Properties

Label 441.4.a.t.1.2
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Defining polynomial: \(x^{3} - x^{2} - 24 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.248072\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.248072 q^{2} -7.93846 q^{4} -12.4346 q^{5} +3.95388 q^{8} +O(q^{10})\) \(q-0.248072 q^{2} -7.93846 q^{4} -12.4346 q^{5} +3.95388 q^{8} +3.08468 q^{10} -60.3115 q^{11} -36.4269 q^{13} +62.5268 q^{16} -48.7461 q^{17} +50.5500 q^{19} +98.7116 q^{20} +14.9616 q^{22} -138.792 q^{23} +29.6194 q^{25} +9.03649 q^{26} +61.1345 q^{29} +1.16935 q^{31} -47.1422 q^{32} +12.0925 q^{34} +69.5268 q^{37} -12.5400 q^{38} -49.1650 q^{40} +308.115 q^{41} +174.443 q^{43} +478.781 q^{44} +34.4305 q^{46} +389.362 q^{47} -7.34774 q^{50} +289.173 q^{52} -314.935 q^{53} +749.950 q^{55} -15.1657 q^{58} +844.526 q^{59} +338.538 q^{61} -0.290084 q^{62} -488.520 q^{64} +452.954 q^{65} -971.550 q^{67} +386.969 q^{68} +98.4698 q^{71} -710.235 q^{73} -17.2477 q^{74} -401.289 q^{76} -486.884 q^{79} -777.496 q^{80} -76.4348 q^{82} +605.688 q^{83} +606.139 q^{85} -43.2743 q^{86} -238.465 q^{88} +218.069 q^{89} +1101.80 q^{92} -96.5897 q^{94} -628.569 q^{95} +782.288 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 25q^{4} + 11q^{5} - 39q^{8} + O(q^{10}) \) \( 3q - q^{2} + 25q^{4} + 11q^{5} - 39q^{8} + 55q^{10} - 35q^{11} - 62q^{13} + 241q^{16} + 48q^{17} + 202q^{19} + 439q^{20} - 7q^{22} - 216q^{23} + 130q^{25} + 274q^{26} - 53q^{29} + 95q^{31} - 683q^{32} + 24q^{34} + 262q^{37} - 398q^{38} - 21q^{40} + 244q^{41} + 360q^{43} + 905q^{44} - 1056q^{46} - 210q^{47} + 1378q^{50} - 324q^{52} - 393q^{53} + 1031q^{55} - 1249q^{58} + 1143q^{59} + 70q^{61} + 1059q^{62} - 399q^{64} + 472q^{65} - 628q^{67} + 1944q^{68} - 318q^{71} - 988q^{73} - 1002q^{74} + 2340q^{76} + 861q^{79} + 175q^{80} - 124q^{82} + 519q^{83} + 1800q^{85} + 3208q^{86} - 891q^{88} + 1766q^{89} + 672q^{92} + 3294q^{94} + 736q^{95} - 19q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.248072 −0.0877067 −0.0438533 0.999038i \(-0.513963\pi\)
−0.0438533 + 0.999038i \(0.513963\pi\)
\(3\) 0 0
\(4\) −7.93846 −0.992308
\(5\) −12.4346 −1.11218 −0.556092 0.831120i \(-0.687700\pi\)
−0.556092 + 0.831120i \(0.687700\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 3.95388 0.174739
\(9\) 0 0
\(10\) 3.08468 0.0975460
\(11\) −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 \(-0.627033\pi\)
−0.388577 + 0.921416i \(0.627033\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 62.5268 0.976982
\(17\) −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 \(0.401282\pi\)
\(20\) 98.7116 1.10363
\(21\) 0 0
\(22\) 14.9616 0.144992
\(23\) −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
\(26\) 9.03649 0.0681616
\(27\) 0 0
\(28\) 0 0
\(29\) 61.1345 0.391462 0.195731 0.980658i \(-0.437292\pi\)
0.195731 + 0.980658i \(0.437292\pi\)
\(30\) 0 0
\(31\) 1.16935 0.00677490 0.00338745 0.999994i \(-0.498922\pi\)
0.00338745 + 0.999994i \(0.498922\pi\)
\(32\) −47.1422 −0.260426
\(33\) 0 0
\(34\) 12.0925 0.0609957
\(35\) 0 0
\(36\) 0 0
\(37\) 69.5268 0.308923 0.154461 0.987999i \(-0.450636\pi\)
0.154461 + 0.987999i \(0.450636\pi\)
\(38\) −12.5400 −0.0535332
\(39\) 0 0
\(40\) −49.1650 −0.194342
\(41\) 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 \(-0.399896\pi\)
0.309329 + 0.950955i \(0.399896\pi\)
\(44\) 478.781 1.64043
\(45\) 0 0
\(46\) 34.4305 0.110359
\(47\) 389.362 1.20839 0.604194 0.796837i \(-0.293495\pi\)
0.604194 + 0.796837i \(0.293495\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −7.34774 −0.0207825
\(51\) 0 0
\(52\) 289.173 0.771176
\(53\) −314.935 −0.816220 −0.408110 0.912933i \(-0.633812\pi\)
−0.408110 + 0.912933i \(0.633812\pi\)
\(54\) 0 0
\(55\) 749.950 1.83860
\(56\) 0 0
\(57\) 0 0
\(58\) −15.1657 −0.0343338
\(59\) 844.526 1.86352 0.931762 0.363068i \(-0.118271\pi\)
0.931762 + 0.363068i \(0.118271\pi\)
\(60\) 0 0
\(61\) 338.538 0.710579 0.355290 0.934756i \(-0.384382\pi\)
0.355290 + 0.934756i \(0.384382\pi\)
\(62\) −0.290084 −0.000594204 0
\(63\) 0 0
\(64\) −488.520 −0.954141
\(65\) 452.954 0.864339
\(66\) 0 0
\(67\) −971.550 −1.77155 −0.885774 0.464117i \(-0.846372\pi\)
−0.885774 + 0.464117i \(0.846372\pi\)
\(68\) 386.969 0.690102
\(69\) 0 0
\(70\) 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 \(-0.692809\pi\)
−0.569361 + 0.822088i \(0.692809\pi\)
\(74\) −17.2477 −0.0270946
\(75\) 0 0
\(76\) −401.289 −0.605671
\(77\) 0 0
\(78\) 0 0
\(79\) −486.884 −0.693402 −0.346701 0.937976i \(-0.612698\pi\)
−0.346701 + 0.937976i \(0.612698\pi\)
\(80\) −777.496 −1.08658
\(81\) 0 0
\(82\) −76.4348 −0.102937
\(83\) 605.688 0.800999 0.400499 0.916297i \(-0.368837\pi\)
0.400499 + 0.916297i \(0.368837\pi\)
\(84\) 0 0
\(85\) 606.139 0.773470
\(86\) −43.2743 −0.0542604
\(87\) 0 0
\(88\) −238.465 −0.288869
\(89\) 218.069 0.259722 0.129861 0.991532i \(-0.458547\pi\)
0.129861 + 0.991532i \(0.458547\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1101.80 1.24859
\(93\) 0 0
\(94\) −96.5897 −0.105984
\(95\) −628.569 −0.678840
\(96\) 0 0
\(97\) 782.288 0.818859 0.409429 0.912342i \(-0.365728\pi\)
0.409429 + 0.912342i \(0.365728\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −235.132 −0.235132
\(101\) −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 \(0.477256\pi\)
\(104\) −144.028 −0.135799
\(105\) 0 0
\(106\) 78.1265 0.0715879
\(107\) −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 \(0.295816\pi\)
\(110\) −186.042 −0.161258
\(111\) 0 0
\(112\) 0 0
\(113\) −1048.55 −0.872917 −0.436459 0.899724i \(-0.643767\pi\)
−0.436459 + 0.899724i \(0.643767\pi\)
\(114\) 0 0
\(115\) 1725.83 1.39943
\(116\) −485.313 −0.388450
\(117\) 0 0
\(118\) −209.503 −0.163444
\(119\) 0 0
\(120\) 0 0
\(121\) 2306.48 1.73289
\(122\) −83.9817 −0.0623225
\(123\) 0 0
\(124\) −9.28286 −0.00672279
\(125\) 1186.02 0.848647
\(126\) 0 0
\(127\) 488.408 0.341254 0.170627 0.985336i \(-0.445421\pi\)
0.170627 + 0.985336i \(0.445421\pi\)
\(128\) 498.326 0.344111
\(129\) 0 0
\(130\) −112.365 −0.0758083
\(131\) −1854.23 −1.23668 −0.618338 0.785912i \(-0.712194\pi\)
−0.618338 + 0.785912i \(0.712194\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 241.014 0.155377
\(135\) 0 0
\(136\) −192.737 −0.121522
\(137\) 511.115 0.318741 0.159370 0.987219i \(-0.449054\pi\)
0.159370 + 0.987219i \(0.449054\pi\)
\(138\) 0 0
\(139\) −2266.10 −1.38279 −0.691397 0.722475i \(-0.743005\pi\)
−0.691397 + 0.722475i \(0.743005\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −24.4276 −0.0144360
\(143\) 2196.96 1.28475
\(144\) 0 0
\(145\) −760.183 −0.435378
\(146\) 176.189 0.0998735
\(147\) 0 0
\(148\) −551.936 −0.306546
\(149\) −1507.90 −0.829074 −0.414537 0.910033i \(-0.636056\pi\)
−0.414537 + 0.910033i \(0.636056\pi\)
\(150\) 0 0
\(151\) 1591.83 0.857887 0.428943 0.903331i \(-0.358886\pi\)
0.428943 + 0.903331i \(0.358886\pi\)
\(152\) 199.869 0.106655
\(153\) 0 0
\(154\) 0 0
\(155\) −14.5404 −0.00753494
\(156\) 0 0
\(157\) −1164.16 −0.591784 −0.295892 0.955221i \(-0.595617\pi\)
−0.295892 + 0.955221i \(0.595617\pi\)
\(158\) 120.782 0.0608160
\(159\) 0 0
\(160\) 586.195 0.289642
\(161\) 0 0
\(162\) 0 0
\(163\) −1155.88 −0.555432 −0.277716 0.960663i \(-0.589577\pi\)
−0.277716 + 0.960663i \(0.589577\pi\)
\(164\) −2445.96 −1.16462
\(165\) 0 0
\(166\) −150.254 −0.0702529
\(167\) −2890.61 −1.33941 −0.669707 0.742626i \(-0.733580\pi\)
−0.669707 + 0.742626i \(0.733580\pi\)
\(168\) 0 0
\(169\) −870.082 −0.396032
\(170\) −150.366 −0.0678385
\(171\) 0 0
\(172\) −1384.81 −0.613898
\(173\) 1894.94 0.832770 0.416385 0.909188i \(-0.363297\pi\)
0.416385 + 0.909188i \(0.363297\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3771.09 −1.61509
\(177\) 0 0
\(178\) −54.0967 −0.0227793
\(179\) 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 \(-0.525106\pi\)
−0.0787917 + 0.996891i \(0.525106\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −548.769 −0.219868
\(185\) −864.539 −0.343579
\(186\) 0 0
\(187\) 2939.95 1.14968
\(188\) −3090.93 −1.19909
\(189\) 0 0
\(190\) 155.930 0.0595388
\(191\) −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 \(0.462504\pi\)
\(194\) −194.064 −0.0718194
\(195\) 0 0
\(196\) 0 0
\(197\) 1250.23 0.452158 0.226079 0.974109i \(-0.427409\pi\)
0.226079 + 0.974109i \(0.427409\pi\)
\(198\) 0 0
\(199\) 1092.24 0.389081 0.194541 0.980894i \(-0.437678\pi\)
0.194541 + 0.980894i \(0.437678\pi\)
\(200\) 117.112 0.0414052
\(201\) 0 0
\(202\) 77.3105 0.0269285
\(203\) 0 0
\(204\) 0 0
\(205\) −3831.29 −1.30531
\(206\) −37.0267 −0.0125232
\(207\) 0 0
\(208\) −2277.66 −0.759265
\(209\) −3048.75 −1.00902
\(210\) 0 0
\(211\) −3620.05 −1.18111 −0.590556 0.806997i \(-0.701091\pi\)
−0.590556 + 0.806997i \(0.701091\pi\)
\(212\) 2500.10 0.809941
\(213\) 0 0
\(214\) 211.238 0.0674763
\(215\) −2169.13 −0.688061
\(216\) 0 0
\(217\) 0 0
\(218\) −337.844 −0.104962
\(219\) 0 0
\(220\) −5953.45 −1.82446
\(221\) 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
\(225\) 0 0
\(226\) 260.117 0.0765607
\(227\) 2279.52 0.666506 0.333253 0.942837i \(-0.391854\pi\)
0.333253 + 0.942837i \(0.391854\pi\)
\(228\) 0 0
\(229\) −5412.67 −1.56192 −0.780960 0.624582i \(-0.785270\pi\)
−0.780960 + 0.624582i \(0.785270\pi\)
\(230\) −428.130 −0.122739
\(231\) 0 0
\(232\) 241.719 0.0684035
\(233\) −1138.37 −0.320073 −0.160036 0.987111i \(-0.551161\pi\)
−0.160036 + 0.987111i \(0.551161\pi\)
\(234\) 0 0
\(235\) −4841.56 −1.34395
\(236\) −6704.24 −1.84919
\(237\) 0 0
\(238\) 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 \(-0.359519\pi\)
0.427147 + 0.904182i \(0.359519\pi\)
\(242\) −572.173 −0.151986
\(243\) 0 0
\(244\) −2687.47 −0.705113
\(245\) 0 0
\(246\) 0 0
\(247\) −1841.38 −0.474349
\(248\) 4.62349 0.00118384
\(249\) 0 0
\(250\) −294.218 −0.0744320
\(251\) 239.608 0.0602546 0.0301273 0.999546i \(-0.490409\pi\)
0.0301273 + 0.999546i \(0.490409\pi\)
\(252\) 0 0
\(253\) 8370.78 2.08010
\(254\) −121.160 −0.0299302
\(255\) 0 0
\(256\) 3784.54 0.923960
\(257\) 699.117 0.169688 0.0848439 0.996394i \(-0.472961\pi\)
0.0848439 + 0.996394i \(0.472961\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3595.76 −0.857690
\(261\) 0 0
\(262\) 459.982 0.108465
\(263\) 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
\(267\) 0 0
\(268\) 7712.61 1.75792
\(269\) −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 \(0.580297\pi\)
\(272\) −3047.94 −0.679443
\(273\) 0 0
\(274\) −126.793 −0.0279557
\(275\) −1786.39 −0.391721
\(276\) 0 0
\(277\) 7307.69 1.58511 0.792557 0.609797i \(-0.208749\pi\)
0.792557 + 0.609797i \(0.208749\pi\)
\(278\) 562.157 0.121280
\(279\) 0 0
\(280\) 0 0
\(281\) −2730.61 −0.579696 −0.289848 0.957073i \(-0.593605\pi\)
−0.289848 + 0.957073i \(0.593605\pi\)
\(282\) 0 0
\(283\) 1769.85 0.371755 0.185878 0.982573i \(-0.440487\pi\)
0.185878 + 0.982573i \(0.440487\pi\)
\(284\) −781.698 −0.163328
\(285\) 0 0
\(286\) −545.004 −0.112681
\(287\) 0 0
\(288\) 0 0
\(289\) −2536.81 −0.516347
\(290\) 188.580 0.0381855
\(291\) 0 0
\(292\) 5638.17 1.12996
\(293\) 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
\(296\) 274.901 0.0539807
\(297\) 0 0
\(298\) 374.068 0.0727153
\(299\) 5055.78 0.977870
\(300\) 0 0
\(301\) 0 0
\(302\) −394.887 −0.0752424
\(303\) 0 0
\(304\) 3160.73 0.596317
\(305\) −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
\(309\) 0 0
\(310\) 3.60707 0.000660865 0
\(311\) 1193.71 0.217650 0.108825 0.994061i \(-0.465291\pi\)
0.108825 + 0.994061i \(0.465291\pi\)
\(312\) 0 0
\(313\) 8846.04 1.59747 0.798734 0.601684i \(-0.205503\pi\)
0.798734 + 0.601684i \(0.205503\pi\)
\(314\) 288.795 0.0519034
\(315\) 0 0
\(316\) 3865.11 0.688068
\(317\) −6081.43 −1.07750 −0.538750 0.842466i \(-0.681103\pi\)
−0.538750 + 0.842466i \(0.681103\pi\)
\(318\) 0 0
\(319\) −3687.11 −0.647143
\(320\) 6074.55 1.06118
\(321\) 0 0
\(322\) 0 0
\(323\) −2464.12 −0.424480
\(324\) 0 0
\(325\) −1078.94 −0.184151
\(326\) 286.741 0.0487151
\(327\) 0 0
\(328\) 1218.25 0.205082
\(329\) 0 0
\(330\) 0 0
\(331\) 3053.30 0.507022 0.253511 0.967333i \(-0.418415\pi\)
0.253511 + 0.967333i \(0.418415\pi\)
\(332\) −4808.23 −0.794837
\(333\) 0 0
\(334\) 717.079 0.117475
\(335\) 12080.8 1.97029
\(336\) 0 0
\(337\) 3865.80 0.624877 0.312438 0.949938i \(-0.398854\pi\)
0.312438 + 0.949938i \(0.398854\pi\)
\(338\) 215.843 0.0347346
\(339\) 0 0
\(340\) −4811.81 −0.767521
\(341\) −70.5255 −0.0111999
\(342\) 0 0
\(343\) 0 0
\(344\) 689.726 0.108103
\(345\) 0 0
\(346\) −470.080 −0.0730395
\(347\) 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 \(-0.410778\pi\)
0.276643 + 0.960973i \(0.410778\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2843.22 0.430523
\(353\) 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
\(356\) −1731.13 −0.257724
\(357\) 0 0
\(358\) −1063.85 −0.157057
\(359\) 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
\(362\) 95.1932 0.0138211
\(363\) 0 0
\(364\) 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 \(0.518684\pi\)
\(368\) −8678.25 −1.22931
\(369\) 0 0
\(370\) 214.468 0.0301342
\(371\) 0 0
\(372\) 0 0
\(373\) 1333.85 0.185159 0.0925793 0.995705i \(-0.470489\pi\)
0.0925793 + 0.995705i \(0.470489\pi\)
\(374\) −729.320 −0.100835
\(375\) 0 0
\(376\) 1539.49 0.211152
\(377\) −2226.94 −0.304226
\(378\) 0 0
\(379\) −1338.29 −0.181380 −0.0906902 0.995879i \(-0.528907\pi\)
−0.0906902 + 0.995879i \(0.528907\pi\)
\(380\) 4989.87 0.673618
\(381\) 0 0
\(382\) 95.5847 0.0128025
\(383\) 353.376 0.0471453 0.0235727 0.999722i \(-0.492496\pi\)
0.0235727 + 0.999722i \(0.492496\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −156.341 −0.0206154
\(387\) 0 0
\(388\) −6210.16 −0.812560
\(389\) −11737.2 −1.52982 −0.764908 0.644139i \(-0.777216\pi\)
−0.764908 + 0.644139i \(0.777216\pi\)
\(390\) 0 0
\(391\) 6765.59 0.875066
\(392\) 0 0
\(393\) 0 0
\(394\) −310.147 −0.0396573
\(395\) 6054.21 0.771191
\(396\) 0 0
\(397\) −13281.4 −1.67903 −0.839516 0.543335i \(-0.817161\pi\)
−0.839516 + 0.543335i \(0.817161\pi\)
\(398\) −270.955 −0.0341250
\(399\) 0 0
\(400\) 1852.01 0.231501
\(401\) 7482.36 0.931798 0.465899 0.884838i \(-0.345731\pi\)
0.465899 + 0.884838i \(0.345731\pi\)
\(402\) 0 0
\(403\) −42.5959 −0.00526514
\(404\) 2473.99 0.304667
\(405\) 0 0
\(406\) 0 0
\(407\) −4193.27 −0.510694
\(408\) 0 0
\(409\) 13796.6 1.66797 0.833983 0.551791i \(-0.186055\pi\)
0.833983 + 0.551791i \(0.186055\pi\)
\(410\) 950.436 0.114485
\(411\) 0 0
\(412\) −1184.88 −0.141686
\(413\) 0 0
\(414\) 0 0
\(415\) −7531.49 −0.890859
\(416\) 1717.24 0.202391
\(417\) 0 0
\(418\) 756.308 0.0884982
\(419\) 9497.56 1.10737 0.553683 0.832728i \(-0.313222\pi\)
0.553683 + 0.832728i \(0.313222\pi\)
\(420\) 0 0
\(421\) 624.367 0.0722797 0.0361399 0.999347i \(-0.488494\pi\)
0.0361399 + 0.999347i \(0.488494\pi\)
\(422\) 898.032 0.103591
\(423\) 0 0
\(424\) −1245.22 −0.142625
\(425\) −1443.83 −0.164791
\(426\) 0 0
\(427\) 0 0
\(428\) 6759.75 0.763423
\(429\) 0 0
\(430\) 538.099 0.0603475
\(431\) 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 \(-0.215179\pi\)
0.780079 + 0.625681i \(0.215179\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10811.2 −1.18753
\(437\) −7015.95 −0.768006
\(438\) 0 0
\(439\) 16368.8 1.77960 0.889798 0.456356i \(-0.150845\pi\)
0.889798 + 0.456356i \(0.150845\pi\)
\(440\) 2965.22 0.321275
\(441\) 0 0
\(442\) −440.494 −0.0474031
\(443\) 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
\(446\) −45.6067 −0.00484201
\(447\) 0 0
\(448\) 0 0
\(449\) 12400.9 1.30342 0.651709 0.758469i \(-0.274052\pi\)
0.651709 + 0.758469i \(0.274052\pi\)
\(450\) 0 0
\(451\) −18582.9 −1.94021
\(452\) 8323.90 0.866202
\(453\) 0 0
\(454\) −565.484 −0.0584570
\(455\) 0 0
\(456\) 0 0
\(457\) 9925.58 1.01597 0.507986 0.861365i \(-0.330390\pi\)
0.507986 + 0.861365i \(0.330390\pi\)
\(458\) 1342.73 0.136991
\(459\) 0 0
\(460\) −13700.4 −1.38866
\(461\) −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 \(-0.162899\pi\)
0.871883 + 0.489714i \(0.162899\pi\)
\(464\) 3822.54 0.382451
\(465\) 0 0
\(466\) 282.397 0.0280725
\(467\) 2108.06 0.208886 0.104443 0.994531i \(-0.466694\pi\)
0.104443 + 0.994531i \(0.466694\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1201.05 0.117873
\(471\) 0 0
\(472\) 3339.16 0.325630
\(473\) −10520.9 −1.02273
\(474\) 0 0
\(475\) 1497.26 0.144629
\(476\) 0 0
\(477\) 0 0
\(478\) −1544.59 −0.147798
\(479\) −2450.04 −0.233706 −0.116853 0.993149i \(-0.537281\pi\)
−0.116853 + 0.993149i \(0.537281\pi\)
\(480\) 0 0
\(481\) −2532.65 −0.240081
\(482\) −792.887 −0.0749274
\(483\) 0 0
\(484\) −18309.9 −1.71956
\(485\) −9727.44 −0.910722
\(486\) 0 0
\(487\) 645.236 0.0600379 0.0300189 0.999549i \(-0.490443\pi\)
0.0300189 + 0.999549i \(0.490443\pi\)
\(488\) 1338.54 0.124166
\(489\) 0 0
\(490\) 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
\(494\) 456.794 0.0416035
\(495\) 0 0
\(496\) 73.1159 0.00661896
\(497\) 0 0
\(498\) 0 0
\(499\) −44.0209 −0.00394919 −0.00197459 0.999998i \(-0.500629\pi\)
−0.00197459 + 0.999998i \(0.500629\pi\)
\(500\) −9415.17 −0.842119
\(501\) 0 0
\(502\) −59.4399 −0.00528473
\(503\) 8290.27 0.734880 0.367440 0.930047i \(-0.380234\pi\)
0.367440 + 0.930047i \(0.380234\pi\)
\(504\) 0 0
\(505\) 3875.19 0.341473
\(506\) −2076.56 −0.182439
\(507\) 0 0
\(508\) −3877.21 −0.338629
\(509\) 6915.04 0.602168 0.301084 0.953598i \(-0.402651\pi\)
0.301084 + 0.953598i \(0.402651\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −4925.45 −0.425148
\(513\) 0 0
\(514\) −173.431 −0.0148827
\(515\) −1855.96 −0.158803
\(516\) 0 0
\(517\) −23483.0 −1.99764
\(518\) 0 0
\(519\) 0 0
\(520\) 1790.93 0.151033
\(521\) 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 \(0.363640\pi\)
\(524\) 14719.7 1.22716
\(525\) 0 0
\(526\) −227.988 −0.0188988
\(527\) −57.0014 −0.00471161
\(528\) 0 0
\(529\) 7096.33 0.583244
\(530\) −971.472 −0.0796190
\(531\) 0 0
\(532\) 0 0
\(533\) −11223.7 −0.912104
\(534\) 0 0
\(535\) 10588.3 0.855649
\(536\) −3841.40 −0.309558
\(537\) 0 0
\(538\) 689.435 0.0552484
\(539\) 0 0
\(540\) 0 0
\(541\) 9286.17 0.737973 0.368987 0.929435i \(-0.379705\pi\)
0.368987 + 0.929435i \(0.379705\pi\)
\(542\) 552.452 0.0437820
\(543\) 0 0
\(544\) 2298.00 0.181114
\(545\) −16934.4 −1.33099
\(546\) 0 0
\(547\) −16821.6 −1.31488 −0.657438 0.753508i \(-0.728360\pi\)
−0.657438 + 0.753508i \(0.728360\pi\)
\(548\) −4057.47 −0.316289
\(549\) 0 0
\(550\) 443.153 0.0343566
\(551\) 3090.35 0.238935
\(552\) 0 0
\(553\) 0 0
\(554\) −1812.83 −0.139025
\(555\) 0 0
\(556\) 17989.4 1.37216
\(557\) −1805.94 −0.137379 −0.0686897 0.997638i \(-0.521882\pi\)
−0.0686897 + 0.997638i \(0.521882\pi\)
\(558\) 0 0
\(559\) −6354.40 −0.480792
\(560\) 0 0
\(561\) 0 0
\(562\) 677.388 0.0508432
\(563\) 12214.9 0.914381 0.457190 0.889369i \(-0.348856\pi\)
0.457190 + 0.889369i \(0.348856\pi\)
\(564\) 0 0
\(565\) 13038.3 0.970845
\(566\) −439.050 −0.0326054
\(567\) 0 0
\(568\) 389.338 0.0287610
\(569\) −4283.77 −0.315615 −0.157808 0.987470i \(-0.550443\pi\)
−0.157808 + 0.987470i \(0.550443\pi\)
\(570\) 0 0
\(571\) 6359.94 0.466121 0.233060 0.972462i \(-0.425126\pi\)
0.233060 + 0.972462i \(0.425126\pi\)
\(572\) −17440.5 −1.27487
\(573\) 0 0
\(574\) 0 0
\(575\) −4110.95 −0.298153
\(576\) 0 0
\(577\) −14468.7 −1.04392 −0.521959 0.852971i \(-0.674799\pi\)
−0.521959 + 0.852971i \(0.674799\pi\)
\(578\) 629.313 0.0452871
\(579\) 0 0
\(580\) 6034.68 0.432028
\(581\) 0 0
\(582\) 0 0
\(583\) 18994.2 1.34933
\(584\) −2808.19 −0.198979
\(585\) 0 0
\(586\) −2041.34 −0.143903
\(587\) −11132.6 −0.782777 −0.391388 0.920226i \(-0.628005\pi\)
−0.391388 + 0.920226i \(0.628005\pi\)
\(588\) 0 0
\(589\) 59.1108 0.00413517
\(590\) 2605.09 0.181779
\(591\) 0 0
\(592\) 4347.29 0.301812
\(593\) −19775.6 −1.36946 −0.684728 0.728799i \(-0.740079\pi\)
−0.684728 + 0.728799i \(0.740079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 11970.4 0.822696
\(597\) 0 0
\(598\) −1254.20 −0.0857657
\(599\) 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 \(-0.728683\pi\)
−0.658204 + 0.752840i \(0.728683\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −12636.6 −0.851288
\(605\) −28680.2 −1.92730
\(606\) 0 0
\(607\) −14596.7 −0.976051 −0.488025 0.872829i \(-0.662283\pi\)
−0.488025 + 0.872829i \(0.662283\pi\)
\(608\) −2383.04 −0.158956
\(609\) 0 0
\(610\) 1044.28 0.0693142
\(611\) −14183.2 −0.939104
\(612\) 0 0
\(613\) 1979.80 0.130446 0.0652229 0.997871i \(-0.479224\pi\)
0.0652229 + 0.997871i \(0.479224\pi\)
\(614\) 1493.30 0.0981509
\(615\) 0 0
\(616\) 0 0
\(617\) −16262.4 −1.06110 −0.530551 0.847653i \(-0.678015\pi\)
−0.530551 + 0.847653i \(0.678015\pi\)
\(618\) 0 0
\(619\) 12021.0 0.780555 0.390278 0.920697i \(-0.372379\pi\)
0.390278 + 0.920697i \(0.372379\pi\)
\(620\) 115.429 0.00747698
\(621\) 0 0
\(622\) −296.127 −0.0190894
\(623\) 0 0
\(624\) 0 0
\(625\) −18450.1 −1.18081
\(626\) −2194.45 −0.140109
\(627\) 0 0
\(628\) 9241.64 0.587232
\(629\) −3389.16 −0.214841
\(630\) 0 0
\(631\) 25347.6 1.59916 0.799582 0.600557i \(-0.205055\pi\)
0.799582 + 0.600557i \(0.205055\pi\)
\(632\) −1925.08 −0.121164
\(633\) 0 0
\(634\) 1508.63 0.0945039
\(635\) −6073.16 −0.379537
\(636\) 0 0
\(637\) 0 0
\(638\) 914.669 0.0567588
\(639\) 0 0
\(640\) −6196.49 −0.382715
\(641\) 5111.60 0.314971 0.157485 0.987521i \(-0.449661\pi\)
0.157485 + 0.987521i \(0.449661\pi\)
\(642\) 0 0
\(643\) 10931.3 0.670435 0.335217 0.942141i \(-0.391190\pi\)
0.335217 + 0.942141i \(0.391190\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 611.278 0.0372297
\(647\) −18406.1 −1.11842 −0.559211 0.829025i \(-0.688896\pi\)
−0.559211 + 0.829025i \(0.688896\pi\)
\(648\) 0 0
\(649\) −50934.7 −3.08068
\(650\) 267.655 0.0161512
\(651\) 0 0
\(652\) 9175.91 0.551160
\(653\) −19921.4 −1.19385 −0.596926 0.802296i \(-0.703611\pi\)
−0.596926 + 0.802296i \(0.703611\pi\)
\(654\) 0 0
\(655\) 23056.6 1.37541
\(656\) 19265.5 1.14663
\(657\) 0 0
\(658\) 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 \(-0.774812\pi\)
−0.760023 + 0.649896i \(0.774812\pi\)
\(662\) −757.437 −0.0444692
\(663\) 0 0
\(664\) 2394.82 0.139965
\(665\) 0 0
\(666\) 0 0
\(667\) −8485.00 −0.492564
\(668\) 22947.0 1.32911
\(669\) 0 0
\(670\) −2996.92 −0.172807
\(671\) −20417.7 −1.17469
\(672\) 0 0
\(673\) −16275.0 −0.932178 −0.466089 0.884738i \(-0.654337\pi\)
−0.466089 + 0.884738i \(0.654337\pi\)
\(674\) −958.996 −0.0548059
\(675\) 0 0
\(676\) 6907.11 0.392985
\(677\) −26271.8 −1.49144 −0.745720 0.666259i \(-0.767895\pi\)
−0.745720 + 0.666259i \(0.767895\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2396.60 0.135155
\(681\) 0 0
\(682\) 17.4954 0.000982306 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
\(687\) 0 0
\(688\) 10907.3 0.604417
\(689\) 11472.1 0.634328
\(690\) 0 0
\(691\) 24485.3 1.34799 0.673997 0.738734i \(-0.264576\pi\)
0.673997 + 0.738734i \(0.264576\pi\)
\(692\) −15042.9 −0.826364
\(693\) 0 0
\(694\) −24.7062 −0.00135135
\(695\) 28178.1 1.53792
\(696\) 0 0
\(697\) −15019.4 −0.816214
\(698\) −894.880 −0.0485268
\(699\) 0 0
\(700\) 0 0
\(701\) −778.448 −0.0419423 −0.0209712 0.999780i \(-0.506676\pi\)
−0.0209712 + 0.999780i \(0.506676\pi\)
\(702\) 0 0
\(703\) 3514.58 0.188556
\(704\) 29463.4 1.57733
\(705\) 0 0
\(706\) −1768.93 −0.0942985
\(707\) 0 0
\(708\) 0 0
\(709\) −24172.0 −1.28039 −0.640197 0.768211i \(-0.721147\pi\)
−0.640197 + 0.768211i \(0.721147\pi\)
\(710\) 303.747 0.0160555
\(711\) 0 0
\(712\) 862.218 0.0453834
\(713\) −162.297 −0.00852466
\(714\) 0 0
\(715\) −27318.3 −1.42888
\(716\) −34044.0 −1.77693
\(717\) 0 0
\(718\) −1612.54 −0.0838153
\(719\) −81.8835 −0.00424720 −0.00212360 0.999998i \(-0.500676\pi\)
−0.00212360 + 0.999998i \(0.500676\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1067.63 0.0550318
\(723\) 0 0
\(724\) 3046.24 0.156371
\(725\) 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
\(729\) 0 0
\(730\) −2190.85 −0.111078
\(731\) −8503.40 −0.430246
\(732\) 0 0
\(733\) 5068.94 0.255424 0.127712 0.991811i \(-0.459237\pi\)
0.127712 + 0.991811i \(0.459237\pi\)
\(734\) 204.631 0.0102903
\(735\) 0 0
\(736\) 6542.98 0.327687
\(737\) 58595.6 2.92863
\(738\) 0 0
\(739\) −38428.5 −1.91287 −0.956437 0.291939i \(-0.905700\pi\)
−0.956437 + 0.291939i \(0.905700\pi\)
\(740\) 6863.11 0.340936
\(741\) 0 0
\(742\) 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
\(746\) −330.891 −0.0162396
\(747\) 0 0
\(748\) −23338.7 −1.14084
\(749\) 0 0
\(750\) 0 0
\(751\) 8112.60 0.394185 0.197093 0.980385i \(-0.436850\pi\)
0.197093 + 0.980385i \(0.436850\pi\)
\(752\) 24345.6 1.18057
\(753\) 0 0
\(754\) 552.441 0.0266826
\(755\) −19793.7 −0.954129
\(756\) 0 0
\(757\) 3108.01 0.149224 0.0746120 0.997213i \(-0.476228\pi\)
0.0746120 + 0.997213i \(0.476228\pi\)
\(758\) 331.992 0.0159083
\(759\) 0 0
\(760\) −2485.29 −0.118620
\(761\) −7211.93 −0.343538 −0.171769 0.985137i \(-0.554948\pi\)
−0.171769 + 0.985137i \(0.554948\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 3058.77 0.144846
\(765\) 0 0
\(766\) −87.6626 −0.00413496
\(767\) −30763.5 −1.44825
\(768\) 0 0
\(769\) 7533.07 0.353250 0.176625 0.984278i \(-0.443482\pi\)
0.176625 + 0.984278i \(0.443482\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5003.00 −0.233241
\(773\) −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
\(776\) 3093.08 0.143086
\(777\) 0 0
\(778\) 2911.66 0.134175
\(779\) 15575.2 0.716355
\(780\) 0 0
\(781\) −5938.86 −0.272099
\(782\) −1678.35 −0.0767491
\(783\) 0 0
\(784\) 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 \(0.807353\pi\)
\(788\) −9924.89 −0.448680
\(789\) 0 0
\(790\) −1501.88 −0.0676386
\(791\) 0 0
\(792\) 0 0
\(793\) −12331.9 −0.552229
\(794\) 3294.75 0.147262
\(795\) 0 0
\(796\) −8670.74 −0.386088
\(797\) 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
\(800\) −1396.32 −0.0617094
\(801\) 0 0
\(802\) −1856.16 −0.0817249
\(803\) 42835.4 1.88247
\(804\) 0 0
\(805\) 0 0
\(806\) 10.5668 0.000461788 0
\(807\) 0 0
\(808\) −1232.21 −0.0536498
\(809\) −12384.6 −0.538219 −0.269110 0.963110i \(-0.586729\pi\)
−0.269110 + 0.963110i \(0.586729\pi\)
\(810\) 0 0
\(811\) −16742.4 −0.724914 −0.362457 0.932000i \(-0.618062\pi\)
−0.362457 + 0.932000i \(0.618062\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1040.23 0.0447913
\(815\) 14372.9 0.617744
\(816\) 0 0
\(817\) 8818.07 0.377607
\(818\) −3422.55 −0.146292
\(819\) 0 0
\(820\) 30414.6 1.29527
\(821\) −26456.3 −1.12464 −0.562322 0.826918i \(-0.690092\pi\)
−0.562322 + 0.826918i \(0.690092\pi\)
\(822\) 0 0
\(823\) 23098.5 0.978328 0.489164 0.872192i \(-0.337302\pi\)
0.489164 + 0.872192i \(0.337302\pi\)
\(824\) 590.148 0.0249500
\(825\) 0 0
\(826\) 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 \(-0.337173\pi\)
0.489519 + 0.871993i \(0.337173\pi\)
\(830\) 1868.35 0.0781343
\(831\) 0 0
\(832\) 17795.3 0.741514
\(833\) 0 0
\(834\) 0 0
\(835\) 35943.6 1.48968
\(836\) 24202.3 1.00126
\(837\) 0 0
\(838\) −2356.08 −0.0971234
\(839\) 16735.5 0.688645 0.344322 0.938851i \(-0.388109\pi\)
0.344322 + 0.938851i \(0.388109\pi\)
\(840\) 0 0
\(841\) −20651.6 −0.846758
\(842\) −154.888 −0.00633941
\(843\) 0 0
\(844\) 28737.6 1.17203
\(845\) 10819.1 0.440460
\(846\) 0 0
\(847\) 0 0
\(848\) −19691.9 −0.797432
\(849\) 0 0
\(850\) 358.174 0.0144532
\(851\) −9649.80 −0.388708
\(852\) 0 0
\(853\) 10294.5 0.413219 0.206609 0.978424i \(-0.433757\pi\)
0.206609 + 0.978424i \(0.433757\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3366.81 −0.134434
\(857\) −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 \(0.468913\pi\)
\(860\) 17219.5 0.682768
\(861\) 0 0
\(862\) −3323.49 −0.131321
\(863\) −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
\(866\) −3487.21 −0.136836
\(867\) 0 0
\(868\) 0 0
\(869\) 29364.7 1.14629
\(870\) 0 0
\(871\) 35390.5 1.37677
\(872\) 5384.71 0.209116
\(873\) 0 0
\(874\) 1740.46 0.0673592
\(875\) 0 0
\(876\) 0 0
\(877\) 34672.2 1.33500 0.667501 0.744609i \(-0.267364\pi\)
0.667501 + 0.744609i \(0.267364\pi\)
\(878\) −4060.65 −0.156082
\(879\) 0 0
\(880\) 46892.0 1.79628
\(881\) 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 \(-0.306034\pi\)
0.572345 + 0.820013i \(0.306034\pi\)
\(884\) −14096.1 −0.536315
\(885\) 0 0
\(886\) −292.404 −0.0110875
\(887\) 33210.7 1.25717 0.628583 0.777742i \(-0.283635\pi\)
0.628583 + 0.777742i \(0.283635\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 672.671 0.0253348
\(891\) 0 0
\(892\) −1459.44 −0.0547822
\(893\) 19682.2 0.737559
\(894\) 0 0
\(895\) −53325.7 −1.99160
\(896\) 0 0
\(897\) 0 0
\(898\) −3076.32 −0.114318
\(899\) 71.4878 0.00265211
\(900\) 0 0
\(901\) 15351.9 0.567641
\(902\) 4609.90 0.170169
\(903\) 0 0
\(904\) −4145.86 −0.152532
\(905\) 4771.56 0.175262
\(906\) 0 0
\(907\) 2497.83 0.0914433 0.0457217 0.998954i \(-0.485441\pi\)
0.0457217 + 0.998954i \(0.485441\pi\)
\(908\) −18095.9 −0.661379
\(909\) 0 0
\(910\) 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
\(914\) −2462.26 −0.0891075
\(915\) 0 0
\(916\) 42968.3 1.54990
\(917\) 0 0
\(918\) 0 0
\(919\) 6270.71 0.225083 0.112542 0.993647i \(-0.464101\pi\)
0.112542 + 0.993647i \(0.464101\pi\)
\(920\) 6823.73 0.244534
\(921\) 0 0
\(922\) 3971.69 0.141866
\(923\) −3586.95 −0.127915
\(924\) 0 0
\(925\) 2059.34 0.0732008
\(926\) −4309.61 −0.152940
\(927\) 0 0
\(928\) −2882.01 −0.101947
\(929\) −31552.6 −1.11432 −0.557161 0.830404i \(-0.688110\pi\)
−0.557161 + 0.830404i \(0.688110\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 9036.89 0.317611
\(933\) 0 0
\(934\) −522.951 −0.0183207
\(935\) −36557.2 −1.27866
\(936\) 0 0
\(937\) 22030.2 0.768084 0.384042 0.923316i \(-0.374532\pi\)
0.384042 + 0.923316i \(0.374532\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 38434.5 1.33361
\(941\) −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
\(944\) 52805.6 1.82063
\(945\) 0 0
\(946\) 2609.94 0.0897003
\(947\) 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
\(950\) −371.428 −0.0126850
\(951\) 0 0
\(952\) 0 0
\(953\) 52516.4 1.78507 0.892536 0.450976i \(-0.148924\pi\)
0.892536 + 0.450976i \(0.148924\pi\)
\(954\) 0 0
\(955\) 4791.18 0.162345
\(956\) −49427.7 −1.67218
\(957\) 0 0
\(958\) 607.786 0.0204976
\(959\) 0 0
\(960\) 0 0
\(961\) −29789.6 −0.999954
\(962\) 628.279 0.0210567
\(963\) 0 0
\(964\) −25372.9 −0.847723
\(965\) −7836.58 −0.261418
\(966\) 0 0
\(967\) 14721.6 0.489570 0.244785 0.969577i \(-0.421283\pi\)
0.244785 + 0.969577i \(0.421283\pi\)
\(968\) 9119.56 0.302803
\(969\) 0 0
\(970\) 2413.10 0.0798764
\(971\) 13772.5 0.455181 0.227590 0.973757i \(-0.426915\pi\)
0.227590 + 0.973757i \(0.426915\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −160.065 −0.00526572
\(975\) 0 0
\(976\) 21167.7 0.694223
\(977\) −24782.1 −0.811513 −0.405757 0.913981i \(-0.632992\pi\)
−0.405757 + 0.913981i \(0.632992\pi\)
\(978\) 0 0
\(979\) −13152.0 −0.429358
\(980\) 0 0
\(981\) 0 0
\(982\) 2918.84 0.0948514
\(983\) 42804.7 1.38887 0.694435 0.719556i \(-0.255655\pi\)
0.694435 + 0.719556i \(0.255655\pi\)
\(984\) 0 0
\(985\) −15546.1 −0.502883
\(986\) 739.271 0.0238775
\(987\) 0 0
\(988\) 14617.7 0.470700
\(989\) −24211.3 −0.778438
\(990\) 0 0
\(991\) 449.862 0.0144201 0.00721006 0.999974i \(-0.497705\pi\)
0.00721006 + 0.999974i \(0.497705\pi\)
\(992\) −55.1259 −0.00176436
\(993\) 0 0
\(994\) 0 0
\(995\) −13581.6 −0.432730
\(996\) 0 0
\(997\) 21473.7 0.682127 0.341063 0.940040i \(-0.389213\pi\)
0.341063 + 0.940040i \(0.389213\pi\)
\(998\) 10.9203 0.000346370 0
\(999\) 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 441.4.a.t.1.2 3
3.2 odd 2 147.4.a.m.1.2 3
7.2 even 3 441.4.e.w.361.2 6
7.3 odd 6 63.4.e.c.37.2 6
7.4 even 3 441.4.e.w.226.2 6
7.5 odd 6 63.4.e.c.46.2 6
7.6 odd 2 441.4.a.s.1.2 3
12.11 even 2 2352.4.a.cg.1.3 3
21.2 odd 6 147.4.e.n.67.2 6
21.5 even 6 21.4.e.b.4.2 6
21.11 odd 6 147.4.e.n.79.2 6
21.17 even 6 21.4.e.b.16.2 yes 6
21.20 even 2 147.4.a.l.1.2 3
84.47 odd 6 336.4.q.k.193.3 6
84.59 odd 6 336.4.q.k.289.3 6
84.83 odd 2 2352.4.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.2 6 21.5 even 6
21.4.e.b.16.2 yes 6 21.17 even 6
63.4.e.c.37.2 6 7.3 odd 6
63.4.e.c.46.2 6 7.5 odd 6
147.4.a.l.1.2 3 21.20 even 2
147.4.a.m.1.2 3 3.2 odd 2
147.4.e.n.67.2 6 21.2 odd 6
147.4.e.n.79.2 6 21.11 odd 6
336.4.q.k.193.3 6 84.47 odd 6
336.4.q.k.289.3 6 84.59 odd 6
441.4.a.s.1.2 3 7.6 odd 2
441.4.a.t.1.2 3 1.1 even 1 trivial
441.4.e.w.226.2 6 7.4 even 3
441.4.e.w.361.2 6 7.2 even 3
2352.4.a.cg.1.3 3 12.11 even 2
2352.4.a.ci.1.1 3 84.83 odd 2