Properties

Label 7623.2.a.dc.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 \(-0.781515\pi\)
−0.773538 + 0.633750i \(0.781515\pi\)
\(6\) 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 \(-0.778746\pi\)
−0.767997 + 0.640454i \(0.778746\pi\)
\(14\) −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 \(-0.107021\pi\)
0.944010 + 0.329918i \(0.107021\pi\)
\(20\) 0.955969 0.213761
\(21\) 0 0
\(22\) 0 0
\(23\) −7.56119 −1.57662 −0.788309 0.615280i \(-0.789043\pi\)
−0.788309 + 0.615280i \(0.789043\pi\)
\(24\) 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 \(-0.358783\pi\)
0.429235 + 0.903193i \(0.358783\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 9.92695 1.46365
\(47\) −0.0417127 −0.00608443 −0.00304221 0.999995i \(-0.500968\pi\)
−0.00304221 + 0.999995i \(0.500968\pi\)
\(48\) 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 \(-0.599359\pi\)
−0.307101 + 0.951677i \(0.599359\pi\)
\(54\) 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 \(-0.562454\pi\)
−0.194949 + 0.980813i \(0.562454\pi\)
\(60\) 0 0
\(61\) −11.6684 −1.49399 −0.746995 0.664830i \(-0.768504\pi\)
−0.746995 + 0.664830i \(0.768504\pi\)
\(62\) −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 \(-0.636206\pi\)
−0.414966 + 0.909837i \(0.636206\pi\)
\(72\) 0 0
\(73\) 7.38320 0.864138 0.432069 0.901841i \(-0.357784\pi\)
0.432069 + 0.901841i \(0.357784\pi\)
\(74\) 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 \(-0.131740\pi\)
0.915570 + 0.402159i \(0.131740\pi\)
\(80\) 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 \(-0.638426\pi\)
−0.421300 + 0.906922i \(0.638426\pi\)
\(90\) 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 \(-0.818749\pi\)
−0.842216 + 0.539140i \(0.818749\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.37095 −0.318525
\(113\) −12.7641 −1.20075 −0.600374 0.799719i \(-0.704982\pi\)
−0.600374 + 0.799719i \(0.704982\pi\)
\(114\) 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 \(-0.443385\pi\)
0.176924 + 0.984224i \(0.443385\pi\)
\(128\) −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 \(-0.273401\pi\)
0.653259 + 0.757135i \(0.273401\pi\)
\(138\) 0 0
\(139\) −21.5228 −1.82554 −0.912772 0.408470i \(-0.866062\pi\)
−0.912772 + 0.408470i \(0.866062\pi\)
\(140\) 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 \(-0.370608\pi\)
0.395395 + 0.918511i \(0.370608\pi\)
\(152\) 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 \(-0.497528\pi\)
0.00776674 + 0.999970i \(0.497528\pi\)
\(180\) 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 \(-0.468767\pi\)
0.0979630 + 0.995190i \(0.468767\pi\)
\(192\) 0 0
\(193\) −16.4899 −1.18697 −0.593484 0.804846i \(-0.702248\pi\)
−0.593484 + 0.804846i \(0.702248\pi\)
\(194\) 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 \(-0.445893\pi\)
0.169165 + 0.985588i \(0.445893\pi\)
\(212\) 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 \(-0.488482\pi\)
0.0361774 + 0.999345i \(0.488482\pi\)
\(242\) 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 \(-0.787244\pi\)
−0.784819 + 0.619725i \(0.787244\pi\)
\(252\) 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 \(-0.367150\pi\)
0.405350 + 0.914161i \(0.367150\pi\)
\(258\) 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 \(-0.120277\pi\)
0.929456 + 0.368933i \(0.120277\pi\)
\(270\) 0 0
\(271\) −0.119754 −0.00727455 −0.00363727 0.999993i \(-0.501158\pi\)
−0.00363727 + 0.999993i \(0.501158\pi\)
\(272\) −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 \(-0.554145\pi\)
−0.169282 + 0.985568i \(0.554145\pi\)
\(278\) 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 \(-0.366771\pi\)
0.406436 + 0.913679i \(0.366771\pi\)
\(284\) 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 \(-0.451395\pi\)
0.152104 + 0.988364i \(0.451395\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 10.9520 0.622031
\(311\) 7.55259 0.428268 0.214134 0.976804i \(-0.431307\pi\)
0.214134 + 0.976804i \(0.431307\pi\)
\(312\) 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 \(-0.162033\pi\)
0.873213 + 0.487339i \(0.162033\pi\)
\(318\) 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 \(-0.445602\pi\)
0.170065 + 0.985433i \(0.445602\pi\)
\(338\) −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 \(-0.548813\pi\)
−0.152751 + 0.988265i \(0.548813\pi\)
\(350\) −9.14714 −0.488935
\(351\) 0 0
\(352\) 0 0
\(353\) −17.0387 −0.906877 −0.453438 0.891288i \(-0.649803\pi\)
−0.453438 + 0.891288i \(0.649803\pi\)
\(354\) 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 \(-0.385611\pi\)
0.351678 + 0.936121i \(0.385611\pi\)
\(374\) 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 \(-0.402968\pi\)
0.300136 + 0.953896i \(0.402968\pi\)
\(384\) 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 \(-0.796970\pi\)
−0.803385 + 0.595460i \(0.796970\pi\)
\(390\) 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 \(-0.436465\pi\)
0.198280 + 0.980146i \(0.436465\pi\)
\(402\) 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 \(-0.512464\pi\)
−0.0391478 + 0.999233i \(0.512464\pi\)
\(410\) 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 \(-0.354718\pi\)
0.440736 + 0.897637i \(0.354718\pi\)
\(420\) 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 \(-0.566673\pi\)
−0.207932 + 0.978143i \(0.566673\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 36.1079 1.71748
\(443\) 25.3205 1.20301 0.601507 0.798868i \(-0.294567\pi\)
0.601507 + 0.798868i \(0.294567\pi\)
\(444\) 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 \(-0.670077\pi\)
−0.509250 + 0.860618i \(0.670077\pi\)
\(450\) 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 \(-0.256696\pi\)
0.692076 + 0.721824i \(0.256696\pi\)
\(458\) 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 \(-0.222843\pi\)
0.764790 + 0.644280i \(0.222843\pi\)
\(468\) 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 \(-0.683216\pi\)
−0.544329 + 0.838872i \(0.683216\pi\)
\(510\) 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 \(-0.414004\pi\)
0.266891 + 0.963727i \(0.414004\pi\)
\(522\) 0 0
\(523\) 29.8506 1.30527 0.652637 0.757671i \(-0.273663\pi\)
0.652637 + 0.757671i \(0.273663\pi\)
\(524\) 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 \(-0.756704\pi\)
−0.721842 + 0.692058i \(0.756704\pi\)
\(542\) 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 \(-0.315607\pi\)
0.547428 + 0.836853i \(0.315607\pi\)
\(548\) −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 \(-0.277932\pi\)
0.642416 + 0.766356i \(0.277932\pi\)
\(572\) 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 \(-0.455321\pi\)
0.139904 + 0.990165i \(0.455321\pi\)
\(588\) 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 \(-0.683612\pi\)
−0.545372 + 0.838194i \(0.683612\pi\)
\(600\) 0 0
\(601\) 4.26250 0.173871 0.0869354 0.996214i \(-0.472293\pi\)
0.0869354 + 0.996214i \(0.472293\pi\)
\(602\) −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 \(-0.454767\pi\)
0.141625 + 0.989920i \(0.454767\pi\)
\(608\) −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 \(-0.671346\pi\)
−0.512677 + 0.858582i \(0.671346\pi\)
\(614\) −6.99787 −0.282411
\(615\) 0 0
\(616\) 0 0
\(617\) 8.20576 0.330351 0.165176 0.986264i \(-0.447181\pi\)
0.165176 + 0.986264i \(0.447181\pi\)
\(618\) 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 \(-0.708819\pi\)
−0.609970 + 0.792424i \(0.708819\pi\)
\(642\) 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 \(-0.622988\pi\)
−0.376835 + 0.926281i \(0.622988\pi\)
\(648\) 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 \(-0.539548\pi\)
−0.123923 + 0.992292i \(0.539548\pi\)
\(654\) 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 \(-0.648113\pi\)
−0.448701 + 0.893682i \(0.648113\pi\)
\(674\) −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 \(-0.631131\pi\)
−0.400405 + 0.916338i \(0.631131\pi\)
\(684\) 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 \(-0.200041\pi\)
0.808941 + 0.587890i \(0.200041\pi\)
\(720\) 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 \(-0.580742\pi\)
−0.250947 + 0.968001i \(0.580742\pi\)
\(734\) −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 \(-0.358982\pi\)
0.428671 + 0.903461i \(0.358982\pi\)
\(740\) −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 \(-0.846929\pi\)
−0.886584 + 0.462567i \(0.846929\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.55685 0.164005
\(773\) −15.4108 −0.554289 −0.277145 0.960828i \(-0.589388\pi\)
−0.277145 + 0.960828i \(0.589388\pi\)
\(774\) 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 \(-0.433603\pi\)
0.207084 + 0.978323i \(0.433603\pi\)
\(788\) 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 \(-0.549064\pi\)
−0.153530 + 0.988144i \(0.549064\pi\)
\(798\) 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 \(-0.474352\pi\)
0.0804871 + 0.996756i \(0.474352\pi\)
\(812\) 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 \(-0.682602\pi\)
−0.542711 + 0.839919i \(0.682602\pi\)
\(840\) 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 \(-0.863149\pi\)
−0.908995 + 0.416807i \(0.863149\pi\)
\(854\) 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 \(-0.581509\pi\)
−0.253278 + 0.967393i \(0.581509\pi\)
\(864\) 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 \(-0.255559\pi\)
0.694652 + 0.719346i \(0.255559\pi\)
\(878\) 11.4396 0.386067
\(879\) 0 0
\(880\) 0 0
\(881\) 9.03118 0.304268 0.152134 0.988360i \(-0.451385\pi\)
0.152134 + 0.988360i \(0.451385\pi\)
\(882\) 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 \(-0.346512\pi\)
0.463727 + 0.885978i \(0.346512\pi\)
\(912\) 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 \(-0.718894\pi\)
−0.634743 + 0.772723i \(0.718894\pi\)
\(920\) 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 \(-0.217804\pi\)
0.774892 + 0.632094i \(0.217804\pi\)
\(930\) 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 \(-0.752160\pi\)
−0.711888 + 0.702293i \(0.752160\pi\)
\(938\) −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 \(-0.519621\pi\)
−0.0616021 + 0.998101i \(0.519621\pi\)
\(948\) 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 \(-0.370075\pi\)
0.396932 + 0.917848i \(0.370075\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −10.6813 −0.342957
\(971\) 20.2070 0.648473 0.324237 0.945976i \(-0.394893\pi\)
0.324237 + 0.945976i \(0.394893\pi\)
\(972\) 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 \(-0.684928\pi\)
−0.548834 + 0.835931i \(0.684928\pi\)
\(978\) 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 \(-0.323272\pi\)
0.527120 + 0.849791i \(0.323272\pi\)
\(984\) 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 \(-0.512176\pi\)
−0.0382432 + 0.999268i \(0.512176\pi\)
\(998\) −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.dc.1.5 16
3.2 odd 2 inner 7623.2.a.dc.1.12 16
11.3 even 5 693.2.m.k.64.3 32
11.4 even 5 693.2.m.k.379.3 yes 32
11.10 odd 2 7623.2.a.db.1.12 16
33.14 odd 10 693.2.m.k.64.6 yes 32
33.26 odd 10 693.2.m.k.379.6 yes 32
33.32 even 2 7623.2.a.db.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.64.3 32 11.3 even 5
693.2.m.k.64.6 yes 32 33.14 odd 10
693.2.m.k.379.3 yes 32 11.4 even 5
693.2.m.k.379.6 yes 32 33.26 odd 10
7623.2.a.db.1.5 16 33.32 even 2
7623.2.a.db.1.12 16 11.10 odd 2
7623.2.a.dc.1.5 16 1.1 even 1 trivial
7623.2.a.dc.1.12 16 3.2 odd 2 inner