Properties

Label 525.2.bf.b.32.1
Level 525
Weight 2
Character 525.32
Analytic conductor 4.192
Analytic rank 0
Dimension 8
CM discriminant -3
Inner twists 16

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Newspace parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 32.1
Root \(0.258819 - 0.965926i\)
Character \(\chi\) = 525.32
Dual form 525.2.bf.b.443.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-1.48356 - 2.19067i) q^{7} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\) \(q+(-0.448288 - 1.67303i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-1.48356 - 2.19067i) q^{7} +(-2.59808 + 1.50000i) q^{9} +(-0.896575 + 3.34607i) q^{12} +(1.22474 - 1.22474i) q^{13} +(2.00000 + 3.46410i) q^{16} +(-6.06218 + 3.50000i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(3.67423 + 3.67423i) q^{27} +(0.378937 + 5.27792i) q^{28} +(-3.50000 + 6.06218i) q^{31} +6.00000 q^{36} +(3.13801 - 11.7112i) q^{37} +(-2.59808 - 1.50000i) q^{39} +(-8.57321 + 8.57321i) q^{43} +(4.89898 - 4.89898i) q^{48} +(-2.59808 + 6.50000i) q^{49} +(-3.34607 + 0.896575i) q^{52} +(8.57321 + 8.57321i) q^{57} +(-7.00000 - 12.1244i) q^{61} +(7.14042 + 3.46618i) q^{63} -8.00000i q^{64} +(-11.7112 + 3.13801i) q^{67} +(-4.03459 - 15.0573i) q^{73} +14.0000 q^{76} +(11.2583 - 6.50000i) q^{79} +(4.50000 - 7.79423i) q^{81} +(8.66025 - 3.00000i) q^{84} +(-4.50000 - 0.866025i) q^{91} +(11.7112 + 3.13801i) q^{93} +(-9.79796 - 9.79796i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 16q^{16} - 24q^{21} - 28q^{31} + 48q^{36} - 56q^{61} + 112q^{76} + 36q^{81} - 36q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) −0.448288 1.67303i −0.258819 0.965926i
\(4\) −1.73205 1.00000i −0.866025 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) −1.48356 2.19067i −0.560734 0.827996i
\(8\) 0 0
\(9\) −2.59808 + 1.50000i −0.866025 + 0.500000i
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −0.896575 + 3.34607i −0.258819 + 0.965926i
\(13\) 1.22474 1.22474i 0.339683 0.339683i −0.516565 0.856248i \(-0.672790\pi\)
0.856248 + 0.516565i \(0.172790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(18\) 0 0
\(19\) −6.06218 + 3.50000i −1.39076 + 0.802955i −0.993399 0.114708i \(-0.963407\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) −3.00000 + 3.46410i −0.654654 + 0.755929i
\(22\) 0 0
\(23\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.707107 + 0.707107i
\(28\) 0.378937 + 5.27792i 0.0716124 + 0.997433i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 3.13801 11.7112i 0.515886 1.92531i 0.178545 0.983932i \(-0.442861\pi\)
0.337342 0.941382i \(-0.390472\pi\)
\(38\) 0 0
\(39\) −2.59808 1.50000i −0.416025 0.240192i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −8.57321 + 8.57321i −1.30740 + 1.30740i −0.384120 + 0.923283i \(0.625495\pi\)
−0.923283 + 0.384120i \(0.874505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(48\) 4.89898 4.89898i 0.707107 0.707107i
\(49\) −2.59808 + 6.50000i −0.371154 + 0.928571i
\(50\) 0 0
\(51\) 0 0
\(52\) −3.34607 + 0.896575i −0.464016 + 0.124333i
\(53\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.57321 + 8.57321i 1.13555 + 1.13555i
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −7.00000 12.1244i −0.896258 1.55236i −0.832240 0.554416i \(-0.812942\pi\)
−0.0640184 0.997949i \(-0.520392\pi\)
\(62\) 0 0
\(63\) 7.14042 + 3.46618i 0.899608 + 0.436698i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −11.7112 + 3.13801i −1.43075 + 0.383369i −0.889286 0.457352i \(-0.848798\pi\)
−0.541468 + 0.840721i \(0.682131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −4.03459 15.0573i −0.472213 1.76232i −0.631792 0.775138i \(-0.717680\pi\)
0.159579 0.987185i \(-0.448986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 14.0000 1.60591
\(77\) 0 0
\(78\) 0 0
\(79\) 11.2583 6.50000i 1.26666 0.731307i 0.292306 0.956325i \(-0.405577\pi\)
0.974355 + 0.225018i \(0.0722440\pi\)
\(80\) 0 0
\(81\) 4.50000 7.79423i 0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 8.66025 3.00000i 0.944911 0.327327i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) −4.50000 0.866025i −0.471728 0.0907841i
\(92\) 0 0
\(93\) 11.7112 + 3.13801i 1.21440 + 0.325397i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.79796 9.79796i −0.994832 0.994832i 0.00515471 0.999987i \(-0.498359\pi\)
−0.999987 + 0.00515471i \(0.998359\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) −18.4034 4.93117i −1.81334 0.485882i −0.817411 0.576055i \(-0.804591\pi\)
−0.995926 + 0.0901732i \(0.971258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(108\) −2.68973 10.0382i −0.258819 0.965926i
\(109\) 14.7224 + 8.50000i 1.41015 + 0.814152i 0.995402 0.0957826i \(-0.0305354\pi\)
0.414751 + 0.909935i \(0.363869\pi\)
\(110\) 0 0
\(111\) −21.0000 −1.99323
\(112\) 4.62158 9.52056i 0.436698 0.899608i
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.34486 + 5.01910i −0.124333 + 0.464016i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.50000 9.52628i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 12.1244 7.00000i 1.08880 0.628619i
\(125\) 0 0
\(126\) 0 0
\(127\) 8.57321 + 8.57321i 0.760750 + 0.760750i 0.976458 0.215708i \(-0.0692060\pi\)
−0.215708 + 0.976458i \(0.569206\pi\)
\(128\) 0 0
\(129\) 18.1865 + 10.5000i 1.60123 + 0.924473i
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 16.6610 + 8.08776i 1.44469 + 0.701298i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(138\) 0 0
\(139\) 7.00000i 0.593732i −0.954919 0.296866i \(-0.904058\pi\)
0.954919 0.296866i \(-0.0959415\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −10.3923 6.00000i −0.866025 0.500000i
\(145\) 0 0
\(146\) 0 0
\(147\) 12.0394 + 1.43280i 0.992993 + 0.118175i
\(148\) −17.1464 + 17.1464i −1.40943 + 1.40943i
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) −2.00000 + 3.46410i −0.162758 + 0.281905i −0.935857 0.352381i \(-0.885372\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 3.00000 + 5.19615i 0.240192 + 0.416025i
\(157\) −20.0764 + 5.37945i −1.60227 + 0.429327i −0.945727 0.324961i \(-0.894649\pi\)
−0.656543 + 0.754288i \(0.727982\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 23.4225 + 6.27603i 1.83459 + 0.491576i 0.998383 0.0568404i \(-0.0181026\pi\)
0.836205 + 0.548417i \(0.184769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) 10.0000i 0.769231i
\(170\) 0 0
\(171\) 10.5000 18.1865i 0.802955 1.39076i
\(172\) 23.4225 6.27603i 1.78595 0.478543i
\(173\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) −17.1464 + 17.1464i −1.26750 + 1.26750i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.59808 13.5000i 0.188982 0.981981i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) −13.3843 + 3.58630i −0.965926 + 0.258819i
\(193\) −3.13801 11.7112i −0.225879 0.842993i −0.982050 0.188619i \(-0.939599\pi\)
0.756171 0.654374i \(-0.227068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 11.0000 8.66025i 0.785714 0.618590i
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) −24.2487 14.0000i −1.71895 0.992434i −0.920864 0.389885i \(-0.872515\pi\)
−0.798082 0.602549i \(-0.794152\pi\)
\(200\) 0 0
\(201\) 10.5000 + 18.1865i 0.740613 + 1.28278i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 6.69213 + 1.79315i 0.464016 + 0.124333i
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 18.4727 1.32628i 1.25401 0.0900338i
\(218\) 0 0
\(219\) −23.3827 + 13.5000i −1.58006 + 0.912245i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.34847 7.34847i 0.492090 0.492090i −0.416874 0.908964i \(-0.636874\pi\)
0.908964 + 0.416874i \(0.136874\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(228\) −6.27603 23.4225i −0.415640 1.55119i
\(229\) −6.06218 + 3.50000i −0.400600 + 0.231287i −0.686743 0.726900i \(-0.740960\pi\)
0.286143 + 0.958187i \(0.407627\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −15.9217 15.9217i −1.03422 1.03422i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −7.00000 + 12.1244i −0.450910 + 0.780998i −0.998443 0.0557856i \(-0.982234\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) −15.0573 4.03459i −0.965926 0.258819i
\(244\) 28.0000i 1.79252i
\(245\) 0 0
\(246\) 0 0
\(247\) −3.13801 + 11.7112i −0.199667 + 0.745168i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −8.90138 13.1440i −0.560734 0.827996i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(258\) 0 0
\(259\) −30.3109 + 10.5000i −1.88343 + 0.652438i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 23.4225 + 6.27603i 1.43075 + 0.383369i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 14.0000 + 24.2487i 0.850439 + 1.47300i 0.880812 + 0.473466i \(0.156997\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) 0.568406 + 7.91688i 0.0344015 + 0.479151i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.7112 + 3.13801i −0.703660 + 0.188545i −0.592869 0.805299i \(-0.702005\pi\)
−0.110790 + 0.993844i \(0.535338\pi\)
\(278\) 0 0
\(279\) 21.0000i 1.25724i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −8.51747 31.7876i −0.506311 1.88958i −0.454120 0.890941i \(-0.650046\pi\)
−0.0521913 0.998637i \(-0.516621\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 14.7224 8.50000i 0.866025 0.500000i
\(290\) 0 0
\(291\) −12.0000 + 20.7846i −0.703452 + 1.21842i
\(292\) −8.06918 + 30.1146i −0.472213 + 1.76232i
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 31.5000 + 6.06218i 1.81563 + 0.349418i
\(302\) 0 0
\(303\) 0 0
\(304\) −24.2487 14.0000i −1.39076 0.802955i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.22474 + 1.22474i 0.0698999 + 0.0698999i 0.741192 0.671293i \(-0.234261\pi\)
−0.671293 + 0.741192i \(0.734261\pi\)
\(308\) 0 0
\(309\) 33.0000i 1.87730i
\(310\) 0 0
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) −5.01910 1.34486i −0.283696 0.0760162i 0.114165 0.993462i \(-0.463581\pi\)
−0.397861 + 0.917446i \(0.630247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −26.0000 −1.46261
\(317\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −15.5885 + 9.00000i −0.866025 + 0.500000i
\(325\) 0 0
\(326\) 0 0
\(327\) 7.62089 28.4416i 0.421436 1.57282i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.500000 + 0.866025i 0.0274825 + 0.0476011i 0.879440 0.476011i \(-0.157918\pi\)
−0.851957 + 0.523612i \(0.824584\pi\)
\(332\) 0 0
\(333\) 9.41404 + 35.1337i 0.515886 + 1.92531i
\(334\) 0 0
\(335\) 0 0
\(336\) −18.0000 3.46410i −0.981981 0.188982i
\(337\) 25.7196 + 25.7196i 1.40104 + 1.40104i 0.796815 + 0.604223i \(0.206516\pi\)
0.604223 + 0.796815i \(0.293484\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 18.0938 3.95164i 0.976972 0.213368i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(348\) 0 0
\(349\) 14.0000i 0.749403i 0.927146 + 0.374701i \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) 15.0000 25.9808i 0.789474 1.36741i
\(362\) 0 0
\(363\) −13.4722 + 13.4722i −0.707107 + 0.707107i
\(364\) 6.92820 + 6.00000i 0.363137 + 0.314485i
\(365\) 0 0
\(366\) 0 0
\(367\) 15.0573 4.03459i 0.785984 0.210604i 0.156563 0.987668i \(-0.449959\pi\)
0.629421 + 0.777064i \(0.283292\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −17.1464 17.1464i −0.889001 0.889001i
\(373\) −35.1337 9.41404i −1.81915 0.487441i −0.822469 0.568810i \(-0.807404\pi\)
−0.996684 + 0.0813690i \(0.974071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 37.0000i 1.90056i −0.311393 0.950281i \(-0.600796\pi\)
0.311393 0.950281i \(-0.399204\pi\)
\(380\) 0 0
\(381\) 10.5000 18.1865i 0.537931 0.931724i
\(382\) 0 0
\(383\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.41404 35.1337i 0.478543 1.78595i
\(388\) 7.17260 + 26.7685i 0.364134 + 1.35897i
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.93117 18.4034i 0.247488 0.923638i −0.724628 0.689140i \(-0.757989\pi\)
0.972117 0.234498i \(-0.0753447\pi\)
\(398\) 0 0
\(399\) 6.06218 31.5000i 0.303488 1.57697i
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 3.13801 + 11.7112i 0.156316 + 0.583378i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.06218 + 3.50000i 0.299755 + 0.173064i 0.642333 0.766426i \(-0.277967\pi\)
−0.342578 + 0.939490i \(0.611300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 26.9444 + 26.9444i 1.32745 + 1.32745i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.7112 + 3.13801i −0.573501 + 0.153669i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −16.1755 + 33.3220i −0.782788 + 1.61256i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) −5.37945 + 20.0764i −0.258819 + 0.965926i
\(433\) 15.9217 15.9217i 0.765147 0.765147i −0.212101 0.977248i \(-0.568030\pi\)
0.977248 + 0.212101i \(0.0680304\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −17.0000 29.4449i −0.814152 1.41015i
\(437\) 0 0
\(438\) 0 0
\(439\) 24.2487 14.0000i 1.15733 0.668184i 0.206666 0.978412i \(-0.433739\pi\)
0.950662 + 0.310228i \(0.100405\pi\)
\(440\) 0 0
\(441\) −3.00000 20.7846i −0.142857 0.989743i
\(442\) 0 0
\(443\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(444\) 36.3731 + 21.0000i 1.72619 + 0.996616i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −17.5254 + 11.8685i −0.827996 + 0.560734i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 6.69213 + 1.79315i 0.314424 + 0.0842496i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.13801 11.7112i 0.146790 0.547828i −0.852879 0.522108i \(-0.825146\pi\)
0.999669 0.0257197i \(-0.00818773\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 25.7196 25.7196i 1.19529 1.19529i 0.219733 0.975560i \(-0.429481\pi\)
0.975560 0.219733i \(-0.0705187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(468\) 7.34847 7.34847i 0.339683 0.339683i
\(469\) 24.2487 + 21.0000i 1.11970 + 0.969690i
\(470\) 0 0
\(471\) 18.0000 + 31.1769i 0.829396 + 1.43656i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) −10.5000 18.1865i −0.478759 0.829235i
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) 0 0
\(486\) 0 0
\(487\) −35.1337 + 9.41404i −1.59206 + 0.426591i −0.942632 0.333833i \(-0.891658\pi\)
−0.649427 + 0.760424i \(0.724991\pi\)
\(488\) 0 0
\(489\) 42.0000i 1.89931i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) 0 0
\(498\) 0 0
\(499\) 37.2391 21.5000i 1.66705 0.962472i 0.697835 0.716258i \(-0.254147\pi\)
0.969216 0.246214i \(-0.0791865\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 16.7303 4.48288i 0.743020 0.199092i
\(508\) −6.27603 23.4225i −0.278454 1.03920i
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) −27.0000 + 31.1769i −1.19441 + 1.37919i
\(512\) 0 0
\(513\) −35.1337 9.41404i −1.55119 0.415640i
\(514\) 0 0
\(515\) 0 0
\(516\) −21.0000 36.3731i −0.924473 1.60123i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 28.4416 + 7.62089i 1.24366 + 0.333238i 0.819885 0.572528i \(-0.194037\pi\)
0.423777 + 0.905766i \(0.360704\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 19.9186 + 11.5000i 0.866025 + 0.500000i
\(530\) 0 0
\(531\) 0 0
\(532\) −20.7699 30.6694i −0.900489 1.32969i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.5000 25.1147i −0.623404 1.07977i −0.988847 0.148933i \(-0.952416\pi\)
0.365444 0.930834i \(-0.380917\pi\)
\(542\) 0 0
\(543\) −3.13801 11.7112i −0.134665 0.502577i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −17.1464 17.1464i −0.733128 0.733128i 0.238110 0.971238i \(-0.423472\pi\)
−0.971238 + 0.238110i \(0.923472\pi\)
\(548\) 0 0
\(549\) 36.3731 + 21.0000i 1.55236 + 0.896258i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −30.9418 15.0201i −1.31578 0.638721i
\(554\) 0 0
\(555\) 0 0
\(556\) −7.00000 + 12.1244i −0.296866 + 0.514187i
\(557\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(558\) 0 0
\(559\) 21.0000i 0.888205i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −23.7506 + 1.70522i −0.997433 + 0.0716124i
\(568\) 0 0
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) 15.5000 26.8468i 0.648655 1.12350i −0.334790 0.942293i \(-0.608665\pi\)
0.983444 0.181210i \(-0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 12.0000 + 20.7846i 0.500000 + 0.866025i
\(577\) 31.7876 8.51747i 1.32334 0.354587i 0.473109 0.881004i \(-0.343132\pi\)
0.850227 + 0.526417i \(0.176465\pi\)
\(578\) 0 0
\(579\) −18.1865 + 10.5000i −0.755807 + 0.436365i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) −19.4201 14.5211i −0.800869 0.598839i
\(589\) 49.0000i 2.01901i
\(590\) 0 0
\(591\) 0 0
\(592\) 46.8449 12.5521i 1.92531 0.515886i
\(593\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.5521 + 46.8449i −0.513721 + 1.91723i
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 49.0000 1.99875 0.999376 0.0353259i \(-0.0112469\pi\)
0.999376 + 0.0353259i \(0.0112469\pi\)
\(602\) 0 0
\(603\) 25.7196 25.7196i 1.04738 1.04738i
\(604\) 6.92820 4.00000i 0.281905 0.162758i
\(605\) 0 0
\(606\) 0 0
\(607\) −1.34486 + 5.01910i −0.0545863 + 0.203719i −0.987833 0.155517i \(-0.950296\pi\)
0.933247 + 0.359235i \(0.116962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 12.5521 + 46.8449i 0.506973 + 1.89205i 0.448553 + 0.893756i \(0.351939\pi\)
0.0584195 + 0.998292i \(0.481394\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) −42.4352 24.5000i −1.70562 0.984738i −0.939829 0.341644i \(-0.889016\pi\)
−0.765787 0.643094i \(-0.777650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 12.0000i 0.480384i
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 40.1528 + 10.7589i 1.60227 + 0.429327i
\(629\) 0 0
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) 7.17260 + 26.7685i 0.285085 + 1.06395i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.77886 + 11.1428i 0.189345 + 0.441495i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) −35.5176 + 35.5176i −1.40068 + 1.40068i −0.602739 + 0.797938i \(0.705924\pi\)
−0.797938 + 0.602739i \(0.794076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −10.5000 30.3109i −0.411527 1.18798i
\(652\) −34.2929 34.2929i −1.34301 1.34301i
\(653\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 33.0681 + 33.0681i 1.29011 + 1.29011i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −24.5000 + 42.4352i −0.952940 + 1.65054i −0.213925 + 0.976850i \(0.568625\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −15.5885 9.00000i −0.602685 0.347960i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25.7196 25.7196i 0.991419 0.991419i −0.00854415 0.999963i \(-0.502720\pi\)
0.999963 + 0.00854415i \(0.00271972\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 10.0000 17.3205i 0.384615 0.666173i
\(677\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(678\) 0 0
\(679\) −6.92820 + 36.0000i −0.265880 + 1.38155i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(684\) −36.3731 + 21.0000i −1.39076 + 0.802955i
\(685\) 0 0
\(686\) 0 0
\(687\) 8.57321 + 8.57321i 0.327089 + 0.327089i
\(688\) −46.8449 12.5521i −1.78595 0.478543i
\(689\) 0 0
\(690\) 0 0
\(691\) −24.5000 42.4352i −0.932024 1.61431i −0.779857 0.625958i \(-0.784708\pi\)
−0.152167 0.988355i \(-0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 21.9661 + 81.9786i 0.828467 + 3.09188i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.0526 + 11.0000i −0.715534 + 0.413114i −0.813107 0.582115i \(-0.802225\pi\)
0.0975728 + 0.995228i \(0.468892\pi\)
\(710\) 0 0
\(711\) −19.5000 + 33.7750i −0.731307 + 1.26666i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 16.5000 + 47.6314i 0.614492 + 1.77389i
\(722\) 0 0
\(723\) 23.4225 + 6.27603i 0.871091 + 0.233408i
\(724\) −12.1244 7.00000i −0.450598 0.260153i
\(725\) 0 0
\(726\) 0 0
\(727\) −15.9217 15.9217i −0.590503 0.590503i 0.347265 0.937767i \(-0.387111\pi\)
−0.937767 + 0.347265i \(0.887111\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 46.8449 12.5521i 1.73144 0.463937i
\(733\) −51.8640 13.8969i −1.91564 0.513294i −0.991279 0.131777i \(-0.957932\pi\)
−0.924362 0.381518i \(-0.875402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −45.8993 26.5000i −1.68843 0.974818i −0.955718 0.294285i \(-0.904919\pi\)
−0.732717 0.680534i \(-0.761748\pi\)
\(740\) 0 0
\(741\) 21.0000 0.771454
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.5000 + 35.5070i 0.748056 + 1.29567i 0.948753 + 0.316017i \(0.102346\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −18.0000 + 20.7846i −0.654654 + 0.755929i
\(757\) −34.2929 34.2929i −1.24640 1.24640i −0.957301 0.289095i \(-0.906646\pi\)
−0.289095 0.957301i \(-0.593354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −3.22097 44.8623i −0.116607 1.62412i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 26.7685 + 7.17260i 0.965926 + 0.258819i
\(769\) 49.0000i 1.76699i 0.468445 + 0.883493i \(0.344814\pi\)
−0.468445 + 0.883493i \(0.655186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.27603 + 23.4225i −0.225879 + 0.842993i
\(773\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 31.1548 + 46.0041i 1.11767 + 1.65039i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −27.7128 + 4.00000i −0.989743 + 0.142857i
\(785\) 0 0
\(786\) 0 0
\(787\) −3.34607 + 0.896575i −0.119274 + 0.0319595i −0.317962 0.948103i \(-0.602999\pi\)
0.198688 + 0.980063i \(0.436332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −23.4225 6.27603i −0.831756 0.222868i
\(794\) 0 0
\(795\) 0 0
\(796\) 28.0000 + 48.4974i 0.992434 + 1.71895i
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 42.0000i 1.48123i
\(805\) 0 0
\(806\) 0 0
\(807\) 0