Properties

Label 1849.4.a.k.1.6
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $60$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(60\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.92818 q^{2} +1.20886 q^{3} +16.2870 q^{4} -11.7477 q^{5} -5.95749 q^{6} -36.0345 q^{7} -40.8396 q^{8} -25.5387 q^{9} +O(q^{10})\) \(q-4.92818 q^{2} +1.20886 q^{3} +16.2870 q^{4} -11.7477 q^{5} -5.95749 q^{6} -36.0345 q^{7} -40.8396 q^{8} -25.5387 q^{9} +57.8946 q^{10} +4.39201 q^{11} +19.6887 q^{12} -77.3553 q^{13} +177.585 q^{14} -14.2013 q^{15} +70.9695 q^{16} +51.7769 q^{17} +125.859 q^{18} -48.2283 q^{19} -191.334 q^{20} -43.5607 q^{21} -21.6446 q^{22} +49.8143 q^{23} -49.3695 q^{24} +13.0077 q^{25} +381.221 q^{26} -63.5119 q^{27} -586.893 q^{28} -101.263 q^{29} +69.9866 q^{30} +189.182 q^{31} -23.0331 q^{32} +5.30933 q^{33} -255.166 q^{34} +423.322 q^{35} -415.947 q^{36} -64.1616 q^{37} +237.678 q^{38} -93.5118 q^{39} +479.771 q^{40} +123.447 q^{41} +214.675 q^{42} +71.5325 q^{44} +300.020 q^{45} -245.494 q^{46} +442.017 q^{47} +85.7922 q^{48} +955.487 q^{49} -64.1043 q^{50} +62.5910 q^{51} -1259.88 q^{52} +396.700 q^{53} +312.998 q^{54} -51.5958 q^{55} +1471.64 q^{56} -58.3014 q^{57} +499.041 q^{58} -408.711 q^{59} -231.296 q^{60} +60.4412 q^{61} -932.323 q^{62} +920.273 q^{63} -454.244 q^{64} +908.744 q^{65} -26.1653 q^{66} +586.165 q^{67} +843.288 q^{68} +60.2185 q^{69} -2086.21 q^{70} +138.373 q^{71} +1042.99 q^{72} +572.504 q^{73} +316.200 q^{74} +15.7245 q^{75} -785.493 q^{76} -158.264 q^{77} +460.843 q^{78} -1043.69 q^{79} -833.726 q^{80} +612.767 q^{81} -608.367 q^{82} -521.618 q^{83} -709.472 q^{84} -608.257 q^{85} -122.413 q^{87} -179.368 q^{88} -101.737 q^{89} -1478.55 q^{90} +2787.46 q^{91} +811.323 q^{92} +228.695 q^{93} -2178.34 q^{94} +566.571 q^{95} -27.8438 q^{96} +289.303 q^{97} -4708.81 q^{98} -112.166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} - 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} - 625 q^{18} - 610 q^{19} - 345 q^{20} + 611 q^{21} - 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} - 1071 q^{26} - 1609 q^{27} - 46 q^{28} - 773 q^{29} - 375 q^{30} - 97 q^{31} - 1967 q^{32} - 500 q^{33} - 217 q^{34} + 247 q^{35} + 175 q^{36} - 228 q^{37} + 1253 q^{38} - 1493 q^{39} + 2220 q^{40} - 951 q^{41} - 2643 q^{42} - 1378 q^{44} - 1086 q^{45} + 565 q^{46} - 2 q^{47} - 2303 q^{48} + 1264 q^{49} - 3273 q^{50} - 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} - 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} - 2999 q^{61} - 5569 q^{62} - 2377 q^{63} + 2082 q^{64} - 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} - 1817 q^{69} - 2738 q^{70} - 8003 q^{71} - 1412 q^{72} + 1011 q^{73} - 1413 q^{74} - 7457 q^{75} - 5516 q^{76} - 4052 q^{77} + 1091 q^{78} - 4422 q^{79} - 1610 q^{80} + 2108 q^{81} - 4676 q^{82} - 297 q^{83} - 54 q^{84} - 4333 q^{85} + 1377 q^{87} - 3652 q^{88} - 2480 q^{89} - 1414 q^{90} - 4551 q^{91} - 3286 q^{92} - 4 q^{93} - 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} - 753 q^{98} + 5072 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.92818 −1.74237 −0.871187 0.490951i \(-0.836650\pi\)
−0.871187 + 0.490951i \(0.836650\pi\)
\(3\) 1.20886 0.232645 0.116323 0.993211i \(-0.462889\pi\)
0.116323 + 0.993211i \(0.462889\pi\)
\(4\) 16.2870 2.03587
\(5\) −11.7477 −1.05074 −0.525372 0.850873i \(-0.676074\pi\)
−0.525372 + 0.850873i \(0.676074\pi\)
\(6\) −5.95749 −0.405356
\(7\) −36.0345 −1.94568 −0.972841 0.231475i \(-0.925645\pi\)
−0.972841 + 0.231475i \(0.925645\pi\)
\(8\) −40.8396 −1.80487
\(9\) −25.5387 −0.945876
\(10\) 57.8946 1.83079
\(11\) 4.39201 0.120385 0.0601927 0.998187i \(-0.480828\pi\)
0.0601927 + 0.998187i \(0.480828\pi\)
\(12\) 19.6887 0.473636
\(13\) −77.3553 −1.65035 −0.825173 0.564880i \(-0.808922\pi\)
−0.825173 + 0.564880i \(0.808922\pi\)
\(14\) 177.585 3.39011
\(15\) −14.2013 −0.244451
\(16\) 70.9695 1.10890
\(17\) 51.7769 0.738690 0.369345 0.929292i \(-0.379582\pi\)
0.369345 + 0.929292i \(0.379582\pi\)
\(18\) 125.859 1.64807
\(19\) −48.2283 −0.582334 −0.291167 0.956672i \(-0.594043\pi\)
−0.291167 + 0.956672i \(0.594043\pi\)
\(20\) −191.334 −2.13918
\(21\) −43.5607 −0.452654
\(22\) −21.6446 −0.209757
\(23\) 49.8143 0.451608 0.225804 0.974173i \(-0.427499\pi\)
0.225804 + 0.974173i \(0.427499\pi\)
\(24\) −49.3695 −0.419896
\(25\) 13.0077 0.104062
\(26\) 381.221 2.87552
\(27\) −63.5119 −0.452699
\(28\) −586.893 −3.96116
\(29\) −101.263 −0.648414 −0.324207 0.945986i \(-0.605098\pi\)
−0.324207 + 0.945986i \(0.605098\pi\)
\(30\) 69.9866 0.425925
\(31\) 189.182 1.09607 0.548034 0.836456i \(-0.315376\pi\)
0.548034 + 0.836456i \(0.315376\pi\)
\(32\) −23.0331 −0.127241
\(33\) 5.30933 0.0280071
\(34\) −255.166 −1.28708
\(35\) 423.322 2.04441
\(36\) −415.947 −1.92568
\(37\) −64.1616 −0.285084 −0.142542 0.989789i \(-0.545528\pi\)
−0.142542 + 0.989789i \(0.545528\pi\)
\(38\) 237.678 1.01464
\(39\) −93.5118 −0.383945
\(40\) 479.771 1.89646
\(41\) 123.447 0.470222 0.235111 0.971968i \(-0.424455\pi\)
0.235111 + 0.971968i \(0.424455\pi\)
\(42\) 214.675 0.788693
\(43\) 0 0
\(44\) 71.5325 0.245089
\(45\) 300.020 0.993873
\(46\) −245.494 −0.786871
\(47\) 442.017 1.37180 0.685902 0.727694i \(-0.259408\pi\)
0.685902 + 0.727694i \(0.259408\pi\)
\(48\) 85.7922 0.257980
\(49\) 955.487 2.78568
\(50\) −64.1043 −0.181314
\(51\) 62.5910 0.171853
\(52\) −1259.88 −3.35989
\(53\) 396.700 1.02813 0.514066 0.857751i \(-0.328139\pi\)
0.514066 + 0.857751i \(0.328139\pi\)
\(54\) 312.998 0.788772
\(55\) −51.5958 −0.126494
\(56\) 1471.64 3.51171
\(57\) −58.3014 −0.135477
\(58\) 499.041 1.12978
\(59\) −408.711 −0.901858 −0.450929 0.892560i \(-0.648907\pi\)
−0.450929 + 0.892560i \(0.648907\pi\)
\(60\) −231.296 −0.497670
\(61\) 60.4412 0.126864 0.0634320 0.997986i \(-0.479795\pi\)
0.0634320 + 0.997986i \(0.479795\pi\)
\(62\) −932.323 −1.90976
\(63\) 920.273 1.84037
\(64\) −454.244 −0.887196
\(65\) 908.744 1.73409
\(66\) −26.1653 −0.0487989
\(67\) 586.165 1.06883 0.534414 0.845223i \(-0.320532\pi\)
0.534414 + 0.845223i \(0.320532\pi\)
\(68\) 843.288 1.50388
\(69\) 60.2185 0.105065
\(70\) −2086.21 −3.56213
\(71\) 138.373 0.231293 0.115647 0.993290i \(-0.463106\pi\)
0.115647 + 0.993290i \(0.463106\pi\)
\(72\) 1042.99 1.70719
\(73\) 572.504 0.917897 0.458948 0.888463i \(-0.348226\pi\)
0.458948 + 0.888463i \(0.348226\pi\)
\(74\) 316.200 0.496723
\(75\) 15.7245 0.0242094
\(76\) −785.493 −1.18556
\(77\) −158.264 −0.234232
\(78\) 460.843 0.668977
\(79\) −1043.69 −1.48638 −0.743190 0.669080i \(-0.766688\pi\)
−0.743190 + 0.669080i \(0.766688\pi\)
\(80\) −833.726 −1.16517
\(81\) 612.767 0.840558
\(82\) −608.367 −0.819304
\(83\) −521.618 −0.689819 −0.344909 0.938636i \(-0.612090\pi\)
−0.344909 + 0.938636i \(0.612090\pi\)
\(84\) −709.472 −0.921545
\(85\) −608.257 −0.776174
\(86\) 0 0
\(87\) −122.413 −0.150851
\(88\) −179.368 −0.217281
\(89\) −101.737 −0.121169 −0.0605846 0.998163i \(-0.519297\pi\)
−0.0605846 + 0.998163i \(0.519297\pi\)
\(90\) −1478.55 −1.73170
\(91\) 2787.46 3.21105
\(92\) 811.323 0.919416
\(93\) 228.695 0.254995
\(94\) −2178.34 −2.39020
\(95\) 566.571 0.611883
\(96\) −27.8438 −0.0296021
\(97\) 289.303 0.302827 0.151414 0.988470i \(-0.451617\pi\)
0.151414 + 0.988470i \(0.451617\pi\)
\(98\) −4708.81 −4.85369
\(99\) −112.166 −0.113870
\(100\) 211.856 0.211856
\(101\) −1791.73 −1.76519 −0.882594 0.470137i \(-0.844205\pi\)
−0.882594 + 0.470137i \(0.844205\pi\)
\(102\) −308.460 −0.299432
\(103\) 618.181 0.591370 0.295685 0.955285i \(-0.404452\pi\)
0.295685 + 0.955285i \(0.404452\pi\)
\(104\) 3159.16 2.97867
\(105\) 511.737 0.475623
\(106\) −1955.01 −1.79139
\(107\) −242.781 −0.219351 −0.109675 0.993967i \(-0.534981\pi\)
−0.109675 + 0.993967i \(0.534981\pi\)
\(108\) −1034.42 −0.921637
\(109\) −664.305 −0.583751 −0.291876 0.956456i \(-0.594279\pi\)
−0.291876 + 0.956456i \(0.594279\pi\)
\(110\) 254.274 0.220400
\(111\) −77.5625 −0.0663234
\(112\) −2557.35 −2.15756
\(113\) −1281.76 −1.06706 −0.533532 0.845780i \(-0.679136\pi\)
−0.533532 + 0.845780i \(0.679136\pi\)
\(114\) 287.320 0.236052
\(115\) −585.202 −0.474525
\(116\) −1649.26 −1.32009
\(117\) 1975.55 1.56102
\(118\) 2014.20 1.57137
\(119\) −1865.76 −1.43726
\(120\) 579.976 0.441203
\(121\) −1311.71 −0.985507
\(122\) −297.865 −0.221045
\(123\) 149.230 0.109395
\(124\) 3081.20 2.23145
\(125\) 1315.65 0.941401
\(126\) −4535.27 −3.20662
\(127\) 828.066 0.578575 0.289287 0.957242i \(-0.406582\pi\)
0.289287 + 0.957242i \(0.406582\pi\)
\(128\) 2422.86 1.67307
\(129\) 0 0
\(130\) −4478.46 −3.02143
\(131\) 1094.85 0.730212 0.365106 0.930966i \(-0.381033\pi\)
0.365106 + 0.930966i \(0.381033\pi\)
\(132\) 86.4728 0.0570189
\(133\) 1737.89 1.13304
\(134\) −2888.73 −1.86230
\(135\) 746.117 0.475671
\(136\) −2114.55 −1.33324
\(137\) −182.197 −0.113622 −0.0568108 0.998385i \(-0.518093\pi\)
−0.0568108 + 0.998385i \(0.518093\pi\)
\(138\) −296.768 −0.183062
\(139\) −210.240 −0.128290 −0.0641451 0.997941i \(-0.520432\pi\)
−0.0641451 + 0.997941i \(0.520432\pi\)
\(140\) 6894.62 4.16216
\(141\) 534.337 0.319144
\(142\) −681.926 −0.403000
\(143\) −339.745 −0.198678
\(144\) −1812.46 −1.04888
\(145\) 1189.60 0.681317
\(146\) −2821.40 −1.59932
\(147\) 1155.05 0.648075
\(148\) −1045.00 −0.580394
\(149\) −1439.06 −0.791224 −0.395612 0.918418i \(-0.629468\pi\)
−0.395612 + 0.918418i \(0.629468\pi\)
\(150\) −77.4931 −0.0421819
\(151\) 1023.21 0.551439 0.275720 0.961238i \(-0.411084\pi\)
0.275720 + 0.961238i \(0.411084\pi\)
\(152\) 1969.63 1.05104
\(153\) −1322.31 −0.698709
\(154\) 779.953 0.408120
\(155\) −2222.45 −1.15169
\(156\) −1523.02 −0.781663
\(157\) −51.6584 −0.0262598 −0.0131299 0.999914i \(-0.504179\pi\)
−0.0131299 + 0.999914i \(0.504179\pi\)
\(158\) 5143.48 2.58983
\(159\) 479.556 0.239190
\(160\) 270.585 0.133698
\(161\) −1795.03 −0.878686
\(162\) −3019.82 −1.46457
\(163\) 2910.24 1.39845 0.699225 0.714901i \(-0.253528\pi\)
0.699225 + 0.714901i \(0.253528\pi\)
\(164\) 2010.57 0.957312
\(165\) −62.3722 −0.0294283
\(166\) 2570.63 1.20192
\(167\) −3365.95 −1.55967 −0.779836 0.625984i \(-0.784698\pi\)
−0.779836 + 0.625984i \(0.784698\pi\)
\(168\) 1779.01 0.816984
\(169\) 3786.84 1.72364
\(170\) 2997.60 1.35239
\(171\) 1231.69 0.550816
\(172\) 0 0
\(173\) −2181.52 −0.958715 −0.479358 0.877620i \(-0.659130\pi\)
−0.479358 + 0.877620i \(0.659130\pi\)
\(174\) 603.271 0.262838
\(175\) −468.726 −0.202471
\(176\) 311.698 0.133495
\(177\) −494.074 −0.209813
\(178\) 501.377 0.211122
\(179\) 386.165 0.161248 0.0806238 0.996745i \(-0.474309\pi\)
0.0806238 + 0.996745i \(0.474309\pi\)
\(180\) 4886.41 2.02340
\(181\) −1249.25 −0.513017 −0.256509 0.966542i \(-0.582572\pi\)
−0.256509 + 0.966542i \(0.582572\pi\)
\(182\) −13737.1 −5.59485
\(183\) 73.0650 0.0295143
\(184\) −2034.40 −0.815097
\(185\) 753.749 0.299550
\(186\) −1127.05 −0.444297
\(187\) 227.404 0.0889276
\(188\) 7199.12 2.79282
\(189\) 2288.62 0.880809
\(190\) −2792.16 −1.06613
\(191\) 2266.57 0.858655 0.429327 0.903149i \(-0.358751\pi\)
0.429327 + 0.903149i \(0.358751\pi\)
\(192\) −549.118 −0.206402
\(193\) 3168.70 1.18180 0.590901 0.806744i \(-0.298772\pi\)
0.590901 + 0.806744i \(0.298772\pi\)
\(194\) −1425.74 −0.527639
\(195\) 1098.55 0.403428
\(196\) 15562.0 5.67128
\(197\) 4300.78 1.55542 0.777710 0.628623i \(-0.216381\pi\)
0.777710 + 0.628623i \(0.216381\pi\)
\(198\) 552.774 0.198404
\(199\) −345.668 −0.123134 −0.0615672 0.998103i \(-0.519610\pi\)
−0.0615672 + 0.998103i \(0.519610\pi\)
\(200\) −531.230 −0.187818
\(201\) 708.592 0.248658
\(202\) 8829.97 3.07562
\(203\) 3648.96 1.26161
\(204\) 1019.42 0.349870
\(205\) −1450.21 −0.494083
\(206\) −3046.51 −1.03039
\(207\) −1272.19 −0.427166
\(208\) −5489.86 −1.83006
\(209\) −211.819 −0.0701045
\(210\) −2521.93 −0.828714
\(211\) 1209.20 0.394524 0.197262 0.980351i \(-0.436795\pi\)
0.197262 + 0.980351i \(0.436795\pi\)
\(212\) 6461.05 2.09314
\(213\) 167.274 0.0538094
\(214\) 1196.47 0.382191
\(215\) 0 0
\(216\) 2593.81 0.817065
\(217\) −6817.09 −2.13260
\(218\) 3273.82 1.01711
\(219\) 692.077 0.213545
\(220\) −840.340 −0.257526
\(221\) −4005.21 −1.21909
\(222\) 382.242 0.115560
\(223\) −113.234 −0.0340031 −0.0170016 0.999855i \(-0.505412\pi\)
−0.0170016 + 0.999855i \(0.505412\pi\)
\(224\) 829.987 0.247571
\(225\) −332.199 −0.0984293
\(226\) 6316.76 1.85923
\(227\) 4807.72 1.40572 0.702862 0.711326i \(-0.251905\pi\)
0.702862 + 0.711326i \(0.251905\pi\)
\(228\) −949.552 −0.275814
\(229\) 3584.27 1.03430 0.517151 0.855894i \(-0.326992\pi\)
0.517151 + 0.855894i \(0.326992\pi\)
\(230\) 2883.98 0.826800
\(231\) −191.319 −0.0544930
\(232\) 4135.54 1.17031
\(233\) 3418.57 0.961194 0.480597 0.876942i \(-0.340420\pi\)
0.480597 + 0.876942i \(0.340420\pi\)
\(234\) −9735.87 −2.71989
\(235\) −5192.67 −1.44141
\(236\) −6656.66 −1.83607
\(237\) −1261.67 −0.345800
\(238\) 9194.78 2.50424
\(239\) −2254.44 −0.610156 −0.305078 0.952327i \(-0.598683\pi\)
−0.305078 + 0.952327i \(0.598683\pi\)
\(240\) −1007.86 −0.271071
\(241\) −6812.13 −1.82078 −0.910390 0.413752i \(-0.864218\pi\)
−0.910390 + 0.413752i \(0.864218\pi\)
\(242\) 6464.34 1.71712
\(243\) 2455.57 0.648251
\(244\) 984.404 0.258279
\(245\) −11224.7 −2.92703
\(246\) −735.431 −0.190607
\(247\) 3730.72 0.961052
\(248\) −7726.13 −1.97826
\(249\) −630.563 −0.160483
\(250\) −6483.75 −1.64027
\(251\) 5005.14 1.25865 0.629326 0.777141i \(-0.283331\pi\)
0.629326 + 0.777141i \(0.283331\pi\)
\(252\) 14988.5 3.74676
\(253\) 218.785 0.0543671
\(254\) −4080.86 −1.00809
\(255\) −735.299 −0.180573
\(256\) −8306.35 −2.02792
\(257\) 5169.88 1.25482 0.627409 0.778690i \(-0.284115\pi\)
0.627409 + 0.778690i \(0.284115\pi\)
\(258\) 0 0
\(259\) 2312.03 0.554682
\(260\) 14800.7 3.53038
\(261\) 2586.11 0.613320
\(262\) −5395.63 −1.27230
\(263\) −2888.77 −0.677298 −0.338649 0.940913i \(-0.609970\pi\)
−0.338649 + 0.940913i \(0.609970\pi\)
\(264\) −216.831 −0.0505494
\(265\) −4660.30 −1.08030
\(266\) −8564.61 −1.97417
\(267\) −122.986 −0.0281895
\(268\) 9546.85 2.17600
\(269\) −3530.65 −0.800250 −0.400125 0.916460i \(-0.631033\pi\)
−0.400125 + 0.916460i \(0.631033\pi\)
\(270\) −3677.00 −0.828797
\(271\) 2873.34 0.644069 0.322034 0.946728i \(-0.395633\pi\)
0.322034 + 0.946728i \(0.395633\pi\)
\(272\) 3674.58 0.819132
\(273\) 3369.65 0.747036
\(274\) 897.900 0.197971
\(275\) 57.1299 0.0125275
\(276\) 980.777 0.213898
\(277\) −450.892 −0.0978032 −0.0489016 0.998804i \(-0.515572\pi\)
−0.0489016 + 0.998804i \(0.515572\pi\)
\(278\) 1036.10 0.223529
\(279\) −4831.46 −1.03674
\(280\) −17288.3 −3.68991
\(281\) −7939.94 −1.68561 −0.842806 0.538217i \(-0.819098\pi\)
−0.842806 + 0.538217i \(0.819098\pi\)
\(282\) −2633.31 −0.556069
\(283\) −65.3796 −0.0137329 −0.00686645 0.999976i \(-0.502186\pi\)
−0.00686645 + 0.999976i \(0.502186\pi\)
\(284\) 2253.67 0.470884
\(285\) 684.905 0.142352
\(286\) 1674.32 0.346171
\(287\) −4448.34 −0.914903
\(288\) 588.235 0.120354
\(289\) −2232.16 −0.454337
\(290\) −5862.57 −1.18711
\(291\) 349.727 0.0704514
\(292\) 9324.34 1.86872
\(293\) −4525.53 −0.902336 −0.451168 0.892439i \(-0.648993\pi\)
−0.451168 + 0.892439i \(0.648993\pi\)
\(294\) −5692.30 −1.12919
\(295\) 4801.40 0.947621
\(296\) 2620.34 0.514540
\(297\) −278.945 −0.0544984
\(298\) 7091.95 1.37861
\(299\) −3853.40 −0.745310
\(300\) 256.104 0.0492873
\(301\) 0 0
\(302\) −5042.54 −0.960814
\(303\) −2165.95 −0.410663
\(304\) −3422.74 −0.645749
\(305\) −710.043 −0.133301
\(306\) 6516.59 1.21741
\(307\) 812.297 0.151010 0.0755052 0.997145i \(-0.475943\pi\)
0.0755052 + 0.997145i \(0.475943\pi\)
\(308\) −2577.64 −0.476866
\(309\) 747.295 0.137580
\(310\) 10952.6 2.00667
\(311\) −316.376 −0.0576850 −0.0288425 0.999584i \(-0.509182\pi\)
−0.0288425 + 0.999584i \(0.509182\pi\)
\(312\) 3818.99 0.692973
\(313\) −4792.06 −0.865377 −0.432689 0.901543i \(-0.642435\pi\)
−0.432689 + 0.901543i \(0.642435\pi\)
\(314\) 254.582 0.0457544
\(315\) −10811.1 −1.93376
\(316\) −16998.5 −3.02608
\(317\) 4766.60 0.844538 0.422269 0.906470i \(-0.361234\pi\)
0.422269 + 0.906470i \(0.361234\pi\)
\(318\) −2363.34 −0.416759
\(319\) −444.747 −0.0780597
\(320\) 5336.31 0.932215
\(321\) −293.489 −0.0510310
\(322\) 8846.25 1.53100
\(323\) −2497.11 −0.430164
\(324\) 9980.11 1.71127
\(325\) −1006.21 −0.171738
\(326\) −14342.2 −2.43663
\(327\) −803.053 −0.135807
\(328\) −5041.52 −0.848693
\(329\) −15927.9 −2.66910
\(330\) 307.382 0.0512751
\(331\) 6872.71 1.14126 0.570632 0.821206i \(-0.306698\pi\)
0.570632 + 0.821206i \(0.306698\pi\)
\(332\) −8495.57 −1.40438
\(333\) 1638.60 0.269654
\(334\) 16588.0 2.71753
\(335\) −6886.08 −1.12306
\(336\) −3091.48 −0.501947
\(337\) 7556.73 1.22149 0.610744 0.791828i \(-0.290871\pi\)
0.610744 + 0.791828i \(0.290871\pi\)
\(338\) −18662.2 −3.00323
\(339\) −1549.47 −0.248248
\(340\) −9906.67 −1.58019
\(341\) 830.889 0.131951
\(342\) −6069.98 −0.959727
\(343\) −22070.7 −3.47436
\(344\) 0 0
\(345\) −707.427 −0.110396
\(346\) 10750.9 1.67044
\(347\) 8236.05 1.27416 0.637081 0.770797i \(-0.280141\pi\)
0.637081 + 0.770797i \(0.280141\pi\)
\(348\) −1993.73 −0.307112
\(349\) 7553.43 1.15853 0.579264 0.815140i \(-0.303340\pi\)
0.579264 + 0.815140i \(0.303340\pi\)
\(350\) 2309.97 0.352780
\(351\) 4912.98 0.747110
\(352\) −101.162 −0.0153180
\(353\) −752.070 −0.113396 −0.0566978 0.998391i \(-0.518057\pi\)
−0.0566978 + 0.998391i \(0.518057\pi\)
\(354\) 2434.89 0.365573
\(355\) −1625.56 −0.243030
\(356\) −1656.98 −0.246685
\(357\) −2255.44 −0.334371
\(358\) −1903.09 −0.280954
\(359\) 2146.31 0.315538 0.157769 0.987476i \(-0.449570\pi\)
0.157769 + 0.987476i \(0.449570\pi\)
\(360\) −12252.7 −1.79382
\(361\) −4533.03 −0.660887
\(362\) 6156.54 0.893868
\(363\) −1585.68 −0.229274
\(364\) 45399.3 6.53728
\(365\) −6725.58 −0.964474
\(366\) −360.078 −0.0514250
\(367\) −3278.23 −0.466273 −0.233137 0.972444i \(-0.574899\pi\)
−0.233137 + 0.972444i \(0.574899\pi\)
\(368\) 3535.29 0.500788
\(369\) −3152.66 −0.444772
\(370\) −3714.61 −0.521928
\(371\) −14294.9 −2.00042
\(372\) 3724.74 0.519137
\(373\) 2443.41 0.339183 0.169591 0.985514i \(-0.445755\pi\)
0.169591 + 0.985514i \(0.445755\pi\)
\(374\) −1120.69 −0.154945
\(375\) 1590.44 0.219013
\(376\) −18051.8 −2.47594
\(377\) 7833.21 1.07011
\(378\) −11278.7 −1.53470
\(379\) 10011.0 1.35680 0.678401 0.734692i \(-0.262673\pi\)
0.678401 + 0.734692i \(0.262673\pi\)
\(380\) 9227.71 1.24572
\(381\) 1001.02 0.134603
\(382\) −11170.1 −1.49610
\(383\) −12291.6 −1.63987 −0.819936 0.572455i \(-0.805991\pi\)
−0.819936 + 0.572455i \(0.805991\pi\)
\(384\) 2928.91 0.389232
\(385\) 1859.23 0.246118
\(386\) −15615.9 −2.05914
\(387\) 0 0
\(388\) 4711.87 0.616518
\(389\) −6820.52 −0.888982 −0.444491 0.895783i \(-0.646615\pi\)
−0.444491 + 0.895783i \(0.646615\pi\)
\(390\) −5413.83 −0.702923
\(391\) 2579.23 0.333599
\(392\) −39021.8 −5.02780
\(393\) 1323.53 0.169880
\(394\) −21195.0 −2.71012
\(395\) 12260.9 1.56180
\(396\) −1826.84 −0.231824
\(397\) 151.591 0.0191640 0.00958202 0.999954i \(-0.496950\pi\)
0.00958202 + 0.999954i \(0.496950\pi\)
\(398\) 1703.51 0.214546
\(399\) 2100.86 0.263596
\(400\) 923.149 0.115394
\(401\) −8900.01 −1.10834 −0.554171 0.832403i \(-0.686965\pi\)
−0.554171 + 0.832403i \(0.686965\pi\)
\(402\) −3492.07 −0.433255
\(403\) −14634.2 −1.80889
\(404\) −29181.9 −3.59369
\(405\) −7198.58 −0.883210
\(406\) −17982.7 −2.19819
\(407\) −281.798 −0.0343199
\(408\) −2556.20 −0.310173
\(409\) 9621.19 1.16317 0.581586 0.813485i \(-0.302432\pi\)
0.581586 + 0.813485i \(0.302432\pi\)
\(410\) 7146.90 0.860878
\(411\) −220.251 −0.0264335
\(412\) 10068.3 1.20395
\(413\) 14727.7 1.75473
\(414\) 6269.58 0.744283
\(415\) 6127.79 0.724823
\(416\) 1781.73 0.209992
\(417\) −254.151 −0.0298461
\(418\) 1043.88 0.122148
\(419\) −4299.21 −0.501265 −0.250632 0.968082i \(-0.580639\pi\)
−0.250632 + 0.968082i \(0.580639\pi\)
\(420\) 8334.64 0.968307
\(421\) −2833.15 −0.327979 −0.163990 0.986462i \(-0.552436\pi\)
−0.163990 + 0.986462i \(0.552436\pi\)
\(422\) −5959.14 −0.687409
\(423\) −11288.5 −1.29756
\(424\) −16201.1 −1.85565
\(425\) 673.498 0.0768692
\(426\) −824.354 −0.0937561
\(427\) −2177.97 −0.246837
\(428\) −3954.17 −0.446570
\(429\) −410.705 −0.0462214
\(430\) 0 0
\(431\) −5988.00 −0.669215 −0.334608 0.942358i \(-0.608604\pi\)
−0.334608 + 0.942358i \(0.608604\pi\)
\(432\) −4507.41 −0.501997
\(433\) 5090.17 0.564937 0.282469 0.959277i \(-0.408847\pi\)
0.282469 + 0.959277i \(0.408847\pi\)
\(434\) 33595.8 3.71579
\(435\) 1438.06 0.158505
\(436\) −10819.5 −1.18844
\(437\) −2402.46 −0.262987
\(438\) −3410.68 −0.372075
\(439\) −6863.79 −0.746220 −0.373110 0.927787i \(-0.621709\pi\)
−0.373110 + 0.927787i \(0.621709\pi\)
\(440\) 2107.16 0.228306
\(441\) −24401.9 −2.63491
\(442\) 19738.4 2.12412
\(443\) −12487.5 −1.33928 −0.669640 0.742686i \(-0.733552\pi\)
−0.669640 + 0.742686i \(0.733552\pi\)
\(444\) −1263.26 −0.135026
\(445\) 1195.17 0.127318
\(446\) 558.036 0.0592462
\(447\) −1739.62 −0.184075
\(448\) 16368.5 1.72620
\(449\) −6436.15 −0.676483 −0.338241 0.941059i \(-0.609832\pi\)
−0.338241 + 0.941059i \(0.609832\pi\)
\(450\) 1637.14 0.171501
\(451\) 542.178 0.0566080
\(452\) −20876.0 −2.17240
\(453\) 1236.91 0.128290
\(454\) −23693.3 −2.44930
\(455\) −32746.2 −3.37399
\(456\) 2381.01 0.244520
\(457\) 1479.45 0.151435 0.0757176 0.997129i \(-0.475875\pi\)
0.0757176 + 0.997129i \(0.475875\pi\)
\(458\) −17663.9 −1.80214
\(459\) −3288.45 −0.334404
\(460\) −9531.16 −0.966071
\(461\) 9925.64 1.00278 0.501392 0.865220i \(-0.332821\pi\)
0.501392 + 0.865220i \(0.332821\pi\)
\(462\) 942.855 0.0949472
\(463\) −5956.67 −0.597905 −0.298953 0.954268i \(-0.596637\pi\)
−0.298953 + 0.954268i \(0.596637\pi\)
\(464\) −7186.56 −0.719025
\(465\) −2686.63 −0.267934
\(466\) −16847.4 −1.67476
\(467\) 10264.0 1.01705 0.508526 0.861047i \(-0.330191\pi\)
0.508526 + 0.861047i \(0.330191\pi\)
\(468\) 32175.7 3.17804
\(469\) −21122.2 −2.07960
\(470\) 25590.4 2.51148
\(471\) −62.4478 −0.00610922
\(472\) 16691.6 1.62774
\(473\) 0 0
\(474\) 6217.76 0.602513
\(475\) −627.339 −0.0605985
\(476\) −30387.5 −2.92607
\(477\) −10131.2 −0.972485
\(478\) 11110.3 1.06312
\(479\) 10147.3 0.967941 0.483971 0.875084i \(-0.339194\pi\)
0.483971 + 0.875084i \(0.339194\pi\)
\(480\) 327.100 0.0311042
\(481\) 4963.24 0.470487
\(482\) 33571.4 3.17248
\(483\) −2169.95 −0.204422
\(484\) −21363.8 −2.00637
\(485\) −3398.64 −0.318194
\(486\) −12101.5 −1.12950
\(487\) −8735.65 −0.812834 −0.406417 0.913688i \(-0.633222\pi\)
−0.406417 + 0.913688i \(0.633222\pi\)
\(488\) −2468.40 −0.228974
\(489\) 3518.07 0.325343
\(490\) 55317.6 5.09999
\(491\) −7578.86 −0.696597 −0.348299 0.937384i \(-0.613240\pi\)
−0.348299 + 0.937384i \(0.613240\pi\)
\(492\) 2430.50 0.222714
\(493\) −5243.07 −0.478977
\(494\) −18385.6 −1.67451
\(495\) 1317.69 0.119648
\(496\) 13426.1 1.21543
\(497\) −4986.20 −0.450024
\(498\) 3107.53 0.279622
\(499\) 15869.8 1.42371 0.711854 0.702328i \(-0.247856\pi\)
0.711854 + 0.702328i \(0.247856\pi\)
\(500\) 21427.9 1.91657
\(501\) −4068.97 −0.362851
\(502\) −24666.2 −2.19304
\(503\) 6653.57 0.589797 0.294899 0.955529i \(-0.404714\pi\)
0.294899 + 0.955529i \(0.404714\pi\)
\(504\) −37583.6 −3.32164
\(505\) 21048.7 1.85476
\(506\) −1078.21 −0.0947279
\(507\) 4577.76 0.400997
\(508\) 13486.7 1.17790
\(509\) 10218.1 0.889800 0.444900 0.895580i \(-0.353239\pi\)
0.444900 + 0.895580i \(0.353239\pi\)
\(510\) 3623.68 0.314626
\(511\) −20629.9 −1.78594
\(512\) 21552.3 1.86032
\(513\) 3063.08 0.263622
\(514\) −25478.1 −2.18636
\(515\) −7262.18 −0.621378
\(516\) 0 0
\(517\) 1941.34 0.165145
\(518\) −11394.1 −0.966465
\(519\) −2637.15 −0.223041
\(520\) −37112.8 −3.12981
\(521\) −12801.7 −1.07650 −0.538248 0.842787i \(-0.680914\pi\)
−0.538248 + 0.842787i \(0.680914\pi\)
\(522\) −12744.8 −1.06863
\(523\) −3708.23 −0.310038 −0.155019 0.987912i \(-0.549544\pi\)
−0.155019 + 0.987912i \(0.549544\pi\)
\(524\) 17831.8 1.48662
\(525\) −566.625 −0.0471039
\(526\) 14236.4 1.18011
\(527\) 9795.25 0.809655
\(528\) 376.800 0.0310570
\(529\) −9685.54 −0.796050
\(530\) 22966.8 1.88229
\(531\) 10437.9 0.853046
\(532\) 28304.9 2.30671
\(533\) −9549.25 −0.776030
\(534\) 606.095 0.0491166
\(535\) 2852.11 0.230481
\(536\) −23938.8 −1.92910
\(537\) 466.820 0.0375135
\(538\) 17399.7 1.39434
\(539\) 4196.51 0.335355
\(540\) 12152.0 0.968404
\(541\) 18444.5 1.46579 0.732893 0.680344i \(-0.238170\pi\)
0.732893 + 0.680344i \(0.238170\pi\)
\(542\) −14160.3 −1.12221
\(543\) −1510.17 −0.119351
\(544\) −1192.58 −0.0939918
\(545\) 7804.04 0.613373
\(546\) −16606.3 −1.30162
\(547\) −2665.35 −0.208340 −0.104170 0.994559i \(-0.533219\pi\)
−0.104170 + 0.994559i \(0.533219\pi\)
\(548\) −2967.44 −0.231319
\(549\) −1543.59 −0.119998
\(550\) −281.546 −0.0218276
\(551\) 4883.73 0.377594
\(552\) −2459.30 −0.189629
\(553\) 37608.8 2.89202
\(554\) 2222.08 0.170410
\(555\) 911.178 0.0696889
\(556\) −3424.17 −0.261182
\(557\) 6223.14 0.473398 0.236699 0.971583i \(-0.423934\pi\)
0.236699 + 0.971583i \(0.423934\pi\)
\(558\) 23810.3 1.80640
\(559\) 0 0
\(560\) 30042.9 2.26704
\(561\) 274.900 0.0206886
\(562\) 39129.5 2.93697
\(563\) 16282.3 1.21886 0.609431 0.792839i \(-0.291398\pi\)
0.609431 + 0.792839i \(0.291398\pi\)
\(564\) 8702.73 0.649736
\(565\) 15057.7 1.12121
\(566\) 322.202 0.0239279
\(567\) −22080.8 −1.63546
\(568\) −5651.10 −0.417456
\(569\) 8186.09 0.603126 0.301563 0.953446i \(-0.402492\pi\)
0.301563 + 0.953446i \(0.402492\pi\)
\(570\) −3375.34 −0.248030
\(571\) 22181.2 1.62567 0.812834 0.582496i \(-0.197924\pi\)
0.812834 + 0.582496i \(0.197924\pi\)
\(572\) −5533.41 −0.404482
\(573\) 2739.97 0.199762
\(574\) 21922.2 1.59410
\(575\) 647.969 0.0469951
\(576\) 11600.8 0.839178
\(577\) −8781.20 −0.633563 −0.316782 0.948499i \(-0.602602\pi\)
−0.316782 + 0.948499i \(0.602602\pi\)
\(578\) 11000.5 0.791625
\(579\) 3830.52 0.274941
\(580\) 19375.0 1.38707
\(581\) 18796.2 1.34217
\(582\) −1723.52 −0.122753
\(583\) 1742.31 0.123772
\(584\) −23380.8 −1.65669
\(585\) −23208.1 −1.64023
\(586\) 22302.7 1.57221
\(587\) 15965.9 1.12263 0.561313 0.827604i \(-0.310296\pi\)
0.561313 + 0.827604i \(0.310296\pi\)
\(588\) 18812.3 1.31940
\(589\) −9123.94 −0.638277
\(590\) −23662.2 −1.65111
\(591\) 5199.04 0.361861
\(592\) −4553.51 −0.316129
\(593\) 4928.88 0.341324 0.170662 0.985330i \(-0.445409\pi\)
0.170662 + 0.985330i \(0.445409\pi\)
\(594\) 1374.69 0.0949567
\(595\) 21918.3 1.51019
\(596\) −23437.9 −1.61083
\(597\) −417.865 −0.0286467
\(598\) 18990.2 1.29861
\(599\) −12943.4 −0.882891 −0.441446 0.897288i \(-0.645534\pi\)
−0.441446 + 0.897288i \(0.645534\pi\)
\(600\) −642.183 −0.0436950
\(601\) 4989.56 0.338650 0.169325 0.985560i \(-0.445841\pi\)
0.169325 + 0.985560i \(0.445841\pi\)
\(602\) 0 0
\(603\) −14969.9 −1.01098
\(604\) 16664.9 1.12266
\(605\) 15409.5 1.03552
\(606\) 10674.2 0.715528
\(607\) 16584.3 1.10896 0.554478 0.832198i \(-0.312918\pi\)
0.554478 + 0.832198i \(0.312918\pi\)
\(608\) 1110.85 0.0740968
\(609\) 4411.08 0.293507
\(610\) 3499.22 0.232261
\(611\) −34192.4 −2.26395
\(612\) −21536.4 −1.42248
\(613\) 2154.47 0.141955 0.0709773 0.997478i \(-0.477388\pi\)
0.0709773 + 0.997478i \(0.477388\pi\)
\(614\) −4003.14 −0.263117
\(615\) −1753.10 −0.114946
\(616\) 6463.44 0.422759
\(617\) −4703.27 −0.306883 −0.153441 0.988158i \(-0.549036\pi\)
−0.153441 + 0.988158i \(0.549036\pi\)
\(618\) −3682.80 −0.239715
\(619\) −12587.5 −0.817340 −0.408670 0.912682i \(-0.634007\pi\)
−0.408670 + 0.912682i \(0.634007\pi\)
\(620\) −36196.9 −2.34468
\(621\) −3163.80 −0.204443
\(622\) 1559.16 0.100509
\(623\) 3666.03 0.235757
\(624\) −6636.48 −0.425756
\(625\) −17081.8 −1.09323
\(626\) 23616.1 1.50781
\(627\) −256.060 −0.0163095
\(628\) −841.358 −0.0534615
\(629\) −3322.09 −0.210589
\(630\) 53278.9 3.36934
\(631\) 19165.7 1.20915 0.604576 0.796547i \(-0.293342\pi\)
0.604576 + 0.796547i \(0.293342\pi\)
\(632\) 42623.8 2.68273
\(633\) 1461.75 0.0917842
\(634\) −23490.6 −1.47150
\(635\) −9727.84 −0.607933
\(636\) 7810.51 0.486960
\(637\) −73912.0 −4.59733
\(638\) 2191.79 0.136009
\(639\) −3533.86 −0.218775
\(640\) −28463.0 −1.75797
\(641\) 15441.3 0.951475 0.475738 0.879587i \(-0.342181\pi\)
0.475738 + 0.879587i \(0.342181\pi\)
\(642\) 1446.36 0.0889151
\(643\) −12967.4 −0.795307 −0.397654 0.917536i \(-0.630175\pi\)
−0.397654 + 0.917536i \(0.630175\pi\)
\(644\) −29235.7 −1.78889
\(645\) 0 0
\(646\) 12306.2 0.749507
\(647\) −1957.15 −0.118924 −0.0594618 0.998231i \(-0.518938\pi\)
−0.0594618 + 0.998231i \(0.518938\pi\)
\(648\) −25025.2 −1.51710
\(649\) −1795.06 −0.108571
\(650\) 4958.80 0.299231
\(651\) −8240.91 −0.496139
\(652\) 47398.9 2.84706
\(653\) 22721.6 1.36166 0.680830 0.732442i \(-0.261619\pi\)
0.680830 + 0.732442i \(0.261619\pi\)
\(654\) 3957.59 0.236627
\(655\) −12862.0 −0.767265
\(656\) 8760.94 0.521429
\(657\) −14621.0 −0.868217
\(658\) 78495.5 4.65056
\(659\) −14970.8 −0.884946 −0.442473 0.896782i \(-0.645899\pi\)
−0.442473 + 0.896782i \(0.645899\pi\)
\(660\) −1015.85 −0.0599122
\(661\) −13729.4 −0.807886 −0.403943 0.914784i \(-0.632361\pi\)
−0.403943 + 0.914784i \(0.632361\pi\)
\(662\) −33869.9 −1.98851
\(663\) −4841.75 −0.283617
\(664\) 21302.7 1.24504
\(665\) −20416.1 −1.19053
\(666\) −8075.32 −0.469838
\(667\) −5044.33 −0.292829
\(668\) −54821.1 −3.17529
\(669\) −136.884 −0.00791067
\(670\) 33935.8 1.95680
\(671\) 265.458 0.0152726
\(672\) 1003.34 0.0575962
\(673\) −33165.2 −1.89959 −0.949795 0.312874i \(-0.898708\pi\)
−0.949795 + 0.312874i \(0.898708\pi\)
\(674\) −37240.9 −2.12829
\(675\) −826.144 −0.0471086
\(676\) 61676.1 3.50911
\(677\) −24360.0 −1.38291 −0.691455 0.722419i \(-0.743030\pi\)
−0.691455 + 0.722419i \(0.743030\pi\)
\(678\) 7636.09 0.432540
\(679\) −10424.9 −0.589206
\(680\) 24841.0 1.40090
\(681\) 5811.86 0.327035
\(682\) −4094.77 −0.229907
\(683\) −21088.4 −1.18144 −0.590720 0.806876i \(-0.701156\pi\)
−0.590720 + 0.806876i \(0.701156\pi\)
\(684\) 20060.4 1.12139
\(685\) 2140.39 0.119387
\(686\) 108768. 6.05364
\(687\) 4332.89 0.240626
\(688\) 0 0
\(689\) −30686.9 −1.69677
\(690\) 3486.33 0.192351
\(691\) −2625.67 −0.144551 −0.0722757 0.997385i \(-0.523026\pi\)
−0.0722757 + 0.997385i \(0.523026\pi\)
\(692\) −35530.3 −1.95182
\(693\) 4041.85 0.221554
\(694\) −40588.8 −2.22007
\(695\) 2469.83 0.134800
\(696\) 4999.29 0.272267
\(697\) 6391.68 0.347349
\(698\) −37224.7 −2.01859
\(699\) 4132.58 0.223617
\(700\) −7634.12 −0.412204
\(701\) −32287.1 −1.73961 −0.869805 0.493396i \(-0.835755\pi\)
−0.869805 + 0.493396i \(0.835755\pi\)
\(702\) −24212.1 −1.30175
\(703\) 3094.41 0.166014
\(704\) −1995.04 −0.106806
\(705\) −6277.22 −0.335338
\(706\) 3706.34 0.197578
\(707\) 64564.2 3.43449
\(708\) −8046.97 −0.427152
\(709\) 24228.5 1.28338 0.641692 0.766962i \(-0.278233\pi\)
0.641692 + 0.766962i \(0.278233\pi\)
\(710\) 8011.05 0.423450
\(711\) 26654.4 1.40593
\(712\) 4154.89 0.218695
\(713\) 9423.97 0.494994
\(714\) 11115.2 0.582600
\(715\) 3991.21 0.208759
\(716\) 6289.45 0.328279
\(717\) −2725.30 −0.141950
\(718\) −10577.4 −0.549786
\(719\) 26115.0 1.35455 0.677277 0.735729i \(-0.263160\pi\)
0.677277 + 0.735729i \(0.263160\pi\)
\(720\) 21292.2 1.10210
\(721\) −22275.8 −1.15062
\(722\) 22339.6 1.15151
\(723\) −8234.92 −0.423596
\(724\) −20346.5 −1.04444
\(725\) −1317.19 −0.0674750
\(726\) 7814.50 0.399481
\(727\) 11600.5 0.591799 0.295899 0.955219i \(-0.404381\pi\)
0.295899 + 0.955219i \(0.404381\pi\)
\(728\) −113839. −5.79554
\(729\) −13576.3 −0.689745
\(730\) 33144.9 1.68048
\(731\) 0 0
\(732\) 1190.01 0.0600873
\(733\) −1893.82 −0.0954296 −0.0477148 0.998861i \(-0.515194\pi\)
−0.0477148 + 0.998861i \(0.515194\pi\)
\(734\) 16155.7 0.812422
\(735\) −13569.2 −0.680961
\(736\) −1147.38 −0.0574632
\(737\) 2574.44 0.128671
\(738\) 15536.9 0.774960
\(739\) −32380.7 −1.61183 −0.805916 0.592030i \(-0.798327\pi\)
−0.805916 + 0.592030i \(0.798327\pi\)
\(740\) 12276.3 0.609845
\(741\) 4509.92 0.223584
\(742\) 70447.9 3.48548
\(743\) −14950.3 −0.738187 −0.369093 0.929392i \(-0.620332\pi\)
−0.369093 + 0.929392i \(0.620332\pi\)
\(744\) −9339.82 −0.460234
\(745\) 16905.6 0.831374
\(746\) −12041.6 −0.590984
\(747\) 13321.4 0.652483
\(748\) 3703.73 0.181045
\(749\) 8748.50 0.426787
\(750\) −7837.96 −0.381602
\(751\) −38699.6 −1.88039 −0.940193 0.340643i \(-0.889355\pi\)
−0.940193 + 0.340643i \(0.889355\pi\)
\(752\) 31369.7 1.52119
\(753\) 6050.52 0.292820
\(754\) −38603.5 −1.86453
\(755\) −12020.3 −0.579421
\(756\) 37274.7 1.79321
\(757\) 555.655 0.0266785 0.0133392 0.999911i \(-0.495754\pi\)
0.0133392 + 0.999911i \(0.495754\pi\)
\(758\) −49335.8 −2.36406
\(759\) 264.480 0.0126483
\(760\) −23138.5 −1.10437
\(761\) −30618.8 −1.45852 −0.729259 0.684238i \(-0.760135\pi\)
−0.729259 + 0.684238i \(0.760135\pi\)
\(762\) −4933.19 −0.234528
\(763\) 23937.9 1.13579
\(764\) 36915.5 1.74811
\(765\) 15534.1 0.734164
\(766\) 60575.2 2.85727
\(767\) 31615.9 1.48838
\(768\) −10041.2 −0.471786
\(769\) 10000.6 0.468962 0.234481 0.972121i \(-0.424661\pi\)
0.234481 + 0.972121i \(0.424661\pi\)
\(770\) −9162.63 −0.428829
\(771\) 6249.67 0.291928
\(772\) 51608.5 2.40600
\(773\) 14348.4 0.667626 0.333813 0.942639i \(-0.391665\pi\)
0.333813 + 0.942639i \(0.391665\pi\)
\(774\) 0 0
\(775\) 2460.82 0.114059
\(776\) −11815.0 −0.546566
\(777\) 2794.93 0.129044
\(778\) 33612.7 1.54894
\(779\) −5953.63 −0.273826
\(780\) 17892.0 0.821327
\(781\) 607.735 0.0278444
\(782\) −12710.9 −0.581254
\(783\) 6431.39 0.293537
\(784\) 67810.4 3.08903
\(785\) 606.866 0.0275923
\(786\) −6522.57 −0.295995
\(787\) 42907.3 1.94343 0.971716 0.236151i \(-0.0758860\pi\)
0.971716 + 0.236151i \(0.0758860\pi\)
\(788\) 70046.6 3.16663
\(789\) −3492.13 −0.157570
\(790\) −60423.9 −2.72125
\(791\) 46187.8 2.07617
\(792\) 4580.82 0.205521
\(793\) −4675.45 −0.209369
\(794\) −747.067 −0.0333909
\(795\) −5633.66 −0.251328
\(796\) −5629.88 −0.250686
\(797\) −23001.4 −1.02227 −0.511136 0.859500i \(-0.670775\pi\)
−0.511136 + 0.859500i \(0.670775\pi\)
\(798\) −10353.4 −0.459283
\(799\) 22886.3 1.01334
\(800\) −299.608 −0.0132409
\(801\) 2598.22 0.114611
\(802\) 43860.8 1.93115
\(803\) 2514.44 0.110501
\(804\) 11540.8 0.506235
\(805\) 21087.5 0.923274
\(806\) 72120.1 3.15177
\(807\) −4268.06 −0.186175
\(808\) 73173.7 3.18594
\(809\) 28862.5 1.25433 0.627164 0.778888i \(-0.284216\pi\)
0.627164 + 0.778888i \(0.284216\pi\)
\(810\) 35475.9 1.53888
\(811\) −14854.2 −0.643159 −0.321580 0.946883i \(-0.604214\pi\)
−0.321580 + 0.946883i \(0.604214\pi\)
\(812\) 59430.4 2.56847
\(813\) 3473.46 0.149840
\(814\) 1388.75 0.0597982
\(815\) −34188.5 −1.46941
\(816\) 4442.05 0.190567
\(817\) 0 0
\(818\) −47415.0 −2.02668
\(819\) −71188.0 −3.03725
\(820\) −23619.5 −1.00589
\(821\) 1806.78 0.0768052 0.0384026 0.999262i \(-0.487773\pi\)
0.0384026 + 0.999262i \(0.487773\pi\)
\(822\) 1085.44 0.0460571
\(823\) 22648.5 0.959269 0.479634 0.877468i \(-0.340769\pi\)
0.479634 + 0.877468i \(0.340769\pi\)
\(824\) −25246.3 −1.06735
\(825\) 69.0621 0.00291447
\(826\) −72580.8 −3.05739
\(827\) 17580.0 0.739198 0.369599 0.929191i \(-0.379495\pi\)
0.369599 + 0.929191i \(0.379495\pi\)
\(828\) −20720.1 −0.869654
\(829\) −778.570 −0.0326186 −0.0163093 0.999867i \(-0.505192\pi\)
−0.0163093 + 0.999867i \(0.505192\pi\)
\(830\) −30198.9 −1.26291
\(831\) −545.066 −0.0227535
\(832\) 35138.2 1.46418
\(833\) 49472.1 2.05775
\(834\) 1252.50 0.0520031
\(835\) 39542.1 1.63881
\(836\) −3449.89 −0.142724
\(837\) −12015.3 −0.496189
\(838\) 21187.3 0.873391
\(839\) −9935.89 −0.408850 −0.204425 0.978882i \(-0.565532\pi\)
−0.204425 + 0.978882i \(0.565532\pi\)
\(840\) −20899.2 −0.858440
\(841\) −14134.9 −0.579559
\(842\) 13962.3 0.571463
\(843\) −9598.29 −0.392150
\(844\) 19694.1 0.803200
\(845\) −44486.5 −1.81110
\(846\) 55631.9 2.26083
\(847\) 47266.9 1.91748
\(848\) 28153.6 1.14009
\(849\) −79.0348 −0.00319490
\(850\) −3319.12 −0.133935
\(851\) −3196.16 −0.128746
\(852\) 2724.38 0.109549
\(853\) −25751.3 −1.03365 −0.516827 0.856090i \(-0.672887\pi\)
−0.516827 + 0.856090i \(0.672887\pi\)
\(854\) 10733.4 0.430082
\(855\) −14469.4 −0.578766
\(856\) 9915.09 0.395901
\(857\) −16458.1 −0.656007 −0.328004 0.944676i \(-0.606376\pi\)
−0.328004 + 0.944676i \(0.606376\pi\)
\(858\) 2024.03 0.0805351
\(859\) 14331.0 0.569231 0.284615 0.958642i \(-0.408134\pi\)
0.284615 + 0.958642i \(0.408134\pi\)
\(860\) 0 0
\(861\) −5377.43 −0.212848
\(862\) 29509.9 1.16602
\(863\) −13078.3 −0.515864 −0.257932 0.966163i \(-0.583041\pi\)
−0.257932 + 0.966163i \(0.583041\pi\)
\(864\) 1462.88 0.0576020
\(865\) 25627.7 1.00736
\(866\) −25085.3 −0.984333
\(867\) −2698.37 −0.105699
\(868\) −111030. −4.34170
\(869\) −4583.88 −0.178939
\(870\) −7087.03 −0.276176
\(871\) −45343.0 −1.76394
\(872\) 27130.0 1.05360
\(873\) −7388.41 −0.286437
\(874\) 11839.8 0.458222
\(875\) −47408.8 −1.83167
\(876\) 11271.8 0.434749
\(877\) 3287.88 0.126595 0.0632974 0.997995i \(-0.479838\pi\)
0.0632974 + 0.997995i \(0.479838\pi\)
\(878\) 33826.0 1.30020
\(879\) −5470.74 −0.209924
\(880\) −3661.73 −0.140269
\(881\) 34036.1 1.30160 0.650798 0.759251i \(-0.274435\pi\)
0.650798 + 0.759251i \(0.274435\pi\)
\(882\) 120257. 4.59099
\(883\) 26799.0 1.02136 0.510678 0.859772i \(-0.329395\pi\)
0.510678 + 0.859772i \(0.329395\pi\)
\(884\) −65232.8 −2.48192
\(885\) 5804.22 0.220460
\(886\) 61540.9 2.33353
\(887\) −44323.8 −1.67784 −0.838921 0.544253i \(-0.816813\pi\)
−0.838921 + 0.544253i \(0.816813\pi\)
\(888\) 3167.62 0.119705
\(889\) −29839.0 −1.12572
\(890\) −5890.01 −0.221835
\(891\) 2691.28 0.101191
\(892\) −1844.23 −0.0692259
\(893\) −21317.8 −0.798848
\(894\) 8573.18 0.320727
\(895\) −4536.54 −0.169430
\(896\) −87306.7 −3.25526
\(897\) −4658.22 −0.173393
\(898\) 31718.5 1.17869
\(899\) −19157.1 −0.710706
\(900\) −5410.51 −0.200389
\(901\) 20539.9 0.759471
\(902\) −2671.95 −0.0986323
\(903\) 0 0
\(904\) 52346.8 1.92592
\(905\) 14675.8 0.539049
\(906\) −6095.73 −0.223529
\(907\) −4653.88 −0.170374 −0.0851872 0.996365i \(-0.527149\pi\)
−0.0851872 + 0.996365i \(0.527149\pi\)
\(908\) 78303.1 2.86187
\(909\) 45758.4 1.66965
\(910\) 161379. 5.87875
\(911\) 25937.6 0.943305 0.471653 0.881784i \(-0.343658\pi\)
0.471653 + 0.881784i \(0.343658\pi\)
\(912\) −4137.62 −0.150230
\(913\) −2290.95 −0.0830442
\(914\) −7291.01 −0.263857
\(915\) −858.343 −0.0310120
\(916\) 58376.9 2.10571
\(917\) −39452.5 −1.42076
\(918\) 16206.1 0.582658
\(919\) −34390.1 −1.23441 −0.617206 0.786802i \(-0.711736\pi\)
−0.617206 + 0.786802i \(0.711736\pi\)
\(920\) 23899.4 0.856457
\(921\) 981.954 0.0351319
\(922\) −48915.4 −1.74723
\(923\) −10703.9 −0.381714
\(924\) −3116.01 −0.110941
\(925\) −834.594 −0.0296663
\(926\) 29355.6 1.04178
\(927\) −15787.5 −0.559363
\(928\) 2332.40 0.0825050
\(929\) −4086.19 −0.144310 −0.0721548 0.997393i \(-0.522988\pi\)
−0.0721548 + 0.997393i \(0.522988\pi\)
\(930\) 13240.2 0.466842
\(931\) −46081.6 −1.62219
\(932\) 55678.2 1.95687
\(933\) −382.455 −0.0134202
\(934\) −50583.0 −1.77209
\(935\) −2671.47 −0.0934401
\(936\) −80680.8 −2.81745
\(937\) −42025.6 −1.46523 −0.732613 0.680646i \(-0.761699\pi\)
−0.732613 + 0.680646i \(0.761699\pi\)
\(938\) 104094. 3.62344
\(939\) −5792.93 −0.201326
\(940\) −84572.8 −2.93453
\(941\) −7204.96 −0.249602 −0.124801 0.992182i \(-0.539829\pi\)
−0.124801 + 0.992182i \(0.539829\pi\)
\(942\) 307.754 0.0106446
\(943\) 6149.40 0.212356
\(944\) −29006.0 −1.00007
\(945\) −26886.0 −0.925504
\(946\) 0 0
\(947\) 38797.3 1.33130 0.665651 0.746263i \(-0.268154\pi\)
0.665651 + 0.746263i \(0.268154\pi\)
\(948\) −20548.8 −0.704003
\(949\) −44286.2 −1.51485
\(950\) 3091.64 0.105585
\(951\) 5762.15 0.196478
\(952\) 76196.8 2.59407
\(953\) −43711.5 −1.48579 −0.742893 0.669410i \(-0.766547\pi\)
−0.742893 + 0.669410i \(0.766547\pi\)
\(954\) 49928.4 1.69443
\(955\) −26626.9 −0.902226
\(956\) −36717.9 −1.24220
\(957\) −537.637 −0.0181602
\(958\) −50007.9 −1.68652
\(959\) 6565.39 0.221071
\(960\) 6450.86 0.216876
\(961\) 5998.85 0.201365
\(962\) −24459.7 −0.819764
\(963\) 6200.30 0.207479
\(964\) −110949. −3.70687
\(965\) −37224.8 −1.24177
\(966\) 10693.9 0.356180
\(967\) −9219.24 −0.306588 −0.153294 0.988181i \(-0.548988\pi\)
−0.153294 + 0.988181i \(0.548988\pi\)
\(968\) 53569.8 1.77872
\(969\) −3018.66 −0.100076
\(970\) 16749.1 0.554413
\(971\) −3770.19 −0.124605 −0.0623023 0.998057i \(-0.519844\pi\)
−0.0623023 + 0.998057i \(0.519844\pi\)
\(972\) 39993.8 1.31976
\(973\) 7575.90 0.249612
\(974\) 43050.9 1.41626
\(975\) −1216.37 −0.0399540
\(976\) 4289.48 0.140679
\(977\) 40226.1 1.31724 0.658622 0.752474i \(-0.271140\pi\)
0.658622 + 0.752474i \(0.271140\pi\)
\(978\) −17337.7 −0.566870
\(979\) −446.828 −0.0145870
\(980\) −182817. −5.95906
\(981\) 16965.5 0.552157
\(982\) 37350.0 1.21373
\(983\) −8280.68 −0.268680 −0.134340 0.990935i \(-0.542891\pi\)
−0.134340 + 0.990935i \(0.542891\pi\)
\(984\) −6094.49 −0.197444
\(985\) −50524.1 −1.63435
\(986\) 25838.8 0.834558
\(987\) −19254.6 −0.620953
\(988\) 60762.1 1.95658
\(989\) 0 0
\(990\) −6493.81 −0.208471
\(991\) 14631.2 0.468997 0.234498 0.972117i \(-0.424655\pi\)
0.234498 + 0.972117i \(0.424655\pi\)
\(992\) −4357.45 −0.139465
\(993\) 8308.15 0.265510
\(994\) 24572.9 0.784110
\(995\) 4060.79 0.129383
\(996\) −10270.0 −0.326723
\(997\) 7486.54 0.237814 0.118907 0.992905i \(-0.462061\pi\)
0.118907 + 0.992905i \(0.462061\pi\)
\(998\) −78209.3 −2.48063
\(999\) 4075.03 0.129057
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.k.1.6 60
43.30 odd 42 43.4.g.a.40.1 yes 120
43.33 odd 42 43.4.g.a.14.1 120
43.42 odd 2 1849.4.a.l.1.55 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.g.a.14.1 120 43.33 odd 42
43.4.g.a.40.1 yes 120 43.30 odd 42
1849.4.a.k.1.6 60 1.1 even 1 trivial
1849.4.a.l.1.55 60 43.42 odd 2