Properties

Label 441.4.a.v.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
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: \(1\)
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 \(-0.353682\pi\)
0.443654 + 0.896198i \(0.353682\pi\)
\(6\) 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 \(-0.563418\pi\)
−0.197919 + 0.980218i \(0.563418\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −60.2311 −0.941111
\(17\) 93.7102 1.33695 0.668473 0.743737i \(-0.266948\pi\)
0.668473 + 0.743737i \(0.266948\pi\)
\(18\) 0 0
\(19\) −131.908 −1.59273 −0.796365 0.604816i \(-0.793247\pi\)
−0.796365 + 0.604816i \(0.793247\pi\)
\(20\) 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 \(-0.421830\pi\)
0.243117 + 0.969997i \(0.421830\pi\)
\(32\) 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 \(-0.237261\pi\)
0.734833 + 0.678249i \(0.237261\pi\)
\(42\) 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 \(-0.638850\pi\)
−0.422508 + 0.906359i \(0.638850\pi\)
\(48\) 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 \(-0.643827\pi\)
−0.436628 + 0.899642i \(0.643827\pi\)
\(60\) 0 0
\(61\) −13.4738 −0.0282810 −0.0141405 0.999900i \(-0.504501\pi\)
−0.0141405 + 0.999900i \(0.504501\pi\)
\(62\) −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 \(-0.627422\pi\)
−0.389702 + 0.920941i \(0.627422\pi\)
\(74\) −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 \(-0.700184\pi\)
−0.588253 + 0.808677i \(0.700184\pi\)
\(84\) 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 \(-0.738661\pi\)
−0.681474 + 0.731843i \(0.738661\pi\)
\(90\) 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 \(-0.758007\pi\)
−0.724667 + 0.689099i \(0.758007\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −224.549 −0.224549
\(101\) −1785.99 −1.75953 −0.879765 0.475409i \(-0.842300\pi\)
−0.879765 + 0.475409i \(0.842300\pi\)
\(102\) 0 0
\(103\) −489.661 −0.468425 −0.234212 0.972185i \(-0.575251\pi\)
−0.234212 + 0.972185i \(0.575251\pi\)
\(104\) 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 \(-0.629962\pi\)
−0.397037 + 0.917803i \(0.629962\pi\)
\(132\) 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 \(-0.391388\pi\)
0.334631 + 0.942349i \(0.391388\pi\)
\(140\) 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 \(-0.321379\pi\)
0.532163 + 0.846642i \(0.321379\pi\)
\(158\) −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 \(-0.327140\pi\)
0.516754 + 0.856134i \(0.327140\pi\)
\(168\) 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 \(-0.460508\pi\)
0.123749 + 0.992314i \(0.460508\pi\)
\(174\) 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 \(-0.658396\pi\)
−0.477332 + 0.878723i \(0.658396\pi\)
\(182\) 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 \(-0.706900\pi\)
−0.605182 + 0.796087i \(0.706900\pi\)
\(200\) 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 \(-0.390061\pi\)
0.338558 + 0.940946i \(0.390061\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7685.84 −2.26219
\(227\) 3390.80 0.991433 0.495716 0.868484i \(-0.334906\pi\)
0.495716 + 0.868484i \(0.334906\pi\)
\(228\) 0 0
\(229\) −2587.85 −0.746768 −0.373384 0.927677i \(-0.621803\pi\)
−0.373384 + 0.927677i \(0.621803\pi\)
\(230\) 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 \(-0.721372\pi\)
−0.640739 + 0.767759i \(0.721372\pi\)
\(242\) 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 \(-0.787757\pi\)
−0.785816 + 0.618460i \(0.787757\pi\)
\(252\) 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 \(-0.345782\pi\)
0.465756 + 0.884913i \(0.345782\pi\)
\(258\) 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 \(-0.683259\pi\)
−0.544442 + 0.838798i \(0.683259\pi\)
\(270\) 0 0
\(271\) −3215.75 −0.720823 −0.360411 0.932793i \(-0.617364\pi\)
−0.360411 + 0.932793i \(0.617364\pi\)
\(272\) −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 \(-0.579055\pi\)
−0.245813 + 0.969317i \(0.579055\pi\)
\(284\) −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 \(-0.382014\pi\)
0.362235 + 0.932087i \(0.382014\pi\)
\(294\) 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 \(-0.686697\pi\)
−0.553471 + 0.832868i \(0.686697\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3376.37 −0.618597
\(311\) −1180.09 −0.215167 −0.107584 0.994196i \(-0.534311\pi\)
−0.107584 + 0.994196i \(0.534311\pi\)
\(312\) 0 0
\(313\) 9746.25 1.76003 0.880017 0.474943i \(-0.157531\pi\)
0.880017 + 0.474943i \(0.157531\pi\)
\(314\) −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 \(-0.544354\pi\)
−0.138893 + 0.990307i \(0.544354\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3503.21 −0.530459
\(353\) −3327.76 −0.501753 −0.250876 0.968019i \(-0.580719\pi\)
−0.250876 + 0.968019i \(0.580719\pi\)
\(354\) 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 \(-0.526623\pi\)
−0.0835418 + 0.996504i \(0.526623\pi\)
\(368\) 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 \(-0.336769\pi\)
0.490623 + 0.871372i \(0.336769\pi\)
\(384\) 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 \(-0.654409\pi\)
−0.466288 + 0.884633i \(0.654409\pi\)
\(398\) 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 \(-0.738304\pi\)
−0.680652 + 0.732607i \(0.738304\pi\)
\(410\) −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 \(-0.319336\pi\)
0.537587 + 0.843208i \(0.319336\pi\)
\(420\) 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 \(-0.291531\pi\)
0.609100 + 0.793093i \(0.291531\pi\)
\(434\) 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 \(-0.569472\pi\)
−0.216523 + 0.976278i \(0.569472\pi\)
\(440\) 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 \(-0.514105\pi\)
−0.0442987 + 0.999018i \(0.514105\pi\)
\(462\) 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 \(-0.111338\pi\)
0.939449 + 0.342689i \(0.111338\pi\)
\(468\) 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 \(-0.369159\pi\)
0.399572 + 0.916702i \(0.369159\pi\)
\(480\) 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 \(-0.581597\pi\)
−0.253545 + 0.967323i \(0.581597\pi\)
\(504\) 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 \(-0.268059\pi\)
0.665873 + 0.746065i \(0.268059\pi\)
\(510\) 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 \(-0.558801\pi\)
−0.183681 + 0.982986i \(0.558801\pi\)
\(522\) 0 0
\(523\) 2422.17 0.202512 0.101256 0.994860i \(-0.467714\pi\)
0.101256 + 0.994860i \(0.467714\pi\)
\(524\) −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 \(-0.670529\pi\)
−0.510471 + 0.859895i \(0.670529\pi\)
\(564\) 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 \(-0.436667\pi\)
0.197657 + 0.980271i \(0.436667\pi\)
\(578\) −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 \(-0.549531\pi\)
−0.154980 + 0.987918i \(0.549531\pi\)
\(588\) 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 \(-0.531078\pi\)
−0.0974779 + 0.995238i \(0.531078\pi\)
\(594\) 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 \(-0.342623\pi\)
0.474517 + 0.880247i \(0.342623\pi\)
\(602\) 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 \(-0.647729\pi\)
−0.447622 + 0.894223i \(0.647729\pi\)
\(608\) −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 \(-0.511297\pi\)
−0.0354837 + 0.999370i \(0.511297\pi\)
\(620\) 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 \(-0.506717\pi\)
−0.0211019 + 0.999777i \(0.506717\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 50129.4 3.05312
\(647\) −10966.2 −0.666345 −0.333173 0.942866i \(-0.608119\pi\)
−0.333173 + 0.942866i \(0.608119\pi\)
\(648\) 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 \(-0.236373\pi\)
0.736721 + 0.676196i \(0.236373\pi\)
\(662\) 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 \(-0.612112\pi\)
−0.344973 + 0.938613i \(0.612112\pi\)
\(678\) 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 \(-0.646086\pi\)
−0.443000 + 0.896521i \(0.646086\pi\)
\(692\) 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 \(-0.444706\pi\)
0.172839 + 0.984950i \(0.444706\pi\)
\(720\) 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 \(-0.460625\pi\)
0.123386 + 0.992359i \(0.460625\pi\)
\(728\) 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 \(-0.206827\pi\)
0.796225 + 0.605001i \(0.206827\pi\)
\(734\) 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 \(-0.783691\pi\)
−0.777853 + 0.628447i \(0.783691\pi\)
\(762\) 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 \(-0.416924\pi\)
0.258037 + 0.966135i \(0.416924\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −31329.5 −1.46059
\(773\) 3038.18 0.141366 0.0706829 0.997499i \(-0.477482\pi\)
0.0706829 + 0.997499i \(0.477482\pi\)
\(774\) 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 \(-0.410093\pi\)
0.278710 + 0.960375i \(0.410093\pi\)
\(788\) 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 \(-0.521286\pi\)
−0.0668228 + 0.997765i \(0.521286\pi\)
\(798\) 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 \(-0.370953\pi\)
0.394398 + 0.918940i \(0.370953\pi\)
\(812\) 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 \(-0.273761\pi\)
0.652402 + 0.757873i \(0.273761\pi\)
\(830\) 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 \(-0.400290\pi\)
0.308150 + 0.951338i \(0.400290\pi\)
\(840\) 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 \(-0.829682\pi\)
−0.860233 + 0.509901i \(0.829682\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −510.432 −0.0203811
\(857\) 9390.00 0.374278 0.187139 0.982333i \(-0.440079\pi\)
0.187139 + 0.982333i \(0.440079\pi\)
\(858\) 0 0
\(859\) −33561.2 −1.33305 −0.666526 0.745482i \(-0.732219\pi\)
−0.666526 + 0.745482i \(0.732219\pi\)
\(860\) −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 \(-0.273016\pi\)
0.654174 + 0.756344i \(0.273016\pi\)
\(882\) 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 \(-0.524248\pi\)
−0.0761035 + 0.997100i \(0.524248\pi\)
\(888\) 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 \(-0.415238\pi\)
0.263153 + 0.964754i \(0.415238\pi\)
\(930\) 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 \(-0.375105\pi\)
0.382378 + 0.924006i \(0.375105\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −22814.1 −0.791610
\(941\) −14521.5 −0.503069 −0.251535 0.967848i \(-0.580935\pi\)
−0.251535 + 0.967848i \(0.580935\pi\)
\(942\) 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 \(-0.558663\pi\)
−0.183253 + 0.983066i \(0.558663\pi\)
\(972\) 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 \(-0.630454\pi\)
−0.398456 + 0.917188i \(0.630454\pi\)
\(984\) 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 \(-0.260926\pi\)
0.682424 + 0.730957i \(0.260926\pi\)
\(998\) −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.v.1.1 4
3.2 odd 2 inner 441.4.a.v.1.4 4
7.2 even 3 441.4.e.x.361.4 8
7.3 odd 6 63.4.e.d.37.4 yes 8
7.4 even 3 441.4.e.x.226.4 8
7.5 odd 6 63.4.e.d.46.4 yes 8
7.6 odd 2 441.4.a.w.1.1 4
21.2 odd 6 441.4.e.x.361.1 8
21.5 even 6 63.4.e.d.46.1 yes 8
21.11 odd 6 441.4.e.x.226.1 8
21.17 even 6 63.4.e.d.37.1 8
21.20 even 2 441.4.a.w.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.e.d.37.1 8 21.17 even 6
63.4.e.d.37.4 yes 8 7.3 odd 6
63.4.e.d.46.1 yes 8 21.5 even 6
63.4.e.d.46.4 yes 8 7.5 odd 6
441.4.a.v.1.1 4 1.1 even 1 trivial
441.4.a.v.1.4 4 3.2 odd 2 inner
441.4.a.w.1.1 4 7.6 odd 2
441.4.a.w.1.4 4 21.20 even 2
441.4.e.x.226.1 8 21.11 odd 6
441.4.e.x.226.4 8 7.4 even 3
441.4.e.x.361.1 8 21.2 odd 6
441.4.e.x.361.4 8 7.2 even 3