Properties

Label 5520.2.be.a.1471.2
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.2
Root \(0.476829 - 0.476829i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.a.1471.10

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} -3.79952 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} -3.79952 q^{7} -1.00000 q^{9} -3.69013 q^{11} +2.60963 q^{13} +1.00000 q^{15} -1.60151i q^{17} +8.20535 q^{19} +3.79952i q^{21} +(-1.19801 - 4.64379i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +0.706420 q^{29} -6.46598i q^{31} +3.69013i q^{33} -3.79952i q^{35} +5.01787i q^{37} -2.60963i q^{39} -10.6396 q^{41} +5.94134 q^{43} -1.00000i q^{45} -4.46270i q^{47} +7.43635 q^{49} -1.60151 q^{51} +7.25579i q^{53} -3.69013i q^{55} -8.20535i q^{57} +6.04730i q^{59} -3.36597i q^{61} +3.79952 q^{63} +2.60963i q^{65} -13.5383 q^{67} +(-4.64379 + 1.19801i) q^{69} -8.69292i q^{71} -0.516947 q^{73} +1.00000i q^{75} +14.0207 q^{77} +3.23175 q^{79} +1.00000 q^{81} -4.10586 q^{83} +1.60151 q^{85} -0.706420i q^{87} +5.11034i q^{89} -9.91534 q^{91} -6.46598 q^{93} +8.20535i q^{95} +19.4018i q^{97} +3.69013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q - 8q^{7} - 16q^{9} + 8q^{11} + 8q^{13} + 16q^{15} + 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} - 4q^{51} + 8q^{63} - 16q^{67} + 40q^{73} + 24q^{77} + 32q^{79} + 16q^{81} + 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −3.79952 −1.43608 −0.718042 0.696000i \(-0.754961\pi\)
−0.718042 + 0.696000i \(0.754961\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.69013 −1.11262 −0.556308 0.830976i \(-0.687783\pi\)
−0.556308 + 0.830976i \(0.687783\pi\)
\(12\) 0 0
\(13\) 2.60963 0.723781 0.361891 0.932221i \(-0.382131\pi\)
0.361891 + 0.932221i \(0.382131\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 1.60151i 0.388424i −0.980960 0.194212i \(-0.937785\pi\)
0.980960 0.194212i \(-0.0622150\pi\)
\(18\) 0 0
\(19\) 8.20535 1.88244 0.941218 0.337801i \(-0.109683\pi\)
0.941218 + 0.337801i \(0.109683\pi\)
\(20\) 0 0
\(21\) 3.79952i 0.829123i
\(22\) 0 0
\(23\) −1.19801 4.64379i −0.249802 0.968297i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0.706420 0.131179 0.0655894 0.997847i \(-0.479107\pi\)
0.0655894 + 0.997847i \(0.479107\pi\)
\(30\) 0 0
\(31\) 6.46598i 1.16132i −0.814145 0.580662i \(-0.802794\pi\)
0.814145 0.580662i \(-0.197206\pi\)
\(32\) 0 0
\(33\) 3.69013i 0.642369i
\(34\) 0 0
\(35\) 3.79952i 0.642236i
\(36\) 0 0
\(37\) 5.01787i 0.824933i 0.910973 + 0.412466i \(0.135333\pi\)
−0.910973 + 0.412466i \(0.864667\pi\)
\(38\) 0 0
\(39\) 2.60963i 0.417875i
\(40\) 0 0
\(41\) −10.6396 −1.66162 −0.830812 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(42\) 0 0
\(43\) 5.94134 0.906045 0.453023 0.891499i \(-0.350346\pi\)
0.453023 + 0.891499i \(0.350346\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 4.46270i 0.650952i −0.945550 0.325476i \(-0.894476\pi\)
0.945550 0.325476i \(-0.105524\pi\)
\(48\) 0 0
\(49\) 7.43635 1.06234
\(50\) 0 0
\(51\) −1.60151 −0.224257
\(52\) 0 0
\(53\) 7.25579i 0.996660i 0.866987 + 0.498330i \(0.166053\pi\)
−0.866987 + 0.498330i \(0.833947\pi\)
\(54\) 0 0
\(55\) 3.69013i 0.497577i
\(56\) 0 0
\(57\) 8.20535i 1.08682i
\(58\) 0 0
\(59\) 6.04730i 0.787291i 0.919262 + 0.393646i \(0.128786\pi\)
−0.919262 + 0.393646i \(0.871214\pi\)
\(60\) 0 0
\(61\) 3.36597i 0.430968i −0.976507 0.215484i \(-0.930867\pi\)
0.976507 0.215484i \(-0.0691329\pi\)
\(62\) 0 0
\(63\) 3.79952 0.478695
\(64\) 0 0
\(65\) 2.60963i 0.323685i
\(66\) 0 0
\(67\) −13.5383 −1.65397 −0.826984 0.562226i \(-0.809945\pi\)
−0.826984 + 0.562226i \(0.809945\pi\)
\(68\) 0 0
\(69\) −4.64379 + 1.19801i −0.559047 + 0.144223i
\(70\) 0 0
\(71\) 8.69292i 1.03166i −0.856691 0.515830i \(-0.827484\pi\)
0.856691 0.515830i \(-0.172516\pi\)
\(72\) 0 0
\(73\) −0.516947 −0.0605040 −0.0302520 0.999542i \(-0.509631\pi\)
−0.0302520 + 0.999542i \(0.509631\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 14.0207 1.59781
\(78\) 0 0
\(79\) 3.23175 0.363600 0.181800 0.983336i \(-0.441808\pi\)
0.181800 + 0.983336i \(0.441808\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.10586 −0.450677 −0.225338 0.974281i \(-0.572349\pi\)
−0.225338 + 0.974281i \(0.572349\pi\)
\(84\) 0 0
\(85\) 1.60151 0.173709
\(86\) 0 0
\(87\) 0.706420i 0.0757361i
\(88\) 0 0
\(89\) 5.11034i 0.541695i 0.962622 + 0.270848i \(0.0873040\pi\)
−0.962622 + 0.270848i \(0.912696\pi\)
\(90\) 0 0
\(91\) −9.91534 −1.03941
\(92\) 0 0
\(93\) −6.46598 −0.670491
\(94\) 0 0
\(95\) 8.20535i 0.841851i
\(96\) 0 0
\(97\) 19.4018i 1.96996i 0.172675 + 0.984979i \(0.444759\pi\)
−0.172675 + 0.984979i \(0.555241\pi\)
\(98\) 0 0
\(99\) 3.69013 0.370872
\(100\) 0 0
\(101\) 13.1514 1.30861 0.654305 0.756231i \(-0.272961\pi\)
0.654305 + 0.756231i \(0.272961\pi\)
\(102\) 0 0
\(103\) −5.37041 −0.529162 −0.264581 0.964363i \(-0.585234\pi\)
−0.264581 + 0.964363i \(0.585234\pi\)
\(104\) 0 0
\(105\) −3.79952 −0.370795
\(106\) 0 0
\(107\) −17.9219 −1.73258 −0.866290 0.499542i \(-0.833502\pi\)
−0.866290 + 0.499542i \(0.833502\pi\)
\(108\) 0 0
\(109\) 1.27328i 0.121958i 0.998139 + 0.0609792i \(0.0194223\pi\)
−0.998139 + 0.0609792i \(0.980578\pi\)
\(110\) 0 0
\(111\) 5.01787 0.476275
\(112\) 0 0
\(113\) 3.32431i 0.312725i 0.987700 + 0.156362i \(0.0499768\pi\)
−0.987700 + 0.156362i \(0.950023\pi\)
\(114\) 0 0
\(115\) 4.64379 1.19801i 0.433036 0.111715i
\(116\) 0 0
\(117\) −2.60963 −0.241260
\(118\) 0 0
\(119\) 6.08498i 0.557810i
\(120\) 0 0
\(121\) 2.61707 0.237915
\(122\) 0 0
\(123\) 10.6396i 0.959339i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 15.7160i 1.39457i 0.716793 + 0.697286i \(0.245609\pi\)
−0.716793 + 0.697286i \(0.754391\pi\)
\(128\) 0 0
\(129\) 5.94134i 0.523106i
\(130\) 0 0
\(131\) 10.5445i 0.921280i −0.887587 0.460640i \(-0.847620\pi\)
0.887587 0.460640i \(-0.152380\pi\)
\(132\) 0 0
\(133\) −31.1764 −2.70333
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 1.67864i 0.143416i −0.997426 0.0717081i \(-0.977155\pi\)
0.997426 0.0717081i \(-0.0228450\pi\)
\(138\) 0 0
\(139\) 13.7161i 1.16338i 0.813410 + 0.581691i \(0.197609\pi\)
−0.813410 + 0.581691i \(0.802391\pi\)
\(140\) 0 0
\(141\) −4.46270 −0.375827
\(142\) 0 0
\(143\) −9.62988 −0.805291
\(144\) 0 0
\(145\) 0.706420i 0.0586650i
\(146\) 0 0
\(147\) 7.43635i 0.613340i
\(148\) 0 0
\(149\) 22.3923i 1.83445i 0.398368 + 0.917226i \(0.369577\pi\)
−0.398368 + 0.917226i \(0.630423\pi\)
\(150\) 0 0
\(151\) 22.9206i 1.86526i 0.360840 + 0.932628i \(0.382490\pi\)
−0.360840 + 0.932628i \(0.617510\pi\)
\(152\) 0 0
\(153\) 1.60151i 0.129475i
\(154\) 0 0
\(155\) 6.46598 0.519360
\(156\) 0 0
\(157\) 14.5054i 1.15765i 0.815450 + 0.578827i \(0.196489\pi\)
−0.815450 + 0.578827i \(0.803511\pi\)
\(158\) 0 0
\(159\) 7.25579 0.575422
\(160\) 0 0
\(161\) 4.55185 + 17.6442i 0.358736 + 1.39056i
\(162\) 0 0
\(163\) 6.70939i 0.525520i 0.964861 + 0.262760i \(0.0846327\pi\)
−0.964861 + 0.262760i \(0.915367\pi\)
\(164\) 0 0
\(165\) −3.69013 −0.287276
\(166\) 0 0
\(167\) 5.17703i 0.400611i 0.979734 + 0.200305i \(0.0641934\pi\)
−0.979734 + 0.200305i \(0.935807\pi\)
\(168\) 0 0
\(169\) −6.18983 −0.476141
\(170\) 0 0
\(171\) −8.20535 −0.627478
\(172\) 0 0
\(173\) 14.0638 1.06925 0.534624 0.845090i \(-0.320453\pi\)
0.534624 + 0.845090i \(0.320453\pi\)
\(174\) 0 0
\(175\) 3.79952 0.287217
\(176\) 0 0
\(177\) 6.04730 0.454543
\(178\) 0 0
\(179\) 11.9606i 0.893975i 0.894540 + 0.446988i \(0.147503\pi\)
−0.894540 + 0.446988i \(0.852497\pi\)
\(180\) 0 0
\(181\) 1.90385i 0.141512i −0.997494 0.0707561i \(-0.977459\pi\)
0.997494 0.0707561i \(-0.0225412\pi\)
\(182\) 0 0
\(183\) −3.36597 −0.248819
\(184\) 0 0
\(185\) −5.01787 −0.368921
\(186\) 0 0
\(187\) 5.90980i 0.432167i
\(188\) 0 0
\(189\) 3.79952i 0.276374i
\(190\) 0 0
\(191\) −16.6906 −1.20769 −0.603845 0.797102i \(-0.706365\pi\)
−0.603845 + 0.797102i \(0.706365\pi\)
\(192\) 0 0
\(193\) −11.9024 −0.856751 −0.428376 0.903601i \(-0.640914\pi\)
−0.428376 + 0.903601i \(0.640914\pi\)
\(194\) 0 0
\(195\) 2.60963 0.186879
\(196\) 0 0
\(197\) 15.3515 1.09375 0.546876 0.837214i \(-0.315817\pi\)
0.546876 + 0.837214i \(0.315817\pi\)
\(198\) 0 0
\(199\) 17.4361 1.23601 0.618006 0.786173i \(-0.287941\pi\)
0.618006 + 0.786173i \(0.287941\pi\)
\(200\) 0 0
\(201\) 13.5383i 0.954919i
\(202\) 0 0
\(203\) −2.68406 −0.188384
\(204\) 0 0
\(205\) 10.6396i 0.743101i
\(206\) 0 0
\(207\) 1.19801 + 4.64379i 0.0832672 + 0.322766i
\(208\) 0 0
\(209\) −30.2788 −2.09443
\(210\) 0 0
\(211\) 7.11509i 0.489823i −0.969545 0.244911i \(-0.921241\pi\)
0.969545 0.244911i \(-0.0787589\pi\)
\(212\) 0 0
\(213\) −8.69292 −0.595629
\(214\) 0 0
\(215\) 5.94134i 0.405196i
\(216\) 0 0
\(217\) 24.5676i 1.66776i
\(218\) 0 0
\(219\) 0.516947i 0.0349320i
\(220\) 0 0
\(221\) 4.17936i 0.281134i
\(222\) 0 0
\(223\) 2.92403i 0.195808i 0.995196 + 0.0979038i \(0.0312137\pi\)
−0.995196 + 0.0979038i \(0.968786\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −8.86711 −0.588531 −0.294265 0.955724i \(-0.595075\pi\)
−0.294265 + 0.955724i \(0.595075\pi\)
\(228\) 0 0
\(229\) 7.81117i 0.516177i −0.966121 0.258088i \(-0.916907\pi\)
0.966121 0.258088i \(-0.0830926\pi\)
\(230\) 0 0
\(231\) 14.0207i 0.922496i
\(232\) 0 0
\(233\) −5.09372 −0.333701 −0.166850 0.985982i \(-0.553360\pi\)
−0.166850 + 0.985982i \(0.553360\pi\)
\(234\) 0 0
\(235\) 4.46270 0.291114
\(236\) 0 0
\(237\) 3.23175i 0.209925i
\(238\) 0 0
\(239\) 19.5551i 1.26491i −0.774595 0.632457i \(-0.782046\pi\)
0.774595 0.632457i \(-0.217954\pi\)
\(240\) 0 0
\(241\) 8.39986i 0.541083i 0.962708 + 0.270541i \(0.0872027\pi\)
−0.962708 + 0.270541i \(0.912797\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 7.43635i 0.475091i
\(246\) 0 0
\(247\) 21.4129 1.36247
\(248\) 0 0
\(249\) 4.10586i 0.260198i
\(250\) 0 0
\(251\) −7.84091 −0.494914 −0.247457 0.968899i \(-0.579595\pi\)
−0.247457 + 0.968899i \(0.579595\pi\)
\(252\) 0 0
\(253\) 4.42080 + 17.1362i 0.277933 + 1.07734i
\(254\) 0 0
\(255\) 1.60151i 0.100291i
\(256\) 0 0
\(257\) −5.18308 −0.323312 −0.161656 0.986847i \(-0.551683\pi\)
−0.161656 + 0.986847i \(0.551683\pi\)
\(258\) 0 0
\(259\) 19.0655i 1.18467i
\(260\) 0 0
\(261\) −0.706420 −0.0437263
\(262\) 0 0
\(263\) −6.41083 −0.395309 −0.197654 0.980272i \(-0.563332\pi\)
−0.197654 + 0.980272i \(0.563332\pi\)
\(264\) 0 0
\(265\) −7.25579 −0.445720
\(266\) 0 0
\(267\) 5.11034 0.312748
\(268\) 0 0
\(269\) 28.1418 1.71584 0.857919 0.513785i \(-0.171757\pi\)
0.857919 + 0.513785i \(0.171757\pi\)
\(270\) 0 0
\(271\) 23.8196i 1.44694i −0.690356 0.723470i \(-0.742546\pi\)
0.690356 0.723470i \(-0.257454\pi\)
\(272\) 0 0
\(273\) 9.91534i 0.600104i
\(274\) 0 0
\(275\) 3.69013 0.222523
\(276\) 0 0
\(277\) −22.1168 −1.32887 −0.664436 0.747345i \(-0.731328\pi\)
−0.664436 + 0.747345i \(0.731328\pi\)
\(278\) 0 0
\(279\) 6.46598i 0.387108i
\(280\) 0 0
\(281\) 10.5452i 0.629075i 0.949245 + 0.314538i \(0.101849\pi\)
−0.949245 + 0.314538i \(0.898151\pi\)
\(282\) 0 0
\(283\) −13.3512 −0.793648 −0.396824 0.917895i \(-0.629888\pi\)
−0.396824 + 0.917895i \(0.629888\pi\)
\(284\) 0 0
\(285\) 8.20535 0.486043
\(286\) 0 0
\(287\) 40.4253 2.38623
\(288\) 0 0
\(289\) 14.4352 0.849127
\(290\) 0 0
\(291\) 19.4018 1.13736
\(292\) 0 0
\(293\) 13.0456i 0.762129i −0.924548 0.381065i \(-0.875557\pi\)
0.924548 0.381065i \(-0.124443\pi\)
\(294\) 0 0
\(295\) −6.04730 −0.352087
\(296\) 0 0
\(297\) 3.69013i 0.214123i
\(298\) 0 0
\(299\) −3.12635 12.1186i −0.180802 0.700835i
\(300\) 0 0
\(301\) −22.5742 −1.30116
\(302\) 0 0
\(303\) 13.1514i 0.755526i
\(304\) 0 0
\(305\) 3.36597 0.192735
\(306\) 0 0
\(307\) 24.4269i 1.39412i −0.717015 0.697058i \(-0.754492\pi\)
0.717015 0.697058i \(-0.245508\pi\)
\(308\) 0 0
\(309\) 5.37041i 0.305512i
\(310\) 0 0
\(311\) 2.74305i 0.155544i −0.996971 0.0777721i \(-0.975219\pi\)
0.996971 0.0777721i \(-0.0247807\pi\)
\(312\) 0 0
\(313\) 3.88307i 0.219484i −0.993960 0.109742i \(-0.964998\pi\)
0.993960 0.109742i \(-0.0350025\pi\)
\(314\) 0 0
\(315\) 3.79952i 0.214079i
\(316\) 0 0
\(317\) −26.5346 −1.49033 −0.745166 0.666879i \(-0.767630\pi\)
−0.745166 + 0.666879i \(0.767630\pi\)
\(318\) 0 0
\(319\) −2.60678 −0.145952
\(320\) 0 0
\(321\) 17.9219i 1.00031i
\(322\) 0 0
\(323\) 13.1410i 0.731183i
\(324\) 0 0
\(325\) −2.60963 −0.144756
\(326\) 0 0
\(327\) 1.27328 0.0704127
\(328\) 0 0
\(329\) 16.9561i 0.934821i
\(330\) 0 0
\(331\) 28.5918i 1.57155i 0.618515 + 0.785773i \(0.287735\pi\)
−0.618515 + 0.785773i \(0.712265\pi\)
\(332\) 0 0
\(333\) 5.01787i 0.274978i
\(334\) 0 0
\(335\) 13.5383i 0.739677i
\(336\) 0 0
\(337\) 6.76479i 0.368501i −0.982879 0.184251i \(-0.941014\pi\)
0.982879 0.184251i \(-0.0589858\pi\)
\(338\) 0 0
\(339\) 3.32431 0.180552
\(340\) 0 0
\(341\) 23.8603i 1.29211i
\(342\) 0 0
\(343\) −1.65793 −0.0895200
\(344\) 0 0
\(345\) −1.19801 4.64379i −0.0644985 0.250013i
\(346\) 0 0
\(347\) 29.0461i 1.55927i 0.626231 + 0.779637i \(0.284597\pi\)
−0.626231 + 0.779637i \(0.715403\pi\)
\(348\) 0 0
\(349\) −20.4208 −1.09310 −0.546550 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(350\) 0 0
\(351\) 2.60963i 0.139292i
\(352\) 0 0
\(353\) −18.6709 −0.993751 −0.496876 0.867822i \(-0.665519\pi\)
−0.496876 + 0.867822i \(0.665519\pi\)
\(354\) 0 0
\(355\) 8.69292 0.461372
\(356\) 0 0
\(357\) 6.08498 0.322052
\(358\) 0 0
\(359\) −5.96281 −0.314705 −0.157353 0.987542i \(-0.550296\pi\)
−0.157353 + 0.987542i \(0.550296\pi\)
\(360\) 0 0
\(361\) 48.3277 2.54356
\(362\) 0 0
\(363\) 2.61707i 0.137360i
\(364\) 0 0
\(365\) 0.516947i 0.0270582i
\(366\) 0 0
\(367\) 23.4914 1.22624 0.613121 0.789989i \(-0.289914\pi\)
0.613121 + 0.789989i \(0.289914\pi\)
\(368\) 0 0
\(369\) 10.6396 0.553875
\(370\) 0 0
\(371\) 27.5685i 1.43129i
\(372\) 0 0
\(373\) 20.8948i 1.08189i −0.841057 0.540946i \(-0.818066\pi\)
0.841057 0.540946i \(-0.181934\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 1.84349 0.0949447
\(378\) 0 0
\(379\) −8.21337 −0.421892 −0.210946 0.977498i \(-0.567654\pi\)
−0.210946 + 0.977498i \(0.567654\pi\)
\(380\) 0 0
\(381\) 15.7160 0.805156
\(382\) 0 0
\(383\) −7.84776 −0.401002 −0.200501 0.979694i \(-0.564257\pi\)
−0.200501 + 0.979694i \(0.564257\pi\)
\(384\) 0 0
\(385\) 14.0207i 0.714562i
\(386\) 0 0
\(387\) −5.94134 −0.302015
\(388\) 0 0
\(389\) 25.0255i 1.26884i 0.772988 + 0.634421i \(0.218761\pi\)
−0.772988 + 0.634421i \(0.781239\pi\)
\(390\) 0 0
\(391\) −7.43709 + 1.91862i −0.376110 + 0.0970290i
\(392\) 0 0
\(393\) −10.5445 −0.531901
\(394\) 0 0
\(395\) 3.23175i 0.162607i
\(396\) 0 0
\(397\) 19.5249 0.979928 0.489964 0.871743i \(-0.337010\pi\)
0.489964 + 0.871743i \(0.337010\pi\)
\(398\) 0 0
\(399\) 31.1764i 1.56077i
\(400\) 0 0
\(401\) 27.8201i 1.38927i −0.719363 0.694634i \(-0.755566\pi\)
0.719363 0.694634i \(-0.244434\pi\)
\(402\) 0 0
\(403\) 16.8738i 0.840545i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 18.5166i 0.917834i
\(408\) 0 0
\(409\) −18.0574 −0.892882 −0.446441 0.894813i \(-0.647309\pi\)
−0.446441 + 0.894813i \(0.647309\pi\)
\(410\) 0 0
\(411\) −1.67864 −0.0828014
\(412\) 0 0
\(413\) 22.9768i 1.13062i
\(414\) 0 0
\(415\) 4.10586i 0.201549i
\(416\) 0 0
\(417\) 13.7161 0.671679
\(418\) 0 0
\(419\) 11.4005 0.556952 0.278476 0.960443i \(-0.410171\pi\)
0.278476 + 0.960443i \(0.410171\pi\)
\(420\) 0 0
\(421\) 20.5611i 1.00209i −0.865422 0.501044i \(-0.832949\pi\)
0.865422 0.501044i \(-0.167051\pi\)
\(422\) 0 0
\(423\) 4.46270i 0.216984i
\(424\) 0 0
\(425\) 1.60151i 0.0776848i
\(426\) 0 0
\(427\) 12.7891i 0.618906i
\(428\) 0 0
\(429\) 9.62988i 0.464935i
\(430\) 0 0
\(431\) 3.10368 0.149499 0.0747496 0.997202i \(-0.476184\pi\)
0.0747496 + 0.997202i \(0.476184\pi\)
\(432\) 0 0
\(433\) 9.48758i 0.455944i 0.973668 + 0.227972i \(0.0732095\pi\)
−0.973668 + 0.227972i \(0.926791\pi\)
\(434\) 0 0
\(435\) 0.706420 0.0338702
\(436\) 0 0
\(437\) −9.83005 38.1039i −0.470235 1.82276i
\(438\) 0 0
\(439\) 4.16012i 0.198551i 0.995060 + 0.0992757i \(0.0316526\pi\)
−0.995060 + 0.0992757i \(0.968347\pi\)
\(440\) 0 0
\(441\) −7.43635 −0.354112
\(442\) 0 0
\(443\) 31.5136i 1.49726i 0.662990 + 0.748629i \(0.269287\pi\)
−0.662990 + 0.748629i \(0.730713\pi\)
\(444\) 0 0
\(445\) −5.11034 −0.242254
\(446\) 0 0
\(447\) 22.3923 1.05912
\(448\) 0 0
\(449\) −15.5729 −0.734930 −0.367465 0.930037i \(-0.619774\pi\)
−0.367465 + 0.930037i \(0.619774\pi\)
\(450\) 0 0
\(451\) 39.2615 1.84875
\(452\) 0 0
\(453\) 22.9206 1.07691
\(454\) 0 0
\(455\) 9.91534i 0.464838i
\(456\) 0 0
\(457\) 30.9828i 1.44931i 0.689110 + 0.724657i \(0.258002\pi\)
−0.689110 + 0.724657i \(0.741998\pi\)
\(458\) 0 0
\(459\) 1.60151 0.0747523
\(460\) 0 0
\(461\) 15.7446 0.733301 0.366651 0.930359i \(-0.380504\pi\)
0.366651 + 0.930359i \(0.380504\pi\)
\(462\) 0 0
\(463\) 1.71750i 0.0798191i −0.999203 0.0399096i \(-0.987293\pi\)
0.999203 0.0399096i \(-0.0127070\pi\)
\(464\) 0 0
\(465\) 6.46598i 0.299853i
\(466\) 0 0
\(467\) −35.3495 −1.63578 −0.817891 0.575373i \(-0.804857\pi\)
−0.817891 + 0.575373i \(0.804857\pi\)
\(468\) 0 0
\(469\) 51.4391 2.37524
\(470\) 0 0
\(471\) 14.5054 0.668372
\(472\) 0 0
\(473\) −21.9243 −1.00808
\(474\) 0 0
\(475\) −8.20535 −0.376487
\(476\) 0 0
\(477\) 7.25579i 0.332220i
\(478\) 0 0
\(479\) −27.0253 −1.23482 −0.617408 0.786643i \(-0.711817\pi\)
−0.617408 + 0.786643i \(0.711817\pi\)
\(480\) 0 0
\(481\) 13.0948i 0.597071i
\(482\) 0 0
\(483\) 17.6442 4.55185i 0.802838 0.207116i
\(484\) 0 0
\(485\) −19.4018 −0.880992
\(486\) 0 0
\(487\) 42.0999i 1.90773i 0.300237 + 0.953865i \(0.402934\pi\)
−0.300237 + 0.953865i \(0.597066\pi\)
\(488\) 0 0
\(489\) 6.70939 0.303409
\(490\) 0 0
\(491\) 14.3694i 0.648480i 0.945975 + 0.324240i \(0.105109\pi\)
−0.945975 + 0.324240i \(0.894891\pi\)
\(492\) 0 0
\(493\) 1.13134i 0.0509530i
\(494\) 0 0
\(495\) 3.69013i 0.165859i
\(496\) 0 0
\(497\) 33.0289i 1.48155i
\(498\) 0 0
\(499\) 11.1409i 0.498734i 0.968409 + 0.249367i \(0.0802226\pi\)
−0.968409 + 0.249367i \(0.919777\pi\)
\(500\) 0 0
\(501\) 5.17703 0.231293
\(502\) 0 0
\(503\) −15.8161 −0.705207 −0.352603 0.935773i \(-0.614703\pi\)
−0.352603 + 0.935773i \(0.614703\pi\)
\(504\) 0 0
\(505\) 13.1514i 0.585228i
\(506\) 0 0
\(507\) 6.18983i 0.274900i
\(508\) 0 0
\(509\) −31.3061 −1.38762 −0.693809 0.720159i \(-0.744069\pi\)
−0.693809 + 0.720159i \(0.744069\pi\)
\(510\) 0 0
\(511\) 1.96415 0.0868888
\(512\) 0 0
\(513\) 8.20535i 0.362275i
\(514\) 0 0
\(515\) 5.37041i 0.236648i
\(516\) 0 0
\(517\) 16.4679i 0.724259i
\(518\) 0 0
\(519\) 14.0638i 0.617331i
\(520\) 0 0
\(521\) 23.5026i 1.02967i 0.857290 + 0.514833i \(0.172146\pi\)
−0.857290 + 0.514833i \(0.827854\pi\)
\(522\) 0 0
\(523\) 35.4909 1.55191 0.775954 0.630789i \(-0.217269\pi\)
0.775954 + 0.630789i \(0.217269\pi\)
\(524\) 0 0
\(525\) 3.79952i 0.165825i
\(526\) 0 0
\(527\) −10.3554 −0.451087
\(528\) 0 0
\(529\) −20.1296 + 11.1266i −0.875198 + 0.483764i
\(530\) 0 0
\(531\) 6.04730i 0.262430i
\(532\) 0 0
\(533\) −27.7654 −1.20265
\(534\) 0 0
\(535\) 17.9219i 0.774833i
\(536\) 0 0
\(537\) 11.9606 0.516137
\(538\) 0 0
\(539\) −27.4411 −1.18197
\(540\) 0 0
\(541\) 31.7968 1.36705 0.683525 0.729927i \(-0.260446\pi\)
0.683525 + 0.729927i \(0.260446\pi\)
\(542\) 0 0
\(543\) −1.90385 −0.0817022
\(544\) 0 0
\(545\) −1.27328 −0.0545415
\(546\) 0 0
\(547\) 21.8699i 0.935090i −0.883969 0.467545i \(-0.845139\pi\)
0.883969 0.467545i \(-0.154861\pi\)
\(548\) 0 0
\(549\) 3.36597i 0.143656i
\(550\) 0 0
\(551\) 5.79642 0.246936
\(552\) 0 0
\(553\) −12.2791 −0.522160
\(554\) 0 0
\(555\) 5.01787i 0.212997i
\(556\) 0 0
\(557\) 1.91446i 0.0811182i 0.999177 + 0.0405591i \(0.0129139\pi\)
−0.999177 + 0.0405591i \(0.987086\pi\)
\(558\) 0 0
\(559\) 15.5047 0.655779
\(560\) 0 0
\(561\) 5.90980 0.249512
\(562\) 0 0
\(563\) −33.0420 −1.39255 −0.696277 0.717773i \(-0.745162\pi\)
−0.696277 + 0.717773i \(0.745162\pi\)
\(564\) 0 0
\(565\) −3.32431 −0.139855
\(566\) 0 0
\(567\) −3.79952 −0.159565
\(568\) 0 0
\(569\) 10.1677i 0.426250i 0.977025 + 0.213125i \(0.0683642\pi\)
−0.977025 + 0.213125i \(0.931636\pi\)
\(570\) 0 0
\(571\) 22.6137 0.946355 0.473177 0.880967i \(-0.343107\pi\)
0.473177 + 0.880967i \(0.343107\pi\)
\(572\) 0 0
\(573\) 16.6906i 0.697260i
\(574\) 0 0
\(575\) 1.19801 + 4.64379i 0.0499603 + 0.193659i
\(576\) 0 0
\(577\) −10.0069 −0.416591 −0.208295 0.978066i \(-0.566792\pi\)
−0.208295 + 0.978066i \(0.566792\pi\)
\(578\) 0 0
\(579\) 11.9024i 0.494646i
\(580\) 0 0
\(581\) 15.6003 0.647210
\(582\) 0 0
\(583\) 26.7748i 1.10890i
\(584\) 0 0
\(585\) 2.60963i 0.107895i
\(586\) 0 0
\(587\) 20.6747i 0.853336i −0.904408 0.426668i \(-0.859687\pi\)
0.904408 0.426668i \(-0.140313\pi\)
\(588\) 0 0
\(589\) 53.0556i 2.18612i
\(590\) 0 0
\(591\) 15.3515i 0.631478i
\(592\) 0 0
\(593\) −34.2092 −1.40480 −0.702402 0.711780i \(-0.747889\pi\)
−0.702402 + 0.711780i \(0.747889\pi\)
\(594\) 0 0
\(595\) −6.08498 −0.249460
\(596\) 0 0
\(597\) 17.4361i 0.713612i
\(598\) 0 0
\(599\) 29.2726i 1.19605i 0.801479 + 0.598023i \(0.204047\pi\)
−0.801479 + 0.598023i \(0.795953\pi\)
\(600\) 0 0
\(601\) 32.7471 1.33578 0.667891 0.744259i \(-0.267197\pi\)
0.667891 + 0.744259i \(0.267197\pi\)
\(602\) 0 0
\(603\) 13.5383 0.551323
\(604\) 0 0
\(605\) 2.61707i 0.106399i
\(606\) 0 0
\(607\) 29.9681i 1.21637i 0.793796 + 0.608185i \(0.208102\pi\)
−0.793796 + 0.608185i \(0.791898\pi\)
\(608\) 0 0
\(609\) 2.68406i 0.108763i
\(610\) 0 0
\(611\) 11.6460i 0.471146i
\(612\) 0 0
\(613\) 12.6769i 0.512014i 0.966675 + 0.256007i \(0.0824070\pi\)
−0.966675 + 0.256007i \(0.917593\pi\)
\(614\) 0 0
\(615\) −10.6396 −0.429029
\(616\) 0 0
\(617\) 6.68220i 0.269015i 0.990913 + 0.134508i \(0.0429453\pi\)
−0.990913 + 0.134508i \(0.957055\pi\)
\(618\) 0 0
\(619\) 12.1774 0.489451 0.244725 0.969592i \(-0.421302\pi\)
0.244725 + 0.969592i \(0.421302\pi\)
\(620\) 0 0
\(621\) 4.64379 1.19801i 0.186349 0.0480743i
\(622\) 0 0
\(623\) 19.4169i 0.777920i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 30.2788i 1.20922i
\(628\) 0 0
\(629\) 8.03619 0.320424
\(630\) 0 0
\(631\) 20.6325 0.821365 0.410683 0.911778i \(-0.365290\pi\)
0.410683 + 0.911778i \(0.365290\pi\)
\(632\) 0 0
\(633\) −7.11509 −0.282799
\(634\) 0 0
\(635\) −15.7160 −0.623671
\(636\) 0 0
\(637\) 19.4061 0.768899
\(638\) 0 0
\(639\) 8.69292i 0.343887i
\(640\) 0 0
\(641\) 18.8025i 0.742655i −0.928502 0.371328i \(-0.878903\pi\)
0.928502 0.371328i \(-0.121097\pi\)
\(642\) 0 0
\(643\) −6.17101 −0.243361 −0.121680 0.992569i \(-0.538828\pi\)
−0.121680 + 0.992569i \(0.538828\pi\)
\(644\) 0 0
\(645\) 5.94134 0.233940
\(646\) 0 0
\(647\) 11.0800i 0.435601i 0.975993 + 0.217801i \(0.0698882\pi\)
−0.975993 + 0.217801i \(0.930112\pi\)
\(648\) 0 0
\(649\) 22.3153i 0.875953i
\(650\) 0 0
\(651\) 24.5676 0.962881
\(652\) 0 0
\(653\) −14.6281 −0.572441 −0.286221 0.958164i \(-0.592399\pi\)
−0.286221 + 0.958164i \(0.592399\pi\)
\(654\) 0 0
\(655\) 10.5445 0.412009
\(656\) 0 0
\(657\) 0.516947 0.0201680
\(658\) 0 0
\(659\) 19.1684 0.746694 0.373347 0.927692i \(-0.378210\pi\)
0.373347 + 0.927692i \(0.378210\pi\)
\(660\) 0 0
\(661\) 32.1112i 1.24898i 0.781033 + 0.624490i \(0.214693\pi\)
−0.781033 + 0.624490i \(0.785307\pi\)
\(662\) 0 0
\(663\) −4.17936 −0.162313
\(664\) 0 0
\(665\) 31.1764i 1.20897i
\(666\) 0 0
\(667\) −0.846295 3.28046i −0.0327687 0.127020i
\(668\) 0 0
\(669\) 2.92403 0.113050
\(670\) 0 0
\(671\) 12.4209i 0.479502i
\(672\) 0 0
\(673\) 0.150913 0.00581729 0.00290864 0.999996i \(-0.499074\pi\)
0.00290864 + 0.999996i \(0.499074\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 41.2608i 1.58578i −0.609365 0.792890i \(-0.708575\pi\)
0.609365 0.792890i \(-0.291425\pi\)
\(678\) 0 0
\(679\) 73.7177i 2.82902i
\(680\) 0 0
\(681\) 8.86711i 0.339788i
\(682\) 0 0
\(683\) 20.7486i 0.793925i −0.917835 0.396963i \(-0.870064\pi\)
0.917835 0.396963i \(-0.129936\pi\)
\(684\) 0 0
\(685\) 1.67864 0.0641377
\(686\) 0 0
\(687\) −7.81117 −0.298015
\(688\) 0 0
\(689\) 18.9349i 0.721364i
\(690\) 0 0
\(691\) 4.48560i 0.170640i −0.996354 0.0853202i \(-0.972809\pi\)
0.996354 0.0853202i \(-0.0271913\pi\)
\(692\) 0 0
\(693\) −14.0207 −0.532603
\(694\) 0 0
\(695\) −13.7161 −0.520280
\(696\) 0 0
\(697\) 17.0394i 0.645415i
\(698\) 0 0
\(699\) 5.09372i 0.192662i
\(700\) 0 0
\(701\) 1.49486i 0.0564601i −0.999601 0.0282301i \(-0.991013\pi\)
0.999601 0.0282301i \(-0.00898710\pi\)
\(702\) 0 0
\(703\) 41.1734i 1.55288i
\(704\) 0 0
\(705\) 4.46270i 0.168075i
\(706\) 0 0
\(707\) −49.9689 −1.87927
\(708\) 0 0
\(709\) 48.1753i 1.80926i −0.426194 0.904632i \(-0.640146\pi\)
0.426194 0.904632i \(-0.359854\pi\)
\(710\) 0 0
\(711\) −3.23175 −0.121200
\(712\) 0 0
\(713\) −30.0267 + 7.74629i −1.12451 + 0.290101i
\(714\) 0 0
\(715\) 9.62988i 0.360137i
\(716\) 0 0
\(717\) −19.5551 −0.730299
\(718\) 0 0
\(719\) 13.1017i 0.488610i 0.969698 + 0.244305i \(0.0785599\pi\)
−0.969698 + 0.244305i \(0.921440\pi\)
\(720\) 0 0
\(721\) 20.4050 0.759921
\(722\) 0 0
\(723\) 8.39986 0.312394
\(724\) 0 0
\(725\) −0.706420 −0.0262358
\(726\) 0 0
\(727\) −7.30395 −0.270888 −0.135444 0.990785i \(-0.543246\pi\)
−0.135444 + 0.990785i \(0.543246\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 9.51513i 0.351930i
\(732\) 0 0
\(733\) 17.8028i 0.657563i −0.944406 0.328781i \(-0.893362\pi\)
0.944406 0.328781i \(-0.106638\pi\)
\(734\) 0 0
\(735\) 7.43635 0.274294
\(736\) 0 0
\(737\) 49.9581 1.84023
\(738\) 0 0
\(739\) 30.6415i 1.12716i 0.826060 + 0.563582i \(0.190577\pi\)
−0.826060 + 0.563582i \(0.809423\pi\)
\(740\) 0 0
\(741\) 21.4129i 0.786623i
\(742\) 0 0
\(743\) 23.7573 0.871572 0.435786 0.900050i \(-0.356470\pi\)
0.435786 + 0.900050i \(0.356470\pi\)
\(744\) 0 0
\(745\) −22.3923 −0.820392
\(746\) 0 0
\(747\) 4.10586 0.150226
\(748\) 0 0
\(749\) 68.0948 2.48813
\(750\) 0 0
\(751\) −11.6741 −0.425996 −0.212998 0.977053i \(-0.568323\pi\)
−0.212998 + 0.977053i \(0.568323\pi\)
\(752\) 0 0
\(753\) 7.84091i 0.285739i
\(754\) 0 0
\(755\) −22.9206 −0.834167
\(756\) 0 0
\(757\) 16.7159i 0.607551i 0.952744 + 0.303776i \(0.0982473\pi\)
−0.952744 + 0.303776i \(0.901753\pi\)
\(758\) 0 0
\(759\) 17.1362 4.42080i 0.622004 0.160465i
\(760\) 0 0
\(761\) 25.0602 0.908430 0.454215 0.890892i \(-0.349920\pi\)
0.454215 + 0.890892i \(0.349920\pi\)
\(762\) 0 0
\(763\) 4.83786i 0.175142i
\(764\) 0 0
\(765\) −1.60151 −0.0579029
\(766\) 0 0
\(767\) 15.7812i 0.569826i
\(768\) 0 0
\(769\) 21.2483i 0.766234i 0.923700 + 0.383117i \(0.125149\pi\)
−0.923700 + 0.383117i \(0.874851\pi\)
\(770\) 0 0
\(771\) 5.18308i 0.186664i
\(772\) 0 0
\(773\) 19.8965i 0.715628i 0.933793 + 0.357814i \(0.116478\pi\)
−0.933793 + 0.357814i \(0.883522\pi\)
\(774\) 0 0
\(775\) 6.46598i 0.232265i
\(776\) 0 0
\(777\) −19.0655 −0.683971
\(778\) 0 0
\(779\) −87.3015 −3.12790
\(780\) 0 0
\(781\) 32.0780i 1.14784i
\(782\) 0 0
\(783\) 0.706420i 0.0252454i
\(784\) 0 0
\(785\) −14.5054 −0.517719
\(786\) 0 0
\(787\) 9.36272 0.333745 0.166873 0.985978i \(-0.446633\pi\)
0.166873 + 0.985978i \(0.446633\pi\)
\(788\) 0 0
\(789\) 6.41083i 0.228232i
\(790\) 0 0
\(791\) 12.6308i 0.449099i
\(792\) 0 0
\(793\) 8.78393i 0.311926i
\(794\) 0 0
\(795\) 7.25579i 0.257336i
\(796\) 0 0
\(797\) 43.4609i 1.53946i −0.638367 0.769732i \(-0.720390\pi\)
0.638367 0.769732i \(-0.279610\pi\)
\(798\) 0 0
\(799\) −7.14708 −0.252845
\(800\) 0 0
\(801\) 5.11034i 0.180565i
\(802\) 0 0
\(803\) 1.90760 0.0673178
\(804\) 0 0
\(805\)