Properties

Label 147.4.a.l.1.2
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
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 \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.248072 q^{2} -3.00000 q^{3} -7.93846 q^{4} -12.4346 q^{5} -0.744216 q^{6} -3.95388 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+0.248072 q^{2} -3.00000 q^{3} -7.93846 q^{4} -12.4346 q^{5} -0.744216 q^{6} -3.95388 q^{8} +9.00000 q^{9} -3.08468 q^{10} +60.3115 q^{11} +23.8154 q^{12} +36.4269 q^{13} +37.3038 q^{15} +62.5268 q^{16} -48.7461 q^{17} +2.23265 q^{18} -50.5500 q^{19} +98.7116 q^{20} +14.9616 q^{22} +138.792 q^{23} +11.8617 q^{24} +29.6194 q^{25} +9.03649 q^{26} -27.0000 q^{27} -61.1345 q^{29} +9.25403 q^{30} -1.16935 q^{31} +47.1422 q^{32} -180.935 q^{33} -12.0925 q^{34} -71.4461 q^{36} +69.5268 q^{37} -12.5400 q^{38} -109.281 q^{39} +49.1650 q^{40} +308.115 q^{41} +174.443 q^{43} -478.781 q^{44} -111.911 q^{45} +34.4305 q^{46} +389.362 q^{47} -187.581 q^{48} +7.34774 q^{50} +146.238 q^{51} -289.173 q^{52} +314.935 q^{53} -6.69794 q^{54} -749.950 q^{55} +151.650 q^{57} -15.1657 q^{58} +844.526 q^{59} -296.135 q^{60} -338.538 q^{61} -0.290084 q^{62} -488.520 q^{64} -452.954 q^{65} -44.8848 q^{66} -971.550 q^{67} +386.969 q^{68} -416.377 q^{69} -98.4698 q^{71} -35.5850 q^{72} +710.235 q^{73} +17.2477 q^{74} -88.8581 q^{75} +401.289 q^{76} -27.1095 q^{78} -486.884 q^{79} -777.496 q^{80} +81.0000 q^{81} +76.4348 q^{82} +605.688 q^{83} +606.139 q^{85} +43.2743 q^{86} +183.403 q^{87} -238.465 q^{88} +218.069 q^{89} -27.7621 q^{90} -1101.80 q^{92} +3.50806 q^{93} +96.5897 q^{94} +628.569 q^{95} -141.427 q^{96} -782.288 q^{97} +542.804 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 9q^{3} + 25q^{4} + 11q^{5} - 3q^{6} + 39q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + q^{2} - 9q^{3} + 25q^{4} + 11q^{5} - 3q^{6} + 39q^{8} + 27q^{9} - 55q^{10} + 35q^{11} - 75q^{12} + 62q^{13} - 33q^{15} + 241q^{16} + 48q^{17} + 9q^{18} - 202q^{19} + 439q^{20} - 7q^{22} + 216q^{23} - 117q^{24} + 130q^{25} + 274q^{26} - 81q^{27} + 53q^{29} + 165q^{30} - 95q^{31} + 683q^{32} - 105q^{33} - 24q^{34} + 225q^{36} + 262q^{37} - 398q^{38} - 186q^{39} + 21q^{40} + 244q^{41} + 360q^{43} - 905q^{44} + 99q^{45} - 1056q^{46} - 210q^{47} - 723q^{48} - 1378q^{50} - 144q^{51} + 324q^{52} + 393q^{53} - 27q^{54} - 1031q^{55} + 606q^{57} - 1249q^{58} + 1143q^{59} - 1317q^{60} - 70q^{61} + 1059q^{62} - 399q^{64} - 472q^{65} + 21q^{66} - 628q^{67} + 1944q^{68} - 648q^{69} + 318q^{71} + 351q^{72} + 988q^{73} + 1002q^{74} - 390q^{75} - 2340q^{76} - 822q^{78} + 861q^{79} + 175q^{80} + 243q^{81} + 124q^{82} + 519q^{83} + 1800q^{85} - 3208q^{86} - 159q^{87} - 891q^{88} + 1766q^{89} - 495q^{90} - 672q^{92} + 285q^{93} - 3294q^{94} - 736q^{95} - 2049q^{96} + 19q^{97} + 315q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.248072 0.0877067 0.0438533 0.999038i \(-0.486037\pi\)
0.0438533 + 0.999038i \(0.486037\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.93846 −0.992308
\(5\) −12.4346 −1.11218 −0.556092 0.831120i \(-0.687700\pi\)
−0.556092 + 0.831120i \(0.687700\pi\)
\(6\) −0.744216 −0.0506375
\(7\) 0 0
\(8\) −3.95388 −0.174739
\(9\) 9.00000 0.333333
\(10\) −3.08468 −0.0975460
\(11\) 60.3115 1.65315 0.826573 0.562829i \(-0.190287\pi\)
0.826573 + 0.562829i \(0.190287\pi\)
\(12\) 23.8154 0.572909
\(13\) 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
\(16\) 62.5268 0.976982
\(17\) −48.7461 −0.695451 −0.347726 0.937596i \(-0.613046\pi\)
−0.347726 + 0.937596i \(0.613046\pi\)
\(18\) 2.23265 0.0292356
\(19\) −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 \(-0.283409\pi\)
0.629135 + 0.777296i \(0.283409\pi\)
\(24\) 11.8617 0.100885
\(25\) 29.6194 0.236955
\(26\) 9.03649 0.0681616
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −61.1345 −0.391462 −0.195731 0.980658i \(-0.562708\pi\)
−0.195731 + 0.980658i \(0.562708\pi\)
\(30\) 9.25403 0.0563182
\(31\) −1.16935 −0.00677490 −0.00338745 0.999994i \(-0.501078\pi\)
−0.00338745 + 0.999994i \(0.501078\pi\)
\(32\) 47.1422 0.260426
\(33\) −180.935 −0.954444
\(34\) −12.0925 −0.0609957
\(35\) 0 0
\(36\) −71.4461 −0.330769
\(37\) 69.5268 0.308923 0.154461 0.987999i \(-0.450636\pi\)
0.154461 + 0.987999i \(0.450636\pi\)
\(38\) −12.5400 −0.0535332
\(39\) −109.281 −0.448690
\(40\) 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
\(45\) −111.911 −0.370728
\(46\) 34.4305 0.110359
\(47\) 389.362 1.20839 0.604194 0.796837i \(-0.293495\pi\)
0.604194 + 0.796837i \(0.293495\pi\)
\(48\) −187.581 −0.564061
\(49\) 0 0
\(50\) 7.34774 0.0207825
\(51\) 146.238 0.401519
\(52\) −289.173 −0.771176
\(53\) 314.935 0.816220 0.408110 0.912933i \(-0.366188\pi\)
0.408110 + 0.912933i \(0.366188\pi\)
\(54\) −6.69794 −0.0168792
\(55\) −749.950 −1.83860
\(56\) 0 0
\(57\) 151.650 0.352395
\(58\) −15.1657 −0.0343338
\(59\) 844.526 1.86352 0.931762 0.363068i \(-0.118271\pi\)
0.931762 + 0.363068i \(0.118271\pi\)
\(60\) −296.135 −0.637181
\(61\) −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
\(66\) −44.8848 −0.0837111
\(67\) −971.550 −1.77155 −0.885774 0.464117i \(-0.846372\pi\)
−0.885774 + 0.464117i \(0.846372\pi\)
\(68\) 386.969 0.690102
\(69\) −416.377 −0.726463
\(70\) 0 0
\(71\) −98.4698 −0.164595 −0.0822973 0.996608i \(-0.526226\pi\)
−0.0822973 + 0.996608i \(0.526226\pi\)
\(72\) −35.5850 −0.0582462
\(73\) 710.235 1.13872 0.569361 0.822088i \(-0.307191\pi\)
0.569361 + 0.822088i \(0.307191\pi\)
\(74\) 17.2477 0.0270946
\(75\) −88.8581 −0.136806
\(76\) 401.289 0.605671
\(77\) 0 0
\(78\) −27.1095 −0.0393531
\(79\) −486.884 −0.693402 −0.346701 0.937976i \(-0.612698\pi\)
−0.346701 + 0.937976i \(0.612698\pi\)
\(80\) −777.496 −1.08658
\(81\) 81.0000 0.111111
\(82\) 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
\(87\) 183.403 0.226010
\(88\) −238.465 −0.288869
\(89\) 218.069 0.259722 0.129861 0.991532i \(-0.458547\pi\)
0.129861 + 0.991532i \(0.458547\pi\)
\(90\) −27.7621 −0.0325153
\(91\) 0 0
\(92\) −1101.80 −1.24859
\(93\) 3.50806 0.00391149
\(94\) 96.5897 0.105984
\(95\) 628.569 0.678840
\(96\) −141.427 −0.150357
\(97\) −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
\(100\) −235.132 −0.235132
\(101\) −311.646 −0.307029 −0.153514 0.988146i \(-0.549059\pi\)
−0.153514 + 0.988146i \(0.549059\pi\)
\(102\) 36.2776 0.0352159
\(103\) −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 \(-0.374315\pi\)
0.384670 + 0.923054i \(0.374315\pi\)
\(108\) 214.338 0.190970
\(109\) 1361.88 1.19674 0.598369 0.801221i \(-0.295816\pi\)
0.598369 + 0.801221i \(0.295816\pi\)
\(110\) −186.042 −0.161258
\(111\) −208.581 −0.178357
\(112\) 0 0
\(113\) 1048.55 0.872917 0.436459 0.899724i \(-0.356233\pi\)
0.436459 + 0.899724i \(0.356233\pi\)
\(114\) 37.6201 0.0309074
\(115\) −1725.83 −1.39943
\(116\) 485.313 0.388450
\(117\) 327.842 0.259051
\(118\) 209.503 0.163444
\(119\) 0 0
\(120\) −147.495 −0.112203
\(121\) 2306.48 1.73289
\(122\) −83.9817 −0.0623225
\(123\) −924.346 −0.677605
\(124\) 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
\(129\) −523.328 −0.357182
\(130\) −112.365 −0.0758083
\(131\) −1854.23 −1.23668 −0.618338 0.785912i \(-0.712194\pi\)
−0.618338 + 0.785912i \(0.712194\pi\)
\(132\) 1436.34 0.947102
\(133\) 0 0
\(134\) −241.014 −0.155377
\(135\) 335.734 0.214040
\(136\) 192.737 0.121522
\(137\) −511.115 −0.318741 −0.159370 0.987219i \(-0.550946\pi\)
−0.159370 + 0.987219i \(0.550946\pi\)
\(138\) −103.292 −0.0637156
\(139\) 2266.10 1.38279 0.691397 0.722475i \(-0.256995\pi\)
0.691397 + 0.722475i \(0.256995\pi\)
\(140\) 0 0
\(141\) −1168.09 −0.697663
\(142\) −24.4276 −0.0144360
\(143\) 2196.96 1.28475
\(144\) 562.742 0.325661
\(145\) 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 \(-0.363944\pi\)
0.414537 + 0.910033i \(0.363944\pi\)
\(150\) −22.0432 −0.0119988
\(151\) 1591.83 0.857887 0.428943 0.903331i \(-0.358886\pi\)
0.428943 + 0.903331i \(0.358886\pi\)
\(152\) 199.869 0.106655
\(153\) −438.715 −0.231817
\(154\) 0 0
\(155\) 14.5404 0.00753494
\(156\) 867.520 0.445239
\(157\) 1164.16 0.591784 0.295892 0.955221i \(-0.404383\pi\)
0.295892 + 0.955221i \(0.404383\pi\)
\(158\) −120.782 −0.0608160
\(159\) −944.805 −0.471245
\(160\) −586.195 −0.289642
\(161\) 0 0
\(162\) 20.0938 0.00974519
\(163\) −1155.88 −0.555432 −0.277716 0.960663i \(-0.589577\pi\)
−0.277716 + 0.960663i \(0.589577\pi\)
\(164\) −2445.96 −1.16462
\(165\) 2249.85 1.06152
\(166\) 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
\(171\) −454.950 −0.203455
\(172\) −1384.81 −0.613898
\(173\) 1894.94 0.832770 0.416385 0.909188i \(-0.363297\pi\)
0.416385 + 0.909188i \(0.363297\pi\)
\(174\) 45.4972 0.0198226
\(175\) 0 0
\(176\) 3771.09 1.61509
\(177\) −2533.58 −1.07591
\(178\) 54.0967 0.0227793
\(179\) −4288.49 −1.79071 −0.895355 0.445354i \(-0.853078\pi\)
−0.895355 + 0.445354i \(0.853078\pi\)
\(180\) 888.405 0.367876
\(181\) 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
\(184\) −548.769 −0.219868
\(185\) −864.539 −0.343579
\(186\) 0.870251 0.000343064 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 \(-0.476748\pi\)
0.0729845 + 0.997333i \(0.476748\pi\)
\(192\) 1465.56 0.550873
\(193\) 630.224 0.235049 0.117525 0.993070i \(-0.462504\pi\)
0.117525 + 0.993070i \(0.462504\pi\)
\(194\) −194.064 −0.0718194
\(195\) 1358.86 0.499026
\(196\) 0 0
\(197\) −1250.23 −0.452158 −0.226079 0.974109i \(-0.572591\pi\)
−0.226079 + 0.974109i \(0.572591\pi\)
\(198\) 134.654 0.0483307
\(199\) −1092.24 −0.389081 −0.194541 0.980894i \(-0.562322\pi\)
−0.194541 + 0.980894i \(0.562322\pi\)
\(200\) −117.112 −0.0414052
\(201\) 2914.65 1.02280
\(202\) −77.3105 −0.0269285
\(203\) 0 0
\(204\) −1160.91 −0.398430
\(205\) −3831.29 −1.30531
\(206\) −37.0267 −0.0125232
\(207\) 1249.13 0.419423
\(208\) 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
\(213\) 295.409 0.0950287
\(214\) 211.238 0.0674763
\(215\) −2169.13 −0.688061
\(216\) 106.755 0.0336285
\(217\) 0 0
\(218\) 337.844 0.104962
\(219\) −2130.70 −0.657442
\(220\) 5953.45 1.82446
\(221\) −1775.67 −0.540473
\(222\) −51.7430 −0.0156431
\(223\) −183.844 −0.0552069 −0.0276034 0.999619i \(-0.508788\pi\)
−0.0276034 + 0.999619i \(0.508788\pi\)
\(224\) 0 0
\(225\) 266.574 0.0789850
\(226\) 260.117 0.0765607
\(227\) 2279.52 0.666506 0.333253 0.942837i \(-0.391854\pi\)
0.333253 + 0.942837i \(0.391854\pi\)
\(228\) −1203.87 −0.349684
\(229\) 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 \(-0.448839\pi\)
0.160036 + 0.987111i \(0.448839\pi\)
\(234\) 81.3284 0.0227205
\(235\) −4841.56 −1.34395
\(236\) −6704.24 −1.84919
\(237\) 1460.65 0.400336
\(238\) 0 0
\(239\) −6226.36 −1.68515 −0.842573 0.538583i \(-0.818960\pi\)
−0.842573 + 0.538583i \(0.818960\pi\)
\(240\) 2332.49 0.627340
\(241\) −3196.20 −0.854295 −0.427147 0.904182i \(-0.640481\pi\)
−0.427147 + 0.904182i \(0.640481\pi\)
\(242\) 572.173 0.151986
\(243\) −243.000 −0.0641500
\(244\) 2687.47 0.705113
\(245\) 0 0
\(246\) −229.304 −0.0594305
\(247\) −1841.38 −0.474349
\(248\) 4.62349 0.00118384
\(249\) −1817.06 −0.462457
\(250\) 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
\(255\) −1818.42 −0.446563
\(256\) 3784.54 0.923960
\(257\) 699.117 0.169688 0.0848439 0.996394i \(-0.472961\pi\)
0.0848439 + 0.996394i \(0.472961\pi\)
\(258\) −129.823 −0.0313272
\(259\) 0 0
\(260\) 3595.76 0.857690
\(261\) −550.210 −0.130487
\(262\) −459.982 −0.108465
\(263\) −919.040 −0.215477 −0.107738 0.994179i \(-0.534361\pi\)
−0.107738 + 0.994179i \(0.534361\pi\)
\(264\) 715.394 0.166778
\(265\) −3916.09 −0.907787
\(266\) 0 0
\(267\) −654.206 −0.149950
\(268\) 7712.61 1.75792
\(269\) −2779.17 −0.629923 −0.314961 0.949104i \(-0.601992\pi\)
−0.314961 + 0.949104i \(0.601992\pi\)
\(270\) 83.2863 0.0187727
\(271\) 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
\(276\) 3305.39 0.720874
\(277\) 7307.69 1.58511 0.792557 0.609797i \(-0.208749\pi\)
0.792557 + 0.609797i \(0.208749\pi\)
\(278\) 562.157 0.121280
\(279\) −10.5242 −0.00225830
\(280\) 0 0
\(281\) 2730.61 0.579696 0.289848 0.957073i \(-0.406395\pi\)
0.289848 + 0.957073i \(0.406395\pi\)
\(282\) −289.769 −0.0611897
\(283\) −1769.85 −0.371755 −0.185878 0.982573i \(-0.559513\pi\)
−0.185878 + 0.982573i \(0.559513\pi\)
\(284\) 781.698 0.163328
\(285\) −1885.71 −0.391928
\(286\) 545.004 0.112681
\(287\) 0 0
\(288\) 424.280 0.0868088
\(289\) −2536.81 −0.516347
\(290\) 188.580 0.0381855
\(291\) 2346.86 0.472768
\(292\) −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
\(297\) −1628.41 −0.318148
\(298\) 374.068 0.0727153
\(299\) 5055.78 0.977870
\(300\) 705.397 0.135754
\(301\) 0 0
\(302\) 394.887 0.0752424
\(303\) 934.937 0.177263
\(304\) −3160.73 −0.596317
\(305\) 4209.58 0.790295
\(306\) −108.833 −0.0203319
\(307\) 6019.62 1.11908 0.559541 0.828803i \(-0.310977\pi\)
0.559541 + 0.828803i \(0.310977\pi\)
\(308\) 0 0
\(309\) 447.773 0.0824366
\(310\) 3.60707 0.000660865 0
\(311\) 1193.71 0.217650 0.108825 0.994061i \(-0.465291\pi\)
0.108825 + 0.994061i \(0.465291\pi\)
\(312\) 432.083 0.0784035
\(313\) −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 \(-0.318897\pi\)
0.538750 + 0.842466i \(0.318897\pi\)
\(318\) −234.380 −0.0413313
\(319\) −3687.11 −0.647143
\(320\) 6074.55 1.06118
\(321\) −2554.56 −0.444179
\(322\) 0 0
\(323\) 2464.12 0.424480
\(324\) −643.015 −0.110256
\(325\) 1078.94 0.184151
\(326\) −286.741 −0.0487151
\(327\) −4085.64 −0.690936
\(328\) −1218.25 −0.205082
\(329\) 0 0
\(330\) 558.125 0.0931023
\(331\) 3053.30 0.507022 0.253511 0.967333i \(-0.418415\pi\)
0.253511 + 0.967333i \(0.418415\pi\)
\(332\) −4808.23 −0.794837
\(333\) 625.742 0.102974
\(334\) −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
\(339\) −3145.66 −0.503979
\(340\) −4811.81 −0.767521
\(341\) −70.5255 −0.0111999
\(342\) −112.860 −0.0178444
\(343\) 0 0
\(344\) −689.726 −0.108103
\(345\) 5177.49 0.807961
\(346\) 470.080 0.0730395
\(347\) −99.5931 −0.0154076 −0.00770380 0.999970i \(-0.502452\pi\)
−0.00770380 + 0.999970i \(0.502452\pi\)
\(348\) −1455.94 −0.224272
\(349\) −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
\(352\) 2843.22 0.430523
\(353\) 7130.73 1.07516 0.537579 0.843214i \(-0.319339\pi\)
0.537579 + 0.843214i \(0.319339\pi\)
\(354\) −628.510 −0.0943642
\(355\) 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 \(-0.658572\pi\)
−0.477816 + 0.878460i \(0.658572\pi\)
\(360\) 442.485 0.0647806
\(361\) −4303.70 −0.627453
\(362\) 95.1932 0.0138211
\(363\) −6919.44 −1.00049
\(364\) 0 0
\(365\) −8831.49 −1.26647
\(366\) 251.945 0.0359819
\(367\) 824.886 0.117326 0.0586631 0.998278i \(-0.481316\pi\)
0.0586631 + 0.998278i \(0.481316\pi\)
\(368\) 8678.25 1.22931
\(369\) 2773.04 0.391216
\(370\) −214.468 −0.0301342
\(371\) 0 0
\(372\) −27.8486 −0.00388140
\(373\) 1333.85 0.185159 0.0925793 0.995705i \(-0.470489\pi\)
0.0925793 + 0.995705i \(0.470489\pi\)
\(374\) −729.320 −0.100835
\(375\) −3558.06 −0.489967
\(376\) −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
\(381\) −1465.22 −0.197023
\(382\) 95.5847 0.0128025
\(383\) 353.376 0.0471453 0.0235727 0.999722i \(-0.492496\pi\)
0.0235727 + 0.999722i \(0.492496\pi\)
\(384\) 1494.98 0.198673
\(385\) 0 0
\(386\) 156.341 0.0206154
\(387\) 1569.98 0.206219
\(388\) 6210.16 0.812560
\(389\) 11737.2 1.52982 0.764908 0.644139i \(-0.222784\pi\)
0.764908 + 0.644139i \(0.222784\pi\)
\(390\) 337.096 0.0437679
\(391\) −6765.59 −0.875066
\(392\) 0 0
\(393\) 5562.68 0.713996
\(394\) −310.147 −0.0396573
\(395\) 6054.21 0.771191
\(396\) −4309.03 −0.546810
\(397\) 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 \(-0.654269\pi\)
−0.465899 + 0.884838i \(0.654269\pi\)
\(402\) 723.043 0.0897067
\(403\) −42.5959 −0.00526514
\(404\) 2473.99 0.304667
\(405\) −1007.20 −0.123576
\(406\) 0 0
\(407\) 4193.27 0.510694
\(408\) −578.210 −0.0701609
\(409\) −13796.6 −1.66797 −0.833983 0.551791i \(-0.813945\pi\)
−0.833983 + 0.551791i \(0.813945\pi\)
\(410\) −950.436 −0.114485
\(411\) 1533.35 0.184025
\(412\) 1184.88 0.141686
\(413\) 0 0
\(414\) 309.875 0.0367862
\(415\) −7531.49 −0.890859
\(416\) 1717.24 0.202391
\(417\) −6798.31 −0.798357
\(418\) −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
\(423\) 3504.26 0.402796
\(424\) −1245.22 −0.142625
\(425\) −1443.83 −0.164791
\(426\) 73.2827 0.00833465
\(427\) 0 0
\(428\) −6759.75 −0.763423
\(429\) −6590.88 −0.741750
\(430\) −538.099 −0.0603475
\(431\) −13397.3 −1.49727 −0.748636 0.662981i \(-0.769291\pi\)
−0.748636 + 0.662981i \(0.769291\pi\)
\(432\) −1688.22 −0.188020
\(433\) −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
\(436\) −10811.2 −1.18753
\(437\) −7015.95 −0.768006
\(438\) −528.568 −0.0576620
\(439\) −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 \(-0.520133\pi\)
−0.0632078 + 0.998000i \(0.520133\pi\)
\(444\) 1655.81 0.176985
\(445\) −2711.60 −0.288858
\(446\) −45.6067 −0.00484201
\(447\) −4523.70 −0.478666
\(448\) 0 0
\(449\) −12400.9 −1.30342 −0.651709 0.758469i \(-0.725948\pi\)
−0.651709 + 0.758469i \(0.725948\pi\)
\(450\) 66.1296 0.00692751
\(451\) 18582.9 1.94021
\(452\) −8323.90 −0.866202
\(453\) −4775.48 −0.495301
\(454\) 565.484 0.0584570
\(455\) 0 0
\(456\) −599.606 −0.0615771
\(457\) 9925.58 1.01597 0.507986 0.861365i \(-0.330390\pi\)
0.507986 + 0.861365i \(0.330390\pi\)
\(458\) 1342.73 0.136991
\(459\) 1316.15 0.133840
\(460\) 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
\(465\) −43.6213 −0.00435030
\(466\) 282.397 0.0280725
\(467\) 2108.06 0.208886 0.104443 0.994531i \(-0.466694\pi\)
0.104443 + 0.994531i \(0.466694\pi\)
\(468\) −2602.56 −0.257059
\(469\) 0 0
\(470\) −1201.05 −0.117873
\(471\) −3492.48 −0.341667
\(472\) −3339.16 −0.325630
\(473\) 10520.9 1.02273
\(474\) 362.347 0.0351121
\(475\) −1497.26 −0.144629
\(476\) 0 0
\(477\) 2834.41 0.272073
\(478\) −1544.59 −0.147798
\(479\) −2450.04 −0.233706 −0.116853 0.993149i \(-0.537281\pi\)
−0.116853 + 0.993149i \(0.537281\pi\)
\(480\) 1758.58 0.167225
\(481\) 2532.65 0.240081
\(482\) −792.887 −0.0749274
\(483\) 0 0
\(484\) −18309.9 −1.71956
\(485\) 9727.44 0.910722
\(486\) −60.2815 −0.00562639
\(487\) 645.236 0.0600379 0.0300189 0.999549i \(-0.490443\pi\)
0.0300189 + 0.999549i \(0.490443\pi\)
\(488\) 1338.54 0.124166
\(489\) 3467.64 0.320679
\(490\) 0 0
\(491\) 11766.1 1.08146 0.540731 0.841196i \(-0.318148\pi\)
0.540731 + 0.841196i \(0.318148\pi\)
\(492\) 7337.88 0.672393
\(493\) 2980.07 0.272242
\(494\) −456.794 −0.0416035
\(495\) −6749.55 −0.612868
\(496\) −73.1159 −0.00661896
\(497\) 0 0
\(498\) −450.763 −0.0405606
\(499\) −44.0209 −0.00394919 −0.00197459 0.999998i \(-0.500629\pi\)
−0.00197459 + 0.999998i \(0.500629\pi\)
\(500\) −9415.17 −0.842119
\(501\) 8671.83 0.773311
\(502\) 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
\(507\) 2610.24 0.228649
\(508\) −3877.21 −0.338629
\(509\) 6915.04 0.602168 0.301084 0.953598i \(-0.402651\pi\)
0.301084 + 0.953598i \(0.402651\pi\)
\(510\) −451.098 −0.0391666
\(511\) 0 0
\(512\) 4925.45 0.425148
\(513\) 1364.85 0.117465
\(514\) 173.431 0.0148827
\(515\) 1855.96 0.158803
\(516\) 4154.42 0.354434
\(517\) 23483.0 1.99764
\(518\) 0 0
\(519\) −5684.81 −0.480800
\(520\) 1790.93 0.151033
\(521\) 13399.3 1.12674 0.563371 0.826204i \(-0.309504\pi\)
0.563371 + 0.826204i \(0.309504\pi\)
\(522\) −136.492 −0.0114446
\(523\) −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
\(528\) −11313.3 −0.932475
\(529\) 7096.33 0.583244
\(530\) −971.472 −0.0796190
\(531\) 7600.74 0.621175
\(532\) 0 0
\(533\) 11223.7 0.912104
\(534\) −162.290 −0.0131516
\(535\) −10588.3 −0.855649
\(536\) 3841.40 0.309558
\(537\) 12865.5 1.03387
\(538\) −689.435 −0.0552484
\(539\) 0 0
\(540\) −2665.21 −0.212394
\(541\) 9286.17 0.737973 0.368987 0.929435i \(-0.379705\pi\)
0.368987 + 0.929435i \(0.379705\pi\)
\(542\) 552.452 0.0437820
\(543\) −1151.20 −0.0909809
\(544\) −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
\(549\) −3046.84 −0.236860
\(550\) 443.153 0.0343566
\(551\) 3090.35 0.238935
\(552\) 1646.31 0.126941
\(553\) 0 0
\(554\) 1812.83 0.139025
\(555\) 2593.62 0.198366
\(556\) −17989.4 −1.37216
\(557\) 1805.94 0.137379 0.0686897 0.997638i \(-0.478118\pi\)
0.0686897 + 0.997638i \(0.478118\pi\)
\(558\) −2.61075 −0.000198068 0
\(559\) 6354.40 0.480792
\(560\) 0 0
\(561\) 8819.86 0.663770
\(562\) 677.388 0.0508432
\(563\) 12214.9 0.914381 0.457190 0.889369i \(-0.348856\pi\)
0.457190 + 0.889369i \(0.348856\pi\)
\(564\) 9272.80 0.692296
\(565\) −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 \(-0.449557\pi\)
0.157808 + 0.987470i \(0.449557\pi\)
\(570\) −467.791 −0.0343747
\(571\) 6359.94 0.466121 0.233060 0.972462i \(-0.425126\pi\)
0.233060 + 0.972462i \(0.425126\pi\)
\(572\) −17440.5 −1.27487
\(573\) −1155.93 −0.0842753
\(574\) 0 0
\(575\) 4110.95 0.298153
\(576\) −4396.68 −0.318047
\(577\) 14468.7 1.04392 0.521959 0.852971i \(-0.325201\pi\)
0.521959 + 0.852971i \(0.325201\pi\)
\(578\) −629.313 −0.0452871
\(579\) −1890.67 −0.135706
\(580\) −6034.68 −0.432028
\(581\) 0 0
\(582\) 582.191 0.0414649
\(583\) 18994.2 1.34933
\(584\) −2808.19 −0.198979
\(585\) −4076.59 −0.288113
\(586\) 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
\(591\) 3750.69 0.261054
\(592\) 4347.29 0.301812
\(593\) −19775.6 −1.36946 −0.684728 0.728799i \(-0.740079\pi\)
−0.684728 + 0.728799i \(0.740079\pi\)
\(594\) −403.963 −0.0279037
\(595\) 0 0
\(596\) −11970.4 −0.822696
\(597\) 3276.73 0.224636
\(598\) 1254.20 0.0857657
\(599\) −23891.0 −1.62965 −0.814825 0.579707i \(-0.803167\pi\)
−0.814825 + 0.579707i \(0.803167\pi\)
\(600\) 351.335 0.0239053
\(601\) 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
\(604\) −12636.6 −0.851288
\(605\) −28680.2 −1.92730
\(606\) 231.932 0.0155472
\(607\) 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
\(612\) 3482.72 0.230034
\(613\) 1979.80 0.130446 0.0652229 0.997871i \(-0.479224\pi\)
0.0652229 + 0.997871i \(0.479224\pi\)
\(614\) 1493.30 0.0981509
\(615\) 11493.9 0.753622
\(616\) 0 0
\(617\) 16262.4 1.06110 0.530551 0.847653i \(-0.321985\pi\)
0.530551 + 0.847653i \(0.321985\pi\)
\(618\) 111.080 0.00723024
\(619\) −12021.0 −0.780555 −0.390278 0.920697i \(-0.627621\pi\)
−0.390278 + 0.920697i \(0.627621\pi\)
\(620\) −115.429 −0.00747698
\(621\) −3747.39 −0.242154
\(622\) 296.127 0.0190894
\(623\) 0 0
\(624\) −6832.97 −0.438362
\(625\) −18450.1 −1.18081
\(626\) −2194.45 −0.140109
\(627\) 9146.24 0.582561
\(628\) −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
\(633\) 10860.1 0.681915
\(634\) 1508.63 0.0945039
\(635\) −6073.16 −0.379537
\(636\) 7500.29 0.467620
\(637\) 0 0
\(638\) −914.669 −0.0567588
\(639\) −886.228 −0.0548648
\(640\) 6196.49 0.382715
\(641\) −5111.60 −0.314971 −0.157485 0.987521i \(-0.550339\pi\)
−0.157485 + 0.987521i \(0.550339\pi\)
\(642\) −633.714 −0.0389575
\(643\) −10931.3 −0.670435 −0.335217 0.942141i \(-0.608810\pi\)
−0.335217 + 0.942141i \(0.608810\pi\)
\(644\) 0 0
\(645\) 6507.38 0.397252
\(646\) 611.278 0.0372297
\(647\) −18406.1 −1.11842 −0.559211 0.829025i \(-0.688896\pi\)
−0.559211 + 0.829025i \(0.688896\pi\)
\(648\) −320.265 −0.0194154
\(649\) 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 \(-0.296389\pi\)
0.596926 + 0.802296i \(0.296389\pi\)
\(654\) −1013.53 −0.0605997
\(655\) 23056.6 1.37541
\(656\) 19265.5 1.14663
\(657\) 6392.11 0.379574
\(658\) 0 0
\(659\) −18858.8 −1.11477 −0.557385 0.830254i \(-0.688195\pi\)
−0.557385 + 0.830254i \(0.688195\pi\)
\(660\) −17860.3 −1.05335
\(661\) 25832.1 1.52005 0.760023 0.649896i \(-0.225188\pi\)
0.760023 + 0.649896i \(0.225188\pi\)
\(662\) 757.437 0.0444692
\(663\) 5327.01 0.312042
\(664\) −2394.82 −0.139965
\(665\) 0 0
\(666\) 155.229 0.00903153
\(667\) −8485.00 −0.492564
\(668\) 22947.0 1.32911
\(669\) 551.533 0.0318737
\(670\) 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
\(675\) −799.723 −0.0456020
\(676\) 6907.11 0.392985
\(677\) −26271.8 −1.49144 −0.745720 0.666259i \(-0.767895\pi\)
−0.745720 + 0.666259i \(0.767895\pi\)
\(678\) −780.350 −0.0442023
\(679\) 0 0
\(680\) −2396.60 −0.135155
\(681\) −6838.55 −0.384808
\(682\) −17.4954 −0.000982306 0
\(683\) −8072.29 −0.452237 −0.226118 0.974100i \(-0.572604\pi\)
−0.226118 + 0.974100i \(0.572604\pi\)
\(684\) 3611.60 0.201890
\(685\) 6355.51 0.354499
\(686\) 0 0
\(687\) −16238.0 −0.901774
\(688\) 10907.3 0.604417
\(689\) 11472.1 0.634328
\(690\) 1284.39 0.0708636
\(691\) −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
\(696\) −725.156 −0.0394928
\(697\) −15019.4 −0.816214
\(698\) −894.880 −0.0485268
\(699\) −3415.10 −0.184794
\(700\) 0 0
\(701\) 778.448 0.0419423 0.0209712 0.999780i \(-0.493324\pi\)
0.0209712 + 0.999780i \(0.493324\pi\)
\(702\) −243.985 −0.0131177
\(703\) −3514.58 −0.188556
\(704\) −29463.4 −1.57733
\(705\) 14524.7 0.775930
\(706\) 1768.93 0.0942985
\(707\) 0 0
\(708\) 20112.7 1.06763
\(709\) −24172.0 −1.28039 −0.640197 0.768211i \(-0.721147\pi\)
−0.640197 + 0.768211i \(0.721147\pi\)
\(710\) 303.747 0.0160555
\(711\) −4381.96 −0.231134
\(712\) −862.218 −0.0453834
\(713\) −162.297 −0.00852466
\(714\) 0 0
\(715\) −27318.3 −1.42888
\(716\) 34044.0 1.77693
\(717\) 18679.1 0.972919
\(718\) −1612.54 −0.0838153
\(719\) −81.8835 −0.00424720 −0.00212360 0.999998i \(-0.500676\pi\)
−0.00212360 + 0.999998i \(0.500676\pi\)
\(720\) −6997.47 −0.362195
\(721\) 0 0
\(722\) −1067.63 −0.0550318
\(723\) 9588.59 0.493227
\(724\) −3046.24 −0.156371
\(725\) −1810.76 −0.0927588
\(726\) −1716.52 −0.0877493
\(727\) −32542.9 −1.66018 −0.830088 0.557632i \(-0.811710\pi\)
−0.830088 + 0.557632i \(0.811710\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −2190.85 −0.111078
\(731\) −8503.40 −0.430246
\(732\) −8062.41 −0.407097
\(733\) −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
\(738\) 687.913 0.0343122
\(739\) −38428.5 −1.91287 −0.956437 0.291939i \(-0.905700\pi\)
−0.956437 + 0.291939i \(0.905700\pi\)
\(740\) 6863.11 0.340936
\(741\) 5524.14 0.273865
\(742\) 0 0
\(743\) 21592.9 1.06617 0.533086 0.846061i \(-0.321032\pi\)
0.533086 + 0.846061i \(0.321032\pi\)
\(744\) −13.8705 −0.000683489 0
\(745\) −18750.1 −0.922083
\(746\) 330.891 0.0162396
\(747\) 5451.19 0.267000
\(748\) 23338.7 1.14084
\(749\) 0 0
\(750\) −882.655 −0.0429733
\(751\) 8112.60 0.394185 0.197093 0.980385i \(-0.436850\pi\)
0.197093 + 0.980385i \(0.436850\pi\)
\(752\) 24345.6 1.18057
\(753\) −718.823 −0.0347880
\(754\) −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
\(759\) −25112.3 −1.20095
\(760\) −2485.29 −0.118620
\(761\) −7211.93 −0.343538 −0.171769 0.985137i \(-0.554948\pi\)
−0.171769 + 0.985137i \(0.554948\pi\)
\(762\) −363.481 −0.0172802
\(763\) 0 0
\(764\) −3058.77 −0.144846
\(765\) 5455.25 0.257823
\(766\) 87.6626 0.00413496
\(767\) 30763.5 1.44825
\(768\) −11353.6 −0.533448
\(769\) −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
\(772\) −5003.00 −0.233241
\(773\) −24832.6 −1.15546 −0.577728 0.816229i \(-0.696060\pi\)
−0.577728 + 0.816229i \(0.696060\pi\)
\(774\) 389.469 0.0180868
\(775\) −34.6355 −0.00160535
\(776\) 3093.08 0.143086
\(777\) 0 0
\(778\) 2911.66 0.134175
\(779\) −15575.2 −0.716355
\(780\) −10787.3 −0.495188
\(781\) −5938.86 −0.272099
\(782\) −1678.35 −0.0767491
\(783\) 1650.63 0.0753368
\(784\) 0 0
\(785\) −14475.9 −0.658173
\(786\) 1379.95 0.0626222
\(787\) 36313.1 1.64476 0.822378 0.568941i \(-0.192647\pi\)
0.822378 + 0.568941i \(0.192647\pi\)
\(788\) 9924.89 0.448680
\(789\) 2757.12 0.124406
\(790\) 1501.88 0.0676386
\(791\) 0 0
\(792\) −2146.18 −0.0962895
\(793\) −12331.9 −0.552229
\(794\) 3294.75 0.147262
\(795\) 11748.3 0.524111
\(796\) 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
\(801\) 1962.62 0.0865738
\(802\) −1856.16 −0.0817249
\(803\) 42835.4 1.88247
\(804\) −23137.8 −1.01494
\(805\) 0 0
\(806\) −10.5668 −0.000461788 0
\(807\) 8337.52 0.363686
\(808\) 1232.21 0.0536498
\(809\) 12384.6 0.538219 0.269110 0.963110i \(-0.413271\pi\)
0.269110 + 0.963110i \(0.413271\pi\)
\(810\) −249.859 −0.0108384
\(811\) 16742.4 0.724914 0.362457 0.932000i \(-0.381938\pi\)
0.362457 + 0.932000i \(0.381938\pi\)
\(812\) 0 0
\(813\) −6680.94 −0.288205
\(814\) 1040.23 0.0447913
\(815\) 14372.9 0.617744
\(816\) 9143.82 0.392277
\(817\) −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 \(-0.309908\pi\)
0.562322 + 0.826918i \(0.309908\pi\)
\(822\) 380.380 0.0161402
\(823\) 23098.5 0.978328 0.489164 0.872192i \(-0.337302\pi\)
0.489164 + 0.872192i \(0.337302\pi\)
\(824\) 590.148 0.0249500
\(825\) −5359.17 −0.226160
\(826\) 0 0
\(827\) 20647.6 0.868183 0.434092 0.900869i \(-0.357069\pi\)
0.434092 + 0.900869i \(0.357069\pi\)
\(828\) −9916.18 −0.416197
\(829\) −23368.5 −0.979037 −0.489519 0.871993i \(-0.662827\pi\)
−0.489519 + 0.871993i \(0.662827\pi\)
\(830\) −1868.35 −0.0781343
\(831\) −21923.1 −0.915166
\(832\) −17795.3 −0.741514
\(833\) 0 0
\(834\) −1686.47 −0.0700212
\(835\) 35943.6 1.48968
\(836\) 24202.3 1.00126
\(837\) 31.5725 0.00130383
\(838\) 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
\(843\) −8191.84 −0.334688
\(844\) 28737.6 1.17203
\(845\) 10819.1 0.440460
\(846\) 869.307 0.0353279
\(847\) 0 0
\(848\) 19691.9 0.797432
\(849\) 5309.55 0.214633
\(850\) −358.174 −0.0144532
\(851\) 9649.80 0.388708
\(852\) −2345.09 −0.0942977
\(853\) −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
\(856\) −3366.81 −0.134434
\(857\) −32788.6 −1.30693 −0.653463 0.756958i \(-0.726685\pi\)
−0.653463 + 0.756958i \(0.726685\pi\)
\(858\) −1635.01 −0.0650565
\(859\) −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 \(-0.385857\pi\)
0.350954 + 0.936393i \(0.385857\pi\)
\(864\) −1272.84 −0.0501191
\(865\) −23562.8 −0.926195
\(866\) −3487.21 −0.136836
\(867\) 7610.44 0.298113
\(868\) 0 0
\(869\) −29364.7 −1.14629
\(870\) −565.740 −0.0220464
\(871\) −35390.5 −1.37677
\(872\) −5384.71 −0.209116
\(873\) −7040.59 −0.272953
\(874\) −1740.46 −0.0673592
\(875\) 0 0
\(876\) 16914.5 0.652384
\(877\) 34672.2 1.33500 0.667501 0.744609i \(-0.267364\pi\)
0.667501 + 0.744609i \(0.267364\pi\)
\(878\) −4060.65 −0.156082
\(879\) −24686.4 −0.947273
\(880\) −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
\(885\) 31504.1 1.19661
\(886\) −292.404 −0.0110875
\(887\) 33210.7 1.25717 0.628583 0.777742i \(-0.283635\pi\)
0.628583 + 0.777742i \(0.283635\pi\)
\(888\) 824.703 0.0311658
\(889\) 0 0
\(890\) −672.671 −0.0253348
\(891\) 4885.23 0.183683
\(892\) 1459.44 0.0547822
\(893\) −19682.2 −0.737559
\(894\) −1122.20 −0.0419822
\(895\) 53325.7 1.99160
\(896\) 0 0
\(897\) −15167.3 −0.564573
\(898\) −3076.32 −0.114318
\(899\) 71.4878 0.00265211
\(900\) −2116.19 −0.0783774
\(901\) −15351.9 −0.567641
\(902\) 4609.90 0.170169
\(903\) 0 0
\(904\) −4145.86 −0.152532
\(905\) −4771.56 −0.175262
\(906\) −1184.66 −0.0434412
\(907\) 2497.83 0.0914433 0.0457217 0.998954i \(-0.485441\pi\)
0.0457217 + 0.998954i \(0.485441\pi\)
\(908\) −18095.9 −0.661379
\(909\) −2804.81 −0.102343
\(910\) 0 0
\(911\) −1895.00 −0.0689180 −0.0344590 0.999406i \(-0.510971\pi\)
−0.0344590 + 0.999406i \(0.510971\pi\)
\(912\) 9482.19 0.344284
\(913\) 36530.0 1.32417
\(914\) 2462.26 0.0891075
\(915\) −12628.8 −0.456277
\(916\) −42968.3 −1.54990
\(917\) 0 0
\(918\) 326.499 0.0117386
\(919\) 6270.71 0.225083 0.112542 0.993647i \(-0.464101\pi\)
0.112542 + 0.993647i \(0.464101\pi\)
\(920\) 6823.73 0.244534
\(921\) −18058.9 −0.646102
\(922\) −3971.69 −0.141866
\(923\) −3586.95 −0.127915
\(924\) 0 0
\(925\) 2059.34 0.0732008
\(926\) 4309.61 0.152940
\(927\) −1343.32 −0.0475948
\(928\) −2882.01 −0.101947
\(929\) −31552.6 −1.11432 −0.557161 0.830404i \(-0.688110\pi\)
−0.557161 + 0.830404i \(0.688110\pi\)
\(930\) −10.8212 −0.000381550 0
\(931\) 0 0
\(932\) −9036.89 −0.317611
\(933\) −3581.14 −0.125661
\(934\) 522.951 0.0183207
\(935\) 36557.2 1.27866
\(936\) −1296.25 −0.0452663
\(937\) −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
\(940\) 38434.5 1.33361
\(941\) −32538.6 −1.12724 −0.563618 0.826036i \(-0.690591\pi\)
−0.563618 + 0.826036i \(0.690591\pi\)
\(942\) −866.386 −0.0299664
\(943\) 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 \(-0.746142\pi\)
−0.698485 + 0.715625i \(0.746142\pi\)
\(948\) −11595.3 −0.397256
\(949\) 25871.7 0.884962
\(950\) −371.428 −0.0126850
\(951\) −18244.3 −0.622095
\(952\) 0 0
\(953\) −52516.4 −1.78507 −0.892536 0.450976i \(-0.851076\pi\)
−0.892536 + 0.450976i \(0.851076\pi\)
\(954\) 703.139 0.0238626
\(955\) −4791.18 −0.162345
\(956\) 49427.7 1.67218
\(957\) 11061.3 0.373628
\(958\) −607.786 −0.0204976
\(959\) 0 0
\(960\) −18223.7 −0.612673
\(961\) −29789.6 −0.999954
\(962\) 628.279 0.0210567
\(963\) 7663.67 0.256447
\(964\) 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
\(969\) −7392.35 −0.245074
\(970\) 2413.10 0.0798764
\(971\) 13772.5 0.455181 0.227590 0.973757i \(-0.426915\pi\)
0.227590 + 0.973757i \(0.426915\pi\)
\(972\) 1929.05 0.0636566
\(973\) 0 0
\(974\) 160.065 0.00526572
\(975\) −3236.83 −0.106319
\(976\) −21167.7 −0.694223
\(977\) 24782.1 0.811513 0.405757 0.913981i \(-0.367008\pi\)
0.405757 + 0.913981i \(0.367008\pi\)
\(978\) 860.224 0.0281257
\(979\) 13152.0 0.429358
\(980\) 0 0
\(981\) 12256.9 0.398912
\(982\) 2918.84 0.0948514
\(983\) 42804.7 1.38887 0.694435 0.719556i \(-0.255655\pi\)
0.694435 + 0.719556i \(0.255655\pi\)
\(984\) 3654.76 0.118404
\(985\) 15546.1 0.502883
\(986\) 739.271 0.0238775
\(987\) 0 0
\(988\) 14617.7 0.470700
\(989\) 24211.3 0.778438
\(990\) −1674.37 −0.0537526
\(991\) 449.862 0.0144201 0.00721006 0.999974i \(-0.497705\pi\)
0.00721006 + 0.999974i \(0.497705\pi\)
\(992\) −55.1259 −0.00176436
\(993\) −9159.89 −0.292729
\(994\) 0 0
\(995\) 13581.6 0.432730
\(996\) 14424.7 0.458899
\(997\) −21473.7 −0.682127 −0.341063 0.940040i \(-0.610787\pi\)
−0.341063 + 0.940040i \(0.610787\pi\)
\(998\) −10.9203 −0.000346370 0
\(999\) −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 147.4.a.l.1.2 3
3.2 odd 2 441.4.a.s.1.2 3
4.3 odd 2 2352.4.a.ci.1.1 3
7.2 even 3 21.4.e.b.4.2 6
7.3 odd 6 147.4.e.n.79.2 6
7.4 even 3 21.4.e.b.16.2 yes 6
7.5 odd 6 147.4.e.n.67.2 6
7.6 odd 2 147.4.a.m.1.2 3
21.2 odd 6 63.4.e.c.46.2 6
21.5 even 6 441.4.e.w.361.2 6
21.11 odd 6 63.4.e.c.37.2 6
21.17 even 6 441.4.e.w.226.2 6
21.20 even 2 441.4.a.t.1.2 3
28.11 odd 6 336.4.q.k.289.3 6
28.23 odd 6 336.4.q.k.193.3 6
28.27 even 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 7.2 even 3
21.4.e.b.16.2 yes 6 7.4 even 3
63.4.e.c.37.2 6 21.11 odd 6
63.4.e.c.46.2 6 21.2 odd 6
147.4.a.l.1.2 3 1.1 even 1 trivial
147.4.a.m.1.2 3 7.6 odd 2
147.4.e.n.67.2 6 7.5 odd 6
147.4.e.n.79.2 6 7.3 odd 6
336.4.q.k.193.3 6 28.23 odd 6
336.4.q.k.289.3 6 28.11 odd 6
441.4.a.s.1.2 3 3.2 odd 2
441.4.a.t.1.2 3 21.20 even 2
441.4.e.w.226.2 6 21.17 even 6
441.4.e.w.361.2 6 21.5 even 6
2352.4.a.cg.1.3 3 28.27 even 2
2352.4.a.ci.1.1 3 4.3 odd 2