Properties

Label 1216.2.c.h.609.6
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.6
Root \(-0.396143 + 1.68614i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.h.609.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.37228i q^{3} +0.792287i q^{5} +3.31662 q^{7} -2.62772 q^{9} +O(q^{10})\) \(q+2.37228i q^{3} +0.792287i q^{5} +3.31662 q^{7} -2.62772 q^{9} -3.37228i q^{11} +4.10891i q^{13} -1.87953 q^{15} +5.00000 q^{17} +1.00000i q^{19} +7.86797i q^{21} +2.52434 q^{23} +4.37228 q^{25} +0.883156i q^{27} +5.98844i q^{29} -3.46410 q^{31} +8.00000 q^{33} +2.62772i q^{35} +3.16915i q^{37} -9.74749 q^{39} -9.37228i q^{43} -2.08191i q^{45} -9.30506 q^{47} +4.00000 q^{49} +11.8614i q^{51} -9.45254i q^{53} +2.67181 q^{55} -2.37228 q^{57} -4.37228i q^{59} +4.55134i q^{61} -8.71516 q^{63} -3.25544 q^{65} +8.37228i q^{67} +5.98844i q^{69} -6.63325 q^{71} -14.4891 q^{73} +10.3723i q^{75} -11.1846i q^{77} +13.5615 q^{79} -9.97825 q^{81} -8.00000i q^{83} +3.96143i q^{85} -14.2063 q^{87} +7.48913 q^{89} +13.6277i q^{91} -8.21782i q^{93} -0.792287 q^{95} +8.74456 q^{97} +8.86141i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 44q^{9} + O(q^{10}) \) \( 8q - 44q^{9} + 40q^{17} + 12q^{25} + 64q^{33} + 32q^{49} + 4q^{57} - 72q^{65} - 24q^{73} + 104q^{81} - 32q^{89} + 24q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.37228i 1.36964i 0.728714 + 0.684819i \(0.240119\pi\)
−0.728714 + 0.684819i \(0.759881\pi\)
\(4\) 0 0
\(5\) 0.792287i 0.354322i 0.984182 + 0.177161i \(0.0566913\pi\)
−0.984182 + 0.177161i \(0.943309\pi\)
\(6\) 0 0
\(7\) 3.31662 1.25357 0.626783 0.779194i \(-0.284371\pi\)
0.626783 + 0.779194i \(0.284371\pi\)
\(8\) 0 0
\(9\) −2.62772 −0.875906
\(10\) 0 0
\(11\) − 3.37228i − 1.01678i −0.861127 0.508391i \(-0.830241\pi\)
0.861127 0.508391i \(-0.169759\pi\)
\(12\) 0 0
\(13\) 4.10891i 1.13961i 0.821781 + 0.569804i \(0.192981\pi\)
−0.821781 + 0.569804i \(0.807019\pi\)
\(14\) 0 0
\(15\) −1.87953 −0.485292
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 7.86797i 1.71693i
\(22\) 0 0
\(23\) 2.52434 0.526361 0.263180 0.964747i \(-0.415229\pi\)
0.263180 + 0.964747i \(0.415229\pi\)
\(24\) 0 0
\(25\) 4.37228 0.874456
\(26\) 0 0
\(27\) 0.883156i 0.169963i
\(28\) 0 0
\(29\) 5.98844i 1.11203i 0.831174 + 0.556013i \(0.187669\pi\)
−0.831174 + 0.556013i \(0.812331\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 2.62772i 0.444166i
\(36\) 0 0
\(37\) 3.16915i 0.521005i 0.965473 + 0.260502i \(0.0838882\pi\)
−0.965473 + 0.260502i \(0.916112\pi\)
\(38\) 0 0
\(39\) −9.74749 −1.56085
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 9.37228i − 1.42926i −0.699503 0.714630i \(-0.746595\pi\)
0.699503 0.714630i \(-0.253405\pi\)
\(44\) 0 0
\(45\) − 2.08191i − 0.310352i
\(46\) 0 0
\(47\) −9.30506 −1.35728 −0.678642 0.734470i \(-0.737431\pi\)
−0.678642 + 0.734470i \(0.737431\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 11.8614i 1.66093i
\(52\) 0 0
\(53\) − 9.45254i − 1.29841i −0.760615 0.649203i \(-0.775102\pi\)
0.760615 0.649203i \(-0.224898\pi\)
\(54\) 0 0
\(55\) 2.67181 0.360267
\(56\) 0 0
\(57\) −2.37228 −0.314216
\(58\) 0 0
\(59\) − 4.37228i − 0.569223i −0.958643 0.284611i \(-0.908135\pi\)
0.958643 0.284611i \(-0.0918645\pi\)
\(60\) 0 0
\(61\) 4.55134i 0.582740i 0.956610 + 0.291370i \(0.0941110\pi\)
−0.956610 + 0.291370i \(0.905889\pi\)
\(62\) 0 0
\(63\) −8.71516 −1.09801
\(64\) 0 0
\(65\) −3.25544 −0.403787
\(66\) 0 0
\(67\) 8.37228i 1.02284i 0.859332 + 0.511418i \(0.170880\pi\)
−0.859332 + 0.511418i \(0.829120\pi\)
\(68\) 0 0
\(69\) 5.98844i 0.720923i
\(70\) 0 0
\(71\) −6.63325 −0.787222 −0.393611 0.919277i \(-0.628774\pi\)
−0.393611 + 0.919277i \(0.628774\pi\)
\(72\) 0 0
\(73\) −14.4891 −1.69582 −0.847912 0.530137i \(-0.822140\pi\)
−0.847912 + 0.530137i \(0.822140\pi\)
\(74\) 0 0
\(75\) 10.3723i 1.19769i
\(76\) 0 0
\(77\) − 11.1846i − 1.27460i
\(78\) 0 0
\(79\) 13.5615 1.52578 0.762891 0.646527i \(-0.223779\pi\)
0.762891 + 0.646527i \(0.223779\pi\)
\(80\) 0 0
\(81\) −9.97825 −1.10869
\(82\) 0 0
\(83\) − 8.00000i − 0.878114i −0.898459 0.439057i \(-0.855313\pi\)
0.898459 0.439057i \(-0.144687\pi\)
\(84\) 0 0
\(85\) 3.96143i 0.429678i
\(86\) 0 0
\(87\) −14.2063 −1.52307
\(88\) 0 0
\(89\) 7.48913 0.793846 0.396923 0.917852i \(-0.370078\pi\)
0.396923 + 0.917852i \(0.370078\pi\)
\(90\) 0 0
\(91\) 13.6277i 1.42857i
\(92\) 0 0
\(93\) − 8.21782i − 0.852149i
\(94\) 0 0
\(95\) −0.792287 −0.0812869
\(96\) 0 0
\(97\) 8.74456 0.887876 0.443938 0.896058i \(-0.353581\pi\)
0.443938 + 0.896058i \(0.353581\pi\)
\(98\) 0 0
\(99\) 8.86141i 0.890605i
\(100\) 0 0
\(101\) 13.8564i 1.37876i 0.724398 + 0.689382i \(0.242118\pi\)
−0.724398 + 0.689382i \(0.757882\pi\)
\(102\) 0 0
\(103\) −5.34363 −0.526523 −0.263262 0.964724i \(-0.584798\pi\)
−0.263262 + 0.964724i \(0.584798\pi\)
\(104\) 0 0
\(105\) −6.23369 −0.608346
\(106\) 0 0
\(107\) − 7.62772i − 0.737399i −0.929549 0.368700i \(-0.879803\pi\)
0.929549 0.368700i \(-0.120197\pi\)
\(108\) 0 0
\(109\) 12.6217i 1.20894i 0.796628 + 0.604469i \(0.206615\pi\)
−0.796628 + 0.604469i \(0.793385\pi\)
\(110\) 0 0
\(111\) −7.51811 −0.713587
\(112\) 0 0
\(113\) −3.25544 −0.306246 −0.153123 0.988207i \(-0.548933\pi\)
−0.153123 + 0.988207i \(0.548933\pi\)
\(114\) 0 0
\(115\) 2.00000i 0.186501i
\(116\) 0 0
\(117\) − 10.7971i − 0.998189i
\(118\) 0 0
\(119\) 16.5831 1.52017
\(120\) 0 0
\(121\) −0.372281 −0.0338438
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.42554i 0.664160i
\(126\) 0 0
\(127\) −18.9051 −1.67755 −0.838777 0.544475i \(-0.816729\pi\)
−0.838777 + 0.544475i \(0.816729\pi\)
\(128\) 0 0
\(129\) 22.2337 1.95757
\(130\) 0 0
\(131\) 1.37228i 0.119897i 0.998201 + 0.0599484i \(0.0190936\pi\)
−0.998201 + 0.0599484i \(0.980906\pi\)
\(132\) 0 0
\(133\) 3.31662i 0.287588i
\(134\) 0 0
\(135\) −0.699713 −0.0602217
\(136\) 0 0
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) − 9.37228i − 0.794947i −0.917614 0.397473i \(-0.869887\pi\)
0.917614 0.397473i \(-0.130113\pi\)
\(140\) 0 0
\(141\) − 22.0742i − 1.85899i
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) −4.74456 −0.394014
\(146\) 0 0
\(147\) 9.48913i 0.782650i
\(148\) 0 0
\(149\) − 21.2819i − 1.74348i −0.489965 0.871742i \(-0.662990\pi\)
0.489965 0.871742i \(-0.337010\pi\)
\(150\) 0 0
\(151\) −22.3692 −1.82038 −0.910189 0.414193i \(-0.864064\pi\)
−0.910189 + 0.414193i \(0.864064\pi\)
\(152\) 0 0
\(153\) −13.1386 −1.06219
\(154\) 0 0
\(155\) − 2.74456i − 0.220449i
\(156\) 0 0
\(157\) − 1.28962i − 0.102923i −0.998675 0.0514615i \(-0.983612\pi\)
0.998675 0.0514615i \(-0.0163879\pi\)
\(158\) 0 0
\(159\) 22.4241 1.77835
\(160\) 0 0
\(161\) 8.37228 0.659828
\(162\) 0 0
\(163\) − 0.744563i − 0.0583186i −0.999575 0.0291593i \(-0.990717\pi\)
0.999575 0.0291593i \(-0.00928302\pi\)
\(164\) 0 0
\(165\) 6.33830i 0.493436i
\(166\) 0 0
\(167\) 20.1947 1.56271 0.781356 0.624085i \(-0.214528\pi\)
0.781356 + 0.624085i \(0.214528\pi\)
\(168\) 0 0
\(169\) −3.88316 −0.298704
\(170\) 0 0
\(171\) − 2.62772i − 0.200947i
\(172\) 0 0
\(173\) 4.75372i 0.361419i 0.983537 + 0.180709i \(0.0578393\pi\)
−0.983537 + 0.180709i \(0.942161\pi\)
\(174\) 0 0
\(175\) 14.5012 1.09619
\(176\) 0 0
\(177\) 10.3723 0.779628
\(178\) 0 0
\(179\) − 22.9783i − 1.71748i −0.512416 0.858738i \(-0.671249\pi\)
0.512416 0.858738i \(-0.328751\pi\)
\(180\) 0 0
\(181\) 11.6819i 0.868311i 0.900838 + 0.434155i \(0.142953\pi\)
−0.900838 + 0.434155i \(0.857047\pi\)
\(182\) 0 0
\(183\) −10.7971 −0.798142
\(184\) 0 0
\(185\) −2.51087 −0.184603
\(186\) 0 0
\(187\) − 16.8614i − 1.23303i
\(188\) 0 0
\(189\) 2.92910i 0.213060i
\(190\) 0 0
\(191\) 13.7089 0.991943 0.495972 0.868339i \(-0.334812\pi\)
0.495972 + 0.868339i \(0.334812\pi\)
\(192\) 0 0
\(193\) 14.7446 1.06134 0.530668 0.847580i \(-0.321941\pi\)
0.530668 + 0.847580i \(0.321941\pi\)
\(194\) 0 0
\(195\) − 7.72281i − 0.553042i
\(196\) 0 0
\(197\) − 20.1947i − 1.43881i −0.694589 0.719406i \(-0.744414\pi\)
0.694589 0.719406i \(-0.255586\pi\)
\(198\) 0 0
\(199\) 6.78073 0.480673 0.240336 0.970690i \(-0.422742\pi\)
0.240336 + 0.970690i \(0.422742\pi\)
\(200\) 0 0
\(201\) −19.8614 −1.40092
\(202\) 0 0
\(203\) 19.8614i 1.39400i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.63325 −0.461043
\(208\) 0 0
\(209\) 3.37228 0.233266
\(210\) 0 0
\(211\) − 8.37228i − 0.576372i −0.957575 0.288186i \(-0.906948\pi\)
0.957575 0.288186i \(-0.0930521\pi\)
\(212\) 0 0
\(213\) − 15.7359i − 1.07821i
\(214\) 0 0
\(215\) 7.42554 0.506417
\(216\) 0 0
\(217\) −11.4891 −0.779933
\(218\) 0 0
\(219\) − 34.3723i − 2.32266i
\(220\) 0 0
\(221\) 20.5446i 1.38198i
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) −11.4891 −0.765942
\(226\) 0 0
\(227\) 3.11684i 0.206872i 0.994636 + 0.103436i \(0.0329837\pi\)
−0.994636 + 0.103436i \(0.967016\pi\)
\(228\) 0 0
\(229\) 3.96143i 0.261779i 0.991397 + 0.130889i \(0.0417833\pi\)
−0.991397 + 0.130889i \(0.958217\pi\)
\(230\) 0 0
\(231\) 26.5330 1.74574
\(232\) 0 0
\(233\) −28.1168 −1.84200 −0.920998 0.389568i \(-0.872624\pi\)
−0.920998 + 0.389568i \(0.872624\pi\)
\(234\) 0 0
\(235\) − 7.37228i − 0.480915i
\(236\) 0 0
\(237\) 32.1716i 2.08977i
\(238\) 0 0
\(239\) 24.1012 1.55898 0.779490 0.626415i \(-0.215478\pi\)
0.779490 + 0.626415i \(0.215478\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) − 21.0217i − 1.34855i
\(244\) 0 0
\(245\) 3.16915i 0.202469i
\(246\) 0 0
\(247\) −4.10891 −0.261444
\(248\) 0 0
\(249\) 18.9783 1.20270
\(250\) 0 0
\(251\) 10.8614i 0.685566i 0.939415 + 0.342783i \(0.111370\pi\)
−0.939415 + 0.342783i \(0.888630\pi\)
\(252\) 0 0
\(253\) − 8.51278i − 0.535194i
\(254\) 0 0
\(255\) −9.39764 −0.588503
\(256\) 0 0
\(257\) 18.2337 1.13739 0.568693 0.822550i \(-0.307449\pi\)
0.568693 + 0.822550i \(0.307449\pi\)
\(258\) 0 0
\(259\) 10.5109i 0.653114i
\(260\) 0 0
\(261\) − 15.7359i − 0.974030i
\(262\) 0 0
\(263\) 24.1561 1.48953 0.744766 0.667326i \(-0.232561\pi\)
0.744766 + 0.667326i \(0.232561\pi\)
\(264\) 0 0
\(265\) 7.48913 0.460053
\(266\) 0 0
\(267\) 17.7663i 1.08728i
\(268\) 0 0
\(269\) − 8.51278i − 0.519033i −0.965739 0.259517i \(-0.916437\pi\)
0.965739 0.259517i \(-0.0835632\pi\)
\(270\) 0 0
\(271\) 7.57301 0.460028 0.230014 0.973187i \(-0.426123\pi\)
0.230014 + 0.973187i \(0.426123\pi\)
\(272\) 0 0
\(273\) −32.3288 −1.95663
\(274\) 0 0
\(275\) − 14.7446i − 0.889131i
\(276\) 0 0
\(277\) 13.0641i 0.784947i 0.919763 + 0.392473i \(0.128381\pi\)
−0.919763 + 0.392473i \(0.871619\pi\)
\(278\) 0 0
\(279\) 9.10268 0.544963
\(280\) 0 0
\(281\) −28.2337 −1.68428 −0.842140 0.539258i \(-0.818705\pi\)
−0.842140 + 0.539258i \(0.818705\pi\)
\(282\) 0 0
\(283\) 23.3723i 1.38934i 0.719330 + 0.694669i \(0.244449\pi\)
−0.719330 + 0.694669i \(0.755551\pi\)
\(284\) 0 0
\(285\) − 1.87953i − 0.111334i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 20.7446i 1.21607i
\(292\) 0 0
\(293\) − 28.0627i − 1.63944i −0.572765 0.819719i \(-0.694129\pi\)
0.572765 0.819719i \(-0.305871\pi\)
\(294\) 0 0
\(295\) 3.46410 0.201688
\(296\) 0 0
\(297\) 2.97825 0.172816
\(298\) 0 0
\(299\) 10.3723i 0.599845i
\(300\) 0 0
\(301\) − 31.0843i − 1.79167i
\(302\) 0 0
\(303\) −32.8713 −1.88841
\(304\) 0 0
\(305\) −3.60597 −0.206477
\(306\) 0 0
\(307\) − 26.7446i − 1.52639i −0.646167 0.763196i \(-0.723629\pi\)
0.646167 0.763196i \(-0.276371\pi\)
\(308\) 0 0
\(309\) − 12.6766i − 0.721146i
\(310\) 0 0
\(311\) 23.2164 1.31648 0.658240 0.752808i \(-0.271301\pi\)
0.658240 + 0.752808i \(0.271301\pi\)
\(312\) 0 0
\(313\) 21.1168 1.19359 0.596797 0.802392i \(-0.296440\pi\)
0.596797 + 0.802392i \(0.296440\pi\)
\(314\) 0 0
\(315\) − 6.90491i − 0.389047i
\(316\) 0 0
\(317\) − 18.2603i − 1.02560i −0.858508 0.512800i \(-0.828608\pi\)
0.858508 0.512800i \(-0.171392\pi\)
\(318\) 0 0
\(319\) 20.1947 1.13069
\(320\) 0 0
\(321\) 18.0951 1.00997
\(322\) 0 0
\(323\) 5.00000i 0.278207i
\(324\) 0 0
\(325\) 17.9653i 0.996537i
\(326\) 0 0
\(327\) −29.9422 −1.65581
\(328\) 0 0
\(329\) −30.8614 −1.70144
\(330\) 0 0
\(331\) 17.8614i 0.981752i 0.871230 + 0.490876i \(0.163323\pi\)
−0.871230 + 0.490876i \(0.836677\pi\)
\(332\) 0 0
\(333\) − 8.32763i − 0.456351i
\(334\) 0 0
\(335\) −6.63325 −0.362413
\(336\) 0 0
\(337\) −24.7446 −1.34792 −0.673961 0.738767i \(-0.735408\pi\)
−0.673961 + 0.738767i \(0.735408\pi\)
\(338\) 0 0
\(339\) − 7.72281i − 0.419446i
\(340\) 0 0
\(341\) 11.6819i 0.632612i
\(342\) 0 0
\(343\) −9.94987 −0.537243
\(344\) 0 0
\(345\) −4.74456 −0.255439
\(346\) 0 0
\(347\) − 19.8832i − 1.06738i −0.845679 0.533692i \(-0.820804\pi\)
0.845679 0.533692i \(-0.179196\pi\)
\(348\) 0 0
\(349\) − 23.1615i − 1.23981i −0.784679 0.619903i \(-0.787172\pi\)
0.784679 0.619903i \(-0.212828\pi\)
\(350\) 0 0
\(351\) −3.62881 −0.193692
\(352\) 0 0
\(353\) 21.1168 1.12394 0.561968 0.827159i \(-0.310044\pi\)
0.561968 + 0.827159i \(0.310044\pi\)
\(354\) 0 0
\(355\) − 5.25544i − 0.278930i
\(356\) 0 0
\(357\) 39.3398i 2.08208i
\(358\) 0 0
\(359\) −16.5831 −0.875224 −0.437612 0.899164i \(-0.644176\pi\)
−0.437612 + 0.899164i \(0.644176\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) − 0.883156i − 0.0463537i
\(364\) 0 0
\(365\) − 11.4795i − 0.600867i
\(366\) 0 0
\(367\) −30.5870 −1.59663 −0.798314 0.602241i \(-0.794275\pi\)
−0.798314 + 0.602241i \(0.794275\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 31.3505i − 1.62764i
\(372\) 0 0
\(373\) − 24.8935i − 1.28894i −0.764631 0.644469i \(-0.777079\pi\)
0.764631 0.644469i \(-0.222921\pi\)
\(374\) 0 0
\(375\) −17.6155 −0.909659
\(376\) 0 0
\(377\) −24.6060 −1.26727
\(378\) 0 0
\(379\) 15.3505i 0.788504i 0.919002 + 0.394252i \(0.128996\pi\)
−0.919002 + 0.394252i \(0.871004\pi\)
\(380\) 0 0
\(381\) − 44.8482i − 2.29764i
\(382\) 0 0
\(383\) −7.92287 −0.404840 −0.202420 0.979299i \(-0.564881\pi\)
−0.202420 + 0.979299i \(0.564881\pi\)
\(384\) 0 0
\(385\) 8.86141 0.451619
\(386\) 0 0
\(387\) 24.6277i 1.25190i
\(388\) 0 0
\(389\) 12.1793i 0.617513i 0.951141 + 0.308756i \(0.0999128\pi\)
−0.951141 + 0.308756i \(0.900087\pi\)
\(390\) 0 0
\(391\) 12.6217 0.638306
\(392\) 0 0
\(393\) −3.25544 −0.164215
\(394\) 0 0
\(395\) 10.7446i 0.540618i
\(396\) 0 0
\(397\) 9.30506i 0.467008i 0.972356 + 0.233504i \(0.0750192\pi\)
−0.972356 + 0.233504i \(0.924981\pi\)
\(398\) 0 0
\(399\) −7.86797 −0.393891
\(400\) 0 0
\(401\) −21.7228 −1.08479 −0.542393 0.840125i \(-0.682482\pi\)
−0.542393 + 0.840125i \(0.682482\pi\)
\(402\) 0 0
\(403\) − 14.2337i − 0.709030i
\(404\) 0 0
\(405\) − 7.90564i − 0.392834i
\(406\) 0 0
\(407\) 10.6873 0.529748
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 11.8614i 0.585080i
\(412\) 0 0
\(413\) − 14.5012i − 0.713558i
\(414\) 0 0
\(415\) 6.33830 0.311135
\(416\) 0 0
\(417\) 22.2337 1.08879
\(418\) 0 0
\(419\) − 34.2337i − 1.67243i −0.548405 0.836213i \(-0.684765\pi\)
0.548405 0.836213i \(-0.315235\pi\)
\(420\) 0 0
\(421\) − 35.9855i − 1.75383i −0.480647 0.876914i \(-0.659598\pi\)
0.480647 0.876914i \(-0.340402\pi\)
\(422\) 0 0
\(423\) 24.4511 1.18885
\(424\) 0 0
\(425\) 21.8614 1.06043
\(426\) 0 0
\(427\) 15.0951i 0.730503i
\(428\) 0 0
\(429\) 32.8713i 1.58704i
\(430\) 0 0
\(431\) −11.3870 −0.548491 −0.274246 0.961660i \(-0.588428\pi\)
−0.274246 + 0.961660i \(0.588428\pi\)
\(432\) 0 0
\(433\) 21.4891 1.03270 0.516351 0.856377i \(-0.327290\pi\)
0.516351 + 0.856377i \(0.327290\pi\)
\(434\) 0 0
\(435\) − 11.2554i − 0.539657i
\(436\) 0 0
\(437\) 2.52434i 0.120755i
\(438\) 0 0
\(439\) 6.92820 0.330665 0.165333 0.986238i \(-0.447130\pi\)
0.165333 + 0.986238i \(0.447130\pi\)
\(440\) 0 0
\(441\) −10.5109 −0.500518
\(442\) 0 0
\(443\) 10.3505i 0.491769i 0.969299 + 0.245884i \(0.0790783\pi\)
−0.969299 + 0.245884i \(0.920922\pi\)
\(444\) 0 0
\(445\) 5.93354i 0.281277i
\(446\) 0 0
\(447\) 50.4868 2.38794
\(448\) 0 0
\(449\) 14.5109 0.684811 0.342405 0.939552i \(-0.388758\pi\)
0.342405 + 0.939552i \(0.388758\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 53.0660i − 2.49326i
\(454\) 0 0
\(455\) −10.7971 −0.506174
\(456\) 0 0
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 0 0
\(459\) 4.41578i 0.206111i
\(460\) 0 0
\(461\) − 11.1846i − 0.520918i −0.965485 0.260459i \(-0.916126\pi\)
0.965485 0.260459i \(-0.0838739\pi\)
\(462\) 0 0
\(463\) −8.31040 −0.386217 −0.193108 0.981177i \(-0.561857\pi\)
−0.193108 + 0.981177i \(0.561857\pi\)
\(464\) 0 0
\(465\) 6.51087 0.301935
\(466\) 0 0
\(467\) 28.1168i 1.30109i 0.759467 + 0.650546i \(0.225460\pi\)
−0.759467 + 0.650546i \(0.774540\pi\)
\(468\) 0 0
\(469\) 27.7677i 1.28219i
\(470\) 0 0
\(471\) 3.05934 0.140967
\(472\) 0 0
\(473\) −31.6060 −1.45324
\(474\) 0 0
\(475\) 4.37228i 0.200614i
\(476\) 0 0
\(477\) 24.8386i 1.13728i
\(478\) 0 0
\(479\) −14.1514 −0.646592 −0.323296 0.946298i \(-0.604791\pi\)
−0.323296 + 0.946298i \(0.604791\pi\)
\(480\) 0 0
\(481\) −13.0217 −0.593741
\(482\) 0 0
\(483\) 19.8614i 0.903725i
\(484\) 0 0
\(485\) 6.92820i 0.314594i
\(486\) 0 0
\(487\) −29.2974 −1.32759 −0.663796 0.747914i \(-0.731056\pi\)
−0.663796 + 0.747914i \(0.731056\pi\)
\(488\) 0 0
\(489\) 1.76631 0.0798754
\(490\) 0 0
\(491\) 14.9783i 0.675959i 0.941153 + 0.337979i \(0.109743\pi\)
−0.941153 + 0.337979i \(0.890257\pi\)
\(492\) 0 0
\(493\) 29.9422i 1.34853i
\(494\) 0 0
\(495\) −7.02078 −0.315560
\(496\) 0 0
\(497\) −22.0000 −0.986835
\(498\) 0 0
\(499\) 1.37228i 0.0614317i 0.999528 + 0.0307159i \(0.00977870\pi\)
−0.999528 + 0.0307159i \(0.990221\pi\)
\(500\) 0 0
\(501\) 47.9075i 2.14035i
\(502\) 0 0
\(503\) 21.4294 0.955491 0.477745 0.878498i \(-0.341454\pi\)
0.477745 + 0.878498i \(0.341454\pi\)
\(504\) 0 0
\(505\) −10.9783 −0.488526
\(506\) 0 0
\(507\) − 9.21194i − 0.409117i
\(508\) 0 0
\(509\) 24.5437i 1.08788i 0.839124 + 0.543939i \(0.183068\pi\)
−0.839124 + 0.543939i \(0.816932\pi\)
\(510\) 0 0
\(511\) −48.0550 −2.12583
\(512\) 0 0
\(513\) −0.883156 −0.0389923
\(514\) 0 0
\(515\) − 4.23369i − 0.186559i
\(516\) 0 0
\(517\) 31.3793i 1.38006i
\(518\) 0 0
\(519\) −11.2772 −0.495013
\(520\) 0 0
\(521\) −40.2337 −1.76267 −0.881335 0.472492i \(-0.843355\pi\)
−0.881335 + 0.472492i \(0.843355\pi\)
\(522\) 0 0
\(523\) 19.6277i 0.858260i 0.903243 + 0.429130i \(0.141180\pi\)
−0.903243 + 0.429130i \(0.858820\pi\)
\(524\) 0 0
\(525\) 34.4010i 1.50138i
\(526\) 0 0
\(527\) −17.3205 −0.754493
\(528\) 0 0
\(529\) −16.6277 −0.722944
\(530\) 0 0
\(531\) 11.4891i 0.498586i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.04334 0.261276
\(536\) 0 0
\(537\) 54.5109 2.35232
\(538\) 0 0
\(539\) − 13.4891i − 0.581018i
\(540\) 0 0
\(541\) 33.5538i 1.44259i 0.692628 + 0.721295i \(0.256453\pi\)
−0.692628 + 0.721295i \(0.743547\pi\)
\(542\) 0 0
\(543\) −27.7128 −1.18927
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) − 25.4891i − 1.08984i −0.838489 0.544918i \(-0.816561\pi\)
0.838489 0.544918i \(-0.183439\pi\)
\(548\) 0 0
\(549\) − 11.9596i − 0.510425i
\(550\) 0 0
\(551\) −5.98844 −0.255116
\(552\) 0 0
\(553\) 44.9783 1.91267
\(554\) 0 0
\(555\) − 5.95650i − 0.252839i
\(556\) 0 0
\(557\) − 28.5051i − 1.20780i −0.797060 0.603900i \(-0.793613\pi\)
0.797060 0.603900i \(-0.206387\pi\)
\(558\) 0 0
\(559\) 38.5099 1.62879
\(560\) 0 0
\(561\) 40.0000 1.68880
\(562\) 0 0
\(563\) 9.48913i 0.399919i 0.979804 + 0.199959i \(0.0640811\pi\)
−0.979804 + 0.199959i \(0.935919\pi\)
\(564\) 0 0
\(565\) − 2.57924i − 0.108509i
\(566\) 0 0
\(567\) −33.0941 −1.38982
\(568\) 0 0
\(569\) 18.9783 0.795610 0.397805 0.917470i \(-0.369772\pi\)
0.397805 + 0.917470i \(0.369772\pi\)
\(570\) 0 0
\(571\) 2.97825i 0.124636i 0.998056 + 0.0623180i \(0.0198493\pi\)
−0.998056 + 0.0623180i \(0.980151\pi\)
\(572\) 0 0
\(573\) 32.5214i 1.35860i
\(574\) 0 0
\(575\) 11.0371 0.460280
\(576\) 0 0
\(577\) −35.4674 −1.47653 −0.738263 0.674513i \(-0.764354\pi\)
−0.738263 + 0.674513i \(0.764354\pi\)
\(578\) 0 0
\(579\) 34.9783i 1.45365i
\(580\) 0 0
\(581\) − 26.5330i − 1.10077i
\(582\) 0 0
\(583\) −31.8766 −1.32020
\(584\) 0 0
\(585\) 8.55437 0.353680
\(586\) 0 0
\(587\) 3.13859i 0.129544i 0.997900 + 0.0647718i \(0.0206319\pi\)
−0.997900 + 0.0647718i \(0.979368\pi\)
\(588\) 0 0
\(589\) − 3.46410i − 0.142736i
\(590\) 0 0
\(591\) 47.9075 1.97065
\(592\) 0 0
\(593\) −40.9783 −1.68278 −0.841388 0.540432i \(-0.818261\pi\)
−0.841388 + 0.540432i \(0.818261\pi\)
\(594\) 0 0
\(595\) 13.1386i 0.538630i
\(596\) 0 0
\(597\) 16.0858i 0.658348i
\(598\) 0 0
\(599\) 33.0564 1.35065 0.675325 0.737520i \(-0.264003\pi\)
0.675325 + 0.737520i \(0.264003\pi\)
\(600\) 0 0
\(601\) 46.4674 1.89544 0.947722 0.319097i \(-0.103380\pi\)
0.947722 + 0.319097i \(0.103380\pi\)
\(602\) 0 0
\(603\) − 22.0000i − 0.895909i
\(604\) 0 0
\(605\) − 0.294954i − 0.0119916i
\(606\) 0 0
\(607\) −7.62792 −0.309608 −0.154804 0.987945i \(-0.549475\pi\)
−0.154804 + 0.987945i \(0.549475\pi\)
\(608\) 0 0
\(609\) −47.1168 −1.90927
\(610\) 0 0
\(611\) − 38.2337i − 1.54677i
\(612\) 0 0
\(613\) 35.1383i 1.41922i 0.704592 + 0.709612i \(0.251130\pi\)
−0.704592 + 0.709612i \(0.748870\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.8397 1.36233 0.681167 0.732128i \(-0.261473\pi\)
0.681167 + 0.732128i \(0.261473\pi\)
\(618\) 0 0
\(619\) − 18.2337i − 0.732874i −0.930443 0.366437i \(-0.880578\pi\)
0.930443 0.366437i \(-0.119422\pi\)
\(620\) 0 0
\(621\) 2.22938i 0.0894621i
\(622\) 0 0
\(623\) 24.8386 0.995138
\(624\) 0 0
\(625\) 15.9783 0.639130
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) 15.8457i 0.631811i
\(630\) 0 0
\(631\) −11.8843 −0.473107 −0.236553 0.971619i \(-0.576018\pi\)
−0.236553 + 0.971619i \(0.576018\pi\)
\(632\) 0 0
\(633\) 19.8614 0.789420
\(634\) 0 0
\(635\) − 14.9783i − 0.594394i
\(636\) 0 0
\(637\) 16.4356i 0.651204i
\(638\) 0 0
\(639\) 17.4303 0.689533
\(640\) 0 0
\(641\) 34.9783 1.38156 0.690779 0.723066i \(-0.257268\pi\)
0.690779 + 0.723066i \(0.257268\pi\)
\(642\) 0 0
\(643\) 30.6277i 1.20784i 0.797045 + 0.603920i \(0.206395\pi\)
−0.797045 + 0.603920i \(0.793605\pi\)
\(644\) 0 0
\(645\) 17.6155i 0.693608i
\(646\) 0 0
\(647\) −16.5831 −0.651950 −0.325975 0.945378i \(-0.605693\pi\)
−0.325975 + 0.945378i \(0.605693\pi\)
\(648\) 0 0
\(649\) −14.7446 −0.578775
\(650\) 0 0
\(651\) − 27.2554i − 1.06822i
\(652\) 0 0
\(653\) − 47.5200i − 1.85960i −0.368064 0.929800i \(-0.619979\pi\)
0.368064 0.929800i \(-0.380021\pi\)
\(654\) 0 0
\(655\) −1.08724 −0.0424820
\(656\) 0 0
\(657\) 38.0733 1.48538
\(658\) 0 0
\(659\) − 18.1386i − 0.706579i −0.935514 0.353290i \(-0.885063\pi\)
0.935514 0.353290i \(-0.114937\pi\)
\(660\) 0 0
\(661\) 19.2549i 0.748930i 0.927241 + 0.374465i \(0.122174\pi\)
−0.927241 + 0.374465i \(0.877826\pi\)
\(662\) 0 0
\(663\) −48.7375 −1.89281
\(664\) 0 0
\(665\) −2.62772 −0.101899
\(666\) 0 0
\(667\) 15.1168i 0.585327i
\(668\) 0 0
\(669\) 16.4356i 0.635439i
\(670\) 0 0
\(671\) 15.3484 0.592519
\(672\) 0 0
\(673\) 5.25544 0.202582 0.101291 0.994857i \(-0.467703\pi\)
0.101291 + 0.994857i \(0.467703\pi\)
\(674\) 0 0
\(675\) 3.86141i 0.148626i
\(676\) 0 0
\(677\) − 4.10891i − 0.157918i −0.996878 0.0789592i \(-0.974840\pi\)
0.996878 0.0789592i \(-0.0251597\pi\)
\(678\) 0 0
\(679\) 29.0024 1.11301
\(680\) 0 0
\(681\) −7.39403 −0.283340
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 3.96143i 0.151359i
\(686\) 0 0
\(687\) −9.39764 −0.358542
\(688\) 0 0
\(689\) 38.8397 1.47967
\(690\) 0 0
\(691\) − 10.8614i − 0.413187i −0.978427 0.206594i \(-0.933762\pi\)
0.978427 0.206594i \(-0.0662378\pi\)
\(692\) 0 0
\(693\) 29.3900i 1.11643i
\(694\) 0 0
\(695\) 7.42554 0.281667
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) − 66.7011i − 2.52287i
\(700\) 0 0
\(701\) − 27.7128i − 1.04670i −0.852118 0.523349i \(-0.824682\pi\)
0.852118 0.523349i \(-0.175318\pi\)
\(702\) 0 0
\(703\) −3.16915 −0.119527
\(704\) 0 0
\(705\) 17.4891 0.658679
\(706\) 0 0
\(707\) 45.9565i 1.72837i
\(708\) 0 0
\(709\) − 0.699713i − 0.0262783i −0.999914 0.0131391i \(-0.995818\pi\)
0.999914 0.0131391i \(-0.00418244\pi\)
\(710\) 0 0
\(711\) −35.6357 −1.33644
\(712\) 0 0
\(713\) −8.74456 −0.327486
\(714\) 0 0
\(715\) 10.9783i 0.410563i
\(716\) 0 0
\(717\) 57.1749i 2.13524i
\(718\) 0 0
\(719\) −5.89587 −0.219879 −0.109939 0.993938i \(-0.535066\pi\)
−0.109939 + 0.993938i \(0.535066\pi\)
\(720\) 0 0
\(721\) −17.7228 −0.660032
\(722\) 0 0
\(723\) 18.9783i 0.705809i
\(724\) 0 0
\(725\) 26.1831i 0.972417i
\(726\) 0 0
\(727\) 20.6371 0.765389 0.382694 0.923875i \(-0.374996\pi\)
0.382694 + 0.923875i \(0.374996\pi\)
\(728\) 0 0
\(729\) 19.9348 0.738324
\(730\) 0 0
\(731\) − 46.8614i − 1.73323i
\(732\) 0 0
\(733\) − 38.5099i − 1.42239i −0.702992 0.711197i \(-0.748153\pi\)
0.702992 0.711197i \(-0.251847\pi\)
\(734\) 0 0
\(735\) −7.51811 −0.277310
\(736\) 0 0
\(737\) 28.2337 1.04000
\(738\) 0 0
\(739\) 27.6060i 1.01550i 0.861504 + 0.507751i \(0.169523\pi\)
−0.861504 + 0.507751i \(0.830477\pi\)
\(740\) 0 0
\(741\) − 9.74749i − 0.358083i
\(742\) 0 0
\(743\) −6.63325 −0.243350 −0.121675 0.992570i \(-0.538827\pi\)
−0.121675 + 0.992570i \(0.538827\pi\)
\(744\) 0 0
\(745\) 16.8614 0.617754
\(746\) 0 0
\(747\) 21.0217i 0.769146i
\(748\) 0 0
\(749\) − 25.2983i − 0.924379i
\(750\) 0 0
\(751\) −30.2921 −1.10537 −0.552686 0.833389i \(-0.686397\pi\)
−0.552686 + 0.833389i \(0.686397\pi\)
\(752\) 0 0
\(753\) −25.7663 −0.938977
\(754\) 0 0
\(755\) − 17.7228i − 0.644999i
\(756\) 0 0
\(757\) − 48.6998i − 1.77002i −0.465568 0.885012i \(-0.654150\pi\)
0.465568 0.885012i \(-0.345850\pi\)
\(758\) 0 0
\(759\) 20.1947 0.733021
\(760\) 0 0
\(761\) −11.0000 −0.398750 −0.199375 0.979923i \(-0.563891\pi\)
−0.199375 + 0.979923i \(0.563891\pi\)
\(762\) 0 0
\(763\) 41.8614i 1.51548i
\(764\) 0 0
\(765\) − 10.4095i − 0.376358i
\(766\) 0 0
\(767\) 17.9653 0.648690
\(768\) 0 0
\(769\) −31.4674 −1.13474 −0.567371 0.823462i \(-0.692040\pi\)
−0.567371 + 0.823462i \(0.692040\pi\)
\(770\) 0 0
\(771\) 43.2554i 1.55781i
\(772\) 0 0
\(773\) − 18.9600i − 0.681943i −0.940074 0.340972i \(-0.889244\pi\)
0.940074 0.340972i \(-0.110756\pi\)
\(774\) 0 0
\(775\) −15.1460 −0.544061
\(776\) 0 0
\(777\) −24.9348 −0.894529
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 22.3692i 0.800432i
\(782\) 0 0
\(783\) −5.28873 −0.189004
\(784\) 0 0
\(785\) 1.02175 0.0364678
\(786\) 0 0
\(787\) 20.6060i 0.734523i 0.930118 + 0.367262i \(0.119705\pi\)
−0.930118 + 0.367262i \(0.880295\pi\)
\(788\) 0 0
\(789\) 57.3052i 2.04012i
\(790\) 0 0
\(791\) −10.7971 −0.383899
\(792\) 0 0
\(793\) −18.7011 −0.664094
\(794\) 0 0
\(795\) 17.7663i 0.630106i
\(796\) 0 0
\(797\) 11.3321i 0.401402i 0.979652 + 0.200701i \(0.0643220\pi\)
−0.979652 + 0.200701i \(0.935678\pi\)
\(798\) 0 0
\(799\) −46.5253 −1.64595
\(800\) 0 0
\(801\) −19.6793 −0.695334
\(802\) 0 0
\(803\) 48.8614i 1.72428i
\(804\) 0 0
\(805\) 6.63325i 0.233791i
\(806\) 0 0
\(807\) 20.1947 0.710887
\(808\) 0 0
\(809\) 7.97825 0.280500 0.140250 0.990116i \(-0.455209\pi\)
0.140250 + 0.990116i \(0.455209\pi\)
\(810\) 0 0
\(811\) − 0.138593i − 0.00486667i −0.999997 0.00243334i \(-0.999225\pi\)
0.999997 0.00243334i \(-0.000774556\pi\)
\(812\) 0 0
\(813\) 17.9653i 0.630071i
\(814\) 0 0
\(815\) 0.589907 0.0206636
\(816\) 0 0
\(817\) 9.37228 0.327895
\(818\) 0 0
\(819\) − 35.8098i − 1.25130i
\(820\) 0 0
\(821\) − 37.0179i − 1.29193i −0.763366 0.645966i \(-0.776455\pi\)
0.763366 0.645966i \(-0.223545\pi\)
\(822\) 0 0
\(823\) −27.2704 −0.950586 −0.475293 0.879828i \(-0.657658\pi\)
−0.475293 + 0.879828i \(0.657658\pi\)
\(824\) 0 0
\(825\) 34.9783 1.21779
\(826\) 0 0
\(827\) 11.3505i 0.394697i 0.980333 + 0.197348i \(0.0632330\pi\)
−0.980333 + 0.197348i \(0.936767\pi\)
\(828\) 0 0
\(829\) − 33.1113i − 1.15000i −0.818152 0.575002i \(-0.805001\pi\)
0.818152 0.575002i \(-0.194999\pi\)
\(830\) 0 0
\(831\) −30.9918 −1.07509
\(832\) 0 0
\(833\) 20.0000 0.692959
\(834\) 0 0
\(835\) 16.0000i 0.553703i
\(836\) 0 0
\(837\) − 3.05934i − 0.105746i
\(838\) 0 0
\(839\) −16.7306 −0.577604 −0.288802 0.957389i \(-0.593257\pi\)
−0.288802 + 0.957389i \(0.593257\pi\)
\(840\) 0 0
\(841\) −6.86141 −0.236600
\(842\) 0 0
\(843\) − 66.9783i − 2.30685i
\(844\) 0 0
\(845\) − 3.07657i − 0.105837i
\(846\) 0 0
\(847\) −1.23472 −0.0424254
\(848\) 0 0
\(849\) −55.4456 −1.90289
\(850\) 0 0
\(851\) 8.00000i 0.274236i
\(852\) 0 0
\(853\) − 12.6766i − 0.434038i −0.976167 0.217019i \(-0.930367\pi\)
0.976167 0.217019i \(-0.0696334\pi\)
\(854\) 0 0
\(855\) 2.08191 0.0711997
\(856\) 0 0
\(857\) −12.2337 −0.417895 −0.208947 0.977927i \(-0.567004\pi\)
−0.208947 + 0.977927i \(0.567004\pi\)
\(858\) 0 0
\(859\) − 31.6060i − 1.07838i −0.842184 0.539191i \(-0.818730\pi\)
0.842184 0.539191i \(-0.181270\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −49.1971 −1.67469 −0.837345 0.546675i \(-0.815893\pi\)
−0.837345 + 0.546675i \(0.815893\pi\)
\(864\) 0 0
\(865\) −3.76631 −0.128058
\(866\) 0 0
\(867\) 18.9783i 0.644535i
\(868\) 0 0
\(869\) − 45.7330i − 1.55139i
\(870\) 0 0
\(871\) −34.4010 −1.16563
\(872\) 0 0
\(873\) −22.9783 −0.777696
\(874\) 0 0
\(875\) 24.6277i 0.832569i
\(876\) 0 0
\(877\) 24.5986i 0.830635i 0.909677 + 0.415317i \(0.136329\pi\)
−0.909677 + 0.415317i \(0.863671\pi\)
\(878\) 0 0
\(879\) 66.5725 2.24544
\(880\) 0 0
\(881\) 1.37228 0.0462333 0.0231167 0.999733i \(-0.492641\pi\)
0.0231167 + 0.999733i \(0.492641\pi\)
\(882\) 0 0
\(883\) − 36.1168i − 1.21543i −0.794156 0.607714i \(-0.792087\pi\)
0.794156 0.607714i \(-0.207913\pi\)
\(884\) 0 0
\(885\) 8.21782i 0.276239i
\(886\) 0 0
\(887\) −45.0333 −1.51207 −0.756035 0.654531i \(-0.772866\pi\)
−0.756035 + 0.654531i \(0.772866\pi\)
\(888\) 0 0
\(889\) −62.7011 −2.10293
\(890\) 0 0
\(891\) 33.6495i 1.12730i
\(892\) 0 0
\(893\) − 9.30506i − 0.311382i
\(894\) 0 0
\(895\) 18.2054 0.608538
\(896\) 0 0
\(897\) −24.6060 −0.821569
\(898\) 0 0
\(899\) − 20.7446i − 0.691870i
\(900\) 0 0
\(901\) − 47.2627i − 1.57455i
\(902\) 0 0
\(903\) 73.7408 2.45394
\(904\) 0 0
\(905\) −9.25544 −0.307661
\(906\) 0 0
\(907\) 18.1386i 0.602282i 0.953580 + 0.301141i \(0.0973675\pi\)
−0.953580 + 0.301141i \(0.902632\pi\)
\(908\) 0 0
\(909\) − 36.4107i − 1.20767i
\(910\) 0 0
\(911\) −7.81306 −0.258858 −0.129429 0.991589i \(-0.541314\pi\)
−0.129429 + 0.991589i \(0.541314\pi\)
\(912\) 0 0
\(913\) −26.9783 −0.892850
\(914\) 0 0
\(915\) − 8.55437i − 0.282799i
\(916\) 0 0
\(917\) 4.55134i 0.150299i
\(918\) 0 0
\(919\) −27.7677 −0.915972 −0.457986 0.888959i \(-0.651429\pi\)
−0.457986 + 0.888959i \(0.651429\pi\)
\(920\) 0 0
\(921\) 63.4456 2.09060
\(922\) 0 0
\(923\) − 27.2554i − 0.897124i
\(924\) 0 0
\(925\) 13.8564i 0.455596i
\(926\) 0 0
\(927\) 14.0416 0.461185
\(928\) 0 0
\(929\) −45.1168 −1.48024 −0.740118 0.672477i \(-0.765230\pi\)
−0.740118 + 0.672477i \(0.765230\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 55.0758i 1.80310i
\(934\) 0 0
\(935\) 13.3591 0.436888
\(936\) 0 0
\(937\) 23.9783 0.783335 0.391668 0.920107i \(-0.371898\pi\)
0.391668 + 0.920107i \(0.371898\pi\)
\(938\) 0 0
\(939\) 50.0951i 1.63479i
\(940\) 0 0
\(941\) 5.69349i 0.185602i 0.995685 + 0.0928012i \(0.0295821\pi\)
−0.995685 + 0.0928012i \(0.970418\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2.32069 −0.0754919
\(946\) 0 0
\(947\) 37.2119i 1.20923i 0.796520 + 0.604613i \(0.206672\pi\)
−0.796520 + 0.604613i \(0.793328\pi\)
\(948\) 0 0
\(949\) − 59.5345i − 1.93257i
\(950\) 0 0
\(951\) 43.3185 1.40470
\(952\) 0 0
\(953\) −8.23369 −0.266715 −0.133358 0.991068i \(-0.542576\pi\)
−0.133358 + 0.991068i \(0.542576\pi\)
\(954\) 0 0
\(955\) 10.8614i 0.351467i
\(956\) 0 0
\(957\) 47.9075i 1.54863i
\(958\) 0 0
\(959\) 16.5831 0.535497
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 20.0435i 0.645893i
\(964\) 0 0
\(965\) 11.6819i 0.376054i
\(966\) 0 0
\(967\) 41.8642 1.34626 0.673131 0.739524i \(-0.264949\pi\)
0.673131 + 0.739524i \(0.264949\pi\)
\(968\) 0 0
\(969\) −11.8614 −0.381043
\(970\) 0 0
\(971\) − 4.00000i − 0.128366i −0.997938 0.0641831i \(-0.979556\pi\)
0.997938 0.0641831i \(-0.0204442\pi\)
\(972\) 0 0
\(973\) − 31.0843i − 0.996518i
\(974\) 0 0
\(975\) −42.6188 −1.36489
\(976\) 0 0
\(977\) 4.51087 0.144316 0.0721578 0.997393i \(-0.477011\pi\)
0.0721578 + 0.997393i \(0.477011\pi\)
\(978\) 0 0
\(979\) − 25.2554i − 0.807167i
\(980\) 0 0
\(981\) − 33.1662i − 1.05892i
\(982\) 0 0
\(983\) 6.22849 0.198658 0.0993290 0.995055i \(-0.468330\pi\)
0.0993290 + 0.995055i \(0.468330\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) − 73.2119i − 2.33036i
\(988\) 0 0
\(989\) − 23.6588i − 0.752306i
\(990\) 0 0
\(991\) 25.2434 0.801882 0.400941 0.916104i \(-0.368683\pi\)
0.400941 + 0.916104i \(0.368683\pi\)
\(992\) 0 0
\(993\) −42.3723 −1.34464
\(994\) 0 0
\(995\) 5.37228i 0.170313i
\(996\) 0 0
\(997\) − 10.2997i − 0.326196i −0.986610 0.163098i \(-0.947851\pi\)
0.986610 0.163098i \(-0.0521486\pi\)
\(998\) 0 0
\(999\) −2.79885 −0.0885518
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 1216.2.c.h.609.6 yes 8
4.3 odd 2 inner 1216.2.c.h.609.4 yes 8
8.3 odd 2 inner 1216.2.c.h.609.5 yes 8
8.5 even 2 inner 1216.2.c.h.609.3 8
16.3 odd 4 4864.2.a.bi.1.4 4
16.5 even 4 4864.2.a.bi.1.3 4
16.11 odd 4 4864.2.a.bl.1.1 4
16.13 even 4 4864.2.a.bl.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.h.609.3 8 8.5 even 2 inner
1216.2.c.h.609.4 yes 8 4.3 odd 2 inner
1216.2.c.h.609.5 yes 8 8.3 odd 2 inner
1216.2.c.h.609.6 yes 8 1.1 even 1 trivial
4864.2.a.bi.1.3 4 16.5 even 4
4864.2.a.bi.1.4 4 16.3 odd 4
4864.2.a.bl.1.1 4 16.11 odd 4
4864.2.a.bl.1.2 4 16.13 even 4