Properties

Label 1856.4.a.s.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.19816.1
Defining polynomial: \( x^{3} - x^{2} - 42x - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.53003\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.53003 q^{3} -20.9205 q^{5} -8.55839 q^{7} +15.6413 q^{9} +O(q^{10})\) \(q-6.53003 q^{3} -20.9205 q^{5} -8.55839 q^{7} +15.6413 q^{9} +10.8092 q^{11} -54.7046 q^{13} +136.611 q^{15} -106.127 q^{17} -113.636 q^{19} +55.8865 q^{21} +112.855 q^{23} +312.666 q^{25} +74.1729 q^{27} -29.0000 q^{29} +102.805 q^{31} -70.5845 q^{33} +179.045 q^{35} +105.665 q^{37} +357.223 q^{39} +216.958 q^{41} -102.230 q^{43} -327.222 q^{45} -455.212 q^{47} -269.754 q^{49} +693.011 q^{51} +593.714 q^{53} -226.134 q^{55} +742.048 q^{57} -558.141 q^{59} +473.986 q^{61} -133.864 q^{63} +1144.45 q^{65} +193.132 q^{67} -736.949 q^{69} +2.38155 q^{71} +119.013 q^{73} -2041.71 q^{75} -92.5096 q^{77} +964.306 q^{79} -906.665 q^{81} +1068.19 q^{83} +2220.22 q^{85} +189.371 q^{87} +772.544 q^{89} +468.184 q^{91} -671.318 q^{93} +2377.32 q^{95} +1344.03 q^{97} +169.070 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9} + 10 q^{11} + 4 q^{13} + 130 q^{15} - 66 q^{17} - 164 q^{19} + 88 q^{21} + 204 q^{23} + 79 q^{25} - 142 q^{27} - 87 q^{29} + 86 q^{31} - 130 q^{33} + 24 q^{35} + 42 q^{37} + 394 q^{39} + 562 q^{41} + 18 q^{43} - 422 q^{45} - 654 q^{47} + 539 q^{49} + 556 q^{51} - 712 q^{53} - 142 q^{55} + 828 q^{57} + 184 q^{59} - 322 q^{61} + 784 q^{63} + 1494 q^{65} - 228 q^{67} - 684 q^{69} + 52 q^{71} - 494 q^{73} - 3048 q^{75} - 872 q^{77} + 2110 q^{79} - 1513 q^{81} - 288 q^{83} + 2704 q^{85} - 58 q^{87} + 914 q^{89} - 2984 q^{91} + 62 q^{93} + 1900 q^{95} + 218 q^{97} - 304 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −6.53003 −1.25670 −0.628352 0.777929i \(-0.716270\pi\)
−0.628352 + 0.777929i \(0.716270\pi\)
\(4\) 0 0
\(5\) −20.9205 −1.87118 −0.935591 0.353085i \(-0.885133\pi\)
−0.935591 + 0.353085i \(0.885133\pi\)
\(6\) 0 0
\(7\) −8.55839 −0.462110 −0.231055 0.972941i \(-0.574218\pi\)
−0.231055 + 0.972941i \(0.574218\pi\)
\(8\) 0 0
\(9\) 15.6413 0.579306
\(10\) 0 0
\(11\) 10.8092 0.296282 0.148141 0.988966i \(-0.452671\pi\)
0.148141 + 0.988966i \(0.452671\pi\)
\(12\) 0 0
\(13\) −54.7046 −1.16710 −0.583551 0.812076i \(-0.698337\pi\)
−0.583551 + 0.812076i \(0.698337\pi\)
\(14\) 0 0
\(15\) 136.611 2.35152
\(16\) 0 0
\(17\) −106.127 −1.51409 −0.757045 0.653363i \(-0.773358\pi\)
−0.757045 + 0.653363i \(0.773358\pi\)
\(18\) 0 0
\(19\) −113.636 −1.37210 −0.686051 0.727554i \(-0.740657\pi\)
−0.686051 + 0.727554i \(0.740657\pi\)
\(20\) 0 0
\(21\) 55.8865 0.580735
\(22\) 0 0
\(23\) 112.855 1.02313 0.511564 0.859245i \(-0.329066\pi\)
0.511564 + 0.859245i \(0.329066\pi\)
\(24\) 0 0
\(25\) 312.666 2.50132
\(26\) 0 0
\(27\) 74.1729 0.528688
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 102.805 0.595622 0.297811 0.954625i \(-0.403743\pi\)
0.297811 + 0.954625i \(0.403743\pi\)
\(32\) 0 0
\(33\) −70.5845 −0.372339
\(34\) 0 0
\(35\) 179.045 0.864692
\(36\) 0 0
\(37\) 105.665 0.469493 0.234746 0.972057i \(-0.424574\pi\)
0.234746 + 0.972057i \(0.424574\pi\)
\(38\) 0 0
\(39\) 357.223 1.46670
\(40\) 0 0
\(41\) 216.958 0.826417 0.413209 0.910636i \(-0.364408\pi\)
0.413209 + 0.910636i \(0.364408\pi\)
\(42\) 0 0
\(43\) −102.230 −0.362555 −0.181278 0.983432i \(-0.558023\pi\)
−0.181278 + 0.983432i \(0.558023\pi\)
\(44\) 0 0
\(45\) −327.222 −1.08399
\(46\) 0 0
\(47\) −455.212 −1.41275 −0.706377 0.707836i \(-0.749672\pi\)
−0.706377 + 0.707836i \(0.749672\pi\)
\(48\) 0 0
\(49\) −269.754 −0.786455
\(50\) 0 0
\(51\) 693.011 1.90276
\(52\) 0 0
\(53\) 593.714 1.53873 0.769367 0.638807i \(-0.220572\pi\)
0.769367 + 0.638807i \(0.220572\pi\)
\(54\) 0 0
\(55\) −226.134 −0.554398
\(56\) 0 0
\(57\) 742.048 1.72433
\(58\) 0 0
\(59\) −558.141 −1.23159 −0.615794 0.787907i \(-0.711165\pi\)
−0.615794 + 0.787907i \(0.711165\pi\)
\(60\) 0 0
\(61\) 473.986 0.994880 0.497440 0.867498i \(-0.334273\pi\)
0.497440 + 0.867498i \(0.334273\pi\)
\(62\) 0 0
\(63\) −133.864 −0.267703
\(64\) 0 0
\(65\) 1144.45 2.18386
\(66\) 0 0
\(67\) 193.132 0.352162 0.176081 0.984376i \(-0.443658\pi\)
0.176081 + 0.984376i \(0.443658\pi\)
\(68\) 0 0
\(69\) −736.949 −1.28577
\(70\) 0 0
\(71\) 2.38155 0.00398082 0.00199041 0.999998i \(-0.499366\pi\)
0.00199041 + 0.999998i \(0.499366\pi\)
\(72\) 0 0
\(73\) 119.013 0.190814 0.0954071 0.995438i \(-0.469585\pi\)
0.0954071 + 0.995438i \(0.469585\pi\)
\(74\) 0 0
\(75\) −2041.71 −3.14343
\(76\) 0 0
\(77\) −92.5096 −0.136915
\(78\) 0 0
\(79\) 964.306 1.37333 0.686664 0.726975i \(-0.259074\pi\)
0.686664 + 0.726975i \(0.259074\pi\)
\(80\) 0 0
\(81\) −906.665 −1.24371
\(82\) 0 0
\(83\) 1068.19 1.41264 0.706319 0.707893i \(-0.250354\pi\)
0.706319 + 0.707893i \(0.250354\pi\)
\(84\) 0 0
\(85\) 2220.22 2.83314
\(86\) 0 0
\(87\) 189.371 0.233364
\(88\) 0 0
\(89\) 772.544 0.920106 0.460053 0.887891i \(-0.347830\pi\)
0.460053 + 0.887891i \(0.347830\pi\)
\(90\) 0 0
\(91\) 468.184 0.539330
\(92\) 0 0
\(93\) −671.318 −0.748521
\(94\) 0 0
\(95\) 2377.32 2.56745
\(96\) 0 0
\(97\) 1344.03 1.40686 0.703431 0.710763i \(-0.251650\pi\)
0.703431 + 0.710763i \(0.251650\pi\)
\(98\) 0 0
\(99\) 169.070 0.171638
\(100\) 0 0
\(101\) −986.733 −0.972115 −0.486057 0.873927i \(-0.661565\pi\)
−0.486057 + 0.873927i \(0.661565\pi\)
\(102\) 0 0
\(103\) −548.272 −0.524493 −0.262247 0.965001i \(-0.584463\pi\)
−0.262247 + 0.965001i \(0.584463\pi\)
\(104\) 0 0
\(105\) −1169.17 −1.08666
\(106\) 0 0
\(107\) 1387.51 1.25361 0.626803 0.779178i \(-0.284363\pi\)
0.626803 + 0.779178i \(0.284363\pi\)
\(108\) 0 0
\(109\) 1293.32 1.13649 0.568246 0.822859i \(-0.307622\pi\)
0.568246 + 0.822859i \(0.307622\pi\)
\(110\) 0 0
\(111\) −689.996 −0.590014
\(112\) 0 0
\(113\) 302.883 0.252149 0.126075 0.992021i \(-0.459762\pi\)
0.126075 + 0.992021i \(0.459762\pi\)
\(114\) 0 0
\(115\) −2360.99 −1.91446
\(116\) 0 0
\(117\) −855.650 −0.676110
\(118\) 0 0
\(119\) 908.274 0.699676
\(120\) 0 0
\(121\) −1214.16 −0.912217
\(122\) 0 0
\(123\) −1416.74 −1.03856
\(124\) 0 0
\(125\) −3926.05 −2.80925
\(126\) 0 0
\(127\) −2021.68 −1.41256 −0.706281 0.707931i \(-0.749629\pi\)
−0.706281 + 0.707931i \(0.749629\pi\)
\(128\) 0 0
\(129\) 667.562 0.455625
\(130\) 0 0
\(131\) 854.726 0.570059 0.285030 0.958519i \(-0.407997\pi\)
0.285030 + 0.958519i \(0.407997\pi\)
\(132\) 0 0
\(133\) 972.543 0.634062
\(134\) 0 0
\(135\) −1551.73 −0.989272
\(136\) 0 0
\(137\) 365.024 0.227636 0.113818 0.993502i \(-0.463692\pi\)
0.113818 + 0.993502i \(0.463692\pi\)
\(138\) 0 0
\(139\) −1010.10 −0.616370 −0.308185 0.951326i \(-0.599722\pi\)
−0.308185 + 0.951326i \(0.599722\pi\)
\(140\) 0 0
\(141\) 2972.55 1.77541
\(142\) 0 0
\(143\) −591.315 −0.345792
\(144\) 0 0
\(145\) 606.693 0.347470
\(146\) 0 0
\(147\) 1761.50 0.988341
\(148\) 0 0
\(149\) −819.765 −0.450723 −0.225362 0.974275i \(-0.572356\pi\)
−0.225362 + 0.974275i \(0.572356\pi\)
\(150\) 0 0
\(151\) −1000.25 −0.539068 −0.269534 0.962991i \(-0.586870\pi\)
−0.269534 + 0.962991i \(0.586870\pi\)
\(152\) 0 0
\(153\) −1659.96 −0.877121
\(154\) 0 0
\(155\) −2150.72 −1.11452
\(156\) 0 0
\(157\) 1702.38 0.865382 0.432691 0.901542i \(-0.357564\pi\)
0.432691 + 0.901542i \(0.357564\pi\)
\(158\) 0 0
\(159\) −3876.97 −1.93373
\(160\) 0 0
\(161\) −965.860 −0.472798
\(162\) 0 0
\(163\) −3451.76 −1.65867 −0.829334 0.558753i \(-0.811280\pi\)
−0.829334 + 0.558753i \(0.811280\pi\)
\(164\) 0 0
\(165\) 1476.66 0.696714
\(166\) 0 0
\(167\) −1409.23 −0.652990 −0.326495 0.945199i \(-0.605868\pi\)
−0.326495 + 0.945199i \(0.605868\pi\)
\(168\) 0 0
\(169\) 795.598 0.362129
\(170\) 0 0
\(171\) −1777.41 −0.794867
\(172\) 0 0
\(173\) −1358.39 −0.596973 −0.298487 0.954414i \(-0.596482\pi\)
−0.298487 + 0.954414i \(0.596482\pi\)
\(174\) 0 0
\(175\) −2675.91 −1.15589
\(176\) 0 0
\(177\) 3644.67 1.54774
\(178\) 0 0
\(179\) −1702.56 −0.710924 −0.355462 0.934691i \(-0.615677\pi\)
−0.355462 + 0.934691i \(0.615677\pi\)
\(180\) 0 0
\(181\) 7.33698 0.00301300 0.00150650 0.999999i \(-0.499520\pi\)
0.00150650 + 0.999999i \(0.499520\pi\)
\(182\) 0 0
\(183\) −3095.14 −1.25027
\(184\) 0 0
\(185\) −2210.56 −0.878507
\(186\) 0 0
\(187\) −1147.15 −0.448598
\(188\) 0 0
\(189\) −634.801 −0.244312
\(190\) 0 0
\(191\) −1324.78 −0.501873 −0.250936 0.968004i \(-0.580738\pi\)
−0.250936 + 0.968004i \(0.580738\pi\)
\(192\) 0 0
\(193\) 1834.47 0.684187 0.342094 0.939666i \(-0.388864\pi\)
0.342094 + 0.939666i \(0.388864\pi\)
\(194\) 0 0
\(195\) −7473.26 −2.74447
\(196\) 0 0
\(197\) 4949.99 1.79021 0.895107 0.445851i \(-0.147099\pi\)
0.895107 + 0.445851i \(0.147099\pi\)
\(198\) 0 0
\(199\) −3554.04 −1.26603 −0.633014 0.774140i \(-0.718182\pi\)
−0.633014 + 0.774140i \(0.718182\pi\)
\(200\) 0 0
\(201\) −1261.16 −0.442563
\(202\) 0 0
\(203\) 248.193 0.0858116
\(204\) 0 0
\(205\) −4538.85 −1.54638
\(206\) 0 0
\(207\) 1765.20 0.592705
\(208\) 0 0
\(209\) −1228.32 −0.406529
\(210\) 0 0
\(211\) 2475.23 0.807590 0.403795 0.914849i \(-0.367691\pi\)
0.403795 + 0.914849i \(0.367691\pi\)
\(212\) 0 0
\(213\) −15.5516 −0.00500271
\(214\) 0 0
\(215\) 2138.69 0.678407
\(216\) 0 0
\(217\) −879.844 −0.275243
\(218\) 0 0
\(219\) −777.159 −0.239797
\(220\) 0 0
\(221\) 5805.63 1.76710
\(222\) 0 0
\(223\) −2381.72 −0.715211 −0.357605 0.933873i \(-0.616407\pi\)
−0.357605 + 0.933873i \(0.616407\pi\)
\(224\) 0 0
\(225\) 4890.48 1.44903
\(226\) 0 0
\(227\) 5452.74 1.59432 0.797160 0.603768i \(-0.206334\pi\)
0.797160 + 0.603768i \(0.206334\pi\)
\(228\) 0 0
\(229\) −596.232 −0.172053 −0.0860264 0.996293i \(-0.527417\pi\)
−0.0860264 + 0.996293i \(0.527417\pi\)
\(230\) 0 0
\(231\) 604.090 0.172062
\(232\) 0 0
\(233\) −5623.04 −1.58102 −0.790509 0.612450i \(-0.790184\pi\)
−0.790509 + 0.612450i \(0.790184\pi\)
\(234\) 0 0
\(235\) 9523.24 2.64352
\(236\) 0 0
\(237\) −6296.95 −1.72587
\(238\) 0 0
\(239\) −1564.27 −0.423366 −0.211683 0.977338i \(-0.567894\pi\)
−0.211683 + 0.977338i \(0.567894\pi\)
\(240\) 0 0
\(241\) −730.326 −0.195205 −0.0976026 0.995225i \(-0.531117\pi\)
−0.0976026 + 0.995225i \(0.531117\pi\)
\(242\) 0 0
\(243\) 3917.88 1.03429
\(244\) 0 0
\(245\) 5643.38 1.47160
\(246\) 0 0
\(247\) 6216.43 1.60138
\(248\) 0 0
\(249\) −6975.31 −1.77527
\(250\) 0 0
\(251\) −4244.39 −1.06735 −0.533673 0.845691i \(-0.679189\pi\)
−0.533673 + 0.845691i \(0.679189\pi\)
\(252\) 0 0
\(253\) 1219.88 0.303135
\(254\) 0 0
\(255\) −14498.1 −3.56042
\(256\) 0 0
\(257\) −235.374 −0.0571292 −0.0285646 0.999592i \(-0.509094\pi\)
−0.0285646 + 0.999592i \(0.509094\pi\)
\(258\) 0 0
\(259\) −904.323 −0.216957
\(260\) 0 0
\(261\) −453.597 −0.107574
\(262\) 0 0
\(263\) −3453.63 −0.809734 −0.404867 0.914376i \(-0.632682\pi\)
−0.404867 + 0.914376i \(0.632682\pi\)
\(264\) 0 0
\(265\) −12420.8 −2.87925
\(266\) 0 0
\(267\) −5044.73 −1.15630
\(268\) 0 0
\(269\) 1921.31 0.435481 0.217741 0.976007i \(-0.430131\pi\)
0.217741 + 0.976007i \(0.430131\pi\)
\(270\) 0 0
\(271\) 2480.09 0.555921 0.277961 0.960592i \(-0.410342\pi\)
0.277961 + 0.960592i \(0.410342\pi\)
\(272\) 0 0
\(273\) −3057.25 −0.677778
\(274\) 0 0
\(275\) 3379.67 0.741098
\(276\) 0 0
\(277\) 4766.90 1.03399 0.516995 0.855989i \(-0.327051\pi\)
0.516995 + 0.855989i \(0.327051\pi\)
\(278\) 0 0
\(279\) 1608.00 0.345047
\(280\) 0 0
\(281\) 194.329 0.0412552 0.0206276 0.999787i \(-0.493434\pi\)
0.0206276 + 0.999787i \(0.493434\pi\)
\(282\) 0 0
\(283\) −2817.42 −0.591797 −0.295898 0.955219i \(-0.595619\pi\)
−0.295898 + 0.955219i \(0.595619\pi\)
\(284\) 0 0
\(285\) −15524.0 −3.22653
\(286\) 0 0
\(287\) −1856.81 −0.381895
\(288\) 0 0
\(289\) 6349.89 1.29247
\(290\) 0 0
\(291\) −8776.56 −1.76801
\(292\) 0 0
\(293\) −4059.12 −0.809339 −0.404670 0.914463i \(-0.632614\pi\)
−0.404670 + 0.914463i \(0.632614\pi\)
\(294\) 0 0
\(295\) 11676.6 2.30453
\(296\) 0 0
\(297\) 801.751 0.156641
\(298\) 0 0
\(299\) −6173.71 −1.19410
\(300\) 0 0
\(301\) 874.921 0.167540
\(302\) 0 0
\(303\) 6443.39 1.22166
\(304\) 0 0
\(305\) −9916.01 −1.86160
\(306\) 0 0
\(307\) −131.572 −0.0244600 −0.0122300 0.999925i \(-0.503893\pi\)
−0.0122300 + 0.999925i \(0.503893\pi\)
\(308\) 0 0
\(309\) 3580.23 0.659133
\(310\) 0 0
\(311\) −1517.84 −0.276748 −0.138374 0.990380i \(-0.544188\pi\)
−0.138374 + 0.990380i \(0.544188\pi\)
\(312\) 0 0
\(313\) 4244.17 0.766436 0.383218 0.923658i \(-0.374816\pi\)
0.383218 + 0.923658i \(0.374816\pi\)
\(314\) 0 0
\(315\) 2800.50 0.500921
\(316\) 0 0
\(317\) −5596.56 −0.991590 −0.495795 0.868439i \(-0.665123\pi\)
−0.495795 + 0.868439i \(0.665123\pi\)
\(318\) 0 0
\(319\) −313.467 −0.0550182
\(320\) 0 0
\(321\) −9060.50 −1.57541
\(322\) 0 0
\(323\) 12059.8 2.07748
\(324\) 0 0
\(325\) −17104.3 −2.91930
\(326\) 0 0
\(327\) −8445.42 −1.42824
\(328\) 0 0
\(329\) 3895.88 0.652848
\(330\) 0 0
\(331\) −8383.85 −1.39220 −0.696100 0.717945i \(-0.745083\pi\)
−0.696100 + 0.717945i \(0.745083\pi\)
\(332\) 0 0
\(333\) 1652.73 0.271980
\(334\) 0 0
\(335\) −4040.41 −0.658959
\(336\) 0 0
\(337\) 9887.12 1.59818 0.799089 0.601213i \(-0.205316\pi\)
0.799089 + 0.601213i \(0.205316\pi\)
\(338\) 0 0
\(339\) −1977.84 −0.316877
\(340\) 0 0
\(341\) 1111.24 0.176472
\(342\) 0 0
\(343\) 5244.19 0.825538
\(344\) 0 0
\(345\) 15417.3 2.40591
\(346\) 0 0
\(347\) 10678.8 1.65206 0.826032 0.563623i \(-0.190593\pi\)
0.826032 + 0.563623i \(0.190593\pi\)
\(348\) 0 0
\(349\) 1457.88 0.223605 0.111803 0.993730i \(-0.464338\pi\)
0.111803 + 0.993730i \(0.464338\pi\)
\(350\) 0 0
\(351\) −4057.60 −0.617033
\(352\) 0 0
\(353\) 6737.20 1.01582 0.507911 0.861410i \(-0.330418\pi\)
0.507911 + 0.861410i \(0.330418\pi\)
\(354\) 0 0
\(355\) −49.8232 −0.00744884
\(356\) 0 0
\(357\) −5931.06 −0.879285
\(358\) 0 0
\(359\) 3539.85 0.520407 0.260204 0.965554i \(-0.416210\pi\)
0.260204 + 0.965554i \(0.416210\pi\)
\(360\) 0 0
\(361\) 6054.19 0.882663
\(362\) 0 0
\(363\) 7928.50 1.14639
\(364\) 0 0
\(365\) −2489.81 −0.357048
\(366\) 0 0
\(367\) 4917.53 0.699436 0.349718 0.936855i \(-0.386277\pi\)
0.349718 + 0.936855i \(0.386277\pi\)
\(368\) 0 0
\(369\) 3393.49 0.478748
\(370\) 0 0
\(371\) −5081.24 −0.711064
\(372\) 0 0
\(373\) −2032.31 −0.282115 −0.141057 0.990001i \(-0.545050\pi\)
−0.141057 + 0.990001i \(0.545050\pi\)
\(374\) 0 0
\(375\) 25637.2 3.53040
\(376\) 0 0
\(377\) 1586.43 0.216726
\(378\) 0 0
\(379\) −7051.47 −0.955699 −0.477849 0.878442i \(-0.658584\pi\)
−0.477849 + 0.878442i \(0.658584\pi\)
\(380\) 0 0
\(381\) 13201.7 1.77517
\(382\) 0 0
\(383\) 9334.72 1.24538 0.622691 0.782467i \(-0.286039\pi\)
0.622691 + 0.782467i \(0.286039\pi\)
\(384\) 0 0
\(385\) 1935.34 0.256193
\(386\) 0 0
\(387\) −1599.00 −0.210030
\(388\) 0 0
\(389\) 1901.32 0.247816 0.123908 0.992294i \(-0.460457\pi\)
0.123908 + 0.992294i \(0.460457\pi\)
\(390\) 0 0
\(391\) −11977.0 −1.54911
\(392\) 0 0
\(393\) −5581.38 −0.716396
\(394\) 0 0
\(395\) −20173.7 −2.56975
\(396\) 0 0
\(397\) −1995.81 −0.252309 −0.126155 0.992011i \(-0.540264\pi\)
−0.126155 + 0.992011i \(0.540264\pi\)
\(398\) 0 0
\(399\) −6350.73 −0.796828
\(400\) 0 0
\(401\) 12920.6 1.60904 0.804520 0.593925i \(-0.202422\pi\)
0.804520 + 0.593925i \(0.202422\pi\)
\(402\) 0 0
\(403\) −5623.90 −0.695152
\(404\) 0 0
\(405\) 18967.8 2.32721
\(406\) 0 0
\(407\) 1142.16 0.139102
\(408\) 0 0
\(409\) −9713.54 −1.17434 −0.587168 0.809465i \(-0.699757\pi\)
−0.587168 + 0.809465i \(0.699757\pi\)
\(410\) 0 0
\(411\) −2383.62 −0.286071
\(412\) 0 0
\(413\) 4776.79 0.569129
\(414\) 0 0
\(415\) −22347.0 −2.64330
\(416\) 0 0
\(417\) 6595.97 0.774595
\(418\) 0 0
\(419\) 15925.4 1.85682 0.928411 0.371555i \(-0.121175\pi\)
0.928411 + 0.371555i \(0.121175\pi\)
\(420\) 0 0
\(421\) −10849.9 −1.25604 −0.628019 0.778198i \(-0.716134\pi\)
−0.628019 + 0.778198i \(0.716134\pi\)
\(422\) 0 0
\(423\) −7120.09 −0.818417
\(424\) 0 0
\(425\) −33182.2 −3.78723
\(426\) 0 0
\(427\) −4056.56 −0.459744
\(428\) 0 0
\(429\) 3861.30 0.434558
\(430\) 0 0
\(431\) 532.335 0.0594934 0.0297467 0.999557i \(-0.490530\pi\)
0.0297467 + 0.999557i \(0.490530\pi\)
\(432\) 0 0
\(433\) −6995.94 −0.776451 −0.388225 0.921564i \(-0.626912\pi\)
−0.388225 + 0.921564i \(0.626912\pi\)
\(434\) 0 0
\(435\) −3961.72 −0.436667
\(436\) 0 0
\(437\) −12824.5 −1.40384
\(438\) 0 0
\(439\) −3272.22 −0.355750 −0.177875 0.984053i \(-0.556922\pi\)
−0.177875 + 0.984053i \(0.556922\pi\)
\(440\) 0 0
\(441\) −4219.29 −0.455598
\(442\) 0 0
\(443\) −3818.14 −0.409493 −0.204746 0.978815i \(-0.565637\pi\)
−0.204746 + 0.978815i \(0.565637\pi\)
\(444\) 0 0
\(445\) −16162.0 −1.72169
\(446\) 0 0
\(447\) 5353.09 0.566426
\(448\) 0 0
\(449\) −4323.19 −0.454396 −0.227198 0.973849i \(-0.572957\pi\)
−0.227198 + 0.973849i \(0.572957\pi\)
\(450\) 0 0
\(451\) 2345.14 0.244853
\(452\) 0 0
\(453\) 6531.66 0.677449
\(454\) 0 0
\(455\) −9794.62 −1.00918
\(456\) 0 0
\(457\) −8367.43 −0.856481 −0.428240 0.903665i \(-0.640866\pi\)
−0.428240 + 0.903665i \(0.640866\pi\)
\(458\) 0 0
\(459\) −7871.73 −0.800481
\(460\) 0 0
\(461\) 17249.0 1.74266 0.871328 0.490701i \(-0.163259\pi\)
0.871328 + 0.490701i \(0.163259\pi\)
\(462\) 0 0
\(463\) 16774.3 1.68373 0.841864 0.539690i \(-0.181458\pi\)
0.841864 + 0.539690i \(0.181458\pi\)
\(464\) 0 0
\(465\) 14044.3 1.40062
\(466\) 0 0
\(467\) 7701.05 0.763088 0.381544 0.924351i \(-0.375392\pi\)
0.381544 + 0.924351i \(0.375392\pi\)
\(468\) 0 0
\(469\) −1652.90 −0.162737
\(470\) 0 0
\(471\) −11116.6 −1.08753
\(472\) 0 0
\(473\) −1105.02 −0.107419
\(474\) 0 0
\(475\) −35530.1 −3.43207
\(476\) 0 0
\(477\) 9286.44 0.891398
\(478\) 0 0
\(479\) −5988.77 −0.571261 −0.285630 0.958340i \(-0.592203\pi\)
−0.285630 + 0.958340i \(0.592203\pi\)
\(480\) 0 0
\(481\) −5780.37 −0.547946
\(482\) 0 0
\(483\) 6307.09 0.594167
\(484\) 0 0
\(485\) −28117.7 −2.63250
\(486\) 0 0
\(487\) −5790.34 −0.538779 −0.269390 0.963031i \(-0.586822\pi\)
−0.269390 + 0.963031i \(0.586822\pi\)
\(488\) 0 0
\(489\) 22540.1 2.08446
\(490\) 0 0
\(491\) 19228.8 1.76738 0.883692 0.468069i \(-0.155050\pi\)
0.883692 + 0.468069i \(0.155050\pi\)
\(492\) 0 0
\(493\) 3077.68 0.281159
\(494\) 0 0
\(495\) −3537.02 −0.321166
\(496\) 0 0
\(497\) −20.3823 −0.00183958
\(498\) 0 0
\(499\) −7081.65 −0.635307 −0.317653 0.948207i \(-0.602895\pi\)
−0.317653 + 0.948207i \(0.602895\pi\)
\(500\) 0 0
\(501\) 9202.29 0.820615
\(502\) 0 0
\(503\) 5312.64 0.470932 0.235466 0.971883i \(-0.424338\pi\)
0.235466 + 0.971883i \(0.424338\pi\)
\(504\) 0 0
\(505\) 20642.9 1.81900
\(506\) 0 0
\(507\) −5195.28 −0.455089
\(508\) 0 0
\(509\) 13862.7 1.20717 0.603587 0.797297i \(-0.293738\pi\)
0.603587 + 0.797297i \(0.293738\pi\)
\(510\) 0 0
\(511\) −1018.56 −0.0881771
\(512\) 0 0
\(513\) −8428.72 −0.725414
\(514\) 0 0
\(515\) 11470.1 0.981423
\(516\) 0 0
\(517\) −4920.49 −0.418574
\(518\) 0 0
\(519\) 8870.32 0.750219
\(520\) 0 0
\(521\) 15447.5 1.29897 0.649487 0.760373i \(-0.274984\pi\)
0.649487 + 0.760373i \(0.274984\pi\)
\(522\) 0 0
\(523\) 4349.54 0.363656 0.181828 0.983330i \(-0.441799\pi\)
0.181828 + 0.983330i \(0.441799\pi\)
\(524\) 0 0
\(525\) 17473.8 1.45261
\(526\) 0 0
\(527\) −10910.3 −0.901825
\(528\) 0 0
\(529\) 569.329 0.0467928
\(530\) 0 0
\(531\) −8730.03 −0.713467
\(532\) 0 0
\(533\) −11868.6 −0.964514
\(534\) 0 0
\(535\) −29027.4 −2.34573
\(536\) 0 0
\(537\) 11117.8 0.893422
\(538\) 0 0
\(539\) −2915.83 −0.233012
\(540\) 0 0
\(541\) −3206.84 −0.254848 −0.127424 0.991848i \(-0.540671\pi\)
−0.127424 + 0.991848i \(0.540671\pi\)
\(542\) 0 0
\(543\) −47.9107 −0.00378645
\(544\) 0 0
\(545\) −27056.9 −2.12658
\(546\) 0 0
\(547\) −3289.81 −0.257152 −0.128576 0.991700i \(-0.541041\pi\)
−0.128576 + 0.991700i \(0.541041\pi\)
\(548\) 0 0
\(549\) 7413.74 0.576340
\(550\) 0 0
\(551\) 3295.45 0.254793
\(552\) 0 0
\(553\) −8252.91 −0.634628
\(554\) 0 0
\(555\) 14435.0 1.10402
\(556\) 0 0
\(557\) 313.140 0.0238207 0.0119104 0.999929i \(-0.496209\pi\)
0.0119104 + 0.999929i \(0.496209\pi\)
\(558\) 0 0
\(559\) 5592.43 0.423139
\(560\) 0 0
\(561\) 7490.91 0.563755
\(562\) 0 0
\(563\) −3425.84 −0.256451 −0.128225 0.991745i \(-0.540928\pi\)
−0.128225 + 0.991745i \(0.540928\pi\)
\(564\) 0 0
\(565\) −6336.46 −0.471818
\(566\) 0 0
\(567\) 7759.60 0.574731
\(568\) 0 0
\(569\) 17763.8 1.30878 0.654390 0.756158i \(-0.272926\pi\)
0.654390 + 0.756158i \(0.272926\pi\)
\(570\) 0 0
\(571\) 5741.80 0.420818 0.210409 0.977613i \(-0.432520\pi\)
0.210409 + 0.977613i \(0.432520\pi\)
\(572\) 0 0
\(573\) 8650.85 0.630706
\(574\) 0 0
\(575\) 35286.0 2.55918
\(576\) 0 0
\(577\) 14477.5 1.04455 0.522275 0.852777i \(-0.325083\pi\)
0.522275 + 0.852777i \(0.325083\pi\)
\(578\) 0 0
\(579\) −11979.1 −0.859821
\(580\) 0 0
\(581\) −9141.98 −0.652794
\(582\) 0 0
\(583\) 6417.59 0.455899
\(584\) 0 0
\(585\) 17900.6 1.26512
\(586\) 0 0
\(587\) −18082.8 −1.27148 −0.635738 0.771905i \(-0.719304\pi\)
−0.635738 + 0.771905i \(0.719304\pi\)
\(588\) 0 0
\(589\) −11682.3 −0.817254
\(590\) 0 0
\(591\) −32323.6 −2.24977
\(592\) 0 0
\(593\) 1731.57 0.119911 0.0599553 0.998201i \(-0.480904\pi\)
0.0599553 + 0.998201i \(0.480904\pi\)
\(594\) 0 0
\(595\) −19001.5 −1.30922
\(596\) 0 0
\(597\) 23208.0 1.59102
\(598\) 0 0
\(599\) 11480.5 0.783105 0.391552 0.920156i \(-0.371938\pi\)
0.391552 + 0.920156i \(0.371938\pi\)
\(600\) 0 0
\(601\) 2924.56 0.198495 0.0992474 0.995063i \(-0.468356\pi\)
0.0992474 + 0.995063i \(0.468356\pi\)
\(602\) 0 0
\(603\) 3020.83 0.204009
\(604\) 0 0
\(605\) 25400.8 1.70692
\(606\) 0 0
\(607\) −3586.09 −0.239794 −0.119897 0.992786i \(-0.538256\pi\)
−0.119897 + 0.992786i \(0.538256\pi\)
\(608\) 0 0
\(609\) −1620.71 −0.107840
\(610\) 0 0
\(611\) 24902.2 1.64883
\(612\) 0 0
\(613\) 11679.8 0.769561 0.384781 0.923008i \(-0.374277\pi\)
0.384781 + 0.923008i \(0.374277\pi\)
\(614\) 0 0
\(615\) 29638.8 1.94334
\(616\) 0 0
\(617\) 11063.0 0.721847 0.360924 0.932595i \(-0.382461\pi\)
0.360924 + 0.932595i \(0.382461\pi\)
\(618\) 0 0
\(619\) −2463.60 −0.159969 −0.0799843 0.996796i \(-0.525487\pi\)
−0.0799843 + 0.996796i \(0.525487\pi\)
\(620\) 0 0
\(621\) 8370.81 0.540916
\(622\) 0 0
\(623\) −6611.73 −0.425190
\(624\) 0 0
\(625\) 43051.5 2.75530
\(626\) 0 0
\(627\) 8020.96 0.510887
\(628\) 0 0
\(629\) −11213.9 −0.710854
\(630\) 0 0
\(631\) −1032.43 −0.0651352 −0.0325676 0.999470i \(-0.510368\pi\)
−0.0325676 + 0.999470i \(0.510368\pi\)
\(632\) 0 0
\(633\) −16163.3 −1.01490
\(634\) 0 0
\(635\) 42294.6 2.64316
\(636\) 0 0
\(637\) 14756.8 0.917873
\(638\) 0 0
\(639\) 37.2505 0.00230611
\(640\) 0 0
\(641\) 12841.6 0.791282 0.395641 0.918405i \(-0.370522\pi\)
0.395641 + 0.918405i \(0.370522\pi\)
\(642\) 0 0
\(643\) 8449.14 0.518198 0.259099 0.965851i \(-0.416574\pi\)
0.259099 + 0.965851i \(0.416574\pi\)
\(644\) 0 0
\(645\) −13965.7 −0.852557
\(646\) 0 0
\(647\) 27036.8 1.64285 0.821426 0.570315i \(-0.193179\pi\)
0.821426 + 0.570315i \(0.193179\pi\)
\(648\) 0 0
\(649\) −6033.07 −0.364898
\(650\) 0 0
\(651\) 5745.40 0.345899
\(652\) 0 0
\(653\) 27105.2 1.62436 0.812181 0.583405i \(-0.198280\pi\)
0.812181 + 0.583405i \(0.198280\pi\)
\(654\) 0 0
\(655\) −17881.3 −1.06668
\(656\) 0 0
\(657\) 1861.52 0.110540
\(658\) 0 0
\(659\) −22622.4 −1.33724 −0.668621 0.743603i \(-0.733115\pi\)
−0.668621 + 0.743603i \(0.733115\pi\)
\(660\) 0 0
\(661\) 21000.2 1.23572 0.617862 0.786287i \(-0.287999\pi\)
0.617862 + 0.786287i \(0.287999\pi\)
\(662\) 0 0
\(663\) −37910.9 −2.22072
\(664\) 0 0
\(665\) −20346.0 −1.18645
\(666\) 0 0
\(667\) −3272.80 −0.189990
\(668\) 0 0
\(669\) 15552.7 0.898809
\(670\) 0 0
\(671\) 5123.42 0.294765
\(672\) 0 0
\(673\) −13691.2 −0.784186 −0.392093 0.919926i \(-0.628249\pi\)
−0.392093 + 0.919926i \(0.628249\pi\)
\(674\) 0 0
\(675\) 23191.3 1.32242
\(676\) 0 0
\(677\) −9694.60 −0.550360 −0.275180 0.961393i \(-0.588737\pi\)
−0.275180 + 0.961393i \(0.588737\pi\)
\(678\) 0 0
\(679\) −11502.7 −0.650125
\(680\) 0 0
\(681\) −35606.5 −2.00359
\(682\) 0 0
\(683\) 6012.25 0.336826 0.168413 0.985717i \(-0.446136\pi\)
0.168413 + 0.985717i \(0.446136\pi\)
\(684\) 0 0
\(685\) −7636.47 −0.425948
\(686\) 0 0
\(687\) 3893.41 0.216220
\(688\) 0 0
\(689\) −32478.9 −1.79586
\(690\) 0 0
\(691\) −10304.9 −0.567317 −0.283659 0.958925i \(-0.591548\pi\)
−0.283659 + 0.958925i \(0.591548\pi\)
\(692\) 0 0
\(693\) −1446.97 −0.0793156
\(694\) 0 0
\(695\) 21131.7 1.15334
\(696\) 0 0
\(697\) −23025.0 −1.25127
\(698\) 0 0
\(699\) 36718.6 1.98687
\(700\) 0 0
\(701\) −11785.8 −0.635011 −0.317506 0.948256i \(-0.602845\pi\)
−0.317506 + 0.948256i \(0.602845\pi\)
\(702\) 0 0
\(703\) −12007.4 −0.644192
\(704\) 0 0
\(705\) −62187.0 −3.32212
\(706\) 0 0
\(707\) 8444.85 0.449224
\(708\) 0 0
\(709\) −29226.1 −1.54811 −0.774054 0.633120i \(-0.781774\pi\)
−0.774054 + 0.633120i \(0.781774\pi\)
\(710\) 0 0
\(711\) 15083.0 0.795577
\(712\) 0 0
\(713\) 11602.1 0.609398
\(714\) 0 0
\(715\) 12370.6 0.647040
\(716\) 0 0
\(717\) 10214.7 0.532045
\(718\) 0 0
\(719\) −14706.5 −0.762809 −0.381404 0.924408i \(-0.624559\pi\)
−0.381404 + 0.924408i \(0.624559\pi\)
\(720\) 0 0
\(721\) 4692.32 0.242374
\(722\) 0 0
\(723\) 4769.05 0.245315
\(724\) 0 0
\(725\) −9067.30 −0.464484
\(726\) 0 0
\(727\) 35975.1 1.83527 0.917637 0.397419i \(-0.130094\pi\)
0.917637 + 0.397419i \(0.130094\pi\)
\(728\) 0 0
\(729\) −1103.91 −0.0560843
\(730\) 0 0
\(731\) 10849.3 0.548941
\(732\) 0 0
\(733\) −10872.8 −0.547879 −0.273939 0.961747i \(-0.588327\pi\)
−0.273939 + 0.961747i \(0.588327\pi\)
\(734\) 0 0
\(735\) −36851.4 −1.84937
\(736\) 0 0
\(737\) 2087.61 0.104339
\(738\) 0 0
\(739\) −1078.25 −0.0536727 −0.0268363 0.999640i \(-0.508543\pi\)
−0.0268363 + 0.999640i \(0.508543\pi\)
\(740\) 0 0
\(741\) −40593.4 −2.01247
\(742\) 0 0
\(743\) 23176.4 1.14436 0.572180 0.820128i \(-0.306098\pi\)
0.572180 + 0.820128i \(0.306098\pi\)
\(744\) 0 0
\(745\) 17149.9 0.843385
\(746\) 0 0
\(747\) 16707.8 0.818350
\(748\) 0 0
\(749\) −11874.9 −0.579304
\(750\) 0 0
\(751\) −8738.85 −0.424614 −0.212307 0.977203i \(-0.568098\pi\)
−0.212307 + 0.977203i \(0.568098\pi\)
\(752\) 0 0
\(753\) 27716.0 1.34134
\(754\) 0 0
\(755\) 20925.7 1.00869
\(756\) 0 0
\(757\) −4121.15 −0.197868 −0.0989338 0.995094i \(-0.531543\pi\)
−0.0989338 + 0.995094i \(0.531543\pi\)
\(758\) 0 0
\(759\) −7965.84 −0.380951
\(760\) 0 0
\(761\) 8706.54 0.414733 0.207367 0.978263i \(-0.433511\pi\)
0.207367 + 0.978263i \(0.433511\pi\)
\(762\) 0 0
\(763\) −11068.7 −0.525184
\(764\) 0 0
\(765\) 34727.0 1.64125
\(766\) 0 0
\(767\) 30532.9 1.43739
\(768\) 0 0
\(769\) −32258.8 −1.51272 −0.756361 0.654154i \(-0.773025\pi\)
−0.756361 + 0.654154i \(0.773025\pi\)
\(770\) 0 0
\(771\) 1537.00 0.0717945
\(772\) 0 0
\(773\) −24794.1 −1.15366 −0.576832 0.816863i \(-0.695711\pi\)
−0.576832 + 0.816863i \(0.695711\pi\)
\(774\) 0 0
\(775\) 32143.5 1.48984
\(776\) 0 0
\(777\) 5905.25 0.272651
\(778\) 0 0
\(779\) −24654.2 −1.13393
\(780\) 0 0
\(781\) 25.7427 0.00117945
\(782\) 0 0
\(783\) −2151.01 −0.0981749
\(784\) 0 0
\(785\) −35614.6 −1.61929
\(786\) 0 0
\(787\) −11114.7 −0.503427 −0.251714 0.967802i \(-0.580994\pi\)
−0.251714 + 0.967802i \(0.580994\pi\)
\(788\) 0 0
\(789\) 22552.3 1.01760
\(790\) 0 0
\(791\) −2592.20 −0.116521
\(792\) 0 0
\(793\) −25929.2 −1.16113
\(794\) 0 0
\(795\) 81108.0 3.61837
\(796\) 0 0
\(797\) 12329.8 0.547986 0.273993 0.961732i \(-0.411656\pi\)
0.273993 + 0.961732i \(0.411656\pi\)
\(798\) 0 0
\(799\) 48310.1 2.13904
\(800\) 0 0
\(801\) 12083.6 0.533023
\(802\) 0 0
\(803\) 1286.44 0.0565348
\(804\) 0 0
\(805\) 20206.2 0.884691
\(806\) 0 0
\(807\) −12546.2 −0.547271
\(808\) 0 0
\(809\) −36175.5 −1.57214 −0.786070 0.618138i \(-0.787887\pi\)
−0.786070 + 0.618138i \(0.787887\pi\)
\(810\) 0 0
\(811\) −26581.0 −1.15091 −0.575453 0.817835i \(-0.695174\pi\)
−0.575453 + 0.817835i \(0.695174\pi\)
\(812\) 0 0
\(813\) −16195.0 −0.698629
\(814\) 0 0
\(815\) 72212.5 3.10367
\(816\) 0 0
\(817\) 11617.0 0.497462
\(818\) 0 0
\(819\) 7322.98 0.312437
\(820\) 0 0
\(821\) −10811.1 −0.459574 −0.229787 0.973241i \(-0.573803\pi\)
−0.229787 + 0.973241i \(0.573803\pi\)
\(822\) 0 0
\(823\) −28625.1 −1.21240 −0.606202 0.795311i \(-0.707308\pi\)
−0.606202 + 0.795311i \(0.707308\pi\)
\(824\) 0 0
\(825\) −22069.3 −0.931341
\(826\) 0 0
\(827\) −9970.68 −0.419244 −0.209622 0.977783i \(-0.567223\pi\)
−0.209622 + 0.977783i \(0.567223\pi\)
\(828\) 0 0
\(829\) −20201.0 −0.846334 −0.423167 0.906052i \(-0.639082\pi\)
−0.423167 + 0.906052i \(0.639082\pi\)
\(830\) 0 0
\(831\) −31128.0 −1.29942
\(832\) 0 0
\(833\) 28628.1 1.19076
\(834\) 0 0
\(835\) 29481.7 1.22186
\(836\) 0 0
\(837\) 7625.33 0.314898
\(838\) 0 0
\(839\) −28023.4 −1.15313 −0.576564 0.817052i \(-0.695607\pi\)
−0.576564 + 0.817052i \(0.695607\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −1268.98 −0.0518456
\(844\) 0 0
\(845\) −16644.3 −0.677610
\(846\) 0 0
\(847\) 10391.3 0.421544
\(848\) 0 0
\(849\) 18397.9 0.743714
\(850\) 0 0
\(851\) 11924.9 0.480352
\(852\) 0 0
\(853\) 26333.8 1.05704 0.528518 0.848922i \(-0.322748\pi\)
0.528518 + 0.848922i \(0.322748\pi\)
\(854\) 0 0
\(855\) 37184.3 1.48734
\(856\) 0 0
\(857\) 36388.6 1.45042 0.725211 0.688527i \(-0.241742\pi\)
0.725211 + 0.688527i \(0.241742\pi\)
\(858\) 0 0
\(859\) 14974.0 0.594770 0.297385 0.954758i \(-0.403885\pi\)
0.297385 + 0.954758i \(0.403885\pi\)
\(860\) 0 0
\(861\) 12125.0 0.479930
\(862\) 0 0
\(863\) 40168.2 1.58440 0.792202 0.610259i \(-0.208935\pi\)
0.792202 + 0.610259i \(0.208935\pi\)
\(864\) 0 0
\(865\) 28418.1 1.11705
\(866\) 0 0
\(867\) −41465.0 −1.62425
\(868\) 0 0
\(869\) 10423.4 0.406893
\(870\) 0 0
\(871\) −10565.2 −0.411009
\(872\) 0 0
\(873\) 21022.3 0.815004
\(874\) 0 0
\(875\) 33600.7 1.29818
\(876\) 0 0
\(877\) −32161.7 −1.23834 −0.619169 0.785258i \(-0.712531\pi\)
−0.619169 + 0.785258i \(0.712531\pi\)
\(878\) 0 0
\(879\) 26506.2 1.01710
\(880\) 0 0
\(881\) −8870.49 −0.339222 −0.169611 0.985511i \(-0.554251\pi\)
−0.169611 + 0.985511i \(0.554251\pi\)
\(882\) 0 0
\(883\) −24647.3 −0.939351 −0.469675 0.882839i \(-0.655629\pi\)
−0.469675 + 0.882839i \(0.655629\pi\)
\(884\) 0 0
\(885\) −76248.3 −2.89611
\(886\) 0 0
\(887\) −32371.7 −1.22541 −0.612703 0.790313i \(-0.709918\pi\)
−0.612703 + 0.790313i \(0.709918\pi\)
\(888\) 0 0
\(889\) 17302.4 0.652759
\(890\) 0 0
\(891\) −9800.35 −0.368489
\(892\) 0 0
\(893\) 51728.5 1.93844
\(894\) 0 0
\(895\) 35618.4 1.33027
\(896\) 0 0
\(897\) 40314.5 1.50063
\(898\) 0 0
\(899\) −2981.34 −0.110604
\(900\) 0 0
\(901\) −63008.9 −2.32978
\(902\) 0 0
\(903\) −5713.26 −0.210549
\(904\) 0 0
\(905\) −153.493 −0.00563787
\(906\) 0 0
\(907\) −43728.0 −1.60084 −0.800422 0.599437i \(-0.795391\pi\)
−0.800422 + 0.599437i \(0.795391\pi\)
\(908\) 0 0
\(909\) −15433.7 −0.563152
\(910\) 0 0
\(911\) −16033.6 −0.583115 −0.291557 0.956553i \(-0.594173\pi\)
−0.291557 + 0.956553i \(0.594173\pi\)
\(912\) 0 0
\(913\) 11546.3 0.418540
\(914\) 0 0
\(915\) 64751.8 2.33948
\(916\) 0 0
\(917\) −7315.08 −0.263430
\(918\) 0 0
\(919\) −29331.1 −1.05282 −0.526411 0.850230i \(-0.676463\pi\)
−0.526411 + 0.850230i \(0.676463\pi\)
\(920\) 0 0
\(921\) 859.170 0.0307390
\(922\) 0 0
\(923\) −130.282 −0.00464603
\(924\) 0 0
\(925\) 33037.8 1.17435
\(926\) 0 0
\(927\) −8575.66 −0.303842
\(928\) 0 0
\(929\) −10024.6 −0.354034 −0.177017 0.984208i \(-0.556645\pi\)
−0.177017 + 0.984208i \(0.556645\pi\)
\(930\) 0 0
\(931\) 30653.8 1.07910
\(932\) 0 0
\(933\) 9911.51 0.347790
\(934\) 0 0
\(935\) 23998.9 0.839408
\(936\) 0 0
\(937\) 21241.3 0.740578 0.370289 0.928917i \(-0.379259\pi\)
0.370289 + 0.928917i \(0.379259\pi\)
\(938\) 0 0
\(939\) −27714.5 −0.963184
\(940\) 0 0
\(941\) −11953.8 −0.414115 −0.207057 0.978329i \(-0.566389\pi\)
−0.207057 + 0.978329i \(0.566389\pi\)
\(942\) 0 0
\(943\) 24484.8 0.845531
\(944\) 0 0
\(945\) 13280.3 0.457152
\(946\) 0 0
\(947\) −31650.6 −1.08607 −0.543034 0.839711i \(-0.682725\pi\)
−0.543034 + 0.839711i \(0.682725\pi\)
\(948\) 0 0
\(949\) −6510.57 −0.222700
\(950\) 0 0
\(951\) 36545.7 1.24614
\(952\) 0 0
\(953\) −17373.5 −0.590539 −0.295269 0.955414i \(-0.595409\pi\)
−0.295269 + 0.955414i \(0.595409\pi\)
\(954\) 0 0
\(955\) 27715.0 0.939095
\(956\) 0 0
\(957\) 2046.95 0.0691416
\(958\) 0 0
\(959\) −3124.02 −0.105193
\(960\) 0 0
\(961\) −19222.2 −0.645234
\(962\) 0 0
\(963\) 21702.4 0.726222
\(964\) 0 0
\(965\) −38378.0 −1.28024
\(966\) 0 0
\(967\) 11017.6 0.366394 0.183197 0.983076i \(-0.441355\pi\)
0.183197 + 0.983076i \(0.441355\pi\)
\(968\) 0 0
\(969\) −78751.1 −2.61078
\(970\) 0 0
\(971\) 17365.0 0.573912 0.286956 0.957944i \(-0.407357\pi\)
0.286956 + 0.957944i \(0.407357\pi\)
\(972\) 0 0
\(973\) 8644.82 0.284831
\(974\) 0 0
\(975\) 111691. 3.66870
\(976\) 0 0
\(977\) −29193.4 −0.955967 −0.477983 0.878369i \(-0.658632\pi\)
−0.477983 + 0.878369i \(0.658632\pi\)
\(978\) 0 0
\(979\) 8350.60 0.272611
\(980\) 0 0
\(981\) 20229.2 0.658377
\(982\) 0 0
\(983\) −50359.8 −1.63401 −0.817003 0.576634i \(-0.804366\pi\)
−0.817003 + 0.576634i \(0.804366\pi\)
\(984\) 0 0
\(985\) −103556. −3.34982
\(986\) 0 0
\(987\) −25440.2 −0.820436
\(988\) 0 0
\(989\) −11537.2 −0.370941
\(990\) 0 0
\(991\) 20823.5 0.667488 0.333744 0.942664i \(-0.391688\pi\)
0.333744 + 0.942664i \(0.391688\pi\)
\(992\) 0 0
\(993\) 54746.8 1.74958
\(994\) 0 0
\(995\) 74352.2 2.36897
\(996\) 0 0
\(997\) 27651.2 0.878356 0.439178 0.898400i \(-0.355270\pi\)
0.439178 + 0.898400i \(0.355270\pi\)
\(998\) 0 0
\(999\) 7837.48 0.248215
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 1856.4.a.s.1.1 3
4.3 odd 2 1856.4.a.r.1.3 3
8.3 odd 2 58.4.a.d.1.1 3
8.5 even 2 464.4.a.i.1.3 3
24.11 even 2 522.4.a.k.1.1 3
40.19 odd 2 1450.4.a.h.1.3 3
232.115 odd 2 1682.4.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.4.a.d.1.1 3 8.3 odd 2
464.4.a.i.1.3 3 8.5 even 2
522.4.a.k.1.1 3 24.11 even 2
1450.4.a.h.1.3 3 40.19 odd 2
1682.4.a.d.1.3 3 232.115 odd 2
1856.4.a.r.1.3 3 4.3 odd 2
1856.4.a.s.1.1 3 1.1 even 1 trivial