Properties

Label 2205.4.a.by.1.3
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
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: \(0\)
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 1.3
Root \(-0.525584\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.525584 q^{2} -7.72376 q^{4} +5.00000 q^{5} +8.26415 q^{8} +O(q^{10})\) \(q-0.525584 q^{2} -7.72376 q^{4} +5.00000 q^{5} +8.26415 q^{8} -2.62792 q^{10} -48.9415 q^{11} -40.1517 q^{13} +57.4466 q^{16} +59.1704 q^{17} -129.508 q^{19} -38.6188 q^{20} +25.7228 q^{22} -93.5609 q^{23} +25.0000 q^{25} +21.1031 q^{26} -117.184 q^{29} +162.945 q^{31} -96.3062 q^{32} -31.0990 q^{34} -3.72097 q^{37} +68.0673 q^{38} +41.3208 q^{40} -222.816 q^{41} -62.2753 q^{43} +378.012 q^{44} +49.1741 q^{46} -460.327 q^{47} -13.1396 q^{50} +310.122 q^{52} +492.977 q^{53} -244.707 q^{55} +61.5902 q^{58} +153.928 q^{59} -194.916 q^{61} -85.6412 q^{62} -408.956 q^{64} -200.759 q^{65} +667.084 q^{67} -457.018 q^{68} +256.941 q^{71} -1219.59 q^{73} +1.95568 q^{74} +1000.29 q^{76} +381.236 q^{79} +287.233 q^{80} +117.108 q^{82} -281.766 q^{83} +295.852 q^{85} +32.7309 q^{86} -404.460 q^{88} -926.242 q^{89} +722.642 q^{92} +241.940 q^{94} -647.540 q^{95} +634.683 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 20 q^{4} + 30 q^{5}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.525584 −0.185822 −0.0929109 0.995674i \(-0.529617\pi\)
−0.0929109 + 0.995674i \(0.529617\pi\)
\(3\) 0 0
\(4\) −7.72376 −0.965470
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 8.26415 0.365227
\(9\) 0 0
\(10\) −2.62792 −0.0831021
\(11\) −48.9415 −1.34149 −0.670746 0.741687i \(-0.734026\pi\)
−0.670746 + 0.741687i \(0.734026\pi\)
\(12\) 0 0
\(13\) −40.1517 −0.856622 −0.428311 0.903631i \(-0.640891\pi\)
−0.428311 + 0.903631i \(0.640891\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 57.4466 0.897603
\(17\) 59.1704 0.844172 0.422086 0.906556i \(-0.361298\pi\)
0.422086 + 0.906556i \(0.361298\pi\)
\(18\) 0 0
\(19\) −129.508 −1.56375 −0.781873 0.623438i \(-0.785735\pi\)
−0.781873 + 0.623438i \(0.785735\pi\)
\(20\) −38.6188 −0.431771
\(21\) 0 0
\(22\) 25.7228 0.249278
\(23\) −93.5609 −0.848209 −0.424104 0.905613i \(-0.639411\pi\)
−0.424104 + 0.905613i \(0.639411\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 21.1031 0.159179
\(27\) 0 0
\(28\) 0 0
\(29\) −117.184 −0.750365 −0.375183 0.926951i \(-0.622420\pi\)
−0.375183 + 0.926951i \(0.622420\pi\)
\(30\) 0 0
\(31\) 162.945 0.944057 0.472029 0.881583i \(-0.343522\pi\)
0.472029 + 0.881583i \(0.343522\pi\)
\(32\) −96.3062 −0.532022
\(33\) 0 0
\(34\) −31.0990 −0.156866
\(35\) 0 0
\(36\) 0 0
\(37\) −3.72097 −0.0165331 −0.00826654 0.999966i \(-0.502631\pi\)
−0.00826654 + 0.999966i \(0.502631\pi\)
\(38\) 68.0673 0.290578
\(39\) 0 0
\(40\) 41.3208 0.163335
\(41\) −222.816 −0.848732 −0.424366 0.905491i \(-0.639503\pi\)
−0.424366 + 0.905491i \(0.639503\pi\)
\(42\) 0 0
\(43\) −62.2753 −0.220858 −0.110429 0.993884i \(-0.535222\pi\)
−0.110429 + 0.993884i \(0.535222\pi\)
\(44\) 378.012 1.29517
\(45\) 0 0
\(46\) 49.1741 0.157616
\(47\) −460.327 −1.42863 −0.714315 0.699824i \(-0.753262\pi\)
−0.714315 + 0.699824i \(0.753262\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −13.1396 −0.0371644
\(51\) 0 0
\(52\) 310.122 0.827043
\(53\) 492.977 1.27765 0.638826 0.769351i \(-0.279420\pi\)
0.638826 + 0.769351i \(0.279420\pi\)
\(54\) 0 0
\(55\) −244.707 −0.599933
\(56\) 0 0
\(57\) 0 0
\(58\) 61.5902 0.139434
\(59\) 153.928 0.339657 0.169829 0.985474i \(-0.445679\pi\)
0.169829 + 0.985474i \(0.445679\pi\)
\(60\) 0 0
\(61\) −194.916 −0.409122 −0.204561 0.978854i \(-0.565577\pi\)
−0.204561 + 0.978854i \(0.565577\pi\)
\(62\) −85.6412 −0.175427
\(63\) 0 0
\(64\) −408.956 −0.798742
\(65\) −200.759 −0.383093
\(66\) 0 0
\(67\) 667.084 1.21638 0.608189 0.793793i \(-0.291896\pi\)
0.608189 + 0.793793i \(0.291896\pi\)
\(68\) −457.018 −0.815023
\(69\) 0 0
\(70\) 0 0
\(71\) 256.941 0.429483 0.214741 0.976671i \(-0.431109\pi\)
0.214741 + 0.976671i \(0.431109\pi\)
\(72\) 0 0
\(73\) −1219.59 −1.95537 −0.977686 0.210073i \(-0.932630\pi\)
−0.977686 + 0.210073i \(0.932630\pi\)
\(74\) 1.95568 0.00307221
\(75\) 0 0
\(76\) 1000.29 1.50975
\(77\) 0 0
\(78\) 0 0
\(79\) 381.236 0.542941 0.271471 0.962447i \(-0.412490\pi\)
0.271471 + 0.962447i \(0.412490\pi\)
\(80\) 287.233 0.401420
\(81\) 0 0
\(82\) 117.108 0.157713
\(83\) −281.766 −0.372625 −0.186313 0.982491i \(-0.559654\pi\)
−0.186313 + 0.982491i \(0.559654\pi\)
\(84\) 0 0
\(85\) 295.852 0.377525
\(86\) 32.7309 0.0410402
\(87\) 0 0
\(88\) −404.460 −0.489949
\(89\) −926.242 −1.10316 −0.551581 0.834121i \(-0.685975\pi\)
−0.551581 + 0.834121i \(0.685975\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 722.642 0.818920
\(93\) 0 0
\(94\) 241.940 0.265471
\(95\) −647.540 −0.699328
\(96\) 0 0
\(97\) 634.683 0.664354 0.332177 0.943217i \(-0.392217\pi\)
0.332177 + 0.943217i \(0.392217\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −193.094 −0.193094
\(101\) 1921.81 1.89334 0.946670 0.322205i \(-0.104424\pi\)
0.946670 + 0.322205i \(0.104424\pi\)
\(102\) 0 0
\(103\) 1310.57 1.25373 0.626865 0.779128i \(-0.284338\pi\)
0.626865 + 0.779128i \(0.284338\pi\)
\(104\) −331.820 −0.312862
\(105\) 0 0
\(106\) −259.101 −0.237416
\(107\) −1866.73 −1.68657 −0.843286 0.537465i \(-0.819382\pi\)
−0.843286 + 0.537465i \(0.819382\pi\)
\(108\) 0 0
\(109\) −649.011 −0.570312 −0.285156 0.958481i \(-0.592045\pi\)
−0.285156 + 0.958481i \(0.592045\pi\)
\(110\) 128.614 0.111481
\(111\) 0 0
\(112\) 0 0
\(113\) −857.830 −0.714140 −0.357070 0.934078i \(-0.616224\pi\)
−0.357070 + 0.934078i \(0.616224\pi\)
\(114\) 0 0
\(115\) −467.805 −0.379330
\(116\) 905.104 0.724455
\(117\) 0 0
\(118\) −80.9022 −0.0631157
\(119\) 0 0
\(120\) 0 0
\(121\) 1064.27 0.799600
\(122\) 102.445 0.0760238
\(123\) 0 0
\(124\) −1258.55 −0.911459
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 972.678 0.679616 0.339808 0.940495i \(-0.389638\pi\)
0.339808 + 0.940495i \(0.389638\pi\)
\(128\) 985.390 0.680445
\(129\) 0 0
\(130\) 105.515 0.0711871
\(131\) −131.175 −0.0874873 −0.0437436 0.999043i \(-0.513928\pi\)
−0.0437436 + 0.999043i \(0.513928\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −350.608 −0.226029
\(135\) 0 0
\(136\) 488.993 0.308315
\(137\) −2160.79 −1.34751 −0.673754 0.738956i \(-0.735319\pi\)
−0.673754 + 0.738956i \(0.735319\pi\)
\(138\) 0 0
\(139\) −1791.95 −1.09346 −0.546732 0.837308i \(-0.684128\pi\)
−0.546732 + 0.837308i \(0.684128\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −135.044 −0.0798073
\(143\) 1965.09 1.14915
\(144\) 0 0
\(145\) −585.922 −0.335573
\(146\) 640.996 0.363351
\(147\) 0 0
\(148\) 28.7399 0.0159622
\(149\) 3022.74 1.66196 0.830981 0.556301i \(-0.187780\pi\)
0.830981 + 0.556301i \(0.187780\pi\)
\(150\) 0 0
\(151\) 908.211 0.489464 0.244732 0.969591i \(-0.421300\pi\)
0.244732 + 0.969591i \(0.421300\pi\)
\(152\) −1070.27 −0.571123
\(153\) 0 0
\(154\) 0 0
\(155\) 814.725 0.422195
\(156\) 0 0
\(157\) −1387.49 −0.705310 −0.352655 0.935753i \(-0.614721\pi\)
−0.352655 + 0.935753i \(0.614721\pi\)
\(158\) −200.371 −0.100890
\(159\) 0 0
\(160\) −481.531 −0.237927
\(161\) 0 0
\(162\) 0 0
\(163\) −2637.35 −1.26732 −0.633660 0.773612i \(-0.718448\pi\)
−0.633660 + 0.773612i \(0.718448\pi\)
\(164\) 1720.98 0.819425
\(165\) 0 0
\(166\) 148.092 0.0692419
\(167\) 2525.54 1.17025 0.585126 0.810943i \(-0.301045\pi\)
0.585126 + 0.810943i \(0.301045\pi\)
\(168\) 0 0
\(169\) −584.838 −0.266198
\(170\) −155.495 −0.0701524
\(171\) 0 0
\(172\) 481.000 0.213232
\(173\) −649.874 −0.285601 −0.142801 0.989751i \(-0.545611\pi\)
−0.142801 + 0.989751i \(0.545611\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2811.52 −1.20413
\(177\) 0 0
\(178\) 486.817 0.204992
\(179\) 3434.35 1.43405 0.717026 0.697047i \(-0.245503\pi\)
0.717026 + 0.697047i \(0.245503\pi\)
\(180\) 0 0
\(181\) −1187.20 −0.487537 −0.243768 0.969834i \(-0.578384\pi\)
−0.243768 + 0.969834i \(0.578384\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −773.201 −0.309789
\(185\) −18.6049 −0.00739382
\(186\) 0 0
\(187\) −2895.88 −1.13245
\(188\) 3555.46 1.37930
\(189\) 0 0
\(190\) 340.336 0.129950
\(191\) 180.895 0.0685293 0.0342646 0.999413i \(-0.489091\pi\)
0.0342646 + 0.999413i \(0.489091\pi\)
\(192\) 0 0
\(193\) −3100.23 −1.15627 −0.578134 0.815942i \(-0.696219\pi\)
−0.578134 + 0.815942i \(0.696219\pi\)
\(194\) −333.579 −0.123451
\(195\) 0 0
\(196\) 0 0
\(197\) 611.701 0.221228 0.110614 0.993863i \(-0.464718\pi\)
0.110614 + 0.993863i \(0.464718\pi\)
\(198\) 0 0
\(199\) 1387.39 0.494218 0.247109 0.968988i \(-0.420519\pi\)
0.247109 + 0.968988i \(0.420519\pi\)
\(200\) 206.604 0.0730455
\(201\) 0 0
\(202\) −1010.07 −0.351824
\(203\) 0 0
\(204\) 0 0
\(205\) −1114.08 −0.379564
\(206\) −688.814 −0.232970
\(207\) 0 0
\(208\) −2306.58 −0.768907
\(209\) 6338.31 2.09775
\(210\) 0 0
\(211\) −1415.49 −0.461830 −0.230915 0.972974i \(-0.574172\pi\)
−0.230915 + 0.972974i \(0.574172\pi\)
\(212\) −3807.64 −1.23354
\(213\) 0 0
\(214\) 981.121 0.313402
\(215\) −311.377 −0.0987707
\(216\) 0 0
\(217\) 0 0
\(218\) 341.109 0.105976
\(219\) 0 0
\(220\) 1890.06 0.579218
\(221\) −2375.79 −0.723136
\(222\) 0 0
\(223\) 4213.75 1.26535 0.632676 0.774417i \(-0.281957\pi\)
0.632676 + 0.774417i \(0.281957\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 450.861 0.132703
\(227\) 6180.32 1.80706 0.903529 0.428526i \(-0.140967\pi\)
0.903529 + 0.428526i \(0.140967\pi\)
\(228\) 0 0
\(229\) 1196.83 0.345365 0.172683 0.984978i \(-0.444756\pi\)
0.172683 + 0.984978i \(0.444756\pi\)
\(230\) 245.870 0.0704879
\(231\) 0 0
\(232\) −968.429 −0.274054
\(233\) −927.278 −0.260721 −0.130360 0.991467i \(-0.541613\pi\)
−0.130360 + 0.991467i \(0.541613\pi\)
\(234\) 0 0
\(235\) −2301.64 −0.638903
\(236\) −1188.91 −0.327929
\(237\) 0 0
\(238\) 0 0
\(239\) 3429.57 0.928202 0.464101 0.885782i \(-0.346378\pi\)
0.464101 + 0.885782i \(0.346378\pi\)
\(240\) 0 0
\(241\) −4748.69 −1.26925 −0.634626 0.772819i \(-0.718846\pi\)
−0.634626 + 0.772819i \(0.718846\pi\)
\(242\) −559.362 −0.148583
\(243\) 0 0
\(244\) 1505.49 0.394995
\(245\) 0 0
\(246\) 0 0
\(247\) 5199.97 1.33954
\(248\) 1346.60 0.344796
\(249\) 0 0
\(250\) −65.6979 −0.0166204
\(251\) 7273.04 1.82896 0.914482 0.404626i \(-0.132598\pi\)
0.914482 + 0.404626i \(0.132598\pi\)
\(252\) 0 0
\(253\) 4579.01 1.13786
\(254\) −511.223 −0.126287
\(255\) 0 0
\(256\) 2753.74 0.672300
\(257\) 1613.28 0.391570 0.195785 0.980647i \(-0.437275\pi\)
0.195785 + 0.980647i \(0.437275\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1550.61 0.369865
\(261\) 0 0
\(262\) 68.9436 0.0162571
\(263\) 2593.26 0.608013 0.304007 0.952670i \(-0.401676\pi\)
0.304007 + 0.952670i \(0.401676\pi\)
\(264\) 0 0
\(265\) 2464.88 0.571384
\(266\) 0 0
\(267\) 0 0
\(268\) −5152.40 −1.17438
\(269\) 5922.82 1.34246 0.671228 0.741251i \(-0.265767\pi\)
0.671228 + 0.741251i \(0.265767\pi\)
\(270\) 0 0
\(271\) 1783.59 0.399798 0.199899 0.979816i \(-0.435939\pi\)
0.199899 + 0.979816i \(0.435939\pi\)
\(272\) 3399.14 0.757731
\(273\) 0 0
\(274\) 1135.68 0.250397
\(275\) −1223.54 −0.268298
\(276\) 0 0
\(277\) −3042.89 −0.660035 −0.330017 0.943975i \(-0.607055\pi\)
−0.330017 + 0.943975i \(0.607055\pi\)
\(278\) 941.821 0.203190
\(279\) 0 0
\(280\) 0 0
\(281\) 1812.14 0.384708 0.192354 0.981326i \(-0.438388\pi\)
0.192354 + 0.981326i \(0.438388\pi\)
\(282\) 0 0
\(283\) 7062.64 1.48350 0.741750 0.670677i \(-0.233996\pi\)
0.741750 + 0.670677i \(0.233996\pi\)
\(284\) −1984.55 −0.414653
\(285\) 0 0
\(286\) −1032.82 −0.213537
\(287\) 0 0
\(288\) 0 0
\(289\) −1411.87 −0.287374
\(290\) 307.951 0.0623569
\(291\) 0 0
\(292\) 9419.82 1.88785
\(293\) 6252.90 1.24675 0.623376 0.781922i \(-0.285760\pi\)
0.623376 + 0.781922i \(0.285760\pi\)
\(294\) 0 0
\(295\) 769.642 0.151899
\(296\) −30.7507 −0.00603833
\(297\) 0 0
\(298\) −1588.70 −0.308829
\(299\) 3756.63 0.726594
\(300\) 0 0
\(301\) 0 0
\(302\) −477.341 −0.0909532
\(303\) 0 0
\(304\) −7439.79 −1.40362
\(305\) −974.580 −0.182965
\(306\) 0 0
\(307\) 946.955 0.176044 0.0880221 0.996119i \(-0.471945\pi\)
0.0880221 + 0.996119i \(0.471945\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −428.206 −0.0784531
\(311\) 7114.71 1.29723 0.648614 0.761117i \(-0.275349\pi\)
0.648614 + 0.761117i \(0.275349\pi\)
\(312\) 0 0
\(313\) −2357.90 −0.425803 −0.212901 0.977074i \(-0.568291\pi\)
−0.212901 + 0.977074i \(0.568291\pi\)
\(314\) 729.241 0.131062
\(315\) 0 0
\(316\) −2944.57 −0.524194
\(317\) 9260.69 1.64079 0.820397 0.571794i \(-0.193752\pi\)
0.820397 + 0.571794i \(0.193752\pi\)
\(318\) 0 0
\(319\) 5735.17 1.00661
\(320\) −2044.78 −0.357208
\(321\) 0 0
\(322\) 0 0
\(323\) −7663.03 −1.32007
\(324\) 0 0
\(325\) −1003.79 −0.171324
\(326\) 1386.15 0.235496
\(327\) 0 0
\(328\) −1841.38 −0.309980
\(329\) 0 0
\(330\) 0 0
\(331\) 10664.3 1.77089 0.885443 0.464748i \(-0.153855\pi\)
0.885443 + 0.464748i \(0.153855\pi\)
\(332\) 2176.30 0.359758
\(333\) 0 0
\(334\) −1327.38 −0.217458
\(335\) 3335.42 0.543980
\(336\) 0 0
\(337\) −865.728 −0.139938 −0.0699691 0.997549i \(-0.522290\pi\)
−0.0699691 + 0.997549i \(0.522290\pi\)
\(338\) 307.381 0.0494655
\(339\) 0 0
\(340\) −2285.09 −0.364489
\(341\) −7974.77 −1.26645
\(342\) 0 0
\(343\) 0 0
\(344\) −514.653 −0.0806634
\(345\) 0 0
\(346\) 341.563 0.0530710
\(347\) 8164.72 1.26313 0.631563 0.775324i \(-0.282414\pi\)
0.631563 + 0.775324i \(0.282414\pi\)
\(348\) 0 0
\(349\) 6227.30 0.955128 0.477564 0.878597i \(-0.341520\pi\)
0.477564 + 0.878597i \(0.341520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4713.37 0.713703
\(353\) 11121.3 1.67685 0.838425 0.545017i \(-0.183477\pi\)
0.838425 + 0.545017i \(0.183477\pi\)
\(354\) 0 0
\(355\) 1284.70 0.192071
\(356\) 7154.07 1.06507
\(357\) 0 0
\(358\) −1805.04 −0.266478
\(359\) −3362.46 −0.494328 −0.247164 0.968974i \(-0.579499\pi\)
−0.247164 + 0.968974i \(0.579499\pi\)
\(360\) 0 0
\(361\) 9913.32 1.44530
\(362\) 623.974 0.0905949
\(363\) 0 0
\(364\) 0 0
\(365\) −6097.95 −0.874469
\(366\) 0 0
\(367\) 1447.20 0.205839 0.102920 0.994690i \(-0.467182\pi\)
0.102920 + 0.994690i \(0.467182\pi\)
\(368\) −5374.76 −0.761355
\(369\) 0 0
\(370\) 9.77841 0.00137393
\(371\) 0 0
\(372\) 0 0
\(373\) −13117.6 −1.82092 −0.910460 0.413597i \(-0.864272\pi\)
−0.910460 + 0.413597i \(0.864272\pi\)
\(374\) 1522.03 0.210434
\(375\) 0 0
\(376\) −3804.21 −0.521775
\(377\) 4705.15 0.642779
\(378\) 0 0
\(379\) −4102.23 −0.555983 −0.277991 0.960584i \(-0.589669\pi\)
−0.277991 + 0.960584i \(0.589669\pi\)
\(380\) 5001.44 0.675181
\(381\) 0 0
\(382\) −95.0754 −0.0127342
\(383\) −7371.61 −0.983477 −0.491739 0.870743i \(-0.663638\pi\)
−0.491739 + 0.870743i \(0.663638\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1629.43 0.214860
\(387\) 0 0
\(388\) −4902.14 −0.641414
\(389\) −8529.66 −1.11175 −0.555875 0.831266i \(-0.687617\pi\)
−0.555875 + 0.831266i \(0.687617\pi\)
\(390\) 0 0
\(391\) −5536.03 −0.716034
\(392\) 0 0
\(393\) 0 0
\(394\) −321.500 −0.0411090
\(395\) 1906.18 0.242811
\(396\) 0 0
\(397\) −10274.2 −1.29886 −0.649428 0.760423i \(-0.724992\pi\)
−0.649428 + 0.760423i \(0.724992\pi\)
\(398\) −729.189 −0.0918365
\(399\) 0 0
\(400\) 1436.16 0.179521
\(401\) −254.845 −0.0317366 −0.0158683 0.999874i \(-0.505051\pi\)
−0.0158683 + 0.999874i \(0.505051\pi\)
\(402\) 0 0
\(403\) −6542.52 −0.808701
\(404\) −14843.6 −1.82796
\(405\) 0 0
\(406\) 0 0
\(407\) 182.110 0.0221790
\(408\) 0 0
\(409\) 6790.11 0.820903 0.410451 0.911882i \(-0.365371\pi\)
0.410451 + 0.911882i \(0.365371\pi\)
\(410\) 585.542 0.0705314
\(411\) 0 0
\(412\) −10122.5 −1.21044
\(413\) 0 0
\(414\) 0 0
\(415\) −1408.83 −0.166643
\(416\) 3866.86 0.455741
\(417\) 0 0
\(418\) −3331.31 −0.389808
\(419\) 2717.47 0.316843 0.158421 0.987372i \(-0.449360\pi\)
0.158421 + 0.987372i \(0.449360\pi\)
\(420\) 0 0
\(421\) −3455.49 −0.400024 −0.200012 0.979793i \(-0.564098\pi\)
−0.200012 + 0.979793i \(0.564098\pi\)
\(422\) 743.957 0.0858181
\(423\) 0 0
\(424\) 4074.04 0.466634
\(425\) 1479.26 0.168834
\(426\) 0 0
\(427\) 0 0
\(428\) 14418.1 1.62834
\(429\) 0 0
\(430\) 163.654 0.0183538
\(431\) −5333.95 −0.596119 −0.298060 0.954547i \(-0.596339\pi\)
−0.298060 + 0.954547i \(0.596339\pi\)
\(432\) 0 0
\(433\) −14032.3 −1.55739 −0.778696 0.627401i \(-0.784119\pi\)
−0.778696 + 0.627401i \(0.784119\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5012.81 0.550619
\(437\) 12116.9 1.32638
\(438\) 0 0
\(439\) 13158.0 1.43052 0.715258 0.698860i \(-0.246309\pi\)
0.715258 + 0.698860i \(0.246309\pi\)
\(440\) −2022.30 −0.219112
\(441\) 0 0
\(442\) 1248.68 0.134375
\(443\) −14922.9 −1.60047 −0.800237 0.599683i \(-0.795293\pi\)
−0.800237 + 0.599683i \(0.795293\pi\)
\(444\) 0 0
\(445\) −4631.21 −0.493349
\(446\) −2214.68 −0.235130
\(447\) 0 0
\(448\) 0 0
\(449\) −640.776 −0.0673498 −0.0336749 0.999433i \(-0.510721\pi\)
−0.0336749 + 0.999433i \(0.510721\pi\)
\(450\) 0 0
\(451\) 10904.9 1.13857
\(452\) 6625.67 0.689481
\(453\) 0 0
\(454\) −3248.28 −0.335791
\(455\) 0 0
\(456\) 0 0
\(457\) 14469.0 1.48103 0.740515 0.672039i \(-0.234581\pi\)
0.740515 + 0.672039i \(0.234581\pi\)
\(458\) −629.033 −0.0641764
\(459\) 0 0
\(460\) 3613.21 0.366232
\(461\) −10275.8 −1.03816 −0.519082 0.854725i \(-0.673726\pi\)
−0.519082 + 0.854725i \(0.673726\pi\)
\(462\) 0 0
\(463\) 4435.77 0.445244 0.222622 0.974905i \(-0.428538\pi\)
0.222622 + 0.974905i \(0.428538\pi\)
\(464\) −6731.84 −0.673530
\(465\) 0 0
\(466\) 487.362 0.0484476
\(467\) −12909.3 −1.27917 −0.639585 0.768720i \(-0.720894\pi\)
−0.639585 + 0.768720i \(0.720894\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1209.70 0.118722
\(471\) 0 0
\(472\) 1272.09 0.124052
\(473\) 3047.85 0.296279
\(474\) 0 0
\(475\) −3237.70 −0.312749
\(476\) 0 0
\(477\) 0 0
\(478\) −1802.52 −0.172480
\(479\) 15622.8 1.49023 0.745117 0.666934i \(-0.232394\pi\)
0.745117 + 0.666934i \(0.232394\pi\)
\(480\) 0 0
\(481\) 149.403 0.0141626
\(482\) 2495.83 0.235855
\(483\) 0 0
\(484\) −8220.15 −0.771990
\(485\) 3173.42 0.297108
\(486\) 0 0
\(487\) 7817.06 0.727361 0.363680 0.931524i \(-0.381520\pi\)
0.363680 + 0.931524i \(0.381520\pi\)
\(488\) −1610.82 −0.149423
\(489\) 0 0
\(490\) 0 0
\(491\) −1057.33 −0.0971822 −0.0485911 0.998819i \(-0.515473\pi\)
−0.0485911 + 0.998819i \(0.515473\pi\)
\(492\) 0 0
\(493\) −6933.84 −0.633437
\(494\) −2733.02 −0.248916
\(495\) 0 0
\(496\) 9360.63 0.847389
\(497\) 0 0
\(498\) 0 0
\(499\) −4434.30 −0.397809 −0.198904 0.980019i \(-0.563738\pi\)
−0.198904 + 0.980019i \(0.563738\pi\)
\(500\) −965.470 −0.0863543
\(501\) 0 0
\(502\) −3822.59 −0.339862
\(503\) −3705.19 −0.328442 −0.164221 0.986424i \(-0.552511\pi\)
−0.164221 + 0.986424i \(0.552511\pi\)
\(504\) 0 0
\(505\) 9609.05 0.846727
\(506\) −2406.65 −0.211440
\(507\) 0 0
\(508\) −7512.73 −0.656149
\(509\) 1670.83 0.145498 0.0727490 0.997350i \(-0.476823\pi\)
0.0727490 + 0.997350i \(0.476823\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −9330.44 −0.805373
\(513\) 0 0
\(514\) −847.911 −0.0727622
\(515\) 6552.85 0.560685
\(516\) 0 0
\(517\) 22529.1 1.91649
\(518\) 0 0
\(519\) 0 0
\(520\) −1659.10 −0.139916
\(521\) 9521.12 0.800629 0.400315 0.916378i \(-0.368901\pi\)
0.400315 + 0.916378i \(0.368901\pi\)
\(522\) 0 0
\(523\) 14460.0 1.20897 0.604487 0.796615i \(-0.293378\pi\)
0.604487 + 0.796615i \(0.293378\pi\)
\(524\) 1013.17 0.0844664
\(525\) 0 0
\(526\) −1362.98 −0.112982
\(527\) 9641.51 0.796947
\(528\) 0 0
\(529\) −3413.36 −0.280542
\(530\) −1295.50 −0.106176
\(531\) 0 0
\(532\) 0 0
\(533\) 8946.45 0.727042
\(534\) 0 0
\(535\) −9333.63 −0.754258
\(536\) 5512.88 0.444254
\(537\) 0 0
\(538\) −3112.93 −0.249458
\(539\) 0 0
\(540\) 0 0
\(541\) 5461.17 0.434000 0.217000 0.976172i \(-0.430373\pi\)
0.217000 + 0.976172i \(0.430373\pi\)
\(542\) −937.425 −0.0742913
\(543\) 0 0
\(544\) −5698.47 −0.449118
\(545\) −3245.05 −0.255051
\(546\) 0 0
\(547\) 20483.5 1.60112 0.800559 0.599254i \(-0.204536\pi\)
0.800559 + 0.599254i \(0.204536\pi\)
\(548\) 16689.4 1.30098
\(549\) 0 0
\(550\) 643.071 0.0498557
\(551\) 15176.3 1.17338
\(552\) 0 0
\(553\) 0 0
\(554\) 1599.29 0.122649
\(555\) 0 0
\(556\) 13840.6 1.05571
\(557\) 8382.62 0.637671 0.318836 0.947810i \(-0.396708\pi\)
0.318836 + 0.947810i \(0.396708\pi\)
\(558\) 0 0
\(559\) 2500.46 0.189192
\(560\) 0 0
\(561\) 0 0
\(562\) −952.429 −0.0714872
\(563\) 10678.5 0.799369 0.399685 0.916653i \(-0.369120\pi\)
0.399685 + 0.916653i \(0.369120\pi\)
\(564\) 0 0
\(565\) −4289.15 −0.319373
\(566\) −3712.01 −0.275667
\(567\) 0 0
\(568\) 2123.40 0.156859
\(569\) −17416.3 −1.28318 −0.641590 0.767048i \(-0.721725\pi\)
−0.641590 + 0.767048i \(0.721725\pi\)
\(570\) 0 0
\(571\) 7072.55 0.518348 0.259174 0.965831i \(-0.416550\pi\)
0.259174 + 0.965831i \(0.416550\pi\)
\(572\) −15177.8 −1.10947
\(573\) 0 0
\(574\) 0 0
\(575\) −2339.02 −0.169642
\(576\) 0 0
\(577\) 2399.09 0.173094 0.0865471 0.996248i \(-0.472417\pi\)
0.0865471 + 0.996248i \(0.472417\pi\)
\(578\) 742.055 0.0534004
\(579\) 0 0
\(580\) 4525.52 0.323986
\(581\) 0 0
\(582\) 0 0
\(583\) −24127.0 −1.71396
\(584\) −10078.9 −0.714155
\(585\) 0 0
\(586\) −3286.42 −0.231674
\(587\) −10345.7 −0.727451 −0.363725 0.931506i \(-0.618495\pi\)
−0.363725 + 0.931506i \(0.618495\pi\)
\(588\) 0 0
\(589\) −21102.7 −1.47627
\(590\) −404.511 −0.0282262
\(591\) 0 0
\(592\) −213.757 −0.0148401
\(593\) −3622.84 −0.250881 −0.125440 0.992101i \(-0.540034\pi\)
−0.125440 + 0.992101i \(0.540034\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −23346.9 −1.60457
\(597\) 0 0
\(598\) −1974.42 −0.135017
\(599\) 1509.74 0.102982 0.0514910 0.998673i \(-0.483603\pi\)
0.0514910 + 0.998673i \(0.483603\pi\)
\(600\) 0 0
\(601\) 10522.7 0.714194 0.357097 0.934067i \(-0.383767\pi\)
0.357097 + 0.934067i \(0.383767\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −7014.80 −0.472563
\(605\) 5321.34 0.357592
\(606\) 0 0
\(607\) 15525.1 1.03813 0.519065 0.854735i \(-0.326280\pi\)
0.519065 + 0.854735i \(0.326280\pi\)
\(608\) 12472.4 0.831946
\(609\) 0 0
\(610\) 512.223 0.0339989
\(611\) 18482.9 1.22380
\(612\) 0 0
\(613\) −10339.1 −0.681227 −0.340614 0.940203i \(-0.610635\pi\)
−0.340614 + 0.940203i \(0.610635\pi\)
\(614\) −497.704 −0.0327129
\(615\) 0 0
\(616\) 0 0
\(617\) 12652.5 0.825562 0.412781 0.910830i \(-0.364558\pi\)
0.412781 + 0.910830i \(0.364558\pi\)
\(618\) 0 0
\(619\) 4188.31 0.271959 0.135979 0.990712i \(-0.456582\pi\)
0.135979 + 0.990712i \(0.456582\pi\)
\(620\) −6292.74 −0.407617
\(621\) 0 0
\(622\) −3739.37 −0.241053
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 1239.27 0.0791235
\(627\) 0 0
\(628\) 10716.6 0.680956
\(629\) −220.171 −0.0139568
\(630\) 0 0
\(631\) −12886.7 −0.813016 −0.406508 0.913647i \(-0.633254\pi\)
−0.406508 + 0.913647i \(0.633254\pi\)
\(632\) 3150.59 0.198297
\(633\) 0 0
\(634\) −4867.26 −0.304896
\(635\) 4863.39 0.303933
\(636\) 0 0
\(637\) 0 0
\(638\) −3014.31 −0.187050
\(639\) 0 0
\(640\) 4926.95 0.304304
\(641\) −23195.4 −1.42927 −0.714635 0.699498i \(-0.753407\pi\)
−0.714635 + 0.699498i \(0.753407\pi\)
\(642\) 0 0
\(643\) 9691.46 0.594391 0.297196 0.954817i \(-0.403949\pi\)
0.297196 + 0.954817i \(0.403949\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4027.56 0.245298
\(647\) −15711.6 −0.954694 −0.477347 0.878715i \(-0.658401\pi\)
−0.477347 + 0.878715i \(0.658401\pi\)
\(648\) 0 0
\(649\) −7533.48 −0.455647
\(650\) 527.577 0.0318358
\(651\) 0 0
\(652\) 20370.3 1.22356
\(653\) 5596.58 0.335392 0.167696 0.985839i \(-0.446367\pi\)
0.167696 + 0.985839i \(0.446367\pi\)
\(654\) 0 0
\(655\) −655.876 −0.0391255
\(656\) −12800.0 −0.761824
\(657\) 0 0
\(658\) 0 0
\(659\) 490.979 0.0290225 0.0145112 0.999895i \(-0.495381\pi\)
0.0145112 + 0.999895i \(0.495381\pi\)
\(660\) 0 0
\(661\) −26400.3 −1.55348 −0.776741 0.629820i \(-0.783128\pi\)
−0.776741 + 0.629820i \(0.783128\pi\)
\(662\) −5604.98 −0.329069
\(663\) 0 0
\(664\) −2328.56 −0.136093
\(665\) 0 0
\(666\) 0 0
\(667\) 10963.9 0.636466
\(668\) −19506.7 −1.12984
\(669\) 0 0
\(670\) −1753.04 −0.101083
\(671\) 9539.48 0.548834
\(672\) 0 0
\(673\) −22727.4 −1.30175 −0.650875 0.759185i \(-0.725598\pi\)
−0.650875 + 0.759185i \(0.725598\pi\)
\(674\) 455.012 0.0260036
\(675\) 0 0
\(676\) 4517.15 0.257007
\(677\) 26717.2 1.51673 0.758365 0.651830i \(-0.225999\pi\)
0.758365 + 0.651830i \(0.225999\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2444.96 0.137882
\(681\) 0 0
\(682\) 4191.41 0.235333
\(683\) −19578.8 −1.09687 −0.548433 0.836194i \(-0.684776\pi\)
−0.548433 + 0.836194i \(0.684776\pi\)
\(684\) 0 0
\(685\) −10803.9 −0.602624
\(686\) 0 0
\(687\) 0 0
\(688\) −3577.50 −0.198243
\(689\) −19793.9 −1.09447
\(690\) 0 0
\(691\) 13856.7 0.762854 0.381427 0.924399i \(-0.375433\pi\)
0.381427 + 0.924399i \(0.375433\pi\)
\(692\) 5019.47 0.275740
\(693\) 0 0
\(694\) −4291.24 −0.234717
\(695\) −8959.77 −0.489012
\(696\) 0 0
\(697\) −13184.1 −0.716475
\(698\) −3272.97 −0.177484
\(699\) 0 0
\(700\) 0 0
\(701\) 28106.3 1.51435 0.757176 0.653210i \(-0.226578\pi\)
0.757176 + 0.653210i \(0.226578\pi\)
\(702\) 0 0
\(703\) 481.896 0.0258535
\(704\) 20014.9 1.07151
\(705\) 0 0
\(706\) −5845.18 −0.311595
\(707\) 0 0
\(708\) 0 0
\(709\) −15866.5 −0.840449 −0.420225 0.907420i \(-0.638049\pi\)
−0.420225 + 0.907420i \(0.638049\pi\)
\(710\) −675.220 −0.0356909
\(711\) 0 0
\(712\) −7654.60 −0.402905
\(713\) −15245.3 −0.800758
\(714\) 0 0
\(715\) 9825.43 0.513916
\(716\) −26526.1 −1.38453
\(717\) 0 0
\(718\) 1767.25 0.0918569
\(719\) 9593.84 0.497621 0.248811 0.968552i \(-0.419960\pi\)
0.248811 + 0.968552i \(0.419960\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −5210.28 −0.268568
\(723\) 0 0
\(724\) 9169.67 0.470702
\(725\) −2929.61 −0.150073
\(726\) 0 0
\(727\) −29844.5 −1.52252 −0.761259 0.648448i \(-0.775419\pi\)
−0.761259 + 0.648448i \(0.775419\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3204.98 0.162495
\(731\) −3684.85 −0.186442
\(732\) 0 0
\(733\) −26065.3 −1.31343 −0.656714 0.754139i \(-0.728054\pi\)
−0.656714 + 0.754139i \(0.728054\pi\)
\(734\) −760.623 −0.0382495
\(735\) 0 0
\(736\) 9010.49 0.451265
\(737\) −32648.1 −1.63176
\(738\) 0 0
\(739\) 11751.1 0.584941 0.292470 0.956275i \(-0.405523\pi\)
0.292470 + 0.956275i \(0.405523\pi\)
\(740\) 143.699 0.00713851
\(741\) 0 0
\(742\) 0 0
\(743\) −31874.5 −1.57384 −0.786919 0.617057i \(-0.788325\pi\)
−0.786919 + 0.617057i \(0.788325\pi\)
\(744\) 0 0
\(745\) 15113.7 0.743252
\(746\) 6894.39 0.338367
\(747\) 0 0
\(748\) 22367.1 1.09335
\(749\) 0 0
\(750\) 0 0
\(751\) −12981.1 −0.630741 −0.315370 0.948969i \(-0.602129\pi\)
−0.315370 + 0.948969i \(0.602129\pi\)
\(752\) −26444.2 −1.28234
\(753\) 0 0
\(754\) −2472.95 −0.119442
\(755\) 4541.05 0.218895
\(756\) 0 0
\(757\) 31766.3 1.52518 0.762592 0.646880i \(-0.223927\pi\)
0.762592 + 0.646880i \(0.223927\pi\)
\(758\) 2156.06 0.103314
\(759\) 0 0
\(760\) −5351.37 −0.255414
\(761\) 21164.7 1.00817 0.504087 0.863653i \(-0.331829\pi\)
0.504087 + 0.863653i \(0.331829\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1397.19 −0.0661630
\(765\) 0 0
\(766\) 3874.40 0.182752
\(767\) −6180.49 −0.290958
\(768\) 0 0
\(769\) −29643.8 −1.39009 −0.695047 0.718965i \(-0.744616\pi\)
−0.695047 + 0.718965i \(0.744616\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 23945.5 1.11634
\(773\) −15627.2 −0.727128 −0.363564 0.931569i \(-0.618440\pi\)
−0.363564 + 0.931569i \(0.618440\pi\)
\(774\) 0 0
\(775\) 4073.62 0.188811
\(776\) 5245.12 0.242640
\(777\) 0 0
\(778\) 4483.05 0.206587
\(779\) 28856.4 1.32720
\(780\) 0 0
\(781\) −12575.1 −0.576148
\(782\) 2909.65 0.133055
\(783\) 0 0
\(784\) 0 0
\(785\) −6937.44 −0.315424
\(786\) 0 0
\(787\) −4641.64 −0.210237 −0.105118 0.994460i \(-0.533522\pi\)
−0.105118 + 0.994460i \(0.533522\pi\)
\(788\) −4724.63 −0.213589
\(789\) 0 0
\(790\) −1001.86 −0.0451196
\(791\) 0 0
\(792\) 0 0
\(793\) 7826.22 0.350463
\(794\) 5399.93 0.241356
\(795\) 0 0
\(796\) −10715.9 −0.477153
\(797\) 4097.25 0.182098 0.0910489 0.995846i \(-0.470978\pi\)
0.0910489 + 0.995846i \(0.470978\pi\)
\(798\) 0 0
\(799\) −27237.7 −1.20601
\(800\) −2407.65 −0.106404
\(801\) 0 0
\(802\) 133.943 0.00589735
\(803\) 59688.5 2.62311
\(804\) 0 0
\(805\) 0 0
\(806\) 3438.64 0.150274
\(807\) 0 0
\(808\) 15882.1 0.691499
\(809\) 36787.7 1.59875 0.799374 0.600834i \(-0.205165\pi\)
0.799374 + 0.600834i \(0.205165\pi\)
\(810\) 0 0
\(811\) −23507.4 −1.01782 −0.508912 0.860819i \(-0.669952\pi\)
−0.508912 + 0.860819i \(0.669952\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −95.7139 −0.00412134
\(815\) −13186.7 −0.566762
\(816\) 0 0
\(817\) 8065.15 0.345366
\(818\) −3568.77 −0.152542
\(819\) 0 0
\(820\) 8604.89 0.366458
\(821\) 34565.8 1.46937 0.734686 0.678407i \(-0.237330\pi\)
0.734686 + 0.678407i \(0.237330\pi\)
\(822\) 0 0
\(823\) 26917.7 1.14009 0.570044 0.821614i \(-0.306926\pi\)
0.570044 + 0.821614i \(0.306926\pi\)
\(824\) 10830.7 0.457896
\(825\) 0 0
\(826\) 0 0
\(827\) 35448.4 1.49052 0.745261 0.666773i \(-0.232325\pi\)
0.745261 + 0.666773i \(0.232325\pi\)
\(828\) 0 0
\(829\) −24726.0 −1.03591 −0.517955 0.855408i \(-0.673306\pi\)
−0.517955 + 0.855408i \(0.673306\pi\)
\(830\) 740.459 0.0309659
\(831\) 0 0
\(832\) 16420.3 0.684220
\(833\) 0 0
\(834\) 0 0
\(835\) 12627.7 0.523352
\(836\) −48955.6 −2.02532
\(837\) 0 0
\(838\) −1428.26 −0.0588764
\(839\) −21114.3 −0.868829 −0.434415 0.900713i \(-0.643045\pi\)
−0.434415 + 0.900713i \(0.643045\pi\)
\(840\) 0 0
\(841\) −10656.8 −0.436952
\(842\) 1816.15 0.0743332
\(843\) 0 0
\(844\) 10932.9 0.445883
\(845\) −2924.19 −0.119048
\(846\) 0 0
\(847\) 0 0
\(848\) 28319.8 1.14682
\(849\) 0 0
\(850\) −777.474 −0.0313731
\(851\) 348.138 0.0140235
\(852\) 0 0
\(853\) 19801.4 0.794828 0.397414 0.917640i \(-0.369908\pi\)
0.397414 + 0.917640i \(0.369908\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −15426.9 −0.615982
\(857\) 13201.0 0.526180 0.263090 0.964771i \(-0.415258\pi\)
0.263090 + 0.964771i \(0.415258\pi\)
\(858\) 0 0
\(859\) −43020.3 −1.70877 −0.854385 0.519641i \(-0.826066\pi\)
−0.854385 + 0.519641i \(0.826066\pi\)
\(860\) 2405.00 0.0953602
\(861\) 0 0
\(862\) 2803.44 0.110772
\(863\) −6633.09 −0.261637 −0.130819 0.991406i \(-0.541761\pi\)
−0.130819 + 0.991406i \(0.541761\pi\)
\(864\) 0 0
\(865\) −3249.37 −0.127725
\(866\) 7375.16 0.289397
\(867\) 0 0
\(868\) 0 0
\(869\) −18658.2 −0.728351
\(870\) 0 0
\(871\) −26784.6 −1.04198
\(872\) −5363.52 −0.208293
\(873\) 0 0
\(874\) −6368.44 −0.246471
\(875\) 0 0
\(876\) 0 0
\(877\) 41119.1 1.58323 0.791616 0.611019i \(-0.209240\pi\)
0.791616 + 0.611019i \(0.209240\pi\)
\(878\) −6915.62 −0.265821
\(879\) 0 0
\(880\) −14057.6 −0.538502
\(881\) −37501.8 −1.43413 −0.717064 0.697007i \(-0.754515\pi\)
−0.717064 + 0.697007i \(0.754515\pi\)
\(882\) 0 0
\(883\) −10396.0 −0.396208 −0.198104 0.980181i \(-0.563478\pi\)
−0.198104 + 0.980181i \(0.563478\pi\)
\(884\) 18350.1 0.698166
\(885\) 0 0
\(886\) 7843.25 0.297403
\(887\) 42954.0 1.62599 0.812996 0.582270i \(-0.197835\pi\)
0.812996 + 0.582270i \(0.197835\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2434.09 0.0916750
\(891\) 0 0
\(892\) −32546.0 −1.22166
\(893\) 59616.0 2.23401
\(894\) 0 0
\(895\) 17171.7 0.641327
\(896\) 0 0
\(897\) 0 0
\(898\) 336.781 0.0125151
\(899\) −19094.6 −0.708388
\(900\) 0 0
\(901\) 29169.6 1.07856
\(902\) −5731.46 −0.211571
\(903\) 0 0
\(904\) −7089.23 −0.260823
\(905\) −5936.02 −0.218033
\(906\) 0 0
\(907\) −45162.6 −1.65336 −0.826681 0.562670i \(-0.809774\pi\)
−0.826681 + 0.562670i \(0.809774\pi\)
\(908\) −47735.3 −1.74466
\(909\) 0 0
\(910\) 0 0
\(911\) −40669.4 −1.47908 −0.739538 0.673115i \(-0.764956\pi\)
−0.739538 + 0.673115i \(0.764956\pi\)
\(912\) 0 0
\(913\) 13790.1 0.499873
\(914\) −7604.67 −0.275208
\(915\) 0 0
\(916\) −9244.02 −0.333440
\(917\) 0 0
\(918\) 0 0
\(919\) −50310.3 −1.80586 −0.902930 0.429788i \(-0.858588\pi\)
−0.902930 + 0.429788i \(0.858588\pi\)
\(920\) −3866.01 −0.138542
\(921\) 0 0
\(922\) 5400.81 0.192914
\(923\) −10316.6 −0.367905
\(924\) 0 0
\(925\) −93.0243 −0.00330662
\(926\) −2331.37 −0.0827360
\(927\) 0 0
\(928\) 11285.6 0.399210
\(929\) 24058.7 0.849668 0.424834 0.905271i \(-0.360332\pi\)
0.424834 + 0.905271i \(0.360332\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 7162.07 0.251718
\(933\) 0 0
\(934\) 6784.93 0.237698
\(935\) −14479.4 −0.506447
\(936\) 0 0
\(937\) 11252.2 0.392310 0.196155 0.980573i \(-0.437154\pi\)
0.196155 + 0.980573i \(0.437154\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 17777.3 0.616841
\(941\) −40002.5 −1.38581 −0.692903 0.721030i \(-0.743669\pi\)
−0.692903 + 0.721030i \(0.743669\pi\)
\(942\) 0 0
\(943\) 20846.9 0.719902
\(944\) 8842.66 0.304877
\(945\) 0 0
\(946\) −1601.90 −0.0550552
\(947\) −42875.0 −1.47123 −0.735613 0.677402i \(-0.763106\pi\)
−0.735613 + 0.677402i \(0.763106\pi\)
\(948\) 0 0
\(949\) 48968.6 1.67501
\(950\) 1701.68 0.0581156
\(951\) 0 0
\(952\) 0 0
\(953\) −7241.41 −0.246141 −0.123070 0.992398i \(-0.539274\pi\)
−0.123070 + 0.992398i \(0.539274\pi\)
\(954\) 0 0
\(955\) 904.474 0.0306472
\(956\) −26489.2 −0.896151
\(957\) 0 0
\(958\) −8211.06 −0.276918
\(959\) 0 0
\(960\) 0 0
\(961\) −3239.93 −0.108755
\(962\) −78.5240 −0.00263172
\(963\) 0 0
\(964\) 36677.7 1.22543
\(965\) −15501.2 −0.517099
\(966\) 0 0
\(967\) 32573.3 1.08323 0.541617 0.840625i \(-0.317812\pi\)
0.541617 + 0.840625i \(0.317812\pi\)
\(968\) 8795.27 0.292036
\(969\) 0 0
\(970\) −1667.90 −0.0552092
\(971\) −1233.95 −0.0407820 −0.0203910 0.999792i \(-0.506491\pi\)
−0.0203910 + 0.999792i \(0.506491\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −4108.52 −0.135159
\(975\) 0 0
\(976\) −11197.3 −0.367229
\(977\) −35891.5 −1.17530 −0.587651 0.809115i \(-0.699947\pi\)
−0.587651 + 0.809115i \(0.699947\pi\)
\(978\) 0 0
\(979\) 45331.6 1.47988
\(980\) 0 0
\(981\) 0 0
\(982\) 555.713 0.0180586
\(983\) −26119.1 −0.847476 −0.423738 0.905785i \(-0.639282\pi\)
−0.423738 + 0.905785i \(0.639282\pi\)
\(984\) 0 0
\(985\) 3058.51 0.0989361
\(986\) 3644.31 0.117706
\(987\) 0 0
\(988\) −40163.3 −1.29329
\(989\) 5826.53 0.187334
\(990\) 0 0
\(991\) −7910.14 −0.253556 −0.126778 0.991931i \(-0.540464\pi\)
−0.126778 + 0.991931i \(0.540464\pi\)
\(992\) −15692.6 −0.502259
\(993\) 0 0
\(994\) 0 0
\(995\) 6936.95 0.221021
\(996\) 0 0
\(997\) 22952.3 0.729094 0.364547 0.931185i \(-0.381224\pi\)
0.364547 + 0.931185i \(0.381224\pi\)
\(998\) 2330.60 0.0739215
\(999\) 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.by.1.3 yes 6
3.2 odd 2 2205.4.a.bx.1.4 yes 6
7.6 odd 2 2205.4.a.bx.1.3 6
21.20 even 2 inner 2205.4.a.by.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2205.4.a.bx.1.3 6 7.6 odd 2
2205.4.a.bx.1.4 yes 6 3.2 odd 2
2205.4.a.by.1.3 yes 6 1.1 even 1 trivial
2205.4.a.by.1.4 yes 6 21.20 even 2 inner