Properties

Label 441.4.a.s.1.2
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
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: \(1\)
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 \(-0.312300\pi\)
0.556092 + 0.831120i \(0.312300\pi\)
\(6\) 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 \(-0.372967\pi\)
0.388577 + 0.921416i \(0.372967\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 62.5268 0.976982
\(17\) 48.7461 0.695451 0.347726 0.937596i \(-0.386954\pi\)
0.347726 + 0.937596i \(0.386954\pi\)
\(18\) 0 0
\(19\) −50.5500 −0.610366 −0.305183 0.952294i \(-0.598718\pi\)
−0.305183 + 0.952294i \(0.598718\pi\)
\(20\) −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 \(-0.501078\pi\)
−0.00338745 + 0.999994i \(0.501078\pi\)
\(32\) −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 \(-0.699622\pi\)
−0.586823 + 0.809715i \(0.699622\pi\)
\(42\) 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 \(-0.706505\pi\)
−0.604194 + 0.796837i \(0.706505\pi\)
\(48\) 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 \(-0.881729\pi\)
−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 \(0.615618\pi\)
\(62\) 0.290084 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 \(-0.307191\pi\)
0.569361 + 0.822088i \(0.307191\pi\)
\(74\) −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 \(-0.631163\pi\)
−0.400499 + 0.916297i \(0.631163\pi\)
\(84\) 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 \(-0.541453\pi\)
−0.129861 + 0.991532i \(0.541453\pi\)
\(90\) 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 \(-0.634272\pi\)
−0.409429 + 0.912342i \(0.634272\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −235.132 −0.235132
\(101\) 311.646 0.307029 0.153514 0.988146i \(-0.450941\pi\)
0.153514 + 0.988146i \(0.450941\pi\)
\(102\) 0 0
\(103\) −149.258 −0.142784 −0.0713922 0.997448i \(-0.522744\pi\)
−0.0713922 + 0.997448i \(0.522744\pi\)
\(104\) 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 \(-0.287806\pi\)
0.618338 + 0.785912i \(0.287806\pi\)
\(132\) 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 \(-0.256995\pi\)
0.691397 + 0.722475i \(0.256995\pi\)
\(140\) 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 \(-0.404383\pi\)
0.295892 + 0.955221i \(0.404383\pi\)
\(158\) 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 \(-0.266420\pi\)
0.669707 + 0.742626i \(0.266420\pi\)
\(168\) 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 \(-0.636703\pi\)
−0.416385 + 0.909188i \(0.636703\pi\)
\(174\) 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 \(-0.474894\pi\)
0.0787917 + 0.996891i \(0.474894\pi\)
\(182\) 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 \(-0.562322\pi\)
−0.194541 + 0.980894i \(0.562322\pi\)
\(200\) 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 \(-0.508788\pi\)
−0.0276034 + 0.999619i \(0.508788\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 260.117 0.0765607
\(227\) −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 \(0.214730\pi\)
\(230\) 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 \(-0.640481\pi\)
−0.427147 + 0.904182i \(0.640481\pi\)
\(242\) −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 \(-0.509591\pi\)
−0.0301273 + 0.999546i \(0.509591\pi\)
\(252\) 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 \(-0.527039\pi\)
−0.0848439 + 0.996394i \(0.527039\pi\)
\(258\) 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 \(-0.398008\pi\)
0.314961 + 0.949104i \(0.398008\pi\)
\(270\) 0 0
\(271\) 2226.98 0.499186 0.249593 0.968351i \(-0.419703\pi\)
0.249593 + 0.968351i \(0.419703\pi\)
\(272\) 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 \(-0.559513\pi\)
−0.185878 + 0.982573i \(0.559513\pi\)
\(284\) −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 \(-0.806228\pi\)
−0.820362 + 0.571844i \(0.806228\pi\)
\(294\) 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 \(-0.310977\pi\)
0.559541 + 0.828803i \(0.310977\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3.60707 0.000660865 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 \(0.794497\pi\)
\(314\) −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 \(-0.589222\pi\)
−0.276643 + 0.960973i \(0.589222\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2843.22 0.430523
\(353\) −7130.73 −1.07516 −0.537579 0.843214i \(-0.680661\pi\)
−0.537579 + 0.843214i \(0.680661\pi\)
\(354\) 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 \(-0.481316\pi\)
0.0586631 + 0.998278i \(0.481316\pi\)
\(368\) −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 \(-0.507504\pi\)
−0.0235727 + 0.999722i \(0.507504\pi\)
\(384\) 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 \(-0.182839\pi\)
0.839516 + 0.543335i \(0.182839\pi\)
\(398\) 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 \(-0.813945\pi\)
−0.833983 + 0.551791i \(0.813945\pi\)
\(410\) 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 \(-0.686778\pi\)
−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 \(0.488494\pi\)
\(422\) 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 \(-0.784821\pi\)
−0.780079 + 0.625681i \(0.784821\pi\)
\(434\) 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 \(-0.849155\pi\)
−0.889798 + 0.456356i \(0.849155\pi\)
\(440\) −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 \(-0.200142\pi\)
0.808755 + 0.588146i \(0.200142\pi\)
\(462\) 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 \(-0.533306\pi\)
−0.104443 + 0.994531i \(0.533306\pi\)
\(468\) 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 \(-0.462719\pi\)
0.116853 + 0.993149i \(0.462719\pi\)
\(480\) 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 \(-0.619766\pi\)
−0.367440 + 0.930047i \(0.619766\pi\)
\(504\) 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 \(-0.597349\pi\)
−0.301084 + 0.953598i \(0.597349\pi\)
\(510\) 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 \(-0.690496\pi\)
−0.563371 + 0.826204i \(0.690496\pi\)
\(522\) 0 0
\(523\) −9936.99 −0.830811 −0.415406 0.909636i \(-0.636360\pi\)
−0.415406 + 0.909636i \(0.636360\pi\)
\(524\) −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 \(-0.651144\pi\)
−0.457190 + 0.889369i \(0.651144\pi\)
\(564\) 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 \(-0.325201\pi\)
0.521959 + 0.852971i \(0.325201\pi\)
\(578\) 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 \(-0.371995\pi\)
0.391388 + 0.920226i \(0.371995\pi\)
\(588\) 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 \(-0.259921\pi\)
0.684728 + 0.728799i \(0.259921\pi\)
\(594\) 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 \(-0.271317\pi\)
0.658204 + 0.752840i \(0.271317\pi\)
\(602\) 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 \(-0.337717\pi\)
0.488025 + 0.872829i \(0.337717\pi\)
\(608\) 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 \(-0.627621\pi\)
−0.390278 + 0.920697i \(0.627621\pi\)
\(620\) 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 \(-0.608810\pi\)
−0.335217 + 0.942141i \(0.608810\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 611.278 0.0372297
\(647\) 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
\(650\) −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 \(-0.225188\pi\)
0.760023 + 0.649896i \(0.225188\pi\)
\(662\) −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 \(-0.232105\pi\)
0.745720 + 0.666259i \(0.232105\pi\)
\(678\) 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 \(-0.735424\pi\)
−0.673997 + 0.738734i \(0.735424\pi\)
\(692\) 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 \(-0.499324\pi\)
0.00212360 + 0.999998i \(0.499324\pi\)
\(720\) 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 \(-0.811710\pi\)
−0.830088 + 0.557632i \(0.811710\pi\)
\(728\) 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 \(-0.540763\pi\)
−0.127712 + 0.991811i \(0.540763\pi\)
\(734\) −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 \(-0.445052\pi\)
0.171769 + 0.985137i \(0.445052\pi\)
\(762\) 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 \(-0.556518\pi\)
−0.176625 + 0.984278i \(0.556518\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5003.00 −0.233241
\(773\) 24832.6 1.15546 0.577728 0.816229i \(-0.303940\pi\)
0.577728 + 0.816229i \(0.303940\pi\)
\(774\) 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 \(-0.192647\pi\)
0.822378 + 0.568941i \(0.192647\pi\)
\(788\) −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 \(-0.748459\pi\)
−0.703675 + 0.710522i \(0.748459\pi\)
\(798\) 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 \(-0.381938\pi\)
0.362457 + 0.932000i \(0.381938\pi\)
\(812\) 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 \(-0.662827\pi\)
−0.489519 + 0.871993i \(0.662827\pi\)
\(830\) 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 \(-0.611891\pi\)
−0.344322 + 0.938851i \(0.611891\pi\)
\(840\) 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 \(-0.566243\pi\)
−0.206609 + 0.978424i \(0.566243\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3366.81 −0.134434
\(857\) 32788.6 1.30693 0.653463 0.756958i \(-0.273315\pi\)
0.653463 + 0.756958i \(0.273315\pi\)
\(858\) 0 0
\(859\) −4909.76 −0.195016 −0.0975081 0.995235i \(-0.531087\pi\)
−0.0975081 + 0.995235i \(0.531087\pi\)
\(860\) −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 \(-0.785316\pi\)
−0.781051 + 0.624467i \(0.785316\pi\)
\(882\) 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 \(-0.716365\pi\)
−0.628583 + 0.777742i \(0.716365\pi\)
\(888\) 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 \(-0.311890\pi\)
0.557161 + 0.830404i \(0.311890\pi\)
\(930\) 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 \(-0.625468\pi\)
−0.384042 + 0.923316i \(0.625468\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 38434.5 1.33361
\(941\) 32538.6 1.12724 0.563618 0.826036i \(-0.309409\pi\)
0.563618 + 0.826036i \(0.309409\pi\)
\(942\) 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 \(-0.573085\pi\)
−0.227590 + 0.973757i \(0.573085\pi\)
\(972\) 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 \(-0.744345\pi\)
−0.694435 + 0.719556i \(0.744345\pi\)
\(984\) 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 \(-0.610787\pi\)
−0.341063 + 0.940040i \(0.610787\pi\)
\(998\) 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.s.1.2 3
3.2 odd 2 147.4.a.l.1.2 3
7.2 even 3 63.4.e.c.46.2 6
7.3 odd 6 441.4.e.w.226.2 6
7.4 even 3 63.4.e.c.37.2 6
7.5 odd 6 441.4.e.w.361.2 6
7.6 odd 2 441.4.a.t.1.2 3
12.11 even 2 2352.4.a.ci.1.1 3
21.2 odd 6 21.4.e.b.4.2 6
21.5 even 6 147.4.e.n.67.2 6
21.11 odd 6 21.4.e.b.16.2 yes 6
21.17 even 6 147.4.e.n.79.2 6
21.20 even 2 147.4.a.m.1.2 3
84.11 even 6 336.4.q.k.289.3 6
84.23 even 6 336.4.q.k.193.3 6
84.83 odd 2 2352.4.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.b.4.2 6 21.2 odd 6
21.4.e.b.16.2 yes 6 21.11 odd 6
63.4.e.c.37.2 6 7.4 even 3
63.4.e.c.46.2 6 7.2 even 3
147.4.a.l.1.2 3 3.2 odd 2
147.4.a.m.1.2 3 21.20 even 2
147.4.e.n.67.2 6 21.5 even 6
147.4.e.n.79.2 6 21.17 even 6
336.4.q.k.193.3 6 84.23 even 6
336.4.q.k.289.3 6 84.11 even 6
441.4.a.s.1.2 3 1.1 even 1 trivial
441.4.a.t.1.2 3 7.6 odd 2
441.4.e.w.226.2 6 7.3 odd 6
441.4.e.w.361.2 6 7.5 odd 6
2352.4.a.cg.1.3 3 84.83 odd 2
2352.4.a.ci.1.1 3 12.11 even 2