Properties

Label 441.4.a.x.1.4
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 74 x^{6} + 1469 x^{4} - 8828 x^{2} + 2500\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.37406\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.95985 q^{2} -4.15899 q^{4} +11.8897 q^{5} +23.8298 q^{8} +O(q^{10})\) \(q-1.95985 q^{2} -4.15899 q^{4} +11.8897 q^{5} +23.8298 q^{8} -23.3019 q^{10} -36.4130 q^{11} -0.964525 q^{13} -13.4309 q^{16} +98.2514 q^{17} +106.001 q^{19} -49.4490 q^{20} +71.3640 q^{22} -54.3633 q^{23} +16.3640 q^{25} +1.89032 q^{26} -229.725 q^{29} -127.729 q^{31} -164.316 q^{32} -192.558 q^{34} +311.816 q^{37} -207.746 q^{38} +283.328 q^{40} -419.919 q^{41} +523.180 q^{43} +151.441 q^{44} +106.544 q^{46} +270.328 q^{47} -32.0711 q^{50} +4.01145 q^{52} +251.082 q^{53} -432.938 q^{55} +450.226 q^{58} +408.029 q^{59} +860.546 q^{61} +250.329 q^{62} +429.481 q^{64} -11.4679 q^{65} +506.360 q^{67} -408.626 q^{68} +523.702 q^{71} -629.964 q^{73} -611.112 q^{74} -440.856 q^{76} +319.548 q^{79} -159.689 q^{80} +822.978 q^{82} +1309.16 q^{83} +1168.18 q^{85} -1025.35 q^{86} -867.715 q^{88} -348.581 q^{89} +226.096 q^{92} -529.803 q^{94} +1260.31 q^{95} +161.996 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 68q^{4} + O(q^{10}) \) \( 8q + 68q^{4} + 804q^{16} + 976q^{22} + 536q^{25} + 64q^{37} + 2160q^{43} - 768q^{46} + 2184q^{58} + 7588q^{64} + 5392q^{79} + 2864q^{85} + 5616q^{88} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.95985 −0.692912 −0.346456 0.938066i \(-0.612615\pi\)
−0.346456 + 0.938066i \(0.612615\pi\)
\(3\) 0 0
\(4\) −4.15899 −0.519874
\(5\) 11.8897 1.06344 0.531722 0.846919i \(-0.321545\pi\)
0.531722 + 0.846919i \(0.321545\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 23.8298 1.05314
\(9\) 0 0
\(10\) −23.3019 −0.736872
\(11\) −36.4130 −0.998085 −0.499043 0.866577i \(-0.666315\pi\)
−0.499043 + 0.866577i \(0.666315\pi\)
\(12\) 0 0
\(13\) −0.964525 −0.0205778 −0.0102889 0.999947i \(-0.503275\pi\)
−0.0102889 + 0.999947i \(0.503275\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −13.4309 −0.209858
\(17\) 98.2514 1.40173 0.700866 0.713293i \(-0.252797\pi\)
0.700866 + 0.713293i \(0.252797\pi\)
\(18\) 0 0
\(19\) 106.001 1.27991 0.639954 0.768413i \(-0.278953\pi\)
0.639954 + 0.768413i \(0.278953\pi\)
\(20\) −49.4490 −0.552856
\(21\) 0 0
\(22\) 71.3640 0.691585
\(23\) −54.3633 −0.492849 −0.246424 0.969162i \(-0.579256\pi\)
−0.246424 + 0.969162i \(0.579256\pi\)
\(24\) 0 0
\(25\) 16.3640 0.130912
\(26\) 1.89032 0.0142586
\(27\) 0 0
\(28\) 0 0
\(29\) −229.725 −1.47099 −0.735497 0.677528i \(-0.763051\pi\)
−0.735497 + 0.677528i \(0.763051\pi\)
\(30\) 0 0
\(31\) −127.729 −0.740025 −0.370013 0.929027i \(-0.620647\pi\)
−0.370013 + 0.929027i \(0.620647\pi\)
\(32\) −164.316 −0.907725
\(33\) 0 0
\(34\) −192.558 −0.971277
\(35\) 0 0
\(36\) 0 0
\(37\) 311.816 1.38546 0.692732 0.721195i \(-0.256407\pi\)
0.692732 + 0.721195i \(0.256407\pi\)
\(38\) −207.746 −0.886864
\(39\) 0 0
\(40\) 283.328 1.11995
\(41\) −419.919 −1.59952 −0.799760 0.600320i \(-0.795040\pi\)
−0.799760 + 0.600320i \(0.795040\pi\)
\(42\) 0 0
\(43\) 523.180 1.85545 0.927723 0.373270i \(-0.121763\pi\)
0.927723 + 0.373270i \(0.121763\pi\)
\(44\) 151.441 0.518878
\(45\) 0 0
\(46\) 106.544 0.341501
\(47\) 270.328 0.838966 0.419483 0.907763i \(-0.362211\pi\)
0.419483 + 0.907763i \(0.362211\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −32.0711 −0.0907107
\(51\) 0 0
\(52\) 4.01145 0.0106978
\(53\) 251.082 0.650732 0.325366 0.945588i \(-0.394513\pi\)
0.325366 + 0.945588i \(0.394513\pi\)
\(54\) 0 0
\(55\) −432.938 −1.06141
\(56\) 0 0
\(57\) 0 0
\(58\) 450.226 1.01927
\(59\) 408.029 0.900354 0.450177 0.892939i \(-0.351361\pi\)
0.450177 + 0.892939i \(0.351361\pi\)
\(60\) 0 0
\(61\) 860.546 1.80626 0.903128 0.429372i \(-0.141265\pi\)
0.903128 + 0.429372i \(0.141265\pi\)
\(62\) 250.329 0.512772
\(63\) 0 0
\(64\) 429.481 0.838831
\(65\) −11.4679 −0.0218833
\(66\) 0 0
\(67\) 506.360 0.923308 0.461654 0.887060i \(-0.347256\pi\)
0.461654 + 0.887060i \(0.347256\pi\)
\(68\) −408.626 −0.728724
\(69\) 0 0
\(70\) 0 0
\(71\) 523.702 0.875380 0.437690 0.899126i \(-0.355797\pi\)
0.437690 + 0.899126i \(0.355797\pi\)
\(72\) 0 0
\(73\) −629.964 −1.01002 −0.505012 0.863113i \(-0.668512\pi\)
−0.505012 + 0.863113i \(0.668512\pi\)
\(74\) −611.112 −0.960004
\(75\) 0 0
\(76\) −440.856 −0.665391
\(77\) 0 0
\(78\) 0 0
\(79\) 319.548 0.455089 0.227544 0.973768i \(-0.426930\pi\)
0.227544 + 0.973768i \(0.426930\pi\)
\(80\) −159.689 −0.223172
\(81\) 0 0
\(82\) 822.978 1.10833
\(83\) 1309.16 1.73131 0.865654 0.500643i \(-0.166903\pi\)
0.865654 + 0.500643i \(0.166903\pi\)
\(84\) 0 0
\(85\) 1168.18 1.49066
\(86\) −1025.35 −1.28566
\(87\) 0 0
\(88\) −867.715 −1.05112
\(89\) −348.581 −0.415163 −0.207581 0.978218i \(-0.566559\pi\)
−0.207581 + 0.978218i \(0.566559\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 226.096 0.256219
\(93\) 0 0
\(94\) −529.803 −0.581329
\(95\) 1260.31 1.36111
\(96\) 0 0
\(97\) 161.996 0.169569 0.0847844 0.996399i \(-0.472980\pi\)
0.0847844 + 0.996399i \(0.472980\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −68.0579 −0.0680579
\(101\) −218.342 −0.215108 −0.107554 0.994199i \(-0.534302\pi\)
−0.107554 + 0.994199i \(0.534302\pi\)
\(102\) 0 0
\(103\) 231.090 0.221068 0.110534 0.993872i \(-0.464744\pi\)
0.110534 + 0.993872i \(0.464744\pi\)
\(104\) −22.9844 −0.0216712
\(105\) 0 0
\(106\) −492.083 −0.450899
\(107\) 1865.12 1.68513 0.842563 0.538598i \(-0.181046\pi\)
0.842563 + 0.538598i \(0.181046\pi\)
\(108\) 0 0
\(109\) 601.004 0.528127 0.264063 0.964505i \(-0.414937\pi\)
0.264063 + 0.964505i \(0.414937\pi\)
\(110\) 848.494 0.735461
\(111\) 0 0
\(112\) 0 0
\(113\) 475.349 0.395727 0.197863 0.980230i \(-0.436600\pi\)
0.197863 + 0.980230i \(0.436600\pi\)
\(114\) 0 0
\(115\) −646.361 −0.524117
\(116\) 955.422 0.764731
\(117\) 0 0
\(118\) −799.676 −0.623865
\(119\) 0 0
\(120\) 0 0
\(121\) −5.09213 −0.00382580
\(122\) −1686.54 −1.25158
\(123\) 0 0
\(124\) 531.223 0.384720
\(125\) −1291.64 −0.924226
\(126\) 0 0
\(127\) −29.0876 −0.0203237 −0.0101619 0.999948i \(-0.503235\pi\)
−0.0101619 + 0.999948i \(0.503235\pi\)
\(128\) 472.807 0.326489
\(129\) 0 0
\(130\) 22.4753 0.0151632
\(131\) −2115.26 −1.41077 −0.705387 0.708822i \(-0.749227\pi\)
−0.705387 + 0.708822i \(0.749227\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −992.389 −0.639771
\(135\) 0 0
\(136\) 2341.31 1.47622
\(137\) −177.342 −0.110594 −0.0552968 0.998470i \(-0.517610\pi\)
−0.0552968 + 0.998470i \(0.517610\pi\)
\(138\) 0 0
\(139\) 2369.18 1.44569 0.722846 0.691009i \(-0.242834\pi\)
0.722846 + 0.691009i \(0.242834\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1026.38 −0.606561
\(143\) 35.1213 0.0205384
\(144\) 0 0
\(145\) −2731.35 −1.56432
\(146\) 1234.63 0.699857
\(147\) 0 0
\(148\) −1296.84 −0.720266
\(149\) −1491.03 −0.819801 −0.409900 0.912130i \(-0.634437\pi\)
−0.409900 + 0.912130i \(0.634437\pi\)
\(150\) 0 0
\(151\) 410.461 0.221211 0.110605 0.993864i \(-0.464721\pi\)
0.110605 + 0.993864i \(0.464721\pi\)
\(152\) 2525.98 1.34792
\(153\) 0 0
\(154\) 0 0
\(155\) −1518.65 −0.786975
\(156\) 0 0
\(157\) 453.500 0.230530 0.115265 0.993335i \(-0.463228\pi\)
0.115265 + 0.993335i \(0.463228\pi\)
\(158\) −626.267 −0.315336
\(159\) 0 0
\(160\) −1953.66 −0.965314
\(161\) 0 0
\(162\) 0 0
\(163\) 748.166 0.359515 0.179757 0.983711i \(-0.442469\pi\)
0.179757 + 0.983711i \(0.442469\pi\)
\(164\) 1746.44 0.831548
\(165\) 0 0
\(166\) −2565.75 −1.19964
\(167\) −518.269 −0.240149 −0.120074 0.992765i \(-0.538313\pi\)
−0.120074 + 0.992765i \(0.538313\pi\)
\(168\) 0 0
\(169\) −2196.07 −0.999577
\(170\) −2289.45 −1.03290
\(171\) 0 0
\(172\) −2175.90 −0.964597
\(173\) −1313.48 −0.577239 −0.288620 0.957444i \(-0.593196\pi\)
−0.288620 + 0.957444i \(0.593196\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 489.060 0.209456
\(177\) 0 0
\(178\) 683.166 0.287671
\(179\) −4133.15 −1.72584 −0.862921 0.505339i \(-0.831368\pi\)
−0.862921 + 0.505339i \(0.831368\pi\)
\(180\) 0 0
\(181\) 3714.04 1.52521 0.762603 0.646867i \(-0.223921\pi\)
0.762603 + 0.646867i \(0.223921\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1295.47 −0.519038
\(185\) 3707.38 1.47336
\(186\) 0 0
\(187\) −3577.63 −1.39905
\(188\) −1124.29 −0.436156
\(189\) 0 0
\(190\) −2470.03 −0.943129
\(191\) 3068.37 1.16241 0.581203 0.813758i \(-0.302582\pi\)
0.581203 + 0.813758i \(0.302582\pi\)
\(192\) 0 0
\(193\) 2089.53 0.779314 0.389657 0.920960i \(-0.372593\pi\)
0.389657 + 0.920960i \(0.372593\pi\)
\(194\) −317.487 −0.117496
\(195\) 0 0
\(196\) 0 0
\(197\) 3729.89 1.34895 0.674476 0.738297i \(-0.264370\pi\)
0.674476 + 0.738297i \(0.264370\pi\)
\(198\) 0 0
\(199\) 2108.85 0.751217 0.375608 0.926778i \(-0.377434\pi\)
0.375608 + 0.926778i \(0.377434\pi\)
\(200\) 389.952 0.137869
\(201\) 0 0
\(202\) 427.918 0.149051
\(203\) 0 0
\(204\) 0 0
\(205\) −4992.69 −1.70100
\(206\) −452.903 −0.153181
\(207\) 0 0
\(208\) 12.9544 0.00431841
\(209\) −3859.81 −1.27746
\(210\) 0 0
\(211\) −1546.53 −0.504584 −0.252292 0.967651i \(-0.581184\pi\)
−0.252292 + 0.967651i \(0.581184\pi\)
\(212\) −1044.25 −0.338298
\(213\) 0 0
\(214\) −3655.36 −1.16764
\(215\) 6220.43 1.97316
\(216\) 0 0
\(217\) 0 0
\(218\) −1177.88 −0.365945
\(219\) 0 0
\(220\) 1800.59 0.551798
\(221\) −94.7659 −0.0288445
\(222\) 0 0
\(223\) 4860.40 1.45954 0.729768 0.683695i \(-0.239628\pi\)
0.729768 + 0.683695i \(0.239628\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −931.614 −0.274204
\(227\) −700.942 −0.204948 −0.102474 0.994736i \(-0.532676\pi\)
−0.102474 + 0.994736i \(0.532676\pi\)
\(228\) 0 0
\(229\) −3694.24 −1.06604 −0.533018 0.846104i \(-0.678942\pi\)
−0.533018 + 0.846104i \(0.678942\pi\)
\(230\) 1266.77 0.363167
\(231\) 0 0
\(232\) −5474.29 −1.54916
\(233\) −344.322 −0.0968124 −0.0484062 0.998828i \(-0.515414\pi\)
−0.0484062 + 0.998828i \(0.515414\pi\)
\(234\) 0 0
\(235\) 3214.11 0.892193
\(236\) −1696.99 −0.468070
\(237\) 0 0
\(238\) 0 0
\(239\) −6144.21 −1.66291 −0.831456 0.555590i \(-0.812492\pi\)
−0.831456 + 0.555590i \(0.812492\pi\)
\(240\) 0 0
\(241\) −5626.11 −1.50378 −0.751888 0.659291i \(-0.770856\pi\)
−0.751888 + 0.659291i \(0.770856\pi\)
\(242\) 9.97982 0.00265094
\(243\) 0 0
\(244\) −3579.00 −0.939025
\(245\) 0 0
\(246\) 0 0
\(247\) −102.240 −0.0263377
\(248\) −3043.75 −0.779349
\(249\) 0 0
\(250\) 2531.43 0.640407
\(251\) 520.460 0.130881 0.0654405 0.997856i \(-0.479155\pi\)
0.0654405 + 0.997856i \(0.479155\pi\)
\(252\) 0 0
\(253\) 1979.53 0.491905
\(254\) 57.0074 0.0140825
\(255\) 0 0
\(256\) −4362.48 −1.06506
\(257\) −4009.50 −0.973174 −0.486587 0.873632i \(-0.661758\pi\)
−0.486587 + 0.873632i \(0.661758\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 47.6948 0.0113766
\(261\) 0 0
\(262\) 4145.60 0.977542
\(263\) 3585.19 0.840579 0.420290 0.907390i \(-0.361928\pi\)
0.420290 + 0.907390i \(0.361928\pi\)
\(264\) 0 0
\(265\) 2985.28 0.692016
\(266\) 0 0
\(267\) 0 0
\(268\) −2105.94 −0.480004
\(269\) −5509.19 −1.24870 −0.624352 0.781143i \(-0.714637\pi\)
−0.624352 + 0.781143i \(0.714637\pi\)
\(270\) 0 0
\(271\) −5819.89 −1.30455 −0.652276 0.757982i \(-0.726186\pi\)
−0.652276 + 0.757982i \(0.726186\pi\)
\(272\) −1319.60 −0.294165
\(273\) 0 0
\(274\) 347.563 0.0766316
\(275\) −595.864 −0.130662
\(276\) 0 0
\(277\) −1150.45 −0.249545 −0.124772 0.992185i \(-0.539820\pi\)
−0.124772 + 0.992185i \(0.539820\pi\)
\(278\) −4643.24 −1.00174
\(279\) 0 0
\(280\) 0 0
\(281\) 895.631 0.190138 0.0950692 0.995471i \(-0.469693\pi\)
0.0950692 + 0.995471i \(0.469693\pi\)
\(282\) 0 0
\(283\) 122.450 0.0257205 0.0128603 0.999917i \(-0.495906\pi\)
0.0128603 + 0.999917i \(0.495906\pi\)
\(284\) −2178.07 −0.455087
\(285\) 0 0
\(286\) −68.8324 −0.0142313
\(287\) 0 0
\(288\) 0 0
\(289\) 4740.33 0.964854
\(290\) 5353.03 1.08393
\(291\) 0 0
\(292\) 2620.01 0.525084
\(293\) 7601.77 1.51570 0.757850 0.652429i \(-0.226250\pi\)
0.757850 + 0.652429i \(0.226250\pi\)
\(294\) 0 0
\(295\) 4851.33 0.957475
\(296\) 7430.50 1.45909
\(297\) 0 0
\(298\) 2922.20 0.568050
\(299\) 52.4347 0.0101417
\(300\) 0 0
\(301\) 0 0
\(302\) −804.441 −0.153279
\(303\) 0 0
\(304\) −1423.69 −0.268599
\(305\) 10231.6 1.92085
\(306\) 0 0
\(307\) −3539.05 −0.657929 −0.328964 0.944342i \(-0.606700\pi\)
−0.328964 + 0.944342i \(0.606700\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2976.33 0.545304
\(311\) 3894.93 0.710165 0.355083 0.934835i \(-0.384453\pi\)
0.355083 + 0.934835i \(0.384453\pi\)
\(312\) 0 0
\(313\) 8682.58 1.56795 0.783975 0.620793i \(-0.213189\pi\)
0.783975 + 0.620793i \(0.213189\pi\)
\(314\) −888.792 −0.159737
\(315\) 0 0
\(316\) −1329.00 −0.236589
\(317\) −8495.52 −1.50522 −0.752612 0.658464i \(-0.771206\pi\)
−0.752612 + 0.658464i \(0.771206\pi\)
\(318\) 0 0
\(319\) 8364.97 1.46818
\(320\) 5106.39 0.892049
\(321\) 0 0
\(322\) 0 0
\(323\) 10414.7 1.79409
\(324\) 0 0
\(325\) −15.7835 −0.00269389
\(326\) −1466.29 −0.249112
\(327\) 0 0
\(328\) −10006.6 −1.68451
\(329\) 0 0
\(330\) 0 0
\(331\) 1270.71 0.211011 0.105505 0.994419i \(-0.466354\pi\)
0.105505 + 0.994419i \(0.466354\pi\)
\(332\) −5444.76 −0.900061
\(333\) 0 0
\(334\) 1015.73 0.166402
\(335\) 6020.44 0.981886
\(336\) 0 0
\(337\) 4695.47 0.758986 0.379493 0.925195i \(-0.376098\pi\)
0.379493 + 0.925195i \(0.376098\pi\)
\(338\) 4303.97 0.692618
\(339\) 0 0
\(340\) −4858.43 −0.774957
\(341\) 4651.00 0.738609
\(342\) 0 0
\(343\) 0 0
\(344\) 12467.3 1.95404
\(345\) 0 0
\(346\) 2574.23 0.399976
\(347\) −6513.97 −1.00775 −0.503874 0.863777i \(-0.668092\pi\)
−0.503874 + 0.863777i \(0.668092\pi\)
\(348\) 0 0
\(349\) 7184.75 1.10198 0.550990 0.834512i \(-0.314250\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5983.23 0.905987
\(353\) −6068.55 −0.915004 −0.457502 0.889209i \(-0.651256\pi\)
−0.457502 + 0.889209i \(0.651256\pi\)
\(354\) 0 0
\(355\) 6226.64 0.930918
\(356\) 1449.74 0.215832
\(357\) 0 0
\(358\) 8100.34 1.19586
\(359\) −1743.71 −0.256350 −0.128175 0.991752i \(-0.540912\pi\)
−0.128175 + 0.991752i \(0.540912\pi\)
\(360\) 0 0
\(361\) 4377.18 0.638167
\(362\) −7278.96 −1.05683
\(363\) 0 0
\(364\) 0 0
\(365\) −7490.06 −1.07410
\(366\) 0 0
\(367\) −9000.12 −1.28012 −0.640058 0.768327i \(-0.721090\pi\)
−0.640058 + 0.768327i \(0.721090\pi\)
\(368\) 730.148 0.103428
\(369\) 0 0
\(370\) −7265.91 −1.02091
\(371\) 0 0
\(372\) 0 0
\(373\) 5271.75 0.731799 0.365899 0.930654i \(-0.380761\pi\)
0.365899 + 0.930654i \(0.380761\pi\)
\(374\) 7011.61 0.969417
\(375\) 0 0
\(376\) 6441.86 0.883547
\(377\) 221.575 0.0302698
\(378\) 0 0
\(379\) 10480.0 1.42037 0.710185 0.704015i \(-0.248611\pi\)
0.710185 + 0.704015i \(0.248611\pi\)
\(380\) −5241.63 −0.707606
\(381\) 0 0
\(382\) −6013.55 −0.805445
\(383\) −1374.03 −0.183315 −0.0916574 0.995791i \(-0.529216\pi\)
−0.0916574 + 0.995791i \(0.529216\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4095.17 −0.539996
\(387\) 0 0
\(388\) −673.739 −0.0881544
\(389\) −1898.27 −0.247419 −0.123710 0.992318i \(-0.539479\pi\)
−0.123710 + 0.992318i \(0.539479\pi\)
\(390\) 0 0
\(391\) −5341.26 −0.690842
\(392\) 0 0
\(393\) 0 0
\(394\) −7310.02 −0.934704
\(395\) 3799.32 0.483961
\(396\) 0 0
\(397\) 5572.30 0.704448 0.352224 0.935916i \(-0.385426\pi\)
0.352224 + 0.935916i \(0.385426\pi\)
\(398\) −4133.02 −0.520527
\(399\) 0 0
\(400\) −219.784 −0.0274730
\(401\) 5716.38 0.711877 0.355938 0.934509i \(-0.384161\pi\)
0.355938 + 0.934509i \(0.384161\pi\)
\(402\) 0 0
\(403\) 123.198 0.0152281
\(404\) 908.083 0.111829
\(405\) 0 0
\(406\) 0 0
\(407\) −11354.2 −1.38281
\(408\) 0 0
\(409\) −10348.5 −1.25111 −0.625553 0.780182i \(-0.715126\pi\)
−0.625553 + 0.780182i \(0.715126\pi\)
\(410\) 9784.93 1.17864
\(411\) 0 0
\(412\) −961.103 −0.114927
\(413\) 0 0
\(414\) 0 0
\(415\) 15565.4 1.84115
\(416\) 158.487 0.0186790
\(417\) 0 0
\(418\) 7564.65 0.885165
\(419\) −3251.80 −0.379142 −0.189571 0.981867i \(-0.560710\pi\)
−0.189571 + 0.981867i \(0.560710\pi\)
\(420\) 0 0
\(421\) 3708.63 0.429329 0.214664 0.976688i \(-0.431134\pi\)
0.214664 + 0.976688i \(0.431134\pi\)
\(422\) 3030.96 0.349632
\(423\) 0 0
\(424\) 5983.23 0.685310
\(425\) 1607.79 0.183504
\(426\) 0 0
\(427\) 0 0
\(428\) −7757.03 −0.876052
\(429\) 0 0
\(430\) −12191.1 −1.36723
\(431\) −4148.74 −0.463661 −0.231831 0.972756i \(-0.574471\pi\)
−0.231831 + 0.972756i \(0.574471\pi\)
\(432\) 0 0
\(433\) −1985.64 −0.220378 −0.110189 0.993911i \(-0.535146\pi\)
−0.110189 + 0.993911i \(0.535146\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2499.57 −0.274559
\(437\) −5762.55 −0.630802
\(438\) 0 0
\(439\) 6914.42 0.751725 0.375863 0.926675i \(-0.377346\pi\)
0.375863 + 0.926675i \(0.377346\pi\)
\(440\) −10316.8 −1.11781
\(441\) 0 0
\(442\) 185.727 0.0199867
\(443\) −4237.47 −0.454466 −0.227233 0.973840i \(-0.572968\pi\)
−0.227233 + 0.973840i \(0.572968\pi\)
\(444\) 0 0
\(445\) −4144.51 −0.441502
\(446\) −9525.65 −1.01133
\(447\) 0 0
\(448\) 0 0
\(449\) 2097.02 0.220411 0.110206 0.993909i \(-0.464849\pi\)
0.110206 + 0.993909i \(0.464849\pi\)
\(450\) 0 0
\(451\) 15290.5 1.59646
\(452\) −1976.97 −0.205728
\(453\) 0 0
\(454\) 1373.74 0.142011
\(455\) 0 0
\(456\) 0 0
\(457\) −18676.3 −1.91168 −0.955842 0.293880i \(-0.905053\pi\)
−0.955842 + 0.293880i \(0.905053\pi\)
\(458\) 7240.15 0.738668
\(459\) 0 0
\(460\) 2688.21 0.272475
\(461\) −13879.5 −1.40224 −0.701121 0.713043i \(-0.747317\pi\)
−0.701121 + 0.713043i \(0.747317\pi\)
\(462\) 0 0
\(463\) 4780.89 0.479885 0.239942 0.970787i \(-0.422871\pi\)
0.239942 + 0.970787i \(0.422871\pi\)
\(464\) 3085.41 0.308699
\(465\) 0 0
\(466\) 674.819 0.0670824
\(467\) 8376.19 0.829986 0.414993 0.909824i \(-0.363784\pi\)
0.414993 + 0.909824i \(0.363784\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −6299.17 −0.618211
\(471\) 0 0
\(472\) 9723.25 0.948197
\(473\) −19050.6 −1.85189
\(474\) 0 0
\(475\) 1734.60 0.167556
\(476\) 0 0
\(477\) 0 0
\(478\) 12041.7 1.15225
\(479\) −17011.4 −1.62269 −0.811346 0.584567i \(-0.801265\pi\)
−0.811346 + 0.584567i \(0.801265\pi\)
\(480\) 0 0
\(481\) −300.754 −0.0285098
\(482\) 11026.3 1.04198
\(483\) 0 0
\(484\) 21.1781 0.00198893
\(485\) 1926.07 0.180327
\(486\) 0 0
\(487\) 6017.65 0.559930 0.279965 0.960010i \(-0.409677\pi\)
0.279965 + 0.960010i \(0.409677\pi\)
\(488\) 20506.6 1.90224
\(489\) 0 0
\(490\) 0 0
\(491\) 12306.6 1.13113 0.565567 0.824702i \(-0.308657\pi\)
0.565567 + 0.824702i \(0.308657\pi\)
\(492\) 0 0
\(493\) −22570.8 −2.06194
\(494\) 200.376 0.0182497
\(495\) 0 0
\(496\) 1715.51 0.155300
\(497\) 0 0
\(498\) 0 0
\(499\) 8565.93 0.768464 0.384232 0.923237i \(-0.374466\pi\)
0.384232 + 0.923237i \(0.374466\pi\)
\(500\) 5371.94 0.480481
\(501\) 0 0
\(502\) −1020.02 −0.0906890
\(503\) −10520.4 −0.932568 −0.466284 0.884635i \(-0.654408\pi\)
−0.466284 + 0.884635i \(0.654408\pi\)
\(504\) 0 0
\(505\) −2596.02 −0.228755
\(506\) −3879.58 −0.340847
\(507\) 0 0
\(508\) 120.975 0.0105658
\(509\) 5445.41 0.474192 0.237096 0.971486i \(-0.423804\pi\)
0.237096 + 0.971486i \(0.423804\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 4767.35 0.411502
\(513\) 0 0
\(514\) 7858.02 0.674324
\(515\) 2747.59 0.235094
\(516\) 0 0
\(517\) −9843.46 −0.837360
\(518\) 0 0
\(519\) 0 0
\(520\) −273.277 −0.0230461
\(521\) 22217.9 1.86830 0.934149 0.356883i \(-0.116160\pi\)
0.934149 + 0.356883i \(0.116160\pi\)
\(522\) 0 0
\(523\) −22109.9 −1.84856 −0.924281 0.381712i \(-0.875335\pi\)
−0.924281 + 0.381712i \(0.875335\pi\)
\(524\) 8797.36 0.733425
\(525\) 0 0
\(526\) −7026.44 −0.582447
\(527\) −12549.5 −1.03732
\(528\) 0 0
\(529\) −9211.64 −0.757100
\(530\) −5850.70 −0.479506
\(531\) 0 0
\(532\) 0 0
\(533\) 405.022 0.0329146
\(534\) 0 0
\(535\) 22175.7 1.79204
\(536\) 12066.4 0.972371
\(537\) 0 0
\(538\) 10797.2 0.865241
\(539\) 0 0
\(540\) 0 0
\(541\) −8631.34 −0.685934 −0.342967 0.939347i \(-0.611432\pi\)
−0.342967 + 0.939347i \(0.611432\pi\)
\(542\) 11406.1 0.903939
\(543\) 0 0
\(544\) −16144.2 −1.27239
\(545\) 7145.74 0.561633
\(546\) 0 0
\(547\) −17846.7 −1.39501 −0.697505 0.716580i \(-0.745707\pi\)
−0.697505 + 0.716580i \(0.745707\pi\)
\(548\) 737.562 0.0574947
\(549\) 0 0
\(550\) 1167.80 0.0905370
\(551\) −24351.0 −1.88274
\(552\) 0 0
\(553\) 0 0
\(554\) 2254.71 0.172913
\(555\) 0 0
\(556\) −9853.39 −0.751577
\(557\) −15703.3 −1.19456 −0.597279 0.802033i \(-0.703752\pi\)
−0.597279 + 0.802033i \(0.703752\pi\)
\(558\) 0 0
\(559\) −504.620 −0.0381810
\(560\) 0 0
\(561\) 0 0
\(562\) −1755.30 −0.131749
\(563\) −673.679 −0.0504302 −0.0252151 0.999682i \(-0.508027\pi\)
−0.0252151 + 0.999682i \(0.508027\pi\)
\(564\) 0 0
\(565\) 5651.74 0.420833
\(566\) −239.984 −0.0178221
\(567\) 0 0
\(568\) 12479.7 0.921896
\(569\) 15186.2 1.11887 0.559436 0.828873i \(-0.311018\pi\)
0.559436 + 0.828873i \(0.311018\pi\)
\(570\) 0 0
\(571\) 2526.66 0.185180 0.0925898 0.995704i \(-0.470485\pi\)
0.0925898 + 0.995704i \(0.470485\pi\)
\(572\) −146.069 −0.0106774
\(573\) 0 0
\(574\) 0 0
\(575\) −889.603 −0.0645200
\(576\) 0 0
\(577\) 21680.4 1.56424 0.782119 0.623129i \(-0.214139\pi\)
0.782119 + 0.623129i \(0.214139\pi\)
\(578\) −9290.33 −0.668559
\(579\) 0 0
\(580\) 11359.6 0.813248
\(581\) 0 0
\(582\) 0 0
\(583\) −9142.66 −0.649486
\(584\) −15011.9 −1.06369
\(585\) 0 0
\(586\) −14898.3 −1.05025
\(587\) 17458.1 1.22755 0.613776 0.789481i \(-0.289650\pi\)
0.613776 + 0.789481i \(0.289650\pi\)
\(588\) 0 0
\(589\) −13539.4 −0.947165
\(590\) −9507.87 −0.663446
\(591\) 0 0
\(592\) −4187.97 −0.290751
\(593\) −5487.00 −0.379973 −0.189987 0.981787i \(-0.560844\pi\)
−0.189987 + 0.981787i \(0.560844\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6201.20 0.426193
\(597\) 0 0
\(598\) −102.764 −0.00702732
\(599\) 22997.3 1.56869 0.784343 0.620327i \(-0.213000\pi\)
0.784343 + 0.620327i \(0.213000\pi\)
\(600\) 0 0
\(601\) 18027.1 1.22353 0.611764 0.791040i \(-0.290460\pi\)
0.611764 + 0.791040i \(0.290460\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1707.10 −0.115002
\(605\) −60.5437 −0.00406852
\(606\) 0 0
\(607\) −10285.1 −0.687744 −0.343872 0.939017i \(-0.611739\pi\)
−0.343872 + 0.939017i \(0.611739\pi\)
\(608\) −17417.6 −1.16181
\(609\) 0 0
\(610\) −20052.4 −1.33098
\(611\) −260.738 −0.0172641
\(612\) 0 0
\(613\) −1596.93 −0.105219 −0.0526096 0.998615i \(-0.516754\pi\)
−0.0526096 + 0.998615i \(0.516754\pi\)
\(614\) 6936.00 0.455886
\(615\) 0 0
\(616\) 0 0
\(617\) −5850.74 −0.381753 −0.190877 0.981614i \(-0.561133\pi\)
−0.190877 + 0.981614i \(0.561133\pi\)
\(618\) 0 0
\(619\) −1721.62 −0.111789 −0.0558947 0.998437i \(-0.517801\pi\)
−0.0558947 + 0.998437i \(0.517801\pi\)
\(620\) 6316.06 0.409128
\(621\) 0 0
\(622\) −7633.48 −0.492082
\(623\) 0 0
\(624\) 0 0
\(625\) −17402.7 −1.11377
\(626\) −17016.5 −1.08645
\(627\) 0 0
\(628\) −1886.10 −0.119847
\(629\) 30636.3 1.94205
\(630\) 0 0
\(631\) −19533.5 −1.23235 −0.616177 0.787608i \(-0.711319\pi\)
−0.616177 + 0.787608i \(0.711319\pi\)
\(632\) 7614.77 0.479271
\(633\) 0 0
\(634\) 16649.9 1.04299
\(635\) −345.842 −0.0216131
\(636\) 0 0
\(637\) 0 0
\(638\) −16394.1 −1.01732
\(639\) 0 0
\(640\) 5621.52 0.347203
\(641\) 23496.2 1.44781 0.723903 0.689902i \(-0.242346\pi\)
0.723903 + 0.689902i \(0.242346\pi\)
\(642\) 0 0
\(643\) −1537.40 −0.0942913 −0.0471456 0.998888i \(-0.515012\pi\)
−0.0471456 + 0.998888i \(0.515012\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −20411.3 −1.24315
\(647\) −28273.0 −1.71797 −0.858985 0.512000i \(-0.828905\pi\)
−0.858985 + 0.512000i \(0.828905\pi\)
\(648\) 0 0
\(649\) −14857.6 −0.898630
\(650\) 30.9333 0.00186662
\(651\) 0 0
\(652\) −3111.62 −0.186902
\(653\) 2668.97 0.159946 0.0799730 0.996797i \(-0.474517\pi\)
0.0799730 + 0.996797i \(0.474517\pi\)
\(654\) 0 0
\(655\) −25149.8 −1.50028
\(656\) 5639.89 0.335672
\(657\) 0 0
\(658\) 0 0
\(659\) 6343.74 0.374988 0.187494 0.982266i \(-0.439964\pi\)
0.187494 + 0.982266i \(0.439964\pi\)
\(660\) 0 0
\(661\) −8409.15 −0.494823 −0.247412 0.968910i \(-0.579580\pi\)
−0.247412 + 0.968910i \(0.579580\pi\)
\(662\) −2490.40 −0.146212
\(663\) 0 0
\(664\) 31196.9 1.82331
\(665\) 0 0
\(666\) 0 0
\(667\) 12488.6 0.724977
\(668\) 2155.48 0.124847
\(669\) 0 0
\(670\) −11799.2 −0.680360
\(671\) −31335.1 −1.80280
\(672\) 0 0
\(673\) −15326.7 −0.877862 −0.438931 0.898521i \(-0.644643\pi\)
−0.438931 + 0.898521i \(0.644643\pi\)
\(674\) −9202.41 −0.525910
\(675\) 0 0
\(676\) 9133.43 0.519653
\(677\) 13011.6 0.738666 0.369333 0.929297i \(-0.379586\pi\)
0.369333 + 0.929297i \(0.379586\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 27837.4 1.56987
\(681\) 0 0
\(682\) −9115.25 −0.511790
\(683\) 17472.0 0.978840 0.489420 0.872048i \(-0.337209\pi\)
0.489420 + 0.872048i \(0.337209\pi\)
\(684\) 0 0
\(685\) −2108.53 −0.117610
\(686\) 0 0
\(687\) 0 0
\(688\) −7026.78 −0.389380
\(689\) −242.175 −0.0133906
\(690\) 0 0
\(691\) −7317.10 −0.402830 −0.201415 0.979506i \(-0.564554\pi\)
−0.201415 + 0.979506i \(0.564554\pi\)
\(692\) 5462.77 0.300091
\(693\) 0 0
\(694\) 12766.4 0.698280
\(695\) 28168.7 1.53741
\(696\) 0 0
\(697\) −41257.6 −2.24210
\(698\) −14081.0 −0.763575
\(699\) 0 0
\(700\) 0 0
\(701\) −7874.65 −0.424282 −0.212141 0.977239i \(-0.568044\pi\)
−0.212141 + 0.977239i \(0.568044\pi\)
\(702\) 0 0
\(703\) 33052.7 1.77327
\(704\) −15638.7 −0.837225
\(705\) 0 0
\(706\) 11893.4 0.634017
\(707\) 0 0
\(708\) 0 0
\(709\) −25236.8 −1.33680 −0.668398 0.743804i \(-0.733020\pi\)
−0.668398 + 0.743804i \(0.733020\pi\)
\(710\) −12203.3 −0.645044
\(711\) 0 0
\(712\) −8306.61 −0.437224
\(713\) 6943.76 0.364721
\(714\) 0 0
\(715\) 417.580 0.0218414
\(716\) 17189.7 0.897220
\(717\) 0 0
\(718\) 3417.42 0.177628
\(719\) −19673.0 −1.02041 −0.510207 0.860052i \(-0.670431\pi\)
−0.510207 + 0.860052i \(0.670431\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −8578.62 −0.442193
\(723\) 0 0
\(724\) −15446.6 −0.792914
\(725\) −3759.22 −0.192571
\(726\) 0 0
\(727\) −31921.7 −1.62849 −0.814243 0.580525i \(-0.802848\pi\)
−0.814243 + 0.580525i \(0.802848\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14679.4 0.744258
\(731\) 51403.1 2.60084
\(732\) 0 0
\(733\) 13545.3 0.682547 0.341274 0.939964i \(-0.389142\pi\)
0.341274 + 0.939964i \(0.389142\pi\)
\(734\) 17638.9 0.887007
\(735\) 0 0
\(736\) 8932.74 0.447371
\(737\) −18438.1 −0.921541
\(738\) 0 0
\(739\) 25497.6 1.26921 0.634604 0.772837i \(-0.281163\pi\)
0.634604 + 0.772837i \(0.281163\pi\)
\(740\) −15419.0 −0.765963
\(741\) 0 0
\(742\) 0 0
\(743\) 11146.9 0.550390 0.275195 0.961388i \(-0.411258\pi\)
0.275195 + 0.961388i \(0.411258\pi\)
\(744\) 0 0
\(745\) −17727.9 −0.871812
\(746\) −10331.8 −0.507072
\(747\) 0 0
\(748\) 14879.3 0.727328
\(749\) 0 0
\(750\) 0 0
\(751\) 21412.5 1.04042 0.520208 0.854040i \(-0.325854\pi\)
0.520208 + 0.854040i \(0.325854\pi\)
\(752\) −3630.75 −0.176064
\(753\) 0 0
\(754\) −434.254 −0.0209743
\(755\) 4880.24 0.235245
\(756\) 0 0
\(757\) −33963.9 −1.63070 −0.815350 0.578968i \(-0.803456\pi\)
−0.815350 + 0.578968i \(0.803456\pi\)
\(758\) −20539.2 −0.984191
\(759\) 0 0
\(760\) 30033.0 1.43344
\(761\) 19920.0 0.948880 0.474440 0.880288i \(-0.342651\pi\)
0.474440 + 0.880288i \(0.342651\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12761.3 −0.604305
\(765\) 0 0
\(766\) 2692.89 0.127021
\(767\) −393.554 −0.0185273
\(768\) 0 0
\(769\) 10039.2 0.470769 0.235385 0.971902i \(-0.424365\pi\)
0.235385 + 0.971902i \(0.424365\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8690.33 −0.405145
\(773\) 21385.3 0.995054 0.497527 0.867449i \(-0.334242\pi\)
0.497527 + 0.867449i \(0.334242\pi\)
\(774\) 0 0
\(775\) −2090.16 −0.0968785
\(776\) 3860.33 0.178579
\(777\) 0 0
\(778\) 3720.32 0.171440
\(779\) −44511.8 −2.04724
\(780\) 0 0
\(781\) −19069.6 −0.873704
\(782\) 10468.1 0.478693
\(783\) 0 0
\(784\) 0 0
\(785\) 5391.96 0.245156
\(786\) 0 0
\(787\) 37421.3 1.69495 0.847475 0.530835i \(-0.178122\pi\)
0.847475 + 0.530835i \(0.178122\pi\)
\(788\) −15512.6 −0.701284
\(789\) 0 0
\(790\) −7446.10 −0.335342
\(791\) 0 0
\(792\) 0 0
\(793\) −830.018 −0.0371687
\(794\) −10920.9 −0.488120
\(795\) 0 0
\(796\) −8770.67 −0.390538
\(797\) −30888.7 −1.37281 −0.686407 0.727217i \(-0.740813\pi\)
−0.686407 + 0.727217i \(0.740813\pi\)
\(798\) 0 0
\(799\) 26560.1 1.17601
\(800\) −2688.87 −0.118832
\(801\) 0 0
\(802\) −11203.3 −0.493268
\(803\) 22938.9 1.00809
\(804\) 0 0
\(805\) 0 0
\(806\) −241.449 −0.0105517
\(807\) 0 0
\(808\) −5203.05 −0.226538
\(809\) −7238.50 −0.314576 −0.157288 0.987553i \(-0.550275\pi\)
−0.157288 + 0.987553i \(0.550275\pi\)
\(810\) 0 0
\(811\) −35466.4 −1.53563 −0.767814 0.640673i \(-0.778656\pi\)
−0.767814 + 0.640673i \(0.778656\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 22252.4 0.958166
\(815\) 8895.44 0.382324
\(816\) 0 0
\(817\) 55457.5 2.37480
\(818\) 20281.6 0.866905
\(819\) 0 0
\(820\) 20764.6 0.884304
\(821\) 37947.3 1.61312 0.806558 0.591154i \(-0.201328\pi\)
0.806558 + 0.591154i \(0.201328\pi\)
\(822\) 0 0
\(823\) 26932.2 1.14070 0.570351 0.821401i \(-0.306807\pi\)
0.570351 + 0.821401i \(0.306807\pi\)
\(824\) 5506.84 0.232815
\(825\) 0 0
\(826\) 0 0
\(827\) 1913.98 0.0804783 0.0402391 0.999190i \(-0.487188\pi\)
0.0402391 + 0.999190i \(0.487188\pi\)
\(828\) 0 0
\(829\) −15281.9 −0.640245 −0.320122 0.947376i \(-0.603724\pi\)
−0.320122 + 0.947376i \(0.603724\pi\)
\(830\) −30505.9 −1.27575
\(831\) 0 0
\(832\) −414.246 −0.0172613
\(833\) 0 0
\(834\) 0 0
\(835\) −6162.04 −0.255385
\(836\) 16052.9 0.664117
\(837\) 0 0
\(838\) 6373.03 0.262712
\(839\) −25779.5 −1.06079 −0.530397 0.847749i \(-0.677957\pi\)
−0.530397 + 0.847749i \(0.677957\pi\)
\(840\) 0 0
\(841\) 28384.4 1.16382
\(842\) −7268.35 −0.297487
\(843\) 0 0
\(844\) 6431.98 0.262320
\(845\) −26110.5 −1.06299
\(846\) 0 0
\(847\) 0 0
\(848\) −3372.26 −0.136561
\(849\) 0 0
\(850\) −3151.03 −0.127152
\(851\) −16951.3 −0.682825
\(852\) 0 0
\(853\) −6452.21 −0.258991 −0.129496 0.991580i \(-0.541336\pi\)
−0.129496 + 0.991580i \(0.541336\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 44445.5 1.77467
\(857\) −32061.8 −1.27796 −0.638980 0.769224i \(-0.720643\pi\)
−0.638980 + 0.769224i \(0.720643\pi\)
\(858\) 0 0
\(859\) −3934.64 −0.156284 −0.0781421 0.996942i \(-0.524899\pi\)
−0.0781421 + 0.996942i \(0.524899\pi\)
\(860\) −25870.7 −1.02579
\(861\) 0 0
\(862\) 8130.91 0.321276
\(863\) 14043.1 0.553921 0.276961 0.960881i \(-0.410673\pi\)
0.276961 + 0.960881i \(0.410673\pi\)
\(864\) 0 0
\(865\) −15616.9 −0.613861
\(866\) 3891.56 0.152703
\(867\) 0 0
\(868\) 0 0
\(869\) −11635.7 −0.454217
\(870\) 0 0
\(871\) −488.396 −0.0189996
\(872\) 14321.8 0.556190
\(873\) 0 0
\(874\) 11293.7 0.437090
\(875\) 0 0
\(876\) 0 0
\(877\) 2091.31 0.0805228 0.0402614 0.999189i \(-0.487181\pi\)
0.0402614 + 0.999189i \(0.487181\pi\)
\(878\) −13551.2 −0.520879
\(879\) 0 0
\(880\) 5814.75 0.222745
\(881\) −11548.6 −0.441639 −0.220819 0.975315i \(-0.570873\pi\)
−0.220819 + 0.975315i \(0.570873\pi\)
\(882\) 0 0
\(883\) 7531.24 0.287029 0.143514 0.989648i \(-0.454160\pi\)
0.143514 + 0.989648i \(0.454160\pi\)
\(884\) 394.130 0.0149955
\(885\) 0 0
\(886\) 8304.80 0.314904
\(887\) 38502.7 1.45749 0.728745 0.684785i \(-0.240104\pi\)
0.728745 + 0.684785i \(0.240104\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8122.61 0.305922
\(891\) 0 0
\(892\) −20214.3 −0.758774
\(893\) 28655.0 1.07380
\(894\) 0 0
\(895\) −49141.7 −1.83534
\(896\) 0 0
\(897\) 0 0
\(898\) −4109.85 −0.152725
\(899\) 29342.5 1.08857
\(900\) 0 0
\(901\) 24669.2 0.912152
\(902\) −29967.1 −1.10620
\(903\) 0 0
\(904\) 11327.5 0.416755
\(905\) 44158.7 1.62197
\(906\) 0 0
\(907\) −31306.7 −1.14611 −0.573054 0.819517i \(-0.694242\pi\)
−0.573054 + 0.819517i \(0.694242\pi\)
\(908\) 2915.21 0.106547
\(909\) 0 0
\(910\) 0 0
\(911\) −34154.3 −1.24213 −0.621067 0.783758i \(-0.713300\pi\)
−0.621067 + 0.783758i \(0.713300\pi\)
\(912\) 0 0
\(913\) −47670.3 −1.72799
\(914\) 36602.7 1.32463
\(915\) 0 0
\(916\) 15364.3 0.554204
\(917\) 0 0
\(918\) 0 0
\(919\) 35988.9 1.29180 0.645899 0.763423i \(-0.276483\pi\)
0.645899 + 0.763423i \(0.276483\pi\)
\(920\) −15402.6 −0.551967
\(921\) 0 0
\(922\) 27201.8 0.971629
\(923\) −505.124 −0.0180134
\(924\) 0 0
\(925\) 5102.57 0.181374
\(926\) −9369.82 −0.332518
\(927\) 0 0
\(928\) 37747.4 1.33526
\(929\) −13906.0 −0.491109 −0.245554 0.969383i \(-0.578970\pi\)
−0.245554 + 0.969383i \(0.578970\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1432.03 0.0503302
\(933\) 0 0
\(934\) −16416.1 −0.575107
\(935\) −42536.8 −1.48781
\(936\) 0 0
\(937\) −40905.3 −1.42617 −0.713083 0.701079i \(-0.752702\pi\)
−0.713083 + 0.701079i \(0.752702\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −13367.4 −0.463828
\(941\) 8067.50 0.279482 0.139741 0.990188i \(-0.455373\pi\)
0.139741 + 0.990188i \(0.455373\pi\)
\(942\) 0 0
\(943\) 22828.2 0.788321
\(944\) −5480.20 −0.188946
\(945\) 0 0
\(946\) 37336.2 1.28320
\(947\) −39781.2 −1.36506 −0.682531 0.730856i \(-0.739121\pi\)
−0.682531 + 0.730856i \(0.739121\pi\)
\(948\) 0 0
\(949\) 607.616 0.0207840
\(950\) −3399.56 −0.116101
\(951\) 0 0
\(952\) 0 0
\(953\) −32354.0 −1.09974 −0.549868 0.835252i \(-0.685322\pi\)
−0.549868 + 0.835252i \(0.685322\pi\)
\(954\) 0 0
\(955\) 36481.9 1.23615
\(956\) 25553.7 0.864504
\(957\) 0 0
\(958\) 33339.7 1.12438
\(959\) 0 0
\(960\) 0 0
\(961\) −13476.3 −0.452362
\(962\) 589.433 0.0197548
\(963\) 0 0
\(964\) 23398.9 0.781773
\(965\) 24843.8 0.828757
\(966\) 0 0
\(967\) 17389.2 0.578282 0.289141 0.957287i \(-0.406630\pi\)
0.289141 + 0.957287i \(0.406630\pi\)
\(968\) −121.344 −0.00402909
\(969\) 0 0
\(970\) −3774.82 −0.124951
\(971\) −52406.2 −1.73202 −0.866012 0.500023i \(-0.833325\pi\)
−0.866012 + 0.500023i \(0.833325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −11793.7 −0.387982
\(975\) 0 0
\(976\) −11557.9 −0.379057
\(977\) −6846.70 −0.224202 −0.112101 0.993697i \(-0.535758\pi\)
−0.112101 + 0.993697i \(0.535758\pi\)
\(978\) 0 0
\(979\) 12692.9 0.414368
\(980\) 0 0
\(981\) 0 0
\(982\) −24119.0 −0.783776
\(983\) −37647.9 −1.22155 −0.610774 0.791805i \(-0.709142\pi\)
−0.610774 + 0.791805i \(0.709142\pi\)
\(984\) 0 0
\(985\) 44347.1 1.43453
\(986\) 44235.3 1.42874
\(987\) 0 0
\(988\) 425.217 0.0136923
\(989\) −28441.8 −0.914454
\(990\) 0 0
\(991\) 2036.17 0.0652686 0.0326343 0.999467i \(-0.489610\pi\)
0.0326343 + 0.999467i \(0.489610\pi\)
\(992\) 20987.9 0.671740
\(993\) 0 0
\(994\) 0 0
\(995\) 25073.5 0.798877
\(996\) 0 0
\(997\) −13540.6 −0.430127 −0.215064 0.976600i \(-0.568996\pi\)
−0.215064 + 0.976600i \(0.568996\pi\)
\(998\) −16787.9 −0.532478
\(999\) 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.x.1.4 yes 8
3.2 odd 2 inner 441.4.a.x.1.5 yes 8
7.2 even 3 441.4.e.z.361.5 16
7.3 odd 6 441.4.e.z.226.6 16
7.4 even 3 441.4.e.z.226.5 16
7.5 odd 6 441.4.e.z.361.6 16
7.6 odd 2 inner 441.4.a.x.1.3 8
21.2 odd 6 441.4.e.z.361.4 16
21.5 even 6 441.4.e.z.361.3 16
21.11 odd 6 441.4.e.z.226.4 16
21.17 even 6 441.4.e.z.226.3 16
21.20 even 2 inner 441.4.a.x.1.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.4.a.x.1.3 8 7.6 odd 2 inner
441.4.a.x.1.4 yes 8 1.1 even 1 trivial
441.4.a.x.1.5 yes 8 3.2 odd 2 inner
441.4.a.x.1.6 yes 8 21.20 even 2 inner
441.4.e.z.226.3 16 21.17 even 6
441.4.e.z.226.4 16 21.11 odd 6
441.4.e.z.226.5 16 7.4 even 3
441.4.e.z.226.6 16 7.3 odd 6
441.4.e.z.361.3 16 21.5 even 6
441.4.e.z.361.4 16 21.2 odd 6
441.4.e.z.361.5 16 7.2 even 3
441.4.e.z.361.6 16 7.5 odd 6