Properties

Label 300.2.e.a.251.4
Level $300$
Weight $2$
Character 300.251
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.4
Root \(0.866025 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 300.251
Dual form 300.2.e.a.251.3

$q$-expansion

\(f(q)\) \(=\) \(q+(0.866025 + 1.11803i) q^{2} +1.73205 q^{3} +(-0.500000 + 1.93649i) q^{4} +(1.50000 + 1.93649i) q^{6} +(-2.59808 + 1.11803i) q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+(0.866025 + 1.11803i) q^{2} +1.73205 q^{3} +(-0.500000 + 1.93649i) q^{4} +(1.50000 + 1.93649i) q^{6} +(-2.59808 + 1.11803i) q^{8} +3.00000 q^{9} +(-0.866025 + 3.35410i) q^{12} +(-3.50000 - 1.93649i) q^{16} +4.47214i q^{17} +(2.59808 + 3.35410i) q^{18} -7.74597i q^{19} -3.46410 q^{23} +(-4.50000 + 1.93649i) q^{24} +5.19615 q^{27} -7.74597i q^{31} +(-0.866025 - 5.59017i) q^{32} +(-5.00000 + 3.87298i) q^{34} +(-1.50000 + 5.80948i) q^{36} +(8.66025 - 6.70820i) q^{38} +(-3.00000 - 3.87298i) q^{46} -10.3923 q^{47} +(-6.06218 - 3.35410i) q^{48} +7.00000 q^{49} +7.74597i q^{51} +4.47214i q^{53} +(4.50000 + 5.80948i) q^{54} -13.4164i q^{57} -2.00000 q^{61} +(8.66025 - 6.70820i) q^{62} +(5.50000 - 5.80948i) q^{64} +(-8.66025 - 2.23607i) q^{68} -6.00000 q^{69} +(-7.79423 + 3.35410i) q^{72} +(15.0000 + 3.87298i) q^{76} +7.74597i q^{79} +9.00000 q^{81} +3.46410 q^{83} +(1.73205 - 6.70820i) q^{92} -13.4164i q^{93} +(-9.00000 - 11.6190i) q^{94} +(-1.50000 - 9.68246i) q^{96} +(6.06218 + 7.82624i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{4} + 6q^{6} + 12q^{9} + O(q^{10}) \) \( 4q - 2q^{4} + 6q^{6} + 12q^{9} - 14q^{16} - 18q^{24} - 20q^{34} - 6q^{36} - 12q^{46} + 28q^{49} + 18q^{54} - 8q^{61} + 22q^{64} - 24q^{69} + 60q^{76} + 36q^{81} - 36q^{94} - 6q^{96} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 1.11803i 0.612372 + 0.790569i
\(3\) 1.73205 1.00000
\(4\) −0.500000 + 1.93649i −0.250000 + 0.968246i
\(5\) 0 0
\(6\) 1.50000 + 1.93649i 0.612372 + 0.790569i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −2.59808 + 1.11803i −0.918559 + 0.395285i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −0.866025 + 3.35410i −0.250000 + 0.968246i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.50000 1.93649i −0.875000 0.484123i
\(17\) 4.47214i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 2.59808 + 3.35410i 0.612372 + 0.790569i
\(19\) 7.74597i 1.77705i −0.458831 0.888523i \(-0.651732\pi\)
0.458831 0.888523i \(-0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) −4.50000 + 1.93649i −0.918559 + 0.395285i
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.74597i 1.39122i −0.718421 0.695608i \(-0.755135\pi\)
0.718421 0.695608i \(-0.244865\pi\)
\(32\) −0.866025 5.59017i −0.153093 0.988212i
\(33\) 0 0
\(34\) −5.00000 + 3.87298i −0.857493 + 0.664211i
\(35\) 0 0
\(36\) −1.50000 + 5.80948i −0.250000 + 0.968246i
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 8.66025 6.70820i 1.40488 1.08821i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 3.87298i −0.442326 0.571040i
\(47\) −10.3923 −1.51587 −0.757937 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) −6.06218 3.35410i −0.875000 0.484123i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 7.74597i 1.08465i
\(52\) 0 0
\(53\) 4.47214i 0.614295i 0.951662 + 0.307148i \(0.0993745\pi\)
−0.951662 + 0.307148i \(0.900625\pi\)
\(54\) 4.50000 + 5.80948i 0.612372 + 0.790569i
\(55\) 0 0
\(56\) 0 0
\(57\) 13.4164i 1.77705i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.66025 6.70820i 1.09985 0.851943i
\(63\) 0 0
\(64\) 5.50000 5.80948i 0.687500 0.726184i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −8.66025 2.23607i −1.05021 0.271163i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −7.79423 + 3.35410i −0.918559 + 0.395285i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 15.0000 + 3.87298i 1.72062 + 0.444262i
\(77\) 0 0
\(78\) 0 0
\(79\) 7.74597i 0.871489i 0.900070 + 0.435745i \(0.143515\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 3.46410 0.380235 0.190117 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.73205 6.70820i 0.180579 0.699379i
\(93\) 13.4164i 1.39122i
\(94\) −9.00000 11.6190i −0.928279 1.19840i
\(95\) 0 0
\(96\) −1.50000 9.68246i −0.153093 0.988212i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 6.06218 + 7.82624i 0.612372 + 0.790569i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −8.66025 + 6.70820i −0.857493 + 0.664211i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −5.00000 + 3.87298i −0.485643 + 0.376177i
\(107\) 10.3923 1.00466 0.502331 0.864675i \(-0.332476\pi\)
0.502331 + 0.864675i \(0.332476\pi\)
\(108\) −2.59808 + 10.0623i −0.250000 + 0.968246i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.47214i 0.420703i −0.977626 0.210352i \(-0.932539\pi\)
0.977626 0.210352i \(-0.0674609\pi\)
\(114\) 15.0000 11.6190i 1.40488 1.08821i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −1.73205 2.23607i −0.156813 0.202444i
\(123\) 0 0
\(124\) 15.0000 + 3.87298i 1.34704 + 0.347804i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.2583 + 1.11803i 0.995105 + 0.0988212i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −5.00000 11.6190i −0.428746 0.996317i
\(137\) 22.3607i 1.91040i 0.295958 + 0.955201i \(0.404361\pi\)
−0.295958 + 0.955201i \(0.595639\pi\)
\(138\) −5.19615 6.70820i −0.442326 0.571040i
\(139\) 23.2379i 1.97101i 0.169638 + 0.985506i \(0.445740\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −18.0000 −1.51587
\(142\) 0 0
\(143\) 0 0
\(144\) −10.5000 5.80948i −0.875000 0.484123i
\(145\) 0 0
\(146\) 0 0
\(147\) 12.1244 1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 23.2379i 1.89107i 0.325515 + 0.945537i \(0.394462\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 8.66025 + 20.1246i 0.702439 + 1.63232i
\(153\) 13.4164i 1.08465i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −8.66025 + 6.70820i −0.688973 + 0.533676i
\(159\) 7.74597i 0.614295i
\(160\) 0 0
\(161\) 0 0
\(162\) 7.79423 + 10.0623i 0.612372 + 0.790569i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 3.00000 + 3.87298i 0.232845 + 0.300602i
\(167\) −24.2487 −1.87642 −0.938211 0.346064i \(-0.887518\pi\)
−0.938211 + 0.346064i \(0.887518\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 23.2379i 1.77705i
\(172\) 0 0
\(173\) 22.3607i 1.70005i 0.526742 + 0.850026i \(0.323414\pi\)
−0.526742 + 0.850026i \(0.676586\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −3.46410 −0.256074
\(184\) 9.00000 3.87298i 0.663489 0.285520i
\(185\) 0 0
\(186\) 15.0000 11.6190i 1.09985 0.851943i
\(187\) 0 0
\(188\) 5.19615 20.1246i 0.378968 1.46774i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 9.52628 10.0623i 0.687500 0.726184i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −3.50000 + 13.5554i −0.250000 + 0.968246i
\(197\) 4.47214i 0.318626i −0.987228 0.159313i \(-0.949072\pi\)
0.987228 0.159313i \(-0.0509280\pi\)
\(198\) 0 0
\(199\) 23.2379i 1.64729i −0.567105 0.823646i \(-0.691937\pi\)
0.567105 0.823646i \(-0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) −15.0000 3.87298i −1.05021 0.271163i
\(205\) 0 0
\(206\) 0 0
\(207\) −10.3923 −0.722315
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 7.74597i 0.533254i 0.963800 + 0.266627i \(0.0859092\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −8.66025 2.23607i −0.594789 0.153574i
\(213\) 0 0
\(214\) 9.00000 + 11.6190i 0.615227 + 0.794255i
\(215\) 0 0
\(216\) −13.5000 + 5.80948i −0.918559 + 0.395285i
\(217\) 0 0
\(218\) −12.1244 15.6525i −0.821165 1.06012i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 5.00000 3.87298i 0.332595 0.257627i
\(227\) 24.2487 1.60944 0.804722 0.593652i \(-0.202314\pi\)
0.804722 + 0.593652i \(0.202314\pi\)
\(228\) 25.9808 + 6.70820i 1.72062 + 0.444262i
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.3607i 1.46490i −0.680823 0.732448i \(-0.738378\pi\)
0.680823 0.732448i \(-0.261622\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.4164i 0.871489i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −9.52628 12.2984i −0.612372 0.790569i
\(243\) 15.5885 1.00000
\(244\) 1.00000 3.87298i 0.0640184 0.247942i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 8.66025 + 20.1246i 0.549927 + 1.27791i
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 8.50000 + 13.5554i 0.531250 + 0.847215i
\(257\) 31.3050i 1.95275i −0.216085 0.976375i \(-0.569329\pi\)
0.216085 0.976375i \(-0.430671\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −31.1769 −1.92245 −0.961225 0.275764i \(-0.911069\pi\)
−0.961225 + 0.275764i \(0.911069\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 7.74597i 0.470534i −0.971931 0.235267i \(-0.924404\pi\)
0.971931 0.235267i \(-0.0755965\pi\)
\(272\) 8.66025 15.6525i 0.525105 0.949071i
\(273\) 0 0
\(274\) −25.0000 + 19.3649i −1.51031 + 1.16988i
\(275\) 0 0
\(276\) 3.00000 11.6190i 0.180579 0.699379i
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −25.9808 + 20.1246i −1.55822 + 1.20699i
\(279\) 23.2379i 1.39122i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −15.5885 20.1246i −0.928279 1.19840i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −2.59808 16.7705i −0.153093 0.988212i
\(289\) −3.00000 −0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 31.3050i 1.82885i −0.404750 0.914427i \(-0.632641\pi\)
0.404750 0.914427i \(-0.367359\pi\)
\(294\) 10.5000 + 13.5554i 0.612372 + 0.790569i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −25.9808 + 20.1246i −1.49502 + 1.15804i
\(303\) 0 0
\(304\) −15.0000 + 27.1109i −0.860309 + 1.55492i
\(305\) 0 0
\(306\) −15.0000 + 11.6190i −0.857493 + 0.664211i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −15.0000 3.87298i −0.843816 0.217872i
\(317\) 22.3607i 1.25590i −0.778253 0.627950i \(-0.783894\pi\)
0.778253 0.627950i \(-0.216106\pi\)
\(318\) −8.66025 + 6.70820i −0.485643 + 0.376177i
\(319\) 0 0
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 34.6410 1.92748
\(324\) −4.50000 + 17.4284i −0.250000 + 0.968246i
\(325\) 0 0
\(326\) 0 0
\(327\) −24.2487 −1.34096
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 23.2379i 1.27727i −0.769510 0.638635i \(-0.779499\pi\)
0.769510 0.638635i \(-0.220501\pi\)
\(332\) −1.73205 + 6.70820i −0.0950586 + 0.368161i
\(333\) 0 0
\(334\) −21.0000 27.1109i −1.14907 1.48344i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) −11.2583 14.5344i −0.612372 0.790569i
\(339\) 7.74597i 0.420703i
\(340\) 0 0
\(341\) 0 0
\(342\) 25.9808 20.1246i 1.40488 1.08821i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −25.0000 + 19.3649i −1.34401 + 1.04106i
\(347\) 10.3923 0.557888 0.278944 0.960307i \(-0.410016\pi\)
0.278944 + 0.960307i \(0.410016\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.3050i 1.66619i 0.553127 + 0.833097i \(0.313435\pi\)
−0.553127 + 0.833097i \(0.686565\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −41.0000 −2.15789
\(362\) 19.0526 + 24.5967i 1.00138 + 1.29278i
\(363\) −19.0526 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) −3.00000 3.87298i −0.156813 0.202444i
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 12.1244 + 6.70820i 0.632026 + 0.349689i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 25.9808 + 6.70820i 1.34704 + 0.347804i
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 27.0000 11.6190i 1.39242 0.599202i
\(377\) 0 0
\(378\) 0 0
\(379\) 38.7298i 1.98942i −0.102733 0.994709i \(-0.532759\pi\)
0.102733 0.994709i \(-0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 38.1051 1.94708 0.973540 0.228515i \(-0.0733872\pi\)
0.973540 + 0.228515i \(0.0733872\pi\)
\(384\) 19.5000 + 1.93649i 0.995105 + 0.0988212i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 15.4919i 0.783461i
\(392\) −18.1865 + 7.82624i −0.918559 + 0.395285i
\(393\) 0 0
\(394\) 5.00000 3.87298i 0.251896 0.195118i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 25.9808 20.1246i 1.30230 1.00876i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −8.66025 20.1246i −0.428746 0.996317i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 38.7298i 1.91040i
\(412\) 0 0
\(413\) 0 0
\(414\) −9.00000 11.6190i −0.442326 0.571040i
\(415\) 0 0
\(416\) 0 0
\(417\) 40.2492i 1.97101i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −8.66025 + 6.70820i −0.421575 + 0.326550i
\(423\) −31.1769 −1.51587
\(424\) −5.00000 11.6190i −0.242821 0.564266i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −5.19615 + 20.1246i −0.251166 + 0.972760i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −18.1865 10.0623i −0.875000 0.484123i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.00000 27.1109i 0.335239 1.29838i
\(437\) 26.8328i 1.28359i
\(438\) 0 0
\(439\) 38.7298i 1.84847i 0.381819 + 0.924237i \(0.375298\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 21.0000 1.00000
\(442\) 0 0
\(443\) −38.1051 −1.81043 −0.905214 0.424955i \(-0.860290\pi\)
−0.905214 + 0.424955i \(0.860290\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 8.66025 + 2.23607i 0.407344 + 0.105176i
\(453\) 40.2492i 1.89107i
\(454\) 21.0000 + 27.1109i 0.985579 + 1.27238i
\(455\) 0 0
\(456\) 15.0000 + 34.8569i 0.702439 + 1.63232i
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 22.5167 + 29.0689i 1.05213 + 1.35830i
\(459\) 23.2379i 1.08465i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 25.0000 19.3649i 1.15810 0.897062i
\(467\) 24.2487 1.12210 0.561048 0.827783i \(-0.310398\pi\)
0.561048 + 0.827783i \(0.310398\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) −15.0000 + 11.6190i −0.688973 + 0.533676i
\(475\) 0 0
\(476\) 0 0
\(477\) 13.4164i 0.614295i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.73205 + 2.23607i 0.0788928 + 0.101850i
\(483\) 0 0
\(484\) 5.50000 21.3014i 0.250000 0.968246i
\(485\) 0 0
\(486\) 13.5000 + 17.4284i 0.612372 + 0.790569i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 5.19615 2.23607i 0.235219 0.101222i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −15.0000 + 27.1109i −0.673520 + 1.21731i
\(497\) 0 0
\(498\) 5.19615 + 6.70820i 0.232845 + 0.300602i
\(499\) 7.74597i 0.346757i −0.984855 0.173379i \(-0.944532\pi\)
0.984855 0.173379i \(-0.0554684\pi\)
\(500\) 0 0
\(501\) −42.0000 −1.87642
\(502\) 0 0
\(503\) −3.46410 −0.154457 −0.0772283 0.997013i \(-0.524607\pi\)
−0.0772283 + 0.997013i \(0.524607\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −22.5167 −1.00000
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −7.79423 + 21.2426i −0.344459 + 0.938801i
\(513\) 40.2492i 1.77705i
\(514\) 35.0000 27.1109i 1.54378 1.19581i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 38.7298i 1.70005i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −27.0000 34.8569i −1.17726 1.51983i
\(527\) 34.6410 1.50899
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 8.66025 6.70820i 0.371990 0.288142i
\(543\) 38.1051 1.63525
\(544\) 25.0000 3.87298i 1.07187 0.166053i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −43.3013 11.1803i −1.84974 0.477600i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) 0 0
\(552\) 15.5885 6.70820i 0.663489 0.285520i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −45.0000 11.6190i −1.90843 0.492753i
\(557\) 22.3607i 0.947452i −0.880672 0.473726i \(-0.842909\pi\)
0.880672 0.473726i \(-0.157091\pi\)
\(558\) 25.9808 20.1246i 1.09985 0.851943i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.1769 1.31395 0.656975 0.753912i \(-0.271836\pi\)
0.656975 + 0.753912i \(0.271836\pi\)
\(564\) 9.00000 34.8569i 0.378968 1.46774i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 38.7298i 1.62079i 0.585882 + 0.810397i \(0.300748\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 16.5000 17.4284i 0.687500 0.726184i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −2.59808 3.35410i −0.108066 0.139512i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 35.0000 27.1109i 1.44584 1.11994i
\(587\) −45.0333 −1.85872 −0.929362 0.369170i \(-0.879642\pi\)
−0.929362 + 0.369170i \(0.879642\pi\)
\(588\) −6.06218 + 23.4787i −0.250000 + 0.968246i
\(589\) −60.0000 −2.47226
\(590\) 0 0
\(591\) 7.74597i 0.318626i
\(592\) 0 0
\(593\) 4.47214i 0.183649i −0.995775 0.0918243i \(-0.970730\pi\)
0.995775 0.0918243i \(-0.0292698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 40.2492i 1.64729i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −45.0000 11.6190i −1.83102 0.472768i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −43.3013 + 6.70820i −1.75610 + 0.272054i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −25.9808 6.70820i −1.05021 0.271163i
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.1935i 1.98046i −0.139459 0.990228i \(-0.544536\pi\)
0.139459 0.990228i \(-0.455464\pi\)
\(618\) 0 0
\(619\) 23.2379i 0.934010i 0.884255 + 0.467005i \(0.154667\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 38.7298i 1.54181i −0.636950 0.770905i \(-0.719804\pi\)
0.636950 0.770905i \(-0.280196\pi\)
\(632\) −8.66025 20.1246i −0.344486 0.800514i
\(633\) 13.4164i 0.533254i
\(634\) 25.0000 19.3649i 0.992877 0.769079i
\(635\) 0 0
\(636\) −15.0000 3.87298i −0.594789 0.153574i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 15.5885 + 20.1246i 0.615227 + 0.794255i
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 30.0000 + 38.7298i 1.18033 + 1.52380i
\(647\) −24.2487 −0.953315 −0.476658 0.879089i \(-0.658152\pi\)
−0.476658 + 0.879089i \(0.658152\pi\)
\(648\) −23.3827 + 10.0623i −0.918559 + 0.395285i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 49.1935i 1.92509i −0.271122 0.962545i \(-0.587395\pi\)
0.271122 0.962545i \(-0.412605\pi\)
\(654\) −21.0000 27.1109i −0.821165 1.06012i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 25.9808 20.1246i 1.00977 0.782165i
\(663\) 0 0
\(664\) −9.00000 + 3.87298i −0.349268 + 0.150301i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 12.1244 46.9574i 0.469105 1.81684i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 6.50000 25.1744i 0.250000 0.968246i
\(677\) 31.3050i 1.20315i 0.798817 + 0.601574i \(0.205459\pi\)
−0.798817 + 0.601574i \(0.794541\pi\)
\(678\) 8.66025 6.70820i 0.332595 0.257627i
\(679\) 0 0
\(680\) 0 0
\(681\) 42.0000 1.60944
\(682\) 0 0
\(683\) −38.1051 −1.45805 −0.729026 0.684486i \(-0.760027\pi\)
−0.729026 + 0.684486i \(0.760027\pi\)
\(684\) 45.0000 + 11.6190i 1.72062 + 0.444262i
\(685\) 0 0
\(686\) 0 0
\(687\) 45.0333 1.71813
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 7.74597i 0.294670i 0.989087 + 0.147335i \(0.0470696\pi\)
−0.989087 + 0.147335i \(0.952930\pi\)
\(692\) −43.3013 11.1803i −1.64607 0.425013i
\(693\) 0 0
\(694\) 9.00000 + 11.6190i 0.341635 + 0.441049i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 29.4449 + 38.0132i 1.11450 + 1.43882i
\(699\) 38.7298i 1.46490i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −35.0000 + 27.1109i −1.31724 + 1.02033i
\(707\) 0 0
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 23.2379i 0.871489i
\(712\) 0 0
\(713\) 26.8328i 1.00490i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −35.5070 45.8394i −1.32144 1.70597i
\(723\) 3.46410 0.128831
\(724\) −11.0000 + 42.6028i −0.408812 + 1.58332i
\(725\) 0 0
\(726\) −16.5000 21.3014i −0.612372 0.790569i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 1.73205 6.70820i 0.0640184 0.247942i
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 3.00000 + 19.3649i 0.110581 + 0.713800i
\(737\) 0 0
\(738\) 0 0
\(739\) 54.2218i 1.99458i 0.0735712 + 0.997290i \(0.476560\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.1769 −1.14377 −0.571885 0.820334i \(-0.693788\pi\)
−0.571885 + 0.820334i \(0.693788\pi\)
\(744\) 15.0000 + 34.8569i 0.549927 + 1.27791i
\(745\) 0 0
\(746\) 0 0
\(747\) 10.3923 0.380235
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 54.2218i 1.97858i 0.145962 + 0.989290i \(0.453372\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 36.3731 + 20.1246i 1.32639 + 0.733869i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 43.3013 33.5410i 1.57277 1.21826i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 33.0000 + 42.6028i 1.19234 + 1.53930i
\(767\) 0 0
\(768\) 14.7224 + 23.4787i 0.531250 + 0.847215i
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 54.2218i 1.95275i
\(772\) 0 0
\(773\) 4.47214i 0.160852i 0.996761 + 0.0804258i \(0.0256280\pi\)
−0.996761 + 0.0804258i \(0.974372\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 17.3205 13.4164i 0.619380 0.479770i
\(783\) 0 0
\(784\) −24.5000 13.5554i −0.875000 0.484123i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 8.66025 + 2.23607i 0.308509 + 0.0796566i
\(789\) −54.0000 −1.92245
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 45.0000 + 11.6190i 1.59498 + 0.411823i
\(797\) 49.1935i 1.74252i 0.490819 + 0.871262i \(0.336698\pi\)
−0.490819 + 0.871262i \(0.663302\pi\)
\(798\) 0 0
\(799\) 46.4758i 1.64420i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0