Properties

Label 60.2.h.a.59.1
Level $60$
Weight $2$
Character 60.59
Analytic conductor $0.479$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 59.1
Root \(1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 60.59
Dual form 60.2.h.a.59.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} +(-1.58114 - 0.707107i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-1.00000 + 2.23607i) q^{6} +3.16228 q^{7} +2.82843i q^{8} +(2.00000 + 2.23607i) q^{9} +O(q^{10})\) \(q-1.41421i q^{2} +(-1.58114 - 0.707107i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-1.00000 + 2.23607i) q^{6} +3.16228 q^{7} +2.82843i q^{8} +(2.00000 + 2.23607i) q^{9} -3.16228 q^{10} +(3.16228 + 1.41421i) q^{12} -4.47214i q^{14} +(-1.58114 + 3.53553i) q^{15} +4.00000 q^{16} +(3.16228 - 2.82843i) q^{18} +4.47214i q^{20} +(-5.00000 - 2.23607i) q^{21} +1.41421i q^{23} +(2.00000 - 4.47214i) q^{24} -5.00000 q^{25} +(-1.58114 - 4.94975i) q^{27} -6.32456 q^{28} +8.94427i q^{29} +(5.00000 + 2.23607i) q^{30} -5.65685i q^{32} -7.07107i q^{35} +(-4.00000 - 4.47214i) q^{36} +6.32456 q^{40} -4.47214i q^{41} +(-3.16228 + 7.07107i) q^{42} +3.16228 q^{43} +(5.00000 - 4.47214i) q^{45} +2.00000 q^{46} +9.89949i q^{47} +(-6.32456 - 2.82843i) q^{48} +3.00000 q^{49} +7.07107i q^{50} +(-7.00000 + 2.23607i) q^{54} +8.94427i q^{56} +12.6491 q^{58} +(3.16228 - 7.07107i) q^{60} +8.00000 q^{61} +(6.32456 + 7.07107i) q^{63} -8.00000 q^{64} -15.8114 q^{67} +(1.00000 - 2.23607i) q^{69} -10.0000 q^{70} +(-6.32456 + 5.65685i) q^{72} +(7.90569 + 3.53553i) q^{75} -8.94427i q^{80} +(-1.00000 + 8.94427i) q^{81} -6.32456 q^{82} -15.5563i q^{83} +(10.0000 + 4.47214i) q^{84} -4.47214i q^{86} +(6.32456 - 14.1421i) q^{87} -17.8885i q^{89} +(-6.32456 - 7.07107i) q^{90} -2.82843i q^{92} +14.0000 q^{94} +(-4.00000 + 8.94427i) q^{96} -4.24264i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 4q^{6} + 8q^{9} + O(q^{10}) \) \( 4q - 8q^{4} - 4q^{6} + 8q^{9} + 16q^{16} - 20q^{21} + 8q^{24} - 20q^{25} + 20q^{30} - 16q^{36} + 20q^{45} + 8q^{46} + 12q^{49} - 28q^{54} + 32q^{61} - 32q^{64} + 4q^{69} - 40q^{70} - 4q^{81} + 40q^{84} + 56q^{94} - 16q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) −1.58114 0.707107i −0.912871 0.408248i
\(4\) −2.00000 −1.00000
\(5\) 2.23607i 1.00000i
\(6\) −1.00000 + 2.23607i −0.408248 + 0.912871i
\(7\) 3.16228 1.19523 0.597614 0.801784i \(-0.296115\pi\)
0.597614 + 0.801784i \(0.296115\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 2.00000 + 2.23607i 0.666667 + 0.745356i
\(10\) −3.16228 −1.00000
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 3.16228 + 1.41421i 0.912871 + 0.408248i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 4.47214i 1.19523i
\(15\) −1.58114 + 3.53553i −0.408248 + 0.912871i
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 3.16228 2.82843i 0.745356 0.666667i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 4.47214i 1.00000i
\(21\) −5.00000 2.23607i −1.09109 0.487950i
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 2.00000 4.47214i 0.408248 0.912871i
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.58114 4.94975i −0.304290 0.952579i
\(28\) −6.32456 −1.19523
\(29\) 8.94427i 1.66091i 0.557086 + 0.830455i \(0.311919\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 5.00000 + 2.23607i 0.912871 + 0.408248i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 7.07107i 1.19523i
\(36\) −4.00000 4.47214i −0.666667 0.745356i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 6.32456 1.00000
\(41\) 4.47214i 0.698430i −0.937043 0.349215i \(-0.886448\pi\)
0.937043 0.349215i \(-0.113552\pi\)
\(42\) −3.16228 + 7.07107i −0.487950 + 1.09109i
\(43\) 3.16228 0.482243 0.241121 0.970495i \(-0.422485\pi\)
0.241121 + 0.970495i \(0.422485\pi\)
\(44\) 0 0
\(45\) 5.00000 4.47214i 0.745356 0.666667i
\(46\) 2.00000 0.294884
\(47\) 9.89949i 1.44399i 0.691898 + 0.721995i \(0.256775\pi\)
−0.691898 + 0.721995i \(0.743225\pi\)
\(48\) −6.32456 2.82843i −0.912871 0.408248i
\(49\) 3.00000 0.428571
\(50\) 7.07107i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −7.00000 + 2.23607i −0.952579 + 0.304290i
\(55\) 0 0
\(56\) 8.94427i 1.19523i
\(57\) 0 0
\(58\) 12.6491 1.66091
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 3.16228 7.07107i 0.408248 0.912871i
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 6.32456 + 7.07107i 0.796819 + 0.890871i
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −15.8114 −1.93167 −0.965834 0.259161i \(-0.916554\pi\)
−0.965834 + 0.259161i \(0.916554\pi\)
\(68\) 0 0
\(69\) 1.00000 2.23607i 0.120386 0.269191i
\(70\) −10.0000 −1.19523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −6.32456 + 5.65685i −0.745356 + 0.666667i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 7.90569 + 3.53553i 0.912871 + 0.408248i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.94427i 1.00000i
\(81\) −1.00000 + 8.94427i −0.111111 + 0.993808i
\(82\) −6.32456 −0.698430
\(83\) 15.5563i 1.70753i −0.520658 0.853766i \(-0.674313\pi\)
0.520658 0.853766i \(-0.325687\pi\)
\(84\) 10.0000 + 4.47214i 1.09109 + 0.487950i
\(85\) 0 0
\(86\) 4.47214i 0.482243i
\(87\) 6.32456 14.1421i 0.678064 1.51620i
\(88\) 0 0
\(89\) 17.8885i 1.89618i −0.317999 0.948091i \(-0.603011\pi\)
0.317999 0.948091i \(-0.396989\pi\)
\(90\) −6.32456 7.07107i −0.666667 0.745356i
\(91\) 0 0
\(92\) 2.82843i 0.294884i
\(93\) 0 0
\(94\) 14.0000 1.44399
\(95\) 0 0
\(96\) −4.00000 + 8.94427i −0.408248 + 0.912871i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 4.24264i 0.428571i
\(99\) 0 0
\(100\) 10.0000 1.00000
\(101\) 8.94427i 0.889988i 0.895533 + 0.444994i \(0.146794\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −15.8114 −1.55794 −0.778971 0.627060i \(-0.784258\pi\)
−0.778971 + 0.627060i \(0.784258\pi\)
\(104\) 0 0
\(105\) −5.00000 + 11.1803i −0.487950 + 1.09109i
\(106\) 0 0
\(107\) 18.3848i 1.77732i 0.458563 + 0.888662i \(0.348364\pi\)
−0.458563 + 0.888662i \(0.651636\pi\)
\(108\) 3.16228 + 9.89949i 0.304290 + 0.952579i
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12.6491 1.19523
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 3.16228 0.294884
\(116\) 17.8885i 1.66091i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −10.0000 4.47214i −0.912871 0.408248i
\(121\) −11.0000 −1.00000
\(122\) 11.3137i 1.02430i
\(123\) −3.16228 + 7.07107i −0.285133 + 0.637577i
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 10.0000 8.94427i 0.890871 0.796819i
\(127\) 22.1359 1.96425 0.982124 0.188237i \(-0.0602772\pi\)
0.982124 + 0.188237i \(0.0602772\pi\)
\(128\) 11.3137i 1.00000i
\(129\) −5.00000 2.23607i −0.440225 0.196875i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 22.3607i 1.93167i
\(135\) −11.0680 + 3.53553i −0.952579 + 0.304290i
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −3.16228 1.41421i −0.269191 0.120386i
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 14.1421i 1.19523i
\(141\) 7.00000 15.6525i 0.589506 1.31818i
\(142\) 0 0
\(143\) 0 0
\(144\) 8.00000 + 8.94427i 0.666667 + 0.745356i
\(145\) 20.0000 1.66091
\(146\) 0 0
\(147\) −4.74342 2.12132i −0.391230 0.174964i
\(148\) 0 0
\(149\) 4.47214i 0.366372i −0.983078 0.183186i \(-0.941359\pi\)
0.983078 0.183186i \(-0.0586410\pi\)
\(150\) 5.00000 11.1803i 0.408248 0.912871i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −12.6491 −1.00000
\(161\) 4.47214i 0.352454i
\(162\) 12.6491 + 1.41421i 0.993808 + 0.111111i
\(163\) 22.1359 1.73382 0.866910 0.498464i \(-0.166102\pi\)
0.866910 + 0.498464i \(0.166102\pi\)
\(164\) 8.94427i 0.698430i
\(165\) 0 0
\(166\) −22.0000 −1.70753
\(167\) 24.0416i 1.86040i −0.367057 0.930199i \(-0.619634\pi\)
0.367057 0.930199i \(-0.380366\pi\)
\(168\) 6.32456 14.1421i 0.487950 1.09109i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −6.32456 −0.482243
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) −20.0000 8.94427i −1.51620 0.678064i
\(175\) −15.8114 −1.19523
\(176\) 0 0
\(177\) 0 0
\(178\) −25.2982 −1.89618
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −10.0000 + 8.94427i −0.745356 + 0.666667i
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −12.6491 5.65685i −0.935049 0.418167i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 19.7990i 1.44399i
\(189\) −5.00000 15.6525i −0.363696 1.13855i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 12.6491 + 5.65685i 0.912871 + 0.408248i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 14.1421i 1.00000i
\(201\) 25.0000 + 11.1803i 1.76336 + 0.788600i
\(202\) 12.6491 0.889988
\(203\) 28.2843i 1.98517i
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 22.3607i 1.55794i
\(207\) −3.16228 + 2.82843i −0.219793 + 0.196589i
\(208\) 0 0
\(209\) 0 0
\(210\) 15.8114 + 7.07107i 1.09109 + 0.487950i
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 26.0000 1.77732
\(215\) 7.07107i 0.482243i
\(216\) 14.0000 4.47214i 0.952579 0.304290i
\(217\) 0 0
\(218\) 22.6274i 1.53252i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.16228 0.211762 0.105881 0.994379i \(-0.466234\pi\)
0.105881 + 0.994379i \(0.466234\pi\)
\(224\) 17.8885i 1.19523i
\(225\) −10.0000 11.1803i −0.666667 0.745356i
\(226\) 0 0
\(227\) 9.89949i 0.657053i 0.944495 + 0.328526i \(0.106552\pi\)
−0.944495 + 0.328526i \(0.893448\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 4.47214i 0.294884i
\(231\) 0 0
\(232\) −25.2982 −1.66091
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 22.1359 1.44399
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −6.32456 + 14.1421i −0.408248 + 0.912871i
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 15.5563i 1.00000i
\(243\) 7.90569 13.4350i 0.507151 0.861858i
\(244\) −16.0000 −1.02430
\(245\) 6.70820i 0.428571i
\(246\) 10.0000 + 4.47214i 0.637577 + 0.285133i
\(247\) 0 0
\(248\) 0 0
\(249\) −11.0000 + 24.5967i −0.697097 + 1.55876i
\(250\) 15.8114 1.00000
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −12.6491 14.1421i −0.796819 0.890871i
\(253\) 0 0
\(254\) 31.3050i 1.96425i
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) −3.16228 + 7.07107i −0.196875 + 0.440225i
\(259\) 0 0
\(260\) 0 0
\(261\) −20.0000 + 17.8885i −1.23797 + 1.10727i
\(262\) 0 0
\(263\) 15.5563i 0.959246i −0.877475 0.479623i \(-0.840774\pi\)
0.877475 0.479623i \(-0.159226\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.6491 + 28.2843i −0.774113 + 1.73097i
\(268\) 31.6228 1.93167
\(269\) 22.3607i 1.36335i 0.731653 + 0.681677i \(0.238749\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 5.00000 + 15.6525i 0.304290 + 0.952579i
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −2.00000 + 4.47214i −0.120386 + 0.269191i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 20.0000 1.19523
\(281\) 31.3050i 1.86750i −0.357930 0.933748i \(-0.616517\pi\)
0.357930 0.933748i \(-0.383483\pi\)
\(282\) −22.1359 9.89949i −1.31818 0.589506i
\(283\) −15.8114 −0.939889 −0.469945 0.882696i \(-0.655726\pi\)
−0.469945 + 0.882696i \(0.655726\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.1421i 0.834784i
\(288\) 12.6491 11.3137i 0.745356 0.666667i
\(289\) −17.0000 −1.00000
\(290\) 28.2843i 1.66091i
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −3.00000 + 6.70820i −0.174964 + 0.391230i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −6.32456 −0.366372
\(299\) 0 0
\(300\) −15.8114 7.07107i −0.912871 0.408248i
\(301\) 10.0000 0.576390
\(302\) 0 0
\(303\) 6.32456 14.1421i 0.363336 0.812444i
\(304\) 0 0
\(305\) 17.8885i 1.02430i
\(306\) 0 0
\(307\) −34.7851 −1.98529 −0.992644 0.121070i \(-0.961367\pi\)
−0.992644 + 0.121070i \(0.961367\pi\)
\(308\) 0 0
\(309\) 25.0000 + 11.1803i 1.42220 + 0.636027i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 15.8114 14.1421i 0.890871 0.796819i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 17.8885i 1.00000i
\(321\) 13.0000 29.0689i 0.725589 1.62247i
\(322\) 6.32456 0.352454
\(323\) 0 0
\(324\) 2.00000 17.8885i 0.111111 0.993808i
\(325\) 0 0
\(326\) 31.3050i 1.73382i
\(327\) 25.2982 + 11.3137i 1.39899 + 0.625650i
\(328\) 12.6491 0.698430
\(329\) 31.3050i 1.72590i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 31.1127i 1.70753i
\(333\) 0 0
\(334\) −34.0000 −1.86040
\(335\) 35.3553i 1.93167i
\(336\) −20.0000 8.94427i −1.09109 0.487950i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 18.3848i 1.00000i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −12.6491 −0.682988
\(344\) 8.94427i 0.482243i
\(345\) −5.00000 2.23607i −0.269191 0.120386i
\(346\) 0 0
\(347\) 24.0416i 1.29062i −0.763920 0.645311i \(-0.776728\pi\)
0.763920 0.645311i \(-0.223272\pi\)
\(348\) −12.6491 + 28.2843i −0.678064 + 1.51620i
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 22.3607i 1.19523i
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 35.7771i 1.89618i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 12.6491 + 14.1421i 0.666667 + 0.745356i
\(361\) 19.0000 1.00000
\(362\) 2.82843i 0.148659i
\(363\) 17.3925 + 7.77817i 0.912871 + 0.408248i
\(364\) 0 0
\(365\) 0 0
\(366\) −8.00000 + 17.8885i −0.418167 + 0.935049i
\(367\) 3.16228 0.165070 0.0825348 0.996588i \(-0.473698\pi\)
0.0825348 + 0.996588i \(0.473698\pi\)
\(368\) 5.65685i 0.294884i
\(369\) 10.0000 8.94427i 0.520579 0.465620i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 7.90569 17.6777i 0.408248 0.912871i
\(376\) −28.0000 −1.44399
\(377\) 0 0
\(378\) −22.1359 + 7.07107i −1.13855 + 0.363696i
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −35.0000 15.6525i −1.79310 0.801901i
\(382\) 0 0
\(383\) 26.8701i 1.37300i 0.727132 + 0.686498i \(0.240853\pi\)
−0.727132 + 0.686498i \(0.759147\pi\)
\(384\) 8.00000 17.8885i 0.408248 0.912871i
\(385\) 0 0
\(386\) 0 0
\(387\) 6.32456 + 7.07107i 0.321495 + 0.359443i
\(388\) 0 0
\(389\) 31.3050i 1.58722i −0.608424 0.793612i \(-0.708198\pi\)
0.608424 0.793612i \(-0.291802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 8.48528i 0.428571i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −20.0000 −1.00000
\(401\) 35.7771i 1.78662i 0.449439 + 0.893311i \(0.351624\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 15.8114 35.3553i 0.788600 1.76336i
\(403\) 0 0
\(404\) 17.8885i 0.889988i
\(405\) 20.0000 + 2.23607i 0.993808 + 0.111111i
\(406\) 40.0000 1.98517
\(407\) 0 0
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 14.1421i 0.698430i
\(411\) 0 0
\(412\) 31.6228 1.55794
\(413\) 0 0
\(414\) 4.00000 + 4.47214i 0.196589 + 0.219793i
\(415\) −34.7851 −1.70753
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 10.0000 22.3607i 0.487950 1.09109i
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) −22.1359 + 19.7990i −1.07629 + 0.962660i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 25.2982 1.22427
\(428\) 36.7696i 1.77732i
\(429\) 0 0
\(430\) −10.0000 −0.482243
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −6.32456 19.7990i −0.304290 0.952579i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −31.6228 14.1421i −1.51620 0.678064i
\(436\) 32.0000 1.53252
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 6.00000 + 6.70820i 0.285714 + 0.319438i
\(442\) 0 0
\(443\) 41.0122i 1.94855i −0.225367 0.974274i \(-0.572358\pi\)
0.225367 0.974274i \(-0.427642\pi\)
\(444\) 0 0
\(445\) −40.0000 −1.89618
\(446\) 4.47214i 0.211762i
\(447\) −3.16228 + 7.07107i −0.149571 + 0.334450i
\(448\) −25.2982 −1.19523
\(449\) 22.3607i 1.05527i 0.849473 + 0.527633i \(0.176920\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) −15.8114 + 14.1421i −0.745356 + 0.666667i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 14.0000 0.657053
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 19.7990i 0.925146i
\(459\) 0 0
\(460\) −6.32456 −0.294884
\(461\) 8.94427i 0.416576i 0.978068 + 0.208288i \(0.0667892\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) 41.1096 1.91053 0.955263 0.295758i \(-0.0955723\pi\)
0.955263 + 0.295758i \(0.0955723\pi\)
\(464\) 35.7771i 1.66091i
\(465\) 0 0
\(466\) 0 0
\(467\) 32.5269i 1.50517i −0.658497 0.752583i \(-0.728808\pi\)
0.658497 0.752583i \(-0.271192\pi\)
\(468\) 0 0
\(469\) −50.0000 −2.30879
\(470\) 31.3050i 1.44399i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 20.0000 + 8.94427i 0.912871 + 0.408248i
\(481\) 0 0
\(482\) 39.5980i 1.80364i
\(483\) 3.16228 7.07107i 0.143889 0.321745i
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) −19.0000 11.1803i −0.861858 0.507151i
\(487\) 22.1359 1.00308 0.501538 0.865136i \(-0.332768\pi\)
0.501538 + 0.865136i \(0.332768\pi\)
\(488\) 22.6274i 1.02430i
\(489\) −35.0000 15.6525i −1.58275 0.707829i
\(490\) −9.48683 −0.428571
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.32456 14.1421i 0.285133 0.637577i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 34.7851 + 15.5563i 1.55876 + 0.697097i
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 22.3607i 1.00000i
\(501\) −17.0000 + 38.0132i −0.759504 + 1.69830i
\(502\) 0 0
\(503\) 43.8406i 1.95476i 0.211498 + 0.977378i \(0.432166\pi\)
−0.211498 + 0.977378i \(0.567834\pi\)
\(504\) −20.0000 + 17.8885i −0.890871 + 0.796819i
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) −20.5548 9.19239i −0.912871 0.408248i
\(508\) −44.2719 −1.96425
\(509\) 44.7214i 1.98224i −0.132973 0.991120i \(-0.542452\pi\)
0.132973 0.991120i \(-0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 35.3553i 1.55794i
\(516\) 10.0000 + 4.47214i 0.440225 + 0.196875i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.8885i 0.783711i −0.920027 0.391856i \(-0.871833\pi\)
0.920027 0.391856i \(-0.128167\pi\)
\(522\) 25.2982 + 28.2843i 1.10727 + 1.23797i
\(523\) −34.7851 −1.52104 −0.760522 0.649312i \(-0.775057\pi\)
−0.760522 + 0.649312i \(0.775057\pi\)
\(524\) 0 0
\(525\) 25.0000 + 11.1803i 1.09109 + 0.487950i
\(526\) −22.0000 −0.959246
\(527\) 0 0
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 40.0000 + 17.8885i 1.73097 + 0.774113i
\(535\) 41.1096 1.77732
\(536\) 44.7214i 1.93167i
\(537\) 0 0
\(538\) 31.6228 1.36335
\(539\) 0 0
\(540\) 22.1359 7.07107i 0.952579 0.304290i
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) −3.16228 1.41421i −0.135706 0.0606897i
\(544\) 0 0
\(545\) 35.7771i 1.53252i
\(546\) 0 0
\(547\) 3.16228 0.135209 0.0676046 0.997712i \(-0.478464\pi\)
0.0676046 + 0.997712i \(0.478464\pi\)
\(548\) 0 0
\(549\) 16.0000 + 17.8885i 0.682863 + 0.763464i
\(550\) 0 0
\(551\) 0 0
\(552\) 6.32456 + 2.82843i 0.269191 + 0.120386i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 28.2843i 1.19523i
\(561\) 0 0
\(562\) −44.2719 −1.86750
\(563\) 1.41421i 0.0596020i 0.999556 + 0.0298010i \(0.00948736\pi\)
−0.999556 + 0.0298010i \(0.990513\pi\)
\(564\) −14.0000 + 31.3050i −0.589506 + 1.31818i
\(565\) 0 0
\(566\) 22.3607i 0.939889i
\(567\) −3.16228 + 28.2843i −0.132803 + 1.18783i
\(568\) 0 0
\(569\) 31.3050i 1.31237i −0.754599 0.656186i \(-0.772169\pi\)
0.754599 0.656186i \(-0.227831\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −20.0000 −0.834784
\(575\) 7.07107i 0.294884i
\(576\) −16.0000 17.8885i −0.666667 0.745356i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 24.0416i 1.00000i
\(579\) 0 0
\(580\) −40.0000 −1.66091
\(581\) 49.1935i 2.04089i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.89949i 0.408596i 0.978909 + 0.204298i \(0.0654911\pi\)
−0.978909 + 0.204298i \(0.934509\pi\)
\(588\) 9.48683 + 4.24264i 0.391230 + 0.174964i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.94427i 0.366372i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −10.0000 + 22.3607i −0.408248 + 0.912871i
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 14.1421i 0.576390i
\(603\) −31.6228 35.3553i −1.28778 1.43978i
\(604\) 0 0
\(605\) 24.5967i 1.00000i
\(606\) −20.0000 8.94427i −0.812444 0.363336i
\(607\) −15.8114 −0.641764 −0.320882 0.947119i \(-0.603979\pi\)
−0.320882 + 0.947119i \(0.603979\pi\)
\(608\) 0 0
\(609\) 20.0000 44.7214i 0.810441 1.81220i
\(610\) −25.2982 −1.02430
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 49.1935i 1.98529i
\(615\) 15.8114 + 7.07107i 0.637577 + 0.285133i
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 15.8114 35.3553i 0.636027 1.42220i
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 7.00000 2.23607i 0.280900 0.0897303i
\(622\) 0 0
\(623\) 56.5685i 2.26637i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −20.0000 22.3607i −0.796819 0.890871i
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 49.4975i 1.96425i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 25.2982 1.00000
\(641\) 49.1935i 1.94303i 0.236986 + 0.971513i \(0.423841\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) −41.1096 18.3848i −1.62247 0.725589i
\(643\) 41.1096 1.62120 0.810602 0.585597i \(-0.199140\pi\)
0.810602 + 0.585597i \(0.199140\pi\)
\(644\) 8.94427i 0.352454i
\(645\) −5.00000 + 11.1803i −0.196875 + 0.440225i
\(646\) 0 0
\(647\) 18.3848i 0.722780i 0.932415 + 0.361390i \(0.117698\pi\)
−0.932415 + 0.361390i \(0.882302\pi\)
\(648\) −25.2982 2.82843i −0.993808 0.111111i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −44.2719 −1.73382
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 16.0000 35.7771i 0.625650 1.39899i
\(655\) 0 0
\(656\) 17.8885i 0.698430i
\(657\) 0 0
\(658\) 44.2719 1.72590
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 44.0000 1.70753
\(665\) 0 0
\(666\) 0 0
\(667\) −12.6491 −0.489776
\(668\) 48.0833i 1.86040i
\(669\) −5.00000 2.23607i −0.193311 0.0864514i
\(670\) 50.0000 1.93167
\(671\) 0 0
\(672\) −12.6491 + 28.2843i −0.487950 + 1.09109i
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 7.90569 + 24.7487i 0.304290 + 0.952579i
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.00000 15.6525i 0.268241 0.599804i
\(682\) 0 0
\(683\) 43.8406i 1.67751i 0.544505 + 0.838757i \(0.316717\pi\)
−0.544505 + 0.838757i \(0.683283\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 17.8885i 0.682988i
\(687\) −22.1359 9.89949i −0.844539 0.377689i
\(688\) 12.6491 0.482243
\(689\) 0 0
\(690\) −3.16228 + 7.07107i −0.120386 + 0.269191i
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −34.0000 −1.29062
\(695\) 0 0
\(696\) 40.0000 + 17.8885i 1.51620 + 0.678064i
\(697\) 0 0
\(698\) 36.7696i 1.39175i
\(699\) 0 0
\(700\) 31.6228 1.19523
\(701\) 22.3607i 0.844551i 0.906467 + 0.422276i \(0.138769\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −35.0000 15.6525i −1.31818 0.589506i
\(706\) 0 0
\(707\) 28.2843i 1.06374i
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 50.5964 1.89618
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 20.0000 17.8885i 0.745356 0.666667i
\(721\) −50.0000 −1.86210
\(722\) 26.8701i 1.00000i
\(723\) 44.2719 + 19.7990i 1.64649 + 0.736332i
\(724\) −4.00000 −0.148659
\(725\) 44.7214i 1.66091i
\(726\) 11.0000 24.5967i 0.408248 0.912871i
\(727\) −53.7587 −1.99380 −0.996900 0.0786754i \(-0.974931\pi\)
−0.996900 + 0.0786754i \(0.974931\pi\)
\(728\) 0 0
\(729\) −22.0000 + 15.6525i −0.814815 + 0.579721i
\(730\) 0 0
\(731\) 0 0
\(732\) 25.2982 + 11.3137i 0.935049 + 0.418167i
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 4.47214i 0.165070i
\(735\) −4.74342 + 10.6066i −0.174964 + 0.391230i
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) −12.6491 14.1421i −0.465620 0.520579i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 26.8701i 0.985767i 0.870095 + 0.492883i \(0.164057\pi\)
−0.870095 + 0.492883i \(0.835943\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) 34.7851 31.1127i 1.27272 1.13835i
\(748\) 0 0
\(749\) 58.1378i 2.12431i
\(750\) −25.0000 11.1803i −0.912871 0.408248i
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 39.5980i 1.44399i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 10.0000 + 31.3050i 0.363696 + 1.13855i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.7771i 1.29692i 0.761249 + 0.648459i \(0.224586\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −22.1359 + 49.4975i −0.801901 + 1.79310i
\(763\) −50.5964 −1.83171
\(764\) 0 0
\(765\) 0 0
\(766\) 38.0000 1.37300
\(767\) 0 0
\(768\) −25.2982 11.3137i −0.912871 0.408248i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 10.0000 8.94427i 0.359443 0.321495i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −44.2719 −1.58722
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 44.2719 14.1421i 1.58215 0.505399i
\(784\) 12.0000 0.428571
\(785\) 0 0
\(786\) 0 0
\(787\) 41.1096 1.46540 0.732700 0.680552i \(-0.238260\pi\)
0.732700 + 0.680552i \(0.238260\pi\)
\(788\) 0 0
\(789\) −11.0000 + 24.5967i −0.391610 + 0.875667i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 28.2843i 1.00000i
\(801\) 40.0000 35.7771i 1.41333 1.26412i
\(802\) 50.5964 1.78662
\(803\) 0 0