Properties

Label 5733.2.a.be.1.1
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
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: \(0\)
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 \(-0.562045\pi\)
−0.193688 + 0.981063i \(0.562045\pi\)
\(6\) 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 \(-0.366884\pi\)
0.406113 + 0.913823i \(0.366884\pi\)
\(18\) 0 0
\(19\) −5.38350 −1.23506 −0.617530 0.786547i \(-0.711867\pi\)
−0.617530 + 0.786547i \(0.711867\pi\)
\(20\) −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 \(-0.262298\pi\)
0.679266 + 0.733892i \(0.262298\pi\)
\(32\) 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 \(-0.397072\pi\)
0.317752 + 0.948174i \(0.397072\pi\)
\(42\) 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 \(-0.414204\pi\)
0.266284 + 0.963895i \(0.414204\pi\)
\(48\) 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 \(-0.441975\pi\)
0.181282 + 0.983431i \(0.441975\pi\)
\(60\) 0 0
\(61\) 9.03461 1.15676 0.578382 0.815766i \(-0.303685\pi\)
0.578382 + 0.815766i \(0.303685\pi\)
\(62\) −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 \(-0.136261\pi\)
0.909766 + 0.415122i \(0.136261\pi\)
\(74\) 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 \(-0.427771\pi\)
0.224972 + 0.974365i \(0.427771\pi\)
\(84\) 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 \(-0.492946\pi\)
0.0221598 + 0.999754i \(0.492946\pi\)
\(90\) 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 \(-0.617652\pi\)
−0.361255 + 0.932467i \(0.617652\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −6.30101 −0.630101
\(101\) 14.1159 1.40458 0.702292 0.711889i \(-0.252160\pi\)
0.702292 + 0.711889i \(0.252160\pi\)
\(102\) 0 0
\(103\) −16.8604 −1.66130 −0.830651 0.556794i \(-0.812031\pi\)
−0.830651 + 0.556794i \(0.812031\pi\)
\(104\) 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 \(-0.646425\pi\)
−0.443954 + 0.896049i \(0.646425\pi\)
\(132\) 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 \(-0.569136\pi\)
−0.215495 + 0.976505i \(0.569136\pi\)
\(140\) 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 \(-0.509151\pi\)
−0.0287440 + 0.999587i \(0.509151\pi\)
\(158\) −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 \(-0.714853\pi\)
−0.624882 + 0.780719i \(0.714853\pi\)
\(168\) 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 \(-0.294187\pi\)
0.602462 + 0.798148i \(0.294187\pi\)
\(174\) 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 \(-0.578661\pi\)
−0.244614 + 0.969621i \(0.578661\pi\)
\(182\) 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 \(-0.250517\pi\)
0.705957 + 0.708254i \(0.250517\pi\)
\(200\) −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 \(-0.437446\pi\)
0.195256 + 0.980752i \(0.437446\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 19.1926 1.27667
\(227\) 15.7324 1.04420 0.522098 0.852886i \(-0.325150\pi\)
0.522098 + 0.852886i \(0.325150\pi\)
\(228\) 0 0
\(229\) −8.87827 −0.586693 −0.293346 0.956006i \(-0.594769\pi\)
−0.293346 + 0.956006i \(0.594769\pi\)
\(230\) 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 \(-0.449186\pi\)
0.158959 + 0.987285i \(0.449186\pi\)
\(242\) −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 \(-0.659760\pi\)
−0.481092 + 0.876670i \(0.659760\pi\)
\(252\) 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 \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(258\) 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 \(-0.665946\pi\)
−0.498037 + 0.867156i \(0.665946\pi\)
\(270\) 0 0
\(271\) −12.4994 −0.759285 −0.379642 0.925133i \(-0.623953\pi\)
−0.379642 + 0.925133i \(0.623953\pi\)
\(272\) −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 \(-0.393394\pi\)
0.328687 + 0.944439i \(0.393394\pi\)
\(284\) −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 \(-0.212010\pi\)
0.786269 + 0.617884i \(0.212010\pi\)
\(294\) 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 \(-0.642388\pi\)
−0.432555 + 0.901608i \(0.642388\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12.2272 0.694456
\(311\) −4.24507 −0.240716 −0.120358 0.992731i \(-0.538404\pi\)
−0.120358 + 0.992731i \(0.538404\pi\)
\(312\) 0 0
\(313\) 17.9533 1.01478 0.507391 0.861716i \(-0.330610\pi\)
0.507391 + 0.861716i \(0.330610\pi\)
\(314\) 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 \(-0.0970622\pi\)
0.953868 + 0.300226i \(0.0970622\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 26.9296 1.43535
\(353\) −5.41348 −0.288130 −0.144065 0.989568i \(-0.546018\pi\)
−0.144065 + 0.989568i \(0.546018\pi\)
\(354\) 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 \(-0.706517\pi\)
−0.604223 + 0.796815i \(0.706517\pi\)
\(368\) −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 \(-0.501885\pi\)
−0.00592215 + 0.999982i \(0.501885\pi\)
\(384\) 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 \(-0.421366\pi\)
0.244530 + 0.969642i \(0.421366\pi\)
\(398\) −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 \(-0.598959\pi\)
−0.305904 + 0.952062i \(0.598959\pi\)
\(410\) 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 \(-0.327594\pi\)
0.515534 + 0.856869i \(0.327594\pi\)
\(420\) 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 \(-0.234102\pi\)
0.741527 + 0.670923i \(0.234102\pi\)
\(434\) 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 \(-0.651476\pi\)
−0.458116 + 0.888892i \(0.651476\pi\)
\(440\) −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 \(-0.385403\pi\)
0.352290 + 0.935891i \(0.385403\pi\)
\(462\) 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 \(-0.322088\pi\)
0.530276 + 0.847825i \(0.322088\pi\)
\(468\) 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 \(-0.202719\pi\)
0.803967 + 0.594674i \(0.202719\pi\)
\(480\) 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 \(-0.343133\pi\)
0.473106 + 0.881005i \(0.343133\pi\)
\(504\) 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 \(-0.856353\pi\)
−0.899889 + 0.436119i \(0.856353\pi\)
\(510\) 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 \(-0.475818\pi\)
0.0758977 + 0.997116i \(0.475818\pi\)
\(522\) 0 0
\(523\) −5.56982 −0.243551 −0.121776 0.992558i \(-0.538859\pi\)
−0.121776 + 0.992558i \(0.538859\pi\)
\(524\) −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 \(-0.604916\pi\)
−0.323669 + 0.946171i \(0.604916\pi\)
\(564\) 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 \(-0.339076\pi\)
0.484297 + 0.874904i \(0.339076\pi\)
\(578\) 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 \(-0.107349\pi\)
0.943669 + 0.330892i \(0.107349\pi\)
\(588\) 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 \(-0.293794\pi\)
0.603446 + 0.797404i \(0.293794\pi\)
\(594\) 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 \(-0.283296\pi\)
0.629410 + 0.777073i \(0.283296\pi\)
\(602\) 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 \(-0.234606\pi\)
0.740463 + 0.672097i \(0.234606\pi\)
\(608\) −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 \(-0.419176\pi\)
0.251197 + 0.967936i \(0.419176\pi\)
\(620\) −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 \(-0.422755\pi\)
0.240299 + 0.970699i \(0.422755\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 33.6453 1.32376
\(647\) 3.72152 0.146308 0.0731540 0.997321i \(-0.476694\pi\)
0.0731540 + 0.997321i \(0.476694\pi\)
\(648\) 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 \(-0.809617\pi\)
−0.826404 + 0.563077i \(0.809617\pi\)
\(662\) −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 \(-0.185205\pi\)
0.835453 + 0.549562i \(0.185205\pi\)
\(678\) 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 \(-0.417707\pi\)
0.255661 + 0.966766i \(0.417707\pi\)
\(692\) 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 \(-0.601077\pi\)
−0.312232 + 0.950006i \(0.601077\pi\)
\(720\) 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 \(-0.0958748\pi\)
0.954981 + 0.296666i \(0.0958748\pi\)
\(728\) 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 \(-0.285181\pi\)
0.624799 + 0.780786i \(0.285181\pi\)
\(734\) 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 \(-0.813376\pi\)
−0.832995 + 0.553281i \(0.813376\pi\)
\(762\) 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 \(-0.523203\pi\)
−0.0728293 + 0.997344i \(0.523203\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.36217 0.121007
\(773\) 36.1175 1.29906 0.649528 0.760337i \(-0.274966\pi\)
0.649528 + 0.760337i \(0.274966\pi\)
\(774\) 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 \(-0.347493\pi\)
0.460994 + 0.887403i \(0.347493\pi\)
\(788\) 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 \(-0.501645\pi\)
−0.00516720 + 0.999987i \(0.501645\pi\)
\(798\) 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 \(-0.268625\pi\)
0.664545 + 0.747248i \(0.268625\pi\)
\(812\) 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 \(-0.478462\pi\)
0.0676110 + 0.997712i \(0.478462\pi\)
\(830\) 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 \(-0.242961\pi\)
0.722570 + 0.691298i \(0.242961\pi\)
\(840\) 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 \(-0.509278\pi\)
−0.0291449 + 0.999575i \(0.509278\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.82816 −0.335919
\(857\) −9.57908 −0.327215 −0.163608 0.986526i \(-0.552313\pi\)
−0.163608 + 0.986526i \(0.552313\pi\)
\(858\) 0 0
\(859\) −6.17424 −0.210662 −0.105331 0.994437i \(-0.533590\pi\)
−0.105331 + 0.994437i \(0.533590\pi\)
\(860\) −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 \(-0.357700\pi\)
0.432307 + 0.901726i \(0.357700\pi\)
\(882\) 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 \(-0.595057\pi\)
−0.294212 + 0.955740i \(0.595057\pi\)
\(888\) 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 \(-0.346551\pi\)
0.463619 + 0.886034i \(0.346551\pi\)
\(930\) 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 \(-0.546112\pi\)
−0.144360 + 0.989525i \(0.546112\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4.68915 −0.152943
\(941\) −42.7853 −1.39476 −0.697381 0.716701i \(-0.745651\pi\)
−0.697381 + 0.716701i \(0.745651\pi\)
\(942\) 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 \(-0.525081\pi\)
−0.0787115 + 0.996897i \(0.525081\pi\)
\(972\) 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 \(-0.131818\pi\)
0.915472 + 0.402382i \(0.131818\pi\)
\(984\) 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 \(-0.693436\pi\)
−0.570977 + 0.820966i \(0.693436\pi\)
\(998\) 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.be.1.1 3
3.2 odd 2 637.2.a.h.1.3 3
7.6 odd 2 5733.2.a.bd.1.1 3
21.2 odd 6 637.2.e.l.508.1 6
21.5 even 6 637.2.e.k.508.1 6
21.11 odd 6 637.2.e.l.79.1 6
21.17 even 6 637.2.e.k.79.1 6
21.20 even 2 637.2.a.i.1.3 yes 3
39.38 odd 2 8281.2.a.bh.1.1 3
273.272 even 2 8281.2.a.bk.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.3 3 3.2 odd 2
637.2.a.i.1.3 yes 3 21.20 even 2
637.2.e.k.79.1 6 21.17 even 6
637.2.e.k.508.1 6 21.5 even 6
637.2.e.l.79.1 6 21.11 odd 6
637.2.e.l.508.1 6 21.2 odd 6
5733.2.a.bd.1.1 3 7.6 odd 2
5733.2.a.be.1.1 3 1.1 even 1 trivial
8281.2.a.bh.1.1 3 39.38 odd 2
8281.2.a.bk.1.1 3 273.272 even 2