Properties

Label 4560.2.d.e.2431.5
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9821011968.3
Defining polynomial: \(x^{6} + 20 x^{4} + 118 x^{2} + 192\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.5
Root \(2.60808i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.e.2431.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +2.60808i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +2.60808i q^{7} +1.00000 q^{9} -1.74632i q^{11} +2.60808i q^{13} +1.00000 q^{15} -0.802094 q^{17} +(0.197906 - 4.35440i) q^{19} -2.60808i q^{21} +1.23017i q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.74632i q^{29} +8.15874 q^{31} +1.74632i q^{33} -2.60808i q^{35} +2.97649i q^{37} -2.60808i q^{39} +6.96249i q^{41} +2.97649i q^{43} -1.00000 q^{45} -9.93897i q^{47} +0.197906 q^{49} +0.802094 q^{51} -1.23017i q^{53} +1.74632i q^{55} +(-0.197906 + 4.35440i) q^{57} +1.60419 q^{59} +10.1587 q^{61} +2.60808i q^{63} -2.60808i q^{65} -9.35664 q^{67} -1.23017i q^{69} +1.60419 q^{71} -10.5650 q^{73} -1.00000 q^{75} +4.55455 q^{77} +16.9608 q^{79} +1.00000 q^{81} -4.72281i q^{83} +0.802094 q^{85} +1.74632i q^{87} +12.5471i q^{89} -6.80209 q^{91} -8.15874 q^{93} +(-0.197906 + 4.35440i) q^{95} +8.85656i q^{97} -1.74632i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{5} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{5} + 6q^{9} + 6q^{15} - 4q^{17} + 2q^{19} + 6q^{25} - 6q^{27} - 12q^{31} - 6q^{45} + 2q^{49} + 4q^{51} - 2q^{57} + 8q^{59} + 4q^{67} + 8q^{71} - 6q^{75} - 32q^{77} + 40q^{79} + 6q^{81} + 4q^{85} - 40q^{91} + 12q^{93} - 2q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.60808i 0.985763i 0.870097 + 0.492881i \(0.164056\pi\)
−0.870097 + 0.492881i \(0.835944\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.74632i 0.526536i −0.964723 0.263268i \(-0.915200\pi\)
0.964723 0.263268i \(-0.0848003\pi\)
\(12\) 0 0
\(13\) 2.60808i 0.723352i 0.932304 + 0.361676i \(0.117795\pi\)
−0.932304 + 0.361676i \(0.882205\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −0.802094 −0.194536 −0.0972682 0.995258i \(-0.531010\pi\)
−0.0972682 + 0.995258i \(0.531010\pi\)
\(18\) 0 0
\(19\) 0.197906 4.35440i 0.0454027 0.998969i
\(20\) 0 0
\(21\) 2.60808i 0.569130i
\(22\) 0 0
\(23\) 1.23017i 0.256508i 0.991741 + 0.128254i \(0.0409372\pi\)
−0.991741 + 0.128254i \(0.959063\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.74632i 0.324284i −0.986767 0.162142i \(-0.948160\pi\)
0.986767 0.162142i \(-0.0518402\pi\)
\(30\) 0 0
\(31\) 8.15874 1.46535 0.732676 0.680577i \(-0.238271\pi\)
0.732676 + 0.680577i \(0.238271\pi\)
\(32\) 0 0
\(33\) 1.74632i 0.303996i
\(34\) 0 0
\(35\) 2.60808i 0.440846i
\(36\) 0 0
\(37\) 2.97649i 0.489332i 0.969607 + 0.244666i \(0.0786783\pi\)
−0.969607 + 0.244666i \(0.921322\pi\)
\(38\) 0 0
\(39\) 2.60808i 0.417627i
\(40\) 0 0
\(41\) 6.96249i 1.08736i 0.839293 + 0.543679i \(0.182969\pi\)
−0.839293 + 0.543679i \(0.817031\pi\)
\(42\) 0 0
\(43\) 2.97649i 0.453910i 0.973905 + 0.226955i \(0.0728771\pi\)
−0.973905 + 0.226955i \(0.927123\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 9.93897i 1.44975i −0.688881 0.724874i \(-0.741898\pi\)
0.688881 0.724874i \(-0.258102\pi\)
\(48\) 0 0
\(49\) 0.197906 0.0282722
\(50\) 0 0
\(51\) 0.802094 0.112316
\(52\) 0 0
\(53\) 1.23017i 0.168976i −0.996424 0.0844882i \(-0.973074\pi\)
0.996424 0.0844882i \(-0.0269255\pi\)
\(54\) 0 0
\(55\) 1.74632i 0.235474i
\(56\) 0 0
\(57\) −0.197906 + 4.35440i −0.0262132 + 0.576755i
\(58\) 0 0
\(59\) 1.60419 0.208848 0.104424 0.994533i \(-0.466700\pi\)
0.104424 + 0.994533i \(0.466700\pi\)
\(60\) 0 0
\(61\) 10.1587 1.30069 0.650347 0.759638i \(-0.274624\pi\)
0.650347 + 0.759638i \(0.274624\pi\)
\(62\) 0 0
\(63\) 2.60808i 0.328588i
\(64\) 0 0
\(65\) 2.60808i 0.323493i
\(66\) 0 0
\(67\) −9.35664 −1.14310 −0.571548 0.820569i \(-0.693657\pi\)
−0.571548 + 0.820569i \(0.693657\pi\)
\(68\) 0 0
\(69\) 1.23017i 0.148095i
\(70\) 0 0
\(71\) 1.60419 0.190382 0.0951911 0.995459i \(-0.469654\pi\)
0.0951911 + 0.995459i \(0.469654\pi\)
\(72\) 0 0
\(73\) −10.5650 −1.23654 −0.618271 0.785965i \(-0.712167\pi\)
−0.618271 + 0.785965i \(0.712167\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 4.55455 0.519039
\(78\) 0 0
\(79\) 16.9608 1.90824 0.954121 0.299420i \(-0.0967932\pi\)
0.954121 + 0.299420i \(0.0967932\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.72281i 0.518396i −0.965824 0.259198i \(-0.916542\pi\)
0.965824 0.259198i \(-0.0834582\pi\)
\(84\) 0 0
\(85\) 0.802094 0.0869993
\(86\) 0 0
\(87\) 1.74632i 0.187225i
\(88\) 0 0
\(89\) 12.5471i 1.32999i 0.746850 + 0.664993i \(0.231565\pi\)
−0.746850 + 0.664993i \(0.768435\pi\)
\(90\) 0 0
\(91\) −6.80209 −0.713053
\(92\) 0 0
\(93\) −8.15874 −0.846022
\(94\) 0 0
\(95\) −0.197906 + 4.35440i −0.0203047 + 0.446752i
\(96\) 0 0
\(97\) 8.85656i 0.899247i 0.893218 + 0.449624i \(0.148442\pi\)
−0.893218 + 0.449624i \(0.851558\pi\)
\(98\) 0 0
\(99\) 1.74632i 0.175512i
\(100\) 0 0
\(101\) −18.3175 −1.82266 −0.911329 0.411680i \(-0.864942\pi\)
−0.911329 + 0.411680i \(0.864942\pi\)
\(102\) 0 0
\(103\) 2.39581 0.236066 0.118033 0.993010i \(-0.462341\pi\)
0.118033 + 0.993010i \(0.462341\pi\)
\(104\) 0 0
\(105\) 2.60808i 0.254523i
\(106\) 0 0
\(107\) −9.75246 −0.942806 −0.471403 0.881918i \(-0.656252\pi\)
−0.471403 + 0.881918i \(0.656252\pi\)
\(108\) 0 0
\(109\) 2.75583i 0.263961i −0.991252 0.131980i \(-0.957866\pi\)
0.991252 0.131980i \(-0.0421336\pi\)
\(110\) 0 0
\(111\) 2.97649i 0.282516i
\(112\) 0 0
\(113\) 13.4316i 1.26354i 0.775156 + 0.631770i \(0.217671\pi\)
−0.775156 + 0.631770i \(0.782329\pi\)
\(114\) 0 0
\(115\) 1.23017i 0.114714i
\(116\) 0 0
\(117\) 2.60808i 0.241117i
\(118\) 0 0
\(119\) 2.09193i 0.191767i
\(120\) 0 0
\(121\) 7.95036 0.722760
\(122\) 0 0
\(123\) 6.96249i 0.627786i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.5650 −1.11497 −0.557483 0.830189i \(-0.688233\pi\)
−0.557483 + 0.830189i \(0.688233\pi\)
\(128\) 0 0
\(129\) 2.97649i 0.262065i
\(130\) 0 0
\(131\) 15.3029i 1.33702i −0.743703 0.668510i \(-0.766932\pi\)
0.743703 0.668510i \(-0.233068\pi\)
\(132\) 0 0
\(133\) 11.3566 + 0.516154i 0.984746 + 0.0447562i
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −8.56502 −0.731759 −0.365880 0.930662i \(-0.619232\pi\)
−0.365880 + 0.930662i \(0.619232\pi\)
\(138\) 0 0
\(139\) 13.9250i 1.18110i 0.807001 + 0.590550i \(0.201089\pi\)
−0.807001 + 0.590550i \(0.798911\pi\)
\(140\) 0 0
\(141\) 9.93897i 0.837013i
\(142\) 0 0
\(143\) 4.55455 0.380871
\(144\) 0 0
\(145\) 1.74632i 0.145024i
\(146\) 0 0
\(147\) −0.197906 −0.0163230
\(148\) 0 0
\(149\) 7.60419 0.622959 0.311480 0.950253i \(-0.399175\pi\)
0.311480 + 0.950253i \(0.399175\pi\)
\(150\) 0 0
\(151\) −13.5154 −1.09987 −0.549933 0.835209i \(-0.685347\pi\)
−0.549933 + 0.835209i \(0.685347\pi\)
\(152\) 0 0
\(153\) −0.802094 −0.0648455
\(154\) 0 0
\(155\) −8.15874 −0.655326
\(156\) 0 0
\(157\) 15.3566 1.22559 0.612797 0.790240i \(-0.290044\pi\)
0.612797 + 0.790240i \(0.290044\pi\)
\(158\) 0 0
\(159\) 1.23017i 0.0975586i
\(160\) 0 0
\(161\) −3.20838 −0.252856
\(162\) 0 0
\(163\) 14.0727i 1.10226i 0.834419 + 0.551130i \(0.185803\pi\)
−0.834419 + 0.551130i \(0.814197\pi\)
\(164\) 0 0
\(165\) 1.74632i 0.135951i
\(166\) 0 0
\(167\) −5.19791 −0.402226 −0.201113 0.979568i \(-0.564456\pi\)
−0.201113 + 0.979568i \(0.564456\pi\)
\(168\) 0 0
\(169\) 6.19791 0.476762
\(170\) 0 0
\(171\) 0.197906 4.35440i 0.0151342 0.332990i
\(172\) 0 0
\(173\) 22.1404i 1.68331i 0.540019 + 0.841653i \(0.318417\pi\)
−0.540019 + 0.841653i \(0.681583\pi\)
\(174\) 0 0
\(175\) 2.60808i 0.197153i
\(176\) 0 0
\(177\) −1.60419 −0.120578
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 6.93969i 0.515823i 0.966169 + 0.257911i \(0.0830343\pi\)
−0.966169 + 0.257911i \(0.916966\pi\)
\(182\) 0 0
\(183\) −10.1587 −0.750956
\(184\) 0 0
\(185\) 2.97649i 0.218836i
\(186\) 0 0
\(187\) 1.40071i 0.102430i
\(188\) 0 0
\(189\) 2.60808i 0.189710i
\(190\) 0 0
\(191\) 1.00951i 0.0730455i −0.999333 0.0365228i \(-0.988372\pi\)
0.999333 0.0365228i \(-0.0116281\pi\)
\(192\) 0 0
\(193\) 26.9654i 1.94101i 0.241079 + 0.970505i \(0.422499\pi\)
−0.241079 + 0.970505i \(0.577501\pi\)
\(194\) 0 0
\(195\) 2.60808i 0.186769i
\(196\) 0 0
\(197\) 11.1979 0.797818 0.398909 0.916991i \(-0.369389\pi\)
0.398909 + 0.916991i \(0.369389\pi\)
\(198\) 0 0
\(199\) 5.95298i 0.421995i 0.977487 + 0.210998i \(0.0676713\pi\)
−0.977487 + 0.210998i \(0.932329\pi\)
\(200\) 0 0
\(201\) 9.35664 0.659967
\(202\) 0 0
\(203\) 4.55455 0.319667
\(204\) 0 0
\(205\) 6.96249i 0.486281i
\(206\) 0 0
\(207\) 1.23017i 0.0855025i
\(208\) 0 0
\(209\) −7.60419 0.345607i −0.525993 0.0239061i
\(210\) 0 0
\(211\) 1.35664 0.0933953 0.0466976 0.998909i \(-0.485130\pi\)
0.0466976 + 0.998909i \(0.485130\pi\)
\(212\) 0 0
\(213\) −1.60419 −0.109917
\(214\) 0 0
\(215\) 2.97649i 0.202995i
\(216\) 0 0
\(217\) 21.2787i 1.44449i
\(218\) 0 0
\(219\) 10.5650 0.713918
\(220\) 0 0
\(221\) 2.09193i 0.140718i
\(222\) 0 0
\(223\) −0.812566 −0.0544135 −0.0272067 0.999630i \(-0.508661\pi\)
−0.0272067 + 0.999630i \(0.508661\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 22.3279 1.48196 0.740979 0.671528i \(-0.234362\pi\)
0.740979 + 0.671528i \(0.234362\pi\)
\(228\) 0 0
\(229\) −16.8021 −1.11031 −0.555157 0.831746i \(-0.687342\pi\)
−0.555157 + 0.831746i \(0.687342\pi\)
\(230\) 0 0
\(231\) −4.55455 −0.299667
\(232\) 0 0
\(233\) 25.5049 1.67088 0.835441 0.549580i \(-0.185212\pi\)
0.835441 + 0.549580i \(0.185212\pi\)
\(234\) 0 0
\(235\) 9.93897i 0.648347i
\(236\) 0 0
\(237\) −16.9608 −1.10172
\(238\) 0 0
\(239\) 8.36320i 0.540970i −0.962724 0.270485i \(-0.912816\pi\)
0.962724 0.270485i \(-0.0871841\pi\)
\(240\) 0 0
\(241\) 26.8632i 1.73041i 0.501416 + 0.865207i \(0.332813\pi\)
−0.501416 + 0.865207i \(0.667187\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.197906 −0.0126437
\(246\) 0 0
\(247\) 11.3566 + 0.516154i 0.722606 + 0.0328421i
\(248\) 0 0
\(249\) 4.72281i 0.299296i
\(250\) 0 0
\(251\) 3.10144i 0.195761i 0.995198 + 0.0978805i \(0.0312063\pi\)
−0.995198 + 0.0978805i \(0.968794\pi\)
\(252\) 0 0
\(253\) 2.14827 0.135060
\(254\) 0 0
\(255\) −0.802094 −0.0502291
\(256\) 0 0
\(257\) 17.6155i 1.09882i 0.835552 + 0.549412i \(0.185148\pi\)
−0.835552 + 0.549412i \(0.814852\pi\)
\(258\) 0 0
\(259\) −7.76293 −0.482365
\(260\) 0 0
\(261\) 1.74632i 0.108095i
\(262\) 0 0
\(263\) 17.6155i 1.08622i 0.839663 + 0.543108i \(0.182753\pi\)
−0.839663 + 0.543108i \(0.817247\pi\)
\(264\) 0 0
\(265\) 1.23017i 0.0755686i
\(266\) 0 0
\(267\) 12.5471i 0.767867i
\(268\) 0 0
\(269\) 13.2839i 0.809932i −0.914332 0.404966i \(-0.867283\pi\)
0.914332 0.404966i \(-0.132717\pi\)
\(270\) 0 0
\(271\) 22.6338i 1.37490i 0.726230 + 0.687452i \(0.241271\pi\)
−0.726230 + 0.687452i \(0.758729\pi\)
\(272\) 0 0
\(273\) 6.80209 0.411681
\(274\) 0 0
\(275\) 1.74632i 0.105307i
\(276\) 0 0
\(277\) −10.3175 −0.619917 −0.309959 0.950750i \(-0.600315\pi\)
−0.309959 + 0.950750i \(0.600315\pi\)
\(278\) 0 0
\(279\) 8.15874 0.488451
\(280\) 0 0
\(281\) 27.5043i 1.64077i 0.571811 + 0.820386i \(0.306241\pi\)
−0.571811 + 0.820386i \(0.693759\pi\)
\(282\) 0 0
\(283\) 0.884560i 0.0525817i 0.999654 + 0.0262908i \(0.00836959\pi\)
−0.999654 + 0.0262908i \(0.991630\pi\)
\(284\) 0 0
\(285\) 0.197906 4.35440i 0.0117229 0.257933i
\(286\) 0 0
\(287\) −18.1587 −1.07188
\(288\) 0 0
\(289\) −16.3566 −0.962156
\(290\) 0 0
\(291\) 8.85656i 0.519181i
\(292\) 0 0
\(293\) 2.26248i 0.132175i 0.997814 + 0.0660876i \(0.0210517\pi\)
−0.997814 + 0.0660876i \(0.978948\pi\)
\(294\) 0 0
\(295\) −1.60419 −0.0933995
\(296\) 0 0
\(297\) 1.74632i 0.101332i
\(298\) 0 0
\(299\) −3.20838 −0.185545
\(300\) 0 0
\(301\) −7.76293 −0.447448
\(302\) 0 0
\(303\) 18.3175 1.05231
\(304\) 0 0
\(305\) −10.1587 −0.581688
\(306\) 0 0
\(307\) 14.3958 0.821612 0.410806 0.911723i \(-0.365247\pi\)
0.410806 + 0.911723i \(0.365247\pi\)
\(308\) 0 0
\(309\) −2.39581 −0.136293
\(310\) 0 0
\(311\) 8.06770i 0.457478i −0.973488 0.228739i \(-0.926540\pi\)
0.973488 0.228739i \(-0.0734602\pi\)
\(312\) 0 0
\(313\) −12.1692 −0.687844 −0.343922 0.938998i \(-0.611756\pi\)
−0.343922 + 0.938998i \(0.611756\pi\)
\(314\) 0 0
\(315\) 2.60808i 0.146949i
\(316\) 0 0
\(317\) 10.9257i 0.613648i 0.951766 + 0.306824i \(0.0992662\pi\)
−0.951766 + 0.306824i \(0.900734\pi\)
\(318\) 0 0
\(319\) −3.04964 −0.170747
\(320\) 0 0
\(321\) 9.75246 0.544329
\(322\) 0 0
\(323\) −0.158739 + 3.49264i −0.00883247 + 0.194336i
\(324\) 0 0
\(325\) 2.60808i 0.144670i
\(326\) 0 0
\(327\) 2.75583i 0.152398i
\(328\) 0 0
\(329\) 25.9217 1.42911
\(330\) 0 0
\(331\) 7.59372 0.417388 0.208694 0.977981i \(-0.433079\pi\)
0.208694 + 0.977981i \(0.433079\pi\)
\(332\) 0 0
\(333\) 2.97649i 0.163111i
\(334\) 0 0
\(335\) 9.35664 0.511208
\(336\) 0 0
\(337\) 27.0383i 1.47287i −0.676508 0.736435i \(-0.736508\pi\)
0.676508 0.736435i \(-0.263492\pi\)
\(338\) 0 0
\(339\) 13.4316i 0.729505i
\(340\) 0 0
\(341\) 14.2478i 0.771561i
\(342\) 0 0
\(343\) 18.7727i 1.01363i
\(344\) 0 0
\(345\) 1.23017i 0.0662300i
\(346\) 0 0
\(347\) 31.8359i 1.70904i −0.519416 0.854521i \(-0.673850\pi\)
0.519416 0.854521i \(-0.326150\pi\)
\(348\) 0 0
\(349\) −6.14827 −0.329109 −0.164555 0.986368i \(-0.552619\pi\)
−0.164555 + 0.986368i \(0.552619\pi\)
\(350\) 0 0
\(351\) 2.60808i 0.139209i
\(352\) 0 0
\(353\) 2.79162 0.148583 0.0742915 0.997237i \(-0.476330\pi\)
0.0742915 + 0.997237i \(0.476330\pi\)
\(354\) 0 0
\(355\) −1.60419 −0.0851415
\(356\) 0 0
\(357\) 2.09193i 0.110717i
\(358\) 0 0
\(359\) 2.43754i 0.128648i −0.997929 0.0643241i \(-0.979511\pi\)
0.997929 0.0643241i \(-0.0204891\pi\)
\(360\) 0 0
\(361\) −18.9217 1.72352i −0.995877 0.0907117i
\(362\) 0 0
\(363\) −7.95036 −0.417286
\(364\) 0 0
\(365\) 10.5650 0.552999
\(366\) 0 0
\(367\) 29.0573i 1.51678i 0.651801 + 0.758390i \(0.274014\pi\)
−0.651801 + 0.758390i \(0.725986\pi\)
\(368\) 0 0
\(369\) 6.96249i 0.362453i
\(370\) 0 0
\(371\) 3.20838 0.166571
\(372\) 0 0
\(373\) 17.2699i 0.894200i 0.894484 + 0.447100i \(0.147543\pi\)
−0.894484 + 0.447100i \(0.852457\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 4.55455 0.234571
\(378\) 0 0
\(379\) 7.59372 0.390063 0.195032 0.980797i \(-0.437519\pi\)
0.195032 + 0.980797i \(0.437519\pi\)
\(380\) 0 0
\(381\) 12.5650 0.643726
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −4.55455 −0.232121
\(386\) 0 0
\(387\) 2.97649i 0.151303i
\(388\) 0 0
\(389\) −1.18743 −0.0602053 −0.0301026 0.999547i \(-0.509583\pi\)
−0.0301026 + 0.999547i \(0.509583\pi\)
\(390\) 0 0
\(391\) 0.986710i 0.0499001i
\(392\) 0 0
\(393\) 15.3029i 0.771929i
\(394\) 0 0
\(395\) −16.9608 −0.853392
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) −11.3566 0.516154i −0.568543 0.0258400i
\(400\) 0 0
\(401\) 11.4418i 0.571378i 0.958322 + 0.285689i \(0.0922225\pi\)
−0.958322 + 0.285689i \(0.907778\pi\)
\(402\) 0 0
\(403\) 21.2787i 1.05997i
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 5.19791 0.257651
\(408\) 0 0
\(409\) 4.22945i 0.209133i −0.994518 0.104567i \(-0.966654\pi\)
0.994518 0.104567i \(-0.0333455\pi\)
\(410\) 0 0
\(411\) 8.56502 0.422481
\(412\) 0 0
\(413\) 4.18386i 0.205874i
\(414\) 0 0
\(415\) 4.72281i 0.231834i
\(416\) 0 0
\(417\) 13.9250i 0.681909i
\(418\) 0 0
\(419\) 18.1316i 0.885788i 0.896574 + 0.442894i \(0.146048\pi\)
−0.896574 + 0.442894i \(0.853952\pi\)
\(420\) 0 0
\(421\) 9.44562i 0.460351i −0.973149 0.230176i \(-0.926070\pi\)
0.973149 0.230176i \(-0.0739301\pi\)
\(422\) 0 0
\(423\) 9.93897i 0.483249i
\(424\) 0 0
\(425\) −0.802094 −0.0389073
\(426\) 0 0
\(427\) 26.4948i 1.28217i
\(428\) 0 0
\(429\) −4.55455 −0.219896
\(430\) 0 0
\(431\) 1.60419 0.0772711 0.0386355 0.999253i \(-0.487699\pi\)
0.0386355 + 0.999253i \(0.487699\pi\)
\(432\) 0 0
\(433\) 0.884560i 0.0425093i 0.999774 + 0.0212546i \(0.00676607\pi\)
−0.999774 + 0.0212546i \(0.993234\pi\)
\(434\) 0 0
\(435\) 1.74632i 0.0837297i
\(436\) 0 0
\(437\) 5.35664 + 0.243457i 0.256243 + 0.0116461i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0.197906 0.00942408
\(442\) 0 0
\(443\) 14.4639i 0.687202i 0.939116 + 0.343601i \(0.111647\pi\)
−0.939116 + 0.343601i \(0.888353\pi\)
\(444\) 0 0
\(445\) 12.5471i 0.594788i
\(446\) 0 0
\(447\) −7.60419 −0.359666
\(448\) 0 0
\(449\) 33.3844i 1.57551i −0.615990 0.787754i \(-0.711244\pi\)
0.615990 0.787754i \(-0.288756\pi\)
\(450\) 0 0
\(451\) 12.1587 0.572533
\(452\) 0 0
\(453\) 13.5154 0.635008
\(454\) 0 0
\(455\) 6.80209 0.318887
\(456\) 0 0
\(457\) −7.35664 −0.344129 −0.172065 0.985086i \(-0.555044\pi\)
−0.172065 + 0.985086i \(0.555044\pi\)
\(458\) 0 0
\(459\) 0.802094 0.0374386
\(460\) 0 0
\(461\) −30.6350 −1.42681 −0.713406 0.700751i \(-0.752848\pi\)
−0.713406 + 0.700751i \(0.752848\pi\)
\(462\) 0 0
\(463\) 15.1779i 0.705379i 0.935740 + 0.352689i \(0.114733\pi\)
−0.935740 + 0.352689i \(0.885267\pi\)
\(464\) 0 0
\(465\) 8.15874 0.378352
\(466\) 0 0
\(467\) 13.7271i 0.635215i −0.948222 0.317608i \(-0.897121\pi\)
0.948222 0.317608i \(-0.102879\pi\)
\(468\) 0 0
\(469\) 24.4029i 1.12682i
\(470\) 0 0
\(471\) −15.3566 −0.707597
\(472\) 0 0
\(473\) 5.19791 0.239000
\(474\) 0 0
\(475\) 0.197906 4.35440i 0.00908053 0.199794i
\(476\) 0 0
\(477\) 1.23017i 0.0563255i
\(478\) 0 0
\(479\) 14.2706i 0.652039i 0.945363 + 0.326020i \(0.105708\pi\)
−0.945363 + 0.326020i \(0.894292\pi\)
\(480\) 0 0
\(481\) −7.76293 −0.353959
\(482\) 0 0
\(483\) 3.20838 0.145986
\(484\) 0 0
\(485\) 8.85656i 0.402155i
\(486\) 0 0
\(487\) 11.7525 0.532555 0.266277 0.963896i \(-0.414206\pi\)
0.266277 + 0.963896i \(0.414206\pi\)
\(488\) 0 0
\(489\) 14.0727i 0.636390i
\(490\) 0 0
\(491\) 5.23896i 0.236431i −0.992988 0.118216i \(-0.962283\pi\)
0.992988 0.118216i \(-0.0377174\pi\)
\(492\) 0 0
\(493\) 1.40071i 0.0630850i
\(494\) 0 0
\(495\) 1.74632i 0.0784913i
\(496\) 0 0
\(497\) 4.18386i 0.187672i
\(498\) 0 0
\(499\) 14.6618i 0.656352i −0.944617 0.328176i \(-0.893566\pi\)
0.944617 0.328176i \(-0.106434\pi\)
\(500\) 0 0
\(501\) 5.19791 0.232225
\(502\) 0 0
\(503\) 17.6155i 0.785435i −0.919659 0.392718i \(-0.871535\pi\)
0.919659 0.392718i \(-0.128465\pi\)
\(504\) 0 0
\(505\) 18.3175 0.815117
\(506\) 0 0
\(507\) −6.19791 −0.275259
\(508\) 0 0
\(509\) 27.1359i 1.20278i −0.798956 0.601390i \(-0.794614\pi\)
0.798956 0.601390i \(-0.205386\pi\)
\(510\) 0 0
\(511\) 27.5544i 1.21894i
\(512\) 0 0
\(513\) −0.197906 + 4.35440i −0.00873775 + 0.192252i
\(514\) 0 0
\(515\) −2.39581 −0.105572
\(516\) 0 0
\(517\) −17.3566 −0.763344
\(518\) 0 0
\(519\) 22.1404i 0.971857i
\(520\) 0 0
\(521\) 40.3970i 1.76982i 0.465757 + 0.884912i \(0.345782\pi\)
−0.465757 + 0.884912i \(0.654218\pi\)
\(522\) 0 0
\(523\) −17.3776 −0.759869 −0.379934 0.925013i \(-0.624053\pi\)
−0.379934 + 0.925013i \(0.624053\pi\)
\(524\) 0 0
\(525\) 2.60808i 0.113826i
\(526\) 0 0
\(527\) −6.54408 −0.285065
\(528\) 0 0
\(529\) 21.4867 0.934204
\(530\) 0 0
\(531\) 1.60419 0.0696159
\(532\) 0 0
\(533\) −18.1587 −0.786542
\(534\) 0 0
\(535\) 9.75246 0.421635
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.345607i 0.0148863i
\(540\) 0 0
\(541\) 28.6350 1.23111 0.615556 0.788093i \(-0.288931\pi\)
0.615556 + 0.788093i \(0.288931\pi\)
\(542\) 0 0
\(543\) 6.93969i 0.297810i
\(544\) 0 0
\(545\) 2.75583i 0.118047i
\(546\) 0 0
\(547\) 16.3175 0.697685 0.348842 0.937181i \(-0.386575\pi\)
0.348842 + 0.937181i \(0.386575\pi\)
\(548\) 0 0
\(549\) 10.1587 0.433564
\(550\) 0 0
\(551\) −7.60419 0.345607i −0.323949 0.0147233i
\(552\) 0 0
\(553\) 44.2353i 1.88107i
\(554\) 0 0
\(555\) 2.97649i 0.126345i
\(556\) 0 0
\(557\) 22.4867 0.952792 0.476396 0.879231i \(-0.341943\pi\)
0.476396 + 0.879231i \(0.341943\pi\)
\(558\) 0 0
\(559\) −7.76293 −0.328337
\(560\) 0 0
\(561\) 1.40071i 0.0591382i
\(562\) 0 0
\(563\) 19.1196 0.805794 0.402897 0.915245i \(-0.368003\pi\)
0.402897 + 0.915245i \(0.368003\pi\)
\(564\) 0 0
\(565\) 13.4316i 0.565072i
\(566\) 0 0
\(567\) 2.60808i 0.109529i
\(568\) 0 0
\(569\) 29.3008i 1.22835i −0.789169 0.614176i \(-0.789488\pi\)
0.789169 0.614176i \(-0.210512\pi\)
\(570\) 0 0
\(571\) 26.0808i 1.09145i 0.837965 + 0.545724i \(0.183745\pi\)
−0.837965 + 0.545724i \(0.816255\pi\)
\(572\) 0 0
\(573\) 1.00951i 0.0421729i
\(574\) 0 0
\(575\) 1.23017i 0.0513015i
\(576\) 0 0
\(577\) −5.75246 −0.239478 −0.119739 0.992805i \(-0.538206\pi\)
−0.119739 + 0.992805i \(0.538206\pi\)
\(578\) 0 0
\(579\) 26.9654i 1.12064i
\(580\) 0 0
\(581\) 12.3175 0.511015
\(582\) 0 0
\(583\) −2.14827 −0.0889721
\(584\) 0 0
\(585\) 2.60808i 0.107831i
\(586\) 0 0
\(587\) 1.48007i 0.0610888i 0.999533 + 0.0305444i \(0.00972410\pi\)
−0.999533 + 0.0305444i \(0.990276\pi\)
\(588\) 0 0
\(589\) 1.61466 35.5264i 0.0665309 1.46384i
\(590\) 0 0
\(591\) −11.1979 −0.460620
\(592\) 0 0
\(593\) 2.14827 0.0882188 0.0441094 0.999027i \(-0.485955\pi\)
0.0441094 + 0.999027i \(0.485955\pi\)
\(594\) 0 0
\(595\) 2.09193i 0.0857607i
\(596\) 0 0
\(597\) 5.95298i 0.243639i
\(598\) 0 0
\(599\) −36.3175 −1.48389 −0.741946 0.670460i \(-0.766097\pi\)
−0.741946 + 0.670460i \(0.766097\pi\)
\(600\) 0 0
\(601\) 19.1867i 0.782643i 0.920254 + 0.391322i \(0.127982\pi\)
−0.920254 + 0.391322i \(0.872018\pi\)
\(602\) 0 0
\(603\) −9.35664 −0.381032
\(604\) 0 0
\(605\) −7.95036 −0.323228
\(606\) 0 0
\(607\) −15.7734 −0.640223 −0.320111 0.947380i \(-0.603720\pi\)
−0.320111 + 0.947380i \(0.603720\pi\)
\(608\) 0 0
\(609\) −4.55455 −0.184560
\(610\) 0 0
\(611\) 25.9217 1.04868
\(612\) 0 0
\(613\) −19.6741 −0.794630 −0.397315 0.917682i \(-0.630058\pi\)
−0.397315 + 0.917682i \(0.630058\pi\)
\(614\) 0 0
\(615\) 6.96249i 0.280755i
\(616\) 0 0
\(617\) 2.40628 0.0968733 0.0484367 0.998826i \(-0.484576\pi\)
0.0484367 + 0.998826i \(0.484576\pi\)
\(618\) 0 0
\(619\) 23.6205i 0.949388i 0.880151 + 0.474694i \(0.157441\pi\)
−0.880151 + 0.474694i \(0.842559\pi\)
\(620\) 0 0
\(621\) 1.23017i 0.0493649i
\(622\) 0 0
\(623\) −32.7238 −1.31105
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.60419 + 0.345607i 0.303682 + 0.0138022i
\(628\) 0 0
\(629\) 2.38742i 0.0951929i
\(630\) 0 0
\(631\) 6.64419i 0.264501i −0.991216 0.132251i \(-0.957780\pi\)
0.991216 0.132251i \(-0.0422203\pi\)
\(632\) 0 0
\(633\) −1.35664 −0.0539218
\(634\) 0 0
\(635\) 12.5650 0.498628
\(636\) 0 0
\(637\) 0.516154i 0.0204508i
\(638\) 0 0
\(639\) 1.60419 0.0634607
\(640\) 0 0
\(641\) 30.9514i 1.22251i −0.791435 0.611253i \(-0.790666\pi\)
0.791435 0.611253i \(-0.209334\pi\)
\(642\) 0 0
\(643\) 13.4088i 0.528792i −0.964414 0.264396i \(-0.914827\pi\)
0.964414 0.264396i \(-0.0851726\pi\)
\(644\) 0 0
\(645\) 2.97649i 0.117199i
\(646\) 0 0
\(647\) 23.1271i 0.909222i −0.890690 0.454611i \(-0.849778\pi\)
0.890690 0.454611i \(-0.150222\pi\)
\(648\) 0 0
\(649\) 2.80143i 0.109966i
\(650\) 0 0
\(651\) 21.2787i 0.833977i
\(652\) 0 0
\(653\) −17.9321 −0.701739 −0.350869 0.936424i \(-0.614114\pi\)
−0.350869 + 0.936424i \(0.614114\pi\)
\(654\) 0 0
\(655\) 15.3029i 0.597933i
\(656\) 0 0
\(657\) −10.5650 −0.412181
\(658\) 0 0
\(659\) −37.9217 −1.47722 −0.738609 0.674134i \(-0.764517\pi\)
−0.738609 + 0.674134i \(0.764517\pi\)
\(660\) 0 0
\(661\) 0.295496i 0.0114935i −0.999983 0.00574674i \(-0.998171\pi\)
0.999983 0.00574674i \(-0.00182925\pi\)
\(662\) 0 0
\(663\) 2.09193i 0.0812438i
\(664\) 0 0
\(665\) −11.3566 0.516154i −0.440392 0.0200156i
\(666\) 0 0
\(667\) 2.14827 0.0831812
\(668\) 0 0
\(669\) 0.812566 0.0314156
\(670\) 0 0
\(671\) 17.7404i 0.684861i
\(672\) 0 0
\(673\) 39.4896i 1.52221i −0.648626 0.761107i \(-0.724656\pi\)
0.648626 0.761107i \(-0.275344\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 13.1361i 0.504862i −0.967615 0.252431i \(-0.918770\pi\)
0.967615 0.252431i \(-0.0812301\pi\)
\(678\) 0 0
\(679\) −23.0986 −0.886444
\(680\) 0 0
\(681\) −22.3279 −0.855609
\(682\) 0 0
\(683\) −49.5958 −1.89773 −0.948865 0.315682i \(-0.897767\pi\)
−0.948865 + 0.315682i \(0.897767\pi\)
\(684\) 0 0
\(685\) 8.56502 0.327253
\(686\) 0 0
\(687\) 16.8021 0.641040
\(688\) 0 0
\(689\) 3.20838 0.122229
\(690\) 0 0
\(691\) 28.2913i 1.07625i 0.842865 + 0.538125i \(0.180867\pi\)
−0.842865 + 0.538125i \(0.819133\pi\)
\(692\) 0 0
\(693\) 4.55455 0.173013
\(694\) 0 0
\(695\) 13.9250i 0.528204i
\(696\) 0 0
\(697\) 5.58457i 0.211531i
\(698\) 0 0
\(699\) −25.5049 −0.964684
\(700\) 0 0
\(701\) −1.18743 −0.0448488 −0.0224244 0.999749i \(-0.507138\pi\)
−0.0224244 + 0.999749i \(0.507138\pi\)
\(702\) 0 0
\(703\) 12.9608 + 0.589064i 0.488827 + 0.0222170i
\(704\) 0 0
\(705\) 9.93897i 0.374323i
\(706\) 0 0
\(707\) 47.7735i 1.79671i
\(708\) 0 0
\(709\) −7.74198 −0.290756 −0.145378 0.989376i \(-0.546440\pi\)
−0.145378 + 0.989376i \(0.546440\pi\)
\(710\) 0 0
\(711\) 16.9608 0.636081
\(712\) 0 0
\(713\) 10.0366i 0.375874i
\(714\) 0 0
\(715\) −4.55455 −0.170331
\(716\) 0 0
\(717\) 8.36320i 0.312329i
\(718\) 0 0
\(719\) 36.9044i 1.37630i 0.725568 + 0.688150i \(0.241577\pi\)
−0.725568 + 0.688150i \(0.758423\pi\)
\(720\) 0 0
\(721\) 6.24847i 0.232705i
\(722\) 0 0
\(723\) 26.8632i 0.999054i
\(724\) 0 0
\(725\) 1.74632i 0.0648567i
\(726\) 0 0
\(727\) 0.147748i 0.00547968i −0.999996 0.00273984i \(-0.999128\pi\)
0.999996 0.00273984i \(-0.000872119\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.38742i 0.0883021i
\(732\) 0 0
\(733\) 30.5650 1.12894 0.564472 0.825452i \(-0.309080\pi\)
0.564472 + 0.825452i \(0.309080\pi\)
\(734\) 0 0
\(735\) 0.197906 0.00729986
\(736\) 0 0
\(737\) 16.3397i 0.601881i
\(738\) 0 0
\(739\) 38.0324i 1.39904i 0.714611 + 0.699522i \(0.246604\pi\)
−0.714611 + 0.699522i \(0.753396\pi\)
\(740\) 0 0
\(741\) −11.3566 0.516154i −0.417197 0.0189614i
\(742\) 0 0
\(743\) −10.0783 −0.369738 −0.184869 0.982763i \(-0.559186\pi\)
−0.184869 + 0.982763i \(0.559186\pi\)
\(744\) 0 0
\(745\) −7.60419 −0.278596
\(746\) 0 0
\(747\) 4.72281i 0.172799i
\(748\) 0 0
\(749\) 25.4352i 0.929382i
\(750\) 0 0
\(751\) −10.3070 −0.376108 −0.188054 0.982159i \(-0.560218\pi\)
−0.188054 + 0.982159i \(0.560218\pi\)
\(752\) 0 0
\(753\) 3.10144i 0.113023i
\(754\) 0 0
\(755\) 13.5154 0.491875
\(756\) 0 0
\(757\) 2.31748 0.0842302 0.0421151 0.999113i \(-0.486590\pi\)
0.0421151 + 0.999113i \(0.486590\pi\)
\(758\) 0 0
\(759\) −2.14827 −0.0779772
\(760\) 0 0
\(761\) −39.9426 −1.44792 −0.723959 0.689843i \(-0.757680\pi\)
−0.723959 + 0.689843i \(0.757680\pi\)
\(762\) 0 0
\(763\) 7.18743 0.260203
\(764\) 0 0
\(765\) 0.802094 0.0289998
\(766\) 0 0
\(767\) 4.18386i 0.151070i
\(768\) 0 0
\(769\) −15.5049 −0.559121 −0.279561 0.960128i \(-0.590189\pi\)
−0.279561 + 0.960128i \(0.590189\pi\)
\(770\) 0 0
\(771\) 17.6155i 0.634406i
\(772\) 0 0
\(773\) 32.5272i 1.16992i 0.811062 + 0.584960i \(0.198890\pi\)
−0.811062 + 0.584960i \(0.801110\pi\)
\(774\) 0 0
\(775\) 8.15874 0.293071
\(776\) 0 0
\(777\) 7.76293 0.278493
\(778\) 0 0
\(779\) 30.3175 + 1.37792i 1.08624 + 0.0493689i
\(780\) 0 0
\(781\) 2.80143i 0.100243i
\(782\) 0 0
\(783\) 1.74632i 0.0624084i
\(784\) 0 0
\(785\) −15.3566 −0.548102
\(786\) 0 0
\(787\) 1.03917 0.0370423 0.0185211 0.999828i \(-0.494104\pi\)
0.0185211 + 0.999828i \(0.494104\pi\)
\(788\) 0 0
\(789\) 17.6155i 0.627127i
\(790\) 0 0
\(791\) −35.0308 −1.24555
\(792\) 0 0
\(793\) 26.4948i 0.940859i
\(794\) 0 0
\(795\) 1.23017i 0.0436295i
\(796\) 0 0
\(797\) 31.0991i 1.10159i −0.834641 0.550794i \(-0.814325\pi\)
0.834641 0.550794i \(-0.185675\pi\)
\(798\) 0 0
\(799\) 7.97200i 0.282029i
\(800\) 0 0
\(801\) 12.5471i 0.443328i
\(802\) 0 0
\(803\) 18.4499i 0.651084i