Properties

Label 1840.2.e.a.369.1
Level $1840$
Weight $2$
Character 1840.369
Analytic conductor $14.692$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 369.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1840.369
Dual form 1840.2.e.a.369.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +3.00000i q^{7} -1.00000 q^{9} +6.00000i q^{13} +(-2.00000 - 4.00000i) q^{15} -7.00000i q^{17} -4.00000 q^{19} +6.00000 q^{21} -1.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000i q^{27} +9.00000 q^{29} +3.00000 q^{31} +(3.00000 + 6.00000i) q^{35} -7.00000i q^{37} +12.0000 q^{39} +9.00000 q^{41} -4.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} +2.00000i q^{47} -2.00000 q^{49} -14.0000 q^{51} +7.00000i q^{53} +8.00000i q^{57} +9.00000 q^{59} -2.00000 q^{61} -3.00000i q^{63} +(6.00000 + 12.0000i) q^{65} -13.0000i q^{67} -2.00000 q^{69} +13.0000 q^{71} +4.00000i q^{73} +(-8.00000 - 6.00000i) q^{75} -2.00000 q^{79} -11.0000 q^{81} -11.0000i q^{83} +(-7.00000 - 14.0000i) q^{85} -18.0000i q^{87} +10.0000 q^{89} -18.0000 q^{91} -6.00000i q^{93} +(-8.00000 + 4.00000i) q^{95} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 2 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{5} - 2 q^{9} - 4 q^{15} - 8 q^{19} + 12 q^{21} + 6 q^{25} + 18 q^{29} + 6 q^{31} + 6 q^{35} + 24 q^{39} + 18 q^{41} - 4 q^{45} - 4 q^{49} - 28 q^{51} + 18 q^{59} - 4 q^{61} + 12 q^{65} - 4 q^{69} + 26 q^{71} - 16 q^{75} - 4 q^{79} - 22 q^{81} - 14 q^{85} + 20 q^{89} - 36 q^{91} - 16 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 2.00000 1.00000i 0.894427 0.447214i
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −2.00000 4.00000i −0.516398 1.03280i
\(16\) 0 0
\(17\) 7.00000i 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 + 6.00000i 0.507093 + 1.01419i
\(36\) 0 0
\(37\) 7.00000i 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 0 0
\(39\) 12.0000 1.92154
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) −2.00000 + 1.00000i −0.298142 + 0.149071i
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −14.0000 −1.96039
\(52\) 0 0
\(53\) 7.00000i 0.961524i 0.876851 + 0.480762i \(0.159640\pi\)
−0.876851 + 0.480762i \(0.840360\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) 6.00000 + 12.0000i 0.744208 + 1.48842i
\(66\) 0 0
\(67\) 13.0000i 1.58820i −0.607785 0.794101i \(-0.707942\pi\)
0.607785 0.794101i \(-0.292058\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 13.0000 1.54282 0.771408 0.636341i \(-0.219553\pi\)
0.771408 + 0.636341i \(0.219553\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) −8.00000 6.00000i −0.923760 0.692820i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 11.0000i 1.20741i −0.797209 0.603703i \(-0.793691\pi\)
0.797209 0.603703i \(-0.206309\pi\)
\(84\) 0 0
\(85\) −7.00000 14.0000i −0.759257 1.51851i
\(86\) 0 0
\(87\) 18.0000i 1.92980i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) 0 0
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) −8.00000 + 4.00000i −0.820783 + 0.410391i
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −13.0000 −1.29355 −0.646774 0.762682i \(-0.723882\pi\)
−0.646774 + 0.762682i \(0.723882\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 12.0000 6.00000i 1.17108 0.585540i
\(106\) 0 0
\(107\) 15.0000i 1.45010i 0.688694 + 0.725052i \(0.258184\pi\)
−0.688694 + 0.725052i \(0.741816\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −14.0000 −1.32882
\(112\) 0 0
\(113\) 5.00000i 0.470360i −0.971952 0.235180i \(-0.924432\pi\)
0.971952 0.235180i \(-0.0755680\pi\)
\(114\) 0 0
\(115\) −1.00000 2.00000i −0.0932505 0.186501i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 21.0000 1.92507
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 18.0000i 1.62301i
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 12.0000i 1.04053i
\(134\) 0 0
\(135\) −4.00000 8.00000i −0.344265 0.688530i
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 18.0000 9.00000i 1.49482 0.747409i
\(146\) 0 0
\(147\) 4.00000i 0.329914i
\(148\) 0 0
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 7.00000i 0.565916i
\(154\) 0 0
\(155\) 6.00000 3.00000i 0.481932 0.240966i
\(156\) 0 0
\(157\) 13.0000i 1.03751i −0.854922 0.518756i \(-0.826395\pi\)
0.854922 0.518756i \(-0.173605\pi\)
\(158\) 0 0
\(159\) 14.0000 1.11027
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) 12.0000 + 9.00000i 0.907115 + 0.680336i
\(176\) 0 0
\(177\) 18.0000i 1.35296i
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 4.00000i 0.295689i
\(184\) 0 0
\(185\) −7.00000 14.0000i −0.514650 1.02930i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 12.0000 0.872872
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) 0 0
\(195\) 24.0000 12.0000i 1.71868 0.859338i
\(196\) 0 0
\(197\) 8.00000i 0.569976i −0.958531 0.284988i \(-0.908010\pi\)
0.958531 0.284988i \(-0.0919897\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) −26.0000 −1.83390
\(202\) 0 0
\(203\) 27.0000i 1.89503i
\(204\) 0 0
\(205\) 18.0000 9.00000i 1.25717 0.628587i
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) 0 0
\(213\) 26.0000i 1.78149i
\(214\) 0 0
\(215\) −4.00000 8.00000i −0.272798 0.545595i
\(216\) 0 0
\(217\) 9.00000i 0.610960i
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 42.0000 2.82523
\(222\) 0 0
\(223\) 2.00000i 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) 16.0000i 1.06196i −0.847385 0.530979i \(-0.821824\pi\)
0.847385 0.530979i \(-0.178176\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.00000i 0.131024i −0.997852 0.0655122i \(-0.979132\pi\)
0.997852 0.0655122i \(-0.0208681\pi\)
\(234\) 0 0
\(235\) 2.00000 + 4.00000i 0.130466 + 0.260931i
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) −4.00000 + 2.00000i −0.255551 + 0.127775i
\(246\) 0 0
\(247\) 24.0000i 1.52708i
\(248\) 0 0
\(249\) −22.0000 −1.39419
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −28.0000 + 14.0000i −1.75343 + 0.876714i
\(256\) 0 0
\(257\) 26.0000i 1.62184i 0.585160 + 0.810918i \(0.301032\pi\)
−0.585160 + 0.810918i \(0.698968\pi\)
\(258\) 0 0
\(259\) 21.0000 1.30488
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) 9.00000i 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) 0 0
\(265\) 7.00000 + 14.0000i 0.430007 + 0.860013i
\(266\) 0 0
\(267\) 20.0000i 1.22398i
\(268\) 0 0
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) 25.0000 1.51864 0.759321 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(272\) 0 0
\(273\) 36.0000i 2.17882i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) 19.0000i 1.12943i 0.825285 + 0.564716i \(0.191014\pi\)
−0.825285 + 0.564716i \(0.808986\pi\)
\(284\) 0 0
\(285\) 8.00000 + 16.0000i 0.473879 + 0.947758i
\(286\) 0 0
\(287\) 27.0000i 1.59376i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) 3.00000i 0.175262i −0.996153 0.0876309i \(-0.972070\pi\)
0.996153 0.0876309i \(-0.0279296\pi\)
\(294\) 0 0
\(295\) 18.0000 9.00000i 1.04800 0.524000i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 26.0000i 1.49366i
\(304\) 0 0
\(305\) −4.00000 + 2.00000i −0.229039 + 0.114520i
\(306\) 0 0
\(307\) 10.0000i 0.570730i 0.958419 + 0.285365i \(0.0921148\pi\)
−0.958419 + 0.285365i \(0.907885\pi\)
\(308\) 0 0
\(309\) 32.0000 1.82042
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) 1.00000i 0.0565233i 0.999601 + 0.0282617i \(0.00899717\pi\)
−0.999601 + 0.0282617i \(0.991003\pi\)
\(314\) 0 0
\(315\) −3.00000 6.00000i −0.169031 0.338062i
\(316\) 0 0
\(317\) 12.0000i 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 30.0000 1.67444
\(322\) 0 0
\(323\) 28.0000i 1.55796i
\(324\) 0 0
\(325\) 24.0000 + 18.0000i 1.33128 + 0.998460i
\(326\) 0 0
\(327\) 4.00000i 0.221201i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 7.00000i 0.383598i
\(334\) 0 0
\(335\) −13.0000 26.0000i −0.710266 1.42053i
\(336\) 0 0
\(337\) 30.0000i 1.63420i 0.576493 + 0.817102i \(0.304421\pi\)
−0.576493 + 0.817102i \(0.695579\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) −4.00000 + 2.00000i −0.215353 + 0.107676i
\(346\) 0 0
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 0 0
\(349\) −27.0000 −1.44528 −0.722638 0.691226i \(-0.757071\pi\)
−0.722638 + 0.691226i \(0.757071\pi\)
\(350\) 0 0
\(351\) 24.0000 1.28103
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 26.0000 13.0000i 1.37994 0.689968i
\(356\) 0 0
\(357\) 42.0000i 2.22288i
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 22.0000i 1.15470i
\(364\) 0 0
\(365\) 4.00000 + 8.00000i 0.209370 + 0.418739i
\(366\) 0 0
\(367\) 1.00000i 0.0521996i 0.999659 + 0.0260998i \(0.00830876\pi\)
−0.999659 + 0.0260998i \(0.991691\pi\)
\(368\) 0 0
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) −21.0000 −1.09027
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 0 0
\(375\) −22.0000 4.00000i −1.13608 0.206559i
\(376\) 0 0
\(377\) 54.0000i 2.78114i
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 9.00000i 0.459879i −0.973205 0.229939i \(-0.926147\pi\)
0.973205 0.229939i \(-0.0738528\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) 0 0
\(393\) 16.0000i 0.807093i
\(394\) 0 0
\(395\) −4.00000 + 2.00000i −0.201262 + 0.100631i
\(396\) 0 0
\(397\) 20.0000i 1.00377i −0.864934 0.501886i \(-0.832640\pi\)
0.864934 0.501886i \(-0.167360\pi\)
\(398\) 0 0
\(399\) −24.0000 −1.20150
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 18.0000i 0.896644i
\(404\) 0 0
\(405\) −22.0000 + 11.0000i −1.09319 + 0.546594i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −21.0000 −1.03838 −0.519192 0.854658i \(-0.673767\pi\)
−0.519192 + 0.854658i \(0.673767\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 0 0
\(413\) 27.0000i 1.32858i
\(414\) 0 0
\(415\) −11.0000 22.0000i −0.539969 1.07994i
\(416\) 0 0
\(417\) 22.0000i 1.07734i
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 0 0
\(423\) 2.00000i 0.0972433i
\(424\) 0 0
\(425\) −28.0000 21.0000i −1.35820 1.01865i
\(426\) 0 0
\(427\) 6.00000i 0.290360i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 25.0000i 1.20142i −0.799466 0.600712i \(-0.794884\pi\)
0.799466 0.600712i \(-0.205116\pi\)
\(434\) 0 0
\(435\) −18.0000 36.0000i −0.863034 1.72607i
\(436\) 0 0
\(437\) 4.00000i 0.191346i
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 28.0000i 1.33032i −0.746701 0.665160i \(-0.768363\pi\)
0.746701 0.665160i \(-0.231637\pi\)
\(444\) 0 0
\(445\) 20.0000 10.0000i 0.948091 0.474045i
\(446\) 0 0
\(447\) 40.0000i 1.89194i
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −36.0000 + 18.0000i −1.68771 + 0.843853i
\(456\) 0 0
\(457\) 13.0000i 0.608114i −0.952654 0.304057i \(-0.901659\pi\)
0.952654 0.304057i \(-0.0983414\pi\)
\(458\) 0 0
\(459\) −28.0000 −1.30693
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) −6.00000 12.0000i −0.278243 0.556487i
\(466\) 0 0
\(467\) 29.0000i 1.34196i −0.741475 0.670980i \(-0.765874\pi\)
0.741475 0.670980i \(-0.234126\pi\)
\(468\) 0 0
\(469\) 39.0000 1.80085
\(470\) 0 0
\(471\) −26.0000 −1.19802
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12.0000 + 16.0000i −0.550598 + 0.734130i
\(476\) 0 0
\(477\) 7.00000i 0.320508i
\(478\) 0 0
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) 42.0000 1.91504
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 2.00000 + 4.00000i 0.0908153 + 0.181631i
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) 29.0000 1.30875 0.654376 0.756169i \(-0.272931\pi\)
0.654376 + 0.756169i \(0.272931\pi\)
\(492\) 0 0
\(493\) 63.0000i 2.83738i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 39.0000i 1.74939i
\(498\) 0 0
\(499\) −25.0000 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.0000i 0.490466i 0.969464 + 0.245233i \(0.0788644\pi\)
−0.969464 + 0.245233i \(0.921136\pi\)
\(504\) 0 0
\(505\) −26.0000 + 13.0000i −1.15698 + 0.578492i
\(506\) 0 0
\(507\) 46.0000i 2.04293i
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 0 0
\(513\) 16.0000i 0.706417i
\(514\) 0 0
\(515\) 16.0000 + 32.0000i 0.705044 + 1.41009i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 36.0000 1.58022
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 0 0
\(525\) 18.0000 24.0000i 0.785584 1.04745i
\(526\) 0 0
\(527\) 21.0000i 0.914774i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) 54.0000i 2.33900i
\(534\) 0 0
\(535\) 15.0000 + 30.0000i 0.648507 + 1.29701i
\(536\) 0 0
\(537\) 40.0000i 1.72613i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 20.0000i 0.858282i
\(544\) 0 0
\(545\) 4.00000 2.00000i 0.171341 0.0856706i
\(546\) 0 0
\(547\) 4.00000i 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) 6.00000i 0.255146i
\(554\) 0 0
\(555\) −28.0000 + 14.0000i −1.18853 + 0.594267i
\(556\) 0 0
\(557\) 15.0000i 0.635570i −0.948163 0.317785i \(-0.897061\pi\)
0.948163 0.317785i \(-0.102939\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 17.0000i 0.716465i −0.933632 0.358232i \(-0.883380\pi\)
0.933632 0.358232i \(-0.116620\pi\)
\(564\) 0 0
\(565\) −5.00000 10.0000i −0.210352 0.420703i
\(566\) 0 0
\(567\) 33.0000i 1.38587i
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) 32.0000i 1.33682i
\(574\) 0 0
\(575\) −4.00000 3.00000i −0.166812 0.125109i
\(576\) 0 0
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) 0 0
\(579\) 32.0000 1.32987
\(580\) 0 0
\(581\) 33.0000 1.36907
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.00000 12.0000i −0.248069 0.496139i
\(586\) 0 0
\(587\) 36.0000i 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 0 0
\(593\) 42.0000i 1.72473i 0.506284 + 0.862367i \(0.331019\pi\)
−0.506284 + 0.862367i \(0.668981\pi\)
\(594\) 0 0
\(595\) 42.0000 21.0000i 1.72183 0.860916i
\(596\) 0 0
\(597\) 36.0000i 1.47338i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 7.00000 0.285536 0.142768 0.989756i \(-0.454400\pi\)
0.142768 + 0.989756i \(0.454400\pi\)
\(602\) 0 0
\(603\) 13.0000i 0.529401i
\(604\) 0 0
\(605\) −22.0000 + 11.0000i −0.894427 + 0.447214i
\(606\) 0 0
\(607\) 30.0000i 1.21766i −0.793300 0.608831i \(-0.791639\pi\)
0.793300 0.608831i \(-0.208361\pi\)
\(608\) 0 0
\(609\) 54.0000 2.18819
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) 0 0
\(615\) −18.0000 36.0000i −0.725830 1.45166i
\(616\) 0 0
\(617\) 3.00000i 0.120775i 0.998175 + 0.0603877i \(0.0192337\pi\)
−0.998175 + 0.0603877i \(0.980766\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 30.0000i 1.20192i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −49.0000 −1.95376
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 22.0000i 0.874421i
\(634\) 0 0
\(635\) 8.00000 + 16.0000i 0.317470 + 0.634941i
\(636\) 0 0
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) −13.0000 −0.514272
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 5.00000i 0.197181i 0.995128 + 0.0985904i \(0.0314334\pi\)
−0.995128 + 0.0985904i \(0.968567\pi\)
\(644\) 0 0
\(645\) −16.0000 + 8.00000i −0.629999 + 0.315000i
\(646\) 0 0
\(647\) 2.00000i 0.0786281i −0.999227 0.0393141i \(-0.987483\pi\)
0.999227 0.0393141i \(-0.0125173\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) 0 0
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 0 0
\(655\) −16.0000 + 8.00000i −0.625172 + 0.312586i
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 84.0000i 3.26229i
\(664\) 0 0
\(665\) −12.0000 24.0000i −0.465340 0.930680i
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) 0 0
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.00000i 0.308377i 0.988041 + 0.154189i \(0.0492764\pi\)
−0.988041 + 0.154189i \(0.950724\pi\)
\(674\) 0 0
\(675\) −16.0000 12.0000i −0.615840 0.461880i
\(676\) 0 0
\(677\) 3.00000i 0.115299i 0.998337 + 0.0576497i \(0.0183606\pi\)
−0.998337 + 0.0576497i \(0.981639\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −32.0000 −1.22624
\(682\) 0 0
\(683\) 18.0000i 0.688751i 0.938832 + 0.344375i \(0.111909\pi\)
−0.938832 + 0.344375i \(0.888091\pi\)
\(684\) 0 0
\(685\) 2.00000 + 4.00000i 0.0764161 + 0.152832i
\(686\) 0 0
\(687\) 16.0000i 0.610438i
\(688\) 0 0
\(689\) −42.0000 −1.60007
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 22.0000 11.0000i 0.834508 0.417254i
\(696\) 0 0
\(697\) 63.0000i 2.38630i
\(698\) 0 0
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) 28.0000i 1.05604i
\(704\) 0 0
\(705\) 8.00000 4.00000i 0.301297 0.150649i
\(706\) 0 0
\(707\) 39.0000i 1.46675i
\(708\) 0 0
\(709\) 48.0000 1.80268 0.901339 0.433114i \(-0.142585\pi\)
0.901339 + 0.433114i \(0.142585\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) 3.00000i 0.112351i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 30.0000i 1.12037i
\(718\) 0 0
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) 44.0000i 1.63638i
\(724\) 0 0
\(725\) 27.0000 36.0000i 1.00275 1.33701i
\(726\) 0 0
\(727\) 1.00000i 0.0370879i 0.999828 + 0.0185440i \(0.00590307\pi\)
−0.999828 + 0.0185440i \(0.994097\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −28.0000 −1.03562
\(732\) 0 0
\(733\) 31.0000i 1.14501i 0.819901 + 0.572506i \(0.194029\pi\)
−0.819901 + 0.572506i \(0.805971\pi\)
\(734\) 0 0
\(735\) 4.00000 + 8.00000i 0.147542 + 0.295084i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) 0 0
\(741\) −48.0000 −1.76332
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) −40.0000 + 20.0000i −1.46549 + 0.732743i
\(746\) 0 0
\(747\) 11.0000i 0.402469i
\(748\) 0 0
\(749\) −45.0000 −1.64426
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.0000i 0.472493i −0.971693 0.236247i \(-0.924083\pi\)
0.971693 0.236247i \(-0.0759173\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) 6.00000i 0.217215i
\(764\) 0 0
\(765\) 7.00000 + 14.0000i 0.253086 + 0.506171i
\(766\) 0 0
\(767\) 54.0000i 1.94983i
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 52.0000 1.87273
\(772\) 0 0
\(773\) 26.0000i 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) 0 0
\(775\) 9.00000 12.0000i 0.323290 0.431053i
\(776\) 0 0
\(777\) 42.0000i 1.50674i
\(778\) 0 0
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 36.0000i 1.28654i
\(784\) 0 0
\(785\) −13.0000 26.0000i −0.463990 0.927980i
\(786\) 0 0
\(787\) 7.00000i 0.249523i 0.992187 + 0.124762i \(0.0398166\pi\)
−0.992187 + 0.124762i \(0.960183\pi\)
\(788\) 0 0
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 15.0000 0.533339
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 28.0000 14.0000i 0.993058 0.496529i
\(796\) 0 0
\(797\) 39.0000i 1.38145i 0.723117 + 0.690725i \(0.242709\pi\)
−0.723117 + 0.690725i \(0.757291\pi\)
\(798\) 0 0
\(799\) 14.0000 0.495284
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\)