Properties

Label 6930.2.a.bv.1.2
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +5.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +3.26795 q^{19} +1.00000 q^{20} +1.00000 q^{22} -2.19615 q^{23} +1.00000 q^{25} -5.46410 q^{26} +1.00000 q^{28} +1.26795 q^{29} +2.00000 q^{31} -1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} -2.73205 q^{37} -3.26795 q^{38} -1.00000 q^{40} -8.19615 q^{41} +2.00000 q^{43} -1.00000 q^{44} +2.19615 q^{46} +6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} +5.46410 q^{52} +10.7321 q^{53} -1.00000 q^{55} -1.00000 q^{56} -1.26795 q^{58} +6.92820 q^{59} +8.92820 q^{61} -2.00000 q^{62} +1.00000 q^{64} +5.46410 q^{65} -4.00000 q^{67} -3.46410 q^{68} -1.00000 q^{70} -2.53590 q^{71} +6.39230 q^{73} +2.73205 q^{74} +3.26795 q^{76} -1.00000 q^{77} -1.80385 q^{79} +1.00000 q^{80} +8.19615 q^{82} -4.39230 q^{83} -3.46410 q^{85} -2.00000 q^{86} +1.00000 q^{88} +3.46410 q^{89} +5.46410 q^{91} -2.19615 q^{92} -6.92820 q^{94} +3.26795 q^{95} -16.5885 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{19} + 2 q^{20} + 2 q^{22} + 6 q^{23} + 2 q^{25} - 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 2 q^{32} + 2 q^{35} - 2 q^{37} - 10 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 18 q^{53} - 2 q^{55} - 2 q^{56} - 6 q^{58} + 4 q^{61} - 4 q^{62} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} - 8 q^{73} + 2 q^{74} + 10 q^{76} - 2 q^{77} - 14 q^{79} + 2 q^{80} + 6 q^{82} + 12 q^{83} - 4 q^{86} + 2 q^{88} + 4 q^{91} + 6 q^{92} + 10 q^{95} - 2 q^{97} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 3.26795 0.749719 0.374859 0.927082i \(-0.377691\pi\)
0.374859 + 0.927082i \(0.377691\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.19615 −0.457929 −0.228965 0.973435i \(-0.573534\pi\)
−0.228965 + 0.973435i \(0.573534\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.46410 −1.07160
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 1.26795 0.235452 0.117726 0.993046i \(-0.462440\pi\)
0.117726 + 0.993046i \(0.462440\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.46410 0.594089
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.73205 −0.449146 −0.224573 0.974457i \(-0.572099\pi\)
−0.224573 + 0.974457i \(0.572099\pi\)
\(38\) −3.26795 −0.530131
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −8.19615 −1.28002 −0.640012 0.768365i \(-0.721071\pi\)
−0.640012 + 0.768365i \(0.721071\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 2.19615 0.323805
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 5.46410 0.757735
\(53\) 10.7321 1.47416 0.737080 0.675805i \(-0.236204\pi\)
0.737080 + 0.675805i \(0.236204\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −1.26795 −0.166490
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.46410 0.677738
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −2.53590 −0.300956 −0.150478 0.988613i \(-0.548081\pi\)
−0.150478 + 0.988613i \(0.548081\pi\)
\(72\) 0 0
\(73\) 6.39230 0.748163 0.374081 0.927396i \(-0.377958\pi\)
0.374081 + 0.927396i \(0.377958\pi\)
\(74\) 2.73205 0.317594
\(75\) 0 0
\(76\) 3.26795 0.374859
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −1.80385 −0.202949 −0.101474 0.994838i \(-0.532356\pi\)
−0.101474 + 0.994838i \(0.532356\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 8.19615 0.905114
\(83\) −4.39230 −0.482118 −0.241059 0.970510i \(-0.577495\pi\)
−0.241059 + 0.970510i \(0.577495\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) −2.19615 −0.228965
\(93\) 0 0
\(94\) −6.92820 −0.714590
\(95\) 3.26795 0.335285
\(96\) 0 0
\(97\) −16.5885 −1.68430 −0.842151 0.539241i \(-0.818711\pi\)
−0.842151 + 0.539241i \(0.818711\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −19.8564 −1.97579 −0.987893 0.155136i \(-0.950418\pi\)
−0.987893 + 0.155136i \(0.950418\pi\)
\(102\) 0 0
\(103\) −8.39230 −0.826918 −0.413459 0.910523i \(-0.635680\pi\)
−0.413459 + 0.910523i \(0.635680\pi\)
\(104\) −5.46410 −0.535799
\(105\) 0 0
\(106\) −10.7321 −1.04239
\(107\) 19.8564 1.91959 0.959796 0.280700i \(-0.0905665\pi\)
0.959796 + 0.280700i \(0.0905665\pi\)
\(108\) 0 0
\(109\) 1.66025 0.159023 0.0795117 0.996834i \(-0.474664\pi\)
0.0795117 + 0.996834i \(0.474664\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 19.8564 1.86793 0.933967 0.357360i \(-0.116323\pi\)
0.933967 + 0.357360i \(0.116323\pi\)
\(114\) 0 0
\(115\) −2.19615 −0.204792
\(116\) 1.26795 0.117726
\(117\) 0 0
\(118\) −6.92820 −0.637793
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −8.92820 −0.808322
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.9282 1.32466 0.662332 0.749211i \(-0.269567\pi\)
0.662332 + 0.749211i \(0.269567\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −5.46410 −0.479233
\(131\) 11.6603 1.01876 0.509381 0.860541i \(-0.329875\pi\)
0.509381 + 0.860541i \(0.329875\pi\)
\(132\) 0 0
\(133\) 3.26795 0.283367
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) −12.9282 −1.10453 −0.552265 0.833668i \(-0.686237\pi\)
−0.552265 + 0.833668i \(0.686237\pi\)
\(138\) 0 0
\(139\) −3.66025 −0.310459 −0.155229 0.987878i \(-0.549612\pi\)
−0.155229 + 0.987878i \(0.549612\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 2.53590 0.212808
\(143\) −5.46410 −0.456931
\(144\) 0 0
\(145\) 1.26795 0.105297
\(146\) −6.39230 −0.529031
\(147\) 0 0
\(148\) −2.73205 −0.224573
\(149\) 10.7321 0.879204 0.439602 0.898193i \(-0.355120\pi\)
0.439602 + 0.898193i \(0.355120\pi\)
\(150\) 0 0
\(151\) −13.1244 −1.06804 −0.534022 0.845470i \(-0.679320\pi\)
−0.534022 + 0.845470i \(0.679320\pi\)
\(152\) −3.26795 −0.265066
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 11.4641 0.914935 0.457467 0.889226i \(-0.348757\pi\)
0.457467 + 0.889226i \(0.348757\pi\)
\(158\) 1.80385 0.143506
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −2.19615 −0.173081
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −8.19615 −0.640012
\(165\) 0 0
\(166\) 4.39230 0.340909
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 3.46410 0.265684
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 12.9282 0.982913 0.491457 0.870902i \(-0.336465\pi\)
0.491457 + 0.870902i \(0.336465\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −3.46410 −0.259645
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −5.46410 −0.405026
\(183\) 0 0
\(184\) 2.19615 0.161903
\(185\) −2.73205 −0.200864
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) 6.92820 0.505291
\(189\) 0 0
\(190\) −3.26795 −0.237082
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −8.39230 −0.604091 −0.302046 0.953293i \(-0.597670\pi\)
−0.302046 + 0.953293i \(0.597670\pi\)
\(194\) 16.5885 1.19098
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −24.2487 −1.72765 −0.863825 0.503793i \(-0.831938\pi\)
−0.863825 + 0.503793i \(0.831938\pi\)
\(198\) 0 0
\(199\) −10.9282 −0.774680 −0.387340 0.921937i \(-0.626606\pi\)
−0.387340 + 0.921937i \(0.626606\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 19.8564 1.39709
\(203\) 1.26795 0.0889926
\(204\) 0 0
\(205\) −8.19615 −0.572444
\(206\) 8.39230 0.584720
\(207\) 0 0
\(208\) 5.46410 0.378867
\(209\) −3.26795 −0.226049
\(210\) 0 0
\(211\) 13.0718 0.899900 0.449950 0.893054i \(-0.351442\pi\)
0.449950 + 0.893054i \(0.351442\pi\)
\(212\) 10.7321 0.737080
\(213\) 0 0
\(214\) −19.8564 −1.35736
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −1.66025 −0.112447
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −18.9282 −1.27325
\(222\) 0 0
\(223\) −18.5359 −1.24126 −0.620628 0.784105i \(-0.713122\pi\)
−0.620628 + 0.784105i \(0.713122\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −19.8564 −1.32083
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) 0 0
\(229\) 3.60770 0.238403 0.119202 0.992870i \(-0.461967\pi\)
0.119202 + 0.992870i \(0.461967\pi\)
\(230\) 2.19615 0.144810
\(231\) 0 0
\(232\) −1.26795 −0.0832449
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) 6.92820 0.450988
\(237\) 0 0
\(238\) 3.46410 0.224544
\(239\) −4.73205 −0.306091 −0.153045 0.988219i \(-0.548908\pi\)
−0.153045 + 0.988219i \(0.548908\pi\)
\(240\) 0 0
\(241\) 6.73205 0.433650 0.216825 0.976211i \(-0.430430\pi\)
0.216825 + 0.976211i \(0.430430\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 8.92820 0.571570
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 17.8564 1.13618
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.19615 0.138071
\(254\) −14.9282 −0.936679
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.33975 0.395462 0.197731 0.980256i \(-0.436643\pi\)
0.197731 + 0.980256i \(0.436643\pi\)
\(258\) 0 0
\(259\) −2.73205 −0.169761
\(260\) 5.46410 0.338869
\(261\) 0 0
\(262\) −11.6603 −0.720373
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 10.7321 0.659265
\(266\) −3.26795 −0.200371
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −7.60770 −0.463849 −0.231925 0.972734i \(-0.574502\pi\)
−0.231925 + 0.972734i \(0.574502\pi\)
\(270\) 0 0
\(271\) −20.3923 −1.23874 −0.619372 0.785098i \(-0.712613\pi\)
−0.619372 + 0.785098i \(0.712613\pi\)
\(272\) −3.46410 −0.210042
\(273\) 0 0
\(274\) 12.9282 0.781021
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 20.9282 1.25745 0.628727 0.777626i \(-0.283576\pi\)
0.628727 + 0.777626i \(0.283576\pi\)
\(278\) 3.66025 0.219527
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) −1.60770 −0.0959071 −0.0479535 0.998850i \(-0.515270\pi\)
−0.0479535 + 0.998850i \(0.515270\pi\)
\(282\) 0 0
\(283\) 23.7128 1.40958 0.704790 0.709416i \(-0.251041\pi\)
0.704790 + 0.709416i \(0.251041\pi\)
\(284\) −2.53590 −0.150478
\(285\) 0 0
\(286\) 5.46410 0.323099
\(287\) −8.19615 −0.483804
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) −1.26795 −0.0744565
\(291\) 0 0
\(292\) 6.39230 0.374081
\(293\) −14.5359 −0.849196 −0.424598 0.905382i \(-0.639585\pi\)
−0.424598 + 0.905382i \(0.639585\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) 2.73205 0.158797
\(297\) 0 0
\(298\) −10.7321 −0.621691
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 13.1244 0.755222
\(303\) 0 0
\(304\) 3.26795 0.187430
\(305\) 8.92820 0.511227
\(306\) 0 0
\(307\) 3.60770 0.205902 0.102951 0.994686i \(-0.467172\pi\)
0.102951 + 0.994686i \(0.467172\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) −11.0718 −0.627824 −0.313912 0.949452i \(-0.601640\pi\)
−0.313912 + 0.949452i \(0.601640\pi\)
\(312\) 0 0
\(313\) 11.8038 0.667193 0.333596 0.942716i \(-0.391738\pi\)
0.333596 + 0.942716i \(0.391738\pi\)
\(314\) −11.4641 −0.646957
\(315\) 0 0
\(316\) −1.80385 −0.101474
\(317\) 0.588457 0.0330511 0.0165255 0.999863i \(-0.494740\pi\)
0.0165255 + 0.999863i \(0.494740\pi\)
\(318\) 0 0
\(319\) −1.26795 −0.0709915
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 2.19615 0.122387
\(323\) −11.3205 −0.629890
\(324\) 0 0
\(325\) 5.46410 0.303094
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 8.19615 0.452557
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) 22.7846 1.25236 0.626178 0.779680i \(-0.284618\pi\)
0.626178 + 0.779680i \(0.284618\pi\)
\(332\) −4.39230 −0.241059
\(333\) 0 0
\(334\) −13.8564 −0.758189
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −18.7846 −1.02326 −0.511631 0.859205i \(-0.670959\pi\)
−0.511631 + 0.859205i \(0.670959\pi\)
\(338\) −16.8564 −0.916868
\(339\) 0 0
\(340\) −3.46410 −0.187867
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −12.9282 −0.695025
\(347\) 36.9282 1.98241 0.991205 0.132336i \(-0.0422478\pi\)
0.991205 + 0.132336i \(0.0422478\pi\)
\(348\) 0 0
\(349\) −26.3923 −1.41275 −0.706374 0.707839i \(-0.749670\pi\)
−0.706374 + 0.707839i \(0.749670\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −3.80385 −0.202458 −0.101229 0.994863i \(-0.532278\pi\)
−0.101229 + 0.994863i \(0.532278\pi\)
\(354\) 0 0
\(355\) −2.53590 −0.134592
\(356\) 3.46410 0.183597
\(357\) 0 0
\(358\) −6.00000 −0.317110
\(359\) −4.05256 −0.213886 −0.106943 0.994265i \(-0.534106\pi\)
−0.106943 + 0.994265i \(0.534106\pi\)
\(360\) 0 0
\(361\) −8.32051 −0.437921
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) 5.46410 0.286397
\(365\) 6.39230 0.334589
\(366\) 0 0
\(367\) −8.39230 −0.438075 −0.219037 0.975716i \(-0.570292\pi\)
−0.219037 + 0.975716i \(0.570292\pi\)
\(368\) −2.19615 −0.114482
\(369\) 0 0
\(370\) 2.73205 0.142033
\(371\) 10.7321 0.557180
\(372\) 0 0
\(373\) −2.39230 −0.123869 −0.0619344 0.998080i \(-0.519727\pi\)
−0.0619344 + 0.998080i \(0.519727\pi\)
\(374\) −3.46410 −0.179124
\(375\) 0 0
\(376\) −6.92820 −0.357295
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) 33.8564 1.73909 0.869543 0.493857i \(-0.164413\pi\)
0.869543 + 0.493857i \(0.164413\pi\)
\(380\) 3.26795 0.167642
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 26.5359 1.35592 0.677961 0.735098i \(-0.262864\pi\)
0.677961 + 0.735098i \(0.262864\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 8.39230 0.427157
\(387\) 0 0
\(388\) −16.5885 −0.842151
\(389\) 22.3923 1.13533 0.567667 0.823258i \(-0.307846\pi\)
0.567667 + 0.823258i \(0.307846\pi\)
\(390\) 0 0
\(391\) 7.60770 0.384738
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 24.2487 1.22163
\(395\) −1.80385 −0.0907614
\(396\) 0 0
\(397\) 9.60770 0.482196 0.241098 0.970501i \(-0.422492\pi\)
0.241098 + 0.970501i \(0.422492\pi\)
\(398\) 10.9282 0.547781
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −9.46410 −0.472615 −0.236307 0.971678i \(-0.575937\pi\)
−0.236307 + 0.971678i \(0.575937\pi\)
\(402\) 0 0
\(403\) 10.9282 0.544373
\(404\) −19.8564 −0.987893
\(405\) 0 0
\(406\) −1.26795 −0.0629273
\(407\) 2.73205 0.135423
\(408\) 0 0
\(409\) 35.1244 1.73679 0.868394 0.495875i \(-0.165153\pi\)
0.868394 + 0.495875i \(0.165153\pi\)
\(410\) 8.19615 0.404779
\(411\) 0 0
\(412\) −8.39230 −0.413459
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) −4.39230 −0.215610
\(416\) −5.46410 −0.267900
\(417\) 0 0
\(418\) 3.26795 0.159841
\(419\) −17.0718 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(420\) 0 0
\(421\) −8.14359 −0.396894 −0.198447 0.980112i \(-0.563590\pi\)
−0.198447 + 0.980112i \(0.563590\pi\)
\(422\) −13.0718 −0.636325
\(423\) 0 0
\(424\) −10.7321 −0.521194
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 8.92820 0.432066
\(428\) 19.8564 0.959796
\(429\) 0 0
\(430\) −2.00000 −0.0964486
\(431\) 21.1244 1.01752 0.508762 0.860907i \(-0.330103\pi\)
0.508762 + 0.860907i \(0.330103\pi\)
\(432\) 0 0
\(433\) −7.80385 −0.375029 −0.187514 0.982262i \(-0.560043\pi\)
−0.187514 + 0.982262i \(0.560043\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 1.66025 0.0795117
\(437\) −7.17691 −0.343318
\(438\) 0 0
\(439\) −22.2487 −1.06187 −0.530937 0.847412i \(-0.678160\pi\)
−0.530937 + 0.847412i \(0.678160\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 18.9282 0.900323
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 3.46410 0.164214
\(446\) 18.5359 0.877700
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −19.6077 −0.925344 −0.462672 0.886529i \(-0.653109\pi\)
−0.462672 + 0.886529i \(0.653109\pi\)
\(450\) 0 0
\(451\) 8.19615 0.385942
\(452\) 19.8564 0.933967
\(453\) 0 0
\(454\) −6.92820 −0.325157
\(455\) 5.46410 0.256161
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −3.60770 −0.168577
\(459\) 0 0
\(460\) −2.19615 −0.102396
\(461\) −1.60770 −0.0748778 −0.0374389 0.999299i \(-0.511920\pi\)
−0.0374389 + 0.999299i \(0.511920\pi\)
\(462\) 0 0
\(463\) −17.5167 −0.814068 −0.407034 0.913413i \(-0.633437\pi\)
−0.407034 + 0.913413i \(0.633437\pi\)
\(464\) 1.26795 0.0588631
\(465\) 0 0
\(466\) 19.8564 0.919830
\(467\) −4.05256 −0.187530 −0.0937650 0.995594i \(-0.529890\pi\)
−0.0937650 + 0.995594i \(0.529890\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −6.92820 −0.319574
\(471\) 0 0
\(472\) −6.92820 −0.318896
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) 3.26795 0.149944
\(476\) −3.46410 −0.158777
\(477\) 0 0
\(478\) 4.73205 0.216439
\(479\) −8.78461 −0.401379 −0.200690 0.979655i \(-0.564318\pi\)
−0.200690 + 0.979655i \(0.564318\pi\)
\(480\) 0 0
\(481\) −14.9282 −0.680667
\(482\) −6.73205 −0.306637
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −16.5885 −0.753243
\(486\) 0 0
\(487\) −6.19615 −0.280774 −0.140387 0.990097i \(-0.544835\pi\)
−0.140387 + 0.990097i \(0.544835\pi\)
\(488\) −8.92820 −0.404161
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −27.7128 −1.25066 −0.625331 0.780360i \(-0.715036\pi\)
−0.625331 + 0.780360i \(0.715036\pi\)
\(492\) 0 0
\(493\) −4.39230 −0.197819
\(494\) −17.8564 −0.803398
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −2.53590 −0.113751
\(498\) 0 0
\(499\) 39.8564 1.78422 0.892109 0.451820i \(-0.149225\pi\)
0.892109 + 0.451820i \(0.149225\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 32.7846 1.46179 0.730897 0.682488i \(-0.239102\pi\)
0.730897 + 0.682488i \(0.239102\pi\)
\(504\) 0 0
\(505\) −19.8564 −0.883598
\(506\) −2.19615 −0.0976309
\(507\) 0 0
\(508\) 14.9282 0.662332
\(509\) −24.9282 −1.10492 −0.552462 0.833538i \(-0.686311\pi\)
−0.552462 + 0.833538i \(0.686311\pi\)
\(510\) 0 0
\(511\) 6.39230 0.282779
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.33975 −0.279634
\(515\) −8.39230 −0.369809
\(516\) 0 0
\(517\) −6.92820 −0.304702
\(518\) 2.73205 0.120039
\(519\) 0 0
\(520\) −5.46410 −0.239617
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) 0 0
\(523\) 33.1769 1.45073 0.725363 0.688367i \(-0.241672\pi\)
0.725363 + 0.688367i \(0.241672\pi\)
\(524\) 11.6603 0.509381
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −6.92820 −0.301797
\(528\) 0 0
\(529\) −18.1769 −0.790301
\(530\) −10.7321 −0.466170
\(531\) 0 0
\(532\) 3.26795 0.141684
\(533\) −44.7846 −1.93984
\(534\) 0 0
\(535\) 19.8564 0.858467
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 7.60770 0.327991
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −17.2679 −0.742407 −0.371204 0.928552i \(-0.621055\pi\)
−0.371204 + 0.928552i \(0.621055\pi\)
\(542\) 20.3923 0.875924
\(543\) 0 0
\(544\) 3.46410 0.148522
\(545\) 1.66025 0.0711175
\(546\) 0 0
\(547\) −12.7846 −0.546630 −0.273315 0.961925i \(-0.588120\pi\)
−0.273315 + 0.961925i \(0.588120\pi\)
\(548\) −12.9282 −0.552265
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 4.14359 0.176523
\(552\) 0 0
\(553\) −1.80385 −0.0767074
\(554\) −20.9282 −0.889154
\(555\) 0 0
\(556\) −3.66025 −0.155229
\(557\) 46.3923 1.96571 0.982853 0.184393i \(-0.0590320\pi\)
0.982853 + 0.184393i \(0.0590320\pi\)
\(558\) 0 0
\(559\) 10.9282 0.462214
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 1.60770 0.0678165
\(563\) −18.9282 −0.797729 −0.398864 0.917010i \(-0.630596\pi\)
−0.398864 + 0.917010i \(0.630596\pi\)
\(564\) 0 0
\(565\) 19.8564 0.835365
\(566\) −23.7128 −0.996724
\(567\) 0 0
\(568\) 2.53590 0.106404
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) −5.46410 −0.228466
\(573\) 0 0
\(574\) 8.19615 0.342101
\(575\) −2.19615 −0.0915859
\(576\) 0 0
\(577\) −3.41154 −0.142024 −0.0710122 0.997475i \(-0.522623\pi\)
−0.0710122 + 0.997475i \(0.522623\pi\)
\(578\) 5.00000 0.207973
\(579\) 0 0
\(580\) 1.26795 0.0526487
\(581\) −4.39230 −0.182224
\(582\) 0 0
\(583\) −10.7321 −0.444476
\(584\) −6.39230 −0.264515
\(585\) 0 0
\(586\) 14.5359 0.600472
\(587\) 42.5885 1.75781 0.878907 0.476993i \(-0.158273\pi\)
0.878907 + 0.476993i \(0.158273\pi\)
\(588\) 0 0
\(589\) 6.53590 0.269307
\(590\) −6.92820 −0.285230
\(591\) 0 0
\(592\) −2.73205 −0.112287
\(593\) 48.2487 1.98134 0.990669 0.136293i \(-0.0435189\pi\)
0.990669 + 0.136293i \(0.0435189\pi\)
\(594\) 0 0
\(595\) −3.46410 −0.142014
\(596\) 10.7321 0.439602
\(597\) 0 0
\(598\) 12.0000 0.490716
\(599\) 25.1769 1.02870 0.514350 0.857580i \(-0.328033\pi\)
0.514350 + 0.857580i \(0.328033\pi\)
\(600\) 0 0
\(601\) 39.5167 1.61192 0.805959 0.591971i \(-0.201650\pi\)
0.805959 + 0.591971i \(0.201650\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 0 0
\(604\) −13.1244 −0.534022
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 7.07180 0.287035 0.143518 0.989648i \(-0.454159\pi\)
0.143518 + 0.989648i \(0.454159\pi\)
\(608\) −3.26795 −0.132533
\(609\) 0 0
\(610\) −8.92820 −0.361492
\(611\) 37.8564 1.53151
\(612\) 0 0
\(613\) 27.1769 1.09767 0.548833 0.835932i \(-0.315072\pi\)
0.548833 + 0.835932i \(0.315072\pi\)
\(614\) −3.60770 −0.145595
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 17.3205 0.697297 0.348649 0.937253i \(-0.386641\pi\)
0.348649 + 0.937253i \(0.386641\pi\)
\(618\) 0 0
\(619\) −12.7846 −0.513857 −0.256928 0.966430i \(-0.582710\pi\)
−0.256928 + 0.966430i \(0.582710\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 11.0718 0.443939
\(623\) 3.46410 0.138786
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −11.8038 −0.471777
\(627\) 0 0
\(628\) 11.4641 0.457467
\(629\) 9.46410 0.377358
\(630\) 0 0
\(631\) −33.0718 −1.31657 −0.658284 0.752770i \(-0.728717\pi\)
−0.658284 + 0.752770i \(0.728717\pi\)
\(632\) 1.80385 0.0717532
\(633\) 0 0
\(634\) −0.588457 −0.0233706
\(635\) 14.9282 0.592408
\(636\) 0 0
\(637\) 5.46410 0.216496
\(638\) 1.26795 0.0501986
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −47.5692 −1.87887 −0.939436 0.342725i \(-0.888650\pi\)
−0.939436 + 0.342725i \(0.888650\pi\)
\(642\) 0 0
\(643\) 36.7321 1.44857 0.724285 0.689500i \(-0.242170\pi\)
0.724285 + 0.689500i \(0.242170\pi\)
\(644\) −2.19615 −0.0865405
\(645\) 0 0
\(646\) 11.3205 0.445399
\(647\) 33.4641 1.31561 0.657805 0.753188i \(-0.271485\pi\)
0.657805 + 0.753188i \(0.271485\pi\)
\(648\) 0 0
\(649\) −6.92820 −0.271956
\(650\) −5.46410 −0.214320
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 5.66025 0.221503 0.110751 0.993848i \(-0.464674\pi\)
0.110751 + 0.993848i \(0.464674\pi\)
\(654\) 0 0
\(655\) 11.6603 0.455604
\(656\) −8.19615 −0.320006
\(657\) 0 0
\(658\) −6.92820 −0.270089
\(659\) 18.2487 0.710869 0.355434 0.934701i \(-0.384333\pi\)
0.355434 + 0.934701i \(0.384333\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −22.7846 −0.885549
\(663\) 0 0
\(664\) 4.39230 0.170454
\(665\) 3.26795 0.126726
\(666\) 0 0
\(667\) −2.78461 −0.107821
\(668\) 13.8564 0.536120
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −8.92820 −0.344669
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 18.7846 0.723556
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 31.8564 1.22434 0.612171 0.790726i \(-0.290297\pi\)
0.612171 + 0.790726i \(0.290297\pi\)
\(678\) 0 0
\(679\) −16.5885 −0.636607
\(680\) 3.46410 0.132842
\(681\) 0 0
\(682\) 2.00000 0.0765840
\(683\) 42.2487 1.61660 0.808301 0.588769i \(-0.200387\pi\)
0.808301 + 0.588769i \(0.200387\pi\)
\(684\) 0 0
\(685\) −12.9282 −0.493961
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) 58.6410 2.23404
\(690\) 0 0
\(691\) −6.53590 −0.248637 −0.124319 0.992242i \(-0.539675\pi\)
−0.124319 + 0.992242i \(0.539675\pi\)
\(692\) 12.9282 0.491457
\(693\) 0 0
\(694\) −36.9282 −1.40178
\(695\) −3.66025 −0.138841
\(696\) 0 0
\(697\) 28.3923 1.07544
\(698\) 26.3923 0.998963
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 23.9090 0.903029 0.451515 0.892264i \(-0.350884\pi\)
0.451515 + 0.892264i \(0.350884\pi\)
\(702\) 0 0
\(703\) −8.92820 −0.336734
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 3.80385 0.143160
\(707\) −19.8564 −0.746777
\(708\) 0 0
\(709\) −9.32051 −0.350039 −0.175020 0.984565i \(-0.555999\pi\)
−0.175020 + 0.984565i \(0.555999\pi\)
\(710\) 2.53590 0.0951706
\(711\) 0 0
\(712\) −3.46410 −0.129823
\(713\) −4.39230 −0.164493
\(714\) 0 0
\(715\) −5.46410 −0.204346
\(716\) 6.00000 0.224231
\(717\) 0 0
\(718\) 4.05256 0.151240
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) −8.39230 −0.312546
\(722\) 8.32051 0.309657
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 1.26795 0.0470905
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) −5.46410 −0.202513
\(729\) 0 0
\(730\) −6.39230 −0.236590
\(731\) −6.92820 −0.256249
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 8.39230 0.309766
\(735\) 0 0
\(736\) 2.19615 0.0809513
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −43.7128 −1.60800 −0.804001 0.594628i \(-0.797299\pi\)
−0.804001 + 0.594628i \(0.797299\pi\)
\(740\) −2.73205 −0.100432
\(741\) 0 0
\(742\) −10.7321 −0.393986
\(743\) −8.78461 −0.322276 −0.161138 0.986932i \(-0.551516\pi\)
−0.161138 + 0.986932i \(0.551516\pi\)
\(744\) 0 0
\(745\) 10.7321 0.393192
\(746\) 2.39230 0.0875885
\(747\) 0 0
\(748\) 3.46410 0.126660
\(749\) 19.8564 0.725537
\(750\) 0 0
\(751\) −32.3923 −1.18201 −0.591006 0.806667i \(-0.701269\pi\)
−0.591006 + 0.806667i \(0.701269\pi\)
\(752\) 6.92820 0.252646
\(753\) 0 0
\(754\) −6.92820 −0.252310
\(755\) −13.1244 −0.477644
\(756\) 0 0
\(757\) 29.3731 1.06758 0.533791 0.845616i \(-0.320767\pi\)
0.533791 + 0.845616i \(0.320767\pi\)
\(758\) −33.8564 −1.22972
\(759\) 0 0
\(760\) −3.26795 −0.118541
\(761\) 29.6603 1.07518 0.537592 0.843205i \(-0.319334\pi\)
0.537592 + 0.843205i \(0.319334\pi\)
\(762\) 0 0
\(763\) 1.66025 0.0601052
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −26.5359 −0.958781
\(767\) 37.8564 1.36692
\(768\) 0 0
\(769\) −7.80385 −0.281414 −0.140707 0.990051i \(-0.544938\pi\)
−0.140707 + 0.990051i \(0.544938\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −8.39230 −0.302046
\(773\) −12.9282 −0.464995 −0.232498 0.972597i \(-0.574690\pi\)
−0.232498 + 0.972597i \(0.574690\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 16.5885 0.595491
\(777\) 0 0
\(778\) −22.3923 −0.802803
\(779\) −26.7846 −0.959658
\(780\) 0 0
\(781\) 2.53590 0.0907416
\(782\) −7.60770 −0.272051
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 11.4641 0.409171
\(786\) 0 0
\(787\) 45.8564 1.63460 0.817302 0.576209i \(-0.195469\pi\)
0.817302 + 0.576209i \(0.195469\pi\)
\(788\) −24.2487 −0.863825
\(789\) 0 0
\(790\) 1.80385 0.0641780
\(791\) 19.8564 0.706013
\(792\) 0 0
\(793\) 48.7846 1.73239
\(794\) −9.60770 −0.340964
\(795\) 0 0
\(796\) −10.9282 −0.387340
\(797\) 46.3923 1.64330 0.821650 0.569993i \(-0.193054\pi\)
0.821650 + 0.569993i \(0.193054\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 9.46410 0.334189
\(803\) −6.39230 −0.225580
\(804\) 0 0
\(805\) −2.19615 −0.0774042
\(806\) −10.9282 −0.384930
\(807\) 0 0
\(808\) 19.8564 0.698546
\(809\) −31.1769 −1.09612 −0.548061 0.836438i \(-0.684634\pi\)
−0.548061 + 0.836438i \(0.684634\pi\)
\(810\) 0 0
\(811\) −18.8756 −0.662814 −0.331407 0.943488i \(-0.607523\pi\)
−0.331407 + 0.943488i \(0.607523\pi\)
\(812\) 1.26795 0.0444963
\(813\) 0 0
\(814\) −2.73205 −0.0957583
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 6.53590 0.228662
\(818\) −35.1244 −1.22809
\(819\) 0 0
\(820\) −8.19615 −0.286222
\(821\) 47.9090 1.67203 0.836017 0.548703i \(-0.184878\pi\)
0.836017 + 0.548703i \(0.184878\pi\)
\(822\) 0 0
\(823\) −25.1244 −0.875780 −0.437890 0.899029i \(-0.644274\pi\)
−0.437890 + 0.899029i \(0.644274\pi\)
\(824\) 8.39230 0.292360
\(825\) 0 0
\(826\) −6.92820 −0.241063
\(827\) −1.85641 −0.0645536 −0.0322768 0.999479i \(-0.510276\pi\)
−0.0322768 + 0.999479i \(0.510276\pi\)
\(828\) 0 0
\(829\) −46.2487 −1.60628 −0.803142 0.595788i \(-0.796840\pi\)
−0.803142 + 0.595788i \(0.796840\pi\)
\(830\) 4.39230 0.152459
\(831\) 0 0
\(832\) 5.46410 0.189434
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) 13.8564 0.479521
\(836\) −3.26795 −0.113024
\(837\) 0 0
\(838\) 17.0718 0.589735
\(839\) 4.14359 0.143053 0.0715264 0.997439i \(-0.477213\pi\)
0.0715264 + 0.997439i \(0.477213\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) 8.14359 0.280647
\(843\) 0 0
\(844\) 13.0718 0.449950
\(845\) 16.8564 0.579878
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 10.7321 0.368540
\(849\) 0 0
\(850\) 3.46410 0.118818
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −13.2154 −0.452486 −0.226243 0.974071i \(-0.572644\pi\)
−0.226243 + 0.974071i \(0.572644\pi\)
\(854\) −8.92820 −0.305517
\(855\) 0 0
\(856\) −19.8564 −0.678678
\(857\) 20.5359 0.701493 0.350746 0.936470i \(-0.385928\pi\)
0.350746 + 0.936470i \(0.385928\pi\)
\(858\) 0 0
\(859\) 28.7846 0.982118 0.491059 0.871126i \(-0.336610\pi\)
0.491059 + 0.871126i \(0.336610\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) −21.1244 −0.719498
\(863\) −37.5167 −1.27708 −0.638541 0.769588i \(-0.720462\pi\)
−0.638541 + 0.769588i \(0.720462\pi\)
\(864\) 0 0
\(865\) 12.9282 0.439572
\(866\) 7.80385 0.265186
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 1.80385 0.0611913
\(870\) 0 0
\(871\) −21.8564 −0.740576
\(872\) −1.66025 −0.0562233
\(873\) 0 0
\(874\) 7.17691 0.242763
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 13.3205 0.449802 0.224901 0.974382i \(-0.427794\pi\)
0.224901 + 0.974382i \(0.427794\pi\)
\(878\) 22.2487 0.750858
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −24.9282 −0.839853 −0.419926 0.907558i \(-0.637944\pi\)
−0.419926 + 0.907558i \(0.637944\pi\)
\(882\) 0 0
\(883\) −56.3923 −1.89775 −0.948876 0.315649i \(-0.897778\pi\)
−0.948876 + 0.315649i \(0.897778\pi\)
\(884\) −18.9282 −0.636624
\(885\) 0 0
\(886\) 0 0
\(887\) 13.8564 0.465253 0.232626 0.972566i \(-0.425268\pi\)
0.232626 + 0.972566i \(0.425268\pi\)
\(888\) 0 0
\(889\) 14.9282 0.500676
\(890\) −3.46410 −0.116117
\(891\) 0 0
\(892\) −18.5359 −0.620628
\(893\) 22.6410 0.757653
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 19.6077 0.654317
\(899\) 2.53590 0.0845769
\(900\) 0 0
\(901\) −37.1769 −1.23854
\(902\) −8.19615 −0.272902
\(903\) 0 0
\(904\) −19.8564 −0.660414
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 59.0333 1.96017 0.980085 0.198580i \(-0.0636331\pi\)
0.980085 + 0.198580i \(0.0636331\pi\)
\(908\) 6.92820 0.229920
\(909\) 0 0
\(910\) −5.46410 −0.181133
\(911\) −2.53590 −0.0840181 −0.0420090 0.999117i \(-0.513376\pi\)
−0.0420090 + 0.999117i \(0.513376\pi\)
\(912\) 0 0
\(913\) 4.39230 0.145364
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 3.60770 0.119202
\(917\) 11.6603 0.385056
\(918\) 0 0
\(919\) 47.3731 1.56269 0.781347 0.624097i \(-0.214533\pi\)
0.781347 + 0.624097i \(0.214533\pi\)
\(920\) 2.19615 0.0724050
\(921\) 0 0
\(922\) 1.60770 0.0529466
\(923\) −13.8564 −0.456089
\(924\) 0 0
\(925\) −2.73205 −0.0898293
\(926\) 17.5167 0.575633
\(927\) 0 0
\(928\) −1.26795 −0.0416225
\(929\) 46.3923 1.52208 0.761041 0.648704i \(-0.224689\pi\)
0.761041 + 0.648704i \(0.224689\pi\)
\(930\) 0 0
\(931\) 3.26795 0.107103
\(932\) −19.8564 −0.650418
\(933\) 0 0
\(934\) 4.05256 0.132604
\(935\) 3.46410 0.113288
\(936\) 0 0
\(937\) −42.7846 −1.39771 −0.698856 0.715262i \(-0.746307\pi\)
−0.698856 + 0.715262i \(0.746307\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 6.92820 0.225973
\(941\) −26.7846 −0.873153 −0.436577 0.899667i \(-0.643809\pi\)
−0.436577 + 0.899667i \(0.643809\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 6.92820 0.225494
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) −12.6795 −0.412028 −0.206014 0.978549i \(-0.566049\pi\)
−0.206014 + 0.978549i \(0.566049\pi\)
\(948\) 0 0
\(949\) 34.9282 1.13382
\(950\) −3.26795 −0.106026
\(951\) 0 0
\(952\) 3.46410 0.112272
\(953\) −33.4641 −1.08401 −0.542004 0.840376i \(-0.682334\pi\)
−0.542004 + 0.840376i \(0.682334\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) −4.73205 −0.153045
\(957\) 0 0
\(958\) 8.78461 0.283818
\(959\) −12.9282 −0.417473
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 14.9282 0.481305
\(963\) 0 0
\(964\) 6.73205 0.216825
\(965\) −8.39230 −0.270158
\(966\) 0 0
\(967\) 13.0718 0.420361 0.210180 0.977663i \(-0.432595\pi\)
0.210180 + 0.977663i \(0.432595\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 16.5885 0.532623
\(971\) −29.0718 −0.932958 −0.466479 0.884532i \(-0.654478\pi\)
−0.466479 + 0.884532i \(0.654478\pi\)
\(972\) 0 0
\(973\) −3.66025 −0.117342
\(974\) 6.19615 0.198538
\(975\) 0 0
\(976\) 8.92820 0.285785
\(977\) −21.7128 −0.694654 −0.347327 0.937744i \(-0.612911\pi\)
−0.347327 + 0.937744i \(0.612911\pi\)
\(978\) 0 0
\(979\) −3.46410 −0.110713
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 27.7128 0.884351
\(983\) −25.1769 −0.803019 −0.401509 0.915855i \(-0.631514\pi\)
−0.401509 + 0.915855i \(0.631514\pi\)
\(984\) 0 0
\(985\) −24.2487 −0.772628
\(986\) 4.39230 0.139879
\(987\) 0 0
\(988\) 17.8564 0.568088
\(989\) −4.39230 −0.139667
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 2.53590 0.0804338
\(995\) −10.9282 −0.346447
\(996\) 0 0
\(997\) −37.7128 −1.19438 −0.597188 0.802101i \(-0.703716\pi\)
−0.597188 + 0.802101i \(0.703716\pi\)
\(998\) −39.8564 −1.26163
\(999\) 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 6930.2.a.bv.1.2 2
3.2 odd 2 770.2.a.j.1.1 2
12.11 even 2 6160.2.a.t.1.2 2
15.2 even 4 3850.2.c.x.1849.4 4
15.8 even 4 3850.2.c.x.1849.1 4
15.14 odd 2 3850.2.a.bd.1.2 2
21.20 even 2 5390.2.a.bs.1.2 2
33.32 even 2 8470.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.1 2 3.2 odd 2
3850.2.a.bd.1.2 2 15.14 odd 2
3850.2.c.x.1849.1 4 15.8 even 4
3850.2.c.x.1849.4 4 15.2 even 4
5390.2.a.bs.1.2 2 21.20 even 2
6160.2.a.t.1.2 2 12.11 even 2
6930.2.a.bv.1.2 2 1.1 even 1 trivial
8470.2.a.br.1.1 2 33.32 even 2