Properties

Label 128.4.a.g.1.2
Level $128$
Weight $4$
Character 128.1
Self dual yes
Analytic conductor $7.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 128.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.92820 q^{3} +11.8564 q^{5} -9.85641 q^{7} +52.7128 q^{9} +O(q^{10})\) \(q+8.92820 q^{3} +11.8564 q^{5} -9.85641 q^{7} +52.7128 q^{9} +39.0718 q^{11} -91.5692 q^{13} +105.856 q^{15} -37.1384 q^{17} -46.4974 q^{19} -88.0000 q^{21} +120.708 q^{23} +15.5744 q^{25} +229.569 q^{27} +27.2820 q^{29} +81.1487 q^{31} +348.841 q^{33} -116.862 q^{35} -10.9948 q^{37} -817.549 q^{39} -205.426 q^{41} +115.359 q^{43} +624.985 q^{45} +312.841 q^{47} -245.851 q^{49} -331.580 q^{51} -90.9948 q^{53} +463.251 q^{55} -415.138 q^{57} -550.631 q^{59} -630.974 q^{61} -519.559 q^{63} -1085.68 q^{65} -661.041 q^{67} +1077.70 q^{69} -494.985 q^{71} +566.267 q^{73} +139.051 q^{75} -385.108 q^{77} -49.4153 q^{79} +626.395 q^{81} +564.067 q^{83} -440.328 q^{85} +243.580 q^{87} +1089.67 q^{89} +902.543 q^{91} +724.513 q^{93} -551.292 q^{95} +464.605 q^{97} +2059.58 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 4q^{5} + 8q^{7} + 50q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 4q^{5} + 8q^{7} + 50q^{9} + 92q^{11} - 100q^{13} + 184q^{15} + 92q^{17} + 4q^{19} - 176q^{21} - 8q^{23} + 142q^{25} + 376q^{27} - 84q^{29} + 384q^{31} + 88q^{33} - 400q^{35} + 172q^{37} - 776q^{39} - 300q^{41} + 300q^{43} + 668q^{45} + 16q^{47} - 270q^{49} - 968q^{51} + 12q^{53} - 376q^{55} - 664q^{57} - 644q^{59} - 292q^{61} - 568q^{63} - 952q^{65} - 172q^{67} + 1712q^{69} - 408q^{71} + 412q^{73} - 484q^{75} + 560q^{77} + 400q^{79} - 22q^{81} + 948q^{83} - 2488q^{85} + 792q^{87} + 572q^{89} + 752q^{91} - 768q^{93} - 1352q^{95} + 2204q^{97} + 1916q^{99} + O(q^{100}) \)

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
\(3\) 8.92820 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(4\) 0 0
\(5\) 11.8564 1.06047 0.530235 0.847851i \(-0.322104\pi\)
0.530235 + 0.847851i \(0.322104\pi\)
\(6\) 0 0
\(7\) −9.85641 −0.532196 −0.266098 0.963946i \(-0.585734\pi\)
−0.266098 + 0.963946i \(0.585734\pi\)
\(8\) 0 0
\(9\) 52.7128 1.95233
\(10\) 0 0
\(11\) 39.0718 1.07096 0.535481 0.844547i \(-0.320130\pi\)
0.535481 + 0.844547i \(0.320130\pi\)
\(12\) 0 0
\(13\) −91.5692 −1.95359 −0.976797 0.214166i \(-0.931297\pi\)
−0.976797 + 0.214166i \(0.931297\pi\)
\(14\) 0 0
\(15\) 105.856 1.82213
\(16\) 0 0
\(17\) −37.1384 −0.529847 −0.264923 0.964269i \(-0.585347\pi\)
−0.264923 + 0.964269i \(0.585347\pi\)
\(18\) 0 0
\(19\) −46.4974 −0.561434 −0.280717 0.959791i \(-0.590572\pi\)
−0.280717 + 0.959791i \(0.590572\pi\)
\(20\) 0 0
\(21\) −88.0000 −0.914437
\(22\) 0 0
\(23\) 120.708 1.09432 0.547158 0.837029i \(-0.315710\pi\)
0.547158 + 0.837029i \(0.315710\pi\)
\(24\) 0 0
\(25\) 15.5744 0.124595
\(26\) 0 0
\(27\) 229.569 1.63632
\(28\) 0 0
\(29\) 27.2820 0.174695 0.0873473 0.996178i \(-0.472161\pi\)
0.0873473 + 0.996178i \(0.472161\pi\)
\(30\) 0 0
\(31\) 81.1487 0.470153 0.235077 0.971977i \(-0.424466\pi\)
0.235077 + 0.971977i \(0.424466\pi\)
\(32\) 0 0
\(33\) 348.841 1.84016
\(34\) 0 0
\(35\) −116.862 −0.564377
\(36\) 0 0
\(37\) −10.9948 −0.0488525 −0.0244262 0.999702i \(-0.507776\pi\)
−0.0244262 + 0.999702i \(0.507776\pi\)
\(38\) 0 0
\(39\) −817.549 −3.35673
\(40\) 0 0
\(41\) −205.426 −0.782490 −0.391245 0.920287i \(-0.627956\pi\)
−0.391245 + 0.920287i \(0.627956\pi\)
\(42\) 0 0
\(43\) 115.359 0.409118 0.204559 0.978854i \(-0.434424\pi\)
0.204559 + 0.978854i \(0.434424\pi\)
\(44\) 0 0
\(45\) 624.985 2.07038
\(46\) 0 0
\(47\) 312.841 0.970905 0.485453 0.874263i \(-0.338655\pi\)
0.485453 + 0.874263i \(0.338655\pi\)
\(48\) 0 0
\(49\) −245.851 −0.716767
\(50\) 0 0
\(51\) −331.580 −0.910400
\(52\) 0 0
\(53\) −90.9948 −0.235832 −0.117916 0.993024i \(-0.537621\pi\)
−0.117916 + 0.993024i \(0.537621\pi\)
\(54\) 0 0
\(55\) 463.251 1.13572
\(56\) 0 0
\(57\) −415.138 −0.964674
\(58\) 0 0
\(59\) −550.631 −1.21502 −0.607509 0.794313i \(-0.707831\pi\)
−0.607509 + 0.794313i \(0.707831\pi\)
\(60\) 0 0
\(61\) −630.974 −1.32439 −0.662196 0.749330i \(-0.730376\pi\)
−0.662196 + 0.749330i \(0.730376\pi\)
\(62\) 0 0
\(63\) −519.559 −1.03902
\(64\) 0 0
\(65\) −1085.68 −2.07173
\(66\) 0 0
\(67\) −661.041 −1.20536 −0.602679 0.797984i \(-0.705900\pi\)
−0.602679 + 0.797984i \(0.705900\pi\)
\(68\) 0 0
\(69\) 1077.70 1.88029
\(70\) 0 0
\(71\) −494.985 −0.827378 −0.413689 0.910418i \(-0.635760\pi\)
−0.413689 + 0.910418i \(0.635760\pi\)
\(72\) 0 0
\(73\) 566.267 0.907897 0.453949 0.891028i \(-0.350015\pi\)
0.453949 + 0.891028i \(0.350015\pi\)
\(74\) 0 0
\(75\) 139.051 0.214083
\(76\) 0 0
\(77\) −385.108 −0.569962
\(78\) 0 0
\(79\) −49.4153 −0.0703754 −0.0351877 0.999381i \(-0.511203\pi\)
−0.0351877 + 0.999381i \(0.511203\pi\)
\(80\) 0 0
\(81\) 626.395 0.859252
\(82\) 0 0
\(83\) 564.067 0.745956 0.372978 0.927840i \(-0.378337\pi\)
0.372978 + 0.927840i \(0.378337\pi\)
\(84\) 0 0
\(85\) −440.328 −0.561886
\(86\) 0 0
\(87\) 243.580 0.300166
\(88\) 0 0
\(89\) 1089.67 1.29781 0.648904 0.760870i \(-0.275228\pi\)
0.648904 + 0.760870i \(0.275228\pi\)
\(90\) 0 0
\(91\) 902.543 1.03970
\(92\) 0 0
\(93\) 724.513 0.807833
\(94\) 0 0
\(95\) −551.292 −0.595383
\(96\) 0 0
\(97\) 464.605 0.486325 0.243162 0.969986i \(-0.421815\pi\)
0.243162 + 0.969986i \(0.421815\pi\)
\(98\) 0 0
\(99\) 2059.58 2.09087
\(100\) 0 0
\(101\) 965.005 0.950709 0.475354 0.879794i \(-0.342320\pi\)
0.475354 + 0.879794i \(0.342320\pi\)
\(102\) 0 0
\(103\) 1828.99 1.74967 0.874836 0.484419i \(-0.160969\pi\)
0.874836 + 0.484419i \(0.160969\pi\)
\(104\) 0 0
\(105\) −1043.36 −0.969732
\(106\) 0 0
\(107\) −1726.59 −1.55996 −0.779980 0.625805i \(-0.784771\pi\)
−0.779980 + 0.625805i \(0.784771\pi\)
\(108\) 0 0
\(109\) −423.015 −0.371720 −0.185860 0.982576i \(-0.559507\pi\)
−0.185860 + 0.982576i \(0.559507\pi\)
\(110\) 0 0
\(111\) −98.1642 −0.0839400
\(112\) 0 0
\(113\) 763.108 0.635284 0.317642 0.948211i \(-0.397109\pi\)
0.317642 + 0.948211i \(0.397109\pi\)
\(114\) 0 0
\(115\) 1431.16 1.16049
\(116\) 0 0
\(117\) −4826.87 −3.81405
\(118\) 0 0
\(119\) 366.052 0.281982
\(120\) 0 0
\(121\) 195.605 0.146961
\(122\) 0 0
\(123\) −1834.08 −1.34450
\(124\) 0 0
\(125\) −1297.39 −0.928340
\(126\) 0 0
\(127\) 2159.38 1.50877 0.754387 0.656430i \(-0.227934\pi\)
0.754387 + 0.656430i \(0.227934\pi\)
\(128\) 0 0
\(129\) 1029.95 0.702961
\(130\) 0 0
\(131\) −710.723 −0.474016 −0.237008 0.971508i \(-0.576167\pi\)
−0.237008 + 0.971508i \(0.576167\pi\)
\(132\) 0 0
\(133\) 458.297 0.298793
\(134\) 0 0
\(135\) 2721.87 1.73527
\(136\) 0 0
\(137\) −987.128 −0.615592 −0.307796 0.951452i \(-0.599591\pi\)
−0.307796 + 0.951452i \(0.599591\pi\)
\(138\) 0 0
\(139\) 2334.15 1.42432 0.712158 0.702019i \(-0.247718\pi\)
0.712158 + 0.702019i \(0.247718\pi\)
\(140\) 0 0
\(141\) 2793.11 1.66824
\(142\) 0 0
\(143\) −3577.77 −2.09223
\(144\) 0 0
\(145\) 323.467 0.185258
\(146\) 0 0
\(147\) −2195.01 −1.23157
\(148\) 0 0
\(149\) 1792.29 0.985435 0.492718 0.870189i \(-0.336004\pi\)
0.492718 + 0.870189i \(0.336004\pi\)
\(150\) 0 0
\(151\) −2864.32 −1.54367 −0.771837 0.635820i \(-0.780662\pi\)
−0.771837 + 0.635820i \(0.780662\pi\)
\(152\) 0 0
\(153\) −1957.67 −1.03443
\(154\) 0 0
\(155\) 962.133 0.498583
\(156\) 0 0
\(157\) −2123.57 −1.07949 −0.539743 0.841830i \(-0.681479\pi\)
−0.539743 + 0.841830i \(0.681479\pi\)
\(158\) 0 0
\(159\) −812.420 −0.405215
\(160\) 0 0
\(161\) −1189.74 −0.582391
\(162\) 0 0
\(163\) 3164.21 1.52049 0.760245 0.649636i \(-0.225079\pi\)
0.760245 + 0.649636i \(0.225079\pi\)
\(164\) 0 0
\(165\) 4136.00 1.95144
\(166\) 0 0
\(167\) 1340.34 0.621069 0.310534 0.950562i \(-0.399492\pi\)
0.310534 + 0.950562i \(0.399492\pi\)
\(168\) 0 0
\(169\) 6187.92 2.81653
\(170\) 0 0
\(171\) −2451.01 −1.09610
\(172\) 0 0
\(173\) 2802.13 1.23146 0.615729 0.787958i \(-0.288862\pi\)
0.615729 + 0.787958i \(0.288862\pi\)
\(174\) 0 0
\(175\) −153.507 −0.0663089
\(176\) 0 0
\(177\) −4916.14 −2.08768
\(178\) 0 0
\(179\) 1074.84 0.448810 0.224405 0.974496i \(-0.427956\pi\)
0.224405 + 0.974496i \(0.427956\pi\)
\(180\) 0 0
\(181\) −571.036 −0.234502 −0.117251 0.993102i \(-0.537408\pi\)
−0.117251 + 0.993102i \(0.537408\pi\)
\(182\) 0 0
\(183\) −5633.47 −2.27562
\(184\) 0 0
\(185\) −130.359 −0.0518065
\(186\) 0 0
\(187\) −1451.07 −0.567446
\(188\) 0 0
\(189\) −2262.73 −0.870842
\(190\) 0 0
\(191\) −872.657 −0.330593 −0.165296 0.986244i \(-0.552858\pi\)
−0.165296 + 0.986244i \(0.552858\pi\)
\(192\) 0 0
\(193\) 672.523 0.250825 0.125413 0.992105i \(-0.459975\pi\)
0.125413 + 0.992105i \(0.459975\pi\)
\(194\) 0 0
\(195\) −9693.19 −3.55971
\(196\) 0 0
\(197\) 3129.56 1.13184 0.565918 0.824461i \(-0.308522\pi\)
0.565918 + 0.824461i \(0.308522\pi\)
\(198\) 0 0
\(199\) −1502.37 −0.535176 −0.267588 0.963533i \(-0.586227\pi\)
−0.267588 + 0.963533i \(0.586227\pi\)
\(200\) 0 0
\(201\) −5901.91 −2.07109
\(202\) 0 0
\(203\) −268.903 −0.0929718
\(204\) 0 0
\(205\) −2435.61 −0.829807
\(206\) 0 0
\(207\) 6362.84 2.13646
\(208\) 0 0
\(209\) −1816.74 −0.601275
\(210\) 0 0
\(211\) −1773.70 −0.578703 −0.289352 0.957223i \(-0.593440\pi\)
−0.289352 + 0.957223i \(0.593440\pi\)
\(212\) 0 0
\(213\) −4419.32 −1.42163
\(214\) 0 0
\(215\) 1367.74 0.433857
\(216\) 0 0
\(217\) −799.835 −0.250214
\(218\) 0 0
\(219\) 5055.74 1.55998
\(220\) 0 0
\(221\) 3400.74 1.03511
\(222\) 0 0
\(223\) 5413.66 1.62568 0.812838 0.582490i \(-0.197922\pi\)
0.812838 + 0.582490i \(0.197922\pi\)
\(224\) 0 0
\(225\) 820.969 0.243250
\(226\) 0 0
\(227\) 3744.27 1.09478 0.547392 0.836876i \(-0.315621\pi\)
0.547392 + 0.836876i \(0.315621\pi\)
\(228\) 0 0
\(229\) −680.739 −0.196439 −0.0982194 0.995165i \(-0.531315\pi\)
−0.0982194 + 0.995165i \(0.531315\pi\)
\(230\) 0 0
\(231\) −3438.32 −0.979328
\(232\) 0 0
\(233\) −679.271 −0.190989 −0.0954947 0.995430i \(-0.530443\pi\)
−0.0954947 + 0.995430i \(0.530443\pi\)
\(234\) 0 0
\(235\) 3709.17 1.02962
\(236\) 0 0
\(237\) −441.190 −0.120921
\(238\) 0 0
\(239\) −3901.27 −1.05587 −0.527934 0.849286i \(-0.677033\pi\)
−0.527934 + 0.849286i \(0.677033\pi\)
\(240\) 0 0
\(241\) 5859.72 1.56622 0.783108 0.621886i \(-0.213633\pi\)
0.783108 + 0.621886i \(0.213633\pi\)
\(242\) 0 0
\(243\) −605.790 −0.159924
\(244\) 0 0
\(245\) −2914.91 −0.760110
\(246\) 0 0
\(247\) 4257.73 1.09681
\(248\) 0 0
\(249\) 5036.10 1.28173
\(250\) 0 0
\(251\) −3381.48 −0.850348 −0.425174 0.905112i \(-0.639787\pi\)
−0.425174 + 0.905112i \(0.639787\pi\)
\(252\) 0 0
\(253\) 4716.27 1.17197
\(254\) 0 0
\(255\) −3931.34 −0.965452
\(256\) 0 0
\(257\) 3652.22 0.886455 0.443227 0.896409i \(-0.353833\pi\)
0.443227 + 0.896409i \(0.353833\pi\)
\(258\) 0 0
\(259\) 108.370 0.0259991
\(260\) 0 0
\(261\) 1438.11 0.341061
\(262\) 0 0
\(263\) −2462.74 −0.577410 −0.288705 0.957418i \(-0.593225\pi\)
−0.288705 + 0.957418i \(0.593225\pi\)
\(264\) 0 0
\(265\) −1078.87 −0.250093
\(266\) 0 0
\(267\) 9728.81 2.22994
\(268\) 0 0
\(269\) −482.503 −0.109363 −0.0546816 0.998504i \(-0.517414\pi\)
−0.0546816 + 0.998504i \(0.517414\pi\)
\(270\) 0 0
\(271\) 3602.48 0.807510 0.403755 0.914867i \(-0.367705\pi\)
0.403755 + 0.914867i \(0.367705\pi\)
\(272\) 0 0
\(273\) 8058.09 1.78644
\(274\) 0 0
\(275\) 608.519 0.133437
\(276\) 0 0
\(277\) −6177.72 −1.34001 −0.670006 0.742356i \(-0.733709\pi\)
−0.670006 + 0.742356i \(0.733709\pi\)
\(278\) 0 0
\(279\) 4277.58 0.917892
\(280\) 0 0
\(281\) −543.477 −0.115378 −0.0576888 0.998335i \(-0.518373\pi\)
−0.0576888 + 0.998335i \(0.518373\pi\)
\(282\) 0 0
\(283\) 6582.11 1.38256 0.691282 0.722585i \(-0.257046\pi\)
0.691282 + 0.722585i \(0.257046\pi\)
\(284\) 0 0
\(285\) −4922.05 −1.02301
\(286\) 0 0
\(287\) 2024.76 0.416438
\(288\) 0 0
\(289\) −3533.74 −0.719262
\(290\) 0 0
\(291\) 4148.09 0.835620
\(292\) 0 0
\(293\) −6753.68 −1.34660 −0.673301 0.739369i \(-0.735124\pi\)
−0.673301 + 0.739369i \(0.735124\pi\)
\(294\) 0 0
\(295\) −6528.50 −1.28849
\(296\) 0 0
\(297\) 8969.68 1.75244
\(298\) 0 0
\(299\) −11053.1 −2.13785
\(300\) 0 0
\(301\) −1137.03 −0.217731
\(302\) 0 0
\(303\) 8615.76 1.63354
\(304\) 0 0
\(305\) −7481.09 −1.40448
\(306\) 0 0
\(307\) −1821.88 −0.338698 −0.169349 0.985556i \(-0.554167\pi\)
−0.169349 + 0.985556i \(0.554167\pi\)
\(308\) 0 0
\(309\) 16329.6 3.00635
\(310\) 0 0
\(311\) 10598.2 1.93237 0.966184 0.257852i \(-0.0830148\pi\)
0.966184 + 0.257852i \(0.0830148\pi\)
\(312\) 0 0
\(313\) −7969.69 −1.43921 −0.719606 0.694382i \(-0.755678\pi\)
−0.719606 + 0.694382i \(0.755678\pi\)
\(314\) 0 0
\(315\) −6160.10 −1.10185
\(316\) 0 0
\(317\) −7773.66 −1.37733 −0.688663 0.725082i \(-0.741802\pi\)
−0.688663 + 0.725082i \(0.741802\pi\)
\(318\) 0 0
\(319\) 1065.96 0.187092
\(320\) 0 0
\(321\) −15415.3 −2.68038
\(322\) 0 0
\(323\) 1726.84 0.297474
\(324\) 0 0
\(325\) −1426.13 −0.243408
\(326\) 0 0
\(327\) −3776.77 −0.638703
\(328\) 0 0
\(329\) −3083.49 −0.516712
\(330\) 0 0
\(331\) −8212.87 −1.36381 −0.681903 0.731442i \(-0.738848\pi\)
−0.681903 + 0.731442i \(0.738848\pi\)
\(332\) 0 0
\(333\) −579.569 −0.0953760
\(334\) 0 0
\(335\) −7837.57 −1.27825
\(336\) 0 0
\(337\) 5260.17 0.850267 0.425133 0.905131i \(-0.360227\pi\)
0.425133 + 0.905131i \(0.360227\pi\)
\(338\) 0 0
\(339\) 6813.18 1.09157
\(340\) 0 0
\(341\) 3170.63 0.503516
\(342\) 0 0
\(343\) 5803.96 0.913657
\(344\) 0 0
\(345\) 12777.7 1.99399
\(346\) 0 0
\(347\) −2162.12 −0.334492 −0.167246 0.985915i \(-0.553487\pi\)
−0.167246 + 0.985915i \(0.553487\pi\)
\(348\) 0 0
\(349\) −2888.90 −0.443093 −0.221546 0.975150i \(-0.571110\pi\)
−0.221546 + 0.975150i \(0.571110\pi\)
\(350\) 0 0
\(351\) −21021.5 −3.19670
\(352\) 0 0
\(353\) 11550.2 1.74152 0.870760 0.491709i \(-0.163628\pi\)
0.870760 + 0.491709i \(0.163628\pi\)
\(354\) 0 0
\(355\) −5868.74 −0.877409
\(356\) 0 0
\(357\) 3268.18 0.484511
\(358\) 0 0
\(359\) 6523.97 0.959114 0.479557 0.877511i \(-0.340797\pi\)
0.479557 + 0.877511i \(0.340797\pi\)
\(360\) 0 0
\(361\) −4696.99 −0.684792
\(362\) 0 0
\(363\) 1746.40 0.252514
\(364\) 0 0
\(365\) 6713.89 0.962797
\(366\) 0 0
\(367\) −6856.96 −0.975287 −0.487644 0.873043i \(-0.662143\pi\)
−0.487644 + 0.873043i \(0.662143\pi\)
\(368\) 0 0
\(369\) −10828.6 −1.52768
\(370\) 0 0
\(371\) 896.882 0.125509
\(372\) 0 0
\(373\) 5001.68 0.694309 0.347155 0.937808i \(-0.387148\pi\)
0.347155 + 0.937808i \(0.387148\pi\)
\(374\) 0 0
\(375\) −11583.4 −1.59511
\(376\) 0 0
\(377\) −2498.19 −0.341283
\(378\) 0 0
\(379\) 84.9775 0.0115172 0.00575858 0.999983i \(-0.498167\pi\)
0.00575858 + 0.999983i \(0.498167\pi\)
\(380\) 0 0
\(381\) 19279.4 2.59243
\(382\) 0 0
\(383\) −2596.76 −0.346444 −0.173222 0.984883i \(-0.555418\pi\)
−0.173222 + 0.984883i \(0.555418\pi\)
\(384\) 0 0
\(385\) −4565.99 −0.604427
\(386\) 0 0
\(387\) 6080.90 0.798732
\(388\) 0 0
\(389\) −9213.21 −1.20084 −0.600422 0.799683i \(-0.705001\pi\)
−0.600422 + 0.799683i \(0.705001\pi\)
\(390\) 0 0
\(391\) −4482.89 −0.579820
\(392\) 0 0
\(393\) −6345.48 −0.814471
\(394\) 0 0
\(395\) −585.888 −0.0746310
\(396\) 0 0
\(397\) 6614.19 0.836163 0.418082 0.908409i \(-0.362703\pi\)
0.418082 + 0.908409i \(0.362703\pi\)
\(398\) 0 0
\(399\) 4091.77 0.513396
\(400\) 0 0
\(401\) 3434.94 0.427763 0.213881 0.976860i \(-0.431389\pi\)
0.213881 + 0.976860i \(0.431389\pi\)
\(402\) 0 0
\(403\) −7430.73 −0.918489
\(404\) 0 0
\(405\) 7426.79 0.911210
\(406\) 0 0
\(407\) −429.588 −0.0523192
\(408\) 0 0
\(409\) −6435.82 −0.778071 −0.389035 0.921223i \(-0.627192\pi\)
−0.389035 + 0.921223i \(0.627192\pi\)
\(410\) 0 0
\(411\) −8813.28 −1.05773
\(412\) 0 0
\(413\) 5427.24 0.646627
\(414\) 0 0
\(415\) 6687.80 0.791064
\(416\) 0 0
\(417\) 20839.8 2.44731
\(418\) 0 0
\(419\) 4222.04 0.492267 0.246134 0.969236i \(-0.420840\pi\)
0.246134 + 0.969236i \(0.420840\pi\)
\(420\) 0 0
\(421\) −5164.92 −0.597917 −0.298958 0.954266i \(-0.596639\pi\)
−0.298958 + 0.954266i \(0.596639\pi\)
\(422\) 0 0
\(423\) 16490.7 1.89552
\(424\) 0 0
\(425\) −578.408 −0.0660163
\(426\) 0 0
\(427\) 6219.14 0.704837
\(428\) 0 0
\(429\) −31943.1 −3.59493
\(430\) 0 0
\(431\) −7476.41 −0.835559 −0.417780 0.908548i \(-0.637192\pi\)
−0.417780 + 0.908548i \(0.637192\pi\)
\(432\) 0 0
\(433\) −11289.7 −1.25299 −0.626497 0.779424i \(-0.715512\pi\)
−0.626497 + 0.779424i \(0.715512\pi\)
\(434\) 0 0
\(435\) 2887.98 0.318317
\(436\) 0 0
\(437\) −5612.59 −0.614386
\(438\) 0 0
\(439\) 1800.79 0.195779 0.0978895 0.995197i \(-0.468791\pi\)
0.0978895 + 0.995197i \(0.468791\pi\)
\(440\) 0 0
\(441\) −12959.5 −1.39936
\(442\) 0 0
\(443\) −7020.76 −0.752972 −0.376486 0.926422i \(-0.622868\pi\)
−0.376486 + 0.926422i \(0.622868\pi\)
\(444\) 0 0
\(445\) 12919.6 1.37629
\(446\) 0 0
\(447\) 16001.9 1.69321
\(448\) 0 0
\(449\) −3318.29 −0.348774 −0.174387 0.984677i \(-0.555794\pi\)
−0.174387 + 0.984677i \(0.555794\pi\)
\(450\) 0 0
\(451\) −8026.35 −0.838018
\(452\) 0 0
\(453\) −25573.2 −2.65239
\(454\) 0 0
\(455\) 10700.9 1.10256
\(456\) 0 0
\(457\) 4468.38 0.457378 0.228689 0.973500i \(-0.426556\pi\)
0.228689 + 0.973500i \(0.426556\pi\)
\(458\) 0 0
\(459\) −8525.84 −0.866998
\(460\) 0 0
\(461\) −1179.45 −0.119159 −0.0595795 0.998224i \(-0.518976\pi\)
−0.0595795 + 0.998224i \(0.518976\pi\)
\(462\) 0 0
\(463\) −13763.1 −1.38148 −0.690742 0.723101i \(-0.742716\pi\)
−0.690742 + 0.723101i \(0.742716\pi\)
\(464\) 0 0
\(465\) 8590.11 0.856682
\(466\) 0 0
\(467\) 17014.1 1.68591 0.842955 0.537984i \(-0.180814\pi\)
0.842955 + 0.537984i \(0.180814\pi\)
\(468\) 0 0
\(469\) 6515.49 0.641487
\(470\) 0 0
\(471\) −18959.7 −1.85481
\(472\) 0 0
\(473\) 4507.28 0.438150
\(474\) 0 0
\(475\) −724.168 −0.0699518
\(476\) 0 0
\(477\) −4796.59 −0.460421
\(478\) 0 0
\(479\) 18976.7 1.81017 0.905083 0.425236i \(-0.139809\pi\)
0.905083 + 0.425236i \(0.139809\pi\)
\(480\) 0 0
\(481\) 1006.79 0.0954379
\(482\) 0 0
\(483\) −10622.3 −1.00068
\(484\) 0 0
\(485\) 5508.55 0.515733
\(486\) 0 0
\(487\) −10579.6 −0.984408 −0.492204 0.870480i \(-0.663809\pi\)
−0.492204 + 0.870480i \(0.663809\pi\)
\(488\) 0 0
\(489\) 28250.7 2.61256
\(490\) 0 0
\(491\) 15200.9 1.39716 0.698579 0.715533i \(-0.253816\pi\)
0.698579 + 0.715533i \(0.253816\pi\)
\(492\) 0 0
\(493\) −1013.21 −0.0925614
\(494\) 0 0
\(495\) 24419.3 2.21730
\(496\) 0 0
\(497\) 4878.77 0.440327
\(498\) 0 0
\(499\) −2553.92 −0.229117 −0.114558 0.993417i \(-0.536545\pi\)
−0.114558 + 0.993417i \(0.536545\pi\)
\(500\) 0 0
\(501\) 11966.8 1.06714
\(502\) 0 0
\(503\) −4863.70 −0.431137 −0.215568 0.976489i \(-0.569160\pi\)
−0.215568 + 0.976489i \(0.569160\pi\)
\(504\) 0 0
\(505\) 11441.5 1.00820
\(506\) 0 0
\(507\) 55247.0 4.83946
\(508\) 0 0
\(509\) −17659.5 −1.53781 −0.768903 0.639366i \(-0.779197\pi\)
−0.768903 + 0.639366i \(0.779197\pi\)
\(510\) 0 0
\(511\) −5581.35 −0.483179
\(512\) 0 0
\(513\) −10674.4 −0.918685
\(514\) 0 0
\(515\) 21685.3 1.85547
\(516\) 0 0
\(517\) 12223.3 1.03980
\(518\) 0 0
\(519\) 25018.0 2.11593
\(520\) 0 0
\(521\) −3778.19 −0.317707 −0.158853 0.987302i \(-0.550780\pi\)
−0.158853 + 0.987302i \(0.550780\pi\)
\(522\) 0 0
\(523\) 2792.67 0.233489 0.116745 0.993162i \(-0.462754\pi\)
0.116745 + 0.993162i \(0.462754\pi\)
\(524\) 0 0
\(525\) −1370.54 −0.113934
\(526\) 0 0
\(527\) −3013.74 −0.249109
\(528\) 0 0
\(529\) 2403.34 0.197529
\(530\) 0 0
\(531\) −29025.3 −2.37211
\(532\) 0 0
\(533\) 18810.7 1.52867
\(534\) 0 0
\(535\) −20471.1 −1.65429
\(536\) 0 0
\(537\) 9596.35 0.771160
\(538\) 0 0
\(539\) −9605.85 −0.767631
\(540\) 0 0
\(541\) 2062.12 0.163877 0.0819383 0.996637i \(-0.473889\pi\)
0.0819383 + 0.996637i \(0.473889\pi\)
\(542\) 0 0
\(543\) −5098.33 −0.402928
\(544\) 0 0
\(545\) −5015.44 −0.394198
\(546\) 0 0
\(547\) −16562.0 −1.29459 −0.647296 0.762239i \(-0.724100\pi\)
−0.647296 + 0.762239i \(0.724100\pi\)
\(548\) 0 0
\(549\) −33260.4 −2.58565
\(550\) 0 0
\(551\) −1268.54 −0.0980795
\(552\) 0 0
\(553\) 487.057 0.0374535
\(554\) 0 0
\(555\) −1163.87 −0.0890157
\(556\) 0 0
\(557\) 9272.94 0.705399 0.352699 0.935737i \(-0.385264\pi\)
0.352699 + 0.935737i \(0.385264\pi\)
\(558\) 0 0
\(559\) −10563.3 −0.799251
\(560\) 0 0
\(561\) −12955.4 −0.975005
\(562\) 0 0
\(563\) 62.0181 0.00464254 0.00232127 0.999997i \(-0.499261\pi\)
0.00232127 + 0.999997i \(0.499261\pi\)
\(564\) 0 0
\(565\) 9047.71 0.673699
\(566\) 0 0
\(567\) −6174.00 −0.457290
\(568\) 0 0
\(569\) 10015.9 0.737938 0.368969 0.929442i \(-0.379711\pi\)
0.368969 + 0.929442i \(0.379711\pi\)
\(570\) 0 0
\(571\) −15543.3 −1.13917 −0.569587 0.821931i \(-0.692897\pi\)
−0.569587 + 0.821931i \(0.692897\pi\)
\(572\) 0 0
\(573\) −7791.26 −0.568036
\(574\) 0 0
\(575\) 1879.95 0.136346
\(576\) 0 0
\(577\) 3776.77 0.272494 0.136247 0.990675i \(-0.456496\pi\)
0.136247 + 0.990675i \(0.456496\pi\)
\(578\) 0 0
\(579\) 6004.42 0.430976
\(580\) 0 0
\(581\) −5559.67 −0.396995
\(582\) 0 0
\(583\) −3555.33 −0.252567
\(584\) 0 0
\(585\) −57229.3 −4.04469
\(586\) 0 0
\(587\) 596.285 0.0419273 0.0209637 0.999780i \(-0.493327\pi\)
0.0209637 + 0.999780i \(0.493327\pi\)
\(588\) 0 0
\(589\) −3773.21 −0.263960
\(590\) 0 0
\(591\) 27941.3 1.94476
\(592\) 0 0
\(593\) 20179.6 1.39743 0.698717 0.715398i \(-0.253755\pi\)
0.698717 + 0.715398i \(0.253755\pi\)
\(594\) 0 0
\(595\) 4340.06 0.299034
\(596\) 0 0
\(597\) −13413.5 −0.919558
\(598\) 0 0
\(599\) 13076.3 0.891960 0.445980 0.895043i \(-0.352855\pi\)
0.445980 + 0.895043i \(0.352855\pi\)
\(600\) 0 0
\(601\) −16866.9 −1.14479 −0.572393 0.819980i \(-0.693985\pi\)
−0.572393 + 0.819980i \(0.693985\pi\)
\(602\) 0 0
\(603\) −34845.3 −2.35325
\(604\) 0 0
\(605\) 2319.18 0.155848
\(606\) 0 0
\(607\) 19953.0 1.33421 0.667107 0.744962i \(-0.267532\pi\)
0.667107 + 0.744962i \(0.267532\pi\)
\(608\) 0 0
\(609\) −2400.82 −0.159747
\(610\) 0 0
\(611\) −28646.6 −1.89676
\(612\) 0 0
\(613\) 7214.32 0.475340 0.237670 0.971346i \(-0.423616\pi\)
0.237670 + 0.971346i \(0.423616\pi\)
\(614\) 0 0
\(615\) −21745.6 −1.42580
\(616\) 0 0
\(617\) 22212.9 1.44936 0.724681 0.689084i \(-0.241987\pi\)
0.724681 + 0.689084i \(0.241987\pi\)
\(618\) 0 0
\(619\) 4349.02 0.282394 0.141197 0.989982i \(-0.454905\pi\)
0.141197 + 0.989982i \(0.454905\pi\)
\(620\) 0 0
\(621\) 27710.8 1.79065
\(622\) 0 0
\(623\) −10740.2 −0.690688
\(624\) 0 0
\(625\) −17329.2 −1.10907
\(626\) 0 0
\(627\) −16220.2 −1.03313
\(628\) 0 0
\(629\) 408.331 0.0258843
\(630\) 0 0
\(631\) −10127.9 −0.638966 −0.319483 0.947592i \(-0.603509\pi\)
−0.319483 + 0.947592i \(0.603509\pi\)
\(632\) 0 0
\(633\) −15835.9 −0.994347
\(634\) 0 0
\(635\) 25602.5 1.60001
\(636\) 0 0
\(637\) 22512.4 1.40027
\(638\) 0 0
\(639\) −26092.0 −1.61531
\(640\) 0 0
\(641\) −17624.6 −1.08601 −0.543003 0.839731i \(-0.682713\pi\)
−0.543003 + 0.839731i \(0.682713\pi\)
\(642\) 0 0
\(643\) 27238.1 1.67055 0.835277 0.549829i \(-0.185307\pi\)
0.835277 + 0.549829i \(0.185307\pi\)
\(644\) 0 0
\(645\) 12211.5 0.745468
\(646\) 0 0
\(647\) −28702.1 −1.74404 −0.872021 0.489469i \(-0.837191\pi\)
−0.872021 + 0.489469i \(0.837191\pi\)
\(648\) 0 0
\(649\) −21514.1 −1.30124
\(650\) 0 0
\(651\) −7141.09 −0.429925
\(652\) 0 0
\(653\) −7937.84 −0.475700 −0.237850 0.971302i \(-0.576443\pi\)
−0.237850 + 0.971302i \(0.576443\pi\)
\(654\) 0 0
\(655\) −8426.62 −0.502680
\(656\) 0 0
\(657\) 29849.5 1.77251
\(658\) 0 0
\(659\) 3675.29 0.217252 0.108626 0.994083i \(-0.465355\pi\)
0.108626 + 0.994083i \(0.465355\pi\)
\(660\) 0 0
\(661\) 4205.49 0.247466 0.123733 0.992316i \(-0.460513\pi\)
0.123733 + 0.992316i \(0.460513\pi\)
\(662\) 0 0
\(663\) 30362.5 1.77855
\(664\) 0 0
\(665\) 5433.76 0.316860
\(666\) 0 0
\(667\) 3293.15 0.191171
\(668\) 0 0
\(669\) 48334.3 2.79329
\(670\) 0 0
\(671\) −24653.3 −1.41838
\(672\) 0 0
\(673\) −4634.55 −0.265452 −0.132726 0.991153i \(-0.542373\pi\)
−0.132726 + 0.991153i \(0.542373\pi\)
\(674\) 0 0
\(675\) 3575.40 0.203877
\(676\) 0 0
\(677\) −5327.42 −0.302437 −0.151218 0.988500i \(-0.548320\pi\)
−0.151218 + 0.988500i \(0.548320\pi\)
\(678\) 0 0
\(679\) −4579.34 −0.258820
\(680\) 0 0
\(681\) 33429.6 1.88109
\(682\) 0 0
\(683\) −2816.05 −0.157764 −0.0788822 0.996884i \(-0.525135\pi\)
−0.0788822 + 0.996884i \(0.525135\pi\)
\(684\) 0 0
\(685\) −11703.8 −0.652816
\(686\) 0 0
\(687\) −6077.77 −0.337528
\(688\) 0 0
\(689\) 8332.33 0.460720
\(690\) 0 0
\(691\) 16910.1 0.930955 0.465477 0.885060i \(-0.345883\pi\)
0.465477 + 0.885060i \(0.345883\pi\)
\(692\) 0 0
\(693\) −20300.1 −1.11275
\(694\) 0 0
\(695\) 27674.6 1.51044
\(696\) 0 0
\(697\) 7629.19 0.414600
\(698\) 0 0
\(699\) −6064.67 −0.328164
\(700\) 0 0
\(701\) −30588.4 −1.64808 −0.824042 0.566528i \(-0.808286\pi\)
−0.824042 + 0.566528i \(0.808286\pi\)
\(702\) 0 0
\(703\) 511.232 0.0274274
\(704\) 0 0
\(705\) 33116.2 1.76912
\(706\) 0 0
\(707\) −9511.48 −0.505963
\(708\) 0 0
\(709\) 21242.9 1.12524 0.562619 0.826716i \(-0.309794\pi\)
0.562619 + 0.826716i \(0.309794\pi\)
\(710\) 0 0
\(711\) −2604.82 −0.137396
\(712\) 0 0
\(713\) 9795.28 0.514496
\(714\) 0 0
\(715\) −42419.5 −2.21874
\(716\) 0 0
\(717\) −34831.3 −1.81423
\(718\) 0 0
\(719\) −12452.7 −0.645907 −0.322954 0.946415i \(-0.604676\pi\)
−0.322954 + 0.946415i \(0.604676\pi\)
\(720\) 0 0
\(721\) −18027.3 −0.931168
\(722\) 0 0
\(723\) 52316.8 2.69112
\(724\) 0 0
\(725\) 424.901 0.0217661
\(726\) 0 0
\(727\) 27178.3 1.38650 0.693251 0.720696i \(-0.256178\pi\)
0.693251 + 0.720696i \(0.256178\pi\)
\(728\) 0 0
\(729\) −22321.3 −1.13404
\(730\) 0 0
\(731\) −4284.25 −0.216770
\(732\) 0 0
\(733\) −21999.7 −1.10856 −0.554282 0.832329i \(-0.687007\pi\)
−0.554282 + 0.832329i \(0.687007\pi\)
\(734\) 0 0
\(735\) −26024.9 −1.30605
\(736\) 0 0
\(737\) −25828.1 −1.29089
\(738\) 0 0
\(739\) −13644.0 −0.679166 −0.339583 0.940576i \(-0.610286\pi\)
−0.339583 + 0.940576i \(0.610286\pi\)
\(740\) 0 0
\(741\) 38013.9 1.88458
\(742\) 0 0
\(743\) 13980.2 0.690286 0.345143 0.938550i \(-0.387830\pi\)
0.345143 + 0.938550i \(0.387830\pi\)
\(744\) 0 0
\(745\) 21250.1 1.04502
\(746\) 0 0
\(747\) 29733.5 1.45635
\(748\) 0 0
\(749\) 17018.0 0.830204
\(750\) 0 0
\(751\) 21138.7 1.02711 0.513557 0.858055i \(-0.328327\pi\)
0.513557 + 0.858055i \(0.328327\pi\)
\(752\) 0 0
\(753\) −30190.6 −1.46110
\(754\) 0 0
\(755\) −33960.5 −1.63702
\(756\) 0 0
\(757\) −28328.1 −1.36011 −0.680054 0.733162i \(-0.738044\pi\)
−0.680054 + 0.733162i \(0.738044\pi\)
\(758\) 0 0
\(759\) 42107.8 2.01372
\(760\) 0 0
\(761\) −1093.79 −0.0521023 −0.0260512 0.999661i \(-0.508293\pi\)
−0.0260512 + 0.999661i \(0.508293\pi\)
\(762\) 0 0
\(763\) 4169.41 0.197828
\(764\) 0 0
\(765\) −23210.9 −1.09699
\(766\) 0 0
\(767\) 50420.8 2.37365
\(768\) 0 0
\(769\) −1659.21 −0.0778059 −0.0389029 0.999243i \(-0.512386\pi\)
−0.0389029 + 0.999243i \(0.512386\pi\)
\(770\) 0 0
\(771\) 32607.7 1.52314
\(772\) 0 0
\(773\) −7537.89 −0.350736 −0.175368 0.984503i \(-0.556112\pi\)
−0.175368 + 0.984503i \(0.556112\pi\)
\(774\) 0 0
\(775\) 1263.84 0.0585787
\(776\) 0 0
\(777\) 967.546 0.0446725
\(778\) 0 0
\(779\) 9551.76 0.439316
\(780\) 0 0
\(781\) −19339.9 −0.886091
\(782\) 0 0
\(783\) 6263.11 0.285856
\(784\) 0 0
\(785\) −25177.9 −1.14476
\(786\) 0 0
\(787\) −9304.20 −0.421422 −0.210711 0.977548i \(-0.567578\pi\)
−0.210711 + 0.977548i \(0.567578\pi\)
\(788\) 0 0
\(789\) −21987.8 −0.992126
\(790\) 0 0
\(791\) −7521.50 −0.338096
\(792\) 0 0
\(793\) 57777.8 2.58733
\(794\) 0 0
\(795\) −9632.39 −0.429718
\(796\) 0 0
\(797\) −38619.5 −1.71640 −0.858201 0.513314i \(-0.828418\pi\)
−0.858201 + 0.513314i \(0.828418\pi\)
\(798\) 0 0
\(799\) −11618.4 −0.514431
\(800\) 0 0
\(801\) 57439.7 2.53375
\(802\) 0 0
\(803\) 22125.1 0.972324
\(804\) 0 0
\(805\) −14106.1 −0.617608
\(806\) 0 0
\(807\) −4307.88 −0.187912
\(808\) 0 0
\(809\) −18550.5 −0.806179 −0.403090 0.915160i \(-0.632064\pi\)
−0.403090 + 0.915160i \(0.632064\pi\)
\(810\) 0 0
\(811\) −3704.80 −0.160411 −0.0802054 0.996778i \(-0.525558\pi\)
−0.0802054 + 0.996778i \(0.525558\pi\)
\(812\) 0 0
\(813\) 32163.7 1.38749
\(814\) 0 0
\(815\) 37516.2 1.61243
\(816\) 0 0
\(817\) −5363.90 −0.229693
\(818\) 0 0
\(819\) 47575.6 2.02982
\(820\) 0 0
\(821\) 5544.86 0.235709 0.117854 0.993031i \(-0.462398\pi\)
0.117854 + 0.993031i \(0.462398\pi\)
\(822\) 0 0
\(823\) −3915.39 −0.165835 −0.0829174 0.996556i \(-0.526424\pi\)
−0.0829174 + 0.996556i \(0.526424\pi\)
\(824\) 0 0
\(825\) 5432.98 0.229275
\(826\) 0 0
\(827\) −35091.2 −1.47550 −0.737751 0.675073i \(-0.764112\pi\)
−0.737751 + 0.675073i \(0.764112\pi\)
\(828\) 0 0
\(829\) 3532.71 0.148005 0.0740025 0.997258i \(-0.476423\pi\)
0.0740025 + 0.997258i \(0.476423\pi\)
\(830\) 0 0
\(831\) −55156.0 −2.30245
\(832\) 0 0
\(833\) 9130.53 0.379777
\(834\) 0 0
\(835\) 15891.6 0.658624
\(836\) 0 0
\(837\) 18629.3 0.769320
\(838\) 0 0
\(839\) −30081.1 −1.23780 −0.618900 0.785470i \(-0.712421\pi\)
−0.618900 + 0.785470i \(0.712421\pi\)
\(840\) 0 0
\(841\) −23644.7 −0.969482
\(842\) 0 0
\(843\) −4852.27 −0.198246
\(844\) 0 0
\(845\) 73366.5 2.98685
\(846\) 0 0
\(847\) −1927.97 −0.0782121
\(848\) 0 0
\(849\) 58766.4 2.37557
\(850\) 0 0
\(851\) −1327.16 −0.0534601
\(852\) 0 0
\(853\) 11222.1 0.450453 0.225226 0.974306i \(-0.427688\pi\)
0.225226 + 0.974306i \(0.427688\pi\)
\(854\) 0 0
\(855\) −29060.2 −1.16238
\(856\) 0 0
\(857\) −27486.0 −1.09557 −0.547785 0.836619i \(-0.684529\pi\)
−0.547785 + 0.836619i \(0.684529\pi\)
\(858\) 0 0
\(859\) 14976.2 0.594855 0.297427 0.954744i \(-0.403871\pi\)
0.297427 + 0.954744i \(0.403871\pi\)
\(860\) 0 0
\(861\) 18077.5 0.715538
\(862\) 0 0
\(863\) −9419.49 −0.371545 −0.185772 0.982593i \(-0.559479\pi\)
−0.185772 + 0.982593i \(0.559479\pi\)
\(864\) 0 0
\(865\) 33223.2 1.30592
\(866\) 0 0
\(867\) −31549.9 −1.23586
\(868\) 0 0
\(869\) −1930.75 −0.0753694
\(870\) 0 0
\(871\) 60531.0 2.35478
\(872\) 0 0
\(873\) 24490.7 0.949465
\(874\) 0 0
\(875\) 12787.6 0.494059
\(876\) 0 0
\(877\) −21069.8 −0.811263 −0.405631 0.914037i \(-0.632948\pi\)
−0.405631 + 0.914037i \(0.632948\pi\)
\(878\) 0 0
\(879\) −60298.2 −2.31378
\(880\) 0 0
\(881\) −14511.7 −0.554952 −0.277476 0.960733i \(-0.589498\pi\)
−0.277476 + 0.960733i \(0.589498\pi\)
\(882\) 0 0
\(883\) −26515.9 −1.01057 −0.505284 0.862953i \(-0.668612\pi\)
−0.505284 + 0.862953i \(0.668612\pi\)
\(884\) 0 0
\(885\) −58287.8 −2.21392
\(886\) 0 0
\(887\) −29223.6 −1.10624 −0.553119 0.833102i \(-0.686563\pi\)
−0.553119 + 0.833102i \(0.686563\pi\)
\(888\) 0 0
\(889\) −21283.8 −0.802964
\(890\) 0 0
\(891\) 24474.4 0.920227
\(892\) 0 0
\(893\) −14546.3 −0.545099
\(894\) 0 0
\(895\) 12743.7 0.475949
\(896\) 0 0
\(897\) −98684.4 −3.67333
\(898\) 0 0
\(899\) 2213.90 0.0821332
\(900\) 0 0
\(901\) 3379.41 0.124955
\(902\) 0 0
\(903\) −10151.6 −0.374113
\(904\) 0 0
\(905\) −6770.44 −0.248682
\(906\) 0 0
\(907\) 32139.3 1.17659 0.588296 0.808646i \(-0.299799\pi\)
0.588296 + 0.808646i \(0.299799\pi\)
\(908\) 0 0
\(909\) 50868.1 1.85609
\(910\) 0 0
\(911\) 21430.7 0.779396 0.389698 0.920943i \(-0.372579\pi\)
0.389698 + 0.920943i \(0.372579\pi\)
\(912\) 0 0
\(913\) 22039.1 0.798891
\(914\) 0 0
\(915\) −66792.7 −2.41322
\(916\) 0 0
\(917\) 7005.17 0.252270
\(918\) 0 0
\(919\) 11676.1 0.419106 0.209553 0.977797i \(-0.432799\pi\)
0.209553 + 0.977797i \(0.432799\pi\)
\(920\) 0 0
\(921\) −16266.1 −0.581962
\(922\) 0 0
\(923\) 45325.3 1.61636
\(924\) 0 0
\(925\) −171.238 −0.00608677
\(926\) 0 0
\(927\) 96411.5 3.41593
\(928\) 0 0
\(929\) 41969.7 1.48222 0.741110 0.671383i \(-0.234300\pi\)
0.741110 + 0.671383i \(0.234300\pi\)
\(930\) 0 0
\(931\) 11431.4 0.402417
\(932\) 0 0
\(933\) 94622.6 3.32026
\(934\) 0 0
\(935\) −17204.4 −0.601759
\(936\) 0 0
\(937\) −45948.6 −1.60200 −0.801001 0.598663i \(-0.795699\pi\)
−0.801001 + 0.598663i \(0.795699\pi\)
\(938\) 0 0
\(939\) −71155.0 −2.47290
\(940\) 0 0
\(941\) −15000.1 −0.519648 −0.259824 0.965656i \(-0.583665\pi\)
−0.259824 + 0.965656i \(0.583665\pi\)
\(942\) 0 0
\(943\) −24796.4 −0.856292
\(944\) 0 0
\(945\) −26827.8 −0.923502
\(946\) 0 0
\(947\) −37423.2 −1.28415 −0.642075 0.766642i \(-0.721926\pi\)
−0.642075 + 0.766642i \(0.721926\pi\)
\(948\) 0 0
\(949\) −51852.6 −1.77366
\(950\) 0 0
\(951\) −69404.8 −2.36657
\(952\) 0 0
\(953\) −55403.0 −1.88319 −0.941594 0.336750i \(-0.890672\pi\)
−0.941594 + 0.336750i \(0.890672\pi\)
\(954\) 0 0
\(955\) −10346.6 −0.350584
\(956\) 0 0
\(957\) 9517.09 0.321467
\(958\) 0 0
\(959\) 9729.54 0.327615
\(960\) 0 0
\(961\) −23205.9 −0.778956
\(962\) 0 0
\(963\) −91013.4 −3.04555
\(964\) 0 0
\(965\) 7973.70 0.265992
\(966\) 0 0
\(967\) 31440.5 1.04556 0.522780 0.852467i \(-0.324895\pi\)
0.522780 + 0.852467i \(0.324895\pi\)
\(968\) 0 0
\(969\) 15417.6 0.511129
\(970\) 0 0
\(971\) −19010.3 −0.628290 −0.314145 0.949375i \(-0.601718\pi\)
−0.314145 + 0.949375i \(0.601718\pi\)
\(972\) 0 0
\(973\) −23006.3 −0.758015
\(974\) 0 0
\(975\) −12732.8 −0.418232
\(976\) 0 0
\(977\) −8504.01 −0.278472 −0.139236 0.990259i \(-0.544465\pi\)
−0.139236 + 0.990259i \(0.544465\pi\)
\(978\) 0 0
\(979\) 42575.4 1.38990
\(980\) 0 0
\(981\) −22298.3 −0.725720
\(982\) 0 0
\(983\) −52828.3 −1.71410 −0.857051 0.515232i \(-0.827706\pi\)
−0.857051 + 0.515232i \(0.827706\pi\)
\(984\) 0 0
\(985\) 37105.3 1.20028
\(986\) 0 0
\(987\) −27530.0 −0.887831
\(988\) 0 0
\(989\) 13924.7 0.447705
\(990\) 0 0
\(991\) −13275.5 −0.425541 −0.212770 0.977102i \(-0.568249\pi\)
−0.212770 + 0.977102i \(0.568249\pi\)
\(992\) 0 0
\(993\) −73326.1 −2.34334
\(994\) 0 0
\(995\) −17812.7 −0.567538
\(996\) 0 0
\(997\) −20858.1 −0.662570 −0.331285 0.943531i \(-0.607482\pi\)
−0.331285 + 0.943531i \(0.607482\pi\)
\(998\) 0 0
\(999\) −2524.08 −0.0799382
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 128.4.a.g.1.2 yes 2
3.2 odd 2 1152.4.a.t.1.1 2
4.3 odd 2 128.4.a.e.1.1 2
8.3 odd 2 128.4.a.h.1.2 yes 2
8.5 even 2 128.4.a.f.1.1 yes 2
12.11 even 2 1152.4.a.s.1.1 2
16.3 odd 4 256.4.b.i.129.1 4
16.5 even 4 256.4.b.h.129.1 4
16.11 odd 4 256.4.b.i.129.4 4
16.13 even 4 256.4.b.h.129.4 4
24.5 odd 2 1152.4.a.r.1.2 2
24.11 even 2 1152.4.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.a.e.1.1 2 4.3 odd 2
128.4.a.f.1.1 yes 2 8.5 even 2
128.4.a.g.1.2 yes 2 1.1 even 1 trivial
128.4.a.h.1.2 yes 2 8.3 odd 2
256.4.b.h.129.1 4 16.5 even 4
256.4.b.h.129.4 4 16.13 even 4
256.4.b.i.129.1 4 16.3 odd 4
256.4.b.i.129.4 4 16.11 odd 4
1152.4.a.q.1.2 2 24.11 even 2
1152.4.a.r.1.2 2 24.5 odd 2
1152.4.a.s.1.1 2 12.11 even 2
1152.4.a.t.1.1 2 3.2 odd 2