Properties

Label 1840.4.a.m.1.1
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 68 x^{2} - 111 x + 342\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.73081\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +O(q^{10})\) \(q-7.73081 q^{3} -5.00000 q^{5} -23.5622 q^{7} +32.7654 q^{9} +32.1352 q^{11} +40.0811 q^{13} +38.6540 q^{15} +126.165 q^{17} -0.232742 q^{19} +182.155 q^{21} +23.0000 q^{23} +25.0000 q^{25} -44.5711 q^{27} -137.226 q^{29} -112.866 q^{31} -248.431 q^{33} +117.811 q^{35} +45.7057 q^{37} -309.859 q^{39} -135.385 q^{41} -543.528 q^{43} -163.827 q^{45} -26.4344 q^{47} +212.180 q^{49} -975.359 q^{51} +43.6958 q^{53} -160.676 q^{55} +1.79928 q^{57} -202.248 q^{59} +150.279 q^{61} -772.026 q^{63} -200.405 q^{65} +420.722 q^{67} -177.809 q^{69} -667.381 q^{71} +602.960 q^{73} -193.270 q^{75} -757.178 q^{77} +1378.88 q^{79} -540.095 q^{81} +485.178 q^{83} -630.826 q^{85} +1060.86 q^{87} -1127.71 q^{89} -944.400 q^{91} +872.545 q^{93} +1.16371 q^{95} -1486.24 q^{97} +1052.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} - 20 q^{5} + q^{7} + 32 q^{9} + 39 q^{11} - 20 q^{13} - 20 q^{15} - 23 q^{17} - 53 q^{19} + 300 q^{21} + 92 q^{23} + 100 q^{25} - 137 q^{27} + 161 q^{29} - 388 q^{31} + 87 q^{33} - 5 q^{35} + 466 q^{37} - 1047 q^{39} + 484 q^{41} - 894 q^{43} - 160 q^{45} + 265 q^{47} + 1643 q^{49} - 1825 q^{51} + 576 q^{53} - 195 q^{55} + 178 q^{57} + 94 q^{59} + 1153 q^{61} - 60 q^{63} + 100 q^{65} + 1472 q^{67} + 92 q^{69} - 200 q^{71} + 1147 q^{73} + 100 q^{75} - 2176 q^{77} + 908 q^{79} - 1056 q^{81} + 1048 q^{83} + 115 q^{85} + 2167 q^{87} - 1784 q^{89} - 2329 q^{91} + 1483 q^{93} + 265 q^{95} - 2047 q^{97} + 2665 q^{99} + O(q^{100}) \)

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
\(3\) −7.73081 −1.48779 −0.743897 0.668294i \(-0.767025\pi\)
−0.743897 + 0.668294i \(0.767025\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −23.5622 −1.27224 −0.636121 0.771589i \(-0.719462\pi\)
−0.636121 + 0.771589i \(0.719462\pi\)
\(8\) 0 0
\(9\) 32.7654 1.21353
\(10\) 0 0
\(11\) 32.1352 0.880830 0.440415 0.897794i \(-0.354831\pi\)
0.440415 + 0.897794i \(0.354831\pi\)
\(12\) 0 0
\(13\) 40.0811 0.855115 0.427557 0.903988i \(-0.359374\pi\)
0.427557 + 0.903988i \(0.359374\pi\)
\(14\) 0 0
\(15\) 38.6540 0.665362
\(16\) 0 0
\(17\) 126.165 1.79997 0.899987 0.435916i \(-0.143576\pi\)
0.899987 + 0.435916i \(0.143576\pi\)
\(18\) 0 0
\(19\) −0.232742 −0.00281024 −0.00140512 0.999999i \(-0.500447\pi\)
−0.00140512 + 0.999999i \(0.500447\pi\)
\(20\) 0 0
\(21\) 182.155 1.89283
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −44.5711 −0.317693
\(28\) 0 0
\(29\) −137.226 −0.878695 −0.439347 0.898317i \(-0.644790\pi\)
−0.439347 + 0.898317i \(0.644790\pi\)
\(30\) 0 0
\(31\) −112.866 −0.653914 −0.326957 0.945039i \(-0.606023\pi\)
−0.326957 + 0.945039i \(0.606023\pi\)
\(32\) 0 0
\(33\) −248.431 −1.31049
\(34\) 0 0
\(35\) 117.811 0.568964
\(36\) 0 0
\(37\) 45.7057 0.203080 0.101540 0.994831i \(-0.467623\pi\)
0.101540 + 0.994831i \(0.467623\pi\)
\(38\) 0 0
\(39\) −309.859 −1.27223
\(40\) 0 0
\(41\) −135.385 −0.515696 −0.257848 0.966186i \(-0.583013\pi\)
−0.257848 + 0.966186i \(0.583013\pi\)
\(42\) 0 0
\(43\) −543.528 −1.92761 −0.963805 0.266608i \(-0.914097\pi\)
−0.963805 + 0.266608i \(0.914097\pi\)
\(44\) 0 0
\(45\) −163.827 −0.542708
\(46\) 0 0
\(47\) −26.4344 −0.0820394 −0.0410197 0.999158i \(-0.513061\pi\)
−0.0410197 + 0.999158i \(0.513061\pi\)
\(48\) 0 0
\(49\) 212.180 0.618599
\(50\) 0 0
\(51\) −975.359 −2.67799
\(52\) 0 0
\(53\) 43.6958 0.113247 0.0566235 0.998396i \(-0.481967\pi\)
0.0566235 + 0.998396i \(0.481967\pi\)
\(54\) 0 0
\(55\) −160.676 −0.393919
\(56\) 0 0
\(57\) 1.79928 0.00418106
\(58\) 0 0
\(59\) −202.248 −0.446278 −0.223139 0.974787i \(-0.571630\pi\)
−0.223139 + 0.974787i \(0.571630\pi\)
\(60\) 0 0
\(61\) 150.279 0.315430 0.157715 0.987485i \(-0.449587\pi\)
0.157715 + 0.987485i \(0.449587\pi\)
\(62\) 0 0
\(63\) −772.026 −1.54391
\(64\) 0 0
\(65\) −200.405 −0.382419
\(66\) 0 0
\(67\) 420.722 0.767155 0.383577 0.923509i \(-0.374692\pi\)
0.383577 + 0.923509i \(0.374692\pi\)
\(68\) 0 0
\(69\) −177.809 −0.310227
\(70\) 0 0
\(71\) −667.381 −1.11554 −0.557771 0.829995i \(-0.688343\pi\)
−0.557771 + 0.829995i \(0.688343\pi\)
\(72\) 0 0
\(73\) 602.960 0.966728 0.483364 0.875420i \(-0.339415\pi\)
0.483364 + 0.875420i \(0.339415\pi\)
\(74\) 0 0
\(75\) −193.270 −0.297559
\(76\) 0 0
\(77\) −757.178 −1.12063
\(78\) 0 0
\(79\) 1378.88 1.96374 0.981872 0.189545i \(-0.0607013\pi\)
0.981872 + 0.189545i \(0.0607013\pi\)
\(80\) 0 0
\(81\) −540.095 −0.740871
\(82\) 0 0
\(83\) 485.178 0.641629 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(84\) 0 0
\(85\) −630.826 −0.804973
\(86\) 0 0
\(87\) 1060.86 1.30732
\(88\) 0 0
\(89\) −1127.71 −1.34311 −0.671556 0.740954i \(-0.734374\pi\)
−0.671556 + 0.740954i \(0.734374\pi\)
\(90\) 0 0
\(91\) −944.400 −1.08791
\(92\) 0 0
\(93\) 872.545 0.972890
\(94\) 0 0
\(95\) 1.16371 0.00125678
\(96\) 0 0
\(97\) −1486.24 −1.55572 −0.777858 0.628441i \(-0.783694\pi\)
−0.777858 + 0.628441i \(0.783694\pi\)
\(98\) 0 0
\(99\) 1052.92 1.06892
\(100\) 0 0
\(101\) 1888.86 1.86087 0.930437 0.366451i \(-0.119427\pi\)
0.930437 + 0.366451i \(0.119427\pi\)
\(102\) 0 0
\(103\) 1497.17 1.43224 0.716118 0.697979i \(-0.245917\pi\)
0.716118 + 0.697979i \(0.245917\pi\)
\(104\) 0 0
\(105\) −910.776 −0.846501
\(106\) 0 0
\(107\) 585.150 0.528678 0.264339 0.964430i \(-0.414846\pi\)
0.264339 + 0.964430i \(0.414846\pi\)
\(108\) 0 0
\(109\) −139.166 −0.122290 −0.0611452 0.998129i \(-0.519475\pi\)
−0.0611452 + 0.998129i \(0.519475\pi\)
\(110\) 0 0
\(111\) −353.342 −0.302142
\(112\) 0 0
\(113\) −1293.18 −1.07656 −0.538282 0.842765i \(-0.680927\pi\)
−0.538282 + 0.842765i \(0.680927\pi\)
\(114\) 0 0
\(115\) −115.000 −0.0932505
\(116\) 0 0
\(117\) 1313.27 1.03771
\(118\) 0 0
\(119\) −2972.74 −2.29000
\(120\) 0 0
\(121\) −298.328 −0.224138
\(122\) 0 0
\(123\) 1046.63 0.767250
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1298.43 0.907221 0.453610 0.891200i \(-0.350136\pi\)
0.453610 + 0.891200i \(0.350136\pi\)
\(128\) 0 0
\(129\) 4201.91 2.86789
\(130\) 0 0
\(131\) 525.247 0.350313 0.175157 0.984541i \(-0.443957\pi\)
0.175157 + 0.984541i \(0.443957\pi\)
\(132\) 0 0
\(133\) 5.48392 0.00357531
\(134\) 0 0
\(135\) 222.855 0.142077
\(136\) 0 0
\(137\) −2428.12 −1.51422 −0.757109 0.653288i \(-0.773389\pi\)
−0.757109 + 0.653288i \(0.773389\pi\)
\(138\) 0 0
\(139\) −2456.46 −1.49895 −0.749476 0.662031i \(-0.769695\pi\)
−0.749476 + 0.662031i \(0.769695\pi\)
\(140\) 0 0
\(141\) 204.359 0.122058
\(142\) 0 0
\(143\) 1288.01 0.753211
\(144\) 0 0
\(145\) 686.128 0.392964
\(146\) 0 0
\(147\) −1640.32 −0.920349
\(148\) 0 0
\(149\) −2405.39 −1.32253 −0.661266 0.750151i \(-0.729981\pi\)
−0.661266 + 0.750151i \(0.729981\pi\)
\(150\) 0 0
\(151\) 649.276 0.349916 0.174958 0.984576i \(-0.444021\pi\)
0.174958 + 0.984576i \(0.444021\pi\)
\(152\) 0 0
\(153\) 4133.85 2.18433
\(154\) 0 0
\(155\) 564.330 0.292439
\(156\) 0 0
\(157\) 3665.04 1.86307 0.931536 0.363650i \(-0.118470\pi\)
0.931536 + 0.363650i \(0.118470\pi\)
\(158\) 0 0
\(159\) −337.804 −0.168488
\(160\) 0 0
\(161\) −541.932 −0.265281
\(162\) 0 0
\(163\) 1055.27 0.507088 0.253544 0.967324i \(-0.418404\pi\)
0.253544 + 0.967324i \(0.418404\pi\)
\(164\) 0 0
\(165\) 1242.16 0.586071
\(166\) 0 0
\(167\) 731.805 0.339095 0.169547 0.985522i \(-0.445769\pi\)
0.169547 + 0.985522i \(0.445769\pi\)
\(168\) 0 0
\(169\) −590.507 −0.268779
\(170\) 0 0
\(171\) −7.62587 −0.00341032
\(172\) 0 0
\(173\) −521.773 −0.229304 −0.114652 0.993406i \(-0.536575\pi\)
−0.114652 + 0.993406i \(0.536575\pi\)
\(174\) 0 0
\(175\) −589.056 −0.254448
\(176\) 0 0
\(177\) 1563.54 0.663970
\(178\) 0 0
\(179\) 1183.07 0.494005 0.247002 0.969015i \(-0.420554\pi\)
0.247002 + 0.969015i \(0.420554\pi\)
\(180\) 0 0
\(181\) 1723.92 0.707946 0.353973 0.935256i \(-0.384830\pi\)
0.353973 + 0.935256i \(0.384830\pi\)
\(182\) 0 0
\(183\) −1161.78 −0.469295
\(184\) 0 0
\(185\) −228.529 −0.0908204
\(186\) 0 0
\(187\) 4054.35 1.58547
\(188\) 0 0
\(189\) 1050.20 0.404182
\(190\) 0 0
\(191\) 2798.12 1.06002 0.530012 0.847990i \(-0.322187\pi\)
0.530012 + 0.847990i \(0.322187\pi\)
\(192\) 0 0
\(193\) −497.686 −0.185618 −0.0928089 0.995684i \(-0.529585\pi\)
−0.0928089 + 0.995684i \(0.529585\pi\)
\(194\) 0 0
\(195\) 1549.30 0.568961
\(196\) 0 0
\(197\) 296.489 0.107228 0.0536141 0.998562i \(-0.482926\pi\)
0.0536141 + 0.998562i \(0.482926\pi\)
\(198\) 0 0
\(199\) −2347.43 −0.836207 −0.418103 0.908399i \(-0.637305\pi\)
−0.418103 + 0.908399i \(0.637305\pi\)
\(200\) 0 0
\(201\) −3252.52 −1.14137
\(202\) 0 0
\(203\) 3233.34 1.11791
\(204\) 0 0
\(205\) 676.923 0.230626
\(206\) 0 0
\(207\) 753.604 0.253039
\(208\) 0 0
\(209\) −7.47920 −0.00247535
\(210\) 0 0
\(211\) −2838.61 −0.926153 −0.463076 0.886318i \(-0.653254\pi\)
−0.463076 + 0.886318i \(0.653254\pi\)
\(212\) 0 0
\(213\) 5159.39 1.65970
\(214\) 0 0
\(215\) 2717.64 0.862053
\(216\) 0 0
\(217\) 2659.38 0.831937
\(218\) 0 0
\(219\) −4661.37 −1.43829
\(220\) 0 0
\(221\) 5056.84 1.53918
\(222\) 0 0
\(223\) 2124.68 0.638024 0.319012 0.947751i \(-0.396649\pi\)
0.319012 + 0.947751i \(0.396649\pi\)
\(224\) 0 0
\(225\) 819.135 0.242707
\(226\) 0 0
\(227\) −1551.97 −0.453780 −0.226890 0.973920i \(-0.572856\pi\)
−0.226890 + 0.973920i \(0.572856\pi\)
\(228\) 0 0
\(229\) −158.531 −0.0457469 −0.0228735 0.999738i \(-0.507281\pi\)
−0.0228735 + 0.999738i \(0.507281\pi\)
\(230\) 0 0
\(231\) 5853.60 1.66727
\(232\) 0 0
\(233\) −728.999 −0.204971 −0.102486 0.994734i \(-0.532680\pi\)
−0.102486 + 0.994734i \(0.532680\pi\)
\(234\) 0 0
\(235\) 132.172 0.0366891
\(236\) 0 0
\(237\) −10659.8 −2.92165
\(238\) 0 0
\(239\) 6201.60 1.67844 0.839222 0.543789i \(-0.183011\pi\)
0.839222 + 0.543789i \(0.183011\pi\)
\(240\) 0 0
\(241\) −7223.04 −1.93061 −0.965305 0.261126i \(-0.915906\pi\)
−0.965305 + 0.261126i \(0.915906\pi\)
\(242\) 0 0
\(243\) 5378.79 1.41996
\(244\) 0 0
\(245\) −1060.90 −0.276646
\(246\) 0 0
\(247\) −9.32853 −0.00240308
\(248\) 0 0
\(249\) −3750.82 −0.954612
\(250\) 0 0
\(251\) −5042.78 −1.26812 −0.634059 0.773285i \(-0.718612\pi\)
−0.634059 + 0.773285i \(0.718612\pi\)
\(252\) 0 0
\(253\) 739.110 0.183666
\(254\) 0 0
\(255\) 4876.80 1.19763
\(256\) 0 0
\(257\) 2981.15 0.723577 0.361788 0.932260i \(-0.382166\pi\)
0.361788 + 0.932260i \(0.382166\pi\)
\(258\) 0 0
\(259\) −1076.93 −0.258367
\(260\) 0 0
\(261\) −4496.25 −1.06632
\(262\) 0 0
\(263\) −7242.86 −1.69815 −0.849076 0.528271i \(-0.822841\pi\)
−0.849076 + 0.528271i \(0.822841\pi\)
\(264\) 0 0
\(265\) −218.479 −0.0506456
\(266\) 0 0
\(267\) 8718.10 1.99827
\(268\) 0 0
\(269\) −2567.58 −0.581962 −0.290981 0.956729i \(-0.593982\pi\)
−0.290981 + 0.956729i \(0.593982\pi\)
\(270\) 0 0
\(271\) −8066.27 −1.80808 −0.904042 0.427443i \(-0.859414\pi\)
−0.904042 + 0.427443i \(0.859414\pi\)
\(272\) 0 0
\(273\) 7300.98 1.61859
\(274\) 0 0
\(275\) 803.380 0.176166
\(276\) 0 0
\(277\) 8991.54 1.95036 0.975179 0.221418i \(-0.0710684\pi\)
0.975179 + 0.221418i \(0.0710684\pi\)
\(278\) 0 0
\(279\) −3698.10 −0.793546
\(280\) 0 0
\(281\) 968.130 0.205529 0.102765 0.994706i \(-0.467231\pi\)
0.102765 + 0.994706i \(0.467231\pi\)
\(282\) 0 0
\(283\) 2252.65 0.473167 0.236583 0.971611i \(-0.423972\pi\)
0.236583 + 0.971611i \(0.423972\pi\)
\(284\) 0 0
\(285\) −8.99640 −0.00186983
\(286\) 0 0
\(287\) 3189.97 0.656090
\(288\) 0 0
\(289\) 11004.7 2.23991
\(290\) 0 0
\(291\) 11489.8 2.31458
\(292\) 0 0
\(293\) 2734.35 0.545196 0.272598 0.962128i \(-0.412117\pi\)
0.272598 + 0.962128i \(0.412117\pi\)
\(294\) 0 0
\(295\) 1011.24 0.199582
\(296\) 0 0
\(297\) −1432.30 −0.279834
\(298\) 0 0
\(299\) 921.865 0.178304
\(300\) 0 0
\(301\) 12806.7 2.45239
\(302\) 0 0
\(303\) −14602.4 −2.76860
\(304\) 0 0
\(305\) −751.394 −0.141065
\(306\) 0 0
\(307\) 7977.09 1.48299 0.741493 0.670961i \(-0.234118\pi\)
0.741493 + 0.670961i \(0.234118\pi\)
\(308\) 0 0
\(309\) −11574.3 −2.13087
\(310\) 0 0
\(311\) −7729.43 −1.40931 −0.704655 0.709550i \(-0.748898\pi\)
−0.704655 + 0.709550i \(0.748898\pi\)
\(312\) 0 0
\(313\) 5506.25 0.994350 0.497175 0.867650i \(-0.334371\pi\)
0.497175 + 0.867650i \(0.334371\pi\)
\(314\) 0 0
\(315\) 3860.13 0.690456
\(316\) 0 0
\(317\) 5231.37 0.926887 0.463443 0.886126i \(-0.346614\pi\)
0.463443 + 0.886126i \(0.346614\pi\)
\(318\) 0 0
\(319\) −4409.77 −0.773981
\(320\) 0 0
\(321\) −4523.68 −0.786565
\(322\) 0 0
\(323\) −29.3639 −0.00505836
\(324\) 0 0
\(325\) 1002.03 0.171023
\(326\) 0 0
\(327\) 1075.86 0.181943
\(328\) 0 0
\(329\) 622.854 0.104374
\(330\) 0 0
\(331\) 3551.61 0.589771 0.294885 0.955533i \(-0.404719\pi\)
0.294885 + 0.955533i \(0.404719\pi\)
\(332\) 0 0
\(333\) 1497.57 0.246445
\(334\) 0 0
\(335\) −2103.61 −0.343082
\(336\) 0 0
\(337\) −7002.99 −1.13198 −0.565990 0.824412i \(-0.691506\pi\)
−0.565990 + 0.824412i \(0.691506\pi\)
\(338\) 0 0
\(339\) 9997.29 1.60171
\(340\) 0 0
\(341\) −3626.97 −0.575987
\(342\) 0 0
\(343\) 3082.42 0.485234
\(344\) 0 0
\(345\) 889.043 0.138738
\(346\) 0 0
\(347\) 10268.2 1.58854 0.794272 0.607562i \(-0.207852\pi\)
0.794272 + 0.607562i \(0.207852\pi\)
\(348\) 0 0
\(349\) 4515.58 0.692588 0.346294 0.938126i \(-0.387440\pi\)
0.346294 + 0.938126i \(0.387440\pi\)
\(350\) 0 0
\(351\) −1786.46 −0.271664
\(352\) 0 0
\(353\) 7838.63 1.18189 0.590947 0.806711i \(-0.298754\pi\)
0.590947 + 0.806711i \(0.298754\pi\)
\(354\) 0 0
\(355\) 3336.90 0.498886
\(356\) 0 0
\(357\) 22981.7 3.40705
\(358\) 0 0
\(359\) 12464.8 1.83250 0.916250 0.400606i \(-0.131201\pi\)
0.916250 + 0.400606i \(0.131201\pi\)
\(360\) 0 0
\(361\) −6858.95 −0.999992
\(362\) 0 0
\(363\) 2306.32 0.333472
\(364\) 0 0
\(365\) −3014.80 −0.432334
\(366\) 0 0
\(367\) −3936.14 −0.559850 −0.279925 0.960022i \(-0.590310\pi\)
−0.279925 + 0.960022i \(0.590310\pi\)
\(368\) 0 0
\(369\) −4435.93 −0.625814
\(370\) 0 0
\(371\) −1029.57 −0.144077
\(372\) 0 0
\(373\) 1920.72 0.266625 0.133313 0.991074i \(-0.457439\pi\)
0.133313 + 0.991074i \(0.457439\pi\)
\(374\) 0 0
\(375\) 966.351 0.133072
\(376\) 0 0
\(377\) −5500.15 −0.751385
\(378\) 0 0
\(379\) 13074.5 1.77201 0.886004 0.463678i \(-0.153470\pi\)
0.886004 + 0.463678i \(0.153470\pi\)
\(380\) 0 0
\(381\) −10037.9 −1.34976
\(382\) 0 0
\(383\) −8838.22 −1.17914 −0.589571 0.807716i \(-0.700703\pi\)
−0.589571 + 0.807716i \(0.700703\pi\)
\(384\) 0 0
\(385\) 3785.89 0.501160
\(386\) 0 0
\(387\) −17808.9 −2.33922
\(388\) 0 0
\(389\) −12687.3 −1.65365 −0.826827 0.562456i \(-0.809856\pi\)
−0.826827 + 0.562456i \(0.809856\pi\)
\(390\) 0 0
\(391\) 2901.80 0.375321
\(392\) 0 0
\(393\) −4060.58 −0.521194
\(394\) 0 0
\(395\) −6894.38 −0.878213
\(396\) 0 0
\(397\) 8060.49 1.01900 0.509501 0.860470i \(-0.329830\pi\)
0.509501 + 0.860470i \(0.329830\pi\)
\(398\) 0 0
\(399\) −42.3951 −0.00531932
\(400\) 0 0
\(401\) 12325.1 1.53488 0.767440 0.641121i \(-0.221530\pi\)
0.767440 + 0.641121i \(0.221530\pi\)
\(402\) 0 0
\(403\) −4523.79 −0.559171
\(404\) 0 0
\(405\) 2700.47 0.331328
\(406\) 0 0
\(407\) 1468.76 0.178879
\(408\) 0 0
\(409\) −10806.7 −1.30650 −0.653250 0.757143i \(-0.726595\pi\)
−0.653250 + 0.757143i \(0.726595\pi\)
\(410\) 0 0
\(411\) 18771.3 2.25285
\(412\) 0 0
\(413\) 4765.41 0.567774
\(414\) 0 0
\(415\) −2425.89 −0.286945
\(416\) 0 0
\(417\) 18990.4 2.23013
\(418\) 0 0
\(419\) 1823.74 0.212639 0.106319 0.994332i \(-0.466093\pi\)
0.106319 + 0.994332i \(0.466093\pi\)
\(420\) 0 0
\(421\) 5995.43 0.694060 0.347030 0.937854i \(-0.387190\pi\)
0.347030 + 0.937854i \(0.387190\pi\)
\(422\) 0 0
\(423\) −866.133 −0.0995575
\(424\) 0 0
\(425\) 3154.13 0.359995
\(426\) 0 0
\(427\) −3540.91 −0.401303
\(428\) 0 0
\(429\) −9957.39 −1.12062
\(430\) 0 0
\(431\) −3887.80 −0.434499 −0.217249 0.976116i \(-0.569708\pi\)
−0.217249 + 0.976116i \(0.569708\pi\)
\(432\) 0 0
\(433\) −4422.93 −0.490884 −0.245442 0.969411i \(-0.578933\pi\)
−0.245442 + 0.969411i \(0.578933\pi\)
\(434\) 0 0
\(435\) −5304.32 −0.584650
\(436\) 0 0
\(437\) −5.35306 −0.000585976 0
\(438\) 0 0
\(439\) 1748.08 0.190048 0.0950242 0.995475i \(-0.469707\pi\)
0.0950242 + 0.995475i \(0.469707\pi\)
\(440\) 0 0
\(441\) 6952.14 0.750690
\(442\) 0 0
\(443\) −3371.31 −0.361571 −0.180785 0.983523i \(-0.557864\pi\)
−0.180785 + 0.983523i \(0.557864\pi\)
\(444\) 0 0
\(445\) 5638.55 0.600658
\(446\) 0 0
\(447\) 18595.6 1.96766
\(448\) 0 0
\(449\) 13314.7 1.39947 0.699734 0.714404i \(-0.253302\pi\)
0.699734 + 0.714404i \(0.253302\pi\)
\(450\) 0 0
\(451\) −4350.61 −0.454240
\(452\) 0 0
\(453\) −5019.43 −0.520603
\(454\) 0 0
\(455\) 4722.00 0.486529
\(456\) 0 0
\(457\) 3767.20 0.385607 0.192803 0.981237i \(-0.438242\pi\)
0.192803 + 0.981237i \(0.438242\pi\)
\(458\) 0 0
\(459\) −5623.32 −0.571839
\(460\) 0 0
\(461\) 9674.06 0.977366 0.488683 0.872461i \(-0.337477\pi\)
0.488683 + 0.872461i \(0.337477\pi\)
\(462\) 0 0
\(463\) −2977.73 −0.298892 −0.149446 0.988770i \(-0.547749\pi\)
−0.149446 + 0.988770i \(0.547749\pi\)
\(464\) 0 0
\(465\) −4362.73 −0.435089
\(466\) 0 0
\(467\) 10701.0 1.06035 0.530176 0.847888i \(-0.322126\pi\)
0.530176 + 0.847888i \(0.322126\pi\)
\(468\) 0 0
\(469\) −9913.16 −0.976006
\(470\) 0 0
\(471\) −28333.7 −2.77187
\(472\) 0 0
\(473\) −17466.4 −1.69790
\(474\) 0 0
\(475\) −5.81854 −0.000562048 0
\(476\) 0 0
\(477\) 1431.71 0.137429
\(478\) 0 0
\(479\) −2337.15 −0.222937 −0.111469 0.993768i \(-0.535555\pi\)
−0.111469 + 0.993768i \(0.535555\pi\)
\(480\) 0 0
\(481\) 1831.94 0.173657
\(482\) 0 0
\(483\) 4189.57 0.394683
\(484\) 0 0
\(485\) 7431.18 0.695737
\(486\) 0 0
\(487\) 7183.85 0.668442 0.334221 0.942495i \(-0.391527\pi\)
0.334221 + 0.942495i \(0.391527\pi\)
\(488\) 0 0
\(489\) −8158.12 −0.754443
\(490\) 0 0
\(491\) 12084.2 1.11070 0.555350 0.831617i \(-0.312584\pi\)
0.555350 + 0.831617i \(0.312584\pi\)
\(492\) 0 0
\(493\) −17313.1 −1.58163
\(494\) 0 0
\(495\) −5264.61 −0.478034
\(496\) 0 0
\(497\) 15725.0 1.41924
\(498\) 0 0
\(499\) −7145.78 −0.641060 −0.320530 0.947238i \(-0.603861\pi\)
−0.320530 + 0.947238i \(0.603861\pi\)
\(500\) 0 0
\(501\) −5657.45 −0.504503
\(502\) 0 0
\(503\) −20436.2 −1.81154 −0.905770 0.423771i \(-0.860706\pi\)
−0.905770 + 0.423771i \(0.860706\pi\)
\(504\) 0 0
\(505\) −9444.29 −0.832208
\(506\) 0 0
\(507\) 4565.10 0.399888
\(508\) 0 0
\(509\) 19721.7 1.71738 0.858690 0.512495i \(-0.171279\pi\)
0.858690 + 0.512495i \(0.171279\pi\)
\(510\) 0 0
\(511\) −14207.1 −1.22991
\(512\) 0 0
\(513\) 10.3735 0.000892794 0
\(514\) 0 0
\(515\) −7485.84 −0.640516
\(516\) 0 0
\(517\) −849.475 −0.0722628
\(518\) 0 0
\(519\) 4033.73 0.341158
\(520\) 0 0
\(521\) 3483.23 0.292904 0.146452 0.989218i \(-0.453215\pi\)
0.146452 + 0.989218i \(0.453215\pi\)
\(522\) 0 0
\(523\) −15689.0 −1.31172 −0.655862 0.754881i \(-0.727695\pi\)
−0.655862 + 0.754881i \(0.727695\pi\)
\(524\) 0 0
\(525\) 4553.88 0.378567
\(526\) 0 0
\(527\) −14239.8 −1.17703
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −6626.72 −0.541573
\(532\) 0 0
\(533\) −5426.36 −0.440979
\(534\) 0 0
\(535\) −2925.75 −0.236432
\(536\) 0 0
\(537\) −9146.09 −0.734977
\(538\) 0 0
\(539\) 6818.43 0.544881
\(540\) 0 0
\(541\) 7620.73 0.605621 0.302810 0.953051i \(-0.402075\pi\)
0.302810 + 0.953051i \(0.402075\pi\)
\(542\) 0 0
\(543\) −13327.3 −1.05328
\(544\) 0 0
\(545\) 695.828 0.0546899
\(546\) 0 0
\(547\) −5925.62 −0.463183 −0.231592 0.972813i \(-0.574393\pi\)
−0.231592 + 0.972813i \(0.574393\pi\)
\(548\) 0 0
\(549\) 4923.94 0.382784
\(550\) 0 0
\(551\) 31.9381 0.00246934
\(552\) 0 0
\(553\) −32489.4 −2.49836
\(554\) 0 0
\(555\) 1766.71 0.135122
\(556\) 0 0
\(557\) −5456.16 −0.415054 −0.207527 0.978229i \(-0.566541\pi\)
−0.207527 + 0.978229i \(0.566541\pi\)
\(558\) 0 0
\(559\) −21785.2 −1.64833
\(560\) 0 0
\(561\) −31343.4 −2.35886
\(562\) 0 0
\(563\) 9194.29 0.688265 0.344132 0.938921i \(-0.388173\pi\)
0.344132 + 0.938921i \(0.388173\pi\)
\(564\) 0 0
\(565\) 6465.88 0.481454
\(566\) 0 0
\(567\) 12725.8 0.942567
\(568\) 0 0
\(569\) 338.831 0.0249640 0.0124820 0.999922i \(-0.496027\pi\)
0.0124820 + 0.999922i \(0.496027\pi\)
\(570\) 0 0
\(571\) 1725.34 0.126451 0.0632254 0.997999i \(-0.479861\pi\)
0.0632254 + 0.997999i \(0.479861\pi\)
\(572\) 0 0
\(573\) −21631.7 −1.57710
\(574\) 0 0
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) 23300.0 1.68109 0.840547 0.541738i \(-0.182234\pi\)
0.840547 + 0.541738i \(0.182234\pi\)
\(578\) 0 0
\(579\) 3847.52 0.276161
\(580\) 0 0
\(581\) −11431.9 −0.816307
\(582\) 0 0
\(583\) 1404.18 0.0997513
\(584\) 0 0
\(585\) −6566.36 −0.464078
\(586\) 0 0
\(587\) −15653.3 −1.10065 −0.550325 0.834950i \(-0.685496\pi\)
−0.550325 + 0.834950i \(0.685496\pi\)
\(588\) 0 0
\(589\) 26.2686 0.00183766
\(590\) 0 0
\(591\) −2292.10 −0.159533
\(592\) 0 0
\(593\) −2658.08 −0.184072 −0.0920358 0.995756i \(-0.529337\pi\)
−0.0920358 + 0.995756i \(0.529337\pi\)
\(594\) 0 0
\(595\) 14863.7 1.02412
\(596\) 0 0
\(597\) 18147.6 1.24410
\(598\) 0 0
\(599\) −19417.6 −1.32451 −0.662256 0.749278i \(-0.730401\pi\)
−0.662256 + 0.749278i \(0.730401\pi\)
\(600\) 0 0
\(601\) −18469.0 −1.25352 −0.626760 0.779213i \(-0.715619\pi\)
−0.626760 + 0.779213i \(0.715619\pi\)
\(602\) 0 0
\(603\) 13785.1 0.930968
\(604\) 0 0
\(605\) 1491.64 0.100238
\(606\) 0 0
\(607\) −3968.56 −0.265369 −0.132684 0.991158i \(-0.542360\pi\)
−0.132684 + 0.991158i \(0.542360\pi\)
\(608\) 0 0
\(609\) −24996.4 −1.66322
\(610\) 0 0
\(611\) −1059.52 −0.0701531
\(612\) 0 0
\(613\) −11478.7 −0.756311 −0.378156 0.925742i \(-0.623442\pi\)
−0.378156 + 0.925742i \(0.623442\pi\)
\(614\) 0 0
\(615\) −5233.16 −0.343124
\(616\) 0 0
\(617\) −15691.1 −1.02382 −0.511911 0.859039i \(-0.671062\pi\)
−0.511911 + 0.859039i \(0.671062\pi\)
\(618\) 0 0
\(619\) 18249.4 1.18499 0.592494 0.805575i \(-0.298144\pi\)
0.592494 + 0.805575i \(0.298144\pi\)
\(620\) 0 0
\(621\) −1025.14 −0.0662436
\(622\) 0 0
\(623\) 26571.4 1.70876
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 57.8203 0.00368281
\(628\) 0 0
\(629\) 5766.48 0.365540
\(630\) 0 0
\(631\) 23528.8 1.48442 0.742208 0.670169i \(-0.233778\pi\)
0.742208 + 0.670169i \(0.233778\pi\)
\(632\) 0 0
\(633\) 21944.8 1.37793
\(634\) 0 0
\(635\) −6492.15 −0.405722
\(636\) 0 0
\(637\) 8504.38 0.528973
\(638\) 0 0
\(639\) −21867.0 −1.35375
\(640\) 0 0
\(641\) 3748.79 0.230996 0.115498 0.993308i \(-0.463154\pi\)
0.115498 + 0.993308i \(0.463154\pi\)
\(642\) 0 0
\(643\) −15924.3 −0.976660 −0.488330 0.872659i \(-0.662394\pi\)
−0.488330 + 0.872659i \(0.662394\pi\)
\(644\) 0 0
\(645\) −21009.5 −1.28256
\(646\) 0 0
\(647\) 2854.11 0.173426 0.0867129 0.996233i \(-0.472364\pi\)
0.0867129 + 0.996233i \(0.472364\pi\)
\(648\) 0 0
\(649\) −6499.27 −0.393095
\(650\) 0 0
\(651\) −20559.1 −1.23775
\(652\) 0 0
\(653\) 7925.97 0.474988 0.237494 0.971389i \(-0.423674\pi\)
0.237494 + 0.971389i \(0.423674\pi\)
\(654\) 0 0
\(655\) −2626.24 −0.156665
\(656\) 0 0
\(657\) 19756.2 1.17316
\(658\) 0 0
\(659\) 5799.43 0.342813 0.171406 0.985200i \(-0.445169\pi\)
0.171406 + 0.985200i \(0.445169\pi\)
\(660\) 0 0
\(661\) 22324.8 1.31367 0.656833 0.754036i \(-0.271896\pi\)
0.656833 + 0.754036i \(0.271896\pi\)
\(662\) 0 0
\(663\) −39093.4 −2.28999
\(664\) 0 0
\(665\) −27.4196 −0.00159893
\(666\) 0 0
\(667\) −3156.19 −0.183221
\(668\) 0 0
\(669\) −16425.5 −0.949249
\(670\) 0 0
\(671\) 4829.24 0.277840
\(672\) 0 0
\(673\) −23253.5 −1.33188 −0.665940 0.746005i \(-0.731969\pi\)
−0.665940 + 0.746005i \(0.731969\pi\)
\(674\) 0 0
\(675\) −1114.28 −0.0635386
\(676\) 0 0
\(677\) −19003.7 −1.07884 −0.539419 0.842038i \(-0.681356\pi\)
−0.539419 + 0.842038i \(0.681356\pi\)
\(678\) 0 0
\(679\) 35019.1 1.97925
\(680\) 0 0
\(681\) 11998.0 0.675131
\(682\) 0 0
\(683\) 11222.5 0.628722 0.314361 0.949304i \(-0.398210\pi\)
0.314361 + 0.949304i \(0.398210\pi\)
\(684\) 0 0
\(685\) 12140.6 0.677179
\(686\) 0 0
\(687\) 1225.58 0.0680620
\(688\) 0 0
\(689\) 1751.38 0.0968391
\(690\) 0 0
\(691\) −4297.68 −0.236601 −0.118301 0.992978i \(-0.537745\pi\)
−0.118301 + 0.992978i \(0.537745\pi\)
\(692\) 0 0
\(693\) −24809.2 −1.35992
\(694\) 0 0
\(695\) 12282.3 0.670352
\(696\) 0 0
\(697\) −17080.8 −0.928240
\(698\) 0 0
\(699\) 5635.75 0.304955
\(700\) 0 0
\(701\) 25769.6 1.38845 0.694225 0.719758i \(-0.255747\pi\)
0.694225 + 0.719758i \(0.255747\pi\)
\(702\) 0 0
\(703\) −10.6376 −0.000570705 0
\(704\) 0 0
\(705\) −1021.80 −0.0545859
\(706\) 0 0
\(707\) −44505.7 −2.36748
\(708\) 0 0
\(709\) −4900.01 −0.259554 −0.129777 0.991543i \(-0.541426\pi\)
−0.129777 + 0.991543i \(0.541426\pi\)
\(710\) 0 0
\(711\) 45179.4 2.38307
\(712\) 0 0
\(713\) −2595.92 −0.136350
\(714\) 0 0
\(715\) −6440.07 −0.336846
\(716\) 0 0
\(717\) −47943.4 −2.49718
\(718\) 0 0
\(719\) −7631.85 −0.395855 −0.197928 0.980217i \(-0.563421\pi\)
−0.197928 + 0.980217i \(0.563421\pi\)
\(720\) 0 0
\(721\) −35276.6 −1.82215
\(722\) 0 0
\(723\) 55839.9 2.87235
\(724\) 0 0
\(725\) −3430.64 −0.175739
\(726\) 0 0
\(727\) 36985.6 1.88682 0.943410 0.331628i \(-0.107598\pi\)
0.943410 + 0.331628i \(0.107598\pi\)
\(728\) 0 0
\(729\) −26999.8 −1.37173
\(730\) 0 0
\(731\) −68574.3 −3.46965
\(732\) 0 0
\(733\) −22227.3 −1.12003 −0.560015 0.828482i \(-0.689205\pi\)
−0.560015 + 0.828482i \(0.689205\pi\)
\(734\) 0 0
\(735\) 8201.60 0.411592
\(736\) 0 0
\(737\) 13520.0 0.675733
\(738\) 0 0
\(739\) 17899.1 0.890974 0.445487 0.895288i \(-0.353030\pi\)
0.445487 + 0.895288i \(0.353030\pi\)
\(740\) 0 0
\(741\) 72.1171 0.00357529
\(742\) 0 0
\(743\) 29843.4 1.47355 0.736776 0.676137i \(-0.236347\pi\)
0.736776 + 0.676137i \(0.236347\pi\)
\(744\) 0 0
\(745\) 12027.0 0.591455
\(746\) 0 0
\(747\) 15897.0 0.778638
\(748\) 0 0
\(749\) −13787.4 −0.672606
\(750\) 0 0
\(751\) 9399.65 0.456722 0.228361 0.973577i \(-0.426663\pi\)
0.228361 + 0.973577i \(0.426663\pi\)
\(752\) 0 0
\(753\) 38984.8 1.88670
\(754\) 0 0
\(755\) −3246.38 −0.156487
\(756\) 0 0
\(757\) 29871.6 1.43422 0.717109 0.696961i \(-0.245465\pi\)
0.717109 + 0.696961i \(0.245465\pi\)
\(758\) 0 0
\(759\) −5713.92 −0.273257
\(760\) 0 0
\(761\) −273.955 −0.0130497 −0.00652487 0.999979i \(-0.502077\pi\)
−0.00652487 + 0.999979i \(0.502077\pi\)
\(762\) 0 0
\(763\) 3279.05 0.155583
\(764\) 0 0
\(765\) −20669.3 −0.976861
\(766\) 0 0
\(767\) −8106.31 −0.381619
\(768\) 0 0
\(769\) −13273.8 −0.622450 −0.311225 0.950336i \(-0.600739\pi\)
−0.311225 + 0.950336i \(0.600739\pi\)
\(770\) 0 0
\(771\) −23046.7 −1.07653
\(772\) 0 0
\(773\) 34161.1 1.58951 0.794753 0.606933i \(-0.207600\pi\)
0.794753 + 0.606933i \(0.207600\pi\)
\(774\) 0 0
\(775\) −2821.65 −0.130783
\(776\) 0 0
\(777\) 8325.54 0.384398
\(778\) 0 0
\(779\) 31.5096 0.00144923
\(780\) 0 0
\(781\) −21446.4 −0.982603
\(782\) 0 0
\(783\) 6116.29 0.279155
\(784\) 0 0
\(785\) −18325.2 −0.833191
\(786\) 0 0
\(787\) 38489.6 1.74334 0.871669 0.490095i \(-0.163038\pi\)
0.871669 + 0.490095i \(0.163038\pi\)
\(788\) 0 0
\(789\) 55993.2 2.52650
\(790\) 0 0
\(791\) 30470.1 1.36965
\(792\) 0 0
\(793\) 6023.33 0.269729
\(794\) 0 0
\(795\) 1689.02 0.0753502
\(796\) 0 0
\(797\) 17176.8 0.763405 0.381702 0.924285i \(-0.375338\pi\)
0.381702 + 0.924285i \(0.375338\pi\)
\(798\) 0 0
\(799\) −3335.10 −0.147669
\(800\) 0 0
\(801\) −36949.8 −1.62991
\(802\) 0 0
\(803\) 19376.2 0.851523
\(804\) 0 0
\(805\) 2709.66 0.118637
\(806\) 0 0
\(807\) 19849.4 0.865841
\(808\) 0 0
\(809\) 8226.61 0.357518 0.178759 0.983893i \(-0.442792\pi\)
0.178759 + 0.983893i \(0.442792\pi\)
\(810\) 0 0
\(811\) −27867.9 −1.20662 −0.603312 0.797505i \(-0.706153\pi\)
−0.603312 + 0.797505i \(0.706153\pi\)
\(812\) 0 0
\(813\) 62358.8 2.69006
\(814\) 0 0
\(815\) −5276.37 −0.226777
\(816\) 0 0
\(817\) 126.502 0.00541705
\(818\) 0 0
\(819\) −30943.6 −1.32022
\(820\) 0 0
\(821\) −17637.5 −0.749761 −0.374880 0.927073i \(-0.622316\pi\)
−0.374880 + 0.927073i \(0.622316\pi\)
\(822\) 0 0
\(823\) −28048.6 −1.18799 −0.593993 0.804470i \(-0.702449\pi\)
−0.593993 + 0.804470i \(0.702449\pi\)
\(824\) 0 0
\(825\) −6210.78 −0.262099
\(826\) 0 0
\(827\) 32592.9 1.37046 0.685228 0.728329i \(-0.259703\pi\)
0.685228 + 0.728329i \(0.259703\pi\)
\(828\) 0 0
\(829\) 18815.9 0.788303 0.394152 0.919045i \(-0.371038\pi\)
0.394152 + 0.919045i \(0.371038\pi\)
\(830\) 0 0
\(831\) −69511.9 −2.90173
\(832\) 0 0
\(833\) 26769.7 1.11346
\(834\) 0 0
\(835\) −3659.03 −0.151648
\(836\) 0 0
\(837\) 5030.56 0.207744
\(838\) 0 0
\(839\) −7612.72 −0.313254 −0.156627 0.987658i \(-0.550062\pi\)
−0.156627 + 0.987658i \(0.550062\pi\)
\(840\) 0 0
\(841\) −5558.14 −0.227896
\(842\) 0 0
\(843\) −7484.43 −0.305786
\(844\) 0 0
\(845\) 2952.54 0.120202
\(846\) 0 0
\(847\) 7029.28 0.285158
\(848\) 0 0
\(849\) −17414.8 −0.703975
\(850\) 0 0
\(851\) 1051.23 0.0423452
\(852\) 0 0
\(853\) −31421.4 −1.26125 −0.630627 0.776086i \(-0.717202\pi\)
−0.630627 + 0.776086i \(0.717202\pi\)
\(854\) 0 0
\(855\) 38.1293 0.00152514
\(856\) 0 0
\(857\) 20909.5 0.833435 0.416718 0.909036i \(-0.363180\pi\)
0.416718 + 0.909036i \(0.363180\pi\)
\(858\) 0 0
\(859\) −23304.5 −0.925655 −0.462828 0.886448i \(-0.653165\pi\)
−0.462828 + 0.886448i \(0.653165\pi\)
\(860\) 0 0
\(861\) −24661.0 −0.976127
\(862\) 0 0
\(863\) 19146.3 0.755211 0.377606 0.925966i \(-0.376747\pi\)
0.377606 + 0.925966i \(0.376747\pi\)
\(864\) 0 0
\(865\) 2608.86 0.102548
\(866\) 0 0
\(867\) −85075.0 −3.33252
\(868\) 0 0
\(869\) 44310.5 1.72972
\(870\) 0 0
\(871\) 16863.0 0.656005
\(872\) 0 0
\(873\) −48697.1 −1.88791
\(874\) 0 0
\(875\) 2945.28 0.113793
\(876\) 0 0
\(877\) 31389.0 1.20859 0.604293 0.796762i \(-0.293456\pi\)
0.604293 + 0.796762i \(0.293456\pi\)
\(878\) 0 0
\(879\) −21138.7 −0.811139
\(880\) 0 0
\(881\) 44496.3 1.70161 0.850806 0.525480i \(-0.176114\pi\)
0.850806 + 0.525480i \(0.176114\pi\)
\(882\) 0 0
\(883\) −8906.65 −0.339448 −0.169724 0.985492i \(-0.554288\pi\)
−0.169724 + 0.985492i \(0.554288\pi\)
\(884\) 0 0
\(885\) −7817.69 −0.296936
\(886\) 0 0
\(887\) 36584.7 1.38489 0.692444 0.721472i \(-0.256534\pi\)
0.692444 + 0.721472i \(0.256534\pi\)
\(888\) 0 0
\(889\) −30593.9 −1.15420
\(890\) 0 0
\(891\) −17356.1 −0.652581
\(892\) 0 0
\(893\) 6.15238 0.000230551 0
\(894\) 0 0
\(895\) −5915.35 −0.220926
\(896\) 0 0
\(897\) −7126.76 −0.265279
\(898\) 0 0
\(899\) 15488.1 0.574591
\(900\) 0 0
\(901\) 5512.90 0.203842
\(902\) 0 0
\(903\) −99006.4 −3.64865
\(904\) 0 0
\(905\) −8619.62 −0.316603
\(906\) 0 0
\(907\) 21346.1 0.781463 0.390731 0.920505i \(-0.372222\pi\)
0.390731 + 0.920505i \(0.372222\pi\)
\(908\) 0 0
\(909\) 61889.1 2.25823
\(910\) 0 0
\(911\) 10208.0 0.371246 0.185623 0.982621i \(-0.440570\pi\)
0.185623 + 0.982621i \(0.440570\pi\)
\(912\) 0 0
\(913\) 15591.3 0.565166
\(914\) 0 0
\(915\) 5808.88 0.209875
\(916\) 0 0
\(917\) −12376.0 −0.445683
\(918\) 0 0
\(919\) −1758.00 −0.0631025 −0.0315513 0.999502i \(-0.510045\pi\)
−0.0315513 + 0.999502i \(0.510045\pi\)
\(920\) 0 0
\(921\) −61669.3 −2.20638
\(922\) 0 0
\(923\) −26749.3 −0.953917
\(924\) 0 0
\(925\) 1142.64 0.0406161
\(926\) 0 0
\(927\) 49055.3 1.73807
\(928\) 0 0
\(929\) −845.008 −0.0298426 −0.0149213 0.999889i \(-0.504750\pi\)
−0.0149213 + 0.999889i \(0.504750\pi\)
\(930\) 0 0
\(931\) −49.3830 −0.00173841
\(932\) 0 0
\(933\) 59754.7 2.09676
\(934\) 0 0
\(935\) −20271.7 −0.709045
\(936\) 0 0
\(937\) 3851.61 0.134287 0.0671433 0.997743i \(-0.478612\pi\)
0.0671433 + 0.997743i \(0.478612\pi\)
\(938\) 0 0
\(939\) −42567.7 −1.47939
\(940\) 0 0
\(941\) −4379.36 −0.151714 −0.0758571 0.997119i \(-0.524169\pi\)
−0.0758571 + 0.997119i \(0.524169\pi\)
\(942\) 0 0
\(943\) −3113.85 −0.107530
\(944\) 0 0
\(945\) −5250.98 −0.180756
\(946\) 0 0
\(947\) −29746.9 −1.02074 −0.510371 0.859954i \(-0.670492\pi\)
−0.510371 + 0.859954i \(0.670492\pi\)
\(948\) 0 0
\(949\) 24167.3 0.826663
\(950\) 0 0
\(951\) −40442.7 −1.37902
\(952\) 0 0
\(953\) 4306.61 0.146385 0.0731924 0.997318i \(-0.476681\pi\)
0.0731924 + 0.997318i \(0.476681\pi\)
\(954\) 0 0
\(955\) −13990.6 −0.474057
\(956\) 0 0
\(957\) 34091.1 1.15152
\(958\) 0 0
\(959\) 57211.9 1.92645
\(960\) 0 0
\(961\) −17052.3 −0.572397
\(962\) 0 0
\(963\) 19172.7 0.641568
\(964\) 0 0
\(965\) 2488.43 0.0830108
\(966\) 0 0
\(967\) 12233.4 0.406824 0.203412 0.979093i \(-0.434797\pi\)
0.203412 + 0.979093i \(0.434797\pi\)
\(968\) 0 0
\(969\) 227.007 0.00752581
\(970\) 0 0
\(971\) −48207.7 −1.59326 −0.796631 0.604465i \(-0.793387\pi\)
−0.796631 + 0.604465i \(0.793387\pi\)
\(972\) 0 0
\(973\) 57879.8 1.90703
\(974\) 0 0
\(975\) −7746.48 −0.254447
\(976\) 0 0
\(977\) 28662.9 0.938595 0.469298 0.883040i \(-0.344507\pi\)
0.469298 + 0.883040i \(0.344507\pi\)
\(978\) 0 0
\(979\) −36239.2 −1.18305
\(980\) 0 0
\(981\) −4559.81 −0.148403
\(982\) 0 0
\(983\) 20944.1 0.679567 0.339783 0.940504i \(-0.389646\pi\)
0.339783 + 0.940504i \(0.389646\pi\)
\(984\) 0 0
\(985\) −1482.44 −0.0479539
\(986\) 0 0
\(987\) −4815.16 −0.155287
\(988\) 0 0
\(989\) −12501.1 −0.401934
\(990\) 0 0
\(991\) 36440.2 1.16808 0.584038 0.811727i \(-0.301472\pi\)
0.584038 + 0.811727i \(0.301472\pi\)
\(992\) 0 0
\(993\) −27456.8 −0.877458
\(994\) 0 0
\(995\) 11737.2 0.373963
\(996\) 0 0
\(997\) −11945.0 −0.379439 −0.189720 0.981838i \(-0.560758\pi\)
−0.189720 + 0.981838i \(0.560758\pi\)
\(998\) 0 0
\(999\) −2037.15 −0.0645172
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 1840.4.a.m.1.1 4
4.3 odd 2 230.4.a.h.1.4 4
12.11 even 2 2070.4.a.bj.1.3 4
20.3 even 4 1150.4.b.n.599.8 8
20.7 even 4 1150.4.b.n.599.1 8
20.19 odd 2 1150.4.a.p.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.4 4 4.3 odd 2
1150.4.a.p.1.1 4 20.19 odd 2
1150.4.b.n.599.1 8 20.7 even 4
1150.4.b.n.599.8 8 20.3 even 4
1840.4.a.m.1.1 4 1.1 even 1 trivial
2070.4.a.bj.1.3 4 12.11 even 2