Properties

Label 1386.2.a.r.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1304.1
Defining polynomial: \(x^{3} - 11 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.182370\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.89219 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.89219 q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.89219 q^{10} +1.00000 q^{11} -0.364739 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.89219 q^{17} +7.25693 q^{19} -2.89219 q^{20} +1.00000 q^{22} +8.14911 q^{23} +3.36474 q^{25} -0.364739 q^{26} +1.00000 q^{28} -2.36474 q^{29} +10.6766 q^{31} +1.00000 q^{32} -2.89219 q^{34} -2.89219 q^{35} -3.78437 q^{37} +7.25693 q^{38} -2.89219 q^{40} +2.89219 q^{41} +11.4196 q^{43} +1.00000 q^{44} +8.14911 q^{46} -6.52745 q^{47} +1.00000 q^{49} +3.36474 q^{50} -0.364739 q^{52} +11.7844 q^{53} -2.89219 q^{55} +1.00000 q^{56} -2.36474 q^{58} -10.5139 q^{59} -11.9335 q^{61} +10.6766 q^{62} +1.00000 q^{64} +1.05489 q^{65} -6.14911 q^{67} -2.89219 q^{68} -2.89219 q^{70} +13.9335 q^{71} -1.10781 q^{73} -3.78437 q^{74} +7.25693 q^{76} +1.00000 q^{77} +10.1491 q^{79} -2.89219 q^{80} +2.89219 q^{82} -3.10781 q^{83} +8.36474 q^{85} +11.4196 q^{86} +1.00000 q^{88} +2.14911 q^{89} -0.364739 q^{91} +8.14911 q^{92} -6.52745 q^{94} -20.9884 q^{95} +6.72948 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 2q^{5} + 3q^{7} + 3q^{8} + 2q^{10} + 3q^{11} + 3q^{14} + 3q^{16} + 2q^{17} + 10q^{19} + 2q^{20} + 3q^{22} + 2q^{23} + 9q^{25} + 3q^{28} - 6q^{29} + 3q^{32} + 2q^{34} + 2q^{35} + 10q^{37} + 10q^{38} + 2q^{40} - 2q^{41} + 14q^{43} + 3q^{44} + 2q^{46} - 10q^{47} + 3q^{49} + 9q^{50} + 14q^{53} + 2q^{55} + 3q^{56} - 6q^{58} - 8q^{59} + 8q^{61} + 3q^{64} - 16q^{65} + 4q^{67} + 2q^{68} + 2q^{70} - 2q^{71} - 14q^{73} + 10q^{74} + 10q^{76} + 3q^{77} + 8q^{79} + 2q^{80} - 2q^{82} - 20q^{83} + 24q^{85} + 14q^{86} + 3q^{88} - 16q^{89} + 2q^{92} - 10q^{94} + 18q^{97} + 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.89219 −1.29342 −0.646712 0.762734i \(-0.723857\pi\)
−0.646712 + 0.762734i \(0.723857\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.89219 −0.914589
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −0.364739 −0.101160 −0.0505802 0.998720i \(-0.516107\pi\)
−0.0505802 + 0.998720i \(0.516107\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.89219 −0.701458 −0.350729 0.936477i \(-0.614066\pi\)
−0.350729 + 0.936477i \(0.614066\pi\)
\(18\) 0 0
\(19\) 7.25693 1.66485 0.832426 0.554136i \(-0.186951\pi\)
0.832426 + 0.554136i \(0.186951\pi\)
\(20\) −2.89219 −0.646712
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 8.14911 1.69921 0.849604 0.527422i \(-0.176841\pi\)
0.849604 + 0.527422i \(0.176841\pi\)
\(24\) 0 0
\(25\) 3.36474 0.672948
\(26\) −0.364739 −0.0715312
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −2.36474 −0.439121 −0.219561 0.975599i \(-0.570462\pi\)
−0.219561 + 0.975599i \(0.570462\pi\)
\(30\) 0 0
\(31\) 10.6766 1.91757 0.958783 0.284139i \(-0.0917076\pi\)
0.958783 + 0.284139i \(0.0917076\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.89219 −0.496006
\(35\) −2.89219 −0.488869
\(36\) 0 0
\(37\) −3.78437 −0.622147 −0.311073 0.950386i \(-0.600689\pi\)
−0.311073 + 0.950386i \(0.600689\pi\)
\(38\) 7.25693 1.17723
\(39\) 0 0
\(40\) −2.89219 −0.457295
\(41\) 2.89219 0.451684 0.225842 0.974164i \(-0.427487\pi\)
0.225842 + 0.974164i \(0.427487\pi\)
\(42\) 0 0
\(43\) 11.4196 1.74148 0.870739 0.491746i \(-0.163641\pi\)
0.870739 + 0.491746i \(0.163641\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 8.14911 1.20152
\(47\) −6.52745 −0.952126 −0.476063 0.879411i \(-0.657937\pi\)
−0.476063 + 0.879411i \(0.657937\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.36474 0.475846
\(51\) 0 0
\(52\) −0.364739 −0.0505802
\(53\) 11.7844 1.61871 0.809354 0.587321i \(-0.199817\pi\)
0.809354 + 0.587321i \(0.199817\pi\)
\(54\) 0 0
\(55\) −2.89219 −0.389982
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −2.36474 −0.310505
\(59\) −10.5139 −1.36879 −0.684393 0.729113i \(-0.739933\pi\)
−0.684393 + 0.729113i \(0.739933\pi\)
\(60\) 0 0
\(61\) −11.9335 −1.52793 −0.763963 0.645260i \(-0.776749\pi\)
−0.763963 + 0.645260i \(0.776749\pi\)
\(62\) 10.6766 1.35592
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.05489 0.130843
\(66\) 0 0
\(67\) −6.14911 −0.751233 −0.375617 0.926775i \(-0.622569\pi\)
−0.375617 + 0.926775i \(0.622569\pi\)
\(68\) −2.89219 −0.350729
\(69\) 0 0
\(70\) −2.89219 −0.345682
\(71\) 13.9335 1.65360 0.826800 0.562496i \(-0.190159\pi\)
0.826800 + 0.562496i \(0.190159\pi\)
\(72\) 0 0
\(73\) −1.10781 −0.129660 −0.0648299 0.997896i \(-0.520650\pi\)
−0.0648299 + 0.997896i \(0.520650\pi\)
\(74\) −3.78437 −0.439924
\(75\) 0 0
\(76\) 7.25693 0.832426
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 10.1491 1.14186 0.570932 0.820997i \(-0.306582\pi\)
0.570932 + 0.820997i \(0.306582\pi\)
\(80\) −2.89219 −0.323356
\(81\) 0 0
\(82\) 2.89219 0.319389
\(83\) −3.10781 −0.341127 −0.170563 0.985347i \(-0.554559\pi\)
−0.170563 + 0.985347i \(0.554559\pi\)
\(84\) 0 0
\(85\) 8.36474 0.907283
\(86\) 11.4196 1.23141
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 2.14911 0.227805 0.113903 0.993492i \(-0.463665\pi\)
0.113903 + 0.993492i \(0.463665\pi\)
\(90\) 0 0
\(91\) −0.364739 −0.0382351
\(92\) 8.14911 0.849604
\(93\) 0 0
\(94\) −6.52745 −0.673255
\(95\) −20.9884 −2.15336
\(96\) 0 0
\(97\) 6.72948 0.683275 0.341638 0.939832i \(-0.389018\pi\)
0.341638 + 0.939832i \(0.389018\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 3.36474 0.336474
\(101\) −1.27052 −0.126422 −0.0632108 0.998000i \(-0.520134\pi\)
−0.0632108 + 0.998000i \(0.520134\pi\)
\(102\) 0 0
\(103\) −0.892186 −0.0879097 −0.0439548 0.999034i \(-0.513996\pi\)
−0.0439548 + 0.999034i \(0.513996\pi\)
\(104\) −0.364739 −0.0357656
\(105\) 0 0
\(106\) 11.7844 1.14460
\(107\) 5.78437 0.559196 0.279598 0.960117i \(-0.409799\pi\)
0.279598 + 0.960117i \(0.409799\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) −2.89219 −0.275759
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −19.9335 −1.87518 −0.937592 0.347737i \(-0.886950\pi\)
−0.937592 + 0.347737i \(0.886950\pi\)
\(114\) 0 0
\(115\) −23.5687 −2.19780
\(116\) −2.36474 −0.219561
\(117\) 0 0
\(118\) −10.5139 −0.967878
\(119\) −2.89219 −0.265126
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −11.9335 −1.08041
\(123\) 0 0
\(124\) 10.6766 0.958783
\(125\) 4.72948 0.423017
\(126\) 0 0
\(127\) 10.1491 0.900588 0.450294 0.892880i \(-0.351319\pi\)
0.450294 + 0.892880i \(0.351319\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.05489 0.0925203
\(131\) −20.8922 −1.82536 −0.912679 0.408676i \(-0.865990\pi\)
−0.912679 + 0.408676i \(0.865990\pi\)
\(132\) 0 0
\(133\) 7.25693 0.629255
\(134\) −6.14911 −0.531202
\(135\) 0 0
\(136\) −2.89219 −0.248003
\(137\) −7.93348 −0.677803 −0.338902 0.940822i \(-0.610055\pi\)
−0.338902 + 0.940822i \(0.610055\pi\)
\(138\) 0 0
\(139\) 18.8257 1.59677 0.798386 0.602146i \(-0.205687\pi\)
0.798386 + 0.602146i \(0.205687\pi\)
\(140\) −2.89219 −0.244434
\(141\) 0 0
\(142\) 13.9335 1.16927
\(143\) −0.364739 −0.0305010
\(144\) 0 0
\(145\) 6.83927 0.567970
\(146\) −1.10781 −0.0916833
\(147\) 0 0
\(148\) −3.78437 −0.311073
\(149\) −1.27052 −0.104085 −0.0520426 0.998645i \(-0.516573\pi\)
−0.0520426 + 0.998645i \(0.516573\pi\)
\(150\) 0 0
\(151\) −8.29822 −0.675300 −0.337650 0.941272i \(-0.609632\pi\)
−0.337650 + 0.941272i \(0.609632\pi\)
\(152\) 7.25693 0.588614
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −30.8786 −2.48023
\(156\) 0 0
\(157\) −18.8922 −1.50776 −0.753880 0.657012i \(-0.771820\pi\)
−0.753880 + 0.657012i \(0.771820\pi\)
\(158\) 10.1491 0.807420
\(159\) 0 0
\(160\) −2.89219 −0.228647
\(161\) 8.14911 0.642240
\(162\) 0 0
\(163\) −20.2982 −1.58988 −0.794940 0.606688i \(-0.792498\pi\)
−0.794940 + 0.606688i \(0.792498\pi\)
\(164\) 2.89219 0.225842
\(165\) 0 0
\(166\) −3.10781 −0.241213
\(167\) 1.05489 0.0816301 0.0408151 0.999167i \(-0.487005\pi\)
0.0408151 + 0.999167i \(0.487005\pi\)
\(168\) 0 0
\(169\) −12.8670 −0.989767
\(170\) 8.36474 0.641546
\(171\) 0 0
\(172\) 11.4196 0.870739
\(173\) −16.5139 −1.25552 −0.627762 0.778405i \(-0.716029\pi\)
−0.627762 + 0.778405i \(0.716029\pi\)
\(174\) 0 0
\(175\) 3.36474 0.254350
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 2.14911 0.161083
\(179\) 20.9884 1.56874 0.784372 0.620290i \(-0.212985\pi\)
0.784372 + 0.620290i \(0.212985\pi\)
\(180\) 0 0
\(181\) −1.10781 −0.0823432 −0.0411716 0.999152i \(-0.513109\pi\)
−0.0411716 + 0.999152i \(0.513109\pi\)
\(182\) −0.364739 −0.0270363
\(183\) 0 0
\(184\) 8.14911 0.600760
\(185\) 10.9451 0.804700
\(186\) 0 0
\(187\) −2.89219 −0.211498
\(188\) −6.52745 −0.476063
\(189\) 0 0
\(190\) −20.9884 −1.52266
\(191\) −10.6902 −0.773512 −0.386756 0.922182i \(-0.626405\pi\)
−0.386756 + 0.922182i \(0.626405\pi\)
\(192\) 0 0
\(193\) −21.5687 −1.55255 −0.776276 0.630393i \(-0.782894\pi\)
−0.776276 + 0.630393i \(0.782894\pi\)
\(194\) 6.72948 0.483148
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −4.90578 −0.349523 −0.174761 0.984611i \(-0.555915\pi\)
−0.174761 + 0.984611i \(0.555915\pi\)
\(198\) 0 0
\(199\) −7.10781 −0.503860 −0.251930 0.967746i \(-0.581065\pi\)
−0.251930 + 0.967746i \(0.581065\pi\)
\(200\) 3.36474 0.237923
\(201\) 0 0
\(202\) −1.27052 −0.0893936
\(203\) −2.36474 −0.165972
\(204\) 0 0
\(205\) −8.36474 −0.584219
\(206\) −0.892186 −0.0621615
\(207\) 0 0
\(208\) −0.364739 −0.0252901
\(209\) 7.25693 0.501972
\(210\) 0 0
\(211\) 8.87859 0.611227 0.305614 0.952156i \(-0.401138\pi\)
0.305614 + 0.952156i \(0.401138\pi\)
\(212\) 11.7844 0.809354
\(213\) 0 0
\(214\) 5.78437 0.395412
\(215\) −33.0277 −2.25247
\(216\) 0 0
\(217\) 10.6766 0.724772
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) −2.89219 −0.194991
\(221\) 1.05489 0.0709598
\(222\) 0 0
\(223\) 22.2453 1.48966 0.744828 0.667257i \(-0.232532\pi\)
0.744828 + 0.667257i \(0.232532\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −19.9335 −1.32596
\(227\) 8.46093 0.561572 0.280786 0.959770i \(-0.409405\pi\)
0.280786 + 0.959770i \(0.409405\pi\)
\(228\) 0 0
\(229\) −2.16271 −0.142916 −0.0714579 0.997444i \(-0.522765\pi\)
−0.0714579 + 0.997444i \(0.522765\pi\)
\(230\) −23.5687 −1.55408
\(231\) 0 0
\(232\) −2.36474 −0.155253
\(233\) −3.45896 −0.226604 −0.113302 0.993561i \(-0.536143\pi\)
−0.113302 + 0.993561i \(0.536143\pi\)
\(234\) 0 0
\(235\) 18.8786 1.23150
\(236\) −10.5139 −0.684393
\(237\) 0 0
\(238\) −2.89219 −0.187473
\(239\) 16.2982 1.05424 0.527122 0.849790i \(-0.323271\pi\)
0.527122 + 0.849790i \(0.323271\pi\)
\(240\) 0 0
\(241\) 4.67656 0.301244 0.150622 0.988591i \(-0.451872\pi\)
0.150622 + 0.988591i \(0.451872\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −11.9335 −0.763963
\(245\) −2.89219 −0.184775
\(246\) 0 0
\(247\) −2.64688 −0.168417
\(248\) 10.6766 0.677962
\(249\) 0 0
\(250\) 4.72948 0.299118
\(251\) −10.0826 −0.636408 −0.318204 0.948022i \(-0.603080\pi\)
−0.318204 + 0.948022i \(0.603080\pi\)
\(252\) 0 0
\(253\) 8.14911 0.512330
\(254\) 10.1491 0.636812
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.41963 0.587581 0.293790 0.955870i \(-0.405083\pi\)
0.293790 + 0.955870i \(0.405083\pi\)
\(258\) 0 0
\(259\) −3.78437 −0.235149
\(260\) 1.05489 0.0654217
\(261\) 0 0
\(262\) −20.8922 −1.29072
\(263\) 6.87859 0.424152 0.212076 0.977253i \(-0.431978\pi\)
0.212076 + 0.977253i \(0.431978\pi\)
\(264\) 0 0
\(265\) −34.0826 −2.09368
\(266\) 7.25693 0.444951
\(267\) 0 0
\(268\) −6.14911 −0.375617
\(269\) 26.8922 1.63965 0.819823 0.572617i \(-0.194072\pi\)
0.819823 + 0.572617i \(0.194072\pi\)
\(270\) 0 0
\(271\) −21.7844 −1.32331 −0.661653 0.749810i \(-0.730145\pi\)
−0.661653 + 0.749810i \(0.730145\pi\)
\(272\) −2.89219 −0.175365
\(273\) 0 0
\(274\) −7.93348 −0.479279
\(275\) 3.36474 0.202901
\(276\) 0 0
\(277\) 3.27052 0.196507 0.0982533 0.995161i \(-0.468674\pi\)
0.0982533 + 0.995161i \(0.468674\pi\)
\(278\) 18.8257 1.12909
\(279\) 0 0
\(280\) −2.89219 −0.172841
\(281\) 10.2982 0.614340 0.307170 0.951655i \(-0.400618\pi\)
0.307170 + 0.951655i \(0.400618\pi\)
\(282\) 0 0
\(283\) −28.3118 −1.68296 −0.841481 0.540286i \(-0.818316\pi\)
−0.841481 + 0.540286i \(0.818316\pi\)
\(284\) 13.9335 0.826800
\(285\) 0 0
\(286\) −0.364739 −0.0215675
\(287\) 2.89219 0.170720
\(288\) 0 0
\(289\) −8.63526 −0.507957
\(290\) 6.83927 0.401615
\(291\) 0 0
\(292\) −1.10781 −0.0648299
\(293\) 11.7844 0.688450 0.344225 0.938887i \(-0.388142\pi\)
0.344225 + 0.938887i \(0.388142\pi\)
\(294\) 0 0
\(295\) 30.4080 1.77042
\(296\) −3.78437 −0.219962
\(297\) 0 0
\(298\) −1.27052 −0.0735993
\(299\) −2.97230 −0.171893
\(300\) 0 0
\(301\) 11.4196 0.658217
\(302\) −8.29822 −0.477509
\(303\) 0 0
\(304\) 7.25693 0.416213
\(305\) 34.5139 1.97626
\(306\) 0 0
\(307\) 4.01360 0.229068 0.114534 0.993419i \(-0.463463\pi\)
0.114534 + 0.993419i \(0.463463\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −30.8786 −1.75379
\(311\) −6.52745 −0.370138 −0.185069 0.982726i \(-0.559251\pi\)
−0.185069 + 0.982726i \(0.559251\pi\)
\(312\) 0 0
\(313\) 0.513850 0.0290445 0.0145223 0.999895i \(-0.495377\pi\)
0.0145223 + 0.999895i \(0.495377\pi\)
\(314\) −18.8922 −1.06615
\(315\) 0 0
\(316\) 10.1491 0.570932
\(317\) 7.48615 0.420464 0.210232 0.977652i \(-0.432578\pi\)
0.210232 + 0.977652i \(0.432578\pi\)
\(318\) 0 0
\(319\) −2.36474 −0.132400
\(320\) −2.89219 −0.161678
\(321\) 0 0
\(322\) 8.14911 0.454132
\(323\) −20.9884 −1.16782
\(324\) 0 0
\(325\) −1.22725 −0.0680757
\(326\) −20.2982 −1.12421
\(327\) 0 0
\(328\) 2.89219 0.159694
\(329\) −6.52745 −0.359870
\(330\) 0 0
\(331\) −13.4196 −0.737610 −0.368805 0.929507i \(-0.620233\pi\)
−0.368805 + 0.929507i \(0.620233\pi\)
\(332\) −3.10781 −0.170563
\(333\) 0 0
\(334\) 1.05489 0.0577212
\(335\) 17.7844 0.971664
\(336\) 0 0
\(337\) −8.94511 −0.487271 −0.243636 0.969867i \(-0.578340\pi\)
−0.243636 + 0.969867i \(0.578340\pi\)
\(338\) −12.8670 −0.699871
\(339\) 0 0
\(340\) 8.36474 0.453642
\(341\) 10.6766 0.578168
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 11.4196 0.615705
\(345\) 0 0
\(346\) −16.5139 −0.887790
\(347\) 5.78437 0.310521 0.155261 0.987874i \(-0.450378\pi\)
0.155261 + 0.987874i \(0.450378\pi\)
\(348\) 0 0
\(349\) 22.1491 1.18561 0.592807 0.805344i \(-0.298020\pi\)
0.592807 + 0.805344i \(0.298020\pi\)
\(350\) 3.36474 0.179853
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 9.41963 0.501356 0.250678 0.968070i \(-0.419346\pi\)
0.250678 + 0.968070i \(0.419346\pi\)
\(354\) 0 0
\(355\) −40.2982 −2.13881
\(356\) 2.14911 0.113903
\(357\) 0 0
\(358\) 20.9884 1.10927
\(359\) −18.4473 −0.973613 −0.486806 0.873510i \(-0.661838\pi\)
−0.486806 + 0.873510i \(0.661838\pi\)
\(360\) 0 0
\(361\) 33.6630 1.77173
\(362\) −1.10781 −0.0582254
\(363\) 0 0
\(364\) −0.364739 −0.0191175
\(365\) 3.20400 0.167705
\(366\) 0 0
\(367\) −18.6766 −0.974908 −0.487454 0.873149i \(-0.662074\pi\)
−0.487454 + 0.873149i \(0.662074\pi\)
\(368\) 8.14911 0.424802
\(369\) 0 0
\(370\) 10.9451 0.569009
\(371\) 11.7844 0.611814
\(372\) 0 0
\(373\) −19.2040 −0.994346 −0.497173 0.867652i \(-0.665629\pi\)
−0.497173 + 0.867652i \(0.665629\pi\)
\(374\) −2.89219 −0.149551
\(375\) 0 0
\(376\) −6.52745 −0.336627
\(377\) 0.862513 0.0444217
\(378\) 0 0
\(379\) −31.8670 −1.63690 −0.818448 0.574581i \(-0.805165\pi\)
−0.818448 + 0.574581i \(0.805165\pi\)
\(380\) −20.9884 −1.07668
\(381\) 0 0
\(382\) −10.6902 −0.546956
\(383\) 29.0413 1.48394 0.741970 0.670433i \(-0.233891\pi\)
0.741970 + 0.670433i \(0.233891\pi\)
\(384\) 0 0
\(385\) −2.89219 −0.147399
\(386\) −21.5687 −1.09782
\(387\) 0 0
\(388\) 6.72948 0.341638
\(389\) −26.5964 −1.34849 −0.674247 0.738506i \(-0.735532\pi\)
−0.674247 + 0.738506i \(0.735532\pi\)
\(390\) 0 0
\(391\) −23.5687 −1.19192
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −4.90578 −0.247150
\(395\) −29.3531 −1.47692
\(396\) 0 0
\(397\) −8.37834 −0.420497 −0.210248 0.977648i \(-0.567427\pi\)
−0.210248 + 0.977648i \(0.567427\pi\)
\(398\) −7.10781 −0.356283
\(399\) 0 0
\(400\) 3.36474 0.168237
\(401\) −7.93348 −0.396179 −0.198090 0.980184i \(-0.563474\pi\)
−0.198090 + 0.980184i \(0.563474\pi\)
\(402\) 0 0
\(403\) −3.89416 −0.193982
\(404\) −1.27052 −0.0632108
\(405\) 0 0
\(406\) −2.36474 −0.117360
\(407\) −3.78437 −0.187584
\(408\) 0 0
\(409\) −24.6766 −1.22018 −0.610089 0.792333i \(-0.708866\pi\)
−0.610089 + 0.792333i \(0.708866\pi\)
\(410\) −8.36474 −0.413105
\(411\) 0 0
\(412\) −0.892186 −0.0439548
\(413\) −10.5139 −0.517353
\(414\) 0 0
\(415\) 8.98838 0.441222
\(416\) −0.364739 −0.0178828
\(417\) 0 0
\(418\) 7.25693 0.354948
\(419\) −2.54104 −0.124138 −0.0620690 0.998072i \(-0.519770\pi\)
−0.0620690 + 0.998072i \(0.519770\pi\)
\(420\) 0 0
\(421\) 2.43126 0.118492 0.0592461 0.998243i \(-0.481130\pi\)
0.0592461 + 0.998243i \(0.481130\pi\)
\(422\) 8.87859 0.432203
\(423\) 0 0
\(424\) 11.7844 0.572300
\(425\) −9.73145 −0.472045
\(426\) 0 0
\(427\) −11.9335 −0.577502
\(428\) 5.78437 0.279598
\(429\) 0 0
\(430\) −33.0277 −1.59274
\(431\) 2.58037 0.124292 0.0621460 0.998067i \(-0.480206\pi\)
0.0621460 + 0.998067i \(0.480206\pi\)
\(432\) 0 0
\(433\) 37.1375 1.78471 0.892357 0.451331i \(-0.149050\pi\)
0.892357 + 0.451331i \(0.149050\pi\)
\(434\) 10.6766 0.512491
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 59.1375 2.82893
\(438\) 0 0
\(439\) 18.5139 0.883618 0.441809 0.897109i \(-0.354337\pi\)
0.441809 + 0.897109i \(0.354337\pi\)
\(440\) −2.89219 −0.137880
\(441\) 0 0
\(442\) 1.05489 0.0501762
\(443\) −21.4196 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(444\) 0 0
\(445\) −6.21563 −0.294649
\(446\) 22.2453 1.05335
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 4.06652 0.191911 0.0959554 0.995386i \(-0.469409\pi\)
0.0959554 + 0.995386i \(0.469409\pi\)
\(450\) 0 0
\(451\) 2.89219 0.136188
\(452\) −19.9335 −0.937592
\(453\) 0 0
\(454\) 8.46093 0.397091
\(455\) 1.05489 0.0494542
\(456\) 0 0
\(457\) −16.8393 −0.787708 −0.393854 0.919173i \(-0.628858\pi\)
−0.393854 + 0.919173i \(0.628858\pi\)
\(458\) −2.16271 −0.101057
\(459\) 0 0
\(460\) −23.5687 −1.09890
\(461\) −15.4590 −0.719995 −0.359998 0.932953i \(-0.617223\pi\)
−0.359998 + 0.932953i \(0.617223\pi\)
\(462\) 0 0
\(463\) 19.5687 0.909437 0.454718 0.890635i \(-0.349740\pi\)
0.454718 + 0.890635i \(0.349740\pi\)
\(464\) −2.36474 −0.109780
\(465\) 0 0
\(466\) −3.45896 −0.160233
\(467\) 16.2982 0.754192 0.377096 0.926174i \(-0.376923\pi\)
0.377096 + 0.926174i \(0.376923\pi\)
\(468\) 0 0
\(469\) −6.14911 −0.283940
\(470\) 18.8786 0.870804
\(471\) 0 0
\(472\) −10.5139 −0.483939
\(473\) 11.4196 0.525075
\(474\) 0 0
\(475\) 24.4177 1.12036
\(476\) −2.89219 −0.132563
\(477\) 0 0
\(478\) 16.2982 0.745463
\(479\) −0.431256 −0.0197046 −0.00985231 0.999951i \(-0.503136\pi\)
−0.00985231 + 0.999951i \(0.503136\pi\)
\(480\) 0 0
\(481\) 1.38031 0.0629367
\(482\) 4.67656 0.213011
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −19.4629 −0.883765
\(486\) 0 0
\(487\) −14.0826 −0.638143 −0.319072 0.947731i \(-0.603371\pi\)
−0.319072 + 0.947731i \(0.603371\pi\)
\(488\) −11.9335 −0.540203
\(489\) 0 0
\(490\) −2.89219 −0.130656
\(491\) −23.5687 −1.06364 −0.531821 0.846857i \(-0.678492\pi\)
−0.531821 + 0.846857i \(0.678492\pi\)
\(492\) 0 0
\(493\) 6.83927 0.308025
\(494\) −2.64688 −0.119089
\(495\) 0 0
\(496\) 10.6766 0.479392
\(497\) 13.9335 0.625002
\(498\) 0 0
\(499\) 28.9884 1.29770 0.648849 0.760917i \(-0.275251\pi\)
0.648849 + 0.760917i \(0.275251\pi\)
\(500\) 4.72948 0.211509
\(501\) 0 0
\(502\) −10.0826 −0.450008
\(503\) 25.0549 1.11714 0.558571 0.829457i \(-0.311350\pi\)
0.558571 + 0.829457i \(0.311350\pi\)
\(504\) 0 0
\(505\) 3.67458 0.163517
\(506\) 8.14911 0.362272
\(507\) 0 0
\(508\) 10.1491 0.450294
\(509\) 29.0020 1.28549 0.642745 0.766080i \(-0.277796\pi\)
0.642745 + 0.766080i \(0.277796\pi\)
\(510\) 0 0
\(511\) −1.10781 −0.0490068
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 9.41963 0.415482
\(515\) 2.58037 0.113705
\(516\) 0 0
\(517\) −6.52745 −0.287077
\(518\) −3.78437 −0.166276
\(519\) 0 0
\(520\) 1.05489 0.0462601
\(521\) 20.9884 0.919517 0.459759 0.888044i \(-0.347936\pi\)
0.459759 + 0.888044i \(0.347936\pi\)
\(522\) 0 0
\(523\) −15.8806 −0.694408 −0.347204 0.937790i \(-0.612869\pi\)
−0.347204 + 0.937790i \(0.612869\pi\)
\(524\) −20.8922 −0.912679
\(525\) 0 0
\(526\) 6.87859 0.299921
\(527\) −30.8786 −1.34509
\(528\) 0 0
\(529\) 43.4080 1.88730
\(530\) −34.0826 −1.48045
\(531\) 0 0
\(532\) 7.25693 0.314628
\(533\) −1.05489 −0.0456925
\(534\) 0 0
\(535\) −16.7295 −0.723278
\(536\) −6.14911 −0.265601
\(537\) 0 0
\(538\) 26.8922 1.15940
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −14.4745 −0.622308 −0.311154 0.950359i \(-0.600716\pi\)
−0.311154 + 0.950359i \(0.600716\pi\)
\(542\) −21.7844 −0.935719
\(543\) 0 0
\(544\) −2.89219 −0.124001
\(545\) −23.1375 −0.991101
\(546\) 0 0
\(547\) −29.0710 −1.24298 −0.621492 0.783420i \(-0.713473\pi\)
−0.621492 + 0.783420i \(0.713473\pi\)
\(548\) −7.93348 −0.338902
\(549\) 0 0
\(550\) 3.36474 0.143473
\(551\) −17.1607 −0.731072
\(552\) 0 0
\(553\) 10.1491 0.431584
\(554\) 3.27052 0.138951
\(555\) 0 0
\(556\) 18.8257 0.798386
\(557\) 37.5022 1.58902 0.794510 0.607251i \(-0.207728\pi\)
0.794510 + 0.607251i \(0.207728\pi\)
\(558\) 0 0
\(559\) −4.16519 −0.176169
\(560\) −2.89219 −0.122217
\(561\) 0 0
\(562\) 10.2982 0.434404
\(563\) −27.1078 −1.14246 −0.571229 0.820791i \(-0.693533\pi\)
−0.571229 + 0.820791i \(0.693533\pi\)
\(564\) 0 0
\(565\) 57.6513 2.42541
\(566\) −28.3118 −1.19003
\(567\) 0 0
\(568\) 13.9335 0.584636
\(569\) 46.2982 1.94092 0.970461 0.241257i \(-0.0775597\pi\)
0.970461 + 0.241257i \(0.0775597\pi\)
\(570\) 0 0
\(571\) 5.63526 0.235828 0.117914 0.993024i \(-0.462379\pi\)
0.117914 + 0.993024i \(0.462379\pi\)
\(572\) −0.364739 −0.0152505
\(573\) 0 0
\(574\) 2.89219 0.120718
\(575\) 27.4196 1.14348
\(576\) 0 0
\(577\) 0.513850 0.0213919 0.0106959 0.999943i \(-0.496595\pi\)
0.0106959 + 0.999943i \(0.496595\pi\)
\(578\) −8.63526 −0.359179
\(579\) 0 0
\(580\) 6.83927 0.283985
\(581\) −3.10781 −0.128934
\(582\) 0 0
\(583\) 11.7844 0.488059
\(584\) −1.10781 −0.0458417
\(585\) 0 0
\(586\) 11.7844 0.486808
\(587\) 17.7844 0.734040 0.367020 0.930213i \(-0.380378\pi\)
0.367020 + 0.930213i \(0.380378\pi\)
\(588\) 0 0
\(589\) 77.4790 3.19247
\(590\) 30.4080 1.25188
\(591\) 0 0
\(592\) −3.78437 −0.155537
\(593\) 26.0297 1.06891 0.534455 0.845197i \(-0.320517\pi\)
0.534455 + 0.845197i \(0.320517\pi\)
\(594\) 0 0
\(595\) 8.36474 0.342921
\(596\) −1.27052 −0.0520426
\(597\) 0 0
\(598\) −2.97230 −0.121546
\(599\) 25.9335 1.05961 0.529807 0.848118i \(-0.322264\pi\)
0.529807 + 0.848118i \(0.322264\pi\)
\(600\) 0 0
\(601\) −33.2730 −1.35723 −0.678617 0.734492i \(-0.737420\pi\)
−0.678617 + 0.734492i \(0.737420\pi\)
\(602\) 11.4196 0.465429
\(603\) 0 0
\(604\) −8.29822 −0.337650
\(605\) −2.89219 −0.117584
\(606\) 0 0
\(607\) −34.4080 −1.39658 −0.698289 0.715816i \(-0.746055\pi\)
−0.698289 + 0.715816i \(0.746055\pi\)
\(608\) 7.25693 0.294307
\(609\) 0 0
\(610\) 34.5139 1.39742
\(611\) 2.38082 0.0963175
\(612\) 0 0
\(613\) 7.33704 0.296340 0.148170 0.988962i \(-0.452662\pi\)
0.148170 + 0.988962i \(0.452662\pi\)
\(614\) 4.01360 0.161976
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −7.70178 −0.310062 −0.155031 0.987910i \(-0.549548\pi\)
−0.155031 + 0.987910i \(0.549548\pi\)
\(618\) 0 0
\(619\) −20.2982 −0.815854 −0.407927 0.913014i \(-0.633748\pi\)
−0.407927 + 0.913014i \(0.633748\pi\)
\(620\) −30.8786 −1.24011
\(621\) 0 0
\(622\) −6.52745 −0.261727
\(623\) 2.14911 0.0861023
\(624\) 0 0
\(625\) −30.5022 −1.22009
\(626\) 0.513850 0.0205376
\(627\) 0 0
\(628\) −18.8922 −0.753880
\(629\) 10.9451 0.436410
\(630\) 0 0
\(631\) 35.2433 1.40301 0.701507 0.712662i \(-0.252511\pi\)
0.701507 + 0.712662i \(0.252511\pi\)
\(632\) 10.1491 0.403710
\(633\) 0 0
\(634\) 7.48615 0.297313
\(635\) −29.3531 −1.16484
\(636\) 0 0
\(637\) −0.364739 −0.0144515
\(638\) −2.36474 −0.0936209
\(639\) 0 0
\(640\) −2.89219 −0.114324
\(641\) 25.7572 1.01735 0.508674 0.860959i \(-0.330136\pi\)
0.508674 + 0.860959i \(0.330136\pi\)
\(642\) 0 0
\(643\) −5.48615 −0.216353 −0.108176 0.994132i \(-0.534501\pi\)
−0.108176 + 0.994132i \(0.534501\pi\)
\(644\) 8.14911 0.321120
\(645\) 0 0
\(646\) −20.9884 −0.825777
\(647\) 17.4726 0.686917 0.343458 0.939168i \(-0.388402\pi\)
0.343458 + 0.939168i \(0.388402\pi\)
\(648\) 0 0
\(649\) −10.5139 −0.412705
\(650\) −1.22725 −0.0481368
\(651\) 0 0
\(652\) −20.2982 −0.794940
\(653\) −8.54104 −0.334237 −0.167118 0.985937i \(-0.553446\pi\)
−0.167118 + 0.985937i \(0.553446\pi\)
\(654\) 0 0
\(655\) 60.4241 2.36096
\(656\) 2.89219 0.112921
\(657\) 0 0
\(658\) −6.52745 −0.254466
\(659\) −43.5416 −1.69614 −0.848069 0.529886i \(-0.822235\pi\)
−0.848069 + 0.529886i \(0.822235\pi\)
\(660\) 0 0
\(661\) 18.8650 0.733763 0.366882 0.930268i \(-0.380425\pi\)
0.366882 + 0.930268i \(0.380425\pi\)
\(662\) −13.4196 −0.521569
\(663\) 0 0
\(664\) −3.10781 −0.120607
\(665\) −20.9884 −0.813894
\(666\) 0 0
\(667\) −19.2705 −0.746158
\(668\) 1.05489 0.0408151
\(669\) 0 0
\(670\) 17.7844 0.687070
\(671\) −11.9335 −0.460687
\(672\) 0 0
\(673\) −8.08259 −0.311561 −0.155781 0.987792i \(-0.549789\pi\)
−0.155781 + 0.987792i \(0.549789\pi\)
\(674\) −8.94511 −0.344553
\(675\) 0 0
\(676\) −12.8670 −0.494883
\(677\) −6.62364 −0.254567 −0.127284 0.991866i \(-0.540626\pi\)
−0.127284 + 0.991866i \(0.540626\pi\)
\(678\) 0 0
\(679\) 6.72948 0.258254
\(680\) 8.36474 0.320773
\(681\) 0 0
\(682\) 10.6766 0.408827
\(683\) 13.7179 0.524899 0.262450 0.964946i \(-0.415470\pi\)
0.262450 + 0.964946i \(0.415470\pi\)
\(684\) 0 0
\(685\) 22.9451 0.876687
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 11.4196 0.435369
\(689\) −4.29822 −0.163749
\(690\) 0 0
\(691\) −18.3808 −0.699239 −0.349620 0.936892i \(-0.613689\pi\)
−0.349620 + 0.936892i \(0.613689\pi\)
\(692\) −16.5139 −0.627762
\(693\) 0 0
\(694\) 5.78437 0.219572
\(695\) −54.4473 −2.06531
\(696\) 0 0
\(697\) −8.36474 −0.316837
\(698\) 22.1491 0.838356
\(699\) 0 0
\(700\) 3.36474 0.127175
\(701\) −31.3259 −1.18316 −0.591582 0.806245i \(-0.701496\pi\)
−0.591582 + 0.806245i \(0.701496\pi\)
\(702\) 0 0
\(703\) −27.4629 −1.03578
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 9.41963 0.354513
\(707\) −1.27052 −0.0477829
\(708\) 0 0
\(709\) −41.1103 −1.54393 −0.771965 0.635665i \(-0.780726\pi\)
−0.771965 + 0.635665i \(0.780726\pi\)
\(710\) −40.2982 −1.51237
\(711\) 0 0
\(712\) 2.14911 0.0805413
\(713\) 87.0045 3.25834
\(714\) 0 0
\(715\) 1.05489 0.0394508
\(716\) 20.9884 0.784372
\(717\) 0 0
\(718\) −18.4473 −0.688448
\(719\) 1.17433 0.0437952 0.0218976 0.999760i \(-0.493029\pi\)
0.0218976 + 0.999760i \(0.493029\pi\)
\(720\) 0 0
\(721\) −0.892186 −0.0332267
\(722\) 33.6630 1.25281
\(723\) 0 0
\(724\) −1.10781 −0.0411716
\(725\) −7.95673 −0.295506
\(726\) 0 0
\(727\) 43.2730 1.60491 0.802453 0.596715i \(-0.203528\pi\)
0.802453 + 0.596715i \(0.203528\pi\)
\(728\) −0.364739 −0.0135181
\(729\) 0 0
\(730\) 3.20400 0.118586
\(731\) −33.0277 −1.22157
\(732\) 0 0
\(733\) −14.9058 −0.550558 −0.275279 0.961364i \(-0.588770\pi\)
−0.275279 + 0.961364i \(0.588770\pi\)
\(734\) −18.6766 −0.689364
\(735\) 0 0
\(736\) 8.14911 0.300380
\(737\) −6.14911 −0.226505
\(738\) 0 0
\(739\) 24.9058 0.916174 0.458087 0.888907i \(-0.348535\pi\)
0.458087 + 0.888907i \(0.348535\pi\)
\(740\) 10.9451 0.402350
\(741\) 0 0
\(742\) 11.7844 0.432618
\(743\) −11.5687 −0.424416 −0.212208 0.977225i \(-0.568065\pi\)
−0.212208 + 0.977225i \(0.568065\pi\)
\(744\) 0 0
\(745\) 3.67458 0.134626
\(746\) −19.2040 −0.703109
\(747\) 0 0
\(748\) −2.89219 −0.105749
\(749\) 5.78437 0.211356
\(750\) 0 0
\(751\) 34.3808 1.25457 0.627287 0.778788i \(-0.284165\pi\)
0.627287 + 0.778788i \(0.284165\pi\)
\(752\) −6.52745 −0.238031
\(753\) 0 0
\(754\) 0.862513 0.0314109
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) 2.62364 0.0953577 0.0476789 0.998863i \(-0.484818\pi\)
0.0476789 + 0.998863i \(0.484818\pi\)
\(758\) −31.8670 −1.15746
\(759\) 0 0
\(760\) −20.9884 −0.761328
\(761\) −45.7315 −1.65776 −0.828882 0.559424i \(-0.811023\pi\)
−0.828882 + 0.559424i \(0.811023\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −10.6902 −0.386756
\(765\) 0 0
\(766\) 29.0413 1.04930
\(767\) 3.83481 0.138467
\(768\) 0 0
\(769\) 14.5668 0.525291 0.262646 0.964892i \(-0.415405\pi\)
0.262646 + 0.964892i \(0.415405\pi\)
\(770\) −2.89219 −0.104227
\(771\) 0 0
\(772\) −21.5687 −0.776276
\(773\) −38.0297 −1.36783 −0.683916 0.729561i \(-0.739725\pi\)
−0.683916 + 0.729561i \(0.739725\pi\)
\(774\) 0 0
\(775\) 35.9238 1.29042
\(776\) 6.72948 0.241574
\(777\) 0 0
\(778\) −26.5964 −0.953529
\(779\) 20.9884 0.751987
\(780\) 0 0
\(781\) 13.9335 0.498579
\(782\) −23.5687 −0.842817
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 54.6397 1.95017
\(786\) 0 0
\(787\) −26.2020 −0.934002 −0.467001 0.884257i \(-0.654665\pi\)
−0.467001 + 0.884257i \(0.654665\pi\)
\(788\) −4.90578 −0.174761
\(789\) 0 0
\(790\) −29.3531 −1.04434
\(791\) −19.9335 −0.708753
\(792\) 0 0
\(793\) 4.35261 0.154566
\(794\) −8.37834 −0.297336
\(795\) 0 0
\(796\) −7.10781 −0.251930
\(797\) 13.8373 0.490142 0.245071 0.969505i \(-0.421189\pi\)
0.245071 + 0.969505i \(0.421189\pi\)
\(798\) 0 0
\(799\) 18.8786 0.667876
\(800\) 3.36474 0.118961
\(801\) 0 0
\(802\) −7.93348 −0.280141
\(803\) −1.10781 −0.0390939
\(804\) 0 0
\(805\) −23.5687 −0.830689
\(806\) −3.89416 −0.137166
\(807\) 0 0
\(808\) −1.27052 −0.0446968
\(809\) −28.0826 −0.987331 −0.493666 0.869652i \(-0.664343\pi\)
−0.493666 + 0.869652i \(0.664343\pi\)
\(810\) 0 0
\(811\) 56.1516 1.97175 0.985875 0.167486i \(-0.0535648\pi\)
0.985875 + 0.167486i \(0.0535648\pi\)
\(812\) −2.36474 −0.0829861
\(813\) 0 0
\(814\) −3.78437 −0.132642
\(815\) 58.7062 2.05639
\(816\) 0 0
\(817\) 82.8714 2.89930
\(818\) −24.6766 −0.862796
\(819\) 0 0
\(820\) −8.36474 −0.292109
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 35.2433 1.22851 0.614253 0.789109i \(-0.289458\pi\)
0.614253 + 0.789109i \(0.289458\pi\)
\(824\) −0.892186 −0.0310808
\(825\) 0 0
\(826\) −10.5139 −0.365824
\(827\) −0.431256 −0.0149963 −0.00749813 0.999972i \(-0.502387\pi\)
−0.00749813 + 0.999972i \(0.502387\pi\)
\(828\) 0 0
\(829\) −2.86499 −0.0995053 −0.0497527 0.998762i \(-0.515843\pi\)
−0.0497527 + 0.998762i \(0.515843\pi\)
\(830\) 8.98838 0.311991
\(831\) 0 0
\(832\) −0.364739 −0.0126451
\(833\) −2.89219 −0.100208
\(834\) 0 0
\(835\) −3.05095 −0.105582
\(836\) 7.25693 0.250986
\(837\) 0 0
\(838\) −2.54104 −0.0877789
\(839\) 1.17433 0.0405424 0.0202712 0.999795i \(-0.493547\pi\)
0.0202712 + 0.999795i \(0.493547\pi\)
\(840\) 0 0
\(841\) −23.4080 −0.807173
\(842\) 2.43126 0.0837866
\(843\) 0 0
\(844\) 8.87859 0.305614
\(845\) 37.2137 1.28019
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 11.7844 0.404677
\(849\) 0 0
\(850\) −9.73145 −0.333786
\(851\) −30.8393 −1.05716
\(852\) 0 0
\(853\) −51.8004 −1.77361 −0.886807 0.462140i \(-0.847082\pi\)
−0.886807 + 0.462140i \(0.847082\pi\)
\(854\) −11.9335 −0.408355
\(855\) 0 0
\(856\) 5.78437 0.197706
\(857\) 12.7824 0.436638 0.218319 0.975877i \(-0.429943\pi\)
0.218319 + 0.975877i \(0.429943\pi\)
\(858\) 0 0
\(859\) 16.7567 0.571730 0.285865 0.958270i \(-0.407719\pi\)
0.285865 + 0.958270i \(0.407719\pi\)
\(860\) −33.0277 −1.12624
\(861\) 0 0
\(862\) 2.58037 0.0878877
\(863\) 26.5571 0.904015 0.452007 0.892014i \(-0.350708\pi\)
0.452007 + 0.892014i \(0.350708\pi\)
\(864\) 0 0
\(865\) 47.7611 1.62393
\(866\) 37.1375 1.26198
\(867\) 0 0
\(868\) 10.6766 0.362386
\(869\) 10.1491 0.344285
\(870\) 0 0
\(871\) 2.24282 0.0759951
\(872\) 8.00000 0.270914
\(873\) 0 0
\(874\) 59.1375 2.00036
\(875\) 4.72948 0.159886
\(876\) 0 0
\(877\) −15.5687 −0.525719 −0.262860 0.964834i \(-0.584666\pi\)
−0.262860 + 0.964834i \(0.584666\pi\)
\(878\) 18.5139 0.624812
\(879\) 0 0
\(880\) −2.89219 −0.0974956
\(881\) −45.6907 −1.53936 −0.769679 0.638431i \(-0.779584\pi\)
−0.769679 + 0.638431i \(0.779584\pi\)
\(882\) 0 0
\(883\) −55.0045 −1.85105 −0.925524 0.378690i \(-0.876375\pi\)
−0.925524 + 0.378690i \(0.876375\pi\)
\(884\) 1.05489 0.0354799
\(885\) 0 0
\(886\) −21.4196 −0.719607
\(887\) 30.8393 1.03548 0.517741 0.855538i \(-0.326773\pi\)
0.517741 + 0.855538i \(0.326773\pi\)
\(888\) 0 0
\(889\) 10.1491 0.340390
\(890\) −6.21563 −0.208348
\(891\) 0 0
\(892\) 22.2453 0.744828
\(893\) −47.3692 −1.58515
\(894\) 0 0
\(895\) −60.7023 −2.02905
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 4.06652 0.135701
\(899\) −25.2473 −0.842044
\(900\) 0 0
\(901\) −34.0826 −1.13546
\(902\) 2.89219 0.0962993
\(903\) 0 0
\(904\) −19.9335 −0.662978
\(905\) 3.20400 0.106505
\(906\) 0 0
\(907\) −25.4196 −0.844045 −0.422023 0.906585i \(-0.638680\pi\)
−0.422023 + 0.906585i \(0.638680\pi\)
\(908\) 8.46093 0.280786
\(909\) 0 0
\(910\) 1.05489 0.0349694
\(911\) 50.5571 1.67503 0.837516 0.546413i \(-0.184007\pi\)
0.837516 + 0.546413i \(0.184007\pi\)
\(912\) 0 0
\(913\) −3.10781 −0.102854
\(914\) −16.8393 −0.556993
\(915\) 0 0
\(916\) −2.16271 −0.0714579
\(917\) −20.8922 −0.689921
\(918\) 0 0
\(919\) 12.7295 0.419907 0.209953 0.977711i \(-0.432669\pi\)
0.209953 + 0.977711i \(0.432669\pi\)
\(920\) −23.5687 −0.777038
\(921\) 0 0
\(922\) −15.4590 −0.509114
\(923\) −5.08209 −0.167279
\(924\) 0 0
\(925\) −12.7334 −0.418672
\(926\) 19.5687 0.643069
\(927\) 0 0
\(928\) −2.36474 −0.0776264
\(929\) −6.44733 −0.211530 −0.105765 0.994391i \(-0.533729\pi\)
−0.105765 + 0.994391i \(0.533729\pi\)
\(930\) 0 0
\(931\) 7.25693 0.237836
\(932\) −3.45896 −0.113302
\(933\) 0 0
\(934\) 16.2982 0.533294
\(935\) 8.36474 0.273556
\(936\) 0 0
\(937\) 39.1904 1.28029 0.640147 0.768252i \(-0.278873\pi\)
0.640147 + 0.768252i \(0.278873\pi\)
\(938\) −6.14911 −0.200776
\(939\) 0 0
\(940\) 18.8786 0.615752
\(941\) −37.1103 −1.20976 −0.604881 0.796316i \(-0.706779\pi\)
−0.604881 + 0.796316i \(0.706779\pi\)
\(942\) 0 0
\(943\) 23.5687 0.767504
\(944\) −10.5139 −0.342197
\(945\) 0 0
\(946\) 11.4196 0.371284
\(947\) −49.3259 −1.60288 −0.801439 0.598077i \(-0.795932\pi\)
−0.801439 + 0.598077i \(0.795932\pi\)
\(948\) 0 0
\(949\) 0.404063 0.0131164
\(950\) 24.4177 0.792213
\(951\) 0 0
\(952\) −2.89219 −0.0937363
\(953\) 24.2156 0.784421 0.392211 0.919875i \(-0.371710\pi\)
0.392211 + 0.919875i \(0.371710\pi\)
\(954\) 0 0
\(955\) 30.9179 1.00048
\(956\) 16.2982 0.527122
\(957\) 0 0
\(958\) −0.431256 −0.0139333
\(959\) −7.93348 −0.256186
\(960\) 0 0
\(961\) 82.9889 2.67706
\(962\) 1.38031 0.0445029
\(963\) 0 0
\(964\) 4.67656 0.150622
\(965\) 62.3808 2.00811
\(966\) 0 0
\(967\) −43.8670 −1.41067 −0.705333 0.708876i \(-0.749203\pi\)
−0.705333 + 0.708876i \(0.749203\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −19.4629 −0.624916
\(971\) 28.2982 0.908133 0.454067 0.890968i \(-0.349973\pi\)
0.454067 + 0.890968i \(0.349973\pi\)
\(972\) 0 0
\(973\) 18.8257 0.603523
\(974\) −14.0826 −0.451235
\(975\) 0 0
\(976\) −11.9335 −0.381981
\(977\) −28.2982 −0.905340 −0.452670 0.891678i \(-0.649528\pi\)
−0.452670 + 0.891678i \(0.649528\pi\)
\(978\) 0 0
\(979\) 2.14911 0.0686859
\(980\) −2.89219 −0.0923875
\(981\) 0 0
\(982\) −23.5687 −0.752109
\(983\) −32.6372 −1.04097 −0.520483 0.853872i \(-0.674248\pi\)
−0.520483 + 0.853872i \(0.674248\pi\)
\(984\) 0 0
\(985\) 14.1884 0.452081
\(986\) 6.83927 0.217807
\(987\) 0 0
\(988\) −2.64688 −0.0842086
\(989\) 93.0599 2.95913
\(990\) 0 0
\(991\) −31.8670 −1.01229 −0.506144 0.862449i \(-0.668929\pi\)
−0.506144 + 0.862449i \(0.668929\pi\)
\(992\) 10.6766 0.338981
\(993\) 0 0
\(994\) 13.9335 0.441943
\(995\) 20.5571 0.651705
\(996\) 0 0
\(997\) −37.4196 −1.18509 −0.592546 0.805537i \(-0.701877\pi\)
−0.592546 + 0.805537i \(0.701877\pi\)
\(998\) 28.9884 0.917611
\(999\) 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 1386.2.a.r.1.1 yes 3
3.2 odd 2 1386.2.a.q.1.3 3
7.6 odd 2 9702.2.a.dy.1.3 3
21.20 even 2 9702.2.a.dx.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.a.q.1.3 3 3.2 odd 2
1386.2.a.r.1.1 yes 3 1.1 even 1 trivial
9702.2.a.dx.1.1 3 21.20 even 2
9702.2.a.dy.1.3 3 7.6 odd 2