Properties

Label 147.4.a.m.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 \(-0.312300\pi\)
0.556092 + 0.831120i \(0.312300\pi\)
\(6\) 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 \(-0.627033\pi\)
−0.388577 + 0.921416i \(0.627033\pi\)
\(14\) 0 0
\(15\) 37.3038 0.642120
\(16\) 62.5268 0.976982
\(17\) 48.7461 0.695451 0.347726 0.937596i \(-0.386954\pi\)
0.347726 + 0.937596i \(0.386954\pi\)
\(18\) 2.23265 0.0292356
\(19\) 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 \(-0.283409\pi\)
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 \(-0.498922\pi\)
0.00338745 + 0.999994i \(0.498922\pi\)
\(32\) 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 \(-0.699622\pi\)
−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
\(45\) 111.911 0.370728
\(46\) 34.4305 0.110359
\(47\) −389.362 −1.20839 −0.604194 0.796837i \(-0.706505\pi\)
−0.604194 + 0.796837i \(0.706505\pi\)
\(48\) 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 \(-0.881729\pi\)
−0.931762 + 0.363068i \(0.881729\pi\)
\(60\) −296.135 −0.637181
\(61\) 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
\(66\) 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 \(-0.692809\pi\)
−0.569361 + 0.822088i \(0.692809\pi\)
\(74\) 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 \(-0.631163\pi\)
−0.400499 + 0.916297i \(0.631163\pi\)
\(84\) 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 \(-0.541453\pi\)
−0.129861 + 0.991532i \(0.541453\pi\)
\(90\) 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 \(-0.365728\pi\)
0.409429 + 0.912342i \(0.365728\pi\)
\(98\) 0 0
\(99\) 542.804 0.551049
\(100\) −235.132 −0.235132
\(101\) 311.646 0.307029 0.153514 0.988146i \(-0.450941\pi\)
0.153514 + 0.988146i \(0.450941\pi\)
\(102\) 36.2776 0.0352159
\(103\) 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 \(-0.374315\pi\)
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 \(-0.287806\pi\)
0.618338 + 0.785912i \(0.287806\pi\)
\(132\) −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 \(-0.743005\pi\)
−0.691397 + 0.722475i \(0.743005\pi\)
\(140\) 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 \(-0.595617\pi\)
−0.295892 + 0.955221i \(0.595617\pi\)
\(158\) −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 \(-0.266420\pi\)
0.669707 + 0.742626i \(0.266420\pi\)
\(168\) 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 \(-0.636703\pi\)
−0.416385 + 0.909188i \(0.636703\pi\)
\(174\) −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 \(-0.525106\pi\)
−0.0787917 + 0.996891i \(0.525106\pi\)
\(182\) 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 \(-0.437678\pi\)
0.194541 + 0.980894i \(0.437678\pi\)
\(200\) −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 \(-0.491212\pi\)
0.0276034 + 0.999619i \(0.491212\pi\)
\(224\) 0 0
\(225\) 266.574 0.0789850
\(226\) 260.117 0.0765607
\(227\) −2279.52 −0.666506 −0.333253 0.942837i \(-0.608146\pi\)
−0.333253 + 0.942837i \(0.608146\pi\)
\(228\) −1203.87 −0.349684
\(229\) −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 \(-0.448839\pi\)
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 \(-0.359519\pi\)
0.427147 + 0.904182i \(0.359519\pi\)
\(242\) 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 \(-0.509591\pi\)
−0.0301273 + 0.999546i \(0.509591\pi\)
\(252\) 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 \(-0.527039\pi\)
−0.0848439 + 0.996394i \(0.527039\pi\)
\(258\) 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 \(-0.398008\pi\)
0.314961 + 0.949104i \(0.398008\pi\)
\(270\) 83.2863 0.0187727
\(271\) −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
\(276\) −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 \(-0.440487\pi\)
0.185878 + 0.982573i \(0.440487\pi\)
\(284\) 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 \(-0.806228\pi\)
−0.820362 + 0.571844i \(0.806228\pi\)
\(294\) 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 \(-0.689023\pi\)
−0.559541 + 0.828803i \(0.689023\pi\)
\(308\) 0 0
\(309\) 447.773 0.0824366
\(310\) 3.60707 0.000660865 0
\(311\) −1193.71 −0.217650 −0.108825 0.994061i \(-0.534709\pi\)
−0.108825 + 0.994061i \(0.534709\pi\)
\(312\) 432.083 0.0784035
\(313\) 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 \(-0.318897\pi\)
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 \(-0.410778\pi\)
0.276643 + 0.960973i \(0.410778\pi\)
\(350\) 0 0
\(351\) −983.526 −0.149563
\(352\) 2843.22 0.430523
\(353\) −7130.73 −1.07516 −0.537579 0.843214i \(-0.680661\pi\)
−0.537579 + 0.843214i \(0.680661\pi\)
\(354\) −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 \(-0.518684\pi\)
−0.0586631 + 0.998278i \(0.518684\pi\)
\(368\) 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 \(-0.507504\pi\)
−0.0235727 + 0.999722i \(0.507504\pi\)
\(384\) −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 \(-0.817161\pi\)
−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 \(-0.654269\pi\)
−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 \(-0.186055\pi\)
0.833983 + 0.551791i \(0.186055\pi\)
\(410\) −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 \(-0.686778\pi\)
−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
\(423\) −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 \(-0.215179\pi\)
0.780079 + 0.625681i \(0.215179\pi\)
\(434\) 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 \(-0.150845\pi\)
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 \(-0.520133\pi\)
−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 \(-0.200142\pi\)
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
\(465\) 43.6213 0.00435030
\(466\) 282.397 0.0280725
\(467\) −2108.06 −0.208886 −0.104443 0.994531i \(-0.533306\pi\)
−0.104443 + 0.994531i \(0.533306\pi\)
\(468\) 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 \(-0.462719\pi\)
0.116853 + 0.993149i \(0.462719\pi\)
\(480\) 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 \(-0.619766\pi\)
−0.367440 + 0.930047i \(0.619766\pi\)
\(504\) 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 \(-0.597349\pi\)
−0.301084 + 0.953598i \(0.597349\pi\)
\(510\) 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 \(-0.690496\pi\)
−0.563371 + 0.826204i \(0.690496\pi\)
\(522\) −136.492 −0.0114446
\(523\) 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
\(528\) 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 \(-0.651144\pi\)
−0.457190 + 0.889369i \(0.651144\pi\)
\(564\) 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 \(-0.674799\pi\)
−0.521959 + 0.852971i \(0.674799\pi\)
\(578\) −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 \(-0.371995\pi\)
0.391388 + 0.920226i \(0.371995\pi\)
\(588\) 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 \(-0.259921\pi\)
0.684728 + 0.728799i \(0.259921\pi\)
\(594\) 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 \(-0.728683\pi\)
−0.658204 + 0.752840i \(0.728683\pi\)
\(602\) 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 \(-0.662283\pi\)
−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
\(612\) −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 \(-0.372379\pi\)
0.390278 + 0.920697i \(0.372379\pi\)
\(620\) −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 \(-0.391190\pi\)
0.335217 + 0.942141i \(0.391190\pi\)
\(644\) 0 0
\(645\) 6507.38 0.397252
\(646\) 611.278 0.0372297
\(647\) 18406.1 1.11842 0.559211 0.829025i \(-0.311104\pi\)
0.559211 + 0.829025i \(0.311104\pi\)
\(648\) −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 \(-0.774812\pi\)
−0.760023 + 0.649896i \(0.774812\pi\)
\(662\) 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 \(-0.232105\pi\)
0.745720 + 0.666259i \(0.232105\pi\)
\(678\) 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 \(-0.264576\pi\)
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
\(696\) 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 \(-0.499324\pi\)
0.00212360 + 0.999998i \(0.499324\pi\)
\(720\) 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 \(-0.188290\pi\)
0.830088 + 0.557632i \(0.188290\pi\)
\(728\) 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 \(-0.459237\pi\)
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
\(738\) −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 \(-0.445052\pi\)
0.171769 + 0.985137i \(0.445052\pi\)
\(762\) 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 \(-0.443482\pi\)
0.176625 + 0.984278i \(0.443482\pi\)
\(770\) 0 0
\(771\) −2097.35 −0.0979693
\(772\) −5003.00 −0.233241
\(773\) 24832.6 1.15546 0.577728 0.816229i \(-0.303940\pi\)
0.577728 + 0.816229i \(0.303940\pi\)
\(774\) 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 \(-0.807353\pi\)
−0.822378 + 0.568941i \(0.807353\pi\)
\(788\) 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 \(-0.748459\pi\)
−0.703675 + 0.710522i \(0.748459\pi\)
\(798\) 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 \(-0.618062\pi\)
−0.362457 + 0.932000i \(0.618062\pi\)
\(812\) 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 \(-0.337173\pi\)
0.489519 + 0.871993i \(0.337173\pi\)
\(830\) −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 \(-0.611891\pi\)
−0.344322 + 0.938851i \(0.611891\pi\)
\(840\) 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 \(-0.433757\pi\)
0.206609 + 0.978424i \(0.433757\pi\)
\(854\) 0 0
\(855\) 5657.12 0.226280
\(856\) −3366.81 −0.134434
\(857\) 32788.6 1.30693 0.653463 0.756958i \(-0.273315\pi\)
0.653463 + 0.756958i \(0.273315\pi\)
\(858\) −1635.01 −0.0650565
\(859\) 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 \(-0.385857\pi\)
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 \(-0.785316\pi\)
−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
\(885\) −31504.1 −1.19661
\(886\) −292.404 −0.0110875
\(887\) −33210.7 −1.25717 −0.628583 0.777742i \(-0.716365\pi\)
−0.628583 + 0.777742i \(0.716365\pi\)
\(888\) −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 \(-0.311890\pi\)
0.557161 + 0.830404i \(0.311890\pi\)
\(930\) 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 \(-0.374532\pi\)
0.384042 + 0.923316i \(0.374532\pi\)
\(938\) 0 0
\(939\) 26538.1 0.922299
\(940\) 38434.5 1.33361
\(941\) 32538.6 1.12724 0.563618 0.826036i \(-0.309409\pi\)
0.563618 + 0.826036i \(0.309409\pi\)
\(942\) −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 \(-0.573085\pi\)
−0.227590 + 0.973757i \(0.573085\pi\)
\(972\) −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 \(-0.744345\pi\)
−0.694435 + 0.719556i \(0.744345\pi\)
\(984\) 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 \(-0.389213\pi\)
0.341063 + 0.940040i \(0.389213\pi\)
\(998\) −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.m.1.2 3
3.2 odd 2 441.4.a.t.1.2 3
4.3 odd 2 2352.4.a.cg.1.3 3
7.2 even 3 147.4.e.n.67.2 6
7.3 odd 6 21.4.e.b.16.2 yes 6
7.4 even 3 147.4.e.n.79.2 6
7.5 odd 6 21.4.e.b.4.2 6
7.6 odd 2 147.4.a.l.1.2 3
21.2 odd 6 441.4.e.w.361.2 6
21.5 even 6 63.4.e.c.46.2 6
21.11 odd 6 441.4.e.w.226.2 6
21.17 even 6 63.4.e.c.37.2 6
21.20 even 2 441.4.a.s.1.2 3
28.3 even 6 336.4.q.k.289.3 6
28.19 even 6 336.4.q.k.193.3 6
28.27 even 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 7.5 odd 6
21.4.e.b.16.2 yes 6 7.3 odd 6
63.4.e.c.37.2 6 21.17 even 6
63.4.e.c.46.2 6 21.5 even 6
147.4.a.l.1.2 3 7.6 odd 2
147.4.a.m.1.2 3 1.1 even 1 trivial
147.4.e.n.67.2 6 7.2 even 3
147.4.e.n.79.2 6 7.4 even 3
336.4.q.k.193.3 6 28.19 even 6
336.4.q.k.289.3 6 28.3 even 6
441.4.a.s.1.2 3 21.20 even 2
441.4.a.t.1.2 3 3.2 odd 2
441.4.e.w.226.2 6 21.11 odd 6
441.4.e.w.361.2 6 21.2 odd 6
2352.4.a.cg.1.3 3 4.3 odd 2
2352.4.a.ci.1.1 3 28.27 even 2