Properties

Label 3344.2.a.k.1.1
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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155 q^{3} +2.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} +2.00000 q^{5} +0.561553 q^{7} +3.56155 q^{9} -1.00000 q^{11} +0.561553 q^{13} -5.12311 q^{15} -0.561553 q^{17} +1.00000 q^{19} -1.43845 q^{21} -1.43845 q^{23} -1.00000 q^{25} -1.43845 q^{27} -5.68466 q^{29} -2.00000 q^{31} +2.56155 q^{33} +1.12311 q^{35} -5.12311 q^{37} -1.43845 q^{39} +2.00000 q^{41} +7.12311 q^{45} -8.00000 q^{47} -6.68466 q^{49} +1.43845 q^{51} +12.8078 q^{53} -2.00000 q^{55} -2.56155 q^{57} +7.68466 q^{59} +6.24621 q^{61} +2.00000 q^{63} +1.12311 q^{65} +7.68466 q^{67} +3.68466 q^{69} -6.00000 q^{71} -9.68466 q^{73} +2.56155 q^{75} -0.561553 q^{77} +4.00000 q^{79} -7.00000 q^{81} -14.2462 q^{83} -1.12311 q^{85} +14.5616 q^{87} -0.876894 q^{89} +0.315342 q^{91} +5.12311 q^{93} +2.00000 q^{95} -7.12311 q^{97} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 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
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0.561553 0.212247 0.106124 0.994353i \(-0.466156\pi\)
0.106124 + 0.994353i \(0.466156\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) 0 0
\(15\) −5.12311 −1.32278
\(16\) 0 0
\(17\) −0.561553 −0.136197 −0.0680983 0.997679i \(-0.521693\pi\)
−0.0680983 + 0.997679i \(0.521693\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.43845 −0.313895
\(22\) 0 0
\(23\) −1.43845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 2.56155 0.445909
\(34\) 0 0
\(35\) 1.12311 0.189839
\(36\) 0 0
\(37\) −5.12311 −0.842233 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(38\) 0 0
\(39\) −1.43845 −0.230336
\(40\) 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
\(45\) 7.12311 1.06185
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) 0 0
\(51\) 1.43845 0.201423
\(52\) 0 0
\(53\) 12.8078 1.75928 0.879641 0.475638i \(-0.157783\pi\)
0.879641 + 0.475638i \(0.157783\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −2.56155 −0.339286
\(58\) 0 0
\(59\) 7.68466 1.00046 0.500229 0.865893i \(-0.333249\pi\)
0.500229 + 0.865893i \(0.333249\pi\)
\(60\) 0 0
\(61\) 6.24621 0.799745 0.399873 0.916571i \(-0.369054\pi\)
0.399873 + 0.916571i \(0.369054\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 1.12311 0.139304
\(66\) 0 0
\(67\) 7.68466 0.938830 0.469415 0.882978i \(-0.344465\pi\)
0.469415 + 0.882978i \(0.344465\pi\)
\(68\) 0 0
\(69\) 3.68466 0.443581
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −9.68466 −1.13350 −0.566752 0.823889i \(-0.691800\pi\)
−0.566752 + 0.823889i \(0.691800\pi\)
\(74\) 0 0
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) −0.561553 −0.0639949
\(78\) 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
\(83\) −14.2462 −1.56372 −0.781862 0.623451i \(-0.785730\pi\)
−0.781862 + 0.623451i \(0.785730\pi\)
\(84\) 0 0
\(85\) −1.12311 −0.121818
\(86\) 0 0
\(87\) 14.5616 1.56116
\(88\) 0 0
\(89\) −0.876894 −0.0929506 −0.0464753 0.998919i \(-0.514799\pi\)
−0.0464753 + 0.998919i \(0.514799\pi\)
\(90\) 0 0
\(91\) 0.315342 0.0330568
\(92\) 0 0
\(93\) 5.12311 0.531241
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −7.12311 −0.723242 −0.361621 0.932325i \(-0.617777\pi\)
−0.361621 + 0.932325i \(0.617777\pi\)
\(98\) 0 0
\(99\) −3.56155 −0.357950
\(100\) 0 0
\(101\) −6.87689 −0.684277 −0.342138 0.939650i \(-0.611151\pi\)
−0.342138 + 0.939650i \(0.611151\pi\)
\(102\) 0 0
\(103\) 13.3693 1.31732 0.658659 0.752442i \(-0.271124\pi\)
0.658659 + 0.752442i \(0.271124\pi\)
\(104\) 0 0
\(105\) −2.87689 −0.280756
\(106\) 0 0
\(107\) 9.93087 0.960053 0.480027 0.877254i \(-0.340627\pi\)
0.480027 + 0.877254i \(0.340627\pi\)
\(108\) 0 0
\(109\) −6.31534 −0.604900 −0.302450 0.953165i \(-0.597805\pi\)
−0.302450 + 0.953165i \(0.597805\pi\)
\(110\) 0 0
\(111\) 13.1231 1.24559
\(112\) 0 0
\(113\) −3.12311 −0.293797 −0.146899 0.989152i \(-0.546929\pi\)
−0.146899 + 0.989152i \(0.546929\pi\)
\(114\) 0 0
\(115\) −2.87689 −0.268272
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −0.315342 −0.0289073
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −5.12311 −0.461935
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −2.24621 −0.199319 −0.0996595 0.995022i \(-0.531775\pi\)
−0.0996595 + 0.995022i \(0.531775\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.12311 −0.0981262 −0.0490631 0.998796i \(-0.515624\pi\)
−0.0490631 + 0.998796i \(0.515624\pi\)
\(132\) 0 0
\(133\) 0.561553 0.0486928
\(134\) 0 0
\(135\) −2.87689 −0.247604
\(136\) 0 0
\(137\) 12.5616 1.07321 0.536603 0.843835i \(-0.319707\pi\)
0.536603 + 0.843835i \(0.319707\pi\)
\(138\) 0 0
\(139\) −7.36932 −0.625057 −0.312529 0.949908i \(-0.601176\pi\)
−0.312529 + 0.949908i \(0.601176\pi\)
\(140\) 0 0
\(141\) 20.4924 1.72577
\(142\) 0 0
\(143\) −0.561553 −0.0469594
\(144\) 0 0
\(145\) −11.3693 −0.944170
\(146\) 0 0
\(147\) 17.1231 1.41229
\(148\) 0 0
\(149\) −6.87689 −0.563377 −0.281689 0.959506i \(-0.590894\pi\)
−0.281689 + 0.959506i \(0.590894\pi\)
\(150\) 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
\(157\) −14.4924 −1.15662 −0.578311 0.815817i \(-0.696288\pi\)
−0.578311 + 0.815817i \(0.696288\pi\)
\(158\) 0 0
\(159\) −32.8078 −2.60182
\(160\) 0 0
\(161\) −0.807764 −0.0636607
\(162\) 0 0
\(163\) −11.3693 −0.890514 −0.445257 0.895403i \(-0.646888\pi\)
−0.445257 + 0.895403i \(0.646888\pi\)
\(164\) 0 0
\(165\) 5.12311 0.398833
\(166\) 0 0
\(167\) 0.630683 0.0488037 0.0244019 0.999702i \(-0.492232\pi\)
0.0244019 + 0.999702i \(0.492232\pi\)
\(168\) 0 0
\(169\) −12.6847 −0.975743
\(170\) 0 0
\(171\) 3.56155 0.272359
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −0.561553 −0.0424494
\(176\) 0 0
\(177\) −19.6847 −1.47959
\(178\) 0 0
\(179\) −5.75379 −0.430058 −0.215029 0.976608i \(-0.568985\pi\)
−0.215029 + 0.976608i \(0.568985\pi\)
\(180\) 0 0
\(181\) −10.8769 −0.808473 −0.404237 0.914654i \(-0.632463\pi\)
−0.404237 + 0.914654i \(0.632463\pi\)
\(182\) 0 0
\(183\) −16.0000 −1.18275
\(184\) 0 0
\(185\) −10.2462 −0.753316
\(186\) 0 0
\(187\) 0.561553 0.0410648
\(188\) 0 0
\(189\) −0.807764 −0.0587562
\(190\) 0 0
\(191\) 8.31534 0.601677 0.300838 0.953675i \(-0.402733\pi\)
0.300838 + 0.953675i \(0.402733\pi\)
\(192\) 0 0
\(193\) 7.12311 0.512732 0.256366 0.966580i \(-0.417475\pi\)
0.256366 + 0.966580i \(0.417475\pi\)
\(194\) 0 0
\(195\) −2.87689 −0.206019
\(196\) 0 0
\(197\) 18.2462 1.29999 0.649994 0.759939i \(-0.274771\pi\)
0.649994 + 0.759939i \(0.274771\pi\)
\(198\) 0 0
\(199\) −11.6847 −0.828303 −0.414152 0.910208i \(-0.635922\pi\)
−0.414152 + 0.910208i \(0.635922\pi\)
\(200\) 0 0
\(201\) −19.6847 −1.38845
\(202\) 0 0
\(203\) −3.19224 −0.224051
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) −5.12311 −0.356080
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 0.315342 0.0217090 0.0108545 0.999941i \(-0.496545\pi\)
0.0108545 + 0.999941i \(0.496545\pi\)
\(212\) 0 0
\(213\) 15.3693 1.05309
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.12311 −0.0762414
\(218\) 0 0
\(219\) 24.8078 1.67635
\(220\) 0 0
\(221\) −0.315342 −0.0212122
\(222\) 0 0
\(223\) −24.7386 −1.65662 −0.828311 0.560269i \(-0.810698\pi\)
−0.828311 + 0.560269i \(0.810698\pi\)
\(224\) 0 0
\(225\) −3.56155 −0.237437
\(226\) 0 0
\(227\) −28.1771 −1.87018 −0.935089 0.354412i \(-0.884681\pi\)
−0.935089 + 0.354412i \(0.884681\pi\)
\(228\) 0 0
\(229\) 12.8769 0.850929 0.425465 0.904975i \(-0.360111\pi\)
0.425465 + 0.904975i \(0.360111\pi\)
\(230\) 0 0
\(231\) 1.43845 0.0946429
\(232\) 0 0
\(233\) −28.7386 −1.88273 −0.941365 0.337389i \(-0.890456\pi\)
−0.941365 + 0.337389i \(0.890456\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) −10.2462 −0.665563
\(238\) 0 0
\(239\) −23.9309 −1.54796 −0.773980 0.633210i \(-0.781737\pi\)
−0.773980 + 0.633210i \(0.781737\pi\)
\(240\) 0 0
\(241\) −19.6155 −1.26355 −0.631774 0.775153i \(-0.717673\pi\)
−0.631774 + 0.775153i \(0.717673\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) −13.3693 −0.854134
\(246\) 0 0
\(247\) 0.561553 0.0357307
\(248\) 0 0
\(249\) 36.4924 2.31261
\(250\) 0 0
\(251\) −17.1231 −1.08080 −0.540400 0.841408i \(-0.681727\pi\)
−0.540400 + 0.841408i \(0.681727\pi\)
\(252\) 0 0
\(253\) 1.43845 0.0904344
\(254\) 0 0
\(255\) 2.87689 0.180158
\(256\) 0 0
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) −2.87689 −0.178762
\(260\) 0 0
\(261\) −20.2462 −1.25321
\(262\) 0 0
\(263\) −8.87689 −0.547373 −0.273686 0.961819i \(-0.588243\pi\)
−0.273686 + 0.961819i \(0.588243\pi\)
\(264\) 0 0
\(265\) 25.6155 1.57355
\(266\) 0 0
\(267\) 2.24621 0.137466
\(268\) 0 0
\(269\) 2.87689 0.175407 0.0877037 0.996147i \(-0.472047\pi\)
0.0877037 + 0.996147i \(0.472047\pi\)
\(270\) 0 0
\(271\) −25.6847 −1.56023 −0.780116 0.625635i \(-0.784840\pi\)
−0.780116 + 0.625635i \(0.784840\pi\)
\(272\) 0 0
\(273\) −0.807764 −0.0488881
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 8.49242 0.510260 0.255130 0.966907i \(-0.417882\pi\)
0.255130 + 0.966907i \(0.417882\pi\)
\(278\) 0 0
\(279\) −7.12311 −0.426449
\(280\) 0 0
\(281\) −25.3693 −1.51341 −0.756703 0.653758i \(-0.773191\pi\)
−0.756703 + 0.653758i \(0.773191\pi\)
\(282\) 0 0
\(283\) −11.3693 −0.675836 −0.337918 0.941176i \(-0.609723\pi\)
−0.337918 + 0.941176i \(0.609723\pi\)
\(284\) 0 0
\(285\) −5.12311 −0.303467
\(286\) 0 0
\(287\) 1.12311 0.0662948
\(288\) 0 0
\(289\) −16.6847 −0.981450
\(290\) 0 0
\(291\) 18.2462 1.06961
\(292\) 0 0
\(293\) 3.93087 0.229644 0.114822 0.993386i \(-0.463370\pi\)
0.114822 + 0.993386i \(0.463370\pi\)
\(294\) 0 0
\(295\) 15.3693 0.894836
\(296\) 0 0
\(297\) 1.43845 0.0834672
\(298\) 0 0
\(299\) −0.807764 −0.0467142
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 17.6155 1.01199
\(304\) 0 0
\(305\) 12.4924 0.715314
\(306\) 0 0
\(307\) 22.2462 1.26966 0.634829 0.772653i \(-0.281070\pi\)
0.634829 + 0.772653i \(0.281070\pi\)
\(308\) 0 0
\(309\) −34.2462 −1.94820
\(310\) 0 0
\(311\) 4.80776 0.272623 0.136312 0.990666i \(-0.456475\pi\)
0.136312 + 0.990666i \(0.456475\pi\)
\(312\) 0 0
\(313\) −0.561553 −0.0317408 −0.0158704 0.999874i \(-0.505052\pi\)
−0.0158704 + 0.999874i \(0.505052\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) −7.05398 −0.396191 −0.198095 0.980183i \(-0.563476\pi\)
−0.198095 + 0.980183i \(0.563476\pi\)
\(318\) 0 0
\(319\) 5.68466 0.318280
\(320\) 0 0
\(321\) −25.4384 −1.41984
\(322\) 0 0
\(323\) −0.561553 −0.0312456
\(324\) 0 0
\(325\) −0.561553 −0.0311493
\(326\) 0 0
\(327\) 16.1771 0.894595
\(328\) 0 0
\(329\) −4.49242 −0.247675
\(330\) 0 0
\(331\) 14.5616 0.800375 0.400188 0.916433i \(-0.368945\pi\)
0.400188 + 0.916433i \(0.368945\pi\)
\(332\) 0 0
\(333\) −18.2462 −0.999886
\(334\) 0 0
\(335\) 15.3693 0.839715
\(336\) 0 0
\(337\) 11.1231 0.605914 0.302957 0.953004i \(-0.402026\pi\)
0.302957 + 0.953004i \(0.402026\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −7.68466 −0.414933
\(344\) 0 0
\(345\) 7.36932 0.396751
\(346\) 0 0
\(347\) −9.61553 −0.516189 −0.258094 0.966120i \(-0.583095\pi\)
−0.258094 + 0.966120i \(0.583095\pi\)
\(348\) 0 0
\(349\) 21.6155 1.15705 0.578526 0.815664i \(-0.303628\pi\)
0.578526 + 0.815664i \(0.303628\pi\)
\(350\) 0 0
\(351\) −0.807764 −0.0431153
\(352\) 0 0
\(353\) 22.1771 1.18037 0.590183 0.807269i \(-0.299055\pi\)
0.590183 + 0.807269i \(0.299055\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0.807764 0.0427514
\(358\) 0 0
\(359\) 15.9309 0.840799 0.420400 0.907339i \(-0.361890\pi\)
0.420400 + 0.907339i \(0.361890\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.56155 −0.134447
\(364\) 0 0
\(365\) −19.3693 −1.01384
\(366\) 0 0
\(367\) −30.2462 −1.57884 −0.789420 0.613854i \(-0.789618\pi\)
−0.789420 + 0.613854i \(0.789618\pi\)
\(368\) 0 0
\(369\) 7.12311 0.370814
\(370\) 0 0
\(371\) 7.19224 0.373402
\(372\) 0 0
\(373\) 3.43845 0.178036 0.0890180 0.996030i \(-0.471627\pi\)
0.0890180 + 0.996030i \(0.471627\pi\)
\(374\) 0 0
\(375\) 30.7386 1.58734
\(376\) 0 0
\(377\) −3.19224 −0.164409
\(378\) 0 0
\(379\) −6.56155 −0.337044 −0.168522 0.985698i \(-0.553899\pi\)
−0.168522 + 0.985698i \(0.553899\pi\)
\(380\) 0 0
\(381\) 5.75379 0.294776
\(382\) 0 0
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) −1.12311 −0.0572388
\(386\) 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
\(391\) 0.807764 0.0408504
\(392\) 0 0
\(393\) 2.87689 0.145120
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 38.9848 1.95659 0.978297 0.207209i \(-0.0664381\pi\)
0.978297 + 0.207209i \(0.0664381\pi\)
\(398\) 0 0
\(399\) −1.43845 −0.0720124
\(400\) 0 0
\(401\) −7.75379 −0.387206 −0.193603 0.981080i \(-0.562017\pi\)
−0.193603 + 0.981080i \(0.562017\pi\)
\(402\) 0 0
\(403\) −1.12311 −0.0559459
\(404\) 0 0
\(405\) −14.0000 −0.695666
\(406\) 0 0
\(407\) 5.12311 0.253943
\(408\) 0 0
\(409\) 16.2462 0.803323 0.401662 0.915788i \(-0.368433\pi\)
0.401662 + 0.915788i \(0.368433\pi\)
\(410\) 0 0
\(411\) −32.1771 −1.58718
\(412\) 0 0
\(413\) 4.31534 0.212344
\(414\) 0 0
\(415\) −28.4924 −1.39864
\(416\) 0 0
\(417\) 18.8769 0.924405
\(418\) 0 0
\(419\) 14.8769 0.726784 0.363392 0.931636i \(-0.381619\pi\)
0.363392 + 0.931636i \(0.381619\pi\)
\(420\) 0 0
\(421\) −29.9309 −1.45874 −0.729371 0.684119i \(-0.760187\pi\)
−0.729371 + 0.684119i \(0.760187\pi\)
\(422\) 0 0
\(423\) −28.4924 −1.38535
\(424\) 0 0
\(425\) 0.561553 0.0272393
\(426\) 0 0
\(427\) 3.50758 0.169744
\(428\) 0 0
\(429\) 1.43845 0.0694489
\(430\) 0 0
\(431\) −31.8617 −1.53473 −0.767363 0.641213i \(-0.778432\pi\)
−0.767363 + 0.641213i \(0.778432\pi\)
\(432\) 0 0
\(433\) 6.63068 0.318650 0.159325 0.987226i \(-0.449068\pi\)
0.159325 + 0.987226i \(0.449068\pi\)
\(434\) 0 0
\(435\) 29.1231 1.39635
\(436\) 0 0
\(437\) −1.43845 −0.0688103
\(438\) 0 0
\(439\) 9.61553 0.458924 0.229462 0.973318i \(-0.426303\pi\)
0.229462 + 0.973318i \(0.426303\pi\)
\(440\) 0 0
\(441\) −23.8078 −1.13370
\(442\) 0 0
\(443\) −39.8617 −1.89389 −0.946944 0.321398i \(-0.895847\pi\)
−0.946944 + 0.321398i \(0.895847\pi\)
\(444\) 0 0
\(445\) −1.75379 −0.0831376
\(446\) 0 0
\(447\) 17.6155 0.833186
\(448\) 0 0
\(449\) 23.6155 1.11449 0.557243 0.830350i \(-0.311859\pi\)
0.557243 + 0.830350i \(0.311859\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) 10.2462 0.481409
\(454\) 0 0
\(455\) 0.630683 0.0295669
\(456\) 0 0
\(457\) 36.5616 1.71028 0.855139 0.518399i \(-0.173472\pi\)
0.855139 + 0.518399i \(0.173472\pi\)
\(458\) 0 0
\(459\) 0.807764 0.0377032
\(460\) 0 0
\(461\) 26.7386 1.24534 0.622671 0.782484i \(-0.286047\pi\)
0.622671 + 0.782484i \(0.286047\pi\)
\(462\) 0 0
\(463\) −3.50758 −0.163011 −0.0815055 0.996673i \(-0.525973\pi\)
−0.0815055 + 0.996673i \(0.525973\pi\)
\(464\) 0 0
\(465\) 10.2462 0.475157
\(466\) 0 0
\(467\) −9.75379 −0.451352 −0.225676 0.974202i \(-0.572459\pi\)
−0.225676 + 0.974202i \(0.572459\pi\)
\(468\) 0 0
\(469\) 4.31534 0.199264
\(470\) 0 0
\(471\) 37.1231 1.71054
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 45.6155 2.08859
\(478\) 0 0
\(479\) 13.3693 0.610860 0.305430 0.952215i \(-0.401200\pi\)
0.305430 + 0.952215i \(0.401200\pi\)
\(480\) 0 0
\(481\) −2.87689 −0.131175
\(482\) 0 0
\(483\) 2.06913 0.0941487
\(484\) 0 0
\(485\) −14.2462 −0.646887
\(486\) 0 0
\(487\) −31.1231 −1.41032 −0.705161 0.709047i \(-0.749125\pi\)
−0.705161 + 0.709047i \(0.749125\pi\)
\(488\) 0 0
\(489\) 29.1231 1.31699
\(490\) 0 0
\(491\) 39.3693 1.77671 0.888356 0.459155i \(-0.151848\pi\)
0.888356 + 0.459155i \(0.151848\pi\)
\(492\) 0 0
\(493\) 3.19224 0.143771
\(494\) 0 0
\(495\) −7.12311 −0.320160
\(496\) 0 0
\(497\) −3.36932 −0.151135
\(498\) 0 0
\(499\) −9.75379 −0.436640 −0.218320 0.975877i \(-0.570058\pi\)
−0.218320 + 0.975877i \(0.570058\pi\)
\(500\) 0 0
\(501\) −1.61553 −0.0721765
\(502\) 0 0
\(503\) −21.6847 −0.966871 −0.483436 0.875380i \(-0.660611\pi\)
−0.483436 + 0.875380i \(0.660611\pi\)
\(504\) 0 0
\(505\) −13.7538 −0.612036
\(506\) 0 0
\(507\) 32.4924 1.44304
\(508\) 0 0
\(509\) −31.3693 −1.39042 −0.695210 0.718806i \(-0.744689\pi\)
−0.695210 + 0.718806i \(0.744689\pi\)
\(510\) 0 0
\(511\) −5.43845 −0.240583
\(512\) 0 0
\(513\) −1.43845 −0.0635090
\(514\) 0 0
\(515\) 26.7386 1.17824
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −46.1080 −2.02391
\(520\) 0 0
\(521\) −7.12311 −0.312069 −0.156034 0.987752i \(-0.549871\pi\)
−0.156034 + 0.987752i \(0.549871\pi\)
\(522\) 0 0
\(523\) −31.0540 −1.35790 −0.678948 0.734187i \(-0.737564\pi\)
−0.678948 + 0.734187i \(0.737564\pi\)
\(524\) 0 0
\(525\) 1.43845 0.0627790
\(526\) 0 0
\(527\) 1.12311 0.0489232
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) 0 0
\(531\) 27.3693 1.18773
\(532\) 0 0
\(533\) 1.12311 0.0486471
\(534\) 0 0
\(535\) 19.8617 0.858698
\(536\) 0 0
\(537\) 14.7386 0.636019
\(538\) 0 0
\(539\) 6.68466 0.287929
\(540\) 0 0
\(541\) −39.2311 −1.68667 −0.843337 0.537384i \(-0.819412\pi\)
−0.843337 + 0.537384i \(0.819412\pi\)
\(542\) 0 0
\(543\) 27.8617 1.19566
\(544\) 0 0
\(545\) −12.6307 −0.541039
\(546\) 0 0
\(547\) 42.7386 1.82737 0.913686 0.406421i \(-0.133223\pi\)
0.913686 + 0.406421i \(0.133223\pi\)
\(548\) 0 0
\(549\) 22.2462 0.949445
\(550\) 0 0
\(551\) −5.68466 −0.242175
\(552\) 0 0
\(553\) 2.24621 0.0955186
\(554\) 0 0
\(555\) 26.2462 1.11409
\(556\) 0 0
\(557\) 17.1231 0.725529 0.362765 0.931881i \(-0.381833\pi\)
0.362765 + 0.931881i \(0.381833\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1.43845 −0.0607313
\(562\) 0 0
\(563\) 42.7386 1.80122 0.900609 0.434630i \(-0.143121\pi\)
0.900609 + 0.434630i \(0.143121\pi\)
\(564\) 0 0
\(565\) −6.24621 −0.262780
\(566\) 0 0
\(567\) −3.93087 −0.165081
\(568\) 0 0
\(569\) −21.3693 −0.895848 −0.447924 0.894072i \(-0.647837\pi\)
−0.447924 + 0.894072i \(0.647837\pi\)
\(570\) 0 0
\(571\) 5.61553 0.235003 0.117501 0.993073i \(-0.462512\pi\)
0.117501 + 0.993073i \(0.462512\pi\)
\(572\) 0 0
\(573\) −21.3002 −0.889828
\(574\) 0 0
\(575\) 1.43845 0.0599874
\(576\) 0 0
\(577\) 41.0540 1.70910 0.854550 0.519370i \(-0.173833\pi\)
0.854550 + 0.519370i \(0.173833\pi\)
\(578\) 0 0
\(579\) −18.2462 −0.758287
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) −12.8078 −0.530443
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) 40.4924 1.67130 0.835651 0.549261i \(-0.185091\pi\)
0.835651 + 0.549261i \(0.185091\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −46.7386 −1.92257
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) −0.630683 −0.0258555
\(596\) 0 0
\(597\) 29.9309 1.22499
\(598\) 0 0
\(599\) 17.8617 0.729811 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(600\) 0 0
\(601\) 21.3693 0.871673 0.435836 0.900026i \(-0.356453\pi\)
0.435836 + 0.900026i \(0.356453\pi\)
\(602\) 0 0
\(603\) 27.3693 1.11456
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 13.6155 0.552637 0.276319 0.961066i \(-0.410885\pi\)
0.276319 + 0.961066i \(0.410885\pi\)
\(608\) 0 0
\(609\) 8.17708 0.331352
\(610\) 0 0
\(611\) −4.49242 −0.181744
\(612\) 0 0
\(613\) 18.2462 0.736958 0.368479 0.929636i \(-0.379879\pi\)
0.368479 + 0.929636i \(0.379879\pi\)
\(614\) 0 0
\(615\) −10.2462 −0.413167
\(616\) 0 0
\(617\) 44.7386 1.80111 0.900555 0.434743i \(-0.143161\pi\)
0.900555 + 0.434743i \(0.143161\pi\)
\(618\) 0 0
\(619\) 9.75379 0.392038 0.196019 0.980600i \(-0.437199\pi\)
0.196019 + 0.980600i \(0.437199\pi\)
\(620\) 0 0
\(621\) 2.06913 0.0830313
\(622\) 0 0
\(623\) −0.492423 −0.0197285
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 2.56155 0.102299
\(628\) 0 0
\(629\) 2.87689 0.114709
\(630\) 0 0
\(631\) −3.50758 −0.139634 −0.0698172 0.997560i \(-0.522242\pi\)
−0.0698172 + 0.997560i \(0.522242\pi\)
\(632\) 0 0
\(633\) −0.807764 −0.0321057
\(634\) 0 0
\(635\) −4.49242 −0.178276
\(636\) 0 0
\(637\) −3.75379 −0.148731
\(638\) 0 0
\(639\) −21.3693 −0.845357
\(640\) 0 0
\(641\) 31.6155 1.24874 0.624369 0.781129i \(-0.285356\pi\)
0.624369 + 0.781129i \(0.285356\pi\)
\(642\) 0 0
\(643\) 12.6307 0.498106 0.249053 0.968490i \(-0.419881\pi\)
0.249053 + 0.968490i \(0.419881\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.5464 1.86924 0.934621 0.355646i \(-0.115739\pi\)
0.934621 + 0.355646i \(0.115739\pi\)
\(648\) 0 0
\(649\) −7.68466 −0.301649
\(650\) 0 0
\(651\) 2.87689 0.112754
\(652\) 0 0
\(653\) −15.6155 −0.611083 −0.305541 0.952179i \(-0.598837\pi\)
−0.305541 + 0.952179i \(0.598837\pi\)
\(654\) 0 0
\(655\) −2.24621 −0.0877667
\(656\) 0 0
\(657\) −34.4924 −1.34568
\(658\) 0 0
\(659\) 14.4233 0.561852 0.280926 0.959729i \(-0.409359\pi\)
0.280926 + 0.959729i \(0.409359\pi\)
\(660\) 0 0
\(661\) −26.5616 −1.03312 −0.516562 0.856250i \(-0.672789\pi\)
−0.516562 + 0.856250i \(0.672789\pi\)
\(662\) 0 0
\(663\) 0.807764 0.0313710
\(664\) 0 0
\(665\) 1.12311 0.0435522
\(666\) 0 0
\(667\) 8.17708 0.316618
\(668\) 0 0
\(669\) 63.3693 2.45000
\(670\) 0 0
\(671\) −6.24621 −0.241132
\(672\) 0 0
\(673\) −27.1231 −1.04552 −0.522759 0.852480i \(-0.675097\pi\)
−0.522759 + 0.852480i \(0.675097\pi\)
\(674\) 0 0
\(675\) 1.43845 0.0553659
\(676\) 0 0
\(677\) 6.17708 0.237405 0.118702 0.992930i \(-0.462127\pi\)
0.118702 + 0.992930i \(0.462127\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 72.1771 2.76583
\(682\) 0 0
\(683\) −28.4924 −1.09023 −0.545116 0.838361i \(-0.683514\pi\)
−0.545116 + 0.838361i \(0.683514\pi\)
\(684\) 0 0
\(685\) 25.1231 0.959905
\(686\) 0 0
\(687\) −32.9848 −1.25845
\(688\) 0 0
\(689\) 7.19224 0.274002
\(690\) 0 0
\(691\) 2.38447 0.0907096 0.0453548 0.998971i \(-0.485558\pi\)
0.0453548 + 0.998971i \(0.485558\pi\)
\(692\) 0 0
\(693\) −2.00000 −0.0759737
\(694\) 0 0
\(695\) −14.7386 −0.559068
\(696\) 0 0
\(697\) −1.12311 −0.0425407
\(698\) 0 0
\(699\) 73.6155 2.78439
\(700\) 0 0
\(701\) −40.9848 −1.54798 −0.773988 0.633200i \(-0.781741\pi\)
−0.773988 + 0.633200i \(0.781741\pi\)
\(702\) 0 0
\(703\) −5.12311 −0.193222
\(704\) 0 0
\(705\) 40.9848 1.54358
\(706\) 0 0
\(707\) −3.86174 −0.145236
\(708\) 0 0
\(709\) −42.4924 −1.59584 −0.797918 0.602766i \(-0.794065\pi\)
−0.797918 + 0.602766i \(0.794065\pi\)
\(710\) 0 0
\(711\) 14.2462 0.534275
\(712\) 0 0
\(713\) 2.87689 0.107741
\(714\) 0 0
\(715\) −1.12311 −0.0420018
\(716\) 0 0
\(717\) 61.3002 2.28930
\(718\) 0 0
\(719\) 10.5616 0.393879 0.196940 0.980416i \(-0.436900\pi\)
0.196940 + 0.980416i \(0.436900\pi\)
\(720\) 0 0
\(721\) 7.50758 0.279597
\(722\) 0 0
\(723\) 50.2462 1.86868
\(724\) 0 0
\(725\) 5.68466 0.211123
\(726\) 0 0
\(727\) −22.4233 −0.831634 −0.415817 0.909448i \(-0.636504\pi\)
−0.415817 + 0.909448i \(0.636504\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 26.1080 0.964319 0.482160 0.876083i \(-0.339853\pi\)
0.482160 + 0.876083i \(0.339853\pi\)
\(734\) 0 0
\(735\) 34.2462 1.26319
\(736\) 0 0
\(737\) −7.68466 −0.283068
\(738\) 0 0
\(739\) 44.9848 1.65479 0.827397 0.561617i \(-0.189821\pi\)
0.827397 + 0.561617i \(0.189821\pi\)
\(740\) 0 0
\(741\) −1.43845 −0.0528427
\(742\) 0 0
\(743\) 42.2462 1.54986 0.774932 0.632045i \(-0.217784\pi\)
0.774932 + 0.632045i \(0.217784\pi\)
\(744\) 0 0
\(745\) −13.7538 −0.503900
\(746\) 0 0
\(747\) −50.7386 −1.85643
\(748\) 0 0
\(749\) 5.57671 0.203768
\(750\) 0 0
\(751\) −41.2311 −1.50454 −0.752271 0.658853i \(-0.771042\pi\)
−0.752271 + 0.658853i \(0.771042\pi\)
\(752\) 0 0
\(753\) 43.8617 1.59841
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −42.4924 −1.54441 −0.772207 0.635371i \(-0.780847\pi\)
−0.772207 + 0.635371i \(0.780847\pi\)
\(758\) 0 0
\(759\) −3.68466 −0.133745
\(760\) 0 0
\(761\) −42.8078 −1.55178 −0.775890 0.630868i \(-0.782699\pi\)
−0.775890 + 0.630868i \(0.782699\pi\)
\(762\) 0 0
\(763\) −3.54640 −0.128388
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 4.31534 0.155818
\(768\) 0 0
\(769\) −13.6847 −0.493481 −0.246741 0.969082i \(-0.579360\pi\)
−0.246741 + 0.969082i \(0.579360\pi\)
\(770\) 0 0
\(771\) 56.3542 2.02955
\(772\) 0 0
\(773\) 37.9309 1.36428 0.682139 0.731222i \(-0.261050\pi\)
0.682139 + 0.731222i \(0.261050\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 7.36932 0.264373
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 8.17708 0.292225
\(784\) 0 0
\(785\) −28.9848 −1.03451
\(786\) 0 0
\(787\) −15.6847 −0.559098 −0.279549 0.960131i \(-0.590185\pi\)
−0.279549 + 0.960131i \(0.590185\pi\)
\(788\) 0 0
\(789\) 22.7386 0.809517
\(790\) 0 0
\(791\) −1.75379 −0.0623575
\(792\) 0 0
\(793\) 3.50758 0.124558
\(794\) 0 0
\(795\) −65.6155 −2.32714
\(796\) 0 0
\(797\) −15.1922 −0.538137 −0.269068 0.963121i \(-0.586716\pi\)
−0.269068 + 0.963121i \(0.586716\pi\)
\(798\) 0 0
\(799\) 4.49242 0.158930
\(800\) 0 0
\(801\) −3.12311 −0.110350
\(802\) 0 0
\(803\) 9.68466 0.341764
\(804\) 0 0
\(805\) −1.61553 −0.0569399
\(806\) 0 0
\(807\) −7.36932 −0.259412
\(808\) 0 0
\(809\) 8.06913 0.283696 0.141848 0.989888i \(-0.454696\pi\)
0.141848 + 0.989888i \(0.454696\pi\)
\(810\) 0 0
\(811\) −8.31534 −0.291991 −0.145996 0.989285i \(-0.546639\pi\)
−0.145996 + 0.989285i \(0.546639\pi\)
\(812\) 0 0
\(813\) 65.7926 2.30745
\(814\) 0 0
\(815\) −22.7386 −0.796500
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 1.12311 0.0392445
\(820\) 0 0
\(821\) −20.9848 −0.732376 −0.366188 0.930541i \(-0.619337\pi\)
−0.366188 + 0.930541i \(0.619337\pi\)
\(822\) 0 0
\(823\) 13.9309 0.485600 0.242800 0.970076i \(-0.421934\pi\)
0.242800 + 0.970076i \(0.421934\pi\)
\(824\) 0 0
\(825\) −2.56155 −0.0891818
\(826\) 0 0
\(827\) −49.9309 −1.73627 −0.868133 0.496331i \(-0.834680\pi\)
−0.868133 + 0.496331i \(0.834680\pi\)
\(828\) 0 0
\(829\) 12.1771 0.422928 0.211464 0.977386i \(-0.432177\pi\)
0.211464 + 0.977386i \(0.432177\pi\)
\(830\) 0 0
\(831\) −21.7538 −0.754631
\(832\) 0 0
\(833\) 3.75379 0.130061
\(834\) 0 0
\(835\) 1.26137 0.0436514
\(836\) 0 0
\(837\) 2.87689 0.0994400
\(838\) 0 0
\(839\) 7.12311 0.245917 0.122958 0.992412i \(-0.460762\pi\)
0.122958 + 0.992412i \(0.460762\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) 0 0
\(843\) 64.9848 2.23820
\(844\) 0 0
\(845\) −25.3693 −0.872731
\(846\) 0 0
\(847\) 0.561553 0.0192952
\(848\) 0 0
\(849\) 29.1231 0.999502
\(850\) 0 0
\(851\) 7.36932 0.252617
\(852\) 0 0
\(853\) −52.3542 −1.79257 −0.896286 0.443476i \(-0.853745\pi\)
−0.896286 + 0.443476i \(0.853745\pi\)
\(854\) 0 0
\(855\) 7.12311 0.243605
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) −15.8617 −0.541196 −0.270598 0.962692i \(-0.587221\pi\)
−0.270598 + 0.962692i \(0.587221\pi\)
\(860\) 0 0
\(861\) −2.87689 −0.0980443
\(862\) 0 0
\(863\) −0.246211 −0.00838113 −0.00419056 0.999991i \(-0.501334\pi\)
−0.00419056 + 0.999991i \(0.501334\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) 42.7386 1.45148
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) 4.31534 0.146220
\(872\) 0 0
\(873\) −25.3693 −0.858621
\(874\) 0 0
\(875\) −6.73863 −0.227807
\(876\) 0 0
\(877\) −44.5616 −1.50474 −0.752368 0.658743i \(-0.771089\pi\)
−0.752368 + 0.658743i \(0.771089\pi\)
\(878\) 0 0
\(879\) −10.0691 −0.339623
\(880\) 0 0
\(881\) 20.7386 0.698702 0.349351 0.936992i \(-0.386402\pi\)
0.349351 + 0.936992i \(0.386402\pi\)
\(882\) 0 0
\(883\) 5.26137 0.177059 0.0885295 0.996074i \(-0.471783\pi\)
0.0885295 + 0.996074i \(0.471783\pi\)
\(884\) 0 0
\(885\) −39.3693 −1.32339
\(886\) 0 0
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) −1.26137 −0.0423049
\(890\) 0 0
\(891\) 7.00000 0.234509
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −11.5076 −0.384656
\(896\) 0 0
\(897\) 2.06913 0.0690863
\(898\) 0 0
\(899\) 11.3693 0.379188
\(900\) 0 0
\(901\) −7.19224 −0.239608
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −21.7538 −0.723120
\(906\) 0 0
\(907\) 29.7926 0.989247 0.494624 0.869107i \(-0.335306\pi\)
0.494624 + 0.869107i \(0.335306\pi\)
\(908\) 0 0
\(909\) −24.4924 −0.812362
\(910\) 0 0
\(911\) −18.9848 −0.628996 −0.314498 0.949258i \(-0.601836\pi\)
−0.314498 + 0.949258i \(0.601836\pi\)
\(912\) 0 0
\(913\) 14.2462 0.471481
\(914\) 0 0
\(915\) −32.0000 −1.05789
\(916\) 0 0
\(917\) −0.630683 −0.0208270
\(918\) 0 0
\(919\) −32.5616 −1.07411 −0.537053 0.843548i \(-0.680463\pi\)
−0.537053 + 0.843548i \(0.680463\pi\)
\(920\) 0 0
\(921\) −56.9848 −1.87771
\(922\) 0 0
\(923\) −3.36932 −0.110902
\(924\) 0 0
\(925\) 5.12311 0.168447
\(926\) 0 0
\(927\) 47.6155 1.56390
\(928\) 0 0
\(929\) 11.9309 0.391439 0.195720 0.980660i \(-0.437296\pi\)
0.195720 + 0.980660i \(0.437296\pi\)
\(930\) 0 0
\(931\) −6.68466 −0.219081
\(932\) 0 0
\(933\) −12.3153 −0.403186
\(934\) 0 0
\(935\) 1.12311 0.0367295
\(936\) 0 0
\(937\) −0.699813 −0.0228619 −0.0114310 0.999935i \(-0.503639\pi\)
−0.0114310 + 0.999935i \(0.503639\pi\)
\(938\) 0 0
\(939\) 1.43845 0.0469419
\(940\) 0 0
\(941\) 24.5616 0.800684 0.400342 0.916366i \(-0.368891\pi\)
0.400342 + 0.916366i \(0.368891\pi\)
\(942\) 0 0
\(943\) −2.87689 −0.0936846
\(944\) 0 0
\(945\) −1.61553 −0.0525531
\(946\) 0 0
\(947\) 36.9848 1.20185 0.600923 0.799307i \(-0.294800\pi\)
0.600923 + 0.799307i \(0.294800\pi\)
\(948\) 0 0
\(949\) −5.43845 −0.176539
\(950\) 0 0
\(951\) 18.0691 0.585932
\(952\) 0 0
\(953\) −0.876894 −0.0284054 −0.0142027 0.999899i \(-0.504521\pi\)
−0.0142027 + 0.999899i \(0.504521\pi\)
\(954\) 0 0
\(955\) 16.6307 0.538156
\(956\) 0 0
\(957\) −14.5616 −0.470708
\(958\) 0 0
\(959\) 7.05398 0.227785
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 35.3693 1.13976
\(964\) 0 0
\(965\) 14.2462 0.458602
\(966\) 0 0
\(967\) −53.8617 −1.73208 −0.866038 0.499978i \(-0.833342\pi\)
−0.866038 + 0.499978i \(0.833342\pi\)
\(968\) 0 0
\(969\) 1.43845 0.0462096
\(970\) 0 0
\(971\) −2.24621 −0.0720843 −0.0360422 0.999350i \(-0.511475\pi\)
−0.0360422 + 0.999350i \(0.511475\pi\)
\(972\) 0 0
\(973\) −4.13826 −0.132667
\(974\) 0 0
\(975\) 1.43845 0.0460672
\(976\) 0 0
\(977\) 30.6307 0.979962 0.489981 0.871733i \(-0.337004\pi\)
0.489981 + 0.871733i \(0.337004\pi\)
\(978\) 0 0
\(979\) 0.876894 0.0280257
\(980\) 0 0
\(981\) −22.4924 −0.718128
\(982\) 0 0
\(983\) 31.7538 1.01279 0.506394 0.862302i \(-0.330978\pi\)
0.506394 + 0.862302i \(0.330978\pi\)
\(984\) 0 0
\(985\) 36.4924 1.16275
\(986\) 0 0
\(987\) 11.5076 0.366290
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −12.8769 −0.409048 −0.204524 0.978862i \(-0.565565\pi\)
−0.204524 + 0.978862i \(0.565565\pi\)
\(992\) 0 0
\(993\) −37.3002 −1.18369
\(994\) 0 0
\(995\) −23.3693 −0.740857
\(996\) 0 0
\(997\) −28.6307 −0.906743 −0.453371 0.891322i \(-0.649779\pi\)
−0.453371 + 0.891322i \(0.649779\pi\)
\(998\) 0 0
\(999\) 7.36932 0.233155
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.1.1 2
4.3 odd 2 418.2.a.e.1.2 2
12.11 even 2 3762.2.a.y.1.1 2
44.43 even 2 4598.2.a.bj.1.2 2
76.75 even 2 7942.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.e.1.2 2 4.3 odd 2
3344.2.a.k.1.1 2 1.1 even 1 trivial
3762.2.a.y.1.1 2 12.11 even 2
4598.2.a.bj.1.2 2 44.43 even 2
7942.2.a.x.1.1 2 76.75 even 2