Properties

Label 525.2.bf.b.32.2
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: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
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 32.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 525.32
Dual form 525.2.bf.b.443.2

$q$-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 −0.856248 0.516565i \(-0.827210\pi\)
0.516565 + 0.856248i \(0.327210\pi\)
\(14\) 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 \(0.557139\pi\)
−0.337342 + 0.941382i \(0.609528\pi\)
\(38\) 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 \(-0.374505\pi\)
0.923283 0.384120i \(-0.125495\pi\)
\(44\) 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 0.541468 0.840721i \(-0.317869\pi\)
0.889286 + 0.457352i \(0.151202\pi\)
\(68\) 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 \(0.282320\pi\)
−0.159579 + 0.987185i \(0.551014\pi\)
\(74\) 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 0.999987 0.00515471i \(-0.00164080\pi\)
−0.00515471 + 0.999987i \(0.501641\pi\)
\(98\) 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 0.995926 0.0901732i \(-0.0287421\pi\)
0.817411 + 0.576055i \(0.195409\pi\)
\(104\) 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 0.215708 0.976458i \(-0.430794\pi\)
−0.976458 + 0.215708i \(0.930794\pi\)
\(128\) 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 0.656543 0.754288i \(-0.272018\pi\)
0.945727 + 0.324961i \(0.105351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −23.4225 6.27603i −1.83459 0.491576i −0.836205 0.548417i \(-0.815231\pi\)
−0.998383 + 0.0568404i \(0.981897\pi\)
\(164\) 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 \(0.0604011\pi\)
−0.756171 + 0.654374i \(0.772932\pi\)
\(194\) 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 −0.908964 0.416874i \(-0.863126\pi\)
0.416874 + 0.908964i \(0.363126\pi\)
\(224\) 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 0.110790 0.993844i \(-0.464662\pi\)
0.592869 + 0.805299i \(0.297995\pi\)
\(278\) 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 \(0.349954\pi\)
0.0521913 + 0.998637i \(0.483379\pi\)
\(284\) 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 0.671293 0.741192i \(-0.265739\pi\)
−0.741192 + 0.671293i \(0.765739\pi\)
\(308\) 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 0.397861 0.917446i \(-0.369753\pi\)
−0.114165 + 0.993462i \(0.536419\pi\)
\(314\) 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 \(-0.793484\pi\)
−0.604223 0.796815i \(-0.706516\pi\)
\(338\) 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 −0.629421 0.777064i \(-0.716708\pi\)
−0.156563 + 0.987668i \(0.550041\pi\)
\(368\) 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 0.996684 0.0813690i \(-0.0259292\pi\)
0.822469 + 0.568810i \(0.192596\pi\)
\(374\) 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 \(0.242011\pi\)
−0.972117 + 0.234498i \(0.924655\pi\)
\(398\) 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 −0.977248 0.212101i \(-0.931970\pi\)
0.212101 + 0.977248i \(0.431970\pi\)
\(434\) 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 \(0.174854\pi\)
−0.999669 + 0.0257197i \(0.991812\pi\)
\(458\) 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 \(0.570519\pi\)
−0.975560 + 0.219733i \(0.929481\pi\)
\(464\) 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 0.649427 0.760424i \(-0.275009\pi\)
0.942632 + 0.333833i \(0.108342\pi\)
\(488\) 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 −0.423777 0.905766i \(-0.639296\pi\)
−0.819885 + 0.572528i \(0.805963\pi\)
\(524\) 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 0.971238 0.238110i \(-0.0765278\pi\)
−0.238110 + 0.971238i \(0.576528\pi\)
\(548\) 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 −0.850227 0.526417i \(-0.823535\pi\)
−0.473109 + 0.881004i \(0.656868\pi\)
\(578\) 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 −0.933247 0.359235i \(-0.883038\pi\)
0.987833 + 0.155517i \(0.0497042\pi\)
\(608\) 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 \(-0.648061\pi\)
−0.0584195 0.998292i \(-0.518606\pi\)
\(614\) 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 \(-0.294076\pi\)
0.797938 0.602739i \(-0.205924\pi\)
\(644\) 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 −0.999963 0.00854415i \(-0.997280\pi\)
0.00854415 + 0.999963i \(0.497280\pi\)
\(674\) 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 0.937767 0.347265i \(-0.112889\pi\)
−0.347265 + 0.937767i \(0.612889\pi\)
\(728\) 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 \(0.0420682\pi\)
0.924362 + 0.381518i \(0.124598\pi\)
\(734\) 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 \(0.0933542\pi\)
0.289095 + 0.957301i \(0.406646\pi\)
\(758\) 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 −0.198688 0.980063i \(-0.563668\pi\)
0.317962 + 0.948103i \(0.397001\pi\)
\(788\) 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 0
\(808\) 0 0