Properties

Label 8281.2.a.bk.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
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 \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.86620 q^{2} +3.34889 q^{3} +1.48270 q^{4} +0.866198 q^{5} -6.24970 q^{6} +0.965392 q^{8} +8.21509 q^{9} +O(q^{10})\) \(q-1.86620 q^{2} +3.34889 q^{3} +1.48270 q^{4} +0.866198 q^{5} -6.24970 q^{6} +0.965392 q^{8} +8.21509 q^{9} -1.61650 q^{10} +3.86620 q^{11} +4.96539 q^{12} +2.90081 q^{15} -4.76700 q^{16} +3.34889 q^{17} -15.3310 q^{18} -5.38350 q^{19} +1.28431 q^{20} -7.21509 q^{22} -5.24970 q^{23} +3.23300 q^{24} -4.24970 q^{25} +17.4648 q^{27} +1.69779 q^{29} -5.41348 q^{30} +7.56399 q^{31} +6.96539 q^{32} +12.9475 q^{33} -6.24970 q^{34} +12.1805 q^{36} +4.83159 q^{37} +10.0467 q^{38} +0.836221 q^{40} -4.06922 q^{41} +4.03461 q^{43} +5.73240 q^{44} +7.11590 q^{45} +9.79698 q^{46} -3.65111 q^{47} -15.9642 q^{48} +7.93078 q^{50} +11.2151 q^{51} -0.215092 q^{53} -32.5928 q^{54} +3.34889 q^{55} -18.0288 q^{57} -3.16841 q^{58} -2.78491 q^{59} +4.30101 q^{60} -9.03461 q^{61} -14.1159 q^{62} -3.46479 q^{64} -24.1626 q^{66} +7.66318 q^{67} +4.96539 q^{68} -17.5807 q^{69} -4.90081 q^{71} +7.93078 q^{72} +15.5461 q^{73} -9.01671 q^{74} -14.2318 q^{75} -7.98210 q^{76} +9.43018 q^{79} -4.12917 q^{80} +33.8425 q^{81} +7.59396 q^{82} -4.09919 q^{83} +2.90081 q^{85} -7.52938 q^{86} +5.68571 q^{87} +3.73240 q^{88} -0.418110 q^{89} -13.2797 q^{90} -7.78371 q^{92} +25.3310 q^{93} +6.81369 q^{94} -4.66318 q^{95} +23.3264 q^{96} -7.11590 q^{97} +31.7612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{8} + 11 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{2} + 4 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{8} + 11 q^{9} - 14 q^{10} + 4 q^{11} + 18 q^{12} - 2 q^{15} + 4 q^{16} + 4 q^{17} - 8 q^{18} - 7 q^{19} - 16 q^{20} - 8 q^{22} + q^{23} + 28 q^{24} + 4 q^{25} + 22 q^{27} - 7 q^{29} - 24 q^{30} + 3 q^{31} + 24 q^{32} + 10 q^{33} - 2 q^{34} + 26 q^{36} + 10 q^{37} + 12 q^{38} - 22 q^{40} - 6 q^{41} + 9 q^{43} + 2 q^{44} - 3 q^{45} + 28 q^{46} - 17 q^{47} + 16 q^{48} + 30 q^{50} + 20 q^{51} + 13 q^{53} - 28 q^{54} + 4 q^{55} - 4 q^{57} - 14 q^{58} - 22 q^{59} - 42 q^{60} - 24 q^{61} - 18 q^{62} + 20 q^{64} - 30 q^{66} + 14 q^{67} + 18 q^{68} + 2 q^{69} - 4 q^{71} + 30 q^{72} - 5 q^{73} + 8 q^{74} + 6 q^{75} + 8 q^{76} + q^{79} - 40 q^{80} + 15 q^{81} + 20 q^{82} - 23 q^{83} - 2 q^{85} - 6 q^{86} + 20 q^{87} - 4 q^{88} + 11 q^{89} - 40 q^{90} + 30 q^{92} + 38 q^{93} - 16 q^{94} - 5 q^{95} + 52 q^{96} + 3 q^{97} + 30 q^{99} + O(q^{100}) \)

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 \(0.729359\pi\)
\(3\) 3.34889 1.93348 0.966742 0.255752i \(-0.0823230\pi\)
0.966742 + 0.255752i \(0.0823230\pi\)
\(4\) 1.48270 0.741348
\(5\) 0.866198 0.387376 0.193688 0.981063i \(-0.437955\pi\)
0.193688 + 0.981063i \(0.437955\pi\)
\(6\) −6.24970 −2.55143
\(7\) 0 0
\(8\) 0.965392 0.341318
\(9\) 8.21509 2.73836
\(10\) −1.61650 −0.511181
\(11\) 3.86620 1.16570 0.582851 0.812579i \(-0.301937\pi\)
0.582851 + 0.812579i \(0.301937\pi\)
\(12\) 4.96539 1.43339
\(13\) 0 0
\(14\) 0 0
\(15\) 2.90081 0.748985
\(16\) −4.76700 −1.19175
\(17\) 3.34889 0.812226 0.406113 0.913823i \(-0.366884\pi\)
0.406113 + 0.913823i \(0.366884\pi\)
\(18\) −15.3310 −3.61355
\(19\) −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 \(-0.684352\pi\)
−0.547319 + 0.836924i \(0.684352\pi\)
\(24\) 3.23300 0.659932
\(25\) −4.24970 −0.849940
\(26\) 0 0
\(27\) 17.4648 3.36110
\(28\) 0 0
\(29\) 1.69779 0.315271 0.157636 0.987497i \(-0.449613\pi\)
0.157636 + 0.987497i \(0.449613\pi\)
\(30\) −5.41348 −0.988362
\(31\) 7.56399 1.35853 0.679266 0.733892i \(-0.262298\pi\)
0.679266 + 0.733892i \(0.262298\pi\)
\(32\) 6.96539 1.23132
\(33\) 12.9475 2.25387
\(34\) −6.24970 −1.07181
\(35\) 0 0
\(36\) 12.1805 2.03008
\(37\) 4.83159 0.794309 0.397154 0.917752i \(-0.369998\pi\)
0.397154 + 0.917752i \(0.369998\pi\)
\(38\) 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
\(45\) 7.11590 1.06078
\(46\) 9.79698 1.44449
\(47\) −3.65111 −0.532569 −0.266284 0.963895i \(-0.585796\pi\)
−0.266284 + 0.963895i \(0.585796\pi\)
\(48\) −15.9642 −2.30423
\(49\) 0 0
\(50\) 7.93078 1.12158
\(51\) 11.2151 1.57043
\(52\) 0 0
\(53\) −0.215092 −0.0295452 −0.0147726 0.999891i \(-0.504702\pi\)
−0.0147726 + 0.999891i \(0.504702\pi\)
\(54\) −32.5928 −4.43531
\(55\) 3.34889 0.451565
\(56\) 0 0
\(57\) −18.0288 −2.38797
\(58\) −3.16841 −0.416033
\(59\) −2.78491 −0.362564 −0.181282 0.983431i \(-0.558025\pi\)
−0.181282 + 0.983431i \(0.558025\pi\)
\(60\) 4.30101 0.555258
\(61\) −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 0
\(66\) −24.1626 −2.97421
\(67\) 7.66318 0.936206 0.468103 0.883674i \(-0.344938\pi\)
0.468103 + 0.883674i \(0.344938\pi\)
\(68\) 4.96539 0.602142
\(69\) −17.5807 −2.11647
\(70\) 0 0
\(71\) −4.90081 −0.581619 −0.290809 0.956781i \(-0.593925\pi\)
−0.290809 + 0.956781i \(0.593925\pi\)
\(72\) 7.93078 0.934652
\(73\) 15.5461 1.81953 0.909766 0.415122i \(-0.136261\pi\)
0.909766 + 0.415122i \(0.136261\pi\)
\(74\) −9.01671 −1.04817
\(75\) −14.2318 −1.64335
\(76\) −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
\(81\) 33.8425 3.76027
\(82\) 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
\(87\) 5.68571 0.609573
\(88\) 3.73240 0.397875
\(89\) −0.418110 −0.0443196 −0.0221598 0.999754i \(-0.507054\pi\)
−0.0221598 + 0.999754i \(0.507054\pi\)
\(90\) −13.2797 −1.39980
\(91\) 0 0
\(92\) −7.78371 −0.811508
\(93\) 25.3310 2.62670
\(94\) 6.81369 0.702778
\(95\) −4.66318 −0.478432
\(96\) 23.3264 2.38074
\(97\) −7.11590 −0.722510 −0.361255 0.932467i \(-0.617652\pi\)
−0.361255 + 0.932467i \(0.617652\pi\)
\(98\) 0 0
\(99\) 31.7612 3.19212
\(100\) −6.30101 −0.630101
\(101\) 14.1159 1.40458 0.702292 0.711889i \(-0.252160\pi\)
0.702292 + 0.711889i \(0.252160\pi\)
\(102\) −20.9296 −2.07234
\(103\) 16.8604 1.66130 0.830651 0.556794i \(-0.187969\pi\)
0.830651 + 0.556794i \(0.187969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.401405 0.0389879
\(107\) 10.1805 0.984185 0.492092 0.870543i \(-0.336232\pi\)
0.492092 + 0.870543i \(0.336232\pi\)
\(108\) 25.8950 2.49175
\(109\) 6.20302 0.594141 0.297071 0.954855i \(-0.403990\pi\)
0.297071 + 0.954855i \(0.403990\pi\)
\(110\) −6.24970 −0.595886
\(111\) 16.1805 1.53578
\(112\) 0 0
\(113\) 10.2843 0.967466 0.483733 0.875216i \(-0.339281\pi\)
0.483733 + 0.875216i \(0.339281\pi\)
\(114\) 33.6453 3.15117
\(115\) −4.54728 −0.424036
\(116\) 2.51730 0.233726
\(117\) 0 0
\(118\) 5.19719 0.478440
\(119\) 0 0
\(120\) 2.80041 0.255642
\(121\) 3.94749 0.358863
\(122\) 16.8604 1.52647
\(123\) −13.6274 −1.22874
\(124\) 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
\(129\) 13.5115 1.18962
\(130\) 0 0
\(131\) −10.1626 −0.887909 −0.443954 0.896049i \(-0.646425\pi\)
−0.443954 + 0.896049i \(0.646425\pi\)
\(132\) 19.1972 1.67090
\(133\) 0 0
\(134\) −14.3010 −1.23542
\(135\) 15.1280 1.30201
\(136\) 3.23300 0.277227
\(137\) −7.79698 −0.666141 −0.333071 0.942902i \(-0.608085\pi\)
−0.333071 + 0.942902i \(0.608085\pi\)
\(138\) 32.8091 2.79289
\(139\) 5.08129 0.430989 0.215495 0.976505i \(-0.430864\pi\)
0.215495 + 0.976505i \(0.430864\pi\)
\(140\) 0 0
\(141\) −12.2272 −1.02971
\(142\) 9.14588 0.767505
\(143\) 0 0
\(144\) −39.1614 −3.26345
\(145\) 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 \(0.467415\pi\)
\(150\) 26.5594 2.16856
\(151\) 3.26178 0.265439 0.132720 0.991154i \(-0.457629\pi\)
0.132720 + 0.991154i \(0.457629\pi\)
\(152\) −5.19719 −0.421548
\(153\) 27.5115 2.22417
\(154\) 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
\(159\) −0.720322 −0.0571252
\(160\) 6.03341 0.476983
\(161\) 0 0
\(162\) −63.1568 −4.96206
\(163\) −1.30221 −0.101997 −0.0509985 0.998699i \(-0.516240\pi\)
−0.0509985 + 0.998699i \(0.516240\pi\)
\(164\) −6.03341 −0.471130
\(165\) 11.2151 0.873094
\(166\) 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
\(169\) 0 0
\(170\) −5.41348 −0.415195
\(171\) −44.2260 −3.38204
\(172\) 5.98210 0.456131
\(173\) 15.8483 1.20492 0.602462 0.798148i \(-0.294187\pi\)
0.602462 + 0.798148i \(0.294187\pi\)
\(174\) −10.6107 −0.804393
\(175\) 0 0
\(176\) −18.4302 −1.38923
\(177\) −9.32636 −0.701012
\(178\) 0.780277 0.0584842
\(179\) −20.4648 −1.52961 −0.764805 0.644262i \(-0.777165\pi\)
−0.764805 + 0.644262i \(0.777165\pi\)
\(180\) 10.5507 0.786404
\(181\) 6.58189 0.489228 0.244614 0.969621i \(-0.421339\pi\)
0.244614 + 0.969621i \(0.421339\pi\)
\(182\) 0 0
\(183\) −30.2559 −2.23658
\(184\) −5.06802 −0.373619
\(185\) 4.18512 0.307696
\(186\) −47.2727 −3.46620
\(187\) 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 \(-0.347402\pi\)
0.461246 + 0.887272i \(0.347402\pi\)
\(192\) −11.6032 −0.837391
\(193\) −2.26760 −0.163226 −0.0816128 0.996664i \(-0.526007\pi\)
−0.0816128 + 0.996664i \(0.526007\pi\)
\(194\) 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 \(0.267970\pi\)
\(198\) −59.2727 −4.21232
\(199\) −19.9175 −1.41191 −0.705957 0.708254i \(-0.749483\pi\)
−0.705957 + 0.708254i \(0.749483\pi\)
\(200\) −4.10263 −0.290100
\(201\) 25.6632 1.81014
\(202\) −26.3431 −1.85349
\(203\) 0 0
\(204\) 16.6286 1.16423
\(205\) −3.52475 −0.246179
\(206\) −31.4648 −2.19226
\(207\) −43.1268 −2.99752
\(208\) 0 0
\(209\) −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
\(213\) −16.4123 −1.12455
\(214\) −18.9988 −1.29873
\(215\) 3.49477 0.238341
\(216\) 16.8604 1.14720
\(217\) 0 0
\(218\) −11.5761 −0.784030
\(219\) 52.0622 3.51804
\(220\) 4.96539 0.334767
\(221\) 0 0
\(222\) −30.1960 −2.02662
\(223\) 5.83159 0.390512 0.195256 0.980752i \(-0.437446\pi\)
0.195256 + 0.980752i \(0.437446\pi\)
\(224\) 0 0
\(225\) −34.9117 −2.32745
\(226\) −19.1926 −1.27667
\(227\) −15.7324 −1.04420 −0.522098 0.852886i \(-0.674850\pi\)
−0.522098 + 0.852886i \(0.674850\pi\)
\(228\) −26.7312 −1.77032
\(229\) −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 \(-0.626224\pi\)
−0.386232 + 0.922402i \(0.626224\pi\)
\(234\) 0 0
\(235\) −3.16258 −0.206304
\(236\) −4.12917 −0.268786
\(237\) 31.5807 2.05139
\(238\) 0 0
\(239\) 16.0692 1.03943 0.519716 0.854339i \(-0.326038\pi\)
0.519716 + 0.854339i \(0.326038\pi\)
\(240\) −13.8282 −0.892604
\(241\) 4.93541 0.317918 0.158959 0.987285i \(-0.449186\pi\)
0.158959 + 0.987285i \(0.449186\pi\)
\(242\) −7.36680 −0.473556
\(243\) 60.9405 3.90933
\(244\) −13.3956 −0.857564
\(245\) 0 0
\(246\) 25.4314 1.62145
\(247\) 0 0
\(248\) 7.30221 0.463691
\(249\) −13.7278 −0.869962
\(250\) 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
\(255\) 9.71449 0.608345
\(256\) 20.8604 1.30377
\(257\) 15.6165 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(258\) −25.2151 −1.56982
\(259\) 0 0
\(260\) 0 0
\(261\) 13.9475 0.863328
\(262\) 18.9654 1.17169
\(263\) 17.0934 1.05402 0.527011 0.849858i \(-0.323313\pi\)
0.527011 + 0.849858i \(0.323313\pi\)
\(264\) 12.4994 0.769285
\(265\) −0.186313 −0.0114451
\(266\) 0 0
\(267\) −1.40021 −0.0856913
\(268\) 11.3622 0.694055
\(269\) −16.3368 −0.996073 −0.498037 0.867156i \(-0.665946\pi\)
−0.498037 + 0.867156i \(0.665946\pi\)
\(270\) −28.2318 −1.71813
\(271\) −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
\(276\) −26.0668 −1.56904
\(277\) −3.00000 −0.180253 −0.0901263 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) −9.48270 −0.568734
\(279\) 62.1389 3.72016
\(280\) 0 0
\(281\) −0.831590 −0.0496085 −0.0248043 0.999692i \(-0.507896\pi\)
−0.0248043 + 0.999692i \(0.507896\pi\)
\(282\) 22.8183 1.35881
\(283\) −11.0588 −0.657375 −0.328687 0.944439i \(-0.606606\pi\)
−0.328687 + 0.944439i \(0.606606\pi\)
\(284\) −7.26641 −0.431182
\(285\) −15.6165 −0.925041
\(286\) 0 0
\(287\) 0 0
\(288\) 57.2213 3.37180
\(289\) −5.78491 −0.340289
\(290\) −2.74447 −0.161161
\(291\) −23.8304 −1.39696
\(292\) 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
\(297\) 67.5224 3.91804
\(298\) −4.65574 −0.269700
\(299\) 0 0
\(300\) −21.1014 −1.21829
\(301\) 0 0
\(302\) −6.08712 −0.350274
\(303\) 47.2727 2.71574
\(304\) 25.6632 1.47188
\(305\) −7.82576 −0.448102
\(306\) −51.3419 −2.93502
\(307\) −15.1580 −0.865110 −0.432555 0.901608i \(-0.642388\pi\)
−0.432555 + 0.901608i \(0.642388\pi\)
\(308\) 0 0
\(309\) 56.4636 3.21210
\(310\) −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 \(-0.669390\pi\)
−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 \(0.358963\pi\)
\(318\) 1.34426 0.0753826
\(319\) 6.56399 0.367513
\(320\) −3.00120 −0.167772
\(321\) 34.0934 1.90291
\(322\) 0 0
\(323\) −18.0288 −1.00315
\(324\) 50.1781 2.78767
\(325\) 0 0
\(326\) 2.43018 0.134595
\(327\) 20.7733 1.14876
\(328\) −3.92839 −0.216909
\(329\) 0 0
\(330\) −20.9296 −1.15214
\(331\) −17.2664 −0.949047 −0.474524 0.880243i \(-0.657380\pi\)
−0.474524 + 0.880243i \(0.657380\pi\)
\(332\) −6.07786 −0.333566
\(333\) 39.6920 2.17511
\(334\) −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 \(0.744891\pi\)
\(338\) 0 0
\(339\) 34.4411 1.87058
\(340\) 4.30101 0.233255
\(341\) 29.2439 1.58364
\(342\) 82.5344 4.46295
\(343\) 0 0
\(344\) 3.89498 0.210003
\(345\) −15.2284 −0.819868
\(346\) −29.5761 −1.59002
\(347\) −12.6286 −0.677937 −0.338969 0.940798i \(-0.610078\pi\)
−0.338969 + 0.940798i \(0.610078\pi\)
\(348\) 8.43018 0.451905
\(349\) 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 \(-0.453982\pi\)
0.144065 + 0.989568i \(0.453982\pi\)
\(354\) 17.4048 0.925057
\(355\) −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 \(0.334454\pi\)
\(360\) 6.86963 0.362061
\(361\) 9.98210 0.525374
\(362\) −12.2831 −0.645586
\(363\) 13.2197 0.693856
\(364\) 0 0
\(365\) 13.4660 0.704842
\(366\) 56.4636 2.95140
\(367\) 23.1505 1.20845 0.604223 0.796815i \(-0.293483\pi\)
0.604223 + 0.796815i \(0.293483\pi\)
\(368\) 25.0253 1.30454
\(369\) −33.4290 −1.74024
\(370\) −7.81025 −0.406036
\(371\) 0 0
\(372\) 37.5582 1.94730
\(373\) −33.1793 −1.71796 −0.858979 0.512011i \(-0.828901\pi\)
−0.858979 + 0.512011i \(0.828901\pi\)
\(374\) −24.1626 −1.24942
\(375\) −26.8316 −1.38558
\(376\) −3.52475 −0.181775
\(377\) 0 0
\(378\) 0 0
\(379\) −37.7853 −1.94090 −0.970451 0.241299i \(-0.922427\pi\)
−0.970451 + 0.241299i \(0.922427\pi\)
\(380\) −6.91408 −0.354685
\(381\) −6.40604 −0.328191
\(382\) −23.7924 −1.21732
\(383\) 0.231798 0.0118443 0.00592215 0.999982i \(-0.498115\pi\)
0.00592215 + 0.999982i \(0.498115\pi\)
\(384\) −24.9988 −1.27571
\(385\) 0 0
\(386\) 4.23180 0.215393
\(387\) 33.1447 1.68484
\(388\) −10.5507 −0.535631
\(389\) 9.35352 0.474243 0.237121 0.971480i \(-0.423796\pi\)
0.237121 + 0.971480i \(0.423796\pi\)
\(390\) 0 0
\(391\) −17.5807 −0.889094
\(392\) 0 0
\(393\) −34.0334 −1.71676
\(394\) −34.8938 −1.75792
\(395\) 8.16841 0.410997
\(396\) 47.0922 2.36647
\(397\) 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 \(0.578134\pi\)
\(402\) −47.8926 −2.38866
\(403\) 0 0
\(404\) 20.9296 1.04129
\(405\) 29.3143 1.45664
\(406\) 0 0
\(407\) 18.6799 0.925928
\(408\) 10.8270 0.536014
\(409\) −12.3730 −0.611808 −0.305904 0.952062i \(-0.598959\pi\)
−0.305904 + 0.952062i \(0.598959\pi\)
\(410\) 6.57788 0.324858
\(411\) −26.1113 −1.28797
\(412\) 24.9988 1.23160
\(413\) 0 0
\(414\) 80.4831 3.95553
\(415\) −3.55071 −0.174298
\(416\) 0 0
\(417\) 17.0167 0.833312
\(418\) 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 \(-0.691840\pi\)
−0.566855 + 0.823818i \(0.691840\pi\)
\(422\) −1.20422 −0.0586203
\(423\) −29.9942 −1.45837
\(424\) −0.207649 −0.0100843
\(425\) −14.2318 −0.690344
\(426\) 30.6286 1.48396
\(427\) 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 \(0.365815\pi\)
\(432\) −83.2547 −4.00560
\(433\) −30.8604 −1.48305 −0.741527 0.670923i \(-0.765898\pi\)
−0.741527 + 0.670923i \(0.765898\pi\)
\(434\) 0 0
\(435\) 4.92496 0.236134
\(436\) 9.19719 0.440465
\(437\) 28.2618 1.35194
\(438\) −97.1584 −4.64241
\(439\) 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
\(442\) 0 0
\(443\) −34.3777 −1.63333 −0.816666 0.577110i \(-0.804180\pi\)
−0.816666 + 0.577110i \(0.804180\pi\)
\(444\) 23.9907 1.13855
\(445\) −0.362166 −0.0171683
\(446\) −10.8829 −0.515320
\(447\) 8.35472 0.395165
\(448\) 0 0
\(449\) −29.6274 −1.39820 −0.699101 0.715023i \(-0.746416\pi\)
−0.699101 + 0.715023i \(0.746416\pi\)
\(450\) 65.1521 3.07130
\(451\) −15.7324 −0.740810
\(452\) 15.2485 0.717229
\(453\) 10.9233 0.513223
\(454\) 29.3598 1.37792
\(455\) 0 0
\(456\) −17.4048 −0.815056
\(457\) 17.0392 0.797062 0.398531 0.917155i \(-0.369520\pi\)
0.398531 + 0.917155i \(0.369520\pi\)
\(458\) 16.5686 0.774201
\(459\) 58.4877 2.72997
\(460\) −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 \(-0.707627\pi\)
−0.607000 + 0.794702i \(0.707627\pi\)
\(464\) −8.09337 −0.375725
\(465\) 21.9417 1.01752
\(466\) 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
\(471\) 2.41228 0.111152
\(472\) −2.68853 −0.123749
\(473\) 15.5986 0.717224
\(474\) −58.9358 −2.70701
\(475\) 22.8783 1.04973
\(476\) 0 0
\(477\) −1.76700 −0.0809056
\(478\) −29.9883 −1.37163
\(479\) −35.1914 −1.60793 −0.803967 0.594674i \(-0.797281\pi\)
−0.803967 + 0.594674i \(0.797281\pi\)
\(480\) 20.2053 0.922239
\(481\) 0 0
\(482\) −9.21046 −0.419525
\(483\) 0 0
\(484\) 5.85293 0.266042
\(485\) −6.16378 −0.279883
\(486\) −113.727 −5.15876
\(487\) 28.3010 1.28244 0.641221 0.767357i \(-0.278428\pi\)
0.641221 + 0.767357i \(0.278428\pi\)
\(488\) −8.72194 −0.394824
\(489\) −4.36097 −0.197210
\(490\) 0 0
\(491\) 8.24970 0.372304 0.186152 0.982521i \(-0.440398\pi\)
0.186152 + 0.982521i \(0.440398\pi\)
\(492\) −20.2053 −0.910923
\(493\) 5.68571 0.256072
\(494\) 0 0
\(495\) 27.5115 1.23655
\(496\) −36.0576 −1.61903
\(497\) 0 0
\(498\) 25.6187 1.14800
\(499\) 16.7266 0.748784 0.374392 0.927271i \(-0.377851\pi\)
0.374392 + 0.927271i \(0.377851\pi\)
\(500\) −11.8795 −0.531266
\(501\) 54.0863 2.41640
\(502\) 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 \(-0.143647\pi\)
0.899889 + 0.436119i \(0.143647\pi\)
\(510\) −18.1292 −0.802773
\(511\) 0 0
\(512\) −24.0000 −1.06066
\(513\) −94.0218 −4.15116
\(514\) −29.1435 −1.28546
\(515\) 14.6044 0.643548
\(516\) 20.0334 0.881922
\(517\) −14.1159 −0.620817
\(518\) 0 0
\(519\) 53.0743 2.32970
\(520\) 0 0
\(521\) 3.46479 0.151795 0.0758977 0.997116i \(-0.475818\pi\)
0.0758977 + 0.997116i \(0.475818\pi\)
\(522\) −26.0288 −1.13925
\(523\) 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
\(528\) −61.7207 −2.68605
\(529\) 4.55936 0.198233
\(530\) 0.347696 0.0151030
\(531\) −22.8783 −0.992832
\(532\) 0 0
\(533\) 0 0
\(534\) 2.61306 0.113078
\(535\) 8.81832 0.381249
\(536\) 7.39797 0.319544
\(537\) −68.5344 −2.95748
\(538\) 30.4877 1.31442
\(539\) 0 0
\(540\) 22.4302 0.965241
\(541\) −24.2364 −1.04201 −0.521003 0.853555i \(-0.674442\pi\)
−0.521003 + 0.853555i \(0.674442\pi\)
\(542\) 23.3264 1.00195
\(543\) 22.0421 0.945915
\(544\) 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
\(549\) −74.2201 −3.16764
\(550\) 30.6620 1.30743
\(551\) −9.14005 −0.389379
\(552\) −16.9723 −0.722387
\(553\) 0 0
\(554\) 5.59859 0.237861
\(555\) 14.0155 0.594925
\(556\) 7.53401 0.319513
\(557\) 17.2213 0.729692 0.364846 0.931068i \(-0.381122\pi\)
0.364846 + 0.931068i \(0.381122\pi\)
\(558\) −115.963 −4.90912
\(559\) 0 0
\(560\) 0 0
\(561\) 43.3598 1.83065
\(562\) 1.55191 0.0654635
\(563\) −15.3598 −0.647337 −0.323669 0.946171i \(-0.604916\pi\)
−0.323669 + 0.946171i \(0.604916\pi\)
\(564\) −18.1292 −0.763376
\(565\) 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 \(-0.334348\pi\)
0.497238 + 0.867614i \(0.334348\pi\)
\(570\) 29.1435 1.22069
\(571\) −27.3189 −1.14326 −0.571631 0.820511i \(-0.693689\pi\)
−0.571631 + 0.820511i \(0.693689\pi\)
\(572\) 0 0
\(573\) 42.6954 1.78363
\(574\) 0 0
\(575\) 22.3097 0.930377
\(576\) −28.4636 −1.18598
\(577\) 23.2664 0.968593 0.484297 0.874904i \(-0.339076\pi\)
0.484297 + 0.874904i \(0.339076\pi\)
\(578\) 10.7958 0.449045
\(579\) −7.59396 −0.315594
\(580\) 2.18048 0.0905397
\(581\) 0 0
\(582\) 44.4722 1.84343
\(583\) −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
\(591\) 62.6169 2.57572
\(592\) −23.0322 −0.946618
\(593\) −29.3897 −1.20689 −0.603446 0.797404i \(-0.706206\pi\)
−0.603446 + 0.797404i \(0.706206\pi\)
\(594\) −126.010 −5.17026
\(595\) 0 0
\(596\) 3.69899 0.151516
\(597\) −66.7016 −2.72992
\(598\) 0 0
\(599\) 22.0588 0.901296 0.450648 0.892702i \(-0.351193\pi\)
0.450648 + 0.892702i \(0.351193\pi\)
\(600\) −13.7393 −0.560903
\(601\) −30.8604 −1.25882 −0.629410 0.777073i \(-0.716704\pi\)
−0.629410 + 0.777073i \(0.716704\pi\)
\(602\) 0 0
\(603\) 62.9537 2.56367
\(604\) 4.83622 0.196783
\(605\) 3.41931 0.139015
\(606\) −88.2201 −3.58370
\(607\) −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
\(611\) 0 0
\(612\) 40.7912 1.64888
\(613\) −28.9988 −1.17125 −0.585625 0.810582i \(-0.699151\pi\)
−0.585625 + 0.810582i \(0.699151\pi\)
\(614\) 28.2877 1.14160
\(615\) −11.8040 −0.475984
\(616\) 0 0
\(617\) 41.6515 1.67683 0.838414 0.545035i \(-0.183483\pi\)
0.838414 + 0.545035i \(0.183483\pi\)
\(618\) −105.372 −4.23869
\(619\) 12.4994 0.502393 0.251197 0.967936i \(-0.419176\pi\)
0.251197 + 0.967936i \(0.419176\pi\)
\(620\) 9.71449 0.390143
\(621\) −91.6849 −3.67919
\(622\) 7.92214 0.317649
\(623\) 0 0
\(624\) 0 0
\(625\) 14.3085 0.572338
\(626\) 33.5044 1.33911
\(627\) −69.7028 −2.78366
\(628\) 1.06802 0.0426186
\(629\) 16.1805 0.645158
\(630\) 0 0
\(631\) 35.5582 1.41555 0.707774 0.706439i \(-0.249700\pi\)
0.707774 + 0.706439i \(0.249700\pi\)
\(632\) 9.10382 0.362131
\(633\) 2.16097 0.0858907
\(634\) −28.4901 −1.13149
\(635\) −1.65693 −0.0657534
\(636\) −1.06802 −0.0423497
\(637\) 0 0
\(638\) −12.2497 −0.484970
\(639\) −40.2606 −1.59268
\(640\) −6.46599 −0.255591
\(641\) 1.36097 0.0537550 0.0268775 0.999639i \(-0.491444\pi\)
0.0268775 + 0.999639i \(0.491444\pi\)
\(642\) −63.6250 −2.51108
\(643\) 12.1867 0.480598 0.240299 0.970699i \(-0.422755\pi\)
0.240299 + 0.970699i \(0.422755\pi\)
\(644\) 0 0
\(645\) 11.7036 0.460829
\(646\) 33.6453 1.32376
\(647\) 3.72152 0.146308 0.0731540 0.997321i \(-0.476694\pi\)
0.0731540 + 0.997321i \(0.476694\pi\)
\(648\) 32.6712 1.28345
\(649\) −10.7670 −0.422642
\(650\) 0 0
\(651\) 0 0
\(652\) −1.93078 −0.0756153
\(653\) −45.0588 −1.76329 −0.881643 0.471917i \(-0.843562\pi\)
−0.881643 + 0.471917i \(0.843562\pi\)
\(654\) −38.7670 −1.51591
\(655\) −8.80281 −0.343954
\(656\) 19.3980 0.757364
\(657\) 127.713 4.98254
\(658\) 0 0
\(659\) 5.37887 0.209531 0.104766 0.994497i \(-0.466591\pi\)
0.104766 + 0.994497i \(0.466591\pi\)
\(660\) 16.6286 0.647266
\(661\) −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
\(666\) −74.0731 −2.87027
\(667\) −8.91288 −0.345108
\(668\) 23.9463 0.926510
\(669\) 19.5294 0.755049
\(670\) −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
\(675\) −74.2201 −2.85673
\(676\) 0 0
\(677\) 43.4757 1.67091 0.835453 0.549562i \(-0.185205\pi\)
0.835453 + 0.549562i \(0.185205\pi\)
\(678\) −64.2738 −2.46842
\(679\) 0 0
\(680\) 2.80041 0.107391
\(681\) −52.6861 −2.01894
\(682\) −54.5749 −2.08978
\(683\) −31.3956 −1.20132 −0.600659 0.799505i \(-0.705095\pi\)
−0.600659 + 0.799505i \(0.705095\pi\)
\(684\) −65.5737 −2.50727
\(685\) −6.75373 −0.258047
\(686\) 0 0
\(687\) −29.7324 −1.13436
\(688\) −19.2330 −0.733251
\(689\) 0 0
\(690\) 28.4191 1.08190
\(691\) 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
\(696\) 5.48894 0.208058
\(697\) −13.6274 −0.516174
\(698\) −66.5103 −2.51745
\(699\) −39.4873 −1.49355
\(700\) 0 0
\(701\) −28.0346 −1.05885 −0.529426 0.848356i \(-0.677593\pi\)
−0.529426 + 0.848356i \(0.677593\pi\)
\(702\) 0 0
\(703\) −26.0109 −0.981019
\(704\) −13.3956 −0.504865
\(705\) −10.5912 −0.398886
\(706\) −10.1026 −0.380217
\(707\) 0 0
\(708\) −13.8282 −0.519694
\(709\) 40.2847 1.51292 0.756462 0.654037i \(-0.226926\pi\)
0.756462 + 0.654037i \(0.226926\pi\)
\(710\) 7.92214 0.297313
\(711\) 77.4698 2.90535
\(712\) −0.403640 −0.0151271
\(713\) −39.7087 −1.48710
\(714\) 0 0
\(715\) 0 0
\(716\) −30.3431 −1.13397
\(717\) 53.8141 2.00972
\(718\) −35.1435 −1.31154
\(719\) −16.7445 −0.624463 −0.312232 0.950006i \(-0.601077\pi\)
−0.312232 + 0.950006i \(0.601077\pi\)
\(720\) −33.9215 −1.26418
\(721\) 0 0
\(722\) −18.6286 −0.693284
\(723\) 16.5282 0.614690
\(724\) 9.75894 0.362688
\(725\) −7.21509 −0.267962
\(726\) −24.6706 −0.915613
\(727\) −51.4982 −1.90996 −0.954981 0.296666i \(-0.904125\pi\)
−0.954981 + 0.296666i \(0.904125\pi\)
\(728\) 0 0
\(729\) 102.556 3.79836
\(730\) −25.1302 −0.930111
\(731\) 13.5115 0.499740
\(732\) −44.8604 −1.65809
\(733\) 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
\(738\) 62.3851 2.29643
\(739\) −20.6690 −0.760322 −0.380161 0.924920i \(-0.624131\pi\)
−0.380161 + 0.924920i \(0.624131\pi\)
\(740\) 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 \(0.316632\pi\)
\(744\) 24.4543 0.896539
\(745\) 2.16097 0.0791717
\(746\) 61.9191 2.26702
\(747\) −33.6753 −1.23211
\(748\) 19.1972 0.701919
\(749\) 0 0
\(750\) 50.0731 1.82841
\(751\) −12.0230 −0.438724 −0.219362 0.975644i \(-0.570398\pi\)
−0.219362 + 0.975644i \(0.570398\pi\)
\(752\) 17.4048 0.634689
\(753\) −51.0501 −1.86037
\(754\) 0 0
\(755\) 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
\(759\) −67.9704 −2.46717
\(760\) −4.50180 −0.163297
\(761\) 45.9584 1.66599 0.832995 0.553281i \(-0.186624\pi\)
0.832995 + 0.553281i \(0.186624\pi\)
\(762\) 11.9549 0.433082
\(763\) 0 0
\(764\) 18.9030 0.683888
\(765\) 23.8304 0.861590
\(766\) −0.432580 −0.0156298
\(767\) 0 0
\(768\) 69.8592 2.52083
\(769\) −4.03924 −0.145659 −0.0728293 0.997344i \(-0.523203\pi\)
−0.0728293 + 0.997344i \(0.523203\pi\)
\(770\) 0 0
\(771\) 52.2980 1.88347
\(772\) −3.36217 −0.121007
\(773\) −36.1175 −1.29906 −0.649528 0.760337i \(-0.725034\pi\)
−0.649528 + 0.760337i \(0.725034\pi\)
\(774\) −61.8545 −2.22332
\(775\) −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
\(783\) 29.6515 1.05966
\(784\) 0 0
\(785\) 0.623942 0.0222694
\(786\) 63.5131 2.26544
\(787\) 25.8650 0.921988 0.460994 0.887403i \(-0.347493\pi\)
0.460994 + 0.887403i \(0.347493\pi\)
\(788\) 27.7231 0.987596
\(789\) 57.2439 2.03794
\(790\) −15.2439 −0.542353
\(791\) 0 0
\(792\) 30.6620 1.08953
\(793\) 0 0
\(794\) −18.1851 −0.645366
\(795\) −0.623942 −0.0221289
\(796\) −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
\(801\) −3.43482 −0.121363
\(802\) 18.1626 0.641343
\(803\) 60.1042 2.12103
\(804\) 38.0507 1.34194
\(805\) 0 0
\(806\) 0 0
\(807\) −54.7103 −1.92589
\(808\) 13.6274 0.479409
\(809\) −5.25595 −0.184789 −0.0923946 0.995722i \(-0.529452\pi\)
−0.0923946 + 0.995722i \(0.529452\pi\)
\(810\) −54.7063 −1.92218
\(811\) 37.8499 1.32909 0.664545 0.747248i \(-0.268625\pi\)
0.664545 + 0.747248i \(0.268625\pi\)
\(812\) 0 0
\(813\) −41.8592 −1.46807
\(814\) −34.8604 −1.22186
\(815\) −1.12797 −0.0395112
\(816\) −53.4624 −1.87156
\(817\) −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 \(0.351841\pi\)
\(822\) 48.7288 1.69961
\(823\) −19.4827 −0.679124 −0.339562 0.940584i \(-0.610279\pi\)
−0.339562 + 0.940584i \(0.610279\pi\)
\(824\) 16.2769 0.567031
\(825\) −55.0230 −1.91565
\(826\) 0 0
\(827\) 32.0934 1.11600 0.557998 0.829842i \(-0.311570\pi\)
0.557998 + 0.829842i \(0.311570\pi\)
\(828\) −63.9439 −2.22220
\(829\) −3.89336 −0.135222 −0.0676110 0.997712i \(-0.521538\pi\)
−0.0676110 + 0.997712i \(0.521538\pi\)
\(830\) 6.62634 0.230004
\(831\) −10.0467 −0.348516
\(832\) 0 0
\(833\) 0 0
\(834\) −31.7565 −1.09964
\(835\) 13.9895 0.484128
\(836\) −30.8604 −1.06733
\(837\) 132.103 4.56616
\(838\) −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
\(843\) −2.78491 −0.0959173
\(844\) 0.956750 0.0329327
\(845\) 0 0
\(846\) 55.9751 1.92446
\(847\) 0 0
\(848\) 1.02535 0.0352106
\(849\) −37.0346 −1.27102
\(850\) 26.5594 0.910978
\(851\) −25.3644 −0.869480
\(852\) −24.3344 −0.833684
\(853\) −1.70242 −0.0582897 −0.0291449 0.999575i \(-0.509278\pi\)
−0.0291449 + 0.999575i \(0.509278\pi\)
\(854\) 0 0
\(855\) −38.3085 −1.31012
\(856\) 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 \(-0.466410\pi\)
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 \(0.639068\pi\)
\(864\) 121.649 4.13859
\(865\) 13.7278 0.466758
\(866\) 57.5916 1.95704
\(867\) −19.3730 −0.657943
\(868\) 0 0
\(869\) 36.4590 1.23679
\(870\) −9.19094 −0.311602
\(871\) 0 0
\(872\) 5.98834 0.202791
\(873\) −58.4578 −1.97850
\(874\) −52.7421 −1.78403
\(875\) 0 0
\(876\) 77.1924 2.60809
\(877\) −25.6562 −0.866347 −0.433173 0.901311i \(-0.642606\pi\)
−0.433173 + 0.901311i \(0.642606\pi\)
\(878\) −35.8258 −1.20906
\(879\) −90.1439 −3.04048
\(880\) −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 \(0.194580\pi\)
\(884\) 0 0
\(885\) −8.07848 −0.271555
\(886\) 64.1556 2.15535
\(887\) −17.5247 −0.588423 −0.294212 0.955740i \(-0.595057\pi\)
−0.294212 + 0.955740i \(0.595057\pi\)
\(888\) 15.6205 0.524190
\(889\) 0 0
\(890\) 0.675874 0.0226554
\(891\) 130.842 4.38336
\(892\) 8.64648 0.289505
\(893\) 19.6557 0.657754
\(894\) −15.5916 −0.521460
\(895\) −17.7266 −0.592534
\(896\) 0 0
\(897\) 0 0
\(898\) 55.2906 1.84507
\(899\) 12.8420 0.428306
\(900\) −51.7634 −1.72545
\(901\) −0.720322 −0.0239974
\(902\) 29.3598 0.977573
\(903\) 0 0
\(904\) 9.92839 0.330213
\(905\) 5.70122 0.189515
\(906\) −20.3851 −0.677250
\(907\) 57.3765 1.90515 0.952577 0.304297i \(-0.0984214\pi\)
0.952577 + 0.304297i \(0.0984214\pi\)
\(908\) −23.3264 −0.774112
\(909\) 115.963 3.84626
\(910\) 0 0
\(911\) 7.87203 0.260812 0.130406 0.991461i \(-0.458372\pi\)
0.130406 + 0.991461i \(0.458372\pi\)
\(912\) 85.9433 2.84587
\(913\) −15.8483 −0.524502
\(914\) −31.7986 −1.05180
\(915\) −26.2076 −0.866398
\(916\) −13.1638 −0.434944
\(917\) 0 0
\(918\) −109.150 −3.60248
\(919\) −47.0230 −1.55114 −0.775572 0.631259i \(-0.782538\pi\)
−0.775572 + 0.631259i \(0.782538\pi\)
\(920\) −4.38991 −0.144731
\(921\) −50.7624 −1.67268
\(922\) 28.2318 0.929765
\(923\) 0 0
\(924\) 0 0
\(925\) −20.5328 −0.675115
\(926\) 48.7491 1.60199
\(927\) 138.509 4.54925
\(928\) 11.8258 0.388200
\(929\) −28.2618 −0.927239 −0.463619 0.886034i \(-0.653449\pi\)
−0.463619 + 0.886034i \(0.653449\pi\)
\(930\) −40.9475 −1.34272
\(931\) 0 0
\(932\) −17.4827 −0.572665
\(933\) −14.2163 −0.465420
\(934\) −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
\(939\) −60.1238 −1.96206
\(940\) −4.68915 −0.152943
\(941\) 42.7853 1.39476 0.697381 0.716701i \(-0.254349\pi\)
0.697381 + 0.716701i \(0.254349\pi\)
\(942\) −4.50180 −0.146676
\(943\) 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 \(0.657473\pi\)
\(948\) 46.8246 1.52079
\(949\) 0 0
\(950\) −42.6954 −1.38522
\(951\) 51.1256 1.65786
\(952\) 0 0
\(953\) 3.66198 0.118623 0.0593116 0.998240i \(-0.481109\pi\)
0.0593116 + 0.998240i \(0.481109\pi\)
\(954\) 3.29758 0.106763
\(955\) 11.0432 0.357351
\(956\) 23.8258 0.770580
\(957\) 21.9821 0.710580
\(958\) 65.6741 2.12183
\(959\) 0 0
\(960\) −10.0507 −0.324385
\(961\) 26.2139 0.845610
\(962\) 0 0
\(963\) 83.6336 2.69506
\(964\) 7.31772 0.235688
\(965\) −1.96419 −0.0632296
\(966\) 0 0
\(967\) 30.0288 0.965660 0.482830 0.875714i \(-0.339609\pi\)
0.482830 + 0.875714i \(0.339609\pi\)
\(968\) 3.81087 0.122486
\(969\) −60.3765 −1.93957
\(970\) 11.5028 0.369334
\(971\) −4.90544 −0.157423 −0.0787115 0.996897i \(-0.525081\pi\)
−0.0787115 + 0.996897i \(0.525081\pi\)
\(972\) 90.3562 2.89818
\(973\) 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 \(0.617923\pi\)
\(978\) 8.13843 0.260238
\(979\) −1.61650 −0.0516635
\(980\) 0 0
\(981\) 50.9584 1.62698
\(982\) −15.3956 −0.491293
\(983\) −57.4053 −1.83094 −0.915472 0.402382i \(-0.868182\pi\)
−0.915472 + 0.402382i \(0.868182\pi\)
\(984\) −13.1558 −0.419390
\(985\) 16.1960 0.516047
\(986\) −10.6107 −0.337913
\(987\) 0 0
\(988\) 0 0
\(989\) −21.1805 −0.673500
\(990\) −51.3419 −1.63175
\(991\) 31.1793 0.990443 0.495221 0.868767i \(-0.335087\pi\)
0.495221 + 0.868767i \(0.335087\pi\)
\(992\) 52.6861 1.67279
\(993\) −57.8234 −1.83497
\(994\) 0 0
\(995\) −17.2525 −0.546941
\(996\) −20.3541 −0.644944
\(997\) 36.0576 1.14195 0.570977 0.820966i \(-0.306564\pi\)
0.570977 + 0.820966i \(0.306564\pi\)
\(998\) −31.2151 −0.988096
\(999\) 84.3827 2.66975
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 8281.2.a.bk.1.1 3
7.6 odd 2 8281.2.a.bh.1.1 3
13.12 even 2 637.2.a.i.1.3 yes 3
39.38 odd 2 5733.2.a.bd.1.1 3
91.12 odd 6 637.2.e.l.508.1 6
91.25 even 6 637.2.e.k.79.1 6
91.38 odd 6 637.2.e.l.79.1 6
91.51 even 6 637.2.e.k.508.1 6
91.90 odd 2 637.2.a.h.1.3 3
273.272 even 2 5733.2.a.be.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.3 3 91.90 odd 2
637.2.a.i.1.3 yes 3 13.12 even 2
637.2.e.k.79.1 6 91.25 even 6
637.2.e.k.508.1 6 91.51 even 6
637.2.e.l.79.1 6 91.38 odd 6
637.2.e.l.508.1 6 91.12 odd 6
5733.2.a.bd.1.1 3 39.38 odd 2
5733.2.a.be.1.1 3 273.272 even 2
8281.2.a.bh.1.1 3 7.6 odd 2
8281.2.a.bk.1.1 3 1.1 even 1 trivial