Properties

Label 65.4.a.c.1.1
Level $65$
Weight $4$
Character 65.1
Self dual yes
Analytic conductor $3.835$
Analytic rank $0$
Dimension $2$
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] = [65,4,Mod(1,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 65.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: \(3.83512415037\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{2} -8.24621 q^{3} -5.56155 q^{4} -5.00000 q^{5} +12.8769 q^{6} +33.1231 q^{7} +21.1771 q^{8} +41.0000 q^{9} +O(q^{10})\) \(q-1.56155 q^{2} -8.24621 q^{3} -5.56155 q^{4} -5.00000 q^{5} +12.8769 q^{6} +33.1231 q^{7} +21.1771 q^{8} +41.0000 q^{9} +7.80776 q^{10} -47.1231 q^{11} +45.8617 q^{12} -13.0000 q^{13} -51.7235 q^{14} +41.2311 q^{15} +11.4233 q^{16} +51.9697 q^{17} -64.0237 q^{18} +37.6458 q^{19} +27.8078 q^{20} -273.140 q^{21} +73.5852 q^{22} +161.939 q^{23} -174.631 q^{24} +25.0000 q^{25} +20.3002 q^{26} -115.447 q^{27} -184.216 q^{28} +2.27652 q^{29} -64.3845 q^{30} +220.756 q^{31} -187.255 q^{32} +388.587 q^{33} -81.1534 q^{34} -165.616 q^{35} -228.024 q^{36} +105.784 q^{37} -58.7860 q^{38} +107.201 q^{39} -105.885 q^{40} -291.633 q^{41} +426.523 q^{42} +60.4621 q^{43} +262.078 q^{44} -205.000 q^{45} -252.877 q^{46} +53.1837 q^{47} -94.1989 q^{48} +754.140 q^{49} -39.0388 q^{50} -428.553 q^{51} +72.3002 q^{52} -395.909 q^{53} +180.277 q^{54} +235.616 q^{55} +701.451 q^{56} -310.436 q^{57} -3.55491 q^{58} +367.710 q^{59} -229.309 q^{60} +442.000 q^{61} -344.722 q^{62} +1358.05 q^{63} +201.022 q^{64} +65.0000 q^{65} -606.799 q^{66} +1093.93 q^{67} -289.032 q^{68} -1335.39 q^{69} +258.617 q^{70} -330.328 q^{71} +868.260 q^{72} -722.682 q^{73} -165.187 q^{74} -206.155 q^{75} -209.369 q^{76} -1560.86 q^{77} -167.400 q^{78} -114.739 q^{79} -57.1165 q^{80} -155.000 q^{81} +455.400 q^{82} +806.419 q^{83} +1519.08 q^{84} -259.848 q^{85} -94.4148 q^{86} -18.7727 q^{87} -997.930 q^{88} -673.386 q^{89} +320.118 q^{90} -430.600 q^{91} -900.634 q^{92} -1820.40 q^{93} -83.0492 q^{94} -188.229 q^{95} +1544.14 q^{96} +1739.46 q^{97} -1177.63 q^{98} -1932.05 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} - 10 q^{5} + 34 q^{6} + 58 q^{7} - 3 q^{8} + 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 7 q^{4} - 10 q^{5} + 34 q^{6} + 58 q^{7} - 3 q^{8} + 82 q^{9} - 5 q^{10} - 86 q^{11} + 34 q^{12} - 26 q^{13} + 12 q^{14} - 39 q^{16} - 28 q^{17} + 41 q^{18} + 166 q^{19} + 35 q^{20} - 68 q^{21} - 26 q^{22} + 60 q^{23} - 374 q^{24} + 50 q^{25} - 13 q^{26} - 220 q^{28} + 120 q^{29} - 170 q^{30} - 78 q^{31} - 123 q^{32} + 68 q^{33} - 286 q^{34} - 290 q^{35} - 287 q^{36} + 360 q^{37} + 270 q^{38} + 15 q^{40} - 72 q^{41} + 952 q^{42} - 44 q^{43} + 318 q^{44} - 410 q^{45} - 514 q^{46} + 362 q^{47} - 510 q^{48} + 1030 q^{49} + 25 q^{50} - 1088 q^{51} + 91 q^{52} - 396 q^{53} + 476 q^{54} + 430 q^{55} + 100 q^{56} + 748 q^{57} + 298 q^{58} + 18 q^{59} - 170 q^{60} + 884 q^{61} - 1110 q^{62} + 2378 q^{63} + 769 q^{64} + 130 q^{65} - 1428 q^{66} + 1322 q^{67} - 174 q^{68} - 2176 q^{69} - 60 q^{70} + 634 q^{71} - 123 q^{72} - 60 q^{73} + 486 q^{74} - 394 q^{76} - 2528 q^{77} - 442 q^{78} - 180 q^{79} + 195 q^{80} - 310 q^{81} + 1018 q^{82} + 714 q^{83} + 1224 q^{84} + 140 q^{85} - 362 q^{86} + 952 q^{87} - 58 q^{88} - 852 q^{89} - 205 q^{90} - 754 q^{91} - 754 q^{92} - 4284 q^{93} + 708 q^{94} - 830 q^{95} + 2074 q^{96} + 32 q^{97} - 471 q^{98} - 3526 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −0.552092 −0.276046 0.961144i \(-0.589024\pi\)
−0.276046 + 0.961144i \(0.589024\pi\)
\(3\) −8.24621 −1.58698 −0.793492 0.608581i \(-0.791739\pi\)
−0.793492 + 0.608581i \(0.791739\pi\)
\(4\) −5.56155 −0.695194
\(5\) −5.00000 −0.447214
\(6\) 12.8769 0.876162
\(7\) 33.1231 1.78848 0.894240 0.447588i \(-0.147717\pi\)
0.894240 + 0.447588i \(0.147717\pi\)
\(8\) 21.1771 0.935904
\(9\) 41.0000 1.51852
\(10\) 7.80776 0.246903
\(11\) −47.1231 −1.29165 −0.645825 0.763485i \(-0.723486\pi\)
−0.645825 + 0.763485i \(0.723486\pi\)
\(12\) 45.8617 1.10326
\(13\) −13.0000 −0.277350
\(14\) −51.7235 −0.987406
\(15\) 41.2311 0.709721
\(16\) 11.4233 0.178489
\(17\) 51.9697 0.741441 0.370721 0.928744i \(-0.379111\pi\)
0.370721 + 0.928744i \(0.379111\pi\)
\(18\) −64.0237 −0.838362
\(19\) 37.6458 0.454555 0.227278 0.973830i \(-0.427018\pi\)
0.227278 + 0.973830i \(0.427018\pi\)
\(20\) 27.8078 0.310900
\(21\) −273.140 −2.83829
\(22\) 73.5852 0.713110
\(23\) 161.939 1.46812 0.734059 0.679086i \(-0.237624\pi\)
0.734059 + 0.679086i \(0.237624\pi\)
\(24\) −174.631 −1.48526
\(25\) 25.0000 0.200000
\(26\) 20.3002 0.153123
\(27\) −115.447 −0.822881
\(28\) −184.216 −1.24334
\(29\) 2.27652 0.0145772 0.00728861 0.999973i \(-0.497680\pi\)
0.00728861 + 0.999973i \(0.497680\pi\)
\(30\) −64.3845 −0.391831
\(31\) 220.756 1.27900 0.639498 0.768793i \(-0.279142\pi\)
0.639498 + 0.768793i \(0.279142\pi\)
\(32\) −187.255 −1.03445
\(33\) 388.587 2.04983
\(34\) −81.1534 −0.409344
\(35\) −165.616 −0.799832
\(36\) −228.024 −1.05567
\(37\) 105.784 0.470022 0.235011 0.971993i \(-0.424487\pi\)
0.235011 + 0.971993i \(0.424487\pi\)
\(38\) −58.7860 −0.250956
\(39\) 107.201 0.440150
\(40\) −105.885 −0.418549
\(41\) −291.633 −1.11086 −0.555431 0.831563i \(-0.687447\pi\)
−0.555431 + 0.831563i \(0.687447\pi\)
\(42\) 426.523 1.56700
\(43\) 60.4621 0.214428 0.107214 0.994236i \(-0.465807\pi\)
0.107214 + 0.994236i \(0.465807\pi\)
\(44\) 262.078 0.897948
\(45\) −205.000 −0.679102
\(46\) −252.877 −0.810536
\(47\) 53.1837 0.165056 0.0825281 0.996589i \(-0.473701\pi\)
0.0825281 + 0.996589i \(0.473701\pi\)
\(48\) −94.1989 −0.283259
\(49\) 754.140 2.19866
\(50\) −39.0388 −0.110418
\(51\) −428.553 −1.17666
\(52\) 72.3002 0.192812
\(53\) −395.909 −1.02608 −0.513041 0.858364i \(-0.671481\pi\)
−0.513041 + 0.858364i \(0.671481\pi\)
\(54\) 180.277 0.454306
\(55\) 235.616 0.577643
\(56\) 701.451 1.67384
\(57\) −310.436 −0.721372
\(58\) −3.55491 −0.00804797
\(59\) 367.710 0.811386 0.405693 0.914009i \(-0.367030\pi\)
0.405693 + 0.914009i \(0.367030\pi\)
\(60\) −229.309 −0.493394
\(61\) 442.000 0.927743 0.463871 0.885903i \(-0.346460\pi\)
0.463871 + 0.885903i \(0.346460\pi\)
\(62\) −344.722 −0.706124
\(63\) 1358.05 2.71584
\(64\) 201.022 0.392621
\(65\) 65.0000 0.124035
\(66\) −606.799 −1.13169
\(67\) 1093.93 1.99469 0.997346 0.0728098i \(-0.0231966\pi\)
0.997346 + 0.0728098i \(0.0231966\pi\)
\(68\) −289.032 −0.515446
\(69\) −1335.39 −2.32988
\(70\) 258.617 0.441581
\(71\) −330.328 −0.552150 −0.276075 0.961136i \(-0.589034\pi\)
−0.276075 + 0.961136i \(0.589034\pi\)
\(72\) 868.260 1.42119
\(73\) −722.682 −1.15868 −0.579339 0.815087i \(-0.696689\pi\)
−0.579339 + 0.815087i \(0.696689\pi\)
\(74\) −165.187 −0.259495
\(75\) −206.155 −0.317397
\(76\) −209.369 −0.316004
\(77\) −1560.86 −2.31009
\(78\) −167.400 −0.243004
\(79\) −114.739 −0.163406 −0.0817032 0.996657i \(-0.526036\pi\)
−0.0817032 + 0.996657i \(0.526036\pi\)
\(80\) −57.1165 −0.0798227
\(81\) −155.000 −0.212620
\(82\) 455.400 0.613298
\(83\) 806.419 1.06646 0.533229 0.845971i \(-0.320979\pi\)
0.533229 + 0.845971i \(0.320979\pi\)
\(84\) 1519.08 1.97316
\(85\) −259.848 −0.331583
\(86\) −94.4148 −0.118384
\(87\) −18.7727 −0.0231338
\(88\) −997.930 −1.20886
\(89\) −673.386 −0.802009 −0.401005 0.916076i \(-0.631339\pi\)
−0.401005 + 0.916076i \(0.631339\pi\)
\(90\) 320.118 0.374927
\(91\) −430.600 −0.496035
\(92\) −900.634 −1.02063
\(93\) −1820.40 −2.02975
\(94\) −83.0492 −0.0911263
\(95\) −188.229 −0.203283
\(96\) 1544.14 1.64165
\(97\) 1739.46 1.82078 0.910388 0.413756i \(-0.135783\pi\)
0.910388 + 0.413756i \(0.135783\pi\)
\(98\) −1177.63 −1.21386
\(99\) −1932.05 −1.96139
\(100\) −139.039 −0.139039
\(101\) 648.038 0.638437 0.319219 0.947681i \(-0.396580\pi\)
0.319219 + 0.947681i \(0.396580\pi\)
\(102\) 669.208 0.649622
\(103\) 1080.83 1.03396 0.516979 0.855998i \(-0.327057\pi\)
0.516979 + 0.855998i \(0.327057\pi\)
\(104\) −275.302 −0.259573
\(105\) 1365.70 1.26932
\(106\) 618.233 0.566491
\(107\) 1002.09 0.905385 0.452692 0.891667i \(-0.350464\pi\)
0.452692 + 0.891667i \(0.350464\pi\)
\(108\) 642.064 0.572062
\(109\) −734.932 −0.645814 −0.322907 0.946431i \(-0.604660\pi\)
−0.322907 + 0.946431i \(0.604660\pi\)
\(110\) −367.926 −0.318913
\(111\) −872.318 −0.745917
\(112\) 378.375 0.319224
\(113\) −1167.11 −0.971612 −0.485806 0.874067i \(-0.661474\pi\)
−0.485806 + 0.874067i \(0.661474\pi\)
\(114\) 484.761 0.398264
\(115\) −809.697 −0.656562
\(116\) −12.6610 −0.0101340
\(117\) −533.000 −0.421161
\(118\) −574.199 −0.447960
\(119\) 1721.40 1.32605
\(120\) 873.153 0.664230
\(121\) 889.587 0.668360
\(122\) −690.206 −0.512200
\(123\) 2404.86 1.76292
\(124\) −1227.74 −0.889151
\(125\) −125.000 −0.0894427
\(126\) −2120.66 −1.49939
\(127\) −2737.33 −1.91259 −0.956295 0.292402i \(-0.905545\pi\)
−0.956295 + 0.292402i \(0.905545\pi\)
\(128\) 1184.13 0.817683
\(129\) −498.583 −0.340293
\(130\) −101.501 −0.0684786
\(131\) 1447.49 0.965402 0.482701 0.875785i \(-0.339656\pi\)
0.482701 + 0.875785i \(0.339656\pi\)
\(132\) −2161.15 −1.42503
\(133\) 1246.95 0.812963
\(134\) −1708.22 −1.10125
\(135\) 577.235 0.368003
\(136\) 1100.57 0.693918
\(137\) 1955.20 1.21930 0.609650 0.792671i \(-0.291310\pi\)
0.609650 + 0.792671i \(0.291310\pi\)
\(138\) 2085.28 1.28631
\(139\) 53.5113 0.0326530 0.0163265 0.999867i \(-0.494803\pi\)
0.0163265 + 0.999867i \(0.494803\pi\)
\(140\) 921.080 0.556039
\(141\) −438.564 −0.261942
\(142\) 515.824 0.304838
\(143\) 612.600 0.358239
\(144\) 468.355 0.271039
\(145\) −11.3826 −0.00651913
\(146\) 1128.51 0.639697
\(147\) −6218.80 −3.48924
\(148\) −588.324 −0.326756
\(149\) 108.901 0.0598762 0.0299381 0.999552i \(-0.490469\pi\)
0.0299381 + 0.999552i \(0.490469\pi\)
\(150\) 321.922 0.175232
\(151\) −359.710 −0.193860 −0.0969298 0.995291i \(-0.530902\pi\)
−0.0969298 + 0.995291i \(0.530902\pi\)
\(152\) 797.229 0.425420
\(153\) 2130.76 1.12589
\(154\) 2437.37 1.27538
\(155\) −1103.78 −0.571985
\(156\) −596.203 −0.305990
\(157\) −249.682 −0.126922 −0.0634611 0.997984i \(-0.520214\pi\)
−0.0634611 + 0.997984i \(0.520214\pi\)
\(158\) 179.170 0.0902154
\(159\) 3264.75 1.62837
\(160\) 936.274 0.462618
\(161\) 5363.94 2.62570
\(162\) 242.041 0.117386
\(163\) 883.505 0.424549 0.212275 0.977210i \(-0.431913\pi\)
0.212275 + 0.977210i \(0.431913\pi\)
\(164\) 1621.93 0.772265
\(165\) −1942.94 −0.916711
\(166\) −1259.27 −0.588783
\(167\) −945.036 −0.437899 −0.218949 0.975736i \(-0.570263\pi\)
−0.218949 + 0.975736i \(0.570263\pi\)
\(168\) −5784.31 −2.65636
\(169\) 169.000 0.0769231
\(170\) 405.767 0.183064
\(171\) 1543.48 0.690250
\(172\) −336.263 −0.149069
\(173\) −2125.83 −0.934240 −0.467120 0.884194i \(-0.654708\pi\)
−0.467120 + 0.884194i \(0.654708\pi\)
\(174\) 29.3145 0.0127720
\(175\) 828.078 0.357696
\(176\) −538.301 −0.230545
\(177\) −3032.22 −1.28766
\(178\) 1051.53 0.442783
\(179\) −3448.13 −1.43981 −0.719903 0.694075i \(-0.755814\pi\)
−0.719903 + 0.694075i \(0.755814\pi\)
\(180\) 1140.12 0.472108
\(181\) 3273.24 1.34419 0.672094 0.740466i \(-0.265395\pi\)
0.672094 + 0.740466i \(0.265395\pi\)
\(182\) 672.405 0.273857
\(183\) −3644.83 −1.47231
\(184\) 3429.40 1.37402
\(185\) −528.920 −0.210200
\(186\) 2842.65 1.12061
\(187\) −2448.97 −0.957683
\(188\) −295.784 −0.114746
\(189\) −3823.96 −1.47171
\(190\) 293.930 0.112231
\(191\) −443.973 −0.168193 −0.0840963 0.996458i \(-0.526800\pi\)
−0.0840963 + 0.996458i \(0.526800\pi\)
\(192\) −1657.67 −0.623083
\(193\) −4188.38 −1.56211 −0.781053 0.624465i \(-0.785317\pi\)
−0.781053 + 0.624465i \(0.785317\pi\)
\(194\) −2716.26 −1.00524
\(195\) −536.004 −0.196841
\(196\) −4194.19 −1.52849
\(197\) 481.204 0.174032 0.0870162 0.996207i \(-0.472267\pi\)
0.0870162 + 0.996207i \(0.472267\pi\)
\(198\) 3016.99 1.08287
\(199\) −3500.02 −1.24678 −0.623392 0.781909i \(-0.714246\pi\)
−0.623392 + 0.781909i \(0.714246\pi\)
\(200\) 529.427 0.187181
\(201\) −9020.75 −3.16554
\(202\) −1011.95 −0.352476
\(203\) 75.4055 0.0260711
\(204\) 2383.42 0.818004
\(205\) 1458.16 0.496793
\(206\) −1687.78 −0.570840
\(207\) 6639.51 2.22936
\(208\) −148.503 −0.0495039
\(209\) −1773.99 −0.587126
\(210\) −2132.61 −0.700782
\(211\) 1206.57 0.393667 0.196834 0.980437i \(-0.436934\pi\)
0.196834 + 0.980437i \(0.436934\pi\)
\(212\) 2201.87 0.713325
\(213\) 2723.95 0.876254
\(214\) −1564.82 −0.499856
\(215\) −302.311 −0.0958949
\(216\) −2444.83 −0.770137
\(217\) 7312.11 2.28746
\(218\) 1147.63 0.356549
\(219\) 5959.39 1.83880
\(220\) −1310.39 −0.401574
\(221\) −675.606 −0.205639
\(222\) 1362.17 0.411815
\(223\) 202.733 0.0608790 0.0304395 0.999537i \(-0.490309\pi\)
0.0304395 + 0.999537i \(0.490309\pi\)
\(224\) −6202.46 −1.85009
\(225\) 1025.00 0.303704
\(226\) 1822.50 0.536419
\(227\) 5345.87 1.56307 0.781536 0.623860i \(-0.214436\pi\)
0.781536 + 0.623860i \(0.214436\pi\)
\(228\) 1726.50 0.501493
\(229\) 1003.45 0.289563 0.144781 0.989464i \(-0.453752\pi\)
0.144781 + 0.989464i \(0.453752\pi\)
\(230\) 1264.38 0.362483
\(231\) 12871.2 3.66608
\(232\) 48.2101 0.0136429
\(233\) 1948.83 0.547950 0.273975 0.961737i \(-0.411661\pi\)
0.273975 + 0.961737i \(0.411661\pi\)
\(234\) 832.308 0.232520
\(235\) −265.919 −0.0738154
\(236\) −2045.04 −0.564071
\(237\) 946.159 0.259323
\(238\) −2688.05 −0.732103
\(239\) 602.725 0.163126 0.0815629 0.996668i \(-0.474009\pi\)
0.0815629 + 0.996668i \(0.474009\pi\)
\(240\) 470.994 0.126677
\(241\) −492.171 −0.131550 −0.0657749 0.997834i \(-0.520952\pi\)
−0.0657749 + 0.997834i \(0.520952\pi\)
\(242\) −1389.14 −0.368996
\(243\) 4395.23 1.16031
\(244\) −2458.21 −0.644961
\(245\) −3770.70 −0.983270
\(246\) −3755.32 −0.973295
\(247\) −489.396 −0.126071
\(248\) 4674.96 1.19702
\(249\) −6649.90 −1.69245
\(250\) 195.194 0.0493806
\(251\) 2990.29 0.751973 0.375987 0.926625i \(-0.377304\pi\)
0.375987 + 0.926625i \(0.377304\pi\)
\(252\) −7552.85 −1.88804
\(253\) −7631.09 −1.89629
\(254\) 4274.49 1.05593
\(255\) 2142.77 0.526216
\(256\) −3457.26 −0.844057
\(257\) 4254.24 1.03258 0.516288 0.856415i \(-0.327314\pi\)
0.516288 + 0.856415i \(0.327314\pi\)
\(258\) 778.564 0.187873
\(259\) 3503.90 0.840624
\(260\) −361.501 −0.0862282
\(261\) 93.3374 0.0221358
\(262\) −2260.33 −0.532991
\(263\) −4166.84 −0.976953 −0.488476 0.872577i \(-0.662447\pi\)
−0.488476 + 0.872577i \(0.662447\pi\)
\(264\) 8229.14 1.91844
\(265\) 1979.55 0.458877
\(266\) −1947.17 −0.448830
\(267\) 5552.89 1.27278
\(268\) −6083.93 −1.38670
\(269\) −4209.91 −0.954212 −0.477106 0.878846i \(-0.658314\pi\)
−0.477106 + 0.878846i \(0.658314\pi\)
\(270\) −901.383 −0.203172
\(271\) −4951.69 −1.10994 −0.554970 0.831871i \(-0.687270\pi\)
−0.554970 + 0.831871i \(0.687270\pi\)
\(272\) 593.665 0.132339
\(273\) 3550.82 0.787200
\(274\) −3053.15 −0.673166
\(275\) −1178.08 −0.258330
\(276\) 7426.82 1.61972
\(277\) 4348.35 0.943202 0.471601 0.881812i \(-0.343676\pi\)
0.471601 + 0.881812i \(0.343676\pi\)
\(278\) −83.5607 −0.0180275
\(279\) 9050.98 1.94218
\(280\) −3507.25 −0.748566
\(281\) 1217.69 0.258509 0.129254 0.991611i \(-0.458742\pi\)
0.129254 + 0.991611i \(0.458742\pi\)
\(282\) 684.841 0.144616
\(283\) −1142.98 −0.240081 −0.120041 0.992769i \(-0.538302\pi\)
−0.120041 + 0.992769i \(0.538302\pi\)
\(284\) 1837.13 0.383852
\(285\) 1552.18 0.322607
\(286\) −956.608 −0.197781
\(287\) −9659.78 −1.98675
\(288\) −7677.44 −1.57083
\(289\) −2212.15 −0.450265
\(290\) 17.7745 0.00359916
\(291\) −14343.9 −2.88954
\(292\) 4019.23 0.805506
\(293\) 8222.16 1.63940 0.819699 0.572795i \(-0.194141\pi\)
0.819699 + 0.572795i \(0.194141\pi\)
\(294\) 9710.98 1.92638
\(295\) −1838.55 −0.362863
\(296\) 2240.20 0.439895
\(297\) 5440.22 1.06287
\(298\) −170.055 −0.0330572
\(299\) −2105.21 −0.407182
\(300\) 1146.54 0.220652
\(301\) 2002.69 0.383499
\(302\) 561.706 0.107028
\(303\) −5343.86 −1.01319
\(304\) 430.039 0.0811331
\(305\) −2210.00 −0.414899
\(306\) −3327.29 −0.621596
\(307\) −2543.07 −0.472771 −0.236385 0.971659i \(-0.575963\pi\)
−0.236385 + 0.971659i \(0.575963\pi\)
\(308\) 8680.83 1.60596
\(309\) −8912.78 −1.64087
\(310\) 1723.61 0.315788
\(311\) 2028.16 0.369795 0.184898 0.982758i \(-0.440805\pi\)
0.184898 + 0.982758i \(0.440805\pi\)
\(312\) 2270.20 0.411938
\(313\) 7321.83 1.32222 0.661109 0.750290i \(-0.270086\pi\)
0.661109 + 0.750290i \(0.270086\pi\)
\(314\) 389.892 0.0700728
\(315\) −6790.24 −1.21456
\(316\) 638.125 0.113599
\(317\) −2484.31 −0.440167 −0.220084 0.975481i \(-0.570633\pi\)
−0.220084 + 0.975481i \(0.570633\pi\)
\(318\) −5098.08 −0.899013
\(319\) −107.277 −0.0188287
\(320\) −1005.11 −0.175585
\(321\) −8263.48 −1.43683
\(322\) −8376.07 −1.44963
\(323\) 1956.44 0.337026
\(324\) 862.041 0.147812
\(325\) −325.000 −0.0554700
\(326\) −1379.64 −0.234390
\(327\) 6060.40 1.02490
\(328\) −6175.93 −1.03966
\(329\) 1761.61 0.295200
\(330\) 3034.00 0.506109
\(331\) −6914.29 −1.14817 −0.574085 0.818796i \(-0.694642\pi\)
−0.574085 + 0.818796i \(0.694642\pi\)
\(332\) −4484.94 −0.741395
\(333\) 4337.15 0.713736
\(334\) 1475.72 0.241760
\(335\) −5469.63 −0.892053
\(336\) −3120.16 −0.506603
\(337\) −5456.98 −0.882080 −0.441040 0.897488i \(-0.645390\pi\)
−0.441040 + 0.897488i \(0.645390\pi\)
\(338\) −263.902 −0.0424686
\(339\) 9624.20 1.54193
\(340\) 1445.16 0.230514
\(341\) −10402.7 −1.65202
\(342\) −2410.22 −0.381082
\(343\) 13618.2 2.14378
\(344\) 1280.41 0.200684
\(345\) 6676.93 1.04195
\(346\) 3319.59 0.515787
\(347\) −710.943 −0.109987 −0.0549934 0.998487i \(-0.517514\pi\)
−0.0549934 + 0.998487i \(0.517514\pi\)
\(348\) 104.405 0.0160825
\(349\) −5108.22 −0.783486 −0.391743 0.920075i \(-0.628128\pi\)
−0.391743 + 0.920075i \(0.628128\pi\)
\(350\) −1293.09 −0.197481
\(351\) 1500.81 0.228226
\(352\) 8824.02 1.33614
\(353\) 3605.90 0.543691 0.271845 0.962341i \(-0.412366\pi\)
0.271845 + 0.962341i \(0.412366\pi\)
\(354\) 4734.97 0.710906
\(355\) 1651.64 0.246929
\(356\) 3745.07 0.557552
\(357\) −14195.0 −2.10442
\(358\) 5384.44 0.794906
\(359\) 87.8655 0.0129174 0.00645872 0.999979i \(-0.497944\pi\)
0.00645872 + 0.999979i \(0.497944\pi\)
\(360\) −4341.30 −0.635574
\(361\) −5441.79 −0.793380
\(362\) −5111.33 −0.742115
\(363\) −7335.72 −1.06068
\(364\) 2394.81 0.344841
\(365\) 3613.41 0.518177
\(366\) 5691.59 0.812853
\(367\) −2379.95 −0.338508 −0.169254 0.985572i \(-0.554136\pi\)
−0.169254 + 0.985572i \(0.554136\pi\)
\(368\) 1849.88 0.262043
\(369\) −11956.9 −1.68686
\(370\) 825.937 0.116050
\(371\) −13113.7 −1.83512
\(372\) 10124.2 1.41107
\(373\) −7291.14 −1.01212 −0.506060 0.862498i \(-0.668899\pi\)
−0.506060 + 0.862498i \(0.668899\pi\)
\(374\) 3824.20 0.528729
\(375\) 1030.78 0.141944
\(376\) 1126.28 0.154477
\(377\) −29.5948 −0.00404299
\(378\) 5971.32 0.812517
\(379\) 5827.67 0.789834 0.394917 0.918717i \(-0.370773\pi\)
0.394917 + 0.918717i \(0.370773\pi\)
\(380\) 1046.85 0.141321
\(381\) 22572.6 3.03525
\(382\) 693.288 0.0928578
\(383\) 6029.95 0.804481 0.402240 0.915534i \(-0.368232\pi\)
0.402240 + 0.915534i \(0.368232\pi\)
\(384\) −9764.60 −1.29765
\(385\) 7804.32 1.03310
\(386\) 6540.38 0.862426
\(387\) 2478.95 0.325612
\(388\) −9674.09 −1.26579
\(389\) 7929.38 1.03351 0.516755 0.856133i \(-0.327140\pi\)
0.516755 + 0.856133i \(0.327140\pi\)
\(390\) 836.998 0.108674
\(391\) 8415.94 1.08852
\(392\) 15970.5 2.05773
\(393\) −11936.3 −1.53208
\(394\) −751.426 −0.0960820
\(395\) 573.693 0.0730776
\(396\) 10745.2 1.36355
\(397\) −3677.91 −0.464959 −0.232480 0.972601i \(-0.574684\pi\)
−0.232480 + 0.972601i \(0.574684\pi\)
\(398\) 5465.47 0.688340
\(399\) −10282.6 −1.29016
\(400\) 285.582 0.0356978
\(401\) 7770.69 0.967705 0.483852 0.875150i \(-0.339237\pi\)
0.483852 + 0.875150i \(0.339237\pi\)
\(402\) 14086.4 1.74767
\(403\) −2869.82 −0.354730
\(404\) −3604.10 −0.443838
\(405\) 775.000 0.0950866
\(406\) −117.750 −0.0143936
\(407\) −4984.88 −0.607103
\(408\) −9075.50 −1.10124
\(409\) 7102.25 0.858639 0.429320 0.903153i \(-0.358753\pi\)
0.429320 + 0.903153i \(0.358753\pi\)
\(410\) −2277.00 −0.274275
\(411\) −16123.0 −1.93501
\(412\) −6011.11 −0.718801
\(413\) 12179.7 1.45115
\(414\) −10368.0 −1.23081
\(415\) −4032.09 −0.476934
\(416\) 2434.31 0.286904
\(417\) −441.266 −0.0518198
\(418\) 2770.18 0.324148
\(419\) −9131.61 −1.06470 −0.532349 0.846525i \(-0.678691\pi\)
−0.532349 + 0.846525i \(0.678691\pi\)
\(420\) −7595.42 −0.882425
\(421\) −9425.45 −1.09114 −0.545568 0.838066i \(-0.683686\pi\)
−0.545568 + 0.838066i \(0.683686\pi\)
\(422\) −1884.13 −0.217341
\(423\) 2180.53 0.250641
\(424\) −8384.20 −0.960313
\(425\) 1299.24 0.148288
\(426\) −4253.59 −0.483773
\(427\) 14640.4 1.65925
\(428\) −5573.20 −0.629418
\(429\) −5051.63 −0.568520
\(430\) 472.074 0.0529428
\(431\) 6543.74 0.731325 0.365662 0.930748i \(-0.380843\pi\)
0.365662 + 0.930748i \(0.380843\pi\)
\(432\) −1318.78 −0.146875
\(433\) −9189.31 −1.01988 −0.509942 0.860209i \(-0.670333\pi\)
−0.509942 + 0.860209i \(0.670333\pi\)
\(434\) −11418.3 −1.26289
\(435\) 93.8634 0.0103458
\(436\) 4087.36 0.448966
\(437\) 6096.34 0.667340
\(438\) −9305.90 −1.01519
\(439\) −3024.75 −0.328846 −0.164423 0.986390i \(-0.552576\pi\)
−0.164423 + 0.986390i \(0.552576\pi\)
\(440\) 4989.65 0.540619
\(441\) 30919.7 3.33870
\(442\) 1054.99 0.113532
\(443\) −7883.72 −0.845524 −0.422762 0.906241i \(-0.638939\pi\)
−0.422762 + 0.906241i \(0.638939\pi\)
\(444\) 4851.44 0.518557
\(445\) 3366.93 0.358669
\(446\) −316.579 −0.0336108
\(447\) −898.024 −0.0950225
\(448\) 6658.47 0.702194
\(449\) 3739.78 0.393076 0.196538 0.980496i \(-0.437030\pi\)
0.196538 + 0.980496i \(0.437030\pi\)
\(450\) −1600.59 −0.167672
\(451\) 13742.6 1.43485
\(452\) 6490.92 0.675459
\(453\) 2966.25 0.307652
\(454\) −8347.85 −0.862961
\(455\) 2153.00 0.221834
\(456\) −6574.12 −0.675134
\(457\) −7986.51 −0.817491 −0.408745 0.912648i \(-0.634034\pi\)
−0.408745 + 0.912648i \(0.634034\pi\)
\(458\) −1566.94 −0.159865
\(459\) −5999.74 −0.610118
\(460\) 4503.17 0.456438
\(461\) −80.4439 −0.00812722 −0.00406361 0.999992i \(-0.501293\pi\)
−0.00406361 + 0.999992i \(0.501293\pi\)
\(462\) −20099.1 −2.02401
\(463\) 15159.0 1.52159 0.760796 0.648991i \(-0.224809\pi\)
0.760796 + 0.648991i \(0.224809\pi\)
\(464\) 26.0054 0.00260187
\(465\) 9101.99 0.907730
\(466\) −3043.21 −0.302519
\(467\) −18397.2 −1.82296 −0.911479 0.411347i \(-0.865059\pi\)
−0.911479 + 0.411347i \(0.865059\pi\)
\(468\) 2964.31 0.292789
\(469\) 36234.2 3.56747
\(470\) 415.246 0.0407529
\(471\) 2058.93 0.201424
\(472\) 7787.03 0.759379
\(473\) −2849.16 −0.276965
\(474\) −1477.48 −0.143170
\(475\) 941.146 0.0909110
\(476\) −9573.64 −0.921864
\(477\) −16232.3 −1.55812
\(478\) −941.187 −0.0900605
\(479\) −3047.92 −0.290737 −0.145368 0.989378i \(-0.546437\pi\)
−0.145368 + 0.989378i \(0.546437\pi\)
\(480\) −7720.71 −0.734168
\(481\) −1375.19 −0.130361
\(482\) 768.550 0.0726276
\(483\) −44232.1 −4.16694
\(484\) −4947.49 −0.464640
\(485\) −8697.29 −0.814276
\(486\) −6863.38 −0.640596
\(487\) 3729.62 0.347033 0.173517 0.984831i \(-0.444487\pi\)
0.173517 + 0.984831i \(0.444487\pi\)
\(488\) 9360.27 0.868278
\(489\) −7285.57 −0.673753
\(490\) 5888.15 0.542856
\(491\) −15519.2 −1.42641 −0.713207 0.700953i \(-0.752758\pi\)
−0.713207 + 0.700953i \(0.752758\pi\)
\(492\) −13374.8 −1.22557
\(493\) 118.310 0.0108082
\(494\) 764.218 0.0696028
\(495\) 9660.24 0.877162
\(496\) 2521.76 0.228287
\(497\) −10941.5 −0.987510
\(498\) 10384.2 0.934389
\(499\) 10975.7 0.984652 0.492326 0.870411i \(-0.336147\pi\)
0.492326 + 0.870411i \(0.336147\pi\)
\(500\) 695.194 0.0621801
\(501\) 7792.97 0.694938
\(502\) −4669.49 −0.415159
\(503\) 13173.2 1.16772 0.583861 0.811854i \(-0.301541\pi\)
0.583861 + 0.811854i \(0.301541\pi\)
\(504\) 28759.5 2.54176
\(505\) −3240.19 −0.285518
\(506\) 11916.3 1.04693
\(507\) −1393.61 −0.122076
\(508\) 15223.8 1.32962
\(509\) 950.065 0.0827326 0.0413663 0.999144i \(-0.486829\pi\)
0.0413663 + 0.999144i \(0.486829\pi\)
\(510\) −3346.04 −0.290520
\(511\) −23937.5 −2.07227
\(512\) −4074.36 −0.351686
\(513\) −4346.10 −0.374045
\(514\) −6643.21 −0.570077
\(515\) −5404.17 −0.462400
\(516\) 2772.90 0.236570
\(517\) −2506.18 −0.213195
\(518\) −5471.52 −0.464102
\(519\) 17530.0 1.48262
\(520\) 1376.51 0.116085
\(521\) −16813.2 −1.41382 −0.706908 0.707305i \(-0.749911\pi\)
−0.706908 + 0.707305i \(0.749911\pi\)
\(522\) −145.751 −0.0122210
\(523\) −6938.06 −0.580077 −0.290038 0.957015i \(-0.593668\pi\)
−0.290038 + 0.957015i \(0.593668\pi\)
\(524\) −8050.28 −0.671142
\(525\) −6828.50 −0.567658
\(526\) 6506.75 0.539368
\(527\) 11472.6 0.948301
\(528\) 4438.94 0.365872
\(529\) 14057.4 1.15537
\(530\) −3091.16 −0.253343
\(531\) 15076.1 1.23211
\(532\) −6934.96 −0.565167
\(533\) 3791.22 0.308098
\(534\) −8671.12 −0.702690
\(535\) −5010.47 −0.404900
\(536\) 23166.2 1.86684
\(537\) 28434.0 2.28495
\(538\) 6574.00 0.526813
\(539\) −35537.4 −2.83990
\(540\) −3210.32 −0.255834
\(541\) 23180.2 1.84213 0.921066 0.389407i \(-0.127320\pi\)
0.921066 + 0.389407i \(0.127320\pi\)
\(542\) 7732.32 0.612789
\(543\) −26991.8 −2.13320
\(544\) −9731.57 −0.766981
\(545\) 3674.66 0.288817
\(546\) −5544.80 −0.434607
\(547\) −20408.1 −1.59522 −0.797612 0.603171i \(-0.793904\pi\)
−0.797612 + 0.603171i \(0.793904\pi\)
\(548\) −10874.0 −0.847650
\(549\) 18122.0 1.40879
\(550\) 1839.63 0.142622
\(551\) 85.7016 0.00662615
\(552\) −28279.6 −2.18054
\(553\) −3800.50 −0.292249
\(554\) −6790.18 −0.520735
\(555\) 4361.59 0.333584
\(556\) −297.606 −0.0227002
\(557\) −4882.71 −0.371431 −0.185715 0.982604i \(-0.559460\pi\)
−0.185715 + 0.982604i \(0.559460\pi\)
\(558\) −14133.6 −1.07226
\(559\) −786.007 −0.0594715
\(560\) −1891.87 −0.142761
\(561\) 20194.8 1.51983
\(562\) −1901.48 −0.142721
\(563\) 18374.1 1.37545 0.687724 0.725972i \(-0.258610\pi\)
0.687724 + 0.725972i \(0.258610\pi\)
\(564\) 2439.10 0.182100
\(565\) 5835.53 0.434518
\(566\) 1784.82 0.132547
\(567\) −5134.08 −0.380267
\(568\) −6995.37 −0.516759
\(569\) −5478.72 −0.403655 −0.201828 0.979421i \(-0.564688\pi\)
−0.201828 + 0.979421i \(0.564688\pi\)
\(570\) −2423.81 −0.178109
\(571\) −9766.61 −0.715797 −0.357898 0.933761i \(-0.616507\pi\)
−0.357898 + 0.933761i \(0.616507\pi\)
\(572\) −3407.01 −0.249046
\(573\) 3661.10 0.266919
\(574\) 15084.2 1.09687
\(575\) 4048.48 0.293623
\(576\) 8241.89 0.596202
\(577\) 18623.4 1.34368 0.671840 0.740696i \(-0.265504\pi\)
0.671840 + 0.740696i \(0.265504\pi\)
\(578\) 3454.39 0.248588
\(579\) 34538.3 2.47904
\(580\) 63.3050 0.00453206
\(581\) 26711.1 1.90734
\(582\) 22398.8 1.59529
\(583\) 18656.5 1.32534
\(584\) −15304.3 −1.08441
\(585\) 2665.00 0.188349
\(586\) −12839.3 −0.905099
\(587\) 16290.0 1.14542 0.572710 0.819758i \(-0.305892\pi\)
0.572710 + 0.819758i \(0.305892\pi\)
\(588\) 34586.2 2.42570
\(589\) 8310.53 0.581374
\(590\) 2870.99 0.200334
\(591\) −3968.11 −0.276187
\(592\) 1208.40 0.0838937
\(593\) 8431.78 0.583898 0.291949 0.956434i \(-0.405696\pi\)
0.291949 + 0.956434i \(0.405696\pi\)
\(594\) −8495.19 −0.586804
\(595\) −8606.99 −0.593029
\(596\) −605.661 −0.0416256
\(597\) 28861.9 1.97863
\(598\) 3287.40 0.224802
\(599\) −26227.6 −1.78903 −0.894516 0.447035i \(-0.852480\pi\)
−0.894516 + 0.447035i \(0.852480\pi\)
\(600\) −4365.77 −0.297053
\(601\) −24169.8 −1.64045 −0.820223 0.572044i \(-0.806151\pi\)
−0.820223 + 0.572044i \(0.806151\pi\)
\(602\) −3127.31 −0.211727
\(603\) 44851.0 3.02898
\(604\) 2000.55 0.134770
\(605\) −4447.94 −0.298900
\(606\) 8344.71 0.559374
\(607\) −11155.6 −0.745953 −0.372977 0.927841i \(-0.621663\pi\)
−0.372977 + 0.927841i \(0.621663\pi\)
\(608\) −7049.36 −0.470213
\(609\) −621.809 −0.0413744
\(610\) 3451.03 0.229063
\(611\) −691.388 −0.0457784
\(612\) −11850.3 −0.782714
\(613\) −24816.7 −1.63513 −0.817567 0.575834i \(-0.804677\pi\)
−0.817567 + 0.575834i \(0.804677\pi\)
\(614\) 3971.14 0.261013
\(615\) −12024.3 −0.788402
\(616\) −33054.5 −2.16202
\(617\) −5299.65 −0.345795 −0.172898 0.984940i \(-0.555313\pi\)
−0.172898 + 0.984940i \(0.555313\pi\)
\(618\) 13917.8 0.905914
\(619\) 4104.30 0.266504 0.133252 0.991082i \(-0.457458\pi\)
0.133252 + 0.991082i \(0.457458\pi\)
\(620\) 6138.72 0.397640
\(621\) −18695.4 −1.20809
\(622\) −3167.08 −0.204161
\(623\) −22304.6 −1.43438
\(624\) 1224.59 0.0785619
\(625\) 625.000 0.0400000
\(626\) −11433.4 −0.729986
\(627\) 14628.7 0.931760
\(628\) 1388.62 0.0882356
\(629\) 5497.57 0.348493
\(630\) 10603.3 0.670549
\(631\) 13060.9 0.824004 0.412002 0.911183i \(-0.364830\pi\)
0.412002 + 0.911183i \(0.364830\pi\)
\(632\) −2429.83 −0.152933
\(633\) −9949.65 −0.624744
\(634\) 3879.39 0.243013
\(635\) 13686.7 0.855337
\(636\) −18157.1 −1.13204
\(637\) −9803.82 −0.609798
\(638\) 167.518 0.0103952
\(639\) −13543.4 −0.838450
\(640\) −5920.66 −0.365679
\(641\) 995.223 0.0613244 0.0306622 0.999530i \(-0.490238\pi\)
0.0306622 + 0.999530i \(0.490238\pi\)
\(642\) 12903.9 0.793263
\(643\) −499.256 −0.0306201 −0.0153100 0.999883i \(-0.504874\pi\)
−0.0153100 + 0.999883i \(0.504874\pi\)
\(644\) −29831.8 −1.82537
\(645\) 2492.92 0.152184
\(646\) −3055.09 −0.186069
\(647\) 30522.4 1.85465 0.927325 0.374256i \(-0.122102\pi\)
0.927325 + 0.374256i \(0.122102\pi\)
\(648\) −3282.45 −0.198992
\(649\) −17327.6 −1.04803
\(650\) 507.505 0.0306246
\(651\) −60297.2 −3.63016
\(652\) −4913.66 −0.295144
\(653\) −9837.63 −0.589550 −0.294775 0.955567i \(-0.595245\pi\)
−0.294775 + 0.955567i \(0.595245\pi\)
\(654\) −9463.64 −0.565837
\(655\) −7237.44 −0.431741
\(656\) −3331.40 −0.198277
\(657\) −29630.0 −1.75947
\(658\) −2750.85 −0.162977
\(659\) −6309.40 −0.372958 −0.186479 0.982459i \(-0.559708\pi\)
−0.186479 + 0.982459i \(0.559708\pi\)
\(660\) 10805.7 0.637292
\(661\) 10067.3 0.592393 0.296196 0.955127i \(-0.404282\pi\)
0.296196 + 0.955127i \(0.404282\pi\)
\(662\) 10797.0 0.633895
\(663\) 5571.19 0.326346
\(664\) 17077.6 0.998101
\(665\) −6234.74 −0.363568
\(666\) −6772.69 −0.394048
\(667\) 368.658 0.0214011
\(668\) 5255.87 0.304425
\(669\) −1671.78 −0.0966141
\(670\) 8541.12 0.492496
\(671\) −20828.4 −1.19832
\(672\) 51146.8 2.93606
\(673\) 23938.7 1.37113 0.685564 0.728012i \(-0.259556\pi\)
0.685564 + 0.728012i \(0.259556\pi\)
\(674\) 8521.37 0.486989
\(675\) −2886.17 −0.164576
\(676\) −939.902 −0.0534765
\(677\) −5102.53 −0.289669 −0.144835 0.989456i \(-0.546265\pi\)
−0.144835 + 0.989456i \(0.546265\pi\)
\(678\) −15028.7 −0.851289
\(679\) 57616.3 3.25642
\(680\) −5502.83 −0.310329
\(681\) −44083.1 −2.48057
\(682\) 16244.4 0.912065
\(683\) −27531.8 −1.54242 −0.771210 0.636580i \(-0.780348\pi\)
−0.771210 + 0.636580i \(0.780348\pi\)
\(684\) −8584.14 −0.479858
\(685\) −9776.00 −0.545287
\(686\) −21265.6 −1.18356
\(687\) −8274.67 −0.459532
\(688\) 690.676 0.0382730
\(689\) 5146.82 0.284584
\(690\) −10426.4 −0.575254
\(691\) 31475.4 1.73282 0.866412 0.499330i \(-0.166421\pi\)
0.866412 + 0.499330i \(0.166421\pi\)
\(692\) 11822.9 0.649478
\(693\) −63995.4 −3.50791
\(694\) 1110.18 0.0607229
\(695\) −267.557 −0.0146029
\(696\) −397.550 −0.0216510
\(697\) −15156.1 −0.823639
\(698\) 7976.75 0.432556
\(699\) −16070.5 −0.869588
\(700\) −4605.40 −0.248668
\(701\) 34138.6 1.83937 0.919684 0.392660i \(-0.128445\pi\)
0.919684 + 0.392660i \(0.128445\pi\)
\(702\) −2343.59 −0.126002
\(703\) 3982.33 0.213651
\(704\) −9472.77 −0.507129
\(705\) 2192.82 0.117144
\(706\) −5630.81 −0.300167
\(707\) 21465.0 1.14183
\(708\) 16863.8 0.895172
\(709\) −6114.73 −0.323898 −0.161949 0.986799i \(-0.551778\pi\)
−0.161949 + 0.986799i \(0.551778\pi\)
\(710\) −2579.12 −0.136328
\(711\) −4704.28 −0.248136
\(712\) −14260.4 −0.750603
\(713\) 35749.0 1.87772
\(714\) 22166.3 1.16184
\(715\) −3063.00 −0.160209
\(716\) 19176.9 1.00094
\(717\) −4970.20 −0.258878
\(718\) −137.207 −0.00713162
\(719\) 26415.4 1.37014 0.685069 0.728478i \(-0.259772\pi\)
0.685069 + 0.728478i \(0.259772\pi\)
\(720\) −2341.77 −0.121212
\(721\) 35800.6 1.84921
\(722\) 8497.64 0.438019
\(723\) 4058.54 0.208767
\(724\) −18204.3 −0.934471
\(725\) 56.9130 0.00291544
\(726\) 11455.1 0.585591
\(727\) 36222.5 1.84789 0.923946 0.382524i \(-0.124945\pi\)
0.923946 + 0.382524i \(0.124945\pi\)
\(728\) −9118.86 −0.464241
\(729\) −32059.0 −1.62877
\(730\) −5642.53 −0.286081
\(731\) 3142.20 0.158985
\(732\) 20270.9 1.02354
\(733\) −14555.7 −0.733463 −0.366731 0.930327i \(-0.619523\pi\)
−0.366731 + 0.930327i \(0.619523\pi\)
\(734\) 3716.42 0.186888
\(735\) 31094.0 1.56043
\(736\) −30323.9 −1.51869
\(737\) −51549.2 −2.57644
\(738\) 18671.4 0.931305
\(739\) −27876.6 −1.38763 −0.693814 0.720155i \(-0.744071\pi\)
−0.693814 + 0.720155i \(0.744071\pi\)
\(740\) 2941.62 0.146130
\(741\) 4035.66 0.200073
\(742\) 20477.8 1.01316
\(743\) 3226.10 0.159292 0.0796460 0.996823i \(-0.474621\pi\)
0.0796460 + 0.996823i \(0.474621\pi\)
\(744\) −38550.7 −1.89965
\(745\) −544.507 −0.0267774
\(746\) 11385.5 0.558784
\(747\) 33063.2 1.61943
\(748\) 13620.1 0.665775
\(749\) 33192.5 1.61926
\(750\) −1609.61 −0.0783663
\(751\) 27690.7 1.34547 0.672735 0.739884i \(-0.265120\pi\)
0.672735 + 0.739884i \(0.265120\pi\)
\(752\) 607.533 0.0294607
\(753\) −24658.5 −1.19337
\(754\) 46.2138 0.00223211
\(755\) 1798.55 0.0866967
\(756\) 21267.2 1.02312
\(757\) −12300.3 −0.590571 −0.295285 0.955409i \(-0.595415\pi\)
−0.295285 + 0.955409i \(0.595415\pi\)
\(758\) −9100.21 −0.436061
\(759\) 62927.6 3.00939
\(760\) −3986.14 −0.190254
\(761\) −25188.7 −1.19986 −0.599928 0.800054i \(-0.704804\pi\)
−0.599928 + 0.800054i \(0.704804\pi\)
\(762\) −35248.4 −1.67574
\(763\) −24343.2 −1.15502
\(764\) 2469.18 0.116927
\(765\) −10653.8 −0.503514
\(766\) −9416.09 −0.444148
\(767\) −4780.23 −0.225038
\(768\) 28509.3 1.33951
\(769\) 4955.84 0.232396 0.116198 0.993226i \(-0.462929\pi\)
0.116198 + 0.993226i \(0.462929\pi\)
\(770\) −12186.9 −0.570369
\(771\) −35081.3 −1.63868
\(772\) 23293.9 1.08597
\(773\) −14186.7 −0.660106 −0.330053 0.943962i \(-0.607067\pi\)
−0.330053 + 0.943962i \(0.607067\pi\)
\(774\) −3871.01 −0.179768
\(775\) 5518.89 0.255799
\(776\) 36836.6 1.70407
\(777\) −28893.9 −1.33406
\(778\) −12382.1 −0.570593
\(779\) −10978.8 −0.504948
\(780\) 2981.01 0.136843
\(781\) 15566.1 0.713185
\(782\) −13141.9 −0.600965
\(783\) −262.817 −0.0119953
\(784\) 8614.76 0.392436
\(785\) 1248.41 0.0567613
\(786\) 18639.2 0.845848
\(787\) −27311.8 −1.23705 −0.618526 0.785764i \(-0.712270\pi\)
−0.618526 + 0.785764i \(0.712270\pi\)
\(788\) −2676.24 −0.120986
\(789\) 34360.7 1.55041
\(790\) −895.852 −0.0403456
\(791\) −38658.2 −1.73771
\(792\) −40915.1 −1.83568
\(793\) −5746.00 −0.257310
\(794\) 5743.24 0.256700
\(795\) −16323.7 −0.728231
\(796\) 19465.6 0.866757
\(797\) −3313.27 −0.147255 −0.0736274 0.997286i \(-0.523458\pi\)
−0.0736274 + 0.997286i \(0.523458\pi\)
\(798\) 16056.8 0.712287
\(799\) 2763.94 0.122380
\(800\) −4681.37 −0.206889
\(801\) −27608.8 −1.21787
\(802\) −12134.3 −0.534262
\(803\) 34055.0 1.49661
\(804\) 50169.4 2.20067
\(805\) −26819.7 −1.17425
\(806\) 4481.38 0.195844
\(807\) 34715.8 1.51432
\(808\) 13723.5 0.597516
\(809\) 3606.84 0.156749 0.0783744 0.996924i \(-0.475027\pi\)
0.0783744 + 0.996924i \(0.475027\pi\)
\(810\) −1210.20 −0.0524966
\(811\) 4908.14 0.212513 0.106257 0.994339i \(-0.466113\pi\)
0.106257 + 0.994339i \(0.466113\pi\)
\(812\) −419.371 −0.0181244
\(813\) 40832.7 1.76146
\(814\) 7784.15 0.335177
\(815\) −4417.53 −0.189864
\(816\) −4895.49 −0.210020
\(817\) 2276.15 0.0974692
\(818\) −11090.5 −0.474048
\(819\) −17654.6 −0.753238
\(820\) −8109.65 −0.345367
\(821\) 25125.8 1.06808 0.534042 0.845458i \(-0.320672\pi\)
0.534042 + 0.845458i \(0.320672\pi\)
\(822\) 25176.9 1.06830
\(823\) −32032.1 −1.35671 −0.678353 0.734736i \(-0.737306\pi\)
−0.678353 + 0.734736i \(0.737306\pi\)
\(824\) 22888.9 0.967685
\(825\) 9714.68 0.409966
\(826\) −19019.3 −0.801167
\(827\) 4182.06 0.175846 0.0879229 0.996127i \(-0.471977\pi\)
0.0879229 + 0.996127i \(0.471977\pi\)
\(828\) −36926.0 −1.54984
\(829\) −39983.0 −1.67511 −0.837556 0.546351i \(-0.816017\pi\)
−0.837556 + 0.546351i \(0.816017\pi\)
\(830\) 6296.33 0.263312
\(831\) −35857.4 −1.49685
\(832\) −2613.28 −0.108893
\(833\) 39192.4 1.63018
\(834\) 689.059 0.0286093
\(835\) 4725.18 0.195834
\(836\) 9866.13 0.408167
\(837\) −25485.6 −1.05246
\(838\) 14259.5 0.587811
\(839\) 28100.7 1.15631 0.578155 0.815927i \(-0.303773\pi\)
0.578155 + 0.815927i \(0.303773\pi\)
\(840\) 28921.6 1.18796
\(841\) −24383.8 −0.999788
\(842\) 14718.3 0.602408
\(843\) −10041.3 −0.410250
\(844\) −6710.41 −0.273675
\(845\) −845.000 −0.0344010
\(846\) −3405.02 −0.138377
\(847\) 29465.9 1.19535
\(848\) −4522.58 −0.183144
\(849\) 9425.23 0.381005
\(850\) −2028.84 −0.0818688
\(851\) 17130.6 0.690047
\(852\) −15149.4 −0.609166
\(853\) −10845.1 −0.435321 −0.217661 0.976024i \(-0.569843\pi\)
−0.217661 + 0.976024i \(0.569843\pi\)
\(854\) −22861.8 −0.916058
\(855\) −7717.40 −0.308689
\(856\) 21221.4 0.847353
\(857\) −20977.4 −0.836144 −0.418072 0.908414i \(-0.637294\pi\)
−0.418072 + 0.908414i \(0.637294\pi\)
\(858\) 7888.39 0.313876
\(859\) −18440.5 −0.732458 −0.366229 0.930525i \(-0.619351\pi\)
−0.366229 + 0.930525i \(0.619351\pi\)
\(860\) 1681.32 0.0666656
\(861\) 79656.6 3.15295
\(862\) −10218.4 −0.403759
\(863\) 28960.1 1.14231 0.571154 0.820843i \(-0.306496\pi\)
0.571154 + 0.820843i \(0.306496\pi\)
\(864\) 21618.0 0.851226
\(865\) 10629.1 0.417805
\(866\) 14349.6 0.563071
\(867\) 18241.9 0.714563
\(868\) −40666.7 −1.59023
\(869\) 5406.84 0.211064
\(870\) −146.573 −0.00571181
\(871\) −14221.0 −0.553228
\(872\) −15563.7 −0.604419
\(873\) 71317.8 2.76488
\(874\) −9519.76 −0.368433
\(875\) −4140.39 −0.159966
\(876\) −33143.4 −1.27833
\(877\) −34736.4 −1.33747 −0.668737 0.743499i \(-0.733165\pi\)
−0.668737 + 0.743499i \(0.733165\pi\)
\(878\) 4723.31 0.181553
\(879\) −67801.6 −2.60170
\(880\) 2691.51 0.103103
\(881\) −4957.71 −0.189591 −0.0947953 0.995497i \(-0.530220\pi\)
−0.0947953 + 0.995497i \(0.530220\pi\)
\(882\) −48282.8 −1.84327
\(883\) 10146.6 0.386704 0.193352 0.981129i \(-0.438064\pi\)
0.193352 + 0.981129i \(0.438064\pi\)
\(884\) 3757.42 0.142959
\(885\) 15161.1 0.575858
\(886\) 12310.8 0.466807
\(887\) 29481.3 1.11599 0.557997 0.829843i \(-0.311570\pi\)
0.557997 + 0.829843i \(0.311570\pi\)
\(888\) −18473.1 −0.698106
\(889\) −90669.0 −3.42063
\(890\) −5257.64 −0.198019
\(891\) 7304.08 0.274631
\(892\) −1127.51 −0.0423227
\(893\) 2002.15 0.0750272
\(894\) 1402.31 0.0524612
\(895\) 17240.6 0.643901
\(896\) 39222.1 1.46241
\(897\) 17360.0 0.646192
\(898\) −5839.86 −0.217014
\(899\) 502.555 0.0186442
\(900\) −5700.59 −0.211133
\(901\) −20575.3 −0.760779
\(902\) −21459.8 −0.792167
\(903\) −16514.6 −0.608607
\(904\) −24715.9 −0.909335
\(905\) −16366.2 −0.601139
\(906\) −4631.95 −0.169852
\(907\) −34640.7 −1.26817 −0.634083 0.773265i \(-0.718622\pi\)
−0.634083 + 0.773265i \(0.718622\pi\)
\(908\) −29731.3 −1.08664
\(909\) 26569.5 0.969479
\(910\) −3362.03 −0.122473
\(911\) −14409.8 −0.524059 −0.262029 0.965060i \(-0.584392\pi\)
−0.262029 + 0.965060i \(0.584392\pi\)
\(912\) −3546.20 −0.128757
\(913\) −38000.9 −1.37749
\(914\) 12471.4 0.451330
\(915\) 18224.1 0.658438
\(916\) −5580.74 −0.201302
\(917\) 47945.3 1.72660
\(918\) 9368.91 0.336841
\(919\) −17403.1 −0.624672 −0.312336 0.949972i \(-0.601112\pi\)
−0.312336 + 0.949972i \(0.601112\pi\)
\(920\) −17147.0 −0.614479
\(921\) 20970.7 0.750280
\(922\) 125.617 0.00448697
\(923\) 4294.26 0.153139
\(924\) −71583.9 −2.54863
\(925\) 2644.60 0.0940043
\(926\) −23671.5 −0.840060
\(927\) 44314.2 1.57008
\(928\) −426.289 −0.0150793
\(929\) −11474.1 −0.405223 −0.202612 0.979259i \(-0.564943\pi\)
−0.202612 + 0.979259i \(0.564943\pi\)
\(930\) −14213.2 −0.501151
\(931\) 28390.2 0.999412
\(932\) −10838.5 −0.380932
\(933\) −16724.6 −0.586859
\(934\) 28728.2 1.00644
\(935\) 12244.9 0.428289
\(936\) −11287.4 −0.394166
\(937\) 15121.6 0.527216 0.263608 0.964630i \(-0.415087\pi\)
0.263608 + 0.964630i \(0.415087\pi\)
\(938\) −56581.7 −1.96957
\(939\) −60377.3 −2.09834
\(940\) 1478.92 0.0513160
\(941\) 18152.2 0.628846 0.314423 0.949283i \(-0.398189\pi\)
0.314423 + 0.949283i \(0.398189\pi\)
\(942\) −3215.13 −0.111204
\(943\) −47226.8 −1.63088
\(944\) 4200.46 0.144823
\(945\) 19119.8 0.658167
\(946\) 4449.12 0.152910
\(947\) 19041.0 0.653378 0.326689 0.945132i \(-0.394067\pi\)
0.326689 + 0.945132i \(0.394067\pi\)
\(948\) −5262.11 −0.180280
\(949\) 9394.86 0.321359
\(950\) −1469.65 −0.0501913
\(951\) 20486.2 0.698538
\(952\) 36454.2 1.24106
\(953\) −22836.0 −0.776213 −0.388106 0.921615i \(-0.626871\pi\)
−0.388106 + 0.921615i \(0.626871\pi\)
\(954\) 25347.5 0.860228
\(955\) 2219.87 0.0752180
\(956\) −3352.09 −0.113404
\(957\) 884.627 0.0298808
\(958\) 4759.48 0.160513
\(959\) 64762.3 2.18069
\(960\) 8288.34 0.278651
\(961\) 18942.1 0.635832
\(962\) 2147.44 0.0719710
\(963\) 41085.9 1.37484
\(964\) 2737.23 0.0914526
\(965\) 20941.9 0.698595
\(966\) 69070.8 2.30054
\(967\) −32917.6 −1.09468 −0.547342 0.836909i \(-0.684360\pi\)
−0.547342 + 0.836909i \(0.684360\pi\)
\(968\) 18838.9 0.625520
\(969\) −16133.2 −0.534855
\(970\) 13581.3 0.449555
\(971\) −52645.6 −1.73994 −0.869968 0.493108i \(-0.835861\pi\)
−0.869968 + 0.493108i \(0.835861\pi\)
\(972\) −24444.3 −0.806637
\(973\) 1772.46 0.0583993
\(974\) −5824.00 −0.191594
\(975\) 2680.02 0.0880300
\(976\) 5049.10 0.165592
\(977\) −11301.4 −0.370075 −0.185038 0.982731i \(-0.559241\pi\)
−0.185038 + 0.982731i \(0.559241\pi\)
\(978\) 11376.8 0.371974
\(979\) 31732.1 1.03592
\(980\) 20971.0 0.683564
\(981\) −30132.2 −0.980680
\(982\) 24234.0 0.787512
\(983\) −57289.3 −1.85884 −0.929422 0.369018i \(-0.879694\pi\)
−0.929422 + 0.369018i \(0.879694\pi\)
\(984\) 50928.0 1.64992
\(985\) −2406.02 −0.0778297
\(986\) −184.747 −0.00596710
\(987\) −14526.6 −0.468477
\(988\) 2721.80 0.0876438
\(989\) 9791.20 0.314805
\(990\) −15085.0 −0.484275
\(991\) 3413.94 0.109432 0.0547162 0.998502i \(-0.482575\pi\)
0.0547162 + 0.998502i \(0.482575\pi\)
\(992\) −41337.5 −1.32305
\(993\) 57016.7 1.82213
\(994\) 17085.7 0.545196
\(995\) 17500.1 0.557579
\(996\) 36983.8 1.17658
\(997\) −46243.7 −1.46896 −0.734480 0.678630i \(-0.762574\pi\)
−0.734480 + 0.678630i \(0.762574\pi\)
\(998\) −17139.2 −0.543619
\(999\) −12212.5 −0.386772
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 65.4.a.c.1.1 2
3.2 odd 2 585.4.a.h.1.2 2
4.3 odd 2 1040.4.a.k.1.2 2
5.2 odd 4 325.4.b.f.274.2 4
5.3 odd 4 325.4.b.f.274.3 4
5.4 even 2 325.4.a.g.1.2 2
13.12 even 2 845.4.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.4.a.c.1.1 2 1.1 even 1 trivial
325.4.a.g.1.2 2 5.4 even 2
325.4.b.f.274.2 4 5.2 odd 4
325.4.b.f.274.3 4 5.3 odd 4
585.4.a.h.1.2 2 3.2 odd 2
845.4.a.d.1.2 2 13.12 even 2
1040.4.a.k.1.2 2 4.3 odd 2