Properties

Label 1560.2.l.a.1249.1
Level $1560$
Weight $2$
Character 1560.1249
Analytic conductor $12.457$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1560.1249
Dual form 1560.2.l.a.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -5.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -5.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -1.00000i q^{13} +(-2.00000 + 1.00000i) q^{15} -3.00000i q^{17} +4.00000 q^{19} -5.00000 q^{21} -5.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000i q^{27} +4.00000 q^{29} -5.00000i q^{33} +(-10.0000 + 5.00000i) q^{35} -7.00000i q^{37} -1.00000 q^{39} +11.0000 q^{41} +12.0000i q^{43} +(1.00000 + 2.00000i) q^{45} +6.00000i q^{47} -18.0000 q^{49} -3.00000 q^{51} -1.00000i q^{53} +(-5.00000 - 10.0000i) q^{55} -4.00000i q^{57} -12.0000 q^{59} -7.00000 q^{61} +5.00000i q^{63} +(-2.00000 + 1.00000i) q^{65} +4.00000i q^{67} -5.00000 q^{69} -7.00000 q^{71} +14.0000i q^{73} +(4.00000 + 3.00000i) q^{75} -25.0000i q^{77} +5.00000 q^{79} +1.00000 q^{81} +2.00000i q^{83} +(-6.00000 + 3.00000i) q^{85} -4.00000i q^{87} +3.00000 q^{89} -5.00000 q^{91} +(-4.00000 - 8.00000i) q^{95} -1.00000i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{9} + 10 q^{11} - 4 q^{15} + 8 q^{19} - 10 q^{21} - 6 q^{25} + 8 q^{29} - 20 q^{35} - 2 q^{39} + 22 q^{41} + 2 q^{45} - 36 q^{49} - 6 q^{51} - 10 q^{55} - 24 q^{59} - 14 q^{61} - 4 q^{65} - 10 q^{69} - 14 q^{71} + 8 q^{75} + 10 q^{79} + 2 q^{81} - 12 q^{85} + 6 q^{89} - 10 q^{91} - 8 q^{95} - 10 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −1.00000 2.00000i −0.447214 0.894427i
\(6\) 0 0
\(7\) 5.00000i 1.88982i −0.327327 0.944911i \(-0.606148\pi\)
0.327327 0.944911i \(-0.393852\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −2.00000 + 1.00000i −0.516398 + 0.258199i
\(16\) 0 0
\(17\) 3.00000i 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −5.00000 −1.09109
\(22\) 0 0
\(23\) 5.00000i 1.04257i −0.853382 0.521286i \(-0.825452\pi\)
0.853382 0.521286i \(-0.174548\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 5.00000i 0.870388i
\(34\) 0 0
\(35\) −10.0000 + 5.00000i −1.69031 + 0.845154i
\(36\) 0 0
\(37\) 7.00000i 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 0 0
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 0 0
\(45\) 1.00000 + 2.00000i 0.149071 + 0.298142i
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −18.0000 −2.57143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 1.00000i 0.137361i −0.997639 0.0686803i \(-0.978121\pi\)
0.997639 0.0686803i \(-0.0218788\pi\)
\(54\) 0 0
\(55\) −5.00000 10.0000i −0.674200 1.34840i
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 5.00000i 0.629941i
\(64\) 0 0
\(65\) −2.00000 + 1.00000i −0.248069 + 0.124035i
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −7.00000 −0.830747 −0.415374 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 4.00000 + 3.00000i 0.461880 + 0.346410i
\(76\) 0 0
\(77\) 25.0000i 2.84901i
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) −6.00000 + 3.00000i −0.650791 + 0.325396i
\(86\) 0 0
\(87\) 4.00000i 0.428845i
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 8.00000i −0.410391 0.820783i
\(96\) 0 0
\(97\) 1.00000i 0.101535i −0.998711 0.0507673i \(-0.983833\pi\)
0.998711 0.0507673i \(-0.0161667\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 0 0
\(105\) 5.00000 + 10.0000i 0.487950 + 0.975900i
\(106\) 0 0
\(107\) 11.0000i 1.06341i −0.846930 0.531705i \(-0.821551\pi\)
0.846930 0.531705i \(-0.178449\pi\)
\(108\) 0 0
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 0 0
\(113\) 14.0000i 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) −10.0000 + 5.00000i −0.932505 + 0.466252i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) −15.0000 −1.37505
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 11.0000i 0.991837i
\(124\) 0 0
\(125\) 11.0000 + 2.00000i 0.983870 + 0.178885i
\(126\) 0 0
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) 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
\(133\) 20.0000i 1.73422i
\(134\) 0 0
\(135\) 2.00000 1.00000i 0.172133 0.0860663i
\(136\) 0 0
\(137\) 14.0000i 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 5.00000i 0.418121i
\(144\) 0 0
\(145\) −4.00000 8.00000i −0.332182 0.664364i
\(146\) 0 0
\(147\) 18.0000i 1.48461i
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) 3.00000i 0.242536i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 0 0
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) −25.0000 −1.97028
\(162\) 0 0
\(163\) 7.00000i 0.548282i −0.961689 0.274141i \(-0.911606\pi\)
0.961689 0.274141i \(-0.0883936\pi\)
\(164\) 0 0
\(165\) −10.0000 + 5.00000i −0.778499 + 0.389249i
\(166\) 0 0
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 6.00000i 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 20.0000 + 15.0000i 1.51186 + 1.13389i
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −23.0000 −1.70958 −0.854788 0.518977i \(-0.826313\pi\)
−0.854788 + 0.518977i \(0.826313\pi\)
\(182\) 0 0
\(183\) 7.00000i 0.517455i
\(184\) 0 0
\(185\) −14.0000 + 7.00000i −1.02930 + 0.514650i
\(186\) 0 0
\(187\) 15.0000i 1.09691i
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) 23.0000i 1.65558i 0.561041 + 0.827788i \(0.310401\pi\)
−0.561041 + 0.827788i \(0.689599\pi\)
\(194\) 0 0
\(195\) 1.00000 + 2.00000i 0.0716115 + 0.143223i
\(196\) 0 0
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 20.0000i 1.40372i
\(204\) 0 0
\(205\) −11.0000 22.0000i −0.768273 1.53655i
\(206\) 0 0
\(207\) 5.00000i 0.347524i
\(208\) 0 0
\(209\) 20.0000 1.38343
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 7.00000i 0.479632i
\(214\) 0 0
\(215\) 24.0000 12.0000i 1.63679 0.818393i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) 24.0000i 1.60716i 0.595198 + 0.803579i \(0.297074\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(224\) 0 0
\(225\) 3.00000 4.00000i 0.200000 0.266667i
\(226\) 0 0
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −25.0000 −1.64488
\(232\) 0 0
\(233\) 9.00000i 0.589610i 0.955557 + 0.294805i \(0.0952546\pi\)
−0.955557 + 0.294805i \(0.904745\pi\)
\(234\) 0 0
\(235\) 12.0000 6.00000i 0.782794 0.391397i
\(236\) 0 0
\(237\) 5.00000i 0.324785i
\(238\) 0 0
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 18.0000 + 36.0000i 1.14998 + 2.29996i
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 25.0000i 1.57174i
\(254\) 0 0
\(255\) 3.00000 + 6.00000i 0.187867 + 0.375735i
\(256\) 0 0
\(257\) 6.00000i 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) −35.0000 −2.17479
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 0 0
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) −2.00000 + 1.00000i −0.122859 + 0.0614295i
\(266\) 0 0
\(267\) 3.00000i 0.183597i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) 5.00000i 0.302614i
\(274\) 0 0
\(275\) −15.0000 + 20.0000i −0.904534 + 1.20605i
\(276\) 0 0
\(277\) 30.0000i 1.80253i −0.433273 0.901263i \(-0.642641\pi\)
0.433273 0.901263i \(-0.357359\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 18.0000i 1.06999i 0.844856 + 0.534994i \(0.179686\pi\)
−0.844856 + 0.534994i \(0.820314\pi\)
\(284\) 0 0
\(285\) −8.00000 + 4.00000i −0.473879 + 0.236940i
\(286\) 0 0
\(287\) 55.0000i 3.24655i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) 0 0
\(293\) 20.0000i 1.16841i −0.811605 0.584206i \(-0.801406\pi\)
0.811605 0.584206i \(-0.198594\pi\)
\(294\) 0 0
\(295\) 12.0000 + 24.0000i 0.698667 + 1.39733i
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) 0 0
\(299\) −5.00000 −0.289157
\(300\) 0 0
\(301\) 60.0000 3.45834
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) 7.00000 + 14.0000i 0.400819 + 0.801638i
\(306\) 0 0
\(307\) 25.0000i 1.42683i −0.700744 0.713413i \(-0.747149\pi\)
0.700744 0.713413i \(-0.252851\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 20.0000i 1.13047i 0.824931 + 0.565233i \(0.191214\pi\)
−0.824931 + 0.565233i \(0.808786\pi\)
\(314\) 0 0
\(315\) 10.0000 5.00000i 0.563436 0.281718i
\(316\) 0 0
\(317\) 24.0000i 1.34797i 0.738743 + 0.673987i \(0.235420\pi\)
−0.738743 + 0.673987i \(0.764580\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) −11.0000 −0.613960
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 4.00000 + 3.00000i 0.221880 + 0.166410i
\(326\) 0 0
\(327\) 12.0000i 0.663602i
\(328\) 0 0
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 7.00000i 0.383598i
\(334\) 0 0
\(335\) 8.00000 4.00000i 0.437087 0.218543i
\(336\) 0 0
\(337\) 8.00000i 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 0 0
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 55.0000i 2.96972i
\(344\) 0 0
\(345\) 5.00000 + 10.0000i 0.269191 + 0.538382i
\(346\) 0 0
\(347\) 7.00000i 0.375780i −0.982190 0.187890i \(-0.939835\pi\)
0.982190 0.187890i \(-0.0601648\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 30.0000i 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 7.00000 + 14.0000i 0.371521 + 0.743043i
\(356\) 0 0
\(357\) 15.0000i 0.793884i
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 28.0000 14.0000i 1.46559 0.732793i
\(366\) 0 0
\(367\) 22.0000i 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) 0 0
\(369\) −11.0000 −0.572637
\(370\) 0 0
\(371\) −5.00000 −0.259587
\(372\) 0 0
\(373\) 14.0000i 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 0 0
\(375\) 2.00000 11.0000i 0.103280 0.568038i
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 0 0
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) 0 0
\(385\) −50.0000 + 25.0000i −2.54824 + 1.27412i
\(386\) 0 0
\(387\) 12.0000i 0.609994i
\(388\) 0 0
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) −15.0000 −0.758583
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) −5.00000 10.0000i −0.251577 0.503155i
\(396\) 0 0
\(397\) 3.00000i 0.150566i −0.997162 0.0752828i \(-0.976014\pi\)
0.997162 0.0752828i \(-0.0239860\pi\)
\(398\) 0 0
\(399\) −20.0000 −1.00125
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 2.00000i −0.0496904 0.0993808i
\(406\) 0 0
\(407\) 35.0000i 1.73489i
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 0 0
\(413\) 60.0000i 2.95241i
\(414\) 0 0
\(415\) 4.00000 2.00000i 0.196352 0.0981761i
\(416\) 0 0
\(417\) 11.0000i 0.538672i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 12.0000 + 9.00000i 0.582086 + 0.436564i
\(426\) 0 0
\(427\) 35.0000i 1.69377i
\(428\) 0 0
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) 4.00000 0.192673 0.0963366 0.995349i \(-0.469287\pi\)
0.0963366 + 0.995349i \(0.469287\pi\)
\(432\) 0 0
\(433\) 18.0000i 0.865025i 0.901628 + 0.432512i \(0.142373\pi\)
−0.901628 + 0.432512i \(0.857627\pi\)
\(434\) 0 0
\(435\) −8.00000 + 4.00000i −0.383571 + 0.191785i
\(436\) 0 0
\(437\) 20.0000i 0.956730i
\(438\) 0 0
\(439\) 27.0000 1.28864 0.644320 0.764756i \(-0.277141\pi\)
0.644320 + 0.764756i \(0.277141\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 21.0000i 0.997740i −0.866677 0.498870i \(-0.833748\pi\)
0.866677 0.498870i \(-0.166252\pi\)
\(444\) 0 0
\(445\) −3.00000 6.00000i −0.142214 0.284427i
\(446\) 0 0
\(447\) 15.0000i 0.709476i
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 55.0000 2.58985
\(452\) 0 0
\(453\) 2.00000i 0.0939682i
\(454\) 0 0
\(455\) 5.00000 + 10.0000i 0.234404 + 0.468807i
\(456\) 0 0
\(457\) 39.0000i 1.82434i 0.409809 + 0.912172i \(0.365595\pi\)
−0.409809 + 0.912172i \(0.634405\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) 1.00000i 0.0464739i 0.999730 + 0.0232370i \(0.00739722\pi\)
−0.999730 + 0.0232370i \(0.992603\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15.0000i 0.694117i −0.937843 0.347059i \(-0.887180\pi\)
0.937843 0.347059i \(-0.112820\pi\)
\(468\) 0 0
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 60.0000i 2.75880i
\(474\) 0 0
\(475\) −12.0000 + 16.0000i −0.550598 + 0.734130i
\(476\) 0 0
\(477\) 1.00000i 0.0457869i
\(478\) 0 0
\(479\) −11.0000 −0.502603 −0.251301 0.967909i \(-0.580859\pi\)
−0.251301 + 0.967909i \(0.580859\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 0 0
\(483\) 25.0000i 1.13754i
\(484\) 0 0
\(485\) −2.00000 + 1.00000i −0.0908153 + 0.0454077i
\(486\) 0 0
\(487\) 23.0000i 1.04223i −0.853487 0.521115i \(-0.825516\pi\)
0.853487 0.521115i \(-0.174484\pi\)
\(488\) 0 0
\(489\) −7.00000 −0.316551
\(490\) 0 0
\(491\) −22.0000 −0.992846 −0.496423 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 5.00000 + 10.0000i 0.224733 + 0.449467i
\(496\) 0 0
\(497\) 35.0000i 1.56996i
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 0 0
\(503\) 8.00000i 0.356702i 0.983967 + 0.178351i \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) 0 0
\(505\) 6.00000 + 12.0000i 0.266996 + 0.533993i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −13.0000 −0.576215 −0.288107 0.957598i \(-0.593026\pi\)
−0.288107 + 0.957598i \(0.593026\pi\)
\(510\) 0 0
\(511\) 70.0000 3.09662
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) 28.0000 14.0000i 1.23383 0.616914i
\(516\) 0 0
\(517\) 30.0000i 1.31940i
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 38.0000i 1.66162i −0.556553 0.830812i \(-0.687876\pi\)
0.556553 0.830812i \(-0.312124\pi\)
\(524\) 0 0
\(525\) 15.0000 20.0000i 0.654654 0.872872i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 11.0000i 0.476463i
\(534\) 0 0
\(535\) −22.0000 + 11.0000i −0.951143 + 0.475571i
\(536\) 0 0
\(537\) 10.0000i 0.431532i
\(538\) 0 0
\(539\) −90.0000 −3.87657
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) 23.0000i 0.987024i
\(544\) 0 0
\(545\) −12.0000 24.0000i −0.514024 1.02805i
\(546\) 0 0
\(547\) 20.0000i 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 7.00000 0.298753
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 25.0000i 1.06311i
\(554\) 0 0
\(555\) 7.00000 + 14.0000i 0.297133 + 0.594267i
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 39.0000i 1.64365i 0.569737 + 0.821827i \(0.307045\pi\)
−0.569737 + 0.821827i \(0.692955\pi\)
\(564\) 0 0
\(565\) −28.0000 + 14.0000i −1.17797 + 0.588984i
\(566\) 0 0
\(567\) 5.00000i 0.209980i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 43.0000 1.79949 0.899747 0.436412i \(-0.143751\pi\)
0.899747 + 0.436412i \(0.143751\pi\)
\(572\) 0 0
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 20.0000 + 15.0000i 0.834058 + 0.625543i
\(576\) 0 0
\(577\) 17.0000i 0.707719i −0.935299 0.353860i \(-0.884869\pi\)
0.935299 0.353860i \(-0.115131\pi\)
\(578\) 0 0
\(579\) 23.0000 0.955847
\(580\) 0 0
\(581\) 10.0000 0.414870
\(582\) 0 0
\(583\) 5.00000i 0.207079i
\(584\) 0 0
\(585\) 2.00000 1.00000i 0.0826898 0.0413449i
\(586\) 0 0
\(587\) 34.0000i 1.40333i 0.712507 + 0.701665i \(0.247560\pi\)
−0.712507 + 0.701665i \(0.752440\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 16.0000i 0.657041i −0.944497 0.328521i \(-0.893450\pi\)
0.944497 0.328521i \(-0.106550\pi\)
\(594\) 0 0
\(595\) 15.0000 + 30.0000i 0.614940 + 1.22988i
\(596\) 0 0
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) −34.0000 −1.38920 −0.694601 0.719395i \(-0.744419\pi\)
−0.694601 + 0.719395i \(0.744419\pi\)
\(600\) 0 0
\(601\) 15.0000 0.611863 0.305931 0.952054i \(-0.401032\pi\)
0.305931 + 0.952054i \(0.401032\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) −14.0000 28.0000i −0.569181 1.13836i
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 0 0
\(609\) −20.0000 −0.810441
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 11.0000i 0.444286i −0.975014 0.222143i \(-0.928695\pi\)
0.975014 0.222143i \(-0.0713052\pi\)
\(614\) 0 0
\(615\) −22.0000 + 11.0000i −0.887126 + 0.443563i
\(616\) 0 0
\(617\) 4.00000i 0.161034i −0.996753 0.0805170i \(-0.974343\pi\)
0.996753 0.0805170i \(-0.0256571\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) 15.0000i 0.600962i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 20.0000i 0.798723i
\(628\) 0 0
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 12.0000 6.00000i 0.476205 0.238103i
\(636\) 0 0
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) 7.00000 0.276916
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) 0 0
\(643\) 39.0000i 1.53801i −0.639243 0.769005i \(-0.720752\pi\)
0.639243 0.769005i \(-0.279248\pi\)
\(644\) 0 0
\(645\) −12.0000 24.0000i −0.472500 0.944999i
\(646\) 0 0
\(647\) 29.0000i 1.14011i −0.821607 0.570054i \(-0.806922\pi\)
0.821607 0.570054i \(-0.193078\pi\)
\(648\) 0 0
\(649\) −60.0000 −2.35521
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000i 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) −12.0000 24.0000i −0.468879 0.937758i
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 0 0
\(663\) 3.00000i 0.116510i
\(664\) 0 0
\(665\) −40.0000 + 20.0000i −1.55113 + 0.775567i
\(666\) 0 0
\(667\) 20.0000i 0.774403i
\(668\) 0 0
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) −35.0000 −1.35116
\(672\) 0 0
\(673\) 10.0000i 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) 0 0
\(675\) −4.00000 3.00000i −0.153960 0.115470i
\(676\) 0 0
\(677\) 15.0000i 0.576497i −0.957556 0.288248i \(-0.906927\pi\)
0.957556 0.288248i \(-0.0930729\pi\)
\(678\) 0 0
\(679\) −5.00000 −0.191882
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 14.0000i 0.535695i −0.963461 0.267848i \(-0.913688\pi\)
0.963461 0.267848i \(-0.0863124\pi\)
\(684\) 0 0
\(685\) −28.0000 + 14.0000i −1.06983 + 0.534913i
\(686\) 0 0
\(687\) 6.00000i 0.228914i
\(688\) 0 0
\(689\) −1.00000 −0.0380970
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 0 0
\(693\) 25.0000i 0.949671i
\(694\) 0 0
\(695\) −11.0000 22.0000i −0.417254 0.834508i
\(696\) 0 0
\(697\) 33.0000i 1.24996i
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) 4.00000 0.151078 0.0755390 0.997143i \(-0.475932\pi\)
0.0755390 + 0.997143i \(0.475932\pi\)
\(702\) 0 0
\(703\) 28.0000i 1.05604i
\(704\) 0 0
\(705\) −6.00000 12.0000i −0.225973 0.451946i
\(706\) 0 0
\(707\) 30.0000i 1.12827i
\(708\) 0 0
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −10.0000 + 5.00000i −0.373979 + 0.186989i
\(716\) 0 0
\(717\) 5.00000i 0.186728i
\(718\) 0 0
\(719\) −22.0000 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(720\) 0 0
\(721\) 70.0000 2.60694
\(722\) 0 0
\(723\) 4.00000i 0.148762i
\(724\) 0 0
\(725\) −12.0000 + 16.0000i −0.445669 + 0.594225i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) 3.00000i 0.110808i 0.998464 + 0.0554038i \(0.0176446\pi\)
−0.998464 + 0.0554038i \(0.982355\pi\)
\(734\) 0 0
\(735\) 36.0000 18.0000i 1.32788 0.663940i
\(736\) 0 0
\(737\) 20.0000i 0.736709i
\(738\) 0 0
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 36.0000i 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) −15.0000 30.0000i −0.549557 1.09911i
\(746\) 0 0
\(747\) 2.00000i 0.0731762i
\(748\) 0 0
\(749\) −55.0000 −2.00966
\(750\) 0 0
\(751\) 17.0000 0.620339 0.310169 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) 0 0
\(753\) 2.00000i 0.0728841i
\(754\) 0 0
\(755\) 2.00000 + 4.00000i 0.0727875 + 0.145575i
\(756\) 0 0
\(757\) 22.0000i 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 0 0
\(759\) −25.0000 −0.907443
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 60.0000i 2.17215i
\(764\) 0 0
\(765\) 6.00000 3.00000i 0.216930 0.108465i
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) 2.00000i 0.0719350i −0.999353 0.0359675i \(-0.988549\pi\)
0.999353 0.0359675i \(-0.0114513\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 35.0000i 1.25562i
\(778\) 0 0
\(779\) 44.0000 1.57646
\(780\) 0 0
\(781\) −35.0000 −1.25240
\(782\) 0 0
\(783\) 4.00000i 0.142948i
\(784\) 0 0
\(785\) 8.00000 4.00000i 0.285532 0.142766i
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −70.0000 −2.48891
\(792\) 0 0
\(793\) 7.00000i 0.248577i
\(794\) 0 0
\(795\) 1.00000 + 2.00000i 0.0354663 + 0.0709327i
\(796\) 0 0
\(797\) 45.0000i 1.59398i −0.603991 0.796991i \(-0.706424\pi\)
0.603991 0.796991i \(-0.293576\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) −3.00000 −0.106000