Properties

Label 1792.2.f.a.1791.3
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 1791.3
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1791
Dual form 1792.2.f.a.1791.4

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{3} -2.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} -2.46410 q^{9} +O(q^{10})\) \(q+0.732051 q^{3} -2.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} -2.46410 q^{9} +1.46410i q^{11} +1.26795i q^{13} -2.00000i q^{15} +4.00000i q^{17} -4.73205 q^{19} +(-1.46410 - 1.26795i) q^{21} -1.46410i q^{23} -2.46410 q^{25} -4.00000 q^{27} +6.92820 q^{29} -6.92820 q^{31} +1.07180i q^{33} +(-4.73205 + 5.46410i) q^{35} -4.00000 q^{37} +0.928203i q^{39} -10.9282i q^{41} +9.46410i q^{43} +6.73205i q^{45} +6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} +2.92820i q^{51} -6.92820 q^{53} +4.00000 q^{55} -3.46410 q^{57} -14.1962 q^{59} +5.66025i q^{61} +(4.92820 + 4.26795i) q^{63} +3.46410 q^{65} +9.46410i q^{67} -1.07180i q^{69} +11.4641i q^{71} +6.92820i q^{73} -1.80385 q^{75} +(2.53590 - 2.92820i) q^{77} -3.46410i q^{79} +4.46410 q^{81} -7.26795 q^{83} +10.9282 q^{85} +5.07180 q^{87} -1.07180i q^{89} +(2.19615 - 2.53590i) q^{91} -5.07180 q^{93} +12.9282i q^{95} -12.0000i q^{97} -3.60770i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 8q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 8q^{7} + 4q^{9} - 12q^{19} + 8q^{21} + 4q^{25} - 16q^{27} - 12q^{35} - 16q^{37} + 4q^{49} + 16q^{55} - 36q^{59} - 8q^{63} - 28q^{75} + 24q^{77} + 4q^{81} - 36q^{83} + 16q^{85} + 48q^{87} - 12q^{91} - 48q^{93} + O(q^{100}) \)

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
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 0 0
\(5\) 2.73205i 1.22181i −0.791704 0.610905i \(-0.790806\pi\)
0.791704 0.610905i \(-0.209194\pi\)
\(6\) 0 0
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 1.46410i 0.441443i 0.975337 + 0.220722i \(0.0708412\pi\)
−0.975337 + 0.220722i \(0.929159\pi\)
\(12\) 0 0
\(13\) 1.26795i 0.351666i 0.984420 + 0.175833i \(0.0562618\pi\)
−0.984420 + 0.175833i \(0.943738\pi\)
\(14\) 0 0
\(15\) 2.00000i 0.516398i
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) −4.73205 −1.08561 −0.542803 0.839860i \(-0.682637\pi\)
−0.542803 + 0.839860i \(0.682637\pi\)
\(20\) 0 0
\(21\) −1.46410 1.26795i −0.319493 0.276689i
\(22\) 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
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 6.92820 1.28654 0.643268 0.765641i \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 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
\(35\) −4.73205 + 5.46410i −0.799863 + 0.923602i
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 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
\(43\) 9.46410i 1.44326i 0.692278 + 0.721631i \(0.256607\pi\)
−0.692278 + 0.721631i \(0.743393\pi\)
\(44\) 0 0
\(45\) 6.73205i 1.00355i
\(46\) 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
\(51\) 2.92820i 0.410030i
\(52\) 0 0
\(53\) −6.92820 −0.951662 −0.475831 0.879537i \(-0.657853\pi\)
−0.475831 + 0.879537i \(0.657853\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −3.46410 −0.458831
\(58\) 0 0
\(59\) −14.1962 −1.84818 −0.924091 0.382173i \(-0.875176\pi\)
−0.924091 + 0.382173i \(0.875176\pi\)
\(60\) 0 0
\(61\) 5.66025i 0.724721i 0.932038 + 0.362361i \(0.118029\pi\)
−0.932038 + 0.362361i \(0.881971\pi\)
\(62\) 0 0
\(63\) 4.92820 + 4.26795i 0.620895 + 0.537711i
\(64\) 0 0
\(65\) 3.46410 0.429669
\(66\) 0 0
\(67\) 9.46410i 1.15622i 0.815957 + 0.578112i \(0.196210\pi\)
−0.815957 + 0.578112i \(0.803790\pi\)
\(68\) 0 0
\(69\) 1.07180i 0.129029i
\(70\) 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
\(75\) −1.80385 −0.208290
\(76\) 0 0
\(77\) 2.53590 2.92820i 0.288992 0.333700i
\(78\) 0 0
\(79\) 3.46410i 0.389742i −0.980829 0.194871i \(-0.937571\pi\)
0.980829 0.194871i \(-0.0624288\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −7.26795 −0.797761 −0.398881 0.917003i \(-0.630601\pi\)
−0.398881 + 0.917003i \(0.630601\pi\)
\(84\) 0 0
\(85\) 10.9282 1.18533
\(86\) 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
\(91\) 2.19615 2.53590i 0.230219 0.265834i
\(92\) 0 0
\(93\) −5.07180 −0.525921
\(94\) 0 0
\(95\) 12.9282i 1.32641i
\(96\) 0 0
\(97\) 12.0000i 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) 0 0
\(99\) 3.60770i 0.362587i
\(100\) 0 0
\(101\) 4.19615i 0.417533i −0.977966 0.208766i \(-0.933055\pi\)
0.977966 0.208766i \(-0.0669448\pi\)
\(102\) 0 0
\(103\) −5.07180 −0.499739 −0.249869 0.968280i \(-0.580388\pi\)
−0.249869 + 0.968280i \(0.580388\pi\)
\(104\) 0 0
\(105\) −3.46410 + 4.00000i −0.338062 + 0.390360i
\(106\) 0 0
\(107\) 12.3923i 1.19801i 0.800746 + 0.599005i \(0.204437\pi\)
−0.800746 + 0.599005i \(0.795563\pi\)
\(108\) 0 0
\(109\) −6.92820 −0.663602 −0.331801 0.943349i \(-0.607656\pi\)
−0.331801 + 0.943349i \(0.607656\pi\)
\(110\) 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
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 3.12436i 0.288847i
\(118\) 0 0
\(119\) 6.92820 8.00000i 0.635107 0.733359i
\(120\) 0 0
\(121\) 8.85641 0.805128
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 14.5359i 1.28985i −0.764245 0.644926i \(-0.776888\pi\)
0.764245 0.644926i \(-0.223112\pi\)
\(128\) 0 0
\(129\) 6.92820i 0.609994i
\(130\) 0 0
\(131\) −7.26795 −0.635004 −0.317502 0.948258i \(-0.602844\pi\)
−0.317502 + 0.948258i \(0.602844\pi\)
\(132\) 0 0
\(133\) 9.46410 + 8.19615i 0.820642 + 0.710697i
\(134\) 0 0
\(135\) 10.9282i 0.940550i
\(136\) 0 0
\(137\) 7.85641 0.671218 0.335609 0.942001i \(-0.391058\pi\)
0.335609 + 0.942001i \(0.391058\pi\)
\(138\) 0 0
\(139\) −6.19615 −0.525551 −0.262775 0.964857i \(-0.584638\pi\)
−0.262775 + 0.964857i \(0.584638\pi\)
\(140\) 0 0
\(141\) 5.07180 0.427122
\(142\) 0 0
\(143\) −1.85641 −0.155241
\(144\) 0 0
\(145\) 18.9282i 1.57190i
\(146\) 0 0
\(147\) 0.732051 + 5.07180i 0.0603785 + 0.418315i
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 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
\(155\) 18.9282i 1.52035i
\(156\) 0 0
\(157\) 22.7321i 1.81422i −0.420899 0.907108i \(-0.638285\pi\)
0.420899 0.907108i \(-0.361715\pi\)
\(158\) 0 0
\(159\) −5.07180 −0.402220
\(160\) 0 0
\(161\) −2.53590 + 2.92820i −0.199857 + 0.230775i
\(162\) 0 0
\(163\) 4.39230i 0.344032i −0.985094 0.172016i \(-0.944972\pi\)
0.985094 0.172016i \(-0.0550281\pi\)
\(164\) 0 0
\(165\) 2.92820 0.227960
\(166\) 0 0
\(167\) −18.9282 −1.46471 −0.732354 0.680924i \(-0.761578\pi\)
−0.732354 + 0.680924i \(0.761578\pi\)
\(168\) 0 0
\(169\) 11.3923 0.876331
\(170\) 0 0
\(171\) 11.6603 0.891682
\(172\) 0 0
\(173\) 6.73205i 0.511828i −0.966700 0.255914i \(-0.917624\pi\)
0.966700 0.255914i \(-0.0823764\pi\)
\(174\) 0 0
\(175\) 4.92820 + 4.26795i 0.372537 + 0.322627i
\(176\) 0 0
\(177\) −10.3923 −0.781133
\(178\) 0 0
\(179\) 20.3923i 1.52419i 0.647464 + 0.762096i \(0.275830\pi\)
−0.647464 + 0.762096i \(0.724170\pi\)
\(180\) 0 0
\(181\) 13.2679i 0.986199i 0.869973 + 0.493099i \(0.164136\pi\)
−0.869973 + 0.493099i \(0.835864\pi\)
\(182\) 0 0
\(183\) 4.14359i 0.306303i
\(184\) 0 0
\(185\) 10.9282i 0.803457i
\(186\) 0 0
\(187\) −5.85641 −0.428263
\(188\) 0 0
\(189\) 8.00000 + 6.92820i 0.581914 + 0.503953i
\(190\) 0 0
\(191\) 4.53590i 0.328206i −0.986443 0.164103i \(-0.947527\pi\)
0.986443 0.164103i \(-0.0524730\pi\)
\(192\) 0 0
\(193\) −16.3923 −1.17994 −0.589972 0.807424i \(-0.700861\pi\)
−0.589972 + 0.807424i \(0.700861\pi\)
\(194\) 0 0
\(195\) 2.53590 0.181599
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) 0 0
\(201\) 6.92820i 0.488678i
\(202\) 0 0
\(203\) −13.8564 12.0000i −0.972529 0.842235i
\(204\) 0 0
\(205\) −29.8564 −2.08526
\(206\) 0 0
\(207\) 3.60770i 0.250752i
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) 28.3923i 1.95461i −0.211844 0.977303i \(-0.567947\pi\)
0.211844 0.977303i \(-0.432053\pi\)
\(212\) 0 0
\(213\) 8.39230i 0.575031i
\(214\) 0 0
\(215\) 25.8564 1.76339
\(216\) 0 0
\(217\) 13.8564 + 12.0000i 0.940634 + 0.814613i
\(218\) 0 0
\(219\) 5.07180i 0.342720i
\(220\) 0 0
\(221\) −5.07180 −0.341166
\(222\) 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
\(227\) −4.73205 −0.314077 −0.157039 0.987592i \(-0.550195\pi\)
−0.157039 + 0.987592i \(0.550195\pi\)
\(228\) 0 0
\(229\) 17.6603i 1.16702i 0.812105 + 0.583511i \(0.198322\pi\)
−0.812105 + 0.583511i \(0.801678\pi\)
\(230\) 0 0
\(231\) 1.85641 2.14359i 0.122143 0.141038i
\(232\) 0 0
\(233\) −7.85641 −0.514690 −0.257345 0.966320i \(-0.582848\pi\)
−0.257345 + 0.966320i \(0.582848\pi\)
\(234\) 0 0
\(235\) 18.9282i 1.23474i
\(236\) 0 0
\(237\) 2.53590i 0.164724i
\(238\) 0 0
\(239\) 20.3923i 1.31907i −0.751674 0.659534i \(-0.770754\pi\)
0.751674 0.659534i \(-0.229246\pi\)
\(240\) 0 0
\(241\) 12.0000i 0.772988i −0.922292 0.386494i \(-0.873686\pi\)
0.922292 0.386494i \(-0.126314\pi\)
\(242\) 0 0
\(243\) 15.2679 0.979439
\(244\) 0 0
\(245\) 18.9282 2.73205i 1.20928 0.174544i
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) −5.32051 −0.337173
\(250\) 0 0
\(251\) 16.0526 1.01323 0.506614 0.862173i \(-0.330897\pi\)
0.506614 + 0.862173i \(0.330897\pi\)
\(252\) 0 0
\(253\) 2.14359 0.134767
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 5.85641i 0.365313i 0.983177 + 0.182656i \(0.0584696\pi\)
−0.983177 + 0.182656i \(0.941530\pi\)
\(258\) 0 0
\(259\) 8.00000 + 6.92820i 0.497096 + 0.430498i
\(260\) 0 0
\(261\) −17.0718 −1.05672
\(262\) 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
\(267\) 0.784610i 0.0480173i
\(268\) 0 0
\(269\) 23.1244i 1.40992i 0.709249 + 0.704958i \(0.249034\pi\)
−0.709249 + 0.704958i \(0.750966\pi\)
\(270\) 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
\(275\) 3.60770i 0.217552i
\(276\) 0 0
\(277\) 12.7846 0.768153 0.384076 0.923301i \(-0.374520\pi\)
0.384076 + 0.923301i \(0.374520\pi\)
\(278\) 0 0
\(279\) 17.0718 1.02206
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 26.1962 1.55720 0.778600 0.627521i \(-0.215930\pi\)
0.778600 + 0.627521i \(0.215930\pi\)
\(284\) 0 0
\(285\) 9.46410i 0.560605i
\(286\) 0 0
\(287\) −18.9282 + 21.8564i −1.11730 + 1.29014i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 8.78461i 0.514963i
\(292\) 0 0
\(293\) 12.1962i 0.712507i −0.934389 0.356253i \(-0.884054\pi\)
0.934389 0.356253i \(-0.115946\pi\)
\(294\) 0 0
\(295\) 38.7846i 2.25813i
\(296\) 0 0
\(297\) 5.85641i 0.339823i
\(298\) 0 0
\(299\) 1.85641 0.107359
\(300\) 0 0
\(301\) 16.3923 18.9282i 0.944837 1.09100i
\(302\) 0 0
\(303\) 3.07180i 0.176470i
\(304\) 0 0
\(305\) 15.4641 0.885472
\(306\) 0 0
\(307\) 16.7321 0.954949 0.477474 0.878646i \(-0.341552\pi\)
0.477474 + 0.878646i \(0.341552\pi\)
\(308\) 0 0
\(309\) −3.71281 −0.211215
\(310\) 0 0
\(311\) −1.85641 −0.105267 −0.0526336 0.998614i \(-0.516762\pi\)
−0.0526336 + 0.998614i \(0.516762\pi\)
\(312\) 0 0
\(313\) 8.78461i 0.496535i −0.968691 0.248268i \(-0.920139\pi\)
0.968691 0.248268i \(-0.0798613\pi\)
\(314\) 0 0
\(315\) 11.6603 13.4641i 0.656981 0.758616i
\(316\) 0 0
\(317\) −25.8564 −1.45224 −0.726120 0.687568i \(-0.758678\pi\)
−0.726120 + 0.687568i \(0.758678\pi\)
\(318\) 0 0
\(319\) 10.1436i 0.567932i
\(320\) 0 0
\(321\) 9.07180i 0.506338i
\(322\) 0 0
\(323\) 18.9282i 1.05319i
\(324\) 0 0
\(325\) 3.12436i 0.173308i
\(326\) 0 0
\(327\) −5.07180 −0.280471
\(328\) 0 0
\(329\) −13.8564 12.0000i −0.763928 0.661581i
\(330\) 0 0
\(331\) 23.3205i 1.28181i −0.767620 0.640906i \(-0.778559\pi\)
0.767620 0.640906i \(-0.221441\pi\)
\(332\) 0 0
\(333\) 9.85641 0.540128
\(334\) 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
\(339\) −1.85641 −0.100826
\(340\) 0 0
\(341\) 10.1436i 0.549306i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) −2.92820 −0.157649
\(346\) 0 0
\(347\) 6.53590i 0.350865i −0.984491 0.175433i \(-0.943868\pi\)
0.984491 0.175433i \(-0.0561324\pi\)
\(348\) 0 0
\(349\) 5.66025i 0.302986i 0.988458 + 0.151493i \(0.0484082\pi\)
−0.988458 + 0.151493i \(0.951592\pi\)
\(350\) 0 0
\(351\) 5.07180i 0.270712i
\(352\) 0 0
\(353\) 2.14359i 0.114092i 0.998372 + 0.0570460i \(0.0181682\pi\)
−0.998372 + 0.0570460i \(0.981832\pi\)
\(354\) 0 0
\(355\) 31.3205 1.66232
\(356\) 0 0
\(357\) 5.07180 5.85641i 0.268428 0.309954i
\(358\) 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
\(363\) 6.48334 0.340287
\(364\) 0 0
\(365\) 18.9282 0.990747
\(366\) 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
\(371\) 13.8564 + 12.0000i 0.719389 + 0.623009i
\(372\) 0 0
\(373\) −30.9282 −1.60140 −0.800701 0.599064i \(-0.795539\pi\)
−0.800701 + 0.599064i \(0.795539\pi\)
\(374\) 0 0
\(375\) 5.07180i 0.261906i
\(376\) 0 0
\(377\) 8.78461i 0.452430i
\(378\) 0 0
\(379\) 14.5359i 0.746659i 0.927699 + 0.373329i \(0.121784\pi\)
−0.927699 + 0.373329i \(0.878216\pi\)
\(380\) 0 0
\(381\) 10.6410i 0.545156i
\(382\) 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
\(387\) 23.3205i 1.18545i
\(388\) 0 0
\(389\) −25.8564 −1.31097 −0.655486 0.755207i \(-0.727536\pi\)
−0.655486 + 0.755207i \(0.727536\pi\)
\(390\) 0 0
\(391\) 5.85641 0.296171
\(392\) 0 0
\(393\) −5.32051 −0.268384
\(394\) 0 0
\(395\) −9.46410 −0.476191
\(396\) 0 0
\(397\) 35.9090i 1.80222i −0.433591 0.901110i \(-0.642754\pi\)
0.433591 0.901110i \(-0.357246\pi\)
\(398\) 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
\(403\) 8.78461i 0.437593i
\(404\) 0 0
\(405\) 12.1962i 0.606032i
\(406\) 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
\(411\) 5.75129 0.283690
\(412\) 0 0
\(413\) 28.3923 + 24.5885i 1.39709 + 1.20992i
\(414\) 0 0
\(415\) 19.8564i 0.974713i
\(416\) 0 0
\(417\) −4.53590 −0.222124
\(418\) 0 0
\(419\) −21.1244 −1.03199 −0.515996 0.856591i \(-0.672578\pi\)
−0.515996 + 0.856591i \(0.672578\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) −17.0718 −0.830059
\(424\) 0 0
\(425\) 9.85641i 0.478106i
\(426\) 0 0
\(427\) 9.80385 11.3205i 0.474441 0.547838i
\(428\) 0 0
\(429\) −1.35898 −0.0656124
\(430\) 0 0
\(431\) 1.46410i 0.0705233i 0.999378 + 0.0352616i \(0.0112265\pi\)
−0.999378 + 0.0352616i \(0.988774\pi\)
\(432\) 0 0
\(433\) 25.8564i 1.24258i 0.783581 + 0.621290i \(0.213391\pi\)
−0.783581 + 0.621290i \(0.786609\pi\)
\(434\) 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
\(441\) −2.46410 17.0718i −0.117338 0.812943i
\(442\) 0 0
\(443\) 1.46410i 0.0695616i −0.999395 0.0347808i \(-0.988927\pi\)
0.999395 0.0347808i \(-0.0110733\pi\)
\(444\) 0 0
\(445\) −2.92820 −0.138810
\(446\) 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
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) 17.0718i 0.802103i
\(454\) 0 0
\(455\) −6.92820 6.00000i −0.324799 0.281284i
\(456\) 0 0
\(457\) 8.39230 0.392575 0.196288 0.980546i \(-0.437111\pi\)
0.196288 + 0.980546i \(0.437111\pi\)
\(458\) 0 0
\(459\) 16.0000i 0.746816i
\(460\) 0 0
\(461\) 27.5167i 1.28158i 0.767717 + 0.640789i \(0.221393\pi\)
−0.767717 + 0.640789i \(0.778607\pi\)
\(462\) 0 0
\(463\) 20.5359i 0.954384i −0.878799 0.477192i \(-0.841655\pi\)
0.878799 0.477192i \(-0.158345\pi\)
\(464\) 0 0
\(465\) 13.8564i 0.642575i
\(466\) 0 0
\(467\) −18.5885 −0.860171 −0.430086 0.902788i \(-0.641517\pi\)
−0.430086 + 0.902788i \(0.641517\pi\)
\(468\) 0 0
\(469\) 16.3923 18.9282i 0.756926 0.874023i
\(470\) 0 0
\(471\) 16.6410i 0.766778i
\(472\) 0 0
\(473\) −13.8564 −0.637118
\(474\) 0 0
\(475\) 11.6603 0.535009
\(476\) 0 0
\(477\) 17.0718 0.781664
\(478\) 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
\(483\) −1.85641 + 2.14359i −0.0844694 + 0.0975369i
\(484\) 0 0
\(485\) −32.7846 −1.48867
\(486\) 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
\(491\) 15.3205i 0.691405i 0.938344 + 0.345702i \(0.112359\pi\)
−0.938344 + 0.345702i \(0.887641\pi\)
\(492\) 0 0
\(493\) 27.7128i 1.24812i
\(494\) 0 0
\(495\) −9.85641 −0.443013
\(496\) 0 0
\(497\) 19.8564 22.9282i 0.890682 1.02847i
\(498\) 0 0
\(499\) 4.39230i 0.196627i −0.995156 0.0983133i \(-0.968655\pi\)
0.995156 0.0983133i \(-0.0313447\pi\)
\(500\) 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
\(505\) −11.4641 −0.510146
\(506\) 0 0
\(507\) 8.33975 0.370381
\(508\) 0 0
\(509\) 31.8038i 1.40968i 0.709366 + 0.704840i \(0.248981\pi\)
−0.709366 + 0.704840i \(0.751019\pi\)
\(510\) 0 0
\(511\) 12.0000 13.8564i 0.530849 0.612971i
\(512\) 0 0
\(513\) 18.9282 0.835701
\(514\) 0 0
\(515\) 13.8564i 0.610586i
\(516\) 0 0
\(517\) 10.1436i 0.446115i
\(518\) 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
\(523\) 43.3731 1.89657 0.948286 0.317417i \(-0.102816\pi\)
0.948286 + 0.317417i \(0.102816\pi\)
\(524\) 0 0
\(525\) 3.60770 + 3.12436i 0.157453 + 0.136358i
\(526\) 0 0
\(527\) 27.7128i 1.20719i
\(528\) 0 0
\(529\) 20.8564 0.906800
\(530\) 0 0
\(531\) 34.9808 1.51804
\(532\) 0 0
\(533\) 13.8564 0.600188
\(534\) 0 0
\(535\) 33.8564 1.46374
\(536\) 0 0
\(537\) 14.9282i 0.644200i
\(538\) 0 0
\(539\) −10.1436 + 1.46410i −0.436916 + 0.0630633i
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) 0 0
\(543\) 9.71281i 0.416817i
\(544\) 0 0
\(545\) 18.9282i 0.810795i
\(546\) 0 0
\(547\) 14.5359i 0.621510i −0.950490 0.310755i \(-0.899418\pi\)
0.950490 0.310755i \(-0.100582\pi\)
\(548\) 0 0
\(549\) 13.9474i 0.595262i
\(550\) 0 0
\(551\) −32.7846 −1.39667
\(552\) 0 0
\(553\) −6.00000 + 6.92820i −0.255146 + 0.294617i
\(554\) 0 0
\(555\) 8.00000i 0.339581i
\(556\) 0 0
\(557\) −17.0718 −0.723355 −0.361678 0.932303i \(-0.617796\pi\)
−0.361678 + 0.932303i \(0.617796\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −4.28719 −0.181005
\(562\) 0 0
\(563\) 22.9808 0.968524 0.484262 0.874923i \(-0.339088\pi\)
0.484262 + 0.874923i \(0.339088\pi\)
\(564\) 0 0
\(565\) 6.92820i 0.291472i
\(566\) 0 0
\(567\) −8.92820 7.73205i −0.374949 0.324716i
\(568\) 0 0
\(569\) −7.60770 −0.318931 −0.159466 0.987203i \(-0.550977\pi\)
−0.159466 + 0.987203i \(0.550977\pi\)
\(570\) 0 0
\(571\) 18.2487i 0.763685i −0.924227 0.381842i \(-0.875290\pi\)
0.924227 0.381842i \(-0.124710\pi\)
\(572\) 0 0
\(573\) 3.32051i 0.138716i
\(574\) 0 0
\(575\) 3.60770i 0.150451i
\(576\) 0 0
\(577\) 37.8564i 1.57598i −0.615686 0.787991i \(-0.711121\pi\)
0.615686 0.787991i \(-0.288879\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) 14.5359 + 12.5885i 0.603051 + 0.522257i
\(582\) 0 0
\(583\) 10.1436i 0.420105i
\(584\) 0 0
\(585\) −8.53590 −0.352916
\(586\) 0 0
\(587\) −14.1962 −0.585938 −0.292969 0.956122i \(-0.594643\pi\)
−0.292969 + 0.956122i \(0.594643\pi\)
\(588\) 0 0
\(589\) 32.7846 1.35087
\(590\) 0 0
\(591\) 8.78461 0.361351
\(592\) 0 0
\(593\) 35.7128i 1.46655i 0.679932 + 0.733275i \(0.262009\pi\)
−0.679932 + 0.733275i \(0.737991\pi\)
\(594\) 0 0
\(595\) −21.8564 18.9282i −0.896025 0.775981i
\(596\) 0 0
\(597\) −18.1436 −0.742568
\(598\) 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
\(603\) 23.3205i 0.949685i
\(604\) 0 0
\(605\) 24.1962i 0.983713i
\(606\) 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
\(611\) 8.78461i 0.355387i
\(612\) 0 0
\(613\) −1.85641 −0.0749796 −0.0374898 0.999297i \(-0.511936\pi\)
−0.0374898 + 0.999297i \(0.511936\pi\)
\(614\) 0 0
\(615\) −21.8564 −0.881335
\(616\) 0 0
\(617\) 16.3923 0.659929 0.329965 0.943993i \(-0.392963\pi\)
0.329965 + 0.943993i \(0.392963\pi\)
\(618\) 0 0
\(619\) −13.8038 −0.554823 −0.277412 0.960751i \(-0.589477\pi\)
−0.277412 + 0.960751i \(0.589477\pi\)
\(620\) 0 0
\(621\) 5.85641i 0.235009i
\(622\) 0 0
\(623\) −1.85641 + 2.14359i −0.0743754 + 0.0858813i
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) 0 0
\(627\) 5.07180i 0.202548i
\(628\) 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
\(635\) −39.7128 −1.57595
\(636\) 0 0
\(637\) −8.78461 + 1.26795i −0.348059 + 0.0502380i
\(638\) 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
\(643\) 19.2679 0.759854 0.379927 0.925017i \(-0.375949\pi\)
0.379927 + 0.925017i \(0.375949\pi\)
\(644\) 0 0
\(645\) 18.9282 0.745297
\(646\) 0 0
\(647\) 46.6410 1.83365 0.916824 0.399292i \(-0.130744\pi\)
0.916824 + 0.399292i \(0.130744\pi\)
\(648\) 0 0
\(649\) 20.7846i 0.815867i
\(650\) 0 0
\(651\) 10.1436 + 8.78461i 0.397559 + 0.344296i
\(652\) 0 0
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) 0 0
\(655\) 19.8564i 0.775854i
\(656\) 0 0
\(657\) 17.0718i 0.666034i
\(658\) 0 0
\(659\) 40.1051i 1.56227i 0.624360 + 0.781137i \(0.285360\pi\)
−0.624360 + 0.781137i \(0.714640\pi\)
\(660\) 0 0
\(661\) 3.80385i 0.147953i 0.997260 + 0.0739763i \(0.0235689\pi\)
−0.997260 + 0.0739763i \(0.976431\pi\)
\(662\) 0 0
\(663\) −3.71281 −0.144194
\(664\) 0 0
\(665\) 22.3923 25.8564i 0.868336 1.00267i
\(666\) 0 0
\(667\) 10.1436i 0.392762i
\(668\) 0 0
\(669\) −5.85641 −0.226422
\(670\) 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
\(675\) 9.85641 0.379373
\(676\) 0 0
\(677\) 43.1244i 1.65740i 0.559690 + 0.828702i \(0.310920\pi\)
−0.559690 + 0.828702i \(0.689080\pi\)
\(678\) 0 0
\(679\) −20.7846 + 24.0000i −0.797640 + 0.921035i
\(680\) 0 0
\(681\) −3.46410 −0.132745
\(682\) 0 0
\(683\) 22.5359i 0.862312i 0.902277 + 0.431156i \(0.141894\pi\)
−0.902277 + 0.431156i \(0.858106\pi\)
\(684\) 0 0
\(685\) 21.4641i 0.820101i
\(686\) 0 0
\(687\) 12.9282i 0.493242i
\(688\) 0 0
\(689\) 8.78461i 0.334667i
\(690\) 0 0
\(691\) −18.9808 −0.722062 −0.361031 0.932554i \(-0.617575\pi\)
−0.361031 + 0.932554i \(0.617575\pi\)
\(692\) 0 0
\(693\) −6.24871 + 7.21539i −0.237369 + 0.274090i
\(694\) 0 0
\(695\) 16.9282i 0.642123i
\(696\) 0 0
\(697\) 43.7128 1.65574
\(698\) 0 0
\(699\) −5.75129 −0.217534
\(700\) 0 0
\(701\) 25.8564 0.976583 0.488291 0.872681i \(-0.337620\pi\)
0.488291 + 0.872681i \(0.337620\pi\)
\(702\) 0 0
\(703\) 18.9282 0.713891
\(704\) 0 0
\(705\) 13.8564i 0.521862i
\(706\) 0 0
\(707\) −7.26795 + 8.39230i −0.273339 + 0.315625i
\(708\) 0 0
\(709\) −17.0718 −0.641145 −0.320572 0.947224i \(-0.603875\pi\)
−0.320572 + 0.947224i \(0.603875\pi\)
\(710\) 0 0
\(711\) 8.53590i 0.320121i
\(712\) 0 0
\(713\) 10.1436i 0.379881i
\(714\) 0 0
\(715\) 5.07180i 0.189674i
\(716\) 0 0
\(717\) 14.9282i 0.557504i
\(718\) 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
\(723\) 8.78461i 0.326703i
\(724\) 0 0
\(725\) −17.0718 −0.634031
\(726\) 0 0
\(727\) 32.7846 1.21591 0.607957 0.793970i \(-0.291989\pi\)
0.607957 + 0.793970i \(0.291989\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) −37.8564 −1.40017
\(732\) 0 0
\(733\) 8.19615i 0.302732i −0.988478 0.151366i \(-0.951633\pi\)
0.988478 0.151366i \(-0.0483672\pi\)
\(734\) 0 0
\(735\) 13.8564 2.00000i 0.511101 0.0737711i
\(736\) 0 0
\(737\) −13.8564 −0.510407
\(738\) 0 0
\(739\) 23.3205i 0.857859i 0.903338 + 0.428929i \(0.141109\pi\)
−0.903338 + 0.428929i \(0.858891\pi\)
\(740\) 0 0
\(741\) 4.39230i 0.161355i
\(742\) 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
\(747\) 17.9090 0.655255
\(748\) 0 0
\(749\) 21.4641 24.7846i 0.784281 0.905610i
\(750\) 0 0
\(751\) 33.4641i 1.22112i 0.791969 + 0.610561i \(0.209056\pi\)
−0.791969 + 0.610561i \(0.790944\pi\)
\(752\) 0 0
\(753\) 11.7513 0.428241
\(754\) 0 0
\(755\) −63.7128 −2.31875
\(756\) 0 0
\(757\) 53.5692 1.94701 0.973503 0.228673i \(-0.0734388\pi\)
0.973503 + 0.228673i \(0.0734388\pi\)
\(758\) 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
\(763\) 13.8564 + 12.0000i 0.501636 + 0.434429i
\(764\) 0 0
\(765\) −26.9282 −0.973591
\(766\) 0 0
\(767\) 18.0000i 0.649942i
\(768\) 0 0
\(769\) 25.8564i 0.932406i 0.884678 + 0.466203i \(0.154378\pi\)
−0.884678 + 0.466203i \(0.845622\pi\)
\(770\) 0 0
\(771\) 4.28719i 0.154399i
\(772\) 0 0
\(773\) 22.4449i 0.807286i −0.914917 0.403643i \(-0.867744\pi\)
0.914917 0.403643i \(-0.132256\pi\)
\(774\) 0 0
\(775\) 17.0718 0.613237
\(776\) 0 0
\(777\) 5.85641 + 5.07180i 0.210097 + 0.181950i
\(778\) 0 0
\(779\) 51.7128i 1.85280i
\(780\) 0 0
\(781\) −16.7846 −0.600601
\(782\) 0 0
\(783\) −27.7128 −0.990375
\(784\) 0 0
\(785\) −62.1051 −2.21663
\(786\) 0 0
\(787\) −10.5885 −0.377438 −0.188719 0.982031i \(-0.560433\pi\)
−0.188719 + 0.982031i \(0.560433\pi\)
\(788\) 0 0
\(789\) 3.32051i 0.118213i
\(790\) 0 0
\(791\) 5.07180 + 4.39230i 0.180332 + 0.156172i
\(792\) 0 0
\(793\) −7.17691 −0.254860
\(794\) 0 0
\(795\) 13.8564i 0.491436i
\(796\) 0 0
\(797\) 16.8756i 0.597766i −0.954290 0.298883i \(-0.903386\pi\)
0.954290 0.298883i \(-0.0966140\pi\)
\(798\) 0 0
\(799\) 27.7128i 0.980409i
\(800\) 0 0
\(801\) 2.64102i 0.0933157i
\(802\)