Properties

Label 3344.2.a.k.1.2
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{3} +2.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} +2.00000 q^{5} -3.56155 q^{7} -0.561553 q^{9} -1.00000 q^{11} -3.56155 q^{13} +3.12311 q^{15} +3.56155 q^{17} +1.00000 q^{19} -5.56155 q^{21} -5.56155 q^{23} -1.00000 q^{25} -5.56155 q^{27} +6.68466 q^{29} -2.00000 q^{31} -1.56155 q^{33} -7.12311 q^{35} +3.12311 q^{37} -5.56155 q^{39} +2.00000 q^{41} -1.12311 q^{45} -8.00000 q^{47} +5.68466 q^{49} +5.56155 q^{51} -7.80776 q^{53} -2.00000 q^{55} +1.56155 q^{57} -4.68466 q^{59} -10.2462 q^{61} +2.00000 q^{63} -7.12311 q^{65} -4.68466 q^{67} -8.68466 q^{69} -6.00000 q^{71} +2.68466 q^{73} -1.56155 q^{75} +3.56155 q^{77} +4.00000 q^{79} -7.00000 q^{81} +2.24621 q^{83} +7.12311 q^{85} +10.4384 q^{87} -9.12311 q^{89} +12.6847 q^{91} -3.12311 q^{93} +2.00000 q^{95} +1.12311 q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{13} - 2 q^{15} + 3 q^{17} + 2 q^{19} - 7 q^{21} - 7 q^{23} - 2 q^{25} - 7 q^{27} + q^{29} - 4 q^{31} + q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 4 q^{41} + 6 q^{45} - 16 q^{47} - q^{49} + 7 q^{51} + 5 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} - 4 q^{61} + 4 q^{63} - 6 q^{65} + 3 q^{67} - 5 q^{69} - 12 q^{71} - 7 q^{73} + q^{75} + 3 q^{77} + 8 q^{79} - 14 q^{81} - 12 q^{83} + 6 q^{85} + 25 q^{87} - 10 q^{89} + 13 q^{91} + 2 q^{93} + 4 q^{95} - 6 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −3.56155 −1.34614 −0.673070 0.739579i \(-0.735025\pi\)
−0.673070 + 0.739579i \(0.735025\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) 3.12311 0.806382
\(16\) 0 0
\(17\) 3.56155 0.863803 0.431902 0.901921i \(-0.357843\pi\)
0.431902 + 0.901921i \(0.357843\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −5.56155 −1.21363
\(22\) 0 0
\(23\) −5.56155 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −1.56155 −0.271831
\(34\) 0 0
\(35\) −7.12311 −1.20402
\(36\) 0 0
\(37\) 3.12311 0.513435 0.256718 0.966486i \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(38\) 0 0
\(39\) −5.56155 −0.890561
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −1.12311 −0.167423
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) 0 0
\(51\) 5.56155 0.778773
\(52\) 0 0
\(53\) −7.80776 −1.07248 −0.536239 0.844066i \(-0.680156\pi\)
−0.536239 + 0.844066i \(0.680156\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 1.56155 0.206833
\(58\) 0 0
\(59\) −4.68466 −0.609891 −0.304945 0.952370i \(-0.598638\pi\)
−0.304945 + 0.952370i \(0.598638\pi\)
\(60\) 0 0
\(61\) −10.2462 −1.31189 −0.655946 0.754807i \(-0.727730\pi\)
−0.655946 + 0.754807i \(0.727730\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −7.12311 −0.883513
\(66\) 0 0
\(67\) −4.68466 −0.572322 −0.286161 0.958182i \(-0.592379\pi\)
−0.286161 + 0.958182i \(0.592379\pi\)
\(68\) 0 0
\(69\) −8.68466 −1.04551
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 2.68466 0.314216 0.157108 0.987581i \(-0.449783\pi\)
0.157108 + 0.987581i \(0.449783\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 3.56155 0.405877
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 2.24621 0.246554 0.123277 0.992372i \(-0.460660\pi\)
0.123277 + 0.992372i \(0.460660\pi\)
\(84\) 0 0
\(85\) 7.12311 0.772609
\(86\) 0 0
\(87\) 10.4384 1.11912
\(88\) 0 0
\(89\) −9.12311 −0.967047 −0.483524 0.875331i \(-0.660643\pi\)
−0.483524 + 0.875331i \(0.660643\pi\)
\(90\) 0 0
\(91\) 12.6847 1.32971
\(92\) 0 0
\(93\) −3.12311 −0.323851
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 1.12311 0.114034 0.0570170 0.998373i \(-0.481841\pi\)
0.0570170 + 0.998373i \(0.481841\pi\)
\(98\) 0 0
\(99\) 0.561553 0.0564382
\(100\) 0 0
\(101\) −15.1231 −1.50481 −0.752403 0.658703i \(-0.771105\pi\)
−0.752403 + 0.658703i \(0.771105\pi\)
\(102\) 0 0
\(103\) −11.3693 −1.12025 −0.560126 0.828407i \(-0.689247\pi\)
−0.560126 + 0.828407i \(0.689247\pi\)
\(104\) 0 0
\(105\) −11.1231 −1.08550
\(106\) 0 0
\(107\) −18.9309 −1.83012 −0.915058 0.403322i \(-0.867855\pi\)
−0.915058 + 0.403322i \(0.867855\pi\)
\(108\) 0 0
\(109\) −18.6847 −1.78967 −0.894833 0.446401i \(-0.852705\pi\)
−0.894833 + 0.446401i \(0.852705\pi\)
\(110\) 0 0
\(111\) 4.87689 0.462894
\(112\) 0 0
\(113\) 5.12311 0.481941 0.240971 0.970532i \(-0.422534\pi\)
0.240971 + 0.970532i \(0.422534\pi\)
\(114\) 0 0
\(115\) −11.1231 −1.03723
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −12.6847 −1.16280
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 3.12311 0.281601
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 14.2462 1.26415 0.632073 0.774909i \(-0.282204\pi\)
0.632073 + 0.774909i \(0.282204\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.12311 0.622349 0.311174 0.950353i \(-0.399278\pi\)
0.311174 + 0.950353i \(0.399278\pi\)
\(132\) 0 0
\(133\) −3.56155 −0.308826
\(134\) 0 0
\(135\) −11.1231 −0.957325
\(136\) 0 0
\(137\) 8.43845 0.720945 0.360473 0.932770i \(-0.382615\pi\)
0.360473 + 0.932770i \(0.382615\pi\)
\(138\) 0 0
\(139\) 17.3693 1.47325 0.736623 0.676303i \(-0.236419\pi\)
0.736623 + 0.676303i \(0.236419\pi\)
\(140\) 0 0
\(141\) −12.4924 −1.05205
\(142\) 0 0
\(143\) 3.56155 0.297832
\(144\) 0 0
\(145\) 13.3693 1.11026
\(146\) 0 0
\(147\) 8.87689 0.732154
\(148\) 0 0
\(149\) −15.1231 −1.23893 −0.619467 0.785023i \(-0.712651\pi\)
−0.619467 + 0.785023i \(0.712651\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 18.4924 1.47586 0.737928 0.674879i \(-0.235804\pi\)
0.737928 + 0.674879i \(0.235804\pi\)
\(158\) 0 0
\(159\) −12.1922 −0.966907
\(160\) 0 0
\(161\) 19.8078 1.56107
\(162\) 0 0
\(163\) 13.3693 1.04717 0.523583 0.851975i \(-0.324595\pi\)
0.523583 + 0.851975i \(0.324595\pi\)
\(164\) 0 0
\(165\) −3.12311 −0.243133
\(166\) 0 0
\(167\) 25.3693 1.96314 0.981568 0.191111i \(-0.0612092\pi\)
0.981568 + 0.191111i \(0.0612092\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 0 0
\(171\) −0.561553 −0.0429430
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 3.56155 0.269228
\(176\) 0 0
\(177\) −7.31534 −0.549855
\(178\) 0 0
\(179\) −22.2462 −1.66276 −0.831380 0.555704i \(-0.812449\pi\)
−0.831380 + 0.555704i \(0.812449\pi\)
\(180\) 0 0
\(181\) −19.1231 −1.42141 −0.710705 0.703491i \(-0.751624\pi\)
−0.710705 + 0.703491i \(0.751624\pi\)
\(182\) 0 0
\(183\) −16.0000 −1.18275
\(184\) 0 0
\(185\) 6.24621 0.459231
\(186\) 0 0
\(187\) −3.56155 −0.260447
\(188\) 0 0
\(189\) 19.8078 1.44080
\(190\) 0 0
\(191\) 20.6847 1.49669 0.748345 0.663310i \(-0.230849\pi\)
0.748345 + 0.663310i \(0.230849\pi\)
\(192\) 0 0
\(193\) −1.12311 −0.0808429 −0.0404215 0.999183i \(-0.512870\pi\)
−0.0404215 + 0.999183i \(0.512870\pi\)
\(194\) 0 0
\(195\) −11.1231 −0.796542
\(196\) 0 0
\(197\) 1.75379 0.124952 0.0624761 0.998046i \(-0.480100\pi\)
0.0624761 + 0.998046i \(0.480100\pi\)
\(198\) 0 0
\(199\) 0.684658 0.0485341 0.0242671 0.999706i \(-0.492275\pi\)
0.0242671 + 0.999706i \(0.492275\pi\)
\(200\) 0 0
\(201\) −7.31534 −0.515984
\(202\) 0 0
\(203\) −23.8078 −1.67098
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) 3.12311 0.217071
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 12.6847 0.873248 0.436624 0.899644i \(-0.356174\pi\)
0.436624 + 0.899644i \(0.356174\pi\)
\(212\) 0 0
\(213\) −9.36932 −0.641975
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.12311 0.483548
\(218\) 0 0
\(219\) 4.19224 0.283285
\(220\) 0 0
\(221\) −12.6847 −0.853262
\(222\) 0 0
\(223\) 24.7386 1.65662 0.828311 0.560269i \(-0.189302\pi\)
0.828311 + 0.560269i \(0.189302\pi\)
\(224\) 0 0
\(225\) 0.561553 0.0374369
\(226\) 0 0
\(227\) 17.1771 1.14008 0.570041 0.821616i \(-0.306927\pi\)
0.570041 + 0.821616i \(0.306927\pi\)
\(228\) 0 0
\(229\) 21.1231 1.39585 0.697927 0.716169i \(-0.254106\pi\)
0.697927 + 0.716169i \(0.254106\pi\)
\(230\) 0 0
\(231\) 5.56155 0.365923
\(232\) 0 0
\(233\) 20.7386 1.35863 0.679317 0.733845i \(-0.262276\pi\)
0.679317 + 0.733845i \(0.262276\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) 6.24621 0.405735
\(238\) 0 0
\(239\) 4.93087 0.318951 0.159476 0.987202i \(-0.449020\pi\)
0.159476 + 0.987202i \(0.449020\pi\)
\(240\) 0 0
\(241\) 21.6155 1.39238 0.696189 0.717858i \(-0.254877\pi\)
0.696189 + 0.717858i \(0.254877\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 11.3693 0.726359
\(246\) 0 0
\(247\) −3.56155 −0.226616
\(248\) 0 0
\(249\) 3.50758 0.222284
\(250\) 0 0
\(251\) −8.87689 −0.560305 −0.280152 0.959956i \(-0.590385\pi\)
−0.280152 + 0.959956i \(0.590385\pi\)
\(252\) 0 0
\(253\) 5.56155 0.349652
\(254\) 0 0
\(255\) 11.1231 0.696556
\(256\) 0 0
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) −11.1231 −0.691156
\(260\) 0 0
\(261\) −3.75379 −0.232354
\(262\) 0 0
\(263\) −17.1231 −1.05586 −0.527928 0.849289i \(-0.677031\pi\)
−0.527928 + 0.849289i \(0.677031\pi\)
\(264\) 0 0
\(265\) −15.6155 −0.959254
\(266\) 0 0
\(267\) −14.2462 −0.871854
\(268\) 0 0
\(269\) 11.1231 0.678188 0.339094 0.940753i \(-0.389880\pi\)
0.339094 + 0.940753i \(0.389880\pi\)
\(270\) 0 0
\(271\) −13.3153 −0.808849 −0.404425 0.914571i \(-0.632528\pi\)
−0.404425 + 0.914571i \(0.632528\pi\)
\(272\) 0 0
\(273\) 19.8078 1.19882
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −24.4924 −1.47161 −0.735804 0.677195i \(-0.763195\pi\)
−0.735804 + 0.677195i \(0.763195\pi\)
\(278\) 0 0
\(279\) 1.12311 0.0672386
\(280\) 0 0
\(281\) −0.630683 −0.0376234 −0.0188117 0.999823i \(-0.505988\pi\)
−0.0188117 + 0.999823i \(0.505988\pi\)
\(282\) 0 0
\(283\) 13.3693 0.794723 0.397362 0.917662i \(-0.369926\pi\)
0.397362 + 0.917662i \(0.369926\pi\)
\(284\) 0 0
\(285\) 3.12311 0.184997
\(286\) 0 0
\(287\) −7.12311 −0.420464
\(288\) 0 0
\(289\) −4.31534 −0.253844
\(290\) 0 0
\(291\) 1.75379 0.102809
\(292\) 0 0
\(293\) −24.9309 −1.45648 −0.728238 0.685324i \(-0.759661\pi\)
−0.728238 + 0.685324i \(0.759661\pi\)
\(294\) 0 0
\(295\) −9.36932 −0.545503
\(296\) 0 0
\(297\) 5.56155 0.322714
\(298\) 0 0
\(299\) 19.8078 1.14551
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −23.6155 −1.35668
\(304\) 0 0
\(305\) −20.4924 −1.17339
\(306\) 0 0
\(307\) 5.75379 0.328386 0.164193 0.986428i \(-0.447498\pi\)
0.164193 + 0.986428i \(0.447498\pi\)
\(308\) 0 0
\(309\) −17.7538 −1.00998
\(310\) 0 0
\(311\) −15.8078 −0.896376 −0.448188 0.893939i \(-0.647930\pi\)
−0.448188 + 0.893939i \(0.647930\pi\)
\(312\) 0 0
\(313\) 3.56155 0.201311 0.100655 0.994921i \(-0.467906\pi\)
0.100655 + 0.994921i \(0.467906\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 30.0540 1.68800 0.844000 0.536344i \(-0.180195\pi\)
0.844000 + 0.536344i \(0.180195\pi\)
\(318\) 0 0
\(319\) −6.68466 −0.374269
\(320\) 0 0
\(321\) −29.5616 −1.64996
\(322\) 0 0
\(323\) 3.56155 0.198170
\(324\) 0 0
\(325\) 3.56155 0.197559
\(326\) 0 0
\(327\) −29.1771 −1.61350
\(328\) 0 0
\(329\) 28.4924 1.57084
\(330\) 0 0
\(331\) 10.4384 0.573749 0.286874 0.957968i \(-0.407384\pi\)
0.286874 + 0.957968i \(0.407384\pi\)
\(332\) 0 0
\(333\) −1.75379 −0.0961070
\(334\) 0 0
\(335\) −9.36932 −0.511900
\(336\) 0 0
\(337\) 2.87689 0.156714 0.0783572 0.996925i \(-0.475033\pi\)
0.0783572 + 0.996925i \(0.475033\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 4.68466 0.252948
\(344\) 0 0
\(345\) −17.3693 −0.935133
\(346\) 0 0
\(347\) 31.6155 1.69721 0.848605 0.529027i \(-0.177443\pi\)
0.848605 + 0.529027i \(0.177443\pi\)
\(348\) 0 0
\(349\) −19.6155 −1.05000 −0.524998 0.851104i \(-0.675934\pi\)
−0.524998 + 0.851104i \(0.675934\pi\)
\(350\) 0 0
\(351\) 19.8078 1.05726
\(352\) 0 0
\(353\) −23.1771 −1.23359 −0.616796 0.787123i \(-0.711570\pi\)
−0.616796 + 0.787123i \(0.711570\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) −19.8078 −1.04834
\(358\) 0 0
\(359\) −12.9309 −0.682465 −0.341233 0.939979i \(-0.610844\pi\)
−0.341233 + 0.939979i \(0.610844\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.56155 0.0819603
\(364\) 0 0
\(365\) 5.36932 0.281043
\(366\) 0 0
\(367\) −13.7538 −0.717942 −0.358971 0.933349i \(-0.616872\pi\)
−0.358971 + 0.933349i \(0.616872\pi\)
\(368\) 0 0
\(369\) −1.12311 −0.0584665
\(370\) 0 0
\(371\) 27.8078 1.44371
\(372\) 0 0
\(373\) 7.56155 0.391522 0.195761 0.980652i \(-0.437282\pi\)
0.195761 + 0.980652i \(0.437282\pi\)
\(374\) 0 0
\(375\) −18.7386 −0.967659
\(376\) 0 0
\(377\) −23.8078 −1.22616
\(378\) 0 0
\(379\) −2.43845 −0.125255 −0.0626273 0.998037i \(-0.519948\pi\)
−0.0626273 + 0.998037i \(0.519948\pi\)
\(380\) 0 0
\(381\) 22.2462 1.13971
\(382\) 0 0
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) 7.12311 0.363027
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −19.8078 −1.00172
\(392\) 0 0
\(393\) 11.1231 0.561086
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −26.9848 −1.35433 −0.677165 0.735831i \(-0.736792\pi\)
−0.677165 + 0.735831i \(0.736792\pi\)
\(398\) 0 0
\(399\) −5.56155 −0.278426
\(400\) 0 0
\(401\) −24.2462 −1.21080 −0.605399 0.795922i \(-0.706986\pi\)
−0.605399 + 0.795922i \(0.706986\pi\)
\(402\) 0 0
\(403\) 7.12311 0.354827
\(404\) 0 0
\(405\) −14.0000 −0.695666
\(406\) 0 0
\(407\) −3.12311 −0.154807
\(408\) 0 0
\(409\) −0.246211 −0.0121744 −0.00608718 0.999981i \(-0.501938\pi\)
−0.00608718 + 0.999981i \(0.501938\pi\)
\(410\) 0 0
\(411\) 13.1771 0.649977
\(412\) 0 0
\(413\) 16.6847 0.820998
\(414\) 0 0
\(415\) 4.49242 0.220524
\(416\) 0 0
\(417\) 27.1231 1.32822
\(418\) 0 0
\(419\) 23.1231 1.12964 0.564819 0.825215i \(-0.308946\pi\)
0.564819 + 0.825215i \(0.308946\pi\)
\(420\) 0 0
\(421\) −1.06913 −0.0521062 −0.0260531 0.999661i \(-0.508294\pi\)
−0.0260531 + 0.999661i \(0.508294\pi\)
\(422\) 0 0
\(423\) 4.49242 0.218429
\(424\) 0 0
\(425\) −3.56155 −0.172761
\(426\) 0 0
\(427\) 36.4924 1.76599
\(428\) 0 0
\(429\) 5.56155 0.268514
\(430\) 0 0
\(431\) 25.8617 1.24572 0.622858 0.782335i \(-0.285971\pi\)
0.622858 + 0.782335i \(0.285971\pi\)
\(432\) 0 0
\(433\) 31.3693 1.50751 0.753757 0.657154i \(-0.228240\pi\)
0.753757 + 0.657154i \(0.228240\pi\)
\(434\) 0 0
\(435\) 20.8769 1.00097
\(436\) 0 0
\(437\) −5.56155 −0.266045
\(438\) 0 0
\(439\) −31.6155 −1.50893 −0.754463 0.656342i \(-0.772103\pi\)
−0.754463 + 0.656342i \(0.772103\pi\)
\(440\) 0 0
\(441\) −3.19224 −0.152011
\(442\) 0 0
\(443\) 17.8617 0.848637 0.424318 0.905513i \(-0.360514\pi\)
0.424318 + 0.905513i \(0.360514\pi\)
\(444\) 0 0
\(445\) −18.2462 −0.864953
\(446\) 0 0
\(447\) −23.6155 −1.11698
\(448\) 0 0
\(449\) −17.6155 −0.831328 −0.415664 0.909518i \(-0.636451\pi\)
−0.415664 + 0.909518i \(0.636451\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) −6.24621 −0.293473
\(454\) 0 0
\(455\) 25.3693 1.18933
\(456\) 0 0
\(457\) 32.4384 1.51741 0.758703 0.651436i \(-0.225833\pi\)
0.758703 + 0.651436i \(0.225833\pi\)
\(458\) 0 0
\(459\) −19.8078 −0.924547
\(460\) 0 0
\(461\) −22.7386 −1.05904 −0.529522 0.848296i \(-0.677629\pi\)
−0.529522 + 0.848296i \(0.677629\pi\)
\(462\) 0 0
\(463\) −36.4924 −1.69595 −0.847973 0.530039i \(-0.822177\pi\)
−0.847973 + 0.530039i \(0.822177\pi\)
\(464\) 0 0
\(465\) −6.24621 −0.289661
\(466\) 0 0
\(467\) −26.2462 −1.21453 −0.607265 0.794499i \(-0.707733\pi\)
−0.607265 + 0.794499i \(0.707733\pi\)
\(468\) 0 0
\(469\) 16.6847 0.770426
\(470\) 0 0
\(471\) 28.8769 1.33058
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 4.38447 0.200751
\(478\) 0 0
\(479\) −11.3693 −0.519477 −0.259739 0.965679i \(-0.583636\pi\)
−0.259739 + 0.965679i \(0.583636\pi\)
\(480\) 0 0
\(481\) −11.1231 −0.507170
\(482\) 0 0
\(483\) 30.9309 1.40740
\(484\) 0 0
\(485\) 2.24621 0.101995
\(486\) 0 0
\(487\) −22.8769 −1.03665 −0.518326 0.855183i \(-0.673444\pi\)
−0.518326 + 0.855183i \(0.673444\pi\)
\(488\) 0 0
\(489\) 20.8769 0.944086
\(490\) 0 0
\(491\) 14.6307 0.660273 0.330137 0.943933i \(-0.392905\pi\)
0.330137 + 0.943933i \(0.392905\pi\)
\(492\) 0 0
\(493\) 23.8078 1.07225
\(494\) 0 0
\(495\) 1.12311 0.0504798
\(496\) 0 0
\(497\) 21.3693 0.958545
\(498\) 0 0
\(499\) −26.2462 −1.17494 −0.587471 0.809245i \(-0.699876\pi\)
−0.587471 + 0.809245i \(0.699876\pi\)
\(500\) 0 0
\(501\) 39.6155 1.76989
\(502\) 0 0
\(503\) −9.31534 −0.415351 −0.207675 0.978198i \(-0.566590\pi\)
−0.207675 + 0.978198i \(0.566590\pi\)
\(504\) 0 0
\(505\) −30.2462 −1.34594
\(506\) 0 0
\(507\) −0.492423 −0.0218693
\(508\) 0 0
\(509\) −6.63068 −0.293900 −0.146950 0.989144i \(-0.546946\pi\)
−0.146950 + 0.989144i \(0.546946\pi\)
\(510\) 0 0
\(511\) −9.56155 −0.422978
\(512\) 0 0
\(513\) −5.56155 −0.245549
\(514\) 0 0
\(515\) −22.7386 −1.00198
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 28.1080 1.23380
\(520\) 0 0
\(521\) 1.12311 0.0492042 0.0246021 0.999697i \(-0.492168\pi\)
0.0246021 + 0.999697i \(0.492168\pi\)
\(522\) 0 0
\(523\) 6.05398 0.264722 0.132361 0.991202i \(-0.457744\pi\)
0.132361 + 0.991202i \(0.457744\pi\)
\(524\) 0 0
\(525\) 5.56155 0.242726
\(526\) 0 0
\(527\) −7.12311 −0.310287
\(528\) 0 0
\(529\) 7.93087 0.344820
\(530\) 0 0
\(531\) 2.63068 0.114162
\(532\) 0 0
\(533\) −7.12311 −0.308536
\(534\) 0 0
\(535\) −37.8617 −1.63691
\(536\) 0 0
\(537\) −34.7386 −1.49908
\(538\) 0 0
\(539\) −5.68466 −0.244856
\(540\) 0 0
\(541\) 43.2311 1.85865 0.929324 0.369265i \(-0.120391\pi\)
0.929324 + 0.369265i \(0.120391\pi\)
\(542\) 0 0
\(543\) −29.8617 −1.28149
\(544\) 0 0
\(545\) −37.3693 −1.60073
\(546\) 0 0
\(547\) −6.73863 −0.288123 −0.144062 0.989569i \(-0.546016\pi\)
−0.144062 + 0.989569i \(0.546016\pi\)
\(548\) 0 0
\(549\) 5.75379 0.245566
\(550\) 0 0
\(551\) 6.68466 0.284776
\(552\) 0 0
\(553\) −14.2462 −0.605811
\(554\) 0 0
\(555\) 9.75379 0.414025
\(556\) 0 0
\(557\) 8.87689 0.376126 0.188063 0.982157i \(-0.439779\pi\)
0.188063 + 0.982157i \(0.439779\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −5.56155 −0.234809
\(562\) 0 0
\(563\) −6.73863 −0.284000 −0.142000 0.989867i \(-0.545353\pi\)
−0.142000 + 0.989867i \(0.545353\pi\)
\(564\) 0 0
\(565\) 10.2462 0.431061
\(566\) 0 0
\(567\) 24.9309 1.04700
\(568\) 0 0
\(569\) 3.36932 0.141249 0.0706246 0.997503i \(-0.477501\pi\)
0.0706246 + 0.997503i \(0.477501\pi\)
\(570\) 0 0
\(571\) −35.6155 −1.49046 −0.745232 0.666806i \(-0.767661\pi\)
−0.745232 + 0.666806i \(0.767661\pi\)
\(572\) 0 0
\(573\) 32.3002 1.34936
\(574\) 0 0
\(575\) 5.56155 0.231933
\(576\) 0 0
\(577\) 3.94602 0.164275 0.0821376 0.996621i \(-0.473825\pi\)
0.0821376 + 0.996621i \(0.473825\pi\)
\(578\) 0 0
\(579\) −1.75379 −0.0728850
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 7.80776 0.323365
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) 7.50758 0.309871 0.154935 0.987925i \(-0.450483\pi\)
0.154935 + 0.987925i \(0.450483\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 2.73863 0.112652
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) −25.3693 −1.04004
\(596\) 0 0
\(597\) 1.06913 0.0437566
\(598\) 0 0
\(599\) −39.8617 −1.62871 −0.814353 0.580370i \(-0.802908\pi\)
−0.814353 + 0.580370i \(0.802908\pi\)
\(600\) 0 0
\(601\) −3.36932 −0.137437 −0.0687187 0.997636i \(-0.521891\pi\)
−0.0687187 + 0.997636i \(0.521891\pi\)
\(602\) 0 0
\(603\) 2.63068 0.107130
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −27.6155 −1.12088 −0.560440 0.828195i \(-0.689368\pi\)
−0.560440 + 0.828195i \(0.689368\pi\)
\(608\) 0 0
\(609\) −37.1771 −1.50649
\(610\) 0 0
\(611\) 28.4924 1.15268
\(612\) 0 0
\(613\) 1.75379 0.0708349 0.0354174 0.999373i \(-0.488724\pi\)
0.0354174 + 0.999373i \(0.488724\pi\)
\(614\) 0 0
\(615\) 6.24621 0.251872
\(616\) 0 0
\(617\) −4.73863 −0.190770 −0.0953851 0.995440i \(-0.530408\pi\)
−0.0953851 + 0.995440i \(0.530408\pi\)
\(618\) 0 0
\(619\) 26.2462 1.05492 0.527462 0.849579i \(-0.323144\pi\)
0.527462 + 0.849579i \(0.323144\pi\)
\(620\) 0 0
\(621\) 30.9309 1.24121
\(622\) 0 0
\(623\) 32.4924 1.30178
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −1.56155 −0.0623624
\(628\) 0 0
\(629\) 11.1231 0.443507
\(630\) 0 0
\(631\) −36.4924 −1.45274 −0.726370 0.687304i \(-0.758794\pi\)
−0.726370 + 0.687304i \(0.758794\pi\)
\(632\) 0 0
\(633\) 19.8078 0.787288
\(634\) 0 0
\(635\) 28.4924 1.13069
\(636\) 0 0
\(637\) −20.2462 −0.802184
\(638\) 0 0
\(639\) 3.36932 0.133288
\(640\) 0 0
\(641\) −9.61553 −0.379791 −0.189895 0.981804i \(-0.560815\pi\)
−0.189895 + 0.981804i \(0.560815\pi\)
\(642\) 0 0
\(643\) 37.3693 1.47370 0.736851 0.676055i \(-0.236312\pi\)
0.736851 + 0.676055i \(0.236312\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.5464 −0.886390 −0.443195 0.896425i \(-0.646155\pi\)
−0.443195 + 0.896425i \(0.646155\pi\)
\(648\) 0 0
\(649\) 4.68466 0.183889
\(650\) 0 0
\(651\) 11.1231 0.435949
\(652\) 0 0
\(653\) 25.6155 1.00241 0.501207 0.865328i \(-0.332890\pi\)
0.501207 + 0.865328i \(0.332890\pi\)
\(654\) 0 0
\(655\) 14.2462 0.556646
\(656\) 0 0
\(657\) −1.50758 −0.0588162
\(658\) 0 0
\(659\) −47.4233 −1.84735 −0.923675 0.383178i \(-0.874830\pi\)
−0.923675 + 0.383178i \(0.874830\pi\)
\(660\) 0 0
\(661\) −22.4384 −0.872754 −0.436377 0.899764i \(-0.643739\pi\)
−0.436377 + 0.899764i \(0.643739\pi\)
\(662\) 0 0
\(663\) −19.8078 −0.769270
\(664\) 0 0
\(665\) −7.12311 −0.276222
\(666\) 0 0
\(667\) −37.1771 −1.43950
\(668\) 0 0
\(669\) 38.6307 1.49355
\(670\) 0 0
\(671\) 10.2462 0.395551
\(672\) 0 0
\(673\) −18.8769 −0.727651 −0.363825 0.931467i \(-0.618530\pi\)
−0.363825 + 0.931467i \(0.618530\pi\)
\(674\) 0 0
\(675\) 5.56155 0.214064
\(676\) 0 0
\(677\) −39.1771 −1.50570 −0.752849 0.658194i \(-0.771321\pi\)
−0.752849 + 0.658194i \(0.771321\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 26.8229 1.02786
\(682\) 0 0
\(683\) 4.49242 0.171898 0.0859489 0.996300i \(-0.472608\pi\)
0.0859489 + 0.996300i \(0.472608\pi\)
\(684\) 0 0
\(685\) 16.8769 0.644833
\(686\) 0 0
\(687\) 32.9848 1.25845
\(688\) 0 0
\(689\) 27.8078 1.05939
\(690\) 0 0
\(691\) 43.6155 1.65921 0.829606 0.558349i \(-0.188565\pi\)
0.829606 + 0.558349i \(0.188565\pi\)
\(692\) 0 0
\(693\) −2.00000 −0.0759737
\(694\) 0 0
\(695\) 34.7386 1.31771
\(696\) 0 0
\(697\) 7.12311 0.269807
\(698\) 0 0
\(699\) 32.3845 1.22489
\(700\) 0 0
\(701\) 24.9848 0.943665 0.471832 0.881688i \(-0.343593\pi\)
0.471832 + 0.881688i \(0.343593\pi\)
\(702\) 0 0
\(703\) 3.12311 0.117790
\(704\) 0 0
\(705\) −24.9848 −0.940984
\(706\) 0 0
\(707\) 53.8617 2.02568
\(708\) 0 0
\(709\) −9.50758 −0.357065 −0.178532 0.983934i \(-0.557135\pi\)
−0.178532 + 0.983934i \(0.557135\pi\)
\(710\) 0 0
\(711\) −2.24621 −0.0842395
\(712\) 0 0
\(713\) 11.1231 0.416564
\(714\) 0 0
\(715\) 7.12311 0.266389
\(716\) 0 0
\(717\) 7.69981 0.287555
\(718\) 0 0
\(719\) 6.43845 0.240114 0.120057 0.992767i \(-0.461692\pi\)
0.120057 + 0.992767i \(0.461692\pi\)
\(720\) 0 0
\(721\) 40.4924 1.50802
\(722\) 0 0
\(723\) 33.7538 1.25532
\(724\) 0 0
\(725\) −6.68466 −0.248262
\(726\) 0 0
\(727\) 39.4233 1.46213 0.731064 0.682308i \(-0.239024\pi\)
0.731064 + 0.682308i \(0.239024\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −48.1080 −1.77691 −0.888454 0.458966i \(-0.848220\pi\)
−0.888454 + 0.458966i \(0.848220\pi\)
\(734\) 0 0
\(735\) 17.7538 0.654858
\(736\) 0 0
\(737\) 4.68466 0.172562
\(738\) 0 0
\(739\) −20.9848 −0.771940 −0.385970 0.922511i \(-0.626133\pi\)
−0.385970 + 0.922511i \(0.626133\pi\)
\(740\) 0 0
\(741\) −5.56155 −0.204309
\(742\) 0 0
\(743\) 25.7538 0.944815 0.472407 0.881380i \(-0.343385\pi\)
0.472407 + 0.881380i \(0.343385\pi\)
\(744\) 0 0
\(745\) −30.2462 −1.10814
\(746\) 0 0
\(747\) −1.26137 −0.0461510
\(748\) 0 0
\(749\) 67.4233 2.46359
\(750\) 0 0
\(751\) 41.2311 1.50454 0.752271 0.658853i \(-0.228958\pi\)
0.752271 + 0.658853i \(0.228958\pi\)
\(752\) 0 0
\(753\) −13.8617 −0.505150
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −9.50758 −0.345559 −0.172779 0.984961i \(-0.555275\pi\)
−0.172779 + 0.984961i \(0.555275\pi\)
\(758\) 0 0
\(759\) 8.68466 0.315233
\(760\) 0 0
\(761\) −22.1922 −0.804468 −0.402234 0.915537i \(-0.631766\pi\)
−0.402234 + 0.915537i \(0.631766\pi\)
\(762\) 0 0
\(763\) 66.5464 2.40914
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 16.6847 0.602448
\(768\) 0 0
\(769\) −1.31534 −0.0474324 −0.0237162 0.999719i \(-0.507550\pi\)
−0.0237162 + 0.999719i \(0.507550\pi\)
\(770\) 0 0
\(771\) −34.3542 −1.23723
\(772\) 0 0
\(773\) 9.06913 0.326194 0.163097 0.986610i \(-0.447852\pi\)
0.163097 + 0.986610i \(0.447852\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) −17.3693 −0.623121
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) −37.1771 −1.32860
\(784\) 0 0
\(785\) 36.9848 1.32005
\(786\) 0 0
\(787\) −3.31534 −0.118179 −0.0590896 0.998253i \(-0.518820\pi\)
−0.0590896 + 0.998253i \(0.518820\pi\)
\(788\) 0 0
\(789\) −26.7386 −0.951921
\(790\) 0 0
\(791\) −18.2462 −0.648761
\(792\) 0 0
\(793\) 36.4924 1.29588
\(794\) 0 0
\(795\) −24.3845 −0.864828
\(796\) 0 0
\(797\) −35.8078 −1.26838 −0.634188 0.773179i \(-0.718665\pi\)
−0.634188 + 0.773179i \(0.718665\pi\)
\(798\) 0 0
\(799\) −28.4924 −1.00799
\(800\) 0 0
\(801\) 5.12311 0.181016
\(802\) 0 0
\(803\) −2.68466 −0.0947395
\(804\) 0 0
\(805\) 39.6155 1.39626
\(806\) 0 0
\(807\) 17.3693 0.611429
\(808\) 0 0
\(809\) 36.9309 1.29842 0.649210 0.760609i \(-0.275100\pi\)
0.649210 + 0.760609i \(0.275100\pi\)
\(810\) 0 0
\(811\) −20.6847 −0.726337 −0.363168 0.931724i \(-0.618305\pi\)
−0.363168 + 0.931724i \(0.618305\pi\)
\(812\) 0 0
\(813\) −20.7926 −0.729229
\(814\) 0 0
\(815\) 26.7386 0.936613
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −7.12311 −0.248901
\(820\) 0 0
\(821\) 44.9848 1.56998 0.784991 0.619507i \(-0.212668\pi\)
0.784991 + 0.619507i \(0.212668\pi\)
\(822\) 0 0
\(823\) −14.9309 −0.520457 −0.260229 0.965547i \(-0.583798\pi\)
−0.260229 + 0.965547i \(0.583798\pi\)
\(824\) 0 0
\(825\) 1.56155 0.0543663
\(826\) 0 0
\(827\) −21.0691 −0.732645 −0.366323 0.930488i \(-0.619383\pi\)
−0.366323 + 0.930488i \(0.619383\pi\)
\(828\) 0 0
\(829\) −33.1771 −1.15229 −0.576144 0.817348i \(-0.695443\pi\)
−0.576144 + 0.817348i \(0.695443\pi\)
\(830\) 0 0
\(831\) −38.2462 −1.32675
\(832\) 0 0
\(833\) 20.2462 0.701490
\(834\) 0 0
\(835\) 50.7386 1.75588
\(836\) 0 0
\(837\) 11.1231 0.384471
\(838\) 0 0
\(839\) −1.12311 −0.0387739 −0.0193870 0.999812i \(-0.506171\pi\)
−0.0193870 + 0.999812i \(0.506171\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 0 0
\(843\) −0.984845 −0.0339199
\(844\) 0 0
\(845\) −0.630683 −0.0216962
\(846\) 0 0
\(847\) −3.56155 −0.122376
\(848\) 0 0
\(849\) 20.8769 0.716493
\(850\) 0 0
\(851\) −17.3693 −0.595413
\(852\) 0 0
\(853\) 38.3542 1.31322 0.656611 0.754230i \(-0.271989\pi\)
0.656611 + 0.754230i \(0.271989\pi\)
\(854\) 0 0
\(855\) −1.12311 −0.0384094
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) 41.8617 1.42830 0.714152 0.699991i \(-0.246812\pi\)
0.714152 + 0.699991i \(0.246812\pi\)
\(860\) 0 0
\(861\) −11.1231 −0.379074
\(862\) 0 0
\(863\) 16.2462 0.553027 0.276514 0.961010i \(-0.410821\pi\)
0.276514 + 0.961010i \(0.410821\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) −6.73863 −0.228856
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) 16.6847 0.565338
\(872\) 0 0
\(873\) −0.630683 −0.0213454
\(874\) 0 0
\(875\) 42.7386 1.44483
\(876\) 0 0
\(877\) −40.4384 −1.36551 −0.682755 0.730648i \(-0.739218\pi\)
−0.682755 + 0.730648i \(0.739218\pi\)
\(878\) 0 0
\(879\) −38.9309 −1.31311
\(880\) 0 0
\(881\) −28.7386 −0.968229 −0.484115 0.875005i \(-0.660858\pi\)
−0.484115 + 0.875005i \(0.660858\pi\)
\(882\) 0 0
\(883\) 54.7386 1.84210 0.921051 0.389442i \(-0.127332\pi\)
0.921051 + 0.389442i \(0.127332\pi\)
\(884\) 0 0
\(885\) −14.6307 −0.491805
\(886\) 0 0
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) −50.7386 −1.70172
\(890\) 0 0
\(891\) 7.00000 0.234509
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −44.4924 −1.48722
\(896\) 0 0
\(897\) 30.9309 1.03275
\(898\) 0 0
\(899\) −13.3693 −0.445892
\(900\) 0 0
\(901\) −27.8078 −0.926411
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −38.2462 −1.27135
\(906\) 0 0
\(907\) −56.7926 −1.88577 −0.942884 0.333122i \(-0.891898\pi\)
−0.942884 + 0.333122i \(0.891898\pi\)
\(908\) 0 0
\(909\) 8.49242 0.281676
\(910\) 0 0
\(911\) 46.9848 1.55668 0.778339 0.627845i \(-0.216063\pi\)
0.778339 + 0.627845i \(0.216063\pi\)
\(912\) 0 0
\(913\) −2.24621 −0.0743387
\(914\) 0 0
\(915\) −32.0000 −1.05789
\(916\) 0 0
\(917\) −25.3693 −0.837769
\(918\) 0 0
\(919\) −28.4384 −0.938098 −0.469049 0.883172i \(-0.655403\pi\)
−0.469049 + 0.883172i \(0.655403\pi\)
\(920\) 0 0
\(921\) 8.98485 0.296061
\(922\) 0 0
\(923\) 21.3693 0.703380
\(924\) 0 0
\(925\) −3.12311 −0.102687
\(926\) 0 0
\(927\) 6.38447 0.209694
\(928\) 0 0
\(929\) −16.9309 −0.555484 −0.277742 0.960656i \(-0.589586\pi\)
−0.277742 + 0.960656i \(0.589586\pi\)
\(930\) 0 0
\(931\) 5.68466 0.186307
\(932\) 0 0
\(933\) −24.6847 −0.808139
\(934\) 0 0
\(935\) −7.12311 −0.232950
\(936\) 0 0
\(937\) −54.3002 −1.77391 −0.886955 0.461856i \(-0.847184\pi\)
−0.886955 + 0.461856i \(0.847184\pi\)
\(938\) 0 0
\(939\) 5.56155 0.181494
\(940\) 0 0
\(941\) 20.4384 0.666274 0.333137 0.942878i \(-0.391893\pi\)
0.333137 + 0.942878i \(0.391893\pi\)
\(942\) 0 0
\(943\) −11.1231 −0.362218
\(944\) 0 0
\(945\) 39.6155 1.28869
\(946\) 0 0
\(947\) −28.9848 −0.941881 −0.470940 0.882165i \(-0.656085\pi\)
−0.470940 + 0.882165i \(0.656085\pi\)
\(948\) 0 0
\(949\) −9.56155 −0.310381
\(950\) 0 0
\(951\) 46.9309 1.52184
\(952\) 0 0
\(953\) −9.12311 −0.295526 −0.147763 0.989023i \(-0.547207\pi\)
−0.147763 + 0.989023i \(0.547207\pi\)
\(954\) 0 0
\(955\) 41.3693 1.33868
\(956\) 0 0
\(957\) −10.4384 −0.337427
\(958\) 0 0
\(959\) −30.0540 −0.970493
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 10.6307 0.342569
\(964\) 0 0
\(965\) −2.24621 −0.0723081
\(966\) 0 0
\(967\) 3.86174 0.124185 0.0620926 0.998070i \(-0.480223\pi\)
0.0620926 + 0.998070i \(0.480223\pi\)
\(968\) 0 0
\(969\) 5.56155 0.178663
\(970\) 0 0
\(971\) 14.2462 0.457183 0.228591 0.973522i \(-0.426588\pi\)
0.228591 + 0.973522i \(0.426588\pi\)
\(972\) 0 0
\(973\) −61.8617 −1.98320
\(974\) 0 0
\(975\) 5.56155 0.178112
\(976\) 0 0
\(977\) 55.3693 1.77142 0.885711 0.464238i \(-0.153672\pi\)
0.885711 + 0.464238i \(0.153672\pi\)
\(978\) 0 0
\(979\) 9.12311 0.291576
\(980\) 0 0
\(981\) 10.4924 0.334997
\(982\) 0 0
\(983\) 48.2462 1.53882 0.769408 0.638758i \(-0.220552\pi\)
0.769408 + 0.638758i \(0.220552\pi\)
\(984\) 0 0
\(985\) 3.50758 0.111761
\(986\) 0 0
\(987\) 44.4924 1.41621
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −21.1231 −0.670998 −0.335499 0.942041i \(-0.608905\pi\)
−0.335499 + 0.942041i \(0.608905\pi\)
\(992\) 0 0
\(993\) 16.3002 0.517271
\(994\) 0 0
\(995\) 1.36932 0.0434103
\(996\) 0 0
\(997\) −53.3693 −1.69022 −0.845112 0.534590i \(-0.820466\pi\)
−0.845112 + 0.534590i \(0.820466\pi\)
\(998\) 0 0
\(999\) −17.3693 −0.549541
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 3344.2.a.k.1.2 2
4.3 odd 2 418.2.a.e.1.1 2
12.11 even 2 3762.2.a.y.1.2 2
44.43 even 2 4598.2.a.bj.1.1 2
76.75 even 2 7942.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.1 2 4.3 odd 2
3344.2.a.k.1.2 2 1.1 even 1 trivial
3762.2.a.y.1.2 2 12.11 even 2
4598.2.a.bj.1.1 2 44.43 even 2
7942.2.a.x.1.2 2 76.75 even 2