Properties

Label 546.8.a.i.1.2
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 1.2
Root \(-108.730\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -114.198 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -114.198 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +913.581 q^{10} -4322.64 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -3083.34 q^{15} +4096.00 q^{16} -5472.07 q^{17} -5832.00 q^{18} +31062.8 q^{19} -7308.65 q^{20} +9261.00 q^{21} +34581.1 q^{22} -40956.7 q^{23} -13824.0 q^{24} -65083.9 q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} +161624. q^{29} +24666.7 q^{30} +151931. q^{31} -32768.0 q^{32} -116711. q^{33} +43776.6 q^{34} -39169.8 q^{35} +46656.0 q^{36} -201642. q^{37} -248502. q^{38} +59319.0 q^{39} +58469.2 q^{40} -620702. q^{41} -74088.0 q^{42} +498168. q^{43} -276649. q^{44} -83250.1 q^{45} +327654. q^{46} +1.15999e6 q^{47} +110592. q^{48} +117649. q^{49} +520671. q^{50} -147746. q^{51} +140608. q^{52} -1.47959e6 q^{53} -157464. q^{54} +493636. q^{55} -175616. q^{56} +838694. q^{57} -1.29299e6 q^{58} -2.10534e6 q^{59} -197334. q^{60} +1.77641e6 q^{61} -1.21545e6 q^{62} +250047. q^{63} +262144. q^{64} -250892. q^{65} +933691. q^{66} -171815. q^{67} -350213. q^{68} -1.10583e6 q^{69} +313358. q^{70} -1.49004e6 q^{71} -373248. q^{72} -2.29869e6 q^{73} +1.61314e6 q^{74} -1.75727e6 q^{75} +1.98802e6 q^{76} -1.48267e6 q^{77} -474552. q^{78} +4.20237e6 q^{79} -467754. q^{80} +531441. q^{81} +4.96562e6 q^{82} +6.53999e6 q^{83} +592704. q^{84} +624898. q^{85} -3.98534e6 q^{86} +4.36384e6 q^{87} +2.21319e6 q^{88} -4.71042e6 q^{89} +666001. q^{90} +753571. q^{91} -2.62123e6 q^{92} +4.10214e6 q^{93} -9.27989e6 q^{94} -3.54729e6 q^{95} -884736. q^{96} -1.12411e7 q^{97} -941192. q^{98} -3.15121e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99} + O(q^{100}) \)

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
\(5\) −114.198 −0.408566 −0.204283 0.978912i \(-0.565486\pi\)
−0.204283 + 0.978912i \(0.565486\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 913.581 0.288900
\(11\) −4322.64 −0.979208 −0.489604 0.871945i \(-0.662859\pi\)
−0.489604 + 0.871945i \(0.662859\pi\)
\(12\) 1728.00 0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −3083.34 −0.235886
\(16\) 4096.00 0.250000
\(17\) −5472.07 −0.270135 −0.135067 0.990836i \(-0.543125\pi\)
−0.135067 + 0.990836i \(0.543125\pi\)
\(18\) −5832.00 −0.235702
\(19\) 31062.8 1.03897 0.519484 0.854480i \(-0.326124\pi\)
0.519484 + 0.854480i \(0.326124\pi\)
\(20\) −7308.65 −0.204283
\(21\) 9261.00 0.218218
\(22\) 34581.1 0.692405
\(23\) −40956.7 −0.701904 −0.350952 0.936393i \(-0.614142\pi\)
−0.350952 + 0.936393i \(0.614142\pi\)
\(24\) −13824.0 −0.204124
\(25\) −65083.9 −0.833074
\(26\) −17576.0 −0.196116
\(27\) 19683.0 0.192450
\(28\) 21952.0 0.188982
\(29\) 161624. 1.23059 0.615293 0.788299i \(-0.289038\pi\)
0.615293 + 0.788299i \(0.289038\pi\)
\(30\) 24666.7 0.166796
\(31\) 151931. 0.915968 0.457984 0.888960i \(-0.348572\pi\)
0.457984 + 0.888960i \(0.348572\pi\)
\(32\) −32768.0 −0.176777
\(33\) −116711. −0.565346
\(34\) 43776.6 0.191014
\(35\) −39169.8 −0.154423
\(36\) 46656.0 0.166667
\(37\) −201642. −0.654449 −0.327225 0.944947i \(-0.606113\pi\)
−0.327225 + 0.944947i \(0.606113\pi\)
\(38\) −248502. −0.734662
\(39\) 59319.0 0.160128
\(40\) 58469.2 0.144450
\(41\) −620702. −1.40650 −0.703250 0.710942i \(-0.748269\pi\)
−0.703250 + 0.710942i \(0.748269\pi\)
\(42\) −74088.0 −0.154303
\(43\) 498168. 0.955512 0.477756 0.878493i \(-0.341450\pi\)
0.477756 + 0.878493i \(0.341450\pi\)
\(44\) −276649. −0.489604
\(45\) −83250.1 −0.136189
\(46\) 327654. 0.496321
\(47\) 1.15999e6 1.62971 0.814855 0.579665i \(-0.196816\pi\)
0.814855 + 0.579665i \(0.196816\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) 520671. 0.589072
\(51\) −147746. −0.155962
\(52\) 140608. 0.138675
\(53\) −1.47959e6 −1.36514 −0.682569 0.730821i \(-0.739138\pi\)
−0.682569 + 0.730821i \(0.739138\pi\)
\(54\) −157464. −0.136083
\(55\) 493636. 0.400071
\(56\) −175616. −0.133631
\(57\) 838694. 0.599849
\(58\) −1.29299e6 −0.870155
\(59\) −2.10534e6 −1.33457 −0.667283 0.744804i \(-0.732543\pi\)
−0.667283 + 0.744804i \(0.732543\pi\)
\(60\) −197334. −0.117943
\(61\) 1.77641e6 1.00205 0.501025 0.865433i \(-0.332956\pi\)
0.501025 + 0.865433i \(0.332956\pi\)
\(62\) −1.21545e6 −0.647687
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −250892. −0.113316
\(66\) 933691. 0.399760
\(67\) −171815. −0.0697908 −0.0348954 0.999391i \(-0.511110\pi\)
−0.0348954 + 0.999391i \(0.511110\pi\)
\(68\) −350213. −0.135067
\(69\) −1.10583e6 −0.405244
\(70\) 313358. 0.109194
\(71\) −1.49004e6 −0.494074 −0.247037 0.969006i \(-0.579457\pi\)
−0.247037 + 0.969006i \(0.579457\pi\)
\(72\) −373248. −0.117851
\(73\) −2.29869e6 −0.691593 −0.345797 0.938309i \(-0.612391\pi\)
−0.345797 + 0.938309i \(0.612391\pi\)
\(74\) 1.61314e6 0.462765
\(75\) −1.75727e6 −0.480975
\(76\) 1.98802e6 0.519484
\(77\) −1.48267e6 −0.370106
\(78\) −474552. −0.113228
\(79\) 4.20237e6 0.958959 0.479479 0.877553i \(-0.340826\pi\)
0.479479 + 0.877553i \(0.340826\pi\)
\(80\) −467754. −0.102141
\(81\) 531441. 0.111111
\(82\) 4.96562e6 0.994546
\(83\) 6.53999e6 1.25546 0.627732 0.778430i \(-0.283984\pi\)
0.627732 + 0.778430i \(0.283984\pi\)
\(84\) 592704. 0.109109
\(85\) 624898. 0.110368
\(86\) −3.98534e6 −0.675649
\(87\) 4.36384e6 0.710479
\(88\) 2.21319e6 0.346202
\(89\) −4.71042e6 −0.708263 −0.354132 0.935196i \(-0.615223\pi\)
−0.354132 + 0.935196i \(0.615223\pi\)
\(90\) 666001. 0.0962999
\(91\) 753571. 0.104828
\(92\) −2.62123e6 −0.350952
\(93\) 4.10214e6 0.528835
\(94\) −9.27989e6 −1.15238
\(95\) −3.54729e6 −0.424487
\(96\) −884736. −0.102062
\(97\) −1.12411e7 −1.25057 −0.625283 0.780398i \(-0.715016\pi\)
−0.625283 + 0.780398i \(0.715016\pi\)
\(98\) −941192. −0.101015
\(99\) −3.15121e6 −0.326403
\(100\) −4.16537e6 −0.416537
\(101\) −7.60490e6 −0.734461 −0.367231 0.930130i \(-0.619694\pi\)
−0.367231 + 0.930130i \(0.619694\pi\)
\(102\) 1.18197e6 0.110282
\(103\) −1.85115e7 −1.66921 −0.834607 0.550846i \(-0.814305\pi\)
−0.834607 + 0.550846i \(0.814305\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −1.05758e6 −0.0891564
\(106\) 1.18367e7 0.965299
\(107\) −7.08860e6 −0.559393 −0.279697 0.960088i \(-0.590234\pi\)
−0.279697 + 0.960088i \(0.590234\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 5.65530e6 0.418276 0.209138 0.977886i \(-0.432934\pi\)
0.209138 + 0.977886i \(0.432934\pi\)
\(110\) −3.94908e6 −0.282893
\(111\) −5.44435e6 −0.377846
\(112\) 1.40493e6 0.0944911
\(113\) 1.95653e7 1.27559 0.637797 0.770204i \(-0.279846\pi\)
0.637797 + 0.770204i \(0.279846\pi\)
\(114\) −6.70955e6 −0.424157
\(115\) 4.67716e6 0.286774
\(116\) 1.03439e7 0.615293
\(117\) 1.60161e6 0.0924500
\(118\) 1.68427e7 0.943681
\(119\) −1.87692e6 −0.102101
\(120\) 1.57867e6 0.0833982
\(121\) −801936. −0.0411520
\(122\) −1.42113e7 −0.708557
\(123\) −1.67590e7 −0.812044
\(124\) 9.72358e6 0.457984
\(125\) 1.63541e7 0.748932
\(126\) −2.00038e6 −0.0890871
\(127\) 4.75787e6 0.206110 0.103055 0.994676i \(-0.467138\pi\)
0.103055 + 0.994676i \(0.467138\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 1.34505e7 0.551665
\(130\) 2.00714e6 0.0801264
\(131\) −1.09360e7 −0.425018 −0.212509 0.977159i \(-0.568164\pi\)
−0.212509 + 0.977159i \(0.568164\pi\)
\(132\) −7.46953e6 −0.282673
\(133\) 1.06545e7 0.392693
\(134\) 1.37452e6 0.0493496
\(135\) −2.24775e6 −0.0786286
\(136\) 2.80170e6 0.0955071
\(137\) −164680. −0.00547164 −0.00273582 0.999996i \(-0.500871\pi\)
−0.00273582 + 0.999996i \(0.500871\pi\)
\(138\) 8.84665e6 0.286551
\(139\) −4.29877e7 −1.35766 −0.678832 0.734294i \(-0.737514\pi\)
−0.678832 + 0.734294i \(0.737514\pi\)
\(140\) −2.50687e6 −0.0772117
\(141\) 3.13196e7 0.940914
\(142\) 1.19203e7 0.349363
\(143\) −9.49684e6 −0.271583
\(144\) 2.98598e6 0.0833333
\(145\) −1.84570e7 −0.502775
\(146\) 1.83895e7 0.489030
\(147\) 3.17652e6 0.0824786
\(148\) −1.29051e7 −0.327225
\(149\) −4.65198e7 −1.15209 −0.576044 0.817419i \(-0.695404\pi\)
−0.576044 + 0.817419i \(0.695404\pi\)
\(150\) 1.40581e7 0.340101
\(151\) 6.75745e7 1.59721 0.798607 0.601853i \(-0.205571\pi\)
0.798607 + 0.601853i \(0.205571\pi\)
\(152\) −1.59041e7 −0.367331
\(153\) −3.98914e6 −0.0900449
\(154\) 1.18613e7 0.261704
\(155\) −1.73502e7 −0.374233
\(156\) 3.79642e6 0.0800641
\(157\) 5.56786e7 1.14826 0.574129 0.818765i \(-0.305341\pi\)
0.574129 + 0.818765i \(0.305341\pi\)
\(158\) −3.36190e7 −0.678086
\(159\) −3.99490e7 −0.788163
\(160\) 3.74203e6 0.0722249
\(161\) −1.40482e7 −0.265295
\(162\) −4.25153e6 −0.0785674
\(163\) 5.39876e7 0.976422 0.488211 0.872726i \(-0.337650\pi\)
0.488211 + 0.872726i \(0.337650\pi\)
\(164\) −3.97250e7 −0.703250
\(165\) 1.33282e7 0.230981
\(166\) −5.23199e7 −0.887746
\(167\) 4.34162e7 0.721347 0.360674 0.932692i \(-0.382547\pi\)
0.360674 + 0.932692i \(0.382547\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −4.99918e6 −0.0780419
\(171\) 2.26447e7 0.346323
\(172\) 3.18827e7 0.477756
\(173\) 3.16989e7 0.465461 0.232730 0.972541i \(-0.425234\pi\)
0.232730 + 0.972541i \(0.425234\pi\)
\(174\) −3.49107e7 −0.502385
\(175\) −2.23238e7 −0.314872
\(176\) −1.77055e7 −0.244802
\(177\) −5.68442e7 −0.770512
\(178\) 3.76834e7 0.500818
\(179\) 9.72037e7 1.26677 0.633384 0.773838i \(-0.281665\pi\)
0.633384 + 0.773838i \(0.281665\pi\)
\(180\) −5.32801e6 −0.0680943
\(181\) 5.37158e6 0.0673328 0.0336664 0.999433i \(-0.489282\pi\)
0.0336664 + 0.999433i \(0.489282\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 4.79632e7 0.578534
\(184\) 2.09698e7 0.248161
\(185\) 2.30271e7 0.267386
\(186\) −3.28171e7 −0.373942
\(187\) 2.36538e7 0.264518
\(188\) 7.42391e7 0.814855
\(189\) 6.75127e6 0.0727393
\(190\) 2.83783e7 0.300158
\(191\) 1.02776e8 1.06727 0.533635 0.845715i \(-0.320826\pi\)
0.533635 + 0.845715i \(0.320826\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −2.52063e7 −0.252382 −0.126191 0.992006i \(-0.540275\pi\)
−0.126191 + 0.992006i \(0.540275\pi\)
\(194\) 8.99285e7 0.884284
\(195\) −6.77409e6 −0.0654229
\(196\) 7.52954e6 0.0714286
\(197\) −1.68194e8 −1.56740 −0.783698 0.621142i \(-0.786669\pi\)
−0.783698 + 0.621142i \(0.786669\pi\)
\(198\) 2.52096e7 0.230802
\(199\) −1.67863e8 −1.50998 −0.754988 0.655739i \(-0.772357\pi\)
−0.754988 + 0.655739i \(0.772357\pi\)
\(200\) 3.33230e7 0.294536
\(201\) −4.63900e6 −0.0402938
\(202\) 6.08392e7 0.519343
\(203\) 5.54369e7 0.465118
\(204\) −9.45574e6 −0.0779812
\(205\) 7.08828e7 0.574648
\(206\) 1.48092e8 1.18031
\(207\) −2.98574e7 −0.233968
\(208\) 8.99891e6 0.0693375
\(209\) −1.34273e8 −1.01737
\(210\) 8.46068e6 0.0630431
\(211\) −2.02999e8 −1.48766 −0.743832 0.668367i \(-0.766994\pi\)
−0.743832 + 0.668367i \(0.766994\pi\)
\(212\) −9.46940e7 −0.682569
\(213\) −4.02309e7 −0.285254
\(214\) 5.67088e7 0.395551
\(215\) −5.68896e7 −0.390390
\(216\) −1.00777e7 −0.0680414
\(217\) 5.21123e7 0.346203
\(218\) −4.52424e7 −0.295766
\(219\) −6.20647e7 −0.399292
\(220\) 3.15927e7 0.200035
\(221\) −1.20221e7 −0.0749219
\(222\) 4.35548e7 0.267178
\(223\) 4.14075e7 0.250041 0.125021 0.992154i \(-0.460100\pi\)
0.125021 + 0.992154i \(0.460100\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −4.74462e7 −0.277691
\(226\) −1.56523e8 −0.901982
\(227\) −3.40587e8 −1.93258 −0.966290 0.257458i \(-0.917115\pi\)
−0.966290 + 0.257458i \(0.917115\pi\)
\(228\) 5.36764e7 0.299924
\(229\) 6.31916e7 0.347725 0.173862 0.984770i \(-0.444375\pi\)
0.173862 + 0.984770i \(0.444375\pi\)
\(230\) −3.74173e7 −0.202780
\(231\) −4.00320e7 −0.213681
\(232\) −8.27513e7 −0.435078
\(233\) −1.10790e8 −0.573792 −0.286896 0.957962i \(-0.592623\pi\)
−0.286896 + 0.957962i \(0.592623\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) −1.32468e8 −0.665844
\(236\) −1.34742e8 −0.667283
\(237\) 1.13464e8 0.553655
\(238\) 1.50154e7 0.0721966
\(239\) −9.61311e7 −0.455482 −0.227741 0.973722i \(-0.573134\pi\)
−0.227741 + 0.973722i \(0.573134\pi\)
\(240\) −1.26293e7 −0.0589714
\(241\) −3.14902e8 −1.44916 −0.724580 0.689191i \(-0.757966\pi\)
−0.724580 + 0.689191i \(0.757966\pi\)
\(242\) 6.41549e6 0.0290988
\(243\) 1.43489e7 0.0641500
\(244\) 1.13690e8 0.501025
\(245\) −1.34352e7 −0.0583666
\(246\) 1.34072e8 0.574201
\(247\) 6.82449e7 0.288158
\(248\) −7.77887e7 −0.323844
\(249\) 1.76580e8 0.724842
\(250\) −1.30833e8 −0.529575
\(251\) −3.54522e8 −1.41509 −0.707547 0.706666i \(-0.750198\pi\)
−0.707547 + 0.706666i \(0.750198\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 1.77041e8 0.687310
\(254\) −3.80629e7 −0.145742
\(255\) 1.68722e7 0.0637209
\(256\) 1.67772e7 0.0625000
\(257\) 7.77123e7 0.285577 0.142789 0.989753i \(-0.454393\pi\)
0.142789 + 0.989753i \(0.454393\pi\)
\(258\) −1.07604e8 −0.390086
\(259\) −6.91633e7 −0.247358
\(260\) −1.60571e7 −0.0566579
\(261\) 1.17824e8 0.410195
\(262\) 8.74877e7 0.300533
\(263\) −2.18870e8 −0.741894 −0.370947 0.928654i \(-0.620967\pi\)
−0.370947 + 0.928654i \(0.620967\pi\)
\(264\) 5.97562e7 0.199880
\(265\) 1.68966e8 0.557749
\(266\) −8.52362e7 −0.277676
\(267\) −1.27181e8 −0.408916
\(268\) −1.09961e7 −0.0348954
\(269\) −3.37329e8 −1.05663 −0.528313 0.849050i \(-0.677175\pi\)
−0.528313 + 0.849050i \(0.677175\pi\)
\(270\) 1.79820e7 0.0555988
\(271\) −4.92865e8 −1.50430 −0.752152 0.658990i \(-0.770984\pi\)
−0.752152 + 0.658990i \(0.770984\pi\)
\(272\) −2.24136e7 −0.0675337
\(273\) 2.03464e7 0.0605228
\(274\) 1.31744e6 0.00386904
\(275\) 2.81334e8 0.815752
\(276\) −7.07732e7 −0.202622
\(277\) −4.95338e7 −0.140030 −0.0700152 0.997546i \(-0.522305\pi\)
−0.0700152 + 0.997546i \(0.522305\pi\)
\(278\) 3.43902e8 0.960014
\(279\) 1.10758e8 0.305323
\(280\) 2.00549e7 0.0545969
\(281\) −4.36151e8 −1.17264 −0.586320 0.810080i \(-0.699424\pi\)
−0.586320 + 0.810080i \(0.699424\pi\)
\(282\) −2.50557e8 −0.665327
\(283\) −5.47947e8 −1.43710 −0.718548 0.695477i \(-0.755193\pi\)
−0.718548 + 0.695477i \(0.755193\pi\)
\(284\) −9.53623e7 −0.247037
\(285\) −9.57769e7 −0.245078
\(286\) 7.59748e7 0.192038
\(287\) −2.12901e8 −0.531607
\(288\) −2.38879e7 −0.0589256
\(289\) −3.80395e8 −0.927027
\(290\) 1.47656e8 0.355516
\(291\) −3.03509e8 −0.722014
\(292\) −1.47116e8 −0.345797
\(293\) −6.76978e8 −1.57231 −0.786154 0.618030i \(-0.787931\pi\)
−0.786154 + 0.618030i \(0.787931\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 2.40425e8 0.545258
\(296\) 1.03241e8 0.231383
\(297\) −8.50826e7 −0.188449
\(298\) 3.72158e8 0.814649
\(299\) −8.99819e7 −0.194673
\(300\) −1.12465e8 −0.240488
\(301\) 1.70872e8 0.361150
\(302\) −5.40596e8 −1.12940
\(303\) −2.05332e8 −0.424041
\(304\) 1.27233e8 0.259742
\(305\) −2.02862e8 −0.409404
\(306\) 3.19131e7 0.0636714
\(307\) −4.39325e8 −0.866566 −0.433283 0.901258i \(-0.642645\pi\)
−0.433283 + 0.901258i \(0.642645\pi\)
\(308\) −9.48906e7 −0.185053
\(309\) −4.99811e8 −0.963721
\(310\) 1.38801e8 0.264623
\(311\) 9.91159e8 1.86845 0.934226 0.356681i \(-0.116092\pi\)
0.934226 + 0.356681i \(0.116092\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) 9.12174e8 1.68140 0.840702 0.541497i \(-0.182142\pi\)
0.840702 + 0.541497i \(0.182142\pi\)
\(314\) −4.45429e8 −0.811941
\(315\) −2.85548e7 −0.0514745
\(316\) 2.68952e8 0.479479
\(317\) −1.08134e9 −1.90658 −0.953290 0.302058i \(-0.902327\pi\)
−0.953290 + 0.302058i \(0.902327\pi\)
\(318\) 3.19592e8 0.557316
\(319\) −6.98641e8 −1.20500
\(320\) −2.99362e7 −0.0510707
\(321\) −1.91392e8 −0.322966
\(322\) 1.12385e8 0.187592
\(323\) −1.69978e8 −0.280662
\(324\) 3.40122e7 0.0555556
\(325\) −1.42989e8 −0.231053
\(326\) −4.31901e8 −0.690435
\(327\) 1.52693e8 0.241492
\(328\) 3.17800e8 0.497273
\(329\) 3.97875e8 0.615973
\(330\) −1.06625e8 −0.163328
\(331\) −5.18637e8 −0.786077 −0.393039 0.919522i \(-0.628576\pi\)
−0.393039 + 0.919522i \(0.628576\pi\)
\(332\) 4.18560e8 0.627732
\(333\) −1.46997e8 −0.218150
\(334\) −3.47330e8 −0.510069
\(335\) 1.96208e7 0.0285142
\(336\) 3.79331e7 0.0545545
\(337\) −1.70997e8 −0.243379 −0.121690 0.992568i \(-0.538831\pi\)
−0.121690 + 0.992568i \(0.538831\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) 5.28264e8 0.736465
\(340\) 3.99934e7 0.0551839
\(341\) −6.56743e8 −0.896923
\(342\) −1.81158e8 −0.244887
\(343\) 4.03536e7 0.0539949
\(344\) −2.55062e8 −0.337824
\(345\) 1.26283e8 0.165569
\(346\) −2.53591e8 −0.329130
\(347\) −1.48931e8 −0.191352 −0.0956759 0.995413i \(-0.530501\pi\)
−0.0956759 + 0.995413i \(0.530501\pi\)
\(348\) 2.79286e8 0.355239
\(349\) −1.40034e9 −1.76338 −0.881689 0.471832i \(-0.843593\pi\)
−0.881689 + 0.471832i \(0.843593\pi\)
\(350\) 1.78590e8 0.222648
\(351\) 4.32436e7 0.0533761
\(352\) 1.41644e8 0.173101
\(353\) −6.44976e8 −0.780426 −0.390213 0.920725i \(-0.627599\pi\)
−0.390213 + 0.920725i \(0.627599\pi\)
\(354\) 4.54753e8 0.544834
\(355\) 1.70159e8 0.201862
\(356\) −3.01467e8 −0.354132
\(357\) −5.06768e7 −0.0589482
\(358\) −7.77629e8 −0.895740
\(359\) 5.02766e7 0.0573503 0.0286751 0.999589i \(-0.490871\pi\)
0.0286751 + 0.999589i \(0.490871\pi\)
\(360\) 4.26240e7 0.0481500
\(361\) 7.10229e7 0.0794554
\(362\) −4.29726e7 −0.0476115
\(363\) −2.16523e7 −0.0237591
\(364\) 4.82285e7 0.0524142
\(365\) 2.62505e8 0.282561
\(366\) −3.83705e8 −0.409085
\(367\) 1.69410e9 1.78899 0.894495 0.447078i \(-0.147535\pi\)
0.894495 + 0.447078i \(0.147535\pi\)
\(368\) −1.67759e8 −0.175476
\(369\) −4.52492e8 −0.468834
\(370\) −1.84217e8 −0.189070
\(371\) −5.07501e8 −0.515974
\(372\) 2.62537e8 0.264417
\(373\) 1.45350e8 0.145022 0.0725109 0.997368i \(-0.476899\pi\)
0.0725109 + 0.997368i \(0.476899\pi\)
\(374\) −1.89230e8 −0.187043
\(375\) 4.41561e8 0.432396
\(376\) −5.93913e8 −0.576190
\(377\) 3.55087e8 0.341303
\(378\) −5.40102e7 −0.0514344
\(379\) −8.76746e8 −0.827249 −0.413625 0.910448i \(-0.635737\pi\)
−0.413625 + 0.910448i \(0.635737\pi\)
\(380\) −2.27027e8 −0.212244
\(381\) 1.28462e8 0.118998
\(382\) −8.22207e8 −0.754674
\(383\) 6.02849e8 0.548293 0.274146 0.961688i \(-0.411605\pi\)
0.274146 + 0.961688i \(0.411605\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 1.69317e8 0.151213
\(386\) 2.01650e8 0.178461
\(387\) 3.63164e8 0.318504
\(388\) −7.19428e8 −0.625283
\(389\) −6.85576e8 −0.590517 −0.295258 0.955417i \(-0.595406\pi\)
−0.295258 + 0.955417i \(0.595406\pi\)
\(390\) 5.41927e7 0.0462610
\(391\) 2.24118e8 0.189609
\(392\) −6.02363e7 −0.0505076
\(393\) −2.95271e8 −0.245384
\(394\) 1.34555e9 1.10832
\(395\) −4.79901e8 −0.391798
\(396\) −2.01677e8 −0.163201
\(397\) 6.33806e8 0.508381 0.254191 0.967154i \(-0.418191\pi\)
0.254191 + 0.967154i \(0.418191\pi\)
\(398\) 1.34291e9 1.06771
\(399\) 2.87672e8 0.226721
\(400\) −2.66584e8 −0.208268
\(401\) 1.39697e9 1.08189 0.540944 0.841058i \(-0.318067\pi\)
0.540944 + 0.841058i \(0.318067\pi\)
\(402\) 3.71120e7 0.0284920
\(403\) 3.33792e8 0.254044
\(404\) −4.86714e8 −0.367231
\(405\) −6.06893e7 −0.0453962
\(406\) −4.43495e8 −0.328888
\(407\) 8.71628e8 0.640842
\(408\) 7.56459e7 0.0551410
\(409\) 3.71443e8 0.268448 0.134224 0.990951i \(-0.457146\pi\)
0.134224 + 0.990951i \(0.457146\pi\)
\(410\) −5.67062e8 −0.406338
\(411\) −4.44635e6 −0.00315906
\(412\) −1.18474e9 −0.834607
\(413\) −7.22131e8 −0.504419
\(414\) 2.38860e8 0.165440
\(415\) −7.46852e8 −0.512939
\(416\) −7.19913e7 −0.0490290
\(417\) −1.16067e9 −0.783848
\(418\) 1.07419e9 0.719386
\(419\) −1.02350e9 −0.679736 −0.339868 0.940473i \(-0.610382\pi\)
−0.339868 + 0.940473i \(0.610382\pi\)
\(420\) −6.76854e7 −0.0445782
\(421\) −6.29529e8 −0.411176 −0.205588 0.978639i \(-0.565911\pi\)
−0.205588 + 0.978639i \(0.565911\pi\)
\(422\) 1.62399e9 1.05194
\(423\) 8.45630e8 0.543237
\(424\) 7.57552e8 0.482649
\(425\) 3.56144e8 0.225042
\(426\) 3.21848e8 0.201705
\(427\) 6.09310e8 0.378740
\(428\) −4.53670e8 −0.279697
\(429\) −2.56415e8 −0.156799
\(430\) 4.55117e8 0.276047
\(431\) 1.63007e9 0.980700 0.490350 0.871526i \(-0.336869\pi\)
0.490350 + 0.871526i \(0.336869\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.68735e9 0.998845 0.499422 0.866359i \(-0.333546\pi\)
0.499422 + 0.866359i \(0.333546\pi\)
\(434\) −4.16899e8 −0.244803
\(435\) −4.98340e8 −0.290278
\(436\) 3.61939e8 0.209138
\(437\) −1.27223e9 −0.729256
\(438\) 4.96518e8 0.282342
\(439\) 2.61267e9 1.47387 0.736935 0.675964i \(-0.236273\pi\)
0.736935 + 0.675964i \(0.236273\pi\)
\(440\) −2.52741e8 −0.141446
\(441\) 8.57661e7 0.0476190
\(442\) 9.61771e7 0.0529778
\(443\) 2.79195e8 0.152579 0.0762896 0.997086i \(-0.475693\pi\)
0.0762896 + 0.997086i \(0.475693\pi\)
\(444\) −3.48438e8 −0.188923
\(445\) 5.37919e8 0.289372
\(446\) −3.31260e8 −0.176806
\(447\) −1.25603e9 −0.665158
\(448\) 8.99154e7 0.0472456
\(449\) −3.47767e9 −1.81312 −0.906560 0.422077i \(-0.861301\pi\)
−0.906560 + 0.422077i \(0.861301\pi\)
\(450\) 3.79569e8 0.196357
\(451\) 2.68307e9 1.37726
\(452\) 1.25218e9 0.637797
\(453\) 1.82451e9 0.922152
\(454\) 2.72469e9 1.36654
\(455\) −8.60560e7 −0.0428293
\(456\) −4.29412e8 −0.212079
\(457\) −3.44211e9 −1.68701 −0.843506 0.537120i \(-0.819512\pi\)
−0.843506 + 0.537120i \(0.819512\pi\)
\(458\) −5.05533e8 −0.245878
\(459\) −1.07707e8 −0.0519875
\(460\) 2.99338e8 0.143387
\(461\) −7.03893e8 −0.334621 −0.167311 0.985904i \(-0.553508\pi\)
−0.167311 + 0.985904i \(0.553508\pi\)
\(462\) 3.20256e8 0.151095
\(463\) −9.67830e8 −0.453175 −0.226587 0.973991i \(-0.572757\pi\)
−0.226587 + 0.973991i \(0.572757\pi\)
\(464\) 6.62010e8 0.307646
\(465\) −4.68454e8 −0.216064
\(466\) 8.86319e8 0.405732
\(467\) −3.35799e9 −1.52570 −0.762852 0.646573i \(-0.776202\pi\)
−0.762852 + 0.646573i \(0.776202\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) −5.89324e7 −0.0263785
\(470\) 1.05974e9 0.470823
\(471\) 1.50332e9 0.662947
\(472\) 1.07793e9 0.471840
\(473\) −2.15340e9 −0.935645
\(474\) −9.07713e8 −0.391493
\(475\) −2.02169e9 −0.865537
\(476\) −1.20123e8 −0.0510507
\(477\) −1.07862e9 −0.455046
\(478\) 7.69049e8 0.322075
\(479\) −1.59041e9 −0.661202 −0.330601 0.943771i \(-0.607252\pi\)
−0.330601 + 0.943771i \(0.607252\pi\)
\(480\) 1.01035e8 0.0416991
\(481\) −4.43008e8 −0.181512
\(482\) 2.51922e9 1.02471
\(483\) −3.79300e8 −0.153168
\(484\) −5.13239e7 −0.0205760
\(485\) 1.28370e9 0.510939
\(486\) −1.14791e8 −0.0453609
\(487\) −2.72444e8 −0.106887 −0.0534436 0.998571i \(-0.517020\pi\)
−0.0534436 + 0.998571i \(0.517020\pi\)
\(488\) −9.09524e8 −0.354278
\(489\) 1.45767e9 0.563738
\(490\) 1.07482e8 0.0412714
\(491\) 4.30608e9 1.64171 0.820857 0.571134i \(-0.193496\pi\)
0.820857 + 0.571134i \(0.193496\pi\)
\(492\) −1.07257e9 −0.406022
\(493\) −8.84416e8 −0.332424
\(494\) −5.45959e8 −0.203758
\(495\) 3.59860e8 0.133357
\(496\) 6.22309e8 0.228992
\(497\) −5.11082e8 −0.186743
\(498\) −1.41264e9 −0.512541
\(499\) 4.06792e9 1.46562 0.732808 0.680435i \(-0.238209\pi\)
0.732808 + 0.680435i \(0.238209\pi\)
\(500\) 1.04666e9 0.374466
\(501\) 1.17224e9 0.416470
\(502\) 2.83618e9 1.00062
\(503\) −4.88902e9 −1.71291 −0.856454 0.516223i \(-0.827337\pi\)
−0.856454 + 0.516223i \(0.827337\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 8.68462e8 0.300076
\(506\) −1.41633e9 −0.486002
\(507\) 1.30324e8 0.0444116
\(508\) 3.04504e8 0.103055
\(509\) 3.71435e9 1.24845 0.624224 0.781246i \(-0.285415\pi\)
0.624224 + 0.781246i \(0.285415\pi\)
\(510\) −1.34978e8 −0.0450575
\(511\) −7.88451e8 −0.261398
\(512\) −1.34218e8 −0.0441942
\(513\) 6.11408e8 0.199950
\(514\) −6.21698e8 −0.201934
\(515\) 2.11397e9 0.681984
\(516\) 8.60834e8 0.275833
\(517\) −5.01421e9 −1.59583
\(518\) 5.53307e8 0.174909
\(519\) 8.55871e8 0.268734
\(520\) 1.28457e8 0.0400632
\(521\) 3.08190e9 0.954743 0.477372 0.878702i \(-0.341590\pi\)
0.477372 + 0.878702i \(0.341590\pi\)
\(522\) −9.42589e8 −0.290052
\(523\) 5.15353e9 1.57525 0.787623 0.616157i \(-0.211311\pi\)
0.787623 + 0.616157i \(0.211311\pi\)
\(524\) −6.99902e8 −0.212509
\(525\) −6.02742e8 −0.181792
\(526\) 1.75096e9 0.524598
\(527\) −8.31377e8 −0.247435
\(528\) −4.78050e8 −0.141336
\(529\) −1.72737e9 −0.507331
\(530\) −1.35173e9 −0.394388
\(531\) −1.53479e9 −0.444855
\(532\) 6.81890e8 0.196347
\(533\) −1.36368e9 −0.390093
\(534\) 1.01745e9 0.289147
\(535\) 8.09501e8 0.228549
\(536\) 8.79691e7 0.0246748
\(537\) 2.62450e9 0.731369
\(538\) 2.69863e9 0.747147
\(539\) −5.08555e8 −0.139887
\(540\) −1.43856e8 −0.0393143
\(541\) −1.68695e9 −0.458049 −0.229024 0.973421i \(-0.573554\pi\)
−0.229024 + 0.973421i \(0.573554\pi\)
\(542\) 3.94292e9 1.06370
\(543\) 1.45033e8 0.0388746
\(544\) 1.79309e8 0.0477535
\(545\) −6.45822e8 −0.170893
\(546\) −1.62771e8 −0.0427960
\(547\) 1.65150e9 0.431443 0.215721 0.976455i \(-0.430790\pi\)
0.215721 + 0.976455i \(0.430790\pi\)
\(548\) −1.05395e7 −0.00273582
\(549\) 1.29501e9 0.334017
\(550\) −2.25068e9 −0.576824
\(551\) 5.02048e9 1.27854
\(552\) 5.66186e8 0.143276
\(553\) 1.44141e9 0.362452
\(554\) 3.96270e8 0.0990164
\(555\) 6.21731e8 0.154375
\(556\) −2.75121e9 −0.678832
\(557\) −5.84669e9 −1.43356 −0.716782 0.697297i \(-0.754386\pi\)
−0.716782 + 0.697297i \(0.754386\pi\)
\(558\) −8.86062e8 −0.215896
\(559\) 1.09447e9 0.265011
\(560\) −1.60439e8 −0.0386059
\(561\) 6.38653e8 0.152720
\(562\) 3.48921e9 0.829181
\(563\) −7.96471e9 −1.88101 −0.940503 0.339785i \(-0.889646\pi\)
−0.940503 + 0.339785i \(0.889646\pi\)
\(564\) 2.00446e9 0.470457
\(565\) −2.23432e9 −0.521165
\(566\) 4.38358e9 1.01618
\(567\) 1.82284e8 0.0419961
\(568\) 7.62898e8 0.174682
\(569\) 2.60228e9 0.592190 0.296095 0.955158i \(-0.404315\pi\)
0.296095 + 0.955158i \(0.404315\pi\)
\(570\) 7.66215e8 0.173296
\(571\) 4.56539e9 1.02625 0.513123 0.858315i \(-0.328488\pi\)
0.513123 + 0.858315i \(0.328488\pi\)
\(572\) −6.07798e8 −0.135792
\(573\) 2.77495e9 0.616188
\(574\) 1.70321e9 0.375903
\(575\) 2.66562e9 0.584738
\(576\) 1.91103e8 0.0416667
\(577\) 2.14968e9 0.465863 0.232931 0.972493i \(-0.425168\pi\)
0.232931 + 0.972493i \(0.425168\pi\)
\(578\) 3.04316e9 0.655507
\(579\) −6.80569e8 −0.145713
\(580\) −1.18125e9 −0.251388
\(581\) 2.24322e9 0.474520
\(582\) 2.42807e9 0.510541
\(583\) 6.39575e9 1.33675
\(584\) 1.17693e9 0.244515
\(585\) −1.82900e8 −0.0377719
\(586\) 5.41582e9 1.11179
\(587\) 2.85256e9 0.582106 0.291053 0.956707i \(-0.405994\pi\)
0.291053 + 0.956707i \(0.405994\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) 4.71940e9 0.951662
\(590\) −1.92340e9 −0.385556
\(591\) −4.54124e9 −0.904937
\(592\) −8.25927e8 −0.163612
\(593\) −1.68085e9 −0.331007 −0.165504 0.986209i \(-0.552925\pi\)
−0.165504 + 0.986209i \(0.552925\pi\)
\(594\) 6.80661e8 0.133253
\(595\) 2.14340e8 0.0417151
\(596\) −2.97727e9 −0.576044
\(597\) −4.53231e9 −0.871785
\(598\) 7.19855e8 0.137655
\(599\) −3.02602e9 −0.575278 −0.287639 0.957739i \(-0.592870\pi\)
−0.287639 + 0.957739i \(0.592870\pi\)
\(600\) 8.99720e8 0.170050
\(601\) 1.94179e9 0.364873 0.182437 0.983218i \(-0.441602\pi\)
0.182437 + 0.983218i \(0.441602\pi\)
\(602\) −1.36697e9 −0.255371
\(603\) −1.25253e8 −0.0232636
\(604\) 4.32476e9 0.798607
\(605\) 9.15792e7 0.0168133
\(606\) 1.64266e9 0.299843
\(607\) −6.18712e9 −1.12287 −0.561433 0.827522i \(-0.689750\pi\)
−0.561433 + 0.827522i \(0.689750\pi\)
\(608\) −1.01786e9 −0.183665
\(609\) 1.49680e9 0.268536
\(610\) 1.62290e9 0.289492
\(611\) 2.54849e9 0.452000
\(612\) −2.55305e8 −0.0450225
\(613\) 7.13692e9 1.25141 0.625705 0.780060i \(-0.284812\pi\)
0.625705 + 0.780060i \(0.284812\pi\)
\(614\) 3.51460e9 0.612755
\(615\) 1.91383e9 0.331773
\(616\) 7.59125e8 0.130852
\(617\) −1.39551e8 −0.0239185 −0.0119593 0.999928i \(-0.503807\pi\)
−0.0119593 + 0.999928i \(0.503807\pi\)
\(618\) 3.99849e9 0.681454
\(619\) 7.28641e9 1.23480 0.617400 0.786650i \(-0.288186\pi\)
0.617400 + 0.786650i \(0.288186\pi\)
\(620\) −1.11041e9 −0.187117
\(621\) −8.06151e8 −0.135081
\(622\) −7.92927e9 −1.32120
\(623\) −1.61567e9 −0.267698
\(624\) 2.42971e8 0.0400320
\(625\) 3.21708e9 0.527086
\(626\) −7.29739e9 −1.18893
\(627\) −3.62538e9 −0.587377
\(628\) 3.56343e9 0.574129
\(629\) 1.10340e9 0.176789
\(630\) 2.28438e8 0.0363979
\(631\) −5.92039e9 −0.938097 −0.469049 0.883172i \(-0.655403\pi\)
−0.469049 + 0.883172i \(0.655403\pi\)
\(632\) −2.15162e9 −0.339043
\(633\) −5.48097e9 −0.858903
\(634\) 8.65072e9 1.34816
\(635\) −5.43337e8 −0.0842096
\(636\) −2.55674e9 −0.394082
\(637\) 2.58475e8 0.0396214
\(638\) 5.58913e9 0.852063
\(639\) −1.08624e9 −0.164691
\(640\) 2.39490e8 0.0361125
\(641\) −7.92497e9 −1.18849 −0.594244 0.804285i \(-0.702549\pi\)
−0.594244 + 0.804285i \(0.702549\pi\)
\(642\) 1.53114e9 0.228371
\(643\) −6.09314e9 −0.903863 −0.451932 0.892053i \(-0.649265\pi\)
−0.451932 + 0.892053i \(0.649265\pi\)
\(644\) −8.99082e8 −0.132647
\(645\) −1.53602e9 −0.225392
\(646\) 1.35982e9 0.198458
\(647\) −3.68128e9 −0.534360 −0.267180 0.963647i \(-0.586092\pi\)
−0.267180 + 0.963647i \(0.586092\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) 9.10063e9 1.30682
\(650\) 1.14391e9 0.163379
\(651\) 1.40703e9 0.199881
\(652\) 3.45521e9 0.488211
\(653\) 6.67391e9 0.937960 0.468980 0.883209i \(-0.344622\pi\)
0.468980 + 0.883209i \(0.344622\pi\)
\(654\) −1.22155e9 −0.170761
\(655\) 1.24886e9 0.173648
\(656\) −2.54240e9 −0.351625
\(657\) −1.67575e9 −0.230531
\(658\) −3.18300e9 −0.435558
\(659\) −1.21524e9 −0.165410 −0.0827052 0.996574i \(-0.526356\pi\)
−0.0827052 + 0.996574i \(0.526356\pi\)
\(660\) 8.53002e8 0.115491
\(661\) −3.59173e9 −0.483725 −0.241862 0.970311i \(-0.577758\pi\)
−0.241862 + 0.970311i \(0.577758\pi\)
\(662\) 4.14909e9 0.555841
\(663\) −3.24598e8 −0.0432562
\(664\) −3.34848e9 −0.443873
\(665\) −1.21672e9 −0.160441
\(666\) 1.17598e9 0.154255
\(667\) −6.61957e9 −0.863753
\(668\) 2.77864e9 0.360674
\(669\) 1.11800e9 0.144362
\(670\) −1.56967e8 −0.0201626
\(671\) −7.67880e9 −0.981216
\(672\) −3.03464e8 −0.0385758
\(673\) −1.39440e9 −0.176333 −0.0881665 0.996106i \(-0.528101\pi\)
−0.0881665 + 0.996106i \(0.528101\pi\)
\(674\) 1.36797e9 0.172095
\(675\) −1.28105e9 −0.160325
\(676\) 3.08916e8 0.0384615
\(677\) −5.81817e9 −0.720653 −0.360326 0.932826i \(-0.617335\pi\)
−0.360326 + 0.932826i \(0.617335\pi\)
\(678\) −4.22611e9 −0.520759
\(679\) −3.85569e9 −0.472669
\(680\) −3.19948e8 −0.0390209
\(681\) −9.19584e9 −1.11578
\(682\) 5.25395e9 0.634221
\(683\) 6.91715e9 0.830720 0.415360 0.909657i \(-0.363656\pi\)
0.415360 + 0.909657i \(0.363656\pi\)
\(684\) 1.44926e9 0.173161
\(685\) 1.88060e7 0.00223553
\(686\) −3.22829e8 −0.0381802
\(687\) 1.70617e9 0.200759
\(688\) 2.04050e9 0.238878
\(689\) −3.25067e9 −0.378621
\(690\) −1.01027e9 −0.117075
\(691\) −8.40248e9 −0.968800 −0.484400 0.874847i \(-0.660962\pi\)
−0.484400 + 0.874847i \(0.660962\pi\)
\(692\) 2.02873e9 0.232730
\(693\) −1.08086e9 −0.123369
\(694\) 1.19145e9 0.135306
\(695\) 4.90909e9 0.554695
\(696\) −2.23429e9 −0.251192
\(697\) 3.39653e9 0.379945
\(698\) 1.12027e10 1.24690
\(699\) −2.99133e9 −0.331279
\(700\) −1.42872e9 −0.157436
\(701\) 5.37740e9 0.589603 0.294801 0.955559i \(-0.404747\pi\)
0.294801 + 0.955559i \(0.404747\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) −6.26357e9 −0.679952
\(704\) −1.13315e9 −0.122401
\(705\) −3.57663e9 −0.384425
\(706\) 5.15981e9 0.551845
\(707\) −2.60848e9 −0.277600
\(708\) −3.63803e9 −0.385256
\(709\) −7.21484e9 −0.760265 −0.380132 0.924932i \(-0.624122\pi\)
−0.380132 + 0.924932i \(0.624122\pi\)
\(710\) −1.36127e9 −0.142738
\(711\) 3.06353e9 0.319653
\(712\) 2.41173e9 0.250409
\(713\) −6.22260e9 −0.642922
\(714\) 4.05415e8 0.0416827
\(715\) 1.08452e9 0.110960
\(716\) 6.22103e9 0.633384
\(717\) −2.59554e9 −0.262973
\(718\) −4.02213e8 −0.0405528
\(719\) −1.17559e9 −0.117952 −0.0589758 0.998259i \(-0.518783\pi\)
−0.0589758 + 0.998259i \(0.518783\pi\)
\(720\) −3.40992e8 −0.0340472
\(721\) −6.34945e9 −0.630903
\(722\) −5.68183e8 −0.0561834
\(723\) −8.50236e9 −0.836673
\(724\) 3.43781e8 0.0336664
\(725\) −1.05191e10 −1.02517
\(726\) 1.73218e8 0.0168002
\(727\) −5.44033e9 −0.525116 −0.262558 0.964916i \(-0.584566\pi\)
−0.262558 + 0.964916i \(0.584566\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −2.10004e9 −0.199801
\(731\) −2.72601e9 −0.258117
\(732\) 3.06964e9 0.289267
\(733\) 1.46661e10 1.37547 0.687733 0.725963i \(-0.258606\pi\)
0.687733 + 0.725963i \(0.258606\pi\)
\(734\) −1.35528e10 −1.26501
\(735\) −3.62751e8 −0.0336980
\(736\) 1.34207e9 0.124080
\(737\) 7.42693e8 0.0683397
\(738\) 3.61994e9 0.331515
\(739\) 1.91373e10 1.74431 0.872157 0.489226i \(-0.162721\pi\)
0.872157 + 0.489226i \(0.162721\pi\)
\(740\) 1.47373e9 0.133693
\(741\) 1.84261e9 0.166368
\(742\) 4.06000e9 0.364849
\(743\) −1.00201e9 −0.0896214 −0.0448107 0.998995i \(-0.514268\pi\)
−0.0448107 + 0.998995i \(0.514268\pi\)
\(744\) −2.10029e9 −0.186971
\(745\) 5.31245e9 0.470704
\(746\) −1.16280e9 −0.102546
\(747\) 4.76765e9 0.418488
\(748\) 1.51384e9 0.132259
\(749\) −2.43139e9 −0.211431
\(750\) −3.53249e9 −0.305750
\(751\) 1.34446e10 1.15826 0.579131 0.815234i \(-0.303392\pi\)
0.579131 + 0.815234i \(0.303392\pi\)
\(752\) 4.75130e9 0.407428
\(753\) −9.57210e9 −0.817005
\(754\) −2.84070e9 −0.241338
\(755\) −7.71684e9 −0.652567
\(756\) 4.32081e8 0.0363696
\(757\) 1.47442e10 1.23534 0.617668 0.786439i \(-0.288077\pi\)
0.617668 + 0.786439i \(0.288077\pi\)
\(758\) 7.01396e9 0.584954
\(759\) 4.78011e9 0.396819
\(760\) 1.81621e9 0.150079
\(761\) −3.18559e9 −0.262026 −0.131013 0.991381i \(-0.541823\pi\)
−0.131013 + 0.991381i \(0.541823\pi\)
\(762\) −1.02770e9 −0.0841441
\(763\) 1.93977e9 0.158094
\(764\) 6.57765e9 0.533635
\(765\) 4.55550e8 0.0367893
\(766\) −4.82279e9 −0.387702
\(767\) −4.62543e9 −0.370142
\(768\) 4.52985e8 0.0360844
\(769\) 7.70782e9 0.611209 0.305604 0.952159i \(-0.401142\pi\)
0.305604 + 0.952159i \(0.401142\pi\)
\(770\) −1.35454e9 −0.106923
\(771\) 2.09823e9 0.164878
\(772\) −1.61320e9 −0.126191
\(773\) 3.98146e9 0.310037 0.155019 0.987912i \(-0.450456\pi\)
0.155019 + 0.987912i \(0.450456\pi\)
\(774\) −2.90531e9 −0.225216
\(775\) −9.88826e9 −0.763069
\(776\) 5.75543e9 0.442142
\(777\) −1.86741e9 −0.142812
\(778\) 5.48461e9 0.417558
\(779\) −1.92807e10 −1.46131
\(780\) −4.33542e8 −0.0327115
\(781\) 6.44089e9 0.483801
\(782\) −1.79294e9 −0.134074
\(783\) 3.18124e9 0.236826
\(784\) 4.81890e8 0.0357143
\(785\) −6.35836e9 −0.469139
\(786\) 2.36217e9 0.173513
\(787\) 2.27353e10 1.66261 0.831303 0.555820i \(-0.187596\pi\)
0.831303 + 0.555820i \(0.187596\pi\)
\(788\) −1.07644e10 −0.783698
\(789\) −5.90950e9 −0.428332
\(790\) 3.83921e9 0.277043
\(791\) 6.71091e9 0.482130
\(792\) 1.61342e9 0.115401
\(793\) 3.90278e9 0.277919
\(794\) −5.07045e9 −0.359480
\(795\) 4.56208e9 0.322017
\(796\) −1.07433e10 −0.754988
\(797\) 6.69299e9 0.468291 0.234146 0.972202i \(-0.424771\pi\)
0.234146 + 0.972202i \(0.424771\pi\)
\(798\) −2.30138e9 −0.160316
\(799\) −6.34753e9 −0.440241
\(800\) 2.13267e9 0.147268
\(801\) −3.43390e9 −0.236088
\(802\) −1.11758e10 −0.765011
\(803\) 9.93642e9 0.677214
\(804\) −2.96896e8 −0.0201469
\(805\) 1.60427e9 0.108390
\(806\) −2.67034e9 −0.179636
\(807\) −9.10789e9 −0.610043
\(808\) 3.89371e9 0.259671
\(809\) 7.75441e9 0.514907 0.257454 0.966291i \(-0.417117\pi\)
0.257454 + 0.966291i \(0.417117\pi\)
\(810\) 4.85515e8 0.0321000
\(811\) 3.37599e9 0.222243 0.111122 0.993807i \(-0.464556\pi\)
0.111122 + 0.993807i \(0.464556\pi\)
\(812\) 3.54796e9 0.232559
\(813\) −1.33074e10 −0.868510
\(814\) −6.97302e9 −0.453143
\(815\) −6.16526e9 −0.398933
\(816\) −6.05167e8 −0.0389906
\(817\) 1.54745e10 0.992747
\(818\) −2.97155e9 −0.189822
\(819\) 5.49353e8 0.0349428
\(820\) 4.53650e9 0.287324
\(821\) 1.33626e10 0.842733 0.421367 0.906890i \(-0.361551\pi\)
0.421367 + 0.906890i \(0.361551\pi\)
\(822\) 3.55708e7 0.00223379
\(823\) −2.47508e9 −0.154771 −0.0773854 0.997001i \(-0.524657\pi\)
−0.0773854 + 0.997001i \(0.524657\pi\)
\(824\) 9.47790e9 0.590156
\(825\) 7.59603e9 0.470975
\(826\) 5.77705e9 0.356678
\(827\) 7.93862e8 0.0488063 0.0244031 0.999702i \(-0.492231\pi\)
0.0244031 + 0.999702i \(0.492231\pi\)
\(828\) −1.91088e9 −0.116984
\(829\) −4.64935e9 −0.283433 −0.141717 0.989907i \(-0.545262\pi\)
−0.141717 + 0.989907i \(0.545262\pi\)
\(830\) 5.97481e9 0.362703
\(831\) −1.33741e9 −0.0808466
\(832\) 5.75930e8 0.0346688
\(833\) −6.43784e8 −0.0385907
\(834\) 9.28534e9 0.554264
\(835\) −4.95803e9 −0.294718
\(836\) −8.59348e9 −0.508683
\(837\) 2.99046e9 0.176278
\(838\) 8.18803e9 0.480646
\(839\) −2.60994e9 −0.152568 −0.0762842 0.997086i \(-0.524306\pi\)
−0.0762842 + 0.997086i \(0.524306\pi\)
\(840\) 5.41483e8 0.0315215
\(841\) 8.87232e9 0.514341
\(842\) 5.03623e9 0.290746
\(843\) −1.17761e10 −0.677024
\(844\) −1.29919e10 −0.743832
\(845\) −5.51210e8 −0.0314281
\(846\) −6.76504e9 −0.384126
\(847\) −2.75064e8 −0.0155540
\(848\) −6.06041e9 −0.341285
\(849\) −1.47946e10 −0.829708
\(850\) −2.84915e9 −0.159129
\(851\) 8.25861e9 0.459360
\(852\) −2.57478e9 −0.142627
\(853\) −1.99945e10 −1.10303 −0.551516 0.834164i \(-0.685951\pi\)
−0.551516 + 0.834164i \(0.685951\pi\)
\(854\) −4.87448e9 −0.267809
\(855\) −2.58598e9 −0.141496
\(856\) 3.62936e9 0.197775
\(857\) −1.92128e10 −1.04270 −0.521348 0.853344i \(-0.674571\pi\)
−0.521348 + 0.853344i \(0.674571\pi\)
\(858\) 2.05132e9 0.110873
\(859\) 3.27625e9 0.176360 0.0881802 0.996105i \(-0.471895\pi\)
0.0881802 + 0.996105i \(0.471895\pi\)
\(860\) −3.64093e9 −0.195195
\(861\) −5.74833e9 −0.306924
\(862\) −1.30406e10 −0.693460
\(863\) 1.71851e10 0.910154 0.455077 0.890452i \(-0.349612\pi\)
0.455077 + 0.890452i \(0.349612\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −3.61994e9 −0.190171
\(866\) −1.34988e10 −0.706290
\(867\) −1.02707e10 −0.535219
\(868\) 3.33519e9 0.173102
\(869\) −1.81654e10 −0.939020
\(870\) 3.98672e9 0.205257
\(871\) −3.77477e8 −0.0193565
\(872\) −2.89552e9 −0.147883
\(873\) −8.19474e9 −0.416855
\(874\) 1.01778e10 0.515662
\(875\) 5.60946e9 0.283070
\(876\) −3.97214e9 −0.199646
\(877\) 7.48053e9 0.374484 0.187242 0.982314i \(-0.440045\pi\)
0.187242 + 0.982314i \(0.440045\pi\)
\(878\) −2.09014e10 −1.04218
\(879\) −1.82784e10 −0.907773
\(880\) 2.02193e9 0.100018
\(881\) −1.65454e10 −0.815194 −0.407597 0.913162i \(-0.633633\pi\)
−0.407597 + 0.913162i \(0.633633\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −1.45834e9 −0.0712845 −0.0356422 0.999365i \(-0.511348\pi\)
−0.0356422 + 0.999365i \(0.511348\pi\)
\(884\) −7.69417e8 −0.0374610
\(885\) 6.49147e9 0.314805
\(886\) −2.23356e9 −0.107890
\(887\) −4.46088e8 −0.0214629 −0.0107314 0.999942i \(-0.503416\pi\)
−0.0107314 + 0.999942i \(0.503416\pi\)
\(888\) 2.78750e9 0.133589
\(889\) 1.63195e9 0.0779023
\(890\) −4.30335e9 −0.204617
\(891\) −2.29723e9 −0.108801
\(892\) 2.65008e9 0.125021
\(893\) 3.60324e10 1.69322
\(894\) 1.00483e10 0.470338
\(895\) −1.11004e10 −0.517558
\(896\) −7.19323e8 −0.0334077
\(897\) −2.42951e9 −0.112395
\(898\) 2.78214e10 1.28207
\(899\) 2.45556e10 1.12718
\(900\) −3.03655e9 −0.138846
\(901\) 8.09644e9 0.368771
\(902\) −2.14646e10 −0.973867
\(903\) 4.61353e9 0.208510
\(904\) −1.00175e10 −0.450991
\(905\) −6.13421e8 −0.0275099
\(906\) −1.45961e10 −0.652060
\(907\) −3.28950e10 −1.46387 −0.731937 0.681372i \(-0.761384\pi\)
−0.731937 + 0.681372i \(0.761384\pi\)
\(908\) −2.17976e10 −0.966290
\(909\) −5.54398e9 −0.244820
\(910\) 6.88448e8 0.0302849
\(911\) 3.14561e10 1.37845 0.689225 0.724547i \(-0.257951\pi\)
0.689225 + 0.724547i \(0.257951\pi\)
\(912\) 3.43529e9 0.149962
\(913\) −2.82700e10 −1.22936
\(914\) 2.75369e10 1.19290
\(915\) −5.47728e9 −0.236369
\(916\) 4.04426e9 0.173862
\(917\) −3.75104e9 −0.160642
\(918\) 8.61654e8 0.0367607
\(919\) 2.61818e10 1.11274 0.556371 0.830934i \(-0.312193\pi\)
0.556371 + 0.830934i \(0.312193\pi\)
\(920\) −2.39471e9 −0.101390
\(921\) −1.18618e10 −0.500312
\(922\) 5.63114e9 0.236613
\(923\) −3.27361e9 −0.137032
\(924\) −2.56205e9 −0.106840
\(925\) 1.31237e10 0.545204
\(926\) 7.74264e9 0.320443
\(927\) −1.34949e10 −0.556404
\(928\) −5.29608e9 −0.217539
\(929\) −2.59091e10 −1.06022 −0.530112 0.847928i \(-0.677850\pi\)
−0.530112 + 0.847928i \(0.677850\pi\)
\(930\) 3.74764e9 0.152780
\(931\) 3.65450e9 0.148424
\(932\) −7.09055e9 −0.286896
\(933\) 2.67613e10 1.07875
\(934\) 2.68639e10 1.07884
\(935\) −2.70121e9 −0.108073
\(936\) −8.20026e8 −0.0326860
\(937\) 2.95456e10 1.17329 0.586644 0.809845i \(-0.300449\pi\)
0.586644 + 0.809845i \(0.300449\pi\)
\(938\) 4.71459e8 0.0186524
\(939\) 2.46287e10 0.970760
\(940\) −8.47794e9 −0.332922
\(941\) 3.61995e10 1.41625 0.708123 0.706090i \(-0.249542\pi\)
0.708123 + 0.706090i \(0.249542\pi\)
\(942\) −1.20266e10 −0.468774
\(943\) 2.54219e10 0.987228
\(944\) −8.62347e9 −0.333642
\(945\) −7.70979e8 −0.0297188
\(946\) 1.72272e10 0.661601
\(947\) −3.81117e10 −1.45825 −0.729127 0.684379i \(-0.760074\pi\)
−0.729127 + 0.684379i \(0.760074\pi\)
\(948\) 7.26170e9 0.276828
\(949\) −5.05023e9 −0.191813
\(950\) 1.61735e10 0.612027
\(951\) −2.91962e10 −1.10076
\(952\) 9.60983e8 0.0360983
\(953\) 3.23305e10 1.21001 0.605003 0.796223i \(-0.293172\pi\)
0.605003 + 0.796223i \(0.293172\pi\)
\(954\) 8.62899e9 0.321766
\(955\) −1.17368e10 −0.436050
\(956\) −6.15239e9 −0.227741
\(957\) −1.88633e10 −0.695707
\(958\) 1.27233e10 0.467541
\(959\) −5.64851e7 −0.00206809
\(960\) −8.08278e8 −0.0294857
\(961\) −4.42959e9 −0.161002
\(962\) 3.54407e9 0.128348
\(963\) −5.16759e9 −0.186464
\(964\) −2.01538e10 −0.724580
\(965\) 2.87850e9 0.103115
\(966\) 3.03440e9 0.108306
\(967\) −1.79638e10 −0.638860 −0.319430 0.947610i \(-0.603491\pi\)
−0.319430 + 0.947610i \(0.603491\pi\)
\(968\) 4.10591e8 0.0145494
\(969\) −4.58940e9 −0.162040
\(970\) −1.02696e10 −0.361288
\(971\) −8.89365e8 −0.0311754 −0.0155877 0.999879i \(-0.504962\pi\)
−0.0155877 + 0.999879i \(0.504962\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −1.47448e10 −0.513149
\(974\) 2.17955e9 0.0755807
\(975\) −3.86071e9 −0.133399
\(976\) 7.27619e9 0.250513
\(977\) −3.01359e10 −1.03384 −0.516920 0.856034i \(-0.672922\pi\)
−0.516920 + 0.856034i \(0.672922\pi\)
\(978\) −1.16613e10 −0.398623
\(979\) 2.03615e10 0.693537
\(980\) −8.59855e8 −0.0291833
\(981\) 4.12272e9 0.139425
\(982\) −3.44487e10 −1.16087
\(983\) 3.12543e10 1.04948 0.524739 0.851263i \(-0.324163\pi\)
0.524739 + 0.851263i \(0.324163\pi\)
\(984\) 8.58059e9 0.287101
\(985\) 1.92074e10 0.640385
\(986\) 7.07533e9 0.235059
\(987\) 1.07426e10 0.355632
\(988\) 4.36767e9 0.144079
\(989\) −2.04033e10 −0.670678
\(990\) −2.87888e9 −0.0942976
\(991\) 3.68129e10 1.20155 0.600776 0.799418i \(-0.294859\pi\)
0.600776 + 0.799418i \(0.294859\pi\)
\(992\) −4.97847e9 −0.161922
\(993\) −1.40032e10 −0.453842
\(994\) 4.08866e9 0.132047
\(995\) 1.91696e10 0.616925
\(996\) 1.13011e10 0.362421
\(997\) −6.07848e9 −0.194250 −0.0971252 0.995272i \(-0.530965\pi\)
−0.0971252 + 0.995272i \(0.530965\pi\)
\(998\) −3.25434e10 −1.03635
\(999\) −3.96893e9 −0.125949
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.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.i.1.2 5 1.1 even 1 trivial