Properties

Label 225.4.b.i.199.3
Level $225$
Weight $4$
Character 225.199
Analytic conductor $13.275$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(1.58114 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.4.b.i.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+3.16228i q^{2} -2.00000 q^{4} -15.0000i q^{7} +18.9737i q^{8} +O(q^{10})\) \(q+3.16228i q^{2} -2.00000 q^{4} -15.0000i q^{7} +18.9737i q^{8} -63.2456 q^{11} +35.0000i q^{13} +47.4342 q^{14} -76.0000 q^{16} +88.5438i q^{17} -91.0000 q^{19} -200.000i q^{22} +113.842i q^{23} -110.680 q^{26} +30.0000i q^{28} -63.2456 q^{29} -147.000 q^{31} -88.5438i q^{32} -280.000 q^{34} -370.000i q^{37} -287.767i q^{38} +442.719 q^{41} +335.000i q^{43} +126.491 q^{44} -360.000 q^{46} -177.088i q^{47} +118.000 q^{49} -70.0000i q^{52} -88.5438i q^{53} +284.605 q^{56} -200.000i q^{58} -885.438 q^{59} +427.000 q^{61} -464.855i q^{62} -328.000 q^{64} -15.0000i q^{67} -177.088i q^{68} +63.2456 q^{71} -70.0000i q^{73} +1170.04 q^{74} +182.000 q^{76} +948.683i q^{77} +876.000 q^{79} +1400.00i q^{82} +531.263i q^{83} -1059.36 q^{86} -1200.00i q^{88} +525.000 q^{91} -227.684i q^{92} +560.000 q^{94} +1085.00i q^{97} +373.149i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 304 q^{16} - 364 q^{19} - 588 q^{31} - 1120 q^{34} - 1440 q^{46} + 472 q^{49} + 1708 q^{61} - 1312 q^{64} + 728 q^{76} + 3504 q^{79} + 2100 q^{91} + 2240 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.16228i 1.11803i 0.829156 + 0.559017i \(0.188821\pi\)
−0.829156 + 0.559017i \(0.811179\pi\)
\(3\) 0 0
\(4\) −2.00000 −0.250000
\(5\) 0 0
\(6\) 0 0
\(7\) − 15.0000i − 0.809924i −0.914334 0.404962i \(-0.867285\pi\)
0.914334 0.404962i \(-0.132715\pi\)
\(8\) 18.9737i 0.838525i
\(9\) 0 0
\(10\) 0 0
\(11\) −63.2456 −1.73357 −0.866784 0.498683i \(-0.833817\pi\)
−0.866784 + 0.498683i \(0.833817\pi\)
\(12\) 0 0
\(13\) 35.0000i 0.746712i 0.927688 + 0.373356i \(0.121793\pi\)
−0.927688 + 0.373356i \(0.878207\pi\)
\(14\) 47.4342 0.905522
\(15\) 0 0
\(16\) −76.0000 −1.18750
\(17\) 88.5438i 1.26324i 0.775280 + 0.631618i \(0.217609\pi\)
−0.775280 + 0.631618i \(0.782391\pi\)
\(18\) 0 0
\(19\) −91.0000 −1.09878 −0.549390 0.835566i \(-0.685140\pi\)
−0.549390 + 0.835566i \(0.685140\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 200.000i − 1.93819i
\(23\) 113.842i 1.03207i 0.856566 + 0.516037i \(0.172593\pi\)
−0.856566 + 0.516037i \(0.827407\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −110.680 −0.834849
\(27\) 0 0
\(28\) 30.0000i 0.202481i
\(29\) −63.2456 −0.404979 −0.202490 0.979284i \(-0.564903\pi\)
−0.202490 + 0.979284i \(0.564903\pi\)
\(30\) 0 0
\(31\) −147.000 −0.851677 −0.425838 0.904799i \(-0.640021\pi\)
−0.425838 + 0.904799i \(0.640021\pi\)
\(32\) − 88.5438i − 0.489140i
\(33\) 0 0
\(34\) −280.000 −1.41234
\(35\) 0 0
\(36\) 0 0
\(37\) − 370.000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) − 287.767i − 1.22847i
\(39\) 0 0
\(40\) 0 0
\(41\) 442.719 1.68637 0.843184 0.537625i \(-0.180679\pi\)
0.843184 + 0.537625i \(0.180679\pi\)
\(42\) 0 0
\(43\) 335.000i 1.18807i 0.804439 + 0.594035i \(0.202466\pi\)
−0.804439 + 0.594035i \(0.797534\pi\)
\(44\) 126.491 0.433392
\(45\) 0 0
\(46\) −360.000 −1.15389
\(47\) − 177.088i − 0.549593i −0.961502 0.274797i \(-0.911390\pi\)
0.961502 0.274797i \(-0.0886105\pi\)
\(48\) 0 0
\(49\) 118.000 0.344023
\(50\) 0 0
\(51\) 0 0
\(52\) − 70.0000i − 0.186678i
\(53\) − 88.5438i − 0.229480i −0.993396 0.114740i \(-0.963397\pi\)
0.993396 0.114740i \(-0.0366034\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 284.605 0.679142
\(57\) 0 0
\(58\) − 200.000i − 0.452781i
\(59\) −885.438 −1.95380 −0.976900 0.213698i \(-0.931449\pi\)
−0.976900 + 0.213698i \(0.931449\pi\)
\(60\) 0 0
\(61\) 427.000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) − 464.855i − 0.952204i
\(63\) 0 0
\(64\) −328.000 −0.640625
\(65\) 0 0
\(66\) 0 0
\(67\) − 15.0000i − 0.0273514i −0.999906 0.0136757i \(-0.995647\pi\)
0.999906 0.0136757i \(-0.00435324\pi\)
\(68\) − 177.088i − 0.315809i
\(69\) 0 0
\(70\) 0 0
\(71\) 63.2456 0.105716 0.0528582 0.998602i \(-0.483167\pi\)
0.0528582 + 0.998602i \(0.483167\pi\)
\(72\) 0 0
\(73\) − 70.0000i − 0.112231i −0.998424 0.0561156i \(-0.982128\pi\)
0.998424 0.0561156i \(-0.0178715\pi\)
\(74\) 1170.04 1.83804
\(75\) 0 0
\(76\) 182.000 0.274695
\(77\) 948.683i 1.40406i
\(78\) 0 0
\(79\) 876.000 1.24757 0.623783 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1400.00i 1.88542i
\(83\) 531.263i 0.702574i 0.936268 + 0.351287i \(0.114256\pi\)
−0.936268 + 0.351287i \(0.885744\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1059.36 −1.32830
\(87\) 0 0
\(88\) − 1200.00i − 1.45364i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 525.000 0.604780
\(92\) − 227.684i − 0.258018i
\(93\) 0 0
\(94\) 560.000 0.614464
\(95\) 0 0
\(96\) 0 0
\(97\) 1085.00i 1.13572i 0.823124 + 0.567861i \(0.192229\pi\)
−0.823124 + 0.567861i \(0.807771\pi\)
\(98\) 373.149i 0.384630i
\(99\) 0 0
\(100\) 0 0
\(101\) −1328.16 −1.30848 −0.654240 0.756287i \(-0.727011\pi\)
−0.654240 + 0.756287i \(0.727011\pi\)
\(102\) 0 0
\(103\) 1540.00i 1.47321i 0.676323 + 0.736605i \(0.263572\pi\)
−0.676323 + 0.736605i \(0.736428\pi\)
\(104\) −664.078 −0.626137
\(105\) 0 0
\(106\) 280.000 0.256566
\(107\) 796.894i 0.719987i 0.932955 + 0.359994i \(0.117221\pi\)
−0.932955 + 0.359994i \(0.882779\pi\)
\(108\) 0 0
\(109\) 811.000 0.712658 0.356329 0.934361i \(-0.384028\pi\)
0.356329 + 0.934361i \(0.384028\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1140.00i 0.961785i
\(113\) 1619.09i 1.34788i 0.738785 + 0.673942i \(0.235400\pi\)
−0.738785 + 0.673942i \(0.764600\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 126.491 0.101245
\(117\) 0 0
\(118\) − 2800.00i − 2.18441i
\(119\) 1328.16 1.02313
\(120\) 0 0
\(121\) 2669.00 2.00526
\(122\) 1350.29i 1.00205i
\(123\) 0 0
\(124\) 294.000 0.212919
\(125\) 0 0
\(126\) 0 0
\(127\) − 260.000i − 0.181664i −0.995866 0.0908318i \(-0.971047\pi\)
0.995866 0.0908318i \(-0.0289526\pi\)
\(128\) − 1745.58i − 1.20538i
\(129\) 0 0
\(130\) 0 0
\(131\) 1328.16 0.885814 0.442907 0.896568i \(-0.353947\pi\)
0.442907 + 0.896568i \(0.353947\pi\)
\(132\) 0 0
\(133\) 1365.00i 0.889929i
\(134\) 47.4342 0.0305798
\(135\) 0 0
\(136\) −1680.00 −1.05926
\(137\) − 2656.31i − 1.65653i −0.560339 0.828263i \(-0.689329\pi\)
0.560339 0.828263i \(-0.310671\pi\)
\(138\) 0 0
\(139\) 784.000 0.478403 0.239201 0.970970i \(-0.423114\pi\)
0.239201 + 0.970970i \(0.423114\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 200.000i 0.118195i
\(143\) − 2213.59i − 1.29448i
\(144\) 0 0
\(145\) 0 0
\(146\) 221.359 0.125478
\(147\) 0 0
\(148\) 740.000i 0.410997i
\(149\) −2150.35 −1.18230 −0.591152 0.806560i \(-0.701327\pi\)
−0.591152 + 0.806560i \(0.701327\pi\)
\(150\) 0 0
\(151\) −797.000 −0.429529 −0.214765 0.976666i \(-0.568898\pi\)
−0.214765 + 0.976666i \(0.568898\pi\)
\(152\) − 1726.60i − 0.921356i
\(153\) 0 0
\(154\) −3000.00 −1.56978
\(155\) 0 0
\(156\) 0 0
\(157\) − 1225.00i − 0.622711i −0.950293 0.311356i \(-0.899217\pi\)
0.950293 0.311356i \(-0.100783\pi\)
\(158\) 2770.16i 1.39482i
\(159\) 0 0
\(160\) 0 0
\(161\) 1707.63 0.835901
\(162\) 0 0
\(163\) − 275.000i − 0.132145i −0.997815 0.0660726i \(-0.978953\pi\)
0.997815 0.0660726i \(-0.0210469\pi\)
\(164\) −885.438 −0.421592
\(165\) 0 0
\(166\) −1680.00 −0.785502
\(167\) 265.631i 0.123085i 0.998104 + 0.0615424i \(0.0196019\pi\)
−0.998104 + 0.0615424i \(0.980398\pi\)
\(168\) 0 0
\(169\) 972.000 0.442421
\(170\) 0 0
\(171\) 0 0
\(172\) − 670.000i − 0.297018i
\(173\) − 3984.47i − 1.75106i −0.483162 0.875531i \(-0.660512\pi\)
0.483162 0.875531i \(-0.339488\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4806.66 2.05861
\(177\) 0 0
\(178\) 0 0
\(179\) 1770.88 0.739449 0.369725 0.929141i \(-0.379452\pi\)
0.369725 + 0.929141i \(0.379452\pi\)
\(180\) 0 0
\(181\) −1687.00 −0.692783 −0.346391 0.938090i \(-0.612593\pi\)
−0.346391 + 0.938090i \(0.612593\pi\)
\(182\) 1660.20i 0.676164i
\(183\) 0 0
\(184\) −2160.00 −0.865420
\(185\) 0 0
\(186\) 0 0
\(187\) − 5600.00i − 2.18991i
\(188\) 354.175i 0.137398i
\(189\) 0 0
\(190\) 0 0
\(191\) −2213.59 −0.838587 −0.419293 0.907851i \(-0.637722\pi\)
−0.419293 + 0.907851i \(0.637722\pi\)
\(192\) 0 0
\(193\) 3625.00i 1.35199i 0.736908 + 0.675993i \(0.236285\pi\)
−0.736908 + 0.675993i \(0.763715\pi\)
\(194\) −3431.07 −1.26978
\(195\) 0 0
\(196\) −236.000 −0.0860058
\(197\) − 1745.58i − 0.631306i −0.948875 0.315653i \(-0.897776\pi\)
0.948875 0.315653i \(-0.102224\pi\)
\(198\) 0 0
\(199\) −3199.00 −1.13955 −0.569777 0.821800i \(-0.692970\pi\)
−0.569777 + 0.821800i \(0.692970\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 4200.00i − 1.46293i
\(203\) 948.683i 0.328003i
\(204\) 0 0
\(205\) 0 0
\(206\) −4869.91 −1.64710
\(207\) 0 0
\(208\) − 2660.00i − 0.886720i
\(209\) 5755.35 1.90481
\(210\) 0 0
\(211\) 887.000 0.289401 0.144700 0.989476i \(-0.453778\pi\)
0.144700 + 0.989476i \(0.453778\pi\)
\(212\) 177.088i 0.0573699i
\(213\) 0 0
\(214\) −2520.00 −0.804970
\(215\) 0 0
\(216\) 0 0
\(217\) 2205.00i 0.689793i
\(218\) 2564.61i 0.796776i
\(219\) 0 0
\(220\) 0 0
\(221\) −3099.03 −0.943274
\(222\) 0 0
\(223\) 3535.00i 1.06153i 0.847519 + 0.530765i \(0.178095\pi\)
−0.847519 + 0.530765i \(0.821905\pi\)
\(224\) −1328.16 −0.396166
\(225\) 0 0
\(226\) −5120.00 −1.50698
\(227\) − 88.5438i − 0.0258892i −0.999916 0.0129446i \(-0.995879\pi\)
0.999916 0.0129446i \(-0.00412052\pi\)
\(228\) 0 0
\(229\) 5019.00 1.44832 0.724159 0.689633i \(-0.242228\pi\)
0.724159 + 0.689633i \(0.242228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 1200.00i − 0.339586i
\(233\) 3605.00i 1.01361i 0.862061 + 0.506805i \(0.169174\pi\)
−0.862061 + 0.506805i \(0.830826\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1770.88 0.488450
\(237\) 0 0
\(238\) 4200.00i 1.14389i
\(239\) 442.719 0.119821 0.0599103 0.998204i \(-0.480919\pi\)
0.0599103 + 0.998204i \(0.480919\pi\)
\(240\) 0 0
\(241\) −623.000 −0.166518 −0.0832592 0.996528i \(-0.526533\pi\)
−0.0832592 + 0.996528i \(0.526533\pi\)
\(242\) 8440.12i 2.24195i
\(243\) 0 0
\(244\) −854.000 −0.224065
\(245\) 0 0
\(246\) 0 0
\(247\) − 3185.00i − 0.820472i
\(248\) − 2789.13i − 0.714153i
\(249\) 0 0
\(250\) 0 0
\(251\) −2656.31 −0.667988 −0.333994 0.942575i \(-0.608397\pi\)
−0.333994 + 0.942575i \(0.608397\pi\)
\(252\) 0 0
\(253\) − 7200.00i − 1.78917i
\(254\) 822.192 0.203106
\(255\) 0 0
\(256\) 2896.00 0.707031
\(257\) 3718.84i 0.902626i 0.892366 + 0.451313i \(0.149044\pi\)
−0.892366 + 0.451313i \(0.850956\pi\)
\(258\) 0 0
\(259\) −5550.00 −1.33151
\(260\) 0 0
\(261\) 0 0
\(262\) 4200.00i 0.990370i
\(263\) 2896.65i 0.679144i 0.940580 + 0.339572i \(0.110282\pi\)
−0.940580 + 0.339572i \(0.889718\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4316.51 −0.994970
\(267\) 0 0
\(268\) 30.0000i 0.00683784i
\(269\) 442.719 0.100346 0.0501729 0.998741i \(-0.484023\pi\)
0.0501729 + 0.998741i \(0.484023\pi\)
\(270\) 0 0
\(271\) −308.000 −0.0690394 −0.0345197 0.999404i \(-0.510990\pi\)
−0.0345197 + 0.999404i \(0.510990\pi\)
\(272\) − 6729.33i − 1.50009i
\(273\) 0 0
\(274\) 8400.00 1.85205
\(275\) 0 0
\(276\) 0 0
\(277\) 7485.00i 1.62357i 0.583953 + 0.811787i \(0.301505\pi\)
−0.583953 + 0.811787i \(0.698495\pi\)
\(278\) 2479.23i 0.534871i
\(279\) 0 0
\(280\) 0 0
\(281\) 948.683 0.201401 0.100701 0.994917i \(-0.467892\pi\)
0.100701 + 0.994917i \(0.467892\pi\)
\(282\) 0 0
\(283\) 525.000i 0.110276i 0.998479 + 0.0551378i \(0.0175598\pi\)
−0.998479 + 0.0551378i \(0.982440\pi\)
\(284\) −126.491 −0.0264291
\(285\) 0 0
\(286\) 7000.00 1.44727
\(287\) − 6640.78i − 1.36583i
\(288\) 0 0
\(289\) −2927.00 −0.595766
\(290\) 0 0
\(291\) 0 0
\(292\) 140.000i 0.0280578i
\(293\) − 6375.15i − 1.27113i −0.772048 0.635564i \(-0.780768\pi\)
0.772048 0.635564i \(-0.219232\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7020.26 1.37853
\(297\) 0 0
\(298\) − 6800.00i − 1.32186i
\(299\) −3984.47 −0.770662
\(300\) 0 0
\(301\) 5025.00 0.962246
\(302\) − 2520.34i − 0.480228i
\(303\) 0 0
\(304\) 6916.00 1.30480
\(305\) 0 0
\(306\) 0 0
\(307\) 2695.00i 0.501016i 0.968115 + 0.250508i \(0.0805976\pi\)
−0.968115 + 0.250508i \(0.919402\pi\)
\(308\) − 1897.37i − 0.351015i
\(309\) 0 0
\(310\) 0 0
\(311\) 5312.63 0.968654 0.484327 0.874887i \(-0.339064\pi\)
0.484327 + 0.874887i \(0.339064\pi\)
\(312\) 0 0
\(313\) − 2555.00i − 0.461397i −0.973025 0.230698i \(-0.925899\pi\)
0.973025 0.230698i \(-0.0741010\pi\)
\(314\) 3873.79 0.696212
\(315\) 0 0
\(316\) −1752.00 −0.311891
\(317\) 7462.98i 1.32228i 0.750263 + 0.661140i \(0.229927\pi\)
−0.750263 + 0.661140i \(0.770073\pi\)
\(318\) 0 0
\(319\) 4000.00 0.702060
\(320\) 0 0
\(321\) 0 0
\(322\) 5400.00i 0.934566i
\(323\) − 8057.48i − 1.38802i
\(324\) 0 0
\(325\) 0 0
\(326\) 869.626 0.147743
\(327\) 0 0
\(328\) 8400.00i 1.41406i
\(329\) −2656.31 −0.445129
\(330\) 0 0
\(331\) −5088.00 −0.844900 −0.422450 0.906386i \(-0.638830\pi\)
−0.422450 + 0.906386i \(0.638830\pi\)
\(332\) − 1062.53i − 0.175644i
\(333\) 0 0
\(334\) −840.000 −0.137613
\(335\) 0 0
\(336\) 0 0
\(337\) 11485.0i 1.85646i 0.372004 + 0.928231i \(0.378671\pi\)
−0.372004 + 0.928231i \(0.621329\pi\)
\(338\) 3073.73i 0.494642i
\(339\) 0 0
\(340\) 0 0
\(341\) 9297.10 1.47644
\(342\) 0 0
\(343\) − 6915.00i − 1.08856i
\(344\) −6356.18 −0.996227
\(345\) 0 0
\(346\) 12600.0 1.95775
\(347\) − 6400.45i − 0.990185i −0.868840 0.495092i \(-0.835134\pi\)
0.868840 0.495092i \(-0.164866\pi\)
\(348\) 0 0
\(349\) 9674.00 1.48377 0.741887 0.670525i \(-0.233931\pi\)
0.741887 + 0.670525i \(0.233931\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5600.00i 0.847957i
\(353\) 5578.26i 0.841078i 0.907274 + 0.420539i \(0.138159\pi\)
−0.907274 + 0.420539i \(0.861841\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 5600.00i 0.826730i
\(359\) −8917.62 −1.31101 −0.655507 0.755189i \(-0.727545\pi\)
−0.655507 + 0.755189i \(0.727545\pi\)
\(360\) 0 0
\(361\) 1422.00 0.207319
\(362\) − 5334.76i − 0.774555i
\(363\) 0 0
\(364\) −1050.00 −0.151195
\(365\) 0 0
\(366\) 0 0
\(367\) − 8505.00i − 1.20969i −0.796342 0.604847i \(-0.793234\pi\)
0.796342 0.604847i \(-0.206766\pi\)
\(368\) − 8651.99i − 1.22559i
\(369\) 0 0
\(370\) 0 0
\(371\) −1328.16 −0.185861
\(372\) 0 0
\(373\) − 6775.00i − 0.940472i −0.882541 0.470236i \(-0.844169\pi\)
0.882541 0.470236i \(-0.155831\pi\)
\(374\) 17708.8 2.44839
\(375\) 0 0
\(376\) 3360.00 0.460848
\(377\) − 2213.59i − 0.302403i
\(378\) 0 0
\(379\) −9241.00 −1.25245 −0.626225 0.779643i \(-0.715401\pi\)
−0.626225 + 0.779643i \(0.715401\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 7000.00i − 0.937568i
\(383\) 9031.46i 1.20493i 0.798147 + 0.602463i \(0.205814\pi\)
−0.798147 + 0.602463i \(0.794186\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −11463.3 −1.51157
\(387\) 0 0
\(388\) − 2170.00i − 0.283931i
\(389\) −4363.94 −0.568794 −0.284397 0.958707i \(-0.591793\pi\)
−0.284397 + 0.958707i \(0.591793\pi\)
\(390\) 0 0
\(391\) −10080.0 −1.30375
\(392\) 2238.89i 0.288472i
\(393\) 0 0
\(394\) 5520.00 0.705821
\(395\) 0 0
\(396\) 0 0
\(397\) 4795.00i 0.606182i 0.952962 + 0.303091i \(0.0980186\pi\)
−0.952962 + 0.303091i \(0.901981\pi\)
\(398\) − 10116.1i − 1.27406i
\(399\) 0 0
\(400\) 0 0
\(401\) −7020.26 −0.874252 −0.437126 0.899400i \(-0.644004\pi\)
−0.437126 + 0.899400i \(0.644004\pi\)
\(402\) 0 0
\(403\) − 5145.00i − 0.635957i
\(404\) 2656.31 0.327120
\(405\) 0 0
\(406\) −3000.00 −0.366718
\(407\) 23400.9i 2.84997i
\(408\) 0 0
\(409\) −6881.00 −0.831891 −0.415946 0.909389i \(-0.636549\pi\)
−0.415946 + 0.909389i \(0.636549\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 3080.00i − 0.368303i
\(413\) 13281.6i 1.58243i
\(414\) 0 0
\(415\) 0 0
\(416\) 3099.03 0.365247
\(417\) 0 0
\(418\) 18200.0i 2.12964i
\(419\) 12838.8 1.49694 0.748471 0.663167i \(-0.230788\pi\)
0.748471 + 0.663167i \(0.230788\pi\)
\(420\) 0 0
\(421\) 6418.00 0.742979 0.371490 0.928437i \(-0.378847\pi\)
0.371490 + 0.928437i \(0.378847\pi\)
\(422\) 2804.94i 0.323560i
\(423\) 0 0
\(424\) 1680.00 0.192425
\(425\) 0 0
\(426\) 0 0
\(427\) − 6405.00i − 0.725901i
\(428\) − 1593.79i − 0.179997i
\(429\) 0 0
\(430\) 0 0
\(431\) −1328.16 −0.148434 −0.0742170 0.997242i \(-0.523646\pi\)
−0.0742170 + 0.997242i \(0.523646\pi\)
\(432\) 0 0
\(433\) − 8155.00i − 0.905091i −0.891741 0.452545i \(-0.850516\pi\)
0.891741 0.452545i \(-0.149484\pi\)
\(434\) −6972.82 −0.771212
\(435\) 0 0
\(436\) −1622.00 −0.178164
\(437\) − 10359.6i − 1.13402i
\(438\) 0 0
\(439\) −14749.0 −1.60349 −0.801744 0.597667i \(-0.796094\pi\)
−0.801744 + 0.597667i \(0.796094\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 9800.00i − 1.05461i
\(443\) 11156.5i 1.19653i 0.801299 + 0.598264i \(0.204143\pi\)
−0.801299 + 0.598264i \(0.795857\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11178.7 −1.18683
\(447\) 0 0
\(448\) 4920.00i 0.518857i
\(449\) −9233.85 −0.970540 −0.485270 0.874364i \(-0.661279\pi\)
−0.485270 + 0.874364i \(0.661279\pi\)
\(450\) 0 0
\(451\) −28000.0 −2.92343
\(452\) − 3238.17i − 0.336971i
\(453\) 0 0
\(454\) 280.000 0.0289450
\(455\) 0 0
\(456\) 0 0
\(457\) 1030.00i 0.105430i 0.998610 + 0.0527148i \(0.0167874\pi\)
−0.998610 + 0.0527148i \(0.983213\pi\)
\(458\) 15871.5i 1.61927i
\(459\) 0 0
\(460\) 0 0
\(461\) −9297.10 −0.939282 −0.469641 0.882858i \(-0.655617\pi\)
−0.469641 + 0.882858i \(0.655617\pi\)
\(462\) 0 0
\(463\) 11940.0i 1.19849i 0.800567 + 0.599243i \(0.204532\pi\)
−0.800567 + 0.599243i \(0.795468\pi\)
\(464\) 4806.66 0.480913
\(465\) 0 0
\(466\) −11400.0 −1.13325
\(467\) 12130.5i 1.20200i 0.799250 + 0.600998i \(0.205230\pi\)
−0.799250 + 0.600998i \(0.794770\pi\)
\(468\) 0 0
\(469\) −225.000 −0.0221525
\(470\) 0 0
\(471\) 0 0
\(472\) − 16800.0i − 1.63831i
\(473\) − 21187.3i − 2.05960i
\(474\) 0 0
\(475\) 0 0
\(476\) −2656.31 −0.255781
\(477\) 0 0
\(478\) 1400.00i 0.133963i
\(479\) −2656.31 −0.253382 −0.126691 0.991942i \(-0.540436\pi\)
−0.126691 + 0.991942i \(0.540436\pi\)
\(480\) 0 0
\(481\) 12950.0 1.22759
\(482\) − 1970.10i − 0.186173i
\(483\) 0 0
\(484\) −5338.00 −0.501315
\(485\) 0 0
\(486\) 0 0
\(487\) − 14125.0i − 1.31430i −0.753759 0.657151i \(-0.771761\pi\)
0.753759 0.657151i \(-0.228239\pi\)
\(488\) 8101.76i 0.751535i
\(489\) 0 0
\(490\) 0 0
\(491\) 4047.72 0.372038 0.186019 0.982546i \(-0.440441\pi\)
0.186019 + 0.982546i \(0.440441\pi\)
\(492\) 0 0
\(493\) − 5600.00i − 0.511585i
\(494\) 10071.9 0.917316
\(495\) 0 0
\(496\) 11172.0 1.01137
\(497\) − 948.683i − 0.0856223i
\(498\) 0 0
\(499\) −1279.00 −0.114741 −0.0573706 0.998353i \(-0.518272\pi\)
−0.0573706 + 0.998353i \(0.518272\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 8400.00i − 0.746833i
\(503\) − 16380.6i − 1.45204i −0.687675 0.726019i \(-0.741369\pi\)
0.687675 0.726019i \(-0.258631\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 22768.4 2.00035
\(507\) 0 0
\(508\) 520.000i 0.0454159i
\(509\) 4427.19 0.385524 0.192762 0.981246i \(-0.438255\pi\)
0.192762 + 0.981246i \(0.438255\pi\)
\(510\) 0 0
\(511\) −1050.00 −0.0908988
\(512\) − 4806.66i − 0.414895i
\(513\) 0 0
\(514\) −11760.0 −1.00917
\(515\) 0 0
\(516\) 0 0
\(517\) 11200.0i 0.952757i
\(518\) − 17550.6i − 1.48867i
\(519\) 0 0
\(520\) 0 0
\(521\) 11068.0 0.930704 0.465352 0.885126i \(-0.345928\pi\)
0.465352 + 0.885126i \(0.345928\pi\)
\(522\) 0 0
\(523\) 10745.0i 0.898367i 0.893439 + 0.449184i \(0.148285\pi\)
−0.893439 + 0.449184i \(0.851715\pi\)
\(524\) −2656.31 −0.221453
\(525\) 0 0
\(526\) −9160.00 −0.759306
\(527\) − 13015.9i − 1.07587i
\(528\) 0 0
\(529\) −793.000 −0.0651763
\(530\) 0 0
\(531\) 0 0
\(532\) − 2730.00i − 0.222482i
\(533\) 15495.2i 1.25923i
\(534\) 0 0
\(535\) 0 0
\(536\) 284.605 0.0229348
\(537\) 0 0
\(538\) 1400.00i 0.112190i
\(539\) −7462.98 −0.596388
\(540\) 0 0
\(541\) −9423.00 −0.748847 −0.374424 0.927258i \(-0.622159\pi\)
−0.374424 + 0.927258i \(0.622159\pi\)
\(542\) − 973.982i − 0.0771884i
\(543\) 0 0
\(544\) 7840.00 0.617899
\(545\) 0 0
\(546\) 0 0
\(547\) − 9080.00i − 0.709749i −0.934914 0.354875i \(-0.884524\pi\)
0.934914 0.354875i \(-0.115476\pi\)
\(548\) 5312.63i 0.414132i
\(549\) 0 0
\(550\) 0 0
\(551\) 5755.35 0.444984
\(552\) 0 0
\(553\) − 13140.0i − 1.01043i
\(554\) −23669.6 −1.81521
\(555\) 0 0
\(556\) −1568.00 −0.119601
\(557\) − 8348.41i − 0.635069i −0.948247 0.317535i \(-0.897145\pi\)
0.948247 0.317535i \(-0.102855\pi\)
\(558\) 0 0
\(559\) −11725.0 −0.887146
\(560\) 0 0
\(561\) 0 0
\(562\) 3000.00i 0.225173i
\(563\) − 7703.31i − 0.576653i −0.957532 0.288327i \(-0.906901\pi\)
0.957532 0.288327i \(-0.0930989\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1660.20 −0.123292
\(567\) 0 0
\(568\) 1200.00i 0.0886459i
\(569\) −6261.31 −0.461314 −0.230657 0.973035i \(-0.574088\pi\)
−0.230657 + 0.973035i \(0.574088\pi\)
\(570\) 0 0
\(571\) 587.000 0.0430213 0.0215107 0.999769i \(-0.493152\pi\)
0.0215107 + 0.999769i \(0.493152\pi\)
\(572\) 4427.19i 0.323619i
\(573\) 0 0
\(574\) 21000.0 1.52704
\(575\) 0 0
\(576\) 0 0
\(577\) 15995.0i 1.15404i 0.816730 + 0.577020i \(0.195784\pi\)
−0.816730 + 0.577020i \(0.804216\pi\)
\(578\) − 9255.99i − 0.666087i
\(579\) 0 0
\(580\) 0 0
\(581\) 7968.94 0.569032
\(582\) 0 0
\(583\) 5600.00i 0.397819i
\(584\) 1328.16 0.0941088
\(585\) 0 0
\(586\) 20160.0 1.42116
\(587\) − 20453.6i − 1.43818i −0.694918 0.719089i \(-0.744559\pi\)
0.694918 0.719089i \(-0.255441\pi\)
\(588\) 0 0
\(589\) 13377.0 0.935806
\(590\) 0 0
\(591\) 0 0
\(592\) 28120.0i 1.95224i
\(593\) − 11599.2i − 0.803244i −0.915806 0.401622i \(-0.868447\pi\)
0.915806 0.401622i \(-0.131553\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4300.70 0.295576
\(597\) 0 0
\(598\) − 12600.0i − 0.861626i
\(599\) 19100.2 1.30286 0.651428 0.758710i \(-0.274170\pi\)
0.651428 + 0.758710i \(0.274170\pi\)
\(600\) 0 0
\(601\) 3143.00 0.213321 0.106660 0.994296i \(-0.465984\pi\)
0.106660 + 0.994296i \(0.465984\pi\)
\(602\) 15890.4i 1.07582i
\(603\) 0 0
\(604\) 1594.00 0.107382
\(605\) 0 0
\(606\) 0 0
\(607\) − 2660.00i − 0.177868i −0.996038 0.0889342i \(-0.971654\pi\)
0.996038 0.0889342i \(-0.0283461\pi\)
\(608\) 8057.48i 0.537457i
\(609\) 0 0
\(610\) 0 0
\(611\) 6198.06 0.410388
\(612\) 0 0
\(613\) − 6670.00i − 0.439476i −0.975559 0.219738i \(-0.929480\pi\)
0.975559 0.219738i \(-0.0705202\pi\)
\(614\) −8522.34 −0.560152
\(615\) 0 0
\(616\) −18000.0 −1.17734
\(617\) 11042.7i 0.720521i 0.932852 + 0.360260i \(0.117312\pi\)
−0.932852 + 0.360260i \(0.882688\pi\)
\(618\) 0 0
\(619\) −5579.00 −0.362260 −0.181130 0.983459i \(-0.557975\pi\)
−0.181130 + 0.983459i \(0.557975\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16800.0i 1.08299i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 8079.62 0.515857
\(627\) 0 0
\(628\) 2450.00i 0.155678i
\(629\) 32761.2 2.07675
\(630\) 0 0
\(631\) 23617.0 1.48998 0.744990 0.667075i \(-0.232454\pi\)
0.744990 + 0.667075i \(0.232454\pi\)
\(632\) 16620.9i 1.04612i
\(633\) 0 0
\(634\) −23600.0 −1.47835
\(635\) 0 0
\(636\) 0 0
\(637\) 4130.00i 0.256886i
\(638\) 12649.1i 0.784926i
\(639\) 0 0
\(640\) 0 0
\(641\) −31496.3 −1.94076 −0.970381 0.241579i \(-0.922335\pi\)
−0.970381 + 0.241579i \(0.922335\pi\)
\(642\) 0 0
\(643\) − 14280.0i − 0.875814i −0.899020 0.437907i \(-0.855720\pi\)
0.899020 0.437907i \(-0.144280\pi\)
\(644\) −3415.26 −0.208975
\(645\) 0 0
\(646\) 25480.0 1.55185
\(647\) − 12396.1i − 0.753234i −0.926369 0.376617i \(-0.877087\pi\)
0.926369 0.376617i \(-0.122913\pi\)
\(648\) 0 0
\(649\) 56000.0 3.38705
\(650\) 0 0
\(651\) 0 0
\(652\) 550.000i 0.0330363i
\(653\) 12042.0i 0.721651i 0.932633 + 0.360825i \(0.117505\pi\)
−0.932633 + 0.360825i \(0.882495\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −33646.6 −2.00256
\(657\) 0 0
\(658\) − 8400.00i − 0.497669i
\(659\) 5755.35 0.340207 0.170104 0.985426i \(-0.445590\pi\)
0.170104 + 0.985426i \(0.445590\pi\)
\(660\) 0 0
\(661\) 25438.0 1.49686 0.748429 0.663215i \(-0.230808\pi\)
0.748429 + 0.663215i \(0.230808\pi\)
\(662\) − 16089.7i − 0.944626i
\(663\) 0 0
\(664\) −10080.0 −0.589126
\(665\) 0 0
\(666\) 0 0
\(667\) − 7200.00i − 0.417969i
\(668\) − 531.263i − 0.0307712i
\(669\) 0 0
\(670\) 0 0
\(671\) −27005.9 −1.55372
\(672\) 0 0
\(673\) 15150.0i 0.867741i 0.900975 + 0.433870i \(0.142852\pi\)
−0.900975 + 0.433870i \(0.857148\pi\)
\(674\) −36318.8 −2.07559
\(675\) 0 0
\(676\) −1944.00 −0.110605
\(677\) 20984.9i 1.19131i 0.803242 + 0.595653i \(0.203107\pi\)
−0.803242 + 0.595653i \(0.796893\pi\)
\(678\) 0 0
\(679\) 16275.0 0.919849
\(680\) 0 0
\(681\) 0 0
\(682\) 29400.0i 1.65071i
\(683\) 1973.26i 0.110549i 0.998471 + 0.0552743i \(0.0176033\pi\)
−0.998471 + 0.0552743i \(0.982397\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 21867.2 1.21704
\(687\) 0 0
\(688\) − 25460.0i − 1.41083i
\(689\) 3099.03 0.171355
\(690\) 0 0
\(691\) −20552.0 −1.13145 −0.565727 0.824592i \(-0.691404\pi\)
−0.565727 + 0.824592i \(0.691404\pi\)
\(692\) 7968.94i 0.437765i
\(693\) 0 0
\(694\) 20240.0 1.10706
\(695\) 0 0
\(696\) 0 0
\(697\) 39200.0i 2.13028i
\(698\) 30591.9i 1.65891i
\(699\) 0 0
\(700\) 0 0
\(701\) 8411.66 0.453215 0.226608 0.973986i \(-0.427236\pi\)
0.226608 + 0.973986i \(0.427236\pi\)
\(702\) 0 0
\(703\) 33670.0i 1.80638i
\(704\) 20744.5 1.11057
\(705\) 0 0
\(706\) −17640.0 −0.940354
\(707\) 19922.3i 1.05977i
\(708\) 0 0
\(709\) −17281.0 −0.915376 −0.457688 0.889113i \(-0.651322\pi\)
−0.457688 + 0.889113i \(0.651322\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 16734.8i − 0.878993i
\(714\) 0 0
\(715\) 0 0
\(716\) −3541.75 −0.184862
\(717\) 0 0
\(718\) − 28200.0i − 1.46576i
\(719\) −9297.10 −0.482230 −0.241115 0.970497i \(-0.577513\pi\)
−0.241115 + 0.970497i \(0.577513\pi\)
\(720\) 0 0
\(721\) 23100.0 1.19319
\(722\) 4496.76i 0.231790i
\(723\) 0 0
\(724\) 3374.00 0.173196
\(725\) 0 0
\(726\) 0 0
\(727\) − 30415.0i − 1.55162i −0.630965 0.775811i \(-0.717341\pi\)
0.630965 0.775811i \(-0.282659\pi\)
\(728\) 9961.17i 0.507123i
\(729\) 0 0
\(730\) 0 0
\(731\) −29662.2 −1.50081
\(732\) 0 0
\(733\) 18550.0i 0.934734i 0.884063 + 0.467367i \(0.154797\pi\)
−0.884063 + 0.467367i \(0.845203\pi\)
\(734\) 26895.2 1.35248
\(735\) 0 0
\(736\) 10080.0 0.504828
\(737\) 948.683i 0.0474155i
\(738\) 0 0
\(739\) −24016.0 −1.19546 −0.597729 0.801699i \(-0.703930\pi\)
−0.597729 + 0.801699i \(0.703930\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 4200.00i − 0.207799i
\(743\) 12396.1i 0.612072i 0.952020 + 0.306036i \(0.0990029\pi\)
−0.952020 + 0.306036i \(0.900997\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 21424.4 1.05148
\(747\) 0 0
\(748\) 11200.0i 0.547477i
\(749\) 11953.4 0.583135
\(750\) 0 0
\(751\) −7772.00 −0.377636 −0.188818 0.982012i \(-0.560466\pi\)
−0.188818 + 0.982012i \(0.560466\pi\)
\(752\) 13458.7i 0.652642i
\(753\) 0 0
\(754\) 7000.00 0.338097
\(755\) 0 0
\(756\) 0 0
\(757\) 3865.00i 0.185569i 0.995686 + 0.0927846i \(0.0295768\pi\)
−0.995686 + 0.0927846i \(0.970423\pi\)
\(758\) − 29222.6i − 1.40028i
\(759\) 0 0
\(760\) 0 0
\(761\) 15937.9 0.759195 0.379598 0.925152i \(-0.376062\pi\)
0.379598 + 0.925152i \(0.376062\pi\)
\(762\) 0 0
\(763\) − 12165.0i − 0.577199i
\(764\) 4427.19 0.209647
\(765\) 0 0
\(766\) −28560.0 −1.34715
\(767\) − 30990.3i − 1.45893i
\(768\) 0 0
\(769\) 7441.00 0.348933 0.174466 0.984663i \(-0.444180\pi\)
0.174466 + 0.984663i \(0.444180\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 7250.00i − 0.337996i
\(773\) − 25766.2i − 1.19890i −0.800413 0.599448i \(-0.795387\pi\)
0.800413 0.599448i \(-0.204613\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −20586.4 −0.952332
\(777\) 0 0
\(778\) − 13800.0i − 0.635931i
\(779\) −40287.4 −1.85295
\(780\) 0 0
\(781\) −4000.00 −0.183267
\(782\) − 31875.8i − 1.45764i
\(783\) 0 0
\(784\) −8968.00 −0.408528
\(785\) 0 0
\(786\) 0 0
\(787\) − 6125.00i − 0.277424i −0.990333 0.138712i \(-0.955704\pi\)
0.990333 0.138712i \(-0.0442962\pi\)
\(788\) 3491.15i 0.157826i
\(789\) 0 0
\(790\) 0 0
\(791\) 24286.3 1.09168
\(792\) 0 0
\(793\) 14945.0i 0.669247i
\(794\) −15163.1 −0.677732
\(795\) 0 0
\(796\) 6398.00 0.284888
\(797\) − 20010.9i − 0.889363i −0.895689 0.444681i \(-0.853317\pi\)
0.895689 0.444681i \(-0.146683\pi\)
\(798\) 0 0
\(799\) 15680.0 0.694266
\(800\) 0 0
\(801\) 0 0
\(802\) − 22200.0i − 0.977443i
\(803\) 4427.19i 0.194561i
\(804\) 0 0
\(805\) 0 0
\(806\) 16269.9 0.711022
\(807\) 0 0
\(808\) − 25200.0i − 1.09719i
\(809\) −2719.56 −0.118189 −0.0590943 0.998252i \(-0.518821\pi\)
−0.0590943 + 0.998252i \(0.518821\pi\)
\(810\) 0 0
\(811\) 28623.0 1.23932 0.619661 0.784870i \(-0.287270\pi\)
0.619661 + 0.784870i \(0.287270\pi\)
\(812\) − 1897.37i − 0.0820006i
\(813\) 0 0
\(814\) −74000.0 −3.18636
\(815\) 0 0
\(816\) 0 0
\(817\) − 30485.0i − 1.30543i
\(818\) − 21759.6i − 0.930083i
\(819\) 0 0
\(820\) 0 0
\(821\) 26563.1 1.12918 0.564592 0.825370i \(-0.309034\pi\)
0.564592 + 0.825370i \(0.309034\pi\)
\(822\) 0 0
\(823\) 1135.00i 0.0480724i 0.999711 + 0.0240362i \(0.00765170\pi\)
−0.999711 + 0.0240362i \(0.992348\pi\)
\(824\) −29219.4 −1.23532
\(825\) 0 0
\(826\) −42000.0 −1.76921
\(827\) 999.280i 0.0420174i 0.999779 + 0.0210087i \(0.00668776\pi\)
−0.999779 + 0.0210087i \(0.993312\pi\)
\(828\) 0 0
\(829\) 1974.00 0.0827019 0.0413509 0.999145i \(-0.486834\pi\)
0.0413509 + 0.999145i \(0.486834\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 11480.0i − 0.478362i
\(833\) 10448.2i 0.434583i
\(834\) 0 0
\(835\) 0 0
\(836\) −11510.7 −0.476203
\(837\) 0 0
\(838\) 40600.0i 1.67363i
\(839\) 42058.3 1.73065 0.865324 0.501213i \(-0.167113\pi\)
0.865324 + 0.501213i \(0.167113\pi\)
\(840\) 0 0
\(841\) −20389.0 −0.835992
\(842\) 20295.5i 0.830676i
\(843\) 0 0
\(844\) −1774.00 −0.0723502
\(845\) 0 0
\(846\) 0 0
\(847\) − 40035.0i − 1.62411i
\(848\) 6729.33i 0.272507i
\(849\) 0 0
\(850\) 0 0
\(851\) 42121.5 1.69672
\(852\) 0 0
\(853\) 36645.0i 1.47093i 0.677564 + 0.735464i \(0.263036\pi\)
−0.677564 + 0.735464i \(0.736964\pi\)
\(854\) 20254.4 0.811582
\(855\) 0 0
\(856\) −15120.0 −0.603728
\(857\) − 10802.3i − 0.430573i −0.976551 0.215286i \(-0.930931\pi\)
0.976551 0.215286i \(-0.0690685\pi\)
\(858\) 0 0
\(859\) −20104.0 −0.798533 −0.399266 0.916835i \(-0.630735\pi\)
−0.399266 + 0.916835i \(0.630735\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 4200.00i − 0.165954i
\(863\) − 2150.35i − 0.0848189i −0.999100 0.0424095i \(-0.986497\pi\)
0.999100 0.0424095i \(-0.0135034\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 25788.4 1.01192
\(867\) 0 0
\(868\) − 4410.00i − 0.172448i
\(869\) −55403.1 −2.16274
\(870\) 0 0
\(871\) 525.000 0.0204236
\(872\) 15387.6i 0.597582i
\(873\) 0 0
\(874\) 32760.0 1.26788
\(875\) 0 0
\(876\) 0 0
\(877\) 21975.0i 0.846115i 0.906103 + 0.423058i \(0.139043\pi\)
−0.906103 + 0.423058i \(0.860957\pi\)
\(878\) − 46640.4i − 1.79275i
\(879\) 0 0
\(880\) 0 0
\(881\) 885.438 0.0338606 0.0169303 0.999857i \(-0.494611\pi\)
0.0169303 + 0.999857i \(0.494611\pi\)
\(882\) 0 0
\(883\) 34915.0i 1.33067i 0.746544 + 0.665336i \(0.231712\pi\)
−0.746544 + 0.665336i \(0.768288\pi\)
\(884\) 6198.06 0.235818
\(885\) 0 0
\(886\) −35280.0 −1.33776
\(887\) 49407.4i 1.87028i 0.354277 + 0.935140i \(0.384727\pi\)
−0.354277 + 0.935140i \(0.615273\pi\)
\(888\) 0 0
\(889\) −3900.00 −0.147134
\(890\) 0 0
\(891\) 0 0
\(892\) − 7070.00i − 0.265382i
\(893\) 16115.0i 0.603882i
\(894\) 0 0
\(895\) 0 0
\(896\) −26183.7 −0.976266
\(897\) 0 0
\(898\) − 29200.0i − 1.08510i
\(899\) 9297.10 0.344912
\(900\) 0 0
\(901\) 7840.00 0.289887
\(902\) − 88543.8i − 3.26850i
\(903\) 0 0
\(904\) −30720.0 −1.13023
\(905\) 0 0
\(906\) 0 0
\(907\) − 17880.0i − 0.654571i −0.944926 0.327285i \(-0.893866\pi\)
0.944926 0.327285i \(-0.106134\pi\)
\(908\) 177.088i 0.00647231i
\(909\) 0 0
\(910\) 0 0
\(911\) −17329.3 −0.630236 −0.315118 0.949053i \(-0.602044\pi\)
−0.315118 + 0.949053i \(0.602044\pi\)
\(912\) 0 0
\(913\) − 33600.0i − 1.21796i
\(914\) −3257.15 −0.117874
\(915\) 0 0
\(916\) −10038.0 −0.362080
\(917\) − 19922.3i − 0.717442i
\(918\) 0 0
\(919\) 25829.0 0.927117 0.463558 0.886066i \(-0.346572\pi\)
0.463558 + 0.886066i \(0.346572\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 29400.0i − 1.05015i
\(923\) 2213.59i 0.0789397i
\(924\) 0 0
\(925\) 0 0
\(926\) −37757.6 −1.33995
\(927\) 0 0
\(928\) 5600.00i 0.198092i
\(929\) −3099.03 −0.109447 −0.0547233 0.998502i \(-0.517428\pi\)
−0.0547233 + 0.998502i \(0.517428\pi\)
\(930\) 0 0
\(931\) −10738.0 −0.378006
\(932\) − 7209.99i − 0.253403i
\(933\) 0 0
\(934\) −38360.0 −1.34387
\(935\) 0 0
\(936\) 0 0
\(937\) − 40005.0i − 1.39478i −0.716693 0.697389i \(-0.754345\pi\)
0.716693 0.697389i \(-0.245655\pi\)
\(938\) − 711.512i − 0.0247673i
\(939\) 0 0
\(940\) 0 0
\(941\) −30104.9 −1.04292 −0.521462 0.853275i \(-0.674613\pi\)
−0.521462 + 0.853275i \(0.674613\pi\)
\(942\) 0 0
\(943\) 50400.0i 1.74046i
\(944\) 67293.3 2.32014
\(945\) 0 0
\(946\) 67000.0 2.30270
\(947\) 43297.9i 1.48574i 0.669437 + 0.742868i \(0.266535\pi\)
−0.669437 + 0.742868i \(0.733465\pi\)
\(948\) 0 0
\(949\) 2450.00 0.0838044
\(950\) 0 0
\(951\) 0 0
\(952\) 25200.0i 0.857917i
\(953\) 53682.8i 1.82472i 0.409390 + 0.912360i \(0.365742\pi\)
−0.409390 + 0.912360i \(0.634258\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −885.438 −0.0299551
\(957\) 0 0
\(958\) − 8400.00i − 0.283290i
\(959\) −39844.7 −1.34166
\(960\) 0 0
\(961\) −8182.00 −0.274647
\(962\) 40951.5i 1.37248i
\(963\) 0 0
\(964\) 1246.00 0.0416296
\(965\) 0 0
\(966\) 0 0
\(967\) 20540.0i 0.683063i 0.939870 + 0.341531i \(0.110946\pi\)
−0.939870 + 0.341531i \(0.889054\pi\)
\(968\) 50640.7i 1.68146i
\(969\) 0 0
\(970\) 0 0
\(971\) −10625.3 −0.351164 −0.175582 0.984465i \(-0.556181\pi\)
−0.175582 + 0.984465i \(0.556181\pi\)
\(972\) 0 0
\(973\) − 11760.0i − 0.387470i
\(974\) 44667.2 1.46943
\(975\) 0 0
\(976\) −32452.0 −1.06431
\(977\) 28359.3i 0.928654i 0.885664 + 0.464327i \(0.153704\pi\)
−0.885664 + 0.464327i \(0.846296\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 12800.0i 0.415952i
\(983\) − 35063.3i − 1.13769i −0.822446 0.568844i \(-0.807391\pi\)
0.822446 0.568844i \(-0.192609\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 17708.8 0.571969
\(987\) 0 0
\(988\) 6370.00i 0.205118i
\(989\) −38137.1 −1.22618
\(990\) 0 0
\(991\) −18283.0 −0.586053 −0.293027 0.956104i \(-0.594662\pi\)
−0.293027 + 0.956104i \(0.594662\pi\)
\(992\) 13015.9i 0.416589i
\(993\) 0 0
\(994\) 3000.00 0.0957286
\(995\) 0 0
\(996\) 0 0
\(997\) 13230.0i 0.420259i 0.977674 + 0.210130i \(0.0673886\pi\)
−0.977674 + 0.210130i \(0.932611\pi\)
\(998\) − 4044.55i − 0.128285i
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 225.4.b.i.199.3 4
3.2 odd 2 inner 225.4.b.i.199.1 4
5.2 odd 4 225.4.a.m.1.1 yes 2
5.3 odd 4 225.4.a.l.1.2 yes 2
5.4 even 2 inner 225.4.b.i.199.2 4
15.2 even 4 225.4.a.m.1.2 yes 2
15.8 even 4 225.4.a.l.1.1 2
15.14 odd 2 inner 225.4.b.i.199.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.4.a.l.1.1 2 15.8 even 4
225.4.a.l.1.2 yes 2 5.3 odd 4
225.4.a.m.1.1 yes 2 5.2 odd 4
225.4.a.m.1.2 yes 2 15.2 even 4
225.4.b.i.199.1 4 3.2 odd 2 inner
225.4.b.i.199.2 4 5.4 even 2 inner
225.4.b.i.199.3 4 1.1 even 1 trivial
225.4.b.i.199.4 4 15.14 odd 2 inner