Properties

Label 441.4.a.u.1.3
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.94534\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.53113 q^{2} +4.46887 q^{4} -2.07730 q^{5} -12.4689 q^{8} +O(q^{10})\) \(q+3.53113 q^{2} +4.46887 q^{4} -2.07730 q^{5} -12.4689 q^{8} -7.33521 q^{10} -49.1868 q^{11} +44.8559 q^{13} -79.7802 q^{16} +26.5179 q^{17} -77.7350 q^{19} -9.28317 q^{20} -173.685 q^{22} -55.7510 q^{23} -120.685 q^{25} +158.392 q^{26} -121.436 q^{29} -305.553 q^{31} -181.963 q^{32} +93.6380 q^{34} +77.1868 q^{37} -274.492 q^{38} +25.9016 q^{40} +248.720 q^{41} -147.179 q^{43} -219.809 q^{44} -196.864 q^{46} +269.851 q^{47} -426.154 q^{50} +200.455 q^{52} +141.121 q^{53} +102.176 q^{55} -428.805 q^{58} -424.834 q^{59} +587.996 q^{61} -1078.95 q^{62} -4.29373 q^{64} -93.1790 q^{65} -179.634 q^{67} +118.505 q^{68} -674.872 q^{71} -237.489 q^{73} +272.556 q^{74} -347.388 q^{76} +495.852 q^{79} +165.727 q^{80} +878.262 q^{82} -24.4406 q^{83} -55.0855 q^{85} -519.708 q^{86} +613.304 q^{88} +1072.29 q^{89} -249.144 q^{92} +952.877 q^{94} +161.479 q^{95} -1667.43 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 34q^{4} - 66q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 34q^{4} - 66q^{8} - 100q^{11} - 174q^{16} - 340q^{22} - 352q^{23} - 128q^{25} - 260q^{29} + 30q^{32} + 212q^{37} + 540q^{43} - 460q^{44} + 696q^{46} - 1366q^{50} - 16q^{53} - 780q^{58} - 1678q^{64} + 756q^{65} - 1944q^{67} - 2248q^{71} + 284q^{74} - 1048q^{79} - 3284q^{85} - 4820q^{86} + 1260q^{88} - 3512q^{92} - 2192q^{95} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.53113 1.24844 0.624221 0.781248i \(-0.285416\pi\)
0.624221 + 0.781248i \(0.285416\pi\)
\(3\) 0 0
\(4\) 4.46887 0.558609
\(5\) −2.07730 −0.185799 −0.0928996 0.995675i \(-0.529614\pi\)
−0.0928996 + 0.995675i \(0.529614\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −12.4689 −0.551051
\(9\) 0 0
\(10\) −7.33521 −0.231960
\(11\) −49.1868 −1.34822 −0.674108 0.738633i \(-0.735472\pi\)
−0.674108 + 0.738633i \(0.735472\pi\)
\(12\) 0 0
\(13\) 44.8559 0.956983 0.478492 0.878092i \(-0.341184\pi\)
0.478492 + 0.878092i \(0.341184\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −79.7802 −1.24656
\(17\) 26.5179 0.378325 0.189163 0.981946i \(-0.439423\pi\)
0.189163 + 0.981946i \(0.439423\pi\)
\(18\) 0 0
\(19\) −77.7350 −0.938612 −0.469306 0.883036i \(-0.655496\pi\)
−0.469306 + 0.883036i \(0.655496\pi\)
\(20\) −9.28317 −0.103789
\(21\) 0 0
\(22\) −173.685 −1.68317
\(23\) −55.7510 −0.505430 −0.252715 0.967541i \(-0.581323\pi\)
−0.252715 + 0.967541i \(0.581323\pi\)
\(24\) 0 0
\(25\) −120.685 −0.965479
\(26\) 158.392 1.19474
\(27\) 0 0
\(28\) 0 0
\(29\) −121.436 −0.777588 −0.388794 0.921325i \(-0.627108\pi\)
−0.388794 + 0.921325i \(0.627108\pi\)
\(30\) 0 0
\(31\) −305.553 −1.77029 −0.885143 0.465319i \(-0.845940\pi\)
−0.885143 + 0.465319i \(0.845940\pi\)
\(32\) −181.963 −1.00521
\(33\) 0 0
\(34\) 93.6380 0.472317
\(35\) 0 0
\(36\) 0 0
\(37\) 77.1868 0.342957 0.171479 0.985188i \(-0.445146\pi\)
0.171479 + 0.985188i \(0.445146\pi\)
\(38\) −274.492 −1.17180
\(39\) 0 0
\(40\) 25.9016 0.102385
\(41\) 248.720 0.947403 0.473702 0.880685i \(-0.342917\pi\)
0.473702 + 0.880685i \(0.342917\pi\)
\(42\) 0 0
\(43\) −147.179 −0.521967 −0.260984 0.965343i \(-0.584047\pi\)
−0.260984 + 0.965343i \(0.584047\pi\)
\(44\) −219.809 −0.753125
\(45\) 0 0
\(46\) −196.864 −0.631000
\(47\) 269.851 0.837484 0.418742 0.908105i \(-0.362471\pi\)
0.418742 + 0.908105i \(0.362471\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −426.154 −1.20534
\(51\) 0 0
\(52\) 200.455 0.534579
\(53\) 141.121 0.365744 0.182872 0.983137i \(-0.441461\pi\)
0.182872 + 0.983137i \(0.441461\pi\)
\(54\) 0 0
\(55\) 102.176 0.250497
\(56\) 0 0
\(57\) 0 0
\(58\) −428.805 −0.970774
\(59\) −424.834 −0.937434 −0.468717 0.883348i \(-0.655284\pi\)
−0.468717 + 0.883348i \(0.655284\pi\)
\(60\) 0 0
\(61\) 587.996 1.23418 0.617092 0.786891i \(-0.288311\pi\)
0.617092 + 0.786891i \(0.288311\pi\)
\(62\) −1078.95 −2.21010
\(63\) 0 0
\(64\) −4.29373 −0.00838618
\(65\) −93.1790 −0.177807
\(66\) 0 0
\(67\) −179.634 −0.327549 −0.163775 0.986498i \(-0.552367\pi\)
−0.163775 + 0.986498i \(0.552367\pi\)
\(68\) 118.505 0.211336
\(69\) 0 0
\(70\) 0 0
\(71\) −674.872 −1.12806 −0.564032 0.825753i \(-0.690750\pi\)
−0.564032 + 0.825753i \(0.690750\pi\)
\(72\) 0 0
\(73\) −237.489 −0.380767 −0.190383 0.981710i \(-0.560973\pi\)
−0.190383 + 0.981710i \(0.560973\pi\)
\(74\) 272.556 0.428163
\(75\) 0 0
\(76\) −347.388 −0.524317
\(77\) 0 0
\(78\) 0 0
\(79\) 495.852 0.706174 0.353087 0.935591i \(-0.385132\pi\)
0.353087 + 0.935591i \(0.385132\pi\)
\(80\) 165.727 0.231611
\(81\) 0 0
\(82\) 878.262 1.18278
\(83\) −24.4406 −0.0323217 −0.0161609 0.999869i \(-0.505144\pi\)
−0.0161609 + 0.999869i \(0.505144\pi\)
\(84\) 0 0
\(85\) −55.0855 −0.0702925
\(86\) −519.708 −0.651646
\(87\) 0 0
\(88\) 613.304 0.742936
\(89\) 1072.29 1.27710 0.638552 0.769579i \(-0.279534\pi\)
0.638552 + 0.769579i \(0.279534\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −249.144 −0.282337
\(93\) 0 0
\(94\) 952.877 1.04555
\(95\) 161.479 0.174393
\(96\) 0 0
\(97\) −1667.43 −1.74538 −0.872690 0.488275i \(-0.837626\pi\)
−0.872690 + 0.488275i \(0.837626\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −539.325 −0.539325
\(101\) −77.1187 −0.0759762 −0.0379881 0.999278i \(-0.512095\pi\)
−0.0379881 + 0.999278i \(0.512095\pi\)
\(102\) 0 0
\(103\) −164.693 −0.157550 −0.0787749 0.996892i \(-0.525101\pi\)
−0.0787749 + 0.996892i \(0.525101\pi\)
\(104\) −559.302 −0.527347
\(105\) 0 0
\(106\) 498.315 0.456610
\(107\) −1022.62 −0.923931 −0.461966 0.886898i \(-0.652856\pi\)
−0.461966 + 0.886898i \(0.652856\pi\)
\(108\) 0 0
\(109\) 1362.52 1.19730 0.598649 0.801011i \(-0.295704\pi\)
0.598649 + 0.801011i \(0.295704\pi\)
\(110\) 360.795 0.312731
\(111\) 0 0
\(112\) 0 0
\(113\) 1538.41 1.28072 0.640360 0.768075i \(-0.278785\pi\)
0.640360 + 0.768075i \(0.278785\pi\)
\(114\) 0 0
\(115\) 115.811 0.0939084
\(116\) −542.681 −0.434368
\(117\) 0 0
\(118\) −1500.14 −1.17033
\(119\) 0 0
\(120\) 0 0
\(121\) 1088.34 0.817685
\(122\) 2076.29 1.54081
\(123\) 0 0
\(124\) −1365.48 −0.988898
\(125\) 510.360 0.365184
\(126\) 0 0
\(127\) −170.358 −0.119030 −0.0595151 0.998227i \(-0.518955\pi\)
−0.0595151 + 0.998227i \(0.518955\pi\)
\(128\) 1440.54 0.994744
\(129\) 0 0
\(130\) −329.027 −0.221981
\(131\) −751.935 −0.501503 −0.250751 0.968051i \(-0.580678\pi\)
−0.250751 + 0.968051i \(0.580678\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −634.312 −0.408927
\(135\) 0 0
\(136\) −330.648 −0.208477
\(137\) −518.623 −0.323423 −0.161711 0.986838i \(-0.551701\pi\)
−0.161711 + 0.986838i \(0.551701\pi\)
\(138\) 0 0
\(139\) 2975.72 1.81581 0.907905 0.419177i \(-0.137681\pi\)
0.907905 + 0.419177i \(0.137681\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2383.06 −1.40832
\(143\) −2206.32 −1.29022
\(144\) 0 0
\(145\) 252.258 0.144475
\(146\) −838.604 −0.475365
\(147\) 0 0
\(148\) 344.938 0.191579
\(149\) 2717.94 1.49438 0.747188 0.664612i \(-0.231403\pi\)
0.747188 + 0.664612i \(0.231403\pi\)
\(150\) 0 0
\(151\) 707.650 0.381376 0.190688 0.981651i \(-0.438928\pi\)
0.190688 + 0.981651i \(0.438928\pi\)
\(152\) 969.267 0.517224
\(153\) 0 0
\(154\) 0 0
\(155\) 634.724 0.328918
\(156\) 0 0
\(157\) −3117.91 −1.58495 −0.792473 0.609906i \(-0.791207\pi\)
−0.792473 + 0.609906i \(0.791207\pi\)
\(158\) 1750.92 0.881618
\(159\) 0 0
\(160\) 377.991 0.186768
\(161\) 0 0
\(162\) 0 0
\(163\) 1808.77 0.869167 0.434583 0.900632i \(-0.356896\pi\)
0.434583 + 0.900632i \(0.356896\pi\)
\(164\) 1111.50 0.529228
\(165\) 0 0
\(166\) −86.3028 −0.0403518
\(167\) 3147.38 1.45839 0.729197 0.684303i \(-0.239894\pi\)
0.729197 + 0.684303i \(0.239894\pi\)
\(168\) 0 0
\(169\) −184.949 −0.0841827
\(170\) −194.514 −0.0877562
\(171\) 0 0
\(172\) −657.724 −0.291576
\(173\) −3284.36 −1.44338 −0.721691 0.692215i \(-0.756635\pi\)
−0.721691 + 0.692215i \(0.756635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3924.13 1.68064
\(177\) 0 0
\(178\) 3786.39 1.59439
\(179\) −2798.83 −1.16868 −0.584341 0.811508i \(-0.698647\pi\)
−0.584341 + 0.811508i \(0.698647\pi\)
\(180\) 0 0
\(181\) 3723.04 1.52890 0.764451 0.644682i \(-0.223010\pi\)
0.764451 + 0.644682i \(0.223010\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 695.152 0.278518
\(185\) −160.340 −0.0637212
\(186\) 0 0
\(187\) −1304.33 −0.510064
\(188\) 1205.93 0.467826
\(189\) 0 0
\(190\) 570.202 0.217720
\(191\) −959.650 −0.363549 −0.181774 0.983340i \(-0.558184\pi\)
−0.181774 + 0.983340i \(0.558184\pi\)
\(192\) 0 0
\(193\) −3790.25 −1.41362 −0.706808 0.707406i \(-0.749865\pi\)
−0.706808 + 0.707406i \(0.749865\pi\)
\(194\) −5887.91 −2.17901
\(195\) 0 0
\(196\) 0 0
\(197\) −5117.99 −1.85097 −0.925487 0.378779i \(-0.876344\pi\)
−0.925487 + 0.378779i \(0.876344\pi\)
\(198\) 0 0
\(199\) −864.855 −0.308080 −0.154040 0.988065i \(-0.549228\pi\)
−0.154040 + 0.988065i \(0.549228\pi\)
\(200\) 1504.80 0.532028
\(201\) 0 0
\(202\) −272.316 −0.0948519
\(203\) 0 0
\(204\) 0 0
\(205\) −516.665 −0.176027
\(206\) −581.550 −0.196692
\(207\) 0 0
\(208\) −3578.61 −1.19294
\(209\) 3823.53 1.26545
\(210\) 0 0
\(211\) −1344.61 −0.438707 −0.219353 0.975645i \(-0.570395\pi\)
−0.219353 + 0.975645i \(0.570395\pi\)
\(212\) 630.650 0.204308
\(213\) 0 0
\(214\) −3611.01 −1.15348
\(215\) 305.735 0.0969811
\(216\) 0 0
\(217\) 0 0
\(218\) 4811.23 1.49476
\(219\) 0 0
\(220\) 456.609 0.139930
\(221\) 1189.48 0.362051
\(222\) 0 0
\(223\) 864.916 0.259727 0.129863 0.991532i \(-0.458546\pi\)
0.129863 + 0.991532i \(0.458546\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 5432.32 1.59890
\(227\) 1715.34 0.501548 0.250774 0.968046i \(-0.419315\pi\)
0.250774 + 0.968046i \(0.419315\pi\)
\(228\) 0 0
\(229\) −1045.46 −0.301685 −0.150842 0.988558i \(-0.548199\pi\)
−0.150842 + 0.988558i \(0.548199\pi\)
\(230\) 408.945 0.117239
\(231\) 0 0
\(232\) 1514.17 0.428491
\(233\) −1448.67 −0.407320 −0.203660 0.979042i \(-0.565284\pi\)
−0.203660 + 0.979042i \(0.565284\pi\)
\(234\) 0 0
\(235\) −560.560 −0.155604
\(236\) −1898.53 −0.523659
\(237\) 0 0
\(238\) 0 0
\(239\) 3153.12 0.853383 0.426691 0.904397i \(-0.359679\pi\)
0.426691 + 0.904397i \(0.359679\pi\)
\(240\) 0 0
\(241\) −381.012 −0.101839 −0.0509194 0.998703i \(-0.516215\pi\)
−0.0509194 + 0.998703i \(0.516215\pi\)
\(242\) 3843.06 1.02083
\(243\) 0 0
\(244\) 2627.68 0.689426
\(245\) 0 0
\(246\) 0 0
\(247\) −3486.87 −0.898236
\(248\) 3809.90 0.975519
\(249\) 0 0
\(250\) 1802.15 0.455912
\(251\) −3776.23 −0.949617 −0.474808 0.880089i \(-0.657483\pi\)
−0.474808 + 0.880089i \(0.657483\pi\)
\(252\) 0 0
\(253\) 2742.21 0.681428
\(254\) −601.556 −0.148602
\(255\) 0 0
\(256\) 5121.09 1.25027
\(257\) −4258.42 −1.03359 −0.516795 0.856109i \(-0.672875\pi\)
−0.516795 + 0.856109i \(0.672875\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −416.405 −0.0993244
\(261\) 0 0
\(262\) −2655.18 −0.626098
\(263\) −4198.83 −0.984451 −0.492226 0.870468i \(-0.663816\pi\)
−0.492226 + 0.870468i \(0.663816\pi\)
\(264\) 0 0
\(265\) −293.150 −0.0679548
\(266\) 0 0
\(267\) 0 0
\(268\) −802.762 −0.182972
\(269\) −3740.59 −0.847835 −0.423917 0.905701i \(-0.639345\pi\)
−0.423917 + 0.905701i \(0.639345\pi\)
\(270\) 0 0
\(271\) 4356.30 0.976480 0.488240 0.872709i \(-0.337639\pi\)
0.488240 + 0.872709i \(0.337639\pi\)
\(272\) −2115.60 −0.471607
\(273\) 0 0
\(274\) −1831.32 −0.403775
\(275\) 5936.10 1.30167
\(276\) 0 0
\(277\) −1344.30 −0.291593 −0.145797 0.989315i \(-0.546575\pi\)
−0.145797 + 0.989315i \(0.546575\pi\)
\(278\) 10507.7 2.26693
\(279\) 0 0
\(280\) 0 0
\(281\) −4205.54 −0.892817 −0.446408 0.894829i \(-0.647297\pi\)
−0.446408 + 0.894829i \(0.647297\pi\)
\(282\) 0 0
\(283\) 4752.03 0.998159 0.499079 0.866556i \(-0.333672\pi\)
0.499079 + 0.866556i \(0.333672\pi\)
\(284\) −3015.91 −0.630146
\(285\) 0 0
\(286\) −7790.79 −1.61077
\(287\) 0 0
\(288\) 0 0
\(289\) −4209.80 −0.856870
\(290\) 890.757 0.180369
\(291\) 0 0
\(292\) −1061.31 −0.212700
\(293\) 4961.17 0.989196 0.494598 0.869122i \(-0.335315\pi\)
0.494598 + 0.869122i \(0.335315\pi\)
\(294\) 0 0
\(295\) 882.506 0.174174
\(296\) −962.432 −0.188987
\(297\) 0 0
\(298\) 9597.39 1.86564
\(299\) −2500.76 −0.483688
\(300\) 0 0
\(301\) 0 0
\(302\) 2498.80 0.476126
\(303\) 0 0
\(304\) 6201.71 1.17004
\(305\) −1221.44 −0.229310
\(306\) 0 0
\(307\) −4234.00 −0.787124 −0.393562 0.919298i \(-0.628757\pi\)
−0.393562 + 0.919298i \(0.628757\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2241.29 0.410635
\(311\) 684.700 0.124842 0.0624209 0.998050i \(-0.480118\pi\)
0.0624209 + 0.998050i \(0.480118\pi\)
\(312\) 0 0
\(313\) −5944.07 −1.07341 −0.536707 0.843768i \(-0.680332\pi\)
−0.536707 + 0.843768i \(0.680332\pi\)
\(314\) −11009.8 −1.97872
\(315\) 0 0
\(316\) 2215.90 0.394475
\(317\) 2823.89 0.500333 0.250166 0.968203i \(-0.419515\pi\)
0.250166 + 0.968203i \(0.419515\pi\)
\(318\) 0 0
\(319\) 5973.04 1.04836
\(320\) 8.91935 0.00155815
\(321\) 0 0
\(322\) 0 0
\(323\) −2061.37 −0.355101
\(324\) 0 0
\(325\) −5413.43 −0.923947
\(326\) 6387.02 1.08510
\(327\) 0 0
\(328\) −3101.26 −0.522068
\(329\) 0 0
\(330\) 0 0
\(331\) −2812.97 −0.467114 −0.233557 0.972343i \(-0.575037\pi\)
−0.233557 + 0.972343i \(0.575037\pi\)
\(332\) −109.222 −0.0180552
\(333\) 0 0
\(334\) 11113.8 1.82072
\(335\) 373.154 0.0608584
\(336\) 0 0
\(337\) 4260.10 0.688612 0.344306 0.938857i \(-0.388114\pi\)
0.344306 + 0.938857i \(0.388114\pi\)
\(338\) −653.080 −0.105097
\(339\) 0 0
\(340\) −246.170 −0.0392660
\(341\) 15029.2 2.38673
\(342\) 0 0
\(343\) 0 0
\(344\) 1835.16 0.287631
\(345\) 0 0
\(346\) −11597.5 −1.80198
\(347\) −36.0584 −0.00557843 −0.00278922 0.999996i \(-0.500888\pi\)
−0.00278922 + 0.999996i \(0.500888\pi\)
\(348\) 0 0
\(349\) −242.692 −0.0372236 −0.0186118 0.999827i \(-0.505925\pi\)
−0.0186118 + 0.999827i \(0.505925\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 8950.18 1.35524
\(353\) 109.990 0.0165840 0.00829201 0.999966i \(-0.497361\pi\)
0.00829201 + 0.999966i \(0.497361\pi\)
\(354\) 0 0
\(355\) 1401.91 0.209593
\(356\) 4791.92 0.713402
\(357\) 0 0
\(358\) −9883.01 −1.45903
\(359\) −12404.5 −1.82363 −0.911814 0.410604i \(-0.865318\pi\)
−0.911814 + 0.410604i \(0.865318\pi\)
\(360\) 0 0
\(361\) −816.273 −0.119008
\(362\) 13146.5 1.90875
\(363\) 0 0
\(364\) 0 0
\(365\) 493.335 0.0707461
\(366\) 0 0
\(367\) 13859.6 1.97130 0.985649 0.168807i \(-0.0539913\pi\)
0.985649 + 0.168807i \(0.0539913\pi\)
\(368\) 4447.82 0.630051
\(369\) 0 0
\(370\) −566.181 −0.0795523
\(371\) 0 0
\(372\) 0 0
\(373\) 4898.06 0.679925 0.339963 0.940439i \(-0.389586\pi\)
0.339963 + 0.940439i \(0.389586\pi\)
\(374\) −4605.75 −0.636786
\(375\) 0 0
\(376\) −3364.73 −0.461497
\(377\) −5447.11 −0.744139
\(378\) 0 0
\(379\) −9806.25 −1.32906 −0.664530 0.747262i \(-0.731368\pi\)
−0.664530 + 0.747262i \(0.731368\pi\)
\(380\) 721.627 0.0974176
\(381\) 0 0
\(382\) −3388.65 −0.453870
\(383\) −10729.7 −1.43149 −0.715746 0.698361i \(-0.753913\pi\)
−0.715746 + 0.698361i \(0.753913\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −13383.8 −1.76482
\(387\) 0 0
\(388\) −7451.53 −0.974984
\(389\) −5264.05 −0.686113 −0.343057 0.939315i \(-0.611462\pi\)
−0.343057 + 0.939315i \(0.611462\pi\)
\(390\) 0 0
\(391\) −1478.40 −0.191217
\(392\) 0 0
\(393\) 0 0
\(394\) −18072.3 −2.31083
\(395\) −1030.03 −0.131206
\(396\) 0 0
\(397\) 1214.90 0.153587 0.0767935 0.997047i \(-0.475532\pi\)
0.0767935 + 0.997047i \(0.475532\pi\)
\(398\) −3053.91 −0.384620
\(399\) 0 0
\(400\) 9628.26 1.20353
\(401\) −2295.45 −0.285859 −0.142929 0.989733i \(-0.545652\pi\)
−0.142929 + 0.989733i \(0.545652\pi\)
\(402\) 0 0
\(403\) −13705.8 −1.69413
\(404\) −344.633 −0.0424410
\(405\) 0 0
\(406\) 0 0
\(407\) −3796.57 −0.462381
\(408\) 0 0
\(409\) 4646.54 0.561753 0.280876 0.959744i \(-0.409375\pi\)
0.280876 + 0.959744i \(0.409375\pi\)
\(410\) −1824.41 −0.219759
\(411\) 0 0
\(412\) −735.990 −0.0880087
\(413\) 0 0
\(414\) 0 0
\(415\) 50.7703 0.00600535
\(416\) −8162.11 −0.961973
\(417\) 0 0
\(418\) 13501.4 1.57984
\(419\) −7541.24 −0.879269 −0.439634 0.898177i \(-0.644892\pi\)
−0.439634 + 0.898177i \(0.644892\pi\)
\(420\) 0 0
\(421\) −6243.63 −0.722794 −0.361397 0.932412i \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(422\) −4748.01 −0.547700
\(423\) 0 0
\(424\) −1759.62 −0.201544
\(425\) −3200.31 −0.365265
\(426\) 0 0
\(427\) 0 0
\(428\) −4569.97 −0.516116
\(429\) 0 0
\(430\) 1079.59 0.121075
\(431\) −11465.8 −1.28141 −0.640706 0.767786i \(-0.721358\pi\)
−0.640706 + 0.767786i \(0.721358\pi\)
\(432\) 0 0
\(433\) 5156.40 0.572289 0.286144 0.958187i \(-0.407626\pi\)
0.286144 + 0.958187i \(0.407626\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6088.92 0.668822
\(437\) 4333.80 0.474402
\(438\) 0 0
\(439\) −5064.25 −0.550577 −0.275289 0.961362i \(-0.588773\pi\)
−0.275289 + 0.961362i \(0.588773\pi\)
\(440\) −1274.01 −0.138037
\(441\) 0 0
\(442\) 4200.22 0.452000
\(443\) −12703.6 −1.36246 −0.681228 0.732071i \(-0.738554\pi\)
−0.681228 + 0.732071i \(0.738554\pi\)
\(444\) 0 0
\(445\) −2227.46 −0.237285
\(446\) 3054.13 0.324254
\(447\) 0 0
\(448\) 0 0
\(449\) −13942.2 −1.46542 −0.732709 0.680542i \(-0.761744\pi\)
−0.732709 + 0.680542i \(0.761744\pi\)
\(450\) 0 0
\(451\) −12233.7 −1.27730
\(452\) 6874.95 0.715421
\(453\) 0 0
\(454\) 6057.10 0.626154
\(455\) 0 0
\(456\) 0 0
\(457\) −15214.0 −1.55729 −0.778646 0.627464i \(-0.784093\pi\)
−0.778646 + 0.627464i \(0.784093\pi\)
\(458\) −3691.65 −0.376636
\(459\) 0 0
\(460\) 517.546 0.0524581
\(461\) 11430.2 1.15479 0.577394 0.816465i \(-0.304070\pi\)
0.577394 + 0.816465i \(0.304070\pi\)
\(462\) 0 0
\(463\) −9347.88 −0.938300 −0.469150 0.883119i \(-0.655440\pi\)
−0.469150 + 0.883119i \(0.655440\pi\)
\(464\) 9688.17 0.969314
\(465\) 0 0
\(466\) −5115.44 −0.508515
\(467\) −3630.84 −0.359776 −0.179888 0.983687i \(-0.557573\pi\)
−0.179888 + 0.983687i \(0.557573\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1979.41 −0.194262
\(471\) 0 0
\(472\) 5297.20 0.516575
\(473\) 7239.26 0.703724
\(474\) 0 0
\(475\) 9381.43 0.906210
\(476\) 0 0
\(477\) 0 0
\(478\) 11134.1 1.06540
\(479\) −6521.25 −0.622053 −0.311027 0.950401i \(-0.600673\pi\)
−0.311027 + 0.950401i \(0.600673\pi\)
\(480\) 0 0
\(481\) 3462.28 0.328205
\(482\) −1345.40 −0.127140
\(483\) 0 0
\(484\) 4863.65 0.456766
\(485\) 3463.75 0.324290
\(486\) 0 0
\(487\) −3666.29 −0.341140 −0.170570 0.985346i \(-0.554561\pi\)
−0.170570 + 0.985346i \(0.554561\pi\)
\(488\) −7331.65 −0.680099
\(489\) 0 0
\(490\) 0 0
\(491\) 12470.7 1.14623 0.573113 0.819476i \(-0.305736\pi\)
0.573113 + 0.819476i \(0.305736\pi\)
\(492\) 0 0
\(493\) −3220.22 −0.294181
\(494\) −12312.6 −1.12140
\(495\) 0 0
\(496\) 24377.0 2.20678
\(497\) 0 0
\(498\) 0 0
\(499\) −2303.93 −0.206690 −0.103345 0.994646i \(-0.532955\pi\)
−0.103345 + 0.994646i \(0.532955\pi\)
\(500\) 2280.74 0.203995
\(501\) 0 0
\(502\) −13334.4 −1.18554
\(503\) 10520.4 0.932570 0.466285 0.884635i \(-0.345592\pi\)
0.466285 + 0.884635i \(0.345592\pi\)
\(504\) 0 0
\(505\) 160.198 0.0141163
\(506\) 9683.10 0.850724
\(507\) 0 0
\(508\) −761.308 −0.0664913
\(509\) 9662.22 0.841395 0.420698 0.907201i \(-0.361785\pi\)
0.420698 + 0.907201i \(0.361785\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 6558.89 0.566142
\(513\) 0 0
\(514\) −15037.0 −1.29038
\(515\) 342.115 0.0292726
\(516\) 0 0
\(517\) −13273.1 −1.12911
\(518\) 0 0
\(519\) 0 0
\(520\) 1161.84 0.0979806
\(521\) −8607.81 −0.723829 −0.361914 0.932211i \(-0.617877\pi\)
−0.361914 + 0.932211i \(0.617877\pi\)
\(522\) 0 0
\(523\) −10482.7 −0.876439 −0.438219 0.898868i \(-0.644391\pi\)
−0.438219 + 0.898868i \(0.644391\pi\)
\(524\) −3360.30 −0.280144
\(525\) 0 0
\(526\) −14826.6 −1.22903
\(527\) −8102.61 −0.669744
\(528\) 0 0
\(529\) −9058.83 −0.744541
\(530\) −1035.15 −0.0848377
\(531\) 0 0
\(532\) 0 0
\(533\) 11156.6 0.906649
\(534\) 0 0
\(535\) 2124.29 0.171666
\(536\) 2239.84 0.180497
\(537\) 0 0
\(538\) −13208.5 −1.05847
\(539\) 0 0
\(540\) 0 0
\(541\) 20722.6 1.64683 0.823416 0.567438i \(-0.192066\pi\)
0.823416 + 0.567438i \(0.192066\pi\)
\(542\) 15382.6 1.21908
\(543\) 0 0
\(544\) −4825.27 −0.380298
\(545\) −2830.35 −0.222457
\(546\) 0 0
\(547\) −4175.09 −0.326351 −0.163176 0.986597i \(-0.552174\pi\)
−0.163176 + 0.986597i \(0.552174\pi\)
\(548\) −2317.66 −0.180667
\(549\) 0 0
\(550\) 20961.1 1.62506
\(551\) 9439.81 0.729854
\(552\) 0 0
\(553\) 0 0
\(554\) −4746.91 −0.364038
\(555\) 0 0
\(556\) 13298.1 1.01433
\(557\) 10161.7 0.773011 0.386505 0.922287i \(-0.373682\pi\)
0.386505 + 0.922287i \(0.373682\pi\)
\(558\) 0 0
\(559\) −6601.85 −0.499514
\(560\) 0 0
\(561\) 0 0
\(562\) −14850.3 −1.11463
\(563\) 17104.4 1.28040 0.640201 0.768208i \(-0.278851\pi\)
0.640201 + 0.768208i \(0.278851\pi\)
\(564\) 0 0
\(565\) −3195.73 −0.237957
\(566\) 16780.0 1.24614
\(567\) 0 0
\(568\) 8414.89 0.621621
\(569\) 18257.6 1.34516 0.672581 0.740023i \(-0.265185\pi\)
0.672581 + 0.740023i \(0.265185\pi\)
\(570\) 0 0
\(571\) 13630.5 0.998982 0.499491 0.866319i \(-0.333520\pi\)
0.499491 + 0.866319i \(0.333520\pi\)
\(572\) −9859.74 −0.720728
\(573\) 0 0
\(574\) 0 0
\(575\) 6728.30 0.487981
\(576\) 0 0
\(577\) 4442.08 0.320496 0.160248 0.987077i \(-0.448771\pi\)
0.160248 + 0.987077i \(0.448771\pi\)
\(578\) −14865.4 −1.06975
\(579\) 0 0
\(580\) 1127.31 0.0807052
\(581\) 0 0
\(582\) 0 0
\(583\) −6941.27 −0.493101
\(584\) 2961.22 0.209822
\(585\) 0 0
\(586\) 17518.5 1.23495
\(587\) −3103.38 −0.218211 −0.109106 0.994030i \(-0.534799\pi\)
−0.109106 + 0.994030i \(0.534799\pi\)
\(588\) 0 0
\(589\) 23752.1 1.66161
\(590\) 3116.24 0.217447
\(591\) 0 0
\(592\) −6157.97 −0.427519
\(593\) −5937.71 −0.411185 −0.205592 0.978638i \(-0.565912\pi\)
−0.205592 + 0.978638i \(0.565912\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12146.1 0.834772
\(597\) 0 0
\(598\) −8830.50 −0.603856
\(599\) 2600.33 0.177373 0.0886866 0.996060i \(-0.471733\pi\)
0.0886866 + 0.996060i \(0.471733\pi\)
\(600\) 0 0
\(601\) −13881.4 −0.942156 −0.471078 0.882092i \(-0.656135\pi\)
−0.471078 + 0.882092i \(0.656135\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3162.40 0.213040
\(605\) −2260.80 −0.151925
\(606\) 0 0
\(607\) −12284.6 −0.821442 −0.410721 0.911761i \(-0.634723\pi\)
−0.410721 + 0.911761i \(0.634723\pi\)
\(608\) 14144.9 0.943505
\(609\) 0 0
\(610\) −4313.07 −0.286281
\(611\) 12104.4 0.801459
\(612\) 0 0
\(613\) 22062.0 1.45363 0.726815 0.686833i \(-0.241000\pi\)
0.726815 + 0.686833i \(0.241000\pi\)
\(614\) −14950.8 −0.982679
\(615\) 0 0
\(616\) 0 0
\(617\) 12182.2 0.794871 0.397436 0.917630i \(-0.369900\pi\)
0.397436 + 0.917630i \(0.369900\pi\)
\(618\) 0 0
\(619\) 23248.6 1.50960 0.754799 0.655956i \(-0.227734\pi\)
0.754799 + 0.655956i \(0.227734\pi\)
\(620\) 2836.50 0.183736
\(621\) 0 0
\(622\) 2417.76 0.155858
\(623\) 0 0
\(624\) 0 0
\(625\) 14025.4 0.897628
\(626\) −20989.3 −1.34010
\(627\) 0 0
\(628\) −13933.6 −0.885365
\(629\) 2046.83 0.129749
\(630\) 0 0
\(631\) 19184.4 1.21033 0.605165 0.796100i \(-0.293107\pi\)
0.605165 + 0.796100i \(0.293107\pi\)
\(632\) −6182.72 −0.389138
\(633\) 0 0
\(634\) 9971.52 0.624637
\(635\) 353.884 0.0221157
\(636\) 0 0
\(637\) 0 0
\(638\) 21091.6 1.30881
\(639\) 0 0
\(640\) −2992.44 −0.184823
\(641\) 19433.4 1.19746 0.598730 0.800951i \(-0.295672\pi\)
0.598730 + 0.800951i \(0.295672\pi\)
\(642\) 0 0
\(643\) 5777.47 0.354341 0.177170 0.984180i \(-0.443306\pi\)
0.177170 + 0.984180i \(0.443306\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −7278.95 −0.443323
\(647\) 29231.5 1.77621 0.888106 0.459640i \(-0.152021\pi\)
0.888106 + 0.459640i \(0.152021\pi\)
\(648\) 0 0
\(649\) 20896.2 1.26386
\(650\) −19115.5 −1.15349
\(651\) 0 0
\(652\) 8083.18 0.485524
\(653\) 7093.27 0.425086 0.212543 0.977152i \(-0.431826\pi\)
0.212543 + 0.977152i \(0.431826\pi\)
\(654\) 0 0
\(655\) 1561.99 0.0931788
\(656\) −19842.9 −1.18100
\(657\) 0 0
\(658\) 0 0
\(659\) −19014.2 −1.12396 −0.561980 0.827151i \(-0.689960\pi\)
−0.561980 + 0.827151i \(0.689960\pi\)
\(660\) 0 0
\(661\) −21058.4 −1.23915 −0.619573 0.784939i \(-0.712694\pi\)
−0.619573 + 0.784939i \(0.712694\pi\)
\(662\) −9932.96 −0.583165
\(663\) 0 0
\(664\) 304.746 0.0178109
\(665\) 0 0
\(666\) 0 0
\(667\) 6770.16 0.393016
\(668\) 14065.3 0.814672
\(669\) 0 0
\(670\) 1317.65 0.0759782
\(671\) −28921.6 −1.66395
\(672\) 0 0
\(673\) 9634.87 0.551853 0.275926 0.961179i \(-0.411015\pi\)
0.275926 + 0.961179i \(0.411015\pi\)
\(674\) 15043.0 0.859693
\(675\) 0 0
\(676\) −826.515 −0.0470252
\(677\) −8371.31 −0.475237 −0.237619 0.971359i \(-0.576367\pi\)
−0.237619 + 0.971359i \(0.576367\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 686.854 0.0387348
\(681\) 0 0
\(682\) 53069.9 2.97969
\(683\) 12068.8 0.676137 0.338069 0.941121i \(-0.390226\pi\)
0.338069 + 0.941121i \(0.390226\pi\)
\(684\) 0 0
\(685\) 1077.33 0.0600917
\(686\) 0 0
\(687\) 0 0
\(688\) 11742.0 0.650666
\(689\) 6330.09 0.350011
\(690\) 0 0
\(691\) 2981.29 0.164130 0.0820648 0.996627i \(-0.473849\pi\)
0.0820648 + 0.996627i \(0.473849\pi\)
\(692\) −14677.4 −0.806286
\(693\) 0 0
\(694\) −127.327 −0.00696435
\(695\) −6181.46 −0.337376
\(696\) 0 0
\(697\) 6595.53 0.358427
\(698\) −856.978 −0.0464715
\(699\) 0 0
\(700\) 0 0
\(701\) 28978.0 1.56132 0.780660 0.624956i \(-0.214883\pi\)
0.780660 + 0.624956i \(0.214883\pi\)
\(702\) 0 0
\(703\) −6000.11 −0.321904
\(704\) 211.195 0.0113064
\(705\) 0 0
\(706\) 388.388 0.0207042
\(707\) 0 0
\(708\) 0 0
\(709\) −16372.4 −0.867249 −0.433625 0.901094i \(-0.642766\pi\)
−0.433625 + 0.901094i \(0.642766\pi\)
\(710\) 4950.32 0.261665
\(711\) 0 0
\(712\) −13370.2 −0.703750
\(713\) 17034.9 0.894755
\(714\) 0 0
\(715\) 4583.18 0.239722
\(716\) −12507.6 −0.652836
\(717\) 0 0
\(718\) −43801.7 −2.27669
\(719\) −23010.5 −1.19353 −0.596765 0.802416i \(-0.703547\pi\)
−0.596765 + 0.802416i \(0.703547\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2882.36 −0.148574
\(723\) 0 0
\(724\) 16637.8 0.854058
\(725\) 14655.5 0.750745
\(726\) 0 0
\(727\) −24636.8 −1.25685 −0.628423 0.777872i \(-0.716299\pi\)
−0.628423 + 0.777872i \(0.716299\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1742.03 0.0883225
\(731\) −3902.87 −0.197473
\(732\) 0 0
\(733\) 6904.76 0.347931 0.173965 0.984752i \(-0.444342\pi\)
0.173965 + 0.984752i \(0.444342\pi\)
\(734\) 48940.1 2.46105
\(735\) 0 0
\(736\) 10144.6 0.508065
\(737\) 8835.63 0.441607
\(738\) 0 0
\(739\) −9234.89 −0.459690 −0.229845 0.973227i \(-0.573822\pi\)
−0.229845 + 0.973227i \(0.573822\pi\)
\(740\) −716.538 −0.0355952
\(741\) 0 0
\(742\) 0 0
\(743\) −20216.9 −0.998232 −0.499116 0.866535i \(-0.666342\pi\)
−0.499116 + 0.866535i \(0.666342\pi\)
\(744\) 0 0
\(745\) −5645.97 −0.277654
\(746\) 17295.7 0.848847
\(747\) 0 0
\(748\) −5828.88 −0.284926
\(749\) 0 0
\(750\) 0 0
\(751\) 24054.9 1.16881 0.584405 0.811462i \(-0.301328\pi\)
0.584405 + 0.811462i \(0.301328\pi\)
\(752\) −21528.7 −1.04398
\(753\) 0 0
\(754\) −19234.5 −0.929015
\(755\) −1470.00 −0.0708593
\(756\) 0 0
\(757\) −30328.2 −1.45614 −0.728069 0.685504i \(-0.759582\pi\)
−0.728069 + 0.685504i \(0.759582\pi\)
\(758\) −34627.1 −1.65925
\(759\) 0 0
\(760\) −2013.46 −0.0960997
\(761\) 33834.1 1.61168 0.805839 0.592135i \(-0.201715\pi\)
0.805839 + 0.592135i \(0.201715\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −4288.55 −0.203082
\(765\) 0 0
\(766\) −37887.9 −1.78713
\(767\) −19056.3 −0.897109
\(768\) 0 0
\(769\) 31738.1 1.48830 0.744151 0.668011i \(-0.232854\pi\)
0.744151 + 0.668011i \(0.232854\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16938.1 −0.789658
\(773\) −27494.2 −1.27930 −0.639650 0.768667i \(-0.720920\pi\)
−0.639650 + 0.768667i \(0.720920\pi\)
\(774\) 0 0
\(775\) 36875.6 1.70917
\(776\) 20791.0 0.961794
\(777\) 0 0
\(778\) −18588.0 −0.856573
\(779\) −19334.2 −0.889244
\(780\) 0 0
\(781\) 33194.8 1.52087
\(782\) −5220.41 −0.238723
\(783\) 0 0
\(784\) 0 0
\(785\) 6476.84 0.294482
\(786\) 0 0
\(787\) −468.356 −0.0212136 −0.0106068 0.999944i \(-0.503376\pi\)
−0.0106068 + 0.999944i \(0.503376\pi\)
\(788\) −22871.6 −1.03397
\(789\) 0 0
\(790\) −3637.18 −0.163804
\(791\) 0 0
\(792\) 0 0
\(793\) 26375.1 1.18109
\(794\) 4289.97 0.191745
\(795\) 0 0
\(796\) −3864.93 −0.172096
\(797\) −37723.8 −1.67659 −0.838297 0.545214i \(-0.816449\pi\)
−0.838297 + 0.545214i \(0.816449\pi\)
\(798\) 0 0
\(799\) 7155.87 0.316841
\(800\) 21960.2 0.970512
\(801\) 0 0
\(802\) −8105.53 −0.356878
\(803\) 11681.3 0.513356
\(804\) 0 0
\(805\) 0 0
\(806\) −48397.1 −2.11503
\(807\) 0 0
\(808\) 961.583 0.0418668
\(809\) 7797.13 0.338854 0.169427 0.985543i \(-0.445808\pi\)
0.169427 + 0.985543i \(0.445808\pi\)
\(810\) 0 0
\(811\) 16925.9 0.732860 0.366430 0.930446i \(-0.380580\pi\)
0.366430 + 0.930446i \(0.380580\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −13406.2 −0.577256
\(815\) −3757.36 −0.161490
\(816\) 0 0
\(817\) 11441.0 0.489925
\(818\) 16407.5 0.701316
\(819\) 0 0
\(820\) −2308.91 −0.0983301
\(821\) 30009.3 1.27568 0.637840 0.770169i \(-0.279828\pi\)
0.637840 + 0.770169i \(0.279828\pi\)
\(822\) 0 0
\(823\) −23385.6 −0.990486 −0.495243 0.868754i \(-0.664921\pi\)
−0.495243 + 0.868754i \(0.664921\pi\)
\(824\) 2053.53 0.0868181
\(825\) 0 0
\(826\) 0 0
\(827\) 37325.9 1.56947 0.784734 0.619833i \(-0.212800\pi\)
0.784734 + 0.619833i \(0.212800\pi\)
\(828\) 0 0
\(829\) −24671.3 −1.03362 −0.516809 0.856100i \(-0.672880\pi\)
−0.516809 + 0.856100i \(0.672880\pi\)
\(830\) 179.277 0.00749733
\(831\) 0 0
\(832\) −192.599 −0.00802544
\(833\) 0 0
\(834\) 0 0
\(835\) −6538.05 −0.270969
\(836\) 17086.9 0.706892
\(837\) 0 0
\(838\) −26629.1 −1.09772
\(839\) −14147.4 −0.582147 −0.291074 0.956701i \(-0.594013\pi\)
−0.291074 + 0.956701i \(0.594013\pi\)
\(840\) 0 0
\(841\) −9642.35 −0.395356
\(842\) −22047.1 −0.902366
\(843\) 0 0
\(844\) −6008.91 −0.245065
\(845\) 384.195 0.0156411
\(846\) 0 0
\(847\) 0 0
\(848\) −11258.6 −0.455923
\(849\) 0 0
\(850\) −11300.7 −0.456012
\(851\) −4303.24 −0.173341
\(852\) 0 0
\(853\) 27963.6 1.12246 0.561229 0.827661i \(-0.310329\pi\)
0.561229 + 0.827661i \(0.310329\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12750.9 0.509134
\(857\) −33855.8 −1.34947 −0.674734 0.738061i \(-0.735741\pi\)
−0.674734 + 0.738061i \(0.735741\pi\)
\(858\) 0 0
\(859\) −24282.7 −0.964511 −0.482255 0.876031i \(-0.660182\pi\)
−0.482255 + 0.876031i \(0.660182\pi\)
\(860\) 1366.29 0.0541745
\(861\) 0 0
\(862\) −40487.2 −1.59977
\(863\) 19667.0 0.775750 0.387875 0.921712i \(-0.373209\pi\)
0.387875 + 0.921712i \(0.373209\pi\)
\(864\) 0 0
\(865\) 6822.59 0.268179
\(866\) 18207.9 0.714469
\(867\) 0 0
\(868\) 0 0
\(869\) −24389.4 −0.952075
\(870\) 0 0
\(871\) −8057.65 −0.313459
\(872\) −16989.1 −0.659773
\(873\) 0 0
\(874\) 15303.2 0.592264
\(875\) 0 0
\(876\) 0 0
\(877\) −36061.0 −1.38848 −0.694238 0.719745i \(-0.744259\pi\)
−0.694238 + 0.719745i \(0.744259\pi\)
\(878\) −17882.5 −0.687364
\(879\) 0 0
\(880\) −8151.58 −0.312261
\(881\) 15889.7 0.607646 0.303823 0.952728i \(-0.401737\pi\)
0.303823 + 0.952728i \(0.401737\pi\)
\(882\) 0 0
\(883\) 14861.3 0.566390 0.283195 0.959062i \(-0.408606\pi\)
0.283195 + 0.959062i \(0.408606\pi\)
\(884\) 5315.65 0.202245
\(885\) 0 0
\(886\) −44858.2 −1.70095
\(887\) 38189.9 1.44565 0.722824 0.691032i \(-0.242844\pi\)
0.722824 + 0.691032i \(0.242844\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −7865.45 −0.296237
\(891\) 0 0
\(892\) 3865.20 0.145086
\(893\) −20976.8 −0.786073
\(894\) 0 0
\(895\) 5813.99 0.217140
\(896\) 0 0
\(897\) 0 0
\(898\) −49231.7 −1.82949
\(899\) 37105.0 1.37655
\(900\) 0 0
\(901\) 3742.22 0.138370
\(902\) −43198.9 −1.59464
\(903\) 0 0
\(904\) −19182.2 −0.705742
\(905\) −7733.86 −0.284069
\(906\) 0 0
\(907\) 16865.5 0.617431 0.308715 0.951154i \(-0.400101\pi\)
0.308715 + 0.951154i \(0.400101\pi\)
\(908\) 7665.65 0.280169
\(909\) 0 0
\(910\) 0 0
\(911\) −26754.1 −0.973000 −0.486500 0.873681i \(-0.661727\pi\)
−0.486500 + 0.873681i \(0.661727\pi\)
\(912\) 0 0
\(913\) 1202.15 0.0435766
\(914\) −53722.7 −1.94419
\(915\) 0 0
\(916\) −4672.02 −0.168524
\(917\) 0 0
\(918\) 0 0
\(919\) −41527.5 −1.49061 −0.745303 0.666726i \(-0.767695\pi\)
−0.745303 + 0.666726i \(0.767695\pi\)
\(920\) −1444.04 −0.0517484
\(921\) 0 0
\(922\) 40361.5 1.44169
\(923\) −30272.0 −1.07954
\(924\) 0 0
\(925\) −9315.27 −0.331118
\(926\) −33008.6 −1.17141
\(927\) 0 0
\(928\) 22096.8 0.781642
\(929\) −4584.68 −0.161914 −0.0809572 0.996718i \(-0.525798\pi\)
−0.0809572 + 0.996718i \(0.525798\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6473.92 −0.227532
\(933\) 0 0
\(934\) −12821.0 −0.449159
\(935\) 2709.48 0.0947694
\(936\) 0 0
\(937\) −6928.18 −0.241552 −0.120776 0.992680i \(-0.538538\pi\)
−0.120776 + 0.992680i \(0.538538\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2505.07 −0.0869217
\(941\) −20944.9 −0.725596 −0.362798 0.931868i \(-0.618178\pi\)
−0.362798 + 0.931868i \(0.618178\pi\)
\(942\) 0 0
\(943\) −13866.4 −0.478846
\(944\) 33893.3 1.16857
\(945\) 0 0
\(946\) 25562.8 0.878559
\(947\) −29278.9 −1.00468 −0.502342 0.864669i \(-0.667528\pi\)
−0.502342 + 0.864669i \(0.667528\pi\)
\(948\) 0 0
\(949\) −10652.8 −0.364387
\(950\) 33127.1 1.13135
\(951\) 0 0
\(952\) 0 0
\(953\) −2136.81 −0.0726316 −0.0363158 0.999340i \(-0.511562\pi\)
−0.0363158 + 0.999340i \(0.511562\pi\)
\(954\) 0 0
\(955\) 1993.48 0.0675470
\(956\) 14090.9 0.476707
\(957\) 0 0
\(958\) −23027.4 −0.776597
\(959\) 0 0
\(960\) 0 0
\(961\) 63571.4 2.13391
\(962\) 12225.8 0.409745
\(963\) 0 0
\(964\) −1702.70 −0.0568881
\(965\) 7873.47 0.262649
\(966\) 0 0
\(967\) −3921.32 −0.130405 −0.0652023 0.997872i \(-0.520769\pi\)
−0.0652023 + 0.997872i \(0.520769\pi\)
\(968\) −13570.4 −0.450586
\(969\) 0 0
\(970\) 12230.9 0.404857
\(971\) −47809.1 −1.58009 −0.790045 0.613049i \(-0.789943\pi\)
−0.790045 + 0.613049i \(0.789943\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −12946.1 −0.425894
\(975\) 0 0
\(976\) −46910.5 −1.53849
\(977\) 51329.1 1.68082 0.840411 0.541949i \(-0.182314\pi\)
0.840411 + 0.541949i \(0.182314\pi\)
\(978\) 0 0
\(979\) −52742.4 −1.72181
\(980\) 0 0
\(981\) 0 0
\(982\) 44035.8 1.43100
\(983\) 16326.3 0.529734 0.264867 0.964285i \(-0.414672\pi\)
0.264867 + 0.964285i \(0.414672\pi\)
\(984\) 0 0
\(985\) 10631.6 0.343909
\(986\) −11371.0 −0.367268
\(987\) 0 0
\(988\) −15582.4 −0.501763
\(989\) 8205.37 0.263818
\(990\) 0 0
\(991\) 33770.0 1.08248 0.541242 0.840867i \(-0.317954\pi\)
0.541242 + 0.840867i \(0.317954\pi\)
\(992\) 55599.3 1.77952
\(993\) 0 0
\(994\) 0 0
\(995\) 1796.56 0.0572410
\(996\) 0 0
\(997\) 50695.4 1.61037 0.805185 0.593023i \(-0.202066\pi\)
0.805185 + 0.593023i \(0.202066\pi\)
\(998\) −8135.48 −0.258040
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 441.4.a.u.1.3 4
3.2 odd 2 49.4.a.e.1.2 yes 4
7.2 even 3 441.4.e.y.361.2 8
7.3 odd 6 441.4.e.y.226.1 8
7.4 even 3 441.4.e.y.226.2 8
7.5 odd 6 441.4.e.y.361.1 8
7.6 odd 2 inner 441.4.a.u.1.4 4
12.11 even 2 784.4.a.bf.1.1 4
15.14 odd 2 1225.4.a.bb.1.3 4
21.2 odd 6 49.4.c.e.18.3 8
21.5 even 6 49.4.c.e.18.4 8
21.11 odd 6 49.4.c.e.30.3 8
21.17 even 6 49.4.c.e.30.4 8
21.20 even 2 49.4.a.e.1.1 4
84.83 odd 2 784.4.a.bf.1.4 4
105.104 even 2 1225.4.a.bb.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.e.1.1 4 21.20 even 2
49.4.a.e.1.2 yes 4 3.2 odd 2
49.4.c.e.18.3 8 21.2 odd 6
49.4.c.e.18.4 8 21.5 even 6
49.4.c.e.30.3 8 21.11 odd 6
49.4.c.e.30.4 8 21.17 even 6
441.4.a.u.1.3 4 1.1 even 1 trivial
441.4.a.u.1.4 4 7.6 odd 2 inner
441.4.e.y.226.1 8 7.3 odd 6
441.4.e.y.226.2 8 7.4 even 3
441.4.e.y.361.1 8 7.5 odd 6
441.4.e.y.361.2 8 7.2 even 3
784.4.a.bf.1.1 4 12.11 even 2
784.4.a.bf.1.4 4 84.83 odd 2
1225.4.a.bb.1.3 4 15.14 odd 2
1225.4.a.bb.1.4 4 105.104 even 2