Properties

Label 1521.4.a.bk.1.6
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.917374\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.917374 q^{2} -7.15843 q^{4} -15.4704 q^{5} +20.5833 q^{7} -13.9059 q^{8} +O(q^{10})\) \(q+0.917374 q^{2} -7.15843 q^{4} -15.4704 q^{5} +20.5833 q^{7} -13.9059 q^{8} -14.1922 q^{10} -65.8420 q^{11} +18.8826 q^{14} +44.5105 q^{16} -44.2956 q^{17} -147.053 q^{19} +110.744 q^{20} -60.4017 q^{22} -53.1586 q^{23} +114.334 q^{25} -147.344 q^{28} +38.6257 q^{29} -88.3894 q^{31} +152.080 q^{32} -40.6357 q^{34} -318.433 q^{35} -78.9587 q^{37} -134.903 q^{38} +215.131 q^{40} -354.966 q^{41} -407.846 q^{43} +471.325 q^{44} -48.7663 q^{46} -67.9674 q^{47} +80.6738 q^{49} +104.887 q^{50} -226.572 q^{53} +1018.60 q^{55} -286.231 q^{56} +35.4342 q^{58} +142.031 q^{59} +266.831 q^{61} -81.0862 q^{62} -216.569 q^{64} +411.187 q^{67} +317.087 q^{68} -292.122 q^{70} -91.5052 q^{71} -63.1328 q^{73} -72.4347 q^{74} +1052.67 q^{76} -1355.25 q^{77} -287.115 q^{79} -688.595 q^{80} -325.637 q^{82} -373.812 q^{83} +685.272 q^{85} -374.147 q^{86} +915.595 q^{88} +119.403 q^{89} +380.532 q^{92} -62.3515 q^{94} +2274.97 q^{95} -554.650 q^{97} +74.0080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.917374 0.324341 0.162170 0.986763i \(-0.448151\pi\)
0.162170 + 0.986763i \(0.448151\pi\)
\(3\) 0 0
\(4\) −7.15843 −0.894803
\(5\) −15.4704 −1.38372 −0.691858 0.722034i \(-0.743208\pi\)
−0.691858 + 0.722034i \(0.743208\pi\)
\(6\) 0 0
\(7\) 20.5833 1.11140 0.555698 0.831384i \(-0.312451\pi\)
0.555698 + 0.831384i \(0.312451\pi\)
\(8\) −13.9059 −0.614562
\(9\) 0 0
\(10\) −14.1922 −0.448795
\(11\) −65.8420 −1.80474 −0.902369 0.430964i \(-0.858173\pi\)
−0.902369 + 0.430964i \(0.858173\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 18.8826 0.360471
\(15\) 0 0
\(16\) 44.5105 0.695476
\(17\) −44.2956 −0.631957 −0.315979 0.948766i \(-0.602333\pi\)
−0.315979 + 0.948766i \(0.602333\pi\)
\(18\) 0 0
\(19\) −147.053 −1.77560 −0.887798 0.460234i \(-0.847766\pi\)
−0.887798 + 0.460234i \(0.847766\pi\)
\(20\) 110.744 1.23815
\(21\) 0 0
\(22\) −60.4017 −0.585350
\(23\) −53.1586 −0.481928 −0.240964 0.970534i \(-0.577464\pi\)
−0.240964 + 0.970534i \(0.577464\pi\)
\(24\) 0 0
\(25\) 114.334 0.914669
\(26\) 0 0
\(27\) 0 0
\(28\) −147.344 −0.994480
\(29\) 38.6257 0.247331 0.123666 0.992324i \(-0.460535\pi\)
0.123666 + 0.992324i \(0.460535\pi\)
\(30\) 0 0
\(31\) −88.3894 −0.512104 −0.256052 0.966663i \(-0.582422\pi\)
−0.256052 + 0.966663i \(0.582422\pi\)
\(32\) 152.080 0.840133
\(33\) 0 0
\(34\) −40.6357 −0.204969
\(35\) −318.433 −1.53786
\(36\) 0 0
\(37\) −78.9587 −0.350831 −0.175415 0.984495i \(-0.556127\pi\)
−0.175415 + 0.984495i \(0.556127\pi\)
\(38\) −134.903 −0.575898
\(39\) 0 0
\(40\) 215.131 0.850379
\(41\) −354.966 −1.35211 −0.676054 0.736852i \(-0.736311\pi\)
−0.676054 + 0.736852i \(0.736311\pi\)
\(42\) 0 0
\(43\) −407.846 −1.44642 −0.723208 0.690630i \(-0.757333\pi\)
−0.723208 + 0.690630i \(0.757333\pi\)
\(44\) 471.325 1.61489
\(45\) 0 0
\(46\) −48.7663 −0.156309
\(47\) −67.9674 −0.210938 −0.105469 0.994423i \(-0.533634\pi\)
−0.105469 + 0.994423i \(0.533634\pi\)
\(48\) 0 0
\(49\) 80.6738 0.235201
\(50\) 104.887 0.296664
\(51\) 0 0
\(52\) 0 0
\(53\) −226.572 −0.587209 −0.293604 0.955927i \(-0.594855\pi\)
−0.293604 + 0.955927i \(0.594855\pi\)
\(54\) 0 0
\(55\) 1018.60 2.49724
\(56\) −286.231 −0.683021
\(57\) 0 0
\(58\) 35.4342 0.0802196
\(59\) 142.031 0.313404 0.156702 0.987646i \(-0.449914\pi\)
0.156702 + 0.987646i \(0.449914\pi\)
\(60\) 0 0
\(61\) 266.831 0.560069 0.280035 0.959990i \(-0.409654\pi\)
0.280035 + 0.959990i \(0.409654\pi\)
\(62\) −81.0862 −0.166096
\(63\) 0 0
\(64\) −216.569 −0.422987
\(65\) 0 0
\(66\) 0 0
\(67\) 411.187 0.749768 0.374884 0.927072i \(-0.377683\pi\)
0.374884 + 0.927072i \(0.377683\pi\)
\(68\) 317.087 0.565477
\(69\) 0 0
\(70\) −292.122 −0.498789
\(71\) −91.5052 −0.152953 −0.0764765 0.997071i \(-0.524367\pi\)
−0.0764765 + 0.997071i \(0.524367\pi\)
\(72\) 0 0
\(73\) −63.1328 −0.101221 −0.0506105 0.998718i \(-0.516117\pi\)
−0.0506105 + 0.998718i \(0.516117\pi\)
\(74\) −72.4347 −0.113789
\(75\) 0 0
\(76\) 1052.67 1.58881
\(77\) −1355.25 −2.00578
\(78\) 0 0
\(79\) −287.115 −0.408899 −0.204449 0.978877i \(-0.565540\pi\)
−0.204449 + 0.978877i \(0.565540\pi\)
\(80\) −688.595 −0.962341
\(81\) 0 0
\(82\) −325.637 −0.438543
\(83\) −373.812 −0.494352 −0.247176 0.968971i \(-0.579503\pi\)
−0.247176 + 0.968971i \(0.579503\pi\)
\(84\) 0 0
\(85\) 685.272 0.874449
\(86\) −374.147 −0.469132
\(87\) 0 0
\(88\) 915.595 1.10912
\(89\) 119.403 0.142209 0.0711047 0.997469i \(-0.477348\pi\)
0.0711047 + 0.997469i \(0.477348\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 380.532 0.431231
\(93\) 0 0
\(94\) −62.3515 −0.0684157
\(95\) 2274.97 2.45692
\(96\) 0 0
\(97\) −554.650 −0.580579 −0.290290 0.956939i \(-0.593752\pi\)
−0.290290 + 0.956939i \(0.593752\pi\)
\(98\) 74.0080 0.0762851
\(99\) 0 0
\(100\) −818.449 −0.818449
\(101\) 462.565 0.455712 0.227856 0.973695i \(-0.426828\pi\)
0.227856 + 0.973695i \(0.426828\pi\)
\(102\) 0 0
\(103\) 1122.07 1.07341 0.536704 0.843771i \(-0.319669\pi\)
0.536704 + 0.843771i \(0.319669\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −207.851 −0.190456
\(107\) −602.756 −0.544585 −0.272293 0.962214i \(-0.587782\pi\)
−0.272293 + 0.962214i \(0.587782\pi\)
\(108\) 0 0
\(109\) −1421.89 −1.24947 −0.624735 0.780837i \(-0.714793\pi\)
−0.624735 + 0.780837i \(0.714793\pi\)
\(110\) 934.440 0.809958
\(111\) 0 0
\(112\) 916.174 0.772949
\(113\) 396.719 0.330267 0.165134 0.986271i \(-0.447194\pi\)
0.165134 + 0.986271i \(0.447194\pi\)
\(114\) 0 0
\(115\) 822.386 0.666851
\(116\) −276.499 −0.221313
\(117\) 0 0
\(118\) 130.295 0.101650
\(119\) −911.752 −0.702354
\(120\) 0 0
\(121\) 3004.17 2.25708
\(122\) 244.784 0.181653
\(123\) 0 0
\(124\) 632.729 0.458232
\(125\) 165.013 0.118074
\(126\) 0 0
\(127\) 437.868 0.305941 0.152970 0.988231i \(-0.451116\pi\)
0.152970 + 0.988231i \(0.451116\pi\)
\(128\) −1415.32 −0.977324
\(129\) 0 0
\(130\) 0 0
\(131\) −1657.44 −1.10543 −0.552715 0.833370i \(-0.686408\pi\)
−0.552715 + 0.833370i \(0.686408\pi\)
\(132\) 0 0
\(133\) −3026.85 −1.97339
\(134\) 377.212 0.243180
\(135\) 0 0
\(136\) 615.973 0.388377
\(137\) 1967.42 1.22692 0.613460 0.789725i \(-0.289777\pi\)
0.613460 + 0.789725i \(0.289777\pi\)
\(138\) 0 0
\(139\) 2825.13 1.72392 0.861960 0.506977i \(-0.169237\pi\)
0.861960 + 0.506977i \(0.169237\pi\)
\(140\) 2279.48 1.37608
\(141\) 0 0
\(142\) −83.9445 −0.0496089
\(143\) 0 0
\(144\) 0 0
\(145\) −597.555 −0.342236
\(146\) −57.9164 −0.0328301
\(147\) 0 0
\(148\) 565.220 0.313924
\(149\) 797.658 0.438569 0.219284 0.975661i \(-0.429628\pi\)
0.219284 + 0.975661i \(0.429628\pi\)
\(150\) 0 0
\(151\) 161.987 0.0873003 0.0436501 0.999047i \(-0.486101\pi\)
0.0436501 + 0.999047i \(0.486101\pi\)
\(152\) 2044.91 1.09121
\(153\) 0 0
\(154\) −1243.27 −0.650555
\(155\) 1367.42 0.708606
\(156\) 0 0
\(157\) −342.000 −0.173851 −0.0869255 0.996215i \(-0.527704\pi\)
−0.0869255 + 0.996215i \(0.527704\pi\)
\(158\) −263.392 −0.132622
\(159\) 0 0
\(160\) −2352.74 −1.16250
\(161\) −1094.18 −0.535613
\(162\) 0 0
\(163\) 556.597 0.267460 0.133730 0.991018i \(-0.457304\pi\)
0.133730 + 0.991018i \(0.457304\pi\)
\(164\) 2541.00 1.20987
\(165\) 0 0
\(166\) −342.926 −0.160339
\(167\) 3136.42 1.45332 0.726658 0.686999i \(-0.241072\pi\)
0.726658 + 0.686999i \(0.241072\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 628.650 0.283619
\(171\) 0 0
\(172\) 2919.53 1.29426
\(173\) −3324.19 −1.46089 −0.730444 0.682972i \(-0.760687\pi\)
−0.730444 + 0.682972i \(0.760687\pi\)
\(174\) 0 0
\(175\) 2353.37 1.01656
\(176\) −2930.66 −1.25515
\(177\) 0 0
\(178\) 109.537 0.0461243
\(179\) 3313.25 1.38349 0.691743 0.722144i \(-0.256843\pi\)
0.691743 + 0.722144i \(0.256843\pi\)
\(180\) 0 0
\(181\) 76.0118 0.0312150 0.0156075 0.999878i \(-0.495032\pi\)
0.0156075 + 0.999878i \(0.495032\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 739.221 0.296174
\(185\) 1221.52 0.485450
\(186\) 0 0
\(187\) 2916.51 1.14052
\(188\) 486.540 0.188748
\(189\) 0 0
\(190\) 2087.00 0.796879
\(191\) 4073.39 1.54314 0.771572 0.636142i \(-0.219471\pi\)
0.771572 + 0.636142i \(0.219471\pi\)
\(192\) 0 0
\(193\) −867.581 −0.323574 −0.161787 0.986826i \(-0.551726\pi\)
−0.161787 + 0.986826i \(0.551726\pi\)
\(194\) −508.821 −0.188305
\(195\) 0 0
\(196\) −577.497 −0.210458
\(197\) −1887.69 −0.682703 −0.341352 0.939936i \(-0.610885\pi\)
−0.341352 + 0.939936i \(0.610885\pi\)
\(198\) 0 0
\(199\) 2786.22 0.992511 0.496256 0.868176i \(-0.334708\pi\)
0.496256 + 0.868176i \(0.334708\pi\)
\(200\) −1589.92 −0.562120
\(201\) 0 0
\(202\) 424.345 0.147806
\(203\) 795.045 0.274883
\(204\) 0 0
\(205\) 5491.47 1.87093
\(206\) 1029.36 0.348150
\(207\) 0 0
\(208\) 0 0
\(209\) 9682.28 3.20448
\(210\) 0 0
\(211\) 708.789 0.231256 0.115628 0.993293i \(-0.463112\pi\)
0.115628 + 0.993293i \(0.463112\pi\)
\(212\) 1621.90 0.525436
\(213\) 0 0
\(214\) −552.953 −0.176631
\(215\) 6309.54 2.00143
\(216\) 0 0
\(217\) −1819.35 −0.569150
\(218\) −1304.40 −0.405254
\(219\) 0 0
\(220\) −7291.59 −2.23454
\(221\) 0 0
\(222\) 0 0
\(223\) −5396.64 −1.62056 −0.810281 0.586041i \(-0.800686\pi\)
−0.810281 + 0.586041i \(0.800686\pi\)
\(224\) 3130.32 0.933720
\(225\) 0 0
\(226\) 363.940 0.107119
\(227\) 3181.79 0.930321 0.465160 0.885226i \(-0.345997\pi\)
0.465160 + 0.885226i \(0.345997\pi\)
\(228\) 0 0
\(229\) −2034.00 −0.586945 −0.293473 0.955967i \(-0.594811\pi\)
−0.293473 + 0.955967i \(0.594811\pi\)
\(230\) 754.435 0.216287
\(231\) 0 0
\(232\) −537.126 −0.152000
\(233\) 2794.22 0.785645 0.392823 0.919614i \(-0.371499\pi\)
0.392823 + 0.919614i \(0.371499\pi\)
\(234\) 0 0
\(235\) 1051.48 0.291878
\(236\) −1016.72 −0.280435
\(237\) 0 0
\(238\) −836.417 −0.227802
\(239\) −5493.81 −1.48688 −0.743442 0.668800i \(-0.766808\pi\)
−0.743442 + 0.668800i \(0.766808\pi\)
\(240\) 0 0
\(241\) −4061.27 −1.08552 −0.542758 0.839889i \(-0.682620\pi\)
−0.542758 + 0.839889i \(0.682620\pi\)
\(242\) 2755.95 0.732062
\(243\) 0 0
\(244\) −1910.09 −0.501152
\(245\) −1248.06 −0.325451
\(246\) 0 0
\(247\) 0 0
\(248\) 1229.14 0.314719
\(249\) 0 0
\(250\) 151.379 0.0382961
\(251\) 3570.88 0.897977 0.448988 0.893538i \(-0.351784\pi\)
0.448988 + 0.893538i \(0.351784\pi\)
\(252\) 0 0
\(253\) 3500.07 0.869753
\(254\) 401.688 0.0992290
\(255\) 0 0
\(256\) 434.179 0.106001
\(257\) 7518.26 1.82481 0.912405 0.409288i \(-0.134223\pi\)
0.912405 + 0.409288i \(0.134223\pi\)
\(258\) 0 0
\(259\) −1625.23 −0.389912
\(260\) 0 0
\(261\) 0 0
\(262\) −1520.49 −0.358536
\(263\) 4660.18 1.09262 0.546310 0.837583i \(-0.316032\pi\)
0.546310 + 0.837583i \(0.316032\pi\)
\(264\) 0 0
\(265\) 3505.16 0.812530
\(266\) −2776.75 −0.640050
\(267\) 0 0
\(268\) −2943.45 −0.670895
\(269\) −5347.44 −1.21204 −0.606021 0.795449i \(-0.707235\pi\)
−0.606021 + 0.795449i \(0.707235\pi\)
\(270\) 0 0
\(271\) −2973.08 −0.666427 −0.333214 0.942851i \(-0.608133\pi\)
−0.333214 + 0.942851i \(0.608133\pi\)
\(272\) −1971.62 −0.439511
\(273\) 0 0
\(274\) 1804.86 0.397940
\(275\) −7527.96 −1.65074
\(276\) 0 0
\(277\) −764.153 −0.165753 −0.0828764 0.996560i \(-0.526411\pi\)
−0.0828764 + 0.996560i \(0.526411\pi\)
\(278\) 2591.70 0.559137
\(279\) 0 0
\(280\) 4428.11 0.945107
\(281\) −7040.34 −1.49463 −0.747316 0.664469i \(-0.768658\pi\)
−0.747316 + 0.664469i \(0.768658\pi\)
\(282\) 0 0
\(283\) −9035.17 −1.89783 −0.948913 0.315537i \(-0.897815\pi\)
−0.948913 + 0.315537i \(0.897815\pi\)
\(284\) 655.033 0.136863
\(285\) 0 0
\(286\) 0 0
\(287\) −7306.39 −1.50273
\(288\) 0 0
\(289\) −2950.90 −0.600630
\(290\) −548.181 −0.111001
\(291\) 0 0
\(292\) 451.931 0.0905729
\(293\) 1785.23 0.355954 0.177977 0.984035i \(-0.443045\pi\)
0.177977 + 0.984035i \(0.443045\pi\)
\(294\) 0 0
\(295\) −2197.27 −0.433662
\(296\) 1098.00 0.215607
\(297\) 0 0
\(298\) 731.751 0.142246
\(299\) 0 0
\(300\) 0 0
\(301\) −8394.83 −1.60754
\(302\) 148.603 0.0283150
\(303\) 0 0
\(304\) −6545.40 −1.23488
\(305\) −4127.99 −0.774977
\(306\) 0 0
\(307\) −5323.13 −0.989600 −0.494800 0.869007i \(-0.664759\pi\)
−0.494800 + 0.869007i \(0.664759\pi\)
\(308\) 9701.45 1.79478
\(309\) 0 0
\(310\) 1254.44 0.229830
\(311\) −6265.64 −1.14242 −0.571209 0.820805i \(-0.693525\pi\)
−0.571209 + 0.820805i \(0.693525\pi\)
\(312\) 0 0
\(313\) 7193.77 1.29909 0.649547 0.760322i \(-0.274959\pi\)
0.649547 + 0.760322i \(0.274959\pi\)
\(314\) −313.742 −0.0563869
\(315\) 0 0
\(316\) 2055.29 0.365884
\(317\) −9576.87 −1.69682 −0.848408 0.529343i \(-0.822439\pi\)
−0.848408 + 0.529343i \(0.822439\pi\)
\(318\) 0 0
\(319\) −2543.19 −0.446368
\(320\) 3350.41 0.585293
\(321\) 0 0
\(322\) −1003.77 −0.173721
\(323\) 6513.81 1.12210
\(324\) 0 0
\(325\) 0 0
\(326\) 510.608 0.0867483
\(327\) 0 0
\(328\) 4936.14 0.830953
\(329\) −1399.00 −0.234435
\(330\) 0 0
\(331\) −5773.13 −0.958670 −0.479335 0.877632i \(-0.659122\pi\)
−0.479335 + 0.877632i \(0.659122\pi\)
\(332\) 2675.91 0.442348
\(333\) 0 0
\(334\) 2877.27 0.471370
\(335\) −6361.23 −1.03747
\(336\) 0 0
\(337\) 1238.09 0.200127 0.100063 0.994981i \(-0.468095\pi\)
0.100063 + 0.994981i \(0.468095\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −4905.47 −0.782460
\(341\) 5819.74 0.924213
\(342\) 0 0
\(343\) −5399.55 −0.849995
\(344\) 5671.48 0.888912
\(345\) 0 0
\(346\) −3049.53 −0.473826
\(347\) −5449.97 −0.843140 −0.421570 0.906796i \(-0.638521\pi\)
−0.421570 + 0.906796i \(0.638521\pi\)
\(348\) 0 0
\(349\) 1374.03 0.210746 0.105373 0.994433i \(-0.466396\pi\)
0.105373 + 0.994433i \(0.466396\pi\)
\(350\) 2158.92 0.329711
\(351\) 0 0
\(352\) −10013.3 −1.51622
\(353\) −5970.44 −0.900211 −0.450106 0.892975i \(-0.648614\pi\)
−0.450106 + 0.892975i \(0.648614\pi\)
\(354\) 0 0
\(355\) 1415.62 0.211644
\(356\) −854.734 −0.127249
\(357\) 0 0
\(358\) 3039.49 0.448720
\(359\) −7813.71 −1.14872 −0.574362 0.818602i \(-0.694750\pi\)
−0.574362 + 0.818602i \(0.694750\pi\)
\(360\) 0 0
\(361\) 14765.6 2.15274
\(362\) 69.7313 0.0101243
\(363\) 0 0
\(364\) 0 0
\(365\) 976.690 0.140061
\(366\) 0 0
\(367\) 1688.39 0.240145 0.120073 0.992765i \(-0.461687\pi\)
0.120073 + 0.992765i \(0.461687\pi\)
\(368\) −2366.11 −0.335169
\(369\) 0 0
\(370\) 1120.59 0.157451
\(371\) −4663.61 −0.652622
\(372\) 0 0
\(373\) 1870.61 0.259670 0.129835 0.991536i \(-0.458555\pi\)
0.129835 + 0.991536i \(0.458555\pi\)
\(374\) 2675.53 0.369916
\(375\) 0 0
\(376\) 945.151 0.129634
\(377\) 0 0
\(378\) 0 0
\(379\) 11667.0 1.58125 0.790627 0.612298i \(-0.209755\pi\)
0.790627 + 0.612298i \(0.209755\pi\)
\(380\) −16285.2 −2.19846
\(381\) 0 0
\(382\) 3736.82 0.500504
\(383\) 6676.60 0.890753 0.445376 0.895343i \(-0.353070\pi\)
0.445376 + 0.895343i \(0.353070\pi\)
\(384\) 0 0
\(385\) 20966.3 2.77543
\(386\) −795.896 −0.104948
\(387\) 0 0
\(388\) 3970.42 0.519504
\(389\) −14285.3 −1.86194 −0.930969 0.365099i \(-0.881035\pi\)
−0.930969 + 0.365099i \(0.881035\pi\)
\(390\) 0 0
\(391\) 2354.69 0.304558
\(392\) −1121.85 −0.144545
\(393\) 0 0
\(394\) −1731.72 −0.221428
\(395\) 4441.79 0.565800
\(396\) 0 0
\(397\) 3569.28 0.451227 0.225614 0.974217i \(-0.427561\pi\)
0.225614 + 0.974217i \(0.427561\pi\)
\(398\) 2556.00 0.321912
\(399\) 0 0
\(400\) 5089.04 0.636130
\(401\) −499.348 −0.0621852 −0.0310926 0.999517i \(-0.509899\pi\)
−0.0310926 + 0.999517i \(0.509899\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3311.23 −0.407772
\(405\) 0 0
\(406\) 729.354 0.0891557
\(407\) 5198.80 0.633158
\(408\) 0 0
\(409\) 10457.5 1.26428 0.632140 0.774855i \(-0.282177\pi\)
0.632140 + 0.774855i \(0.282177\pi\)
\(410\) 5037.73 0.606819
\(411\) 0 0
\(412\) −8032.27 −0.960489
\(413\) 2923.47 0.348316
\(414\) 0 0
\(415\) 5783.03 0.684043
\(416\) 0 0
\(417\) 0 0
\(418\) 8882.27 1.03934
\(419\) 1705.42 0.198843 0.0994215 0.995045i \(-0.468301\pi\)
0.0994215 + 0.995045i \(0.468301\pi\)
\(420\) 0 0
\(421\) −8765.57 −1.01475 −0.507373 0.861727i \(-0.669383\pi\)
−0.507373 + 0.861727i \(0.669383\pi\)
\(422\) 650.225 0.0750058
\(423\) 0 0
\(424\) 3150.70 0.360876
\(425\) −5064.48 −0.578032
\(426\) 0 0
\(427\) 5492.28 0.622459
\(428\) 4314.79 0.487297
\(429\) 0 0
\(430\) 5788.21 0.649145
\(431\) 8080.37 0.903057 0.451529 0.892257i \(-0.350879\pi\)
0.451529 + 0.892257i \(0.350879\pi\)
\(432\) 0 0
\(433\) −4124.48 −0.457760 −0.228880 0.973455i \(-0.573506\pi\)
−0.228880 + 0.973455i \(0.573506\pi\)
\(434\) −1669.02 −0.184598
\(435\) 0 0
\(436\) 10178.5 1.11803
\(437\) 7817.14 0.855709
\(438\) 0 0
\(439\) −6115.52 −0.664870 −0.332435 0.943126i \(-0.607870\pi\)
−0.332435 + 0.943126i \(0.607870\pi\)
\(440\) −14164.6 −1.53471
\(441\) 0 0
\(442\) 0 0
\(443\) 11058.8 1.18605 0.593025 0.805184i \(-0.297933\pi\)
0.593025 + 0.805184i \(0.297933\pi\)
\(444\) 0 0
\(445\) −1847.21 −0.196777
\(446\) −4950.73 −0.525614
\(447\) 0 0
\(448\) −4457.72 −0.470106
\(449\) −242.012 −0.0254371 −0.0127185 0.999919i \(-0.504049\pi\)
−0.0127185 + 0.999919i \(0.504049\pi\)
\(450\) 0 0
\(451\) 23371.7 2.44020
\(452\) −2839.88 −0.295524
\(453\) 0 0
\(454\) 2918.89 0.301741
\(455\) 0 0
\(456\) 0 0
\(457\) 11052.2 1.13129 0.565644 0.824650i \(-0.308628\pi\)
0.565644 + 0.824650i \(0.308628\pi\)
\(458\) −1865.94 −0.190370
\(459\) 0 0
\(460\) −5886.99 −0.596700
\(461\) 273.763 0.0276582 0.0138291 0.999904i \(-0.495598\pi\)
0.0138291 + 0.999904i \(0.495598\pi\)
\(462\) 0 0
\(463\) −11579.2 −1.16227 −0.581134 0.813808i \(-0.697391\pi\)
−0.581134 + 0.813808i \(0.697391\pi\)
\(464\) 1719.25 0.172013
\(465\) 0 0
\(466\) 2563.34 0.254817
\(467\) −902.915 −0.0894688 −0.0447344 0.998999i \(-0.514244\pi\)
−0.0447344 + 0.998999i \(0.514244\pi\)
\(468\) 0 0
\(469\) 8463.59 0.833289
\(470\) 964.604 0.0946678
\(471\) 0 0
\(472\) −1975.07 −0.192606
\(473\) 26853.4 2.61040
\(474\) 0 0
\(475\) −16813.1 −1.62408
\(476\) 6526.71 0.628469
\(477\) 0 0
\(478\) −5039.88 −0.482257
\(479\) −12068.8 −1.15122 −0.575611 0.817723i \(-0.695236\pi\)
−0.575611 + 0.817723i \(0.695236\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −3725.70 −0.352077
\(483\) 0 0
\(484\) −21505.1 −2.01964
\(485\) 8580.66 0.803356
\(486\) 0 0
\(487\) −11492.5 −1.06936 −0.534678 0.845056i \(-0.679567\pi\)
−0.534678 + 0.845056i \(0.679567\pi\)
\(488\) −3710.54 −0.344197
\(489\) 0 0
\(490\) −1144.93 −0.105557
\(491\) −15704.2 −1.44342 −0.721710 0.692196i \(-0.756643\pi\)
−0.721710 + 0.692196i \(0.756643\pi\)
\(492\) 0 0
\(493\) −1710.95 −0.156303
\(494\) 0 0
\(495\) 0 0
\(496\) −3934.25 −0.356156
\(497\) −1883.48 −0.169991
\(498\) 0 0
\(499\) −9019.80 −0.809181 −0.404591 0.914498i \(-0.632586\pi\)
−0.404591 + 0.914498i \(0.632586\pi\)
\(500\) −1181.23 −0.105653
\(501\) 0 0
\(502\) 3275.83 0.291250
\(503\) 6033.84 0.534862 0.267431 0.963577i \(-0.413825\pi\)
0.267431 + 0.963577i \(0.413825\pi\)
\(504\) 0 0
\(505\) −7156.06 −0.630576
\(506\) 3210.87 0.282096
\(507\) 0 0
\(508\) −3134.44 −0.273757
\(509\) 22451.5 1.95510 0.977551 0.210699i \(-0.0675741\pi\)
0.977551 + 0.210699i \(0.0675741\pi\)
\(510\) 0 0
\(511\) −1299.48 −0.112497
\(512\) 11720.8 1.01170
\(513\) 0 0
\(514\) 6897.06 0.591860
\(515\) −17358.9 −1.48529
\(516\) 0 0
\(517\) 4475.11 0.380687
\(518\) −1490.95 −0.126464
\(519\) 0 0
\(520\) 0 0
\(521\) −15674.7 −1.31808 −0.659040 0.752108i \(-0.729037\pi\)
−0.659040 + 0.752108i \(0.729037\pi\)
\(522\) 0 0
\(523\) 13510.8 1.12961 0.564804 0.825225i \(-0.308952\pi\)
0.564804 + 0.825225i \(0.308952\pi\)
\(524\) 11864.7 0.989142
\(525\) 0 0
\(526\) 4275.13 0.354381
\(527\) 3915.27 0.323627
\(528\) 0 0
\(529\) −9341.16 −0.767746
\(530\) 3215.55 0.263537
\(531\) 0 0
\(532\) 21667.4 1.76580
\(533\) 0 0
\(534\) 0 0
\(535\) 9324.89 0.753551
\(536\) −5717.94 −0.460778
\(537\) 0 0
\(538\) −4905.60 −0.393114
\(539\) −5311.73 −0.424475
\(540\) 0 0
\(541\) −12103.6 −0.961875 −0.480937 0.876755i \(-0.659704\pi\)
−0.480937 + 0.876755i \(0.659704\pi\)
\(542\) −2727.43 −0.216149
\(543\) 0 0
\(544\) −6736.49 −0.530928
\(545\) 21997.2 1.72891
\(546\) 0 0
\(547\) −15228.6 −1.19036 −0.595181 0.803592i \(-0.702920\pi\)
−0.595181 + 0.803592i \(0.702920\pi\)
\(548\) −14083.6 −1.09785
\(549\) 0 0
\(550\) −6905.95 −0.535401
\(551\) −5680.03 −0.439160
\(552\) 0 0
\(553\) −5909.79 −0.454448
\(554\) −701.014 −0.0537603
\(555\) 0 0
\(556\) −20223.5 −1.54257
\(557\) 23596.9 1.79503 0.897516 0.440982i \(-0.145370\pi\)
0.897516 + 0.440982i \(0.145370\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −14173.6 −1.06954
\(561\) 0 0
\(562\) −6458.63 −0.484770
\(563\) −7941.62 −0.594493 −0.297246 0.954801i \(-0.596068\pi\)
−0.297246 + 0.954801i \(0.596068\pi\)
\(564\) 0 0
\(565\) −6137.40 −0.456996
\(566\) −8288.63 −0.615542
\(567\) 0 0
\(568\) 1272.47 0.0939991
\(569\) −2274.65 −0.167590 −0.0837948 0.996483i \(-0.526704\pi\)
−0.0837948 + 0.996483i \(0.526704\pi\)
\(570\) 0 0
\(571\) −4499.84 −0.329794 −0.164897 0.986311i \(-0.552729\pi\)
−0.164897 + 0.986311i \(0.552729\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6702.69 −0.487395
\(575\) −6077.82 −0.440804
\(576\) 0 0
\(577\) 25253.3 1.82202 0.911011 0.412381i \(-0.135303\pi\)
0.911011 + 0.412381i \(0.135303\pi\)
\(578\) −2707.08 −0.194809
\(579\) 0 0
\(580\) 4277.55 0.306234
\(581\) −7694.31 −0.549421
\(582\) 0 0
\(583\) 14918.0 1.05976
\(584\) 877.921 0.0622066
\(585\) 0 0
\(586\) 1637.73 0.115450
\(587\) −11285.2 −0.793508 −0.396754 0.917925i \(-0.629863\pi\)
−0.396754 + 0.917925i \(0.629863\pi\)
\(588\) 0 0
\(589\) 12997.9 0.909289
\(590\) −2015.72 −0.140654
\(591\) 0 0
\(592\) −3514.49 −0.243994
\(593\) 12824.5 0.888090 0.444045 0.896005i \(-0.353543\pi\)
0.444045 + 0.896005i \(0.353543\pi\)
\(594\) 0 0
\(595\) 14105.2 0.971859
\(596\) −5709.98 −0.392433
\(597\) 0 0
\(598\) 0 0
\(599\) 26180.3 1.78581 0.892905 0.450245i \(-0.148664\pi\)
0.892905 + 0.450245i \(0.148664\pi\)
\(600\) 0 0
\(601\) 16012.6 1.08680 0.543399 0.839474i \(-0.317137\pi\)
0.543399 + 0.839474i \(0.317137\pi\)
\(602\) −7701.20 −0.521391
\(603\) 0 0
\(604\) −1159.57 −0.0781166
\(605\) −46475.8 −3.12316
\(606\) 0 0
\(607\) −9531.48 −0.637349 −0.318674 0.947864i \(-0.603238\pi\)
−0.318674 + 0.947864i \(0.603238\pi\)
\(608\) −22363.9 −1.49174
\(609\) 0 0
\(610\) −3786.91 −0.251356
\(611\) 0 0
\(612\) 0 0
\(613\) −10187.1 −0.671211 −0.335605 0.942003i \(-0.608941\pi\)
−0.335605 + 0.942003i \(0.608941\pi\)
\(614\) −4883.30 −0.320967
\(615\) 0 0
\(616\) 18846.0 1.23267
\(617\) 8012.62 0.522813 0.261407 0.965229i \(-0.415814\pi\)
0.261407 + 0.965229i \(0.415814\pi\)
\(618\) 0 0
\(619\) −1886.59 −0.122501 −0.0612506 0.998122i \(-0.519509\pi\)
−0.0612506 + 0.998122i \(0.519509\pi\)
\(620\) −9788.58 −0.634063
\(621\) 0 0
\(622\) −5747.94 −0.370533
\(623\) 2457.70 0.158051
\(624\) 0 0
\(625\) −16844.5 −1.07805
\(626\) 6599.38 0.421349
\(627\) 0 0
\(628\) 2448.18 0.155562
\(629\) 3497.53 0.221710
\(630\) 0 0
\(631\) −14956.1 −0.943569 −0.471784 0.881714i \(-0.656390\pi\)
−0.471784 + 0.881714i \(0.656390\pi\)
\(632\) 3992.61 0.251293
\(633\) 0 0
\(634\) −8785.57 −0.550346
\(635\) −6773.99 −0.423335
\(636\) 0 0
\(637\) 0 0
\(638\) −2333.06 −0.144775
\(639\) 0 0
\(640\) 21895.5 1.35234
\(641\) −23691.7 −1.45985 −0.729927 0.683525i \(-0.760446\pi\)
−0.729927 + 0.683525i \(0.760446\pi\)
\(642\) 0 0
\(643\) −13651.3 −0.837254 −0.418627 0.908158i \(-0.637489\pi\)
−0.418627 + 0.908158i \(0.637489\pi\)
\(644\) 7832.62 0.479268
\(645\) 0 0
\(646\) 5975.60 0.363943
\(647\) 18132.3 1.10178 0.550892 0.834577i \(-0.314288\pi\)
0.550892 + 0.834577i \(0.314288\pi\)
\(648\) 0 0
\(649\) −9351.59 −0.565612
\(650\) 0 0
\(651\) 0 0
\(652\) −3984.36 −0.239324
\(653\) −6363.47 −0.381351 −0.190675 0.981653i \(-0.561068\pi\)
−0.190675 + 0.981653i \(0.561068\pi\)
\(654\) 0 0
\(655\) 25641.3 1.52960
\(656\) −15799.7 −0.940358
\(657\) 0 0
\(658\) −1283.40 −0.0760369
\(659\) 1051.43 0.0621518 0.0310759 0.999517i \(-0.490107\pi\)
0.0310759 + 0.999517i \(0.490107\pi\)
\(660\) 0 0
\(661\) −8119.75 −0.477794 −0.238897 0.971045i \(-0.576786\pi\)
−0.238897 + 0.971045i \(0.576786\pi\)
\(662\) −5296.12 −0.310936
\(663\) 0 0
\(664\) 5198.21 0.303810
\(665\) 46826.5 2.73061
\(666\) 0 0
\(667\) −2053.29 −0.119196
\(668\) −22451.9 −1.30043
\(669\) 0 0
\(670\) −5835.62 −0.336492
\(671\) −17568.7 −1.01078
\(672\) 0 0
\(673\) −190.264 −0.0108977 −0.00544885 0.999985i \(-0.501734\pi\)
−0.00544885 + 0.999985i \(0.501734\pi\)
\(674\) 1135.79 0.0649093
\(675\) 0 0
\(676\) 0 0
\(677\) −4861.93 −0.276010 −0.138005 0.990431i \(-0.544069\pi\)
−0.138005 + 0.990431i \(0.544069\pi\)
\(678\) 0 0
\(679\) −11416.5 −0.645253
\(680\) −9529.35 −0.537403
\(681\) 0 0
\(682\) 5338.88 0.299760
\(683\) −14539.6 −0.814556 −0.407278 0.913304i \(-0.633522\pi\)
−0.407278 + 0.913304i \(0.633522\pi\)
\(684\) 0 0
\(685\) −30436.8 −1.69771
\(686\) −4953.40 −0.275688
\(687\) 0 0
\(688\) −18153.4 −1.00595
\(689\) 0 0
\(690\) 0 0
\(691\) 22106.5 1.21703 0.608517 0.793541i \(-0.291765\pi\)
0.608517 + 0.793541i \(0.291765\pi\)
\(692\) 23796.0 1.30721
\(693\) 0 0
\(694\) −4999.66 −0.273465
\(695\) −43706.0 −2.38541
\(696\) 0 0
\(697\) 15723.4 0.854474
\(698\) 1260.50 0.0683534
\(699\) 0 0
\(700\) −16846.4 −0.909620
\(701\) −229.971 −0.0123907 −0.00619535 0.999981i \(-0.501972\pi\)
−0.00619535 + 0.999981i \(0.501972\pi\)
\(702\) 0 0
\(703\) 11611.1 0.622934
\(704\) 14259.4 0.763380
\(705\) 0 0
\(706\) −5477.13 −0.291975
\(707\) 9521.12 0.506476
\(708\) 0 0
\(709\) 4802.22 0.254374 0.127187 0.991879i \(-0.459405\pi\)
0.127187 + 0.991879i \(0.459405\pi\)
\(710\) 1298.66 0.0686446
\(711\) 0 0
\(712\) −1660.40 −0.0873965
\(713\) 4698.66 0.246797
\(714\) 0 0
\(715\) 0 0
\(716\) −23717.6 −1.23795
\(717\) 0 0
\(718\) −7168.09 −0.372578
\(719\) 11016.9 0.571435 0.285718 0.958314i \(-0.407768\pi\)
0.285718 + 0.958314i \(0.407768\pi\)
\(720\) 0 0
\(721\) 23096.0 1.19298
\(722\) 13545.6 0.698221
\(723\) 0 0
\(724\) −544.125 −0.0279313
\(725\) 4416.21 0.226226
\(726\) 0 0
\(727\) −13498.2 −0.688612 −0.344306 0.938857i \(-0.611886\pi\)
−0.344306 + 0.938857i \(0.611886\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 895.990 0.0454275
\(731\) 18065.8 0.914073
\(732\) 0 0
\(733\) 17014.7 0.857368 0.428684 0.903455i \(-0.358977\pi\)
0.428684 + 0.903455i \(0.358977\pi\)
\(734\) 1548.88 0.0778888
\(735\) 0 0
\(736\) −8084.38 −0.404883
\(737\) −27073.4 −1.35313
\(738\) 0 0
\(739\) −13392.6 −0.666651 −0.333326 0.942812i \(-0.608171\pi\)
−0.333326 + 0.942812i \(0.608171\pi\)
\(740\) −8744.19 −0.434382
\(741\) 0 0
\(742\) −4278.27 −0.211672
\(743\) 11796.5 0.582467 0.291233 0.956652i \(-0.405934\pi\)
0.291233 + 0.956652i \(0.405934\pi\)
\(744\) 0 0
\(745\) −12340.1 −0.606854
\(746\) 1716.05 0.0842214
\(747\) 0 0
\(748\) −20877.6 −1.02054
\(749\) −12406.7 −0.605250
\(750\) 0 0
\(751\) −12450.5 −0.604961 −0.302480 0.953156i \(-0.597815\pi\)
−0.302480 + 0.953156i \(0.597815\pi\)
\(752\) −3025.26 −0.146702
\(753\) 0 0
\(754\) 0 0
\(755\) −2506.01 −0.120799
\(756\) 0 0
\(757\) −7029.84 −0.337522 −0.168761 0.985657i \(-0.553977\pi\)
−0.168761 + 0.985657i \(0.553977\pi\)
\(758\) 10703.0 0.512865
\(759\) 0 0
\(760\) −31635.6 −1.50993
\(761\) 17740.6 0.845067 0.422533 0.906347i \(-0.361141\pi\)
0.422533 + 0.906347i \(0.361141\pi\)
\(762\) 0 0
\(763\) −29267.2 −1.38866
\(764\) −29159.1 −1.38081
\(765\) 0 0
\(766\) 6124.94 0.288907
\(767\) 0 0
\(768\) 0 0
\(769\) −31976.5 −1.49948 −0.749741 0.661732i \(-0.769822\pi\)
−0.749741 + 0.661732i \(0.769822\pi\)
\(770\) 19233.9 0.900184
\(771\) 0 0
\(772\) 6210.51 0.289535
\(773\) −2135.11 −0.0993463 −0.0496732 0.998766i \(-0.515818\pi\)
−0.0496732 + 0.998766i \(0.515818\pi\)
\(774\) 0 0
\(775\) −10105.9 −0.468405
\(776\) 7712.93 0.356802
\(777\) 0 0
\(778\) −13105.0 −0.603902
\(779\) 52198.9 2.40080
\(780\) 0 0
\(781\) 6024.89 0.276040
\(782\) 2160.14 0.0987804
\(783\) 0 0
\(784\) 3590.83 0.163576
\(785\) 5290.89 0.240560
\(786\) 0 0
\(787\) 11704.4 0.530134 0.265067 0.964230i \(-0.414606\pi\)
0.265067 + 0.964230i \(0.414606\pi\)
\(788\) 13512.9 0.610885
\(789\) 0 0
\(790\) 4074.78 0.183512
\(791\) 8165.80 0.367057
\(792\) 0 0
\(793\) 0 0
\(794\) 3274.37 0.146351
\(795\) 0 0
\(796\) −19944.9 −0.888102
\(797\) 149.733 0.00665474 0.00332737 0.999994i \(-0.498941\pi\)
0.00332737 + 0.999994i \(0.498941\pi\)
\(798\) 0 0
\(799\) 3010.66 0.133304
\(800\) 17387.9 0.768443
\(801\) 0 0
\(802\) −458.089 −0.0201692
\(803\) 4156.79 0.182677
\(804\) 0 0
\(805\) 16927.4 0.741135
\(806\) 0 0
\(807\) 0 0
\(808\) −6432.40 −0.280063
\(809\) −23520.8 −1.02219 −0.511093 0.859526i \(-0.670759\pi\)
−0.511093 + 0.859526i \(0.670759\pi\)
\(810\) 0 0
\(811\) −29604.8 −1.28183 −0.640915 0.767612i \(-0.721445\pi\)
−0.640915 + 0.767612i \(0.721445\pi\)
\(812\) −5691.27 −0.245966
\(813\) 0 0
\(814\) 4769.25 0.205359
\(815\) −8610.79 −0.370089
\(816\) 0 0
\(817\) 59975.0 2.56825
\(818\) 9593.44 0.410057
\(819\) 0 0
\(820\) −39310.3 −1.67412
\(821\) 43034.7 1.82938 0.914691 0.404154i \(-0.132434\pi\)
0.914691 + 0.404154i \(0.132434\pi\)
\(822\) 0 0
\(823\) 5584.95 0.236548 0.118274 0.992981i \(-0.462264\pi\)
0.118274 + 0.992981i \(0.462264\pi\)
\(824\) −15603.5 −0.659675
\(825\) 0 0
\(826\) 2681.91 0.112973
\(827\) −4788.13 −0.201330 −0.100665 0.994920i \(-0.532097\pi\)
−0.100665 + 0.994920i \(0.532097\pi\)
\(828\) 0 0
\(829\) −32392.2 −1.35709 −0.678546 0.734558i \(-0.737390\pi\)
−0.678546 + 0.734558i \(0.737390\pi\)
\(830\) 5305.20 0.221863
\(831\) 0 0
\(832\) 0 0
\(833\) −3573.50 −0.148637
\(834\) 0 0
\(835\) −48521.8 −2.01098
\(836\) −69309.9 −2.86738
\(837\) 0 0
\(838\) 1564.51 0.0644929
\(839\) 13598.9 0.559577 0.279788 0.960062i \(-0.409736\pi\)
0.279788 + 0.960062i \(0.409736\pi\)
\(840\) 0 0
\(841\) −22897.1 −0.938827
\(842\) −8041.31 −0.329123
\(843\) 0 0
\(844\) −5073.82 −0.206929
\(845\) 0 0
\(846\) 0 0
\(847\) 61835.9 2.50851
\(848\) −10084.8 −0.408390
\(849\) 0 0
\(850\) −4646.02 −0.187479
\(851\) 4197.34 0.169075
\(852\) 0 0
\(853\) −41037.0 −1.64722 −0.823611 0.567155i \(-0.808044\pi\)
−0.823611 + 0.567155i \(0.808044\pi\)
\(854\) 5038.47 0.201889
\(855\) 0 0
\(856\) 8381.89 0.334681
\(857\) 39959.3 1.59275 0.796374 0.604804i \(-0.206749\pi\)
0.796374 + 0.604804i \(0.206749\pi\)
\(858\) 0 0
\(859\) −32570.5 −1.29371 −0.646853 0.762615i \(-0.723915\pi\)
−0.646853 + 0.762615i \(0.723915\pi\)
\(860\) −45166.4 −1.79088
\(861\) 0 0
\(862\) 7412.72 0.292898
\(863\) −16951.8 −0.668652 −0.334326 0.942457i \(-0.608509\pi\)
−0.334326 + 0.942457i \(0.608509\pi\)
\(864\) 0 0
\(865\) 51426.6 2.02145
\(866\) −3783.69 −0.148470
\(867\) 0 0
\(868\) 13023.7 0.509277
\(869\) 18904.3 0.737955
\(870\) 0 0
\(871\) 0 0
\(872\) 19772.7 0.767876
\(873\) 0 0
\(874\) 7171.24 0.277541
\(875\) 3396.52 0.131227
\(876\) 0 0
\(877\) 45939.2 1.76882 0.884411 0.466709i \(-0.154561\pi\)
0.884411 + 0.466709i \(0.154561\pi\)
\(878\) −5610.22 −0.215644
\(879\) 0 0
\(880\) 45338.5 1.73677
\(881\) −37960.5 −1.45167 −0.725836 0.687868i \(-0.758547\pi\)
−0.725836 + 0.687868i \(0.758547\pi\)
\(882\) 0 0
\(883\) 43172.9 1.64539 0.822697 0.568480i \(-0.192468\pi\)
0.822697 + 0.568480i \(0.192468\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 10145.1 0.384684
\(887\) −28706.3 −1.08665 −0.543327 0.839521i \(-0.682836\pi\)
−0.543327 + 0.839521i \(0.682836\pi\)
\(888\) 0 0
\(889\) 9012.78 0.340021
\(890\) −1694.58 −0.0638229
\(891\) 0 0
\(892\) 38631.4 1.45008
\(893\) 9994.83 0.374540
\(894\) 0 0
\(895\) −51257.3 −1.91435
\(896\) −29131.9 −1.08619
\(897\) 0 0
\(898\) −222.015 −0.00825027
\(899\) −3414.10 −0.126659
\(900\) 0 0
\(901\) 10036.2 0.371091
\(902\) 21440.6 0.791456
\(903\) 0 0
\(904\) −5516.75 −0.202969
\(905\) −1175.93 −0.0431927
\(906\) 0 0
\(907\) −22356.1 −0.818435 −0.409218 0.912437i \(-0.634198\pi\)
−0.409218 + 0.912437i \(0.634198\pi\)
\(908\) −22776.6 −0.832454
\(909\) 0 0
\(910\) 0 0
\(911\) −6953.80 −0.252897 −0.126449 0.991973i \(-0.540358\pi\)
−0.126449 + 0.991973i \(0.540358\pi\)
\(912\) 0 0
\(913\) 24612.6 0.892176
\(914\) 10139.0 0.366923
\(915\) 0 0
\(916\) 14560.2 0.525200
\(917\) −34115.7 −1.22857
\(918\) 0 0
\(919\) 40625.2 1.45822 0.729108 0.684399i \(-0.239935\pi\)
0.729108 + 0.684399i \(0.239935\pi\)
\(920\) −11436.0 −0.409821
\(921\) 0 0
\(922\) 251.143 0.00897068
\(923\) 0 0
\(924\) 0 0
\(925\) −9027.64 −0.320894
\(926\) −10622.4 −0.376971
\(927\) 0 0
\(928\) 5874.20 0.207791
\(929\) 42813.8 1.51203 0.756014 0.654555i \(-0.227144\pi\)
0.756014 + 0.654555i \(0.227144\pi\)
\(930\) 0 0
\(931\) −11863.3 −0.417621
\(932\) −20002.2 −0.702998
\(933\) 0 0
\(934\) −828.311 −0.0290184
\(935\) −45119.7 −1.57815
\(936\) 0 0
\(937\) 43484.1 1.51608 0.758038 0.652210i \(-0.226158\pi\)
0.758038 + 0.652210i \(0.226158\pi\)
\(938\) 7764.28 0.270269
\(939\) 0 0
\(940\) −7526.97 −0.261173
\(941\) −7108.58 −0.246263 −0.123131 0.992390i \(-0.539294\pi\)
−0.123131 + 0.992390i \(0.539294\pi\)
\(942\) 0 0
\(943\) 18869.5 0.651618
\(944\) 6321.85 0.217965
\(945\) 0 0
\(946\) 24634.6 0.846660
\(947\) 1938.99 0.0665351 0.0332676 0.999446i \(-0.489409\pi\)
0.0332676 + 0.999446i \(0.489409\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −15423.9 −0.526756
\(951\) 0 0
\(952\) 12678.8 0.431640
\(953\) 47806.7 1.62498 0.812492 0.582972i \(-0.198110\pi\)
0.812492 + 0.582972i \(0.198110\pi\)
\(954\) 0 0
\(955\) −63017.1 −2.13527
\(956\) 39327.1 1.33047
\(957\) 0 0
\(958\) −11071.6 −0.373388
\(959\) 40496.1 1.36359
\(960\) 0 0
\(961\) −21978.3 −0.737750
\(962\) 0 0
\(963\) 0 0
\(964\) 29072.3 0.971323
\(965\) 13421.8 0.447735
\(966\) 0 0
\(967\) 2832.71 0.0942025 0.0471013 0.998890i \(-0.485002\pi\)
0.0471013 + 0.998890i \(0.485002\pi\)
\(968\) −41775.8 −1.38711
\(969\) 0 0
\(970\) 7871.68 0.260561
\(971\) 41276.3 1.36418 0.682090 0.731268i \(-0.261071\pi\)
0.682090 + 0.731268i \(0.261071\pi\)
\(972\) 0 0
\(973\) 58150.7 1.91596
\(974\) −10542.9 −0.346835
\(975\) 0 0
\(976\) 11876.8 0.389515
\(977\) 4278.89 0.140117 0.0700583 0.997543i \(-0.477681\pi\)
0.0700583 + 0.997543i \(0.477681\pi\)
\(978\) 0 0
\(979\) −7861.70 −0.256651
\(980\) 8934.12 0.291214
\(981\) 0 0
\(982\) −14406.6 −0.468159
\(983\) 22652.9 0.735011 0.367505 0.930021i \(-0.380212\pi\)
0.367505 + 0.930021i \(0.380212\pi\)
\(984\) 0 0
\(985\) 29203.4 0.944667
\(986\) −1569.58 −0.0506953
\(987\) 0 0
\(988\) 0 0
\(989\) 21680.5 0.697068
\(990\) 0 0
\(991\) 22060.2 0.707131 0.353565 0.935410i \(-0.384969\pi\)
0.353565 + 0.935410i \(0.384969\pi\)
\(992\) −13442.3 −0.430235
\(993\) 0 0
\(994\) −1727.86 −0.0551351
\(995\) −43103.9 −1.37335
\(996\) 0 0
\(997\) 35635.5 1.13198 0.565992 0.824411i \(-0.308493\pi\)
0.565992 + 0.824411i \(0.308493\pi\)
\(998\) −8274.52 −0.262450
\(999\) 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 1521.4.a.bk.1.6 10
3.2 odd 2 507.4.a.r.1.5 10
13.2 odd 12 117.4.q.e.82.3 10
13.7 odd 12 117.4.q.e.10.3 10
13.12 even 2 inner 1521.4.a.bk.1.5 10
39.2 even 12 39.4.j.c.4.3 10
39.5 even 4 507.4.b.i.337.6 10
39.8 even 4 507.4.b.i.337.5 10
39.20 even 12 39.4.j.c.10.3 yes 10
39.38 odd 2 507.4.a.r.1.6 10
156.59 odd 12 624.4.bv.h.49.4 10
156.119 odd 12 624.4.bv.h.433.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 10 39.2 even 12
39.4.j.c.10.3 yes 10 39.20 even 12
117.4.q.e.10.3 10 13.7 odd 12
117.4.q.e.82.3 10 13.2 odd 12
507.4.a.r.1.5 10 3.2 odd 2
507.4.a.r.1.6 10 39.38 odd 2
507.4.b.i.337.5 10 39.8 even 4
507.4.b.i.337.6 10 39.5 even 4
624.4.bv.h.49.4 10 156.59 odd 12
624.4.bv.h.433.2 10 156.119 odd 12
1521.4.a.bk.1.5 10 13.12 even 2 inner
1521.4.a.bk.1.6 10 1.1 even 1 trivial