Properties

Label 825.4.a.m.1.1
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
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: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

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

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 \(-0.589024\pi\)
−0.276046 + 0.961144i \(0.589024\pi\)
\(3\) −3.00000 −0.577350
\(4\) −5.56155 −0.695194
\(5\) 0 0
\(6\) 4.68466 0.318751
\(7\) 10.2462 0.553243 0.276622 0.960979i \(-0.410785\pi\)
0.276622 + 0.960979i \(0.410785\pi\)
\(8\) 21.1771 0.935904
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 16.6847 0.401371
\(13\) 40.8769 0.872093 0.436047 0.899924i \(-0.356378\pi\)
0.436047 + 0.899924i \(0.356378\pi\)
\(14\) −16.0000 −0.305441
\(15\) 0 0
\(16\) 11.4233 0.178489
\(17\) 98.7083 1.40825 0.704126 0.710075i \(-0.251339\pi\)
0.704126 + 0.710075i \(0.251339\pi\)
\(18\) −14.0540 −0.184031
\(19\) −39.6458 −0.478704 −0.239352 0.970933i \(-0.576935\pi\)
−0.239352 + 0.970933i \(0.576935\pi\)
\(20\) 0 0
\(21\) −30.7386 −0.319415
\(22\) 17.1771 0.166462
\(23\) −61.6932 −0.559301 −0.279650 0.960102i \(-0.590219\pi\)
−0.279650 + 0.960102i \(0.590219\pi\)
\(24\) −63.5312 −0.540344
\(25\) 0 0
\(26\) −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 \(0.658400\pi\)
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) −50.0540 −0.231731
\(37\) −44.8939 −0.199473 −0.0997367 0.995014i \(-0.531800\pi\)
−0.0997367 + 0.995014i \(0.531800\pi\)
\(38\) 61.9091 0.264289
\(39\) −122.631 −0.503503
\(40\) 0 0
\(41\) 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 \(-0.498664\pi\)
0.00419806 + 0.999991i \(0.498664\pi\)
\(44\) 61.1771 0.209609
\(45\) 0 0
\(46\) 96.3371 0.308786
\(47\) 333.295 1.03439 0.517193 0.855869i \(-0.326977\pi\)
0.517193 + 0.855869i \(0.326977\pi\)
\(48\) −34.2699 −0.103051
\(49\) −238.015 −0.693922
\(50\) 0 0
\(51\) −296.125 −0.813055
\(52\) −227.339 −0.606274
\(53\) −640.064 −1.65886 −0.829430 0.558610i \(-0.811335\pi\)
−0.829430 + 0.558610i \(0.811335\pi\)
\(54\) 42.1619 0.106250
\(55\) 0 0
\(56\) 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 \(0.634147\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) −51.5312 −0.0961069
\(67\) 404.985 0.738459 0.369230 0.929338i \(-0.379622\pi\)
0.369230 + 0.929338i \(0.379622\pi\)
\(68\) −548.972 −0.979009
\(69\) 185.080 0.322912
\(70\) 0 0
\(71\) 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 \(-0.406127\pi\)
0.290655 + 0.956828i \(0.406127\pi\)
\(74\) 70.1042 0.110128
\(75\) 0 0
\(76\) 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 \(0.262941\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −525.430 −0.707610
\(83\) −735.221 −0.972302 −0.486151 0.873875i \(-0.661599\pi\)
−0.486151 + 0.873875i \(0.661599\pi\)
\(84\) 170.955 0.222056
\(85\) 0 0
\(86\) −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 \(0.426195\pi\)
\(90\) 0 0
\(91\) 418.833 0.482480
\(92\) 343.110 0.388823
\(93\) −164.216 −0.183101
\(94\) −520.458 −0.571076
\(95\) 0 0
\(96\) 561.764 0.597238
\(97\) 966.345 1.01152 0.505760 0.862674i \(-0.331212\pi\)
0.505760 + 0.862674i \(0.331212\pi\)
\(98\) 371.673 0.383109
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 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 \(-0.237242\pi\)
0.734873 + 0.678204i \(0.237242\pi\)
\(104\) 865.653 0.816195
\(105\) 0 0
\(106\) 999.494 0.915844
\(107\) 779.180 0.703983 0.351991 0.936003i \(-0.385505\pi\)
0.351991 + 0.936003i \(0.385505\pi\)
\(108\) 150.162 0.133790
\(109\) −1501.79 −1.31968 −0.659842 0.751404i \(-0.729377\pi\)
−0.659842 + 0.751404i \(0.729377\pi\)
\(110\) 0 0
\(111\) 134.682 0.115166
\(112\) 117.045 0.0987478
\(113\) −170.000 −0.141524 −0.0707622 0.997493i \(-0.522543\pi\)
−0.0707622 + 0.997493i \(0.522543\pi\)
\(114\) −185.727 −0.152587
\(115\) 0 0
\(116\) 829.187 0.663691
\(117\) 367.892 0.290698
\(118\) 578.981 0.451691
\(119\) 1011.39 0.779106
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1115.58 0.827869
\(123\) −1009.44 −0.739983
\(124\) −304.432 −0.220474
\(125\) 0 0
\(126\) −144.000 −0.101814
\(127\) 1739.82 1.21562 0.607811 0.794082i \(-0.292048\pi\)
0.607811 + 0.794082i \(0.292048\pi\)
\(128\) 1184.13 0.817683
\(129\) −7.10235 −0.00484750
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) 2090.35 1.31799
\(137\) 716.928 0.447090 0.223545 0.974694i \(-0.428237\pi\)
0.223545 + 0.974694i \(0.428237\pi\)
\(138\) −289.011 −0.178277
\(139\) −876.483 −0.534837 −0.267418 0.963581i \(-0.586171\pi\)
−0.267418 + 0.963581i \(0.586171\pi\)
\(140\) 0 0
\(141\) −999.886 −0.597203
\(142\) −1466.75 −0.866811
\(143\) −449.646 −0.262946
\(144\) 102.810 0.0594963
\(145\) 0 0
\(146\) −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 \(0.726610\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) 682.017 0.350032
\(157\) 1881.24 0.956301 0.478150 0.878278i \(-0.341307\pi\)
0.478150 + 0.878278i \(0.341307\pi\)
\(158\) −1486.34 −0.748398
\(159\) 1920.19 0.957743
\(160\) 0 0
\(161\) −632.121 −0.309429
\(162\) −126.486 −0.0613436
\(163\) 2465.49 1.18474 0.592369 0.805667i \(-0.298193\pi\)
0.592369 + 0.805667i \(0.298193\pi\)
\(164\) −1871.35 −0.891022
\(165\) 0 0
\(166\) 1148.09 0.536800
\(167\) 1254.30 0.581200 0.290600 0.956845i \(-0.406145\pi\)
0.290600 + 0.956845i \(0.406145\pi\)
\(168\) −650.955 −0.298942
\(169\) −526.080 −0.239454
\(170\) 0 0
\(171\) −356.813 −0.159568
\(172\) −13.1667 −0.00583693
\(173\) 1206.71 0.530314 0.265157 0.964205i \(-0.414576\pi\)
0.265157 + 0.964205i \(0.414576\pi\)
\(174\) −698.449 −0.304306
\(175\) 0 0
\(176\) −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 \(0.597361\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 256.432 0.101089
\(187\) −1085.79 −0.424604
\(188\) −1853.64 −0.719099
\(189\) −276.648 −0.106472
\(190\) 0 0
\(191\) −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 \(-0.348303\pi\)
0.458734 + 0.888574i \(0.348303\pi\)
\(194\) −1509.00 −0.558452
\(195\) 0 0
\(196\) 1323.73 0.482410
\(197\) −3477.06 −1.25751 −0.628756 0.777602i \(-0.716436\pi\)
−0.628756 + 0.777602i \(0.716436\pi\)
\(198\) 154.594 0.0554874
\(199\) 3995.04 1.42312 0.711560 0.702626i \(-0.247989\pi\)
0.711560 + 0.702626i \(0.247989\pi\)
\(200\) 0 0
\(201\) −1214.95 −0.426350
\(202\) −544.358 −0.189608
\(203\) −1527.64 −0.528173
\(204\) 1646.91 0.565231
\(205\) 0 0
\(206\) −2399.14 −0.811436
\(207\) −555.239 −0.186434
\(208\) 466.949 0.155659
\(209\) 436.104 0.144335
\(210\) 0 0
\(211\) 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
\(215\) 0 0
\(216\) −571.781 −0.180115
\(217\) 560.864 0.175456
\(218\) 2345.13 0.728587
\(219\) −1087.71 −0.335619
\(220\) 0 0
\(221\) 4034.89 1.22813
\(222\) −210.313 −0.0635823
\(223\) 506.265 0.152027 0.0760135 0.997107i \(-0.475781\pi\)
0.0760135 + 0.997107i \(0.475781\pi\)
\(224\) −1918.65 −0.572300
\(225\) 0 0
\(226\) 265.464 0.0781345
\(227\) 4286.29 1.25326 0.626632 0.779315i \(-0.284433\pi\)
0.626632 + 0.779315i \(0.284433\pi\)
\(228\) −661.477 −0.192138
\(229\) 5709.37 1.64754 0.823769 0.566926i \(-0.191867\pi\)
0.823769 + 0.566926i \(0.191867\pi\)
\(230\) 0 0
\(231\) 338.125 0.0963073
\(232\) −3157.35 −0.893492
\(233\) −2946.09 −0.828348 −0.414174 0.910198i \(-0.635929\pi\)
−0.414174 + 0.910198i \(0.635929\pi\)
\(234\) −574.483 −0.160492
\(235\) 0 0
\(236\) 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 \(0.590773\pi\)
\(240\) 0 0
\(241\) 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
\(245\) 0 0
\(246\) 1576.29 0.408539
\(247\) −1620.60 −0.417475
\(248\) 1159.20 0.296813
\(249\) 2205.66 0.561359
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) −3457.26 −0.844057
\(257\) −5519.25 −1.33962 −0.669809 0.742534i \(-0.733624\pi\)
−0.669809 + 0.742534i \(0.733624\pi\)
\(258\) 11.0907 0.00267627
\(259\) −459.993 −0.110357
\(260\) 0 0
\(261\) −1341.84 −0.318228
\(262\) −488.512 −0.115192
\(263\) −2259.65 −0.529795 −0.264898 0.964277i \(-0.585338\pi\)
−0.264898 + 0.964277i \(0.585338\pi\)
\(264\) 698.844 0.162920
\(265\) 0 0
\(266\) 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 \(0.206006\pi\)
\(270\) 0 0
\(271\) 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
\(275\) 0 0
\(276\) −1029.33 −0.224487
\(277\) −9074.52 −1.96836 −0.984179 0.177175i \(-0.943304\pi\)
−0.984179 + 0.177175i \(0.943304\pi\)
\(278\) 1368.67 0.295279
\(279\) 492.648 0.105713
\(280\) 0 0
\(281\) −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 \(-0.122270\pi\)
0.927127 + 0.374746i \(0.122270\pi\)
\(284\) −5223.92 −1.09149
\(285\) 0 0
\(286\) 702.146 0.145170
\(287\) 3447.64 0.709085
\(288\) −1685.29 −0.344815
\(289\) 4830.33 0.983174
\(290\) 0 0
\(291\) −2899.03 −0.584001
\(292\) −2016.45 −0.404123
\(293\) 4528.29 0.902886 0.451443 0.892300i \(-0.350909\pi\)
0.451443 + 0.892300i \(0.350909\pi\)
\(294\) −1115.02 −0.221188
\(295\) 0 0
\(296\) −950.722 −0.186688
\(297\) 297.000 0.0580259
\(298\) 3710.81 0.721347
\(299\) −2521.83 −0.487762
\(300\) 0 0
\(301\) 24.2574 0.00464509
\(302\) 144.985 0.0276256
\(303\) −1045.80 −0.198283
\(304\) −452.886 −0.0854434
\(305\) 0 0
\(306\) −1387.24 −0.259162
\(307\) −568.106 −0.105614 −0.0528071 0.998605i \(-0.516817\pi\)
−0.0528071 + 0.998605i \(0.516817\pi\)
\(308\) 626.833 0.115965
\(309\) −4609.14 −0.848559
\(310\) 0 0
\(311\) −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 \(-0.467208\pi\)
0.102837 + 0.994698i \(0.467208\pi\)
\(314\) −2937.65 −0.527966
\(315\) 0 0
\(316\) −5293.68 −0.942382
\(317\) −3207.48 −0.568297 −0.284148 0.958780i \(-0.591711\pi\)
−0.284148 + 0.958780i \(0.591711\pi\)
\(318\) −2998.48 −0.528763
\(319\) 1640.02 0.287848
\(320\) 0 0
\(321\) −2337.54 −0.406445
\(322\) 987.091 0.170834
\(323\) −3913.37 −0.674136
\(324\) −450.486 −0.0772438
\(325\) 0 0
\(326\) −3850.00 −0.654085
\(327\) 4505.37 0.761920
\(328\) 7125.65 1.19954
\(329\) 3415.02 0.572267
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) −351.136 −0.0570121
\(337\) 3470.05 0.560907 0.280453 0.959868i \(-0.409515\pi\)
0.280453 + 0.959868i \(0.409515\pi\)
\(338\) 821.501 0.132200
\(339\) 510.000 0.0817091
\(340\) 0 0
\(341\) −602.125 −0.0956214
\(342\) 557.182 0.0880963
\(343\) −5953.20 −0.937151
\(344\) 50.1357 0.00785795
\(345\) 0 0
\(346\) −1884.34 −0.292782
\(347\) 89.3315 0.0138201 0.00691004 0.999976i \(-0.497800\pi\)
0.00691004 + 0.999976i \(0.497800\pi\)
\(348\) −2487.56 −0.383182
\(349\) −149.375 −0.0229107 −0.0114554 0.999934i \(-0.503646\pi\)
−0.0114554 + 0.999934i \(0.503646\pi\)
\(350\) 0 0
\(351\) −1103.68 −0.167834
\(352\) 2059.80 0.311897
\(353\) −7867.64 −1.18627 −0.593133 0.805104i \(-0.702109\pi\)
−0.593133 + 0.805104i \(0.702109\pi\)
\(354\) −1736.94 −0.260784
\(355\) 0 0
\(356\) −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 \(0.380850\pi\)
\(360\) 0 0
\(361\) −5287.21 −0.770842
\(362\) −6655.04 −0.966246
\(363\) −363.000 −0.0524864
\(364\) −2329.36 −0.335417
\(365\) 0 0
\(366\) −3346.74 −0.477970
\(367\) 13266.7 1.88696 0.943479 0.331433i \(-0.107532\pi\)
0.943479 + 0.331433i \(0.107532\pi\)
\(368\) −704.739 −0.0998290
\(369\) 3028.31 0.427229
\(370\) 0 0
\(371\) −6558.23 −0.917754
\(372\) 913.295 0.127291
\(373\) 4632.77 0.643099 0.321549 0.946893i \(-0.395796\pi\)
0.321549 + 0.946893i \(0.395796\pi\)
\(374\) 1695.52 0.234421
\(375\) 0 0
\(376\) 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 \(0.354729\pi\)
\(380\) 0 0
\(381\) −5219.45 −0.701839
\(382\) 1330.79 0.178244
\(383\) −12734.5 −1.69897 −0.849484 0.527614i \(-0.823087\pi\)
−0.849484 + 0.527614i \(0.823087\pi\)
\(384\) −3552.39 −0.472090
\(385\) 0 0
\(386\) −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 \(0.213361\pi\)
\(390\) 0 0
\(391\) −6089.63 −0.787636
\(392\) −5040.47 −0.649444
\(393\) −938.511 −0.120462
\(394\) 5429.61 0.694263
\(395\) 0 0
\(396\) 550.594 0.0698696
\(397\) 5223.65 0.660371 0.330186 0.943916i \(-0.392889\pi\)
0.330186 + 0.943916i \(0.392889\pi\)
\(398\) −6238.46 −0.785693
\(399\) 1218.66 0.152905
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 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 \(0.538780\pi\)
\(410\) 0 0
\(411\) −2150.78 −0.258127
\(412\) −8544.65 −1.02176
\(413\) −3799.02 −0.452633
\(414\) 867.034 0.102929
\(415\) 0 0
\(416\) −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 \(0.416750\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) 4400.26 0.500454
\(427\) −7319.95 −0.829595
\(428\) −4333.45 −0.489405
\(429\) 1348.94 0.151812
\(430\) 0 0
\(431\) −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 \(-0.208049\pi\)
0.793898 + 0.608051i \(0.208049\pi\)
\(434\) −875.818 −0.0968678
\(435\) 0 0
\(436\) 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 \(0.423403\pi\)
\(440\) 0 0
\(441\) −2142.14 −0.231307
\(442\) −6300.69 −0.678039
\(443\) 10090.0 1.08214 0.541071 0.840977i \(-0.318019\pi\)
0.541071 + 0.840977i \(0.318019\pi\)
\(444\) −749.040 −0.0800627
\(445\) 0 0
\(446\) −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 \(0.332011\pi\)
\(450\) 0 0
\(451\) −3701.27 −0.386444
\(452\) 945.464 0.0983869
\(453\) 278.540 0.0288895
\(454\) −6693.27 −0.691918
\(455\) 0 0
\(456\) 2518.75 0.258665
\(457\) −9999.34 −1.02352 −0.511761 0.859128i \(-0.671007\pi\)
−0.511761 + 0.859128i \(0.671007\pi\)
\(458\) −8915.49 −0.909593
\(459\) −2665.12 −0.271018
\(460\) 0 0
\(461\) −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 \(-0.474938\pi\)
0.0786524 + 0.996902i \(0.474938\pi\)
\(464\) −1703.13 −0.170401
\(465\) 0 0
\(466\) 4600.48 0.457325
\(467\) −12648.8 −1.25335 −0.626675 0.779281i \(-0.715585\pi\)
−0.626675 + 0.779281i \(0.715585\pi\)
\(468\) −2046.05 −0.202091
\(469\) 4149.56 0.408548
\(470\) 0 0
\(471\) −5643.72 −0.552120
\(472\) −7851.88 −0.765704
\(473\) −26.0420 −0.00253152
\(474\) 4459.02 0.432088
\(475\) 0 0
\(476\) −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 \(0.670816\pi\)
\(480\) 0 0
\(481\) −1835.12 −0.173959
\(482\) −2894.14 −0.273494
\(483\) 1896.36 0.178649
\(484\) −672.948 −0.0631995
\(485\) 0 0
\(486\) 379.457 0.0354167
\(487\) −7161.20 −0.666335 −0.333167 0.942868i \(-0.608117\pi\)
−0.333167 + 0.942868i \(0.608117\pi\)
\(488\) −15129.0 −1.40340
\(489\) −7396.48 −0.684009
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) 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 \(0.566723\pi\)
\(500\) 0 0
\(501\) −3762.89 −0.335556
\(502\) 3682.75 0.327428
\(503\) 12206.3 1.08201 0.541006 0.841019i \(-0.318044\pi\)
0.541006 + 0.841019i \(0.318044\pi\)
\(504\) 1952.86 0.172594
\(505\) 0 0
\(506\) −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 \(0.356319\pi\)
\(510\) 0 0
\(511\) 3714.97 0.321606
\(512\) −4074.36 −0.351686
\(513\) 1070.44 0.0921267
\(514\) 8618.61 0.739592
\(515\) 0 0
\(516\) 39.5001 0.00336995
\(517\) −3666.25 −0.311879
\(518\) 718.303 0.0609274
\(519\) −3620.12 −0.306177
\(520\) 0 0
\(521\) 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 \(-0.262564\pi\)
0.678652 + 0.734460i \(0.262564\pi\)
\(524\) −1739.86 −0.145050
\(525\) 0 0
\(526\) 3528.57 0.292496
\(527\) 5403.16 0.446613
\(528\) 376.969 0.0310709
\(529\) −8360.95 −0.687183
\(530\) 0 0
\(531\) −3336.95 −0.272715
\(532\) 2259.21 0.184115
\(533\) 13754.2 1.11775
\(534\) 1807.71 0.146493
\(535\) 0 0
\(536\) 8576.40 0.691127
\(537\) 4326.86 0.347706
\(538\) −10992.6 −0.880900
\(539\) 2618.17 0.209225
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) 1962.09 0.153791
\(547\) −13058.2 −1.02071 −0.510355 0.859964i \(-0.670486\pi\)
−0.510355 + 0.859964i \(0.670486\pi\)
\(548\) −3987.23 −0.310814
\(549\) −6429.65 −0.499837
\(550\) 0 0
\(551\) 5910.91 0.457011
\(552\) 3919.44 0.302215
\(553\) 9752.70 0.749959
\(554\) 14170.3 1.08672
\(555\) 0 0
\(556\) 4874.61 0.371815
\(557\) 6710.48 0.510471 0.255236 0.966879i \(-0.417847\pi\)
0.255236 + 0.966879i \(0.417847\pi\)
\(558\) −769.295 −0.0583636
\(559\) 96.7741 0.00732219
\(560\) 0 0
\(561\) 3257.37 0.245145
\(562\) 5321.45 0.399416
\(563\) 20820.5 1.55858 0.779288 0.626666i \(-0.215581\pi\)
0.779288 + 0.626666i \(0.215581\pi\)
\(564\) 5560.92 0.415172
\(565\) 0 0
\(566\) −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 \(0.461786\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 1809.20 0.130874
\(577\) −14462.4 −1.04346 −0.521730 0.853111i \(-0.674713\pi\)
−0.521730 + 0.853111i \(0.674713\pi\)
\(578\) −7542.82 −0.542803
\(579\) −7379.86 −0.529700
\(580\) 0 0
\(581\) −7533.23 −0.537920
\(582\) 4526.99 0.322423
\(583\) 7040.71 0.500165
\(584\) 7678.18 0.544050
\(585\) 0 0
\(586\) −7071.17 −0.498476
\(587\) 22759.7 1.60033 0.800166 0.599779i \(-0.204745\pi\)
0.800166 + 0.599779i \(0.204745\pi\)
\(588\) −3971.20 −0.278520
\(589\) −2170.16 −0.151816
\(590\) 0 0
\(591\) 10431.2 0.726025
\(592\) −512.836 −0.0356038
\(593\) 14956.4 1.03573 0.517864 0.855463i \(-0.326727\pi\)
0.517864 + 0.855463i \(0.326727\pi\)
\(594\) −463.781 −0.0320356
\(595\) 0 0
\(596\) 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 \(0.476630\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) 1633.07 0.109470
\(607\) 10991.5 0.734974 0.367487 0.930029i \(-0.380218\pi\)
0.367487 + 0.930029i \(0.380218\pi\)
\(608\) 7423.87 0.495194
\(609\) 4582.91 0.304941
\(610\) 0 0
\(611\) 13624.1 0.902081
\(612\) −4940.74 −0.326336
\(613\) −10646.1 −0.701457 −0.350728 0.936477i \(-0.614066\pi\)
−0.350728 + 0.936477i \(0.614066\pi\)
\(614\) 887.128 0.0583088
\(615\) 0 0
\(616\) −2386.83 −0.156117
\(617\) −7199.92 −0.469786 −0.234893 0.972021i \(-0.575474\pi\)
−0.234893 + 0.972021i \(0.575474\pi\)
\(618\) 7197.41 0.468483
\(619\) 12186.9 0.791332 0.395666 0.918395i \(-0.370514\pi\)
0.395666 + 0.918395i \(0.370514\pi\)
\(620\) 0 0
\(621\) 1665.72 0.107637
\(622\) 10702.2 0.689905
\(623\) 3953.80 0.254262
\(624\) −1400.85 −0.0898698
\(625\) 0 0
\(626\) −1778.49 −0.113551
\(627\) −1308.31 −0.0833317
\(628\) −10462.6 −0.664814
\(629\) −4431.40 −0.280909
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) −10679.3 −0.665818
\(637\) −9729.32 −0.605164
\(638\) −2560.98 −0.158919
\(639\) 8453.63 0.523349
\(640\) 0 0
\(641\) −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 \(-0.268657\pi\)
0.664472 + 0.747313i \(0.268657\pi\)
\(644\) 3515.58 0.215113
\(645\) 0 0
\(646\) 6110.94 0.372185
\(647\) 27625.3 1.67861 0.839305 0.543661i \(-0.182962\pi\)
0.839305 + 0.543661i \(0.182962\pi\)
\(648\) 1715.34 0.103989
\(649\) 4078.50 0.246680
\(650\) 0 0
\(651\) −1682.59 −0.101299
\(652\) −13712.0 −0.823623
\(653\) 14314.0 0.857810 0.428905 0.903350i \(-0.358899\pi\)
0.428905 + 0.903350i \(0.358899\pi\)
\(654\) −7035.38 −0.420650
\(655\) 0 0
\(656\) 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 \(0.815837\pi\)
\(660\) 0 0
\(661\) −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
\(665\) 0 0
\(666\) 630.938 0.0367092
\(667\) 9198.01 0.533955
\(668\) −6975.84 −0.404047
\(669\) −1518.80 −0.0877729
\(670\) 0 0
\(671\) 7858.46 0.452120
\(672\) 5755.95 0.330418
\(673\) 6207.38 0.355538 0.177769 0.984072i \(-0.443112\pi\)
0.177769 + 0.984072i \(0.443112\pi\)
\(674\) −5418.66 −0.309672
\(675\) 0 0
\(676\) 2925.82 0.166467
\(677\) −28831.1 −1.63674 −0.818368 0.574695i \(-0.805121\pi\)
−0.818368 + 0.574695i \(0.805121\pi\)
\(678\) −796.392 −0.0451110
\(679\) 9901.37 0.559617
\(680\) 0 0
\(681\) −12858.9 −0.723573
\(682\) 940.250 0.0527918
\(683\) −3193.10 −0.178888 −0.0894441 0.995992i \(-0.528509\pi\)
−0.0894441 + 0.995992i \(0.528509\pi\)
\(684\) 1984.43 0.110931
\(685\) 0 0
\(686\) 9296.24 0.517394
\(687\) −17128.1 −0.951206
\(688\) 27.0441 0.00149861
\(689\) −26163.8 −1.44668
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) 9472.05 0.515858
\(697\) 33213.3 1.80494
\(698\) 233.256 0.0126488
\(699\) 8838.28 0.478247
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) 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 \(0.642131\pi\)
\(710\) 0 0
\(711\) 8566.52 0.451856
\(712\) 8171.79 0.430127
\(713\) −3377.00 −0.177377
\(714\) 4738.00 0.248341
\(715\) 0 0
\(716\) 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 \(0.532861\pi\)
\(720\) 0 0
\(721\) 15742.1 0.813128
\(722\) 8256.25 0.425576
\(723\) −5560.11 −0.286007
\(724\) −23702.3 −1.21670
\(725\) 0 0
\(726\) 566.844 0.0289773
\(727\) 10780.4 0.549961 0.274980 0.961450i \(-0.411329\pi\)
0.274980 + 0.961450i \(0.411329\pi\)
\(728\) 8869.67 0.451555
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 233.687 0.0118238
\(732\) −11919.6 −0.601860
\(733\) 9211.46 0.464165 0.232083 0.972696i \(-0.425446\pi\)
0.232083 + 0.972696i \(0.425446\pi\)
\(734\) −20716.6 −1.04177
\(735\) 0 0
\(736\) 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 \(0.411029\pi\)
\(740\) 0 0
\(741\) 4861.80 0.241029
\(742\) 10241.0 0.506685
\(743\) 27420.4 1.35391 0.676955 0.736024i \(-0.263299\pi\)
0.676955 + 0.736024i \(0.263299\pi\)
\(744\) −3477.61 −0.171365
\(745\) 0 0
\(746\) −7234.32 −0.355050
\(747\) −6616.99 −0.324101
\(748\) 6038.69 0.295182
\(749\) 7983.64 0.389474
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 1538.59 0.0740185
\(757\) −3739.19 −0.179528 −0.0897642 0.995963i \(-0.528611\pi\)
−0.0897642 + 0.995963i \(0.528611\pi\)
\(758\) −10155.3 −0.486617
\(759\) −2035.87 −0.0973617
\(760\) 0 0
\(761\) −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
\(765\) 0 0
\(766\) 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 \(0.107482\pi\)
\(770\) 0 0
\(771\) 16557.8 0.773428
\(772\) −13681.2 −0.637818
\(773\) −22821.4 −1.06187 −0.530936 0.847412i \(-0.678160\pi\)
−0.530936 + 0.847412i \(0.678160\pi\)
\(774\) −33.2721 −0.00154514
\(775\) 0 0
\(776\) 20464.4 0.946685
\(777\) 1379.98 0.0637148
\(778\) −18777.0 −0.865282
\(779\) −13340.0 −0.613549
\(780\) 0 0
\(781\) −10332.2 −0.473387
\(782\) 9509.28 0.434848
\(783\) 4025.51 0.183729
\(784\) −2718.92 −0.123857
\(785\) 0 0
\(786\) 1465.53 0.0665062
\(787\) 29454.3 1.33410 0.667048 0.745015i \(-0.267558\pi\)
0.667048 + 0.745015i \(0.267558\pi\)
\(788\) 19337.8 0.874216
\(789\) 6778.96 0.305878
\(790\) 0 0
\(791\) −1741.86 −0.0782974
\(792\) −2096.53 −0.0940619
\(793\) −29202.7 −1.30771
\(794\) −8157.00 −0.364586
\(795\) 0 0
\(796\) −22218.6 −0.989344
\(797\) 27440.3 1.21955 0.609777 0.792573i \(-0.291259\pi\)
0.609777 + 0.792573i \(0.291259\pi\)
\(798\) −1903.00 −0.0844179
\(799\) 32899.0 1.45668
\(800\) 0 0
\(801\) 3472.91 0.153195
\(802\) −15066.1 −0.663346
\(803\) −3988.27 −0.175272
\(804\) 6757.03 0.296396
\(805\) 0 0
\(806\) −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 \(0.535070\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) −3382.72 −0.145121
\(817\) −93.8596 −0.00401925
\(818\) 3139.46 0.134192
\(819\) 3769.50 0.160827
\(820\) 0 0
\(821\) 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 \(-0.666867\pi\)
−0.500546 + 0.865710i \(0.666867\pi\)
\(824\) 32536.0 1.37554
\(825\) 0 0
\(826\) 5932.36 0.249895
\(827\) 42634.3 1.79267 0.896336 0.443376i \(-0.146219\pi\)
0.896336 + 0.443376i \(0.146219\pi\)
\(828\) 3087.99 0.129608
\(829\) −45152.5 −1.89169 −0.945845 0.324619i \(-0.894764\pi\)
−0.945845 + 0.324619i \(0.894764\pi\)
\(830\) 0 0
\(831\) 27223.6 1.13643
\(832\) 8217.15 0.342402
\(833\) −23494.1 −0.977217
\(834\) −4106.02 −0.170480
\(835\) 0 0
\(836\) −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 \(0.284646\pi\)
\(840\) 0 0
\(841\) −2160.34 −0.0885784
\(842\) −23763.6 −0.972623
\(843\) 10223.4 0.417689
\(844\) −5818.09 −0.237283
\(845\) 0 0
\(846\) −4684.13 −0.190359
\(847\) 1239.79 0.0502949
\(848\) −7311.64 −0.296088
\(849\) −26483.2 −1.07055
\(850\) 0 0
\(851\) 2769.65 0.111566
\(852\) 15671.8 0.630171
\(853\) −10367.2 −0.416139 −0.208070 0.978114i \(-0.566718\pi\)
−0.208070 + 0.978114i \(0.566718\pi\)
\(854\) 11430.5 0.458013
\(855\) 0 0
\(856\) 16500.8 0.658860
\(857\) −12947.1 −0.516063 −0.258032 0.966136i \(-0.583074\pi\)
−0.258032 + 0.966136i \(0.583074\pi\)
\(858\) −2106.44 −0.0838142
\(859\) −20383.5 −0.809636 −0.404818 0.914397i \(-0.632665\pi\)
−0.404818 + 0.914397i \(0.632665\pi\)
\(860\) 0 0
\(861\) −10342.9 −0.409391
\(862\) 8779.22 0.346893
\(863\) −9056.42 −0.357224 −0.178612 0.983920i \(-0.557161\pi\)
−0.178612 + 0.983920i \(0.557161\pi\)
\(864\) 5055.88 0.199079
\(865\) 0 0
\(866\) −22340.0 −0.876610
\(867\) −14491.0 −0.567636
\(868\) −3119.27 −0.121976
\(869\) −10470.2 −0.408719
\(870\) 0 0
\(871\) 16554.5 0.644005
\(872\) −31803.6 −1.23510
\(873\) 8697.10 0.337173
\(874\) −3819.37 −0.147817
\(875\) 0 0
\(876\) 6049.36 0.233321
\(877\) 2867.88 0.110424 0.0552118 0.998475i \(-0.482417\pi\)
0.0552118 + 0.998475i \(0.482417\pi\)
\(878\) −6846.16 −0.263151
\(879\) −13584.9 −0.521281
\(880\) 0 0
\(881\) −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 \(-0.720107\pi\)
−0.637683 + 0.770299i \(0.720107\pi\)
\(884\) −22440.3 −0.853787
\(885\) 0 0
\(886\) −15756.0 −0.597442
\(887\) −2420.75 −0.0916357 −0.0458178 0.998950i \(-0.514589\pi\)
−0.0458178 + 0.998950i \(0.514589\pi\)
\(888\) 2852.17 0.107784
\(889\) 17826.5 0.672534
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) −2815.62 −0.105688
\(893\) −13213.8 −0.495165
\(894\) −11132.4 −0.416470
\(895\) 0 0
\(896\) 12132.9 0.452378
\(897\) 7565.48 0.281610
\(898\) −14963.6 −0.556060
\(899\) −8161.14 −0.302769
\(900\) 0 0
\(901\) −63179.7 −2.33609
\(902\) 5779.73 0.213352
\(903\) −72.7722 −0.00268185
\(904\) −3600.10 −0.132453
\(905\) 0 0
\(906\) −434.955 −0.0159497
\(907\) 38154.1 1.39679 0.698393 0.715714i \(-0.253899\pi\)
0.698393 + 0.715714i \(0.253899\pi\)
\(908\) −23838.4 −0.871262
\(909\) 3137.40 0.114479
\(910\) 0 0
\(911\) −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
\(915\) 0 0
\(916\) −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 \(0.398982\pi\)
\(920\) 0 0
\(921\) 1704.32 0.0609764
\(922\) 17357.9 0.620013
\(923\) 38395.3 1.36923
\(924\) −1880.50 −0.0669523
\(925\) 0 0
\(926\) −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 \(0.460803\pi\)
\(930\) 0 0
\(931\) 9436.31 0.332183
\(932\) 16384.9 0.575863
\(933\) 20560.8 0.721467
\(934\) 19751.7 0.691965
\(935\) 0 0
\(936\) 7790.88 0.272065
\(937\) 16074.5 0.560438 0.280219 0.959936i \(-0.409593\pi\)
0.280219 + 0.959936i \(0.409593\pi\)
\(938\) −6479.76 −0.225556
\(939\) −3416.77 −0.118746
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) 40.6659 0.00139763
\(947\) −35352.0 −1.21308 −0.606540 0.795053i \(-0.707443\pi\)
−0.606540 + 0.795053i \(0.707443\pi\)
\(948\) 15881.0 0.544085
\(949\) 14820.7 0.506956
\(950\) 0 0
\(951\) 9622.44 0.328106
\(952\) 21418.2 0.729168
\(953\) 19390.7 0.659103 0.329552 0.944138i \(-0.393102\pi\)
0.329552 + 0.944138i \(0.393102\pi\)
\(954\) 8995.45 0.305281
\(955\) 0 0
\(956\) 11561.9 0.391148
\(957\) −4920.06 −0.166189
\(958\) 16738.7 0.564511
\(959\) 7345.80 0.247349
\(960\) 0 0
\(961\) −26794.7 −0.899422
\(962\) 2865.64 0.0960416
\(963\) 7012.62 0.234661
\(964\) −10307.6 −0.344384
\(965\) 0 0
\(966\) −2961.27 −0.0986308
\(967\) −28643.6 −0.952551 −0.476275 0.879296i \(-0.658013\pi\)
−0.476275 + 0.879296i \(0.658013\pi\)
\(968\) 2562.43 0.0850821
\(969\) 11740.1 0.389213
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −8160.86 −0.267646
\(977\) −50095.5 −1.64043 −0.820213 0.572058i \(-0.806145\pi\)
−0.820213 + 0.572058i \(0.806145\pi\)
\(978\) 11550.0 0.377636
\(979\) −4244.67 −0.138570
\(980\) 0 0
\(981\) −13516.1 −0.439895
\(982\) −22747.6 −0.739211
\(983\) −14445.4 −0.468706 −0.234353 0.972152i \(-0.575297\pi\)
−0.234353 + 0.972152i \(0.575297\pi\)
\(984\) −21376.9 −0.692553
\(985\) 0 0
\(986\) 22980.9 0.742253
\(987\) −10245.0 −0.330399
\(988\) 9013.05 0.290226
\(989\) −146.056 −0.00469595
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) −12266.9 −0.390253
\(997\) 9137.45 0.290257 0.145128 0.989413i \(-0.453640\pi\)
0.145128 + 0.989413i \(0.453640\pi\)
\(998\) 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 825.4.a.m.1.1 2
3.2 odd 2 2475.4.a.n.1.2 2
5.2 odd 4 825.4.c.j.199.2 4
5.3 odd 4 825.4.c.j.199.3 4
5.4 even 2 165.4.a.c.1.2 2
15.14 odd 2 495.4.a.d.1.1 2
55.54 odd 2 1815.4.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.2 2 5.4 even 2
495.4.a.d.1.1 2 15.14 odd 2
825.4.a.m.1.1 2 1.1 even 1 trivial
825.4.c.j.199.2 4 5.2 odd 4
825.4.c.j.199.3 4 5.3 odd 4
1815.4.a.n.1.1 2 55.54 odd 2
2475.4.a.n.1.2 2 3.2 odd 2