Properties

Label 165.4.a.c.1.2
Level $165$
Weight $4$
Character 165.1
Self dual yes
Analytic conductor $9.735$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.73531515095\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} +4.68466 q^{6} -10.2462 q^{7} -21.1771 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+1.56155 q^{2} +3.00000 q^{3} -5.56155 q^{4} -5.00000 q^{5} +4.68466 q^{6} -10.2462 q^{7} -21.1771 q^{8} +9.00000 q^{9} -7.80776 q^{10} -11.0000 q^{11} -16.6847 q^{12} -40.8769 q^{13} -16.0000 q^{14} -15.0000 q^{15} +11.4233 q^{16} -98.7083 q^{17} +14.0540 q^{18} -39.6458 q^{19} +27.8078 q^{20} -30.7386 q^{21} -17.1771 q^{22} +61.6932 q^{23} -63.5312 q^{24} +25.0000 q^{25} -63.8314 q^{26} +27.0000 q^{27} +56.9848 q^{28} -149.093 q^{29} -23.4233 q^{30} +54.7386 q^{31} +187.255 q^{32} -33.0000 q^{33} -154.138 q^{34} +51.2311 q^{35} -50.0540 q^{36} +44.8939 q^{37} -61.9091 q^{38} -122.631 q^{39} +105.885 q^{40} +336.479 q^{41} -48.0000 q^{42} -2.36745 q^{43} +61.1771 q^{44} -45.0000 q^{45} +96.3371 q^{46} -333.295 q^{47} +34.2699 q^{48} -238.015 q^{49} +39.0388 q^{50} -296.125 q^{51} +227.339 q^{52} +640.064 q^{53} +42.1619 q^{54} +55.0000 q^{55} +216.985 q^{56} -118.938 q^{57} -232.816 q^{58} -370.773 q^{59} +83.4233 q^{60} -714.405 q^{61} +85.4773 q^{62} -92.2159 q^{63} +201.022 q^{64} +204.384 q^{65} -51.5312 q^{66} -404.985 q^{67} +548.972 q^{68} +185.080 q^{69} +80.0000 q^{70} +939.292 q^{71} -190.594 q^{72} -362.570 q^{73} +70.1042 q^{74} +75.0000 q^{75} +220.492 q^{76} +112.708 q^{77} -191.494 q^{78} +951.835 q^{79} -57.1165 q^{80} +81.0000 q^{81} +525.430 q^{82} +735.221 q^{83} +170.955 q^{84} +493.542 q^{85} -3.69690 q^{86} -447.278 q^{87} +232.948 q^{88} +385.879 q^{89} -70.2699 q^{90} +418.833 q^{91} -343.110 q^{92} +164.216 q^{93} -520.458 q^{94} +198.229 q^{95} +561.764 q^{96} -966.345 q^{97} -371.673 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 6q^{3} - 7q^{4} - 10q^{5} - 3q^{6} - 4q^{7} + 3q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - q^{2} + 6q^{3} - 7q^{4} - 10q^{5} - 3q^{6} - 4q^{7} + 3q^{8} + 18q^{9} + 5q^{10} - 22q^{11} - 21q^{12} - 90q^{13} - 32q^{14} - 30q^{15} - 39q^{16} - 16q^{17} - 9q^{18} - 170q^{19} + 35q^{20} - 12q^{21} + 11q^{22} - 124q^{23} + 9q^{24} + 50q^{25} + 62q^{26} + 54q^{27} + 48q^{28} - 158q^{29} + 15q^{30} + 60q^{31} + 123q^{32} - 66q^{33} - 366q^{34} + 20q^{35} - 63q^{36} - 372q^{37} + 272q^{38} - 270q^{39} - 15q^{40} + 38q^{41} - 96q^{42} - 516q^{43} + 77q^{44} - 90q^{45} + 572q^{46} + 224q^{47} - 117q^{48} - 542q^{49} - 25q^{50} - 48q^{51} + 298q^{52} + 472q^{53} - 27q^{54} + 110q^{55} + 368q^{56} - 510q^{57} - 210q^{58} + 248q^{59} + 105q^{60} + 72q^{61} + 72q^{62} - 36q^{63} + 769q^{64} + 450q^{65} + 33q^{66} - 744q^{67} + 430q^{68} - 372q^{69} + 160q^{70} + 2060q^{71} + 27q^{72} - 486q^{73} + 1138q^{74} + 150q^{75} + 408q^{76} + 44q^{77} + 186q^{78} + 642q^{79} + 195q^{80} + 162q^{81} + 1290q^{82} - 286q^{83} + 144q^{84} + 80q^{85} + 1312q^{86} - 474q^{87} - 33q^{88} + 244q^{89} + 45q^{90} + 112q^{91} - 76q^{92} + 180q^{93} - 1948q^{94} + 850q^{95} + 369q^{96} - 168q^{97} + 407q^{98} - 198q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155 0.552092 0.276046 0.961144i \(-0.410976\pi\)
0.276046 + 0.961144i \(0.410976\pi\)
\(3\) 3.00000 0.577350
\(4\) −5.56155 −0.695194
\(5\) −5.00000 −0.447214
\(6\) 4.68466 0.318751
\(7\) −10.2462 −0.553243 −0.276622 0.960979i \(-0.589215\pi\)
−0.276622 + 0.960979i \(0.589215\pi\)
\(8\) −21.1771 −0.935904
\(9\) 9.00000 0.333333
\(10\) −7.80776 −0.246903
\(11\) −11.0000 −0.301511
\(12\) −16.6847 −0.401371
\(13\) −40.8769 −0.872093 −0.436047 0.899924i \(-0.643622\pi\)
−0.436047 + 0.899924i \(0.643622\pi\)
\(14\) −16.0000 −0.305441
\(15\) −15.0000 −0.258199
\(16\) 11.4233 0.178489
\(17\) −98.7083 −1.40825 −0.704126 0.710075i \(-0.748661\pi\)
−0.704126 + 0.710075i \(0.748661\pi\)
\(18\) 14.0540 0.184031
\(19\) −39.6458 −0.478704 −0.239352 0.970933i \(-0.576935\pi\)
−0.239352 + 0.970933i \(0.576935\pi\)
\(20\) 27.8078 0.310900
\(21\) −30.7386 −0.319415
\(22\) −17.1771 −0.166462
\(23\) 61.6932 0.559301 0.279650 0.960102i \(-0.409781\pi\)
0.279650 + 0.960102i \(0.409781\pi\)
\(24\) −63.5312 −0.540344
\(25\) 25.0000 0.200000
\(26\) −63.8314 −0.481476
\(27\) 27.0000 0.192450
\(28\) 56.9848 0.384612
\(29\) −149.093 −0.954684 −0.477342 0.878718i \(-0.658400\pi\)
−0.477342 + 0.878718i \(0.658400\pi\)
\(30\) −23.4233 −0.142550
\(31\) 54.7386 0.317140 0.158570 0.987348i \(-0.449312\pi\)
0.158570 + 0.987348i \(0.449312\pi\)
\(32\) 187.255 1.03445
\(33\) −33.0000 −0.174078
\(34\) −154.138 −0.777485
\(35\) 51.2311 0.247418
\(36\) −50.0540 −0.231731
\(37\) 44.8939 0.199473 0.0997367 0.995014i \(-0.468200\pi\)
0.0997367 + 0.995014i \(0.468200\pi\)
\(38\) −61.9091 −0.264289
\(39\) −122.631 −0.503503
\(40\) 105.885 0.418549
\(41\) 336.479 1.28169 0.640844 0.767671i \(-0.278585\pi\)
0.640844 + 0.767671i \(0.278585\pi\)
\(42\) −48.0000 −0.176347
\(43\) −2.36745 −0.00839611 −0.00419806 0.999991i \(-0.501336\pi\)
−0.00419806 + 0.999991i \(0.501336\pi\)
\(44\) 61.1771 0.209609
\(45\) −45.0000 −0.149071
\(46\) 96.3371 0.308786
\(47\) −333.295 −1.03439 −0.517193 0.855869i \(-0.673023\pi\)
−0.517193 + 0.855869i \(0.673023\pi\)
\(48\) 34.2699 0.103051
\(49\) −238.015 −0.693922
\(50\) 39.0388 0.110418
\(51\) −296.125 −0.813055
\(52\) 227.339 0.606274
\(53\) 640.064 1.65886 0.829430 0.558610i \(-0.188665\pi\)
0.829430 + 0.558610i \(0.188665\pi\)
\(54\) 42.1619 0.106250
\(55\) 55.0000 0.134840
\(56\) 216.985 0.517782
\(57\) −118.938 −0.276380
\(58\) −232.816 −0.527074
\(59\) −370.773 −0.818144 −0.409072 0.912502i \(-0.634147\pi\)
−0.409072 + 0.912502i \(0.634147\pi\)
\(60\) 83.4233 0.179498
\(61\) −714.405 −1.49951 −0.749756 0.661715i \(-0.769829\pi\)
−0.749756 + 0.661715i \(0.769829\pi\)
\(62\) 85.4773 0.175091
\(63\) −92.2159 −0.184414
\(64\) 201.022 0.392621
\(65\) 204.384 0.390012
\(66\) −51.5312 −0.0961069
\(67\) −404.985 −0.738459 −0.369230 0.929338i \(-0.620378\pi\)
−0.369230 + 0.929338i \(0.620378\pi\)
\(68\) 548.972 0.979009
\(69\) 185.080 0.322912
\(70\) 80.0000 0.136598
\(71\) 939.292 1.57005 0.785024 0.619465i \(-0.212651\pi\)
0.785024 + 0.619465i \(0.212651\pi\)
\(72\) −190.594 −0.311968
\(73\) −362.570 −0.581310 −0.290655 0.956828i \(-0.593873\pi\)
−0.290655 + 0.956828i \(0.593873\pi\)
\(74\) 70.1042 0.110128
\(75\) 75.0000 0.115470
\(76\) 220.492 0.332792
\(77\) 112.708 0.166809
\(78\) −191.494 −0.277980
\(79\) 951.835 1.35557 0.677784 0.735261i \(-0.262941\pi\)
0.677784 + 0.735261i \(0.262941\pi\)
\(80\) −57.1165 −0.0798227
\(81\) 81.0000 0.111111
\(82\) 525.430 0.707610
\(83\) 735.221 0.972302 0.486151 0.873875i \(-0.338401\pi\)
0.486151 + 0.873875i \(0.338401\pi\)
\(84\) 170.955 0.222056
\(85\) 493.542 0.629789
\(86\) −3.69690 −0.00463543
\(87\) −447.278 −0.551187
\(88\) 232.948 0.282186
\(89\) 385.879 0.459585 0.229793 0.973240i \(-0.426195\pi\)
0.229793 + 0.973240i \(0.426195\pi\)
\(90\) −70.2699 −0.0823011
\(91\) 418.833 0.482480
\(92\) −343.110 −0.388823
\(93\) 164.216 0.183101
\(94\) −520.458 −0.571076
\(95\) 198.229 0.214083
\(96\) 561.764 0.597238
\(97\) −966.345 −1.01152 −0.505760 0.862674i \(-0.668788\pi\)
−0.505760 + 0.862674i \(0.668788\pi\)
\(98\) −371.673 −0.383109
\(99\) −99.0000 −0.100504
\(100\) −139.039 −0.139039
\(101\) 348.600 0.343436 0.171718 0.985146i \(-0.445068\pi\)
0.171718 + 0.985146i \(0.445068\pi\)
\(102\) −462.415 −0.448881
\(103\) −1536.38 −1.46975 −0.734873 0.678204i \(-0.762758\pi\)
−0.734873 + 0.678204i \(0.762758\pi\)
\(104\) 865.653 0.816195
\(105\) 153.693 0.142847
\(106\) 999.494 0.915844
\(107\) −779.180 −0.703983 −0.351991 0.936003i \(-0.614495\pi\)
−0.351991 + 0.936003i \(0.614495\pi\)
\(108\) −150.162 −0.133790
\(109\) −1501.79 −1.31968 −0.659842 0.751404i \(-0.729377\pi\)
−0.659842 + 0.751404i \(0.729377\pi\)
\(110\) 85.8854 0.0744441
\(111\) 134.682 0.115166
\(112\) −117.045 −0.0987478
\(113\) 170.000 0.141524 0.0707622 0.997493i \(-0.477457\pi\)
0.0707622 + 0.997493i \(0.477457\pi\)
\(114\) −185.727 −0.152587
\(115\) −308.466 −0.250127
\(116\) 829.187 0.663691
\(117\) −367.892 −0.290698
\(118\) −578.981 −0.451691
\(119\) 1011.39 0.779106
\(120\) 317.656 0.241649
\(121\) 121.000 0.0909091
\(122\) −1115.58 −0.827869
\(123\) 1009.44 0.739983
\(124\) −304.432 −0.220474
\(125\) −125.000 −0.0894427
\(126\) −144.000 −0.101814
\(127\) −1739.82 −1.21562 −0.607811 0.794082i \(-0.707952\pi\)
−0.607811 + 0.794082i \(0.707952\pi\)
\(128\) −1184.13 −0.817683
\(129\) −7.10235 −0.00484750
\(130\) 319.157 0.215323
\(131\) 312.837 0.208647 0.104323 0.994543i \(-0.466732\pi\)
0.104323 + 0.994543i \(0.466732\pi\)
\(132\) 183.531 0.121018
\(133\) 406.220 0.264840
\(134\) −632.405 −0.407698
\(135\) −135.000 −0.0860663
\(136\) 2090.35 1.31799
\(137\) −716.928 −0.447090 −0.223545 0.974694i \(-0.571763\pi\)
−0.223545 + 0.974694i \(0.571763\pi\)
\(138\) 289.011 0.178277
\(139\) −876.483 −0.534837 −0.267418 0.963581i \(-0.586171\pi\)
−0.267418 + 0.963581i \(0.586171\pi\)
\(140\) −284.924 −0.172004
\(141\) −999.886 −0.597203
\(142\) 1466.75 0.866811
\(143\) 449.646 0.262946
\(144\) 102.810 0.0594963
\(145\) 745.464 0.426948
\(146\) −566.172 −0.320937
\(147\) −714.045 −0.400636
\(148\) −249.680 −0.138673
\(149\) −2376.36 −1.30657 −0.653285 0.757112i \(-0.726610\pi\)
−0.653285 + 0.757112i \(0.726610\pi\)
\(150\) 117.116 0.0637501
\(151\) −92.8466 −0.0500381 −0.0250190 0.999687i \(-0.507965\pi\)
−0.0250190 + 0.999687i \(0.507965\pi\)
\(152\) 839.583 0.448021
\(153\) −888.375 −0.469417
\(154\) 176.000 0.0920941
\(155\) −273.693 −0.141829
\(156\) 682.017 0.350032
\(157\) −1881.24 −0.956301 −0.478150 0.878278i \(-0.658693\pi\)
−0.478150 + 0.878278i \(0.658693\pi\)
\(158\) 1486.34 0.748398
\(159\) 1920.19 0.957743
\(160\) −936.274 −0.462618
\(161\) −632.121 −0.309429
\(162\) 126.486 0.0613436
\(163\) −2465.49 −1.18474 −0.592369 0.805667i \(-0.701807\pi\)
−0.592369 + 0.805667i \(0.701807\pi\)
\(164\) −1871.35 −0.891022
\(165\) 165.000 0.0778499
\(166\) 1148.09 0.536800
\(167\) −1254.30 −0.581200 −0.290600 0.956845i \(-0.593855\pi\)
−0.290600 + 0.956845i \(0.593855\pi\)
\(168\) 650.955 0.298942
\(169\) −526.080 −0.239454
\(170\) 770.691 0.347702
\(171\) −356.813 −0.159568
\(172\) 13.1667 0.00583693
\(173\) −1206.71 −0.530314 −0.265157 0.964205i \(-0.585424\pi\)
−0.265157 + 0.964205i \(0.585424\pi\)
\(174\) −698.449 −0.304306
\(175\) −256.155 −0.110649
\(176\) −125.656 −0.0538164
\(177\) −1112.32 −0.472356
\(178\) 602.570 0.253733
\(179\) −1442.29 −0.602244 −0.301122 0.953586i \(-0.597361\pi\)
−0.301122 + 0.953586i \(0.597361\pi\)
\(180\) 250.270 0.103633
\(181\) 4261.81 1.75015 0.875076 0.483985i \(-0.160811\pi\)
0.875076 + 0.483985i \(0.160811\pi\)
\(182\) 654.030 0.266373
\(183\) −2143.22 −0.865744
\(184\) −1306.48 −0.523451
\(185\) −224.470 −0.0892072
\(186\) 256.432 0.101089
\(187\) 1085.79 0.424604
\(188\) 1853.64 0.719099
\(189\) −276.648 −0.106472
\(190\) 309.545 0.118194
\(191\) −852.223 −0.322852 −0.161426 0.986885i \(-0.551609\pi\)
−0.161426 + 0.986885i \(0.551609\pi\)
\(192\) 603.065 0.226680
\(193\) −2459.95 −0.917468 −0.458734 0.888574i \(-0.651697\pi\)
−0.458734 + 0.888574i \(0.651697\pi\)
\(194\) −1509.00 −0.558452
\(195\) 613.153 0.225173
\(196\) 1323.73 0.482410
\(197\) 3477.06 1.25751 0.628756 0.777602i \(-0.283564\pi\)
0.628756 + 0.777602i \(0.283564\pi\)
\(198\) −154.594 −0.0554874
\(199\) 3995.04 1.42312 0.711560 0.702626i \(-0.247989\pi\)
0.711560 + 0.702626i \(0.247989\pi\)
\(200\) −529.427 −0.187181
\(201\) −1214.95 −0.426350
\(202\) 544.358 0.189608
\(203\) 1527.64 0.528173
\(204\) 1646.91 0.565231
\(205\) −1682.40 −0.573188
\(206\) −2399.14 −0.811436
\(207\) 555.239 0.186434
\(208\) −466.949 −0.155659
\(209\) 436.104 0.144335
\(210\) 240.000 0.0788646
\(211\) 1046.13 0.341319 0.170660 0.985330i \(-0.445410\pi\)
0.170660 + 0.985330i \(0.445410\pi\)
\(212\) −3559.75 −1.15323
\(213\) 2817.88 0.906468
\(214\) −1216.73 −0.388664
\(215\) 11.8373 0.00375486
\(216\) −571.781 −0.180115
\(217\) −560.864 −0.175456
\(218\) −2345.13 −0.728587
\(219\) −1087.71 −0.335619
\(220\) −305.885 −0.0937400
\(221\) 4034.89 1.22813
\(222\) 210.313 0.0635823
\(223\) −506.265 −0.152027 −0.0760135 0.997107i \(-0.524219\pi\)
−0.0760135 + 0.997107i \(0.524219\pi\)
\(224\) −1918.65 −0.572300
\(225\) 225.000 0.0666667
\(226\) 265.464 0.0781345
\(227\) −4286.29 −1.25326 −0.626632 0.779315i \(-0.715567\pi\)
−0.626632 + 0.779315i \(0.715567\pi\)
\(228\) 661.477 0.192138
\(229\) 5709.37 1.64754 0.823769 0.566926i \(-0.191867\pi\)
0.823769 + 0.566926i \(0.191867\pi\)
\(230\) −481.686 −0.138093
\(231\) 338.125 0.0963073
\(232\) 3157.35 0.893492
\(233\) 2946.09 0.828348 0.414174 0.910198i \(-0.364071\pi\)
0.414174 + 0.910198i \(0.364071\pi\)
\(234\) −574.483 −0.160492
\(235\) 1666.48 0.462591
\(236\) 2062.07 0.568769
\(237\) 2855.51 0.782637
\(238\) 1579.33 0.430139
\(239\) −2078.89 −0.562646 −0.281323 0.959613i \(-0.590773\pi\)
−0.281323 + 0.959613i \(0.590773\pi\)
\(240\) −171.349 −0.0460856
\(241\) 1853.37 0.495378 0.247689 0.968840i \(-0.420329\pi\)
0.247689 + 0.968840i \(0.420329\pi\)
\(242\) 188.948 0.0501902
\(243\) 243.000 0.0641500
\(244\) 3973.20 1.04245
\(245\) 1190.08 0.310331
\(246\) 1576.29 0.408539
\(247\) 1620.60 0.417475
\(248\) −1159.20 −0.296813
\(249\) 2205.66 0.561359
\(250\) −195.194 −0.0493806
\(251\) −2358.39 −0.593068 −0.296534 0.955022i \(-0.595831\pi\)
−0.296534 + 0.955022i \(0.595831\pi\)
\(252\) 512.864 0.128204
\(253\) −678.625 −0.168635
\(254\) −2716.82 −0.671135
\(255\) 1480.62 0.363609
\(256\) −3457.26 −0.844057
\(257\) 5519.25 1.33962 0.669809 0.742534i \(-0.266376\pi\)
0.669809 + 0.742534i \(0.266376\pi\)
\(258\) −11.0907 −0.00267627
\(259\) −459.993 −0.110357
\(260\) −1136.70 −0.271134
\(261\) −1341.84 −0.318228
\(262\) 488.512 0.115192
\(263\) 2259.65 0.529795 0.264898 0.964277i \(-0.414662\pi\)
0.264898 + 0.964277i \(0.414662\pi\)
\(264\) 698.844 0.162920
\(265\) −3200.32 −0.741865
\(266\) 634.333 0.146216
\(267\) 1157.64 0.265342
\(268\) 2252.34 0.513373
\(269\) 7039.53 1.59557 0.797783 0.602944i \(-0.206006\pi\)
0.797783 + 0.602944i \(0.206006\pi\)
\(270\) −210.810 −0.0475165
\(271\) 5155.08 1.15553 0.577765 0.816203i \(-0.303925\pi\)
0.577765 + 0.816203i \(0.303925\pi\)
\(272\) −1127.57 −0.251357
\(273\) 1256.50 0.278560
\(274\) −1119.52 −0.246835
\(275\) −275.000 −0.0603023
\(276\) −1029.33 −0.224487
\(277\) 9074.52 1.96836 0.984179 0.177175i \(-0.0566960\pi\)
0.984179 + 0.177175i \(0.0566960\pi\)
\(278\) −1368.67 −0.295279
\(279\) 492.648 0.105713
\(280\) −1084.92 −0.231559
\(281\) −3407.79 −0.723459 −0.361729 0.932283i \(-0.617814\pi\)
−0.361729 + 0.932283i \(0.617814\pi\)
\(282\) −1561.38 −0.329711
\(283\) −8827.73 −1.85425 −0.927127 0.374746i \(-0.877730\pi\)
−0.927127 + 0.374746i \(0.877730\pi\)
\(284\) −5223.92 −1.09149
\(285\) 594.688 0.123601
\(286\) 702.146 0.145170
\(287\) −3447.64 −0.709085
\(288\) 1685.29 0.344815
\(289\) 4830.33 0.983174
\(290\) 1164.08 0.235715
\(291\) −2899.03 −0.584001
\(292\) 2016.45 0.404123
\(293\) −4528.29 −0.902886 −0.451443 0.892300i \(-0.649091\pi\)
−0.451443 + 0.892300i \(0.649091\pi\)
\(294\) −1115.02 −0.221188
\(295\) 1853.86 0.365885
\(296\) −950.722 −0.186688
\(297\) −297.000 −0.0580259
\(298\) −3710.81 −0.721347
\(299\) −2521.83 −0.487762
\(300\) −417.116 −0.0802741
\(301\) 24.2574 0.00464509
\(302\) −144.985 −0.0276256
\(303\) 1045.80 0.198283
\(304\) −452.886 −0.0854434
\(305\) 3572.03 0.670602
\(306\) −1387.24 −0.259162
\(307\) 568.106 0.105614 0.0528071 0.998605i \(-0.483183\pi\)
0.0528071 + 0.998605i \(0.483183\pi\)
\(308\) −626.833 −0.115965
\(309\) −4609.14 −0.848559
\(310\) −427.386 −0.0783029
\(311\) −6853.59 −1.24962 −0.624809 0.780778i \(-0.714823\pi\)
−0.624809 + 0.780778i \(0.714823\pi\)
\(312\) 2596.96 0.471230
\(313\) −1138.92 −0.205673 −0.102837 0.994698i \(-0.532792\pi\)
−0.102837 + 0.994698i \(0.532792\pi\)
\(314\) −2937.65 −0.527966
\(315\) 461.080 0.0824727
\(316\) −5293.68 −0.942382
\(317\) 3207.48 0.568297 0.284148 0.958780i \(-0.408289\pi\)
0.284148 + 0.958780i \(0.408289\pi\)
\(318\) 2998.48 0.528763
\(319\) 1640.02 0.287848
\(320\) −1005.11 −0.175585
\(321\) −2337.54 −0.406445
\(322\) −987.091 −0.170834
\(323\) 3913.37 0.674136
\(324\) −450.486 −0.0772438
\(325\) −1021.92 −0.174419
\(326\) −3850.00 −0.654085
\(327\) −4505.37 −0.761920
\(328\) −7125.65 −1.19954
\(329\) 3415.02 0.572267
\(330\) 257.656 0.0429803
\(331\) −9135.12 −1.51695 −0.758477 0.651700i \(-0.774056\pi\)
−0.758477 + 0.651700i \(0.774056\pi\)
\(332\) −4088.97 −0.675938
\(333\) 404.045 0.0664911
\(334\) −1958.65 −0.320876
\(335\) 2024.92 0.330249
\(336\) −351.136 −0.0570121
\(337\) −3470.05 −0.560907 −0.280453 0.959868i \(-0.590485\pi\)
−0.280453 + 0.959868i \(0.590485\pi\)
\(338\) −821.501 −0.132200
\(339\) 510.000 0.0817091
\(340\) −2744.86 −0.437826
\(341\) −602.125 −0.0956214
\(342\) −557.182 −0.0880963
\(343\) 5953.20 0.937151
\(344\) 50.1357 0.00785795
\(345\) −925.398 −0.144411
\(346\) −1884.34 −0.292782
\(347\) −89.3315 −0.0138201 −0.00691004 0.999976i \(-0.502200\pi\)
−0.00691004 + 0.999976i \(0.502200\pi\)
\(348\) 2487.56 0.383182
\(349\) −149.375 −0.0229107 −0.0114554 0.999934i \(-0.503646\pi\)
−0.0114554 + 0.999934i \(0.503646\pi\)
\(350\) −400.000 −0.0610883
\(351\) −1103.68 −0.167834
\(352\) −2059.80 −0.311897
\(353\) 7867.64 1.18627 0.593133 0.805104i \(-0.297891\pi\)
0.593133 + 0.805104i \(0.297891\pi\)
\(354\) −1736.94 −0.260784
\(355\) −4696.46 −0.702147
\(356\) −2146.09 −0.319501
\(357\) 3034.16 0.449817
\(358\) −2252.21 −0.332494
\(359\) 4974.22 0.731279 0.365639 0.930757i \(-0.380850\pi\)
0.365639 + 0.930757i \(0.380850\pi\)
\(360\) 952.969 0.139516
\(361\) −5287.21 −0.770842
\(362\) 6655.04 0.966246
\(363\) 363.000 0.0524864
\(364\) −2329.36 −0.335417
\(365\) 1812.85 0.259970
\(366\) −3346.74 −0.477970
\(367\) −13266.7 −1.88696 −0.943479 0.331433i \(-0.892468\pi\)
−0.943479 + 0.331433i \(0.892468\pi\)
\(368\) 704.739 0.0998290
\(369\) 3028.31 0.427229
\(370\) −350.521 −0.0492506
\(371\) −6558.23 −0.917754
\(372\) −913.295 −0.127291
\(373\) −4632.77 −0.643099 −0.321549 0.946893i \(-0.604204\pi\)
−0.321549 + 0.946893i \(0.604204\pi\)
\(374\) 1695.52 0.234421
\(375\) −375.000 −0.0516398
\(376\) 7058.22 0.968085
\(377\) 6094.45 0.832573
\(378\) −432.000 −0.0587822
\(379\) 6503.31 0.881406 0.440703 0.897653i \(-0.354729\pi\)
0.440703 + 0.897653i \(0.354729\pi\)
\(380\) −1102.46 −0.148829
\(381\) −5219.45 −0.701839
\(382\) −1330.79 −0.178244
\(383\) 12734.5 1.69897 0.849484 0.527614i \(-0.176913\pi\)
0.849484 + 0.527614i \(0.176913\pi\)
\(384\) −3552.39 −0.472090
\(385\) −563.542 −0.0745993
\(386\) −3841.35 −0.506527
\(387\) −21.3071 −0.00279870
\(388\) 5374.38 0.703203
\(389\) 12024.6 1.56728 0.783639 0.621216i \(-0.213361\pi\)
0.783639 + 0.621216i \(0.213361\pi\)
\(390\) 957.471 0.124317
\(391\) −6089.63 −0.787636
\(392\) 5040.47 0.649444
\(393\) 938.511 0.120462
\(394\) 5429.61 0.694263
\(395\) −4759.18 −0.606228
\(396\) 550.594 0.0698696
\(397\) −5223.65 −0.660371 −0.330186 0.943916i \(-0.607111\pi\)
−0.330186 + 0.943916i \(0.607111\pi\)
\(398\) 6238.46 0.785693
\(399\) 1218.66 0.152905
\(400\) 285.582 0.0356978
\(401\) 9648.18 1.20151 0.600757 0.799432i \(-0.294866\pi\)
0.600757 + 0.799432i \(0.294866\pi\)
\(402\) −1897.22 −0.235384
\(403\) −2237.55 −0.276576
\(404\) −1938.76 −0.238755
\(405\) −405.000 −0.0496904
\(406\) 2385.48 0.291600
\(407\) −493.833 −0.0601435
\(408\) 6271.06 0.760941
\(409\) −2010.47 −0.243060 −0.121530 0.992588i \(-0.538780\pi\)
−0.121530 + 0.992588i \(0.538780\pi\)
\(410\) −2627.15 −0.316453
\(411\) −2150.78 −0.258127
\(412\) 8544.65 1.02176
\(413\) 3799.02 0.452633
\(414\) 867.034 0.102929
\(415\) −3676.11 −0.434827
\(416\) −7654.39 −0.902133
\(417\) −2629.45 −0.308788
\(418\) 681.000 0.0796861
\(419\) 4435.27 0.517129 0.258565 0.965994i \(-0.416750\pi\)
0.258565 + 0.965994i \(0.416750\pi\)
\(420\) −854.773 −0.0993063
\(421\) 15217.9 1.76170 0.880852 0.473392i \(-0.156971\pi\)
0.880852 + 0.473392i \(0.156971\pi\)
\(422\) 1633.58 0.188440
\(423\) −2999.66 −0.344795
\(424\) −13554.7 −1.55253
\(425\) −2467.71 −0.281650
\(426\) 4400.26 0.500454
\(427\) 7319.95 0.829595
\(428\) 4333.45 0.489405
\(429\) 1348.94 0.151812
\(430\) 18.4845 0.00207303
\(431\) −5622.11 −0.628324 −0.314162 0.949369i \(-0.601723\pi\)
−0.314162 + 0.949369i \(0.601723\pi\)
\(432\) 308.429 0.0343502
\(433\) −14306.3 −1.58780 −0.793898 0.608051i \(-0.791951\pi\)
−0.793898 + 0.608051i \(0.791951\pi\)
\(434\) −875.818 −0.0968678
\(435\) 2236.39 0.246498
\(436\) 8352.29 0.917436
\(437\) −2445.88 −0.267740
\(438\) −1698.52 −0.185293
\(439\) 4384.20 0.476643 0.238322 0.971186i \(-0.423403\pi\)
0.238322 + 0.971186i \(0.423403\pi\)
\(440\) −1164.74 −0.126197
\(441\) −2142.14 −0.231307
\(442\) 6300.69 0.678039
\(443\) −10090.0 −1.08214 −0.541071 0.840977i \(-0.681981\pi\)
−0.541071 + 0.840977i \(0.681981\pi\)
\(444\) −749.040 −0.0800627
\(445\) −1929.39 −0.205533
\(446\) −790.560 −0.0839330
\(447\) −7129.07 −0.754348
\(448\) −2059.71 −0.217215
\(449\) 9582.52 1.00719 0.503594 0.863941i \(-0.332011\pi\)
0.503594 + 0.863941i \(0.332011\pi\)
\(450\) 351.349 0.0368062
\(451\) −3701.27 −0.386444
\(452\) −945.464 −0.0983869
\(453\) −278.540 −0.0288895
\(454\) −6693.27 −0.691918
\(455\) −2094.17 −0.215772
\(456\) 2518.75 0.258665
\(457\) 9999.34 1.02352 0.511761 0.859128i \(-0.328993\pi\)
0.511761 + 0.859128i \(0.328993\pi\)
\(458\) 8915.49 0.909593
\(459\) −2665.12 −0.271018
\(460\) 1715.55 0.173887
\(461\) −11115.8 −1.12302 −0.561512 0.827468i \(-0.689780\pi\)
−0.561512 + 0.827468i \(0.689780\pi\)
\(462\) 528.000 0.0531705
\(463\) −1567.16 −0.157305 −0.0786524 0.996902i \(-0.525062\pi\)
−0.0786524 + 0.996902i \(0.525062\pi\)
\(464\) −1703.13 −0.170401
\(465\) −821.080 −0.0818853
\(466\) 4600.48 0.457325
\(467\) 12648.8 1.25335 0.626675 0.779281i \(-0.284415\pi\)
0.626675 + 0.779281i \(0.284415\pi\)
\(468\) 2046.05 0.202091
\(469\) 4149.56 0.408548
\(470\) 2602.29 0.255393
\(471\) −5643.72 −0.552120
\(472\) 7851.88 0.765704
\(473\) 26.0420 0.00253152
\(474\) 4459.02 0.432088
\(475\) −991.146 −0.0957408
\(476\) −5624.88 −0.541630
\(477\) 5760.58 0.552953
\(478\) −3246.30 −0.310633
\(479\) −10719.2 −1.02249 −0.511247 0.859434i \(-0.670816\pi\)
−0.511247 + 0.859434i \(0.670816\pi\)
\(480\) −2808.82 −0.267093
\(481\) −1835.12 −0.173959
\(482\) 2894.14 0.273494
\(483\) −1896.36 −0.178649
\(484\) −672.948 −0.0631995
\(485\) 4831.72 0.452365
\(486\) 379.457 0.0354167
\(487\) 7161.20 0.666335 0.333167 0.942868i \(-0.391883\pi\)
0.333167 + 0.942868i \(0.391883\pi\)
\(488\) 15129.0 1.40340
\(489\) −7396.48 −0.684009
\(490\) 1858.37 0.171331
\(491\) 14567.3 1.33893 0.669463 0.742845i \(-0.266524\pi\)
0.669463 + 0.742845i \(0.266524\pi\)
\(492\) −5614.04 −0.514432
\(493\) 14716.7 1.34444
\(494\) 2530.65 0.230485
\(495\) 495.000 0.0449467
\(496\) 625.295 0.0566060
\(497\) −9624.18 −0.868619
\(498\) 3444.26 0.309922
\(499\) −4638.99 −0.416172 −0.208086 0.978111i \(-0.566723\pi\)
−0.208086 + 0.978111i \(0.566723\pi\)
\(500\) 695.194 0.0621801
\(501\) −3762.89 −0.335556
\(502\) −3682.75 −0.327428
\(503\) −12206.3 −1.08201 −0.541006 0.841019i \(-0.681956\pi\)
−0.541006 + 0.841019i \(0.681956\pi\)
\(504\) 1952.86 0.172594
\(505\) −1743.00 −0.153589
\(506\) −1059.71 −0.0931024
\(507\) −1578.24 −0.138249
\(508\) 9676.09 0.845093
\(509\) 10018.6 0.872427 0.436214 0.899843i \(-0.356319\pi\)
0.436214 + 0.899843i \(0.356319\pi\)
\(510\) 2312.07 0.200746
\(511\) 3714.97 0.321606
\(512\) 4074.36 0.351686
\(513\) −1070.44 −0.0921267
\(514\) 8618.61 0.739592
\(515\) 7681.89 0.657291
\(516\) 39.5001 0.00336995
\(517\) 3666.25 0.311879
\(518\) −718.303 −0.0609274
\(519\) −3620.12 −0.306177
\(520\) −4328.27 −0.365014
\(521\) 1054.72 0.0886916 0.0443458 0.999016i \(-0.485880\pi\)
0.0443458 + 0.999016i \(0.485880\pi\)
\(522\) −2095.35 −0.175691
\(523\) −16234.2 −1.35730 −0.678652 0.734460i \(-0.737436\pi\)
−0.678652 + 0.734460i \(0.737436\pi\)
\(524\) −1739.86 −0.145050
\(525\) −768.466 −0.0638830
\(526\) 3528.57 0.292496
\(527\) −5403.16 −0.446613
\(528\) −376.969 −0.0310709
\(529\) −8360.95 −0.687183
\(530\) −4997.47 −0.409578
\(531\) −3336.95 −0.272715
\(532\) −2259.21 −0.184115
\(533\) −13754.2 −1.11775
\(534\) 1807.71 0.146493
\(535\) 3895.90 0.314831
\(536\) 8576.40 0.691127
\(537\) −4326.86 −0.347706
\(538\) 10992.6 0.880900
\(539\) 2618.17 0.209225
\(540\) 750.810 0.0598328
\(541\) 675.936 0.0537167 0.0268584 0.999639i \(-0.491450\pi\)
0.0268584 + 0.999639i \(0.491450\pi\)
\(542\) 8049.93 0.637960
\(543\) 12785.4 1.01045
\(544\) −18483.6 −1.45676
\(545\) 7508.96 0.590181
\(546\) 1962.09 0.153791
\(547\) 13058.2 1.02071 0.510355 0.859964i \(-0.329514\pi\)
0.510355 + 0.859964i \(0.329514\pi\)
\(548\) 3987.23 0.310814
\(549\) −6429.65 −0.499837
\(550\) −429.427 −0.0332924
\(551\) 5910.91 0.457011
\(552\) −3919.44 −0.302215
\(553\) −9752.70 −0.749959
\(554\) 14170.3 1.08672
\(555\) −673.409 −0.0515038
\(556\) 4874.61 0.371815
\(557\) −6710.48 −0.510471 −0.255236 0.966879i \(-0.582153\pi\)
−0.255236 + 0.966879i \(0.582153\pi\)
\(558\) 769.295 0.0583636
\(559\) 96.7741 0.00732219
\(560\) 585.227 0.0441614
\(561\) 3257.37 0.245145
\(562\) −5321.45 −0.399416
\(563\) −20820.5 −1.55858 −0.779288 0.626666i \(-0.784419\pi\)
−0.779288 + 0.626666i \(0.784419\pi\)
\(564\) 5560.92 0.415172
\(565\) −850.000 −0.0632916
\(566\) −13785.0 −1.02372
\(567\) −829.943 −0.0614715
\(568\) −19891.5 −1.46941
\(569\) 3251.08 0.239530 0.119765 0.992802i \(-0.461786\pi\)
0.119765 + 0.992802i \(0.461786\pi\)
\(570\) 928.636 0.0682391
\(571\) −4637.50 −0.339883 −0.169941 0.985454i \(-0.554358\pi\)
−0.169941 + 0.985454i \(0.554358\pi\)
\(572\) −2500.73 −0.182798
\(573\) −2556.67 −0.186399
\(574\) −5383.67 −0.391481
\(575\) 1542.33 0.111860
\(576\) 1809.20 0.130874
\(577\) 14462.4 1.04346 0.521730 0.853111i \(-0.325287\pi\)
0.521730 + 0.853111i \(0.325287\pi\)
\(578\) 7542.82 0.542803
\(579\) −7379.86 −0.529700
\(580\) −4145.94 −0.296812
\(581\) −7533.23 −0.537920
\(582\) −4526.99 −0.322423
\(583\) −7040.71 −0.500165
\(584\) 7678.18 0.544050
\(585\) 1839.46 0.130004
\(586\) −7071.17 −0.498476
\(587\) −22759.7 −1.60033 −0.800166 0.599779i \(-0.795255\pi\)
−0.800166 + 0.599779i \(0.795255\pi\)
\(588\) 3971.20 0.278520
\(589\) −2170.16 −0.151816
\(590\) 2894.91 0.202002
\(591\) 10431.2 0.726025
\(592\) 512.836 0.0356038
\(593\) −14956.4 −1.03573 −0.517864 0.855463i \(-0.673273\pi\)
−0.517864 + 0.855463i \(0.673273\pi\)
\(594\) −463.781 −0.0320356
\(595\) −5056.93 −0.348427
\(596\) 13216.2 0.908319
\(597\) 11985.1 0.821638
\(598\) −3937.96 −0.269290
\(599\) 2150.77 0.146708 0.0733539 0.997306i \(-0.476630\pi\)
0.0733539 + 0.997306i \(0.476630\pi\)
\(600\) −1588.28 −0.108069
\(601\) 27759.8 1.88410 0.942050 0.335472i \(-0.108896\pi\)
0.942050 + 0.335472i \(0.108896\pi\)
\(602\) 37.8792 0.00256452
\(603\) −3644.86 −0.246153
\(604\) 516.371 0.0347862
\(605\) −605.000 −0.0406558
\(606\) 1633.07 0.109470
\(607\) −10991.5 −0.734974 −0.367487 0.930029i \(-0.619782\pi\)
−0.367487 + 0.930029i \(0.619782\pi\)
\(608\) −7423.87 −0.495194
\(609\) 4582.91 0.304941
\(610\) 5577.91 0.370234
\(611\) 13624.1 0.902081
\(612\) 4940.74 0.326336
\(613\) 10646.1 0.701457 0.350728 0.936477i \(-0.385934\pi\)
0.350728 + 0.936477i \(0.385934\pi\)
\(614\) 887.128 0.0583088
\(615\) −5047.19 −0.330930
\(616\) −2386.83 −0.156117
\(617\) 7199.92 0.469786 0.234893 0.972021i \(-0.424526\pi\)
0.234893 + 0.972021i \(0.424526\pi\)
\(618\) −7197.41 −0.468483
\(619\) 12186.9 0.791332 0.395666 0.918395i \(-0.370514\pi\)
0.395666 + 0.918395i \(0.370514\pi\)
\(620\) 1522.16 0.0985990
\(621\) 1665.72 0.107637
\(622\) −10702.2 −0.689905
\(623\) −3953.80 −0.254262
\(624\) −1400.85 −0.0898698
\(625\) 625.000 0.0400000
\(626\) −1778.49 −0.113551
\(627\) 1308.31 0.0833317
\(628\) 10462.6 0.664814
\(629\) −4431.40 −0.280909
\(630\) 720.000 0.0455325
\(631\) 7370.64 0.465009 0.232505 0.972595i \(-0.425308\pi\)
0.232505 + 0.972595i \(0.425308\pi\)
\(632\) −20157.1 −1.26868
\(633\) 3138.38 0.197061
\(634\) 5008.65 0.313752
\(635\) 8699.09 0.543642
\(636\) −10679.3 −0.665818
\(637\) 9729.32 0.605164
\(638\) 2560.98 0.158919
\(639\) 8453.63 0.523349
\(640\) 5920.66 0.365679
\(641\) −25014.9 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(642\) −3650.19 −0.224395
\(643\) −21668.2 −1.32894 −0.664472 0.747313i \(-0.731343\pi\)
−0.664472 + 0.747313i \(0.731343\pi\)
\(644\) 3515.58 0.215113
\(645\) 35.5118 0.00216787
\(646\) 6110.94 0.372185
\(647\) −27625.3 −1.67861 −0.839305 0.543661i \(-0.817038\pi\)
−0.839305 + 0.543661i \(0.817038\pi\)
\(648\) −1715.34 −0.103989
\(649\) 4078.50 0.246680
\(650\) −1595.79 −0.0962952
\(651\) −1682.59 −0.101299
\(652\) 13712.0 0.823623
\(653\) −14314.0 −0.857810 −0.428905 0.903350i \(-0.641101\pi\)
−0.428905 + 0.903350i \(0.641101\pi\)
\(654\) −7035.38 −0.420650
\(655\) −1564.19 −0.0933096
\(656\) 3843.70 0.228767
\(657\) −3263.13 −0.193770
\(658\) 5332.73 0.315944
\(659\) −28327.8 −1.67450 −0.837249 0.546822i \(-0.815837\pi\)
−0.837249 + 0.546822i \(0.815837\pi\)
\(660\) −917.656 −0.0541208
\(661\) −32190.9 −1.89422 −0.947112 0.320905i \(-0.896013\pi\)
−0.947112 + 0.320905i \(0.896013\pi\)
\(662\) −14265.0 −0.837498
\(663\) 12104.7 0.709059
\(664\) −15569.8 −0.909981
\(665\) −2031.10 −0.118440
\(666\) 630.938 0.0367092
\(667\) −9198.01 −0.533955
\(668\) 6975.84 0.404047
\(669\) −1518.80 −0.0877729
\(670\) 3162.03 0.182328
\(671\) 7858.46 0.452120
\(672\) −5755.95 −0.330418
\(673\) −6207.38 −0.355538 −0.177769 0.984072i \(-0.556888\pi\)
−0.177769 + 0.984072i \(0.556888\pi\)
\(674\) −5418.66 −0.309672
\(675\) 675.000 0.0384900
\(676\) 2925.82 0.166467
\(677\) 28831.1 1.63674 0.818368 0.574695i \(-0.194879\pi\)
0.818368 + 0.574695i \(0.194879\pi\)
\(678\) 796.392 0.0451110
\(679\) 9901.37 0.559617
\(680\) −10451.8 −0.589422
\(681\) −12858.9 −0.723573
\(682\) −940.250 −0.0527918
\(683\) 3193.10 0.178888 0.0894441 0.995992i \(-0.471491\pi\)
0.0894441 + 0.995992i \(0.471491\pi\)
\(684\) 1984.43 0.110931
\(685\) 3584.64 0.199945
\(686\) 9296.24 0.517394
\(687\) 17128.1 0.951206
\(688\) −27.0441 −0.00149861
\(689\) −26163.8 −1.44668
\(690\) −1445.06 −0.0797281
\(691\) 7682.49 0.422946 0.211473 0.977384i \(-0.432174\pi\)
0.211473 + 0.977384i \(0.432174\pi\)
\(692\) 6711.17 0.368671
\(693\) 1014.37 0.0556031
\(694\) −139.496 −0.00762996
\(695\) 4382.41 0.239186
\(696\) 9472.05 0.515858
\(697\) −33213.3 −1.80494
\(698\) −233.256 −0.0126488
\(699\) 8838.28 0.478247
\(700\) 1424.62 0.0769223
\(701\) 26551.6 1.43058 0.715292 0.698825i \(-0.246293\pi\)
0.715292 + 0.698825i \(0.246293\pi\)
\(702\) −1723.45 −0.0926601
\(703\) −1779.86 −0.0954887
\(704\) −2211.24 −0.118380
\(705\) 4999.43 0.267077
\(706\) 12285.7 0.654928
\(707\) −3571.83 −0.190004
\(708\) 6186.22 0.328379
\(709\) −16304.6 −0.863655 −0.431828 0.901956i \(-0.642131\pi\)
−0.431828 + 0.901956i \(0.642131\pi\)
\(710\) −7333.77 −0.387650
\(711\) 8566.52 0.451856
\(712\) −8171.79 −0.430127
\(713\) 3377.00 0.177377
\(714\) 4738.00 0.248341
\(715\) −2248.23 −0.117593
\(716\) 8021.36 0.418676
\(717\) −6236.68 −0.324844
\(718\) 7767.50 0.403733
\(719\) −3973.62 −0.206107 −0.103053 0.994676i \(-0.532861\pi\)
−0.103053 + 0.994676i \(0.532861\pi\)
\(720\) −514.048 −0.0266076
\(721\) 15742.1 0.813128
\(722\) −8256.25 −0.425576
\(723\) 5560.11 0.286007
\(724\) −23702.3 −1.21670
\(725\) −3727.32 −0.190937
\(726\) 566.844 0.0289773
\(727\) −10780.4 −0.549961 −0.274980 0.961450i \(-0.588671\pi\)
−0.274980 + 0.961450i \(0.588671\pi\)
\(728\) −8869.67 −0.451555
\(729\) 729.000 0.0370370
\(730\) 2830.86 0.143527
\(731\) 233.687 0.0118238
\(732\) 11919.6 0.601860
\(733\) −9211.46 −0.464165 −0.232083 0.972696i \(-0.574554\pi\)
−0.232083 + 0.972696i \(0.574554\pi\)
\(734\) −20716.6 −1.04177
\(735\) 3570.23 0.179170
\(736\) 11552.3 0.578566
\(737\) 4454.83 0.222654
\(738\) 4728.87 0.235870
\(739\) 11084.7 0.551768 0.275884 0.961191i \(-0.411029\pi\)
0.275884 + 0.961191i \(0.411029\pi\)
\(740\) 1248.40 0.0620163
\(741\) 4861.80 0.241029
\(742\) −10241.0 −0.506685
\(743\) −27420.4 −1.35391 −0.676955 0.736024i \(-0.736701\pi\)
−0.676955 + 0.736024i \(0.736701\pi\)
\(744\) −3477.61 −0.171365
\(745\) 11881.8 0.584316
\(746\) −7234.32 −0.355050
\(747\) 6616.99 0.324101
\(748\) −6038.69 −0.295182
\(749\) 7983.64 0.389474
\(750\) −585.582 −0.0285099
\(751\) −11290.8 −0.548614 −0.274307 0.961642i \(-0.588448\pi\)
−0.274307 + 0.961642i \(0.588448\pi\)
\(752\) −3807.33 −0.184626
\(753\) −7075.16 −0.342408
\(754\) 9516.81 0.459657
\(755\) 464.233 0.0223777
\(756\) 1538.59 0.0740185
\(757\) 3739.19 0.179528 0.0897642 0.995963i \(-0.471389\pi\)
0.0897642 + 0.995963i \(0.471389\pi\)
\(758\) 10155.3 0.486617
\(759\) −2035.87 −0.0973617
\(760\) −4197.92 −0.200361
\(761\) −15621.1 −0.744107 −0.372053 0.928211i \(-0.621346\pi\)
−0.372053 + 0.928211i \(0.621346\pi\)
\(762\) −8150.45 −0.387480
\(763\) 15387.7 0.730106
\(764\) 4739.69 0.224445
\(765\) 4441.87 0.209930
\(766\) 19885.7 0.937987
\(767\) 15156.0 0.713498
\(768\) −10371.8 −0.487317
\(769\) 40241.7 1.88706 0.943531 0.331284i \(-0.107482\pi\)
0.943531 + 0.331284i \(0.107482\pi\)
\(770\) −880.000 −0.0411857
\(771\) 16557.8 0.773428
\(772\) 13681.2 0.637818
\(773\) 22821.4 1.06187 0.530936 0.847412i \(-0.321840\pi\)
0.530936 + 0.847412i \(0.321840\pi\)
\(774\) −33.2721 −0.00154514
\(775\) 1368.47 0.0634281
\(776\) 20464.4 0.946685
\(777\) −1379.98 −0.0637148
\(778\) 18777.0 0.865282
\(779\) −13340.0 −0.613549
\(780\) −3410.09 −0.156539
\(781\) −10332.2 −0.473387
\(782\) −9509.28 −0.434848
\(783\) −4025.51 −0.183729
\(784\) −2718.92 −0.123857
\(785\) 9406.19 0.427671
\(786\) 1465.53 0.0665062
\(787\) −29454.3 −1.33410 −0.667048 0.745015i \(-0.732442\pi\)
−0.667048 + 0.745015i \(0.732442\pi\)
\(788\) −19337.8 −0.874216
\(789\) 6778.96 0.305878
\(790\) −7431.70 −0.334694
\(791\) −1741.86 −0.0782974
\(792\) 2096.53 0.0940619
\(793\) 29202.7 1.30771
\(794\) −8157.00 −0.364586
\(795\) −9600.97 −0.428316
\(796\) −22218.6 −0.989344
\(797\) −27440.3 −1.21955 −0.609777 0.792573i \(-0.708741\pi\)
−0.609777 + 0.792573i \(0.708741\pi\)
\(798\) 1903.00 0.0844179
\(799\) 32899.0 1.45668
\(800\) 4681.37 0.206889
\(801\) 3472.91 0.153195
\(802\) 15066.1 0.663346
\(803\) 3988.27 0.175272
\(804\) 6757.03 0.296396
\(805\) 3160.61 0.138381
\(806\) −3494.05 −0.152695
\(807\) 21118.6 0.921201
\(808\) −7382.34 −0.321423
\(809\) −5060.18 −0.219909 −0.109954 0.993937i \(-0.535070\pi\)
−0.109954 + 0.993937i \(0.535070\pi\)
\(810\) −632.429 −0.0274337
\(811\) 30480.1 1.31973 0.659865 0.751384i \(-0.270613\pi\)
0.659865 + 0.751384i \(0.270613\pi\)
\(812\) −8496.03 −0.367183
\(813\) 15465.2 0.667146
\(814\) −771.146 −0.0332048
\(815\) 12327.5 0.529831
\(816\) −3382.72 −0.145121
\(817\) 93.8596 0.00401925
\(818\) −3139.46 −0.134192
\(819\) 3769.50 0.160827
\(820\) 9356.73 0.398477
\(821\) 37909.0 1.61149 0.805745 0.592263i \(-0.201765\pi\)
0.805745 + 0.592263i \(0.201765\pi\)
\(822\) −3358.56 −0.142510
\(823\) 23636.0 1.00109 0.500546 0.865710i \(-0.333133\pi\)
0.500546 + 0.865710i \(0.333133\pi\)
\(824\) 32536.0 1.37554
\(825\) −825.000 −0.0348155
\(826\) 5932.36 0.249895
\(827\) −42634.3 −1.79267 −0.896336 0.443376i \(-0.853781\pi\)
−0.896336 + 0.443376i \(0.853781\pi\)
\(828\) −3087.99 −0.129608
\(829\) −45152.5 −1.89169 −0.945845 0.324619i \(-0.894764\pi\)
−0.945845 + 0.324619i \(0.894764\pi\)
\(830\) −5740.44 −0.240064
\(831\) 27223.6 1.13643
\(832\) −8217.15 −0.342402
\(833\) 23494.1 0.977217
\(834\) −4106.02 −0.170480
\(835\) 6271.49 0.259921
\(836\) −2425.42 −0.100341
\(837\) 1477.94 0.0610337
\(838\) 6925.91 0.285503
\(839\) 30431.5 1.25222 0.626110 0.779734i \(-0.284646\pi\)
0.626110 + 0.779734i \(0.284646\pi\)
\(840\) −3254.77 −0.133691
\(841\) −2160.34 −0.0885784
\(842\) 23763.6 0.972623
\(843\) −10223.4 −0.417689
\(844\) −5818.09 −0.237283
\(845\) 2630.40 0.107087
\(846\) −4684.13 −0.190359
\(847\) −1239.79 −0.0502949
\(848\) 7311.64 0.296088
\(849\) −26483.2 −1.07055
\(850\) −3853.46 −0.155497
\(851\) 2769.65 0.111566
\(852\) −15671.8 −0.630171
\(853\) 10367.2 0.416139 0.208070 0.978114i \(-0.433282\pi\)
0.208070 + 0.978114i \(0.433282\pi\)
\(854\) 11430.5 0.458013
\(855\) 1784.06 0.0713610
\(856\) 16500.8 0.658860
\(857\) 12947.1 0.516063 0.258032 0.966136i \(-0.416926\pi\)
0.258032 + 0.966136i \(0.416926\pi\)
\(858\) 2106.44 0.0838142
\(859\) −20383.5 −0.809636 −0.404818 0.914397i \(-0.632665\pi\)
−0.404818 + 0.914397i \(0.632665\pi\)
\(860\) −65.8335 −0.00261035
\(861\) −10342.9 −0.409391
\(862\) −8779.22 −0.346893
\(863\) 9056.42 0.357224 0.178612 0.983920i \(-0.442839\pi\)
0.178612 + 0.983920i \(0.442839\pi\)
\(864\) 5055.88 0.199079
\(865\) 6033.54 0.237164
\(866\) −22340.0 −0.876610
\(867\) 14491.0 0.567636
\(868\) 3119.27 0.121976
\(869\) −10470.2 −0.408719
\(870\) 3492.24 0.136090
\(871\) 16554.5 0.644005
\(872\) 31803.6 1.23510
\(873\) −8697.10 −0.337173
\(874\) −3819.37 −0.147817
\(875\) 1280.78 0.0494836
\(876\) 6049.36 0.233321
\(877\) −2867.88 −0.110424 −0.0552118 0.998475i \(-0.517583\pi\)
−0.0552118 + 0.998475i \(0.517583\pi\)
\(878\) 6846.16 0.263151
\(879\) −13584.9 −0.521281
\(880\) 628.281 0.0240674
\(881\) −11862.5 −0.453640 −0.226820 0.973937i \(-0.572833\pi\)
−0.226820 + 0.973937i \(0.572833\pi\)
\(882\) −3345.06 −0.127703
\(883\) 33463.8 1.27537 0.637683 0.770299i \(-0.279893\pi\)
0.637683 + 0.770299i \(0.279893\pi\)
\(884\) −22440.3 −0.853787
\(885\) 5561.59 0.211244
\(886\) −15756.0 −0.597442
\(887\) 2420.75 0.0916357 0.0458178 0.998950i \(-0.485411\pi\)
0.0458178 + 0.998950i \(0.485411\pi\)
\(888\) −2852.17 −0.107784
\(889\) 17826.5 0.672534
\(890\) −3012.85 −0.113473
\(891\) −891.000 −0.0335013
\(892\) 2815.62 0.105688
\(893\) 13213.8 0.495165
\(894\) −11132.4 −0.416470
\(895\) 7211.44 0.269332
\(896\) 12132.9 0.452378
\(897\) −7565.48 −0.281610
\(898\) 14963.6 0.556060
\(899\) −8161.14 −0.302769
\(900\) −1251.35 −0.0463463
\(901\) −63179.7 −2.33609
\(902\) −5779.73 −0.213352
\(903\) 72.7722 0.00268185
\(904\) −3600.10 −0.132453
\(905\) −21309.0 −0.782692
\(906\) −434.955 −0.0159497
\(907\) −38154.1 −1.39679 −0.698393 0.715714i \(-0.746101\pi\)
−0.698393 + 0.715714i \(0.746101\pi\)
\(908\) 23838.4 0.871262
\(909\) 3137.40 0.114479
\(910\) −3270.15 −0.119126
\(911\) −35758.0 −1.30045 −0.650227 0.759740i \(-0.725326\pi\)
−0.650227 + 0.759740i \(0.725326\pi\)
\(912\) −1358.66 −0.0493308
\(913\) −8087.44 −0.293160
\(914\) 15614.5 0.565079
\(915\) 10716.1 0.387172
\(916\) −31753.0 −1.14536
\(917\) −3205.39 −0.115432
\(918\) −4161.73 −0.149627
\(919\) 17387.5 0.624115 0.312058 0.950063i \(-0.398982\pi\)
0.312058 + 0.950063i \(0.398982\pi\)
\(920\) 6532.41 0.234095
\(921\) 1704.32 0.0609764
\(922\) −17357.9 −0.620013
\(923\) −38395.3 −1.36923
\(924\) −1880.50 −0.0669523
\(925\) 1122.35 0.0398947
\(926\) −2447.20 −0.0868467
\(927\) −13827.4 −0.489915
\(928\) −27918.3 −0.987569
\(929\) 6955.93 0.245658 0.122829 0.992428i \(-0.460803\pi\)
0.122829 + 0.992428i \(0.460803\pi\)
\(930\) −1282.16 −0.0452082
\(931\) 9436.31 0.332183
\(932\) −16384.9 −0.575863
\(933\) −20560.8 −0.721467
\(934\) 19751.7 0.691965
\(935\) −5428.96 −0.189889
\(936\) 7790.88 0.272065
\(937\) −16074.5 −0.560438 −0.280219 0.959936i \(-0.590407\pi\)
−0.280219 + 0.959936i \(0.590407\pi\)
\(938\) 6479.76 0.225556
\(939\) −3416.77 −0.118746
\(940\) −9268.20 −0.321591
\(941\) −687.126 −0.0238041 −0.0119021 0.999929i \(-0.503789\pi\)
−0.0119021 + 0.999929i \(0.503789\pi\)
\(942\) −8812.96 −0.304821
\(943\) 20758.5 0.716849
\(944\) −4235.44 −0.146030
\(945\) 1383.24 0.0476156
\(946\) 40.6659 0.00139763
\(947\) 35352.0 1.21308 0.606540 0.795053i \(-0.292557\pi\)
0.606540 + 0.795053i \(0.292557\pi\)
\(948\) −15881.0 −0.544085
\(949\) 14820.7 0.506956
\(950\) −1547.73 −0.0528578
\(951\) 9622.44 0.328106
\(952\) −21418.2 −0.729168
\(953\) −19390.7 −0.659103 −0.329552 0.944138i \(-0.606898\pi\)
−0.329552 + 0.944138i \(0.606898\pi\)
\(954\) 8995.45 0.305281
\(955\) 4261.12 0.144384
\(956\) 11561.9 0.391148
\(957\) 4920.06 0.166189
\(958\) −16738.7 −0.564511
\(959\) 7345.80 0.247349
\(960\) −3015.33 −0.101374
\(961\) −26794.7 −0.899422
\(962\) −2865.64 −0.0960416
\(963\) −7012.62 −0.234661
\(964\) −10307.6 −0.344384
\(965\) 12299.8 0.410304
\(966\) −2961.27 −0.0986308
\(967\) 28643.6 0.952551 0.476275 0.879296i \(-0.341987\pi\)
0.476275 + 0.879296i \(0.341987\pi\)
\(968\) −2562.43 −0.0850821
\(969\) 11740.1 0.389213
\(970\) 7544.99 0.249747
\(971\) −19574.8 −0.646946 −0.323473 0.946237i \(-0.604851\pi\)
−0.323473 + 0.946237i \(0.604851\pi\)
\(972\) −1351.46 −0.0445967
\(973\) 8980.63 0.295895
\(974\) 11182.6 0.367878
\(975\) −3065.77 −0.100701
\(976\) −8160.86 −0.267646
\(977\) 50095.5 1.64043 0.820213 0.572058i \(-0.193855\pi\)
0.820213 + 0.572058i \(0.193855\pi\)
\(978\) −11550.0 −0.377636
\(979\) −4244.67 −0.138570
\(980\) −6618.67 −0.215740
\(981\) −13516.1 −0.439895
\(982\) 22747.6 0.739211
\(983\) 14445.4 0.468706 0.234353 0.972152i \(-0.424703\pi\)
0.234353 + 0.972152i \(0.424703\pi\)
\(984\) −21376.9 −0.692553
\(985\) −17385.3 −0.562377
\(986\) 22980.9 0.742253
\(987\) 10245.0 0.330399
\(988\) −9013.05 −0.290226
\(989\) −146.056 −0.00469595
\(990\) 772.969 0.0248147
\(991\) 29120.1 0.933430 0.466715 0.884408i \(-0.345437\pi\)
0.466715 + 0.884408i \(0.345437\pi\)
\(992\) 10250.1 0.328064
\(993\) −27405.4 −0.875814
\(994\) −15028.7 −0.479558
\(995\) −19975.2 −0.636438
\(996\) −12266.9 −0.390253
\(997\) −9137.45 −0.290257 −0.145128 0.989413i \(-0.546360\pi\)
−0.145128 + 0.989413i \(0.546360\pi\)
\(998\) −7244.03 −0.229765
\(999\) 1212.14 0.0383887
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 165.4.a.c.1.2 2
3.2 odd 2 495.4.a.d.1.1 2
5.2 odd 4 825.4.c.j.199.3 4
5.3 odd 4 825.4.c.j.199.2 4
5.4 even 2 825.4.a.m.1.1 2
11.10 odd 2 1815.4.a.n.1.1 2
15.14 odd 2 2475.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.2 2 1.1 even 1 trivial
495.4.a.d.1.1 2 3.2 odd 2
825.4.a.m.1.1 2 5.4 even 2
825.4.c.j.199.2 4 5.3 odd 4
825.4.c.j.199.3 4 5.2 odd 4
1815.4.a.n.1.1 2 11.10 odd 2
2475.4.a.n.1.2 2 15.14 odd 2