Properties

Label 441.4.a.w.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6257832.1
Defining polynomial: \(x^{4} - 19 x^{2} + 42\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.05539\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.05539 q^{2} +8.44622 q^{4} -9.92039 q^{5} -1.80961 q^{8} +O(q^{10})\) \(q-4.05539 q^{2} +8.44622 q^{4} -9.92039 q^{5} -1.80961 q^{8} +40.2311 q^{10} -13.5396 q^{11} +18.5538 q^{13} -60.2311 q^{16} -93.7102 q^{17} +131.908 q^{19} -83.7899 q^{20} +54.9084 q^{22} -198.278 q^{23} -26.5858 q^{25} -75.2429 q^{26} +188.358 q^{29} -83.9244 q^{31} +258.738 q^{32} +380.032 q^{34} +80.1555 q^{37} -534.941 q^{38} +17.9520 q^{40} -385.828 q^{41} -397.048 q^{43} -114.359 q^{44} +804.096 q^{46} +272.277 q^{47} +107.816 q^{50} +156.709 q^{52} -36.9996 q^{53} +134.318 q^{55} -763.865 q^{58} +395.749 q^{59} +13.4738 q^{61} +340.347 q^{62} -567.435 q^{64} -184.061 q^{65} +340.522 q^{67} -791.498 q^{68} -211.140 q^{71} +486.124 q^{73} -325.062 q^{74} +1114.13 q^{76} +293.741 q^{79} +597.516 q^{80} +1564.69 q^{82} +889.635 q^{83} +929.643 q^{85} +1610.19 q^{86} +24.5014 q^{88} +1144.36 q^{89} -1674.70 q^{92} -1104.19 q^{94} -1308.58 q^{95} +1384.61 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + O(q^{10}) \) \( 4q + 6q^{4} + 22q^{10} + 102q^{13} - 102q^{16} + 222q^{19} - 86q^{22} + 366q^{25} + 220q^{31} + 1020q^{34} - 374q^{37} + 822q^{40} - 838q^{43} + 1716q^{46} - 40q^{52} + 2510q^{55} - 1694q^{58} + 1332q^{61} - 686q^{64} + 1890q^{67} + 1750q^{73} + 2456q^{76} + 8q^{79} + 2480q^{82} - 1116q^{85} + 2682q^{88} - 1416q^{94} + 3010q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.05539 −1.43380 −0.716899 0.697177i \(-0.754439\pi\)
−0.716899 + 0.697177i \(0.754439\pi\)
\(3\) 0 0
\(4\) 8.44622 1.05578
\(5\) −9.92039 −0.887307 −0.443654 0.896198i \(-0.646318\pi\)
−0.443654 + 0.896198i \(0.646318\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.80961 −0.0799740
\(9\) 0 0
\(10\) 40.2311 1.27222
\(11\) −13.5396 −0.371122 −0.185561 0.982633i \(-0.559410\pi\)
−0.185561 + 0.982633i \(0.559410\pi\)
\(12\) 0 0
\(13\) 18.5538 0.395838 0.197919 0.980218i \(-0.436582\pi\)
0.197919 + 0.980218i \(0.436582\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −60.2311 −0.941111
\(17\) −93.7102 −1.33695 −0.668473 0.743737i \(-0.733052\pi\)
−0.668473 + 0.743737i \(0.733052\pi\)
\(18\) 0 0
\(19\) 131.908 1.59273 0.796365 0.604816i \(-0.206753\pi\)
0.796365 + 0.604816i \(0.206753\pi\)
\(20\) −83.7899 −0.936799
\(21\) 0 0
\(22\) 54.9084 0.532115
\(23\) −198.278 −1.79756 −0.898779 0.438401i \(-0.855545\pi\)
−0.898779 + 0.438401i \(0.855545\pi\)
\(24\) 0 0
\(25\) −26.5858 −0.212686
\(26\) −75.2429 −0.567552
\(27\) 0 0
\(28\) 0 0
\(29\) 188.358 1.20611 0.603054 0.797700i \(-0.293950\pi\)
0.603054 + 0.797700i \(0.293950\pi\)
\(30\) 0 0
\(31\) −83.9244 −0.486235 −0.243117 0.969997i \(-0.578170\pi\)
−0.243117 + 0.969997i \(0.578170\pi\)
\(32\) 258.738 1.42934
\(33\) 0 0
\(34\) 380.032 1.91691
\(35\) 0 0
\(36\) 0 0
\(37\) 80.1555 0.356148 0.178074 0.984017i \(-0.443013\pi\)
0.178074 + 0.984017i \(0.443013\pi\)
\(38\) −534.941 −2.28365
\(39\) 0 0
\(40\) 17.9520 0.0709615
\(41\) −385.828 −1.46967 −0.734833 0.678249i \(-0.762739\pi\)
−0.734833 + 0.678249i \(0.762739\pi\)
\(42\) 0 0
\(43\) −397.048 −1.40812 −0.704061 0.710139i \(-0.748632\pi\)
−0.704061 + 0.710139i \(0.748632\pi\)
\(44\) −114.359 −0.391823
\(45\) 0 0
\(46\) 804.096 2.57734
\(47\) 272.277 0.845016 0.422508 0.906359i \(-0.361150\pi\)
0.422508 + 0.906359i \(0.361150\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 107.816 0.304949
\(51\) 0 0
\(52\) 156.709 0.417917
\(53\) −36.9996 −0.0958922 −0.0479461 0.998850i \(-0.515268\pi\)
−0.0479461 + 0.998850i \(0.515268\pi\)
\(54\) 0 0
\(55\) 134.318 0.329299
\(56\) 0 0
\(57\) 0 0
\(58\) −763.865 −1.72932
\(59\) 395.749 0.873256 0.436628 0.899642i \(-0.356173\pi\)
0.436628 + 0.899642i \(0.356173\pi\)
\(60\) 0 0
\(61\) 13.4738 0.0282810 0.0141405 0.999900i \(-0.495499\pi\)
0.0141405 + 0.999900i \(0.495499\pi\)
\(62\) 340.347 0.697162
\(63\) 0 0
\(64\) −567.435 −1.10827
\(65\) −184.061 −0.351230
\(66\) 0 0
\(67\) 340.522 0.620916 0.310458 0.950587i \(-0.399518\pi\)
0.310458 + 0.950587i \(0.399518\pi\)
\(68\) −791.498 −1.41152
\(69\) 0 0
\(70\) 0 0
\(71\) −211.140 −0.352925 −0.176463 0.984307i \(-0.556466\pi\)
−0.176463 + 0.984307i \(0.556466\pi\)
\(72\) 0 0
\(73\) 486.124 0.779404 0.389702 0.920941i \(-0.372578\pi\)
0.389702 + 0.920941i \(0.372578\pi\)
\(74\) −325.062 −0.510645
\(75\) 0 0
\(76\) 1114.13 1.68157
\(77\) 0 0
\(78\) 0 0
\(79\) 293.741 0.418335 0.209168 0.977880i \(-0.432925\pi\)
0.209168 + 0.977880i \(0.432925\pi\)
\(80\) 597.516 0.835055
\(81\) 0 0
\(82\) 1564.69 2.10720
\(83\) 889.635 1.17651 0.588253 0.808677i \(-0.299816\pi\)
0.588253 + 0.808677i \(0.299816\pi\)
\(84\) 0 0
\(85\) 929.643 1.18628
\(86\) 1610.19 2.01896
\(87\) 0 0
\(88\) 24.5014 0.0296801
\(89\) 1144.36 1.36295 0.681474 0.731843i \(-0.261339\pi\)
0.681474 + 0.731843i \(0.261339\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1674.70 −1.89782
\(93\) 0 0
\(94\) −1104.19 −1.21158
\(95\) −1308.58 −1.41324
\(96\) 0 0
\(97\) 1384.61 1.44933 0.724667 0.689099i \(-0.241993\pi\)
0.724667 + 0.689099i \(0.241993\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −224.549 −0.224549
\(101\) 1785.99 1.75953 0.879765 0.475409i \(-0.157700\pi\)
0.879765 + 0.475409i \(0.157700\pi\)
\(102\) 0 0
\(103\) 489.661 0.468425 0.234212 0.972185i \(-0.424749\pi\)
0.234212 + 0.972185i \(0.424749\pi\)
\(104\) −33.5750 −0.0316568
\(105\) 0 0
\(106\) 150.048 0.137490
\(107\) 282.068 0.254846 0.127423 0.991848i \(-0.459329\pi\)
0.127423 + 0.991848i \(0.459329\pi\)
\(108\) 0 0
\(109\) −287.231 −0.252401 −0.126201 0.992005i \(-0.540278\pi\)
−0.126201 + 0.992005i \(0.540278\pi\)
\(110\) −544.713 −0.472149
\(111\) 0 0
\(112\) 0 0
\(113\) 1895.21 1.57776 0.788879 0.614548i \(-0.210662\pi\)
0.788879 + 0.614548i \(0.210662\pi\)
\(114\) 0 0
\(115\) 1967.00 1.59499
\(116\) 1590.91 1.27338
\(117\) 0 0
\(118\) −1604.92 −1.25207
\(119\) 0 0
\(120\) 0 0
\(121\) −1147.68 −0.862268
\(122\) −54.6415 −0.0405493
\(123\) 0 0
\(124\) −708.844 −0.513356
\(125\) 1503.79 1.07603
\(126\) 0 0
\(127\) −1222.92 −0.854463 −0.427231 0.904142i \(-0.640511\pi\)
−0.427231 + 0.904142i \(0.640511\pi\)
\(128\) 231.269 0.159699
\(129\) 0 0
\(130\) 746.439 0.503593
\(131\) 1190.61 0.794074 0.397037 0.917803i \(-0.370038\pi\)
0.397037 + 0.917803i \(0.370038\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1380.95 −0.890268
\(135\) 0 0
\(136\) 169.579 0.106921
\(137\) 106.961 0.0667031 0.0333516 0.999444i \(-0.489382\pi\)
0.0333516 + 0.999444i \(0.489382\pi\)
\(138\) 0 0
\(139\) −1096.78 −0.669262 −0.334631 0.942349i \(-0.608612\pi\)
−0.334631 + 0.942349i \(0.608612\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 856.256 0.506024
\(143\) −251.211 −0.146904
\(144\) 0 0
\(145\) −1868.58 −1.07019
\(146\) −1971.42 −1.11751
\(147\) 0 0
\(148\) 677.012 0.376014
\(149\) 716.662 0.394035 0.197018 0.980400i \(-0.436874\pi\)
0.197018 + 0.980400i \(0.436874\pi\)
\(150\) 0 0
\(151\) 154.368 0.0831939 0.0415970 0.999134i \(-0.486755\pi\)
0.0415970 + 0.999134i \(0.486755\pi\)
\(152\) −238.702 −0.127377
\(153\) 0 0
\(154\) 0 0
\(155\) 832.564 0.431439
\(156\) 0 0
\(157\) −2093.75 −1.06433 −0.532163 0.846642i \(-0.678621\pi\)
−0.532163 + 0.846642i \(0.678621\pi\)
\(158\) −1191.24 −0.599808
\(159\) 0 0
\(160\) −2566.78 −1.26826
\(161\) 0 0
\(162\) 0 0
\(163\) 3007.86 1.44536 0.722680 0.691183i \(-0.242910\pi\)
0.722680 + 0.691183i \(0.242910\pi\)
\(164\) −3258.79 −1.55164
\(165\) 0 0
\(166\) −3607.82 −1.68687
\(167\) −2230.43 −1.03351 −0.516754 0.856134i \(-0.672860\pi\)
−0.516754 + 0.856134i \(0.672860\pi\)
\(168\) 0 0
\(169\) −1852.76 −0.843312
\(170\) −3770.07 −1.70089
\(171\) 0 0
\(172\) −3353.56 −1.48666
\(173\) −563.170 −0.247497 −0.123749 0.992314i \(-0.539492\pi\)
−0.123749 + 0.992314i \(0.539492\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 815.506 0.349267
\(177\) 0 0
\(178\) −4640.85 −1.95419
\(179\) 1839.50 0.768104 0.384052 0.923312i \(-0.374528\pi\)
0.384052 + 0.923312i \(0.374528\pi\)
\(180\) 0 0
\(181\) 2324.71 0.954664 0.477332 0.878723i \(-0.341604\pi\)
0.477332 + 0.878723i \(0.341604\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 358.805 0.143758
\(185\) −795.175 −0.316013
\(186\) 0 0
\(187\) 1268.80 0.496170
\(188\) 2299.71 0.892149
\(189\) 0 0
\(190\) 5306.82 2.02630
\(191\) −3127.27 −1.18472 −0.592360 0.805673i \(-0.701804\pi\)
−0.592360 + 0.805673i \(0.701804\pi\)
\(192\) 0 0
\(193\) −3709.29 −1.38342 −0.691711 0.722175i \(-0.743143\pi\)
−0.691711 + 0.722175i \(0.743143\pi\)
\(194\) −5615.12 −2.07805
\(195\) 0 0
\(196\) 0 0
\(197\) 851.150 0.307827 0.153913 0.988084i \(-0.450812\pi\)
0.153913 + 0.988084i \(0.450812\pi\)
\(198\) 0 0
\(199\) 3397.78 1.21036 0.605182 0.796087i \(-0.293100\pi\)
0.605182 + 0.796087i \(0.293100\pi\)
\(200\) 48.1098 0.0170094
\(201\) 0 0
\(202\) −7242.89 −2.52281
\(203\) 0 0
\(204\) 0 0
\(205\) 3827.57 1.30404
\(206\) −1985.77 −0.671627
\(207\) 0 0
\(208\) −1117.51 −0.372527
\(209\) −1785.99 −0.591098
\(210\) 0 0
\(211\) 216.732 0.0707132 0.0353566 0.999375i \(-0.488743\pi\)
0.0353566 + 0.999375i \(0.488743\pi\)
\(212\) −312.507 −0.101241
\(213\) 0 0
\(214\) −1143.90 −0.365398
\(215\) 3938.87 1.24944
\(216\) 0 0
\(217\) 0 0
\(218\) 1164.84 0.361893
\(219\) 0 0
\(220\) 1134.48 0.347667
\(221\) −1738.68 −0.529214
\(222\) 0 0
\(223\) −2254.86 −0.677115 −0.338558 0.940946i \(-0.609939\pi\)
−0.338558 + 0.940946i \(0.609939\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7685.84 −2.26219
\(227\) −3390.80 −0.991433 −0.495716 0.868484i \(-0.665094\pi\)
−0.495716 + 0.868484i \(0.665094\pi\)
\(228\) 0 0
\(229\) 2587.85 0.746768 0.373384 0.927677i \(-0.378197\pi\)
0.373384 + 0.927677i \(0.378197\pi\)
\(230\) −7976.95 −2.28689
\(231\) 0 0
\(232\) −340.853 −0.0964574
\(233\) −954.420 −0.268353 −0.134176 0.990957i \(-0.542839\pi\)
−0.134176 + 0.990957i \(0.542839\pi\)
\(234\) 0 0
\(235\) −2701.10 −0.749788
\(236\) 3342.58 0.921964
\(237\) 0 0
\(238\) 0 0
\(239\) 199.504 0.0539951 0.0269976 0.999635i \(-0.491405\pi\)
0.0269976 + 0.999635i \(0.491405\pi\)
\(240\) 0 0
\(241\) 4794.43 1.28148 0.640739 0.767759i \(-0.278628\pi\)
0.640739 + 0.767759i \(0.278628\pi\)
\(242\) 4654.29 1.23632
\(243\) 0 0
\(244\) 113.803 0.0298585
\(245\) 0 0
\(246\) 0 0
\(247\) 2447.40 0.630463
\(248\) 151.870 0.0388862
\(249\) 0 0
\(250\) −6098.46 −1.54280
\(251\) 6249.73 1.57163 0.785816 0.618460i \(-0.212243\pi\)
0.785816 + 0.618460i \(0.212243\pi\)
\(252\) 0 0
\(253\) 2684.61 0.667114
\(254\) 4959.43 1.22513
\(255\) 0 0
\(256\) 3601.59 0.879294
\(257\) −3837.85 −0.931512 −0.465756 0.884913i \(-0.654218\pi\)
−0.465756 + 0.884913i \(0.654218\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1554.62 −0.370821
\(261\) 0 0
\(262\) −4828.38 −1.13854
\(263\) 207.203 0.0485806 0.0242903 0.999705i \(-0.492267\pi\)
0.0242903 + 0.999705i \(0.492267\pi\)
\(264\) 0 0
\(265\) 367.051 0.0850858
\(266\) 0 0
\(267\) 0 0
\(268\) 2876.12 0.655549
\(269\) 4804.08 1.08888 0.544442 0.838798i \(-0.316741\pi\)
0.544442 + 0.838798i \(0.316741\pi\)
\(270\) 0 0
\(271\) 3215.75 0.720823 0.360411 0.932793i \(-0.382636\pi\)
0.360411 + 0.932793i \(0.382636\pi\)
\(272\) 5644.27 1.25821
\(273\) 0 0
\(274\) −433.771 −0.0956388
\(275\) 359.961 0.0789326
\(276\) 0 0
\(277\) 2054.39 0.445619 0.222810 0.974862i \(-0.428477\pi\)
0.222810 + 0.974862i \(0.428477\pi\)
\(278\) 4447.87 0.959587
\(279\) 0 0
\(280\) 0 0
\(281\) 1768.61 0.375468 0.187734 0.982220i \(-0.439886\pi\)
0.187734 + 0.982220i \(0.439886\pi\)
\(282\) 0 0
\(283\) 2340.53 0.491625 0.245813 0.969317i \(-0.420945\pi\)
0.245813 + 0.969317i \(0.420945\pi\)
\(284\) −1783.34 −0.372611
\(285\) 0 0
\(286\) 1018.76 0.210631
\(287\) 0 0
\(288\) 0 0
\(289\) 3868.61 0.787423
\(290\) 7577.84 1.53443
\(291\) 0 0
\(292\) 4105.91 0.822877
\(293\) −3633.47 −0.724470 −0.362235 0.932087i \(-0.617986\pi\)
−0.362235 + 0.932087i \(0.617986\pi\)
\(294\) 0 0
\(295\) −3925.98 −0.774846
\(296\) −145.050 −0.0284826
\(297\) 0 0
\(298\) −2906.35 −0.564967
\(299\) −3678.81 −0.711542
\(300\) 0 0
\(301\) 0 0
\(302\) −626.023 −0.119283
\(303\) 0 0
\(304\) −7944.99 −1.49894
\(305\) −133.665 −0.0250939
\(306\) 0 0
\(307\) 5954.32 1.10694 0.553471 0.832868i \(-0.313303\pi\)
0.553471 + 0.832868i \(0.313303\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3376.37 −0.618597
\(311\) 1180.09 0.215167 0.107584 0.994196i \(-0.465689\pi\)
0.107584 + 0.994196i \(0.465689\pi\)
\(312\) 0 0
\(313\) −9746.25 −1.76003 −0.880017 0.474943i \(-0.842469\pi\)
−0.880017 + 0.474943i \(0.842469\pi\)
\(314\) 8490.97 1.52603
\(315\) 0 0
\(316\) 2481.00 0.441669
\(317\) 8591.91 1.52230 0.761151 0.648574i \(-0.224634\pi\)
0.761151 + 0.648574i \(0.224634\pi\)
\(318\) 0 0
\(319\) −2550.29 −0.447614
\(320\) 5629.18 0.983377
\(321\) 0 0
\(322\) 0 0
\(323\) −12361.2 −2.12939
\(324\) 0 0
\(325\) −493.267 −0.0841892
\(326\) −12198.0 −2.07235
\(327\) 0 0
\(328\) 698.197 0.117535
\(329\) 0 0
\(330\) 0 0
\(331\) −5251.20 −0.872000 −0.436000 0.899947i \(-0.643605\pi\)
−0.436000 + 0.899947i \(0.643605\pi\)
\(332\) 7514.05 1.24213
\(333\) 0 0
\(334\) 9045.28 1.48184
\(335\) −3378.11 −0.550943
\(336\) 0 0
\(337\) −8496.45 −1.37339 −0.686693 0.726947i \(-0.740938\pi\)
−0.686693 + 0.726947i \(0.740938\pi\)
\(338\) 7513.66 1.20914
\(339\) 0 0
\(340\) 7851.97 1.25245
\(341\) 1136.30 0.180453
\(342\) 0 0
\(343\) 0 0
\(344\) 718.500 0.112613
\(345\) 0 0
\(346\) 2283.88 0.354861
\(347\) −5830.15 −0.901956 −0.450978 0.892535i \(-0.648925\pi\)
−0.450978 + 0.892535i \(0.648925\pi\)
\(348\) 0 0
\(349\) 1811.13 0.277786 0.138893 0.990307i \(-0.455646\pi\)
0.138893 + 0.990307i \(0.455646\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3503.21 −0.530459
\(353\) 3327.76 0.501753 0.250876 0.968019i \(-0.419281\pi\)
0.250876 + 0.968019i \(0.419281\pi\)
\(354\) 0 0
\(355\) 2094.59 0.313153
\(356\) 9665.55 1.43897
\(357\) 0 0
\(358\) −7459.89 −1.10131
\(359\) −870.861 −0.128029 −0.0640143 0.997949i \(-0.520390\pi\)
−0.0640143 + 0.997949i \(0.520390\pi\)
\(360\) 0 0
\(361\) 10540.8 1.53679
\(362\) −9427.61 −1.36880
\(363\) 0 0
\(364\) 0 0
\(365\) −4822.54 −0.691570
\(366\) 0 0
\(367\) 1174.72 0.167084 0.0835418 0.996504i \(-0.473377\pi\)
0.0835418 + 0.996504i \(0.473377\pi\)
\(368\) 11942.5 1.69170
\(369\) 0 0
\(370\) 3224.75 0.453099
\(371\) 0 0
\(372\) 0 0
\(373\) 3628.33 0.503667 0.251834 0.967771i \(-0.418966\pi\)
0.251834 + 0.967771i \(0.418966\pi\)
\(374\) −5145.48 −0.711408
\(375\) 0 0
\(376\) −492.715 −0.0675793
\(377\) 3494.75 0.477424
\(378\) 0 0
\(379\) 7321.99 0.992362 0.496181 0.868219i \(-0.334735\pi\)
0.496181 + 0.868219i \(0.334735\pi\)
\(380\) −11052.6 −1.49207
\(381\) 0 0
\(382\) 12682.3 1.69865
\(383\) −7354.89 −0.981247 −0.490623 0.871372i \(-0.663231\pi\)
−0.490623 + 0.871372i \(0.663231\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15042.6 1.98355
\(387\) 0 0
\(388\) 11694.7 1.53018
\(389\) 9069.62 1.18213 0.591064 0.806624i \(-0.298708\pi\)
0.591064 + 0.806624i \(0.298708\pi\)
\(390\) 0 0
\(391\) 18580.7 2.40324
\(392\) 0 0
\(393\) 0 0
\(394\) −3451.75 −0.441362
\(395\) −2914.03 −0.371192
\(396\) 0 0
\(397\) 7376.83 0.932575 0.466288 0.884633i \(-0.345591\pi\)
0.466288 + 0.884633i \(0.345591\pi\)
\(398\) −13779.4 −1.73542
\(399\) 0 0
\(400\) 1601.29 0.200161
\(401\) −2853.51 −0.355356 −0.177678 0.984089i \(-0.556858\pi\)
−0.177678 + 0.984089i \(0.556858\pi\)
\(402\) 0 0
\(403\) −1557.12 −0.192470
\(404\) 15084.9 1.85767
\(405\) 0 0
\(406\) 0 0
\(407\) −1085.27 −0.132175
\(408\) 0 0
\(409\) 11260.1 1.36130 0.680652 0.732607i \(-0.261696\pi\)
0.680652 + 0.732607i \(0.261696\pi\)
\(410\) −15522.3 −1.86974
\(411\) 0 0
\(412\) 4135.79 0.494553
\(413\) 0 0
\(414\) 0 0
\(415\) −8825.53 −1.04392
\(416\) 4800.56 0.565786
\(417\) 0 0
\(418\) 7242.89 0.847515
\(419\) −9221.47 −1.07517 −0.537587 0.843208i \(-0.680664\pi\)
−0.537587 + 0.843208i \(0.680664\pi\)
\(420\) 0 0
\(421\) −8520.28 −0.986349 −0.493175 0.869930i \(-0.664164\pi\)
−0.493175 + 0.869930i \(0.664164\pi\)
\(422\) −878.936 −0.101388
\(423\) 0 0
\(424\) 66.9547 0.00766889
\(425\) 2491.36 0.284350
\(426\) 0 0
\(427\) 0 0
\(428\) 2382.41 0.269061
\(429\) 0 0
\(430\) −15973.7 −1.79144
\(431\) 9162.10 1.02395 0.511976 0.859000i \(-0.328914\pi\)
0.511976 + 0.859000i \(0.328914\pi\)
\(432\) 0 0
\(433\) −10976.2 −1.21820 −0.609100 0.793093i \(-0.708469\pi\)
−0.609100 + 0.793093i \(0.708469\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2426.02 −0.266480
\(437\) −26154.6 −2.86303
\(438\) 0 0
\(439\) 3983.18 0.433045 0.216523 0.976278i \(-0.430528\pi\)
0.216523 + 0.976278i \(0.430528\pi\)
\(440\) −243.063 −0.0263354
\(441\) 0 0
\(442\) 7051.03 0.758786
\(443\) 4524.45 0.485244 0.242622 0.970121i \(-0.421993\pi\)
0.242622 + 0.970121i \(0.421993\pi\)
\(444\) 0 0
\(445\) −11352.5 −1.20935
\(446\) 9144.35 0.970847
\(447\) 0 0
\(448\) 0 0
\(449\) −2076.49 −0.218253 −0.109127 0.994028i \(-0.534805\pi\)
−0.109127 + 0.994028i \(0.534805\pi\)
\(450\) 0 0
\(451\) 5223.96 0.545425
\(452\) 16007.4 1.66576
\(453\) 0 0
\(454\) 13751.0 1.42151
\(455\) 0 0
\(456\) 0 0
\(457\) −1847.59 −0.189117 −0.0945587 0.995519i \(-0.530144\pi\)
−0.0945587 + 0.995519i \(0.530144\pi\)
\(458\) −10494.7 −1.07071
\(459\) 0 0
\(460\) 16613.7 1.68395
\(461\) 876.945 0.0885974 0.0442987 0.999018i \(-0.485895\pi\)
0.0442987 + 0.999018i \(0.485895\pi\)
\(462\) 0 0
\(463\) 16245.2 1.63062 0.815310 0.579025i \(-0.196566\pi\)
0.815310 + 0.579025i \(0.196566\pi\)
\(464\) −11345.0 −1.13508
\(465\) 0 0
\(466\) 3870.55 0.384763
\(467\) −18961.8 −1.87890 −0.939449 0.342689i \(-0.888662\pi\)
−0.939449 + 0.342689i \(0.888662\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 10954.0 1.07505
\(471\) 0 0
\(472\) −716.149 −0.0698378
\(473\) 5375.87 0.522585
\(474\) 0 0
\(475\) −3506.89 −0.338752
\(476\) 0 0
\(477\) 0 0
\(478\) −809.067 −0.0774181
\(479\) −8377.76 −0.799143 −0.399572 0.916702i \(-0.630841\pi\)
−0.399572 + 0.916702i \(0.630841\pi\)
\(480\) 0 0
\(481\) 1487.19 0.140977
\(482\) −19443.3 −1.83738
\(483\) 0 0
\(484\) −9693.55 −0.910364
\(485\) −13735.8 −1.28600
\(486\) 0 0
\(487\) −4558.85 −0.424191 −0.212096 0.977249i \(-0.568029\pi\)
−0.212096 + 0.977249i \(0.568029\pi\)
\(488\) −24.3822 −0.00226175
\(489\) 0 0
\(490\) 0 0
\(491\) 15809.9 1.45314 0.726570 0.687092i \(-0.241113\pi\)
0.726570 + 0.687092i \(0.241113\pi\)
\(492\) 0 0
\(493\) −17651.1 −1.61250
\(494\) −9925.17 −0.903957
\(495\) 0 0
\(496\) 5054.86 0.457601
\(497\) 0 0
\(498\) 0 0
\(499\) 13386.1 1.20089 0.600444 0.799667i \(-0.294990\pi\)
0.600444 + 0.799667i \(0.294990\pi\)
\(500\) 12701.3 1.13604
\(501\) 0 0
\(502\) −25345.1 −2.25340
\(503\) 5720.55 0.507091 0.253545 0.967323i \(-0.418403\pi\)
0.253545 + 0.967323i \(0.418403\pi\)
\(504\) 0 0
\(505\) −17717.7 −1.56124
\(506\) −10887.1 −0.956507
\(507\) 0 0
\(508\) −10329.1 −0.902123
\(509\) −15293.2 −1.33175 −0.665873 0.746065i \(-0.731941\pi\)
−0.665873 + 0.746065i \(0.731941\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −16456.0 −1.42043
\(513\) 0 0
\(514\) 15564.0 1.33560
\(515\) −4857.63 −0.415637
\(516\) 0 0
\(517\) −3686.53 −0.313604
\(518\) 0 0
\(519\) 0 0
\(520\) 333.078 0.0280893
\(521\) 4368.69 0.367363 0.183681 0.982986i \(-0.441199\pi\)
0.183681 + 0.982986i \(0.441199\pi\)
\(522\) 0 0
\(523\) −2422.17 −0.202512 −0.101256 0.994860i \(-0.532286\pi\)
−0.101256 + 0.994860i \(0.532286\pi\)
\(524\) 10056.1 0.838366
\(525\) 0 0
\(526\) −840.291 −0.0696548
\(527\) 7864.58 0.650069
\(528\) 0 0
\(529\) 27147.2 2.23122
\(530\) −1488.54 −0.121996
\(531\) 0 0
\(532\) 0 0
\(533\) −7158.57 −0.581749
\(534\) 0 0
\(535\) −2798.23 −0.226127
\(536\) −616.210 −0.0496571
\(537\) 0 0
\(538\) −19482.4 −1.56124
\(539\) 0 0
\(540\) 0 0
\(541\) 13162.5 1.04603 0.523014 0.852324i \(-0.324807\pi\)
0.523014 + 0.852324i \(0.324807\pi\)
\(542\) −13041.1 −1.03351
\(543\) 0 0
\(544\) −24246.4 −1.91095
\(545\) 2849.45 0.223958
\(546\) 0 0
\(547\) 12112.4 0.946778 0.473389 0.880853i \(-0.343031\pi\)
0.473389 + 0.880853i \(0.343031\pi\)
\(548\) 903.420 0.0704237
\(549\) 0 0
\(550\) −1459.78 −0.113173
\(551\) 24846.0 1.92101
\(552\) 0 0
\(553\) 0 0
\(554\) −8331.38 −0.638928
\(555\) 0 0
\(556\) −9263.63 −0.706592
\(557\) −8359.65 −0.635924 −0.317962 0.948103i \(-0.602998\pi\)
−0.317962 + 0.948103i \(0.602998\pi\)
\(558\) 0 0
\(559\) −7366.74 −0.557388
\(560\) 0 0
\(561\) 0 0
\(562\) −7172.42 −0.538346
\(563\) 13638.4 1.02094 0.510471 0.859895i \(-0.329471\pi\)
0.510471 + 0.859895i \(0.329471\pi\)
\(564\) 0 0
\(565\) −18801.3 −1.39996
\(566\) −9491.76 −0.704891
\(567\) 0 0
\(568\) 382.080 0.0282249
\(569\) 15491.3 1.14135 0.570677 0.821175i \(-0.306681\pi\)
0.570677 + 0.821175i \(0.306681\pi\)
\(570\) 0 0
\(571\) −4648.15 −0.340664 −0.170332 0.985387i \(-0.554484\pi\)
−0.170332 + 0.985387i \(0.554484\pi\)
\(572\) −2121.78 −0.155098
\(573\) 0 0
\(574\) 0 0
\(575\) 5271.38 0.382316
\(576\) 0 0
\(577\) −5479.06 −0.395314 −0.197657 0.980271i \(-0.563333\pi\)
−0.197657 + 0.980271i \(0.563333\pi\)
\(578\) −15688.7 −1.12901
\(579\) 0 0
\(580\) −15782.5 −1.12988
\(581\) 0 0
\(582\) 0 0
\(583\) 500.960 0.0355877
\(584\) −879.692 −0.0623321
\(585\) 0 0
\(586\) 14735.2 1.03874
\(587\) 4408.22 0.309960 0.154980 0.987918i \(-0.450469\pi\)
0.154980 + 0.987918i \(0.450469\pi\)
\(588\) 0 0
\(589\) −11070.3 −0.774441
\(590\) 15921.4 1.11097
\(591\) 0 0
\(592\) −4827.86 −0.335175
\(593\) 2815.26 0.194956 0.0974779 0.995238i \(-0.468922\pi\)
0.0974779 + 0.995238i \(0.468922\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6053.09 0.416014
\(597\) 0 0
\(598\) 14919.0 1.02021
\(599\) 19719.1 1.34507 0.672537 0.740064i \(-0.265205\pi\)
0.672537 + 0.740064i \(0.265205\pi\)
\(600\) 0 0
\(601\) −13982.8 −0.949033 −0.474517 0.880247i \(-0.657377\pi\)
−0.474517 + 0.880247i \(0.657377\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1303.83 0.0878343
\(605\) 11385.4 0.765097
\(606\) 0 0
\(607\) 13388.3 0.895245 0.447622 0.894223i \(-0.352271\pi\)
0.447622 + 0.894223i \(0.352271\pi\)
\(608\) 34129.7 2.27655
\(609\) 0 0
\(610\) 542.065 0.0359797
\(611\) 5051.77 0.334489
\(612\) 0 0
\(613\) −27791.9 −1.83116 −0.915582 0.402131i \(-0.868269\pi\)
−0.915582 + 0.402131i \(0.868269\pi\)
\(614\) −24147.1 −1.58713
\(615\) 0 0
\(616\) 0 0
\(617\) −19107.2 −1.24672 −0.623361 0.781935i \(-0.714233\pi\)
−0.623361 + 0.781935i \(0.714233\pi\)
\(618\) 0 0
\(619\) 1092.94 0.0709675 0.0354837 0.999370i \(-0.488703\pi\)
0.0354837 + 0.999370i \(0.488703\pi\)
\(620\) 7032.02 0.455504
\(621\) 0 0
\(622\) −4785.75 −0.308507
\(623\) 0 0
\(624\) 0 0
\(625\) −11595.0 −0.742078
\(626\) 39524.9 2.52353
\(627\) 0 0
\(628\) −17684.3 −1.12369
\(629\) −7511.40 −0.476151
\(630\) 0 0
\(631\) 19235.2 1.21353 0.606767 0.794879i \(-0.292466\pi\)
0.606767 + 0.794879i \(0.292466\pi\)
\(632\) −531.556 −0.0334560
\(633\) 0 0
\(634\) −34843.6 −2.18267
\(635\) 12131.9 0.758171
\(636\) 0 0
\(637\) 0 0
\(638\) 10342.4 0.641788
\(639\) 0 0
\(640\) −2294.28 −0.141702
\(641\) −19950.7 −1.22933 −0.614667 0.788787i \(-0.710710\pi\)
−0.614667 + 0.788787i \(0.710710\pi\)
\(642\) 0 0
\(643\) 688.125 0.0422037 0.0211019 0.999777i \(-0.493283\pi\)
0.0211019 + 0.999777i \(0.493283\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 50129.4 3.05312
\(647\) 10966.2 0.666345 0.333173 0.942866i \(-0.391881\pi\)
0.333173 + 0.942866i \(0.391881\pi\)
\(648\) 0 0
\(649\) −5358.28 −0.324085
\(650\) 2000.39 0.120710
\(651\) 0 0
\(652\) 25405.0 1.52598
\(653\) −12925.1 −0.774575 −0.387287 0.921959i \(-0.626588\pi\)
−0.387287 + 0.921959i \(0.626588\pi\)
\(654\) 0 0
\(655\) −11811.3 −0.704588
\(656\) 23238.9 1.38312
\(657\) 0 0
\(658\) 0 0
\(659\) −11779.0 −0.696273 −0.348137 0.937444i \(-0.613185\pi\)
−0.348137 + 0.937444i \(0.613185\pi\)
\(660\) 0 0
\(661\) −25040.1 −1.47344 −0.736721 0.676196i \(-0.763627\pi\)
−0.736721 + 0.676196i \(0.763627\pi\)
\(662\) 21295.7 1.25027
\(663\) 0 0
\(664\) −1609.89 −0.0940900
\(665\) 0 0
\(666\) 0 0
\(667\) −37347.2 −2.16805
\(668\) −18838.7 −1.09116
\(669\) 0 0
\(670\) 13699.6 0.789941
\(671\) −182.430 −0.0104957
\(672\) 0 0
\(673\) 4104.64 0.235100 0.117550 0.993067i \(-0.462496\pi\)
0.117550 + 0.993067i \(0.462496\pi\)
\(674\) 34456.5 1.96916
\(675\) 0 0
\(676\) −15648.8 −0.890350
\(677\) 12153.4 0.689945 0.344973 0.938613i \(-0.387888\pi\)
0.344973 + 0.938613i \(0.387888\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1682.29 −0.0948717
\(681\) 0 0
\(682\) −4608.16 −0.258733
\(683\) −20414.0 −1.14366 −0.571830 0.820372i \(-0.693766\pi\)
−0.571830 + 0.820372i \(0.693766\pi\)
\(684\) 0 0
\(685\) −1061.10 −0.0591862
\(686\) 0 0
\(687\) 0 0
\(688\) 23914.6 1.32520
\(689\) −686.483 −0.0379578
\(690\) 0 0
\(691\) 16093.5 0.886001 0.443000 0.896521i \(-0.353914\pi\)
0.443000 + 0.896521i \(0.353914\pi\)
\(692\) −4756.66 −0.261302
\(693\) 0 0
\(694\) 23643.6 1.29322
\(695\) 10880.5 0.593841
\(696\) 0 0
\(697\) 36156.1 1.96486
\(698\) −7344.83 −0.398290
\(699\) 0 0
\(700\) 0 0
\(701\) 20803.0 1.12085 0.560426 0.828204i \(-0.310637\pi\)
0.560426 + 0.828204i \(0.310637\pi\)
\(702\) 0 0
\(703\) 10573.2 0.567248
\(704\) 7682.84 0.411304
\(705\) 0 0
\(706\) −13495.4 −0.719412
\(707\) 0 0
\(708\) 0 0
\(709\) 141.492 0.00749484 0.00374742 0.999993i \(-0.498807\pi\)
0.00374742 + 0.999993i \(0.498807\pi\)
\(710\) −8494.40 −0.448999
\(711\) 0 0
\(712\) −2070.85 −0.109000
\(713\) 16640.4 0.874035
\(714\) 0 0
\(715\) 2492.11 0.130349
\(716\) 15536.8 0.810947
\(717\) 0 0
\(718\) 3531.68 0.183567
\(719\) −6664.46 −0.345678 −0.172839 0.984950i \(-0.555294\pi\)
−0.172839 + 0.984950i \(0.555294\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −42747.2 −2.20345
\(723\) 0 0
\(724\) 19635.0 1.00791
\(725\) −5007.64 −0.256523
\(726\) 0 0
\(727\) −4837.23 −0.246772 −0.123386 0.992359i \(-0.539375\pi\)
−0.123386 + 0.992359i \(0.539375\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 19557.3 0.991572
\(731\) 37207.5 1.88258
\(732\) 0 0
\(733\) −31602.5 −1.59245 −0.796225 0.605001i \(-0.793173\pi\)
−0.796225 + 0.605001i \(0.793173\pi\)
\(734\) −4763.93 −0.239564
\(735\) 0 0
\(736\) −51302.0 −2.56932
\(737\) −4610.53 −0.230436
\(738\) 0 0
\(739\) −8227.58 −0.409549 −0.204774 0.978809i \(-0.565646\pi\)
−0.204774 + 0.978809i \(0.565646\pi\)
\(740\) −6716.22 −0.333639
\(741\) 0 0
\(742\) 0 0
\(743\) 37020.3 1.82792 0.913959 0.405805i \(-0.133009\pi\)
0.913959 + 0.405805i \(0.133009\pi\)
\(744\) 0 0
\(745\) −7109.57 −0.349630
\(746\) −14714.3 −0.722158
\(747\) 0 0
\(748\) 10716.6 0.523846
\(749\) 0 0
\(750\) 0 0
\(751\) −25149.9 −1.22202 −0.611008 0.791625i \(-0.709236\pi\)
−0.611008 + 0.791625i \(0.709236\pi\)
\(752\) −16399.6 −0.795254
\(753\) 0 0
\(754\) −14172.6 −0.684529
\(755\) −1531.39 −0.0738186
\(756\) 0 0
\(757\) 20460.8 0.982377 0.491189 0.871053i \(-0.336563\pi\)
0.491189 + 0.871053i \(0.336563\pi\)
\(758\) −29693.6 −1.42285
\(759\) 0 0
\(760\) 2368.02 0.113023
\(761\) 32659.1 1.55571 0.777853 0.628447i \(-0.216309\pi\)
0.777853 + 0.628447i \(0.216309\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −26413.7 −1.25080
\(765\) 0 0
\(766\) 29827.0 1.40691
\(767\) 7342.64 0.345668
\(768\) 0 0
\(769\) −11005.3 −0.516075 −0.258037 0.966135i \(-0.583076\pi\)
−0.258037 + 0.966135i \(0.583076\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −31329.5 −1.46059
\(773\) −3038.18 −0.141366 −0.0706829 0.997499i \(-0.522518\pi\)
−0.0706829 + 0.997499i \(0.522518\pi\)
\(774\) 0 0
\(775\) 2231.20 0.103415
\(776\) −2505.59 −0.115909
\(777\) 0 0
\(778\) −36780.9 −1.69493
\(779\) −50894.0 −2.34078
\(780\) 0 0
\(781\) 2858.75 0.130979
\(782\) −75352.0 −3.44576
\(783\) 0 0
\(784\) 0 0
\(785\) 20770.8 0.944384
\(786\) 0 0
\(787\) −12306.8 −0.557419 −0.278710 0.960375i \(-0.589907\pi\)
−0.278710 + 0.960375i \(0.589907\pi\)
\(788\) 7189.00 0.324997
\(789\) 0 0
\(790\) 11817.5 0.532214
\(791\) 0 0
\(792\) 0 0
\(793\) 249.990 0.0111947
\(794\) −29915.9 −1.33712
\(795\) 0 0
\(796\) 28698.4 1.27788
\(797\) 3007.06 0.133646 0.0668228 0.997765i \(-0.478714\pi\)
0.0668228 + 0.997765i \(0.478714\pi\)
\(798\) 0 0
\(799\) −25515.2 −1.12974
\(800\) −6878.74 −0.304000
\(801\) 0 0
\(802\) 11572.1 0.509508
\(803\) −6581.92 −0.289254
\(804\) 0 0
\(805\) 0 0
\(806\) 6314.72 0.275963
\(807\) 0 0
\(808\) −3231.94 −0.140717
\(809\) −10585.0 −0.460009 −0.230005 0.973190i \(-0.573874\pi\)
−0.230005 + 0.973190i \(0.573874\pi\)
\(810\) 0 0
\(811\) −18217.8 −0.788796 −0.394398 0.918940i \(-0.629047\pi\)
−0.394398 + 0.918940i \(0.629047\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4401.22 0.189512
\(815\) −29839.1 −1.28248
\(816\) 0 0
\(817\) −52374.0 −2.24276
\(818\) −45663.9 −1.95184
\(819\) 0 0
\(820\) 32328.5 1.37678
\(821\) −25596.5 −1.08809 −0.544047 0.839055i \(-0.683109\pi\)
−0.544047 + 0.839055i \(0.683109\pi\)
\(822\) 0 0
\(823\) 43778.9 1.85424 0.927118 0.374769i \(-0.122278\pi\)
0.927118 + 0.374769i \(0.122278\pi\)
\(824\) −886.094 −0.0374618
\(825\) 0 0
\(826\) 0 0
\(827\) 2735.78 0.115033 0.0575166 0.998345i \(-0.481682\pi\)
0.0575166 + 0.998345i \(0.481682\pi\)
\(828\) 0 0
\(829\) −31144.2 −1.30480 −0.652402 0.757873i \(-0.726239\pi\)
−0.652402 + 0.757873i \(0.726239\pi\)
\(830\) 35791.0 1.49677
\(831\) 0 0
\(832\) −10528.1 −0.438696
\(833\) 0 0
\(834\) 0 0
\(835\) 22126.8 0.917039
\(836\) −15084.9 −0.624068
\(837\) 0 0
\(838\) 37396.7 1.54158
\(839\) −14977.3 −0.616300 −0.308150 0.951338i \(-0.599710\pi\)
−0.308150 + 0.951338i \(0.599710\pi\)
\(840\) 0 0
\(841\) 11089.6 0.454698
\(842\) 34553.1 1.41423
\(843\) 0 0
\(844\) 1830.57 0.0746574
\(845\) 18380.1 0.748277
\(846\) 0 0
\(847\) 0 0
\(848\) 2228.53 0.0902452
\(849\) 0 0
\(850\) −10103.4 −0.407700
\(851\) −15893.1 −0.640198
\(852\) 0 0
\(853\) 42861.7 1.72047 0.860233 0.509901i \(-0.170318\pi\)
0.860233 + 0.509901i \(0.170318\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −510.432 −0.0203811
\(857\) −9390.00 −0.374278 −0.187139 0.982333i \(-0.559921\pi\)
−0.187139 + 0.982333i \(0.559921\pi\)
\(858\) 0 0
\(859\) 33561.2 1.33305 0.666526 0.745482i \(-0.267781\pi\)
0.666526 + 0.745482i \(0.267781\pi\)
\(860\) 33268.6 1.31913
\(861\) 0 0
\(862\) −37155.9 −1.46814
\(863\) 25691.6 1.01339 0.506693 0.862127i \(-0.330868\pi\)
0.506693 + 0.862127i \(0.330868\pi\)
\(864\) 0 0
\(865\) 5586.87 0.219606
\(866\) 44512.7 1.74665
\(867\) 0 0
\(868\) 0 0
\(869\) −3977.14 −0.155254
\(870\) 0 0
\(871\) 6317.97 0.245782
\(872\) 519.775 0.0201856
\(873\) 0 0
\(874\) 106067. 4.10500
\(875\) 0 0
\(876\) 0 0
\(877\) −5351.52 −0.206052 −0.103026 0.994679i \(-0.532853\pi\)
−0.103026 + 0.994679i \(0.532853\pi\)
\(878\) −16153.4 −0.620900
\(879\) 0 0
\(880\) −8090.14 −0.309907
\(881\) −34212.7 −1.30835 −0.654174 0.756344i \(-0.726984\pi\)
−0.654174 + 0.756344i \(0.726984\pi\)
\(882\) 0 0
\(883\) 17149.2 0.653587 0.326794 0.945096i \(-0.394032\pi\)
0.326794 + 0.945096i \(0.394032\pi\)
\(884\) −14685.3 −0.558732
\(885\) 0 0
\(886\) −18348.4 −0.695742
\(887\) 4020.87 0.152207 0.0761035 0.997100i \(-0.475752\pi\)
0.0761035 + 0.997100i \(0.475752\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 46039.0 1.73397
\(891\) 0 0
\(892\) −19045.1 −0.714883
\(893\) 35915.7 1.34588
\(894\) 0 0
\(895\) −18248.5 −0.681544
\(896\) 0 0
\(897\) 0 0
\(898\) 8420.99 0.312931
\(899\) −15807.8 −0.586452
\(900\) 0 0
\(901\) 3467.24 0.128203
\(902\) −21185.2 −0.782030
\(903\) 0 0
\(904\) −3429.59 −0.126180
\(905\) −23062.0 −0.847080
\(906\) 0 0
\(907\) −22967.2 −0.840810 −0.420405 0.907337i \(-0.638112\pi\)
−0.420405 + 0.907337i \(0.638112\pi\)
\(908\) −28639.4 −1.04673
\(909\) 0 0
\(910\) 0 0
\(911\) 9860.77 0.358619 0.179309 0.983793i \(-0.442614\pi\)
0.179309 + 0.983793i \(0.442614\pi\)
\(912\) 0 0
\(913\) −12045.3 −0.436628
\(914\) 7492.71 0.271156
\(915\) 0 0
\(916\) 21857.5 0.788421
\(917\) 0 0
\(918\) 0 0
\(919\) −5271.55 −0.189219 −0.0946096 0.995514i \(-0.530160\pi\)
−0.0946096 + 0.995514i \(0.530160\pi\)
\(920\) −3559.49 −0.127558
\(921\) 0 0
\(922\) −3556.36 −0.127031
\(923\) −3917.44 −0.139701
\(924\) 0 0
\(925\) −2131.00 −0.0757478
\(926\) −65880.6 −2.33798
\(927\) 0 0
\(928\) 48735.3 1.72394
\(929\) −14902.6 −0.526307 −0.263153 0.964754i \(-0.584762\pi\)
−0.263153 + 0.964754i \(0.584762\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −8061.24 −0.283321
\(933\) 0 0
\(934\) 76897.4 2.69396
\(935\) −12587.0 −0.440255
\(936\) 0 0
\(937\) −21934.8 −0.764757 −0.382378 0.924006i \(-0.624895\pi\)
−0.382378 + 0.924006i \(0.624895\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −22814.1 −0.791610
\(941\) 14521.5 0.503069 0.251535 0.967848i \(-0.419065\pi\)
0.251535 + 0.967848i \(0.419065\pi\)
\(942\) 0 0
\(943\) 76501.3 2.64181
\(944\) −23836.4 −0.821831
\(945\) 0 0
\(946\) −21801.3 −0.749282
\(947\) 1943.73 0.0666979 0.0333489 0.999444i \(-0.489383\pi\)
0.0333489 + 0.999444i \(0.489383\pi\)
\(948\) 0 0
\(949\) 9019.43 0.308517
\(950\) 14221.8 0.485702
\(951\) 0 0
\(952\) 0 0
\(953\) −16904.1 −0.574583 −0.287292 0.957843i \(-0.592755\pi\)
−0.287292 + 0.957843i \(0.592755\pi\)
\(954\) 0 0
\(955\) 31023.8 1.05121
\(956\) 1685.05 0.0570068
\(957\) 0 0
\(958\) 33975.1 1.14581
\(959\) 0 0
\(960\) 0 0
\(961\) −22747.7 −0.763576
\(962\) −6031.13 −0.202133
\(963\) 0 0
\(964\) 40494.8 1.35296
\(965\) 36797.6 1.22752
\(966\) 0 0
\(967\) 26699.3 0.887891 0.443946 0.896054i \(-0.353578\pi\)
0.443946 + 0.896054i \(0.353578\pi\)
\(968\) 2076.85 0.0689591
\(969\) 0 0
\(970\) 55704.2 1.84387
\(971\) 11089.5 0.366507 0.183253 0.983066i \(-0.441337\pi\)
0.183253 + 0.983066i \(0.441337\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 18487.9 0.608205
\(975\) 0 0
\(976\) −811.541 −0.0266156
\(977\) 51891.9 1.69925 0.849625 0.527387i \(-0.176828\pi\)
0.849625 + 0.527387i \(0.176828\pi\)
\(978\) 0 0
\(979\) −15494.2 −0.505820
\(980\) 0 0
\(981\) 0 0
\(982\) −64115.5 −2.08351
\(983\) 24560.7 0.796911 0.398456 0.917188i \(-0.369546\pi\)
0.398456 + 0.917188i \(0.369546\pi\)
\(984\) 0 0
\(985\) −8443.74 −0.273137
\(986\) 71582.0 2.31200
\(987\) 0 0
\(988\) 20671.3 0.665629
\(989\) 78725.9 2.53118
\(990\) 0 0
\(991\) −18507.5 −0.593250 −0.296625 0.954994i \(-0.595861\pi\)
−0.296625 + 0.954994i \(0.595861\pi\)
\(992\) −21714.4 −0.694993
\(993\) 0 0
\(994\) 0 0
\(995\) −33707.4 −1.07397
\(996\) 0 0
\(997\) −42966.2 −1.36485 −0.682424 0.730957i \(-0.739074\pi\)
−0.682424 + 0.730957i \(0.739074\pi\)
\(998\) −54285.8 −1.72183
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.4.a.w.1.1 4
3.2 odd 2 inner 441.4.a.w.1.4 4
7.2 even 3 63.4.e.d.46.4 yes 8
7.3 odd 6 441.4.e.x.226.4 8
7.4 even 3 63.4.e.d.37.4 yes 8
7.5 odd 6 441.4.e.x.361.4 8
7.6 odd 2 441.4.a.v.1.1 4
21.2 odd 6 63.4.e.d.46.1 yes 8
21.5 even 6 441.4.e.x.361.1 8
21.11 odd 6 63.4.e.d.37.1 8
21.17 even 6 441.4.e.x.226.1 8
21.20 even 2 441.4.a.v.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.e.d.37.1 8 21.11 odd 6
63.4.e.d.37.4 yes 8 7.4 even 3
63.4.e.d.46.1 yes 8 21.2 odd 6
63.4.e.d.46.4 yes 8 7.2 even 3
441.4.a.v.1.1 4 7.6 odd 2
441.4.a.v.1.4 4 21.20 even 2
441.4.a.w.1.1 4 1.1 even 1 trivial
441.4.a.w.1.4 4 3.2 odd 2 inner
441.4.e.x.226.1 8 21.17 even 6
441.4.e.x.226.4 8 7.3 odd 6
441.4.e.x.361.1 8 21.5 even 6
441.4.e.x.361.4 8 7.5 odd 6