Properties

Label 531.8.a.b.1.3
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-17.9442\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-17.9442 q^{2} +193.993 q^{4} -1.22320 q^{5} -719.259 q^{7} -1184.20 q^{8} +O(q^{10})\) \(q-17.9442 q^{2} +193.993 q^{4} -1.22320 q^{5} -719.259 q^{7} -1184.20 q^{8} +21.9494 q^{10} -1110.85 q^{11} +8283.40 q^{13} +12906.5 q^{14} -3581.74 q^{16} -4834.97 q^{17} -31620.9 q^{19} -237.293 q^{20} +19933.3 q^{22} +43588.0 q^{23} -78123.5 q^{25} -148639. q^{26} -139531. q^{28} +232502. q^{29} -85314.0 q^{31} +215848. q^{32} +86759.6 q^{34} +879.800 q^{35} -52796.8 q^{37} +567411. q^{38} +1448.51 q^{40} +589172. q^{41} +367000. q^{43} -215497. q^{44} -782150. q^{46} -739433. q^{47} -306210. q^{49} +1.40186e6 q^{50} +1.60692e6 q^{52} -1.53905e6 q^{53} +1358.79 q^{55} +851743. q^{56} -4.17205e6 q^{58} -205379. q^{59} +325249. q^{61} +1.53089e6 q^{62} -3.41476e6 q^{64} -10132.3 q^{65} -1.72338e6 q^{67} -937953. q^{68} -15787.3 q^{70} +4.29199e6 q^{71} -1.55847e6 q^{73} +947394. q^{74} -6.13424e6 q^{76} +798988. q^{77} -3.39722e6 q^{79} +4381.20 q^{80} -1.05722e7 q^{82} +7.04945e6 q^{83} +5914.16 q^{85} -6.58552e6 q^{86} +1.31546e6 q^{88} -1.14007e7 q^{89} -5.95791e6 q^{91} +8.45577e6 q^{92} +1.32685e7 q^{94} +38678.8 q^{95} +1.36312e7 q^{97} +5.49468e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −17.9442 −1.58606 −0.793028 0.609185i \(-0.791497\pi\)
−0.793028 + 0.609185i \(0.791497\pi\)
\(3\) 0 0
\(4\) 193.993 1.51557
\(5\) −1.22320 −0.00437627 −0.00218813 0.999998i \(-0.500697\pi\)
−0.00218813 + 0.999998i \(0.500697\pi\)
\(6\) 0 0
\(7\) −719.259 −0.792578 −0.396289 0.918126i \(-0.629702\pi\)
−0.396289 + 0.918126i \(0.629702\pi\)
\(8\) −1184.20 −0.817727
\(9\) 0 0
\(10\) 21.9494 0.00694101
\(11\) −1110.85 −0.251640 −0.125820 0.992053i \(-0.540156\pi\)
−0.125820 + 0.992053i \(0.540156\pi\)
\(12\) 0 0
\(13\) 8283.40 1.04570 0.522850 0.852425i \(-0.324869\pi\)
0.522850 + 0.852425i \(0.324869\pi\)
\(14\) 12906.5 1.25707
\(15\) 0 0
\(16\) −3581.74 −0.218612
\(17\) −4834.97 −0.238684 −0.119342 0.992853i \(-0.538078\pi\)
−0.119342 + 0.992853i \(0.538078\pi\)
\(18\) 0 0
\(19\) −31620.9 −1.05764 −0.528818 0.848735i \(-0.677365\pi\)
−0.528818 + 0.848735i \(0.677365\pi\)
\(20\) −237.293 −0.00663255
\(21\) 0 0
\(22\) 19933.3 0.399116
\(23\) 43588.0 0.746997 0.373499 0.927631i \(-0.378158\pi\)
0.373499 + 0.927631i \(0.378158\pi\)
\(24\) 0 0
\(25\) −78123.5 −0.999981
\(26\) −148639. −1.65854
\(27\) 0 0
\(28\) −139531. −1.20121
\(29\) 232502. 1.77025 0.885123 0.465358i \(-0.154074\pi\)
0.885123 + 0.465358i \(0.154074\pi\)
\(30\) 0 0
\(31\) −85314.0 −0.514345 −0.257172 0.966366i \(-0.582791\pi\)
−0.257172 + 0.966366i \(0.582791\pi\)
\(32\) 215848. 1.16446
\(33\) 0 0
\(34\) 86759.6 0.378566
\(35\) 879.800 0.00346853
\(36\) 0 0
\(37\) −52796.8 −0.171357 −0.0856784 0.996323i \(-0.527306\pi\)
−0.0856784 + 0.996323i \(0.527306\pi\)
\(38\) 567411. 1.67747
\(39\) 0 0
\(40\) 1448.51 0.00357859
\(41\) 589172. 1.33505 0.667527 0.744585i \(-0.267353\pi\)
0.667527 + 0.744585i \(0.267353\pi\)
\(42\) 0 0
\(43\) 367000. 0.703926 0.351963 0.936014i \(-0.385514\pi\)
0.351963 + 0.936014i \(0.385514\pi\)
\(44\) −215497. −0.381379
\(45\) 0 0
\(46\) −782150. −1.18478
\(47\) −739433. −1.03886 −0.519429 0.854514i \(-0.673855\pi\)
−0.519429 + 0.854514i \(0.673855\pi\)
\(48\) 0 0
\(49\) −306210. −0.371820
\(50\) 1.40186e6 1.58603
\(51\) 0 0
\(52\) 1.60692e6 1.58483
\(53\) −1.53905e6 −1.42000 −0.710000 0.704201i \(-0.751305\pi\)
−0.710000 + 0.704201i \(0.751305\pi\)
\(54\) 0 0
\(55\) 1358.79 0.00110125
\(56\) 851743. 0.648113
\(57\) 0 0
\(58\) −4.17205e6 −2.80771
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 325249. 0.183469 0.0917343 0.995784i \(-0.470759\pi\)
0.0917343 + 0.995784i \(0.470759\pi\)
\(62\) 1.53089e6 0.815779
\(63\) 0 0
\(64\) −3.41476e6 −1.62828
\(65\) −10132.3 −0.00457626
\(66\) 0 0
\(67\) −1.72338e6 −0.700035 −0.350018 0.936743i \(-0.613824\pi\)
−0.350018 + 0.936743i \(0.613824\pi\)
\(68\) −937953. −0.361743
\(69\) 0 0
\(70\) −15787.3 −0.00550129
\(71\) 4.29199e6 1.42316 0.711582 0.702603i \(-0.247979\pi\)
0.711582 + 0.702603i \(0.247979\pi\)
\(72\) 0 0
\(73\) −1.55847e6 −0.468888 −0.234444 0.972130i \(-0.575327\pi\)
−0.234444 + 0.972130i \(0.575327\pi\)
\(74\) 947394. 0.271781
\(75\) 0 0
\(76\) −6.13424e6 −1.60292
\(77\) 798988. 0.199445
\(78\) 0 0
\(79\) −3.39722e6 −0.775226 −0.387613 0.921822i \(-0.626700\pi\)
−0.387613 + 0.921822i \(0.626700\pi\)
\(80\) 4381.20 0.000956704 0
\(81\) 0 0
\(82\) −1.05722e7 −2.11747
\(83\) 7.04945e6 1.35326 0.676631 0.736322i \(-0.263439\pi\)
0.676631 + 0.736322i \(0.263439\pi\)
\(84\) 0 0
\(85\) 5914.16 0.00104454
\(86\) −6.58552e6 −1.11647
\(87\) 0 0
\(88\) 1.31546e6 0.205773
\(89\) −1.14007e7 −1.71422 −0.857109 0.515136i \(-0.827741\pi\)
−0.857109 + 0.515136i \(0.827741\pi\)
\(90\) 0 0
\(91\) −5.95791e6 −0.828798
\(92\) 8.45577e6 1.13213
\(93\) 0 0
\(94\) 1.32685e7 1.64769
\(95\) 38678.8 0.00462850
\(96\) 0 0
\(97\) 1.36312e7 1.51647 0.758236 0.651980i \(-0.226062\pi\)
0.758236 + 0.651980i \(0.226062\pi\)
\(98\) 5.49468e6 0.589727
\(99\) 0 0
\(100\) −1.51554e7 −1.51554
\(101\) 1.39105e7 1.34344 0.671719 0.740806i \(-0.265556\pi\)
0.671719 + 0.740806i \(0.265556\pi\)
\(102\) 0 0
\(103\) 5.48771e6 0.494835 0.247418 0.968909i \(-0.420418\pi\)
0.247418 + 0.968909i \(0.420418\pi\)
\(104\) −9.80916e6 −0.855097
\(105\) 0 0
\(106\) 2.76171e7 2.25220
\(107\) −735470. −0.0580393 −0.0290196 0.999579i \(-0.509239\pi\)
−0.0290196 + 0.999579i \(0.509239\pi\)
\(108\) 0 0
\(109\) 2.33048e7 1.72366 0.861831 0.507196i \(-0.169318\pi\)
0.861831 + 0.507196i \(0.169318\pi\)
\(110\) −24382.5 −0.00174664
\(111\) 0 0
\(112\) 2.57620e6 0.173267
\(113\) −1.31124e7 −0.854885 −0.427442 0.904043i \(-0.640585\pi\)
−0.427442 + 0.904043i \(0.640585\pi\)
\(114\) 0 0
\(115\) −53317.0 −0.00326906
\(116\) 4.51038e7 2.68294
\(117\) 0 0
\(118\) 3.68536e6 0.206487
\(119\) 3.47760e6 0.189176
\(120\) 0 0
\(121\) −1.82532e7 −0.936677
\(122\) −5.83632e6 −0.290991
\(123\) 0 0
\(124\) −1.65503e7 −0.779527
\(125\) 191124. 0.00875245
\(126\) 0 0
\(127\) 2.57648e7 1.11613 0.558063 0.829799i \(-0.311545\pi\)
0.558063 + 0.829799i \(0.311545\pi\)
\(128\) 3.36464e7 1.41809
\(129\) 0 0
\(130\) 181815. 0.00725820
\(131\) 4.04125e7 1.57060 0.785302 0.619113i \(-0.212508\pi\)
0.785302 + 0.619113i \(0.212508\pi\)
\(132\) 0 0
\(133\) 2.27436e7 0.838259
\(134\) 3.09247e7 1.11029
\(135\) 0 0
\(136\) 5.72555e6 0.195178
\(137\) −3.40595e7 −1.13166 −0.565830 0.824522i \(-0.691444\pi\)
−0.565830 + 0.824522i \(0.691444\pi\)
\(138\) 0 0
\(139\) 3.98464e6 0.125845 0.0629227 0.998018i \(-0.479958\pi\)
0.0629227 + 0.998018i \(0.479958\pi\)
\(140\) 170675. 0.00525682
\(141\) 0 0
\(142\) −7.70163e7 −2.25722
\(143\) −9.20160e6 −0.263140
\(144\) 0 0
\(145\) −284397. −0.00774707
\(146\) 2.79655e7 0.743682
\(147\) 0 0
\(148\) −1.02422e7 −0.259704
\(149\) −5.83891e6 −0.144604 −0.0723020 0.997383i \(-0.523035\pi\)
−0.0723020 + 0.997383i \(0.523035\pi\)
\(150\) 0 0
\(151\) 5.13811e7 1.21446 0.607231 0.794525i \(-0.292280\pi\)
0.607231 + 0.794525i \(0.292280\pi\)
\(152\) 3.74453e7 0.864858
\(153\) 0 0
\(154\) −1.43372e7 −0.316330
\(155\) 104356. 0.00225091
\(156\) 0 0
\(157\) −1.14443e7 −0.236016 −0.118008 0.993013i \(-0.537651\pi\)
−0.118008 + 0.993013i \(0.537651\pi\)
\(158\) 6.09602e7 1.22955
\(159\) 0 0
\(160\) −264027. −0.00509598
\(161\) −3.13510e7 −0.592054
\(162\) 0 0
\(163\) 1.25340e7 0.226690 0.113345 0.993556i \(-0.463844\pi\)
0.113345 + 0.993556i \(0.463844\pi\)
\(164\) 1.14296e8 2.02337
\(165\) 0 0
\(166\) −1.26497e8 −2.14635
\(167\) 3.77486e7 0.627182 0.313591 0.949558i \(-0.398468\pi\)
0.313591 + 0.949558i \(0.398468\pi\)
\(168\) 0 0
\(169\) 5.86614e6 0.0934865
\(170\) −106125. −0.00165671
\(171\) 0 0
\(172\) 7.11956e7 1.06685
\(173\) 2.06219e7 0.302808 0.151404 0.988472i \(-0.451621\pi\)
0.151404 + 0.988472i \(0.451621\pi\)
\(174\) 0 0
\(175\) 5.61910e7 0.792563
\(176\) 3.97877e6 0.0550116
\(177\) 0 0
\(178\) 2.04576e8 2.71884
\(179\) 1.01937e8 1.32845 0.664227 0.747531i \(-0.268761\pi\)
0.664227 + 0.747531i \(0.268761\pi\)
\(180\) 0 0
\(181\) 9.14678e6 0.114655 0.0573275 0.998355i \(-0.481742\pi\)
0.0573275 + 0.998355i \(0.481742\pi\)
\(182\) 1.06910e8 1.31452
\(183\) 0 0
\(184\) −5.16167e7 −0.610840
\(185\) 64581.2 0.000749903 0
\(186\) 0 0
\(187\) 5.37092e6 0.0600625
\(188\) −1.43445e8 −1.57447
\(189\) 0 0
\(190\) −694059. −0.00734106
\(191\) −1.28136e8 −1.33062 −0.665311 0.746566i \(-0.731701\pi\)
−0.665311 + 0.746566i \(0.731701\pi\)
\(192\) 0 0
\(193\) 2.97509e7 0.297885 0.148943 0.988846i \(-0.452413\pi\)
0.148943 + 0.988846i \(0.452413\pi\)
\(194\) −2.44601e8 −2.40521
\(195\) 0 0
\(196\) −5.94027e7 −0.563521
\(197\) 1.48736e7 0.138607 0.0693036 0.997596i \(-0.477922\pi\)
0.0693036 + 0.997596i \(0.477922\pi\)
\(198\) 0 0
\(199\) −1.07194e8 −0.964242 −0.482121 0.876105i \(-0.660133\pi\)
−0.482121 + 0.876105i \(0.660133\pi\)
\(200\) 9.25135e7 0.817712
\(201\) 0 0
\(202\) −2.49612e8 −2.13077
\(203\) −1.67229e8 −1.40306
\(204\) 0 0
\(205\) −720678. −0.00584256
\(206\) −9.84724e7 −0.784836
\(207\) 0 0
\(208\) −2.96690e7 −0.228602
\(209\) 3.51260e7 0.266144
\(210\) 0 0
\(211\) 8.62669e7 0.632201 0.316101 0.948726i \(-0.397626\pi\)
0.316101 + 0.948726i \(0.397626\pi\)
\(212\) −2.98566e8 −2.15211
\(213\) 0 0
\(214\) 1.31974e7 0.0920535
\(215\) −448917. −0.00308057
\(216\) 0 0
\(217\) 6.13628e7 0.407658
\(218\) −4.18185e8 −2.73382
\(219\) 0 0
\(220\) 263597. 0.00166902
\(221\) −4.00500e7 −0.249591
\(222\) 0 0
\(223\) 1.22690e8 0.740871 0.370435 0.928858i \(-0.379208\pi\)
0.370435 + 0.928858i \(0.379208\pi\)
\(224\) −1.55251e8 −0.922924
\(225\) 0 0
\(226\) 2.35291e8 1.35590
\(227\) −1.12105e8 −0.636115 −0.318057 0.948071i \(-0.603030\pi\)
−0.318057 + 0.948071i \(0.603030\pi\)
\(228\) 0 0
\(229\) 1.28324e8 0.706131 0.353066 0.935599i \(-0.385139\pi\)
0.353066 + 0.935599i \(0.385139\pi\)
\(230\) 956729. 0.00518491
\(231\) 0 0
\(232\) −2.75328e8 −1.44758
\(233\) 3.60112e7 0.186505 0.0932527 0.995642i \(-0.470274\pi\)
0.0932527 + 0.995642i \(0.470274\pi\)
\(234\) 0 0
\(235\) 904477. 0.00454632
\(236\) −3.98422e7 −0.197311
\(237\) 0 0
\(238\) −6.24026e7 −0.300043
\(239\) 2.30387e8 1.09160 0.545801 0.837915i \(-0.316225\pi\)
0.545801 + 0.837915i \(0.316225\pi\)
\(240\) 0 0
\(241\) −3.09598e8 −1.42475 −0.712376 0.701798i \(-0.752381\pi\)
−0.712376 + 0.701798i \(0.752381\pi\)
\(242\) 3.27538e8 1.48562
\(243\) 0 0
\(244\) 6.30961e7 0.278060
\(245\) 374557. 0.00162718
\(246\) 0 0
\(247\) −2.61928e8 −1.10597
\(248\) 1.01028e8 0.420594
\(249\) 0 0
\(250\) −3.42956e6 −0.0138819
\(251\) −1.81483e8 −0.724399 −0.362200 0.932101i \(-0.617974\pi\)
−0.362200 + 0.932101i \(0.617974\pi\)
\(252\) 0 0
\(253\) −4.84196e7 −0.187975
\(254\) −4.62327e8 −1.77024
\(255\) 0 0
\(256\) −1.66668e8 −0.620887
\(257\) −4.15191e8 −1.52574 −0.762872 0.646549i \(-0.776212\pi\)
−0.762872 + 0.646549i \(0.776212\pi\)
\(258\) 0 0
\(259\) 3.79745e7 0.135814
\(260\) −1.96560e6 −0.00693566
\(261\) 0 0
\(262\) −7.25169e8 −2.49106
\(263\) 3.81009e8 1.29149 0.645744 0.763554i \(-0.276547\pi\)
0.645744 + 0.763554i \(0.276547\pi\)
\(264\) 0 0
\(265\) 1.88258e6 0.00621430
\(266\) −4.08115e8 −1.32953
\(267\) 0 0
\(268\) −3.34325e8 −1.06095
\(269\) 9.53490e7 0.298664 0.149332 0.988787i \(-0.452288\pi\)
0.149332 + 0.988787i \(0.452288\pi\)
\(270\) 0 0
\(271\) −4.17979e8 −1.27574 −0.637870 0.770144i \(-0.720184\pi\)
−0.637870 + 0.770144i \(0.720184\pi\)
\(272\) 1.73176e7 0.0521791
\(273\) 0 0
\(274\) 6.11169e8 1.79488
\(275\) 8.67834e7 0.251636
\(276\) 0 0
\(277\) −3.26940e8 −0.924249 −0.462125 0.886815i \(-0.652913\pi\)
−0.462125 + 0.886815i \(0.652913\pi\)
\(278\) −7.15011e7 −0.199598
\(279\) 0 0
\(280\) −1.04186e6 −0.00283631
\(281\) −9.47550e7 −0.254759 −0.127380 0.991854i \(-0.540657\pi\)
−0.127380 + 0.991854i \(0.540657\pi\)
\(282\) 0 0
\(283\) −4.46938e8 −1.17218 −0.586091 0.810246i \(-0.699334\pi\)
−0.586091 + 0.810246i \(0.699334\pi\)
\(284\) 8.32618e8 2.15691
\(285\) 0 0
\(286\) 1.65115e8 0.417355
\(287\) −4.23767e8 −1.05813
\(288\) 0 0
\(289\) −3.86962e8 −0.943030
\(290\) 5.10327e6 0.0122873
\(291\) 0 0
\(292\) −3.02333e8 −0.710634
\(293\) −4.04818e8 −0.940207 −0.470103 0.882611i \(-0.655783\pi\)
−0.470103 + 0.882611i \(0.655783\pi\)
\(294\) 0 0
\(295\) 251220. 0.000569742 0
\(296\) 6.25217e7 0.140123
\(297\) 0 0
\(298\) 1.04774e8 0.229350
\(299\) 3.61056e8 0.781134
\(300\) 0 0
\(301\) −2.63968e8 −0.557916
\(302\) −9.21991e8 −1.92620
\(303\) 0 0
\(304\) 1.13258e8 0.231212
\(305\) −397846. −0.000802908 0
\(306\) 0 0
\(307\) −7.95438e8 −1.56900 −0.784498 0.620131i \(-0.787079\pi\)
−0.784498 + 0.620131i \(0.787079\pi\)
\(308\) 1.54998e8 0.302273
\(309\) 0 0
\(310\) −1.87259e6 −0.00357007
\(311\) −3.49530e8 −0.658906 −0.329453 0.944172i \(-0.606864\pi\)
−0.329453 + 0.944172i \(0.606864\pi\)
\(312\) 0 0
\(313\) 9.82107e7 0.181031 0.0905157 0.995895i \(-0.471148\pi\)
0.0905157 + 0.995895i \(0.471148\pi\)
\(314\) 2.05359e8 0.374334
\(315\) 0 0
\(316\) −6.59037e8 −1.17491
\(317\) −9.11476e8 −1.60708 −0.803541 0.595250i \(-0.797053\pi\)
−0.803541 + 0.595250i \(0.797053\pi\)
\(318\) 0 0
\(319\) −2.58274e8 −0.445465
\(320\) 4.17695e6 0.00712580
\(321\) 0 0
\(322\) 5.62568e8 0.939030
\(323\) 1.52886e8 0.252441
\(324\) 0 0
\(325\) −6.47128e8 −1.04568
\(326\) −2.24912e8 −0.359542
\(327\) 0 0
\(328\) −6.97695e8 −1.09171
\(329\) 5.31844e8 0.823376
\(330\) 0 0
\(331\) 8.69089e8 1.31724 0.658622 0.752474i \(-0.271140\pi\)
0.658622 + 0.752474i \(0.271140\pi\)
\(332\) 1.36755e9 2.05097
\(333\) 0 0
\(334\) −6.77368e8 −0.994745
\(335\) 2.10805e6 0.00306354
\(336\) 0 0
\(337\) 3.71807e7 0.0529192 0.0264596 0.999650i \(-0.491577\pi\)
0.0264596 + 0.999650i \(0.491577\pi\)
\(338\) −1.05263e8 −0.148275
\(339\) 0 0
\(340\) 1.14731e6 0.00158308
\(341\) 9.47709e7 0.129430
\(342\) 0 0
\(343\) 8.12585e8 1.08727
\(344\) −4.34600e8 −0.575619
\(345\) 0 0
\(346\) −3.70043e8 −0.480270
\(347\) 5.17499e8 0.664900 0.332450 0.943121i \(-0.392125\pi\)
0.332450 + 0.943121i \(0.392125\pi\)
\(348\) 0 0
\(349\) −7.74296e8 −0.975030 −0.487515 0.873115i \(-0.662097\pi\)
−0.487515 + 0.873115i \(0.662097\pi\)
\(350\) −1.00830e9 −1.25705
\(351\) 0 0
\(352\) −2.39775e8 −0.293025
\(353\) 6.61876e8 0.800876 0.400438 0.916324i \(-0.368858\pi\)
0.400438 + 0.916324i \(0.368858\pi\)
\(354\) 0 0
\(355\) −5.24998e6 −0.00622815
\(356\) −2.21166e9 −2.59802
\(357\) 0 0
\(358\) −1.82918e9 −2.10700
\(359\) −6.26557e8 −0.714711 −0.357355 0.933968i \(-0.616321\pi\)
−0.357355 + 0.933968i \(0.616321\pi\)
\(360\) 0 0
\(361\) 1.06008e8 0.118595
\(362\) −1.64131e8 −0.181849
\(363\) 0 0
\(364\) −1.15579e9 −1.25610
\(365\) 1.90633e6 0.00205198
\(366\) 0 0
\(367\) 8.99146e7 0.0949509 0.0474755 0.998872i \(-0.484882\pi\)
0.0474755 + 0.998872i \(0.484882\pi\)
\(368\) −1.56121e8 −0.163303
\(369\) 0 0
\(370\) −1.15886e6 −0.00118939
\(371\) 1.10698e9 1.12546
\(372\) 0 0
\(373\) 1.93821e8 0.193383 0.0966917 0.995314i \(-0.469174\pi\)
0.0966917 + 0.995314i \(0.469174\pi\)
\(374\) −9.63768e7 −0.0952625
\(375\) 0 0
\(376\) 8.75633e8 0.849503
\(377\) 1.92591e9 1.85114
\(378\) 0 0
\(379\) −2.28105e8 −0.215227 −0.107614 0.994193i \(-0.534321\pi\)
−0.107614 + 0.994193i \(0.534321\pi\)
\(380\) 7.50343e6 0.00701483
\(381\) 0 0
\(382\) 2.29930e9 2.11044
\(383\) 1.16094e8 0.105588 0.0527938 0.998605i \(-0.483187\pi\)
0.0527938 + 0.998605i \(0.483187\pi\)
\(384\) 0 0
\(385\) −977325. −0.000872824 0
\(386\) −5.33855e8 −0.472463
\(387\) 0 0
\(388\) 2.64437e9 2.29832
\(389\) −6.84222e8 −0.589351 −0.294675 0.955597i \(-0.595211\pi\)
−0.294675 + 0.955597i \(0.595211\pi\)
\(390\) 0 0
\(391\) −2.10747e8 −0.178296
\(392\) 3.62612e8 0.304047
\(393\) 0 0
\(394\) −2.66895e8 −0.219839
\(395\) 4.15549e6 0.00339260
\(396\) 0 0
\(397\) −1.73331e9 −1.39031 −0.695153 0.718862i \(-0.744663\pi\)
−0.695153 + 0.718862i \(0.744663\pi\)
\(398\) 1.92351e9 1.52934
\(399\) 0 0
\(400\) 2.79818e8 0.218608
\(401\) −1.39266e9 −1.07855 −0.539274 0.842130i \(-0.681301\pi\)
−0.539274 + 0.842130i \(0.681301\pi\)
\(402\) 0 0
\(403\) −7.06690e8 −0.537850
\(404\) 2.69854e9 2.03608
\(405\) 0 0
\(406\) 3.00079e9 2.22533
\(407\) 5.86492e7 0.0431203
\(408\) 0 0
\(409\) −1.48753e9 −1.07506 −0.537530 0.843244i \(-0.680643\pi\)
−0.537530 + 0.843244i \(0.680643\pi\)
\(410\) 1.29320e7 0.00926662
\(411\) 0 0
\(412\) 1.06458e9 0.749959
\(413\) 1.47721e8 0.103185
\(414\) 0 0
\(415\) −8.62292e6 −0.00592224
\(416\) 1.78796e9 1.21767
\(417\) 0 0
\(418\) −6.30307e8 −0.422119
\(419\) −9.73899e8 −0.646792 −0.323396 0.946264i \(-0.604825\pi\)
−0.323396 + 0.946264i \(0.604825\pi\)
\(420\) 0 0
\(421\) 2.01358e9 1.31517 0.657584 0.753381i \(-0.271578\pi\)
0.657584 + 0.753381i \(0.271578\pi\)
\(422\) −1.54799e9 −1.00271
\(423\) 0 0
\(424\) 1.82254e9 1.16117
\(425\) 3.77725e8 0.238679
\(426\) 0 0
\(427\) −2.33938e8 −0.145413
\(428\) −1.42676e8 −0.0879627
\(429\) 0 0
\(430\) 8.05544e6 0.00488596
\(431\) −2.94920e8 −0.177433 −0.0887164 0.996057i \(-0.528276\pi\)
−0.0887164 + 0.996057i \(0.528276\pi\)
\(432\) 0 0
\(433\) −1.42354e9 −0.842676 −0.421338 0.906904i \(-0.638439\pi\)
−0.421338 + 0.906904i \(0.638439\pi\)
\(434\) −1.10111e9 −0.646569
\(435\) 0 0
\(436\) 4.52097e9 2.61233
\(437\) −1.37829e9 −0.790052
\(438\) 0 0
\(439\) −1.55418e9 −0.876750 −0.438375 0.898792i \(-0.644446\pi\)
−0.438375 + 0.898792i \(0.644446\pi\)
\(440\) −1.60908e6 −0.000900519 0
\(441\) 0 0
\(442\) 7.18664e8 0.395866
\(443\) 6.41286e8 0.350460 0.175230 0.984528i \(-0.443933\pi\)
0.175230 + 0.984528i \(0.443933\pi\)
\(444\) 0 0
\(445\) 1.39454e7 0.00750187
\(446\) −2.20157e9 −1.17506
\(447\) 0 0
\(448\) 2.45609e9 1.29054
\(449\) −2.96725e9 −1.54701 −0.773503 0.633793i \(-0.781497\pi\)
−0.773503 + 0.633793i \(0.781497\pi\)
\(450\) 0 0
\(451\) −6.54481e8 −0.335954
\(452\) −2.54372e9 −1.29564
\(453\) 0 0
\(454\) 2.01164e9 1.00891
\(455\) 7.28774e6 0.00362704
\(456\) 0 0
\(457\) −3.31780e9 −1.62609 −0.813043 0.582203i \(-0.802191\pi\)
−0.813043 + 0.582203i \(0.802191\pi\)
\(458\) −2.30268e9 −1.11996
\(459\) 0 0
\(460\) −1.03431e7 −0.00495450
\(461\) −9.48650e8 −0.450975 −0.225488 0.974246i \(-0.572398\pi\)
−0.225488 + 0.974246i \(0.572398\pi\)
\(462\) 0 0
\(463\) 1.33385e9 0.624561 0.312281 0.949990i \(-0.398907\pi\)
0.312281 + 0.949990i \(0.398907\pi\)
\(464\) −8.32761e8 −0.386997
\(465\) 0 0
\(466\) −6.46191e8 −0.295808
\(467\) 3.76180e9 1.70918 0.854588 0.519306i \(-0.173810\pi\)
0.854588 + 0.519306i \(0.173810\pi\)
\(468\) 0 0
\(469\) 1.23956e9 0.554833
\(470\) −1.62301e7 −0.00721072
\(471\) 0 0
\(472\) 2.43209e8 0.106459
\(473\) −4.07682e8 −0.177136
\(474\) 0 0
\(475\) 2.47033e9 1.05762
\(476\) 6.74631e8 0.286709
\(477\) 0 0
\(478\) −4.13410e9 −1.73134
\(479\) 2.93522e9 1.22030 0.610151 0.792285i \(-0.291109\pi\)
0.610151 + 0.792285i \(0.291109\pi\)
\(480\) 0 0
\(481\) −4.37337e8 −0.179188
\(482\) 5.55549e9 2.25973
\(483\) 0 0
\(484\) −3.54100e9 −1.41960
\(485\) −1.66738e7 −0.00663649
\(486\) 0 0
\(487\) −1.25935e9 −0.494077 −0.247038 0.969006i \(-0.579457\pi\)
−0.247038 + 0.969006i \(0.579457\pi\)
\(488\) −3.85158e8 −0.150027
\(489\) 0 0
\(490\) −6.72112e6 −0.00258081
\(491\) −4.20973e8 −0.160498 −0.0802489 0.996775i \(-0.525572\pi\)
−0.0802489 + 0.996775i \(0.525572\pi\)
\(492\) 0 0
\(493\) −1.12414e9 −0.422529
\(494\) 4.70009e9 1.75413
\(495\) 0 0
\(496\) 3.05572e8 0.112442
\(497\) −3.08705e9 −1.12797
\(498\) 0 0
\(499\) −4.86604e9 −1.75317 −0.876584 0.481248i \(-0.840184\pi\)
−0.876584 + 0.481248i \(0.840184\pi\)
\(500\) 3.70767e7 0.0132650
\(501\) 0 0
\(502\) 3.25656e9 1.14894
\(503\) 3.22489e9 1.12987 0.564933 0.825137i \(-0.308902\pi\)
0.564933 + 0.825137i \(0.308902\pi\)
\(504\) 0 0
\(505\) −1.70154e7 −0.00587925
\(506\) 8.68850e8 0.298138
\(507\) 0 0
\(508\) 4.99819e9 1.69157
\(509\) −1.08431e8 −0.0364452 −0.0182226 0.999834i \(-0.505801\pi\)
−0.0182226 + 0.999834i \(0.505801\pi\)
\(510\) 0 0
\(511\) 1.12094e9 0.371630
\(512\) −1.31602e9 −0.433329
\(513\) 0 0
\(514\) 7.45026e9 2.41992
\(515\) −6.71259e6 −0.00216553
\(516\) 0 0
\(517\) 8.21398e8 0.261419
\(518\) −6.81422e8 −0.215408
\(519\) 0 0
\(520\) 1.19986e7 0.00374213
\(521\) −2.88309e8 −0.0893152 −0.0446576 0.999002i \(-0.514220\pi\)
−0.0446576 + 0.999002i \(0.514220\pi\)
\(522\) 0 0
\(523\) 8.83228e8 0.269971 0.134985 0.990848i \(-0.456901\pi\)
0.134985 + 0.990848i \(0.456901\pi\)
\(524\) 7.83976e9 2.38036
\(525\) 0 0
\(526\) −6.83689e9 −2.04837
\(527\) 4.12491e8 0.122766
\(528\) 0 0
\(529\) −1.50492e9 −0.441995
\(530\) −3.37813e7 −0.00985623
\(531\) 0 0
\(532\) 4.41211e9 1.27044
\(533\) 4.88035e9 1.39607
\(534\) 0 0
\(535\) 899630. 0.000253995 0
\(536\) 2.04082e9 0.572438
\(537\) 0 0
\(538\) −1.71096e9 −0.473698
\(539\) 3.40153e8 0.0935650
\(540\) 0 0
\(541\) −3.92838e9 −1.06665 −0.533326 0.845910i \(-0.679058\pi\)
−0.533326 + 0.845910i \(0.679058\pi\)
\(542\) 7.50029e9 2.02340
\(543\) 0 0
\(544\) −1.04362e9 −0.277937
\(545\) −2.85065e7 −0.00754321
\(546\) 0 0
\(547\) 1.10099e9 0.287624 0.143812 0.989605i \(-0.454064\pi\)
0.143812 + 0.989605i \(0.454064\pi\)
\(548\) −6.60731e9 −1.71511
\(549\) 0 0
\(550\) −1.55726e9 −0.399108
\(551\) −7.35191e9 −1.87228
\(552\) 0 0
\(553\) 2.44348e9 0.614427
\(554\) 5.86667e9 1.46591
\(555\) 0 0
\(556\) 7.72994e8 0.190728
\(557\) 5.42968e9 1.33132 0.665658 0.746257i \(-0.268151\pi\)
0.665658 + 0.746257i \(0.268151\pi\)
\(558\) 0 0
\(559\) 3.04001e9 0.736095
\(560\) −3.15121e6 −0.000758263 0
\(561\) 0 0
\(562\) 1.70030e9 0.404063
\(563\) 7.14917e8 0.168840 0.0844201 0.996430i \(-0.473096\pi\)
0.0844201 + 0.996430i \(0.473096\pi\)
\(564\) 0 0
\(565\) 1.60391e7 0.00374121
\(566\) 8.01993e9 1.85914
\(567\) 0 0
\(568\) −5.08256e9 −1.16376
\(569\) 1.67905e9 0.382094 0.191047 0.981581i \(-0.438812\pi\)
0.191047 + 0.981581i \(0.438812\pi\)
\(570\) 0 0
\(571\) −9.65128e8 −0.216950 −0.108475 0.994099i \(-0.534597\pi\)
−0.108475 + 0.994099i \(0.534597\pi\)
\(572\) −1.78505e9 −0.398808
\(573\) 0 0
\(574\) 7.60416e9 1.67826
\(575\) −3.40524e9 −0.746983
\(576\) 0 0
\(577\) 2.50966e9 0.543875 0.271938 0.962315i \(-0.412336\pi\)
0.271938 + 0.962315i \(0.412336\pi\)
\(578\) 6.94371e9 1.49570
\(579\) 0 0
\(580\) −5.51712e7 −0.0117412
\(581\) −5.07038e9 −1.07257
\(582\) 0 0
\(583\) 1.70966e9 0.357330
\(584\) 1.84554e9 0.383422
\(585\) 0 0
\(586\) 7.26413e9 1.49122
\(587\) −3.24638e9 −0.662470 −0.331235 0.943548i \(-0.607465\pi\)
−0.331235 + 0.943548i \(0.607465\pi\)
\(588\) 0 0
\(589\) 2.69770e9 0.543990
\(590\) −4.50794e6 −0.000903642 0
\(591\) 0 0
\(592\) 1.89104e8 0.0374606
\(593\) −7.03607e9 −1.38560 −0.692801 0.721129i \(-0.743624\pi\)
−0.692801 + 0.721129i \(0.743624\pi\)
\(594\) 0 0
\(595\) −4.25381e6 −0.000827883 0
\(596\) −1.13271e9 −0.219158
\(597\) 0 0
\(598\) −6.47886e9 −1.23892
\(599\) −9.00254e9 −1.71148 −0.855738 0.517409i \(-0.826897\pi\)
−0.855738 + 0.517409i \(0.826897\pi\)
\(600\) 0 0
\(601\) 5.40983e9 1.01654 0.508268 0.861199i \(-0.330286\pi\)
0.508268 + 0.861199i \(0.330286\pi\)
\(602\) 4.73669e9 0.884886
\(603\) 0 0
\(604\) 9.96758e9 1.84061
\(605\) 2.23274e7 0.00409915
\(606\) 0 0
\(607\) −6.20066e9 −1.12532 −0.562662 0.826687i \(-0.690223\pi\)
−0.562662 + 0.826687i \(0.690223\pi\)
\(608\) −6.82531e9 −1.23157
\(609\) 0 0
\(610\) 7.13902e6 0.00127346
\(611\) −6.12502e9 −1.08633
\(612\) 0 0
\(613\) 5.56027e9 0.974955 0.487477 0.873136i \(-0.337917\pi\)
0.487477 + 0.873136i \(0.337917\pi\)
\(614\) 1.42735e10 2.48851
\(615\) 0 0
\(616\) −9.46158e8 −0.163091
\(617\) −8.63571e9 −1.48013 −0.740065 0.672535i \(-0.765205\pi\)
−0.740065 + 0.672535i \(0.765205\pi\)
\(618\) 0 0
\(619\) 4.62517e9 0.783809 0.391904 0.920006i \(-0.371816\pi\)
0.391904 + 0.920006i \(0.371816\pi\)
\(620\) 2.02445e7 0.00341142
\(621\) 0 0
\(622\) 6.27203e9 1.04506
\(623\) 8.20004e9 1.35865
\(624\) 0 0
\(625\) 6.10316e9 0.999943
\(626\) −1.76231e9 −0.287126
\(627\) 0 0
\(628\) −2.22012e9 −0.357699
\(629\) 2.55271e8 0.0409001
\(630\) 0 0
\(631\) −4.78738e9 −0.758568 −0.379284 0.925280i \(-0.623830\pi\)
−0.379284 + 0.925280i \(0.623830\pi\)
\(632\) 4.02297e9 0.633923
\(633\) 0 0
\(634\) 1.63557e10 2.54892
\(635\) −3.15156e7 −0.00488447
\(636\) 0 0
\(637\) −2.53646e9 −0.388812
\(638\) 4.63452e9 0.706533
\(639\) 0 0
\(640\) −4.11564e7 −0.00620594
\(641\) −1.18477e10 −1.77676 −0.888382 0.459104i \(-0.848170\pi\)
−0.888382 + 0.459104i \(0.848170\pi\)
\(642\) 0 0
\(643\) −1.06317e10 −1.57712 −0.788562 0.614956i \(-0.789174\pi\)
−0.788562 + 0.614956i \(0.789174\pi\)
\(644\) −6.08189e9 −0.897300
\(645\) 0 0
\(646\) −2.74342e9 −0.400385
\(647\) 9.21966e9 1.33829 0.669145 0.743132i \(-0.266661\pi\)
0.669145 + 0.743132i \(0.266661\pi\)
\(648\) 0 0
\(649\) 2.28145e8 0.0327608
\(650\) 1.16122e10 1.65851
\(651\) 0 0
\(652\) 2.43150e9 0.343565
\(653\) 1.41112e10 1.98321 0.991605 0.129302i \(-0.0412738\pi\)
0.991605 + 0.129302i \(0.0412738\pi\)
\(654\) 0 0
\(655\) −4.94328e7 −0.00687338
\(656\) −2.11026e9 −0.291859
\(657\) 0 0
\(658\) −9.54349e9 −1.30592
\(659\) −6.80419e9 −0.926141 −0.463071 0.886321i \(-0.653252\pi\)
−0.463071 + 0.886321i \(0.653252\pi\)
\(660\) 0 0
\(661\) 2.76563e8 0.0372468 0.0186234 0.999827i \(-0.494072\pi\)
0.0186234 + 0.999827i \(0.494072\pi\)
\(662\) −1.55951e10 −2.08922
\(663\) 0 0
\(664\) −8.34793e9 −1.10660
\(665\) −2.78201e7 −0.00366845
\(666\) 0 0
\(667\) 1.01343e10 1.32237
\(668\) 7.32298e9 0.950540
\(669\) 0 0
\(670\) −3.78272e7 −0.00485895
\(671\) −3.61302e8 −0.0461681
\(672\) 0 0
\(673\) −7.86611e9 −0.994734 −0.497367 0.867540i \(-0.665700\pi\)
−0.497367 + 0.867540i \(0.665700\pi\)
\(674\) −6.67178e8 −0.0839328
\(675\) 0 0
\(676\) 1.13799e9 0.141686
\(677\) −6.11786e8 −0.0757773 −0.0378886 0.999282i \(-0.512063\pi\)
−0.0378886 + 0.999282i \(0.512063\pi\)
\(678\) 0 0
\(679\) −9.80439e9 −1.20192
\(680\) −7.00352e6 −0.000854152 0
\(681\) 0 0
\(682\) −1.70059e9 −0.205283
\(683\) 1.81400e9 0.217854 0.108927 0.994050i \(-0.465259\pi\)
0.108927 + 0.994050i \(0.465259\pi\)
\(684\) 0 0
\(685\) 4.16617e7 0.00495245
\(686\) −1.45812e10 −1.72448
\(687\) 0 0
\(688\) −1.31450e9 −0.153887
\(689\) −1.27486e10 −1.48489
\(690\) 0 0
\(691\) 8.17172e9 0.942194 0.471097 0.882081i \(-0.343858\pi\)
0.471097 + 0.882081i \(0.343858\pi\)
\(692\) 4.00051e9 0.458927
\(693\) 0 0
\(694\) −9.28610e9 −1.05457
\(695\) −4.87403e6 −0.000550734 0
\(696\) 0 0
\(697\) −2.84863e9 −0.318656
\(698\) 1.38941e10 1.54645
\(699\) 0 0
\(700\) 1.09007e10 1.20119
\(701\) −8.54219e9 −0.936605 −0.468302 0.883568i \(-0.655134\pi\)
−0.468302 + 0.883568i \(0.655134\pi\)
\(702\) 0 0
\(703\) 1.66948e9 0.181233
\(704\) 3.79328e9 0.409742
\(705\) 0 0
\(706\) −1.18768e10 −1.27023
\(707\) −1.00052e10 −1.06478
\(708\) 0 0
\(709\) −8.20988e9 −0.865117 −0.432559 0.901606i \(-0.642389\pi\)
−0.432559 + 0.901606i \(0.642389\pi\)
\(710\) 9.42066e7 0.00987819
\(711\) 0 0
\(712\) 1.35006e10 1.40176
\(713\) −3.71866e9 −0.384214
\(714\) 0 0
\(715\) 1.12554e7 0.00115157
\(716\) 1.97751e10 2.01337
\(717\) 0 0
\(718\) 1.12431e10 1.13357
\(719\) 1.72415e9 0.172991 0.0864957 0.996252i \(-0.472433\pi\)
0.0864957 + 0.996252i \(0.472433\pi\)
\(720\) 0 0
\(721\) −3.94708e9 −0.392196
\(722\) −1.90223e9 −0.188098
\(723\) 0 0
\(724\) 1.77441e9 0.173768
\(725\) −1.81639e10 −1.77021
\(726\) 0 0
\(727\) −1.17427e9 −0.113343 −0.0566717 0.998393i \(-0.518049\pi\)
−0.0566717 + 0.998393i \(0.518049\pi\)
\(728\) 7.05532e9 0.677731
\(729\) 0 0
\(730\) −3.42075e7 −0.00325455
\(731\) −1.77444e9 −0.168016
\(732\) 0 0
\(733\) 1.78077e10 1.67011 0.835053 0.550170i \(-0.185437\pi\)
0.835053 + 0.550170i \(0.185437\pi\)
\(734\) −1.61344e9 −0.150597
\(735\) 0 0
\(736\) 9.40839e9 0.869847
\(737\) 1.91442e9 0.176157
\(738\) 0 0
\(739\) −1.69312e10 −1.54324 −0.771620 0.636084i \(-0.780553\pi\)
−0.771620 + 0.636084i \(0.780553\pi\)
\(740\) 1.25283e7 0.00113653
\(741\) 0 0
\(742\) −1.98638e10 −1.78504
\(743\) −5.43777e9 −0.486362 −0.243181 0.969981i \(-0.578191\pi\)
−0.243181 + 0.969981i \(0.578191\pi\)
\(744\) 0 0
\(745\) 7.14219e6 0.000632826 0
\(746\) −3.47795e9 −0.306717
\(747\) 0 0
\(748\) 1.04192e9 0.0910291
\(749\) 5.28993e8 0.0460006
\(750\) 0 0
\(751\) −1.10319e10 −0.950406 −0.475203 0.879876i \(-0.657625\pi\)
−0.475203 + 0.879876i \(0.657625\pi\)
\(752\) 2.64845e9 0.227107
\(753\) 0 0
\(754\) −3.45588e10 −2.93602
\(755\) −6.28496e7 −0.00531481
\(756\) 0 0
\(757\) 9.41776e9 0.789064 0.394532 0.918882i \(-0.370907\pi\)
0.394532 + 0.918882i \(0.370907\pi\)
\(758\) 4.09315e9 0.341363
\(759\) 0 0
\(760\) −4.58033e7 −0.00378485
\(761\) −1.61763e10 −1.33056 −0.665278 0.746596i \(-0.731687\pi\)
−0.665278 + 0.746596i \(0.731687\pi\)
\(762\) 0 0
\(763\) −1.67622e10 −1.36614
\(764\) −2.48575e10 −2.01665
\(765\) 0 0
\(766\) −2.08321e9 −0.167468
\(767\) −1.70124e9 −0.136138
\(768\) 0 0
\(769\) −1.61743e10 −1.28257 −0.641287 0.767301i \(-0.721599\pi\)
−0.641287 + 0.767301i \(0.721599\pi\)
\(770\) 1.75373e7 0.00138435
\(771\) 0 0
\(772\) 5.77147e9 0.451467
\(773\) 1.43501e9 0.111745 0.0558723 0.998438i \(-0.482206\pi\)
0.0558723 + 0.998438i \(0.482206\pi\)
\(774\) 0 0
\(775\) 6.66503e9 0.514335
\(776\) −1.61421e10 −1.24006
\(777\) 0 0
\(778\) 1.22778e10 0.934743
\(779\) −1.86302e10 −1.41200
\(780\) 0 0
\(781\) −4.76775e9 −0.358125
\(782\) 3.78167e9 0.282788
\(783\) 0 0
\(784\) 1.09676e9 0.0812843
\(785\) 1.39987e7 0.00103287
\(786\) 0 0
\(787\) −1.73282e10 −1.26719 −0.633597 0.773664i \(-0.718422\pi\)
−0.633597 + 0.773664i \(0.718422\pi\)
\(788\) 2.88539e9 0.210069
\(789\) 0 0
\(790\) −7.45668e7 −0.00538085
\(791\) 9.43121e9 0.677563
\(792\) 0 0
\(793\) 2.69417e9 0.191853
\(794\) 3.11029e10 2.20510
\(795\) 0 0
\(796\) −2.07950e10 −1.46138
\(797\) 1.59506e10 1.11602 0.558012 0.829833i \(-0.311564\pi\)
0.558012 + 0.829833i \(0.311564\pi\)
\(798\) 0 0
\(799\) 3.57514e9 0.247959
\(800\) −1.68628e10 −1.16444
\(801\) 0 0
\(802\) 2.49901e10 1.71064
\(803\) 1.73123e9 0.117991
\(804\) 0 0
\(805\) 3.83487e7 0.00259099
\(806\) 1.26810e10 0.853060
\(807\) 0 0
\(808\) −1.64727e10 −1.09857
\(809\) −1.99696e10 −1.32602 −0.663010 0.748611i \(-0.730721\pi\)
−0.663010 + 0.748611i \(0.730721\pi\)
\(810\) 0 0
\(811\) 1.94704e10 1.28175 0.640874 0.767646i \(-0.278572\pi\)
0.640874 + 0.767646i \(0.278572\pi\)
\(812\) −3.24413e10 −2.12644
\(813\) 0 0
\(814\) −1.05241e9 −0.0683912
\(815\) −1.53316e7 −0.000992055 0
\(816\) 0 0
\(817\) −1.16049e10 −0.744498
\(818\) 2.66924e10 1.70511
\(819\) 0 0
\(820\) −1.39807e8 −0.00885482
\(821\) 1.65074e10 1.04107 0.520534 0.853841i \(-0.325733\pi\)
0.520534 + 0.853841i \(0.325733\pi\)
\(822\) 0 0
\(823\) 1.23795e10 0.774113 0.387057 0.922056i \(-0.373492\pi\)
0.387057 + 0.922056i \(0.373492\pi\)
\(824\) −6.49852e9 −0.404640
\(825\) 0 0
\(826\) −2.65072e9 −0.163657
\(827\) −7.73227e9 −0.475376 −0.237688 0.971342i \(-0.576390\pi\)
−0.237688 + 0.971342i \(0.576390\pi\)
\(828\) 0 0
\(829\) 8.83980e9 0.538892 0.269446 0.963016i \(-0.413159\pi\)
0.269446 + 0.963016i \(0.413159\pi\)
\(830\) 1.54731e8 0.00939300
\(831\) 0 0
\(832\) −2.82858e10 −1.70269
\(833\) 1.48052e9 0.0887474
\(834\) 0 0
\(835\) −4.61743e7 −0.00274472
\(836\) 6.81421e9 0.403361
\(837\) 0 0
\(838\) 1.74758e10 1.02585
\(839\) 1.45899e10 0.852877 0.426439 0.904516i \(-0.359768\pi\)
0.426439 + 0.904516i \(0.359768\pi\)
\(840\) 0 0
\(841\) 3.68072e10 2.13377
\(842\) −3.61320e10 −2.08593
\(843\) 0 0
\(844\) 1.67352e10 0.958147
\(845\) −7.17549e6 −0.000409122 0
\(846\) 0 0
\(847\) 1.31288e10 0.742390
\(848\) 5.51249e9 0.310429
\(849\) 0 0
\(850\) −6.77796e9 −0.378559
\(851\) −2.30130e9 −0.128003
\(852\) 0 0
\(853\) −1.83684e10 −1.01333 −0.506665 0.862143i \(-0.669122\pi\)
−0.506665 + 0.862143i \(0.669122\pi\)
\(854\) 4.19783e9 0.230633
\(855\) 0 0
\(856\) 8.70941e8 0.0474603
\(857\) −1.71154e10 −0.928871 −0.464436 0.885607i \(-0.653743\pi\)
−0.464436 + 0.885607i \(0.653743\pi\)
\(858\) 0 0
\(859\) −1.64262e10 −0.884220 −0.442110 0.896961i \(-0.645770\pi\)
−0.442110 + 0.896961i \(0.645770\pi\)
\(860\) −8.70868e7 −0.00466883
\(861\) 0 0
\(862\) 5.29210e9 0.281418
\(863\) 1.57145e10 0.832265 0.416132 0.909304i \(-0.363385\pi\)
0.416132 + 0.909304i \(0.363385\pi\)
\(864\) 0 0
\(865\) −2.52248e7 −0.00132517
\(866\) 2.55442e10 1.33653
\(867\) 0 0
\(868\) 1.19040e10 0.617836
\(869\) 3.77379e9 0.195078
\(870\) 0 0
\(871\) −1.42755e10 −0.732026
\(872\) −2.75974e10 −1.40949
\(873\) 0 0
\(874\) 2.47323e10 1.25307
\(875\) −1.37467e8 −0.00693700
\(876\) 0 0
\(877\) 4.28124e9 0.214324 0.107162 0.994242i \(-0.465824\pi\)
0.107162 + 0.994242i \(0.465824\pi\)
\(878\) 2.78885e10 1.39057
\(879\) 0 0
\(880\) −4.86685e6 −0.000240746 0
\(881\) −1.71858e10 −0.846748 −0.423374 0.905955i \(-0.639154\pi\)
−0.423374 + 0.905955i \(0.639154\pi\)
\(882\) 0 0
\(883\) −1.87937e10 −0.918648 −0.459324 0.888269i \(-0.651908\pi\)
−0.459324 + 0.888269i \(0.651908\pi\)
\(884\) −7.76943e9 −0.378274
\(885\) 0 0
\(886\) −1.15073e10 −0.555849
\(887\) −1.22073e10 −0.587339 −0.293669 0.955907i \(-0.594876\pi\)
−0.293669 + 0.955907i \(0.594876\pi\)
\(888\) 0 0
\(889\) −1.85315e10 −0.884617
\(890\) −2.50238e8 −0.0118984
\(891\) 0 0
\(892\) 2.38011e10 1.12284
\(893\) 2.33815e10 1.09873
\(894\) 0 0
\(895\) −1.24690e8 −0.00581368
\(896\) −2.42005e10 −1.12395
\(897\) 0 0
\(898\) 5.32448e10 2.45364
\(899\) −1.98357e10 −0.910516
\(900\) 0 0
\(901\) 7.44129e9 0.338931
\(902\) 1.17441e10 0.532841
\(903\) 0 0
\(904\) 1.55276e10 0.699063
\(905\) −1.11884e7 −0.000501761 0
\(906\) 0 0
\(907\) 1.54047e9 0.0685534 0.0342767 0.999412i \(-0.489087\pi\)
0.0342767 + 0.999412i \(0.489087\pi\)
\(908\) −2.17477e10 −0.964078
\(909\) 0 0
\(910\) −1.30772e8 −0.00575269
\(911\) −3.19695e10 −1.40094 −0.700472 0.713680i \(-0.747027\pi\)
−0.700472 + 0.713680i \(0.747027\pi\)
\(912\) 0 0
\(913\) −7.83088e9 −0.340536
\(914\) 5.95352e10 2.57906
\(915\) 0 0
\(916\) 2.48941e10 1.07019
\(917\) −2.90671e10 −1.24483
\(918\) 0 0
\(919\) 2.59602e10 1.10333 0.551663 0.834067i \(-0.313993\pi\)
0.551663 + 0.834067i \(0.313993\pi\)
\(920\) 6.31377e7 0.00267320
\(921\) 0 0
\(922\) 1.70227e10 0.715272
\(923\) 3.55523e10 1.48820
\(924\) 0 0
\(925\) 4.12467e9 0.171354
\(926\) −2.39349e10 −0.990589
\(927\) 0 0
\(928\) 5.01851e10 2.06138
\(929\) 6.20074e9 0.253740 0.126870 0.991919i \(-0.459507\pi\)
0.126870 + 0.991919i \(0.459507\pi\)
\(930\) 0 0
\(931\) 9.68263e9 0.393251
\(932\) 6.98593e9 0.282663
\(933\) 0 0
\(934\) −6.75024e10 −2.71085
\(935\) −6.56974e6 −0.000262850 0
\(936\) 0 0
\(937\) −2.50957e10 −0.996575 −0.498288 0.867012i \(-0.666038\pi\)
−0.498288 + 0.867012i \(0.666038\pi\)
\(938\) −2.22428e10 −0.879995
\(939\) 0 0
\(940\) 1.75463e8 0.00689028
\(941\) 8.17848e9 0.319970 0.159985 0.987119i \(-0.448855\pi\)
0.159985 + 0.987119i \(0.448855\pi\)
\(942\) 0 0
\(943\) 2.56808e10 0.997282
\(944\) 7.35614e8 0.0284608
\(945\) 0 0
\(946\) 7.31552e9 0.280948
\(947\) −1.81622e10 −0.694935 −0.347468 0.937692i \(-0.612958\pi\)
−0.347468 + 0.937692i \(0.612958\pi\)
\(948\) 0 0
\(949\) −1.29094e10 −0.490315
\(950\) −4.43281e10 −1.67744
\(951\) 0 0
\(952\) −4.11815e9 −0.154694
\(953\) 4.42866e10 1.65748 0.828738 0.559637i \(-0.189059\pi\)
0.828738 + 0.559637i \(0.189059\pi\)
\(954\) 0 0
\(955\) 1.56737e8 0.00582316
\(956\) 4.46935e10 1.65440
\(957\) 0 0
\(958\) −5.26702e10 −1.93547
\(959\) 2.44976e10 0.896929
\(960\) 0 0
\(961\) −2.02341e10 −0.735449
\(962\) 7.84764e9 0.284202
\(963\) 0 0
\(964\) −6.00600e10 −2.15931
\(965\) −3.63914e7 −0.00130363
\(966\) 0 0
\(967\) 3.34893e10 1.19101 0.595503 0.803353i \(-0.296953\pi\)
0.595503 + 0.803353i \(0.296953\pi\)
\(968\) 2.16153e10 0.765946
\(969\) 0 0
\(970\) 2.99197e8 0.0105258
\(971\) 2.53801e10 0.889663 0.444831 0.895614i \(-0.353264\pi\)
0.444831 + 0.895614i \(0.353264\pi\)
\(972\) 0 0
\(973\) −2.86599e9 −0.0997423
\(974\) 2.25980e10 0.783633
\(975\) 0 0
\(976\) −1.16496e9 −0.0401084
\(977\) 1.10528e10 0.379177 0.189589 0.981864i \(-0.439285\pi\)
0.189589 + 0.981864i \(0.439285\pi\)
\(978\) 0 0
\(979\) 1.26644e10 0.431366
\(980\) 7.26616e7 0.00246612
\(981\) 0 0
\(982\) 7.55401e9 0.254558
\(983\) 3.71529e9 0.124754 0.0623771 0.998053i \(-0.480132\pi\)
0.0623771 + 0.998053i \(0.480132\pi\)
\(984\) 0 0
\(985\) −1.81935e7 −0.000606582 0
\(986\) 2.01718e10 0.670154
\(987\) 0 0
\(988\) −5.08123e10 −1.67618
\(989\) 1.59968e10 0.525831
\(990\) 0 0
\(991\) −3.64285e10 −1.18901 −0.594503 0.804094i \(-0.702651\pi\)
−0.594503 + 0.804094i \(0.702651\pi\)
\(992\) −1.84149e10 −0.598933
\(993\) 0 0
\(994\) 5.53946e10 1.78902
\(995\) 1.31121e8 0.00421978
\(996\) 0 0
\(997\) 4.94528e10 1.58037 0.790183 0.612871i \(-0.209985\pi\)
0.790183 + 0.612871i \(0.209985\pi\)
\(998\) 8.73170e10 2.78062
\(999\) 0 0
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 531.8.a.b.1.3 16
3.2 odd 2 177.8.a.a.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.14 16 3.2 odd 2
531.8.a.b.1.3 16 1.1 even 1 trivial