Properties

Label 1849.4.a.j.1.47
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 1.47
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.05256 q^{2} -9.36203 q^{3} +17.5283 q^{4} -14.1438 q^{5} -47.3022 q^{6} +13.7151 q^{7} +48.1425 q^{8} +60.6475 q^{9} +O(q^{10})\) \(q+5.05256 q^{2} -9.36203 q^{3} +17.5283 q^{4} -14.1438 q^{5} -47.3022 q^{6} +13.7151 q^{7} +48.1425 q^{8} +60.6475 q^{9} -71.4625 q^{10} +10.3705 q^{11} -164.101 q^{12} -61.4205 q^{13} +69.2963 q^{14} +132.415 q^{15} +103.016 q^{16} -24.4390 q^{17} +306.425 q^{18} +65.5106 q^{19} -247.918 q^{20} -128.401 q^{21} +52.3977 q^{22} +23.4730 q^{23} -450.711 q^{24} +75.0480 q^{25} -310.331 q^{26} -315.009 q^{27} +240.403 q^{28} +236.538 q^{29} +669.034 q^{30} -210.270 q^{31} +135.354 q^{32} -97.0891 q^{33} -123.479 q^{34} -193.984 q^{35} +1063.05 q^{36} +316.642 q^{37} +330.996 q^{38} +575.020 q^{39} -680.919 q^{40} +13.2054 q^{41} -648.754 q^{42} +181.778 q^{44} -857.789 q^{45} +118.599 q^{46} +450.216 q^{47} -964.439 q^{48} -154.896 q^{49} +379.185 q^{50} +228.798 q^{51} -1076.60 q^{52} -205.541 q^{53} -1591.60 q^{54} -146.679 q^{55} +660.279 q^{56} -613.312 q^{57} +1195.12 q^{58} +334.558 q^{59} +2321.01 q^{60} -811.774 q^{61} -1062.40 q^{62} +831.787 q^{63} -140.242 q^{64} +868.722 q^{65} -490.548 q^{66} -113.497 q^{67} -428.375 q^{68} -219.755 q^{69} -980.116 q^{70} +53.8087 q^{71} +2919.72 q^{72} +108.423 q^{73} +1599.85 q^{74} -702.602 q^{75} +1148.29 q^{76} +142.233 q^{77} +2905.32 q^{78} -723.417 q^{79} -1457.04 q^{80} +1311.64 q^{81} +66.7209 q^{82} +191.269 q^{83} -2250.66 q^{84} +345.661 q^{85} -2214.47 q^{87} +499.263 q^{88} -419.159 q^{89} -4334.03 q^{90} -842.389 q^{91} +411.443 q^{92} +1968.55 q^{93} +2274.74 q^{94} -926.571 q^{95} -1267.19 q^{96} -953.469 q^{97} -782.621 q^{98} +628.947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.05256 1.78635 0.893174 0.449710i \(-0.148473\pi\)
0.893174 + 0.449710i \(0.148473\pi\)
\(3\) −9.36203 −1.80172 −0.900861 0.434107i \(-0.857064\pi\)
−0.900861 + 0.434107i \(0.857064\pi\)
\(4\) 17.5283 2.19104
\(5\) −14.1438 −1.26506 −0.632531 0.774535i \(-0.717984\pi\)
−0.632531 + 0.774535i \(0.717984\pi\)
\(6\) −47.3022 −3.21851
\(7\) 13.7151 0.740546 0.370273 0.928923i \(-0.379264\pi\)
0.370273 + 0.928923i \(0.379264\pi\)
\(8\) 48.1425 2.12762
\(9\) 60.6475 2.24620
\(10\) −71.4625 −2.25984
\(11\) 10.3705 0.284257 0.142129 0.989848i \(-0.454605\pi\)
0.142129 + 0.989848i \(0.454605\pi\)
\(12\) −164.101 −3.94765
\(13\) −61.4205 −1.31038 −0.655192 0.755463i \(-0.727412\pi\)
−0.655192 + 0.755463i \(0.727412\pi\)
\(14\) 69.2963 1.32287
\(15\) 132.415 2.27929
\(16\) 103.016 1.60963
\(17\) −24.4390 −0.348666 −0.174333 0.984687i \(-0.555777\pi\)
−0.174333 + 0.984687i \(0.555777\pi\)
\(18\) 306.425 4.01251
\(19\) 65.5106 0.791008 0.395504 0.918464i \(-0.370570\pi\)
0.395504 + 0.918464i \(0.370570\pi\)
\(20\) −247.918 −2.77181
\(21\) −128.401 −1.33426
\(22\) 52.3977 0.507783
\(23\) 23.4730 0.212803 0.106401 0.994323i \(-0.466067\pi\)
0.106401 + 0.994323i \(0.466067\pi\)
\(24\) −450.711 −3.83338
\(25\) 75.0480 0.600384
\(26\) −310.331 −2.34080
\(27\) −315.009 −2.24532
\(28\) 240.403 1.62257
\(29\) 236.538 1.51462 0.757310 0.653055i \(-0.226513\pi\)
0.757310 + 0.653055i \(0.226513\pi\)
\(30\) 669.034 4.07161
\(31\) −210.270 −1.21824 −0.609122 0.793076i \(-0.708478\pi\)
−0.609122 + 0.793076i \(0.708478\pi\)
\(32\) 135.354 0.747735
\(33\) −97.0891 −0.512153
\(34\) −123.479 −0.622840
\(35\) −193.984 −0.936837
\(36\) 1063.05 4.92153
\(37\) 316.642 1.40691 0.703455 0.710740i \(-0.251640\pi\)
0.703455 + 0.710740i \(0.251640\pi\)
\(38\) 330.996 1.41302
\(39\) 575.020 2.36095
\(40\) −680.919 −2.69157
\(41\) 13.2054 0.0503008 0.0251504 0.999684i \(-0.491994\pi\)
0.0251504 + 0.999684i \(0.491994\pi\)
\(42\) −648.754 −2.38345
\(43\) 0 0
\(44\) 181.778 0.622820
\(45\) −857.789 −2.84159
\(46\) 118.599 0.380140
\(47\) 450.216 1.39725 0.698625 0.715488i \(-0.253796\pi\)
0.698625 + 0.715488i \(0.253796\pi\)
\(48\) −964.439 −2.90010
\(49\) −154.896 −0.451592
\(50\) 379.185 1.07250
\(51\) 228.798 0.628200
\(52\) −1076.60 −2.87111
\(53\) −205.541 −0.532701 −0.266351 0.963876i \(-0.585818\pi\)
−0.266351 + 0.963876i \(0.585818\pi\)
\(54\) −1591.60 −4.01092
\(55\) −146.679 −0.359603
\(56\) 660.279 1.57560
\(57\) −613.312 −1.42518
\(58\) 1195.12 2.70564
\(59\) 334.558 0.738232 0.369116 0.929383i \(-0.379661\pi\)
0.369116 + 0.929383i \(0.379661\pi\)
\(60\) 2321.01 4.99403
\(61\) −811.774 −1.70389 −0.851943 0.523634i \(-0.824576\pi\)
−0.851943 + 0.523634i \(0.824576\pi\)
\(62\) −1062.40 −2.17621
\(63\) 831.787 1.66342
\(64\) −140.242 −0.273910
\(65\) 868.722 1.65772
\(66\) −490.548 −0.914884
\(67\) −113.497 −0.206952 −0.103476 0.994632i \(-0.532997\pi\)
−0.103476 + 0.994632i \(0.532997\pi\)
\(68\) −428.375 −0.763943
\(69\) −219.755 −0.383411
\(70\) −980.116 −1.67352
\(71\) 53.8087 0.0899425 0.0449712 0.998988i \(-0.485680\pi\)
0.0449712 + 0.998988i \(0.485680\pi\)
\(72\) 2919.72 4.77907
\(73\) 108.423 0.173835 0.0869174 0.996216i \(-0.472298\pi\)
0.0869174 + 0.996216i \(0.472298\pi\)
\(74\) 1599.85 2.51323
\(75\) −702.602 −1.08173
\(76\) 1148.29 1.73313
\(77\) 142.233 0.210506
\(78\) 2905.32 4.21748
\(79\) −723.417 −1.03026 −0.515132 0.857111i \(-0.672257\pi\)
−0.515132 + 0.857111i \(0.672257\pi\)
\(80\) −1457.04 −2.03628
\(81\) 1311.64 1.79923
\(82\) 66.7209 0.0898548
\(83\) 191.269 0.252945 0.126473 0.991970i \(-0.459634\pi\)
0.126473 + 0.991970i \(0.459634\pi\)
\(84\) −2250.66 −2.92342
\(85\) 345.661 0.441085
\(86\) 0 0
\(87\) −2214.47 −2.72893
\(88\) 499.263 0.604791
\(89\) −419.159 −0.499221 −0.249611 0.968346i \(-0.580303\pi\)
−0.249611 + 0.968346i \(0.580303\pi\)
\(90\) −4334.03 −5.07607
\(91\) −842.389 −0.970399
\(92\) 411.443 0.466260
\(93\) 1968.55 2.19494
\(94\) 2274.74 2.49598
\(95\) −926.571 −1.00068
\(96\) −1267.19 −1.34721
\(97\) −953.469 −0.998043 −0.499021 0.866590i \(-0.666307\pi\)
−0.499021 + 0.866590i \(0.666307\pi\)
\(98\) −782.621 −0.806701
\(99\) 628.947 0.638500
\(100\) 1315.47 1.31547
\(101\) −1393.87 −1.37322 −0.686612 0.727024i \(-0.740903\pi\)
−0.686612 + 0.727024i \(0.740903\pi\)
\(102\) 1156.02 1.12218
\(103\) −1470.15 −1.40639 −0.703197 0.710995i \(-0.748245\pi\)
−0.703197 + 0.710995i \(0.748245\pi\)
\(104\) −2956.94 −2.78800
\(105\) 1816.08 1.68792
\(106\) −1038.51 −0.951590
\(107\) −1071.54 −0.968124 −0.484062 0.875034i \(-0.660839\pi\)
−0.484062 + 0.875034i \(0.660839\pi\)
\(108\) −5521.59 −4.91958
\(109\) −1637.65 −1.43907 −0.719536 0.694455i \(-0.755645\pi\)
−0.719536 + 0.694455i \(0.755645\pi\)
\(110\) −741.104 −0.642377
\(111\) −2964.41 −2.53486
\(112\) 1412.88 1.19200
\(113\) −1051.03 −0.874980 −0.437490 0.899223i \(-0.644132\pi\)
−0.437490 + 0.899223i \(0.644132\pi\)
\(114\) −3098.79 −2.54586
\(115\) −331.998 −0.269209
\(116\) 4146.12 3.31860
\(117\) −3725.00 −2.94339
\(118\) 1690.37 1.31874
\(119\) −335.183 −0.258203
\(120\) 6374.79 4.84946
\(121\) −1223.45 −0.919198
\(122\) −4101.54 −3.04374
\(123\) −123.629 −0.0906281
\(124\) −3685.68 −2.66923
\(125\) 706.512 0.505539
\(126\) 4202.65 2.97144
\(127\) 316.221 0.220946 0.110473 0.993879i \(-0.464763\pi\)
0.110473 + 0.993879i \(0.464763\pi\)
\(128\) −1791.42 −1.23703
\(129\) 0 0
\(130\) 4389.27 2.96126
\(131\) 1798.72 1.19965 0.599826 0.800130i \(-0.295236\pi\)
0.599826 + 0.800130i \(0.295236\pi\)
\(132\) −1701.81 −1.12215
\(133\) 898.484 0.585778
\(134\) −573.448 −0.369689
\(135\) 4455.44 2.84047
\(136\) −1176.55 −0.741829
\(137\) −2412.04 −1.50419 −0.752096 0.659054i \(-0.770957\pi\)
−0.752096 + 0.659054i \(0.770957\pi\)
\(138\) −1110.32 −0.684907
\(139\) 2424.03 1.47916 0.739582 0.673066i \(-0.235023\pi\)
0.739582 + 0.673066i \(0.235023\pi\)
\(140\) −3400.22 −2.05265
\(141\) −4214.93 −2.51746
\(142\) 271.871 0.160669
\(143\) −636.963 −0.372486
\(144\) 6247.67 3.61555
\(145\) −3345.55 −1.91609
\(146\) 547.813 0.310530
\(147\) 1450.14 0.813643
\(148\) 5550.21 3.08260
\(149\) 263.130 0.144674 0.0723372 0.997380i \(-0.476954\pi\)
0.0723372 + 0.997380i \(0.476954\pi\)
\(150\) −3549.94 −1.93234
\(151\) −1149.48 −0.619493 −0.309746 0.950819i \(-0.600244\pi\)
−0.309746 + 0.950819i \(0.600244\pi\)
\(152\) 3153.84 1.68296
\(153\) −1482.16 −0.783176
\(154\) 718.639 0.376036
\(155\) 2974.02 1.54116
\(156\) 10079.2 5.17294
\(157\) 945.081 0.480418 0.240209 0.970721i \(-0.422784\pi\)
0.240209 + 0.970721i \(0.422784\pi\)
\(158\) −3655.11 −1.84041
\(159\) 1924.28 0.959780
\(160\) −1914.43 −0.945932
\(161\) 321.935 0.157590
\(162\) 6627.13 3.21405
\(163\) 1316.21 0.632476 0.316238 0.948680i \(-0.397580\pi\)
0.316238 + 0.948680i \(0.397580\pi\)
\(164\) 231.468 0.110211
\(165\) 1373.21 0.647906
\(166\) 966.396 0.451849
\(167\) −1745.74 −0.808917 −0.404459 0.914556i \(-0.632540\pi\)
−0.404459 + 0.914556i \(0.632540\pi\)
\(168\) −6181.55 −2.83879
\(169\) 1575.48 0.717105
\(170\) 1746.47 0.787931
\(171\) 3973.05 1.77677
\(172\) 0 0
\(173\) 1020.88 0.448646 0.224323 0.974515i \(-0.427983\pi\)
0.224323 + 0.974515i \(0.427983\pi\)
\(174\) −11188.8 −4.87482
\(175\) 1029.29 0.444612
\(176\) 1068.33 0.457548
\(177\) −3132.14 −1.33009
\(178\) −2117.82 −0.891784
\(179\) −1448.99 −0.605043 −0.302521 0.953143i \(-0.597828\pi\)
−0.302521 + 0.953143i \(0.597828\pi\)
\(180\) −15035.6 −6.22605
\(181\) −1207.52 −0.495882 −0.247941 0.968775i \(-0.579754\pi\)
−0.247941 + 0.968775i \(0.579754\pi\)
\(182\) −4256.22 −1.73347
\(183\) 7599.85 3.06993
\(184\) 1130.05 0.452763
\(185\) −4478.53 −1.77983
\(186\) 9946.23 3.92093
\(187\) −253.445 −0.0991109
\(188\) 7891.54 3.06143
\(189\) −4320.38 −1.66276
\(190\) −4681.55 −1.78756
\(191\) −4420.48 −1.67463 −0.837316 0.546719i \(-0.815877\pi\)
−0.837316 + 0.546719i \(0.815877\pi\)
\(192\) 1312.95 0.493510
\(193\) 3256.05 1.21438 0.607192 0.794556i \(-0.292296\pi\)
0.607192 + 0.794556i \(0.292296\pi\)
\(194\) −4817.46 −1.78285
\(195\) −8132.99 −2.98675
\(196\) −2715.07 −0.989457
\(197\) −4223.73 −1.52755 −0.763777 0.645480i \(-0.776657\pi\)
−0.763777 + 0.645480i \(0.776657\pi\)
\(198\) 3177.79 1.14058
\(199\) −4105.07 −1.46232 −0.731158 0.682208i \(-0.761020\pi\)
−0.731158 + 0.682208i \(0.761020\pi\)
\(200\) 3613.00 1.27739
\(201\) 1062.56 0.372871
\(202\) −7042.63 −2.45306
\(203\) 3244.14 1.12165
\(204\) 4010.46 1.37641
\(205\) −186.775 −0.0636337
\(206\) −7428.04 −2.51231
\(207\) 1423.58 0.477998
\(208\) −6327.30 −2.10923
\(209\) 679.379 0.224850
\(210\) 9175.87 3.01522
\(211\) 4020.06 1.31162 0.655811 0.754925i \(-0.272327\pi\)
0.655811 + 0.754925i \(0.272327\pi\)
\(212\) −3602.78 −1.16717
\(213\) −503.758 −0.162051
\(214\) −5413.99 −1.72941
\(215\) 0 0
\(216\) −15165.3 −4.77717
\(217\) −2883.87 −0.902166
\(218\) −8274.34 −2.57068
\(219\) −1015.06 −0.313202
\(220\) −2571.04 −0.787906
\(221\) 1501.06 0.456887
\(222\) −14977.9 −4.52815
\(223\) −924.761 −0.277698 −0.138849 0.990314i \(-0.544340\pi\)
−0.138849 + 0.990314i \(0.544340\pi\)
\(224\) 1856.40 0.553732
\(225\) 4551.48 1.34859
\(226\) −5310.40 −1.56302
\(227\) −3472.29 −1.01526 −0.507630 0.861575i \(-0.669478\pi\)
−0.507630 + 0.861575i \(0.669478\pi\)
\(228\) −10750.3 −3.12263
\(229\) 2118.36 0.611289 0.305644 0.952146i \(-0.401128\pi\)
0.305644 + 0.952146i \(0.401128\pi\)
\(230\) −1677.44 −0.480901
\(231\) −1331.59 −0.379273
\(232\) 11387.5 3.22253
\(233\) 3691.16 1.03784 0.518919 0.854824i \(-0.326335\pi\)
0.518919 + 0.854824i \(0.326335\pi\)
\(234\) −18820.8 −5.25792
\(235\) −6367.78 −1.76761
\(236\) 5864.24 1.61750
\(237\) 6772.65 1.85625
\(238\) −1693.53 −0.461241
\(239\) 4854.95 1.31398 0.656988 0.753901i \(-0.271830\pi\)
0.656988 + 0.753901i \(0.271830\pi\)
\(240\) 13640.9 3.66881
\(241\) −2585.98 −0.691193 −0.345597 0.938383i \(-0.612323\pi\)
−0.345597 + 0.938383i \(0.612323\pi\)
\(242\) −6181.56 −1.64201
\(243\) −3774.36 −0.996400
\(244\) −14229.1 −3.73329
\(245\) 2190.82 0.571292
\(246\) −624.643 −0.161893
\(247\) −4023.69 −1.03652
\(248\) −10122.9 −2.59196
\(249\) −1790.66 −0.455737
\(250\) 3569.69 0.903069
\(251\) −2636.94 −0.663116 −0.331558 0.943435i \(-0.607574\pi\)
−0.331558 + 0.943435i \(0.607574\pi\)
\(252\) 14579.8 3.64462
\(253\) 243.427 0.0604907
\(254\) 1597.73 0.394686
\(255\) −3236.09 −0.794712
\(256\) −7929.30 −1.93586
\(257\) 7352.59 1.78460 0.892299 0.451444i \(-0.149091\pi\)
0.892299 + 0.451444i \(0.149091\pi\)
\(258\) 0 0
\(259\) 4342.78 1.04188
\(260\) 15227.2 3.63213
\(261\) 14345.4 3.40215
\(262\) 9088.11 2.14300
\(263\) 298.863 0.0700710 0.0350355 0.999386i \(-0.488846\pi\)
0.0350355 + 0.999386i \(0.488846\pi\)
\(264\) −4674.11 −1.08967
\(265\) 2907.13 0.673901
\(266\) 4539.64 1.04640
\(267\) 3924.17 0.899459
\(268\) −1989.41 −0.453442
\(269\) −3228.02 −0.731657 −0.365828 0.930682i \(-0.619214\pi\)
−0.365828 + 0.930682i \(0.619214\pi\)
\(270\) 22511.3 5.07406
\(271\) −5962.95 −1.33662 −0.668309 0.743883i \(-0.732982\pi\)
−0.668309 + 0.743883i \(0.732982\pi\)
\(272\) −2517.61 −0.561222
\(273\) 7886.46 1.74839
\(274\) −12187.0 −2.68701
\(275\) 778.287 0.170664
\(276\) −3851.94 −0.840071
\(277\) −1771.14 −0.384179 −0.192089 0.981377i \(-0.561526\pi\)
−0.192089 + 0.981377i \(0.561526\pi\)
\(278\) 12247.6 2.64230
\(279\) −12752.4 −2.73643
\(280\) −9338.88 −1.99323
\(281\) −981.025 −0.208267 −0.104134 0.994563i \(-0.533207\pi\)
−0.104134 + 0.994563i \(0.533207\pi\)
\(282\) −21296.2 −4.49706
\(283\) 8076.55 1.69647 0.848235 0.529620i \(-0.177666\pi\)
0.848235 + 0.529620i \(0.177666\pi\)
\(284\) 943.177 0.197068
\(285\) 8674.58 1.80294
\(286\) −3218.29 −0.665390
\(287\) 181.113 0.0372501
\(288\) 8208.91 1.67957
\(289\) −4315.74 −0.878432
\(290\) −16903.6 −3.42281
\(291\) 8926.41 1.79820
\(292\) 1900.47 0.380879
\(293\) −4573.34 −0.911869 −0.455934 0.890013i \(-0.650695\pi\)
−0.455934 + 0.890013i \(0.650695\pi\)
\(294\) 7326.92 1.45345
\(295\) −4731.93 −0.933910
\(296\) 15243.9 2.99337
\(297\) −3266.81 −0.638247
\(298\) 1329.48 0.258439
\(299\) −1441.72 −0.278853
\(300\) −12315.4 −2.37011
\(301\) 0 0
\(302\) −5807.82 −1.10663
\(303\) 13049.5 2.47417
\(304\) 6748.64 1.27323
\(305\) 11481.6 2.15552
\(306\) −7488.72 −1.39903
\(307\) 4509.23 0.838290 0.419145 0.907919i \(-0.362330\pi\)
0.419145 + 0.907919i \(0.362330\pi\)
\(308\) 2493.10 0.461227
\(309\) 13763.6 2.53393
\(310\) 15026.4 2.75304
\(311\) −720.198 −0.131314 −0.0656571 0.997842i \(-0.520914\pi\)
−0.0656571 + 0.997842i \(0.520914\pi\)
\(312\) 27682.9 5.02320
\(313\) −8593.59 −1.55188 −0.775940 0.630807i \(-0.782724\pi\)
−0.775940 + 0.630807i \(0.782724\pi\)
\(314\) 4775.08 0.858195
\(315\) −11764.7 −2.10433
\(316\) −12680.3 −2.25735
\(317\) −57.3605 −0.0101630 −0.00508152 0.999987i \(-0.501618\pi\)
−0.00508152 + 0.999987i \(0.501618\pi\)
\(318\) 9722.51 1.71450
\(319\) 2453.02 0.430542
\(320\) 1983.56 0.346514
\(321\) 10031.7 1.74429
\(322\) 1626.59 0.281511
\(323\) −1601.01 −0.275798
\(324\) 22990.9 3.94219
\(325\) −4609.49 −0.786734
\(326\) 6650.24 1.12982
\(327\) 15331.8 2.59281
\(328\) 635.740 0.107021
\(329\) 6174.76 1.03473
\(330\) 6938.23 1.15739
\(331\) 3284.93 0.545487 0.272743 0.962087i \(-0.412069\pi\)
0.272743 + 0.962087i \(0.412069\pi\)
\(332\) 3352.62 0.554214
\(333\) 19203.6 3.16021
\(334\) −8820.43 −1.44501
\(335\) 1605.28 0.261808
\(336\) −13227.4 −2.14766
\(337\) 4713.15 0.761844 0.380922 0.924607i \(-0.375607\pi\)
0.380922 + 0.924607i \(0.375607\pi\)
\(338\) 7960.20 1.28100
\(339\) 9839.78 1.57647
\(340\) 6058.86 0.966436
\(341\) −2180.61 −0.346295
\(342\) 20074.1 3.17393
\(343\) −6828.69 −1.07497
\(344\) 0 0
\(345\) 3108.18 0.485040
\(346\) 5158.03 0.801438
\(347\) 75.9955 0.0117569 0.00587846 0.999983i \(-0.498129\pi\)
0.00587846 + 0.999983i \(0.498129\pi\)
\(348\) −38816.1 −5.97920
\(349\) 8798.79 1.34954 0.674769 0.738029i \(-0.264243\pi\)
0.674769 + 0.738029i \(0.264243\pi\)
\(350\) 5200.55 0.794232
\(351\) 19348.0 2.94222
\(352\) 1403.70 0.212549
\(353\) 2914.96 0.439511 0.219756 0.975555i \(-0.429474\pi\)
0.219756 + 0.975555i \(0.429474\pi\)
\(354\) −15825.3 −2.37600
\(355\) −761.061 −0.113783
\(356\) −7347.15 −1.09382
\(357\) 3137.99 0.465211
\(358\) −7321.11 −1.08082
\(359\) 4644.90 0.682864 0.341432 0.939906i \(-0.389088\pi\)
0.341432 + 0.939906i \(0.389088\pi\)
\(360\) −41296.1 −6.04582
\(361\) −2567.36 −0.374306
\(362\) −6101.09 −0.885818
\(363\) 11454.0 1.65614
\(364\) −14765.7 −2.12619
\(365\) −1533.52 −0.219912
\(366\) 38398.7 5.48397
\(367\) 285.848 0.0406571 0.0203285 0.999793i \(-0.493529\pi\)
0.0203285 + 0.999793i \(0.493529\pi\)
\(368\) 2418.10 0.342533
\(369\) 800.873 0.112986
\(370\) −22628.1 −3.17940
\(371\) −2819.01 −0.394490
\(372\) 34505.5 4.80921
\(373\) −10360.3 −1.43816 −0.719080 0.694927i \(-0.755437\pi\)
−0.719080 + 0.694927i \(0.755437\pi\)
\(374\) −1280.55 −0.177047
\(375\) −6614.39 −0.910841
\(376\) 21674.5 2.97281
\(377\) −14528.3 −1.98473
\(378\) −21829.0 −2.97027
\(379\) 3278.07 0.444283 0.222141 0.975014i \(-0.428695\pi\)
0.222141 + 0.975014i \(0.428695\pi\)
\(380\) −16241.2 −2.19252
\(381\) −2960.47 −0.398083
\(382\) −22334.7 −2.99148
\(383\) −8473.04 −1.13042 −0.565212 0.824946i \(-0.691206\pi\)
−0.565212 + 0.824946i \(0.691206\pi\)
\(384\) 16771.3 2.22879
\(385\) −2011.72 −0.266303
\(386\) 16451.4 2.16931
\(387\) 0 0
\(388\) −16712.7 −2.18675
\(389\) −4236.19 −0.552142 −0.276071 0.961137i \(-0.589033\pi\)
−0.276071 + 0.961137i \(0.589033\pi\)
\(390\) −41092.4 −5.33537
\(391\) −573.657 −0.0741971
\(392\) −7457.08 −0.960815
\(393\) −16839.6 −2.16144
\(394\) −21340.6 −2.72874
\(395\) 10231.9 1.30335
\(396\) 11024.4 1.39898
\(397\) 11874.3 1.50114 0.750571 0.660789i \(-0.229778\pi\)
0.750571 + 0.660789i \(0.229778\pi\)
\(398\) −20741.1 −2.61221
\(399\) −8411.63 −1.05541
\(400\) 7731.15 0.966394
\(401\) 2340.65 0.291487 0.145744 0.989322i \(-0.453443\pi\)
0.145744 + 0.989322i \(0.453443\pi\)
\(402\) 5368.64 0.666078
\(403\) 12914.9 1.59637
\(404\) −24432.3 −3.00879
\(405\) −18551.6 −2.27614
\(406\) 16391.2 2.00365
\(407\) 3283.74 0.399924
\(408\) 11014.9 1.33657
\(409\) −7241.71 −0.875501 −0.437750 0.899097i \(-0.644225\pi\)
−0.437750 + 0.899097i \(0.644225\pi\)
\(410\) −943.690 −0.113672
\(411\) 22581.6 2.71014
\(412\) −25769.4 −3.08147
\(413\) 4588.49 0.546695
\(414\) 7192.72 0.853872
\(415\) −2705.27 −0.319992
\(416\) −8313.54 −0.979819
\(417\) −22693.9 −2.66505
\(418\) 3432.60 0.401660
\(419\) 7303.95 0.851602 0.425801 0.904817i \(-0.359992\pi\)
0.425801 + 0.904817i \(0.359992\pi\)
\(420\) 31832.9 3.69831
\(421\) 13018.6 1.50710 0.753551 0.657390i \(-0.228339\pi\)
0.753551 + 0.657390i \(0.228339\pi\)
\(422\) 20311.6 2.34302
\(423\) 27304.5 3.13851
\(424\) −9895.23 −1.13338
\(425\) −1834.10 −0.209334
\(426\) −2545.27 −0.289480
\(427\) −11133.6 −1.26181
\(428\) −18782.2 −2.12120
\(429\) 5963.26 0.671117
\(430\) 0 0
\(431\) −6513.96 −0.727996 −0.363998 0.931400i \(-0.618589\pi\)
−0.363998 + 0.931400i \(0.618589\pi\)
\(432\) −32451.0 −3.61412
\(433\) 3136.65 0.348124 0.174062 0.984735i \(-0.444311\pi\)
0.174062 + 0.984735i \(0.444311\pi\)
\(434\) −14570.9 −1.61158
\(435\) 31321.2 3.45226
\(436\) −28705.4 −3.15307
\(437\) 1537.73 0.168329
\(438\) −5128.64 −0.559488
\(439\) 670.745 0.0729223 0.0364612 0.999335i \(-0.488391\pi\)
0.0364612 + 0.999335i \(0.488391\pi\)
\(440\) −7061.49 −0.765099
\(441\) −9394.06 −1.01437
\(442\) 7584.17 0.816159
\(443\) 3012.53 0.323091 0.161546 0.986865i \(-0.448352\pi\)
0.161546 + 0.986865i \(0.448352\pi\)
\(444\) −51961.2 −5.55399
\(445\) 5928.51 0.631547
\(446\) −4672.41 −0.496065
\(447\) −2463.43 −0.260663
\(448\) −1923.43 −0.202843
\(449\) −12542.4 −1.31829 −0.659147 0.752014i \(-0.729083\pi\)
−0.659147 + 0.752014i \(0.729083\pi\)
\(450\) 22996.6 2.40905
\(451\) 136.947 0.0142984
\(452\) −18422.8 −1.91712
\(453\) 10761.5 1.11615
\(454\) −17544.0 −1.81361
\(455\) 11914.6 1.22762
\(456\) −29526.4 −3.03223
\(457\) −9096.37 −0.931094 −0.465547 0.885023i \(-0.654142\pi\)
−0.465547 + 0.885023i \(0.654142\pi\)
\(458\) 10703.1 1.09197
\(459\) 7698.50 0.782866
\(460\) −5819.38 −0.589848
\(461\) 13699.3 1.38403 0.692016 0.721882i \(-0.256723\pi\)
0.692016 + 0.721882i \(0.256723\pi\)
\(462\) −6727.92 −0.677513
\(463\) 18592.6 1.86625 0.933124 0.359555i \(-0.117072\pi\)
0.933124 + 0.359555i \(0.117072\pi\)
\(464\) 24367.2 2.43797
\(465\) −27842.9 −2.77674
\(466\) 18649.8 1.85394
\(467\) 1536.60 0.152260 0.0761299 0.997098i \(-0.475744\pi\)
0.0761299 + 0.997098i \(0.475744\pi\)
\(468\) −65293.1 −6.44909
\(469\) −1556.62 −0.153258
\(470\) −32173.6 −3.15757
\(471\) −8847.87 −0.865581
\(472\) 16106.4 1.57068
\(473\) 0 0
\(474\) 34219.2 3.31591
\(475\) 4916.44 0.474909
\(476\) −5875.21 −0.565735
\(477\) −12465.5 −1.19656
\(478\) 24529.9 2.34722
\(479\) −11184.2 −1.06685 −0.533424 0.845848i \(-0.679095\pi\)
−0.533424 + 0.845848i \(0.679095\pi\)
\(480\) 17923.0 1.70431
\(481\) −19448.3 −1.84359
\(482\) −13065.8 −1.23471
\(483\) −3013.96 −0.283934
\(484\) −21445.1 −2.01400
\(485\) 13485.7 1.26259
\(486\) −19070.2 −1.77992
\(487\) 5611.36 0.522125 0.261063 0.965322i \(-0.415927\pi\)
0.261063 + 0.965322i \(0.415927\pi\)
\(488\) −39080.8 −3.62522
\(489\) −12322.4 −1.13955
\(490\) 11069.3 1.02053
\(491\) −6705.50 −0.616324 −0.308162 0.951334i \(-0.599714\pi\)
−0.308162 + 0.951334i \(0.599714\pi\)
\(492\) −2167.01 −0.198570
\(493\) −5780.75 −0.528097
\(494\) −20329.9 −1.85159
\(495\) −8895.72 −0.807743
\(496\) −21661.2 −1.96092
\(497\) 737.991 0.0666065
\(498\) −9047.42 −0.814106
\(499\) −1528.07 −0.137086 −0.0685430 0.997648i \(-0.521835\pi\)
−0.0685430 + 0.997648i \(0.521835\pi\)
\(500\) 12384.0 1.10766
\(501\) 16343.6 1.45744
\(502\) −13323.3 −1.18456
\(503\) 16420.3 1.45556 0.727779 0.685812i \(-0.240553\pi\)
0.727779 + 0.685812i \(0.240553\pi\)
\(504\) 40044.3 3.53912
\(505\) 19714.7 1.73722
\(506\) 1229.93 0.108058
\(507\) −14749.7 −1.29202
\(508\) 5542.83 0.484101
\(509\) −13005.9 −1.13257 −0.566284 0.824210i \(-0.691620\pi\)
−0.566284 + 0.824210i \(0.691620\pi\)
\(510\) −16350.5 −1.41963
\(511\) 1487.03 0.128733
\(512\) −25731.9 −2.22109
\(513\) −20636.4 −1.77606
\(514\) 37149.4 3.18792
\(515\) 20793.6 1.77918
\(516\) 0 0
\(517\) 4668.97 0.397178
\(518\) 21942.1 1.86116
\(519\) −9557.47 −0.808336
\(520\) 41822.4 3.52699
\(521\) 3433.05 0.288685 0.144342 0.989528i \(-0.453893\pi\)
0.144342 + 0.989528i \(0.453893\pi\)
\(522\) 72481.2 6.07742
\(523\) −14300.7 −1.19565 −0.597826 0.801626i \(-0.703969\pi\)
−0.597826 + 0.801626i \(0.703969\pi\)
\(524\) 31528.5 2.62849
\(525\) −9636.25 −0.801068
\(526\) 1510.02 0.125171
\(527\) 5138.79 0.424761
\(528\) −10001.7 −0.824374
\(529\) −11616.0 −0.954715
\(530\) 14688.4 1.20382
\(531\) 20290.1 1.65822
\(532\) 15748.9 1.28346
\(533\) −811.081 −0.0659134
\(534\) 19827.1 1.60675
\(535\) 15155.6 1.22474
\(536\) −5464.01 −0.440316
\(537\) 13565.5 1.09012
\(538\) −16309.7 −1.30699
\(539\) −1606.35 −0.128368
\(540\) 78096.4 6.22358
\(541\) 2506.39 0.199184 0.0995918 0.995028i \(-0.468246\pi\)
0.0995918 + 0.995028i \(0.468246\pi\)
\(542\) −30128.2 −2.38767
\(543\) 11304.9 0.893441
\(544\) −3307.93 −0.260710
\(545\) 23162.7 1.82052
\(546\) 39846.8 3.12323
\(547\) 3120.71 0.243934 0.121967 0.992534i \(-0.461080\pi\)
0.121967 + 0.992534i \(0.461080\pi\)
\(548\) −42279.0 −3.29575
\(549\) −49232.1 −3.82728
\(550\) 3932.34 0.304865
\(551\) 15495.7 1.19808
\(552\) −10579.6 −0.815753
\(553\) −9921.74 −0.762957
\(554\) −8948.79 −0.686277
\(555\) 41928.2 3.20676
\(556\) 42489.3 3.24091
\(557\) 4921.10 0.374351 0.187176 0.982326i \(-0.440067\pi\)
0.187176 + 0.982326i \(0.440067\pi\)
\(558\) −64432.0 −4.88821
\(559\) 0 0
\(560\) −19983.5 −1.50796
\(561\) 2372.76 0.178570
\(562\) −4956.69 −0.372038
\(563\) −13911.2 −1.04136 −0.520680 0.853752i \(-0.674321\pi\)
−0.520680 + 0.853752i \(0.674321\pi\)
\(564\) −73880.8 −5.51586
\(565\) 14865.6 1.10690
\(566\) 40807.2 3.03049
\(567\) 17989.3 1.33241
\(568\) 2590.48 0.191363
\(569\) −2045.35 −0.150695 −0.0753476 0.997157i \(-0.524007\pi\)
−0.0753476 + 0.997157i \(0.524007\pi\)
\(570\) 43828.8 3.22068
\(571\) 19012.8 1.39345 0.696725 0.717339i \(-0.254640\pi\)
0.696725 + 0.717339i \(0.254640\pi\)
\(572\) −11164.9 −0.816133
\(573\) 41384.7 3.01722
\(574\) 915.084 0.0665416
\(575\) 1761.60 0.127763
\(576\) −8505.33 −0.615258
\(577\) 4056.31 0.292663 0.146331 0.989236i \(-0.453253\pi\)
0.146331 + 0.989236i \(0.453253\pi\)
\(578\) −21805.5 −1.56919
\(579\) −30483.3 −2.18798
\(580\) −58642.0 −4.19824
\(581\) 2623.27 0.187318
\(582\) 45101.2 3.21221
\(583\) −2131.56 −0.151424
\(584\) 5219.75 0.369854
\(585\) 52685.8 3.72357
\(586\) −23107.1 −1.62892
\(587\) −19491.1 −1.37050 −0.685249 0.728309i \(-0.740307\pi\)
−0.685249 + 0.728309i \(0.740307\pi\)
\(588\) 25418.6 1.78273
\(589\) −13774.9 −0.963642
\(590\) −23908.3 −1.66829
\(591\) 39542.6 2.75223
\(592\) 32619.2 2.26460
\(593\) −5465.18 −0.378462 −0.189231 0.981933i \(-0.560600\pi\)
−0.189231 + 0.981933i \(0.560600\pi\)
\(594\) −16505.7 −1.14013
\(595\) 4740.78 0.326643
\(596\) 4612.24 0.316988
\(597\) 38431.8 2.63469
\(598\) −7284.40 −0.498129
\(599\) −25500.9 −1.73946 −0.869731 0.493526i \(-0.835708\pi\)
−0.869731 + 0.493526i \(0.835708\pi\)
\(600\) −33825.0 −2.30150
\(601\) 24308.3 1.64985 0.824923 0.565245i \(-0.191219\pi\)
0.824923 + 0.565245i \(0.191219\pi\)
\(602\) 0 0
\(603\) −6883.29 −0.464858
\(604\) −20148.5 −1.35733
\(605\) 17304.3 1.16284
\(606\) 65933.3 4.41973
\(607\) 2862.31 0.191397 0.0956983 0.995410i \(-0.469492\pi\)
0.0956983 + 0.995410i \(0.469492\pi\)
\(608\) 8867.15 0.591464
\(609\) −30371.7 −2.02090
\(610\) 58011.5 3.85052
\(611\) −27652.5 −1.83093
\(612\) −25979.9 −1.71597
\(613\) 20771.8 1.36862 0.684312 0.729189i \(-0.260103\pi\)
0.684312 + 0.729189i \(0.260103\pi\)
\(614\) 22783.1 1.49748
\(615\) 1748.59 0.114650
\(616\) 6847.44 0.447875
\(617\) −18855.6 −1.23030 −0.615152 0.788408i \(-0.710905\pi\)
−0.615152 + 0.788408i \(0.710905\pi\)
\(618\) 69541.5 4.52649
\(619\) −5019.16 −0.325908 −0.162954 0.986634i \(-0.552102\pi\)
−0.162954 + 0.986634i \(0.552102\pi\)
\(620\) 52129.7 3.37674
\(621\) −7394.21 −0.477809
\(622\) −3638.84 −0.234573
\(623\) −5748.80 −0.369696
\(624\) 59236.3 3.80024
\(625\) −19373.8 −1.23992
\(626\) −43419.6 −2.77220
\(627\) −6360.36 −0.405117
\(628\) 16565.7 1.05262
\(629\) −7738.42 −0.490542
\(630\) −59441.6 −3.75906
\(631\) −7678.40 −0.484425 −0.242213 0.970223i \(-0.577873\pi\)
−0.242213 + 0.970223i \(0.577873\pi\)
\(632\) −34827.1 −2.19201
\(633\) −37635.9 −2.36318
\(634\) −289.817 −0.0181547
\(635\) −4472.58 −0.279510
\(636\) 33729.4 2.10292
\(637\) 9513.79 0.591759
\(638\) 12394.0 0.769098
\(639\) 3263.36 0.202029
\(640\) 25337.5 1.56493
\(641\) −30358.4 −1.87065 −0.935324 0.353792i \(-0.884892\pi\)
−0.935324 + 0.353792i \(0.884892\pi\)
\(642\) 50686.0 3.11591
\(643\) 1310.34 0.0803654 0.0401827 0.999192i \(-0.487206\pi\)
0.0401827 + 0.999192i \(0.487206\pi\)
\(644\) 5642.98 0.345287
\(645\) 0 0
\(646\) −8089.21 −0.492671
\(647\) 16438.0 0.998834 0.499417 0.866362i \(-0.333548\pi\)
0.499417 + 0.866362i \(0.333548\pi\)
\(648\) 63145.6 3.82808
\(649\) 3469.54 0.209848
\(650\) −23289.7 −1.40538
\(651\) 26998.9 1.62545
\(652\) 23071.0 1.38578
\(653\) −8211.20 −0.492081 −0.246041 0.969260i \(-0.579130\pi\)
−0.246041 + 0.969260i \(0.579130\pi\)
\(654\) 77464.6 4.63166
\(655\) −25440.7 −1.51764
\(656\) 1360.37 0.0809655
\(657\) 6575.58 0.390469
\(658\) 31198.3 1.84838
\(659\) −18816.5 −1.11227 −0.556136 0.831091i \(-0.687717\pi\)
−0.556136 + 0.831091i \(0.687717\pi\)
\(660\) 24070.1 1.41959
\(661\) 10768.9 0.633677 0.316838 0.948480i \(-0.397379\pi\)
0.316838 + 0.948480i \(0.397379\pi\)
\(662\) 16597.3 0.974430
\(663\) −14052.9 −0.823183
\(664\) 9208.15 0.538171
\(665\) −12708.0 −0.741046
\(666\) 97027.1 5.64523
\(667\) 5552.26 0.322315
\(668\) −30599.9 −1.77237
\(669\) 8657.63 0.500334
\(670\) 8110.76 0.467680
\(671\) −8418.52 −0.484342
\(672\) −17379.7 −0.997671
\(673\) 24154.4 1.38348 0.691740 0.722146i \(-0.256844\pi\)
0.691740 + 0.722146i \(0.256844\pi\)
\(674\) 23813.4 1.36092
\(675\) −23640.8 −1.34805
\(676\) 27615.6 1.57121
\(677\) −926.147 −0.0525771 −0.0262886 0.999654i \(-0.508369\pi\)
−0.0262886 + 0.999654i \(0.508369\pi\)
\(678\) 49716.1 2.81613
\(679\) −13076.9 −0.739096
\(680\) 16641.0 0.938460
\(681\) 32507.7 1.82922
\(682\) −11017.7 −0.618604
\(683\) −14605.3 −0.818237 −0.409119 0.912481i \(-0.634164\pi\)
−0.409119 + 0.912481i \(0.634164\pi\)
\(684\) 69641.1 3.89297
\(685\) 34115.5 1.90290
\(686\) −34502.4 −1.92027
\(687\) −19832.1 −1.10137
\(688\) 0 0
\(689\) 12624.4 0.698043
\(690\) 15704.2 0.866450
\(691\) 14706.2 0.809627 0.404813 0.914399i \(-0.367336\pi\)
0.404813 + 0.914399i \(0.367336\pi\)
\(692\) 17894.3 0.983003
\(693\) 8626.06 0.472839
\(694\) 383.972 0.0210020
\(695\) −34285.1 −1.87124
\(696\) −106610. −5.80611
\(697\) −322.726 −0.0175382
\(698\) 44456.4 2.41075
\(699\) −34556.8 −1.86990
\(700\) 18041.8 0.974164
\(701\) −6588.69 −0.354995 −0.177497 0.984121i \(-0.556800\pi\)
−0.177497 + 0.984121i \(0.556800\pi\)
\(702\) 97757.0 5.25584
\(703\) 20743.4 1.11288
\(704\) −1454.38 −0.0778610
\(705\) 59615.3 3.18474
\(706\) 14728.0 0.785121
\(707\) −19117.1 −1.01694
\(708\) −54901.2 −2.91428
\(709\) −7624.16 −0.403852 −0.201926 0.979401i \(-0.564720\pi\)
−0.201926 + 0.979401i \(0.564720\pi\)
\(710\) −3845.30 −0.203256
\(711\) −43873.5 −2.31418
\(712\) −20179.3 −1.06215
\(713\) −4935.67 −0.259246
\(714\) 15854.9 0.831029
\(715\) 9009.10 0.471218
\(716\) −25398.4 −1.32567
\(717\) −45452.1 −2.36742
\(718\) 23468.6 1.21983
\(719\) −31629.6 −1.64059 −0.820296 0.571939i \(-0.806192\pi\)
−0.820296 + 0.571939i \(0.806192\pi\)
\(720\) −88366.0 −4.57390
\(721\) −20163.3 −1.04150
\(722\) −12971.8 −0.668641
\(723\) 24210.0 1.24534
\(724\) −21165.9 −1.08650
\(725\) 17751.7 0.909355
\(726\) 57872.0 2.95844
\(727\) −23784.0 −1.21334 −0.606670 0.794954i \(-0.707495\pi\)
−0.606670 + 0.794954i \(0.707495\pi\)
\(728\) −40554.7 −2.06464
\(729\) −78.6220 −0.00399441
\(730\) −7748.18 −0.392840
\(731\) 0 0
\(732\) 133213. 6.72635
\(733\) 13440.2 0.677252 0.338626 0.940921i \(-0.390038\pi\)
0.338626 + 0.940921i \(0.390038\pi\)
\(734\) 1444.26 0.0726277
\(735\) −20510.5 −1.02931
\(736\) 3177.18 0.159120
\(737\) −1177.02 −0.0588278
\(738\) 4046.46 0.201832
\(739\) −28469.7 −1.41715 −0.708575 0.705636i \(-0.750662\pi\)
−0.708575 + 0.705636i \(0.750662\pi\)
\(740\) −78501.3 −3.89968
\(741\) 37669.9 1.86753
\(742\) −14243.2 −0.704696
\(743\) 3370.81 0.166437 0.0832187 0.996531i \(-0.473480\pi\)
0.0832187 + 0.996531i \(0.473480\pi\)
\(744\) 94771.0 4.66999
\(745\) −3721.67 −0.183022
\(746\) −52345.8 −2.56906
\(747\) 11600.0 0.568167
\(748\) −4442.47 −0.217156
\(749\) −14696.2 −0.716940
\(750\) −33419.6 −1.62708
\(751\) 2003.54 0.0973505 0.0486752 0.998815i \(-0.484500\pi\)
0.0486752 + 0.998815i \(0.484500\pi\)
\(752\) 46379.5 2.24905
\(753\) 24687.1 1.19475
\(754\) −73405.0 −3.54543
\(755\) 16258.1 0.783697
\(756\) −75729.1 −3.64318
\(757\) 4412.38 0.211850 0.105925 0.994374i \(-0.466220\pi\)
0.105925 + 0.994374i \(0.466220\pi\)
\(758\) 16562.6 0.793644
\(759\) −2278.97 −0.108988
\(760\) −44607.4 −2.12905
\(761\) −10159.0 −0.483921 −0.241961 0.970286i \(-0.577790\pi\)
−0.241961 + 0.970286i \(0.577790\pi\)
\(762\) −14957.9 −0.711115
\(763\) −22460.6 −1.06570
\(764\) −77483.7 −3.66919
\(765\) 20963.5 0.990767
\(766\) −42810.5 −2.01933
\(767\) −20548.7 −0.967367
\(768\) 74234.3 3.48789
\(769\) 17334.7 0.812881 0.406440 0.913677i \(-0.366770\pi\)
0.406440 + 0.913677i \(0.366770\pi\)
\(770\) −10164.3 −0.475710
\(771\) −68835.1 −3.21535
\(772\) 57073.2 2.66076
\(773\) −11330.7 −0.527213 −0.263607 0.964630i \(-0.584912\pi\)
−0.263607 + 0.964630i \(0.584912\pi\)
\(774\) 0 0
\(775\) −15780.3 −0.731415
\(776\) −45902.4 −2.12345
\(777\) −40657.2 −1.87718
\(778\) −21403.6 −0.986319
\(779\) 865.092 0.0397884
\(780\) −142558. −6.54409
\(781\) 558.024 0.0255668
\(782\) −2898.43 −0.132542
\(783\) −74511.6 −3.40080
\(784\) −15956.8 −0.726894
\(785\) −13367.1 −0.607759
\(786\) −85083.1 −3.86109
\(787\) 14755.8 0.668343 0.334172 0.942512i \(-0.391543\pi\)
0.334172 + 0.942512i \(0.391543\pi\)
\(788\) −74034.9 −3.34694
\(789\) −2797.96 −0.126249
\(790\) 51697.3 2.32824
\(791\) −14415.0 −0.647962
\(792\) 30279.1 1.35848
\(793\) 49859.6 2.23274
\(794\) 59995.6 2.68156
\(795\) −27216.6 −1.21418
\(796\) −71955.1 −3.20400
\(797\) 21567.4 0.958538 0.479269 0.877668i \(-0.340902\pi\)
0.479269 + 0.877668i \(0.340902\pi\)
\(798\) −42500.3 −1.88533
\(799\) −11002.8 −0.487174
\(800\) 10158.1 0.448928
\(801\) −25420.9 −1.12135
\(802\) 11826.3 0.520698
\(803\) 1124.40 0.0494138
\(804\) 18624.9 0.816976
\(805\) −4553.39 −0.199361
\(806\) 65253.2 2.85167
\(807\) 30220.8 1.31824
\(808\) −67104.6 −2.92170
\(809\) 27638.2 1.20112 0.600561 0.799579i \(-0.294944\pi\)
0.600561 + 0.799579i \(0.294944\pi\)
\(810\) −93733.1 −4.06598
\(811\) −5477.83 −0.237180 −0.118590 0.992943i \(-0.537837\pi\)
−0.118590 + 0.992943i \(0.537837\pi\)
\(812\) 56864.4 2.45757
\(813\) 55825.3 2.40822
\(814\) 16591.3 0.714404
\(815\) −18616.3 −0.800122
\(816\) 23569.9 1.01117
\(817\) 0 0
\(818\) −36589.2 −1.56395
\(819\) −51088.8 −2.17972
\(820\) −3273.85 −0.139424
\(821\) −3043.53 −0.129379 −0.0646894 0.997905i \(-0.520606\pi\)
−0.0646894 + 0.997905i \(0.520606\pi\)
\(822\) 114095. 4.84125
\(823\) 22867.8 0.968556 0.484278 0.874914i \(-0.339082\pi\)
0.484278 + 0.874914i \(0.339082\pi\)
\(824\) −70776.9 −2.99227
\(825\) −7286.35 −0.307489
\(826\) 23183.6 0.976587
\(827\) −13967.0 −0.587281 −0.293640 0.955916i \(-0.594867\pi\)
−0.293640 + 0.955916i \(0.594867\pi\)
\(828\) 24953.0 1.04732
\(829\) 4419.68 0.185165 0.0925825 0.995705i \(-0.470488\pi\)
0.0925825 + 0.995705i \(0.470488\pi\)
\(830\) −13668.5 −0.571617
\(831\) 16581.5 0.692183
\(832\) 8613.74 0.358927
\(833\) 3785.50 0.157455
\(834\) −114662. −4.76070
\(835\) 24691.4 1.02333
\(836\) 11908.4 0.492656
\(837\) 66236.9 2.73534
\(838\) 36903.6 1.52126
\(839\) 16617.7 0.683797 0.341898 0.939737i \(-0.388930\pi\)
0.341898 + 0.939737i \(0.388930\pi\)
\(840\) 87430.8 3.59125
\(841\) 31561.2 1.29408
\(842\) 65777.4 2.69221
\(843\) 9184.38 0.375240
\(844\) 70465.0 2.87382
\(845\) −22283.3 −0.907183
\(846\) 137957. 5.60647
\(847\) −16779.8 −0.680708
\(848\) −21174.0 −0.857449
\(849\) −75612.8 −3.05657
\(850\) −9266.89 −0.373943
\(851\) 7432.55 0.299394
\(852\) −8830.04 −0.355061
\(853\) −12302.6 −0.493825 −0.246912 0.969038i \(-0.579416\pi\)
−0.246912 + 0.969038i \(0.579416\pi\)
\(854\) −56253.0 −2.25403
\(855\) −56194.2 −2.24772
\(856\) −51586.4 −2.05980
\(857\) 39311.0 1.56690 0.783452 0.621452i \(-0.213457\pi\)
0.783452 + 0.621452i \(0.213457\pi\)
\(858\) 30129.7 1.19885
\(859\) −3412.44 −0.135542 −0.0677712 0.997701i \(-0.521589\pi\)
−0.0677712 + 0.997701i \(0.521589\pi\)
\(860\) 0 0
\(861\) −1695.58 −0.0671143
\(862\) −32912.1 −1.30045
\(863\) 19362.2 0.763727 0.381863 0.924219i \(-0.375282\pi\)
0.381863 + 0.924219i \(0.375282\pi\)
\(864\) −42637.9 −1.67890
\(865\) −14439.1 −0.567565
\(866\) 15848.1 0.621872
\(867\) 40404.0 1.58269
\(868\) −50549.5 −1.97668
\(869\) −7502.22 −0.292860
\(870\) 158252. 6.16695
\(871\) 6971.02 0.271187
\(872\) −78840.8 −3.06179
\(873\) −57825.6 −2.24181
\(874\) 7769.47 0.300694
\(875\) 9689.89 0.374375
\(876\) −17792.3 −0.686239
\(877\) −16784.7 −0.646272 −0.323136 0.946353i \(-0.604737\pi\)
−0.323136 + 0.946353i \(0.604737\pi\)
\(878\) 3388.98 0.130265
\(879\) 42815.7 1.64293
\(880\) −15110.3 −0.578827
\(881\) −21039.3 −0.804578 −0.402289 0.915513i \(-0.631785\pi\)
−0.402289 + 0.915513i \(0.631785\pi\)
\(882\) −47464.0 −1.81202
\(883\) 20024.2 0.763157 0.381579 0.924336i \(-0.375381\pi\)
0.381579 + 0.924336i \(0.375381\pi\)
\(884\) 26311.0 1.00106
\(885\) 44300.4 1.68265
\(886\) 15221.0 0.577154
\(887\) −16204.3 −0.613401 −0.306700 0.951806i \(-0.599225\pi\)
−0.306700 + 0.951806i \(0.599225\pi\)
\(888\) −142714. −5.39322
\(889\) 4337.00 0.163620
\(890\) 29954.1 1.12816
\(891\) 13602.4 0.511445
\(892\) −16209.5 −0.608447
\(893\) 29493.9 1.10524
\(894\) −12446.6 −0.465635
\(895\) 20494.3 0.765417
\(896\) −24569.5 −0.916080
\(897\) 13497.5 0.502416
\(898\) −63371.4 −2.35493
\(899\) −49736.8 −1.84518
\(900\) 79779.9 2.95481
\(901\) 5023.20 0.185735
\(902\) 691.931 0.0255419
\(903\) 0 0
\(904\) −50599.3 −1.86162
\(905\) 17079.0 0.627322
\(906\) 54372.9 1.99384
\(907\) −13786.2 −0.504699 −0.252350 0.967636i \(-0.581203\pi\)
−0.252350 + 0.967636i \(0.581203\pi\)
\(908\) −60863.5 −2.22448
\(909\) −84535.0 −3.08454
\(910\) 60199.2 2.19295
\(911\) 23933.3 0.870412 0.435206 0.900331i \(-0.356676\pi\)
0.435206 + 0.900331i \(0.356676\pi\)
\(912\) −63180.9 −2.29400
\(913\) 1983.56 0.0719015
\(914\) −45959.9 −1.66326
\(915\) −107491. −3.88366
\(916\) 37131.3 1.33936
\(917\) 24669.6 0.888398
\(918\) 38897.1 1.39847
\(919\) −31897.7 −1.14495 −0.572475 0.819922i \(-0.694017\pi\)
−0.572475 + 0.819922i \(0.694017\pi\)
\(920\) −15983.2 −0.572774
\(921\) −42215.5 −1.51037
\(922\) 69216.4 2.47236
\(923\) −3304.96 −0.117859
\(924\) −23340.5 −0.831002
\(925\) 23763.4 0.844686
\(926\) 93940.3 3.33377
\(927\) −89161.2 −3.15905
\(928\) 32016.5 1.13253
\(929\) 42084.6 1.48628 0.743138 0.669138i \(-0.233337\pi\)
0.743138 + 0.669138i \(0.233337\pi\)
\(930\) −140678. −4.96022
\(931\) −10147.3 −0.357213
\(932\) 64700.0 2.27395
\(933\) 6742.51 0.236592
\(934\) 7763.75 0.271989
\(935\) 3584.69 0.125382
\(936\) −179331. −6.26241
\(937\) 6000.64 0.209213 0.104606 0.994514i \(-0.466642\pi\)
0.104606 + 0.994514i \(0.466642\pi\)
\(938\) −7864.90 −0.273772
\(939\) 80453.4 2.79606
\(940\) −111617. −3.87291
\(941\) −17755.8 −0.615114 −0.307557 0.951530i \(-0.599511\pi\)
−0.307557 + 0.951530i \(0.599511\pi\)
\(942\) −44704.4 −1.54623
\(943\) 309.970 0.0107041
\(944\) 34464.8 1.18828
\(945\) 61106.7 2.10349
\(946\) 0 0
\(947\) 28692.8 0.984572 0.492286 0.870434i \(-0.336161\pi\)
0.492286 + 0.870434i \(0.336161\pi\)
\(948\) 118713. 4.06712
\(949\) −6659.39 −0.227790
\(950\) 24840.6 0.848353
\(951\) 537.010 0.0183110
\(952\) −16136.6 −0.549358
\(953\) 20739.6 0.704956 0.352478 0.935820i \(-0.385339\pi\)
0.352478 + 0.935820i \(0.385339\pi\)
\(954\) −62982.8 −2.13747
\(955\) 62522.6 2.11852
\(956\) 85099.2 2.87898
\(957\) −22965.3 −0.775717
\(958\) −56508.9 −1.90576
\(959\) −33081.3 −1.11392
\(960\) −18570.1 −0.624321
\(961\) 14422.4 0.484121
\(962\) −98263.8 −3.29330
\(963\) −64986.0 −2.17460
\(964\) −45327.9 −1.51443
\(965\) −46053.1 −1.53627
\(966\) −15228.2 −0.507205
\(967\) −35018.1 −1.16454 −0.582268 0.812997i \(-0.697835\pi\)
−0.582268 + 0.812997i \(0.697835\pi\)
\(968\) −58900.0 −1.95570
\(969\) 14988.7 0.496911
\(970\) 68137.4 2.25542
\(971\) −52435.4 −1.73299 −0.866495 0.499186i \(-0.833633\pi\)
−0.866495 + 0.499186i \(0.833633\pi\)
\(972\) −66158.2 −2.18316
\(973\) 33245.9 1.09539
\(974\) 28351.7 0.932697
\(975\) 43154.2 1.41748
\(976\) −83625.8 −2.74262
\(977\) 10021.8 0.328173 0.164086 0.986446i \(-0.447532\pi\)
0.164086 + 0.986446i \(0.447532\pi\)
\(978\) −62259.7 −2.03563
\(979\) −4346.89 −0.141907
\(980\) 38401.5 1.25173
\(981\) −99319.7 −3.23245
\(982\) −33879.9 −1.10097
\(983\) 54750.4 1.77647 0.888233 0.459394i \(-0.151933\pi\)
0.888233 + 0.459394i \(0.151933\pi\)
\(984\) −5951.81 −0.192822
\(985\) 59739.7 1.93245
\(986\) −29207.6 −0.943366
\(987\) −57808.2 −1.86429
\(988\) −70528.7 −2.27107
\(989\) 0 0
\(990\) −44946.1 −1.44291
\(991\) −49955.3 −1.60130 −0.800648 0.599135i \(-0.795511\pi\)
−0.800648 + 0.599135i \(0.795511\pi\)
\(992\) −28461.0 −0.910924
\(993\) −30753.6 −0.982816
\(994\) 3728.74 0.118982
\(995\) 58061.5 1.84992
\(996\) −31387.3 −0.998540
\(997\) 42986.1 1.36548 0.682740 0.730661i \(-0.260788\pi\)
0.682740 + 0.730661i \(0.260788\pi\)
\(998\) −7720.68 −0.244884
\(999\) −99745.1 −3.15896
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.j.1.47 yes 50
43.42 odd 2 1849.4.a.i.1.4 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.4 50 43.42 odd 2
1849.4.a.j.1.47 yes 50 1.1 even 1 trivial