Properties

Label 7623.2.a.db.1.16
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(2.68084\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.68084 q^{2} +5.18691 q^{4} +1.17039 q^{5} -1.00000 q^{7} +8.54361 q^{8} +O(q^{10})\) \(q+2.68084 q^{2} +5.18691 q^{4} +1.17039 q^{5} -1.00000 q^{7} +8.54361 q^{8} +3.13763 q^{10} +1.68268 q^{13} -2.68084 q^{14} +12.5302 q^{16} +5.59675 q^{17} -4.35416 q^{19} +6.07072 q^{20} -0.119612 q^{23} -3.63018 q^{25} +4.51101 q^{26} -5.18691 q^{28} +4.39599 q^{29} -6.31294 q^{31} +16.5044 q^{32} +15.0040 q^{34} -1.17039 q^{35} +5.85418 q^{37} -11.6728 q^{38} +9.99937 q^{40} +11.0165 q^{41} -5.97227 q^{43} -0.320661 q^{46} +12.5812 q^{47} +1.00000 q^{49} -9.73195 q^{50} +8.72794 q^{52} +0.417004 q^{53} -8.54361 q^{56} +11.7849 q^{58} -0.723544 q^{59} +8.85799 q^{61} -16.9240 q^{62} +19.1851 q^{64} +1.96940 q^{65} +8.78741 q^{67} +29.0299 q^{68} -3.13763 q^{70} +3.21286 q^{71} +14.5891 q^{73} +15.6941 q^{74} -22.5846 q^{76} -14.4761 q^{79} +14.6653 q^{80} +29.5335 q^{82} +3.45931 q^{83} +6.55039 q^{85} -16.0107 q^{86} -12.0621 q^{89} -1.68268 q^{91} -0.620417 q^{92} +33.7283 q^{94} -5.09607 q^{95} -5.99575 q^{97} +2.68084 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} - 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} - 16q^{7} + 6q^{10} + 32q^{16} - 10q^{19} + 44q^{25} - 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} + 68q^{40} - 16q^{43} - 80q^{46} + 16q^{49} - 2q^{52} + 18q^{58} + 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} + 14q^{73} + 14q^{76} - 54q^{79} + 64q^{82} - 30q^{85} + 60q^{94} + 108q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.68084 1.89564 0.947821 0.318804i \(-0.103281\pi\)
0.947821 + 0.318804i \(0.103281\pi\)
\(3\) 0 0
\(4\) 5.18691 2.59346
\(5\) 1.17039 0.523415 0.261708 0.965147i \(-0.415714\pi\)
0.261708 + 0.965147i \(0.415714\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 8.54361 3.02062
\(9\) 0 0
\(10\) 3.13763 0.992207
\(11\) 0 0
\(12\) 0 0
\(13\) 1.68268 0.466693 0.233346 0.972394i \(-0.425032\pi\)
0.233346 + 0.972394i \(0.425032\pi\)
\(14\) −2.68084 −0.716485
\(15\) 0 0
\(16\) 12.5302 3.13256
\(17\) 5.59675 1.35741 0.678706 0.734410i \(-0.262541\pi\)
0.678706 + 0.734410i \(0.262541\pi\)
\(18\) 0 0
\(19\) −4.35416 −0.998913 −0.499456 0.866339i \(-0.666467\pi\)
−0.499456 + 0.866339i \(0.666467\pi\)
\(20\) 6.07072 1.35745
\(21\) 0 0
\(22\) 0 0
\(23\) −0.119612 −0.0249408 −0.0124704 0.999922i \(-0.503970\pi\)
−0.0124704 + 0.999922i \(0.503970\pi\)
\(24\) 0 0
\(25\) −3.63018 −0.726037
\(26\) 4.51101 0.884682
\(27\) 0 0
\(28\) −5.18691 −0.980234
\(29\) 4.39599 0.816315 0.408157 0.912912i \(-0.366171\pi\)
0.408157 + 0.912912i \(0.366171\pi\)
\(30\) 0 0
\(31\) −6.31294 −1.13384 −0.566919 0.823774i \(-0.691865\pi\)
−0.566919 + 0.823774i \(0.691865\pi\)
\(32\) 16.5044 2.91759
\(33\) 0 0
\(34\) 15.0040 2.57317
\(35\) −1.17039 −0.197832
\(36\) 0 0
\(37\) 5.85418 0.962421 0.481210 0.876605i \(-0.340197\pi\)
0.481210 + 0.876605i \(0.340197\pi\)
\(38\) −11.6728 −1.89358
\(39\) 0 0
\(40\) 9.99937 1.58104
\(41\) 11.0165 1.72049 0.860244 0.509883i \(-0.170311\pi\)
0.860244 + 0.509883i \(0.170311\pi\)
\(42\) 0 0
\(43\) −5.97227 −0.910762 −0.455381 0.890297i \(-0.650497\pi\)
−0.455381 + 0.890297i \(0.650497\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.320661 −0.0472789
\(47\) 12.5812 1.83516 0.917580 0.397552i \(-0.130140\pi\)
0.917580 + 0.397552i \(0.130140\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −9.73195 −1.37631
\(51\) 0 0
\(52\) 8.72794 1.21035
\(53\) 0.417004 0.0572800 0.0286400 0.999590i \(-0.490882\pi\)
0.0286400 + 0.999590i \(0.490882\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −8.54361 −1.14169
\(57\) 0 0
\(58\) 11.7849 1.54744
\(59\) −0.723544 −0.0941974 −0.0470987 0.998890i \(-0.514998\pi\)
−0.0470987 + 0.998890i \(0.514998\pi\)
\(60\) 0 0
\(61\) 8.85799 1.13415 0.567075 0.823666i \(-0.308075\pi\)
0.567075 + 0.823666i \(0.308075\pi\)
\(62\) −16.9240 −2.14935
\(63\) 0 0
\(64\) 19.1851 2.39814
\(65\) 1.96940 0.244274
\(66\) 0 0
\(67\) 8.78741 1.07355 0.536777 0.843724i \(-0.319642\pi\)
0.536777 + 0.843724i \(0.319642\pi\)
\(68\) 29.0299 3.52039
\(69\) 0 0
\(70\) −3.13763 −0.375019
\(71\) 3.21286 0.381297 0.190648 0.981658i \(-0.438941\pi\)
0.190648 + 0.981658i \(0.438941\pi\)
\(72\) 0 0
\(73\) 14.5891 1.70752 0.853762 0.520664i \(-0.174315\pi\)
0.853762 + 0.520664i \(0.174315\pi\)
\(74\) 15.6941 1.82440
\(75\) 0 0
\(76\) −22.5846 −2.59064
\(77\) 0 0
\(78\) 0 0
\(79\) −14.4761 −1.62869 −0.814347 0.580378i \(-0.802905\pi\)
−0.814347 + 0.580378i \(0.802905\pi\)
\(80\) 14.6653 1.63963
\(81\) 0 0
\(82\) 29.5335 3.26143
\(83\) 3.45931 0.379709 0.189854 0.981812i \(-0.439198\pi\)
0.189854 + 0.981812i \(0.439198\pi\)
\(84\) 0 0
\(85\) 6.55039 0.710490
\(86\) −16.0107 −1.72648
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0621 −1.27858 −0.639292 0.768964i \(-0.720772\pi\)
−0.639292 + 0.768964i \(0.720772\pi\)
\(90\) 0 0
\(91\) −1.68268 −0.176393
\(92\) −0.620417 −0.0646830
\(93\) 0 0
\(94\) 33.7283 3.47880
\(95\) −5.09607 −0.522846
\(96\) 0 0
\(97\) −5.99575 −0.608776 −0.304388 0.952548i \(-0.598452\pi\)
−0.304388 + 0.952548i \(0.598452\pi\)
\(98\) 2.68084 0.270806
\(99\) 0 0
\(100\) −18.8294 −1.88294
\(101\) −0.101612 −0.0101108 −0.00505540 0.999987i \(-0.501609\pi\)
−0.00505540 + 0.999987i \(0.501609\pi\)
\(102\) 0 0
\(103\) −1.73990 −0.171438 −0.0857188 0.996319i \(-0.527319\pi\)
−0.0857188 + 0.996319i \(0.527319\pi\)
\(104\) 14.3762 1.40970
\(105\) 0 0
\(106\) 1.11792 0.108582
\(107\) −11.0032 −1.06372 −0.531861 0.846831i \(-0.678507\pi\)
−0.531861 + 0.846831i \(0.678507\pi\)
\(108\) 0 0
\(109\) −16.5040 −1.58080 −0.790400 0.612592i \(-0.790127\pi\)
−0.790400 + 0.612592i \(0.790127\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −12.5302 −1.18400
\(113\) −6.28108 −0.590875 −0.295437 0.955362i \(-0.595465\pi\)
−0.295437 + 0.955362i \(0.595465\pi\)
\(114\) 0 0
\(115\) −0.139993 −0.0130544
\(116\) 22.8016 2.11708
\(117\) 0 0
\(118\) −1.93971 −0.178564
\(119\) −5.59675 −0.513053
\(120\) 0 0
\(121\) 0 0
\(122\) 23.7469 2.14994
\(123\) 0 0
\(124\) −32.7447 −2.94056
\(125\) −10.1007 −0.903434
\(126\) 0 0
\(127\) −17.2379 −1.52962 −0.764809 0.644257i \(-0.777167\pi\)
−0.764809 + 0.644257i \(0.777167\pi\)
\(128\) 18.4235 1.62843
\(129\) 0 0
\(130\) 5.27965 0.463056
\(131\) 0.324063 0.0283135 0.0141567 0.999900i \(-0.495494\pi\)
0.0141567 + 0.999900i \(0.495494\pi\)
\(132\) 0 0
\(133\) 4.35416 0.377553
\(134\) 23.5577 2.03507
\(135\) 0 0
\(136\) 47.8165 4.10023
\(137\) 10.3726 0.886188 0.443094 0.896475i \(-0.353881\pi\)
0.443094 + 0.896475i \(0.353881\pi\)
\(138\) 0 0
\(139\) −13.3776 −1.13467 −0.567336 0.823486i \(-0.692026\pi\)
−0.567336 + 0.823486i \(0.692026\pi\)
\(140\) −6.07072 −0.513069
\(141\) 0 0
\(142\) 8.61317 0.722801
\(143\) 0 0
\(144\) 0 0
\(145\) 5.14503 0.427271
\(146\) 39.1110 3.23685
\(147\) 0 0
\(148\) 30.3651 2.49600
\(149\) −9.66988 −0.792188 −0.396094 0.918210i \(-0.629635\pi\)
−0.396094 + 0.918210i \(0.629635\pi\)
\(150\) 0 0
\(151\) −2.56082 −0.208396 −0.104198 0.994557i \(-0.533228\pi\)
−0.104198 + 0.994557i \(0.533228\pi\)
\(152\) −37.2002 −3.01734
\(153\) 0 0
\(154\) 0 0
\(155\) −7.38862 −0.593468
\(156\) 0 0
\(157\) 18.5092 1.47719 0.738597 0.674148i \(-0.235489\pi\)
0.738597 + 0.674148i \(0.235489\pi\)
\(158\) −38.8083 −3.08742
\(159\) 0 0
\(160\) 19.3166 1.52711
\(161\) 0.119612 0.00942675
\(162\) 0 0
\(163\) 6.47823 0.507414 0.253707 0.967281i \(-0.418350\pi\)
0.253707 + 0.967281i \(0.418350\pi\)
\(164\) 57.1416 4.46201
\(165\) 0 0
\(166\) 9.27386 0.719791
\(167\) −25.2567 −1.95442 −0.977212 0.212266i \(-0.931916\pi\)
−0.977212 + 0.212266i \(0.931916\pi\)
\(168\) 0 0
\(169\) −10.1686 −0.782198
\(170\) 17.5606 1.34683
\(171\) 0 0
\(172\) −30.9776 −2.36202
\(173\) −9.26804 −0.704636 −0.352318 0.935880i \(-0.614606\pi\)
−0.352318 + 0.935880i \(0.614606\pi\)
\(174\) 0 0
\(175\) 3.63018 0.274416
\(176\) 0 0
\(177\) 0 0
\(178\) −32.3367 −2.42373
\(179\) 15.4438 1.15432 0.577161 0.816630i \(-0.304160\pi\)
0.577161 + 0.816630i \(0.304160\pi\)
\(180\) 0 0
\(181\) 10.5318 0.782824 0.391412 0.920215i \(-0.371987\pi\)
0.391412 + 0.920215i \(0.371987\pi\)
\(182\) −4.51101 −0.334378
\(183\) 0 0
\(184\) −1.02192 −0.0753369
\(185\) 6.85168 0.503746
\(186\) 0 0
\(187\) 0 0
\(188\) 65.2577 4.75940
\(189\) 0 0
\(190\) −13.6618 −0.991128
\(191\) 6.86336 0.496615 0.248308 0.968681i \(-0.420126\pi\)
0.248308 + 0.968681i \(0.420126\pi\)
\(192\) 0 0
\(193\) −22.3297 −1.60733 −0.803663 0.595085i \(-0.797119\pi\)
−0.803663 + 0.595085i \(0.797119\pi\)
\(194\) −16.0737 −1.15402
\(195\) 0 0
\(196\) 5.18691 0.370494
\(197\) 4.45203 0.317194 0.158597 0.987343i \(-0.449303\pi\)
0.158597 + 0.987343i \(0.449303\pi\)
\(198\) 0 0
\(199\) −15.0993 −1.07036 −0.535180 0.844738i \(-0.679756\pi\)
−0.535180 + 0.844738i \(0.679756\pi\)
\(200\) −31.0149 −2.19308
\(201\) 0 0
\(202\) −0.272406 −0.0191665
\(203\) −4.39599 −0.308538
\(204\) 0 0
\(205\) 12.8936 0.900529
\(206\) −4.66440 −0.324984
\(207\) 0 0
\(208\) 21.0844 1.46194
\(209\) 0 0
\(210\) 0 0
\(211\) 22.8486 1.57296 0.786480 0.617616i \(-0.211901\pi\)
0.786480 + 0.617616i \(0.211901\pi\)
\(212\) 2.16297 0.148553
\(213\) 0 0
\(214\) −29.4979 −2.01644
\(215\) −6.98989 −0.476707
\(216\) 0 0
\(217\) 6.31294 0.428551
\(218\) −44.2447 −2.99663
\(219\) 0 0
\(220\) 0 0
\(221\) 9.41757 0.633494
\(222\) 0 0
\(223\) 9.71068 0.650275 0.325137 0.945667i \(-0.394589\pi\)
0.325137 + 0.945667i \(0.394589\pi\)
\(224\) −16.5044 −1.10274
\(225\) 0 0
\(226\) −16.8386 −1.12009
\(227\) −18.6174 −1.23568 −0.617838 0.786305i \(-0.711991\pi\)
−0.617838 + 0.786305i \(0.711991\pi\)
\(228\) 0 0
\(229\) 9.04273 0.597561 0.298780 0.954322i \(-0.403420\pi\)
0.298780 + 0.954322i \(0.403420\pi\)
\(230\) −0.375299 −0.0247465
\(231\) 0 0
\(232\) 37.5576 2.46578
\(233\) 8.62319 0.564924 0.282462 0.959278i \(-0.408849\pi\)
0.282462 + 0.959278i \(0.408849\pi\)
\(234\) 0 0
\(235\) 14.7250 0.960550
\(236\) −3.75296 −0.244297
\(237\) 0 0
\(238\) −15.0040 −0.972565
\(239\) −20.6870 −1.33813 −0.669067 0.743202i \(-0.733306\pi\)
−0.669067 + 0.743202i \(0.733306\pi\)
\(240\) 0 0
\(241\) 24.8415 1.60018 0.800090 0.599880i \(-0.204785\pi\)
0.800090 + 0.599880i \(0.204785\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 45.9456 2.94137
\(245\) 1.17039 0.0747736
\(246\) 0 0
\(247\) −7.32668 −0.466185
\(248\) −53.9353 −3.42490
\(249\) 0 0
\(250\) −27.0784 −1.71259
\(251\) −10.5076 −0.663236 −0.331618 0.943414i \(-0.607595\pi\)
−0.331618 + 0.943414i \(0.607595\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −46.2121 −2.89961
\(255\) 0 0
\(256\) 11.0203 0.688771
\(257\) 3.68093 0.229610 0.114805 0.993388i \(-0.463376\pi\)
0.114805 + 0.993388i \(0.463376\pi\)
\(258\) 0 0
\(259\) −5.85418 −0.363761
\(260\) 10.2151 0.633514
\(261\) 0 0
\(262\) 0.868760 0.0536722
\(263\) −10.1816 −0.627824 −0.313912 0.949452i \(-0.601640\pi\)
−0.313912 + 0.949452i \(0.601640\pi\)
\(264\) 0 0
\(265\) 0.488059 0.0299812
\(266\) 11.6728 0.715706
\(267\) 0 0
\(268\) 45.5795 2.78421
\(269\) 32.0196 1.95227 0.976135 0.217163i \(-0.0696802\pi\)
0.976135 + 0.217163i \(0.0696802\pi\)
\(270\) 0 0
\(271\) 20.7478 1.26034 0.630169 0.776458i \(-0.282986\pi\)
0.630169 + 0.776458i \(0.282986\pi\)
\(272\) 70.1286 4.25217
\(273\) 0 0
\(274\) 27.8072 1.67990
\(275\) 0 0
\(276\) 0 0
\(277\) −15.1464 −0.910057 −0.455028 0.890477i \(-0.650371\pi\)
−0.455028 + 0.890477i \(0.650371\pi\)
\(278\) −35.8632 −2.15093
\(279\) 0 0
\(280\) −9.99937 −0.597577
\(281\) 25.7257 1.53467 0.767335 0.641247i \(-0.221583\pi\)
0.767335 + 0.641247i \(0.221583\pi\)
\(282\) 0 0
\(283\) −17.9944 −1.06966 −0.534828 0.844961i \(-0.679624\pi\)
−0.534828 + 0.844961i \(0.679624\pi\)
\(284\) 16.6648 0.988876
\(285\) 0 0
\(286\) 0 0
\(287\) −11.0165 −0.650283
\(288\) 0 0
\(289\) 14.3236 0.842567
\(290\) 13.7930 0.809953
\(291\) 0 0
\(292\) 75.6723 4.42839
\(293\) −30.9599 −1.80870 −0.904349 0.426794i \(-0.859643\pi\)
−0.904349 + 0.426794i \(0.859643\pi\)
\(294\) 0 0
\(295\) −0.846830 −0.0493043
\(296\) 50.0158 2.90711
\(297\) 0 0
\(298\) −25.9234 −1.50170
\(299\) −0.201269 −0.0116397
\(300\) 0 0
\(301\) 5.97227 0.344236
\(302\) −6.86514 −0.395045
\(303\) 0 0
\(304\) −54.5586 −3.12915
\(305\) 10.3673 0.593631
\(306\) 0 0
\(307\) 10.0193 0.571830 0.285915 0.958255i \(-0.407702\pi\)
0.285915 + 0.958255i \(0.407702\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −19.8077 −1.12500
\(311\) 16.3344 0.926240 0.463120 0.886296i \(-0.346730\pi\)
0.463120 + 0.886296i \(0.346730\pi\)
\(312\) 0 0
\(313\) 4.47058 0.252692 0.126346 0.991986i \(-0.459675\pi\)
0.126346 + 0.991986i \(0.459675\pi\)
\(314\) 49.6202 2.80023
\(315\) 0 0
\(316\) −75.0865 −4.22395
\(317\) −11.9782 −0.672765 −0.336382 0.941725i \(-0.609203\pi\)
−0.336382 + 0.941725i \(0.609203\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 22.4541 1.25522
\(321\) 0 0
\(322\) 0.320661 0.0178697
\(323\) −24.3692 −1.35594
\(324\) 0 0
\(325\) −6.10845 −0.338836
\(326\) 17.3671 0.961876
\(327\) 0 0
\(328\) 94.1206 5.19694
\(329\) −12.5812 −0.693625
\(330\) 0 0
\(331\) −18.0873 −0.994168 −0.497084 0.867702i \(-0.665596\pi\)
−0.497084 + 0.867702i \(0.665596\pi\)
\(332\) 17.9431 0.984758
\(333\) 0 0
\(334\) −67.7093 −3.70489
\(335\) 10.2847 0.561914
\(336\) 0 0
\(337\) −19.0858 −1.03967 −0.519836 0.854266i \(-0.674007\pi\)
−0.519836 + 0.854266i \(0.674007\pi\)
\(338\) −27.2603 −1.48277
\(339\) 0 0
\(340\) 33.9763 1.84262
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −51.0247 −2.75107
\(345\) 0 0
\(346\) −24.8462 −1.33574
\(347\) −5.31766 −0.285467 −0.142733 0.989761i \(-0.545589\pi\)
−0.142733 + 0.989761i \(0.545589\pi\)
\(348\) 0 0
\(349\) 20.6262 1.10410 0.552048 0.833813i \(-0.313847\pi\)
0.552048 + 0.833813i \(0.313847\pi\)
\(350\) 9.73195 0.520194
\(351\) 0 0
\(352\) 0 0
\(353\) −7.31745 −0.389468 −0.194734 0.980856i \(-0.562384\pi\)
−0.194734 + 0.980856i \(0.562384\pi\)
\(354\) 0 0
\(355\) 3.76031 0.199576
\(356\) −62.5652 −3.31595
\(357\) 0 0
\(358\) 41.4023 2.18818
\(359\) 24.2290 1.27876 0.639378 0.768892i \(-0.279192\pi\)
0.639378 + 0.768892i \(0.279192\pi\)
\(360\) 0 0
\(361\) −0.0412986 −0.00217361
\(362\) 28.2342 1.48395
\(363\) 0 0
\(364\) −8.72794 −0.457468
\(365\) 17.0749 0.893744
\(366\) 0 0
\(367\) −22.5059 −1.17480 −0.587399 0.809298i \(-0.699848\pi\)
−0.587399 + 0.809298i \(0.699848\pi\)
\(368\) −1.49877 −0.0781287
\(369\) 0 0
\(370\) 18.3683 0.954921
\(371\) −0.417004 −0.0216498
\(372\) 0 0
\(373\) −6.25290 −0.323763 −0.161881 0.986810i \(-0.551756\pi\)
−0.161881 + 0.986810i \(0.551756\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 107.489 5.54332
\(377\) 7.39706 0.380968
\(378\) 0 0
\(379\) 12.8192 0.658480 0.329240 0.944246i \(-0.393207\pi\)
0.329240 + 0.944246i \(0.393207\pi\)
\(380\) −26.4329 −1.35598
\(381\) 0 0
\(382\) 18.3996 0.941405
\(383\) −6.46736 −0.330467 −0.165233 0.986255i \(-0.552838\pi\)
−0.165233 + 0.986255i \(0.552838\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −59.8624 −3.04691
\(387\) 0 0
\(388\) −31.0994 −1.57883
\(389\) 14.8967 0.755295 0.377647 0.925949i \(-0.376733\pi\)
0.377647 + 0.925949i \(0.376733\pi\)
\(390\) 0 0
\(391\) −0.669439 −0.0338550
\(392\) 8.54361 0.431517
\(393\) 0 0
\(394\) 11.9352 0.601286
\(395\) −16.9428 −0.852483
\(396\) 0 0
\(397\) 1.85862 0.0932813 0.0466407 0.998912i \(-0.485148\pi\)
0.0466407 + 0.998912i \(0.485148\pi\)
\(398\) −40.4788 −2.02902
\(399\) 0 0
\(400\) −45.4871 −2.27435
\(401\) 8.74249 0.436579 0.218290 0.975884i \(-0.429952\pi\)
0.218290 + 0.975884i \(0.429952\pi\)
\(402\) 0 0
\(403\) −10.6227 −0.529154
\(404\) −0.527054 −0.0262219
\(405\) 0 0
\(406\) −11.7849 −0.584877
\(407\) 0 0
\(408\) 0 0
\(409\) −12.7198 −0.628952 −0.314476 0.949265i \(-0.601829\pi\)
−0.314476 + 0.949265i \(0.601829\pi\)
\(410\) 34.5657 1.70708
\(411\) 0 0
\(412\) −9.02472 −0.444616
\(413\) 0.723544 0.0356033
\(414\) 0 0
\(415\) 4.04875 0.198745
\(416\) 27.7716 1.36162
\(417\) 0 0
\(418\) 0 0
\(419\) 8.09885 0.395655 0.197827 0.980237i \(-0.436611\pi\)
0.197827 + 0.980237i \(0.436611\pi\)
\(420\) 0 0
\(421\) −26.4786 −1.29049 −0.645243 0.763977i \(-0.723244\pi\)
−0.645243 + 0.763977i \(0.723244\pi\)
\(422\) 61.2534 2.98177
\(423\) 0 0
\(424\) 3.56272 0.173021
\(425\) −20.3172 −0.985531
\(426\) 0 0
\(427\) −8.85799 −0.428668
\(428\) −57.0728 −2.75872
\(429\) 0 0
\(430\) −18.7388 −0.903665
\(431\) −0.356446 −0.0171694 −0.00858471 0.999963i \(-0.502733\pi\)
−0.00858471 + 0.999963i \(0.502733\pi\)
\(432\) 0 0
\(433\) −18.2884 −0.878886 −0.439443 0.898270i \(-0.644824\pi\)
−0.439443 + 0.898270i \(0.644824\pi\)
\(434\) 16.9240 0.812378
\(435\) 0 0
\(436\) −85.6049 −4.09973
\(437\) 0.520810 0.0249137
\(438\) 0 0
\(439\) −6.67633 −0.318644 −0.159322 0.987227i \(-0.550931\pi\)
−0.159322 + 0.987227i \(0.550931\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 25.2470 1.20088
\(443\) 20.4403 0.971146 0.485573 0.874196i \(-0.338611\pi\)
0.485573 + 0.874196i \(0.338611\pi\)
\(444\) 0 0
\(445\) −14.1174 −0.669230
\(446\) 26.0328 1.23269
\(447\) 0 0
\(448\) −19.1851 −0.906411
\(449\) −5.51429 −0.260236 −0.130118 0.991499i \(-0.541536\pi\)
−0.130118 + 0.991499i \(0.541536\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −32.5794 −1.53241
\(453\) 0 0
\(454\) −49.9102 −2.34240
\(455\) −1.96940 −0.0923269
\(456\) 0 0
\(457\) −2.00275 −0.0936849 −0.0468424 0.998902i \(-0.514916\pi\)
−0.0468424 + 0.998902i \(0.514916\pi\)
\(458\) 24.2421 1.13276
\(459\) 0 0
\(460\) −0.726131 −0.0338561
\(461\) −8.17436 −0.380718 −0.190359 0.981715i \(-0.560965\pi\)
−0.190359 + 0.981715i \(0.560965\pi\)
\(462\) 0 0
\(463\) −4.26686 −0.198298 −0.0991488 0.995073i \(-0.531612\pi\)
−0.0991488 + 0.995073i \(0.531612\pi\)
\(464\) 55.0828 2.55715
\(465\) 0 0
\(466\) 23.1174 1.07089
\(467\) −40.5138 −1.87476 −0.937378 0.348315i \(-0.886754\pi\)
−0.937378 + 0.348315i \(0.886754\pi\)
\(468\) 0 0
\(469\) −8.78741 −0.405765
\(470\) 39.4753 1.82086
\(471\) 0 0
\(472\) −6.18168 −0.284535
\(473\) 0 0
\(474\) 0 0
\(475\) 15.8064 0.725247
\(476\) −29.0299 −1.33058
\(477\) 0 0
\(478\) −55.4587 −2.53662
\(479\) 8.24083 0.376533 0.188267 0.982118i \(-0.439713\pi\)
0.188267 + 0.982118i \(0.439713\pi\)
\(480\) 0 0
\(481\) 9.85073 0.449155
\(482\) 66.5960 3.03337
\(483\) 0 0
\(484\) 0 0
\(485\) −7.01738 −0.318643
\(486\) 0 0
\(487\) −0.964882 −0.0437230 −0.0218615 0.999761i \(-0.506959\pi\)
−0.0218615 + 0.999761i \(0.506959\pi\)
\(488\) 75.6792 3.42584
\(489\) 0 0
\(490\) 3.13763 0.141744
\(491\) −30.4284 −1.37321 −0.686607 0.727029i \(-0.740901\pi\)
−0.686607 + 0.727029i \(0.740901\pi\)
\(492\) 0 0
\(493\) 24.6033 1.10808
\(494\) −19.6417 −0.883720
\(495\) 0 0
\(496\) −79.1027 −3.55181
\(497\) −3.21286 −0.144117
\(498\) 0 0
\(499\) −37.5590 −1.68137 −0.840687 0.541522i \(-0.817848\pi\)
−0.840687 + 0.541522i \(0.817848\pi\)
\(500\) −52.3914 −2.34302
\(501\) 0 0
\(502\) −28.1693 −1.25726
\(503\) 22.9897 1.02506 0.512530 0.858669i \(-0.328708\pi\)
0.512530 + 0.858669i \(0.328708\pi\)
\(504\) 0 0
\(505\) −0.118926 −0.00529214
\(506\) 0 0
\(507\) 0 0
\(508\) −89.4116 −3.96700
\(509\) −4.19559 −0.185966 −0.0929831 0.995668i \(-0.529640\pi\)
−0.0929831 + 0.995668i \(0.529640\pi\)
\(510\) 0 0
\(511\) −14.5891 −0.645383
\(512\) −7.30328 −0.322763
\(513\) 0 0
\(514\) 9.86798 0.435258
\(515\) −2.03637 −0.0897330
\(516\) 0 0
\(517\) 0 0
\(518\) −15.6941 −0.689560
\(519\) 0 0
\(520\) 16.8258 0.737859
\(521\) 36.9418 1.61845 0.809225 0.587499i \(-0.199887\pi\)
0.809225 + 0.587499i \(0.199887\pi\)
\(522\) 0 0
\(523\) 1.84645 0.0807395 0.0403697 0.999185i \(-0.487146\pi\)
0.0403697 + 0.999185i \(0.487146\pi\)
\(524\) 1.68088 0.0734298
\(525\) 0 0
\(526\) −27.2953 −1.19013
\(527\) −35.3320 −1.53909
\(528\) 0 0
\(529\) −22.9857 −0.999378
\(530\) 1.30841 0.0568336
\(531\) 0 0
\(532\) 22.5846 0.979168
\(533\) 18.5373 0.802939
\(534\) 0 0
\(535\) −12.8781 −0.556769
\(536\) 75.0762 3.24280
\(537\) 0 0
\(538\) 85.8395 3.70081
\(539\) 0 0
\(540\) 0 0
\(541\) 9.73546 0.418560 0.209280 0.977856i \(-0.432888\pi\)
0.209280 + 0.977856i \(0.432888\pi\)
\(542\) 55.6215 2.38915
\(543\) 0 0
\(544\) 92.3708 3.96037
\(545\) −19.3162 −0.827414
\(546\) 0 0
\(547\) 1.27745 0.0546199 0.0273100 0.999627i \(-0.491306\pi\)
0.0273100 + 0.999627i \(0.491306\pi\)
\(548\) 53.8016 2.29829
\(549\) 0 0
\(550\) 0 0
\(551\) −19.1408 −0.815427
\(552\) 0 0
\(553\) 14.4761 0.615588
\(554\) −40.6050 −1.72514
\(555\) 0 0
\(556\) −69.3884 −2.94272
\(557\) −36.7874 −1.55873 −0.779366 0.626569i \(-0.784459\pi\)
−0.779366 + 0.626569i \(0.784459\pi\)
\(558\) 0 0
\(559\) −10.0494 −0.425046
\(560\) −14.6653 −0.619721
\(561\) 0 0
\(562\) 68.9667 2.90918
\(563\) 14.6601 0.617849 0.308924 0.951087i \(-0.400031\pi\)
0.308924 + 0.951087i \(0.400031\pi\)
\(564\) 0 0
\(565\) −7.35133 −0.309273
\(566\) −48.2402 −2.02769
\(567\) 0 0
\(568\) 27.4494 1.15175
\(569\) −27.8442 −1.16729 −0.583645 0.812009i \(-0.698374\pi\)
−0.583645 + 0.812009i \(0.698374\pi\)
\(570\) 0 0
\(571\) −14.5879 −0.610485 −0.305243 0.952275i \(-0.598738\pi\)
−0.305243 + 0.952275i \(0.598738\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −29.5335 −1.23270
\(575\) 0.434214 0.0181080
\(576\) 0 0
\(577\) −26.1147 −1.08717 −0.543584 0.839355i \(-0.682933\pi\)
−0.543584 + 0.839355i \(0.682933\pi\)
\(578\) 38.3994 1.59721
\(579\) 0 0
\(580\) 26.6868 1.10811
\(581\) −3.45931 −0.143516
\(582\) 0 0
\(583\) 0 0
\(584\) 124.643 5.15778
\(585\) 0 0
\(586\) −82.9987 −3.42864
\(587\) 14.6634 0.605222 0.302611 0.953114i \(-0.402142\pi\)
0.302611 + 0.953114i \(0.402142\pi\)
\(588\) 0 0
\(589\) 27.4876 1.13261
\(590\) −2.27022 −0.0934633
\(591\) 0 0
\(592\) 73.3542 3.01484
\(593\) −17.7060 −0.727100 −0.363550 0.931575i \(-0.618435\pi\)
−0.363550 + 0.931575i \(0.618435\pi\)
\(594\) 0 0
\(595\) −6.55039 −0.268540
\(596\) −50.1568 −2.05450
\(597\) 0 0
\(598\) −0.539572 −0.0220647
\(599\) 20.2930 0.829148 0.414574 0.910016i \(-0.363931\pi\)
0.414574 + 0.910016i \(0.363931\pi\)
\(600\) 0 0
\(601\) 10.9477 0.446566 0.223283 0.974754i \(-0.428323\pi\)
0.223283 + 0.974754i \(0.428323\pi\)
\(602\) 16.0107 0.652548
\(603\) 0 0
\(604\) −13.2827 −0.540467
\(605\) 0 0
\(606\) 0 0
\(607\) 33.9487 1.37793 0.688967 0.724793i \(-0.258064\pi\)
0.688967 + 0.724793i \(0.258064\pi\)
\(608\) −71.8626 −2.91441
\(609\) 0 0
\(610\) 27.7931 1.12531
\(611\) 21.1702 0.856455
\(612\) 0 0
\(613\) −7.46975 −0.301700 −0.150850 0.988557i \(-0.548201\pi\)
−0.150850 + 0.988557i \(0.548201\pi\)
\(614\) 26.8601 1.08399
\(615\) 0 0
\(616\) 0 0
\(617\) −29.3727 −1.18250 −0.591250 0.806489i \(-0.701365\pi\)
−0.591250 + 0.806489i \(0.701365\pi\)
\(618\) 0 0
\(619\) 36.1344 1.45236 0.726181 0.687503i \(-0.241293\pi\)
0.726181 + 0.687503i \(0.241293\pi\)
\(620\) −38.3241 −1.53913
\(621\) 0 0
\(622\) 43.7900 1.75582
\(623\) 12.0621 0.483259
\(624\) 0 0
\(625\) 6.32915 0.253166
\(626\) 11.9849 0.479014
\(627\) 0 0
\(628\) 96.0055 3.83104
\(629\) 32.7644 1.30640
\(630\) 0 0
\(631\) 39.1360 1.55798 0.778990 0.627036i \(-0.215732\pi\)
0.778990 + 0.627036i \(0.215732\pi\)
\(632\) −123.679 −4.91967
\(633\) 0 0
\(634\) −32.1118 −1.27532
\(635\) −20.1751 −0.800625
\(636\) 0 0
\(637\) 1.68268 0.0666704
\(638\) 0 0
\(639\) 0 0
\(640\) 21.5627 0.852342
\(641\) −27.8586 −1.10035 −0.550174 0.835050i \(-0.685439\pi\)
−0.550174 + 0.835050i \(0.685439\pi\)
\(642\) 0 0
\(643\) 13.1417 0.518258 0.259129 0.965843i \(-0.416565\pi\)
0.259129 + 0.965843i \(0.416565\pi\)
\(644\) 0.620417 0.0244479
\(645\) 0 0
\(646\) −65.3298 −2.57037
\(647\) −3.36604 −0.132333 −0.0661663 0.997809i \(-0.521077\pi\)
−0.0661663 + 0.997809i \(0.521077\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −16.3758 −0.642312
\(651\) 0 0
\(652\) 33.6020 1.31596
\(653\) −24.8798 −0.973622 −0.486811 0.873507i \(-0.661840\pi\)
−0.486811 + 0.873507i \(0.661840\pi\)
\(654\) 0 0
\(655\) 0.379280 0.0148197
\(656\) 138.039 5.38953
\(657\) 0 0
\(658\) −33.7283 −1.31486
\(659\) −30.5058 −1.18834 −0.594169 0.804340i \(-0.702519\pi\)
−0.594169 + 0.804340i \(0.702519\pi\)
\(660\) 0 0
\(661\) 10.2042 0.396899 0.198449 0.980111i \(-0.436409\pi\)
0.198449 + 0.980111i \(0.436409\pi\)
\(662\) −48.4892 −1.88459
\(663\) 0 0
\(664\) 29.5550 1.14696
\(665\) 5.09607 0.197617
\(666\) 0 0
\(667\) −0.525813 −0.0203596
\(668\) −131.004 −5.06871
\(669\) 0 0
\(670\) 27.5717 1.06519
\(671\) 0 0
\(672\) 0 0
\(673\) 48.7196 1.87800 0.939001 0.343914i \(-0.111753\pi\)
0.939001 + 0.343914i \(0.111753\pi\)
\(674\) −51.1661 −1.97085
\(675\) 0 0
\(676\) −52.7435 −2.02860
\(677\) −16.0041 −0.615087 −0.307543 0.951534i \(-0.599507\pi\)
−0.307543 + 0.951534i \(0.599507\pi\)
\(678\) 0 0
\(679\) 5.99575 0.230096
\(680\) 55.9640 2.14612
\(681\) 0 0
\(682\) 0 0
\(683\) −26.6991 −1.02161 −0.510805 0.859696i \(-0.670653\pi\)
−0.510805 + 0.859696i \(0.670653\pi\)
\(684\) 0 0
\(685\) 12.1400 0.463844
\(686\) −2.68084 −0.102355
\(687\) 0 0
\(688\) −74.8339 −2.85302
\(689\) 0.701687 0.0267321
\(690\) 0 0
\(691\) 15.7496 0.599143 0.299571 0.954074i \(-0.403156\pi\)
0.299571 + 0.954074i \(0.403156\pi\)
\(692\) −48.0725 −1.82744
\(693\) 0 0
\(694\) −14.2558 −0.541143
\(695\) −15.6570 −0.593905
\(696\) 0 0
\(697\) 61.6566 2.33541
\(698\) 55.2956 2.09297
\(699\) 0 0
\(700\) 18.8294 0.711686
\(701\) −32.1718 −1.21511 −0.607557 0.794276i \(-0.707850\pi\)
−0.607557 + 0.794276i \(0.707850\pi\)
\(702\) 0 0
\(703\) −25.4900 −0.961374
\(704\) 0 0
\(705\) 0 0
\(706\) −19.6169 −0.738293
\(707\) 0.101612 0.00382152
\(708\) 0 0
\(709\) −3.52842 −0.132513 −0.0662564 0.997803i \(-0.521106\pi\)
−0.0662564 + 0.997803i \(0.521106\pi\)
\(710\) 10.0808 0.378325
\(711\) 0 0
\(712\) −103.054 −3.86212
\(713\) 0.755104 0.0282789
\(714\) 0 0
\(715\) 0 0
\(716\) 80.1055 2.99368
\(717\) 0 0
\(718\) 64.9540 2.42406
\(719\) −37.1145 −1.38414 −0.692069 0.721831i \(-0.743301\pi\)
−0.692069 + 0.721831i \(0.743301\pi\)
\(720\) 0 0
\(721\) 1.73990 0.0647973
\(722\) −0.110715 −0.00412039
\(723\) 0 0
\(724\) 54.6277 2.03022
\(725\) −15.9582 −0.592674
\(726\) 0 0
\(727\) −7.85215 −0.291220 −0.145610 0.989342i \(-0.546514\pi\)
−0.145610 + 0.989342i \(0.546514\pi\)
\(728\) −14.3762 −0.532817
\(729\) 0 0
\(730\) 45.7752 1.69422
\(731\) −33.4253 −1.23628
\(732\) 0 0
\(733\) 29.0960 1.07468 0.537342 0.843365i \(-0.319429\pi\)
0.537342 + 0.843365i \(0.319429\pi\)
\(734\) −60.3347 −2.22699
\(735\) 0 0
\(736\) −1.97412 −0.0727671
\(737\) 0 0
\(738\) 0 0
\(739\) 7.18580 0.264334 0.132167 0.991227i \(-0.457806\pi\)
0.132167 + 0.991227i \(0.457806\pi\)
\(740\) 35.5391 1.30644
\(741\) 0 0
\(742\) −1.11792 −0.0410402
\(743\) 10.6220 0.389684 0.194842 0.980835i \(-0.437581\pi\)
0.194842 + 0.980835i \(0.437581\pi\)
\(744\) 0 0
\(745\) −11.3175 −0.414643
\(746\) −16.7630 −0.613739
\(747\) 0 0
\(748\) 0 0
\(749\) 11.0032 0.402049
\(750\) 0 0
\(751\) 16.0045 0.584014 0.292007 0.956416i \(-0.405677\pi\)
0.292007 + 0.956416i \(0.405677\pi\)
\(752\) 157.646 5.74874
\(753\) 0 0
\(754\) 19.8304 0.722179
\(755\) −2.99716 −0.109078
\(756\) 0 0
\(757\) −8.61194 −0.313006 −0.156503 0.987677i \(-0.550022\pi\)
−0.156503 + 0.987677i \(0.550022\pi\)
\(758\) 34.3664 1.24824
\(759\) 0 0
\(760\) −43.5388 −1.57932
\(761\) 15.5310 0.563000 0.281500 0.959561i \(-0.409168\pi\)
0.281500 + 0.959561i \(0.409168\pi\)
\(762\) 0 0
\(763\) 16.5040 0.597486
\(764\) 35.5997 1.28795
\(765\) 0 0
\(766\) −17.3380 −0.626446
\(767\) −1.21750 −0.0439612
\(768\) 0 0
\(769\) −51.4797 −1.85640 −0.928202 0.372076i \(-0.878646\pi\)
−0.928202 + 0.372076i \(0.878646\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −115.822 −4.16853
\(773\) 20.2078 0.726823 0.363411 0.931629i \(-0.381612\pi\)
0.363411 + 0.931629i \(0.381612\pi\)
\(774\) 0 0
\(775\) 22.9171 0.823208
\(776\) −51.2253 −1.83888
\(777\) 0 0
\(778\) 39.9358 1.43177
\(779\) −47.9676 −1.71862
\(780\) 0 0
\(781\) 0 0
\(782\) −1.79466 −0.0641769
\(783\) 0 0
\(784\) 12.5302 0.447508
\(785\) 21.6630 0.773185
\(786\) 0 0
\(787\) −49.5330 −1.76566 −0.882831 0.469692i \(-0.844365\pi\)
−0.882831 + 0.469692i \(0.844365\pi\)
\(788\) 23.0923 0.822629
\(789\) 0 0
\(790\) −45.4209 −1.61600
\(791\) 6.28108 0.223330
\(792\) 0 0
\(793\) 14.9052 0.529299
\(794\) 4.98266 0.176828
\(795\) 0 0
\(796\) −78.3187 −2.77593
\(797\) 26.6619 0.944414 0.472207 0.881488i \(-0.343457\pi\)
0.472207 + 0.881488i \(0.343457\pi\)
\(798\) 0 0
\(799\) 70.4140 2.49107
\(800\) −59.9139 −2.11827
\(801\) 0 0
\(802\) 23.4372 0.827597
\(803\) 0 0
\(804\) 0 0
\(805\) 0.139993 0.00493411
\(806\) −28.4778 −1.00309
\(807\) 0 0
\(808\) −0.868135 −0.0305409
\(809\) 51.2793 1.80288 0.901442 0.432900i \(-0.142510\pi\)
0.901442 + 0.432900i \(0.142510\pi\)
\(810\) 0 0
\(811\) −26.9340 −0.945782 −0.472891 0.881121i \(-0.656790\pi\)
−0.472891 + 0.881121i \(0.656790\pi\)
\(812\) −22.8016 −0.800180
\(813\) 0 0
\(814\) 0 0
\(815\) 7.58207 0.265588
\(816\) 0 0
\(817\) 26.0042 0.909772
\(818\) −34.0997 −1.19227
\(819\) 0 0
\(820\) 66.8781 2.33548
\(821\) −36.3229 −1.26768 −0.633839 0.773465i \(-0.718522\pi\)
−0.633839 + 0.773465i \(0.718522\pi\)
\(822\) 0 0
\(823\) −11.4540 −0.399262 −0.199631 0.979871i \(-0.563974\pi\)
−0.199631 + 0.979871i \(0.563974\pi\)
\(824\) −14.8650 −0.517848
\(825\) 0 0
\(826\) 1.93971 0.0674910
\(827\) −18.8743 −0.656324 −0.328162 0.944621i \(-0.606429\pi\)
−0.328162 + 0.944621i \(0.606429\pi\)
\(828\) 0 0
\(829\) −12.0160 −0.417331 −0.208666 0.977987i \(-0.566912\pi\)
−0.208666 + 0.977987i \(0.566912\pi\)
\(830\) 10.8541 0.376750
\(831\) 0 0
\(832\) 32.2825 1.11919
\(833\) 5.59675 0.193916
\(834\) 0 0
\(835\) −29.5603 −1.02297
\(836\) 0 0
\(837\) 0 0
\(838\) 21.7117 0.750019
\(839\) −5.34302 −0.184461 −0.0922307 0.995738i \(-0.529400\pi\)
−0.0922307 + 0.995738i \(0.529400\pi\)
\(840\) 0 0
\(841\) −9.67528 −0.333630
\(842\) −70.9849 −2.44630
\(843\) 0 0
\(844\) 118.513 4.07940
\(845\) −11.9012 −0.409414
\(846\) 0 0
\(847\) 0 0
\(848\) 5.22516 0.179433
\(849\) 0 0
\(850\) −54.4673 −1.86821
\(851\) −0.700230 −0.0240036
\(852\) 0 0
\(853\) −1.05041 −0.0359655 −0.0179827 0.999838i \(-0.505724\pi\)
−0.0179827 + 0.999838i \(0.505724\pi\)
\(854\) −23.7469 −0.812601
\(855\) 0 0
\(856\) −94.0073 −3.21310
\(857\) 13.9506 0.476542 0.238271 0.971199i \(-0.423419\pi\)
0.238271 + 0.971199i \(0.423419\pi\)
\(858\) 0 0
\(859\) 51.5493 1.75884 0.879419 0.476048i \(-0.157931\pi\)
0.879419 + 0.476048i \(0.157931\pi\)
\(860\) −36.2560 −1.23632
\(861\) 0 0
\(862\) −0.955576 −0.0325470
\(863\) 33.1894 1.12978 0.564890 0.825166i \(-0.308918\pi\)
0.564890 + 0.825166i \(0.308918\pi\)
\(864\) 0 0
\(865\) −10.8472 −0.368817
\(866\) −49.0284 −1.66605
\(867\) 0 0
\(868\) 32.7447 1.11143
\(869\) 0 0
\(870\) 0 0
\(871\) 14.7864 0.501020
\(872\) −141.004 −4.77500
\(873\) 0 0
\(874\) 1.39621 0.0472275
\(875\) 10.1007 0.341466
\(876\) 0 0
\(877\) 50.2380 1.69642 0.848208 0.529663i \(-0.177682\pi\)
0.848208 + 0.529663i \(0.177682\pi\)
\(878\) −17.8982 −0.604034
\(879\) 0 0
\(880\) 0 0
\(881\) −38.9209 −1.31128 −0.655639 0.755074i \(-0.727601\pi\)
−0.655639 + 0.755074i \(0.727601\pi\)
\(882\) 0 0
\(883\) −10.3534 −0.348420 −0.174210 0.984709i \(-0.555737\pi\)
−0.174210 + 0.984709i \(0.555737\pi\)
\(884\) 48.8481 1.64294
\(885\) 0 0
\(886\) 54.7971 1.84094
\(887\) −24.6430 −0.827432 −0.413716 0.910406i \(-0.635769\pi\)
−0.413716 + 0.910406i \(0.635769\pi\)
\(888\) 0 0
\(889\) 17.2379 0.578141
\(890\) −37.8466 −1.26862
\(891\) 0 0
\(892\) 50.3684 1.68646
\(893\) −54.7806 −1.83316
\(894\) 0 0
\(895\) 18.0753 0.604190
\(896\) −18.4235 −0.615487
\(897\) 0 0
\(898\) −14.7829 −0.493313
\(899\) −27.7516 −0.925569
\(900\) 0 0
\(901\) 2.33387 0.0777525
\(902\) 0 0
\(903\) 0 0
\(904\) −53.6631 −1.78481
\(905\) 12.3264 0.409742
\(906\) 0 0
\(907\) 43.0708 1.43014 0.715071 0.699052i \(-0.246394\pi\)
0.715071 + 0.699052i \(0.246394\pi\)
\(908\) −96.5666 −3.20467
\(909\) 0 0
\(910\) −5.27965 −0.175019
\(911\) 45.4296 1.50515 0.752574 0.658508i \(-0.228812\pi\)
0.752574 + 0.658508i \(0.228812\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −5.36907 −0.177593
\(915\) 0 0
\(916\) 46.9039 1.54975
\(917\) −0.324063 −0.0107015
\(918\) 0 0
\(919\) −16.7647 −0.553015 −0.276508 0.961012i \(-0.589177\pi\)
−0.276508 + 0.961012i \(0.589177\pi\)
\(920\) −1.19605 −0.0394324
\(921\) 0 0
\(922\) −21.9142 −0.721705
\(923\) 5.40623 0.177948
\(924\) 0 0
\(925\) −21.2517 −0.698753
\(926\) −11.4388 −0.375901
\(927\) 0 0
\(928\) 72.5530 2.38167
\(929\) −22.5588 −0.740131 −0.370065 0.929006i \(-0.620665\pi\)
−0.370065 + 0.929006i \(0.620665\pi\)
\(930\) 0 0
\(931\) −4.35416 −0.142702
\(932\) 44.7278 1.46511
\(933\) 0 0
\(934\) −108.611 −3.55386
\(935\) 0 0
\(936\) 0 0
\(937\) −7.56458 −0.247124 −0.123562 0.992337i \(-0.539432\pi\)
−0.123562 + 0.992337i \(0.539432\pi\)
\(938\) −23.5577 −0.769185
\(939\) 0 0
\(940\) 76.3770 2.49114
\(941\) −48.7593 −1.58951 −0.794754 0.606932i \(-0.792400\pi\)
−0.794754 + 0.606932i \(0.792400\pi\)
\(942\) 0 0
\(943\) −1.31771 −0.0429104
\(944\) −9.06618 −0.295079
\(945\) 0 0
\(946\) 0 0
\(947\) 14.4056 0.468119 0.234060 0.972222i \(-0.424799\pi\)
0.234060 + 0.972222i \(0.424799\pi\)
\(948\) 0 0
\(949\) 24.5488 0.796889
\(950\) 42.3744 1.37481
\(951\) 0 0
\(952\) −47.8165 −1.54974
\(953\) −31.4716 −1.01947 −0.509733 0.860333i \(-0.670256\pi\)
−0.509733 + 0.860333i \(0.670256\pi\)
\(954\) 0 0
\(955\) 8.03282 0.259936
\(956\) −107.302 −3.47039
\(957\) 0 0
\(958\) 22.0924 0.713772
\(959\) −10.3726 −0.334948
\(960\) 0 0
\(961\) 8.85325 0.285589
\(962\) 26.4083 0.851436
\(963\) 0 0
\(964\) 128.851 4.15000
\(965\) −26.1345 −0.841299
\(966\) 0 0
\(967\) −19.1109 −0.614566 −0.307283 0.951618i \(-0.599420\pi\)
−0.307283 + 0.951618i \(0.599420\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −18.8125 −0.604032
\(971\) 42.7809 1.37291 0.686453 0.727174i \(-0.259167\pi\)
0.686453 + 0.727174i \(0.259167\pi\)
\(972\) 0 0
\(973\) 13.3776 0.428866
\(974\) −2.58670 −0.0828831
\(975\) 0 0
\(976\) 110.993 3.55279
\(977\) −40.2273 −1.28699 −0.643493 0.765452i \(-0.722515\pi\)
−0.643493 + 0.765452i \(0.722515\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.07072 0.193922
\(981\) 0 0
\(982\) −81.5737 −2.60312
\(983\) 23.9559 0.764074 0.382037 0.924147i \(-0.375223\pi\)
0.382037 + 0.924147i \(0.375223\pi\)
\(984\) 0 0
\(985\) 5.21062 0.166024
\(986\) 65.9574 2.10051
\(987\) 0 0
\(988\) −38.0028 −1.20903
\(989\) 0.714356 0.0227152
\(990\) 0 0
\(991\) −19.7275 −0.626665 −0.313332 0.949644i \(-0.601445\pi\)
−0.313332 + 0.949644i \(0.601445\pi\)
\(992\) −104.191 −3.30807
\(993\) 0 0
\(994\) −8.61317 −0.273193
\(995\) −17.6721 −0.560243
\(996\) 0 0
\(997\) −1.90849 −0.0604426 −0.0302213 0.999543i \(-0.509621\pi\)
−0.0302213 + 0.999543i \(0.509621\pi\)
\(998\) −100.690 −3.18728
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7623.2.a.db.1.16 16
3.2 odd 2 inner 7623.2.a.db.1.1 16
11.2 odd 10 693.2.m.k.631.8 yes 32
11.6 odd 10 693.2.m.k.190.8 yes 32
11.10 odd 2 7623.2.a.dc.1.1 16
33.2 even 10 693.2.m.k.631.1 yes 32
33.17 even 10 693.2.m.k.190.1 32
33.32 even 2 7623.2.a.dc.1.16 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.190.1 32 33.17 even 10
693.2.m.k.190.8 yes 32 11.6 odd 10
693.2.m.k.631.1 yes 32 33.2 even 10
693.2.m.k.631.8 yes 32 11.2 odd 10
7623.2.a.db.1.1 16 3.2 odd 2 inner
7623.2.a.db.1.16 16 1.1 even 1 trivial
7623.2.a.dc.1.1 16 11.10 odd 2
7623.2.a.dc.1.16 16 33.32 even 2