Properties

Label 1792.2.f.a.1791.1
Level $1792$
Weight $2$
Character 1792.1791
Analytic conductor $14.309$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(1\)
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: no (minimal twist has level 896)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1791.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1791
Dual form 1792.2.f.a.1791.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.73205 q^{3} -0.732051i q^{5} +(-2.00000 - 1.73205i) q^{7} +4.46410 q^{9} +O(q^{10})\) \(q-2.73205 q^{3} -0.732051i q^{5} +(-2.00000 - 1.73205i) q^{7} +4.46410 q^{9} +5.46410i q^{11} -4.73205i q^{13} +2.00000i q^{15} -4.00000i q^{17} -1.26795 q^{19} +(5.46410 + 4.73205i) q^{21} -5.46410i q^{23} +4.46410 q^{25} -4.00000 q^{27} -6.92820 q^{29} +6.92820 q^{31} -14.9282i q^{33} +(-1.26795 + 1.46410i) q^{35} -4.00000 q^{37} +12.9282i q^{39} -2.92820i q^{41} -2.53590i q^{43} -3.26795i q^{45} -6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} +10.9282i q^{51} +6.92820 q^{53} +4.00000 q^{55} +3.46410 q^{57} -3.80385 q^{59} +11.6603i q^{61} +(-8.92820 - 7.73205i) q^{63} -3.46410 q^{65} -2.53590i q^{67} +14.9282i q^{69} -4.53590i q^{71} +6.92820i q^{73} -12.1962 q^{75} +(9.46410 - 10.9282i) q^{77} -3.46410i q^{79} -2.46410 q^{81} -10.7321 q^{83} -2.92820 q^{85} +18.9282 q^{87} +14.9282i q^{89} +(-8.19615 + 9.46410i) q^{91} -18.9282 q^{93} +0.928203i q^{95} +12.0000i q^{97} +24.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 8q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 8q^{7} + 4q^{9} - 12q^{19} + 8q^{21} + 4q^{25} - 16q^{27} - 12q^{35} - 16q^{37} + 4q^{49} + 16q^{55} - 36q^{59} - 8q^{63} - 28q^{75} + 24q^{77} + 4q^{81} - 36q^{83} + 16q^{85} + 48q^{87} - 12q^{91} - 48q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 0 0
\(5\) 0.732051i 0.327383i −0.986512 0.163692i \(-0.947660\pi\)
0.986512 0.163692i \(-0.0523402\pi\)
\(6\) 0 0
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 5.46410i 1.64749i 0.566961 + 0.823744i \(0.308119\pi\)
−0.566961 + 0.823744i \(0.691881\pi\)
\(12\) 0 0
\(13\) 4.73205i 1.31243i −0.754572 0.656217i \(-0.772155\pi\)
0.754572 0.656217i \(-0.227845\pi\)
\(14\) 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
\(19\) −1.26795 −0.290887 −0.145444 0.989367i \(-0.546461\pi\)
−0.145444 + 0.989367i \(0.546461\pi\)
\(20\) 0 0
\(21\) 5.46410 + 4.73205i 1.19236 + 1.03262i
\(22\) 0 0
\(23\) 5.46410i 1.13934i −0.821872 0.569672i \(-0.807070\pi\)
0.821872 0.569672i \(-0.192930\pi\)
\(24\) 0 0
\(25\) 4.46410 0.892820
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 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
\(35\) −1.26795 + 1.46410i −0.214323 + 0.247478i
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 12.9282i 2.07017i
\(40\) 0 0
\(41\) 2.92820i 0.457309i −0.973508 0.228654i \(-0.926567\pi\)
0.973508 0.228654i \(-0.0734325\pi\)
\(42\) 0 0
\(43\) 2.53590i 0.386721i −0.981128 0.193360i \(-0.938061\pi\)
0.981128 0.193360i \(-0.0619387\pi\)
\(44\) 0 0
\(45\) 3.26795i 0.487157i
\(46\) 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
\(51\) 10.9282i 1.53025i
\(52\) 0 0
\(53\) 6.92820 0.951662 0.475831 0.879537i \(-0.342147\pi\)
0.475831 + 0.879537i \(0.342147\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) −3.80385 −0.495219 −0.247609 0.968860i \(-0.579645\pi\)
−0.247609 + 0.968860i \(0.579645\pi\)
\(60\) 0 0
\(61\) 11.6603i 1.49294i 0.665418 + 0.746471i \(0.268253\pi\)
−0.665418 + 0.746471i \(0.731747\pi\)
\(62\) 0 0
\(63\) −8.92820 7.73205i −1.12485 0.974147i
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) 2.53590i 0.309809i −0.987929 0.154905i \(-0.950493\pi\)
0.987929 0.154905i \(-0.0495070\pi\)
\(68\) 0 0
\(69\) 14.9282i 1.79714i
\(70\) 0 0
\(71\) 4.53590i 0.538312i −0.963097 0.269156i \(-0.913255\pi\)
0.963097 0.269156i \(-0.0867447\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 0 0
\(75\) −12.1962 −1.40829
\(76\) 0 0
\(77\) 9.46410 10.9282i 1.07853 1.24538i
\(78\) 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
\(83\) −10.7321 −1.17800 −0.588998 0.808135i \(-0.700477\pi\)
−0.588998 + 0.808135i \(0.700477\pi\)
\(84\) 0 0
\(85\) −2.92820 −0.317608
\(86\) 0 0
\(87\) 18.9282 2.02932
\(88\) 0 0
\(89\) 14.9282i 1.58239i 0.611566 + 0.791193i \(0.290540\pi\)
−0.611566 + 0.791193i \(0.709460\pi\)
\(90\) 0 0
\(91\) −8.19615 + 9.46410i −0.859190 + 0.992107i
\(92\) 0 0
\(93\) −18.9282 −1.96276
\(94\) 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
\(99\) 24.3923i 2.45152i
\(100\) 0 0
\(101\) 6.19615i 0.616540i −0.951299 0.308270i \(-0.900250\pi\)
0.951299 0.308270i \(-0.0997501\pi\)
\(102\) 0 0
\(103\) −18.9282 −1.86505 −0.932526 0.361104i \(-0.882400\pi\)
−0.932526 + 0.361104i \(0.882400\pi\)
\(104\) 0 0
\(105\) 3.46410 4.00000i 0.338062 0.390360i
\(106\) 0 0
\(107\) 8.39230i 0.811315i 0.914025 + 0.405657i \(0.132957\pi\)
−0.914025 + 0.405657i \(0.867043\pi\)
\(108\) 0 0
\(109\) 6.92820 0.663602 0.331801 0.943349i \(-0.392344\pi\)
0.331801 + 0.943349i \(0.392344\pi\)
\(110\) 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
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 21.1244i 1.95295i
\(118\) 0 0
\(119\) −6.92820 + 8.00000i −0.635107 + 0.733359i
\(120\) 0 0
\(121\) −18.8564 −1.71422
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 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
\(131\) −10.7321 −0.937664 −0.468832 0.883287i \(-0.655325\pi\)
−0.468832 + 0.883287i \(0.655325\pi\)
\(132\) 0 0
\(133\) 2.53590 + 2.19615i 0.219890 + 0.190431i
\(134\) 0 0
\(135\) 2.92820i 0.252020i
\(136\) 0 0
\(137\) −19.8564 −1.69645 −0.848224 0.529638i \(-0.822328\pi\)
−0.848224 + 0.529638i \(0.822328\pi\)
\(138\) 0 0
\(139\) 4.19615 0.355913 0.177957 0.984038i \(-0.443051\pi\)
0.177957 + 0.984038i \(0.443051\pi\)
\(140\) 0 0
\(141\) 18.9282 1.59404
\(142\) 0 0
\(143\) 25.8564 2.16222
\(144\) 0 0
\(145\) 5.07180i 0.421190i
\(146\) 0 0
\(147\) −2.73205 18.9282i −0.225336 1.56117i
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 11.3205i 0.921250i −0.887595 0.460625i \(-0.847625\pi\)
0.887595 0.460625i \(-0.152375\pi\)
\(152\) 0 0
\(153\) 17.8564i 1.44360i
\(154\) 0 0
\(155\) 5.07180i 0.407377i
\(156\) 0 0
\(157\) 19.2679i 1.53775i 0.639399 + 0.768875i \(0.279183\pi\)
−0.639399 + 0.768875i \(0.720817\pi\)
\(158\) 0 0
\(159\) −18.9282 −1.50110
\(160\) 0 0
\(161\) −9.46410 + 10.9282i −0.745876 + 0.861263i
\(162\) 0 0
\(163\) 16.3923i 1.28394i −0.766728 0.641972i \(-0.778116\pi\)
0.766728 0.641972i \(-0.221884\pi\)
\(164\) 0 0
\(165\) −10.9282 −0.850759
\(166\) 0 0
\(167\) −5.07180 −0.392467 −0.196234 0.980557i \(-0.562871\pi\)
−0.196234 + 0.980557i \(0.562871\pi\)
\(168\) 0 0
\(169\) −9.39230 −0.722485
\(170\) 0 0
\(171\) −5.66025 −0.432850
\(172\) 0 0
\(173\) 3.26795i 0.248458i 0.992254 + 0.124229i \(0.0396457\pi\)
−0.992254 + 0.124229i \(0.960354\pi\)
\(174\) 0 0
\(175\) −8.92820 7.73205i −0.674909 0.584488i
\(176\) 0 0
\(177\) 10.3923 0.781133
\(178\) 0 0
\(179\) 0.392305i 0.0293222i 0.999893 + 0.0146611i \(0.00466695\pi\)
−0.999893 + 0.0146611i \(0.995333\pi\)
\(180\) 0 0
\(181\) 16.7321i 1.24368i −0.783143 0.621842i \(-0.786385\pi\)
0.783143 0.621842i \(-0.213615\pi\)
\(182\) 0 0
\(183\) 31.8564i 2.35489i
\(184\) 0 0
\(185\) 2.92820i 0.215286i
\(186\) 0 0
\(187\) 21.8564 1.59830
\(188\) 0 0
\(189\) 8.00000 + 6.92820i 0.581914 + 0.503953i
\(190\) 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
\(195\) 9.46410 0.677738
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 16.7846 1.18983 0.594915 0.803789i \(-0.297186\pi\)
0.594915 + 0.803789i \(0.297186\pi\)
\(200\) 0 0
\(201\) 6.92820i 0.488678i
\(202\) 0 0
\(203\) 13.8564 + 12.0000i 0.972529 + 0.842235i
\(204\) 0 0
\(205\) −2.14359 −0.149715
\(206\) 0 0
\(207\) 24.3923i 1.69538i
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) 7.60770i 0.523735i 0.965104 + 0.261868i \(0.0843384\pi\)
−0.965104 + 0.261868i \(0.915662\pi\)
\(212\) 0 0
\(213\) 12.3923i 0.849107i
\(214\) 0 0
\(215\) −1.85641 −0.126606
\(216\) 0 0
\(217\) −13.8564 12.0000i −0.940634 0.814613i
\(218\) 0 0
\(219\) 18.9282i 1.27905i
\(220\) 0 0
\(221\) −18.9282 −1.27325
\(222\) 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
\(227\) −1.26795 −0.0841567 −0.0420784 0.999114i \(-0.513398\pi\)
−0.0420784 + 0.999114i \(0.513398\pi\)
\(228\) 0 0
\(229\) 0.339746i 0.0224510i −0.999937 0.0112255i \(-0.996427\pi\)
0.999937 0.0112255i \(-0.00357327\pi\)
\(230\) 0 0
\(231\) −25.8564 + 29.8564i −1.70123 + 1.96441i
\(232\) 0 0
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 0 0
\(235\) 5.07180i 0.330848i
\(236\) 0 0
\(237\) 9.46410i 0.614759i
\(238\) 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
\(243\) 18.7321 1.20166
\(244\) 0 0
\(245\) 5.07180 0.732051i 0.324025 0.0467690i
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) 29.3205 1.85811
\(250\) 0 0
\(251\) −22.0526 −1.39195 −0.695973 0.718068i \(-0.745027\pi\)
−0.695973 + 0.718068i \(0.745027\pi\)
\(252\) 0 0
\(253\) 29.8564 1.87706
\(254\) 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
\(259\) 8.00000 + 6.92820i 0.497096 + 0.430498i
\(260\) 0 0
\(261\) −30.9282 −1.91441
\(262\) 0 0
\(263\) 11.4641i 0.706907i 0.935452 + 0.353453i \(0.114993\pi\)
−0.935452 + 0.353453i \(0.885007\pi\)
\(264\) 0 0
\(265\) 5.07180i 0.311558i
\(266\) 0 0
\(267\) 40.7846i 2.49598i
\(268\) 0 0
\(269\) 1.12436i 0.0685532i 0.999412 + 0.0342766i \(0.0109127\pi\)
−0.999412 + 0.0342766i \(0.989087\pi\)
\(270\) 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
\(275\) 24.3923i 1.47091i
\(276\) 0 0
\(277\) −28.7846 −1.72950 −0.864750 0.502203i \(-0.832523\pi\)
−0.864750 + 0.502203i \(0.832523\pi\)
\(278\) 0 0
\(279\) 30.9282 1.85162
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 15.8038 0.939441 0.469721 0.882815i \(-0.344355\pi\)
0.469721 + 0.882815i \(0.344355\pi\)
\(284\) 0 0
\(285\) 2.53590i 0.150214i
\(286\) 0 0
\(287\) −5.07180 + 5.85641i −0.299379 + 0.345693i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 32.7846i 1.92187i
\(292\) 0 0
\(293\) 1.80385i 0.105382i 0.998611 + 0.0526910i \(0.0167798\pi\)
−0.998611 + 0.0526910i \(0.983220\pi\)
\(294\) 0 0
\(295\) 2.78461i 0.162126i
\(296\) 0 0
\(297\) 21.8564i 1.26824i
\(298\) 0 0
\(299\) −25.8564 −1.49531
\(300\) 0 0
\(301\) −4.39230 + 5.07180i −0.253168 + 0.292334i
\(302\) 0 0
\(303\) 16.9282i 0.972500i
\(304\) 0 0
\(305\) 8.53590 0.488764
\(306\) 0 0
\(307\) 13.2679 0.757242 0.378621 0.925552i \(-0.376398\pi\)
0.378621 + 0.925552i \(0.376398\pi\)
\(308\) 0 0
\(309\) 51.7128 2.94184
\(310\) 0 0
\(311\) 25.8564 1.46618 0.733091 0.680130i \(-0.238077\pi\)
0.733091 + 0.680130i \(0.238077\pi\)
\(312\) 0 0
\(313\) 32.7846i 1.85310i −0.376177 0.926548i \(-0.622762\pi\)
0.376177 0.926548i \(-0.377238\pi\)
\(314\) 0 0
\(315\) −5.66025 + 6.53590i −0.318919 + 0.368256i
\(316\) 0 0
\(317\) 1.85641 0.104266 0.0521331 0.998640i \(-0.483398\pi\)
0.0521331 + 0.998640i \(0.483398\pi\)
\(318\) 0 0
\(319\) 37.8564i 2.11955i
\(320\) 0 0
\(321\) 22.9282i 1.27973i
\(322\) 0 0
\(323\) 5.07180i 0.282202i
\(324\) 0 0
\(325\) 21.1244i 1.17177i
\(326\) 0 0
\(327\) −18.9282 −1.04673
\(328\) 0 0
\(329\) 13.8564 + 12.0000i 0.763928 + 0.661581i
\(330\) 0 0
\(331\) 11.3205i 0.622231i −0.950372 0.311116i \(-0.899297\pi\)
0.950372 0.311116i \(-0.100703\pi\)
\(332\) 0 0
\(333\) −17.8564 −0.978525
\(334\) 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
\(339\) 25.8564 1.40433
\(340\) 0 0
\(341\) 37.8564i 2.05004i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 10.9282 0.588355
\(346\) 0 0
\(347\) 13.4641i 0.722791i 0.932413 + 0.361395i \(0.117700\pi\)
−0.932413 + 0.361395i \(0.882300\pi\)
\(348\) 0 0
\(349\) 11.6603i 0.624159i 0.950056 + 0.312080i \(0.101026\pi\)
−0.950056 + 0.312080i \(0.898974\pi\)
\(350\) 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
\(355\) −3.32051 −0.176234
\(356\) 0 0
\(357\) 18.9282 21.8564i 1.00179 1.15676i
\(358\) 0 0
\(359\) 0.392305i 0.0207051i 0.999946 + 0.0103525i \(0.00329537\pi\)
−0.999946 + 0.0103525i \(0.996705\pi\)
\(360\) 0 0
\(361\) −17.3923 −0.915384
\(362\) 0 0
\(363\) 51.5167 2.70392
\(364\) 0 0
\(365\) 5.07180 0.265470
\(366\) 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
\(371\) −13.8564 12.0000i −0.719389 0.623009i
\(372\) 0 0
\(373\) −17.0718 −0.883944 −0.441972 0.897029i \(-0.645721\pi\)
−0.441972 + 0.897029i \(0.645721\pi\)
\(374\) 0 0
\(375\) 18.9282i 0.977448i
\(376\) 0 0
\(377\) 32.7846i 1.68849i
\(378\) 0 0
\(379\) 21.4641i 1.10254i −0.834328 0.551268i \(-0.814144\pi\)
0.834328 0.551268i \(-0.185856\pi\)
\(380\) 0 0
\(381\) 58.6410i 3.00427i
\(382\) 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
\(387\) 11.3205i 0.575454i
\(388\) 0 0
\(389\) 1.85641 0.0941235 0.0470618 0.998892i \(-0.485014\pi\)
0.0470618 + 0.998892i \(0.485014\pi\)
\(390\) 0 0
\(391\) −21.8564 −1.10533
\(392\) 0 0
\(393\) 29.3205 1.47902
\(394\) 0 0
\(395\) −2.53590 −0.127595
\(396\) 0 0
\(397\) 29.9090i 1.50109i −0.660821 0.750544i \(-0.729792\pi\)
0.660821 0.750544i \(-0.270208\pi\)
\(398\) 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
\(403\) 32.7846i 1.63312i
\(404\) 0 0
\(405\) 1.80385i 0.0896339i
\(406\) 0 0
\(407\) 21.8564i 1.08338i
\(408\) 0 0
\(409\) 5.07180i 0.250784i 0.992107 + 0.125392i \(0.0400189\pi\)
−0.992107 + 0.125392i \(0.959981\pi\)
\(410\) 0 0
\(411\) 54.2487 2.67589
\(412\) 0 0
\(413\) 7.60770 + 6.58846i 0.374350 + 0.324197i
\(414\) 0 0
\(415\) 7.85641i 0.385656i
\(416\) 0 0
\(417\) −11.4641 −0.561399
\(418\) 0 0
\(419\) 3.12436 0.152635 0.0763174 0.997084i \(-0.475684\pi\)
0.0763174 + 0.997084i \(0.475684\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) −30.9282 −1.50378
\(424\) 0 0
\(425\) 17.8564i 0.866163i
\(426\) 0 0
\(427\) 20.1962 23.3205i 0.977360 1.12856i
\(428\) 0 0
\(429\) −70.6410 −3.41058
\(430\) 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
\(435\) 13.8564i 0.664364i
\(436\) 0 0
\(437\) 6.92820i 0.331421i
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 4.46410 + 30.9282i 0.212576 + 1.47277i
\(442\) 0 0
\(443\) 5.46410i 0.259607i −0.991540 0.129804i \(-0.958565\pi\)
0.991540 0.129804i \(-0.0414347\pi\)
\(444\) 0 0
\(445\) 10.9282 0.518047
\(446\) 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
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) 30.9282i 1.45313i
\(454\) 0 0
\(455\) 6.92820 + 6.00000i 0.324799 + 0.281284i
\(456\) 0 0
\(457\) −12.3923 −0.579688 −0.289844 0.957074i \(-0.593603\pi\)
−0.289844 + 0.957074i \(0.593603\pi\)
\(458\) 0 0
\(459\) 16.0000i 0.746816i
\(460\) 0 0
\(461\) 17.5167i 0.815832i 0.913019 + 0.407916i \(0.133744\pi\)
−0.913019 + 0.407916i \(0.866256\pi\)
\(462\) 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
\(467\) 12.5885 0.582524 0.291262 0.956643i \(-0.405925\pi\)
0.291262 + 0.956643i \(0.405925\pi\)
\(468\) 0 0
\(469\) −4.39230 + 5.07180i −0.202818 + 0.234194i
\(470\) 0 0
\(471\) 52.6410i 2.42557i
\(472\) 0 0
\(473\) 13.8564 0.637118
\(474\) 0 0
\(475\) −5.66025 −0.259710
\(476\) 0 0
\(477\) 30.9282 1.41611
\(478\) 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
\(483\) 25.8564 29.8564i 1.17651 1.35851i
\(484\) 0 0
\(485\) 8.78461 0.398889
\(486\) 0 0
\(487\) 16.3923i 0.742806i 0.928472 + 0.371403i \(0.121123\pi\)
−0.928472 + 0.371403i \(0.878877\pi\)
\(488\) 0 0
\(489\) 44.7846i 2.02523i
\(490\) 0 0
\(491\) 19.3205i 0.871922i 0.899965 + 0.435961i \(0.143591\pi\)
−0.899965 + 0.435961i \(0.856409\pi\)
\(492\) 0 0
\(493\) 27.7128i 1.24812i
\(494\) 0 0
\(495\) 17.8564 0.802586
\(496\) 0 0
\(497\) −7.85641 + 9.07180i −0.352408 + 0.406926i
\(498\) 0 0
\(499\) 16.3923i 0.733820i −0.930256 0.366910i \(-0.880416\pi\)
0.930256 0.366910i \(-0.119584\pi\)
\(500\) 0 0
\(501\) 13.8564 0.619059
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −4.53590 −0.201845
\(506\) 0 0
\(507\) 25.6603 1.13961
\(508\) 0 0
\(509\) 42.1962i 1.87031i −0.354237 0.935156i \(-0.615259\pi\)
0.354237 0.935156i \(-0.384741\pi\)
\(510\) 0 0
\(511\) 12.0000 13.8564i 0.530849 0.612971i
\(512\) 0 0
\(513\) 5.07180 0.223925
\(514\) 0 0
\(515\) 13.8564i 0.610586i
\(516\) 0 0
\(517\) 37.8564i 1.66492i
\(518\) 0 0
\(519\) 8.92820i 0.391905i
\(520\) 0 0
\(521\) 13.0718i 0.572686i 0.958127 + 0.286343i \(0.0924397\pi\)
−0.958127 + 0.286343i \(0.907560\pi\)
\(522\) 0 0
\(523\) −29.3731 −1.28439 −0.642197 0.766539i \(-0.721977\pi\)
−0.642197 + 0.766539i \(0.721977\pi\)
\(524\) 0 0
\(525\) 24.3923 + 21.1244i 1.06457 + 0.921942i
\(526\) 0 0
\(527\) 27.7128i 1.20719i
\(528\) 0 0
\(529\) −6.85641 −0.298105
\(530\) 0 0
\(531\) −16.9808 −0.736902
\(532\) 0 0
\(533\) −13.8564 −0.600188
\(534\) 0 0
\(535\) 6.14359 0.265611
\(536\) 0 0
\(537\) 1.07180i 0.0462514i
\(538\) 0 0
\(539\) −37.8564 + 5.46410i −1.63059 + 0.235356i
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) 0 0
\(543\) 45.7128i 1.96172i
\(544\) 0 0
\(545\) 5.07180i 0.217252i
\(546\) 0 0
\(547\) 21.4641i 0.917739i 0.888504 + 0.458869i \(0.151745\pi\)
−0.888504 + 0.458869i \(0.848255\pi\)
\(548\) 0 0
\(549\) 52.0526i 2.22155i
\(550\) 0 0
\(551\) 8.78461 0.374237
\(552\) 0 0
\(553\) −6.00000 + 6.92820i −0.255146 + 0.294617i
\(554\) 0 0
\(555\) 8.00000i 0.339581i
\(556\) 0 0
\(557\) −30.9282 −1.31047 −0.655235 0.755425i \(-0.727430\pi\)
−0.655235 + 0.755425i \(0.727430\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −59.7128 −2.52108
\(562\) 0 0
\(563\) −28.9808 −1.22139 −0.610697 0.791865i \(-0.709111\pi\)
−0.610697 + 0.791865i \(0.709111\pi\)
\(564\) 0 0
\(565\) 6.92820i 0.291472i
\(566\) 0 0
\(567\) 4.92820 + 4.26795i 0.206965 + 0.179237i
\(568\) 0 0
\(569\) −28.3923 −1.19027 −0.595134 0.803627i \(-0.702901\pi\)
−0.595134 + 0.803627i \(0.702901\pi\)
\(570\) 0 0
\(571\) 30.2487i 1.26587i −0.774205 0.632935i \(-0.781850\pi\)
0.774205 0.632935i \(-0.218150\pi\)
\(572\) 0 0
\(573\) 31.3205i 1.30843i
\(574\) 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
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) 21.4641 + 18.5885i 0.890481 + 0.771179i
\(582\) 0 0
\(583\) 37.8564i 1.56785i
\(584\) 0 0
\(585\) −15.4641 −0.639362
\(586\) 0 0
\(587\) −3.80385 −0.157002 −0.0785008 0.996914i \(-0.525013\pi\)
−0.0785008 + 0.996914i \(0.525013\pi\)
\(588\) 0 0
\(589\) −8.78461 −0.361964
\(590\) 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
\(595\) 5.85641 + 5.07180i 0.240089 + 0.207923i
\(596\) 0 0
\(597\) −45.8564 −1.87678
\(598\) 0 0
\(599\) 12.5359i 0.512203i 0.966650 + 0.256101i \(0.0824381\pi\)
−0.966650 + 0.256101i \(0.917562\pi\)
\(600\) 0 0
\(601\) 34.6410i 1.41304i 0.707695 + 0.706518i \(0.249735\pi\)
−0.707695 + 0.706518i \(0.750265\pi\)
\(602\) 0 0
\(603\) 11.3205i 0.461007i
\(604\) 0 0
\(605\) 13.8038i 0.561206i
\(606\) 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
\(611\) 32.7846i 1.32632i
\(612\) 0 0
\(613\) 25.8564 1.04433 0.522165 0.852844i \(-0.325124\pi\)
0.522165 + 0.852844i \(0.325124\pi\)
\(614\) 0 0
\(615\) 5.85641 0.236153
\(616\) 0 0
\(617\) −4.39230 −0.176828 −0.0884138 0.996084i \(-0.528180\pi\)
−0.0884138 + 0.996084i \(0.528180\pi\)
\(618\) 0 0
\(619\) −24.1962 −0.972525 −0.486263 0.873813i \(-0.661640\pi\)
−0.486263 + 0.873813i \(0.661640\pi\)
\(620\) 0 0
\(621\) 21.8564i 0.877067i
\(622\) 0 0
\(623\) 25.8564 29.8564i 1.03592 1.19617i
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) 0 0
\(627\) 18.9282i 0.755920i
\(628\) 0 0
\(629\) 16.0000i 0.637962i
\(630\) 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
\(635\) 15.7128 0.623544
\(636\) 0 0
\(637\) 32.7846 4.73205i 1.29897 0.187491i
\(638\) 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
\(643\) 22.7321 0.896465 0.448232 0.893917i \(-0.352054\pi\)
0.448232 + 0.893917i \(0.352054\pi\)
\(644\) 0 0
\(645\) 5.07180 0.199702
\(646\) 0 0
\(647\) −22.6410 −0.890110 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(648\) 0 0
\(649\) 20.7846i 0.815867i
\(650\) 0 0
\(651\) 37.8564 + 32.7846i 1.48371 + 1.28493i
\(652\) 0 0
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) 0 0
\(655\) 7.85641i 0.306975i
\(656\) 0 0
\(657\) 30.9282i 1.20662i
\(658\) 0 0
\(659\) 36.1051i 1.40646i 0.710965 + 0.703228i \(0.248259\pi\)
−0.710965 + 0.703228i \(0.751741\pi\)
\(660\) 0 0
\(661\) 14.1962i 0.552166i −0.961134 0.276083i \(-0.910963\pi\)
0.961134 0.276083i \(-0.0890365\pi\)
\(662\) 0 0
\(663\) 51.7128 2.00836
\(664\) 0 0
\(665\) 1.60770 1.85641i 0.0623437 0.0719884i
\(666\) 0 0
\(667\) 37.8564i 1.46581i
\(668\) 0 0
\(669\) 21.8564 0.845017
\(670\) 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
\(675\) −17.8564 −0.687293
\(676\) 0 0
\(677\) 18.8756i 0.725450i −0.931896 0.362725i \(-0.881846\pi\)
0.931896 0.362725i \(-0.118154\pi\)
\(678\) 0 0
\(679\) 20.7846 24.0000i 0.797640 0.921035i
\(680\) 0 0
\(681\) 3.46410 0.132745
\(682\) 0 0
\(683\) 29.4641i 1.12741i −0.825975 0.563706i \(-0.809375\pi\)
0.825975 0.563706i \(-0.190625\pi\)
\(684\) 0 0
\(685\) 14.5359i 0.555388i
\(686\) 0 0
\(687\) 0.928203i 0.0354132i
\(688\) 0 0
\(689\) 32.7846i 1.24899i
\(690\) 0 0
\(691\) 32.9808 1.25465 0.627324 0.778759i \(-0.284150\pi\)
0.627324 + 0.778759i \(0.284150\pi\)
\(692\) 0 0
\(693\) 42.2487 48.7846i 1.60490 1.85317i
\(694\) 0 0
\(695\) 3.07180i 0.116520i
\(696\) 0 0
\(697\) −11.7128 −0.443654
\(698\) 0 0
\(699\) −54.2487 −2.05187
\(700\) 0 0
\(701\) −1.85641 −0.0701155 −0.0350578 0.999385i \(-0.511162\pi\)
−0.0350578 + 0.999385i \(0.511162\pi\)
\(702\) 0 0
\(703\) 5.07180 0.191286
\(704\) 0 0
\(705\) 13.8564i 0.521862i
\(706\) 0 0
\(707\) −10.7321 + 12.3923i −0.403620 + 0.466061i
\(708\) 0 0
\(709\) −30.9282 −1.16153 −0.580767 0.814070i \(-0.697247\pi\)
−0.580767 + 0.814070i \(0.697247\pi\)
\(710\) 0 0
\(711\) 15.4641i 0.579949i
\(712\) 0 0
\(713\) 37.8564i 1.41773i
\(714\) 0 0
\(715\) 18.9282i 0.707875i
\(716\) 0 0
\(717\) 1.07180i 0.0400270i
\(718\) 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
\(723\) 32.7846i 1.21927i
\(724\) 0 0
\(725\) −30.9282 −1.14864
\(726\) 0 0
\(727\) −8.78461 −0.325803 −0.162902 0.986642i \(-0.552085\pi\)
−0.162902 + 0.986642i \(0.552085\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −10.1436 −0.375174
\(732\) 0 0
\(733\) 2.19615i 0.0811167i −0.999177 0.0405584i \(-0.987086\pi\)
0.999177 0.0405584i \(-0.0129137\pi\)
\(734\) 0 0
\(735\) −13.8564 + 2.00000i −0.511101 + 0.0737711i
\(736\) 0 0
\(737\) 13.8564 0.510407
\(738\) 0 0
\(739\) 11.3205i 0.416432i 0.978083 + 0.208216i \(0.0667656\pi\)
−0.978083 + 0.208216i \(0.933234\pi\)
\(740\) 0 0
\(741\) 16.3923i 0.602186i
\(742\) 0 0
\(743\) 10.5359i 0.386525i 0.981147 + 0.193262i \(0.0619068\pi\)
−0.981147 + 0.193262i \(0.938093\pi\)
\(744\) 0 0
\(745\) 8.78461i 0.321843i
\(746\) 0 0
\(747\) −47.9090 −1.75290
\(748\) 0 0
\(749\) 14.5359 16.7846i 0.531130 0.613296i
\(750\) 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
\(755\) −8.28719 −0.301602
\(756\) 0 0
\(757\) −29.5692 −1.07471 −0.537356 0.843356i \(-0.680577\pi\)
−0.537356 + 0.843356i \(0.680577\pi\)
\(758\) 0 0
\(759\) −81.5692 −2.96078
\(760\) 0 0
\(761\) 16.7846i 0.608442i 0.952602 + 0.304221i \(0.0983961\pi\)
−0.952602 + 0.304221i \(0.901604\pi\)
\(762\) 0 0
\(763\) −13.8564 12.0000i −0.501636 0.434429i
\(764\) 0 0
\(765\) −13.0718 −0.472612
\(766\) 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
\(771\) 59.7128i 2.15050i
\(772\) 0 0
\(773\) 36.4449i 1.31083i −0.755269 0.655415i \(-0.772494\pi\)
0.755269 0.655415i \(-0.227506\pi\)
\(774\) 0 0
\(775\) 30.9282 1.11097
\(776\) 0 0
\(777\) −21.8564 18.9282i −0.784094 0.679046i
\(778\) 0 0
\(779\) 3.71281i 0.133025i
\(780\) 0 0
\(781\) 24.7846 0.886863
\(782\) 0 0
\(783\) 27.7128 0.990375
\(784\) 0 0
\(785\) 14.1051 0.503433
\(786\) 0 0
\(787\) 20.5885 0.733899 0.366950 0.930241i \(-0.380402\pi\)
0.366950 + 0.930241i \(0.380402\pi\)
\(788\) 0 0
\(789\) 31.3205i 1.11504i
\(790\) 0 0
\(791\) 18.9282 + 16.3923i 0.673009 + 0.582843i
\(792\) 0 0
\(793\) 55.1769 1.95939
\(794\) 0 0
\(795\) 13.8564i 0.491436i
\(796\) 0 0
\(797\) 41.1244i 1.45670i 0.685206 + 0.728350i \(0.259712\pi\)
−0.685206 + 0.728350i \(0.740288\pi\)
\(798\) 0 0
\(799\) 27.7128i 0.980409i
\(800\) 0 0
\(801\) 66.6410i 2.35464i
\(802\) 0