Properties

Label 225.4.a.l.1.1
Level $225$
Weight $4$
Character 225.1
Self dual yes
Analytic conductor $13.275$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2754297513\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.16228\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.16228 q^{2} +2.00000 q^{4} -15.0000 q^{7} +18.9737 q^{8} +O(q^{10})\) \(q-3.16228 q^{2} +2.00000 q^{4} -15.0000 q^{7} +18.9737 q^{8} +63.2456 q^{11} -35.0000 q^{13} +47.4342 q^{14} -76.0000 q^{16} -88.5438 q^{17} +91.0000 q^{19} -200.000 q^{22} +113.842 q^{23} +110.680 q^{26} -30.0000 q^{28} -63.2456 q^{29} -147.000 q^{31} +88.5438 q^{32} +280.000 q^{34} -370.000 q^{37} -287.767 q^{38} -442.719 q^{41} -335.000 q^{43} +126.491 q^{44} -360.000 q^{46} +177.088 q^{47} -118.000 q^{49} -70.0000 q^{52} -88.5438 q^{53} -284.605 q^{56} +200.000 q^{58} -885.438 q^{59} +427.000 q^{61} +464.855 q^{62} +328.000 q^{64} -15.0000 q^{67} -177.088 q^{68} -63.2456 q^{71} +70.0000 q^{73} +1170.04 q^{74} +182.000 q^{76} -948.683 q^{77} -876.000 q^{79} +1400.00 q^{82} +531.263 q^{83} +1059.36 q^{86} +1200.00 q^{88} +525.000 q^{91} +227.684 q^{92} -560.000 q^{94} +1085.00 q^{97} +373.149 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 30 q^{7} - 70 q^{13} - 152 q^{16} + 182 q^{19} - 400 q^{22} - 60 q^{28} - 294 q^{31} + 560 q^{34} - 740 q^{37} - 670 q^{43} - 720 q^{46} - 236 q^{49} - 140 q^{52} + 400 q^{58} + 854 q^{61} + 656 q^{64} - 30 q^{67} + 140 q^{73} + 364 q^{76} - 1752 q^{79} + 2800 q^{82} + 2400 q^{88} + 1050 q^{91} - 1120 q^{94} + 2170 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.16228 −1.11803 −0.559017 0.829156i \(-0.688821\pi\)
−0.559017 + 0.829156i \(0.688821\pi\)
\(3\) 0 0
\(4\) 2.00000 0.250000
\(5\) 0 0
\(6\) 0 0
\(7\) −15.0000 −0.809924 −0.404962 0.914334i \(-0.632715\pi\)
−0.404962 + 0.914334i \(0.632715\pi\)
\(8\) 18.9737 0.838525
\(9\) 0 0
\(10\) 0 0
\(11\) 63.2456 1.73357 0.866784 0.498683i \(-0.166183\pi\)
0.866784 + 0.498683i \(0.166183\pi\)
\(12\) 0 0
\(13\) −35.0000 −0.746712 −0.373356 0.927688i \(-0.621793\pi\)
−0.373356 + 0.927688i \(0.621793\pi\)
\(14\) 47.4342 0.905522
\(15\) 0 0
\(16\) −76.0000 −1.18750
\(17\) −88.5438 −1.26324 −0.631618 0.775280i \(-0.717609\pi\)
−0.631618 + 0.775280i \(0.717609\pi\)
\(18\) 0 0
\(19\) 91.0000 1.09878 0.549390 0.835566i \(-0.314860\pi\)
0.549390 + 0.835566i \(0.314860\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −200.000 −1.93819
\(23\) 113.842 1.03207 0.516037 0.856566i \(-0.327407\pi\)
0.516037 + 0.856566i \(0.327407\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 110.680 0.834849
\(27\) 0 0
\(28\) −30.0000 −0.202481
\(29\) −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 \(0.640021\pi\)
\(32\) 88.5438 0.489140
\(33\) 0 0
\(34\) 280.000 1.41234
\(35\) 0 0
\(36\) 0 0
\(37\) −370.000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −287.767 −1.22847
\(39\) 0 0
\(40\) 0 0
\(41\) −442.719 −1.68637 −0.843184 0.537625i \(-0.819321\pi\)
−0.843184 + 0.537625i \(0.819321\pi\)
\(42\) 0 0
\(43\) −335.000 −1.18807 −0.594035 0.804439i \(-0.702466\pi\)
−0.594035 + 0.804439i \(0.702466\pi\)
\(44\) 126.491 0.433392
\(45\) 0 0
\(46\) −360.000 −1.15389
\(47\) 177.088 0.549593 0.274797 0.961502i \(-0.411390\pi\)
0.274797 + 0.961502i \(0.411390\pi\)
\(48\) 0 0
\(49\) −118.000 −0.344023
\(50\) 0 0
\(51\) 0 0
\(52\) −70.0000 −0.186678
\(53\) −88.5438 −0.229480 −0.114740 0.993396i \(-0.536603\pi\)
−0.114740 + 0.993396i \(0.536603\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −284.605 −0.679142
\(57\) 0 0
\(58\) 200.000 0.452781
\(59\) −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 \(0.352090\pi\)
\(62\) 464.855 0.952204
\(63\) 0 0
\(64\) 328.000 0.640625
\(65\) 0 0
\(66\) 0 0
\(67\) −15.0000 −0.0273514 −0.0136757 0.999906i \(-0.504353\pi\)
−0.0136757 + 0.999906i \(0.504353\pi\)
\(68\) −177.088 −0.315809
\(69\) 0 0
\(70\) 0 0
\(71\) −63.2456 −0.105716 −0.0528582 0.998602i \(-0.516833\pi\)
−0.0528582 + 0.998602i \(0.516833\pi\)
\(72\) 0 0
\(73\) 70.0000 0.112231 0.0561156 0.998424i \(-0.482128\pi\)
0.0561156 + 0.998424i \(0.482128\pi\)
\(74\) 1170.04 1.83804
\(75\) 0 0
\(76\) 182.000 0.274695
\(77\) −948.683 −1.40406
\(78\) 0 0
\(79\) −876.000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1400.00 1.88542
\(83\) 531.263 0.702574 0.351287 0.936268i \(-0.385744\pi\)
0.351287 + 0.936268i \(0.385744\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1059.36 1.32830
\(87\) 0 0
\(88\) 1200.00 1.45364
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 525.000 0.604780
\(92\) 227.684 0.258018
\(93\) 0 0
\(94\) −560.000 −0.614464
\(95\) 0 0
\(96\) 0 0
\(97\) 1085.00 1.13572 0.567861 0.823124i \(-0.307771\pi\)
0.567861 + 0.823124i \(0.307771\pi\)
\(98\) 373.149 0.384630
\(99\) 0 0
\(100\) 0 0
\(101\) 1328.16 1.30848 0.654240 0.756287i \(-0.272989\pi\)
0.654240 + 0.756287i \(0.272989\pi\)
\(102\) 0 0
\(103\) −1540.00 −1.47321 −0.736605 0.676323i \(-0.763572\pi\)
−0.736605 + 0.676323i \(0.763572\pi\)
\(104\) −664.078 −0.626137
\(105\) 0 0
\(106\) 280.000 0.256566
\(107\) −796.894 −0.719987 −0.359994 0.932955i \(-0.617221\pi\)
−0.359994 + 0.932955i \(0.617221\pi\)
\(108\) 0 0
\(109\) −811.000 −0.712658 −0.356329 0.934361i \(-0.615972\pi\)
−0.356329 + 0.934361i \(0.615972\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1140.00 0.961785
\(113\) 1619.09 1.34788 0.673942 0.738785i \(-0.264600\pi\)
0.673942 + 0.738785i \(0.264600\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −126.491 −0.101245
\(117\) 0 0
\(118\) 2800.00 2.18441
\(119\) 1328.16 1.02313
\(120\) 0 0
\(121\) 2669.00 2.00526
\(122\) −1350.29 −1.00205
\(123\) 0 0
\(124\) −294.000 −0.212919
\(125\) 0 0
\(126\) 0 0
\(127\) −260.000 −0.181664 −0.0908318 0.995866i \(-0.528953\pi\)
−0.0908318 + 0.995866i \(0.528953\pi\)
\(128\) −1745.58 −1.20538
\(129\) 0 0
\(130\) 0 0
\(131\) −1328.16 −0.885814 −0.442907 0.896568i \(-0.646053\pi\)
−0.442907 + 0.896568i \(0.646053\pi\)
\(132\) 0 0
\(133\) −1365.00 −0.889929
\(134\) 47.4342 0.0305798
\(135\) 0 0
\(136\) −1680.00 −1.05926
\(137\) 2656.31 1.65653 0.828263 0.560339i \(-0.189329\pi\)
0.828263 + 0.560339i \(0.189329\pi\)
\(138\) 0 0
\(139\) −784.000 −0.478403 −0.239201 0.970970i \(-0.576886\pi\)
−0.239201 + 0.970970i \(0.576886\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 200.000 0.118195
\(143\) −2213.59 −1.29448
\(144\) 0 0
\(145\) 0 0
\(146\) −221.359 −0.125478
\(147\) 0 0
\(148\) −740.000 −0.410997
\(149\) −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 \(0.568898\pi\)
\(152\) 1726.60 0.921356
\(153\) 0 0
\(154\) 3000.00 1.56978
\(155\) 0 0
\(156\) 0 0
\(157\) −1225.00 −0.622711 −0.311356 0.950293i \(-0.600783\pi\)
−0.311356 + 0.950293i \(0.600783\pi\)
\(158\) 2770.16 1.39482
\(159\) 0 0
\(160\) 0 0
\(161\) −1707.63 −0.835901
\(162\) 0 0
\(163\) 275.000 0.132145 0.0660726 0.997815i \(-0.478953\pi\)
0.0660726 + 0.997815i \(0.478953\pi\)
\(164\) −885.438 −0.421592
\(165\) 0 0
\(166\) −1680.00 −0.785502
\(167\) −265.631 −0.123085 −0.0615424 0.998104i \(-0.519602\pi\)
−0.0615424 + 0.998104i \(0.519602\pi\)
\(168\) 0 0
\(169\) −972.000 −0.442421
\(170\) 0 0
\(171\) 0 0
\(172\) −670.000 −0.297018
\(173\) −3984.47 −1.75106 −0.875531 0.483162i \(-0.839488\pi\)
−0.875531 + 0.483162i \(0.839488\pi\)
\(174\) 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 \(0.612593\pi\)
\(182\) −1660.20 −0.676164
\(183\) 0 0
\(184\) 2160.00 0.865420
\(185\) 0 0
\(186\) 0 0
\(187\) −5600.00 −2.18991
\(188\) 354.175 0.137398
\(189\) 0 0
\(190\) 0 0
\(191\) 2213.59 0.838587 0.419293 0.907851i \(-0.362278\pi\)
0.419293 + 0.907851i \(0.362278\pi\)
\(192\) 0 0
\(193\) −3625.00 −1.35199 −0.675993 0.736908i \(-0.736285\pi\)
−0.675993 + 0.736908i \(0.736285\pi\)
\(194\) −3431.07 −1.26978
\(195\) 0 0
\(196\) −236.000 −0.0860058
\(197\) 1745.58 0.631306 0.315653 0.948875i \(-0.397776\pi\)
0.315653 + 0.948875i \(0.397776\pi\)
\(198\) 0 0
\(199\) 3199.00 1.13955 0.569777 0.821800i \(-0.307030\pi\)
0.569777 + 0.821800i \(0.307030\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4200.00 −1.46293
\(203\) 948.683 0.328003
\(204\) 0 0
\(205\) 0 0
\(206\) 4869.91 1.64710
\(207\) 0 0
\(208\) 2660.00 0.886720
\(209\) 5755.35 1.90481
\(210\) 0 0
\(211\) 887.000 0.289401 0.144700 0.989476i \(-0.453778\pi\)
0.144700 + 0.989476i \(0.453778\pi\)
\(212\) −177.088 −0.0573699
\(213\) 0 0
\(214\) 2520.00 0.804970
\(215\) 0 0
\(216\) 0 0
\(217\) 2205.00 0.689793
\(218\) 2564.61 0.796776
\(219\) 0 0
\(220\) 0 0
\(221\) 3099.03 0.943274
\(222\) 0 0
\(223\) −3535.00 −1.06153 −0.530765 0.847519i \(-0.678095\pi\)
−0.530765 + 0.847519i \(0.678095\pi\)
\(224\) −1328.16 −0.396166
\(225\) 0 0
\(226\) −5120.00 −1.50698
\(227\) 88.5438 0.0258892 0.0129446 0.999916i \(-0.495879\pi\)
0.0129446 + 0.999916i \(0.495879\pi\)
\(228\) 0 0
\(229\) −5019.00 −1.44832 −0.724159 0.689633i \(-0.757772\pi\)
−0.724159 + 0.689633i \(0.757772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1200.00 −0.339586
\(233\) 3605.00 1.01361 0.506805 0.862061i \(-0.330826\pi\)
0.506805 + 0.862061i \(0.330826\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1770.88 −0.488450
\(237\) 0 0
\(238\) −4200.00 −1.14389
\(239\) 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 \(0.526533\pi\)
\(242\) −8440.12 −2.24195
\(243\) 0 0
\(244\) 854.000 0.224065
\(245\) 0 0
\(246\) 0 0
\(247\) −3185.00 −0.820472
\(248\) −2789.13 −0.714153
\(249\) 0 0
\(250\) 0 0
\(251\) 2656.31 0.667988 0.333994 0.942575i \(-0.391603\pi\)
0.333994 + 0.942575i \(0.391603\pi\)
\(252\) 0 0
\(253\) 7200.00 1.78917
\(254\) 822.192 0.203106
\(255\) 0 0
\(256\) 2896.00 0.707031
\(257\) −3718.84 −0.902626 −0.451313 0.892366i \(-0.649044\pi\)
−0.451313 + 0.892366i \(0.649044\pi\)
\(258\) 0 0
\(259\) 5550.00 1.33151
\(260\) 0 0
\(261\) 0 0
\(262\) 4200.00 0.990370
\(263\) 2896.65 0.679144 0.339572 0.940580i \(-0.389718\pi\)
0.339572 + 0.940580i \(0.389718\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4316.51 0.994970
\(267\) 0 0
\(268\) −30.0000 −0.00683784
\(269\) 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 \(0.510990\pi\)
\(272\) 6729.33 1.50009
\(273\) 0 0
\(274\) −8400.00 −1.85205
\(275\) 0 0
\(276\) 0 0
\(277\) 7485.00 1.62357 0.811787 0.583953i \(-0.198495\pi\)
0.811787 + 0.583953i \(0.198495\pi\)
\(278\) 2479.23 0.534871
\(279\) 0 0
\(280\) 0 0
\(281\) −948.683 −0.201401 −0.100701 0.994917i \(-0.532108\pi\)
−0.100701 + 0.994917i \(0.532108\pi\)
\(282\) 0 0
\(283\) −525.000 −0.110276 −0.0551378 0.998479i \(-0.517560\pi\)
−0.0551378 + 0.998479i \(0.517560\pi\)
\(284\) −126.491 −0.0264291
\(285\) 0 0
\(286\) 7000.00 1.44727
\(287\) 6640.78 1.36583
\(288\) 0 0
\(289\) 2927.00 0.595766
\(290\) 0 0
\(291\) 0 0
\(292\) 140.000 0.0280578
\(293\) −6375.15 −1.27113 −0.635564 0.772048i \(-0.719232\pi\)
−0.635564 + 0.772048i \(0.719232\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7020.26 −1.37853
\(297\) 0 0
\(298\) 6800.00 1.32186
\(299\) −3984.47 −0.770662
\(300\) 0 0
\(301\) 5025.00 0.962246
\(302\) 2520.34 0.480228
\(303\) 0 0
\(304\) −6916.00 −1.30480
\(305\) 0 0
\(306\) 0 0
\(307\) 2695.00 0.501016 0.250508 0.968115i \(-0.419402\pi\)
0.250508 + 0.968115i \(0.419402\pi\)
\(308\) −1897.37 −0.351015
\(309\) 0 0
\(310\) 0 0
\(311\) −5312.63 −0.968654 −0.484327 0.874887i \(-0.660936\pi\)
−0.484327 + 0.874887i \(0.660936\pi\)
\(312\) 0 0
\(313\) 2555.00 0.461397 0.230698 0.973025i \(-0.425899\pi\)
0.230698 + 0.973025i \(0.425899\pi\)
\(314\) 3873.79 0.696212
\(315\) 0 0
\(316\) −1752.00 −0.311891
\(317\) −7462.98 −1.32228 −0.661140 0.750263i \(-0.729927\pi\)
−0.661140 + 0.750263i \(0.729927\pi\)
\(318\) 0 0
\(319\) −4000.00 −0.702060
\(320\) 0 0
\(321\) 0 0
\(322\) 5400.00 0.934566
\(323\) −8057.48 −1.38802
\(324\) 0 0
\(325\) 0 0
\(326\) −869.626 −0.147743
\(327\) 0 0
\(328\) −8400.00 −1.41406
\(329\) −2656.31 −0.445129
\(330\) 0 0
\(331\) −5088.00 −0.844900 −0.422450 0.906386i \(-0.638830\pi\)
−0.422450 + 0.906386i \(0.638830\pi\)
\(332\) 1062.53 0.175644
\(333\) 0 0
\(334\) 840.000 0.137613
\(335\) 0 0
\(336\) 0 0
\(337\) 11485.0 1.85646 0.928231 0.372004i \(-0.121329\pi\)
0.928231 + 0.372004i \(0.121329\pi\)
\(338\) 3073.73 0.494642
\(339\) 0 0
\(340\) 0 0
\(341\) −9297.10 −1.47644
\(342\) 0 0
\(343\) 6915.00 1.08856
\(344\) −6356.18 −0.996227
\(345\) 0 0
\(346\) 12600.0 1.95775
\(347\) 6400.45 0.990185 0.495092 0.868840i \(-0.335134\pi\)
0.495092 + 0.868840i \(0.335134\pi\)
\(348\) 0 0
\(349\) −9674.00 −1.48377 −0.741887 0.670525i \(-0.766069\pi\)
−0.741887 + 0.670525i \(0.766069\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5600.00 0.847957
\(353\) 5578.26 0.841078 0.420539 0.907274i \(-0.361841\pi\)
0.420539 + 0.907274i \(0.361841\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −5600.00 −0.826730
\(359\) −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
\(362\) 5334.76 0.774555
\(363\) 0 0
\(364\) 1050.00 0.151195
\(365\) 0 0
\(366\) 0 0
\(367\) −8505.00 −1.20969 −0.604847 0.796342i \(-0.706766\pi\)
−0.604847 + 0.796342i \(0.706766\pi\)
\(368\) −8651.99 −1.22559
\(369\) 0 0
\(370\) 0 0
\(371\) 1328.16 0.185861
\(372\) 0 0
\(373\) 6775.00 0.940472 0.470236 0.882541i \(-0.344169\pi\)
0.470236 + 0.882541i \(0.344169\pi\)
\(374\) 17708.8 2.44839
\(375\) 0 0
\(376\) 3360.00 0.460848
\(377\) 2213.59 0.302403
\(378\) 0 0
\(379\) 9241.00 1.25245 0.626225 0.779643i \(-0.284599\pi\)
0.626225 + 0.779643i \(0.284599\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −7000.00 −0.937568
\(383\) 9031.46 1.20493 0.602463 0.798147i \(-0.294186\pi\)
0.602463 + 0.798147i \(0.294186\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 11463.3 1.51157
\(387\) 0 0
\(388\) 2170.00 0.283931
\(389\) −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
\(392\) −2238.89 −0.288472
\(393\) 0 0
\(394\) −5520.00 −0.705821
\(395\) 0 0
\(396\) 0 0
\(397\) 4795.00 0.606182 0.303091 0.952962i \(-0.401981\pi\)
0.303091 + 0.952962i \(0.401981\pi\)
\(398\) −10116.1 −1.27406
\(399\) 0 0
\(400\) 0 0
\(401\) 7020.26 0.874252 0.437126 0.899400i \(-0.355996\pi\)
0.437126 + 0.899400i \(0.355996\pi\)
\(402\) 0 0
\(403\) 5145.00 0.635957
\(404\) 2656.31 0.327120
\(405\) 0 0
\(406\) −3000.00 −0.366718
\(407\) −23400.9 −2.84997
\(408\) 0 0
\(409\) 6881.00 0.831891 0.415946 0.909389i \(-0.363451\pi\)
0.415946 + 0.909389i \(0.363451\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3080.00 −0.368303
\(413\) 13281.6 1.58243
\(414\) 0 0
\(415\) 0 0
\(416\) −3099.03 −0.365247
\(417\) 0 0
\(418\) −18200.0 −2.12964
\(419\) 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 \(0.378847\pi\)
\(422\) −2804.94 −0.323560
\(423\) 0 0
\(424\) −1680.00 −0.192425
\(425\) 0 0
\(426\) 0 0
\(427\) −6405.00 −0.725901
\(428\) −1593.79 −0.179997
\(429\) 0 0
\(430\) 0 0
\(431\) 1328.16 0.148434 0.0742170 0.997242i \(-0.476354\pi\)
0.0742170 + 0.997242i \(0.476354\pi\)
\(432\) 0 0
\(433\) 8155.00 0.905091 0.452545 0.891741i \(-0.350516\pi\)
0.452545 + 0.891741i \(0.350516\pi\)
\(434\) −6972.82 −0.771212
\(435\) 0 0
\(436\) −1622.00 −0.178164
\(437\) 10359.6 1.13402
\(438\) 0 0
\(439\) 14749.0 1.60349 0.801744 0.597667i \(-0.203906\pi\)
0.801744 + 0.597667i \(0.203906\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −9800.00 −1.05461
\(443\) 11156.5 1.19653 0.598264 0.801299i \(-0.295857\pi\)
0.598264 + 0.801299i \(0.295857\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 11178.7 1.18683
\(447\) 0 0
\(448\) −4920.00 −0.518857
\(449\) −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
\(452\) 3238.17 0.336971
\(453\) 0 0
\(454\) −280.000 −0.0289450
\(455\) 0 0
\(456\) 0 0
\(457\) 1030.00 0.105430 0.0527148 0.998610i \(-0.483213\pi\)
0.0527148 + 0.998610i \(0.483213\pi\)
\(458\) 15871.5 1.61927
\(459\) 0 0
\(460\) 0 0
\(461\) 9297.10 0.939282 0.469641 0.882858i \(-0.344383\pi\)
0.469641 + 0.882858i \(0.344383\pi\)
\(462\) 0 0
\(463\) −11940.0 −1.19849 −0.599243 0.800567i \(-0.704532\pi\)
−0.599243 + 0.800567i \(0.704532\pi\)
\(464\) 4806.66 0.480913
\(465\) 0 0
\(466\) −11400.0 −1.13325
\(467\) −12130.5 −1.20200 −0.600998 0.799250i \(-0.705230\pi\)
−0.600998 + 0.799250i \(0.705230\pi\)
\(468\) 0 0
\(469\) 225.000 0.0221525
\(470\) 0 0
\(471\) 0 0
\(472\) −16800.0 −1.63831
\(473\) −21187.3 −2.05960
\(474\) 0 0
\(475\) 0 0
\(476\) 2656.31 0.255781
\(477\) 0 0
\(478\) −1400.00 −0.133963
\(479\) −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
\(482\) 1970.10 0.186173
\(483\) 0 0
\(484\) 5338.00 0.501315
\(485\) 0 0
\(486\) 0 0
\(487\) −14125.0 −1.31430 −0.657151 0.753759i \(-0.728239\pi\)
−0.657151 + 0.753759i \(0.728239\pi\)
\(488\) 8101.76 0.751535
\(489\) 0 0
\(490\) 0 0
\(491\) −4047.72 −0.372038 −0.186019 0.982546i \(-0.559559\pi\)
−0.186019 + 0.982546i \(0.559559\pi\)
\(492\) 0 0
\(493\) 5600.00 0.511585
\(494\) 10071.9 0.917316
\(495\) 0 0
\(496\) 11172.0 1.01137
\(497\) 948.683 0.0856223
\(498\) 0 0
\(499\) 1279.00 0.114741 0.0573706 0.998353i \(-0.481728\pi\)
0.0573706 + 0.998353i \(0.481728\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −8400.00 −0.746833
\(503\) −16380.6 −1.45204 −0.726019 0.687675i \(-0.758631\pi\)
−0.726019 + 0.687675i \(0.758631\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −22768.4 −2.00035
\(507\) 0 0
\(508\) −520.000 −0.0454159
\(509\) 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
\(512\) 4806.66 0.414895
\(513\) 0 0
\(514\) 11760.0 1.00917
\(515\) 0 0
\(516\) 0 0
\(517\) 11200.0 0.952757
\(518\) −17550.6 −1.48867
\(519\) 0 0
\(520\) 0 0
\(521\) −11068.0 −0.930704 −0.465352 0.885126i \(-0.654072\pi\)
−0.465352 + 0.885126i \(0.654072\pi\)
\(522\) 0 0
\(523\) −10745.0 −0.898367 −0.449184 0.893439i \(-0.648285\pi\)
−0.449184 + 0.893439i \(0.648285\pi\)
\(524\) −2656.31 −0.221453
\(525\) 0 0
\(526\) −9160.00 −0.759306
\(527\) 13015.9 1.07587
\(528\) 0 0
\(529\) 793.000 0.0651763
\(530\) 0 0
\(531\) 0 0
\(532\) −2730.00 −0.222482
\(533\) 15495.2 1.25923
\(534\) 0 0
\(535\) 0 0
\(536\) −284.605 −0.0229348
\(537\) 0 0
\(538\) −1400.00 −0.112190
\(539\) −7462.98 −0.596388
\(540\) 0 0
\(541\) −9423.00 −0.748847 −0.374424 0.927258i \(-0.622159\pi\)
−0.374424 + 0.927258i \(0.622159\pi\)
\(542\) 973.982 0.0771884
\(543\) 0 0
\(544\) −7840.00 −0.617899
\(545\) 0 0
\(546\) 0 0
\(547\) −9080.00 −0.709749 −0.354875 0.934914i \(-0.615476\pi\)
−0.354875 + 0.934914i \(0.615476\pi\)
\(548\) 5312.63 0.414132
\(549\) 0 0
\(550\) 0 0
\(551\) −5755.35 −0.444984
\(552\) 0 0
\(553\) 13140.0 1.01043
\(554\) −23669.6 −1.81521
\(555\) 0 0
\(556\) −1568.00 −0.119601
\(557\) 8348.41 0.635069 0.317535 0.948247i \(-0.397145\pi\)
0.317535 + 0.948247i \(0.397145\pi\)
\(558\) 0 0
\(559\) 11725.0 0.887146
\(560\) 0 0
\(561\) 0 0
\(562\) 3000.00 0.225173
\(563\) −7703.31 −0.576653 −0.288327 0.957532i \(-0.593099\pi\)
−0.288327 + 0.957532i \(0.593099\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1660.20 0.123292
\(567\) 0 0
\(568\) −1200.00 −0.0886459
\(569\) −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 \(0.493152\pi\)
\(572\) −4427.19 −0.323619
\(573\) 0 0
\(574\) −21000.0 −1.52704
\(575\) 0 0
\(576\) 0 0
\(577\) 15995.0 1.15404 0.577020 0.816730i \(-0.304216\pi\)
0.577020 + 0.816730i \(0.304216\pi\)
\(578\) −9255.99 −0.666087
\(579\) 0 0
\(580\) 0 0
\(581\) −7968.94 −0.569032
\(582\) 0 0
\(583\) −5600.00 −0.397819
\(584\) 1328.16 0.0941088
\(585\) 0 0
\(586\) 20160.0 1.42116
\(587\) 20453.6 1.43818 0.719089 0.694918i \(-0.244559\pi\)
0.719089 + 0.694918i \(0.244559\pi\)
\(588\) 0 0
\(589\) −13377.0 −0.935806
\(590\) 0 0
\(591\) 0 0
\(592\) 28120.0 1.95224
\(593\) −11599.2 −0.803244 −0.401622 0.915806i \(-0.631553\pi\)
−0.401622 + 0.915806i \(0.631553\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4300.70 −0.295576
\(597\) 0 0
\(598\) 12600.0 0.861626
\(599\) 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 \(0.465984\pi\)
\(602\) −15890.4 −1.07582
\(603\) 0 0
\(604\) −1594.00 −0.107382
\(605\) 0 0
\(606\) 0 0
\(607\) −2660.00 −0.177868 −0.0889342 0.996038i \(-0.528346\pi\)
−0.0889342 + 0.996038i \(0.528346\pi\)
\(608\) 8057.48 0.537457
\(609\) 0 0
\(610\) 0 0
\(611\) −6198.06 −0.410388
\(612\) 0 0
\(613\) 6670.00 0.439476 0.219738 0.975559i \(-0.429480\pi\)
0.219738 + 0.975559i \(0.429480\pi\)
\(614\) −8522.34 −0.560152
\(615\) 0 0
\(616\) −18000.0 −1.17734
\(617\) −11042.7 −0.720521 −0.360260 0.932852i \(-0.617312\pi\)
−0.360260 + 0.932852i \(0.617312\pi\)
\(618\) 0 0
\(619\) 5579.00 0.362260 0.181130 0.983459i \(-0.442025\pi\)
0.181130 + 0.983459i \(0.442025\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16800.0 1.08299
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −8079.62 −0.515857
\(627\) 0 0
\(628\) −2450.00 −0.155678
\(629\) 32761.2 2.07675
\(630\) 0 0
\(631\) 23617.0 1.48998 0.744990 0.667075i \(-0.232454\pi\)
0.744990 + 0.667075i \(0.232454\pi\)
\(632\) −16620.9 −1.04612
\(633\) 0 0
\(634\) 23600.0 1.47835
\(635\) 0 0
\(636\) 0 0
\(637\) 4130.00 0.256886
\(638\) 12649.1 0.784926
\(639\) 0 0
\(640\) 0 0
\(641\) 31496.3 1.94076 0.970381 0.241579i \(-0.0776655\pi\)
0.970381 + 0.241579i \(0.0776655\pi\)
\(642\) 0 0
\(643\) 14280.0 0.875814 0.437907 0.899020i \(-0.355720\pi\)
0.437907 + 0.899020i \(0.355720\pi\)
\(644\) −3415.26 −0.208975
\(645\) 0 0
\(646\) 25480.0 1.55185
\(647\) 12396.1 0.753234 0.376617 0.926369i \(-0.377087\pi\)
0.376617 + 0.926369i \(0.377087\pi\)
\(648\) 0 0
\(649\) −56000.0 −3.38705
\(650\) 0 0
\(651\) 0 0
\(652\) 550.000 0.0330363
\(653\) 12042.0 0.721651 0.360825 0.932633i \(-0.382495\pi\)
0.360825 + 0.932633i \(0.382495\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 33646.6 2.00256
\(657\) 0 0
\(658\) 8400.00 0.497669
\(659\) 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 \(0.230808\pi\)
\(662\) 16089.7 0.944626
\(663\) 0 0
\(664\) 10080.0 0.589126
\(665\) 0 0
\(666\) 0 0
\(667\) −7200.00 −0.417969
\(668\) −531.263 −0.0307712
\(669\) 0 0
\(670\) 0 0
\(671\) 27005.9 1.55372
\(672\) 0 0
\(673\) −15150.0 −0.867741 −0.433870 0.900975i \(-0.642852\pi\)
−0.433870 + 0.900975i \(0.642852\pi\)
\(674\) −36318.8 −2.07559
\(675\) 0 0
\(676\) −1944.00 −0.110605
\(677\) −20984.9 −1.19131 −0.595653 0.803242i \(-0.703107\pi\)
−0.595653 + 0.803242i \(0.703107\pi\)
\(678\) 0 0
\(679\) −16275.0 −0.919849
\(680\) 0 0
\(681\) 0 0
\(682\) 29400.0 1.65071
\(683\) 1973.26 0.110549 0.0552743 0.998471i \(-0.482397\pi\)
0.0552743 + 0.998471i \(0.482397\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −21867.2 −1.21704
\(687\) 0 0
\(688\) 25460.0 1.41083
\(689\) 3099.03 0.171355
\(690\) 0 0
\(691\) −20552.0 −1.13145 −0.565727 0.824592i \(-0.691404\pi\)
−0.565727 + 0.824592i \(0.691404\pi\)
\(692\) −7968.94 −0.437765
\(693\) 0 0
\(694\) −20240.0 −1.10706
\(695\) 0 0
\(696\) 0 0
\(697\) 39200.0 2.13028
\(698\) 30591.9 1.65891
\(699\) 0 0
\(700\) 0 0
\(701\) −8411.66 −0.453215 −0.226608 0.973986i \(-0.572764\pi\)
−0.226608 + 0.973986i \(0.572764\pi\)
\(702\) 0 0
\(703\) −33670.0 −1.80638
\(704\) 20744.5 1.11057
\(705\) 0 0
\(706\) −17640.0 −0.940354
\(707\) −19922.3 −1.05977
\(708\) 0 0
\(709\) 17281.0 0.915376 0.457688 0.889113i \(-0.348678\pi\)
0.457688 + 0.889113i \(0.348678\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16734.8 −0.878993
\(714\) 0 0
\(715\) 0 0
\(716\) 3541.75 0.184862
\(717\) 0 0
\(718\) 28200.0 1.46576
\(719\) −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
\(722\) −4496.76 −0.231790
\(723\) 0 0
\(724\) −3374.00 −0.173196
\(725\) 0 0
\(726\) 0 0
\(727\) −30415.0 −1.55162 −0.775811 0.630965i \(-0.782659\pi\)
−0.775811 + 0.630965i \(0.782659\pi\)
\(728\) 9961.17 0.507123
\(729\) 0 0
\(730\) 0 0
\(731\) 29662.2 1.50081
\(732\) 0 0
\(733\) −18550.0 −0.934734 −0.467367 0.884063i \(-0.654797\pi\)
−0.467367 + 0.884063i \(0.654797\pi\)
\(734\) 26895.2 1.35248
\(735\) 0 0
\(736\) 10080.0 0.504828
\(737\) −948.683 −0.0474155
\(738\) 0 0
\(739\) 24016.0 1.19546 0.597729 0.801699i \(-0.296070\pi\)
0.597729 + 0.801699i \(0.296070\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4200.00 −0.207799
\(743\) 12396.1 0.612072 0.306036 0.952020i \(-0.400997\pi\)
0.306036 + 0.952020i \(0.400997\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −21424.4 −1.05148
\(747\) 0 0
\(748\) −11200.0 −0.547477
\(749\) 11953.4 0.583135
\(750\) 0 0
\(751\) −7772.00 −0.377636 −0.188818 0.982012i \(-0.560466\pi\)
−0.188818 + 0.982012i \(0.560466\pi\)
\(752\) −13458.7 −0.652642
\(753\) 0 0
\(754\) −7000.00 −0.338097
\(755\) 0 0
\(756\) 0 0
\(757\) 3865.00 0.185569 0.0927846 0.995686i \(-0.470423\pi\)
0.0927846 + 0.995686i \(0.470423\pi\)
\(758\) −29222.6 −1.40028
\(759\) 0 0
\(760\) 0 0
\(761\) −15937.9 −0.759195 −0.379598 0.925152i \(-0.623938\pi\)
−0.379598 + 0.925152i \(0.623938\pi\)
\(762\) 0 0
\(763\) 12165.0 0.577199
\(764\) 4427.19 0.209647
\(765\) 0 0
\(766\) −28560.0 −1.34715
\(767\) 30990.3 1.45893
\(768\) 0 0
\(769\) −7441.00 −0.348933 −0.174466 0.984663i \(-0.555820\pi\)
−0.174466 + 0.984663i \(0.555820\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7250.00 −0.337996
\(773\) −25766.2 −1.19890 −0.599448 0.800413i \(-0.704613\pi\)
−0.599448 + 0.800413i \(0.704613\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 20586.4 0.952332
\(777\) 0 0
\(778\) 13800.0 0.635931
\(779\) −40287.4 −1.85295
\(780\) 0 0
\(781\) −4000.00 −0.183267
\(782\) 31875.8 1.45764
\(783\) 0 0
\(784\) 8968.00 0.408528
\(785\) 0 0
\(786\) 0 0
\(787\) −6125.00 −0.277424 −0.138712 0.990333i \(-0.544296\pi\)
−0.138712 + 0.990333i \(0.544296\pi\)
\(788\) 3491.15 0.157826
\(789\) 0 0
\(790\) 0 0
\(791\) −24286.3 −1.09168
\(792\) 0 0
\(793\) −14945.0 −0.669247
\(794\) −15163.1 −0.677732
\(795\) 0 0
\(796\) 6398.00 0.284888
\(797\) 20010.9 0.889363 0.444681 0.895689i \(-0.353317\pi\)
0.444681 + 0.895689i \(0.353317\pi\)
\(798\) 0 0
\(799\) −15680.0 −0.694266
\(800\) 0 0
\(801\) 0 0
\(802\) −22200.0 −0.977443
\(803\) 4427.19 0.194561
\(804\) 0 0
\(805\) 0 0
\(806\) −16269.9 −0.711022
\(807\) 0 0
\(808\) 25200.0 1.09719
\(809\) −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 \(0.287270\pi\)
\(812\) 1897.37 0.0820006
\(813\) 0 0
\(814\) 74000.0 3.18636
\(815\) 0 0
\(816\) 0 0
\(817\) −30485.0 −1.30543
\(818\) −21759.6 −0.930083
\(819\) 0 0
\(820\) 0 0
\(821\) −26563.1 −1.12918 −0.564592 0.825370i \(-0.690966\pi\)
−0.564592 + 0.825370i \(0.690966\pi\)
\(822\) 0 0
\(823\) −1135.00 −0.0480724 −0.0240362 0.999711i \(-0.507652\pi\)
−0.0240362 + 0.999711i \(0.507652\pi\)
\(824\) −29219.4 −1.23532
\(825\) 0 0
\(826\) −42000.0 −1.76921
\(827\) −999.280 −0.0420174 −0.0210087 0.999779i \(-0.506688\pi\)
−0.0210087 + 0.999779i \(0.506688\pi\)
\(828\) 0 0
\(829\) −1974.00 −0.0827019 −0.0413509 0.999145i \(-0.513166\pi\)
−0.0413509 + 0.999145i \(0.513166\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −11480.0 −0.478362
\(833\) 10448.2 0.434583
\(834\) 0 0
\(835\) 0 0
\(836\) 11510.7 0.476203
\(837\) 0 0
\(838\) −40600.0 −1.67363
\(839\) 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
\(842\) −20295.5 −0.830676
\(843\) 0 0
\(844\) 1774.00 0.0723502
\(845\) 0 0
\(846\) 0 0
\(847\) −40035.0 −1.62411
\(848\) 6729.33 0.272507
\(849\) 0 0
\(850\) 0 0
\(851\) −42121.5 −1.69672
\(852\) 0 0
\(853\) −36645.0 −1.47093 −0.735464 0.677564i \(-0.763036\pi\)
−0.735464 + 0.677564i \(0.763036\pi\)
\(854\) 20254.4 0.811582
\(855\) 0 0
\(856\) −15120.0 −0.603728
\(857\) 10802.3 0.430573 0.215286 0.976551i \(-0.430931\pi\)
0.215286 + 0.976551i \(0.430931\pi\)
\(858\) 0 0
\(859\) 20104.0 0.798533 0.399266 0.916835i \(-0.369265\pi\)
0.399266 + 0.916835i \(0.369265\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4200.00 −0.165954
\(863\) −2150.35 −0.0848189 −0.0424095 0.999100i \(-0.513503\pi\)
−0.0424095 + 0.999100i \(0.513503\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −25788.4 −1.01192
\(867\) 0 0
\(868\) 4410.00 0.172448
\(869\) −55403.1 −2.16274
\(870\) 0 0
\(871\) 525.000 0.0204236
\(872\) −15387.6 −0.597582
\(873\) 0 0
\(874\) −32760.0 −1.26788
\(875\) 0 0
\(876\) 0 0
\(877\) 21975.0 0.846115 0.423058 0.906103i \(-0.360957\pi\)
0.423058 + 0.906103i \(0.360957\pi\)
\(878\) −46640.4 −1.79275
\(879\) 0 0
\(880\) 0 0
\(881\) −885.438 −0.0338606 −0.0169303 0.999857i \(-0.505389\pi\)
−0.0169303 + 0.999857i \(0.505389\pi\)
\(882\) 0 0
\(883\) −34915.0 −1.33067 −0.665336 0.746544i \(-0.731712\pi\)
−0.665336 + 0.746544i \(0.731712\pi\)
\(884\) 6198.06 0.235818
\(885\) 0 0
\(886\) −35280.0 −1.33776
\(887\) −49407.4 −1.87028 −0.935140 0.354277i \(-0.884727\pi\)
−0.935140 + 0.354277i \(0.884727\pi\)
\(888\) 0 0
\(889\) 3900.00 0.147134
\(890\) 0 0
\(891\) 0 0
\(892\) −7070.00 −0.265382
\(893\) 16115.0 0.603882
\(894\) 0 0
\(895\) 0 0
\(896\) 26183.7 0.976266
\(897\) 0 0
\(898\) 29200.0 1.08510
\(899\) 9297.10 0.344912
\(900\) 0 0
\(901\) 7840.00 0.289887
\(902\) 88543.8 3.26850
\(903\) 0 0
\(904\) 30720.0 1.13023
\(905\) 0 0
\(906\) 0 0
\(907\) −17880.0 −0.654571 −0.327285 0.944926i \(-0.606134\pi\)
−0.327285 + 0.944926i \(0.606134\pi\)
\(908\) 177.088 0.00647231
\(909\) 0 0
\(910\) 0 0
\(911\) 17329.3 0.630236 0.315118 0.949053i \(-0.397956\pi\)
0.315118 + 0.949053i \(0.397956\pi\)
\(912\) 0 0
\(913\) 33600.0 1.21796
\(914\) −3257.15 −0.117874
\(915\) 0 0
\(916\) −10038.0 −0.362080
\(917\) 19922.3 0.717442
\(918\) 0 0
\(919\) −25829.0 −0.927117 −0.463558 0.886066i \(-0.653428\pi\)
−0.463558 + 0.886066i \(0.653428\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −29400.0 −1.05015
\(923\) 2213.59 0.0789397
\(924\) 0 0
\(925\) 0 0
\(926\) 37757.6 1.33995
\(927\) 0 0
\(928\) −5600.00 −0.198092
\(929\) −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
\(932\) 7209.99 0.253403
\(933\) 0 0
\(934\) 38360.0 1.34387
\(935\) 0 0
\(936\) 0 0
\(937\) −40005.0 −1.39478 −0.697389 0.716693i \(-0.745655\pi\)
−0.697389 + 0.716693i \(0.745655\pi\)
\(938\) −711.512 −0.0247673
\(939\) 0 0
\(940\) 0 0
\(941\) 30104.9 1.04292 0.521462 0.853275i \(-0.325387\pi\)
0.521462 + 0.853275i \(0.325387\pi\)
\(942\) 0 0
\(943\) −50400.0 −1.74046
\(944\) 67293.3 2.32014
\(945\) 0 0
\(946\) 67000.0 2.30270
\(947\) −43297.9 −1.48574 −0.742868 0.669437i \(-0.766535\pi\)
−0.742868 + 0.669437i \(0.766535\pi\)
\(948\) 0 0
\(949\) −2450.00 −0.0838044
\(950\) 0 0
\(951\) 0 0
\(952\) 25200.0 0.857917
\(953\) 53682.8 1.82472 0.912360 0.409390i \(-0.134258\pi\)
0.912360 + 0.409390i \(0.134258\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 885.438 0.0299551
\(957\) 0 0
\(958\) 8400.00 0.283290
\(959\) −39844.7 −1.34166
\(960\) 0 0
\(961\) −8182.00 −0.274647
\(962\) −40951.5 −1.37248
\(963\) 0 0
\(964\) −1246.00 −0.0416296
\(965\) 0 0
\(966\) 0 0
\(967\) 20540.0 0.683063 0.341531 0.939870i \(-0.389054\pi\)
0.341531 + 0.939870i \(0.389054\pi\)
\(968\) 50640.7 1.68146
\(969\) 0 0
\(970\) 0 0
\(971\) 10625.3 0.351164 0.175582 0.984465i \(-0.443819\pi\)
0.175582 + 0.984465i \(0.443819\pi\)
\(972\) 0 0
\(973\) 11760.0 0.387470
\(974\) 44667.2 1.46943
\(975\) 0 0
\(976\) −32452.0 −1.06431
\(977\) −28359.3 −0.928654 −0.464327 0.885664i \(-0.653704\pi\)
−0.464327 + 0.885664i \(0.653704\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 12800.0 0.415952
\(983\) −35063.3 −1.13769 −0.568844 0.822446i \(-0.692609\pi\)
−0.568844 + 0.822446i \(0.692609\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −17708.8 −0.571969
\(987\) 0 0
\(988\) −6370.00 −0.205118
\(989\) −38137.1 −1.22618
\(990\) 0 0
\(991\) −18283.0 −0.586053 −0.293027 0.956104i \(-0.594662\pi\)
−0.293027 + 0.956104i \(0.594662\pi\)
\(992\) −13015.9 −0.416589
\(993\) 0 0
\(994\) −3000.00 −0.0957286
\(995\) 0 0
\(996\) 0 0
\(997\) 13230.0 0.420259 0.210130 0.977674i \(-0.432611\pi\)
0.210130 + 0.977674i \(0.432611\pi\)
\(998\) −4044.55 −0.128285
\(999\) 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.a.l.1.1 2
3.2 odd 2 inner 225.4.a.l.1.2 yes 2
5.2 odd 4 225.4.b.i.199.1 4
5.3 odd 4 225.4.b.i.199.4 4
5.4 even 2 225.4.a.m.1.2 yes 2
15.2 even 4 225.4.b.i.199.3 4
15.8 even 4 225.4.b.i.199.2 4
15.14 odd 2 225.4.a.m.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.4.a.l.1.1 2 1.1 even 1 trivial
225.4.a.l.1.2 yes 2 3.2 odd 2 inner
225.4.a.m.1.1 yes 2 15.14 odd 2
225.4.a.m.1.2 yes 2 5.4 even 2
225.4.b.i.199.1 4 5.2 odd 4
225.4.b.i.199.2 4 15.8 even 4
225.4.b.i.199.3 4 15.2 even 4
225.4.b.i.199.4 4 5.3 odd 4