Properties

Label 5390.2.a.bs.1.2
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.0393666895\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 5390.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{10} +1.00000 q^{11} +0.732051 q^{12} -5.46410 q^{13} +0.732051 q^{15} +1.00000 q^{16} -3.46410 q^{17} -2.46410 q^{18} -3.26795 q^{19} +1.00000 q^{20} +1.00000 q^{22} +2.19615 q^{23} +0.732051 q^{24} +1.00000 q^{25} -5.46410 q^{26} -4.00000 q^{27} -1.26795 q^{29} +0.732051 q^{30} -2.00000 q^{31} +1.00000 q^{32} +0.732051 q^{33} -3.46410 q^{34} -2.46410 q^{36} -2.73205 q^{37} -3.26795 q^{38} -4.00000 q^{39} +1.00000 q^{40} -8.19615 q^{41} +2.00000 q^{43} +1.00000 q^{44} -2.46410 q^{45} +2.19615 q^{46} +6.92820 q^{47} +0.732051 q^{48} +1.00000 q^{50} -2.53590 q^{51} -5.46410 q^{52} -10.7321 q^{53} -4.00000 q^{54} +1.00000 q^{55} -2.39230 q^{57} -1.26795 q^{58} +6.92820 q^{59} +0.732051 q^{60} -8.92820 q^{61} -2.00000 q^{62} +1.00000 q^{64} -5.46410 q^{65} +0.732051 q^{66} -4.00000 q^{67} -3.46410 q^{68} +1.60770 q^{69} +2.53590 q^{71} -2.46410 q^{72} -6.39230 q^{73} -2.73205 q^{74} +0.732051 q^{75} -3.26795 q^{76} -4.00000 q^{78} -1.80385 q^{79} +1.00000 q^{80} +4.46410 q^{81} -8.19615 q^{82} -4.39230 q^{83} -3.46410 q^{85} +2.00000 q^{86} -0.928203 q^{87} +1.00000 q^{88} +3.46410 q^{89} -2.46410 q^{90} +2.19615 q^{92} -1.46410 q^{93} +6.92820 q^{94} -3.26795 q^{95} +0.732051 q^{96} +16.5885 q^{97} -2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{15} + 2 q^{16} + 2 q^{18} - 10 q^{19} + 2 q^{20} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 2 q^{25} - 4 q^{26} - 8 q^{27} - 6 q^{29} - 2 q^{30} - 4 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{36} - 2 q^{37} - 10 q^{38} - 8 q^{39} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 2 q^{44} + 2 q^{45} - 6 q^{46} - 2 q^{48} + 2 q^{50} - 12 q^{51} - 4 q^{52} - 18 q^{53} - 8 q^{54} + 2 q^{55} + 16 q^{57} - 6 q^{58} - 2 q^{60} - 4 q^{61} - 4 q^{62} + 2 q^{64} - 4 q^{65} - 2 q^{66} - 8 q^{67} + 24 q^{69} + 12 q^{71} + 2 q^{72} + 8 q^{73} - 2 q^{74} - 2 q^{75} - 10 q^{76} - 8 q^{78} - 14 q^{79} + 2 q^{80} + 2 q^{81} - 6 q^{82} + 12 q^{83} + 4 q^{86} + 12 q^{87} + 2 q^{88} + 2 q^{90} - 6 q^{92} + 4 q^{93} - 10 q^{95} - 2 q^{96} + 2 q^{97} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.732051 0.298858
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.46410 −0.821367
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 0.732051 0.211325
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 0 0
\(15\) 0.732051 0.189015
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) −2.46410 −0.580794
\(19\) −3.26795 −0.749719 −0.374859 0.927082i \(-0.622309\pi\)
−0.374859 + 0.927082i \(0.622309\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 2.19615 0.457929 0.228965 0.973435i \(-0.426466\pi\)
0.228965 + 0.973435i \(0.426466\pi\)
\(24\) 0.732051 0.149429
\(25\) 1.00000 0.200000
\(26\) −5.46410 −1.07160
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −1.26795 −0.235452 −0.117726 0.993046i \(-0.537560\pi\)
−0.117726 + 0.993046i \(0.537560\pi\)
\(30\) 0.732051 0.133654
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.732051 0.127434
\(34\) −3.46410 −0.594089
\(35\) 0 0
\(36\) −2.46410 −0.410684
\(37\) −2.73205 −0.449146 −0.224573 0.974457i \(-0.572099\pi\)
−0.224573 + 0.974457i \(0.572099\pi\)
\(38\) −3.26795 −0.530131
\(39\) −4.00000 −0.640513
\(40\) 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
\(45\) −2.46410 −0.367327
\(46\) 2.19615 0.323805
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0.732051 0.105662
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −2.53590 −0.355097
\(52\) −5.46410 −0.757735
\(53\) −10.7321 −1.47416 −0.737080 0.675805i \(-0.763796\pi\)
−0.737080 + 0.675805i \(0.763796\pi\)
\(54\) −4.00000 −0.544331
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −2.39230 −0.316869
\(58\) −1.26795 −0.166490
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0.732051 0.0945074
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.46410 −0.677738
\(66\) 0.732051 0.0901092
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.46410 −0.420084
\(69\) 1.60770 0.193544
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) −2.46410 −0.290397
\(73\) −6.39230 −0.748163 −0.374081 0.927396i \(-0.622042\pi\)
−0.374081 + 0.927396i \(0.622042\pi\)
\(74\) −2.73205 −0.317594
\(75\) 0.732051 0.0845299
\(76\) −3.26795 −0.374859
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −1.80385 −0.202949 −0.101474 0.994838i \(-0.532356\pi\)
−0.101474 + 0.994838i \(0.532356\pi\)
\(80\) 1.00000 0.111803
\(81\) 4.46410 0.496011
\(82\) −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
\(87\) −0.928203 −0.0995138
\(88\) 1.00000 0.106600
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) −2.46410 −0.259739
\(91\) 0 0
\(92\) 2.19615 0.228965
\(93\) −1.46410 −0.151820
\(94\) 6.92820 0.714590
\(95\) −3.26795 −0.335285
\(96\) 0.732051 0.0747146
\(97\) 16.5885 1.68430 0.842151 0.539241i \(-0.181289\pi\)
0.842151 + 0.539241i \(0.181289\pi\)
\(98\) 0 0
\(99\) −2.46410 −0.247652
\(100\) 1.00000 0.100000
\(101\) −19.8564 −1.97579 −0.987893 0.155136i \(-0.950418\pi\)
−0.987893 + 0.155136i \(0.950418\pi\)
\(102\) −2.53590 −0.251091
\(103\) 8.39230 0.826918 0.413459 0.910523i \(-0.364320\pi\)
0.413459 + 0.910523i \(0.364320\pi\)
\(104\) −5.46410 −0.535799
\(105\) 0 0
\(106\) −10.7321 −1.04239
\(107\) −19.8564 −1.91959 −0.959796 0.280700i \(-0.909433\pi\)
−0.959796 + 0.280700i \(0.909433\pi\)
\(108\) −4.00000 −0.384900
\(109\) 1.66025 0.159023 0.0795117 0.996834i \(-0.474664\pi\)
0.0795117 + 0.996834i \(0.474664\pi\)
\(110\) 1.00000 0.0953463
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) −2.39230 −0.224060
\(115\) 2.19615 0.204792
\(116\) −1.26795 −0.117726
\(117\) 13.4641 1.24476
\(118\) 6.92820 0.637793
\(119\) 0 0
\(120\) 0.732051 0.0668268
\(121\) 1.00000 0.0909091
\(122\) −8.92820 −0.808322
\(123\) −6.00000 −0.541002
\(124\) −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
\(129\) 1.46410 0.128907
\(130\) −5.46410 −0.479233
\(131\) 11.6603 1.01876 0.509381 0.860541i \(-0.329875\pi\)
0.509381 + 0.860541i \(0.329875\pi\)
\(132\) 0.732051 0.0637168
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −4.00000 −0.344265
\(136\) −3.46410 −0.297044
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 1.60770 0.136856
\(139\) 3.66025 0.310459 0.155229 0.987878i \(-0.450388\pi\)
0.155229 + 0.987878i \(0.450388\pi\)
\(140\) 0 0
\(141\) 5.07180 0.427122
\(142\) 2.53590 0.212808
\(143\) −5.46410 −0.456931
\(144\) −2.46410 −0.205342
\(145\) −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 \(-0.644880\pi\)
−0.439602 + 0.898193i \(0.644880\pi\)
\(150\) 0.732051 0.0597717
\(151\) −13.1244 −1.06804 −0.534022 0.845470i \(-0.679320\pi\)
−0.534022 + 0.845470i \(0.679320\pi\)
\(152\) −3.26795 −0.265066
\(153\) 8.53590 0.690086
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) −4.00000 −0.320256
\(157\) −11.4641 −0.914935 −0.457467 0.889226i \(-0.651243\pi\)
−0.457467 + 0.889226i \(0.651243\pi\)
\(158\) −1.80385 −0.143506
\(159\) −7.85641 −0.623054
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 4.46410 0.350733
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −8.19615 −0.640012
\(165\) 0.732051 0.0569901
\(166\) −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
\(171\) 8.05256 0.615795
\(172\) 2.00000 0.152499
\(173\) 12.9282 0.982913 0.491457 0.870902i \(-0.336465\pi\)
0.491457 + 0.870902i \(0.336465\pi\)
\(174\) −0.928203 −0.0703669
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 5.07180 0.381220
\(178\) 3.46410 0.259645
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) −2.46410 −0.183663
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −6.53590 −0.483148
\(184\) 2.19615 0.161903
\(185\) −2.73205 −0.200864
\(186\) −1.46410 −0.107353
\(187\) −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 \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0.732051 0.0528312
\(193\) −8.39230 −0.604091 −0.302046 0.953293i \(-0.597670\pi\)
−0.302046 + 0.953293i \(0.597670\pi\)
\(194\) 16.5885 1.19098
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 24.2487 1.72765 0.863825 0.503793i \(-0.168062\pi\)
0.863825 + 0.503793i \(0.168062\pi\)
\(198\) −2.46410 −0.175116
\(199\) 10.9282 0.774680 0.387340 0.921937i \(-0.373394\pi\)
0.387340 + 0.921937i \(0.373394\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.92820 −0.206540
\(202\) −19.8564 −1.39709
\(203\) 0 0
\(204\) −2.53590 −0.177548
\(205\) −8.19615 −0.572444
\(206\) 8.39230 0.584720
\(207\) −5.41154 −0.376128
\(208\) −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
\(213\) 1.85641 0.127199
\(214\) −19.8564 −1.35736
\(215\) 2.00000 0.136399
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) 1.66025 0.112447
\(219\) −4.67949 −0.316211
\(220\) 1.00000 0.0674200
\(221\) 18.9282 1.27325
\(222\) −2.00000 −0.134231
\(223\) 18.5359 1.24126 0.620628 0.784105i \(-0.286878\pi\)
0.620628 + 0.784105i \(0.286878\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) −19.8564 −1.32083
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) −2.39230 −0.158434
\(229\) −3.60770 −0.238403 −0.119202 0.992870i \(-0.538033\pi\)
−0.119202 + 0.992870i \(0.538033\pi\)
\(230\) 2.19615 0.144810
\(231\) 0 0
\(232\) −1.26795 −0.0832449
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 13.4641 0.880176
\(235\) 6.92820 0.451946
\(236\) 6.92820 0.450988
\(237\) −1.32051 −0.0857762
\(238\) 0 0
\(239\) 4.73205 0.306091 0.153045 0.988219i \(-0.451092\pi\)
0.153045 + 0.988219i \(0.451092\pi\)
\(240\) 0.732051 0.0472537
\(241\) −6.73205 −0.433650 −0.216825 0.976211i \(-0.569570\pi\)
−0.216825 + 0.976211i \(0.569570\pi\)
\(242\) 1.00000 0.0642824
\(243\) 15.2679 0.979439
\(244\) −8.92820 −0.571570
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 17.8564 1.13618
\(248\) −2.00000 −0.127000
\(249\) −3.21539 −0.203767
\(250\) 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
\(255\) −2.53590 −0.158804
\(256\) 1.00000 0.0625000
\(257\) 6.33975 0.395462 0.197731 0.980256i \(-0.436643\pi\)
0.197731 + 0.980256i \(0.436643\pi\)
\(258\) 1.46410 0.0911510
\(259\) 0 0
\(260\) −5.46410 −0.338869
\(261\) 3.12436 0.193393
\(262\) 11.6603 0.720373
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0.732051 0.0450546
\(265\) −10.7321 −0.659265
\(266\) 0 0
\(267\) 2.53590 0.155194
\(268\) −4.00000 −0.244339
\(269\) −7.60770 −0.463849 −0.231925 0.972734i \(-0.574502\pi\)
−0.231925 + 0.972734i \(0.574502\pi\)
\(270\) −4.00000 −0.243432
\(271\) 20.3923 1.23874 0.619372 0.785098i \(-0.287387\pi\)
0.619372 + 0.785098i \(0.287387\pi\)
\(272\) −3.46410 −0.210042
\(273\) 0 0
\(274\) 12.9282 0.781021
\(275\) 1.00000 0.0603023
\(276\) 1.60770 0.0967719
\(277\) 20.9282 1.25745 0.628727 0.777626i \(-0.283576\pi\)
0.628727 + 0.777626i \(0.283576\pi\)
\(278\) 3.66025 0.219527
\(279\) 4.92820 0.295044
\(280\) 0 0
\(281\) 1.60770 0.0959071 0.0479535 0.998850i \(-0.484730\pi\)
0.0479535 + 0.998850i \(0.484730\pi\)
\(282\) 5.07180 0.302021
\(283\) −23.7128 −1.40958 −0.704790 0.709416i \(-0.748959\pi\)
−0.704790 + 0.709416i \(0.748959\pi\)
\(284\) 2.53590 0.150478
\(285\) −2.39230 −0.141708
\(286\) −5.46410 −0.323099
\(287\) 0 0
\(288\) −2.46410 −0.145199
\(289\) −5.00000 −0.294118
\(290\) −1.26795 −0.0744565
\(291\) 12.1436 0.711870
\(292\) −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
\(297\) −4.00000 −0.232104
\(298\) −10.7321 −0.621691
\(299\) −12.0000 −0.693978
\(300\) 0.732051 0.0422650
\(301\) 0 0
\(302\) −13.1244 −0.755222
\(303\) −14.5359 −0.835066
\(304\) −3.26795 −0.187430
\(305\) −8.92820 −0.511227
\(306\) 8.53590 0.487965
\(307\) −3.60770 −0.205902 −0.102951 0.994686i \(-0.532828\pi\)
−0.102951 + 0.994686i \(0.532828\pi\)
\(308\) 0 0
\(309\) 6.14359 0.349497
\(310\) −2.00000 −0.113592
\(311\) −11.0718 −0.627824 −0.313912 0.949452i \(-0.601640\pi\)
−0.313912 + 0.949452i \(0.601640\pi\)
\(312\) −4.00000 −0.226455
\(313\) −11.8038 −0.667193 −0.333596 0.942716i \(-0.608262\pi\)
−0.333596 + 0.942716i \(0.608262\pi\)
\(314\) −11.4641 −0.646957
\(315\) 0 0
\(316\) −1.80385 −0.101474
\(317\) −0.588457 −0.0330511 −0.0165255 0.999863i \(-0.505260\pi\)
−0.0165255 + 0.999863i \(0.505260\pi\)
\(318\) −7.85641 −0.440565
\(319\) −1.26795 −0.0709915
\(320\) 1.00000 0.0559017
\(321\) −14.5359 −0.811315
\(322\) 0 0
\(323\) 11.3205 0.629890
\(324\) 4.46410 0.248006
\(325\) −5.46410 −0.303094
\(326\) −4.00000 −0.221540
\(327\) 1.21539 0.0672112
\(328\) −8.19615 −0.452557
\(329\) 0 0
\(330\) 0.732051 0.0402981
\(331\) 22.7846 1.25236 0.626178 0.779680i \(-0.284618\pi\)
0.626178 + 0.779680i \(0.284618\pi\)
\(332\) −4.39230 −0.241059
\(333\) 6.73205 0.368914
\(334\) 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
\(339\) −14.5359 −0.789482
\(340\) −3.46410 −0.187867
\(341\) −2.00000 −0.108306
\(342\) 8.05256 0.435433
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 1.60770 0.0865554
\(346\) 12.9282 0.695025
\(347\) −36.9282 −1.98241 −0.991205 0.132336i \(-0.957752\pi\)
−0.991205 + 0.132336i \(0.957752\pi\)
\(348\) −0.928203 −0.0497569
\(349\) 26.3923 1.41275 0.706374 0.707839i \(-0.250330\pi\)
0.706374 + 0.707839i \(0.250330\pi\)
\(350\) 0 0
\(351\) 21.8564 1.16661
\(352\) 1.00000 0.0533002
\(353\) −3.80385 −0.202458 −0.101229 0.994863i \(-0.532278\pi\)
−0.101229 + 0.994863i \(0.532278\pi\)
\(354\) 5.07180 0.269563
\(355\) 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 \(-0.465894\pi\)
0.106943 + 0.994265i \(0.465894\pi\)
\(360\) −2.46410 −0.129870
\(361\) −8.32051 −0.437921
\(362\) −14.0000 −0.735824
\(363\) 0.732051 0.0384227
\(364\) 0 0
\(365\) −6.39230 −0.334589
\(366\) −6.53590 −0.341637
\(367\) 8.39230 0.438075 0.219037 0.975716i \(-0.429708\pi\)
0.219037 + 0.975716i \(0.429708\pi\)
\(368\) 2.19615 0.114482
\(369\) 20.1962 1.05137
\(370\) −2.73205 −0.142033
\(371\) 0 0
\(372\) −1.46410 −0.0759101
\(373\) −2.39230 −0.123869 −0.0619344 0.998080i \(-0.519727\pi\)
−0.0619344 + 0.998080i \(0.519727\pi\)
\(374\) −3.46410 −0.179124
\(375\) 0.732051 0.0378029
\(376\) 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
\(381\) 10.9282 0.559869
\(382\) −12.0000 −0.613973
\(383\) 26.5359 1.35592 0.677961 0.735098i \(-0.262864\pi\)
0.677961 + 0.735098i \(0.262864\pi\)
\(384\) 0.732051 0.0373573
\(385\) 0 0
\(386\) −8.39230 −0.427157
\(387\) −4.92820 −0.250515
\(388\) 16.5885 0.842151
\(389\) −22.3923 −1.13533 −0.567667 0.823258i \(-0.692154\pi\)
−0.567667 + 0.823258i \(0.692154\pi\)
\(390\) −4.00000 −0.202548
\(391\) −7.60770 −0.384738
\(392\) 0 0
\(393\) 8.53590 0.430579
\(394\) 24.2487 1.22163
\(395\) −1.80385 −0.0907614
\(396\) −2.46410 −0.123826
\(397\) −9.60770 −0.482196 −0.241098 0.970501i \(-0.577508\pi\)
−0.241098 + 0.970501i \(0.577508\pi\)
\(398\) 10.9282 0.547781
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 9.46410 0.472615 0.236307 0.971678i \(-0.424063\pi\)
0.236307 + 0.971678i \(0.424063\pi\)
\(402\) −2.92820 −0.146046
\(403\) 10.9282 0.544373
\(404\) −19.8564 −0.987893
\(405\) 4.46410 0.221823
\(406\) 0 0
\(407\) −2.73205 −0.135423
\(408\) −2.53590 −0.125546
\(409\) −35.1244 −1.73679 −0.868394 0.495875i \(-0.834847\pi\)
−0.868394 + 0.495875i \(0.834847\pi\)
\(410\) −8.19615 −0.404779
\(411\) 9.46410 0.466830
\(412\) 8.39230 0.413459
\(413\) 0 0
\(414\) −5.41154 −0.265963
\(415\) −4.39230 −0.215610
\(416\) −5.46410 −0.267900
\(417\) 2.67949 0.131215
\(418\) −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
\(423\) −17.0718 −0.830059
\(424\) −10.7321 −0.521194
\(425\) −3.46410 −0.168034
\(426\) 1.85641 0.0899432
\(427\) 0 0
\(428\) −19.8564 −0.959796
\(429\) −4.00000 −0.193122
\(430\) 2.00000 0.0964486
\(431\) −21.1244 −1.01752 −0.508762 0.860907i \(-0.669897\pi\)
−0.508762 + 0.860907i \(0.669897\pi\)
\(432\) −4.00000 −0.192450
\(433\) 7.80385 0.375029 0.187514 0.982262i \(-0.439957\pi\)
0.187514 + 0.982262i \(0.439957\pi\)
\(434\) 0 0
\(435\) −0.928203 −0.0445039
\(436\) 1.66025 0.0795117
\(437\) −7.17691 −0.343318
\(438\) −4.67949 −0.223595
\(439\) 22.2487 1.06187 0.530937 0.847412i \(-0.321840\pi\)
0.530937 + 0.847412i \(0.321840\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) 18.9282 0.900323
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 3.46410 0.164214
\(446\) 18.5359 0.877700
\(447\) −7.85641 −0.371595
\(448\) 0 0
\(449\) 19.6077 0.925344 0.462672 0.886529i \(-0.346891\pi\)
0.462672 + 0.886529i \(0.346891\pi\)
\(450\) −2.46410 −0.116159
\(451\) −8.19615 −0.385942
\(452\) −19.8564 −0.933967
\(453\) −9.60770 −0.451409
\(454\) 6.92820 0.325157
\(455\) 0 0
\(456\) −2.39230 −0.112030
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −3.60770 −0.168577
\(459\) 13.8564 0.646762
\(460\) 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
\(465\) −1.46410 −0.0678961
\(466\) 19.8564 0.919830
\(467\) −4.05256 −0.187530 −0.0937650 0.995594i \(-0.529890\pi\)
−0.0937650 + 0.995594i \(0.529890\pi\)
\(468\) 13.4641 0.622378
\(469\) 0 0
\(470\) 6.92820 0.319574
\(471\) −8.39230 −0.386697
\(472\) 6.92820 0.318896
\(473\) 2.00000 0.0919601
\(474\) −1.32051 −0.0606529
\(475\) −3.26795 −0.149944
\(476\) 0 0
\(477\) 26.4449 1.21083
\(478\) 4.73205 0.216439
\(479\) −8.78461 −0.401379 −0.200690 0.979655i \(-0.564318\pi\)
−0.200690 + 0.979655i \(0.564318\pi\)
\(480\) 0.732051 0.0334134
\(481\) 14.9282 0.680667
\(482\) −6.73205 −0.306637
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 16.5885 0.753243
\(486\) 15.2679 0.692568
\(487\) −6.19615 −0.280774 −0.140387 0.990097i \(-0.544835\pi\)
−0.140387 + 0.990097i \(0.544835\pi\)
\(488\) −8.92820 −0.404161
\(489\) −2.92820 −0.132418
\(490\) 0 0
\(491\) 27.7128 1.25066 0.625331 0.780360i \(-0.284964\pi\)
0.625331 + 0.780360i \(0.284964\pi\)
\(492\) −6.00000 −0.270501
\(493\) 4.39230 0.197819
\(494\) 17.8564 0.803398
\(495\) −2.46410 −0.110753
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) −3.21539 −0.144085
\(499\) 39.8564 1.78422 0.892109 0.451820i \(-0.149225\pi\)
0.892109 + 0.451820i \(0.149225\pi\)
\(500\) 1.00000 0.0447214
\(501\) 10.1436 0.453182
\(502\) −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
\(507\) 12.3397 0.548027
\(508\) 14.9282 0.662332
\(509\) −24.9282 −1.10492 −0.552462 0.833538i \(-0.686311\pi\)
−0.552462 + 0.833538i \(0.686311\pi\)
\(510\) −2.53590 −0.112291
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 13.0718 0.577134
\(514\) 6.33975 0.279634
\(515\) 8.39230 0.369809
\(516\) 1.46410 0.0644535
\(517\) 6.92820 0.304702
\(518\) 0 0
\(519\) 9.46410 0.415428
\(520\) −5.46410 −0.239617
\(521\) 10.3923 0.455295 0.227648 0.973744i \(-0.426897\pi\)
0.227648 + 0.973744i \(0.426897\pi\)
\(522\) 3.12436 0.136749
\(523\) −33.1769 −1.45073 −0.725363 0.688367i \(-0.758328\pi\)
−0.725363 + 0.688367i \(0.758328\pi\)
\(524\) 11.6603 0.509381
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 6.92820 0.301797
\(528\) 0.732051 0.0318584
\(529\) −18.1769 −0.790301
\(530\) −10.7321 −0.466170
\(531\) −17.0718 −0.740853
\(532\) 0 0
\(533\) 44.7846 1.93984
\(534\) 2.53590 0.109739
\(535\) −19.8564 −0.858467
\(536\) −4.00000 −0.172774
\(537\) −4.39230 −0.189542
\(538\) −7.60770 −0.327991
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) −17.2679 −0.742407 −0.371204 0.928552i \(-0.621055\pi\)
−0.371204 + 0.928552i \(0.621055\pi\)
\(542\) 20.3923 0.875924
\(543\) −10.2487 −0.439814
\(544\) −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
\(549\) 22.0000 0.938937
\(550\) 1.00000 0.0426401
\(551\) 4.14359 0.176523
\(552\) 1.60770 0.0684280
\(553\) 0 0
\(554\) 20.9282 0.889154
\(555\) −2.00000 −0.0848953
\(556\) 3.66025 0.155229
\(557\) −46.3923 −1.96571 −0.982853 0.184393i \(-0.940968\pi\)
−0.982853 + 0.184393i \(0.940968\pi\)
\(558\) 4.92820 0.208627
\(559\) −10.9282 −0.462214
\(560\) 0 0
\(561\) −2.53590 −0.107066
\(562\) 1.60770 0.0678165
\(563\) −18.9282 −0.797729 −0.398864 0.917010i \(-0.630596\pi\)
−0.398864 + 0.917010i \(0.630596\pi\)
\(564\) 5.07180 0.213561
\(565\) −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 \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −2.39230 −0.100203
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) −5.46410 −0.228466
\(573\) −8.78461 −0.366982
\(574\) 0 0
\(575\) 2.19615 0.0915859
\(576\) −2.46410 −0.102671
\(577\) 3.41154 0.142024 0.0710122 0.997475i \(-0.477377\pi\)
0.0710122 + 0.997475i \(0.477377\pi\)
\(578\) −5.00000 −0.207973
\(579\) −6.14359 −0.255319
\(580\) −1.26795 −0.0526487
\(581\) 0 0
\(582\) 12.1436 0.503368
\(583\) −10.7321 −0.444476
\(584\) −6.39230 −0.264515
\(585\) 13.4641 0.556672
\(586\) −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
\(591\) 17.7513 0.730190
\(592\) −2.73205 −0.112287
\(593\) 48.2487 1.98134 0.990669 0.136293i \(-0.0435189\pi\)
0.990669 + 0.136293i \(0.0435189\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.7321 −0.439602
\(597\) 8.00000 0.327418
\(598\) −12.0000 −0.490716
\(599\) −25.1769 −1.02870 −0.514350 0.857580i \(-0.671967\pi\)
−0.514350 + 0.857580i \(0.671967\pi\)
\(600\) 0.732051 0.0298858
\(601\) −39.5167 −1.61192 −0.805959 0.591971i \(-0.798350\pi\)
−0.805959 + 0.591971i \(0.798350\pi\)
\(602\) 0 0
\(603\) 9.85641 0.401384
\(604\) −13.1244 −0.534022
\(605\) 1.00000 0.0406558
\(606\) −14.5359 −0.590481
\(607\) −7.07180 −0.287035 −0.143518 0.989648i \(-0.545841\pi\)
−0.143518 + 0.989648i \(0.545841\pi\)
\(608\) −3.26795 −0.132533
\(609\) 0 0
\(610\) −8.92820 −0.361492
\(611\) −37.8564 −1.53151
\(612\) 8.53590 0.345043
\(613\) 27.1769 1.09767 0.548833 0.835932i \(-0.315072\pi\)
0.548833 + 0.835932i \(0.315072\pi\)
\(614\) −3.60770 −0.145595
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −17.3205 −0.697297 −0.348649 0.937253i \(-0.613359\pi\)
−0.348649 + 0.937253i \(0.613359\pi\)
\(618\) 6.14359 0.247132
\(619\) 12.7846 0.513857 0.256928 0.966430i \(-0.417290\pi\)
0.256928 + 0.966430i \(0.417290\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −8.78461 −0.352514
\(622\) −11.0718 −0.443939
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −11.8038 −0.471777
\(627\) −2.39230 −0.0955395
\(628\) −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
\(633\) 9.56922 0.380342
\(634\) −0.588457 −0.0233706
\(635\) 14.9282 0.592408
\(636\) −7.85641 −0.311527
\(637\) 0 0
\(638\) −1.26795 −0.0501986
\(639\) −6.24871 −0.247195
\(640\) 1.00000 0.0395285
\(641\) 47.5692 1.87887 0.939436 0.342725i \(-0.111350\pi\)
0.939436 + 0.342725i \(0.111350\pi\)
\(642\) −14.5359 −0.573686
\(643\) −36.7321 −1.44857 −0.724285 0.689500i \(-0.757830\pi\)
−0.724285 + 0.689500i \(0.757830\pi\)
\(644\) 0 0
\(645\) 1.46410 0.0576489
\(646\) 11.3205 0.445399
\(647\) 33.4641 1.31561 0.657805 0.753188i \(-0.271485\pi\)
0.657805 + 0.753188i \(0.271485\pi\)
\(648\) 4.46410 0.175366
\(649\) 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 \(-0.535326\pi\)
−0.110751 + 0.993848i \(0.535326\pi\)
\(654\) 1.21539 0.0475255
\(655\) 11.6603 0.455604
\(656\) −8.19615 −0.320006
\(657\) 15.7513 0.614516
\(658\) 0 0
\(659\) −18.2487 −0.710869 −0.355434 0.934701i \(-0.615667\pi\)
−0.355434 + 0.934701i \(0.615667\pi\)
\(660\) 0.732051 0.0284950
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 22.7846 0.885549
\(663\) 13.8564 0.538138
\(664\) −4.39230 −0.170454
\(665\) 0 0
\(666\) 6.73205 0.260862
\(667\) −2.78461 −0.107821
\(668\) 13.8564 0.536120
\(669\) 13.5692 0.524616
\(670\) −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
\(675\) −4.00000 −0.153960
\(676\) 16.8564 0.648323
\(677\) 31.8564 1.22434 0.612171 0.790726i \(-0.290297\pi\)
0.612171 + 0.790726i \(0.290297\pi\)
\(678\) −14.5359 −0.558248
\(679\) 0 0
\(680\) −3.46410 −0.132842
\(681\) 5.07180 0.194352
\(682\) −2.00000 −0.0765840
\(683\) −42.2487 −1.61660 −0.808301 0.588769i \(-0.799613\pi\)
−0.808301 + 0.588769i \(0.799613\pi\)
\(684\) 8.05256 0.307897
\(685\) 12.9282 0.493961
\(686\) 0 0
\(687\) −2.64102 −0.100761
\(688\) 2.00000 0.0762493
\(689\) 58.6410 2.23404
\(690\) 1.60770 0.0612039
\(691\) 6.53590 0.248637 0.124319 0.992242i \(-0.460325\pi\)
0.124319 + 0.992242i \(0.460325\pi\)
\(692\) 12.9282 0.491457
\(693\) 0 0
\(694\) −36.9282 −1.40178
\(695\) 3.66025 0.138841
\(696\) −0.928203 −0.0351835
\(697\) 28.3923 1.07544
\(698\) 26.3923 0.998963
\(699\) 14.5359 0.549798
\(700\) 0 0
\(701\) −23.9090 −0.903029 −0.451515 0.892264i \(-0.649116\pi\)
−0.451515 + 0.892264i \(0.649116\pi\)
\(702\) 21.8564 0.824917
\(703\) 8.92820 0.336734
\(704\) 1.00000 0.0376889
\(705\) 5.07180 0.191015
\(706\) −3.80385 −0.143160
\(707\) 0 0
\(708\) 5.07180 0.190610
\(709\) −9.32051 −0.350039 −0.175020 0.984565i \(-0.555999\pi\)
−0.175020 + 0.984565i \(0.555999\pi\)
\(710\) 2.53590 0.0951706
\(711\) 4.44486 0.166695
\(712\) 3.46410 0.129823
\(713\) −4.39230 −0.164493
\(714\) 0 0
\(715\) −5.46410 −0.204346
\(716\) −6.00000 −0.224231
\(717\) 3.46410 0.129369
\(718\) 4.05256 0.151240
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) −2.46410 −0.0918316
\(721\) 0 0
\(722\) −8.32051 −0.309657
\(723\) −4.92820 −0.183282
\(724\) −14.0000 −0.520306
\(725\) −1.26795 −0.0470905
\(726\) 0.732051 0.0271690
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) −6.39230 −0.236590
\(731\) −6.92820 −0.256249
\(732\) −6.53590 −0.241574
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 8.39230 0.309766
\(735\) 0 0
\(736\) 2.19615 0.0809513
\(737\) −4.00000 −0.147342
\(738\) 20.1962 0.743431
\(739\) −43.7128 −1.60800 −0.804001 0.594628i \(-0.797299\pi\)
−0.804001 + 0.594628i \(0.797299\pi\)
\(740\) −2.73205 −0.100432
\(741\) 13.0718 0.480204
\(742\) 0 0
\(743\) 8.78461 0.322276 0.161138 0.986932i \(-0.448484\pi\)
0.161138 + 0.986932i \(0.448484\pi\)
\(744\) −1.46410 −0.0536766
\(745\) −10.7321 −0.393192
\(746\) −2.39230 −0.0875885
\(747\) 10.8231 0.395996
\(748\) −3.46410 −0.126660
\(749\) 0 0
\(750\) 0.732051 0.0267307
\(751\) −32.3923 −1.18201 −0.591006 0.806667i \(-0.701269\pi\)
−0.591006 + 0.806667i \(0.701269\pi\)
\(752\) 6.92820 0.252646
\(753\) −8.78461 −0.320129
\(754\) 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
\(759\) 1.60770 0.0583556
\(760\) −3.26795 −0.118541
\(761\) 29.6603 1.07518 0.537592 0.843205i \(-0.319334\pi\)
0.537592 + 0.843205i \(0.319334\pi\)
\(762\) 10.9282 0.395887
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 8.53590 0.308616
\(766\) 26.5359 0.958781
\(767\) −37.8564 −1.36692
\(768\) 0.732051 0.0264156
\(769\) 7.80385 0.281414 0.140707 0.990051i \(-0.455062\pi\)
0.140707 + 0.990051i \(0.455062\pi\)
\(770\) 0 0
\(771\) 4.64102 0.167142
\(772\) −8.39230 −0.302046
\(773\) −12.9282 −0.464995 −0.232498 0.972597i \(-0.574690\pi\)
−0.232498 + 0.972597i \(0.574690\pi\)
\(774\) −4.92820 −0.177141
\(775\) −2.00000 −0.0718421
\(776\) 16.5885 0.595491
\(777\) 0 0
\(778\) −22.3923 −0.802803
\(779\) 26.7846 0.959658
\(780\) −4.00000 −0.143223
\(781\) 2.53590 0.0907416
\(782\) −7.60770 −0.272051
\(783\) 5.07180 0.181251
\(784\) 0 0
\(785\) −11.4641 −0.409171
\(786\) 8.53590 0.304465
\(787\) −45.8564 −1.63460 −0.817302 0.576209i \(-0.804531\pi\)
−0.817302 + 0.576209i \(0.804531\pi\)
\(788\) 24.2487 0.863825
\(789\) −17.5692 −0.625481
\(790\) −1.80385 −0.0641780
\(791\) 0 0
\(792\) −2.46410 −0.0875580
\(793\) 48.7846 1.73239
\(794\) −9.60770 −0.340964
\(795\) −7.85641 −0.278638
\(796\) 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
\(801\) −8.53590 −0.301601
\(802\) 9.46410 0.334189
\(803\) −6.39230 −0.225580
\(804\) −2.92820 −0.103270
\(805\) 0 0
\(806\) 10.9282 0.384930
\(807\) −5.56922 −0.196046
\(808\) −19.8564 −0.698546
\(809\) 31.1769 1.09612 0.548061 0.836438i \(-0.315366\pi\)
0.548061 + 0.836438i \(0.315366\pi\)
\(810\) 4.46410 0.156853
\(811\) 18.8756 0.662814 0.331407 0.943488i \(-0.392477\pi\)
0.331407 + 0.943488i \(0.392477\pi\)
\(812\) 0 0
\(813\) 14.9282 0.523555
\(814\) −2.73205 −0.0957583
\(815\) −4.00000 −0.140114
\(816\) −2.53590 −0.0887742
\(817\) −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 \(-0.815122\pi\)
−0.836017 + 0.548703i \(0.815122\pi\)
\(822\) 9.46410 0.330098
\(823\) −25.1244 −0.875780 −0.437890 0.899029i \(-0.644274\pi\)
−0.437890 + 0.899029i \(0.644274\pi\)
\(824\) 8.39230 0.292360
\(825\) 0.732051 0.0254867
\(826\) 0 0
\(827\) 1.85641 0.0645536 0.0322768 0.999479i \(-0.489724\pi\)
0.0322768 + 0.999479i \(0.489724\pi\)
\(828\) −5.41154 −0.188064
\(829\) 46.2487 1.60628 0.803142 0.595788i \(-0.203160\pi\)
0.803142 + 0.595788i \(0.203160\pi\)
\(830\) −4.39230 −0.152459
\(831\) 15.3205 0.531463
\(832\) −5.46410 −0.189434
\(833\) 0 0
\(834\) 2.67949 0.0927832
\(835\) 13.8564 0.479521
\(836\) −3.26795 −0.113024
\(837\) 8.00000 0.276520
\(838\) −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
\(843\) 1.17691 0.0405351
\(844\) 13.0718 0.449950
\(845\) 16.8564 0.579878
\(846\) −17.0718 −0.586940
\(847\) 0 0
\(848\) −10.7321 −0.368540
\(849\) −17.3590 −0.595759
\(850\) −3.46410 −0.118818
\(851\) −6.00000 −0.205677
\(852\) 1.85641 0.0635994
\(853\) 13.2154 0.452486 0.226243 0.974071i \(-0.427356\pi\)
0.226243 + 0.974071i \(0.427356\pi\)
\(854\) 0 0
\(855\) 8.05256 0.275392
\(856\) −19.8564 −0.678678
\(857\) 20.5359 0.701493 0.350746 0.936470i \(-0.385928\pi\)
0.350746 + 0.936470i \(0.385928\pi\)
\(858\) −4.00000 −0.136558
\(859\) −28.7846 −0.982118 −0.491059 0.871126i \(-0.663390\pi\)
−0.491059 + 0.871126i \(0.663390\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) −21.1244 −0.719498
\(863\) 37.5167 1.27708 0.638541 0.769588i \(-0.279538\pi\)
0.638541 + 0.769588i \(0.279538\pi\)
\(864\) −4.00000 −0.136083
\(865\) 12.9282 0.439572
\(866\) 7.80385 0.265186
\(867\) −3.66025 −0.124309
\(868\) 0 0
\(869\) −1.80385 −0.0611913
\(870\) −0.928203 −0.0314690
\(871\) 21.8564 0.740576
\(872\) 1.66025 0.0562233
\(873\) −40.8756 −1.38343
\(874\) −7.17691 −0.242763
\(875\) 0 0
\(876\) −4.67949 −0.158105
\(877\) 13.3205 0.449802 0.224901 0.974382i \(-0.427794\pi\)
0.224901 + 0.974382i \(0.427794\pi\)
\(878\) 22.2487 0.750858
\(879\) −10.6410 −0.358913
\(880\) 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
\(885\) 5.07180 0.170487
\(886\) 0 0
\(887\) 13.8564 0.465253 0.232626 0.972566i \(-0.425268\pi\)
0.232626 + 0.972566i \(0.425268\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 3.46410 0.116117
\(891\) 4.46410 0.149553
\(892\) 18.5359 0.620628
\(893\) −22.6410 −0.757653
\(894\) −7.85641 −0.262758
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) −8.78461 −0.293310
\(898\) 19.6077 0.654317
\(899\) 2.53590 0.0845769
\(900\) −2.46410 −0.0821367
\(901\) 37.1769 1.23854
\(902\) −8.19615 −0.272902
\(903\) 0 0
\(904\) −19.8564 −0.660414
\(905\) −14.0000 −0.465376
\(906\) −9.60770 −0.319194
\(907\) 59.0333 1.96017 0.980085 0.198580i \(-0.0636331\pi\)
0.980085 + 0.198580i \(0.0636331\pi\)
\(908\) 6.92820 0.229920
\(909\) 48.9282 1.62285
\(910\) 0 0
\(911\) 2.53590 0.0840181 0.0420090 0.999117i \(-0.486624\pi\)
0.0420090 + 0.999117i \(0.486624\pi\)
\(912\) −2.39230 −0.0792171
\(913\) −4.39230 −0.145364
\(914\) 2.00000 0.0661541
\(915\) −6.53590 −0.216070
\(916\) −3.60770 −0.119202
\(917\) 0 0
\(918\) 13.8564 0.457330
\(919\) 47.3731 1.56269 0.781347 0.624097i \(-0.214533\pi\)
0.781347 + 0.624097i \(0.214533\pi\)
\(920\) 2.19615 0.0724050
\(921\) −2.64102 −0.0870244
\(922\) −1.60770 −0.0529466
\(923\) −13.8564 −0.456089
\(924\) 0 0
\(925\) −2.73205 −0.0898293
\(926\) −17.5167 −0.575633
\(927\) −20.6795 −0.679204
\(928\) −1.26795 −0.0416225
\(929\) 46.3923 1.52208 0.761041 0.648704i \(-0.224689\pi\)
0.761041 + 0.648704i \(0.224689\pi\)
\(930\) −1.46410 −0.0480098
\(931\) 0 0
\(932\) 19.8564 0.650418
\(933\) −8.10512 −0.265350
\(934\) −4.05256 −0.132604
\(935\) −3.46410 −0.113288
\(936\) 13.4641 0.440088
\(937\) 42.7846 1.39771 0.698856 0.715262i \(-0.253693\pi\)
0.698856 + 0.715262i \(0.253693\pi\)
\(938\) 0 0
\(939\) −8.64102 −0.281989
\(940\) 6.92820 0.225973
\(941\) −26.7846 −0.873153 −0.436577 0.899667i \(-0.643809\pi\)
−0.436577 + 0.899667i \(0.643809\pi\)
\(942\) −8.39230 −0.273436
\(943\) −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 \(-0.433951\pi\)
0.206014 + 0.978549i \(0.433951\pi\)
\(948\) −1.32051 −0.0428881
\(949\) 34.9282 1.13382
\(950\) −3.26795 −0.106026
\(951\) −0.430781 −0.0139690
\(952\) 0 0
\(953\) 33.4641 1.08401 0.542004 0.840376i \(-0.317666\pi\)
0.542004 + 0.840376i \(0.317666\pi\)
\(954\) 26.4449 0.856184
\(955\) −12.0000 −0.388311
\(956\) 4.73205 0.153045
\(957\) −0.928203 −0.0300045
\(958\) −8.78461 −0.283818
\(959\) 0 0
\(960\) 0.732051 0.0236268
\(961\) −27.0000 −0.870968
\(962\) 14.9282 0.481305
\(963\) 48.9282 1.57669
\(964\) −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
\(969\) 8.28719 0.266223
\(970\) 16.5885 0.532623
\(971\) −29.0718 −0.932958 −0.466479 0.884532i \(-0.654478\pi\)
−0.466479 + 0.884532i \(0.654478\pi\)
\(972\) 15.2679 0.489720
\(973\) 0 0
\(974\) −6.19615 −0.198538
\(975\) −4.00000 −0.128103
\(976\) −8.92820 −0.285785
\(977\) 21.7128 0.694654 0.347327 0.937744i \(-0.387089\pi\)
0.347327 + 0.937744i \(0.387089\pi\)
\(978\) −2.92820 −0.0936336
\(979\) 3.46410 0.110713
\(980\) 0 0
\(981\) −4.09103 −0.130617
\(982\) 27.7128 0.884351
\(983\) −25.1769 −0.803019 −0.401509 0.915855i \(-0.631514\pi\)
−0.401509 + 0.915855i \(0.631514\pi\)
\(984\) −6.00000 −0.191273
\(985\) 24.2487 0.772628
\(986\) 4.39230 0.139879
\(987\) 0 0
\(988\) 17.8564 0.568088
\(989\) 4.39230 0.139667
\(990\) −2.46410 −0.0783143
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 16.6795 0.529308
\(994\) 0 0
\(995\) 10.9282 0.346447
\(996\) −3.21539 −0.101884
\(997\) 37.7128 1.19438 0.597188 0.802101i \(-0.296284\pi\)
0.597188 + 0.802101i \(0.296284\pi\)
\(998\) 39.8564 1.26163
\(999\) 10.9282 0.345753
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 5390.2.a.bs.1.2 2
7.6 odd 2 770.2.a.j.1.1 2
21.20 even 2 6930.2.a.bv.1.2 2
28.27 even 2 6160.2.a.t.1.2 2
35.13 even 4 3850.2.c.x.1849.1 4
35.27 even 4 3850.2.c.x.1849.4 4
35.34 odd 2 3850.2.a.bd.1.2 2
77.76 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 7.6 odd 2
3850.2.a.bd.1.2 2 35.34 odd 2
3850.2.c.x.1849.1 4 35.13 even 4
3850.2.c.x.1849.4 4 35.27 even 4
5390.2.a.bs.1.2 2 1.1 even 1 trivial
6160.2.a.t.1.2 2 28.27 even 2
6930.2.a.bv.1.2 2 21.20 even 2
8470.2.a.br.1.1 2 77.76 even 2