Properties

Label 300.3.g.d.101.1
Level $300$
Weight $3$
Character 300.101
Analytic conductor $8.174$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.1
Root \(-2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 300.101
Dual form 300.3.g.d.101.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.00000 - 2.23607i) q^{3} -2.00000 q^{7} +(-1.00000 + 8.94427i) q^{9} +O(q^{10})\) \(q+(-2.00000 - 2.23607i) q^{3} -2.00000 q^{7} +(-1.00000 + 8.94427i) q^{9} -13.4164i q^{11} -8.00000 q^{13} +13.4164i q^{17} -34.0000 q^{19} +(4.00000 + 4.47214i) q^{21} +40.2492i q^{23} +(22.0000 - 15.6525i) q^{27} +40.2492i q^{29} +14.0000 q^{31} +(-30.0000 + 26.8328i) q^{33} -56.0000 q^{37} +(16.0000 + 17.8885i) q^{39} +26.8328i q^{41} -8.00000 q^{43} -40.2492i q^{47} -45.0000 q^{49} +(30.0000 - 26.8328i) q^{51} -40.2492i q^{53} +(68.0000 + 76.0263i) q^{57} -13.4164i q^{59} -46.0000 q^{61} +(2.00000 - 17.8885i) q^{63} -32.0000 q^{67} +(90.0000 - 80.4984i) q^{69} -53.6656i q^{71} +106.000 q^{73} +26.8328i q^{77} -22.0000 q^{79} +(-79.0000 - 17.8885i) q^{81} -120.748i q^{83} +(90.0000 - 80.4984i) q^{87} -107.331i q^{89} +16.0000 q^{91} +(-28.0000 - 31.3050i) q^{93} -122.000 q^{97} +(120.000 + 13.4164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 4q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 4q^{7} - 2q^{9} - 16q^{13} - 68q^{19} + 8q^{21} + 44q^{27} + 28q^{31} - 60q^{33} - 112q^{37} + 32q^{39} - 16q^{43} - 90q^{49} + 60q^{51} + 136q^{57} - 92q^{61} + 4q^{63} - 64q^{67} + 180q^{69} + 212q^{73} - 44q^{79} - 158q^{81} + 180q^{87} + 32q^{91} - 56q^{93} - 244q^{97} + 240q^{99} + O(q^{100}) \)

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
\(3\) −2.00000 2.23607i −0.666667 0.745356i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.285714 −0.142857 0.989743i \(-0.545629\pi\)
−0.142857 + 0.989743i \(0.545629\pi\)
\(8\) 0 0
\(9\) −1.00000 + 8.94427i −0.111111 + 0.993808i
\(10\) 0 0
\(11\) 13.4164i 1.21967i −0.792527 0.609837i \(-0.791235\pi\)
0.792527 0.609837i \(-0.208765\pi\)
\(12\) 0 0
\(13\) −8.00000 −0.615385 −0.307692 0.951486i \(-0.599557\pi\)
−0.307692 + 0.951486i \(0.599557\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 13.4164i 0.789200i 0.918853 + 0.394600i \(0.129117\pi\)
−0.918853 + 0.394600i \(0.870883\pi\)
\(18\) 0 0
\(19\) −34.0000 −1.78947 −0.894737 0.446594i \(-0.852637\pi\)
−0.894737 + 0.446594i \(0.852637\pi\)
\(20\) 0 0
\(21\) 4.00000 + 4.47214i 0.190476 + 0.212959i
\(22\) 0 0
\(23\) 40.2492i 1.74997i 0.484153 + 0.874983i \(0.339128\pi\)
−0.484153 + 0.874983i \(0.660872\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 22.0000 15.6525i 0.814815 0.579721i
\(28\) 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
\(33\) −30.0000 + 26.8328i −0.909091 + 0.813116i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −56.0000 −1.51351 −0.756757 0.653697i \(-0.773217\pi\)
−0.756757 + 0.653697i \(0.773217\pi\)
\(38\) 0 0
\(39\) 16.0000 + 17.8885i 0.410256 + 0.458681i
\(40\) 0 0
\(41\) 26.8328i 0.654459i 0.944945 + 0.327229i \(0.106115\pi\)
−0.944945 + 0.327229i \(0.893885\pi\)
\(42\) 0 0
\(43\) −8.00000 −0.186047 −0.0930233 0.995664i \(-0.529653\pi\)
−0.0930233 + 0.995664i \(0.529653\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 40.2492i 0.856366i −0.903692 0.428183i \(-0.859154\pi\)
0.903692 0.428183i \(-0.140846\pi\)
\(48\) 0 0
\(49\) −45.0000 −0.918367
\(50\) 0 0
\(51\) 30.0000 26.8328i 0.588235 0.526134i
\(52\) 0 0
\(53\) 40.2492i 0.759419i −0.925106 0.379710i \(-0.876024\pi\)
0.925106 0.379710i \(-0.123976\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 68.0000 + 76.0263i 1.19298 + 1.33379i
\(58\) 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
\(63\) 2.00000 17.8885i 0.0317460 0.283945i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −32.0000 −0.477612 −0.238806 0.971067i \(-0.576756\pi\)
−0.238806 + 0.971067i \(0.576756\pi\)
\(68\) 0 0
\(69\) 90.0000 80.4984i 1.30435 1.16664i
\(70\) 0 0
\(71\) 53.6656i 0.755854i −0.925835 0.377927i \(-0.876637\pi\)
0.925835 0.377927i \(-0.123363\pi\)
\(72\) 0 0
\(73\) 106.000 1.45205 0.726027 0.687666i \(-0.241365\pi\)
0.726027 + 0.687666i \(0.241365\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 26.8328i 0.348478i
\(78\) 0 0
\(79\) −22.0000 −0.278481 −0.139241 0.990259i \(-0.544466\pi\)
−0.139241 + 0.990259i \(0.544466\pi\)
\(80\) 0 0
\(81\) −79.0000 17.8885i −0.975309 0.220846i
\(82\) 0 0
\(83\) 120.748i 1.45479i −0.686218 0.727396i \(-0.740731\pi\)
0.686218 0.727396i \(-0.259269\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 90.0000 80.4984i 1.03448 0.925270i
\(88\) 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
\(93\) −28.0000 31.3050i −0.301075 0.336612i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −122.000 −1.25773 −0.628866 0.777514i \(-0.716481\pi\)
−0.628866 + 0.777514i \(0.716481\pi\)
\(98\) 0 0
\(99\) 120.000 + 13.4164i 1.21212 + 0.135519i
\(100\) 0 0
\(101\) 174.413i 1.72686i 0.504465 + 0.863432i \(0.331690\pi\)
−0.504465 + 0.863432i \(0.668310\pi\)
\(102\) 0 0
\(103\) 46.0000 0.446602 0.223301 0.974750i \(-0.428317\pi\)
0.223301 + 0.974750i \(0.428317\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.4164i 0.125387i 0.998033 + 0.0626935i \(0.0199691\pi\)
−0.998033 + 0.0626935i \(0.980031\pi\)
\(108\) 0 0
\(109\) 86.0000 0.788991 0.394495 0.918898i \(-0.370919\pi\)
0.394495 + 0.918898i \(0.370919\pi\)
\(110\) 0 0
\(111\) 112.000 + 125.220i 1.00901 + 1.12811i
\(112\) 0 0
\(113\) 93.9149i 0.831105i 0.909569 + 0.415552i \(0.136412\pi\)
−0.909569 + 0.415552i \(0.863588\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.00000 71.5542i 0.0683761 0.611574i
\(118\) 0 0
\(119\) 26.8328i 0.225486i
\(120\) 0 0
\(121\) −59.0000 −0.487603
\(122\) 0 0
\(123\) 60.0000 53.6656i 0.487805 0.436306i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 34.0000 0.267717 0.133858 0.991000i \(-0.457263\pi\)
0.133858 + 0.991000i \(0.457263\pi\)
\(128\) 0 0
\(129\) 16.0000 + 17.8885i 0.124031 + 0.138671i
\(130\) 0 0
\(131\) 147.580i 1.12657i −0.826263 0.563284i \(-0.809538\pi\)
0.826263 0.563284i \(-0.190462\pi\)
\(132\) 0 0
\(133\) 68.0000 0.511278
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 40.2492i 0.293790i −0.989152 0.146895i \(-0.953072\pi\)
0.989152 0.146895i \(-0.0469279\pi\)
\(138\) 0 0
\(139\) −82.0000 −0.589928 −0.294964 0.955508i \(-0.595308\pi\)
−0.294964 + 0.955508i \(0.595308\pi\)
\(140\) 0 0
\(141\) −90.0000 + 80.4984i −0.638298 + 0.570911i
\(142\) 0 0
\(143\) 107.331i 0.750568i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 90.0000 + 100.623i 0.612245 + 0.684511i
\(148\) 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
\(153\) −120.000 13.4164i −0.784314 0.0876889i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −92.0000 −0.585987 −0.292994 0.956114i \(-0.594651\pi\)
−0.292994 + 0.956114i \(0.594651\pi\)
\(158\) 0 0
\(159\) −90.0000 + 80.4984i −0.566038 + 0.506280i
\(160\) 0 0
\(161\) 80.4984i 0.499990i
\(162\) 0 0
\(163\) −68.0000 −0.417178 −0.208589 0.978003i \(-0.566887\pi\)
−0.208589 + 0.978003i \(0.566887\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 67.0820i 0.401689i 0.979623 + 0.200844i \(0.0643686\pi\)
−0.979623 + 0.200844i \(0.935631\pi\)
\(168\) 0 0
\(169\) −105.000 −0.621302
\(170\) 0 0
\(171\) 34.0000 304.105i 0.198830 1.77839i
\(172\) 0 0
\(173\) 120.748i 0.697963i −0.937130 0.348982i \(-0.886528\pi\)
0.937130 0.348982i \(-0.113472\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −30.0000 + 26.8328i −0.169492 + 0.151598i
\(178\) 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
\(183\) 92.0000 + 102.859i 0.502732 + 0.562072i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 180.000 0.962567
\(188\) 0 0
\(189\) −44.0000 + 31.3050i −0.232804 + 0.165635i
\(190\) 0 0
\(191\) 80.4984i 0.421458i 0.977545 + 0.210729i \(0.0675837\pi\)
−0.977545 + 0.210729i \(0.932416\pi\)
\(192\) 0 0
\(193\) −218.000 −1.12953 −0.564767 0.825251i \(-0.691034\pi\)
−0.564767 + 0.825251i \(0.691034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 93.9149i 0.476725i 0.971176 + 0.238363i \(0.0766107\pi\)
−0.971176 + 0.238363i \(0.923389\pi\)
\(198\) 0 0
\(199\) −34.0000 −0.170854 −0.0854271 0.996344i \(-0.527225\pi\)
−0.0854271 + 0.996344i \(0.527225\pi\)
\(200\) 0 0
\(201\) 64.0000 + 71.5542i 0.318408 + 0.355991i
\(202\) 0 0
\(203\) 80.4984i 0.396544i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −360.000 40.2492i −1.73913 0.194441i
\(208\) 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
\(213\) −120.000 + 107.331i −0.563380 + 0.503903i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −28.0000 −0.129032
\(218\) 0 0
\(219\) −212.000 237.023i −0.968037 1.08230i
\(220\) 0 0
\(221\) 107.331i 0.485662i
\(222\) 0 0
\(223\) −398.000 −1.78475 −0.892377 0.451291i \(-0.850964\pi\)
−0.892377 + 0.451291i \(0.850964\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 335.410i 1.47758i −0.673937 0.738789i \(-0.735398\pi\)
0.673937 0.738789i \(-0.264602\pi\)
\(228\) 0 0
\(229\) 86.0000 0.375546 0.187773 0.982212i \(-0.439873\pi\)
0.187773 + 0.982212i \(0.439873\pi\)
\(230\) 0 0
\(231\) 60.0000 53.6656i 0.259740 0.232319i
\(232\) 0 0
\(233\) 308.577i 1.32437i 0.749342 + 0.662183i \(0.230370\pi\)
−0.749342 + 0.662183i \(0.769630\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 44.0000 + 49.1935i 0.185654 + 0.207567i
\(238\) 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
\(243\) 118.000 + 212.426i 0.485597 + 0.874183i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 272.000 1.10121
\(248\) 0 0
\(249\) −270.000 + 241.495i −1.08434 + 0.969861i
\(250\) 0 0
\(251\) 308.577i 1.22939i 0.788764 + 0.614696i \(0.210721\pi\)
−0.788764 + 0.614696i \(0.789279\pi\)
\(252\) 0 0
\(253\) 540.000 2.13439
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 174.413i 0.678651i 0.940669 + 0.339325i \(0.110199\pi\)
−0.940669 + 0.339325i \(0.889801\pi\)
\(258\) 0 0
\(259\) 112.000 0.432432
\(260\) 0 0
\(261\) −360.000 40.2492i −1.37931 0.154212i
\(262\) 0 0
\(263\) 254.912i 0.969246i 0.874723 + 0.484623i \(0.161043\pi\)
−0.874723 + 0.484623i \(0.838957\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −240.000 + 214.663i −0.898876 + 0.803979i
\(268\) 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
\(273\) −32.0000 35.7771i −0.117216 0.131052i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 448.000 1.61733 0.808664 0.588270i \(-0.200191\pi\)
0.808664 + 0.588270i \(0.200191\pi\)
\(278\) 0 0
\(279\) −14.0000 + 125.220i −0.0501792 + 0.448817i
\(280\) 0 0
\(281\) 187.830i 0.668433i −0.942496 0.334217i \(-0.891528\pi\)
0.942496 0.334217i \(-0.108472\pi\)
\(282\) 0 0
\(283\) −248.000 −0.876325 −0.438163 0.898896i \(-0.644371\pi\)
−0.438163 + 0.898896i \(0.644371\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 53.6656i 0.186988i
\(288\) 0 0
\(289\) 109.000 0.377163
\(290\) 0 0
\(291\) 244.000 + 272.800i 0.838488 + 0.937458i
\(292\) 0 0
\(293\) 147.580i 0.503688i −0.967768 0.251844i \(-0.918963\pi\)
0.967768 0.251844i \(-0.0810369\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −210.000 295.161i −0.707071 0.993808i
\(298\) 0 0
\(299\) 321.994i 1.07690i
\(300\) 0 0
\(301\) 16.0000 0.0531561
\(302\) 0 0
\(303\) 390.000 348.827i 1.28713 1.15124i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −272.000 −0.885993 −0.442997 0.896523i \(-0.646085\pi\)
−0.442997 + 0.896523i \(0.646085\pi\)
\(308\) 0 0
\(309\) −92.0000 102.859i −0.297735 0.332877i
\(310\) 0 0
\(311\) 53.6656i 0.172558i −0.996271 0.0862792i \(-0.972502\pi\)
0.996271 0.0862792i \(-0.0274977\pi\)
\(312\) 0 0
\(313\) −278.000 −0.888179 −0.444089 0.895982i \(-0.646473\pi\)
−0.444089 + 0.895982i \(0.646473\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 120.748i 0.380907i 0.981696 + 0.190454i \(0.0609959\pi\)
−0.981696 + 0.190454i \(0.939004\pi\)
\(318\) 0 0
\(319\) 540.000 1.69279
\(320\) 0 0
\(321\) 30.0000 26.8328i 0.0934579 0.0835913i
\(322\) 0 0
\(323\) 456.158i 1.41225i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −172.000 192.302i −0.525994 0.588079i
\(328\) 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
\(333\) 56.0000 500.879i 0.168168 1.50414i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 358.000 1.06231 0.531157 0.847273i \(-0.321757\pi\)
0.531157 + 0.847273i \(0.321757\pi\)
\(338\) 0 0
\(339\) 210.000 187.830i 0.619469 0.554070i
\(340\) 0 0
\(341\) 187.830i 0.550820i
\(342\) 0 0
\(343\) 188.000 0.548105
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 254.912i 0.734616i 0.930099 + 0.367308i \(0.119720\pi\)
−0.930099 + 0.367308i \(0.880280\pi\)
\(348\) 0 0
\(349\) 518.000 1.48424 0.742120 0.670267i \(-0.233820\pi\)
0.742120 + 0.670267i \(0.233820\pi\)
\(350\) 0 0
\(351\) −176.000 + 125.220i −0.501425 + 0.356752i
\(352\) 0 0
\(353\) 281.745i 0.798143i −0.916920 0.399072i \(-0.869332\pi\)
0.916920 0.399072i \(-0.130668\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −60.0000 + 53.6656i −0.168067 + 0.150324i
\(358\) 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
\(363\) 118.000 + 131.928i 0.325069 + 0.363438i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 178.000 0.485014 0.242507 0.970150i \(-0.422030\pi\)
0.242507 + 0.970150i \(0.422030\pi\)
\(368\) 0 0
\(369\) −240.000 26.8328i −0.650407 0.0727177i
\(370\) 0 0
\(371\) 80.4984i 0.216977i
\(372\) 0 0
\(373\) 532.000 1.42627 0.713137 0.701025i \(-0.247274\pi\)
0.713137 + 0.701025i \(0.247274\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 321.994i 0.854095i
\(378\) 0 0
\(379\) 86.0000 0.226913 0.113456 0.993543i \(-0.463808\pi\)
0.113456 + 0.993543i \(0.463808\pi\)
\(380\) 0 0
\(381\) −68.0000 76.0263i −0.178478 0.199544i
\(382\) 0 0
\(383\) 120.748i 0.315268i −0.987498 0.157634i \(-0.949613\pi\)
0.987498 0.157634i \(-0.0503866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 71.5542i 0.0206718 0.184895i
\(388\) 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
\(393\) −330.000 + 295.161i −0.839695 + 0.751046i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000 0.0705290 0.0352645 0.999378i \(-0.488773\pi\)
0.0352645 + 0.999378i \(0.488773\pi\)
\(398\) 0 0
\(399\) −136.000 152.053i −0.340852 0.381084i
\(400\) 0 0
\(401\) 268.328i 0.669148i 0.942370 + 0.334574i \(0.108592\pi\)
−0.942370 + 0.334574i \(0.891408\pi\)
\(402\) 0 0
\(403\) −112.000 −0.277916
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 751.319i 1.84599i
\(408\) 0 0
\(409\) −214.000 −0.523227 −0.261614 0.965173i \(-0.584255\pi\)
−0.261614 + 0.965173i \(0.584255\pi\)
\(410\) 0 0
\(411\) −90.0000 + 80.4984i −0.218978 + 0.195860i
\(412\) 0 0
\(413\) 26.8328i 0.0649705i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 164.000 + 183.358i 0.393285 + 0.439706i
\(418\) 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
\(423\) 360.000 + 40.2492i 0.851064 + 0.0951518i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 92.0000 0.215457
\(428\) 0 0
\(429\) 240.000 214.663i 0.559441 0.500379i
\(430\) 0 0
\(431\) 241.495i 0.560314i 0.959954 + 0.280157i \(0.0903865\pi\)
−0.959954 + 0.280157i \(0.909613\pi\)
\(432\) 0 0
\(433\) 382.000 0.882217 0.441109 0.897454i \(-0.354585\pi\)
0.441109 + 0.897454i \(0.354585\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1368.47i 3.13152i
\(438\) 0 0
\(439\) −274.000 −0.624146 −0.312073 0.950058i \(-0.601023\pi\)
−0.312073 + 0.950058i \(0.601023\pi\)
\(440\) 0 0
\(441\) 45.0000 402.492i 0.102041 0.912681i
\(442\) 0 0
\(443\) 764.735i 1.72626i 0.504978 + 0.863132i \(0.331501\pi\)
−0.504978 + 0.863132i \(0.668499\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −330.000 + 295.161i −0.738255 + 0.660315i
\(448\) 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
\(453\) 92.0000 + 102.859i 0.203091 + 0.227062i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −446.000 −0.975930 −0.487965 0.872863i \(-0.662261\pi\)
−0.487965 + 0.872863i \(0.662261\pi\)
\(458\) 0 0
\(459\) 210.000 + 295.161i 0.457516 + 0.643052i
\(460\) 0 0
\(461\) 93.9149i 0.203720i 0.994799 + 0.101860i \(0.0324794\pi\)
−0.994799 + 0.101860i \(0.967521\pi\)
\(462\) 0 0
\(463\) −854.000 −1.84449 −0.922246 0.386603i \(-0.873648\pi\)
−0.922246 + 0.386603i \(0.873648\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.4164i 0.0287289i 0.999897 + 0.0143645i \(0.00457251\pi\)
−0.999897 + 0.0143645i \(0.995427\pi\)
\(468\) 0 0
\(469\) 64.0000 0.136461
\(470\) 0 0
\(471\) 184.000 + 205.718i 0.390658 + 0.436769i
\(472\) 0 0
\(473\) 107.331i 0.226916i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 360.000 + 40.2492i 0.754717 + 0.0843799i
\(478\) 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
\(483\) −180.000 + 160.997i −0.372671 + 0.333327i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.00410678 −0.00205339 0.999998i \(-0.500654\pi\)
−0.00205339 + 0.999998i \(0.500654\pi\)
\(488\) 0 0
\(489\) 136.000 + 152.053i 0.278119 + 0.310946i
\(490\) 0 0
\(491\) 469.574i 0.956363i 0.878261 + 0.478182i \(0.158704\pi\)
−0.878261 + 0.478182i \(0.841296\pi\)
\(492\) 0 0
\(493\) −540.000 −1.09533
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 107.331i 0.215958i
\(498\) 0 0
\(499\) −514.000 −1.03006 −0.515030 0.857172i \(-0.672219\pi\)
−0.515030 + 0.857172i \(0.672219\pi\)
\(500\) 0 0
\(501\) 150.000 134.164i 0.299401 0.267793i
\(502\) 0 0
\(503\) 657.404i 1.30697i −0.756941 0.653483i \(-0.773307\pi\)
0.756941 0.653483i \(-0.226693\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 210.000 + 234.787i 0.414201 + 0.463091i
\(508\) 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
\(513\) −748.000 + 532.184i −1.45809 + 1.03740i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −540.000 −1.04449
\(518\) 0 0
\(519\) −270.000 + 241.495i −0.520231 + 0.465309i
\(520\) 0 0
\(521\) 670.820i 1.28756i −0.765209 0.643782i \(-0.777365\pi\)
0.765209 0.643782i \(-0.222635\pi\)
\(522\) 0 0
\(523\) 832.000 1.59082 0.795411 0.606070i \(-0.207255\pi\)
0.795411 + 0.606070i \(0.207255\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 187.830i 0.356413i
\(528\) 0 0
\(529\) −1091.00 −2.06238
\(530\) 0 0
\(531\) 120.000 + 13.4164i 0.225989 + 0.0252663i
\(532\) 0 0
\(533\) 214.663i 0.402744i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 630.000 563.489i 1.17318 1.04933i
\(538\) 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
\(543\) −388.000 433.797i −0.714549 0.798890i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −536.000 −0.979890 −0.489945 0.871753i \(-0.662983\pi\)
−0.489945 + 0.871753i \(0.662983\pi\)
\(548\) 0 0
\(549\) 46.0000 411.437i 0.0837887 0.749429i
\(550\) 0 0
\(551\) 1368.47i 2.48362i
\(552\) 0 0
\(553\) 44.0000 0.0795660
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 898.899i 1.61382i 0.590672 + 0.806911i \(0.298863\pi\)
−0.590672 + 0.806911i \(0.701137\pi\)
\(558\) 0 0
\(559\) 64.0000 0.114490
\(560\) 0 0
\(561\) −360.000 402.492i −0.641711 0.717455i
\(562\) 0 0
\(563\) 818.401i 1.45364i 0.686827 + 0.726821i \(0.259003\pi\)
−0.686827 + 0.726821i \(0.740997\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 158.000 + 35.7771i 0.278660 + 0.0630989i
\(568\) 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
\(573\) 180.000 160.997i 0.314136 0.280972i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −122.000 −0.211438 −0.105719 0.994396i \(-0.533714\pi\)
−0.105719 + 0.994396i \(0.533714\pi\)
\(578\) 0 0
\(579\) 436.000 + 487.463i 0.753022 + 0.841905i
\(580\) 0 0
\(581\) 241.495i 0.415655i
\(582\) 0 0
\(583\) −540.000 −0.926244
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 228.079i 0.388550i 0.980947 + 0.194275i \(0.0622354\pi\)
−0.980947 + 0.194275i \(0.937765\pi\)
\(588\) 0 0
\(589\) −476.000 −0.808149
\(590\) 0 0
\(591\) 210.000 187.830i 0.355330 0.317817i
\(592\) 0 0
\(593\) 737.902i 1.24435i 0.782876 + 0.622177i \(0.213752\pi\)
−0.782876 + 0.622177i \(0.786248\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 68.0000 + 76.0263i 0.113903 + 0.127347i
\(598\) 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
\(603\) 32.0000 286.217i 0.0530680 0.474655i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −206.000 −0.339374 −0.169687 0.985498i \(-0.554276\pi\)
−0.169687 + 0.985498i \(0.554276\pi\)
\(608\) 0 0
\(609\) −180.000 + 160.997i −0.295567 + 0.264363i
\(610\) 0 0
\(611\) 321.994i 0.526995i
\(612\) 0 0
\(613\) 556.000 0.907015 0.453507 0.891253i \(-0.350173\pi\)
0.453507 + 0.891253i \(0.350173\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 389.076i 0.630593i 0.948993 + 0.315296i \(0.102104\pi\)
−0.948993 + 0.315296i \(0.897896\pi\)
\(618\) 0 0
\(619\) −514.000 −0.830372 −0.415186 0.909737i \(-0.636283\pi\)
−0.415186 + 0.909737i \(0.636283\pi\)
\(620\) 0 0
\(621\) 630.000 + 885.483i 1.01449 + 1.42590i
\(622\) 0 0
\(623\) 214.663i 0.344563i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1020.00 912.316i 1.62679 1.45505i
\(628\) 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
\(633\) 92.0000 + 102.859i 0.145340 + 0.162495i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 360.000 0.565149
\(638\) 0 0
\(639\) 480.000 + 53.6656i 0.751174 + 0.0839838i
\(640\) 0 0
\(641\) 160.997i 0.251165i −0.992083 0.125583i \(-0.959920\pi\)
0.992083 0.125583i \(-0.0400800\pi\)
\(642\) 0 0
\(643\) −404.000 −0.628305 −0.314152 0.949373i \(-0.601720\pi\)
−0.314152 + 0.949373i \(0.601720\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1167.23i 1.80406i −0.431672 0.902031i \(-0.642076\pi\)
0.431672 0.902031i \(-0.357924\pi\)
\(648\) 0 0
\(649\) −180.000 −0.277350
\(650\) 0 0
\(651\) 56.0000 + 62.6099i 0.0860215 + 0.0961750i
\(652\) 0 0
\(653\) 764.735i 1.17111i −0.810632 0.585555i \(-0.800876\pi\)
0.810632 0.585555i \(-0.199124\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −106.000 + 948.093i −0.161339 + 1.44306i
\(658\) 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
\(663\) −240.000 + 214.663i −0.361991 + 0.323775i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1620.00 −2.42879
\(668\) 0 0
\(669\) 796.000 + 889.955i 1.18984 + 1.33028i
\(670\) 0 0
\(671\) 617.155i 0.919754i
\(672\) 0 0
\(673\) −194.000 −0.288262 −0.144131 0.989559i \(-0.546039\pi\)
−0.144131 + 0.989559i \(0.546039\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 415.909i 0.614341i −0.951655 0.307170i \(-0.900618\pi\)
0.951655 0.307170i \(-0.0993821\pi\)
\(678\) 0 0
\(679\) 244.000 0.359352
\(680\) 0 0
\(681\) −750.000 + 670.820i −1.10132 + 0.985052i
\(682\) 0 0
\(683\) 93.9149i 0.137503i 0.997634 + 0.0687517i \(0.0219016\pi\)
−0.997634 + 0.0687517i \(0.978098\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −172.000 192.302i −0.250364 0.279915i
\(688\) 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
\(693\) −240.000 26.8328i −0.346320 0.0387198i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −360.000 −0.516499
\(698\) 0 0
\(699\) 690.000 617.155i 0.987124 0.882911i
\(700\) 0 0
\(701\) 576.906i 0.822975i 0.911415 + 0.411488i \(0.134991\pi\)
−0.911415 + 0.411488i \(0.865009\pi\)
\(702\) 0 0
\(703\) 1904.00 2.70839
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 348.827i 0.493390i
\(708\) 0 0
\(709\) 1166.00 1.64457 0.822285 0.569076i \(-0.192699\pi\)
0.822285 + 0.569076i \(0.192699\pi\)
\(710\) 0 0
\(711\) 22.0000 196.774i 0.0309423 0.276757i
\(712\) 0 0
\(713\) 563.489i 0.790307i
\(714\) 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
\(723\) −268.000 299.633i −0.370678 0.414430i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1318.00 1.81293 0.906465 0.422281i \(-0.138770\pi\)
0.906465 + 0.422281i \(0.138770\pi\)
\(728\) 0 0
\(729\) 239.000 688.709i 0.327846 0.944731i
\(730\) 0 0
\(731\) 107.331i 0.146828i
\(732\) 0 0
\(733\) 136.000 0.185539 0.0927694 0.995688i \(-0.470428\pi\)
0.0927694 + 0.995688i \(0.470428\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 429.325i 0.582531i
\(738\) 0 0
\(739\) −682.000 −0.922869 −0.461434 0.887174i \(-0.652665\pi\)
−0.461434 + 0.887174i \(0.652665\pi\)
\(740\) 0 0
\(741\) −544.000 608.210i −0.734143 0.820797i
\(742\) 0 0
\(743\) 872.067i 1.17371i −0.809692 0.586855i \(-0.800366\pi\)
0.809692 0.586855i \(-0.199634\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1080.00 + 120.748i 1.44578 + 0.161643i
\(748\) 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
\(753\) 690.000 617.155i 0.916335 0.819595i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1112.00 −1.46896 −0.734478 0.678632i \(-0.762573\pi\)
−0.734478 + 0.678632i \(0.762573\pi\)
\(758\) 0 0
\(759\) −1080.00 1207.48i −1.42292 1.59088i
\(760\) 0 0
\(761\) 80.4984i 0.105780i 0.998600 + 0.0528899i \(0.0168432\pi\)
−0.998600 + 0.0528899i \(0.983157\pi\)
\(762\) 0 0
\(763\) −172.000 −0.225426
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 107.331i 0.139936i
\(768\) 0 0
\(769\) 206.000 0.267880 0.133940 0.990989i \(-0.457237\pi\)
0.133940 + 0.990989i \(0.457237\pi\)
\(770\) 0 0
\(771\) 390.000 348.827i 0.505837 0.452434i
\(772\) 0 0
\(773\) 1086.73i 1.40586i −0.711260 0.702930i \(-0.751875\pi\)
0.711260 0.702930i \(-0.248125\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −224.000 250.440i −0.288288 0.322316i
\(778\) 0 0
\(779\) 912.316i 1.17114i
\(780\) 0 0
\(781\) −720.000 −0.921895
\(782\) 0 0
\(783\) 630.000 + 885.483i 0.804598 + 1.13088i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1168.00 1.48412 0.742058 0.670335i \(-0.233850\pi\)
0.742058 + 0.670335i \(0.233850\pi\)
\(788\) 0 0
\(789\) 570.000 509.823i 0.722433 0.646164i
\(790\) 0 0
\(791\) 187.830i 0.237459i
\(792\) 0 0
\(793\) 368.000 0.464061
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1194.06i 1.49819i 0.662461 + 0.749097i \(0.269512\pi\)
−0.662461 + 0.749097i \(0.730488\pi\)
\(798\) 0 0
\(799\) 540.000 0.675845
\(800\) 0 0
\(801\) 960.000 + 107.331i 1.19850 + 0.133997i