.126940
\(657\) 1.25557e9 0.172727
\(658\) 1.24804e9 0.170780
\(659\) −2.22581e9 −0.302963 −0.151482 0.988460i \(-0.548404\pi\)
−0.151482 + 0.988460i \(0.548404\pi\)
\(660\) 4.18166e9 0.566167
\(661\) 5.49601e8 0.0740189 0.0370094 0.999315i \(-0.488217\pi\)
0.0370094 + 0.999315i \(0.488217\pi\)
\(662\) −2.35916e9 −0.316049
\(663\) 2.98397e8 0.0397647
\(664\) 4.16639e9 0.552296
\(665\) 1.08855e9 0.143539
\(666\) −6.57738e8 −0.0862766
\(667\) −4.32627e9 −0.564512
\(668\) −3.33794e9 −0.433272
\(669\) −3.40377e9 −0.439510
\(670\) 1.88867e8 0.0242602
\(671\) −1.21247e10 −1.54932
\(672\) −3.03464e8 −0.0385758
\(673\) 1.49304e10 1.88807 0.944036 0.329844i \(-0.106996\pi\)
0.944036 + 0.329844i \(0.106996\pi\)
\(674\) 5.66098e9 0.712168
\(675\) 2.08355e9 0.260759
\(676\) 3.08916e8 0.0384615
\(677\) 1.16981e10 1.44895 0.724476 0.689300i \(-0.242082\pi\)
0.724476 + 0.689300i \(0.242082\pi\)
\(678\) 4.28878e9 0.528481
\(679\) 2.36006e9 0.289320
\(680\) 1.10473e9 0.134733
\(681\) −7.21162e9 −0.875020
\(682\) −5.14381e9 −0.620925
\(683\) 3.79059e9 0.455234 0.227617 0.973751i \(-0.426907\pi\)
0.227617 + 0.973751i \(0.426907\pi\)
\(684\) −3.45203e8 −0.0412456
\(685\) −1.22117e10 −1.45164
\(686\) −3.22829e8 −0.0381802
\(687\) 3.05639e9 0.359633
\(688\) 2.08658e9 0.244273
\(689\) 1.91057e9 0.222534
\(690\) 6.75468e9 0.782768
\(691\) 1.05651e10 1.21815 0.609077 0.793111i \(-0.291540\pi\)
0.609077 + 0.793111i \(0.291540\pi\)
\(692\) −6.84377e8 −0.0785098
\(693\) −1.41072e9 −0.161018
\(694\) 3.82290e9 0.434145
\(695\) −1.53407e10 −1.73340
\(696\) 8.20317e8 0.0922252
\(697\) 1.12721e9 0.126093
\(698\) 5.97422e9 0.664947
\(699\) 1.23343e9 0.136598
\(700\) 2.32373e9 0.256061
\(701\) −1.41529e10 −1.55179 −0.775896 0.630860i \(-0.782702\pi\)
−0.775896 + 0.630860i \(0.782702\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) −8.34453e8 −0.0905855
\(704\) −1.47897e9 −0.159755
\(705\) 5.26735e9 0.566148
\(706\) −6.05317e9 −0.647390
\(707\) 2.61138e9 0.277909
\(708\) 2.83242e9 0.299945
\(709\) −6.21578e9 −0.654989 −0.327494 0.944853i \(-0.606204\pi\)
−0.327494 + 0.944853i \(0.606204\pi\)
\(710\) −1.44824e10 −1.51857
\(711\) −2.61834e9 −0.273201
\(712\) 2.48933e9 0.258466
\(713\) −8.30886e9 −0.858476
\(714\) −3.72691e8 −0.0383182
\(715\) 5.31661e9 0.543956
\(716\) 1.81272e9 0.184559
\(717\) −1.00902e10 −1.02231
\(718\) 8.09740e8 0.0816414
\(719\) −1.18293e10 −1.18688 −0.593439 0.804879i \(-0.702230\pi\)
−0.593439 + 0.804879i \(0.702230\pi\)
\(720\) −1.28078e9 −0.127882
\(721\) 2.29620e9 0.228158
\(722\) 6.71302e9 0.663801
\(723\) −4.25391e9 −0.418605
\(724\) 3.45469e9 0.338318
\(725\) −6.28145e9 −0.612177
\(726\) −2.66607e9 −0.258579
\(727\) −1.83987e10 −1.77589 −0.887947 0.459945i \(-0.847869\pi\)
−0.887947 + 0.459945i \(0.847869\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 5.91001e9 0.562287
\(731\) 2.56258e9 0.242642
\(732\) 3.71359e9 0.349949
\(733\) 1.19751e10 1.12309 0.561544 0.827447i \(-0.310207\pi\)
0.561544 + 0.827447i \(0.310207\pi\)
\(734\) 9.80725e9 0.915400
\(735\) −1.36250e9 −0.126570
\(736\) −2.38900e9 −0.220873
\(737\) −3.10527e8 −0.0285735
\(738\) −1.30684e9 −0.119681
\(739\) 1.22577e10 1.11726 0.558631 0.829416i \(-0.311327\pi\)
0.558631 + 0.829416i \(0.311327\pi\)
\(740\) −3.09600e9 −0.280860
\(741\) −4.38895e8 −0.0396275
\(742\) −2.38625e9 −0.214439
\(743\) 1.69423e10 1.51534 0.757671 0.652636i \(-0.226337\pi\)
0.757671 + 0.652636i \(0.226337\pi\)
\(744\) 1.57547e9 0.140250
\(745\) −3.72822e8 −0.0330335
\(746\) 4.20274e9 0.370635
\(747\) −5.93223e9 −0.520710
\(748\) −1.81635e9 −0.158688
\(749\) −4.82987e9 −0.420000
\(750\) 2.56916e9 0.222370
\(751\) −1.50990e9 −0.130080 −0.0650399 0.997883i \(-0.520717\pi\)
−0.0650399 + 0.997883i \(0.520717\pi\)
\(752\) −1.86296e9 −0.159750
\(753\) 2.27094e9 0.193831
\(754\) 1.04296e9 0.0886071
\(755\) 8.41841e9 0.711894
\(756\) 4.32081e8 0.0363696
\(757\) 1.06891e10 0.895586 0.447793 0.894137i \(-0.352210\pi\)
0.447793 + 0.894137i \(0.352210\pi\)
\(758\) −4.37633e9 −0.364979
\(759\) −1.11058e10 −0.921939
\(760\) −1.62488e9 −0.134269
\(761\) 7.92640e9 0.651973 0.325987 0.945374i \(-0.394304\pi\)
0.325987 + 0.945374i \(0.394304\pi\)
\(762\) −2.42831e9 −0.198820
\(763\) 5.73548e9 0.467449
\(764\) 2.43471e9 0.197524
\(765\) −1.57295e9 −0.127028
\(766\) −3.60829e9 −0.290069
\(767\) 3.60118e9 0.288178
\(768\) 4.52985e8 0.0360844
\(769\) −5.60784e9 −0.444686 −0.222343 0.974969i \(-0.571370\pi\)
−0.222343 + 0.974969i \(0.571370\pi\)
\(770\) −6.64031e9 −0.524169
\(771\) −7.50511e9 −0.589748
\(772\) 5.63989e9 0.441174
\(773\) 7.85923e9 0.612000 0.306000 0.952031i \(-0.401009\pi\)
0.306000 + 0.952031i \(0.401009\pi\)
\(774\) −2.97093e9 −0.230303
\(775\) −1.20639e10 −0.930961
\(776\) −3.52289e9 −0.270634
\(777\) 1.04446e9 0.0798766
\(778\) 3.00134e9 0.228500
\(779\) −1.65795e9 −0.125658
\(780\) −1.62839e9 −0.122865
\(781\) 2.38113e10 1.78857
\(782\) −2.93398e9 −0.219398
\(783\) −1.16799e9 −0.0869507
\(784\) 4.81890e8 0.0357143
\(785\) 2.94394e10 2.17213
\(786\) 3.21896e9 0.236449
\(787\) −8.00008e9 −0.585036 −0.292518 0.956260i \(-0.594493\pi\)
−0.292518 + 0.956260i \(0.594493\pi\)
\(788\) 5.46390e9 0.397797
\(789\) −8.77279e9 −0.635870
\(790\) −1.23246e10 −0.889363
\(791\) −6.81042e9 −0.489279
\(792\) 2.10580e9 0.150619
\(793\) 4.72150e9 0.336220
\(794\) 6.90398e9 0.489472
\(795\) −1.00712e10 −0.710881
\(796\) −2.64103e9 −0.185600
\(797\) −1.46243e10 −1.02323 −0.511614 0.859215i \(-0.670952\pi\)
−0.511614 + 0.859215i \(0.670952\pi\)
\(798\) 5.48169e8 0.0381860
\(799\) −2.28794e9 −0.158683
\(800\) −3.46866e9 −0.239523
\(801\) −3.54438e9 −0.243684
\(802\) −9.74998e9 −0.667411
\(803\) −9.71699e9 −0.662258
\(804\) 9.51096e7 0.00645399
\(805\) −1.07262e10 −0.724702
\(806\) 2.00307e9 0.134748
\(807\) −1.93706e9 −0.129743
\(808\) −3.89803e9 −0.259960
\(809\) −8.61170e9 −0.571833 −0.285916 0.958255i \(-0.592298\pi\)
−0.285916 + 0.958255i \(0.592298\pi\)
\(810\) 1.82360e9 0.120568
\(811\) 1.28149e10 0.843609 0.421804 0.906687i \(-0.361397\pi\)
0.421804 + 0.906687i \(0.361397\pi\)
\(812\) −1.30263e9 −0.0853839
\(813\) 5.76862e9 0.376491
\(814\) 5.09031e9 0.330795
\(815\) 2.63214e10 1.70317
\(816\) 5.56320e8 0.0358434
\(817\) −3.76914e9 −0.241805
\(818\) −5.02741e9 −0.321150
\(819\) 5.49353e8 0.0349428
\(820\) −6.15133e9 −0.389601
\(821\) 1.55861e10 0.982964 0.491482 0.870888i \(-0.336455\pi\)
0.491482 + 0.870888i \(0.336455\pi\)
\(822\) −6.14957e9 −0.386184
\(823\) 1.84906e10 1.15625 0.578125 0.815948i \(-0.303785\pi\)
0.578125 + 0.815948i \(0.303785\pi\)
\(824\) −3.42757e9 −0.213423
\(825\) −1.61248e10 −0.999783
\(826\) −4.49778e9 −0.277695
\(827\) −3.22049e10 −1.97994 −0.989970 0.141275i \(-0.954880\pi\)
−0.989970 + 0.141275i \(0.954880\pi\)
\(828\) 3.40152e9 0.208241
\(829\) 2.14528e10 1.30781 0.653903 0.756579i \(-0.273131\pi\)
0.653903 + 0.756579i \(0.273131\pi\)
\(830\) −2.79232e10 −1.69509
\(831\) −1.41000e10 −0.852346
\(832\) 5.75930e8 0.0346688
\(833\) 5.91820e8 0.0354758
\(834\) −7.72527e9 −0.461139
\(835\) 2.23709e10 1.32978
\(836\) 2.67157e9 0.158141
\(837\) −2.24319e9 −0.132229
\(838\) −1.82534e10 −1.07149
\(839\) 1.62577e9 0.0950372 0.0475186 0.998870i \(-0.484869\pi\)
0.0475186 + 0.998870i \(0.484869\pi\)
\(840\) 2.03382e9 0.118396
\(841\) −1.37286e10 −0.795868
\(842\) 2.37445e10 1.37079
\(843\) −1.02638e10 −0.590080
\(844\) −6.12421e9 −0.350632
\(845\) −2.07036e9 −0.118045
\(846\) 2.65253e9 0.150614
\(847\) 4.23363e9 0.239398
\(848\) 3.56199e9 0.200589
\(849\) 1.60869e9 0.0902183
\(850\) −4.25994e9 −0.237923
\(851\) 8.22244e9 0.457349
\(852\) −7.29303e9 −0.403989
\(853\) −2.46400e10 −1.35931 −0.679657 0.733531i \(-0.737871\pi\)
−0.679657 + 0.733531i \(0.737871\pi\)
\(854\) −5.89704e9 −0.323990
\(855\) 2.31356e9 0.126590
\(856\) 7.20961e9 0.392874
\(857\) 1.99504e10 1.08273 0.541364 0.840788i \(-0.317908\pi\)
0.541364 + 0.840788i \(0.317908\pi\)
\(858\) 2.67734e9 0.144710
\(859\) 2.30206e10 1.23920 0.619598 0.784919i \(-0.287296\pi\)
0.619598 + 0.784919i \(0.287296\pi\)
\(860\) −1.39843e10 −0.749715
\(861\) 2.07521e9 0.110803
\(862\) −7.96974e9 −0.423808
\(863\) 2.84131e10 1.50481 0.752403 0.658704i \(-0.228895\pi\)
0.752403 + 0.658704i \(0.228895\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 4.58670e9 0.240960
\(866\) 2.20005e10 1.15112
\(867\) −1.03959e10 −0.541746
\(868\) −2.50178e9 −0.129847
\(869\) 2.02636e10 1.04749
\(870\) −5.49778e9 −0.283054
\(871\) 1.20923e8 0.00620079
\(872\) −8.56142e9 −0.437258
\(873\) 5.01599e9 0.255157
\(874\) 4.31541e9 0.218641
\(875\) −4.07973e9 −0.205875
\(876\) 2.97616e9 0.149586
\(877\) −2.43482e10 −1.21890 −0.609450 0.792825i \(-0.708610\pi\)
−0.609450 + 0.792825i \(0.708610\pi\)
\(878\) 3.39958e9 0.169509
\(879\) −1.00569e10 −0.499463
\(880\) 9.91207e9 0.490315
\(881\) −2.41724e10 −1.19098 −0.595490 0.803363i \(-0.703042\pi\)
−0.595490 + 0.803363i \(0.703042\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −1.97895e10 −0.967324 −0.483662 0.875255i \(-0.660694\pi\)
−0.483662 + 0.875255i \(0.660694\pi\)
\(884\) 7.07313e8 0.0344372
\(885\) −1.89829e10 −0.920581
\(886\) −1.54211e10 −0.744901
\(887\) −3.75324e10 −1.80582 −0.902908 0.429833i \(-0.858572\pi\)
−0.902908 + 0.429833i \(0.858572\pi\)
\(888\) −1.55908e9 −0.0747177
\(889\) 3.85606e9 0.184072
\(890\) −1.66835e10 −0.793274
\(891\) −2.99829e9 −0.142005
\(892\) −8.06820e9 −0.380627
\(893\) 3.36519e9 0.158136
\(894\) −1.87746e8 −0.00878798
\(895\) −1.21489e10 −0.566443
\(896\) −7.19323e8 −0.0334077
\(897\) 4.32473e9 0.200072
\(898\) 1.58286e10 0.729417
\(899\) 6.76276e9 0.310431
\(900\) 4.93878e9 0.225824
\(901\) 4.37456e9 0.199250
\(902\) 1.01138e10 0.458870
\(903\) 4.71773e9 0.213219
\(904\) 1.01660e10 0.457678
\(905\) −2.31534e10 −1.03835
\(906\) 4.23934e9 0.189387
\(907\) 3.67145e10 1.63385 0.816924 0.576745i \(-0.195678\pi\)
0.816924 + 0.576745i \(0.195678\pi\)
\(908\) −1.70942e10 −0.757790
\(909\) 5.55013e9 0.245092
\(910\) 2.58583e9 0.113751
\(911\) 3.24643e10 1.42263 0.711315 0.702874i \(-0.248100\pi\)
0.711315 + 0.702874i \(0.248100\pi\)
\(912\) −8.18259e8 −0.0357198
\(913\) 4.59102e10 1.99646
\(914\) −2.13540e10 −0.925054
\(915\) −2.48885e10 −1.07405
\(916\) 7.24477e9 0.311452
\(917\) −5.11159e9 −0.218909
\(918\) −7.92105e8 −0.0337935
\(919\) −2.65751e10 −1.12946 −0.564730 0.825276i \(-0.691020\pi\)
−0.564730 + 0.825276i \(0.691020\pi\)
\(920\) 1.60111e10 0.677897
\(921\) −1.19290e10 −0.503148
\(922\) 1.75528e10 0.737543
\(923\) −9.27245e9 −0.388140
\(924\) −3.34393e9 −0.139446
\(925\) 1.19384e10 0.495965
\(926\) −3.48604e9 −0.144276
\(927\) 4.88026e9 0.201217
\(928\) 1.94446e9 0.0798693
\(929\) 3.05820e10 1.25144 0.625722 0.780046i \(-0.284805\pi\)
0.625722 + 0.780046i \(0.284805\pi\)
\(930\) −1.05588e10 −0.430452
\(931\) −8.70473e8 −0.0353534
\(932\) 2.92368e9 0.118297
\(933\) −3.23678e9 −0.130475
\(934\) −3.01470e9 −0.121068
\(935\) 1.21732e10 0.487041
\(936\) −8.20026e8 −0.0326860
\(937\) 2.98201e10 1.18419 0.592095 0.805868i \(-0.298301\pi\)
0.592095 + 0.805868i \(0.298301\pi\)
\(938\) −1.51031e8 −0.00597523
\(939\) −3.20391e9 −0.126285
\(940\) 1.24856e10 0.490299
\(941\) 3.92420e10 1.53528 0.767639 0.640882i \(-0.221431\pi\)
0.767639 + 0.640882i \(0.221431\pi\)
\(942\) 1.48251e10 0.577855
\(943\) 1.63369e10 0.634422
\(944\) 6.71389e9 0.259760
\(945\) −2.89582e9 −0.111625
\(946\) 2.29924e10 0.883010
\(947\) −3.38049e10 −1.29346 −0.646732 0.762718i \(-0.723865\pi\)
−0.646732 + 0.762718i \(0.723865\pi\)
\(948\) −6.20643e9 −0.236599
\(949\) 3.78393e9 0.143718
\(950\) 6.26569e9 0.237103
\(951\) 2.30270e9 0.0868173
\(952\) −8.83416e8 −0.0331846
\(953\) −8.95517e9 −0.335157 −0.167579 0.985859i \(-0.553595\pi\)
−0.167579 + 0.985859i \(0.553595\pi\)
\(954\) −5.07166e9 −0.189117
\(955\) −1.63174e10 −0.606234
\(956\) −2.39176e10 −0.885349
\(957\) 9.03922e9 0.333380
\(958\) 1.40587e9 0.0516612
\(959\) 9.76530e9 0.357536
\(960\) −3.03591e9 −0.110749
\(961\) −1.45243e10 −0.527916
\(962\) −1.98224e9 −0.0717865
\(963\) −1.02652e10 −0.370405
\(964\) −1.00833e10 −0.362522
\(965\) −3.77986e10 −1.35404
\(966\) −5.40149e9 −0.192794
\(967\) 2.29472e10 0.816089 0.408044 0.912962i \(-0.366211\pi\)
0.408044 + 0.912962i \(0.366211\pi\)
\(968\) −6.31958e9 −0.223936
\(969\) −1.00492e9 −0.0354812
\(970\) 2.36105e10 0.830622
\(971\) 3.46493e9 0.121458 0.0607292 0.998154i \(-0.480657\pi\)
0.0607292 + 0.998154i \(0.480657\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 1.22674e10 0.426932
\(974\) 3.13028e10 1.08549
\(975\) 6.27922e9 0.216965
\(976\) 8.80258e9 0.303065
\(977\) −2.44027e9 −0.0837158 −0.0418579 0.999124i \(-0.513328\pi\)
−0.0418579 + 0.999124i \(0.513328\pi\)
\(978\) 1.32549e10 0.453098
\(979\) 2.74304e10 0.934313
\(980\) −3.22964e9 −0.109613
\(981\) 1.21900e10 0.412251
\(982\) 3.14004e9 0.105814
\(983\) −1.38431e10 −0.464833 −0.232417 0.972616i \(-0.574663\pi\)
−0.232417 + 0.972616i \(0.574663\pi\)
\(984\) −3.09768e9 −0.103646
\(985\) −3.66191e10 −1.22090
\(986\) 2.38803e9 0.0793360
\(987\) −4.21212e9 −0.139441
\(988\) −1.04034e9 −0.0343184
\(989\) 3.71399e10 1.22083
\(990\) −1.41131e10 −0.462273
\(991\) −4.50663e10 −1.47094 −0.735469 0.677558i \(-0.763038\pi\)
−0.735469 + 0.677558i \(0.763038\pi\)
\(992\) 3.73444e9 0.121460
\(993\) 7.96218e9 0.258053
\(994\) 1.15811e10 0.374021
\(995\) 1.77002e10 0.569636
\(996\) −1.40616e10 −0.450948
\(997\) −2.19579e10 −0.701710 −0.350855 0.936430i \(-0.614109\pi\)
−0.350855 + 0.936430i \(0.614109\pi\)
\(998\) 1.86018e10 0.592377
\(999\) 2.21986e9 0.0704445
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.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.i.1.1 5 1.1 even 1 trivial