Properties

Label 4864.2.a.be.1.3
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.59774\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.59774 q^{3} +1.59774 q^{5} +4.34596 q^{7} +3.74823 q^{9} +O(q^{10})\) \(q+2.59774 q^{3} +1.59774 q^{5} +4.34596 q^{7} +3.74823 q^{9} +1.74823 q^{11} +0.748228 q^{13} +4.15049 q^{15} +3.00000 q^{17} -1.00000 q^{19} +11.2897 q^{21} -1.94370 q^{23} -2.44724 q^{25} +1.94370 q^{27} -7.28966 q^{29} +0.654037 q^{31} +4.54143 q^{33} +6.94370 q^{35} +3.84951 q^{37} +1.94370 q^{39} +4.54143 q^{41} -10.4402 q^{43} +5.98868 q^{45} +7.59774 q^{47} +11.8874 q^{49} +7.79321 q^{51} +6.59774 q^{53} +2.79321 q^{55} -2.59774 q^{57} -6.74823 q^{59} +9.09419 q^{61} +16.2897 q^{63} +1.19547 q^{65} +7.40226 q^{67} -5.04921 q^{69} -1.69901 q^{71} -16.6848 q^{73} -6.35729 q^{75} +7.59774 q^{77} +0.654037 q^{79} -6.19547 q^{81} -12.5793 q^{83} +4.79321 q^{85} -18.9366 q^{87} -14.5414 q^{89} +3.25177 q^{91} +1.69901 q^{93} -1.59774 q^{95} -10.5793 q^{97} +6.55276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} - 2q^{5} + 3q^{7} + 8q^{9} + O(q^{10}) \) \( 3q + q^{3} - 2q^{5} + 3q^{7} + 8q^{9} + 2q^{11} - q^{13} + 16q^{15} + 9q^{17} - 3q^{19} + 7q^{21} + 11q^{23} + 3q^{25} - 11q^{27} + 5q^{29} + 12q^{31} - 10q^{33} + 4q^{35} + 8q^{37} - 11q^{39} - 10q^{41} - 8q^{43} - 16q^{45} + 16q^{47} + 2q^{49} + 3q^{51} + 13q^{53} - 12q^{55} - q^{57} - 17q^{59} + 14q^{61} + 22q^{63} - 10q^{65} + 29q^{67} - 19q^{69} + 2q^{71} - 17q^{73} - 43q^{75} + 16q^{77} + 12q^{79} - 5q^{81} + 16q^{83} - 6q^{85} - 27q^{87} - 20q^{89} + 13q^{91} - 2q^{93} + 2q^{95} + 22q^{97} + 30q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.59774 1.49980 0.749902 0.661549i \(-0.230101\pi\)
0.749902 + 0.661549i \(0.230101\pi\)
\(4\) 0 0
\(5\) 1.59774 0.714529 0.357264 0.934003i \(-0.383709\pi\)
0.357264 + 0.934003i \(0.383709\pi\)
\(6\) 0 0
\(7\) 4.34596 1.64262 0.821310 0.570482i \(-0.193244\pi\)
0.821310 + 0.570482i \(0.193244\pi\)
\(8\) 0 0
\(9\) 3.74823 1.24941
\(10\) 0 0
\(11\) 1.74823 0.527111 0.263555 0.964644i \(-0.415105\pi\)
0.263555 + 0.964644i \(0.415105\pi\)
\(12\) 0 0
\(13\) 0.748228 0.207521 0.103761 0.994602i \(-0.466912\pi\)
0.103761 + 0.994602i \(0.466912\pi\)
\(14\) 0 0
\(15\) 4.15049 1.07165
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 11.2897 2.46361
\(22\) 0 0
\(23\) −1.94370 −0.405289 −0.202645 0.979252i \(-0.564954\pi\)
−0.202645 + 0.979252i \(0.564954\pi\)
\(24\) 0 0
\(25\) −2.44724 −0.489448
\(26\) 0 0
\(27\) 1.94370 0.374065
\(28\) 0 0
\(29\) −7.28966 −1.35366 −0.676828 0.736141i \(-0.736646\pi\)
−0.676828 + 0.736141i \(0.736646\pi\)
\(30\) 0 0
\(31\) 0.654037 0.117468 0.0587342 0.998274i \(-0.481294\pi\)
0.0587342 + 0.998274i \(0.481294\pi\)
\(32\) 0 0
\(33\) 4.54143 0.790562
\(34\) 0 0
\(35\) 6.94370 1.17370
\(36\) 0 0
\(37\) 3.84951 0.632855 0.316428 0.948617i \(-0.397517\pi\)
0.316428 + 0.948617i \(0.397517\pi\)
\(38\) 0 0
\(39\) 1.94370 0.311241
\(40\) 0 0
\(41\) 4.54143 0.709253 0.354626 0.935008i \(-0.384608\pi\)
0.354626 + 0.935008i \(0.384608\pi\)
\(42\) 0 0
\(43\) −10.4402 −1.59211 −0.796054 0.605225i \(-0.793083\pi\)
−0.796054 + 0.605225i \(0.793083\pi\)
\(44\) 0 0
\(45\) 5.98868 0.892739
\(46\) 0 0
\(47\) 7.59774 1.10824 0.554122 0.832436i \(-0.313054\pi\)
0.554122 + 0.832436i \(0.313054\pi\)
\(48\) 0 0
\(49\) 11.8874 1.69820
\(50\) 0 0
\(51\) 7.79321 1.09127
\(52\) 0 0
\(53\) 6.59774 0.906269 0.453134 0.891442i \(-0.350306\pi\)
0.453134 + 0.891442i \(0.350306\pi\)
\(54\) 0 0
\(55\) 2.79321 0.376636
\(56\) 0 0
\(57\) −2.59774 −0.344078
\(58\) 0 0
\(59\) −6.74823 −0.878544 −0.439272 0.898354i \(-0.644764\pi\)
−0.439272 + 0.898354i \(0.644764\pi\)
\(60\) 0 0
\(61\) 9.09419 1.16439 0.582196 0.813049i \(-0.302194\pi\)
0.582196 + 0.813049i \(0.302194\pi\)
\(62\) 0 0
\(63\) 16.2897 2.05230
\(64\) 0 0
\(65\) 1.19547 0.148280
\(66\) 0 0
\(67\) 7.40226 0.904331 0.452165 0.891934i \(-0.350652\pi\)
0.452165 + 0.891934i \(0.350652\pi\)
\(68\) 0 0
\(69\) −5.04921 −0.607854
\(70\) 0 0
\(71\) −1.69901 −0.201636 −0.100818 0.994905i \(-0.532146\pi\)
−0.100818 + 0.994905i \(0.532146\pi\)
\(72\) 0 0
\(73\) −16.6848 −1.95281 −0.976406 0.215941i \(-0.930718\pi\)
−0.976406 + 0.215941i \(0.930718\pi\)
\(74\) 0 0
\(75\) −6.35729 −0.734076
\(76\) 0 0
\(77\) 7.59774 0.865842
\(78\) 0 0
\(79\) 0.654037 0.0735849 0.0367924 0.999323i \(-0.488286\pi\)
0.0367924 + 0.999323i \(0.488286\pi\)
\(80\) 0 0
\(81\) −6.19547 −0.688386
\(82\) 0 0
\(83\) −12.5793 −1.38076 −0.690380 0.723447i \(-0.742557\pi\)
−0.690380 + 0.723447i \(0.742557\pi\)
\(84\) 0 0
\(85\) 4.79321 0.519896
\(86\) 0 0
\(87\) −18.9366 −2.03022
\(88\) 0 0
\(89\) −14.5414 −1.54139 −0.770694 0.637205i \(-0.780091\pi\)
−0.770694 + 0.637205i \(0.780091\pi\)
\(90\) 0 0
\(91\) 3.25177 0.340878
\(92\) 0 0
\(93\) 1.69901 0.176180
\(94\) 0 0
\(95\) −1.59774 −0.163924
\(96\) 0 0
\(97\) −10.5793 −1.07417 −0.537084 0.843529i \(-0.680474\pi\)
−0.537084 + 0.843529i \(0.680474\pi\)
\(98\) 0 0
\(99\) 6.55276 0.658577
\(100\) 0 0
\(101\) −4.80453 −0.478069 −0.239034 0.971011i \(-0.576831\pi\)
−0.239034 + 0.971011i \(0.576831\pi\)
\(102\) 0 0
\(103\) −12.5793 −1.23948 −0.619739 0.784808i \(-0.712761\pi\)
−0.619739 + 0.784808i \(0.712761\pi\)
\(104\) 0 0
\(105\) 18.0379 1.76032
\(106\) 0 0
\(107\) 5.44015 0.525920 0.262960 0.964807i \(-0.415301\pi\)
0.262960 + 0.964807i \(0.415301\pi\)
\(108\) 0 0
\(109\) 14.7861 1.41625 0.708127 0.706085i \(-0.249541\pi\)
0.708127 + 0.706085i \(0.249541\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 14.0379 1.32057 0.660287 0.751014i \(-0.270435\pi\)
0.660287 + 0.751014i \(0.270435\pi\)
\(114\) 0 0
\(115\) −3.10552 −0.289591
\(116\) 0 0
\(117\) 2.80453 0.259279
\(118\) 0 0
\(119\) 13.0379 1.19518
\(120\) 0 0
\(121\) −7.94370 −0.722154
\(122\) 0 0
\(123\) 11.7974 1.06374
\(124\) 0 0
\(125\) −11.8987 −1.06425
\(126\) 0 0
\(127\) 7.54852 0.669823 0.334911 0.942250i \(-0.391294\pi\)
0.334911 + 0.942250i \(0.391294\pi\)
\(128\) 0 0
\(129\) −27.1208 −2.38785
\(130\) 0 0
\(131\) 0.0492139 0.00429984 0.00214992 0.999998i \(-0.499316\pi\)
0.00214992 + 0.999998i \(0.499316\pi\)
\(132\) 0 0
\(133\) −4.34596 −0.376843
\(134\) 0 0
\(135\) 3.10552 0.267280
\(136\) 0 0
\(137\) 17.3909 1.48581 0.742904 0.669398i \(-0.233448\pi\)
0.742904 + 0.669398i \(0.233448\pi\)
\(138\) 0 0
\(139\) −8.25177 −0.699906 −0.349953 0.936767i \(-0.613802\pi\)
−0.349953 + 0.936767i \(0.613802\pi\)
\(140\) 0 0
\(141\) 19.7369 1.66215
\(142\) 0 0
\(143\) 1.30807 0.109387
\(144\) 0 0
\(145\) −11.6469 −0.967226
\(146\) 0 0
\(147\) 30.8803 2.54696
\(148\) 0 0
\(149\) −6.28966 −0.515269 −0.257635 0.966242i \(-0.582943\pi\)
−0.257635 + 0.966242i \(0.582943\pi\)
\(150\) 0 0
\(151\) 22.1884 1.80566 0.902832 0.429992i \(-0.141484\pi\)
0.902832 + 0.429992i \(0.141484\pi\)
\(152\) 0 0
\(153\) 11.2447 0.909079
\(154\) 0 0
\(155\) 1.04498 0.0839346
\(156\) 0 0
\(157\) −8.99291 −0.717713 −0.358856 0.933393i \(-0.616833\pi\)
−0.358856 + 0.933393i \(0.616833\pi\)
\(158\) 0 0
\(159\) 17.1392 1.35922
\(160\) 0 0
\(161\) −8.44724 −0.665736
\(162\) 0 0
\(163\) −16.9929 −1.33099 −0.665494 0.746403i \(-0.731779\pi\)
−0.665494 + 0.746403i \(0.731779\pi\)
\(164\) 0 0
\(165\) 7.25601 0.564879
\(166\) 0 0
\(167\) −16.3389 −1.26434 −0.632170 0.774830i \(-0.717836\pi\)
−0.632170 + 0.774830i \(0.717836\pi\)
\(168\) 0 0
\(169\) −12.4402 −0.956935
\(170\) 0 0
\(171\) −3.74823 −0.286634
\(172\) 0 0
\(173\) −23.3839 −1.77784 −0.888921 0.458061i \(-0.848544\pi\)
−0.888921 + 0.458061i \(0.848544\pi\)
\(174\) 0 0
\(175\) −10.6356 −0.803978
\(176\) 0 0
\(177\) −17.5301 −1.31764
\(178\) 0 0
\(179\) −4.54143 −0.339443 −0.169721 0.985492i \(-0.554287\pi\)
−0.169721 + 0.985492i \(0.554287\pi\)
\(180\) 0 0
\(181\) −12.0900 −0.898639 −0.449320 0.893371i \(-0.648334\pi\)
−0.449320 + 0.893371i \(0.648334\pi\)
\(182\) 0 0
\(183\) 23.6243 1.74636
\(184\) 0 0
\(185\) 6.15049 0.452193
\(186\) 0 0
\(187\) 5.24468 0.383529
\(188\) 0 0
\(189\) 8.44724 0.614446
\(190\) 0 0
\(191\) 12.5485 0.907979 0.453990 0.891007i \(-0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(192\) 0 0
\(193\) 10.7298 0.772349 0.386175 0.922426i \(-0.373796\pi\)
0.386175 + 0.922426i \(0.373796\pi\)
\(194\) 0 0
\(195\) 3.10552 0.222391
\(196\) 0 0
\(197\) 9.38385 0.668572 0.334286 0.942472i \(-0.391505\pi\)
0.334286 + 0.942472i \(0.391505\pi\)
\(198\) 0 0
\(199\) 11.2404 0.796814 0.398407 0.917209i \(-0.369563\pi\)
0.398407 + 0.917209i \(0.369563\pi\)
\(200\) 0 0
\(201\) 19.2291 1.35632
\(202\) 0 0
\(203\) −31.6806 −2.22354
\(204\) 0 0
\(205\) 7.25601 0.506782
\(206\) 0 0
\(207\) −7.28543 −0.506372
\(208\) 0 0
\(209\) −1.74823 −0.120927
\(210\) 0 0
\(211\) −5.44015 −0.374516 −0.187258 0.982311i \(-0.559960\pi\)
−0.187258 + 0.982311i \(0.559960\pi\)
\(212\) 0 0
\(213\) −4.41359 −0.302414
\(214\) 0 0
\(215\) −16.6806 −1.13761
\(216\) 0 0
\(217\) 2.84242 0.192956
\(218\) 0 0
\(219\) −43.3428 −2.92883
\(220\) 0 0
\(221\) 2.24468 0.150994
\(222\) 0 0
\(223\) 25.4217 1.70237 0.851183 0.524869i \(-0.175886\pi\)
0.851183 + 0.524869i \(0.175886\pi\)
\(224\) 0 0
\(225\) −9.17282 −0.611521
\(226\) 0 0
\(227\) 3.74114 0.248308 0.124154 0.992263i \(-0.460378\pi\)
0.124154 + 0.992263i \(0.460378\pi\)
\(228\) 0 0
\(229\) −5.29675 −0.350019 −0.175010 0.984567i \(-0.555996\pi\)
−0.175010 + 0.984567i \(0.555996\pi\)
\(230\) 0 0
\(231\) 19.7369 1.29859
\(232\) 0 0
\(233\) 9.93661 0.650969 0.325485 0.945547i \(-0.394473\pi\)
0.325485 + 0.945547i \(0.394473\pi\)
\(234\) 0 0
\(235\) 12.1392 0.791872
\(236\) 0 0
\(237\) 1.69901 0.110363
\(238\) 0 0
\(239\) 27.8424 1.80098 0.900488 0.434880i \(-0.143209\pi\)
0.900488 + 0.434880i \(0.143209\pi\)
\(240\) 0 0
\(241\) 5.84951 0.376800 0.188400 0.982092i \(-0.439670\pi\)
0.188400 + 0.982092i \(0.439670\pi\)
\(242\) 0 0
\(243\) −21.9253 −1.40651
\(244\) 0 0
\(245\) 18.9929 1.21341
\(246\) 0 0
\(247\) −0.748228 −0.0476086
\(248\) 0 0
\(249\) −32.6778 −2.07087
\(250\) 0 0
\(251\) 15.2447 0.962236 0.481118 0.876656i \(-0.340231\pi\)
0.481118 + 0.876656i \(0.340231\pi\)
\(252\) 0 0
\(253\) −3.39803 −0.213632
\(254\) 0 0
\(255\) 12.4515 0.779742
\(256\) 0 0
\(257\) 7.58641 0.473227 0.236614 0.971604i \(-0.423962\pi\)
0.236614 + 0.971604i \(0.423962\pi\)
\(258\) 0 0
\(259\) 16.7298 1.03954
\(260\) 0 0
\(261\) −27.3233 −1.69127
\(262\) 0 0
\(263\) −2.79321 −0.172236 −0.0861182 0.996285i \(-0.527446\pi\)
−0.0861182 + 0.996285i \(0.527446\pi\)
\(264\) 0 0
\(265\) 10.5414 0.647555
\(266\) 0 0
\(267\) −37.7748 −2.31178
\(268\) 0 0
\(269\) 25.0308 1.52615 0.763077 0.646307i \(-0.223687\pi\)
0.763077 + 0.646307i \(0.223687\pi\)
\(270\) 0 0
\(271\) −27.5301 −1.67234 −0.836168 0.548474i \(-0.815209\pi\)
−0.836168 + 0.548474i \(0.815209\pi\)
\(272\) 0 0
\(273\) 8.44724 0.511250
\(274\) 0 0
\(275\) −4.27834 −0.257993
\(276\) 0 0
\(277\) −10.4023 −0.625012 −0.312506 0.949916i \(-0.601168\pi\)
−0.312506 + 0.949916i \(0.601168\pi\)
\(278\) 0 0
\(279\) 2.45148 0.146766
\(280\) 0 0
\(281\) −13.8874 −0.828453 −0.414226 0.910174i \(-0.635948\pi\)
−0.414226 + 0.910174i \(0.635948\pi\)
\(282\) 0 0
\(283\) 27.3346 1.62488 0.812438 0.583048i \(-0.198140\pi\)
0.812438 + 0.583048i \(0.198140\pi\)
\(284\) 0 0
\(285\) −4.15049 −0.245854
\(286\) 0 0
\(287\) 19.7369 1.16503
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −27.4823 −1.61104
\(292\) 0 0
\(293\) −10.1321 −0.591923 −0.295961 0.955200i \(-0.595640\pi\)
−0.295961 + 0.955200i \(0.595640\pi\)
\(294\) 0 0
\(295\) −10.7819 −0.627745
\(296\) 0 0
\(297\) 3.39803 0.197174
\(298\) 0 0
\(299\) −1.45433 −0.0841061
\(300\) 0 0
\(301\) −45.3725 −2.61523
\(302\) 0 0
\(303\) −12.4809 −0.717009
\(304\) 0 0
\(305\) 14.5301 0.831992
\(306\) 0 0
\(307\) 17.1208 0.977133 0.488567 0.872527i \(-0.337520\pi\)
0.488567 + 0.872527i \(0.337520\pi\)
\(308\) 0 0
\(309\) −32.6778 −1.85897
\(310\) 0 0
\(311\) 31.2404 1.77148 0.885742 0.464179i \(-0.153650\pi\)
0.885742 + 0.464179i \(0.153650\pi\)
\(312\) 0 0
\(313\) −25.2291 −1.42603 −0.713017 0.701147i \(-0.752672\pi\)
−0.713017 + 0.701147i \(0.752672\pi\)
\(314\) 0 0
\(315\) 26.0266 1.46643
\(316\) 0 0
\(317\) −11.6427 −0.653920 −0.326960 0.945038i \(-0.606024\pi\)
−0.326960 + 0.945038i \(0.606024\pi\)
\(318\) 0 0
\(319\) −12.7440 −0.713527
\(320\) 0 0
\(321\) 14.1321 0.788776
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −1.83110 −0.101571
\(326\) 0 0
\(327\) 38.4104 2.12410
\(328\) 0 0
\(329\) 33.0195 1.82042
\(330\) 0 0
\(331\) 12.5977 0.692434 0.346217 0.938154i \(-0.387466\pi\)
0.346217 + 0.938154i \(0.387466\pi\)
\(332\) 0 0
\(333\) 14.4288 0.790695
\(334\) 0 0
\(335\) 11.8269 0.646170
\(336\) 0 0
\(337\) −5.49646 −0.299411 −0.149706 0.988731i \(-0.547833\pi\)
−0.149706 + 0.988731i \(0.547833\pi\)
\(338\) 0 0
\(339\) 36.4667 1.98060
\(340\) 0 0
\(341\) 1.14341 0.0619189
\(342\) 0 0
\(343\) 21.2404 1.14688
\(344\) 0 0
\(345\) −8.06731 −0.434329
\(346\) 0 0
\(347\) 10.3417 0.555173 0.277586 0.960701i \(-0.410465\pi\)
0.277586 + 0.960701i \(0.410465\pi\)
\(348\) 0 0
\(349\) −30.6806 −1.64230 −0.821148 0.570716i \(-0.806666\pi\)
−0.821148 + 0.570716i \(0.806666\pi\)
\(350\) 0 0
\(351\) 1.45433 0.0776264
\(352\) 0 0
\(353\) 23.3276 1.24160 0.620800 0.783969i \(-0.286808\pi\)
0.620800 + 0.783969i \(0.286808\pi\)
\(354\) 0 0
\(355\) −2.71457 −0.144075
\(356\) 0 0
\(357\) 33.8690 1.79254
\(358\) 0 0
\(359\) 12.1576 0.641653 0.320826 0.947138i \(-0.396039\pi\)
0.320826 + 0.947138i \(0.396039\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −20.6356 −1.08309
\(364\) 0 0
\(365\) −26.6580 −1.39534
\(366\) 0 0
\(367\) 9.08287 0.474122 0.237061 0.971495i \(-0.423816\pi\)
0.237061 + 0.971495i \(0.423816\pi\)
\(368\) 0 0
\(369\) 17.0223 0.886147
\(370\) 0 0
\(371\) 28.6735 1.48865
\(372\) 0 0
\(373\) −18.1321 −0.938844 −0.469422 0.882974i \(-0.655538\pi\)
−0.469422 + 0.882974i \(0.655538\pi\)
\(374\) 0 0
\(375\) −30.9097 −1.59617
\(376\) 0 0
\(377\) −5.45433 −0.280912
\(378\) 0 0
\(379\) 23.6427 1.21444 0.607222 0.794532i \(-0.292284\pi\)
0.607222 + 0.794532i \(0.292284\pi\)
\(380\) 0 0
\(381\) 19.6091 1.00460
\(382\) 0 0
\(383\) −21.9253 −1.12033 −0.560165 0.828381i \(-0.689262\pi\)
−0.560165 + 0.828381i \(0.689262\pi\)
\(384\) 0 0
\(385\) 12.1392 0.618669
\(386\) 0 0
\(387\) −39.1321 −1.98920
\(388\) 0 0
\(389\) −29.2599 −1.48354 −0.741769 0.670656i \(-0.766013\pi\)
−0.741769 + 0.670656i \(0.766013\pi\)
\(390\) 0 0
\(391\) −5.83110 −0.294891
\(392\) 0 0
\(393\) 0.127845 0.00644892
\(394\) 0 0
\(395\) 1.04498 0.0525785
\(396\) 0 0
\(397\) −6.10128 −0.306214 −0.153107 0.988210i \(-0.548928\pi\)
−0.153107 + 0.988210i \(0.548928\pi\)
\(398\) 0 0
\(399\) −11.2897 −0.565190
\(400\) 0 0
\(401\) 16.0379 0.800894 0.400447 0.916320i \(-0.368855\pi\)
0.400447 + 0.916320i \(0.368855\pi\)
\(402\) 0 0
\(403\) 0.489369 0.0243772
\(404\) 0 0
\(405\) −9.89872 −0.491871
\(406\) 0 0
\(407\) 6.72982 0.333585
\(408\) 0 0
\(409\) −25.5343 −1.26259 −0.631296 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(410\) 0 0
\(411\) 45.1771 2.22842
\(412\) 0 0
\(413\) −29.3276 −1.44311
\(414\) 0 0
\(415\) −20.0984 −0.986593
\(416\) 0 0
\(417\) −21.4359 −1.04972
\(418\) 0 0
\(419\) −35.1586 −1.71761 −0.858806 0.512301i \(-0.828793\pi\)
−0.858806 + 0.512301i \(0.828793\pi\)
\(420\) 0 0
\(421\) 2.89872 0.141275 0.0706375 0.997502i \(-0.477497\pi\)
0.0706375 + 0.997502i \(0.477497\pi\)
\(422\) 0 0
\(423\) 28.4780 1.38465
\(424\) 0 0
\(425\) −7.34173 −0.356126
\(426\) 0 0
\(427\) 39.5230 1.91265
\(428\) 0 0
\(429\) 3.39803 0.164058
\(430\) 0 0
\(431\) 25.9774 1.25129 0.625643 0.780110i \(-0.284837\pi\)
0.625643 + 0.780110i \(0.284837\pi\)
\(432\) 0 0
\(433\) 30.9324 1.48652 0.743258 0.669005i \(-0.233280\pi\)
0.743258 + 0.669005i \(0.233280\pi\)
\(434\) 0 0
\(435\) −30.2557 −1.45065
\(436\) 0 0
\(437\) 1.94370 0.0929797
\(438\) 0 0
\(439\) 10.1278 0.483376 0.241688 0.970354i \(-0.422299\pi\)
0.241688 + 0.970354i \(0.422299\pi\)
\(440\) 0 0
\(441\) 44.5567 2.12175
\(442\) 0 0
\(443\) −14.3276 −0.680723 −0.340361 0.940295i \(-0.610549\pi\)
−0.340361 + 0.940295i \(0.610549\pi\)
\(444\) 0 0
\(445\) −23.2334 −1.10137
\(446\) 0 0
\(447\) −16.3389 −0.772802
\(448\) 0 0
\(449\) 6.61615 0.312235 0.156118 0.987738i \(-0.450102\pi\)
0.156118 + 0.987738i \(0.450102\pi\)
\(450\) 0 0
\(451\) 7.93946 0.373855
\(452\) 0 0
\(453\) 57.6395 2.70814
\(454\) 0 0
\(455\) 5.19547 0.243567
\(456\) 0 0
\(457\) 38.5864 1.80500 0.902498 0.430694i \(-0.141731\pi\)
0.902498 + 0.430694i \(0.141731\pi\)
\(458\) 0 0
\(459\) 5.83110 0.272172
\(460\) 0 0
\(461\) −33.1841 −1.54554 −0.772770 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(462\) 0 0
\(463\) 13.2826 0.617294 0.308647 0.951177i \(-0.400124\pi\)
0.308647 + 0.951177i \(0.400124\pi\)
\(464\) 0 0
\(465\) 2.71457 0.125885
\(466\) 0 0
\(467\) 20.8311 0.963948 0.481974 0.876185i \(-0.339920\pi\)
0.481974 + 0.876185i \(0.339920\pi\)
\(468\) 0 0
\(469\) 32.1700 1.48547
\(470\) 0 0
\(471\) −23.3612 −1.07643
\(472\) 0 0
\(473\) −18.2518 −0.839217
\(474\) 0 0
\(475\) 2.44724 0.112287
\(476\) 0 0
\(477\) 24.7298 1.13230
\(478\) 0 0
\(479\) 25.6848 1.17357 0.586785 0.809743i \(-0.300393\pi\)
0.586785 + 0.809743i \(0.300393\pi\)
\(480\) 0 0
\(481\) 2.88031 0.131331
\(482\) 0 0
\(483\) −21.9437 −0.998473
\(484\) 0 0
\(485\) −16.9030 −0.767524
\(486\) 0 0
\(487\) 9.51063 0.430968 0.215484 0.976507i \(-0.430867\pi\)
0.215484 + 0.976507i \(0.430867\pi\)
\(488\) 0 0
\(489\) −44.1431 −1.99622
\(490\) 0 0
\(491\) −6.89448 −0.311144 −0.155572 0.987825i \(-0.549722\pi\)
−0.155572 + 0.987825i \(0.549722\pi\)
\(492\) 0 0
\(493\) −21.8690 −0.984930
\(494\) 0 0
\(495\) 10.4696 0.470572
\(496\) 0 0
\(497\) −7.38385 −0.331211
\(498\) 0 0
\(499\) −29.1321 −1.30413 −0.652066 0.758163i \(-0.726097\pi\)
−0.652066 + 0.758163i \(0.726097\pi\)
\(500\) 0 0
\(501\) −42.4441 −1.89626
\(502\) 0 0
\(503\) 13.1533 0.586479 0.293239 0.956039i \(-0.405267\pi\)
0.293239 + 0.956039i \(0.405267\pi\)
\(504\) 0 0
\(505\) −7.67637 −0.341594
\(506\) 0 0
\(507\) −32.3162 −1.43521
\(508\) 0 0
\(509\) 32.5651 1.44342 0.721712 0.692193i \(-0.243355\pi\)
0.721712 + 0.692193i \(0.243355\pi\)
\(510\) 0 0
\(511\) −72.5117 −3.20773
\(512\) 0 0
\(513\) −1.94370 −0.0858164
\(514\) 0 0
\(515\) −20.0984 −0.885643
\(516\) 0 0
\(517\) 13.2826 0.584167
\(518\) 0 0
\(519\) −60.7451 −2.66641
\(520\) 0 0
\(521\) 13.0450 0.571511 0.285755 0.958303i \(-0.407756\pi\)
0.285755 + 0.958303i \(0.407756\pi\)
\(522\) 0 0
\(523\) −11.9437 −0.522261 −0.261131 0.965303i \(-0.584095\pi\)
−0.261131 + 0.965303i \(0.584095\pi\)
\(524\) 0 0
\(525\) −27.6285 −1.20581
\(526\) 0 0
\(527\) 1.96211 0.0854709
\(528\) 0 0
\(529\) −19.2220 −0.835741
\(530\) 0 0
\(531\) −25.2939 −1.09766
\(532\) 0 0
\(533\) 3.39803 0.147185
\(534\) 0 0
\(535\) 8.69193 0.375785
\(536\) 0 0
\(537\) −11.7974 −0.509097
\(538\) 0 0
\(539\) 20.7819 0.895139
\(540\) 0 0
\(541\) 1.48513 0.0638508 0.0319254 0.999490i \(-0.489836\pi\)
0.0319254 + 0.999490i \(0.489836\pi\)
\(542\) 0 0
\(543\) −31.4065 −1.34778
\(544\) 0 0
\(545\) 23.6243 1.01195
\(546\) 0 0
\(547\) −6.01418 −0.257148 −0.128574 0.991700i \(-0.541040\pi\)
−0.128574 + 0.991700i \(0.541040\pi\)
\(548\) 0 0
\(549\) 34.0871 1.45480
\(550\) 0 0
\(551\) 7.28966 0.310550
\(552\) 0 0
\(553\) 2.84242 0.120872
\(554\) 0 0
\(555\) 15.9774 0.678201
\(556\) 0 0
\(557\) −27.3584 −1.15921 −0.579605 0.814897i \(-0.696793\pi\)
−0.579605 + 0.814897i \(0.696793\pi\)
\(558\) 0 0
\(559\) −7.81162 −0.330396
\(560\) 0 0
\(561\) 13.6243 0.575218
\(562\) 0 0
\(563\) 21.1434 0.891088 0.445544 0.895260i \(-0.353010\pi\)
0.445544 + 0.895260i \(0.353010\pi\)
\(564\) 0 0
\(565\) 22.4288 0.943588
\(566\) 0 0
\(567\) −26.9253 −1.13076
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −9.18130 −0.384225 −0.192113 0.981373i \(-0.561534\pi\)
−0.192113 + 0.981373i \(0.561534\pi\)
\(572\) 0 0
\(573\) 32.5977 1.36179
\(574\) 0 0
\(575\) 4.75670 0.198368
\(576\) 0 0
\(577\) −4.79744 −0.199720 −0.0998601 0.995001i \(-0.531840\pi\)
−0.0998601 + 0.995001i \(0.531840\pi\)
\(578\) 0 0
\(579\) 27.8732 1.15837
\(580\) 0 0
\(581\) −54.6693 −2.26806
\(582\) 0 0
\(583\) 11.5343 0.477704
\(584\) 0 0
\(585\) 4.48090 0.185262
\(586\) 0 0
\(587\) 1.64980 0.0680945 0.0340473 0.999420i \(-0.489160\pi\)
0.0340473 + 0.999420i \(0.489160\pi\)
\(588\) 0 0
\(589\) −0.654037 −0.0269491
\(590\) 0 0
\(591\) 24.3768 1.00273
\(592\) 0 0
\(593\) −19.8874 −0.816678 −0.408339 0.912830i \(-0.633892\pi\)
−0.408339 + 0.912830i \(0.633892\pi\)
\(594\) 0 0
\(595\) 20.8311 0.853992
\(596\) 0 0
\(597\) 29.1997 1.19506
\(598\) 0 0
\(599\) −2.25462 −0.0921214 −0.0460607 0.998939i \(-0.514667\pi\)
−0.0460607 + 0.998939i \(0.514667\pi\)
\(600\) 0 0
\(601\) 24.7677 1.01030 0.505148 0.863033i \(-0.331438\pi\)
0.505148 + 0.863033i \(0.331438\pi\)
\(602\) 0 0
\(603\) 27.7454 1.12988
\(604\) 0 0
\(605\) −12.6919 −0.516000
\(606\) 0 0
\(607\) −45.9774 −1.86616 −0.933082 0.359665i \(-0.882891\pi\)
−0.933082 + 0.359665i \(0.882891\pi\)
\(608\) 0 0
\(609\) −82.2978 −3.33488
\(610\) 0 0
\(611\) 5.68484 0.229984
\(612\) 0 0
\(613\) −0.326485 −0.0131866 −0.00659330 0.999978i \(-0.502099\pi\)
−0.00659330 + 0.999978i \(0.502099\pi\)
\(614\) 0 0
\(615\) 18.8492 0.760073
\(616\) 0 0
\(617\) −37.9140 −1.52636 −0.763179 0.646187i \(-0.776363\pi\)
−0.763179 + 0.646187i \(0.776363\pi\)
\(618\) 0 0
\(619\) 4.76771 0.191630 0.0958151 0.995399i \(-0.469454\pi\)
0.0958151 + 0.995399i \(0.469454\pi\)
\(620\) 0 0
\(621\) −3.77796 −0.151604
\(622\) 0 0
\(623\) −63.1965 −2.53192
\(624\) 0 0
\(625\) −6.77479 −0.270992
\(626\) 0 0
\(627\) −4.54143 −0.181367
\(628\) 0 0
\(629\) 11.5485 0.460470
\(630\) 0 0
\(631\) 10.9958 0.437734 0.218867 0.975755i \(-0.429764\pi\)
0.218867 + 0.975755i \(0.429764\pi\)
\(632\) 0 0
\(633\) −14.1321 −0.561700
\(634\) 0 0
\(635\) 12.0605 0.478608
\(636\) 0 0
\(637\) 8.89448 0.352412
\(638\) 0 0
\(639\) −6.36829 −0.251926
\(640\) 0 0
\(641\) −25.1586 −0.993707 −0.496853 0.867834i \(-0.665511\pi\)
−0.496853 + 0.867834i \(0.665511\pi\)
\(642\) 0 0
\(643\) 36.9069 1.45547 0.727733 0.685861i \(-0.240574\pi\)
0.727733 + 0.685861i \(0.240574\pi\)
\(644\) 0 0
\(645\) −43.3318 −1.70619
\(646\) 0 0
\(647\) 17.3531 0.682219 0.341109 0.940024i \(-0.389197\pi\)
0.341109 + 0.940024i \(0.389197\pi\)
\(648\) 0 0
\(649\) −11.7974 −0.463090
\(650\) 0 0
\(651\) 7.38385 0.289396
\(652\) 0 0
\(653\) 14.4780 0.566570 0.283285 0.959036i \(-0.408576\pi\)
0.283285 + 0.959036i \(0.408576\pi\)
\(654\) 0 0
\(655\) 0.0786309 0.00307236
\(656\) 0 0
\(657\) −62.5386 −2.43986
\(658\) 0 0
\(659\) 47.5301 1.85151 0.925755 0.378124i \(-0.123431\pi\)
0.925755 + 0.378124i \(0.123431\pi\)
\(660\) 0 0
\(661\) −8.22204 −0.319800 −0.159900 0.987133i \(-0.551117\pi\)
−0.159900 + 0.987133i \(0.551117\pi\)
\(662\) 0 0
\(663\) 5.83110 0.226461
\(664\) 0 0
\(665\) −6.94370 −0.269265
\(666\) 0 0
\(667\) 14.1689 0.548622
\(668\) 0 0
\(669\) 66.0390 2.55321
\(670\) 0 0
\(671\) 15.8987 0.613763
\(672\) 0 0
\(673\) 15.9016 0.612961 0.306480 0.951877i \(-0.400849\pi\)
0.306480 + 0.951877i \(0.400849\pi\)
\(674\) 0 0
\(675\) −4.75670 −0.183086
\(676\) 0 0
\(677\) 21.9816 0.844821 0.422411 0.906405i \(-0.361184\pi\)
0.422411 + 0.906405i \(0.361184\pi\)
\(678\) 0 0
\(679\) −45.9774 −1.76445
\(680\) 0 0
\(681\) 9.71849 0.372413
\(682\) 0 0
\(683\) 1.30807 0.0500520 0.0250260 0.999687i \(-0.492033\pi\)
0.0250260 + 0.999687i \(0.492033\pi\)
\(684\) 0 0
\(685\) 27.7861 1.06165
\(686\) 0 0
\(687\) −13.7596 −0.524960
\(688\) 0 0
\(689\) 4.93661 0.188070
\(690\) 0 0
\(691\) −44.2291 −1.68256 −0.841278 0.540603i \(-0.818196\pi\)
−0.841278 + 0.540603i \(0.818196\pi\)
\(692\) 0 0
\(693\) 28.4780 1.08179
\(694\) 0 0
\(695\) −13.1841 −0.500103
\(696\) 0 0
\(697\) 13.6243 0.516057
\(698\) 0 0
\(699\) 25.8127 0.976325
\(700\) 0 0
\(701\) 24.5793 0.928348 0.464174 0.885744i \(-0.346351\pi\)
0.464174 + 0.885744i \(0.346351\pi\)
\(702\) 0 0
\(703\) −3.84951 −0.145187
\(704\) 0 0
\(705\) 31.5343 1.18765
\(706\) 0 0
\(707\) −20.8803 −0.785285
\(708\) 0 0
\(709\) 9.39803 0.352950 0.176475 0.984305i \(-0.443530\pi\)
0.176475 + 0.984305i \(0.443530\pi\)
\(710\) 0 0
\(711\) 2.45148 0.0919376
\(712\) 0 0
\(713\) −1.27125 −0.0476087
\(714\) 0 0
\(715\) 2.08995 0.0781599
\(716\) 0 0
\(717\) 72.3272 2.70111
\(718\) 0 0
\(719\) 24.3460 0.907951 0.453976 0.891014i \(-0.350005\pi\)
0.453976 + 0.891014i \(0.350005\pi\)
\(720\) 0 0
\(721\) −54.6693 −2.03599
\(722\) 0 0
\(723\) 15.1955 0.565126
\(724\) 0 0
\(725\) 17.8396 0.662545
\(726\) 0 0
\(727\) −29.1137 −1.07977 −0.539883 0.841740i \(-0.681532\pi\)
−0.539883 + 0.841740i \(0.681532\pi\)
\(728\) 0 0
\(729\) −38.3697 −1.42110
\(730\) 0 0
\(731\) −31.3205 −1.15843
\(732\) 0 0
\(733\) 4.91713 0.181618 0.0908092 0.995868i \(-0.471055\pi\)
0.0908092 + 0.995868i \(0.471055\pi\)
\(734\) 0 0
\(735\) 49.3386 1.81988
\(736\) 0 0
\(737\) 12.9408 0.476682
\(738\) 0 0
\(739\) −51.7114 −1.90223 −0.951117 0.308830i \(-0.900063\pi\)
−0.951117 + 0.308830i \(0.900063\pi\)
\(740\) 0 0
\(741\) −1.94370 −0.0714035
\(742\) 0 0
\(743\) 18.0379 0.661746 0.330873 0.943675i \(-0.392657\pi\)
0.330873 + 0.943675i \(0.392657\pi\)
\(744\) 0 0
\(745\) −10.0492 −0.368175
\(746\) 0 0
\(747\) −47.1502 −1.72513
\(748\) 0 0
\(749\) 23.6427 0.863886
\(750\) 0 0
\(751\) −10.9171 −0.398372 −0.199186 0.979962i \(-0.563830\pi\)
−0.199186 + 0.979962i \(0.563830\pi\)
\(752\) 0 0
\(753\) 39.6017 1.44316
\(754\) 0 0
\(755\) 35.4512 1.29020
\(756\) 0 0
\(757\) −33.4710 −1.21652 −0.608261 0.793737i \(-0.708133\pi\)
−0.608261 + 0.793737i \(0.708133\pi\)
\(758\) 0 0
\(759\) −8.82718 −0.320406
\(760\) 0 0
\(761\) −51.3399 −1.86107 −0.930536 0.366201i \(-0.880658\pi\)
−0.930536 + 0.366201i \(0.880658\pi\)
\(762\) 0 0
\(763\) 64.2599 2.32637
\(764\) 0 0
\(765\) 17.9660 0.649563
\(766\) 0 0
\(767\) −5.04921 −0.182317
\(768\) 0 0
\(769\) 16.7974 0.605731 0.302866 0.953033i \(-0.402057\pi\)
0.302866 + 0.953033i \(0.402057\pi\)
\(770\) 0 0
\(771\) 19.7075 0.709748
\(772\) 0 0
\(773\) −8.62145 −0.310092 −0.155046 0.987907i \(-0.549553\pi\)
−0.155046 + 0.987907i \(0.549553\pi\)
\(774\) 0 0
\(775\) −1.60059 −0.0574948
\(776\) 0 0
\(777\) 43.4596 1.55911
\(778\) 0 0
\(779\) −4.54143 −0.162714
\(780\) 0 0
\(781\) −2.97026 −0.106284
\(782\) 0 0
\(783\) −14.1689 −0.506355
\(784\) 0 0
\(785\) −14.3683 −0.512826
\(786\) 0 0
\(787\) −25.0124 −0.891595 −0.445798 0.895134i \(-0.647080\pi\)
−0.445798 + 0.895134i \(0.647080\pi\)
\(788\) 0 0
\(789\) −7.25601 −0.258321
\(790\) 0 0
\(791\) 61.0082 2.16920
\(792\) 0 0
\(793\) 6.80453 0.241636
\(794\) 0 0
\(795\) 27.3839 0.971205
\(796\) 0 0
\(797\) 48.0574 1.70228 0.851140 0.524939i \(-0.175912\pi\)
0.851140 + 0.524939i \(0.175912\pi\)
\(798\) 0 0
\(799\) 22.7932 0.806366
\(800\) 0 0
\(801\) −54.5046 −1.92583
\(802\) 0 0
\(803\) −29.1689 −1.02935
\(804\) 0 0
\(805\) −13.4965 −0.475688
\(806\) 0 0
\(807\) 65.0234 2.28893
\(808\) 0 0
\(809\) 6.20965 0.218320 0.109160 0.994024i \(-0.465184\pi\)
0.109160 + 0.994024i \(0.465184\pi\)
\(810\) 0 0
\(811\) −26.6130 −0.934508 −0.467254 0.884123i \(-0.654757\pi\)
−0.467254 + 0.884123i \(0.654757\pi\)
\(812\) 0 0
\(813\) −71.5159 −2.50817
\(814\) 0 0
\(815\) −27.1502 −0.951029
\(816\) 0 0
\(817\) 10.4402 0.365255
\(818\) 0 0
\(819\) 12.1884 0.425897
\(820\) 0 0
\(821\) −25.4993 −0.889932 −0.444966 0.895547i \(-0.646784\pi\)
−0.444966 + 0.895547i \(0.646784\pi\)
\(822\) 0 0
\(823\) 34.0166 1.18575 0.592873 0.805296i \(-0.297994\pi\)
0.592873 + 0.805296i \(0.297994\pi\)
\(824\) 0 0
\(825\) −11.1140 −0.386939
\(826\) 0 0
\(827\) −25.6048 −0.890367 −0.445183 0.895439i \(-0.646862\pi\)
−0.445183 + 0.895439i \(0.646862\pi\)
\(828\) 0 0
\(829\) −19.0350 −0.661114 −0.330557 0.943786i \(-0.607237\pi\)
−0.330557 + 0.943786i \(0.607237\pi\)
\(830\) 0 0
\(831\) −27.0223 −0.937394
\(832\) 0 0
\(833\) 35.6622 1.23562
\(834\) 0 0
\(835\) −26.1052 −0.903408
\(836\) 0 0
\(837\) 1.27125 0.0439408
\(838\) 0 0
\(839\) −15.6848 −0.541501 −0.270750 0.962650i \(-0.587272\pi\)
−0.270750 + 0.962650i \(0.587272\pi\)
\(840\) 0 0
\(841\) 24.1392 0.832385
\(842\) 0 0
\(843\) −36.0758 −1.24252
\(844\) 0 0
\(845\) −19.8761 −0.683758
\(846\) 0 0
\(847\) −34.5230 −1.18623
\(848\) 0 0
\(849\) 71.0082 2.43699
\(850\) 0 0
\(851\) −7.48228 −0.256489
\(852\) 0 0
\(853\) −3.57224 −0.122311 −0.0611555 0.998128i \(-0.519479\pi\)
−0.0611555 + 0.998128i \(0.519479\pi\)
\(854\) 0 0
\(855\) −5.98868 −0.204808
\(856\) 0 0
\(857\) 56.7309 1.93789 0.968945 0.247276i \(-0.0795355\pi\)
0.968945 + 0.247276i \(0.0795355\pi\)
\(858\) 0 0
\(859\) −18.6059 −0.634825 −0.317412 0.948288i \(-0.602814\pi\)
−0.317412 + 0.948288i \(0.602814\pi\)
\(860\) 0 0
\(861\) 51.2713 1.74732
\(862\) 0 0
\(863\) −17.9395 −0.610666 −0.305333 0.952246i \(-0.598768\pi\)
−0.305333 + 0.952246i \(0.598768\pi\)
\(864\) 0 0
\(865\) −37.3612 −1.27032
\(866\) 0 0
\(867\) −20.7819 −0.705790
\(868\) 0 0
\(869\) 1.14341 0.0387874
\(870\) 0 0
\(871\) 5.53858 0.187668
\(872\) 0 0
\(873\) −39.6537 −1.34208
\(874\) 0 0
\(875\) −51.7114 −1.74816
\(876\) 0 0
\(877\) 57.6201 1.94569 0.972846 0.231455i \(-0.0743485\pi\)
0.972846 + 0.231455i \(0.0743485\pi\)
\(878\) 0 0
\(879\) −26.3205 −0.887767
\(880\) 0 0
\(881\) 15.3431 0.516923 0.258461 0.966022i \(-0.416785\pi\)
0.258461 + 0.966022i \(0.416785\pi\)
\(882\) 0 0
\(883\) −17.8608 −0.601065 −0.300532 0.953772i \(-0.597164\pi\)
−0.300532 + 0.953772i \(0.597164\pi\)
\(884\) 0 0
\(885\) −28.0085 −0.941495
\(886\) 0 0
\(887\) −49.1445 −1.65011 −0.825055 0.565053i \(-0.808856\pi\)
−0.825055 + 0.565053i \(0.808856\pi\)
\(888\) 0 0
\(889\) 32.8056 1.10026
\(890\) 0 0
\(891\) −10.8311 −0.362855
\(892\) 0 0
\(893\) −7.59774 −0.254249
\(894\) 0 0
\(895\) −7.25601 −0.242542
\(896\) 0 0
\(897\) −3.77796 −0.126143
\(898\) 0 0
\(899\) −4.76771 −0.159012
\(900\) 0 0
\(901\) 19.7932 0.659407
\(902\) 0 0
\(903\) −117.866 −3.92233
\(904\) 0 0
\(905\) −19.3165 −0.642104
\(906\) 0 0
\(907\) 9.19971 0.305471 0.152736 0.988267i \(-0.451192\pi\)
0.152736 + 0.988267i \(0.451192\pi\)
\(908\) 0 0
\(909\) −18.0085 −0.597303
\(910\) 0 0
\(911\) 25.2939 0.838024 0.419012 0.907981i \(-0.362376\pi\)
0.419012 + 0.907981i \(0.362376\pi\)
\(912\) 0 0
\(913\) −21.9915 −0.727813
\(914\) 0 0
\(915\) 37.7454 1.24782
\(916\) 0 0
\(917\) 0.213882 0.00706301
\(918\) 0 0
\(919\) 44.8155 1.47833 0.739164 0.673525i \(-0.235221\pi\)
0.739164 + 0.673525i \(0.235221\pi\)
\(920\) 0 0
\(921\) 44.4752 1.46551
\(922\) 0 0
\(923\) −1.27125 −0.0418437
\(924\) 0 0
\(925\) −9.42068 −0.309750
\(926\) 0 0
\(927\) −47.1502 −1.54861
\(928\) 0 0
\(929\) −5.65119 −0.185409 −0.0927047 0.995694i \(-0.529551\pi\)
−0.0927047 + 0.995694i \(0.529551\pi\)
\(930\) 0 0
\(931\) −11.8874 −0.389594
\(932\) 0 0
\(933\) 81.1544 2.65688
\(934\) 0 0
\(935\) 8.37962 0.274043
\(936\) 0 0
\(937\) −8.58641 −0.280506 −0.140253 0.990116i \(-0.544792\pi\)
−0.140253 + 0.990116i \(0.544792\pi\)
\(938\) 0 0
\(939\) −65.5386 −2.13877
\(940\) 0 0
\(941\) 34.8155 1.13495 0.567477 0.823389i \(-0.307920\pi\)
0.567477 + 0.823389i \(0.307920\pi\)
\(942\) 0 0
\(943\) −8.82718 −0.287452
\(944\) 0 0
\(945\) 13.4965 0.439040
\(946\) 0 0
\(947\) −18.6551 −0.606209 −0.303105 0.952957i \(-0.598023\pi\)
−0.303105 + 0.952957i \(0.598023\pi\)
\(948\) 0 0
\(949\) −12.4841 −0.405250
\(950\) 0 0
\(951\) −30.2447 −0.980751
\(952\) 0 0
\(953\) 37.7227 1.22196 0.610980 0.791646i \(-0.290776\pi\)
0.610980 + 0.791646i \(0.290776\pi\)
\(954\) 0 0
\(955\) 20.0492 0.648777
\(956\) 0 0
\(957\) −33.1055 −1.07015
\(958\) 0 0
\(959\) 75.5804 2.44062
\(960\) 0 0
\(961\) −30.5722 −0.986201
\(962\) 0 0
\(963\) 20.3909 0.657089
\(964\) 0 0
\(965\) 17.1434 0.551866
\(966\) 0 0
\(967\) −26.3683 −0.847947 −0.423974 0.905675i \(-0.639365\pi\)
−0.423974 + 0.905675i \(0.639365\pi\)
\(968\) 0 0
\(969\) −7.79321 −0.250354
\(970\) 0 0
\(971\) 0.978737 0.0314092 0.0157046 0.999877i \(-0.495001\pi\)
0.0157046 + 0.999877i \(0.495001\pi\)
\(972\) 0 0
\(973\) −35.8619 −1.14968
\(974\) 0 0
\(975\) −4.75670 −0.152336
\(976\) 0 0
\(977\) −0.254623 −0.00814611 −0.00407306 0.999992i \(-0.501296\pi\)
−0.00407306 + 0.999992i \(0.501296\pi\)
\(978\) 0 0
\(979\) −25.4217 −0.812482
\(980\) 0 0
\(981\) 55.4217 1.76948
\(982\) 0 0
\(983\) −49.3754 −1.57483 −0.787415 0.616423i \(-0.788581\pi\)
−0.787415 + 0.616423i \(0.788581\pi\)
\(984\) 0 0
\(985\) 14.9929 0.477714
\(986\) 0 0
\(987\) 85.7759 2.73028
\(988\) 0 0
\(989\) 20.2925 0.645264
\(990\) 0 0
\(991\) 37.6101 1.19473 0.597363 0.801971i \(-0.296215\pi\)
0.597363 + 0.801971i \(0.296215\pi\)
\(992\) 0 0
\(993\) 32.7256 1.03851
\(994\) 0 0
\(995\) 17.9593 0.569347
\(996\) 0 0
\(997\) 19.3867 0.613983 0.306992 0.951712i \(-0.400678\pi\)
0.306992 + 0.951712i \(0.400678\pi\)
\(998\) 0 0
\(999\) 7.48228 0.236729
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 4864.2.a.be.1.3 3
4.3 odd 2 4864.2.a.bc.1.1 3
8.3 odd 2 4864.2.a.bf.1.3 3
8.5 even 2 4864.2.a.bd.1.1 3
16.3 odd 4 2432.2.c.g.1217.2 yes 6
16.5 even 4 2432.2.c.f.1217.2 6
16.11 odd 4 2432.2.c.g.1217.5 yes 6
16.13 even 4 2432.2.c.f.1217.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.f.1217.2 6 16.5 even 4
2432.2.c.f.1217.5 yes 6 16.13 even 4
2432.2.c.g.1217.2 yes 6 16.3 odd 4
2432.2.c.g.1217.5 yes 6 16.11 odd 4
4864.2.a.bc.1.1 3 4.3 odd 2
4864.2.a.bd.1.1 3 8.5 even 2
4864.2.a.be.1.3 3 1.1 even 1 trivial
4864.2.a.bf.1.3 3 8.3 odd 2