Properties

Label 1216.4.a.m.1.2
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.27492 q^{3} -1.27492 q^{5} +12.0997 q^{7} -8.72508 q^{9} +O(q^{10})\) \(q+4.27492 q^{3} -1.27492 q^{5} +12.0997 q^{7} -8.72508 q^{9} -18.9244 q^{11} +55.9244 q^{13} -5.45017 q^{15} -89.7492 q^{17} +19.0000 q^{19} +51.7251 q^{21} -135.072 q^{23} -123.375 q^{25} -152.722 q^{27} +102.474 q^{29} -103.698 q^{31} -80.9003 q^{33} -15.4261 q^{35} -29.6977 q^{37} +239.072 q^{39} +234.743 q^{41} +53.3231 q^{43} +11.1238 q^{45} -33.3713 q^{47} -196.598 q^{49} -383.670 q^{51} -93.1752 q^{53} +24.1271 q^{55} +81.2234 q^{57} -637.320 q^{59} +125.571 q^{61} -105.571 q^{63} -71.2990 q^{65} -119.375 q^{67} -577.423 q^{69} +18.5083 q^{71} -394.794 q^{73} -527.416 q^{75} -228.979 q^{77} +303.341 q^{79} -417.296 q^{81} +394.337 q^{83} +114.423 q^{85} +438.069 q^{87} -1021.88 q^{89} +676.667 q^{91} -443.299 q^{93} -24.2234 q^{95} +1322.82 q^{97} +165.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 5q^{5} - 6q^{7} - 25q^{9} + O(q^{10}) \) \( 2q + q^{3} + 5q^{5} - 6q^{7} - 25q^{9} + 15q^{11} + 59q^{13} - 26q^{15} - 104q^{17} + 38q^{19} + 111q^{21} - 21q^{23} - 209q^{25} - 11q^{27} + 137q^{29} + 4q^{31} - 192q^{33} - 129q^{35} + 152q^{37} + 229q^{39} - 210q^{41} - 67q^{43} - 91q^{45} + 273q^{47} - 212q^{49} - 337q^{51} - 209q^{53} + 237q^{55} + 19q^{57} - 799q^{59} - 149q^{61} + 189q^{63} - 52q^{65} - 201q^{67} - 951q^{69} + 792q^{71} - 246q^{73} - 247q^{75} - 843q^{77} - 254q^{79} - 442q^{81} - 374q^{83} + 25q^{85} + 325q^{87} - 564q^{89} + 621q^{91} - 796q^{93} + 95q^{95} - 178q^{97} - 387q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 4.27492 0.822708 0.411354 0.911476i \(-0.365056\pi\)
0.411354 + 0.911476i \(0.365056\pi\)
\(4\) 0 0
\(5\) −1.27492 −0.114032 −0.0570160 0.998373i \(-0.518159\pi\)
−0.0570160 + 0.998373i \(0.518159\pi\)
\(6\) 0 0
\(7\) 12.0997 0.653321 0.326660 0.945142i \(-0.394077\pi\)
0.326660 + 0.945142i \(0.394077\pi\)
\(8\) 0 0
\(9\) −8.72508 −0.323151
\(10\) 0 0
\(11\) −18.9244 −0.518721 −0.259360 0.965781i \(-0.583512\pi\)
−0.259360 + 0.965781i \(0.583512\pi\)
\(12\) 0 0
\(13\) 55.9244 1.19313 0.596563 0.802566i \(-0.296532\pi\)
0.596563 + 0.802566i \(0.296532\pi\)
\(14\) 0 0
\(15\) −5.45017 −0.0938151
\(16\) 0 0
\(17\) −89.7492 −1.28043 −0.640217 0.768194i \(-0.721155\pi\)
−0.640217 + 0.768194i \(0.721155\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 51.7251 0.537492
\(22\) 0 0
\(23\) −135.072 −1.22454 −0.612272 0.790647i \(-0.709744\pi\)
−0.612272 + 0.790647i \(0.709744\pi\)
\(24\) 0 0
\(25\) −123.375 −0.986997
\(26\) 0 0
\(27\) −152.722 −1.08857
\(28\) 0 0
\(29\) 102.474 0.656172 0.328086 0.944648i \(-0.393596\pi\)
0.328086 + 0.944648i \(0.393596\pi\)
\(30\) 0 0
\(31\) −103.698 −0.600795 −0.300398 0.953814i \(-0.597119\pi\)
−0.300398 + 0.953814i \(0.597119\pi\)
\(32\) 0 0
\(33\) −80.9003 −0.426756
\(34\) 0 0
\(35\) −15.4261 −0.0744995
\(36\) 0 0
\(37\) −29.6977 −0.131953 −0.0659766 0.997821i \(-0.521016\pi\)
−0.0659766 + 0.997821i \(0.521016\pi\)
\(38\) 0 0
\(39\) 239.072 0.981595
\(40\) 0 0
\(41\) 234.743 0.894162 0.447081 0.894494i \(-0.352464\pi\)
0.447081 + 0.894494i \(0.352464\pi\)
\(42\) 0 0
\(43\) 53.3231 0.189109 0.0945546 0.995520i \(-0.469857\pi\)
0.0945546 + 0.995520i \(0.469857\pi\)
\(44\) 0 0
\(45\) 11.1238 0.0368496
\(46\) 0 0
\(47\) −33.3713 −0.103568 −0.0517841 0.998658i \(-0.516491\pi\)
−0.0517841 + 0.998658i \(0.516491\pi\)
\(48\) 0 0
\(49\) −196.598 −0.573172
\(50\) 0 0
\(51\) −383.670 −1.05342
\(52\) 0 0
\(53\) −93.1752 −0.241483 −0.120742 0.992684i \(-0.538527\pi\)
−0.120742 + 0.992684i \(0.538527\pi\)
\(54\) 0 0
\(55\) 24.1271 0.0591508
\(56\) 0 0
\(57\) 81.2234 0.188742
\(58\) 0 0
\(59\) −637.320 −1.40630 −0.703152 0.711039i \(-0.748225\pi\)
−0.703152 + 0.711039i \(0.748225\pi\)
\(60\) 0 0
\(61\) 125.571 0.263568 0.131784 0.991278i \(-0.457929\pi\)
0.131784 + 0.991278i \(0.457929\pi\)
\(62\) 0 0
\(63\) −105.571 −0.211121
\(64\) 0 0
\(65\) −71.2990 −0.136055
\(66\) 0 0
\(67\) −119.375 −0.217671 −0.108835 0.994060i \(-0.534712\pi\)
−0.108835 + 0.994060i \(0.534712\pi\)
\(68\) 0 0
\(69\) −577.423 −1.00744
\(70\) 0 0
\(71\) 18.5083 0.0309370 0.0154685 0.999880i \(-0.495076\pi\)
0.0154685 + 0.999880i \(0.495076\pi\)
\(72\) 0 0
\(73\) −394.794 −0.632975 −0.316487 0.948597i \(-0.602503\pi\)
−0.316487 + 0.948597i \(0.602503\pi\)
\(74\) 0 0
\(75\) −527.416 −0.812010
\(76\) 0 0
\(77\) −228.979 −0.338891
\(78\) 0 0
\(79\) 303.341 0.432006 0.216003 0.976393i \(-0.430698\pi\)
0.216003 + 0.976393i \(0.430698\pi\)
\(80\) 0 0
\(81\) −417.296 −0.572422
\(82\) 0 0
\(83\) 394.337 0.521496 0.260748 0.965407i \(-0.416031\pi\)
0.260748 + 0.965407i \(0.416031\pi\)
\(84\) 0 0
\(85\) 114.423 0.146010
\(86\) 0 0
\(87\) 438.069 0.539838
\(88\) 0 0
\(89\) −1021.88 −1.21707 −0.608536 0.793526i \(-0.708243\pi\)
−0.608536 + 0.793526i \(0.708243\pi\)
\(90\) 0 0
\(91\) 676.667 0.779494
\(92\) 0 0
\(93\) −443.299 −0.494279
\(94\) 0 0
\(95\) −24.2234 −0.0261607
\(96\) 0 0
\(97\) 1322.82 1.38466 0.692330 0.721581i \(-0.256584\pi\)
0.692330 + 0.721581i \(0.256584\pi\)
\(98\) 0 0
\(99\) 165.117 0.167625
\(100\) 0 0
\(101\) −1923.06 −1.89457 −0.947285 0.320391i \(-0.896186\pi\)
−0.947285 + 0.320391i \(0.896186\pi\)
\(102\) 0 0
\(103\) 467.774 0.447487 0.223743 0.974648i \(-0.428172\pi\)
0.223743 + 0.974648i \(0.428172\pi\)
\(104\) 0 0
\(105\) −65.9452 −0.0612914
\(106\) 0 0
\(107\) −260.468 −0.235330 −0.117665 0.993053i \(-0.537541\pi\)
−0.117665 + 0.993053i \(0.537541\pi\)
\(108\) 0 0
\(109\) 511.856 0.449789 0.224894 0.974383i \(-0.427796\pi\)
0.224894 + 0.974383i \(0.427796\pi\)
\(110\) 0 0
\(111\) −126.955 −0.108559
\(112\) 0 0
\(113\) −1453.48 −1.21002 −0.605008 0.796220i \(-0.706830\pi\)
−0.605008 + 0.796220i \(0.706830\pi\)
\(114\) 0 0
\(115\) 172.206 0.139637
\(116\) 0 0
\(117\) −487.945 −0.385560
\(118\) 0 0
\(119\) −1085.94 −0.836534
\(120\) 0 0
\(121\) −972.866 −0.730929
\(122\) 0 0
\(123\) 1003.50 0.735634
\(124\) 0 0
\(125\) 316.657 0.226581
\(126\) 0 0
\(127\) −2166.29 −1.51360 −0.756799 0.653647i \(-0.773238\pi\)
−0.756799 + 0.653647i \(0.773238\pi\)
\(128\) 0 0
\(129\) 227.952 0.155582
\(130\) 0 0
\(131\) −329.310 −0.219633 −0.109817 0.993952i \(-0.535026\pi\)
−0.109817 + 0.993952i \(0.535026\pi\)
\(132\) 0 0
\(133\) 229.894 0.149882
\(134\) 0 0
\(135\) 194.708 0.124132
\(136\) 0 0
\(137\) 736.919 0.459556 0.229778 0.973243i \(-0.426200\pi\)
0.229778 + 0.973243i \(0.426200\pi\)
\(138\) 0 0
\(139\) 3041.10 1.85571 0.927853 0.372947i \(-0.121653\pi\)
0.927853 + 0.372947i \(0.121653\pi\)
\(140\) 0 0
\(141\) −142.659 −0.0852063
\(142\) 0 0
\(143\) −1058.34 −0.618899
\(144\) 0 0
\(145\) −130.646 −0.0748247
\(146\) 0 0
\(147\) −840.440 −0.471553
\(148\) 0 0
\(149\) −2156.31 −1.18558 −0.592790 0.805357i \(-0.701974\pi\)
−0.592790 + 0.805357i \(0.701974\pi\)
\(150\) 0 0
\(151\) −1816.60 −0.979024 −0.489512 0.871997i \(-0.662825\pi\)
−0.489512 + 0.871997i \(0.662825\pi\)
\(152\) 0 0
\(153\) 783.069 0.413774
\(154\) 0 0
\(155\) 132.206 0.0685099
\(156\) 0 0
\(157\) 1118.10 0.568368 0.284184 0.958770i \(-0.408277\pi\)
0.284184 + 0.958770i \(0.408277\pi\)
\(158\) 0 0
\(159\) −398.316 −0.198670
\(160\) 0 0
\(161\) −1634.33 −0.800020
\(162\) 0 0
\(163\) 3304.95 1.58812 0.794060 0.607839i \(-0.207963\pi\)
0.794060 + 0.607839i \(0.207963\pi\)
\(164\) 0 0
\(165\) 103.141 0.0486638
\(166\) 0 0
\(167\) 1750.74 0.811237 0.405618 0.914043i \(-0.367056\pi\)
0.405618 + 0.914043i \(0.367056\pi\)
\(168\) 0 0
\(169\) 930.541 0.423551
\(170\) 0 0
\(171\) −165.777 −0.0741360
\(172\) 0 0
\(173\) −2698.18 −1.18577 −0.592887 0.805286i \(-0.702012\pi\)
−0.592887 + 0.805286i \(0.702012\pi\)
\(174\) 0 0
\(175\) −1492.79 −0.644825
\(176\) 0 0
\(177\) −2724.49 −1.15698
\(178\) 0 0
\(179\) −3867.80 −1.61505 −0.807523 0.589836i \(-0.799192\pi\)
−0.807523 + 0.589836i \(0.799192\pi\)
\(180\) 0 0
\(181\) −3858.74 −1.58463 −0.792315 0.610112i \(-0.791125\pi\)
−0.792315 + 0.610112i \(0.791125\pi\)
\(182\) 0 0
\(183\) 536.804 0.216840
\(184\) 0 0
\(185\) 37.8621 0.0150469
\(186\) 0 0
\(187\) 1698.45 0.664187
\(188\) 0 0
\(189\) −1847.88 −0.711184
\(190\) 0 0
\(191\) 4111.06 1.55741 0.778706 0.627389i \(-0.215876\pi\)
0.778706 + 0.627389i \(0.215876\pi\)
\(192\) 0 0
\(193\) −1825.36 −0.680789 −0.340395 0.940283i \(-0.610561\pi\)
−0.340395 + 0.940283i \(0.610561\pi\)
\(194\) 0 0
\(195\) −304.797 −0.111933
\(196\) 0 0
\(197\) 767.088 0.277425 0.138713 0.990333i \(-0.455704\pi\)
0.138713 + 0.990333i \(0.455704\pi\)
\(198\) 0 0
\(199\) 3176.43 1.13151 0.565757 0.824572i \(-0.308584\pi\)
0.565757 + 0.824572i \(0.308584\pi\)
\(200\) 0 0
\(201\) −510.316 −0.179079
\(202\) 0 0
\(203\) 1239.90 0.428691
\(204\) 0 0
\(205\) −299.277 −0.101963
\(206\) 0 0
\(207\) 1178.52 0.395713
\(208\) 0 0
\(209\) −359.564 −0.119003
\(210\) 0 0
\(211\) −2460.54 −0.802797 −0.401399 0.915903i \(-0.631476\pi\)
−0.401399 + 0.915903i \(0.631476\pi\)
\(212\) 0 0
\(213\) 79.1214 0.0254521
\(214\) 0 0
\(215\) −67.9825 −0.0215645
\(216\) 0 0
\(217\) −1254.71 −0.392512
\(218\) 0 0
\(219\) −1687.71 −0.520753
\(220\) 0 0
\(221\) −5019.17 −1.52772
\(222\) 0 0
\(223\) 3731.73 1.12061 0.560304 0.828287i \(-0.310684\pi\)
0.560304 + 0.828287i \(0.310684\pi\)
\(224\) 0 0
\(225\) 1076.45 0.318949
\(226\) 0 0
\(227\) −653.188 −0.190985 −0.0954926 0.995430i \(-0.530443\pi\)
−0.0954926 + 0.995430i \(0.530443\pi\)
\(228\) 0 0
\(229\) −340.511 −0.0982602 −0.0491301 0.998792i \(-0.515645\pi\)
−0.0491301 + 0.998792i \(0.515645\pi\)
\(230\) 0 0
\(231\) −978.867 −0.278808
\(232\) 0 0
\(233\) −3936.99 −1.10696 −0.553478 0.832864i \(-0.686700\pi\)
−0.553478 + 0.832864i \(0.686700\pi\)
\(234\) 0 0
\(235\) 42.5456 0.0118101
\(236\) 0 0
\(237\) 1296.76 0.355415
\(238\) 0 0
\(239\) 4762.18 1.28887 0.644434 0.764660i \(-0.277093\pi\)
0.644434 + 0.764660i \(0.277093\pi\)
\(240\) 0 0
\(241\) −3893.55 −1.04069 −0.520343 0.853957i \(-0.674196\pi\)
−0.520343 + 0.853957i \(0.674196\pi\)
\(242\) 0 0
\(243\) 2339.58 0.617631
\(244\) 0 0
\(245\) 250.646 0.0653600
\(246\) 0 0
\(247\) 1062.56 0.273722
\(248\) 0 0
\(249\) 1685.76 0.429039
\(250\) 0 0
\(251\) −1384.13 −0.348069 −0.174034 0.984740i \(-0.555680\pi\)
−0.174034 + 0.984740i \(0.555680\pi\)
\(252\) 0 0
\(253\) 2556.16 0.635196
\(254\) 0 0
\(255\) 489.148 0.120124
\(256\) 0 0
\(257\) −4645.01 −1.12742 −0.563711 0.825972i \(-0.690627\pi\)
−0.563711 + 0.825972i \(0.690627\pi\)
\(258\) 0 0
\(259\) −359.332 −0.0862078
\(260\) 0 0
\(261\) −894.096 −0.212043
\(262\) 0 0
\(263\) −2151.41 −0.504416 −0.252208 0.967673i \(-0.581157\pi\)
−0.252208 + 0.967673i \(0.581157\pi\)
\(264\) 0 0
\(265\) 118.791 0.0275368
\(266\) 0 0
\(267\) −4368.47 −1.00130
\(268\) 0 0
\(269\) 5768.66 1.30752 0.653758 0.756704i \(-0.273192\pi\)
0.653758 + 0.756704i \(0.273192\pi\)
\(270\) 0 0
\(271\) −6859.23 −1.53752 −0.768761 0.639537i \(-0.779126\pi\)
−0.768761 + 0.639537i \(0.779126\pi\)
\(272\) 0 0
\(273\) 2892.70 0.641296
\(274\) 0 0
\(275\) 2334.79 0.511976
\(276\) 0 0
\(277\) −1237.24 −0.268371 −0.134185 0.990956i \(-0.542842\pi\)
−0.134185 + 0.990956i \(0.542842\pi\)
\(278\) 0 0
\(279\) 904.771 0.194148
\(280\) 0 0
\(281\) 4355.21 0.924592 0.462296 0.886726i \(-0.347026\pi\)
0.462296 + 0.886726i \(0.347026\pi\)
\(282\) 0 0
\(283\) 3651.29 0.766949 0.383474 0.923551i \(-0.374727\pi\)
0.383474 + 0.923551i \(0.374727\pi\)
\(284\) 0 0
\(285\) −103.553 −0.0215227
\(286\) 0 0
\(287\) 2840.31 0.584174
\(288\) 0 0
\(289\) 3141.91 0.639510
\(290\) 0 0
\(291\) 5654.94 1.13917
\(292\) 0 0
\(293\) 3391.25 0.676174 0.338087 0.941115i \(-0.390220\pi\)
0.338087 + 0.941115i \(0.390220\pi\)
\(294\) 0 0
\(295\) 812.530 0.160364
\(296\) 0 0
\(297\) 2890.17 0.564662
\(298\) 0 0
\(299\) −7553.84 −1.46104
\(300\) 0 0
\(301\) 645.192 0.123549
\(302\) 0 0
\(303\) −8220.92 −1.55868
\(304\) 0 0
\(305\) −160.092 −0.0300552
\(306\) 0 0
\(307\) −4343.35 −0.807452 −0.403726 0.914880i \(-0.632285\pi\)
−0.403726 + 0.914880i \(0.632285\pi\)
\(308\) 0 0
\(309\) 1999.70 0.368151
\(310\) 0 0
\(311\) 5671.28 1.03405 0.517024 0.855971i \(-0.327040\pi\)
0.517024 + 0.855971i \(0.327040\pi\)
\(312\) 0 0
\(313\) −9449.71 −1.70648 −0.853241 0.521516i \(-0.825367\pi\)
−0.853241 + 0.521516i \(0.825367\pi\)
\(314\) 0 0
\(315\) 134.594 0.0240746
\(316\) 0 0
\(317\) 5390.75 0.955125 0.477562 0.878598i \(-0.341520\pi\)
0.477562 + 0.878598i \(0.341520\pi\)
\(318\) 0 0
\(319\) −1939.27 −0.340370
\(320\) 0 0
\(321\) −1113.48 −0.193608
\(322\) 0 0
\(323\) −1705.23 −0.293752
\(324\) 0 0
\(325\) −6899.65 −1.17761
\(326\) 0 0
\(327\) 2188.14 0.370045
\(328\) 0 0
\(329\) −403.781 −0.0676632
\(330\) 0 0
\(331\) 9230.14 1.53273 0.766366 0.642404i \(-0.222063\pi\)
0.766366 + 0.642404i \(0.222063\pi\)
\(332\) 0 0
\(333\) 259.115 0.0426408
\(334\) 0 0
\(335\) 152.193 0.0248214
\(336\) 0 0
\(337\) −7815.90 −1.26338 −0.631690 0.775221i \(-0.717638\pi\)
−0.631690 + 0.775221i \(0.717638\pi\)
\(338\) 0 0
\(339\) −6213.50 −0.995490
\(340\) 0 0
\(341\) 1962.42 0.311645
\(342\) 0 0
\(343\) −6528.96 −1.02779
\(344\) 0 0
\(345\) 736.166 0.114881
\(346\) 0 0
\(347\) −680.076 −0.105211 −0.0526057 0.998615i \(-0.516753\pi\)
−0.0526057 + 0.998615i \(0.516753\pi\)
\(348\) 0 0
\(349\) −3641.72 −0.558558 −0.279279 0.960210i \(-0.590095\pi\)
−0.279279 + 0.960210i \(0.590095\pi\)
\(350\) 0 0
\(351\) −8540.88 −1.29880
\(352\) 0 0
\(353\) −7885.46 −1.18895 −0.594477 0.804113i \(-0.702641\pi\)
−0.594477 + 0.804113i \(0.702641\pi\)
\(354\) 0 0
\(355\) −23.5965 −0.00352781
\(356\) 0 0
\(357\) −4642.28 −0.688223
\(358\) 0 0
\(359\) −2862.17 −0.420779 −0.210389 0.977618i \(-0.567473\pi\)
−0.210389 + 0.977618i \(0.567473\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) −4158.92 −0.601341
\(364\) 0 0
\(365\) 503.330 0.0721794
\(366\) 0 0
\(367\) −9783.29 −1.39151 −0.695754 0.718280i \(-0.744930\pi\)
−0.695754 + 0.718280i \(0.744930\pi\)
\(368\) 0 0
\(369\) −2048.15 −0.288949
\(370\) 0 0
\(371\) −1127.39 −0.157766
\(372\) 0 0
\(373\) −6551.84 −0.909495 −0.454747 0.890621i \(-0.650270\pi\)
−0.454747 + 0.890621i \(0.650270\pi\)
\(374\) 0 0
\(375\) 1353.68 0.186410
\(376\) 0 0
\(377\) 5730.81 0.782896
\(378\) 0 0
\(379\) 2228.78 0.302071 0.151035 0.988528i \(-0.451739\pi\)
0.151035 + 0.988528i \(0.451739\pi\)
\(380\) 0 0
\(381\) −9260.71 −1.24525
\(382\) 0 0
\(383\) −5223.85 −0.696935 −0.348468 0.937321i \(-0.613298\pi\)
−0.348468 + 0.937321i \(0.613298\pi\)
\(384\) 0 0
\(385\) 291.930 0.0386444
\(386\) 0 0
\(387\) −465.248 −0.0611109
\(388\) 0 0
\(389\) 7672.26 0.999998 0.499999 0.866026i \(-0.333334\pi\)
0.499999 + 0.866026i \(0.333334\pi\)
\(390\) 0 0
\(391\) 12122.6 1.56795
\(392\) 0 0
\(393\) −1407.77 −0.180694
\(394\) 0 0
\(395\) −386.734 −0.0492625
\(396\) 0 0
\(397\) 13564.0 1.71475 0.857375 0.514692i \(-0.172094\pi\)
0.857375 + 0.514692i \(0.172094\pi\)
\(398\) 0 0
\(399\) 982.777 0.123309
\(400\) 0 0
\(401\) −6488.39 −0.808017 −0.404009 0.914755i \(-0.632383\pi\)
−0.404009 + 0.914755i \(0.632383\pi\)
\(402\) 0 0
\(403\) −5799.23 −0.716825
\(404\) 0 0
\(405\) 532.017 0.0652745
\(406\) 0 0
\(407\) 562.011 0.0684469
\(408\) 0 0
\(409\) 14696.2 1.77672 0.888360 0.459147i \(-0.151845\pi\)
0.888360 + 0.459147i \(0.151845\pi\)
\(410\) 0 0
\(411\) 3150.27 0.378081
\(412\) 0 0
\(413\) −7711.36 −0.918768
\(414\) 0 0
\(415\) −502.747 −0.0594672
\(416\) 0 0
\(417\) 13000.5 1.52670
\(418\) 0 0
\(419\) −9492.08 −1.10673 −0.553363 0.832940i \(-0.686656\pi\)
−0.553363 + 0.832940i \(0.686656\pi\)
\(420\) 0 0
\(421\) 13028.4 1.50823 0.754116 0.656741i \(-0.228065\pi\)
0.754116 + 0.656741i \(0.228065\pi\)
\(422\) 0 0
\(423\) 291.167 0.0334682
\(424\) 0 0
\(425\) 11072.8 1.26378
\(426\) 0 0
\(427\) 1519.36 0.172195
\(428\) 0 0
\(429\) −4524.30 −0.509174
\(430\) 0 0
\(431\) −5404.99 −0.604058 −0.302029 0.953299i \(-0.597664\pi\)
−0.302029 + 0.953299i \(0.597664\pi\)
\(432\) 0 0
\(433\) 16745.4 1.85850 0.929252 0.369448i \(-0.120453\pi\)
0.929252 + 0.369448i \(0.120453\pi\)
\(434\) 0 0
\(435\) −558.502 −0.0615589
\(436\) 0 0
\(437\) −2566.37 −0.280930
\(438\) 0 0
\(439\) 13422.6 1.45928 0.729641 0.683831i \(-0.239687\pi\)
0.729641 + 0.683831i \(0.239687\pi\)
\(440\) 0 0
\(441\) 1715.33 0.185221
\(442\) 0 0
\(443\) 14048.7 1.50672 0.753358 0.657611i \(-0.228433\pi\)
0.753358 + 0.657611i \(0.228433\pi\)
\(444\) 0 0
\(445\) 1302.82 0.138785
\(446\) 0 0
\(447\) −9218.03 −0.975387
\(448\) 0 0
\(449\) 9841.11 1.03437 0.517183 0.855875i \(-0.326980\pi\)
0.517183 + 0.855875i \(0.326980\pi\)
\(450\) 0 0
\(451\) −4442.37 −0.463820
\(452\) 0 0
\(453\) −7765.81 −0.805451
\(454\) 0 0
\(455\) −862.694 −0.0888873
\(456\) 0 0
\(457\) 4443.15 0.454796 0.227398 0.973802i \(-0.426978\pi\)
0.227398 + 0.973802i \(0.426978\pi\)
\(458\) 0 0
\(459\) 13706.7 1.39384
\(460\) 0 0
\(461\) 12288.1 1.24146 0.620728 0.784026i \(-0.286837\pi\)
0.620728 + 0.784026i \(0.286837\pi\)
\(462\) 0 0
\(463\) 4814.38 0.483246 0.241623 0.970370i \(-0.422320\pi\)
0.241623 + 0.970370i \(0.422320\pi\)
\(464\) 0 0
\(465\) 565.170 0.0563637
\(466\) 0 0
\(467\) −9863.03 −0.977316 −0.488658 0.872475i \(-0.662513\pi\)
−0.488658 + 0.872475i \(0.662513\pi\)
\(468\) 0 0
\(469\) −1444.39 −0.142209
\(470\) 0 0
\(471\) 4779.77 0.467601
\(472\) 0 0
\(473\) −1009.11 −0.0980949
\(474\) 0 0
\(475\) −2344.12 −0.226433
\(476\) 0 0
\(477\) 812.962 0.0780356
\(478\) 0 0
\(479\) 12743.3 1.21557 0.607785 0.794102i \(-0.292058\pi\)
0.607785 + 0.794102i \(0.292058\pi\)
\(480\) 0 0
\(481\) −1660.83 −0.157437
\(482\) 0 0
\(483\) −6986.62 −0.658183
\(484\) 0 0
\(485\) −1686.48 −0.157896
\(486\) 0 0
\(487\) −4200.07 −0.390807 −0.195404 0.980723i \(-0.562602\pi\)
−0.195404 + 0.980723i \(0.562602\pi\)
\(488\) 0 0
\(489\) 14128.4 1.30656
\(490\) 0 0
\(491\) 11292.3 1.03791 0.518955 0.854802i \(-0.326321\pi\)
0.518955 + 0.854802i \(0.326321\pi\)
\(492\) 0 0
\(493\) −9196.98 −0.840185
\(494\) 0 0
\(495\) −210.511 −0.0191146
\(496\) 0 0
\(497\) 223.944 0.0202118
\(498\) 0 0
\(499\) 12126.6 1.08790 0.543948 0.839119i \(-0.316929\pi\)
0.543948 + 0.839119i \(0.316929\pi\)
\(500\) 0 0
\(501\) 7484.28 0.667411
\(502\) 0 0
\(503\) −2033.61 −0.180267 −0.0901336 0.995930i \(-0.528729\pi\)
−0.0901336 + 0.995930i \(0.528729\pi\)
\(504\) 0 0
\(505\) 2451.74 0.216042
\(506\) 0 0
\(507\) 3977.98 0.348459
\(508\) 0 0
\(509\) 1367.41 0.119076 0.0595379 0.998226i \(-0.481037\pi\)
0.0595379 + 0.998226i \(0.481037\pi\)
\(510\) 0 0
\(511\) −4776.88 −0.413535
\(512\) 0 0
\(513\) −2901.71 −0.249734
\(514\) 0 0
\(515\) −596.373 −0.0510279
\(516\) 0 0
\(517\) 631.532 0.0537229
\(518\) 0 0
\(519\) −11534.5 −0.975545
\(520\) 0 0
\(521\) −9613.98 −0.808438 −0.404219 0.914662i \(-0.632457\pi\)
−0.404219 + 0.914662i \(0.632457\pi\)
\(522\) 0 0
\(523\) 8185.79 0.684397 0.342199 0.939628i \(-0.388828\pi\)
0.342199 + 0.939628i \(0.388828\pi\)
\(524\) 0 0
\(525\) −6381.56 −0.530503
\(526\) 0 0
\(527\) 9306.78 0.769278
\(528\) 0 0
\(529\) 6077.52 0.499508
\(530\) 0 0
\(531\) 5560.67 0.454449
\(532\) 0 0
\(533\) 13127.8 1.06685
\(534\) 0 0
\(535\) 332.075 0.0268352
\(536\) 0 0
\(537\) −16534.5 −1.32871
\(538\) 0 0
\(539\) 3720.50 0.297316
\(540\) 0 0
\(541\) 1869.73 0.148587 0.0742937 0.997236i \(-0.476330\pi\)
0.0742937 + 0.997236i \(0.476330\pi\)
\(542\) 0 0
\(543\) −16495.8 −1.30369
\(544\) 0 0
\(545\) −652.575 −0.0512903
\(546\) 0 0
\(547\) −4024.62 −0.314589 −0.157295 0.987552i \(-0.550277\pi\)
−0.157295 + 0.987552i \(0.550277\pi\)
\(548\) 0 0
\(549\) −1095.61 −0.0851724
\(550\) 0 0
\(551\) 1947.01 0.150536
\(552\) 0 0
\(553\) 3670.32 0.282239
\(554\) 0 0
\(555\) 161.857 0.0123792
\(556\) 0 0
\(557\) 15750.7 1.19816 0.599082 0.800688i \(-0.295532\pi\)
0.599082 + 0.800688i \(0.295532\pi\)
\(558\) 0 0
\(559\) 2982.06 0.225631
\(560\) 0 0
\(561\) 7260.74 0.546432
\(562\) 0 0
\(563\) 641.091 0.0479907 0.0239954 0.999712i \(-0.492361\pi\)
0.0239954 + 0.999712i \(0.492361\pi\)
\(564\) 0 0
\(565\) 1853.06 0.137981
\(566\) 0 0
\(567\) −5049.14 −0.373975
\(568\) 0 0
\(569\) −18392.3 −1.35509 −0.677544 0.735482i \(-0.736956\pi\)
−0.677544 + 0.735482i \(0.736956\pi\)
\(570\) 0 0
\(571\) −1400.26 −0.102625 −0.0513126 0.998683i \(-0.516340\pi\)
−0.0513126 + 0.998683i \(0.516340\pi\)
\(572\) 0 0
\(573\) 17574.4 1.28130
\(574\) 0 0
\(575\) 16664.5 1.20862
\(576\) 0 0
\(577\) 23464.7 1.69298 0.846489 0.532406i \(-0.178712\pi\)
0.846489 + 0.532406i \(0.178712\pi\)
\(578\) 0 0
\(579\) −7803.26 −0.560091
\(580\) 0 0
\(581\) 4771.35 0.340704
\(582\) 0 0
\(583\) 1763.29 0.125262
\(584\) 0 0
\(585\) 622.090 0.0439662
\(586\) 0 0
\(587\) 1336.76 0.0939934 0.0469967 0.998895i \(-0.485035\pi\)
0.0469967 + 0.998895i \(0.485035\pi\)
\(588\) 0 0
\(589\) −1970.26 −0.137832
\(590\) 0 0
\(591\) 3279.24 0.228240
\(592\) 0 0
\(593\) −14890.5 −1.03116 −0.515581 0.856841i \(-0.672424\pi\)
−0.515581 + 0.856841i \(0.672424\pi\)
\(594\) 0 0
\(595\) 1384.48 0.0953917
\(596\) 0 0
\(597\) 13579.0 0.930906
\(598\) 0 0
\(599\) 22648.7 1.54491 0.772456 0.635068i \(-0.219028\pi\)
0.772456 + 0.635068i \(0.219028\pi\)
\(600\) 0 0
\(601\) 8930.64 0.606137 0.303069 0.952969i \(-0.401989\pi\)
0.303069 + 0.952969i \(0.401989\pi\)
\(602\) 0 0
\(603\) 1041.55 0.0703405
\(604\) 0 0
\(605\) 1240.32 0.0833493
\(606\) 0 0
\(607\) −5611.75 −0.375245 −0.187623 0.982241i \(-0.560078\pi\)
−0.187623 + 0.982241i \(0.560078\pi\)
\(608\) 0 0
\(609\) 5300.49 0.352687
\(610\) 0 0
\(611\) −1866.27 −0.123570
\(612\) 0 0
\(613\) 16212.8 1.06823 0.534117 0.845410i \(-0.320644\pi\)
0.534117 + 0.845410i \(0.320644\pi\)
\(614\) 0 0
\(615\) −1279.39 −0.0838859
\(616\) 0 0
\(617\) −26562.7 −1.73318 −0.866592 0.499017i \(-0.833694\pi\)
−0.866592 + 0.499017i \(0.833694\pi\)
\(618\) 0 0
\(619\) −816.376 −0.0530096 −0.0265048 0.999649i \(-0.508438\pi\)
−0.0265048 + 0.999649i \(0.508438\pi\)
\(620\) 0 0
\(621\) 20628.5 1.33300
\(622\) 0 0
\(623\) −12364.5 −0.795139
\(624\) 0 0
\(625\) 15018.1 0.961159
\(626\) 0 0
\(627\) −1537.11 −0.0979045
\(628\) 0 0
\(629\) 2665.34 0.168957
\(630\) 0 0
\(631\) 29218.6 1.84338 0.921690 0.387928i \(-0.126809\pi\)
0.921690 + 0.387928i \(0.126809\pi\)
\(632\) 0 0
\(633\) −10518.6 −0.660468
\(634\) 0 0
\(635\) 2761.84 0.172599
\(636\) 0 0
\(637\) −10994.6 −0.683867
\(638\) 0 0
\(639\) −161.486 −0.00999734
\(640\) 0 0
\(641\) −4831.98 −0.297740 −0.148870 0.988857i \(-0.547564\pi\)
−0.148870 + 0.988857i \(0.547564\pi\)
\(642\) 0 0
\(643\) −25259.5 −1.54920 −0.774602 0.632449i \(-0.782050\pi\)
−0.774602 + 0.632449i \(0.782050\pi\)
\(644\) 0 0
\(645\) −290.620 −0.0177413
\(646\) 0 0
\(647\) 11004.6 0.668676 0.334338 0.942453i \(-0.391487\pi\)
0.334338 + 0.942453i \(0.391487\pi\)
\(648\) 0 0
\(649\) 12060.9 0.729479
\(650\) 0 0
\(651\) −5363.77 −0.322923
\(652\) 0 0
\(653\) −5034.69 −0.301720 −0.150860 0.988555i \(-0.548204\pi\)
−0.150860 + 0.988555i \(0.548204\pi\)
\(654\) 0 0
\(655\) 419.843 0.0250452
\(656\) 0 0
\(657\) 3444.61 0.204547
\(658\) 0 0
\(659\) 5027.20 0.297165 0.148583 0.988900i \(-0.452529\pi\)
0.148583 + 0.988900i \(0.452529\pi\)
\(660\) 0 0
\(661\) 28126.0 1.65503 0.827515 0.561444i \(-0.189754\pi\)
0.827515 + 0.561444i \(0.189754\pi\)
\(662\) 0 0
\(663\) −21456.5 −1.25687
\(664\) 0 0
\(665\) −293.095 −0.0170914
\(666\) 0 0
\(667\) −13841.4 −0.803512
\(668\) 0 0
\(669\) 15952.9 0.921933
\(670\) 0 0
\(671\) −2376.35 −0.136718
\(672\) 0 0
\(673\) 15864.9 0.908689 0.454344 0.890826i \(-0.349874\pi\)
0.454344 + 0.890826i \(0.349874\pi\)
\(674\) 0 0
\(675\) 18842.0 1.07441
\(676\) 0 0
\(677\) −15911.8 −0.903308 −0.451654 0.892193i \(-0.649166\pi\)
−0.451654 + 0.892193i \(0.649166\pi\)
\(678\) 0 0
\(679\) 16005.7 0.904626
\(680\) 0 0
\(681\) −2792.33 −0.157125
\(682\) 0 0
\(683\) 30172.0 1.69033 0.845167 0.534502i \(-0.179501\pi\)
0.845167 + 0.534502i \(0.179501\pi\)
\(684\) 0 0
\(685\) −939.510 −0.0524042
\(686\) 0 0
\(687\) −1455.65 −0.0808394
\(688\) 0 0
\(689\) −5210.77 −0.288120
\(690\) 0 0
\(691\) 14213.9 0.782522 0.391261 0.920280i \(-0.372039\pi\)
0.391261 + 0.920280i \(0.372039\pi\)
\(692\) 0 0
\(693\) 1997.86 0.109513
\(694\) 0 0
\(695\) −3877.16 −0.211610
\(696\) 0 0
\(697\) −21067.9 −1.14491
\(698\) 0 0
\(699\) −16830.3 −0.910703
\(700\) 0 0
\(701\) −25215.2 −1.35858 −0.679292 0.733869i \(-0.737713\pi\)
−0.679292 + 0.733869i \(0.737713\pi\)
\(702\) 0 0
\(703\) −564.256 −0.0302721
\(704\) 0 0
\(705\) 181.879 0.00971625
\(706\) 0 0
\(707\) −23268.4 −1.23776
\(708\) 0 0
\(709\) −34384.4 −1.82134 −0.910672 0.413130i \(-0.864436\pi\)
−0.910672 + 0.413130i \(0.864436\pi\)
\(710\) 0 0
\(711\) −2646.67 −0.139603
\(712\) 0 0
\(713\) 14006.7 0.735700
\(714\) 0 0
\(715\) 1349.29 0.0705744
\(716\) 0 0
\(717\) 20357.9 1.06036
\(718\) 0 0
\(719\) 4418.46 0.229180 0.114590 0.993413i \(-0.463445\pi\)
0.114590 + 0.993413i \(0.463445\pi\)
\(720\) 0 0
\(721\) 5659.91 0.292353
\(722\) 0 0
\(723\) −16644.6 −0.856181
\(724\) 0 0
\(725\) −12642.7 −0.647640
\(726\) 0 0
\(727\) 805.044 0.0410694 0.0205347 0.999789i \(-0.493463\pi\)
0.0205347 + 0.999789i \(0.493463\pi\)
\(728\) 0 0
\(729\) 21268.5 1.08055
\(730\) 0 0
\(731\) −4785.70 −0.242142
\(732\) 0 0
\(733\) 1662.37 0.0837668 0.0418834 0.999123i \(-0.486664\pi\)
0.0418834 + 0.999123i \(0.486664\pi\)
\(734\) 0 0
\(735\) 1071.49 0.0537722
\(736\) 0 0
\(737\) 2259.09 0.112910
\(738\) 0 0
\(739\) −3145.59 −0.156580 −0.0782899 0.996931i \(-0.524946\pi\)
−0.0782899 + 0.996931i \(0.524946\pi\)
\(740\) 0 0
\(741\) 4542.37 0.225193
\(742\) 0 0
\(743\) 35469.5 1.75134 0.875672 0.482906i \(-0.160419\pi\)
0.875672 + 0.482906i \(0.160419\pi\)
\(744\) 0 0
\(745\) 2749.11 0.135194
\(746\) 0 0
\(747\) −3440.63 −0.168522
\(748\) 0 0
\(749\) −3151.57 −0.153746
\(750\) 0 0
\(751\) −29978.8 −1.45665 −0.728324 0.685233i \(-0.759700\pi\)
−0.728324 + 0.685233i \(0.759700\pi\)
\(752\) 0 0
\(753\) −5917.02 −0.286359
\(754\) 0 0
\(755\) 2316.01 0.111640
\(756\) 0 0
\(757\) 33882.6 1.62680 0.813398 0.581707i \(-0.197615\pi\)
0.813398 + 0.581707i \(0.197615\pi\)
\(758\) 0 0
\(759\) 10927.4 0.522581
\(760\) 0 0
\(761\) 9023.05 0.429810 0.214905 0.976635i \(-0.431056\pi\)
0.214905 + 0.976635i \(0.431056\pi\)
\(762\) 0 0
\(763\) 6193.29 0.293856
\(764\) 0 0
\(765\) −998.348 −0.0471835
\(766\) 0 0
\(767\) −35641.7 −1.67790
\(768\) 0 0
\(769\) −773.311 −0.0362631 −0.0181315 0.999836i \(-0.505772\pi\)
−0.0181315 + 0.999836i \(0.505772\pi\)
\(770\) 0 0
\(771\) −19857.0 −0.927540
\(772\) 0 0
\(773\) −19778.7 −0.920299 −0.460150 0.887841i \(-0.652204\pi\)
−0.460150 + 0.887841i \(0.652204\pi\)
\(774\) 0 0
\(775\) 12793.7 0.592983
\(776\) 0 0
\(777\) −1536.12 −0.0709238
\(778\) 0 0
\(779\) 4460.11 0.205135
\(780\) 0 0
\(781\) −350.258 −0.0160477
\(782\) 0 0
\(783\) −15650.0 −0.714288
\(784\) 0 0
\(785\) −1425.48 −0.0648122
\(786\) 0 0
\(787\) −32308.3 −1.46336 −0.731681 0.681648i \(-0.761264\pi\)
−0.731681 + 0.681648i \(0.761264\pi\)
\(788\) 0 0
\(789\) −9197.09 −0.414987
\(790\) 0 0
\(791\) −17586.6 −0.790528
\(792\) 0 0
\(793\) 7022.46 0.314470
\(794\) 0 0
\(795\) 507.821 0.0226548
\(796\) 0 0
\(797\) −40290.5 −1.79067 −0.895335 0.445394i \(-0.853064\pi\)
−0.895335 + 0.445394i \(0.853064\pi\)
\(798\) 0 0
\(799\) 2995.04 0.132612
\(800\) 0 0
\(801\) 8916.02 0.393298
\(802\) 0 0
\(803\) 7471.25 0.328337
\(804\) 0 0
\(805\) 2083.64 0.0912279
\(806\) 0 0
\(807\) 24660.6 1.07570
\(808\) 0 0
\(809\) −16600.1 −0.721417 −0.360709 0.932679i \(-0.617465\pi\)
−0.360709 + 0.932679i \(0.617465\pi\)
\(810\) 0 0
\(811\) −9586.41 −0.415073 −0.207537 0.978227i \(-0.566545\pi\)
−0.207537 + 0.978227i \(0.566545\pi\)
\(812\) 0 0
\(813\) −29322.6 −1.26493
\(814\) 0 0
\(815\) −4213.54 −0.181097
\(816\) 0 0
\(817\) 1013.14 0.0433846
\(818\) 0 0
\(819\) −5903.98 −0.251894
\(820\) 0 0
\(821\) 33381.4 1.41902 0.709512 0.704693i \(-0.248915\pi\)
0.709512 + 0.704693i \(0.248915\pi\)
\(822\) 0 0
\(823\) −3953.76 −0.167460 −0.0837299 0.996488i \(-0.526683\pi\)
−0.0837299 + 0.996488i \(0.526683\pi\)
\(824\) 0 0
\(825\) 9981.04 0.421207
\(826\) 0 0
\(827\) 33403.5 1.40454 0.702270 0.711910i \(-0.252170\pi\)
0.702270 + 0.711910i \(0.252170\pi\)
\(828\) 0 0
\(829\) −41612.3 −1.74337 −0.871686 0.490065i \(-0.836973\pi\)
−0.871686 + 0.490065i \(0.836973\pi\)
\(830\) 0 0
\(831\) −5289.11 −0.220791
\(832\) 0 0
\(833\) 17644.5 0.733909
\(834\) 0 0
\(835\) −2232.05 −0.0925070
\(836\) 0 0
\(837\) 15836.9 0.654006
\(838\) 0 0
\(839\) 10843.5 0.446199 0.223099 0.974796i \(-0.428383\pi\)
0.223099 + 0.974796i \(0.428383\pi\)
\(840\) 0 0
\(841\) −13888.0 −0.569438
\(842\) 0 0
\(843\) 18618.2 0.760669
\(844\) 0 0
\(845\) −1186.36 −0.0482984
\(846\) 0 0
\(847\) −11771.4 −0.477531
\(848\) 0 0
\(849\) 15609.0 0.630975
\(850\) 0 0
\(851\) 4011.33 0.161583
\(852\) 0 0
\(853\) −13530.7 −0.543119 −0.271560 0.962422i \(-0.587539\pi\)
−0.271560 + 0.962422i \(0.587539\pi\)
\(854\) 0 0
\(855\) 211.351 0.00845388
\(856\) 0 0
\(857\) −3510.58 −0.139929 −0.0699645 0.997549i \(-0.522289\pi\)
−0.0699645 + 0.997549i \(0.522289\pi\)
\(858\) 0 0
\(859\) 4945.74 0.196445 0.0982226 0.995164i \(-0.468684\pi\)
0.0982226 + 0.995164i \(0.468684\pi\)
\(860\) 0 0
\(861\) 12142.1 0.480605
\(862\) 0 0
\(863\) −38957.6 −1.53665 −0.768326 0.640059i \(-0.778910\pi\)
−0.768326 + 0.640059i \(0.778910\pi\)
\(864\) 0 0
\(865\) 3439.96 0.135216
\(866\) 0 0
\(867\) 13431.4 0.526130
\(868\) 0 0
\(869\) −5740.54 −0.224090
\(870\) 0 0
\(871\) −6675.95 −0.259708
\(872\) 0 0
\(873\) −11541.7 −0.447454
\(874\) 0 0
\(875\) 3831.45 0.148030
\(876\) 0 0
\(877\) −544.671 −0.0209718 −0.0104859 0.999945i \(-0.503338\pi\)
−0.0104859 + 0.999945i \(0.503338\pi\)
\(878\) 0 0
\(879\) 14497.3 0.556294
\(880\) 0 0
\(881\) −19771.1 −0.756079 −0.378040 0.925789i \(-0.623402\pi\)
−0.378040 + 0.925789i \(0.623402\pi\)
\(882\) 0 0
\(883\) 12249.6 0.466854 0.233427 0.972374i \(-0.425006\pi\)
0.233427 + 0.972374i \(0.425006\pi\)
\(884\) 0 0
\(885\) 3473.50 0.131933
\(886\) 0 0
\(887\) −21817.3 −0.825877 −0.412938 0.910759i \(-0.635497\pi\)
−0.412938 + 0.910759i \(0.635497\pi\)
\(888\) 0 0
\(889\) −26211.4 −0.988866
\(890\) 0 0
\(891\) 7897.08 0.296927
\(892\) 0 0
\(893\) −634.054 −0.0237602
\(894\) 0 0
\(895\) 4931.13 0.184167
\(896\) 0 0
\(897\) −32292.0 −1.20201
\(898\) 0 0
\(899\) −10626.3 −0.394225
\(900\) 0 0
\(901\) 8362.40 0.309203
\(902\) 0 0
\(903\) 2758.14 0.101645
\(904\) 0 0
\(905\) 4919.58 0.180699
\(906\) 0 0
\(907\) 35592.0 1.30299 0.651496 0.758652i \(-0.274142\pi\)
0.651496 + 0.758652i \(0.274142\pi\)
\(908\) 0 0
\(909\) 16778.9 0.612233
\(910\) 0 0
\(911\) 952.654 0.0346464 0.0173232 0.999850i \(-0.494486\pi\)
0.0173232 + 0.999850i \(0.494486\pi\)
\(912\) 0 0
\(913\) −7462.60 −0.270511
\(914\) 0 0
\(915\) −684.381 −0.0247267
\(916\) 0 0
\(917\) −3984.54 −0.143491
\(918\) 0 0
\(919\) −42326.8 −1.51929 −0.759647 0.650335i \(-0.774628\pi\)
−0.759647 + 0.650335i \(0.774628\pi\)
\(920\) 0 0
\(921\) −18567.4 −0.664298
\(922\) 0 0
\(923\) 1035.06 0.0369118
\(924\) 0 0
\(925\) 3663.94 0.130237
\(926\) 0 0
\(927\) −4081.37 −0.144606
\(928\) 0 0
\(929\) −27446.4 −0.969309 −0.484654 0.874706i \(-0.661055\pi\)
−0.484654 + 0.874706i \(0.661055\pi\)
\(930\) 0 0
\(931\) −3735.36 −0.131495
\(932\) 0 0
\(933\) 24244.3 0.850720
\(934\) 0 0
\(935\) −2165.38 −0.0757387
\(936\) 0 0
\(937\) 15560.3 0.542511 0.271256 0.962507i \(-0.412561\pi\)
0.271256 + 0.962507i \(0.412561\pi\)
\(938\) 0 0
\(939\) −40396.7 −1.40394
\(940\) 0 0
\(941\) −17725.4 −0.614060 −0.307030 0.951700i \(-0.599335\pi\)
−0.307030 + 0.951700i \(0.599335\pi\)
\(942\) 0 0
\(943\) −31707.2 −1.09494
\(944\) 0 0
\(945\) 2355.90 0.0810977
\(946\) 0 0
\(947\) 588.382 0.0201899 0.0100950 0.999949i \(-0.496787\pi\)
0.0100950 + 0.999949i \(0.496787\pi\)
\(948\) 0 0
\(949\) −22078.6 −0.755219
\(950\) 0 0
\(951\) 23045.0 0.785789
\(952\) 0 0
\(953\) 28644.2 0.973638 0.486819 0.873503i \(-0.338157\pi\)
0.486819 + 0.873503i \(0.338157\pi\)
\(954\) 0 0
\(955\) −5241.26 −0.177595
\(956\) 0 0
\(957\) −8290.20 −0.280025
\(958\) 0 0
\(959\) 8916.47 0.300238
\(960\) 0 0
\(961\) −19037.8 −0.639045
\(962\) 0 0
\(963\) 2272.60 0.0760473
\(964\) 0 0
\(965\) 2327.18 0.0776318
\(966\) 0 0
\(967\) 44523.4 1.48064 0.740318 0.672257i \(-0.234675\pi\)
0.740318 + 0.672257i \(0.234675\pi\)
\(968\) 0 0
\(969\) −7289.74 −0.241672
\(970\) 0 0
\(971\) −18242.7 −0.602922 −0.301461 0.953478i \(-0.597474\pi\)
−0.301461 + 0.953478i \(0.597474\pi\)
\(972\) 0 0
\(973\) 36796.4 1.21237
\(974\) 0 0
\(975\) −29495.4 −0.968831
\(976\) 0 0
\(977\) 23434.6 0.767390 0.383695 0.923460i \(-0.374651\pi\)
0.383695 + 0.923460i \(0.374651\pi\)
\(978\) 0 0
\(979\) 19338.6 0.631321
\(980\) 0 0
\(981\) −4465.99 −0.145350
\(982\) 0 0
\(983\) −59519.4 −1.93121 −0.965603 0.260022i \(-0.916270\pi\)
−0.965603 + 0.260022i \(0.916270\pi\)
\(984\) 0 0
\(985\) −977.974 −0.0316354
\(986\) 0 0
\(987\) −1726.13 −0.0556671
\(988\) 0 0
\(989\) −7202.47 −0.231573
\(990\) 0 0
\(991\) 8466.93 0.271404 0.135702 0.990750i \(-0.456671\pi\)
0.135702 + 0.990750i \(0.456671\pi\)
\(992\) 0 0
\(993\) 39458.1 1.26099
\(994\) 0 0
\(995\) −4049.69 −0.129029
\(996\) 0 0
\(997\) −27474.3 −0.872738 −0.436369 0.899768i \(-0.643736\pi\)
−0.436369 + 0.899768i \(0.643736\pi\)
\(998\) 0 0
\(999\) 4535.48 0.143640
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 1216.4.a.m.1.2 2
4.3 odd 2 1216.4.a.k.1.1 2
8.3 odd 2 304.4.a.e.1.2 2
8.5 even 2 152.4.a.a.1.1 2
24.5 odd 2 1368.4.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.a.1.1 2 8.5 even 2
304.4.a.e.1.2 2 8.3 odd 2
1216.4.a.k.1.1 2 4.3 odd 2
1216.4.a.m.1.2 2 1.1 even 1 trivial
1368.4.a.a.1.1 2 24.5 odd 2