Properties

Label 1216.2.c.h.609.4
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 609.4
Root \(0.396143 - 1.68614i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.h.609.5

$q$-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 \(-0.759881\pi\)
0.728714 0.684819i \(-0.240119\pi\)
\(4\) 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 \(-0.715629\pi\)
−0.626783 + 0.779194i \(0.715629\pi\)
\(8\) 0 0
\(9\) −2.62772 −0.875906
\(10\) 0 0
\(11\) 3.37228i 1.01678i 0.861127 + 0.508391i \(0.169759\pi\)
−0.861127 + 0.508391i \(0.830241\pi\)
\(12\) 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 \(-0.584771\pi\)
−0.263180 + 0.964747i \(0.584771\pi\)
\(24\) 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 \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 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 \(0.253405\pi\)
−0.699503 + 0.714630i \(0.746595\pi\)
\(44\) 0 0
\(45\) − 2.08191i − 0.310352i
\(46\) 0 0
\(47\) 9.30506 1.35728 0.678642 0.734470i \(-0.262569\pi\)
0.678642 + 0.734470i \(0.262569\pi\)
\(48\) 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 \(0.0918645\pi\)
−0.958643 + 0.284611i \(0.908135\pi\)
\(60\) 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 \(-0.829120\pi\)
0.859332 0.511418i \(-0.170880\pi\)
\(68\) 0 0
\(69\) 5.98844i 0.720923i
\(70\) 0 0
\(71\) 6.63325 0.787222 0.393611 0.919277i \(-0.371226\pi\)
0.393611 + 0.919277i \(0.371226\pi\)
\(72\) 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 \(-0.776221\pi\)
−0.762891 + 0.646527i \(0.776221\pi\)
\(80\) 0 0
\(81\) −9.97825 −1.10869
\(82\) 0 0
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 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 \(-0.415202\pi\)
0.263262 + 0.964724i \(0.415202\pi\)
\(104\) 0 0
\(105\) −6.23369 −0.608346
\(106\) 0 0
\(107\) 7.62772i 0.737399i 0.929549 + 0.368700i \(0.120197\pi\)
−0.929549 + 0.368700i \(0.879803\pi\)
\(108\) 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 \(-0.183271\pi\)
0.838777 + 0.544475i \(0.183271\pi\)
\(128\) 0 0
\(129\) 22.2337 1.95757
\(130\) 0 0
\(131\) − 1.37228i − 0.119897i −0.998201 0.0599484i \(-0.980906\pi\)
0.998201 0.0599484i \(-0.0190936\pi\)
\(132\) 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 \(0.130113\pi\)
−0.917614 + 0.397473i \(0.869887\pi\)
\(140\) 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 \(-0.135936\pi\)
0.910189 + 0.414193i \(0.135936\pi\)
\(152\) 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 \(0.00928302\pi\)
−0.999575 + 0.0291593i \(0.990717\pi\)
\(164\) 0 0
\(165\) 6.33830i 0.493436i
\(166\) 0 0
\(167\) −20.1947 −1.56271 −0.781356 0.624085i \(-0.785472\pi\)
−0.781356 + 0.624085i \(0.785472\pi\)
\(168\) 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 \(0.328751\pi\)
−0.512416 + 0.858738i \(0.671249\pi\)
\(180\) 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 \(-0.665188\pi\)
−0.495972 + 0.868339i \(0.665188\pi\)
\(192\) 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 \(-0.577258\pi\)
−0.240336 + 0.970690i \(0.577258\pi\)
\(200\) 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 \(0.0930521\pi\)
−0.957575 + 0.288186i \(0.906948\pi\)
\(212\) 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 \(-0.574518\pi\)
−0.231973 + 0.972722i \(0.574518\pi\)
\(224\) 0 0
\(225\) −11.4891 −0.765942
\(226\) 0 0
\(227\) − 3.11684i − 0.206872i −0.994636 0.103436i \(-0.967016\pi\)
0.994636 0.103436i \(-0.0329837\pi\)
\(228\) 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 \(-0.784522\pi\)
−0.779490 + 0.626415i \(0.784522\pi\)
\(240\) 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 \(-0.888630\pi\)
0.939415 0.342783i \(-0.111370\pi\)
\(252\) 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 \(-0.767439\pi\)
−0.744766 + 0.667326i \(0.767439\pi\)
\(264\) 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 \(-0.573877\pi\)
−0.230014 + 0.973187i \(0.573877\pi\)
\(272\) 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 \(-0.755551\pi\)
0.719330 0.694669i \(-0.244449\pi\)
\(284\) 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 \(0.276371\pi\)
−0.646167 + 0.763196i \(0.723629\pi\)
\(308\) 0 0
\(309\) − 12.6766i − 0.721146i
\(310\) 0 0
\(311\) −23.2164 −1.31648 −0.658240 0.752808i \(-0.728699\pi\)
−0.658240 + 0.752808i \(0.728699\pi\)
\(312\) 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 \(-0.836677\pi\)
0.871230 0.490876i \(-0.163323\pi\)
\(332\) 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 \(0.179196\pi\)
−0.845679 + 0.533692i \(0.820804\pi\)
\(348\) 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 \(-0.355824\pi\)
0.437612 + 0.899164i \(0.355824\pi\)
\(360\) 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 \(-0.205725\pi\)
0.798314 + 0.602241i \(0.205725\pi\)
\(368\) 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 \(-0.871004\pi\)
0.919002 0.394252i \(-0.128996\pi\)
\(380\) 0 0
\(381\) − 44.8482i − 2.29764i
\(382\) 0 0
\(383\) 7.92287 0.404840 0.202420 0.979299i \(-0.435119\pi\)
0.202420 + 0.979299i \(0.435119\pi\)
\(384\) 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 \(0.315235\pi\)
−0.548405 + 0.836213i \(0.684765\pi\)
\(420\) 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 \(-0.411572\pi\)
0.274246 + 0.961660i \(0.411572\pi\)
\(432\) 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 \(-0.552870\pi\)
−0.165333 + 0.986238i \(0.552870\pi\)
\(440\) 0 0
\(441\) −10.5109 −0.500518
\(442\) 0 0
\(443\) − 10.3505i − 0.491769i −0.969299 0.245884i \(-0.920922\pi\)
0.969299 0.245884i \(-0.0790783\pi\)
\(444\) 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 \(-0.438143\pi\)
0.193108 + 0.981177i \(0.438143\pi\)
\(464\) 0 0
\(465\) 6.51087 0.301935
\(466\) 0 0
\(467\) − 28.1168i − 1.30109i −0.759467 0.650546i \(-0.774540\pi\)
0.759467 0.650546i \(-0.225460\pi\)
\(468\) 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 \(-0.395209\pi\)
0.323296 + 0.946298i \(0.395209\pi\)
\(480\) 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 \(-0.268944\pi\)
0.663796 + 0.747914i \(0.268944\pi\)
\(488\) 0 0
\(489\) 1.76631 0.0798754
\(490\) 0 0
\(491\) − 14.9783i − 0.675959i −0.941153 0.337979i \(-0.890257\pi\)
0.941153 0.337979i \(-0.109743\pi\)
\(492\) 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 \(-0.990221\pi\)
0.999528 0.0307159i \(-0.00977870\pi\)
\(500\) 0 0
\(501\) 47.9075i 2.14035i
\(502\) 0 0
\(503\) −21.4294 −0.955491 −0.477745 0.878498i \(-0.658546\pi\)
−0.477745 + 0.878498i \(0.658546\pi\)
\(504\) 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 \(-0.858820\pi\)
0.903243 0.429130i \(-0.141180\pi\)
\(524\) 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 \(0.183439\pi\)
−0.838489 + 0.544918i \(0.816561\pi\)
\(548\) 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 \(-0.935919\pi\)
0.979804 0.199959i \(-0.0640811\pi\)
\(564\) 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 \(-0.980151\pi\)
0.998056 0.0623180i \(-0.0198493\pi\)
\(572\) 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 \(-0.979368\pi\)
0.997900 0.0647718i \(-0.0206319\pi\)
\(588\) 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 \(-0.735997\pi\)
−0.675325 + 0.737520i \(0.735997\pi\)
\(600\) 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 \(-0.450525\pi\)
0.154804 + 0.987945i \(0.450525\pi\)
\(608\) 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 \(0.119422\pi\)
−0.930443 + 0.366437i \(0.880578\pi\)
\(620\) 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 \(-0.423982\pi\)
0.236553 + 0.971619i \(0.423982\pi\)
\(632\) 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 \(-0.793605\pi\)
0.797045 0.603920i \(-0.206395\pi\)
\(644\) 0 0
\(645\) 17.6155i 0.693608i
\(646\) 0 0
\(647\) 16.5831 0.651950 0.325975 0.945378i \(-0.394307\pi\)
0.325975 + 0.945378i \(0.394307\pi\)
\(648\) 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 \(0.114937\pi\)
−0.935514 + 0.353290i \(0.885063\pi\)
\(660\) 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 \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 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 \(0.0662378\pi\)
−0.978427 + 0.206594i \(0.933762\pi\)
\(692\) 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 \(-0.464934\pi\)
0.109939 + 0.993938i \(0.464934\pi\)
\(720\) 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 \(-0.625004\pi\)
−0.382694 + 0.923875i \(0.625004\pi\)
\(728\) 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 \(-0.830477\pi\)
0.861504 0.507751i \(-0.169523\pi\)
\(740\) 0 0
\(741\) − 9.74749i − 0.358083i
\(742\) 0 0
\(743\) 6.63325 0.243350 0.121675 0.992570i \(-0.461173\pi\)
0.121675 + 0.992570i \(0.461173\pi\)
\(744\) 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 \(-0.313603\pi\)
0.552686 + 0.833389i \(0.313603\pi\)
\(752\) 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 \(-0.880295\pi\)
0.930118 0.367262i \(-0.119705\pi\)
\(788\) 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 \(0.000774556\pi\)
−0.999997 + 0.00243334i \(0.999225\pi\)
\(812\) 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 \(-0.342342\pi\)
0.475293 + 0.879828i \(0.342342\pi\)
\(824\) 0 0
\(825\) 34.9783 1.21779
\(826\) 0 0
\(827\) − 11.3505i − 0.394697i −0.980333 0.197348i \(-0.936767\pi\)
0.980333 0.197348i \(-0.0632330\pi\)
\(828\) 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 \(-0.406743\pi\)
0.288802 + 0.957389i \(0.406743\pi\)
\(840\) 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 \(0.181270\pi\)
−0.842184 + 0.539191i \(0.818730\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 49.1971 1.67469 0.837345 0.546675i \(-0.184107\pi\)
0.837345 + 0.546675i \(0.184107\pi\)
\(864\) 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 \(0.207913\pi\)
−0.794156 + 0.607714i \(0.792087\pi\)
\(884\) 0 0
\(885\) 8.21782i 0.276239i
\(886\) 0 0
\(887\) 45.0333 1.51207 0.756035 0.654531i \(-0.227134\pi\)
0.756035 + 0.654531i \(0.227134\pi\)
\(888\) 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 \(-0.902632\pi\)
0.953580 0.301141i \(-0.0973675\pi\)
\(908\) 0 0
\(909\) − 36.4107i − 1.20767i
\(910\) 0 0
\(911\) 7.81306 0.258858 0.129429 0.991589i \(-0.458686\pi\)
0.129429 + 0.991589i \(0.458686\pi\)
\(912\) 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 \(-0.348571\pi\)
0.457986 + 0.888959i \(0.348571\pi\)
\(920\) 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 \(-0.793328\pi\)
0.796520 0.604613i \(-0.206672\pi\)
\(948\) 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 \(-0.735051\pi\)
−0.673131 + 0.739524i \(0.735051\pi\)
\(968\) 0 0
\(969\) −11.8614 −0.381043
\(970\) 0 0
\(971\) 4.00000i 0.128366i 0.997938 + 0.0641831i \(0.0204442\pi\)
−0.997938 + 0.0641831i \(0.979556\pi\)
\(972\) 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 \(-0.531670\pi\)
−0.0993290 + 0.995055i \(0.531670\pi\)
\(984\) 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 \(-0.631317\pi\)
−0.400941 + 0.916104i \(0.631317\pi\)
\(992\) 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.609.4 yes 8
4.3 odd 2 inner 1216.2.c.h.609.6 yes 8
8.3 odd 2 inner 1216.2.c.h.609.3 8
8.5 even 2 inner 1216.2.c.h.609.5 yes 8
16.3 odd 4 4864.2.a.bl.1.2 4
16.5 even 4 4864.2.a.bl.1.1 4
16.11 odd 4 4864.2.a.bi.1.3 4
16.13 even 4 4864.2.a.bi.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.h.609.3 8 8.3 odd 2 inner
1216.2.c.h.609.4 yes 8 1.1 even 1 trivial
1216.2.c.h.609.5 yes 8 8.5 even 2 inner
1216.2.c.h.609.6 yes 8 4.3 odd 2 inner
4864.2.a.bi.1.3 4 16.11 odd 4
4864.2.a.bi.1.4 4 16.13 even 4
4864.2.a.bl.1.1 4 16.5 even 4
4864.2.a.bl.1.2 4 16.3 odd 4