Properties

Label 5733.2.a.bd.1.1
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.7782354788\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.86620 q^{2} +1.48270 q^{4} +0.866198 q^{5} +0.965392 q^{8} +O(q^{10})\) \(q-1.86620 q^{2} +1.48270 q^{4} +0.866198 q^{5} +0.965392 q^{8} -1.61650 q^{10} +3.86620 q^{11} -1.00000 q^{13} -4.76700 q^{16} -3.34889 q^{17} +5.38350 q^{19} +1.28431 q^{20} -7.21509 q^{22} +5.24970 q^{23} -4.24970 q^{25} +1.86620 q^{26} -1.69779 q^{29} -7.56399 q^{31} +6.96539 q^{32} +6.24970 q^{34} -4.83159 q^{37} -10.0467 q^{38} +0.836221 q^{40} -4.06922 q^{41} +4.03461 q^{43} +5.73240 q^{44} -9.79698 q^{46} -3.65111 q^{47} +7.93078 q^{50} -1.48270 q^{52} +0.215092 q^{53} +3.34889 q^{55} +3.16841 q^{58} -2.78491 q^{59} -9.03461 q^{61} +14.1159 q^{62} -3.46479 q^{64} -0.866198 q^{65} -7.66318 q^{67} -4.96539 q^{68} -4.90081 q^{71} -15.5461 q^{73} +9.01671 q^{74} +7.98210 q^{76} +9.43018 q^{79} -4.12917 q^{80} +7.59396 q^{82} -4.09919 q^{83} -2.90081 q^{85} -7.52938 q^{86} +3.73240 q^{88} -0.418110 q^{89} +7.78371 q^{92} +6.81369 q^{94} +4.66318 q^{95} +7.11590 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 6 q^{8} + O(q^{10}) \) \( 3 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 6 q^{8} - 14 q^{10} + 4 q^{11} - 3 q^{13} + 4 q^{16} - 4 q^{17} + 7 q^{19} - 16 q^{20} - 8 q^{22} - q^{23} + 4 q^{25} - 2 q^{26} + 7 q^{29} - 3 q^{31} + 24 q^{32} + 2 q^{34} - 10 q^{37} - 12 q^{38} - 22 q^{40} - 6 q^{41} + 9 q^{43} + 2 q^{44} - 28 q^{46} - 17 q^{47} + 30 q^{50} - 6 q^{52} - 13 q^{53} + 4 q^{55} + 14 q^{58} - 22 q^{59} - 24 q^{61} + 18 q^{62} + 20 q^{64} + 5 q^{65} - 14 q^{67} - 18 q^{68} - 4 q^{71} + 5 q^{73} - 8 q^{74} - 8 q^{76} + q^{79} - 40 q^{80} + 20 q^{82} - 23 q^{83} + 2 q^{85} - 6 q^{86} - 4 q^{88} + 11 q^{89} - 30 q^{92} - 16 q^{94} + 5 q^{95} - 3 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.86620 −1.31960 −0.659801 0.751441i \(-0.729359\pi\)
−0.659801 + 0.751441i \(0.729359\pi\)
\(3\) 0 0
\(4\) 1.48270 0.741348
\(5\) 0.866198 0.387376 0.193688 0.981063i \(-0.437955\pi\)
0.193688 + 0.981063i \(0.437955\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0.965392 0.341318
\(9\) 0 0
\(10\) −1.61650 −0.511181
\(11\) 3.86620 1.16570 0.582851 0.812579i \(-0.301937\pi\)
0.582851 + 0.812579i \(0.301937\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) −4.76700 −1.19175
\(17\) −3.34889 −0.812226 −0.406113 0.913823i \(-0.633116\pi\)
−0.406113 + 0.913823i \(0.633116\pi\)
\(18\) 0 0
\(19\) 5.38350 1.23506 0.617530 0.786547i \(-0.288133\pi\)
0.617530 + 0.786547i \(0.288133\pi\)
\(20\) 1.28431 0.287180
\(21\) 0 0
\(22\) −7.21509 −1.53826
\(23\) 5.24970 1.09464 0.547319 0.836924i \(-0.315648\pi\)
0.547319 + 0.836924i \(0.315648\pi\)
\(24\) 0 0
\(25\) −4.24970 −0.849940
\(26\) 1.86620 0.365992
\(27\) 0 0
\(28\) 0 0
\(29\) −1.69779 −0.315271 −0.157636 0.987497i \(-0.550387\pi\)
−0.157636 + 0.987497i \(0.550387\pi\)
\(30\) 0 0
\(31\) −7.56399 −1.35853 −0.679266 0.733892i \(-0.737702\pi\)
−0.679266 + 0.733892i \(0.737702\pi\)
\(32\) 6.96539 1.23132
\(33\) 0 0
\(34\) 6.24970 1.07181
\(35\) 0 0
\(36\) 0 0
\(37\) −4.83159 −0.794309 −0.397154 0.917752i \(-0.630002\pi\)
−0.397154 + 0.917752i \(0.630002\pi\)
\(38\) −10.0467 −1.62979
\(39\) 0 0
\(40\) 0.836221 0.132218
\(41\) −4.06922 −0.635505 −0.317752 0.948174i \(-0.602928\pi\)
−0.317752 + 0.948174i \(0.602928\pi\)
\(42\) 0 0
\(43\) 4.03461 0.615272 0.307636 0.951504i \(-0.400462\pi\)
0.307636 + 0.951504i \(0.400462\pi\)
\(44\) 5.73240 0.864191
\(45\) 0 0
\(46\) −9.79698 −1.44449
\(47\) −3.65111 −0.532569 −0.266284 0.963895i \(-0.585796\pi\)
−0.266284 + 0.963895i \(0.585796\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 7.93078 1.12158
\(51\) 0 0
\(52\) −1.48270 −0.205613
\(53\) 0.215092 0.0295452 0.0147726 0.999891i \(-0.495298\pi\)
0.0147726 + 0.999891i \(0.495298\pi\)
\(54\) 0 0
\(55\) 3.34889 0.451565
\(56\) 0 0
\(57\) 0 0
\(58\) 3.16841 0.416033
\(59\) −2.78491 −0.362564 −0.181282 0.983431i \(-0.558025\pi\)
−0.181282 + 0.983431i \(0.558025\pi\)
\(60\) 0 0
\(61\) −9.03461 −1.15676 −0.578382 0.815766i \(-0.696315\pi\)
−0.578382 + 0.815766i \(0.696315\pi\)
\(62\) 14.1159 1.79272
\(63\) 0 0
\(64\) −3.46479 −0.433099
\(65\) −0.866198 −0.107439
\(66\) 0 0
\(67\) −7.66318 −0.936206 −0.468103 0.883674i \(-0.655062\pi\)
−0.468103 + 0.883674i \(0.655062\pi\)
\(68\) −4.96539 −0.602142
\(69\) 0 0
\(70\) 0 0
\(71\) −4.90081 −0.581619 −0.290809 0.956781i \(-0.593925\pi\)
−0.290809 + 0.956781i \(0.593925\pi\)
\(72\) 0 0
\(73\) −15.5461 −1.81953 −0.909766 0.415122i \(-0.863739\pi\)
−0.909766 + 0.415122i \(0.863739\pi\)
\(74\) 9.01671 1.04817
\(75\) 0 0
\(76\) 7.98210 0.915609
\(77\) 0 0
\(78\) 0 0
\(79\) 9.43018 1.06098 0.530489 0.847692i \(-0.322008\pi\)
0.530489 + 0.847692i \(0.322008\pi\)
\(80\) −4.12917 −0.461655
\(81\) 0 0
\(82\) 7.59396 0.838613
\(83\) −4.09919 −0.449945 −0.224972 0.974365i \(-0.572229\pi\)
−0.224972 + 0.974365i \(0.572229\pi\)
\(84\) 0 0
\(85\) −2.90081 −0.314637
\(86\) −7.52938 −0.811914
\(87\) 0 0
\(88\) 3.73240 0.397875
\(89\) −0.418110 −0.0443196 −0.0221598 0.999754i \(-0.507054\pi\)
−0.0221598 + 0.999754i \(0.507054\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.78371 0.811508
\(93\) 0 0
\(94\) 6.81369 0.702778
\(95\) 4.66318 0.478432
\(96\) 0 0
\(97\) 7.11590 0.722510 0.361255 0.932467i \(-0.382348\pi\)
0.361255 + 0.932467i \(0.382348\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −6.30101 −0.630101
\(101\) −14.1159 −1.40458 −0.702292 0.711889i \(-0.747840\pi\)
−0.702292 + 0.711889i \(0.747840\pi\)
\(102\) 0 0
\(103\) 16.8604 1.66130 0.830651 0.556794i \(-0.187969\pi\)
0.830651 + 0.556794i \(0.187969\pi\)
\(104\) −0.965392 −0.0946645
\(105\) 0 0
\(106\) −0.401405 −0.0389879
\(107\) −10.1805 −0.984185 −0.492092 0.870543i \(-0.663768\pi\)
−0.492092 + 0.870543i \(0.663768\pi\)
\(108\) 0 0
\(109\) −6.20302 −0.594141 −0.297071 0.954855i \(-0.596010\pi\)
−0.297071 + 0.954855i \(0.596010\pi\)
\(110\) −6.24970 −0.595886
\(111\) 0 0
\(112\) 0 0
\(113\) −10.2843 −0.967466 −0.483733 0.875216i \(-0.660719\pi\)
−0.483733 + 0.875216i \(0.660719\pi\)
\(114\) 0 0
\(115\) 4.54728 0.424036
\(116\) −2.51730 −0.233726
\(117\) 0 0
\(118\) 5.19719 0.478440
\(119\) 0 0
\(120\) 0 0
\(121\) 3.94749 0.358863
\(122\) 16.8604 1.52647
\(123\) 0 0
\(124\) −11.2151 −1.00715
\(125\) −8.01207 −0.716622
\(126\) 0 0
\(127\) −1.91288 −0.169741 −0.0848704 0.996392i \(-0.527048\pi\)
−0.0848704 + 0.996392i \(0.527048\pi\)
\(128\) −7.46479 −0.659801
\(129\) 0 0
\(130\) 1.61650 0.141776
\(131\) 10.1626 0.887909 0.443954 0.896049i \(-0.353575\pi\)
0.443954 + 0.896049i \(0.353575\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 14.3010 1.23542
\(135\) 0 0
\(136\) −3.23300 −0.277227
\(137\) −7.79698 −0.666141 −0.333071 0.942902i \(-0.608085\pi\)
−0.333071 + 0.942902i \(0.608085\pi\)
\(138\) 0 0
\(139\) 5.08129 0.430989 0.215495 0.976505i \(-0.430864\pi\)
0.215495 + 0.976505i \(0.430864\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.14588 0.767505
\(143\) −3.86620 −0.323308
\(144\) 0 0
\(145\) −1.47062 −0.122128
\(146\) 29.0121 2.40106
\(147\) 0 0
\(148\) −7.16378 −0.588859
\(149\) 2.49477 0.204380 0.102190 0.994765i \(-0.467415\pi\)
0.102190 + 0.994765i \(0.467415\pi\)
\(150\) 0 0
\(151\) −3.26178 −0.265439 −0.132720 0.991154i \(-0.542371\pi\)
−0.132720 + 0.991154i \(0.542371\pi\)
\(152\) 5.19719 0.421548
\(153\) 0 0
\(154\) 0 0
\(155\) −6.55191 −0.526262
\(156\) 0 0
\(157\) 0.720322 0.0574880 0.0287440 0.999587i \(-0.490849\pi\)
0.0287440 + 0.999587i \(0.490849\pi\)
\(158\) −17.5986 −1.40007
\(159\) 0 0
\(160\) 6.03341 0.476983
\(161\) 0 0
\(162\) 0 0
\(163\) 1.30221 0.101997 0.0509985 0.998699i \(-0.483760\pi\)
0.0509985 + 0.998699i \(0.483760\pi\)
\(164\) −6.03341 −0.471130
\(165\) 0 0
\(166\) 7.64991 0.593748
\(167\) 16.1505 1.24976 0.624882 0.780719i \(-0.285147\pi\)
0.624882 + 0.780719i \(0.285147\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 5.41348 0.415195
\(171\) 0 0
\(172\) 5.98210 0.456131
\(173\) −15.8483 −1.20492 −0.602462 0.798148i \(-0.705813\pi\)
−0.602462 + 0.798148i \(0.705813\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −18.4302 −1.38923
\(177\) 0 0
\(178\) 0.780277 0.0584842
\(179\) 20.4648 1.52961 0.764805 0.644262i \(-0.222835\pi\)
0.764805 + 0.644262i \(0.222835\pi\)
\(180\) 0 0
\(181\) 6.58189 0.489228 0.244614 0.969621i \(-0.421339\pi\)
0.244614 + 0.969621i \(0.421339\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.06802 0.373619
\(185\) −4.18512 −0.307696
\(186\) 0 0
\(187\) −12.9475 −0.946814
\(188\) −5.41348 −0.394819
\(189\) 0 0
\(190\) −8.70242 −0.631340
\(191\) −12.7491 −0.922493 −0.461246 0.887272i \(-0.652598\pi\)
−0.461246 + 0.887272i \(0.652598\pi\)
\(192\) 0 0
\(193\) 2.26760 0.163226 0.0816128 0.996664i \(-0.473993\pi\)
0.0816128 + 0.996664i \(0.473993\pi\)
\(194\) −13.2797 −0.953425
\(195\) 0 0
\(196\) 0 0
\(197\) 18.6978 1.33216 0.666081 0.745879i \(-0.267970\pi\)
0.666081 + 0.745879i \(0.267970\pi\)
\(198\) 0 0
\(199\) −19.9175 −1.41191 −0.705957 0.708254i \(-0.749483\pi\)
−0.705957 + 0.708254i \(0.749483\pi\)
\(200\) −4.10263 −0.290100
\(201\) 0 0
\(202\) 26.3431 1.85349
\(203\) 0 0
\(204\) 0 0
\(205\) −3.52475 −0.246179
\(206\) −31.4648 −2.19226
\(207\) 0 0
\(208\) 4.76700 0.330532
\(209\) 20.8137 1.43971
\(210\) 0 0
\(211\) 0.645277 0.0444227 0.0222114 0.999753i \(-0.492929\pi\)
0.0222114 + 0.999753i \(0.492929\pi\)
\(212\) 0.318917 0.0219033
\(213\) 0 0
\(214\) 18.9988 1.29873
\(215\) 3.49477 0.238341
\(216\) 0 0
\(217\) 0 0
\(218\) 11.5761 0.784030
\(219\) 0 0
\(220\) 4.96539 0.334767
\(221\) 3.34889 0.225271
\(222\) 0 0
\(223\) −5.83159 −0.390512 −0.195256 0.980752i \(-0.562554\pi\)
−0.195256 + 0.980752i \(0.562554\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 19.1926 1.27667
\(227\) −15.7324 −1.04420 −0.522098 0.852886i \(-0.674850\pi\)
−0.522098 + 0.852886i \(0.674850\pi\)
\(228\) 0 0
\(229\) 8.87827 0.586693 0.293346 0.956006i \(-0.405231\pi\)
0.293346 + 0.956006i \(0.405231\pi\)
\(230\) −8.48613 −0.559559
\(231\) 0 0
\(232\) −1.63903 −0.107608
\(233\) 11.7912 0.772464 0.386232 0.922402i \(-0.373776\pi\)
0.386232 + 0.922402i \(0.373776\pi\)
\(234\) 0 0
\(235\) −3.16258 −0.206304
\(236\) −4.12917 −0.268786
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0692 1.03943 0.519716 0.854339i \(-0.326038\pi\)
0.519716 + 0.854339i \(0.326038\pi\)
\(240\) 0 0
\(241\) −4.93541 −0.317918 −0.158959 0.987285i \(-0.550814\pi\)
−0.158959 + 0.987285i \(0.550814\pi\)
\(242\) −7.36680 −0.473556
\(243\) 0 0
\(244\) −13.3956 −0.857564
\(245\) 0 0
\(246\) 0 0
\(247\) −5.38350 −0.342544
\(248\) −7.30221 −0.463691
\(249\) 0 0
\(250\) 14.9521 0.945655
\(251\) 15.2439 0.962185 0.481092 0.876670i \(-0.340240\pi\)
0.481092 + 0.876670i \(0.340240\pi\)
\(252\) 0 0
\(253\) 20.2964 1.27602
\(254\) 3.56982 0.223990
\(255\) 0 0
\(256\) 20.8604 1.30377
\(257\) −15.6165 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.28431 −0.0796494
\(261\) 0 0
\(262\) −18.9654 −1.17169
\(263\) −17.0934 −1.05402 −0.527011 0.849858i \(-0.676687\pi\)
−0.527011 + 0.849858i \(0.676687\pi\)
\(264\) 0 0
\(265\) 0.186313 0.0114451
\(266\) 0 0
\(267\) 0 0
\(268\) −11.3622 −0.694055
\(269\) 16.3368 0.996073 0.498037 0.867156i \(-0.334054\pi\)
0.498037 + 0.867156i \(0.334054\pi\)
\(270\) 0 0
\(271\) 12.4994 0.759285 0.379642 0.925133i \(-0.376047\pi\)
0.379642 + 0.925133i \(0.376047\pi\)
\(272\) 15.9642 0.967971
\(273\) 0 0
\(274\) 14.5507 0.879041
\(275\) −16.4302 −0.990777
\(276\) 0 0
\(277\) −3.00000 −0.180253 −0.0901263 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) −9.48270 −0.568734
\(279\) 0 0
\(280\) 0 0
\(281\) −0.831590 −0.0496085 −0.0248043 0.999692i \(-0.507896\pi\)
−0.0248043 + 0.999692i \(0.507896\pi\)
\(282\) 0 0
\(283\) −11.0588 −0.657375 −0.328687 0.944439i \(-0.606606\pi\)
−0.328687 + 0.944439i \(0.606606\pi\)
\(284\) −7.26641 −0.431182
\(285\) 0 0
\(286\) 7.21509 0.426637
\(287\) 0 0
\(288\) 0 0
\(289\) −5.78491 −0.340289
\(290\) 2.74447 0.161161
\(291\) 0 0
\(292\) −23.0501 −1.34891
\(293\) −26.9175 −1.57254 −0.786269 0.617884i \(-0.787990\pi\)
−0.786269 + 0.617884i \(0.787990\pi\)
\(294\) 0 0
\(295\) −2.41228 −0.140448
\(296\) −4.66438 −0.271111
\(297\) 0 0
\(298\) −4.65574 −0.269700
\(299\) −5.24970 −0.303598
\(300\) 0 0
\(301\) 0 0
\(302\) 6.08712 0.350274
\(303\) 0 0
\(304\) −25.6632 −1.47188
\(305\) −7.82576 −0.448102
\(306\) 0 0
\(307\) 15.1580 0.865110 0.432555 0.901608i \(-0.357612\pi\)
0.432555 + 0.901608i \(0.357612\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12.2272 0.694456
\(311\) 4.24507 0.240716 0.120358 0.992731i \(-0.461596\pi\)
0.120358 + 0.992731i \(0.461596\pi\)
\(312\) 0 0
\(313\) −17.9533 −1.01478 −0.507391 0.861716i \(-0.669390\pi\)
−0.507391 + 0.861716i \(0.669390\pi\)
\(314\) −1.34426 −0.0758612
\(315\) 0 0
\(316\) 13.9821 0.786554
\(317\) 15.2664 0.857447 0.428723 0.903436i \(-0.358963\pi\)
0.428723 + 0.903436i \(0.358963\pi\)
\(318\) 0 0
\(319\) −6.56399 −0.367513
\(320\) −3.00120 −0.167772
\(321\) 0 0
\(322\) 0 0
\(323\) −18.0288 −1.00315
\(324\) 0 0
\(325\) 4.24970 0.235731
\(326\) −2.43018 −0.134595
\(327\) 0 0
\(328\) −3.92839 −0.216909
\(329\) 0 0
\(330\) 0 0
\(331\) 17.2664 0.949047 0.474524 0.880243i \(-0.342620\pi\)
0.474524 + 0.880243i \(0.342620\pi\)
\(332\) −6.07786 −0.333566
\(333\) 0 0
\(334\) −30.1400 −1.64919
\(335\) −6.63783 −0.362664
\(336\) 0 0
\(337\) −25.5415 −1.39133 −0.695666 0.718366i \(-0.744891\pi\)
−0.695666 + 0.718366i \(0.744891\pi\)
\(338\) −1.86620 −0.101508
\(339\) 0 0
\(340\) −4.30101 −0.233255
\(341\) −29.2439 −1.58364
\(342\) 0 0
\(343\) 0 0
\(344\) 3.89498 0.210003
\(345\) 0 0
\(346\) 29.5761 1.59002
\(347\) 12.6286 0.677937 0.338969 0.940798i \(-0.389922\pi\)
0.338969 + 0.940798i \(0.389922\pi\)
\(348\) 0 0
\(349\) −35.6394 −1.90774 −0.953868 0.300226i \(-0.902938\pi\)
−0.953868 + 0.300226i \(0.902938\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 26.9296 1.43535
\(353\) 5.41348 0.288130 0.144065 0.989568i \(-0.453982\pi\)
0.144065 + 0.989568i \(0.453982\pi\)
\(354\) 0 0
\(355\) −4.24507 −0.225305
\(356\) −0.619931 −0.0328563
\(357\) 0 0
\(358\) −38.1914 −2.01848
\(359\) 18.8316 0.993893 0.496947 0.867781i \(-0.334454\pi\)
0.496947 + 0.867781i \(0.334454\pi\)
\(360\) 0 0
\(361\) 9.98210 0.525374
\(362\) −12.2831 −0.645586
\(363\) 0 0
\(364\) 0 0
\(365\) −13.4660 −0.704842
\(366\) 0 0
\(367\) 23.1505 1.20845 0.604223 0.796815i \(-0.293483\pi\)
0.604223 + 0.796815i \(0.293483\pi\)
\(368\) −25.0253 −1.30454
\(369\) 0 0
\(370\) 7.81025 0.406036
\(371\) 0 0
\(372\) 0 0
\(373\) −33.1793 −1.71796 −0.858979 0.512011i \(-0.828901\pi\)
−0.858979 + 0.512011i \(0.828901\pi\)
\(374\) 24.1626 1.24942
\(375\) 0 0
\(376\) −3.52475 −0.181775
\(377\) 1.69779 0.0874406
\(378\) 0 0
\(379\) 37.7853 1.94090 0.970451 0.241299i \(-0.0775733\pi\)
0.970451 + 0.241299i \(0.0775733\pi\)
\(380\) 6.91408 0.354685
\(381\) 0 0
\(382\) 23.7924 1.21732
\(383\) 0.231798 0.0118443 0.00592215 0.999982i \(-0.498115\pi\)
0.00592215 + 0.999982i \(0.498115\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.23180 −0.215393
\(387\) 0 0
\(388\) 10.5507 0.535631
\(389\) −9.35352 −0.474243 −0.237121 0.971480i \(-0.576204\pi\)
−0.237121 + 0.971480i \(0.576204\pi\)
\(390\) 0 0
\(391\) −17.5807 −0.889094
\(392\) 0 0
\(393\) 0 0
\(394\) −34.8938 −1.75792
\(395\) 8.16841 0.410997
\(396\) 0 0
\(397\) −9.74447 −0.489061 −0.244530 0.969642i \(-0.578634\pi\)
−0.244530 + 0.969642i \(0.578634\pi\)
\(398\) 37.1700 1.86317
\(399\) 0 0
\(400\) 20.2583 1.01292
\(401\) −9.73240 −0.486013 −0.243006 0.970025i \(-0.578134\pi\)
−0.243006 + 0.970025i \(0.578134\pi\)
\(402\) 0 0
\(403\) 7.56399 0.376789
\(404\) −20.9296 −1.04129
\(405\) 0 0
\(406\) 0 0
\(407\) −18.6799 −0.925928
\(408\) 0 0
\(409\) 12.3730 0.611808 0.305904 0.952062i \(-0.401041\pi\)
0.305904 + 0.952062i \(0.401041\pi\)
\(410\) 6.57788 0.324858
\(411\) 0 0
\(412\) 24.9988 1.23160
\(413\) 0 0
\(414\) 0 0
\(415\) −3.55071 −0.174298
\(416\) −6.96539 −0.341506
\(417\) 0 0
\(418\) −38.8425 −1.89985
\(419\) −21.1054 −1.03107 −0.515534 0.856869i \(-0.672406\pi\)
−0.515534 + 0.856869i \(0.672406\pi\)
\(420\) 0 0
\(421\) 23.2618 1.13371 0.566855 0.823818i \(-0.308160\pi\)
0.566855 + 0.823818i \(0.308160\pi\)
\(422\) −1.20422 −0.0586203
\(423\) 0 0
\(424\) 0.207649 0.0100843
\(425\) 14.2318 0.690344
\(426\) 0 0
\(427\) 0 0
\(428\) −15.0946 −0.729623
\(429\) 0 0
\(430\) −6.52193 −0.314516
\(431\) 16.9895 0.818357 0.409179 0.912454i \(-0.365815\pi\)
0.409179 + 0.912454i \(0.365815\pi\)
\(432\) 0 0
\(433\) −30.8604 −1.48305 −0.741527 0.670923i \(-0.765898\pi\)
−0.741527 + 0.670923i \(0.765898\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.19719 −0.440465
\(437\) 28.2618 1.35194
\(438\) 0 0
\(439\) 19.1972 0.916232 0.458116 0.888892i \(-0.348524\pi\)
0.458116 + 0.888892i \(0.348524\pi\)
\(440\) 3.23300 0.154127
\(441\) 0 0
\(442\) −6.24970 −0.297268
\(443\) 34.3777 1.63333 0.816666 0.577110i \(-0.195820\pi\)
0.816666 + 0.577110i \(0.195820\pi\)
\(444\) 0 0
\(445\) −0.362166 −0.0171683
\(446\) 10.8829 0.515320
\(447\) 0 0
\(448\) 0 0
\(449\) −29.6274 −1.39820 −0.699101 0.715023i \(-0.746416\pi\)
−0.699101 + 0.715023i \(0.746416\pi\)
\(450\) 0 0
\(451\) −15.7324 −0.740810
\(452\) −15.2485 −0.717229
\(453\) 0 0
\(454\) 29.3598 1.37792
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0392 −0.797062 −0.398531 0.917155i \(-0.630480\pi\)
−0.398531 + 0.917155i \(0.630480\pi\)
\(458\) −16.5686 −0.774201
\(459\) 0 0
\(460\) 6.74224 0.314358
\(461\) −15.1280 −0.704580 −0.352290 0.935891i \(-0.614597\pi\)
−0.352290 + 0.935891i \(0.614597\pi\)
\(462\) 0 0
\(463\) 26.1221 1.21400 0.607000 0.794702i \(-0.292373\pi\)
0.607000 + 0.794702i \(0.292373\pi\)
\(464\) 8.09337 0.375725
\(465\) 0 0
\(466\) −22.0046 −1.01934
\(467\) −22.9187 −1.06055 −0.530276 0.847825i \(-0.677912\pi\)
−0.530276 + 0.847825i \(0.677912\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 5.90200 0.272239
\(471\) 0 0
\(472\) −2.68853 −0.123749
\(473\) 15.5986 0.717224
\(474\) 0 0
\(475\) −22.8783 −1.04973
\(476\) 0 0
\(477\) 0 0
\(478\) −29.9883 −1.37163
\(479\) −35.1914 −1.60793 −0.803967 0.594674i \(-0.797281\pi\)
−0.803967 + 0.594674i \(0.797281\pi\)
\(480\) 0 0
\(481\) 4.83159 0.220302
\(482\) 9.21046 0.419525
\(483\) 0 0
\(484\) 5.85293 0.266042
\(485\) 6.16378 0.279883
\(486\) 0 0
\(487\) −28.3010 −1.28244 −0.641221 0.767357i \(-0.721572\pi\)
−0.641221 + 0.767357i \(0.721572\pi\)
\(488\) −8.72194 −0.394824
\(489\) 0 0
\(490\) 0 0
\(491\) −8.24970 −0.372304 −0.186152 0.982521i \(-0.559602\pi\)
−0.186152 + 0.982521i \(0.559602\pi\)
\(492\) 0 0
\(493\) 5.68571 0.256072
\(494\) 10.0467 0.452022
\(495\) 0 0
\(496\) 36.0576 1.61903
\(497\) 0 0
\(498\) 0 0
\(499\) −16.7266 −0.748784 −0.374392 0.927271i \(-0.622149\pi\)
−0.374392 + 0.927271i \(0.622149\pi\)
\(500\) −11.8795 −0.531266
\(501\) 0 0
\(502\) −28.4481 −1.26970
\(503\) −21.2213 −0.946213 −0.473106 0.881005i \(-0.656867\pi\)
−0.473106 + 0.881005i \(0.656867\pi\)
\(504\) 0 0
\(505\) −12.2272 −0.544102
\(506\) −37.8771 −1.68384
\(507\) 0 0
\(508\) −2.83622 −0.125837
\(509\) 40.6048 1.79978 0.899889 0.436119i \(-0.143647\pi\)
0.899889 + 0.436119i \(0.143647\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −24.0000 −1.06066
\(513\) 0 0
\(514\) 29.1435 1.28546
\(515\) 14.6044 0.643548
\(516\) 0 0
\(517\) −14.1159 −0.620817
\(518\) 0 0
\(519\) 0 0
\(520\) −0.836221 −0.0366707
\(521\) −3.46479 −0.151795 −0.0758977 0.997116i \(-0.524182\pi\)
−0.0758977 + 0.997116i \(0.524182\pi\)
\(522\) 0 0
\(523\) 5.56982 0.243551 0.121776 0.992558i \(-0.461141\pi\)
0.121776 + 0.992558i \(0.461141\pi\)
\(524\) 15.0680 0.658249
\(525\) 0 0
\(526\) 31.8996 1.39089
\(527\) 25.3310 1.10344
\(528\) 0 0
\(529\) 4.55936 0.198233
\(530\) −0.347696 −0.0151030
\(531\) 0 0
\(532\) 0 0
\(533\) 4.06922 0.176257
\(534\) 0 0
\(535\) −8.81832 −0.381249
\(536\) −7.39797 −0.319544
\(537\) 0 0
\(538\) −30.4877 −1.31442
\(539\) 0 0
\(540\) 0 0
\(541\) 24.2364 1.04201 0.521003 0.853555i \(-0.325558\pi\)
0.521003 + 0.853555i \(0.325558\pi\)
\(542\) −23.3264 −1.00195
\(543\) 0 0
\(544\) −23.3264 −1.00011
\(545\) −5.37304 −0.230156
\(546\) 0 0
\(547\) 15.7733 0.674416 0.337208 0.941430i \(-0.390518\pi\)
0.337208 + 0.941430i \(0.390518\pi\)
\(548\) −11.5606 −0.493842
\(549\) 0 0
\(550\) 30.6620 1.30743
\(551\) −9.14005 −0.389379
\(552\) 0 0
\(553\) 0 0
\(554\) 5.59859 0.237861
\(555\) 0 0
\(556\) 7.53401 0.319513
\(557\) 17.2213 0.729692 0.364846 0.931068i \(-0.381122\pi\)
0.364846 + 0.931068i \(0.381122\pi\)
\(558\) 0 0
\(559\) −4.03461 −0.170646
\(560\) 0 0
\(561\) 0 0
\(562\) 1.55191 0.0654635
\(563\) 15.3598 0.647337 0.323669 0.946171i \(-0.395084\pi\)
0.323669 + 0.946171i \(0.395084\pi\)
\(564\) 0 0
\(565\) −8.90825 −0.374773
\(566\) 20.6378 0.867473
\(567\) 0 0
\(568\) −4.73120 −0.198517
\(569\) −23.7219 −0.994475 −0.497238 0.867614i \(-0.665652\pi\)
−0.497238 + 0.867614i \(0.665652\pi\)
\(570\) 0 0
\(571\) −27.3189 −1.14326 −0.571631 0.820511i \(-0.693689\pi\)
−0.571631 + 0.820511i \(0.693689\pi\)
\(572\) −5.73240 −0.239684
\(573\) 0 0
\(574\) 0 0
\(575\) −22.3097 −0.930377
\(576\) 0 0
\(577\) −23.2664 −0.968593 −0.484297 0.874904i \(-0.660924\pi\)
−0.484297 + 0.874904i \(0.660924\pi\)
\(578\) 10.7958 0.449045
\(579\) 0 0
\(580\) −2.18048 −0.0905397
\(581\) 0 0
\(582\) 0 0
\(583\) 0.831590 0.0344409
\(584\) −15.0081 −0.621038
\(585\) 0 0
\(586\) 50.2334 2.07512
\(587\) −45.7266 −1.88734 −0.943669 0.330892i \(-0.892651\pi\)
−0.943669 + 0.330892i \(0.892651\pi\)
\(588\) 0 0
\(589\) −40.7207 −1.67787
\(590\) 4.50180 0.185336
\(591\) 0 0
\(592\) 23.0322 0.946618
\(593\) −29.3897 −1.20689 −0.603446 0.797404i \(-0.706206\pi\)
−0.603446 + 0.797404i \(0.706206\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.69899 0.151516
\(597\) 0 0
\(598\) 9.79698 0.400628
\(599\) −22.0588 −0.901296 −0.450648 0.892702i \(-0.648807\pi\)
−0.450648 + 0.892702i \(0.648807\pi\)
\(600\) 0 0
\(601\) −30.8604 −1.25882 −0.629410 0.777073i \(-0.716704\pi\)
−0.629410 + 0.777073i \(0.716704\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.83622 −0.196783
\(605\) 3.41931 0.139015
\(606\) 0 0
\(607\) −36.4861 −1.48093 −0.740463 0.672097i \(-0.765394\pi\)
−0.740463 + 0.672097i \(0.765394\pi\)
\(608\) 37.4982 1.52075
\(609\) 0 0
\(610\) 14.6044 0.591316
\(611\) 3.65111 0.147708
\(612\) 0 0
\(613\) 28.9988 1.17125 0.585625 0.810582i \(-0.300849\pi\)
0.585625 + 0.810582i \(0.300849\pi\)
\(614\) −28.2877 −1.14160
\(615\) 0 0
\(616\) 0 0
\(617\) 41.6515 1.67683 0.838414 0.545035i \(-0.183483\pi\)
0.838414 + 0.545035i \(0.183483\pi\)
\(618\) 0 0
\(619\) −12.4994 −0.502393 −0.251197 0.967936i \(-0.580824\pi\)
−0.251197 + 0.967936i \(0.580824\pi\)
\(620\) −9.71449 −0.390143
\(621\) 0 0
\(622\) −7.92214 −0.317649
\(623\) 0 0
\(624\) 0 0
\(625\) 14.3085 0.572338
\(626\) 33.5044 1.33911
\(627\) 0 0
\(628\) 1.06802 0.0426186
\(629\) 16.1805 0.645158
\(630\) 0 0
\(631\) −35.5582 −1.41555 −0.707774 0.706439i \(-0.750300\pi\)
−0.707774 + 0.706439i \(0.750300\pi\)
\(632\) 9.10382 0.362131
\(633\) 0 0
\(634\) −28.4901 −1.13149
\(635\) −1.65693 −0.0657534
\(636\) 0 0
\(637\) 0 0
\(638\) 12.2497 0.484970
\(639\) 0 0
\(640\) −6.46599 −0.255591
\(641\) −1.36097 −0.0537550 −0.0268775 0.999639i \(-0.508556\pi\)
−0.0268775 + 0.999639i \(0.508556\pi\)
\(642\) 0 0
\(643\) −12.1867 −0.480598 −0.240299 0.970699i \(-0.577245\pi\)
−0.240299 + 0.970699i \(0.577245\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 33.6453 1.32376
\(647\) −3.72152 −0.146308 −0.0731540 0.997321i \(-0.523306\pi\)
−0.0731540 + 0.997321i \(0.523306\pi\)
\(648\) 0 0
\(649\) −10.7670 −0.422642
\(650\) −7.93078 −0.311071
\(651\) 0 0
\(652\) 1.93078 0.0756153
\(653\) 45.0588 1.76329 0.881643 0.471917i \(-0.156438\pi\)
0.881643 + 0.471917i \(0.156438\pi\)
\(654\) 0 0
\(655\) 8.80281 0.343954
\(656\) 19.3980 0.757364
\(657\) 0 0
\(658\) 0 0
\(659\) −5.37887 −0.209531 −0.104766 0.994497i \(-0.533409\pi\)
−0.104766 + 0.994497i \(0.533409\pi\)
\(660\) 0 0
\(661\) 42.4936 1.65281 0.826404 0.563077i \(-0.190383\pi\)
0.826404 + 0.563077i \(0.190383\pi\)
\(662\) −32.2225 −1.25236
\(663\) 0 0
\(664\) −3.95733 −0.153574
\(665\) 0 0
\(666\) 0 0
\(667\) −8.91288 −0.345108
\(668\) 23.9463 0.926510
\(669\) 0 0
\(670\) 12.3875 0.478571
\(671\) −34.9296 −1.34844
\(672\) 0 0
\(673\) −37.3765 −1.44076 −0.720379 0.693581i \(-0.756032\pi\)
−0.720379 + 0.693581i \(0.756032\pi\)
\(674\) 47.6654 1.83600
\(675\) 0 0
\(676\) 1.48270 0.0570268
\(677\) −43.4757 −1.67091 −0.835453 0.549562i \(-0.814795\pi\)
−0.835453 + 0.549562i \(0.814795\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.80041 −0.107391
\(681\) 0 0
\(682\) 54.5749 2.08978
\(683\) −31.3956 −1.20132 −0.600659 0.799505i \(-0.705095\pi\)
−0.600659 + 0.799505i \(0.705095\pi\)
\(684\) 0 0
\(685\) −6.75373 −0.258047
\(686\) 0 0
\(687\) 0 0
\(688\) −19.2330 −0.733251
\(689\) −0.215092 −0.00819437
\(690\) 0 0
\(691\) −13.4411 −0.511322 −0.255661 0.966766i \(-0.582293\pi\)
−0.255661 + 0.966766i \(0.582293\pi\)
\(692\) −23.4982 −0.893268
\(693\) 0 0
\(694\) −23.5674 −0.894607
\(695\) 4.40141 0.166955
\(696\) 0 0
\(697\) 13.6274 0.516174
\(698\) 66.5103 2.51745
\(699\) 0 0
\(700\) 0 0
\(701\) 28.0346 1.05885 0.529426 0.848356i \(-0.322407\pi\)
0.529426 + 0.848356i \(0.322407\pi\)
\(702\) 0 0
\(703\) −26.0109 −0.981019
\(704\) −13.3956 −0.504865
\(705\) 0 0
\(706\) −10.1026 −0.380217
\(707\) 0 0
\(708\) 0 0
\(709\) −40.2847 −1.51292 −0.756462 0.654037i \(-0.773074\pi\)
−0.756462 + 0.654037i \(0.773074\pi\)
\(710\) 7.92214 0.297313
\(711\) 0 0
\(712\) −0.403640 −0.0151271
\(713\) −39.7087 −1.48710
\(714\) 0 0
\(715\) −3.34889 −0.125242
\(716\) 30.3431 1.13397
\(717\) 0 0
\(718\) −35.1435 −1.31154
\(719\) 16.7445 0.624463 0.312232 0.950006i \(-0.398923\pi\)
0.312232 + 0.950006i \(0.398923\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.6286 −0.693284
\(723\) 0 0
\(724\) 9.75894 0.362688
\(725\) 7.21509 0.267962
\(726\) 0 0
\(727\) −51.4982 −1.90996 −0.954981 0.296666i \(-0.904125\pi\)
−0.954981 + 0.296666i \(0.904125\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 25.1302 0.930111
\(731\) −13.5115 −0.499740
\(732\) 0 0
\(733\) −33.8316 −1.24960 −0.624799 0.780786i \(-0.714819\pi\)
−0.624799 + 0.780786i \(0.714819\pi\)
\(734\) −43.2034 −1.59467
\(735\) 0 0
\(736\) 36.5662 1.34785
\(737\) −29.6274 −1.09134
\(738\) 0 0
\(739\) 20.6690 0.760322 0.380161 0.924920i \(-0.375869\pi\)
0.380161 + 0.924920i \(0.375869\pi\)
\(740\) −6.20525 −0.228110
\(741\) 0 0
\(742\) 0 0
\(743\) 29.6966 1.08946 0.544731 0.838611i \(-0.316632\pi\)
0.544731 + 0.838611i \(0.316632\pi\)
\(744\) 0 0
\(745\) 2.16097 0.0791717
\(746\) 61.9191 2.26702
\(747\) 0 0
\(748\) −19.1972 −0.701919
\(749\) 0 0
\(750\) 0 0
\(751\) −12.0230 −0.438724 −0.219362 0.975644i \(-0.570398\pi\)
−0.219362 + 0.975644i \(0.570398\pi\)
\(752\) 17.4048 0.634689
\(753\) 0 0
\(754\) −3.16841 −0.115387
\(755\) −2.82534 −0.102825
\(756\) 0 0
\(757\) −30.2906 −1.10093 −0.550464 0.834859i \(-0.685549\pi\)
−0.550464 + 0.834859i \(0.685549\pi\)
\(758\) −70.5149 −2.56122
\(759\) 0 0
\(760\) 4.50180 0.163297
\(761\) 45.9584 1.66599 0.832995 0.553281i \(-0.186624\pi\)
0.832995 + 0.553281i \(0.186624\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −18.9030 −0.683888
\(765\) 0 0
\(766\) −0.432580 −0.0156298
\(767\) 2.78491 0.100557
\(768\) 0 0
\(769\) 4.03924 0.145659 0.0728293 0.997344i \(-0.476797\pi\)
0.0728293 + 0.997344i \(0.476797\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.36217 0.121007
\(773\) −36.1175 −1.29906 −0.649528 0.760337i \(-0.725034\pi\)
−0.649528 + 0.760337i \(0.725034\pi\)
\(774\) 0 0
\(775\) 32.1447 1.15467
\(776\) 6.86963 0.246605
\(777\) 0 0
\(778\) 17.4555 0.625811
\(779\) −21.9066 −0.784887
\(780\) 0 0
\(781\) −18.9475 −0.677994
\(782\) 32.8091 1.17325
\(783\) 0 0
\(784\) 0 0
\(785\) 0.623942 0.0222694
\(786\) 0 0
\(787\) −25.8650 −0.921988 −0.460994 0.887403i \(-0.652507\pi\)
−0.460994 + 0.887403i \(0.652507\pi\)
\(788\) 27.7231 0.987596
\(789\) 0 0
\(790\) −15.2439 −0.542353
\(791\) 0 0
\(792\) 0 0
\(793\) 9.03461 0.320828
\(794\) 18.1851 0.645366
\(795\) 0 0
\(796\) −29.5316 −1.04672
\(797\) 0.291753 0.0103344 0.00516720 0.999987i \(-0.498355\pi\)
0.00516720 + 0.999987i \(0.498355\pi\)
\(798\) 0 0
\(799\) 12.2272 0.432566
\(800\) −29.6008 −1.04655
\(801\) 0 0
\(802\) 18.1626 0.641343
\(803\) −60.1042 −2.12103
\(804\) 0 0
\(805\) 0 0
\(806\) −14.1159 −0.497211
\(807\) 0 0
\(808\) −13.6274 −0.479409
\(809\) 5.25595 0.184789 0.0923946 0.995722i \(-0.470548\pi\)
0.0923946 + 0.995722i \(0.470548\pi\)
\(810\) 0 0
\(811\) −37.8499 −1.32909 −0.664545 0.747248i \(-0.731375\pi\)
−0.664545 + 0.747248i \(0.731375\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 34.8604 1.22186
\(815\) 1.12797 0.0395112
\(816\) 0 0
\(817\) 21.7203 0.759898
\(818\) −23.0906 −0.807342
\(819\) 0 0
\(820\) −5.22613 −0.182504
\(821\) 25.7207 0.897660 0.448830 0.893617i \(-0.351841\pi\)
0.448830 + 0.893617i \(0.351841\pi\)
\(822\) 0 0
\(823\) −19.4827 −0.679124 −0.339562 0.940584i \(-0.610279\pi\)
−0.339562 + 0.940584i \(0.610279\pi\)
\(824\) 16.2769 0.567031
\(825\) 0 0
\(826\) 0 0
\(827\) 32.0934 1.11600 0.557998 0.829842i \(-0.311570\pi\)
0.557998 + 0.829842i \(0.311570\pi\)
\(828\) 0 0
\(829\) −3.89336 −0.135222 −0.0676110 0.997712i \(-0.521538\pi\)
−0.0676110 + 0.997712i \(0.521538\pi\)
\(830\) 6.62634 0.230004
\(831\) 0 0
\(832\) 3.46479 0.120120
\(833\) 0 0
\(834\) 0 0
\(835\) 13.9895 0.484128
\(836\) 30.8604 1.06733
\(837\) 0 0
\(838\) 39.3869 1.36060
\(839\) −41.8592 −1.44514 −0.722570 0.691298i \(-0.757039\pi\)
−0.722570 + 0.691298i \(0.757039\pi\)
\(840\) 0 0
\(841\) −26.1175 −0.900604
\(842\) −43.4111 −1.49604
\(843\) 0 0
\(844\) 0.956750 0.0329327
\(845\) 0.866198 0.0297981
\(846\) 0 0
\(847\) 0 0
\(848\) −1.02535 −0.0352106
\(849\) 0 0
\(850\) −26.5594 −0.910978
\(851\) −25.3644 −0.869480
\(852\) 0 0
\(853\) 1.70242 0.0582897 0.0291449 0.999575i \(-0.490722\pi\)
0.0291449 + 0.999575i \(0.490722\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.82816 −0.335919
\(857\) 9.57908 0.327215 0.163608 0.986526i \(-0.447687\pi\)
0.163608 + 0.986526i \(0.447687\pi\)
\(858\) 0 0
\(859\) 6.17424 0.210662 0.105331 0.994437i \(-0.466410\pi\)
0.105331 + 0.994437i \(0.466410\pi\)
\(860\) 5.18168 0.176694
\(861\) 0 0
\(862\) −31.7059 −1.07991
\(863\) −24.8604 −0.846257 −0.423128 0.906070i \(-0.639068\pi\)
−0.423128 + 0.906070i \(0.639068\pi\)
\(864\) 0 0
\(865\) −13.7278 −0.466758
\(866\) 57.5916 1.95704
\(867\) 0 0
\(868\) 0 0
\(869\) 36.4590 1.23679
\(870\) 0 0
\(871\) 7.66318 0.259657
\(872\) −5.98834 −0.202791
\(873\) 0 0
\(874\) −52.7421 −1.78403
\(875\) 0 0
\(876\) 0 0
\(877\) 25.6562 0.866347 0.433173 0.901311i \(-0.357394\pi\)
0.433173 + 0.901311i \(0.357394\pi\)
\(878\) −35.8258 −1.20906
\(879\) 0 0
\(880\) −15.9642 −0.538153
\(881\) −25.6632 −0.864615 −0.432307 0.901726i \(-0.642300\pi\)
−0.432307 + 0.901726i \(0.642300\pi\)
\(882\) 0 0
\(883\) 48.6682 1.63782 0.818908 0.573925i \(-0.194580\pi\)
0.818908 + 0.573925i \(0.194580\pi\)
\(884\) 4.96539 0.167004
\(885\) 0 0
\(886\) −64.1556 −2.15535
\(887\) 17.5247 0.588423 0.294212 0.955740i \(-0.404943\pi\)
0.294212 + 0.955740i \(0.404943\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0.675874 0.0226554
\(891\) 0 0
\(892\) −8.64648 −0.289505
\(893\) −19.6557 −0.657754
\(894\) 0 0
\(895\) 17.7266 0.592534
\(896\) 0 0
\(897\) 0 0
\(898\) 55.2906 1.84507
\(899\) 12.8420 0.428306
\(900\) 0 0
\(901\) −0.720322 −0.0239974
\(902\) 29.3598 0.977573
\(903\) 0 0
\(904\) −9.92839 −0.330213
\(905\) 5.70122 0.189515
\(906\) 0 0
\(907\) 57.3765 1.90515 0.952577 0.304297i \(-0.0984214\pi\)
0.952577 + 0.304297i \(0.0984214\pi\)
\(908\) −23.3264 −0.774112
\(909\) 0 0
\(910\) 0 0
\(911\) −7.87203 −0.260812 −0.130406 0.991461i \(-0.541628\pi\)
−0.130406 + 0.991461i \(0.541628\pi\)
\(912\) 0 0
\(913\) −15.8483 −0.524502
\(914\) 31.7986 1.05180
\(915\) 0 0
\(916\) 13.1638 0.434944
\(917\) 0 0
\(918\) 0 0
\(919\) −47.0230 −1.55114 −0.775572 0.631259i \(-0.782538\pi\)
−0.775572 + 0.631259i \(0.782538\pi\)
\(920\) 4.38991 0.144731
\(921\) 0 0
\(922\) 28.2318 0.929765
\(923\) 4.90081 0.161312
\(924\) 0 0
\(925\) 20.5328 0.675115
\(926\) −48.7491 −1.60199
\(927\) 0 0
\(928\) −11.8258 −0.388200
\(929\) −28.2618 −0.927239 −0.463619 0.886034i \(-0.653449\pi\)
−0.463619 + 0.886034i \(0.653449\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 17.4827 0.572665
\(933\) 0 0
\(934\) 42.7709 1.39951
\(935\) −11.2151 −0.366773
\(936\) 0 0
\(937\) 8.83784 0.288720 0.144360 0.989525i \(-0.453888\pi\)
0.144360 + 0.989525i \(0.453888\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4.68915 −0.152943
\(941\) 42.7853 1.39476 0.697381 0.716701i \(-0.254349\pi\)
0.697381 + 0.716701i \(0.254349\pi\)
\(942\) 0 0
\(943\) −21.3622 −0.695648
\(944\) 13.2757 0.432086
\(945\) 0 0
\(946\) −29.1101 −0.946450
\(947\) −29.2213 −0.949566 −0.474783 0.880103i \(-0.657473\pi\)
−0.474783 + 0.880103i \(0.657473\pi\)
\(948\) 0 0
\(949\) 15.5461 0.504647
\(950\) 42.6954 1.38522
\(951\) 0 0
\(952\) 0 0
\(953\) −3.66198 −0.118623 −0.0593116 0.998240i \(-0.518891\pi\)
−0.0593116 + 0.998240i \(0.518891\pi\)
\(954\) 0 0
\(955\) −11.0432 −0.357351
\(956\) 23.8258 0.770580
\(957\) 0 0
\(958\) 65.6741 2.12183
\(959\) 0 0
\(960\) 0 0
\(961\) 26.2139 0.845610
\(962\) −9.01671 −0.290710
\(963\) 0 0
\(964\) −7.31772 −0.235688
\(965\) 1.96419 0.0632296
\(966\) 0 0
\(967\) −30.0288 −0.965660 −0.482830 0.875714i \(-0.660391\pi\)
−0.482830 + 0.875714i \(0.660391\pi\)
\(968\) 3.81087 0.122486
\(969\) 0 0
\(970\) −11.5028 −0.369334
\(971\) 4.90544 0.157423 0.0787115 0.996897i \(-0.474919\pi\)
0.0787115 + 0.996897i \(0.474919\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 52.8153 1.69231
\(975\) 0 0
\(976\) 43.0680 1.37857
\(977\) −22.6332 −0.724100 −0.362050 0.932159i \(-0.617923\pi\)
−0.362050 + 0.932159i \(0.617923\pi\)
\(978\) 0 0
\(979\) −1.61650 −0.0516635
\(980\) 0 0
\(981\) 0 0
\(982\) 15.3956 0.491293
\(983\) −57.4053 −1.83094 −0.915472 0.402382i \(-0.868182\pi\)
−0.915472 + 0.402382i \(0.868182\pi\)
\(984\) 0 0
\(985\) 16.1960 0.516047
\(986\) −10.6107 −0.337913
\(987\) 0 0
\(988\) −7.98210 −0.253944
\(989\) 21.1805 0.673500
\(990\) 0 0
\(991\) 31.1793 0.990443 0.495221 0.868767i \(-0.335087\pi\)
0.495221 + 0.868767i \(0.335087\pi\)
\(992\) −52.6861 −1.67279
\(993\) 0 0
\(994\) 0 0
\(995\) −17.2525 −0.546941
\(996\) 0 0
\(997\) 36.0576 1.14195 0.570977 0.820966i \(-0.306564\pi\)
0.570977 + 0.820966i \(0.306564\pi\)
\(998\) 31.2151 0.988096
\(999\) 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 5733.2.a.bd.1.1 3
3.2 odd 2 637.2.a.i.1.3 yes 3
7.6 odd 2 5733.2.a.be.1.1 3
21.2 odd 6 637.2.e.k.508.1 6
21.5 even 6 637.2.e.l.508.1 6
21.11 odd 6 637.2.e.k.79.1 6
21.17 even 6 637.2.e.l.79.1 6
21.20 even 2 637.2.a.h.1.3 3
39.38 odd 2 8281.2.a.bk.1.1 3
273.272 even 2 8281.2.a.bh.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.3 3 21.20 even 2
637.2.a.i.1.3 yes 3 3.2 odd 2
637.2.e.k.79.1 6 21.11 odd 6
637.2.e.k.508.1 6 21.2 odd 6
637.2.e.l.79.1 6 21.17 even 6
637.2.e.l.508.1 6 21.5 even 6
5733.2.a.bd.1.1 3 1.1 even 1 trivial
5733.2.a.be.1.1 3 7.6 odd 2
8281.2.a.bh.1.1 3 273.272 even 2
8281.2.a.bk.1.1 3 39.38 odd 2