Properties

Label 896.2.e.f.447.3
Level $896$
Weight $2$
Character 896.447
Analytic conductor $7.155$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 447.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 896.447
Dual form 896.2.e.f.447.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051i q^{3} +2.73205 q^{5} +(2.00000 - 1.73205i) q^{7} +2.46410 q^{9} +O(q^{10})\) \(q+0.732051i q^{3} +2.73205 q^{5} +(2.00000 - 1.73205i) q^{7} +2.46410 q^{9} -1.46410 q^{11} +1.26795 q^{13} +2.00000i q^{15} -4.00000i q^{17} -4.73205i q^{19} +(1.26795 + 1.46410i) q^{21} -1.46410i q^{23} +2.46410 q^{25} +4.00000i q^{27} +6.92820i q^{29} -6.92820 q^{31} -1.07180i q^{33} +(5.46410 - 4.73205i) q^{35} +4.00000i q^{37} +0.928203i q^{39} -10.9282i q^{41} -9.46410 q^{43} +6.73205 q^{45} +6.92820 q^{47} +(1.00000 - 6.92820i) q^{49} +2.92820 q^{51} +6.92820i q^{53} -4.00000 q^{55} +3.46410 q^{57} +14.1962i q^{59} +5.66025 q^{61} +(4.92820 - 4.26795i) q^{63} +3.46410 q^{65} +9.46410 q^{67} +1.07180 q^{69} +11.4641i q^{71} +6.92820i q^{73} +1.80385i q^{75} +(-2.92820 + 2.53590i) q^{77} +3.46410i q^{79} +4.46410 q^{81} -7.26795i q^{83} -10.9282i q^{85} -5.07180 q^{87} -1.07180i q^{89} +(2.53590 - 2.19615i) q^{91} -5.07180i q^{93} -12.9282i q^{95} +12.0000i q^{97} -3.60770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{5} + 8q^{7} - 4q^{9} + 8q^{11} + 12q^{13} + 12q^{21} - 4q^{25} + 8q^{35} - 24q^{43} + 20q^{45} + 4q^{49} - 16q^{51} - 16q^{55} - 12q^{61} - 8q^{63} + 24q^{67} + 32q^{69} + 16q^{77} + 4q^{81} - 48q^{87} + 24q^{91} - 56q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0.732051i 0.422650i 0.977416 + 0.211325i \(0.0677778\pi\)
−0.977416 + 0.211325i \(0.932222\pi\)
\(4\) 0 0
\(5\) 2.73205 1.22181 0.610905 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) 0 0
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 0 0
\(9\) 2.46410 0.821367
\(10\) 0 0
\(11\) −1.46410 −0.441443 −0.220722 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(12\) 0 0
\(13\) 1.26795 0.351666 0.175833 0.984420i \(-0.443738\pi\)
0.175833 + 0.984420i \(0.443738\pi\)
\(14\) 0 0
\(15\) 2.00000i 0.516398i
\(16\) 0 0
\(17\) 4.00000i 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 4.73205i 1.08561i −0.839860 0.542803i \(-0.817363\pi\)
0.839860 0.542803i \(-0.182637\pi\)
\(20\) 0 0
\(21\) 1.26795 + 1.46410i 0.276689 + 0.319493i
\(22\) 0 0
\(23\) 1.46410i 0.305286i −0.988281 0.152643i \(-0.951221\pi\)
0.988281 0.152643i \(-0.0487785\pi\)
\(24\) 0 0
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) 1.07180i 0.186576i
\(34\) 0 0
\(35\) 5.46410 4.73205i 0.923602 0.799863i
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 0.928203i 0.148631i
\(40\) 0 0
\(41\) 10.9282i 1.70670i −0.521340 0.853349i \(-0.674568\pi\)
0.521340 0.853349i \(-0.325432\pi\)
\(42\) 0 0
\(43\) −9.46410 −1.44326 −0.721631 0.692278i \(-0.756607\pi\)
−0.721631 + 0.692278i \(0.756607\pi\)
\(44\) 0 0
\(45\) 6.73205 1.00355
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 2.92820 0.410030
\(52\) 0 0
\(53\) 6.92820i 0.951662i 0.879537 + 0.475831i \(0.157853\pi\)
−0.879537 + 0.475831i \(0.842147\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) 14.1962i 1.84818i 0.382173 + 0.924091i \(0.375176\pi\)
−0.382173 + 0.924091i \(0.624824\pi\)
\(60\) 0 0
\(61\) 5.66025 0.724721 0.362361 0.932038i \(-0.381971\pi\)
0.362361 + 0.932038i \(0.381971\pi\)
\(62\) 0 0
\(63\) 4.92820 4.26795i 0.620895 0.537711i
\(64\) 0 0
\(65\) 3.46410 0.429669
\(66\) 0 0
\(67\) 9.46410 1.15622 0.578112 0.815957i \(-0.303790\pi\)
0.578112 + 0.815957i \(0.303790\pi\)
\(68\) 0 0
\(69\) 1.07180 0.129029
\(70\) 0 0
\(71\) 11.4641i 1.36054i 0.732962 + 0.680269i \(0.238137\pi\)
−0.732962 + 0.680269i \(0.761863\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 0 0
\(75\) 1.80385i 0.208290i
\(76\) 0 0
\(77\) −2.92820 + 2.53590i −0.333700 + 0.288992i
\(78\) 0 0
\(79\) 3.46410i 0.389742i 0.980829 + 0.194871i \(0.0624288\pi\)
−0.980829 + 0.194871i \(0.937571\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) 7.26795i 0.797761i −0.917003 0.398881i \(-0.869399\pi\)
0.917003 0.398881i \(-0.130601\pi\)
\(84\) 0 0
\(85\) 10.9282i 1.18533i
\(86\) 0 0
\(87\) −5.07180 −0.543754
\(88\) 0 0
\(89\) 1.07180i 0.113610i −0.998385 0.0568051i \(-0.981909\pi\)
0.998385 0.0568051i \(-0.0180914\pi\)
\(90\) 0 0
\(91\) 2.53590 2.19615i 0.265834 0.230219i
\(92\) 0 0
\(93\) 5.07180i 0.525921i
\(94\) 0 0
\(95\) 12.9282i 1.32641i
\(96\) 0 0
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 0 0
\(99\) −3.60770 −0.362587
\(100\) 0 0
\(101\) 4.19615 0.417533 0.208766 0.977966i \(-0.433055\pi\)
0.208766 + 0.977966i \(0.433055\pi\)
\(102\) 0 0
\(103\) 5.07180 0.499739 0.249869 0.968280i \(-0.419612\pi\)
0.249869 + 0.968280i \(0.419612\pi\)
\(104\) 0 0
\(105\) 3.46410 + 4.00000i 0.338062 + 0.390360i
\(106\) 0 0
\(107\) −12.3923 −1.19801 −0.599005 0.800746i \(-0.704437\pi\)
−0.599005 + 0.800746i \(0.704437\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i −0.943349 0.331801i \(-0.892344\pi\)
0.943349 0.331801i \(-0.107656\pi\)
\(110\) 0 0
\(111\) −2.92820 −0.277933
\(112\) 0 0
\(113\) −2.53590 −0.238557 −0.119279 0.992861i \(-0.538058\pi\)
−0.119279 + 0.992861i \(0.538058\pi\)
\(114\) 0 0
\(115\) 4.00000i 0.373002i
\(116\) 0 0
\(117\) 3.12436 0.288847
\(118\) 0 0
\(119\) −6.92820 8.00000i −0.635107 0.733359i
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) 14.5359i 1.28985i 0.764245 + 0.644926i \(0.223112\pi\)
−0.764245 + 0.644926i \(0.776888\pi\)
\(128\) 0 0
\(129\) 6.92820i 0.609994i
\(130\) 0 0
\(131\) 7.26795i 0.635004i −0.948258 0.317502i \(-0.897156\pi\)
0.948258 0.317502i \(-0.102844\pi\)
\(132\) 0 0
\(133\) −8.19615 9.46410i −0.710697 0.820642i
\(134\) 0 0
\(135\) 10.9282i 0.940550i
\(136\) 0 0
\(137\) −7.85641 −0.671218 −0.335609 0.942001i \(-0.608942\pi\)
−0.335609 + 0.942001i \(0.608942\pi\)
\(138\) 0 0
\(139\) 6.19615i 0.525551i 0.964857 + 0.262775i \(0.0846378\pi\)
−0.964857 + 0.262775i \(0.915362\pi\)
\(140\) 0 0
\(141\) 5.07180i 0.427122i
\(142\) 0 0
\(143\) −1.85641 −0.155241
\(144\) 0 0
\(145\) 18.9282i 1.57190i
\(146\) 0 0
\(147\) 5.07180 + 0.732051i 0.418315 + 0.0603785i
\(148\) 0 0
\(149\) 12.0000i 0.983078i 0.870855 + 0.491539i \(0.163566\pi\)
−0.870855 + 0.491539i \(0.836434\pi\)
\(150\) 0 0
\(151\) 23.3205i 1.89780i −0.315583 0.948898i \(-0.602200\pi\)
0.315583 0.948898i \(-0.397800\pi\)
\(152\) 0 0
\(153\) 9.85641i 0.796843i
\(154\) 0 0
\(155\) −18.9282 −1.52035
\(156\) 0 0
\(157\) −22.7321 −1.81422 −0.907108 0.420899i \(-0.861715\pi\)
−0.907108 + 0.420899i \(0.861715\pi\)
\(158\) 0 0
\(159\) −5.07180 −0.402220
\(160\) 0 0
\(161\) −2.53590 2.92820i −0.199857 0.230775i
\(162\) 0 0
\(163\) −4.39230 −0.344032 −0.172016 0.985094i \(-0.555028\pi\)
−0.172016 + 0.985094i \(0.555028\pi\)
\(164\) 0 0
\(165\) 2.92820i 0.227960i
\(166\) 0 0
\(167\) 18.9282 1.46471 0.732354 0.680924i \(-0.238422\pi\)
0.732354 + 0.680924i \(0.238422\pi\)
\(168\) 0 0
\(169\) −11.3923 −0.876331
\(170\) 0 0
\(171\) 11.6603i 0.891682i
\(172\) 0 0
\(173\) −6.73205 −0.511828 −0.255914 0.966700i \(-0.582376\pi\)
−0.255914 + 0.966700i \(0.582376\pi\)
\(174\) 0 0
\(175\) 4.92820 4.26795i 0.372537 0.322627i
\(176\) 0 0
\(177\) −10.3923 −0.781133
\(178\) 0 0
\(179\) 20.3923 1.52419 0.762096 0.647464i \(-0.224170\pi\)
0.762096 + 0.647464i \(0.224170\pi\)
\(180\) 0 0
\(181\) −13.2679 −0.986199 −0.493099 0.869973i \(-0.664136\pi\)
−0.493099 + 0.869973i \(0.664136\pi\)
\(182\) 0 0
\(183\) 4.14359i 0.306303i
\(184\) 0 0
\(185\) 10.9282i 0.803457i
\(186\) 0 0
\(187\) 5.85641i 0.428263i
\(188\) 0 0
\(189\) 6.92820 + 8.00000i 0.503953 + 0.581914i
\(190\) 0 0
\(191\) 4.53590i 0.328206i 0.986443 + 0.164103i \(0.0524730\pi\)
−0.986443 + 0.164103i \(0.947527\pi\)
\(192\) 0 0
\(193\) −16.3923 −1.17994 −0.589972 0.807424i \(-0.700861\pi\)
−0.589972 + 0.807424i \(0.700861\pi\)
\(194\) 0 0
\(195\) 2.53590i 0.181599i
\(196\) 0 0
\(197\) 12.0000i 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) 24.7846 1.75693 0.878467 0.477803i \(-0.158567\pi\)
0.878467 + 0.477803i \(0.158567\pi\)
\(200\) 0 0
\(201\) 6.92820i 0.488678i
\(202\) 0 0
\(203\) 12.0000 + 13.8564i 0.842235 + 0.972529i
\(204\) 0 0
\(205\) 29.8564i 2.08526i
\(206\) 0 0
\(207\) 3.60770i 0.250752i
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) −28.3923 −1.95461 −0.977303 0.211844i \(-0.932053\pi\)
−0.977303 + 0.211844i \(0.932053\pi\)
\(212\) 0 0
\(213\) −8.39230 −0.575031
\(214\) 0 0
\(215\) −25.8564 −1.76339
\(216\) 0 0
\(217\) −13.8564 + 12.0000i −0.940634 + 0.814613i
\(218\) 0 0
\(219\) −5.07180 −0.342720
\(220\) 0 0
\(221\) 5.07180i 0.341166i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 6.07180 0.404786
\(226\) 0 0
\(227\) 4.73205i 0.314077i −0.987592 0.157039i \(-0.949805\pi\)
0.987592 0.157039i \(-0.0501947\pi\)
\(228\) 0 0
\(229\) −17.6603 −1.16702 −0.583511 0.812105i \(-0.698322\pi\)
−0.583511 + 0.812105i \(0.698322\pi\)
\(230\) 0 0
\(231\) −1.85641 2.14359i −0.122143 0.141038i
\(232\) 0 0
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) 0 0
\(235\) 18.9282 1.23474
\(236\) 0 0
\(237\) −2.53590 −0.164724
\(238\) 0 0
\(239\) 20.3923i 1.31907i 0.751674 + 0.659534i \(0.229246\pi\)
−0.751674 + 0.659534i \(0.770754\pi\)
\(240\) 0 0
\(241\) 12.0000i 0.772988i 0.922292 + 0.386494i \(0.126314\pi\)
−0.922292 + 0.386494i \(0.873686\pi\)
\(242\) 0 0
\(243\) 15.2679i 0.979439i
\(244\) 0 0
\(245\) 2.73205 18.9282i 0.174544 1.20928i
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) 5.32051 0.337173
\(250\) 0 0
\(251\) 16.0526i 1.01323i −0.862173 0.506614i \(-0.830897\pi\)
0.862173 0.506614i \(-0.169103\pi\)
\(252\) 0 0
\(253\) 2.14359i 0.134767i
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 5.85641i 0.365313i −0.983177 0.182656i \(-0.941530\pi\)
0.983177 0.182656i \(-0.0584696\pi\)
\(258\) 0 0
\(259\) 6.92820 + 8.00000i 0.430498 + 0.497096i
\(260\) 0 0
\(261\) 17.0718i 1.05672i
\(262\) 0 0
\(263\) 4.53590i 0.279695i −0.990173 0.139848i \(-0.955339\pi\)
0.990173 0.139848i \(-0.0446613\pi\)
\(264\) 0 0
\(265\) 18.9282i 1.16275i
\(266\) 0 0
\(267\) 0.784610 0.0480173
\(268\) 0 0
\(269\) 23.1244 1.40992 0.704958 0.709249i \(-0.250966\pi\)
0.704958 + 0.709249i \(0.250966\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 1.60770 + 1.85641i 0.0973021 + 0.112355i
\(274\) 0 0
\(275\) −3.60770 −0.217552
\(276\) 0 0
\(277\) 12.7846i 0.768153i −0.923301 0.384076i \(-0.874520\pi\)
0.923301 0.384076i \(-0.125480\pi\)
\(278\) 0 0
\(279\) −17.0718 −1.02206
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 26.1962i 1.55720i −0.627521 0.778600i \(-0.715930\pi\)
0.627521 0.778600i \(-0.284070\pi\)
\(284\) 0 0
\(285\) 9.46410 0.560605
\(286\) 0 0
\(287\) −18.9282 21.8564i −1.11730 1.29014i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −8.78461 −0.514963
\(292\) 0 0
\(293\) 12.1962 0.712507 0.356253 0.934389i \(-0.384054\pi\)
0.356253 + 0.934389i \(0.384054\pi\)
\(294\) 0 0
\(295\) 38.7846i 2.25813i
\(296\) 0 0
\(297\) 5.85641i 0.339823i
\(298\) 0 0
\(299\) 1.85641i 0.107359i
\(300\) 0 0
\(301\) −18.9282 + 16.3923i −1.09100 + 0.944837i
\(302\) 0 0
\(303\) 3.07180i 0.176470i
\(304\) 0 0
\(305\) 15.4641 0.885472
\(306\) 0 0
\(307\) 16.7321i 0.954949i 0.878646 + 0.477474i \(0.158448\pi\)
−0.878646 + 0.477474i \(0.841552\pi\)
\(308\) 0 0
\(309\) 3.71281i 0.211215i
\(310\) 0 0
\(311\) 1.85641 0.105267 0.0526336 0.998614i \(-0.483238\pi\)
0.0526336 + 0.998614i \(0.483238\pi\)
\(312\) 0 0
\(313\) 8.78461i 0.496535i −0.968691 0.248268i \(-0.920139\pi\)
0.968691 0.248268i \(-0.0798613\pi\)
\(314\) 0 0
\(315\) 13.4641 11.6603i 0.758616 0.656981i
\(316\) 0 0
\(317\) 25.8564i 1.45224i −0.687568 0.726120i \(-0.741322\pi\)
0.687568 0.726120i \(-0.258678\pi\)
\(318\) 0 0
\(319\) 10.1436i 0.567932i
\(320\) 0 0
\(321\) 9.07180i 0.506338i
\(322\) 0 0
\(323\) −18.9282 −1.05319
\(324\) 0 0
\(325\) 3.12436 0.173308
\(326\) 0 0
\(327\) 5.07180 0.280471
\(328\) 0 0
\(329\) 13.8564 12.0000i 0.763928 0.661581i
\(330\) 0 0
\(331\) 23.3205 1.28181 0.640906 0.767620i \(-0.278559\pi\)
0.640906 + 0.767620i \(0.278559\pi\)
\(332\) 0 0
\(333\) 9.85641i 0.540128i
\(334\) 0 0
\(335\) 25.8564 1.41269
\(336\) 0 0
\(337\) −24.3923 −1.32873 −0.664367 0.747407i \(-0.731299\pi\)
−0.664367 + 0.747407i \(0.731299\pi\)
\(338\) 0 0
\(339\) 1.85641i 0.100826i
\(340\) 0 0
\(341\) 10.1436 0.549306
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 2.92820 0.157649
\(346\) 0 0
\(347\) 6.53590 0.350865 0.175433 0.984491i \(-0.443868\pi\)
0.175433 + 0.984491i \(0.443868\pi\)
\(348\) 0 0
\(349\) 5.66025 0.302986 0.151493 0.988458i \(-0.451592\pi\)
0.151493 + 0.988458i \(0.451592\pi\)
\(350\) 0 0
\(351\) 5.07180i 0.270712i
\(352\) 0 0
\(353\) 2.14359i 0.114092i −0.998372 0.0570460i \(-0.981832\pi\)
0.998372 0.0570460i \(-0.0181682\pi\)
\(354\) 0 0
\(355\) 31.3205i 1.66232i
\(356\) 0 0
\(357\) 5.85641 5.07180i 0.309954 0.268428i
\(358\) 0 0
\(359\) 20.3923i 1.07626i 0.842860 + 0.538132i \(0.180870\pi\)
−0.842860 + 0.538132i \(0.819130\pi\)
\(360\) 0 0
\(361\) −3.39230 −0.178542
\(362\) 0 0
\(363\) 6.48334i 0.340287i
\(364\) 0 0
\(365\) 18.9282i 0.990747i
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 26.9282i 1.40183i
\(370\) 0 0
\(371\) 12.0000 + 13.8564i 0.623009 + 0.719389i
\(372\) 0 0
\(373\) 30.9282i 1.60140i 0.599064 + 0.800701i \(0.295539\pi\)
−0.599064 + 0.800701i \(0.704461\pi\)
\(374\) 0 0
\(375\) 5.07180i 0.261906i
\(376\) 0 0
\(377\) 8.78461i 0.452430i
\(378\) 0 0
\(379\) −14.5359 −0.746659 −0.373329 0.927699i \(-0.621784\pi\)
−0.373329 + 0.927699i \(0.621784\pi\)
\(380\) 0 0
\(381\) −10.6410 −0.545156
\(382\) 0 0
\(383\) 6.92820 0.354015 0.177007 0.984210i \(-0.443358\pi\)
0.177007 + 0.984210i \(0.443358\pi\)
\(384\) 0 0
\(385\) −8.00000 + 6.92820i −0.407718 + 0.353094i
\(386\) 0 0
\(387\) −23.3205 −1.18545
\(388\) 0 0
\(389\) 25.8564i 1.31097i 0.755207 + 0.655486i \(0.227536\pi\)
−0.755207 + 0.655486i \(0.772464\pi\)
\(390\) 0 0
\(391\) −5.85641 −0.296171
\(392\) 0 0
\(393\) 5.32051 0.268384
\(394\) 0 0
\(395\) 9.46410i 0.476191i
\(396\) 0 0
\(397\) −35.9090 −1.80222 −0.901110 0.433591i \(-0.857246\pi\)
−0.901110 + 0.433591i \(0.857246\pi\)
\(398\) 0 0
\(399\) 6.92820 6.00000i 0.346844 0.300376i
\(400\) 0 0
\(401\) 11.3205 0.565319 0.282660 0.959220i \(-0.408783\pi\)
0.282660 + 0.959220i \(0.408783\pi\)
\(402\) 0 0
\(403\) −8.78461 −0.437593
\(404\) 0 0
\(405\) 12.1962 0.606032
\(406\) 0 0
\(407\) 5.85641i 0.290291i
\(408\) 0 0
\(409\) 18.9282i 0.935939i −0.883745 0.467970i \(-0.844986\pi\)
0.883745 0.467970i \(-0.155014\pi\)
\(410\) 0 0
\(411\) 5.75129i 0.283690i
\(412\) 0 0
\(413\) 24.5885 + 28.3923i 1.20992 + 1.39709i
\(414\) 0 0
\(415\) 19.8564i 0.974713i
\(416\) 0 0
\(417\) −4.53590 −0.222124
\(418\) 0 0
\(419\) 21.1244i 1.03199i −0.856591 0.515996i \(-0.827422\pi\)
0.856591 0.515996i \(-0.172578\pi\)
\(420\) 0 0
\(421\) 20.0000i 0.974740i −0.873195 0.487370i \(-0.837956\pi\)
0.873195 0.487370i \(-0.162044\pi\)
\(422\) 0 0
\(423\) 17.0718 0.830059
\(424\) 0 0
\(425\) 9.85641i 0.478106i
\(426\) 0 0
\(427\) 11.3205 9.80385i 0.547838 0.474441i
\(428\) 0 0
\(429\) 1.35898i 0.0656124i
\(430\) 0 0
\(431\) 1.46410i 0.0705233i −0.999378 0.0352616i \(-0.988774\pi\)
0.999378 0.0352616i \(-0.0112265\pi\)
\(432\) 0 0
\(433\) 25.8564i 1.24258i −0.783581 0.621290i \(-0.786609\pi\)
0.783581 0.621290i \(-0.213391\pi\)
\(434\) 0 0
\(435\) −13.8564 −0.664364
\(436\) 0 0
\(437\) −6.92820 −0.331421
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 2.46410 17.0718i 0.117338 0.812943i
\(442\) 0 0
\(443\) 1.46410 0.0695616 0.0347808 0.999395i \(-0.488927\pi\)
0.0347808 + 0.999395i \(0.488927\pi\)
\(444\) 0 0
\(445\) 2.92820i 0.138810i
\(446\) 0 0
\(447\) −8.78461 −0.415498
\(448\) 0 0
\(449\) 31.8564 1.50340 0.751698 0.659507i \(-0.229235\pi\)
0.751698 + 0.659507i \(0.229235\pi\)
\(450\) 0 0
\(451\) 16.0000i 0.753411i
\(452\) 0 0
\(453\) 17.0718 0.802103
\(454\) 0 0
\(455\) 6.92820 6.00000i 0.324799 0.281284i
\(456\) 0 0
\(457\) −8.39230 −0.392575 −0.196288 0.980546i \(-0.562889\pi\)
−0.196288 + 0.980546i \(0.562889\pi\)
\(458\) 0 0
\(459\) 16.0000 0.746816
\(460\) 0 0
\(461\) 27.5167 1.28158 0.640789 0.767717i \(-0.278607\pi\)
0.640789 + 0.767717i \(0.278607\pi\)
\(462\) 0 0
\(463\) 20.5359i 0.954384i 0.878799 + 0.477192i \(0.158345\pi\)
−0.878799 + 0.477192i \(0.841655\pi\)
\(464\) 0 0
\(465\) 13.8564i 0.642575i
\(466\) 0 0
\(467\) 18.5885i 0.860171i −0.902788 0.430086i \(-0.858483\pi\)
0.902788 0.430086i \(-0.141517\pi\)
\(468\) 0 0
\(469\) 18.9282 16.3923i 0.874023 0.756926i
\(470\) 0 0
\(471\) 16.6410i 0.766778i
\(472\) 0 0
\(473\) 13.8564 0.637118
\(474\) 0 0
\(475\) 11.6603i 0.535009i
\(476\) 0 0
\(477\) 17.0718i 0.781664i
\(478\) 0 0
\(479\) −30.9282 −1.41315 −0.706573 0.707640i \(-0.749760\pi\)
−0.706573 + 0.707640i \(0.749760\pi\)
\(480\) 0 0
\(481\) 5.07180i 0.231254i
\(482\) 0 0
\(483\) 2.14359 1.85641i 0.0975369 0.0844694i
\(484\) 0 0
\(485\) 32.7846i 1.48867i
\(486\) 0 0
\(487\) 4.39230i 0.199034i 0.995036 + 0.0995172i \(0.0317298\pi\)
−0.995036 + 0.0995172i \(0.968270\pi\)
\(488\) 0 0
\(489\) 3.21539i 0.145405i
\(490\) 0 0
\(491\) −15.3205 −0.691405 −0.345702 0.938344i \(-0.612359\pi\)
−0.345702 + 0.938344i \(0.612359\pi\)
\(492\) 0 0
\(493\) 27.7128 1.24812
\(494\) 0 0
\(495\) −9.85641 −0.443013
\(496\) 0 0
\(497\) 19.8564 + 22.9282i 0.890682 + 1.02847i
\(498\) 0 0
\(499\) −4.39230 −0.196627 −0.0983133 0.995156i \(-0.531345\pi\)
−0.0983133 + 0.995156i \(0.531345\pi\)
\(500\) 0 0
\(501\) 13.8564i 0.619059i
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 11.4641 0.510146
\(506\) 0 0
\(507\) 8.33975i 0.370381i
\(508\) 0 0
\(509\) 31.8038 1.40968 0.704840 0.709366i \(-0.251019\pi\)
0.704840 + 0.709366i \(0.251019\pi\)
\(510\) 0 0
\(511\) 12.0000 + 13.8564i 0.530849 + 0.612971i
\(512\) 0 0
\(513\) 18.9282 0.835701
\(514\) 0 0
\(515\) 13.8564 0.610586
\(516\) 0 0
\(517\) −10.1436 −0.446115
\(518\) 0 0
\(519\) 4.92820i 0.216324i
\(520\) 0 0
\(521\) 26.9282i 1.17975i −0.807496 0.589873i \(-0.799178\pi\)
0.807496 0.589873i \(-0.200822\pi\)
\(522\) 0 0
\(523\) 43.3731i 1.89657i −0.317417 0.948286i \(-0.602816\pi\)
0.317417 0.948286i \(-0.397184\pi\)
\(524\) 0 0
\(525\) 3.12436 + 3.60770i 0.136358 + 0.157453i
\(526\) 0 0
\(527\) 27.7128i 1.20719i
\(528\) 0 0
\(529\) 20.8564 0.906800
\(530\) 0 0
\(531\) 34.9808i 1.51804i
\(532\) 0 0
\(533\) 13.8564i 0.600188i
\(534\) 0 0
\(535\) −33.8564 −1.46374
\(536\) 0 0
\(537\) 14.9282i 0.644200i
\(538\) 0 0
\(539\) −1.46410 + 10.1436i −0.0630633 + 0.436916i
\(540\) 0 0
\(541\) 28.0000i 1.20381i −0.798566 0.601907i \(-0.794408\pi\)
0.798566 0.601907i \(-0.205592\pi\)
\(542\) 0 0
\(543\) 9.71281i 0.416817i
\(544\) 0 0
\(545\) 18.9282i 0.810795i
\(546\) 0 0
\(547\) −14.5359 −0.621510 −0.310755 0.950490i \(-0.600582\pi\)
−0.310755 + 0.950490i \(0.600582\pi\)
\(548\) 0 0
\(549\) 13.9474 0.595262
\(550\) 0 0
\(551\) 32.7846 1.39667
\(552\) 0 0
\(553\) 6.00000 + 6.92820i 0.255146 + 0.294617i
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 17.0718i 0.723355i −0.932303 0.361678i \(-0.882204\pi\)
0.932303 0.361678i \(-0.117796\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −4.28719 −0.181005
\(562\) 0 0
\(563\) 22.9808i 0.968524i 0.874923 + 0.484262i \(0.160912\pi\)
−0.874923 + 0.484262i \(0.839088\pi\)
\(564\) 0 0
\(565\) −6.92820 −0.291472
\(566\) 0 0
\(567\) 8.92820 7.73205i 0.374949 0.324716i
\(568\) 0 0
\(569\) 7.60770 0.318931 0.159466 0.987203i \(-0.449023\pi\)
0.159466 + 0.987203i \(0.449023\pi\)
\(570\) 0 0
\(571\) 18.2487 0.763685 0.381842 0.924227i \(-0.375290\pi\)
0.381842 + 0.924227i \(0.375290\pi\)
\(572\) 0 0
\(573\) −3.32051 −0.138716
\(574\) 0 0
\(575\) 3.60770i 0.150451i
\(576\) 0 0
\(577\) 37.8564i 1.57598i 0.615686 + 0.787991i \(0.288879\pi\)
−0.615686 + 0.787991i \(0.711121\pi\)
\(578\) 0 0
\(579\) 12.0000i 0.498703i
\(580\) 0 0
\(581\) −12.5885 14.5359i −0.522257 0.603051i
\(582\) 0 0
\(583\) 10.1436i 0.420105i
\(584\) 0 0
\(585\) 8.53590 0.352916
\(586\) 0 0
\(587\) 14.1962i 0.585938i 0.956122 + 0.292969i \(0.0946433\pi\)
−0.956122 + 0.292969i \(0.905357\pi\)
\(588\) 0 0
\(589\) 32.7846i 1.35087i
\(590\) 0 0
\(591\) 8.78461 0.361351
\(592\) 0 0
\(593\) 35.7128i 1.46655i −0.679932 0.733275i \(-0.737991\pi\)
0.679932 0.733275i \(-0.262009\pi\)
\(594\) 0 0
\(595\) −18.9282 21.8564i −0.775981 0.896025i
\(596\) 0 0
\(597\) 18.1436i 0.742568i
\(598\) 0 0
\(599\) 19.4641i 0.795282i −0.917541 0.397641i \(-0.869829\pi\)
0.917541 0.397641i \(-0.130171\pi\)
\(600\) 0 0
\(601\) 34.6410i 1.41304i 0.707695 + 0.706518i \(0.249735\pi\)
−0.707695 + 0.706518i \(0.750265\pi\)
\(602\) 0 0
\(603\) 23.3205 0.949685
\(604\) 0 0
\(605\) −24.1962 −0.983713
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) −10.1436 + 8.78461i −0.411039 + 0.355970i
\(610\) 0 0
\(611\) 8.78461 0.355387
\(612\) 0 0
\(613\) 1.85641i 0.0749796i 0.999297 + 0.0374898i \(0.0119362\pi\)
−0.999297 + 0.0374898i \(0.988064\pi\)
\(614\) 0 0
\(615\) 21.8564 0.881335
\(616\) 0 0
\(617\) −16.3923 −0.659929 −0.329965 0.943993i \(-0.607037\pi\)
−0.329965 + 0.943993i \(0.607037\pi\)
\(618\) 0 0
\(619\) 13.8038i 0.554823i 0.960751 + 0.277412i \(0.0894766\pi\)
−0.960751 + 0.277412i \(0.910523\pi\)
\(620\) 0 0
\(621\) 5.85641 0.235009
\(622\) 0 0
\(623\) −1.85641 2.14359i −0.0743754 0.0858813i
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) 0 0
\(627\) −5.07180 −0.202548
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 24.2487i 0.965326i 0.875806 + 0.482663i \(0.160330\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) 20.7846i 0.826114i
\(634\) 0 0
\(635\) 39.7128i 1.57595i
\(636\) 0 0
\(637\) 1.26795 8.78461i 0.0502380 0.348059i
\(638\) 0 0
\(639\) 28.2487i 1.11750i
\(640\) 0 0
\(641\) −26.5359 −1.04810 −0.524052 0.851686i \(-0.675580\pi\)
−0.524052 + 0.851686i \(0.675580\pi\)
\(642\) 0 0
\(643\) 19.2679i 0.759854i 0.925017 + 0.379927i \(0.124051\pi\)
−0.925017 + 0.379927i \(0.875949\pi\)
\(644\) 0 0
\(645\) 18.9282i 0.745297i
\(646\) 0 0
\(647\) −46.6410 −1.83365 −0.916824 0.399292i \(-0.869256\pi\)
−0.916824 + 0.399292i \(0.869256\pi\)
\(648\) 0 0
\(649\) 20.7846i 0.815867i
\(650\) 0 0
\(651\) −8.78461 10.1436i −0.344296 0.397559i
\(652\) 0 0
\(653\) 12.0000i 0.469596i −0.972044 0.234798i \(-0.924557\pi\)
0.972044 0.234798i \(-0.0754429\pi\)
\(654\) 0 0
\(655\) 19.8564i 0.775854i
\(656\) 0 0
\(657\) 17.0718i 0.666034i
\(658\) 0 0
\(659\) 40.1051 1.56227 0.781137 0.624360i \(-0.214640\pi\)
0.781137 + 0.624360i \(0.214640\pi\)
\(660\) 0 0
\(661\) −3.80385 −0.147953 −0.0739763 0.997260i \(-0.523569\pi\)
−0.0739763 + 0.997260i \(0.523569\pi\)
\(662\) 0 0
\(663\) 3.71281 0.144194
\(664\) 0 0
\(665\) −22.3923 25.8564i −0.868336 1.00267i
\(666\) 0 0
\(667\) 10.1436 0.392762
\(668\) 0 0
\(669\) 5.85641i 0.226422i
\(670\) 0 0
\(671\) −8.28719 −0.319923
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 0 0
\(675\) 9.85641i 0.379373i
\(676\) 0 0
\(677\) −43.1244 −1.65740 −0.828702 0.559690i \(-0.810920\pi\)
−0.828702 + 0.559690i \(0.810920\pi\)
\(678\) 0 0
\(679\) 20.7846 + 24.0000i 0.797640 + 0.921035i
\(680\) 0 0
\(681\) 3.46410 0.132745
\(682\) 0 0
\(683\) −22.5359 −0.862312 −0.431156 0.902277i \(-0.641894\pi\)
−0.431156 + 0.902277i \(0.641894\pi\)
\(684\) 0 0
\(685\) −21.4641 −0.820101
\(686\) 0 0
\(687\) 12.9282i 0.493242i
\(688\) 0 0
\(689\) 8.78461i 0.334667i
\(690\) 0 0
\(691\) 18.9808i 0.722062i −0.932554 0.361031i \(-0.882425\pi\)
0.932554 0.361031i \(-0.117575\pi\)
\(692\) 0 0
\(693\) −7.21539 + 6.24871i −0.274090 + 0.237369i
\(694\) 0 0
\(695\) 16.9282i 0.642123i
\(696\) 0 0
\(697\) −43.7128 −1.65574
\(698\) 0 0
\(699\) 5.75129i 0.217534i
\(700\) 0 0
\(701\) 25.8564i 0.976583i 0.872681 + 0.488291i \(0.162380\pi\)
−0.872681 + 0.488291i \(0.837620\pi\)
\(702\) 0 0
\(703\) 18.9282 0.713891
\(704\) 0 0
\(705\) 13.8564i 0.521862i
\(706\) 0 0
\(707\) 8.39230 7.26795i 0.315625 0.273339i
\(708\) 0 0
\(709\) 17.0718i 0.641145i 0.947224 + 0.320572i \(0.103875\pi\)
−0.947224 + 0.320572i \(0.896125\pi\)
\(710\) 0 0
\(711\) 8.53590i 0.320121i
\(712\) 0 0
\(713\) 10.1436i 0.379881i
\(714\) 0 0
\(715\) −5.07180 −0.189674
\(716\) 0 0
\(717\) −14.9282 −0.557504
\(718\) 0 0
\(719\) 30.9282 1.15343 0.576714 0.816946i \(-0.304335\pi\)
0.576714 + 0.816946i \(0.304335\pi\)
\(720\) 0 0
\(721\) 10.1436 8.78461i 0.377767 0.327156i
\(722\) 0 0
\(723\) −8.78461 −0.326703
\(724\) 0 0
\(725\) 17.0718i 0.634031i
\(726\) 0 0
\(727\) −32.7846 −1.21591 −0.607957 0.793970i \(-0.708011\pi\)
−0.607957 + 0.793970i \(0.708011\pi\)
\(728\) 0 0
\(729\) 2.21539 0.0820515
\(730\) 0 0
\(731\) 37.8564i 1.40017i
\(732\) 0 0
\(733\) −8.19615 −0.302732 −0.151366 0.988478i \(-0.548367\pi\)
−0.151366 + 0.988478i \(0.548367\pi\)
\(734\) 0 0
\(735\) 13.8564 + 2.00000i 0.511101 + 0.0737711i
\(736\) 0 0
\(737\) −13.8564 −0.510407
\(738\) 0 0
\(739\) 23.3205 0.857859 0.428929 0.903338i \(-0.358891\pi\)
0.428929 + 0.903338i \(0.358891\pi\)
\(740\) 0 0
\(741\) 4.39230 0.161355
\(742\) 0 0
\(743\) 17.4641i 0.640696i −0.947300 0.320348i \(-0.896200\pi\)
0.947300 0.320348i \(-0.103800\pi\)
\(744\) 0 0
\(745\) 32.7846i 1.20114i
\(746\) 0 0
\(747\) 17.9090i 0.655255i
\(748\) 0 0
\(749\) −24.7846 + 21.4641i −0.905610 + 0.784281i
\(750\) 0 0
\(751\) 33.4641i 1.22112i −0.791969 0.610561i \(-0.790944\pi\)
0.791969 0.610561i \(-0.209056\pi\)
\(752\) 0 0
\(753\) 11.7513 0.428241
\(754\) 0 0
\(755\) 63.7128i 2.31875i
\(756\) 0 0
\(757\) 53.5692i 1.94701i −0.228673 0.973503i \(-0.573439\pi\)
0.228673 0.973503i \(-0.426561\pi\)
\(758\) 0 0
\(759\) −1.56922 −0.0569591
\(760\) 0 0
\(761\) 24.7846i 0.898441i 0.893421 + 0.449221i \(0.148298\pi\)
−0.893421 + 0.449221i \(0.851702\pi\)
\(762\) 0 0
\(763\) −12.0000 13.8564i −0.434429 0.501636i
\(764\) 0 0
\(765\) 26.9282i 0.973591i
\(766\) 0 0
\(767\) 18.0000i 0.649942i
\(768\) 0 0
\(769\) 25.8564i 0.932406i −0.884678 0.466203i \(-0.845622\pi\)
0.884678 0.466203i \(-0.154378\pi\)
\(770\) 0 0
\(771\) 4.28719 0.154399
\(772\) 0 0
\(773\) 22.4449 0.807286 0.403643 0.914917i \(-0.367744\pi\)
0.403643 + 0.914917i \(0.367744\pi\)
\(774\) 0 0
\(775\) −17.0718 −0.613237
\(776\) 0 0
\(777\) −5.85641 + 5.07180i −0.210097 + 0.181950i
\(778\) 0 0
\(779\) −51.7128 −1.85280
\(780\) 0 0
\(781\) 16.7846i 0.600601i
\(782\) 0 0
\(783\) −27.7128 −0.990375
\(784\) 0 0
\(785\) −62.1051 −2.21663
\(786\) 0 0
\(787\) 10.5885i 0.377438i −0.982031 0.188719i \(-0.939567\pi\)
0.982031 0.188719i \(-0.0604335\pi\)
\(788\) 0 0
\(789\) 3.32051 0.118213
\(790\) 0 0
\(791\) −5.07180 + 4.39230i −0.180332 + 0.156172i
\(792\) 0 0
\(793\) 7.17691 0.254860
\(794\) 0 0
\(795\) −13.8564 −0.491436
\(796\) 0 0
\(797\) −16.8756 −0.597766 −0.298883 0.954290i \(-0.596614\pi\)
−0.298883 + 0.954290i \(0.596614\pi\)
\(798\) 0 0
\(799\) 27.7128i 0.980409i
\(800\) 0 0
\(801\) 2.64102i 0.0933157i
\(802\) 0 0