Properties

Label 637.2.a.h.1.3
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.86620 q^{2} -3.34889 q^{3} +1.48270 q^{4} +0.866198 q^{5} -6.24970 q^{6} -0.965392 q^{8} +8.21509 q^{9} +O(q^{10})\) \(q+1.86620 q^{2} -3.34889 q^{3} +1.48270 q^{4} +0.866198 q^{5} -6.24970 q^{6} -0.965392 q^{8} +8.21509 q^{9} +1.61650 q^{10} -3.86620 q^{11} -4.96539 q^{12} +1.00000 q^{13} -2.90081 q^{15} -4.76700 q^{16} -3.34889 q^{17} +15.3310 q^{18} -5.38350 q^{19} +1.28431 q^{20} -7.21509 q^{22} -5.24970 q^{23} +3.23300 q^{24} -4.24970 q^{25} +1.86620 q^{26} -17.4648 q^{27} +1.69779 q^{29} -5.41348 q^{30} +7.56399 q^{31} -6.96539 q^{32} +12.9475 q^{33} -6.24970 q^{34} +12.1805 q^{36} -4.83159 q^{37} -10.0467 q^{38} -3.34889 q^{39} -0.836221 q^{40} -4.06922 q^{41} +4.03461 q^{43} -5.73240 q^{44} +7.11590 q^{45} -9.79698 q^{46} -3.65111 q^{47} +15.9642 q^{48} -7.93078 q^{50} +11.2151 q^{51} +1.48270 q^{52} -0.215092 q^{53} -32.5928 q^{54} -3.34889 q^{55} +18.0288 q^{57} +3.16841 q^{58} -2.78491 q^{59} -4.30101 q^{60} +9.03461 q^{61} +14.1159 q^{62} -3.46479 q^{64} +0.866198 q^{65} +24.1626 q^{66} -7.66318 q^{67} -4.96539 q^{68} +17.5807 q^{69} +4.90081 q^{71} -7.93078 q^{72} +15.5461 q^{73} -9.01671 q^{74} +14.2318 q^{75} -7.98210 q^{76} -6.24970 q^{78} +9.43018 q^{79} -4.12917 q^{80} +33.8425 q^{81} -7.59396 q^{82} -4.09919 q^{83} -2.90081 q^{85} +7.52938 q^{86} -5.68571 q^{87} +3.73240 q^{88} -0.418110 q^{89} +13.2797 q^{90} -7.78371 q^{92} -25.3310 q^{93} -6.81369 q^{94} -4.66318 q^{95} +23.3264 q^{96} -7.11590 q^{97} -31.7612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} - 4q^{3} + 6q^{4} - 5q^{5} - 2q^{6} - 6q^{8} + 11q^{9} + O(q^{10}) \) \( 3q - 2q^{2} - 4q^{3} + 6q^{4} - 5q^{5} - 2q^{6} - 6q^{8} + 11q^{9} + 14q^{10} - 4q^{11} - 18q^{12} + 3q^{13} + 2q^{15} + 4q^{16} - 4q^{17} + 8q^{18} - 7q^{19} - 16q^{20} - 8q^{22} + q^{23} + 28q^{24} + 4q^{25} - 2q^{26} - 22q^{27} - 7q^{29} - 24q^{30} + 3q^{31} - 24q^{32} + 10q^{33} - 2q^{34} + 26q^{36} - 10q^{37} - 12q^{38} - 4q^{39} + 22q^{40} - 6q^{41} + 9q^{43} - 2q^{44} - 3q^{45} - 28q^{46} - 17q^{47} - 16q^{48} - 30q^{50} + 20q^{51} + 6q^{52} + 13q^{53} - 28q^{54} - 4q^{55} + 4q^{57} + 14q^{58} - 22q^{59} + 42q^{60} + 24q^{61} + 18q^{62} + 20q^{64} - 5q^{65} + 30q^{66} - 14q^{67} - 18q^{68} - 2q^{69} + 4q^{71} - 30q^{72} - 5q^{73} + 8q^{74} - 6q^{75} + 8q^{76} - 2q^{78} + q^{79} - 40q^{80} + 15q^{81} - 20q^{82} - 23q^{83} + 2q^{85} + 6q^{86} - 20q^{87} - 4q^{88} + 11q^{89} + 40q^{90} + 30q^{92} - 38q^{93} + 16q^{94} - 5q^{95} + 52q^{96} + 3q^{97} - 30q^{99} + O(q^{100}) \)

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 \(-0.270641\pi\)
0.659801 + 0.751441i \(0.270641\pi\)
\(3\) −3.34889 −1.93348 −0.966742 0.255752i \(-0.917677\pi\)
−0.966742 + 0.255752i \(0.917677\pi\)
\(4\) 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 \(-0.698063\pi\)
−0.582851 + 0.812579i \(0.698063\pi\)
\(12\) −4.96539 −1.43339
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −2.90081 −0.748985
\(16\) −4.76700 −1.19175
\(17\) −3.34889 −0.812226 −0.406113 0.913823i \(-0.633116\pi\)
−0.406113 + 0.913823i \(0.633116\pi\)
\(18\) 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
\(26\) 1.86620 0.365992
\(27\) −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 \(-0.630002\pi\)
−0.397154 + 0.917752i \(0.630002\pi\)
\(38\) −10.0467 −1.62979
\(39\) −3.34889 −0.536252
\(40\) −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
\(52\) 1.48270 0.205613
\(53\) −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 \(-0.303685\pi\)
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
\(66\) 24.1626 2.97421
\(67\) −7.66318 −0.936206 −0.468103 0.883674i \(-0.655062\pi\)
−0.468103 + 0.883674i \(0.655062\pi\)
\(68\) −4.96539 −0.602142
\(69\) 17.5807 2.11647
\(70\) 0 0
\(71\) 4.90081 0.581619 0.290809 0.956781i \(-0.406075\pi\)
0.290809 + 0.956781i \(0.406075\pi\)
\(72\) −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
\(78\) −6.24970 −0.707639
\(79\) 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 \(-0.747840\pi\)
−0.702292 + 0.711889i \(0.747840\pi\)
\(102\) 20.9296 2.07234
\(103\) −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 \(-0.336232\pi\)
0.492092 + 0.870543i \(0.336232\pi\)
\(108\) −25.8950 −2.49175
\(109\) −6.20302 −0.594141 −0.297071 0.954855i \(-0.596010\pi\)
−0.297071 + 0.954855i \(0.596010\pi\)
\(110\) −6.24970 −0.595886
\(111\) 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
\(117\) 8.21509 0.759486
\(118\) −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
\(130\) 1.61650 0.141776
\(131\) 10.1626 0.887909 0.443954 0.896049i \(-0.353575\pi\)
0.443954 + 0.896049i \(0.353575\pi\)
\(132\) 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 \(-0.391915\pi\)
0.333071 + 0.942902i \(0.391915\pi\)
\(138\) 32.8091 2.79289
\(139\) −5.08129 −0.430989 −0.215495 0.976505i \(-0.569136\pi\)
−0.215495 + 0.976505i \(0.569136\pi\)
\(140\) 0 0
\(141\) 12.2272 1.02971
\(142\) 9.14588 0.767505
\(143\) −3.86620 −0.323308
\(144\) −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 \(-0.532585\pi\)
−0.102190 + 0.994765i \(0.532585\pi\)
\(150\) 26.5594 2.16856
\(151\) −3.26178 −0.265439 −0.132720 0.991154i \(-0.542371\pi\)
−0.132720 + 0.991154i \(0.542371\pi\)
\(152\) 5.19719 0.421548
\(153\) −27.5115 −2.22417
\(154\) 0 0
\(155\) 6.55191 0.526262
\(156\) −4.96539 −0.397550
\(157\) −0.720322 −0.0574880 −0.0287440 0.999587i \(-0.509151\pi\)
−0.0287440 + 0.999587i \(0.509151\pi\)
\(158\) 17.5986 1.40007
\(159\) 0.720322 0.0571252
\(160\) −6.03341 −0.476983
\(161\) 0 0
\(162\) 63.1568 4.96206
\(163\) 1.30221 0.101997 0.0509985 0.998699i \(-0.483760\pi\)
0.0509985 + 0.998699i \(0.483760\pi\)
\(164\) −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
\(169\) 1.00000 0.0769231
\(170\) −5.41348 −0.415195
\(171\) −44.2260 −3.38204
\(172\) 5.98210 0.456131
\(173\) −15.8483 −1.20492 −0.602462 0.798148i \(-0.705813\pi\)
−0.602462 + 0.798148i \(0.705813\pi\)
\(174\) −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 \(-0.578661\pi\)
−0.244614 + 0.969621i \(0.578661\pi\)
\(182\) 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 \(-0.473993\pi\)
0.0816128 + 0.996664i \(0.473993\pi\)
\(194\) −13.2797 −0.953425
\(195\) −2.90081 −0.207731
\(196\) 0 0
\(197\) −18.6978 −1.33216 −0.666081 0.745879i \(-0.732030\pi\)
−0.666081 + 0.745879i \(0.732030\pi\)
\(198\) −59.2727 −4.21232
\(199\) 19.9175 1.41191 0.705957 0.708254i \(-0.250517\pi\)
0.705957 + 0.708254i \(0.250517\pi\)
\(200\) 4.10263 0.290100
\(201\) 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
\(208\) −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
\(213\) −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
\(221\) −3.34889 −0.225271
\(222\) 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 \(0.626224\pi\)
\(234\) 15.3310 1.00222
\(235\) −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 \(-0.673962\pi\)
−0.519716 + 0.854339i \(0.673962\pi\)
\(240\) 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
\(247\) −5.38350 −0.342544
\(248\) −7.30221 −0.463691
\(249\) 13.7278 0.869962
\(250\) −14.9521 −0.945655
\(251\) 15.2439 0.962185 0.481092 0.876670i \(-0.340240\pi\)
0.481092 + 0.876670i \(0.340240\pi\)
\(252\) 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 \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(258\) −25.2151 −1.56982
\(259\) 0 0
\(260\) 1.28431 0.0796494
\(261\) 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 \(-0.334054\pi\)
0.498037 + 0.867156i \(0.334054\pi\)
\(270\) −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 \(-0.492104\pi\)
0.0248043 + 0.999692i \(0.492104\pi\)
\(282\) 22.8183 1.35881
\(283\) 11.0588 0.657375 0.328687 0.944439i \(-0.393394\pi\)
0.328687 + 0.944439i \(0.393394\pi\)
\(284\) 7.26641 0.431182
\(285\) 15.6165 0.925041
\(286\) −7.21509 −0.426637
\(287\) 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
\(299\) −5.24970 −0.303598
\(300\) 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 \(-0.461596\pi\)
0.120358 + 0.992731i \(0.461596\pi\)
\(312\) 3.23300 0.183032
\(313\) 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 \(-0.641037\pi\)
−0.428723 + 0.903436i \(0.641037\pi\)
\(318\) 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
\(325\) −4.24970 −0.235731
\(326\) 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 \(-0.342620\pi\)
0.474524 + 0.880243i \(0.342620\pi\)
\(332\) −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 \(0.744891\pi\)
\(338\) 1.86620 0.101508
\(339\) −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
\(351\) −17.4648 −0.932202
\(352\) 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 \(-0.665546\pi\)
−0.496947 + 0.867781i \(0.665546\pi\)
\(360\) −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 \(-0.706517\pi\)
−0.604223 + 0.796815i \(0.706517\pi\)
\(368\) 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
\(377\) 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
\(381\) 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 \(0.423796\pi\)
\(390\) −5.41348 −0.274122
\(391\) 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 \(-0.421866\pi\)
0.243006 + 0.970025i \(0.421866\pi\)
\(402\) 47.8926 2.38866
\(403\) 7.56399 0.376789
\(404\) −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
\(416\) −6.96539 −0.341506
\(417\) 17.0167 0.833312
\(418\) 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
\(423\) −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
\(429\) 12.9475 0.625111
\(430\) 6.52193 0.314516
\(431\) −16.9895 −0.818357 −0.409179 0.912454i \(-0.634185\pi\)
−0.409179 + 0.912454i \(0.634185\pi\)
\(432\) 83.2547 4.00560
\(433\) 30.8604 1.48305 0.741527 0.670923i \(-0.234102\pi\)
0.741527 + 0.670923i \(0.234102\pi\)
\(434\) 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 \(-0.651476\pi\)
−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 \(-0.804180\pi\)
−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 \(-0.253584\pi\)
0.699101 + 0.715023i \(0.253584\pi\)
\(450\) −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 \(-0.630480\pi\)
−0.398531 + 0.917155i \(0.630480\pi\)
\(458\) −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 \(-0.292373\pi\)
0.607000 + 0.794702i \(0.292373\pi\)
\(464\) −8.09337 −0.375725
\(465\) −21.9417 −1.01752
\(466\) −22.0046 −1.01934
\(467\) −22.9187 −1.06055 −0.530276 0.847825i \(-0.677912\pi\)
−0.530276 + 0.847825i \(0.677912\pi\)
\(468\) 12.1805 0.563043
\(469\) 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
\(481\) −4.83159 −0.220302
\(482\) 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 \(-0.721572\pi\)
−0.641221 + 0.767357i \(0.721572\pi\)
\(488\) −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
\(494\) −10.0467 −0.452022
\(495\) −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 \(-0.622149\pi\)
−0.374392 + 0.927271i \(0.622149\pi\)
\(500\) −11.8795 −0.531266
\(501\) −54.0863 −2.41640
\(502\) 28.4481 1.26970
\(503\) −21.2213 −0.946213 −0.473106 0.881005i \(-0.656867\pi\)
−0.473106 + 0.881005i \(0.656867\pi\)
\(504\) 0 0
\(505\) −12.2272 −0.544102
\(506\) 37.8771 1.68384
\(507\) −3.34889 −0.148730
\(508\) −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
\(520\) −0.836221 −0.0366707
\(521\) −3.46479 −0.151795 −0.0758977 0.997116i \(-0.524182\pi\)
−0.0758977 + 0.997116i \(0.524182\pi\)
\(522\) 26.0288 1.13925
\(523\) −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
\(528\) −61.7207 −2.68605
\(529\) 4.55936 0.198233
\(530\) −0.347696 −0.0151030
\(531\) −22.8783 −0.992832
\(532\) 0 0
\(533\) −4.06922 −0.176257
\(534\) 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 \(-0.325558\pi\)
0.521003 + 0.853555i \(0.325558\pi\)
\(542\) −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 \(-0.618878\pi\)
−0.364846 + 0.931068i \(0.618878\pi\)
\(558\) 115.963 4.90912
\(559\) 4.03461 0.170646
\(560\) 0 0
\(561\) −43.3598 −1.83065
\(562\) 1.55191 0.0654635
\(563\) 15.3598 0.647337 0.323669 0.946171i \(-0.395084\pi\)
0.323669 + 0.946171i \(0.395084\pi\)
\(564\) 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 \(0.693689\pi\)
\(572\) −5.73240 −0.239684
\(573\) −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
\(585\) 7.11590 0.294206
\(586\) −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
\(598\) −9.79698 −0.400628
\(599\) 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 \(-0.283296\pi\)
0.629410 + 0.777073i \(0.283296\pi\)
\(602\) 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 \(-0.234606\pi\)
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
\(612\) −40.7912 −1.64888
\(613\) 28.9988 1.17125 0.585625 0.810582i \(-0.300849\pi\)
0.585625 + 0.810582i \(0.300849\pi\)
\(614\) −28.2877 −1.14160
\(615\) 11.8040 0.475984
\(616\) 0 0
\(617\) −41.6515 −1.67683 −0.838414 0.545035i \(-0.816517\pi\)
−0.838414 + 0.545035i \(0.816517\pi\)
\(618\) 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
\(624\) 15.9642 0.639079
\(625\) 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 \(-0.750300\pi\)
−0.707774 + 0.706439i \(0.750300\pi\)
\(632\) −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 \(-0.523306\pi\)
−0.0731540 + 0.997321i \(0.523306\pi\)
\(648\) −32.6712 −1.28345
\(649\) 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 \(-0.843562\pi\)
−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
\(663\) 11.2151 0.435558
\(664\) 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
\(676\) 1.48270 0.0570268
\(677\) −43.4757 −1.67091 −0.835453 0.549562i \(-0.814795\pi\)
−0.835453 + 0.549562i \(0.814795\pi\)
\(678\) −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 \(-0.294905\pi\)
0.600659 + 0.799505i \(0.294905\pi\)
\(684\) −65.5737 −2.50727
\(685\) 6.75373 0.258047
\(686\) 0 0
\(687\) 29.7324 1.13436
\(688\) −19.2330 −0.733251
\(689\) −0.215092 −0.00819437
\(690\) 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 \(0.677593\pi\)
\(702\) −32.5928 −1.23013
\(703\) 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 \(-0.773074\pi\)
−0.756462 + 0.654037i \(0.773074\pi\)
\(710\) 7.92214 0.297313
\(711\) 77.4698 2.90535
\(712\) 0.403640 0.0151271
\(713\) −39.7087 −1.48710
\(714\) 0 0
\(715\) −3.34889 −0.125242
\(716\) −30.3431 −1.13397
\(717\) 53.8141 2.00972
\(718\) −35.1435 −1.31154
\(719\) 16.7445 0.624463 0.312232 0.950006i \(-0.398923\pi\)
0.312232 + 0.950006i \(0.398923\pi\)
\(720\) −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 \(-0.0958748\pi\)
0.954981 + 0.296666i \(0.0958748\pi\)
\(728\) 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 \(-0.375869\pi\)
0.380161 + 0.924920i \(0.375869\pi\)
\(740\) −6.20525 −0.228110
\(741\) 18.0288 0.662304
\(742\) 0 0
\(743\) −29.6966 −1.08946 −0.544731 0.838611i \(-0.683368\pi\)
−0.544731 + 0.838611i \(0.683368\pi\)
\(744\) 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
\(754\) 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
\(759\) −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
\(767\) −2.78491 −0.100557
\(768\) −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
\(780\) −4.30101 −0.154001
\(781\) −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
\(793\) 9.03461 0.320828
\(794\) 18.1851 0.645366
\(795\) 0.623942 0.0221289
\(796\) 29.5316 1.04672
\(797\) 0.291753 0.0103344 0.00516720 0.999987i \(-0.498355\pi\)
0.00516720 + 0.999987i \(0.498355\pi\)
\(798\) 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
\(806\) 14.1159 0.497211
\(807\) −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 \(-0.648159\pi\)
−0.448830 + 0.893617i \(0.648159\pi\)
\(822\) −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 \(-0.688430\pi\)
−0.557998 + 0.829842i \(0.688430\pi\)
\(828\) −63.9439 −2.22220
\(829\) 3.89336 0.135222 0.0676110 0.997712i \(-0.478462\pi\)
0.0676110 + 0.997712i \(0.478462\pi\)
\(830\) −6.62634 −0.230004
\(831\) 10.0467 0.348516
\(832\) −3.46479 −0.120120
\(833\) 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
\(845\) 0.866198 0.0297981
\(846\) −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 \(-0.447687\pi\)
0.163608 + 0.986526i \(0.447687\pi\)
\(858\) 24.1626 0.824897
\(859\) −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 \(-0.360932\pi\)
0.423128 + 0.906070i \(0.360932\pi\)
\(864\) 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
\(871\) −7.66318 −0.259657
\(872\) 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 \(-0.357394\pi\)
0.433173 + 0.901311i \(0.357394\pi\)
\(878\) −35.8258 −1.20906
\(879\) 90.1439 3.04048
\(880\) 15.9642 0.538153
\(881\) −25.6632 −0.864615 −0.432307 0.901726i \(-0.642300\pi\)
−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
\(885\) 8.07848 0.271555
\(886\) −64.1556 −2.15535
\(887\) 17.5247 0.588423 0.294212 0.955740i \(-0.404943\pi\)
0.294212 + 0.955740i \(0.404943\pi\)
\(888\) −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
\(897\) 17.5807 0.587002
\(898\) 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
\(923\) 4.90081 0.161312
\(924\) 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
\(936\) −7.93078 −0.259226
\(937\) −8.83784 −0.288720 −0.144360 0.989525i \(-0.546112\pi\)
−0.144360 + 0.989525i \(0.546112\pi\)
\(938\) 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 \(-0.342527\pi\)
0.474783 + 0.880103i \(0.342527\pi\)
\(948\) −46.8246 −1.52079
\(949\) 15.5461 0.504647
\(950\) 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
\(962\) −9.01671 −0.290710
\(963\) 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 \(-0.660391\pi\)
−0.482830 + 0.875714i \(0.660391\pi\)
\(968\) −3.81087 −0.122486
\(969\) −60.3765 −1.93957
\(970\) −11.5028 −0.369334
\(971\) 4.90544 0.157423 0.0787115 0.996897i \(-0.474919\pi\)
0.0787115 + 0.996897i \(0.474919\pi\)
\(972\) −90.3562 −2.89818
\(973\) 0 0
\(974\) −52.8153 −1.69231
\(975\) 14.2318 0.455782
\(976\) −43.0680 −1.37857
\(977\) 22.6332 0.724100 0.362050 0.932159i \(-0.382077\pi\)
0.362050 + 0.932159i \(0.382077\pi\)
\(978\) −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
\(988\) −7.98210 −0.253944
\(989\) −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 \(-0.693436\pi\)
−0.570977 + 0.820966i \(0.693436\pi\)
\(998\) −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 637.2.a.h.1.3 3
3.2 odd 2 5733.2.a.be.1.1 3
7.2 even 3 637.2.e.l.508.1 6
7.3 odd 6 637.2.e.k.79.1 6
7.4 even 3 637.2.e.l.79.1 6
7.5 odd 6 637.2.e.k.508.1 6
7.6 odd 2 637.2.a.i.1.3 yes 3
13.12 even 2 8281.2.a.bh.1.1 3
21.20 even 2 5733.2.a.bd.1.1 3
91.90 odd 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 1.1 even 1 trivial
637.2.a.i.1.3 yes 3 7.6 odd 2
637.2.e.k.79.1 6 7.3 odd 6
637.2.e.k.508.1 6 7.5 odd 6
637.2.e.l.79.1 6 7.4 even 3
637.2.e.l.508.1 6 7.2 even 3
5733.2.a.bd.1.1 3 21.20 even 2
5733.2.a.be.1.1 3 3.2 odd 2
8281.2.a.bh.1.1 3 13.12 even 2
8281.2.a.bk.1.1 3 91.90 odd 2