Properties

Label 7623.2.a.db.1.5
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 1.5
Root \(-1.31288\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.31288 q^{2} -0.276342 q^{4} +3.45937 q^{5} -1.00000 q^{7} +2.98857 q^{8} +O(q^{10})\) \(q-1.31288 q^{2} -0.276342 q^{4} +3.45937 q^{5} -1.00000 q^{7} +2.98857 q^{8} -4.54174 q^{10} +5.53810 q^{13} +1.31288 q^{14} -3.37095 q^{16} +4.96611 q^{17} -8.22969 q^{19} -0.955969 q^{20} +7.56119 q^{23} +6.96722 q^{25} -7.27087 q^{26} +0.276342 q^{28} -1.47011 q^{29} +2.41141 q^{31} -1.55148 q^{32} -6.51991 q^{34} -3.45937 q^{35} -2.83639 q^{37} +10.8046 q^{38} +10.3386 q^{40} +6.93310 q^{41} -5.62936 q^{43} -9.92695 q^{46} +0.0417127 q^{47} +1.00000 q^{49} -9.14714 q^{50} -1.53041 q^{52} +4.47145 q^{53} -2.98857 q^{56} +1.93008 q^{58} +2.99486 q^{59} +11.6684 q^{61} -3.16589 q^{62} +8.77881 q^{64} +19.1583 q^{65} +9.95852 q^{67} -1.37234 q^{68} +4.54174 q^{70} +6.99312 q^{71} -7.38320 q^{73} +3.72385 q^{74} +2.27421 q^{76} -16.2755 q^{79} -11.6614 q^{80} -9.10234 q^{82} -9.64377 q^{83} +17.1796 q^{85} +7.39068 q^{86} +7.94906 q^{89} -5.53810 q^{91} -2.08948 q^{92} -0.0547639 q^{94} -28.4695 q^{95} -2.35182 q^{97} -1.31288 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} - 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} - 16q^{7} + 6q^{10} + 32q^{16} - 10q^{19} + 44q^{25} - 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} + 68q^{40} - 16q^{43} - 80q^{46} + 16q^{49} - 2q^{52} + 18q^{58} + 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} + 14q^{73} + 14q^{76} - 54q^{79} + 64q^{82} - 30q^{85} + 60q^{94} + 108q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.31288 −0.928347 −0.464174 0.885744i \(-0.653649\pi\)
−0.464174 + 0.885744i \(0.653649\pi\)
\(3\) 0 0
\(4\) −0.276342 −0.138171
\(5\) 3.45937 1.54708 0.773538 0.633750i \(-0.218485\pi\)
0.773538 + 0.633750i \(0.218485\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.98857 1.05662
\(9\) 0 0
\(10\) −4.54174 −1.43622
\(11\) 0 0
\(12\) 0 0
\(13\) 5.53810 1.53599 0.767997 0.640454i \(-0.221254\pi\)
0.767997 + 0.640454i \(0.221254\pi\)
\(14\) 1.31288 0.350882
\(15\) 0 0
\(16\) −3.37095 −0.842738
\(17\) 4.96611 1.20446 0.602229 0.798324i \(-0.294280\pi\)
0.602229 + 0.798324i \(0.294280\pi\)
\(18\) 0 0
\(19\) −8.22969 −1.88802 −0.944010 0.329918i \(-0.892979\pi\)
−0.944010 + 0.329918i \(0.892979\pi\)
\(20\) −0.955969 −0.213761
\(21\) 0 0
\(22\) 0 0
\(23\) 7.56119 1.57662 0.788309 0.615280i \(-0.210957\pi\)
0.788309 + 0.615280i \(0.210957\pi\)
\(24\) 0 0
\(25\) 6.96722 1.39344
\(26\) −7.27087 −1.42594
\(27\) 0 0
\(28\) 0.276342 0.0522238
\(29\) −1.47011 −0.272993 −0.136496 0.990641i \(-0.543584\pi\)
−0.136496 + 0.990641i \(0.543584\pi\)
\(30\) 0 0
\(31\) 2.41141 0.433102 0.216551 0.976271i \(-0.430519\pi\)
0.216551 + 0.976271i \(0.430519\pi\)
\(32\) −1.55148 −0.274265
\(33\) 0 0
\(34\) −6.51991 −1.11815
\(35\) −3.45937 −0.584740
\(36\) 0 0
\(37\) −2.83639 −0.466300 −0.233150 0.972441i \(-0.574903\pi\)
−0.233150 + 0.972441i \(0.574903\pi\)
\(38\) 10.8046 1.75274
\(39\) 0 0
\(40\) 10.3386 1.63467
\(41\) 6.93310 1.08277 0.541384 0.840775i \(-0.317900\pi\)
0.541384 + 0.840775i \(0.317900\pi\)
\(42\) 0 0
\(43\) −5.62936 −0.858470 −0.429235 0.903193i \(-0.641217\pi\)
−0.429235 + 0.903193i \(0.641217\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −9.92695 −1.46365
\(47\) 0.0417127 0.00608443 0.00304221 0.999995i \(-0.499032\pi\)
0.00304221 + 0.999995i \(0.499032\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −9.14714 −1.29360
\(51\) 0 0
\(52\) −1.53041 −0.212230
\(53\) 4.47145 0.614201 0.307101 0.951677i \(-0.400641\pi\)
0.307101 + 0.951677i \(0.400641\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.98857 −0.399364
\(57\) 0 0
\(58\) 1.93008 0.253432
\(59\) 2.99486 0.389898 0.194949 0.980813i \(-0.437546\pi\)
0.194949 + 0.980813i \(0.437546\pi\)
\(60\) 0 0
\(61\) 11.6684 1.49399 0.746995 0.664830i \(-0.231496\pi\)
0.746995 + 0.664830i \(0.231496\pi\)
\(62\) −3.16589 −0.402069
\(63\) 0 0
\(64\) 8.77881 1.09735
\(65\) 19.1583 2.37630
\(66\) 0 0
\(67\) 9.95852 1.21663 0.608314 0.793697i \(-0.291846\pi\)
0.608314 + 0.793697i \(0.291846\pi\)
\(68\) −1.37234 −0.166421
\(69\) 0 0
\(70\) 4.54174 0.542842
\(71\) 6.99312 0.829931 0.414966 0.909837i \(-0.363794\pi\)
0.414966 + 0.909837i \(0.363794\pi\)
\(72\) 0 0
\(73\) −7.38320 −0.864138 −0.432069 0.901841i \(-0.642216\pi\)
−0.432069 + 0.901841i \(0.642216\pi\)
\(74\) 3.72385 0.432888
\(75\) 0 0
\(76\) 2.27421 0.260870
\(77\) 0 0
\(78\) 0 0
\(79\) −16.2755 −1.83114 −0.915570 0.402159i \(-0.868260\pi\)
−0.915570 + 0.402159i \(0.868260\pi\)
\(80\) −11.6614 −1.30378
\(81\) 0 0
\(82\) −9.10234 −1.00519
\(83\) −9.64377 −1.05854 −0.529271 0.848453i \(-0.677534\pi\)
−0.529271 + 0.848453i \(0.677534\pi\)
\(84\) 0 0
\(85\) 17.1796 1.86339
\(86\) 7.39068 0.796958
\(87\) 0 0
\(88\) 0 0
\(89\) 7.94906 0.842599 0.421300 0.906922i \(-0.361574\pi\)
0.421300 + 0.906922i \(0.361574\pi\)
\(90\) 0 0
\(91\) −5.53810 −0.580551
\(92\) −2.08948 −0.217843
\(93\) 0 0
\(94\) −0.0547639 −0.00564846
\(95\) −28.4695 −2.92091
\(96\) 0 0
\(97\) −2.35182 −0.238791 −0.119395 0.992847i \(-0.538096\pi\)
−0.119395 + 0.992847i \(0.538096\pi\)
\(98\) −1.31288 −0.132621
\(99\) 0 0
\(100\) −1.92534 −0.192534
\(101\) −13.8502 −1.37814 −0.689072 0.724693i \(-0.741982\pi\)
−0.689072 + 0.724693i \(0.741982\pi\)
\(102\) 0 0
\(103\) 3.79732 0.374161 0.187081 0.982345i \(-0.440097\pi\)
0.187081 + 0.982345i \(0.440097\pi\)
\(104\) 16.5510 1.62296
\(105\) 0 0
\(106\) −5.87049 −0.570192
\(107\) −1.34165 −0.129702 −0.0648512 0.997895i \(-0.520657\pi\)
−0.0648512 + 0.997895i \(0.520657\pi\)
\(108\) 0 0
\(109\) 17.5860 1.68443 0.842216 0.539140i \(-0.181251\pi\)
0.842216 + 0.539140i \(0.181251\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.37095 0.318525
\(113\) 12.7641 1.20075 0.600374 0.799719i \(-0.295018\pi\)
0.600374 + 0.799719i \(0.295018\pi\)
\(114\) 0 0
\(115\) 26.1569 2.43915
\(116\) 0.406254 0.0377197
\(117\) 0 0
\(118\) −3.93190 −0.361961
\(119\) −4.96611 −0.455242
\(120\) 0 0
\(121\) 0 0
\(122\) −15.3193 −1.38694
\(123\) 0 0
\(124\) −0.666374 −0.0598422
\(125\) 6.80535 0.608689
\(126\) 0 0
\(127\) −3.98767 −0.353848 −0.176924 0.984224i \(-0.556615\pi\)
−0.176924 + 0.984224i \(0.556615\pi\)
\(128\) −8.42258 −0.744458
\(129\) 0 0
\(130\) −25.1526 −2.20603
\(131\) −2.02685 −0.177087 −0.0885433 0.996072i \(-0.528221\pi\)
−0.0885433 + 0.996072i \(0.528221\pi\)
\(132\) 0 0
\(133\) 8.22969 0.713604
\(134\) −13.0744 −1.12945
\(135\) 0 0
\(136\) 14.8415 1.27265
\(137\) −15.2924 −1.30652 −0.653259 0.757135i \(-0.726599\pi\)
−0.653259 + 0.757135i \(0.726599\pi\)
\(138\) 0 0
\(139\) 21.5228 1.82554 0.912772 0.408470i \(-0.133938\pi\)
0.912772 + 0.408470i \(0.133938\pi\)
\(140\) 0.955969 0.0807941
\(141\) 0 0
\(142\) −9.18114 −0.770464
\(143\) 0 0
\(144\) 0 0
\(145\) −5.08565 −0.422340
\(146\) 9.69326 0.802220
\(147\) 0 0
\(148\) 0.783815 0.0644292
\(149\) 1.21143 0.0992440 0.0496220 0.998768i \(-0.484198\pi\)
0.0496220 + 0.998768i \(0.484198\pi\)
\(150\) 0 0
\(151\) −9.71739 −0.790790 −0.395395 0.918511i \(-0.629392\pi\)
−0.395395 + 0.918511i \(0.629392\pi\)
\(152\) −24.5950 −1.99492
\(153\) 0 0
\(154\) 0 0
\(155\) 8.34195 0.670042
\(156\) 0 0
\(157\) 16.5385 1.31991 0.659957 0.751303i \(-0.270574\pi\)
0.659957 + 0.751303i \(0.270574\pi\)
\(158\) 21.3678 1.69993
\(159\) 0 0
\(160\) −5.36713 −0.424309
\(161\) −7.56119 −0.595905
\(162\) 0 0
\(163\) −7.60022 −0.595295 −0.297648 0.954676i \(-0.596202\pi\)
−0.297648 + 0.954676i \(0.596202\pi\)
\(164\) −1.91591 −0.149607
\(165\) 0 0
\(166\) 12.6611 0.982694
\(167\) −4.08561 −0.316154 −0.158077 0.987427i \(-0.550529\pi\)
−0.158077 + 0.987427i \(0.550529\pi\)
\(168\) 0 0
\(169\) 17.6706 1.35928
\(170\) −22.5548 −1.72987
\(171\) 0 0
\(172\) 1.55563 0.118616
\(173\) 3.21279 0.244264 0.122132 0.992514i \(-0.461027\pi\)
0.122132 + 0.992514i \(0.461027\pi\)
\(174\) 0 0
\(175\) −6.96722 −0.526673
\(176\) 0 0
\(177\) 0 0
\(178\) −10.4362 −0.782225
\(179\) −0.207824 −0.0155335 −0.00776674 0.999970i \(-0.502472\pi\)
−0.00776674 + 0.999970i \(0.502472\pi\)
\(180\) 0 0
\(181\) −5.25519 −0.390615 −0.195307 0.980742i \(-0.562570\pi\)
−0.195307 + 0.980742i \(0.562570\pi\)
\(182\) 7.27087 0.538953
\(183\) 0 0
\(184\) 22.5971 1.66588
\(185\) −9.81212 −0.721402
\(186\) 0 0
\(187\) 0 0
\(188\) −0.0115270 −0.000840692 0
\(189\) 0 0
\(190\) 37.3771 2.71162
\(191\) −2.70775 −0.195926 −0.0979630 0.995190i \(-0.531233\pi\)
−0.0979630 + 0.995190i \(0.531233\pi\)
\(192\) 0 0
\(193\) 16.4899 1.18697 0.593484 0.804846i \(-0.297752\pi\)
0.593484 + 0.804846i \(0.297752\pi\)
\(194\) 3.08766 0.221681
\(195\) 0 0
\(196\) −0.276342 −0.0197387
\(197\) −3.21597 −0.229128 −0.114564 0.993416i \(-0.536547\pi\)
−0.114564 + 0.993416i \(0.536547\pi\)
\(198\) 0 0
\(199\) 8.63547 0.612152 0.306076 0.952007i \(-0.400984\pi\)
0.306076 + 0.952007i \(0.400984\pi\)
\(200\) 20.8220 1.47234
\(201\) 0 0
\(202\) 18.1836 1.27940
\(203\) 1.47011 0.103182
\(204\) 0 0
\(205\) 23.9842 1.67513
\(206\) −4.98543 −0.347351
\(207\) 0 0
\(208\) −18.6687 −1.29444
\(209\) 0 0
\(210\) 0 0
\(211\) −4.91452 −0.338329 −0.169165 0.985588i \(-0.554107\pi\)
−0.169165 + 0.985588i \(0.554107\pi\)
\(212\) −1.23565 −0.0848649
\(213\) 0 0
\(214\) 1.76143 0.120409
\(215\) −19.4740 −1.32812
\(216\) 0 0
\(217\) −2.41141 −0.163697
\(218\) −23.0883 −1.56374
\(219\) 0 0
\(220\) 0 0
\(221\) 27.5028 1.85004
\(222\) 0 0
\(223\) −1.15583 −0.0774000 −0.0387000 0.999251i \(-0.512322\pi\)
−0.0387000 + 0.999251i \(0.512322\pi\)
\(224\) 1.55148 0.103662
\(225\) 0 0
\(226\) −16.7578 −1.11471
\(227\) 2.62410 0.174168 0.0870839 0.996201i \(-0.472245\pi\)
0.0870839 + 0.996201i \(0.472245\pi\)
\(228\) 0 0
\(229\) −10.8283 −0.715555 −0.357777 0.933807i \(-0.616465\pi\)
−0.357777 + 0.933807i \(0.616465\pi\)
\(230\) −34.3410 −2.26438
\(231\) 0 0
\(232\) −4.39352 −0.288449
\(233\) −16.7906 −1.09999 −0.549995 0.835168i \(-0.685370\pi\)
−0.549995 + 0.835168i \(0.685370\pi\)
\(234\) 0 0
\(235\) 0.144300 0.00941307
\(236\) −0.827607 −0.0538726
\(237\) 0 0
\(238\) 6.51991 0.422623
\(239\) −12.8564 −0.831613 −0.415806 0.909453i \(-0.636501\pi\)
−0.415806 + 0.909453i \(0.636501\pi\)
\(240\) 0 0
\(241\) −1.12325 −0.0723548 −0.0361774 0.999345i \(-0.511518\pi\)
−0.0361774 + 0.999345i \(0.511518\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −3.22448 −0.206426
\(245\) 3.45937 0.221011
\(246\) 0 0
\(247\) −45.5769 −2.89999
\(248\) 7.20666 0.457623
\(249\) 0 0
\(250\) −8.93461 −0.565075
\(251\) 24.8678 1.56964 0.784819 0.619725i \(-0.212756\pi\)
0.784819 + 0.619725i \(0.212756\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 5.23533 0.328494
\(255\) 0 0
\(256\) −6.49976 −0.406235
\(257\) −12.9965 −0.810701 −0.405350 0.914161i \(-0.632850\pi\)
−0.405350 + 0.914161i \(0.632850\pi\)
\(258\) 0 0
\(259\) 2.83639 0.176245
\(260\) −5.29426 −0.328336
\(261\) 0 0
\(262\) 2.66101 0.164398
\(263\) 28.0645 1.73053 0.865266 0.501313i \(-0.167149\pi\)
0.865266 + 0.501313i \(0.167149\pi\)
\(264\) 0 0
\(265\) 15.4684 0.950216
\(266\) −10.8046 −0.662473
\(267\) 0 0
\(268\) −2.75196 −0.168103
\(269\) −30.4884 −1.85891 −0.929456 0.368933i \(-0.879723\pi\)
−0.929456 + 0.368933i \(0.879723\pi\)
\(270\) 0 0
\(271\) 0.119754 0.00727455 0.00363727 0.999993i \(-0.498842\pi\)
0.00363727 + 0.999993i \(0.498842\pi\)
\(272\) −16.7405 −1.01504
\(273\) 0 0
\(274\) 20.0771 1.21290
\(275\) 0 0
\(276\) 0 0
\(277\) 5.63483 0.338564 0.169282 0.985568i \(-0.445855\pi\)
0.169282 + 0.985568i \(0.445855\pi\)
\(278\) −28.2569 −1.69474
\(279\) 0 0
\(280\) −10.3386 −0.617847
\(281\) −4.16605 −0.248526 −0.124263 0.992249i \(-0.539657\pi\)
−0.124263 + 0.992249i \(0.539657\pi\)
\(282\) 0 0
\(283\) −13.6746 −0.812872 −0.406436 0.913679i \(-0.633229\pi\)
−0.406436 + 0.913679i \(0.633229\pi\)
\(284\) −1.93250 −0.114673
\(285\) 0 0
\(286\) 0 0
\(287\) −6.93310 −0.409248
\(288\) 0 0
\(289\) 7.66220 0.450718
\(290\) 6.67686 0.392079
\(291\) 0 0
\(292\) 2.04029 0.119399
\(293\) 3.09005 0.180523 0.0902614 0.995918i \(-0.471230\pi\)
0.0902614 + 0.995918i \(0.471230\pi\)
\(294\) 0 0
\(295\) 10.3603 0.603201
\(296\) −8.47675 −0.492701
\(297\) 0 0
\(298\) −1.59046 −0.0921329
\(299\) 41.8747 2.42167
\(300\) 0 0
\(301\) 5.62936 0.324471
\(302\) 12.7578 0.734128
\(303\) 0 0
\(304\) 27.7419 1.59111
\(305\) 40.3654 2.31132
\(306\) 0 0
\(307\) −5.33016 −0.304208 −0.152104 0.988364i \(-0.548605\pi\)
−0.152104 + 0.988364i \(0.548605\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −10.9520 −0.622031
\(311\) −7.55259 −0.428268 −0.214134 0.976804i \(-0.568693\pi\)
−0.214134 + 0.976804i \(0.568693\pi\)
\(312\) 0 0
\(313\) 4.84740 0.273991 0.136996 0.990572i \(-0.456255\pi\)
0.136996 + 0.990572i \(0.456255\pi\)
\(314\) −21.7131 −1.22534
\(315\) 0 0
\(316\) 4.49761 0.253011
\(317\) −31.0942 −1.74643 −0.873213 0.487339i \(-0.837967\pi\)
−0.873213 + 0.487339i \(0.837967\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 30.3691 1.69769
\(321\) 0 0
\(322\) 9.92695 0.553207
\(323\) −40.8695 −2.27404
\(324\) 0 0
\(325\) 38.5852 2.14032
\(326\) 9.97819 0.552641
\(327\) 0 0
\(328\) 20.7201 1.14407
\(329\) −0.0417127 −0.00229970
\(330\) 0 0
\(331\) −27.0887 −1.48893 −0.744465 0.667661i \(-0.767295\pi\)
−0.744465 + 0.667661i \(0.767295\pi\)
\(332\) 2.66498 0.146260
\(333\) 0 0
\(334\) 5.36392 0.293500
\(335\) 34.4502 1.88222
\(336\) 0 0
\(337\) −6.24394 −0.340129 −0.170065 0.985433i \(-0.554398\pi\)
−0.170065 + 0.985433i \(0.554398\pi\)
\(338\) −23.1994 −1.26188
\(339\) 0 0
\(340\) −4.74744 −0.257466
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −16.8237 −0.907075
\(345\) 0 0
\(346\) −4.21801 −0.226762
\(347\) 1.55233 0.0833332 0.0416666 0.999132i \(-0.486733\pi\)
0.0416666 + 0.999132i \(0.486733\pi\)
\(348\) 0 0
\(349\) 5.70726 0.305503 0.152751 0.988265i \(-0.451187\pi\)
0.152751 + 0.988265i \(0.451187\pi\)
\(350\) 9.14714 0.488935
\(351\) 0 0
\(352\) 0 0
\(353\) 17.0387 0.906877 0.453438 0.891288i \(-0.350197\pi\)
0.453438 + 0.891288i \(0.350197\pi\)
\(354\) 0 0
\(355\) 24.1918 1.28397
\(356\) −2.19666 −0.116423
\(357\) 0 0
\(358\) 0.272848 0.0144205
\(359\) 16.5696 0.874512 0.437256 0.899337i \(-0.355950\pi\)
0.437256 + 0.899337i \(0.355950\pi\)
\(360\) 0 0
\(361\) 48.7277 2.56462
\(362\) 6.89944 0.362626
\(363\) 0 0
\(364\) 1.53041 0.0802154
\(365\) −25.5412 −1.33689
\(366\) 0 0
\(367\) 20.3623 1.06290 0.531450 0.847089i \(-0.321647\pi\)
0.531450 + 0.847089i \(0.321647\pi\)
\(368\) −25.4884 −1.32867
\(369\) 0 0
\(370\) 12.8822 0.669711
\(371\) −4.47145 −0.232146
\(372\) 0 0
\(373\) −13.5841 −0.703356 −0.351678 0.936121i \(-0.614389\pi\)
−0.351678 + 0.936121i \(0.614389\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.124661 0.00642892
\(377\) −8.14163 −0.419315
\(378\) 0 0
\(379\) 12.7562 0.655244 0.327622 0.944809i \(-0.393753\pi\)
0.327622 + 0.944809i \(0.393753\pi\)
\(380\) 7.86733 0.403585
\(381\) 0 0
\(382\) 3.55496 0.181887
\(383\) −11.7476 −0.600273 −0.300136 0.953896i \(-0.597032\pi\)
−0.300136 + 0.953896i \(0.597032\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −21.6493 −1.10192
\(387\) 0 0
\(388\) 0.649907 0.0329940
\(389\) 31.6905 1.60677 0.803385 0.595460i \(-0.203030\pi\)
0.803385 + 0.595460i \(0.203030\pi\)
\(390\) 0 0
\(391\) 37.5497 1.89897
\(392\) 2.98857 0.150945
\(393\) 0 0
\(394\) 4.22219 0.212711
\(395\) −56.3030 −2.83291
\(396\) 0 0
\(397\) 24.2723 1.21819 0.609097 0.793096i \(-0.291532\pi\)
0.609097 + 0.793096i \(0.291532\pi\)
\(398\) −11.3374 −0.568290
\(399\) 0 0
\(400\) −23.4862 −1.17431
\(401\) −7.94109 −0.396559 −0.198280 0.980146i \(-0.563535\pi\)
−0.198280 + 0.980146i \(0.563535\pi\)
\(402\) 0 0
\(403\) 13.3546 0.665242
\(404\) 3.82739 0.190420
\(405\) 0 0
\(406\) −1.93008 −0.0957883
\(407\) 0 0
\(408\) 0 0
\(409\) 1.58343 0.0782956 0.0391478 0.999233i \(-0.487536\pi\)
0.0391478 + 0.999233i \(0.487536\pi\)
\(410\) −31.4884 −1.55510
\(411\) 0 0
\(412\) −1.04936 −0.0516982
\(413\) −2.99486 −0.147367
\(414\) 0 0
\(415\) −33.3613 −1.63764
\(416\) −8.59224 −0.421269
\(417\) 0 0
\(418\) 0 0
\(419\) −18.0433 −0.881472 −0.440736 0.897637i \(-0.645282\pi\)
−0.440736 + 0.897637i \(0.645282\pi\)
\(420\) 0 0
\(421\) −6.96477 −0.339442 −0.169721 0.985492i \(-0.554287\pi\)
−0.169721 + 0.985492i \(0.554287\pi\)
\(422\) 6.45218 0.314087
\(423\) 0 0
\(424\) 13.3632 0.648976
\(425\) 34.6000 1.67834
\(426\) 0 0
\(427\) −11.6684 −0.564675
\(428\) 0.370755 0.0179211
\(429\) 0 0
\(430\) 25.5671 1.23295
\(431\) 24.0781 1.15980 0.579901 0.814687i \(-0.303091\pi\)
0.579901 + 0.814687i \(0.303091\pi\)
\(432\) 0 0
\(433\) −24.2687 −1.16628 −0.583141 0.812371i \(-0.698176\pi\)
−0.583141 + 0.812371i \(0.698176\pi\)
\(434\) 3.16589 0.151968
\(435\) 0 0
\(436\) −4.85975 −0.232740
\(437\) −62.2262 −2.97668
\(438\) 0 0
\(439\) 8.71332 0.415864 0.207932 0.978143i \(-0.433327\pi\)
0.207932 + 0.978143i \(0.433327\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −36.1079 −1.71748
\(443\) −25.3205 −1.20301 −0.601507 0.798868i \(-0.705433\pi\)
−0.601507 + 0.798868i \(0.705433\pi\)
\(444\) 0 0
\(445\) 27.4987 1.30357
\(446\) 1.51747 0.0718541
\(447\) 0 0
\(448\) −8.77881 −0.414760
\(449\) 21.5816 1.01850 0.509250 0.860618i \(-0.329923\pi\)
0.509250 + 0.860618i \(0.329923\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −3.52727 −0.165909
\(453\) 0 0
\(454\) −3.44514 −0.161688
\(455\) −19.1583 −0.898157
\(456\) 0 0
\(457\) −29.5898 −1.38415 −0.692076 0.721824i \(-0.743304\pi\)
−0.692076 + 0.721824i \(0.743304\pi\)
\(458\) 14.2163 0.664283
\(459\) 0 0
\(460\) −7.22827 −0.337020
\(461\) −18.2856 −0.851643 −0.425821 0.904807i \(-0.640015\pi\)
−0.425821 + 0.904807i \(0.640015\pi\)
\(462\) 0 0
\(463\) 21.5147 0.999875 0.499937 0.866062i \(-0.333356\pi\)
0.499937 + 0.866062i \(0.333356\pi\)
\(464\) 4.95567 0.230061
\(465\) 0 0
\(466\) 22.0441 1.02117
\(467\) −33.0545 −1.52958 −0.764790 0.644280i \(-0.777157\pi\)
−0.764790 + 0.644280i \(0.777157\pi\)
\(468\) 0 0
\(469\) −9.95852 −0.459842
\(470\) −0.189448 −0.00873860
\(471\) 0 0
\(472\) 8.95035 0.411973
\(473\) 0 0
\(474\) 0 0
\(475\) −57.3381 −2.63085
\(476\) 1.37234 0.0629013
\(477\) 0 0
\(478\) 16.8790 0.772026
\(479\) −8.22730 −0.375915 −0.187957 0.982177i \(-0.560187\pi\)
−0.187957 + 0.982177i \(0.560187\pi\)
\(480\) 0 0
\(481\) −15.7082 −0.716234
\(482\) 1.47469 0.0671704
\(483\) 0 0
\(484\) 0 0
\(485\) −8.13580 −0.369428
\(486\) 0 0
\(487\) 16.7692 0.759884 0.379942 0.925010i \(-0.375944\pi\)
0.379942 + 0.925010i \(0.375944\pi\)
\(488\) 34.8719 1.57858
\(489\) 0 0
\(490\) −4.54174 −0.205175
\(491\) 21.7641 0.982199 0.491100 0.871103i \(-0.336595\pi\)
0.491100 + 0.871103i \(0.336595\pi\)
\(492\) 0 0
\(493\) −7.30072 −0.328808
\(494\) 59.8370 2.69219
\(495\) 0 0
\(496\) −8.12874 −0.364991
\(497\) −6.99312 −0.313684
\(498\) 0 0
\(499\) 27.4664 1.22956 0.614781 0.788698i \(-0.289244\pi\)
0.614781 + 0.788698i \(0.289244\pi\)
\(500\) −1.88060 −0.0841032
\(501\) 0 0
\(502\) −32.6484 −1.45717
\(503\) 15.4162 0.687375 0.343688 0.939084i \(-0.388324\pi\)
0.343688 + 0.939084i \(0.388324\pi\)
\(504\) 0 0
\(505\) −47.9129 −2.13209
\(506\) 0 0
\(507\) 0 0
\(508\) 1.10196 0.0488916
\(509\) 24.5612 1.08866 0.544329 0.838872i \(-0.316784\pi\)
0.544329 + 0.838872i \(0.316784\pi\)
\(510\) 0 0
\(511\) 7.38320 0.326613
\(512\) 25.3786 1.12159
\(513\) 0 0
\(514\) 17.0629 0.752612
\(515\) 13.1363 0.578856
\(516\) 0 0
\(517\) 0 0
\(518\) −3.72385 −0.163616
\(519\) 0 0
\(520\) 57.2560 2.51084
\(521\) −12.1838 −0.533782 −0.266891 0.963727i \(-0.585996\pi\)
−0.266891 + 0.963727i \(0.585996\pi\)
\(522\) 0 0
\(523\) −29.8506 −1.30527 −0.652637 0.757671i \(-0.726337\pi\)
−0.652637 + 0.757671i \(0.726337\pi\)
\(524\) 0.560104 0.0244682
\(525\) 0 0
\(526\) −36.8454 −1.60653
\(527\) 11.9753 0.521653
\(528\) 0 0
\(529\) 34.1716 1.48572
\(530\) −20.3082 −0.882131
\(531\) 0 0
\(532\) −2.27421 −0.0985995
\(533\) 38.3963 1.66313
\(534\) 0 0
\(535\) −4.64127 −0.200660
\(536\) 29.7617 1.28551
\(537\) 0 0
\(538\) 40.0277 1.72572
\(539\) 0 0
\(540\) 0 0
\(541\) 33.5792 1.44368 0.721842 0.692058i \(-0.243296\pi\)
0.721842 + 0.692058i \(0.243296\pi\)
\(542\) −0.157223 −0.00675331
\(543\) 0 0
\(544\) −7.70480 −0.330340
\(545\) 60.8364 2.60594
\(546\) 0 0
\(547\) −25.6065 −1.09486 −0.547428 0.836853i \(-0.684393\pi\)
−0.547428 + 0.836853i \(0.684393\pi\)
\(548\) 4.22593 0.180523
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0985 0.515415
\(552\) 0 0
\(553\) 16.2755 0.692106
\(554\) −7.39787 −0.314305
\(555\) 0 0
\(556\) −5.94767 −0.252237
\(557\) −5.85837 −0.248227 −0.124114 0.992268i \(-0.539609\pi\)
−0.124114 + 0.992268i \(0.539609\pi\)
\(558\) 0 0
\(559\) −31.1760 −1.31860
\(560\) 11.6614 0.492782
\(561\) 0 0
\(562\) 5.46953 0.230718
\(563\) 6.17238 0.260135 0.130067 0.991505i \(-0.458481\pi\)
0.130067 + 0.991505i \(0.458481\pi\)
\(564\) 0 0
\(565\) 44.1558 1.85765
\(566\) 17.9532 0.754628
\(567\) 0 0
\(568\) 20.8994 0.876920
\(569\) 27.0150 1.13253 0.566264 0.824224i \(-0.308388\pi\)
0.566264 + 0.824224i \(0.308388\pi\)
\(570\) 0 0
\(571\) −30.7018 −1.28483 −0.642416 0.766356i \(-0.722068\pi\)
−0.642416 + 0.766356i \(0.722068\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.10234 0.379925
\(575\) 52.6805 2.19693
\(576\) 0 0
\(577\) −9.06613 −0.377428 −0.188714 0.982032i \(-0.560432\pi\)
−0.188714 + 0.982032i \(0.560432\pi\)
\(578\) −10.0596 −0.418423
\(579\) 0 0
\(580\) 1.40538 0.0583552
\(581\) 9.64377 0.400091
\(582\) 0 0
\(583\) 0 0
\(584\) −22.0652 −0.913064
\(585\) 0 0
\(586\) −4.05687 −0.167588
\(587\) −6.77919 −0.279807 −0.139904 0.990165i \(-0.544679\pi\)
−0.139904 + 0.990165i \(0.544679\pi\)
\(588\) 0 0
\(589\) −19.8451 −0.817705
\(590\) −13.6019 −0.559981
\(591\) 0 0
\(592\) 9.56134 0.392969
\(593\) 32.2361 1.32378 0.661890 0.749601i \(-0.269755\pi\)
0.661890 + 0.749601i \(0.269755\pi\)
\(594\) 0 0
\(595\) −17.1796 −0.704294
\(596\) −0.334769 −0.0137127
\(597\) 0 0
\(598\) −54.9765 −2.24816
\(599\) 26.6954 1.09074 0.545372 0.838194i \(-0.316388\pi\)
0.545372 + 0.838194i \(0.316388\pi\)
\(600\) 0 0
\(601\) −4.26250 −0.173871 −0.0869354 0.996214i \(-0.527707\pi\)
−0.0869354 + 0.996214i \(0.527707\pi\)
\(602\) −7.39068 −0.301222
\(603\) 0 0
\(604\) 2.68532 0.109264
\(605\) 0 0
\(606\) 0 0
\(607\) −6.97851 −0.283249 −0.141625 0.989920i \(-0.545233\pi\)
−0.141625 + 0.989920i \(0.545233\pi\)
\(608\) 12.7682 0.517817
\(609\) 0 0
\(610\) −52.9950 −2.14570
\(611\) 0.231009 0.00934564
\(612\) 0 0
\(613\) 25.3866 1.02535 0.512677 0.858582i \(-0.328654\pi\)
0.512677 + 0.858582i \(0.328654\pi\)
\(614\) 6.99787 0.282411
\(615\) 0 0
\(616\) 0 0
\(617\) −8.20576 −0.330351 −0.165176 0.986264i \(-0.552819\pi\)
−0.165176 + 0.986264i \(0.552819\pi\)
\(618\) 0 0
\(619\) −4.24253 −0.170522 −0.0852609 0.996359i \(-0.527172\pi\)
−0.0852609 + 0.996359i \(0.527172\pi\)
\(620\) −2.30523 −0.0925804
\(621\) 0 0
\(622\) 9.91565 0.397581
\(623\) −7.94906 −0.318473
\(624\) 0 0
\(625\) −11.2939 −0.451757
\(626\) −6.36406 −0.254359
\(627\) 0 0
\(628\) −4.57028 −0.182374
\(629\) −14.0858 −0.561639
\(630\) 0 0
\(631\) 35.8595 1.42755 0.713773 0.700377i \(-0.246985\pi\)
0.713773 + 0.700377i \(0.246985\pi\)
\(632\) −48.6405 −1.93482
\(633\) 0 0
\(634\) 40.8230 1.62129
\(635\) −13.7948 −0.547430
\(636\) 0 0
\(637\) 5.53810 0.219428
\(638\) 0 0
\(639\) 0 0
\(640\) −29.1368 −1.15173
\(641\) 30.8864 1.21994 0.609970 0.792424i \(-0.291181\pi\)
0.609970 + 0.792424i \(0.291181\pi\)
\(642\) 0 0
\(643\) 11.1381 0.439243 0.219621 0.975585i \(-0.429518\pi\)
0.219621 + 0.975585i \(0.429518\pi\)
\(644\) 2.08948 0.0823369
\(645\) 0 0
\(646\) 53.6568 2.11110
\(647\) 19.1705 0.753669 0.376835 0.926281i \(-0.377012\pi\)
0.376835 + 0.926281i \(0.377012\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −50.6578 −1.98696
\(651\) 0 0
\(652\) 2.10026 0.0822526
\(653\) 6.33344 0.247847 0.123923 0.992292i \(-0.460452\pi\)
0.123923 + 0.992292i \(0.460452\pi\)
\(654\) 0 0
\(655\) −7.01161 −0.273966
\(656\) −23.3712 −0.912490
\(657\) 0 0
\(658\) 0.0547639 0.00213492
\(659\) 33.0256 1.28650 0.643248 0.765658i \(-0.277586\pi\)
0.643248 + 0.765658i \(0.277586\pi\)
\(660\) 0 0
\(661\) −20.1870 −0.785181 −0.392591 0.919713i \(-0.628421\pi\)
−0.392591 + 0.919713i \(0.628421\pi\)
\(662\) 35.5643 1.38224
\(663\) 0 0
\(664\) −28.8211 −1.11847
\(665\) 28.4695 1.10400
\(666\) 0 0
\(667\) −11.1158 −0.430405
\(668\) 1.12903 0.0436833
\(669\) 0 0
\(670\) −45.2290 −1.74735
\(671\) 0 0
\(672\) 0 0
\(673\) 23.2806 0.897403 0.448701 0.893682i \(-0.351887\pi\)
0.448701 + 0.893682i \(0.351887\pi\)
\(674\) 8.19755 0.315758
\(675\) 0 0
\(676\) −4.88313 −0.187813
\(677\) 11.7124 0.450143 0.225072 0.974342i \(-0.427738\pi\)
0.225072 + 0.974342i \(0.427738\pi\)
\(678\) 0 0
\(679\) 2.35182 0.0902545
\(680\) 51.3423 1.96889
\(681\) 0 0
\(682\) 0 0
\(683\) 20.9286 0.800811 0.400405 0.916338i \(-0.368869\pi\)
0.400405 + 0.916338i \(0.368869\pi\)
\(684\) 0 0
\(685\) −52.9020 −2.02128
\(686\) 1.31288 0.0501260
\(687\) 0 0
\(688\) 18.9763 0.723465
\(689\) 24.7634 0.943409
\(690\) 0 0
\(691\) 5.06534 0.192695 0.0963473 0.995348i \(-0.469284\pi\)
0.0963473 + 0.995348i \(0.469284\pi\)
\(692\) −0.887829 −0.0337502
\(693\) 0 0
\(694\) −2.03802 −0.0773622
\(695\) 74.4554 2.82425
\(696\) 0 0
\(697\) 34.4305 1.30415
\(698\) −7.49296 −0.283613
\(699\) 0 0
\(700\) 1.92534 0.0727709
\(701\) 32.5293 1.22861 0.614307 0.789067i \(-0.289436\pi\)
0.614307 + 0.789067i \(0.289436\pi\)
\(702\) 0 0
\(703\) 23.3426 0.880384
\(704\) 0 0
\(705\) 0 0
\(706\) −22.3698 −0.841897
\(707\) 13.8502 0.520890
\(708\) 0 0
\(709\) 13.7164 0.515129 0.257564 0.966261i \(-0.417080\pi\)
0.257564 + 0.966261i \(0.417080\pi\)
\(710\) −31.7609 −1.19197
\(711\) 0 0
\(712\) 23.7563 0.890306
\(713\) 18.2331 0.682836
\(714\) 0 0
\(715\) 0 0
\(716\) 0.0574305 0.00214628
\(717\) 0 0
\(718\) −21.7540 −0.811851
\(719\) −43.3822 −1.61788 −0.808941 0.587890i \(-0.799959\pi\)
−0.808941 + 0.587890i \(0.799959\pi\)
\(720\) 0 0
\(721\) −3.79732 −0.141420
\(722\) −63.9737 −2.38086
\(723\) 0 0
\(724\) 1.45223 0.0539717
\(725\) −10.2426 −0.380400
\(726\) 0 0
\(727\) −2.79407 −0.103626 −0.0518132 0.998657i \(-0.516500\pi\)
−0.0518132 + 0.998657i \(0.516500\pi\)
\(728\) −16.5510 −0.613421
\(729\) 0 0
\(730\) 33.5325 1.24110
\(731\) −27.9560 −1.03399
\(732\) 0 0
\(733\) 13.5883 0.501894 0.250947 0.968001i \(-0.419258\pi\)
0.250947 + 0.968001i \(0.419258\pi\)
\(734\) −26.7332 −0.986741
\(735\) 0 0
\(736\) −11.7310 −0.432411
\(737\) 0 0
\(738\) 0 0
\(739\) −23.3065 −0.857342 −0.428671 0.903461i \(-0.641018\pi\)
−0.428671 + 0.903461i \(0.641018\pi\)
\(740\) 2.71150 0.0996769
\(741\) 0 0
\(742\) 5.87049 0.215512
\(743\) 21.3263 0.782387 0.391193 0.920308i \(-0.372062\pi\)
0.391193 + 0.920308i \(0.372062\pi\)
\(744\) 0 0
\(745\) 4.19077 0.153538
\(746\) 17.8343 0.652959
\(747\) 0 0
\(748\) 0 0
\(749\) 1.34165 0.0490229
\(750\) 0 0
\(751\) 8.57812 0.313020 0.156510 0.987676i \(-0.449976\pi\)
0.156510 + 0.987676i \(0.449976\pi\)
\(752\) −0.140612 −0.00512758
\(753\) 0 0
\(754\) 10.6890 0.389270
\(755\) −33.6160 −1.22341
\(756\) 0 0
\(757\) −50.2751 −1.82728 −0.913639 0.406526i \(-0.866740\pi\)
−0.913639 + 0.406526i \(0.866740\pi\)
\(758\) −16.7474 −0.608294
\(759\) 0 0
\(760\) −85.0830 −3.08629
\(761\) 10.5311 0.381753 0.190876 0.981614i \(-0.438867\pi\)
0.190876 + 0.981614i \(0.438867\pi\)
\(762\) 0 0
\(763\) −17.5860 −0.636655
\(764\) 0.748266 0.0270713
\(765\) 0 0
\(766\) 15.4232 0.557262
\(767\) 16.5859 0.598881
\(768\) 0 0
\(769\) 49.1715 1.77317 0.886584 0.462567i \(-0.153071\pi\)
0.886584 + 0.462567i \(0.153071\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.55685 −0.164005
\(773\) 15.4108 0.554289 0.277145 0.960828i \(-0.410612\pi\)
0.277145 + 0.960828i \(0.410612\pi\)
\(774\) 0 0
\(775\) 16.8008 0.603503
\(776\) −7.02857 −0.252311
\(777\) 0 0
\(778\) −41.6058 −1.49164
\(779\) −57.0573 −2.04429
\(780\) 0 0
\(781\) 0 0
\(782\) −49.2983 −1.76290
\(783\) 0 0
\(784\) −3.37095 −0.120391
\(785\) 57.2127 2.04201
\(786\) 0 0
\(787\) −11.6189 −0.414169 −0.207084 0.978323i \(-0.566397\pi\)
−0.207084 + 0.978323i \(0.566397\pi\)
\(788\) 0.888709 0.0316589
\(789\) 0 0
\(790\) 73.9192 2.62993
\(791\) −12.7641 −0.453840
\(792\) 0 0
\(793\) 64.6210 2.29476
\(794\) −31.8667 −1.13091
\(795\) 0 0
\(796\) −2.38635 −0.0845818
\(797\) 8.66866 0.307060 0.153530 0.988144i \(-0.450936\pi\)
0.153530 + 0.988144i \(0.450936\pi\)
\(798\) 0 0
\(799\) 0.207150 0.00732843
\(800\) −10.8095 −0.382173
\(801\) 0 0
\(802\) 10.4257 0.368145
\(803\) 0 0
\(804\) 0 0
\(805\) −26.1569 −0.921911
\(806\) −17.5331 −0.617576
\(807\) 0 0
\(808\) −41.3922 −1.45617
\(809\) 16.1800 0.568857 0.284429 0.958697i \(-0.408196\pi\)
0.284429 + 0.958697i \(0.408196\pi\)
\(810\) 0 0
\(811\) −4.58424 −0.160974 −0.0804871 0.996756i \(-0.525648\pi\)
−0.0804871 + 0.996756i \(0.525648\pi\)
\(812\) −0.406254 −0.0142567
\(813\) 0 0
\(814\) 0 0
\(815\) −26.2920 −0.920967
\(816\) 0 0
\(817\) 46.3279 1.62081
\(818\) −2.07886 −0.0726855
\(819\) 0 0
\(820\) −6.62783 −0.231454
\(821\) 46.8787 1.63608 0.818040 0.575162i \(-0.195061\pi\)
0.818040 + 0.575162i \(0.195061\pi\)
\(822\) 0 0
\(823\) −0.814184 −0.0283807 −0.0141903 0.999899i \(-0.504517\pi\)
−0.0141903 + 0.999899i \(0.504517\pi\)
\(824\) 11.3485 0.395345
\(825\) 0 0
\(826\) 3.93190 0.136808
\(827\) 26.4971 0.921395 0.460698 0.887557i \(-0.347599\pi\)
0.460698 + 0.887557i \(0.347599\pi\)
\(828\) 0 0
\(829\) 32.9292 1.14368 0.571839 0.820366i \(-0.306230\pi\)
0.571839 + 0.820366i \(0.306230\pi\)
\(830\) 43.7995 1.52030
\(831\) 0 0
\(832\) 48.6179 1.68552
\(833\) 4.96611 0.172065
\(834\) 0 0
\(835\) −14.1336 −0.489114
\(836\) 0 0
\(837\) 0 0
\(838\) 23.6887 0.818312
\(839\) 31.4398 1.08542 0.542711 0.839919i \(-0.317398\pi\)
0.542711 + 0.839919i \(0.317398\pi\)
\(840\) 0 0
\(841\) −26.8388 −0.925475
\(842\) 9.14391 0.315120
\(843\) 0 0
\(844\) 1.35809 0.0467473
\(845\) 61.1291 2.10290
\(846\) 0 0
\(847\) 0 0
\(848\) −15.0730 −0.517610
\(849\) 0 0
\(850\) −45.4256 −1.55809
\(851\) −21.4465 −0.735177
\(852\) 0 0
\(853\) 53.0965 1.81799 0.908995 0.416807i \(-0.136851\pi\)
0.908995 + 0.416807i \(0.136851\pi\)
\(854\) 15.3193 0.524215
\(855\) 0 0
\(856\) −4.00962 −0.137046
\(857\) −38.6964 −1.32185 −0.660923 0.750454i \(-0.729835\pi\)
−0.660923 + 0.750454i \(0.729835\pi\)
\(858\) 0 0
\(859\) −4.54428 −0.155049 −0.0775245 0.996990i \(-0.524702\pi\)
−0.0775245 + 0.996990i \(0.524702\pi\)
\(860\) 5.38150 0.183508
\(861\) 0 0
\(862\) −31.6117 −1.07670
\(863\) 14.8810 0.506557 0.253278 0.967393i \(-0.418491\pi\)
0.253278 + 0.967393i \(0.418491\pi\)
\(864\) 0 0
\(865\) 11.1142 0.377895
\(866\) 31.8620 1.08271
\(867\) 0 0
\(868\) 0.666374 0.0226182
\(869\) 0 0
\(870\) 0 0
\(871\) 55.1513 1.86873
\(872\) 52.5569 1.77980
\(873\) 0 0
\(874\) 81.6957 2.76340
\(875\) −6.80535 −0.230063
\(876\) 0 0
\(877\) −41.1431 −1.38930 −0.694652 0.719346i \(-0.744441\pi\)
−0.694652 + 0.719346i \(0.744441\pi\)
\(878\) −11.4396 −0.386067
\(879\) 0 0
\(880\) 0 0
\(881\) −9.03118 −0.304268 −0.152134 0.988360i \(-0.548615\pi\)
−0.152134 + 0.988360i \(0.548615\pi\)
\(882\) 0 0
\(883\) 30.5916 1.02949 0.514745 0.857343i \(-0.327886\pi\)
0.514745 + 0.857343i \(0.327886\pi\)
\(884\) −7.60019 −0.255622
\(885\) 0 0
\(886\) 33.2428 1.11681
\(887\) −29.7478 −0.998833 −0.499416 0.866362i \(-0.666452\pi\)
−0.499416 + 0.866362i \(0.666452\pi\)
\(888\) 0 0
\(889\) 3.98767 0.133742
\(890\) −36.1026 −1.21016
\(891\) 0 0
\(892\) 0.319404 0.0106945
\(893\) −0.343283 −0.0114875
\(894\) 0 0
\(895\) −0.718939 −0.0240315
\(896\) 8.42258 0.281379
\(897\) 0 0
\(898\) −28.3341 −0.945523
\(899\) −3.54504 −0.118234
\(900\) 0 0
\(901\) 22.2057 0.739779
\(902\) 0 0
\(903\) 0 0
\(904\) 38.1465 1.26873
\(905\) −18.1796 −0.604311
\(906\) 0 0
\(907\) 39.1485 1.29991 0.649953 0.759975i \(-0.274789\pi\)
0.649953 + 0.759975i \(0.274789\pi\)
\(908\) −0.725150 −0.0240650
\(909\) 0 0
\(910\) 25.1526 0.833801
\(911\) −27.9931 −0.927454 −0.463727 0.885978i \(-0.653488\pi\)
−0.463727 + 0.885978i \(0.653488\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 38.8479 1.28497
\(915\) 0 0
\(916\) 2.99232 0.0988690
\(917\) 2.02685 0.0669324
\(918\) 0 0
\(919\) 38.4845 1.26949 0.634743 0.772723i \(-0.281106\pi\)
0.634743 + 0.772723i \(0.281106\pi\)
\(920\) 78.1718 2.57725
\(921\) 0 0
\(922\) 24.0068 0.790621
\(923\) 38.7287 1.27477
\(924\) 0 0
\(925\) −19.7618 −0.649763
\(926\) −28.2463 −0.928231
\(927\) 0 0
\(928\) 2.28084 0.0748723
\(929\) −47.2366 −1.54978 −0.774892 0.632094i \(-0.782196\pi\)
−0.774892 + 0.632094i \(0.782196\pi\)
\(930\) 0 0
\(931\) −8.22969 −0.269717
\(932\) 4.63996 0.151987
\(933\) 0 0
\(934\) 43.3966 1.41998
\(935\) 0 0
\(936\) 0 0
\(937\) 43.5825 1.42378 0.711888 0.702293i \(-0.247840\pi\)
0.711888 + 0.702293i \(0.247840\pi\)
\(938\) 13.0744 0.426893
\(939\) 0 0
\(940\) −0.0398761 −0.00130061
\(941\) 33.8165 1.10239 0.551193 0.834378i \(-0.314173\pi\)
0.551193 + 0.834378i \(0.314173\pi\)
\(942\) 0 0
\(943\) 52.4225 1.70711
\(944\) −10.0955 −0.328582
\(945\) 0 0
\(946\) 0 0
\(947\) 3.79141 0.123204 0.0616021 0.998101i \(-0.480379\pi\)
0.0616021 + 0.998101i \(0.480379\pi\)
\(948\) 0 0
\(949\) −40.8889 −1.32731
\(950\) 75.2781 2.44234
\(951\) 0 0
\(952\) −14.8415 −0.481017
\(953\) 3.16992 0.102684 0.0513419 0.998681i \(-0.483650\pi\)
0.0513419 + 0.998681i \(0.483650\pi\)
\(954\) 0 0
\(955\) −9.36711 −0.303113
\(956\) 3.55277 0.114905
\(957\) 0 0
\(958\) 10.8015 0.348980
\(959\) 15.2924 0.493817
\(960\) 0 0
\(961\) −25.1851 −0.812423
\(962\) 20.6231 0.664914
\(963\) 0 0
\(964\) 0.310401 0.00999734
\(965\) 57.0446 1.83633
\(966\) 0 0
\(967\) −24.6865 −0.793865 −0.396932 0.917848i \(-0.629925\pi\)
−0.396932 + 0.917848i \(0.629925\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 10.6813 0.342957
\(971\) −20.2070 −0.648473 −0.324237 0.945976i \(-0.605107\pi\)
−0.324237 + 0.945976i \(0.605107\pi\)
\(972\) 0 0
\(973\) −21.5228 −0.689991
\(974\) −22.0159 −0.705436
\(975\) 0 0
\(976\) −39.3337 −1.25904
\(977\) 34.3098 1.09767 0.548834 0.835931i \(-0.315072\pi\)
0.548834 + 0.835931i \(0.315072\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.955969 −0.0305373
\(981\) 0 0
\(982\) −28.5737 −0.911822
\(983\) −33.0534 −1.05424 −0.527120 0.849791i \(-0.676728\pi\)
−0.527120 + 0.849791i \(0.676728\pi\)
\(984\) 0 0
\(985\) −11.1252 −0.354479
\(986\) 9.58498 0.305248
\(987\) 0 0
\(988\) 12.5948 0.400694
\(989\) −42.5647 −1.35348
\(990\) 0 0
\(991\) 39.3191 1.24901 0.624506 0.781020i \(-0.285300\pi\)
0.624506 + 0.781020i \(0.285300\pi\)
\(992\) −3.74124 −0.118785
\(993\) 0 0
\(994\) 9.18114 0.291208
\(995\) 29.8733 0.947046
\(996\) 0 0
\(997\) 2.41508 0.0764865 0.0382432 0.999268i \(-0.487824\pi\)
0.0382432 + 0.999268i \(0.487824\pi\)
\(998\) −36.0601 −1.14146
\(999\) 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.1.5 16
3.2 odd 2 inner 7623.2.a.db.1.12 16
11.7 odd 10 693.2.m.k.379.6 yes 32
11.8 odd 10 693.2.m.k.64.6 yes 32
11.10 odd 2 7623.2.a.dc.1.12 16
33.8 even 10 693.2.m.k.64.3 32
33.29 even 10 693.2.m.k.379.3 yes 32
33.32 even 2 7623.2.a.dc.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.64.3 32 33.8 even 10
693.2.m.k.64.6 yes 32 11.8 odd 10
693.2.m.k.379.3 yes 32 33.29 even 10
693.2.m.k.379.6 yes 32 11.7 odd 10
7623.2.a.db.1.5 16 1.1 even 1 trivial
7623.2.a.db.1.12 16 3.2 odd 2 inner
7623.2.a.dc.1.5 16 33.32 even 2
7623.2.a.dc.1.12 16 11.10 odd 2