Properties

Label 7623.2.a.dc.1.1
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 1.1
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 \(-0.896719\pi\)
−0.947821 + 0.318804i \(0.896719\pi\)
\(3\) 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 \(-0.574968\pi\)
−0.233346 + 0.972394i \(0.574968\pi\)
\(14\) −2.68084 −0.716485
\(15\) 0 0
\(16\) 12.5302 3.13256
\(17\) −5.59675 −1.35741 −0.678706 0.734410i \(-0.737459\pi\)
−0.678706 + 0.734410i \(0.737459\pi\)
\(18\) 0 0
\(19\) 4.35416 0.998913 0.499456 0.866339i \(-0.333533\pi\)
0.499456 + 0.866339i \(0.333533\pi\)
\(20\) 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 \(-0.633829\pi\)
−0.408157 + 0.912912i \(0.633829\pi\)
\(30\) 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 \(-0.829689\pi\)
−0.860244 + 0.509883i \(0.829689\pi\)
\(42\) 0 0
\(43\) 5.97227 0.910762 0.455381 0.890297i \(-0.349503\pi\)
0.455381 + 0.890297i \(0.349503\pi\)
\(44\) 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 \(-0.691925\pi\)
−0.567075 + 0.823666i \(0.691925\pi\)
\(62\) 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 \(-0.825685\pi\)
−0.853762 + 0.520664i \(0.825685\pi\)
\(74\) −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 \(-0.197095\pi\)
0.814347 + 0.580378i \(0.197095\pi\)
\(80\) 14.6653 1.63963
\(81\) 0 0
\(82\) 29.5335 3.26143
\(83\) −3.45931 −0.379709 −0.189854 0.981812i \(-0.560802\pi\)
−0.189854 + 0.981812i \(0.560802\pi\)
\(84\) 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 \(-0.498391\pi\)
0.00505540 + 0.999987i \(0.498391\pi\)
\(102\) 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 \(-0.321493\pi\)
0.531861 + 0.846831i \(0.321493\pi\)
\(108\) 0 0
\(109\) 16.5040 1.58080 0.790400 0.612592i \(-0.209873\pi\)
0.790400 + 0.612592i \(0.209873\pi\)
\(110\) 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 \(-0.222833\pi\)
0.764809 + 0.644257i \(0.222833\pi\)
\(128\) −18.4235 −1.62843
\(129\) 0 0
\(130\) 5.27965 0.463056
\(131\) −0.324063 −0.0283135 −0.0141567 0.999900i \(-0.504506\pi\)
−0.0141567 + 0.999900i \(0.504506\pi\)
\(132\) 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 \(-0.307974\pi\)
0.567336 + 0.823486i \(0.307974\pi\)
\(140\) 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 \(-0.370365\pi\)
0.396094 + 0.918210i \(0.370365\pi\)
\(150\) 0 0
\(151\) 2.56082 0.208396 0.104198 0.994557i \(-0.466772\pi\)
0.104198 + 0.994557i \(0.466772\pi\)
\(152\) −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 \(-0.0680843\pi\)
0.977212 + 0.212266i \(0.0680843\pi\)
\(168\) 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 \(-0.385394\pi\)
0.352318 + 0.935880i \(0.385394\pi\)
\(174\) 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 \(-0.202881\pi\)
0.803663 + 0.595085i \(0.202881\pi\)
\(194\) 16.0737 1.15402
\(195\) 0 0
\(196\) 5.18691 0.370494
\(197\) −4.45203 −0.317194 −0.158597 0.987343i \(-0.550697\pi\)
−0.158597 + 0.987343i \(0.550697\pi\)
\(198\) 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 \(-0.788099\pi\)
−0.786480 + 0.617616i \(0.788099\pi\)
\(212\) 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 \(-0.288009\pi\)
0.617838 + 0.786305i \(0.288009\pi\)
\(228\) 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 \(-0.591151\pi\)
−0.282462 + 0.959278i \(0.591151\pi\)
\(234\) 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 \(-0.266694\pi\)
0.669067 + 0.743202i \(0.266694\pi\)
\(240\) 0 0
\(241\) −24.8415 −1.60018 −0.800090 0.599880i \(-0.795215\pi\)
−0.800090 + 0.599880i \(0.795215\pi\)
\(242\) 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 \(-0.398360\pi\)
0.313912 + 0.949452i \(0.398360\pi\)
\(264\) 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 \(-0.717014\pi\)
−0.630169 + 0.776458i \(0.717014\pi\)
\(272\) −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 \(-0.349629\pi\)
0.455028 + 0.890477i \(0.349629\pi\)
\(278\) −35.8632 −2.15093
\(279\) 0 0
\(280\) −9.99937 −0.597577
\(281\) −25.7257 −1.53467 −0.767335 0.641247i \(-0.778417\pi\)
−0.767335 + 0.641247i \(0.778417\pi\)
\(282\) 0 0
\(283\) 17.9944 1.06966 0.534828 0.844961i \(-0.320376\pi\)
0.534828 + 0.844961i \(0.320376\pi\)
\(284\) 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 \(-0.140357\pi\)
0.904349 + 0.426794i \(0.140357\pi\)
\(294\) 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 \(-0.592298\pi\)
−0.285915 + 0.958255i \(0.592298\pi\)
\(308\) 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 \(-0.325993\pi\)
0.519836 + 0.854266i \(0.325993\pi\)
\(338\) 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 \(-0.454411\pi\)
0.142733 + 0.989761i \(0.454411\pi\)
\(348\) 0 0
\(349\) −20.6262 −1.10410 −0.552048 0.833813i \(-0.686153\pi\)
−0.552048 + 0.833813i \(0.686153\pi\)
\(350\) 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 \(-0.720808\pi\)
−0.639378 + 0.768892i \(0.720808\pi\)
\(360\) 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 \(-0.448244\pi\)
0.161881 + 0.986810i \(0.448244\pi\)
\(374\) 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 \(-0.398171\pi\)
0.314476 + 0.949265i \(0.398171\pi\)
\(410\) 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 \(-0.497267\pi\)
0.00858471 + 0.999963i \(0.497267\pi\)
\(432\) 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 \(-0.449069\pi\)
0.159322 + 0.987227i \(0.449069\pi\)
\(440\) 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 \(-0.485084\pi\)
0.0468424 + 0.998902i \(0.485084\pi\)
\(458\) −24.2421 −1.13276
\(459\) 0 0
\(460\) −0.726131 −0.0338561
\(461\) 8.17436 0.380718 0.190359 0.981715i \(-0.439035\pi\)
0.190359 + 0.981715i \(0.439035\pi\)
\(462\) 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 \(-0.560287\pi\)
−0.188267 + 0.982118i \(0.560287\pi\)
\(480\) 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 \(-0.259099\pi\)
0.686607 + 0.727029i \(0.259099\pi\)
\(492\) 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 \(-0.671292\pi\)
−0.512530 + 0.858669i \(0.671292\pi\)
\(504\) 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 \(-0.512854\pi\)
−0.0403697 + 0.999185i \(0.512854\pi\)
\(524\) −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 \(-0.567112\pi\)
−0.209280 + 0.977856i \(0.567112\pi\)
\(542\) 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 \(-0.508694\pi\)
−0.0273100 + 0.999627i \(0.508694\pi\)
\(548\) 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 \(-0.215541\pi\)
0.779366 + 0.626569i \(0.215541\pi\)
\(558\) 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 \(-0.599969\pi\)
−0.308924 + 0.951087i \(0.599969\pi\)
\(564\) 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 \(-0.301626\pi\)
0.583645 + 0.812009i \(0.301626\pi\)
\(570\) 0 0
\(571\) 14.5879 0.610485 0.305243 0.952275i \(-0.401262\pi\)
0.305243 + 0.952275i \(0.401262\pi\)
\(572\) 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 \(-0.381565\pi\)
0.363550 + 0.931575i \(0.381565\pi\)
\(594\) 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 \(-0.571677\pi\)
−0.223283 + 0.974754i \(0.571677\pi\)
\(602\) −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 \(-0.741936\pi\)
−0.688967 + 0.724793i \(0.741936\pi\)
\(608\) −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 \(-0.451799\pi\)
0.150850 + 0.988557i \(0.451799\pi\)
\(614\) 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 \(-0.297481\pi\)
0.594169 + 0.804340i \(0.297481\pi\)
\(660\) 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 \(-0.888247\pi\)
−0.939001 + 0.343914i \(0.888247\pi\)
\(674\) −51.1661 −1.97085
\(675\) 0 0
\(676\) −52.7435 −2.02860
\(677\) 16.0041 0.615087 0.307543 0.951534i \(-0.400493\pi\)
0.307543 + 0.951534i \(0.400493\pi\)
\(678\) 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 \(-0.292150\pi\)
0.607557 + 0.794276i \(0.292150\pi\)
\(702\) 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 \(-0.680571\pi\)
−0.537342 + 0.843365i \(0.680571\pi\)
\(734\) 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 \(-0.542194\pi\)
−0.132167 + 0.991227i \(0.542194\pi\)
\(740\) 35.5391 1.30644
\(741\) 0 0
\(742\) −1.11792 −0.0410402
\(743\) −10.6220 −0.389684 −0.194842 0.980835i \(-0.562419\pi\)
−0.194842 + 0.980835i \(0.562419\pi\)
\(744\) 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 \(-0.590832\pi\)
−0.281500 + 0.959561i \(0.590832\pi\)
\(762\) 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 \(-0.121354\pi\)
0.928202 + 0.372076i \(0.121354\pi\)
\(770\) 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 \(-0.155635\pi\)
0.882831 + 0.469692i \(0.155635\pi\)
\(788\) −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 \(-0.857490\pi\)
−0.901442 + 0.432900i \(0.857490\pi\)
\(810\) 0 0
\(811\) 26.9340 0.945782 0.472891 0.881121i \(-0.343210\pi\)
0.472891 + 0.881121i \(0.343210\pi\)
\(812\) −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 \(-0.281478\pi\)
0.633839 + 0.773465i \(0.281478\pi\)
\(822\) 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 \(-0.393571\pi\)
0.328162 + 0.944621i \(0.393571\pi\)
\(828\) 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 \(-0.494276\pi\)
0.0179827 + 0.999838i \(0.494276\pi\)
\(854\) 23.7469 0.812601
\(855\) 0 0
\(856\) −94.0073 −3.21310
\(857\) −13.9506 −0.476542 −0.238271 0.971199i \(-0.576581\pi\)
−0.238271 + 0.971199i \(0.576581\pi\)
\(858\) 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 \(-0.822318\pi\)
−0.848208 + 0.529663i \(0.822318\pi\)
\(878\) −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 \(-0.364231\pi\)
0.413716 + 0.910406i \(0.364231\pi\)
\(888\) 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 \(-0.410823\pi\)
0.276508 + 0.961012i \(0.410823\pi\)
\(920\) 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 \(-0.460568\pi\)
0.123562 + 0.992337i \(0.460568\pi\)
\(938\) −23.5577 −0.769185
\(939\) 0 0
\(940\) 76.3770 2.49114
\(941\) 48.7593 1.58951 0.794754 0.606932i \(-0.207600\pi\)
0.794754 + 0.606932i \(0.207600\pi\)
\(942\) 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 \(-0.329744\pi\)
0.509733 + 0.860333i \(0.329744\pi\)
\(954\) 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 \(-0.400580\pi\)
0.307283 + 0.951618i \(0.400580\pi\)
\(968\) 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 \(-0.490379\pi\)
0.0302213 + 0.999543i \(0.490379\pi\)
\(998\) 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.dc.1.1 16
3.2 odd 2 inner 7623.2.a.dc.1.16 16
11.5 even 5 693.2.m.k.190.8 yes 32
11.9 even 5 693.2.m.k.631.8 yes 32
11.10 odd 2 7623.2.a.db.1.16 16
33.5 odd 10 693.2.m.k.190.1 32
33.20 odd 10 693.2.m.k.631.1 yes 32
33.32 even 2 7623.2.a.db.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.190.1 32 33.5 odd 10
693.2.m.k.190.8 yes 32 11.5 even 5
693.2.m.k.631.1 yes 32 33.20 odd 10
693.2.m.k.631.8 yes 32 11.9 even 5
7623.2.a.db.1.1 16 33.32 even 2
7623.2.a.db.1.16 16 11.10 odd 2
7623.2.a.dc.1.1 16 1.1 even 1 trivial
7623.2.a.dc.1.16 16 3.2 odd 2 inner