Properties

Label 3648.2.a.bt.1.2
Level $3648$
Weight $2$
Character 3648.1
Self dual yes
Analytic conductor $29.129$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.1294266574\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1824)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 3648.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.37228 q^{5} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.37228 q^{5} -3.37228 q^{7} +1.00000 q^{9} -0.627719 q^{11} +4.00000 q^{13} +3.37228 q^{15} -5.37228 q^{17} -1.00000 q^{19} -3.37228 q^{21} -4.74456 q^{23} +6.37228 q^{25} +1.00000 q^{27} +8.74456 q^{29} +6.00000 q^{31} -0.627719 q^{33} -11.3723 q^{35} +4.00000 q^{37} +4.00000 q^{39} +8.74456 q^{41} +7.37228 q^{43} +3.37228 q^{45} +8.11684 q^{47} +4.37228 q^{49} -5.37228 q^{51} +10.0000 q^{53} -2.11684 q^{55} -1.00000 q^{57} -2.74456 q^{59} -9.37228 q^{61} -3.37228 q^{63} +13.4891 q^{65} -6.74456 q^{67} -4.74456 q^{69} +14.7446 q^{71} +2.62772 q^{73} +6.37228 q^{75} +2.11684 q^{77} +2.00000 q^{79} +1.00000 q^{81} -18.1168 q^{85} +8.74456 q^{87} -7.48913 q^{89} -13.4891 q^{91} +6.00000 q^{93} -3.37228 q^{95} +7.25544 q^{97} -0.627719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} - q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} - q^{7} + 2q^{9} - 7q^{11} + 8q^{13} + q^{15} - 5q^{17} - 2q^{19} - q^{21} + 2q^{23} + 7q^{25} + 2q^{27} + 6q^{29} + 12q^{31} - 7q^{33} - 17q^{35} + 8q^{37} + 8q^{39} + 6q^{41} + 9q^{43} + q^{45} - q^{47} + 3q^{49} - 5q^{51} + 20q^{53} + 13q^{55} - 2q^{57} + 6q^{59} - 13q^{61} - q^{63} + 4q^{65} - 2q^{67} + 2q^{69} + 18q^{71} + 11q^{73} + 7q^{75} - 13q^{77} + 4q^{79} + 2q^{81} - 19q^{85} + 6q^{87} + 8q^{89} - 4q^{91} + 12q^{93} - q^{95} + 26q^{97} - 7q^{99} + O(q^{100}) \)

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
\(5\) 3.37228 1.50813 0.754065 0.656800i \(-0.228090\pi\)
0.754065 + 0.656800i \(0.228090\pi\)
\(6\) 0 0
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.627719 −0.189264 −0.0946322 0.995512i \(-0.530167\pi\)
−0.0946322 + 0.995512i \(0.530167\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 3.37228 0.870719
\(16\) 0 0
\(17\) −5.37228 −1.30297 −0.651485 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.37228 −0.735892
\(22\) 0 0
\(23\) −4.74456 −0.989310 −0.494655 0.869090i \(-0.664706\pi\)
−0.494655 + 0.869090i \(0.664706\pi\)
\(24\) 0 0
\(25\) 6.37228 1.27446
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.74456 1.62382 0.811912 0.583779i \(-0.198427\pi\)
0.811912 + 0.583779i \(0.198427\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) −0.627719 −0.109272
\(34\) 0 0
\(35\) −11.3723 −1.92227
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 8.74456 1.36567 0.682836 0.730572i \(-0.260747\pi\)
0.682836 + 0.730572i \(0.260747\pi\)
\(42\) 0 0
\(43\) 7.37228 1.12426 0.562131 0.827048i \(-0.309982\pi\)
0.562131 + 0.827048i \(0.309982\pi\)
\(44\) 0 0
\(45\) 3.37228 0.502710
\(46\) 0 0
\(47\) 8.11684 1.18396 0.591982 0.805951i \(-0.298346\pi\)
0.591982 + 0.805951i \(0.298346\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) −5.37228 −0.752270
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −2.11684 −0.285435
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −2.74456 −0.357312 −0.178656 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) −9.37228 −1.20000 −0.599999 0.800001i \(-0.704832\pi\)
−0.599999 + 0.800001i \(0.704832\pi\)
\(62\) 0 0
\(63\) −3.37228 −0.424868
\(64\) 0 0
\(65\) 13.4891 1.67312
\(66\) 0 0
\(67\) −6.74456 −0.823979 −0.411990 0.911188i \(-0.635166\pi\)
−0.411990 + 0.911188i \(0.635166\pi\)
\(68\) 0 0
\(69\) −4.74456 −0.571178
\(70\) 0 0
\(71\) 14.7446 1.74986 0.874929 0.484252i \(-0.160908\pi\)
0.874929 + 0.484252i \(0.160908\pi\)
\(72\) 0 0
\(73\) 2.62772 0.307551 0.153776 0.988106i \(-0.450857\pi\)
0.153776 + 0.988106i \(0.450857\pi\)
\(74\) 0 0
\(75\) 6.37228 0.735808
\(76\) 0 0
\(77\) 2.11684 0.241237
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −18.1168 −1.96505
\(86\) 0 0
\(87\) 8.74456 0.937516
\(88\) 0 0
\(89\) −7.48913 −0.793846 −0.396923 0.917852i \(-0.629922\pi\)
−0.396923 + 0.917852i \(0.629922\pi\)
\(90\) 0 0
\(91\) −13.4891 −1.41404
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) −3.37228 −0.345989
\(96\) 0 0
\(97\) 7.25544 0.736678 0.368339 0.929692i \(-0.379927\pi\)
0.368339 + 0.929692i \(0.379927\pi\)
\(98\) 0 0
\(99\) −0.627719 −0.0630881
\(100\) 0 0
\(101\) 14.7446 1.46714 0.733569 0.679615i \(-0.237853\pi\)
0.733569 + 0.679615i \(0.237853\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 0 0
\(105\) −11.3723 −1.10982
\(106\) 0 0
\(107\) 2.74456 0.265327 0.132663 0.991161i \(-0.457647\pi\)
0.132663 + 0.991161i \(0.457647\pi\)
\(108\) 0 0
\(109\) −6.74456 −0.646012 −0.323006 0.946397i \(-0.604693\pi\)
−0.323006 + 0.946397i \(0.604693\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −4.74456 −0.446331 −0.223165 0.974781i \(-0.571639\pi\)
−0.223165 + 0.974781i \(0.571639\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 18.1168 1.66077
\(120\) 0 0
\(121\) −10.6060 −0.964179
\(122\) 0 0
\(123\) 8.74456 0.788471
\(124\) 0 0
\(125\) 4.62772 0.413916
\(126\) 0 0
\(127\) 18.2337 1.61798 0.808989 0.587824i \(-0.200015\pi\)
0.808989 + 0.587824i \(0.200015\pi\)
\(128\) 0 0
\(129\) 7.37228 0.649093
\(130\) 0 0
\(131\) 2.11684 0.184950 0.0924748 0.995715i \(-0.470522\pi\)
0.0924748 + 0.995715i \(0.470522\pi\)
\(132\) 0 0
\(133\) 3.37228 0.292414
\(134\) 0 0
\(135\) 3.37228 0.290240
\(136\) 0 0
\(137\) −21.3723 −1.82596 −0.912979 0.408007i \(-0.866224\pi\)
−0.912979 + 0.408007i \(0.866224\pi\)
\(138\) 0 0
\(139\) −8.62772 −0.731794 −0.365897 0.930655i \(-0.619238\pi\)
−0.365897 + 0.930655i \(0.619238\pi\)
\(140\) 0 0
\(141\) 8.11684 0.683562
\(142\) 0 0
\(143\) −2.51087 −0.209970
\(144\) 0 0
\(145\) 29.4891 2.44894
\(146\) 0 0
\(147\) 4.37228 0.360620
\(148\) 0 0
\(149\) −12.8614 −1.05365 −0.526824 0.849975i \(-0.676617\pi\)
−0.526824 + 0.849975i \(0.676617\pi\)
\(150\) 0 0
\(151\) −11.4891 −0.934972 −0.467486 0.884001i \(-0.654840\pi\)
−0.467486 + 0.884001i \(0.654840\pi\)
\(152\) 0 0
\(153\) −5.37228 −0.434323
\(154\) 0 0
\(155\) 20.2337 1.62521
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) −2.11684 −0.164796
\(166\) 0 0
\(167\) −25.4891 −1.97241 −0.986204 0.165535i \(-0.947065\pi\)
−0.986204 + 0.165535i \(0.947065\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −21.4891 −1.62443
\(176\) 0 0
\(177\) −2.74456 −0.206294
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 6.74456 0.501319 0.250660 0.968075i \(-0.419352\pi\)
0.250660 + 0.968075i \(0.419352\pi\)
\(182\) 0 0
\(183\) −9.37228 −0.692819
\(184\) 0 0
\(185\) 13.4891 0.991740
\(186\) 0 0
\(187\) 3.37228 0.246606
\(188\) 0 0
\(189\) −3.37228 −0.245297
\(190\) 0 0
\(191\) −21.6060 −1.56335 −0.781677 0.623684i \(-0.785635\pi\)
−0.781677 + 0.623684i \(0.785635\pi\)
\(192\) 0 0
\(193\) 24.9783 1.79797 0.898987 0.437975i \(-0.144304\pi\)
0.898987 + 0.437975i \(0.144304\pi\)
\(194\) 0 0
\(195\) 13.4891 0.965976
\(196\) 0 0
\(197\) −8.23369 −0.586626 −0.293313 0.956016i \(-0.594758\pi\)
−0.293313 + 0.956016i \(0.594758\pi\)
\(198\) 0 0
\(199\) −19.3723 −1.37326 −0.686632 0.727005i \(-0.740912\pi\)
−0.686632 + 0.727005i \(0.740912\pi\)
\(200\) 0 0
\(201\) −6.74456 −0.475725
\(202\) 0 0
\(203\) −29.4891 −2.06973
\(204\) 0 0
\(205\) 29.4891 2.05961
\(206\) 0 0
\(207\) −4.74456 −0.329770
\(208\) 0 0
\(209\) 0.627719 0.0434202
\(210\) 0 0
\(211\) −16.2337 −1.11757 −0.558787 0.829311i \(-0.688733\pi\)
−0.558787 + 0.829311i \(0.688733\pi\)
\(212\) 0 0
\(213\) 14.7446 1.01028
\(214\) 0 0
\(215\) 24.8614 1.69553
\(216\) 0 0
\(217\) −20.2337 −1.37355
\(218\) 0 0
\(219\) 2.62772 0.177565
\(220\) 0 0
\(221\) −21.4891 −1.44551
\(222\) 0 0
\(223\) 20.7446 1.38916 0.694579 0.719416i \(-0.255591\pi\)
0.694579 + 0.719416i \(0.255591\pi\)
\(224\) 0 0
\(225\) 6.37228 0.424819
\(226\) 0 0
\(227\) 1.48913 0.0988367 0.0494184 0.998778i \(-0.484263\pi\)
0.0494184 + 0.998778i \(0.484263\pi\)
\(228\) 0 0
\(229\) −2.62772 −0.173645 −0.0868223 0.996224i \(-0.527671\pi\)
−0.0868223 + 0.996224i \(0.527671\pi\)
\(230\) 0 0
\(231\) 2.11684 0.139278
\(232\) 0 0
\(233\) 13.3723 0.876047 0.438024 0.898963i \(-0.355679\pi\)
0.438024 + 0.898963i \(0.355679\pi\)
\(234\) 0 0
\(235\) 27.3723 1.78557
\(236\) 0 0
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) −20.1168 −1.30125 −0.650625 0.759399i \(-0.725493\pi\)
−0.650625 + 0.759399i \(0.725493\pi\)
\(240\) 0 0
\(241\) −22.2337 −1.43220 −0.716099 0.697999i \(-0.754074\pi\)
−0.716099 + 0.697999i \(0.754074\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 14.7446 0.941996
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.8614 −1.06428 −0.532141 0.846656i \(-0.678612\pi\)
−0.532141 + 0.846656i \(0.678612\pi\)
\(252\) 0 0
\(253\) 2.97825 0.187241
\(254\) 0 0
\(255\) −18.1168 −1.13452
\(256\) 0 0
\(257\) 15.2554 0.951608 0.475804 0.879551i \(-0.342157\pi\)
0.475804 + 0.879551i \(0.342157\pi\)
\(258\) 0 0
\(259\) −13.4891 −0.838173
\(260\) 0 0
\(261\) 8.74456 0.541275
\(262\) 0 0
\(263\) −13.3723 −0.824570 −0.412285 0.911055i \(-0.635269\pi\)
−0.412285 + 0.911055i \(0.635269\pi\)
\(264\) 0 0
\(265\) 33.7228 2.07158
\(266\) 0 0
\(267\) −7.48913 −0.458327
\(268\) 0 0
\(269\) −20.9783 −1.27907 −0.639533 0.768763i \(-0.720872\pi\)
−0.639533 + 0.768763i \(0.720872\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −13.4891 −0.816399
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −32.1168 −1.92971 −0.964857 0.262775i \(-0.915362\pi\)
−0.964857 + 0.262775i \(0.915362\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −8.51087 −0.507716 −0.253858 0.967241i \(-0.581700\pi\)
−0.253858 + 0.967241i \(0.581700\pi\)
\(282\) 0 0
\(283\) 4.86141 0.288981 0.144490 0.989506i \(-0.453846\pi\)
0.144490 + 0.989506i \(0.453846\pi\)
\(284\) 0 0
\(285\) −3.37228 −0.199757
\(286\) 0 0
\(287\) −29.4891 −1.74069
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) 7.25544 0.425321
\(292\) 0 0
\(293\) 28.7446 1.67928 0.839638 0.543147i \(-0.182767\pi\)
0.839638 + 0.543147i \(0.182767\pi\)
\(294\) 0 0
\(295\) −9.25544 −0.538872
\(296\) 0 0
\(297\) −0.627719 −0.0364239
\(298\) 0 0
\(299\) −18.9783 −1.09754
\(300\) 0 0
\(301\) −24.8614 −1.43299
\(302\) 0 0
\(303\) 14.7446 0.847053
\(304\) 0 0
\(305\) −31.6060 −1.80975
\(306\) 0 0
\(307\) −20.2337 −1.15480 −0.577399 0.816462i \(-0.695932\pi\)
−0.577399 + 0.816462i \(0.695932\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) −2.86141 −0.162255 −0.0811277 0.996704i \(-0.525852\pi\)
−0.0811277 + 0.996704i \(0.525852\pi\)
\(312\) 0 0
\(313\) −20.9783 −1.18576 −0.592880 0.805291i \(-0.702009\pi\)
−0.592880 + 0.805291i \(0.702009\pi\)
\(314\) 0 0
\(315\) −11.3723 −0.640755
\(316\) 0 0
\(317\) 0.510875 0.0286936 0.0143468 0.999897i \(-0.495433\pi\)
0.0143468 + 0.999897i \(0.495433\pi\)
\(318\) 0 0
\(319\) −5.48913 −0.307332
\(320\) 0 0
\(321\) 2.74456 0.153187
\(322\) 0 0
\(323\) 5.37228 0.298922
\(324\) 0 0
\(325\) 25.4891 1.41388
\(326\) 0 0
\(327\) −6.74456 −0.372975
\(328\) 0 0
\(329\) −27.3723 −1.50908
\(330\) 0 0
\(331\) 28.2337 1.55186 0.775932 0.630817i \(-0.217280\pi\)
0.775932 + 0.630817i \(0.217280\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) −22.7446 −1.24267
\(336\) 0 0
\(337\) 22.2337 1.21115 0.605573 0.795790i \(-0.292944\pi\)
0.605573 + 0.795790i \(0.292944\pi\)
\(338\) 0 0
\(339\) −4.74456 −0.257689
\(340\) 0 0
\(341\) −3.76631 −0.203957
\(342\) 0 0
\(343\) 8.86141 0.478471
\(344\) 0 0
\(345\) −16.0000 −0.861411
\(346\) 0 0
\(347\) −1.88316 −0.101093 −0.0505466 0.998722i \(-0.516096\pi\)
−0.0505466 + 0.998722i \(0.516096\pi\)
\(348\) 0 0
\(349\) 1.37228 0.0734565 0.0367283 0.999325i \(-0.488306\pi\)
0.0367283 + 0.999325i \(0.488306\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 20.9783 1.11656 0.558280 0.829653i \(-0.311462\pi\)
0.558280 + 0.829653i \(0.311462\pi\)
\(354\) 0 0
\(355\) 49.7228 2.63901
\(356\) 0 0
\(357\) 18.1168 0.958845
\(358\) 0 0
\(359\) 16.3505 0.862948 0.431474 0.902125i \(-0.357994\pi\)
0.431474 + 0.902125i \(0.357994\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −10.6060 −0.556669
\(364\) 0 0
\(365\) 8.86141 0.463827
\(366\) 0 0
\(367\) 32.4674 1.69478 0.847392 0.530968i \(-0.178172\pi\)
0.847392 + 0.530968i \(0.178172\pi\)
\(368\) 0 0
\(369\) 8.74456 0.455224
\(370\) 0 0
\(371\) −33.7228 −1.75080
\(372\) 0 0
\(373\) −9.48913 −0.491328 −0.245664 0.969355i \(-0.579006\pi\)
−0.245664 + 0.969355i \(0.579006\pi\)
\(374\) 0 0
\(375\) 4.62772 0.238974
\(376\) 0 0
\(377\) 34.9783 1.80147
\(378\) 0 0
\(379\) 30.7446 1.57924 0.789621 0.613595i \(-0.210277\pi\)
0.789621 + 0.613595i \(0.210277\pi\)
\(380\) 0 0
\(381\) 18.2337 0.934140
\(382\) 0 0
\(383\) −17.2554 −0.881712 −0.440856 0.897578i \(-0.645325\pi\)
−0.440856 + 0.897578i \(0.645325\pi\)
\(384\) 0 0
\(385\) 7.13859 0.363816
\(386\) 0 0
\(387\) 7.37228 0.374754
\(388\) 0 0
\(389\) 13.8832 0.703904 0.351952 0.936018i \(-0.385518\pi\)
0.351952 + 0.936018i \(0.385518\pi\)
\(390\) 0 0
\(391\) 25.4891 1.28904
\(392\) 0 0
\(393\) 2.11684 0.106781
\(394\) 0 0
\(395\) 6.74456 0.339356
\(396\) 0 0
\(397\) 36.3505 1.82438 0.912190 0.409766i \(-0.134390\pi\)
0.912190 + 0.409766i \(0.134390\pi\)
\(398\) 0 0
\(399\) 3.37228 0.168825
\(400\) 0 0
\(401\) −3.48913 −0.174239 −0.0871193 0.996198i \(-0.527766\pi\)
−0.0871193 + 0.996198i \(0.527766\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) 0 0
\(405\) 3.37228 0.167570
\(406\) 0 0
\(407\) −2.51087 −0.124459
\(408\) 0 0
\(409\) 32.9783 1.63067 0.815335 0.578990i \(-0.196553\pi\)
0.815335 + 0.578990i \(0.196553\pi\)
\(410\) 0 0
\(411\) −21.3723 −1.05422
\(412\) 0 0
\(413\) 9.25544 0.455430
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.62772 −0.422501
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 18.7446 0.913554 0.456777 0.889581i \(-0.349004\pi\)
0.456777 + 0.889581i \(0.349004\pi\)
\(422\) 0 0
\(423\) 8.11684 0.394654
\(424\) 0 0
\(425\) −34.2337 −1.66058
\(426\) 0 0
\(427\) 31.6060 1.52952
\(428\) 0 0
\(429\) −2.51087 −0.121226
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 0 0
\(433\) 1.76631 0.0848835 0.0424418 0.999099i \(-0.486486\pi\)
0.0424418 + 0.999099i \(0.486486\pi\)
\(434\) 0 0
\(435\) 29.4891 1.41390
\(436\) 0 0
\(437\) 4.74456 0.226963
\(438\) 0 0
\(439\) −6.23369 −0.297518 −0.148759 0.988874i \(-0.547528\pi\)
−0.148759 + 0.988874i \(0.547528\pi\)
\(440\) 0 0
\(441\) 4.37228 0.208204
\(442\) 0 0
\(443\) 30.1168 1.43089 0.715447 0.698667i \(-0.246223\pi\)
0.715447 + 0.698667i \(0.246223\pi\)
\(444\) 0 0
\(445\) −25.2554 −1.19722
\(446\) 0 0
\(447\) −12.8614 −0.608324
\(448\) 0 0
\(449\) 7.25544 0.342405 0.171203 0.985236i \(-0.445235\pi\)
0.171203 + 0.985236i \(0.445235\pi\)
\(450\) 0 0
\(451\) −5.48913 −0.258473
\(452\) 0 0
\(453\) −11.4891 −0.539806
\(454\) 0 0
\(455\) −45.4891 −2.13256
\(456\) 0 0
\(457\) −32.1168 −1.50236 −0.751181 0.660096i \(-0.770516\pi\)
−0.751181 + 0.660096i \(0.770516\pi\)
\(458\) 0 0
\(459\) −5.37228 −0.250757
\(460\) 0 0
\(461\) −17.8832 −0.832902 −0.416451 0.909158i \(-0.636726\pi\)
−0.416451 + 0.909158i \(0.636726\pi\)
\(462\) 0 0
\(463\) −20.6277 −0.958651 −0.479326 0.877637i \(-0.659119\pi\)
−0.479326 + 0.877637i \(0.659119\pi\)
\(464\) 0 0
\(465\) 20.2337 0.938315
\(466\) 0 0
\(467\) −19.3723 −0.896442 −0.448221 0.893923i \(-0.647942\pi\)
−0.448221 + 0.893923i \(0.647942\pi\)
\(468\) 0 0
\(469\) 22.7446 1.05025
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) −4.62772 −0.212783
\(474\) 0 0
\(475\) −6.37228 −0.292380
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) −1.76631 −0.0807049 −0.0403524 0.999186i \(-0.512848\pi\)
−0.0403524 + 0.999186i \(0.512848\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) 24.4674 1.11101
\(486\) 0 0
\(487\) −30.4674 −1.38061 −0.690304 0.723519i \(-0.742523\pi\)
−0.690304 + 0.723519i \(0.742523\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −5.48913 −0.247721 −0.123860 0.992300i \(-0.539527\pi\)
−0.123860 + 0.992300i \(0.539527\pi\)
\(492\) 0 0
\(493\) −46.9783 −2.11579
\(494\) 0 0
\(495\) −2.11684 −0.0951451
\(496\) 0 0
\(497\) −49.7228 −2.23037
\(498\) 0 0
\(499\) 11.6060 0.519555 0.259777 0.965669i \(-0.416351\pi\)
0.259777 + 0.965669i \(0.416351\pi\)
\(500\) 0 0
\(501\) −25.4891 −1.13877
\(502\) 0 0
\(503\) 31.7228 1.41445 0.707225 0.706988i \(-0.249947\pi\)
0.707225 + 0.706988i \(0.249947\pi\)
\(504\) 0 0
\(505\) 49.7228 2.21264
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −8.74456 −0.387596 −0.193798 0.981041i \(-0.562081\pi\)
−0.193798 + 0.981041i \(0.562081\pi\)
\(510\) 0 0
\(511\) −8.86141 −0.392006
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 33.7228 1.48600
\(516\) 0 0
\(517\) −5.09509 −0.224082
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 17.7228 0.774965 0.387482 0.921877i \(-0.373345\pi\)
0.387482 + 0.921877i \(0.373345\pi\)
\(524\) 0 0
\(525\) −21.4891 −0.937862
\(526\) 0 0
\(527\) −32.2337 −1.40412
\(528\) 0 0
\(529\) −0.489125 −0.0212663
\(530\) 0 0
\(531\) −2.74456 −0.119104
\(532\) 0 0
\(533\) 34.9783 1.51508
\(534\) 0 0
\(535\) 9.25544 0.400147
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) −2.74456 −0.118217
\(540\) 0 0
\(541\) −32.3505 −1.39086 −0.695429 0.718595i \(-0.744786\pi\)
−0.695429 + 0.718595i \(0.744786\pi\)
\(542\) 0 0
\(543\) 6.74456 0.289437
\(544\) 0 0
\(545\) −22.7446 −0.974270
\(546\) 0 0
\(547\) −2.51087 −0.107357 −0.0536786 0.998558i \(-0.517095\pi\)
−0.0536786 + 0.998558i \(0.517095\pi\)
\(548\) 0 0
\(549\) −9.37228 −0.399999
\(550\) 0 0
\(551\) −8.74456 −0.372531
\(552\) 0 0
\(553\) −6.74456 −0.286808
\(554\) 0 0
\(555\) 13.4891 0.572581
\(556\) 0 0
\(557\) −44.8614 −1.90084 −0.950419 0.310971i \(-0.899346\pi\)
−0.950419 + 0.310971i \(0.899346\pi\)
\(558\) 0 0
\(559\) 29.4891 1.24726
\(560\) 0 0
\(561\) 3.37228 0.142378
\(562\) 0 0
\(563\) −33.4891 −1.41140 −0.705699 0.708512i \(-0.749367\pi\)
−0.705699 + 0.708512i \(0.749367\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) 0 0
\(567\) −3.37228 −0.141623
\(568\) 0 0
\(569\) −19.2554 −0.807230 −0.403615 0.914929i \(-0.632246\pi\)
−0.403615 + 0.914929i \(0.632246\pi\)
\(570\) 0 0
\(571\) −17.4891 −0.731897 −0.365949 0.930635i \(-0.619255\pi\)
−0.365949 + 0.930635i \(0.619255\pi\)
\(572\) 0 0
\(573\) −21.6060 −0.902602
\(574\) 0 0
\(575\) −30.2337 −1.26083
\(576\) 0 0
\(577\) −5.13859 −0.213922 −0.106961 0.994263i \(-0.534112\pi\)
−0.106961 + 0.994263i \(0.534112\pi\)
\(578\) 0 0
\(579\) 24.9783 1.03806
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.27719 −0.259975
\(584\) 0 0
\(585\) 13.4891 0.557707
\(586\) 0 0
\(587\) −42.3505 −1.74799 −0.873997 0.485932i \(-0.838480\pi\)
−0.873997 + 0.485932i \(0.838480\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) −8.23369 −0.338689
\(592\) 0 0
\(593\) −11.4891 −0.471802 −0.235901 0.971777i \(-0.575804\pi\)
−0.235901 + 0.971777i \(0.575804\pi\)
\(594\) 0 0
\(595\) 61.0951 2.50465
\(596\) 0 0
\(597\) −19.3723 −0.792855
\(598\) 0 0
\(599\) −8.23369 −0.336419 −0.168210 0.985751i \(-0.553799\pi\)
−0.168210 + 0.985751i \(0.553799\pi\)
\(600\) 0 0
\(601\) −40.9783 −1.67154 −0.835769 0.549081i \(-0.814978\pi\)
−0.835769 + 0.549081i \(0.814978\pi\)
\(602\) 0 0
\(603\) −6.74456 −0.274660
\(604\) 0 0
\(605\) −35.7663 −1.45411
\(606\) 0 0
\(607\) 41.2119 1.67274 0.836370 0.548165i \(-0.184673\pi\)
0.836370 + 0.548165i \(0.184673\pi\)
\(608\) 0 0
\(609\) −29.4891 −1.19496
\(610\) 0 0
\(611\) 32.4674 1.31349
\(612\) 0 0
\(613\) −22.6277 −0.913925 −0.456962 0.889486i \(-0.651063\pi\)
−0.456962 + 0.889486i \(0.651063\pi\)
\(614\) 0 0
\(615\) 29.4891 1.18912
\(616\) 0 0
\(617\) 12.1168 0.487806 0.243903 0.969800i \(-0.421572\pi\)
0.243903 + 0.969800i \(0.421572\pi\)
\(618\) 0 0
\(619\) −14.9783 −0.602027 −0.301013 0.953620i \(-0.597325\pi\)
−0.301013 + 0.953620i \(0.597325\pi\)
\(620\) 0 0
\(621\) −4.74456 −0.190393
\(622\) 0 0
\(623\) 25.2554 1.01184
\(624\) 0 0
\(625\) −16.2554 −0.650217
\(626\) 0 0
\(627\) 0.627719 0.0250687
\(628\) 0 0
\(629\) −21.4891 −0.856828
\(630\) 0 0
\(631\) 14.1168 0.561983 0.280991 0.959710i \(-0.409337\pi\)
0.280991 + 0.959710i \(0.409337\pi\)
\(632\) 0 0
\(633\) −16.2337 −0.645231
\(634\) 0 0
\(635\) 61.4891 2.44012
\(636\) 0 0
\(637\) 17.4891 0.692944
\(638\) 0 0
\(639\) 14.7446 0.583286
\(640\) 0 0
\(641\) 2.23369 0.0882254 0.0441127 0.999027i \(-0.485954\pi\)
0.0441127 + 0.999027i \(0.485954\pi\)
\(642\) 0 0
\(643\) 4.39403 0.173284 0.0866418 0.996240i \(-0.472386\pi\)
0.0866418 + 0.996240i \(0.472386\pi\)
\(644\) 0 0
\(645\) 24.8614 0.978917
\(646\) 0 0
\(647\) 40.3505 1.58634 0.793172 0.608998i \(-0.208428\pi\)
0.793172 + 0.608998i \(0.208428\pi\)
\(648\) 0 0
\(649\) 1.72281 0.0676263
\(650\) 0 0
\(651\) −20.2337 −0.793021
\(652\) 0 0
\(653\) 11.1386 0.435887 0.217943 0.975961i \(-0.430065\pi\)
0.217943 + 0.975961i \(0.430065\pi\)
\(654\) 0 0
\(655\) 7.13859 0.278928
\(656\) 0 0
\(657\) 2.62772 0.102517
\(658\) 0 0
\(659\) 8.23369 0.320739 0.160369 0.987057i \(-0.448731\pi\)
0.160369 + 0.987057i \(0.448731\pi\)
\(660\) 0 0
\(661\) 6.51087 0.253244 0.126622 0.991951i \(-0.459587\pi\)
0.126622 + 0.991951i \(0.459587\pi\)
\(662\) 0 0
\(663\) −21.4891 −0.834568
\(664\) 0 0
\(665\) 11.3723 0.440998
\(666\) 0 0
\(667\) −41.4891 −1.60647
\(668\) 0 0
\(669\) 20.7446 0.802031
\(670\) 0 0
\(671\) 5.88316 0.227117
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 0 0
\(675\) 6.37228 0.245269
\(676\) 0 0
\(677\) −39.7228 −1.52667 −0.763336 0.646002i \(-0.776440\pi\)
−0.763336 + 0.646002i \(0.776440\pi\)
\(678\) 0 0
\(679\) −24.4674 −0.938972
\(680\) 0 0
\(681\) 1.48913 0.0570634
\(682\) 0 0
\(683\) 17.4891 0.669203 0.334601 0.942360i \(-0.391398\pi\)
0.334601 + 0.942360i \(0.391398\pi\)
\(684\) 0 0
\(685\) −72.0733 −2.75378
\(686\) 0 0
\(687\) −2.62772 −0.100254
\(688\) 0 0
\(689\) 40.0000 1.52388
\(690\) 0 0
\(691\) 8.62772 0.328214 0.164107 0.986443i \(-0.447526\pi\)
0.164107 + 0.986443i \(0.447526\pi\)
\(692\) 0 0
\(693\) 2.11684 0.0804123
\(694\) 0 0
\(695\) −29.0951 −1.10364
\(696\) 0 0
\(697\) −46.9783 −1.77943
\(698\) 0 0
\(699\) 13.3723 0.505786
\(700\) 0 0
\(701\) −1.25544 −0.0474172 −0.0237086 0.999719i \(-0.507547\pi\)
−0.0237086 + 0.999719i \(0.507547\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 27.3723 1.03090
\(706\) 0 0
\(707\) −49.7228 −1.87002
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) −28.4674 −1.06611
\(714\) 0 0
\(715\) −8.46738 −0.316662
\(716\) 0 0
\(717\) −20.1168 −0.751277
\(718\) 0 0
\(719\) −22.6277 −0.843872 −0.421936 0.906626i \(-0.638649\pi\)
−0.421936 + 0.906626i \(0.638649\pi\)
\(720\) 0 0
\(721\) −33.7228 −1.25590
\(722\) 0 0
\(723\) −22.2337 −0.826880
\(724\) 0 0
\(725\) 55.7228 2.06949
\(726\) 0 0
\(727\) 39.8397 1.47757 0.738786 0.673941i \(-0.235400\pi\)
0.738786 + 0.673941i \(0.235400\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −39.6060 −1.46488
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 14.7446 0.543861
\(736\) 0 0
\(737\) 4.23369 0.155950
\(738\) 0 0
\(739\) −30.1168 −1.10787 −0.553933 0.832561i \(-0.686874\pi\)
−0.553933 + 0.832561i \(0.686874\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −13.7228 −0.503441 −0.251721 0.967800i \(-0.580996\pi\)
−0.251721 + 0.967800i \(0.580996\pi\)
\(744\) 0 0
\(745\) −43.3723 −1.58904
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.25544 −0.338186
\(750\) 0 0
\(751\) 8.74456 0.319094 0.159547 0.987190i \(-0.448997\pi\)
0.159547 + 0.987190i \(0.448997\pi\)
\(752\) 0 0
\(753\) −16.8614 −0.614464
\(754\) 0 0
\(755\) −38.7446 −1.41006
\(756\) 0 0
\(757\) 35.0951 1.27555 0.637776 0.770222i \(-0.279854\pi\)
0.637776 + 0.770222i \(0.279854\pi\)
\(758\) 0 0
\(759\) 2.97825 0.108104
\(760\) 0 0
\(761\) −13.3723 −0.484745 −0.242372 0.970183i \(-0.577926\pi\)
−0.242372 + 0.970183i \(0.577926\pi\)
\(762\) 0 0
\(763\) 22.7446 0.823408
\(764\) 0 0
\(765\) −18.1168 −0.655016
\(766\) 0 0
\(767\) −10.9783 −0.396402
\(768\) 0 0
\(769\) −31.0951 −1.12132 −0.560659 0.828047i \(-0.689452\pi\)
−0.560659 + 0.828047i \(0.689452\pi\)
\(770\) 0 0
\(771\) 15.2554 0.549411
\(772\) 0 0
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) 0 0
\(775\) 38.2337 1.37339
\(776\) 0 0
\(777\) −13.4891 −0.483920
\(778\) 0 0
\(779\) −8.74456 −0.313306
\(780\) 0 0
\(781\) −9.25544 −0.331186
\(782\) 0 0
\(783\) 8.74456 0.312505
\(784\) 0 0
\(785\) −6.74456 −0.240724
\(786\) 0 0
\(787\) −41.9565 −1.49559 −0.747794 0.663931i \(-0.768887\pi\)
−0.747794 + 0.663931i \(0.768887\pi\)
\(788\) 0 0
\(789\) −13.3723 −0.476066
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) 0 0
\(793\) −37.4891 −1.33128
\(794\) 0 0
\(795\) 33.7228 1.19602
\(796\) 0 0
\(797\) 11.4891 0.406966 0.203483 0.979079i \(-0.434774\pi\)
0.203483 + 0.979079i \(0.434774\pi\)
\(798\) 0 0
\(799\) −43.6060 −1.54267
\(800\) 0 0
\(801\) −7.48913 −0.264615
\(802\) 0 0
\(803\) −1.64947 −0.0582085
\(804\) 0 0
\(805\) 53.9565 1.90172
\(806\) 0 0
\(807\) −20.9783 −0.738469
\(808\) 0 0
\(809\) 27.8832 0.980320 0.490160 0.871633i \(-0.336938\pi\)
0.490160 + 0.871633i \(0.336938\pi\)
\(810\) 0 0
\(811\) −32.0000 −1.12367 −0.561836 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 40.4674 1.41751
\(816\) 0 0
\(817\) −7.37228 −0.257923
\(818\) 0 0
\(819\) −13.4891 −0.471348
\(820\) 0 0
\(821\) −38.3505 −1.33844 −0.669221 0.743063i \(-0.733372\pi\)
−0.669221 + 0.743063i \(0.733372\pi\)
\(822\) 0 0
\(823\) 30.1168 1.04981 0.524904 0.851162i \(-0.324101\pi\)
0.524904 + 0.851162i \(0.324101\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −26.7446 −0.930000 −0.465000 0.885311i \(-0.653946\pi\)
−0.465000 + 0.885311i \(0.653946\pi\)
\(828\) 0 0
\(829\) −18.5109 −0.642909 −0.321455 0.946925i \(-0.604172\pi\)
−0.321455 + 0.946925i \(0.604172\pi\)
\(830\) 0 0
\(831\) −32.1168 −1.11412
\(832\) 0 0
\(833\) −23.4891 −0.813850
\(834\) 0 0
\(835\) −85.9565 −2.97465
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 0 0
\(839\) 2.74456 0.0947528 0.0473764 0.998877i \(-0.484914\pi\)
0.0473764 + 0.998877i \(0.484914\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) 0 0
\(843\) −8.51087 −0.293130
\(844\) 0 0
\(845\) 10.1168 0.348030
\(846\) 0 0
\(847\) 35.7663 1.22895
\(848\) 0 0
\(849\) 4.86141 0.166843
\(850\) 0 0
\(851\) −18.9783 −0.650566
\(852\) 0 0
\(853\) 15.4891 0.530338 0.265169 0.964202i \(-0.414572\pi\)
0.265169 + 0.964202i \(0.414572\pi\)
\(854\) 0 0
\(855\) −3.37228 −0.115330
\(856\) 0 0
\(857\) 0.978251 0.0334164 0.0167082 0.999860i \(-0.494681\pi\)
0.0167082 + 0.999860i \(0.494681\pi\)
\(858\) 0 0
\(859\) 34.3505 1.17203 0.586013 0.810302i \(-0.300697\pi\)
0.586013 + 0.810302i \(0.300697\pi\)
\(860\) 0 0
\(861\) −29.4891 −1.00499
\(862\) 0 0
\(863\) 52.4674 1.78601 0.893005 0.450046i \(-0.148593\pi\)
0.893005 + 0.450046i \(0.148593\pi\)
\(864\) 0 0
\(865\) 20.2337 0.687966
\(866\) 0 0
\(867\) 11.8614 0.402834
\(868\) 0 0
\(869\) −1.25544 −0.0425878
\(870\) 0 0
\(871\) −26.9783 −0.914123
\(872\) 0 0
\(873\) 7.25544 0.245559
\(874\) 0 0
\(875\) −15.6060 −0.527578
\(876\) 0 0
\(877\) 15.7663 0.532391 0.266195 0.963919i \(-0.414233\pi\)
0.266195 + 0.963919i \(0.414233\pi\)
\(878\) 0 0
\(879\) 28.7446 0.969530
\(880\) 0 0
\(881\) −25.3723 −0.854814 −0.427407 0.904059i \(-0.640573\pi\)
−0.427407 + 0.904059i \(0.640573\pi\)
\(882\) 0 0
\(883\) 52.8614 1.77893 0.889464 0.457005i \(-0.151078\pi\)
0.889464 + 0.457005i \(0.151078\pi\)
\(884\) 0 0
\(885\) −9.25544 −0.311118
\(886\) 0 0
\(887\) −37.7228 −1.26661 −0.633304 0.773903i \(-0.718302\pi\)
−0.633304 + 0.773903i \(0.718302\pi\)
\(888\) 0 0
\(889\) −61.4891 −2.06228
\(890\) 0 0
\(891\) −0.627719 −0.0210294
\(892\) 0 0
\(893\) −8.11684 −0.271620
\(894\) 0 0
\(895\) 13.4891 0.450892
\(896\) 0 0
\(897\) −18.9783 −0.633665
\(898\) 0 0
\(899\) 52.4674 1.74988
\(900\) 0 0
\(901\) −53.7228 −1.78977
\(902\) 0 0
\(903\) −24.8614 −0.827336
\(904\) 0 0
\(905\) 22.7446 0.756055
\(906\) 0 0
\(907\) −25.2554 −0.838593 −0.419297 0.907849i \(-0.637723\pi\)
−0.419297 + 0.907849i \(0.637723\pi\)
\(908\) 0 0
\(909\) 14.7446 0.489046
\(910\) 0 0
\(911\) 5.25544 0.174120 0.0870602 0.996203i \(-0.472253\pi\)
0.0870602 + 0.996203i \(0.472253\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −31.6060 −1.04486
\(916\) 0 0
\(917\) −7.13859 −0.235737
\(918\) 0 0
\(919\) 25.4891 0.840809 0.420404 0.907337i \(-0.361888\pi\)
0.420404 + 0.907337i \(0.361888\pi\)
\(920\) 0 0
\(921\) −20.2337 −0.666723
\(922\) 0 0
\(923\) 58.9783 1.94129
\(924\) 0 0
\(925\) 25.4891 0.838077
\(926\) 0 0
\(927\) 10.0000 0.328443
\(928\) 0 0
\(929\) 16.9783 0.557038 0.278519 0.960431i \(-0.410156\pi\)
0.278519 + 0.960431i \(0.410156\pi\)
\(930\) 0 0
\(931\) −4.37228 −0.143296
\(932\) 0 0
\(933\) −2.86141 −0.0936782
\(934\) 0 0
\(935\) 11.3723 0.371913
\(936\) 0 0
\(937\) 48.5842 1.58718 0.793589 0.608455i \(-0.208210\pi\)
0.793589 + 0.608455i \(0.208210\pi\)
\(938\) 0 0
\(939\) −20.9783 −0.684599
\(940\) 0 0
\(941\) −28.9783 −0.944664 −0.472332 0.881421i \(-0.656588\pi\)
−0.472332 + 0.881421i \(0.656588\pi\)
\(942\) 0 0
\(943\) −41.4891 −1.35107
\(944\) 0 0
\(945\) −11.3723 −0.369940
\(946\) 0 0
\(947\) −37.4891 −1.21823 −0.609116 0.793081i \(-0.708476\pi\)
−0.609116 + 0.793081i \(0.708476\pi\)
\(948\) 0 0
\(949\) 10.5109 0.341197
\(950\) 0 0
\(951\) 0.510875 0.0165662
\(952\) 0 0
\(953\) 23.4891 0.760887 0.380444 0.924804i \(-0.375771\pi\)
0.380444 + 0.924804i \(0.375771\pi\)
\(954\) 0 0
\(955\) −72.8614 −2.35774
\(956\) 0 0
\(957\) −5.48913 −0.177438
\(958\) 0 0
\(959\) 72.0733 2.32737
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 2.74456 0.0884423
\(964\) 0 0
\(965\) 84.2337 2.71158
\(966\) 0 0
\(967\) 50.9783 1.63935 0.819675 0.572829i \(-0.194154\pi\)
0.819675 + 0.572829i \(0.194154\pi\)
\(968\) 0 0
\(969\) 5.37228 0.172583
\(970\) 0 0
\(971\) 22.9783 0.737407 0.368704 0.929547i \(-0.379802\pi\)
0.368704 + 0.929547i \(0.379802\pi\)
\(972\) 0 0
\(973\) 29.0951 0.932746
\(974\) 0 0
\(975\) 25.4891 0.816305
\(976\) 0 0
\(977\) 20.9783 0.671154 0.335577 0.942013i \(-0.391069\pi\)
0.335577 + 0.942013i \(0.391069\pi\)
\(978\) 0 0
\(979\) 4.70106 0.150247
\(980\) 0 0
\(981\) −6.74456 −0.215337
\(982\) 0 0
\(983\) −2.51087 −0.0800845 −0.0400422 0.999198i \(-0.512749\pi\)
−0.0400422 + 0.999198i \(0.512749\pi\)
\(984\) 0 0
\(985\) −27.7663 −0.884708
\(986\) 0 0
\(987\) −27.3723 −0.871269
\(988\) 0 0
\(989\) −34.9783 −1.11224
\(990\) 0 0
\(991\) 18.2337 0.579212 0.289606 0.957146i \(-0.406476\pi\)
0.289606 + 0.957146i \(0.406476\pi\)
\(992\) 0 0
\(993\) 28.2337 0.895969
\(994\) 0 0
\(995\) −65.3288 −2.07106
\(996\) 0 0
\(997\) −12.1168 −0.383744 −0.191872 0.981420i \(-0.561456\pi\)
−0.191872 + 0.981420i \(0.561456\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 3648.2.a.bt.1.2 2
4.3 odd 2 3648.2.a.bm.1.2 2
8.3 odd 2 1824.2.a.r.1.1 yes 2
8.5 even 2 1824.2.a.o.1.1 2
24.5 odd 2 5472.2.a.bd.1.2 2
24.11 even 2 5472.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1824.2.a.o.1.1 2 8.5 even 2
1824.2.a.r.1.1 yes 2 8.3 odd 2
3648.2.a.bm.1.2 2 4.3 odd 2
3648.2.a.bt.1.2 2 1.1 even 1 trivial
5472.2.a.bd.1.2 2 24.5 odd 2
5472.2.a.be.1.2 2 24.11 even 2