Properties

Label 177.8.a.a.1.14
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(-17.9442\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+17.9442 q^{2} -27.0000 q^{3} +193.993 q^{4} +1.22320 q^{5} -484.493 q^{6} -719.259 q^{7} +1184.20 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+17.9442 q^{2} -27.0000 q^{3} +193.993 q^{4} +1.22320 q^{5} -484.493 q^{6} -719.259 q^{7} +1184.20 q^{8} +729.000 q^{9} +21.9494 q^{10} +1110.85 q^{11} -5237.82 q^{12} +8283.40 q^{13} -12906.5 q^{14} -33.0265 q^{15} -3581.74 q^{16} +4834.97 q^{17} +13081.3 q^{18} -31620.9 q^{19} +237.293 q^{20} +19420.0 q^{21} +19933.3 q^{22} -43588.0 q^{23} -31973.3 q^{24} -78123.5 q^{25} +148639. q^{26} -19683.0 q^{27} -139531. q^{28} -232502. q^{29} -592.634 q^{30} -85314.0 q^{31} -215848. q^{32} -29992.9 q^{33} +86759.6 q^{34} -879.800 q^{35} +141421. q^{36} -52796.8 q^{37} -567411. q^{38} -223652. q^{39} +1448.51 q^{40} -589172. q^{41} +348476. q^{42} +367000. q^{43} +215497. q^{44} +891.716 q^{45} -782150. q^{46} +739433. q^{47} +96706.9 q^{48} -306210. q^{49} -1.40186e6 q^{50} -130544. q^{51} +1.60692e6 q^{52} +1.53905e6 q^{53} -353195. q^{54} +1358.79 q^{55} -851743. q^{56} +853764. q^{57} -4.17205e6 q^{58} +205379. q^{59} -6406.92 q^{60} +325249. q^{61} -1.53089e6 q^{62} -524340. q^{63} -3.41476e6 q^{64} +10132.3 q^{65} -538198. q^{66} -1.72338e6 q^{67} +937953. q^{68} +1.17687e6 q^{69} -15787.3 q^{70} -4.29199e6 q^{71} +863278. q^{72} -1.55847e6 q^{73} -947394. q^{74} +2.10933e6 q^{75} -6.13424e6 q^{76} -798988. q^{77} -4.01324e6 q^{78} -3.39722e6 q^{79} -4381.20 q^{80} +531441. q^{81} -1.05722e7 q^{82} -7.04945e6 q^{83} +3.76735e6 q^{84} +5914.16 q^{85} +6.58552e6 q^{86} +6.27755e6 q^{87} +1.31546e6 q^{88} +1.14007e7 q^{89} +16001.1 q^{90} -5.95791e6 q^{91} -8.45577e6 q^{92} +2.30348e6 q^{93} +1.32685e7 q^{94} -38678.8 q^{95} +5.82790e6 q^{96} +1.36312e7 q^{97} -5.49468e6 q^{98} +809809. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.9442 1.58606 0.793028 0.609185i \(-0.208503\pi\)
0.793028 + 0.609185i \(0.208503\pi\)
\(3\) −27.0000 −0.577350
\(4\) 193.993 1.51557
\(5\) 1.22320 0.00437627 0.00218813 0.999998i \(-0.499303\pi\)
0.00218813 + 0.999998i \(0.499303\pi\)
\(6\) −484.493 −0.915710
\(7\) −719.259 −0.792578 −0.396289 0.918126i \(-0.629702\pi\)
−0.396289 + 0.918126i \(0.629702\pi\)
\(8\) 1184.20 0.817727
\(9\) 729.000 0.333333
\(10\) 21.9494 0.00694101
\(11\) 1110.85 0.251640 0.125820 0.992053i \(-0.459844\pi\)
0.125820 + 0.992053i \(0.459844\pi\)
\(12\) −5237.82 −0.875016
\(13\) 8283.40 1.04570 0.522850 0.852425i \(-0.324869\pi\)
0.522850 + 0.852425i \(0.324869\pi\)
\(14\) −12906.5 −1.25707
\(15\) −33.0265 −0.00252664
\(16\) −3581.74 −0.218612
\(17\) 4834.97 0.238684 0.119342 0.992853i \(-0.461922\pi\)
0.119342 + 0.992853i \(0.461922\pi\)
\(18\) 13081.3 0.528685
\(19\) −31620.9 −1.05764 −0.528818 0.848735i \(-0.677365\pi\)
−0.528818 + 0.848735i \(0.677365\pi\)
\(20\) 237.293 0.00663255
\(21\) 19420.0 0.457595
\(22\) 19933.3 0.399116
\(23\) −43588.0 −0.746997 −0.373499 0.927631i \(-0.621842\pi\)
−0.373499 + 0.927631i \(0.621842\pi\)
\(24\) −31973.3 −0.472115
\(25\) −78123.5 −0.999981
\(26\) 148639. 1.65854
\(27\) −19683.0 −0.192450
\(28\) −139531. −1.20121
\(29\) −232502. −1.77025 −0.885123 0.465358i \(-0.845926\pi\)
−0.885123 + 0.465358i \(0.845926\pi\)
\(30\) −592.634 −0.00400739
\(31\) −85314.0 −0.514345 −0.257172 0.966366i \(-0.582791\pi\)
−0.257172 + 0.966366i \(0.582791\pi\)
\(32\) −215848. −1.16446
\(33\) −29992.9 −0.145285
\(34\) 86759.6 0.378566
\(35\) −879.800 −0.00346853
\(36\) 141421. 0.505191
\(37\) −52796.8 −0.171357 −0.0856784 0.996323i \(-0.527306\pi\)
−0.0856784 + 0.996323i \(0.527306\pi\)
\(38\) −567411. −1.67747
\(39\) −223652. −0.603735
\(40\) 1448.51 0.00357859
\(41\) −589172. −1.33505 −0.667527 0.744585i \(-0.732647\pi\)
−0.667527 + 0.744585i \(0.732647\pi\)
\(42\) 348476. 0.725771
\(43\) 367000. 0.703926 0.351963 0.936014i \(-0.385514\pi\)
0.351963 + 0.936014i \(0.385514\pi\)
\(44\) 215497. 0.381379
\(45\) 891.716 0.00145876
\(46\) −782150. −1.18478
\(47\) 739433. 1.03886 0.519429 0.854514i \(-0.326145\pi\)
0.519429 + 0.854514i \(0.326145\pi\)
\(48\) 96706.9 0.126216
\(49\) −306210. −0.371820
\(50\) −1.40186e6 −1.58603
\(51\) −130544. −0.137804
\(52\) 1.60692e6 1.58483
\(53\) 1.53905e6 1.42000 0.710000 0.704201i \(-0.248695\pi\)
0.710000 + 0.704201i \(0.248695\pi\)
\(54\) −353195. −0.305237
\(55\) 1358.79 0.00110125
\(56\) −851743. −0.648113
\(57\) 853764. 0.610627
\(58\) −4.17205e6 −2.80771
\(59\) 205379. 0.130189
\(60\) −6406.92 −0.00382931
\(61\) 325249. 0.183469 0.0917343 0.995784i \(-0.470759\pi\)
0.0917343 + 0.995784i \(0.470759\pi\)
\(62\) −1.53089e6 −0.815779
\(63\) −524340. −0.264193
\(64\) −3.41476e6 −1.62828
\(65\) 10132.3 0.00457626
\(66\) −538198. −0.230430
\(67\) −1.72338e6 −0.700035 −0.350018 0.936743i \(-0.613824\pi\)
−0.350018 + 0.936743i \(0.613824\pi\)
\(68\) 937953. 0.361743
\(69\) 1.17687e6 0.431279
\(70\) −15787.3 −0.00550129
\(71\) −4.29199e6 −1.42316 −0.711582 0.702603i \(-0.752021\pi\)
−0.711582 + 0.702603i \(0.752021\pi\)
\(72\) 863278. 0.272576
\(73\) −1.55847e6 −0.468888 −0.234444 0.972130i \(-0.575327\pi\)
−0.234444 + 0.972130i \(0.575327\pi\)
\(74\) −947394. −0.271781
\(75\) 2.10933e6 0.577339
\(76\) −6.13424e6 −1.60292
\(77\) −798988. −0.199445
\(78\) −4.01324e6 −0.957557
\(79\) −3.39722e6 −0.775226 −0.387613 0.921822i \(-0.626700\pi\)
−0.387613 + 0.921822i \(0.626700\pi\)
\(80\) −4381.20 −0.000956704 0
\(81\) 531441. 0.111111
\(82\) −1.05722e7 −2.11747
\(83\) −7.04945e6 −1.35326 −0.676631 0.736322i \(-0.736561\pi\)
−0.676631 + 0.736322i \(0.736561\pi\)
\(84\) 3.76735e6 0.693519
\(85\) 5914.16 0.00104454
\(86\) 6.58552e6 1.11647
\(87\) 6.27755e6 1.02205
\(88\) 1.31546e6 0.205773
\(89\) 1.14007e7 1.71422 0.857109 0.515136i \(-0.172259\pi\)
0.857109 + 0.515136i \(0.172259\pi\)
\(90\) 16001.1 0.00231367
\(91\) −5.95791e6 −0.828798
\(92\) −8.45577e6 −1.13213
\(93\) 2.30348e6 0.296957
\(94\) 1.32685e7 1.64769
\(95\) −38678.8 −0.00462850
\(96\) 5.82790e6 0.672300
\(97\) 1.36312e7 1.51647 0.758236 0.651980i \(-0.226062\pi\)
0.758236 + 0.651980i \(0.226062\pi\)
\(98\) −5.49468e6 −0.589727
\(99\) 809809. 0.0838802
\(100\) −1.51554e7 −1.51554
\(101\) −1.39105e7 −1.34344 −0.671719 0.740806i \(-0.734444\pi\)
−0.671719 + 0.740806i \(0.734444\pi\)
\(102\) −2.34251e6 −0.218565
\(103\) 5.48771e6 0.494835 0.247418 0.968909i \(-0.420418\pi\)
0.247418 + 0.968909i \(0.420418\pi\)
\(104\) 9.80916e6 0.855097
\(105\) 23754.6 0.00200256
\(106\) 2.76171e7 2.25220
\(107\) 735470. 0.0580393 0.0290196 0.999579i \(-0.490761\pi\)
0.0290196 + 0.999579i \(0.490761\pi\)
\(108\) −3.81837e6 −0.291672
\(109\) 2.33048e7 1.72366 0.861831 0.507196i \(-0.169318\pi\)
0.861831 + 0.507196i \(0.169318\pi\)
\(110\) 24382.5 0.00174664
\(111\) 1.42551e6 0.0989329
\(112\) 2.57620e6 0.173267
\(113\) 1.31124e7 0.854885 0.427442 0.904043i \(-0.359415\pi\)
0.427442 + 0.904043i \(0.359415\pi\)
\(114\) 1.53201e7 0.968488
\(115\) −53317.0 −0.00326906
\(116\) −4.51038e7 −2.68294
\(117\) 6.03860e6 0.348566
\(118\) 3.68536e6 0.206487
\(119\) −3.47760e6 −0.189176
\(120\) −39109.9 −0.00206610
\(121\) −1.82532e7 −0.936677
\(122\) 5.83632e6 0.290991
\(123\) 1.59077e7 0.770794
\(124\) −1.65503e7 −0.779527
\(125\) −191124. −0.00875245
\(126\) −9.40884e6 −0.419024
\(127\) 2.57648e7 1.11613 0.558063 0.829799i \(-0.311545\pi\)
0.558063 + 0.829799i \(0.311545\pi\)
\(128\) −3.36464e7 −1.41809
\(129\) −9.90901e6 −0.406412
\(130\) 181815. 0.00725820
\(131\) −4.04125e7 −1.57060 −0.785302 0.619113i \(-0.787492\pi\)
−0.785302 + 0.619113i \(0.787492\pi\)
\(132\) −5.81843e6 −0.220190
\(133\) 2.27436e7 0.838259
\(134\) −3.09247e7 −1.11029
\(135\) −24076.3 −0.000842213 0
\(136\) 5.72555e6 0.195178
\(137\) 3.40595e7 1.13166 0.565830 0.824522i \(-0.308556\pi\)
0.565830 + 0.824522i \(0.308556\pi\)
\(138\) 2.11180e7 0.684033
\(139\) 3.98464e6 0.125845 0.0629227 0.998018i \(-0.479958\pi\)
0.0629227 + 0.998018i \(0.479958\pi\)
\(140\) −170675. −0.00525682
\(141\) −1.99647e7 −0.599785
\(142\) −7.70163e7 −2.25722
\(143\) 9.20160e6 0.263140
\(144\) −2.61109e6 −0.0728706
\(145\) −284397. −0.00774707
\(146\) −2.79655e7 −0.743682
\(147\) 8.26767e6 0.214670
\(148\) −1.02422e7 −0.259704
\(149\) 5.83891e6 0.144604 0.0723020 0.997383i \(-0.476965\pi\)
0.0723020 + 0.997383i \(0.476965\pi\)
\(150\) 3.78503e7 0.915692
\(151\) 5.13811e7 1.21446 0.607231 0.794525i \(-0.292280\pi\)
0.607231 + 0.794525i \(0.292280\pi\)
\(152\) −3.74453e7 −0.864858
\(153\) 3.52470e6 0.0795613
\(154\) −1.43372e7 −0.316330
\(155\) −104356. −0.00225091
\(156\) −4.33869e7 −0.915004
\(157\) −1.14443e7 −0.236016 −0.118008 0.993013i \(-0.537651\pi\)
−0.118008 + 0.993013i \(0.537651\pi\)
\(158\) −6.09602e7 −1.22955
\(159\) −4.15545e7 −0.819838
\(160\) −264027. −0.00509598
\(161\) 3.13510e7 0.592054
\(162\) 9.53627e6 0.176228
\(163\) 1.25340e7 0.226690 0.113345 0.993556i \(-0.463844\pi\)
0.113345 + 0.993556i \(0.463844\pi\)
\(164\) −1.14296e8 −2.02337
\(165\) −36687.5 −0.000635805 0
\(166\) −1.26497e8 −2.14635
\(167\) −3.77486e7 −0.627182 −0.313591 0.949558i \(-0.601532\pi\)
−0.313591 + 0.949558i \(0.601532\pi\)
\(168\) 2.29971e7 0.374188
\(169\) 5.86614e6 0.0934865
\(170\) 106125. 0.00165671
\(171\) −2.30516e7 −0.352545
\(172\) 7.11956e7 1.06685
\(173\) −2.06219e7 −0.302808 −0.151404 0.988472i \(-0.548379\pi\)
−0.151404 + 0.988472i \(0.548379\pi\)
\(174\) 1.12645e8 1.62103
\(175\) 5.61910e7 0.792563
\(176\) −3.97877e6 −0.0550116
\(177\) −5.54523e6 −0.0751646
\(178\) 2.04576e8 2.71884
\(179\) −1.01937e8 −1.32845 −0.664227 0.747531i \(-0.731239\pi\)
−0.664227 + 0.747531i \(0.731239\pi\)
\(180\) 172987. 0.00221085
\(181\) 9.14678e6 0.114655 0.0573275 0.998355i \(-0.481742\pi\)
0.0573275 + 0.998355i \(0.481742\pi\)
\(182\) −1.06910e8 −1.31452
\(183\) −8.78172e6 −0.105926
\(184\) −5.16167e7 −0.610840
\(185\) −64581.2 −0.000749903 0
\(186\) 4.13340e7 0.470990
\(187\) 5.37092e6 0.0600625
\(188\) 1.43445e8 1.57447
\(189\) 1.41572e7 0.152532
\(190\) −694059. −0.00734106
\(191\) 1.28136e8 1.33062 0.665311 0.746566i \(-0.268299\pi\)
0.665311 + 0.746566i \(0.268299\pi\)
\(192\) 9.21984e7 0.940090
\(193\) 2.97509e7 0.297885 0.148943 0.988846i \(-0.452413\pi\)
0.148943 + 0.988846i \(0.452413\pi\)
\(194\) 2.44601e8 2.40521
\(195\) −273572. −0.00264211
\(196\) −5.94027e7 −0.563521
\(197\) −1.48736e7 −0.138607 −0.0693036 0.997596i \(-0.522078\pi\)
−0.0693036 + 0.997596i \(0.522078\pi\)
\(198\) 1.45313e7 0.133039
\(199\) −1.07194e8 −0.964242 −0.482121 0.876105i \(-0.660133\pi\)
−0.482121 + 0.876105i \(0.660133\pi\)
\(200\) −9.25135e7 −0.817712
\(201\) 4.65313e7 0.404166
\(202\) −2.49612e8 −2.13077
\(203\) 1.67229e8 1.40306
\(204\) −2.53247e7 −0.208852
\(205\) −720678. −0.00584256
\(206\) 9.84724e7 0.784836
\(207\) −3.17756e7 −0.248999
\(208\) −2.96690e7 −0.228602
\(209\) −3.51260e7 −0.266144
\(210\) 426257. 0.00317617
\(211\) 8.62669e7 0.632201 0.316101 0.948726i \(-0.397626\pi\)
0.316101 + 0.948726i \(0.397626\pi\)
\(212\) 2.98566e8 2.15211
\(213\) 1.15884e8 0.821664
\(214\) 1.31974e7 0.0920535
\(215\) 448917. 0.00308057
\(216\) −2.33085e7 −0.157372
\(217\) 6.13628e7 0.407658
\(218\) 4.18185e8 2.73382
\(219\) 4.20787e7 0.270712
\(220\) 263597. 0.00166902
\(221\) 4.00500e7 0.249591
\(222\) 2.55797e7 0.156913
\(223\) 1.22690e8 0.740871 0.370435 0.928858i \(-0.379208\pi\)
0.370435 + 0.928858i \(0.379208\pi\)
\(224\) 1.55251e8 0.922924
\(225\) −5.69520e7 −0.333327
\(226\) 2.35291e8 1.35590
\(227\) 1.12105e8 0.636115 0.318057 0.948071i \(-0.396970\pi\)
0.318057 + 0.948071i \(0.396970\pi\)
\(228\) 1.65624e8 0.925449
\(229\) 1.28324e8 0.706131 0.353066 0.935599i \(-0.385139\pi\)
0.353066 + 0.935599i \(0.385139\pi\)
\(230\) −956729. −0.00518491
\(231\) 2.15727e7 0.115149
\(232\) −2.75328e8 −1.44758
\(233\) −3.60112e7 −0.186505 −0.0932527 0.995642i \(-0.529726\pi\)
−0.0932527 + 0.995642i \(0.529726\pi\)
\(234\) 1.08358e8 0.552846
\(235\) 904477. 0.00454632
\(236\) 3.98422e7 0.197311
\(237\) 9.17248e7 0.447577
\(238\) −6.24026e7 −0.300043
\(239\) −2.30387e8 −1.09160 −0.545801 0.837915i \(-0.683775\pi\)
−0.545801 + 0.837915i \(0.683775\pi\)
\(240\) 118292. 0.000552354 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
\(243\) −1.43489e7 −0.0641500
\(244\) 6.30961e7 0.278060
\(245\) −374557. −0.00162718
\(246\) 2.85450e8 1.22252
\(247\) −2.61928e8 −1.10597
\(248\) −1.01028e8 −0.420594
\(249\) 1.90335e8 0.781307
\(250\) −3.42956e6 −0.0138819
\(251\) 1.81483e8 0.724399 0.362200 0.932101i \(-0.382026\pi\)
0.362200 + 0.932101i \(0.382026\pi\)
\(252\) −1.01718e8 −0.400403
\(253\) −4.84196e7 −0.187975
\(254\) 4.62327e8 1.77024
\(255\) −159682. −0.000603068 0
\(256\) −1.66668e8 −0.620887
\(257\) 4.15191e8 1.52574 0.762872 0.646549i \(-0.223788\pi\)
0.762872 + 0.646549i \(0.223788\pi\)
\(258\) −1.77809e8 −0.644592
\(259\) 3.79745e7 0.135814
\(260\) 1.96560e6 0.00693566
\(261\) −1.69494e8 −0.590082
\(262\) −7.25169e8 −2.49106
\(263\) −3.81009e8 −1.29149 −0.645744 0.763554i \(-0.723453\pi\)
−0.645744 + 0.763554i \(0.723453\pi\)
\(264\) −3.55175e7 −0.118803
\(265\) 1.88258e6 0.00621430
\(266\) 4.08115e8 1.32953
\(267\) −3.07818e8 −0.989704
\(268\) −3.34325e8 −1.06095
\(269\) −9.53490e7 −0.298664 −0.149332 0.988787i \(-0.547712\pi\)
−0.149332 + 0.988787i \(0.547712\pi\)
\(270\) −432030. −0.00133580
\(271\) −4.17979e8 −1.27574 −0.637870 0.770144i \(-0.720184\pi\)
−0.637870 + 0.770144i \(0.720184\pi\)
\(272\) −1.73176e7 −0.0521791
\(273\) 1.60863e8 0.478507
\(274\) 6.11169e8 1.79488
\(275\) −8.67834e7 −0.251636
\(276\) 2.28306e8 0.653635
\(277\) −3.26940e8 −0.924249 −0.462125 0.886815i \(-0.652913\pi\)
−0.462125 + 0.886815i \(0.652913\pi\)
\(278\) 7.15011e7 0.199598
\(279\) −6.21939e7 −0.171448
\(280\) −1.04186e6 −0.00283631
\(281\) 9.47550e7 0.254759 0.127380 0.991854i \(-0.459343\pi\)
0.127380 + 0.991854i \(0.459343\pi\)
\(282\) −3.58250e8 −0.951292
\(283\) −4.46938e8 −1.17218 −0.586091 0.810246i \(-0.699334\pi\)
−0.586091 + 0.810246i \(0.699334\pi\)
\(284\) −8.32618e8 −2.15691
\(285\) 1.04433e6 0.00267227
\(286\) 1.65115e8 0.417355
\(287\) 4.23767e8 1.05813
\(288\) −1.57353e8 −0.388153
\(289\) −3.86962e8 −0.943030
\(290\) −5.10327e6 −0.0122873
\(291\) −3.68044e8 −0.875535
\(292\) −3.02333e8 −0.710634
\(293\) 4.04818e8 0.940207 0.470103 0.882611i \(-0.344217\pi\)
0.470103 + 0.882611i \(0.344217\pi\)
\(294\) 1.48356e8 0.340479
\(295\) 251220. 0.000569742 0
\(296\) −6.25217e7 −0.140123
\(297\) −2.18648e7 −0.0484282
\(298\) 1.04774e8 0.229350
\(299\) −3.61056e8 −0.781134
\(300\) 4.09197e8 0.875000
\(301\) −2.63968e8 −0.557916
\(302\) 9.21991e8 1.92620
\(303\) 3.75583e8 0.775634
\(304\) 1.13258e8 0.231212
\(305\) 397846. 0.000802908 0
\(306\) 6.32478e7 0.126189
\(307\) −7.95438e8 −1.56900 −0.784498 0.620131i \(-0.787079\pi\)
−0.784498 + 0.620131i \(0.787079\pi\)
\(308\) −1.54998e8 −0.302273
\(309\) −1.48168e8 −0.285693
\(310\) −1.87259e6 −0.00357007
\(311\) 3.49530e8 0.658906 0.329453 0.944172i \(-0.393136\pi\)
0.329453 + 0.944172i \(0.393136\pi\)
\(312\) −2.64847e8 −0.493690
\(313\) 9.82107e7 0.181031 0.0905157 0.995895i \(-0.471148\pi\)
0.0905157 + 0.995895i \(0.471148\pi\)
\(314\) −2.05359e8 −0.374334
\(315\) −641374. −0.00115618
\(316\) −6.59037e8 −1.17491
\(317\) 9.11476e8 1.60708 0.803541 0.595250i \(-0.202947\pi\)
0.803541 + 0.595250i \(0.202947\pi\)
\(318\) −7.45661e8 −1.30031
\(319\) −2.58274e8 −0.445465
\(320\) −4.17695e6 −0.00712580
\(321\) −1.98577e7 −0.0335090
\(322\) 5.62568e8 0.939030
\(323\) −1.52886e8 −0.252441
\(324\) 1.03096e8 0.168397
\(325\) −6.47128e8 −1.04568
\(326\) 2.24912e8 0.359542
\(327\) −6.29229e8 −0.995157
\(328\) −6.97695e8 −1.09171
\(329\) −5.31844e8 −0.823376
\(330\) −658326. −0.00100842
\(331\) 8.69089e8 1.31724 0.658622 0.752474i \(-0.271140\pi\)
0.658622 + 0.752474i \(0.271140\pi\)
\(332\) −1.36755e9 −2.05097
\(333\) −3.84889e7 −0.0571189
\(334\) −6.77368e8 −0.994745
\(335\) −2.10805e6 −0.00306354
\(336\) −6.95573e7 −0.100036
\(337\) 3.71807e7 0.0529192 0.0264596 0.999650i \(-0.491577\pi\)
0.0264596 + 0.999650i \(0.491577\pi\)
\(338\) 1.05263e8 0.148275
\(339\) −3.54035e8 −0.493568
\(340\) 1.14731e6 0.00158308
\(341\) −9.47709e7 −0.129430
\(342\) −4.13642e8 −0.559157
\(343\) 8.12585e8 1.08727
\(344\) 4.34600e8 0.575619
\(345\) 1.43956e6 0.00188739
\(346\) −3.70043e8 −0.480270
\(347\) −5.17499e8 −0.664900 −0.332450 0.943121i \(-0.607875\pi\)
−0.332450 + 0.943121i \(0.607875\pi\)
\(348\) 1.21780e9 1.54899
\(349\) −7.74296e8 −0.975030 −0.487515 0.873115i \(-0.662097\pi\)
−0.487515 + 0.873115i \(0.662097\pi\)
\(350\) 1.00830e9 1.25705
\(351\) −1.63042e8 −0.201245
\(352\) −2.39775e8 −0.293025
\(353\) −6.61876e8 −0.800876 −0.400438 0.916324i \(-0.631142\pi\)
−0.400438 + 0.916324i \(0.631142\pi\)
\(354\) −9.95046e7 −0.119215
\(355\) −5.24998e6 −0.00622815
\(356\) 2.21166e9 2.59802
\(357\) 9.38951e7 0.109221
\(358\) −1.82918e9 −2.10700
\(359\) 6.26557e8 0.714711 0.357355 0.933968i \(-0.383679\pi\)
0.357355 + 0.933968i \(0.383679\pi\)
\(360\) 1.05597e6 0.00119286
\(361\) 1.06008e8 0.118595
\(362\) 1.64131e8 0.181849
\(363\) 4.92836e8 0.540791
\(364\) −1.15579e9 −1.25610
\(365\) −1.90633e6 −0.00205198
\(366\) −1.57581e8 −0.168004
\(367\) 8.99146e7 0.0949509 0.0474755 0.998872i \(-0.484882\pi\)
0.0474755 + 0.998872i \(0.484882\pi\)
\(368\) 1.56121e8 0.163303
\(369\) −4.29507e8 −0.445018
\(370\) −1.15886e6 −0.00118939
\(371\) −1.10698e9 −1.12546
\(372\) 4.46859e8 0.450060
\(373\) 1.93821e8 0.193383 0.0966917 0.995314i \(-0.469174\pi\)
0.0966917 + 0.995314i \(0.469174\pi\)
\(374\) 9.63768e7 0.0952625
\(375\) 5.16034e6 0.00505323
\(376\) 8.75633e8 0.849503
\(377\) −1.92591e9 −1.85114
\(378\) 2.54039e8 0.241924
\(379\) −2.28105e8 −0.215227 −0.107614 0.994193i \(-0.534321\pi\)
−0.107614 + 0.994193i \(0.534321\pi\)
\(380\) −7.50343e6 −0.00701483
\(381\) −6.95649e8 −0.644395
\(382\) 2.29930e9 2.11044
\(383\) −1.16094e8 −0.105588 −0.0527938 0.998605i \(-0.516813\pi\)
−0.0527938 + 0.998605i \(0.516813\pi\)
\(384\) 9.08453e8 0.818735
\(385\) −977325. −0.000872824 0
\(386\) 5.33855e8 0.472463
\(387\) 2.67543e8 0.234642
\(388\) 2.64437e9 2.29832
\(389\) 6.84222e8 0.589351 0.294675 0.955597i \(-0.404789\pi\)
0.294675 + 0.955597i \(0.404789\pi\)
\(390\) −4.90902e6 −0.00419053
\(391\) −2.10747e8 −0.178296
\(392\) −3.62612e8 −0.304047
\(393\) 1.09114e9 0.906788
\(394\) −2.66895e8 −0.219839
\(395\) −4.15549e6 −0.00339260
\(396\) 1.57097e8 0.127126
\(397\) −1.73331e9 −1.39031 −0.695153 0.718862i \(-0.744663\pi\)
−0.695153 + 0.718862i \(0.744663\pi\)
\(398\) −1.92351e9 −1.52934
\(399\) −6.14077e8 −0.483969
\(400\) 2.79818e8 0.218608
\(401\) 1.39266e9 1.07855 0.539274 0.842130i \(-0.318699\pi\)
0.539274 + 0.842130i \(0.318699\pi\)
\(402\) 8.34966e8 0.641029
\(403\) −7.06690e8 −0.537850
\(404\) −2.69854e9 −2.03608
\(405\) 650061. 0.000486252 0
\(406\) 3.00079e9 2.22533
\(407\) −5.86492e7 −0.0431203
\(408\) −1.54590e8 −0.112686
\(409\) −1.48753e9 −1.07506 −0.537530 0.843244i \(-0.680643\pi\)
−0.537530 + 0.843244i \(0.680643\pi\)
\(410\) −1.29320e7 −0.00926662
\(411\) −9.19606e8 −0.653364
\(412\) 1.06458e9 0.749959
\(413\) −1.47721e8 −0.103185
\(414\) −5.70187e8 −0.394927
\(415\) −8.62292e6 −0.00592224
\(416\) −1.78796e9 −1.21767
\(417\) −1.07585e8 −0.0726569
\(418\) −6.30307e8 −0.422119
\(419\) 9.73899e8 0.646792 0.323396 0.946264i \(-0.395175\pi\)
0.323396 + 0.946264i \(0.395175\pi\)
\(420\) 4.60824e6 0.00303502
\(421\) 2.01358e9 1.31517 0.657584 0.753381i \(-0.271578\pi\)
0.657584 + 0.753381i \(0.271578\pi\)
\(422\) 1.54799e9 1.00271
\(423\) 5.39047e8 0.346286
\(424\) 1.82254e9 1.16117
\(425\) −3.77725e8 −0.238679
\(426\) 2.07944e9 1.30320
\(427\) −2.33938e8 −0.145413
\(428\) 1.42676e8 0.0879627
\(429\) −2.48443e8 −0.151924
\(430\) 8.05544e6 0.00488596
\(431\) 2.94920e8 0.177433 0.0887164 0.996057i \(-0.471724\pi\)
0.0887164 + 0.996057i \(0.471724\pi\)
\(432\) 7.04993e7 0.0420719
\(433\) −1.42354e9 −0.842676 −0.421338 0.906904i \(-0.638439\pi\)
−0.421338 + 0.906904i \(0.638439\pi\)
\(434\) 1.10111e9 0.646569
\(435\) 7.67873e6 0.00447277
\(436\) 4.52097e9 2.61233
\(437\) 1.37829e9 0.790052
\(438\) 7.55068e8 0.429365
\(439\) −1.55418e9 −0.876750 −0.438375 0.898792i \(-0.644446\pi\)
−0.438375 + 0.898792i \(0.644446\pi\)
\(440\) 1.60908e6 0.000900519 0
\(441\) −2.23227e8 −0.123940
\(442\) 7.18664e8 0.395866
\(443\) −6.41286e8 −0.350460 −0.175230 0.984528i \(-0.556067\pi\)
−0.175230 + 0.984528i \(0.556067\pi\)
\(444\) 2.76540e8 0.149940
\(445\) 1.39454e7 0.00750187
\(446\) 2.20157e9 1.17506
\(447\) −1.57651e8 −0.0834871
\(448\) 2.45609e9 1.29054
\(449\) 2.96725e9 1.54701 0.773503 0.633793i \(-0.218503\pi\)
0.773503 + 0.633793i \(0.218503\pi\)
\(450\) −1.02196e9 −0.528675
\(451\) −6.54481e8 −0.335954
\(452\) 2.54372e9 1.29564
\(453\) −1.38729e9 −0.701170
\(454\) 2.01164e9 1.00891
\(455\) −7.28774e6 −0.00362704
\(456\) 1.01102e9 0.499326
\(457\) −3.31780e9 −1.62609 −0.813043 0.582203i \(-0.802191\pi\)
−0.813043 + 0.582203i \(0.802191\pi\)
\(458\) 2.30268e9 1.11996
\(459\) −9.51668e7 −0.0459347
\(460\) −1.03431e7 −0.00495450
\(461\) 9.48650e8 0.450975 0.225488 0.974246i \(-0.427602\pi\)
0.225488 + 0.974246i \(0.427602\pi\)
\(462\) 3.87104e8 0.182633
\(463\) 1.33385e9 0.624561 0.312281 0.949990i \(-0.398907\pi\)
0.312281 + 0.949990i \(0.398907\pi\)
\(464\) 8.32761e8 0.386997
\(465\) 2.81762e6 0.00129956
\(466\) −6.46191e8 −0.295808
\(467\) −3.76180e9 −1.70918 −0.854588 0.519306i \(-0.826190\pi\)
−0.854588 + 0.519306i \(0.826190\pi\)
\(468\) 1.17145e9 0.528278
\(469\) 1.23956e9 0.554833
\(470\) 1.62301e7 0.00721072
\(471\) 3.08997e8 0.136264
\(472\) 2.43209e8 0.106459
\(473\) 4.07682e8 0.177136
\(474\) 1.64593e9 0.709882
\(475\) 2.47033e9 1.05762
\(476\) −6.74631e8 −0.286709
\(477\) 1.12197e9 0.473334
\(478\) −4.13410e9 −1.73134
\(479\) −2.93522e9 −1.22030 −0.610151 0.792285i \(-0.708891\pi\)
−0.610151 + 0.792285i \(0.708891\pi\)
\(480\) 7.12872e6 0.00294217
\(481\) −4.37337e8 −0.179188
\(482\) −5.55549e9 −2.25973
\(483\) −8.46478e8 −0.341822
\(484\) −3.54100e9 −1.41960
\(485\) 1.66738e7 0.00663649
\(486\) −2.57479e8 −0.101746
\(487\) −1.25935e9 −0.494077 −0.247038 0.969006i \(-0.579457\pi\)
−0.247038 + 0.969006i \(0.579457\pi\)
\(488\) 3.85158e8 0.150027
\(489\) −3.38417e8 −0.130879
\(490\) −6.72112e6 −0.00258081
\(491\) 4.20973e8 0.160498 0.0802489 0.996775i \(-0.474428\pi\)
0.0802489 + 0.996775i \(0.474428\pi\)
\(492\) 3.08598e9 1.16819
\(493\) −1.12414e9 −0.422529
\(494\) −4.70009e9 −1.75413
\(495\) 990562. 0.000367082 0
\(496\) 3.05572e8 0.112442
\(497\) 3.08705e9 1.12797
\(498\) 3.41541e9 1.23920
\(499\) −4.86604e9 −1.75317 −0.876584 0.481248i \(-0.840184\pi\)
−0.876584 + 0.481248i \(0.840184\pi\)
\(500\) −3.70767e7 −0.0132650
\(501\) 1.01921e9 0.362104
\(502\) 3.25656e9 1.14894
\(503\) −3.22489e9 −1.12987 −0.564933 0.825137i \(-0.691098\pi\)
−0.564933 + 0.825137i \(0.691098\pi\)
\(504\) −6.20921e8 −0.216038
\(505\) −1.70154e7 −0.00587925
\(506\) −8.68850e8 −0.298138
\(507\) −1.58386e8 −0.0539745
\(508\) 4.99819e9 1.69157
\(509\) 1.08431e8 0.0364452 0.0182226 0.999834i \(-0.494199\pi\)
0.0182226 + 0.999834i \(0.494199\pi\)
\(510\) −2.86537e6 −0.000956499 0
\(511\) 1.12094e9 0.371630
\(512\) 1.31602e9 0.433329
\(513\) 6.22394e8 0.203542
\(514\) 7.45026e9 2.41992
\(515\) 6.71259e6 0.00216553
\(516\) −1.92228e9 −0.615947
\(517\) 8.21398e8 0.261419
\(518\) 6.81422e8 0.215408
\(519\) 5.56791e8 0.174826
\(520\) 1.19986e7 0.00374213
\(521\) 2.88309e8 0.0893152 0.0446576 0.999002i \(-0.485780\pi\)
0.0446576 + 0.999002i \(0.485780\pi\)
\(522\) −3.04143e9 −0.935903
\(523\) 8.83228e8 0.269971 0.134985 0.990848i \(-0.456901\pi\)
0.134985 + 0.990848i \(0.456901\pi\)
\(524\) −7.83976e9 −2.38036
\(525\) −1.51716e9 −0.457586
\(526\) −6.83689e9 −2.04837
\(527\) −4.12491e8 −0.122766
\(528\) 1.07427e8 0.0317610
\(529\) −1.50492e9 −0.441995
\(530\) 3.37813e7 0.00985623
\(531\) 1.49721e8 0.0433963
\(532\) 4.41211e9 1.27044
\(533\) −4.88035e9 −1.39607
\(534\) −5.52355e9 −1.56973
\(535\) 899630. 0.000253995 0
\(536\) −2.04082e9 −0.572438
\(537\) 2.75230e9 0.766984
\(538\) −1.71096e9 −0.473698
\(539\) −3.40153e8 −0.0935650
\(540\) −4.67065e6 −0.00127644
\(541\) −3.92838e9 −1.06665 −0.533326 0.845910i \(-0.679058\pi\)
−0.533326 + 0.845910i \(0.679058\pi\)
\(542\) −7.50029e9 −2.02340
\(543\) −2.46963e8 −0.0661961
\(544\) −1.04362e9 −0.277937
\(545\) 2.85065e7 0.00754321
\(546\) 2.88656e9 0.758938
\(547\) 1.10099e9 0.287624 0.143812 0.989605i \(-0.454064\pi\)
0.143812 + 0.989605i \(0.454064\pi\)
\(548\) 6.60731e9 1.71511
\(549\) 2.37107e8 0.0611562
\(550\) −1.55726e9 −0.399108
\(551\) 7.35191e9 1.87228
\(552\) 1.39365e9 0.352669
\(553\) 2.44348e9 0.614427
\(554\) −5.86667e9 −1.46591
\(555\) 1.74369e6 0.000432957 0
\(556\) 7.72994e8 0.190728
\(557\) −5.42968e9 −1.33132 −0.665658 0.746257i \(-0.731849\pi\)
−0.665658 + 0.746257i \(0.731849\pi\)
\(558\) −1.11602e9 −0.271926
\(559\) 3.04001e9 0.736095
\(560\) 3.15121e6 0.000758263 0
\(561\) −1.45015e8 −0.0346771
\(562\) 1.70030e9 0.404063
\(563\) −7.14917e8 −0.168840 −0.0844201 0.996430i \(-0.526904\pi\)
−0.0844201 + 0.996430i \(0.526904\pi\)
\(564\) −3.87302e9 −0.909018
\(565\) 1.60391e7 0.00374121
\(566\) −8.01993e9 −1.85914
\(567\) −3.82244e8 −0.0880642
\(568\) −5.08256e9 −1.16376
\(569\) −1.67905e9 −0.382094 −0.191047 0.981581i \(-0.561188\pi\)
−0.191047 + 0.981581i \(0.561188\pi\)
\(570\) 1.87396e7 0.00423836
\(571\) −9.65128e8 −0.216950 −0.108475 0.994099i \(-0.534597\pi\)
−0.108475 + 0.994099i \(0.534597\pi\)
\(572\) 1.78505e9 0.398808
\(573\) −3.45968e9 −0.768235
\(574\) 7.60416e9 1.67826
\(575\) 3.40524e9 0.746983
\(576\) −2.48936e9 −0.542761
\(577\) 2.50966e9 0.543875 0.271938 0.962315i \(-0.412336\pi\)
0.271938 + 0.962315i \(0.412336\pi\)
\(578\) −6.94371e9 −1.49570
\(579\) −8.03274e8 −0.171984
\(580\) −5.51712e7 −0.0117412
\(581\) 5.07038e9 1.07257
\(582\) −6.60424e9 −1.38865
\(583\) 1.70966e9 0.357330
\(584\) −1.84554e9 −0.383422
\(585\) 7.38644e6 0.00152542
\(586\) 7.26413e9 1.49122
\(587\) 3.24638e9 0.662470 0.331235 0.943548i \(-0.392535\pi\)
0.331235 + 0.943548i \(0.392535\pi\)
\(588\) 1.60387e9 0.325349
\(589\) 2.69770e9 0.543990
\(590\) 4.50794e6 0.000903642 0
\(591\) 4.01588e8 0.0800249
\(592\) 1.89104e8 0.0374606
\(593\) 7.03607e9 1.38560 0.692801 0.721129i \(-0.256376\pi\)
0.692801 + 0.721129i \(0.256376\pi\)
\(594\) −3.92346e8 −0.0768099
\(595\) −4.25381e6 −0.000827883 0
\(596\) 1.13271e9 0.219158
\(597\) 2.89425e9 0.556705
\(598\) −6.47886e9 −1.23892
\(599\) 9.00254e9 1.71148 0.855738 0.517409i \(-0.173103\pi\)
0.855738 + 0.517409i \(0.173103\pi\)
\(600\) 2.49786e9 0.472106
\(601\) 5.40983e9 1.01654 0.508268 0.861199i \(-0.330286\pi\)
0.508268 + 0.861199i \(0.330286\pi\)
\(602\) −4.73669e9 −0.884886
\(603\) −1.25635e9 −0.233345
\(604\) 9.96758e9 1.84061
\(605\) −2.23274e7 −0.00409915
\(606\) 6.73953e9 1.23020
\(607\) −6.20066e9 −1.12532 −0.562662 0.826687i \(-0.690223\pi\)
−0.562662 + 0.826687i \(0.690223\pi\)
\(608\) 6.82531e9 1.23157
\(609\) −4.51518e9 −0.810056
\(610\) 7.13902e6 0.00127346
\(611\) 6.12502e9 1.08633
\(612\) 6.83767e8 0.120581
\(613\) 5.56027e9 0.974955 0.487477 0.873136i \(-0.337917\pi\)
0.487477 + 0.873136i \(0.337917\pi\)
\(614\) −1.42735e10 −2.48851
\(615\) 1.94583e7 0.00337320
\(616\) −9.46158e8 −0.163091
\(617\) 8.63571e9 1.48013 0.740065 0.672535i \(-0.234795\pi\)
0.740065 + 0.672535i \(0.234795\pi\)
\(618\) −2.65875e9 −0.453125
\(619\) 4.62517e9 0.783809 0.391904 0.920006i \(-0.371816\pi\)
0.391904 + 0.920006i \(0.371816\pi\)
\(620\) −2.02445e7 −0.00341142
\(621\) 8.57942e8 0.143760
\(622\) 6.27203e9 1.04506
\(623\) −8.20004e9 −1.35865
\(624\) 8.01062e8 0.131984
\(625\) 6.10316e9 0.999943
\(626\) 1.76231e9 0.287126
\(627\) 9.48402e8 0.153658
\(628\) −2.22012e9 −0.357699
\(629\) −2.55271e8 −0.0409001
\(630\) −1.15089e7 −0.00183376
\(631\) −4.78738e9 −0.758568 −0.379284 0.925280i \(-0.623830\pi\)
−0.379284 + 0.925280i \(0.623830\pi\)
\(632\) −4.02297e9 −0.633923
\(633\) −2.32921e9 −0.365002
\(634\) 1.63557e10 2.54892
\(635\) 3.15156e7 0.00488447
\(636\) −8.06129e9 −1.24252
\(637\) −2.53646e9 −0.388812
\(638\) −4.63452e9 −0.706533
\(639\) −3.12886e9 −0.474388
\(640\) −4.11564e7 −0.00620594
\(641\) 1.18477e10 1.77676 0.888382 0.459104i \(-0.151830\pi\)
0.888382 + 0.459104i \(0.151830\pi\)
\(642\) −3.56330e8 −0.0531471
\(643\) −1.06317e10 −1.57712 −0.788562 0.614956i \(-0.789174\pi\)
−0.788562 + 0.614956i \(0.789174\pi\)
\(644\) 6.08189e9 0.897300
\(645\) −1.21207e7 −0.00177857
\(646\) −2.74342e9 −0.400385
\(647\) −9.21966e9 −1.33829 −0.669145 0.743132i \(-0.733339\pi\)
−0.669145 + 0.743132i \(0.733339\pi\)
\(648\) 6.29330e8 0.0908586
\(649\) 2.28145e8 0.0327608
\(650\) −1.16122e10 −1.65851
\(651\) −1.65680e9 −0.235362
\(652\) 2.43150e9 0.343565
\(653\) −1.41112e10 −1.98321 −0.991605 0.129302i \(-0.958726\pi\)
−0.991605 + 0.129302i \(0.958726\pi\)
\(654\) −1.12910e10 −1.57837
\(655\) −4.94328e7 −0.00687338
\(656\) 2.11026e9 0.291859
\(657\) −1.13613e9 −0.156296
\(658\) −9.54349e9 −1.30592
\(659\) 6.80419e9 0.926141 0.463071 0.886321i \(-0.346748\pi\)
0.463071 + 0.886321i \(0.346748\pi\)
\(660\) −7.11712e6 −0.000963609 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
\(663\) −1.08135e9 −0.144102
\(664\) −8.34793e9 −1.10660
\(665\) 2.78201e7 0.00366845
\(666\) −6.90651e8 −0.0905938
\(667\) 1.01343e10 1.32237
\(668\) −7.32298e9 −0.950540
\(669\) −3.31263e9 −0.427742
\(670\) −3.78272e7 −0.00485895
\(671\) 3.61302e8 0.0461681
\(672\) −4.19177e9 −0.532850
\(673\) −7.86611e9 −0.994734 −0.497367 0.867540i \(-0.665700\pi\)
−0.497367 + 0.867540i \(0.665700\pi\)
\(674\) 6.67178e8 0.0839328
\(675\) 1.53770e9 0.192446
\(676\) 1.13799e9 0.141686
\(677\) 6.11786e8 0.0757773 0.0378886 0.999282i \(-0.487937\pi\)
0.0378886 + 0.999282i \(0.487937\pi\)
\(678\) −6.35286e9 −0.782826
\(679\) −9.80439e9 −1.20192
\(680\) 7.00352e6 0.000854152 0
\(681\) −3.02684e9 −0.367261
\(682\) −1.70059e9 −0.205283
\(683\) −1.81400e9 −0.217854 −0.108927 0.994050i \(-0.534741\pi\)
−0.108927 + 0.994050i \(0.534741\pi\)
\(684\) −4.47186e9 −0.534308
\(685\) 4.16617e7 0.00495245
\(686\) 1.45812e10 1.72448
\(687\) −3.46476e9 −0.407685
\(688\) −1.31450e9 −0.153887
\(689\) 1.27486e10 1.48489
\(690\) 2.58317e7 0.00299351
\(691\) 8.17172e9 0.942194 0.471097 0.882081i \(-0.343858\pi\)
0.471097 + 0.882081i \(0.343858\pi\)
\(692\) −4.00051e9 −0.458927
\(693\) −5.82462e8 −0.0664816
\(694\) −9.28610e9 −1.05457
\(695\) 4.87403e6 0.000550734 0
\(696\) 7.43385e9 0.835759
\(697\) −2.84863e9 −0.318656
\(698\) −1.38941e10 −1.54645
\(699\) 9.72302e8 0.107679
\(700\) 1.09007e10 1.20119
\(701\) 8.54219e9 0.936605 0.468302 0.883568i \(-0.344866\pi\)
0.468302 + 0.883568i \(0.344866\pi\)
\(702\) −2.92566e9 −0.319186
\(703\) 1.66948e9 0.181233
\(704\) −3.79328e9 −0.409742
\(705\) −2.44209e7 −0.00262482
\(706\) −1.18768e10 −1.27023
\(707\) 1.00052e10 1.06478
\(708\) −1.07574e9 −0.113917
\(709\) −8.20988e9 −0.865117 −0.432559 0.901606i \(-0.642389\pi\)
−0.432559 + 0.901606i \(0.642389\pi\)
\(710\) −9.42066e7 −0.00987819
\(711\) −2.47657e9 −0.258409
\(712\) 1.35006e10 1.40176
\(713\) 3.71866e9 0.384214
\(714\) 1.68487e9 0.173230
\(715\) 1.12554e7 0.00115157
\(716\) −1.97751e10 −2.01337
\(717\) 6.22044e9 0.630237
\(718\) 1.12431e10 1.13357
\(719\) −1.72415e9 −0.172991 −0.0864957 0.996252i \(-0.527567\pi\)
−0.0864957 + 0.996252i \(0.527567\pi\)
\(720\) −3.19389e6 −0.000318901 0
\(721\) −3.94708e9 −0.392196
\(722\) 1.90223e9 0.188098
\(723\) 8.35916e9 0.822580
\(724\) 1.77441e9 0.173768
\(725\) 1.81639e10 1.77021
\(726\) 8.84353e9 0.857724
\(727\) −1.17427e9 −0.113343 −0.0566717 0.998393i \(-0.518049\pi\)
−0.0566717 + 0.998393i \(0.518049\pi\)
\(728\) −7.05532e9 −0.677731
\(729\) 3.87420e8 0.0370370
\(730\) −3.42075e7 −0.00325455
\(731\) 1.77444e9 0.168016
\(732\) −1.70360e9 −0.160538
\(733\) 1.78077e10 1.67011 0.835053 0.550170i \(-0.185437\pi\)
0.835053 + 0.550170i \(0.185437\pi\)
\(734\) 1.61344e9 0.150597
\(735\) 1.01130e7 0.000939456 0
\(736\) 9.40839e9 0.869847
\(737\) −1.91442e9 −0.176157
\(738\) −7.70714e9 −0.705824
\(739\) −1.69312e10 −1.54324 −0.771620 0.636084i \(-0.780553\pi\)
−0.771620 + 0.636084i \(0.780553\pi\)
\(740\) −1.25283e7 −0.00113653
\(741\) 7.07206e9 0.638532
\(742\) −1.98638e10 −1.78504
\(743\) 5.43777e9 0.486362 0.243181 0.969981i \(-0.421809\pi\)
0.243181 + 0.969981i \(0.421809\pi\)
\(744\) 2.72777e9 0.242830
\(745\) 7.14219e6 0.000632826 0
\(746\) 3.47795e9 0.306717
\(747\) −5.13905e9 −0.451088
\(748\) 1.04192e9 0.0910291
\(749\) −5.28993e8 −0.0460006
\(750\) 9.25981e7 0.00801471
\(751\) −1.10319e10 −0.950406 −0.475203 0.879876i \(-0.657625\pi\)
−0.475203 + 0.879876i \(0.657625\pi\)
\(752\) −2.64845e9 −0.227107
\(753\) −4.90004e9 −0.418232
\(754\) −3.45588e10 −2.93602
\(755\) 6.28496e7 0.00531481
\(756\) 2.74640e9 0.231173
\(757\) 9.41776e9 0.789064 0.394532 0.918882i \(-0.370907\pi\)
0.394532 + 0.918882i \(0.370907\pi\)
\(758\) −4.09315e9 −0.341363
\(759\) 1.30733e9 0.108527
\(760\) −4.58033e7 −0.00378485
\(761\) 1.61763e10 1.33056 0.665278 0.746596i \(-0.268313\pi\)
0.665278 + 0.746596i \(0.268313\pi\)
\(762\) −1.24828e10 −1.02205
\(763\) −1.67622e10 −1.36614
\(764\) 2.48575e10 2.01665
\(765\) 4.31142e6 0.000348181 0
\(766\) −2.08321e9 −0.167468
\(767\) 1.70124e9 0.136138
\(768\) 4.50004e9 0.358469
\(769\) −1.61743e10 −1.28257 −0.641287 0.767301i \(-0.721599\pi\)
−0.641287 + 0.767301i \(0.721599\pi\)
\(770\) −1.75373e7 −0.00138435
\(771\) −1.12102e10 −0.880889
\(772\) 5.77147e9 0.451467
\(773\) −1.43501e9 −0.111745 −0.0558723 0.998438i \(-0.517794\pi\)
−0.0558723 + 0.998438i \(0.517794\pi\)
\(774\) 4.80084e9 0.372155
\(775\) 6.66503e9 0.514335
\(776\) 1.61421e10 1.24006
\(777\) −1.02531e9 −0.0784120
\(778\) 1.22778e10 0.934743
\(779\) 1.86302e10 1.41200
\(780\) −5.30711e7 −0.00400430
\(781\) −4.76775e9 −0.358125
\(782\) −3.78167e9 −0.282788
\(783\) 4.57633e9 0.340684
\(784\) 1.09676e9 0.0812843
\(785\) −1.39987e7 −0.00103287
\(786\) 1.95796e10 1.43822
\(787\) −1.73282e10 −1.26719 −0.633597 0.773664i \(-0.718422\pi\)
−0.633597 + 0.773664i \(0.718422\pi\)
\(788\) −2.88539e9 −0.210069
\(789\) 1.02872e10 0.745641
\(790\) −7.45668e7 −0.00538085
\(791\) −9.43121e9 −0.677563
\(792\) 9.58972e8 0.0685911
\(793\) 2.69417e9 0.191853
\(794\) −3.11029e10 −2.20510
\(795\) −5.08296e7 −0.00358783
\(796\) −2.07950e10 −1.46138
\(797\) −1.59506e10 −1.11602 −0.558012 0.829833i \(-0.688436\pi\)
−0.558012 + 0.829833i \(0.688436\pi\)
\(798\) −1.10191e10 −0.767602
\(799\) 3.57514e9 0.247959
\(800\) 1.68628e10 1.16444
\(801\) 8.31110e9 0.571406
\(802\) 2.49901e10 1.71064
\(803\) −1.73123e9 −0.117991
\(804\) 9.02677e9 0.612542
\(805\) 3.83487e7 0.00259099
\(806\) −1.26810e10 −0.853060
\(807\) 2.57442e9 0.172434
\(808\) −1.64727e10 −1.09857
\(809\) 1.99696e10 1.32602 0.663010 0.748611i \(-0.269279\pi\)
0.663010 + 0.748611i \(0.269279\pi\)
\(810\) 1.16648e7 0.000771223 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
\(813\) 1.12854e10 0.736549
\(814\) −1.05241e9 −0.0683912
\(815\) 1.53316e7 0.000992055 0
\(816\) 4.67575e8 0.0301256
\(817\) −1.16049e10 −0.744498
\(818\) −2.66924e10 −1.70511
\(819\) −4.34331e9 −0.276266
\(820\) −1.39807e8 −0.00885482
\(821\) −1.65074e10 −1.04107 −0.520534 0.853841i \(-0.674267\pi\)
−0.520534 + 0.853841i \(0.674267\pi\)
\(822\) −1.65016e10 −1.03627
\(823\) 1.23795e10 0.774113 0.387057 0.922056i \(-0.373492\pi\)
0.387057 + 0.922056i \(0.373492\pi\)
\(824\) 6.49852e9 0.404640
\(825\) 2.34315e9 0.145282
\(826\) −2.65072e9 −0.163657
\(827\) 7.73227e9 0.475376 0.237688 0.971342i \(-0.423610\pi\)
0.237688 + 0.971342i \(0.423610\pi\)
\(828\) −6.16426e9 −0.377376
\(829\) 8.83980e9 0.538892 0.269446 0.963016i \(-0.413159\pi\)
0.269446 + 0.963016i \(0.413159\pi\)
\(830\) −1.54731e8 −0.00939300
\(831\) 8.82738e9 0.533615
\(832\) −2.82858e10 −1.70269
\(833\) −1.48052e9 −0.0887474
\(834\) −1.93053e9 −0.115238
\(835\) −4.61743e7 −0.00274472
\(836\) −6.81421e9 −0.403361
\(837\) 1.67924e9 0.0989857
\(838\) 1.74758e10 1.02585
\(839\) −1.45899e10 −0.852877 −0.426439 0.904516i \(-0.640232\pi\)
−0.426439 + 0.904516i \(0.640232\pi\)
\(840\) 2.81301e7 0.00163755
\(841\) 3.68072e10 2.13377
\(842\) 3.61320e10 2.08593
\(843\) −2.55839e9 −0.147085
\(844\) 1.67352e10 0.958147
\(845\) 7.17549e6 0.000409122 0
\(846\) 9.67274e9 0.549229
\(847\) 1.31288e10 0.742390
\(848\) −5.51249e9 −0.310429
\(849\) 1.20673e10 0.676759
\(850\) −6.77796e9 −0.378559
\(851\) 2.30130e9 0.128003
\(852\) 2.24807e10 1.24529
\(853\) −1.83684e10 −1.01333 −0.506665 0.862143i \(-0.669122\pi\)
−0.506665 + 0.862143i \(0.669122\pi\)
\(854\) −4.19783e9 −0.230633
\(855\) −2.81968e7 −0.00154283
\(856\) 8.70941e8 0.0474603
\(857\) 1.71154e10 0.928871 0.464436 0.885607i \(-0.346257\pi\)
0.464436 + 0.885607i \(0.346257\pi\)
\(858\) −4.45811e9 −0.240960
\(859\) −1.64262e10 −0.884220 −0.442110 0.896961i \(-0.645770\pi\)
−0.442110 + 0.896961i \(0.645770\pi\)
\(860\) 8.70868e7 0.00466883
\(861\) −1.14417e10 −0.610914
\(862\) 5.29210e9 0.281418
\(863\) −1.57145e10 −0.832265 −0.416132 0.909304i \(-0.636615\pi\)
−0.416132 + 0.909304i \(0.636615\pi\)
\(864\) 4.24854e9 0.224100
\(865\) −2.52248e7 −0.00132517
\(866\) −2.55442e10 −1.33653
\(867\) 1.04480e10 0.544459
\(868\) 1.19040e10 0.617836
\(869\) −3.77379e9 −0.195078
\(870\) 1.37788e8 0.00709407
\(871\) −1.42755e10 −0.732026
\(872\) 2.75974e10 1.40949
\(873\) 9.93718e9 0.505491
\(874\) 2.47323e10 1.25307
\(875\) 1.37467e8 0.00693700
\(876\) 8.16299e9 0.410284
\(877\) 4.28124e9 0.214324 0.107162 0.994242i \(-0.465824\pi\)
0.107162 + 0.994242i \(0.465824\pi\)
\(878\) −2.78885e10 −1.39057
\(879\) −1.09301e10 −0.542829
\(880\) −4.86685e6 −0.000240746 0
\(881\) 1.71858e10 0.846748 0.423374 0.905955i \(-0.360846\pi\)
0.423374 + 0.905955i \(0.360846\pi\)
\(882\) −4.00562e9 −0.196576
\(883\) −1.87937e10 −0.918648 −0.459324 0.888269i \(-0.651908\pi\)
−0.459324 + 0.888269i \(0.651908\pi\)
\(884\) 7.76943e9 0.378274
\(885\) −6.78295e6 −0.000328941 0
\(886\) −1.15073e10 −0.555849
\(887\) 1.22073e10 0.587339 0.293669 0.955907i \(-0.405124\pi\)
0.293669 + 0.955907i \(0.405124\pi\)
\(888\) 1.68809e9 0.0809001
\(889\) −1.85315e10 −0.884617
\(890\) 2.50238e8 0.0118984
\(891\) 5.90351e8 0.0279601
\(892\) 2.38011e10 1.12284
\(893\) −2.33815e10 −1.09873
\(894\) −2.82891e9 −0.132415
\(895\) −1.24690e8 −0.00581368
\(896\) 2.42005e10 1.12395
\(897\) 9.74852e9 0.450988
\(898\) 5.32448e10 2.45364
\(899\) 1.98357e10 0.910516
\(900\) −1.10483e10 −0.505181
\(901\) 7.44129e9 0.338931
\(902\) −1.17441e10 −0.532841
\(903\) 7.12714e9 0.322113
\(904\) 1.55276e10 0.699063
\(905\) 1.11884e7 0.000501761 0
\(906\) −2.48938e10 −1.11209
\(907\) 1.54047e9 0.0685534 0.0342767 0.999412i \(-0.489087\pi\)
0.0342767 + 0.999412i \(0.489087\pi\)
\(908\) 2.17477e10 0.964078
\(909\) −1.01407e10 −0.447813
\(910\) −1.30772e8 −0.00575269
\(911\) 3.19695e10 1.40094 0.700472 0.713680i \(-0.252973\pi\)
0.700472 + 0.713680i \(0.252973\pi\)
\(912\) −3.05796e9 −0.133490
\(913\) −7.83088e9 −0.340536
\(914\) −5.95352e10 −2.57906
\(915\) −1.07418e7 −0.000463559 0
\(916\) 2.48941e10 1.07019
\(917\) 2.90671e10 1.24483
\(918\) −1.70769e9 −0.0728550
\(919\) 2.59602e10 1.10333 0.551663 0.834067i \(-0.313993\pi\)
0.551663 + 0.834067i \(0.313993\pi\)
\(920\) −6.31377e7 −0.00267320
\(921\) 2.14768e10 0.905860
\(922\) 1.70227e10 0.715272
\(923\) −3.55523e10 −1.48820
\(924\) 4.18495e9 0.174517
\(925\) 4.12467e9 0.171354
\(926\) 2.39349e10 0.990589
\(927\) 4.00054e9 0.164945
\(928\) 5.01851e10 2.06138
\(929\) −6.20074e9 −0.253740 −0.126870 0.991919i \(-0.540493\pi\)
−0.126870 + 0.991919i \(0.540493\pi\)
\(930\) 5.05599e7 0.00206118
\(931\) 9.68263e9 0.393251
\(932\) −6.98593e9 −0.282663
\(933\) −9.43732e9 −0.380419
\(934\) −6.75024e10 −2.71085
\(935\) 6.56974e6 0.000262850 0
\(936\) 7.15088e9 0.285032
\(937\) −2.50957e10 −0.996575 −0.498288 0.867012i \(-0.666038\pi\)
−0.498288 + 0.867012i \(0.666038\pi\)
\(938\) 2.22428e10 0.879995
\(939\) −2.65169e9 −0.104518
\(940\) 1.75463e8 0.00689028
\(941\) −8.17848e9 −0.319970 −0.159985 0.987119i \(-0.551145\pi\)
−0.159985 + 0.987119i \(0.551145\pi\)
\(942\) 5.54469e9 0.216122
\(943\) 2.56808e10 0.997282
\(944\) −7.35614e8 −0.0284608
\(945\) 1.73171e7 0.000667520 0
\(946\) 7.31552e9 0.280948
\(947\) 1.81622e10 0.694935 0.347468 0.937692i \(-0.387042\pi\)
0.347468 + 0.937692i \(0.387042\pi\)
\(948\) 1.77940e10 0.678336
\(949\) −1.29094e10 −0.490315
\(950\) 4.43281e10 1.67744
\(951\) −2.46099e10 −0.927849
\(952\) −4.11815e9 −0.154694
\(953\) −4.42866e10 −1.65748 −0.828738 0.559637i \(-0.810941\pi\)
−0.828738 + 0.559637i \(0.810941\pi\)
\(954\) 2.01328e10 0.750733
\(955\) 1.56737e8 0.00582316
\(956\) −4.46935e10 −1.65440
\(957\) 6.97341e9 0.257190
\(958\) −5.26702e10 −1.93547
\(959\) −2.44976e10 −0.896929
\(960\) 1.12778e8 0.00411409
\(961\) −2.02341e10 −0.735449
\(962\) −7.84764e9 −0.284202
\(963\) 5.36158e8 0.0193464
\(964\) −6.00600e10 −2.15931
\(965\) 3.63914e7 0.00130363
\(966\) −1.51893e10 −0.542149
\(967\) 3.34893e10 1.19101 0.595503 0.803353i \(-0.296953\pi\)
0.595503 + 0.803353i \(0.296953\pi\)
\(968\) −2.16153e10 −0.765946
\(969\) 4.12793e9 0.145747
\(970\) 2.99197e8 0.0105258
\(971\) −2.53801e10 −0.889663 −0.444831 0.895614i \(-0.646736\pi\)
−0.444831 + 0.895614i \(0.646736\pi\)
\(972\) −2.78359e9 −0.0972240
\(973\) −2.86599e9 −0.0997423
\(974\) −2.25980e10 −0.783633
\(975\) 1.74725e10 0.603723
\(976\) −1.16496e9 −0.0401084
\(977\) −1.10528e10 −0.379177 −0.189589 0.981864i \(-0.560715\pi\)
−0.189589 + 0.981864i \(0.560715\pi\)
\(978\) −6.07261e9 −0.207582
\(979\) 1.26644e10 0.431366
\(980\) −7.26616e7 −0.00246612
\(981\) 1.69892e10 0.574554
\(982\) 7.55401e9 0.254558
\(983\) −3.71529e9 −0.124754 −0.0623771 0.998053i \(-0.519868\pi\)
−0.0623771 + 0.998053i \(0.519868\pi\)
\(984\) 1.88378e10 0.630299
\(985\) −1.81935e7 −0.000606582 0
\(986\) −2.01718e10 −0.670154
\(987\) 1.43598e10 0.475376
\(988\) −5.08123e10 −1.67618
\(989\) −1.59968e10 −0.525831
\(990\) 1.77748e7 0.000582213 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
\(993\) −2.34654e10 −0.760511
\(994\) 5.53946e10 1.78902
\(995\) −1.31121e8 −0.00421978
\(996\) 3.69238e10 1.18413
\(997\) 4.94528e10 1.58037 0.790183 0.612871i \(-0.209985\pi\)
0.790183 + 0.612871i \(0.209985\pi\)
\(998\) −8.73170e10 −2.78062
\(999\) 1.03920e9 0.0329776
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 177.8.a.a.1.14 16
3.2 odd 2 531.8.a.b.1.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.14 16 1.1 even 1 trivial
531.8.a.b.1.3 16 3.2 odd 2