Properties

Label 1849.4.a.i.1.22
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 1.22
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.02286 q^{2} +1.11016 q^{3} -6.95376 q^{4} +17.8254 q^{5} -1.13554 q^{6} +19.7420 q^{7} +15.2956 q^{8} -25.7675 q^{9} +O(q^{10})\) \(q-1.02286 q^{2} +1.11016 q^{3} -6.95376 q^{4} +17.8254 q^{5} -1.13554 q^{6} +19.7420 q^{7} +15.2956 q^{8} -25.7675 q^{9} -18.2329 q^{10} +22.7928 q^{11} -7.71979 q^{12} +19.7572 q^{13} -20.1933 q^{14} +19.7891 q^{15} +39.9848 q^{16} -101.707 q^{17} +26.3566 q^{18} -71.2415 q^{19} -123.954 q^{20} +21.9168 q^{21} -23.3139 q^{22} +39.5390 q^{23} +16.9806 q^{24} +192.745 q^{25} -20.2089 q^{26} -58.5805 q^{27} -137.281 q^{28} -100.371 q^{29} -20.2414 q^{30} +188.118 q^{31} -163.264 q^{32} +25.3037 q^{33} +104.032 q^{34} +351.910 q^{35} +179.181 q^{36} -431.553 q^{37} +72.8700 q^{38} +21.9337 q^{39} +272.650 q^{40} -388.086 q^{41} -22.4178 q^{42} -158.496 q^{44} -459.317 q^{45} -40.4428 q^{46} -444.898 q^{47} +44.3896 q^{48} +46.7482 q^{49} -197.151 q^{50} -112.911 q^{51} -137.387 q^{52} -561.950 q^{53} +59.9196 q^{54} +406.292 q^{55} +301.966 q^{56} -79.0895 q^{57} +102.665 q^{58} -582.977 q^{59} -137.608 q^{60} -490.093 q^{61} -192.418 q^{62} -508.704 q^{63} -152.883 q^{64} +352.181 q^{65} -25.8821 q^{66} +638.092 q^{67} +707.245 q^{68} +43.8946 q^{69} -359.955 q^{70} -37.3436 q^{71} -394.130 q^{72} +504.541 q^{73} +441.418 q^{74} +213.978 q^{75} +495.396 q^{76} +449.977 q^{77} -22.4351 q^{78} +690.289 q^{79} +712.746 q^{80} +630.690 q^{81} +396.958 q^{82} -112.701 q^{83} -152.404 q^{84} -1812.97 q^{85} -111.428 q^{87} +348.630 q^{88} -772.926 q^{89} +469.817 q^{90} +390.048 q^{91} -274.945 q^{92} +208.841 q^{93} +455.068 q^{94} -1269.91 q^{95} -181.249 q^{96} -403.914 q^{97} -47.8168 q^{98} -587.315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.02286 −0.361635 −0.180818 0.983517i \(-0.557874\pi\)
−0.180818 + 0.983517i \(0.557874\pi\)
\(3\) 1.11016 0.213651 0.106825 0.994278i \(-0.465931\pi\)
0.106825 + 0.994278i \(0.465931\pi\)
\(4\) −6.95376 −0.869220
\(5\) 17.8254 1.59435 0.797177 0.603746i \(-0.206326\pi\)
0.797177 + 0.603746i \(0.206326\pi\)
\(6\) −1.13554 −0.0772636
\(7\) 19.7420 1.06597 0.532985 0.846125i \(-0.321070\pi\)
0.532985 + 0.846125i \(0.321070\pi\)
\(8\) 15.2956 0.675976
\(9\) −25.7675 −0.954353
\(10\) −18.2329 −0.576575
\(11\) 22.7928 0.624754 0.312377 0.949958i \(-0.398875\pi\)
0.312377 + 0.949958i \(0.398875\pi\)
\(12\) −7.71979 −0.185709
\(13\) 19.7572 0.421513 0.210757 0.977539i \(-0.432407\pi\)
0.210757 + 0.977539i \(0.432407\pi\)
\(14\) −20.1933 −0.385493
\(15\) 19.7891 0.340634
\(16\) 39.9848 0.624763
\(17\) −101.707 −1.45103 −0.725515 0.688206i \(-0.758399\pi\)
−0.725515 + 0.688206i \(0.758399\pi\)
\(18\) 26.3566 0.345128
\(19\) −71.2415 −0.860206 −0.430103 0.902780i \(-0.641523\pi\)
−0.430103 + 0.902780i \(0.641523\pi\)
\(20\) −123.954 −1.38584
\(21\) 21.9168 0.227745
\(22\) −23.3139 −0.225933
\(23\) 39.5390 0.358454 0.179227 0.983808i \(-0.442640\pi\)
0.179227 + 0.983808i \(0.442640\pi\)
\(24\) 16.9806 0.144423
\(25\) 192.745 1.54196
\(26\) −20.2089 −0.152434
\(27\) −58.5805 −0.417549
\(28\) −137.281 −0.926562
\(29\) −100.371 −0.642703 −0.321351 0.946960i \(-0.604137\pi\)
−0.321351 + 0.946960i \(0.604137\pi\)
\(30\) −20.2414 −0.123186
\(31\) 188.118 1.08990 0.544951 0.838468i \(-0.316548\pi\)
0.544951 + 0.838468i \(0.316548\pi\)
\(32\) −163.264 −0.901913
\(33\) 25.3037 0.133479
\(34\) 104.032 0.524744
\(35\) 351.910 1.69953
\(36\) 179.181 0.829543
\(37\) −431.553 −1.91748 −0.958741 0.284282i \(-0.908245\pi\)
−0.958741 + 0.284282i \(0.908245\pi\)
\(38\) 72.8700 0.311081
\(39\) 21.9337 0.0900565
\(40\) 272.650 1.07775
\(41\) −388.086 −1.47827 −0.739133 0.673560i \(-0.764764\pi\)
−0.739133 + 0.673560i \(0.764764\pi\)
\(42\) −22.4178 −0.0823607
\(43\) 0 0
\(44\) −158.496 −0.543049
\(45\) −459.317 −1.52158
\(46\) −40.4428 −0.129630
\(47\) −444.898 −1.38075 −0.690373 0.723453i \(-0.742554\pi\)
−0.690373 + 0.723453i \(0.742554\pi\)
\(48\) 44.3896 0.133481
\(49\) 46.7482 0.136292
\(50\) −197.151 −0.557629
\(51\) −112.911 −0.310014
\(52\) −137.387 −0.366388
\(53\) −561.950 −1.45641 −0.728206 0.685359i \(-0.759645\pi\)
−0.728206 + 0.685359i \(0.759645\pi\)
\(54\) 59.9196 0.151000
\(55\) 406.292 0.996079
\(56\) 301.966 0.720570
\(57\) −79.0895 −0.183783
\(58\) 102.665 0.232424
\(59\) −582.977 −1.28639 −0.643196 0.765702i \(-0.722392\pi\)
−0.643196 + 0.765702i \(0.722392\pi\)
\(60\) −137.608 −0.296086
\(61\) −490.093 −1.02869 −0.514344 0.857584i \(-0.671964\pi\)
−0.514344 + 0.857584i \(0.671964\pi\)
\(62\) −192.418 −0.394147
\(63\) −508.704 −1.01731
\(64\) −152.883 −0.298599
\(65\) 352.181 0.672041
\(66\) −25.8821 −0.0482708
\(67\) 638.092 1.16351 0.581756 0.813363i \(-0.302366\pi\)
0.581756 + 0.813363i \(0.302366\pi\)
\(68\) 707.245 1.26126
\(69\) 43.8946 0.0765839
\(70\) −359.955 −0.614611
\(71\) −37.3436 −0.0624207 −0.0312103 0.999513i \(-0.509936\pi\)
−0.0312103 + 0.999513i \(0.509936\pi\)
\(72\) −394.130 −0.645120
\(73\) 504.541 0.808933 0.404466 0.914553i \(-0.367457\pi\)
0.404466 + 0.914553i \(0.367457\pi\)
\(74\) 441.418 0.693429
\(75\) 213.978 0.329441
\(76\) 495.396 0.747708
\(77\) 449.977 0.665969
\(78\) −22.4351 −0.0325676
\(79\) 690.289 0.983084 0.491542 0.870854i \(-0.336433\pi\)
0.491542 + 0.870854i \(0.336433\pi\)
\(80\) 712.746 0.996093
\(81\) 630.690 0.865144
\(82\) 396.958 0.534593
\(83\) −112.701 −0.149043 −0.0745216 0.997219i \(-0.523743\pi\)
−0.0745216 + 0.997219i \(0.523743\pi\)
\(84\) −152.404 −0.197961
\(85\) −1812.97 −2.31346
\(86\) 0 0
\(87\) −111.428 −0.137314
\(88\) 348.630 0.422319
\(89\) −772.926 −0.920562 −0.460281 0.887773i \(-0.652251\pi\)
−0.460281 + 0.887773i \(0.652251\pi\)
\(90\) 469.817 0.550256
\(91\) 390.048 0.449321
\(92\) −274.945 −0.311576
\(93\) 208.841 0.232858
\(94\) 455.068 0.499327
\(95\) −1269.91 −1.37147
\(96\) −181.249 −0.192694
\(97\) −403.914 −0.422797 −0.211398 0.977400i \(-0.567802\pi\)
−0.211398 + 0.977400i \(0.567802\pi\)
\(98\) −47.8168 −0.0492881
\(99\) −587.315 −0.596236
\(100\) −1340.30 −1.34030
\(101\) −550.505 −0.542350 −0.271175 0.962530i \(-0.587412\pi\)
−0.271175 + 0.962530i \(0.587412\pi\)
\(102\) 115.492 0.112112
\(103\) 1818.49 1.73962 0.869812 0.493384i \(-0.164240\pi\)
0.869812 + 0.493384i \(0.164240\pi\)
\(104\) 302.199 0.284933
\(105\) 390.677 0.363106
\(106\) 574.796 0.526690
\(107\) 78.4386 0.0708686 0.0354343 0.999372i \(-0.488719\pi\)
0.0354343 + 0.999372i \(0.488719\pi\)
\(108\) 407.354 0.362942
\(109\) −157.371 −0.138288 −0.0691440 0.997607i \(-0.522027\pi\)
−0.0691440 + 0.997607i \(0.522027\pi\)
\(110\) −415.579 −0.360218
\(111\) −479.093 −0.409671
\(112\) 789.382 0.665978
\(113\) −348.983 −0.290527 −0.145263 0.989393i \(-0.546403\pi\)
−0.145263 + 0.989393i \(0.546403\pi\)
\(114\) 80.8974 0.0664626
\(115\) 704.799 0.571503
\(116\) 697.954 0.558650
\(117\) −509.096 −0.402273
\(118\) 596.304 0.465205
\(119\) −2007.90 −1.54676
\(120\) 302.686 0.230261
\(121\) −811.487 −0.609682
\(122\) 501.296 0.372010
\(123\) −430.838 −0.315832
\(124\) −1308.12 −0.947364
\(125\) 1207.59 0.864081
\(126\) 520.333 0.367896
\(127\) 1286.15 0.898643 0.449322 0.893370i \(-0.351666\pi\)
0.449322 + 0.893370i \(0.351666\pi\)
\(128\) 1462.49 1.00990
\(129\) 0 0
\(130\) −360.232 −0.243034
\(131\) 2694.05 1.79680 0.898399 0.439181i \(-0.144731\pi\)
0.898399 + 0.439181i \(0.144731\pi\)
\(132\) −175.956 −0.116023
\(133\) −1406.45 −0.916954
\(134\) −652.678 −0.420767
\(135\) −1044.22 −0.665720
\(136\) −1555.67 −0.980862
\(137\) 21.0377 0.0131195 0.00655976 0.999978i \(-0.497912\pi\)
0.00655976 + 0.999978i \(0.497912\pi\)
\(138\) −44.8980 −0.0276955
\(139\) 3054.31 1.86376 0.931882 0.362761i \(-0.118166\pi\)
0.931882 + 0.362761i \(0.118166\pi\)
\(140\) −2447.10 −1.47727
\(141\) −493.909 −0.294997
\(142\) 38.1973 0.0225735
\(143\) 450.323 0.263342
\(144\) −1030.31 −0.596245
\(145\) −1789.15 −1.02470
\(146\) −516.075 −0.292539
\(147\) 51.8980 0.0291189
\(148\) 3000.91 1.66671
\(149\) −1300.51 −0.715047 −0.357524 0.933904i \(-0.616379\pi\)
−0.357524 + 0.933904i \(0.616379\pi\)
\(150\) −218.870 −0.119138
\(151\) −1123.62 −0.605556 −0.302778 0.953061i \(-0.597914\pi\)
−0.302778 + 0.953061i \(0.597914\pi\)
\(152\) −1089.68 −0.581479
\(153\) 2620.73 1.38480
\(154\) −460.263 −0.240838
\(155\) 3353.28 1.73769
\(156\) −152.522 −0.0782789
\(157\) 2420.82 1.23059 0.615294 0.788298i \(-0.289037\pi\)
0.615294 + 0.788298i \(0.289037\pi\)
\(158\) −706.069 −0.355518
\(159\) −623.855 −0.311163
\(160\) −2910.24 −1.43797
\(161\) 780.580 0.382102
\(162\) −645.107 −0.312867
\(163\) −2633.91 −1.26567 −0.632834 0.774288i \(-0.718108\pi\)
−0.632834 + 0.774288i \(0.718108\pi\)
\(164\) 2698.66 1.28494
\(165\) 451.049 0.212813
\(166\) 115.278 0.0538993
\(167\) −2477.45 −1.14797 −0.573984 0.818866i \(-0.694603\pi\)
−0.573984 + 0.818866i \(0.694603\pi\)
\(168\) 335.231 0.153950
\(169\) −1806.65 −0.822327
\(170\) 1854.41 0.836628
\(171\) 1835.72 0.820941
\(172\) 0 0
\(173\) 1641.92 0.721575 0.360788 0.932648i \(-0.382508\pi\)
0.360788 + 0.932648i \(0.382508\pi\)
\(174\) 113.975 0.0496575
\(175\) 3805.19 1.64369
\(176\) 911.367 0.390323
\(177\) −647.198 −0.274838
\(178\) 790.595 0.332908
\(179\) 197.274 0.0823742 0.0411871 0.999151i \(-0.486886\pi\)
0.0411871 + 0.999151i \(0.486886\pi\)
\(180\) 3193.98 1.32258
\(181\) −2120.28 −0.870714 −0.435357 0.900258i \(-0.643378\pi\)
−0.435357 + 0.900258i \(0.643378\pi\)
\(182\) −398.965 −0.162490
\(183\) −544.081 −0.219780
\(184\) 604.772 0.242307
\(185\) −7692.61 −3.05714
\(186\) −213.615 −0.0842097
\(187\) −2318.19 −0.906538
\(188\) 3093.71 1.20017
\(189\) −1156.50 −0.445094
\(190\) 1298.94 0.495973
\(191\) −3247.98 −1.23045 −0.615223 0.788353i \(-0.710934\pi\)
−0.615223 + 0.788353i \(0.710934\pi\)
\(192\) −169.724 −0.0637959
\(193\) 4635.04 1.72869 0.864345 0.502899i \(-0.167733\pi\)
0.864345 + 0.502899i \(0.167733\pi\)
\(194\) 413.147 0.152898
\(195\) 390.978 0.143582
\(196\) −325.076 −0.118468
\(197\) −2130.99 −0.770695 −0.385348 0.922772i \(-0.625918\pi\)
−0.385348 + 0.922772i \(0.625918\pi\)
\(198\) 600.741 0.215620
\(199\) −3441.83 −1.22605 −0.613027 0.790062i \(-0.710048\pi\)
−0.613027 + 0.790062i \(0.710048\pi\)
\(200\) 2948.16 1.04233
\(201\) 708.385 0.248585
\(202\) 563.090 0.196133
\(203\) −1981.52 −0.685102
\(204\) 785.155 0.269470
\(205\) −6917.80 −2.35688
\(206\) −1860.06 −0.629110
\(207\) −1018.82 −0.342092
\(208\) 789.990 0.263346
\(209\) −1623.80 −0.537417
\(210\) −399.607 −0.131312
\(211\) 1835.67 0.598922 0.299461 0.954109i \(-0.403193\pi\)
0.299461 + 0.954109i \(0.403193\pi\)
\(212\) 3907.67 1.26594
\(213\) −41.4574 −0.0133362
\(214\) −80.2317 −0.0256286
\(215\) 0 0
\(216\) −896.023 −0.282253
\(217\) 3713.83 1.16180
\(218\) 160.968 0.0500098
\(219\) 560.122 0.172829
\(220\) −2825.25 −0.865812
\(221\) −2009.45 −0.611629
\(222\) 490.045 0.148152
\(223\) −2382.48 −0.715438 −0.357719 0.933829i \(-0.616445\pi\)
−0.357719 + 0.933829i \(0.616445\pi\)
\(224\) −3223.16 −0.961412
\(225\) −4966.58 −1.47158
\(226\) 356.960 0.105065
\(227\) −1427.19 −0.417294 −0.208647 0.977991i \(-0.566906\pi\)
−0.208647 + 0.977991i \(0.566906\pi\)
\(228\) 549.969 0.159748
\(229\) −3610.07 −1.04175 −0.520874 0.853634i \(-0.674394\pi\)
−0.520874 + 0.853634i \(0.674394\pi\)
\(230\) −720.910 −0.206676
\(231\) 499.547 0.142285
\(232\) −1535.23 −0.434452
\(233\) 844.802 0.237532 0.118766 0.992922i \(-0.462106\pi\)
0.118766 + 0.992922i \(0.462106\pi\)
\(234\) 520.733 0.145476
\(235\) −7930.50 −2.20140
\(236\) 4053.88 1.11816
\(237\) 766.332 0.210036
\(238\) 2053.80 0.559362
\(239\) 3678.94 0.995693 0.497846 0.867265i \(-0.334124\pi\)
0.497846 + 0.867265i \(0.334124\pi\)
\(240\) 791.263 0.212816
\(241\) 2907.74 0.777194 0.388597 0.921408i \(-0.372960\pi\)
0.388597 + 0.921408i \(0.372960\pi\)
\(242\) 830.037 0.220483
\(243\) 2281.84 0.602387
\(244\) 3407.98 0.894155
\(245\) 833.306 0.217298
\(246\) 440.687 0.114216
\(247\) −1407.54 −0.362588
\(248\) 2877.37 0.736747
\(249\) −125.117 −0.0318432
\(250\) −1235.20 −0.312482
\(251\) 7305.43 1.83711 0.918555 0.395294i \(-0.129357\pi\)
0.918555 + 0.395294i \(0.129357\pi\)
\(252\) 3537.40 0.884268
\(253\) 901.205 0.223946
\(254\) −1315.56 −0.324981
\(255\) −2012.68 −0.494271
\(256\) −272.856 −0.0666153
\(257\) −1564.64 −0.379765 −0.189882 0.981807i \(-0.560811\pi\)
−0.189882 + 0.981807i \(0.560811\pi\)
\(258\) 0 0
\(259\) −8519.73 −2.04398
\(260\) −2448.98 −0.584151
\(261\) 2586.31 0.613365
\(262\) −2755.64 −0.649786
\(263\) −1714.84 −0.402060 −0.201030 0.979585i \(-0.564429\pi\)
−0.201030 + 0.979585i \(0.564429\pi\)
\(264\) 387.035 0.0902287
\(265\) −10017.0 −2.32203
\(266\) 1438.60 0.331603
\(267\) −858.072 −0.196678
\(268\) −4437.14 −1.01135
\(269\) −5451.01 −1.23552 −0.617758 0.786368i \(-0.711959\pi\)
−0.617758 + 0.786368i \(0.711959\pi\)
\(270\) 1068.09 0.240748
\(271\) −1315.31 −0.294832 −0.147416 0.989075i \(-0.547096\pi\)
−0.147416 + 0.989075i \(0.547096\pi\)
\(272\) −4066.73 −0.906550
\(273\) 433.016 0.0959976
\(274\) −21.5186 −0.00474448
\(275\) 4393.21 0.963348
\(276\) −305.233 −0.0665683
\(277\) −2640.70 −0.572795 −0.286398 0.958111i \(-0.592458\pi\)
−0.286398 + 0.958111i \(0.592458\pi\)
\(278\) −3124.13 −0.674003
\(279\) −4847.33 −1.04015
\(280\) 5382.67 1.14884
\(281\) −4294.12 −0.911621 −0.455811 0.890077i \(-0.650651\pi\)
−0.455811 + 0.890077i \(0.650651\pi\)
\(282\) 505.199 0.106681
\(283\) −3322.87 −0.697965 −0.348983 0.937129i \(-0.613473\pi\)
−0.348983 + 0.937129i \(0.613473\pi\)
\(284\) 259.678 0.0542573
\(285\) −1409.80 −0.293016
\(286\) −460.618 −0.0952339
\(287\) −7661.62 −1.57579
\(288\) 4206.90 0.860743
\(289\) 5431.28 1.10549
\(290\) 1830.05 0.370566
\(291\) −448.410 −0.0903307
\(292\) −3508.46 −0.703141
\(293\) −190.708 −0.0380249 −0.0190125 0.999819i \(-0.506052\pi\)
−0.0190125 + 0.999819i \(0.506052\pi\)
\(294\) −53.0844 −0.0105304
\(295\) −10391.8 −2.05096
\(296\) −6600.86 −1.29617
\(297\) −1335.21 −0.260865
\(298\) 1330.24 0.258586
\(299\) 781.181 0.151093
\(300\) −1487.95 −0.286357
\(301\) 0 0
\(302\) 1149.31 0.218990
\(303\) −611.149 −0.115873
\(304\) −2848.58 −0.537425
\(305\) −8736.10 −1.64009
\(306\) −2680.64 −0.500792
\(307\) 4341.31 0.807073 0.403536 0.914964i \(-0.367781\pi\)
0.403536 + 0.914964i \(0.367781\pi\)
\(308\) −3129.03 −0.578874
\(309\) 2018.82 0.371672
\(310\) −3429.93 −0.628410
\(311\) −6092.04 −1.11077 −0.555383 0.831595i \(-0.687428\pi\)
−0.555383 + 0.831595i \(0.687428\pi\)
\(312\) 335.489 0.0608761
\(313\) 5612.18 1.01348 0.506740 0.862099i \(-0.330851\pi\)
0.506740 + 0.862099i \(0.330851\pi\)
\(314\) −2476.16 −0.445024
\(315\) −9067.86 −1.62196
\(316\) −4800.10 −0.854516
\(317\) −3963.80 −0.702299 −0.351150 0.936319i \(-0.614209\pi\)
−0.351150 + 0.936319i \(0.614209\pi\)
\(318\) 638.116 0.112528
\(319\) −2287.73 −0.401531
\(320\) −2725.20 −0.476073
\(321\) 87.0794 0.0151411
\(322\) −798.424 −0.138181
\(323\) 7245.74 1.24819
\(324\) −4385.67 −0.752000
\(325\) 3808.12 0.649958
\(326\) 2694.12 0.457710
\(327\) −174.707 −0.0295453
\(328\) −5936.01 −0.999273
\(329\) −8783.20 −1.47183
\(330\) −461.360 −0.0769607
\(331\) 9099.39 1.51102 0.755510 0.655137i \(-0.227389\pi\)
0.755510 + 0.655137i \(0.227389\pi\)
\(332\) 783.698 0.129551
\(333\) 11120.1 1.82996
\(334\) 2534.08 0.415146
\(335\) 11374.3 1.85505
\(336\) 876.341 0.142287
\(337\) −2484.89 −0.401663 −0.200832 0.979626i \(-0.564364\pi\)
−0.200832 + 0.979626i \(0.564364\pi\)
\(338\) 1847.95 0.297382
\(339\) −387.427 −0.0620712
\(340\) 12606.9 2.01090
\(341\) 4287.73 0.680920
\(342\) −1877.68 −0.296881
\(343\) −5848.62 −0.920687
\(344\) 0 0
\(345\) 782.440 0.122102
\(346\) −1679.45 −0.260947
\(347\) −9452.41 −1.46234 −0.731170 0.682196i \(-0.761025\pi\)
−0.731170 + 0.682196i \(0.761025\pi\)
\(348\) 774.841 0.119356
\(349\) 2929.71 0.449352 0.224676 0.974434i \(-0.427868\pi\)
0.224676 + 0.974434i \(0.427868\pi\)
\(350\) −3892.17 −0.594415
\(351\) −1157.39 −0.176002
\(352\) −3721.24 −0.563474
\(353\) 1497.90 0.225851 0.112926 0.993603i \(-0.463978\pi\)
0.112926 + 0.993603i \(0.463978\pi\)
\(354\) 661.993 0.0993913
\(355\) −665.665 −0.0995207
\(356\) 5374.74 0.800170
\(357\) −2229.09 −0.330465
\(358\) −201.784 −0.0297894
\(359\) 11573.3 1.70144 0.850721 0.525618i \(-0.176166\pi\)
0.850721 + 0.525618i \(0.176166\pi\)
\(360\) −7025.53 −1.02855
\(361\) −1783.65 −0.260045
\(362\) 2168.75 0.314881
\(363\) −900.881 −0.130259
\(364\) −2712.30 −0.390558
\(365\) 8993.66 1.28973
\(366\) 556.519 0.0794801
\(367\) −3815.58 −0.542702 −0.271351 0.962480i \(-0.587470\pi\)
−0.271351 + 0.962480i \(0.587470\pi\)
\(368\) 1580.96 0.223949
\(369\) 10000.0 1.41079
\(370\) 7868.46 1.10557
\(371\) −11094.0 −1.55249
\(372\) −1452.23 −0.202405
\(373\) −1969.77 −0.273434 −0.136717 0.990610i \(-0.543655\pi\)
−0.136717 + 0.990610i \(0.543655\pi\)
\(374\) 2371.18 0.327836
\(375\) 1340.62 0.184611
\(376\) −6804.98 −0.933352
\(377\) −1983.05 −0.270908
\(378\) 1182.93 0.160962
\(379\) 1267.89 0.171839 0.0859194 0.996302i \(-0.472617\pi\)
0.0859194 + 0.996302i \(0.472617\pi\)
\(380\) 8830.64 1.19211
\(381\) 1427.84 0.191996
\(382\) 3322.22 0.444973
\(383\) 2523.67 0.336694 0.168347 0.985728i \(-0.446157\pi\)
0.168347 + 0.985728i \(0.446157\pi\)
\(384\) 1623.60 0.215765
\(385\) 8021.03 1.06179
\(386\) −4740.99 −0.625156
\(387\) 0 0
\(388\) 2808.72 0.367503
\(389\) 12576.9 1.63926 0.819631 0.572892i \(-0.194179\pi\)
0.819631 + 0.572892i \(0.194179\pi\)
\(390\) −399.915 −0.0519243
\(391\) −4021.38 −0.520128
\(392\) 715.041 0.0921302
\(393\) 2990.83 0.383887
\(394\) 2179.71 0.278711
\(395\) 12304.7 1.56738
\(396\) 4084.05 0.518260
\(397\) −6253.88 −0.790613 −0.395306 0.918549i \(-0.629362\pi\)
−0.395306 + 0.918549i \(0.629362\pi\)
\(398\) 3520.51 0.443384
\(399\) −1561.39 −0.195908
\(400\) 7706.89 0.963361
\(401\) 1382.58 0.172177 0.0860883 0.996288i \(-0.472563\pi\)
0.0860883 + 0.996288i \(0.472563\pi\)
\(402\) −724.578 −0.0898972
\(403\) 3716.69 0.459408
\(404\) 3828.08 0.471421
\(405\) 11242.3 1.37935
\(406\) 2026.82 0.247757
\(407\) −9836.31 −1.19795
\(408\) −1727.04 −0.209562
\(409\) −13695.7 −1.65577 −0.827885 0.560899i \(-0.810456\pi\)
−0.827885 + 0.560899i \(0.810456\pi\)
\(410\) 7075.94 0.852331
\(411\) 23.3552 0.00280299
\(412\) −12645.3 −1.51212
\(413\) −11509.2 −1.37126
\(414\) 1042.11 0.123713
\(415\) −2008.95 −0.237628
\(416\) −3225.64 −0.380168
\(417\) 3390.78 0.398194
\(418\) 1660.91 0.194349
\(419\) 10610.7 1.23715 0.618573 0.785727i \(-0.287711\pi\)
0.618573 + 0.785727i \(0.287711\pi\)
\(420\) −2716.67 −0.315619
\(421\) −2110.80 −0.244357 −0.122178 0.992508i \(-0.538988\pi\)
−0.122178 + 0.992508i \(0.538988\pi\)
\(422\) −1877.63 −0.216592
\(423\) 11463.9 1.31772
\(424\) −8595.36 −0.984499
\(425\) −19603.5 −2.23744
\(426\) 42.4051 0.00482285
\(427\) −9675.43 −1.09655
\(428\) −545.443 −0.0616004
\(429\) 499.931 0.0562632
\(430\) 0 0
\(431\) 12954.3 1.44777 0.723883 0.689922i \(-0.242355\pi\)
0.723883 + 0.689922i \(0.242355\pi\)
\(432\) −2342.33 −0.260869
\(433\) 1171.99 0.130075 0.0650374 0.997883i \(-0.479283\pi\)
0.0650374 + 0.997883i \(0.479283\pi\)
\(434\) −3798.72 −0.420149
\(435\) −1986.24 −0.218927
\(436\) 1094.32 0.120203
\(437\) −2816.82 −0.308345
\(438\) −572.926 −0.0625011
\(439\) −16402.2 −1.78322 −0.891609 0.452807i \(-0.850423\pi\)
−0.891609 + 0.452807i \(0.850423\pi\)
\(440\) 6214.47 0.673326
\(441\) −1204.59 −0.130071
\(442\) 2055.38 0.221187
\(443\) 10078.9 1.08096 0.540479 0.841358i \(-0.318243\pi\)
0.540479 + 0.841358i \(0.318243\pi\)
\(444\) 3331.50 0.356094
\(445\) −13777.7 −1.46770
\(446\) 2436.94 0.258728
\(447\) −1443.78 −0.152770
\(448\) −3018.22 −0.318298
\(449\) 6527.47 0.686081 0.343040 0.939321i \(-0.388543\pi\)
0.343040 + 0.939321i \(0.388543\pi\)
\(450\) 5080.11 0.532175
\(451\) −8845.59 −0.923553
\(452\) 2426.74 0.252532
\(453\) −1247.40 −0.129377
\(454\) 1459.81 0.150908
\(455\) 6952.77 0.716376
\(456\) −1209.72 −0.124233
\(457\) 5400.42 0.552782 0.276391 0.961045i \(-0.410862\pi\)
0.276391 + 0.961045i \(0.410862\pi\)
\(458\) 3692.60 0.376733
\(459\) 5958.03 0.605876
\(460\) −4901.00 −0.496762
\(461\) −6304.64 −0.636956 −0.318478 0.947930i \(-0.603172\pi\)
−0.318478 + 0.947930i \(0.603172\pi\)
\(462\) −510.966 −0.0514552
\(463\) −19243.6 −1.93158 −0.965792 0.259317i \(-0.916503\pi\)
−0.965792 + 0.259317i \(0.916503\pi\)
\(464\) −4013.30 −0.401537
\(465\) 3722.68 0.371258
\(466\) −864.114 −0.0858998
\(467\) 2714.13 0.268940 0.134470 0.990918i \(-0.457067\pi\)
0.134470 + 0.990918i \(0.457067\pi\)
\(468\) 3540.13 0.349663
\(469\) 12597.2 1.24027
\(470\) 8111.78 0.796104
\(471\) 2687.50 0.262916
\(472\) −8916.98 −0.869570
\(473\) 0 0
\(474\) −783.850 −0.0759566
\(475\) −13731.5 −1.32641
\(476\) 13962.5 1.34447
\(477\) 14480.1 1.38993
\(478\) −3763.03 −0.360078
\(479\) −11210.3 −1.06933 −0.534666 0.845063i \(-0.679563\pi\)
−0.534666 + 0.845063i \(0.679563\pi\)
\(480\) −3230.84 −0.307223
\(481\) −8526.29 −0.808244
\(482\) −2974.21 −0.281061
\(483\) 866.570 0.0816362
\(484\) 5642.88 0.529948
\(485\) −7199.94 −0.674087
\(486\) −2334.00 −0.217845
\(487\) 8701.07 0.809617 0.404808 0.914402i \(-0.367338\pi\)
0.404808 + 0.914402i \(0.367338\pi\)
\(488\) −7496.26 −0.695368
\(489\) −2924.06 −0.270410
\(490\) −852.355 −0.0785826
\(491\) 1624.22 0.149288 0.0746438 0.997210i \(-0.476218\pi\)
0.0746438 + 0.997210i \(0.476218\pi\)
\(492\) 2995.94 0.274528
\(493\) 10208.4 0.932581
\(494\) 1439.71 0.131125
\(495\) −10469.1 −0.950612
\(496\) 7521.85 0.680930
\(497\) −737.239 −0.0665386
\(498\) 127.977 0.0115156
\(499\) 14069.2 1.26217 0.631084 0.775714i \(-0.282610\pi\)
0.631084 + 0.775714i \(0.282610\pi\)
\(500\) −8397.29 −0.751076
\(501\) −2750.37 −0.245264
\(502\) −7472.43 −0.664364
\(503\) −11488.0 −1.01834 −0.509170 0.860666i \(-0.670048\pi\)
−0.509170 + 0.860666i \(0.670048\pi\)
\(504\) −7780.93 −0.687679
\(505\) −9812.98 −0.864697
\(506\) −921.807 −0.0809868
\(507\) −2005.67 −0.175691
\(508\) −8943.60 −0.781119
\(509\) −9359.44 −0.815030 −0.407515 0.913199i \(-0.633605\pi\)
−0.407515 + 0.913199i \(0.633605\pi\)
\(510\) 2058.69 0.178746
\(511\) 9960.68 0.862298
\(512\) −11420.8 −0.985806
\(513\) 4173.36 0.359178
\(514\) 1600.41 0.137336
\(515\) 32415.4 2.77358
\(516\) 0 0
\(517\) −10140.5 −0.862627
\(518\) 8714.49 0.739175
\(519\) 1822.79 0.154165
\(520\) 5386.82 0.454284
\(521\) 1027.52 0.0864036 0.0432018 0.999066i \(-0.486244\pi\)
0.0432018 + 0.999066i \(0.486244\pi\)
\(522\) −2645.43 −0.221815
\(523\) −9348.97 −0.781648 −0.390824 0.920465i \(-0.627810\pi\)
−0.390824 + 0.920465i \(0.627810\pi\)
\(524\) −18733.8 −1.56181
\(525\) 4224.37 0.351175
\(526\) 1754.04 0.145399
\(527\) −19132.9 −1.58148
\(528\) 1011.76 0.0833928
\(529\) −10603.7 −0.871511
\(530\) 10246.0 0.839730
\(531\) 15021.9 1.22767
\(532\) 9780.13 0.797035
\(533\) −7667.51 −0.623109
\(534\) 877.687 0.0711259
\(535\) 1398.20 0.112990
\(536\) 9760.00 0.786507
\(537\) 219.006 0.0175993
\(538\) 5575.61 0.446806
\(539\) 1065.52 0.0851491
\(540\) 7261.26 0.578657
\(541\) −19965.3 −1.58664 −0.793322 0.608802i \(-0.791650\pi\)
−0.793322 + 0.608802i \(0.791650\pi\)
\(542\) 1345.38 0.106622
\(543\) −2353.85 −0.186029
\(544\) 16605.0 1.30870
\(545\) −2805.20 −0.220480
\(546\) −442.915 −0.0347161
\(547\) −15813.1 −1.23605 −0.618027 0.786157i \(-0.712068\pi\)
−0.618027 + 0.786157i \(0.712068\pi\)
\(548\) −146.291 −0.0114037
\(549\) 12628.5 0.981731
\(550\) −4493.64 −0.348381
\(551\) 7150.56 0.552857
\(552\) 671.394 0.0517689
\(553\) 13627.7 1.04794
\(554\) 2701.07 0.207143
\(555\) −8540.03 −0.653160
\(556\) −21238.9 −1.62002
\(557\) −3373.07 −0.256592 −0.128296 0.991736i \(-0.540951\pi\)
−0.128296 + 0.991736i \(0.540951\pi\)
\(558\) 4958.14 0.376155
\(559\) 0 0
\(560\) 14071.1 1.06181
\(561\) −2573.56 −0.193682
\(562\) 4392.28 0.329675
\(563\) 21680.8 1.62298 0.811489 0.584368i \(-0.198657\pi\)
0.811489 + 0.584368i \(0.198657\pi\)
\(564\) 3434.52 0.256417
\(565\) −6220.76 −0.463203
\(566\) 3398.83 0.252409
\(567\) 12451.1 0.922218
\(568\) −571.193 −0.0421949
\(569\) 3600.43 0.265268 0.132634 0.991165i \(-0.457656\pi\)
0.132634 + 0.991165i \(0.457656\pi\)
\(570\) 1442.03 0.105965
\(571\) 15694.5 1.15025 0.575126 0.818065i \(-0.304953\pi\)
0.575126 + 0.818065i \(0.304953\pi\)
\(572\) −3131.44 −0.228902
\(573\) −3605.78 −0.262886
\(574\) 7836.76 0.569861
\(575\) 7620.96 0.552723
\(576\) 3939.41 0.284969
\(577\) 8951.52 0.645852 0.322926 0.946424i \(-0.395334\pi\)
0.322926 + 0.946424i \(0.395334\pi\)
\(578\) −5555.43 −0.399785
\(579\) 5145.64 0.369336
\(580\) 12441.3 0.890685
\(581\) −2224.96 −0.158876
\(582\) 458.660 0.0326668
\(583\) −12808.4 −0.909899
\(584\) 7717.26 0.546819
\(585\) −9074.84 −0.641365
\(586\) 195.068 0.0137512
\(587\) 4071.25 0.286266 0.143133 0.989703i \(-0.454282\pi\)
0.143133 + 0.989703i \(0.454282\pi\)
\(588\) −360.886 −0.0253107
\(589\) −13401.8 −0.937540
\(590\) 10629.4 0.741701
\(591\) −2365.74 −0.164659
\(592\) −17255.6 −1.19797
\(593\) −7458.47 −0.516497 −0.258249 0.966079i \(-0.583145\pi\)
−0.258249 + 0.966079i \(0.583145\pi\)
\(594\) 1365.74 0.0943382
\(595\) −35791.7 −2.46608
\(596\) 9043.44 0.621533
\(597\) −3820.98 −0.261947
\(598\) −799.039 −0.0546407
\(599\) 3861.66 0.263411 0.131705 0.991289i \(-0.457955\pi\)
0.131705 + 0.991289i \(0.457955\pi\)
\(600\) 3272.93 0.222694
\(601\) 3695.64 0.250829 0.125414 0.992104i \(-0.459974\pi\)
0.125414 + 0.992104i \(0.459974\pi\)
\(602\) 0 0
\(603\) −16442.1 −1.11040
\(604\) 7813.38 0.526361
\(605\) −14465.1 −0.972049
\(606\) 625.120 0.0419039
\(607\) 7398.42 0.494716 0.247358 0.968924i \(-0.420438\pi\)
0.247358 + 0.968924i \(0.420438\pi\)
\(608\) 11631.1 0.775831
\(609\) −2199.81 −0.146372
\(610\) 8935.81 0.593115
\(611\) −8789.96 −0.582003
\(612\) −18224.0 −1.20369
\(613\) 18359.2 1.20966 0.604828 0.796356i \(-0.293242\pi\)
0.604828 + 0.796356i \(0.293242\pi\)
\(614\) −4440.55 −0.291866
\(615\) −7679.87 −0.503548
\(616\) 6882.67 0.450179
\(617\) −12505.8 −0.815986 −0.407993 0.912985i \(-0.633771\pi\)
−0.407993 + 0.912985i \(0.633771\pi\)
\(618\) −2064.97 −0.134410
\(619\) 10523.0 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) −23317.9 −1.51043
\(621\) −2316.21 −0.149672
\(622\) 6231.31 0.401692
\(623\) −15259.1 −0.981291
\(624\) 877.015 0.0562640
\(625\) −2567.38 −0.164313
\(626\) −5740.47 −0.366510
\(627\) −1802.67 −0.114820
\(628\) −16833.8 −1.06965
\(629\) 43891.8 2.78233
\(630\) 9275.15 0.586557
\(631\) 28065.1 1.77061 0.885306 0.465010i \(-0.153949\pi\)
0.885306 + 0.465010i \(0.153949\pi\)
\(632\) 10558.4 0.664541
\(633\) 2037.89 0.127960
\(634\) 4054.41 0.253976
\(635\) 22926.2 1.43276
\(636\) 4338.14 0.270469
\(637\) 923.615 0.0574489
\(638\) 2340.03 0.145208
\(639\) 962.253 0.0595714
\(640\) 26069.4 1.61013
\(641\) −20560.3 −1.26690 −0.633450 0.773784i \(-0.718362\pi\)
−0.633450 + 0.773784i \(0.718362\pi\)
\(642\) −89.0700 −0.00547557
\(643\) 1245.51 0.0763889 0.0381944 0.999270i \(-0.487839\pi\)
0.0381944 + 0.999270i \(0.487839\pi\)
\(644\) −5427.97 −0.332130
\(645\) 0 0
\(646\) −7411.38 −0.451388
\(647\) −11323.3 −0.688047 −0.344024 0.938961i \(-0.611790\pi\)
−0.344024 + 0.938961i \(0.611790\pi\)
\(648\) 9646.78 0.584817
\(649\) −13287.7 −0.803679
\(650\) −3895.17 −0.235048
\(651\) 4122.95 0.248220
\(652\) 18315.6 1.10014
\(653\) −4387.92 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(654\) 178.701 0.0106846
\(655\) 48022.6 2.86473
\(656\) −15517.6 −0.923566
\(657\) −13000.8 −0.772008
\(658\) 8983.98 0.532267
\(659\) −4067.28 −0.240423 −0.120211 0.992748i \(-0.538357\pi\)
−0.120211 + 0.992748i \(0.538357\pi\)
\(660\) −3136.49 −0.184981
\(661\) −700.954 −0.0412465 −0.0206233 0.999787i \(-0.506565\pi\)
−0.0206233 + 0.999787i \(0.506565\pi\)
\(662\) −9307.40 −0.546438
\(663\) −2230.81 −0.130675
\(664\) −1723.84 −0.100750
\(665\) −25070.6 −1.46195
\(666\) −11374.3 −0.661777
\(667\) −3968.56 −0.230379
\(668\) 17227.6 0.997837
\(669\) −2644.94 −0.152854
\(670\) −11634.3 −0.670852
\(671\) −11170.6 −0.642677
\(672\) −3578.22 −0.205406
\(673\) −24721.5 −1.41597 −0.707983 0.706229i \(-0.750395\pi\)
−0.707983 + 0.706229i \(0.750395\pi\)
\(674\) 2541.69 0.145256
\(675\) −11291.1 −0.643845
\(676\) 12563.0 0.714783
\(677\) −10400.6 −0.590442 −0.295221 0.955429i \(-0.595393\pi\)
−0.295221 + 0.955429i \(0.595393\pi\)
\(678\) 396.283 0.0224472
\(679\) −7974.09 −0.450688
\(680\) −27730.4 −1.56384
\(681\) −1584.41 −0.0891551
\(682\) −4385.75 −0.246245
\(683\) −18255.8 −1.02275 −0.511376 0.859357i \(-0.670864\pi\)
−0.511376 + 0.859357i \(0.670864\pi\)
\(684\) −12765.1 −0.713578
\(685\) 375.006 0.0209171
\(686\) 5982.31 0.332953
\(687\) −4007.76 −0.222570
\(688\) 0 0
\(689\) −11102.6 −0.613897
\(690\) −800.326 −0.0441564
\(691\) −30571.0 −1.68304 −0.841518 0.540229i \(-0.818337\pi\)
−0.841518 + 0.540229i \(0.818337\pi\)
\(692\) −11417.5 −0.627208
\(693\) −11594.8 −0.635570
\(694\) 9668.49 0.528834
\(695\) 54444.3 2.97150
\(696\) −1704.35 −0.0928208
\(697\) 39471.0 2.14501
\(698\) −2996.68 −0.162502
\(699\) 937.866 0.0507487
\(700\) −26460.4 −1.42872
\(701\) 9418.96 0.507488 0.253744 0.967271i \(-0.418338\pi\)
0.253744 + 0.967271i \(0.418338\pi\)
\(702\) 1183.85 0.0636487
\(703\) 30744.5 1.64943
\(704\) −3484.63 −0.186551
\(705\) −8804.12 −0.470330
\(706\) −1532.15 −0.0816757
\(707\) −10868.1 −0.578129
\(708\) 4500.46 0.238895
\(709\) 21799.2 1.15471 0.577353 0.816495i \(-0.304086\pi\)
0.577353 + 0.816495i \(0.304086\pi\)
\(710\) 680.882 0.0359902
\(711\) −17787.1 −0.938209
\(712\) −11822.4 −0.622278
\(713\) 7437.98 0.390680
\(714\) 2280.05 0.119508
\(715\) 8027.20 0.419861
\(716\) −1371.80 −0.0716013
\(717\) 4084.21 0.212730
\(718\) −11837.9 −0.615302
\(719\) 9463.10 0.490840 0.245420 0.969417i \(-0.421074\pi\)
0.245420 + 0.969417i \(0.421074\pi\)
\(720\) −18365.7 −0.950625
\(721\) 35900.7 1.85439
\(722\) 1824.43 0.0940416
\(723\) 3228.06 0.166048
\(724\) 14743.9 0.756842
\(725\) −19346.0 −0.991024
\(726\) 921.475 0.0471062
\(727\) 7627.15 0.389099 0.194550 0.980893i \(-0.437675\pi\)
0.194550 + 0.980893i \(0.437675\pi\)
\(728\) 5966.02 0.303730
\(729\) −14495.4 −0.736444
\(730\) −9199.25 −0.466410
\(731\) 0 0
\(732\) 3783.41 0.191037
\(733\) 2047.25 0.103161 0.0515805 0.998669i \(-0.483574\pi\)
0.0515805 + 0.998669i \(0.483574\pi\)
\(734\) 3902.80 0.196260
\(735\) 925.104 0.0464258
\(736\) −6455.28 −0.323294
\(737\) 14543.9 0.726909
\(738\) −10228.6 −0.510191
\(739\) 2538.29 0.126350 0.0631749 0.998002i \(-0.479877\pi\)
0.0631749 + 0.998002i \(0.479877\pi\)
\(740\) 53492.5 2.65733
\(741\) −1562.59 −0.0774672
\(742\) 11347.6 0.561436
\(743\) 18805.1 0.928524 0.464262 0.885698i \(-0.346320\pi\)
0.464262 + 0.885698i \(0.346320\pi\)
\(744\) 3194.35 0.157406
\(745\) −23182.2 −1.14004
\(746\) 2014.80 0.0988833
\(747\) 2904.04 0.142240
\(748\) 16120.1 0.787981
\(749\) 1548.54 0.0755438
\(750\) −1371.27 −0.0667620
\(751\) 33700.0 1.63746 0.818729 0.574181i \(-0.194679\pi\)
0.818729 + 0.574181i \(0.194679\pi\)
\(752\) −17789.2 −0.862639
\(753\) 8110.20 0.392499
\(754\) 2028.38 0.0979698
\(755\) −20029.0 −0.965470
\(756\) 8042.01 0.386885
\(757\) 28589.4 1.37265 0.686326 0.727294i \(-0.259222\pi\)
0.686326 + 0.727294i \(0.259222\pi\)
\(758\) −1296.87 −0.0621430
\(759\) 1000.48 0.0478461
\(760\) −19424.0 −0.927083
\(761\) 16653.2 0.793268 0.396634 0.917977i \(-0.370178\pi\)
0.396634 + 0.917977i \(0.370178\pi\)
\(762\) −1460.48 −0.0694324
\(763\) −3106.82 −0.147411
\(764\) 22585.6 1.06953
\(765\) 46715.7 2.20785
\(766\) −2581.36 −0.121760
\(767\) −11518.0 −0.542231
\(768\) −302.914 −0.0142324
\(769\) 19518.3 0.915279 0.457640 0.889138i \(-0.348695\pi\)
0.457640 + 0.889138i \(0.348695\pi\)
\(770\) −8204.38 −0.383981
\(771\) −1737.00 −0.0811370
\(772\) −32230.9 −1.50261
\(773\) −633.350 −0.0294696 −0.0147348 0.999891i \(-0.504690\pi\)
−0.0147348 + 0.999891i \(0.504690\pi\)
\(774\) 0 0
\(775\) 36258.8 1.68059
\(776\) −6178.11 −0.285800
\(777\) −9458.27 −0.436697
\(778\) −12864.4 −0.592815
\(779\) 27647.8 1.27161
\(780\) −2718.76 −0.124804
\(781\) −851.166 −0.0389976
\(782\) 4113.31 0.188097
\(783\) 5879.76 0.268360
\(784\) 1869.22 0.0851502
\(785\) 43152.1 1.96199
\(786\) −3059.20 −0.138827
\(787\) 22023.2 0.997511 0.498755 0.866743i \(-0.333791\pi\)
0.498755 + 0.866743i \(0.333791\pi\)
\(788\) 14818.4 0.669903
\(789\) −1903.75 −0.0859003
\(790\) −12586.0 −0.566821
\(791\) −6889.63 −0.309693
\(792\) −8983.34 −0.403042
\(793\) −9682.88 −0.433605
\(794\) 6396.84 0.285914
\(795\) −11120.5 −0.496104
\(796\) 23933.6 1.06571
\(797\) −20062.0 −0.891635 −0.445817 0.895124i \(-0.647087\pi\)
−0.445817 + 0.895124i \(0.647087\pi\)
\(798\) 1597.08 0.0708472
\(799\) 45249.2 2.00351
\(800\) −31468.3 −1.39072
\(801\) 19916.4 0.878541
\(802\) −1414.19 −0.0622652
\(803\) 11499.9 0.505384
\(804\) −4925.93 −0.216075
\(805\) 13914.2 0.609205
\(806\) −3801.65 −0.166138
\(807\) −6051.49 −0.263969
\(808\) −8420.31 −0.366615
\(809\) 37497.8 1.62961 0.814804 0.579736i \(-0.196845\pi\)
0.814804 + 0.579736i \(0.196845\pi\)
\(810\) −11499.3 −0.498820
\(811\) 1705.17 0.0738304 0.0369152 0.999318i \(-0.488247\pi\)
0.0369152 + 0.999318i \(0.488247\pi\)
\(812\) 13779.0 0.595504
\(813\) −1460.21 −0.0629910
\(814\) 10061.2 0.433223
\(815\) −46950.5 −2.01792
\(816\) −4514.72 −0.193685
\(817\) 0 0
\(818\) 14008.8 0.598785
\(819\) −10050.6 −0.428811
\(820\) 48104.7 2.04865
\(821\) −18361.5 −0.780535 −0.390267 0.920702i \(-0.627617\pi\)
−0.390267 + 0.920702i \(0.627617\pi\)
\(822\) −23.8891 −0.00101366
\(823\) 14800.1 0.626851 0.313426 0.949613i \(-0.398523\pi\)
0.313426 + 0.949613i \(0.398523\pi\)
\(824\) 27814.9 1.17594
\(825\) 4877.17 0.205820
\(826\) 11772.2 0.495895
\(827\) 14542.4 0.611474 0.305737 0.952116i \(-0.401097\pi\)
0.305737 + 0.952116i \(0.401097\pi\)
\(828\) 7084.65 0.297353
\(829\) −32533.1 −1.36299 −0.681496 0.731822i \(-0.738670\pi\)
−0.681496 + 0.731822i \(0.738670\pi\)
\(830\) 2054.87 0.0859346
\(831\) −2931.60 −0.122378
\(832\) −3020.54 −0.125864
\(833\) −4754.61 −0.197764
\(834\) −3468.29 −0.144001
\(835\) −44161.5 −1.83027
\(836\) 11291.5 0.467134
\(837\) −11020.0 −0.455087
\(838\) −10853.2 −0.447396
\(839\) 9279.94 0.381858 0.190929 0.981604i \(-0.438850\pi\)
0.190929 + 0.981604i \(0.438850\pi\)
\(840\) 5975.63 0.245451
\(841\) −14314.7 −0.586933
\(842\) 2159.05 0.0883680
\(843\) −4767.16 −0.194768
\(844\) −12764.8 −0.520595
\(845\) −32204.3 −1.31108
\(846\) −11726.0 −0.476534
\(847\) −16020.4 −0.649903
\(848\) −22469.5 −0.909911
\(849\) −3688.92 −0.149121
\(850\) 20051.7 0.809136
\(851\) −17063.2 −0.687330
\(852\) 288.285 0.0115921
\(853\) −37303.3 −1.49735 −0.748675 0.662937i \(-0.769310\pi\)
−0.748675 + 0.662937i \(0.769310\pi\)
\(854\) 9896.60 0.396551
\(855\) 32722.4 1.30887
\(856\) 1199.76 0.0479055
\(857\) −28599.1 −1.13994 −0.569969 0.821666i \(-0.693045\pi\)
−0.569969 + 0.821666i \(0.693045\pi\)
\(858\) −511.360 −0.0203468
\(859\) −17588.9 −0.698632 −0.349316 0.937005i \(-0.613586\pi\)
−0.349316 + 0.937005i \(0.613586\pi\)
\(860\) 0 0
\(861\) −8505.63 −0.336668
\(862\) −13250.4 −0.523564
\(863\) −16630.5 −0.655978 −0.327989 0.944681i \(-0.606371\pi\)
−0.327989 + 0.944681i \(0.606371\pi\)
\(864\) 9564.06 0.376592
\(865\) 29267.8 1.15045
\(866\) −1198.78 −0.0470397
\(867\) 6029.59 0.236189
\(868\) −25825.1 −1.00986
\(869\) 15733.6 0.614186
\(870\) 2031.65 0.0791716
\(871\) 12606.9 0.490436
\(872\) −2407.08 −0.0934793
\(873\) 10407.9 0.403497
\(874\) 2881.21 0.111508
\(875\) 23840.3 0.921085
\(876\) −3894.95 −0.150226
\(877\) −29568.5 −1.13849 −0.569247 0.822167i \(-0.692765\pi\)
−0.569247 + 0.822167i \(0.692765\pi\)
\(878\) 16777.1 0.644875
\(879\) −211.717 −0.00812405
\(880\) 16245.5 0.622313
\(881\) 3239.15 0.123870 0.0619351 0.998080i \(-0.480273\pi\)
0.0619351 + 0.998080i \(0.480273\pi\)
\(882\) 1232.12 0.0470382
\(883\) 9200.23 0.350637 0.175318 0.984512i \(-0.443904\pi\)
0.175318 + 0.984512i \(0.443904\pi\)
\(884\) 13973.2 0.531640
\(885\) −11536.6 −0.438189
\(886\) −10309.3 −0.390913
\(887\) 11534.9 0.436643 0.218322 0.975877i \(-0.429942\pi\)
0.218322 + 0.975877i \(0.429942\pi\)
\(888\) −7328.01 −0.276928
\(889\) 25391.3 0.957927
\(890\) 14092.7 0.530773
\(891\) 14375.2 0.540502
\(892\) 16567.2 0.621873
\(893\) 31695.2 1.18773
\(894\) 1476.78 0.0552471
\(895\) 3516.50 0.131334
\(896\) 28872.5 1.07652
\(897\) 867.237 0.0322811
\(898\) −6676.68 −0.248111
\(899\) −18881.5 −0.700482
\(900\) 34536.4 1.27912
\(901\) 57154.2 2.11330
\(902\) 9047.79 0.333990
\(903\) 0 0
\(904\) −5337.90 −0.196389
\(905\) −37794.9 −1.38823
\(906\) 1275.91 0.0467874
\(907\) 27277.5 0.998603 0.499302 0.866428i \(-0.333590\pi\)
0.499302 + 0.866428i \(0.333590\pi\)
\(908\) 9924.32 0.362720
\(909\) 14185.2 0.517593
\(910\) −7111.71 −0.259067
\(911\) −676.309 −0.0245962 −0.0122981 0.999924i \(-0.503915\pi\)
−0.0122981 + 0.999924i \(0.503915\pi\)
\(912\) −3162.38 −0.114821
\(913\) −2568.78 −0.0931154
\(914\) −5523.88 −0.199905
\(915\) −9698.48 −0.350406
\(916\) 25103.6 0.905508
\(917\) 53186.1 1.91533
\(918\) −6094.23 −0.219106
\(919\) 1872.82 0.0672239 0.0336119 0.999435i \(-0.489299\pi\)
0.0336119 + 0.999435i \(0.489299\pi\)
\(920\) 10780.3 0.386322
\(921\) 4819.55 0.172432
\(922\) 6448.76 0.230346
\(923\) −737.806 −0.0263112
\(924\) −3473.73 −0.123677
\(925\) −83179.8 −2.95669
\(926\) 19683.5 0.698530
\(927\) −46858.0 −1.66022
\(928\) 16386.9 0.579662
\(929\) 7059.21 0.249306 0.124653 0.992200i \(-0.460218\pi\)
0.124653 + 0.992200i \(0.460218\pi\)
\(930\) −3807.77 −0.134260
\(931\) −3330.41 −0.117239
\(932\) −5874.55 −0.206467
\(933\) −6763.15 −0.237316
\(934\) −2776.17 −0.0972582
\(935\) −41322.6 −1.44534
\(936\) −7786.92 −0.271927
\(937\) 33383.4 1.16392 0.581958 0.813219i \(-0.302287\pi\)
0.581958 + 0.813219i \(0.302287\pi\)
\(938\) −12885.2 −0.448525
\(939\) 6230.42 0.216530
\(940\) 55146.7 1.91350
\(941\) 24876.3 0.861791 0.430895 0.902402i \(-0.358198\pi\)
0.430895 + 0.902402i \(0.358198\pi\)
\(942\) −2748.93 −0.0950797
\(943\) −15344.5 −0.529891
\(944\) −23310.2 −0.803690
\(945\) −20615.1 −0.709638
\(946\) 0 0
\(947\) −44261.5 −1.51880 −0.759400 0.650624i \(-0.774508\pi\)
−0.759400 + 0.650624i \(0.774508\pi\)
\(948\) −5328.89 −0.182568
\(949\) 9968.35 0.340976
\(950\) 14045.4 0.479676
\(951\) −4400.45 −0.150047
\(952\) −30712.0 −1.04557
\(953\) −5408.14 −0.183827 −0.0919133 0.995767i \(-0.529298\pi\)
−0.0919133 + 0.995767i \(0.529298\pi\)
\(954\) −14811.1 −0.502648
\(955\) −57896.5 −1.96177
\(956\) −25582.4 −0.865476
\(957\) −2539.75 −0.0857873
\(958\) 11466.5 0.386709
\(959\) 415.327 0.0139850
\(960\) −3025.41 −0.101713
\(961\) 5597.27 0.187885
\(962\) 8721.20 0.292290
\(963\) −2021.17 −0.0676337
\(964\) −20219.7 −0.675553
\(965\) 82621.5 2.75614
\(966\) −886.379 −0.0295225
\(967\) −3500.75 −0.116418 −0.0582092 0.998304i \(-0.518539\pi\)
−0.0582092 + 0.998304i \(0.518539\pi\)
\(968\) −12412.2 −0.412131
\(969\) 8043.94 0.266676
\(970\) 7364.52 0.243774
\(971\) 988.893 0.0326829 0.0163414 0.999866i \(-0.494798\pi\)
0.0163414 + 0.999866i \(0.494798\pi\)
\(972\) −15867.4 −0.523607
\(973\) 60298.3 1.98672
\(974\) −8899.98 −0.292786
\(975\) 4227.62 0.138864
\(976\) −19596.3 −0.642685
\(977\) 43478.3 1.42374 0.711870 0.702311i \(-0.247848\pi\)
0.711870 + 0.702311i \(0.247848\pi\)
\(978\) 2990.91 0.0977900
\(979\) −17617.2 −0.575125
\(980\) −5794.61 −0.188880
\(981\) 4055.06 0.131976
\(982\) −1661.35 −0.0539877
\(983\) −43582.2 −1.41410 −0.707048 0.707166i \(-0.749973\pi\)
−0.707048 + 0.707166i \(0.749973\pi\)
\(984\) −6589.93 −0.213495
\(985\) −37985.8 −1.22876
\(986\) −10441.7 −0.337254
\(987\) −9750.76 −0.314458
\(988\) 9787.66 0.315169
\(989\) 0 0
\(990\) 10708.5 0.343775
\(991\) 32709.0 1.04847 0.524236 0.851573i \(-0.324351\pi\)
0.524236 + 0.851573i \(0.324351\pi\)
\(992\) −30712.8 −0.982996
\(993\) 10101.8 0.322830
\(994\) 754.092 0.0240627
\(995\) −61352.0 −1.95476
\(996\) 870.031 0.0276787
\(997\) −29341.0 −0.932035 −0.466017 0.884776i \(-0.654312\pi\)
−0.466017 + 0.884776i \(0.654312\pi\)
\(998\) −14390.8 −0.456445
\(999\) 25280.6 0.800642
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.1.22 50
43.42 odd 2 1849.4.a.j.1.29 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.22 50 1.1 even 1 trivial
1849.4.a.j.1.29 yes 50 43.42 odd 2