Properties

Label 585.2.b.f.181.3
Level $585$
Weight $2$
Character 585.181
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.3
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 585.181
Dual form 585.2.b.f.181.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -2.00000 q^{4} -1.00000i q^{5} -3.60555i q^{7} +O(q^{10})\) \(q+2.00000i q^{2} -2.00000 q^{4} -1.00000i q^{5} -3.60555i q^{7} +2.00000 q^{10} -3.00000i q^{11} -3.60555i q^{13} +7.21110 q^{14} -4.00000 q^{16} -3.60555 q^{17} -7.21110i q^{19} +2.00000i q^{20} +6.00000 q^{22} +3.60555 q^{23} -1.00000 q^{25} +7.21110 q^{26} +7.21110i q^{28} +7.21110 q^{29} +7.21110i q^{31} -8.00000i q^{32} -7.21110i q^{34} -3.60555 q^{35} -3.60555i q^{37} +14.4222 q^{38} +11.0000i q^{41} -4.00000 q^{43} +6.00000i q^{44} +7.21110i q^{46} +4.00000i q^{47} -6.00000 q^{49} -2.00000i q^{50} +7.21110i q^{52} -10.8167 q^{53} -3.00000 q^{55} +14.4222i q^{58} -12.0000i q^{59} +13.0000 q^{61} -14.4222 q^{62} +8.00000 q^{64} -3.60555 q^{65} +7.21110 q^{68} -7.21110i q^{70} -5.00000i q^{71} +7.21110i q^{73} +7.21110 q^{74} +14.4222i q^{76} -10.8167 q^{77} +13.0000 q^{79} +4.00000i q^{80} -22.0000 q^{82} +6.00000i q^{83} +3.60555i q^{85} -8.00000i q^{86} +3.00000i q^{89} -13.0000 q^{91} -7.21110 q^{92} -8.00000 q^{94} -7.21110 q^{95} -3.60555i q^{97} -12.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 8 q^{10} - 16 q^{16} + 24 q^{22} - 4 q^{25} - 16 q^{43} - 24 q^{49} - 12 q^{55} + 52 q^{61} + 32 q^{64} + 52 q^{79} - 88 q^{82} - 52 q^{91} - 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) − 3.60555i − 1.36277i −0.731925 0.681385i \(-0.761378\pi\)
0.731925 0.681385i \(-0.238622\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) − 3.00000i − 0.904534i −0.891883 0.452267i \(-0.850615\pi\)
0.891883 0.452267i \(-0.149385\pi\)
\(12\) 0 0
\(13\) − 3.60555i − 1.00000i
\(14\) 7.21110 1.92725
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −3.60555 −0.874475 −0.437237 0.899346i \(-0.644043\pi\)
−0.437237 + 0.899346i \(0.644043\pi\)
\(18\) 0 0
\(19\) − 7.21110i − 1.65434i −0.561951 0.827170i \(-0.689949\pi\)
0.561951 0.827170i \(-0.310051\pi\)
\(20\) 2.00000i 0.447214i
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 3.60555 0.751809 0.375905 0.926658i \(-0.377332\pi\)
0.375905 + 0.926658i \(0.377332\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 7.21110 1.41421
\(27\) 0 0
\(28\) 7.21110i 1.36277i
\(29\) 7.21110 1.33907 0.669534 0.742781i \(-0.266494\pi\)
0.669534 + 0.742781i \(0.266494\pi\)
\(30\) 0 0
\(31\) 7.21110i 1.29515i 0.762001 + 0.647576i \(0.224217\pi\)
−0.762001 + 0.647576i \(0.775783\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) 0 0
\(34\) − 7.21110i − 1.23669i
\(35\) −3.60555 −0.609449
\(36\) 0 0
\(37\) − 3.60555i − 0.592749i −0.955072 0.296374i \(-0.904222\pi\)
0.955072 0.296374i \(-0.0957776\pi\)
\(38\) 14.4222 2.33959
\(39\) 0 0
\(40\) 0 0
\(41\) 11.0000i 1.71791i 0.512050 + 0.858956i \(0.328886\pi\)
−0.512050 + 0.858956i \(0.671114\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 6.00000i 0.904534i
\(45\) 0 0
\(46\) 7.21110i 1.06322i
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) − 2.00000i − 0.282843i
\(51\) 0 0
\(52\) 7.21110i 1.00000i
\(53\) −10.8167 −1.48578 −0.742891 0.669413i \(-0.766546\pi\)
−0.742891 + 0.669413i \(0.766546\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 14.4222i 1.89373i
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) −14.4222 −1.83162
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) −3.60555 −0.447214
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 7.21110 0.874475
\(69\) 0 0
\(70\) − 7.21110i − 0.861892i
\(71\) − 5.00000i − 0.593391i −0.954972 0.296695i \(-0.904115\pi\)
0.954972 0.296695i \(-0.0958846\pi\)
\(72\) 0 0
\(73\) 7.21110i 0.843996i 0.906597 + 0.421998i \(0.138671\pi\)
−0.906597 + 0.421998i \(0.861329\pi\)
\(74\) 7.21110 0.838274
\(75\) 0 0
\(76\) 14.4222i 1.65434i
\(77\) −10.8167 −1.23267
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 4.00000i 0.447214i
\(81\) 0 0
\(82\) −22.0000 −2.42949
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 3.60555i 0.391077i
\(86\) − 8.00000i − 0.862662i
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000i 0.317999i 0.987279 + 0.159000i \(0.0508269\pi\)
−0.987279 + 0.159000i \(0.949173\pi\)
\(90\) 0 0
\(91\) −13.0000 −1.36277
\(92\) −7.21110 −0.751809
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −7.21110 −0.739844
\(96\) 0 0
\(97\) − 3.60555i − 0.366088i −0.983105 0.183044i \(-0.941405\pi\)
0.983105 0.183044i \(-0.0585951\pi\)
\(98\) − 12.0000i − 1.21218i
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) −7.21110 −0.717532 −0.358766 0.933428i \(-0.616802\pi\)
−0.358766 + 0.933428i \(0.616802\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 21.6333i − 2.10121i
\(107\) −3.60555 −0.348562 −0.174281 0.984696i \(-0.555760\pi\)
−0.174281 + 0.984696i \(0.555760\pi\)
\(108\) 0 0
\(109\) − 14.4222i − 1.38140i −0.723143 0.690698i \(-0.757303\pi\)
0.723143 0.690698i \(-0.242697\pi\)
\(110\) − 6.00000i − 0.572078i
\(111\) 0 0
\(112\) 14.4222i 1.36277i
\(113\) −7.21110 −0.678363 −0.339182 0.940721i \(-0.610150\pi\)
−0.339182 + 0.940721i \(0.610150\pi\)
\(114\) 0 0
\(115\) − 3.60555i − 0.336219i
\(116\) −14.4222 −1.33907
\(117\) 0 0
\(118\) 24.0000 2.20938
\(119\) 13.0000i 1.19171i
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 26.0000i 2.35393i
\(123\) 0 0
\(124\) − 14.4222i − 1.29515i
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) − 7.21110i − 0.632456i
\(131\) 21.6333 1.89011 0.945055 0.326910i \(-0.106007\pi\)
0.945055 + 0.326910i \(0.106007\pi\)
\(132\) 0 0
\(133\) −26.0000 −2.25449
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 7.21110 0.609449
\(141\) 0 0
\(142\) 10.0000 0.839181
\(143\) −10.8167 −0.904534
\(144\) 0 0
\(145\) − 7.21110i − 0.598849i
\(146\) −14.4222 −1.19359
\(147\) 0 0
\(148\) 7.21110i 0.592749i
\(149\) 5.00000i 0.409616i 0.978802 + 0.204808i \(0.0656570\pi\)
−0.978802 + 0.204808i \(0.934343\pi\)
\(150\) 0 0
\(151\) − 7.21110i − 0.586831i −0.955985 0.293416i \(-0.905208\pi\)
0.955985 0.293416i \(-0.0947920\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) − 21.6333i − 1.74326i
\(155\) 7.21110 0.579210
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 26.0000i 2.06845i
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) − 13.0000i − 1.02454i
\(162\) 0 0
\(163\) 10.8167i 0.847226i 0.905843 + 0.423613i \(0.139238\pi\)
−0.905843 + 0.423613i \(0.860762\pi\)
\(164\) − 22.0000i − 1.71791i
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) − 10.0000i − 0.773823i −0.922117 0.386912i \(-0.873542\pi\)
0.922117 0.386912i \(-0.126458\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −7.21110 −0.553066
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −7.21110 −0.548250 −0.274125 0.961694i \(-0.588388\pi\)
−0.274125 + 0.961694i \(0.588388\pi\)
\(174\) 0 0
\(175\) 3.60555i 0.272554i
\(176\) 12.0000i 0.904534i
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 3.00000 0.222988 0.111494 0.993765i \(-0.464436\pi\)
0.111494 + 0.993765i \(0.464436\pi\)
\(182\) − 26.0000i − 1.92725i
\(183\) 0 0
\(184\) 0 0
\(185\) −3.60555 −0.265085
\(186\) 0 0
\(187\) 10.8167i 0.790992i
\(188\) − 8.00000i − 0.583460i
\(189\) 0 0
\(190\) − 14.4222i − 1.04630i
\(191\) −7.21110 −0.521777 −0.260889 0.965369i \(-0.584016\pi\)
−0.260889 + 0.965369i \(0.584016\pi\)
\(192\) 0 0
\(193\) − 3.60555i − 0.259533i −0.991545 0.129767i \(-0.958577\pi\)
0.991545 0.129767i \(-0.0414228\pi\)
\(194\) 7.21110 0.517727
\(195\) 0 0
\(196\) 12.0000 0.857143
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 14.4222i − 1.01474i
\(203\) − 26.0000i − 1.82484i
\(204\) 0 0
\(205\) 11.0000 0.768273
\(206\) 20.0000i 1.39347i
\(207\) 0 0
\(208\) 14.4222i 1.00000i
\(209\) −21.6333 −1.49641
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 21.6333 1.48578
\(213\) 0 0
\(214\) − 7.21110i − 0.492941i
\(215\) 4.00000i 0.272798i
\(216\) 0 0
\(217\) 26.0000 1.76500
\(218\) 28.8444 1.95359
\(219\) 0 0
\(220\) 6.00000 0.404520
\(221\) 13.0000i 0.874475i
\(222\) 0 0
\(223\) 14.4222i 0.965782i 0.875680 + 0.482891i \(0.160413\pi\)
−0.875680 + 0.482891i \(0.839587\pi\)
\(224\) −28.8444 −1.92725
\(225\) 0 0
\(226\) − 14.4222i − 0.959351i
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) 14.4222i 0.953046i 0.879162 + 0.476523i \(0.158103\pi\)
−0.879162 + 0.476523i \(0.841897\pi\)
\(230\) 7.21110 0.475486
\(231\) 0 0
\(232\) 0 0
\(233\) −25.2389 −1.65345 −0.826726 0.562604i \(-0.809799\pi\)
−0.826726 + 0.562604i \(0.809799\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 24.0000i 1.56227i
\(237\) 0 0
\(238\) −26.0000 −1.68533
\(239\) 7.00000i 0.452792i 0.974035 + 0.226396i \(0.0726944\pi\)
−0.974035 + 0.226396i \(0.927306\pi\)
\(240\) 0 0
\(241\) 21.6333i 1.39352i 0.717302 + 0.696762i \(0.245377\pi\)
−0.717302 + 0.696762i \(0.754623\pi\)
\(242\) 4.00000i 0.257130i
\(243\) 0 0
\(244\) −26.0000 −1.66448
\(245\) 6.00000i 0.383326i
\(246\) 0 0
\(247\) −26.0000 −1.65434
\(248\) 0 0
\(249\) 0 0
\(250\) −2.00000 −0.126491
\(251\) 21.6333 1.36548 0.682741 0.730660i \(-0.260788\pi\)
0.682741 + 0.730660i \(0.260788\pi\)
\(252\) 0 0
\(253\) − 10.8167i − 0.680037i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 21.6333 1.34945 0.674724 0.738070i \(-0.264263\pi\)
0.674724 + 0.738070i \(0.264263\pi\)
\(258\) 0 0
\(259\) −13.0000 −0.807781
\(260\) 7.21110 0.447214
\(261\) 0 0
\(262\) 43.2666i 2.67302i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 10.8167i 0.664462i
\(266\) − 52.0000i − 3.18832i
\(267\) 0 0
\(268\) 0 0
\(269\) −7.21110 −0.439669 −0.219834 0.975537i \(-0.570552\pi\)
−0.219834 + 0.975537i \(0.570552\pi\)
\(270\) 0 0
\(271\) 28.8444i 1.75217i 0.482154 + 0.876087i \(0.339855\pi\)
−0.482154 + 0.876087i \(0.660145\pi\)
\(272\) 14.4222 0.874475
\(273\) 0 0
\(274\) 24.0000 1.44989
\(275\) 3.00000i 0.180907i
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 14.0000i 0.839664i
\(279\) 0 0
\(280\) 0 0
\(281\) − 30.0000i − 1.78965i −0.446417 0.894825i \(-0.647300\pi\)
0.446417 0.894825i \(-0.352700\pi\)
\(282\) 0 0
\(283\) 26.0000 1.54554 0.772770 0.634686i \(-0.218871\pi\)
0.772770 + 0.634686i \(0.218871\pi\)
\(284\) 10.0000i 0.593391i
\(285\) 0 0
\(286\) − 21.6333i − 1.27920i
\(287\) 39.6611 2.34112
\(288\) 0 0
\(289\) −4.00000 −0.235294
\(290\) 14.4222 0.846901
\(291\) 0 0
\(292\) − 14.4222i − 0.843996i
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) − 13.0000i − 0.751809i
\(300\) 0 0
\(301\) 14.4222i 0.831282i
\(302\) 14.4222 0.829905
\(303\) 0 0
\(304\) 28.8444i 1.65434i
\(305\) − 13.0000i − 0.744378i
\(306\) 0 0
\(307\) − 18.0278i − 1.02890i −0.857521 0.514449i \(-0.827996\pi\)
0.857521 0.514449i \(-0.172004\pi\)
\(308\) 21.6333 1.23267
\(309\) 0 0
\(310\) 14.4222i 0.819126i
\(311\) 14.4222 0.817808 0.408904 0.912577i \(-0.365911\pi\)
0.408904 + 0.912577i \(0.365911\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 4.00000i 0.225733i
\(315\) 0 0
\(316\) −26.0000 −1.46261
\(317\) 32.0000i 1.79730i 0.438667 + 0.898650i \(0.355451\pi\)
−0.438667 + 0.898650i \(0.644549\pi\)
\(318\) 0 0
\(319\) − 21.6333i − 1.21123i
\(320\) − 8.00000i − 0.447214i
\(321\) 0 0
\(322\) 26.0000 1.44892
\(323\) 26.0000i 1.44668i
\(324\) 0 0
\(325\) 3.60555i 0.200000i
\(326\) −21.6333 −1.19816
\(327\) 0 0
\(328\) 0 0
\(329\) 14.4222 0.795122
\(330\) 0 0
\(331\) 28.8444i 1.58543i 0.609591 + 0.792716i \(0.291334\pi\)
−0.609591 + 0.792716i \(0.708666\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) − 26.0000i − 1.41421i
\(339\) 0 0
\(340\) − 7.21110i − 0.391077i
\(341\) 21.6333 1.17151
\(342\) 0 0
\(343\) − 3.60555i − 0.194681i
\(344\) 0 0
\(345\) 0 0
\(346\) − 14.4222i − 0.775343i
\(347\) 25.2389 1.35489 0.677446 0.735572i \(-0.263087\pi\)
0.677446 + 0.735572i \(0.263087\pi\)
\(348\) 0 0
\(349\) 7.21110i 0.386001i 0.981199 + 0.193001i \(0.0618220\pi\)
−0.981199 + 0.193001i \(0.938178\pi\)
\(350\) −7.21110 −0.385450
\(351\) 0 0
\(352\) −24.0000 −1.27920
\(353\) − 24.0000i − 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) 0 0
\(355\) −5.00000 −0.265372
\(356\) − 6.00000i − 0.317999i
\(357\) 0 0
\(358\) 0 0
\(359\) − 4.00000i − 0.211112i −0.994413 0.105556i \(-0.966338\pi\)
0.994413 0.105556i \(-0.0336622\pi\)
\(360\) 0 0
\(361\) −33.0000 −1.73684
\(362\) 6.00000i 0.315353i
\(363\) 0 0
\(364\) 26.0000 1.36277
\(365\) 7.21110 0.377446
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −14.4222 −0.751809
\(369\) 0 0
\(370\) − 7.21110i − 0.374887i
\(371\) 39.0000i 2.02478i
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −21.6333 −1.11863
\(375\) 0 0
\(376\) 0 0
\(377\) − 26.0000i − 1.33907i
\(378\) 0 0
\(379\) − 21.6333i − 1.11123i −0.831440 0.555614i \(-0.812483\pi\)
0.831440 0.555614i \(-0.187517\pi\)
\(380\) 14.4222 0.739844
\(381\) 0 0
\(382\) − 14.4222i − 0.737904i
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) 0 0
\(385\) 10.8167i 0.551268i
\(386\) 7.21110 0.367035
\(387\) 0 0
\(388\) 7.21110i 0.366088i
\(389\) −28.8444 −1.46247 −0.731235 0.682126i \(-0.761056\pi\)
−0.731235 + 0.682126i \(0.761056\pi\)
\(390\) 0 0
\(391\) −13.0000 −0.657438
\(392\) 0 0
\(393\) 0 0
\(394\) −4.00000 −0.201517
\(395\) − 13.0000i − 0.654101i
\(396\) 0 0
\(397\) − 18.0278i − 0.904787i −0.891818 0.452394i \(-0.850570\pi\)
0.891818 0.452394i \(-0.149430\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 10.0000i 0.499376i 0.968326 + 0.249688i \(0.0803281\pi\)
−0.968326 + 0.249688i \(0.919672\pi\)
\(402\) 0 0
\(403\) 26.0000 1.29515
\(404\) 14.4222 0.717532
\(405\) 0 0
\(406\) 52.0000 2.58072
\(407\) −10.8167 −0.536162
\(408\) 0 0
\(409\) 21.6333i 1.06970i 0.844948 + 0.534849i \(0.179632\pi\)
−0.844948 + 0.534849i \(0.820368\pi\)
\(410\) 22.0000i 1.08650i
\(411\) 0 0
\(412\) −20.0000 −0.985329
\(413\) −43.2666 −2.12901
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) −28.8444 −1.41421
\(417\) 0 0
\(418\) − 43.2666i − 2.11624i
\(419\) 21.6333 1.05686 0.528428 0.848978i \(-0.322782\pi\)
0.528428 + 0.848978i \(0.322782\pi\)
\(420\) 0 0
\(421\) − 7.21110i − 0.351448i −0.984440 0.175724i \(-0.943773\pi\)
0.984440 0.175724i \(-0.0562266\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.60555 0.174895
\(426\) 0 0
\(427\) − 46.8722i − 2.26830i
\(428\) 7.21110 0.348562
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 24.0000i 1.15604i 0.816023 + 0.578020i \(0.196174\pi\)
−0.816023 + 0.578020i \(0.803826\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 52.0000i 2.49608i
\(435\) 0 0
\(436\) 28.8444i 1.38140i
\(437\) − 26.0000i − 1.24375i
\(438\) 0 0
\(439\) 25.0000 1.19318 0.596592 0.802544i \(-0.296521\pi\)
0.596592 + 0.802544i \(0.296521\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −26.0000 −1.23669
\(443\) 10.8167 0.513915 0.256957 0.966423i \(-0.417280\pi\)
0.256957 + 0.966423i \(0.417280\pi\)
\(444\) 0 0
\(445\) 3.00000 0.142214
\(446\) −28.8444 −1.36582
\(447\) 0 0
\(448\) − 28.8444i − 1.36277i
\(449\) − 27.0000i − 1.27421i −0.770778 0.637104i \(-0.780132\pi\)
0.770778 0.637104i \(-0.219868\pi\)
\(450\) 0 0
\(451\) 33.0000 1.55391
\(452\) 14.4222 0.678363
\(453\) 0 0
\(454\) −36.0000 −1.68956
\(455\) 13.0000i 0.609449i
\(456\) 0 0
\(457\) − 18.0278i − 0.843303i −0.906758 0.421651i \(-0.861451\pi\)
0.906758 0.421651i \(-0.138549\pi\)
\(458\) −28.8444 −1.34781
\(459\) 0 0
\(460\) 7.21110i 0.336219i
\(461\) − 5.00000i − 0.232873i −0.993198 0.116437i \(-0.962853\pi\)
0.993198 0.116437i \(-0.0371472\pi\)
\(462\) 0 0
\(463\) − 18.0278i − 0.837821i −0.908028 0.418910i \(-0.862412\pi\)
0.908028 0.418910i \(-0.137588\pi\)
\(464\) −28.8444 −1.33907
\(465\) 0 0
\(466\) − 50.4777i − 2.33834i
\(467\) 32.4500 1.50161 0.750803 0.660527i \(-0.229667\pi\)
0.750803 + 0.660527i \(0.229667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8.00000i 0.369012i
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 7.21110i 0.330868i
\(476\) − 26.0000i − 1.19171i
\(477\) 0 0
\(478\) −14.0000 −0.640345
\(479\) 3.00000i 0.137073i 0.997649 + 0.0685367i \(0.0218330\pi\)
−0.997649 + 0.0685367i \(0.978167\pi\)
\(480\) 0 0
\(481\) −13.0000 −0.592749
\(482\) −43.2666 −1.97074
\(483\) 0 0
\(484\) −4.00000 −0.181818
\(485\) −3.60555 −0.163720
\(486\) 0 0
\(487\) 10.8167i 0.490149i 0.969504 + 0.245075i \(0.0788125\pi\)
−0.969504 + 0.245075i \(0.921188\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −12.0000 −0.542105
\(491\) 7.21110 0.325433 0.162716 0.986673i \(-0.447974\pi\)
0.162716 + 0.986673i \(0.447974\pi\)
\(492\) 0 0
\(493\) −26.0000 −1.17098
\(494\) − 52.0000i − 2.33959i
\(495\) 0 0
\(496\) − 28.8444i − 1.29515i
\(497\) −18.0278 −0.808655
\(498\) 0 0
\(499\) − 28.8444i − 1.29125i −0.763653 0.645627i \(-0.776596\pi\)
0.763653 0.645627i \(-0.223404\pi\)
\(500\) − 2.00000i − 0.0894427i
\(501\) 0 0
\(502\) 43.2666i 1.93108i
\(503\) −14.4222 −0.643054 −0.321527 0.946900i \(-0.604196\pi\)
−0.321527 + 0.946900i \(0.604196\pi\)
\(504\) 0 0
\(505\) 7.21110i 0.320890i
\(506\) 21.6333 0.961718
\(507\) 0 0
\(508\) 0 0
\(509\) − 15.0000i − 0.664863i −0.943127 0.332432i \(-0.892131\pi\)
0.943127 0.332432i \(-0.107869\pi\)
\(510\) 0 0
\(511\) 26.0000 1.15017
\(512\) 32.0000i 1.41421i
\(513\) 0 0
\(514\) 43.2666i 1.90841i
\(515\) − 10.0000i − 0.440653i
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) − 26.0000i − 1.14237i
\(519\) 0 0
\(520\) 0 0
\(521\) −28.8444 −1.26370 −0.631848 0.775092i \(-0.717703\pi\)
−0.631848 + 0.775092i \(0.717703\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −43.2666 −1.89011
\(525\) 0 0
\(526\) 0 0
\(527\) − 26.0000i − 1.13258i
\(528\) 0 0
\(529\) −10.0000 −0.434783
\(530\) −21.6333 −0.939691
\(531\) 0 0
\(532\) 52.0000 2.25449
\(533\) 39.6611 1.71791
\(534\) 0 0
\(535\) 3.60555i 0.155882i
\(536\) 0 0
\(537\) 0 0
\(538\) − 14.4222i − 0.621785i
\(539\) 18.0000i 0.775315i
\(540\) 0 0
\(541\) 7.21110i 0.310030i 0.987912 + 0.155015i \(0.0495425\pi\)
−0.987912 + 0.155015i \(0.950457\pi\)
\(542\) −57.6888 −2.47795
\(543\) 0 0
\(544\) 28.8444i 1.23669i
\(545\) −14.4222 −0.617779
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 24.0000i 1.02523i
\(549\) 0 0
\(550\) −6.00000 −0.255841
\(551\) − 52.0000i − 2.21527i
\(552\) 0 0
\(553\) − 46.8722i − 1.99321i
\(554\) 8.00000i 0.339887i
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 42.0000i 1.77960i 0.456354 + 0.889799i \(0.349155\pi\)
−0.456354 + 0.889799i \(0.650845\pi\)
\(558\) 0 0
\(559\) 14.4222i 0.609994i
\(560\) 14.4222 0.609449
\(561\) 0 0
\(562\) 60.0000 2.53095
\(563\) 32.4500 1.36760 0.683801 0.729668i \(-0.260325\pi\)
0.683801 + 0.729668i \(0.260325\pi\)
\(564\) 0 0
\(565\) 7.21110i 0.303373i
\(566\) 52.0000i 2.18572i
\(567\) 0 0
\(568\) 0 0
\(569\) 43.2666 1.81383 0.906915 0.421313i \(-0.138431\pi\)
0.906915 + 0.421313i \(0.138431\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 21.6333 0.904534
\(573\) 0 0
\(574\) 79.3221i 3.31084i
\(575\) −3.60555 −0.150362
\(576\) 0 0
\(577\) − 3.60555i − 0.150101i −0.997180 0.0750505i \(-0.976088\pi\)
0.997180 0.0750505i \(-0.0239118\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 0 0
\(580\) 14.4222i 0.598849i
\(581\) 21.6333 0.897501
\(582\) 0 0
\(583\) 32.4500i 1.34394i
\(584\) 0 0
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) 52.0000 2.14262
\(590\) − 24.0000i − 0.988064i
\(591\) 0 0
\(592\) 14.4222i 0.592749i
\(593\) 4.00000i 0.164260i 0.996622 + 0.0821302i \(0.0261723\pi\)
−0.996622 + 0.0821302i \(0.973828\pi\)
\(594\) 0 0
\(595\) 13.0000 0.532948
\(596\) − 10.0000i − 0.409616i
\(597\) 0 0
\(598\) 26.0000 1.06322
\(599\) −7.21110 −0.294638 −0.147319 0.989089i \(-0.547064\pi\)
−0.147319 + 0.989089i \(0.547064\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) −28.8444 −1.17561
\(603\) 0 0
\(604\) 14.4222i 0.586831i
\(605\) − 2.00000i − 0.0813116i
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) −57.6888 −2.33959
\(609\) 0 0
\(610\) 26.0000 1.05271
\(611\) 14.4222 0.583460
\(612\) 0 0
\(613\) − 3.60555i − 0.145627i −0.997346 0.0728134i \(-0.976802\pi\)
0.997346 0.0728134i \(-0.0231978\pi\)
\(614\) 36.0555 1.45508
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000i 1.36879i 0.729112 + 0.684394i \(0.239933\pi\)
−0.729112 + 0.684394i \(0.760067\pi\)
\(618\) 0 0
\(619\) 14.4222i 0.579677i 0.957076 + 0.289839i \(0.0936017\pi\)
−0.957076 + 0.289839i \(0.906398\pi\)
\(620\) −14.4222 −0.579210
\(621\) 0 0
\(622\) 28.8444i 1.15656i
\(623\) 10.8167 0.433360
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 52.0000i 2.07834i
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 13.0000i 0.518344i
\(630\) 0 0
\(631\) − 28.8444i − 1.14828i −0.818758 0.574139i \(-0.805337\pi\)
0.818758 0.574139i \(-0.194663\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −64.0000 −2.54176
\(635\) 0 0
\(636\) 0 0
\(637\) 21.6333i 0.857143i
\(638\) 43.2666 1.71294
\(639\) 0 0
\(640\) 0 0
\(641\) −43.2666 −1.70893 −0.854464 0.519510i \(-0.826114\pi\)
−0.854464 + 0.519510i \(0.826114\pi\)
\(642\) 0 0
\(643\) − 25.2389i − 0.995323i −0.867371 0.497662i \(-0.834192\pi\)
0.867371 0.497662i \(-0.165808\pi\)
\(644\) 26.0000i 1.02454i
\(645\) 0 0
\(646\) −52.0000 −2.04591
\(647\) −18.0278 −0.708744 −0.354372 0.935104i \(-0.615305\pi\)
−0.354372 + 0.935104i \(0.615305\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) −7.21110 −0.282843
\(651\) 0 0
\(652\) − 21.6333i − 0.847226i
\(653\) −7.21110 −0.282192 −0.141096 0.989996i \(-0.545063\pi\)
−0.141096 + 0.989996i \(0.545063\pi\)
\(654\) 0 0
\(655\) − 21.6333i − 0.845283i
\(656\) − 44.0000i − 1.71791i
\(657\) 0 0
\(658\) 28.8444i 1.12447i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) − 7.21110i − 0.280479i −0.990118 0.140240i \(-0.955213\pi\)
0.990118 0.140240i \(-0.0447873\pi\)
\(662\) −57.6888 −2.24214
\(663\) 0 0
\(664\) 0 0
\(665\) 26.0000i 1.00824i
\(666\) 0 0
\(667\) 26.0000 1.00672
\(668\) 20.0000i 0.773823i
\(669\) 0 0
\(670\) 0 0
\(671\) − 39.0000i − 1.50558i
\(672\) 0 0
\(673\) 40.0000 1.54189 0.770943 0.636904i \(-0.219785\pi\)
0.770943 + 0.636904i \(0.219785\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 26.0000 1.00000
\(677\) −10.8167 −0.415718 −0.207859 0.978159i \(-0.566649\pi\)
−0.207859 + 0.978159i \(0.566649\pi\)
\(678\) 0 0
\(679\) −13.0000 −0.498894
\(680\) 0 0
\(681\) 0 0
\(682\) 43.2666i 1.65676i
\(683\) − 38.0000i − 1.45403i −0.686622 0.727015i \(-0.740907\pi\)
0.686622 0.727015i \(-0.259093\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 7.21110 0.275321
\(687\) 0 0
\(688\) 16.0000 0.609994
\(689\) 39.0000i 1.48578i
\(690\) 0 0
\(691\) 14.4222i 0.548647i 0.961638 + 0.274323i \(0.0884538\pi\)
−0.961638 + 0.274323i \(0.911546\pi\)
\(692\) 14.4222 0.548250
\(693\) 0 0
\(694\) 50.4777i 1.91611i
\(695\) − 7.00000i − 0.265525i
\(696\) 0 0
\(697\) − 39.6611i − 1.50227i
\(698\) −14.4222 −0.545889
\(699\) 0 0
\(700\) − 7.21110i − 0.272554i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −26.0000 −0.980609
\(704\) − 24.0000i − 0.904534i
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) 26.0000i 0.977831i
\(708\) 0 0
\(709\) 28.8444i 1.08327i 0.840612 + 0.541637i \(0.182195\pi\)
−0.840612 + 0.541637i \(0.817805\pi\)
\(710\) − 10.0000i − 0.375293i
\(711\) 0 0
\(712\) 0 0
\(713\) 26.0000i 0.973708i
\(714\) 0 0
\(715\) 10.8167i 0.404520i
\(716\) 0 0
\(717\) 0 0
\(718\) 8.00000 0.298557
\(719\) 21.6333 0.806786 0.403393 0.915027i \(-0.367831\pi\)
0.403393 + 0.915027i \(0.367831\pi\)
\(720\) 0 0
\(721\) − 36.0555i − 1.34278i
\(722\) − 66.0000i − 2.45627i
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) −7.21110 −0.267814
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14.4222i 0.533790i
\(731\) 14.4222 0.533425
\(732\) 0 0
\(733\) 32.4500i 1.19857i 0.800537 + 0.599283i \(0.204548\pi\)
−0.800537 + 0.599283i \(0.795452\pi\)
\(734\) − 36.0000i − 1.32878i
\(735\) 0 0
\(736\) − 28.8444i − 1.06322i
\(737\) 0 0
\(738\) 0 0
\(739\) − 21.6333i − 0.795794i −0.917430 0.397897i \(-0.869740\pi\)
0.917430 0.397897i \(-0.130260\pi\)
\(740\) 7.21110 0.265085
\(741\) 0 0
\(742\) −78.0000 −2.86347
\(743\) 2.00000i 0.0733729i 0.999327 + 0.0366864i \(0.0116803\pi\)
−0.999327 + 0.0366864i \(0.988320\pi\)
\(744\) 0 0
\(745\) 5.00000 0.183186
\(746\) − 52.0000i − 1.90386i
\(747\) 0 0
\(748\) − 21.6333i − 0.790992i
\(749\) 13.0000i 0.475010i
\(750\) 0 0
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) − 16.0000i − 0.583460i
\(753\) 0 0
\(754\) 52.0000 1.89373
\(755\) −7.21110 −0.262439
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 43.2666 1.57151
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0000i 1.37750i 0.724999 + 0.688749i \(0.241840\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 0 0
\(763\) −52.0000 −1.88253
\(764\) 14.4222 0.521777
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −43.2666 −1.56227
\(768\) 0 0
\(769\) 7.21110i 0.260039i 0.991511 + 0.130020i \(0.0415040\pi\)
−0.991511 + 0.130020i \(0.958496\pi\)
\(770\) −21.6333 −0.779610
\(771\) 0 0
\(772\) 7.21110i 0.259533i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) − 7.21110i − 0.259030i
\(776\) 0 0
\(777\) 0 0
\(778\) − 57.6888i − 2.06824i
\(779\) 79.3221 2.84201
\(780\) 0 0
\(781\) −15.0000 −0.536742
\(782\) − 26.0000i − 0.929758i
\(783\) 0 0
\(784\) 24.0000 0.857143
\(785\) − 2.00000i − 0.0713831i
\(786\) 0 0
\(787\) 28.8444i 1.02819i 0.857733 + 0.514096i \(0.171873\pi\)
−0.857733 + 0.514096i \(0.828127\pi\)
\(788\) − 4.00000i − 0.142494i
\(789\) 0 0
\(790\) 26.0000 0.925038
\(791\) 26.0000i 0.924454i
\(792\) 0 0
\(793\) − 46.8722i − 1.66448i
\(794\) 36.0555 1.27956
\(795\) 0 0
\(796\) 0 0
\(797\) −39.6611 −1.40487 −0.702433 0.711749i \(-0.747903\pi\)
−0.702433 + 0.711749i \(0.747903\pi\)
\(798\) 0 0
\(799\) − 14.4222i − 0.510221i
\(800\) 8.00000i 0.282843i
\(801\) 0 0
\(802\) −20.0000 −0.706225
\(803\) 21.6333 0.763423
\(804\) 0 0
\(805\) −13.0000 −0.458190
\(806\) 52.0000i 1.83162i
\(807\) 0 0
\(808\) 0 0
\(809\) −21.6333 −0.760587 −0.380293 0.924866i \(-0.624177\pi\)
−0.380293 + 0.924866i \(0.624177\pi\)
\(810\) 0 0
\(811\) 28.8444i 1.01286i 0.862280 + 0.506432i \(0.169036\pi\)
−0.862280 + 0.506432i \(0.830964\pi\)
\(812\) 52.0000i 1.82484i
\(813\) 0 0
\(814\) − 21.6333i − 0.758247i
\(815\) 10.8167 0.378891
\(816\) 0 0
\(817\) 28.8444i 1.00914i
\(818\) −43.2666 −1.51278
\(819\) 0 0
\(820\) −22.0000 −0.768273
\(821\) 15.0000i 0.523504i 0.965135 + 0.261752i \(0.0843002\pi\)
−0.965135 + 0.261752i \(0.915700\pi\)
\(822\) 0 0
\(823\) 30.0000 1.04573 0.522867 0.852414i \(-0.324862\pi\)
0.522867 + 0.852414i \(0.324862\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) − 86.5332i − 3.01088i
\(827\) 4.00000i 0.139094i 0.997579 + 0.0695468i \(0.0221553\pi\)
−0.997579 + 0.0695468i \(0.977845\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 12.0000i 0.416526i
\(831\) 0 0
\(832\) − 28.8444i − 1.00000i
\(833\) 21.6333 0.749550
\(834\) 0 0
\(835\) −10.0000 −0.346064
\(836\) 43.2666 1.49641
\(837\) 0 0
\(838\) 43.2666i 1.49462i
\(839\) − 51.0000i − 1.76072i −0.474310 0.880358i \(-0.657302\pi\)
0.474310 0.880358i \(-0.342698\pi\)
\(840\) 0 0
\(841\) 23.0000 0.793103
\(842\) 14.4222 0.497022
\(843\) 0 0
\(844\) 0 0
\(845\) 13.0000i 0.447214i
\(846\) 0 0
\(847\) − 7.21110i − 0.247776i
\(848\) 43.2666 1.48578
\(849\) 0 0
\(850\) 7.21110i 0.247339i
\(851\) − 13.0000i − 0.445634i
\(852\) 0 0
\(853\) − 54.0833i − 1.85178i −0.377798 0.925888i \(-0.623319\pi\)
0.377798 0.925888i \(-0.376681\pi\)
\(854\) 93.7443 3.20787
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0278 0.615816 0.307908 0.951416i \(-0.400371\pi\)
0.307908 + 0.951416i \(0.400371\pi\)
\(858\) 0 0
\(859\) −13.0000 −0.443554 −0.221777 0.975097i \(-0.571186\pi\)
−0.221777 + 0.975097i \(0.571186\pi\)
\(860\) − 8.00000i − 0.272798i
\(861\) 0 0
\(862\) −48.0000 −1.63489
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 7.21110i 0.245185i
\(866\) 52.0000i 1.76703i
\(867\) 0 0
\(868\) −52.0000 −1.76500
\(869\) − 39.0000i − 1.32298i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 52.0000 1.75893
\(875\) 3.60555 0.121890
\(876\) 0 0
\(877\) − 50.4777i − 1.70451i −0.523125 0.852256i \(-0.675234\pi\)
0.523125 0.852256i \(-0.324766\pi\)
\(878\) 50.0000i 1.68742i
\(879\) 0 0
\(880\) 12.0000 0.404520
\(881\) 43.2666 1.45769 0.728845 0.684679i \(-0.240058\pi\)
0.728845 + 0.684679i \(0.240058\pi\)
\(882\) 0 0
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) − 26.0000i − 0.874475i
\(885\) 0 0
\(886\) 21.6333i 0.726785i
\(887\) −25.2389 −0.847438 −0.423719 0.905794i \(-0.639276\pi\)
−0.423719 + 0.905794i \(0.639276\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.00000i 0.201120i
\(891\) 0 0
\(892\) − 28.8444i − 0.965782i
\(893\) 28.8444 0.965241
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 54.0000 1.80200
\(899\) 52.0000i 1.73430i
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) 66.0000i 2.19756i
\(903\) 0 0
\(904\) 0 0
\(905\) − 3.00000i − 0.0997234i
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) − 36.0000i − 1.19470i
\(909\) 0 0
\(910\) −26.0000 −0.861892
\(911\) −36.0555 −1.19457 −0.597286 0.802028i \(-0.703754\pi\)
−0.597286 + 0.802028i \(0.703754\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) 36.0555 1.19261
\(915\) 0 0
\(916\) − 28.8444i − 0.953046i
\(917\) − 78.0000i − 2.57579i
\(918\) 0 0
\(919\) 13.0000 0.428830 0.214415 0.976743i \(-0.431215\pi\)
0.214415 + 0.976743i \(0.431215\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 10.0000 0.329332
\(923\) −18.0278 −0.593391
\(924\) 0 0
\(925\) 3.60555i 0.118550i
\(926\) 36.0555 1.18486
\(927\) 0 0
\(928\) − 57.6888i − 1.89373i
\(929\) 5.00000i 0.164045i 0.996630 + 0.0820223i \(0.0261379\pi\)
−0.996630 + 0.0820223i \(0.973862\pi\)
\(930\) 0 0
\(931\) 43.2666i 1.41801i
\(932\) 50.4777 1.65345
\(933\) 0 0
\(934\) 64.8999i 2.12359i
\(935\) 10.8167 0.353742
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) − 59.0000i − 1.92335i −0.274201 0.961673i \(-0.588413\pi\)
0.274201 0.961673i \(-0.411587\pi\)
\(942\) 0 0
\(943\) 39.6611i 1.29154i
\(944\) 48.0000i 1.56227i
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) − 42.0000i − 1.36482i −0.730971 0.682408i \(-0.760933\pi\)
0.730971 0.682408i \(-0.239067\pi\)
\(948\) 0 0
\(949\) 26.0000 0.843996
\(950\) −14.4222 −0.467918
\(951\) 0 0
\(952\) 0 0
\(953\) 10.8167 0.350386 0.175193 0.984534i \(-0.443945\pi\)
0.175193 + 0.984534i \(0.443945\pi\)
\(954\) 0 0
\(955\) 7.21110i 0.233346i
\(956\) − 14.0000i − 0.452792i
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) −43.2666 −1.39715
\(960\) 0 0
\(961\) −21.0000 −0.677419
\(962\) − 26.0000i − 0.838274i
\(963\) 0 0
\(964\) − 43.2666i − 1.39352i
\(965\) −3.60555 −0.116067
\(966\) 0 0
\(967\) − 14.4222i − 0.463787i −0.972741 0.231893i \(-0.925508\pi\)
0.972741 0.231893i \(-0.0744921\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 7.21110i − 0.231535i
\(971\) −21.6333 −0.694246 −0.347123 0.937820i \(-0.612841\pi\)
−0.347123 + 0.937820i \(0.612841\pi\)
\(972\) 0 0
\(973\) − 25.2389i − 0.809121i
\(974\) −21.6333 −0.693176
\(975\) 0 0
\(976\) −52.0000 −1.66448
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) − 12.0000i − 0.383326i
\(981\) 0 0
\(982\) 14.4222i 0.460231i
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) − 52.0000i − 1.65602i
\(987\) 0 0
\(988\) 52.0000 1.65434
\(989\) −14.4222 −0.458599
\(990\) 0 0
\(991\) −5.00000 −0.158830 −0.0794151 0.996842i \(-0.525305\pi\)
−0.0794151 + 0.996842i \(0.525305\pi\)
\(992\) 57.6888 1.83162
\(993\) 0 0
\(994\) − 36.0555i − 1.14361i
\(995\) 0 0
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 57.6888 1.82611
\(999\) 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 585.2.b.f.181.3 yes 4
3.2 odd 2 inner 585.2.b.f.181.1 4
13.5 odd 4 7605.2.a.bl.1.2 2
13.8 odd 4 7605.2.a.w.1.1 2
13.12 even 2 inner 585.2.b.f.181.2 yes 4
39.5 even 4 7605.2.a.w.1.2 2
39.8 even 4 7605.2.a.bl.1.1 2
39.38 odd 2 inner 585.2.b.f.181.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.b.f.181.1 4 3.2 odd 2 inner
585.2.b.f.181.2 yes 4 13.12 even 2 inner
585.2.b.f.181.3 yes 4 1.1 even 1 trivial
585.2.b.f.181.4 yes 4 39.38 odd 2 inner
7605.2.a.w.1.1 2 13.8 odd 4
7605.2.a.w.1.2 2 39.5 even 4
7605.2.a.bl.1.1 2 39.8 even 4
7605.2.a.bl.1.2 2 13.5 odd 4