Properties

Label 1849.4.a.g.1.18
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $30$
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: \(30\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.32038 q^{2} +4.35495 q^{3} -6.25659 q^{4} +15.6111 q^{5} +5.75020 q^{6} -4.97789 q^{7} -18.8242 q^{8} -8.03441 q^{9} +O(q^{10})\) \(q+1.32038 q^{2} +4.35495 q^{3} -6.25659 q^{4} +15.6111 q^{5} +5.75020 q^{6} -4.97789 q^{7} -18.8242 q^{8} -8.03441 q^{9} +20.6126 q^{10} +3.73387 q^{11} -27.2471 q^{12} +34.7034 q^{13} -6.57272 q^{14} +67.9853 q^{15} +25.1976 q^{16} -90.1847 q^{17} -10.6085 q^{18} -19.2643 q^{19} -97.6719 q^{20} -21.6785 q^{21} +4.93014 q^{22} +34.2277 q^{23} -81.9782 q^{24} +118.705 q^{25} +45.8217 q^{26} -152.573 q^{27} +31.1446 q^{28} +29.2747 q^{29} +89.7667 q^{30} -224.399 q^{31} +183.864 q^{32} +16.2608 q^{33} -119.078 q^{34} -77.7101 q^{35} +50.2680 q^{36} -214.118 q^{37} -25.4362 q^{38} +151.131 q^{39} -293.865 q^{40} +487.539 q^{41} -28.6238 q^{42} -23.3613 q^{44} -125.426 q^{45} +45.1936 q^{46} -114.819 q^{47} +109.734 q^{48} -318.221 q^{49} +156.736 q^{50} -392.750 q^{51} -217.125 q^{52} -683.378 q^{53} -201.455 q^{54} +58.2897 q^{55} +93.7045 q^{56} -83.8949 q^{57} +38.6538 q^{58} -370.581 q^{59} -425.356 q^{60} +854.289 q^{61} -296.292 q^{62} +39.9944 q^{63} +41.1893 q^{64} +541.756 q^{65} +21.4705 q^{66} -475.209 q^{67} +564.249 q^{68} +149.060 q^{69} -102.607 q^{70} -879.804 q^{71} +151.241 q^{72} +670.576 q^{73} -282.717 q^{74} +516.954 q^{75} +120.529 q^{76} -18.5868 q^{77} +199.551 q^{78} -115.793 q^{79} +393.362 q^{80} -447.519 q^{81} +643.738 q^{82} -939.943 q^{83} +135.633 q^{84} -1407.88 q^{85} +127.490 q^{87} -70.2870 q^{88} +598.117 q^{89} -165.610 q^{90} -172.749 q^{91} -214.149 q^{92} -977.246 q^{93} -151.605 q^{94} -300.736 q^{95} +800.717 q^{96} +664.593 q^{97} -420.173 q^{98} -29.9995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30q - 6q^{2} - 2q^{3} + 114q^{4} - 27q^{5} + 8q^{6} - 48q^{7} - 90q^{8} + 216q^{9} + O(q^{10}) \) \( 30q - 6q^{2} - 2q^{3} + 114q^{4} - 27q^{5} + 8q^{6} - 48q^{7} - 90q^{8} + 216q^{9} - 27q^{10} + 80q^{11} + 36q^{12} - 13q^{13} + 36q^{14} + 16q^{15} + 318q^{16} + 66q^{17} - 80q^{18} - 254q^{19} - 312q^{20} - 548q^{21} - 305q^{22} - 105q^{23} + 123q^{24} + 523q^{25} - 549q^{26} + 10q^{27} - 578q^{28} - 793q^{29} - 1560q^{30} - 359q^{31} - 676q^{32} - 208q^{33} - 1007q^{34} - 514q^{35} + 776q^{36} - 510q^{37} - 2066q^{38} - 898q^{39} - 1248q^{40} - 270q^{41} + 915q^{42} + 3256q^{44} - 807q^{45} - 1960q^{46} + 1421q^{47} + 632q^{48} + 386q^{49} + 141q^{50} - 209q^{51} + 2825q^{52} - 21q^{53} + 2368q^{54} - 2258q^{55} + 2521q^{56} - 1723q^{57} - 347q^{58} + 1752q^{59} + 2711q^{60} - 1759q^{61} - 395q^{62} - 2204q^{63} + 222q^{64} - 1151q^{65} + 160q^{66} - 3001q^{67} + 1921q^{68} - 1660q^{69} - 1597q^{70} - 727q^{71} - 9100q^{72} - 4623q^{73} - 2649q^{74} - 1027q^{75} - 874q^{76} - 3556q^{77} - 4979q^{78} + 546q^{79} - 5809q^{80} - 410q^{81} + 4397q^{82} - 492q^{83} - 10611q^{84} + 1723q^{85} + 5937q^{87} - 3974q^{88} - 5218q^{89} + 10492q^{90} - 1104q^{91} + 1060q^{92} - 1997q^{93} + 2134q^{94} + 6346q^{95} - 11984q^{96} + 2590q^{97} - 6270q^{98} - 2693q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32038 0.466826 0.233413 0.972378i \(-0.425011\pi\)
0.233413 + 0.972378i \(0.425011\pi\)
\(3\) 4.35495 0.838110 0.419055 0.907961i \(-0.362361\pi\)
0.419055 + 0.907961i \(0.362361\pi\)
\(4\) −6.25659 −0.782074
\(5\) 15.6111 1.39629 0.698147 0.715954i \(-0.254008\pi\)
0.698147 + 0.715954i \(0.254008\pi\)
\(6\) 5.75020 0.391252
\(7\) −4.97789 −0.268781 −0.134390 0.990928i \(-0.542908\pi\)
−0.134390 + 0.990928i \(0.542908\pi\)
\(8\) −18.8242 −0.831918
\(9\) −8.03441 −0.297571
\(10\) 20.6126 0.651826
\(11\) 3.73387 0.102346 0.0511730 0.998690i \(-0.483704\pi\)
0.0511730 + 0.998690i \(0.483704\pi\)
\(12\) −27.2471 −0.655464
\(13\) 34.7034 0.740383 0.370192 0.928955i \(-0.379292\pi\)
0.370192 + 0.928955i \(0.379292\pi\)
\(14\) −6.57272 −0.125474
\(15\) 67.9853 1.17025
\(16\) 25.1976 0.393713
\(17\) −90.1847 −1.28665 −0.643324 0.765594i \(-0.722445\pi\)
−0.643324 + 0.765594i \(0.722445\pi\)
\(18\) −10.6085 −0.138914
\(19\) −19.2643 −0.232607 −0.116303 0.993214i \(-0.537104\pi\)
−0.116303 + 0.993214i \(0.537104\pi\)
\(20\) −97.6719 −1.09201
\(21\) −21.6785 −0.225268
\(22\) 4.93014 0.0477777
\(23\) 34.2277 0.310303 0.155151 0.987891i \(-0.450413\pi\)
0.155151 + 0.987891i \(0.450413\pi\)
\(24\) −81.9782 −0.697239
\(25\) 118.705 0.949639
\(26\) 45.8217 0.345630
\(27\) −152.573 −1.08751
\(28\) 31.1446 0.210206
\(29\) 29.2747 0.187454 0.0937272 0.995598i \(-0.470122\pi\)
0.0937272 + 0.995598i \(0.470122\pi\)
\(30\) 89.7667 0.546302
\(31\) −224.399 −1.30010 −0.650052 0.759890i \(-0.725253\pi\)
−0.650052 + 0.759890i \(0.725253\pi\)
\(32\) 183.864 1.01571
\(33\) 16.2608 0.0857772
\(34\) −119.078 −0.600640
\(35\) −77.7101 −0.375297
\(36\) 50.2680 0.232722
\(37\) −214.118 −0.951371 −0.475685 0.879615i \(-0.657800\pi\)
−0.475685 + 0.879615i \(0.657800\pi\)
\(38\) −25.4362 −0.108587
\(39\) 151.131 0.620523
\(40\) −293.865 −1.16160
\(41\) 487.539 1.85709 0.928546 0.371218i \(-0.121060\pi\)
0.928546 + 0.371218i \(0.121060\pi\)
\(42\) −28.6238 −0.105161
\(43\) 0 0
\(44\) −23.3613 −0.0800421
\(45\) −125.426 −0.415497
\(46\) 45.1936 0.144857
\(47\) −114.819 −0.356343 −0.178171 0.983999i \(-0.557018\pi\)
−0.178171 + 0.983999i \(0.557018\pi\)
\(48\) 109.734 0.329975
\(49\) −318.221 −0.927757
\(50\) 156.736 0.443316
\(51\) −392.750 −1.07835
\(52\) −217.125 −0.579034
\(53\) −683.378 −1.77112 −0.885558 0.464528i \(-0.846224\pi\)
−0.885558 + 0.464528i \(0.846224\pi\)
\(54\) −201.455 −0.507677
\(55\) 58.2897 0.142905
\(56\) 93.7045 0.223603
\(57\) −83.8949 −0.194950
\(58\) 38.6538 0.0875085
\(59\) −370.581 −0.817721 −0.408861 0.912597i \(-0.634074\pi\)
−0.408861 + 0.912597i \(0.634074\pi\)
\(60\) −425.356 −0.915221
\(61\) 854.289 1.79312 0.896562 0.442919i \(-0.146057\pi\)
0.896562 + 0.442919i \(0.146057\pi\)
\(62\) −296.292 −0.606922
\(63\) 39.9944 0.0799813
\(64\) 41.1893 0.0804479
\(65\) 541.756 1.03379
\(66\) 21.4705 0.0400430
\(67\) −475.209 −0.866508 −0.433254 0.901272i \(-0.642635\pi\)
−0.433254 + 0.901272i \(0.642635\pi\)
\(68\) 564.249 1.00625
\(69\) 149.060 0.260068
\(70\) −102.607 −0.175198
\(71\) −879.804 −1.47061 −0.735306 0.677735i \(-0.762962\pi\)
−0.735306 + 0.677735i \(0.762962\pi\)
\(72\) 151.241 0.247555
\(73\) 670.576 1.07514 0.537568 0.843220i \(-0.319343\pi\)
0.537568 + 0.843220i \(0.319343\pi\)
\(74\) −282.717 −0.444124
\(75\) 516.954 0.795903
\(76\) 120.529 0.181916
\(77\) −18.5868 −0.0275086
\(78\) 199.551 0.289676
\(79\) −115.793 −0.164908 −0.0824539 0.996595i \(-0.526276\pi\)
−0.0824539 + 0.996595i \(0.526276\pi\)
\(80\) 393.362 0.549740
\(81\) −447.519 −0.613881
\(82\) 643.738 0.866938
\(83\) −939.943 −1.24304 −0.621519 0.783399i \(-0.713484\pi\)
−0.621519 + 0.783399i \(0.713484\pi\)
\(84\) 135.633 0.176176
\(85\) −1407.88 −1.79654
\(86\) 0 0
\(87\) 127.490 0.157107
\(88\) −70.2870 −0.0851434
\(89\) 598.117 0.712362 0.356181 0.934417i \(-0.384079\pi\)
0.356181 + 0.934417i \(0.384079\pi\)
\(90\) −165.610 −0.193965
\(91\) −172.749 −0.199001
\(92\) −214.149 −0.242680
\(93\) −977.246 −1.08963
\(94\) −151.605 −0.166350
\(95\) −300.736 −0.324788
\(96\) 800.717 0.851280
\(97\) 664.593 0.695661 0.347831 0.937557i \(-0.386918\pi\)
0.347831 + 0.937557i \(0.386918\pi\)
\(98\) −420.173 −0.433101
\(99\) −29.9995 −0.0304552
\(100\) −742.688 −0.742688
\(101\) −1130.18 −1.11344 −0.556719 0.830701i \(-0.687940\pi\)
−0.556719 + 0.830701i \(0.687940\pi\)
\(102\) −518.580 −0.503403
\(103\) −1560.33 −1.49266 −0.746329 0.665577i \(-0.768185\pi\)
−0.746329 + 0.665577i \(0.768185\pi\)
\(104\) −653.261 −0.615938
\(105\) −338.423 −0.314540
\(106\) −902.320 −0.826803
\(107\) 964.626 0.871532 0.435766 0.900060i \(-0.356477\pi\)
0.435766 + 0.900060i \(0.356477\pi\)
\(108\) 954.587 0.850511
\(109\) 1.57769 0.00138637 0.000693187 1.00000i \(-0.499779\pi\)
0.000693187 1.00000i \(0.499779\pi\)
\(110\) 76.9647 0.0667118
\(111\) −932.472 −0.797354
\(112\) −125.431 −0.105822
\(113\) 461.586 0.384269 0.192134 0.981369i \(-0.438459\pi\)
0.192134 + 0.981369i \(0.438459\pi\)
\(114\) −110.773 −0.0910077
\(115\) 534.330 0.433274
\(116\) −183.160 −0.146603
\(117\) −278.821 −0.220316
\(118\) −489.309 −0.381733
\(119\) 448.929 0.345826
\(120\) −1279.77 −0.973551
\(121\) −1317.06 −0.989525
\(122\) 1127.99 0.837076
\(123\) 2123.21 1.55645
\(124\) 1403.97 1.01678
\(125\) −98.2728 −0.0703183
\(126\) 52.8079 0.0373373
\(127\) −1877.59 −1.31188 −0.655942 0.754811i \(-0.727728\pi\)
−0.655942 + 0.754811i \(0.727728\pi\)
\(128\) −1416.52 −0.978158
\(129\) 0 0
\(130\) 715.325 0.482601
\(131\) −812.688 −0.542022 −0.271011 0.962576i \(-0.587358\pi\)
−0.271011 + 0.962576i \(0.587358\pi\)
\(132\) −101.737 −0.0670841
\(133\) 95.8954 0.0625202
\(134\) −627.458 −0.404508
\(135\) −2381.83 −1.51848
\(136\) 1697.65 1.07038
\(137\) −898.575 −0.560368 −0.280184 0.959946i \(-0.590395\pi\)
−0.280184 + 0.959946i \(0.590395\pi\)
\(138\) 196.816 0.121406
\(139\) −423.030 −0.258136 −0.129068 0.991636i \(-0.541199\pi\)
−0.129068 + 0.991636i \(0.541199\pi\)
\(140\) 486.200 0.293510
\(141\) −500.032 −0.298654
\(142\) −1161.68 −0.686520
\(143\) 129.578 0.0757752
\(144\) −202.448 −0.117158
\(145\) 457.009 0.261742
\(146\) 885.417 0.501901
\(147\) −1385.83 −0.777563
\(148\) 1339.65 0.744042
\(149\) 473.514 0.260348 0.130174 0.991491i \(-0.458446\pi\)
0.130174 + 0.991491i \(0.458446\pi\)
\(150\) 682.577 0.371548
\(151\) 1307.13 0.704453 0.352227 0.935915i \(-0.385425\pi\)
0.352227 + 0.935915i \(0.385425\pi\)
\(152\) 362.634 0.193510
\(153\) 724.581 0.382869
\(154\) −24.5417 −0.0128417
\(155\) −3503.10 −1.81533
\(156\) −945.567 −0.485295
\(157\) −2801.81 −1.42426 −0.712129 0.702049i \(-0.752269\pi\)
−0.712129 + 0.702049i \(0.752269\pi\)
\(158\) −152.891 −0.0769832
\(159\) −2976.08 −1.48439
\(160\) 2870.31 1.41824
\(161\) −170.382 −0.0834034
\(162\) −590.896 −0.286575
\(163\) 2190.20 1.05245 0.526227 0.850344i \(-0.323606\pi\)
0.526227 + 0.850344i \(0.323606\pi\)
\(164\) −3050.33 −1.45238
\(165\) 253.849 0.119770
\(166\) −1241.08 −0.580282
\(167\) 2913.56 1.35005 0.675025 0.737795i \(-0.264133\pi\)
0.675025 + 0.737795i \(0.264133\pi\)
\(168\) 408.078 0.187404
\(169\) −992.677 −0.451833
\(170\) −1858.94 −0.838671
\(171\) 154.777 0.0692170
\(172\) 0 0
\(173\) 3189.01 1.40148 0.700740 0.713416i \(-0.252853\pi\)
0.700740 + 0.713416i \(0.252853\pi\)
\(174\) 168.335 0.0733418
\(175\) −590.900 −0.255245
\(176\) 94.0848 0.0402949
\(177\) −1613.86 −0.685341
\(178\) 789.743 0.332549
\(179\) −2902.73 −1.21207 −0.606034 0.795439i \(-0.707240\pi\)
−0.606034 + 0.795439i \(0.707240\pi\)
\(180\) 784.737 0.324949
\(181\) 4262.88 1.75059 0.875296 0.483587i \(-0.160666\pi\)
0.875296 + 0.483587i \(0.160666\pi\)
\(182\) −228.095 −0.0928986
\(183\) 3720.39 1.50284
\(184\) −644.307 −0.258146
\(185\) −3342.60 −1.32839
\(186\) −1290.34 −0.508668
\(187\) −336.738 −0.131683
\(188\) 718.377 0.278686
\(189\) 759.492 0.292301
\(190\) −397.086 −0.151619
\(191\) −1817.08 −0.688374 −0.344187 0.938901i \(-0.611846\pi\)
−0.344187 + 0.938901i \(0.611846\pi\)
\(192\) 179.377 0.0674242
\(193\) −891.204 −0.332385 −0.166192 0.986093i \(-0.553147\pi\)
−0.166192 + 0.986093i \(0.553147\pi\)
\(194\) 877.516 0.324753
\(195\) 2359.32 0.866433
\(196\) 1990.98 0.725574
\(197\) −1697.67 −0.613980 −0.306990 0.951713i \(-0.599322\pi\)
−0.306990 + 0.951713i \(0.599322\pi\)
\(198\) −39.6108 −0.0142173
\(199\) 2475.28 0.881750 0.440875 0.897569i \(-0.354668\pi\)
0.440875 + 0.897569i \(0.354668\pi\)
\(200\) −2234.52 −0.790022
\(201\) −2069.51 −0.726229
\(202\) −1492.27 −0.519781
\(203\) −145.726 −0.0503841
\(204\) 2457.27 0.843351
\(205\) 7610.99 2.59305
\(206\) −2060.23 −0.696811
\(207\) −274.999 −0.0923371
\(208\) 874.443 0.291499
\(209\) −71.9304 −0.0238064
\(210\) −446.848 −0.146836
\(211\) 4547.34 1.48366 0.741829 0.670589i \(-0.233959\pi\)
0.741829 + 0.670589i \(0.233959\pi\)
\(212\) 4275.62 1.38514
\(213\) −3831.50 −1.23254
\(214\) 1273.68 0.406854
\(215\) 0 0
\(216\) 2872.06 0.904717
\(217\) 1117.03 0.349443
\(218\) 2.08315 0.000647195 0
\(219\) 2920.32 0.901083
\(220\) −364.695 −0.111762
\(221\) −3129.71 −0.952612
\(222\) −1231.22 −0.372225
\(223\) −3359.59 −1.00885 −0.504427 0.863454i \(-0.668296\pi\)
−0.504427 + 0.863454i \(0.668296\pi\)
\(224\) −915.253 −0.273004
\(225\) −953.724 −0.282585
\(226\) 609.470 0.179386
\(227\) −3617.71 −1.05778 −0.528889 0.848691i \(-0.677391\pi\)
−0.528889 + 0.848691i \(0.677391\pi\)
\(228\) 524.896 0.152465
\(229\) 2149.30 0.620218 0.310109 0.950701i \(-0.399634\pi\)
0.310109 + 0.950701i \(0.399634\pi\)
\(230\) 705.520 0.202264
\(231\) −80.9446 −0.0230553
\(232\) −551.071 −0.155947
\(233\) 1356.11 0.381296 0.190648 0.981658i \(-0.438941\pi\)
0.190648 + 0.981658i \(0.438941\pi\)
\(234\) −368.151 −0.102849
\(235\) −1792.45 −0.497559
\(236\) 2318.57 0.639518
\(237\) −504.272 −0.138211
\(238\) 592.759 0.161440
\(239\) −4542.30 −1.22936 −0.614679 0.788777i \(-0.710715\pi\)
−0.614679 + 0.788777i \(0.710715\pi\)
\(240\) 1713.07 0.460743
\(241\) −344.024 −0.0919524 −0.0459762 0.998943i \(-0.514640\pi\)
−0.0459762 + 0.998943i \(0.514640\pi\)
\(242\) −1739.02 −0.461936
\(243\) 2170.55 0.573008
\(244\) −5344.94 −1.40236
\(245\) −4967.76 −1.29542
\(246\) 2803.45 0.726590
\(247\) −668.535 −0.172218
\(248\) 4224.12 1.08158
\(249\) −4093.40 −1.04180
\(250\) −129.758 −0.0328264
\(251\) 1574.04 0.395826 0.197913 0.980220i \(-0.436584\pi\)
0.197913 + 0.980220i \(0.436584\pi\)
\(252\) −250.229 −0.0625513
\(253\) 127.802 0.0317582
\(254\) −2479.14 −0.612421
\(255\) −6131.24 −1.50570
\(256\) −2199.87 −0.537077
\(257\) −2254.21 −0.547136 −0.273568 0.961853i \(-0.588204\pi\)
−0.273568 + 0.961853i \(0.588204\pi\)
\(258\) 0 0
\(259\) 1065.85 0.255710
\(260\) −3389.55 −0.808503
\(261\) −235.205 −0.0557809
\(262\) −1073.06 −0.253030
\(263\) 3718.26 0.871777 0.435889 0.900001i \(-0.356434\pi\)
0.435889 + 0.900001i \(0.356434\pi\)
\(264\) −306.096 −0.0713596
\(265\) −10668.3 −2.47300
\(266\) 126.619 0.0291860
\(267\) 2604.77 0.597038
\(268\) 2973.19 0.677673
\(269\) −5900.56 −1.33741 −0.668706 0.743527i \(-0.733151\pi\)
−0.668706 + 0.743527i \(0.733151\pi\)
\(270\) −3144.92 −0.708866
\(271\) −6867.08 −1.53928 −0.769641 0.638476i \(-0.779565\pi\)
−0.769641 + 0.638476i \(0.779565\pi\)
\(272\) −2272.44 −0.506570
\(273\) −752.315 −0.166785
\(274\) −1186.46 −0.261594
\(275\) 443.229 0.0971917
\(276\) −932.606 −0.203392
\(277\) 1041.33 0.225876 0.112938 0.993602i \(-0.463974\pi\)
0.112938 + 0.993602i \(0.463974\pi\)
\(278\) −558.562 −0.120505
\(279\) 1802.91 0.386873
\(280\) 1462.83 0.312216
\(281\) −2022.29 −0.429323 −0.214662 0.976689i \(-0.568865\pi\)
−0.214662 + 0.976689i \(0.568865\pi\)
\(282\) −660.233 −0.139420
\(283\) 8496.85 1.78475 0.892377 0.451292i \(-0.149037\pi\)
0.892377 + 0.451292i \(0.149037\pi\)
\(284\) 5504.57 1.15013
\(285\) −1309.69 −0.272208
\(286\) 171.093 0.0353738
\(287\) −2426.91 −0.499150
\(288\) −1477.24 −0.302247
\(289\) 3220.28 0.655461
\(290\) 603.426 0.122188
\(291\) 2894.27 0.583041
\(292\) −4195.52 −0.840836
\(293\) 1114.98 0.222313 0.111157 0.993803i \(-0.464544\pi\)
0.111157 + 0.993803i \(0.464544\pi\)
\(294\) −1829.83 −0.362986
\(295\) −5785.16 −1.14178
\(296\) 4030.58 0.791462
\(297\) −569.689 −0.111302
\(298\) 625.220 0.121537
\(299\) 1187.82 0.229743
\(300\) −3234.37 −0.622455
\(301\) 0 0
\(302\) 1725.91 0.328857
\(303\) −4921.88 −0.933184
\(304\) −485.414 −0.0915803
\(305\) 13336.4 2.50373
\(306\) 956.724 0.178733
\(307\) −4992.32 −0.928100 −0.464050 0.885809i \(-0.653604\pi\)
−0.464050 + 0.885809i \(0.653604\pi\)
\(308\) 116.290 0.0215138
\(309\) −6795.15 −1.25101
\(310\) −4625.44 −0.847442
\(311\) −7610.06 −1.38755 −0.693773 0.720194i \(-0.744053\pi\)
−0.693773 + 0.720194i \(0.744053\pi\)
\(312\) −2844.92 −0.516224
\(313\) −4478.36 −0.808728 −0.404364 0.914598i \(-0.632507\pi\)
−0.404364 + 0.914598i \(0.632507\pi\)
\(314\) −3699.46 −0.664880
\(315\) 624.355 0.111677
\(316\) 724.469 0.128970
\(317\) −5059.65 −0.896461 −0.448230 0.893918i \(-0.647946\pi\)
−0.448230 + 0.893918i \(0.647946\pi\)
\(318\) −3929.56 −0.692952
\(319\) 109.308 0.0191852
\(320\) 643.009 0.112329
\(321\) 4200.90 0.730440
\(322\) −224.969 −0.0389349
\(323\) 1737.34 0.299283
\(324\) 2799.94 0.480100
\(325\) 4119.46 0.703097
\(326\) 2891.91 0.491312
\(327\) 6.87074 0.00116194
\(328\) −9177.50 −1.54495
\(329\) 571.557 0.0957780
\(330\) 335.177 0.0559118
\(331\) 10063.6 1.67113 0.835566 0.549390i \(-0.185140\pi\)
0.835566 + 0.549390i \(0.185140\pi\)
\(332\) 5880.84 0.972147
\(333\) 1720.31 0.283100
\(334\) 3847.02 0.630238
\(335\) −7418.51 −1.20990
\(336\) −546.246 −0.0886909
\(337\) −961.257 −0.155380 −0.0776899 0.996978i \(-0.524754\pi\)
−0.0776899 + 0.996978i \(0.524754\pi\)
\(338\) −1310.71 −0.210927
\(339\) 2010.18 0.322060
\(340\) 8808.52 1.40503
\(341\) −837.877 −0.133060
\(342\) 204.365 0.0323123
\(343\) 3291.48 0.518144
\(344\) 0 0
\(345\) 2326.98 0.363132
\(346\) 4210.72 0.654247
\(347\) 7415.19 1.14717 0.573585 0.819146i \(-0.305552\pi\)
0.573585 + 0.819146i \(0.305552\pi\)
\(348\) −797.652 −0.122870
\(349\) −7957.18 −1.22045 −0.610226 0.792227i \(-0.708922\pi\)
−0.610226 + 0.792227i \(0.708922\pi\)
\(350\) −780.214 −0.119155
\(351\) −5294.80 −0.805172
\(352\) 686.524 0.103954
\(353\) 4679.48 0.705563 0.352781 0.935706i \(-0.385236\pi\)
0.352781 + 0.935706i \(0.385236\pi\)
\(354\) −2130.91 −0.319935
\(355\) −13734.7 −2.05341
\(356\) −3742.17 −0.557120
\(357\) 1955.06 0.289840
\(358\) −3832.71 −0.565824
\(359\) 4626.84 0.680209 0.340105 0.940388i \(-0.389538\pi\)
0.340105 + 0.940388i \(0.389538\pi\)
\(360\) 2361.03 0.345659
\(361\) −6487.89 −0.945894
\(362\) 5628.63 0.817222
\(363\) −5735.72 −0.829332
\(364\) 1080.82 0.155633
\(365\) 10468.4 1.50121
\(366\) 4912.33 0.701562
\(367\) −7285.27 −1.03621 −0.518103 0.855318i \(-0.673362\pi\)
−0.518103 + 0.855318i \(0.673362\pi\)
\(368\) 862.457 0.122170
\(369\) −3917.09 −0.552616
\(370\) −4413.51 −0.620129
\(371\) 3401.78 0.476042
\(372\) 6114.23 0.852172
\(373\) −9225.62 −1.28066 −0.640328 0.768102i \(-0.721201\pi\)
−0.640328 + 0.768102i \(0.721201\pi\)
\(374\) −444.623 −0.0614731
\(375\) −427.973 −0.0589345
\(376\) 2161.37 0.296448
\(377\) 1015.93 0.138788
\(378\) 1002.82 0.136454
\(379\) −7678.83 −1.04073 −0.520363 0.853945i \(-0.674203\pi\)
−0.520363 + 0.853945i \(0.674203\pi\)
\(380\) 1881.58 0.254008
\(381\) −8176.81 −1.09950
\(382\) −2399.24 −0.321351
\(383\) 12068.6 1.61013 0.805063 0.593189i \(-0.202131\pi\)
0.805063 + 0.593189i \(0.202131\pi\)
\(384\) −6168.89 −0.819805
\(385\) −290.160 −0.0384101
\(386\) −1176.73 −0.155166
\(387\) 0 0
\(388\) −4158.08 −0.544059
\(389\) 779.907 0.101653 0.0508263 0.998708i \(-0.483815\pi\)
0.0508263 + 0.998708i \(0.483815\pi\)
\(390\) 3115.20 0.404473
\(391\) −3086.81 −0.399250
\(392\) 5990.23 0.771818
\(393\) −3539.22 −0.454274
\(394\) −2241.58 −0.286622
\(395\) −1807.65 −0.230260
\(396\) 187.695 0.0238182
\(397\) 6572.49 0.830891 0.415446 0.909618i \(-0.363626\pi\)
0.415446 + 0.909618i \(0.363626\pi\)
\(398\) 3268.32 0.411624
\(399\) 417.620 0.0523988
\(400\) 2991.08 0.373886
\(401\) 8199.25 1.02108 0.510538 0.859855i \(-0.329446\pi\)
0.510538 + 0.859855i \(0.329446\pi\)
\(402\) −2732.55 −0.339023
\(403\) −7787.40 −0.962575
\(404\) 7071.08 0.870790
\(405\) −6986.24 −0.857159
\(406\) −192.414 −0.0235206
\(407\) −799.488 −0.0973690
\(408\) 7393.18 0.897101
\(409\) −2566.69 −0.310305 −0.155152 0.987891i \(-0.549587\pi\)
−0.155152 + 0.987891i \(0.549587\pi\)
\(410\) 10049.4 1.21050
\(411\) −3913.25 −0.469650
\(412\) 9762.34 1.16737
\(413\) 1844.71 0.219788
\(414\) −363.104 −0.0431053
\(415\) −14673.5 −1.73565
\(416\) 6380.69 0.752017
\(417\) −1842.28 −0.216347
\(418\) −94.9756 −0.0111134
\(419\) 3189.67 0.371899 0.185949 0.982559i \(-0.440464\pi\)
0.185949 + 0.982559i \(0.440464\pi\)
\(420\) 2117.38 0.245994
\(421\) 8593.75 0.994855 0.497427 0.867506i \(-0.334278\pi\)
0.497427 + 0.867506i \(0.334278\pi\)
\(422\) 6004.23 0.692610
\(423\) 922.505 0.106037
\(424\) 12864.0 1.47342
\(425\) −10705.4 −1.22185
\(426\) −5059.05 −0.575379
\(427\) −4252.56 −0.481957
\(428\) −6035.27 −0.681603
\(429\) 564.306 0.0635080
\(430\) 0 0
\(431\) 14083.4 1.57396 0.786978 0.616980i \(-0.211644\pi\)
0.786978 + 0.616980i \(0.211644\pi\)
\(432\) −3844.48 −0.428166
\(433\) 4030.11 0.447285 0.223643 0.974671i \(-0.428205\pi\)
0.223643 + 0.974671i \(0.428205\pi\)
\(434\) 1474.91 0.163129
\(435\) 1990.25 0.219368
\(436\) −9.87093 −0.00108425
\(437\) −659.372 −0.0721785
\(438\) 3855.95 0.420649
\(439\) −1907.40 −0.207370 −0.103685 0.994610i \(-0.533063\pi\)
−0.103685 + 0.994610i \(0.533063\pi\)
\(440\) −1097.25 −0.118885
\(441\) 2556.72 0.276073
\(442\) −4132.42 −0.444704
\(443\) −383.394 −0.0411187 −0.0205594 0.999789i \(-0.506545\pi\)
−0.0205594 + 0.999789i \(0.506545\pi\)
\(444\) 5834.09 0.623590
\(445\) 9337.23 0.994668
\(446\) −4435.94 −0.470959
\(447\) 2062.13 0.218200
\(448\) −205.036 −0.0216228
\(449\) −6486.48 −0.681773 −0.340886 0.940105i \(-0.610727\pi\)
−0.340886 + 0.940105i \(0.610727\pi\)
\(450\) −1259.28 −0.131918
\(451\) 1820.41 0.190066
\(452\) −2887.95 −0.300526
\(453\) 5692.47 0.590409
\(454\) −4776.76 −0.493798
\(455\) −2696.80 −0.277864
\(456\) 1579.25 0.162182
\(457\) −6968.83 −0.713322 −0.356661 0.934234i \(-0.616085\pi\)
−0.356661 + 0.934234i \(0.616085\pi\)
\(458\) 2837.90 0.289534
\(459\) 13759.8 1.39924
\(460\) −3343.08 −0.338852
\(461\) −9235.71 −0.933080 −0.466540 0.884500i \(-0.654500\pi\)
−0.466540 + 0.884500i \(0.654500\pi\)
\(462\) −106.878 −0.0107628
\(463\) −6346.35 −0.637019 −0.318510 0.947920i \(-0.603182\pi\)
−0.318510 + 0.947920i \(0.603182\pi\)
\(464\) 737.653 0.0738032
\(465\) −15255.8 −1.52145
\(466\) 1790.59 0.177999
\(467\) 12576.7 1.24621 0.623103 0.782140i \(-0.285872\pi\)
0.623103 + 0.782140i \(0.285872\pi\)
\(468\) 1744.47 0.172304
\(469\) 2365.54 0.232901
\(470\) −2366.72 −0.232274
\(471\) −12201.7 −1.19369
\(472\) 6975.87 0.680277
\(473\) 0 0
\(474\) −665.832 −0.0645204
\(475\) −2286.76 −0.220893
\(476\) −2808.77 −0.270461
\(477\) 5490.54 0.527033
\(478\) −5997.57 −0.573896
\(479\) 10394.1 0.991478 0.495739 0.868471i \(-0.334897\pi\)
0.495739 + 0.868471i \(0.334897\pi\)
\(480\) 12500.0 1.18864
\(481\) −7430.60 −0.704379
\(482\) −454.243 −0.0429257
\(483\) −742.003 −0.0699013
\(484\) 8240.29 0.773882
\(485\) 10375.0 0.971348
\(486\) 2865.96 0.267495
\(487\) 10010.4 0.931449 0.465724 0.884930i \(-0.345794\pi\)
0.465724 + 0.884930i \(0.345794\pi\)
\(488\) −16081.3 −1.49173
\(489\) 9538.23 0.882073
\(490\) −6559.34 −0.604736
\(491\) −7102.70 −0.652832 −0.326416 0.945226i \(-0.605841\pi\)
−0.326416 + 0.945226i \(0.605841\pi\)
\(492\) −13284.0 −1.21726
\(493\) −2640.13 −0.241188
\(494\) −882.722 −0.0803958
\(495\) −468.324 −0.0425244
\(496\) −5654.32 −0.511868
\(497\) 4379.57 0.395272
\(498\) −5404.86 −0.486340
\(499\) −20381.3 −1.82844 −0.914222 0.405213i \(-0.867197\pi\)
−0.914222 + 0.405213i \(0.867197\pi\)
\(500\) 614.852 0.0549941
\(501\) 12688.4 1.13149
\(502\) 2078.33 0.184782
\(503\) 3122.92 0.276827 0.138414 0.990375i \(-0.455800\pi\)
0.138414 + 0.990375i \(0.455800\pi\)
\(504\) −752.861 −0.0665379
\(505\) −17643.3 −1.55469
\(506\) 168.747 0.0148256
\(507\) −4323.06 −0.378686
\(508\) 11747.3 1.02599
\(509\) 16509.6 1.43768 0.718838 0.695178i \(-0.244674\pi\)
0.718838 + 0.695178i \(0.244674\pi\)
\(510\) −8095.58 −0.702899
\(511\) −3338.05 −0.288976
\(512\) 8427.53 0.727437
\(513\) 2939.21 0.252962
\(514\) −2976.42 −0.255417
\(515\) −24358.4 −2.08419
\(516\) 0 0
\(517\) −428.721 −0.0364702
\(518\) 1407.33 0.119372
\(519\) 13888.0 1.17460
\(520\) −10198.1 −0.860031
\(521\) 4809.95 0.404468 0.202234 0.979337i \(-0.435180\pi\)
0.202234 + 0.979337i \(0.435180\pi\)
\(522\) −310.561 −0.0260400
\(523\) 16802.3 1.40481 0.702405 0.711778i \(-0.252110\pi\)
0.702405 + 0.711778i \(0.252110\pi\)
\(524\) 5084.66 0.423901
\(525\) −2573.34 −0.213923
\(526\) 4909.52 0.406968
\(527\) 20237.3 1.67278
\(528\) 409.735 0.0337716
\(529\) −10995.5 −0.903712
\(530\) −14086.2 −1.15446
\(531\) 2977.40 0.243330
\(532\) −599.978 −0.0488954
\(533\) 16919.2 1.37496
\(534\) 3439.29 0.278713
\(535\) 15058.8 1.21692
\(536\) 8945.41 0.720863
\(537\) −12641.2 −1.01585
\(538\) −7791.00 −0.624338
\(539\) −1188.20 −0.0949522
\(540\) 14902.1 1.18756
\(541\) 7216.68 0.573511 0.286755 0.958004i \(-0.407423\pi\)
0.286755 + 0.958004i \(0.407423\pi\)
\(542\) −9067.18 −0.718577
\(543\) 18564.6 1.46719
\(544\) −16581.7 −1.30686
\(545\) 24.6293 0.00193579
\(546\) −993.344 −0.0778593
\(547\) −2417.37 −0.188957 −0.0944784 0.995527i \(-0.530118\pi\)
−0.0944784 + 0.995527i \(0.530118\pi\)
\(548\) 5622.01 0.438249
\(549\) −6863.71 −0.533581
\(550\) 585.232 0.0453716
\(551\) −563.956 −0.0436031
\(552\) −2805.93 −0.216355
\(553\) 576.404 0.0443240
\(554\) 1374.96 0.105445
\(555\) −14556.9 −1.11334
\(556\) 2646.73 0.201882
\(557\) 2933.69 0.223168 0.111584 0.993755i \(-0.464408\pi\)
0.111584 + 0.993755i \(0.464408\pi\)
\(558\) 2380.54 0.180602
\(559\) 0 0
\(560\) −1958.11 −0.147759
\(561\) −1466.48 −0.110365
\(562\) −2670.20 −0.200419
\(563\) 1621.04 0.121348 0.0606740 0.998158i \(-0.480675\pi\)
0.0606740 + 0.998158i \(0.480675\pi\)
\(564\) 3128.49 0.233570
\(565\) 7205.84 0.536552
\(566\) 11219.1 0.833169
\(567\) 2227.70 0.164999
\(568\) 16561.6 1.22343
\(569\) 12415.4 0.914730 0.457365 0.889279i \(-0.348793\pi\)
0.457365 + 0.889279i \(0.348793\pi\)
\(570\) −1729.29 −0.127074
\(571\) 24621.9 1.80454 0.902271 0.431168i \(-0.141899\pi\)
0.902271 + 0.431168i \(0.141899\pi\)
\(572\) −810.716 −0.0592618
\(573\) −7913.31 −0.576934
\(574\) −3204.45 −0.233016
\(575\) 4062.99 0.294676
\(576\) −330.932 −0.0239390
\(577\) −9543.80 −0.688585 −0.344293 0.938862i \(-0.611881\pi\)
−0.344293 + 0.938862i \(0.611881\pi\)
\(578\) 4252.00 0.305986
\(579\) −3881.15 −0.278575
\(580\) −2859.32 −0.204701
\(581\) 4678.93 0.334105
\(582\) 3821.54 0.272179
\(583\) −2551.65 −0.181267
\(584\) −12623.0 −0.894425
\(585\) −4352.69 −0.307627
\(586\) 1472.20 0.103782
\(587\) −10249.6 −0.720692 −0.360346 0.932819i \(-0.617341\pi\)
−0.360346 + 0.932819i \(0.617341\pi\)
\(588\) 8670.60 0.608111
\(589\) 4322.88 0.302413
\(590\) −7638.62 −0.533012
\(591\) −7393.27 −0.514583
\(592\) −5395.26 −0.374567
\(593\) −4080.19 −0.282552 −0.141276 0.989970i \(-0.545121\pi\)
−0.141276 + 0.989970i \(0.545121\pi\)
\(594\) −752.207 −0.0519586
\(595\) 7008.26 0.482875
\(596\) −2962.58 −0.203611
\(597\) 10779.7 0.739004
\(598\) 1568.37 0.107250
\(599\) 18792.0 1.28184 0.640918 0.767609i \(-0.278554\pi\)
0.640918 + 0.767609i \(0.278554\pi\)
\(600\) −9731.22 −0.662126
\(601\) 11550.6 0.783961 0.391981 0.919974i \(-0.371790\pi\)
0.391981 + 0.919974i \(0.371790\pi\)
\(602\) 0 0
\(603\) 3818.03 0.257848
\(604\) −8178.15 −0.550934
\(605\) −20560.7 −1.38167
\(606\) −6498.77 −0.435634
\(607\) −15936.5 −1.06564 −0.532819 0.846229i \(-0.678867\pi\)
−0.532819 + 0.846229i \(0.678867\pi\)
\(608\) −3542.00 −0.236262
\(609\) −634.630 −0.0422274
\(610\) 17609.1 1.16881
\(611\) −3984.61 −0.263830
\(612\) −4533.41 −0.299432
\(613\) −5661.66 −0.373038 −0.186519 0.982451i \(-0.559721\pi\)
−0.186519 + 0.982451i \(0.559721\pi\)
\(614\) −6591.77 −0.433261
\(615\) 33145.5 2.17326
\(616\) 349.881 0.0228849
\(617\) 730.404 0.0476579 0.0238290 0.999716i \(-0.492414\pi\)
0.0238290 + 0.999716i \(0.492414\pi\)
\(618\) −8972.20 −0.584005
\(619\) −19348.7 −1.25636 −0.628181 0.778067i \(-0.716200\pi\)
−0.628181 + 0.778067i \(0.716200\pi\)
\(620\) 21917.5 1.41972
\(621\) −5222.22 −0.337457
\(622\) −10048.2 −0.647742
\(623\) −2977.36 −0.191469
\(624\) 3808.15 0.244308
\(625\) −16372.3 −1.04782
\(626\) −5913.15 −0.377535
\(627\) −313.253 −0.0199524
\(628\) 17529.8 1.11387
\(629\) 19310.1 1.22408
\(630\) 824.387 0.0521339
\(631\) 1817.20 0.114646 0.0573230 0.998356i \(-0.481744\pi\)
0.0573230 + 0.998356i \(0.481744\pi\)
\(632\) 2179.70 0.137190
\(633\) 19803.4 1.24347
\(634\) −6680.67 −0.418491
\(635\) −29311.2 −1.83178
\(636\) 18620.1 1.16090
\(637\) −11043.3 −0.686896
\(638\) 144.328 0.00895614
\(639\) 7068.71 0.437612
\(640\) −22113.4 −1.36580
\(641\) −18827.8 −1.16014 −0.580072 0.814565i \(-0.696976\pi\)
−0.580072 + 0.814565i \(0.696976\pi\)
\(642\) 5546.79 0.340988
\(643\) 29918.2 1.83493 0.917463 0.397822i \(-0.130234\pi\)
0.917463 + 0.397822i \(0.130234\pi\)
\(644\) 1066.01 0.0652276
\(645\) 0 0
\(646\) 2293.96 0.139713
\(647\) −2632.68 −0.159971 −0.0799856 0.996796i \(-0.525487\pi\)
−0.0799856 + 0.996796i \(0.525487\pi\)
\(648\) 8424.17 0.510698
\(649\) −1383.70 −0.0836904
\(650\) 5439.26 0.328224
\(651\) 4864.62 0.292872
\(652\) −13703.2 −0.823096
\(653\) 10367.7 0.621315 0.310658 0.950522i \(-0.399451\pi\)
0.310658 + 0.950522i \(0.399451\pi\)
\(654\) 9.07201 0.000542421 0
\(655\) −12686.9 −0.756823
\(656\) 12284.8 0.731161
\(657\) −5387.68 −0.319929
\(658\) 754.674 0.0447116
\(659\) −14731.7 −0.870815 −0.435407 0.900234i \(-0.643396\pi\)
−0.435407 + 0.900234i \(0.643396\pi\)
\(660\) −1588.23 −0.0936692
\(661\) 17743.4 1.04408 0.522042 0.852920i \(-0.325171\pi\)
0.522042 + 0.852920i \(0.325171\pi\)
\(662\) 13287.8 0.780127
\(663\) −13629.7 −0.798394
\(664\) 17693.6 1.03411
\(665\) 1497.03 0.0872966
\(666\) 2271.47 0.132158
\(667\) 1002.01 0.0581676
\(668\) −18229.0 −1.05584
\(669\) −14630.8 −0.845532
\(670\) −9795.28 −0.564813
\(671\) 3189.81 0.183519
\(672\) −3985.88 −0.228808
\(673\) 16842.3 0.964668 0.482334 0.875987i \(-0.339789\pi\)
0.482334 + 0.875987i \(0.339789\pi\)
\(674\) −1269.23 −0.0725353
\(675\) −18111.2 −1.03274
\(676\) 6210.77 0.353367
\(677\) 7750.69 0.440005 0.220002 0.975499i \(-0.429394\pi\)
0.220002 + 0.975499i \(0.429394\pi\)
\(678\) 2654.21 0.150346
\(679\) −3308.27 −0.186980
\(680\) 26502.1 1.49457
\(681\) −15754.9 −0.886535
\(682\) −1106.32 −0.0621160
\(683\) −3639.03 −0.203871 −0.101935 0.994791i \(-0.532503\pi\)
−0.101935 + 0.994791i \(0.532503\pi\)
\(684\) −968.377 −0.0541328
\(685\) −14027.7 −0.782439
\(686\) 4346.02 0.241883
\(687\) 9360.11 0.519811
\(688\) 0 0
\(689\) −23715.5 −1.31131
\(690\) 3072.50 0.169519
\(691\) −21688.0 −1.19399 −0.596997 0.802243i \(-0.703640\pi\)
−0.596997 + 0.802243i \(0.703640\pi\)
\(692\) −19952.3 −1.09606
\(693\) 149.334 0.00818576
\(694\) 9790.89 0.535529
\(695\) −6603.95 −0.360435
\(696\) −2399.89 −0.130700
\(697\) −43968.5 −2.38942
\(698\) −10506.5 −0.569739
\(699\) 5905.80 0.319568
\(700\) 3697.02 0.199620
\(701\) −5751.17 −0.309870 −0.154935 0.987925i \(-0.549517\pi\)
−0.154935 + 0.987925i \(0.549517\pi\)
\(702\) −6991.16 −0.375875
\(703\) 4124.82 0.221295
\(704\) 153.796 0.00823352
\(705\) −7806.02 −0.417010
\(706\) 6178.70 0.329375
\(707\) 5625.91 0.299271
\(708\) 10097.3 0.535987
\(709\) −23130.5 −1.22522 −0.612612 0.790384i \(-0.709881\pi\)
−0.612612 + 0.790384i \(0.709881\pi\)
\(710\) −18135.0 −0.958584
\(711\) 930.328 0.0490718
\(712\) −11259.0 −0.592627
\(713\) −7680.65 −0.403426
\(714\) 2581.43 0.135305
\(715\) 2022.85 0.105805
\(716\) 18161.2 0.947926
\(717\) −19781.5 −1.03034
\(718\) 6109.19 0.317539
\(719\) −32891.3 −1.70604 −0.853018 0.521882i \(-0.825230\pi\)
−0.853018 + 0.521882i \(0.825230\pi\)
\(720\) −3160.43 −0.163586
\(721\) 7767.14 0.401198
\(722\) −8566.49 −0.441568
\(723\) −1498.21 −0.0770662
\(724\) −26671.1 −1.36909
\(725\) 3475.05 0.178014
\(726\) −7573.35 −0.387153
\(727\) −7446.63 −0.379890 −0.189945 0.981795i \(-0.560831\pi\)
−0.189945 + 0.981795i \(0.560831\pi\)
\(728\) 3251.86 0.165552
\(729\) 21535.7 1.09412
\(730\) 13822.3 0.700802
\(731\) 0 0
\(732\) −23276.9 −1.17533
\(733\) 26197.6 1.32009 0.660047 0.751224i \(-0.270536\pi\)
0.660047 + 0.751224i \(0.270536\pi\)
\(734\) −9619.34 −0.483728
\(735\) −21634.3 −1.08571
\(736\) 6293.23 0.315179
\(737\) −1774.37 −0.0886836
\(738\) −5172.05 −0.257976
\(739\) −9936.92 −0.494635 −0.247318 0.968934i \(-0.579549\pi\)
−0.247318 + 0.968934i \(0.579549\pi\)
\(740\) 20913.3 1.03890
\(741\) −2911.44 −0.144338
\(742\) 4491.65 0.222229
\(743\) 6039.26 0.298195 0.149098 0.988822i \(-0.452363\pi\)
0.149098 + 0.988822i \(0.452363\pi\)
\(744\) 18395.8 0.906483
\(745\) 7392.05 0.363522
\(746\) −12181.3 −0.597843
\(747\) 7551.89 0.369892
\(748\) 2106.83 0.102986
\(749\) −4801.80 −0.234251
\(750\) −565.088 −0.0275121
\(751\) −26396.0 −1.28256 −0.641280 0.767307i \(-0.721596\pi\)
−0.641280 + 0.767307i \(0.721596\pi\)
\(752\) −2893.17 −0.140297
\(753\) 6854.86 0.331746
\(754\) 1341.42 0.0647898
\(755\) 20405.6 0.983624
\(756\) −4751.83 −0.228601
\(757\) 37477.2 1.79938 0.899691 0.436526i \(-0.143791\pi\)
0.899691 + 0.436526i \(0.143791\pi\)
\(758\) −10139.0 −0.485838
\(759\) 556.571 0.0266169
\(760\) 5661.09 0.270197
\(761\) 20551.1 0.978944 0.489472 0.872019i \(-0.337190\pi\)
0.489472 + 0.872019i \(0.337190\pi\)
\(762\) −10796.5 −0.513276
\(763\) −7.85354 −0.000372631 0
\(764\) 11368.7 0.538360
\(765\) 11311.5 0.534598
\(766\) 15935.2 0.751648
\(767\) −12860.4 −0.605427
\(768\) −9580.32 −0.450130
\(769\) 36513.5 1.71224 0.856119 0.516778i \(-0.172869\pi\)
0.856119 + 0.516778i \(0.172869\pi\)
\(770\) −383.122 −0.0179308
\(771\) −9816.98 −0.458560
\(772\) 5575.90 0.259949
\(773\) 21597.5 1.00493 0.502464 0.864598i \(-0.332427\pi\)
0.502464 + 0.864598i \(0.332427\pi\)
\(774\) 0 0
\(775\) −26637.3 −1.23463
\(776\) −12510.4 −0.578733
\(777\) 4641.74 0.214313
\(778\) 1029.78 0.0474540
\(779\) −9392.08 −0.431972
\(780\) −14761.3 −0.677614
\(781\) −3285.08 −0.150511
\(782\) −4075.78 −0.186380
\(783\) −4466.53 −0.203858
\(784\) −8018.41 −0.365270
\(785\) −43739.1 −1.98868
\(786\) −4673.12 −0.212067
\(787\) 24574.8 1.11308 0.556542 0.830820i \(-0.312128\pi\)
0.556542 + 0.830820i \(0.312128\pi\)
\(788\) 10621.6 0.480178
\(789\) 16192.8 0.730646
\(790\) −2386.79 −0.107491
\(791\) −2297.72 −0.103284
\(792\) 564.715 0.0253362
\(793\) 29646.7 1.32760
\(794\) 8678.20 0.387881
\(795\) −46459.7 −2.07265
\(796\) −15486.8 −0.689594
\(797\) 21154.6 0.940194 0.470097 0.882615i \(-0.344219\pi\)
0.470097 + 0.882615i \(0.344219\pi\)
\(798\) 551.418 0.0244611
\(799\) 10354.9 0.458487
\(800\) 21825.5 0.964561
\(801\) −4805.52 −0.211978
\(802\) 10826.2 0.476664
\(803\) 2503.85 0.110036
\(804\) 12948.1 0.567965
\(805\) −2659.84 −0.116456
\(806\) −10282.3 −0.449355
\(807\) −25696.7 −1.12090
\(808\) 21274.7 0.926289
\(809\) 28704.3 1.24745 0.623726 0.781643i \(-0.285618\pi\)
0.623726 + 0.781643i \(0.285618\pi\)
\(810\) −9224.51 −0.400144
\(811\) −17593.2 −0.761753 −0.380876 0.924626i \(-0.624378\pi\)
−0.380876 + 0.924626i \(0.624378\pi\)
\(812\) 911.749 0.0394041
\(813\) −29905.8 −1.29009
\(814\) −1055.63 −0.0454543
\(815\) 34191.4 1.46954
\(816\) −9896.37 −0.424562
\(817\) 0 0
\(818\) −3389.01 −0.144858
\(819\) 1387.94 0.0592168
\(820\) −47618.9 −2.02795
\(821\) 30584.0 1.30011 0.650055 0.759887i \(-0.274746\pi\)
0.650055 + 0.759887i \(0.274746\pi\)
\(822\) −5166.98 −0.219245
\(823\) 15398.4 0.652191 0.326096 0.945337i \(-0.394267\pi\)
0.326096 + 0.945337i \(0.394267\pi\)
\(824\) 29371.9 1.24177
\(825\) 1930.24 0.0814574
\(826\) 2435.72 0.102603
\(827\) 20703.4 0.870529 0.435264 0.900303i \(-0.356655\pi\)
0.435264 + 0.900303i \(0.356655\pi\)
\(828\) 1720.56 0.0722144
\(829\) −6121.19 −0.256451 −0.128225 0.991745i \(-0.540928\pi\)
−0.128225 + 0.991745i \(0.540928\pi\)
\(830\) −19374.6 −0.810245
\(831\) 4534.95 0.189309
\(832\) 1429.41 0.0595623
\(833\) 28698.6 1.19370
\(834\) −2432.51 −0.100996
\(835\) 45483.8 1.88507
\(836\) 450.039 0.0186183
\(837\) 34237.2 1.41387
\(838\) 4211.58 0.173612
\(839\) −5023.30 −0.206703 −0.103351 0.994645i \(-0.532957\pi\)
−0.103351 + 0.994645i \(0.532957\pi\)
\(840\) 6370.53 0.261672
\(841\) −23532.0 −0.964861
\(842\) 11347.0 0.464424
\(843\) −8806.98 −0.359820
\(844\) −28450.8 −1.16033
\(845\) −15496.7 −0.630892
\(846\) 1218.06 0.0495009
\(847\) 6556.17 0.265965
\(848\) −17219.5 −0.697312
\(849\) 37003.3 1.49582
\(850\) −14135.2 −0.570391
\(851\) −7328.75 −0.295213
\(852\) 23972.1 0.963934
\(853\) −15719.7 −0.630986 −0.315493 0.948928i \(-0.602170\pi\)
−0.315493 + 0.948928i \(0.602170\pi\)
\(854\) −5615.00 −0.224990
\(855\) 2416.23 0.0966473
\(856\) −18158.3 −0.725043
\(857\) 49442.8 1.97075 0.985375 0.170398i \(-0.0545055\pi\)
0.985375 + 0.170398i \(0.0545055\pi\)
\(858\) 745.099 0.0296472
\(859\) −1836.36 −0.0729402 −0.0364701 0.999335i \(-0.511611\pi\)
−0.0364701 + 0.999335i \(0.511611\pi\)
\(860\) 0 0
\(861\) −10569.1 −0.418343
\(862\) 18595.5 0.734764
\(863\) 29206.5 1.15203 0.576014 0.817440i \(-0.304607\pi\)
0.576014 + 0.817440i \(0.304607\pi\)
\(864\) −28052.7 −1.10460
\(865\) 49783.8 1.95688
\(866\) 5321.28 0.208804
\(867\) 14024.2 0.549349
\(868\) −6988.82 −0.273290
\(869\) −432.356 −0.0168776
\(870\) 2627.89 0.102407
\(871\) −16491.4 −0.641548
\(872\) −29.6986 −0.00115335
\(873\) −5339.61 −0.207009
\(874\) −870.623 −0.0336948
\(875\) 489.191 0.0189002
\(876\) −18271.3 −0.704714
\(877\) 45448.1 1.74991 0.874957 0.484201i \(-0.160890\pi\)
0.874957 + 0.484201i \(0.160890\pi\)
\(878\) −2518.50 −0.0968055
\(879\) 4855.68 0.186323
\(880\) 1468.76 0.0562636
\(881\) −4104.22 −0.156952 −0.0784760 0.996916i \(-0.525005\pi\)
−0.0784760 + 0.996916i \(0.525005\pi\)
\(882\) 3375.84 0.128878
\(883\) −17943.1 −0.683843 −0.341922 0.939728i \(-0.611078\pi\)
−0.341922 + 0.939728i \(0.611078\pi\)
\(884\) 19581.3 0.745013
\(885\) −25194.1 −0.956938
\(886\) −506.227 −0.0191953
\(887\) 5704.94 0.215956 0.107978 0.994153i \(-0.465562\pi\)
0.107978 + 0.994153i \(0.465562\pi\)
\(888\) 17553.0 0.663333
\(889\) 9346.44 0.352609
\(890\) 12328.7 0.464336
\(891\) −1670.98 −0.0628282
\(892\) 21019.6 0.788999
\(893\) 2211.91 0.0828877
\(894\) 2722.80 0.101861
\(895\) −45314.6 −1.69240
\(896\) 7051.30 0.262910
\(897\) 5172.88 0.192550
\(898\) −8564.63 −0.318269
\(899\) −6569.21 −0.243710
\(900\) 5967.06 0.221002
\(901\) 61630.3 2.27880
\(902\) 2403.64 0.0887276
\(903\) 0 0
\(904\) −8688.96 −0.319680
\(905\) 66548.0 2.44434
\(906\) 7516.24 0.275618
\(907\) −7250.35 −0.265429 −0.132714 0.991154i \(-0.542369\pi\)
−0.132714 + 0.991154i \(0.542369\pi\)
\(908\) 22634.5 0.827261
\(909\) 9080.34 0.331327
\(910\) −3560.81 −0.129714
\(911\) 24710.5 0.898679 0.449340 0.893361i \(-0.351659\pi\)
0.449340 + 0.893361i \(0.351659\pi\)
\(912\) −2113.95 −0.0767544
\(913\) −3509.63 −0.127220
\(914\) −9201.53 −0.332997
\(915\) 58079.2 2.09840
\(916\) −13447.3 −0.485057
\(917\) 4045.47 0.145685
\(918\) 18168.1 0.653201
\(919\) 18905.6 0.678604 0.339302 0.940678i \(-0.389809\pi\)
0.339302 + 0.940678i \(0.389809\pi\)
\(920\) −10058.3 −0.360449
\(921\) −21741.3 −0.777850
\(922\) −12194.7 −0.435586
\(923\) −30532.2 −1.08882
\(924\) 506.437 0.0180309
\(925\) −25416.8 −0.903459
\(926\) −8379.61 −0.297377
\(927\) 12536.3 0.444171
\(928\) 5382.56 0.190400
\(929\) 40793.3 1.44067 0.720336 0.693625i \(-0.243987\pi\)
0.720336 + 0.693625i \(0.243987\pi\)
\(930\) −20143.5 −0.710250
\(931\) 6130.29 0.215803
\(932\) −8484.64 −0.298201
\(933\) −33141.4 −1.16292
\(934\) 16606.0 0.581761
\(935\) −5256.84 −0.183869
\(936\) 5248.57 0.183285
\(937\) 30424.2 1.06074 0.530370 0.847766i \(-0.322053\pi\)
0.530370 + 0.847766i \(0.322053\pi\)
\(938\) 3123.41 0.108724
\(939\) −19503.0 −0.677803
\(940\) 11214.6 0.389128
\(941\) 13717.9 0.475231 0.237615 0.971359i \(-0.423634\pi\)
0.237615 + 0.971359i \(0.423634\pi\)
\(942\) −16110.9 −0.557243
\(943\) 16687.3 0.576261
\(944\) −9337.77 −0.321948
\(945\) 11856.5 0.408138
\(946\) 0 0
\(947\) −6731.42 −0.230984 −0.115492 0.993308i \(-0.536844\pi\)
−0.115492 + 0.993308i \(0.536844\pi\)
\(948\) 3155.02 0.108091
\(949\) 23271.2 0.796013
\(950\) −3019.40 −0.103118
\(951\) −22034.5 −0.751333
\(952\) −8450.72 −0.287699
\(953\) 22468.9 0.763735 0.381868 0.924217i \(-0.375281\pi\)
0.381868 + 0.924217i \(0.375281\pi\)
\(954\) 7249.62 0.246032
\(955\) −28366.6 −0.961174
\(956\) 28419.3 0.961449
\(957\) 476.031 0.0160793
\(958\) 13724.2 0.462848
\(959\) 4473.00 0.150616
\(960\) 2800.27 0.0941441
\(961\) 20563.9 0.690271
\(962\) −9811.24 −0.328822
\(963\) −7750.21 −0.259343
\(964\) 2152.42 0.0719135
\(965\) −13912.6 −0.464107
\(966\) −979.728 −0.0326317
\(967\) 38313.9 1.27414 0.637069 0.770807i \(-0.280147\pi\)
0.637069 + 0.770807i \(0.280147\pi\)
\(968\) 24792.5 0.823204
\(969\) 7566.04 0.250832
\(970\) 13699.0 0.453450
\(971\) −54694.4 −1.80765 −0.903825 0.427902i \(-0.859253\pi\)
−0.903825 + 0.427902i \(0.859253\pi\)
\(972\) −13580.2 −0.448134
\(973\) 2105.80 0.0693821
\(974\) 13217.6 0.434824
\(975\) 17940.0 0.589273
\(976\) 21526.1 0.705976
\(977\) −14933.3 −0.489005 −0.244502 0.969649i \(-0.578625\pi\)
−0.244502 + 0.969649i \(0.578625\pi\)
\(978\) 12594.1 0.411774
\(979\) 2233.29 0.0729074
\(980\) 31081.2 1.01312
\(981\) −12.6758 −0.000412545 0
\(982\) −9378.28 −0.304759
\(983\) 17906.2 0.580997 0.290499 0.956875i \(-0.406179\pi\)
0.290499 + 0.956875i \(0.406179\pi\)
\(984\) −39967.6 −1.29484
\(985\) −26502.4 −0.857297
\(986\) −3485.98 −0.112593
\(987\) 2489.10 0.0802726
\(988\) 4182.75 0.134687
\(989\) 0 0
\(990\) −618.366 −0.0198515
\(991\) −5800.06 −0.185918 −0.0929591 0.995670i \(-0.529633\pi\)
−0.0929591 + 0.995670i \(0.529633\pi\)
\(992\) −41258.8 −1.32053
\(993\) 43826.4 1.40059
\(994\) 5782.70 0.184523
\(995\) 38641.8 1.23118
\(996\) 25610.7 0.814767
\(997\) 29540.8 0.938383 0.469191 0.883097i \(-0.344545\pi\)
0.469191 + 0.883097i \(0.344545\pi\)
\(998\) −26911.2 −0.853565
\(999\) 32668.6 1.03462
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.g.1.18 30
43.32 odd 14 43.4.e.a.35.6 yes 60
43.39 odd 14 43.4.e.a.16.6 60
43.42 odd 2 1849.4.a.h.1.13 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.16.6 60 43.39 odd 14
43.4.e.a.35.6 yes 60 43.32 odd 14
1849.4.a.g.1.18 30 1.1 even 1 trivial
1849.4.a.h.1.13 30 43.42 odd 2