Properties

Label 289.2.b.a.288.2
Level $289$
Weight $2$
Character 289.288
Analytic conductor $2.308$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30767661842\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 288.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 289.288
Dual form 289.2.b.a.288.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +2.00000i q^{5} +4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +2.00000i q^{5} +4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} +2.00000i q^{10} -2.00000 q^{13} +4.00000i q^{14} -1.00000 q^{16} +3.00000 q^{18} +4.00000 q^{19} -2.00000i q^{20} +4.00000i q^{23} +1.00000 q^{25} -2.00000 q^{26} -4.00000i q^{28} -6.00000i q^{29} -4.00000i q^{31} +5.00000 q^{32} -8.00000 q^{35} -3.00000 q^{36} +2.00000i q^{37} +4.00000 q^{38} -6.00000i q^{40} -6.00000i q^{41} -4.00000 q^{43} +6.00000i q^{45} +4.00000i q^{46} -9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -12.0000i q^{56} -6.00000i q^{58} +12.0000 q^{59} -10.0000i q^{61} -4.00000i q^{62} +12.0000i q^{63} +7.00000 q^{64} -4.00000i q^{65} +4.00000 q^{67} -8.00000 q^{70} +4.00000i q^{71} -9.00000 q^{72} +6.00000i q^{73} +2.00000i q^{74} -4.00000 q^{76} +12.0000i q^{79} -2.00000i q^{80} +9.00000 q^{81} -6.00000i q^{82} +4.00000 q^{83} -4.00000 q^{86} +10.0000 q^{89} +6.00000i q^{90} -8.00000i q^{91} -4.00000i q^{92} +8.00000i q^{95} -2.00000i q^{97} -9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9} - 4 q^{13} - 2 q^{16} + 6 q^{18} + 8 q^{19} + 2 q^{25} - 4 q^{26} + 10 q^{32} - 16 q^{35} - 6 q^{36} + 8 q^{38} - 8 q^{43} - 18 q^{49} + 2 q^{50} + 4 q^{52} - 12 q^{53} + 24 q^{59} + 14 q^{64} + 8 q^{67} - 16 q^{70} - 18 q^{72} - 8 q^{76} + 18 q^{81} + 8 q^{83} - 8 q^{86} + 20 q^{89} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −1.00000 −0.500000
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 3.00000 1.00000
\(10\) 2.00000i 0.632456i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000i 1.06904i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0
\(18\) 3.00000 0.707107
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) − 2.00000i − 0.447214i
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) − 4.00000i − 0.755929i
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) − 4.00000i − 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) −3.00000 −0.500000
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) − 6.00000i − 0.948683i
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 6.00000i 0.894427i
\(46\) 4.00000i 0.589768i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 12.0000i − 1.60357i
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) − 10.0000i − 1.28037i −0.768221 0.640184i \(-0.778858\pi\)
0.768221 0.640184i \(-0.221142\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 12.0000i 1.51186i
\(64\) 7.00000 0.875000
\(65\) − 4.00000i − 0.496139i
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 4.00000i 0.474713i 0.971423 + 0.237356i \(0.0762809\pi\)
−0.971423 + 0.237356i \(0.923719\pi\)
\(72\) −9.00000 −1.06066
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 2.00000i 0.232495i
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 12.0000i 1.35011i 0.737769 + 0.675053i \(0.235879\pi\)
−0.737769 + 0.675053i \(0.764121\pi\)
\(80\) − 2.00000i − 0.223607i
\(81\) 9.00000 1.00000
\(82\) − 6.00000i − 0.662589i
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 6.00000i 0.632456i
\(91\) − 8.00000i − 0.838628i
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000i 0.820783i
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i 0.957826 + 0.287348i \(0.0927736\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 4.00000i − 0.377964i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 6.00000i 0.557086i
\(117\) −6.00000 −0.554700
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) − 10.0000i − 0.905357i
\(123\) 0 0
\(124\) 4.00000i 0.359211i
\(125\) 12.0000i 1.07331i
\(126\) 12.0000i 1.06904i
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) − 4.00000i − 0.350823i
\(131\) − 16.0000i − 1.39793i −0.715158 0.698963i \(-0.753645\pi\)
0.715158 0.698963i \(-0.246355\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 8.00000i 0.678551i 0.940687 + 0.339276i \(0.110182\pi\)
−0.940687 + 0.339276i \(0.889818\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) 4.00000i 0.335673i
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) 12.0000 0.996546
\(146\) 6.00000i 0.496564i
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −12.0000 −0.973329
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 12.0000i 0.954669i
\(159\) 0 0
\(160\) 10.0000i 0.790569i
\(161\) −16.0000 −1.26098
\(162\) 9.00000 0.707107
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 6.00000i 0.468521i
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) 4.00000 0.304997
\(173\) − 22.0000i − 1.67263i −0.548250 0.836315i \(-0.684706\pi\)
0.548250 0.836315i \(-0.315294\pi\)
\(174\) 0 0
\(175\) 4.00000i 0.302372i
\(176\) 0 0
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) − 6.00000i − 0.447214i
\(181\) − 2.00000i − 0.148659i −0.997234 0.0743294i \(-0.976318\pi\)
0.997234 0.0743294i \(-0.0236816\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) 0 0
\(184\) − 12.0000i − 0.884652i
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 8.00000i 0.580381i
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) − 2.00000i − 0.143592i
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 20.0000i 1.41776i 0.705328 + 0.708881i \(0.250800\pi\)
−0.705328 + 0.708881i \(0.749200\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 8.00000 0.557386
\(207\) 12.0000i 0.834058i
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000i 0.550743i 0.961338 + 0.275371i \(0.0888008\pi\)
−0.961338 + 0.275371i \(0.911199\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) − 8.00000i − 0.546869i
\(215\) − 8.00000i − 0.545595i
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 6.00000i 0.406371i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 20.0000i 1.33631i
\(225\) 3.00000 0.200000
\(226\) − 14.0000i − 0.931266i
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 18.0000i 1.18176i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) − 18.0000i − 1.15948i −0.814801 0.579741i \(-0.803154\pi\)
0.814801 0.579741i \(-0.196846\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) 10.0000i 0.640184i
\(245\) − 18.0000i − 1.14998i
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 12.0000i 0.762001i
\(249\) 0 0
\(250\) 12.0000i 0.758947i
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 12.0000i − 0.755929i
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 4.00000i 0.248069i
\(261\) − 18.0000i − 1.11417i
\(262\) − 16.0000i − 0.988483i
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) − 12.0000i − 0.737154i
\(266\) 16.0000i 0.981023i
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) − 22.0000i − 1.34136i −0.741745 0.670682i \(-0.766002\pi\)
0.741745 0.670682i \(-0.233998\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 14.0000i − 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 8.00000i 0.479808i
\(279\) − 12.0000i − 0.718421i
\(280\) 24.0000 1.43427
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) − 4.00000i − 0.237356i
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 15.0000 0.883883
\(289\) 0 0
\(290\) 12.0000 0.704664
\(291\) 0 0
\(292\) − 6.00000i − 0.351123i
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 24.0000i 1.39733i
\(296\) − 6.00000i − 0.348743i
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) − 8.00000i − 0.462652i
\(300\) 0 0
\(301\) − 16.0000i − 0.922225i
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) − 28.0000i − 1.58773i −0.608091 0.793867i \(-0.708065\pi\)
0.608091 0.793867i \(-0.291935\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) −2.00000 −0.112867
\(315\) −24.0000 −1.35225
\(316\) − 12.0000i − 0.675053i
\(317\) − 10.0000i − 0.561656i −0.959758 0.280828i \(-0.909391\pi\)
0.959758 0.280828i \(-0.0906090\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 14.0000i 0.782624i
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) 0 0
\(324\) −9.00000 −0.500000
\(325\) −2.00000 −0.110940
\(326\) 24.0000i 1.32924i
\(327\) 0 0
\(328\) 18.0000i 0.993884i
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −4.00000 −0.219529
\(333\) 6.00000i 0.328798i
\(334\) 4.00000i 0.218870i
\(335\) 8.00000i 0.437087i
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 12.0000 0.648886
\(343\) − 8.00000i − 0.431959i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) − 22.0000i − 1.18273i
\(347\) 32.0000i 1.71785i 0.512101 + 0.858925i \(0.328867\pi\)
−0.512101 + 0.858925i \(0.671133\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 4.00000i 0.213809i
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) − 18.0000i − 0.948683i
\(361\) −3.00000 −0.157895
\(362\) − 2.00000i − 0.105118i
\(363\) 0 0
\(364\) 8.00000i 0.419314i
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) 28.0000i 1.46159i 0.682598 + 0.730794i \(0.260850\pi\)
−0.682598 + 0.730794i \(0.739150\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) − 18.0000i − 0.937043i
\(370\) −4.00000 −0.207950
\(371\) − 24.0000i − 1.24602i
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 8.00000i 0.410932i 0.978664 + 0.205466i \(0.0658711\pi\)
−0.978664 + 0.205466i \(0.934129\pi\)
\(380\) − 8.00000i − 0.410391i
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.00000i 0.101797i
\(387\) −12.0000 −0.609994
\(388\) 2.00000i 0.101535i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 27.0000 1.36371
\(393\) 0 0
\(394\) − 18.0000i − 0.906827i
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) 20.0000i 1.00251i
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) − 14.0000i − 0.699127i −0.936913 0.349563i \(-0.886330\pi\)
0.936913 0.349563i \(-0.113670\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) 10.0000 0.497519
\(405\) 18.0000i 0.894427i
\(406\) 24.0000 1.19110
\(407\) 0 0
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 48.0000i 2.36193i
\(414\) 12.0000i 0.589768i
\(415\) 8.00000i 0.392705i
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) 0 0
\(419\) 8.00000i 0.390826i 0.980721 + 0.195413i \(0.0626047\pi\)
−0.980721 + 0.195413i \(0.937395\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 8.00000i 0.389434i
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) 40.0000 1.93574
\(428\) 8.00000i 0.386695i
\(429\) 0 0
\(430\) − 8.00000i − 0.385794i
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) − 6.00000i − 0.287348i
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) 20.0000i 0.954548i 0.878755 + 0.477274i \(0.158375\pi\)
−0.878755 + 0.477274i \(0.841625\pi\)
\(440\) 0 0
\(441\) −27.0000 −1.28571
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) 20.0000i 0.948091i
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) 28.0000i 1.32288i
\(449\) 34.0000i 1.60456i 0.596948 + 0.802280i \(0.296380\pi\)
−0.596948 + 0.802280i \(0.703620\pi\)
\(450\) 3.00000 0.141421
\(451\) 0 0
\(452\) 14.0000i 0.658505i
\(453\) 0 0
\(454\) − 24.0000i − 1.12638i
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 6.00000i 0.278543i
\(465\) 0 0
\(466\) 6.00000i 0.277945i
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 6.00000 0.277350
\(469\) 16.0000i 0.738811i
\(470\) 0 0
\(471\) 0 0
\(472\) −36.0000 −1.65703
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) −16.0000 −0.731823
\(479\) − 36.0000i − 1.64488i −0.568850 0.822441i \(-0.692612\pi\)
0.568850 0.822441i \(-0.307388\pi\)
\(480\) 0 0
\(481\) − 4.00000i − 0.182384i
\(482\) − 18.0000i − 0.819878i
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 30.0000i 1.35804i
\(489\) 0 0
\(490\) − 18.0000i − 0.813157i
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 4.00000i 0.179605i
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) − 40.0000i − 1.79065i −0.445418 0.895323i \(-0.646945\pi\)
0.445418 0.895323i \(-0.353055\pi\)
\(500\) − 12.0000i − 0.536656i
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) − 36.0000i − 1.60357i
\(505\) − 20.0000i − 0.889988i
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −24.0000 −1.06170
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 16.0000i 0.705044i
\(516\) 0 0
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 12.0000i 0.526235i
\(521\) 26.0000i 1.13908i 0.821963 + 0.569540i \(0.192879\pi\)
−0.821963 + 0.569540i \(0.807121\pi\)
\(522\) − 18.0000i − 0.787839i
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 16.0000i 0.698963i
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) − 12.0000i − 0.521247i
\(531\) 36.0000 1.56227
\(532\) − 16.0000i − 0.693688i
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) − 22.0000i − 0.948487i
\(539\) 0 0
\(540\) 0 0
\(541\) − 6.00000i − 0.257960i −0.991647 0.128980i \(-0.958830\pi\)
0.991647 0.128980i \(-0.0411703\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 6.00000 0.256307
\(549\) − 30.0000i − 1.28037i
\(550\) 0 0
\(551\) − 24.0000i − 1.02243i
\(552\) 0 0
\(553\) −48.0000 −2.04117
\(554\) − 14.0000i − 0.594803i
\(555\) 0 0
\(556\) − 8.00000i − 0.339276i
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) − 12.0000i − 0.508001i
\(559\) 8.00000 0.338364
\(560\) 8.00000 0.338062
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 28.0000 1.17797
\(566\) − 16.0000i − 0.672530i
\(567\) 36.0000i 1.51186i
\(568\) − 12.0000i − 0.503509i
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) − 32.0000i − 1.33916i −0.742741 0.669579i \(-0.766474\pi\)
0.742741 0.669579i \(-0.233526\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 4.00000i 0.166812i
\(576\) 21.0000 0.875000
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) 16.0000i 0.663792i
\(582\) 0 0
\(583\) 0 0
\(584\) − 18.0000i − 0.744845i
\(585\) − 12.0000i − 0.496139i
\(586\) 6.00000 0.247858
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 0 0
\(589\) − 16.0000i − 0.659269i
\(590\) 24.0000i 0.988064i
\(591\) 0 0
\(592\) − 2.00000i − 0.0821995i
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) − 8.00000i − 0.327144i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 10.0000i 0.407909i 0.978980 + 0.203954i \(0.0653794\pi\)
−0.978980 + 0.203954i \(0.934621\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) 12.0000 0.488678
\(604\) −16.0000 −0.651031
\(605\) 22.0000i 0.894427i
\(606\) 0 0
\(607\) − 20.0000i − 0.811775i −0.913923 0.405887i \(-0.866962\pi\)
0.913923 0.405887i \(-0.133038\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) 0 0
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) − 48.0000i − 1.92928i −0.263566 0.964641i \(-0.584899\pi\)
0.263566 0.964641i \(-0.415101\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) − 28.0000i − 1.12270i
\(623\) 40.0000i 1.60257i
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) − 22.0000i − 0.879297i
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) −24.0000 −0.956183
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 36.0000i − 1.43200i
\(633\) 0 0
\(634\) − 10.0000i − 0.397151i
\(635\) − 16.0000i − 0.634941i
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 12.0000i 0.474713i
\(640\) − 6.00000i − 0.237171i
\(641\) 30.0000i 1.18493i 0.805597 + 0.592464i \(0.201845\pi\)
−0.805597 + 0.592464i \(0.798155\pi\)
\(642\) 0 0
\(643\) − 32.0000i − 1.26196i −0.775800 0.630978i \(-0.782654\pi\)
0.775800 0.630978i \(-0.217346\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) −27.0000 −1.06066
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) − 24.0000i − 0.939913i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 0 0
\(655\) 32.0000 1.25034
\(656\) 6.00000i 0.234261i
\(657\) 18.0000i 0.702247i
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) −32.0000 −1.24091
\(666\) 6.00000i 0.232495i
\(667\) 24.0000 0.929284
\(668\) − 4.00000i − 0.154765i
\(669\) 0 0
\(670\) 8.00000i 0.309067i
\(671\) 0 0
\(672\) 0 0
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) 14.0000i 0.539260i
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.0000i 1.53056i 0.643699 + 0.765279i \(0.277399\pi\)
−0.643699 + 0.765279i \(0.722601\pi\)
\(684\) −12.0000 −0.458831
\(685\) − 12.0000i − 0.458496i
\(686\) − 8.00000i − 0.305441i
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 22.0000i 0.836315i
\(693\) 0 0
\(694\) 32.0000i 1.21470i
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 18.0000 0.681310
\(699\) 0 0
\(700\) − 4.00000i − 0.151186i
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) − 40.0000i − 1.50435i
\(708\) 0 0
\(709\) 34.0000i 1.27690i 0.769665 + 0.638448i \(0.220423\pi\)
−0.769665 + 0.638448i \(0.779577\pi\)
\(710\) −8.00000 −0.300235
\(711\) 36.0000i 1.35011i
\(712\) −30.0000 −1.12430
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) − 4.00000i − 0.149175i −0.997214 0.0745874i \(-0.976236\pi\)
0.997214 0.0745874i \(-0.0237640\pi\)
\(720\) − 6.00000i − 0.223607i
\(721\) 32.0000i 1.19174i
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 2.00000i 0.0743294i
\(725\) − 6.00000i − 0.222834i
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 24.0000i 0.889499i
\(729\) 27.0000 1.00000
\(730\) −12.0000 −0.444140
\(731\) 0 0
\(732\) 0 0
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 28.0000i 1.03350i
\(735\) 0 0
\(736\) 20.0000i 0.737210i
\(737\) 0 0
\(738\) − 18.0000i − 0.662589i
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) − 24.0000i − 0.881068i
\(743\) − 12.0000i − 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) 0 0
\(745\) − 20.0000i − 0.732743i
\(746\) 6.00000 0.219676
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) − 20.0000i − 0.729810i −0.931045 0.364905i \(-0.881101\pi\)
0.931045 0.364905i \(-0.118899\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 12.0000i 0.437014i
\(755\) 32.0000i 1.16460i
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 8.00000i 0.290573i
\(759\) 0 0
\(760\) − 24.0000i − 0.870572i
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) −24.0000 −0.868858
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 2.00000i − 0.0719816i
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −12.0000 −0.431331
\(775\) − 4.00000i − 0.143684i
\(776\) 6.00000i 0.215387i
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) − 24.0000i − 0.859889i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) − 4.00000i − 0.142766i
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) −24.0000 −0.853882
\(791\) 56.0000 1.99113
\(792\) 0 0
\(793\) 20.0000i 0.710221i
\(794\) 6.00000i 0.212932i
\(795\) 0 0
\(796\) − 20.0000i − 0.708881i
\(797\) 50.0000 1.77109 0.885545 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 30.0000 1.06000
\(802\) − 14.0000i − 0.494357i
\(803\) 0 0
\(804\) 0 0
\(805\) − 32.0000i − 1.12785i
\(806\) 8.00000i 0.281788i
\(807\) 0 0
\(808\) 30.0000 1.05540
\(809\) 26.0000i 0.914111i 0.889438 + 0.457056i \(0.151096\pi\)
−0.889438 + 0.457056i \(0.848904\pi\)
\(810\) 18.0000i 0.632456i
\(811\) − 40.0000i − 1.40459i −0.711886 0.702295i \(-0.752159\pi\)
0.711886 0.702295i \(-0.247841\pi\)
\(812\) −24.0000 −0.842235
\(813\) 0 0
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 26.0000 0.909069
\(819\) − 24.0000i − 0.838628i
\(820\) −12.0000 −0.419058
\(821\) 18.0000i 0.628204i 0.949389 + 0.314102i \(0.101703\pi\)
−0.949389 + 0.314102i \(0.898297\pi\)
\(822\) 0 0
\(823\) 20.0000i 0.697156i 0.937280 + 0.348578i \(0.113335\pi\)
−0.937280 + 0.348578i \(0.886665\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 48.0000i 1.67013i
\(827\) − 48.0000i − 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) − 12.0000i − 0.417029i
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 8.00000i 0.277684i
\(831\) 0 0
\(832\) −14.0000 −0.485363
\(833\) 0 0
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 8.00000i 0.276355i
\(839\) 20.0000i 0.690477i 0.938515 + 0.345238i \(0.112202\pi\)
−0.938515 + 0.345238i \(0.887798\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) − 8.00000i − 0.275371i
\(845\) − 18.0000i − 0.619219i
\(846\) 0 0
\(847\) 44.0000i 1.51186i
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 40.0000 1.36877
\(855\) 24.0000i 0.820783i
\(856\) 24.0000i 0.820303i
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 8.00000i 0.272798i
\(861\) 0 0
\(862\) 12.0000i 0.408722i
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) 44.0000 1.49604
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 18.0000i − 0.609557i
\(873\) − 6.00000i − 0.203069i
\(874\) 16.0000i 0.541208i
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) 6.00000i 0.202606i 0.994856 + 0.101303i \(0.0323011\pi\)
−0.994856 + 0.101303i \(0.967699\pi\)
\(878\) 20.0000i 0.674967i
\(879\) 0 0
\(880\) 0 0
\(881\) 46.0000i 1.54978i 0.632096 + 0.774890i \(0.282195\pi\)
−0.632096 + 0.774890i \(0.717805\pi\)
\(882\) −27.0000 −0.909137
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) 0 0
\(889\) − 32.0000i − 1.07325i
\(890\) 20.0000i 0.670402i
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) 0 0
\(894\) 0 0
\(895\) − 24.0000i − 0.802232i
\(896\) − 12.0000i − 0.400892i
\(897\) 0 0
\(898\) 34.0000i 1.13459i
\(899\) −24.0000 −0.800445
\(900\) −3.00000 −0.100000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 42.0000i 1.39690i
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 32.0000i 1.06254i 0.847202 + 0.531271i \(0.178286\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(908\) 24.0000i 0.796468i
\(909\) −30.0000 −0.995037
\(910\) 16.0000 0.530395
\(911\) − 4.00000i − 0.132526i −0.997802 0.0662630i \(-0.978892\pi\)
0.997802 0.0662630i \(-0.0211076\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 64.0000 2.11347
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 24.0000 0.791257
\(921\) 0 0
\(922\) 2.00000 0.0658665
\(923\) − 8.00000i − 0.263323i
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) 32.0000 1.05159
\(927\) 24.0000 0.788263
\(928\) − 30.0000i − 0.984798i
\(929\) − 30.0000i − 0.984268i −0.870519 0.492134i \(-0.836217\pi\)
0.870519 0.492134i \(-0.163783\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) − 6.00000i − 0.196537i
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 16.0000i 0.522419i
\(939\) 0 0
\(940\) 0 0
\(941\) 6.00000i 0.195594i 0.995206 + 0.0977972i \(0.0311797\pi\)
−0.995206 + 0.0977972i \(0.968820\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) − 32.0000i − 1.03986i −0.854209 0.519930i \(-0.825958\pi\)
0.854209 0.519930i \(-0.174042\pi\)
\(948\) 0 0
\(949\) − 12.0000i − 0.389536i
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −18.0000 −0.582772
\(955\) − 32.0000i − 1.03550i
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) − 36.0000i − 1.16311i
\(959\) − 24.0000i − 0.775000i
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) − 4.00000i − 0.128965i
\(963\) − 24.0000i − 0.773389i
\(964\) 18.0000i 0.579741i
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −33.0000 −1.06066
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 20.0000i 0.640841i
\(975\) 0 0
\(976\) 10.0000i 0.320092i
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 18.0000i 0.574989i
\(981\) 18.0000i 0.574696i
\(982\) −20.0000 −0.638226
\(983\) − 12.0000i − 0.382741i −0.981518 0.191370i \(-0.938707\pi\)
0.981518 0.191370i \(-0.0612931\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) − 16.0000i − 0.508770i
\(990\) 0 0
\(991\) 12.0000i 0.381193i 0.981669 + 0.190596i \(0.0610421\pi\)
−0.981669 + 0.190596i \(0.938958\pi\)
\(992\) − 20.0000i − 0.635001i
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) −40.0000 −1.26809
\(996\) 0 0
\(997\) 46.0000i 1.45683i 0.685134 + 0.728417i \(0.259744\pi\)
−0.685134 + 0.728417i \(0.740256\pi\)
\(998\) − 40.0000i − 1.26618i
\(999\) 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 289.2.b.a.288.2 2
17.2 even 8 289.2.c.a.251.1 4
17.3 odd 16 289.2.d.d.110.1 8
17.4 even 4 289.2.a.a.1.1 1
17.5 odd 16 289.2.d.d.179.2 8
17.6 odd 16 289.2.d.d.134.2 8
17.7 odd 16 289.2.d.d.155.1 8
17.8 even 8 289.2.c.a.38.1 4
17.9 even 8 289.2.c.a.38.2 4
17.10 odd 16 289.2.d.d.155.2 8
17.11 odd 16 289.2.d.d.134.1 8
17.12 odd 16 289.2.d.d.179.1 8
17.13 even 4 17.2.a.a.1.1 1
17.14 odd 16 289.2.d.d.110.2 8
17.15 even 8 289.2.c.a.251.2 4
17.16 even 2 inner 289.2.b.a.288.1 2
51.38 odd 4 2601.2.a.g.1.1 1
51.47 odd 4 153.2.a.c.1.1 1
68.47 odd 4 272.2.a.b.1.1 1
68.55 odd 4 4624.2.a.d.1.1 1
85.4 even 4 7225.2.a.g.1.1 1
85.13 odd 4 425.2.b.b.324.2 2
85.47 odd 4 425.2.b.b.324.1 2
85.64 even 4 425.2.a.d.1.1 1
119.13 odd 4 833.2.a.a.1.1 1
119.30 even 12 833.2.e.b.18.1 2
119.47 odd 12 833.2.e.a.18.1 2
119.81 even 12 833.2.e.b.324.1 2
119.115 odd 12 833.2.e.a.324.1 2
136.13 even 4 1088.2.a.i.1.1 1
136.115 odd 4 1088.2.a.h.1.1 1
187.98 odd 4 2057.2.a.e.1.1 1
204.47 even 4 2448.2.a.o.1.1 1
221.64 even 4 2873.2.a.c.1.1 1
255.149 odd 4 3825.2.a.d.1.1 1
323.132 odd 4 6137.2.a.b.1.1 1
340.319 odd 4 6800.2.a.n.1.1 1
357.251 even 4 7497.2.a.l.1.1 1
391.183 odd 4 8993.2.a.a.1.1 1
408.149 odd 4 9792.2.a.n.1.1 1
408.251 even 4 9792.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.a.a.1.1 1 17.13 even 4
153.2.a.c.1.1 1 51.47 odd 4
272.2.a.b.1.1 1 68.47 odd 4
289.2.a.a.1.1 1 17.4 even 4
289.2.b.a.288.1 2 17.16 even 2 inner
289.2.b.a.288.2 2 1.1 even 1 trivial
289.2.c.a.38.1 4 17.8 even 8
289.2.c.a.38.2 4 17.9 even 8
289.2.c.a.251.1 4 17.2 even 8
289.2.c.a.251.2 4 17.15 even 8
289.2.d.d.110.1 8 17.3 odd 16
289.2.d.d.110.2 8 17.14 odd 16
289.2.d.d.134.1 8 17.11 odd 16
289.2.d.d.134.2 8 17.6 odd 16
289.2.d.d.155.1 8 17.7 odd 16
289.2.d.d.155.2 8 17.10 odd 16
289.2.d.d.179.1 8 17.12 odd 16
289.2.d.d.179.2 8 17.5 odd 16
425.2.a.d.1.1 1 85.64 even 4
425.2.b.b.324.1 2 85.47 odd 4
425.2.b.b.324.2 2 85.13 odd 4
833.2.a.a.1.1 1 119.13 odd 4
833.2.e.a.18.1 2 119.47 odd 12
833.2.e.a.324.1 2 119.115 odd 12
833.2.e.b.18.1 2 119.30 even 12
833.2.e.b.324.1 2 119.81 even 12
1088.2.a.h.1.1 1 136.115 odd 4
1088.2.a.i.1.1 1 136.13 even 4
2057.2.a.e.1.1 1 187.98 odd 4
2448.2.a.o.1.1 1 204.47 even 4
2601.2.a.g.1.1 1 51.38 odd 4
2873.2.a.c.1.1 1 221.64 even 4
3825.2.a.d.1.1 1 255.149 odd 4
4624.2.a.d.1.1 1 68.55 odd 4
6137.2.a.b.1.1 1 323.132 odd 4
6800.2.a.n.1.1 1 340.319 odd 4
7225.2.a.g.1.1 1 85.4 even 4
7497.2.a.l.1.1 1 357.251 even 4
8993.2.a.a.1.1 1 391.183 odd 4
9792.2.a.i.1.1 1 408.251 even 4
9792.2.a.n.1.1 1 408.149 odd 4