Properties

Label 1849.4.a.g.1.2
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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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: \(30\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
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)
 
\(f(q)\) \(=\) \(q-5.07016 q^{2} +5.26544 q^{3} +17.7066 q^{4} +3.42027 q^{5} -26.6966 q^{6} -25.1802 q^{7} -49.2138 q^{8} +0.724844 q^{9} +O(q^{10})\) \(q-5.07016 q^{2} +5.26544 q^{3} +17.7066 q^{4} +3.42027 q^{5} -26.6966 q^{6} -25.1802 q^{7} -49.2138 q^{8} +0.724844 q^{9} -17.3413 q^{10} +69.6830 q^{11} +93.2328 q^{12} +26.4992 q^{13} +127.668 q^{14} +18.0092 q^{15} +107.870 q^{16} +87.9096 q^{17} -3.67508 q^{18} -15.6070 q^{19} +60.5611 q^{20} -132.585 q^{21} -353.304 q^{22} -0.430477 q^{23} -259.133 q^{24} -113.302 q^{25} -134.355 q^{26} -138.350 q^{27} -445.854 q^{28} +68.9739 q^{29} -91.3096 q^{30} -228.219 q^{31} -153.207 q^{32} +366.911 q^{33} -445.716 q^{34} -86.1229 q^{35} +12.8345 q^{36} -276.114 q^{37} +79.1300 q^{38} +139.530 q^{39} -168.324 q^{40} -379.952 q^{41} +672.226 q^{42} +1233.85 q^{44} +2.47916 q^{45} +2.18259 q^{46} -59.2069 q^{47} +567.982 q^{48} +291.041 q^{49} +574.459 q^{50} +462.883 q^{51} +469.209 q^{52} -263.990 q^{53} +701.458 q^{54} +238.334 q^{55} +1239.21 q^{56} -82.1776 q^{57} -349.709 q^{58} +697.350 q^{59} +318.881 q^{60} -782.784 q^{61} +1157.11 q^{62} -18.2517 q^{63} -86.1752 q^{64} +90.6342 q^{65} -1860.30 q^{66} +546.386 q^{67} +1556.58 q^{68} -2.26665 q^{69} +436.657 q^{70} +400.262 q^{71} -35.6724 q^{72} +79.9453 q^{73} +1399.94 q^{74} -596.584 q^{75} -276.346 q^{76} -1754.63 q^{77} -707.439 q^{78} -155.448 q^{79} +368.943 q^{80} -748.045 q^{81} +1926.42 q^{82} +92.2738 q^{83} -2347.62 q^{84} +300.674 q^{85} +363.178 q^{87} -3429.37 q^{88} -924.441 q^{89} -12.5697 q^{90} -667.254 q^{91} -7.62226 q^{92} -1201.67 q^{93} +300.189 q^{94} -53.3800 q^{95} -806.701 q^{96} +190.938 q^{97} -1475.62 q^{98} +50.5093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.07016 −1.79257 −0.896287 0.443475i \(-0.853746\pi\)
−0.896287 + 0.443475i \(0.853746\pi\)
\(3\) 5.26544 1.01333 0.506667 0.862142i \(-0.330877\pi\)
0.506667 + 0.862142i \(0.330877\pi\)
\(4\) 17.7066 2.21332
\(5\) 3.42027 0.305918 0.152959 0.988233i \(-0.451120\pi\)
0.152959 + 0.988233i \(0.451120\pi\)
\(6\) −26.6966 −1.81648
\(7\) −25.1802 −1.35960 −0.679801 0.733397i \(-0.737934\pi\)
−0.679801 + 0.733397i \(0.737934\pi\)
\(8\) −49.2138 −2.17497
\(9\) 0.724844 0.0268461
\(10\) −17.3413 −0.548380
\(11\) 69.6830 1.91002 0.955009 0.296576i \(-0.0958449\pi\)
0.955009 + 0.296576i \(0.0958449\pi\)
\(12\) 93.2328 2.24283
\(13\) 26.4992 0.565350 0.282675 0.959216i \(-0.408778\pi\)
0.282675 + 0.959216i \(0.408778\pi\)
\(14\) 127.668 2.43719
\(15\) 18.0092 0.309997
\(16\) 107.870 1.68547
\(17\) 87.9096 1.25419 0.627095 0.778943i \(-0.284244\pi\)
0.627095 + 0.778943i \(0.284244\pi\)
\(18\) −3.67508 −0.0481236
\(19\) −15.6070 −0.188447 −0.0942234 0.995551i \(-0.530037\pi\)
−0.0942234 + 0.995551i \(0.530037\pi\)
\(20\) 60.5611 0.677094
\(21\) −132.585 −1.37773
\(22\) −353.304 −3.42385
\(23\) −0.430477 −0.00390263 −0.00195132 0.999998i \(-0.500621\pi\)
−0.00195132 + 0.999998i \(0.500621\pi\)
\(24\) −259.133 −2.20397
\(25\) −113.302 −0.906414
\(26\) −134.355 −1.01343
\(27\) −138.350 −0.986130
\(28\) −445.854 −3.00923
\(29\) 68.9739 0.441660 0.220830 0.975312i \(-0.429123\pi\)
0.220830 + 0.975312i \(0.429123\pi\)
\(30\) −91.3096 −0.555692
\(31\) −228.219 −1.32223 −0.661117 0.750283i \(-0.729917\pi\)
−0.661117 + 0.750283i \(0.729917\pi\)
\(32\) −153.207 −0.846355
\(33\) 366.911 1.93549
\(34\) −445.716 −2.24823
\(35\) −86.1229 −0.415926
\(36\) 12.8345 0.0594189
\(37\) −276.114 −1.22683 −0.613417 0.789759i \(-0.710206\pi\)
−0.613417 + 0.789759i \(0.710206\pi\)
\(38\) 79.1300 0.337805
\(39\) 139.530 0.572888
\(40\) −168.324 −0.665361
\(41\) −379.952 −1.44728 −0.723641 0.690176i \(-0.757533\pi\)
−0.723641 + 0.690176i \(0.757533\pi\)
\(42\) 672.226 2.46968
\(43\) 0 0
\(44\) 1233.85 4.22748
\(45\) 2.47916 0.00821269
\(46\) 2.18259 0.00699576
\(47\) −59.2069 −0.183749 −0.0918747 0.995771i \(-0.529286\pi\)
−0.0918747 + 0.995771i \(0.529286\pi\)
\(48\) 567.982 1.70794
\(49\) 291.041 0.848516
\(50\) 574.459 1.62481
\(51\) 462.883 1.27091
\(52\) 469.209 1.25130
\(53\) −263.990 −0.684185 −0.342093 0.939666i \(-0.611136\pi\)
−0.342093 + 0.939666i \(0.611136\pi\)
\(54\) 701.458 1.76771
\(55\) 238.334 0.584309
\(56\) 1239.21 2.95709
\(57\) −82.1776 −0.190959
\(58\) −349.709 −0.791708
\(59\) 697.350 1.53877 0.769383 0.638788i \(-0.220564\pi\)
0.769383 + 0.638788i \(0.220564\pi\)
\(60\) 318.881 0.686123
\(61\) −782.784 −1.64304 −0.821518 0.570182i \(-0.806873\pi\)
−0.821518 + 0.570182i \(0.806873\pi\)
\(62\) 1157.11 2.37020
\(63\) −18.2517 −0.0365000
\(64\) −86.1752 −0.168311
\(65\) 90.6342 0.172951
\(66\) −1860.30 −3.46950
\(67\) 546.386 0.996294 0.498147 0.867093i \(-0.334014\pi\)
0.498147 + 0.867093i \(0.334014\pi\)
\(68\) 1556.58 2.77592
\(69\) −2.26665 −0.00395467
\(70\) 436.657 0.745579
\(71\) 400.262 0.669048 0.334524 0.942387i \(-0.391424\pi\)
0.334524 + 0.942387i \(0.391424\pi\)
\(72\) −35.6724 −0.0583893
\(73\) 79.9453 0.128177 0.0640883 0.997944i \(-0.479586\pi\)
0.0640883 + 0.997944i \(0.479586\pi\)
\(74\) 1399.94 2.19919
\(75\) −596.584 −0.918501
\(76\) −276.346 −0.417093
\(77\) −1754.63 −2.59686
\(78\) −707.439 −1.02694
\(79\) −155.448 −0.221383 −0.110692 0.993855i \(-0.535307\pi\)
−0.110692 + 0.993855i \(0.535307\pi\)
\(80\) 368.943 0.515614
\(81\) −748.045 −1.02613
\(82\) 1926.42 2.59436
\(83\) 92.2738 0.122028 0.0610142 0.998137i \(-0.480566\pi\)
0.0610142 + 0.998137i \(0.480566\pi\)
\(84\) −2347.62 −3.04936
\(85\) 300.674 0.383679
\(86\) 0 0
\(87\) 363.178 0.447549
\(88\) −3429.37 −4.15422
\(89\) −924.441 −1.10102 −0.550509 0.834830i \(-0.685566\pi\)
−0.550509 + 0.834830i \(0.685566\pi\)
\(90\) −12.5697 −0.0147219
\(91\) −667.254 −0.768650
\(92\) −7.62226 −0.00863778
\(93\) −1201.67 −1.33987
\(94\) 300.189 0.329384
\(95\) −53.3800 −0.0576492
\(96\) −806.701 −0.857641
\(97\) 190.938 0.199864 0.0999321 0.994994i \(-0.468137\pi\)
0.0999321 + 0.994994i \(0.468137\pi\)
\(98\) −1475.62 −1.52103
\(99\) 50.5093 0.0512765
\(100\) −2006.18 −2.00618
\(101\) −19.3893 −0.0191021 −0.00955105 0.999954i \(-0.503040\pi\)
−0.00955105 + 0.999954i \(0.503040\pi\)
\(102\) −2346.89 −2.27821
\(103\) 614.919 0.588250 0.294125 0.955767i \(-0.404972\pi\)
0.294125 + 0.955767i \(0.404972\pi\)
\(104\) −1304.13 −1.22962
\(105\) −453.475 −0.421472
\(106\) 1338.47 1.22645
\(107\) 1143.39 1.03304 0.516520 0.856275i \(-0.327227\pi\)
0.516520 + 0.856275i \(0.327227\pi\)
\(108\) −2449.71 −2.18262
\(109\) 399.428 0.350993 0.175496 0.984480i \(-0.443847\pi\)
0.175496 + 0.984480i \(0.443847\pi\)
\(110\) −1208.39 −1.04742
\(111\) −1453.86 −1.24319
\(112\) −2716.18 −2.29156
\(113\) 836.348 0.696257 0.348129 0.937447i \(-0.386817\pi\)
0.348129 + 0.937447i \(0.386817\pi\)
\(114\) 416.654 0.342309
\(115\) −1.47234 −0.00119389
\(116\) 1221.29 0.977535
\(117\) 19.2078 0.0151774
\(118\) −3535.68 −2.75835
\(119\) −2213.58 −1.70520
\(120\) −886.302 −0.674233
\(121\) 3524.71 2.64817
\(122\) 3968.84 2.94526
\(123\) −2000.62 −1.46658
\(124\) −4040.97 −2.92653
\(125\) −815.055 −0.583206
\(126\) 92.5391 0.0654289
\(127\) −1742.03 −1.21716 −0.608582 0.793491i \(-0.708261\pi\)
−0.608582 + 0.793491i \(0.708261\pi\)
\(128\) 1662.58 1.14807
\(129\) 0 0
\(130\) −459.530 −0.310027
\(131\) −592.422 −0.395116 −0.197558 0.980291i \(-0.563301\pi\)
−0.197558 + 0.980291i \(0.563301\pi\)
\(132\) 6496.74 4.28385
\(133\) 392.986 0.256212
\(134\) −2770.27 −1.78593
\(135\) −473.195 −0.301675
\(136\) −4326.37 −2.72782
\(137\) −1517.58 −0.946394 −0.473197 0.880957i \(-0.656900\pi\)
−0.473197 + 0.880957i \(0.656900\pi\)
\(138\) 11.4923 0.00708904
\(139\) −1161.08 −0.708502 −0.354251 0.935150i \(-0.615264\pi\)
−0.354251 + 0.935150i \(0.615264\pi\)
\(140\) −1524.94 −0.920578
\(141\) −311.751 −0.186200
\(142\) −2029.40 −1.19932
\(143\) 1846.54 1.07983
\(144\) 78.1888 0.0452481
\(145\) 235.909 0.135112
\(146\) −405.336 −0.229766
\(147\) 1532.46 0.859830
\(148\) −4889.03 −2.71538
\(149\) −485.816 −0.267111 −0.133556 0.991041i \(-0.542639\pi\)
−0.133556 + 0.991041i \(0.542639\pi\)
\(150\) 3024.78 1.64648
\(151\) 2216.13 1.19434 0.597172 0.802113i \(-0.296291\pi\)
0.597172 + 0.802113i \(0.296291\pi\)
\(152\) 768.080 0.409865
\(153\) 63.7208 0.0336701
\(154\) 8896.25 4.65507
\(155\) −780.568 −0.404495
\(156\) 2470.59 1.26799
\(157\) −432.876 −0.220046 −0.110023 0.993929i \(-0.535092\pi\)
−0.110023 + 0.993929i \(0.535092\pi\)
\(158\) 788.147 0.396846
\(159\) −1390.02 −0.693308
\(160\) −524.008 −0.258915
\(161\) 10.8395 0.00530603
\(162\) 3792.71 1.83941
\(163\) 1441.60 0.692727 0.346364 0.938100i \(-0.387416\pi\)
0.346364 + 0.938100i \(0.387416\pi\)
\(164\) −6727.65 −3.20330
\(165\) 1254.93 0.592100
\(166\) −467.843 −0.218745
\(167\) −1815.41 −0.841201 −0.420601 0.907246i \(-0.638181\pi\)
−0.420601 + 0.907246i \(0.638181\pi\)
\(168\) 6525.00 2.99652
\(169\) −1494.79 −0.680380
\(170\) −1524.47 −0.687773
\(171\) −11.3126 −0.00505905
\(172\) 0 0
\(173\) 1459.79 0.641534 0.320767 0.947158i \(-0.396059\pi\)
0.320767 + 0.947158i \(0.396059\pi\)
\(174\) −1841.37 −0.802265
\(175\) 2852.96 1.23236
\(176\) 7516.69 3.21927
\(177\) 3671.85 1.55928
\(178\) 4687.07 1.97365
\(179\) 4162.49 1.73810 0.869048 0.494728i \(-0.164732\pi\)
0.869048 + 0.494728i \(0.164732\pi\)
\(180\) 43.8974 0.0181773
\(181\) 408.845 0.167896 0.0839481 0.996470i \(-0.473247\pi\)
0.0839481 + 0.996470i \(0.473247\pi\)
\(182\) 3383.08 1.37786
\(183\) −4121.70 −1.66495
\(184\) 21.1854 0.00848810
\(185\) −944.384 −0.375311
\(186\) 6092.67 2.40181
\(187\) 6125.80 2.39552
\(188\) −1048.35 −0.406696
\(189\) 3483.68 1.34074
\(190\) 270.645 0.103340
\(191\) −3385.99 −1.28273 −0.641366 0.767235i \(-0.721632\pi\)
−0.641366 + 0.767235i \(0.721632\pi\)
\(192\) −453.750 −0.170555
\(193\) −198.605 −0.0740722 −0.0370361 0.999314i \(-0.511792\pi\)
−0.0370361 + 0.999314i \(0.511792\pi\)
\(194\) −968.087 −0.358271
\(195\) 477.229 0.175257
\(196\) 5153.33 1.87804
\(197\) −800.217 −0.289407 −0.144703 0.989475i \(-0.546223\pi\)
−0.144703 + 0.989475i \(0.546223\pi\)
\(198\) −256.090 −0.0919169
\(199\) −3022.75 −1.07677 −0.538384 0.842699i \(-0.680965\pi\)
−0.538384 + 0.842699i \(0.680965\pi\)
\(200\) 5576.02 1.97142
\(201\) 2876.96 1.00958
\(202\) 98.3071 0.0342419
\(203\) −1736.78 −0.600481
\(204\) 8196.06 2.81294
\(205\) −1299.54 −0.442750
\(206\) −3117.74 −1.05448
\(207\) −0.312028 −0.000104770 0
\(208\) 2858.46 0.952878
\(209\) −1087.54 −0.359937
\(210\) 2299.19 0.755520
\(211\) −581.734 −0.189802 −0.0949011 0.995487i \(-0.530253\pi\)
−0.0949011 + 0.995487i \(0.530253\pi\)
\(212\) −4674.35 −1.51432
\(213\) 2107.56 0.677969
\(214\) −5797.15 −1.85180
\(215\) 0 0
\(216\) 6808.75 2.14480
\(217\) 5746.58 1.79771
\(218\) −2025.16 −0.629181
\(219\) 420.947 0.129886
\(220\) 4220.08 1.29326
\(221\) 2329.53 0.709056
\(222\) 7371.32 2.22852
\(223\) −2267.15 −0.680805 −0.340403 0.940280i \(-0.610563\pi\)
−0.340403 + 0.940280i \(0.610563\pi\)
\(224\) 3857.77 1.15071
\(225\) −82.1261 −0.0243337
\(226\) −4240.42 −1.24809
\(227\) −3026.65 −0.884958 −0.442479 0.896779i \(-0.645901\pi\)
−0.442479 + 0.896779i \(0.645901\pi\)
\(228\) −1455.08 −0.422654
\(229\) −1509.02 −0.435454 −0.217727 0.976010i \(-0.569864\pi\)
−0.217727 + 0.976010i \(0.569864\pi\)
\(230\) 7.46503 0.00214013
\(231\) −9238.89 −2.63149
\(232\) −3394.47 −0.960595
\(233\) −2927.26 −0.823051 −0.411525 0.911398i \(-0.635004\pi\)
−0.411525 + 0.911398i \(0.635004\pi\)
\(234\) −97.3865 −0.0272066
\(235\) −202.503 −0.0562122
\(236\) 12347.7 3.40578
\(237\) −818.503 −0.224335
\(238\) 11223.2 3.05669
\(239\) −3687.79 −0.998089 −0.499044 0.866576i \(-0.666316\pi\)
−0.499044 + 0.866576i \(0.666316\pi\)
\(240\) 1942.65 0.522489
\(241\) −2219.41 −0.593216 −0.296608 0.954999i \(-0.595855\pi\)
−0.296608 + 0.954999i \(0.595855\pi\)
\(242\) −17870.9 −4.74704
\(243\) −203.331 −0.0536778
\(244\) −13860.4 −3.63657
\(245\) 995.437 0.259576
\(246\) 10143.4 2.62895
\(247\) −413.572 −0.106538
\(248\) 11231.5 2.87581
\(249\) 485.862 0.123656
\(250\) 4132.46 1.04544
\(251\) −3749.36 −0.942859 −0.471430 0.881904i \(-0.656262\pi\)
−0.471430 + 0.881904i \(0.656262\pi\)
\(252\) −323.175 −0.0807861
\(253\) −29.9969 −0.00745410
\(254\) 8832.36 2.18186
\(255\) 1583.18 0.388795
\(256\) −7740.13 −1.88968
\(257\) 3129.98 0.759698 0.379849 0.925048i \(-0.375976\pi\)
0.379849 + 0.925048i \(0.375976\pi\)
\(258\) 0 0
\(259\) 6952.60 1.66801
\(260\) 1604.82 0.382795
\(261\) 49.9953 0.0118568
\(262\) 3003.68 0.708274
\(263\) −2025.82 −0.474971 −0.237485 0.971391i \(-0.576323\pi\)
−0.237485 + 0.971391i \(0.576323\pi\)
\(264\) −18057.1 −4.20962
\(265\) −902.916 −0.209304
\(266\) −1992.51 −0.459280
\(267\) −4867.59 −1.11570
\(268\) 9674.62 2.20512
\(269\) 4608.81 1.04462 0.522312 0.852754i \(-0.325070\pi\)
0.522312 + 0.852754i \(0.325070\pi\)
\(270\) 2399.17 0.540774
\(271\) −5970.94 −1.33841 −0.669204 0.743079i \(-0.733365\pi\)
−0.669204 + 0.743079i \(0.733365\pi\)
\(272\) 9482.80 2.11389
\(273\) −3513.38 −0.778900
\(274\) 7694.40 1.69648
\(275\) −7895.20 −1.73127
\(276\) −40.1346 −0.00875296
\(277\) −2568.36 −0.557104 −0.278552 0.960421i \(-0.589855\pi\)
−0.278552 + 0.960421i \(0.589855\pi\)
\(278\) 5886.88 1.27004
\(279\) −165.423 −0.0354968
\(280\) 4238.44 0.904625
\(281\) −4320.27 −0.917174 −0.458587 0.888650i \(-0.651644\pi\)
−0.458587 + 0.888650i \(0.651644\pi\)
\(282\) 1580.63 0.333776
\(283\) −7668.11 −1.61068 −0.805339 0.592814i \(-0.798017\pi\)
−0.805339 + 0.592814i \(0.798017\pi\)
\(284\) 7087.27 1.48082
\(285\) −281.069 −0.0584179
\(286\) −9362.26 −1.93567
\(287\) 9567.26 1.96773
\(288\) −111.051 −0.0227213
\(289\) 2815.11 0.572991
\(290\) −1196.10 −0.242198
\(291\) 1005.37 0.202529
\(292\) 1415.56 0.283696
\(293\) −3130.46 −0.624176 −0.312088 0.950053i \(-0.601028\pi\)
−0.312088 + 0.950053i \(0.601028\pi\)
\(294\) −7769.81 −1.54131
\(295\) 2385.12 0.470736
\(296\) 13588.6 2.66832
\(297\) −9640.65 −1.88353
\(298\) 2463.16 0.478816
\(299\) −11.4073 −0.00220635
\(300\) −10563.4 −2.03294
\(301\) 0 0
\(302\) −11236.1 −2.14095
\(303\) −102.093 −0.0193568
\(304\) −1683.52 −0.317620
\(305\) −2677.33 −0.502634
\(306\) −323.075 −0.0603561
\(307\) 4928.32 0.916202 0.458101 0.888900i \(-0.348530\pi\)
0.458101 + 0.888900i \(0.348530\pi\)
\(308\) −31068.4 −5.74769
\(309\) 3237.82 0.596094
\(310\) 3957.61 0.725087
\(311\) −7959.42 −1.45125 −0.725623 0.688093i \(-0.758448\pi\)
−0.725623 + 0.688093i \(0.758448\pi\)
\(312\) −6866.80 −1.24601
\(313\) −3887.39 −0.702007 −0.351004 0.936374i \(-0.614159\pi\)
−0.351004 + 0.936374i \(0.614159\pi\)
\(314\) 2194.75 0.394449
\(315\) −62.4256 −0.0111660
\(316\) −2752.45 −0.489992
\(317\) 2937.52 0.520466 0.260233 0.965546i \(-0.416201\pi\)
0.260233 + 0.965546i \(0.416201\pi\)
\(318\) 7047.64 1.24281
\(319\) 4806.31 0.843579
\(320\) −294.742 −0.0514893
\(321\) 6020.43 1.04681
\(322\) −54.9579 −0.00951144
\(323\) −1372.00 −0.236348
\(324\) −13245.3 −2.27114
\(325\) −3002.40 −0.512441
\(326\) −7309.13 −1.24176
\(327\) 2103.16 0.355673
\(328\) 18698.9 3.14779
\(329\) 1490.84 0.249826
\(330\) −6362.72 −1.06138
\(331\) 6056.24 1.00568 0.502841 0.864379i \(-0.332288\pi\)
0.502841 + 0.864379i \(0.332288\pi\)
\(332\) 1633.85 0.270088
\(333\) −200.140 −0.0329357
\(334\) 9204.42 1.50792
\(335\) 1868.79 0.304784
\(336\) −14301.9 −2.32212
\(337\) 706.610 0.114218 0.0571091 0.998368i \(-0.481812\pi\)
0.0571091 + 0.998368i \(0.481812\pi\)
\(338\) 7578.85 1.21963
\(339\) 4403.74 0.705541
\(340\) 5323.91 0.849204
\(341\) −15902.9 −2.52549
\(342\) 57.3569 0.00906873
\(343\) 1308.34 0.205958
\(344\) 0 0
\(345\) −7.75254 −0.00120981
\(346\) −7401.35 −1.15000
\(347\) 343.739 0.0531783 0.0265892 0.999646i \(-0.491535\pi\)
0.0265892 + 0.999646i \(0.491535\pi\)
\(348\) 6430.63 0.990569
\(349\) −7218.47 −1.10715 −0.553576 0.832799i \(-0.686737\pi\)
−0.553576 + 0.832799i \(0.686737\pi\)
\(350\) −14465.0 −2.20910
\(351\) −3666.17 −0.557508
\(352\) −10675.9 −1.61655
\(353\) 3833.43 0.577997 0.288999 0.957329i \(-0.406678\pi\)
0.288999 + 0.957329i \(0.406678\pi\)
\(354\) −18616.9 −2.79513
\(355\) 1369.00 0.204674
\(356\) −16368.7 −2.43690
\(357\) −11655.5 −1.72793
\(358\) −21104.5 −3.11566
\(359\) −6085.37 −0.894634 −0.447317 0.894376i \(-0.647620\pi\)
−0.447317 + 0.894376i \(0.647620\pi\)
\(360\) −122.009 −0.0178623
\(361\) −6615.42 −0.964488
\(362\) −2072.91 −0.300966
\(363\) 18559.2 2.68348
\(364\) −11814.8 −1.70127
\(365\) 273.434 0.0392115
\(366\) 20897.7 2.98454
\(367\) 7608.08 1.08212 0.541061 0.840984i \(-0.318023\pi\)
0.541061 + 0.840984i \(0.318023\pi\)
\(368\) −46.4354 −0.00657776
\(369\) −275.406 −0.0388538
\(370\) 4788.18 0.672772
\(371\) 6647.31 0.930219
\(372\) −21277.5 −2.96555
\(373\) −7669.36 −1.06462 −0.532312 0.846548i \(-0.678677\pi\)
−0.532312 + 0.846548i \(0.678677\pi\)
\(374\) −31058.8 −4.29415
\(375\) −4291.62 −0.590983
\(376\) 2913.80 0.399649
\(377\) 1827.75 0.249692
\(378\) −17662.8 −2.40338
\(379\) 8689.95 1.17776 0.588882 0.808219i \(-0.299568\pi\)
0.588882 + 0.808219i \(0.299568\pi\)
\(380\) −945.177 −0.127596
\(381\) −9172.53 −1.23339
\(382\) 17167.5 2.29939
\(383\) −5710.24 −0.761827 −0.380913 0.924611i \(-0.624390\pi\)
−0.380913 + 0.924611i \(0.624390\pi\)
\(384\) 8754.19 1.16337
\(385\) −6001.30 −0.794427
\(386\) 1006.96 0.132780
\(387\) 0 0
\(388\) 3380.86 0.442363
\(389\) 13486.0 1.75775 0.878877 0.477049i \(-0.158293\pi\)
0.878877 + 0.477049i \(0.158293\pi\)
\(390\) −2419.63 −0.314161
\(391\) −37.8431 −0.00489464
\(392\) −14323.2 −1.84549
\(393\) −3119.36 −0.400384
\(394\) 4057.23 0.518782
\(395\) −531.674 −0.0677251
\(396\) 894.345 0.113491
\(397\) 10965.7 1.38628 0.693139 0.720804i \(-0.256227\pi\)
0.693139 + 0.720804i \(0.256227\pi\)
\(398\) 15325.8 1.93019
\(399\) 2069.25 0.259629
\(400\) −12221.8 −1.52773
\(401\) −11989.3 −1.49307 −0.746533 0.665349i \(-0.768283\pi\)
−0.746533 + 0.665349i \(0.768283\pi\)
\(402\) −14586.7 −1.80974
\(403\) −6047.60 −0.747525
\(404\) −343.319 −0.0422790
\(405\) −2558.51 −0.313910
\(406\) 8805.74 1.07641
\(407\) −19240.4 −2.34328
\(408\) −22780.2 −2.76419
\(409\) 1648.50 0.199298 0.0996491 0.995023i \(-0.468228\pi\)
0.0996491 + 0.995023i \(0.468228\pi\)
\(410\) 6588.87 0.793661
\(411\) −7990.75 −0.959014
\(412\) 10888.1 1.30199
\(413\) −17559.4 −2.09211
\(414\) 1.58204 0.000187809 0
\(415\) 315.601 0.0373307
\(416\) −4059.85 −0.478487
\(417\) −6113.61 −0.717949
\(418\) 5514.01 0.645213
\(419\) −12568.0 −1.46536 −0.732680 0.680573i \(-0.761731\pi\)
−0.732680 + 0.680573i \(0.761731\pi\)
\(420\) −8029.48 −0.932853
\(421\) −1815.80 −0.210205 −0.105103 0.994461i \(-0.533517\pi\)
−0.105103 + 0.994461i \(0.533517\pi\)
\(422\) 2949.49 0.340234
\(423\) −42.9158 −0.00493295
\(424\) 12992.0 1.48808
\(425\) −9960.32 −1.13682
\(426\) −10685.7 −1.21531
\(427\) 19710.6 2.23388
\(428\) 20245.4 2.28645
\(429\) 9722.85 1.09423
\(430\) 0 0
\(431\) 6971.17 0.779093 0.389547 0.921007i \(-0.372632\pi\)
0.389547 + 0.921007i \(0.372632\pi\)
\(432\) −14923.8 −1.66209
\(433\) −10300.9 −1.14326 −0.571629 0.820512i \(-0.693688\pi\)
−0.571629 + 0.820512i \(0.693688\pi\)
\(434\) −29136.1 −3.22253
\(435\) 1242.17 0.136913
\(436\) 7072.49 0.776860
\(437\) 6.71844 0.000735439 0
\(438\) −2134.27 −0.232830
\(439\) −9438.10 −1.02610 −0.513048 0.858360i \(-0.671484\pi\)
−0.513048 + 0.858360i \(0.671484\pi\)
\(440\) −11729.3 −1.27085
\(441\) 210.959 0.0227793
\(442\) −11811.1 −1.27103
\(443\) 14850.9 1.59275 0.796374 0.604805i \(-0.206749\pi\)
0.796374 + 0.604805i \(0.206749\pi\)
\(444\) −25742.9 −2.75158
\(445\) −3161.83 −0.336821
\(446\) 11494.8 1.22039
\(447\) −2558.03 −0.270673
\(448\) 2169.91 0.228836
\(449\) 18024.3 1.89448 0.947240 0.320526i \(-0.103860\pi\)
0.947240 + 0.320526i \(0.103860\pi\)
\(450\) 416.393 0.0436199
\(451\) −26476.2 −2.76434
\(452\) 14808.9 1.54104
\(453\) 11668.9 1.21027
\(454\) 15345.6 1.58635
\(455\) −2282.18 −0.235144
\(456\) 4044.28 0.415330
\(457\) 1076.30 0.110169 0.0550847 0.998482i \(-0.482457\pi\)
0.0550847 + 0.998482i \(0.482457\pi\)
\(458\) 7650.98 0.780582
\(459\) −12162.3 −1.23679
\(460\) −26.0702 −0.00264245
\(461\) −13661.5 −1.38021 −0.690107 0.723708i \(-0.742436\pi\)
−0.690107 + 0.723708i \(0.742436\pi\)
\(462\) 46842.7 4.71714
\(463\) 18341.2 1.84101 0.920504 0.390734i \(-0.127779\pi\)
0.920504 + 0.390734i \(0.127779\pi\)
\(464\) 7440.20 0.744403
\(465\) −4110.03 −0.409889
\(466\) 14841.7 1.47538
\(467\) −2240.22 −0.221980 −0.110990 0.993822i \(-0.535402\pi\)
−0.110990 + 0.993822i \(0.535402\pi\)
\(468\) 340.103 0.0335925
\(469\) −13758.1 −1.35456
\(470\) 1026.73 0.100765
\(471\) −2279.28 −0.222980
\(472\) −34319.3 −3.34676
\(473\) 0 0
\(474\) 4149.94 0.402137
\(475\) 1768.30 0.170811
\(476\) −39194.9 −3.77415
\(477\) −191.352 −0.0183677
\(478\) 18697.7 1.78915
\(479\) −17597.1 −1.67857 −0.839284 0.543694i \(-0.817025\pi\)
−0.839284 + 0.543694i \(0.817025\pi\)
\(480\) −2759.13 −0.262368
\(481\) −7316.79 −0.693591
\(482\) 11252.8 1.06338
\(483\) 57.0746 0.00537678
\(484\) 62410.6 5.86125
\(485\) 653.059 0.0611420
\(486\) 1030.92 0.0962213
\(487\) 9889.77 0.920222 0.460111 0.887861i \(-0.347810\pi\)
0.460111 + 0.887861i \(0.347810\pi\)
\(488\) 38523.8 3.57355
\(489\) 7590.64 0.701964
\(490\) −5047.03 −0.465309
\(491\) −3422.49 −0.314572 −0.157286 0.987553i \(-0.550274\pi\)
−0.157286 + 0.987553i \(0.550274\pi\)
\(492\) −35424.0 −3.24601
\(493\) 6063.47 0.553925
\(494\) 2096.88 0.190978
\(495\) 172.755 0.0156864
\(496\) −24617.9 −2.22858
\(497\) −10078.7 −0.909638
\(498\) −2463.40 −0.221662
\(499\) −15634.3 −1.40258 −0.701291 0.712875i \(-0.747393\pi\)
−0.701291 + 0.712875i \(0.747393\pi\)
\(500\) −14431.8 −1.29082
\(501\) −9558.93 −0.852418
\(502\) 19009.9 1.69014
\(503\) 101.907 0.00903342 0.00451671 0.999990i \(-0.498562\pi\)
0.00451671 + 0.999990i \(0.498562\pi\)
\(504\) 898.236 0.0793861
\(505\) −66.3167 −0.00584367
\(506\) 152.089 0.0133620
\(507\) −7870.75 −0.689452
\(508\) −30845.3 −2.69397
\(509\) 4998.09 0.435239 0.217619 0.976034i \(-0.430171\pi\)
0.217619 + 0.976034i \(0.430171\pi\)
\(510\) −8026.99 −0.696944
\(511\) −2013.04 −0.174269
\(512\) 25943.1 2.23933
\(513\) 2159.23 0.185833
\(514\) −15869.5 −1.36182
\(515\) 2103.19 0.179956
\(516\) 0 0
\(517\) −4125.71 −0.350965
\(518\) −35250.8 −2.99002
\(519\) 7686.41 0.650089
\(520\) −4460.46 −0.376162
\(521\) 1425.38 0.119860 0.0599298 0.998203i \(-0.480912\pi\)
0.0599298 + 0.998203i \(0.480912\pi\)
\(522\) −253.485 −0.0212543
\(523\) 4492.23 0.375586 0.187793 0.982209i \(-0.439867\pi\)
0.187793 + 0.982209i \(0.439867\pi\)
\(524\) −10489.8 −0.874518
\(525\) 15022.1 1.24879
\(526\) 10271.2 0.851420
\(527\) −20062.6 −1.65833
\(528\) 39578.6 3.26220
\(529\) −12166.8 −0.999985
\(530\) 4577.93 0.375194
\(531\) 505.470 0.0413098
\(532\) 6958.44 0.567080
\(533\) −10068.4 −0.818221
\(534\) 24679.5 1.99997
\(535\) 3910.68 0.316025
\(536\) −26889.8 −2.16691
\(537\) 21917.3 1.76127
\(538\) −23367.4 −1.87257
\(539\) 20280.6 1.62068
\(540\) −8378.65 −0.667703
\(541\) −14116.5 −1.12184 −0.560922 0.827869i \(-0.689553\pi\)
−0.560922 + 0.827869i \(0.689553\pi\)
\(542\) 30273.6 2.39920
\(543\) 2152.75 0.170135
\(544\) −13468.3 −1.06149
\(545\) 1366.15 0.107375
\(546\) 17813.4 1.39623
\(547\) −5870.12 −0.458845 −0.229423 0.973327i \(-0.573684\pi\)
−0.229423 + 0.973327i \(0.573684\pi\)
\(548\) −26871.2 −2.09467
\(549\) −567.396 −0.0441091
\(550\) 40030.0 3.10342
\(551\) −1076.48 −0.0832294
\(552\) 111.551 0.00860128
\(553\) 3914.21 0.300993
\(554\) 13022.0 0.998650
\(555\) −4972.59 −0.380315
\(556\) −20558.8 −1.56814
\(557\) 19618.5 1.49239 0.746196 0.665726i \(-0.231878\pi\)
0.746196 + 0.665726i \(0.231878\pi\)
\(558\) 838.721 0.0636306
\(559\) 0 0
\(560\) −9290.06 −0.701029
\(561\) 32255.0 2.42747
\(562\) 21904.5 1.64410
\(563\) −16594.0 −1.24219 −0.621094 0.783736i \(-0.713312\pi\)
−0.621094 + 0.783736i \(0.713312\pi\)
\(564\) −5520.03 −0.412119
\(565\) 2860.53 0.212997
\(566\) 38878.6 2.88726
\(567\) 18835.9 1.39512
\(568\) −19698.4 −1.45516
\(569\) −3166.05 −0.233265 −0.116633 0.993175i \(-0.537210\pi\)
−0.116633 + 0.993175i \(0.537210\pi\)
\(570\) 1425.07 0.104718
\(571\) 9197.67 0.674099 0.337049 0.941487i \(-0.390571\pi\)
0.337049 + 0.941487i \(0.390571\pi\)
\(572\) 32695.9 2.39001
\(573\) −17828.7 −1.29984
\(574\) −48507.6 −3.52730
\(575\) 48.7738 0.00353740
\(576\) −62.4636 −0.00451849
\(577\) 2852.95 0.205840 0.102920 0.994690i \(-0.467181\pi\)
0.102920 + 0.994690i \(0.467181\pi\)
\(578\) −14273.0 −1.02713
\(579\) −1045.74 −0.0750599
\(580\) 4177.14 0.299045
\(581\) −2323.47 −0.165910
\(582\) −5097.40 −0.363048
\(583\) −18395.6 −1.30681
\(584\) −3934.42 −0.278780
\(585\) 65.6957 0.00464304
\(586\) 15872.0 1.11888
\(587\) −1209.16 −0.0850208 −0.0425104 0.999096i \(-0.513536\pi\)
−0.0425104 + 0.999096i \(0.513536\pi\)
\(588\) 27134.6 1.90308
\(589\) 3561.80 0.249171
\(590\) −12093.0 −0.843829
\(591\) −4213.49 −0.293265
\(592\) −29784.4 −2.06779
\(593\) 3344.49 0.231605 0.115803 0.993272i \(-0.463056\pi\)
0.115803 + 0.993272i \(0.463056\pi\)
\(594\) 48879.7 3.37636
\(595\) −7571.03 −0.521650
\(596\) −8602.12 −0.591203
\(597\) −15916.1 −1.09113
\(598\) 57.8368 0.00395505
\(599\) −19435.1 −1.32570 −0.662851 0.748752i \(-0.730654\pi\)
−0.662851 + 0.748752i \(0.730654\pi\)
\(600\) 29360.2 1.99771
\(601\) 3263.80 0.221519 0.110760 0.993847i \(-0.464672\pi\)
0.110760 + 0.993847i \(0.464672\pi\)
\(602\) 0 0
\(603\) 396.045 0.0267466
\(604\) 39240.0 2.64346
\(605\) 12055.5 0.810123
\(606\) 517.630 0.0346985
\(607\) −13079.9 −0.874622 −0.437311 0.899310i \(-0.644069\pi\)
−0.437311 + 0.899310i \(0.644069\pi\)
\(608\) 2391.09 0.159493
\(609\) −9144.88 −0.608488
\(610\) 13574.5 0.901009
\(611\) −1568.94 −0.103883
\(612\) 1128.28 0.0745226
\(613\) −24182.4 −1.59334 −0.796671 0.604413i \(-0.793408\pi\)
−0.796671 + 0.604413i \(0.793408\pi\)
\(614\) −24987.4 −1.64236
\(615\) −6842.64 −0.448653
\(616\) 86352.0 5.64809
\(617\) −4358.65 −0.284396 −0.142198 0.989838i \(-0.545417\pi\)
−0.142198 + 0.989838i \(0.545417\pi\)
\(618\) −16416.3 −1.06854
\(619\) −2248.54 −0.146004 −0.0730019 0.997332i \(-0.523258\pi\)
−0.0730019 + 0.997332i \(0.523258\pi\)
\(620\) −13821.2 −0.895277
\(621\) 59.5566 0.00384851
\(622\) 40355.6 2.60147
\(623\) 23277.6 1.49694
\(624\) 15051.0 0.965583
\(625\) 11375.0 0.728001
\(626\) 19709.7 1.25840
\(627\) −5726.38 −0.364736
\(628\) −7664.74 −0.487032
\(629\) −24273.1 −1.53868
\(630\) 316.508 0.0200159
\(631\) 19871.2 1.25366 0.626829 0.779157i \(-0.284352\pi\)
0.626829 + 0.779157i \(0.284352\pi\)
\(632\) 7650.20 0.481501
\(633\) −3063.09 −0.192333
\(634\) −14893.7 −0.932973
\(635\) −5958.19 −0.372352
\(636\) −24612.5 −1.53451
\(637\) 7712.34 0.479708
\(638\) −24368.8 −1.51218
\(639\) 290.128 0.0179613
\(640\) 5686.45 0.351214
\(641\) −2163.67 −0.133323 −0.0666613 0.997776i \(-0.521235\pi\)
−0.0666613 + 0.997776i \(0.521235\pi\)
\(642\) −30524.6 −1.87649
\(643\) −19895.6 −1.22023 −0.610115 0.792313i \(-0.708877\pi\)
−0.610115 + 0.792313i \(0.708877\pi\)
\(644\) 191.930 0.0117439
\(645\) 0 0
\(646\) 6956.29 0.423671
\(647\) −3789.98 −0.230293 −0.115146 0.993349i \(-0.536734\pi\)
−0.115146 + 0.993349i \(0.536734\pi\)
\(648\) 36814.2 2.23179
\(649\) 48593.4 2.93907
\(650\) 15222.7 0.918588
\(651\) 30258.3 1.82168
\(652\) 25525.7 1.53323
\(653\) 16510.4 0.989436 0.494718 0.869054i \(-0.335271\pi\)
0.494718 + 0.869054i \(0.335271\pi\)
\(654\) −10663.4 −0.637570
\(655\) −2026.24 −0.120873
\(656\) −40985.4 −2.43934
\(657\) 57.9479 0.00344104
\(658\) −7558.81 −0.447831
\(659\) −14640.9 −0.865448 −0.432724 0.901526i \(-0.642448\pi\)
−0.432724 + 0.901526i \(0.642448\pi\)
\(660\) 22220.6 1.31051
\(661\) −30193.4 −1.77668 −0.888341 0.459184i \(-0.848142\pi\)
−0.888341 + 0.459184i \(0.848142\pi\)
\(662\) −30706.1 −1.80276
\(663\) 12266.0 0.718510
\(664\) −4541.15 −0.265408
\(665\) 1344.12 0.0783799
\(666\) 1014.74 0.0590396
\(667\) −29.6917 −0.00172364
\(668\) −32144.7 −1.86185
\(669\) −11937.5 −0.689883
\(670\) −9475.05 −0.546348
\(671\) −54546.7 −3.13823
\(672\) 20312.9 1.16605
\(673\) −9061.92 −0.519036 −0.259518 0.965738i \(-0.583564\pi\)
−0.259518 + 0.965738i \(0.583564\pi\)
\(674\) −3582.63 −0.204744
\(675\) 15675.3 0.893842
\(676\) −26467.7 −1.50590
\(677\) 4678.54 0.265599 0.132800 0.991143i \(-0.457603\pi\)
0.132800 + 0.991143i \(0.457603\pi\)
\(678\) −22327.7 −1.26473
\(679\) −4807.85 −0.271736
\(680\) −14797.3 −0.834489
\(681\) −15936.6 −0.896758
\(682\) 80630.5 4.52713
\(683\) 28517.5 1.59764 0.798822 0.601568i \(-0.205457\pi\)
0.798822 + 0.601568i \(0.205457\pi\)
\(684\) −200.308 −0.0111973
\(685\) −5190.54 −0.289519
\(686\) −6633.49 −0.369195
\(687\) −7945.65 −0.441260
\(688\) 0 0
\(689\) −6995.52 −0.386804
\(690\) 39.3067 0.00216866
\(691\) 18910.7 1.04109 0.520547 0.853833i \(-0.325728\pi\)
0.520547 + 0.853833i \(0.325728\pi\)
\(692\) 25847.8 1.41992
\(693\) −1271.83 −0.0697156
\(694\) −1742.81 −0.0953261
\(695\) −3971.21 −0.216743
\(696\) −17873.4 −0.973404
\(697\) −33401.5 −1.81517
\(698\) 36598.8 1.98465
\(699\) −15413.3 −0.834026
\(700\) 50516.1 2.72761
\(701\) 11193.1 0.603080 0.301540 0.953454i \(-0.402499\pi\)
0.301540 + 0.953454i \(0.402499\pi\)
\(702\) 18588.1 0.999375
\(703\) 4309.31 0.231193
\(704\) −6004.95 −0.321477
\(705\) −1066.27 −0.0569618
\(706\) −19436.1 −1.03610
\(707\) 488.227 0.0259712
\(708\) 65015.9 3.45120
\(709\) 12502.4 0.662251 0.331125 0.943587i \(-0.392572\pi\)
0.331125 + 0.943587i \(0.392572\pi\)
\(710\) −6941.07 −0.366893
\(711\) −112.676 −0.00594327
\(712\) 45495.3 2.39467
\(713\) 98.2428 0.00516020
\(714\) 59095.1 3.09745
\(715\) 6315.66 0.330339
\(716\) 73703.4 3.84696
\(717\) −19417.8 −1.01140
\(718\) 30853.8 1.60370
\(719\) 24760.3 1.28429 0.642143 0.766585i \(-0.278045\pi\)
0.642143 + 0.766585i \(0.278045\pi\)
\(720\) 267.426 0.0138422
\(721\) −15483.8 −0.799786
\(722\) 33541.3 1.72892
\(723\) −11686.2 −0.601126
\(724\) 7239.24 0.371608
\(725\) −7814.87 −0.400327
\(726\) −94098.0 −4.81034
\(727\) 11567.8 0.590131 0.295066 0.955477i \(-0.404658\pi\)
0.295066 + 0.955477i \(0.404658\pi\)
\(728\) 32838.1 1.67179
\(729\) 19126.6 0.971732
\(730\) −1386.36 −0.0702895
\(731\) 0 0
\(732\) −72981.2 −3.68506
\(733\) −4270.99 −0.215215 −0.107607 0.994193i \(-0.534319\pi\)
−0.107607 + 0.994193i \(0.534319\pi\)
\(734\) −38574.2 −1.93978
\(735\) 5241.41 0.263037
\(736\) 65.9519 0.00330302
\(737\) 38073.8 1.90294
\(738\) 1396.35 0.0696484
\(739\) 21299.1 1.06022 0.530109 0.847929i \(-0.322151\pi\)
0.530109 + 0.847929i \(0.322151\pi\)
\(740\) −16721.8 −0.830683
\(741\) −2177.64 −0.107959
\(742\) −33703.0 −1.66749
\(743\) −11126.4 −0.549378 −0.274689 0.961533i \(-0.588575\pi\)
−0.274689 + 0.961533i \(0.588575\pi\)
\(744\) 59138.8 2.91416
\(745\) −1661.62 −0.0817141
\(746\) 38884.9 1.90842
\(747\) 66.8841 0.00327598
\(748\) 108467. 5.30206
\(749\) −28790.7 −1.40452
\(750\) 21759.2 1.05938
\(751\) −29708.9 −1.44354 −0.721768 0.692136i \(-0.756670\pi\)
−0.721768 + 0.692136i \(0.756670\pi\)
\(752\) −6386.64 −0.309703
\(753\) −19742.0 −0.955431
\(754\) −9267.00 −0.447592
\(755\) 7579.74 0.365371
\(756\) 61684.0 2.96749
\(757\) 32375.3 1.55443 0.777213 0.629237i \(-0.216633\pi\)
0.777213 + 0.629237i \(0.216633\pi\)
\(758\) −44059.5 −2.11123
\(759\) −157.947 −0.00755350
\(760\) 2627.04 0.125385
\(761\) 29515.1 1.40594 0.702972 0.711218i \(-0.251856\pi\)
0.702972 + 0.711218i \(0.251856\pi\)
\(762\) 46506.2 2.21095
\(763\) −10057.7 −0.477210
\(764\) −59954.3 −2.83910
\(765\) 217.942 0.0103003
\(766\) 28951.8 1.36563
\(767\) 18479.2 0.869941
\(768\) −40755.2 −1.91488
\(769\) 948.011 0.0444553 0.0222277 0.999753i \(-0.492924\pi\)
0.0222277 + 0.999753i \(0.492924\pi\)
\(770\) 30427.6 1.42407
\(771\) 16480.7 0.769828
\(772\) −3516.62 −0.163945
\(773\) 12922.3 0.601270 0.300635 0.953739i \(-0.402801\pi\)
0.300635 + 0.953739i \(0.402801\pi\)
\(774\) 0 0
\(775\) 25857.6 1.19849
\(776\) −9396.80 −0.434698
\(777\) 36608.5 1.69025
\(778\) −68376.1 −3.15090
\(779\) 5929.91 0.272736
\(780\) 8450.08 0.387899
\(781\) 27891.5 1.27789
\(782\) 191.870 0.00877401
\(783\) −9542.56 −0.435534
\(784\) 31394.5 1.43014
\(785\) −1480.55 −0.0673160
\(786\) 15815.7 0.717719
\(787\) −24604.3 −1.11442 −0.557211 0.830371i \(-0.688128\pi\)
−0.557211 + 0.830371i \(0.688128\pi\)
\(788\) −14169.1 −0.640549
\(789\) −10666.8 −0.481304
\(790\) 2695.67 0.121402
\(791\) −21059.4 −0.946632
\(792\) −2485.76 −0.111525
\(793\) −20743.1 −0.928891
\(794\) −55597.9 −2.48501
\(795\) −4754.25 −0.212095
\(796\) −53522.5 −2.38323
\(797\) 18527.4 0.823432 0.411716 0.911312i \(-0.364930\pi\)
0.411716 + 0.911312i \(0.364930\pi\)
\(798\) −10491.4 −0.465404
\(799\) −5204.86 −0.230457
\(800\) 17358.6 0.767149
\(801\) −670.075 −0.0295580
\(802\) 60787.9 2.67643
\(803\) 5570.83 0.244820
\(804\) 50941.1 2.23452
\(805\) 37.0739 0.00162321
\(806\) 30662.3 1.33999
\(807\) 24267.4 1.05855
\(808\) 954.224 0.0415464
\(809\) −12328.4 −0.535778 −0.267889 0.963450i \(-0.586326\pi\)
−0.267889 + 0.963450i \(0.586326\pi\)
\(810\) 12972.1 0.562707
\(811\) −14172.8 −0.613655 −0.306828 0.951765i \(-0.599268\pi\)
−0.306828 + 0.951765i \(0.599268\pi\)
\(812\) −30752.3 −1.32906
\(813\) −31439.6 −1.35625
\(814\) 97552.2 4.20050
\(815\) 4930.64 0.211918
\(816\) 49931.1 2.14208
\(817\) 0 0
\(818\) −8358.15 −0.357257
\(819\) −483.655 −0.0206352
\(820\) −23010.3 −0.979947
\(821\) 3310.52 0.140728 0.0703642 0.997521i \(-0.477584\pi\)
0.0703642 + 0.997521i \(0.477584\pi\)
\(822\) 40514.4 1.71910
\(823\) −25536.1 −1.08157 −0.540785 0.841161i \(-0.681873\pi\)
−0.540785 + 0.841161i \(0.681873\pi\)
\(824\) −30262.5 −1.27942
\(825\) −41571.7 −1.75435
\(826\) 89029.0 3.75026
\(827\) −32268.2 −1.35680 −0.678401 0.734692i \(-0.737327\pi\)
−0.678401 + 0.734692i \(0.737327\pi\)
\(828\) −5.52495 −0.000231890 0
\(829\) −27155.4 −1.13769 −0.568846 0.822444i \(-0.692610\pi\)
−0.568846 + 0.822444i \(0.692610\pi\)
\(830\) −1600.15 −0.0669180
\(831\) −13523.6 −0.564533
\(832\) −2283.57 −0.0951546
\(833\) 25585.3 1.06420
\(834\) 30997.0 1.28698
\(835\) −6209.18 −0.257338
\(836\) −19256.6 −0.796655
\(837\) 31574.1 1.30390
\(838\) 63721.7 2.62677
\(839\) −30776.5 −1.26642 −0.633208 0.773982i \(-0.718262\pi\)
−0.633208 + 0.773982i \(0.718262\pi\)
\(840\) 22317.2 0.916688
\(841\) −19631.6 −0.804936
\(842\) 9206.38 0.376809
\(843\) −22748.1 −0.929404
\(844\) −10300.5 −0.420093
\(845\) −5112.59 −0.208140
\(846\) 217.590 0.00884267
\(847\) −88752.9 −3.60046
\(848\) −28476.5 −1.15317
\(849\) −40376.0 −1.63216
\(850\) 50500.5 2.03782
\(851\) 118.861 0.00478789
\(852\) 37317.6 1.50056
\(853\) 35765.6 1.43563 0.717813 0.696236i \(-0.245143\pi\)
0.717813 + 0.696236i \(0.245143\pi\)
\(854\) −99936.2 −4.00439
\(855\) −38.6922 −0.00154765
\(856\) −56270.4 −2.24683
\(857\) −9888.39 −0.394143 −0.197072 0.980389i \(-0.563143\pi\)
−0.197072 + 0.980389i \(0.563143\pi\)
\(858\) −49296.4 −1.96148
\(859\) 33387.3 1.32615 0.663073 0.748554i \(-0.269252\pi\)
0.663073 + 0.748554i \(0.269252\pi\)
\(860\) 0 0
\(861\) 50375.8 1.99397
\(862\) −35344.9 −1.39658
\(863\) 12883.2 0.508167 0.254084 0.967182i \(-0.418226\pi\)
0.254084 + 0.967182i \(0.418226\pi\)
\(864\) 21196.2 0.834617
\(865\) 4992.86 0.196257
\(866\) 52227.3 2.04937
\(867\) 14822.8 0.580632
\(868\) 101752. 3.97891
\(869\) −10832.1 −0.422846
\(870\) −6297.98 −0.245427
\(871\) 14478.8 0.563255
\(872\) −19657.4 −0.763397
\(873\) 138.400 0.00536557
\(874\) −34.0636 −0.00131833
\(875\) 20523.2 0.792928
\(876\) 7453.53 0.287479
\(877\) 43358.1 1.66944 0.834719 0.550676i \(-0.185630\pi\)
0.834719 + 0.550676i \(0.185630\pi\)
\(878\) 47852.7 1.83935
\(879\) −16483.3 −0.632499
\(880\) 25709.1 0.984832
\(881\) −50023.1 −1.91296 −0.956482 0.291790i \(-0.905749\pi\)
−0.956482 + 0.291790i \(0.905749\pi\)
\(882\) −1069.60 −0.0408336
\(883\) 12280.1 0.468017 0.234008 0.972235i \(-0.424816\pi\)
0.234008 + 0.972235i \(0.424816\pi\)
\(884\) 41248.0 1.56937
\(885\) 12558.7 0.477013
\(886\) −75296.4 −2.85512
\(887\) 13525.7 0.512004 0.256002 0.966676i \(-0.417595\pi\)
0.256002 + 0.966676i \(0.417595\pi\)
\(888\) 71550.1 2.70390
\(889\) 43864.5 1.65486
\(890\) 16031.0 0.603776
\(891\) −52126.0 −1.95992
\(892\) −40143.4 −1.50684
\(893\) 924.042 0.0346270
\(894\) 12969.6 0.485201
\(895\) 14236.8 0.531715
\(896\) −41863.9 −1.56091
\(897\) −60.0643 −0.00223577
\(898\) −91386.4 −3.39599
\(899\) −15741.1 −0.583978
\(900\) −1454.17 −0.0538582
\(901\) −23207.3 −0.858098
\(902\) 134239. 4.95528
\(903\) 0 0
\(904\) −41159.9 −1.51433
\(905\) 1398.36 0.0513624
\(906\) −59163.1 −2.16950
\(907\) −24879.6 −0.910818 −0.455409 0.890282i \(-0.650507\pi\)
−0.455409 + 0.890282i \(0.650507\pi\)
\(908\) −53591.5 −1.95870
\(909\) −14.0542 −0.000512816 0
\(910\) 11571.0 0.421513
\(911\) −19747.9 −0.718198 −0.359099 0.933299i \(-0.616916\pi\)
−0.359099 + 0.933299i \(0.616916\pi\)
\(912\) −8864.48 −0.321856
\(913\) 6429.91 0.233077
\(914\) −5457.04 −0.197487
\(915\) −14097.3 −0.509337
\(916\) −26719.6 −0.963798
\(917\) 14917.3 0.537200
\(918\) 61664.9 2.21704
\(919\) 9581.27 0.343914 0.171957 0.985104i \(-0.444991\pi\)
0.171957 + 0.985104i \(0.444991\pi\)
\(920\) 72.4598 0.00259666
\(921\) 25949.8 0.928419
\(922\) 69265.9 2.47413
\(923\) 10606.6 0.378246
\(924\) −163589. −5.82433
\(925\) 31284.2 1.11202
\(926\) −92992.7 −3.30014
\(927\) 445.720 0.0157922
\(928\) −10567.3 −0.373801
\(929\) 35400.6 1.25022 0.625111 0.780536i \(-0.285054\pi\)
0.625111 + 0.780536i \(0.285054\pi\)
\(930\) 20838.5 0.734756
\(931\) −4542.27 −0.159900
\(932\) −51831.6 −1.82168
\(933\) −41909.9 −1.47060
\(934\) 11358.3 0.397916
\(935\) 20951.9 0.732834
\(936\) −945.288 −0.0330104
\(937\) −24831.3 −0.865745 −0.432872 0.901455i \(-0.642500\pi\)
−0.432872 + 0.901455i \(0.642500\pi\)
\(938\) 69755.8 2.42815
\(939\) −20468.8 −0.711368
\(940\) −3585.64 −0.124416
\(941\) 18196.9 0.630394 0.315197 0.949026i \(-0.397929\pi\)
0.315197 + 0.949026i \(0.397929\pi\)
\(942\) 11556.3 0.399708
\(943\) 163.561 0.00564822
\(944\) 75223.0 2.59354
\(945\) 11915.1 0.410157
\(946\) 0 0
\(947\) −38685.4 −1.32746 −0.663732 0.747971i \(-0.731028\pi\)
−0.663732 + 0.747971i \(0.731028\pi\)
\(948\) −14492.9 −0.496526
\(949\) 2118.48 0.0724646
\(950\) −8965.56 −0.306191
\(951\) 15467.3 0.527406
\(952\) 108939. 3.70875
\(953\) −41792.1 −1.42054 −0.710272 0.703927i \(-0.751428\pi\)
−0.710272 + 0.703927i \(0.751428\pi\)
\(954\) 970.184 0.0329254
\(955\) −11581.0 −0.392411
\(956\) −65298.1 −2.20909
\(957\) 25307.3 0.854827
\(958\) 89220.4 3.00896
\(959\) 38213.0 1.28672
\(960\) −1551.95 −0.0521759
\(961\) 22292.7 0.748304
\(962\) 37097.3 1.24331
\(963\) 828.776 0.0277331
\(964\) −39298.2 −1.31298
\(965\) −679.283 −0.0226600
\(966\) −289.378 −0.00963827
\(967\) 7635.19 0.253910 0.126955 0.991908i \(-0.459480\pi\)
0.126955 + 0.991908i \(0.459480\pi\)
\(968\) −173465. −5.75968
\(969\) −7224.20 −0.239499
\(970\) −3311.12 −0.109602
\(971\) 14594.5 0.482349 0.241175 0.970482i \(-0.422467\pi\)
0.241175 + 0.970482i \(0.422467\pi\)
\(972\) −3600.29 −0.118806
\(973\) 29236.3 0.963280
\(974\) −50142.7 −1.64957
\(975\) −15809.0 −0.519274
\(976\) −84438.8 −2.76928
\(977\) 21327.7 0.698398 0.349199 0.937049i \(-0.386454\pi\)
0.349199 + 0.937049i \(0.386454\pi\)
\(978\) −38485.8 −1.25832
\(979\) −64417.8 −2.10296
\(980\) 17625.8 0.574525
\(981\) 289.523 0.00942278
\(982\) 17352.6 0.563893
\(983\) −14871.0 −0.482515 −0.241258 0.970461i \(-0.577560\pi\)
−0.241258 + 0.970461i \(0.577560\pi\)
\(984\) 98458.0 3.18976
\(985\) −2736.95 −0.0885346
\(986\) −30742.8 −0.992952
\(987\) 7849.93 0.253157
\(988\) −7322.94 −0.235803
\(989\) 0 0
\(990\) −875.897 −0.0281190
\(991\) 21864.6 0.700860 0.350430 0.936589i \(-0.386036\pi\)
0.350430 + 0.936589i \(0.386036\pi\)
\(992\) 34964.6 1.11908
\(993\) 31888.7 1.01909
\(994\) 51100.5 1.63059
\(995\) −10338.6 −0.329403
\(996\) 8602.94 0.273689
\(997\) 2007.96 0.0637840 0.0318920 0.999491i \(-0.489847\pi\)
0.0318920 + 0.999491i \(0.489847\pi\)
\(998\) 79268.6 2.51423
\(999\) 38200.4 1.20982
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1849.4.a.g.1.2 30
43.32 odd 14 43.4.e.a.35.1 yes 60
43.39 odd 14 43.4.e.a.16.1 60
43.42 odd 2 1849.4.a.h.1.29 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.16.1 60 43.39 odd 14
43.4.e.a.35.1 yes 60 43.32 odd 14
1849.4.a.g.1.2 30 1.1 even 1 trivial
1849.4.a.h.1.29 30 43.42 odd 2