Properties

Label 2004.2.a.b.1.5
Level 2004
Weight 2
Character 2004.1
Self dual yes
Analytic conductor 16.002
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0020205651\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.31991\) of defining polynomial
Character \(\chi\) \(=\) 2004.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.28459 q^{5} -1.96468 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.28459 q^{5} -1.96468 q^{7} +1.00000 q^{9} -2.40916 q^{11} -3.92621 q^{13} +1.28459 q^{15} -3.63982 q^{17} +2.30861 q^{19} -1.96468 q^{21} -4.55057 q^{23} -3.34982 q^{25} +1.00000 q^{27} -5.81156 q^{29} -4.65708 q^{31} -2.40916 q^{33} -2.52381 q^{35} +4.09300 q^{37} -3.92621 q^{39} +7.34227 q^{41} -3.88636 q^{43} +1.28459 q^{45} +7.51721 q^{47} -3.14003 q^{49} -3.63982 q^{51} -10.6471 q^{53} -3.09479 q^{55} +2.30861 q^{57} +10.8022 q^{59} -14.0025 q^{61} -1.96468 q^{63} -5.04358 q^{65} +0.286799 q^{67} -4.55057 q^{69} -10.1569 q^{71} +3.44823 q^{73} -3.34982 q^{75} +4.73323 q^{77} +5.99698 q^{79} +1.00000 q^{81} +10.0713 q^{83} -4.67569 q^{85} -5.81156 q^{87} -9.05989 q^{89} +7.71375 q^{91} -4.65708 q^{93} +2.96563 q^{95} +4.02676 q^{97} -2.40916 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} - 7q^{5} - 2q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} - 7q^{5} - 2q^{7} + 5q^{9} - 5q^{11} - 8q^{13} - 7q^{15} - 7q^{17} + 2q^{19} - 2q^{21} - 13q^{23} + 2q^{25} + 5q^{27} - 11q^{29} - 12q^{31} - 5q^{33} - 12q^{35} - 7q^{37} - 8q^{39} - 12q^{41} - 7q^{45} - 19q^{47} - 9q^{49} - 7q^{51} - 21q^{53} - q^{55} + 2q^{57} - 7q^{59} - 6q^{61} - 2q^{63} + 14q^{65} + 10q^{67} - 13q^{69} - 35q^{71} - 8q^{73} + 2q^{75} - 6q^{77} + 5q^{81} - 11q^{83} + 5q^{85} - 11q^{87} - 32q^{89} + 5q^{91} - 12q^{93} - 19q^{95} + 11q^{97} - 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.28459 0.574487 0.287244 0.957858i \(-0.407261\pi\)
0.287244 + 0.957858i \(0.407261\pi\)
\(6\) 0 0
\(7\) −1.96468 −0.742579 −0.371290 0.928517i \(-0.621084\pi\)
−0.371290 + 0.928517i \(0.621084\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.40916 −0.726390 −0.363195 0.931713i \(-0.618314\pi\)
−0.363195 + 0.931713i \(0.618314\pi\)
\(12\) 0 0
\(13\) −3.92621 −1.08893 −0.544467 0.838782i \(-0.683268\pi\)
−0.544467 + 0.838782i \(0.683268\pi\)
\(14\) 0 0
\(15\) 1.28459 0.331680
\(16\) 0 0
\(17\) −3.63982 −0.882787 −0.441393 0.897314i \(-0.645516\pi\)
−0.441393 + 0.897314i \(0.645516\pi\)
\(18\) 0 0
\(19\) 2.30861 0.529632 0.264816 0.964299i \(-0.414689\pi\)
0.264816 + 0.964299i \(0.414689\pi\)
\(20\) 0 0
\(21\) −1.96468 −0.428728
\(22\) 0 0
\(23\) −4.55057 −0.948860 −0.474430 0.880293i \(-0.657346\pi\)
−0.474430 + 0.880293i \(0.657346\pi\)
\(24\) 0 0
\(25\) −3.34982 −0.669965
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.81156 −1.07918 −0.539590 0.841928i \(-0.681421\pi\)
−0.539590 + 0.841928i \(0.681421\pi\)
\(30\) 0 0
\(31\) −4.65708 −0.836436 −0.418218 0.908347i \(-0.637345\pi\)
−0.418218 + 0.908347i \(0.637345\pi\)
\(32\) 0 0
\(33\) −2.40916 −0.419381
\(34\) 0 0
\(35\) −2.52381 −0.426602
\(36\) 0 0
\(37\) 4.09300 0.672885 0.336442 0.941704i \(-0.390776\pi\)
0.336442 + 0.941704i \(0.390776\pi\)
\(38\) 0 0
\(39\) −3.92621 −0.628697
\(40\) 0 0
\(41\) 7.34227 1.14667 0.573335 0.819321i \(-0.305649\pi\)
0.573335 + 0.819321i \(0.305649\pi\)
\(42\) 0 0
\(43\) −3.88636 −0.592664 −0.296332 0.955085i \(-0.595763\pi\)
−0.296332 + 0.955085i \(0.595763\pi\)
\(44\) 0 0
\(45\) 1.28459 0.191496
\(46\) 0 0
\(47\) 7.51721 1.09650 0.548249 0.836315i \(-0.315295\pi\)
0.548249 + 0.836315i \(0.315295\pi\)
\(48\) 0 0
\(49\) −3.14003 −0.448576
\(50\) 0 0
\(51\) −3.63982 −0.509677
\(52\) 0 0
\(53\) −10.6471 −1.46250 −0.731248 0.682111i \(-0.761062\pi\)
−0.731248 + 0.682111i \(0.761062\pi\)
\(54\) 0 0
\(55\) −3.09479 −0.417302
\(56\) 0 0
\(57\) 2.30861 0.305783
\(58\) 0 0
\(59\) 10.8022 1.40633 0.703164 0.711028i \(-0.251770\pi\)
0.703164 + 0.711028i \(0.251770\pi\)
\(60\) 0 0
\(61\) −14.0025 −1.79284 −0.896418 0.443209i \(-0.853840\pi\)
−0.896418 + 0.443209i \(0.853840\pi\)
\(62\) 0 0
\(63\) −1.96468 −0.247526
\(64\) 0 0
\(65\) −5.04358 −0.625579
\(66\) 0 0
\(67\) 0.286799 0.0350381 0.0175191 0.999847i \(-0.494423\pi\)
0.0175191 + 0.999847i \(0.494423\pi\)
\(68\) 0 0
\(69\) −4.55057 −0.547825
\(70\) 0 0
\(71\) −10.1569 −1.20540 −0.602699 0.797968i \(-0.705908\pi\)
−0.602699 + 0.797968i \(0.705908\pi\)
\(72\) 0 0
\(73\) 3.44823 0.403585 0.201792 0.979428i \(-0.435323\pi\)
0.201792 + 0.979428i \(0.435323\pi\)
\(74\) 0 0
\(75\) −3.34982 −0.386804
\(76\) 0 0
\(77\) 4.73323 0.539402
\(78\) 0 0
\(79\) 5.99698 0.674713 0.337356 0.941377i \(-0.390467\pi\)
0.337356 + 0.941377i \(0.390467\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.0713 1.10547 0.552737 0.833356i \(-0.313583\pi\)
0.552737 + 0.833356i \(0.313583\pi\)
\(84\) 0 0
\(85\) −4.67569 −0.507150
\(86\) 0 0
\(87\) −5.81156 −0.623064
\(88\) 0 0
\(89\) −9.05989 −0.960346 −0.480173 0.877174i \(-0.659426\pi\)
−0.480173 + 0.877174i \(0.659426\pi\)
\(90\) 0 0
\(91\) 7.71375 0.808620
\(92\) 0 0
\(93\) −4.65708 −0.482916
\(94\) 0 0
\(95\) 2.96563 0.304267
\(96\) 0 0
\(97\) 4.02676 0.408855 0.204428 0.978882i \(-0.434467\pi\)
0.204428 + 0.978882i \(0.434467\pi\)
\(98\) 0 0
\(99\) −2.40916 −0.242130
\(100\) 0 0
\(101\) 1.06980 0.106450 0.0532248 0.998583i \(-0.483050\pi\)
0.0532248 + 0.998583i \(0.483050\pi\)
\(102\) 0 0
\(103\) 15.5422 1.53142 0.765709 0.643188i \(-0.222388\pi\)
0.765709 + 0.643188i \(0.222388\pi\)
\(104\) 0 0
\(105\) −2.52381 −0.246299
\(106\) 0 0
\(107\) −2.54641 −0.246171 −0.123085 0.992396i \(-0.539279\pi\)
−0.123085 + 0.992396i \(0.539279\pi\)
\(108\) 0 0
\(109\) 15.5695 1.49129 0.745646 0.666343i \(-0.232141\pi\)
0.745646 + 0.666343i \(0.232141\pi\)
\(110\) 0 0
\(111\) 4.09300 0.388490
\(112\) 0 0
\(113\) −3.22887 −0.303746 −0.151873 0.988400i \(-0.548531\pi\)
−0.151873 + 0.988400i \(0.548531\pi\)
\(114\) 0 0
\(115\) −5.84563 −0.545108
\(116\) 0 0
\(117\) −3.92621 −0.362978
\(118\) 0 0
\(119\) 7.15109 0.655539
\(120\) 0 0
\(121\) −5.19594 −0.472358
\(122\) 0 0
\(123\) 7.34227 0.662030
\(124\) 0 0
\(125\) −10.7261 −0.959373
\(126\) 0 0
\(127\) 0.940660 0.0834701 0.0417351 0.999129i \(-0.486711\pi\)
0.0417351 + 0.999129i \(0.486711\pi\)
\(128\) 0 0
\(129\) −3.88636 −0.342175
\(130\) 0 0
\(131\) −13.7284 −1.19946 −0.599729 0.800203i \(-0.704725\pi\)
−0.599729 + 0.800203i \(0.704725\pi\)
\(132\) 0 0
\(133\) −4.53569 −0.393294
\(134\) 0 0
\(135\) 1.28459 0.110560
\(136\) 0 0
\(137\) −17.2672 −1.47524 −0.737620 0.675216i \(-0.764050\pi\)
−0.737620 + 0.675216i \(0.764050\pi\)
\(138\) 0 0
\(139\) 10.0634 0.853570 0.426785 0.904353i \(-0.359646\pi\)
0.426785 + 0.904353i \(0.359646\pi\)
\(140\) 0 0
\(141\) 7.51721 0.633063
\(142\) 0 0
\(143\) 9.45888 0.790991
\(144\) 0 0
\(145\) −7.46548 −0.619975
\(146\) 0 0
\(147\) −3.14003 −0.258985
\(148\) 0 0
\(149\) −18.8300 −1.54261 −0.771306 0.636464i \(-0.780396\pi\)
−0.771306 + 0.636464i \(0.780396\pi\)
\(150\) 0 0
\(151\) 10.6354 0.865498 0.432749 0.901514i \(-0.357544\pi\)
0.432749 + 0.901514i \(0.357544\pi\)
\(152\) 0 0
\(153\) −3.63982 −0.294262
\(154\) 0 0
\(155\) −5.98244 −0.480521
\(156\) 0 0
\(157\) 22.2179 1.77318 0.886592 0.462553i \(-0.153066\pi\)
0.886592 + 0.462553i \(0.153066\pi\)
\(158\) 0 0
\(159\) −10.6471 −0.844373
\(160\) 0 0
\(161\) 8.94042 0.704604
\(162\) 0 0
\(163\) −21.9987 −1.72307 −0.861535 0.507699i \(-0.830496\pi\)
−0.861535 + 0.507699i \(0.830496\pi\)
\(164\) 0 0
\(165\) −3.09479 −0.240929
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 2.41512 0.185778
\(170\) 0 0
\(171\) 2.30861 0.176544
\(172\) 0 0
\(173\) 15.8426 1.20449 0.602246 0.798311i \(-0.294273\pi\)
0.602246 + 0.798311i \(0.294273\pi\)
\(174\) 0 0
\(175\) 6.58133 0.497502
\(176\) 0 0
\(177\) 10.8022 0.811944
\(178\) 0 0
\(179\) −3.43373 −0.256649 −0.128324 0.991732i \(-0.540960\pi\)
−0.128324 + 0.991732i \(0.540960\pi\)
\(180\) 0 0
\(181\) −11.3243 −0.841725 −0.420863 0.907124i \(-0.638273\pi\)
−0.420863 + 0.907124i \(0.638273\pi\)
\(182\) 0 0
\(183\) −14.0025 −1.03509
\(184\) 0 0
\(185\) 5.25783 0.386564
\(186\) 0 0
\(187\) 8.76893 0.641247
\(188\) 0 0
\(189\) −1.96468 −0.142909
\(190\) 0 0
\(191\) −26.0824 −1.88726 −0.943628 0.331007i \(-0.892612\pi\)
−0.943628 + 0.331007i \(0.892612\pi\)
\(192\) 0 0
\(193\) 25.5074 1.83606 0.918031 0.396508i \(-0.129778\pi\)
0.918031 + 0.396508i \(0.129778\pi\)
\(194\) 0 0
\(195\) −5.04358 −0.361178
\(196\) 0 0
\(197\) −4.73730 −0.337519 −0.168759 0.985657i \(-0.553976\pi\)
−0.168759 + 0.985657i \(0.553976\pi\)
\(198\) 0 0
\(199\) −13.8698 −0.983206 −0.491603 0.870819i \(-0.663589\pi\)
−0.491603 + 0.870819i \(0.663589\pi\)
\(200\) 0 0
\(201\) 0.286799 0.0202293
\(202\) 0 0
\(203\) 11.4179 0.801376
\(204\) 0 0
\(205\) 9.43182 0.658747
\(206\) 0 0
\(207\) −4.55057 −0.316287
\(208\) 0 0
\(209\) −5.56182 −0.384719
\(210\) 0 0
\(211\) 17.5267 1.20659 0.603294 0.797519i \(-0.293854\pi\)
0.603294 + 0.797519i \(0.293854\pi\)
\(212\) 0 0
\(213\) −10.1569 −0.695937
\(214\) 0 0
\(215\) −4.99238 −0.340478
\(216\) 0 0
\(217\) 9.14967 0.621120
\(218\) 0 0
\(219\) 3.44823 0.233010
\(220\) 0 0
\(221\) 14.2907 0.961297
\(222\) 0 0
\(223\) 17.9825 1.20420 0.602099 0.798422i \(-0.294331\pi\)
0.602099 + 0.798422i \(0.294331\pi\)
\(224\) 0 0
\(225\) −3.34982 −0.223322
\(226\) 0 0
\(227\) 4.80164 0.318696 0.159348 0.987222i \(-0.449061\pi\)
0.159348 + 0.987222i \(0.449061\pi\)
\(228\) 0 0
\(229\) −27.8758 −1.84208 −0.921041 0.389465i \(-0.872660\pi\)
−0.921041 + 0.389465i \(0.872660\pi\)
\(230\) 0 0
\(231\) 4.73323 0.311424
\(232\) 0 0
\(233\) 5.52381 0.361877 0.180939 0.983494i \(-0.442086\pi\)
0.180939 + 0.983494i \(0.442086\pi\)
\(234\) 0 0
\(235\) 9.65655 0.629924
\(236\) 0 0
\(237\) 5.99698 0.389546
\(238\) 0 0
\(239\) −23.2120 −1.50146 −0.750729 0.660611i \(-0.770297\pi\)
−0.750729 + 0.660611i \(0.770297\pi\)
\(240\) 0 0
\(241\) 3.27780 0.211142 0.105571 0.994412i \(-0.466333\pi\)
0.105571 + 0.994412i \(0.466333\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.03366 −0.257701
\(246\) 0 0
\(247\) −9.06410 −0.576735
\(248\) 0 0
\(249\) 10.0713 0.638246
\(250\) 0 0
\(251\) −11.2250 −0.708518 −0.354259 0.935147i \(-0.615267\pi\)
−0.354259 + 0.935147i \(0.615267\pi\)
\(252\) 0 0
\(253\) 10.9631 0.689242
\(254\) 0 0
\(255\) −4.67569 −0.292803
\(256\) 0 0
\(257\) 12.9517 0.807907 0.403954 0.914779i \(-0.367636\pi\)
0.403954 + 0.914779i \(0.367636\pi\)
\(258\) 0 0
\(259\) −8.04143 −0.499670
\(260\) 0 0
\(261\) −5.81156 −0.359726
\(262\) 0 0
\(263\) −4.16998 −0.257132 −0.128566 0.991701i \(-0.541037\pi\)
−0.128566 + 0.991701i \(0.541037\pi\)
\(264\) 0 0
\(265\) −13.6772 −0.840185
\(266\) 0 0
\(267\) −9.05989 −0.554456
\(268\) 0 0
\(269\) −31.2329 −1.90430 −0.952152 0.305625i \(-0.901135\pi\)
−0.952152 + 0.305625i \(0.901135\pi\)
\(270\) 0 0
\(271\) 19.9256 1.21039 0.605197 0.796076i \(-0.293095\pi\)
0.605197 + 0.796076i \(0.293095\pi\)
\(272\) 0 0
\(273\) 7.71375 0.466857
\(274\) 0 0
\(275\) 8.07027 0.486655
\(276\) 0 0
\(277\) 2.20031 0.132204 0.0661021 0.997813i \(-0.478944\pi\)
0.0661021 + 0.997813i \(0.478944\pi\)
\(278\) 0 0
\(279\) −4.65708 −0.278812
\(280\) 0 0
\(281\) 26.4717 1.57917 0.789585 0.613641i \(-0.210296\pi\)
0.789585 + 0.613641i \(0.210296\pi\)
\(282\) 0 0
\(283\) 14.2466 0.846870 0.423435 0.905926i \(-0.360824\pi\)
0.423435 + 0.905926i \(0.360824\pi\)
\(284\) 0 0
\(285\) 2.96563 0.175669
\(286\) 0 0
\(287\) −14.4252 −0.851494
\(288\) 0 0
\(289\) −3.75169 −0.220687
\(290\) 0 0
\(291\) 4.02676 0.236053
\(292\) 0 0
\(293\) 12.2255 0.714221 0.357111 0.934062i \(-0.383762\pi\)
0.357111 + 0.934062i \(0.383762\pi\)
\(294\) 0 0
\(295\) 13.8764 0.807917
\(296\) 0 0
\(297\) −2.40916 −0.139794
\(298\) 0 0
\(299\) 17.8665 1.03325
\(300\) 0 0
\(301\) 7.63545 0.440100
\(302\) 0 0
\(303\) 1.06980 0.0614587
\(304\) 0 0
\(305\) −17.9875 −1.02996
\(306\) 0 0
\(307\) 1.04146 0.0594395 0.0297197 0.999558i \(-0.490539\pi\)
0.0297197 + 0.999558i \(0.490539\pi\)
\(308\) 0 0
\(309\) 15.5422 0.884164
\(310\) 0 0
\(311\) −1.61814 −0.0917564 −0.0458782 0.998947i \(-0.514609\pi\)
−0.0458782 + 0.998947i \(0.514609\pi\)
\(312\) 0 0
\(313\) −25.1205 −1.41990 −0.709948 0.704254i \(-0.751281\pi\)
−0.709948 + 0.704254i \(0.751281\pi\)
\(314\) 0 0
\(315\) −2.52381 −0.142201
\(316\) 0 0
\(317\) −25.6518 −1.44075 −0.720374 0.693586i \(-0.756030\pi\)
−0.720374 + 0.693586i \(0.756030\pi\)
\(318\) 0 0
\(319\) 14.0010 0.783905
\(320\) 0 0
\(321\) −2.54641 −0.142127
\(322\) 0 0
\(323\) −8.40294 −0.467552
\(324\) 0 0
\(325\) 13.1521 0.729547
\(326\) 0 0
\(327\) 15.5695 0.860998
\(328\) 0 0
\(329\) −14.7689 −0.814236
\(330\) 0 0
\(331\) 17.7667 0.976544 0.488272 0.872691i \(-0.337627\pi\)
0.488272 + 0.872691i \(0.337627\pi\)
\(332\) 0 0
\(333\) 4.09300 0.224295
\(334\) 0 0
\(335\) 0.368420 0.0201290
\(336\) 0 0
\(337\) −19.2213 −1.04705 −0.523526 0.852010i \(-0.675384\pi\)
−0.523526 + 0.852010i \(0.675384\pi\)
\(338\) 0 0
\(339\) −3.22887 −0.175368
\(340\) 0 0
\(341\) 11.2197 0.607578
\(342\) 0 0
\(343\) 19.9219 1.07568
\(344\) 0 0
\(345\) −5.84563 −0.314718
\(346\) 0 0
\(347\) 14.1326 0.758677 0.379339 0.925258i \(-0.376152\pi\)
0.379339 + 0.925258i \(0.376152\pi\)
\(348\) 0 0
\(349\) −28.1862 −1.50877 −0.754387 0.656429i \(-0.772066\pi\)
−0.754387 + 0.656429i \(0.772066\pi\)
\(350\) 0 0
\(351\) −3.92621 −0.209566
\(352\) 0 0
\(353\) −16.0991 −0.856870 −0.428435 0.903573i \(-0.640935\pi\)
−0.428435 + 0.903573i \(0.640935\pi\)
\(354\) 0 0
\(355\) −13.0474 −0.692486
\(356\) 0 0
\(357\) 7.15109 0.378476
\(358\) 0 0
\(359\) −3.09816 −0.163515 −0.0817574 0.996652i \(-0.526053\pi\)
−0.0817574 + 0.996652i \(0.526053\pi\)
\(360\) 0 0
\(361\) −13.6703 −0.719490
\(362\) 0 0
\(363\) −5.19594 −0.272716
\(364\) 0 0
\(365\) 4.42957 0.231854
\(366\) 0 0
\(367\) −15.5041 −0.809307 −0.404654 0.914470i \(-0.632608\pi\)
−0.404654 + 0.914470i \(0.632608\pi\)
\(368\) 0 0
\(369\) 7.34227 0.382223
\(370\) 0 0
\(371\) 20.9182 1.08602
\(372\) 0 0
\(373\) −8.04456 −0.416532 −0.208266 0.978072i \(-0.566782\pi\)
−0.208266 + 0.978072i \(0.566782\pi\)
\(374\) 0 0
\(375\) −10.7261 −0.553894
\(376\) 0 0
\(377\) 22.8174 1.17516
\(378\) 0 0
\(379\) 18.3818 0.944210 0.472105 0.881542i \(-0.343494\pi\)
0.472105 + 0.881542i \(0.343494\pi\)
\(380\) 0 0
\(381\) 0.940660 0.0481915
\(382\) 0 0
\(383\) 1.38490 0.0707649 0.0353825 0.999374i \(-0.488735\pi\)
0.0353825 + 0.999374i \(0.488735\pi\)
\(384\) 0 0
\(385\) 6.08028 0.309880
\(386\) 0 0
\(387\) −3.88636 −0.197555
\(388\) 0 0
\(389\) 18.6320 0.944680 0.472340 0.881416i \(-0.343409\pi\)
0.472340 + 0.881416i \(0.343409\pi\)
\(390\) 0 0
\(391\) 16.5633 0.837641
\(392\) 0 0
\(393\) −13.7284 −0.692507
\(394\) 0 0
\(395\) 7.70367 0.387614
\(396\) 0 0
\(397\) 36.8598 1.84994 0.924969 0.380042i \(-0.124090\pi\)
0.924969 + 0.380042i \(0.124090\pi\)
\(398\) 0 0
\(399\) −4.53569 −0.227068
\(400\) 0 0
\(401\) −5.06368 −0.252868 −0.126434 0.991975i \(-0.540353\pi\)
−0.126434 + 0.991975i \(0.540353\pi\)
\(402\) 0 0
\(403\) 18.2847 0.910824
\(404\) 0 0
\(405\) 1.28459 0.0638319
\(406\) 0 0
\(407\) −9.86070 −0.488777
\(408\) 0 0
\(409\) 24.0361 1.18851 0.594254 0.804278i \(-0.297447\pi\)
0.594254 + 0.804278i \(0.297447\pi\)
\(410\) 0 0
\(411\) −17.2672 −0.851730
\(412\) 0 0
\(413\) −21.2229 −1.04431
\(414\) 0 0
\(415\) 12.9376 0.635080
\(416\) 0 0
\(417\) 10.0634 0.492809
\(418\) 0 0
\(419\) 10.8318 0.529170 0.264585 0.964362i \(-0.414765\pi\)
0.264585 + 0.964362i \(0.414765\pi\)
\(420\) 0 0
\(421\) 2.18637 0.106557 0.0532785 0.998580i \(-0.483033\pi\)
0.0532785 + 0.998580i \(0.483033\pi\)
\(422\) 0 0
\(423\) 7.51721 0.365499
\(424\) 0 0
\(425\) 12.1928 0.591436
\(426\) 0 0
\(427\) 27.5104 1.33132
\(428\) 0 0
\(429\) 9.45888 0.456679
\(430\) 0 0
\(431\) 19.3902 0.933994 0.466997 0.884259i \(-0.345336\pi\)
0.466997 + 0.884259i \(0.345336\pi\)
\(432\) 0 0
\(433\) 18.7996 0.903449 0.451725 0.892157i \(-0.350809\pi\)
0.451725 + 0.892157i \(0.350809\pi\)
\(434\) 0 0
\(435\) −7.46548 −0.357942
\(436\) 0 0
\(437\) −10.5055 −0.502547
\(438\) 0 0
\(439\) −8.05917 −0.384643 −0.192322 0.981332i \(-0.561602\pi\)
−0.192322 + 0.981332i \(0.561602\pi\)
\(440\) 0 0
\(441\) −3.14003 −0.149525
\(442\) 0 0
\(443\) −28.1425 −1.33709 −0.668546 0.743671i \(-0.733083\pi\)
−0.668546 + 0.743671i \(0.733083\pi\)
\(444\) 0 0
\(445\) −11.6383 −0.551706
\(446\) 0 0
\(447\) −18.8300 −0.890627
\(448\) 0 0
\(449\) 28.9364 1.36559 0.682797 0.730608i \(-0.260763\pi\)
0.682797 + 0.730608i \(0.260763\pi\)
\(450\) 0 0
\(451\) −17.6887 −0.832929
\(452\) 0 0
\(453\) 10.6354 0.499696
\(454\) 0 0
\(455\) 9.90902 0.464542
\(456\) 0 0
\(457\) −7.80085 −0.364909 −0.182454 0.983214i \(-0.558404\pi\)
−0.182454 + 0.983214i \(0.558404\pi\)
\(458\) 0 0
\(459\) −3.63982 −0.169892
\(460\) 0 0
\(461\) −13.6981 −0.637984 −0.318992 0.947757i \(-0.603344\pi\)
−0.318992 + 0.947757i \(0.603344\pi\)
\(462\) 0 0
\(463\) 39.1886 1.82125 0.910625 0.413235i \(-0.135601\pi\)
0.910625 + 0.413235i \(0.135601\pi\)
\(464\) 0 0
\(465\) −5.98244 −0.277429
\(466\) 0 0
\(467\) 17.2581 0.798610 0.399305 0.916818i \(-0.369252\pi\)
0.399305 + 0.916818i \(0.369252\pi\)
\(468\) 0 0
\(469\) −0.563469 −0.0260186
\(470\) 0 0
\(471\) 22.2179 1.02375
\(472\) 0 0
\(473\) 9.36287 0.430505
\(474\) 0 0
\(475\) −7.73344 −0.354835
\(476\) 0 0
\(477\) −10.6471 −0.487499
\(478\) 0 0
\(479\) −2.46912 −0.112817 −0.0564084 0.998408i \(-0.517965\pi\)
−0.0564084 + 0.998408i \(0.517965\pi\)
\(480\) 0 0
\(481\) −16.0700 −0.732727
\(482\) 0 0
\(483\) 8.94042 0.406803
\(484\) 0 0
\(485\) 5.17274 0.234882
\(486\) 0 0
\(487\) 17.6955 0.801861 0.400930 0.916108i \(-0.368687\pi\)
0.400930 + 0.916108i \(0.368687\pi\)
\(488\) 0 0
\(489\) −21.9987 −0.994814
\(490\) 0 0
\(491\) 14.9043 0.672623 0.336312 0.941751i \(-0.390820\pi\)
0.336312 + 0.941751i \(0.390820\pi\)
\(492\) 0 0
\(493\) 21.1530 0.952685
\(494\) 0 0
\(495\) −3.09479 −0.139101
\(496\) 0 0
\(497\) 19.9550 0.895104
\(498\) 0 0
\(499\) 14.7483 0.660226 0.330113 0.943941i \(-0.392913\pi\)
0.330113 + 0.943941i \(0.392913\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 21.6170 0.963856 0.481928 0.876211i \(-0.339937\pi\)
0.481928 + 0.876211i \(0.339937\pi\)
\(504\) 0 0
\(505\) 1.37426 0.0611539
\(506\) 0 0
\(507\) 2.41512 0.107259
\(508\) 0 0
\(509\) −11.3422 −0.502735 −0.251368 0.967892i \(-0.580880\pi\)
−0.251368 + 0.967892i \(0.580880\pi\)
\(510\) 0 0
\(511\) −6.77467 −0.299694
\(512\) 0 0
\(513\) 2.30861 0.101928
\(514\) 0 0
\(515\) 19.9654 0.879779
\(516\) 0 0
\(517\) −18.1102 −0.796484
\(518\) 0 0
\(519\) 15.8426 0.695413
\(520\) 0 0
\(521\) −5.90072 −0.258515 −0.129258 0.991611i \(-0.541259\pi\)
−0.129258 + 0.991611i \(0.541259\pi\)
\(522\) 0 0
\(523\) 7.21958 0.315690 0.157845 0.987464i \(-0.449545\pi\)
0.157845 + 0.987464i \(0.449545\pi\)
\(524\) 0 0
\(525\) 6.58133 0.287233
\(526\) 0 0
\(527\) 16.9509 0.738394
\(528\) 0 0
\(529\) −2.29229 −0.0996648
\(530\) 0 0
\(531\) 10.8022 0.468776
\(532\) 0 0
\(533\) −28.8273 −1.24865
\(534\) 0 0
\(535\) −3.27110 −0.141422
\(536\) 0 0
\(537\) −3.43373 −0.148176
\(538\) 0 0
\(539\) 7.56484 0.325841
\(540\) 0 0
\(541\) −4.15094 −0.178463 −0.0892314 0.996011i \(-0.528441\pi\)
−0.0892314 + 0.996011i \(0.528441\pi\)
\(542\) 0 0
\(543\) −11.3243 −0.485970
\(544\) 0 0
\(545\) 20.0005 0.856728
\(546\) 0 0
\(547\) −5.37704 −0.229906 −0.114953 0.993371i \(-0.536672\pi\)
−0.114953 + 0.993371i \(0.536672\pi\)
\(548\) 0 0
\(549\) −14.0025 −0.597612
\(550\) 0 0
\(551\) −13.4166 −0.571568
\(552\) 0 0
\(553\) −11.7822 −0.501028
\(554\) 0 0
\(555\) 5.25783 0.223183
\(556\) 0 0
\(557\) −13.1551 −0.557399 −0.278700 0.960378i \(-0.589903\pi\)
−0.278700 + 0.960378i \(0.589903\pi\)
\(558\) 0 0
\(559\) 15.2587 0.645372
\(560\) 0 0
\(561\) 8.76893 0.370224
\(562\) 0 0
\(563\) 5.08117 0.214146 0.107073 0.994251i \(-0.465852\pi\)
0.107073 + 0.994251i \(0.465852\pi\)
\(564\) 0 0
\(565\) −4.14778 −0.174498
\(566\) 0 0
\(567\) −1.96468 −0.0825088
\(568\) 0 0
\(569\) −16.5777 −0.694972 −0.347486 0.937685i \(-0.612965\pi\)
−0.347486 + 0.937685i \(0.612965\pi\)
\(570\) 0 0
\(571\) 30.5518 1.27855 0.639277 0.768977i \(-0.279234\pi\)
0.639277 + 0.768977i \(0.279234\pi\)
\(572\) 0 0
\(573\) −26.0824 −1.08961
\(574\) 0 0
\(575\) 15.2436 0.635703
\(576\) 0 0
\(577\) −23.2441 −0.967663 −0.483832 0.875161i \(-0.660755\pi\)
−0.483832 + 0.875161i \(0.660755\pi\)
\(578\) 0 0
\(579\) 25.5074 1.06005
\(580\) 0 0
\(581\) −19.7870 −0.820902
\(582\) 0 0
\(583\) 25.6507 1.06234
\(584\) 0 0
\(585\) −5.04358 −0.208526
\(586\) 0 0
\(587\) −19.3846 −0.800088 −0.400044 0.916496i \(-0.631005\pi\)
−0.400044 + 0.916496i \(0.631005\pi\)
\(588\) 0 0
\(589\) −10.7514 −0.443003
\(590\) 0 0
\(591\) −4.73730 −0.194867
\(592\) 0 0
\(593\) 9.34754 0.383857 0.191929 0.981409i \(-0.438526\pi\)
0.191929 + 0.981409i \(0.438526\pi\)
\(594\) 0 0
\(595\) 9.18623 0.376599
\(596\) 0 0
\(597\) −13.8698 −0.567654
\(598\) 0 0
\(599\) 1.66606 0.0680732 0.0340366 0.999421i \(-0.489164\pi\)
0.0340366 + 0.999421i \(0.489164\pi\)
\(600\) 0 0
\(601\) −26.5815 −1.08428 −0.542141 0.840287i \(-0.682386\pi\)
−0.542141 + 0.840287i \(0.682386\pi\)
\(602\) 0 0
\(603\) 0.286799 0.0116794
\(604\) 0 0
\(605\) −6.67466 −0.271363
\(606\) 0 0
\(607\) −30.8790 −1.25334 −0.626670 0.779285i \(-0.715583\pi\)
−0.626670 + 0.779285i \(0.715583\pi\)
\(608\) 0 0
\(609\) 11.4179 0.462675
\(610\) 0 0
\(611\) −29.5141 −1.19401
\(612\) 0 0
\(613\) 14.9530 0.603947 0.301973 0.953316i \(-0.402355\pi\)
0.301973 + 0.953316i \(0.402355\pi\)
\(614\) 0 0
\(615\) 9.43182 0.380328
\(616\) 0 0
\(617\) 20.3705 0.820087 0.410043 0.912066i \(-0.365514\pi\)
0.410043 + 0.912066i \(0.365514\pi\)
\(618\) 0 0
\(619\) 0.534153 0.0214694 0.0107347 0.999942i \(-0.496583\pi\)
0.0107347 + 0.999942i \(0.496583\pi\)
\(620\) 0 0
\(621\) −4.55057 −0.182608
\(622\) 0 0
\(623\) 17.7998 0.713133
\(624\) 0 0
\(625\) 2.97043 0.118817
\(626\) 0 0
\(627\) −5.56182 −0.222118
\(628\) 0 0
\(629\) −14.8978 −0.594014
\(630\) 0 0
\(631\) −16.3425 −0.650583 −0.325292 0.945614i \(-0.605462\pi\)
−0.325292 + 0.945614i \(0.605462\pi\)
\(632\) 0 0
\(633\) 17.5267 0.696624
\(634\) 0 0
\(635\) 1.20836 0.0479525
\(636\) 0 0
\(637\) 12.3284 0.488470
\(638\) 0 0
\(639\) −10.1569 −0.401800
\(640\) 0 0
\(641\) 45.9146 1.81352 0.906759 0.421649i \(-0.138548\pi\)
0.906759 + 0.421649i \(0.138548\pi\)
\(642\) 0 0
\(643\) −2.16517 −0.0853860 −0.0426930 0.999088i \(-0.513594\pi\)
−0.0426930 + 0.999088i \(0.513594\pi\)
\(644\) 0 0
\(645\) −4.99238 −0.196575
\(646\) 0 0
\(647\) −29.7954 −1.17138 −0.585688 0.810536i \(-0.699176\pi\)
−0.585688 + 0.810536i \(0.699176\pi\)
\(648\) 0 0
\(649\) −26.0243 −1.02154
\(650\) 0 0
\(651\) 9.14967 0.358604
\(652\) 0 0
\(653\) 9.26237 0.362464 0.181232 0.983440i \(-0.441991\pi\)
0.181232 + 0.983440i \(0.441991\pi\)
\(654\) 0 0
\(655\) −17.6354 −0.689073
\(656\) 0 0
\(657\) 3.44823 0.134528
\(658\) 0 0
\(659\) 0.492474 0.0191841 0.00959203 0.999954i \(-0.496947\pi\)
0.00959203 + 0.999954i \(0.496947\pi\)
\(660\) 0 0
\(661\) −8.06786 −0.313804 −0.156902 0.987614i \(-0.550151\pi\)
−0.156902 + 0.987614i \(0.550151\pi\)
\(662\) 0 0
\(663\) 14.2907 0.555005
\(664\) 0 0
\(665\) −5.82651 −0.225942
\(666\) 0 0
\(667\) 26.4459 1.02399
\(668\) 0 0
\(669\) 17.9825 0.695244
\(670\) 0 0
\(671\) 33.7343 1.30230
\(672\) 0 0
\(673\) 6.59943 0.254389 0.127195 0.991878i \(-0.459403\pi\)
0.127195 + 0.991878i \(0.459403\pi\)
\(674\) 0 0
\(675\) −3.34982 −0.128935
\(676\) 0 0
\(677\) −14.6380 −0.562584 −0.281292 0.959622i \(-0.590763\pi\)
−0.281292 + 0.959622i \(0.590763\pi\)
\(678\) 0 0
\(679\) −7.91130 −0.303608
\(680\) 0 0
\(681\) 4.80164 0.183999
\(682\) 0 0
\(683\) −10.3419 −0.395724 −0.197862 0.980230i \(-0.563400\pi\)
−0.197862 + 0.980230i \(0.563400\pi\)
\(684\) 0 0
\(685\) −22.1814 −0.847506
\(686\) 0 0
\(687\) −27.8758 −1.06353
\(688\) 0 0
\(689\) 41.8029 1.59256
\(690\) 0 0
\(691\) −34.5829 −1.31559 −0.657797 0.753195i \(-0.728512\pi\)
−0.657797 + 0.753195i \(0.728512\pi\)
\(692\) 0 0
\(693\) 4.73323 0.179801
\(694\) 0 0
\(695\) 12.9274 0.490365
\(696\) 0 0
\(697\) −26.7246 −1.01227
\(698\) 0 0
\(699\) 5.52381 0.208930
\(700\) 0 0
\(701\) −38.5480 −1.45594 −0.727969 0.685610i \(-0.759535\pi\)
−0.727969 + 0.685610i \(0.759535\pi\)
\(702\) 0 0
\(703\) 9.44915 0.356381
\(704\) 0 0
\(705\) 9.65655 0.363687
\(706\) 0 0
\(707\) −2.10183 −0.0790473
\(708\) 0 0
\(709\) 16.9239 0.635592 0.317796 0.948159i \(-0.397057\pi\)
0.317796 + 0.948159i \(0.397057\pi\)
\(710\) 0 0
\(711\) 5.99698 0.224904
\(712\) 0 0
\(713\) 21.1924 0.793660
\(714\) 0 0
\(715\) 12.1508 0.454414
\(716\) 0 0
\(717\) −23.2120 −0.866867
\(718\) 0 0
\(719\) 16.1218 0.601241 0.300621 0.953744i \(-0.402806\pi\)
0.300621 + 0.953744i \(0.402806\pi\)
\(720\) 0 0
\(721\) −30.5354 −1.13720
\(722\) 0 0
\(723\) 3.27780 0.121903
\(724\) 0 0
\(725\) 19.4677 0.723012
\(726\) 0 0
\(727\) 10.8925 0.403980 0.201990 0.979388i \(-0.435259\pi\)
0.201990 + 0.979388i \(0.435259\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 14.1457 0.523196
\(732\) 0 0
\(733\) 39.2054 1.44809 0.724043 0.689755i \(-0.242282\pi\)
0.724043 + 0.689755i \(0.242282\pi\)
\(734\) 0 0
\(735\) −4.03366 −0.148784
\(736\) 0 0
\(737\) −0.690946 −0.0254513
\(738\) 0 0
\(739\) −46.8075 −1.72184 −0.860921 0.508738i \(-0.830112\pi\)
−0.860921 + 0.508738i \(0.830112\pi\)
\(740\) 0 0
\(741\) −9.06410 −0.332978
\(742\) 0 0
\(743\) −33.4808 −1.22829 −0.614145 0.789193i \(-0.710499\pi\)
−0.614145 + 0.789193i \(0.710499\pi\)
\(744\) 0 0
\(745\) −24.1888 −0.886211
\(746\) 0 0
\(747\) 10.0713 0.368491
\(748\) 0 0
\(749\) 5.00289 0.182801
\(750\) 0 0
\(751\) −24.7166 −0.901922 −0.450961 0.892544i \(-0.648919\pi\)
−0.450961 + 0.892544i \(0.648919\pi\)
\(752\) 0 0
\(753\) −11.2250 −0.409063
\(754\) 0 0
\(755\) 13.6622 0.497218
\(756\) 0 0
\(757\) −26.5140 −0.963667 −0.481834 0.876263i \(-0.660029\pi\)
−0.481834 + 0.876263i \(0.660029\pi\)
\(758\) 0 0
\(759\) 10.9631 0.397934
\(760\) 0 0
\(761\) 25.2602 0.915681 0.457840 0.889034i \(-0.348623\pi\)
0.457840 + 0.889034i \(0.348623\pi\)
\(762\) 0 0
\(763\) −30.5892 −1.10740
\(764\) 0 0
\(765\) −4.67569 −0.169050
\(766\) 0 0
\(767\) −42.4117 −1.53140
\(768\) 0 0
\(769\) −19.6768 −0.709564 −0.354782 0.934949i \(-0.615445\pi\)
−0.354782 + 0.934949i \(0.615445\pi\)
\(770\) 0 0
\(771\) 12.9517 0.466446
\(772\) 0 0
\(773\) −36.8440 −1.32519 −0.662594 0.748979i \(-0.730544\pi\)
−0.662594 + 0.748979i \(0.730544\pi\)
\(774\) 0 0
\(775\) 15.6004 0.560382
\(776\) 0 0
\(777\) −8.04143 −0.288485
\(778\) 0 0
\(779\) 16.9505 0.607313
\(780\) 0 0
\(781\) 24.4695 0.875589
\(782\) 0 0
\(783\) −5.81156 −0.207688
\(784\) 0 0
\(785\) 28.5410 1.01867
\(786\) 0 0
\(787\) 5.36433 0.191218 0.0956088 0.995419i \(-0.469520\pi\)
0.0956088 + 0.995419i \(0.469520\pi\)
\(788\) 0 0
\(789\) −4.16998 −0.148455
\(790\) 0 0
\(791\) 6.34369 0.225556
\(792\) 0 0
\(793\) 54.9767 1.95228
\(794\) 0 0
\(795\) −13.6772 −0.485081
\(796\) 0 0
\(797\) −7.20815 −0.255326 −0.127663 0.991818i \(-0.540748\pi\)
−0.127663 + 0.991818i \(0.540748\pi\)
\(798\) 0 0
\(799\) −27.3613 −0.967973
\(800\) 0 0
\(801\) −9.05989 −0.320115
\(802\) 0 0
\(803\) −8.30734 −0.293160
\(804\) 0 0
\(805\) 11.4848 0.404786
\(806\) 0 0
\(807\) −31.2329 −1.09945
\(808\) 0 0
\(809\) 23.3611 0.821334 0.410667 0.911785i \(-0.365296\pi\)
0.410667 + 0.911785i \(0.365296\pi\)
\(810\) 0 0
\(811\) 47.5681 1.67034 0.835170 0.549991i \(-0.185369\pi\)
0.835170 + 0.549991i \(0.185369\pi\)
\(812\) 0 0
\(813\) 19.9256 0.698821
\(814\) 0 0
\(815\) −28.2593 −0.989881
\(816\) 0 0
\(817\) −8.97209 −0.313894
\(818\) 0 0
\(819\) 7.71375 0.269540
\(820\) 0 0
\(821\) 1.28529 0.0448568 0.0224284 0.999748i \(-0.492860\pi\)
0.0224284 + 0.999748i \(0.492860\pi\)
\(822\) 0 0
\(823\) 15.1103 0.526712 0.263356 0.964699i \(-0.415171\pi\)
0.263356 + 0.964699i \(0.415171\pi\)
\(824\) 0 0
\(825\) 8.07027 0.280971
\(826\) 0 0
\(827\) −32.1009 −1.11626 −0.558129 0.829754i \(-0.688481\pi\)
−0.558129 + 0.829754i \(0.688481\pi\)
\(828\) 0 0
\(829\) −39.5385 −1.37323 −0.686614 0.727022i \(-0.740904\pi\)
−0.686614 + 0.727022i \(0.740904\pi\)
\(830\) 0 0
\(831\) 2.20031 0.0763281
\(832\) 0 0
\(833\) 11.4292 0.395997
\(834\) 0 0
\(835\) −1.28459 −0.0444551
\(836\) 0 0
\(837\) −4.65708 −0.160972
\(838\) 0 0
\(839\) −2.69426 −0.0930161 −0.0465081 0.998918i \(-0.514809\pi\)
−0.0465081 + 0.998918i \(0.514809\pi\)
\(840\) 0 0
\(841\) 4.77421 0.164628
\(842\) 0 0
\(843\) 26.4717 0.911735
\(844\) 0 0
\(845\) 3.10244 0.106727
\(846\) 0 0
\(847\) 10.2084 0.350763
\(848\) 0 0
\(849\) 14.2466 0.488941
\(850\) 0 0
\(851\) −18.6255 −0.638473
\(852\) 0 0
\(853\) 48.3525 1.65556 0.827779 0.561054i \(-0.189604\pi\)
0.827779 + 0.561054i \(0.189604\pi\)
\(854\) 0 0
\(855\) 2.96563 0.101422
\(856\) 0 0
\(857\) −36.9684 −1.26282 −0.631408 0.775450i \(-0.717523\pi\)
−0.631408 + 0.775450i \(0.717523\pi\)
\(858\) 0 0
\(859\) 57.7504 1.97042 0.985210 0.171352i \(-0.0548136\pi\)
0.985210 + 0.171352i \(0.0548136\pi\)
\(860\) 0 0
\(861\) −14.4252 −0.491610
\(862\) 0 0
\(863\) 0.667087 0.0227079 0.0113540 0.999936i \(-0.496386\pi\)
0.0113540 + 0.999936i \(0.496386\pi\)
\(864\) 0 0
\(865\) 20.3513 0.691965
\(866\) 0 0
\(867\) −3.75169 −0.127414
\(868\) 0 0
\(869\) −14.4477 −0.490105
\(870\) 0 0
\(871\) −1.12603 −0.0381542
\(872\) 0 0
\(873\) 4.02676 0.136285
\(874\) 0 0
\(875\) 21.0734 0.712411
\(876\) 0 0
\(877\) −17.1291 −0.578407 −0.289204 0.957268i \(-0.593390\pi\)
−0.289204 + 0.957268i \(0.593390\pi\)
\(878\) 0 0
\(879\) 12.2255 0.412356
\(880\) 0 0
\(881\) −18.5491 −0.624935 −0.312468 0.949928i \(-0.601156\pi\)
−0.312468 + 0.949928i \(0.601156\pi\)
\(882\) 0 0
\(883\) −18.0598 −0.607761 −0.303880 0.952710i \(-0.598282\pi\)
−0.303880 + 0.952710i \(0.598282\pi\)
\(884\) 0 0
\(885\) 13.8764 0.466451
\(886\) 0 0
\(887\) −14.4311 −0.484549 −0.242274 0.970208i \(-0.577893\pi\)
−0.242274 + 0.970208i \(0.577893\pi\)
\(888\) 0 0
\(889\) −1.84810 −0.0619832
\(890\) 0 0
\(891\) −2.40916 −0.0807100
\(892\) 0 0
\(893\) 17.3543 0.580740
\(894\) 0 0
\(895\) −4.41094 −0.147442
\(896\) 0 0
\(897\) 17.8665 0.596545
\(898\) 0 0
\(899\) 27.0649 0.902664
\(900\) 0 0
\(901\) 38.7537 1.29107
\(902\) 0 0
\(903\) 7.63545 0.254092
\(904\) 0 0
\(905\) −14.5470 −0.483560
\(906\) 0 0
\(907\) 53.6448 1.78125 0.890624 0.454741i \(-0.150268\pi\)
0.890624 + 0.454741i \(0.150268\pi\)
\(908\) 0 0
\(909\) 1.06980 0.0354832
\(910\) 0 0
\(911\) −39.7916 −1.31835 −0.659177 0.751988i \(-0.729095\pi\)
−0.659177 + 0.751988i \(0.729095\pi\)
\(912\) 0 0
\(913\) −24.2635 −0.803005
\(914\) 0 0
\(915\) −17.9875 −0.594648
\(916\) 0 0
\(917\) 26.9720 0.890693
\(918\) 0 0
\(919\) −25.7644 −0.849889 −0.424945 0.905219i \(-0.639706\pi\)
−0.424945 + 0.905219i \(0.639706\pi\)
\(920\) 0 0
\(921\) 1.04146 0.0343174
\(922\) 0 0
\(923\) 39.8780 1.31260
\(924\) 0 0
\(925\) −13.7108 −0.450809
\(926\) 0 0
\(927\) 15.5422 0.510472
\(928\) 0 0
\(929\) −24.8575 −0.815549 −0.407775 0.913083i \(-0.633695\pi\)
−0.407775 + 0.913083i \(0.633695\pi\)
\(930\) 0 0
\(931\) −7.24911 −0.237580
\(932\) 0 0
\(933\) −1.61814 −0.0529756
\(934\) 0 0
\(935\) 11.2645 0.368388
\(936\) 0 0
\(937\) 12.4739 0.407505 0.203752 0.979022i \(-0.434686\pi\)
0.203752 + 0.979022i \(0.434686\pi\)
\(938\) 0 0
\(939\) −25.1205 −0.819777
\(940\) 0 0
\(941\) −8.43127 −0.274852 −0.137426 0.990512i \(-0.543883\pi\)
−0.137426 + 0.990512i \(0.543883\pi\)
\(942\) 0 0
\(943\) −33.4115 −1.08803
\(944\) 0 0
\(945\) −2.52381 −0.0820997
\(946\) 0 0
\(947\) −46.1816 −1.50070 −0.750350 0.661041i \(-0.770115\pi\)
−0.750350 + 0.661041i \(0.770115\pi\)
\(948\) 0 0
\(949\) −13.5385 −0.439477
\(950\) 0 0
\(951\) −25.6518 −0.831816
\(952\) 0 0
\(953\) −27.1218 −0.878563 −0.439281 0.898350i \(-0.644767\pi\)
−0.439281 + 0.898350i \(0.644767\pi\)
\(954\) 0 0
\(955\) −33.5053 −1.08420
\(956\) 0 0
\(957\) 14.0010 0.452588
\(958\) 0 0
\(959\) 33.9246 1.09548
\(960\) 0 0
\(961\) −9.31164 −0.300375
\(962\) 0 0
\(963\) −2.54641 −0.0820570
\(964\) 0 0
\(965\) 32.7666 1.05479
\(966\) 0 0
\(967\) −51.9680 −1.67118 −0.835590 0.549354i \(-0.814874\pi\)
−0.835590 + 0.549354i \(0.814874\pi\)
\(968\) 0 0
\(969\) −8.40294 −0.269941
\(970\) 0 0
\(971\) −36.3326 −1.16597 −0.582985 0.812483i \(-0.698115\pi\)
−0.582985 + 0.812483i \(0.698115\pi\)
\(972\) 0 0
\(973\) −19.7714 −0.633843
\(974\) 0 0
\(975\) 13.1521 0.421204
\(976\) 0 0
\(977\) −56.2379 −1.79921 −0.899605 0.436704i \(-0.856146\pi\)
−0.899605 + 0.436704i \(0.856146\pi\)
\(978\) 0 0
\(979\) 21.8267 0.697586
\(980\) 0 0
\(981\) 15.5695 0.497097
\(982\) 0 0
\(983\) −45.9094 −1.46428 −0.732141 0.681153i \(-0.761479\pi\)
−0.732141 + 0.681153i \(0.761479\pi\)
\(984\) 0 0
\(985\) −6.08550 −0.193900
\(986\) 0 0
\(987\) −14.7689 −0.470100
\(988\) 0 0
\(989\) 17.6852 0.562355
\(990\) 0 0
\(991\) −23.0349 −0.731726 −0.365863 0.930669i \(-0.619226\pi\)
−0.365863 + 0.930669i \(0.619226\pi\)
\(992\) 0 0
\(993\) 17.7667 0.563808
\(994\) 0 0
\(995\) −17.8171 −0.564839
\(996\) 0 0
\(997\) −49.2868 −1.56093 −0.780464 0.625201i \(-0.785017\pi\)
−0.780464 + 0.625201i \(0.785017\pi\)
\(998\) 0 0
\(999\) 4.09300 0.129497
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 2004.2.a.b.1.5 5
3.2 odd 2 6012.2.a.f.1.1 5
4.3 odd 2 8016.2.a.q.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.b.1.5 5 1.1 even 1 trivial
6012.2.a.f.1.1 5 3.2 odd 2
8016.2.a.q.1.5 5 4.3 odd 2