Properties

Label 5520.2.be.c.1471.16
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 1471.16
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.15

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} +3.60511 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} +3.60511 q^{7} -1.00000 q^{9} +5.78590 q^{11} +0.896503 q^{13} -1.00000 q^{15} +6.83315i q^{17} +1.90832 q^{19} +3.60511i q^{21} +(4.77275 - 0.469990i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +1.39299 q^{29} +1.12061i q^{31} +5.78590i q^{33} +3.60511i q^{35} -7.48133i q^{37} +0.896503i q^{39} +3.19938 q^{41} +2.53782 q^{43} -1.00000i q^{45} -5.80647i q^{47} +5.99683 q^{49} -6.83315 q^{51} +2.67178i q^{53} +5.78590i q^{55} +1.90832i q^{57} +6.58667i q^{59} +0.597564i q^{61} -3.60511 q^{63} +0.896503i q^{65} -5.41745 q^{67} +(0.469990 + 4.77275i) q^{69} -3.72842i q^{71} +0.479048 q^{73} -1.00000i q^{75} +20.8588 q^{77} +1.03711 q^{79} +1.00000 q^{81} -11.2088 q^{83} -6.83315 q^{85} +1.39299i q^{87} -5.21573i q^{89} +3.23199 q^{91} -1.12061 q^{93} +1.90832i q^{95} -4.73433i q^{97} -5.78590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) 3.60511 1.36260 0.681302 0.732002i \(-0.261414\pi\)
0.681302 + 0.732002i \(0.261414\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.78590 1.74452 0.872258 0.489046i \(-0.162655\pi\)
0.872258 + 0.489046i \(0.162655\pi\)
\(12\) 0 0
\(13\) 0.896503 0.248645 0.124323 0.992242i \(-0.460324\pi\)
0.124323 + 0.992242i \(0.460324\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 6.83315i 1.65728i 0.559781 + 0.828641i \(0.310885\pi\)
−0.559781 + 0.828641i \(0.689115\pi\)
\(18\) 0 0
\(19\) 1.90832 0.437798 0.218899 0.975748i \(-0.429753\pi\)
0.218899 + 0.975748i \(0.429753\pi\)
\(20\) 0 0
\(21\) 3.60511i 0.786700i
\(22\) 0 0
\(23\) 4.77275 0.469990i 0.995186 0.0979997i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 1.39299 0.258672 0.129336 0.991601i \(-0.458715\pi\)
0.129336 + 0.991601i \(0.458715\pi\)
\(30\) 0 0
\(31\) 1.12061i 0.201268i 0.994924 + 0.100634i \(0.0320871\pi\)
−0.994924 + 0.100634i \(0.967913\pi\)
\(32\) 0 0
\(33\) 5.78590i 1.00720i
\(34\) 0 0
\(35\) 3.60511i 0.609375i
\(36\) 0 0
\(37\) 7.48133i 1.22992i −0.788557 0.614961i \(-0.789172\pi\)
0.788557 0.614961i \(-0.210828\pi\)
\(38\) 0 0
\(39\) 0.896503i 0.143555i
\(40\) 0 0
\(41\) 3.19938 0.499660 0.249830 0.968290i \(-0.419625\pi\)
0.249830 + 0.968290i \(0.419625\pi\)
\(42\) 0 0
\(43\) 2.53782 0.387014 0.193507 0.981099i \(-0.438014\pi\)
0.193507 + 0.981099i \(0.438014\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 5.80647i 0.846961i −0.905905 0.423480i \(-0.860808\pi\)
0.905905 0.423480i \(-0.139192\pi\)
\(48\) 0 0
\(49\) 5.99683 0.856690
\(50\) 0 0
\(51\) −6.83315 −0.956832
\(52\) 0 0
\(53\) 2.67178i 0.366998i 0.983020 + 0.183499i \(0.0587424\pi\)
−0.983020 + 0.183499i \(0.941258\pi\)
\(54\) 0 0
\(55\) 5.78590i 0.780171i
\(56\) 0 0
\(57\) 1.90832i 0.252763i
\(58\) 0 0
\(59\) 6.58667i 0.857511i 0.903420 + 0.428756i \(0.141048\pi\)
−0.903420 + 0.428756i \(0.858952\pi\)
\(60\) 0 0
\(61\) 0.597564i 0.0765102i 0.999268 + 0.0382551i \(0.0121800\pi\)
−0.999268 + 0.0382551i \(0.987820\pi\)
\(62\) 0 0
\(63\) −3.60511 −0.454201
\(64\) 0 0
\(65\) 0.896503i 0.111197i
\(66\) 0 0
\(67\) −5.41745 −0.661846 −0.330923 0.943658i \(-0.607360\pi\)
−0.330923 + 0.943658i \(0.607360\pi\)
\(68\) 0 0
\(69\) 0.469990 + 4.77275i 0.0565801 + 0.574571i
\(70\) 0 0
\(71\) 3.72842i 0.442482i −0.975219 0.221241i \(-0.928989\pi\)
0.975219 0.221241i \(-0.0710108\pi\)
\(72\) 0 0
\(73\) 0.479048 0.0560683 0.0280342 0.999607i \(-0.491075\pi\)
0.0280342 + 0.999607i \(0.491075\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 20.8588 2.37708
\(78\) 0 0
\(79\) 1.03711 0.116684 0.0583422 0.998297i \(-0.481419\pi\)
0.0583422 + 0.998297i \(0.481419\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.2088 −1.23033 −0.615163 0.788400i \(-0.710910\pi\)
−0.615163 + 0.788400i \(0.710910\pi\)
\(84\) 0 0
\(85\) −6.83315 −0.741159
\(86\) 0 0
\(87\) 1.39299i 0.149344i
\(88\) 0 0
\(89\) 5.21573i 0.552866i −0.961033 0.276433i \(-0.910848\pi\)
0.961033 0.276433i \(-0.0891524\pi\)
\(90\) 0 0
\(91\) 3.23199 0.338805
\(92\) 0 0
\(93\) −1.12061 −0.116202
\(94\) 0 0
\(95\) 1.90832i 0.195789i
\(96\) 0 0
\(97\) 4.73433i 0.480698i −0.970687 0.240349i \(-0.922738\pi\)
0.970687 0.240349i \(-0.0772619\pi\)
\(98\) 0 0
\(99\) −5.78590 −0.581505
\(100\) 0 0
\(101\) −0.362853 −0.0361052 −0.0180526 0.999837i \(-0.505747\pi\)
−0.0180526 + 0.999837i \(0.505747\pi\)
\(102\) 0 0
\(103\) −11.7189 −1.15470 −0.577348 0.816498i \(-0.695912\pi\)
−0.577348 + 0.816498i \(0.695912\pi\)
\(104\) 0 0
\(105\) −3.60511 −0.351823
\(106\) 0 0
\(107\) 12.6988 1.22764 0.613818 0.789447i \(-0.289633\pi\)
0.613818 + 0.789447i \(0.289633\pi\)
\(108\) 0 0
\(109\) 8.37783i 0.802451i −0.915979 0.401225i \(-0.868584\pi\)
0.915979 0.401225i \(-0.131416\pi\)
\(110\) 0 0
\(111\) 7.48133 0.710096
\(112\) 0 0
\(113\) 7.77619i 0.731522i −0.930709 0.365761i \(-0.880809\pi\)
0.930709 0.365761i \(-0.119191\pi\)
\(114\) 0 0
\(115\) 0.469990 + 4.77275i 0.0438268 + 0.445061i
\(116\) 0 0
\(117\) −0.896503 −0.0828817
\(118\) 0 0
\(119\) 24.6343i 2.25822i
\(120\) 0 0
\(121\) 22.4767 2.04334
\(122\) 0 0
\(123\) 3.19938i 0.288479i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 5.02772i 0.446138i 0.974803 + 0.223069i \(0.0716075\pi\)
−0.974803 + 0.223069i \(0.928392\pi\)
\(128\) 0 0
\(129\) 2.53782i 0.223443i
\(130\) 0 0
\(131\) 16.0105i 1.39885i −0.714707 0.699424i \(-0.753440\pi\)
0.714707 0.699424i \(-0.246560\pi\)
\(132\) 0 0
\(133\) 6.87970 0.596545
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 23.3183i 1.99222i −0.0881386 0.996108i \(-0.528092\pi\)
0.0881386 0.996108i \(-0.471908\pi\)
\(138\) 0 0
\(139\) 2.21510i 0.187882i 0.995578 + 0.0939412i \(0.0299466\pi\)
−0.995578 + 0.0939412i \(0.970053\pi\)
\(140\) 0 0
\(141\) 5.80647 0.488993
\(142\) 0 0
\(143\) 5.18708 0.433765
\(144\) 0 0
\(145\) 1.39299i 0.115682i
\(146\) 0 0
\(147\) 5.99683i 0.494610i
\(148\) 0 0
\(149\) 4.16837i 0.341486i 0.985316 + 0.170743i \(0.0546168\pi\)
−0.985316 + 0.170743i \(0.945383\pi\)
\(150\) 0 0
\(151\) 5.22123i 0.424897i −0.977172 0.212449i \(-0.931856\pi\)
0.977172 0.212449i \(-0.0681439\pi\)
\(152\) 0 0
\(153\) 6.83315i 0.552427i
\(154\) 0 0
\(155\) −1.12061 −0.0900098
\(156\) 0 0
\(157\) 1.91699i 0.152992i −0.997070 0.0764961i \(-0.975627\pi\)
0.997070 0.0764961i \(-0.0243733\pi\)
\(158\) 0 0
\(159\) −2.67178 −0.211886
\(160\) 0 0
\(161\) 17.2063 1.69437i 1.35605 0.133535i
\(162\) 0 0
\(163\) 8.45209i 0.662019i 0.943627 + 0.331010i \(0.107389\pi\)
−0.943627 + 0.331010i \(0.892611\pi\)
\(164\) 0 0
\(165\) −5.78590 −0.450432
\(166\) 0 0
\(167\) 19.2777i 1.49175i −0.666084 0.745877i \(-0.732031\pi\)
0.666084 0.745877i \(-0.267969\pi\)
\(168\) 0 0
\(169\) −12.1963 −0.938176
\(170\) 0 0
\(171\) −1.90832 −0.145933
\(172\) 0 0
\(173\) −1.99198 −0.151447 −0.0757236 0.997129i \(-0.524127\pi\)
−0.0757236 + 0.997129i \(0.524127\pi\)
\(174\) 0 0
\(175\) −3.60511 −0.272521
\(176\) 0 0
\(177\) −6.58667 −0.495084
\(178\) 0 0
\(179\) 22.5461i 1.68517i 0.538563 + 0.842586i \(0.318968\pi\)
−0.538563 + 0.842586i \(0.681032\pi\)
\(180\) 0 0
\(181\) 2.37594i 0.176602i −0.996094 0.0883011i \(-0.971856\pi\)
0.996094 0.0883011i \(-0.0281438\pi\)
\(182\) 0 0
\(183\) −0.597564 −0.0441732
\(184\) 0 0
\(185\) 7.48133 0.550038
\(186\) 0 0
\(187\) 39.5359i 2.89115i
\(188\) 0 0
\(189\) 3.60511i 0.262233i
\(190\) 0 0
\(191\) 2.63031 0.190323 0.0951613 0.995462i \(-0.469663\pi\)
0.0951613 + 0.995462i \(0.469663\pi\)
\(192\) 0 0
\(193\) −20.1946 −1.45364 −0.726818 0.686830i \(-0.759002\pi\)
−0.726818 + 0.686830i \(0.759002\pi\)
\(194\) 0 0
\(195\) −0.896503 −0.0641999
\(196\) 0 0
\(197\) 26.4159 1.88205 0.941027 0.338332i \(-0.109863\pi\)
0.941027 + 0.338332i \(0.109863\pi\)
\(198\) 0 0
\(199\) −21.4192 −1.51837 −0.759185 0.650875i \(-0.774402\pi\)
−0.759185 + 0.650875i \(0.774402\pi\)
\(200\) 0 0
\(201\) 5.41745i 0.382117i
\(202\) 0 0
\(203\) 5.02189 0.352467
\(204\) 0 0
\(205\) 3.19938i 0.223455i
\(206\) 0 0
\(207\) −4.77275 + 0.469990i −0.331729 + 0.0326666i
\(208\) 0 0
\(209\) 11.0413 0.763746
\(210\) 0 0
\(211\) 0.342786i 0.0235984i 0.999930 + 0.0117992i \(0.00375588\pi\)
−0.999930 + 0.0117992i \(0.996244\pi\)
\(212\) 0 0
\(213\) 3.72842 0.255467
\(214\) 0 0
\(215\) 2.53782i 0.173078i
\(216\) 0 0
\(217\) 4.03993i 0.274249i
\(218\) 0 0
\(219\) 0.479048i 0.0323711i
\(220\) 0 0
\(221\) 6.12593i 0.412075i
\(222\) 0 0
\(223\) 6.96417i 0.466355i −0.972434 0.233178i \(-0.925088\pi\)
0.972434 0.233178i \(-0.0749124\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −17.1845 −1.14058 −0.570289 0.821444i \(-0.693169\pi\)
−0.570289 + 0.821444i \(0.693169\pi\)
\(228\) 0 0
\(229\) 15.6155i 1.03190i 0.856619 + 0.515950i \(0.172561\pi\)
−0.856619 + 0.515950i \(0.827439\pi\)
\(230\) 0 0
\(231\) 20.8588i 1.37241i
\(232\) 0 0
\(233\) 12.8244 0.840153 0.420076 0.907489i \(-0.362003\pi\)
0.420076 + 0.907489i \(0.362003\pi\)
\(234\) 0 0
\(235\) 5.80647 0.378772
\(236\) 0 0
\(237\) 1.03711i 0.0673677i
\(238\) 0 0
\(239\) 10.9984i 0.711429i 0.934595 + 0.355714i \(0.115762\pi\)
−0.934595 + 0.355714i \(0.884238\pi\)
\(240\) 0 0
\(241\) 11.3738i 0.732652i 0.930487 + 0.366326i \(0.119384\pi\)
−0.930487 + 0.366326i \(0.880616\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 5.99683i 0.383123i
\(246\) 0 0
\(247\) 1.71081 0.108856
\(248\) 0 0
\(249\) 11.2088i 0.710329i
\(250\) 0 0
\(251\) 23.2561 1.46791 0.733957 0.679195i \(-0.237671\pi\)
0.733957 + 0.679195i \(0.237671\pi\)
\(252\) 0 0
\(253\) 27.6147 2.71932i 1.73612 0.170962i
\(254\) 0 0
\(255\) 6.83315i 0.427908i
\(256\) 0 0
\(257\) −27.6478 −1.72462 −0.862312 0.506378i \(-0.830984\pi\)
−0.862312 + 0.506378i \(0.830984\pi\)
\(258\) 0 0
\(259\) 26.9710i 1.67590i
\(260\) 0 0
\(261\) −1.39299 −0.0862240
\(262\) 0 0
\(263\) −23.5536 −1.45237 −0.726187 0.687497i \(-0.758709\pi\)
−0.726187 + 0.687497i \(0.758709\pi\)
\(264\) 0 0
\(265\) −2.67178 −0.164126
\(266\) 0 0
\(267\) 5.21573 0.319198
\(268\) 0 0
\(269\) −10.7515 −0.655533 −0.327766 0.944759i \(-0.606296\pi\)
−0.327766 + 0.944759i \(0.606296\pi\)
\(270\) 0 0
\(271\) 20.0996i 1.22096i 0.792030 + 0.610482i \(0.209024\pi\)
−0.792030 + 0.610482i \(0.790976\pi\)
\(272\) 0 0
\(273\) 3.23199i 0.195609i
\(274\) 0 0
\(275\) −5.78590 −0.348903
\(276\) 0 0
\(277\) 16.5831 0.996384 0.498192 0.867067i \(-0.333998\pi\)
0.498192 + 0.867067i \(0.333998\pi\)
\(278\) 0 0
\(279\) 1.12061i 0.0670893i
\(280\) 0 0
\(281\) 9.61074i 0.573329i 0.958031 + 0.286664i \(0.0925464\pi\)
−0.958031 + 0.286664i \(0.907454\pi\)
\(282\) 0 0
\(283\) −28.3850 −1.68731 −0.843657 0.536883i \(-0.819602\pi\)
−0.843657 + 0.536883i \(0.819602\pi\)
\(284\) 0 0
\(285\) −1.90832 −0.113039
\(286\) 0 0
\(287\) 11.5341 0.680838
\(288\) 0 0
\(289\) −29.6919 −1.74658
\(290\) 0 0
\(291\) 4.73433 0.277531
\(292\) 0 0
\(293\) 24.9347i 1.45670i 0.685204 + 0.728351i \(0.259713\pi\)
−0.685204 + 0.728351i \(0.740287\pi\)
\(294\) 0 0
\(295\) −6.58667 −0.383491
\(296\) 0 0
\(297\) 5.78590i 0.335732i
\(298\) 0 0
\(299\) 4.27878 0.421347i 0.247448 0.0243671i
\(300\) 0 0
\(301\) 9.14913 0.527347
\(302\) 0 0
\(303\) 0.362853i 0.0208454i
\(304\) 0 0
\(305\) −0.597564 −0.0342164
\(306\) 0 0
\(307\) 9.09513i 0.519086i 0.965731 + 0.259543i \(0.0835720\pi\)
−0.965731 + 0.259543i \(0.916428\pi\)
\(308\) 0 0
\(309\) 11.7189i 0.666664i
\(310\) 0 0
\(311\) 1.14205i 0.0647596i −0.999476 0.0323798i \(-0.989691\pi\)
0.999476 0.0323798i \(-0.0103086\pi\)
\(312\) 0 0
\(313\) 28.0497i 1.58546i 0.609572 + 0.792731i \(0.291341\pi\)
−0.609572 + 0.792731i \(0.708659\pi\)
\(314\) 0 0
\(315\) 3.60511i 0.203125i
\(316\) 0 0
\(317\) 18.0554 1.01409 0.507047 0.861918i \(-0.330737\pi\)
0.507047 + 0.861918i \(0.330737\pi\)
\(318\) 0 0
\(319\) 8.05971 0.451257
\(320\) 0 0
\(321\) 12.6988i 0.708776i
\(322\) 0 0
\(323\) 13.0398i 0.725555i
\(324\) 0 0
\(325\) −0.896503 −0.0497290
\(326\) 0 0
\(327\) 8.37783 0.463295
\(328\) 0 0
\(329\) 20.9330i 1.15407i
\(330\) 0 0
\(331\) 17.0971i 0.939742i 0.882735 + 0.469871i \(0.155700\pi\)
−0.882735 + 0.469871i \(0.844300\pi\)
\(332\) 0 0
\(333\) 7.48133i 0.409974i
\(334\) 0 0
\(335\) 5.41745i 0.295987i
\(336\) 0 0
\(337\) 18.9130i 1.03026i 0.857113 + 0.515129i \(0.172256\pi\)
−0.857113 + 0.515129i \(0.827744\pi\)
\(338\) 0 0
\(339\) 7.77619 0.422345
\(340\) 0 0
\(341\) 6.48376i 0.351115i
\(342\) 0 0
\(343\) −3.61655 −0.195275
\(344\) 0 0
\(345\) −4.77275 + 0.469990i −0.256956 + 0.0253034i
\(346\) 0 0
\(347\) 26.3614i 1.41515i 0.706636 + 0.707577i \(0.250212\pi\)
−0.706636 + 0.707577i \(0.749788\pi\)
\(348\) 0 0
\(349\) −7.87666 −0.421628 −0.210814 0.977526i \(-0.567611\pi\)
−0.210814 + 0.977526i \(0.567611\pi\)
\(350\) 0 0
\(351\) 0.896503i 0.0478518i
\(352\) 0 0
\(353\) 9.97831 0.531092 0.265546 0.964098i \(-0.414448\pi\)
0.265546 + 0.964098i \(0.414448\pi\)
\(354\) 0 0
\(355\) 3.72842 0.197884
\(356\) 0 0
\(357\) −24.6343 −1.30378
\(358\) 0 0
\(359\) −3.21862 −0.169872 −0.0849362 0.996386i \(-0.527069\pi\)
−0.0849362 + 0.996386i \(0.527069\pi\)
\(360\) 0 0
\(361\) −15.3583 −0.808333
\(362\) 0 0
\(363\) 22.4767i 1.17972i
\(364\) 0 0
\(365\) 0.479048i 0.0250745i
\(366\) 0 0
\(367\) −17.7708 −0.927630 −0.463815 0.885932i \(-0.653520\pi\)
−0.463815 + 0.885932i \(0.653520\pi\)
\(368\) 0 0
\(369\) −3.19938 −0.166553
\(370\) 0 0
\(371\) 9.63208i 0.500073i
\(372\) 0 0
\(373\) 0.136042i 0.00704396i −0.999994 0.00352198i \(-0.998879\pi\)
0.999994 0.00352198i \(-0.00112108\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 1.24882 0.0643175
\(378\) 0 0
\(379\) 36.4857 1.87415 0.937073 0.349135i \(-0.113524\pi\)
0.937073 + 0.349135i \(0.113524\pi\)
\(380\) 0 0
\(381\) −5.02772 −0.257578
\(382\) 0 0
\(383\) −9.13045 −0.466544 −0.233272 0.972412i \(-0.574943\pi\)
−0.233272 + 0.972412i \(0.574943\pi\)
\(384\) 0 0
\(385\) 20.8588i 1.06306i
\(386\) 0 0
\(387\) −2.53782 −0.129005
\(388\) 0 0
\(389\) 16.2289i 0.822838i 0.911446 + 0.411419i \(0.134967\pi\)
−0.911446 + 0.411419i \(0.865033\pi\)
\(390\) 0 0
\(391\) 3.21151 + 32.6129i 0.162413 + 1.64930i
\(392\) 0 0
\(393\) 16.0105 0.807625
\(394\) 0 0
\(395\) 1.03711i 0.0521828i
\(396\) 0 0
\(397\) −5.37479 −0.269753 −0.134877 0.990862i \(-0.543064\pi\)
−0.134877 + 0.990862i \(0.543064\pi\)
\(398\) 0 0
\(399\) 6.87970i 0.344416i
\(400\) 0 0
\(401\) 3.28928i 0.164259i 0.996622 + 0.0821294i \(0.0261721\pi\)
−0.996622 + 0.0821294i \(0.973828\pi\)
\(402\) 0 0
\(403\) 1.00463i 0.0500443i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 43.2862i 2.14562i
\(408\) 0 0
\(409\) −20.9778 −1.03728 −0.518642 0.854992i \(-0.673562\pi\)
−0.518642 + 0.854992i \(0.673562\pi\)
\(410\) 0 0
\(411\) 23.3183 1.15021
\(412\) 0 0
\(413\) 23.7457i 1.16845i
\(414\) 0 0
\(415\) 11.2088i 0.550219i
\(416\) 0 0
\(417\) −2.21510 −0.108474
\(418\) 0 0
\(419\) −27.7118 −1.35381 −0.676904 0.736071i \(-0.736679\pi\)
−0.676904 + 0.736071i \(0.736679\pi\)
\(420\) 0 0
\(421\) 36.2909i 1.76871i −0.466813 0.884356i \(-0.654598\pi\)
0.466813 0.884356i \(-0.345402\pi\)
\(422\) 0 0
\(423\) 5.80647i 0.282320i
\(424\) 0 0
\(425\) 6.83315i 0.331456i
\(426\) 0 0
\(427\) 2.15428i 0.104253i
\(428\) 0 0
\(429\) 5.18708i 0.250435i
\(430\) 0 0
\(431\) 20.7066 0.997401 0.498700 0.866774i \(-0.333811\pi\)
0.498700 + 0.866774i \(0.333811\pi\)
\(432\) 0 0
\(433\) 27.4933i 1.32125i −0.750718 0.660623i \(-0.770292\pi\)
0.750718 0.660623i \(-0.229708\pi\)
\(434\) 0 0
\(435\) −1.39299 −0.0667888
\(436\) 0 0
\(437\) 9.10792 0.896890i 0.435691 0.0429041i
\(438\) 0 0
\(439\) 2.92167i 0.139444i −0.997566 0.0697219i \(-0.977789\pi\)
0.997566 0.0697219i \(-0.0222112\pi\)
\(440\) 0 0
\(441\) −5.99683 −0.285563
\(442\) 0 0
\(443\) 37.6891i 1.79066i −0.445401 0.895331i \(-0.646939\pi\)
0.445401 0.895331i \(-0.353061\pi\)
\(444\) 0 0
\(445\) 5.21573 0.247249
\(446\) 0 0
\(447\) −4.16837 −0.197157
\(448\) 0 0
\(449\) 5.73059 0.270443 0.135222 0.990815i \(-0.456825\pi\)
0.135222 + 0.990815i \(0.456825\pi\)
\(450\) 0 0
\(451\) 18.5113 0.871664
\(452\) 0 0
\(453\) 5.22123 0.245315
\(454\) 0 0
\(455\) 3.23199i 0.151518i
\(456\) 0 0
\(457\) 27.3237i 1.27815i −0.769146 0.639073i \(-0.779318\pi\)
0.769146 0.639073i \(-0.220682\pi\)
\(458\) 0 0
\(459\) 6.83315 0.318944
\(460\) 0 0
\(461\) 20.2426 0.942793 0.471397 0.881921i \(-0.343750\pi\)
0.471397 + 0.881921i \(0.343750\pi\)
\(462\) 0 0
\(463\) 15.5699i 0.723596i 0.932257 + 0.361798i \(0.117837\pi\)
−0.932257 + 0.361798i \(0.882163\pi\)
\(464\) 0 0
\(465\) 1.12061i 0.0519672i
\(466\) 0 0
\(467\) 20.1385 0.931898 0.465949 0.884812i \(-0.345713\pi\)
0.465949 + 0.884812i \(0.345713\pi\)
\(468\) 0 0
\(469\) −19.5305 −0.901835
\(470\) 0 0
\(471\) 1.91699 0.0883301
\(472\) 0 0
\(473\) 14.6836 0.675152
\(474\) 0 0
\(475\) −1.90832 −0.0875596
\(476\) 0 0
\(477\) 2.67178i 0.122333i
\(478\) 0 0
\(479\) −27.6339 −1.26263 −0.631313 0.775528i \(-0.717484\pi\)
−0.631313 + 0.775528i \(0.717484\pi\)
\(480\) 0 0
\(481\) 6.70703i 0.305814i
\(482\) 0 0
\(483\) 1.69437 + 17.2063i 0.0770963 + 0.782913i
\(484\) 0 0
\(485\) 4.73433 0.214975
\(486\) 0 0
\(487\) 18.3891i 0.833289i 0.909070 + 0.416644i \(0.136794\pi\)
−0.909070 + 0.416644i \(0.863206\pi\)
\(488\) 0 0
\(489\) −8.45209 −0.382217
\(490\) 0 0
\(491\) 20.8576i 0.941292i 0.882322 + 0.470646i \(0.155979\pi\)
−0.882322 + 0.470646i \(0.844021\pi\)
\(492\) 0 0
\(493\) 9.51851i 0.428692i
\(494\) 0 0
\(495\) 5.78590i 0.260057i
\(496\) 0 0
\(497\) 13.4414i 0.602928i
\(498\) 0 0
\(499\) 19.8572i 0.888930i −0.895796 0.444465i \(-0.853394\pi\)
0.895796 0.444465i \(-0.146606\pi\)
\(500\) 0 0
\(501\) 19.2777 0.861264
\(502\) 0 0
\(503\) 15.9821 0.712609 0.356304 0.934370i \(-0.384037\pi\)
0.356304 + 0.934370i \(0.384037\pi\)
\(504\) 0 0
\(505\) 0.362853i 0.0161468i
\(506\) 0 0
\(507\) 12.1963i 0.541656i
\(508\) 0 0
\(509\) 37.8962 1.67972 0.839861 0.542802i \(-0.182637\pi\)
0.839861 + 0.542802i \(0.182637\pi\)
\(510\) 0 0
\(511\) 1.72702 0.0763989
\(512\) 0 0
\(513\) 1.90832i 0.0842543i
\(514\) 0 0
\(515\) 11.7189i 0.516396i
\(516\) 0 0
\(517\) 33.5957i 1.47754i
\(518\) 0 0
\(519\) 1.99198i 0.0874381i
\(520\) 0 0
\(521\) 5.78555i 0.253470i −0.991937 0.126735i \(-0.959550\pi\)
0.991937 0.126735i \(-0.0404497\pi\)
\(522\) 0 0
\(523\) −7.70649 −0.336981 −0.168491 0.985703i \(-0.553889\pi\)
−0.168491 + 0.985703i \(0.553889\pi\)
\(524\) 0 0
\(525\) 3.60511i 0.157340i
\(526\) 0 0
\(527\) −7.65731 −0.333558
\(528\) 0 0
\(529\) 22.5582 4.48628i 0.980792 0.195056i
\(530\) 0 0
\(531\) 6.58667i 0.285837i
\(532\) 0 0
\(533\) 2.86826 0.124238
\(534\) 0 0
\(535\) 12.6988i 0.549016i
\(536\) 0 0
\(537\) −22.5461 −0.972934
\(538\) 0 0
\(539\) 34.6971 1.49451
\(540\) 0 0
\(541\) −17.0098 −0.731308 −0.365654 0.930751i \(-0.619155\pi\)
−0.365654 + 0.930751i \(0.619155\pi\)
\(542\) 0 0
\(543\) 2.37594 0.101961
\(544\) 0 0
\(545\) 8.37783 0.358867
\(546\) 0 0
\(547\) 42.6426i 1.82327i 0.411006 + 0.911633i \(0.365178\pi\)
−0.411006 + 0.911633i \(0.634822\pi\)
\(548\) 0 0
\(549\) 0.597564i 0.0255034i
\(550\) 0 0
\(551\) 2.65827 0.113246
\(552\) 0 0
\(553\) 3.73891 0.158995
\(554\) 0 0
\(555\) 7.48133i 0.317565i
\(556\) 0 0
\(557\) 13.2254i 0.560380i −0.959945 0.280190i \(-0.909603\pi\)
0.959945 0.280190i \(-0.0903974\pi\)
\(558\) 0 0
\(559\) 2.27516 0.0962292
\(560\) 0 0
\(561\) −39.5359 −1.66921
\(562\) 0 0
\(563\) −39.8356 −1.67887 −0.839436 0.543459i \(-0.817114\pi\)
−0.839436 + 0.543459i \(0.817114\pi\)
\(564\) 0 0
\(565\) 7.77619 0.327147
\(566\) 0 0
\(567\) 3.60511 0.151400
\(568\) 0 0
\(569\) 32.1739i 1.34880i −0.738366 0.674401i \(-0.764402\pi\)
0.738366 0.674401i \(-0.235598\pi\)
\(570\) 0 0
\(571\) −7.64594 −0.319972 −0.159986 0.987119i \(-0.551145\pi\)
−0.159986 + 0.987119i \(0.551145\pi\)
\(572\) 0 0
\(573\) 2.63031i 0.109883i
\(574\) 0 0
\(575\) −4.77275 + 0.469990i −0.199037 + 0.0195999i
\(576\) 0 0
\(577\) 7.02954 0.292643 0.146322 0.989237i \(-0.453257\pi\)
0.146322 + 0.989237i \(0.453257\pi\)
\(578\) 0 0
\(579\) 20.1946i 0.839258i
\(580\) 0 0
\(581\) −40.4090 −1.67645
\(582\) 0 0
\(583\) 15.4587i 0.640233i
\(584\) 0 0
\(585\) 0.896503i 0.0370658i
\(586\) 0 0
\(587\) 30.6939i 1.26687i 0.773794 + 0.633437i \(0.218357\pi\)
−0.773794 + 0.633437i \(0.781643\pi\)
\(588\) 0 0
\(589\) 2.13848i 0.0881147i
\(590\) 0 0
\(591\) 26.4159i 1.08660i
\(592\) 0 0
\(593\) −30.5473 −1.25443 −0.627214 0.778847i \(-0.715805\pi\)
−0.627214 + 0.778847i \(0.715805\pi\)
\(594\) 0 0
\(595\) −24.6343 −1.00991
\(596\) 0 0
\(597\) 21.4192i 0.876631i
\(598\) 0 0
\(599\) 8.82194i 0.360455i 0.983625 + 0.180227i \(0.0576834\pi\)
−0.983625 + 0.180227i \(0.942317\pi\)
\(600\) 0 0
\(601\) −28.7517 −1.17280 −0.586402 0.810020i \(-0.699456\pi\)
−0.586402 + 0.810020i \(0.699456\pi\)
\(602\) 0 0
\(603\) 5.41745 0.220615
\(604\) 0 0
\(605\) 22.4767i 0.913808i
\(606\) 0 0
\(607\) 24.5794i 0.997646i −0.866704 0.498823i \(-0.833766\pi\)
0.866704 0.498823i \(-0.166234\pi\)
\(608\) 0 0
\(609\) 5.02189i 0.203497i
\(610\) 0 0
\(611\) 5.20552i 0.210593i
\(612\) 0 0
\(613\) 28.1696i 1.13776i −0.822420 0.568880i \(-0.807377\pi\)
0.822420 0.568880i \(-0.192623\pi\)
\(614\) 0 0
\(615\) −3.19938 −0.129012
\(616\) 0 0
\(617\) 32.7417i 1.31813i 0.752086 + 0.659065i \(0.229048\pi\)
−0.752086 + 0.659065i \(0.770952\pi\)
\(618\) 0 0
\(619\) −4.77997 −0.192123 −0.0960615 0.995375i \(-0.530625\pi\)
−0.0960615 + 0.995375i \(0.530625\pi\)
\(620\) 0 0
\(621\) −0.469990 4.77275i −0.0188600 0.191524i
\(622\) 0 0
\(623\) 18.8033i 0.753338i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.0413i 0.440949i
\(628\) 0 0
\(629\) 51.1210 2.03833
\(630\) 0 0
\(631\) 46.4708 1.84997 0.924987 0.379998i \(-0.124075\pi\)
0.924987 + 0.379998i \(0.124075\pi\)
\(632\) 0 0
\(633\) −0.342786 −0.0136245
\(634\) 0 0
\(635\) −5.02772 −0.199519
\(636\) 0 0
\(637\) 5.37617 0.213012
\(638\) 0 0
\(639\) 3.72842i 0.147494i
\(640\) 0 0
\(641\) 23.9541i 0.946130i −0.881027 0.473065i \(-0.843147\pi\)
0.881027 0.473065i \(-0.156853\pi\)
\(642\) 0 0
\(643\) −33.7394 −1.33055 −0.665277 0.746597i \(-0.731686\pi\)
−0.665277 + 0.746597i \(0.731686\pi\)
\(644\) 0 0
\(645\) −2.53782 −0.0999266
\(646\) 0 0
\(647\) 45.6913i 1.79631i −0.439679 0.898155i \(-0.644908\pi\)
0.439679 0.898155i \(-0.355092\pi\)
\(648\) 0 0
\(649\) 38.1098i 1.49594i
\(650\) 0 0
\(651\) −4.03993 −0.158337
\(652\) 0 0
\(653\) 45.2004 1.76883 0.884415 0.466701i \(-0.154558\pi\)
0.884415 + 0.466701i \(0.154558\pi\)
\(654\) 0 0
\(655\) 16.0105 0.625584
\(656\) 0 0
\(657\) −0.479048 −0.0186894
\(658\) 0 0
\(659\) −25.6885 −1.00068 −0.500342 0.865828i \(-0.666792\pi\)
−0.500342 + 0.865828i \(0.666792\pi\)
\(660\) 0 0
\(661\) 9.39395i 0.365382i 0.983170 + 0.182691i \(0.0584809\pi\)
−0.983170 + 0.182691i \(0.941519\pi\)
\(662\) 0 0
\(663\) −6.12593 −0.237912
\(664\) 0 0
\(665\) 6.87970i 0.266783i
\(666\) 0 0
\(667\) 6.64839 0.654692i 0.257427 0.0253498i
\(668\) 0 0
\(669\) 6.96417 0.269250
\(670\) 0 0
\(671\) 3.45745i 0.133473i
\(672\) 0 0
\(673\) −23.2138 −0.894826 −0.447413 0.894328i \(-0.647655\pi\)
−0.447413 + 0.894328i \(0.647655\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 46.6887i 1.79439i 0.441631 + 0.897197i \(0.354400\pi\)
−0.441631 + 0.897197i \(0.645600\pi\)
\(678\) 0 0
\(679\) 17.0678i 0.655001i
\(680\) 0 0
\(681\) 17.1845i 0.658513i
\(682\) 0 0
\(683\) 41.3680i 1.58290i −0.611233 0.791451i \(-0.709326\pi\)
0.611233 0.791451i \(-0.290674\pi\)
\(684\) 0 0
\(685\) 23.3183 0.890946
\(686\) 0 0
\(687\) −15.6155 −0.595768
\(688\) 0 0
\(689\) 2.39526i 0.0912522i
\(690\) 0 0
\(691\) 38.1418i 1.45098i −0.688231 0.725492i \(-0.741612\pi\)
0.688231 0.725492i \(-0.258388\pi\)
\(692\) 0 0
\(693\) −20.8588 −0.792361
\(694\) 0 0
\(695\) −2.21510 −0.0840236
\(696\) 0 0
\(697\) 21.8619i 0.828077i
\(698\) 0 0
\(699\) 12.8244i 0.485063i
\(700\) 0 0
\(701\) 20.8658i 0.788091i 0.919091 + 0.394045i \(0.128925\pi\)
−0.919091 + 0.394045i \(0.871075\pi\)
\(702\) 0 0
\(703\) 14.2767i 0.538458i
\(704\) 0 0
\(705\) 5.80647i 0.218684i
\(706\) 0 0
\(707\) −1.30813 −0.0491971
\(708\) 0 0
\(709\) 19.2856i 0.724284i −0.932123 0.362142i \(-0.882045\pi\)
0.932123 0.362142i \(-0.117955\pi\)
\(710\) 0 0
\(711\) −1.03711 −0.0388948
\(712\) 0 0
\(713\) 0.526677 + 5.34840i 0.0197242 + 0.200299i
\(714\) 0 0
\(715\) 5.18708i 0.193986i
\(716\) 0 0
\(717\) −10.9984 −0.410744
\(718\) 0 0
\(719\) 6.24028i 0.232723i −0.993207 0.116362i \(-0.962877\pi\)
0.993207 0.116362i \(-0.0371231\pi\)
\(720\) 0 0
\(721\) −42.2479 −1.57339
\(722\) 0 0
\(723\) −11.3738 −0.422997
\(724\) 0 0
\(725\) −1.39299 −0.0517344
\(726\) 0 0
\(727\) 14.0658 0.521673 0.260836 0.965383i \(-0.416002\pi\)
0.260836 + 0.965383i \(0.416002\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 17.3413i 0.641391i
\(732\) 0 0
\(733\) 28.8930i 1.06719i 0.845740 + 0.533594i \(0.179159\pi\)
−0.845740 + 0.533594i \(0.820841\pi\)
\(734\) 0 0
\(735\) −5.99683 −0.221196
\(736\) 0 0
\(737\) −31.3448 −1.15460
\(738\) 0 0
\(739\) 31.3784i 1.15427i 0.816648 + 0.577136i \(0.195830\pi\)
−0.816648 + 0.577136i \(0.804170\pi\)
\(740\) 0 0
\(741\) 1.71081i 0.0628482i
\(742\) 0 0
\(743\) −41.8112 −1.53390 −0.766952 0.641705i \(-0.778227\pi\)
−0.766952 + 0.641705i \(0.778227\pi\)
\(744\) 0 0
\(745\) −4.16837 −0.152717
\(746\) 0 0
\(747\) 11.2088 0.410109
\(748\) 0 0
\(749\) 45.7805 1.67278
\(750\) 0 0
\(751\) 34.4288 1.25632 0.628162 0.778082i \(-0.283807\pi\)
0.628162 + 0.778082i \(0.283807\pi\)
\(752\) 0 0
\(753\) 23.2561i 0.847501i
\(754\) 0 0
\(755\) 5.22123 0.190020
\(756\) 0 0
\(757\) 29.7038i 1.07960i −0.841792 0.539802i \(-0.818499\pi\)
0.841792 0.539802i \(-0.181501\pi\)
\(758\) 0 0
\(759\) 2.71932 + 27.6147i 0.0987049 + 1.00235i
\(760\) 0 0
\(761\) 20.5094 0.743464 0.371732 0.928340i \(-0.378764\pi\)
0.371732 + 0.928340i \(0.378764\pi\)
\(762\) 0 0
\(763\) 30.2030i 1.09342i
\(764\) 0 0
\(765\) 6.83315 0.247053
\(766\) 0 0
\(767\) 5.90497i 0.213216i
\(768\) 0 0
\(769\) 15.5406i 0.560407i −0.959941 0.280203i \(-0.909598\pi\)
0.959941 0.280203i \(-0.0904019\pi\)
\(770\) 0 0
\(771\) 27.6478i 0.995712i
\(772\) 0 0
\(773\) 4.79100i 0.172320i 0.996281 + 0.0861601i \(0.0274597\pi\)
−0.996281 + 0.0861601i \(0.972540\pi\)
\(774\) 0 0
\(775\) 1.12061i 0.0402536i
\(776\) 0 0
\(777\) 26.9710 0.967580
\(778\) 0 0
\(779\) 6.10544 0.218750
\(780\) 0 0
\(781\) 21.5723i 0.771917i
\(782\) 0 0
\(783\) 1.39299i 0.0497814i
\(784\) 0 0
\(785\) 1.91699 0.0684202
\(786\) 0 0
\(787\) −10.9512 −0.390369 −0.195185 0.980767i \(-0.562531\pi\)
−0.195185 + 0.980767i \(0.562531\pi\)
\(788\) 0 0
\(789\) 23.5536i 0.838529i
\(790\) 0 0
\(791\) 28.0340i 0.996775i
\(792\) 0 0
\(793\) 0.535718i 0.0190239i
\(794\) 0 0
\(795\) 2.67178i 0.0947584i
\(796\) 0 0
\(797\) 25.2060i 0.892842i −0.894823 0.446421i \(-0.852698\pi\)
0.894823 0.446421i \(-0.147302\pi\)
\(798\) 0 0
\(799\) 39.6765 1.40365
\(800\) 0 0
\(801\) 5.21573i 0.184289i
\(802\) 0 0
\(803\) 2.77173 0.0978121
\(804\) 0 0
\(805\) 1.69437 + 17.2063i 0.0597185 + 0.606442i
\(806\) 0 0