Properties

Label 1856.4.a.p.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: 3.3.229.1
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 1.1
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.04399 q^{3} -2.15672 q^{5} +27.7969 q^{7} +54.7938 q^{9} +O(q^{10})\) \(q-9.04399 q^{3} -2.15672 q^{5} +27.7969 q^{7} +54.7938 q^{9} +0.351483 q^{11} +65.3652 q^{13} +19.5054 q^{15} -68.3296 q^{17} -89.8292 q^{19} -251.395 q^{21} +110.709 q^{23} -120.349 q^{25} -251.367 q^{27} +29.0000 q^{29} -258.256 q^{31} -3.17881 q^{33} -59.9501 q^{35} +128.225 q^{37} -591.163 q^{39} +283.005 q^{41} -339.318 q^{43} -118.175 q^{45} -147.455 q^{47} +429.668 q^{49} +617.972 q^{51} -518.354 q^{53} -0.758050 q^{55} +812.414 q^{57} +102.347 q^{59} +791.964 q^{61} +1523.10 q^{63} -140.975 q^{65} -287.326 q^{67} -1001.25 q^{69} -546.837 q^{71} -260.144 q^{73} +1088.43 q^{75} +9.77013 q^{77} -204.498 q^{79} +793.927 q^{81} +949.003 q^{83} +147.368 q^{85} -262.276 q^{87} +1390.02 q^{89} +1816.95 q^{91} +2335.67 q^{93} +193.736 q^{95} +1228.30 q^{97} +19.2591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{3} - 4 q^{5} + 16 q^{7} + 45 q^{9}+O(q^{10}) \) 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
\(3\) −9.04399 −1.74052 −0.870259 0.492595i \(-0.836048\pi\)
−0.870259 + 0.492595i \(0.836048\pi\)
\(4\) 0 0
\(5\) −2.15672 −0.192903 −0.0964515 0.995338i \(-0.530749\pi\)
−0.0964515 + 0.995338i \(0.530749\pi\)
\(6\) 0 0
\(7\) 27.7969 1.50089 0.750446 0.660932i \(-0.229839\pi\)
0.750446 + 0.660932i \(0.229839\pi\)
\(8\) 0 0
\(9\) 54.7938 2.02940
\(10\) 0 0
\(11\) 0.351483 0.00963419 0.00481709 0.999988i \(-0.498467\pi\)
0.00481709 + 0.999988i \(0.498467\pi\)
\(12\) 0 0
\(13\) 65.3652 1.39454 0.697271 0.716807i \(-0.254397\pi\)
0.697271 + 0.716807i \(0.254397\pi\)
\(14\) 0 0
\(15\) 19.5054 0.335751
\(16\) 0 0
\(17\) −68.3296 −0.974845 −0.487422 0.873166i \(-0.662063\pi\)
−0.487422 + 0.873166i \(0.662063\pi\)
\(18\) 0 0
\(19\) −89.8292 −1.08464 −0.542322 0.840171i \(-0.682455\pi\)
−0.542322 + 0.840171i \(0.682455\pi\)
\(20\) 0 0
\(21\) −251.395 −2.61233
\(22\) 0 0
\(23\) 110.709 1.00367 0.501834 0.864964i \(-0.332659\pi\)
0.501834 + 0.864964i \(0.332659\pi\)
\(24\) 0 0
\(25\) −120.349 −0.962788
\(26\) 0 0
\(27\) −251.367 −1.79169
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −258.256 −1.49626 −0.748132 0.663550i \(-0.769049\pi\)
−0.748132 + 0.663550i \(0.769049\pi\)
\(32\) 0 0
\(33\) −3.17881 −0.0167685
\(34\) 0 0
\(35\) −59.9501 −0.289526
\(36\) 0 0
\(37\) 128.225 0.569733 0.284866 0.958567i \(-0.408051\pi\)
0.284866 + 0.958567i \(0.408051\pi\)
\(38\) 0 0
\(39\) −591.163 −2.42723
\(40\) 0 0
\(41\) 283.005 1.07800 0.539000 0.842306i \(-0.318802\pi\)
0.539000 + 0.842306i \(0.318802\pi\)
\(42\) 0 0
\(43\) −339.318 −1.20338 −0.601692 0.798728i \(-0.705507\pi\)
−0.601692 + 0.798728i \(0.705507\pi\)
\(44\) 0 0
\(45\) −118.175 −0.391477
\(46\) 0 0
\(47\) −147.455 −0.457628 −0.228814 0.973470i \(-0.573485\pi\)
−0.228814 + 0.973470i \(0.573485\pi\)
\(48\) 0 0
\(49\) 429.668 1.25268
\(50\) 0 0
\(51\) 617.972 1.69673
\(52\) 0 0
\(53\) −518.354 −1.34342 −0.671712 0.740813i \(-0.734441\pi\)
−0.671712 + 0.740813i \(0.734441\pi\)
\(54\) 0 0
\(55\) −0.758050 −0.00185846
\(56\) 0 0
\(57\) 812.414 1.88784
\(58\) 0 0
\(59\) 102.347 0.225838 0.112919 0.993604i \(-0.463980\pi\)
0.112919 + 0.993604i \(0.463980\pi\)
\(60\) 0 0
\(61\) 791.964 1.66230 0.831152 0.556045i \(-0.187682\pi\)
0.831152 + 0.556045i \(0.187682\pi\)
\(62\) 0 0
\(63\) 1523.10 3.04591
\(64\) 0 0
\(65\) −140.975 −0.269011
\(66\) 0 0
\(67\) −287.326 −0.523918 −0.261959 0.965079i \(-0.584368\pi\)
−0.261959 + 0.965079i \(0.584368\pi\)
\(68\) 0 0
\(69\) −1001.25 −1.74690
\(70\) 0 0
\(71\) −546.837 −0.914050 −0.457025 0.889454i \(-0.651085\pi\)
−0.457025 + 0.889454i \(0.651085\pi\)
\(72\) 0 0
\(73\) −260.144 −0.417089 −0.208545 0.978013i \(-0.566873\pi\)
−0.208545 + 0.978013i \(0.566873\pi\)
\(74\) 0 0
\(75\) 1088.43 1.67575
\(76\) 0 0
\(77\) 9.77013 0.0144599
\(78\) 0 0
\(79\) −204.498 −0.291238 −0.145619 0.989341i \(-0.546517\pi\)
−0.145619 + 0.989341i \(0.546517\pi\)
\(80\) 0 0
\(81\) 793.927 1.08906
\(82\) 0 0
\(83\) 949.003 1.25502 0.627510 0.778609i \(-0.284074\pi\)
0.627510 + 0.778609i \(0.284074\pi\)
\(84\) 0 0
\(85\) 147.368 0.188050
\(86\) 0 0
\(87\) −262.276 −0.323206
\(88\) 0 0
\(89\) 1390.02 1.65553 0.827765 0.561074i \(-0.189612\pi\)
0.827765 + 0.561074i \(0.189612\pi\)
\(90\) 0 0
\(91\) 1816.95 2.09306
\(92\) 0 0
\(93\) 2335.67 2.60427
\(94\) 0 0
\(95\) 193.736 0.209231
\(96\) 0 0
\(97\) 1228.30 1.28572 0.642861 0.765983i \(-0.277747\pi\)
0.642861 + 0.765983i \(0.277747\pi\)
\(98\) 0 0
\(99\) 19.2591 0.0195516
\(100\) 0 0
\(101\) −1090.25 −1.07410 −0.537049 0.843551i \(-0.680461\pi\)
−0.537049 + 0.843551i \(0.680461\pi\)
\(102\) 0 0
\(103\) −1226.95 −1.17374 −0.586869 0.809682i \(-0.699640\pi\)
−0.586869 + 0.809682i \(0.699640\pi\)
\(104\) 0 0
\(105\) 542.189 0.503926
\(106\) 0 0
\(107\) 365.324 0.330067 0.165034 0.986288i \(-0.447227\pi\)
0.165034 + 0.986288i \(0.447227\pi\)
\(108\) 0 0
\(109\) 1759.55 1.54619 0.773093 0.634292i \(-0.218708\pi\)
0.773093 + 0.634292i \(0.218708\pi\)
\(110\) 0 0
\(111\) −1159.67 −0.991629
\(112\) 0 0
\(113\) −1764.47 −1.46891 −0.734457 0.678655i \(-0.762563\pi\)
−0.734457 + 0.678655i \(0.762563\pi\)
\(114\) 0 0
\(115\) −238.768 −0.193610
\(116\) 0 0
\(117\) 3581.61 2.83008
\(118\) 0 0
\(119\) −1899.35 −1.46314
\(120\) 0 0
\(121\) −1330.88 −0.999907
\(122\) 0 0
\(123\) −2559.50 −1.87628
\(124\) 0 0
\(125\) 529.148 0.378628
\(126\) 0 0
\(127\) −1121.29 −0.783453 −0.391726 0.920082i \(-0.628122\pi\)
−0.391726 + 0.920082i \(0.628122\pi\)
\(128\) 0 0
\(129\) 3068.79 2.09451
\(130\) 0 0
\(131\) −1276.32 −0.851240 −0.425620 0.904902i \(-0.639944\pi\)
−0.425620 + 0.904902i \(0.639944\pi\)
\(132\) 0 0
\(133\) −2496.97 −1.62793
\(134\) 0 0
\(135\) 542.128 0.345622
\(136\) 0 0
\(137\) −1564.49 −0.975644 −0.487822 0.872943i \(-0.662208\pi\)
−0.487822 + 0.872943i \(0.662208\pi\)
\(138\) 0 0
\(139\) −2231.83 −1.36188 −0.680940 0.732339i \(-0.738428\pi\)
−0.680940 + 0.732339i \(0.738428\pi\)
\(140\) 0 0
\(141\) 1333.58 0.796509
\(142\) 0 0
\(143\) 22.9748 0.0134353
\(144\) 0 0
\(145\) −62.5449 −0.0358212
\(146\) 0 0
\(147\) −3885.91 −2.18030
\(148\) 0 0
\(149\) 3474.95 1.91060 0.955300 0.295639i \(-0.0955326\pi\)
0.955300 + 0.295639i \(0.0955326\pi\)
\(150\) 0 0
\(151\) 1061.60 0.572131 0.286066 0.958210i \(-0.407652\pi\)
0.286066 + 0.958210i \(0.407652\pi\)
\(152\) 0 0
\(153\) −3744.04 −1.97835
\(154\) 0 0
\(155\) 556.986 0.288634
\(156\) 0 0
\(157\) 1664.40 0.846073 0.423036 0.906113i \(-0.360964\pi\)
0.423036 + 0.906113i \(0.360964\pi\)
\(158\) 0 0
\(159\) 4687.99 2.33825
\(160\) 0 0
\(161\) 3077.36 1.50640
\(162\) 0 0
\(163\) 239.164 0.114925 0.0574624 0.998348i \(-0.481699\pi\)
0.0574624 + 0.998348i \(0.481699\pi\)
\(164\) 0 0
\(165\) 6.85580 0.00323469
\(166\) 0 0
\(167\) 1330.43 0.616477 0.308238 0.951309i \(-0.400261\pi\)
0.308238 + 0.951309i \(0.400261\pi\)
\(168\) 0 0
\(169\) 2075.61 0.944749
\(170\) 0 0
\(171\) −4922.08 −2.20118
\(172\) 0 0
\(173\) −3007.36 −1.32165 −0.660824 0.750541i \(-0.729793\pi\)
−0.660824 + 0.750541i \(0.729793\pi\)
\(174\) 0 0
\(175\) −3345.32 −1.44504
\(176\) 0 0
\(177\) −925.627 −0.393076
\(178\) 0 0
\(179\) −722.292 −0.301601 −0.150801 0.988564i \(-0.548185\pi\)
−0.150801 + 0.988564i \(0.548185\pi\)
\(180\) 0 0
\(181\) −2763.91 −1.13503 −0.567514 0.823364i \(-0.692095\pi\)
−0.567514 + 0.823364i \(0.692095\pi\)
\(182\) 0 0
\(183\) −7162.51 −2.89327
\(184\) 0 0
\(185\) −276.546 −0.109903
\(186\) 0 0
\(187\) −24.0167 −0.00939184
\(188\) 0 0
\(189\) −6987.22 −2.68913
\(190\) 0 0
\(191\) 719.960 0.272746 0.136373 0.990658i \(-0.456455\pi\)
0.136373 + 0.990658i \(0.456455\pi\)
\(192\) 0 0
\(193\) 1020.08 0.380452 0.190226 0.981740i \(-0.439078\pi\)
0.190226 + 0.981740i \(0.439078\pi\)
\(194\) 0 0
\(195\) 1274.97 0.468219
\(196\) 0 0
\(197\) −182.348 −0.0659482 −0.0329741 0.999456i \(-0.510498\pi\)
−0.0329741 + 0.999456i \(0.510498\pi\)
\(198\) 0 0
\(199\) −3150.60 −1.12231 −0.561156 0.827710i \(-0.689643\pi\)
−0.561156 + 0.827710i \(0.689643\pi\)
\(200\) 0 0
\(201\) 2598.58 0.911888
\(202\) 0 0
\(203\) 806.110 0.278709
\(204\) 0 0
\(205\) −610.363 −0.207949
\(206\) 0 0
\(207\) 6066.15 2.03684
\(208\) 0 0
\(209\) −31.5734 −0.0104497
\(210\) 0 0
\(211\) −2028.36 −0.661793 −0.330897 0.943667i \(-0.607351\pi\)
−0.330897 + 0.943667i \(0.607351\pi\)
\(212\) 0 0
\(213\) 4945.59 1.59092
\(214\) 0 0
\(215\) 731.815 0.232136
\(216\) 0 0
\(217\) −7178.72 −2.24573
\(218\) 0 0
\(219\) 2352.74 0.725951
\(220\) 0 0
\(221\) −4466.38 −1.35946
\(222\) 0 0
\(223\) −1942.22 −0.583232 −0.291616 0.956535i \(-0.594193\pi\)
−0.291616 + 0.956535i \(0.594193\pi\)
\(224\) 0 0
\(225\) −6594.35 −1.95388
\(226\) 0 0
\(227\) −3476.41 −1.01646 −0.508232 0.861220i \(-0.669701\pi\)
−0.508232 + 0.861220i \(0.669701\pi\)
\(228\) 0 0
\(229\) −3015.30 −0.870116 −0.435058 0.900402i \(-0.643272\pi\)
−0.435058 + 0.900402i \(0.643272\pi\)
\(230\) 0 0
\(231\) −88.3610 −0.0251677
\(232\) 0 0
\(233\) 5510.20 1.54929 0.774646 0.632395i \(-0.217928\pi\)
0.774646 + 0.632395i \(0.217928\pi\)
\(234\) 0 0
\(235\) 318.019 0.0882778
\(236\) 0 0
\(237\) 1849.48 0.506905
\(238\) 0 0
\(239\) 1742.92 0.471716 0.235858 0.971787i \(-0.424210\pi\)
0.235858 + 0.971787i \(0.424210\pi\)
\(240\) 0 0
\(241\) −1188.65 −0.317707 −0.158854 0.987302i \(-0.550780\pi\)
−0.158854 + 0.987302i \(0.550780\pi\)
\(242\) 0 0
\(243\) −393.366 −0.103846
\(244\) 0 0
\(245\) −926.673 −0.241645
\(246\) 0 0
\(247\) −5871.70 −1.51258
\(248\) 0 0
\(249\) −8582.78 −2.18438
\(250\) 0 0
\(251\) −4679.47 −1.17675 −0.588377 0.808587i \(-0.700233\pi\)
−0.588377 + 0.808587i \(0.700233\pi\)
\(252\) 0 0
\(253\) 38.9122 0.00966952
\(254\) 0 0
\(255\) −1332.79 −0.327305
\(256\) 0 0
\(257\) 2223.42 0.539663 0.269831 0.962908i \(-0.413032\pi\)
0.269831 + 0.962908i \(0.413032\pi\)
\(258\) 0 0
\(259\) 3564.27 0.855107
\(260\) 0 0
\(261\) 1589.02 0.376850
\(262\) 0 0
\(263\) 6720.95 1.57579 0.787893 0.615813i \(-0.211172\pi\)
0.787893 + 0.615813i \(0.211172\pi\)
\(264\) 0 0
\(265\) 1117.95 0.259150
\(266\) 0 0
\(267\) −12571.4 −2.88148
\(268\) 0 0
\(269\) 2842.85 0.644356 0.322178 0.946679i \(-0.395585\pi\)
0.322178 + 0.946679i \(0.395585\pi\)
\(270\) 0 0
\(271\) −3793.02 −0.850219 −0.425109 0.905142i \(-0.639764\pi\)
−0.425109 + 0.905142i \(0.639764\pi\)
\(272\) 0 0
\(273\) −16432.5 −3.64300
\(274\) 0 0
\(275\) −42.3005 −0.00927569
\(276\) 0 0
\(277\) −6698.19 −1.45291 −0.726453 0.687216i \(-0.758833\pi\)
−0.726453 + 0.687216i \(0.758833\pi\)
\(278\) 0 0
\(279\) −14150.8 −3.03652
\(280\) 0 0
\(281\) 8532.95 1.81151 0.905753 0.423807i \(-0.139306\pi\)
0.905753 + 0.423807i \(0.139306\pi\)
\(282\) 0 0
\(283\) 9100.83 1.91162 0.955810 0.293986i \(-0.0949819\pi\)
0.955810 + 0.293986i \(0.0949819\pi\)
\(284\) 0 0
\(285\) −1752.15 −0.364170
\(286\) 0 0
\(287\) 7866.66 1.61796
\(288\) 0 0
\(289\) −244.069 −0.0496781
\(290\) 0 0
\(291\) −11108.7 −2.23782
\(292\) 0 0
\(293\) −6416.32 −1.27934 −0.639668 0.768651i \(-0.720928\pi\)
−0.639668 + 0.768651i \(0.720928\pi\)
\(294\) 0 0
\(295\) −220.734 −0.0435649
\(296\) 0 0
\(297\) −88.3511 −0.0172615
\(298\) 0 0
\(299\) 7236.50 1.39966
\(300\) 0 0
\(301\) −9431.99 −1.80615
\(302\) 0 0
\(303\) 9860.20 1.86948
\(304\) 0 0
\(305\) −1708.04 −0.320663
\(306\) 0 0
\(307\) −9836.12 −1.82859 −0.914295 0.405050i \(-0.867254\pi\)
−0.914295 + 0.405050i \(0.867254\pi\)
\(308\) 0 0
\(309\) 11096.5 2.04291
\(310\) 0 0
\(311\) 6806.95 1.24112 0.620558 0.784161i \(-0.286906\pi\)
0.620558 + 0.784161i \(0.286906\pi\)
\(312\) 0 0
\(313\) 1860.04 0.335896 0.167948 0.985796i \(-0.446286\pi\)
0.167948 + 0.985796i \(0.446286\pi\)
\(314\) 0 0
\(315\) −3284.90 −0.587565
\(316\) 0 0
\(317\) −2326.51 −0.412208 −0.206104 0.978530i \(-0.566078\pi\)
−0.206104 + 0.978530i \(0.566078\pi\)
\(318\) 0 0
\(319\) 10.1930 0.00178902
\(320\) 0 0
\(321\) −3303.98 −0.574487
\(322\) 0 0
\(323\) 6137.99 1.05736
\(324\) 0 0
\(325\) −7866.61 −1.34265
\(326\) 0 0
\(327\) −15913.4 −2.69116
\(328\) 0 0
\(329\) −4098.79 −0.686850
\(330\) 0 0
\(331\) 1668.98 0.277146 0.138573 0.990352i \(-0.455748\pi\)
0.138573 + 0.990352i \(0.455748\pi\)
\(332\) 0 0
\(333\) 7025.95 1.15622
\(334\) 0 0
\(335\) 619.682 0.101065
\(336\) 0 0
\(337\) −9794.95 −1.58328 −0.791639 0.610989i \(-0.790772\pi\)
−0.791639 + 0.610989i \(0.790772\pi\)
\(338\) 0 0
\(339\) 15957.8 2.55667
\(340\) 0 0
\(341\) −90.7726 −0.0144153
\(342\) 0 0
\(343\) 2409.09 0.379238
\(344\) 0 0
\(345\) 2159.41 0.336982
\(346\) 0 0
\(347\) 2296.52 0.355285 0.177642 0.984095i \(-0.443153\pi\)
0.177642 + 0.984095i \(0.443153\pi\)
\(348\) 0 0
\(349\) −9856.35 −1.51174 −0.755871 0.654720i \(-0.772786\pi\)
−0.755871 + 0.654720i \(0.772786\pi\)
\(350\) 0 0
\(351\) −16430.7 −2.49858
\(352\) 0 0
\(353\) −10005.5 −1.50860 −0.754301 0.656528i \(-0.772024\pi\)
−0.754301 + 0.656528i \(0.772024\pi\)
\(354\) 0 0
\(355\) 1179.37 0.176323
\(356\) 0 0
\(357\) 17177.7 2.54661
\(358\) 0 0
\(359\) 176.590 0.0259612 0.0129806 0.999916i \(-0.495868\pi\)
0.0129806 + 0.999916i \(0.495868\pi\)
\(360\) 0 0
\(361\) 1210.28 0.176451
\(362\) 0 0
\(363\) 12036.4 1.74036
\(364\) 0 0
\(365\) 561.057 0.0804577
\(366\) 0 0
\(367\) 50.2297 0.00714433 0.00357216 0.999994i \(-0.498863\pi\)
0.00357216 + 0.999994i \(0.498863\pi\)
\(368\) 0 0
\(369\) 15506.9 2.18769
\(370\) 0 0
\(371\) −14408.6 −2.01633
\(372\) 0 0
\(373\) 10666.6 1.48068 0.740340 0.672232i \(-0.234664\pi\)
0.740340 + 0.672232i \(0.234664\pi\)
\(374\) 0 0
\(375\) −4785.61 −0.659008
\(376\) 0 0
\(377\) 1895.59 0.258960
\(378\) 0 0
\(379\) 4158.76 0.563644 0.281822 0.959467i \(-0.409061\pi\)
0.281822 + 0.959467i \(0.409061\pi\)
\(380\) 0 0
\(381\) 10140.9 1.36361
\(382\) 0 0
\(383\) −10572.2 −1.41049 −0.705243 0.708965i \(-0.749162\pi\)
−0.705243 + 0.708965i \(0.749162\pi\)
\(384\) 0 0
\(385\) −21.0714 −0.00278935
\(386\) 0 0
\(387\) −18592.5 −2.44215
\(388\) 0 0
\(389\) 8211.23 1.07025 0.535123 0.844774i \(-0.320265\pi\)
0.535123 + 0.844774i \(0.320265\pi\)
\(390\) 0 0
\(391\) −7564.68 −0.978420
\(392\) 0 0
\(393\) 11543.0 1.48160
\(394\) 0 0
\(395\) 441.045 0.0561807
\(396\) 0 0
\(397\) −4879.38 −0.616849 −0.308424 0.951249i \(-0.599802\pi\)
−0.308424 + 0.951249i \(0.599802\pi\)
\(398\) 0 0
\(399\) 22582.6 2.83344
\(400\) 0 0
\(401\) 7570.58 0.942785 0.471393 0.881923i \(-0.343751\pi\)
0.471393 + 0.881923i \(0.343751\pi\)
\(402\) 0 0
\(403\) −16881.0 −2.08660
\(404\) 0 0
\(405\) −1712.28 −0.210084
\(406\) 0 0
\(407\) 45.0690 0.00548891
\(408\) 0 0
\(409\) −11758.1 −1.42152 −0.710760 0.703435i \(-0.751649\pi\)
−0.710760 + 0.703435i \(0.751649\pi\)
\(410\) 0 0
\(411\) 14149.2 1.69812
\(412\) 0 0
\(413\) 2844.93 0.338959
\(414\) 0 0
\(415\) −2046.73 −0.242097
\(416\) 0 0
\(417\) 20184.6 2.37037
\(418\) 0 0
\(419\) −10150.6 −1.18351 −0.591755 0.806118i \(-0.701565\pi\)
−0.591755 + 0.806118i \(0.701565\pi\)
\(420\) 0 0
\(421\) −2822.78 −0.326779 −0.163390 0.986562i \(-0.552243\pi\)
−0.163390 + 0.986562i \(0.552243\pi\)
\(422\) 0 0
\(423\) −8079.62 −0.928710
\(424\) 0 0
\(425\) 8223.37 0.938569
\(426\) 0 0
\(427\) 22014.1 2.49494
\(428\) 0 0
\(429\) −207.784 −0.0233843
\(430\) 0 0
\(431\) −6663.84 −0.744747 −0.372373 0.928083i \(-0.621456\pi\)
−0.372373 + 0.928083i \(0.621456\pi\)
\(432\) 0 0
\(433\) −6591.27 −0.731539 −0.365769 0.930706i \(-0.619194\pi\)
−0.365769 + 0.930706i \(0.619194\pi\)
\(434\) 0 0
\(435\) 565.656 0.0623474
\(436\) 0 0
\(437\) −9944.87 −1.08862
\(438\) 0 0
\(439\) 12121.1 1.31778 0.658891 0.752238i \(-0.271026\pi\)
0.658891 + 0.752238i \(0.271026\pi\)
\(440\) 0 0
\(441\) 23543.1 2.54218
\(442\) 0 0
\(443\) −9017.80 −0.967152 −0.483576 0.875302i \(-0.660662\pi\)
−0.483576 + 0.875302i \(0.660662\pi\)
\(444\) 0 0
\(445\) −2997.89 −0.319357
\(446\) 0 0
\(447\) −31427.5 −3.32543
\(448\) 0 0
\(449\) −4485.59 −0.471466 −0.235733 0.971818i \(-0.575749\pi\)
−0.235733 + 0.971818i \(0.575749\pi\)
\(450\) 0 0
\(451\) 99.4714 0.0103856
\(452\) 0 0
\(453\) −9601.11 −0.995805
\(454\) 0 0
\(455\) −3918.66 −0.403757
\(456\) 0 0
\(457\) 7305.52 0.747785 0.373893 0.927472i \(-0.378023\pi\)
0.373893 + 0.927472i \(0.378023\pi\)
\(458\) 0 0
\(459\) 17175.8 1.74662
\(460\) 0 0
\(461\) −1174.43 −0.118652 −0.0593259 0.998239i \(-0.518895\pi\)
−0.0593259 + 0.998239i \(0.518895\pi\)
\(462\) 0 0
\(463\) 7866.43 0.789598 0.394799 0.918768i \(-0.370814\pi\)
0.394799 + 0.918768i \(0.370814\pi\)
\(464\) 0 0
\(465\) −5037.38 −0.502372
\(466\) 0 0
\(467\) −12691.8 −1.25762 −0.628810 0.777559i \(-0.716458\pi\)
−0.628810 + 0.777559i \(0.716458\pi\)
\(468\) 0 0
\(469\) −7986.78 −0.786344
\(470\) 0 0
\(471\) −15052.8 −1.47260
\(472\) 0 0
\(473\) −119.265 −0.0115936
\(474\) 0 0
\(475\) 10810.8 1.04428
\(476\) 0 0
\(477\) −28402.6 −2.72634
\(478\) 0 0
\(479\) 11226.1 1.07084 0.535420 0.844586i \(-0.320153\pi\)
0.535420 + 0.844586i \(0.320153\pi\)
\(480\) 0 0
\(481\) 8381.48 0.794516
\(482\) 0 0
\(483\) −27831.6 −2.62191
\(484\) 0 0
\(485\) −2649.10 −0.248020
\(486\) 0 0
\(487\) 2493.14 0.231981 0.115991 0.993250i \(-0.462996\pi\)
0.115991 + 0.993250i \(0.462996\pi\)
\(488\) 0 0
\(489\) −2163.00 −0.200029
\(490\) 0 0
\(491\) −11346.8 −1.04292 −0.521458 0.853277i \(-0.674612\pi\)
−0.521458 + 0.853277i \(0.674612\pi\)
\(492\) 0 0
\(493\) −1981.56 −0.181024
\(494\) 0 0
\(495\) −41.5364 −0.00377156
\(496\) 0 0
\(497\) −15200.4 −1.37189
\(498\) 0 0
\(499\) 19216.1 1.72391 0.861956 0.506982i \(-0.169239\pi\)
0.861956 + 0.506982i \(0.169239\pi\)
\(500\) 0 0
\(501\) −12032.4 −1.07299
\(502\) 0 0
\(503\) −9284.26 −0.822991 −0.411496 0.911412i \(-0.634993\pi\)
−0.411496 + 0.911412i \(0.634993\pi\)
\(504\) 0 0
\(505\) 2351.36 0.207196
\(506\) 0 0
\(507\) −18771.8 −1.64435
\(508\) 0 0
\(509\) −2987.03 −0.260113 −0.130057 0.991507i \(-0.541516\pi\)
−0.130057 + 0.991507i \(0.541516\pi\)
\(510\) 0 0
\(511\) −7231.19 −0.626006
\(512\) 0 0
\(513\) 22580.1 1.94334
\(514\) 0 0
\(515\) 2646.19 0.226418
\(516\) 0 0
\(517\) −51.8279 −0.00440887
\(518\) 0 0
\(519\) 27198.5 2.30035
\(520\) 0 0
\(521\) 11082.9 0.931962 0.465981 0.884795i \(-0.345702\pi\)
0.465981 + 0.884795i \(0.345702\pi\)
\(522\) 0 0
\(523\) −13718.6 −1.14698 −0.573492 0.819211i \(-0.694412\pi\)
−0.573492 + 0.819211i \(0.694412\pi\)
\(524\) 0 0
\(525\) 30255.0 2.51512
\(526\) 0 0
\(527\) 17646.5 1.45862
\(528\) 0 0
\(529\) 89.4044 0.00734811
\(530\) 0 0
\(531\) 5607.99 0.458316
\(532\) 0 0
\(533\) 18498.7 1.50332
\(534\) 0 0
\(535\) −787.901 −0.0636709
\(536\) 0 0
\(537\) 6532.40 0.524942
\(538\) 0 0
\(539\) 151.021 0.0120685
\(540\) 0 0
\(541\) −9946.06 −0.790415 −0.395207 0.918592i \(-0.629327\pi\)
−0.395207 + 0.918592i \(0.629327\pi\)
\(542\) 0 0
\(543\) 24996.8 1.97553
\(544\) 0 0
\(545\) −3794.86 −0.298264
\(546\) 0 0
\(547\) 14925.2 1.16665 0.583325 0.812239i \(-0.301752\pi\)
0.583325 + 0.812239i \(0.301752\pi\)
\(548\) 0 0
\(549\) 43394.7 3.37348
\(550\) 0 0
\(551\) −2605.05 −0.201413
\(552\) 0 0
\(553\) −5684.40 −0.437117
\(554\) 0 0
\(555\) 2501.08 0.191288
\(556\) 0 0
\(557\) −12187.8 −0.927136 −0.463568 0.886061i \(-0.653431\pi\)
−0.463568 + 0.886061i \(0.653431\pi\)
\(558\) 0 0
\(559\) −22179.6 −1.67817
\(560\) 0 0
\(561\) 217.207 0.0163467
\(562\) 0 0
\(563\) 25198.0 1.88627 0.943133 0.332415i \(-0.107864\pi\)
0.943133 + 0.332415i \(0.107864\pi\)
\(564\) 0 0
\(565\) 3805.47 0.283358
\(566\) 0 0
\(567\) 22068.7 1.63457
\(568\) 0 0
\(569\) −14101.5 −1.03895 −0.519477 0.854485i \(-0.673873\pi\)
−0.519477 + 0.854485i \(0.673873\pi\)
\(570\) 0 0
\(571\) −21762.0 −1.59494 −0.797472 0.603356i \(-0.793830\pi\)
−0.797472 + 0.603356i \(0.793830\pi\)
\(572\) 0 0
\(573\) −6511.31 −0.474719
\(574\) 0 0
\(575\) −13323.6 −0.966319
\(576\) 0 0
\(577\) −17435.4 −1.25796 −0.628982 0.777420i \(-0.716528\pi\)
−0.628982 + 0.777420i \(0.716528\pi\)
\(578\) 0 0
\(579\) −9225.63 −0.662183
\(580\) 0 0
\(581\) 26379.3 1.88365
\(582\) 0 0
\(583\) −182.193 −0.0129428
\(584\) 0 0
\(585\) −7724.53 −0.545932
\(586\) 0 0
\(587\) 18080.4 1.27131 0.635655 0.771974i \(-0.280730\pi\)
0.635655 + 0.771974i \(0.280730\pi\)
\(588\) 0 0
\(589\) 23198.9 1.62291
\(590\) 0 0
\(591\) 1649.16 0.114784
\(592\) 0 0
\(593\) −24509.3 −1.69726 −0.848630 0.528987i \(-0.822572\pi\)
−0.848630 + 0.528987i \(0.822572\pi\)
\(594\) 0 0
\(595\) 4096.37 0.282243
\(596\) 0 0
\(597\) 28494.0 1.95340
\(598\) 0 0
\(599\) −9457.05 −0.645083 −0.322541 0.946555i \(-0.604537\pi\)
−0.322541 + 0.946555i \(0.604537\pi\)
\(600\) 0 0
\(601\) −11759.8 −0.798158 −0.399079 0.916917i \(-0.630670\pi\)
−0.399079 + 0.916917i \(0.630670\pi\)
\(602\) 0 0
\(603\) −15743.7 −1.06324
\(604\) 0 0
\(605\) 2870.33 0.192885
\(606\) 0 0
\(607\) 17929.0 1.19887 0.599435 0.800424i \(-0.295392\pi\)
0.599435 + 0.800424i \(0.295392\pi\)
\(608\) 0 0
\(609\) −7290.45 −0.485097
\(610\) 0 0
\(611\) −9638.43 −0.638182
\(612\) 0 0
\(613\) −334.350 −0.0220298 −0.0110149 0.999939i \(-0.503506\pi\)
−0.0110149 + 0.999939i \(0.503506\pi\)
\(614\) 0 0
\(615\) 5520.12 0.361939
\(616\) 0 0
\(617\) 23637.6 1.54233 0.771163 0.636638i \(-0.219675\pi\)
0.771163 + 0.636638i \(0.219675\pi\)
\(618\) 0 0
\(619\) −12855.8 −0.834763 −0.417381 0.908731i \(-0.637052\pi\)
−0.417381 + 0.908731i \(0.637052\pi\)
\(620\) 0 0
\(621\) −27828.5 −1.79826
\(622\) 0 0
\(623\) 38638.4 2.48477
\(624\) 0 0
\(625\) 13902.3 0.889750
\(626\) 0 0
\(627\) 285.550 0.0181878
\(628\) 0 0
\(629\) −8761.58 −0.555401
\(630\) 0 0
\(631\) −15364.3 −0.969321 −0.484660 0.874702i \(-0.661057\pi\)
−0.484660 + 0.874702i \(0.661057\pi\)
\(632\) 0 0
\(633\) 18344.5 1.15186
\(634\) 0 0
\(635\) 2418.31 0.151130
\(636\) 0 0
\(637\) 28085.3 1.74691
\(638\) 0 0
\(639\) −29963.2 −1.85497
\(640\) 0 0
\(641\) 4238.75 0.261186 0.130593 0.991436i \(-0.458312\pi\)
0.130593 + 0.991436i \(0.458312\pi\)
\(642\) 0 0
\(643\) 2694.68 0.165269 0.0826345 0.996580i \(-0.473667\pi\)
0.0826345 + 0.996580i \(0.473667\pi\)
\(644\) 0 0
\(645\) −6618.53 −0.404037
\(646\) 0 0
\(647\) 4728.45 0.287318 0.143659 0.989627i \(-0.454113\pi\)
0.143659 + 0.989627i \(0.454113\pi\)
\(648\) 0 0
\(649\) 35.9733 0.00217577
\(650\) 0 0
\(651\) 64924.3 3.90873
\(652\) 0 0
\(653\) 2902.87 0.173963 0.0869817 0.996210i \(-0.472278\pi\)
0.0869817 + 0.996210i \(0.472278\pi\)
\(654\) 0 0
\(655\) 2752.66 0.164207
\(656\) 0 0
\(657\) −14254.3 −0.846440
\(658\) 0 0
\(659\) −3791.51 −0.224122 −0.112061 0.993701i \(-0.535745\pi\)
−0.112061 + 0.993701i \(0.535745\pi\)
\(660\) 0 0
\(661\) −7566.63 −0.445246 −0.222623 0.974905i \(-0.571462\pi\)
−0.222623 + 0.974905i \(0.571462\pi\)
\(662\) 0 0
\(663\) 40393.9 2.36617
\(664\) 0 0
\(665\) 5385.27 0.314033
\(666\) 0 0
\(667\) 3210.55 0.186376
\(668\) 0 0
\(669\) 17565.4 1.01513
\(670\) 0 0
\(671\) 278.362 0.0160150
\(672\) 0 0
\(673\) 18220.4 1.04360 0.521801 0.853067i \(-0.325260\pi\)
0.521801 + 0.853067i \(0.325260\pi\)
\(674\) 0 0
\(675\) 30251.6 1.72502
\(676\) 0 0
\(677\) −24827.8 −1.40947 −0.704735 0.709470i \(-0.748934\pi\)
−0.704735 + 0.709470i \(0.748934\pi\)
\(678\) 0 0
\(679\) 34143.0 1.92973
\(680\) 0 0
\(681\) 31440.6 1.76917
\(682\) 0 0
\(683\) 7554.87 0.423249 0.211625 0.977351i \(-0.432125\pi\)
0.211625 + 0.977351i \(0.432125\pi\)
\(684\) 0 0
\(685\) 3374.16 0.188205
\(686\) 0 0
\(687\) 27270.3 1.51445
\(688\) 0 0
\(689\) −33882.4 −1.87346
\(690\) 0 0
\(691\) 11730.1 0.645780 0.322890 0.946437i \(-0.395346\pi\)
0.322890 + 0.946437i \(0.395346\pi\)
\(692\) 0 0
\(693\) 535.343 0.0293449
\(694\) 0 0
\(695\) 4813.43 0.262711
\(696\) 0 0
\(697\) −19337.6 −1.05088
\(698\) 0 0
\(699\) −49834.2 −2.69657
\(700\) 0 0
\(701\) 13791.3 0.743065 0.371533 0.928420i \(-0.378832\pi\)
0.371533 + 0.928420i \(0.378832\pi\)
\(702\) 0 0
\(703\) −11518.4 −0.617957
\(704\) 0 0
\(705\) −2876.16 −0.153649
\(706\) 0 0
\(707\) −30305.5 −1.61210
\(708\) 0 0
\(709\) −3236.05 −0.171414 −0.0857068 0.996320i \(-0.527315\pi\)
−0.0857068 + 0.996320i \(0.527315\pi\)
\(710\) 0 0
\(711\) −11205.2 −0.591038
\(712\) 0 0
\(713\) −28591.2 −1.50175
\(714\) 0 0
\(715\) −49.5501 −0.00259171
\(716\) 0 0
\(717\) −15763.0 −0.821030
\(718\) 0 0
\(719\) −5692.08 −0.295241 −0.147621 0.989044i \(-0.547161\pi\)
−0.147621 + 0.989044i \(0.547161\pi\)
\(720\) 0 0
\(721\) −34105.4 −1.76165
\(722\) 0 0
\(723\) 10750.1 0.552975
\(724\) 0 0
\(725\) −3490.11 −0.178785
\(726\) 0 0
\(727\) −24211.1 −1.23513 −0.617565 0.786520i \(-0.711881\pi\)
−0.617565 + 0.786520i \(0.711881\pi\)
\(728\) 0 0
\(729\) −17878.4 −0.908318
\(730\) 0 0
\(731\) 23185.5 1.17311
\(732\) 0 0
\(733\) −35957.5 −1.81190 −0.905948 0.423390i \(-0.860840\pi\)
−0.905948 + 0.423390i \(0.860840\pi\)
\(734\) 0 0
\(735\) 8380.82 0.420587
\(736\) 0 0
\(737\) −100.990 −0.00504752
\(738\) 0 0
\(739\) −15533.2 −0.773206 −0.386603 0.922246i \(-0.626352\pi\)
−0.386603 + 0.922246i \(0.626352\pi\)
\(740\) 0 0
\(741\) 53103.7 2.63267
\(742\) 0 0
\(743\) 7606.78 0.375593 0.187797 0.982208i \(-0.439865\pi\)
0.187797 + 0.982208i \(0.439865\pi\)
\(744\) 0 0
\(745\) −7494.50 −0.368560
\(746\) 0 0
\(747\) 51999.5 2.54694
\(748\) 0 0
\(749\) 10154.9 0.495395
\(750\) 0 0
\(751\) −29108.6 −1.41437 −0.707183 0.707030i \(-0.750035\pi\)
−0.707183 + 0.707030i \(0.750035\pi\)
\(752\) 0 0
\(753\) 42321.1 2.04816
\(754\) 0 0
\(755\) −2289.58 −0.110366
\(756\) 0 0
\(757\) −3291.98 −0.158057 −0.0790284 0.996872i \(-0.525182\pi\)
−0.0790284 + 0.996872i \(0.525182\pi\)
\(758\) 0 0
\(759\) −351.922 −0.0168300
\(760\) 0 0
\(761\) −4341.98 −0.206829 −0.103414 0.994638i \(-0.532977\pi\)
−0.103414 + 0.994638i \(0.532977\pi\)
\(762\) 0 0
\(763\) 48910.0 2.32066
\(764\) 0 0
\(765\) 8074.84 0.381629
\(766\) 0 0
\(767\) 6689.95 0.314941
\(768\) 0 0
\(769\) 13820.2 0.648075 0.324038 0.946044i \(-0.394960\pi\)
0.324038 + 0.946044i \(0.394960\pi\)
\(770\) 0 0
\(771\) −20108.6 −0.939292
\(772\) 0 0
\(773\) −19762.4 −0.919539 −0.459769 0.888038i \(-0.652068\pi\)
−0.459769 + 0.888038i \(0.652068\pi\)
\(774\) 0 0
\(775\) 31080.8 1.44059
\(776\) 0 0
\(777\) −32235.2 −1.48833
\(778\) 0 0
\(779\) −25422.1 −1.16924
\(780\) 0 0
\(781\) −192.204 −0.00880613
\(782\) 0 0
\(783\) −7289.64 −0.332708
\(784\) 0 0
\(785\) −3589.64 −0.163210
\(786\) 0 0
\(787\) −12090.5 −0.547622 −0.273811 0.961784i \(-0.588284\pi\)
−0.273811 + 0.961784i \(0.588284\pi\)
\(788\) 0 0
\(789\) −60784.2 −2.74268
\(790\) 0 0
\(791\) −49046.8 −2.20468
\(792\) 0 0
\(793\) 51766.9 2.31815
\(794\) 0 0
\(795\) −10110.7 −0.451056
\(796\) 0 0
\(797\) 26490.3 1.17733 0.588667 0.808376i \(-0.299653\pi\)
0.588667 + 0.808376i \(0.299653\pi\)
\(798\) 0 0
\(799\) 10075.5 0.446116
\(800\) 0 0
\(801\) 76164.7 3.35973
\(802\) 0 0
\(803\) −91.4360 −0.00401832
\(804\) 0 0
\(805\) −6637.00 −0.290588
\(806\) 0 0
\(807\) −25710.7 −1.12151
\(808\) 0 0
\(809\) −21825.5 −0.948508 −0.474254 0.880388i \(-0.657282\pi\)
−0.474254 + 0.880388i \(0.657282\pi\)
\(810\) 0 0
\(811\) −28758.7 −1.24520 −0.622599 0.782541i \(-0.713923\pi\)
−0.622599 + 0.782541i \(0.713923\pi\)
\(812\) 0 0
\(813\) 34304.0 1.47982
\(814\) 0 0
\(815\) −515.809 −0.0221693
\(816\) 0 0
\(817\) 30480.7 1.30524
\(818\) 0 0
\(819\) 99557.6 4.24765
\(820\) 0 0
\(821\) −3491.27 −0.148412 −0.0742060 0.997243i \(-0.523642\pi\)
−0.0742060 + 0.997243i \(0.523642\pi\)
\(822\) 0 0
\(823\) −1472.33 −0.0623597 −0.0311799 0.999514i \(-0.509926\pi\)
−0.0311799 + 0.999514i \(0.509926\pi\)
\(824\) 0 0
\(825\) 382.565 0.0161445
\(826\) 0 0
\(827\) 17843.8 0.750289 0.375145 0.926966i \(-0.377593\pi\)
0.375145 + 0.926966i \(0.377593\pi\)
\(828\) 0 0
\(829\) −19574.7 −0.820093 −0.410047 0.912065i \(-0.634488\pi\)
−0.410047 + 0.912065i \(0.634488\pi\)
\(830\) 0 0
\(831\) 60578.4 2.52881
\(832\) 0 0
\(833\) −29359.0 −1.22116
\(834\) 0 0
\(835\) −2869.36 −0.118920
\(836\) 0 0
\(837\) 64917.0 2.68084
\(838\) 0 0
\(839\) −26014.0 −1.07045 −0.535223 0.844711i \(-0.679772\pi\)
−0.535223 + 0.844711i \(0.679772\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −77171.9 −3.15296
\(844\) 0 0
\(845\) −4476.52 −0.182245
\(846\) 0 0
\(847\) −36994.2 −1.50075
\(848\) 0 0
\(849\) −82307.8 −3.32721
\(850\) 0 0
\(851\) 14195.6 0.571822
\(852\) 0 0
\(853\) −17613.3 −0.706996 −0.353498 0.935435i \(-0.615008\pi\)
−0.353498 + 0.935435i \(0.615008\pi\)
\(854\) 0 0
\(855\) 10615.6 0.424613
\(856\) 0 0
\(857\) −40047.5 −1.59626 −0.798130 0.602485i \(-0.794177\pi\)
−0.798130 + 0.602485i \(0.794177\pi\)
\(858\) 0 0
\(859\) −10686.0 −0.424448 −0.212224 0.977221i \(-0.568071\pi\)
−0.212224 + 0.977221i \(0.568071\pi\)
\(860\) 0 0
\(861\) −71146.0 −2.81609
\(862\) 0 0
\(863\) 24243.4 0.956263 0.478131 0.878288i \(-0.341314\pi\)
0.478131 + 0.878288i \(0.341314\pi\)
\(864\) 0 0
\(865\) 6486.02 0.254950
\(866\) 0 0
\(867\) 2207.36 0.0864656
\(868\) 0 0
\(869\) −71.8775 −0.00280584
\(870\) 0 0
\(871\) −18781.1 −0.730626
\(872\) 0 0
\(873\) 67303.3 2.60924
\(874\) 0 0
\(875\) 14708.7 0.568279
\(876\) 0 0
\(877\) −32542.0 −1.25298 −0.626491 0.779429i \(-0.715509\pi\)
−0.626491 + 0.779429i \(0.715509\pi\)
\(878\) 0 0
\(879\) 58029.2 2.22671
\(880\) 0 0
\(881\) 28538.8 1.09137 0.545686 0.837990i \(-0.316269\pi\)
0.545686 + 0.837990i \(0.316269\pi\)
\(882\) 0 0
\(883\) 3707.98 0.141317 0.0706587 0.997501i \(-0.477490\pi\)
0.0706587 + 0.997501i \(0.477490\pi\)
\(884\) 0 0
\(885\) 1996.32 0.0758254
\(886\) 0 0
\(887\) 4461.95 0.168904 0.0844519 0.996428i \(-0.473086\pi\)
0.0844519 + 0.996428i \(0.473086\pi\)
\(888\) 0 0
\(889\) −31168.4 −1.17588
\(890\) 0 0
\(891\) 279.052 0.0104922
\(892\) 0 0
\(893\) 13245.8 0.496363
\(894\) 0 0
\(895\) 1557.78 0.0581798
\(896\) 0 0
\(897\) −65446.8 −2.43613
\(898\) 0 0
\(899\) −7489.43 −0.277849
\(900\) 0 0
\(901\) 35418.9 1.30963
\(902\) 0 0
\(903\) 85302.9 3.14363
\(904\) 0 0
\(905\) 5960.99 0.218950
\(906\) 0 0
\(907\) −30719.9 −1.12463 −0.562313 0.826924i \(-0.690088\pi\)
−0.562313 + 0.826924i \(0.690088\pi\)
\(908\) 0 0
\(909\) −59738.9 −2.17977
\(910\) 0 0
\(911\) 22983.9 0.835886 0.417943 0.908473i \(-0.362751\pi\)
0.417943 + 0.908473i \(0.362751\pi\)
\(912\) 0 0
\(913\) 333.558 0.0120911
\(914\) 0 0
\(915\) 15447.5 0.558120
\(916\) 0 0
\(917\) −35477.7 −1.27762
\(918\) 0 0
\(919\) −52064.7 −1.86883 −0.934416 0.356183i \(-0.884078\pi\)
−0.934416 + 0.356183i \(0.884078\pi\)
\(920\) 0 0
\(921\) 88957.8 3.18269
\(922\) 0 0
\(923\) −35744.1 −1.27468
\(924\) 0 0
\(925\) −15431.7 −0.548532
\(926\) 0 0
\(927\) −67229.3 −2.38199
\(928\) 0 0
\(929\) −48448.1 −1.71101 −0.855506 0.517793i \(-0.826754\pi\)
−0.855506 + 0.517793i \(0.826754\pi\)
\(930\) 0 0
\(931\) −38596.7 −1.35871
\(932\) 0 0
\(933\) −61562.0 −2.16018
\(934\) 0 0
\(935\) 51.7973 0.00181171
\(936\) 0 0
\(937\) 20972.3 0.731199 0.365600 0.930772i \(-0.380864\pi\)
0.365600 + 0.930772i \(0.380864\pi\)
\(938\) 0 0
\(939\) −16822.2 −0.584633
\(940\) 0 0
\(941\) 19883.9 0.688840 0.344420 0.938816i \(-0.388076\pi\)
0.344420 + 0.938816i \(0.388076\pi\)
\(942\) 0 0
\(943\) 31331.1 1.08195
\(944\) 0 0
\(945\) 15069.5 0.518741
\(946\) 0 0
\(947\) 45811.5 1.57199 0.785995 0.618233i \(-0.212151\pi\)
0.785995 + 0.618233i \(0.212151\pi\)
\(948\) 0 0
\(949\) −17004.3 −0.581649
\(950\) 0 0
\(951\) 21040.9 0.717454
\(952\) 0 0
\(953\) −24303.7 −0.826101 −0.413050 0.910708i \(-0.635537\pi\)
−0.413050 + 0.910708i \(0.635537\pi\)
\(954\) 0 0
\(955\) −1552.75 −0.0526135
\(956\) 0 0
\(957\) −92.1854 −0.00311383
\(958\) 0 0
\(959\) −43487.9 −1.46434
\(960\) 0 0
\(961\) 36905.3 1.23881
\(962\) 0 0
\(963\) 20017.5 0.669838
\(964\) 0 0
\(965\) −2200.04 −0.0733903
\(966\) 0 0
\(967\) 46462.1 1.54511 0.772555 0.634948i \(-0.218979\pi\)
0.772555 + 0.634948i \(0.218979\pi\)
\(968\) 0 0
\(969\) −55511.9 −1.84035
\(970\) 0 0
\(971\) 5526.77 0.182660 0.0913298 0.995821i \(-0.470888\pi\)
0.0913298 + 0.995821i \(0.470888\pi\)
\(972\) 0 0
\(973\) −62037.9 −2.04403
\(974\) 0 0
\(975\) 71145.6 2.33690
\(976\) 0 0
\(977\) 12957.3 0.424300 0.212150 0.977237i \(-0.431953\pi\)
0.212150 + 0.977237i \(0.431953\pi\)
\(978\) 0 0
\(979\) 488.570 0.0159497
\(980\) 0 0
\(981\) 96412.4 3.13783
\(982\) 0 0
\(983\) 57988.1 1.88152 0.940759 0.339076i \(-0.110114\pi\)
0.940759 + 0.339076i \(0.110114\pi\)
\(984\) 0 0
\(985\) 393.275 0.0127216
\(986\) 0 0
\(987\) 37069.4 1.19547
\(988\) 0 0
\(989\) −37565.5 −1.20780
\(990\) 0 0
\(991\) −33344.7 −1.06885 −0.534425 0.845216i \(-0.679472\pi\)
−0.534425 + 0.845216i \(0.679472\pi\)
\(992\) 0 0
\(993\) −15094.2 −0.482378
\(994\) 0 0
\(995\) 6794.96 0.216497
\(996\) 0 0
\(997\) 13474.8 0.428034 0.214017 0.976830i \(-0.431345\pi\)
0.214017 + 0.976830i \(0.431345\pi\)
\(998\) 0 0
\(999\) −32231.6 −1.02078
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.1.1 3
4.3 odd 2 1856.4.a.u.1.3 3
8.3 odd 2 464.4.a.g.1.1 3
8.5 even 2 232.4.a.b.1.3 3
24.5 odd 2 2088.4.a.b.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.b.1.3 3 8.5 even 2
464.4.a.g.1.1 3 8.3 odd 2
1856.4.a.p.1.1 3 1.1 even 1 trivial
1856.4.a.u.1.3 3 4.3 odd 2
2088.4.a.b.1.2 3 24.5 odd 2