Properties

Label 1728.4.a.bs.1.2
Level $1728$
Weight $4$
Character 1728.1
Self dual yes
Analytic conductor $101.955$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1728.1

$q$-expansion

\(f(q)\) \(=\) \(q+21.2337 q^{5} -29.2337 q^{7} +O(q^{10})\) \(q+21.2337 q^{5} -29.2337 q^{7} -1.00000 q^{11} +52.9348 q^{13} +96.9348 q^{17} +126.467 q^{19} -22.9348 q^{23} +325.870 q^{25} -133.533 q^{29} -101.832 q^{31} -620.739 q^{35} -105.065 q^{37} -16.3369 q^{41} -201.272 q^{43} -251.870 q^{47} +511.609 q^{49} -148.038 q^{53} -21.2337 q^{55} +73.6085 q^{59} +607.478 q^{61} +1124.00 q^{65} +761.945 q^{67} +701.196 q^{71} -287.000 q^{73} +29.2337 q^{77} +128.522 q^{79} -160.478 q^{83} +2058.28 q^{85} -430.206 q^{89} -1547.48 q^{91} +2685.37 q^{95} -31.1305 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{5} - 24q^{7} + O(q^{10}) \) \( 2q + 8q^{5} - 24q^{7} - 2q^{11} - 32q^{13} + 56q^{17} + 184q^{19} + 92q^{23} + 376q^{25} - 336q^{29} - 376q^{31} - 690q^{35} - 348q^{37} + 312q^{41} + 80q^{43} - 228q^{47} + 196q^{49} + 152q^{53} - 8q^{55} - 680q^{59} + 112q^{61} + 2248q^{65} + 352q^{67} + 1816q^{71} - 574q^{73} + 24q^{77} + 1360q^{79} + 782q^{83} + 2600q^{85} - 240q^{89} - 1992q^{91} + 1924q^{95} - 338q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 21.2337 1.89920 0.949599 0.313466i \(-0.101490\pi\)
0.949599 + 0.313466i \(0.101490\pi\)
\(6\) 0 0
\(7\) −29.2337 −1.57847 −0.789235 0.614091i \(-0.789523\pi\)
−0.789235 + 0.614091i \(0.789523\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.0274101 −0.0137051 0.999906i \(-0.504363\pi\)
−0.0137051 + 0.999906i \(0.504363\pi\)
\(12\) 0 0
\(13\) 52.9348 1.12934 0.564671 0.825316i \(-0.309003\pi\)
0.564671 + 0.825316i \(0.309003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 96.9348 1.38295 0.691474 0.722401i \(-0.256961\pi\)
0.691474 + 0.722401i \(0.256961\pi\)
\(18\) 0 0
\(19\) 126.467 1.52703 0.763516 0.645789i \(-0.223471\pi\)
0.763516 + 0.645789i \(0.223471\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −22.9348 −0.207923 −0.103961 0.994581i \(-0.533152\pi\)
−0.103961 + 0.994581i \(0.533152\pi\)
\(24\) 0 0
\(25\) 325.870 2.60696
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −133.533 −0.855048 −0.427524 0.904004i \(-0.640614\pi\)
−0.427524 + 0.904004i \(0.640614\pi\)
\(30\) 0 0
\(31\) −101.832 −0.589983 −0.294992 0.955500i \(-0.595317\pi\)
−0.294992 + 0.955500i \(0.595317\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −620.739 −2.99783
\(36\) 0 0
\(37\) −105.065 −0.466828 −0.233414 0.972378i \(-0.574990\pi\)
−0.233414 + 0.972378i \(0.574990\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −16.3369 −0.0622291 −0.0311145 0.999516i \(-0.509906\pi\)
−0.0311145 + 0.999516i \(0.509906\pi\)
\(42\) 0 0
\(43\) −201.272 −0.713805 −0.356903 0.934142i \(-0.616167\pi\)
−0.356903 + 0.934142i \(0.616167\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −251.870 −0.781680 −0.390840 0.920459i \(-0.627815\pi\)
−0.390840 + 0.920459i \(0.627815\pi\)
\(48\) 0 0
\(49\) 511.609 1.49157
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −148.038 −0.383671 −0.191836 0.981427i \(-0.561444\pi\)
−0.191836 + 0.981427i \(0.561444\pi\)
\(54\) 0 0
\(55\) −21.2337 −0.0520573
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 73.6085 0.162424 0.0812120 0.996697i \(-0.474121\pi\)
0.0812120 + 0.996697i \(0.474121\pi\)
\(60\) 0 0
\(61\) 607.478 1.27508 0.637538 0.770419i \(-0.279953\pi\)
0.637538 + 0.770419i \(0.279953\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1124.00 2.14485
\(66\) 0 0
\(67\) 761.945 1.38935 0.694675 0.719324i \(-0.255548\pi\)
0.694675 + 0.719324i \(0.255548\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 701.196 1.17207 0.586033 0.810287i \(-0.300689\pi\)
0.586033 + 0.810287i \(0.300689\pi\)
\(72\) 0 0
\(73\) −287.000 −0.460148 −0.230074 0.973173i \(-0.573897\pi\)
−0.230074 + 0.973173i \(0.573897\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 29.2337 0.0432661
\(78\) 0 0
\(79\) 128.522 0.183036 0.0915181 0.995803i \(-0.470828\pi\)
0.0915181 + 0.995803i \(0.470828\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −160.478 −0.212226 −0.106113 0.994354i \(-0.533841\pi\)
−0.106113 + 0.994354i \(0.533841\pi\)
\(84\) 0 0
\(85\) 2058.28 2.62649
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −430.206 −0.512380 −0.256190 0.966626i \(-0.582467\pi\)
−0.256190 + 0.966626i \(0.582467\pi\)
\(90\) 0 0
\(91\) −1547.48 −1.78263
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2685.37 2.90014
\(96\) 0 0
\(97\) −31.1305 −0.0325858 −0.0162929 0.999867i \(-0.505186\pi\)
−0.0162929 + 0.999867i \(0.505186\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −983.103 −0.968539 −0.484269 0.874919i \(-0.660915\pi\)
−0.484269 + 0.874919i \(0.660915\pi\)
\(102\) 0 0
\(103\) 952.152 0.910857 0.455429 0.890272i \(-0.349486\pi\)
0.455429 + 0.890272i \(0.349486\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 272.087 0.245828 0.122914 0.992417i \(-0.460776\pi\)
0.122914 + 0.992417i \(0.460776\pi\)
\(108\) 0 0
\(109\) −1355.76 −1.19136 −0.595680 0.803222i \(-0.703117\pi\)
−0.595680 + 0.803222i \(0.703117\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1938.36 1.61368 0.806838 0.590773i \(-0.201177\pi\)
0.806838 + 0.590773i \(0.201177\pi\)
\(114\) 0 0
\(115\) −486.989 −0.394887
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2833.76 −2.18294
\(120\) 0 0
\(121\) −1330.00 −0.999249
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4265.20 3.05193
\(126\) 0 0
\(127\) −232.375 −0.162362 −0.0811808 0.996699i \(-0.525869\pi\)
−0.0811808 + 0.996699i \(0.525869\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2795.30 1.86433 0.932163 0.362038i \(-0.117919\pi\)
0.932163 + 0.362038i \(0.117919\pi\)
\(132\) 0 0
\(133\) −3697.11 −2.41038
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1261.53 0.786715 0.393358 0.919386i \(-0.371313\pi\)
0.393358 + 0.919386i \(0.371313\pi\)
\(138\) 0 0
\(139\) −307.369 −0.187559 −0.0937794 0.995593i \(-0.529895\pi\)
−0.0937794 + 0.995593i \(0.529895\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −52.9348 −0.0309554
\(144\) 0 0
\(145\) −2835.39 −1.62391
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1597.76 0.878478 0.439239 0.898370i \(-0.355248\pi\)
0.439239 + 0.898370i \(0.355248\pi\)
\(150\) 0 0
\(151\) 3415.41 1.84067 0.920337 0.391126i \(-0.127914\pi\)
0.920337 + 0.391126i \(0.127914\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2162.26 −1.12050
\(156\) 0 0
\(157\) −476.826 −0.242387 −0.121194 0.992629i \(-0.538672\pi\)
−0.121194 + 0.992629i \(0.538672\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 670.467 0.328200
\(162\) 0 0
\(163\) 3304.04 1.58768 0.793842 0.608124i \(-0.208078\pi\)
0.793842 + 0.608124i \(0.208078\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2091.65 0.969203 0.484601 0.874735i \(-0.338965\pi\)
0.484601 + 0.874735i \(0.338965\pi\)
\(168\) 0 0
\(169\) 605.088 0.275416
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2924.56 1.28526 0.642631 0.766176i \(-0.277843\pi\)
0.642631 + 0.766176i \(0.277843\pi\)
\(174\) 0 0
\(175\) −9526.37 −4.11500
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 231.913 0.0968381 0.0484191 0.998827i \(-0.484582\pi\)
0.0484191 + 0.998827i \(0.484582\pi\)
\(180\) 0 0
\(181\) −2291.74 −0.941125 −0.470562 0.882367i \(-0.655949\pi\)
−0.470562 + 0.882367i \(0.655949\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2230.92 −0.886598
\(186\) 0 0
\(187\) −96.9348 −0.0379068
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4217.80 −1.59785 −0.798925 0.601430i \(-0.794598\pi\)
−0.798925 + 0.601430i \(0.794598\pi\)
\(192\) 0 0
\(193\) −1188.35 −0.443208 −0.221604 0.975137i \(-0.571129\pi\)
−0.221604 + 0.975137i \(0.571129\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3130.25 −1.13209 −0.566044 0.824375i \(-0.691527\pi\)
−0.566044 + 0.824375i \(0.691527\pi\)
\(198\) 0 0
\(199\) 659.146 0.234802 0.117401 0.993085i \(-0.462544\pi\)
0.117401 + 0.993085i \(0.462544\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3903.65 1.34967
\(204\) 0 0
\(205\) −346.892 −0.118185
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −126.467 −0.0418561
\(210\) 0 0
\(211\) 3613.95 1.17912 0.589560 0.807725i \(-0.299301\pi\)
0.589560 + 0.807725i \(0.299301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4273.74 −1.35566
\(216\) 0 0
\(217\) 2976.91 0.931272
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5131.22 1.56182
\(222\) 0 0
\(223\) 2054.13 0.616837 0.308419 0.951251i \(-0.400200\pi\)
0.308419 + 0.951251i \(0.400200\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2153.61 −0.629692 −0.314846 0.949143i \(-0.601953\pi\)
−0.314846 + 0.949143i \(0.601953\pi\)
\(228\) 0 0
\(229\) 1819.87 0.525154 0.262577 0.964911i \(-0.415428\pi\)
0.262577 + 0.964911i \(0.415428\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3989.68 1.12177 0.560886 0.827893i \(-0.310461\pi\)
0.560886 + 0.827893i \(0.310461\pi\)
\(234\) 0 0
\(235\) −5348.12 −1.48457
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5704.32 −1.54386 −0.771929 0.635709i \(-0.780708\pi\)
−0.771929 + 0.635709i \(0.780708\pi\)
\(240\) 0 0
\(241\) 6223.95 1.66357 0.831785 0.555099i \(-0.187319\pi\)
0.831785 + 0.555099i \(0.187319\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10863.3 2.83279
\(246\) 0 0
\(247\) 6694.52 1.72454
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4622.52 −1.16243 −0.581217 0.813749i \(-0.697423\pi\)
−0.581217 + 0.813749i \(0.697423\pi\)
\(252\) 0 0
\(253\) 22.9348 0.00569919
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4905.09 1.19055 0.595274 0.803523i \(-0.297043\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(258\) 0 0
\(259\) 3071.44 0.736874
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4432.85 1.03932 0.519660 0.854373i \(-0.326059\pi\)
0.519660 + 0.854373i \(0.326059\pi\)
\(264\) 0 0
\(265\) −3143.39 −0.728668
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4454.81 1.00972 0.504860 0.863201i \(-0.331544\pi\)
0.504860 + 0.863201i \(0.331544\pi\)
\(270\) 0 0
\(271\) −3256.23 −0.729897 −0.364948 0.931028i \(-0.618913\pi\)
−0.364948 + 0.931028i \(0.618913\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −325.870 −0.0714570
\(276\) 0 0
\(277\) 3421.61 0.742182 0.371091 0.928596i \(-0.378984\pi\)
0.371091 + 0.928596i \(0.378984\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1715.11 0.364109 0.182055 0.983288i \(-0.441725\pi\)
0.182055 + 0.983288i \(0.441725\pi\)
\(282\) 0 0
\(283\) −4487.60 −0.942615 −0.471307 0.881969i \(-0.656218\pi\)
−0.471307 + 0.881969i \(0.656218\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 477.587 0.0982268
\(288\) 0 0
\(289\) 4483.35 0.912548
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1685.31 0.336031 0.168016 0.985784i \(-0.446264\pi\)
0.168016 + 0.985784i \(0.446264\pi\)
\(294\) 0 0
\(295\) 1562.98 0.308475
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1214.05 −0.234816
\(300\) 0 0
\(301\) 5883.91 1.12672
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12899.0 2.42162
\(306\) 0 0
\(307\) −8079.07 −1.50194 −0.750972 0.660334i \(-0.770415\pi\)
−0.750972 + 0.660334i \(0.770415\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2557.63 0.466334 0.233167 0.972437i \(-0.425091\pi\)
0.233167 + 0.972437i \(0.425091\pi\)
\(312\) 0 0
\(313\) −4081.43 −0.737049 −0.368524 0.929618i \(-0.620137\pi\)
−0.368524 + 0.929618i \(0.620137\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7701.93 −1.36462 −0.682308 0.731065i \(-0.739024\pi\)
−0.682308 + 0.731065i \(0.739024\pi\)
\(318\) 0 0
\(319\) 133.533 0.0234370
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12259.1 2.11181
\(324\) 0 0
\(325\) 17249.8 2.94415
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7363.07 1.23386
\(330\) 0 0
\(331\) −357.314 −0.0593346 −0.0296673 0.999560i \(-0.509445\pi\)
−0.0296673 + 0.999560i \(0.509445\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 16178.9 2.63865
\(336\) 0 0
\(337\) −6659.09 −1.07639 −0.538195 0.842820i \(-0.680894\pi\)
−0.538195 + 0.842820i \(0.680894\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 101.832 0.0161715
\(342\) 0 0
\(343\) −4929.05 −0.775929
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7894.04 −1.22125 −0.610626 0.791919i \(-0.709082\pi\)
−0.610626 + 0.791919i \(0.709082\pi\)
\(348\) 0 0
\(349\) 1254.46 0.192405 0.0962027 0.995362i \(-0.469330\pi\)
0.0962027 + 0.995362i \(0.469330\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8604.92 −1.29743 −0.648716 0.761030i \(-0.724694\pi\)
−0.648716 + 0.761030i \(0.724694\pi\)
\(354\) 0 0
\(355\) 14889.0 2.22598
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3875.61 −0.569768 −0.284884 0.958562i \(-0.591955\pi\)
−0.284884 + 0.958562i \(0.591955\pi\)
\(360\) 0 0
\(361\) 9135.00 1.33183
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6094.07 −0.873913
\(366\) 0 0
\(367\) 11890.1 1.69117 0.845583 0.533843i \(-0.179253\pi\)
0.845583 + 0.533843i \(0.179253\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4327.70 0.605614
\(372\) 0 0
\(373\) −12384.5 −1.71916 −0.859578 0.511005i \(-0.829273\pi\)
−0.859578 + 0.511005i \(0.829273\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7068.52 −0.965642
\(378\) 0 0
\(379\) 2062.80 0.279576 0.139788 0.990181i \(-0.455358\pi\)
0.139788 + 0.990181i \(0.455358\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11905.2 1.58833 0.794164 0.607704i \(-0.207909\pi\)
0.794164 + 0.607704i \(0.207909\pi\)
\(384\) 0 0
\(385\) 620.739 0.0821709
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12716.6 1.65747 0.828735 0.559641i \(-0.189061\pi\)
0.828735 + 0.559641i \(0.189061\pi\)
\(390\) 0 0
\(391\) −2223.17 −0.287547
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2729.00 0.347622
\(396\) 0 0
\(397\) 4531.59 0.572881 0.286441 0.958098i \(-0.407528\pi\)
0.286441 + 0.958098i \(0.407528\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4829.15 0.601388 0.300694 0.953721i \(-0.402782\pi\)
0.300694 + 0.953721i \(0.402782\pi\)
\(402\) 0 0
\(403\) −5390.43 −0.666294
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 105.065 0.0127958
\(408\) 0 0
\(409\) 11484.9 1.38849 0.694245 0.719739i \(-0.255738\pi\)
0.694245 + 0.719739i \(0.255738\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2151.85 −0.256381
\(414\) 0 0
\(415\) −3407.54 −0.403059
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8211.65 −0.957435 −0.478718 0.877969i \(-0.658898\pi\)
−0.478718 + 0.877969i \(0.658898\pi\)
\(420\) 0 0
\(421\) −9788.13 −1.13312 −0.566561 0.824020i \(-0.691726\pi\)
−0.566561 + 0.824020i \(0.691726\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 31588.1 3.60529
\(426\) 0 0
\(427\) −17758.8 −2.01267
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11206.3 1.25241 0.626207 0.779657i \(-0.284606\pi\)
0.626207 + 0.779657i \(0.284606\pi\)
\(432\) 0 0
\(433\) 719.306 0.0798329 0.0399165 0.999203i \(-0.487291\pi\)
0.0399165 + 0.999203i \(0.487291\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2900.50 −0.317505
\(438\) 0 0
\(439\) −8795.00 −0.956179 −0.478090 0.878311i \(-0.658671\pi\)
−0.478090 + 0.878311i \(0.658671\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3023.65 −0.324285 −0.162142 0.986767i \(-0.551840\pi\)
−0.162142 + 0.986767i \(0.551840\pi\)
\(444\) 0 0
\(445\) −9134.87 −0.973111
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3137.67 −0.329790 −0.164895 0.986311i \(-0.552728\pi\)
−0.164895 + 0.986311i \(0.552728\pi\)
\(450\) 0 0
\(451\) 16.3369 0.00170571
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −32858.7 −3.38558
\(456\) 0 0
\(457\) −9015.74 −0.922841 −0.461421 0.887182i \(-0.652660\pi\)
−0.461421 + 0.887182i \(0.652660\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12262.6 −1.23888 −0.619442 0.785042i \(-0.712641\pi\)
−0.619442 + 0.785042i \(0.712641\pi\)
\(462\) 0 0
\(463\) −3472.30 −0.348534 −0.174267 0.984698i \(-0.555756\pi\)
−0.174267 + 0.984698i \(0.555756\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11825.0 −1.17172 −0.585860 0.810412i \(-0.699243\pi\)
−0.585860 + 0.810412i \(0.699243\pi\)
\(468\) 0 0
\(469\) −22274.5 −2.19305
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 201.272 0.0195655
\(474\) 0 0
\(475\) 41211.9 3.98090
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1703.96 −0.162538 −0.0812692 0.996692i \(-0.525897\pi\)
−0.0812692 + 0.996692i \(0.525897\pi\)
\(480\) 0 0
\(481\) −5561.60 −0.527208
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −661.015 −0.0618869
\(486\) 0 0
\(487\) −7721.95 −0.718512 −0.359256 0.933239i \(-0.616969\pi\)
−0.359256 + 0.933239i \(0.616969\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −20223.6 −1.85882 −0.929410 0.369049i \(-0.879683\pi\)
−0.929410 + 0.369049i \(0.879683\pi\)
\(492\) 0 0
\(493\) −12944.0 −1.18249
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −20498.5 −1.85007
\(498\) 0 0
\(499\) −13702.4 −1.22927 −0.614633 0.788813i \(-0.710696\pi\)
−0.614633 + 0.788813i \(0.710696\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13707.6 1.21509 0.607546 0.794285i \(-0.292154\pi\)
0.607546 + 0.794285i \(0.292154\pi\)
\(504\) 0 0
\(505\) −20874.9 −1.83945
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12339.2 1.07451 0.537254 0.843421i \(-0.319462\pi\)
0.537254 + 0.843421i \(0.319462\pi\)
\(510\) 0 0
\(511\) 8390.07 0.726330
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 20217.7 1.72990
\(516\) 0 0
\(517\) 251.870 0.0214259
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1416.70 −0.119130 −0.0595649 0.998224i \(-0.518971\pi\)
−0.0595649 + 0.998224i \(0.518971\pi\)
\(522\) 0 0
\(523\) 6696.15 0.559851 0.279925 0.960022i \(-0.409690\pi\)
0.279925 + 0.960022i \(0.409690\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9871.02 −0.815917
\(528\) 0 0
\(529\) −11641.0 −0.956768
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −864.789 −0.0702780
\(534\) 0 0
\(535\) 5777.40 0.466876
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −511.609 −0.0408841
\(540\) 0 0
\(541\) −7105.69 −0.564691 −0.282345 0.959313i \(-0.591112\pi\)
−0.282345 + 0.959313i \(0.591112\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −28787.8 −2.26263
\(546\) 0 0
\(547\) 12028.0 0.940185 0.470092 0.882617i \(-0.344221\pi\)
0.470092 + 0.882617i \(0.344221\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −16887.5 −1.30569
\(552\) 0 0
\(553\) −3757.17 −0.288917
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14529.5 1.10527 0.552634 0.833424i \(-0.313623\pi\)
0.552634 + 0.833424i \(0.313623\pi\)
\(558\) 0 0
\(559\) −10654.3 −0.806131
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10233.4 0.766053 0.383027 0.923737i \(-0.374882\pi\)
0.383027 + 0.923737i \(0.374882\pi\)
\(564\) 0 0
\(565\) 41158.5 3.06469
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −319.460 −0.0235368 −0.0117684 0.999931i \(-0.503746\pi\)
−0.0117684 + 0.999931i \(0.503746\pi\)
\(570\) 0 0
\(571\) −6793.96 −0.497931 −0.248965 0.968512i \(-0.580091\pi\)
−0.248965 + 0.968512i \(0.580091\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7473.74 −0.542046
\(576\) 0 0
\(577\) −11145.6 −0.804156 −0.402078 0.915605i \(-0.631712\pi\)
−0.402078 + 0.915605i \(0.631712\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4691.36 0.334992
\(582\) 0 0
\(583\) 148.038 0.0105165
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23817.6 1.67471 0.837356 0.546658i \(-0.184100\pi\)
0.837356 + 0.546658i \(0.184100\pi\)
\(588\) 0 0
\(589\) −12878.4 −0.900924
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16416.2 1.13682 0.568408 0.822747i \(-0.307559\pi\)
0.568408 + 0.822747i \(0.307559\pi\)
\(594\) 0 0
\(595\) −60171.2 −4.14585
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4801.70 −0.327533 −0.163766 0.986499i \(-0.552364\pi\)
−0.163766 + 0.986499i \(0.552364\pi\)
\(600\) 0 0
\(601\) 4524.21 0.307066 0.153533 0.988144i \(-0.450935\pi\)
0.153533 + 0.988144i \(0.450935\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −28240.8 −1.89777
\(606\) 0 0
\(607\) 14978.3 1.00157 0.500783 0.865573i \(-0.333045\pi\)
0.500783 + 0.865573i \(0.333045\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13332.6 −0.882784
\(612\) 0 0
\(613\) −11651.6 −0.767709 −0.383854 0.923394i \(-0.625404\pi\)
−0.383854 + 0.923394i \(0.625404\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 341.120 0.0222576 0.0111288 0.999938i \(-0.496458\pi\)
0.0111288 + 0.999938i \(0.496458\pi\)
\(618\) 0 0
\(619\) 18415.1 1.19575 0.597873 0.801591i \(-0.296013\pi\)
0.597873 + 0.801591i \(0.296013\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12576.5 0.808776
\(624\) 0 0
\(625\) 49832.2 3.18926
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10184.5 −0.645599
\(630\) 0 0
\(631\) −13557.6 −0.855341 −0.427671 0.903935i \(-0.640666\pi\)
−0.427671 + 0.903935i \(0.640666\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4934.17 −0.308357
\(636\) 0 0
\(637\) 27081.9 1.68449
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9682.55 −0.596627 −0.298313 0.954468i \(-0.596424\pi\)
−0.298313 + 0.954468i \(0.596424\pi\)
\(642\) 0 0
\(643\) −3259.50 −0.199910 −0.0999551 0.994992i \(-0.531870\pi\)
−0.0999551 + 0.994992i \(0.531870\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8115.35 −0.493118 −0.246559 0.969128i \(-0.579300\pi\)
−0.246559 + 0.969128i \(0.579300\pi\)
\(648\) 0 0
\(649\) −73.6085 −0.00445206
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3427.74 −0.205418 −0.102709 0.994711i \(-0.532751\pi\)
−0.102709 + 0.994711i \(0.532751\pi\)
\(654\) 0 0
\(655\) 59354.6 3.54073
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13820.2 0.816934 0.408467 0.912773i \(-0.366064\pi\)
0.408467 + 0.912773i \(0.366064\pi\)
\(660\) 0 0
\(661\) 22719.7 1.33690 0.668451 0.743756i \(-0.266957\pi\)
0.668451 + 0.743756i \(0.266957\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −78503.2 −4.57778
\(666\) 0 0
\(667\) 3062.54 0.177784
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −607.478 −0.0349500
\(672\) 0 0
\(673\) −5133.17 −0.294011 −0.147005 0.989136i \(-0.546963\pi\)
−0.147005 + 0.989136i \(0.546963\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9295.88 0.527725 0.263862 0.964560i \(-0.415004\pi\)
0.263862 + 0.964560i \(0.415004\pi\)
\(678\) 0 0
\(679\) 910.059 0.0514357
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −13824.3 −0.774486 −0.387243 0.921978i \(-0.626573\pi\)
−0.387243 + 0.921978i \(0.626573\pi\)
\(684\) 0 0
\(685\) 26787.0 1.49413
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −7836.35 −0.433296
\(690\) 0 0
\(691\) −11015.2 −0.606424 −0.303212 0.952923i \(-0.598059\pi\)
−0.303212 + 0.952923i \(0.598059\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6526.57 −0.356212
\(696\) 0 0
\(697\) −1583.61 −0.0860596
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −460.523 −0.0248127 −0.0124064 0.999923i \(-0.503949\pi\)
−0.0124064 + 0.999923i \(0.503949\pi\)
\(702\) 0 0
\(703\) −13287.3 −0.712861
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28739.7 1.52881
\(708\) 0 0
\(709\) −26445.0 −1.40080 −0.700398 0.713752i \(-0.746994\pi\)
−0.700398 + 0.713752i \(0.746994\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2335.48 0.122671
\(714\) 0 0
\(715\) −1124.00 −0.0587905
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15524.4 0.805235 0.402617 0.915368i \(-0.368101\pi\)
0.402617 + 0.915368i \(0.368101\pi\)
\(720\) 0 0
\(721\) −27834.9 −1.43776
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −43514.2 −2.22907
\(726\) 0 0
\(727\) −18934.0 −0.965922 −0.482961 0.875642i \(-0.660439\pi\)
−0.482961 + 0.875642i \(0.660439\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −19510.2 −0.987156
\(732\) 0 0
\(733\) 16779.9 0.845540 0.422770 0.906237i \(-0.361058\pi\)
0.422770 + 0.906237i \(0.361058\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −761.945 −0.0380823
\(738\) 0 0
\(739\) 30309.0 1.50871 0.754353 0.656469i \(-0.227951\pi\)
0.754353 + 0.656469i \(0.227951\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32098.8 1.58492 0.792458 0.609927i \(-0.208801\pi\)
0.792458 + 0.609927i \(0.208801\pi\)
\(744\) 0 0
\(745\) 33926.2 1.66840
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −7954.09 −0.388032
\(750\) 0 0
\(751\) −9434.07 −0.458394 −0.229197 0.973380i \(-0.573610\pi\)
−0.229197 + 0.973380i \(0.573610\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 72521.7 3.49581
\(756\) 0 0
\(757\) −1280.65 −0.0614876 −0.0307438 0.999527i \(-0.509788\pi\)
−0.0307438 + 0.999527i \(0.509788\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −19624.7 −0.934818 −0.467409 0.884041i \(-0.654812\pi\)
−0.467409 + 0.884041i \(0.654812\pi\)
\(762\) 0 0
\(763\) 39633.9 1.88053
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3896.45 0.183432
\(768\) 0 0
\(769\) 17644.9 0.827426 0.413713 0.910407i \(-0.364232\pi\)
0.413713 + 0.910407i \(0.364232\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −22541.7 −1.04886 −0.524430 0.851454i \(-0.675722\pi\)
−0.524430 + 0.851454i \(0.675722\pi\)
\(774\) 0 0
\(775\) −33183.8 −1.53806
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2066.08 −0.0950258
\(780\) 0 0
\(781\) −701.196 −0.0321264
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10124.8 −0.460342
\(786\) 0 0
\(787\) −39687.8 −1.79761 −0.898804 0.438350i \(-0.855563\pi\)
−0.898804 + 0.438350i \(0.855563\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −56665.4 −2.54714
\(792\) 0 0
\(793\) 32156.7 1.44000
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1991.97 −0.0885309 −0.0442654 0.999020i \(-0.514095\pi\)
−0.0442654 + 0.999020i \(0.514095\pi\)
\(798\) 0 0
\(799\) −24414.9 −1.08102
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 287.000 0.0126127
\(804\) 0 0
\(805\) 14236.5 0.623317
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21679.4 0.942160 0.471080 0.882090i \(-0.343864\pi\)
0.471080 + 0.882090i \(0.343864\pi\)
\(810\) 0 0
\(811\) −15099.7 −0.653786 −0.326893 0.945061i \(-0.606002\pi\)
−0.326893 + 0.945061i \(0.606002\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 70157.0 3.01533
\(816\) 0 0
\(817\) −25454.3 −1.09000
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9902.77 0.420961 0.210480 0.977598i \(-0.432497\pi\)
0.210480 + 0.977598i \(0.432497\pi\)
\(822\) 0 0
\(823\) −11488.7 −0.486600 −0.243300 0.969951i \(-0.578230\pi\)
−0.243300 + 0.969951i \(0.578230\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36938.8 −1.55319 −0.776594 0.630001i \(-0.783055\pi\)
−0.776594 + 0.630001i \(0.783055\pi\)
\(828\) 0 0
\(829\) −3205.00 −0.134275 −0.0671376 0.997744i \(-0.521387\pi\)
−0.0671376 + 0.997744i \(0.521387\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 49592.6 2.06277
\(834\) 0 0
\(835\) 44413.5 1.84071
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1126.63 −0.0463594 −0.0231797 0.999731i \(-0.507379\pi\)
−0.0231797 + 0.999731i \(0.507379\pi\)
\(840\) 0 0
\(841\) −6558.04 −0.268893
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12848.2 0.523069
\(846\) 0 0
\(847\) 38880.8 1.57728
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2409.65 0.0970641
\(852\) 0 0
\(853\) −6254.66 −0.251061 −0.125531 0.992090i \(-0.540063\pi\)
−0.125531 + 0.992090i \(0.540063\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −19069.4 −0.760090 −0.380045 0.924968i \(-0.624092\pi\)
−0.380045 + 0.924968i \(0.624092\pi\)
\(858\) 0 0
\(859\) 26790.3 1.06411 0.532056 0.846709i \(-0.321420\pi\)
0.532056 + 0.846709i \(0.321420\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46056.3 1.81666 0.908329 0.418256i \(-0.137359\pi\)
0.908329 + 0.418256i \(0.137359\pi\)
\(864\) 0 0
\(865\) 62099.2 2.44097
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −128.522 −0.00501704
\(870\) 0 0
\(871\) 40333.4 1.56905
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −124688. −4.81738
\(876\) 0 0
\(877\) −36490.2 −1.40500 −0.702500 0.711684i \(-0.747933\pi\)
−0.702500 + 0.711684i \(0.747933\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 38136.3 1.45839 0.729197 0.684303i \(-0.239894\pi\)
0.729197 + 0.684303i \(0.239894\pi\)
\(882\) 0 0
\(883\) −13336.1 −0.508262 −0.254131 0.967170i \(-0.581789\pi\)
−0.254131 + 0.967170i \(0.581789\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28027.2 −1.06095 −0.530474 0.847701i \(-0.677986\pi\)
−0.530474 + 0.847701i \(0.677986\pi\)
\(888\) 0 0
\(889\) 6793.17 0.256283
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −31853.3 −1.19365
\(894\) 0 0
\(895\) 4924.38 0.183915
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13597.8 0.504464
\(900\) 0 0
\(901\) −14350.0 −0.530598
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −48662.1 −1.78738
\(906\) 0 0
\(907\) −45203.3 −1.65485 −0.827427 0.561573i \(-0.810196\pi\)
−0.827427 + 0.561573i \(0.810196\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −29822.0 −1.08457 −0.542287 0.840194i \(-0.682441\pi\)
−0.542287 + 0.840194i \(0.682441\pi\)
\(912\) 0 0
\(913\) 160.478 0.00581714
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −81717.0 −2.94279
\(918\) 0 0
\(919\) −36967.7 −1.32693 −0.663467 0.748206i \(-0.730916\pi\)
−0.663467 + 0.748206i \(0.730916\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 37117.6 1.32366
\(924\) 0 0
\(925\) −34237.6 −1.21700
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −50409.4 −1.78028 −0.890139 0.455690i \(-0.849393\pi\)
−0.890139 + 0.455690i \(0.849393\pi\)
\(930\) 0 0
\(931\) 64701.8 2.27767
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2058.28 −0.0719925
\(936\) 0 0
\(937\) −30156.5 −1.05141 −0.525705 0.850667i \(-0.676198\pi\)
−0.525705 + 0.850667i \(0.676198\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −42221.1 −1.46267 −0.731333 0.682020i \(-0.761102\pi\)
−0.731333 + 0.682020i \(0.761102\pi\)
\(942\) 0 0
\(943\) 374.682 0.0129388
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −40164.6 −1.37822 −0.689111 0.724656i \(-0.741999\pi\)
−0.689111 + 0.724656i \(0.741999\pi\)
\(948\) 0 0
\(949\) −15192.3 −0.519665
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −7132.10 −0.242425 −0.121213 0.992627i \(-0.538678\pi\)
−0.121213 + 0.992627i \(0.538678\pi\)
\(954\) 0 0
\(955\) −89559.5 −3.03464
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36879.3 −1.24181
\(960\) 0 0
\(961\) −19421.3 −0.651919
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −25233.0 −0.841740
\(966\) 0 0
\(967\) 549.206 0.0182640 0.00913199 0.999958i \(-0.497093\pi\)
0.00913199 + 0.999958i \(0.497093\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −31024.8 −1.02537 −0.512685 0.858577i \(-0.671349\pi\)
−0.512685 + 0.858577i \(0.671349\pi\)
\(972\) 0 0
\(973\) 8985.52 0.296056
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7827.03 −0.256304 −0.128152 0.991755i \(-0.540904\pi\)
−0.128152 + 0.991755i \(0.540904\pi\)
\(978\) 0 0
\(979\) 430.206 0.0140444
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −47421.5 −1.53867 −0.769334 0.638847i \(-0.779412\pi\)
−0.769334 + 0.638847i \(0.779412\pi\)
\(984\) 0 0
\(985\) −66466.8 −2.15006
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4616.11 0.148416
\(990\) 0 0
\(991\) −43260.0 −1.38668 −0.693339 0.720612i \(-0.743861\pi\)
−0.693339 + 0.720612i \(0.743861\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 13996.1 0.445936
\(996\) 0 0
\(997\) 40209.1 1.27727 0.638634 0.769511i \(-0.279500\pi\)
0.638634 + 0.769511i \(0.279500\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.4.a.bs.1.2 2
3.2 odd 2 1728.4.a.bg.1.1 2
4.3 odd 2 1728.4.a.bt.1.2 2
8.3 odd 2 216.4.a.e.1.1 2
8.5 even 2 432.4.a.o.1.1 2
12.11 even 2 1728.4.a.bh.1.1 2
24.5 odd 2 432.4.a.s.1.2 2
24.11 even 2 216.4.a.h.1.2 yes 2
72.11 even 6 648.4.i.m.433.1 4
72.43 odd 6 648.4.i.s.433.2 4
72.59 even 6 648.4.i.m.217.1 4
72.67 odd 6 648.4.i.s.217.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.e.1.1 2 8.3 odd 2
216.4.a.h.1.2 yes 2 24.11 even 2
432.4.a.o.1.1 2 8.5 even 2
432.4.a.s.1.2 2 24.5 odd 2
648.4.i.m.217.1 4 72.59 even 6
648.4.i.m.433.1 4 72.11 even 6
648.4.i.s.217.2 4 72.67 odd 6
648.4.i.s.433.2 4 72.43 odd 6
1728.4.a.bg.1.1 2 3.2 odd 2
1728.4.a.bh.1.1 2 12.11 even 2
1728.4.a.bs.1.2 2 1.1 even 1 trivial
1728.4.a.bt.1.2 2 4.3 odd 2