Properties

Label 1521.4.a.q.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.46410 q^{2} +4.00000 q^{4} +13.8564 q^{5} +22.5167 q^{7} -13.8564 q^{8} +O(q^{10})\) \(q+3.46410 q^{2} +4.00000 q^{4} +13.8564 q^{5} +22.5167 q^{7} -13.8564 q^{8} +48.0000 q^{10} -22.5167 q^{11} +78.0000 q^{14} -80.0000 q^{16} +27.0000 q^{17} +88.3346 q^{19} +55.4256 q^{20} -78.0000 q^{22} +57.0000 q^{23} +67.0000 q^{25} +90.0666 q^{28} +69.0000 q^{29} +72.7461 q^{31} -166.277 q^{32} +93.5307 q^{34} +312.000 q^{35} +39.8372 q^{37} +306.000 q^{38} -192.000 q^{40} +393.176 q^{41} +85.0000 q^{43} -90.0666 q^{44} +197.454 q^{46} +342.946 q^{47} +164.000 q^{49} +232.095 q^{50} -426.000 q^{53} -312.000 q^{55} -312.000 q^{56} +239.023 q^{58} +19.0526 q^{59} -17.0000 q^{61} +252.000 q^{62} +64.0000 q^{64} +164.545 q^{67} +108.000 q^{68} +1080.80 q^{70} +583.701 q^{71} +1004.59 q^{73} +138.000 q^{74} +353.338 q^{76} -507.000 q^{77} -1244.00 q^{79} -1108.51 q^{80} +1362.00 q^{82} -426.084 q^{83} +374.123 q^{85} +294.449 q^{86} +312.000 q^{88} +306.573 q^{89} +228.000 q^{92} +1188.00 q^{94} +1224.00 q^{95} -1234.95 q^{97} +568.113 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{4} + 96 q^{10} + 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} + 134 q^{25} + 138 q^{29} + 624 q^{35} + 612 q^{38} - 384 q^{40} + 170 q^{43} + 328 q^{49} - 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} + 128 q^{64} + 216 q^{68} + 276 q^{74} - 1014 q^{77} - 2488 q^{79} + 2724 q^{82} + 624 q^{88} + 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.46410 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 13.8564 1.23935 0.619677 0.784857i \(-0.287263\pi\)
0.619677 + 0.784857i \(0.287263\pi\)
\(6\) 0 0
\(7\) 22.5167 1.21579 0.607893 0.794019i \(-0.292015\pi\)
0.607893 + 0.794019i \(0.292015\pi\)
\(8\) −13.8564 −0.612372
\(9\) 0 0
\(10\) 48.0000 1.51789
\(11\) −22.5167 −0.617184 −0.308592 0.951194i \(-0.599858\pi\)
−0.308592 + 0.951194i \(0.599858\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 78.0000 1.48903
\(15\) 0 0
\(16\) −80.0000 −1.25000
\(17\) 27.0000 0.385204 0.192602 0.981277i \(-0.438307\pi\)
0.192602 + 0.981277i \(0.438307\pi\)
\(18\) 0 0
\(19\) 88.3346 1.06660 0.533299 0.845927i \(-0.320952\pi\)
0.533299 + 0.845927i \(0.320952\pi\)
\(20\) 55.4256 0.619677
\(21\) 0 0
\(22\) −78.0000 −0.755893
\(23\) 57.0000 0.516753 0.258377 0.966044i \(-0.416812\pi\)
0.258377 + 0.966044i \(0.416812\pi\)
\(24\) 0 0
\(25\) 67.0000 0.536000
\(26\) 0 0
\(27\) 0 0
\(28\) 90.0666 0.607893
\(29\) 69.0000 0.441827 0.220913 0.975293i \(-0.429096\pi\)
0.220913 + 0.975293i \(0.429096\pi\)
\(30\) 0 0
\(31\) 72.7461 0.421471 0.210735 0.977543i \(-0.432414\pi\)
0.210735 + 0.977543i \(0.432414\pi\)
\(32\) −166.277 −0.918559
\(33\) 0 0
\(34\) 93.5307 0.471776
\(35\) 312.000 1.50679
\(36\) 0 0
\(37\) 39.8372 0.177005 0.0885026 0.996076i \(-0.471792\pi\)
0.0885026 + 0.996076i \(0.471792\pi\)
\(38\) 306.000 1.30631
\(39\) 0 0
\(40\) −192.000 −0.758947
\(41\) 393.176 1.49765 0.748826 0.662767i \(-0.230618\pi\)
0.748826 + 0.662767i \(0.230618\pi\)
\(42\) 0 0
\(43\) 85.0000 0.301451 0.150725 0.988576i \(-0.451839\pi\)
0.150725 + 0.988576i \(0.451839\pi\)
\(44\) −90.0666 −0.308592
\(45\) 0 0
\(46\) 197.454 0.632891
\(47\) 342.946 1.06434 0.532168 0.846639i \(-0.321377\pi\)
0.532168 + 0.846639i \(0.321377\pi\)
\(48\) 0 0
\(49\) 164.000 0.478134
\(50\) 232.095 0.656463
\(51\) 0 0
\(52\) 0 0
\(53\) −426.000 −1.10407 −0.552034 0.833822i \(-0.686148\pi\)
−0.552034 + 0.833822i \(0.686148\pi\)
\(54\) 0 0
\(55\) −312.000 −0.764910
\(56\) −312.000 −0.744513
\(57\) 0 0
\(58\) 239.023 0.541125
\(59\) 19.0526 0.0420412 0.0210206 0.999779i \(-0.493308\pi\)
0.0210206 + 0.999779i \(0.493308\pi\)
\(60\) 0 0
\(61\) −17.0000 −0.0356824 −0.0178412 0.999841i \(-0.505679\pi\)
−0.0178412 + 0.999841i \(0.505679\pi\)
\(62\) 252.000 0.516194
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 164.545 0.300035 0.150018 0.988683i \(-0.452067\pi\)
0.150018 + 0.988683i \(0.452067\pi\)
\(68\) 108.000 0.192602
\(69\) 0 0
\(70\) 1080.80 1.84543
\(71\) 583.701 0.975670 0.487835 0.872936i \(-0.337787\pi\)
0.487835 + 0.872936i \(0.337787\pi\)
\(72\) 0 0
\(73\) 1004.59 1.61066 0.805331 0.592826i \(-0.201988\pi\)
0.805331 + 0.592826i \(0.201988\pi\)
\(74\) 138.000 0.216786
\(75\) 0 0
\(76\) 353.338 0.533299
\(77\) −507.000 −0.750364
\(78\) 0 0
\(79\) −1244.00 −1.77166 −0.885829 0.464012i \(-0.846409\pi\)
−0.885829 + 0.464012i \(0.846409\pi\)
\(80\) −1108.51 −1.54919
\(81\) 0 0
\(82\) 1362.00 1.83424
\(83\) −426.084 −0.563480 −0.281740 0.959491i \(-0.590912\pi\)
−0.281740 + 0.959491i \(0.590912\pi\)
\(84\) 0 0
\(85\) 374.123 0.477404
\(86\) 294.449 0.369200
\(87\) 0 0
\(88\) 312.000 0.377947
\(89\) 306.573 0.365131 0.182566 0.983194i \(-0.441560\pi\)
0.182566 + 0.983194i \(0.441560\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 228.000 0.258377
\(93\) 0 0
\(94\) 1188.00 1.30354
\(95\) 1224.00 1.32189
\(96\) 0 0
\(97\) −1234.95 −1.29268 −0.646342 0.763048i \(-0.723702\pi\)
−0.646342 + 0.763048i \(0.723702\pi\)
\(98\) 568.113 0.585592
\(99\) 0 0
\(100\) 268.000 0.268000
\(101\) 1959.00 1.92998 0.964989 0.262290i \(-0.0844778\pi\)
0.964989 + 0.262290i \(0.0844778\pi\)
\(102\) 0 0
\(103\) −1856.00 −1.77551 −0.887753 0.460320i \(-0.847735\pi\)
−0.887753 + 0.460320i \(0.847735\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1475.71 −1.35220
\(107\) 255.000 0.230390 0.115195 0.993343i \(-0.463251\pi\)
0.115195 + 0.993343i \(0.463251\pi\)
\(108\) 0 0
\(109\) 609.682 0.535752 0.267876 0.963453i \(-0.413678\pi\)
0.267876 + 0.963453i \(0.413678\pi\)
\(110\) −1080.80 −0.936820
\(111\) 0 0
\(112\) −1801.33 −1.51973
\(113\) −411.000 −0.342156 −0.171078 0.985257i \(-0.554725\pi\)
−0.171078 + 0.985257i \(0.554725\pi\)
\(114\) 0 0
\(115\) 789.815 0.640440
\(116\) 276.000 0.220913
\(117\) 0 0
\(118\) 66.0000 0.0514898
\(119\) 607.950 0.468325
\(120\) 0 0
\(121\) −824.000 −0.619083
\(122\) −58.8897 −0.0437018
\(123\) 0 0
\(124\) 290.985 0.210735
\(125\) −803.672 −0.575061
\(126\) 0 0
\(127\) 2243.00 1.56720 0.783599 0.621267i \(-0.213382\pi\)
0.783599 + 0.621267i \(0.213382\pi\)
\(128\) 1551.92 1.07165
\(129\) 0 0
\(130\) 0 0
\(131\) 372.000 0.248105 0.124053 0.992276i \(-0.460411\pi\)
0.124053 + 0.992276i \(0.460411\pi\)
\(132\) 0 0
\(133\) 1989.00 1.29675
\(134\) 570.000 0.367466
\(135\) 0 0
\(136\) −374.123 −0.235888
\(137\) 1189.92 0.742056 0.371028 0.928622i \(-0.379005\pi\)
0.371028 + 0.928622i \(0.379005\pi\)
\(138\) 0 0
\(139\) −2545.00 −1.55298 −0.776490 0.630130i \(-0.783002\pi\)
−0.776490 + 0.630130i \(0.783002\pi\)
\(140\) 1248.00 0.753395
\(141\) 0 0
\(142\) 2022.00 1.19495
\(143\) 0 0
\(144\) 0 0
\(145\) 956.092 0.547580
\(146\) 3480.00 1.97265
\(147\) 0 0
\(148\) 159.349 0.0885026
\(149\) 1304.23 0.717094 0.358547 0.933512i \(-0.383272\pi\)
0.358547 + 0.933512i \(0.383272\pi\)
\(150\) 0 0
\(151\) −86.6025 −0.0466729 −0.0233365 0.999728i \(-0.507429\pi\)
−0.0233365 + 0.999728i \(0.507429\pi\)
\(152\) −1224.00 −0.653155
\(153\) 0 0
\(154\) −1756.30 −0.919004
\(155\) 1008.00 0.522352
\(156\) 0 0
\(157\) −1534.00 −0.779787 −0.389893 0.920860i \(-0.627488\pi\)
−0.389893 + 0.920860i \(0.627488\pi\)
\(158\) −4309.34 −2.16983
\(159\) 0 0
\(160\) −2304.00 −1.13842
\(161\) 1283.45 0.628261
\(162\) 0 0
\(163\) −1633.32 −0.784858 −0.392429 0.919782i \(-0.628365\pi\)
−0.392429 + 0.919782i \(0.628365\pi\)
\(164\) 1572.70 0.748826
\(165\) 0 0
\(166\) −1476.00 −0.690119
\(167\) 1626.40 0.753618 0.376809 0.926291i \(-0.377021\pi\)
0.376809 + 0.926291i \(0.377021\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1296.00 0.584698
\(171\) 0 0
\(172\) 340.000 0.150725
\(173\) −873.000 −0.383659 −0.191829 0.981428i \(-0.561442\pi\)
−0.191829 + 0.981428i \(0.561442\pi\)
\(174\) 0 0
\(175\) 1508.62 0.651661
\(176\) 1801.33 0.771481
\(177\) 0 0
\(178\) 1062.00 0.447193
\(179\) −1287.00 −0.537402 −0.268701 0.963224i \(-0.586594\pi\)
−0.268701 + 0.963224i \(0.586594\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.000821319 0 −0.000410660 1.00000i \(-0.500131\pi\)
−0.000410660 1.00000i \(0.500131\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −789.815 −0.316445
\(185\) 552.000 0.219372
\(186\) 0 0
\(187\) −607.950 −0.237742
\(188\) 1371.78 0.532168
\(189\) 0 0
\(190\) 4240.06 1.61898
\(191\) 2841.00 1.07627 0.538135 0.842859i \(-0.319129\pi\)
0.538135 + 0.842859i \(0.319129\pi\)
\(192\) 0 0
\(193\) 4245.26 1.58332 0.791659 0.610964i \(-0.209218\pi\)
0.791659 + 0.610964i \(0.209218\pi\)
\(194\) −4278.00 −1.58321
\(195\) 0 0
\(196\) 656.000 0.239067
\(197\) −2752.23 −0.995371 −0.497686 0.867357i \(-0.665817\pi\)
−0.497686 + 0.867357i \(0.665817\pi\)
\(198\) 0 0
\(199\) 1685.00 0.600234 0.300117 0.953902i \(-0.402974\pi\)
0.300117 + 0.953902i \(0.402974\pi\)
\(200\) −928.379 −0.328232
\(201\) 0 0
\(202\) 6786.18 2.36373
\(203\) 1553.65 0.537167
\(204\) 0 0
\(205\) 5448.00 1.85612
\(206\) −6429.37 −2.17454
\(207\) 0 0
\(208\) 0 0
\(209\) −1989.00 −0.658287
\(210\) 0 0
\(211\) 1681.00 0.548459 0.274229 0.961664i \(-0.411577\pi\)
0.274229 + 0.961664i \(0.411577\pi\)
\(212\) −1704.00 −0.552034
\(213\) 0 0
\(214\) 883.346 0.282170
\(215\) 1177.79 0.373604
\(216\) 0 0
\(217\) 1638.00 0.512418
\(218\) 2112.00 0.656159
\(219\) 0 0
\(220\) −1248.00 −0.382455
\(221\) 0 0
\(222\) 0 0
\(223\) −4096.30 −1.23008 −0.615042 0.788495i \(-0.710861\pi\)
−0.615042 + 0.788495i \(0.710861\pi\)
\(224\) −3744.00 −1.11677
\(225\) 0 0
\(226\) −1423.75 −0.419054
\(227\) 438.209 0.128128 0.0640638 0.997946i \(-0.479594\pi\)
0.0640638 + 0.997946i \(0.479594\pi\)
\(228\) 0 0
\(229\) 180.133 0.0519805 0.0259903 0.999662i \(-0.491726\pi\)
0.0259903 + 0.999662i \(0.491726\pi\)
\(230\) 2736.00 0.784376
\(231\) 0 0
\(232\) −956.092 −0.270563
\(233\) −5778.00 −1.62459 −0.812295 0.583247i \(-0.801782\pi\)
−0.812295 + 0.583247i \(0.801782\pi\)
\(234\) 0 0
\(235\) 4752.00 1.31909
\(236\) 76.2102 0.0210206
\(237\) 0 0
\(238\) 2106.00 0.573579
\(239\) −1860.22 −0.503464 −0.251732 0.967797i \(-0.581000\pi\)
−0.251732 + 0.967797i \(0.581000\pi\)
\(240\) 0 0
\(241\) 2059.41 0.550449 0.275224 0.961380i \(-0.411248\pi\)
0.275224 + 0.961380i \(0.411248\pi\)
\(242\) −2854.42 −0.758219
\(243\) 0 0
\(244\) −68.0000 −0.0178412
\(245\) 2272.45 0.592578
\(246\) 0 0
\(247\) 0 0
\(248\) −1008.00 −0.258097
\(249\) 0 0
\(250\) −2784.00 −0.704302
\(251\) 4491.00 1.12936 0.564680 0.825310i \(-0.309000\pi\)
0.564680 + 0.825310i \(0.309000\pi\)
\(252\) 0 0
\(253\) −1283.45 −0.318932
\(254\) 7769.98 1.91942
\(255\) 0 0
\(256\) 4864.00 1.18750
\(257\) 5451.00 1.32305 0.661525 0.749923i \(-0.269909\pi\)
0.661525 + 0.749923i \(0.269909\pi\)
\(258\) 0 0
\(259\) 897.000 0.215200
\(260\) 0 0
\(261\) 0 0
\(262\) 1288.65 0.303866
\(263\) 783.000 0.183581 0.0917906 0.995778i \(-0.470741\pi\)
0.0917906 + 0.995778i \(0.470741\pi\)
\(264\) 0 0
\(265\) −5902.83 −1.36833
\(266\) 6890.10 1.58819
\(267\) 0 0
\(268\) 658.179 0.150018
\(269\) 5085.00 1.15256 0.576279 0.817253i \(-0.304504\pi\)
0.576279 + 0.817253i \(0.304504\pi\)
\(270\) 0 0
\(271\) 1325.02 0.297008 0.148504 0.988912i \(-0.452554\pi\)
0.148504 + 0.988912i \(0.452554\pi\)
\(272\) −2160.00 −0.481505
\(273\) 0 0
\(274\) 4122.00 0.908829
\(275\) −1508.62 −0.330811
\(276\) 0 0
\(277\) 3421.00 0.742050 0.371025 0.928623i \(-0.379006\pi\)
0.371025 + 0.928623i \(0.379006\pi\)
\(278\) −8816.14 −1.90200
\(279\) 0 0
\(280\) −4323.20 −0.922716
\(281\) −810.600 −0.172087 −0.0860433 0.996291i \(-0.527422\pi\)
−0.0860433 + 0.996291i \(0.527422\pi\)
\(282\) 0 0
\(283\) −7177.00 −1.50752 −0.753760 0.657149i \(-0.771762\pi\)
−0.753760 + 0.657149i \(0.771762\pi\)
\(284\) 2334.80 0.487835
\(285\) 0 0
\(286\) 0 0
\(287\) 8853.00 1.82082
\(288\) 0 0
\(289\) −4184.00 −0.851618
\(290\) 3312.00 0.670646
\(291\) 0 0
\(292\) 4018.36 0.805331
\(293\) −9313.24 −1.85695 −0.928473 0.371400i \(-0.878878\pi\)
−0.928473 + 0.371400i \(0.878878\pi\)
\(294\) 0 0
\(295\) 264.000 0.0521040
\(296\) −552.000 −0.108393
\(297\) 0 0
\(298\) 4518.00 0.878257
\(299\) 0 0
\(300\) 0 0
\(301\) 1913.92 0.366499
\(302\) −300.000 −0.0571625
\(303\) 0 0
\(304\) −7066.77 −1.33325
\(305\) −235.559 −0.0442232
\(306\) 0 0
\(307\) −4777.00 −0.888070 −0.444035 0.896009i \(-0.646453\pi\)
−0.444035 + 0.896009i \(0.646453\pi\)
\(308\) −2028.00 −0.375182
\(309\) 0 0
\(310\) 3491.81 0.639748
\(311\) 6192.00 1.12899 0.564495 0.825436i \(-0.309071\pi\)
0.564495 + 0.825436i \(0.309071\pi\)
\(312\) 0 0
\(313\) −770.000 −0.139051 −0.0695255 0.997580i \(-0.522149\pi\)
−0.0695255 + 0.997580i \(0.522149\pi\)
\(314\) −5313.93 −0.955040
\(315\) 0 0
\(316\) −4976.00 −0.885829
\(317\) 8057.50 1.42762 0.713808 0.700341i \(-0.246969\pi\)
0.713808 + 0.700341i \(0.246969\pi\)
\(318\) 0 0
\(319\) −1553.65 −0.272689
\(320\) 886.810 0.154919
\(321\) 0 0
\(322\) 4446.00 0.769459
\(323\) 2385.03 0.410857
\(324\) 0 0
\(325\) 0 0
\(326\) −5658.00 −0.961250
\(327\) 0 0
\(328\) −5448.00 −0.917120
\(329\) 7722.00 1.29400
\(330\) 0 0
\(331\) 5277.56 0.876377 0.438189 0.898883i \(-0.355620\pi\)
0.438189 + 0.898883i \(0.355620\pi\)
\(332\) −1704.34 −0.281740
\(333\) 0 0
\(334\) 5634.00 0.922990
\(335\) 2280.00 0.371850
\(336\) 0 0
\(337\) −8278.00 −1.33808 −0.669038 0.743228i \(-0.733294\pi\)
−0.669038 + 0.743228i \(0.733294\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1496.49 0.238702
\(341\) −1638.00 −0.260125
\(342\) 0 0
\(343\) −4030.48 −0.634477
\(344\) −1177.79 −0.184600
\(345\) 0 0
\(346\) −3024.16 −0.469884
\(347\) −6867.00 −1.06236 −0.531181 0.847258i \(-0.678252\pi\)
−0.531181 + 0.847258i \(0.678252\pi\)
\(348\) 0 0
\(349\) −12153.8 −1.86412 −0.932060 0.362303i \(-0.881990\pi\)
−0.932060 + 0.362303i \(0.881990\pi\)
\(350\) 5226.00 0.798118
\(351\) 0 0
\(352\) 3744.00 0.566920
\(353\) −5807.57 −0.875653 −0.437827 0.899059i \(-0.644252\pi\)
−0.437827 + 0.899059i \(0.644252\pi\)
\(354\) 0 0
\(355\) 8088.00 1.20920
\(356\) 1226.29 0.182566
\(357\) 0 0
\(358\) −4458.30 −0.658180
\(359\) −1340.61 −0.197088 −0.0985439 0.995133i \(-0.531419\pi\)
−0.0985439 + 0.995133i \(0.531419\pi\)
\(360\) 0 0
\(361\) 944.000 0.137629
\(362\) −6.92820 −0.00100591
\(363\) 0 0
\(364\) 0 0
\(365\) 13920.0 1.99618
\(366\) 0 0
\(367\) 3665.00 0.521285 0.260642 0.965435i \(-0.416066\pi\)
0.260642 + 0.965435i \(0.416066\pi\)
\(368\) −4560.00 −0.645941
\(369\) 0 0
\(370\) 1912.18 0.268675
\(371\) −9592.10 −1.34231
\(372\) 0 0
\(373\) 5371.00 0.745576 0.372788 0.927917i \(-0.378402\pi\)
0.372788 + 0.927917i \(0.378402\pi\)
\(374\) −2106.00 −0.291173
\(375\) 0 0
\(376\) −4752.00 −0.651770
\(377\) 0 0
\(378\) 0 0
\(379\) −11509.5 −1.55990 −0.779950 0.625842i \(-0.784756\pi\)
−0.779950 + 0.625842i \(0.784756\pi\)
\(380\) 4896.00 0.660946
\(381\) 0 0
\(382\) 9841.51 1.31816
\(383\) 2419.67 0.322819 0.161409 0.986888i \(-0.448396\pi\)
0.161409 + 0.986888i \(0.448396\pi\)
\(384\) 0 0
\(385\) −7025.20 −0.929967
\(386\) 14706.0 1.93916
\(387\) 0 0
\(388\) −4939.81 −0.646342
\(389\) −9858.00 −1.28489 −0.642443 0.766334i \(-0.722079\pi\)
−0.642443 + 0.766334i \(0.722079\pi\)
\(390\) 0 0
\(391\) 1539.00 0.199055
\(392\) −2272.45 −0.292796
\(393\) 0 0
\(394\) −9534.00 −1.21908
\(395\) −17237.4 −2.19571
\(396\) 0 0
\(397\) −8720.88 −1.10249 −0.551245 0.834344i \(-0.685847\pi\)
−0.551245 + 0.834344i \(0.685847\pi\)
\(398\) 5837.01 0.735133
\(399\) 0 0
\(400\) −5360.00 −0.670000
\(401\) −7584.65 −0.944537 −0.472269 0.881455i \(-0.656565\pi\)
−0.472269 + 0.881455i \(0.656565\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 7836.00 0.964989
\(405\) 0 0
\(406\) 5382.00 0.657892
\(407\) −897.000 −0.109245
\(408\) 0 0
\(409\) −4304.15 −0.520358 −0.260179 0.965560i \(-0.583782\pi\)
−0.260179 + 0.965560i \(0.583782\pi\)
\(410\) 18872.4 2.27327
\(411\) 0 0
\(412\) −7424.00 −0.887753
\(413\) 429.000 0.0511131
\(414\) 0 0
\(415\) −5904.00 −0.698352
\(416\) 0 0
\(417\) 0 0
\(418\) −6890.10 −0.806234
\(419\) −5397.00 −0.629262 −0.314631 0.949214i \(-0.601881\pi\)
−0.314631 + 0.949214i \(0.601881\pi\)
\(420\) 0 0
\(421\) −7260.76 −0.840541 −0.420270 0.907399i \(-0.638065\pi\)
−0.420270 + 0.907399i \(0.638065\pi\)
\(422\) 5823.15 0.671722
\(423\) 0 0
\(424\) 5902.83 0.676101
\(425\) 1809.00 0.206469
\(426\) 0 0
\(427\) −382.783 −0.0433822
\(428\) 1020.00 0.115195
\(429\) 0 0
\(430\) 4080.00 0.457570
\(431\) 486.706 0.0543940 0.0271970 0.999630i \(-0.491342\pi\)
0.0271970 + 0.999630i \(0.491342\pi\)
\(432\) 0 0
\(433\) −12139.0 −1.34726 −0.673629 0.739069i \(-0.735266\pi\)
−0.673629 + 0.739069i \(0.735266\pi\)
\(434\) 5674.20 0.627581
\(435\) 0 0
\(436\) 2438.73 0.267876
\(437\) 5035.07 0.551167
\(438\) 0 0
\(439\) −461.000 −0.0501192 −0.0250596 0.999686i \(-0.507978\pi\)
−0.0250596 + 0.999686i \(0.507978\pi\)
\(440\) 4323.20 0.468410
\(441\) 0 0
\(442\) 0 0
\(443\) −12156.0 −1.30372 −0.651861 0.758338i \(-0.726012\pi\)
−0.651861 + 0.758338i \(0.726012\pi\)
\(444\) 0 0
\(445\) 4248.00 0.452527
\(446\) −14190.0 −1.50654
\(447\) 0 0
\(448\) 1441.07 0.151973
\(449\) 296.181 0.0311306 0.0155653 0.999879i \(-0.495045\pi\)
0.0155653 + 0.999879i \(0.495045\pi\)
\(450\) 0 0
\(451\) −8853.00 −0.924327
\(452\) −1644.00 −0.171078
\(453\) 0 0
\(454\) 1518.00 0.156924
\(455\) 0 0
\(456\) 0 0
\(457\) 611.414 0.0625837 0.0312918 0.999510i \(-0.490038\pi\)
0.0312918 + 0.999510i \(0.490038\pi\)
\(458\) 624.000 0.0636629
\(459\) 0 0
\(460\) 3159.26 0.320220
\(461\) −13127.2 −1.32624 −0.663119 0.748514i \(-0.730767\pi\)
−0.663119 + 0.748514i \(0.730767\pi\)
\(462\) 0 0
\(463\) −834.848 −0.0837985 −0.0418992 0.999122i \(-0.513341\pi\)
−0.0418992 + 0.999122i \(0.513341\pi\)
\(464\) −5520.00 −0.552284
\(465\) 0 0
\(466\) −20015.6 −1.98971
\(467\) 14496.0 1.43639 0.718196 0.695841i \(-0.244968\pi\)
0.718196 + 0.695841i \(0.244968\pi\)
\(468\) 0 0
\(469\) 3705.00 0.364778
\(470\) 16461.4 1.61555
\(471\) 0 0
\(472\) −264.000 −0.0257449
\(473\) −1913.92 −0.186051
\(474\) 0 0
\(475\) 5918.42 0.571696
\(476\) 2431.80 0.234162
\(477\) 0 0
\(478\) −6444.00 −0.616614
\(479\) −8897.54 −0.848725 −0.424362 0.905492i \(-0.639502\pi\)
−0.424362 + 0.905492i \(0.639502\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 7134.00 0.674159
\(483\) 0 0
\(484\) −3296.00 −0.309542
\(485\) −17112.0 −1.60209
\(486\) 0 0
\(487\) −4754.48 −0.442394 −0.221197 0.975229i \(-0.570996\pi\)
−0.221197 + 0.975229i \(0.570996\pi\)
\(488\) 235.559 0.0218509
\(489\) 0 0
\(490\) 7872.00 0.725757
\(491\) −1635.00 −0.150278 −0.0751390 0.997173i \(-0.523940\pi\)
−0.0751390 + 0.997173i \(0.523940\pi\)
\(492\) 0 0
\(493\) 1863.00 0.170193
\(494\) 0 0
\(495\) 0 0
\(496\) −5819.69 −0.526838
\(497\) 13143.0 1.18621
\(498\) 0 0
\(499\) 14434.9 1.29498 0.647490 0.762074i \(-0.275819\pi\)
0.647490 + 0.762074i \(0.275819\pi\)
\(500\) −3214.69 −0.287530
\(501\) 0 0
\(502\) 15557.3 1.38318
\(503\) −12687.0 −1.12462 −0.562312 0.826925i \(-0.690088\pi\)
−0.562312 + 0.826925i \(0.690088\pi\)
\(504\) 0 0
\(505\) 27144.7 2.39193
\(506\) −4446.00 −0.390610
\(507\) 0 0
\(508\) 8972.00 0.783599
\(509\) 5748.68 0.500600 0.250300 0.968168i \(-0.419471\pi\)
0.250300 + 0.968168i \(0.419471\pi\)
\(510\) 0 0
\(511\) 22620.0 1.95822
\(512\) 4434.05 0.382733
\(513\) 0 0
\(514\) 18882.8 1.62040
\(515\) −25717.5 −2.20048
\(516\) 0 0
\(517\) −7722.00 −0.656892
\(518\) 3107.30 0.263565
\(519\) 0 0
\(520\) 0 0
\(521\) −6054.00 −0.509080 −0.254540 0.967062i \(-0.581924\pi\)
−0.254540 + 0.967062i \(0.581924\pi\)
\(522\) 0 0
\(523\) −14803.0 −1.23765 −0.618824 0.785530i \(-0.712391\pi\)
−0.618824 + 0.785530i \(0.712391\pi\)
\(524\) 1488.00 0.124053
\(525\) 0 0
\(526\) 2712.39 0.224840
\(527\) 1964.15 0.162352
\(528\) 0 0
\(529\) −8918.00 −0.732966
\(530\) −20448.0 −1.67586
\(531\) 0 0
\(532\) 7956.00 0.648377
\(533\) 0 0
\(534\) 0 0
\(535\) 3533.38 0.285536
\(536\) −2280.00 −0.183733
\(537\) 0 0
\(538\) 17615.0 1.41159
\(539\) −3692.73 −0.295097
\(540\) 0 0
\(541\) −21470.5 −1.70626 −0.853132 0.521695i \(-0.825300\pi\)
−0.853132 + 0.521695i \(0.825300\pi\)
\(542\) 4590.00 0.363759
\(543\) 0 0
\(544\) −4489.48 −0.353832
\(545\) 8448.00 0.663986
\(546\) 0 0
\(547\) −13516.0 −1.05649 −0.528247 0.849091i \(-0.677151\pi\)
−0.528247 + 0.849091i \(0.677151\pi\)
\(548\) 4759.68 0.371028
\(549\) 0 0
\(550\) −5226.00 −0.405159
\(551\) 6095.09 0.471251
\(552\) 0 0
\(553\) −28010.7 −2.15396
\(554\) 11850.7 0.908822
\(555\) 0 0
\(556\) −10180.0 −0.776490
\(557\) 2890.79 0.219905 0.109952 0.993937i \(-0.464930\pi\)
0.109952 + 0.993937i \(0.464930\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −24960.0 −1.88349
\(561\) 0 0
\(562\) −2808.00 −0.210762
\(563\) 11583.0 0.867079 0.433539 0.901135i \(-0.357265\pi\)
0.433539 + 0.901135i \(0.357265\pi\)
\(564\) 0 0
\(565\) −5694.98 −0.424053
\(566\) −24861.9 −1.84633
\(567\) 0 0
\(568\) −8088.00 −0.597473
\(569\) 12879.0 0.948885 0.474443 0.880286i \(-0.342650\pi\)
0.474443 + 0.880286i \(0.342650\pi\)
\(570\) 0 0
\(571\) 11636.0 0.852805 0.426402 0.904534i \(-0.359781\pi\)
0.426402 + 0.904534i \(0.359781\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 30667.7 2.23004
\(575\) 3819.00 0.276980
\(576\) 0 0
\(577\) −12311.4 −0.888269 −0.444134 0.895960i \(-0.646489\pi\)
−0.444134 + 0.895960i \(0.646489\pi\)
\(578\) −14493.8 −1.04301
\(579\) 0 0
\(580\) 3824.37 0.273790
\(581\) −9594.00 −0.685071
\(582\) 0 0
\(583\) 9592.10 0.681414
\(584\) −13920.0 −0.986325
\(585\) 0 0
\(586\) −32262.0 −2.27428
\(587\) 15645.6 1.10011 0.550054 0.835129i \(-0.314607\pi\)
0.550054 + 0.835129i \(0.314607\pi\)
\(588\) 0 0
\(589\) 6426.00 0.449539
\(590\) 914.523 0.0638141
\(591\) 0 0
\(592\) −3186.97 −0.221256
\(593\) −25821.4 −1.78813 −0.894063 0.447942i \(-0.852157\pi\)
−0.894063 + 0.447942i \(0.852157\pi\)
\(594\) 0 0
\(595\) 8424.00 0.580421
\(596\) 5216.94 0.358547
\(597\) 0 0
\(598\) 0 0
\(599\) −1668.00 −0.113777 −0.0568887 0.998381i \(-0.518118\pi\)
−0.0568887 + 0.998381i \(0.518118\pi\)
\(600\) 0 0
\(601\) 13699.0 0.929773 0.464887 0.885370i \(-0.346095\pi\)
0.464887 + 0.885370i \(0.346095\pi\)
\(602\) 6630.00 0.448868
\(603\) 0 0
\(604\) −346.410 −0.0233365
\(605\) −11417.7 −0.767264
\(606\) 0 0
\(607\) −23173.0 −1.54953 −0.774764 0.632251i \(-0.782131\pi\)
−0.774764 + 0.632251i \(0.782131\pi\)
\(608\) −14688.0 −0.979732
\(609\) 0 0
\(610\) −816.000 −0.0541621
\(611\) 0 0
\(612\) 0 0
\(613\) 16615.6 1.09477 0.547387 0.836880i \(-0.315623\pi\)
0.547387 + 0.836880i \(0.315623\pi\)
\(614\) −16548.0 −1.08766
\(615\) 0 0
\(616\) 7025.20 0.459502
\(617\) 28393.5 1.85264 0.926321 0.376736i \(-0.122954\pi\)
0.926321 + 0.376736i \(0.122954\pi\)
\(618\) 0 0
\(619\) −6245.78 −0.405556 −0.202778 0.979225i \(-0.564997\pi\)
−0.202778 + 0.979225i \(0.564997\pi\)
\(620\) 4032.00 0.261176
\(621\) 0 0
\(622\) 21449.7 1.38273
\(623\) 6903.00 0.443921
\(624\) 0 0
\(625\) −19511.0 −1.24870
\(626\) −2667.36 −0.170302
\(627\) 0 0
\(628\) −6136.00 −0.389893
\(629\) 1075.60 0.0681830
\(630\) 0 0
\(631\) 22379.8 1.41193 0.705964 0.708247i \(-0.250514\pi\)
0.705964 + 0.708247i \(0.250514\pi\)
\(632\) 17237.4 1.08491
\(633\) 0 0
\(634\) 27912.0 1.74847
\(635\) 31079.9 1.94231
\(636\) 0 0
\(637\) 0 0
\(638\) −5382.00 −0.333974
\(639\) 0 0
\(640\) 21504.0 1.32816
\(641\) 19827.0 1.22172 0.610858 0.791740i \(-0.290825\pi\)
0.610858 + 0.791740i \(0.290825\pi\)
\(642\) 0 0
\(643\) 8450.68 0.518293 0.259146 0.965838i \(-0.416559\pi\)
0.259146 + 0.965838i \(0.416559\pi\)
\(644\) 5133.80 0.314130
\(645\) 0 0
\(646\) 8262.00 0.503195
\(647\) 2949.00 0.179192 0.0895959 0.995978i \(-0.471442\pi\)
0.0895959 + 0.995978i \(0.471442\pi\)
\(648\) 0 0
\(649\) −429.000 −0.0259472
\(650\) 0 0
\(651\) 0 0
\(652\) −6533.30 −0.392429
\(653\) −12039.0 −0.721474 −0.360737 0.932668i \(-0.617475\pi\)
−0.360737 + 0.932668i \(0.617475\pi\)
\(654\) 0 0
\(655\) 5154.58 0.307490
\(656\) −31454.0 −1.87206
\(657\) 0 0
\(658\) 26749.8 1.58483
\(659\) −3363.00 −0.198792 −0.0993960 0.995048i \(-0.531691\pi\)
−0.0993960 + 0.995048i \(0.531691\pi\)
\(660\) 0 0
\(661\) −10158.5 −0.597759 −0.298880 0.954291i \(-0.596613\pi\)
−0.298880 + 0.954291i \(0.596613\pi\)
\(662\) 18282.0 1.07334
\(663\) 0 0
\(664\) 5904.00 0.345060
\(665\) 27560.4 1.60714
\(666\) 0 0
\(667\) 3933.00 0.228315
\(668\) 6505.58 0.376809
\(669\) 0 0
\(670\) 7898.15 0.455421
\(671\) 382.783 0.0220226
\(672\) 0 0
\(673\) 18169.0 1.04066 0.520329 0.853966i \(-0.325809\pi\)
0.520329 + 0.853966i \(0.325809\pi\)
\(674\) −28675.8 −1.63880
\(675\) 0 0
\(676\) 0 0
\(677\) −9042.00 −0.513312 −0.256656 0.966503i \(-0.582621\pi\)
−0.256656 + 0.966503i \(0.582621\pi\)
\(678\) 0 0
\(679\) −27807.0 −1.57163
\(680\) −5184.00 −0.292349
\(681\) 0 0
\(682\) −5674.20 −0.318587
\(683\) 12462.1 0.698169 0.349084 0.937091i \(-0.386493\pi\)
0.349084 + 0.937091i \(0.386493\pi\)
\(684\) 0 0
\(685\) 16488.0 0.919670
\(686\) −13962.0 −0.777072
\(687\) 0 0
\(688\) −6800.00 −0.376813
\(689\) 0 0
\(690\) 0 0
\(691\) −4318.00 −0.237720 −0.118860 0.992911i \(-0.537924\pi\)
−0.118860 + 0.992911i \(0.537924\pi\)
\(692\) −3492.00 −0.191829
\(693\) 0 0
\(694\) −23788.0 −1.30112
\(695\) −35264.6 −1.92469
\(696\) 0 0
\(697\) 10615.7 0.576901
\(698\) −42102.0 −2.28307
\(699\) 0 0
\(700\) 6034.47 0.325830
\(701\) −18270.0 −0.984377 −0.492189 0.870489i \(-0.663803\pi\)
−0.492189 + 0.870489i \(0.663803\pi\)
\(702\) 0 0
\(703\) 3519.00 0.188793
\(704\) −1441.07 −0.0771481
\(705\) 0 0
\(706\) −20118.0 −1.07245
\(707\) 44110.1 2.34644
\(708\) 0 0
\(709\) 1629.86 0.0863338 0.0431669 0.999068i \(-0.486255\pi\)
0.0431669 + 0.999068i \(0.486255\pi\)
\(710\) 28017.7 1.48096
\(711\) 0 0
\(712\) −4248.00 −0.223596
\(713\) 4146.53 0.217796
\(714\) 0 0
\(715\) 0 0
\(716\) −5148.00 −0.268701
\(717\) 0 0
\(718\) −4644.00 −0.241382
\(719\) 9831.00 0.509923 0.254961 0.966951i \(-0.417937\pi\)
0.254961 + 0.966951i \(0.417937\pi\)
\(720\) 0 0
\(721\) −41790.9 −2.15863
\(722\) 3270.11 0.168561
\(723\) 0 0
\(724\) −8.00000 −0.000410660 0
\(725\) 4623.00 0.236819
\(726\) 0 0
\(727\) 15464.0 0.788897 0.394448 0.918918i \(-0.370936\pi\)
0.394448 + 0.918918i \(0.370936\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 48220.3 2.44481
\(731\) 2295.00 0.116120
\(732\) 0 0
\(733\) 12616.3 0.635733 0.317866 0.948136i \(-0.397034\pi\)
0.317866 + 0.948136i \(0.397034\pi\)
\(734\) 12695.9 0.638441
\(735\) 0 0
\(736\) −9477.78 −0.474668
\(737\) −3705.00 −0.185177
\(738\) 0 0
\(739\) 16283.0 0.810528 0.405264 0.914200i \(-0.367180\pi\)
0.405264 + 0.914200i \(0.367180\pi\)
\(740\) 2208.00 0.109686
\(741\) 0 0
\(742\) −33228.0 −1.64399
\(743\) −10806.3 −0.533571 −0.266786 0.963756i \(-0.585962\pi\)
−0.266786 + 0.963756i \(0.585962\pi\)
\(744\) 0 0
\(745\) 18072.0 0.888734
\(746\) 18605.7 0.913140
\(747\) 0 0
\(748\) −2431.80 −0.118871
\(749\) 5741.75 0.280105
\(750\) 0 0
\(751\) 13615.0 0.661542 0.330771 0.943711i \(-0.392691\pi\)
0.330771 + 0.943711i \(0.392691\pi\)
\(752\) −27435.7 −1.33042
\(753\) 0 0
\(754\) 0 0
\(755\) −1200.00 −0.0578443
\(756\) 0 0
\(757\) 5551.00 0.266519 0.133259 0.991081i \(-0.457456\pi\)
0.133259 + 0.991081i \(0.457456\pi\)
\(758\) −39870.0 −1.91048
\(759\) 0 0
\(760\) −16960.2 −0.809490
\(761\) −10082.3 −0.480265 −0.240133 0.970740i \(-0.577191\pi\)
−0.240133 + 0.970740i \(0.577191\pi\)
\(762\) 0 0
\(763\) 13728.0 0.651359
\(764\) 11364.0 0.538135
\(765\) 0 0
\(766\) 8382.00 0.395371
\(767\) 0 0
\(768\) 0 0
\(769\) 29758.4 1.39547 0.697733 0.716357i \(-0.254192\pi\)
0.697733 + 0.716357i \(0.254192\pi\)
\(770\) −24336.0 −1.13897
\(771\) 0 0
\(772\) 16981.0 0.791659
\(773\) 27735.3 1.29052 0.645259 0.763964i \(-0.276749\pi\)
0.645259 + 0.763964i \(0.276749\pi\)
\(774\) 0 0
\(775\) 4873.99 0.225908
\(776\) 17112.0 0.791604
\(777\) 0 0
\(778\) −34149.1 −1.57366
\(779\) 34731.0 1.59739
\(780\) 0 0
\(781\) −13143.0 −0.602168
\(782\) 5331.25 0.243792
\(783\) 0 0
\(784\) −13120.0 −0.597668
\(785\) −21255.7 −0.966432
\(786\) 0 0
\(787\) −31549.3 −1.42899 −0.714493 0.699643i \(-0.753342\pi\)
−0.714493 + 0.699643i \(0.753342\pi\)
\(788\) −11008.9 −0.497686
\(789\) 0 0
\(790\) −59712.0 −2.68919
\(791\) −9254.35 −0.415988
\(792\) 0 0
\(793\) 0 0
\(794\) −30210.0 −1.35027
\(795\) 0 0
\(796\) 6740.00 0.300117
\(797\) 1455.00 0.0646659 0.0323330 0.999477i \(-0.489706\pi\)
0.0323330 + 0.999477i \(0.489706\pi\)
\(798\) 0 0
\(799\) 9259.54 0.409986
\(800\) −11140.6 −0.492347
\(801\) 0 0
\(802\) −26274.0 −1.15682
\(803\) −22620.0 −0.994075
\(804\) 0 0
\(805\) 17784.0 0.778638
\(806\) 0 0
\(807\) 0 0
\(808\) −27144.7 −1.18187
\(809\) −1659.00 −0.0720981 −0.0360490 0.999350i \(-0.511477\pi\)
−0.0360490 + 0.999350i \(0.511477\pi\)
\(810\) 0 0
\(811\) −4402.87 −0.190636 −0.0953180 0.995447i \(-0.530387\pi\)
−0.0953180 + 0.995447i \(0.530387\pi\)
\(812\) 6214.60 0.268583
\(813\) 0 0
\(814\) −3107.30 −0.133797
\(815\) −22632.0 −0.972717
\(816\) 0 0
\(817\) 7508.44 0.321526
\(818\) −14910.0 −0.637306
\(819\) 0 0
\(820\) 21792.0 0.928061
\(821\) 28701.8 1.22010 0.610049 0.792364i \(-0.291150\pi\)
0.610049 + 0.792364i \(0.291150\pi\)
\(822\) 0 0
\(823\) −15779.0 −0.668313 −0.334156 0.942518i \(-0.608451\pi\)
−0.334156 + 0.942518i \(0.608451\pi\)
\(824\) 25717.5 1.08727
\(825\) 0 0
\(826\) 1486.10 0.0626005
\(827\) 7354.29 0.309231 0.154615 0.987975i \(-0.450586\pi\)
0.154615 + 0.987975i \(0.450586\pi\)
\(828\) 0 0
\(829\) −17371.0 −0.727768 −0.363884 0.931444i \(-0.618550\pi\)
−0.363884 + 0.931444i \(0.618550\pi\)
\(830\) −20452.1 −0.855303
\(831\) 0 0
\(832\) 0 0
\(833\) 4428.00 0.184179
\(834\) 0 0
\(835\) 22536.0 0.934001
\(836\) −7956.00 −0.329144
\(837\) 0 0
\(838\) −18695.8 −0.770685
\(839\) −29474.3 −1.21283 −0.606416 0.795148i \(-0.707393\pi\)
−0.606416 + 0.795148i \(0.707393\pi\)
\(840\) 0 0
\(841\) −19628.0 −0.804789
\(842\) −25152.0 −1.02945
\(843\) 0 0
\(844\) 6724.00 0.274229
\(845\) 0 0
\(846\) 0 0
\(847\) −18553.7 −0.752673
\(848\) 34080.0 1.38008
\(849\) 0 0
\(850\) 6266.56 0.252872
\(851\) 2270.72 0.0914680
\(852\) 0 0
\(853\) 2909.85 0.116801 0.0584005 0.998293i \(-0.481400\pi\)
0.0584005 + 0.998293i \(0.481400\pi\)
\(854\) −1326.00 −0.0531321
\(855\) 0 0
\(856\) −3533.38 −0.141085
\(857\) −5346.00 −0.213087 −0.106544 0.994308i \(-0.533978\pi\)
−0.106544 + 0.994308i \(0.533978\pi\)
\(858\) 0 0
\(859\) 24244.0 0.962974 0.481487 0.876453i \(-0.340097\pi\)
0.481487 + 0.876453i \(0.340097\pi\)
\(860\) 4711.18 0.186802
\(861\) 0 0
\(862\) 1686.00 0.0666188
\(863\) 32780.8 1.29301 0.646507 0.762908i \(-0.276229\pi\)
0.646507 + 0.762908i \(0.276229\pi\)
\(864\) 0 0
\(865\) −12096.6 −0.475489
\(866\) −42050.7 −1.65005
\(867\) 0 0
\(868\) 6552.00 0.256209
\(869\) 28010.7 1.09344
\(870\) 0 0
\(871\) 0 0
\(872\) −8448.00 −0.328080
\(873\) 0 0
\(874\) 17442.0 0.675039
\(875\) −18096.0 −0.699150
\(876\) 0 0
\(877\) 4543.17 0.174928 0.0874640 0.996168i \(-0.472124\pi\)
0.0874640 + 0.996168i \(0.472124\pi\)
\(878\) −1596.95 −0.0613832
\(879\) 0 0
\(880\) 24960.0 0.956138
\(881\) −20517.0 −0.784603 −0.392302 0.919837i \(-0.628321\pi\)
−0.392302 + 0.919837i \(0.628321\pi\)
\(882\) 0 0
\(883\) 23852.0 0.909042 0.454521 0.890736i \(-0.349811\pi\)
0.454521 + 0.890736i \(0.349811\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −42109.6 −1.59673
\(887\) −38757.0 −1.46712 −0.733558 0.679626i \(-0.762142\pi\)
−0.733558 + 0.679626i \(0.762142\pi\)
\(888\) 0 0
\(889\) 50504.9 1.90538
\(890\) 14715.5 0.554230
\(891\) 0 0
\(892\) −16385.2 −0.615042
\(893\) 30294.0 1.13522
\(894\) 0 0
\(895\) −17833.2 −0.666031
\(896\) 34944.0 1.30290
\(897\) 0 0
\(898\) 1026.00 0.0381270
\(899\) 5019.48 0.186217
\(900\) 0 0
\(901\) −11502.0 −0.425291
\(902\) −30667.7 −1.13206
\(903\) 0 0
\(904\) 5694.98 0.209527
\(905\) −27.7128 −0.00101791
\(906\) 0 0
\(907\) 39071.0 1.43035 0.715177 0.698943i \(-0.246346\pi\)
0.715177 + 0.698943i \(0.246346\pi\)
\(908\) 1752.84 0.0640638
\(909\) 0 0
\(910\) 0 0
\(911\) 53040.0 1.92897 0.964486 0.264134i \(-0.0850860\pi\)
0.964486 + 0.264134i \(0.0850860\pi\)
\(912\) 0 0
\(913\) 9594.00 0.347771
\(914\) 2118.00 0.0766490
\(915\) 0 0
\(916\) 720.533 0.0259903
\(917\) 8376.20 0.301643
\(918\) 0 0
\(919\) 367.000 0.0131732 0.00658662 0.999978i \(-0.497903\pi\)
0.00658662 + 0.999978i \(0.497903\pi\)
\(920\) −10944.0 −0.392188
\(921\) 0 0
\(922\) −45474.0 −1.62430
\(923\) 0 0
\(924\) 0 0
\(925\) 2669.09 0.0948748
\(926\) −2892.00 −0.102632
\(927\) 0 0
\(928\) −11473.1 −0.405844
\(929\) −29935.0 −1.05720 −0.528599 0.848872i \(-0.677282\pi\)
−0.528599 + 0.848872i \(0.677282\pi\)
\(930\) 0 0
\(931\) 14486.9 0.509976
\(932\) −23112.0 −0.812295
\(933\) 0 0
\(934\) 50215.6 1.75921
\(935\) −8424.00 −0.294646
\(936\) 0 0
\(937\) 42166.0 1.47012 0.735060 0.678002i \(-0.237154\pi\)
0.735060 + 0.678002i \(0.237154\pi\)
\(938\) 12834.5 0.446760
\(939\) 0 0
\(940\) 19008.0 0.659545
\(941\) −35022.1 −1.21327 −0.606635 0.794981i \(-0.707481\pi\)
−0.606635 + 0.794981i \(0.707481\pi\)
\(942\) 0 0
\(943\) 22411.0 0.773916
\(944\) −1524.20 −0.0525515
\(945\) 0 0
\(946\) −6630.00 −0.227865
\(947\) −2599.81 −0.0892106 −0.0446053 0.999005i \(-0.514203\pi\)
−0.0446053 + 0.999005i \(0.514203\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 20502.0 0.700182
\(951\) 0 0
\(952\) −8424.00 −0.286789
\(953\) 10623.0 0.361084 0.180542 0.983567i \(-0.442215\pi\)
0.180542 + 0.983567i \(0.442215\pi\)
\(954\) 0 0
\(955\) 39366.1 1.33388
\(956\) −7440.89 −0.251732
\(957\) 0 0
\(958\) −30822.0 −1.03947
\(959\) 26793.0 0.902180
\(960\) 0 0
\(961\) −24499.0 −0.822362
\(962\) 0 0
\(963\) 0 0
\(964\) 8237.63 0.275224
\(965\) 58824.0 1.96229
\(966\) 0 0
\(967\) −20199.2 −0.671729 −0.335864 0.941910i \(-0.609028\pi\)
−0.335864 + 0.941910i \(0.609028\pi\)
\(968\) 11417.7 0.379110
\(969\) 0 0
\(970\) −59277.7 −1.96216
\(971\) 2325.00 0.0768412 0.0384206 0.999262i \(-0.487767\pi\)
0.0384206 + 0.999262i \(0.487767\pi\)
\(972\) 0 0
\(973\) −57304.9 −1.88809
\(974\) −16470.0 −0.541820
\(975\) 0 0
\(976\) 1360.00 0.0446030
\(977\) −32938.4 −1.07860 −0.539300 0.842113i \(-0.681311\pi\)
−0.539300 + 0.842113i \(0.681311\pi\)
\(978\) 0 0
\(979\) −6903.00 −0.225353
\(980\) 9089.80 0.296289
\(981\) 0 0
\(982\) −5663.81 −0.184052
\(983\) −42702.0 −1.38554 −0.692768 0.721161i \(-0.743609\pi\)
−0.692768 + 0.721161i \(0.743609\pi\)
\(984\) 0 0
\(985\) −38136.0 −1.23362
\(986\) 6453.62 0.208443
\(987\) 0 0
\(988\) 0 0
\(989\) 4845.00 0.155776
\(990\) 0 0
\(991\) −4843.00 −0.155240 −0.0776201 0.996983i \(-0.524732\pi\)
−0.0776201 + 0.996983i \(0.524732\pi\)
\(992\) −12096.0 −0.387146
\(993\) 0 0
\(994\) 45528.7 1.45280
\(995\) 23348.0 0.743902
\(996\) 0 0
\(997\) 10943.0 0.347611 0.173806 0.984780i \(-0.444394\pi\)
0.173806 + 0.984780i \(0.444394\pi\)
\(998\) 50004.0 1.58602
\(999\) 0 0
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 1521.4.a.q.1.2 2
3.2 odd 2 169.4.a.h.1.1 2
13.2 odd 12 117.4.q.c.82.1 2
13.7 odd 12 117.4.q.c.10.1 2
13.12 even 2 inner 1521.4.a.q.1.1 2
39.2 even 12 13.4.e.a.4.1 2
39.5 even 4 169.4.b.b.168.2 2
39.8 even 4 169.4.b.b.168.1 2
39.11 even 12 169.4.e.b.147.1 2
39.17 odd 6 169.4.c.i.146.1 4
39.20 even 12 13.4.e.a.10.1 yes 2
39.23 odd 6 169.4.c.i.22.1 4
39.29 odd 6 169.4.c.i.22.2 4
39.32 even 12 169.4.e.b.23.1 2
39.35 odd 6 169.4.c.i.146.2 4
39.38 odd 2 169.4.a.h.1.2 2
156.59 odd 12 208.4.w.a.49.1 2
156.119 odd 12 208.4.w.a.17.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.e.a.4.1 2 39.2 even 12
13.4.e.a.10.1 yes 2 39.20 even 12
117.4.q.c.10.1 2 13.7 odd 12
117.4.q.c.82.1 2 13.2 odd 12
169.4.a.h.1.1 2 3.2 odd 2
169.4.a.h.1.2 2 39.38 odd 2
169.4.b.b.168.1 2 39.8 even 4
169.4.b.b.168.2 2 39.5 even 4
169.4.c.i.22.1 4 39.23 odd 6
169.4.c.i.22.2 4 39.29 odd 6
169.4.c.i.146.1 4 39.17 odd 6
169.4.c.i.146.2 4 39.35 odd 6
169.4.e.b.23.1 2 39.32 even 12
169.4.e.b.147.1 2 39.11 even 12
208.4.w.a.17.1 2 156.119 odd 12
208.4.w.a.49.1 2 156.59 odd 12
1521.4.a.q.1.1 2 13.12 even 2 inner
1521.4.a.q.1.2 2 1.1 even 1 trivial