Properties

Label 1856.4.a.p.1.3
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.97021 q^{3} +12.4882 q^{5} -28.6773 q^{7} +8.64344 q^{9} +O(q^{10})\) \(q+5.97021 q^{3} +12.4882 q^{5} -28.6773 q^{7} +8.64344 q^{9} +18.2099 q^{11} -37.9978 q^{13} +74.5574 q^{15} -3.95162 q^{17} +36.8211 q^{19} -171.209 q^{21} +42.7480 q^{23} +30.9561 q^{25} -109.593 q^{27} +29.0000 q^{29} -160.731 q^{31} +108.717 q^{33} -358.129 q^{35} -313.040 q^{37} -226.855 q^{39} +496.787 q^{41} +139.195 q^{43} +107.941 q^{45} -417.656 q^{47} +479.386 q^{49} -23.5920 q^{51} +137.116 q^{53} +227.409 q^{55} +219.830 q^{57} -190.033 q^{59} -161.072 q^{61} -247.870 q^{63} -474.525 q^{65} -125.259 q^{67} +255.215 q^{69} -165.110 q^{71} -938.243 q^{73} +184.815 q^{75} -522.209 q^{77} -1315.60 q^{79} -887.664 q^{81} -505.294 q^{83} -49.3487 q^{85} +173.136 q^{87} -769.587 q^{89} +1089.67 q^{91} -959.599 q^{93} +459.830 q^{95} +1333.29 q^{97} +157.396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{3} - 4 q^{5} + 16 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{3} - 4 q^{5} + 16 q^{7} + 45 q^{9} + 2 q^{11} - 28 q^{13} + 136 q^{15} - 66 q^{17} + 66 q^{19} - 472 q^{21} + 176 q^{23} - 9 q^{25} - 228 q^{27} + 87 q^{29} - 190 q^{31} + 154 q^{33} - 660 q^{35} - 442 q^{37} - 656 q^{39} + 1162 q^{41} - 30 q^{43} + 254 q^{45} - 738 q^{47} + 851 q^{49} + 576 q^{51} - 312 q^{53} + 464 q^{55} + 684 q^{57} - 44 q^{59} - 54 q^{61} + 964 q^{63} + 178 q^{65} + 116 q^{67} - 812 q^{69} - 1200 q^{71} - 1118 q^{73} + 1038 q^{75} - 792 q^{77} - 2262 q^{79} + 15 q^{81} + 1804 q^{83} + 8 q^{85} - 174 q^{87} + 1578 q^{89} + 1972 q^{91} + 706 q^{93} - 1052 q^{95} + 1450 q^{97} + 482 q^{99}+O(q^{100}) \) 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
\(3\) 5.97021 1.14897 0.574484 0.818516i \(-0.305203\pi\)
0.574484 + 0.818516i \(0.305203\pi\)
\(4\) 0 0
\(5\) 12.4882 1.11698 0.558491 0.829511i \(-0.311381\pi\)
0.558491 + 0.829511i \(0.311381\pi\)
\(6\) 0 0
\(7\) −28.6773 −1.54843 −0.774213 0.632925i \(-0.781854\pi\)
−0.774213 + 0.632925i \(0.781854\pi\)
\(8\) 0 0
\(9\) 8.64344 0.320127
\(10\) 0 0
\(11\) 18.2099 0.499134 0.249567 0.968358i \(-0.419712\pi\)
0.249567 + 0.968358i \(0.419712\pi\)
\(12\) 0 0
\(13\) −37.9978 −0.810668 −0.405334 0.914169i \(-0.632845\pi\)
−0.405334 + 0.914169i \(0.632845\pi\)
\(14\) 0 0
\(15\) 74.5574 1.28338
\(16\) 0 0
\(17\) −3.95162 −0.0563769 −0.0281885 0.999603i \(-0.508974\pi\)
−0.0281885 + 0.999603i \(0.508974\pi\)
\(18\) 0 0
\(19\) 36.8211 0.444597 0.222298 0.974979i \(-0.428644\pi\)
0.222298 + 0.974979i \(0.428644\pi\)
\(20\) 0 0
\(21\) −171.209 −1.77909
\(22\) 0 0
\(23\) 42.7480 0.387547 0.193773 0.981046i \(-0.437927\pi\)
0.193773 + 0.981046i \(0.437927\pi\)
\(24\) 0 0
\(25\) 30.9561 0.247649
\(26\) 0 0
\(27\) −109.593 −0.781152
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −160.731 −0.931231 −0.465616 0.884987i \(-0.654167\pi\)
−0.465616 + 0.884987i \(0.654167\pi\)
\(32\) 0 0
\(33\) 108.717 0.573489
\(34\) 0 0
\(35\) −358.129 −1.72957
\(36\) 0 0
\(37\) −313.040 −1.39090 −0.695451 0.718573i \(-0.744796\pi\)
−0.695451 + 0.718573i \(0.744796\pi\)
\(38\) 0 0
\(39\) −226.855 −0.931432
\(40\) 0 0
\(41\) 496.787 1.89232 0.946159 0.323703i \(-0.104928\pi\)
0.946159 + 0.323703i \(0.104928\pi\)
\(42\) 0 0
\(43\) 139.195 0.493653 0.246827 0.969060i \(-0.420612\pi\)
0.246827 + 0.969060i \(0.420612\pi\)
\(44\) 0 0
\(45\) 107.941 0.357577
\(46\) 0 0
\(47\) −417.656 −1.29620 −0.648100 0.761556i \(-0.724436\pi\)
−0.648100 + 0.761556i \(0.724436\pi\)
\(48\) 0 0
\(49\) 479.386 1.39763
\(50\) 0 0
\(51\) −23.5920 −0.0647753
\(52\) 0 0
\(53\) 137.116 0.355365 0.177683 0.984088i \(-0.443140\pi\)
0.177683 + 0.984088i \(0.443140\pi\)
\(54\) 0 0
\(55\) 227.409 0.557524
\(56\) 0 0
\(57\) 219.830 0.510827
\(58\) 0 0
\(59\) −190.033 −0.419326 −0.209663 0.977774i \(-0.567237\pi\)
−0.209663 + 0.977774i \(0.567237\pi\)
\(60\) 0 0
\(61\) −161.072 −0.338084 −0.169042 0.985609i \(-0.554067\pi\)
−0.169042 + 0.985609i \(0.554067\pi\)
\(62\) 0 0
\(63\) −247.870 −0.495694
\(64\) 0 0
\(65\) −474.525 −0.905502
\(66\) 0 0
\(67\) −125.259 −0.228400 −0.114200 0.993458i \(-0.536430\pi\)
−0.114200 + 0.993458i \(0.536430\pi\)
\(68\) 0 0
\(69\) 255.215 0.445279
\(70\) 0 0
\(71\) −165.110 −0.275984 −0.137992 0.990433i \(-0.544065\pi\)
−0.137992 + 0.990433i \(0.544065\pi\)
\(72\) 0 0
\(73\) −938.243 −1.50429 −0.752144 0.658999i \(-0.770980\pi\)
−0.752144 + 0.658999i \(0.770980\pi\)
\(74\) 0 0
\(75\) 184.815 0.284541
\(76\) 0 0
\(77\) −522.209 −0.772873
\(78\) 0 0
\(79\) −1315.60 −1.87363 −0.936816 0.349821i \(-0.886242\pi\)
−0.936816 + 0.349821i \(0.886242\pi\)
\(80\) 0 0
\(81\) −887.664 −1.21765
\(82\) 0 0
\(83\) −505.294 −0.668231 −0.334116 0.942532i \(-0.608438\pi\)
−0.334116 + 0.942532i \(0.608438\pi\)
\(84\) 0 0
\(85\) −49.3487 −0.0629720
\(86\) 0 0
\(87\) 173.136 0.213358
\(88\) 0 0
\(89\) −769.587 −0.916584 −0.458292 0.888802i \(-0.651539\pi\)
−0.458292 + 0.888802i \(0.651539\pi\)
\(90\) 0 0
\(91\) 1089.67 1.25526
\(92\) 0 0
\(93\) −959.599 −1.06996
\(94\) 0 0
\(95\) 459.830 0.496606
\(96\) 0 0
\(97\) 1333.29 1.39561 0.697807 0.716286i \(-0.254159\pi\)
0.697807 + 0.716286i \(0.254159\pi\)
\(98\) 0 0
\(99\) 157.396 0.159787
\(100\) 0 0
\(101\) −51.3094 −0.0505493 −0.0252746 0.999681i \(-0.508046\pi\)
−0.0252746 + 0.999681i \(0.508046\pi\)
\(102\) 0 0
\(103\) 1062.26 1.01619 0.508097 0.861300i \(-0.330349\pi\)
0.508097 + 0.861300i \(0.330349\pi\)
\(104\) 0 0
\(105\) −2138.10 −1.98721
\(106\) 0 0
\(107\) −1414.85 −1.27831 −0.639155 0.769078i \(-0.720716\pi\)
−0.639155 + 0.769078i \(0.720716\pi\)
\(108\) 0 0
\(109\) −495.330 −0.435266 −0.217633 0.976031i \(-0.569834\pi\)
−0.217633 + 0.976031i \(0.569834\pi\)
\(110\) 0 0
\(111\) −1868.91 −1.59810
\(112\) 0 0
\(113\) −603.969 −0.502802 −0.251401 0.967883i \(-0.580891\pi\)
−0.251401 + 0.967883i \(0.580891\pi\)
\(114\) 0 0
\(115\) 533.848 0.432883
\(116\) 0 0
\(117\) −328.432 −0.259517
\(118\) 0 0
\(119\) 113.322 0.0872956
\(120\) 0 0
\(121\) −999.401 −0.750865
\(122\) 0 0
\(123\) 2965.92 2.17421
\(124\) 0 0
\(125\) −1174.44 −0.840363
\(126\) 0 0
\(127\) −1282.23 −0.895902 −0.447951 0.894058i \(-0.647846\pi\)
−0.447951 + 0.894058i \(0.647846\pi\)
\(128\) 0 0
\(129\) 831.026 0.567192
\(130\) 0 0
\(131\) −2446.25 −1.63152 −0.815762 0.578388i \(-0.803682\pi\)
−0.815762 + 0.578388i \(0.803682\pi\)
\(132\) 0 0
\(133\) −1055.93 −0.688425
\(134\) 0 0
\(135\) −1368.62 −0.872533
\(136\) 0 0
\(137\) 1259.91 0.785703 0.392852 0.919602i \(-0.371489\pi\)
0.392852 + 0.919602i \(0.371489\pi\)
\(138\) 0 0
\(139\) 822.786 0.502070 0.251035 0.967978i \(-0.419229\pi\)
0.251035 + 0.967978i \(0.419229\pi\)
\(140\) 0 0
\(141\) −2493.49 −1.48929
\(142\) 0 0
\(143\) −691.934 −0.404632
\(144\) 0 0
\(145\) 362.159 0.207418
\(146\) 0 0
\(147\) 2862.03 1.60583
\(148\) 0 0
\(149\) 477.366 0.262466 0.131233 0.991352i \(-0.458106\pi\)
0.131233 + 0.991352i \(0.458106\pi\)
\(150\) 0 0
\(151\) −1740.92 −0.938240 −0.469120 0.883135i \(-0.655429\pi\)
−0.469120 + 0.883135i \(0.655429\pi\)
\(152\) 0 0
\(153\) −34.1556 −0.0180478
\(154\) 0 0
\(155\) −2007.25 −1.04017
\(156\) 0 0
\(157\) 1372.90 0.697892 0.348946 0.937143i \(-0.386540\pi\)
0.348946 + 0.937143i \(0.386540\pi\)
\(158\) 0 0
\(159\) 818.613 0.408304
\(160\) 0 0
\(161\) −1225.90 −0.600088
\(162\) 0 0
\(163\) −3688.03 −1.77220 −0.886100 0.463495i \(-0.846595\pi\)
−0.886100 + 0.463495i \(0.846595\pi\)
\(164\) 0 0
\(165\) 1357.68 0.640577
\(166\) 0 0
\(167\) 4059.56 1.88107 0.940534 0.339701i \(-0.110326\pi\)
0.940534 + 0.339701i \(0.110326\pi\)
\(168\) 0 0
\(169\) −753.169 −0.342817
\(170\) 0 0
\(171\) 318.261 0.142328
\(172\) 0 0
\(173\) −2811.92 −1.23576 −0.617879 0.786274i \(-0.712008\pi\)
−0.617879 + 0.786274i \(0.712008\pi\)
\(174\) 0 0
\(175\) −887.737 −0.383466
\(176\) 0 0
\(177\) −1134.54 −0.481792
\(178\) 0 0
\(179\) −163.787 −0.0683913 −0.0341956 0.999415i \(-0.510887\pi\)
−0.0341956 + 0.999415i \(0.510887\pi\)
\(180\) 0 0
\(181\) 3540.42 1.45391 0.726954 0.686686i \(-0.240935\pi\)
0.726954 + 0.686686i \(0.240935\pi\)
\(182\) 0 0
\(183\) −961.632 −0.388447
\(184\) 0 0
\(185\) −3909.31 −1.55361
\(186\) 0 0
\(187\) −71.9584 −0.0281397
\(188\) 0 0
\(189\) 3142.82 1.20956
\(190\) 0 0
\(191\) 2163.65 0.819665 0.409833 0.912161i \(-0.365587\pi\)
0.409833 + 0.912161i \(0.365587\pi\)
\(192\) 0 0
\(193\) 4632.75 1.72784 0.863919 0.503630i \(-0.168003\pi\)
0.863919 + 0.503630i \(0.168003\pi\)
\(194\) 0 0
\(195\) −2833.02 −1.04039
\(196\) 0 0
\(197\) 1239.91 0.448427 0.224214 0.974540i \(-0.428019\pi\)
0.224214 + 0.974540i \(0.428019\pi\)
\(198\) 0 0
\(199\) 4651.45 1.65695 0.828474 0.560028i \(-0.189210\pi\)
0.828474 + 0.560028i \(0.189210\pi\)
\(200\) 0 0
\(201\) −747.820 −0.262424
\(202\) 0 0
\(203\) −831.641 −0.287536
\(204\) 0 0
\(205\) 6203.99 2.11369
\(206\) 0 0
\(207\) 369.490 0.124064
\(208\) 0 0
\(209\) 670.506 0.221913
\(210\) 0 0
\(211\) −3516.90 −1.14746 −0.573728 0.819046i \(-0.694503\pi\)
−0.573728 + 0.819046i \(0.694503\pi\)
\(212\) 0 0
\(213\) −985.739 −0.317097
\(214\) 0 0
\(215\) 1738.30 0.551402
\(216\) 0 0
\(217\) 4609.33 1.44194
\(218\) 0 0
\(219\) −5601.51 −1.72838
\(220\) 0 0
\(221\) 150.153 0.0457030
\(222\) 0 0
\(223\) 3480.17 1.04506 0.522532 0.852620i \(-0.324988\pi\)
0.522532 + 0.852620i \(0.324988\pi\)
\(224\) 0 0
\(225\) 267.567 0.0792792
\(226\) 0 0
\(227\) −1259.51 −0.368266 −0.184133 0.982901i \(-0.558948\pi\)
−0.184133 + 0.982901i \(0.558948\pi\)
\(228\) 0 0
\(229\) 274.295 0.0791524 0.0395762 0.999217i \(-0.487399\pi\)
0.0395762 + 0.999217i \(0.487399\pi\)
\(230\) 0 0
\(231\) −3117.70 −0.888006
\(232\) 0 0
\(233\) 6517.70 1.83257 0.916285 0.400526i \(-0.131173\pi\)
0.916285 + 0.400526i \(0.131173\pi\)
\(234\) 0 0
\(235\) −5215.79 −1.44783
\(236\) 0 0
\(237\) −7854.43 −2.15274
\(238\) 0 0
\(239\) −6052.89 −1.63820 −0.819098 0.573654i \(-0.805526\pi\)
−0.819098 + 0.573654i \(0.805526\pi\)
\(240\) 0 0
\(241\) 2265.19 0.605452 0.302726 0.953078i \(-0.402103\pi\)
0.302726 + 0.953078i \(0.402103\pi\)
\(242\) 0 0
\(243\) −2340.54 −0.617884
\(244\) 0 0
\(245\) 5986.68 1.56112
\(246\) 0 0
\(247\) −1399.12 −0.360420
\(248\) 0 0
\(249\) −3016.71 −0.767777
\(250\) 0 0
\(251\) −6653.27 −1.67311 −0.836555 0.547882i \(-0.815434\pi\)
−0.836555 + 0.547882i \(0.815434\pi\)
\(252\) 0 0
\(253\) 778.435 0.193438
\(254\) 0 0
\(255\) −294.622 −0.0723528
\(256\) 0 0
\(257\) −3144.58 −0.763244 −0.381622 0.924319i \(-0.624634\pi\)
−0.381622 + 0.924319i \(0.624634\pi\)
\(258\) 0 0
\(259\) 8977.12 2.15371
\(260\) 0 0
\(261\) 250.660 0.0594462
\(262\) 0 0
\(263\) 2409.66 0.564965 0.282483 0.959272i \(-0.408842\pi\)
0.282483 + 0.959272i \(0.408842\pi\)
\(264\) 0 0
\(265\) 1712.34 0.396937
\(266\) 0 0
\(267\) −4594.60 −1.05313
\(268\) 0 0
\(269\) 4542.85 1.02968 0.514838 0.857288i \(-0.327852\pi\)
0.514838 + 0.857288i \(0.327852\pi\)
\(270\) 0 0
\(271\) −6519.35 −1.46134 −0.730668 0.682733i \(-0.760791\pi\)
−0.730668 + 0.682733i \(0.760791\pi\)
\(272\) 0 0
\(273\) 6505.58 1.44225
\(274\) 0 0
\(275\) 563.706 0.123610
\(276\) 0 0
\(277\) 7440.91 1.61401 0.807005 0.590544i \(-0.201087\pi\)
0.807005 + 0.590544i \(0.201087\pi\)
\(278\) 0 0
\(279\) −1389.27 −0.298113
\(280\) 0 0
\(281\) 272.086 0.0577625 0.0288813 0.999583i \(-0.490806\pi\)
0.0288813 + 0.999583i \(0.490806\pi\)
\(282\) 0 0
\(283\) 1716.92 0.360636 0.180318 0.983608i \(-0.442287\pi\)
0.180318 + 0.983608i \(0.442287\pi\)
\(284\) 0 0
\(285\) 2745.29 0.570585
\(286\) 0 0
\(287\) −14246.5 −2.93012
\(288\) 0 0
\(289\) −4897.38 −0.996822
\(290\) 0 0
\(291\) 7960.00 1.60352
\(292\) 0 0
\(293\) −5903.51 −1.17709 −0.588544 0.808465i \(-0.700299\pi\)
−0.588544 + 0.808465i \(0.700299\pi\)
\(294\) 0 0
\(295\) −2373.18 −0.468379
\(296\) 0 0
\(297\) −1995.66 −0.389900
\(298\) 0 0
\(299\) −1624.33 −0.314172
\(300\) 0 0
\(301\) −3991.74 −0.764386
\(302\) 0 0
\(303\) −306.328 −0.0580795
\(304\) 0 0
\(305\) −2011.50 −0.377633
\(306\) 0 0
\(307\) 3769.05 0.700688 0.350344 0.936621i \(-0.386065\pi\)
0.350344 + 0.936621i \(0.386065\pi\)
\(308\) 0 0
\(309\) 6341.94 1.16757
\(310\) 0 0
\(311\) 3141.76 0.572839 0.286419 0.958104i \(-0.407535\pi\)
0.286419 + 0.958104i \(0.407535\pi\)
\(312\) 0 0
\(313\) −1743.33 −0.314820 −0.157410 0.987533i \(-0.550314\pi\)
−0.157410 + 0.987533i \(0.550314\pi\)
\(314\) 0 0
\(315\) −3095.46 −0.553681
\(316\) 0 0
\(317\) 1836.62 0.325410 0.162705 0.986675i \(-0.447978\pi\)
0.162705 + 0.986675i \(0.447978\pi\)
\(318\) 0 0
\(319\) 528.086 0.0926869
\(320\) 0 0
\(321\) −8446.99 −1.46874
\(322\) 0 0
\(323\) −145.503 −0.0250650
\(324\) 0 0
\(325\) −1176.26 −0.200761
\(326\) 0 0
\(327\) −2957.23 −0.500107
\(328\) 0 0
\(329\) 11977.2 2.00707
\(330\) 0 0
\(331\) 106.340 0.0176585 0.00882924 0.999961i \(-0.497190\pi\)
0.00882924 + 0.999961i \(0.497190\pi\)
\(332\) 0 0
\(333\) −2705.74 −0.445266
\(334\) 0 0
\(335\) −1564.26 −0.255118
\(336\) 0 0
\(337\) 347.401 0.0561547 0.0280774 0.999606i \(-0.491062\pi\)
0.0280774 + 0.999606i \(0.491062\pi\)
\(338\) 0 0
\(339\) −3605.82 −0.577703
\(340\) 0 0
\(341\) −2926.89 −0.464809
\(342\) 0 0
\(343\) −3911.17 −0.615694
\(344\) 0 0
\(345\) 3187.18 0.497369
\(346\) 0 0
\(347\) 4319.94 0.668319 0.334159 0.942517i \(-0.391548\pi\)
0.334159 + 0.942517i \(0.391548\pi\)
\(348\) 0 0
\(349\) 9732.19 1.49270 0.746350 0.665554i \(-0.231805\pi\)
0.746350 + 0.665554i \(0.231805\pi\)
\(350\) 0 0
\(351\) 4164.27 0.633255
\(352\) 0 0
\(353\) 8396.92 1.26607 0.633035 0.774123i \(-0.281809\pi\)
0.633035 + 0.774123i \(0.281809\pi\)
\(354\) 0 0
\(355\) −2061.93 −0.308270
\(356\) 0 0
\(357\) 676.554 0.100300
\(358\) 0 0
\(359\) 1306.89 0.192130 0.0960652 0.995375i \(-0.469374\pi\)
0.0960652 + 0.995375i \(0.469374\pi\)
\(360\) 0 0
\(361\) −5503.21 −0.802334
\(362\) 0 0
\(363\) −5966.64 −0.862720
\(364\) 0 0
\(365\) −11717.0 −1.68026
\(366\) 0 0
\(367\) −9859.59 −1.40236 −0.701181 0.712984i \(-0.747343\pi\)
−0.701181 + 0.712984i \(0.747343\pi\)
\(368\) 0 0
\(369\) 4293.95 0.605783
\(370\) 0 0
\(371\) −3932.12 −0.550257
\(372\) 0 0
\(373\) 3393.01 0.471001 0.235500 0.971874i \(-0.424327\pi\)
0.235500 + 0.971874i \(0.424327\pi\)
\(374\) 0 0
\(375\) −7011.67 −0.965550
\(376\) 0 0
\(377\) −1101.94 −0.150537
\(378\) 0 0
\(379\) 2030.82 0.275240 0.137620 0.990485i \(-0.456055\pi\)
0.137620 + 0.990485i \(0.456055\pi\)
\(380\) 0 0
\(381\) −7655.19 −1.02936
\(382\) 0 0
\(383\) 4.81248 0.000642052 0 0.000321026 1.00000i \(-0.499898\pi\)
0.000321026 1.00000i \(0.499898\pi\)
\(384\) 0 0
\(385\) −6521.47 −0.863285
\(386\) 0 0
\(387\) 1203.13 0.158032
\(388\) 0 0
\(389\) −9031.93 −1.17722 −0.588608 0.808418i \(-0.700324\pi\)
−0.588608 + 0.808418i \(0.700324\pi\)
\(390\) 0 0
\(391\) −168.924 −0.0218487
\(392\) 0 0
\(393\) −14604.6 −1.87457
\(394\) 0 0
\(395\) −16429.6 −2.09281
\(396\) 0 0
\(397\) 1443.16 0.182443 0.0912217 0.995831i \(-0.470923\pi\)
0.0912217 + 0.995831i \(0.470923\pi\)
\(398\) 0 0
\(399\) −6304.11 −0.790979
\(400\) 0 0
\(401\) −9877.59 −1.23008 −0.615042 0.788495i \(-0.710861\pi\)
−0.615042 + 0.788495i \(0.710861\pi\)
\(402\) 0 0
\(403\) 6107.43 0.754920
\(404\) 0 0
\(405\) −11085.4 −1.36009
\(406\) 0 0
\(407\) −5700.41 −0.694247
\(408\) 0 0
\(409\) 14530.7 1.75672 0.878358 0.478003i \(-0.158639\pi\)
0.878358 + 0.478003i \(0.158639\pi\)
\(410\) 0 0
\(411\) 7521.93 0.902748
\(412\) 0 0
\(413\) 5449.63 0.649295
\(414\) 0 0
\(415\) −6310.23 −0.746403
\(416\) 0 0
\(417\) 4912.21 0.576863
\(418\) 0 0
\(419\) −3312.84 −0.386259 −0.193130 0.981173i \(-0.561864\pi\)
−0.193130 + 0.981173i \(0.561864\pi\)
\(420\) 0 0
\(421\) 6708.67 0.776629 0.388314 0.921527i \(-0.373057\pi\)
0.388314 + 0.921527i \(0.373057\pi\)
\(422\) 0 0
\(423\) −3609.98 −0.414949
\(424\) 0 0
\(425\) −122.327 −0.0139617
\(426\) 0 0
\(427\) 4619.09 0.523498
\(428\) 0 0
\(429\) −4130.99 −0.464909
\(430\) 0 0
\(431\) −14687.7 −1.64148 −0.820742 0.571299i \(-0.806440\pi\)
−0.820742 + 0.571299i \(0.806440\pi\)
\(432\) 0 0
\(433\) 759.822 0.0843296 0.0421648 0.999111i \(-0.486575\pi\)
0.0421648 + 0.999111i \(0.486575\pi\)
\(434\) 0 0
\(435\) 2162.17 0.238317
\(436\) 0 0
\(437\) 1574.03 0.172302
\(438\) 0 0
\(439\) 4803.78 0.522260 0.261130 0.965304i \(-0.415905\pi\)
0.261130 + 0.965304i \(0.415905\pi\)
\(440\) 0 0
\(441\) 4143.54 0.447418
\(442\) 0 0
\(443\) −10123.3 −1.08572 −0.542858 0.839824i \(-0.682658\pi\)
−0.542858 + 0.839824i \(0.682658\pi\)
\(444\) 0 0
\(445\) −9610.78 −1.02381
\(446\) 0 0
\(447\) 2849.98 0.301565
\(448\) 0 0
\(449\) −636.749 −0.0669266 −0.0334633 0.999440i \(-0.510654\pi\)
−0.0334633 + 0.999440i \(0.510654\pi\)
\(450\) 0 0
\(451\) 9046.41 0.944521
\(452\) 0 0
\(453\) −10393.7 −1.07801
\(454\) 0 0
\(455\) 13608.1 1.40210
\(456\) 0 0
\(457\) 12598.2 1.28954 0.644770 0.764376i \(-0.276953\pi\)
0.644770 + 0.764376i \(0.276953\pi\)
\(458\) 0 0
\(459\) 433.068 0.0440389
\(460\) 0 0
\(461\) −15728.3 −1.58902 −0.794512 0.607249i \(-0.792273\pi\)
−0.794512 + 0.607249i \(0.792273\pi\)
\(462\) 0 0
\(463\) 5504.55 0.552523 0.276262 0.961083i \(-0.410904\pi\)
0.276262 + 0.961083i \(0.410904\pi\)
\(464\) 0 0
\(465\) −11983.7 −1.19512
\(466\) 0 0
\(467\) −14851.2 −1.47159 −0.735796 0.677203i \(-0.763192\pi\)
−0.735796 + 0.677203i \(0.763192\pi\)
\(468\) 0 0
\(469\) 3592.07 0.353660
\(470\) 0 0
\(471\) 8196.48 0.801855
\(472\) 0 0
\(473\) 2534.73 0.246399
\(474\) 0 0
\(475\) 1139.84 0.110104
\(476\) 0 0
\(477\) 1185.16 0.113762
\(478\) 0 0
\(479\) 5778.20 0.551175 0.275587 0.961276i \(-0.411128\pi\)
0.275587 + 0.961276i \(0.411128\pi\)
\(480\) 0 0
\(481\) 11894.8 1.12756
\(482\) 0 0
\(483\) −7318.86 −0.689482
\(484\) 0 0
\(485\) 16650.4 1.55888
\(486\) 0 0
\(487\) −12639.2 −1.17605 −0.588027 0.808841i \(-0.700095\pi\)
−0.588027 + 0.808841i \(0.700095\pi\)
\(488\) 0 0
\(489\) −22018.3 −2.03620
\(490\) 0 0
\(491\) 363.554 0.0334154 0.0167077 0.999860i \(-0.494682\pi\)
0.0167077 + 0.999860i \(0.494682\pi\)
\(492\) 0 0
\(493\) −114.597 −0.0104689
\(494\) 0 0
\(495\) 1965.60 0.178479
\(496\) 0 0
\(497\) 4734.89 0.427342
\(498\) 0 0
\(499\) 4327.18 0.388198 0.194099 0.980982i \(-0.437822\pi\)
0.194099 + 0.980982i \(0.437822\pi\)
\(500\) 0 0
\(501\) 24236.4 2.16129
\(502\) 0 0
\(503\) 21621.8 1.91664 0.958318 0.285704i \(-0.0922273\pi\)
0.958318 + 0.285704i \(0.0922273\pi\)
\(504\) 0 0
\(505\) −640.764 −0.0564626
\(506\) 0 0
\(507\) −4496.58 −0.393886
\(508\) 0 0
\(509\) −12903.3 −1.12363 −0.561817 0.827261i \(-0.689898\pi\)
−0.561817 + 0.827261i \(0.689898\pi\)
\(510\) 0 0
\(511\) 26906.2 2.32928
\(512\) 0 0
\(513\) −4035.32 −0.347297
\(514\) 0 0
\(515\) 13265.8 1.13507
\(516\) 0 0
\(517\) −7605.45 −0.646977
\(518\) 0 0
\(519\) −16787.7 −1.41985
\(520\) 0 0
\(521\) −5015.65 −0.421765 −0.210883 0.977511i \(-0.567634\pi\)
−0.210883 + 0.977511i \(0.567634\pi\)
\(522\) 0 0
\(523\) 10454.3 0.874059 0.437030 0.899447i \(-0.356030\pi\)
0.437030 + 0.899447i \(0.356030\pi\)
\(524\) 0 0
\(525\) −5299.98 −0.440591
\(526\) 0 0
\(527\) 635.148 0.0525000
\(528\) 0 0
\(529\) −10339.6 −0.849807
\(530\) 0 0
\(531\) −1642.54 −0.134238
\(532\) 0 0
\(533\) −18876.8 −1.53404
\(534\) 0 0
\(535\) −17669.0 −1.42785
\(536\) 0 0
\(537\) −977.845 −0.0785794
\(538\) 0 0
\(539\) 8729.54 0.697603
\(540\) 0 0
\(541\) −16028.3 −1.27377 −0.636886 0.770958i \(-0.719778\pi\)
−0.636886 + 0.770958i \(0.719778\pi\)
\(542\) 0 0
\(543\) 21137.1 1.67049
\(544\) 0 0
\(545\) −6185.80 −0.486184
\(546\) 0 0
\(547\) 7590.17 0.593295 0.296647 0.954987i \(-0.404131\pi\)
0.296647 + 0.954987i \(0.404131\pi\)
\(548\) 0 0
\(549\) −1392.21 −0.108230
\(550\) 0 0
\(551\) 1067.81 0.0825595
\(552\) 0 0
\(553\) 37727.9 2.90118
\(554\) 0 0
\(555\) −23339.4 −1.78505
\(556\) 0 0
\(557\) 15598.4 1.18658 0.593291 0.804988i \(-0.297828\pi\)
0.593291 + 0.804988i \(0.297828\pi\)
\(558\) 0 0
\(559\) −5289.11 −0.400189
\(560\) 0 0
\(561\) −429.607 −0.0323316
\(562\) 0 0
\(563\) 18275.4 1.36806 0.684030 0.729454i \(-0.260226\pi\)
0.684030 + 0.729454i \(0.260226\pi\)
\(564\) 0 0
\(565\) −7542.51 −0.561621
\(566\) 0 0
\(567\) 25455.8 1.88544
\(568\) 0 0
\(569\) −11103.6 −0.818081 −0.409040 0.912516i \(-0.634136\pi\)
−0.409040 + 0.912516i \(0.634136\pi\)
\(570\) 0 0
\(571\) −12505.8 −0.916550 −0.458275 0.888810i \(-0.651533\pi\)
−0.458275 + 0.888810i \(0.651533\pi\)
\(572\) 0 0
\(573\) 12917.4 0.941769
\(574\) 0 0
\(575\) 1323.31 0.0959756
\(576\) 0 0
\(577\) 16122.3 1.16322 0.581612 0.813466i \(-0.302422\pi\)
0.581612 + 0.813466i \(0.302422\pi\)
\(578\) 0 0
\(579\) 27658.5 1.98523
\(580\) 0 0
\(581\) 14490.4 1.03471
\(582\) 0 0
\(583\) 2496.87 0.177375
\(584\) 0 0
\(585\) −4101.53 −0.289876
\(586\) 0 0
\(587\) −82.7289 −0.00581701 −0.00290851 0.999996i \(-0.500926\pi\)
−0.00290851 + 0.999996i \(0.500926\pi\)
\(588\) 0 0
\(589\) −5918.29 −0.414022
\(590\) 0 0
\(591\) 7402.55 0.515229
\(592\) 0 0
\(593\) −5049.72 −0.349692 −0.174846 0.984596i \(-0.555943\pi\)
−0.174846 + 0.984596i \(0.555943\pi\)
\(594\) 0 0
\(595\) 1415.19 0.0975076
\(596\) 0 0
\(597\) 27770.1 1.90378
\(598\) 0 0
\(599\) −16393.1 −1.11820 −0.559101 0.829100i \(-0.688853\pi\)
−0.559101 + 0.829100i \(0.688853\pi\)
\(600\) 0 0
\(601\) 22564.2 1.53147 0.765733 0.643158i \(-0.222376\pi\)
0.765733 + 0.643158i \(0.222376\pi\)
\(602\) 0 0
\(603\) −1082.67 −0.0731170
\(604\) 0 0
\(605\) −12480.8 −0.838703
\(606\) 0 0
\(607\) −10441.9 −0.698230 −0.349115 0.937080i \(-0.613518\pi\)
−0.349115 + 0.937080i \(0.613518\pi\)
\(608\) 0 0
\(609\) −4965.07 −0.330369
\(610\) 0 0
\(611\) 15870.0 1.05079
\(612\) 0 0
\(613\) 5153.31 0.339544 0.169772 0.985483i \(-0.445697\pi\)
0.169772 + 0.985483i \(0.445697\pi\)
\(614\) 0 0
\(615\) 37039.1 2.42856
\(616\) 0 0
\(617\) 22893.5 1.49377 0.746886 0.664952i \(-0.231548\pi\)
0.746886 + 0.664952i \(0.231548\pi\)
\(618\) 0 0
\(619\) −1872.55 −0.121590 −0.0607949 0.998150i \(-0.519364\pi\)
−0.0607949 + 0.998150i \(0.519364\pi\)
\(620\) 0 0
\(621\) −4684.87 −0.302733
\(622\) 0 0
\(623\) 22069.6 1.41926
\(624\) 0 0
\(625\) −18536.2 −1.18632
\(626\) 0 0
\(627\) 4003.07 0.254971
\(628\) 0 0
\(629\) 1237.01 0.0784148
\(630\) 0 0
\(631\) 12644.7 0.797745 0.398872 0.917006i \(-0.369402\pi\)
0.398872 + 0.917006i \(0.369402\pi\)
\(632\) 0 0
\(633\) −20996.6 −1.31839
\(634\) 0 0
\(635\) −16012.8 −1.00071
\(636\) 0 0
\(637\) −18215.6 −1.13301
\(638\) 0 0
\(639\) −1427.11 −0.0883502
\(640\) 0 0
\(641\) −17232.7 −1.06186 −0.530929 0.847416i \(-0.678157\pi\)
−0.530929 + 0.847416i \(0.678157\pi\)
\(642\) 0 0
\(643\) −4194.66 −0.257265 −0.128632 0.991692i \(-0.541059\pi\)
−0.128632 + 0.991692i \(0.541059\pi\)
\(644\) 0 0
\(645\) 10378.0 0.633543
\(646\) 0 0
\(647\) −29338.5 −1.78271 −0.891355 0.453305i \(-0.850245\pi\)
−0.891355 + 0.453305i \(0.850245\pi\)
\(648\) 0 0
\(649\) −3460.48 −0.209300
\(650\) 0 0
\(651\) 27518.7 1.65675
\(652\) 0 0
\(653\) 30988.6 1.85709 0.928543 0.371225i \(-0.121062\pi\)
0.928543 + 0.371225i \(0.121062\pi\)
\(654\) 0 0
\(655\) −30549.3 −1.82238
\(656\) 0 0
\(657\) −8109.65 −0.481564
\(658\) 0 0
\(659\) −8159.70 −0.482332 −0.241166 0.970484i \(-0.577530\pi\)
−0.241166 + 0.970484i \(0.577530\pi\)
\(660\) 0 0
\(661\) −24409.7 −1.43635 −0.718174 0.695864i \(-0.755022\pi\)
−0.718174 + 0.695864i \(0.755022\pi\)
\(662\) 0 0
\(663\) 896.443 0.0525113
\(664\) 0 0
\(665\) −13186.7 −0.768959
\(666\) 0 0
\(667\) 1239.69 0.0719657
\(668\) 0 0
\(669\) 20777.3 1.20074
\(670\) 0 0
\(671\) −2933.09 −0.168749
\(672\) 0 0
\(673\) 371.652 0.0212870 0.0106435 0.999943i \(-0.496612\pi\)
0.0106435 + 0.999943i \(0.496612\pi\)
\(674\) 0 0
\(675\) −3392.56 −0.193451
\(676\) 0 0
\(677\) −8788.32 −0.498911 −0.249455 0.968386i \(-0.580252\pi\)
−0.249455 + 0.968386i \(0.580252\pi\)
\(678\) 0 0
\(679\) −38235.0 −2.16101
\(680\) 0 0
\(681\) −7519.51 −0.423125
\(682\) 0 0
\(683\) −8840.50 −0.495274 −0.247637 0.968853i \(-0.579654\pi\)
−0.247637 + 0.968853i \(0.579654\pi\)
\(684\) 0 0
\(685\) 15734.0 0.877616
\(686\) 0 0
\(687\) 1637.60 0.0909436
\(688\) 0 0
\(689\) −5210.11 −0.288083
\(690\) 0 0
\(691\) −34252.5 −1.88571 −0.942856 0.333199i \(-0.891872\pi\)
−0.942856 + 0.333199i \(0.891872\pi\)
\(692\) 0 0
\(693\) −4513.68 −0.247418
\(694\) 0 0
\(695\) 10275.1 0.560804
\(696\) 0 0
\(697\) −1963.11 −0.106683
\(698\) 0 0
\(699\) 38912.1 2.10556
\(700\) 0 0
\(701\) 27978.5 1.50747 0.753733 0.657180i \(-0.228251\pi\)
0.753733 + 0.657180i \(0.228251\pi\)
\(702\) 0 0
\(703\) −11526.5 −0.618391
\(704\) 0 0
\(705\) −31139.3 −1.66351
\(706\) 0 0
\(707\) 1471.41 0.0782719
\(708\) 0 0
\(709\) 378.764 0.0200632 0.0100316 0.999950i \(-0.496807\pi\)
0.0100316 + 0.999950i \(0.496807\pi\)
\(710\) 0 0
\(711\) −11371.3 −0.599801
\(712\) 0 0
\(713\) −6870.94 −0.360896
\(714\) 0 0
\(715\) −8641.03 −0.451967
\(716\) 0 0
\(717\) −36137.0 −1.88223
\(718\) 0 0
\(719\) 9369.50 0.485985 0.242993 0.970028i \(-0.421871\pi\)
0.242993 + 0.970028i \(0.421871\pi\)
\(720\) 0 0
\(721\) −30462.8 −1.57350
\(722\) 0 0
\(723\) 13523.7 0.695645
\(724\) 0 0
\(725\) 897.728 0.0459873
\(726\) 0 0
\(727\) 14672.0 0.748494 0.374247 0.927329i \(-0.377901\pi\)
0.374247 + 0.927329i \(0.377901\pi\)
\(728\) 0 0
\(729\) 9993.38 0.507717
\(730\) 0 0
\(731\) −550.047 −0.0278307
\(732\) 0 0
\(733\) 9764.90 0.492053 0.246027 0.969263i \(-0.420875\pi\)
0.246027 + 0.969263i \(0.420875\pi\)
\(734\) 0 0
\(735\) 35741.8 1.79368
\(736\) 0 0
\(737\) −2280.94 −0.114002
\(738\) 0 0
\(739\) −11150.5 −0.555043 −0.277521 0.960719i \(-0.589513\pi\)
−0.277521 + 0.960719i \(0.589513\pi\)
\(740\) 0 0
\(741\) −8353.04 −0.414111
\(742\) 0 0
\(743\) 17461.6 0.862186 0.431093 0.902307i \(-0.358128\pi\)
0.431093 + 0.902307i \(0.358128\pi\)
\(744\) 0 0
\(745\) 5961.47 0.293169
\(746\) 0 0
\(747\) −4367.48 −0.213919
\(748\) 0 0
\(749\) 40574.2 1.97937
\(750\) 0 0
\(751\) 12692.5 0.616717 0.308358 0.951270i \(-0.400220\pi\)
0.308358 + 0.951270i \(0.400220\pi\)
\(752\) 0 0
\(753\) −39721.5 −1.92235
\(754\) 0 0
\(755\) −21741.0 −1.04800
\(756\) 0 0
\(757\) −35923.3 −1.72477 −0.862387 0.506250i \(-0.831031\pi\)
−0.862387 + 0.506250i \(0.831031\pi\)
\(758\) 0 0
\(759\) 4647.42 0.222254
\(760\) 0 0
\(761\) −27302.4 −1.30054 −0.650270 0.759703i \(-0.725344\pi\)
−0.650270 + 0.759703i \(0.725344\pi\)
\(762\) 0 0
\(763\) 14204.7 0.673978
\(764\) 0 0
\(765\) −426.543 −0.0201591
\(766\) 0 0
\(767\) 7220.84 0.339934
\(768\) 0 0
\(769\) 17941.0 0.841314 0.420657 0.907220i \(-0.361800\pi\)
0.420657 + 0.907220i \(0.361800\pi\)
\(770\) 0 0
\(771\) −18773.8 −0.876943
\(772\) 0 0
\(773\) −6566.84 −0.305553 −0.152777 0.988261i \(-0.548822\pi\)
−0.152777 + 0.988261i \(0.548822\pi\)
\(774\) 0 0
\(775\) −4975.61 −0.230619
\(776\) 0 0
\(777\) 53595.3 2.47455
\(778\) 0 0
\(779\) 18292.2 0.841318
\(780\) 0 0
\(781\) −3006.62 −0.137753
\(782\) 0 0
\(783\) −3178.18 −0.145056
\(784\) 0 0
\(785\) 17145.1 0.779533
\(786\) 0 0
\(787\) 28734.0 1.30147 0.650736 0.759304i \(-0.274461\pi\)
0.650736 + 0.759304i \(0.274461\pi\)
\(788\) 0 0
\(789\) 14386.2 0.649127
\(790\) 0 0
\(791\) 17320.2 0.778552
\(792\) 0 0
\(793\) 6120.36 0.274074
\(794\) 0 0
\(795\) 10223.0 0.456068
\(796\) 0 0
\(797\) 22553.9 1.00238 0.501192 0.865336i \(-0.332895\pi\)
0.501192 + 0.865336i \(0.332895\pi\)
\(798\) 0 0
\(799\) 1650.42 0.0730757
\(800\) 0 0
\(801\) −6651.88 −0.293424
\(802\) 0 0
\(803\) −17085.3 −0.750841
\(804\) 0 0
\(805\) −15309.3 −0.670288
\(806\) 0 0
\(807\) 27121.8 1.18306
\(808\) 0 0
\(809\) 24921.7 1.08307 0.541533 0.840680i \(-0.317844\pi\)
0.541533 + 0.840680i \(0.317844\pi\)
\(810\) 0 0
\(811\) −41652.2 −1.80346 −0.901730 0.432300i \(-0.857702\pi\)
−0.901730 + 0.432300i \(0.857702\pi\)
\(812\) 0 0
\(813\) −38921.9 −1.67903
\(814\) 0 0
\(815\) −46057.0 −1.97952
\(816\) 0 0
\(817\) 5125.32 0.219477
\(818\) 0 0
\(819\) 9418.52 0.401843
\(820\) 0 0
\(821\) 10969.1 0.466288 0.233144 0.972442i \(-0.425099\pi\)
0.233144 + 0.972442i \(0.425099\pi\)
\(822\) 0 0
\(823\) 16669.2 0.706015 0.353008 0.935620i \(-0.385159\pi\)
0.353008 + 0.935620i \(0.385159\pi\)
\(824\) 0 0
\(825\) 3365.45 0.142024
\(826\) 0 0
\(827\) 9424.07 0.396260 0.198130 0.980176i \(-0.436513\pi\)
0.198130 + 0.980176i \(0.436513\pi\)
\(828\) 0 0
\(829\) 23359.5 0.978658 0.489329 0.872099i \(-0.337242\pi\)
0.489329 + 0.872099i \(0.337242\pi\)
\(830\) 0 0
\(831\) 44423.8 1.85445
\(832\) 0 0
\(833\) −1894.35 −0.0787938
\(834\) 0 0
\(835\) 50696.8 2.10112
\(836\) 0 0
\(837\) 17614.9 0.727433
\(838\) 0 0
\(839\) −9260.03 −0.381039 −0.190520 0.981683i \(-0.561017\pi\)
−0.190520 + 0.981683i \(0.561017\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 1624.41 0.0663673
\(844\) 0 0
\(845\) −9405.76 −0.382921
\(846\) 0 0
\(847\) 28660.1 1.16266
\(848\) 0 0
\(849\) 10250.4 0.414359
\(850\) 0 0
\(851\) −13381.8 −0.539040
\(852\) 0 0
\(853\) 43938.1 1.76367 0.881836 0.471556i \(-0.156307\pi\)
0.881836 + 0.471556i \(0.156307\pi\)
\(854\) 0 0
\(855\) 3974.52 0.158977
\(856\) 0 0
\(857\) 44540.6 1.77535 0.887676 0.460469i \(-0.152319\pi\)
0.887676 + 0.460469i \(0.152319\pi\)
\(858\) 0 0
\(859\) 5126.48 0.203624 0.101812 0.994804i \(-0.467536\pi\)
0.101812 + 0.994804i \(0.467536\pi\)
\(860\) 0 0
\(861\) −85054.5 −3.36661
\(862\) 0 0
\(863\) 29711.0 1.17193 0.585964 0.810337i \(-0.300716\pi\)
0.585964 + 0.810337i \(0.300716\pi\)
\(864\) 0 0
\(865\) −35115.9 −1.38032
\(866\) 0 0
\(867\) −29238.4 −1.14532
\(868\) 0 0
\(869\) −23956.9 −0.935194
\(870\) 0 0
\(871\) 4759.55 0.185156
\(872\) 0 0
\(873\) 11524.2 0.446775
\(874\) 0 0
\(875\) 33679.8 1.30124
\(876\) 0 0
\(877\) 24925.9 0.959735 0.479867 0.877341i \(-0.340685\pi\)
0.479867 + 0.877341i \(0.340685\pi\)
\(878\) 0 0
\(879\) −35245.2 −1.35244
\(880\) 0 0
\(881\) 13514.7 0.516825 0.258413 0.966035i \(-0.416801\pi\)
0.258413 + 0.966035i \(0.416801\pi\)
\(882\) 0 0
\(883\) −21759.9 −0.829309 −0.414654 0.909979i \(-0.636097\pi\)
−0.414654 + 0.909979i \(0.636097\pi\)
\(884\) 0 0
\(885\) −14168.4 −0.538153
\(886\) 0 0
\(887\) −15556.4 −0.588877 −0.294439 0.955670i \(-0.595133\pi\)
−0.294439 + 0.955670i \(0.595133\pi\)
\(888\) 0 0
\(889\) 36770.9 1.38724
\(890\) 0 0
\(891\) −16164.2 −0.607769
\(892\) 0 0
\(893\) −15378.5 −0.576286
\(894\) 0 0
\(895\) −2045.42 −0.0763919
\(896\) 0 0
\(897\) −9697.60 −0.360974
\(898\) 0 0
\(899\) −4661.20 −0.172925
\(900\) 0 0
\(901\) −541.831 −0.0200344
\(902\) 0 0
\(903\) −23831.5 −0.878255
\(904\) 0 0
\(905\) 44213.6 1.62399
\(906\) 0 0
\(907\) 36707.3 1.34382 0.671910 0.740633i \(-0.265474\pi\)
0.671910 + 0.740633i \(0.265474\pi\)
\(908\) 0 0
\(909\) −443.490 −0.0161822
\(910\) 0 0
\(911\) −21608.1 −0.785849 −0.392924 0.919571i \(-0.628537\pi\)
−0.392924 + 0.919571i \(0.628537\pi\)
\(912\) 0 0
\(913\) −9201.33 −0.333537
\(914\) 0 0
\(915\) −12009.1 −0.433889
\(916\) 0 0
\(917\) 70151.7 2.52630
\(918\) 0 0
\(919\) −40069.5 −1.43827 −0.719135 0.694870i \(-0.755462\pi\)
−0.719135 + 0.694870i \(0.755462\pi\)
\(920\) 0 0
\(921\) 22502.1 0.805068
\(922\) 0 0
\(923\) 6273.79 0.223732
\(924\) 0 0
\(925\) −9690.49 −0.344456
\(926\) 0 0
\(927\) 9181.61 0.325311
\(928\) 0 0
\(929\) −18663.0 −0.659112 −0.329556 0.944136i \(-0.606899\pi\)
−0.329556 + 0.944136i \(0.606899\pi\)
\(930\) 0 0
\(931\) 17651.5 0.621380
\(932\) 0 0
\(933\) 18757.0 0.658173
\(934\) 0 0
\(935\) −898.633 −0.0314315
\(936\) 0 0
\(937\) 13742.0 0.479115 0.239557 0.970882i \(-0.422998\pi\)
0.239557 + 0.970882i \(0.422998\pi\)
\(938\) 0 0
\(939\) −10408.0 −0.361718
\(940\) 0 0
\(941\) −31953.9 −1.10698 −0.553489 0.832856i \(-0.686704\pi\)
−0.553489 + 0.832856i \(0.686704\pi\)
\(942\) 0 0
\(943\) 21236.6 0.733362
\(944\) 0 0
\(945\) 39248.2 1.35105
\(946\) 0 0
\(947\) −11988.8 −0.411386 −0.205693 0.978617i \(-0.565945\pi\)
−0.205693 + 0.978617i \(0.565945\pi\)
\(948\) 0 0
\(949\) 35651.1 1.21948
\(950\) 0 0
\(951\) 10965.0 0.373886
\(952\) 0 0
\(953\) −6567.87 −0.223247 −0.111623 0.993751i \(-0.535605\pi\)
−0.111623 + 0.993751i \(0.535605\pi\)
\(954\) 0 0
\(955\) 27020.2 0.915552
\(956\) 0 0
\(957\) 3152.78 0.106494
\(958\) 0 0
\(959\) −36130.8 −1.21660
\(960\) 0 0
\(961\) −3956.49 −0.132808
\(962\) 0 0
\(963\) −12229.2 −0.409222
\(964\) 0 0
\(965\) 57854.9 1.92997
\(966\) 0 0
\(967\) 40868.4 1.35909 0.679545 0.733634i \(-0.262177\pi\)
0.679545 + 0.733634i \(0.262177\pi\)
\(968\) 0 0
\(969\) −868.683 −0.0287989
\(970\) 0 0
\(971\) 53372.2 1.76395 0.881976 0.471295i \(-0.156213\pi\)
0.881976 + 0.471295i \(0.156213\pi\)
\(972\) 0 0
\(973\) −23595.3 −0.777419
\(974\) 0 0
\(975\) −7022.54 −0.230668
\(976\) 0 0
\(977\) 32609.8 1.06784 0.533920 0.845535i \(-0.320718\pi\)
0.533920 + 0.845535i \(0.320718\pi\)
\(978\) 0 0
\(979\) −14014.1 −0.457499
\(980\) 0 0
\(981\) −4281.36 −0.139341
\(982\) 0 0
\(983\) 23719.0 0.769602 0.384801 0.923000i \(-0.374270\pi\)
0.384801 + 0.923000i \(0.374270\pi\)
\(984\) 0 0
\(985\) 15484.3 0.500885
\(986\) 0 0
\(987\) 71506.6 2.30606
\(988\) 0 0
\(989\) 5950.33 0.191314
\(990\) 0 0
\(991\) −12328.2 −0.395176 −0.197588 0.980285i \(-0.563311\pi\)
−0.197588 + 0.980285i \(0.563311\pi\)
\(992\) 0 0
\(993\) 634.871 0.0202890
\(994\) 0 0
\(995\) 58088.4 1.85078
\(996\) 0 0
\(997\) −55140.7 −1.75158 −0.875790 0.482693i \(-0.839659\pi\)
−0.875790 + 0.482693i \(0.839659\pi\)
\(998\) 0 0
\(999\) 34306.8 1.08651
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.p.1.3 3
4.3 odd 2 1856.4.a.u.1.1 3
8.3 odd 2 464.4.a.g.1.3 3
8.5 even 2 232.4.a.b.1.1 3
24.5 odd 2 2088.4.a.b.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.b.1.1 3 8.5 even 2
464.4.a.g.1.3 3 8.3 odd 2
1856.4.a.p.1.3 3 1.1 even 1 trivial
1856.4.a.u.1.1 3 4.3 odd 2
2088.4.a.b.1.3 3 24.5 odd 2