Properties

Label 2352.4.a.ci.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Defining polynomial: \(x^{3} - x^{2} - 24 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.248072\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -12.4346 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -12.4346 q^{5} +9.00000 q^{9} -60.3115 q^{11} +36.4269 q^{13} -37.3038 q^{15} -48.7461 q^{17} +50.5500 q^{19} -138.792 q^{23} +29.6194 q^{25} +27.0000 q^{27} -61.1345 q^{29} +1.16935 q^{31} -180.935 q^{33} +69.5268 q^{37} +109.281 q^{39} +308.115 q^{41} -174.443 q^{43} -111.911 q^{45} -389.362 q^{47} -146.238 q^{51} +314.935 q^{53} +749.950 q^{55} +151.650 q^{57} -844.526 q^{59} -338.538 q^{61} -452.954 q^{65} +971.550 q^{67} -416.377 q^{69} +98.4698 q^{71} +710.235 q^{73} +88.8581 q^{75} +486.884 q^{79} +81.0000 q^{81} -605.688 q^{83} +606.139 q^{85} -183.403 q^{87} +218.069 q^{89} +3.50806 q^{93} -628.569 q^{95} -782.288 q^{97} -542.804 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 9q^{3} + 11q^{5} + 27q^{9} + O(q^{10}) \) \( 3q + 9q^{3} + 11q^{5} + 27q^{9} - 35q^{11} + 62q^{13} + 33q^{15} + 48q^{17} + 202q^{19} - 216q^{23} + 130q^{25} + 81q^{27} + 53q^{29} + 95q^{31} - 105q^{33} + 262q^{37} + 186q^{39} + 244q^{41} - 360q^{43} + 99q^{45} + 210q^{47} + 144q^{51} + 393q^{53} + 1031q^{55} + 606q^{57} - 1143q^{59} - 70q^{61} - 472q^{65} + 628q^{67} - 648q^{69} - 318q^{71} + 988q^{73} + 390q^{75} - 861q^{79} + 243q^{81} - 519q^{83} + 1800q^{85} + 159q^{87} + 1766q^{89} + 285q^{93} + 736q^{95} + 19q^{97} - 315q^{99} + O(q^{100}) \)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −12.4346 −1.11218 −0.556092 0.831120i \(-0.687700\pi\)
−0.556092 + 0.831120i \(0.687700\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −60.3115 −1.65315 −0.826573 0.562829i \(-0.809713\pi\)
−0.826573 + 0.562829i \(0.809713\pi\)
\(12\) 0 0
\(13\) 36.4269 0.777154 0.388577 0.921416i \(-0.372967\pi\)
0.388577 + 0.921416i \(0.372967\pi\)
\(14\) 0 0
\(15\) −37.3038 −0.642120
\(16\) 0 0
\(17\) −48.7461 −0.695451 −0.347726 0.937596i \(-0.613046\pi\)
−0.347726 + 0.937596i \(0.613046\pi\)
\(18\) 0 0
\(19\) 50.5500 0.610366 0.305183 0.952294i \(-0.401282\pi\)
0.305183 + 0.952294i \(0.401282\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −138.792 −1.25827 −0.629135 0.777296i \(-0.716591\pi\)
−0.629135 + 0.777296i \(0.716591\pi\)
\(24\) 0 0
\(25\) 29.6194 0.236955
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −61.1345 −0.391462 −0.195731 0.980658i \(-0.562708\pi\)
−0.195731 + 0.980658i \(0.562708\pi\)
\(30\) 0 0
\(31\) 1.16935 0.00677490 0.00338745 0.999994i \(-0.498922\pi\)
0.00338745 + 0.999994i \(0.498922\pi\)
\(32\) 0 0
\(33\) −180.935 −0.954444
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 69.5268 0.308923 0.154461 0.987999i \(-0.450636\pi\)
0.154461 + 0.987999i \(0.450636\pi\)
\(38\) 0 0
\(39\) 109.281 0.448690
\(40\) 0 0
\(41\) 308.115 1.17365 0.586823 0.809715i \(-0.300378\pi\)
0.586823 + 0.809715i \(0.300378\pi\)
\(42\) 0 0
\(43\) −174.443 −0.618657 −0.309329 0.950955i \(-0.600104\pi\)
−0.309329 + 0.950955i \(0.600104\pi\)
\(44\) 0 0
\(45\) −111.911 −0.370728
\(46\) 0 0
\(47\) −389.362 −1.20839 −0.604194 0.796837i \(-0.706505\pi\)
−0.604194 + 0.796837i \(0.706505\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −146.238 −0.401519
\(52\) 0 0
\(53\) 314.935 0.816220 0.408110 0.912933i \(-0.366188\pi\)
0.408110 + 0.912933i \(0.366188\pi\)
\(54\) 0 0
\(55\) 749.950 1.83860
\(56\) 0 0
\(57\) 151.650 0.352395
\(58\) 0 0
\(59\) −844.526 −1.86352 −0.931762 0.363068i \(-0.881729\pi\)
−0.931762 + 0.363068i \(0.881729\pi\)
\(60\) 0 0
\(61\) −338.538 −0.710579 −0.355290 0.934756i \(-0.615618\pi\)
−0.355290 + 0.934756i \(0.615618\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −452.954 −0.864339
\(66\) 0 0
\(67\) 971.550 1.77155 0.885774 0.464117i \(-0.153628\pi\)
0.885774 + 0.464117i \(0.153628\pi\)
\(68\) 0 0
\(69\) −416.377 −0.726463
\(70\) 0 0
\(71\) 98.4698 0.164595 0.0822973 0.996608i \(-0.473774\pi\)
0.0822973 + 0.996608i \(0.473774\pi\)
\(72\) 0 0
\(73\) 710.235 1.13872 0.569361 0.822088i \(-0.307191\pi\)
0.569361 + 0.822088i \(0.307191\pi\)
\(74\) 0 0
\(75\) 88.8581 0.136806
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 486.884 0.693402 0.346701 0.937976i \(-0.387302\pi\)
0.346701 + 0.937976i \(0.387302\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −605.688 −0.800999 −0.400499 0.916297i \(-0.631163\pi\)
−0.400499 + 0.916297i \(0.631163\pi\)
\(84\) 0 0
\(85\) 606.139 0.773470
\(86\) 0 0
\(87\) −183.403 −0.226010
\(88\) 0 0
\(89\) 218.069 0.259722 0.129861 0.991532i \(-0.458547\pi\)
0.129861 + 0.991532i \(0.458547\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.50806 0.00391149
\(94\) 0 0
\(95\) −628.569 −0.678840
\(96\) 0 0
\(97\) −782.288 −0.818859 −0.409429 0.912342i \(-0.634272\pi\)
−0.409429 + 0.912342i \(0.634272\pi\)
\(98\) 0 0
\(99\) −542.804 −0.551049
\(100\) 0 0
\(101\) −311.646 −0.307029 −0.153514 0.988146i \(-0.549059\pi\)
−0.153514 + 0.988146i \(0.549059\pi\)
\(102\) 0 0
\(103\) 149.258 0.142784 0.0713922 0.997448i \(-0.477256\pi\)
0.0713922 + 0.997448i \(0.477256\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −851.519 −0.769341 −0.384670 0.923054i \(-0.625685\pi\)
−0.384670 + 0.923054i \(0.625685\pi\)
\(108\) 0 0
\(109\) 1361.88 1.19674 0.598369 0.801221i \(-0.295816\pi\)
0.598369 + 0.801221i \(0.295816\pi\)
\(110\) 0 0
\(111\) 208.581 0.178357
\(112\) 0 0
\(113\) 1048.55 0.872917 0.436459 0.899724i \(-0.356233\pi\)
0.436459 + 0.899724i \(0.356233\pi\)
\(114\) 0 0
\(115\) 1725.83 1.39943
\(116\) 0 0
\(117\) 327.842 0.259051
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2306.48 1.73289
\(122\) 0 0
\(123\) 924.346 0.677605
\(124\) 0 0
\(125\) 1186.02 0.848647
\(126\) 0 0
\(127\) −488.408 −0.341254 −0.170627 0.985336i \(-0.554579\pi\)
−0.170627 + 0.985336i \(0.554579\pi\)
\(128\) 0 0
\(129\) −523.328 −0.357182
\(130\) 0 0
\(131\) 1854.23 1.23668 0.618338 0.785912i \(-0.287806\pi\)
0.618338 + 0.785912i \(0.287806\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −335.734 −0.214040
\(136\) 0 0
\(137\) −511.115 −0.318741 −0.159370 0.987219i \(-0.550946\pi\)
−0.159370 + 0.987219i \(0.550946\pi\)
\(138\) 0 0
\(139\) −2266.10 −1.38279 −0.691397 0.722475i \(-0.743005\pi\)
−0.691397 + 0.722475i \(0.743005\pi\)
\(140\) 0 0
\(141\) −1168.09 −0.697663
\(142\) 0 0
\(143\) −2196.96 −1.28475
\(144\) 0 0
\(145\) 760.183 0.435378
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1507.90 0.829074 0.414537 0.910033i \(-0.363944\pi\)
0.414537 + 0.910033i \(0.363944\pi\)
\(150\) 0 0
\(151\) −1591.83 −0.857887 −0.428943 0.903331i \(-0.641114\pi\)
−0.428943 + 0.903331i \(0.641114\pi\)
\(152\) 0 0
\(153\) −438.715 −0.231817
\(154\) 0 0
\(155\) −14.5404 −0.00753494
\(156\) 0 0
\(157\) 1164.16 0.591784 0.295892 0.955221i \(-0.404383\pi\)
0.295892 + 0.955221i \(0.404383\pi\)
\(158\) 0 0
\(159\) 944.805 0.471245
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1155.88 0.555432 0.277716 0.960663i \(-0.410423\pi\)
0.277716 + 0.960663i \(0.410423\pi\)
\(164\) 0 0
\(165\) 2249.85 1.06152
\(166\) 0 0
\(167\) 2890.61 1.33941 0.669707 0.742626i \(-0.266420\pi\)
0.669707 + 0.742626i \(0.266420\pi\)
\(168\) 0 0
\(169\) −870.082 −0.396032
\(170\) 0 0
\(171\) 454.950 0.203455
\(172\) 0 0
\(173\) 1894.94 0.832770 0.416385 0.909188i \(-0.363297\pi\)
0.416385 + 0.909188i \(0.363297\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2533.58 −1.07591
\(178\) 0 0
\(179\) 4288.49 1.79071 0.895355 0.445354i \(-0.146922\pi\)
0.895355 + 0.445354i \(0.146922\pi\)
\(180\) 0 0
\(181\) 383.732 0.157583 0.0787917 0.996891i \(-0.474894\pi\)
0.0787917 + 0.996891i \(0.474894\pi\)
\(182\) 0 0
\(183\) −1015.61 −0.410253
\(184\) 0 0
\(185\) −864.539 −0.343579
\(186\) 0 0
\(187\) 2939.95 1.14968
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −385.311 −0.145969 −0.0729845 0.997333i \(-0.523252\pi\)
−0.0729845 + 0.997333i \(0.523252\pi\)
\(192\) 0 0
\(193\) 630.224 0.235049 0.117525 0.993070i \(-0.462504\pi\)
0.117525 + 0.993070i \(0.462504\pi\)
\(194\) 0 0
\(195\) −1358.86 −0.499026
\(196\) 0 0
\(197\) −1250.23 −0.452158 −0.226079 0.974109i \(-0.572591\pi\)
−0.226079 + 0.974109i \(0.572591\pi\)
\(198\) 0 0
\(199\) 1092.24 0.389081 0.194541 0.980894i \(-0.437678\pi\)
0.194541 + 0.980894i \(0.437678\pi\)
\(200\) 0 0
\(201\) 2914.65 1.02280
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3831.29 −1.30531
\(206\) 0 0
\(207\) −1249.13 −0.419423
\(208\) 0 0
\(209\) −3048.75 −1.00902
\(210\) 0 0
\(211\) 3620.05 1.18111 0.590556 0.806997i \(-0.298909\pi\)
0.590556 + 0.806997i \(0.298909\pi\)
\(212\) 0 0
\(213\) 295.409 0.0950287
\(214\) 0 0
\(215\) 2169.13 0.688061
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2130.70 0.657442
\(220\) 0 0
\(221\) −1775.67 −0.540473
\(222\) 0 0
\(223\) 183.844 0.0552069 0.0276034 0.999619i \(-0.491212\pi\)
0.0276034 + 0.999619i \(0.491212\pi\)
\(224\) 0 0
\(225\) 266.574 0.0789850
\(226\) 0 0
\(227\) −2279.52 −0.666506 −0.333253 0.942837i \(-0.608146\pi\)
−0.333253 + 0.942837i \(0.608146\pi\)
\(228\) 0 0
\(229\) 5412.67 1.56192 0.780960 0.624582i \(-0.214730\pi\)
0.780960 + 0.624582i \(0.214730\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1138.37 0.320073 0.160036 0.987111i \(-0.448839\pi\)
0.160036 + 0.987111i \(0.448839\pi\)
\(234\) 0 0
\(235\) 4841.56 1.34395
\(236\) 0 0
\(237\) 1460.65 0.400336
\(238\) 0 0
\(239\) 6226.36 1.68515 0.842573 0.538583i \(-0.181040\pi\)
0.842573 + 0.538583i \(0.181040\pi\)
\(240\) 0 0
\(241\) −3196.20 −0.854295 −0.427147 0.904182i \(-0.640481\pi\)
−0.427147 + 0.904182i \(0.640481\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1841.38 0.474349
\(248\) 0 0
\(249\) −1817.06 −0.462457
\(250\) 0 0
\(251\) −239.608 −0.0602546 −0.0301273 0.999546i \(-0.509591\pi\)
−0.0301273 + 0.999546i \(0.509591\pi\)
\(252\) 0 0
\(253\) 8370.78 2.08010
\(254\) 0 0
\(255\) 1818.42 0.446563
\(256\) 0 0
\(257\) 699.117 0.169688 0.0848439 0.996394i \(-0.472961\pi\)
0.0848439 + 0.996394i \(0.472961\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −550.210 −0.130487
\(262\) 0 0
\(263\) 919.040 0.215477 0.107738 0.994179i \(-0.465639\pi\)
0.107738 + 0.994179i \(0.465639\pi\)
\(264\) 0 0
\(265\) −3916.09 −0.907787
\(266\) 0 0
\(267\) 654.206 0.149950
\(268\) 0 0
\(269\) −2779.17 −0.629923 −0.314961 0.949104i \(-0.601992\pi\)
−0.314961 + 0.949104i \(0.601992\pi\)
\(270\) 0 0
\(271\) −2226.98 −0.499186 −0.249593 0.968351i \(-0.580297\pi\)
−0.249593 + 0.968351i \(0.580297\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1786.39 −0.391721
\(276\) 0 0
\(277\) 7307.69 1.58511 0.792557 0.609797i \(-0.208749\pi\)
0.792557 + 0.609797i \(0.208749\pi\)
\(278\) 0 0
\(279\) 10.5242 0.00225830
\(280\) 0 0
\(281\) 2730.61 0.579696 0.289848 0.957073i \(-0.406395\pi\)
0.289848 + 0.957073i \(0.406395\pi\)
\(282\) 0 0
\(283\) 1769.85 0.371755 0.185878 0.982573i \(-0.440487\pi\)
0.185878 + 0.982573i \(0.440487\pi\)
\(284\) 0 0
\(285\) −1885.71 −0.391928
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2536.81 −0.516347
\(290\) 0 0
\(291\) −2346.86 −0.472768
\(292\) 0 0
\(293\) 8228.81 1.64072 0.820362 0.571844i \(-0.193772\pi\)
0.820362 + 0.571844i \(0.193772\pi\)
\(294\) 0 0
\(295\) 10501.4 2.07258
\(296\) 0 0
\(297\) −1628.41 −0.318148
\(298\) 0 0
\(299\) −5055.78 −0.977870
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −934.937 −0.177263
\(304\) 0 0
\(305\) 4209.58 0.790295
\(306\) 0 0
\(307\) −6019.62 −1.11908 −0.559541 0.828803i \(-0.689023\pi\)
−0.559541 + 0.828803i \(0.689023\pi\)
\(308\) 0 0
\(309\) 447.773 0.0824366
\(310\) 0 0
\(311\) −1193.71 −0.217650 −0.108825 0.994061i \(-0.534709\pi\)
−0.108825 + 0.994061i \(0.534709\pi\)
\(312\) 0 0
\(313\) −8846.04 −1.59747 −0.798734 0.601684i \(-0.794497\pi\)
−0.798734 + 0.601684i \(0.794497\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6081.43 1.07750 0.538750 0.842466i \(-0.318897\pi\)
0.538750 + 0.842466i \(0.318897\pi\)
\(318\) 0 0
\(319\) 3687.11 0.647143
\(320\) 0 0
\(321\) −2554.56 −0.444179
\(322\) 0 0
\(323\) −2464.12 −0.424480
\(324\) 0 0
\(325\) 1078.94 0.184151
\(326\) 0 0
\(327\) 4085.64 0.690936
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3053.30 −0.507022 −0.253511 0.967333i \(-0.581585\pi\)
−0.253511 + 0.967333i \(0.581585\pi\)
\(332\) 0 0
\(333\) 625.742 0.102974
\(334\) 0 0
\(335\) −12080.8 −1.97029
\(336\) 0 0
\(337\) 3865.80 0.624877 0.312438 0.949938i \(-0.398854\pi\)
0.312438 + 0.949938i \(0.398854\pi\)
\(338\) 0 0
\(339\) 3145.66 0.503979
\(340\) 0 0
\(341\) −70.5255 −0.0111999
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5177.49 0.807961
\(346\) 0 0
\(347\) 99.5931 0.0154076 0.00770380 0.999970i \(-0.497548\pi\)
0.00770380 + 0.999970i \(0.497548\pi\)
\(348\) 0 0
\(349\) −3607.34 −0.553285 −0.276643 0.960973i \(-0.589222\pi\)
−0.276643 + 0.960973i \(0.589222\pi\)
\(350\) 0 0
\(351\) 983.526 0.149563
\(352\) 0 0
\(353\) 7130.73 1.07516 0.537579 0.843214i \(-0.319339\pi\)
0.537579 + 0.843214i \(0.319339\pi\)
\(354\) 0 0
\(355\) −1224.43 −0.183060
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6500.29 0.955632 0.477816 0.878460i \(-0.341428\pi\)
0.477816 + 0.878460i \(0.341428\pi\)
\(360\) 0 0
\(361\) −4303.70 −0.627453
\(362\) 0 0
\(363\) 6919.44 1.00049
\(364\) 0 0
\(365\) −8831.49 −1.26647
\(366\) 0 0
\(367\) −824.886 −0.117326 −0.0586631 0.998278i \(-0.518684\pi\)
−0.0586631 + 0.998278i \(0.518684\pi\)
\(368\) 0 0
\(369\) 2773.04 0.391216
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1333.85 0.185159 0.0925793 0.995705i \(-0.470489\pi\)
0.0925793 + 0.995705i \(0.470489\pi\)
\(374\) 0 0
\(375\) 3558.06 0.489967
\(376\) 0 0
\(377\) −2226.94 −0.304226
\(378\) 0 0
\(379\) 1338.29 0.181380 0.0906902 0.995879i \(-0.471093\pi\)
0.0906902 + 0.995879i \(0.471093\pi\)
\(380\) 0 0
\(381\) −1465.22 −0.197023
\(382\) 0 0
\(383\) −353.376 −0.0471453 −0.0235727 0.999722i \(-0.507504\pi\)
−0.0235727 + 0.999722i \(0.507504\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1569.98 −0.206219
\(388\) 0 0
\(389\) 11737.2 1.52982 0.764908 0.644139i \(-0.222784\pi\)
0.764908 + 0.644139i \(0.222784\pi\)
\(390\) 0 0
\(391\) 6765.59 0.875066
\(392\) 0 0
\(393\) 5562.68 0.713996
\(394\) 0 0
\(395\) −6054.21 −0.771191
\(396\) 0 0
\(397\) 13281.4 1.67903 0.839516 0.543335i \(-0.182839\pi\)
0.839516 + 0.543335i \(0.182839\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7482.36 −0.931798 −0.465899 0.884838i \(-0.654269\pi\)
−0.465899 + 0.884838i \(0.654269\pi\)
\(402\) 0 0
\(403\) 42.5959 0.00526514
\(404\) 0 0
\(405\) −1007.20 −0.123576
\(406\) 0 0
\(407\) −4193.27 −0.510694
\(408\) 0 0
\(409\) −13796.6 −1.66797 −0.833983 0.551791i \(-0.813945\pi\)
−0.833983 + 0.551791i \(0.813945\pi\)
\(410\) 0 0
\(411\) −1533.35 −0.184025
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7531.49 0.890859
\(416\) 0 0
\(417\) −6798.31 −0.798357
\(418\) 0 0
\(419\) −9497.56 −1.10737 −0.553683 0.832728i \(-0.686778\pi\)
−0.553683 + 0.832728i \(0.686778\pi\)
\(420\) 0 0
\(421\) 624.367 0.0722797 0.0361399 0.999347i \(-0.488494\pi\)
0.0361399 + 0.999347i \(0.488494\pi\)
\(422\) 0 0
\(423\) −3504.26 −0.402796
\(424\) 0 0
\(425\) −1443.83 −0.164791
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6590.88 −0.741750
\(430\) 0 0
\(431\) 13397.3 1.49727 0.748636 0.662981i \(-0.230709\pi\)
0.748636 + 0.662981i \(0.230709\pi\)
\(432\) 0 0
\(433\) −14057.3 −1.56016 −0.780079 0.625681i \(-0.784821\pi\)
−0.780079 + 0.625681i \(0.784821\pi\)
\(434\) 0 0
\(435\) 2280.55 0.251365
\(436\) 0 0
\(437\) −7015.95 −0.768006
\(438\) 0 0
\(439\) 16368.8 1.77960 0.889798 0.456356i \(-0.150845\pi\)
0.889798 + 0.456356i \(0.150845\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1178.71 0.126416 0.0632078 0.998000i \(-0.479867\pi\)
0.0632078 + 0.998000i \(0.479867\pi\)
\(444\) 0 0
\(445\) −2711.60 −0.288858
\(446\) 0 0
\(447\) 4523.70 0.478666
\(448\) 0 0
\(449\) −12400.9 −1.30342 −0.651709 0.758469i \(-0.725948\pi\)
−0.651709 + 0.758469i \(0.725948\pi\)
\(450\) 0 0
\(451\) −18582.9 −1.94021
\(452\) 0 0
\(453\) −4775.48 −0.495301
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9925.58 1.01597 0.507986 0.861365i \(-0.330390\pi\)
0.507986 + 0.861365i \(0.330390\pi\)
\(458\) 0 0
\(459\) −1316.15 −0.133840
\(460\) 0 0
\(461\) −16010.3 −1.61751 −0.808755 0.588146i \(-0.799858\pi\)
−0.808755 + 0.588146i \(0.799858\pi\)
\(462\) 0 0
\(463\) −17372.4 −1.74377 −0.871883 0.489714i \(-0.837101\pi\)
−0.871883 + 0.489714i \(0.837101\pi\)
\(464\) 0 0
\(465\) −43.6213 −0.00435030
\(466\) 0 0
\(467\) −2108.06 −0.208886 −0.104443 0.994531i \(-0.533306\pi\)
−0.104443 + 0.994531i \(0.533306\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3492.48 0.341667
\(472\) 0 0
\(473\) 10520.9 1.02273
\(474\) 0 0
\(475\) 1497.26 0.144629
\(476\) 0 0
\(477\) 2834.41 0.272073
\(478\) 0 0
\(479\) 2450.04 0.233706 0.116853 0.993149i \(-0.462719\pi\)
0.116853 + 0.993149i \(0.462719\pi\)
\(480\) 0 0
\(481\) 2532.65 0.240081
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9727.44 0.910722
\(486\) 0 0
\(487\) −645.236 −0.0600379 −0.0300189 0.999549i \(-0.509557\pi\)
−0.0300189 + 0.999549i \(0.509557\pi\)
\(488\) 0 0
\(489\) 3467.64 0.320679
\(490\) 0 0
\(491\) −11766.1 −1.08146 −0.540731 0.841196i \(-0.681852\pi\)
−0.540731 + 0.841196i \(0.681852\pi\)
\(492\) 0 0
\(493\) 2980.07 0.272242
\(494\) 0 0
\(495\) 6749.55 0.612868
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 44.0209 0.00394919 0.00197459 0.999998i \(-0.499371\pi\)
0.00197459 + 0.999998i \(0.499371\pi\)
\(500\) 0 0
\(501\) 8671.83 0.773311
\(502\) 0 0
\(503\) −8290.27 −0.734880 −0.367440 0.930047i \(-0.619766\pi\)
−0.367440 + 0.930047i \(0.619766\pi\)
\(504\) 0 0
\(505\) 3875.19 0.341473
\(506\) 0 0
\(507\) −2610.24 −0.228649
\(508\) 0 0
\(509\) 6915.04 0.602168 0.301084 0.953598i \(-0.402651\pi\)
0.301084 + 0.953598i \(0.402651\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1364.85 0.117465
\(514\) 0 0
\(515\) −1855.96 −0.158803
\(516\) 0 0
\(517\) 23483.0 1.99764
\(518\) 0 0
\(519\) 5684.81 0.480800
\(520\) 0 0
\(521\) 13399.3 1.12674 0.563371 0.826204i \(-0.309504\pi\)
0.563371 + 0.826204i \(0.309504\pi\)
\(522\) 0 0
\(523\) 9936.99 0.830811 0.415406 0.909636i \(-0.363640\pi\)
0.415406 + 0.909636i \(0.363640\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −57.0014 −0.00471161
\(528\) 0 0
\(529\) 7096.33 0.583244
\(530\) 0 0
\(531\) −7600.74 −0.621175
\(532\) 0 0
\(533\) 11223.7 0.912104
\(534\) 0 0
\(535\) 10588.3 0.855649
\(536\) 0 0
\(537\) 12865.5 1.03387
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9286.17 0.737973 0.368987 0.929435i \(-0.379705\pi\)
0.368987 + 0.929435i \(0.379705\pi\)
\(542\) 0 0
\(543\) 1151.20 0.0909809
\(544\) 0 0
\(545\) −16934.4 −1.33099
\(546\) 0 0
\(547\) 16821.6 1.31488 0.657438 0.753508i \(-0.271640\pi\)
0.657438 + 0.753508i \(0.271640\pi\)
\(548\) 0 0
\(549\) −3046.84 −0.236860
\(550\) 0 0
\(551\) −3090.35 −0.238935
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2593.62 −0.198366
\(556\) 0 0
\(557\) 1805.94 0.137379 0.0686897 0.997638i \(-0.478118\pi\)
0.0686897 + 0.997638i \(0.478118\pi\)
\(558\) 0 0
\(559\) −6354.40 −0.480792
\(560\) 0 0
\(561\) 8819.86 0.663770
\(562\) 0 0
\(563\) −12214.9 −0.914381 −0.457190 0.889369i \(-0.651144\pi\)
−0.457190 + 0.889369i \(0.651144\pi\)
\(564\) 0 0
\(565\) −13038.3 −0.970845
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4283.77 0.315615 0.157808 0.987470i \(-0.449557\pi\)
0.157808 + 0.987470i \(0.449557\pi\)
\(570\) 0 0
\(571\) −6359.94 −0.466121 −0.233060 0.972462i \(-0.574874\pi\)
−0.233060 + 0.972462i \(0.574874\pi\)
\(572\) 0 0
\(573\) −1155.93 −0.0842753
\(574\) 0 0
\(575\) −4110.95 −0.298153
\(576\) 0 0
\(577\) 14468.7 1.04392 0.521959 0.852971i \(-0.325201\pi\)
0.521959 + 0.852971i \(0.325201\pi\)
\(578\) 0 0
\(579\) 1890.67 0.135706
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18994.2 −1.34933
\(584\) 0 0
\(585\) −4076.59 −0.288113
\(586\) 0 0
\(587\) 11132.6 0.782777 0.391388 0.920226i \(-0.371995\pi\)
0.391388 + 0.920226i \(0.371995\pi\)
\(588\) 0 0
\(589\) 59.1108 0.00413517
\(590\) 0 0
\(591\) −3750.69 −0.261054
\(592\) 0 0
\(593\) −19775.6 −1.36946 −0.684728 0.728799i \(-0.740079\pi\)
−0.684728 + 0.728799i \(0.740079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3276.73 0.224636
\(598\) 0 0
\(599\) 23891.0 1.62965 0.814825 0.579707i \(-0.196833\pi\)
0.814825 + 0.579707i \(0.196833\pi\)
\(600\) 0 0
\(601\) 19395.5 1.31641 0.658204 0.752840i \(-0.271317\pi\)
0.658204 + 0.752840i \(0.271317\pi\)
\(602\) 0 0
\(603\) 8743.95 0.590516
\(604\) 0 0
\(605\) −28680.2 −1.92730
\(606\) 0 0
\(607\) −14596.7 −0.976051 −0.488025 0.872829i \(-0.662283\pi\)
−0.488025 + 0.872829i \(0.662283\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −14183.2 −0.939104
\(612\) 0 0
\(613\) 1979.80 0.130446 0.0652229 0.997871i \(-0.479224\pi\)
0.0652229 + 0.997871i \(0.479224\pi\)
\(614\) 0 0
\(615\) −11493.9 −0.753622
\(616\) 0 0
\(617\) 16262.4 1.06110 0.530551 0.847653i \(-0.321985\pi\)
0.530551 + 0.847653i \(0.321985\pi\)
\(618\) 0 0
\(619\) 12021.0 0.780555 0.390278 0.920697i \(-0.372379\pi\)
0.390278 + 0.920697i \(0.372379\pi\)
\(620\) 0 0
\(621\) −3747.39 −0.242154
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −18450.1 −1.18081
\(626\) 0 0
\(627\) −9146.24 −0.582561
\(628\) 0 0
\(629\) −3389.16 −0.214841
\(630\) 0 0
\(631\) −25347.6 −1.59916 −0.799582 0.600557i \(-0.794945\pi\)
−0.799582 + 0.600557i \(0.794945\pi\)
\(632\) 0 0
\(633\) 10860.1 0.681915
\(634\) 0 0
\(635\) 6073.16 0.379537
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 886.228 0.0548648
\(640\) 0 0
\(641\) −5111.60 −0.314971 −0.157485 0.987521i \(-0.550339\pi\)
−0.157485 + 0.987521i \(0.550339\pi\)
\(642\) 0 0
\(643\) 10931.3 0.670435 0.335217 0.942141i \(-0.391190\pi\)
0.335217 + 0.942141i \(0.391190\pi\)
\(644\) 0 0
\(645\) 6507.38 0.397252
\(646\) 0 0
\(647\) 18406.1 1.11842 0.559211 0.829025i \(-0.311104\pi\)
0.559211 + 0.829025i \(0.311104\pi\)
\(648\) 0 0
\(649\) 50934.7 3.08068
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19921.4 1.19385 0.596926 0.802296i \(-0.296389\pi\)
0.596926 + 0.802296i \(0.296389\pi\)
\(654\) 0 0
\(655\) −23056.6 −1.37541
\(656\) 0 0
\(657\) 6392.11 0.379574
\(658\) 0 0
\(659\) 18858.8 1.11477 0.557385 0.830254i \(-0.311805\pi\)
0.557385 + 0.830254i \(0.311805\pi\)
\(660\) 0 0
\(661\) 25832.1 1.52005 0.760023 0.649896i \(-0.225188\pi\)
0.760023 + 0.649896i \(0.225188\pi\)
\(662\) 0 0
\(663\) −5327.01 −0.312042
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8485.00 0.492564
\(668\) 0 0
\(669\) 551.533 0.0318737
\(670\) 0 0
\(671\) 20417.7 1.17469
\(672\) 0 0
\(673\) −16275.0 −0.932178 −0.466089 0.884738i \(-0.654337\pi\)
−0.466089 + 0.884738i \(0.654337\pi\)
\(674\) 0 0
\(675\) 799.723 0.0456020
\(676\) 0 0
\(677\) −26271.8 −1.49144 −0.745720 0.666259i \(-0.767895\pi\)
−0.745720 + 0.666259i \(0.767895\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6838.55 −0.384808
\(682\) 0 0
\(683\) 8072.29 0.452237 0.226118 0.974100i \(-0.427396\pi\)
0.226118 + 0.974100i \(0.427396\pi\)
\(684\) 0 0
\(685\) 6355.51 0.354499
\(686\) 0 0
\(687\) 16238.0 0.901774
\(688\) 0 0
\(689\) 11472.1 0.634328
\(690\) 0 0
\(691\) 24485.3 1.34799 0.673997 0.738734i \(-0.264576\pi\)
0.673997 + 0.738734i \(0.264576\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 28178.1 1.53792
\(696\) 0 0
\(697\) −15019.4 −0.816214
\(698\) 0 0
\(699\) 3415.10 0.184794
\(700\) 0 0
\(701\) 778.448 0.0419423 0.0209712 0.999780i \(-0.493324\pi\)
0.0209712 + 0.999780i \(0.493324\pi\)
\(702\) 0 0
\(703\) 3514.58 0.188556
\(704\) 0 0
\(705\) 14524.7 0.775930
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −24172.0 −1.28039 −0.640197 0.768211i \(-0.721147\pi\)
−0.640197 + 0.768211i \(0.721147\pi\)
\(710\) 0 0
\(711\) 4381.96 0.231134
\(712\) 0 0
\(713\) −162.297 −0.00852466
\(714\) 0 0
\(715\) 27318.3 1.42888
\(716\) 0 0
\(717\) 18679.1 0.972919
\(718\) 0 0
\(719\) 81.8835 0.00424720 0.00212360 0.999998i \(-0.499324\pi\)
0.00212360 + 0.999998i \(0.499324\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −9588.59 −0.493227
\(724\) 0 0
\(725\) −1810.76 −0.0927588
\(726\) 0 0
\(727\) 32542.9 1.66018 0.830088 0.557632i \(-0.188290\pi\)
0.830088 + 0.557632i \(0.188290\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 8503.40 0.430246
\(732\) 0 0
\(733\) −5068.94 −0.255424 −0.127712 0.991811i \(-0.540763\pi\)
−0.127712 + 0.991811i \(0.540763\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −58595.6 −2.92863
\(738\) 0 0
\(739\) 38428.5 1.91287 0.956437 0.291939i \(-0.0943004\pi\)
0.956437 + 0.291939i \(0.0943004\pi\)
\(740\) 0 0
\(741\) 5524.14 0.273865
\(742\) 0 0
\(743\) −21592.9 −1.06617 −0.533086 0.846061i \(-0.678968\pi\)
−0.533086 + 0.846061i \(0.678968\pi\)
\(744\) 0 0
\(745\) −18750.1 −0.922083
\(746\) 0 0
\(747\) −5451.19 −0.267000
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −8112.60 −0.394185 −0.197093 0.980385i \(-0.563150\pi\)
−0.197093 + 0.980385i \(0.563150\pi\)
\(752\) 0 0
\(753\) −718.823 −0.0347880
\(754\) 0 0
\(755\) 19793.7 0.954129
\(756\) 0 0
\(757\) 3108.01 0.149224 0.0746120 0.997213i \(-0.476228\pi\)
0.0746120 + 0.997213i \(0.476228\pi\)
\(758\) 0 0
\(759\) 25112.3 1.20095
\(760\) 0 0
\(761\) −7211.93 −0.343538 −0.171769 0.985137i \(-0.554948\pi\)
−0.171769 + 0.985137i \(0.554948\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 5455.25 0.257823
\(766\) 0 0
\(767\) −30763.5 −1.44825
\(768\) 0 0
\(769\) −7533.07 −0.353250 −0.176625 0.984278i \(-0.556518\pi\)
−0.176625 + 0.984278i \(0.556518\pi\)
\(770\) 0 0
\(771\) 2097.35 0.0979693
\(772\) 0 0
\(773\) −24832.6 −1.15546 −0.577728 0.816229i \(-0.696060\pi\)
−0.577728 + 0.816229i \(0.696060\pi\)
\(774\) 0 0
\(775\) 34.6355 0.00160535
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 15575.2 0.716355
\(780\) 0 0
\(781\) −5938.86 −0.272099
\(782\) 0 0
\(783\) −1650.63 −0.0753368
\(784\) 0 0
\(785\) −14475.9 −0.658173
\(786\) 0 0
\(787\) −36313.1 −1.64476 −0.822378 0.568941i \(-0.807353\pi\)
−0.822378 + 0.568941i \(0.807353\pi\)
\(788\) 0 0
\(789\) 2757.12 0.124406
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12331.9 −0.552229
\(794\) 0 0
\(795\) −11748.3 −0.524111
\(796\) 0 0
\(797\) 31665.7 1.40735 0.703675 0.710522i \(-0.251541\pi\)
0.703675 + 0.710522i \(0.251541\pi\)
\(798\) 0 0
\(799\) 18979.9 0.840375
\(800\) 0 0
\(801\) 1962.62 0.0865738
\(802\) 0 0
\(803\) −42835.4 −1.88247
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −8337.52 −0.363686
\(808\) 0 0
\(809\) 12384.6 0.538219 0.269110 0.963110i \(-0.413271\pi\)
0.269110 + 0.963110i \(0.413271\pi\)
\(810\) 0 0
\(811\) −16742.4 −0.724914 −0.362457 0.932000i \(-0.618062\pi\)
−0.362457 + 0.932000i \(0.618062\pi\)
\(812\) 0 0
\(813\) −6680.94 −0.288205
\(814\) 0 0
\(815\) −14372.9 −0.617744
\(816\) 0 0
\(817\) −8818.07 −0.377607
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26456.3 1.12464 0.562322 0.826918i \(-0.309908\pi\)
0.562322 + 0.826918i \(0.309908\pi\)
\(822\) 0 0
\(823\) −23098.5 −0.978328 −0.489164 0.872192i \(-0.662698\pi\)
−0.489164 + 0.872192i \(0.662698\pi\)
\(824\) 0 0
\(825\) −5359.17 −0.226160
\(826\) 0 0
\(827\) −20647.6 −0.868183 −0.434092 0.900869i \(-0.642931\pi\)
−0.434092 + 0.900869i \(0.642931\pi\)
\(828\) 0 0
\(829\) −23368.5 −0.979037 −0.489519 0.871993i \(-0.662827\pi\)
−0.489519 + 0.871993i \(0.662827\pi\)
\(830\) 0 0
\(831\) 21923.1 0.915166
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −35943.6 −1.48968
\(836\) 0 0
\(837\) 31.5725 0.00130383
\(838\) 0 0
\(839\) −16735.5 −0.688645 −0.344322 0.938851i \(-0.611891\pi\)
−0.344322 + 0.938851i \(0.611891\pi\)
\(840\) 0 0
\(841\) −20651.6 −0.846758
\(842\) 0 0
\(843\) 8191.84 0.334688
\(844\) 0 0
\(845\) 10819.1 0.440460
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 5309.55 0.214633
\(850\) 0 0
\(851\) −9649.80 −0.388708
\(852\) 0 0
\(853\) −10294.5 −0.413219 −0.206609 0.978424i \(-0.566243\pi\)
−0.206609 + 0.978424i \(0.566243\pi\)
\(854\) 0 0
\(855\) −5657.12 −0.226280
\(856\) 0 0
\(857\) −32788.6 −1.30693 −0.653463 0.756958i \(-0.726685\pi\)
−0.653463 + 0.756958i \(0.726685\pi\)
\(858\) 0 0
\(859\) 4909.76 0.195016 0.0975081 0.995235i \(-0.468913\pi\)
0.0975081 + 0.995235i \(0.468913\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17795.0 −0.701909 −0.350954 0.936393i \(-0.614143\pi\)
−0.350954 + 0.936393i \(0.614143\pi\)
\(864\) 0 0
\(865\) −23562.8 −0.926195
\(866\) 0 0
\(867\) −7610.44 −0.298113
\(868\) 0 0
\(869\) −29364.7 −1.14629
\(870\) 0 0
\(871\) 35390.5 1.37677
\(872\) 0 0
\(873\) −7040.59 −0.272953
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34672.2 1.33500 0.667501 0.744609i \(-0.267364\pi\)
0.667501 + 0.744609i \(0.267364\pi\)
\(878\) 0 0
\(879\) 24686.4 0.947273
\(880\) 0 0
\(881\) 40848.2 1.56210 0.781051 0.624467i \(-0.214684\pi\)
0.781051 + 0.624467i \(0.214684\pi\)
\(882\) 0 0
\(883\) −30035.1 −1.14469 −0.572345 0.820013i \(-0.693966\pi\)
−0.572345 + 0.820013i \(0.693966\pi\)
\(884\) 0 0
\(885\) 31504.1 1.19661
\(886\) 0 0
\(887\) −33210.7 −1.25717 −0.628583 0.777742i \(-0.716365\pi\)
−0.628583 + 0.777742i \(0.716365\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4885.23 −0.183683
\(892\) 0 0
\(893\) −19682.2 −0.737559
\(894\) 0 0
\(895\) −53325.7 −1.99160
\(896\) 0 0
\(897\) −15167.3 −0.564573
\(898\) 0 0
\(899\) −71.4878 −0.00265211
\(900\) 0 0
\(901\) −15351.9 −0.567641
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4771.56 −0.175262
\(906\) 0 0
\(907\) −2497.83 −0.0914433 −0.0457217 0.998954i \(-0.514559\pi\)
−0.0457217 + 0.998954i \(0.514559\pi\)
\(908\) 0 0
\(909\) −2804.81 −0.102343
\(910\) 0 0
\(911\) 1895.00 0.0689180 0.0344590 0.999406i \(-0.489029\pi\)
0.0344590 + 0.999406i \(0.489029\pi\)
\(912\) 0 0
\(913\) 36530.0 1.32417
\(914\) 0 0
\(915\) 12628.8 0.456277
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −6270.71 −0.225083 −0.112542 0.993647i \(-0.535899\pi\)
−0.112542 + 0.993647i \(0.535899\pi\)
\(920\) 0 0
\(921\) −18058.9 −0.646102
\(922\) 0 0
\(923\) 3586.95 0.127915
\(924\) 0 0
\(925\) 2059.34 0.0732008
\(926\) 0 0
\(927\) 1343.32 0.0475948
\(928\) 0 0
\(929\) −31552.6 −1.11432 −0.557161 0.830404i \(-0.688110\pi\)
−0.557161 + 0.830404i \(0.688110\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3581.14 −0.125661
\(934\) 0 0
\(935\) −36557.2 −1.27866
\(936\) 0 0
\(937\) −22030.2 −0.768084 −0.384042 0.923316i \(-0.625468\pi\)
−0.384042 + 0.923316i \(0.625468\pi\)
\(938\) 0 0
\(939\) −26538.1 −0.922299
\(940\) 0 0
\(941\) −32538.6 −1.12724 −0.563618 0.826036i \(-0.690591\pi\)
−0.563618 + 0.826036i \(0.690591\pi\)
\(942\) 0 0
\(943\) −42764.1 −1.47677
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40711.0 1.39697 0.698485 0.715625i \(-0.253858\pi\)
0.698485 + 0.715625i \(0.253858\pi\)
\(948\) 0 0
\(949\) 25871.7 0.884962
\(950\) 0 0
\(951\) 18244.3 0.622095
\(952\) 0 0
\(953\) −52516.4 −1.78507 −0.892536 0.450976i \(-0.851076\pi\)
−0.892536 + 0.450976i \(0.851076\pi\)
\(954\) 0 0
\(955\) 4791.18 0.162345
\(956\) 0 0
\(957\) 11061.3 0.373628
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29789.6 −0.999954
\(962\) 0 0
\(963\) −7663.67 −0.256447
\(964\) 0 0
\(965\) −7836.58 −0.261418
\(966\) 0 0
\(967\) −14721.6 −0.489570 −0.244785 0.969577i \(-0.578717\pi\)
−0.244785 + 0.969577i \(0.578717\pi\)
\(968\) 0 0
\(969\) −7392.35 −0.245074
\(970\) 0 0
\(971\) −13772.5 −0.455181 −0.227590 0.973757i \(-0.573085\pi\)
−0.227590 + 0.973757i \(0.573085\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 3236.83 0.106319
\(976\) 0 0
\(977\) 24782.1 0.811513 0.405757 0.913981i \(-0.367008\pi\)
0.405757 + 0.913981i \(0.367008\pi\)
\(978\) 0 0
\(979\) −13152.0 −0.429358
\(980\) 0 0
\(981\) 12256.9 0.398912
\(982\) 0 0
\(983\) −42804.7 −1.38887 −0.694435 0.719556i \(-0.744345\pi\)
−0.694435 + 0.719556i \(0.744345\pi\)
\(984\) 0 0
\(985\) 15546.1 0.502883
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24211.3 0.778438
\(990\) 0 0
\(991\) −449.862 −0.0144201 −0.00721006 0.999974i \(-0.502295\pi\)
−0.00721006 + 0.999974i \(0.502295\pi\)
\(992\) 0 0
\(993\) −9159.89 −0.292729
\(994\) 0 0
\(995\) −13581.6 −0.432730
\(996\) 0 0
\(997\) −21473.7 −0.682127 −0.341063 0.940040i \(-0.610787\pi\)
−0.341063 + 0.940040i \(0.610787\pi\)
\(998\) 0 0
\(999\) 1877.22 0.0594522
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 2352.4.a.ci.1.1 3
4.3 odd 2 147.4.a.l.1.2 3
7.2 even 3 336.4.q.k.193.3 6
7.4 even 3 336.4.q.k.289.3 6
7.6 odd 2 2352.4.a.cg.1.3 3
12.11 even 2 441.4.a.s.1.2 3
28.3 even 6 147.4.e.n.79.2 6
28.11 odd 6 21.4.e.b.16.2 yes 6
28.19 even 6 147.4.e.n.67.2 6
28.23 odd 6 21.4.e.b.4.2 6
28.27 even 2 147.4.a.m.1.2 3
84.11 even 6 63.4.e.c.37.2 6
84.23 even 6 63.4.e.c.46.2 6
84.47 odd 6 441.4.e.w.361.2 6
84.59 odd 6 441.4.e.w.226.2 6
84.83 odd 2 441.4.a.t.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.2 6 28.23 odd 6
21.4.e.b.16.2 yes 6 28.11 odd 6
63.4.e.c.37.2 6 84.11 even 6
63.4.e.c.46.2 6 84.23 even 6
147.4.a.l.1.2 3 4.3 odd 2
147.4.a.m.1.2 3 28.27 even 2
147.4.e.n.67.2 6 28.19 even 6
147.4.e.n.79.2 6 28.3 even 6
336.4.q.k.193.3 6 7.2 even 3
336.4.q.k.289.3 6 7.4 even 3
441.4.a.s.1.2 3 12.11 even 2
441.4.a.t.1.2 3 84.83 odd 2
441.4.e.w.226.2 6 84.59 odd 6
441.4.e.w.361.2 6 84.47 odd 6
2352.4.a.cg.1.3 3 7.6 odd 2
2352.4.a.ci.1.1 3 1.1 even 1 trivial