Properties

Label 1849.4.a.k.1.8
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $60$
CM no
Inner twists $1$

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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 1.8
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)
 
\(f(q)\) \(=\) \(q-4.85348 q^{2} +5.30735 q^{3} +15.5563 q^{4} +16.1569 q^{5} -25.7591 q^{6} +27.9534 q^{7} -36.6742 q^{8} +1.16797 q^{9} +O(q^{10})\) \(q-4.85348 q^{2} +5.30735 q^{3} +15.5563 q^{4} +16.1569 q^{5} -25.7591 q^{6} +27.9534 q^{7} -36.6742 q^{8} +1.16797 q^{9} -78.4173 q^{10} -13.3801 q^{11} +82.5625 q^{12} -52.0918 q^{13} -135.671 q^{14} +85.7504 q^{15} +53.5471 q^{16} -90.9834 q^{17} -5.66872 q^{18} -10.5083 q^{19} +251.341 q^{20} +148.358 q^{21} +64.9401 q^{22} -195.325 q^{23} -194.643 q^{24} +136.046 q^{25} +252.826 q^{26} -137.100 q^{27} +434.850 q^{28} -168.406 q^{29} -416.188 q^{30} +122.360 q^{31} +33.5033 q^{32} -71.0129 q^{33} +441.586 q^{34} +451.641 q^{35} +18.1693 q^{36} +265.643 q^{37} +51.0018 q^{38} -276.469 q^{39} -592.541 q^{40} -250.166 q^{41} -720.054 q^{42} -208.144 q^{44} +18.8708 q^{45} +948.007 q^{46} +179.800 q^{47} +284.193 q^{48} +438.392 q^{49} -660.297 q^{50} -482.881 q^{51} -810.353 q^{52} -63.6412 q^{53} +665.410 q^{54} -216.181 q^{55} -1025.17 q^{56} -55.7712 q^{57} +817.355 q^{58} -29.8681 q^{59} +1333.96 q^{60} -215.166 q^{61} -593.872 q^{62} +32.6488 q^{63} -590.984 q^{64} -841.643 q^{65} +344.660 q^{66} +16.2324 q^{67} -1415.36 q^{68} -1036.66 q^{69} -2192.03 q^{70} +806.449 q^{71} -42.8344 q^{72} -30.3032 q^{73} -1289.29 q^{74} +722.044 q^{75} -163.470 q^{76} -374.019 q^{77} +1341.84 q^{78} -379.053 q^{79} +865.157 q^{80} -759.171 q^{81} +1214.18 q^{82} -607.128 q^{83} +2307.90 q^{84} -1470.01 q^{85} -893.790 q^{87} +490.704 q^{88} -1314.18 q^{89} -91.5891 q^{90} -1456.14 q^{91} -3038.53 q^{92} +649.408 q^{93} -872.655 q^{94} -169.782 q^{95} +177.814 q^{96} +229.873 q^{97} -2127.72 q^{98} -15.6276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.85348 −1.71596 −0.857982 0.513680i \(-0.828282\pi\)
−0.857982 + 0.513680i \(0.828282\pi\)
\(3\) 5.30735 1.02140 0.510700 0.859759i \(-0.329386\pi\)
0.510700 + 0.859759i \(0.329386\pi\)
\(4\) 15.5563 1.94453
\(5\) 16.1569 1.44512 0.722559 0.691309i \(-0.242966\pi\)
0.722559 + 0.691309i \(0.242966\pi\)
\(6\) −25.7591 −1.75269
\(7\) 27.9534 1.50934 0.754670 0.656104i \(-0.227797\pi\)
0.754670 + 0.656104i \(0.227797\pi\)
\(8\) −36.6742 −1.62078
\(9\) 1.16797 0.0432582
\(10\) −78.4173 −2.47977
\(11\) −13.3801 −0.366750 −0.183375 0.983043i \(-0.558702\pi\)
−0.183375 + 0.983043i \(0.558702\pi\)
\(12\) 82.5625 1.98615
\(13\) −52.0918 −1.11136 −0.555679 0.831397i \(-0.687542\pi\)
−0.555679 + 0.831397i \(0.687542\pi\)
\(14\) −135.671 −2.58997
\(15\) 85.7504 1.47604
\(16\) 53.5471 0.836674
\(17\) −90.9834 −1.29804 −0.649021 0.760770i \(-0.724821\pi\)
−0.649021 + 0.760770i \(0.724821\pi\)
\(18\) −5.66872 −0.0742295
\(19\) −10.5083 −0.126883 −0.0634413 0.997986i \(-0.520208\pi\)
−0.0634413 + 0.997986i \(0.520208\pi\)
\(20\) 251.341 2.81008
\(21\) 148.358 1.54164
\(22\) 64.9401 0.629330
\(23\) −195.325 −1.77079 −0.885394 0.464842i \(-0.846111\pi\)
−0.885394 + 0.464842i \(0.846111\pi\)
\(24\) −194.643 −1.65547
\(25\) 136.046 1.08837
\(26\) 252.826 1.90705
\(27\) −137.100 −0.977216
\(28\) 434.850 2.93496
\(29\) −168.406 −1.07835 −0.539176 0.842193i \(-0.681264\pi\)
−0.539176 + 0.842193i \(0.681264\pi\)
\(30\) −416.188 −2.53284
\(31\) 122.360 0.708920 0.354460 0.935071i \(-0.384665\pi\)
0.354460 + 0.935071i \(0.384665\pi\)
\(32\) 33.5033 0.185081
\(33\) −71.0129 −0.374599
\(34\) 441.586 2.22739
\(35\) 451.641 2.18118
\(36\) 18.1693 0.0841170
\(37\) 265.643 1.18031 0.590155 0.807290i \(-0.299067\pi\)
0.590155 + 0.807290i \(0.299067\pi\)
\(38\) 51.0018 0.217726
\(39\) −276.469 −1.13514
\(40\) −592.541 −2.34223
\(41\) −250.166 −0.952911 −0.476456 0.879198i \(-0.658079\pi\)
−0.476456 + 0.879198i \(0.658079\pi\)
\(42\) −720.054 −2.64540
\(43\) 0 0
\(44\) −208.144 −0.713158
\(45\) 18.8708 0.0625132
\(46\) 948.007 3.03861
\(47\) 179.800 0.558011 0.279005 0.960290i \(-0.409995\pi\)
0.279005 + 0.960290i \(0.409995\pi\)
\(48\) 284.193 0.854579
\(49\) 438.392 1.27811
\(50\) −660.297 −1.86760
\(51\) −482.881 −1.32582
\(52\) −810.353 −2.16107
\(53\) −63.6412 −0.164939 −0.0824697 0.996594i \(-0.526281\pi\)
−0.0824697 + 0.996594i \(0.526281\pi\)
\(54\) 665.410 1.67687
\(55\) −216.181 −0.529998
\(56\) −1025.17 −2.44632
\(57\) −55.7712 −0.129598
\(58\) 817.355 1.85041
\(59\) −29.8681 −0.0659066 −0.0329533 0.999457i \(-0.510491\pi\)
−0.0329533 + 0.999457i \(0.510491\pi\)
\(60\) 1333.96 2.87022
\(61\) −215.166 −0.451626 −0.225813 0.974171i \(-0.572504\pi\)
−0.225813 + 0.974171i \(0.572504\pi\)
\(62\) −593.872 −1.21648
\(63\) 32.6488 0.0652914
\(64\) −590.984 −1.15427
\(65\) −841.643 −1.60605
\(66\) 344.660 0.642798
\(67\) 16.2324 0.0295985 0.0147992 0.999890i \(-0.495289\pi\)
0.0147992 + 0.999890i \(0.495289\pi\)
\(68\) −1415.36 −2.52409
\(69\) −1036.66 −1.80868
\(70\) −2192.03 −3.74282
\(71\) 806.449 1.34800 0.673999 0.738732i \(-0.264575\pi\)
0.673999 + 0.738732i \(0.264575\pi\)
\(72\) −42.8344 −0.0701122
\(73\) −30.3032 −0.0485853 −0.0242926 0.999705i \(-0.507733\pi\)
−0.0242926 + 0.999705i \(0.507733\pi\)
\(74\) −1289.29 −2.02537
\(75\) 722.044 1.11166
\(76\) −163.470 −0.246727
\(77\) −374.019 −0.553551
\(78\) 1341.84 1.94786
\(79\) −379.053 −0.539833 −0.269917 0.962884i \(-0.586996\pi\)
−0.269917 + 0.962884i \(0.586996\pi\)
\(80\) 865.157 1.20909
\(81\) −759.171 −1.04139
\(82\) 1214.18 1.63516
\(83\) −607.128 −0.802904 −0.401452 0.915880i \(-0.631494\pi\)
−0.401452 + 0.915880i \(0.631494\pi\)
\(84\) 2307.90 2.99777
\(85\) −1470.01 −1.87583
\(86\) 0 0
\(87\) −893.790 −1.10143
\(88\) 490.704 0.594423
\(89\) −1314.18 −1.56520 −0.782602 0.622522i \(-0.786108\pi\)
−0.782602 + 0.622522i \(0.786108\pi\)
\(90\) −91.5891 −0.107270
\(91\) −1456.14 −1.67742
\(92\) −3038.53 −3.44335
\(93\) 649.408 0.724091
\(94\) −872.655 −0.957526
\(95\) −169.782 −0.183360
\(96\) 177.814 0.189042
\(97\) 229.873 0.240619 0.120310 0.992736i \(-0.461611\pi\)
0.120310 + 0.992736i \(0.461611\pi\)
\(98\) −2127.72 −2.19319
\(99\) −15.6276 −0.0158650
\(100\) 2116.37 2.11637
\(101\) 1013.21 0.998195 0.499098 0.866546i \(-0.333665\pi\)
0.499098 + 0.866546i \(0.333665\pi\)
\(102\) 2343.65 2.27506
\(103\) 664.636 0.635811 0.317906 0.948122i \(-0.397020\pi\)
0.317906 + 0.948122i \(0.397020\pi\)
\(104\) 1910.42 1.80127
\(105\) 2397.01 2.22785
\(106\) 308.881 0.283030
\(107\) −785.202 −0.709424 −0.354712 0.934976i \(-0.615421\pi\)
−0.354712 + 0.934976i \(0.615421\pi\)
\(108\) −2132.76 −1.90023
\(109\) −1541.98 −1.35500 −0.677500 0.735522i \(-0.736937\pi\)
−0.677500 + 0.735522i \(0.736937\pi\)
\(110\) 1049.23 0.909457
\(111\) 1409.86 1.20557
\(112\) 1496.82 1.26283
\(113\) 1142.64 0.951248 0.475624 0.879649i \(-0.342222\pi\)
0.475624 + 0.879649i \(0.342222\pi\)
\(114\) 270.685 0.222385
\(115\) −3155.85 −2.55900
\(116\) −2619.77 −2.09689
\(117\) −60.8417 −0.0480754
\(118\) 144.964 0.113093
\(119\) −2543.29 −1.95919
\(120\) −3144.82 −2.39235
\(121\) −1151.97 −0.865494
\(122\) 1044.30 0.774974
\(123\) −1327.72 −0.973304
\(124\) 1903.47 1.37852
\(125\) 178.470 0.127703
\(126\) −158.460 −0.112038
\(127\) −2422.91 −1.69290 −0.846452 0.532464i \(-0.821266\pi\)
−0.846452 + 0.532464i \(0.821266\pi\)
\(128\) 2600.30 1.79560
\(129\) 0 0
\(130\) 4084.90 2.75592
\(131\) 1794.68 1.19696 0.598480 0.801137i \(-0.295771\pi\)
0.598480 + 0.801137i \(0.295771\pi\)
\(132\) −1104.70 −0.728420
\(133\) −293.743 −0.191509
\(134\) −78.7834 −0.0507899
\(135\) −2215.11 −1.41219
\(136\) 3336.74 2.10385
\(137\) −2170.60 −1.35362 −0.676812 0.736156i \(-0.736639\pi\)
−0.676812 + 0.736156i \(0.736639\pi\)
\(138\) 5031.40 3.10363
\(139\) −245.580 −0.149855 −0.0749275 0.997189i \(-0.523873\pi\)
−0.0749275 + 0.997189i \(0.523873\pi\)
\(140\) 7025.84 4.24137
\(141\) 954.261 0.569952
\(142\) −3914.08 −2.31312
\(143\) 696.994 0.407591
\(144\) 62.5415 0.0361930
\(145\) −2720.92 −1.55835
\(146\) 147.076 0.0833705
\(147\) 2326.70 1.30546
\(148\) 4132.41 2.29515
\(149\) −1142.62 −0.628237 −0.314119 0.949384i \(-0.601709\pi\)
−0.314119 + 0.949384i \(0.601709\pi\)
\(150\) −3504.43 −1.90757
\(151\) 1393.59 0.751049 0.375525 0.926812i \(-0.377463\pi\)
0.375525 + 0.926812i \(0.377463\pi\)
\(152\) 385.383 0.205649
\(153\) −106.266 −0.0561510
\(154\) 1815.29 0.949874
\(155\) 1976.96 1.02447
\(156\) −4300.83 −2.20732
\(157\) −147.109 −0.0747809 −0.0373904 0.999301i \(-0.511905\pi\)
−0.0373904 + 0.999301i \(0.511905\pi\)
\(158\) 1839.73 0.926335
\(159\) −337.766 −0.168469
\(160\) 541.309 0.267464
\(161\) −5460.00 −2.67272
\(162\) 3684.62 1.78698
\(163\) −161.544 −0.0776262 −0.0388131 0.999246i \(-0.512358\pi\)
−0.0388131 + 0.999246i \(0.512358\pi\)
\(164\) −3891.65 −1.85297
\(165\) −1147.35 −0.541340
\(166\) 2946.68 1.37775
\(167\) 1901.23 0.880970 0.440485 0.897760i \(-0.354807\pi\)
0.440485 + 0.897760i \(0.354807\pi\)
\(168\) −5440.92 −2.49867
\(169\) 516.554 0.235118
\(170\) 7134.67 3.21885
\(171\) −12.2734 −0.00548871
\(172\) 0 0
\(173\) 67.1165 0.0294958 0.0147479 0.999891i \(-0.495305\pi\)
0.0147479 + 0.999891i \(0.495305\pi\)
\(174\) 4337.99 1.89001
\(175\) 3802.95 1.64272
\(176\) −716.467 −0.306851
\(177\) −158.520 −0.0673170
\(178\) 6378.36 2.68583
\(179\) 1615.61 0.674617 0.337308 0.941394i \(-0.390483\pi\)
0.337308 + 0.941394i \(0.390483\pi\)
\(180\) 293.559 0.121559
\(181\) 1922.60 0.789536 0.394768 0.918781i \(-0.370825\pi\)
0.394768 + 0.918781i \(0.370825\pi\)
\(182\) 7067.35 2.87839
\(183\) −1141.96 −0.461291
\(184\) 7163.38 2.87006
\(185\) 4291.98 1.70569
\(186\) −3151.89 −1.24251
\(187\) 1217.37 0.476058
\(188\) 2797.01 1.08507
\(189\) −3832.40 −1.47495
\(190\) 824.032 0.314640
\(191\) −3119.14 −1.18164 −0.590820 0.806803i \(-0.701196\pi\)
−0.590820 + 0.806803i \(0.701196\pi\)
\(192\) −3136.56 −1.17897
\(193\) −1853.76 −0.691381 −0.345691 0.938349i \(-0.612355\pi\)
−0.345691 + 0.938349i \(0.612355\pi\)
\(194\) −1115.68 −0.412894
\(195\) −4466.89 −1.64041
\(196\) 6819.73 2.48533
\(197\) 189.579 0.0685633 0.0342817 0.999412i \(-0.489086\pi\)
0.0342817 + 0.999412i \(0.489086\pi\)
\(198\) 75.8482 0.0272237
\(199\) −57.5715 −0.0205082 −0.0102541 0.999947i \(-0.503264\pi\)
−0.0102541 + 0.999947i \(0.503264\pi\)
\(200\) −4989.37 −1.76401
\(201\) 86.1508 0.0302319
\(202\) −4917.57 −1.71287
\(203\) −4707.52 −1.62760
\(204\) −7511.82 −2.57810
\(205\) −4041.91 −1.37707
\(206\) −3225.80 −1.09103
\(207\) −228.134 −0.0766011
\(208\) −2789.37 −0.929845
\(209\) 140.602 0.0465342
\(210\) −11633.9 −3.82292
\(211\) −719.563 −0.234772 −0.117386 0.993086i \(-0.537451\pi\)
−0.117386 + 0.993086i \(0.537451\pi\)
\(212\) −990.019 −0.320730
\(213\) 4280.11 1.37685
\(214\) 3810.96 1.21735
\(215\) 0 0
\(216\) 5028.01 1.58386
\(217\) 3420.38 1.07000
\(218\) 7483.98 2.32513
\(219\) −160.830 −0.0496250
\(220\) −3362.97 −1.03060
\(221\) 4739.49 1.44259
\(222\) −6842.73 −2.06871
\(223\) −32.0202 −0.00961540 −0.00480770 0.999988i \(-0.501530\pi\)
−0.00480770 + 0.999988i \(0.501530\pi\)
\(224\) 936.529 0.279350
\(225\) 158.898 0.0470809
\(226\) −5545.80 −1.63231
\(227\) −4950.83 −1.44757 −0.723785 0.690026i \(-0.757599\pi\)
−0.723785 + 0.690026i \(0.757599\pi\)
\(228\) −867.592 −0.252007
\(229\) 1165.53 0.336333 0.168166 0.985759i \(-0.446215\pi\)
0.168166 + 0.985759i \(0.446215\pi\)
\(230\) 15316.9 4.39115
\(231\) −1985.05 −0.565397
\(232\) 6176.15 1.74778
\(233\) −1071.01 −0.301135 −0.150567 0.988600i \(-0.548110\pi\)
−0.150567 + 0.988600i \(0.548110\pi\)
\(234\) 295.294 0.0824956
\(235\) 2905.01 0.806392
\(236\) −464.635 −0.128158
\(237\) −2011.77 −0.551386
\(238\) 12343.8 3.36190
\(239\) −4274.37 −1.15685 −0.578423 0.815737i \(-0.696332\pi\)
−0.578423 + 0.815737i \(0.696332\pi\)
\(240\) 4591.69 1.23497
\(241\) 7024.10 1.87744 0.938718 0.344687i \(-0.112015\pi\)
0.938718 + 0.344687i \(0.112015\pi\)
\(242\) 5591.08 1.48516
\(243\) −327.497 −0.0864565
\(244\) −3347.18 −0.878202
\(245\) 7083.06 1.84702
\(246\) 6444.06 1.67015
\(247\) 547.396 0.141012
\(248\) −4487.45 −1.14901
\(249\) −3222.24 −0.820086
\(250\) −866.202 −0.219134
\(251\) 2122.77 0.533816 0.266908 0.963722i \(-0.413998\pi\)
0.266908 + 0.963722i \(0.413998\pi\)
\(252\) 507.892 0.126961
\(253\) 2613.47 0.649437
\(254\) 11759.6 2.90496
\(255\) −7801.87 −1.91597
\(256\) −7892.65 −1.92692
\(257\) −1456.60 −0.353543 −0.176771 0.984252i \(-0.556565\pi\)
−0.176771 + 0.984252i \(0.556565\pi\)
\(258\) 0 0
\(259\) 7425.63 1.78149
\(260\) −13092.8 −3.12301
\(261\) −196.693 −0.0466476
\(262\) −8710.44 −2.05394
\(263\) 4608.02 1.08039 0.540195 0.841540i \(-0.318350\pi\)
0.540195 + 0.841540i \(0.318350\pi\)
\(264\) 2604.34 0.607144
\(265\) −1028.25 −0.238357
\(266\) 1425.67 0.328623
\(267\) −6974.83 −1.59870
\(268\) 252.515 0.0575552
\(269\) −3794.78 −0.860118 −0.430059 0.902801i \(-0.641507\pi\)
−0.430059 + 0.902801i \(0.641507\pi\)
\(270\) 10751.0 2.42327
\(271\) 3683.71 0.825717 0.412859 0.910795i \(-0.364530\pi\)
0.412859 + 0.910795i \(0.364530\pi\)
\(272\) −4871.90 −1.08604
\(273\) −7728.25 −1.71332
\(274\) 10534.9 2.32277
\(275\) −1820.31 −0.399160
\(276\) −16126.5 −3.51704
\(277\) 4917.90 1.06674 0.533372 0.845881i \(-0.320925\pi\)
0.533372 + 0.845881i \(0.320925\pi\)
\(278\) 1191.92 0.257146
\(279\) 142.913 0.0306666
\(280\) −16563.5 −3.53522
\(281\) −212.734 −0.0451625 −0.0225813 0.999745i \(-0.507188\pi\)
−0.0225813 + 0.999745i \(0.507188\pi\)
\(282\) −4631.49 −0.978018
\(283\) −2776.00 −0.583096 −0.291548 0.956556i \(-0.594170\pi\)
−0.291548 + 0.956556i \(0.594170\pi\)
\(284\) 12545.3 2.62123
\(285\) −901.091 −0.187284
\(286\) −3382.84 −0.699412
\(287\) −6992.99 −1.43827
\(288\) 39.1308 0.00800627
\(289\) 3364.98 0.684914
\(290\) 13205.9 2.67407
\(291\) 1220.02 0.245769
\(292\) −471.405 −0.0944756
\(293\) −7115.75 −1.41879 −0.709397 0.704809i \(-0.751033\pi\)
−0.709397 + 0.704809i \(0.751033\pi\)
\(294\) −11292.6 −2.24012
\(295\) −482.576 −0.0952429
\(296\) −9742.24 −1.91303
\(297\) 1834.41 0.358394
\(298\) 5545.70 1.07803
\(299\) 10174.8 1.96798
\(300\) 11232.3 2.16166
\(301\) 0 0
\(302\) −6763.74 −1.28877
\(303\) 5377.44 1.01956
\(304\) −562.689 −0.106159
\(305\) −3476.42 −0.652653
\(306\) 515.760 0.0963531
\(307\) −3663.91 −0.681141 −0.340570 0.940219i \(-0.610620\pi\)
−0.340570 + 0.940219i \(0.610620\pi\)
\(308\) −5818.34 −1.07640
\(309\) 3527.46 0.649418
\(310\) −9595.14 −1.75796
\(311\) −7474.92 −1.36291 −0.681453 0.731862i \(-0.738651\pi\)
−0.681453 + 0.731862i \(0.738651\pi\)
\(312\) 10139.3 1.83982
\(313\) 6355.18 1.14765 0.573827 0.818976i \(-0.305458\pi\)
0.573827 + 0.818976i \(0.305458\pi\)
\(314\) 713.992 0.128321
\(315\) 527.503 0.0943538
\(316\) −5896.65 −1.04972
\(317\) 2434.32 0.431309 0.215654 0.976470i \(-0.430812\pi\)
0.215654 + 0.976470i \(0.430812\pi\)
\(318\) 1639.34 0.289087
\(319\) 2253.29 0.395486
\(320\) −9548.49 −1.66805
\(321\) −4167.34 −0.724606
\(322\) 26500.0 4.58629
\(323\) 956.081 0.164699
\(324\) −11809.9 −2.02501
\(325\) −7086.88 −1.20957
\(326\) 784.049 0.133204
\(327\) −8183.84 −1.38400
\(328\) 9174.63 1.54446
\(329\) 5026.01 0.842228
\(330\) 5568.64 0.928920
\(331\) 2053.35 0.340973 0.170487 0.985360i \(-0.445466\pi\)
0.170487 + 0.985360i \(0.445466\pi\)
\(332\) −9444.65 −1.56127
\(333\) 310.264 0.0510581
\(334\) −9227.60 −1.51171
\(335\) 262.265 0.0427733
\(336\) 7944.17 1.28985
\(337\) −7324.43 −1.18394 −0.591969 0.805961i \(-0.701649\pi\)
−0.591969 + 0.805961i \(0.701649\pi\)
\(338\) −2507.08 −0.403454
\(339\) 6064.42 0.971604
\(340\) −22867.9 −3.64760
\(341\) −1637.19 −0.259997
\(342\) 59.5687 0.00941843
\(343\) 2666.52 0.419762
\(344\) 0 0
\(345\) −16749.2 −2.61376
\(346\) −325.749 −0.0506137
\(347\) 210.302 0.0325348 0.0162674 0.999868i \(-0.494822\pi\)
0.0162674 + 0.999868i \(0.494822\pi\)
\(348\) −13904.0 −2.14176
\(349\) −7982.13 −1.22428 −0.612140 0.790750i \(-0.709691\pi\)
−0.612140 + 0.790750i \(0.709691\pi\)
\(350\) −18457.5 −2.81885
\(351\) 7141.76 1.08604
\(352\) −448.277 −0.0678786
\(353\) −9136.44 −1.37757 −0.688787 0.724964i \(-0.741856\pi\)
−0.688787 + 0.724964i \(0.741856\pi\)
\(354\) 769.375 0.115514
\(355\) 13029.7 1.94802
\(356\) −20443.8 −3.04359
\(357\) −13498.2 −2.00112
\(358\) −7841.33 −1.15762
\(359\) −6539.79 −0.961440 −0.480720 0.876874i \(-0.659625\pi\)
−0.480720 + 0.876874i \(0.659625\pi\)
\(360\) −692.071 −0.101320
\(361\) −6748.58 −0.983901
\(362\) −9331.32 −1.35482
\(363\) −6113.92 −0.884016
\(364\) −22652.1 −3.26180
\(365\) −489.607 −0.0702115
\(366\) 5542.49 0.791559
\(367\) 7503.05 1.06718 0.533592 0.845742i \(-0.320842\pi\)
0.533592 + 0.845742i \(0.320842\pi\)
\(368\) −10459.1 −1.48157
\(369\) −292.187 −0.0412212
\(370\) −20831.0 −2.92690
\(371\) −1778.99 −0.248950
\(372\) 10102.4 1.40802
\(373\) −233.234 −0.0323764 −0.0161882 0.999869i \(-0.505153\pi\)
−0.0161882 + 0.999869i \(0.505153\pi\)
\(374\) −5908.47 −0.816898
\(375\) 947.205 0.130436
\(376\) −6594.01 −0.904415
\(377\) 8772.57 1.19844
\(378\) 18600.5 2.53097
\(379\) 7237.58 0.980922 0.490461 0.871463i \(-0.336828\pi\)
0.490461 + 0.871463i \(0.336828\pi\)
\(380\) −2641.17 −0.356550
\(381\) −12859.3 −1.72913
\(382\) 15138.7 2.02765
\(383\) 7769.85 1.03661 0.518304 0.855197i \(-0.326564\pi\)
0.518304 + 0.855197i \(0.326564\pi\)
\(384\) 13800.7 1.83402
\(385\) −6043.00 −0.799948
\(386\) 8997.19 1.18639
\(387\) 0 0
\(388\) 3575.97 0.467892
\(389\) −8231.01 −1.07282 −0.536412 0.843956i \(-0.680221\pi\)
−0.536412 + 0.843956i \(0.680221\pi\)
\(390\) 21680.0 2.81489
\(391\) 17771.4 2.29856
\(392\) −16077.6 −2.07154
\(393\) 9524.99 1.22258
\(394\) −920.120 −0.117652
\(395\) −6124.34 −0.780124
\(396\) −243.107 −0.0308499
\(397\) −8274.41 −1.04605 −0.523023 0.852318i \(-0.675196\pi\)
−0.523023 + 0.852318i \(0.675196\pi\)
\(398\) 279.422 0.0351914
\(399\) −1558.99 −0.195607
\(400\) 7284.88 0.910610
\(401\) −13688.9 −1.70472 −0.852360 0.522956i \(-0.824829\pi\)
−0.852360 + 0.522956i \(0.824829\pi\)
\(402\) −418.131 −0.0518768
\(403\) −6373.95 −0.787864
\(404\) 15761.7 1.94102
\(405\) −12265.9 −1.50493
\(406\) 22847.8 2.79291
\(407\) −3554.33 −0.432879
\(408\) 17709.2 2.14887
\(409\) 13141.7 1.58879 0.794397 0.607398i \(-0.207787\pi\)
0.794397 + 0.607398i \(0.207787\pi\)
\(410\) 19617.3 2.36300
\(411\) −11520.1 −1.38259
\(412\) 10339.3 1.23636
\(413\) −834.914 −0.0994756
\(414\) 1107.24 0.131445
\(415\) −9809.32 −1.16029
\(416\) −1745.24 −0.205691
\(417\) −1303.38 −0.153062
\(418\) −682.410 −0.0798511
\(419\) 7318.77 0.853330 0.426665 0.904410i \(-0.359688\pi\)
0.426665 + 0.904410i \(0.359688\pi\)
\(420\) 37288.6 4.33214
\(421\) −15041.7 −1.74130 −0.870651 0.491901i \(-0.836302\pi\)
−0.870651 + 0.491901i \(0.836302\pi\)
\(422\) 3492.39 0.402859
\(423\) 210.001 0.0241385
\(424\) 2333.99 0.267331
\(425\) −12377.9 −1.41275
\(426\) −20773.4 −2.36262
\(427\) −6014.62 −0.681658
\(428\) −12214.8 −1.37950
\(429\) 3699.19 0.416314
\(430\) 0 0
\(431\) 4332.46 0.484193 0.242096 0.970252i \(-0.422165\pi\)
0.242096 + 0.970252i \(0.422165\pi\)
\(432\) −7341.29 −0.817612
\(433\) −1688.95 −0.187450 −0.0937249 0.995598i \(-0.529877\pi\)
−0.0937249 + 0.995598i \(0.529877\pi\)
\(434\) −16600.7 −1.83608
\(435\) −14440.9 −1.59170
\(436\) −23987.5 −2.63484
\(437\) 2052.54 0.224682
\(438\) 780.584 0.0851547
\(439\) 14355.9 1.56075 0.780373 0.625314i \(-0.215029\pi\)
0.780373 + 0.625314i \(0.215029\pi\)
\(440\) 7928.27 0.859012
\(441\) 512.029 0.0552887
\(442\) −23003.0 −2.47543
\(443\) −1572.96 −0.168699 −0.0843493 0.996436i \(-0.526881\pi\)
−0.0843493 + 0.996436i \(0.526881\pi\)
\(444\) 21932.2 2.34427
\(445\) −21233.2 −2.26191
\(446\) 155.410 0.0164997
\(447\) −6064.30 −0.641682
\(448\) −16520.0 −1.74218
\(449\) −9925.53 −1.04324 −0.521620 0.853178i \(-0.674672\pi\)
−0.521620 + 0.853178i \(0.674672\pi\)
\(450\) −771.208 −0.0807891
\(451\) 3347.25 0.349481
\(452\) 17775.3 1.84973
\(453\) 7396.25 0.767122
\(454\) 24028.8 2.48398
\(455\) −23526.8 −2.42407
\(456\) 2045.36 0.210050
\(457\) −18820.4 −1.92643 −0.963216 0.268729i \(-0.913397\pi\)
−0.963216 + 0.268729i \(0.913397\pi\)
\(458\) −5656.87 −0.577135
\(459\) 12473.8 1.26847
\(460\) −49093.3 −4.97606
\(461\) 2178.77 0.220120 0.110060 0.993925i \(-0.464896\pi\)
0.110060 + 0.993925i \(0.464896\pi\)
\(462\) 9634.41 0.970202
\(463\) 2774.25 0.278467 0.139234 0.990260i \(-0.455536\pi\)
0.139234 + 0.990260i \(0.455536\pi\)
\(464\) −9017.66 −0.902229
\(465\) 10492.4 1.04640
\(466\) 5198.14 0.516737
\(467\) 4212.31 0.417393 0.208696 0.977980i \(-0.433078\pi\)
0.208696 + 0.977980i \(0.433078\pi\)
\(468\) −946.469 −0.0934841
\(469\) 453.749 0.0446742
\(470\) −14099.4 −1.38374
\(471\) −780.760 −0.0763812
\(472\) 1095.39 0.106820
\(473\) 0 0
\(474\) 9764.08 0.946158
\(475\) −1429.61 −0.138095
\(476\) −39564.2 −3.80971
\(477\) −74.3311 −0.00713498
\(478\) 20745.6 1.98511
\(479\) 17700.2 1.68839 0.844197 0.536033i \(-0.180078\pi\)
0.844197 + 0.536033i \(0.180078\pi\)
\(480\) 2872.92 0.273188
\(481\) −13837.8 −1.31175
\(482\) −34091.3 −3.22161
\(483\) −28978.1 −2.72992
\(484\) −17920.4 −1.68298
\(485\) 3714.04 0.347724
\(486\) 1589.50 0.148356
\(487\) 15614.3 1.45288 0.726439 0.687231i \(-0.241174\pi\)
0.726439 + 0.687231i \(0.241174\pi\)
\(488\) 7891.03 0.731988
\(489\) −857.369 −0.0792874
\(490\) −34377.5 −3.16942
\(491\) −18884.0 −1.73569 −0.867843 0.496838i \(-0.834494\pi\)
−0.867843 + 0.496838i \(0.834494\pi\)
\(492\) −20654.3 −1.89262
\(493\) 15322.2 1.39975
\(494\) −2656.78 −0.241972
\(495\) −252.494 −0.0229268
\(496\) 6552.03 0.593135
\(497\) 22543.0 2.03459
\(498\) 15639.1 1.40724
\(499\) 6710.07 0.601971 0.300986 0.953629i \(-0.402684\pi\)
0.300986 + 0.953629i \(0.402684\pi\)
\(500\) 2776.33 0.248323
\(501\) 10090.5 0.899823
\(502\) −10302.8 −0.916010
\(503\) 8300.85 0.735819 0.367909 0.929862i \(-0.380074\pi\)
0.367909 + 0.929862i \(0.380074\pi\)
\(504\) −1197.37 −0.105823
\(505\) 16370.3 1.44251
\(506\) −12684.4 −1.11441
\(507\) 2741.53 0.240149
\(508\) −37691.5 −3.29191
\(509\) −2569.76 −0.223777 −0.111889 0.993721i \(-0.535690\pi\)
−0.111889 + 0.993721i \(0.535690\pi\)
\(510\) 37866.2 3.28773
\(511\) −847.077 −0.0733317
\(512\) 17504.4 1.51092
\(513\) 1440.68 0.123992
\(514\) 7069.60 0.606667
\(515\) 10738.5 0.918823
\(516\) 0 0
\(517\) −2405.74 −0.204651
\(518\) −36040.1 −3.05697
\(519\) 356.211 0.0301270
\(520\) 30866.5 2.60305
\(521\) −12083.3 −1.01609 −0.508043 0.861332i \(-0.669631\pi\)
−0.508043 + 0.861332i \(0.669631\pi\)
\(522\) 954.648 0.0800456
\(523\) 7343.65 0.613988 0.306994 0.951712i \(-0.400677\pi\)
0.306994 + 0.951712i \(0.400677\pi\)
\(524\) 27918.5 2.32753
\(525\) 20183.6 1.67787
\(526\) −22364.9 −1.85391
\(527\) −11132.7 −0.920208
\(528\) −3802.54 −0.313417
\(529\) 25984.9 2.13569
\(530\) 4990.57 0.409012
\(531\) −34.8851 −0.00285100
\(532\) −4569.54 −0.372396
\(533\) 13031.6 1.05903
\(534\) 33852.2 2.74331
\(535\) −12686.4 −1.02520
\(536\) −595.308 −0.0479727
\(537\) 8574.61 0.689054
\(538\) 18417.9 1.47593
\(539\) −5865.73 −0.468747
\(540\) −34458.8 −2.74606
\(541\) 9216.21 0.732414 0.366207 0.930533i \(-0.380656\pi\)
0.366207 + 0.930533i \(0.380656\pi\)
\(542\) −17878.8 −1.41690
\(543\) 10203.9 0.806432
\(544\) −3048.24 −0.240243
\(545\) −24913.7 −1.95814
\(546\) 37508.9 2.93999
\(547\) −4652.64 −0.363680 −0.181840 0.983328i \(-0.558205\pi\)
−0.181840 + 0.983328i \(0.558205\pi\)
\(548\) −33766.4 −2.63217
\(549\) −251.308 −0.0195365
\(550\) 8834.84 0.684943
\(551\) 1769.66 0.136824
\(552\) 38018.6 2.93148
\(553\) −10595.8 −0.814793
\(554\) −23868.9 −1.83049
\(555\) 22779.0 1.74219
\(556\) −3820.31 −0.291398
\(557\) 14312.2 1.08874 0.544369 0.838846i \(-0.316769\pi\)
0.544369 + 0.838846i \(0.316769\pi\)
\(558\) −693.626 −0.0526228
\(559\) 0 0
\(560\) 24184.1 1.82493
\(561\) 6461.00 0.486245
\(562\) 1032.50 0.0774973
\(563\) −3350.90 −0.250841 −0.125421 0.992104i \(-0.540028\pi\)
−0.125421 + 0.992104i \(0.540028\pi\)
\(564\) 14844.7 1.10829
\(565\) 18461.6 1.37467
\(566\) 13473.3 1.00057
\(567\) −21221.4 −1.57181
\(568\) −29575.8 −2.18481
\(569\) 12015.1 0.885238 0.442619 0.896710i \(-0.354050\pi\)
0.442619 + 0.896710i \(0.354050\pi\)
\(570\) 4373.43 0.321373
\(571\) 3985.38 0.292089 0.146045 0.989278i \(-0.453346\pi\)
0.146045 + 0.989278i \(0.453346\pi\)
\(572\) 10842.6 0.792574
\(573\) −16554.4 −1.20693
\(574\) 33940.3 2.46802
\(575\) −26573.2 −1.92727
\(576\) −690.253 −0.0499315
\(577\) 12302.3 0.887609 0.443804 0.896124i \(-0.353629\pi\)
0.443804 + 0.896124i \(0.353629\pi\)
\(578\) −16331.9 −1.17529
\(579\) −9838.56 −0.706177
\(580\) −42327.4 −3.03026
\(581\) −16971.3 −1.21186
\(582\) −5921.33 −0.421730
\(583\) 851.526 0.0604916
\(584\) 1111.34 0.0787462
\(585\) −983.015 −0.0694746
\(586\) 34536.2 2.43460
\(587\) −2130.30 −0.149791 −0.0748953 0.997191i \(-0.523862\pi\)
−0.0748953 + 0.997191i \(0.523862\pi\)
\(588\) 36194.7 2.53851
\(589\) −1285.80 −0.0899496
\(590\) 2342.17 0.163433
\(591\) 1006.16 0.0700306
\(592\) 14224.4 0.987535
\(593\) 15604.8 1.08063 0.540315 0.841463i \(-0.318305\pi\)
0.540315 + 0.841463i \(0.318305\pi\)
\(594\) −8903.26 −0.614992
\(595\) −41091.8 −2.83126
\(596\) −17774.9 −1.22163
\(597\) −305.552 −0.0209471
\(598\) −49383.4 −3.37698
\(599\) −1690.93 −0.115342 −0.0576709 0.998336i \(-0.518367\pi\)
−0.0576709 + 0.998336i \(0.518367\pi\)
\(600\) −26480.4 −1.80176
\(601\) 8333.53 0.565610 0.282805 0.959177i \(-0.408735\pi\)
0.282805 + 0.959177i \(0.408735\pi\)
\(602\) 0 0
\(603\) 18.9589 0.00128038
\(604\) 21679.0 1.46044
\(605\) −18612.3 −1.25074
\(606\) −26099.3 −1.74952
\(607\) 17800.5 1.19028 0.595141 0.803621i \(-0.297096\pi\)
0.595141 + 0.803621i \(0.297096\pi\)
\(608\) −352.062 −0.0234836
\(609\) −24984.5 −1.66243
\(610\) 16872.7 1.11993
\(611\) −9366.10 −0.620150
\(612\) −1653.10 −0.109187
\(613\) −13015.8 −0.857594 −0.428797 0.903401i \(-0.641062\pi\)
−0.428797 + 0.903401i \(0.641062\pi\)
\(614\) 17782.7 1.16881
\(615\) −21451.8 −1.40654
\(616\) 13716.8 0.897187
\(617\) −14286.8 −0.932198 −0.466099 0.884733i \(-0.654341\pi\)
−0.466099 + 0.884733i \(0.654341\pi\)
\(618\) −17120.4 −1.11438
\(619\) −20191.3 −1.31108 −0.655538 0.755162i \(-0.727558\pi\)
−0.655538 + 0.755162i \(0.727558\pi\)
\(620\) 30754.1 1.99212
\(621\) 26779.0 1.73044
\(622\) 36279.3 2.33870
\(623\) −36735.9 −2.36243
\(624\) −14804.1 −0.949744
\(625\) −14122.2 −0.903823
\(626\) −30844.7 −1.96933
\(627\) 746.225 0.0475301
\(628\) −2288.47 −0.145414
\(629\) −24169.1 −1.53209
\(630\) −2560.23 −0.161908
\(631\) 16981.8 1.07137 0.535685 0.844418i \(-0.320053\pi\)
0.535685 + 0.844418i \(0.320053\pi\)
\(632\) 13901.5 0.874953
\(633\) −3818.98 −0.239796
\(634\) −11814.9 −0.740111
\(635\) −39146.8 −2.44645
\(636\) −5254.38 −0.327594
\(637\) −22836.6 −1.42044
\(638\) −10936.3 −0.678640
\(639\) 941.909 0.0583120
\(640\) 42012.9 2.59485
\(641\) 17639.8 1.08694 0.543471 0.839428i \(-0.317110\pi\)
0.543471 + 0.839428i \(0.317110\pi\)
\(642\) 20226.1 1.24340
\(643\) 23124.0 1.41823 0.709115 0.705093i \(-0.249095\pi\)
0.709115 + 0.705093i \(0.249095\pi\)
\(644\) −84937.2 −5.19719
\(645\) 0 0
\(646\) −4640.32 −0.282618
\(647\) −10314.1 −0.626721 −0.313360 0.949634i \(-0.601455\pi\)
−0.313360 + 0.949634i \(0.601455\pi\)
\(648\) 27842.0 1.68786
\(649\) 399.638 0.0241713
\(650\) 34396.0 2.07557
\(651\) 18153.1 1.09290
\(652\) −2513.01 −0.150947
\(653\) 23231.2 1.39220 0.696099 0.717946i \(-0.254917\pi\)
0.696099 + 0.717946i \(0.254917\pi\)
\(654\) 39720.1 2.37489
\(655\) 28996.5 1.72975
\(656\) −13395.7 −0.797276
\(657\) −35.3933 −0.00210171
\(658\) −24393.7 −1.44523
\(659\) 19494.8 1.15236 0.576182 0.817321i \(-0.304542\pi\)
0.576182 + 0.817321i \(0.304542\pi\)
\(660\) −17848.5 −1.05265
\(661\) 31934.6 1.87914 0.939571 0.342354i \(-0.111224\pi\)
0.939571 + 0.342354i \(0.111224\pi\)
\(662\) −9965.88 −0.585098
\(663\) 25154.1 1.47346
\(664\) 22265.9 1.30133
\(665\) −4745.97 −0.276753
\(666\) −1505.86 −0.0876138
\(667\) 32893.9 1.90953
\(668\) 29576.1 1.71307
\(669\) −169.943 −0.00982117
\(670\) −1272.90 −0.0733974
\(671\) 2878.95 0.165634
\(672\) 4970.49 0.285329
\(673\) −31725.0 −1.81710 −0.908550 0.417776i \(-0.862810\pi\)
−0.908550 + 0.417776i \(0.862810\pi\)
\(674\) 35549.0 2.03159
\(675\) −18651.9 −1.06357
\(676\) 8035.64 0.457194
\(677\) 21791.4 1.23709 0.618545 0.785749i \(-0.287722\pi\)
0.618545 + 0.785749i \(0.287722\pi\)
\(678\) −29433.5 −1.66724
\(679\) 6425.73 0.363177
\(680\) 53911.4 3.04031
\(681\) −26275.8 −1.47855
\(682\) 7946.07 0.446145
\(683\) 5819.39 0.326022 0.163011 0.986624i \(-0.447879\pi\)
0.163011 + 0.986624i \(0.447879\pi\)
\(684\) −190.928 −0.0106730
\(685\) −35070.1 −1.95615
\(686\) −12941.9 −0.720297
\(687\) 6185.87 0.343531
\(688\) 0 0
\(689\) 3315.18 0.183307
\(690\) 81292.0 4.48512
\(691\) 28673.5 1.57857 0.789285 0.614027i \(-0.210452\pi\)
0.789285 + 0.614027i \(0.210452\pi\)
\(692\) 1044.08 0.0573556
\(693\) −436.844 −0.0239456
\(694\) −1020.69 −0.0558286
\(695\) −3967.82 −0.216558
\(696\) 32779.0 1.78518
\(697\) 22761.0 1.23692
\(698\) 38741.1 2.10082
\(699\) −5684.24 −0.307579
\(700\) 59159.6 3.19432
\(701\) 14960.3 0.806054 0.403027 0.915188i \(-0.367958\pi\)
0.403027 + 0.915188i \(0.367958\pi\)
\(702\) −34662.4 −1.86360
\(703\) −2791.46 −0.149761
\(704\) 7907.44 0.423328
\(705\) 15417.9 0.823649
\(706\) 44343.5 2.36387
\(707\) 28322.5 1.50662
\(708\) −2465.98 −0.130900
\(709\) 3891.22 0.206118 0.103059 0.994675i \(-0.467137\pi\)
0.103059 + 0.994675i \(0.467137\pi\)
\(710\) −63239.5 −3.34273
\(711\) −442.724 −0.0233522
\(712\) 48196.6 2.53686
\(713\) −23900.0 −1.25535
\(714\) 65513.0 3.43384
\(715\) 11261.3 0.589018
\(716\) 25132.9 1.31181
\(717\) −22685.6 −1.18160
\(718\) 31740.8 1.64980
\(719\) 32574.3 1.68959 0.844796 0.535088i \(-0.179722\pi\)
0.844796 + 0.535088i \(0.179722\pi\)
\(720\) 1010.48 0.0523032
\(721\) 18578.8 0.959656
\(722\) 32754.1 1.68834
\(723\) 37279.3 1.91761
\(724\) 29908.5 1.53528
\(725\) −22911.0 −1.17364
\(726\) 29673.8 1.51694
\(727\) 22352.4 1.14031 0.570154 0.821538i \(-0.306883\pi\)
0.570154 + 0.821538i \(0.306883\pi\)
\(728\) 53402.8 2.71873
\(729\) 18759.5 0.953080
\(730\) 2376.30 0.120480
\(731\) 0 0
\(732\) −17764.7 −0.896995
\(733\) 7894.97 0.397827 0.198914 0.980017i \(-0.436259\pi\)
0.198914 + 0.980017i \(0.436259\pi\)
\(734\) −36415.9 −1.83125
\(735\) 37592.3 1.88655
\(736\) −6544.03 −0.327739
\(737\) −217.191 −0.0108552
\(738\) 1418.12 0.0707342
\(739\) −19992.9 −0.995196 −0.497598 0.867408i \(-0.665784\pi\)
−0.497598 + 0.867408i \(0.665784\pi\)
\(740\) 66767.1 3.31677
\(741\) 2905.22 0.144030
\(742\) 8634.27 0.427189
\(743\) −29649.9 −1.46400 −0.731999 0.681305i \(-0.761413\pi\)
−0.731999 + 0.681305i \(0.761413\pi\)
\(744\) −23816.5 −1.17359
\(745\) −18461.3 −0.907878
\(746\) 1132.00 0.0555567
\(747\) −709.109 −0.0347322
\(748\) 18937.7 0.925709
\(749\) −21949.1 −1.07076
\(750\) −4597.24 −0.223823
\(751\) 16153.5 0.784887 0.392443 0.919776i \(-0.371630\pi\)
0.392443 + 0.919776i \(0.371630\pi\)
\(752\) 9627.77 0.466873
\(753\) 11266.3 0.545240
\(754\) −42577.5 −2.05647
\(755\) 22516.1 1.08536
\(756\) −59617.8 −2.86809
\(757\) −38932.2 −1.86924 −0.934620 0.355648i \(-0.884260\pi\)
−0.934620 + 0.355648i \(0.884260\pi\)
\(758\) −35127.4 −1.68323
\(759\) 13870.6 0.663335
\(760\) 6226.60 0.297188
\(761\) 40667.5 1.93718 0.968592 0.248656i \(-0.0799889\pi\)
0.968592 + 0.248656i \(0.0799889\pi\)
\(762\) 62412.1 2.96713
\(763\) −43103.6 −2.04516
\(764\) −48522.2 −2.29774
\(765\) −1716.93 −0.0811448
\(766\) −37710.8 −1.77878
\(767\) 1555.88 0.0732459
\(768\) −41889.1 −1.96815
\(769\) 971.746 0.0455684 0.0227842 0.999740i \(-0.492747\pi\)
0.0227842 + 0.999740i \(0.492747\pi\)
\(770\) 29329.6 1.37268
\(771\) −7730.71 −0.361109
\(772\) −28837.6 −1.34441
\(773\) 15680.2 0.729596 0.364798 0.931087i \(-0.381138\pi\)
0.364798 + 0.931087i \(0.381138\pi\)
\(774\) 0 0
\(775\) 16646.6 0.771566
\(776\) −8430.40 −0.389992
\(777\) 39410.4 1.81961
\(778\) 39949.0 1.84093
\(779\) 2628.82 0.120908
\(780\) −69488.2 −3.18984
\(781\) −10790.4 −0.494379
\(782\) −86252.9 −3.94424
\(783\) 23088.4 1.05378
\(784\) 23474.6 1.06936
\(785\) −2376.83 −0.108067
\(786\) −46229.4 −2.09790
\(787\) −9789.83 −0.443418 −0.221709 0.975113i \(-0.571163\pi\)
−0.221709 + 0.975113i \(0.571163\pi\)
\(788\) 2949.15 0.133324
\(789\) 24456.4 1.10351
\(790\) 29724.3 1.33866
\(791\) 31940.8 1.43576
\(792\) 573.128 0.0257137
\(793\) 11208.4 0.501919
\(794\) 40159.7 1.79498
\(795\) −5457.26 −0.243458
\(796\) −895.597 −0.0398789
\(797\) −38193.6 −1.69747 −0.848737 0.528815i \(-0.822636\pi\)
−0.848737 + 0.528815i \(0.822636\pi\)
\(798\) 7566.55 0.335655
\(799\) −16358.8 −0.724322
\(800\) 4557.99 0.201436
\(801\) −1534.93 −0.0677079
\(802\) 66438.9 2.92524
\(803\) 405.460 0.0178187
\(804\) 1340.18 0.0587869
\(805\) −88216.8 −3.86240
\(806\) 30935.9 1.35195
\(807\) −20140.2 −0.878524
\(808\) −37158.5 −1.61786
\(809\) −25984.8 −1.12927 −0.564634 0.825342i \(-0.690983\pi\)
−0.564634 + 0.825342i \(0.690983\pi\)
\(810\) 59532.1 2.58240
\(811\) −12058.7 −0.522120 −0.261060 0.965323i \(-0.584072\pi\)
−0.261060 + 0.965323i \(0.584072\pi\)
\(812\) −73231.4 −3.16492
\(813\) 19550.7 0.843388
\(814\) 17250.9 0.742805
\(815\) −2610.05 −0.112179
\(816\) −25856.9 −1.10928
\(817\) 0 0
\(818\) −63783.1 −2.72631
\(819\) −1700.73 −0.0725621
\(820\) −62877.0 −2.67776
\(821\) 10328.1 0.439040 0.219520 0.975608i \(-0.429551\pi\)
0.219520 + 0.975608i \(0.429551\pi\)
\(822\) 55912.6 2.37248
\(823\) 1549.44 0.0656259 0.0328130 0.999462i \(-0.489553\pi\)
0.0328130 + 0.999462i \(0.489553\pi\)
\(824\) −24375.0 −1.03051
\(825\) −9661.03 −0.407702
\(826\) 4052.24 0.170697
\(827\) −34169.0 −1.43673 −0.718364 0.695668i \(-0.755109\pi\)
−0.718364 + 0.695668i \(0.755109\pi\)
\(828\) −3548.91 −0.148953
\(829\) 41058.6 1.72017 0.860086 0.510148i \(-0.170410\pi\)
0.860086 + 0.510148i \(0.170410\pi\)
\(830\) 47609.4 1.99102
\(831\) 26101.0 1.08957
\(832\) 30785.4 1.28280
\(833\) −39886.4 −1.65904
\(834\) 6325.93 0.262649
\(835\) 30718.1 1.27311
\(836\) 2187.24 0.0904874
\(837\) −16775.5 −0.692768
\(838\) −35521.5 −1.46428
\(839\) −14142.7 −0.581953 −0.290977 0.956730i \(-0.593980\pi\)
−0.290977 + 0.956730i \(0.593980\pi\)
\(840\) −87908.5 −3.61087
\(841\) 3971.60 0.162844
\(842\) 73004.6 2.98801
\(843\) −1129.06 −0.0461290
\(844\) −11193.7 −0.456521
\(845\) 8345.92 0.339773
\(846\) −1019.24 −0.0414209
\(847\) −32201.5 −1.30633
\(848\) −3407.80 −0.138001
\(849\) −14733.2 −0.595574
\(850\) 60076.1 2.42423
\(851\) −51886.8 −2.09008
\(852\) 66582.4 2.67732
\(853\) 37292.5 1.49692 0.748459 0.663181i \(-0.230794\pi\)
0.748459 + 0.663181i \(0.230794\pi\)
\(854\) 29191.8 1.16970
\(855\) −198.300 −0.00793184
\(856\) 28796.6 1.14982
\(857\) −5517.53 −0.219924 −0.109962 0.993936i \(-0.535073\pi\)
−0.109962 + 0.993936i \(0.535073\pi\)
\(858\) −17953.9 −0.714379
\(859\) −35180.3 −1.39737 −0.698683 0.715431i \(-0.746230\pi\)
−0.698683 + 0.715431i \(0.746230\pi\)
\(860\) 0 0
\(861\) −37114.2 −1.46905
\(862\) −21027.5 −0.830857
\(863\) −957.480 −0.0377671 −0.0188835 0.999822i \(-0.506011\pi\)
−0.0188835 + 0.999822i \(0.506011\pi\)
\(864\) −4593.28 −0.180864
\(865\) 1084.40 0.0426249
\(866\) 8197.29 0.321657
\(867\) 17859.1 0.699571
\(868\) 53208.3 2.08065
\(869\) 5071.78 0.197984
\(870\) 70088.6 2.73129
\(871\) −845.572 −0.0328945
\(872\) 56550.9 2.19616
\(873\) 268.485 0.0104088
\(874\) −9961.94 −0.385546
\(875\) 4988.85 0.192747
\(876\) −2501.91 −0.0964974
\(877\) 2707.29 0.104240 0.0521201 0.998641i \(-0.483402\pi\)
0.0521201 + 0.998641i \(0.483402\pi\)
\(878\) −69675.9 −2.67818
\(879\) −37765.8 −1.44916
\(880\) −11575.9 −0.443436
\(881\) 10685.3 0.408621 0.204311 0.978906i \(-0.434505\pi\)
0.204311 + 0.978906i \(0.434505\pi\)
\(882\) −2485.12 −0.0948735
\(883\) 7131.73 0.271803 0.135901 0.990722i \(-0.456607\pi\)
0.135901 + 0.990722i \(0.456607\pi\)
\(884\) 73728.7 2.80516
\(885\) −2561.20 −0.0972811
\(886\) 7634.31 0.289481
\(887\) −8191.54 −0.310085 −0.155042 0.987908i \(-0.549551\pi\)
−0.155042 + 0.987908i \(0.549551\pi\)
\(888\) −51705.5 −1.95397
\(889\) −67728.7 −2.55517
\(890\) 103055. 3.88135
\(891\) 10157.8 0.381929
\(892\) −498.115 −0.0186975
\(893\) −1889.39 −0.0708019
\(894\) 29433.0 1.10110
\(895\) 26103.3 0.974901
\(896\) 72687.3 2.71017
\(897\) 54001.4 2.01009
\(898\) 48173.3 1.79016
\(899\) −20606.2 −0.764466
\(900\) 2471.86 0.0915503
\(901\) 5790.29 0.214098
\(902\) −16245.8 −0.599696
\(903\) 0 0
\(904\) −41905.5 −1.54177
\(905\) 31063.4 1.14097
\(906\) −35897.6 −1.31635
\(907\) −33159.0 −1.21392 −0.606960 0.794732i \(-0.707611\pi\)
−0.606960 + 0.794732i \(0.707611\pi\)
\(908\) −77016.5 −2.81485
\(909\) 1183.40 0.0431801
\(910\) 114187. 4.15962
\(911\) −22979.0 −0.835705 −0.417852 0.908515i \(-0.637217\pi\)
−0.417852 + 0.908515i \(0.637217\pi\)
\(912\) −2986.39 −0.108431
\(913\) 8123.44 0.294465
\(914\) 91344.2 3.30569
\(915\) −18450.6 −0.666620
\(916\) 18131.3 0.654010
\(917\) 50167.4 1.80662
\(918\) −60541.3 −2.17665
\(919\) −28118.8 −1.00931 −0.504655 0.863321i \(-0.668380\pi\)
−0.504655 + 0.863321i \(0.668380\pi\)
\(920\) 115738. 4.14758
\(921\) −19445.6 −0.695717
\(922\) −10574.6 −0.377719
\(923\) −42009.3 −1.49811
\(924\) −30880.0 −1.09943
\(925\) 36139.7 1.28461
\(926\) −13464.8 −0.477840
\(927\) 776.276 0.0275041
\(928\) −5642.15 −0.199583
\(929\) 34555.2 1.22037 0.610183 0.792261i \(-0.291096\pi\)
0.610183 + 0.792261i \(0.291096\pi\)
\(930\) −50924.8 −1.79558
\(931\) −4606.75 −0.162170
\(932\) −16661.0 −0.585567
\(933\) −39672.0 −1.39207
\(934\) −20444.4 −0.716231
\(935\) 19668.9 0.687960
\(936\) 2231.32 0.0779198
\(937\) −24391.7 −0.850420 −0.425210 0.905095i \(-0.639800\pi\)
−0.425210 + 0.905095i \(0.639800\pi\)
\(938\) −2202.26 −0.0766593
\(939\) 33729.2 1.17221
\(940\) 45191.1 1.56806
\(941\) −17916.6 −0.620684 −0.310342 0.950625i \(-0.600444\pi\)
−0.310342 + 0.950625i \(0.600444\pi\)
\(942\) 3789.40 0.131067
\(943\) 48863.7 1.68740
\(944\) −1599.35 −0.0551424
\(945\) −61919.8 −2.13148
\(946\) 0 0
\(947\) 21343.8 0.732398 0.366199 0.930537i \(-0.380659\pi\)
0.366199 + 0.930537i \(0.380659\pi\)
\(948\) −31295.6 −1.07219
\(949\) 1578.55 0.0539956
\(950\) 6938.60 0.236966
\(951\) 12919.8 0.440539
\(952\) 93273.2 3.17542
\(953\) −37998.5 −1.29160 −0.645799 0.763508i \(-0.723476\pi\)
−0.645799 + 0.763508i \(0.723476\pi\)
\(954\) 360.764 0.0122434
\(955\) −50395.7 −1.70761
\(956\) −66493.3 −2.24953
\(957\) 11959.0 0.403950
\(958\) −85907.3 −2.89722
\(959\) −60675.5 −2.04308
\(960\) −50677.2 −1.70375
\(961\) −14819.0 −0.497432
\(962\) 67161.6 2.25091
\(963\) −917.094 −0.0306884
\(964\) 109269. 3.65073
\(965\) −29951.1 −0.999128
\(966\) 140645. 4.68444
\(967\) 30226.2 1.00518 0.502590 0.864525i \(-0.332381\pi\)
0.502590 + 0.864525i \(0.332381\pi\)
\(968\) 42247.6 1.40278
\(969\) 5074.26 0.168224
\(970\) −18026.0 −0.596681
\(971\) 18868.7 0.623611 0.311805 0.950146i \(-0.399066\pi\)
0.311805 + 0.950146i \(0.399066\pi\)
\(972\) −5094.63 −0.168118
\(973\) −6864.80 −0.226182
\(974\) −75783.7 −2.49309
\(975\) −37612.6 −1.23545
\(976\) −11521.5 −0.377864
\(977\) 14033.2 0.459531 0.229766 0.973246i \(-0.426204\pi\)
0.229766 + 0.973246i \(0.426204\pi\)
\(978\) 4161.22 0.136054
\(979\) 17583.9 0.574039
\(980\) 110186. 3.59159
\(981\) −1800.99 −0.0586149
\(982\) 91653.0 2.97838
\(983\) −29136.6 −0.945386 −0.472693 0.881227i \(-0.656718\pi\)
−0.472693 + 0.881227i \(0.656718\pi\)
\(984\) 48693.0 1.57752
\(985\) 3063.02 0.0990821
\(986\) −74365.8 −2.40192
\(987\) 26674.8 0.860252
\(988\) 8515.44 0.274203
\(989\) 0 0
\(990\) 1225.47 0.0393415
\(991\) 14864.1 0.476463 0.238232 0.971208i \(-0.423432\pi\)
0.238232 + 0.971208i \(0.423432\pi\)
\(992\) 4099.46 0.131208
\(993\) 10897.8 0.348270
\(994\) −109412. −3.49128
\(995\) −930.178 −0.0296368
\(996\) −50126.1 −1.59468
\(997\) 47590.6 1.51174 0.755872 0.654720i \(-0.227213\pi\)
0.755872 + 0.654720i \(0.227213\pi\)
\(998\) −32567.2 −1.03296
\(999\) −36419.6 −1.15342
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.1.8 60
43.3 odd 42 43.4.g.a.9.9 120
43.29 odd 42 43.4.g.a.24.9 yes 120
43.42 odd 2 1849.4.a.l.1.53 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.g.a.9.9 120 43.3 odd 42
43.4.g.a.24.9 yes 120 43.29 odd 42
1849.4.a.k.1.8 60 1.1 even 1 trivial
1849.4.a.l.1.53 60 43.42 odd 2