Properties

Label 300.3.b.c.149.4
Level $300$
Weight $3$
Character 300.149
Analytic conductor $8.174$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.4
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 300.149
Dual form 300.3.b.c.149.3

$q$-expansion

\(f(q)\) \(=\) \(q+(2.23607 + 2.00000i) q^{3} -2.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})\) \(q+(2.23607 + 2.00000i) q^{3} -2.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +13.4164i q^{11} +8.00000i q^{13} +13.4164 q^{17} +34.0000 q^{19} +(4.00000 - 4.47214i) q^{21} -40.2492 q^{23} +(-15.6525 + 22.0000i) q^{27} +40.2492i q^{29} +14.0000 q^{31} +(-26.8328 + 30.0000i) q^{33} -56.0000i q^{37} +(-16.0000 + 17.8885i) q^{39} -26.8328i q^{41} +8.00000i q^{43} -40.2492 q^{47} +45.0000 q^{49} +(30.0000 + 26.8328i) q^{51} +40.2492 q^{53} +(76.0263 + 68.0000i) q^{57} -13.4164i q^{59} -46.0000 q^{61} +(17.8885 - 2.00000i) q^{63} -32.0000i q^{67} +(-90.0000 - 80.4984i) q^{69} +53.6656i q^{71} -106.000i q^{73} +26.8328 q^{77} +22.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} +120.748 q^{83} +(-80.4984 + 90.0000i) q^{87} -107.331i q^{89} +16.0000 q^{91} +(31.3050 + 28.0000i) q^{93} -122.000i q^{97} +(-120.000 + 13.4164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} + 136q^{19} + 16q^{21} + 56q^{31} - 64q^{39} + 180q^{49} + 120q^{51} - 184q^{61} - 360q^{69} + 88q^{79} - 316q^{81} + 64q^{91} - 480q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.23607 + 2.00000i 0.745356 + 0.666667i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.285714i −0.989743 0.142857i \(-0.954371\pi\)
0.989743 0.142857i \(-0.0456289\pi\)
\(8\) 0 0
\(9\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(10\) 0 0
\(11\) 13.4164i 1.21967i 0.792527 + 0.609837i \(0.208765\pi\)
−0.792527 + 0.609837i \(0.791235\pi\)
\(12\) 0 0
\(13\) 8.00000i 0.615385i 0.951486 + 0.307692i \(0.0995567\pi\)
−0.951486 + 0.307692i \(0.900443\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 13.4164 0.789200 0.394600 0.918853i \(-0.370883\pi\)
0.394600 + 0.918853i \(0.370883\pi\)
\(18\) 0 0
\(19\) 34.0000 1.78947 0.894737 0.446594i \(-0.147363\pi\)
0.894737 + 0.446594i \(0.147363\pi\)
\(20\) 0 0
\(21\) 4.00000 4.47214i 0.190476 0.212959i
\(22\) 0 0
\(23\) −40.2492 −1.74997 −0.874983 0.484153i \(-0.839128\pi\)
−0.874983 + 0.484153i \(0.839128\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −15.6525 + 22.0000i −0.579721 + 0.814815i
\(28\) 0 0
\(29\) 40.2492i 1.38790i 0.720021 + 0.693952i \(0.244132\pi\)
−0.720021 + 0.693952i \(0.755868\pi\)
\(30\) 0 0
\(31\) 14.0000 0.451613 0.225806 0.974172i \(-0.427498\pi\)
0.225806 + 0.974172i \(0.427498\pi\)
\(32\) 0 0
\(33\) −26.8328 + 30.0000i −0.813116 + 0.909091i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 56.0000i 1.51351i −0.653697 0.756757i \(-0.726783\pi\)
0.653697 0.756757i \(-0.273217\pi\)
\(38\) 0 0
\(39\) −16.0000 + 17.8885i −0.410256 + 0.458681i
\(40\) 0 0
\(41\) 26.8328i 0.654459i −0.944945 0.327229i \(-0.893885\pi\)
0.944945 0.327229i \(-0.106115\pi\)
\(42\) 0 0
\(43\) 8.00000i 0.186047i 0.995664 + 0.0930233i \(0.0296531\pi\)
−0.995664 + 0.0930233i \(0.970347\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −40.2492 −0.856366 −0.428183 0.903692i \(-0.640846\pi\)
−0.428183 + 0.903692i \(0.640846\pi\)
\(48\) 0 0
\(49\) 45.0000 0.918367
\(50\) 0 0
\(51\) 30.0000 + 26.8328i 0.588235 + 0.526134i
\(52\) 0 0
\(53\) 40.2492 0.759419 0.379710 0.925106i \(-0.376024\pi\)
0.379710 + 0.925106i \(0.376024\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 76.0263 + 68.0000i 1.33379 + 1.19298i
\(58\) 0 0
\(59\) 13.4164i 0.227397i −0.993515 0.113698i \(-0.963730\pi\)
0.993515 0.113698i \(-0.0362697\pi\)
\(60\) 0 0
\(61\) −46.0000 −0.754098 −0.377049 0.926193i \(-0.623061\pi\)
−0.377049 + 0.926193i \(0.623061\pi\)
\(62\) 0 0
\(63\) 17.8885 2.00000i 0.283945 0.0317460i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 32.0000i 0.477612i −0.971067 0.238806i \(-0.923244\pi\)
0.971067 0.238806i \(-0.0767560\pi\)
\(68\) 0 0
\(69\) −90.0000 80.4984i −1.30435 1.16664i
\(70\) 0 0
\(71\) 53.6656i 0.755854i 0.925835 + 0.377927i \(0.123363\pi\)
−0.925835 + 0.377927i \(0.876637\pi\)
\(72\) 0 0
\(73\) 106.000i 1.45205i −0.687666 0.726027i \(-0.741365\pi\)
0.687666 0.726027i \(-0.258635\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 26.8328 0.348478
\(78\) 0 0
\(79\) 22.0000 0.278481 0.139241 0.990259i \(-0.455534\pi\)
0.139241 + 0.990259i \(0.455534\pi\)
\(80\) 0 0
\(81\) −79.0000 + 17.8885i −0.975309 + 0.220846i
\(82\) 0 0
\(83\) 120.748 1.45479 0.727396 0.686218i \(-0.240731\pi\)
0.727396 + 0.686218i \(0.240731\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −80.4984 + 90.0000i −0.925270 + 1.03448i
\(88\) 0 0
\(89\) 107.331i 1.20597i −0.797753 0.602985i \(-0.793978\pi\)
0.797753 0.602985i \(-0.206022\pi\)
\(90\) 0 0
\(91\) 16.0000 0.175824
\(92\) 0 0
\(93\) 31.3050 + 28.0000i 0.336612 + 0.301075i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 122.000i 1.25773i −0.777514 0.628866i \(-0.783519\pi\)
0.777514 0.628866i \(-0.216481\pi\)
\(98\) 0 0
\(99\) −120.000 + 13.4164i −1.21212 + 0.135519i
\(100\) 0 0
\(101\) 174.413i 1.72686i −0.504465 0.863432i \(-0.668310\pi\)
0.504465 0.863432i \(-0.331690\pi\)
\(102\) 0 0
\(103\) 46.0000i 0.446602i −0.974750 0.223301i \(-0.928317\pi\)
0.974750 0.223301i \(-0.0716833\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.4164 0.125387 0.0626935 0.998033i \(-0.480031\pi\)
0.0626935 + 0.998033i \(0.480031\pi\)
\(108\) 0 0
\(109\) −86.0000 −0.788991 −0.394495 0.918898i \(-0.629081\pi\)
−0.394495 + 0.918898i \(0.629081\pi\)
\(110\) 0 0
\(111\) 112.000 125.220i 1.00901 1.12811i
\(112\) 0 0
\(113\) −93.9149 −0.831105 −0.415552 0.909569i \(-0.636412\pi\)
−0.415552 + 0.909569i \(0.636412\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −71.5542 + 8.00000i −0.611574 + 0.0683761i
\(118\) 0 0
\(119\) 26.8328i 0.225486i
\(120\) 0 0
\(121\) −59.0000 −0.487603
\(122\) 0 0
\(123\) 53.6656 60.0000i 0.436306 0.487805i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 34.0000i 0.267717i 0.991000 + 0.133858i \(0.0427367\pi\)
−0.991000 + 0.133858i \(0.957263\pi\)
\(128\) 0 0
\(129\) −16.0000 + 17.8885i −0.124031 + 0.138671i
\(130\) 0 0
\(131\) 147.580i 1.12657i 0.826263 + 0.563284i \(0.190462\pi\)
−0.826263 + 0.563284i \(0.809538\pi\)
\(132\) 0 0
\(133\) 68.0000i 0.511278i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −40.2492 −0.293790 −0.146895 0.989152i \(-0.546928\pi\)
−0.146895 + 0.989152i \(0.546928\pi\)
\(138\) 0 0
\(139\) 82.0000 0.589928 0.294964 0.955508i \(-0.404692\pi\)
0.294964 + 0.955508i \(0.404692\pi\)
\(140\) 0 0
\(141\) −90.0000 80.4984i −0.638298 0.570911i
\(142\) 0 0
\(143\) −107.331 −0.750568
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 100.623 + 90.0000i 0.684511 + 0.612245i
\(148\) 0 0
\(149\) 147.580i 0.990473i −0.868758 0.495237i \(-0.835081\pi\)
0.868758 0.495237i \(-0.164919\pi\)
\(150\) 0 0
\(151\) −46.0000 −0.304636 −0.152318 0.988332i \(-0.548674\pi\)
−0.152318 + 0.988332i \(0.548674\pi\)
\(152\) 0 0
\(153\) 13.4164 + 120.000i 0.0876889 + 0.784314i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 92.0000i 0.585987i −0.956114 0.292994i \(-0.905349\pi\)
0.956114 0.292994i \(-0.0946515\pi\)
\(158\) 0 0
\(159\) 90.0000 + 80.4984i 0.566038 + 0.506280i
\(160\) 0 0
\(161\) 80.4984i 0.499990i
\(162\) 0 0
\(163\) 68.0000i 0.417178i 0.978003 + 0.208589i \(0.0668871\pi\)
−0.978003 + 0.208589i \(0.933113\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 67.0820 0.401689 0.200844 0.979623i \(-0.435631\pi\)
0.200844 + 0.979623i \(0.435631\pi\)
\(168\) 0 0
\(169\) 105.000 0.621302
\(170\) 0 0
\(171\) 34.0000 + 304.105i 0.198830 + 1.77839i
\(172\) 0 0
\(173\) 120.748 0.697963 0.348982 0.937130i \(-0.386528\pi\)
0.348982 + 0.937130i \(0.386528\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 26.8328 30.0000i 0.151598 0.169492i
\(178\) 0 0
\(179\) 281.745i 1.57399i 0.616958 + 0.786996i \(0.288365\pi\)
−0.616958 + 0.786996i \(0.711635\pi\)
\(180\) 0 0
\(181\) 194.000 1.07182 0.535912 0.844274i \(-0.319968\pi\)
0.535912 + 0.844274i \(0.319968\pi\)
\(182\) 0 0
\(183\) −102.859 92.0000i −0.562072 0.502732i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 180.000i 0.962567i
\(188\) 0 0
\(189\) 44.0000 + 31.3050i 0.232804 + 0.165635i
\(190\) 0 0
\(191\) 80.4984i 0.421458i −0.977545 0.210729i \(-0.932416\pi\)
0.977545 0.210729i \(-0.0675837\pi\)
\(192\) 0 0
\(193\) 218.000i 1.12953i 0.825251 + 0.564767i \(0.191034\pi\)
−0.825251 + 0.564767i \(0.808966\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 93.9149 0.476725 0.238363 0.971176i \(-0.423389\pi\)
0.238363 + 0.971176i \(0.423389\pi\)
\(198\) 0 0
\(199\) 34.0000 0.170854 0.0854271 0.996344i \(-0.472775\pi\)
0.0854271 + 0.996344i \(0.472775\pi\)
\(200\) 0 0
\(201\) 64.0000 71.5542i 0.318408 0.355991i
\(202\) 0 0
\(203\) 80.4984 0.396544
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −40.2492 360.000i −0.194441 1.73913i
\(208\) 0 0
\(209\) 456.158i 2.18257i
\(210\) 0 0
\(211\) −46.0000 −0.218009 −0.109005 0.994041i \(-0.534766\pi\)
−0.109005 + 0.994041i \(0.534766\pi\)
\(212\) 0 0
\(213\) −107.331 + 120.000i −0.503903 + 0.563380i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 28.0000i 0.129032i
\(218\) 0 0
\(219\) 212.000 237.023i 0.968037 1.08230i
\(220\) 0 0
\(221\) 107.331i 0.485662i
\(222\) 0 0
\(223\) 398.000i 1.78475i 0.451291 + 0.892377i \(0.350964\pi\)
−0.451291 + 0.892377i \(0.649036\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −335.410 −1.47758 −0.738789 0.673937i \(-0.764602\pi\)
−0.738789 + 0.673937i \(0.764602\pi\)
\(228\) 0 0
\(229\) −86.0000 −0.375546 −0.187773 0.982212i \(-0.560127\pi\)
−0.187773 + 0.982212i \(0.560127\pi\)
\(230\) 0 0
\(231\) 60.0000 + 53.6656i 0.259740 + 0.232319i
\(232\) 0 0
\(233\) −308.577 −1.32437 −0.662183 0.749342i \(-0.730370\pi\)
−0.662183 + 0.749342i \(0.730370\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 49.1935 + 44.0000i 0.207567 + 0.185654i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 134.000 0.556017 0.278008 0.960579i \(-0.410326\pi\)
0.278008 + 0.960579i \(0.410326\pi\)
\(242\) 0 0
\(243\) −212.426 118.000i −0.874183 0.485597i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 272.000i 1.10121i
\(248\) 0 0
\(249\) 270.000 + 241.495i 1.08434 + 0.969861i
\(250\) 0 0
\(251\) 308.577i 1.22939i −0.788764 0.614696i \(-0.789279\pi\)
0.788764 0.614696i \(-0.210721\pi\)
\(252\) 0 0
\(253\) 540.000i 2.13439i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 174.413 0.678651 0.339325 0.940669i \(-0.389801\pi\)
0.339325 + 0.940669i \(0.389801\pi\)
\(258\) 0 0
\(259\) −112.000 −0.432432
\(260\) 0 0
\(261\) −360.000 + 40.2492i −1.37931 + 0.154212i
\(262\) 0 0
\(263\) −254.912 −0.969246 −0.484623 0.874723i \(-0.661043\pi\)
−0.484623 + 0.874723i \(0.661043\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 214.663 240.000i 0.803979 0.898876i
\(268\) 0 0
\(269\) 335.410i 1.24688i −0.781872 0.623439i \(-0.785735\pi\)
0.781872 0.623439i \(-0.214265\pi\)
\(270\) 0 0
\(271\) 2.00000 0.00738007 0.00369004 0.999993i \(-0.498825\pi\)
0.00369004 + 0.999993i \(0.498825\pi\)
\(272\) 0 0
\(273\) 35.7771 + 32.0000i 0.131052 + 0.117216i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 448.000i 1.61733i 0.588270 + 0.808664i \(0.299809\pi\)
−0.588270 + 0.808664i \(0.700191\pi\)
\(278\) 0 0
\(279\) 14.0000 + 125.220i 0.0501792 + 0.448817i
\(280\) 0 0
\(281\) 187.830i 0.668433i 0.942496 + 0.334217i \(0.108472\pi\)
−0.942496 + 0.334217i \(0.891528\pi\)
\(282\) 0 0
\(283\) 248.000i 0.876325i 0.898896 + 0.438163i \(0.144371\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −53.6656 −0.186988
\(288\) 0 0
\(289\) −109.000 −0.377163
\(290\) 0 0
\(291\) 244.000 272.800i 0.838488 0.937458i
\(292\) 0 0
\(293\) 147.580 0.503688 0.251844 0.967768i \(-0.418963\pi\)
0.251844 + 0.967768i \(0.418963\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −295.161 210.000i −0.993808 0.707071i
\(298\) 0 0
\(299\) 321.994i 1.07690i
\(300\) 0 0
\(301\) 16.0000 0.0531561
\(302\) 0 0
\(303\) 348.827 390.000i 1.15124 1.28713i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 272.000i 0.885993i −0.896523 0.442997i \(-0.853915\pi\)
0.896523 0.442997i \(-0.146085\pi\)
\(308\) 0 0
\(309\) 92.0000 102.859i 0.297735 0.332877i
\(310\) 0 0
\(311\) 53.6656i 0.172558i 0.996271 + 0.0862792i \(0.0274977\pi\)
−0.996271 + 0.0862792i \(0.972502\pi\)
\(312\) 0 0
\(313\) 278.000i 0.888179i 0.895982 + 0.444089i \(0.146473\pi\)
−0.895982 + 0.444089i \(0.853527\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 120.748 0.380907 0.190454 0.981696i \(-0.439004\pi\)
0.190454 + 0.981696i \(0.439004\pi\)
\(318\) 0 0
\(319\) −540.000 −1.69279
\(320\) 0 0
\(321\) 30.0000 + 26.8328i 0.0934579 + 0.0835913i
\(322\) 0 0
\(323\) 456.158 1.41225
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −192.302 172.000i −0.588079 0.525994i
\(328\) 0 0
\(329\) 80.4984i 0.244676i
\(330\) 0 0
\(331\) −598.000 −1.80665 −0.903323 0.428960i \(-0.858880\pi\)
−0.903323 + 0.428960i \(0.858880\pi\)
\(332\) 0 0
\(333\) 500.879 56.0000i 1.50414 0.168168i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 358.000i 1.06231i 0.847273 + 0.531157i \(0.178243\pi\)
−0.847273 + 0.531157i \(0.821757\pi\)
\(338\) 0 0
\(339\) −210.000 187.830i −0.619469 0.554070i
\(340\) 0 0
\(341\) 187.830i 0.550820i
\(342\) 0 0
\(343\) 188.000i 0.548105i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 254.912 0.734616 0.367308 0.930099i \(-0.380280\pi\)
0.367308 + 0.930099i \(0.380280\pi\)
\(348\) 0 0
\(349\) −518.000 −1.48424 −0.742120 0.670267i \(-0.766180\pi\)
−0.742120 + 0.670267i \(0.766180\pi\)
\(350\) 0 0
\(351\) −176.000 125.220i −0.501425 0.356752i
\(352\) 0 0
\(353\) 281.745 0.798143 0.399072 0.916920i \(-0.369332\pi\)
0.399072 + 0.916920i \(0.369332\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 53.6656 60.0000i 0.150324 0.168067i
\(358\) 0 0
\(359\) 348.827i 0.971662i −0.874053 0.485831i \(-0.838517\pi\)
0.874053 0.485831i \(-0.161483\pi\)
\(360\) 0 0
\(361\) 795.000 2.20222
\(362\) 0 0
\(363\) −131.928 118.000i −0.363438 0.325069i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 178.000i 0.485014i 0.970150 + 0.242507i \(0.0779697\pi\)
−0.970150 + 0.242507i \(0.922030\pi\)
\(368\) 0 0
\(369\) 240.000 26.8328i 0.650407 0.0727177i
\(370\) 0 0
\(371\) 80.4984i 0.216977i
\(372\) 0 0
\(373\) 532.000i 1.42627i −0.701025 0.713137i \(-0.747274\pi\)
0.701025 0.713137i \(-0.252726\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −321.994 −0.854095
\(378\) 0 0
\(379\) −86.0000 −0.226913 −0.113456 0.993543i \(-0.536192\pi\)
−0.113456 + 0.993543i \(0.536192\pi\)
\(380\) 0 0
\(381\) −68.0000 + 76.0263i −0.178478 + 0.199544i
\(382\) 0 0
\(383\) 120.748 0.315268 0.157634 0.987498i \(-0.449613\pi\)
0.157634 + 0.987498i \(0.449613\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −71.5542 + 8.00000i −0.184895 + 0.0206718i
\(388\) 0 0
\(389\) 415.909i 1.06917i −0.845113 0.534587i \(-0.820467\pi\)
0.845113 0.534587i \(-0.179533\pi\)
\(390\) 0 0
\(391\) −540.000 −1.38107
\(392\) 0 0
\(393\) −295.161 + 330.000i −0.751046 + 0.839695i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000i 0.0705290i 0.999378 + 0.0352645i \(0.0112274\pi\)
−0.999378 + 0.0352645i \(0.988773\pi\)
\(398\) 0 0
\(399\) 136.000 152.053i 0.340852 0.381084i
\(400\) 0 0
\(401\) 268.328i 0.669148i −0.942370 0.334574i \(-0.891408\pi\)
0.942370 0.334574i \(-0.108592\pi\)
\(402\) 0 0
\(403\) 112.000i 0.277916i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 751.319 1.84599
\(408\) 0 0
\(409\) 214.000 0.523227 0.261614 0.965173i \(-0.415745\pi\)
0.261614 + 0.965173i \(0.415745\pi\)
\(410\) 0 0
\(411\) −90.0000 80.4984i −0.218978 0.195860i
\(412\) 0 0
\(413\) −26.8328 −0.0649705
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 183.358 + 164.000i 0.439706 + 0.393285i
\(418\) 0 0
\(419\) 174.413i 0.416261i 0.978101 + 0.208130i \(0.0667379\pi\)
−0.978101 + 0.208130i \(0.933262\pi\)
\(420\) 0 0
\(421\) −238.000 −0.565321 −0.282660 0.959220i \(-0.591217\pi\)
−0.282660 + 0.959220i \(0.591217\pi\)
\(422\) 0 0
\(423\) −40.2492 360.000i −0.0951518 0.851064i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 92.0000i 0.215457i
\(428\) 0 0
\(429\) −240.000 214.663i −0.559441 0.500379i
\(430\) 0 0
\(431\) 241.495i 0.560314i −0.959954 0.280157i \(-0.909613\pi\)
0.959954 0.280157i \(-0.0903865\pi\)
\(432\) 0 0
\(433\) 382.000i 0.882217i −0.897454 0.441109i \(-0.854585\pi\)
0.897454 0.441109i \(-0.145415\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1368.47 −3.13152
\(438\) 0 0
\(439\) 274.000 0.624146 0.312073 0.950058i \(-0.398977\pi\)
0.312073 + 0.950058i \(0.398977\pi\)
\(440\) 0 0
\(441\) 45.0000 + 402.492i 0.102041 + 0.912681i
\(442\) 0 0
\(443\) −764.735 −1.72626 −0.863132 0.504978i \(-0.831501\pi\)
−0.863132 + 0.504978i \(0.831501\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 295.161 330.000i 0.660315 0.738255i
\(448\) 0 0
\(449\) 778.152i 1.73308i 0.499110 + 0.866539i \(0.333660\pi\)
−0.499110 + 0.866539i \(0.666340\pi\)
\(450\) 0 0
\(451\) 360.000 0.798226
\(452\) 0 0
\(453\) −102.859 92.0000i −0.227062 0.203091i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 446.000i 0.975930i −0.872863 0.487965i \(-0.837739\pi\)
0.872863 0.487965i \(-0.162261\pi\)
\(458\) 0 0
\(459\) −210.000 + 295.161i −0.457516 + 0.643052i
\(460\) 0 0
\(461\) 93.9149i 0.203720i −0.994799 0.101860i \(-0.967521\pi\)
0.994799 0.101860i \(-0.0324794\pi\)
\(462\) 0 0
\(463\) 854.000i 1.84449i 0.386603 + 0.922246i \(0.373648\pi\)
−0.386603 + 0.922246i \(0.626352\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.4164 0.0287289 0.0143645 0.999897i \(-0.495427\pi\)
0.0143645 + 0.999897i \(0.495427\pi\)
\(468\) 0 0
\(469\) −64.0000 −0.136461
\(470\) 0 0
\(471\) 184.000 205.718i 0.390658 0.436769i
\(472\) 0 0
\(473\) −107.331 −0.226916
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 40.2492 + 360.000i 0.0843799 + 0.754717i
\(478\) 0 0
\(479\) 53.6656i 0.112037i −0.998430 0.0560184i \(-0.982159\pi\)
0.998430 0.0560184i \(-0.0178406\pi\)
\(480\) 0 0
\(481\) 448.000 0.931393
\(482\) 0 0
\(483\) −160.997 + 180.000i −0.333327 + 0.372671i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000i 0.00410678i −0.999998 0.00205339i \(-0.999346\pi\)
0.999998 0.00205339i \(-0.000653614\pi\)
\(488\) 0 0
\(489\) −136.000 + 152.053i −0.278119 + 0.310946i
\(490\) 0 0
\(491\) 469.574i 0.956363i −0.878261 0.478182i \(-0.841296\pi\)
0.878261 0.478182i \(-0.158704\pi\)
\(492\) 0 0
\(493\) 540.000i 1.09533i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 107.331 0.215958
\(498\) 0 0
\(499\) 514.000 1.03006 0.515030 0.857172i \(-0.327781\pi\)
0.515030 + 0.857172i \(0.327781\pi\)
\(500\) 0 0
\(501\) 150.000 + 134.164i 0.299401 + 0.267793i
\(502\) 0 0
\(503\) 657.404 1.30697 0.653483 0.756941i \(-0.273307\pi\)
0.653483 + 0.756941i \(0.273307\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 234.787 + 210.000i 0.463091 + 0.414201i
\(508\) 0 0
\(509\) 308.577i 0.606242i 0.952952 + 0.303121i \(0.0980287\pi\)
−0.952952 + 0.303121i \(0.901971\pi\)
\(510\) 0 0
\(511\) −212.000 −0.414873
\(512\) 0 0
\(513\) −532.184 + 748.000i −1.03740 + 1.45809i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 540.000i 1.04449i
\(518\) 0 0
\(519\) 270.000 + 241.495i 0.520231 + 0.465309i
\(520\) 0 0
\(521\) 670.820i 1.28756i 0.765209 + 0.643782i \(0.222635\pi\)
−0.765209 + 0.643782i \(0.777365\pi\)
\(522\) 0 0
\(523\) 832.000i 1.59082i −0.606070 0.795411i \(-0.707255\pi\)
0.606070 0.795411i \(-0.292745\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 187.830 0.356413
\(528\) 0 0
\(529\) 1091.00 2.06238
\(530\) 0 0
\(531\) 120.000 13.4164i 0.225989 0.0252663i
\(532\) 0 0
\(533\) 214.663 0.402744
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −563.489 + 630.000i −1.04933 + 1.17318i
\(538\) 0 0
\(539\) 603.738i 1.12011i
\(540\) 0 0
\(541\) 314.000 0.580407 0.290203 0.956965i \(-0.406277\pi\)
0.290203 + 0.956965i \(0.406277\pi\)
\(542\) 0 0
\(543\) 433.797 + 388.000i 0.798890 + 0.714549i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 536.000i 0.979890i −0.871753 0.489945i \(-0.837017\pi\)
0.871753 0.489945i \(-0.162983\pi\)
\(548\) 0 0
\(549\) −46.0000 411.437i −0.0837887 0.749429i
\(550\) 0 0
\(551\) 1368.47i 2.48362i
\(552\) 0 0
\(553\) 44.0000i 0.0795660i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 898.899 1.61382 0.806911 0.590672i \(-0.201137\pi\)
0.806911 + 0.590672i \(0.201137\pi\)
\(558\) 0 0
\(559\) −64.0000 −0.114490
\(560\) 0 0
\(561\) −360.000 + 402.492i −0.641711 + 0.717455i
\(562\) 0 0
\(563\) −818.401 −1.45364 −0.726821 0.686827i \(-0.759003\pi\)
−0.726821 + 0.686827i \(0.759003\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 35.7771 + 158.000i 0.0630989 + 0.278660i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −526.000 −0.921191 −0.460595 0.887610i \(-0.652364\pi\)
−0.460595 + 0.887610i \(0.652364\pi\)
\(572\) 0 0
\(573\) 160.997 180.000i 0.280972 0.314136i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 122.000i 0.211438i −0.994396 0.105719i \(-0.966286\pi\)
0.994396 0.105719i \(-0.0337145\pi\)
\(578\) 0 0
\(579\) −436.000 + 487.463i −0.753022 + 0.841905i
\(580\) 0 0
\(581\) 241.495i 0.415655i
\(582\) 0 0
\(583\) 540.000i 0.926244i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 228.079 0.388550 0.194275 0.980947i \(-0.437765\pi\)
0.194275 + 0.980947i \(0.437765\pi\)
\(588\) 0 0
\(589\) 476.000 0.808149
\(590\) 0 0
\(591\) 210.000 + 187.830i 0.355330 + 0.317817i
\(592\) 0 0
\(593\) −737.902 −1.24435 −0.622177 0.782876i \(-0.713752\pi\)
−0.622177 + 0.782876i \(0.713752\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 76.0263 + 68.0000i 0.127347 + 0.113903i
\(598\) 0 0
\(599\) 80.4984i 0.134388i 0.997740 + 0.0671940i \(0.0214047\pi\)
−0.997740 + 0.0671940i \(0.978595\pi\)
\(600\) 0 0
\(601\) −766.000 −1.27454 −0.637271 0.770640i \(-0.719937\pi\)
−0.637271 + 0.770640i \(0.719937\pi\)
\(602\) 0 0
\(603\) 286.217 32.0000i 0.474655 0.0530680i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 206.000i 0.339374i −0.985498 0.169687i \(-0.945724\pi\)
0.985498 0.169687i \(-0.0542757\pi\)
\(608\) 0 0
\(609\) 180.000 + 160.997i 0.295567 + 0.264363i
\(610\) 0 0
\(611\) 321.994i 0.526995i
\(612\) 0 0
\(613\) 556.000i 0.907015i −0.891253 0.453507i \(-0.850173\pi\)
0.891253 0.453507i \(-0.149827\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 389.076 0.630593 0.315296 0.948993i \(-0.397896\pi\)
0.315296 + 0.948993i \(0.397896\pi\)
\(618\) 0 0
\(619\) 514.000 0.830372 0.415186 0.909737i \(-0.363717\pi\)
0.415186 + 0.909737i \(0.363717\pi\)
\(620\) 0 0
\(621\) 630.000 885.483i 1.01449 1.42590i
\(622\) 0 0
\(623\) −214.663 −0.344563
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −912.316 + 1020.00i −1.45505 + 1.62679i
\(628\) 0 0
\(629\) 751.319i 1.19447i
\(630\) 0 0
\(631\) 1094.00 1.73376 0.866878 0.498520i \(-0.166123\pi\)
0.866878 + 0.498520i \(0.166123\pi\)
\(632\) 0 0
\(633\) −102.859 92.0000i −0.162495 0.145340i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 360.000i 0.565149i
\(638\) 0 0
\(639\) −480.000 + 53.6656i −0.751174 + 0.0839838i
\(640\) 0 0
\(641\) 160.997i 0.251165i 0.992083 + 0.125583i \(0.0400800\pi\)
−0.992083 + 0.125583i \(0.959920\pi\)
\(642\) 0 0
\(643\) 404.000i 0.628305i 0.949373 + 0.314152i \(0.101720\pi\)
−0.949373 + 0.314152i \(0.898280\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1167.23 −1.80406 −0.902031 0.431672i \(-0.857924\pi\)
−0.902031 + 0.431672i \(0.857924\pi\)
\(648\) 0 0
\(649\) 180.000 0.277350
\(650\) 0 0
\(651\) 56.0000 62.6099i 0.0860215 0.0961750i
\(652\) 0 0
\(653\) 764.735 1.17111 0.585555 0.810632i \(-0.300876\pi\)
0.585555 + 0.810632i \(0.300876\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 948.093 106.000i 1.44306 0.161339i
\(658\) 0 0
\(659\) 791.568i 1.20117i −0.799563 0.600583i \(-0.794935\pi\)
0.799563 0.600583i \(-0.205065\pi\)
\(660\) 0 0
\(661\) −118.000 −0.178517 −0.0892587 0.996008i \(-0.528450\pi\)
−0.0892587 + 0.996008i \(0.528450\pi\)
\(662\) 0 0
\(663\) −214.663 + 240.000i −0.323775 + 0.361991i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1620.00i 2.42879i
\(668\) 0 0
\(669\) −796.000 + 889.955i −1.18984 + 1.33028i
\(670\) 0 0
\(671\) 617.155i 0.919754i
\(672\) 0 0
\(673\) 194.000i 0.288262i 0.989559 + 0.144131i \(0.0460386\pi\)
−0.989559 + 0.144131i \(0.953961\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −415.909 −0.614341 −0.307170 0.951655i \(-0.599382\pi\)
−0.307170 + 0.951655i \(0.599382\pi\)
\(678\) 0 0
\(679\) −244.000 −0.359352
\(680\) 0 0
\(681\) −750.000 670.820i −1.10132 0.985052i
\(682\) 0 0
\(683\) −93.9149 −0.137503 −0.0687517 0.997634i \(-0.521902\pi\)
−0.0687517 + 0.997634i \(0.521902\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −192.302 172.000i −0.279915 0.250364i
\(688\) 0 0
\(689\) 321.994i 0.467335i
\(690\) 0 0
\(691\) 122.000 0.176556 0.0882779 0.996096i \(-0.471864\pi\)
0.0882779 + 0.996096i \(0.471864\pi\)
\(692\) 0 0
\(693\) 26.8328 + 240.000i 0.0387198 + 0.346320i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 360.000i 0.516499i
\(698\) 0 0
\(699\) −690.000 617.155i −0.987124 0.882911i
\(700\) 0 0
\(701\) 576.906i 0.822975i −0.911415 0.411488i \(-0.865009\pi\)
0.911415 0.411488i \(-0.134991\pi\)
\(702\) 0 0
\(703\) 1904.00i 2.70839i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −348.827 −0.493390
\(708\) 0 0
\(709\) −1166.00 −1.64457 −0.822285 0.569076i \(-0.807301\pi\)
−0.822285 + 0.569076i \(0.807301\pi\)
\(710\) 0 0
\(711\) 22.0000 + 196.774i 0.0309423 + 0.276757i
\(712\) 0 0
\(713\) −563.489 −0.790307
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 214.663i 0.298557i 0.988795 + 0.149279i \(0.0476951\pi\)
−0.988795 + 0.149279i \(0.952305\pi\)
\(720\) 0 0
\(721\) −92.0000 −0.127601
\(722\) 0 0
\(723\) 299.633 + 268.000i 0.414430 + 0.370678i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1318.00i 1.81293i 0.422281 + 0.906465i \(0.361230\pi\)
−0.422281 + 0.906465i \(0.638770\pi\)
\(728\) 0 0
\(729\) −239.000 688.709i −0.327846 0.944731i
\(730\) 0 0
\(731\) 107.331i 0.146828i
\(732\) 0 0
\(733\) 136.000i 0.185539i −0.995688 0.0927694i \(-0.970428\pi\)
0.995688 0.0927694i \(-0.0295720\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 429.325 0.582531
\(738\) 0 0
\(739\) 682.000 0.922869 0.461434 0.887174i \(-0.347335\pi\)
0.461434 + 0.887174i \(0.347335\pi\)
\(740\) 0 0
\(741\) −544.000 + 608.210i −0.734143 + 0.820797i
\(742\) 0 0
\(743\) 872.067 1.17371 0.586855 0.809692i \(-0.300366\pi\)
0.586855 + 0.809692i \(0.300366\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 120.748 + 1080.00i 0.161643 + 1.44578i
\(748\) 0 0
\(749\) 26.8328i 0.0358249i
\(750\) 0 0
\(751\) −658.000 −0.876165 −0.438083 0.898935i \(-0.644342\pi\)
−0.438083 + 0.898935i \(0.644342\pi\)
\(752\) 0 0
\(753\) 617.155 690.000i 0.819595 0.916335i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1112.00i 1.46896i −0.678632 0.734478i \(-0.737427\pi\)
0.678632 0.734478i \(-0.262573\pi\)
\(758\) 0 0
\(759\) 1080.00 1207.48i 1.42292 1.59088i
\(760\) 0 0
\(761\) 80.4984i 0.105780i −0.998600 0.0528899i \(-0.983157\pi\)
0.998600 0.0528899i \(-0.0168432\pi\)
\(762\) 0 0
\(763\) 172.000i 0.225426i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 107.331 0.139936
\(768\) 0 0
\(769\) −206.000 −0.267880 −0.133940 0.990989i \(-0.542763\pi\)
−0.133940 + 0.990989i \(0.542763\pi\)
\(770\) 0 0
\(771\) 390.000 + 348.827i 0.505837 + 0.452434i
\(772\) 0 0
\(773\) 1086.73 1.40586 0.702930 0.711260i \(-0.251875\pi\)
0.702930 + 0.711260i \(0.251875\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −250.440 224.000i −0.322316 0.288288i
\(778\) 0 0
\(779\) 912.316i 1.17114i
\(780\) 0 0
\(781\) −720.000 −0.921895
\(782\) 0 0
\(783\) −885.483 630.000i −1.13088 0.804598i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1168.00i 1.48412i 0.670335 + 0.742058i \(0.266150\pi\)
−0.670335 + 0.742058i \(0.733850\pi\)
\(788\) 0 0
\(789\) −570.000 509.823i −0.722433 0.646164i
\(790\) 0 0
\(791\) 187.830i 0.237459i
\(792\) 0 0
\(793\) 368.000i 0.464061i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1194.06 1.49819 0.749097 0.662461i \(-0.230488\pi\)
0.749097 + 0.662461i \(0.230488\pi\)
\(798\) 0 0
\(799\) −540.000 −0.675845
\(800\) 0 0
\(801\) 960.000 107.331i 1.19850 0.133997i
\(802\) 0 0
\(803\) 1422.14