Properties

Label 69.6.a.b.1.1
Level $69$
Weight $6$
Character 69.1
Self dual yes
Analytic conductor $11.066$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.714018\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.2042 q^{2} +9.00000 q^{3} +72.1256 q^{4} -55.5168 q^{5} -91.8378 q^{6} +2.50462 q^{7} -409.450 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-10.2042 q^{2} +9.00000 q^{3} +72.1256 q^{4} -55.5168 q^{5} -91.8378 q^{6} +2.50462 q^{7} -409.450 q^{8} +81.0000 q^{9} +566.504 q^{10} +228.550 q^{11} +649.131 q^{12} +658.703 q^{13} -25.5577 q^{14} -499.651 q^{15} +1870.09 q^{16} -1443.81 q^{17} -826.540 q^{18} -982.167 q^{19} -4004.18 q^{20} +22.5416 q^{21} -2332.17 q^{22} +529.000 q^{23} -3685.05 q^{24} -42.8875 q^{25} -6721.54 q^{26} +729.000 q^{27} +180.648 q^{28} -7157.56 q^{29} +5098.54 q^{30} -9259.98 q^{31} -5980.33 q^{32} +2056.95 q^{33} +14732.9 q^{34} -139.049 q^{35} +5842.18 q^{36} +2422.50 q^{37} +10022.2 q^{38} +5928.33 q^{39} +22731.3 q^{40} -4075.27 q^{41} -230.019 q^{42} -10417.3 q^{43} +16484.3 q^{44} -4496.86 q^{45} -5398.02 q^{46} +9358.08 q^{47} +16830.8 q^{48} -16800.7 q^{49} +437.632 q^{50} -12994.3 q^{51} +47509.4 q^{52} -34280.3 q^{53} -7438.86 q^{54} -12688.4 q^{55} -1025.52 q^{56} -8839.51 q^{57} +73037.2 q^{58} -7268.79 q^{59} -36037.6 q^{60} +26611.7 q^{61} +94490.6 q^{62} +202.874 q^{63} +1181.70 q^{64} -36569.1 q^{65} -20989.5 q^{66} +53450.8 q^{67} -104136. q^{68} +4761.00 q^{69} +1418.88 q^{70} +21673.7 q^{71} -33165.4 q^{72} -82856.3 q^{73} -24719.7 q^{74} -385.987 q^{75} -70839.4 q^{76} +572.432 q^{77} -60493.9 q^{78} -23960.0 q^{79} -103821. q^{80} +6561.00 q^{81} +41584.9 q^{82} +81187.7 q^{83} +1625.83 q^{84} +80155.6 q^{85} +106300. q^{86} -64418.0 q^{87} -93579.7 q^{88} +100115. q^{89} +45886.8 q^{90} +1649.80 q^{91} +38154.5 q^{92} -83339.8 q^{93} -95491.7 q^{94} +54526.8 q^{95} -53822.9 q^{96} +36122.1 q^{97} +171438. q^{98} +18512.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} - 510q^{8} + 243q^{9} + O(q^{10}) \) \( 3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} - 510q^{8} + 243q^{9} + 282q^{10} - 376q^{11} + 198q^{12} - 858q^{13} + 588q^{14} - 504q^{15} + 2738q^{16} - 2548q^{17} - 648q^{18} - 2846q^{19} - 4618q^{20} - 1026q^{21} - 5050q^{22} + 1587q^{23} - 4590q^{24} + 753q^{25} - 7788q^{26} + 2187q^{27} + 4736q^{28} - 16370q^{29} + 2538q^{30} - 14756q^{31} - 3878q^{32} - 3384q^{33} + 16520q^{34} - 18520q^{35} + 1782q^{36} + 15874q^{37} + 12438q^{38} - 7722q^{39} + 38270q^{40} + 12606q^{41} + 5292q^{42} + 3154q^{43} + 27114q^{44} - 4536q^{45} - 4232q^{46} + 29928q^{47} + 24642q^{48} + 4471q^{49} + 1452q^{50} - 22932q^{51} + 86856q^{52} - 44084q^{53} - 5832q^{54} + 38360q^{55} - 35704q^{56} - 25614q^{57} + 73316q^{58} - 29300q^{59} - 41562q^{60} + 54010q^{61} + 99908q^{62} - 9234q^{63} - 1582q^{64} - 51216q^{65} - 45450q^{66} + 43390q^{67} - 69840q^{68} + 14283q^{69} - 2476q^{70} + 23424q^{71} - 41310q^{72} - 91402q^{73} - 2294q^{74} + 6777q^{75} - 14274q^{76} - 97208q^{77} - 70092q^{78} - 49398q^{79} - 52626q^{80} + 19683q^{81} + 40152q^{82} - 103936q^{83} + 42624q^{84} + 5888q^{85} + 133634q^{86} - 147330q^{87} + 48898q^{88} + 96112q^{89} + 22842q^{90} + 129228q^{91} + 11638q^{92} - 132804q^{93} - 133688q^{94} - 55928q^{95} - 34902q^{96} - 135318q^{97} + 108440q^{98} - 30456q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.2042 −1.80386 −0.901932 0.431878i \(-0.857851\pi\)
−0.901932 + 0.431878i \(0.857851\pi\)
\(3\) 9.00000 0.577350
\(4\) 72.1256 2.25393
\(5\) −55.5168 −0.993114 −0.496557 0.868004i \(-0.665403\pi\)
−0.496557 + 0.868004i \(0.665403\pi\)
\(6\) −91.8378 −1.04146
\(7\) 2.50462 0.0193196 0.00965978 0.999953i \(-0.496925\pi\)
0.00965978 + 0.999953i \(0.496925\pi\)
\(8\) −409.450 −2.26191
\(9\) 81.0000 0.333333
\(10\) 566.504 1.79144
\(11\) 228.550 0.569508 0.284754 0.958601i \(-0.408088\pi\)
0.284754 + 0.958601i \(0.408088\pi\)
\(12\) 649.131 1.30130
\(13\) 658.703 1.08101 0.540507 0.841339i \(-0.318232\pi\)
0.540507 + 0.841339i \(0.318232\pi\)
\(14\) −25.5577 −0.0348499
\(15\) −499.651 −0.573375
\(16\) 1870.09 1.82626
\(17\) −1443.81 −1.21168 −0.605839 0.795587i \(-0.707162\pi\)
−0.605839 + 0.795587i \(0.707162\pi\)
\(18\) −826.540 −0.601288
\(19\) −982.167 −0.624168 −0.312084 0.950055i \(-0.601027\pi\)
−0.312084 + 0.950055i \(0.601027\pi\)
\(20\) −4004.18 −2.23841
\(21\) 22.5416 0.0111542
\(22\) −2332.17 −1.02731
\(23\) 529.000 0.208514
\(24\) −3685.05 −1.30592
\(25\) −42.8875 −0.0137240
\(26\) −6721.54 −1.95000
\(27\) 729.000 0.192450
\(28\) 180.648 0.0435449
\(29\) −7157.56 −1.58041 −0.790205 0.612842i \(-0.790026\pi\)
−0.790205 + 0.612842i \(0.790026\pi\)
\(30\) 5098.54 1.03429
\(31\) −9259.98 −1.73064 −0.865318 0.501223i \(-0.832884\pi\)
−0.865318 + 0.501223i \(0.832884\pi\)
\(32\) −5980.33 −1.03240
\(33\) 2056.95 0.328805
\(34\) 14732.9 2.18570
\(35\) −139.049 −0.0191865
\(36\) 5842.18 0.751309
\(37\) 2422.50 0.290911 0.145455 0.989365i \(-0.453535\pi\)
0.145455 + 0.989365i \(0.453535\pi\)
\(38\) 10022.2 1.12591
\(39\) 5928.33 0.624124
\(40\) 22731.3 2.24634
\(41\) −4075.27 −0.378614 −0.189307 0.981918i \(-0.560624\pi\)
−0.189307 + 0.981918i \(0.560624\pi\)
\(42\) −230.019 −0.0201206
\(43\) −10417.3 −0.859178 −0.429589 0.903024i \(-0.641342\pi\)
−0.429589 + 0.903024i \(0.641342\pi\)
\(44\) 16484.3 1.28363
\(45\) −4496.86 −0.331038
\(46\) −5398.02 −0.376132
\(47\) 9358.08 0.617934 0.308967 0.951073i \(-0.400017\pi\)
0.308967 + 0.951073i \(0.400017\pi\)
\(48\) 16830.8 1.05439
\(49\) −16800.7 −0.999627
\(50\) 437.632 0.0247562
\(51\) −12994.3 −0.699563
\(52\) 47509.4 2.43653
\(53\) −34280.3 −1.67631 −0.838157 0.545428i \(-0.816367\pi\)
−0.838157 + 0.545428i \(0.816367\pi\)
\(54\) −7438.86 −0.347154
\(55\) −12688.4 −0.565586
\(56\) −1025.52 −0.0436991
\(57\) −8839.51 −0.360364
\(58\) 73037.2 2.85085
\(59\) −7268.79 −0.271852 −0.135926 0.990719i \(-0.543401\pi\)
−0.135926 + 0.990719i \(0.543401\pi\)
\(60\) −36037.6 −1.29234
\(61\) 26611.7 0.915687 0.457844 0.889033i \(-0.348622\pi\)
0.457844 + 0.889033i \(0.348622\pi\)
\(62\) 94490.6 3.12183
\(63\) 202.874 0.00643985
\(64\) 1181.70 0.0360625
\(65\) −36569.1 −1.07357
\(66\) −20989.5 −0.593120
\(67\) 53450.8 1.45468 0.727339 0.686278i \(-0.240757\pi\)
0.727339 + 0.686278i \(0.240757\pi\)
\(68\) −104136. −2.73103
\(69\) 4761.00 0.120386
\(70\) 1418.88 0.0346099
\(71\) 21673.7 0.510255 0.255128 0.966907i \(-0.417882\pi\)
0.255128 + 0.966907i \(0.417882\pi\)
\(72\) −33165.4 −0.753970
\(73\) −82856.3 −1.81978 −0.909889 0.414851i \(-0.863834\pi\)
−0.909889 + 0.414851i \(0.863834\pi\)
\(74\) −24719.7 −0.524764
\(75\) −385.987 −0.00792355
\(76\) −70839.4 −1.40683
\(77\) 572.432 0.0110026
\(78\) −60493.9 −1.12584
\(79\) −23960.0 −0.431936 −0.215968 0.976400i \(-0.569291\pi\)
−0.215968 + 0.976400i \(0.569291\pi\)
\(80\) −103821. −1.81368
\(81\) 6561.00 0.111111
\(82\) 41584.9 0.682968
\(83\) 81187.7 1.29359 0.646793 0.762666i \(-0.276110\pi\)
0.646793 + 0.762666i \(0.276110\pi\)
\(84\) 1625.83 0.0251406
\(85\) 80155.6 1.20334
\(86\) 106300. 1.54984
\(87\) −64418.0 −0.912451
\(88\) −93579.7 −1.28818
\(89\) 100115. 1.33975 0.669876 0.742473i \(-0.266347\pi\)
0.669876 + 0.742473i \(0.266347\pi\)
\(90\) 45886.8 0.597148
\(91\) 1649.80 0.0208847
\(92\) 38154.5 0.469976
\(93\) −83339.8 −0.999183
\(94\) −95491.7 −1.11467
\(95\) 54526.8 0.619870
\(96\) −53822.9 −0.596059
\(97\) 36122.1 0.389802 0.194901 0.980823i \(-0.437562\pi\)
0.194901 + 0.980823i \(0.437562\pi\)
\(98\) 171438. 1.80319
\(99\) 18512.6 0.189836
\(100\) −3093.29 −0.0309329
\(101\) −41151.8 −0.401408 −0.200704 0.979652i \(-0.564323\pi\)
−0.200704 + 0.979652i \(0.564323\pi\)
\(102\) 132596. 1.26192
\(103\) −172534. −1.60244 −0.801221 0.598369i \(-0.795816\pi\)
−0.801221 + 0.598369i \(0.795816\pi\)
\(104\) −269706. −2.44516
\(105\) −1251.44 −0.0110774
\(106\) 349803. 3.02384
\(107\) 178228. 1.50493 0.752464 0.658633i \(-0.228865\pi\)
0.752464 + 0.658633i \(0.228865\pi\)
\(108\) 52579.6 0.433768
\(109\) −138352. −1.11537 −0.557686 0.830052i \(-0.688311\pi\)
−0.557686 + 0.830052i \(0.688311\pi\)
\(110\) 129475. 1.02024
\(111\) 21802.5 0.167958
\(112\) 4683.86 0.0352825
\(113\) −13523.5 −0.0996307 −0.0498153 0.998758i \(-0.515863\pi\)
−0.0498153 + 0.998758i \(0.515863\pi\)
\(114\) 90200.0 0.650047
\(115\) −29368.4 −0.207079
\(116\) −516243. −3.56213
\(117\) 53355.0 0.360338
\(118\) 74172.1 0.490383
\(119\) −3616.20 −0.0234091
\(120\) 204582. 1.29692
\(121\) −108816. −0.675661
\(122\) −271551. −1.65178
\(123\) −36677.4 −0.218593
\(124\) −667882. −3.90073
\(125\) 175871. 1.00674
\(126\) −2070.17 −0.0116166
\(127\) 158567. 0.872376 0.436188 0.899856i \(-0.356328\pi\)
0.436188 + 0.899856i \(0.356328\pi\)
\(128\) 179312. 0.967353
\(129\) −93755.6 −0.496047
\(130\) 373158. 1.93658
\(131\) −383514. −1.95255 −0.976277 0.216526i \(-0.930527\pi\)
−0.976277 + 0.216526i \(0.930527\pi\)
\(132\) 148359. 0.741103
\(133\) −2459.96 −0.0120587
\(134\) −545422. −2.62404
\(135\) −40471.7 −0.191125
\(136\) 591167. 2.74071
\(137\) −33171.1 −0.150994 −0.0754968 0.997146i \(-0.524054\pi\)
−0.0754968 + 0.997146i \(0.524054\pi\)
\(138\) −48582.2 −0.217160
\(139\) −128488. −0.564060 −0.282030 0.959406i \(-0.591008\pi\)
−0.282030 + 0.959406i \(0.591008\pi\)
\(140\) −10029.0 −0.0432450
\(141\) 84222.7 0.356764
\(142\) −221163. −0.920431
\(143\) 150547. 0.615646
\(144\) 151477. 0.608752
\(145\) 397365. 1.56953
\(146\) 845482. 3.28263
\(147\) −151207. −0.577135
\(148\) 174725. 0.655692
\(149\) −184228. −0.679814 −0.339907 0.940459i \(-0.610396\pi\)
−0.339907 + 0.940459i \(0.610396\pi\)
\(150\) 3938.69 0.0142930
\(151\) −19798.1 −0.0706610 −0.0353305 0.999376i \(-0.511248\pi\)
−0.0353305 + 0.999376i \(0.511248\pi\)
\(152\) 402148. 1.41181
\(153\) −116948. −0.403893
\(154\) −5841.21 −0.0198473
\(155\) 514084. 1.71872
\(156\) 427585. 1.40673
\(157\) 193317. 0.625923 0.312962 0.949766i \(-0.398679\pi\)
0.312962 + 0.949766i \(0.398679\pi\)
\(158\) 244493. 0.779154
\(159\) −308523. −0.967821
\(160\) 332008. 1.02530
\(161\) 1324.95 0.00402841
\(162\) −66949.7 −0.200429
\(163\) 106286. 0.313334 0.156667 0.987651i \(-0.449925\pi\)
0.156667 + 0.987651i \(0.449925\pi\)
\(164\) −293931. −0.853368
\(165\) −114195. −0.326541
\(166\) −828456. −2.33345
\(167\) −77715.5 −0.215634 −0.107817 0.994171i \(-0.534386\pi\)
−0.107817 + 0.994171i \(0.534386\pi\)
\(168\) −9229.66 −0.0252297
\(169\) 62597.3 0.168593
\(170\) −817923. −2.17065
\(171\) −79555.5 −0.208056
\(172\) −751353. −1.93652
\(173\) 218524. 0.555115 0.277558 0.960709i \(-0.410475\pi\)
0.277558 + 0.960709i \(0.410475\pi\)
\(174\) 657334. 1.64594
\(175\) −107.417 −0.000265142 0
\(176\) 427408. 1.04007
\(177\) −65419.1 −0.156954
\(178\) −1.02159e6 −2.41673
\(179\) −545493. −1.27250 −0.636248 0.771485i \(-0.719514\pi\)
−0.636248 + 0.771485i \(0.719514\pi\)
\(180\) −324339. −0.746135
\(181\) 510878. 1.15910 0.579549 0.814937i \(-0.303229\pi\)
0.579549 + 0.814937i \(0.303229\pi\)
\(182\) −16834.9 −0.0376732
\(183\) 239505. 0.528672
\(184\) −216599. −0.471641
\(185\) −134490. −0.288908
\(186\) 850416. 1.80239
\(187\) −329982. −0.690060
\(188\) 674957. 1.39278
\(189\) 1825.87 0.00371805
\(190\) −556402. −1.11816
\(191\) −57685.2 −0.114415 −0.0572073 0.998362i \(-0.518220\pi\)
−0.0572073 + 0.998362i \(0.518220\pi\)
\(192\) 10635.3 0.0208207
\(193\) −822091. −1.58864 −0.794322 0.607497i \(-0.792174\pi\)
−0.794322 + 0.607497i \(0.792174\pi\)
\(194\) −368597. −0.703149
\(195\) −329122. −0.619827
\(196\) −1.21176e6 −2.25308
\(197\) 971571. 1.78365 0.891824 0.452383i \(-0.149426\pi\)
0.891824 + 0.452383i \(0.149426\pi\)
\(198\) −188906. −0.342438
\(199\) 806110. 1.44298 0.721492 0.692423i \(-0.243457\pi\)
0.721492 + 0.692423i \(0.243457\pi\)
\(200\) 17560.3 0.0310425
\(201\) 481057. 0.839859
\(202\) 419921. 0.724085
\(203\) −17927.0 −0.0305329
\(204\) −937220. −1.57676
\(205\) 226246. 0.376007
\(206\) 1.76057e6 2.89059
\(207\) 42849.0 0.0695048
\(208\) 1.23183e6 1.97421
\(209\) −224474. −0.355468
\(210\) 12769.9 0.0199820
\(211\) −821913. −1.27092 −0.635462 0.772132i \(-0.719190\pi\)
−0.635462 + 0.772132i \(0.719190\pi\)
\(212\) −2.47249e6 −3.77829
\(213\) 195063. 0.294596
\(214\) −1.81867e6 −2.71469
\(215\) 578334. 0.853262
\(216\) −298489. −0.435305
\(217\) −23192.8 −0.0334351
\(218\) 1.41177e6 2.01198
\(219\) −745707. −1.05065
\(220\) −915156. −1.27479
\(221\) −951042. −1.30984
\(222\) −222477. −0.302973
\(223\) 511948. 0.689388 0.344694 0.938715i \(-0.387983\pi\)
0.344694 + 0.938715i \(0.387983\pi\)
\(224\) −14978.5 −0.0199456
\(225\) −3473.89 −0.00457467
\(226\) 137996. 0.179720
\(227\) −1.45074e6 −1.86864 −0.934321 0.356433i \(-0.883993\pi\)
−0.934321 + 0.356433i \(0.883993\pi\)
\(228\) −637555. −0.812233
\(229\) 621169. 0.782747 0.391373 0.920232i \(-0.372000\pi\)
0.391373 + 0.920232i \(0.372000\pi\)
\(230\) 299681. 0.373542
\(231\) 5151.89 0.00635238
\(232\) 2.93066e6 3.57475
\(233\) 490334. 0.591701 0.295851 0.955234i \(-0.404397\pi\)
0.295851 + 0.955234i \(0.404397\pi\)
\(234\) −544445. −0.650001
\(235\) −519530. −0.613679
\(236\) −524266. −0.612733
\(237\) −215640. −0.249378
\(238\) 36900.4 0.0422268
\(239\) 109503. 0.124003 0.0620014 0.998076i \(-0.480252\pi\)
0.0620014 + 0.998076i \(0.480252\pi\)
\(240\) −934390. −1.04713
\(241\) 1.36171e6 1.51022 0.755111 0.655597i \(-0.227583\pi\)
0.755111 + 0.655597i \(0.227583\pi\)
\(242\) 1.11038e6 1.21880
\(243\) 59049.0 0.0641500
\(244\) 1.91938e6 2.06389
\(245\) 932722. 0.992744
\(246\) 374264. 0.394312
\(247\) −646957. −0.674735
\(248\) 3.79150e6 3.91455
\(249\) 730690. 0.746852
\(250\) −1.79462e6 −1.81603
\(251\) 8042.27 0.00805739 0.00402869 0.999992i \(-0.498718\pi\)
0.00402869 + 0.999992i \(0.498718\pi\)
\(252\) 14632.4 0.0145150
\(253\) 120903. 0.118751
\(254\) −1.61805e6 −1.57365
\(255\) 721400. 0.694746
\(256\) −1.86755e6 −1.78104
\(257\) 1.16730e6 1.10243 0.551214 0.834364i \(-0.314165\pi\)
0.551214 + 0.834364i \(0.314165\pi\)
\(258\) 956700. 0.894801
\(259\) 6067.46 0.00562027
\(260\) −2.63757e6 −2.41975
\(261\) −579762. −0.526804
\(262\) 3.91345e6 3.52214
\(263\) −558051. −0.497490 −0.248745 0.968569i \(-0.580018\pi\)
−0.248745 + 0.968569i \(0.580018\pi\)
\(264\) −842218. −0.743729
\(265\) 1.90313e6 1.66477
\(266\) 25101.9 0.0217522
\(267\) 901036. 0.773506
\(268\) 3.85517e6 3.27874
\(269\) −1.21947e6 −1.02752 −0.513759 0.857935i \(-0.671747\pi\)
−0.513759 + 0.857935i \(0.671747\pi\)
\(270\) 412981. 0.344763
\(271\) 1.40581e6 1.16280 0.581399 0.813618i \(-0.302505\pi\)
0.581399 + 0.813618i \(0.302505\pi\)
\(272\) −2.70005e6 −2.21283
\(273\) 14848.2 0.0120578
\(274\) 338484. 0.272372
\(275\) −9801.94 −0.00781592
\(276\) 343390. 0.271341
\(277\) 1.87799e6 1.47060 0.735300 0.677742i \(-0.237041\pi\)
0.735300 + 0.677742i \(0.237041\pi\)
\(278\) 1.31112e6 1.01749
\(279\) −750058. −0.576879
\(280\) 56933.4 0.0433982
\(281\) −2.11759e6 −1.59984 −0.799920 0.600106i \(-0.795125\pi\)
−0.799920 + 0.600106i \(0.795125\pi\)
\(282\) −859425. −0.643555
\(283\) 1.16042e6 0.861288 0.430644 0.902522i \(-0.358286\pi\)
0.430644 + 0.902522i \(0.358286\pi\)
\(284\) 1.56323e6 1.15008
\(285\) 490741. 0.357882
\(286\) −1.53621e6 −1.11054
\(287\) −10207.0 −0.00731466
\(288\) −484406. −0.344135
\(289\) 664726. 0.468164
\(290\) −4.05479e6 −2.83122
\(291\) 325099. 0.225052
\(292\) −5.97606e6 −4.10165
\(293\) −1.27325e6 −0.866449 −0.433225 0.901286i \(-0.642624\pi\)
−0.433225 + 0.901286i \(0.642624\pi\)
\(294\) 1.54294e6 1.04107
\(295\) 403540. 0.269980
\(296\) −991893. −0.658015
\(297\) 166613. 0.109602
\(298\) 1.87990e6 1.22629
\(299\) 348454. 0.225407
\(300\) −27839.6 −0.0178591
\(301\) −26091.4 −0.0165990
\(302\) 202023. 0.127463
\(303\) −370367. −0.231753
\(304\) −1.83674e6 −1.13989
\(305\) −1.47739e6 −0.909382
\(306\) 1.19337e6 0.728568
\(307\) −2.26175e6 −1.36961 −0.684807 0.728724i \(-0.740114\pi\)
−0.684807 + 0.728724i \(0.740114\pi\)
\(308\) 41287.0 0.0247991
\(309\) −1.55281e6 −0.925170
\(310\) −5.24582e6 −3.10034
\(311\) 1.20263e6 0.705066 0.352533 0.935799i \(-0.385320\pi\)
0.352533 + 0.935799i \(0.385320\pi\)
\(312\) −2.42735e6 −1.41171
\(313\) −484311. −0.279424 −0.139712 0.990192i \(-0.544618\pi\)
−0.139712 + 0.990192i \(0.544618\pi\)
\(314\) −1.97265e6 −1.12908
\(315\) −11262.9 −0.00639551
\(316\) −1.72813e6 −0.973551
\(317\) 796624. 0.445251 0.222626 0.974904i \(-0.428537\pi\)
0.222626 + 0.974904i \(0.428537\pi\)
\(318\) 3.14823e6 1.74582
\(319\) −1.63586e6 −0.900056
\(320\) −65603.9 −0.0358142
\(321\) 1.60405e6 0.868871
\(322\) −13520.0 −0.00726670
\(323\) 1.41806e6 0.756291
\(324\) 473216. 0.250436
\(325\) −28250.1 −0.0148358
\(326\) −1.08456e6 −0.565212
\(327\) −1.24517e6 −0.643961
\(328\) 1.66862e6 0.856391
\(329\) 23438.5 0.0119382
\(330\) 1.16527e6 0.589036
\(331\) −262856. −0.131871 −0.0659353 0.997824i \(-0.521003\pi\)
−0.0659353 + 0.997824i \(0.521003\pi\)
\(332\) 5.85572e6 2.91565
\(333\) 196223. 0.0969703
\(334\) 793024. 0.388974
\(335\) −2.96741e6 −1.44466
\(336\) 42154.7 0.0203703
\(337\) 1.23123e6 0.590558 0.295279 0.955411i \(-0.404587\pi\)
0.295279 + 0.955411i \(0.404587\pi\)
\(338\) −638755. −0.304118
\(339\) −121712. −0.0575218
\(340\) 5.78127e6 2.71223
\(341\) −2.11637e6 −0.985611
\(342\) 811800. 0.375305
\(343\) −84174.7 −0.0386319
\(344\) 4.26535e6 1.94339
\(345\) −264315. −0.119557
\(346\) −2.22986e6 −1.00135
\(347\) −3.70344e6 −1.65113 −0.825567 0.564305i \(-0.809144\pi\)
−0.825567 + 0.564305i \(0.809144\pi\)
\(348\) −4.64619e6 −2.05660
\(349\) 1.62884e6 0.715837 0.357918 0.933753i \(-0.383487\pi\)
0.357918 + 0.933753i \(0.383487\pi\)
\(350\) 1096.10 0.000478279 0
\(351\) 480195. 0.208041
\(352\) −1.36680e6 −0.587962
\(353\) 2.56589e6 1.09598 0.547989 0.836486i \(-0.315394\pi\)
0.547989 + 0.836486i \(0.315394\pi\)
\(354\) 667549. 0.283123
\(355\) −1.20326e6 −0.506742
\(356\) 7.22086e6 3.01970
\(357\) −32545.8 −0.0135152
\(358\) 5.56631e6 2.29541
\(359\) −3.46895e6 −1.42057 −0.710284 0.703916i \(-0.751433\pi\)
−0.710284 + 0.703916i \(0.751433\pi\)
\(360\) 1.84124e6 0.748779
\(361\) −1.51145e6 −0.610414
\(362\) −5.21310e6 −2.09086
\(363\) −979343. −0.390093
\(364\) 118993. 0.0470726
\(365\) 4.59992e6 1.80725
\(366\) −2.44395e6 −0.953653
\(367\) 1.64135e6 0.636115 0.318058 0.948071i \(-0.396969\pi\)
0.318058 + 0.948071i \(0.396969\pi\)
\(368\) 989275. 0.380801
\(369\) −330097. −0.126205
\(370\) 1.37236e6 0.521151
\(371\) −85859.4 −0.0323857
\(372\) −6.01094e6 −2.25208
\(373\) 1.58607e6 0.590268 0.295134 0.955456i \(-0.404636\pi\)
0.295134 + 0.955456i \(0.404636\pi\)
\(374\) 3.36721e6 1.24477
\(375\) 1.58284e6 0.581244
\(376\) −3.83166e6 −1.39771
\(377\) −4.71471e6 −1.70845
\(378\) −18631.5 −0.00670686
\(379\) −966030. −0.345456 −0.172728 0.984970i \(-0.555258\pi\)
−0.172728 + 0.984970i \(0.555258\pi\)
\(380\) 3.93278e6 1.39714
\(381\) 1.42710e6 0.503667
\(382\) 588631. 0.206388
\(383\) −3.10322e6 −1.08097 −0.540487 0.841352i \(-0.681760\pi\)
−0.540487 + 0.841352i \(0.681760\pi\)
\(384\) 1.61381e6 0.558501
\(385\) −31779.6 −0.0109269
\(386\) 8.38878e6 2.86570
\(387\) −843800. −0.286393
\(388\) 2.60533e6 0.878584
\(389\) 624568. 0.209269 0.104635 0.994511i \(-0.466633\pi\)
0.104635 + 0.994511i \(0.466633\pi\)
\(390\) 3.35842e6 1.11808
\(391\) −763775. −0.252652
\(392\) 6.87905e6 2.26107
\(393\) −3.45163e6 −1.12731
\(394\) −9.91410e6 −3.21746
\(395\) 1.33018e6 0.428962
\(396\) 1.33523e6 0.427876
\(397\) 2.80814e6 0.894216 0.447108 0.894480i \(-0.352454\pi\)
0.447108 + 0.894480i \(0.352454\pi\)
\(398\) −8.22570e6 −2.60295
\(399\) −22139.6 −0.00696207
\(400\) −80203.3 −0.0250635
\(401\) 3.81287e6 1.18411 0.592053 0.805899i \(-0.298318\pi\)
0.592053 + 0.805899i \(0.298318\pi\)
\(402\) −4.90880e6 −1.51499
\(403\) −6.09958e6 −1.87084
\(404\) −2.96810e6 −0.904743
\(405\) −364246. −0.110346
\(406\) 182931. 0.0550771
\(407\) 553663. 0.165676
\(408\) 5.32050e6 1.58235
\(409\) −3.86268e6 −1.14178 −0.570888 0.821028i \(-0.693401\pi\)
−0.570888 + 0.821028i \(0.693401\pi\)
\(410\) −2.30866e6 −0.678266
\(411\) −298540. −0.0871762
\(412\) −1.24441e7 −3.61178
\(413\) −18205.6 −0.00525206
\(414\) −437240. −0.125377
\(415\) −4.50728e6 −1.28468
\(416\) −3.93926e6 −1.11604
\(417\) −1.15639e6 −0.325660
\(418\) 2.29058e6 0.641217
\(419\) −4.00521e6 −1.11453 −0.557263 0.830336i \(-0.688149\pi\)
−0.557263 + 0.830336i \(0.688149\pi\)
\(420\) −90260.7 −0.0249675
\(421\) 4.89697e6 1.34655 0.673275 0.739392i \(-0.264887\pi\)
0.673275 + 0.739392i \(0.264887\pi\)
\(422\) 8.38696e6 2.29257
\(423\) 758005. 0.205978
\(424\) 1.40361e7 3.79168
\(425\) 61921.3 0.0166291
\(426\) −1.99047e6 −0.531411
\(427\) 66652.2 0.0176907
\(428\) 1.28548e7 3.39200
\(429\) 1.35492e6 0.355443
\(430\) −5.90143e6 −1.53917
\(431\) 1.81874e6 0.471605 0.235803 0.971801i \(-0.424228\pi\)
0.235803 + 0.971801i \(0.424228\pi\)
\(432\) 1.36329e6 0.351463
\(433\) 545672. 0.139866 0.0699330 0.997552i \(-0.477721\pi\)
0.0699330 + 0.997552i \(0.477721\pi\)
\(434\) 236663. 0.0603124
\(435\) 3.57628e6 0.906168
\(436\) −9.97874e6 −2.51397
\(437\) −519566. −0.130148
\(438\) 7.60934e6 1.89523
\(439\) 2.81114e6 0.696179 0.348090 0.937461i \(-0.386830\pi\)
0.348090 + 0.937461i \(0.386830\pi\)
\(440\) 5.19525e6 1.27931
\(441\) −1.36086e6 −0.333209
\(442\) 9.70462e6 2.36278
\(443\) −2.81428e6 −0.681332 −0.340666 0.940184i \(-0.610652\pi\)
−0.340666 + 0.940184i \(0.610652\pi\)
\(444\) 1.57252e6 0.378564
\(445\) −5.55807e6 −1.33053
\(446\) −5.22402e6 −1.24356
\(447\) −1.65805e6 −0.392491
\(448\) 2959.70 0.000696712 0
\(449\) 4.00644e6 0.937871 0.468936 0.883232i \(-0.344638\pi\)
0.468936 + 0.883232i \(0.344638\pi\)
\(450\) 35448.2 0.00825207
\(451\) −931403. −0.215624
\(452\) −975391. −0.224560
\(453\) −178182. −0.0407962
\(454\) 1.48037e7 3.37078
\(455\) −91591.8 −0.0207409
\(456\) 3.61933e6 0.815110
\(457\) −5.43801e6 −1.21801 −0.609003 0.793168i \(-0.708430\pi\)
−0.609003 + 0.793168i \(0.708430\pi\)
\(458\) −6.33853e6 −1.41197
\(459\) −1.05254e6 −0.233188
\(460\) −2.11821e6 −0.466740
\(461\) 2.50966e6 0.550001 0.275000 0.961444i \(-0.411322\pi\)
0.275000 + 0.961444i \(0.411322\pi\)
\(462\) −52570.9 −0.0114588
\(463\) 108662. 0.0235573 0.0117787 0.999931i \(-0.496251\pi\)
0.0117787 + 0.999931i \(0.496251\pi\)
\(464\) −1.33852e7 −2.88623
\(465\) 4.62676e6 0.992303
\(466\) −5.00347e6 −1.06735
\(467\) 3.09125e6 0.655906 0.327953 0.944694i \(-0.393641\pi\)
0.327953 + 0.944694i \(0.393641\pi\)
\(468\) 3.84826e6 0.812176
\(469\) 133874. 0.0281037
\(470\) 5.30139e6 1.10699
\(471\) 1.73985e6 0.361377
\(472\) 2.97620e6 0.614904
\(473\) −2.38087e6 −0.489309
\(474\) 2.20043e6 0.449845
\(475\) 42122.7 0.00856608
\(476\) −260820. −0.0527624
\(477\) −2.77671e6 −0.558772
\(478\) −1.11739e6 −0.223684
\(479\) −7.74743e6 −1.54283 −0.771417 0.636330i \(-0.780452\pi\)
−0.771417 + 0.636330i \(0.780452\pi\)
\(480\) 2.98808e6 0.591955
\(481\) 1.59571e6 0.314479
\(482\) −1.38951e7 −2.72424
\(483\) 11924.5 0.00232580
\(484\) −7.84841e6 −1.52289
\(485\) −2.00538e6 −0.387118
\(486\) −602548. −0.115718
\(487\) 6.60929e6 1.26279 0.631397 0.775460i \(-0.282482\pi\)
0.631397 + 0.775460i \(0.282482\pi\)
\(488\) −1.08961e7 −2.07120
\(489\) 956575. 0.180903
\(490\) −9.51768e6 −1.79077
\(491\) 1.68453e6 0.315337 0.157669 0.987492i \(-0.449602\pi\)
0.157669 + 0.987492i \(0.449602\pi\)
\(492\) −2.64538e6 −0.492692
\(493\) 1.03341e7 1.91495
\(494\) 6.60168e6 1.21713
\(495\) −1.02776e6 −0.188529
\(496\) −1.73170e7 −3.16058
\(497\) 54284.5 0.00985791
\(498\) −7.45610e6 −1.34722
\(499\) 6.29003e6 1.13084 0.565420 0.824803i \(-0.308714\pi\)
0.565420 + 0.824803i \(0.308714\pi\)
\(500\) 1.26848e7 2.26913
\(501\) −699440. −0.124496
\(502\) −82064.9 −0.0145344
\(503\) 3.68350e6 0.649143 0.324572 0.945861i \(-0.394780\pi\)
0.324572 + 0.945861i \(0.394780\pi\)
\(504\) −83066.9 −0.0145664
\(505\) 2.28462e6 0.398644
\(506\) −1.23372e6 −0.214210
\(507\) 563376. 0.0973371
\(508\) 1.14367e7 1.96627
\(509\) 3.33612e6 0.570752 0.285376 0.958416i \(-0.407882\pi\)
0.285376 + 0.958416i \(0.407882\pi\)
\(510\) −7.36131e6 −1.25323
\(511\) −207524. −0.0351573
\(512\) 1.33189e7 2.24539
\(513\) −716000. −0.120121
\(514\) −1.19114e7 −1.98863
\(515\) 9.57854e6 1.59141
\(516\) −6.76218e6 −1.11805
\(517\) 2.13879e6 0.351918
\(518\) −61913.5 −0.0101382
\(519\) 1.96671e6 0.320496
\(520\) 1.49732e7 2.42832
\(521\) −987399. −0.159367 −0.0796835 0.996820i \(-0.525391\pi\)
−0.0796835 + 0.996820i \(0.525391\pi\)
\(522\) 5.91601e6 0.950282
\(523\) 6.57834e6 1.05163 0.525814 0.850600i \(-0.323761\pi\)
0.525814 + 0.850600i \(0.323761\pi\)
\(524\) −2.76612e7 −4.40091
\(525\) −966.753 −0.000153080 0
\(526\) 5.69446e6 0.897404
\(527\) 1.33696e7 2.09697
\(528\) 3.84667e6 0.600483
\(529\) 279841. 0.0434783
\(530\) −1.94200e7 −3.00302
\(531\) −588772. −0.0906172
\(532\) −177426. −0.0271793
\(533\) −2.68439e6 −0.409287
\(534\) −9.19435e6 −1.39530
\(535\) −9.89463e6 −1.49457
\(536\) −2.18854e7 −3.29035
\(537\) −4.90943e6 −0.734676
\(538\) 1.24437e7 1.85350
\(539\) −3.83981e6 −0.569295
\(540\) −2.91905e6 −0.430781
\(541\) −6.47470e6 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(542\) −1.43452e7 −2.09753
\(543\) 4.59790e6 0.669206
\(544\) 8.63445e6 1.25094
\(545\) 7.68087e6 1.10769
\(546\) −151514. −0.0217506
\(547\) −495799. −0.0708496 −0.0354248 0.999372i \(-0.511278\pi\)
−0.0354248 + 0.999372i \(0.511278\pi\)
\(548\) −2.39249e6 −0.340328
\(549\) 2.15554e6 0.305229
\(550\) 100021. 0.0140989
\(551\) 7.02992e6 0.986442
\(552\) −1.94939e6 −0.272302
\(553\) −60010.8 −0.00834481
\(554\) −1.91634e7 −2.65276
\(555\) −1.21041e6 −0.166801
\(556\) −9.26727e6 −1.27135
\(557\) 3.94887e6 0.539306 0.269653 0.962958i \(-0.413091\pi\)
0.269653 + 0.962958i \(0.413091\pi\)
\(558\) 7.65374e6 1.04061
\(559\) −6.86190e6 −0.928784
\(560\) −260033. −0.0350395
\(561\) −2.96984e6 −0.398406
\(562\) 2.16083e7 2.88590
\(563\) −2.70237e6 −0.359314 −0.179657 0.983729i \(-0.557499\pi\)
−0.179657 + 0.983729i \(0.557499\pi\)
\(564\) 6.07462e6 0.804120
\(565\) 750781. 0.0989447
\(566\) −1.18411e7 −1.55365
\(567\) 16432.8 0.00214662
\(568\) −8.87430e6 −1.15415
\(569\) 1.33108e7 1.72355 0.861773 0.507294i \(-0.169354\pi\)
0.861773 + 0.507294i \(0.169354\pi\)
\(570\) −5.00762e6 −0.645571
\(571\) −1.10053e7 −1.41257 −0.706286 0.707926i \(-0.749631\pi\)
−0.706286 + 0.707926i \(0.749631\pi\)
\(572\) 1.08583e7 1.38762
\(573\) −519167. −0.0660572
\(574\) 104154. 0.0131947
\(575\) −22687.5 −0.00286165
\(576\) 95717.3 0.0120208
\(577\) −480544. −0.0600888 −0.0300444 0.999549i \(-0.509565\pi\)
−0.0300444 + 0.999549i \(0.509565\pi\)
\(578\) −6.78300e6 −0.844505
\(579\) −7.39882e6 −0.917204
\(580\) 2.86602e7 3.53760
\(581\) 203345. 0.0249915
\(582\) −3.31737e6 −0.405963
\(583\) −7.83477e6 −0.954674
\(584\) 3.39255e7 4.11618
\(585\) −2.96210e6 −0.357857
\(586\) 1.29924e7 1.56296
\(587\) 9.12805e6 1.09341 0.546705 0.837325i \(-0.315882\pi\)
0.546705 + 0.837325i \(0.315882\pi\)
\(588\) −1.09059e7 −1.30082
\(589\) 9.09485e6 1.08021
\(590\) −4.11780e6 −0.487007
\(591\) 8.74414e6 1.02979
\(592\) 4.53029e6 0.531278
\(593\) −2.66114e6 −0.310765 −0.155382 0.987854i \(-0.549661\pi\)
−0.155382 + 0.987854i \(0.549661\pi\)
\(594\) −1.70015e6 −0.197707
\(595\) 200760. 0.0232479
\(596\) −1.32876e7 −1.53225
\(597\) 7.25499e6 0.833107
\(598\) −3.55569e6 −0.406604
\(599\) 6.44915e6 0.734405 0.367202 0.930141i \(-0.380316\pi\)
0.367202 + 0.930141i \(0.380316\pi\)
\(600\) 158042. 0.0179224
\(601\) −1.48491e6 −0.167693 −0.0838463 0.996479i \(-0.526720\pi\)
−0.0838463 + 0.996479i \(0.526720\pi\)
\(602\) 266241. 0.0299423
\(603\) 4.32951e6 0.484893
\(604\) −1.42795e6 −0.159265
\(605\) 6.04111e6 0.671009
\(606\) 3.77929e6 0.418051
\(607\) −6.56656e6 −0.723379 −0.361690 0.932299i \(-0.617800\pi\)
−0.361690 + 0.932299i \(0.617800\pi\)
\(608\) 5.87368e6 0.644394
\(609\) −161343. −0.0176281
\(610\) 1.50756e7 1.64040
\(611\) 6.16420e6 0.667996
\(612\) −8.43498e6 −0.910344
\(613\) −9.33817e6 −1.00372 −0.501858 0.864950i \(-0.667350\pi\)
−0.501858 + 0.864950i \(0.667350\pi\)
\(614\) 2.30793e7 2.47060
\(615\) 2.03621e6 0.217088
\(616\) −234382. −0.0248870
\(617\) 7.56128e6 0.799617 0.399809 0.916599i \(-0.369077\pi\)
0.399809 + 0.916599i \(0.369077\pi\)
\(618\) 1.58452e7 1.66888
\(619\) 1.16180e7 1.21872 0.609359 0.792895i \(-0.291427\pi\)
0.609359 + 0.792895i \(0.291427\pi\)
\(620\) 3.70786e7 3.87387
\(621\) 385641. 0.0401286
\(622\) −1.22718e7 −1.27184
\(623\) 250751. 0.0258834
\(624\) 1.10865e7 1.13981
\(625\) −9.62976e6 −0.986088
\(626\) 4.94201e6 0.504043
\(627\) −2.02027e6 −0.205230
\(628\) 1.39431e7 1.41078
\(629\) −3.49763e6 −0.352491
\(630\) 114929. 0.0115366
\(631\) −8.43872e6 −0.843730 −0.421865 0.906659i \(-0.638624\pi\)
−0.421865 + 0.906659i \(0.638624\pi\)
\(632\) 9.81042e6 0.977001
\(633\) −7.39722e6 −0.733768
\(634\) −8.12891e6 −0.803173
\(635\) −8.80313e6 −0.866369
\(636\) −2.22524e7 −2.18140
\(637\) −1.10667e7 −1.08061
\(638\) 1.66926e7 1.62358
\(639\) 1.75557e6 0.170085
\(640\) −9.95483e6 −0.960692
\(641\) 1.32010e7 1.26900 0.634500 0.772923i \(-0.281206\pi\)
0.634500 + 0.772923i \(0.281206\pi\)
\(642\) −1.63680e7 −1.56733
\(643\) −1.57321e7 −1.50058 −0.750290 0.661109i \(-0.770086\pi\)
−0.750290 + 0.661109i \(0.770086\pi\)
\(644\) 95562.5 0.00907973
\(645\) 5.20501e6 0.492631
\(646\) −1.44702e7 −1.36425
\(647\) −6.22066e6 −0.584219 −0.292109 0.956385i \(-0.594357\pi\)
−0.292109 + 0.956385i \(0.594357\pi\)
\(648\) −2.68640e6 −0.251323
\(649\) −1.66128e6 −0.154822
\(650\) 288270. 0.0267618
\(651\) −208735. −0.0193038
\(652\) 7.66595e6 0.706231
\(653\) −1.57628e7 −1.44660 −0.723302 0.690532i \(-0.757376\pi\)
−0.723302 + 0.690532i \(0.757376\pi\)
\(654\) 1.27060e7 1.16162
\(655\) 2.12915e7 1.93911
\(656\) −7.62110e6 −0.691446
\(657\) −6.71136e6 −0.606593
\(658\) −239171. −0.0215349
\(659\) −5.77008e6 −0.517569 −0.258784 0.965935i \(-0.583322\pi\)
−0.258784 + 0.965935i \(0.583322\pi\)
\(660\) −8.23640e6 −0.736000
\(661\) −1.48192e7 −1.31924 −0.659618 0.751601i \(-0.729282\pi\)
−0.659618 + 0.751601i \(0.729282\pi\)
\(662\) 2.68223e6 0.237877
\(663\) −8.55938e6 −0.756238
\(664\) −3.32423e7 −2.92598
\(665\) 136569. 0.0119756
\(666\) −2.00230e6 −0.174921
\(667\) −3.78635e6 −0.329538
\(668\) −5.60528e6 −0.486022
\(669\) 4.60753e6 0.398018
\(670\) 3.02801e7 2.60597
\(671\) 6.08209e6 0.521491
\(672\) −134806. −0.0115156
\(673\) −1.74615e7 −1.48609 −0.743045 0.669242i \(-0.766619\pi\)
−0.743045 + 0.669242i \(0.766619\pi\)
\(674\) −1.25637e7 −1.06529
\(675\) −31265.0 −0.00264118
\(676\) 4.51487e6 0.379995
\(677\) −2.10975e7 −1.76913 −0.884564 0.466418i \(-0.845544\pi\)
−0.884564 + 0.466418i \(0.845544\pi\)
\(678\) 1.24197e6 0.103762
\(679\) 90472.2 0.00753080
\(680\) −3.28197e7 −2.72184
\(681\) −1.30567e7 −1.07886
\(682\) 2.15958e7 1.77791
\(683\) −1.43654e7 −1.17833 −0.589163 0.808014i \(-0.700542\pi\)
−0.589163 + 0.808014i \(0.700542\pi\)
\(684\) −5.73799e6 −0.468943
\(685\) 1.84155e6 0.149954
\(686\) 858935. 0.0696867
\(687\) 5.59052e6 0.451919
\(688\) −1.94812e7 −1.56908
\(689\) −2.25806e7 −1.81212
\(690\) 2.69713e6 0.215664
\(691\) 1.64154e7 1.30785 0.653923 0.756561i \(-0.273122\pi\)
0.653923 + 0.756561i \(0.273122\pi\)
\(692\) 1.57612e7 1.25119
\(693\) 46367.0 0.00366755
\(694\) 3.77907e7 2.97842
\(695\) 7.13324e6 0.560176
\(696\) 2.63759e7 2.06388
\(697\) 5.88391e6 0.458758
\(698\) −1.66210e7 −1.29127
\(699\) 4.41301e6 0.341619
\(700\) −7747.52 −0.000597609 0
\(701\) −1.07940e7 −0.829635 −0.414818 0.909905i \(-0.636155\pi\)
−0.414818 + 0.909905i \(0.636155\pi\)
\(702\) −4.90000e6 −0.375278
\(703\) −2.37930e6 −0.181577
\(704\) 270077. 0.0205379
\(705\) −4.67577e6 −0.354308
\(706\) −2.61829e7 −1.97700
\(707\) −103070. −0.00775502
\(708\) −4.71839e6 −0.353762
\(709\) 2.44390e7 1.82586 0.912930 0.408115i \(-0.133814\pi\)
0.912930 + 0.408115i \(0.133814\pi\)
\(710\) 1.22783e7 0.914093
\(711\) −1.94076e6 −0.143979
\(712\) −4.09921e7 −3.03040
\(713\) −4.89853e6 −0.360863
\(714\) 332103. 0.0243797
\(715\) −8.35787e6 −0.611407
\(716\) −3.93440e7 −2.86811
\(717\) 985527. 0.0715930
\(718\) 3.53978e7 2.56251
\(719\) −6.41280e6 −0.462621 −0.231310 0.972880i \(-0.574301\pi\)
−0.231310 + 0.972880i \(0.574301\pi\)
\(720\) −8.40951e6 −0.604560
\(721\) −432133. −0.0309585
\(722\) 1.54231e7 1.10110
\(723\) 1.22554e7 0.871928
\(724\) 3.68474e7 2.61252
\(725\) 306970. 0.0216896
\(726\) 9.99341e6 0.703675
\(727\) −1.58788e7 −1.11424 −0.557122 0.830431i \(-0.688095\pi\)
−0.557122 + 0.830431i \(0.688095\pi\)
\(728\) −675512. −0.0472394
\(729\) 531441. 0.0370370
\(730\) −4.69385e7 −3.26003
\(731\) 1.50406e7 1.04105
\(732\) 1.72744e7 1.19159
\(733\) 2.52517e7 1.73592 0.867962 0.496631i \(-0.165430\pi\)
0.867962 + 0.496631i \(0.165430\pi\)
\(734\) −1.67487e7 −1.14747
\(735\) 8.39450e6 0.573161
\(736\) −3.16359e6 −0.215271
\(737\) 1.22162e7 0.828450
\(738\) 3.36837e6 0.227656
\(739\) −1.56260e7 −1.05253 −0.526267 0.850319i \(-0.676409\pi\)
−0.526267 + 0.850319i \(0.676409\pi\)
\(740\) −9.70015e6 −0.651177
\(741\) −5.82261e6 −0.389558
\(742\) 876126. 0.0584193
\(743\) 1.42696e7 0.948284 0.474142 0.880448i \(-0.342758\pi\)
0.474142 + 0.880448i \(0.342758\pi\)
\(744\) 3.41235e7 2.26006
\(745\) 1.02278e7 0.675133
\(746\) −1.61845e7 −1.06476
\(747\) 6.57621e6 0.431195
\(748\) −2.38002e7 −1.55534
\(749\) 446393. 0.0290746
\(750\) −1.61516e7 −1.04848
\(751\) 2.22359e7 1.43865 0.719323 0.694676i \(-0.244452\pi\)
0.719323 + 0.694676i \(0.244452\pi\)
\(752\) 1.75004e7 1.12851
\(753\) 72380.4 0.00465193
\(754\) 4.81098e7 3.08181
\(755\) 1.09912e6 0.0701745
\(756\) 131692. 0.00838021
\(757\) −1.09113e7 −0.692048 −0.346024 0.938226i \(-0.612468\pi\)
−0.346024 + 0.938226i \(0.612468\pi\)
\(758\) 9.85756e6 0.623155
\(759\) 1.08813e6 0.0685607
\(760\) −2.23260e7 −1.40209
\(761\) 2.42444e6 0.151758 0.0758788 0.997117i \(-0.475824\pi\)
0.0758788 + 0.997117i \(0.475824\pi\)
\(762\) −1.45624e7 −0.908546
\(763\) −346520. −0.0215485
\(764\) −4.16058e6 −0.257882
\(765\) 6.49260e6 0.401112
\(766\) 3.16659e7 1.94993
\(767\) −4.78798e6 −0.293876
\(768\) −1.68080e7 −1.02828
\(769\) 2.00363e6 0.122181 0.0610903 0.998132i \(-0.480542\pi\)
0.0610903 + 0.998132i \(0.480542\pi\)
\(770\) 324285. 0.0197106
\(771\) 1.05057e7 0.636487
\(772\) −5.92938e7 −3.58069
\(773\) 3.91181e6 0.235466 0.117733 0.993045i \(-0.462437\pi\)
0.117733 + 0.993045i \(0.462437\pi\)
\(774\) 8.61030e6 0.516614
\(775\) 397137. 0.0237512
\(776\) −1.47902e7 −0.881697
\(777\) 54607.1 0.00324487
\(778\) −6.37322e6 −0.377494
\(779\) 4.00260e6 0.236319
\(780\) −2.37381e7 −1.39704
\(781\) 4.95353e6 0.290594
\(782\) 7.79371e6 0.455751
\(783\) −5.21786e6 −0.304150
\(784\) −3.14188e7 −1.82557
\(785\) −1.07323e7 −0.621614
\(786\) 3.52211e7 2.03351
\(787\) −2.37516e7 −1.36696 −0.683479 0.729970i \(-0.739534\pi\)
−0.683479 + 0.729970i \(0.739534\pi\)
\(788\) 7.00752e7 4.02021
\(789\) −5.02246e6 −0.287226
\(790\) −1.35734e7 −0.773789
\(791\) −33871.3 −0.00192482
\(792\) −7.57996e6 −0.429392
\(793\) 1.75292e7 0.989872
\(794\) −2.86548e7 −1.61304
\(795\) 1.71282e7 0.961157
\(796\) 5.81412e7 3.25238
\(797\) −2.50486e7 −1.39681 −0.698405 0.715703i \(-0.746106\pi\)
−0.698405 + 0.715703i \(0.746106\pi\)
\(798\) 225917. 0.0125586
\(799\) −1.35113e7 −0.748737
\(800\) 256481. 0.0141687
\(801\) 8.10932e6 0.446584
\(802\) −3.89072e7 −2.13597
\(803\) −1.89368e7 −1.03638
\(804\) 3.46965e7 1.89298
\(805\) −73556.7 −0.00400067
\(806\) 6.22413e7 3.37475
\(807\) −1.09752e7 −0.593237
\(808\) 1.68496e7 0.907949
\(809\) −2.26709e7 −1.21786 −0.608931 0.793223i \(-0.708402\pi\)
−0.608931 + 0.793223i \(0.708402\pi\)
\(810\) 3.71683e6 0.199049
\(811\) −1.66885e7 −0.890974 −0.445487 0.895288i \(-0.646969\pi\)
−0.445487 + 0.895288i \(0.646969\pi\)
\(812\) −1.29300e6 −0.0688188
\(813\) 1.26523e7 0.671342
\(814\) −5.64969e6 −0.298857
\(815\) −5.90066e6 −0.311176
\(816\) −2.43004e7 −1.27758
\(817\) 1.02315e7 0.536272
\(818\) 3.94156e7 2.05961
\(819\) 133634. 0.00696158
\(820\) 1.63181e7 0.847492
\(821\) −2.41094e7 −1.24833 −0.624164 0.781293i \(-0.714560\pi\)
−0.624164 + 0.781293i \(0.714560\pi\)
\(822\) 3.04636e6 0.157254
\(823\) −9.03070e6 −0.464753 −0.232376 0.972626i \(-0.574650\pi\)
−0.232376 + 0.972626i \(0.574650\pi\)
\(824\) 7.06441e7 3.62458
\(825\) −88217.4 −0.00451252
\(826\) 185773. 0.00947399
\(827\) 1.22307e7 0.621852 0.310926 0.950434i \(-0.399361\pi\)
0.310926 + 0.950434i \(0.399361\pi\)
\(828\) 3.09051e6 0.156659
\(829\) 2.01683e7 1.01925 0.509627 0.860395i \(-0.329783\pi\)
0.509627 + 0.860395i \(0.329783\pi\)
\(830\) 4.59932e7 2.31739
\(831\) 1.69019e7 0.849051
\(832\) 778387. 0.0389841
\(833\) 2.42570e7 1.21123
\(834\) 1.18000e7 0.587447
\(835\) 4.31452e6 0.214149
\(836\) −1.61904e7 −0.801199
\(837\) −6.75053e6 −0.333061
\(838\) 4.08700e7 2.01045
\(839\) 1.80995e7 0.887690 0.443845 0.896103i \(-0.353614\pi\)
0.443845 + 0.896103i \(0.353614\pi\)
\(840\) 512401. 0.0250560
\(841\) 3.07195e7 1.49770
\(842\) −4.99697e7 −2.42899
\(843\) −1.90583e7 −0.923668
\(844\) −5.92810e7 −2.86457
\(845\) −3.47520e6 −0.167432
\(846\) −7.73483e6 −0.371556
\(847\) −272543. −0.0130535
\(848\) −6.41072e7 −3.06138
\(849\) 1.04438e7 0.497265
\(850\) −631857. −0.0299966
\(851\) 1.28150e6 0.0606591
\(852\) 1.40691e7 0.663998
\(853\) −1.82781e7 −0.860121 −0.430060 0.902800i \(-0.641508\pi\)
−0.430060 + 0.902800i \(0.641508\pi\)
\(854\) −680132. −0.0319116
\(855\) 4.41667e6 0.206623
\(856\) −7.29753e7 −3.40402
\(857\) −1.47267e7 −0.684940 −0.342470 0.939529i \(-0.611264\pi\)
−0.342470 + 0.939529i \(0.611264\pi\)
\(858\) −1.38259e7 −0.641172
\(859\) −1.82990e7 −0.846142 −0.423071 0.906096i \(-0.639048\pi\)
−0.423071 + 0.906096i \(0.639048\pi\)
\(860\) 4.17127e7 1.92319
\(861\) −91863.1 −0.00422312
\(862\) −1.85588e7 −0.850712
\(863\) −3.92569e6 −0.179427 −0.0897137 0.995968i \(-0.528595\pi\)
−0.0897137 + 0.995968i \(0.528595\pi\)
\(864\) −4.35966e6 −0.198686
\(865\) −1.21317e7 −0.551293
\(866\) −5.56814e6 −0.252299
\(867\) 5.98254e6 0.270295
\(868\) −1.67279e6 −0.0753603
\(869\) −5.47606e6 −0.245991
\(870\) −3.64931e7 −1.63460
\(871\) 3.52082e7 1.57253
\(872\) 5.66483e7 2.52287
\(873\) 2.92589e6 0.129934
\(874\) 5.30176e6 0.234769
\(875\) 440490. 0.0194499
\(876\) −5.37846e7 −2.36809
\(877\) −3.49463e7 −1.53427 −0.767135 0.641486i \(-0.778318\pi\)
−0.767135 + 0.641486i \(0.778318\pi\)
\(878\) −2.86854e7 −1.25581
\(879\) −1.14592e7 −0.500245
\(880\) −2.37283e7 −1.03290
\(881\) −3.45541e7 −1.49989 −0.749945 0.661501i \(-0.769920\pi\)
−0.749945 + 0.661501i \(0.769920\pi\)
\(882\) 1.38865e7 0.601064
\(883\) 2.29538e6 0.0990722 0.0495361 0.998772i \(-0.484226\pi\)
0.0495361 + 0.998772i \(0.484226\pi\)
\(884\) −6.85945e7 −2.95229
\(885\) 3.63186e6 0.155873
\(886\) 2.87175e7 1.22903
\(887\) 2.48902e7 1.06223 0.531116 0.847299i \(-0.321773\pi\)
0.531116 + 0.847299i \(0.321773\pi\)
\(888\) −8.92704e6 −0.379905
\(889\) 397151. 0.0168539
\(890\) 5.67156e7 2.40009
\(891\) 1.49952e6 0.0632786
\(892\) 3.69246e7 1.55383
\(893\) −9.19120e6 −0.385695
\(894\) 1.69191e7 0.708000
\(895\) 3.02840e7 1.26373
\(896\) 449109. 0.0186888
\(897\) 3.13609e6 0.130139
\(898\) −4.08826e7 −1.69179
\(899\) 6.62789e7 2.73512
\(900\) −250556. −0.0103110
\(901\) 4.94943e7 2.03115
\(902\) 9.50422e6 0.388956
\(903\) −234822. −0.00958341
\(904\) 5.53719e6 0.225356
\(905\) −2.83623e7 −1.15112
\(906\) 1.81821e6 0.0735908
\(907\) −1.12581e7 −0.454411 −0.227205 0.973847i \(-0.572959\pi\)
−0.227205 + 0.973847i \(0.572959\pi\)
\(908\) −1.04636e8 −4.21178
\(909\) −3.33330e6 −0.133803
\(910\) 934621. 0.0374138
\(911\) −2.07869e7 −0.829837 −0.414919 0.909859i \(-0.636190\pi\)
−0.414919 + 0.909859i \(0.636190\pi\)
\(912\) −1.65306e7 −0.658116
\(913\) 1.85555e7 0.736707
\(914\) 5.54905e7 2.19712
\(915\) −1.32965e7 −0.525032
\(916\) 4.48022e7 1.76425
\(917\) −960558. −0.0377225
\(918\) 1.07403e7 0.420639
\(919\) −1.17836e7 −0.460244 −0.230122 0.973162i \(-0.573913\pi\)
−0.230122 + 0.973162i \(0.573913\pi\)
\(920\) 1.20249e7 0.468394
\(921\) −2.03557e7 −0.790747
\(922\) −2.56091e7 −0.992126
\(923\) 1.42766e7 0.551594
\(924\) 371583. 0.0143178
\(925\) −103895. −0.00399246
\(926\) −1.10881e6 −0.0424942
\(927\) −1.39753e7 −0.534147
\(928\) 4.28045e7 1.63162
\(929\) −2.82219e7 −1.07287 −0.536435 0.843942i \(-0.680229\pi\)
−0.536435 + 0.843942i \(0.680229\pi\)
\(930\) −4.72123e7 −1.78998
\(931\) 1.65011e7 0.623935
\(932\) 3.53657e7 1.33365
\(933\) 1.08236e7 0.407070
\(934\) −3.15437e7 −1.18316
\(935\) 1.83196e7 0.685308
\(936\) −2.18462e7 −0.815053
\(937\) 1.34325e7 0.499814 0.249907 0.968270i \(-0.419600\pi\)
0.249907 + 0.968270i \(0.419600\pi\)
\(938\) −1.36608e6 −0.0506953
\(939\) −4.35880e6 −0.161325
\(940\) −3.74715e7 −1.38319
\(941\) 4.06264e7 1.49566 0.747832 0.663888i \(-0.231095\pi\)
0.747832 + 0.663888i \(0.231095\pi\)
\(942\) −1.77538e7 −0.651875
\(943\) −2.15582e6 −0.0789465
\(944\) −1.35933e7 −0.496470
\(945\) −101366. −0.00369245
\(946\) 2.42949e7 0.882646
\(947\) −3.34341e7 −1.21148 −0.605738 0.795664i \(-0.707122\pi\)
−0.605738 + 0.795664i \(0.707122\pi\)
\(948\) −1.55532e7 −0.562080
\(949\) −5.45778e7 −1.96721
\(950\) −429828. −0.0154520
\(951\) 7.16962e6 0.257066
\(952\) 1.48065e6 0.0529493
\(953\) −8.95158e6 −0.319277 −0.159638 0.987176i \(-0.551033\pi\)
−0.159638 + 0.987176i \(0.551033\pi\)
\(954\) 2.83341e7 1.00795
\(955\) 3.20250e6 0.113627
\(956\) 7.89797e6 0.279493
\(957\) −1.47227e7 −0.519648
\(958\) 7.90563e7 2.78306
\(959\) −83081.1 −0.00291713
\(960\) −590435. −0.0206773
\(961\) 5.71181e7 1.99510
\(962\) −1.62830e7 −0.567277
\(963\) 1.44365e7 0.501643
\(964\) 9.82140e7 3.40393
\(965\) 4.56399e7 1.57771
\(966\) −121680. −0.00419543
\(967\) −2.94663e7 −1.01335 −0.506675 0.862137i \(-0.669126\pi\)
−0.506675 + 0.862137i \(0.669126\pi\)
\(968\) 4.45546e7 1.52829
\(969\) 1.27626e7 0.436645
\(970\) 2.04633e7 0.698307
\(971\) −3.03617e7 −1.03342 −0.516712 0.856159i \(-0.672844\pi\)
−0.516712 + 0.856159i \(0.672844\pi\)
\(972\) 4.25895e6 0.144589
\(973\) −321814. −0.0108974
\(974\) −6.74425e7 −2.27791
\(975\) −254251. −0.00856548
\(976\) 4.97661e7 1.67228
\(977\) 3.77587e6 0.126555 0.0632777 0.997996i \(-0.479845\pi\)
0.0632777 + 0.997996i \(0.479845\pi\)
\(978\) −9.76107e6 −0.326325
\(979\) 2.28813e7 0.762999
\(980\) 6.72732e7 2.23757
\(981\) −1.12065e7 −0.371791
\(982\) −1.71893e7 −0.568826
\(983\) 1.74788e7 0.576938 0.288469 0.957489i \(-0.406854\pi\)
0.288469 + 0.957489i \(0.406854\pi\)
\(984\) 1.50176e7 0.494438
\(985\) −5.39385e7 −1.77137
\(986\) −1.05452e8 −3.45431
\(987\) 210946. 0.00689253
\(988\) −4.66622e7 −1.52080
\(989\) −5.51074e6 −0.179151
\(990\) 1.04874e7 0.340080
\(991\) 4.83380e7 1.56352 0.781762 0.623576i \(-0.214321\pi\)
0.781762 + 0.623576i \(0.214321\pi\)
\(992\) 5.53777e7 1.78672
\(993\) −2.36570e6 −0.0761355
\(994\) −553930. −0.0177823
\(995\) −4.47526e7 −1.43305
\(996\) 5.27014e7 1.68335
\(997\) −1.74335e7 −0.555452 −0.277726 0.960660i \(-0.589581\pi\)
−0.277726 + 0.960660i \(0.589581\pi\)
\(998\) −6.41847e7 −2.03988
\(999\) 1.76601e6 0.0559858
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 69.6.a.b.1.1 3
3.2 odd 2 207.6.a.c.1.3 3
4.3 odd 2 1104.6.a.i.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.b.1.1 3 1.1 even 1 trivial
207.6.a.c.1.3 3 3.2 odd 2
1104.6.a.i.1.2 3 4.3 odd 2