Properties

Label 896.2.e.f.447.1
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 447.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 896.447
Dual form 896.2.e.f.447.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205i q^{3} -0.732051 q^{5} +(2.00000 + 1.73205i) q^{7} -4.46410 q^{9} +O(q^{10})\) \(q-2.73205i q^{3} -0.732051 q^{5} +(2.00000 + 1.73205i) q^{7} -4.46410 q^{9} +5.46410 q^{11} +4.73205 q^{13} +2.00000i q^{15} -4.00000i q^{17} -1.26795i q^{19} +(4.73205 - 5.46410i) q^{21} +5.46410i q^{23} -4.46410 q^{25} +4.00000i q^{27} -6.92820i q^{29} +6.92820 q^{31} -14.9282i q^{33} +(-1.46410 - 1.26795i) q^{35} +4.00000i q^{37} -12.9282i q^{39} +2.92820i q^{41} -2.53590 q^{43} +3.26795 q^{45} -6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} -10.9282 q^{51} -6.92820i q^{53} -4.00000 q^{55} -3.46410 q^{57} +3.80385i q^{59} -11.6603 q^{61} +(-8.92820 - 7.73205i) q^{63} -3.46410 q^{65} +2.53590 q^{67} +14.9282 q^{69} +4.53590i q^{71} -6.92820i q^{73} +12.1962i q^{75} +(10.9282 + 9.46410i) q^{77} -3.46410i q^{79} -2.46410 q^{81} -10.7321i q^{83} +2.92820i q^{85} -18.9282 q^{87} -14.9282i q^{89} +(9.46410 + 8.19615i) q^{91} -18.9282i q^{93} +0.928203i q^{95} +12.0000i q^{97} -24.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{5} + 8q^{7} - 4q^{9} + 8q^{11} + 12q^{13} + 12q^{21} - 4q^{25} + 8q^{35} - 24q^{43} + 20q^{45} + 4q^{49} - 16q^{51} - 16q^{55} - 12q^{61} - 8q^{63} + 24q^{67} + 32q^{69} + 16q^{77} + 4q^{81} - 48q^{87} + 24q^{91} - 56q^{99} + O(q^{100}) \)

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
\(3\) 2.73205i 1.57735i −0.614810 0.788675i \(-0.710767\pi\)
0.614810 0.788675i \(-0.289233\pi\)
\(4\) 0 0
\(5\) −0.732051 −0.327383 −0.163692 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) −4.46410 −1.48803
\(10\) 0 0
\(11\) 5.46410 1.64749 0.823744 0.566961i \(-0.191881\pi\)
0.823744 + 0.566961i \(0.191881\pi\)
\(12\) 0 0
\(13\) 4.73205 1.31243 0.656217 0.754572i \(-0.272155\pi\)
0.656217 + 0.754572i \(0.272155\pi\)
\(14\) 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
\(19\) 1.26795i 0.290887i −0.989367 0.145444i \(-0.953539\pi\)
0.989367 0.145444i \(-0.0464610\pi\)
\(20\) 0 0
\(21\) 4.73205 5.46410i 1.03262 1.19236i
\(22\) 0 0
\(23\) 5.46410i 1.13934i 0.821872 + 0.569672i \(0.192930\pi\)
−0.821872 + 0.569672i \(0.807070\pi\)
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.92820i 1.28654i −0.765641 0.643268i \(-0.777578\pi\)
0.765641 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) 14.9282i 2.59867i
\(34\) 0 0
\(35\) −1.46410 1.26795i −0.247478 0.214323i
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 12.9282i 2.07017i
\(40\) 0 0
\(41\) 2.92820i 0.457309i 0.973508 + 0.228654i \(0.0734325\pi\)
−0.973508 + 0.228654i \(0.926567\pi\)
\(42\) 0 0
\(43\) −2.53590 −0.386721 −0.193360 0.981128i \(-0.561939\pi\)
−0.193360 + 0.981128i \(0.561939\pi\)
\(44\) 0 0
\(45\) 3.26795 0.487157
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) −10.9282 −1.53025
\(52\) 0 0
\(53\) 6.92820i 0.951662i −0.879537 0.475831i \(-0.842147\pi\)
0.879537 0.475831i \(-0.157853\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −3.46410 −0.458831
\(58\) 0 0
\(59\) 3.80385i 0.495219i 0.968860 + 0.247609i \(0.0796450\pi\)
−0.968860 + 0.247609i \(0.920355\pi\)
\(60\) 0 0
\(61\) −11.6603 −1.49294 −0.746471 0.665418i \(-0.768253\pi\)
−0.746471 + 0.665418i \(0.768253\pi\)
\(62\) 0 0
\(63\) −8.92820 7.73205i −1.12485 0.974147i
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) 2.53590 0.309809 0.154905 0.987929i \(-0.450493\pi\)
0.154905 + 0.987929i \(0.450493\pi\)
\(68\) 0 0
\(69\) 14.9282 1.79714
\(70\) 0 0
\(71\) 4.53590i 0.538312i 0.963097 + 0.269156i \(0.0867447\pi\)
−0.963097 + 0.269156i \(0.913255\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 0 0
\(75\) 12.1962i 1.40829i
\(76\) 0 0
\(77\) 10.9282 + 9.46410i 1.24538 + 1.07853i
\(78\) 0 0
\(79\) 3.46410i 0.389742i −0.980829 0.194871i \(-0.937571\pi\)
0.980829 0.194871i \(-0.0624288\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) 10.7321i 1.17800i −0.808135 0.588998i \(-0.799523\pi\)
0.808135 0.588998i \(-0.200477\pi\)
\(84\) 0 0
\(85\) 2.92820i 0.317608i
\(86\) 0 0
\(87\) −18.9282 −2.02932
\(88\) 0 0
\(89\) 14.9282i 1.58239i −0.611566 0.791193i \(-0.709460\pi\)
0.611566 0.791193i \(-0.290540\pi\)
\(90\) 0 0
\(91\) 9.46410 + 8.19615i 0.992107 + 0.859190i
\(92\) 0 0
\(93\) 18.9282i 1.96276i
\(94\) 0 0
\(95\) 0.928203i 0.0952316i
\(96\) 0 0
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 0 0
\(99\) −24.3923 −2.45152
\(100\) 0 0
\(101\) −6.19615 −0.616540 −0.308270 0.951299i \(-0.599750\pi\)
−0.308270 + 0.951299i \(0.599750\pi\)
\(102\) 0 0
\(103\) 18.9282 1.86505 0.932526 0.361104i \(-0.117600\pi\)
0.932526 + 0.361104i \(0.117600\pi\)
\(104\) 0 0
\(105\) −3.46410 + 4.00000i −0.338062 + 0.390360i
\(106\) 0 0
\(107\) 8.39230 0.811315 0.405657 0.914025i \(-0.367043\pi\)
0.405657 + 0.914025i \(0.367043\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i 0.943349 + 0.331801i \(0.107656\pi\)
−0.943349 + 0.331801i \(0.892344\pi\)
\(110\) 0 0
\(111\) 10.9282 1.03726
\(112\) 0 0
\(113\) −9.46410 −0.890308 −0.445154 0.895454i \(-0.646851\pi\)
−0.445154 + 0.895454i \(0.646851\pi\)
\(114\) 0 0
\(115\) 4.00000i 0.373002i
\(116\) 0 0
\(117\) −21.1244 −1.95295
\(118\) 0 0
\(119\) 6.92820 8.00000i 0.635107 0.733359i
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 21.4641i 1.90463i 0.305115 + 0.952316i \(0.401305\pi\)
−0.305115 + 0.952316i \(0.598695\pi\)
\(128\) 0 0
\(129\) 6.92820i 0.609994i
\(130\) 0 0
\(131\) 10.7321i 0.937664i −0.883287 0.468832i \(-0.844675\pi\)
0.883287 0.468832i \(-0.155325\pi\)
\(132\) 0 0
\(133\) 2.19615 2.53590i 0.190431 0.219890i
\(134\) 0 0
\(135\) 2.92820i 0.252020i
\(136\) 0 0
\(137\) 19.8564 1.69645 0.848224 0.529638i \(-0.177672\pi\)
0.848224 + 0.529638i \(0.177672\pi\)
\(138\) 0 0
\(139\) 4.19615i 0.355913i −0.984038 0.177957i \(-0.943051\pi\)
0.984038 0.177957i \(-0.0569486\pi\)
\(140\) 0 0
\(141\) 18.9282i 1.59404i
\(142\) 0 0
\(143\) 25.8564 2.16222
\(144\) 0 0
\(145\) 5.07180i 0.421190i
\(146\) 0 0
\(147\) 18.9282 2.73205i 1.56117 0.225336i
\(148\) 0 0
\(149\) 12.0000i 0.983078i 0.870855 + 0.491539i \(0.163566\pi\)
−0.870855 + 0.491539i \(0.836434\pi\)
\(150\) 0 0
\(151\) 11.3205i 0.921250i 0.887595 + 0.460625i \(0.152375\pi\)
−0.887595 + 0.460625i \(0.847625\pi\)
\(152\) 0 0
\(153\) 17.8564i 1.44360i
\(154\) 0 0
\(155\) −5.07180 −0.407377
\(156\) 0 0
\(157\) −19.2679 −1.53775 −0.768875 0.639399i \(-0.779183\pi\)
−0.768875 + 0.639399i \(0.779183\pi\)
\(158\) 0 0
\(159\) −18.9282 −1.50110
\(160\) 0 0
\(161\) −9.46410 + 10.9282i −0.745876 + 0.861263i
\(162\) 0 0
\(163\) 16.3923 1.28394 0.641972 0.766728i \(-0.278116\pi\)
0.641972 + 0.766728i \(0.278116\pi\)
\(164\) 0 0
\(165\) 10.9282i 0.850759i
\(166\) 0 0
\(167\) 5.07180 0.392467 0.196234 0.980557i \(-0.437129\pi\)
0.196234 + 0.980557i \(0.437129\pi\)
\(168\) 0 0
\(169\) 9.39230 0.722485
\(170\) 0 0
\(171\) 5.66025i 0.432850i
\(172\) 0 0
\(173\) −3.26795 −0.248458 −0.124229 0.992254i \(-0.539646\pi\)
−0.124229 + 0.992254i \(0.539646\pi\)
\(174\) 0 0
\(175\) −8.92820 7.73205i −0.674909 0.584488i
\(176\) 0 0
\(177\) 10.3923 0.781133
\(178\) 0 0
\(179\) −0.392305 −0.0293222 −0.0146611 0.999893i \(-0.504667\pi\)
−0.0146611 + 0.999893i \(0.504667\pi\)
\(180\) 0 0
\(181\) −16.7321 −1.24368 −0.621842 0.783143i \(-0.713615\pi\)
−0.621842 + 0.783143i \(0.713615\pi\)
\(182\) 0 0
\(183\) 31.8564i 2.35489i
\(184\) 0 0
\(185\) 2.92820i 0.215286i
\(186\) 0 0
\(187\) 21.8564i 1.59830i
\(188\) 0 0
\(189\) −6.92820 + 8.00000i −0.503953 + 0.581914i
\(190\) 0 0
\(191\) 11.4641i 0.829513i 0.909932 + 0.414757i \(0.136133\pi\)
−0.909932 + 0.414757i \(0.863867\pi\)
\(192\) 0 0
\(193\) 4.39230 0.316165 0.158083 0.987426i \(-0.449469\pi\)
0.158083 + 0.987426i \(0.449469\pi\)
\(194\) 0 0
\(195\) 9.46410i 0.677738i
\(196\) 0 0
\(197\) 12.0000i 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) 0 0
\(201\) 6.92820i 0.488678i
\(202\) 0 0
\(203\) 12.0000 13.8564i 0.842235 0.972529i
\(204\) 0 0
\(205\) 2.14359i 0.149715i
\(206\) 0 0
\(207\) 24.3923i 1.69538i
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) −7.60770 −0.523735 −0.261868 0.965104i \(-0.584338\pi\)
−0.261868 + 0.965104i \(0.584338\pi\)
\(212\) 0 0
\(213\) 12.3923 0.849107
\(214\) 0 0
\(215\) 1.85641 0.126606
\(216\) 0 0
\(217\) 13.8564 + 12.0000i 0.940634 + 0.814613i
\(218\) 0 0
\(219\) −18.9282 −1.27905
\(220\) 0 0
\(221\) 18.9282i 1.27325i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 19.9282 1.32855
\(226\) 0 0
\(227\) 1.26795i 0.0841567i −0.999114 0.0420784i \(-0.986602\pi\)
0.999114 0.0420784i \(-0.0133979\pi\)
\(228\) 0 0
\(229\) −0.339746 −0.0224510 −0.0112255 0.999937i \(-0.503573\pi\)
−0.0112255 + 0.999937i \(0.503573\pi\)
\(230\) 0 0
\(231\) 25.8564 29.8564i 1.70123 1.96441i
\(232\) 0 0
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) 0 0
\(235\) 5.07180 0.330848
\(236\) 0 0
\(237\) −9.46410 −0.614759
\(238\) 0 0
\(239\) 0.392305i 0.0253761i −0.999920 0.0126880i \(-0.995961\pi\)
0.999920 0.0126880i \(-0.00403884\pi\)
\(240\) 0 0
\(241\) 12.0000i 0.772988i 0.922292 + 0.386494i \(0.126314\pi\)
−0.922292 + 0.386494i \(0.873686\pi\)
\(242\) 0 0
\(243\) 18.7321i 1.20166i
\(244\) 0 0
\(245\) −0.732051 5.07180i −0.0467690 0.324025i
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) −29.3205 −1.85811
\(250\) 0 0
\(251\) 22.0526i 1.39195i 0.718068 + 0.695973i \(0.245027\pi\)
−0.718068 + 0.695973i \(0.754973\pi\)
\(252\) 0 0
\(253\) 29.8564i 1.87706i
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 21.8564i 1.36337i 0.731648 + 0.681683i \(0.238751\pi\)
−0.731648 + 0.681683i \(0.761249\pi\)
\(258\) 0 0
\(259\) −6.92820 + 8.00000i −0.430498 + 0.497096i
\(260\) 0 0
\(261\) 30.9282i 1.91441i
\(262\) 0 0
\(263\) 11.4641i 0.706907i −0.935452 0.353453i \(-0.885007\pi\)
0.935452 0.353453i \(-0.114993\pi\)
\(264\) 0 0
\(265\) 5.07180i 0.311558i
\(266\) 0 0
\(267\) −40.7846 −2.49598
\(268\) 0 0
\(269\) −1.12436 −0.0685532 −0.0342766 0.999412i \(-0.510913\pi\)
−0.0342766 + 0.999412i \(0.510913\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 22.3923 25.8564i 1.35524 1.56490i
\(274\) 0 0
\(275\) −24.3923 −1.47091
\(276\) 0 0
\(277\) 28.7846i 1.72950i 0.502203 + 0.864750i \(0.332523\pi\)
−0.502203 + 0.864750i \(0.667477\pi\)
\(278\) 0 0
\(279\) −30.9282 −1.85162
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 15.8038i 0.939441i −0.882815 0.469721i \(-0.844355\pi\)
0.882815 0.469721i \(-0.155645\pi\)
\(284\) 0 0
\(285\) 2.53590 0.150214
\(286\) 0 0
\(287\) −5.07180 + 5.85641i −0.299379 + 0.345693i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 32.7846 1.92187
\(292\) 0 0
\(293\) 1.80385 0.105382 0.0526910 0.998611i \(-0.483220\pi\)
0.0526910 + 0.998611i \(0.483220\pi\)
\(294\) 0 0
\(295\) 2.78461i 0.162126i
\(296\) 0 0
\(297\) 21.8564i 1.26824i
\(298\) 0 0
\(299\) 25.8564i 1.49531i
\(300\) 0 0
\(301\) −5.07180 4.39230i −0.292334 0.253168i
\(302\) 0 0
\(303\) 16.9282i 0.972500i
\(304\) 0 0
\(305\) 8.53590 0.488764
\(306\) 0 0
\(307\) 13.2679i 0.757242i 0.925552 + 0.378621i \(0.123602\pi\)
−0.925552 + 0.378621i \(0.876398\pi\)
\(308\) 0 0
\(309\) 51.7128i 2.94184i
\(310\) 0 0
\(311\) −25.8564 −1.46618 −0.733091 0.680130i \(-0.761923\pi\)
−0.733091 + 0.680130i \(0.761923\pi\)
\(312\) 0 0
\(313\) 32.7846i 1.85310i 0.376177 + 0.926548i \(0.377238\pi\)
−0.376177 + 0.926548i \(0.622762\pi\)
\(314\) 0 0
\(315\) 6.53590 + 5.66025i 0.368256 + 0.318919i
\(316\) 0 0
\(317\) 1.85641i 0.104266i 0.998640 + 0.0521331i \(0.0166020\pi\)
−0.998640 + 0.0521331i \(0.983398\pi\)
\(318\) 0 0
\(319\) 37.8564i 2.11955i
\(320\) 0 0
\(321\) 22.9282i 1.27973i
\(322\) 0 0
\(323\) −5.07180 −0.282202
\(324\) 0 0
\(325\) −21.1244 −1.17177
\(326\) 0 0
\(327\) 18.9282 1.04673
\(328\) 0 0
\(329\) −13.8564 12.0000i −0.763928 0.661581i
\(330\) 0 0
\(331\) −11.3205 −0.622231 −0.311116 0.950372i \(-0.600703\pi\)
−0.311116 + 0.950372i \(0.600703\pi\)
\(332\) 0 0
\(333\) 17.8564i 0.978525i
\(334\) 0 0
\(335\) −1.85641 −0.101426
\(336\) 0 0
\(337\) −3.60770 −0.196524 −0.0982618 0.995161i \(-0.531328\pi\)
−0.0982618 + 0.995161i \(0.531328\pi\)
\(338\) 0 0
\(339\) 25.8564i 1.40433i
\(340\) 0 0
\(341\) 37.8564 2.05004
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) −10.9282 −0.588355
\(346\) 0 0
\(347\) 13.4641 0.722791 0.361395 0.932413i \(-0.382300\pi\)
0.361395 + 0.932413i \(0.382300\pi\)
\(348\) 0 0
\(349\) −11.6603 −0.624159 −0.312080 0.950056i \(-0.601026\pi\)
−0.312080 + 0.950056i \(0.601026\pi\)
\(350\) 0 0
\(351\) 18.9282i 1.01031i
\(352\) 0 0
\(353\) 29.8564i 1.58910i −0.607201 0.794548i \(-0.707708\pi\)
0.607201 0.794548i \(-0.292292\pi\)
\(354\) 0 0
\(355\) 3.32051i 0.176234i
\(356\) 0 0
\(357\) −21.8564 18.9282i −1.15676 1.00179i
\(358\) 0 0
\(359\) 0.392305i 0.0207051i −0.999946 0.0103525i \(-0.996705\pi\)
0.999946 0.0103525i \(-0.00329537\pi\)
\(360\) 0 0
\(361\) 17.3923 0.915384
\(362\) 0 0
\(363\) 51.5167i 2.70392i
\(364\) 0 0
\(365\) 5.07180i 0.265470i
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 13.0718i 0.680491i
\(370\) 0 0
\(371\) 12.0000 13.8564i 0.623009 0.719389i
\(372\) 0 0
\(373\) 17.0718i 0.883944i 0.897029 + 0.441972i \(0.145721\pi\)
−0.897029 + 0.441972i \(0.854279\pi\)
\(374\) 0 0
\(375\) 18.9282i 0.977448i
\(376\) 0 0
\(377\) 32.7846i 1.68849i
\(378\) 0 0
\(379\) −21.4641 −1.10254 −0.551268 0.834328i \(-0.685856\pi\)
−0.551268 + 0.834328i \(0.685856\pi\)
\(380\) 0 0
\(381\) 58.6410 3.00427
\(382\) 0 0
\(383\) −6.92820 −0.354015 −0.177007 0.984210i \(-0.556642\pi\)
−0.177007 + 0.984210i \(0.556642\pi\)
\(384\) 0 0
\(385\) −8.00000 6.92820i −0.407718 0.353094i
\(386\) 0 0
\(387\) 11.3205 0.575454
\(388\) 0 0
\(389\) 1.85641i 0.0941235i −0.998892 0.0470618i \(-0.985014\pi\)
0.998892 0.0470618i \(-0.0149858\pi\)
\(390\) 0 0
\(391\) 21.8564 1.10533
\(392\) 0 0
\(393\) −29.3205 −1.47902
\(394\) 0 0
\(395\) 2.53590i 0.127595i
\(396\) 0 0
\(397\) 29.9090 1.50109 0.750544 0.660821i \(-0.229792\pi\)
0.750544 + 0.660821i \(0.229792\pi\)
\(398\) 0 0
\(399\) −6.92820 6.00000i −0.346844 0.300376i
\(400\) 0 0
\(401\) −23.3205 −1.16457 −0.582285 0.812985i \(-0.697841\pi\)
−0.582285 + 0.812985i \(0.697841\pi\)
\(402\) 0 0
\(403\) 32.7846 1.63312
\(404\) 0 0
\(405\) 1.80385 0.0896339
\(406\) 0 0
\(407\) 21.8564i 1.08338i
\(408\) 0 0
\(409\) 5.07180i 0.250784i −0.992107 0.125392i \(-0.959981\pi\)
0.992107 0.125392i \(-0.0400189\pi\)
\(410\) 0 0
\(411\) 54.2487i 2.67589i
\(412\) 0 0
\(413\) −6.58846 + 7.60770i −0.324197 + 0.374350i
\(414\) 0 0
\(415\) 7.85641i 0.385656i
\(416\) 0 0
\(417\) −11.4641 −0.561399
\(418\) 0 0
\(419\) 3.12436i 0.152635i 0.997084 + 0.0763174i \(0.0243162\pi\)
−0.997084 + 0.0763174i \(0.975684\pi\)
\(420\) 0 0
\(421\) 20.0000i 0.974740i −0.873195 0.487370i \(-0.837956\pi\)
0.873195 0.487370i \(-0.162044\pi\)
\(422\) 0 0
\(423\) 30.9282 1.50378
\(424\) 0 0
\(425\) 17.8564i 0.866163i
\(426\) 0 0
\(427\) −23.3205 20.1962i −1.12856 0.977360i
\(428\) 0 0
\(429\) 70.6410i 3.41058i
\(430\) 0 0
\(431\) 5.46410i 0.263197i 0.991303 + 0.131598i \(0.0420109\pi\)
−0.991303 + 0.131598i \(0.957989\pi\)
\(432\) 0 0
\(433\) 1.85641i 0.0892132i 0.999005 + 0.0446066i \(0.0142034\pi\)
−0.999005 + 0.0446066i \(0.985797\pi\)
\(434\) 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
\(441\) −4.46410 30.9282i −0.212576 1.47277i
\(442\) 0 0
\(443\) −5.46410 −0.259607 −0.129804 0.991540i \(-0.541435\pi\)
−0.129804 + 0.991540i \(0.541435\pi\)
\(444\) 0 0
\(445\) 10.9282i 0.518047i
\(446\) 0 0
\(447\) 32.7846 1.55066
\(448\) 0 0
\(449\) 4.14359 0.195548 0.0977741 0.995209i \(-0.468828\pi\)
0.0977741 + 0.995209i \(0.468828\pi\)
\(450\) 0 0
\(451\) 16.0000i 0.753411i
\(452\) 0 0
\(453\) 30.9282 1.45313
\(454\) 0 0
\(455\) −6.92820 6.00000i −0.324799 0.281284i
\(456\) 0 0
\(457\) 12.3923 0.579688 0.289844 0.957074i \(-0.406397\pi\)
0.289844 + 0.957074i \(0.406397\pi\)
\(458\) 0 0
\(459\) 16.0000 0.746816
\(460\) 0 0
\(461\) −17.5167 −0.815832 −0.407916 0.913019i \(-0.633744\pi\)
−0.407916 + 0.913019i \(0.633744\pi\)
\(462\) 0 0
\(463\) 27.4641i 1.27637i 0.769885 + 0.638183i \(0.220313\pi\)
−0.769885 + 0.638183i \(0.779687\pi\)
\(464\) 0 0
\(465\) 13.8564i 0.642575i
\(466\) 0 0
\(467\) 12.5885i 0.582524i 0.956643 + 0.291262i \(0.0940752\pi\)
−0.956643 + 0.291262i \(0.905925\pi\)
\(468\) 0 0
\(469\) 5.07180 + 4.39230i 0.234194 + 0.202818i
\(470\) 0 0
\(471\) 52.6410i 2.42557i
\(472\) 0 0
\(473\) −13.8564 −0.637118
\(474\) 0 0
\(475\) 5.66025i 0.259710i
\(476\) 0 0
\(477\) 30.9282i 1.41611i
\(478\) 0 0
\(479\) −17.0718 −0.780030 −0.390015 0.920808i \(-0.627530\pi\)
−0.390015 + 0.920808i \(0.627530\pi\)
\(480\) 0 0
\(481\) 18.9282i 0.863052i
\(482\) 0 0
\(483\) 29.8564 + 25.8564i 1.35851 + 1.17651i
\(484\) 0 0
\(485\) 8.78461i 0.398889i
\(486\) 0 0
\(487\) 16.3923i 0.742806i −0.928472 0.371403i \(-0.878877\pi\)
0.928472 0.371403i \(-0.121123\pi\)
\(488\) 0 0
\(489\) 44.7846i 2.02523i
\(490\) 0 0
\(491\) 19.3205 0.871922 0.435961 0.899965i \(-0.356409\pi\)
0.435961 + 0.899965i \(0.356409\pi\)
\(492\) 0 0
\(493\) −27.7128 −1.24812
\(494\) 0 0
\(495\) 17.8564 0.802586
\(496\) 0 0
\(497\) −7.85641 + 9.07180i −0.352408 + 0.406926i
\(498\) 0 0
\(499\) 16.3923 0.733820 0.366910 0.930256i \(-0.380416\pi\)
0.366910 + 0.930256i \(0.380416\pi\)
\(500\) 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
\(505\) 4.53590 0.201845
\(506\) 0 0
\(507\) 25.6603i 1.13961i
\(508\) 0 0
\(509\) 42.1962 1.87031 0.935156 0.354237i \(-0.115259\pi\)
0.935156 + 0.354237i \(0.115259\pi\)
\(510\) 0 0
\(511\) 12.0000 13.8564i 0.530849 0.612971i
\(512\) 0 0
\(513\) 5.07180 0.223925
\(514\) 0 0
\(515\) −13.8564 −0.610586
\(516\) 0 0
\(517\) −37.8564 −1.66492
\(518\) 0 0
\(519\) 8.92820i 0.391905i
\(520\) 0 0
\(521\) 13.0718i 0.572686i −0.958127 0.286343i \(-0.907560\pi\)
0.958127 0.286343i \(-0.0924397\pi\)
\(522\) 0 0
\(523\) 29.3731i 1.28439i 0.766539 + 0.642197i \(0.221977\pi\)
−0.766539 + 0.642197i \(0.778023\pi\)
\(524\) 0 0
\(525\) −21.1244 + 24.3923i −0.921942 + 1.06457i
\(526\) 0 0
\(527\) 27.7128i 1.20719i
\(528\) 0 0
\(529\) −6.85641 −0.298105
\(530\) 0 0
\(531\) 16.9808i 0.736902i
\(532\) 0 0
\(533\) 13.8564i 0.600188i
\(534\) 0 0
\(535\) −6.14359 −0.265611
\(536\) 0 0
\(537\) 1.07180i 0.0462514i
\(538\) 0 0
\(539\) 5.46410 + 37.8564i 0.235356 + 1.63059i
\(540\) 0 0
\(541\) 28.0000i 1.20381i −0.798566 0.601907i \(-0.794408\pi\)
0.798566 0.601907i \(-0.205592\pi\)
\(542\) 0 0
\(543\) 45.7128i 1.96172i
\(544\) 0 0
\(545\) 5.07180i 0.217252i
\(546\) 0 0
\(547\) −21.4641 −0.917739 −0.458869 0.888504i \(-0.651745\pi\)
−0.458869 + 0.888504i \(0.651745\pi\)
\(548\) 0 0
\(549\) 52.0526 2.22155
\(550\) 0 0
\(551\) −8.78461 −0.374237
\(552\) 0 0
\(553\) 6.00000 6.92820i 0.255146 0.294617i
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 30.9282i 1.31047i −0.755425 0.655235i \(-0.772570\pi\)
0.755425 0.655235i \(-0.227430\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −59.7128 −2.52108
\(562\) 0 0
\(563\) 28.9808i 1.22139i −0.791865 0.610697i \(-0.790889\pi\)
0.791865 0.610697i \(-0.209111\pi\)
\(564\) 0 0
\(565\) 6.92820 0.291472
\(566\) 0 0
\(567\) −4.92820 4.26795i −0.206965 0.179237i
\(568\) 0 0
\(569\) 28.3923 1.19027 0.595134 0.803627i \(-0.297099\pi\)
0.595134 + 0.803627i \(0.297099\pi\)
\(570\) 0 0
\(571\) −30.2487 −1.26587 −0.632935 0.774205i \(-0.718150\pi\)
−0.632935 + 0.774205i \(0.718150\pi\)
\(572\) 0 0
\(573\) 31.3205 1.30843
\(574\) 0 0
\(575\) 24.3923i 1.01723i
\(576\) 0 0
\(577\) 10.1436i 0.422283i 0.977455 + 0.211142i \(0.0677181\pi\)
−0.977455 + 0.211142i \(0.932282\pi\)
\(578\) 0 0
\(579\) 12.0000i 0.498703i
\(580\) 0 0
\(581\) 18.5885 21.4641i 0.771179 0.890481i
\(582\) 0 0
\(583\) 37.8564i 1.56785i
\(584\) 0 0
\(585\) 15.4641 0.639362
\(586\) 0 0
\(587\) 3.80385i 0.157002i 0.996914 + 0.0785008i \(0.0250133\pi\)
−0.996914 + 0.0785008i \(0.974987\pi\)
\(588\) 0 0
\(589\) 8.78461i 0.361964i
\(590\) 0 0
\(591\) −32.7846 −1.34858
\(592\) 0 0
\(593\) 19.7128i 0.809508i 0.914426 + 0.404754i \(0.132643\pi\)
−0.914426 + 0.404754i \(0.867357\pi\)
\(594\) 0 0
\(595\) −5.07180 + 5.85641i −0.207923 + 0.240089i
\(596\) 0 0
\(597\) 45.8564i 1.87678i
\(598\) 0 0
\(599\) 12.5359i 0.512203i −0.966650 0.256101i \(-0.917562\pi\)
0.966650 0.256101i \(-0.0824381\pi\)
\(600\) 0 0
\(601\) 34.6410i 1.41304i −0.707695 0.706518i \(-0.750265\pi\)
0.707695 0.706518i \(-0.249735\pi\)
\(602\) 0 0
\(603\) −11.3205 −0.461007
\(604\) 0 0
\(605\) −13.8038 −0.561206
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) −37.8564 32.7846i −1.53402 1.32850i
\(610\) 0 0
\(611\) −32.7846 −1.32632
\(612\) 0 0
\(613\) 25.8564i 1.04433i −0.852844 0.522165i \(-0.825124\pi\)
0.852844 0.522165i \(-0.174876\pi\)
\(614\) 0 0
\(615\) −5.85641 −0.236153
\(616\) 0 0
\(617\) 4.39230 0.176828 0.0884138 0.996084i \(-0.471820\pi\)
0.0884138 + 0.996084i \(0.471820\pi\)
\(618\) 0 0
\(619\) 24.1962i 0.972525i 0.873813 + 0.486263i \(0.161640\pi\)
−0.873813 + 0.486263i \(0.838360\pi\)
\(620\) 0 0
\(621\) −21.8564 −0.877067
\(622\) 0 0
\(623\) 25.8564 29.8564i 1.03592 1.19617i
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) 0 0
\(627\) −18.9282 −0.755920
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 24.2487i 0.965326i −0.875806 0.482663i \(-0.839670\pi\)
0.875806 0.482663i \(-0.160330\pi\)
\(632\) 0 0
\(633\) 20.7846i 0.826114i
\(634\) 0 0
\(635\) 15.7128i 0.623544i
\(636\) 0 0
\(637\) 4.73205 + 32.7846i 0.187491 + 1.29897i
\(638\) 0 0
\(639\) 20.2487i 0.801027i
\(640\) 0 0
\(641\) −33.4641 −1.32175 −0.660876 0.750495i \(-0.729815\pi\)
−0.660876 + 0.750495i \(0.729815\pi\)
\(642\) 0 0
\(643\) 22.7321i 0.896465i 0.893917 + 0.448232i \(0.147946\pi\)
−0.893917 + 0.448232i \(0.852054\pi\)
\(644\) 0 0
\(645\) 5.07180i 0.199702i
\(646\) 0 0
\(647\) 22.6410 0.890110 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(648\) 0 0
\(649\) 20.7846i 0.815867i
\(650\) 0 0
\(651\) 32.7846 37.8564i 1.28493 1.48371i
\(652\) 0 0
\(653\) 12.0000i 0.469596i −0.972044 0.234798i \(-0.924557\pi\)
0.972044 0.234798i \(-0.0754429\pi\)
\(654\) 0 0
\(655\) 7.85641i 0.306975i
\(656\) 0 0
\(657\) 30.9282i 1.20662i
\(658\) 0 0
\(659\) −36.1051 −1.40646 −0.703228 0.710965i \(-0.748259\pi\)
−0.703228 + 0.710965i \(0.748259\pi\)
\(660\) 0 0
\(661\) −14.1962 −0.552166 −0.276083 0.961134i \(-0.589037\pi\)
−0.276083 + 0.961134i \(0.589037\pi\)
\(662\) 0 0
\(663\) −51.7128 −2.00836
\(664\) 0 0
\(665\) −1.60770 + 1.85641i −0.0623437 + 0.0719884i
\(666\) 0 0
\(667\) 37.8564 1.46581
\(668\) 0 0
\(669\) 21.8564i 0.845017i
\(670\) 0 0
\(671\) −63.7128 −2.45961
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 0 0
\(675\) 17.8564i 0.687293i
\(676\) 0 0
\(677\) −18.8756 −0.725450 −0.362725 0.931896i \(-0.618154\pi\)
−0.362725 + 0.931896i \(0.618154\pi\)
\(678\) 0 0
\(679\) −20.7846 + 24.0000i −0.797640 + 0.921035i
\(680\) 0 0
\(681\) −3.46410 −0.132745
\(682\) 0 0
\(683\) −29.4641 −1.12741 −0.563706 0.825975i \(-0.690625\pi\)
−0.563706 + 0.825975i \(0.690625\pi\)
\(684\) 0 0
\(685\) −14.5359 −0.555388
\(686\) 0 0
\(687\) 0.928203i 0.0354132i
\(688\) 0 0
\(689\) 32.7846i 1.24899i
\(690\) 0 0
\(691\) 32.9808i 1.25465i 0.778759 + 0.627324i \(0.215850\pi\)
−0.778759 + 0.627324i \(0.784150\pi\)
\(692\) 0 0
\(693\) −48.7846 42.2487i −1.85317 1.60490i
\(694\) 0 0
\(695\) 3.07180i 0.116520i
\(696\) 0 0
\(697\) 11.7128 0.443654
\(698\) 0 0
\(699\) 54.2487i 2.05187i
\(700\) 0 0
\(701\) 1.85641i 0.0701155i −0.999385 0.0350578i \(-0.988838\pi\)
0.999385 0.0350578i \(-0.0111615\pi\)
\(702\) 0 0
\(703\) 5.07180 0.191286
\(704\) 0 0
\(705\) 13.8564i 0.521862i
\(706\) 0 0
\(707\) −12.3923 10.7321i −0.466061 0.403620i
\(708\) 0 0
\(709\) 30.9282i 1.16153i 0.814070 + 0.580767i \(0.197247\pi\)
−0.814070 + 0.580767i \(0.802753\pi\)
\(710\) 0 0
\(711\) 15.4641i 0.579949i
\(712\) 0 0
\(713\) 37.8564i 1.41773i
\(714\) 0 0
\(715\) −18.9282 −0.707875
\(716\) 0 0
\(717\) −1.07180 −0.0400270
\(718\) 0 0
\(719\) 17.0718 0.636671 0.318335 0.947978i \(-0.396876\pi\)
0.318335 + 0.947978i \(0.396876\pi\)
\(720\) 0 0
\(721\) 37.8564 + 32.7846i 1.40985 + 1.22096i
\(722\) 0 0
\(723\) 32.7846 1.21927
\(724\) 0 0
\(725\) 30.9282i 1.14864i
\(726\) 0 0
\(727\) 8.78461 0.325803 0.162902 0.986642i \(-0.447915\pi\)
0.162902 + 0.986642i \(0.447915\pi\)
\(728\) 0 0
\(729\) 43.7846 1.62165
\(730\) 0 0
\(731\) 10.1436i 0.375174i
\(732\) 0 0
\(733\) 2.19615 0.0811167 0.0405584 0.999177i \(-0.487086\pi\)
0.0405584 + 0.999177i \(0.487086\pi\)
\(734\) 0 0
\(735\) −13.8564 + 2.00000i −0.511101 + 0.0737711i
\(736\) 0 0
\(737\) 13.8564 0.510407
\(738\) 0 0
\(739\) −11.3205 −0.416432 −0.208216 0.978083i \(-0.566766\pi\)
−0.208216 + 0.978083i \(0.566766\pi\)
\(740\) 0 0
\(741\) −16.3923 −0.602186
\(742\) 0 0
\(743\) 10.5359i 0.386525i −0.981147 0.193262i \(-0.938093\pi\)
0.981147 0.193262i \(-0.0619068\pi\)
\(744\) 0 0
\(745\) 8.78461i 0.321843i
\(746\) 0 0
\(747\) 47.9090i 1.75290i
\(748\) 0 0
\(749\) 16.7846 + 14.5359i 0.613296 + 0.531130i
\(750\) 0 0
\(751\) 26.5359i 0.968309i −0.874983 0.484154i \(-0.839127\pi\)
0.874983 0.484154i \(-0.160873\pi\)
\(752\) 0 0
\(753\) 60.2487 2.19559
\(754\) 0 0
\(755\) 8.28719i 0.301602i
\(756\) 0 0
\(757\) 29.5692i 1.07471i 0.843356 + 0.537356i \(0.180577\pi\)
−0.843356 + 0.537356i \(0.819423\pi\)
\(758\) 0 0
\(759\) 81.5692 2.96078
\(760\) 0 0
\(761\) 16.7846i 0.608442i −0.952602 0.304221i \(-0.901604\pi\)
0.952602 0.304221i \(-0.0983961\pi\)
\(762\) 0 0
\(763\) −12.0000 + 13.8564i −0.434429 + 0.501636i
\(764\) 0 0
\(765\) 13.0718i 0.472612i
\(766\) 0 0
\(767\) 18.0000i 0.649942i
\(768\) 0 0
\(769\) 1.85641i 0.0669437i 0.999440 + 0.0334719i \(0.0106564\pi\)
−0.999440 + 0.0334719i \(0.989344\pi\)
\(770\) 0 0
\(771\) 59.7128 2.15050
\(772\) 0 0
\(773\) −36.4449 −1.31083 −0.655415 0.755269i \(-0.727506\pi\)
−0.655415 + 0.755269i \(0.727506\pi\)
\(774\) 0 0
\(775\) −30.9282 −1.11097
\(776\) 0 0
\(777\) 21.8564 + 18.9282i 0.784094 + 0.679046i
\(778\) 0 0
\(779\) 3.71281 0.133025
\(780\) 0 0
\(781\) 24.7846i 0.886863i
\(782\) 0 0
\(783\) 27.7128 0.990375
\(784\) 0 0
\(785\) 14.1051 0.503433
\(786\) 0 0
\(787\) 20.5885i 0.733899i 0.930241 + 0.366950i \(0.119598\pi\)
−0.930241 + 0.366950i \(0.880402\pi\)
\(788\) 0 0
\(789\) −31.3205 −1.11504
\(790\) 0 0
\(791\) −18.9282 16.3923i −0.673009 0.582843i
\(792\) 0 0
\(793\) −55.1769 −1.95939
\(794\) 0 0
\(795\) 13.8564 0.491436
\(796\) 0 0
\(797\) −41.1244 −1.45670 −0.728350 0.685206i \(-0.759712\pi\)
−0.728350 + 0.685206i \(0.759712\pi\)
\(798\) 0 0
\(799\) 27.7128i 0.980409i
\(800\) 0 0
\(801\) 66.6410i 2.35464i
\(802\) 0 0