Properties

Label 1040.2.f.b.129.1
Level $1040$
Weight $2$
Character 1040.129
Analytic conductor $8.304$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1040.129
Dual form 1040.2.f.b.129.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +2.00000i q^{11} +(-3.00000 - 2.00000i) q^{13} +(-4.00000 - 2.00000i) q^{15} -6.00000i q^{19} -6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -4.00000i q^{27} +6.00000 q^{29} +6.00000i q^{31} +4.00000 q^{33} -6.00000 q^{37} +(-4.00000 + 6.00000i) q^{39} +8.00000i q^{41} +6.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -8.00000 q^{47} -7.00000 q^{49} -12.0000i q^{53} +(4.00000 + 2.00000i) q^{55} -12.0000 q^{57} +2.00000i q^{59} +6.00000 q^{61} +(-7.00000 + 4.00000i) q^{65} +12.0000 q^{67} -12.0000 q^{69} -2.00000i q^{71} +6.00000 q^{73} +(-8.00000 + 6.00000i) q^{75} -11.0000 q^{81} -4.00000 q^{83} -12.0000i q^{87} +8.00000i q^{89} +12.0000 q^{93} +(-12.0000 - 6.00000i) q^{95} +6.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{9} - 6 q^{13} - 8 q^{15} - 6 q^{25} + 12 q^{29} + 8 q^{33} - 12 q^{37} - 8 q^{39} - 2 q^{45} - 16 q^{47} - 14 q^{49} + 8 q^{55} - 24 q^{57} + 12 q^{61} - 14 q^{65} + 24 q^{67} - 24 q^{69} + 12 q^{73} - 16 q^{75} - 22 q^{81} - 8 q^{83} + 24 q^{93} - 24 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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
\(3\) 2.00000i 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 1.00000 2.00000i 0.447214 0.894427i
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) −3.00000 2.00000i −0.832050 0.554700i
\(14\) 0 0
\(15\) −4.00000 2.00000i −1.03280 0.516398i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 6.00000i 1.07763i 0.842424 + 0.538816i \(0.181128\pi\)
−0.842424 + 0.538816i \(0.818872\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −4.00000 + 6.00000i −0.640513 + 0.960769i
\(40\) 0 0
\(41\) 8.00000i 1.24939i 0.780869 + 0.624695i \(0.214777\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) −1.00000 + 2.00000i −0.149071 + 0.298142i
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 4.00000 + 2.00000i 0.539360 + 0.269680i
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) 2.00000i 0.260378i 0.991489 + 0.130189i \(0.0415584\pi\)
−0.991489 + 0.130189i \(0.958442\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.00000 + 4.00000i −0.868243 + 0.496139i
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 2.00000i 0.237356i −0.992933 0.118678i \(-0.962134\pi\)
0.992933 0.118678i \(-0.0378657\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −8.00000 + 6.00000i −0.923760 + 0.692820i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.0000i 1.28654i
\(88\) 0 0
\(89\) 8.00000i 0.847998i 0.905663 + 0.423999i \(0.139374\pi\)
−0.905663 + 0.423999i \(0.860626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) −12.0000 6.00000i −1.23117 0.615587i
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) 12.0000i 1.14939i −0.818367 0.574696i \(-0.805120\pi\)
0.818367 0.574696i \(-0.194880\pi\)
\(110\) 0 0
\(111\) 12.0000i 1.13899i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −12.0000 6.00000i −1.11901 0.559503i
\(116\) 0 0
\(117\) 3.00000 + 2.00000i 0.277350 + 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 16.0000 1.44267
\(124\) 0 0
\(125\) −11.0000 + 2.00000i −0.983870 + 0.178885i
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −8.00000 4.00000i −0.688530 0.344265i
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 16.0000i 1.34744i
\(142\) 0 0
\(143\) 4.00000 6.00000i 0.334497 0.501745i
\(144\) 0 0
\(145\) 6.00000 12.0000i 0.498273 0.996546i
\(146\) 0 0
\(147\) 14.0000i 1.15470i
\(148\) 0 0
\(149\) 20.0000i 1.63846i −0.573462 0.819232i \(-0.694400\pi\)
0.573462 0.819232i \(-0.305600\pi\)
\(150\) 0 0
\(151\) 18.0000i 1.46482i −0.680864 0.732410i \(-0.738396\pi\)
0.680864 0.732410i \(-0.261604\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 + 6.00000i 0.963863 + 0.481932i
\(156\) 0 0
\(157\) 12.0000i 0.957704i 0.877896 + 0.478852i \(0.158947\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) −24.0000 −1.90332
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 4.00000 8.00000i 0.311400 0.622799i
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 5.00000 + 12.0000i 0.384615 + 0.923077i
\(170\) 0 0
\(171\) 6.00000i 0.458831i
\(172\) 0 0
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 12.0000i 0.887066i
\(184\) 0 0
\(185\) −6.00000 + 12.0000i −0.441129 + 0.882258i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 8.00000 + 14.0000i 0.572892 + 1.00256i
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 24.0000i 1.69283i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 16.0000 + 8.00000i 1.11749 + 0.558744i
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 12.0000 + 6.00000i 0.818393 + 0.409197i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.0000i 0.810885i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 3.00000 + 4.00000i 0.200000 + 0.266667i
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 12.0000i 0.792982i 0.918039 + 0.396491i \(0.129772\pi\)
−0.918039 + 0.396491i \(0.870228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000i 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) −8.00000 + 16.0000i −0.521862 + 1.04372i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.0000i 0.646846i −0.946254 0.323423i \(-0.895166\pi\)
0.946254 0.323423i \(-0.104834\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) −7.00000 + 14.0000i −0.447214 + 0.894427i
\(246\) 0 0
\(247\) −12.0000 + 18.0000i −0.763542 + 1.14531i
\(248\) 0 0
\(249\) 8.00000i 0.506979i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 6.00000i 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 0 0
\(265\) −24.0000 12.0000i −1.47431 0.737154i
\(266\) 0 0
\(267\) 16.0000 0.979184
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 6.00000i 0.364474i 0.983255 + 0.182237i \(0.0583338\pi\)
−0.983255 + 0.182237i \(0.941666\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.00000 6.00000i 0.482418 0.361814i
\(276\) 0 0
\(277\) 12.0000i 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 0 0
\(279\) 6.00000i 0.359211i
\(280\) 0 0
\(281\) 8.00000i 0.477240i −0.971113 0.238620i \(-0.923305\pi\)
0.971113 0.238620i \(-0.0766950\pi\)
\(282\) 0 0
\(283\) 22.0000i 1.30776i 0.756596 + 0.653882i \(0.226861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) 0 0
\(285\) −12.0000 + 24.0000i −0.710819 + 1.42164i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 12.0000i 0.703452i
\(292\) 0 0
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) 4.00000 + 2.00000i 0.232889 + 0.116445i
\(296\) 0 0
\(297\) 8.00000 0.464207
\(298\) 0 0
\(299\) −12.0000 + 18.0000i −0.693978 + 1.04097i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) 6.00000 12.0000i 0.343559 0.687118i
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i 0.974106 + 0.226093i \(0.0725954\pi\)
−0.974106 + 0.226093i \(0.927405\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 12.0000i 0.671871i
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.00000 + 18.0000i 0.0554700 + 0.998460i
\(326\) 0 0
\(327\) −24.0000 −1.32720
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 30.0000i 1.64895i −0.565899 0.824475i \(-0.691471\pi\)
0.565899 0.824475i \(-0.308529\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 12.0000 24.0000i 0.655630 1.31126i
\(336\) 0 0
\(337\) 32.0000i 1.74315i 0.490261 + 0.871576i \(0.336901\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −12.0000 + 24.0000i −0.646058 + 1.29212i
\(346\) 0 0
\(347\) 6.00000i 0.322097i 0.986947 + 0.161048i \(0.0514875\pi\)
−0.986947 + 0.161048i \(0.948512\pi\)
\(348\) 0 0
\(349\) 12.0000i 0.642345i −0.947021 0.321173i \(-0.895923\pi\)
0.947021 0.321173i \(-0.104077\pi\)
\(350\) 0 0
\(351\) −8.00000 + 12.0000i −0.427008 + 0.640513i
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −4.00000 2.00000i −0.212298 0.106149i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.00000i 0.105556i −0.998606 0.0527780i \(-0.983192\pi\)
0.998606 0.0527780i \(-0.0168076\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 6.00000 12.0000i 0.314054 0.628109i
\(366\) 0 0
\(367\) 18.0000i 0.939592i 0.882775 + 0.469796i \(0.155673\pi\)
−0.882775 + 0.469796i \(0.844327\pi\)
\(368\) 0 0
\(369\) 8.00000i 0.416463i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 4.00000 + 22.0000i 0.206559 + 1.13608i
\(376\) 0 0
\(377\) −18.0000 12.0000i −0.927047 0.618031i
\(378\) 0 0
\(379\) 18.0000i 0.924598i 0.886724 + 0.462299i \(0.152975\pi\)
−0.886724 + 0.462299i \(0.847025\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.00000i 0.304997i
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 24.0000i 1.21064i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.0000i 0.799002i −0.916733 0.399501i \(-0.869183\pi\)
0.916733 0.399501i \(-0.130817\pi\)
\(402\) 0 0
\(403\) 12.0000 18.0000i 0.597763 0.896644i
\(404\) 0 0
\(405\) −11.0000 + 22.0000i −0.546594 + 1.09319i
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 24.0000i 1.18672i 0.804936 + 0.593362i \(0.202200\pi\)
−0.804936 + 0.593362i \(0.797800\pi\)
\(410\) 0 0
\(411\) 4.00000i 0.197305i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 + 8.00000i −0.196352 + 0.392705i
\(416\) 0 0
\(417\) 8.00000i 0.391762i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 36.0000i 1.75453i −0.480004 0.877266i \(-0.659365\pi\)
0.480004 0.877266i \(-0.340635\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −12.0000 8.00000i −0.579365 0.386244i
\(430\) 0 0
\(431\) 10.0000i 0.481683i −0.970564 0.240842i \(-0.922577\pi\)
0.970564 0.240842i \(-0.0774234\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) −24.0000 12.0000i −1.15071 0.575356i
\(436\) 0 0
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) 6.00000i 0.285069i 0.989790 + 0.142534i \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 0 0
\(445\) 16.0000 + 8.00000i 0.758473 + 0.379236i
\(446\) 0 0
\(447\) −40.0000 −1.89194
\(448\) 0 0
\(449\) 16.0000i 0.755087i 0.925992 + 0.377543i \(0.123231\pi\)
−0.925992 + 0.377543i \(0.876769\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 0 0
\(453\) −36.0000 −1.69143
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.00000i 0.186299i 0.995652 + 0.0931493i \(0.0296934\pi\)
−0.995652 + 0.0931493i \(0.970307\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 12.0000 24.0000i 0.556487 1.11297i
\(466\) 0 0
\(467\) 18.0000i 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 24.0000 1.10586
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) −24.0000 + 18.0000i −1.10120 + 0.825897i
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) 22.0000i 1.00521i 0.864517 + 0.502603i \(0.167624\pi\)
−0.864517 + 0.502603i \(0.832376\pi\)
\(480\) 0 0
\(481\) 18.0000 + 12.0000i 0.820729 + 0.547153i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 12.0000i 0.272446 0.544892i
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 24.0000i 1.08532i
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.00000 2.00000i −0.179787 0.0898933i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.00000i 0.268597i −0.990941 0.134298i \(-0.957122\pi\)
0.990941 0.134298i \(-0.0428781\pi\)
\(500\) 0 0
\(501\) 32.0000i 1.42965i
\(502\) 0 0
\(503\) 6.00000i 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 6.00000 12.0000i 0.266996 0.533993i
\(506\) 0 0
\(507\) 24.0000 10.0000i 1.06588 0.444116i
\(508\) 0 0
\(509\) 20.0000i 0.886484i 0.896402 + 0.443242i \(0.146172\pi\)
−0.896402 + 0.443242i \(0.853828\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −24.0000 −1.05963
\(514\) 0 0
\(515\) −12.0000 6.00000i −0.528783 0.264392i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 42.0000i 1.83653i −0.395964 0.918266i \(-0.629590\pi\)
0.395964 0.918266i \(-0.370410\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 2.00000i 0.0867926i
\(532\) 0 0
\(533\) 16.0000 24.0000i 0.693037 1.03956i
\(534\) 0 0
\(535\) 12.0000 + 6.00000i 0.518805 + 0.259403i
\(536\) 0 0
\(537\) 24.0000i 1.03568i
\(538\) 0 0
\(539\) 14.0000i 0.603023i
\(540\) 0 0
\(541\) 12.0000i 0.515920i −0.966156 0.257960i \(-0.916950\pi\)
0.966156 0.257960i \(-0.0830503\pi\)
\(542\) 0 0
\(543\) 4.00000i 0.171656i
\(544\) 0 0
\(545\) −24.0000 12.0000i −1.02805 0.514024i
\(546\) 0 0
\(547\) 18.0000i 0.769624i −0.922995 0.384812i \(-0.874266\pi\)
0.922995 0.384812i \(-0.125734\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 36.0000i 1.53365i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 24.0000 + 12.0000i 1.01874 + 0.509372i
\(556\) 0 0
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 12.0000 18.0000i 0.507546 0.761319i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.0000i 1.26435i 0.774826 + 0.632175i \(0.217837\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 + 18.0000i −1.00087 + 0.750652i
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 12.0000i 0.498703i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 7.00000 4.00000i 0.289414 0.165380i
\(586\) 0 0
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 0 0
\(589\) 36.0000 1.48335
\(590\) 0 0
\(591\) 4.00000i 0.164538i
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 48.0000i 1.96451i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) 7.00000 14.0000i 0.284590 0.569181i
\(606\) 0 0
\(607\) 18.0000i 0.730597i 0.930890 + 0.365299i \(0.119033\pi\)
−0.930890 + 0.365299i \(0.880967\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 + 16.0000i 0.970936 + 0.647291i
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) 16.0000 32.0000i 0.645182 1.29036i
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) 18.0000i 0.723481i 0.932279 + 0.361741i \(0.117817\pi\)
−0.932279 + 0.361741i \(0.882183\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 24.0000i 0.958468i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 30.0000i 1.19428i 0.802137 + 0.597141i \(0.203697\pi\)
−0.802137 + 0.597141i \(0.796303\pi\)
\(632\) 0 0
\(633\) 24.0000i 0.953914i
\(634\) 0 0
\(635\) 4.00000 + 2.00000i 0.158735 + 0.0793676i
\(636\) 0 0
\(637\) 21.0000 + 14.0000i 0.832050 + 0.554700i
\(638\) 0 0
\(639\) 2.00000i 0.0791188i
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 12.0000 24.0000i 0.472500 0.944999i
\(646\) 0 0
\(647\) 6.00000i 0.235884i −0.993020 0.117942i \(-0.962370\pi\)
0.993020 0.117942i \(-0.0376297\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.0000i 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 0 0
\(655\) 12.0000 24.0000i 0.468879 0.937758i
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 12.0000i 0.466746i 0.972387 + 0.233373i \(0.0749763\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.0000i 1.39393i
\(668\) 0 0
\(669\) 48.0000i 1.85579i
\(670\) 0 0
\(671\) 12.0000i 0.463255i
\(672\) 0 0
\(673\) 48.0000i 1.85026i 0.379646 + 0.925132i \(0.376046\pi\)
−0.379646 + 0.925132i \(0.623954\pi\)
\(674\) 0 0
\(675\) −16.0000 + 12.0000i −0.615840 + 0.461880i
\(676\) 0 0
\(677\) 36.0000i 1.38359i 0.722093 + 0.691796i \(0.243180\pi\)
−0.722093 + 0.691796i \(0.756820\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000i 0.306561i
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) −2.00000 + 4.00000i −0.0764161 + 0.152832i
\(686\) 0 0
\(687\) 24.0000 0.915657
\(688\) 0 0
\(689\) −24.0000 + 36.0000i −0.914327 + 1.37149i
\(690\) 0 0
\(691\) 42.0000i 1.59776i 0.601494 + 0.798878i \(0.294573\pi\)
−0.601494 + 0.798878i \(0.705427\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 8.00000i 0.151729 0.303457i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −48.0000 −1.81553
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 36.0000i 1.35777i
\(704\) 0 0
\(705\) 32.0000 + 16.0000i 1.20519 + 0.602595i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12.0000i 0.450669i 0.974281 + 0.225335i \(0.0723476\pi\)
−0.974281 + 0.225335i \(0.927652\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 36.0000 1.34821
\(714\) 0 0
\(715\) −8.00000 14.0000i −0.299183 0.523570i
\(716\) 0 0
\(717\) −20.0000 −0.746914
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −18.0000 24.0000i −0.668503 0.891338i
\(726\) 0 0
\(727\) 26.0000i 0.964287i 0.876092 + 0.482143i \(0.160142\pi\)
−0.876092 + 0.482143i \(0.839858\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 0 0
\(735\) 28.0000 + 14.0000i 1.03280 + 0.516398i
\(736\) 0 0
\(737\) 24.0000i 0.884051i
\(738\) 0 0
\(739\) 6.00000i 0.220714i −0.993892 0.110357i \(-0.964801\pi\)
0.993892 0.110357i \(-0.0351994\pi\)
\(740\) 0 0
\(741\) 36.0000 + 24.0000i 1.32249 + 0.881662i
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −40.0000 20.0000i −1.46549 0.732743i
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 24.0000i 0.874609i
\(754\) 0 0
\(755\) −36.0000 18.0000i −1.31017 0.655087i
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 0 0
\(759\) 24.0000i 0.871145i
\(760\) 0 0
\(761\) 40.0000i 1.45000i 0.688749 + 0.724999i \(0.258160\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.00000 6.00000i 0.144432 0.216647i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) 0 0
\(775\) 24.0000 18.0000i 0.862105 0.646579i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) 0 0
\(785\) 24.0000 + 12.0000i 0.856597 + 0.428298i
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −18.0000 12.0000i −0.639199 0.426132i
\(794\) 0 0
\(795\) −24.0000 + 48.0000i −0.851192 + 1.70238i
\(796\) 0 0
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 8.00000i 0.282666i
\(802\) 0 0
\(803\) 12.0000i