Properties

Label 7623.2.a.cr.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.3829849.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.63651\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.297484 q^{2} -1.91150 q^{4} +3.06404 q^{5} +1.00000 q^{7} +1.16361 q^{8} +O(q^{10})\) \(q-0.297484 q^{2} -1.91150 q^{4} +3.06404 q^{5} +1.00000 q^{7} +1.16361 q^{8} -0.911503 q^{10} +5.38835 q^{13} -0.297484 q^{14} +3.47685 q^{16} -6.72305 q^{17} -0.434652 q^{19} -5.85693 q^{20} -2.92219 q^{23} +4.38835 q^{25} -1.60295 q^{26} -1.91150 q^{28} +6.86491 q^{29} +4.95370 q^{31} -3.36153 q^{32} +2.00000 q^{34} +3.06404 q^{35} -3.38835 q^{37} +0.129302 q^{38} +3.56535 q^{40} +9.92895 q^{41} -1.82301 q^{43} +0.869304 q^{46} -7.45988 q^{47} +1.00000 q^{49} -1.30547 q^{50} -10.2999 q^{52} +9.33398 q^{53} +1.16361 q^{56} -2.04220 q^{58} +7.17616 q^{59} -6.77671 q^{61} -1.47365 q^{62} -5.95370 q^{64} +16.5101 q^{65} +3.56535 q^{67} +12.8511 q^{68} -0.911503 q^{70} -8.50796 q^{71} +2.61165 q^{73} +1.00798 q^{74} +0.830838 q^{76} +11.6460 q^{79} +10.6532 q^{80} -2.95370 q^{82} -1.73225 q^{83} -20.5997 q^{85} +0.542315 q^{86} -7.31802 q^{89} +5.38835 q^{91} +5.58577 q^{92} +2.21919 q^{94} -1.33179 q^{95} -6.95370 q^{97} -0.297484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{4} + 6q^{7} + O(q^{10}) \) \( 6q + 4q^{4} + 6q^{7} + 10q^{10} + 4q^{13} + 8q^{16} - 2q^{25} + 4q^{28} + 4q^{31} + 12q^{34} + 8q^{37} + 24q^{40} + 20q^{43} + 6q^{49} - 18q^{52} - 2q^{58} + 16q^{61} - 10q^{64} + 24q^{67} + 10q^{70} + 44q^{73} + 54q^{76} + 8q^{79} + 8q^{82} - 36q^{85} + 4q^{91} + 34q^{94} - 16q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.297484 −0.210353 −0.105176 0.994454i \(-0.533541\pi\)
−0.105176 + 0.994454i \(0.533541\pi\)
\(3\) 0 0
\(4\) −1.91150 −0.955752
\(5\) 3.06404 1.37028 0.685141 0.728411i \(-0.259741\pi\)
0.685141 + 0.728411i \(0.259741\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.16361 0.411398
\(9\) 0 0
\(10\) −0.911503 −0.288243
\(11\) 0 0
\(12\) 0 0
\(13\) 5.38835 1.49446 0.747230 0.664565i \(-0.231383\pi\)
0.747230 + 0.664565i \(0.231383\pi\)
\(14\) −0.297484 −0.0795059
\(15\) 0 0
\(16\) 3.47685 0.869213
\(17\) −6.72305 −1.63058 −0.815290 0.579053i \(-0.803422\pi\)
−0.815290 + 0.579053i \(0.803422\pi\)
\(18\) 0 0
\(19\) −0.434652 −0.0997160 −0.0498580 0.998756i \(-0.515877\pi\)
−0.0498580 + 0.998756i \(0.515877\pi\)
\(20\) −5.85693 −1.30965
\(21\) 0 0
\(22\) 0 0
\(23\) −2.92219 −0.609318 −0.304659 0.952461i \(-0.598543\pi\)
−0.304659 + 0.952461i \(0.598543\pi\)
\(24\) 0 0
\(25\) 4.38835 0.877671
\(26\) −1.60295 −0.314364
\(27\) 0 0
\(28\) −1.91150 −0.361240
\(29\) 6.86491 1.27478 0.637391 0.770541i \(-0.280014\pi\)
0.637391 + 0.770541i \(0.280014\pi\)
\(30\) 0 0
\(31\) 4.95370 0.889711 0.444856 0.895602i \(-0.353255\pi\)
0.444856 + 0.895602i \(0.353255\pi\)
\(32\) −3.36153 −0.594239
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 3.06404 0.517918
\(36\) 0 0
\(37\) −3.38835 −0.557042 −0.278521 0.960430i \(-0.589844\pi\)
−0.278521 + 0.960430i \(0.589844\pi\)
\(38\) 0.129302 0.0209755
\(39\) 0 0
\(40\) 3.56535 0.563731
\(41\) 9.92895 1.55064 0.775321 0.631568i \(-0.217588\pi\)
0.775321 + 0.631568i \(0.217588\pi\)
\(42\) 0 0
\(43\) −1.82301 −0.278006 −0.139003 0.990292i \(-0.544390\pi\)
−0.139003 + 0.990292i \(0.544390\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.869304 0.128172
\(47\) −7.45988 −1.08813 −0.544067 0.839042i \(-0.683116\pi\)
−0.544067 + 0.839042i \(0.683116\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.30547 −0.184621
\(51\) 0 0
\(52\) −10.2999 −1.42833
\(53\) 9.33398 1.28212 0.641061 0.767490i \(-0.278495\pi\)
0.641061 + 0.767490i \(0.278495\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.16361 0.155494
\(57\) 0 0
\(58\) −2.04220 −0.268154
\(59\) 7.17616 0.934257 0.467129 0.884189i \(-0.345289\pi\)
0.467129 + 0.884189i \(0.345289\pi\)
\(60\) 0 0
\(61\) −6.77671 −0.867669 −0.433834 0.900993i \(-0.642840\pi\)
−0.433834 + 0.900993i \(0.642840\pi\)
\(62\) −1.47365 −0.187153
\(63\) 0 0
\(64\) −5.95370 −0.744213
\(65\) 16.5101 2.04783
\(66\) 0 0
\(67\) 3.56535 0.435577 0.217788 0.975996i \(-0.430116\pi\)
0.217788 + 0.975996i \(0.430116\pi\)
\(68\) 12.8511 1.55843
\(69\) 0 0
\(70\) −0.911503 −0.108945
\(71\) −8.50796 −1.00971 −0.504854 0.863205i \(-0.668454\pi\)
−0.504854 + 0.863205i \(0.668454\pi\)
\(72\) 0 0
\(73\) 2.61165 0.305670 0.152835 0.988252i \(-0.451160\pi\)
0.152835 + 0.988252i \(0.451160\pi\)
\(74\) 1.00798 0.117175
\(75\) 0 0
\(76\) 0.830838 0.0953037
\(77\) 0 0
\(78\) 0 0
\(79\) 11.6460 1.31028 0.655139 0.755508i \(-0.272610\pi\)
0.655139 + 0.755508i \(0.272610\pi\)
\(80\) 10.6532 1.19107
\(81\) 0 0
\(82\) −2.95370 −0.326182
\(83\) −1.73225 −0.190139 −0.0950696 0.995471i \(-0.530307\pi\)
−0.0950696 + 0.995471i \(0.530307\pi\)
\(84\) 0 0
\(85\) −20.5997 −2.23435
\(86\) 0.542315 0.0584793
\(87\) 0 0
\(88\) 0 0
\(89\) −7.31802 −0.775709 −0.387854 0.921721i \(-0.626784\pi\)
−0.387854 + 0.921721i \(0.626784\pi\)
\(90\) 0 0
\(91\) 5.38835 0.564853
\(92\) 5.58577 0.582357
\(93\) 0 0
\(94\) 2.21919 0.228892
\(95\) −1.33179 −0.136639
\(96\) 0 0
\(97\) −6.95370 −0.706042 −0.353021 0.935615i \(-0.614846\pi\)
−0.353021 + 0.935615i \(0.614846\pi\)
\(98\) −0.297484 −0.0300504
\(99\) 0 0
\(100\) −8.38835 −0.838835
\(101\) 7.91299 0.787372 0.393686 0.919245i \(-0.371200\pi\)
0.393686 + 0.919245i \(0.371200\pi\)
\(102\) 0 0
\(103\) 18.4227 1.81524 0.907622 0.419788i \(-0.137895\pi\)
0.907622 + 0.419788i \(0.137895\pi\)
\(104\) 6.26994 0.614818
\(105\) 0 0
\(106\) −2.77671 −0.269698
\(107\) 13.3042 1.28617 0.643085 0.765795i \(-0.277654\pi\)
0.643085 + 0.765795i \(0.277654\pi\)
\(108\) 0 0
\(109\) −0.953703 −0.0913482 −0.0456741 0.998956i \(-0.514544\pi\)
−0.0456741 + 0.998956i \(0.514544\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.47685 0.328532
\(113\) 14.9198 1.40353 0.701766 0.712407i \(-0.252395\pi\)
0.701766 + 0.712407i \(0.252395\pi\)
\(114\) 0 0
\(115\) −8.95370 −0.834937
\(116\) −13.1223 −1.21837
\(117\) 0 0
\(118\) −2.13479 −0.196524
\(119\) −6.72305 −0.616301
\(120\) 0 0
\(121\) 0 0
\(122\) 2.01596 0.182517
\(123\) 0 0
\(124\) −9.46902 −0.850343
\(125\) −1.87411 −0.167625
\(126\) 0 0
\(127\) 19.7304 1.75079 0.875396 0.483407i \(-0.160601\pi\)
0.875396 + 0.483407i \(0.160601\pi\)
\(128\) 8.49418 0.750787
\(129\) 0 0
\(130\) −4.91150 −0.430767
\(131\) −7.31802 −0.639378 −0.319689 0.947522i \(-0.603578\pi\)
−0.319689 + 0.947522i \(0.603578\pi\)
\(132\) 0 0
\(133\) −0.434652 −0.0376891
\(134\) −1.06063 −0.0916248
\(135\) 0 0
\(136\) −7.82301 −0.670817
\(137\) −17.5582 −1.50010 −0.750050 0.661381i \(-0.769971\pi\)
−0.750050 + 0.661381i \(0.769971\pi\)
\(138\) 0 0
\(139\) 19.6460 1.66635 0.833177 0.553007i \(-0.186520\pi\)
0.833177 + 0.553007i \(0.186520\pi\)
\(140\) −5.85693 −0.495001
\(141\) 0 0
\(142\) 2.53098 0.212395
\(143\) 0 0
\(144\) 0 0
\(145\) 21.0344 1.74681
\(146\) −0.776922 −0.0642986
\(147\) 0 0
\(148\) 6.47685 0.532394
\(149\) 11.5193 0.943702 0.471851 0.881678i \(-0.343586\pi\)
0.471851 + 0.881678i \(0.343586\pi\)
\(150\) 0 0
\(151\) −20.5997 −1.67638 −0.838191 0.545377i \(-0.816386\pi\)
−0.838191 + 0.545377i \(0.816386\pi\)
\(152\) −0.505765 −0.0410229
\(153\) 0 0
\(154\) 0 0
\(155\) 15.1784 1.21915
\(156\) 0 0
\(157\) −5.13070 −0.409474 −0.204737 0.978817i \(-0.565634\pi\)
−0.204737 + 0.978817i \(0.565634\pi\)
\(158\) −3.46450 −0.275621
\(159\) 0 0
\(160\) −10.2999 −0.814275
\(161\) −2.92219 −0.230301
\(162\) 0 0
\(163\) 3.56535 0.279260 0.139630 0.990204i \(-0.455409\pi\)
0.139630 + 0.990204i \(0.455409\pi\)
\(164\) −18.9792 −1.48203
\(165\) 0 0
\(166\) 0.515317 0.0399963
\(167\) −21.0227 −1.62679 −0.813394 0.581713i \(-0.802383\pi\)
−0.813394 + 0.581713i \(0.802383\pi\)
\(168\) 0 0
\(169\) 16.0344 1.23341
\(170\) 6.12808 0.470003
\(171\) 0 0
\(172\) 3.48468 0.265705
\(173\) −3.80087 −0.288974 −0.144487 0.989507i \(-0.546153\pi\)
−0.144487 + 0.989507i \(0.546153\pi\)
\(174\) 0 0
\(175\) 4.38835 0.331728
\(176\) 0 0
\(177\) 0 0
\(178\) 2.17699 0.163173
\(179\) −24.5123 −1.83214 −0.916069 0.401021i \(-0.868656\pi\)
−0.916069 + 0.401021i \(0.868656\pi\)
\(180\) 0 0
\(181\) −5.04630 −0.375088 −0.187544 0.982256i \(-0.560053\pi\)
−0.187544 + 0.982256i \(0.560053\pi\)
\(182\) −1.60295 −0.118818
\(183\) 0 0
\(184\) −3.40028 −0.250672
\(185\) −10.3821 −0.763304
\(186\) 0 0
\(187\) 0 0
\(188\) 14.2596 1.03999
\(189\) 0 0
\(190\) 0.396187 0.0287424
\(191\) −0.826027 −0.0597692 −0.0298846 0.999553i \(-0.509514\pi\)
−0.0298846 + 0.999553i \(0.509514\pi\)
\(192\) 0 0
\(193\) 7.04630 0.507204 0.253602 0.967309i \(-0.418385\pi\)
0.253602 + 0.967309i \(0.418385\pi\)
\(194\) 2.06861 0.148518
\(195\) 0 0
\(196\) −1.91150 −0.136536
\(197\) 14.5834 1.03902 0.519512 0.854463i \(-0.326114\pi\)
0.519512 + 0.854463i \(0.326114\pi\)
\(198\) 0 0
\(199\) −15.7304 −1.11510 −0.557550 0.830144i \(-0.688258\pi\)
−0.557550 + 0.830144i \(0.688258\pi\)
\(200\) 5.10633 0.361072
\(201\) 0 0
\(202\) −2.35399 −0.165626
\(203\) 6.86491 0.481822
\(204\) 0 0
\(205\) 30.4227 2.12482
\(206\) −5.48046 −0.381842
\(207\) 0 0
\(208\) 18.7345 1.29900
\(209\) 0 0
\(210\) 0 0
\(211\) −13.5534 −0.933056 −0.466528 0.884506i \(-0.654495\pi\)
−0.466528 + 0.884506i \(0.654495\pi\)
\(212\) −17.8419 −1.22539
\(213\) 0 0
\(214\) −3.95780 −0.270550
\(215\) −5.58577 −0.380946
\(216\) 0 0
\(217\) 4.95370 0.336279
\(218\) 0.283711 0.0192154
\(219\) 0 0
\(220\) 0 0
\(221\) −36.2262 −2.43684
\(222\) 0 0
\(223\) −15.6460 −1.04773 −0.523867 0.851800i \(-0.675511\pi\)
−0.523867 + 0.851800i \(0.675511\pi\)
\(224\) −3.36153 −0.224601
\(225\) 0 0
\(226\) −4.43839 −0.295237
\(227\) 0.906224 0.0601482 0.0300741 0.999548i \(-0.490426\pi\)
0.0300741 + 0.999548i \(0.490426\pi\)
\(228\) 0 0
\(229\) 15.4690 1.02222 0.511111 0.859515i \(-0.329234\pi\)
0.511111 + 0.859515i \(0.329234\pi\)
\(230\) 2.66358 0.175631
\(231\) 0 0
\(232\) 7.98807 0.524443
\(233\) −17.5307 −1.14847 −0.574237 0.818689i \(-0.694701\pi\)
−0.574237 + 0.818689i \(0.694701\pi\)
\(234\) 0 0
\(235\) −22.8574 −1.49105
\(236\) −13.7173 −0.892918
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) −2.23802 −0.144765 −0.0723826 0.997377i \(-0.523060\pi\)
−0.0723826 + 0.997377i \(0.523060\pi\)
\(240\) 0 0
\(241\) 17.0344 1.09728 0.548640 0.836059i \(-0.315146\pi\)
0.548640 + 0.836059i \(0.315146\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 12.9537 0.829276
\(245\) 3.06404 0.195754
\(246\) 0 0
\(247\) −2.34206 −0.149022
\(248\) 5.76418 0.366025
\(249\) 0 0
\(250\) 0.557517 0.0352604
\(251\) −19.4323 −1.22656 −0.613279 0.789866i \(-0.710150\pi\)
−0.613279 + 0.789866i \(0.710150\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −5.86948 −0.368284
\(255\) 0 0
\(256\) 9.38052 0.586283
\(257\) −0.400459 −0.0249800 −0.0124900 0.999922i \(-0.503976\pi\)
−0.0124900 + 0.999922i \(0.503976\pi\)
\(258\) 0 0
\(259\) −3.38835 −0.210542
\(260\) −31.5592 −1.95722
\(261\) 0 0
\(262\) 2.17699 0.134495
\(263\) −5.98623 −0.369127 −0.184563 0.982821i \(-0.559087\pi\)
−0.184563 + 0.982821i \(0.559087\pi\)
\(264\) 0 0
\(265\) 28.5997 1.75687
\(266\) 0.129302 0.00792801
\(267\) 0 0
\(268\) −6.81517 −0.416303
\(269\) 11.3499 0.692018 0.346009 0.938231i \(-0.387537\pi\)
0.346009 + 0.938231i \(0.387537\pi\)
\(270\) 0 0
\(271\) 23.5653 1.43149 0.715746 0.698360i \(-0.246087\pi\)
0.715746 + 0.698360i \(0.246087\pi\)
\(272\) −23.3751 −1.41732
\(273\) 0 0
\(274\) 5.22329 0.315551
\(275\) 0 0
\(276\) 0 0
\(277\) 15.1307 0.909115 0.454558 0.890717i \(-0.349797\pi\)
0.454558 + 0.890717i \(0.349797\pi\)
\(278\) −5.84437 −0.350522
\(279\) 0 0
\(280\) 3.56535 0.213070
\(281\) 26.1554 1.56030 0.780150 0.625593i \(-0.215143\pi\)
0.780150 + 0.625593i \(0.215143\pi\)
\(282\) 0 0
\(283\) −20.0807 −1.19367 −0.596836 0.802363i \(-0.703576\pi\)
−0.596836 + 0.802363i \(0.703576\pi\)
\(284\) 16.2630 0.965031
\(285\) 0 0
\(286\) 0 0
\(287\) 9.92895 0.586087
\(288\) 0 0
\(289\) 28.1994 1.65879
\(290\) −6.25739 −0.367446
\(291\) 0 0
\(292\) −4.99217 −0.292145
\(293\) 19.5215 1.14046 0.570230 0.821485i \(-0.306854\pi\)
0.570230 + 0.821485i \(0.306854\pi\)
\(294\) 0 0
\(295\) 21.9881 1.28020
\(296\) −3.94272 −0.229166
\(297\) 0 0
\(298\) −3.42682 −0.198510
\(299\) −15.7458 −0.910602
\(300\) 0 0
\(301\) −1.82301 −0.105076
\(302\) 6.12808 0.352632
\(303\) 0 0
\(304\) −1.51122 −0.0866744
\(305\) −20.7641 −1.18895
\(306\) 0 0
\(307\) 23.2920 1.32935 0.664673 0.747134i \(-0.268571\pi\)
0.664673 + 0.747134i \(0.268571\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.51532 −0.256453
\(311\) 8.79167 0.498530 0.249265 0.968435i \(-0.419811\pi\)
0.249265 + 0.968435i \(0.419811\pi\)
\(312\) 0 0
\(313\) 25.3764 1.43436 0.717180 0.696888i \(-0.245432\pi\)
0.717180 + 0.696888i \(0.245432\pi\)
\(314\) 1.52630 0.0861341
\(315\) 0 0
\(316\) −22.2614 −1.25230
\(317\) −0.258604 −0.0145246 −0.00726232 0.999974i \(-0.502312\pi\)
−0.00726232 + 0.999974i \(0.502312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −18.2424 −1.01978
\(321\) 0 0
\(322\) 0.869304 0.0484444
\(323\) 2.92219 0.162595
\(324\) 0 0
\(325\) 23.6460 1.31164
\(326\) −1.06063 −0.0587431
\(327\) 0 0
\(328\) 11.5534 0.637931
\(329\) −7.45988 −0.411276
\(330\) 0 0
\(331\) −27.2920 −1.50011 −0.750053 0.661378i \(-0.769972\pi\)
−0.750053 + 0.661378i \(0.769972\pi\)
\(332\) 3.31120 0.181726
\(333\) 0 0
\(334\) 6.25392 0.342199
\(335\) 10.9244 0.596863
\(336\) 0 0
\(337\) 22.3383 1.21685 0.608423 0.793613i \(-0.291802\pi\)
0.608423 + 0.793613i \(0.291802\pi\)
\(338\) −4.76997 −0.259452
\(339\) 0 0
\(340\) 39.3764 2.13549
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.12127 −0.114371
\(345\) 0 0
\(346\) 1.13070 0.0607866
\(347\) 35.6839 1.91561 0.957805 0.287417i \(-0.0927967\pi\)
0.957805 + 0.287417i \(0.0927967\pi\)
\(348\) 0 0
\(349\) 9.03437 0.483599 0.241799 0.970326i \(-0.422262\pi\)
0.241799 + 0.970326i \(0.422262\pi\)
\(350\) −1.30547 −0.0697800
\(351\) 0 0
\(352\) 0 0
\(353\) 36.0843 1.92058 0.960288 0.279012i \(-0.0900068\pi\)
0.960288 + 0.279012i \(0.0900068\pi\)
\(354\) 0 0
\(355\) −26.0687 −1.38358
\(356\) 13.9884 0.741385
\(357\) 0 0
\(358\) 7.29203 0.385396
\(359\) −23.3224 −1.23091 −0.615455 0.788172i \(-0.711028\pi\)
−0.615455 + 0.788172i \(0.711028\pi\)
\(360\) 0 0
\(361\) −18.8111 −0.990057
\(362\) 1.50119 0.0789009
\(363\) 0 0
\(364\) −10.2999 −0.539859
\(365\) 8.00219 0.418854
\(366\) 0 0
\(367\) −21.1994 −1.10660 −0.553301 0.832982i \(-0.686632\pi\)
−0.553301 + 0.832982i \(0.686632\pi\)
\(368\) −10.1600 −0.529627
\(369\) 0 0
\(370\) 3.08850 0.160563
\(371\) 9.33398 0.484596
\(372\) 0 0
\(373\) 9.82301 0.508616 0.254308 0.967123i \(-0.418152\pi\)
0.254308 + 0.967123i \(0.418152\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.68038 −0.447656
\(377\) 36.9906 1.90511
\(378\) 0 0
\(379\) 24.0807 1.23694 0.618470 0.785808i \(-0.287753\pi\)
0.618470 + 0.785808i \(0.287753\pi\)
\(380\) 2.54572 0.130593
\(381\) 0 0
\(382\) 0.245730 0.0125726
\(383\) 32.7366 1.67276 0.836381 0.548149i \(-0.184667\pi\)
0.836381 + 0.548149i \(0.184667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.09616 −0.106692
\(387\) 0 0
\(388\) 13.2920 0.674800
\(389\) −6.69551 −0.339476 −0.169738 0.985489i \(-0.554292\pi\)
−0.169738 + 0.985489i \(0.554292\pi\)
\(390\) 0 0
\(391\) 19.6460 0.993542
\(392\) 1.16361 0.0587711
\(393\) 0 0
\(394\) −4.33832 −0.218562
\(395\) 35.6839 1.79545
\(396\) 0 0
\(397\) 29.0224 1.45659 0.728297 0.685261i \(-0.240312\pi\)
0.728297 + 0.685261i \(0.240312\pi\)
\(398\) 4.67954 0.234564
\(399\) 0 0
\(400\) 15.2577 0.762883
\(401\) −32.4780 −1.62187 −0.810936 0.585134i \(-0.801042\pi\)
−0.810936 + 0.585134i \(0.801042\pi\)
\(402\) 0 0
\(403\) 26.6923 1.32964
\(404\) −15.1257 −0.752532
\(405\) 0 0
\(406\) −2.04220 −0.101353
\(407\) 0 0
\(408\) 0 0
\(409\) −10.4227 −0.515370 −0.257685 0.966229i \(-0.582960\pi\)
−0.257685 + 0.966229i \(0.582960\pi\)
\(410\) −9.05027 −0.446961
\(411\) 0 0
\(412\) −35.2151 −1.73492
\(413\) 7.17616 0.353116
\(414\) 0 0
\(415\) −5.30769 −0.260544
\(416\) −18.1131 −0.888068
\(417\) 0 0
\(418\) 0 0
\(419\) 28.5077 1.39269 0.696346 0.717706i \(-0.254808\pi\)
0.696346 + 0.717706i \(0.254808\pi\)
\(420\) 0 0
\(421\) −6.16506 −0.300467 −0.150233 0.988651i \(-0.548003\pi\)
−0.150233 + 0.988651i \(0.548003\pi\)
\(422\) 4.03192 0.196271
\(423\) 0 0
\(424\) 10.8611 0.527462
\(425\) −29.5031 −1.43111
\(426\) 0 0
\(427\) −6.77671 −0.327948
\(428\) −25.4311 −1.22926
\(429\) 0 0
\(430\) 1.66168 0.0801332
\(431\) −26.4666 −1.27485 −0.637427 0.770511i \(-0.720001\pi\)
−0.637427 + 0.770511i \(0.720001\pi\)
\(432\) 0 0
\(433\) −18.8693 −0.906801 −0.453400 0.891307i \(-0.649789\pi\)
−0.453400 + 0.891307i \(0.649789\pi\)
\(434\) −1.47365 −0.0707373
\(435\) 0 0
\(436\) 1.82301 0.0873062
\(437\) 1.27013 0.0607587
\(438\) 0 0
\(439\) 14.8574 0.709104 0.354552 0.935036i \(-0.384633\pi\)
0.354552 + 0.935036i \(0.384633\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.7767 0.512596
\(443\) 20.7641 0.986533 0.493267 0.869878i \(-0.335803\pi\)
0.493267 + 0.869878i \(0.335803\pi\)
\(444\) 0 0
\(445\) −22.4227 −1.06294
\(446\) 4.65444 0.220394
\(447\) 0 0
\(448\) −5.95370 −0.281286
\(449\) −33.5877 −1.58510 −0.792551 0.609805i \(-0.791248\pi\)
−0.792551 + 0.609805i \(0.791248\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −28.5192 −1.34143
\(453\) 0 0
\(454\) −0.269587 −0.0126524
\(455\) 16.5101 0.774008
\(456\) 0 0
\(457\) 30.5071 1.42706 0.713532 0.700623i \(-0.247095\pi\)
0.713532 + 0.700623i \(0.247095\pi\)
\(458\) −4.60178 −0.215027
\(459\) 0 0
\(460\) 17.1150 0.797993
\(461\) −10.5515 −0.491431 −0.245715 0.969342i \(-0.579023\pi\)
−0.245715 + 0.969342i \(0.579023\pi\)
\(462\) 0 0
\(463\) −14.6960 −0.682983 −0.341492 0.939885i \(-0.610932\pi\)
−0.341492 + 0.939885i \(0.610932\pi\)
\(464\) 23.8683 1.10806
\(465\) 0 0
\(466\) 5.21510 0.241585
\(467\) −9.83975 −0.455329 −0.227665 0.973740i \(-0.573109\pi\)
−0.227665 + 0.973740i \(0.573109\pi\)
\(468\) 0 0
\(469\) 3.56535 0.164632
\(470\) 6.79970 0.313647
\(471\) 0 0
\(472\) 8.35025 0.384352
\(473\) 0 0
\(474\) 0 0
\(475\) −1.90741 −0.0875178
\(476\) 12.8511 0.589031
\(477\) 0 0
\(478\) 0.665774 0.0304518
\(479\) −13.8100 −0.630996 −0.315498 0.948926i \(-0.602171\pi\)
−0.315498 + 0.948926i \(0.602171\pi\)
\(480\) 0 0
\(481\) −18.2577 −0.832478
\(482\) −5.06745 −0.230816
\(483\) 0 0
\(484\) 0 0
\(485\) −21.3064 −0.967476
\(486\) 0 0
\(487\) −6.26139 −0.283731 −0.141865 0.989886i \(-0.545310\pi\)
−0.141865 + 0.989886i \(0.545310\pi\)
\(488\) −7.88544 −0.356957
\(489\) 0 0
\(490\) −0.911503 −0.0411775
\(491\) 13.8717 0.626020 0.313010 0.949750i \(-0.398663\pi\)
0.313010 + 0.949750i \(0.398663\pi\)
\(492\) 0 0
\(493\) −46.1531 −2.07863
\(494\) 0.696725 0.0313471
\(495\) 0 0
\(496\) 17.2233 0.773349
\(497\) −8.50796 −0.381634
\(498\) 0 0
\(499\) 18.6960 0.836950 0.418475 0.908228i \(-0.362565\pi\)
0.418475 + 0.908228i \(0.362565\pi\)
\(500\) 3.58236 0.160208
\(501\) 0 0
\(502\) 5.78081 0.258010
\(503\) −11.4301 −0.509645 −0.254822 0.966988i \(-0.582017\pi\)
−0.254822 + 0.966988i \(0.582017\pi\)
\(504\) 0 0
\(505\) 24.2457 1.07892
\(506\) 0 0
\(507\) 0 0
\(508\) −37.7147 −1.67332
\(509\) −14.8144 −0.656639 −0.328319 0.944567i \(-0.606482\pi\)
−0.328319 + 0.944567i \(0.606482\pi\)
\(510\) 0 0
\(511\) 2.61165 0.115532
\(512\) −19.7789 −0.874113
\(513\) 0 0
\(514\) 0.119130 0.00525461
\(515\) 56.4480 2.48740
\(516\) 0 0
\(517\) 0 0
\(518\) 1.00798 0.0442881
\(519\) 0 0
\(520\) 19.2114 0.842474
\(521\) 19.3521 0.847832 0.423916 0.905701i \(-0.360655\pi\)
0.423916 + 0.905701i \(0.360655\pi\)
\(522\) 0 0
\(523\) 23.7267 1.03750 0.518748 0.854927i \(-0.326398\pi\)
0.518748 + 0.854927i \(0.326398\pi\)
\(524\) 13.9884 0.611087
\(525\) 0 0
\(526\) 1.78081 0.0776469
\(527\) −33.3040 −1.45075
\(528\) 0 0
\(529\) −14.4608 −0.628732
\(530\) −8.50796 −0.369562
\(531\) 0 0
\(532\) 0.830838 0.0360214
\(533\) 53.5007 2.31737
\(534\) 0 0
\(535\) 40.7648 1.76242
\(536\) 4.14867 0.179195
\(537\) 0 0
\(538\) −3.37643 −0.145568
\(539\) 0 0
\(540\) 0 0
\(541\) 20.5153 0.882022 0.441011 0.897502i \(-0.354620\pi\)
0.441011 + 0.897502i \(0.354620\pi\)
\(542\) −7.01031 −0.301119
\(543\) 0 0
\(544\) 22.5997 0.968955
\(545\) −2.92219 −0.125173
\(546\) 0 0
\(547\) 28.6841 1.22644 0.613222 0.789911i \(-0.289873\pi\)
0.613222 + 0.789911i \(0.289873\pi\)
\(548\) 33.5626 1.43372
\(549\) 0 0
\(550\) 0 0
\(551\) −2.98384 −0.127116
\(552\) 0 0
\(553\) 11.6460 0.495239
\(554\) −4.50114 −0.191235
\(555\) 0 0
\(556\) −37.5534 −1.59262
\(557\) 26.7228 1.13228 0.566141 0.824309i \(-0.308436\pi\)
0.566141 + 0.824309i \(0.308436\pi\)
\(558\) 0 0
\(559\) −9.82301 −0.415469
\(560\) 10.6532 0.450181
\(561\) 0 0
\(562\) −7.78081 −0.328214
\(563\) −20.6839 −0.871724 −0.435862 0.900014i \(-0.643556\pi\)
−0.435862 + 0.900014i \(0.643556\pi\)
\(564\) 0 0
\(565\) 45.7147 1.92323
\(566\) 5.97368 0.251092
\(567\) 0 0
\(568\) −9.89994 −0.415392
\(569\) −7.54908 −0.316474 −0.158237 0.987401i \(-0.550581\pi\)
−0.158237 + 0.987401i \(0.550581\pi\)
\(570\) 0 0
\(571\) −26.7767 −1.12057 −0.560285 0.828300i \(-0.689308\pi\)
−0.560285 + 0.828300i \(0.689308\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2.95370 −0.123285
\(575\) −12.8236 −0.534781
\(576\) 0 0
\(577\) −13.5616 −0.564577 −0.282289 0.959330i \(-0.591094\pi\)
−0.282289 + 0.959330i \(0.591094\pi\)
\(578\) −8.38888 −0.348931
\(579\) 0 0
\(580\) −40.2073 −1.66952
\(581\) −1.73225 −0.0718659
\(582\) 0 0
\(583\) 0 0
\(584\) 3.03893 0.125752
\(585\) 0 0
\(586\) −5.80734 −0.239899
\(587\) 16.2515 0.670773 0.335386 0.942081i \(-0.391133\pi\)
0.335386 + 0.942081i \(0.391133\pi\)
\(588\) 0 0
\(589\) −2.15314 −0.0887184
\(590\) −6.54110 −0.269293
\(591\) 0 0
\(592\) −11.7808 −0.484188
\(593\) 25.6496 1.05330 0.526652 0.850081i \(-0.323447\pi\)
0.526652 + 0.850081i \(0.323447\pi\)
\(594\) 0 0
\(595\) −20.5997 −0.844506
\(596\) −22.0193 −0.901944
\(597\) 0 0
\(598\) 4.68412 0.191548
\(599\) −6.41180 −0.261979 −0.130989 0.991384i \(-0.541815\pi\)
−0.130989 + 0.991384i \(0.541815\pi\)
\(600\) 0 0
\(601\) 20.5191 0.836990 0.418495 0.908219i \(-0.362558\pi\)
0.418495 + 0.908219i \(0.362558\pi\)
\(602\) 0.542315 0.0221031
\(603\) 0 0
\(604\) 39.3764 1.60220
\(605\) 0 0
\(606\) 0 0
\(607\) −10.1733 −0.412920 −0.206460 0.978455i \(-0.566194\pi\)
−0.206460 + 0.978455i \(0.566194\pi\)
\(608\) 1.46109 0.0592552
\(609\) 0 0
\(610\) 6.17699 0.250099
\(611\) −40.1965 −1.62617
\(612\) 0 0
\(613\) −5.46902 −0.220892 −0.110446 0.993882i \(-0.535228\pi\)
−0.110446 + 0.993882i \(0.535228\pi\)
\(614\) −6.92900 −0.279632
\(615\) 0 0
\(616\) 0 0
\(617\) −3.46450 −0.139476 −0.0697378 0.997565i \(-0.522216\pi\)
−0.0697378 + 0.997565i \(0.522216\pi\)
\(618\) 0 0
\(619\) 8.08440 0.324939 0.162470 0.986714i \(-0.448054\pi\)
0.162470 + 0.986714i \(0.448054\pi\)
\(620\) −29.0135 −1.16521
\(621\) 0 0
\(622\) −2.61538 −0.104867
\(623\) −7.31802 −0.293190
\(624\) 0 0
\(625\) −27.6841 −1.10736
\(626\) −7.54908 −0.301722
\(627\) 0 0
\(628\) 9.80734 0.391356
\(629\) 22.7801 0.908302
\(630\) 0 0
\(631\) 23.2920 0.927241 0.463620 0.886034i \(-0.346550\pi\)
0.463620 + 0.886034i \(0.346550\pi\)
\(632\) 13.5514 0.539046
\(633\) 0 0
\(634\) 0.0769305 0.00305530
\(635\) 60.4548 2.39908
\(636\) 0 0
\(637\) 5.38835 0.213494
\(638\) 0 0
\(639\) 0 0
\(640\) 26.0265 1.02879
\(641\) −36.2262 −1.43085 −0.715424 0.698690i \(-0.753767\pi\)
−0.715424 + 0.698690i \(0.753767\pi\)
\(642\) 0 0
\(643\) −27.5616 −1.08692 −0.543462 0.839434i \(-0.682887\pi\)
−0.543462 + 0.839434i \(0.682887\pi\)
\(644\) 5.58577 0.220110
\(645\) 0 0
\(646\) −0.869304 −0.0342023
\(647\) −29.3086 −1.15224 −0.576121 0.817365i \(-0.695434\pi\)
−0.576121 + 0.817365i \(0.695434\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −7.03431 −0.275908
\(651\) 0 0
\(652\) −6.81517 −0.266903
\(653\) −39.1484 −1.53199 −0.765997 0.642844i \(-0.777754\pi\)
−0.765997 + 0.642844i \(0.777754\pi\)
\(654\) 0 0
\(655\) −22.4227 −0.876128
\(656\) 34.5215 1.34784
\(657\) 0 0
\(658\) 2.21919 0.0865132
\(659\) −1.33179 −0.0518792 −0.0259396 0.999664i \(-0.508258\pi\)
−0.0259396 + 0.999664i \(0.508258\pi\)
\(660\) 0 0
\(661\) 31.6378 1.23057 0.615284 0.788305i \(-0.289041\pi\)
0.615284 + 0.788305i \(0.289041\pi\)
\(662\) 8.11894 0.315552
\(663\) 0 0
\(664\) −2.01566 −0.0782229
\(665\) −1.33179 −0.0516447
\(666\) 0 0
\(667\) −20.0605 −0.776747
\(668\) 40.1850 1.55480
\(669\) 0 0
\(670\) −3.24983 −0.125552
\(671\) 0 0
\(672\) 0 0
\(673\) 32.0844 1.23676 0.618381 0.785878i \(-0.287789\pi\)
0.618381 + 0.785878i \(0.287789\pi\)
\(674\) −6.64529 −0.255967
\(675\) 0 0
\(676\) −30.6497 −1.17884
\(677\) 15.1508 0.582293 0.291146 0.956678i \(-0.405963\pi\)
0.291146 + 0.956678i \(0.405963\pi\)
\(678\) 0 0
\(679\) −6.95370 −0.266859
\(680\) −23.9700 −0.919208
\(681\) 0 0
\(682\) 0 0
\(683\) −30.0981 −1.15167 −0.575836 0.817565i \(-0.695323\pi\)
−0.575836 + 0.817565i \(0.695323\pi\)
\(684\) 0 0
\(685\) −53.7991 −2.05556
\(686\) −0.297484 −0.0113580
\(687\) 0 0
\(688\) −6.33832 −0.241646
\(689\) 50.2948 1.91608
\(690\) 0 0
\(691\) 25.1994 0.958632 0.479316 0.877643i \(-0.340885\pi\)
0.479316 + 0.877643i \(0.340885\pi\)
\(692\) 7.26537 0.276188
\(693\) 0 0
\(694\) −10.6154 −0.402954
\(695\) 60.1962 2.28337
\(696\) 0 0
\(697\) −66.7529 −2.52844
\(698\) −2.68758 −0.101726
\(699\) 0 0
\(700\) −8.38835 −0.317050
\(701\) −39.8235 −1.50411 −0.752057 0.659098i \(-0.770938\pi\)
−0.752057 + 0.659098i \(0.770938\pi\)
\(702\) 0 0
\(703\) 1.47275 0.0555460
\(704\) 0 0
\(705\) 0 0
\(706\) −10.7345 −0.403999
\(707\) 7.91299 0.297599
\(708\) 0 0
\(709\) 40.9418 1.53760 0.768800 0.639489i \(-0.220854\pi\)
0.768800 + 0.639489i \(0.220854\pi\)
\(710\) 7.75503 0.291041
\(711\) 0 0
\(712\) −8.51532 −0.319125
\(713\) −14.4756 −0.542117
\(714\) 0 0
\(715\) 0 0
\(716\) 46.8554 1.75107
\(717\) 0 0
\(718\) 6.93804 0.258925
\(719\) −3.14424 −0.117260 −0.0586302 0.998280i \(-0.518673\pi\)
−0.0586302 + 0.998280i \(0.518673\pi\)
\(720\) 0 0
\(721\) 18.4227 0.686098
\(722\) 5.59599 0.208261
\(723\) 0 0
\(724\) 9.64601 0.358491
\(725\) 30.1257 1.11884
\(726\) 0 0
\(727\) 45.9843 1.70546 0.852732 0.522348i \(-0.174944\pi\)
0.852732 + 0.522348i \(0.174944\pi\)
\(728\) 6.26994 0.232379
\(729\) 0 0
\(730\) −2.38052 −0.0881071
\(731\) 12.2562 0.453311
\(732\) 0 0
\(733\) −14.7767 −0.545790 −0.272895 0.962044i \(-0.587981\pi\)
−0.272895 + 0.962044i \(0.587981\pi\)
\(734\) 6.30649 0.232777
\(735\) 0 0
\(736\) 9.82301 0.362081
\(737\) 0 0
\(738\) 0 0
\(739\) 51.3526 1.88903 0.944517 0.328461i \(-0.106530\pi\)
0.944517 + 0.328461i \(0.106530\pi\)
\(740\) 19.8453 0.729529
\(741\) 0 0
\(742\) −2.77671 −0.101936
\(743\) −16.4299 −0.602756 −0.301378 0.953505i \(-0.597447\pi\)
−0.301378 + 0.953505i \(0.597447\pi\)
\(744\) 0 0
\(745\) 35.2958 1.29314
\(746\) −2.92219 −0.106989
\(747\) 0 0
\(748\) 0 0
\(749\) 13.3042 0.486127
\(750\) 0 0
\(751\) 28.9500 1.05640 0.528200 0.849120i \(-0.322867\pi\)
0.528200 + 0.849120i \(0.322867\pi\)
\(752\) −25.9369 −0.945821
\(753\) 0 0
\(754\) −11.0041 −0.400746
\(755\) −63.1184 −2.29711
\(756\) 0 0
\(757\) 36.0725 1.31108 0.655538 0.755162i \(-0.272442\pi\)
0.655538 + 0.755162i \(0.272442\pi\)
\(758\) −7.16361 −0.260194
\(759\) 0 0
\(760\) −1.54968 −0.0562130
\(761\) 34.8303 1.26260 0.631299 0.775540i \(-0.282522\pi\)
0.631299 + 0.775540i \(0.282522\pi\)
\(762\) 0 0
\(763\) −0.953703 −0.0345264
\(764\) 1.57895 0.0571245
\(765\) 0 0
\(766\) −9.73861 −0.351870
\(767\) 38.6677 1.39621
\(768\) 0 0
\(769\) −4.35025 −0.156874 −0.0784371 0.996919i \(-0.524993\pi\)
−0.0784371 + 0.996919i \(0.524993\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13.4690 −0.484761
\(773\) 1.41199 0.0507857 0.0253929 0.999678i \(-0.491916\pi\)
0.0253929 + 0.999678i \(0.491916\pi\)
\(774\) 0 0
\(775\) 21.7386 0.780874
\(776\) −8.09139 −0.290464
\(777\) 0 0
\(778\) 1.99181 0.0714097
\(779\) −4.31564 −0.154624
\(780\) 0 0
\(781\) 0 0
\(782\) −5.84437 −0.208994
\(783\) 0 0
\(784\) 3.47685 0.124173
\(785\) −15.7207 −0.561095
\(786\) 0 0
\(787\) 12.7886 0.455866 0.227933 0.973677i \(-0.426803\pi\)
0.227933 + 0.973677i \(0.426803\pi\)
\(788\) −27.8762 −0.993048
\(789\) 0 0
\(790\) −10.6154 −0.377678
\(791\) 14.9198 0.530485
\(792\) 0 0
\(793\) −36.5153 −1.29670
\(794\) −8.63371 −0.306399
\(795\) 0 0
\(796\) 30.0687 1.06576
\(797\) −8.90842 −0.315552 −0.157776 0.987475i \(-0.550432\pi\)
−0.157776 + 0.987475i \(0.550432\pi\)
\(798\) 0 0
\(799\) 50.1531 1.77429
\(800\) −14.7516 −0.521547
\(801\) 0 0
\(802\) 9.66168 0.341166
\(803\) 0 0
\(804\) 0 0
\(805\) −8.95370 −0.315577
\(806\) −7.94053 −0.279693
\(807\) 0 0
\(808\) 9.20763 0.323923
\(809\) −54.9834 −1.93311 −0.966556 0.256456i \(-0.917445\pi\)
−0.966556 + 0.256456i \(0.917445\pi\)
\(810\) 0 0
\(811\) −3.21136 −0.112766 −0.0563831 0.998409i \(-0.517957\pi\)
−0.0563831 + 0.998409i \(0.517957\pi\)
\(812\) −13.1223 −0.460502
\(813\) 0 0
\(814\) 0 0
\(815\) 10.9244 0.382664
\(816\) 0 0
\(817\) 0.792373 0.0277216
\(818\) 3.10059 0.108410
\(819\) 0 0
\(820\) −58.1531 −2.03080
\(821\) 25.2492 0.881202 0.440601 0.897703i \(-0.354765\pi\)
0.440601 + 0.897703i \(0.354765\pi\)
\(822\) 0 0
\(823\) 5.82674 0.203107 0.101554 0.994830i \(-0.467619\pi\)
0.101554 + 0.994830i \(0.467619\pi\)
\(824\) 21.4369 0.746788
\(825\) 0 0
\(826\) −2.13479 −0.0742790
\(827\) 19.7160 0.685594 0.342797 0.939409i \(-0.388626\pi\)
0.342797 + 0.939409i \(0.388626\pi\)
\(828\) 0 0
\(829\) 7.03810 0.244443 0.122222 0.992503i \(-0.460998\pi\)
0.122222 + 0.992503i \(0.460998\pi\)
\(830\) 1.57895 0.0548062
\(831\) 0 0
\(832\) −32.0807 −1.11220
\(833\) −6.72305 −0.232940
\(834\) 0 0
\(835\) −64.4145 −2.22916
\(836\) 0 0
\(837\) 0 0
\(838\) −8.48059 −0.292957
\(839\) 22.9471 0.792220 0.396110 0.918203i \(-0.370360\pi\)
0.396110 + 0.918203i \(0.370360\pi\)
\(840\) 0 0
\(841\) 18.1270 0.625068
\(842\) 1.83401 0.0632041
\(843\) 0 0
\(844\) 25.9074 0.891770
\(845\) 49.1300 1.69012
\(846\) 0 0
\(847\) 0 0
\(848\) 32.4529 1.11444
\(849\) 0 0
\(850\) 8.77671 0.301039
\(851\) 9.90141 0.339416
\(852\) 0 0
\(853\) 42.5915 1.45831 0.729153 0.684351i \(-0.239914\pi\)
0.729153 + 0.684351i \(0.239914\pi\)
\(854\) 2.01596 0.0689848
\(855\) 0 0
\(856\) 15.4809 0.529128
\(857\) 4.42338 0.151100 0.0755499 0.997142i \(-0.475929\pi\)
0.0755499 + 0.997142i \(0.475929\pi\)
\(858\) 0 0
\(859\) −10.4466 −0.356433 −0.178216 0.983991i \(-0.557033\pi\)
−0.178216 + 0.983991i \(0.557033\pi\)
\(860\) 10.6772 0.364090
\(861\) 0 0
\(862\) 7.87340 0.268169
\(863\) −0.258604 −0.00880298 −0.00440149 0.999990i \(-0.501401\pi\)
−0.00440149 + 0.999990i \(0.501401\pi\)
\(864\) 0 0
\(865\) −11.6460 −0.395976
\(866\) 5.61331 0.190748
\(867\) 0 0
\(868\) −9.46902 −0.321399
\(869\) 0 0
\(870\) 0 0
\(871\) 19.2114 0.650952
\(872\) −1.10974 −0.0375805
\(873\) 0 0
\(874\) −0.377844 −0.0127808
\(875\) −1.87411 −0.0633564
\(876\) 0 0
\(877\) −38.7529 −1.30859 −0.654295 0.756239i \(-0.727035\pi\)
−0.654295 + 0.756239i \(0.727035\pi\)
\(878\) −4.41983 −0.149162
\(879\) 0 0
\(880\) 0 0
\(881\) 29.3888 0.990135 0.495067 0.868855i \(-0.335143\pi\)
0.495067 + 0.868855i \(0.335143\pi\)
\(882\) 0 0
\(883\) 0.788639 0.0265398 0.0132699 0.999912i \(-0.495776\pi\)
0.0132699 + 0.999912i \(0.495776\pi\)
\(884\) 69.2465 2.32901
\(885\) 0 0
\(886\) −6.17699 −0.207520
\(887\) 18.4094 0.618126 0.309063 0.951042i \(-0.399985\pi\)
0.309063 + 0.951042i \(0.399985\pi\)
\(888\) 0 0
\(889\) 19.7304 0.661737
\(890\) 6.67040 0.223592
\(891\) 0 0
\(892\) 29.9074 1.00137
\(893\) 3.24245 0.108504
\(894\) 0 0
\(895\) −75.1068 −2.51054
\(896\) 8.49418 0.283771
\(897\) 0 0
\(898\) 9.99181 0.333431
\(899\) 34.0067 1.13419
\(900\) 0 0
\(901\) −62.7529 −2.09060
\(902\) 0 0
\(903\) 0 0
\(904\) 17.3608 0.577410
\(905\) −15.4621 −0.513976
\(906\) 0 0
\(907\) −25.3846 −0.842882 −0.421441 0.906856i \(-0.638476\pi\)
−0.421441 + 0.906856i \(0.638476\pi\)
\(908\) −1.73225 −0.0574868
\(909\) 0 0
\(910\) −4.91150 −0.162815
\(911\) 31.9357 1.05808 0.529038 0.848598i \(-0.322553\pi\)
0.529038 + 0.848598i \(0.322553\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −9.07538 −0.300187
\(915\) 0 0
\(916\) −29.5691 −0.976990
\(917\) −7.31802 −0.241662
\(918\) 0 0
\(919\) −29.7991 −0.982983 −0.491492 0.870882i \(-0.663548\pi\)
−0.491492 + 0.870882i \(0.663548\pi\)
\(920\) −10.4186 −0.343491
\(921\) 0 0
\(922\) 3.13889 0.103374
\(923\) −45.8439 −1.50897
\(924\) 0 0
\(925\) −14.8693 −0.488900
\(926\) 4.37184 0.143667
\(927\) 0 0
\(928\) −23.0766 −0.757525
\(929\) −30.5237 −1.00145 −0.500725 0.865607i \(-0.666933\pi\)
−0.500725 + 0.865607i \(0.666933\pi\)
\(930\) 0 0
\(931\) −0.434652 −0.0142451
\(932\) 33.5100 1.09766
\(933\) 0 0
\(934\) 2.92717 0.0957798
\(935\) 0 0
\(936\) 0 0
\(937\) −11.2995 −0.369138 −0.184569 0.982820i \(-0.559089\pi\)
−0.184569 + 0.982820i \(0.559089\pi\)
\(938\) −1.06063 −0.0346309
\(939\) 0 0
\(940\) 43.6919 1.42507
\(941\) −7.99319 −0.260570 −0.130285 0.991477i \(-0.541589\pi\)
−0.130285 + 0.991477i \(0.541589\pi\)
\(942\) 0 0
\(943\) −29.0142 −0.944834
\(944\) 24.9505 0.812068
\(945\) 0 0
\(946\) 0 0
\(947\) −49.5921 −1.61153 −0.805763 0.592238i \(-0.798245\pi\)
−0.805763 + 0.592238i \(0.798245\pi\)
\(948\) 0 0
\(949\) 14.0725 0.456812
\(950\) 0.567423 0.0184096
\(951\) 0 0
\(952\) −7.82301 −0.253545
\(953\) 13.7939 0.446829 0.223414 0.974724i \(-0.428280\pi\)
0.223414 + 0.974724i \(0.428280\pi\)
\(954\) 0 0
\(955\) −2.53098 −0.0819006
\(956\) 4.27797 0.138360
\(957\) 0 0
\(958\) 4.10826 0.132732
\(959\) −17.5582 −0.566985
\(960\) 0 0
\(961\) −6.46083 −0.208414
\(962\) 5.43136 0.175114
\(963\) 0 0
\(964\) −32.5613 −1.04873
\(965\) 21.5902 0.695012
\(966\) 0 0
\(967\) −32.1613 −1.03424 −0.517119 0.855913i \(-0.672996\pi\)
−0.517119 + 0.855913i \(0.672996\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.33832 0.203511
\(971\) −2.41642 −0.0775467 −0.0387733 0.999248i \(-0.512345\pi\)
−0.0387733 + 0.999248i \(0.512345\pi\)
\(972\) 0 0
\(973\) 19.6460 0.629822
\(974\) 1.86266 0.0596836
\(975\) 0 0
\(976\) −23.5616 −0.754189
\(977\) 5.30206 0.169628 0.0848139 0.996397i \(-0.472970\pi\)
0.0848139 + 0.996397i \(0.472970\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −5.85693 −0.187093
\(981\) 0 0
\(982\) −4.12660 −0.131685
\(983\) −4.03192 −0.128598 −0.0642992 0.997931i \(-0.520481\pi\)
−0.0642992 + 0.997931i \(0.520481\pi\)
\(984\) 0 0
\(985\) 44.6841 1.42375
\(986\) 13.7298 0.437246
\(987\) 0 0
\(988\) 4.47685 0.142428
\(989\) 5.32717 0.169394
\(990\) 0 0
\(991\) 29.1188 0.924988 0.462494 0.886622i \(-0.346955\pi\)
0.462494 + 0.886622i \(0.346955\pi\)
\(992\) −16.6520 −0.528702
\(993\) 0 0
\(994\) 2.53098 0.0802778
\(995\) −48.1986 −1.52800
\(996\) 0 0
\(997\) −11.1307 −0.352513 −0.176256 0.984344i \(-0.556399\pi\)
−0.176256 + 0.984344i \(0.556399\pi\)
\(998\) −5.56177 −0.176055
\(999\) 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 7623.2.a.cr.1.3 yes 6
3.2 odd 2 inner 7623.2.a.cr.1.4 yes 6
11.10 odd 2 7623.2.a.cq.1.4 yes 6
33.32 even 2 7623.2.a.cq.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.cq.1.3 6 33.32 even 2
7623.2.a.cq.1.4 yes 6 11.10 odd 2
7623.2.a.cr.1.3 yes 6 1.1 even 1 trivial
7623.2.a.cr.1.4 yes 6 3.2 odd 2 inner