Properties

Label 7728.2.a.bx.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.167449 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.167449 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.80451 q^{11} -3.63706 q^{13} -0.167449 q^{15} +4.13941 q^{17} -5.80451 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.97196 q^{25} +1.00000 q^{27} +5.00000 q^{29} -0.195488 q^{31} +1.80451 q^{33} +0.167449 q^{35} -3.33490 q^{37} -3.63706 q^{39} -4.94392 q^{41} -1.30216 q^{43} -0.167449 q^{45} +3.13941 q^{47} +1.00000 q^{49} +4.13941 q^{51} +3.30686 q^{53} -0.302164 q^{55} -5.80451 q^{57} +4.91588 q^{59} +8.77647 q^{61} -1.00000 q^{63} +0.609023 q^{65} -13.2508 q^{67} +1.00000 q^{69} -8.77647 q^{71} -12.4182 q^{73} -4.97196 q^{75} -1.80451 q^{77} -1.52569 q^{79} +1.00000 q^{81} -14.7531 q^{83} -0.693141 q^{85} +5.00000 q^{87} +8.77647 q^{89} +3.63706 q^{91} -0.195488 q^{93} +0.971961 q^{95} -14.2741 q^{97} +1.80451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.167449 −0.0748856 −0.0374428 0.999299i \(-0.511921\pi\)
−0.0374428 + 0.999299i \(0.511921\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.80451 0.544081 0.272040 0.962286i \(-0.412302\pi\)
0.272040 + 0.962286i \(0.412302\pi\)
\(12\) 0 0
\(13\) −3.63706 −1.00874 −0.504370 0.863488i \(-0.668275\pi\)
−0.504370 + 0.863488i \(0.668275\pi\)
\(14\) 0 0
\(15\) −0.167449 −0.0432352
\(16\) 0 0
\(17\) 4.13941 1.00395 0.501977 0.864881i \(-0.332606\pi\)
0.501977 + 0.864881i \(0.332606\pi\)
\(18\) 0 0
\(19\) −5.80451 −1.33165 −0.665823 0.746110i \(-0.731919\pi\)
−0.665823 + 0.746110i \(0.731919\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.97196 −0.994392
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −0.195488 −0.0351108 −0.0175554 0.999846i \(-0.505588\pi\)
−0.0175554 + 0.999846i \(0.505588\pi\)
\(32\) 0 0
\(33\) 1.80451 0.314125
\(34\) 0 0
\(35\) 0.167449 0.0283041
\(36\) 0 0
\(37\) −3.33490 −0.548254 −0.274127 0.961694i \(-0.588389\pi\)
−0.274127 + 0.961694i \(0.588389\pi\)
\(38\) 0 0
\(39\) −3.63706 −0.582396
\(40\) 0 0
\(41\) −4.94392 −0.772111 −0.386055 0.922476i \(-0.626163\pi\)
−0.386055 + 0.922476i \(0.626163\pi\)
\(42\) 0 0
\(43\) −1.30216 −0.198578 −0.0992891 0.995059i \(-0.531657\pi\)
−0.0992891 + 0.995059i \(0.531657\pi\)
\(44\) 0 0
\(45\) −0.167449 −0.0249619
\(46\) 0 0
\(47\) 3.13941 0.457930 0.228965 0.973435i \(-0.426466\pi\)
0.228965 + 0.973435i \(0.426466\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.13941 0.579633
\(52\) 0 0
\(53\) 3.30686 0.454232 0.227116 0.973868i \(-0.427070\pi\)
0.227116 + 0.973868i \(0.427070\pi\)
\(54\) 0 0
\(55\) −0.302164 −0.0407438
\(56\) 0 0
\(57\) −5.80451 −0.768826
\(58\) 0 0
\(59\) 4.91588 0.639993 0.319997 0.947419i \(-0.396318\pi\)
0.319997 + 0.947419i \(0.396318\pi\)
\(60\) 0 0
\(61\) 8.77647 1.12371 0.561856 0.827235i \(-0.310087\pi\)
0.561856 + 0.827235i \(0.310087\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0.609023 0.0755400
\(66\) 0 0
\(67\) −13.2508 −1.61884 −0.809420 0.587230i \(-0.800218\pi\)
−0.809420 + 0.587230i \(0.800218\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.77647 −1.04158 −0.520788 0.853686i \(-0.674362\pi\)
−0.520788 + 0.853686i \(0.674362\pi\)
\(72\) 0 0
\(73\) −12.4182 −1.45344 −0.726722 0.686932i \(-0.758957\pi\)
−0.726722 + 0.686932i \(0.758957\pi\)
\(74\) 0 0
\(75\) −4.97196 −0.574113
\(76\) 0 0
\(77\) −1.80451 −0.205643
\(78\) 0 0
\(79\) −1.52569 −0.171654 −0.0858269 0.996310i \(-0.527353\pi\)
−0.0858269 + 0.996310i \(0.527353\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.7531 −1.61937 −0.809683 0.586867i \(-0.800361\pi\)
−0.809683 + 0.586867i \(0.800361\pi\)
\(84\) 0 0
\(85\) −0.693141 −0.0751817
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 8.77647 0.930304 0.465152 0.885231i \(-0.346000\pi\)
0.465152 + 0.885231i \(0.346000\pi\)
\(90\) 0 0
\(91\) 3.63706 0.381268
\(92\) 0 0
\(93\) −0.195488 −0.0202712
\(94\) 0 0
\(95\) 0.971961 0.0997211
\(96\) 0 0
\(97\) −14.2741 −1.44932 −0.724659 0.689108i \(-0.758003\pi\)
−0.724659 + 0.689108i \(0.758003\pi\)
\(98\) 0 0
\(99\) 1.80451 0.181360
\(100\) 0 0
\(101\) 8.63706 0.859420 0.429710 0.902967i \(-0.358616\pi\)
0.429710 + 0.902967i \(0.358616\pi\)
\(102\) 0 0
\(103\) −8.13471 −0.801537 −0.400769 0.916179i \(-0.631257\pi\)
−0.400769 + 0.916179i \(0.631257\pi\)
\(104\) 0 0
\(105\) 0.167449 0.0163414
\(106\) 0 0
\(107\) −7.16745 −0.692903 −0.346452 0.938068i \(-0.612614\pi\)
−0.346452 + 0.938068i \(0.612614\pi\)
\(108\) 0 0
\(109\) 2.44627 0.234310 0.117155 0.993114i \(-0.462623\pi\)
0.117155 + 0.993114i \(0.462623\pi\)
\(110\) 0 0
\(111\) −3.33490 −0.316535
\(112\) 0 0
\(113\) 5.10668 0.480396 0.240198 0.970724i \(-0.422788\pi\)
0.240198 + 0.970724i \(0.422788\pi\)
\(114\) 0 0
\(115\) −0.167449 −0.0156147
\(116\) 0 0
\(117\) −3.63706 −0.336247
\(118\) 0 0
\(119\) −4.13941 −0.379459
\(120\) 0 0
\(121\) −7.74374 −0.703976
\(122\) 0 0
\(123\) −4.94392 −0.445778
\(124\) 0 0
\(125\) 1.66980 0.149351
\(126\) 0 0
\(127\) 8.97666 0.796549 0.398275 0.917266i \(-0.369609\pi\)
0.398275 + 0.917266i \(0.369609\pi\)
\(128\) 0 0
\(129\) −1.30216 −0.114649
\(130\) 0 0
\(131\) −9.86059 −0.861524 −0.430762 0.902466i \(-0.641755\pi\)
−0.430762 + 0.902466i \(0.641755\pi\)
\(132\) 0 0
\(133\) 5.80451 0.503315
\(134\) 0 0
\(135\) −0.167449 −0.0144117
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) 14.3902 1.22056 0.610280 0.792186i \(-0.291057\pi\)
0.610280 + 0.792186i \(0.291057\pi\)
\(140\) 0 0
\(141\) 3.13941 0.264386
\(142\) 0 0
\(143\) −6.56312 −0.548836
\(144\) 0 0
\(145\) −0.837246 −0.0695295
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −2.33490 −0.191282 −0.0956412 0.995416i \(-0.530490\pi\)
−0.0956412 + 0.995416i \(0.530490\pi\)
\(150\) 0 0
\(151\) −11.8092 −0.961020 −0.480510 0.876989i \(-0.659548\pi\)
−0.480510 + 0.876989i \(0.659548\pi\)
\(152\) 0 0
\(153\) 4.13941 0.334651
\(154\) 0 0
\(155\) 0.0327344 0.00262929
\(156\) 0 0
\(157\) 2.72588 0.217549 0.108774 0.994066i \(-0.465307\pi\)
0.108774 + 0.994066i \(0.465307\pi\)
\(158\) 0 0
\(159\) 3.30686 0.262251
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −14.6371 −1.14646 −0.573232 0.819393i \(-0.694310\pi\)
−0.573232 + 0.819393i \(0.694310\pi\)
\(164\) 0 0
\(165\) −0.302164 −0.0235234
\(166\) 0 0
\(167\) −1.20018 −0.0928730 −0.0464365 0.998921i \(-0.514787\pi\)
−0.0464365 + 0.998921i \(0.514787\pi\)
\(168\) 0 0
\(169\) 0.228223 0.0175556
\(170\) 0 0
\(171\) −5.80451 −0.443882
\(172\) 0 0
\(173\) 19.0880 1.45124 0.725618 0.688098i \(-0.241554\pi\)
0.725618 + 0.688098i \(0.241554\pi\)
\(174\) 0 0
\(175\) 4.97196 0.375845
\(176\) 0 0
\(177\) 4.91588 0.369500
\(178\) 0 0
\(179\) −7.49765 −0.560401 −0.280200 0.959942i \(-0.590401\pi\)
−0.280200 + 0.959942i \(0.590401\pi\)
\(180\) 0 0
\(181\) 16.4088 1.21966 0.609830 0.792532i \(-0.291238\pi\)
0.609830 + 0.792532i \(0.291238\pi\)
\(182\) 0 0
\(183\) 8.77647 0.648776
\(184\) 0 0
\(185\) 0.558426 0.0410563
\(186\) 0 0
\(187\) 7.46961 0.546232
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 1.55294 0.112367 0.0561836 0.998420i \(-0.482107\pi\)
0.0561836 + 0.998420i \(0.482107\pi\)
\(192\) 0 0
\(193\) −13.1394 −0.945795 −0.472898 0.881117i \(-0.656792\pi\)
−0.472898 + 0.881117i \(0.656792\pi\)
\(194\) 0 0
\(195\) 0.609023 0.0436131
\(196\) 0 0
\(197\) −13.8598 −0.987470 −0.493735 0.869612i \(-0.664369\pi\)
−0.493735 + 0.869612i \(0.664369\pi\)
\(198\) 0 0
\(199\) −19.0600 −1.35113 −0.675563 0.737302i \(-0.736100\pi\)
−0.675563 + 0.737302i \(0.736100\pi\)
\(200\) 0 0
\(201\) −13.2508 −0.934638
\(202\) 0 0
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 0.827856 0.0578199
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −10.4743 −0.724523
\(210\) 0 0
\(211\) 7.46961 0.514229 0.257115 0.966381i \(-0.417228\pi\)
0.257115 + 0.966381i \(0.417228\pi\)
\(212\) 0 0
\(213\) −8.77647 −0.601354
\(214\) 0 0
\(215\) 0.218046 0.0148706
\(216\) 0 0
\(217\) 0.195488 0.0132706
\(218\) 0 0
\(219\) −12.4182 −0.839146
\(220\) 0 0
\(221\) −15.0553 −1.01273
\(222\) 0 0
\(223\) −7.64176 −0.511730 −0.255865 0.966713i \(-0.582360\pi\)
−0.255865 + 0.966713i \(0.582360\pi\)
\(224\) 0 0
\(225\) −4.97196 −0.331464
\(226\) 0 0
\(227\) −20.1114 −1.33484 −0.667419 0.744682i \(-0.732601\pi\)
−0.667419 + 0.744682i \(0.732601\pi\)
\(228\) 0 0
\(229\) 7.30686 0.482851 0.241425 0.970419i \(-0.422385\pi\)
0.241425 + 0.970419i \(0.422385\pi\)
\(230\) 0 0
\(231\) −1.80451 −0.118728
\(232\) 0 0
\(233\) 2.30216 0.150820 0.0754099 0.997153i \(-0.475973\pi\)
0.0754099 + 0.997153i \(0.475973\pi\)
\(234\) 0 0
\(235\) −0.525692 −0.0342923
\(236\) 0 0
\(237\) −1.52569 −0.0991043
\(238\) 0 0
\(239\) −11.9767 −0.774705 −0.387353 0.921932i \(-0.626610\pi\)
−0.387353 + 0.921932i \(0.626610\pi\)
\(240\) 0 0
\(241\) 30.1620 1.94290 0.971452 0.237238i \(-0.0762421\pi\)
0.971452 + 0.237238i \(0.0762421\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.167449 −0.0106979
\(246\) 0 0
\(247\) 21.1114 1.34328
\(248\) 0 0
\(249\) −14.7531 −0.934942
\(250\) 0 0
\(251\) −18.8606 −1.19047 −0.595235 0.803552i \(-0.702941\pi\)
−0.595235 + 0.803552i \(0.702941\pi\)
\(252\) 0 0
\(253\) 1.80451 0.113449
\(254\) 0 0
\(255\) −0.693141 −0.0434062
\(256\) 0 0
\(257\) −0.613718 −0.0382827 −0.0191413 0.999817i \(-0.506093\pi\)
−0.0191413 + 0.999817i \(0.506093\pi\)
\(258\) 0 0
\(259\) 3.33490 0.207221
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 0 0
\(263\) −19.8831 −1.22605 −0.613024 0.790065i \(-0.710047\pi\)
−0.613024 + 0.790065i \(0.710047\pi\)
\(264\) 0 0
\(265\) −0.553731 −0.0340154
\(266\) 0 0
\(267\) 8.77647 0.537111
\(268\) 0 0
\(269\) 13.1020 0.798842 0.399421 0.916768i \(-0.369211\pi\)
0.399421 + 0.916768i \(0.369211\pi\)
\(270\) 0 0
\(271\) 4.53039 0.275201 0.137601 0.990488i \(-0.456061\pi\)
0.137601 + 0.990488i \(0.456061\pi\)
\(272\) 0 0
\(273\) 3.63706 0.220125
\(274\) 0 0
\(275\) −8.97196 −0.541030
\(276\) 0 0
\(277\) 1.22353 0.0735147 0.0367573 0.999324i \(-0.488297\pi\)
0.0367573 + 0.999324i \(0.488297\pi\)
\(278\) 0 0
\(279\) −0.195488 −0.0117036
\(280\) 0 0
\(281\) 16.3575 0.975804 0.487902 0.872898i \(-0.337762\pi\)
0.487902 + 0.872898i \(0.337762\pi\)
\(282\) 0 0
\(283\) −0.637062 −0.0378694 −0.0189347 0.999821i \(-0.506027\pi\)
−0.0189347 + 0.999821i \(0.506027\pi\)
\(284\) 0 0
\(285\) 0.971961 0.0575740
\(286\) 0 0
\(287\) 4.94392 0.291830
\(288\) 0 0
\(289\) 0.134715 0.00792440
\(290\) 0 0
\(291\) −14.2741 −0.836764
\(292\) 0 0
\(293\) 13.4969 0.788495 0.394248 0.919004i \(-0.371005\pi\)
0.394248 + 0.919004i \(0.371005\pi\)
\(294\) 0 0
\(295\) −0.823161 −0.0479263
\(296\) 0 0
\(297\) 1.80451 0.104708
\(298\) 0 0
\(299\) −3.63706 −0.210337
\(300\) 0 0
\(301\) 1.30216 0.0750555
\(302\) 0 0
\(303\) 8.63706 0.496186
\(304\) 0 0
\(305\) −1.46961 −0.0841498
\(306\) 0 0
\(307\) −18.6184 −1.06261 −0.531304 0.847181i \(-0.678298\pi\)
−0.531304 + 0.847181i \(0.678298\pi\)
\(308\) 0 0
\(309\) −8.13471 −0.462768
\(310\) 0 0
\(311\) −17.9945 −1.02038 −0.510188 0.860063i \(-0.670424\pi\)
−0.510188 + 0.860063i \(0.670424\pi\)
\(312\) 0 0
\(313\) 33.3847 1.88702 0.943508 0.331351i \(-0.107504\pi\)
0.943508 + 0.331351i \(0.107504\pi\)
\(314\) 0 0
\(315\) 0.167449 0.00943469
\(316\) 0 0
\(317\) −3.77647 −0.212108 −0.106054 0.994360i \(-0.533822\pi\)
−0.106054 + 0.994360i \(0.533822\pi\)
\(318\) 0 0
\(319\) 9.02256 0.505166
\(320\) 0 0
\(321\) −7.16745 −0.400048
\(322\) 0 0
\(323\) −24.0273 −1.33691
\(324\) 0 0
\(325\) 18.0833 1.00308
\(326\) 0 0
\(327\) 2.44627 0.135279
\(328\) 0 0
\(329\) −3.13941 −0.173081
\(330\) 0 0
\(331\) 7.60902 0.418230 0.209115 0.977891i \(-0.432942\pi\)
0.209115 + 0.977891i \(0.432942\pi\)
\(332\) 0 0
\(333\) −3.33490 −0.182751
\(334\) 0 0
\(335\) 2.21883 0.121228
\(336\) 0 0
\(337\) −22.3855 −1.21942 −0.609708 0.792626i \(-0.708713\pi\)
−0.609708 + 0.792626i \(0.708713\pi\)
\(338\) 0 0
\(339\) 5.10668 0.277357
\(340\) 0 0
\(341\) −0.352761 −0.0191031
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −0.167449 −0.00901516
\(346\) 0 0
\(347\) −25.4922 −1.36849 −0.684246 0.729252i \(-0.739868\pi\)
−0.684246 + 0.729252i \(0.739868\pi\)
\(348\) 0 0
\(349\) 10.1900 0.545459 0.272729 0.962091i \(-0.412074\pi\)
0.272729 + 0.962091i \(0.412074\pi\)
\(350\) 0 0
\(351\) −3.63706 −0.194132
\(352\) 0 0
\(353\) 19.2134 1.02262 0.511312 0.859395i \(-0.329160\pi\)
0.511312 + 0.859395i \(0.329160\pi\)
\(354\) 0 0
\(355\) 1.46961 0.0779990
\(356\) 0 0
\(357\) −4.13941 −0.219081
\(358\) 0 0
\(359\) −1.11607 −0.0589037 −0.0294519 0.999566i \(-0.509376\pi\)
−0.0294519 + 0.999566i \(0.509376\pi\)
\(360\) 0 0
\(361\) 14.6924 0.773282
\(362\) 0 0
\(363\) −7.74374 −0.406441
\(364\) 0 0
\(365\) 2.07942 0.108842
\(366\) 0 0
\(367\) −5.69314 −0.297180 −0.148590 0.988899i \(-0.547473\pi\)
−0.148590 + 0.988899i \(0.547473\pi\)
\(368\) 0 0
\(369\) −4.94392 −0.257370
\(370\) 0 0
\(371\) −3.30686 −0.171684
\(372\) 0 0
\(373\) −28.2967 −1.46515 −0.732573 0.680688i \(-0.761681\pi\)
−0.732573 + 0.680688i \(0.761681\pi\)
\(374\) 0 0
\(375\) 1.66980 0.0862279
\(376\) 0 0
\(377\) −18.1853 −0.936591
\(378\) 0 0
\(379\) −13.3622 −0.686368 −0.343184 0.939268i \(-0.611505\pi\)
−0.343184 + 0.939268i \(0.611505\pi\)
\(380\) 0 0
\(381\) 8.97666 0.459888
\(382\) 0 0
\(383\) 2.75313 0.140678 0.0703391 0.997523i \(-0.477592\pi\)
0.0703391 + 0.997523i \(0.477592\pi\)
\(384\) 0 0
\(385\) 0.302164 0.0153997
\(386\) 0 0
\(387\) −1.30216 −0.0661927
\(388\) 0 0
\(389\) 6.92136 0.350927 0.175464 0.984486i \(-0.443858\pi\)
0.175464 + 0.984486i \(0.443858\pi\)
\(390\) 0 0
\(391\) 4.13941 0.209339
\(392\) 0 0
\(393\) −9.86059 −0.497401
\(394\) 0 0
\(395\) 0.255476 0.0128544
\(396\) 0 0
\(397\) −2.60433 −0.130707 −0.0653537 0.997862i \(-0.520818\pi\)
−0.0653537 + 0.997862i \(0.520818\pi\)
\(398\) 0 0
\(399\) 5.80451 0.290589
\(400\) 0 0
\(401\) −2.13941 −0.106837 −0.0534185 0.998572i \(-0.517012\pi\)
−0.0534185 + 0.998572i \(0.517012\pi\)
\(402\) 0 0
\(403\) 0.711004 0.0354176
\(404\) 0 0
\(405\) −0.167449 −0.00832062
\(406\) 0 0
\(407\) −6.01786 −0.298294
\(408\) 0 0
\(409\) −34.2500 −1.69355 −0.846777 0.531949i \(-0.821460\pi\)
−0.846777 + 0.531949i \(0.821460\pi\)
\(410\) 0 0
\(411\) −13.0000 −0.641243
\(412\) 0 0
\(413\) −4.91588 −0.241895
\(414\) 0 0
\(415\) 2.47040 0.121267
\(416\) 0 0
\(417\) 14.3902 0.704691
\(418\) 0 0
\(419\) −3.91119 −0.191074 −0.0955370 0.995426i \(-0.530457\pi\)
−0.0955370 + 0.995426i \(0.530457\pi\)
\(420\) 0 0
\(421\) −31.5249 −1.53643 −0.768215 0.640192i \(-0.778855\pi\)
−0.768215 + 0.640192i \(0.778855\pi\)
\(422\) 0 0
\(423\) 3.13941 0.152643
\(424\) 0 0
\(425\) −20.5810 −0.998324
\(426\) 0 0
\(427\) −8.77647 −0.424723
\(428\) 0 0
\(429\) −6.56312 −0.316870
\(430\) 0 0
\(431\) 19.5716 0.942730 0.471365 0.881938i \(-0.343761\pi\)
0.471365 + 0.881938i \(0.343761\pi\)
\(432\) 0 0
\(433\) −16.0880 −0.773141 −0.386571 0.922260i \(-0.626340\pi\)
−0.386571 + 0.922260i \(0.626340\pi\)
\(434\) 0 0
\(435\) −0.837246 −0.0401429
\(436\) 0 0
\(437\) −5.80451 −0.277667
\(438\) 0 0
\(439\) −20.0833 −0.958525 −0.479263 0.877672i \(-0.659096\pi\)
−0.479263 + 0.877672i \(0.659096\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 11.0833 0.526585 0.263292 0.964716i \(-0.415192\pi\)
0.263292 + 0.964716i \(0.415192\pi\)
\(444\) 0 0
\(445\) −1.46961 −0.0696663
\(446\) 0 0
\(447\) −2.33490 −0.110437
\(448\) 0 0
\(449\) −30.8598 −1.45636 −0.728182 0.685384i \(-0.759634\pi\)
−0.728182 + 0.685384i \(0.759634\pi\)
\(450\) 0 0
\(451\) −8.92136 −0.420091
\(452\) 0 0
\(453\) −11.8092 −0.554845
\(454\) 0 0
\(455\) −0.609023 −0.0285514
\(456\) 0 0
\(457\) 6.91588 0.323511 0.161756 0.986831i \(-0.448284\pi\)
0.161756 + 0.986831i \(0.448284\pi\)
\(458\) 0 0
\(459\) 4.13941 0.193211
\(460\) 0 0
\(461\) −30.4688 −1.41907 −0.709537 0.704668i \(-0.751096\pi\)
−0.709537 + 0.704668i \(0.751096\pi\)
\(462\) 0 0
\(463\) 29.9392 1.39139 0.695697 0.718335i \(-0.255096\pi\)
0.695697 + 0.718335i \(0.255096\pi\)
\(464\) 0 0
\(465\) 0.0327344 0.00151802
\(466\) 0 0
\(467\) 6.21805 0.287737 0.143868 0.989597i \(-0.454046\pi\)
0.143868 + 0.989597i \(0.454046\pi\)
\(468\) 0 0
\(469\) 13.2508 0.611864
\(470\) 0 0
\(471\) 2.72588 0.125602
\(472\) 0 0
\(473\) −2.34977 −0.108043
\(474\) 0 0
\(475\) 28.8598 1.32418
\(476\) 0 0
\(477\) 3.30686 0.151411
\(478\) 0 0
\(479\) −43.6457 −1.99422 −0.997111 0.0759622i \(-0.975797\pi\)
−0.997111 + 0.0759622i \(0.975797\pi\)
\(480\) 0 0
\(481\) 12.1292 0.553045
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 2.39019 0.108533
\(486\) 0 0
\(487\) 26.1059 1.18297 0.591485 0.806316i \(-0.298542\pi\)
0.591485 + 0.806316i \(0.298542\pi\)
\(488\) 0 0
\(489\) −14.6371 −0.661911
\(490\) 0 0
\(491\) 4.99921 0.225611 0.112806 0.993617i \(-0.464016\pi\)
0.112806 + 0.993617i \(0.464016\pi\)
\(492\) 0 0
\(493\) 20.6970 0.932148
\(494\) 0 0
\(495\) −0.302164 −0.0135813
\(496\) 0 0
\(497\) 8.77647 0.393679
\(498\) 0 0
\(499\) 27.2508 1.21991 0.609956 0.792435i \(-0.291187\pi\)
0.609956 + 0.792435i \(0.291187\pi\)
\(500\) 0 0
\(501\) −1.20018 −0.0536202
\(502\) 0 0
\(503\) 39.4174 1.75754 0.878768 0.477248i \(-0.158366\pi\)
0.878768 + 0.477248i \(0.158366\pi\)
\(504\) 0 0
\(505\) −1.44627 −0.0643581
\(506\) 0 0
\(507\) 0.228223 0.0101357
\(508\) 0 0
\(509\) −36.7710 −1.62985 −0.814923 0.579570i \(-0.803221\pi\)
−0.814923 + 0.579570i \(0.803221\pi\)
\(510\) 0 0
\(511\) 12.4182 0.549350
\(512\) 0 0
\(513\) −5.80451 −0.256275
\(514\) 0 0
\(515\) 1.36215 0.0600236
\(516\) 0 0
\(517\) 5.66510 0.249151
\(518\) 0 0
\(519\) 19.0880 0.837871
\(520\) 0 0
\(521\) 21.8878 0.958924 0.479462 0.877563i \(-0.340832\pi\)
0.479462 + 0.877563i \(0.340832\pi\)
\(522\) 0 0
\(523\) 17.2180 0.752893 0.376446 0.926438i \(-0.377146\pi\)
0.376446 + 0.926438i \(0.377146\pi\)
\(524\) 0 0
\(525\) 4.97196 0.216994
\(526\) 0 0
\(527\) −0.809207 −0.0352496
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.91588 0.213331
\(532\) 0 0
\(533\) 17.9814 0.778859
\(534\) 0 0
\(535\) 1.20018 0.0518885
\(536\) 0 0
\(537\) −7.49765 −0.323548
\(538\) 0 0
\(539\) 1.80451 0.0777258
\(540\) 0 0
\(541\) −23.6745 −1.01785 −0.508923 0.860812i \(-0.669956\pi\)
−0.508923 + 0.860812i \(0.669956\pi\)
\(542\) 0 0
\(543\) 16.4088 0.704171
\(544\) 0 0
\(545\) −0.409626 −0.0175464
\(546\) 0 0
\(547\) 39.9206 1.70688 0.853440 0.521191i \(-0.174512\pi\)
0.853440 + 0.521191i \(0.174512\pi\)
\(548\) 0 0
\(549\) 8.77647 0.374571
\(550\) 0 0
\(551\) −29.0226 −1.23640
\(552\) 0 0
\(553\) 1.52569 0.0648790
\(554\) 0 0
\(555\) 0.558426 0.0237039
\(556\) 0 0
\(557\) 23.2835 0.986554 0.493277 0.869872i \(-0.335799\pi\)
0.493277 + 0.869872i \(0.335799\pi\)
\(558\) 0 0
\(559\) 4.73605 0.200314
\(560\) 0 0
\(561\) 7.46961 0.315367
\(562\) 0 0
\(563\) 32.4735 1.36860 0.684298 0.729203i \(-0.260109\pi\)
0.684298 + 0.729203i \(0.260109\pi\)
\(564\) 0 0
\(565\) −0.855109 −0.0359747
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −14.1441 −0.592952 −0.296476 0.955040i \(-0.595811\pi\)
−0.296476 + 0.955040i \(0.595811\pi\)
\(570\) 0 0
\(571\) −32.3715 −1.35471 −0.677353 0.735658i \(-0.736873\pi\)
−0.677353 + 0.735658i \(0.736873\pi\)
\(572\) 0 0
\(573\) 1.55294 0.0648752
\(574\) 0 0
\(575\) −4.97196 −0.207345
\(576\) 0 0
\(577\) −13.7757 −0.573489 −0.286745 0.958007i \(-0.592573\pi\)
−0.286745 + 0.958007i \(0.592573\pi\)
\(578\) 0 0
\(579\) −13.1394 −0.546055
\(580\) 0 0
\(581\) 14.7531 0.612063
\(582\) 0 0
\(583\) 5.96727 0.247139
\(584\) 0 0
\(585\) 0.609023 0.0251800
\(586\) 0 0
\(587\) 22.1721 0.915142 0.457571 0.889173i \(-0.348719\pi\)
0.457571 + 0.889173i \(0.348719\pi\)
\(588\) 0 0
\(589\) 1.13471 0.0467551
\(590\) 0 0
\(591\) −13.8598 −0.570116
\(592\) 0 0
\(593\) 26.3668 1.08276 0.541378 0.840779i \(-0.317903\pi\)
0.541378 + 0.840779i \(0.317903\pi\)
\(594\) 0 0
\(595\) 0.693141 0.0284160
\(596\) 0 0
\(597\) −19.0600 −0.780073
\(598\) 0 0
\(599\) 1.89332 0.0773591 0.0386796 0.999252i \(-0.487685\pi\)
0.0386796 + 0.999252i \(0.487685\pi\)
\(600\) 0 0
\(601\) −2.96257 −0.120846 −0.0604229 0.998173i \(-0.519245\pi\)
−0.0604229 + 0.998173i \(0.519245\pi\)
\(602\) 0 0
\(603\) −13.2508 −0.539614
\(604\) 0 0
\(605\) 1.29668 0.0527176
\(606\) 0 0
\(607\) 1.67058 0.0678069 0.0339034 0.999425i \(-0.489206\pi\)
0.0339034 + 0.999425i \(0.489206\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) −11.4182 −0.461932
\(612\) 0 0
\(613\) −14.1059 −0.569732 −0.284866 0.958567i \(-0.591949\pi\)
−0.284866 + 0.958567i \(0.591949\pi\)
\(614\) 0 0
\(615\) 0.827856 0.0333824
\(616\) 0 0
\(617\) 12.4510 0.501257 0.250628 0.968083i \(-0.419363\pi\)
0.250628 + 0.968083i \(0.419363\pi\)
\(618\) 0 0
\(619\) −15.9945 −0.642874 −0.321437 0.946931i \(-0.604166\pi\)
−0.321437 + 0.946931i \(0.604166\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −8.77647 −0.351622
\(624\) 0 0
\(625\) 24.5802 0.983208
\(626\) 0 0
\(627\) −10.4743 −0.418304
\(628\) 0 0
\(629\) −13.8045 −0.550422
\(630\) 0 0
\(631\) −44.6970 −1.77936 −0.889681 0.456583i \(-0.849073\pi\)
−0.889681 + 0.456583i \(0.849073\pi\)
\(632\) 0 0
\(633\) 7.46961 0.296890
\(634\) 0 0
\(635\) −1.50313 −0.0596500
\(636\) 0 0
\(637\) −3.63706 −0.144106
\(638\) 0 0
\(639\) −8.77647 −0.347192
\(640\) 0 0
\(641\) −6.37233 −0.251692 −0.125846 0.992050i \(-0.540164\pi\)
−0.125846 + 0.992050i \(0.540164\pi\)
\(642\) 0 0
\(643\) −2.11607 −0.0834495 −0.0417247 0.999129i \(-0.513285\pi\)
−0.0417247 + 0.999129i \(0.513285\pi\)
\(644\) 0 0
\(645\) 0.218046 0.00858557
\(646\) 0 0
\(647\) −6.98135 −0.274465 −0.137233 0.990539i \(-0.543821\pi\)
−0.137233 + 0.990539i \(0.543821\pi\)
\(648\) 0 0
\(649\) 8.87077 0.348208
\(650\) 0 0
\(651\) 0.195488 0.00766180
\(652\) 0 0
\(653\) 47.5522 1.86086 0.930430 0.366470i \(-0.119434\pi\)
0.930430 + 0.366470i \(0.119434\pi\)
\(654\) 0 0
\(655\) 1.65115 0.0645157
\(656\) 0 0
\(657\) −12.4182 −0.484481
\(658\) 0 0
\(659\) 39.5708 1.54146 0.770730 0.637162i \(-0.219892\pi\)
0.770730 + 0.637162i \(0.219892\pi\)
\(660\) 0 0
\(661\) 32.4088 1.26056 0.630279 0.776369i \(-0.282941\pi\)
0.630279 + 0.776369i \(0.282941\pi\)
\(662\) 0 0
\(663\) −15.0553 −0.584699
\(664\) 0 0
\(665\) −0.971961 −0.0376910
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 0 0
\(669\) −7.64176 −0.295447
\(670\) 0 0
\(671\) 15.8372 0.611390
\(672\) 0 0
\(673\) −8.05138 −0.310358 −0.155179 0.987886i \(-0.549595\pi\)
−0.155179 + 0.987886i \(0.549595\pi\)
\(674\) 0 0
\(675\) −4.97196 −0.191371
\(676\) 0 0
\(677\) −15.0647 −0.578983 −0.289491 0.957181i \(-0.593486\pi\)
−0.289491 + 0.957181i \(0.593486\pi\)
\(678\) 0 0
\(679\) 14.2741 0.547791
\(680\) 0 0
\(681\) −20.1114 −0.770669
\(682\) 0 0
\(683\) −33.3481 −1.27603 −0.638014 0.770025i \(-0.720244\pi\)
−0.638014 + 0.770025i \(0.720244\pi\)
\(684\) 0 0
\(685\) 2.17684 0.0831728
\(686\) 0 0
\(687\) 7.30686 0.278774
\(688\) 0 0
\(689\) −12.0273 −0.458202
\(690\) 0 0
\(691\) 24.1853 0.920053 0.460026 0.887905i \(-0.347840\pi\)
0.460026 + 0.887905i \(0.347840\pi\)
\(692\) 0 0
\(693\) −1.80451 −0.0685477
\(694\) 0 0
\(695\) −2.40963 −0.0914023
\(696\) 0 0
\(697\) −20.4649 −0.775164
\(698\) 0 0
\(699\) 2.30216 0.0870758
\(700\) 0 0
\(701\) 8.69314 0.328335 0.164168 0.986432i \(-0.447506\pi\)
0.164168 + 0.986432i \(0.447506\pi\)
\(702\) 0 0
\(703\) 19.3575 0.730080
\(704\) 0 0
\(705\) −0.525692 −0.0197987
\(706\) 0 0
\(707\) −8.63706 −0.324830
\(708\) 0 0
\(709\) −14.0973 −0.529435 −0.264717 0.964326i \(-0.585279\pi\)
−0.264717 + 0.964326i \(0.585279\pi\)
\(710\) 0 0
\(711\) −1.52569 −0.0572179
\(712\) 0 0
\(713\) −0.195488 −0.00732110
\(714\) 0 0
\(715\) 1.09899 0.0410999
\(716\) 0 0
\(717\) −11.9767 −0.447276
\(718\) 0 0
\(719\) 0.0607736 0.00226647 0.00113324 0.999999i \(-0.499639\pi\)
0.00113324 + 0.999999i \(0.499639\pi\)
\(720\) 0 0
\(721\) 8.13471 0.302953
\(722\) 0 0
\(723\) 30.1620 1.12174
\(724\) 0 0
\(725\) −24.8598 −0.923270
\(726\) 0 0
\(727\) −0.743738 −0.0275837 −0.0137919 0.999905i \(-0.504390\pi\)
−0.0137919 + 0.999905i \(0.504390\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.39019 −0.199363
\(732\) 0 0
\(733\) −36.6043 −1.35201 −0.676006 0.736896i \(-0.736291\pi\)
−0.676006 + 0.736896i \(0.736291\pi\)
\(734\) 0 0
\(735\) −0.167449 −0.00617646
\(736\) 0 0
\(737\) −23.9112 −0.880780
\(738\) 0 0
\(739\) 13.8606 0.509870 0.254935 0.966958i \(-0.417946\pi\)
0.254935 + 0.966958i \(0.417946\pi\)
\(740\) 0 0
\(741\) 21.1114 0.775546
\(742\) 0 0
\(743\) 5.96727 0.218918 0.109459 0.993991i \(-0.465088\pi\)
0.109459 + 0.993991i \(0.465088\pi\)
\(744\) 0 0
\(745\) 0.390977 0.0143243
\(746\) 0 0
\(747\) −14.7531 −0.539789
\(748\) 0 0
\(749\) 7.16745 0.261893
\(750\) 0 0
\(751\) 5.19627 0.189615 0.0948074 0.995496i \(-0.469776\pi\)
0.0948074 + 0.995496i \(0.469776\pi\)
\(752\) 0 0
\(753\) −18.8606 −0.687318
\(754\) 0 0
\(755\) 1.97744 0.0719665
\(756\) 0 0
\(757\) 16.4088 0.596389 0.298195 0.954505i \(-0.403616\pi\)
0.298195 + 0.954505i \(0.403616\pi\)
\(758\) 0 0
\(759\) 1.80451 0.0654996
\(760\) 0 0
\(761\) 20.2882 0.735447 0.367724 0.929935i \(-0.380137\pi\)
0.367724 + 0.929935i \(0.380137\pi\)
\(762\) 0 0
\(763\) −2.44627 −0.0885609
\(764\) 0 0
\(765\) −0.693141 −0.0250606
\(766\) 0 0
\(767\) −17.8794 −0.645587
\(768\) 0 0
\(769\) −9.92136 −0.357774 −0.178887 0.983870i \(-0.557250\pi\)
−0.178887 + 0.983870i \(0.557250\pi\)
\(770\) 0 0
\(771\) −0.613718 −0.0221025
\(772\) 0 0
\(773\) −45.7055 −1.64391 −0.821957 0.569550i \(-0.807117\pi\)
−0.821957 + 0.569550i \(0.807117\pi\)
\(774\) 0 0
\(775\) 0.971961 0.0349139
\(776\) 0 0
\(777\) 3.33490 0.119639
\(778\) 0 0
\(779\) 28.6970 1.02818
\(780\) 0 0
\(781\) −15.8372 −0.566701
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) −0.456446 −0.0162912
\(786\) 0 0
\(787\) 37.8084 1.34772 0.673862 0.738857i \(-0.264634\pi\)
0.673862 + 0.738857i \(0.264634\pi\)
\(788\) 0 0
\(789\) −19.8831 −0.707859
\(790\) 0 0
\(791\) −5.10668 −0.181572
\(792\) 0 0
\(793\) −31.9206 −1.13353
\(794\) 0 0
\(795\) −0.553731 −0.0196388
\(796\) 0 0
\(797\) 10.9151 0.386633 0.193316 0.981136i \(-0.438076\pi\)
0.193316 + 0.981136i \(0.438076\pi\)
\(798\) 0 0
\(799\) 12.9953 0.459741
\(800\) 0 0
\(801\) 8.77647 0.310101
\(802\) 0 0
\(803\) −22.4088 −0.790791
\(804\) 0 0
\(805\) 0.167449 0.00590181
\(806\) 0 0
\(807\) 13.1020 0.461212
\(808\) 0 0
\(809\) 14.2733 0.501824 0.250912 0.968010i \(-0.419270\pi\)
0.250912 + 0.968010i \(0.419270\pi\)
\(810\) 0 0
\(811\) 43.7804 1.53734 0.768669 0.639647i \(-0.220920\pi\)
0.768669 + 0.639647i \(0.220920\pi\)
\(812\) 0 0
\(813\) 4.53039 0.158888
\(814\) 0 0
\(815\) 2.45096 0.0858535
\(816\) 0 0
\(817\) 7.55843 0.264436
\(818\) 0 0
\(819\) 3.63706 0.127089
\(820\) 0 0
\(821\) 31.1620 1.08756 0.543780 0.839228i \(-0.316993\pi\)
0.543780 + 0.839228i \(0.316993\pi\)
\(822\) 0 0
\(823\) −15.7859 −0.550261 −0.275130 0.961407i \(-0.588721\pi\)
−0.275130 + 0.961407i \(0.588721\pi\)
\(824\) 0 0
\(825\) −8.97196 −0.312364
\(826\) 0 0
\(827\) −40.3661 −1.40367 −0.701833 0.712342i \(-0.747635\pi\)
−0.701833 + 0.712342i \(0.747635\pi\)
\(828\) 0 0
\(829\) 9.42293 0.327272 0.163636 0.986521i \(-0.447678\pi\)
0.163636 + 0.986521i \(0.447678\pi\)
\(830\) 0 0
\(831\) 1.22353 0.0424437
\(832\) 0 0
\(833\) 4.13941 0.143422
\(834\) 0 0
\(835\) 0.200970 0.00695485
\(836\) 0 0
\(837\) −0.195488 −0.00675707
\(838\) 0 0
\(839\) −5.83725 −0.201524 −0.100762 0.994911i \(-0.532128\pi\)
−0.100762 + 0.994911i \(0.532128\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 16.3575 0.563381
\(844\) 0 0
\(845\) −0.0382157 −0.00131466
\(846\) 0 0
\(847\) 7.74374 0.266078
\(848\) 0 0
\(849\) −0.637062 −0.0218639
\(850\) 0 0
\(851\) −3.33490 −0.114319
\(852\) 0 0
\(853\) 11.9120 0.407858 0.203929 0.978986i \(-0.434629\pi\)
0.203929 + 0.978986i \(0.434629\pi\)
\(854\) 0 0
\(855\) 0.971961 0.0332404
\(856\) 0 0
\(857\) 17.6970 0.604520 0.302260 0.953226i \(-0.402259\pi\)
0.302260 + 0.953226i \(0.402259\pi\)
\(858\) 0 0
\(859\) 22.6924 0.774253 0.387127 0.922027i \(-0.373468\pi\)
0.387127 + 0.922027i \(0.373468\pi\)
\(860\) 0 0
\(861\) 4.94392 0.168488
\(862\) 0 0
\(863\) 19.8412 0.675401 0.337700 0.941254i \(-0.390351\pi\)
0.337700 + 0.941254i \(0.390351\pi\)
\(864\) 0 0
\(865\) −3.19627 −0.108677
\(866\) 0 0
\(867\) 0.134715 0.00457515
\(868\) 0 0
\(869\) −2.75313 −0.0933935
\(870\) 0 0
\(871\) 48.1939 1.63299
\(872\) 0 0
\(873\) −14.2741 −0.483106
\(874\) 0 0
\(875\) −1.66980 −0.0564494
\(876\) 0 0
\(877\) −33.0965 −1.11759 −0.558795 0.829306i \(-0.688736\pi\)
−0.558795 + 0.829306i \(0.688736\pi\)
\(878\) 0 0
\(879\) 13.4969 0.455238
\(880\) 0 0
\(881\) −46.2134 −1.55697 −0.778484 0.627665i \(-0.784011\pi\)
−0.778484 + 0.627665i \(0.784011\pi\)
\(882\) 0 0
\(883\) 10.7671 0.362341 0.181171 0.983452i \(-0.442011\pi\)
0.181171 + 0.983452i \(0.442011\pi\)
\(884\) 0 0
\(885\) −0.823161 −0.0276702
\(886\) 0 0
\(887\) −35.3153 −1.18577 −0.592886 0.805286i \(-0.702012\pi\)
−0.592886 + 0.805286i \(0.702012\pi\)
\(888\) 0 0
\(889\) −8.97666 −0.301067
\(890\) 0 0
\(891\) 1.80451 0.0604534
\(892\) 0 0
\(893\) −18.2227 −0.609801
\(894\) 0 0
\(895\) 1.25548 0.0419659
\(896\) 0 0
\(897\) −3.63706 −0.121438
\(898\) 0 0
\(899\) −0.977442 −0.0325995
\(900\) 0 0
\(901\) 13.6884 0.456028
\(902\) 0 0
\(903\) 1.30216 0.0433333
\(904\) 0 0
\(905\) −2.74765 −0.0913349
\(906\) 0 0
\(907\) 26.7157 0.887080 0.443540 0.896255i \(-0.353722\pi\)
0.443540 + 0.896255i \(0.353722\pi\)
\(908\) 0 0
\(909\) 8.63706 0.286473
\(910\) 0 0
\(911\) 14.2453 0.471968 0.235984 0.971757i \(-0.424169\pi\)
0.235984 + 0.971757i \(0.424169\pi\)
\(912\) 0 0
\(913\) −26.6222 −0.881066
\(914\) 0 0
\(915\) −1.46961 −0.0485839
\(916\) 0 0
\(917\) 9.86059 0.325625
\(918\) 0 0
\(919\) 9.20018 0.303486 0.151743 0.988420i \(-0.451511\pi\)
0.151743 + 0.988420i \(0.451511\pi\)
\(920\) 0 0
\(921\) −18.6184 −0.613498
\(922\) 0 0
\(923\) 31.9206 1.05068
\(924\) 0 0
\(925\) 16.5810 0.545179
\(926\) 0 0
\(927\) −8.13471 −0.267179
\(928\) 0 0
\(929\) −27.4416 −0.900329 −0.450164 0.892946i \(-0.648635\pi\)
−0.450164 + 0.892946i \(0.648635\pi\)
\(930\) 0 0
\(931\) −5.80451 −0.190235
\(932\) 0 0
\(933\) −17.9945 −0.589114
\(934\) 0 0
\(935\) −1.25078 −0.0409049
\(936\) 0 0
\(937\) 20.8785 0.682069 0.341035 0.940051i \(-0.389223\pi\)
0.341035 + 0.940051i \(0.389223\pi\)
\(938\) 0 0
\(939\) 33.3847 1.08947
\(940\) 0 0
\(941\) 24.2469 0.790425 0.395213 0.918590i \(-0.370671\pi\)
0.395213 + 0.918590i \(0.370671\pi\)
\(942\) 0 0
\(943\) −4.94392 −0.160996
\(944\) 0 0
\(945\) 0.167449 0.00544712
\(946\) 0 0
\(947\) −4.41353 −0.143421 −0.0717103 0.997426i \(-0.522846\pi\)
−0.0717103 + 0.997426i \(0.522846\pi\)
\(948\) 0 0
\(949\) 45.1659 1.46615
\(950\) 0 0
\(951\) −3.77647 −0.122461
\(952\) 0 0
\(953\) −8.21883 −0.266234 −0.133117 0.991100i \(-0.542499\pi\)
−0.133117 + 0.991100i \(0.542499\pi\)
\(954\) 0 0
\(955\) −0.260039 −0.00841468
\(956\) 0 0
\(957\) 9.02256 0.291658
\(958\) 0 0
\(959\) 13.0000 0.419792
\(960\) 0 0
\(961\) −30.9618 −0.998767
\(962\) 0 0
\(963\) −7.16745 −0.230968
\(964\) 0 0
\(965\) 2.20018 0.0708264
\(966\) 0 0
\(967\) −5.48748 −0.176465 −0.0882327 0.996100i \(-0.528122\pi\)
−0.0882327 + 0.996100i \(0.528122\pi\)
\(968\) 0 0
\(969\) −24.0273 −0.771867
\(970\) 0 0
\(971\) −51.2686 −1.64529 −0.822645 0.568556i \(-0.807502\pi\)
−0.822645 + 0.568556i \(0.807502\pi\)
\(972\) 0 0
\(973\) −14.3902 −0.461328
\(974\) 0 0
\(975\) 18.0833 0.579130
\(976\) 0 0
\(977\) 3.25548 0.104152 0.0520759 0.998643i \(-0.483416\pi\)
0.0520759 + 0.998643i \(0.483416\pi\)
\(978\) 0 0
\(979\) 15.8372 0.506161
\(980\) 0 0
\(981\) 2.44627 0.0781034
\(982\) 0 0
\(983\) 9.03195 0.288074 0.144037 0.989572i \(-0.453992\pi\)
0.144037 + 0.989572i \(0.453992\pi\)
\(984\) 0 0
\(985\) 2.32081 0.0739472
\(986\) 0 0
\(987\) −3.13941 −0.0999285
\(988\) 0 0
\(989\) −1.30216 −0.0414064
\(990\) 0 0
\(991\) −42.4510 −1.34850 −0.674250 0.738503i \(-0.735533\pi\)
−0.674250 + 0.738503i \(0.735533\pi\)
\(992\) 0 0
\(993\) 7.60902 0.241465
\(994\) 0 0
\(995\) 3.19158 0.101180
\(996\) 0 0
\(997\) 40.6222 1.28652 0.643259 0.765649i \(-0.277582\pi\)
0.643259 + 0.765649i \(0.277582\pi\)
\(998\) 0 0
\(999\) −3.33490 −0.105512
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 7728.2.a.bx.1.2 3
4.3 odd 2 3864.2.a.m.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.m.1.2 3 4.3 odd 2
7728.2.a.bx.1.2 3 1.1 even 1 trivial