Properties

Label 1849.4.a.i.1.24
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.24
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.902811 q^{2} -9.58399 q^{3} -7.18493 q^{4} -2.03345 q^{5} +8.65253 q^{6} +26.8278 q^{7} +13.7091 q^{8} +64.8528 q^{9} +O(q^{10})\) \(q-0.902811 q^{2} -9.58399 q^{3} -7.18493 q^{4} -2.03345 q^{5} +8.65253 q^{6} +26.8278 q^{7} +13.7091 q^{8} +64.8528 q^{9} +1.83582 q^{10} +38.6799 q^{11} +68.8603 q^{12} -37.4866 q^{13} -24.2205 q^{14} +19.4886 q^{15} +45.1027 q^{16} -12.0881 q^{17} -58.5498 q^{18} -64.0712 q^{19} +14.6102 q^{20} -257.118 q^{21} -34.9206 q^{22} -85.2739 q^{23} -131.388 q^{24} -120.865 q^{25} +33.8433 q^{26} -362.781 q^{27} -192.756 q^{28} +129.469 q^{29} -17.5945 q^{30} +261.836 q^{31} -150.392 q^{32} -370.707 q^{33} +10.9133 q^{34} -54.5531 q^{35} -465.963 q^{36} +310.147 q^{37} +57.8442 q^{38} +359.271 q^{39} -27.8768 q^{40} -333.686 q^{41} +232.129 q^{42} -277.912 q^{44} -131.875 q^{45} +76.9862 q^{46} -475.663 q^{47} -432.264 q^{48} +376.733 q^{49} +109.118 q^{50} +115.852 q^{51} +269.339 q^{52} +14.1574 q^{53} +327.523 q^{54} -78.6536 q^{55} +367.786 q^{56} +614.057 q^{57} -116.886 q^{58} -652.202 q^{59} -140.024 q^{60} -233.602 q^{61} -236.388 q^{62} +1739.86 q^{63} -225.046 q^{64} +76.2271 q^{65} +334.679 q^{66} +963.472 q^{67} +86.8523 q^{68} +817.264 q^{69} +49.2511 q^{70} +887.185 q^{71} +889.075 q^{72} -345.150 q^{73} -280.004 q^{74} +1158.37 q^{75} +460.347 q^{76} +1037.70 q^{77} -324.354 q^{78} -396.269 q^{79} -91.7141 q^{80} +1725.86 q^{81} +301.255 q^{82} +539.877 q^{83} +1847.37 q^{84} +24.5806 q^{85} -1240.83 q^{87} +530.267 q^{88} +295.412 q^{89} +119.058 q^{90} -1005.68 q^{91} +612.687 q^{92} -2509.43 q^{93} +429.433 q^{94} +130.286 q^{95} +1441.36 q^{96} +191.963 q^{97} -340.119 q^{98} +2508.50 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.902811 −0.319192 −0.159596 0.987182i \(-0.551019\pi\)
−0.159596 + 0.987182i \(0.551019\pi\)
\(3\) −9.58399 −1.84444 −0.922220 0.386667i \(-0.873626\pi\)
−0.922220 + 0.386667i \(0.873626\pi\)
\(4\) −7.18493 −0.898117
\(5\) −2.03345 −0.181877 −0.0909387 0.995856i \(-0.528987\pi\)
−0.0909387 + 0.995856i \(0.528987\pi\)
\(6\) 8.65253 0.588730
\(7\) 26.8278 1.44857 0.724284 0.689502i \(-0.242171\pi\)
0.724284 + 0.689502i \(0.242171\pi\)
\(8\) 13.7091 0.605863
\(9\) 64.8528 2.40196
\(10\) 1.83582 0.0580538
\(11\) 38.6799 1.06022 0.530110 0.847929i \(-0.322151\pi\)
0.530110 + 0.847929i \(0.322151\pi\)
\(12\) 68.8603 1.65652
\(13\) −37.4866 −0.799762 −0.399881 0.916567i \(-0.630949\pi\)
−0.399881 + 0.916567i \(0.630949\pi\)
\(14\) −24.2205 −0.462371
\(15\) 19.4886 0.335462
\(16\) 45.1027 0.704730
\(17\) −12.0881 −0.172459 −0.0862294 0.996275i \(-0.527482\pi\)
−0.0862294 + 0.996275i \(0.527482\pi\)
\(18\) −58.5498 −0.766685
\(19\) −64.0712 −0.773628 −0.386814 0.922158i \(-0.626424\pi\)
−0.386814 + 0.922158i \(0.626424\pi\)
\(20\) 14.6102 0.163347
\(21\) −257.118 −2.67179
\(22\) −34.9206 −0.338414
\(23\) −85.2739 −0.773080 −0.386540 0.922273i \(-0.626330\pi\)
−0.386540 + 0.922273i \(0.626330\pi\)
\(24\) −131.388 −1.11748
\(25\) −120.865 −0.966921
\(26\) 33.8433 0.255278
\(27\) −362.781 −2.58582
\(28\) −192.756 −1.30098
\(29\) 129.469 0.829029 0.414514 0.910043i \(-0.363951\pi\)
0.414514 + 0.910043i \(0.363951\pi\)
\(30\) −17.5945 −0.107077
\(31\) 261.836 1.51700 0.758501 0.651671i \(-0.225932\pi\)
0.758501 + 0.651671i \(0.225932\pi\)
\(32\) −150.392 −0.830808
\(33\) −370.707 −1.95551
\(34\) 10.9133 0.0550474
\(35\) −54.5531 −0.263462
\(36\) −465.963 −2.15724
\(37\) 310.147 1.37805 0.689024 0.724739i \(-0.258039\pi\)
0.689024 + 0.724739i \(0.258039\pi\)
\(38\) 57.8442 0.246936
\(39\) 359.271 1.47511
\(40\) −27.8768 −0.110193
\(41\) −333.686 −1.27105 −0.635524 0.772081i \(-0.719216\pi\)
−0.635524 + 0.772081i \(0.719216\pi\)
\(42\) 232.129 0.852815
\(43\) 0 0
\(44\) −277.912 −0.952201
\(45\) −131.875 −0.436861
\(46\) 76.9862 0.246761
\(47\) −475.663 −1.47622 −0.738112 0.674678i \(-0.764282\pi\)
−0.738112 + 0.674678i \(0.764282\pi\)
\(48\) −432.264 −1.29983
\(49\) 376.733 1.09835
\(50\) 109.118 0.308633
\(51\) 115.852 0.318090
\(52\) 269.339 0.718280
\(53\) 14.1574 0.0366919 0.0183460 0.999832i \(-0.494160\pi\)
0.0183460 + 0.999832i \(0.494160\pi\)
\(54\) 327.523 0.825374
\(55\) −78.6536 −0.192830
\(56\) 367.786 0.877634
\(57\) 614.057 1.42691
\(58\) −116.886 −0.264619
\(59\) −652.202 −1.43914 −0.719571 0.694419i \(-0.755662\pi\)
−0.719571 + 0.694419i \(0.755662\pi\)
\(60\) −140.024 −0.301284
\(61\) −233.602 −0.490323 −0.245162 0.969482i \(-0.578841\pi\)
−0.245162 + 0.969482i \(0.578841\pi\)
\(62\) −236.388 −0.484215
\(63\) 1739.86 3.47939
\(64\) −225.046 −0.439543
\(65\) 76.2271 0.145459
\(66\) 334.679 0.624183
\(67\) 963.472 1.75682 0.878409 0.477909i \(-0.158605\pi\)
0.878409 + 0.477909i \(0.158605\pi\)
\(68\) 86.8523 0.154888
\(69\) 817.264 1.42590
\(70\) 49.2511 0.0840948
\(71\) 887.185 1.48295 0.741475 0.670980i \(-0.234126\pi\)
0.741475 + 0.670980i \(0.234126\pi\)
\(72\) 889.075 1.45526
\(73\) −345.150 −0.553380 −0.276690 0.960959i \(-0.589238\pi\)
−0.276690 + 0.960959i \(0.589238\pi\)
\(74\) −280.004 −0.439862
\(75\) 1158.37 1.78343
\(76\) 460.347 0.694808
\(77\) 1037.70 1.53580
\(78\) −324.354 −0.470844
\(79\) −396.269 −0.564351 −0.282176 0.959363i \(-0.591056\pi\)
−0.282176 + 0.959363i \(0.591056\pi\)
\(80\) −91.7141 −0.128174
\(81\) 1725.86 2.36744
\(82\) 301.255 0.405708
\(83\) 539.877 0.713966 0.356983 0.934111i \(-0.383805\pi\)
0.356983 + 0.934111i \(0.383805\pi\)
\(84\) 1847.37 2.39958
\(85\) 24.5806 0.0313663
\(86\) 0 0
\(87\) −1240.83 −1.52909
\(88\) 530.267 0.642349
\(89\) 295.412 0.351838 0.175919 0.984405i \(-0.443710\pi\)
0.175919 + 0.984405i \(0.443710\pi\)
\(90\) 119.058 0.139443
\(91\) −1005.68 −1.15851
\(92\) 612.687 0.694316
\(93\) −2509.43 −2.79802
\(94\) 429.433 0.471199
\(95\) 130.286 0.140705
\(96\) 1441.36 1.53237
\(97\) 191.963 0.200937 0.100469 0.994940i \(-0.467966\pi\)
0.100469 + 0.994940i \(0.467966\pi\)
\(98\) −340.119 −0.350584
\(99\) 2508.50 2.54660
\(100\) 868.407 0.868407
\(101\) −1717.20 −1.69176 −0.845878 0.533377i \(-0.820923\pi\)
−0.845878 + 0.533377i \(0.820923\pi\)
\(102\) −104.593 −0.101532
\(103\) −31.9437 −0.0305583 −0.0152792 0.999883i \(-0.504864\pi\)
−0.0152792 + 0.999883i \(0.504864\pi\)
\(104\) −513.908 −0.484547
\(105\) 522.836 0.485939
\(106\) −12.7815 −0.0117118
\(107\) 1275.01 1.15196 0.575980 0.817463i \(-0.304620\pi\)
0.575980 + 0.817463i \(0.304620\pi\)
\(108\) 2606.56 2.32237
\(109\) −1181.87 −1.03856 −0.519278 0.854606i \(-0.673799\pi\)
−0.519278 + 0.854606i \(0.673799\pi\)
\(110\) 71.0094 0.0615498
\(111\) −2972.44 −2.54173
\(112\) 1210.01 1.02085
\(113\) 1007.21 0.838495 0.419248 0.907872i \(-0.362294\pi\)
0.419248 + 0.907872i \(0.362294\pi\)
\(114\) −554.378 −0.455458
\(115\) 173.400 0.140606
\(116\) −930.228 −0.744564
\(117\) −2431.11 −1.92099
\(118\) 588.815 0.459363
\(119\) −324.298 −0.249818
\(120\) 267.171 0.203244
\(121\) 165.132 0.124066
\(122\) 210.899 0.156507
\(123\) 3198.04 2.34437
\(124\) −1881.27 −1.36245
\(125\) 499.955 0.357738
\(126\) −1570.77 −1.11059
\(127\) −694.035 −0.484926 −0.242463 0.970161i \(-0.577955\pi\)
−0.242463 + 0.970161i \(0.577955\pi\)
\(128\) 1406.31 0.971106
\(129\) 0 0
\(130\) −68.8187 −0.0464292
\(131\) −1014.00 −0.676286 −0.338143 0.941095i \(-0.609799\pi\)
−0.338143 + 0.941095i \(0.609799\pi\)
\(132\) 2663.51 1.75628
\(133\) −1718.89 −1.12065
\(134\) −869.833 −0.560762
\(135\) 737.697 0.470303
\(136\) −165.717 −0.104486
\(137\) 633.186 0.394867 0.197433 0.980316i \(-0.436739\pi\)
0.197433 + 0.980316i \(0.436739\pi\)
\(138\) −737.835 −0.455135
\(139\) −437.269 −0.266825 −0.133412 0.991061i \(-0.542594\pi\)
−0.133412 + 0.991061i \(0.542594\pi\)
\(140\) 391.960 0.236619
\(141\) 4558.74 2.72280
\(142\) −800.961 −0.473346
\(143\) −1449.98 −0.847924
\(144\) 2925.04 1.69273
\(145\) −263.269 −0.150782
\(146\) 311.605 0.176635
\(147\) −3610.60 −2.02583
\(148\) −2228.38 −1.23765
\(149\) 2052.33 1.12841 0.564205 0.825635i \(-0.309183\pi\)
0.564205 + 0.825635i \(0.309183\pi\)
\(150\) −1045.79 −0.569255
\(151\) 1021.62 0.550583 0.275292 0.961361i \(-0.411226\pi\)
0.275292 + 0.961361i \(0.411226\pi\)
\(152\) −878.360 −0.468713
\(153\) −783.948 −0.414238
\(154\) −936.845 −0.490215
\(155\) −532.430 −0.275909
\(156\) −2581.34 −1.32482
\(157\) 183.529 0.0932942 0.0466471 0.998911i \(-0.485146\pi\)
0.0466471 + 0.998911i \(0.485146\pi\)
\(158\) 357.756 0.180136
\(159\) −135.685 −0.0676760
\(160\) 305.815 0.151105
\(161\) −2287.71 −1.11986
\(162\) −1558.13 −0.755666
\(163\) 832.248 0.399918 0.199959 0.979804i \(-0.435919\pi\)
0.199959 + 0.979804i \(0.435919\pi\)
\(164\) 2397.51 1.14155
\(165\) 753.815 0.355663
\(166\) −487.407 −0.227892
\(167\) −319.899 −0.148231 −0.0741153 0.997250i \(-0.523613\pi\)
−0.0741153 + 0.997250i \(0.523613\pi\)
\(168\) −3524.86 −1.61874
\(169\) −791.756 −0.360380
\(170\) −22.1916 −0.0100119
\(171\) −4155.20 −1.85822
\(172\) 0 0
\(173\) −3014.11 −1.32462 −0.662309 0.749231i \(-0.730423\pi\)
−0.662309 + 0.749231i \(0.730423\pi\)
\(174\) 1120.24 0.488074
\(175\) −3242.55 −1.40065
\(176\) 1744.57 0.747169
\(177\) 6250.69 2.65441
\(178\) −266.701 −0.112304
\(179\) 3477.58 1.45211 0.726053 0.687639i \(-0.241353\pi\)
0.726053 + 0.687639i \(0.241353\pi\)
\(180\) 947.513 0.392352
\(181\) 7.73635 0.00317701 0.00158850 0.999999i \(-0.499494\pi\)
0.00158850 + 0.999999i \(0.499494\pi\)
\(182\) 907.943 0.369787
\(183\) 2238.84 0.904372
\(184\) −1169.03 −0.468381
\(185\) −630.668 −0.250636
\(186\) 2265.54 0.893105
\(187\) −467.567 −0.182844
\(188\) 3417.60 1.32582
\(189\) −9732.63 −3.74574
\(190\) −117.623 −0.0449121
\(191\) −1301.75 −0.493147 −0.246574 0.969124i \(-0.579305\pi\)
−0.246574 + 0.969124i \(0.579305\pi\)
\(192\) 2156.84 0.810710
\(193\) −760.523 −0.283646 −0.141823 0.989892i \(-0.545296\pi\)
−0.141823 + 0.989892i \(0.545296\pi\)
\(194\) −173.306 −0.0641375
\(195\) −730.560 −0.268290
\(196\) −2706.80 −0.986444
\(197\) −200.537 −0.0725263 −0.0362632 0.999342i \(-0.511545\pi\)
−0.0362632 + 0.999342i \(0.511545\pi\)
\(198\) −2264.70 −0.812855
\(199\) −1878.65 −0.669217 −0.334608 0.942357i \(-0.608604\pi\)
−0.334608 + 0.942357i \(0.608604\pi\)
\(200\) −1656.95 −0.585822
\(201\) −9233.90 −3.24034
\(202\) 1550.30 0.539995
\(203\) 3473.38 1.20090
\(204\) −832.391 −0.285682
\(205\) 678.534 0.231175
\(206\) 28.8391 0.00975397
\(207\) −5530.25 −1.85690
\(208\) −1690.75 −0.563616
\(209\) −2478.26 −0.820216
\(210\) −472.022 −0.155108
\(211\) −907.815 −0.296192 −0.148096 0.988973i \(-0.547315\pi\)
−0.148096 + 0.988973i \(0.547315\pi\)
\(212\) −101.720 −0.0329536
\(213\) −8502.77 −2.73521
\(214\) −1151.09 −0.367697
\(215\) 0 0
\(216\) −4973.41 −1.56666
\(217\) 7024.49 2.19748
\(218\) 1067.00 0.331498
\(219\) 3307.91 1.02068
\(220\) 565.121 0.173184
\(221\) 453.142 0.137926
\(222\) 2683.55 0.811298
\(223\) −2724.67 −0.818193 −0.409097 0.912491i \(-0.634156\pi\)
−0.409097 + 0.912491i \(0.634156\pi\)
\(224\) −4034.70 −1.20348
\(225\) −7838.44 −2.32250
\(226\) −909.317 −0.267641
\(227\) 2297.13 0.671655 0.335828 0.941923i \(-0.390984\pi\)
0.335828 + 0.941923i \(0.390984\pi\)
\(228\) −4411.96 −1.28153
\(229\) 3034.87 0.875763 0.437882 0.899033i \(-0.355729\pi\)
0.437882 + 0.899033i \(0.355729\pi\)
\(230\) −156.548 −0.0448802
\(231\) −9945.28 −2.83269
\(232\) 1774.91 0.502278
\(233\) 4994.74 1.40436 0.702180 0.711999i \(-0.252210\pi\)
0.702180 + 0.711999i \(0.252210\pi\)
\(234\) 2194.83 0.613166
\(235\) 967.237 0.268492
\(236\) 4686.02 1.29252
\(237\) 3797.84 1.04091
\(238\) 292.780 0.0797399
\(239\) −2939.14 −0.795470 −0.397735 0.917500i \(-0.630204\pi\)
−0.397735 + 0.917500i \(0.630204\pi\)
\(240\) 878.987 0.236410
\(241\) −4981.47 −1.33147 −0.665736 0.746187i \(-0.731882\pi\)
−0.665736 + 0.746187i \(0.731882\pi\)
\(242\) −149.083 −0.0396010
\(243\) −6745.54 −1.78077
\(244\) 1678.42 0.440368
\(245\) −766.068 −0.199765
\(246\) −2887.23 −0.748304
\(247\) 2401.81 0.618719
\(248\) 3589.54 0.919097
\(249\) −5174.17 −1.31687
\(250\) −451.365 −0.114187
\(251\) 5028.93 1.26463 0.632317 0.774709i \(-0.282104\pi\)
0.632317 + 0.774709i \(0.282104\pi\)
\(252\) −12500.8 −3.12490
\(253\) −3298.38 −0.819634
\(254\) 626.583 0.154785
\(255\) −235.580 −0.0578533
\(256\) 530.733 0.129573
\(257\) −846.260 −0.205402 −0.102701 0.994712i \(-0.532748\pi\)
−0.102701 + 0.994712i \(0.532748\pi\)
\(258\) 0 0
\(259\) 8320.56 1.99620
\(260\) −547.687 −0.130639
\(261\) 8396.44 1.99129
\(262\) 915.449 0.215865
\(263\) −7725.74 −1.81137 −0.905684 0.423954i \(-0.860642\pi\)
−0.905684 + 0.423954i \(0.860642\pi\)
\(264\) −5082.07 −1.18477
\(265\) −28.7884 −0.00667343
\(266\) 1551.83 0.357703
\(267\) −2831.23 −0.648944
\(268\) −6922.48 −1.57783
\(269\) −1281.93 −0.290559 −0.145279 0.989391i \(-0.546408\pi\)
−0.145279 + 0.989391i \(0.546408\pi\)
\(270\) −666.001 −0.150117
\(271\) 1156.22 0.259171 0.129585 0.991568i \(-0.458635\pi\)
0.129585 + 0.991568i \(0.458635\pi\)
\(272\) −545.207 −0.121537
\(273\) 9638.46 2.13680
\(274\) −571.648 −0.126038
\(275\) −4675.05 −1.02515
\(276\) −5871.98 −1.28062
\(277\) 2086.27 0.452533 0.226266 0.974065i \(-0.427348\pi\)
0.226266 + 0.974065i \(0.427348\pi\)
\(278\) 394.771 0.0851684
\(279\) 16980.8 3.64377
\(280\) −747.875 −0.159622
\(281\) 1154.08 0.245006 0.122503 0.992468i \(-0.460908\pi\)
0.122503 + 0.992468i \(0.460908\pi\)
\(282\) −4115.68 −0.869097
\(283\) 5288.51 1.11085 0.555423 0.831568i \(-0.312556\pi\)
0.555423 + 0.831568i \(0.312556\pi\)
\(284\) −6374.37 −1.33186
\(285\) −1248.66 −0.259523
\(286\) 1309.05 0.270650
\(287\) −8952.07 −1.84120
\(288\) −9753.36 −1.99556
\(289\) −4766.88 −0.970258
\(290\) 237.683 0.0481283
\(291\) −1839.77 −0.370616
\(292\) 2479.88 0.497000
\(293\) −3079.73 −0.614060 −0.307030 0.951700i \(-0.599335\pi\)
−0.307030 + 0.951700i \(0.599335\pi\)
\(294\) 3259.69 0.646630
\(295\) 1326.22 0.261747
\(296\) 4251.84 0.834909
\(297\) −14032.3 −2.74154
\(298\) −1852.86 −0.360179
\(299\) 3196.63 0.618280
\(300\) −8322.80 −1.60172
\(301\) 0 0
\(302\) −922.328 −0.175742
\(303\) 16457.6 3.12034
\(304\) −2889.78 −0.545199
\(305\) 475.019 0.0891787
\(306\) 707.757 0.132222
\(307\) −639.484 −0.118884 −0.0594419 0.998232i \(-0.518932\pi\)
−0.0594419 + 0.998232i \(0.518932\pi\)
\(308\) −7455.79 −1.37933
\(309\) 306.148 0.0563630
\(310\) 480.684 0.0880678
\(311\) 7068.39 1.28878 0.644391 0.764696i \(-0.277111\pi\)
0.644391 + 0.764696i \(0.277111\pi\)
\(312\) 4925.29 0.893717
\(313\) 5479.55 0.989528 0.494764 0.869027i \(-0.335254\pi\)
0.494764 + 0.869027i \(0.335254\pi\)
\(314\) −165.692 −0.0297787
\(315\) −3537.92 −0.632823
\(316\) 2847.17 0.506853
\(317\) 6514.68 1.15426 0.577131 0.816652i \(-0.304172\pi\)
0.577131 + 0.816652i \(0.304172\pi\)
\(318\) 122.498 0.0216016
\(319\) 5007.85 0.878953
\(320\) 457.620 0.0799429
\(321\) −12219.7 −2.12472
\(322\) 2065.37 0.357450
\(323\) 774.500 0.133419
\(324\) −12400.2 −2.12623
\(325\) 4530.82 0.773307
\(326\) −751.363 −0.127651
\(327\) 11327.0 1.91555
\(328\) −4574.54 −0.770081
\(329\) −12761.0 −2.13841
\(330\) −680.553 −0.113525
\(331\) −3774.59 −0.626799 −0.313399 0.949621i \(-0.601468\pi\)
−0.313399 + 0.949621i \(0.601468\pi\)
\(332\) −3878.98 −0.641225
\(333\) 20113.9 3.31001
\(334\) 288.808 0.0473140
\(335\) −1959.17 −0.319526
\(336\) −11596.7 −1.88289
\(337\) 11690.7 1.88972 0.944859 0.327478i \(-0.106199\pi\)
0.944859 + 0.327478i \(0.106199\pi\)
\(338\) 714.806 0.115031
\(339\) −9653.05 −1.54655
\(340\) −176.610 −0.0281706
\(341\) 10127.8 1.60836
\(342\) 3751.36 0.593129
\(343\) 904.985 0.142462
\(344\) 0 0
\(345\) −1661.87 −0.259339
\(346\) 2721.18 0.422807
\(347\) −11140.5 −1.72349 −0.861746 0.507340i \(-0.830629\pi\)
−0.861746 + 0.507340i \(0.830629\pi\)
\(348\) 8915.29 1.37330
\(349\) 1226.84 0.188170 0.0940852 0.995564i \(-0.470007\pi\)
0.0940852 + 0.995564i \(0.470007\pi\)
\(350\) 2927.41 0.447076
\(351\) 13599.4 2.06804
\(352\) −5817.15 −0.880839
\(353\) −1582.18 −0.238558 −0.119279 0.992861i \(-0.538058\pi\)
−0.119279 + 0.992861i \(0.538058\pi\)
\(354\) −5643.19 −0.847267
\(355\) −1804.05 −0.269715
\(356\) −2122.52 −0.315992
\(357\) 3108.07 0.460774
\(358\) −3139.60 −0.463500
\(359\) −613.266 −0.0901586 −0.0450793 0.998983i \(-0.514354\pi\)
−0.0450793 + 0.998983i \(0.514354\pi\)
\(360\) −1807.89 −0.264678
\(361\) −2753.88 −0.401499
\(362\) −6.98446 −0.00101407
\(363\) −1582.63 −0.228833
\(364\) 7225.77 1.04048
\(365\) 701.846 0.100647
\(366\) −2021.25 −0.288668
\(367\) 2202.45 0.313261 0.156631 0.987657i \(-0.449937\pi\)
0.156631 + 0.987657i \(0.449937\pi\)
\(368\) −3846.08 −0.544812
\(369\) −21640.5 −3.05300
\(370\) 569.374 0.0800009
\(371\) 379.813 0.0531507
\(372\) 18030.1 2.51295
\(373\) 6268.78 0.870201 0.435101 0.900382i \(-0.356713\pi\)
0.435101 + 0.900382i \(0.356713\pi\)
\(374\) 422.124 0.0583624
\(375\) −4791.56 −0.659827
\(376\) −6520.92 −0.894390
\(377\) −4853.36 −0.663026
\(378\) 8786.72 1.19561
\(379\) 3247.19 0.440097 0.220049 0.975489i \(-0.429378\pi\)
0.220049 + 0.975489i \(0.429378\pi\)
\(380\) −936.093 −0.126370
\(381\) 6651.62 0.894417
\(382\) 1175.23 0.157409
\(383\) −8938.80 −1.19256 −0.596281 0.802776i \(-0.703356\pi\)
−0.596281 + 0.802776i \(0.703356\pi\)
\(384\) −13478.1 −1.79115
\(385\) −2110.11 −0.279327
\(386\) 686.609 0.0905375
\(387\) 0 0
\(388\) −1379.24 −0.180465
\(389\) −4626.84 −0.603060 −0.301530 0.953457i \(-0.597497\pi\)
−0.301530 + 0.953457i \(0.597497\pi\)
\(390\) 659.558 0.0856359
\(391\) 1030.80 0.133324
\(392\) 5164.68 0.665448
\(393\) 9718.15 1.24737
\(394\) 181.047 0.0231498
\(395\) 805.794 0.102643
\(396\) −18023.4 −2.28714
\(397\) 10840.9 1.37051 0.685253 0.728305i \(-0.259692\pi\)
0.685253 + 0.728305i \(0.259692\pi\)
\(398\) 1696.07 0.213609
\(399\) 16473.8 2.06698
\(400\) −5451.34 −0.681418
\(401\) −12391.8 −1.54319 −0.771594 0.636116i \(-0.780540\pi\)
−0.771594 + 0.636116i \(0.780540\pi\)
\(402\) 8336.47 1.03429
\(403\) −9815.33 −1.21324
\(404\) 12337.9 1.51939
\(405\) −3509.45 −0.430583
\(406\) −3135.81 −0.383319
\(407\) 11996.4 1.46103
\(408\) 1588.23 0.192719
\(409\) 7910.71 0.956380 0.478190 0.878256i \(-0.341293\pi\)
0.478190 + 0.878256i \(0.341293\pi\)
\(410\) −612.588 −0.0737891
\(411\) −6068.45 −0.728308
\(412\) 229.513 0.0274449
\(413\) −17497.2 −2.08469
\(414\) 4992.77 0.592708
\(415\) −1097.81 −0.129854
\(416\) 5637.69 0.664448
\(417\) 4190.78 0.492142
\(418\) 2237.41 0.261806
\(419\) −12901.8 −1.50428 −0.752142 0.659001i \(-0.770979\pi\)
−0.752142 + 0.659001i \(0.770979\pi\)
\(420\) −3756.54 −0.436430
\(421\) −16075.1 −1.86093 −0.930464 0.366382i \(-0.880596\pi\)
−0.930464 + 0.366382i \(0.880596\pi\)
\(422\) 819.586 0.0945422
\(423\) −30848.1 −3.54582
\(424\) 194.086 0.0222303
\(425\) 1461.03 0.166754
\(426\) 7676.40 0.873058
\(427\) −6267.05 −0.710266
\(428\) −9160.86 −1.03460
\(429\) 13896.6 1.56394
\(430\) 0 0
\(431\) −9277.47 −1.03684 −0.518422 0.855125i \(-0.673480\pi\)
−0.518422 + 0.855125i \(0.673480\pi\)
\(432\) −16362.4 −1.82231
\(433\) −182.499 −0.0202548 −0.0101274 0.999949i \(-0.503224\pi\)
−0.0101274 + 0.999949i \(0.503224\pi\)
\(434\) −6341.79 −0.701418
\(435\) 2523.17 0.278107
\(436\) 8491.65 0.932744
\(437\) 5463.60 0.598076
\(438\) −2986.42 −0.325792
\(439\) −5806.09 −0.631229 −0.315614 0.948888i \(-0.602211\pi\)
−0.315614 + 0.948888i \(0.602211\pi\)
\(440\) −1078.27 −0.116829
\(441\) 24432.2 2.63818
\(442\) −409.102 −0.0440249
\(443\) 13300.1 1.42643 0.713214 0.700946i \(-0.247239\pi\)
0.713214 + 0.700946i \(0.247239\pi\)
\(444\) 21356.8 2.28277
\(445\) −600.706 −0.0639914
\(446\) 2459.86 0.261161
\(447\) −19669.5 −2.08128
\(448\) −6037.49 −0.636707
\(449\) 7189.18 0.755631 0.377815 0.925881i \(-0.376675\pi\)
0.377815 + 0.925881i \(0.376675\pi\)
\(450\) 7076.63 0.741323
\(451\) −12906.9 −1.34759
\(452\) −7236.71 −0.753066
\(453\) −9791.17 −1.01552
\(454\) −2073.87 −0.214387
\(455\) 2045.01 0.210707
\(456\) 8418.19 0.864513
\(457\) −4199.15 −0.429820 −0.214910 0.976634i \(-0.568946\pi\)
−0.214910 + 0.976634i \(0.568946\pi\)
\(458\) −2739.91 −0.279537
\(459\) 4385.34 0.445948
\(460\) −1245.87 −0.126280
\(461\) −1185.60 −0.119781 −0.0598905 0.998205i \(-0.519075\pi\)
−0.0598905 + 0.998205i \(0.519075\pi\)
\(462\) 8978.71 0.904172
\(463\) −2522.44 −0.253192 −0.126596 0.991954i \(-0.540405\pi\)
−0.126596 + 0.991954i \(0.540405\pi\)
\(464\) 5839.41 0.584241
\(465\) 5102.80 0.508897
\(466\) −4509.30 −0.448261
\(467\) 9629.56 0.954182 0.477091 0.878854i \(-0.341691\pi\)
0.477091 + 0.878854i \(0.341691\pi\)
\(468\) 17467.4 1.72528
\(469\) 25847.9 2.54487
\(470\) −873.232 −0.0857004
\(471\) −1758.94 −0.172075
\(472\) −8941.11 −0.871924
\(473\) 0 0
\(474\) −3428.73 −0.332251
\(475\) 7743.97 0.748037
\(476\) 2330.06 0.224366
\(477\) 918.149 0.0881324
\(478\) 2653.49 0.253908
\(479\) 12358.5 1.17886 0.589431 0.807819i \(-0.299352\pi\)
0.589431 + 0.807819i \(0.299352\pi\)
\(480\) −2930.93 −0.278704
\(481\) −11626.3 −1.10211
\(482\) 4497.33 0.424995
\(483\) 21925.4 2.06551
\(484\) −1186.47 −0.111426
\(485\) −390.348 −0.0365459
\(486\) 6089.95 0.568407
\(487\) 2723.31 0.253398 0.126699 0.991941i \(-0.459562\pi\)
0.126699 + 0.991941i \(0.459562\pi\)
\(488\) −3202.49 −0.297069
\(489\) −7976.25 −0.737625
\(490\) 691.615 0.0637632
\(491\) −4199.46 −0.385986 −0.192993 0.981200i \(-0.561819\pi\)
−0.192993 + 0.981200i \(0.561819\pi\)
\(492\) −22977.7 −2.10552
\(493\) −1565.04 −0.142973
\(494\) −2168.38 −0.197490
\(495\) −5100.91 −0.463169
\(496\) 11809.5 1.06908
\(497\) 23801.3 2.14815
\(498\) 4671.30 0.420333
\(499\) 19985.3 1.79292 0.896458 0.443129i \(-0.146132\pi\)
0.896458 + 0.443129i \(0.146132\pi\)
\(500\) −3592.14 −0.321291
\(501\) 3065.90 0.273402
\(502\) −4540.18 −0.403661
\(503\) 520.717 0.0461583 0.0230792 0.999734i \(-0.492653\pi\)
0.0230792 + 0.999734i \(0.492653\pi\)
\(504\) 23852.0 2.10804
\(505\) 3491.83 0.307692
\(506\) 2977.82 0.261621
\(507\) 7588.18 0.664700
\(508\) 4986.60 0.435520
\(509\) 1746.69 0.152103 0.0760517 0.997104i \(-0.475769\pi\)
0.0760517 + 0.997104i \(0.475769\pi\)
\(510\) 212.684 0.0184663
\(511\) −9259.63 −0.801609
\(512\) −11729.6 −1.01246
\(513\) 23243.8 2.00046
\(514\) 764.013 0.0655626
\(515\) 64.9560 0.00555787
\(516\) 0 0
\(517\) −18398.6 −1.56512
\(518\) −7511.90 −0.637169
\(519\) 28887.2 2.44318
\(520\) 1045.01 0.0881281
\(521\) −12223.8 −1.02790 −0.513948 0.857821i \(-0.671818\pi\)
−0.513948 + 0.857821i \(0.671818\pi\)
\(522\) −7580.40 −0.635604
\(523\) −12923.8 −1.08053 −0.540265 0.841495i \(-0.681676\pi\)
−0.540265 + 0.841495i \(0.681676\pi\)
\(524\) 7285.51 0.607383
\(525\) 31076.5 2.58341
\(526\) 6974.88 0.578174
\(527\) −3165.10 −0.261620
\(528\) −16719.9 −1.37811
\(529\) −4895.37 −0.402348
\(530\) 25.9905 0.00213011
\(531\) −42297.1 −3.45676
\(532\) 12350.1 1.00648
\(533\) 12508.7 1.01654
\(534\) 2556.06 0.207138
\(535\) −2592.67 −0.209516
\(536\) 13208.4 1.06439
\(537\) −33329.1 −2.67832
\(538\) 1157.34 0.0927441
\(539\) 14572.0 1.16449
\(540\) −5300.30 −0.422387
\(541\) −10061.3 −0.799574 −0.399787 0.916608i \(-0.630916\pi\)
−0.399787 + 0.916608i \(0.630916\pi\)
\(542\) −1043.85 −0.0827252
\(543\) −74.1451 −0.00585980
\(544\) 1817.96 0.143280
\(545\) 2403.27 0.188890
\(546\) −8701.71 −0.682049
\(547\) −23951.3 −1.87218 −0.936090 0.351761i \(-0.885583\pi\)
−0.936090 + 0.351761i \(0.885583\pi\)
\(548\) −4549.40 −0.354636
\(549\) −15149.8 −1.17774
\(550\) 4220.68 0.327219
\(551\) −8295.25 −0.641360
\(552\) 11204.0 0.863900
\(553\) −10631.0 −0.817501
\(554\) −1883.50 −0.144445
\(555\) 6044.31 0.462282
\(556\) 3141.75 0.239640
\(557\) 8803.19 0.669665 0.334832 0.942278i \(-0.391320\pi\)
0.334832 + 0.942278i \(0.391320\pi\)
\(558\) −15330.4 −1.16306
\(559\) 0 0
\(560\) −2460.49 −0.185669
\(561\) 4481.15 0.337245
\(562\) −1041.92 −0.0782038
\(563\) −21876.5 −1.63762 −0.818812 0.574061i \(-0.805367\pi\)
−0.818812 + 0.574061i \(0.805367\pi\)
\(564\) −32754.3 −2.44540
\(565\) −2048.10 −0.152503
\(566\) −4774.53 −0.354573
\(567\) 46301.1 3.42939
\(568\) 12162.5 0.898466
\(569\) −22198.4 −1.63551 −0.817756 0.575565i \(-0.804782\pi\)
−0.817756 + 0.575565i \(0.804782\pi\)
\(570\) 1127.30 0.0828376
\(571\) −13270.8 −0.972619 −0.486310 0.873787i \(-0.661657\pi\)
−0.486310 + 0.873787i \(0.661657\pi\)
\(572\) 10418.0 0.761534
\(573\) 12475.9 0.909580
\(574\) 8082.03 0.587696
\(575\) 10306.6 0.747507
\(576\) −14594.9 −1.05576
\(577\) −15373.6 −1.10921 −0.554603 0.832115i \(-0.687130\pi\)
−0.554603 + 0.832115i \(0.687130\pi\)
\(578\) 4303.59 0.309699
\(579\) 7288.85 0.523168
\(580\) 1891.57 0.135419
\(581\) 14483.7 1.03423
\(582\) 1660.97 0.118298
\(583\) 547.607 0.0389015
\(584\) −4731.71 −0.335273
\(585\) 4943.54 0.349385
\(586\) 2780.41 0.196003
\(587\) 863.918 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(588\) 25942.0 1.81944
\(589\) −16776.1 −1.17360
\(590\) −1197.33 −0.0835477
\(591\) 1921.95 0.133770
\(592\) 13988.4 0.971151
\(593\) −1387.83 −0.0961071 −0.0480536 0.998845i \(-0.515302\pi\)
−0.0480536 + 0.998845i \(0.515302\pi\)
\(594\) 12668.5 0.875078
\(595\) 659.444 0.0454363
\(596\) −14745.8 −1.01344
\(597\) 18005.0 1.23433
\(598\) −2885.95 −0.197350
\(599\) −23119.8 −1.57704 −0.788521 0.615008i \(-0.789153\pi\)
−0.788521 + 0.615008i \(0.789153\pi\)
\(600\) 15880.2 1.08051
\(601\) 29375.9 1.99379 0.996897 0.0787233i \(-0.0250844\pi\)
0.996897 + 0.0787233i \(0.0250844\pi\)
\(602\) 0 0
\(603\) 62483.9 4.21980
\(604\) −7340.26 −0.494488
\(605\) −335.789 −0.0225649
\(606\) −14858.1 −0.995987
\(607\) −10354.5 −0.692383 −0.346191 0.938164i \(-0.612525\pi\)
−0.346191 + 0.938164i \(0.612525\pi\)
\(608\) 9635.81 0.642736
\(609\) −33288.8 −2.21499
\(610\) −428.853 −0.0284651
\(611\) 17831.0 1.18063
\(612\) 5632.61 0.372034
\(613\) −9960.96 −0.656312 −0.328156 0.944624i \(-0.606427\pi\)
−0.328156 + 0.944624i \(0.606427\pi\)
\(614\) 577.334 0.0379467
\(615\) −6503.06 −0.426388
\(616\) 14225.9 0.930485
\(617\) −13796.7 −0.900218 −0.450109 0.892974i \(-0.648615\pi\)
−0.450109 + 0.892974i \(0.648615\pi\)
\(618\) −276.394 −0.0179906
\(619\) 25631.5 1.66432 0.832162 0.554533i \(-0.187103\pi\)
0.832162 + 0.554533i \(0.187103\pi\)
\(620\) 3825.47 0.247798
\(621\) 30935.7 1.99905
\(622\) −6381.42 −0.411369
\(623\) 7925.27 0.509662
\(624\) 16204.1 1.03956
\(625\) 14091.5 0.901856
\(626\) −4947.00 −0.315849
\(627\) 23751.7 1.51284
\(628\) −1318.64 −0.0837890
\(629\) −3749.09 −0.237656
\(630\) 3194.08 0.201992
\(631\) −17655.1 −1.11385 −0.556925 0.830563i \(-0.688019\pi\)
−0.556925 + 0.830563i \(0.688019\pi\)
\(632\) −5432.50 −0.341920
\(633\) 8700.49 0.546309
\(634\) −5881.53 −0.368431
\(635\) 1411.29 0.0881972
\(636\) 974.884 0.0607809
\(637\) −14122.4 −0.878417
\(638\) −4521.15 −0.280555
\(639\) 57536.5 3.56198
\(640\) −2859.67 −0.176622
\(641\) 374.249 0.0230608 0.0115304 0.999934i \(-0.496330\pi\)
0.0115304 + 0.999934i \(0.496330\pi\)
\(642\) 11032.1 0.678194
\(643\) −18264.0 −1.12016 −0.560080 0.828438i \(-0.689230\pi\)
−0.560080 + 0.828438i \(0.689230\pi\)
\(644\) 16437.1 1.00576
\(645\) 0 0
\(646\) −699.227 −0.0425863
\(647\) −1121.95 −0.0681735 −0.0340868 0.999419i \(-0.510852\pi\)
−0.0340868 + 0.999419i \(0.510852\pi\)
\(648\) 23660.0 1.43434
\(649\) −25227.1 −1.52581
\(650\) −4090.47 −0.246833
\(651\) −67322.6 −4.05312
\(652\) −5979.65 −0.359173
\(653\) −14019.2 −0.840142 −0.420071 0.907491i \(-0.637995\pi\)
−0.420071 + 0.907491i \(0.637995\pi\)
\(654\) −10226.2 −0.611429
\(655\) 2061.92 0.123001
\(656\) −15050.1 −0.895745
\(657\) −22384.0 −1.32920
\(658\) 11520.8 0.682563
\(659\) −26897.3 −1.58994 −0.794969 0.606650i \(-0.792513\pi\)
−0.794969 + 0.606650i \(0.792513\pi\)
\(660\) −5416.11 −0.319427
\(661\) 4456.60 0.262241 0.131121 0.991366i \(-0.458142\pi\)
0.131121 + 0.991366i \(0.458142\pi\)
\(662\) 3407.74 0.200069
\(663\) −4342.91 −0.254396
\(664\) 7401.24 0.432566
\(665\) 3495.28 0.203821
\(666\) −18159.0 −1.05653
\(667\) −11040.3 −0.640905
\(668\) 2298.45 0.133128
\(669\) 26113.2 1.50911
\(670\) 1768.76 0.101990
\(671\) −9035.71 −0.519851
\(672\) 38668.5 2.21975
\(673\) 12270.1 0.702787 0.351394 0.936228i \(-0.385708\pi\)
0.351394 + 0.936228i \(0.385708\pi\)
\(674\) −10554.5 −0.603183
\(675\) 43847.5 2.50028
\(676\) 5688.71 0.323664
\(677\) 5587.24 0.317186 0.158593 0.987344i \(-0.449304\pi\)
0.158593 + 0.987344i \(0.449304\pi\)
\(678\) 8714.88 0.493647
\(679\) 5149.96 0.291071
\(680\) 336.978 0.0190037
\(681\) −22015.7 −1.23883
\(682\) −9143.47 −0.513375
\(683\) −16419.2 −0.919860 −0.459930 0.887955i \(-0.652126\pi\)
−0.459930 + 0.887955i \(0.652126\pi\)
\(684\) 29854.8 1.66890
\(685\) −1287.55 −0.0718173
\(686\) −817.031 −0.0454729
\(687\) −29086.1 −1.61529
\(688\) 0 0
\(689\) −530.714 −0.0293448
\(690\) 1500.35 0.0827788
\(691\) −21390.2 −1.17760 −0.588801 0.808278i \(-0.700400\pi\)
−0.588801 + 0.808278i \(0.700400\pi\)
\(692\) 21656.2 1.18966
\(693\) 67297.6 3.68892
\(694\) 10057.7 0.550125
\(695\) 889.165 0.0485294
\(696\) −17010.7 −0.926422
\(697\) 4033.63 0.219203
\(698\) −1107.61 −0.0600625
\(699\) −47869.5 −2.59026
\(700\) 23297.5 1.25795
\(701\) 2029.82 0.109365 0.0546827 0.998504i \(-0.482585\pi\)
0.0546827 + 0.998504i \(0.482585\pi\)
\(702\) −12277.7 −0.660103
\(703\) −19871.5 −1.06610
\(704\) −8704.74 −0.466012
\(705\) −9269.98 −0.495217
\(706\) 1428.41 0.0761459
\(707\) −46068.6 −2.45062
\(708\) −44910.8 −2.38397
\(709\) 17269.2 0.914751 0.457376 0.889274i \(-0.348789\pi\)
0.457376 + 0.889274i \(0.348789\pi\)
\(710\) 1628.71 0.0860909
\(711\) −25699.2 −1.35555
\(712\) 4049.84 0.213166
\(713\) −22327.7 −1.17276
\(714\) −2806.00 −0.147075
\(715\) 2948.46 0.154218
\(716\) −24986.2 −1.30416
\(717\) 28168.7 1.46720
\(718\) 553.663 0.0287779
\(719\) 13317.0 0.690739 0.345369 0.938467i \(-0.387754\pi\)
0.345369 + 0.938467i \(0.387754\pi\)
\(720\) −5947.92 −0.307869
\(721\) −856.981 −0.0442658
\(722\) 2486.24 0.128155
\(723\) 47742.4 2.45582
\(724\) −55.5851 −0.00285332
\(725\) −15648.3 −0.801605
\(726\) 1428.81 0.0730416
\(727\) −22136.1 −1.12927 −0.564636 0.825340i \(-0.690983\pi\)
−0.564636 + 0.825340i \(0.690983\pi\)
\(728\) −13787.1 −0.701899
\(729\) 18051.0 0.917084
\(730\) −633.634 −0.0321258
\(731\) 0 0
\(732\) −16085.9 −0.812231
\(733\) −17545.9 −0.884139 −0.442069 0.896981i \(-0.645756\pi\)
−0.442069 + 0.896981i \(0.645756\pi\)
\(734\) −1988.39 −0.0999904
\(735\) 7341.99 0.368453
\(736\) 12824.5 0.642280
\(737\) 37267.0 1.86261
\(738\) 19537.2 0.974493
\(739\) −13702.8 −0.682094 −0.341047 0.940046i \(-0.610781\pi\)
−0.341047 + 0.940046i \(0.610781\pi\)
\(740\) 4531.31 0.225100
\(741\) −23018.9 −1.14119
\(742\) −342.900 −0.0169653
\(743\) −12371.8 −0.610871 −0.305435 0.952213i \(-0.598802\pi\)
−0.305435 + 0.952213i \(0.598802\pi\)
\(744\) −34402.1 −1.69522
\(745\) −4173.30 −0.205232
\(746\) −5659.52 −0.277761
\(747\) 35012.5 1.71492
\(748\) 3359.44 0.164215
\(749\) 34205.7 1.66869
\(750\) 4325.87 0.210611
\(751\) 9938.30 0.482895 0.241447 0.970414i \(-0.422378\pi\)
0.241447 + 0.970414i \(0.422378\pi\)
\(752\) −21453.7 −1.04034
\(753\) −48197.2 −2.33254
\(754\) 4381.67 0.211633
\(755\) −2077.41 −0.100139
\(756\) 69928.3 3.36411
\(757\) −10311.6 −0.495088 −0.247544 0.968877i \(-0.579623\pi\)
−0.247544 + 0.968877i \(0.579623\pi\)
\(758\) −2931.60 −0.140475
\(759\) 31611.7 1.51177
\(760\) 1786.10 0.0852483
\(761\) 19234.7 0.916238 0.458119 0.888891i \(-0.348523\pi\)
0.458119 + 0.888891i \(0.348523\pi\)
\(762\) −6005.16 −0.285491
\(763\) −31707.0 −1.50442
\(764\) 9352.97 0.442904
\(765\) 1594.12 0.0753406
\(766\) 8070.05 0.380656
\(767\) 24448.8 1.15097
\(768\) −5086.54 −0.238990
\(769\) 13634.6 0.639370 0.319685 0.947524i \(-0.396423\pi\)
0.319685 + 0.947524i \(0.396423\pi\)
\(770\) 1905.03 0.0891590
\(771\) 8110.55 0.378851
\(772\) 5464.31 0.254747
\(773\) 11204.3 0.521335 0.260667 0.965429i \(-0.416057\pi\)
0.260667 + 0.965429i \(0.416057\pi\)
\(774\) 0 0
\(775\) −31646.8 −1.46682
\(776\) 2631.65 0.121740
\(777\) −79744.2 −3.68186
\(778\) 4177.17 0.192492
\(779\) 21379.6 0.983318
\(780\) 5249.02 0.240955
\(781\) 34316.2 1.57225
\(782\) −930.618 −0.0425561
\(783\) −46969.0 −2.14372
\(784\) 16991.7 0.774038
\(785\) −373.197 −0.0169681
\(786\) −8773.65 −0.398150
\(787\) −40995.2 −1.85683 −0.928413 0.371551i \(-0.878826\pi\)
−0.928413 + 0.371551i \(0.878826\pi\)
\(788\) 1440.85 0.0651371
\(789\) 74043.4 3.34096
\(790\) −727.480 −0.0327627
\(791\) 27021.2 1.21462
\(792\) 34389.3 1.54289
\(793\) 8756.96 0.392142
\(794\) −9787.31 −0.437454
\(795\) 275.908 0.0123087
\(796\) 13498.0 0.601034
\(797\) −1380.09 −0.0613368 −0.0306684 0.999530i \(-0.509764\pi\)
−0.0306684 + 0.999530i \(0.509764\pi\)
\(798\) −14872.8 −0.659762
\(799\) 5749.86 0.254588
\(800\) 18177.2 0.803325
\(801\) 19158.3 0.845100
\(802\) 11187.5 0.492573
\(803\) −13350.4 −0.586705
\(804\) 66345.0 2.91021
\(805\) 4651.95 0.203677
\(806\) 8861.39 0.387257
\(807\) 12286.0 0.535918
\(808\) −23541.2 −1.02497
\(809\) −18158.2 −0.789131 −0.394566 0.918868i \(-0.629105\pi\)
−0.394566 + 0.918868i \(0.629105\pi\)
\(810\) 3168.37 0.137439
\(811\) −478.154 −0.0207031 −0.0103516 0.999946i \(-0.503295\pi\)
−0.0103516 + 0.999946i \(0.503295\pi\)
\(812\) −24956.0 −1.07855
\(813\) −11081.2 −0.478024
\(814\) −10830.5 −0.466350
\(815\) −1692.34 −0.0727361
\(816\) 5225.25 0.224167
\(817\) 0 0
\(818\) −7141.88 −0.305269
\(819\) −65221.4 −2.78269
\(820\) −4875.22 −0.207622
\(821\) 7722.38 0.328274 0.164137 0.986438i \(-0.447516\pi\)
0.164137 + 0.986438i \(0.447516\pi\)
\(822\) 5478.66 0.232470
\(823\) −1765.67 −0.0747842 −0.0373921 0.999301i \(-0.511905\pi\)
−0.0373921 + 0.999301i \(0.511905\pi\)
\(824\) −437.920 −0.0185142
\(825\) 44805.6 1.89082
\(826\) 15796.6 0.665418
\(827\) −19472.6 −0.818776 −0.409388 0.912360i \(-0.634258\pi\)
−0.409388 + 0.912360i \(0.634258\pi\)
\(828\) 39734.5 1.66772
\(829\) 26318.0 1.10261 0.551304 0.834304i \(-0.314130\pi\)
0.551304 + 0.834304i \(0.314130\pi\)
\(830\) 991.118 0.0414485
\(831\) −19994.7 −0.834669
\(832\) 8436.20 0.351530
\(833\) −4553.99 −0.189420
\(834\) −3783.48 −0.157088
\(835\) 650.498 0.0269598
\(836\) 17806.2 0.736650
\(837\) −94989.0 −3.92270
\(838\) 11647.9 0.480156
\(839\) 6566.60 0.270208 0.135104 0.990831i \(-0.456863\pi\)
0.135104 + 0.990831i \(0.456863\pi\)
\(840\) 7167.63 0.294413
\(841\) −7626.71 −0.312711
\(842\) 14512.8 0.593994
\(843\) −11060.7 −0.451898
\(844\) 6522.59 0.266015
\(845\) 1610.00 0.0655450
\(846\) 27850.0 1.13180
\(847\) 4430.15 0.179719
\(848\) 638.538 0.0258579
\(849\) −50685.0 −2.04889
\(850\) −1319.04 −0.0532265
\(851\) −26447.4 −1.06534
\(852\) 61091.8 2.45654
\(853\) 39036.9 1.56694 0.783468 0.621432i \(-0.213449\pi\)
0.783468 + 0.621432i \(0.213449\pi\)
\(854\) 5657.96 0.226711
\(855\) 8449.39 0.337968
\(856\) 17479.3 0.697931
\(857\) −6060.17 −0.241554 −0.120777 0.992680i \(-0.538539\pi\)
−0.120777 + 0.992680i \(0.538539\pi\)
\(858\) −12546.0 −0.499198
\(859\) 13664.8 0.542767 0.271383 0.962471i \(-0.412519\pi\)
0.271383 + 0.962471i \(0.412519\pi\)
\(860\) 0 0
\(861\) 85796.5 3.39598
\(862\) 8375.80 0.330952
\(863\) −8805.11 −0.347311 −0.173655 0.984806i \(-0.555558\pi\)
−0.173655 + 0.984806i \(0.555558\pi\)
\(864\) 54559.4 2.14832
\(865\) 6129.05 0.240918
\(866\) 164.762 0.00646517
\(867\) 45685.7 1.78958
\(868\) −50470.5 −1.97359
\(869\) −15327.6 −0.598337
\(870\) −2277.95 −0.0887697
\(871\) −36117.3 −1.40504
\(872\) −16202.4 −0.629223
\(873\) 12449.3 0.482642
\(874\) −4932.60 −0.190901
\(875\) 13412.7 0.518208
\(876\) −23767.1 −0.916686
\(877\) 24442.4 0.941120 0.470560 0.882368i \(-0.344052\pi\)
0.470560 + 0.882368i \(0.344052\pi\)
\(878\) 5241.80 0.201483
\(879\) 29516.1 1.13260
\(880\) −3547.49 −0.135893
\(881\) −21839.8 −0.835190 −0.417595 0.908633i \(-0.637127\pi\)
−0.417595 + 0.908633i \(0.637127\pi\)
\(882\) −22057.7 −0.842086
\(883\) 17757.9 0.676785 0.338393 0.941005i \(-0.390117\pi\)
0.338393 + 0.941005i \(0.390117\pi\)
\(884\) −3255.80 −0.123874
\(885\) −12710.5 −0.482777
\(886\) −12007.5 −0.455304
\(887\) −42106.9 −1.59392 −0.796962 0.604030i \(-0.793561\pi\)
−0.796962 + 0.604030i \(0.793561\pi\)
\(888\) −40749.6 −1.53994
\(889\) −18619.5 −0.702449
\(890\) 542.324 0.0204256
\(891\) 66756.1 2.51000
\(892\) 19576.5 0.734833
\(893\) 30476.3 1.14205
\(894\) 17757.8 0.664329
\(895\) −7071.50 −0.264105
\(896\) 37728.3 1.40671
\(897\) −30636.4 −1.14038
\(898\) −6490.47 −0.241191
\(899\) 33899.7 1.25764
\(900\) 56318.7 2.08588
\(901\) −171.137 −0.00632784
\(902\) 11652.5 0.430140
\(903\) 0 0
\(904\) 13807.9 0.508014
\(905\) −15.7315 −0.000577826 0
\(906\) 8839.58 0.324145
\(907\) 1377.33 0.0504227 0.0252114 0.999682i \(-0.491974\pi\)
0.0252114 + 0.999682i \(0.491974\pi\)
\(908\) −16504.7 −0.603225
\(909\) −111365. −4.06352
\(910\) −1846.26 −0.0672559
\(911\) −39864.1 −1.44979 −0.724895 0.688860i \(-0.758112\pi\)
−0.724895 + 0.688860i \(0.758112\pi\)
\(912\) 27695.6 1.00559
\(913\) 20882.4 0.756961
\(914\) 3791.04 0.137195
\(915\) −4552.58 −0.164485
\(916\) −21805.3 −0.786537
\(917\) −27203.4 −0.979646
\(918\) −3959.13 −0.142343
\(919\) 5830.88 0.209296 0.104648 0.994509i \(-0.466628\pi\)
0.104648 + 0.994509i \(0.466628\pi\)
\(920\) 2377.17 0.0851879
\(921\) 6128.81 0.219274
\(922\) 1070.38 0.0382331
\(923\) −33257.5 −1.18601
\(924\) 71456.2 2.54409
\(925\) −37485.9 −1.33246
\(926\) 2277.29 0.0808169
\(927\) −2071.64 −0.0733998
\(928\) −19471.2 −0.688763
\(929\) −4121.82 −0.145568 −0.0727840 0.997348i \(-0.523188\pi\)
−0.0727840 + 0.997348i \(0.523188\pi\)
\(930\) −4606.87 −0.162436
\(931\) −24137.7 −0.849712
\(932\) −35886.8 −1.26128
\(933\) −67743.3 −2.37708
\(934\) −8693.67 −0.304567
\(935\) 950.774 0.0332552
\(936\) −33328.4 −1.16386
\(937\) 33608.5 1.17176 0.585881 0.810397i \(-0.300749\pi\)
0.585881 + 0.810397i \(0.300749\pi\)
\(938\) −23335.7 −0.812302
\(939\) −52515.9 −1.82512
\(940\) −6949.53 −0.241137
\(941\) −30075.8 −1.04192 −0.520958 0.853582i \(-0.674425\pi\)
−0.520958 + 0.853582i \(0.674425\pi\)
\(942\) 1587.99 0.0549251
\(943\) 28454.7 0.982621
\(944\) −29416.1 −1.01421
\(945\) 19790.8 0.681265
\(946\) 0 0
\(947\) 8945.08 0.306944 0.153472 0.988153i \(-0.450955\pi\)
0.153472 + 0.988153i \(0.450955\pi\)
\(948\) −27287.2 −0.934860
\(949\) 12938.5 0.442573
\(950\) −6991.34 −0.238767
\(951\) −62436.6 −2.12897
\(952\) −4445.84 −0.151356
\(953\) −51373.5 −1.74622 −0.873112 0.487519i \(-0.837902\pi\)
−0.873112 + 0.487519i \(0.837902\pi\)
\(954\) −828.915 −0.0281311
\(955\) 2647.04 0.0896924
\(956\) 21117.5 0.714425
\(957\) −47995.2 −1.62118
\(958\) −11157.4 −0.376283
\(959\) 16987.0 0.571991
\(960\) −4385.82 −0.147450
\(961\) 38767.0 1.30130
\(962\) 10496.4 0.351785
\(963\) 82687.9 2.76696
\(964\) 35791.5 1.19582
\(965\) 1546.49 0.0515888
\(966\) −19794.5 −0.659294
\(967\) 22479.1 0.747549 0.373775 0.927520i \(-0.378063\pi\)
0.373775 + 0.927520i \(0.378063\pi\)
\(968\) 2263.82 0.0751673
\(969\) −7422.80 −0.246083
\(970\) 352.410 0.0116652
\(971\) 45003.4 1.48736 0.743680 0.668536i \(-0.233079\pi\)
0.743680 + 0.668536i \(0.233079\pi\)
\(972\) 48466.3 1.59934
\(973\) −11731.0 −0.386514
\(974\) −2458.63 −0.0808826
\(975\) −43423.3 −1.42632
\(976\) −10536.1 −0.345545
\(977\) −21946.4 −0.718657 −0.359329 0.933211i \(-0.616994\pi\)
−0.359329 + 0.933211i \(0.616994\pi\)
\(978\) 7201.05 0.235444
\(979\) 11426.5 0.373026
\(980\) 5504.15 0.179412
\(981\) −76647.5 −2.49456
\(982\) 3791.32 0.123204
\(983\) 950.865 0.0308524 0.0154262 0.999881i \(-0.495089\pi\)
0.0154262 + 0.999881i \(0.495089\pi\)
\(984\) 43842.3 1.42037
\(985\) 407.783 0.0131909
\(986\) 1412.93 0.0456359
\(987\) 122301. 3.94417
\(988\) −17256.8 −0.555681
\(989\) 0 0
\(990\) 4605.16 0.147840
\(991\) 51696.8 1.65712 0.828559 0.559901i \(-0.189161\pi\)
0.828559 + 0.559901i \(0.189161\pi\)
\(992\) −39378.1 −1.26034
\(993\) 36175.6 1.15609
\(994\) −21488.0 −0.685674
\(995\) 3820.15 0.121715
\(996\) 37176.1 1.18270
\(997\) 44474.6 1.41276 0.706381 0.707832i \(-0.250326\pi\)
0.706381 + 0.707832i \(0.250326\pi\)
\(998\) −18042.9 −0.572284
\(999\) −112515. −3.56339
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 1849.4.a.i.1.24 50
43.42 odd 2 1849.4.a.j.1.27 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.24 50 1.1 even 1 trivial
1849.4.a.j.1.27 yes 50 43.42 odd 2