Properties

Label 47.4.a.a.1.2
Level $47$
Weight $4$
Character 47.1
Self dual yes
Analytic conductor $2.773$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [47,4,Mod(1,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, 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("47.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 47.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: \(2.77308977027\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.11903\) of defining polynomial
Character \(\chi\) \(=\) 47.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.60930 q^{2} +1.72833 q^{3} -5.41015 q^{4} -9.01945 q^{5} -2.78140 q^{6} -11.3182 q^{7} +21.5810 q^{8} -24.0129 q^{9} +O(q^{10})\) \(q-1.60930 q^{2} +1.72833 q^{3} -5.41015 q^{4} -9.01945 q^{5} -2.78140 q^{6} -11.3182 q^{7} +21.5810 q^{8} -24.0129 q^{9} +14.5150 q^{10} -40.6469 q^{11} -9.35051 q^{12} -12.3983 q^{13} +18.2143 q^{14} -15.5886 q^{15} +8.55095 q^{16} +59.6717 q^{17} +38.6440 q^{18} +26.4089 q^{19} +48.7966 q^{20} -19.5615 q^{21} +65.4131 q^{22} +107.097 q^{23} +37.2990 q^{24} -43.6495 q^{25} +19.9526 q^{26} -88.1670 q^{27} +61.2330 q^{28} +173.657 q^{29} +25.0867 q^{30} -332.301 q^{31} -186.409 q^{32} -70.2512 q^{33} -96.0296 q^{34} +102.084 q^{35} +129.913 q^{36} -172.339 q^{37} -42.4998 q^{38} -21.4283 q^{39} -194.648 q^{40} -178.190 q^{41} +31.4804 q^{42} +63.2928 q^{43} +219.906 q^{44} +216.583 q^{45} -172.351 q^{46} +47.0000 q^{47} +14.7788 q^{48} -214.899 q^{49} +70.2452 q^{50} +103.132 q^{51} +67.0767 q^{52} -402.256 q^{53} +141.887 q^{54} +366.613 q^{55} -244.257 q^{56} +45.6432 q^{57} -279.466 q^{58} +305.280 q^{59} +84.3365 q^{60} -86.2464 q^{61} +534.772 q^{62} +271.782 q^{63} +231.580 q^{64} +111.826 q^{65} +113.055 q^{66} -681.333 q^{67} -322.833 q^{68} +185.098 q^{69} -164.283 q^{70} +726.348 q^{71} -518.221 q^{72} -79.8711 q^{73} +277.345 q^{74} -75.4406 q^{75} -142.876 q^{76} +460.049 q^{77} +34.4846 q^{78} +279.707 q^{79} -77.1248 q^{80} +495.966 q^{81} +286.761 q^{82} +556.598 q^{83} +105.831 q^{84} -538.206 q^{85} -101.857 q^{86} +300.135 q^{87} -877.200 q^{88} -342.069 q^{89} -348.547 q^{90} +140.326 q^{91} -579.410 q^{92} -574.325 q^{93} -75.6371 q^{94} -238.194 q^{95} -322.175 q^{96} -1637.35 q^{97} +345.837 q^{98} +976.050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 6 q^{5} - 8 q^{6} - 45 q^{7} - 39 q^{8} - 42 q^{9} - 40 q^{10} + 2 q^{11} + 12 q^{12} - 80 q^{13} + 162 q^{14} + 14 q^{15} + 89 q^{16} - 39 q^{17} + 181 q^{18} - 24 q^{19} + 232 q^{20} + 24 q^{21} - 14 q^{22} + 120 q^{23} + 192 q^{24} - 171 q^{25} + 316 q^{26} + 64 q^{27} - 408 q^{28} - 184 q^{29} + 116 q^{30} - 4 q^{31} - 7 q^{32} - 208 q^{33} + 218 q^{34} - 156 q^{35} - 343 q^{36} - 589 q^{37} + 42 q^{38} + 60 q^{39} - 432 q^{40} - 92 q^{41} - 54 q^{42} - 250 q^{43} + 466 q^{44} - 78 q^{45} - 816 q^{46} + 141 q^{47} - 120 q^{48} + 30 q^{49} + 137 q^{50} + 317 q^{51} - 900 q^{52} + 459 q^{53} + 106 q^{54} + 448 q^{55} + 1032 q^{56} + 216 q^{57} + 684 q^{58} + 579 q^{59} - 240 q^{60} + 267 q^{61} - 244 q^{62} + 1044 q^{63} - 87 q^{64} - 424 q^{65} + 16 q^{66} - 540 q^{67} - 1334 q^{68} + 642 q^{69} + 1236 q^{70} + 749 q^{71} + 357 q^{72} - 1924 q^{73} + 950 q^{74} + 473 q^{75} - 402 q^{76} - 288 q^{77} - 152 q^{78} + 805 q^{79} + 448 q^{80} + 291 q^{81} - 938 q^{82} + 712 q^{83} + 372 q^{84} - 1038 q^{85} - 1294 q^{86} + 1216 q^{87} - 2190 q^{88} + 835 q^{89} + 764 q^{90} + 2040 q^{91} + 1596 q^{92} - 1500 q^{93} - 235 q^{94} - 312 q^{95} - 1432 q^{96} - 2243 q^{97} - 2989 q^{98} + 554 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.60930 −0.568974 −0.284487 0.958680i \(-0.591823\pi\)
−0.284487 + 0.958680i \(0.591823\pi\)
\(3\) 1.72833 0.332617 0.166308 0.986074i \(-0.446815\pi\)
0.166308 + 0.986074i \(0.446815\pi\)
\(4\) −5.41015 −0.676269
\(5\) −9.01945 −0.806724 −0.403362 0.915040i \(-0.632159\pi\)
−0.403362 + 0.915040i \(0.632159\pi\)
\(6\) −2.78140 −0.189250
\(7\) −11.3182 −0.611124 −0.305562 0.952172i \(-0.598844\pi\)
−0.305562 + 0.952172i \(0.598844\pi\)
\(8\) 21.5810 0.953753
\(9\) −24.0129 −0.889366
\(10\) 14.5150 0.459005
\(11\) −40.6469 −1.11414 −0.557069 0.830467i \(-0.688074\pi\)
−0.557069 + 0.830467i \(0.688074\pi\)
\(12\) −9.35051 −0.224938
\(13\) −12.3983 −0.264513 −0.132257 0.991216i \(-0.542222\pi\)
−0.132257 + 0.991216i \(0.542222\pi\)
\(14\) 18.2143 0.347714
\(15\) −15.5886 −0.268330
\(16\) 8.55095 0.133609
\(17\) 59.6717 0.851324 0.425662 0.904882i \(-0.360041\pi\)
0.425662 + 0.904882i \(0.360041\pi\)
\(18\) 38.6440 0.506026
\(19\) 26.4089 0.318874 0.159437 0.987208i \(-0.449032\pi\)
0.159437 + 0.987208i \(0.449032\pi\)
\(20\) 48.7966 0.545563
\(21\) −19.5615 −0.203270
\(22\) 65.4131 0.633915
\(23\) 107.097 0.970923 0.485461 0.874258i \(-0.338652\pi\)
0.485461 + 0.874258i \(0.338652\pi\)
\(24\) 37.2990 0.317234
\(25\) −43.6495 −0.349196
\(26\) 19.9526 0.150501
\(27\) −88.1670 −0.628435
\(28\) 61.2330 0.413284
\(29\) 173.657 1.11197 0.555987 0.831191i \(-0.312341\pi\)
0.555987 + 0.831191i \(0.312341\pi\)
\(30\) 25.0867 0.152673
\(31\) −332.301 −1.92526 −0.962629 0.270823i \(-0.912704\pi\)
−0.962629 + 0.270823i \(0.912704\pi\)
\(32\) −186.409 −1.02977
\(33\) −70.2512 −0.370581
\(34\) −96.0296 −0.484381
\(35\) 102.084 0.493009
\(36\) 129.913 0.601451
\(37\) −172.339 −0.765739 −0.382869 0.923802i \(-0.625064\pi\)
−0.382869 + 0.923802i \(0.625064\pi\)
\(38\) −42.4998 −0.181431
\(39\) −21.4283 −0.0879815
\(40\) −194.648 −0.769416
\(41\) −178.190 −0.678746 −0.339373 0.940652i \(-0.610215\pi\)
−0.339373 + 0.940652i \(0.610215\pi\)
\(42\) 31.4804 0.115655
\(43\) 63.2928 0.224467 0.112233 0.993682i \(-0.464200\pi\)
0.112233 + 0.993682i \(0.464200\pi\)
\(44\) 219.906 0.753456
\(45\) 216.583 0.717473
\(46\) −172.351 −0.552429
\(47\) 47.0000 0.145865
\(48\) 14.7788 0.0444404
\(49\) −214.899 −0.626527
\(50\) 70.2452 0.198683
\(51\) 103.132 0.283165
\(52\) 67.0767 0.178882
\(53\) −402.256 −1.04253 −0.521266 0.853395i \(-0.674540\pi\)
−0.521266 + 0.853395i \(0.674540\pi\)
\(54\) 141.887 0.357563
\(55\) 366.613 0.898801
\(56\) −244.257 −0.582861
\(57\) 45.6432 0.106063
\(58\) −279.466 −0.632684
\(59\) 305.280 0.673628 0.336814 0.941571i \(-0.390651\pi\)
0.336814 + 0.941571i \(0.390651\pi\)
\(60\) 84.3365 0.181463
\(61\) −86.2464 −0.181028 −0.0905141 0.995895i \(-0.528851\pi\)
−0.0905141 + 0.995895i \(0.528851\pi\)
\(62\) 534.772 1.09542
\(63\) 271.782 0.543513
\(64\) 231.580 0.452305
\(65\) 111.826 0.213389
\(66\) 113.055 0.210851
\(67\) −681.333 −1.24236 −0.621179 0.783668i \(-0.713346\pi\)
−0.621179 + 0.783668i \(0.713346\pi\)
\(68\) −322.833 −0.575724
\(69\) 185.098 0.322945
\(70\) −164.283 −0.280509
\(71\) 726.348 1.21411 0.607054 0.794661i \(-0.292351\pi\)
0.607054 + 0.794661i \(0.292351\pi\)
\(72\) −518.221 −0.848236
\(73\) −79.8711 −0.128058 −0.0640288 0.997948i \(-0.520395\pi\)
−0.0640288 + 0.997948i \(0.520395\pi\)
\(74\) 277.345 0.435685
\(75\) −75.4406 −0.116148
\(76\) −142.876 −0.215645
\(77\) 460.049 0.680876
\(78\) 34.4846 0.0500591
\(79\) 279.707 0.398348 0.199174 0.979964i \(-0.436174\pi\)
0.199174 + 0.979964i \(0.436174\pi\)
\(80\) −77.1248 −0.107785
\(81\) 495.966 0.680338
\(82\) 286.761 0.386189
\(83\) 556.598 0.736080 0.368040 0.929810i \(-0.380029\pi\)
0.368040 + 0.929810i \(0.380029\pi\)
\(84\) 105.831 0.137465
\(85\) −538.206 −0.686784
\(86\) −101.857 −0.127716
\(87\) 300.135 0.369861
\(88\) −877.200 −1.06261
\(89\) −342.069 −0.407408 −0.203704 0.979033i \(-0.565298\pi\)
−0.203704 + 0.979033i \(0.565298\pi\)
\(90\) −348.547 −0.408223
\(91\) 140.326 0.161650
\(92\) −579.410 −0.656605
\(93\) −574.325 −0.640373
\(94\) −75.6371 −0.0829933
\(95\) −238.194 −0.257244
\(96\) −322.175 −0.342520
\(97\) −1637.35 −1.71390 −0.856949 0.515402i \(-0.827643\pi\)
−0.856949 + 0.515402i \(0.827643\pi\)
\(98\) 345.837 0.356478
\(99\) 976.050 0.990876
\(100\) 236.150 0.236150
\(101\) −1453.29 −1.43176 −0.715879 0.698224i \(-0.753974\pi\)
−0.715879 + 0.698224i \(0.753974\pi\)
\(102\) −165.971 −0.161113
\(103\) 1363.10 1.30398 0.651992 0.758226i \(-0.273933\pi\)
0.651992 + 0.758226i \(0.273933\pi\)
\(104\) −267.567 −0.252280
\(105\) 176.434 0.163983
\(106\) 647.351 0.593173
\(107\) 274.746 0.248231 0.124116 0.992268i \(-0.460391\pi\)
0.124116 + 0.992268i \(0.460391\pi\)
\(108\) 476.997 0.424991
\(109\) 1448.56 1.27291 0.636454 0.771315i \(-0.280401\pi\)
0.636454 + 0.771315i \(0.280401\pi\)
\(110\) −589.990 −0.511394
\(111\) −297.858 −0.254698
\(112\) −96.7811 −0.0816514
\(113\) −1591.30 −1.32475 −0.662376 0.749171i \(-0.730452\pi\)
−0.662376 + 0.749171i \(0.730452\pi\)
\(114\) −73.4536 −0.0603470
\(115\) −965.954 −0.783267
\(116\) −939.509 −0.751993
\(117\) 297.719 0.235249
\(118\) −491.287 −0.383277
\(119\) −675.374 −0.520264
\(120\) −336.416 −0.255921
\(121\) 321.172 0.241301
\(122\) 138.796 0.103000
\(123\) −307.970 −0.225762
\(124\) 1797.80 1.30199
\(125\) 1521.13 1.08843
\(126\) −437.379 −0.309245
\(127\) 747.668 0.522400 0.261200 0.965285i \(-0.415882\pi\)
0.261200 + 0.965285i \(0.415882\pi\)
\(128\) 1118.59 0.772423
\(129\) 109.391 0.0746614
\(130\) −179.961 −0.121413
\(131\) −624.528 −0.416529 −0.208264 0.978073i \(-0.566781\pi\)
−0.208264 + 0.978073i \(0.566781\pi\)
\(132\) 380.070 0.250612
\(133\) −298.900 −0.194872
\(134\) 1096.47 0.706869
\(135\) 795.218 0.506974
\(136\) 1287.77 0.811953
\(137\) −336.418 −0.209796 −0.104898 0.994483i \(-0.533452\pi\)
−0.104898 + 0.994483i \(0.533452\pi\)
\(138\) −297.879 −0.183747
\(139\) 1940.28 1.18397 0.591987 0.805948i \(-0.298344\pi\)
0.591987 + 0.805948i \(0.298344\pi\)
\(140\) −552.288 −0.333406
\(141\) 81.2314 0.0485171
\(142\) −1168.91 −0.690795
\(143\) 503.953 0.294704
\(144\) −205.333 −0.118827
\(145\) −1566.29 −0.897056
\(146\) 128.537 0.0728614
\(147\) −371.416 −0.208394
\(148\) 932.380 0.517845
\(149\) 2810.71 1.54538 0.772692 0.634781i \(-0.218910\pi\)
0.772692 + 0.634781i \(0.218910\pi\)
\(150\) 121.407 0.0660854
\(151\) 1710.82 0.922014 0.461007 0.887396i \(-0.347488\pi\)
0.461007 + 0.887396i \(0.347488\pi\)
\(152\) 569.929 0.304127
\(153\) −1432.89 −0.757138
\(154\) −740.357 −0.387401
\(155\) 2997.17 1.55315
\(156\) 115.930 0.0594991
\(157\) −1378.36 −0.700669 −0.350334 0.936625i \(-0.613932\pi\)
−0.350334 + 0.936625i \(0.613932\pi\)
\(158\) −450.133 −0.226650
\(159\) −695.231 −0.346763
\(160\) 1681.30 0.830743
\(161\) −1212.14 −0.593354
\(162\) −798.159 −0.387095
\(163\) −3255.46 −1.56434 −0.782170 0.623066i \(-0.785887\pi\)
−0.782170 + 0.623066i \(0.785887\pi\)
\(164\) 964.034 0.459015
\(165\) 633.627 0.298956
\(166\) −895.734 −0.418810
\(167\) −2337.08 −1.08293 −0.541463 0.840725i \(-0.682129\pi\)
−0.541463 + 0.840725i \(0.682129\pi\)
\(168\) −422.156 −0.193869
\(169\) −2043.28 −0.930033
\(170\) 866.135 0.390762
\(171\) −634.153 −0.283596
\(172\) −342.424 −0.151800
\(173\) −3269.43 −1.43682 −0.718412 0.695618i \(-0.755130\pi\)
−0.718412 + 0.695618i \(0.755130\pi\)
\(174\) −483.008 −0.210441
\(175\) 494.033 0.213402
\(176\) −347.570 −0.148858
\(177\) 527.624 0.224060
\(178\) 550.492 0.231804
\(179\) −2203.96 −0.920290 −0.460145 0.887844i \(-0.652203\pi\)
−0.460145 + 0.887844i \(0.652203\pi\)
\(180\) −1171.75 −0.485205
\(181\) 1522.63 0.625282 0.312641 0.949871i \(-0.398786\pi\)
0.312641 + 0.949871i \(0.398786\pi\)
\(182\) −225.827 −0.0919748
\(183\) −149.062 −0.0602130
\(184\) 2311.25 0.926020
\(185\) 1554.40 0.617740
\(186\) 924.261 0.364356
\(187\) −2425.47 −0.948491
\(188\) −254.277 −0.0986440
\(189\) 997.889 0.384052
\(190\) 383.325 0.146365
\(191\) −2019.27 −0.764968 −0.382484 0.923962i \(-0.624931\pi\)
−0.382484 + 0.923962i \(0.624931\pi\)
\(192\) 400.246 0.150444
\(193\) −4447.95 −1.65891 −0.829456 0.558571i \(-0.811350\pi\)
−0.829456 + 0.558571i \(0.811350\pi\)
\(194\) 2634.99 0.975163
\(195\) 193.272 0.0709768
\(196\) 1162.64 0.423701
\(197\) 1790.04 0.647388 0.323694 0.946162i \(-0.395075\pi\)
0.323694 + 0.946162i \(0.395075\pi\)
\(198\) −1570.76 −0.563782
\(199\) 3481.42 1.24016 0.620079 0.784540i \(-0.287101\pi\)
0.620079 + 0.784540i \(0.287101\pi\)
\(200\) −941.998 −0.333047
\(201\) −1177.57 −0.413229
\(202\) 2338.78 0.814633
\(203\) −1965.48 −0.679554
\(204\) −557.961 −0.191495
\(205\) 1607.18 0.547561
\(206\) −2193.64 −0.741933
\(207\) −2571.70 −0.863506
\(208\) −106.017 −0.0353412
\(209\) −1073.44 −0.355270
\(210\) −283.936 −0.0933020
\(211\) 5216.72 1.70206 0.851028 0.525120i \(-0.175979\pi\)
0.851028 + 0.525120i \(0.175979\pi\)
\(212\) 2176.27 0.705031
\(213\) 1255.37 0.403833
\(214\) −442.150 −0.141237
\(215\) −570.867 −0.181083
\(216\) −1902.73 −0.599372
\(217\) 3761.04 1.17657
\(218\) −2331.17 −0.724251
\(219\) −138.043 −0.0425941
\(220\) −1983.43 −0.607831
\(221\) −739.827 −0.225186
\(222\) 479.343 0.144916
\(223\) 5075.44 1.52411 0.762055 0.647512i \(-0.224190\pi\)
0.762055 + 0.647512i \(0.224190\pi\)
\(224\) 2109.81 0.629319
\(225\) 1048.15 0.310563
\(226\) 2560.88 0.753749
\(227\) 1752.66 0.512460 0.256230 0.966616i \(-0.417520\pi\)
0.256230 + 0.966616i \(0.417520\pi\)
\(228\) −246.936 −0.0717270
\(229\) −4676.77 −1.34956 −0.674781 0.738018i \(-0.735762\pi\)
−0.674781 + 0.738018i \(0.735762\pi\)
\(230\) 1554.51 0.445658
\(231\) 795.115 0.226471
\(232\) 3747.68 1.06055
\(233\) 4261.89 1.19831 0.599154 0.800634i \(-0.295504\pi\)
0.599154 + 0.800634i \(0.295504\pi\)
\(234\) −479.119 −0.133850
\(235\) −423.914 −0.117673
\(236\) −1651.61 −0.455554
\(237\) 483.426 0.132497
\(238\) 1086.88 0.296017
\(239\) −3227.38 −0.873479 −0.436740 0.899588i \(-0.643867\pi\)
−0.436740 + 0.899588i \(0.643867\pi\)
\(240\) −133.297 −0.0358512
\(241\) −1310.37 −0.350242 −0.175121 0.984547i \(-0.556032\pi\)
−0.175121 + 0.984547i \(0.556032\pi\)
\(242\) −516.863 −0.137294
\(243\) 3237.70 0.854727
\(244\) 466.606 0.122424
\(245\) 1938.27 0.505435
\(246\) 495.617 0.128453
\(247\) −327.425 −0.0843464
\(248\) −7171.38 −1.83622
\(249\) 961.984 0.244832
\(250\) −2447.95 −0.619288
\(251\) 4906.61 1.23387 0.616937 0.787013i \(-0.288373\pi\)
0.616937 + 0.787013i \(0.288373\pi\)
\(252\) −1470.38 −0.367561
\(253\) −4353.15 −1.08174
\(254\) −1203.22 −0.297232
\(255\) −930.196 −0.228436
\(256\) −3652.79 −0.891793
\(257\) 2103.21 0.510485 0.255243 0.966877i \(-0.417845\pi\)
0.255243 + 0.966877i \(0.417845\pi\)
\(258\) −176.043 −0.0424804
\(259\) 1950.56 0.467962
\(260\) −604.995 −0.144308
\(261\) −4170.00 −0.988951
\(262\) 1005.05 0.236994
\(263\) −6993.90 −1.63978 −0.819890 0.572521i \(-0.805965\pi\)
−0.819890 + 0.572521i \(0.805965\pi\)
\(264\) −1516.09 −0.353442
\(265\) 3628.13 0.841035
\(266\) 481.020 0.110877
\(267\) −591.208 −0.135511
\(268\) 3686.11 0.840168
\(269\) 6466.63 1.46571 0.732857 0.680382i \(-0.238186\pi\)
0.732857 + 0.680382i \(0.238186\pi\)
\(270\) −1279.74 −0.288455
\(271\) −8154.65 −1.82790 −0.913948 0.405831i \(-0.866982\pi\)
−0.913948 + 0.405831i \(0.866982\pi\)
\(272\) 510.249 0.113744
\(273\) 242.530 0.0537676
\(274\) 541.397 0.119369
\(275\) 1774.22 0.389052
\(276\) −1001.41 −0.218398
\(277\) 3722.71 0.807495 0.403748 0.914870i \(-0.367707\pi\)
0.403748 + 0.914870i \(0.367707\pi\)
\(278\) −3122.49 −0.673650
\(279\) 7979.50 1.71226
\(280\) 2203.07 0.470208
\(281\) −6046.33 −1.28361 −0.641804 0.766869i \(-0.721814\pi\)
−0.641804 + 0.766869i \(0.721814\pi\)
\(282\) −130.726 −0.0276050
\(283\) 268.000 0.0562931 0.0281466 0.999604i \(-0.491039\pi\)
0.0281466 + 0.999604i \(0.491039\pi\)
\(284\) −3929.65 −0.821063
\(285\) −411.676 −0.0855635
\(286\) −811.011 −0.167679
\(287\) 2016.78 0.414798
\(288\) 4476.21 0.915845
\(289\) −1352.29 −0.275248
\(290\) 2520.63 0.510401
\(291\) −2829.88 −0.570071
\(292\) 432.115 0.0866014
\(293\) 2731.03 0.544535 0.272267 0.962222i \(-0.412226\pi\)
0.272267 + 0.962222i \(0.412226\pi\)
\(294\) 597.719 0.118570
\(295\) −2753.46 −0.543432
\(296\) −3719.24 −0.730326
\(297\) 3583.72 0.700163
\(298\) −4523.28 −0.879283
\(299\) −1327.82 −0.256822
\(300\) 408.145 0.0785476
\(301\) −716.360 −0.137177
\(302\) −2753.22 −0.524602
\(303\) −2511.76 −0.476227
\(304\) 225.821 0.0426043
\(305\) 777.896 0.146040
\(306\) 2305.95 0.430792
\(307\) −7009.57 −1.30312 −0.651559 0.758598i \(-0.725885\pi\)
−0.651559 + 0.758598i \(0.725885\pi\)
\(308\) −2488.93 −0.460455
\(309\) 2355.89 0.433727
\(310\) −4823.35 −0.883703
\(311\) −867.418 −0.158157 −0.0790784 0.996868i \(-0.525198\pi\)
−0.0790784 + 0.996868i \(0.525198\pi\)
\(312\) −462.444 −0.0839126
\(313\) 5724.10 1.03369 0.516845 0.856079i \(-0.327106\pi\)
0.516845 + 0.856079i \(0.327106\pi\)
\(314\) 2218.19 0.398662
\(315\) −2451.33 −0.438465
\(316\) −1513.26 −0.269391
\(317\) 5821.24 1.03140 0.515700 0.856769i \(-0.327532\pi\)
0.515700 + 0.856769i \(0.327532\pi\)
\(318\) 1118.84 0.197299
\(319\) −7058.61 −1.23889
\(320\) −2088.73 −0.364886
\(321\) 474.852 0.0825659
\(322\) 1950.70 0.337603
\(323\) 1575.86 0.271465
\(324\) −2683.25 −0.460092
\(325\) 541.179 0.0923669
\(326\) 5239.01 0.890068
\(327\) 2503.59 0.423390
\(328\) −3845.51 −0.647356
\(329\) −531.954 −0.0891416
\(330\) −1019.70 −0.170098
\(331\) −2300.44 −0.382005 −0.191002 0.981590i \(-0.561174\pi\)
−0.191002 + 0.981590i \(0.561174\pi\)
\(332\) −3011.28 −0.497788
\(333\) 4138.35 0.681022
\(334\) 3761.06 0.616156
\(335\) 6145.25 1.00224
\(336\) −167.269 −0.0271586
\(337\) −6399.32 −1.03440 −0.517201 0.855864i \(-0.673026\pi\)
−0.517201 + 0.855864i \(0.673026\pi\)
\(338\) 3288.26 0.529164
\(339\) −2750.29 −0.440635
\(340\) 2911.77 0.464450
\(341\) 13507.0 2.14500
\(342\) 1020.54 0.161359
\(343\) 6314.40 0.994010
\(344\) 1365.92 0.214086
\(345\) −1669.48 −0.260528
\(346\) 5261.50 0.817515
\(347\) −9188.96 −1.42158 −0.710791 0.703403i \(-0.751663\pi\)
−0.710791 + 0.703403i \(0.751663\pi\)
\(348\) −1623.78 −0.250125
\(349\) 6327.24 0.970457 0.485229 0.874387i \(-0.338736\pi\)
0.485229 + 0.874387i \(0.338736\pi\)
\(350\) −795.047 −0.121420
\(351\) 1093.12 0.166229
\(352\) 7576.94 1.14731
\(353\) 8212.76 1.23830 0.619152 0.785271i \(-0.287477\pi\)
0.619152 + 0.785271i \(0.287477\pi\)
\(354\) −849.105 −0.127484
\(355\) −6551.26 −0.979450
\(356\) 1850.65 0.275517
\(357\) −1167.27 −0.173049
\(358\) 3546.84 0.523621
\(359\) −5614.24 −0.825371 −0.412685 0.910874i \(-0.635409\pi\)
−0.412685 + 0.910874i \(0.635409\pi\)
\(360\) 4674.07 0.684292
\(361\) −6161.57 −0.898319
\(362\) −2450.37 −0.355769
\(363\) 555.091 0.0802609
\(364\) −759.186 −0.109319
\(365\) 720.394 0.103307
\(366\) 239.886 0.0342596
\(367\) −3282.91 −0.466938 −0.233469 0.972364i \(-0.575008\pi\)
−0.233469 + 0.972364i \(0.575008\pi\)
\(368\) 915.778 0.129724
\(369\) 4278.85 0.603654
\(370\) −2501.50 −0.351478
\(371\) 4552.81 0.637116
\(372\) 3107.18 0.433065
\(373\) −9749.67 −1.35340 −0.676701 0.736258i \(-0.736591\pi\)
−0.676701 + 0.736258i \(0.736591\pi\)
\(374\) 3903.31 0.539667
\(375\) 2629.00 0.362030
\(376\) 1014.31 0.139119
\(377\) −2153.05 −0.294131
\(378\) −1605.90 −0.218515
\(379\) −7875.26 −1.06735 −0.533674 0.845690i \(-0.679189\pi\)
−0.533674 + 0.845690i \(0.679189\pi\)
\(380\) 1288.66 0.173966
\(381\) 1292.21 0.173759
\(382\) 3249.61 0.435247
\(383\) −1599.65 −0.213416 −0.106708 0.994290i \(-0.534031\pi\)
−0.106708 + 0.994290i \(0.534031\pi\)
\(384\) 1933.29 0.256921
\(385\) −4149.39 −0.549279
\(386\) 7158.08 0.943878
\(387\) −1519.84 −0.199633
\(388\) 8858.33 1.15906
\(389\) −5101.31 −0.664901 −0.332451 0.943121i \(-0.607876\pi\)
−0.332451 + 0.943121i \(0.607876\pi\)
\(390\) −311.032 −0.0403839
\(391\) 6390.64 0.826569
\(392\) −4637.73 −0.597552
\(393\) −1079.39 −0.138544
\(394\) −2880.72 −0.368347
\(395\) −2522.81 −0.321357
\(396\) −5280.58 −0.670098
\(397\) −8835.01 −1.11692 −0.558459 0.829532i \(-0.688607\pi\)
−0.558459 + 0.829532i \(0.688607\pi\)
\(398\) −5602.65 −0.705617
\(399\) −516.597 −0.0648176
\(400\) −373.244 −0.0466555
\(401\) −15444.1 −1.92330 −0.961649 0.274284i \(-0.911559\pi\)
−0.961649 + 0.274284i \(0.911559\pi\)
\(402\) 1895.06 0.235117
\(403\) 4119.97 0.509256
\(404\) 7862.51 0.968253
\(405\) −4473.35 −0.548845
\(406\) 3163.04 0.386648
\(407\) 7005.05 0.853138
\(408\) 2225.69 0.270069
\(409\) −13309.7 −1.60910 −0.804552 0.593882i \(-0.797595\pi\)
−0.804552 + 0.593882i \(0.797595\pi\)
\(410\) −2586.43 −0.311548
\(411\) −581.440 −0.0697818
\(412\) −7374.59 −0.881844
\(413\) −3455.21 −0.411671
\(414\) 4138.64 0.491312
\(415\) −5020.21 −0.593813
\(416\) 2311.15 0.272388
\(417\) 3353.44 0.393809
\(418\) 1727.49 0.202139
\(419\) −2074.44 −0.241869 −0.120934 0.992661i \(-0.538589\pi\)
−0.120934 + 0.992661i \(0.538589\pi\)
\(420\) −954.535 −0.110897
\(421\) 14903.2 1.72526 0.862631 0.505833i \(-0.168815\pi\)
0.862631 + 0.505833i \(0.168815\pi\)
\(422\) −8395.27 −0.968426
\(423\) −1128.61 −0.129727
\(424\) −8681.08 −0.994317
\(425\) −2604.64 −0.297279
\(426\) −2020.26 −0.229770
\(427\) 976.152 0.110631
\(428\) −1486.42 −0.167871
\(429\) 870.995 0.0980234
\(430\) 918.696 0.103031
\(431\) 7434.74 0.830902 0.415451 0.909616i \(-0.363624\pi\)
0.415451 + 0.909616i \(0.363624\pi\)
\(432\) −753.911 −0.0839642
\(433\) 4385.47 0.486725 0.243363 0.969935i \(-0.421749\pi\)
0.243363 + 0.969935i \(0.421749\pi\)
\(434\) −6052.65 −0.669438
\(435\) −2707.06 −0.298376
\(436\) −7836.93 −0.860828
\(437\) 2828.30 0.309602
\(438\) 222.153 0.0242349
\(439\) −10300.7 −1.11988 −0.559939 0.828534i \(-0.689175\pi\)
−0.559939 + 0.828534i \(0.689175\pi\)
\(440\) 7911.86 0.857235
\(441\) 5160.34 0.557212
\(442\) 1190.60 0.128125
\(443\) 5700.48 0.611372 0.305686 0.952132i \(-0.401114\pi\)
0.305686 + 0.952132i \(0.401114\pi\)
\(444\) 1611.46 0.172244
\(445\) 3085.28 0.328666
\(446\) −8167.91 −0.867179
\(447\) 4857.83 0.514021
\(448\) −2621.07 −0.276415
\(449\) −8946.30 −0.940317 −0.470158 0.882582i \(-0.655803\pi\)
−0.470158 + 0.882582i \(0.655803\pi\)
\(450\) −1686.79 −0.176702
\(451\) 7242.87 0.756216
\(452\) 8609.18 0.895889
\(453\) 2956.85 0.306677
\(454\) −2820.56 −0.291576
\(455\) −1265.66 −0.130407
\(456\) 985.024 0.101158
\(457\) −16498.3 −1.68875 −0.844374 0.535755i \(-0.820027\pi\)
−0.844374 + 0.535755i \(0.820027\pi\)
\(458\) 7526.33 0.767865
\(459\) −5261.07 −0.535001
\(460\) 5225.96 0.529699
\(461\) 9016.08 0.910891 0.455445 0.890264i \(-0.349480\pi\)
0.455445 + 0.890264i \(0.349480\pi\)
\(462\) −1279.58 −0.128856
\(463\) 3241.41 0.325359 0.162679 0.986679i \(-0.447986\pi\)
0.162679 + 0.986679i \(0.447986\pi\)
\(464\) 1484.93 0.148569
\(465\) 5180.10 0.516605
\(466\) −6858.67 −0.681806
\(467\) 12186.1 1.20751 0.603755 0.797170i \(-0.293671\pi\)
0.603755 + 0.797170i \(0.293671\pi\)
\(468\) −1610.70 −0.159092
\(469\) 7711.44 0.759235
\(470\) 682.205 0.0669528
\(471\) −2382.25 −0.233054
\(472\) 6588.24 0.642475
\(473\) −2572.66 −0.250087
\(474\) −777.977 −0.0753875
\(475\) −1152.73 −0.111350
\(476\) 3653.88 0.351839
\(477\) 9659.33 0.927192
\(478\) 5193.82 0.496987
\(479\) −9110.87 −0.869074 −0.434537 0.900654i \(-0.643088\pi\)
−0.434537 + 0.900654i \(0.643088\pi\)
\(480\) 2905.85 0.276319
\(481\) 2136.71 0.202548
\(482\) 2108.78 0.199279
\(483\) −2094.97 −0.197360
\(484\) −1737.59 −0.163185
\(485\) 14768.0 1.38264
\(486\) −5210.43 −0.486317
\(487\) 535.218 0.0498009 0.0249004 0.999690i \(-0.492073\pi\)
0.0249004 + 0.999690i \(0.492073\pi\)
\(488\) −1861.28 −0.172656
\(489\) −5626.50 −0.520325
\(490\) −3119.26 −0.287579
\(491\) 10810.8 0.993658 0.496829 0.867848i \(-0.334498\pi\)
0.496829 + 0.867848i \(0.334498\pi\)
\(492\) 1666.17 0.152676
\(493\) 10362.4 0.946649
\(494\) 526.925 0.0479909
\(495\) −8803.43 −0.799364
\(496\) −2841.49 −0.257231
\(497\) −8220.93 −0.741970
\(498\) −1548.12 −0.139303
\(499\) 16736.4 1.50146 0.750728 0.660612i \(-0.229703\pi\)
0.750728 + 0.660612i \(0.229703\pi\)
\(500\) −8229.52 −0.736071
\(501\) −4039.24 −0.360199
\(502\) −7896.21 −0.702042
\(503\) 6612.65 0.586170 0.293085 0.956086i \(-0.405318\pi\)
0.293085 + 0.956086i \(0.405318\pi\)
\(504\) 5865.32 0.518377
\(505\) 13107.9 1.15503
\(506\) 7005.53 0.615482
\(507\) −3531.46 −0.309345
\(508\) −4045.00 −0.353283
\(509\) 14953.5 1.30216 0.651081 0.759008i \(-0.274316\pi\)
0.651081 + 0.759008i \(0.274316\pi\)
\(510\) 1496.96 0.129974
\(511\) 903.995 0.0782591
\(512\) −3070.27 −0.265016
\(513\) −2328.39 −0.200392
\(514\) −3384.70 −0.290453
\(515\) −12294.4 −1.05196
\(516\) −591.820 −0.0504912
\(517\) −1910.41 −0.162514
\(518\) −3139.04 −0.266258
\(519\) −5650.65 −0.477911
\(520\) 2413.31 0.203520
\(521\) 822.169 0.0691361 0.0345680 0.999402i \(-0.488994\pi\)
0.0345680 + 0.999402i \(0.488994\pi\)
\(522\) 6710.78 0.562687
\(523\) 97.2265 0.00812890 0.00406445 0.999992i \(-0.498706\pi\)
0.00406445 + 0.999992i \(0.498706\pi\)
\(524\) 3378.79 0.281685
\(525\) 853.850 0.0709811
\(526\) 11255.3 0.932992
\(527\) −19828.9 −1.63902
\(528\) −600.714 −0.0495127
\(529\) −697.284 −0.0573095
\(530\) −5838.75 −0.478527
\(531\) −7330.65 −0.599102
\(532\) 1617.10 0.131786
\(533\) 2209.25 0.179537
\(534\) 951.431 0.0771020
\(535\) −2478.06 −0.200254
\(536\) −14703.8 −1.18490
\(537\) −3809.17 −0.306104
\(538\) −10406.7 −0.833953
\(539\) 8734.98 0.698037
\(540\) −4302.25 −0.342851
\(541\) −3856.31 −0.306462 −0.153231 0.988190i \(-0.548968\pi\)
−0.153231 + 0.988190i \(0.548968\pi\)
\(542\) 13123.3 1.04003
\(543\) 2631.60 0.207979
\(544\) −11123.3 −0.876670
\(545\) −13065.2 −1.02689
\(546\) −390.303 −0.0305923
\(547\) −4147.40 −0.324186 −0.162093 0.986775i \(-0.551825\pi\)
−0.162093 + 0.986775i \(0.551825\pi\)
\(548\) 1820.07 0.141879
\(549\) 2071.03 0.161000
\(550\) −2855.25 −0.221360
\(551\) 4586.07 0.354580
\(552\) 3994.60 0.308010
\(553\) −3165.78 −0.243440
\(554\) −5990.96 −0.459443
\(555\) 2686.52 0.205471
\(556\) −10497.2 −0.800684
\(557\) 85.2280 0.00648335 0.00324168 0.999995i \(-0.498968\pi\)
0.00324168 + 0.999995i \(0.498968\pi\)
\(558\) −12841.4 −0.974231
\(559\) −784.724 −0.0593744
\(560\) 872.913 0.0658702
\(561\) −4192.00 −0.315484
\(562\) 9730.37 0.730340
\(563\) 12824.3 0.960003 0.480001 0.877268i \(-0.340636\pi\)
0.480001 + 0.877268i \(0.340636\pi\)
\(564\) −439.474 −0.0328106
\(565\) 14352.7 1.06871
\(566\) −431.293 −0.0320293
\(567\) −5613.44 −0.415771
\(568\) 15675.3 1.15796
\(569\) −12096.8 −0.891253 −0.445626 0.895219i \(-0.647019\pi\)
−0.445626 + 0.895219i \(0.647019\pi\)
\(570\) 662.511 0.0486834
\(571\) 5688.49 0.416910 0.208455 0.978032i \(-0.433156\pi\)
0.208455 + 0.978032i \(0.433156\pi\)
\(572\) −2726.46 −0.199299
\(573\) −3489.95 −0.254441
\(574\) −3245.61 −0.236009
\(575\) −4674.72 −0.339042
\(576\) −5560.91 −0.402265
\(577\) −9878.78 −0.712754 −0.356377 0.934342i \(-0.615988\pi\)
−0.356377 + 0.934342i \(0.615988\pi\)
\(578\) 2176.25 0.156609
\(579\) −7687.51 −0.551782
\(580\) 8473.85 0.606651
\(581\) −6299.68 −0.449836
\(582\) 4554.13 0.324355
\(583\) 16350.5 1.16152
\(584\) −1723.70 −0.122135
\(585\) −2685.26 −0.189781
\(586\) −4395.05 −0.309826
\(587\) −17342.9 −1.21945 −0.609726 0.792613i \(-0.708720\pi\)
−0.609726 + 0.792613i \(0.708720\pi\)
\(588\) 2009.41 0.140930
\(589\) −8775.69 −0.613915
\(590\) 4431.14 0.309199
\(591\) 3093.78 0.215332
\(592\) −1473.66 −0.102309
\(593\) 698.313 0.0483580 0.0241790 0.999708i \(-0.492303\pi\)
0.0241790 + 0.999708i \(0.492303\pi\)
\(594\) −5767.28 −0.398374
\(595\) 6091.51 0.419710
\(596\) −15206.4 −1.04510
\(597\) 6017.03 0.412497
\(598\) 2136.86 0.146125
\(599\) 9549.46 0.651386 0.325693 0.945476i \(-0.394402\pi\)
0.325693 + 0.945476i \(0.394402\pi\)
\(600\) −1628.08 −0.110777
\(601\) 27369.5 1.85761 0.928805 0.370568i \(-0.120837\pi\)
0.928805 + 0.370568i \(0.120837\pi\)
\(602\) 1152.84 0.0780501
\(603\) 16360.8 1.10491
\(604\) −9255.77 −0.623530
\(605\) −2896.80 −0.194664
\(606\) 4042.17 0.270960
\(607\) 18179.8 1.21564 0.607821 0.794074i \(-0.292044\pi\)
0.607821 + 0.794074i \(0.292044\pi\)
\(608\) −4922.84 −0.328368
\(609\) −3396.99 −0.226031
\(610\) −1251.87 −0.0830929
\(611\) −582.720 −0.0385832
\(612\) 7752.14 0.512029
\(613\) −3148.69 −0.207462 −0.103731 0.994605i \(-0.533078\pi\)
−0.103731 + 0.994605i \(0.533078\pi\)
\(614\) 11280.5 0.741440
\(615\) 2777.72 0.182128
\(616\) 9928.30 0.649387
\(617\) −24002.0 −1.56610 −0.783050 0.621959i \(-0.786337\pi\)
−0.783050 + 0.621959i \(0.786337\pi\)
\(618\) −3791.33 −0.246779
\(619\) 17997.2 1.16861 0.584304 0.811535i \(-0.301367\pi\)
0.584304 + 0.811535i \(0.301367\pi\)
\(620\) −16215.2 −1.05035
\(621\) −9442.40 −0.610162
\(622\) 1395.94 0.0899871
\(623\) 3871.60 0.248977
\(624\) −183.232 −0.0117551
\(625\) −8263.54 −0.528866
\(626\) −9211.80 −0.588143
\(627\) −1855.25 −0.118169
\(628\) 7457.13 0.473840
\(629\) −10283.7 −0.651892
\(630\) 3944.92 0.249475
\(631\) 5508.64 0.347536 0.173768 0.984787i \(-0.444406\pi\)
0.173768 + 0.984787i \(0.444406\pi\)
\(632\) 6036.35 0.379926
\(633\) 9016.20 0.566133
\(634\) −9368.13 −0.586839
\(635\) −6743.55 −0.421433
\(636\) 3761.30 0.234505
\(637\) 2664.38 0.165725
\(638\) 11359.4 0.704896
\(639\) −17441.7 −1.07979
\(640\) −10089.1 −0.623132
\(641\) 11605.9 0.715142 0.357571 0.933886i \(-0.383605\pi\)
0.357571 + 0.933886i \(0.383605\pi\)
\(642\) −764.179 −0.0469778
\(643\) 7998.88 0.490583 0.245292 0.969449i \(-0.421116\pi\)
0.245292 + 0.969449i \(0.421116\pi\)
\(644\) 6557.86 0.401267
\(645\) −986.645 −0.0602312
\(646\) −2536.03 −0.154457
\(647\) −5421.64 −0.329438 −0.164719 0.986341i \(-0.552672\pi\)
−0.164719 + 0.986341i \(0.552672\pi\)
\(648\) 10703.4 0.648875
\(649\) −12408.7 −0.750514
\(650\) −870.920 −0.0525543
\(651\) 6500.31 0.391348
\(652\) 17612.5 1.05791
\(653\) −21419.1 −1.28361 −0.641804 0.766869i \(-0.721814\pi\)
−0.641804 + 0.766869i \(0.721814\pi\)
\(654\) −4029.02 −0.240898
\(655\) 5632.90 0.336024
\(656\) −1523.69 −0.0906862
\(657\) 1917.94 0.113890
\(658\) 856.074 0.0507192
\(659\) −11435.7 −0.675982 −0.337991 0.941149i \(-0.609747\pi\)
−0.337991 + 0.941149i \(0.609747\pi\)
\(660\) −3428.02 −0.202175
\(661\) 27123.2 1.59602 0.798011 0.602643i \(-0.205886\pi\)
0.798011 + 0.602643i \(0.205886\pi\)
\(662\) 3702.10 0.217351
\(663\) −1278.66 −0.0749007
\(664\) 12011.9 0.702038
\(665\) 2695.92 0.157208
\(666\) −6659.86 −0.387484
\(667\) 18598.1 1.07964
\(668\) 12644.0 0.732349
\(669\) 8772.02 0.506945
\(670\) −9889.55 −0.570249
\(671\) 3505.65 0.201690
\(672\) 3646.44 0.209322
\(673\) −11097.2 −0.635612 −0.317806 0.948156i \(-0.602946\pi\)
−0.317806 + 0.948156i \(0.602946\pi\)
\(674\) 10298.4 0.588547
\(675\) 3848.44 0.219447
\(676\) 11054.5 0.628952
\(677\) −1647.83 −0.0935468 −0.0467734 0.998906i \(-0.514894\pi\)
−0.0467734 + 0.998906i \(0.514894\pi\)
\(678\) 4426.04 0.250710
\(679\) 18531.9 1.04740
\(680\) −11615.0 −0.655022
\(681\) 3029.18 0.170453
\(682\) −21736.8 −1.22045
\(683\) 9256.41 0.518575 0.259287 0.965800i \(-0.416512\pi\)
0.259287 + 0.965800i \(0.416512\pi\)
\(684\) 3430.86 0.191787
\(685\) 3034.30 0.169248
\(686\) −10161.8 −0.565566
\(687\) −8082.99 −0.448887
\(688\) 541.214 0.0299907
\(689\) 4987.29 0.275763
\(690\) 2686.70 0.148233
\(691\) −2404.88 −0.132397 −0.0661983 0.997806i \(-0.521087\pi\)
−0.0661983 + 0.997806i \(0.521087\pi\)
\(692\) 17688.1 0.971679
\(693\) −11047.1 −0.605548
\(694\) 14787.8 0.808843
\(695\) −17500.3 −0.955140
\(696\) 6477.21 0.352756
\(697\) −10632.9 −0.577833
\(698\) −10182.4 −0.552165
\(699\) 7365.94 0.398577
\(700\) −2672.79 −0.144317
\(701\) 1158.57 0.0624232 0.0312116 0.999513i \(-0.490063\pi\)
0.0312116 + 0.999513i \(0.490063\pi\)
\(702\) −1759.16 −0.0945801
\(703\) −4551.28 −0.244174
\(704\) −9413.02 −0.503930
\(705\) −732.663 −0.0391400
\(706\) −13216.8 −0.704562
\(707\) 16448.6 0.874982
\(708\) −2854.52 −0.151525
\(709\) −26868.0 −1.42320 −0.711600 0.702585i \(-0.752029\pi\)
−0.711600 + 0.702585i \(0.752029\pi\)
\(710\) 10542.9 0.557281
\(711\) −6716.58 −0.354278
\(712\) −7382.19 −0.388566
\(713\) −35588.4 −1.86928
\(714\) 1878.49 0.0984601
\(715\) −4545.38 −0.237745
\(716\) 11923.8 0.622363
\(717\) −5577.96 −0.290534
\(718\) 9035.00 0.469614
\(719\) 12313.1 0.638666 0.319333 0.947643i \(-0.396541\pi\)
0.319333 + 0.947643i \(0.396541\pi\)
\(720\) 1851.99 0.0958605
\(721\) −15427.8 −0.796896
\(722\) 9915.82 0.511120
\(723\) −2264.75 −0.116496
\(724\) −8237.65 −0.422859
\(725\) −7580.02 −0.388296
\(726\) −893.308 −0.0456663
\(727\) 4104.02 0.209367 0.104683 0.994506i \(-0.466617\pi\)
0.104683 + 0.994506i \(0.466617\pi\)
\(728\) 3028.37 0.154174
\(729\) −7795.29 −0.396042
\(730\) −1159.33 −0.0587791
\(731\) 3776.79 0.191094
\(732\) 806.448 0.0407202
\(733\) 36113.9 1.81978 0.909888 0.414854i \(-0.136167\pi\)
0.909888 + 0.414854i \(0.136167\pi\)
\(734\) 5283.18 0.265675
\(735\) 3349.97 0.168116
\(736\) −19963.8 −0.999829
\(737\) 27694.1 1.38416
\(738\) −6885.96 −0.343463
\(739\) −4185.77 −0.208357 −0.104179 0.994559i \(-0.533221\pi\)
−0.104179 + 0.994559i \(0.533221\pi\)
\(740\) −8409.55 −0.417759
\(741\) −565.898 −0.0280550
\(742\) −7326.84 −0.362502
\(743\) −24708.4 −1.22001 −0.610003 0.792399i \(-0.708832\pi\)
−0.610003 + 0.792399i \(0.708832\pi\)
\(744\) −12394.5 −0.610758
\(745\) −25351.1 −1.24670
\(746\) 15690.2 0.770050
\(747\) −13365.5 −0.654644
\(748\) 13122.2 0.641435
\(749\) −3109.63 −0.151700
\(750\) −4230.86 −0.205985
\(751\) 24154.5 1.17365 0.586824 0.809714i \(-0.300378\pi\)
0.586824 + 0.809714i \(0.300378\pi\)
\(752\) 401.894 0.0194888
\(753\) 8480.22 0.410407
\(754\) 3464.90 0.167353
\(755\) −15430.6 −0.743811
\(756\) −5398.73 −0.259722
\(757\) −29215.5 −1.40272 −0.701358 0.712809i \(-0.747423\pi\)
−0.701358 + 0.712809i \(0.747423\pi\)
\(758\) 12673.7 0.607293
\(759\) −7523.67 −0.359805
\(760\) −5140.45 −0.245347
\(761\) 29582.7 1.40916 0.704581 0.709624i \(-0.251135\pi\)
0.704581 + 0.709624i \(0.251135\pi\)
\(762\) −2079.56 −0.0988643
\(763\) −16395.1 −0.777905
\(764\) 10924.5 0.517324
\(765\) 12923.9 0.610802
\(766\) 2574.32 0.121428
\(767\) −3784.95 −0.178183
\(768\) −6313.21 −0.296625
\(769\) −30737.1 −1.44136 −0.720682 0.693265i \(-0.756171\pi\)
−0.720682 + 0.693265i \(0.756171\pi\)
\(770\) 6677.62 0.312525
\(771\) 3635.04 0.169796
\(772\) 24064.1 1.12187
\(773\) −13581.6 −0.631947 −0.315973 0.948768i \(-0.602331\pi\)
−0.315973 + 0.948768i \(0.602331\pi\)
\(774\) 2445.89 0.113586
\(775\) 14504.8 0.672292
\(776\) −35335.7 −1.63463
\(777\) 3371.21 0.155652
\(778\) 8209.54 0.378311
\(779\) −4705.79 −0.216435
\(780\) −1045.63 −0.0479994
\(781\) −29523.8 −1.35268
\(782\) −10284.5 −0.470296
\(783\) −15310.8 −0.698803
\(784\) −1837.59 −0.0837094
\(785\) 12432.0 0.565246
\(786\) 1737.06 0.0788281
\(787\) −37723.8 −1.70865 −0.854325 0.519739i \(-0.826029\pi\)
−0.854325 + 0.519739i \(0.826029\pi\)
\(788\) −9684.41 −0.437808
\(789\) −12087.7 −0.545418
\(790\) 4059.95 0.182844
\(791\) 18010.6 0.809588
\(792\) 21064.1 0.945051
\(793\) 1069.31 0.0478843
\(794\) 14218.2 0.635497
\(795\) 6270.60 0.279742
\(796\) −18835.0 −0.838680
\(797\) −6349.61 −0.282202 −0.141101 0.989995i \(-0.545064\pi\)
−0.141101 + 0.989995i \(0.545064\pi\)
\(798\) 831.361 0.0368795
\(799\) 2804.57 0.124178
\(800\) 8136.65 0.359592
\(801\) 8214.07 0.362334
\(802\) 24854.2 1.09431
\(803\) 3246.51 0.142674
\(804\) 6370.81 0.279454
\(805\) 10932.8 0.478673
\(806\) −6630.27 −0.289753
\(807\) 11176.4 0.487521
\(808\) −31363.4 −1.36554
\(809\) 32315.3 1.40438 0.702191 0.711988i \(-0.252205\pi\)
0.702191 + 0.711988i \(0.252205\pi\)
\(810\) 7198.96 0.312279
\(811\) −27000.2 −1.16906 −0.584528 0.811373i \(-0.698720\pi\)
−0.584528 + 0.811373i \(0.698720\pi\)
\(812\) 10633.5 0.459561
\(813\) −14093.9 −0.607989
\(814\) −11273.2 −0.485413
\(815\) 29362.5 1.26199
\(816\) 881.877 0.0378332
\(817\) 1671.49 0.0715766
\(818\) 21419.4 0.915538
\(819\) −3369.64 −0.143766
\(820\) −8695.06 −0.370298
\(821\) −8734.23 −0.371287 −0.185644 0.982617i \(-0.559437\pi\)
−0.185644 + 0.982617i \(0.559437\pi\)
\(822\) 935.712 0.0397040
\(823\) 37165.8 1.57414 0.787070 0.616864i \(-0.211597\pi\)
0.787070 + 0.616864i \(0.211597\pi\)
\(824\) 29417.1 1.24368
\(825\) 3066.43 0.129405
\(826\) 5560.48 0.234230
\(827\) 44504.9 1.87133 0.935664 0.352892i \(-0.114802\pi\)
0.935664 + 0.352892i \(0.114802\pi\)
\(828\) 13913.3 0.583962
\(829\) 8435.36 0.353404 0.176702 0.984264i \(-0.443457\pi\)
0.176702 + 0.984264i \(0.443457\pi\)
\(830\) 8079.03 0.337864
\(831\) 6434.07 0.268586
\(832\) −2871.20 −0.119641
\(833\) −12823.4 −0.533378
\(834\) −5396.69 −0.224067
\(835\) 21079.2 0.873623
\(836\) 5807.47 0.240258
\(837\) 29298.0 1.20990
\(838\) 3338.40 0.137617
\(839\) −15830.0 −0.651384 −0.325692 0.945476i \(-0.605597\pi\)
−0.325692 + 0.945476i \(0.605597\pi\)
\(840\) 3807.62 0.156399
\(841\) 5767.62 0.236484
\(842\) −23983.7 −0.981629
\(843\) −10450.0 −0.426950
\(844\) −28223.3 −1.15105
\(845\) 18429.3 0.750280
\(846\) 1816.27 0.0738115
\(847\) −3635.08 −0.147465
\(848\) −3439.67 −0.139291
\(849\) 463.192 0.0187240
\(850\) 4191.64 0.169144
\(851\) −18456.9 −0.743473
\(852\) −6791.73 −0.273099
\(853\) −25517.9 −1.02429 −0.512144 0.858900i \(-0.671148\pi\)
−0.512144 + 0.858900i \(0.671148\pi\)
\(854\) −1570.92 −0.0629460
\(855\) 5719.71 0.228784
\(856\) 5929.29 0.236751
\(857\) 1353.59 0.0539530 0.0269765 0.999636i \(-0.491412\pi\)
0.0269765 + 0.999636i \(0.491412\pi\)
\(858\) −1401.69 −0.0557728
\(859\) 44829.9 1.78065 0.890325 0.455326i \(-0.150477\pi\)
0.890325 + 0.455326i \(0.150477\pi\)
\(860\) 3088.48 0.122461
\(861\) 3485.66 0.137969
\(862\) −11964.7 −0.472761
\(863\) −22467.5 −0.886216 −0.443108 0.896468i \(-0.646124\pi\)
−0.443108 + 0.896468i \(0.646124\pi\)
\(864\) 16435.1 0.647145
\(865\) 29488.5 1.15912
\(866\) −7057.53 −0.276934
\(867\) −2337.21 −0.0915521
\(868\) −20347.8 −0.795679
\(869\) −11369.2 −0.443815
\(870\) 4356.47 0.169768
\(871\) 8447.37 0.328620
\(872\) 31261.3 1.21404
\(873\) 39317.6 1.52428
\(874\) −4551.59 −0.176155
\(875\) −17216.4 −0.665165
\(876\) 746.836 0.0288051
\(877\) 408.941 0.0157457 0.00787283 0.999969i \(-0.497494\pi\)
0.00787283 + 0.999969i \(0.497494\pi\)
\(878\) 16576.9 0.637181
\(879\) 4720.12 0.181121
\(880\) 3134.89 0.120088
\(881\) −20458.5 −0.782366 −0.391183 0.920313i \(-0.627934\pi\)
−0.391183 + 0.920313i \(0.627934\pi\)
\(882\) −8304.54 −0.317039
\(883\) 3746.69 0.142793 0.0713965 0.997448i \(-0.477254\pi\)
0.0713965 + 0.997448i \(0.477254\pi\)
\(884\) 4002.58 0.152286
\(885\) −4758.88 −0.180755
\(886\) −9173.78 −0.347855
\(887\) 5289.38 0.200225 0.100113 0.994976i \(-0.468080\pi\)
0.100113 + 0.994976i \(0.468080\pi\)
\(888\) −6428.06 −0.242919
\(889\) −8462.24 −0.319251
\(890\) −4965.14 −0.187002
\(891\) −20159.5 −0.757990
\(892\) −27458.9 −1.03071
\(893\) 1241.22 0.0465126
\(894\) −7817.70 −0.292464
\(895\) 19878.5 0.742420
\(896\) −12660.4 −0.472046
\(897\) −2294.90 −0.0854232
\(898\) 14397.3 0.535015
\(899\) −57706.3 −2.14084
\(900\) −5670.65 −0.210024
\(901\) −24003.3 −0.887531
\(902\) −11656.0 −0.430267
\(903\) −1238.10 −0.0456274
\(904\) −34341.8 −1.26349
\(905\) −13733.3 −0.504430
\(906\) −4758.46 −0.174491
\(907\) −10720.8 −0.392480 −0.196240 0.980556i \(-0.562873\pi\)
−0.196240 + 0.980556i \(0.562873\pi\)
\(908\) −9482.18 −0.346561
\(909\) 34897.6 1.27336
\(910\) 2036.84 0.0741983
\(911\) 3405.66 0.123858 0.0619290 0.998081i \(-0.480275\pi\)
0.0619290 + 0.998081i \(0.480275\pi\)
\(912\) 390.292 0.0141709
\(913\) −22624.0 −0.820094
\(914\) 26550.7 0.960853
\(915\) 1344.46 0.0485753
\(916\) 25302.0 0.912666
\(917\) 7068.52 0.254551
\(918\) 8466.64 0.304402
\(919\) 3184.34 0.114300 0.0571500 0.998366i \(-0.481799\pi\)
0.0571500 + 0.998366i \(0.481799\pi\)
\(920\) −20846.2 −0.747043
\(921\) −12114.8 −0.433439
\(922\) −14509.6 −0.518273
\(923\) −9005.48 −0.321147
\(924\) −4301.69 −0.153155
\(925\) 7522.51 0.267393
\(926\) −5216.40 −0.185121
\(927\) −32732.0 −1.15972
\(928\) −32371.1 −1.14508
\(929\) 24847.1 0.877510 0.438755 0.898607i \(-0.355420\pi\)
0.438755 + 0.898607i \(0.355420\pi\)
\(930\) −8336.33 −0.293934
\(931\) −5675.24 −0.199783
\(932\) −23057.5 −0.810379
\(933\) −1499.18 −0.0526056
\(934\) −19611.2 −0.687041
\(935\) 21876.4 0.765171
\(936\) 6425.06 0.224369
\(937\) 16072.8 0.560381 0.280191 0.959944i \(-0.409602\pi\)
0.280191 + 0.959944i \(0.409602\pi\)
\(938\) −12410.0 −0.431985
\(939\) 9893.12 0.343823
\(940\) 2293.44 0.0795785
\(941\) 49359.1 1.70995 0.854974 0.518672i \(-0.173573\pi\)
0.854974 + 0.518672i \(0.173573\pi\)
\(942\) 3833.76 0.132602
\(943\) −19083.6 −0.659010
\(944\) 2610.43 0.0900025
\(945\) −9000.41 −0.309824
\(946\) 4140.18 0.142293
\(947\) 19624.5 0.673400 0.336700 0.941612i \(-0.390689\pi\)
0.336700 + 0.941612i \(0.390689\pi\)
\(948\) −2615.41 −0.0896038
\(949\) 990.266 0.0338729
\(950\) 1855.09 0.0633550
\(951\) 10061.0 0.343061
\(952\) −14575.2 −0.496204
\(953\) 39172.6 1.33151 0.665753 0.746172i \(-0.268110\pi\)
0.665753 + 0.746172i \(0.268110\pi\)
\(954\) −15544.8 −0.527548
\(955\) 18212.7 0.617118
\(956\) 17460.6 0.590707
\(957\) −12199.6 −0.412076
\(958\) 14662.1 0.494480
\(959\) 3807.64 0.128212
\(960\) −3610.00 −0.121367
\(961\) 80632.9 2.70662
\(962\) −3438.61 −0.115244
\(963\) −6597.46 −0.220768
\(964\) 7089.30 0.236858
\(965\) 40118.0 1.33829
\(966\) 3371.44 0.112292
\(967\) −50809.7 −1.68969 −0.844845 0.535010i \(-0.820308\pi\)
−0.844845 + 0.535010i \(0.820308\pi\)
\(968\) 6931.21 0.230142
\(969\) 2723.60 0.0902939
\(970\) −23766.2 −0.786687
\(971\) −49995.3 −1.65235 −0.826173 0.563417i \(-0.809486\pi\)
−0.826173 + 0.563417i \(0.809486\pi\)
\(972\) −17516.4 −0.578025
\(973\) −21960.4 −0.723555
\(974\) −861.326 −0.0283354
\(975\) 935.335 0.0307228
\(976\) −737.488 −0.0241869
\(977\) −6507.03 −0.213079 −0.106540 0.994308i \(-0.533977\pi\)
−0.106540 + 0.994308i \(0.533977\pi\)
\(978\) 9054.73 0.296052
\(979\) 13904.1 0.453908
\(980\) −10486.3 −0.341810
\(981\) −34784.1 −1.13208
\(982\) −17397.9 −0.565365
\(983\) −37044.4 −1.20197 −0.600983 0.799262i \(-0.705224\pi\)
−0.600983 + 0.799262i \(0.705224\pi\)
\(984\) −6646.30 −0.215321
\(985\) −16145.2 −0.522263
\(986\) −16676.2 −0.538618
\(987\) −919.391 −0.0296500
\(988\) 1771.42 0.0570408
\(989\) 6778.46 0.217940
\(990\) 14167.4 0.454817
\(991\) −17372.8 −0.556877 −0.278439 0.960454i \(-0.589817\pi\)
−0.278439 + 0.960454i \(0.589817\pi\)
\(992\) 61943.8 1.98258
\(993\) −3975.91 −0.127061
\(994\) 13230.0 0.422162
\(995\) −31400.5 −1.00046
\(996\) −5204.48 −0.165573
\(997\) −45015.4 −1.42994 −0.714971 0.699154i \(-0.753560\pi\)
−0.714971 + 0.699154i \(0.753560\pi\)
\(998\) −26934.0 −0.854289
\(999\) 15194.6 0.481217
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 47.4.a.a.1.2 3
3.2 odd 2 423.4.a.b.1.2 3
4.3 odd 2 752.4.a.c.1.1 3
5.4 even 2 1175.4.a.a.1.2 3
7.6 odd 2 2303.4.a.a.1.2 3
47.46 odd 2 2209.4.a.a.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
47.4.a.a.1.2 3 1.1 even 1 trivial
423.4.a.b.1.2 3 3.2 odd 2
752.4.a.c.1.1 3 4.3 odd 2
1175.4.a.a.1.2 3 5.4 even 2
2209.4.a.a.1.2 3 47.46 odd 2
2303.4.a.a.1.2 3 7.6 odd 2