Properties

Label 5520.2.be.c.1471.10
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
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: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.10
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.9

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -1.53388 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -1.53388 q^{7} -1.00000 q^{9} +3.80458 q^{11} +2.39402 q^{13} -1.00000 q^{15} +2.51901i q^{17} -3.13707 q^{19} -1.53388i q^{21} +(-3.77440 - 2.95870i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -1.76846 q^{29} -3.33803i q^{31} +3.80458i q^{33} -1.53388i q^{35} +3.84305i q^{37} +2.39402i q^{39} -12.2015 q^{41} -0.753300 q^{43} -1.00000i q^{45} +9.86269i q^{47} -4.64720 q^{49} -2.51901 q^{51} -5.02467i q^{53} +3.80458i q^{55} -3.13707i q^{57} -11.1175i q^{59} +13.1210i q^{61} +1.53388 q^{63} +2.39402i q^{65} -1.33054 q^{67} +(2.95870 - 3.77440i) q^{69} +7.54359i q^{71} +8.62861 q^{73} -1.00000i q^{75} -5.83578 q^{77} +2.23001 q^{79} +1.00000 q^{81} +16.1380 q^{83} -2.51901 q^{85} -1.76846i q^{87} +11.4152i q^{89} -3.67215 q^{91} +3.33803 q^{93} -3.13707i q^{95} +1.26542i q^{97} -3.80458 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) −1.53388 −0.579754 −0.289877 0.957064i \(-0.593614\pi\)
−0.289877 + 0.957064i \(0.593614\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.80458 1.14712 0.573562 0.819162i \(-0.305561\pi\)
0.573562 + 0.819162i \(0.305561\pi\)
\(12\) 0 0
\(13\) 2.39402 0.663982 0.331991 0.943283i \(-0.392280\pi\)
0.331991 + 0.943283i \(0.392280\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.51901i 0.610950i 0.952200 + 0.305475i \(0.0988152\pi\)
−0.952200 + 0.305475i \(0.901185\pi\)
\(18\) 0 0
\(19\) −3.13707 −0.719694 −0.359847 0.933011i \(-0.617171\pi\)
−0.359847 + 0.933011i \(0.617171\pi\)
\(20\) 0 0
\(21\) 1.53388i 0.334721i
\(22\) 0 0
\(23\) −3.77440 2.95870i −0.787018 0.616931i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −1.76846 −0.328395 −0.164198 0.986427i \(-0.552504\pi\)
−0.164198 + 0.986427i \(0.552504\pi\)
\(30\) 0 0
\(31\) 3.33803i 0.599528i −0.954013 0.299764i \(-0.903092\pi\)
0.954013 0.299764i \(-0.0969080\pi\)
\(32\) 0 0
\(33\) 3.80458i 0.662292i
\(34\) 0 0
\(35\) 1.53388i 0.259274i
\(36\) 0 0
\(37\) 3.84305i 0.631794i 0.948794 + 0.315897i \(0.102305\pi\)
−0.948794 + 0.315897i \(0.897695\pi\)
\(38\) 0 0
\(39\) 2.39402i 0.383350i
\(40\) 0 0
\(41\) −12.2015 −1.90555 −0.952774 0.303680i \(-0.901785\pi\)
−0.952774 + 0.303680i \(0.901785\pi\)
\(42\) 0 0
\(43\) −0.753300 −0.114877 −0.0574386 0.998349i \(-0.518293\pi\)
−0.0574386 + 0.998349i \(0.518293\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 9.86269i 1.43862i 0.694688 + 0.719311i \(0.255542\pi\)
−0.694688 + 0.719311i \(0.744458\pi\)
\(48\) 0 0
\(49\) −4.64720 −0.663886
\(50\) 0 0
\(51\) −2.51901 −0.352732
\(52\) 0 0
\(53\) 5.02467i 0.690191i −0.938568 0.345096i \(-0.887846\pi\)
0.938568 0.345096i \(-0.112154\pi\)
\(54\) 0 0
\(55\) 3.80458i 0.513009i
\(56\) 0 0
\(57\) 3.13707i 0.415516i
\(58\) 0 0
\(59\) 11.1175i 1.44738i −0.690124 0.723691i \(-0.742444\pi\)
0.690124 0.723691i \(-0.257556\pi\)
\(60\) 0 0
\(61\) 13.1210i 1.67997i 0.542612 + 0.839983i \(0.317435\pi\)
−0.542612 + 0.839983i \(0.682565\pi\)
\(62\) 0 0
\(63\) 1.53388 0.193251
\(64\) 0 0
\(65\) 2.39402i 0.296942i
\(66\) 0 0
\(67\) −1.33054 −0.162551 −0.0812757 0.996692i \(-0.525899\pi\)
−0.0812757 + 0.996692i \(0.525899\pi\)
\(68\) 0 0
\(69\) 2.95870 3.77440i 0.356185 0.454385i
\(70\) 0 0
\(71\) 7.54359i 0.895259i 0.894219 + 0.447630i \(0.147732\pi\)
−0.894219 + 0.447630i \(0.852268\pi\)
\(72\) 0 0
\(73\) 8.62861 1.00990 0.504951 0.863148i \(-0.331511\pi\)
0.504951 + 0.863148i \(0.331511\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −5.83578 −0.665049
\(78\) 0 0
\(79\) 2.23001 0.250895 0.125448 0.992100i \(-0.459963\pi\)
0.125448 + 0.992100i \(0.459963\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.1380 1.77138 0.885689 0.464279i \(-0.153686\pi\)
0.885689 + 0.464279i \(0.153686\pi\)
\(84\) 0 0
\(85\) −2.51901 −0.273225
\(86\) 0 0
\(87\) 1.76846i 0.189599i
\(88\) 0 0
\(89\) 11.4152i 1.21001i 0.796222 + 0.605005i \(0.206829\pi\)
−0.796222 + 0.605005i \(0.793171\pi\)
\(90\) 0 0
\(91\) −3.67215 −0.384946
\(92\) 0 0
\(93\) 3.33803 0.346138
\(94\) 0 0
\(95\) 3.13707i 0.321857i
\(96\) 0 0
\(97\) 1.26542i 0.128484i 0.997934 + 0.0642420i \(0.0204629\pi\)
−0.997934 + 0.0642420i \(0.979537\pi\)
\(98\) 0 0
\(99\) −3.80458 −0.382375
\(100\) 0 0
\(101\) −9.29932 −0.925317 −0.462659 0.886537i \(-0.653104\pi\)
−0.462659 + 0.886537i \(0.653104\pi\)
\(102\) 0 0
\(103\) −15.9393 −1.57055 −0.785275 0.619148i \(-0.787478\pi\)
−0.785275 + 0.619148i \(0.787478\pi\)
\(104\) 0 0
\(105\) 1.53388 0.149692
\(106\) 0 0
\(107\) −14.4419 −1.39616 −0.698078 0.716022i \(-0.745961\pi\)
−0.698078 + 0.716022i \(0.745961\pi\)
\(108\) 0 0
\(109\) 10.4690i 1.00275i 0.865231 + 0.501374i \(0.167172\pi\)
−0.865231 + 0.501374i \(0.832828\pi\)
\(110\) 0 0
\(111\) −3.84305 −0.364766
\(112\) 0 0
\(113\) 17.2391i 1.62171i −0.585244 0.810857i \(-0.699001\pi\)
0.585244 0.810857i \(-0.300999\pi\)
\(114\) 0 0
\(115\) 2.95870 3.77440i 0.275900 0.351965i
\(116\) 0 0
\(117\) −2.39402 −0.221327
\(118\) 0 0
\(119\) 3.86387i 0.354200i
\(120\) 0 0
\(121\) 3.47482 0.315892
\(122\) 0 0
\(123\) 12.2015i 1.10017i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 5.16262i 0.458109i −0.973414 0.229054i \(-0.926437\pi\)
0.973414 0.229054i \(-0.0735634\pi\)
\(128\) 0 0
\(129\) 0.753300i 0.0663243i
\(130\) 0 0
\(131\) 14.7628i 1.28983i −0.764255 0.644914i \(-0.776893\pi\)
0.764255 0.644914i \(-0.223107\pi\)
\(132\) 0 0
\(133\) 4.81191 0.417245
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 20.7112i 1.76948i 0.466088 + 0.884739i \(0.345663\pi\)
−0.466088 + 0.884739i \(0.654337\pi\)
\(138\) 0 0
\(139\) 5.33288i 0.452329i −0.974089 0.226165i \(-0.927381\pi\)
0.974089 0.226165i \(-0.0726187\pi\)
\(140\) 0 0
\(141\) −9.86269 −0.830589
\(142\) 0 0
\(143\) 9.10824 0.761669
\(144\) 0 0
\(145\) 1.76846i 0.146863i
\(146\) 0 0
\(147\) 4.64720i 0.383295i
\(148\) 0 0
\(149\) 10.2771i 0.841936i 0.907075 + 0.420968i \(0.138310\pi\)
−0.907075 + 0.420968i \(0.861690\pi\)
\(150\) 0 0
\(151\) 11.9498i 0.972465i 0.873830 + 0.486232i \(0.161629\pi\)
−0.873830 + 0.486232i \(0.838371\pi\)
\(152\) 0 0
\(153\) 2.51901i 0.203650i
\(154\) 0 0
\(155\) 3.33803 0.268117
\(156\) 0 0
\(157\) 10.9153i 0.871135i 0.900156 + 0.435567i \(0.143452\pi\)
−0.900156 + 0.435567i \(0.856548\pi\)
\(158\) 0 0
\(159\) 5.02467 0.398482
\(160\) 0 0
\(161\) 5.78950 + 4.53830i 0.456276 + 0.357668i
\(162\) 0 0
\(163\) 20.7616i 1.62617i −0.582142 0.813087i \(-0.697785\pi\)
0.582142 0.813087i \(-0.302215\pi\)
\(164\) 0 0
\(165\) −3.80458 −0.296186
\(166\) 0 0
\(167\) 12.0149i 0.929738i −0.885380 0.464869i \(-0.846101\pi\)
0.885380 0.464869i \(-0.153899\pi\)
\(168\) 0 0
\(169\) −7.26867 −0.559128
\(170\) 0 0
\(171\) 3.13707 0.239898
\(172\) 0 0
\(173\) −12.7396 −0.968571 −0.484285 0.874910i \(-0.660920\pi\)
−0.484285 + 0.874910i \(0.660920\pi\)
\(174\) 0 0
\(175\) 1.53388 0.115951
\(176\) 0 0
\(177\) 11.1175 0.835646
\(178\) 0 0
\(179\) 20.9807i 1.56817i 0.620653 + 0.784085i \(0.286868\pi\)
−0.620653 + 0.784085i \(0.713132\pi\)
\(180\) 0 0
\(181\) 11.2594i 0.836904i −0.908239 0.418452i \(-0.862573\pi\)
0.908239 0.418452i \(-0.137427\pi\)
\(182\) 0 0
\(183\) −13.1210 −0.969929
\(184\) 0 0
\(185\) −3.84305 −0.282547
\(186\) 0 0
\(187\) 9.58377i 0.700835i
\(188\) 0 0
\(189\) 1.53388i 0.111574i
\(190\) 0 0
\(191\) −20.2870 −1.46792 −0.733959 0.679194i \(-0.762330\pi\)
−0.733959 + 0.679194i \(0.762330\pi\)
\(192\) 0 0
\(193\) −19.5055 −1.40404 −0.702020 0.712158i \(-0.747718\pi\)
−0.702020 + 0.712158i \(0.747718\pi\)
\(194\) 0 0
\(195\) −2.39402 −0.171439
\(196\) 0 0
\(197\) −14.6532 −1.04400 −0.522000 0.852946i \(-0.674814\pi\)
−0.522000 + 0.852946i \(0.674814\pi\)
\(198\) 0 0
\(199\) −10.0280 −0.710866 −0.355433 0.934702i \(-0.615667\pi\)
−0.355433 + 0.934702i \(0.615667\pi\)
\(200\) 0 0
\(201\) 1.33054i 0.0938491i
\(202\) 0 0
\(203\) 2.71262 0.190388
\(204\) 0 0
\(205\) 12.2015i 0.852187i
\(206\) 0 0
\(207\) 3.77440 + 2.95870i 0.262339 + 0.205644i
\(208\) 0 0
\(209\) −11.9352 −0.825578
\(210\) 0 0
\(211\) 2.41236i 0.166073i 0.996546 + 0.0830367i \(0.0264619\pi\)
−0.996546 + 0.0830367i \(0.973538\pi\)
\(212\) 0 0
\(213\) −7.54359 −0.516878
\(214\) 0 0
\(215\) 0.753300i 0.0513746i
\(216\) 0 0
\(217\) 5.12016i 0.347579i
\(218\) 0 0
\(219\) 8.62861i 0.583067i
\(220\) 0 0
\(221\) 6.03056i 0.405659i
\(222\) 0 0
\(223\) 26.5580i 1.77845i −0.457466 0.889227i \(-0.651243\pi\)
0.457466 0.889227i \(-0.348757\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0.381119 0.0252957 0.0126479 0.999920i \(-0.495974\pi\)
0.0126479 + 0.999920i \(0.495974\pi\)
\(228\) 0 0
\(229\) 14.2631i 0.942532i −0.881991 0.471266i \(-0.843797\pi\)
0.881991 0.471266i \(-0.156203\pi\)
\(230\) 0 0
\(231\) 5.83578i 0.383966i
\(232\) 0 0
\(233\) 9.77344 0.640279 0.320140 0.947370i \(-0.396270\pi\)
0.320140 + 0.947370i \(0.396270\pi\)
\(234\) 0 0
\(235\) −9.86269 −0.643371
\(236\) 0 0
\(237\) 2.23001i 0.144854i
\(238\) 0 0
\(239\) 21.5599i 1.39459i −0.716784 0.697295i \(-0.754387\pi\)
0.716784 0.697295i \(-0.245613\pi\)
\(240\) 0 0
\(241\) 9.31459i 0.600005i −0.953938 0.300003i \(-0.903012\pi\)
0.953938 0.300003i \(-0.0969876\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 4.64720i 0.296899i
\(246\) 0 0
\(247\) −7.51022 −0.477864
\(248\) 0 0
\(249\) 16.1380i 1.02271i
\(250\) 0 0
\(251\) −22.6642 −1.43055 −0.715275 0.698843i \(-0.753699\pi\)
−0.715275 + 0.698843i \(0.753699\pi\)
\(252\) 0 0
\(253\) −14.3600 11.2566i −0.902806 0.707696i
\(254\) 0 0
\(255\) 2.51901i 0.157746i
\(256\) 0 0
\(257\) 2.37700 0.148273 0.0741367 0.997248i \(-0.476380\pi\)
0.0741367 + 0.997248i \(0.476380\pi\)
\(258\) 0 0
\(259\) 5.89479i 0.366285i
\(260\) 0 0
\(261\) 1.76846 0.109465
\(262\) 0 0
\(263\) −16.9954 −1.04798 −0.523991 0.851724i \(-0.675557\pi\)
−0.523991 + 0.851724i \(0.675557\pi\)
\(264\) 0 0
\(265\) 5.02467 0.308663
\(266\) 0 0
\(267\) −11.4152 −0.698599
\(268\) 0 0
\(269\) −22.4080 −1.36624 −0.683121 0.730305i \(-0.739378\pi\)
−0.683121 + 0.730305i \(0.739378\pi\)
\(270\) 0 0
\(271\) 0.440168i 0.0267383i 0.999911 + 0.0133692i \(0.00425566\pi\)
−0.999911 + 0.0133692i \(0.995744\pi\)
\(272\) 0 0
\(273\) 3.67215i 0.222249i
\(274\) 0 0
\(275\) −3.80458 −0.229425
\(276\) 0 0
\(277\) 24.5485 1.47498 0.737489 0.675360i \(-0.236012\pi\)
0.737489 + 0.675360i \(0.236012\pi\)
\(278\) 0 0
\(279\) 3.33803i 0.199843i
\(280\) 0 0
\(281\) 5.24108i 0.312656i −0.987705 0.156328i \(-0.950034\pi\)
0.987705 0.156328i \(-0.0499657\pi\)
\(282\) 0 0
\(283\) −0.477815 −0.0284032 −0.0142016 0.999899i \(-0.504521\pi\)
−0.0142016 + 0.999899i \(0.504521\pi\)
\(284\) 0 0
\(285\) 3.13707 0.185824
\(286\) 0 0
\(287\) 18.7156 1.10475
\(288\) 0 0
\(289\) 10.6546 0.626741
\(290\) 0 0
\(291\) −1.26542 −0.0741802
\(292\) 0 0
\(293\) 13.3598i 0.780486i 0.920712 + 0.390243i \(0.127609\pi\)
−0.920712 + 0.390243i \(0.872391\pi\)
\(294\) 0 0
\(295\) 11.1175 0.647289
\(296\) 0 0
\(297\) 3.80458i 0.220764i
\(298\) 0 0
\(299\) −9.03600 7.08317i −0.522565 0.409631i
\(300\) 0 0
\(301\) 1.15547 0.0666004
\(302\) 0 0
\(303\) 9.29932i 0.534232i
\(304\) 0 0
\(305\) −13.1210 −0.751304
\(306\) 0 0
\(307\) 29.7216i 1.69630i −0.529757 0.848150i \(-0.677717\pi\)
0.529757 0.848150i \(-0.322283\pi\)
\(308\) 0 0
\(309\) 15.9393i 0.906757i
\(310\) 0 0
\(311\) 28.2256i 1.60053i 0.599650 + 0.800263i \(0.295307\pi\)
−0.599650 + 0.800263i \(0.704693\pi\)
\(312\) 0 0
\(313\) 4.82571i 0.272765i 0.990656 + 0.136383i \(0.0435477\pi\)
−0.990656 + 0.136383i \(0.956452\pi\)
\(314\) 0 0
\(315\) 1.53388i 0.0864246i
\(316\) 0 0
\(317\) −5.23453 −0.294001 −0.147000 0.989136i \(-0.546962\pi\)
−0.147000 + 0.989136i \(0.546962\pi\)
\(318\) 0 0
\(319\) −6.72826 −0.376710
\(320\) 0 0
\(321\) 14.4419i 0.806071i
\(322\) 0 0
\(323\) 7.90232i 0.439697i
\(324\) 0 0
\(325\) −2.39402 −0.132796
\(326\) 0 0
\(327\) −10.4690 −0.578937
\(328\) 0 0
\(329\) 15.1282i 0.834046i
\(330\) 0 0
\(331\) 21.6135i 1.18799i −0.804470 0.593993i \(-0.797551\pi\)
0.804470 0.593993i \(-0.202449\pi\)
\(332\) 0 0
\(333\) 3.84305i 0.210598i
\(334\) 0 0
\(335\) 1.33054i 0.0726952i
\(336\) 0 0
\(337\) 2.81143i 0.153148i −0.997064 0.0765742i \(-0.975602\pi\)
0.997064 0.0765742i \(-0.0243982\pi\)
\(338\) 0 0
\(339\) 17.2391 0.936297
\(340\) 0 0
\(341\) 12.6998i 0.687733i
\(342\) 0 0
\(343\) 17.8655 0.964644
\(344\) 0 0
\(345\) 3.77440 + 2.95870i 0.203207 + 0.159291i
\(346\) 0 0
\(347\) 11.8025i 0.633591i −0.948494 0.316796i \(-0.897393\pi\)
0.948494 0.316796i \(-0.102607\pi\)
\(348\) 0 0
\(349\) −32.7930 −1.75537 −0.877685 0.479237i \(-0.840913\pi\)
−0.877685 + 0.479237i \(0.840913\pi\)
\(350\) 0 0
\(351\) 2.39402i 0.127783i
\(352\) 0 0
\(353\) 14.1618 0.753755 0.376877 0.926263i \(-0.376998\pi\)
0.376877 + 0.926263i \(0.376998\pi\)
\(354\) 0 0
\(355\) −7.54359 −0.400372
\(356\) 0 0
\(357\) 3.86387 0.204498
\(358\) 0 0
\(359\) 3.37945 0.178360 0.0891802 0.996016i \(-0.471575\pi\)
0.0891802 + 0.996016i \(0.471575\pi\)
\(360\) 0 0
\(361\) −9.15877 −0.482040
\(362\) 0 0
\(363\) 3.47482i 0.182381i
\(364\) 0 0
\(365\) 8.62861i 0.451642i
\(366\) 0 0
\(367\) 3.36610 0.175709 0.0878546 0.996133i \(-0.471999\pi\)
0.0878546 + 0.996133i \(0.471999\pi\)
\(368\) 0 0
\(369\) 12.2015 0.635183
\(370\) 0 0
\(371\) 7.70726i 0.400141i
\(372\) 0 0
\(373\) 28.8077i 1.49160i 0.666168 + 0.745802i \(0.267934\pi\)
−0.666168 + 0.745802i \(0.732066\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −4.23374 −0.218049
\(378\) 0 0
\(379\) 21.3903 1.09875 0.549374 0.835577i \(-0.314866\pi\)
0.549374 + 0.835577i \(0.314866\pi\)
\(380\) 0 0
\(381\) 5.16262 0.264489
\(382\) 0 0
\(383\) 37.4579 1.91401 0.957005 0.290072i \(-0.0936793\pi\)
0.957005 + 0.290072i \(0.0936793\pi\)
\(384\) 0 0
\(385\) 5.83578i 0.297419i
\(386\) 0 0
\(387\) 0.753300 0.0382924
\(388\) 0 0
\(389\) 33.0282i 1.67460i 0.546745 + 0.837299i \(0.315867\pi\)
−0.546745 + 0.837299i \(0.684133\pi\)
\(390\) 0 0
\(391\) 7.45298 9.50776i 0.376913 0.480828i
\(392\) 0 0
\(393\) 14.7628 0.744683
\(394\) 0 0
\(395\) 2.23001i 0.112204i
\(396\) 0 0
\(397\) 6.74610 0.338577 0.169289 0.985567i \(-0.445853\pi\)
0.169289 + 0.985567i \(0.445853\pi\)
\(398\) 0 0
\(399\) 4.81191i 0.240897i
\(400\) 0 0
\(401\) 24.2243i 1.20971i 0.796337 + 0.604853i \(0.206768\pi\)
−0.796337 + 0.604853i \(0.793232\pi\)
\(402\) 0 0
\(403\) 7.99132i 0.398076i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 14.6212i 0.724745i
\(408\) 0 0
\(409\) 32.7223 1.61802 0.809008 0.587798i \(-0.200005\pi\)
0.809008 + 0.587798i \(0.200005\pi\)
\(410\) 0 0
\(411\) −20.7112 −1.02161
\(412\) 0 0
\(413\) 17.0530i 0.839125i
\(414\) 0 0
\(415\) 16.1380i 0.792185i
\(416\) 0 0
\(417\) 5.33288 0.261152
\(418\) 0 0
\(419\) −15.7062 −0.767299 −0.383649 0.923479i \(-0.625333\pi\)
−0.383649 + 0.923479i \(0.625333\pi\)
\(420\) 0 0
\(421\) 8.90543i 0.434024i 0.976169 + 0.217012i \(0.0696311\pi\)
−0.976169 + 0.217012i \(0.930369\pi\)
\(422\) 0 0
\(423\) 9.86269i 0.479541i
\(424\) 0 0
\(425\) 2.51901i 0.122190i
\(426\) 0 0
\(427\) 20.1260i 0.973967i
\(428\) 0 0
\(429\) 9.10824i 0.439750i
\(430\) 0 0
\(431\) −14.1215 −0.680208 −0.340104 0.940388i \(-0.610462\pi\)
−0.340104 + 0.940388i \(0.610462\pi\)
\(432\) 0 0
\(433\) 20.3867i 0.979722i −0.871801 0.489861i \(-0.837048\pi\)
0.871801 0.489861i \(-0.162952\pi\)
\(434\) 0 0
\(435\) 1.76846 0.0847913
\(436\) 0 0
\(437\) 11.8406 + 9.28164i 0.566412 + 0.444001i
\(438\) 0 0
\(439\) 22.5795i 1.07766i −0.842414 0.538830i \(-0.818866\pi\)
0.842414 0.538830i \(-0.181134\pi\)
\(440\) 0 0
\(441\) 4.64720 0.221295
\(442\) 0 0
\(443\) 13.2144i 0.627833i 0.949451 + 0.313916i \(0.101641\pi\)
−0.949451 + 0.313916i \(0.898359\pi\)
\(444\) 0 0
\(445\) −11.4152 −0.541133
\(446\) 0 0
\(447\) −10.2771 −0.486092
\(448\) 0 0
\(449\) 40.8158 1.92621 0.963107 0.269118i \(-0.0867320\pi\)
0.963107 + 0.269118i \(0.0867320\pi\)
\(450\) 0 0
\(451\) −46.4214 −2.18590
\(452\) 0 0
\(453\) −11.9498 −0.561453
\(454\) 0 0
\(455\) 3.67215i 0.172153i
\(456\) 0 0
\(457\) 36.8494i 1.72374i −0.507126 0.861872i \(-0.669292\pi\)
0.507126 0.861872i \(-0.330708\pi\)
\(458\) 0 0
\(459\) 2.51901 0.117577
\(460\) 0 0
\(461\) 12.2462 0.570363 0.285182 0.958473i \(-0.407946\pi\)
0.285182 + 0.958473i \(0.407946\pi\)
\(462\) 0 0
\(463\) 0.407169i 0.0189228i 0.999955 + 0.00946138i \(0.00301170\pi\)
−0.999955 + 0.00946138i \(0.996988\pi\)
\(464\) 0 0
\(465\) 3.33803i 0.154798i
\(466\) 0 0
\(467\) −12.3451 −0.571262 −0.285631 0.958340i \(-0.592203\pi\)
−0.285631 + 0.958340i \(0.592203\pi\)
\(468\) 0 0
\(469\) 2.04089 0.0942398
\(470\) 0 0
\(471\) −10.9153 −0.502950
\(472\) 0 0
\(473\) −2.86599 −0.131778
\(474\) 0 0
\(475\) 3.13707 0.143939
\(476\) 0 0
\(477\) 5.02467i 0.230064i
\(478\) 0 0
\(479\) 13.8805 0.634218 0.317109 0.948389i \(-0.397288\pi\)
0.317109 + 0.948389i \(0.397288\pi\)
\(480\) 0 0
\(481\) 9.20034i 0.419499i
\(482\) 0 0
\(483\) −4.53830 + 5.78950i −0.206500 + 0.263431i
\(484\) 0 0
\(485\) −1.26542 −0.0574598
\(486\) 0 0
\(487\) 0.727009i 0.0329439i −0.999864 0.0164720i \(-0.994757\pi\)
0.999864 0.0164720i \(-0.00524343\pi\)
\(488\) 0 0
\(489\) 20.7616 0.938872
\(490\) 0 0
\(491\) 4.39551i 0.198367i −0.995069 0.0991833i \(-0.968377\pi\)
0.995069 0.0991833i \(-0.0316230\pi\)
\(492\) 0 0
\(493\) 4.45478i 0.200633i
\(494\) 0 0
\(495\) 3.80458i 0.171003i
\(496\) 0 0
\(497\) 11.5710i 0.519030i
\(498\) 0 0
\(499\) 17.7313i 0.793764i 0.917870 + 0.396882i \(0.129908\pi\)
−0.917870 + 0.396882i \(0.870092\pi\)
\(500\) 0 0
\(501\) 12.0149 0.536784
\(502\) 0 0
\(503\) −5.13524 −0.228969 −0.114485 0.993425i \(-0.536522\pi\)
−0.114485 + 0.993425i \(0.536522\pi\)
\(504\) 0 0
\(505\) 9.29932i 0.413814i
\(506\) 0 0
\(507\) 7.26867i 0.322813i
\(508\) 0 0
\(509\) −28.7319 −1.27352 −0.636760 0.771062i \(-0.719726\pi\)
−0.636760 + 0.771062i \(0.719726\pi\)
\(510\) 0 0
\(511\) −13.2353 −0.585495
\(512\) 0 0
\(513\) 3.13707i 0.138505i
\(514\) 0 0
\(515\) 15.9393i 0.702371i
\(516\) 0 0
\(517\) 37.5234i 1.65028i
\(518\) 0 0
\(519\) 12.7396i 0.559205i
\(520\) 0 0
\(521\) 1.48894i 0.0652317i 0.999468 + 0.0326159i \(0.0103838\pi\)
−0.999468 + 0.0326159i \(0.989616\pi\)
\(522\) 0 0
\(523\) 22.5119 0.984376 0.492188 0.870489i \(-0.336197\pi\)
0.492188 + 0.870489i \(0.336197\pi\)
\(524\) 0 0
\(525\) 1.53388i 0.0669442i
\(526\) 0 0
\(527\) 8.40854 0.366282
\(528\) 0 0
\(529\) 5.49224 + 22.3346i 0.238793 + 0.971070i
\(530\) 0 0
\(531\) 11.1175i 0.482461i
\(532\) 0 0
\(533\) −29.2105 −1.26525
\(534\) 0 0
\(535\) 14.4419i 0.624380i
\(536\) 0 0
\(537\) −20.9807 −0.905384
\(538\) 0 0
\(539\) −17.6806 −0.761559
\(540\) 0 0
\(541\) 4.42320 0.190168 0.0950841 0.995469i \(-0.469688\pi\)
0.0950841 + 0.995469i \(0.469688\pi\)
\(542\) 0 0
\(543\) 11.2594 0.483187
\(544\) 0 0
\(545\) −10.4690 −0.448443
\(546\) 0 0
\(547\) 19.9051i 0.851079i 0.904940 + 0.425540i \(0.139916\pi\)
−0.904940 + 0.425540i \(0.860084\pi\)
\(548\) 0 0
\(549\) 13.1210i 0.559989i
\(550\) 0 0
\(551\) 5.54780 0.236344
\(552\) 0 0
\(553\) −3.42057 −0.145457
\(554\) 0 0
\(555\) 3.84305i 0.163128i
\(556\) 0 0
\(557\) 23.5795i 0.999094i 0.866287 + 0.499547i \(0.166500\pi\)
−0.866287 + 0.499547i \(0.833500\pi\)
\(558\) 0 0
\(559\) −1.80341 −0.0762763
\(560\) 0 0
\(561\) −9.58377 −0.404627
\(562\) 0 0
\(563\) 19.7524 0.832462 0.416231 0.909259i \(-0.363351\pi\)
0.416231 + 0.909259i \(0.363351\pi\)
\(564\) 0 0
\(565\) 17.2391 0.725253
\(566\) 0 0
\(567\) −1.53388 −0.0644171
\(568\) 0 0
\(569\) 20.6533i 0.865833i 0.901434 + 0.432916i \(0.142516\pi\)
−0.901434 + 0.432916i \(0.857484\pi\)
\(570\) 0 0
\(571\) −15.4386 −0.646084 −0.323042 0.946385i \(-0.604706\pi\)
−0.323042 + 0.946385i \(0.604706\pi\)
\(572\) 0 0
\(573\) 20.2870i 0.847503i
\(574\) 0 0
\(575\) 3.77440 + 2.95870i 0.157404 + 0.123386i
\(576\) 0 0
\(577\) 45.8416 1.90841 0.954206 0.299150i \(-0.0967031\pi\)
0.954206 + 0.299150i \(0.0967031\pi\)
\(578\) 0 0
\(579\) 19.5055i 0.810623i
\(580\) 0 0
\(581\) −24.7539 −1.02696
\(582\) 0 0
\(583\) 19.1167i 0.791735i
\(584\) 0 0
\(585\) 2.39402i 0.0989805i
\(586\) 0 0
\(587\) 8.31516i 0.343203i 0.985166 + 0.171602i \(0.0548942\pi\)
−0.985166 + 0.171602i \(0.945106\pi\)
\(588\) 0 0
\(589\) 10.4717i 0.431477i
\(590\) 0 0
\(591\) 14.6532i 0.602753i
\(592\) 0 0
\(593\) −43.8609 −1.80115 −0.900577 0.434697i \(-0.856855\pi\)
−0.900577 + 0.434697i \(0.856855\pi\)
\(594\) 0 0
\(595\) 3.86387 0.158403
\(596\) 0 0
\(597\) 10.0280i 0.410419i
\(598\) 0 0
\(599\) 30.3057i 1.23826i 0.785290 + 0.619128i \(0.212514\pi\)
−0.785290 + 0.619128i \(0.787486\pi\)
\(600\) 0 0
\(601\) 46.2015 1.88460 0.942300 0.334770i \(-0.108659\pi\)
0.942300 + 0.334770i \(0.108659\pi\)
\(602\) 0 0
\(603\) 1.33054 0.0541838
\(604\) 0 0
\(605\) 3.47482i 0.141271i
\(606\) 0 0
\(607\) 29.0800i 1.18032i 0.807286 + 0.590161i \(0.200936\pi\)
−0.807286 + 0.590161i \(0.799064\pi\)
\(608\) 0 0
\(609\) 2.71262i 0.109921i
\(610\) 0 0
\(611\) 23.6115i 0.955218i
\(612\) 0 0
\(613\) 21.3595i 0.862700i −0.902185 0.431350i \(-0.858037\pi\)
0.902185 0.431350i \(-0.141963\pi\)
\(614\) 0 0
\(615\) 12.2015 0.492010
\(616\) 0 0
\(617\) 7.10553i 0.286058i −0.989719 0.143029i \(-0.954316\pi\)
0.989719 0.143029i \(-0.0456842\pi\)
\(618\) 0 0
\(619\) −43.7859 −1.75990 −0.879952 0.475063i \(-0.842425\pi\)
−0.879952 + 0.475063i \(0.842425\pi\)
\(620\) 0 0
\(621\) −2.95870 + 3.77440i −0.118728 + 0.151462i
\(622\) 0 0
\(623\) 17.5096i 0.701508i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.9352i 0.476648i
\(628\) 0 0
\(629\) −9.68068 −0.385994
\(630\) 0 0
\(631\) −23.3043 −0.927728 −0.463864 0.885906i \(-0.653537\pi\)
−0.463864 + 0.885906i \(0.653537\pi\)
\(632\) 0 0
\(633\) −2.41236 −0.0958826
\(634\) 0 0
\(635\) 5.16262 0.204872
\(636\) 0 0
\(637\) −11.1255 −0.440808
\(638\) 0 0
\(639\) 7.54359i 0.298420i
\(640\) 0 0
\(641\) 16.9408i 0.669123i −0.942374 0.334562i \(-0.891412\pi\)
0.942374 0.334562i \(-0.108588\pi\)
\(642\) 0 0
\(643\) 12.8234 0.505705 0.252853 0.967505i \(-0.418631\pi\)
0.252853 + 0.967505i \(0.418631\pi\)
\(644\) 0 0
\(645\) 0.753300 0.0296611
\(646\) 0 0
\(647\) 3.68104i 0.144717i −0.997379 0.0723583i \(-0.976948\pi\)
0.997379 0.0723583i \(-0.0230525\pi\)
\(648\) 0 0
\(649\) 42.2976i 1.66033i
\(650\) 0 0
\(651\) −5.12016 −0.200675
\(652\) 0 0
\(653\) −16.5223 −0.646567 −0.323284 0.946302i \(-0.604787\pi\)
−0.323284 + 0.946302i \(0.604787\pi\)
\(654\) 0 0
\(655\) 14.7628 0.576829
\(656\) 0 0
\(657\) −8.62861 −0.336634
\(658\) 0 0
\(659\) −25.6540 −0.999338 −0.499669 0.866216i \(-0.666545\pi\)
−0.499669 + 0.866216i \(0.666545\pi\)
\(660\) 0 0
\(661\) 1.84288i 0.0716795i −0.999358 0.0358398i \(-0.988589\pi\)
0.999358 0.0358398i \(-0.0114106\pi\)
\(662\) 0 0
\(663\) −6.03056 −0.234207
\(664\) 0 0
\(665\) 4.81191i 0.186598i
\(666\) 0 0
\(667\) 6.67490 + 5.23235i 0.258453 + 0.202597i
\(668\) 0 0
\(669\) 26.5580 1.02679
\(670\) 0 0
\(671\) 49.9197i 1.92713i
\(672\) 0 0
\(673\) −14.6701 −0.565489 −0.282745 0.959195i \(-0.591245\pi\)
−0.282745 + 0.959195i \(0.591245\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 21.5348i 0.827649i 0.910357 + 0.413825i \(0.135807\pi\)
−0.910357 + 0.413825i \(0.864193\pi\)
\(678\) 0 0
\(679\) 1.94101i 0.0744890i
\(680\) 0 0
\(681\) 0.381119i 0.0146045i
\(682\) 0 0
\(683\) 2.94492i 0.112684i −0.998412 0.0563421i \(-0.982056\pi\)
0.998412 0.0563421i \(-0.0179437\pi\)
\(684\) 0 0
\(685\) −20.7112 −0.791334
\(686\) 0 0
\(687\) 14.2631 0.544171
\(688\) 0 0
\(689\) 12.0292i 0.458274i
\(690\) 0 0
\(691\) 39.4261i 1.49984i 0.661529 + 0.749920i \(0.269908\pi\)
−0.661529 + 0.749920i \(0.730092\pi\)
\(692\) 0 0
\(693\) 5.83578 0.221683
\(694\) 0 0
\(695\) 5.33288 0.202288
\(696\) 0 0
\(697\) 30.7356i 1.16419i
\(698\) 0 0
\(699\) 9.77344i 0.369665i
\(700\) 0 0
\(701\) 31.2217i 1.17923i 0.807686 + 0.589613i \(0.200720\pi\)
−0.807686 + 0.589613i \(0.799280\pi\)
\(702\) 0 0
\(703\) 12.0559i 0.454698i
\(704\) 0 0
\(705\) 9.86269i 0.371451i
\(706\) 0 0
\(707\) 14.2641 0.536456
\(708\) 0 0
\(709\) 32.9346i 1.23689i −0.785830 0.618443i \(-0.787764\pi\)
0.785830 0.618443i \(-0.212236\pi\)
\(710\) 0 0
\(711\) −2.23001 −0.0836317
\(712\) 0 0
\(713\) −9.87622 + 12.5991i −0.369867 + 0.471839i
\(714\) 0 0
\(715\) 9.10824i 0.340629i
\(716\) 0 0
\(717\) 21.5599 0.805167
\(718\) 0 0
\(719\) 2.50314i 0.0933514i −0.998910 0.0466757i \(-0.985137\pi\)
0.998910 0.0466757i \(-0.0148627\pi\)
\(720\) 0 0
\(721\) 24.4491 0.910532
\(722\) 0 0
\(723\) 9.31459 0.346413
\(724\) 0 0
\(725\) 1.76846 0.0656791
\(726\) 0 0
\(727\) −38.9275 −1.44374 −0.721871 0.692028i \(-0.756718\pi\)
−0.721871 + 0.692028i \(0.756718\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 1.89757i 0.0701841i
\(732\) 0 0
\(733\) 39.0232i 1.44135i −0.693271 0.720677i \(-0.743831\pi\)
0.693271 0.720677i \(-0.256169\pi\)
\(734\) 0 0
\(735\) 4.64720 0.171415
\(736\) 0 0
\(737\) −5.06215 −0.186467
\(738\) 0 0
\(739\) 7.48023i 0.275165i 0.990490 + 0.137582i \(0.0439332\pi\)
−0.990490 + 0.137582i \(0.956067\pi\)
\(740\) 0 0
\(741\) 7.51022i 0.275895i
\(742\) 0 0
\(743\) 23.9955 0.880310 0.440155 0.897922i \(-0.354924\pi\)
0.440155 + 0.897922i \(0.354924\pi\)
\(744\) 0 0
\(745\) −10.2771 −0.376525
\(746\) 0 0
\(747\) −16.1380 −0.590459
\(748\) 0 0
\(749\) 22.1523 0.809426
\(750\) 0 0
\(751\) 0.141614 0.00516756 0.00258378 0.999997i \(-0.499178\pi\)
0.00258378 + 0.999997i \(0.499178\pi\)
\(752\) 0 0
\(753\) 22.6642i 0.825929i
\(754\) 0 0
\(755\) −11.9498 −0.434899
\(756\) 0 0
\(757\) 6.84512i 0.248790i −0.992233 0.124395i \(-0.960301\pi\)
0.992233 0.124395i \(-0.0396990\pi\)
\(758\) 0 0
\(759\) 11.2566 14.3600i 0.408588 0.521235i
\(760\) 0 0
\(761\) −30.8852 −1.11959 −0.559794 0.828632i \(-0.689120\pi\)
−0.559794 + 0.828632i \(0.689120\pi\)
\(762\) 0 0
\(763\) 16.0582i 0.581347i
\(764\) 0 0
\(765\) 2.51901 0.0910750
\(766\) 0 0
\(767\) 26.6156i 0.961035i
\(768\) 0 0
\(769\) 12.3453i 0.445183i −0.974912 0.222592i \(-0.928548\pi\)
0.974912 0.222592i \(-0.0714516\pi\)
\(770\) 0 0
\(771\) 2.37700i 0.0856057i
\(772\) 0 0
\(773\) 19.8084i 0.712457i 0.934399 + 0.356228i \(0.115938\pi\)
−0.934399 + 0.356228i \(0.884062\pi\)
\(774\) 0 0
\(775\) 3.33803i 0.119906i
\(776\) 0 0
\(777\) 5.89479 0.211475
\(778\) 0 0
\(779\) 38.2769 1.37141
\(780\) 0 0
\(781\) 28.7002i 1.02697i
\(782\) 0 0
\(783\) 1.76846i 0.0631997i
\(784\) 0 0
\(785\) −10.9153 −0.389583
\(786\) 0 0
\(787\) 15.0357 0.535965 0.267983 0.963424i \(-0.413643\pi\)
0.267983 + 0.963424i \(0.413643\pi\)
\(788\) 0 0
\(789\) 16.9954i 0.605052i
\(790\) 0 0
\(791\) 26.4427i 0.940195i
\(792\) 0 0
\(793\) 31.4118i 1.11547i
\(794\) 0 0
\(795\) 5.02467i 0.178207i
\(796\) 0 0
\(797\) 28.5067i 1.00976i 0.863190 + 0.504880i \(0.168463\pi\)
−0.863190 + 0.504880i \(0.831537\pi\)
\(798\) 0 0
\(799\) −24.8442 −0.878925
\(800\) 0 0
\(801\) 11.4152i 0.403337i
\(802\) 0 0
\(803\) 32.8282 1.15848
\(804\) 0 0
\(805\) −4.53830 + 5.78950i −0.159954 + 0.204053i
\(806\) 0