Properties

Label 2205.4.a.bx.1.2
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 34x^{4} + 241x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.09945\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.09945 q^{2} +1.60662 q^{4} -5.00000 q^{5} +19.8160 q^{8} +O(q^{10})\) \(q-3.09945 q^{2} +1.60662 q^{4} -5.00000 q^{5} +19.8160 q^{8} +15.4973 q^{10} +31.1878 q^{11} -54.1032 q^{13} -74.2717 q^{16} +81.8784 q^{17} +37.3653 q^{19} -8.03310 q^{20} -96.6651 q^{22} -116.264 q^{23} +25.0000 q^{25} +167.691 q^{26} -22.5290 q^{29} -89.9535 q^{31} +71.6739 q^{32} -253.778 q^{34} +344.782 q^{37} -115.812 q^{38} -99.0800 q^{40} -245.889 q^{41} +41.4519 q^{43} +50.1069 q^{44} +360.356 q^{46} -431.524 q^{47} -77.4864 q^{50} -86.9233 q^{52} +263.673 q^{53} -155.939 q^{55} +69.8276 q^{58} -627.005 q^{59} +715.666 q^{61} +278.807 q^{62} +372.024 q^{64} +270.516 q^{65} -809.809 q^{67} +131.547 q^{68} -54.8762 q^{71} +508.548 q^{73} -1068.64 q^{74} +60.0319 q^{76} +61.8168 q^{79} +371.359 q^{80} +762.122 q^{82} -560.154 q^{83} -409.392 q^{85} -128.478 q^{86} +618.017 q^{88} -102.286 q^{89} -186.792 q^{92} +1337.49 q^{94} -186.827 q^{95} -253.910 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 20 q^{4} - 30 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.09945 −1.09582 −0.547911 0.836536i \(-0.684577\pi\)
−0.547911 + 0.836536i \(0.684577\pi\)
\(3\) 0 0
\(4\) 1.60662 0.200827
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 19.8160 0.875751
\(9\) 0 0
\(10\) 15.4973 0.490067
\(11\) 31.1878 0.854861 0.427430 0.904048i \(-0.359419\pi\)
0.427430 + 0.904048i \(0.359419\pi\)
\(12\) 0 0
\(13\) −54.1032 −1.15427 −0.577136 0.816648i \(-0.695830\pi\)
−0.577136 + 0.816648i \(0.695830\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −74.2717 −1.16050
\(17\) 81.8784 1.16814 0.584071 0.811703i \(-0.301459\pi\)
0.584071 + 0.811703i \(0.301459\pi\)
\(18\) 0 0
\(19\) 37.3653 0.451168 0.225584 0.974224i \(-0.427571\pi\)
0.225584 + 0.974224i \(0.427571\pi\)
\(20\) −8.03310 −0.0898128
\(21\) 0 0
\(22\) −96.6651 −0.936776
\(23\) −116.264 −1.05403 −0.527017 0.849855i \(-0.676689\pi\)
−0.527017 + 0.849855i \(0.676689\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 167.691 1.26488
\(27\) 0 0
\(28\) 0 0
\(29\) −22.5290 −0.144260 −0.0721298 0.997395i \(-0.522980\pi\)
−0.0721298 + 0.997395i \(0.522980\pi\)
\(30\) 0 0
\(31\) −89.9535 −0.521165 −0.260583 0.965452i \(-0.583915\pi\)
−0.260583 + 0.965452i \(0.583915\pi\)
\(32\) 71.6739 0.395946
\(33\) 0 0
\(34\) −253.778 −1.28008
\(35\) 0 0
\(36\) 0 0
\(37\) 344.782 1.53194 0.765971 0.642876i \(-0.222259\pi\)
0.765971 + 0.642876i \(0.222259\pi\)
\(38\) −115.812 −0.494400
\(39\) 0 0
\(40\) −99.0800 −0.391648
\(41\) −245.889 −0.936620 −0.468310 0.883564i \(-0.655137\pi\)
−0.468310 + 0.883564i \(0.655137\pi\)
\(42\) 0 0
\(43\) 41.4519 0.147008 0.0735041 0.997295i \(-0.476582\pi\)
0.0735041 + 0.997295i \(0.476582\pi\)
\(44\) 50.1069 0.171680
\(45\) 0 0
\(46\) 360.356 1.15503
\(47\) −431.524 −1.33924 −0.669620 0.742704i \(-0.733543\pi\)
−0.669620 + 0.742704i \(0.733543\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −77.4864 −0.219165
\(51\) 0 0
\(52\) −86.9233 −0.231810
\(53\) 263.673 0.683365 0.341682 0.939815i \(-0.389003\pi\)
0.341682 + 0.939815i \(0.389003\pi\)
\(54\) 0 0
\(55\) −155.939 −0.382305
\(56\) 0 0
\(57\) 0 0
\(58\) 69.8276 0.158083
\(59\) −627.005 −1.38354 −0.691772 0.722116i \(-0.743170\pi\)
−0.691772 + 0.722116i \(0.743170\pi\)
\(60\) 0 0
\(61\) 715.666 1.50216 0.751079 0.660212i \(-0.229534\pi\)
0.751079 + 0.660212i \(0.229534\pi\)
\(62\) 278.807 0.571105
\(63\) 0 0
\(64\) 372.024 0.726609
\(65\) 270.516 0.516206
\(66\) 0 0
\(67\) −809.809 −1.47663 −0.738313 0.674459i \(-0.764377\pi\)
−0.738313 + 0.674459i \(0.764377\pi\)
\(68\) 131.547 0.234595
\(69\) 0 0
\(70\) 0 0
\(71\) −54.8762 −0.0917268 −0.0458634 0.998948i \(-0.514604\pi\)
−0.0458634 + 0.998948i \(0.514604\pi\)
\(72\) 0 0
\(73\) 508.548 0.815356 0.407678 0.913126i \(-0.366339\pi\)
0.407678 + 0.913126i \(0.366339\pi\)
\(74\) −1068.64 −1.67874
\(75\) 0 0
\(76\) 60.0319 0.0906070
\(77\) 0 0
\(78\) 0 0
\(79\) 61.8168 0.0880372 0.0440186 0.999031i \(-0.485984\pi\)
0.0440186 + 0.999031i \(0.485984\pi\)
\(80\) 371.359 0.518989
\(81\) 0 0
\(82\) 762.122 1.02637
\(83\) −560.154 −0.740782 −0.370391 0.928876i \(-0.620776\pi\)
−0.370391 + 0.928876i \(0.620776\pi\)
\(84\) 0 0
\(85\) −409.392 −0.522409
\(86\) −128.478 −0.161095
\(87\) 0 0
\(88\) 618.017 0.748646
\(89\) −102.286 −0.121824 −0.0609119 0.998143i \(-0.519401\pi\)
−0.0609119 + 0.998143i \(0.519401\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −186.792 −0.211679
\(93\) 0 0
\(94\) 1337.49 1.46757
\(95\) −186.827 −0.201769
\(96\) 0 0
\(97\) −253.910 −0.265780 −0.132890 0.991131i \(-0.542426\pi\)
−0.132890 + 0.991131i \(0.542426\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 40.1655 0.0401655
\(101\) 612.694 0.603617 0.301809 0.953369i \(-0.402410\pi\)
0.301809 + 0.953369i \(0.402410\pi\)
\(102\) 0 0
\(103\) 1865.47 1.78457 0.892283 0.451477i \(-0.149103\pi\)
0.892283 + 0.451477i \(0.149103\pi\)
\(104\) −1072.11 −1.01086
\(105\) 0 0
\(106\) −817.244 −0.748847
\(107\) 997.858 0.901557 0.450779 0.892636i \(-0.351146\pi\)
0.450779 + 0.892636i \(0.351146\pi\)
\(108\) 0 0
\(109\) 1553.97 1.36554 0.682768 0.730635i \(-0.260776\pi\)
0.682768 + 0.730635i \(0.260776\pi\)
\(110\) 483.326 0.418939
\(111\) 0 0
\(112\) 0 0
\(113\) −268.484 −0.223512 −0.111756 0.993736i \(-0.535647\pi\)
−0.111756 + 0.993736i \(0.535647\pi\)
\(114\) 0 0
\(115\) 581.321 0.471378
\(116\) −36.1955 −0.0289713
\(117\) 0 0
\(118\) 1943.37 1.51612
\(119\) 0 0
\(120\) 0 0
\(121\) −358.322 −0.269213
\(122\) −2218.18 −1.64610
\(123\) 0 0
\(124\) −144.521 −0.104664
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −43.7474 −0.0305665 −0.0152833 0.999883i \(-0.504865\pi\)
−0.0152833 + 0.999883i \(0.504865\pi\)
\(128\) −1726.46 −1.19218
\(129\) 0 0
\(130\) −838.453 −0.565670
\(131\) 1641.52 1.09481 0.547407 0.836867i \(-0.315615\pi\)
0.547407 + 0.836867i \(0.315615\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2509.97 1.61812
\(135\) 0 0
\(136\) 1622.50 1.02300
\(137\) 2465.15 1.53731 0.768656 0.639662i \(-0.220926\pi\)
0.768656 + 0.639662i \(0.220926\pi\)
\(138\) 0 0
\(139\) −2954.45 −1.80283 −0.901416 0.432955i \(-0.857471\pi\)
−0.901416 + 0.432955i \(0.857471\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 170.086 0.100516
\(143\) −1687.36 −0.986742
\(144\) 0 0
\(145\) 112.645 0.0645148
\(146\) −1576.22 −0.893486
\(147\) 0 0
\(148\) 553.934 0.307656
\(149\) 16.8760 0.00927875 0.00463938 0.999989i \(-0.498523\pi\)
0.00463938 + 0.999989i \(0.498523\pi\)
\(150\) 0 0
\(151\) −1792.06 −0.965797 −0.482899 0.875676i \(-0.660416\pi\)
−0.482899 + 0.875676i \(0.660416\pi\)
\(152\) 740.431 0.395111
\(153\) 0 0
\(154\) 0 0
\(155\) 449.767 0.233072
\(156\) 0 0
\(157\) 3543.34 1.80121 0.900603 0.434643i \(-0.143125\pi\)
0.900603 + 0.434643i \(0.143125\pi\)
\(158\) −191.598 −0.0964731
\(159\) 0 0
\(160\) −358.370 −0.177073
\(161\) 0 0
\(162\) 0 0
\(163\) −70.4005 −0.0338294 −0.0169147 0.999857i \(-0.505384\pi\)
−0.0169147 + 0.999857i \(0.505384\pi\)
\(164\) −395.050 −0.188099
\(165\) 0 0
\(166\) 1736.17 0.811765
\(167\) −2493.09 −1.15522 −0.577609 0.816313i \(-0.696014\pi\)
−0.577609 + 0.816313i \(0.696014\pi\)
\(168\) 0 0
\(169\) 730.159 0.332344
\(170\) 1268.89 0.572468
\(171\) 0 0
\(172\) 66.5974 0.0295233
\(173\) −487.259 −0.214137 −0.107068 0.994252i \(-0.534146\pi\)
−0.107068 + 0.994252i \(0.534146\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2316.37 −0.992062
\(177\) 0 0
\(178\) 317.032 0.133497
\(179\) 340.024 0.141981 0.0709904 0.997477i \(-0.477384\pi\)
0.0709904 + 0.997477i \(0.477384\pi\)
\(180\) 0 0
\(181\) −1863.23 −0.765152 −0.382576 0.923924i \(-0.624963\pi\)
−0.382576 + 0.923924i \(0.624963\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2303.89 −0.923071
\(185\) −1723.91 −0.685105
\(186\) 0 0
\(187\) 2553.60 0.998599
\(188\) −693.295 −0.268956
\(189\) 0 0
\(190\) 579.061 0.221103
\(191\) −1557.97 −0.590214 −0.295107 0.955464i \(-0.595355\pi\)
−0.295107 + 0.955464i \(0.595355\pi\)
\(192\) 0 0
\(193\) −338.441 −0.126225 −0.0631127 0.998006i \(-0.520103\pi\)
−0.0631127 + 0.998006i \(0.520103\pi\)
\(194\) 786.982 0.291248
\(195\) 0 0
\(196\) 0 0
\(197\) −903.950 −0.326923 −0.163461 0.986550i \(-0.552266\pi\)
−0.163461 + 0.986550i \(0.552266\pi\)
\(198\) 0 0
\(199\) 1987.44 0.707971 0.353985 0.935251i \(-0.384826\pi\)
0.353985 + 0.935251i \(0.384826\pi\)
\(200\) 495.400 0.175150
\(201\) 0 0
\(202\) −1899.02 −0.661457
\(203\) 0 0
\(204\) 0 0
\(205\) 1229.45 0.418869
\(206\) −5781.94 −1.95557
\(207\) 0 0
\(208\) 4018.34 1.33953
\(209\) 1165.34 0.385686
\(210\) 0 0
\(211\) 3749.01 1.22319 0.611594 0.791172i \(-0.290529\pi\)
0.611594 + 0.791172i \(0.290529\pi\)
\(212\) 423.623 0.137238
\(213\) 0 0
\(214\) −3092.82 −0.987947
\(215\) −207.259 −0.0657441
\(216\) 0 0
\(217\) 0 0
\(218\) −4816.47 −1.49639
\(219\) 0 0
\(220\) −250.534 −0.0767774
\(221\) −4429.88 −1.34835
\(222\) 0 0
\(223\) −3939.34 −1.18295 −0.591475 0.806324i \(-0.701454\pi\)
−0.591475 + 0.806324i \(0.701454\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 832.154 0.244929
\(227\) −2808.27 −0.821107 −0.410554 0.911836i \(-0.634665\pi\)
−0.410554 + 0.911836i \(0.634665\pi\)
\(228\) 0 0
\(229\) 1809.85 0.522264 0.261132 0.965303i \(-0.415904\pi\)
0.261132 + 0.965303i \(0.415904\pi\)
\(230\) −1801.78 −0.516547
\(231\) 0 0
\(232\) −446.434 −0.126336
\(233\) −973.995 −0.273856 −0.136928 0.990581i \(-0.543723\pi\)
−0.136928 + 0.990581i \(0.543723\pi\)
\(234\) 0 0
\(235\) 2157.62 0.598926
\(236\) −1007.36 −0.277854
\(237\) 0 0
\(238\) 0 0
\(239\) 6015.41 1.62805 0.814027 0.580828i \(-0.197271\pi\)
0.814027 + 0.580828i \(0.197271\pi\)
\(240\) 0 0
\(241\) 384.989 0.102902 0.0514509 0.998676i \(-0.483615\pi\)
0.0514509 + 0.998676i \(0.483615\pi\)
\(242\) 1110.60 0.295010
\(243\) 0 0
\(244\) 1149.80 0.301675
\(245\) 0 0
\(246\) 0 0
\(247\) −2021.59 −0.520771
\(248\) −1782.52 −0.456411
\(249\) 0 0
\(250\) 387.432 0.0980134
\(251\) −1857.04 −0.466994 −0.233497 0.972357i \(-0.575017\pi\)
−0.233497 + 0.972357i \(0.575017\pi\)
\(252\) 0 0
\(253\) −3626.02 −0.901052
\(254\) 135.593 0.0334955
\(255\) 0 0
\(256\) 2374.90 0.579810
\(257\) 4825.03 1.17112 0.585559 0.810630i \(-0.300875\pi\)
0.585559 + 0.810630i \(0.300875\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 434.617 0.103668
\(261\) 0 0
\(262\) −5087.83 −1.19972
\(263\) −836.796 −0.196194 −0.0980971 0.995177i \(-0.531276\pi\)
−0.0980971 + 0.995177i \(0.531276\pi\)
\(264\) 0 0
\(265\) −1318.37 −0.305610
\(266\) 0 0
\(267\) 0 0
\(268\) −1301.05 −0.296547
\(269\) −5454.11 −1.23622 −0.618110 0.786092i \(-0.712101\pi\)
−0.618110 + 0.786092i \(0.712101\pi\)
\(270\) 0 0
\(271\) 2580.16 0.578352 0.289176 0.957276i \(-0.406619\pi\)
0.289176 + 0.957276i \(0.406619\pi\)
\(272\) −6081.25 −1.35562
\(273\) 0 0
\(274\) −7640.61 −1.68462
\(275\) 779.694 0.170972
\(276\) 0 0
\(277\) 591.932 0.128396 0.0641981 0.997937i \(-0.479551\pi\)
0.0641981 + 0.997937i \(0.479551\pi\)
\(278\) 9157.20 1.97558
\(279\) 0 0
\(280\) 0 0
\(281\) −6365.13 −1.35129 −0.675644 0.737228i \(-0.736134\pi\)
−0.675644 + 0.737228i \(0.736134\pi\)
\(282\) 0 0
\(283\) 1074.93 0.225787 0.112894 0.993607i \(-0.463988\pi\)
0.112894 + 0.993607i \(0.463988\pi\)
\(284\) −88.1651 −0.0184213
\(285\) 0 0
\(286\) 5229.89 1.08129
\(287\) 0 0
\(288\) 0 0
\(289\) 1791.06 0.364556
\(290\) −349.138 −0.0706968
\(291\) 0 0
\(292\) 817.042 0.163746
\(293\) −5914.66 −1.17931 −0.589656 0.807655i \(-0.700737\pi\)
−0.589656 + 0.807655i \(0.700737\pi\)
\(294\) 0 0
\(295\) 3135.03 0.618740
\(296\) 6832.20 1.34160
\(297\) 0 0
\(298\) −52.3063 −0.0101679
\(299\) 6290.27 1.21664
\(300\) 0 0
\(301\) 0 0
\(302\) 5554.39 1.05834
\(303\) 0 0
\(304\) −2775.19 −0.523579
\(305\) −3578.33 −0.671786
\(306\) 0 0
\(307\) 3162.52 0.587929 0.293965 0.955816i \(-0.405025\pi\)
0.293965 + 0.955816i \(0.405025\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1394.03 −0.255406
\(311\) −6941.22 −1.26560 −0.632798 0.774317i \(-0.718094\pi\)
−0.632798 + 0.774317i \(0.718094\pi\)
\(312\) 0 0
\(313\) −8789.17 −1.58720 −0.793600 0.608440i \(-0.791795\pi\)
−0.793600 + 0.608440i \(0.791795\pi\)
\(314\) −10982.4 −1.97380
\(315\) 0 0
\(316\) 99.3161 0.0176803
\(317\) −4837.69 −0.857135 −0.428568 0.903510i \(-0.640982\pi\)
−0.428568 + 0.903510i \(0.640982\pi\)
\(318\) 0 0
\(319\) −702.629 −0.123322
\(320\) −1860.12 −0.324949
\(321\) 0 0
\(322\) 0 0
\(323\) 3059.41 0.527029
\(324\) 0 0
\(325\) −1352.58 −0.230854
\(326\) 218.203 0.0370710
\(327\) 0 0
\(328\) −4872.54 −0.820246
\(329\) 0 0
\(330\) 0 0
\(331\) −4122.43 −0.684560 −0.342280 0.939598i \(-0.611199\pi\)
−0.342280 + 0.939598i \(0.611199\pi\)
\(332\) −899.954 −0.148769
\(333\) 0 0
\(334\) 7727.23 1.26591
\(335\) 4049.04 0.660367
\(336\) 0 0
\(337\) 9671.74 1.56336 0.781681 0.623678i \(-0.214362\pi\)
0.781681 + 0.623678i \(0.214362\pi\)
\(338\) −2263.10 −0.364190
\(339\) 0 0
\(340\) −657.737 −0.104914
\(341\) −2805.45 −0.445524
\(342\) 0 0
\(343\) 0 0
\(344\) 821.410 0.128743
\(345\) 0 0
\(346\) 1510.24 0.234656
\(347\) 6169.21 0.954411 0.477205 0.878792i \(-0.341650\pi\)
0.477205 + 0.878792i \(0.341650\pi\)
\(348\) 0 0
\(349\) 7346.44 1.12678 0.563390 0.826191i \(-0.309497\pi\)
0.563390 + 0.826191i \(0.309497\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2235.35 0.338479
\(353\) −10850.5 −1.63602 −0.818008 0.575206i \(-0.804922\pi\)
−0.818008 + 0.575206i \(0.804922\pi\)
\(354\) 0 0
\(355\) 274.381 0.0410215
\(356\) −164.335 −0.0244656
\(357\) 0 0
\(358\) −1053.89 −0.155586
\(359\) 4365.52 0.641792 0.320896 0.947114i \(-0.396016\pi\)
0.320896 + 0.947114i \(0.396016\pi\)
\(360\) 0 0
\(361\) −5462.83 −0.796447
\(362\) 5774.99 0.838471
\(363\) 0 0
\(364\) 0 0
\(365\) −2542.74 −0.364638
\(366\) 0 0
\(367\) 7197.13 1.02367 0.511835 0.859084i \(-0.328966\pi\)
0.511835 + 0.859084i \(0.328966\pi\)
\(368\) 8635.14 1.22320
\(369\) 0 0
\(370\) 5343.18 0.750754
\(371\) 0 0
\(372\) 0 0
\(373\) 12300.6 1.70751 0.853753 0.520678i \(-0.174321\pi\)
0.853753 + 0.520678i \(0.174321\pi\)
\(374\) −7914.78 −1.09429
\(375\) 0 0
\(376\) −8551.08 −1.17284
\(377\) 1218.89 0.166515
\(378\) 0 0
\(379\) −888.207 −0.120380 −0.0601901 0.998187i \(-0.519171\pi\)
−0.0601901 + 0.998187i \(0.519171\pi\)
\(380\) −300.159 −0.0405207
\(381\) 0 0
\(382\) 4828.86 0.646770
\(383\) −10524.1 −1.40406 −0.702031 0.712147i \(-0.747723\pi\)
−0.702031 + 0.712147i \(0.747723\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1048.98 0.138321
\(387\) 0 0
\(388\) −407.937 −0.0533759
\(389\) 6006.81 0.782924 0.391462 0.920194i \(-0.371970\pi\)
0.391462 + 0.920194i \(0.371970\pi\)
\(390\) 0 0
\(391\) −9519.52 −1.23126
\(392\) 0 0
\(393\) 0 0
\(394\) 2801.75 0.358249
\(395\) −309.084 −0.0393714
\(396\) 0 0
\(397\) −14706.6 −1.85920 −0.929602 0.368565i \(-0.879849\pi\)
−0.929602 + 0.368565i \(0.879849\pi\)
\(398\) −6159.99 −0.775810
\(399\) 0 0
\(400\) −1856.79 −0.232099
\(401\) −2513.22 −0.312978 −0.156489 0.987680i \(-0.550018\pi\)
−0.156489 + 0.987680i \(0.550018\pi\)
\(402\) 0 0
\(403\) 4866.77 0.601566
\(404\) 984.366 0.121223
\(405\) 0 0
\(406\) 0 0
\(407\) 10753.0 1.30960
\(408\) 0 0
\(409\) −4288.99 −0.518526 −0.259263 0.965807i \(-0.583480\pi\)
−0.259263 + 0.965807i \(0.583480\pi\)
\(410\) −3810.61 −0.459006
\(411\) 0 0
\(412\) 2997.10 0.358390
\(413\) 0 0
\(414\) 0 0
\(415\) 2800.77 0.331288
\(416\) −3877.79 −0.457030
\(417\) 0 0
\(418\) −3611.92 −0.422643
\(419\) −6467.11 −0.754031 −0.377016 0.926207i \(-0.623050\pi\)
−0.377016 + 0.926207i \(0.623050\pi\)
\(420\) 0 0
\(421\) −13647.0 −1.57984 −0.789921 0.613209i \(-0.789878\pi\)
−0.789921 + 0.613209i \(0.789878\pi\)
\(422\) −11619.9 −1.34040
\(423\) 0 0
\(424\) 5224.95 0.598458
\(425\) 2046.96 0.233628
\(426\) 0 0
\(427\) 0 0
\(428\) 1603.18 0.181057
\(429\) 0 0
\(430\) 642.391 0.0720438
\(431\) −17252.7 −1.92815 −0.964074 0.265634i \(-0.914419\pi\)
−0.964074 + 0.265634i \(0.914419\pi\)
\(432\) 0 0
\(433\) 4033.03 0.447610 0.223805 0.974634i \(-0.428152\pi\)
0.223805 + 0.974634i \(0.428152\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2496.64 0.274237
\(437\) −4344.25 −0.475546
\(438\) 0 0
\(439\) −16593.8 −1.80405 −0.902025 0.431684i \(-0.857920\pi\)
−0.902025 + 0.431684i \(0.857920\pi\)
\(440\) −3090.08 −0.334804
\(441\) 0 0
\(442\) 13730.2 1.47756
\(443\) 5587.14 0.599216 0.299608 0.954062i \(-0.403144\pi\)
0.299608 + 0.954062i \(0.403144\pi\)
\(444\) 0 0
\(445\) 511.431 0.0544813
\(446\) 12209.8 1.29630
\(447\) 0 0
\(448\) 0 0
\(449\) −13090.2 −1.37587 −0.687935 0.725772i \(-0.741483\pi\)
−0.687935 + 0.725772i \(0.741483\pi\)
\(450\) 0 0
\(451\) −7668.73 −0.800680
\(452\) −431.352 −0.0448873
\(453\) 0 0
\(454\) 8704.10 0.899788
\(455\) 0 0
\(456\) 0 0
\(457\) 7594.11 0.777324 0.388662 0.921380i \(-0.372937\pi\)
0.388662 + 0.921380i \(0.372937\pi\)
\(458\) −5609.55 −0.572308
\(459\) 0 0
\(460\) 933.962 0.0946656
\(461\) −15260.2 −1.54173 −0.770865 0.636998i \(-0.780176\pi\)
−0.770865 + 0.636998i \(0.780176\pi\)
\(462\) 0 0
\(463\) −4278.64 −0.429472 −0.214736 0.976672i \(-0.568889\pi\)
−0.214736 + 0.976672i \(0.568889\pi\)
\(464\) 1673.27 0.167413
\(465\) 0 0
\(466\) 3018.85 0.300098
\(467\) −9102.32 −0.901938 −0.450969 0.892540i \(-0.648922\pi\)
−0.450969 + 0.892540i \(0.648922\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −6687.45 −0.656317
\(471\) 0 0
\(472\) −12424.7 −1.21164
\(473\) 1292.79 0.125672
\(474\) 0 0
\(475\) 934.133 0.0902336
\(476\) 0 0
\(477\) 0 0
\(478\) −18644.5 −1.78406
\(479\) −4226.58 −0.403167 −0.201584 0.979471i \(-0.564609\pi\)
−0.201584 + 0.979471i \(0.564609\pi\)
\(480\) 0 0
\(481\) −18653.8 −1.76828
\(482\) −1193.26 −0.112762
\(483\) 0 0
\(484\) −575.688 −0.0540653
\(485\) 1269.55 0.118860
\(486\) 0 0
\(487\) −9576.48 −0.891071 −0.445536 0.895264i \(-0.646987\pi\)
−0.445536 + 0.895264i \(0.646987\pi\)
\(488\) 14181.6 1.31552
\(489\) 0 0
\(490\) 0 0
\(491\) −21412.2 −1.96806 −0.984030 0.178000i \(-0.943037\pi\)
−0.984030 + 0.178000i \(0.943037\pi\)
\(492\) 0 0
\(493\) −1844.64 −0.168516
\(494\) 6265.81 0.570672
\(495\) 0 0
\(496\) 6681.00 0.604810
\(497\) 0 0
\(498\) 0 0
\(499\) −19706.9 −1.76794 −0.883970 0.467543i \(-0.845139\pi\)
−0.883970 + 0.467543i \(0.845139\pi\)
\(500\) −200.827 −0.0179626
\(501\) 0 0
\(502\) 5755.82 0.511743
\(503\) −176.759 −0.0156685 −0.00783427 0.999969i \(-0.502494\pi\)
−0.00783427 + 0.999969i \(0.502494\pi\)
\(504\) 0 0
\(505\) −3063.47 −0.269946
\(506\) 11238.7 0.987393
\(507\) 0 0
\(508\) −70.2854 −0.00613860
\(509\) −776.451 −0.0676142 −0.0338071 0.999428i \(-0.510763\pi\)
−0.0338071 + 0.999428i \(0.510763\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 6450.80 0.556812
\(513\) 0 0
\(514\) −14955.0 −1.28334
\(515\) −9327.35 −0.798082
\(516\) 0 0
\(517\) −13458.3 −1.14486
\(518\) 0 0
\(519\) 0 0
\(520\) 5360.55 0.452068
\(521\) −4248.01 −0.357215 −0.178607 0.983920i \(-0.557159\pi\)
−0.178607 + 0.983920i \(0.557159\pi\)
\(522\) 0 0
\(523\) −13490.8 −1.12794 −0.563968 0.825797i \(-0.690726\pi\)
−0.563968 + 0.825797i \(0.690726\pi\)
\(524\) 2637.30 0.219869
\(525\) 0 0
\(526\) 2593.61 0.214994
\(527\) −7365.24 −0.608795
\(528\) 0 0
\(529\) 1350.36 0.110986
\(530\) 4086.22 0.334894
\(531\) 0 0
\(532\) 0 0
\(533\) 13303.4 1.08111
\(534\) 0 0
\(535\) −4989.29 −0.403189
\(536\) −16047.2 −1.29316
\(537\) 0 0
\(538\) 16904.8 1.35468
\(539\) 0 0
\(540\) 0 0
\(541\) −10582.4 −0.840984 −0.420492 0.907296i \(-0.638143\pi\)
−0.420492 + 0.907296i \(0.638143\pi\)
\(542\) −7997.08 −0.633771
\(543\) 0 0
\(544\) 5868.54 0.462521
\(545\) −7769.86 −0.610687
\(546\) 0 0
\(547\) −6182.43 −0.483257 −0.241629 0.970369i \(-0.577682\pi\)
−0.241629 + 0.970369i \(0.577682\pi\)
\(548\) 3960.56 0.308735
\(549\) 0 0
\(550\) −2416.63 −0.187355
\(551\) −841.803 −0.0650853
\(552\) 0 0
\(553\) 0 0
\(554\) −1834.67 −0.140699
\(555\) 0 0
\(556\) −4746.68 −0.362058
\(557\) 20848.5 1.58596 0.792979 0.609249i \(-0.208529\pi\)
0.792979 + 0.609249i \(0.208529\pi\)
\(558\) 0 0
\(559\) −2242.68 −0.169687
\(560\) 0 0
\(561\) 0 0
\(562\) 19728.4 1.48077
\(563\) 13326.0 0.997557 0.498779 0.866729i \(-0.333782\pi\)
0.498779 + 0.866729i \(0.333782\pi\)
\(564\) 0 0
\(565\) 1342.42 0.0999576
\(566\) −3331.69 −0.247423
\(567\) 0 0
\(568\) −1087.43 −0.0803299
\(569\) −11592.9 −0.854126 −0.427063 0.904222i \(-0.640452\pi\)
−0.427063 + 0.904222i \(0.640452\pi\)
\(570\) 0 0
\(571\) 1603.77 0.117541 0.0587704 0.998272i \(-0.481282\pi\)
0.0587704 + 0.998272i \(0.481282\pi\)
\(572\) −2710.94 −0.198165
\(573\) 0 0
\(574\) 0 0
\(575\) −2906.60 −0.210807
\(576\) 0 0
\(577\) −12475.4 −0.900097 −0.450048 0.893004i \(-0.648593\pi\)
−0.450048 + 0.893004i \(0.648593\pi\)
\(578\) −5551.32 −0.399489
\(579\) 0 0
\(580\) 180.978 0.0129564
\(581\) 0 0
\(582\) 0 0
\(583\) 8223.39 0.584182
\(584\) 10077.4 0.714049
\(585\) 0 0
\(586\) 18332.2 1.29232
\(587\) −11367.0 −0.799264 −0.399632 0.916676i \(-0.630862\pi\)
−0.399632 + 0.916676i \(0.630862\pi\)
\(588\) 0 0
\(589\) −3361.14 −0.235133
\(590\) −9716.87 −0.678029
\(591\) 0 0
\(592\) −25607.6 −1.77781
\(593\) −2463.42 −0.170591 −0.0852955 0.996356i \(-0.527183\pi\)
−0.0852955 + 0.996356i \(0.527183\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 27.1133 0.00186343
\(597\) 0 0
\(598\) −19496.4 −1.33322
\(599\) 26493.7 1.80719 0.903593 0.428393i \(-0.140920\pi\)
0.903593 + 0.428393i \(0.140920\pi\)
\(600\) 0 0
\(601\) −5703.89 −0.387132 −0.193566 0.981087i \(-0.562005\pi\)
−0.193566 + 0.981087i \(0.562005\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2879.15 −0.193959
\(605\) 1791.61 0.120396
\(606\) 0 0
\(607\) −6111.92 −0.408691 −0.204345 0.978899i \(-0.565507\pi\)
−0.204345 + 0.978899i \(0.565507\pi\)
\(608\) 2678.12 0.178638
\(609\) 0 0
\(610\) 11090.9 0.736158
\(611\) 23346.8 1.54585
\(612\) 0 0
\(613\) 15496.0 1.02101 0.510504 0.859875i \(-0.329459\pi\)
0.510504 + 0.859875i \(0.329459\pi\)
\(614\) −9802.08 −0.644266
\(615\) 0 0
\(616\) 0 0
\(617\) 23071.0 1.50535 0.752676 0.658391i \(-0.228763\pi\)
0.752676 + 0.658391i \(0.228763\pi\)
\(618\) 0 0
\(619\) 9160.95 0.594846 0.297423 0.954746i \(-0.403873\pi\)
0.297423 + 0.954746i \(0.403873\pi\)
\(620\) 722.605 0.0468073
\(621\) 0 0
\(622\) 21514.0 1.38687
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 27241.6 1.73929
\(627\) 0 0
\(628\) 5692.80 0.361732
\(629\) 28230.2 1.78953
\(630\) 0 0
\(631\) 26685.9 1.68360 0.841798 0.539793i \(-0.181497\pi\)
0.841798 + 0.539793i \(0.181497\pi\)
\(632\) 1224.96 0.0770987
\(633\) 0 0
\(634\) 14994.2 0.939268
\(635\) 218.737 0.0136698
\(636\) 0 0
\(637\) 0 0
\(638\) 2177.77 0.135139
\(639\) 0 0
\(640\) 8632.31 0.533159
\(641\) −22763.0 −1.40263 −0.701314 0.712852i \(-0.747403\pi\)
−0.701314 + 0.712852i \(0.747403\pi\)
\(642\) 0 0
\(643\) −13560.1 −0.831660 −0.415830 0.909442i \(-0.636509\pi\)
−0.415830 + 0.909442i \(0.636509\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −9482.51 −0.577530
\(647\) −24534.7 −1.49082 −0.745408 0.666609i \(-0.767745\pi\)
−0.745408 + 0.666609i \(0.767745\pi\)
\(648\) 0 0
\(649\) −19554.9 −1.18274
\(650\) 4192.26 0.252975
\(651\) 0 0
\(652\) −113.107 −0.00679388
\(653\) 14176.0 0.849539 0.424769 0.905302i \(-0.360355\pi\)
0.424769 + 0.905302i \(0.360355\pi\)
\(654\) 0 0
\(655\) −8207.62 −0.489616
\(656\) 18262.6 1.08694
\(657\) 0 0
\(658\) 0 0
\(659\) 20212.3 1.19478 0.597391 0.801950i \(-0.296204\pi\)
0.597391 + 0.801950i \(0.296204\pi\)
\(660\) 0 0
\(661\) −23850.7 −1.40346 −0.701728 0.712445i \(-0.747588\pi\)
−0.701728 + 0.712445i \(0.747588\pi\)
\(662\) 12777.3 0.750156
\(663\) 0 0
\(664\) −11100.0 −0.648741
\(665\) 0 0
\(666\) 0 0
\(667\) 2619.31 0.152054
\(668\) −4005.45 −0.232000
\(669\) 0 0
\(670\) −12549.8 −0.723645
\(671\) 22320.0 1.28414
\(672\) 0 0
\(673\) 395.817 0.0226711 0.0113355 0.999936i \(-0.496392\pi\)
0.0113355 + 0.999936i \(0.496392\pi\)
\(674\) −29977.1 −1.71317
\(675\) 0 0
\(676\) 1173.09 0.0667437
\(677\) −8729.25 −0.495557 −0.247779 0.968817i \(-0.579701\pi\)
−0.247779 + 0.968817i \(0.579701\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −8112.50 −0.457500
\(681\) 0 0
\(682\) 8695.36 0.488215
\(683\) −7531.85 −0.421959 −0.210980 0.977490i \(-0.567665\pi\)
−0.210980 + 0.977490i \(0.567665\pi\)
\(684\) 0 0
\(685\) −12325.7 −0.687507
\(686\) 0 0
\(687\) 0 0
\(688\) −3078.70 −0.170602
\(689\) −14265.6 −0.788789
\(690\) 0 0
\(691\) −163.381 −0.00899465 −0.00449732 0.999990i \(-0.501432\pi\)
−0.00449732 + 0.999990i \(0.501432\pi\)
\(692\) −782.840 −0.0430045
\(693\) 0 0
\(694\) −19121.2 −1.04587
\(695\) 14772.3 0.806251
\(696\) 0 0
\(697\) −20133.0 −1.09411
\(698\) −22770.0 −1.23475
\(699\) 0 0
\(700\) 0 0
\(701\) −29759.6 −1.60343 −0.801716 0.597705i \(-0.796079\pi\)
−0.801716 + 0.597705i \(0.796079\pi\)
\(702\) 0 0
\(703\) 12882.9 0.691163
\(704\) 11602.6 0.621150
\(705\) 0 0
\(706\) 33630.6 1.79278
\(707\) 0 0
\(708\) 0 0
\(709\) −15454.6 −0.818633 −0.409316 0.912393i \(-0.634233\pi\)
−0.409316 + 0.912393i \(0.634233\pi\)
\(710\) −850.431 −0.0449523
\(711\) 0 0
\(712\) −2026.90 −0.106687
\(713\) 10458.4 0.549325
\(714\) 0 0
\(715\) 8436.80 0.441284
\(716\) 546.289 0.0285136
\(717\) 0 0
\(718\) −13530.7 −0.703290
\(719\) −13452.1 −0.697744 −0.348872 0.937170i \(-0.613435\pi\)
−0.348872 + 0.937170i \(0.613435\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 16931.8 0.872765
\(723\) 0 0
\(724\) −2993.50 −0.153664
\(725\) −563.225 −0.0288519
\(726\) 0 0
\(727\) 6644.23 0.338956 0.169478 0.985534i \(-0.445792\pi\)
0.169478 + 0.985534i \(0.445792\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 7881.10 0.399579
\(731\) 3394.01 0.171726
\(732\) 0 0
\(733\) 12532.8 0.631525 0.315762 0.948838i \(-0.397740\pi\)
0.315762 + 0.948838i \(0.397740\pi\)
\(734\) −22307.2 −1.12176
\(735\) 0 0
\(736\) −8333.11 −0.417340
\(737\) −25256.1 −1.26231
\(738\) 0 0
\(739\) 16509.4 0.821799 0.410900 0.911681i \(-0.365215\pi\)
0.410900 + 0.911681i \(0.365215\pi\)
\(740\) −2769.67 −0.137588
\(741\) 0 0
\(742\) 0 0
\(743\) 11768.9 0.581102 0.290551 0.956859i \(-0.406161\pi\)
0.290551 + 0.956859i \(0.406161\pi\)
\(744\) 0 0
\(745\) −84.3799 −0.00414958
\(746\) −38125.1 −1.87112
\(747\) 0 0
\(748\) 4102.67 0.200546
\(749\) 0 0
\(750\) 0 0
\(751\) 18336.1 0.890938 0.445469 0.895297i \(-0.353037\pi\)
0.445469 + 0.895297i \(0.353037\pi\)
\(752\) 32050.0 1.55418
\(753\) 0 0
\(754\) −3777.90 −0.182471
\(755\) 8960.28 0.431918
\(756\) 0 0
\(757\) 24340.5 1.16865 0.584326 0.811519i \(-0.301359\pi\)
0.584326 + 0.811519i \(0.301359\pi\)
\(758\) 2752.96 0.131915
\(759\) 0 0
\(760\) −3702.16 −0.176699
\(761\) −28022.5 −1.33484 −0.667421 0.744681i \(-0.732602\pi\)
−0.667421 + 0.744681i \(0.732602\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2503.07 −0.118531
\(765\) 0 0
\(766\) 32618.9 1.53860
\(767\) 33923.0 1.59699
\(768\) 0 0
\(769\) −33595.9 −1.57542 −0.787712 0.616044i \(-0.788734\pi\)
−0.787712 + 0.616044i \(0.788734\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −543.745 −0.0253495
\(773\) −668.213 −0.0310918 −0.0155459 0.999879i \(-0.504949\pi\)
−0.0155459 + 0.999879i \(0.504949\pi\)
\(774\) 0 0
\(775\) −2248.84 −0.104233
\(776\) −5031.48 −0.232757
\(777\) 0 0
\(778\) −18617.8 −0.857945
\(779\) −9187.73 −0.422573
\(780\) 0 0
\(781\) −1711.47 −0.0784136
\(782\) 29505.3 1.34924
\(783\) 0 0
\(784\) 0 0
\(785\) −17716.7 −0.805524
\(786\) 0 0
\(787\) 16719.9 0.757305 0.378653 0.925539i \(-0.376388\pi\)
0.378653 + 0.925539i \(0.376388\pi\)
\(788\) −1452.30 −0.0656550
\(789\) 0 0
\(790\) 957.992 0.0431441
\(791\) 0 0
\(792\) 0 0
\(793\) −38719.9 −1.73390
\(794\) 45582.5 2.03736
\(795\) 0 0
\(796\) 3193.07 0.142180
\(797\) 40821.1 1.81425 0.907126 0.420860i \(-0.138272\pi\)
0.907126 + 0.420860i \(0.138272\pi\)
\(798\) 0 0
\(799\) −35332.5 −1.56442
\(800\) 1791.85 0.0791892
\(801\) 0 0
\(802\) 7789.60 0.342968
\(803\) 15860.5 0.697016
\(804\) 0 0
\(805\) 0 0
\(806\) −15084.3 −0.659210
\(807\) 0 0
\(808\) 12141.1 0.528619
\(809\) 10822.7 0.470340 0.235170 0.971954i \(-0.424435\pi\)
0.235170 + 0.971954i \(0.424435\pi\)
\(810\) 0 0
\(811\) −19650.6 −0.850833 −0.425417 0.904998i \(-0.639872\pi\)
−0.425417 + 0.904998i \(0.639872\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −33328.4 −1.43509
\(815\) 352.003 0.0151290
\(816\) 0 0
\(817\) 1548.86 0.0663254
\(818\) 13293.5 0.568212
\(819\) 0 0
\(820\) 1975.25 0.0841204
\(821\) 24495.9 1.04130 0.520652 0.853769i \(-0.325689\pi\)
0.520652 + 0.853769i \(0.325689\pi\)
\(822\) 0 0
\(823\) 21651.7 0.917048 0.458524 0.888682i \(-0.348378\pi\)
0.458524 + 0.888682i \(0.348378\pi\)
\(824\) 36966.1 1.56284
\(825\) 0 0
\(826\) 0 0
\(827\) 4550.97 0.191358 0.0956788 0.995412i \(-0.469498\pi\)
0.0956788 + 0.995412i \(0.469498\pi\)
\(828\) 0 0
\(829\) 43235.4 1.81137 0.905686 0.423949i \(-0.139356\pi\)
0.905686 + 0.423949i \(0.139356\pi\)
\(830\) −8680.86 −0.363033
\(831\) 0 0
\(832\) −20127.7 −0.838704
\(833\) 0 0
\(834\) 0 0
\(835\) 12465.5 0.516629
\(836\) 1872.26 0.0774563
\(837\) 0 0
\(838\) 20044.5 0.826285
\(839\) 139.264 0.00573053 0.00286527 0.999996i \(-0.499088\pi\)
0.00286527 + 0.999996i \(0.499088\pi\)
\(840\) 0 0
\(841\) −23881.4 −0.979189
\(842\) 42298.2 1.73123
\(843\) 0 0
\(844\) 6023.24 0.245650
\(845\) −3650.80 −0.148629
\(846\) 0 0
\(847\) 0 0
\(848\) −19583.5 −0.793042
\(849\) 0 0
\(850\) −6344.46 −0.256015
\(851\) −40085.8 −1.61472
\(852\) 0 0
\(853\) 13239.6 0.531437 0.265718 0.964051i \(-0.414391\pi\)
0.265718 + 0.964051i \(0.414391\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 19773.6 0.789540
\(857\) 44722.7 1.78261 0.891305 0.453404i \(-0.149791\pi\)
0.891305 + 0.453404i \(0.149791\pi\)
\(858\) 0 0
\(859\) 19350.8 0.768617 0.384308 0.923205i \(-0.374440\pi\)
0.384308 + 0.923205i \(0.374440\pi\)
\(860\) −332.987 −0.0132032
\(861\) 0 0
\(862\) 53473.9 2.11291
\(863\) −34706.8 −1.36899 −0.684493 0.729020i \(-0.739976\pi\)
−0.684493 + 0.729020i \(0.739976\pi\)
\(864\) 0 0
\(865\) 2436.30 0.0957648
\(866\) −12500.2 −0.490501
\(867\) 0 0
\(868\) 0 0
\(869\) 1927.93 0.0752595
\(870\) 0 0
\(871\) 43813.3 1.70443
\(872\) 30793.5 1.19587
\(873\) 0 0
\(874\) 13464.8 0.521114
\(875\) 0 0
\(876\) 0 0
\(877\) 33566.2 1.29242 0.646209 0.763160i \(-0.276353\pi\)
0.646209 + 0.763160i \(0.276353\pi\)
\(878\) 51431.7 1.97692
\(879\) 0 0
\(880\) 11581.9 0.443664
\(881\) 32706.1 1.25074 0.625368 0.780330i \(-0.284949\pi\)
0.625368 + 0.780330i \(0.284949\pi\)
\(882\) 0 0
\(883\) −19854.3 −0.756683 −0.378342 0.925666i \(-0.623506\pi\)
−0.378342 + 0.925666i \(0.623506\pi\)
\(884\) −7117.14 −0.270786
\(885\) 0 0
\(886\) −17317.1 −0.656635
\(887\) −29902.1 −1.13192 −0.565960 0.824433i \(-0.691494\pi\)
−0.565960 + 0.824433i \(0.691494\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1585.16 −0.0597018
\(891\) 0 0
\(892\) −6329.02 −0.237569
\(893\) −16124.0 −0.604222
\(894\) 0 0
\(895\) −1700.12 −0.0634957
\(896\) 0 0
\(897\) 0 0
\(898\) 40572.5 1.50771
\(899\) 2026.56 0.0751831
\(900\) 0 0
\(901\) 21589.1 0.798267
\(902\) 23768.9 0.877403
\(903\) 0 0
\(904\) −5320.28 −0.195741
\(905\) 9316.14 0.342187
\(906\) 0 0
\(907\) 38973.5 1.42679 0.713393 0.700764i \(-0.247158\pi\)
0.713393 + 0.700764i \(0.247158\pi\)
\(908\) −4511.82 −0.164901
\(909\) 0 0
\(910\) 0 0
\(911\) −41356.0 −1.50404 −0.752022 0.659138i \(-0.770922\pi\)
−0.752022 + 0.659138i \(0.770922\pi\)
\(912\) 0 0
\(913\) −17470.0 −0.633265
\(914\) −23537.6 −0.851810
\(915\) 0 0
\(916\) 2907.74 0.104885
\(917\) 0 0
\(918\) 0 0
\(919\) 14009.2 0.502852 0.251426 0.967876i \(-0.419100\pi\)
0.251426 + 0.967876i \(0.419100\pi\)
\(920\) 11519.5 0.412810
\(921\) 0 0
\(922\) 47298.3 1.68946
\(923\) 2968.98 0.105878
\(924\) 0 0
\(925\) 8619.55 0.306388
\(926\) 13261.5 0.470625
\(927\) 0 0
\(928\) −1614.74 −0.0571190
\(929\) −39996.1 −1.41252 −0.706259 0.707954i \(-0.749619\pi\)
−0.706259 + 0.707954i \(0.749619\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1564.84 −0.0549979
\(933\) 0 0
\(934\) 28212.2 0.988365
\(935\) −12768.0 −0.446587
\(936\) 0 0
\(937\) −14964.6 −0.521743 −0.260872 0.965374i \(-0.584010\pi\)
−0.260872 + 0.965374i \(0.584010\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 3466.48 0.120281
\(941\) −31615.3 −1.09525 −0.547624 0.836725i \(-0.684468\pi\)
−0.547624 + 0.836725i \(0.684468\pi\)
\(942\) 0 0
\(943\) 28588.1 0.987229
\(944\) 46568.8 1.60560
\(945\) 0 0
\(946\) −4006.95 −0.137714
\(947\) −29872.2 −1.02504 −0.512522 0.858674i \(-0.671289\pi\)
−0.512522 + 0.858674i \(0.671289\pi\)
\(948\) 0 0
\(949\) −27514.1 −0.941143
\(950\) −2895.30 −0.0988801
\(951\) 0 0
\(952\) 0 0
\(953\) 41530.2 1.41164 0.705821 0.708390i \(-0.250578\pi\)
0.705821 + 0.708390i \(0.250578\pi\)
\(954\) 0 0
\(955\) 7789.85 0.263952
\(956\) 9664.48 0.326958
\(957\) 0 0
\(958\) 13100.1 0.441800
\(959\) 0 0
\(960\) 0 0
\(961\) −21699.4 −0.728387
\(962\) 57816.7 1.93772
\(963\) 0 0
\(964\) 618.531 0.0206655
\(965\) 1692.20 0.0564497
\(966\) 0 0
\(967\) 53414.9 1.77633 0.888164 0.459527i \(-0.151981\pi\)
0.888164 + 0.459527i \(0.151981\pi\)
\(968\) −7100.51 −0.235764
\(969\) 0 0
\(970\) −3934.91 −0.130250
\(971\) −38471.8 −1.27149 −0.635747 0.771898i \(-0.719308\pi\)
−0.635747 + 0.771898i \(0.719308\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 29681.9 0.976456
\(975\) 0 0
\(976\) −53153.8 −1.74325
\(977\) 53414.8 1.74912 0.874561 0.484916i \(-0.161150\pi\)
0.874561 + 0.484916i \(0.161150\pi\)
\(978\) 0 0
\(979\) −3190.08 −0.104142
\(980\) 0 0
\(981\) 0 0
\(982\) 66366.1 2.15665
\(983\) −13893.2 −0.450788 −0.225394 0.974268i \(-0.572367\pi\)
−0.225394 + 0.974268i \(0.572367\pi\)
\(984\) 0 0
\(985\) 4519.75 0.146204
\(986\) 5717.37 0.184663
\(987\) 0 0
\(988\) −3247.92 −0.104585
\(989\) −4819.37 −0.154951
\(990\) 0 0
\(991\) 10389.5 0.333031 0.166516 0.986039i \(-0.446748\pi\)
0.166516 + 0.986039i \(0.446748\pi\)
\(992\) −6447.32 −0.206353
\(993\) 0 0
\(994\) 0 0
\(995\) −9937.22 −0.316614
\(996\) 0 0
\(997\) 36396.7 1.15616 0.578082 0.815979i \(-0.303801\pi\)
0.578082 + 0.815979i \(0.303801\pi\)
\(998\) 61080.7 1.93735
\(999\) 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 2205.4.a.bx.1.2 6
3.2 odd 2 2205.4.a.by.1.5 yes 6
7.6 odd 2 2205.4.a.by.1.2 yes 6
21.20 even 2 inner 2205.4.a.bx.1.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2205.4.a.bx.1.2 6 1.1 even 1 trivial
2205.4.a.bx.1.5 yes 6 21.20 even 2 inner
2205.4.a.by.1.2 yes 6 7.6 odd 2
2205.4.a.by.1.5 yes 6 3.2 odd 2