Properties

Label 8047.2.a.d.1.7
Level $8047$
Weight $2$
Character 8047.1
Self dual yes
Analytic conductor $64.256$
Analytic rank $0$
Dimension $156$
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] = [8047,2,Mod(1,8047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8047, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8047 = 13 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8047.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: \(64.2556185065\)
Analytic rank: \(0\)
Dimension: \(156\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 8047.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.53163 q^{2} -1.76740 q^{3} +4.40915 q^{4} +3.70059 q^{5} +4.47441 q^{6} +1.97275 q^{7} -6.09908 q^{8} +0.123713 q^{9} +O(q^{10})\) \(q-2.53163 q^{2} -1.76740 q^{3} +4.40915 q^{4} +3.70059 q^{5} +4.47441 q^{6} +1.97275 q^{7} -6.09908 q^{8} +0.123713 q^{9} -9.36852 q^{10} +0.845819 q^{11} -7.79275 q^{12} -1.00000 q^{13} -4.99427 q^{14} -6.54043 q^{15} +6.62232 q^{16} -1.87515 q^{17} -0.313196 q^{18} +0.327217 q^{19} +16.3164 q^{20} -3.48664 q^{21} -2.14130 q^{22} +4.68118 q^{23} +10.7795 q^{24} +8.69433 q^{25} +2.53163 q^{26} +5.08356 q^{27} +8.69816 q^{28} +5.72085 q^{29} +16.5579 q^{30} +4.15100 q^{31} -4.56710 q^{32} -1.49490 q^{33} +4.74719 q^{34} +7.30033 q^{35} +0.545471 q^{36} +4.26358 q^{37} -0.828394 q^{38} +1.76740 q^{39} -22.5702 q^{40} -6.77134 q^{41} +8.82690 q^{42} -8.94374 q^{43} +3.72934 q^{44} +0.457812 q^{45} -11.8510 q^{46} +3.76653 q^{47} -11.7043 q^{48} -3.10826 q^{49} -22.0108 q^{50} +3.31415 q^{51} -4.40915 q^{52} -3.11599 q^{53} -12.8697 q^{54} +3.13002 q^{55} -12.0320 q^{56} -0.578325 q^{57} -14.4831 q^{58} +4.25737 q^{59} -28.8377 q^{60} +13.3881 q^{61} -10.5088 q^{62} +0.244056 q^{63} -1.68242 q^{64} -3.70059 q^{65} +3.78454 q^{66} +8.99342 q^{67} -8.26783 q^{68} -8.27353 q^{69} -18.4817 q^{70} -11.6754 q^{71} -0.754538 q^{72} +6.09426 q^{73} -10.7938 q^{74} -15.3664 q^{75} +1.44275 q^{76} +1.66859 q^{77} -4.47441 q^{78} -4.87010 q^{79} +24.5065 q^{80} -9.35584 q^{81} +17.1425 q^{82} +14.6972 q^{83} -15.3731 q^{84} -6.93916 q^{85} +22.6422 q^{86} -10.1110 q^{87} -5.15872 q^{88} -6.08270 q^{89} -1.15901 q^{90} -1.97275 q^{91} +20.6400 q^{92} -7.33650 q^{93} -9.53546 q^{94} +1.21090 q^{95} +8.07191 q^{96} -5.79937 q^{97} +7.86896 q^{98} +0.104639 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.53163 −1.79013 −0.895067 0.445933i \(-0.852872\pi\)
−0.895067 + 0.445933i \(0.852872\pi\)
\(3\) −1.76740 −1.02041 −0.510205 0.860053i \(-0.670431\pi\)
−0.510205 + 0.860053i \(0.670431\pi\)
\(4\) 4.40915 2.20458
\(5\) 3.70059 1.65495 0.827476 0.561501i \(-0.189776\pi\)
0.827476 + 0.561501i \(0.189776\pi\)
\(6\) 4.47441 1.82667
\(7\) 1.97275 0.745629 0.372815 0.927906i \(-0.378393\pi\)
0.372815 + 0.927906i \(0.378393\pi\)
\(8\) −6.09908 −2.15635
\(9\) 0.123713 0.0412378
\(10\) −9.36852 −2.96258
\(11\) 0.845819 0.255024 0.127512 0.991837i \(-0.459301\pi\)
0.127512 + 0.991837i \(0.459301\pi\)
\(12\) −7.79275 −2.24957
\(13\) −1.00000 −0.277350
\(14\) −4.99427 −1.33478
\(15\) −6.54043 −1.68873
\(16\) 6.62232 1.65558
\(17\) −1.87515 −0.454791 −0.227396 0.973802i \(-0.573021\pi\)
−0.227396 + 0.973802i \(0.573021\pi\)
\(18\) −0.313196 −0.0738211
\(19\) 0.327217 0.0750688 0.0375344 0.999295i \(-0.488050\pi\)
0.0375344 + 0.999295i \(0.488050\pi\)
\(20\) 16.3164 3.64847
\(21\) −3.48664 −0.760848
\(22\) −2.14130 −0.456527
\(23\) 4.68118 0.976093 0.488047 0.872818i \(-0.337710\pi\)
0.488047 + 0.872818i \(0.337710\pi\)
\(24\) 10.7795 2.20036
\(25\) 8.69433 1.73887
\(26\) 2.53163 0.496494
\(27\) 5.08356 0.978331
\(28\) 8.69816 1.64380
\(29\) 5.72085 1.06233 0.531167 0.847267i \(-0.321754\pi\)
0.531167 + 0.847267i \(0.321754\pi\)
\(30\) 16.5579 3.02305
\(31\) 4.15100 0.745542 0.372771 0.927923i \(-0.378408\pi\)
0.372771 + 0.927923i \(0.378408\pi\)
\(32\) −4.56710 −0.807358
\(33\) −1.49490 −0.260229
\(34\) 4.74719 0.814137
\(35\) 7.30033 1.23398
\(36\) 0.545471 0.0909118
\(37\) 4.26358 0.700928 0.350464 0.936576i \(-0.386024\pi\)
0.350464 + 0.936576i \(0.386024\pi\)
\(38\) −0.828394 −0.134383
\(39\) 1.76740 0.283011
\(40\) −22.5702 −3.56866
\(41\) −6.77134 −1.05750 −0.528752 0.848776i \(-0.677340\pi\)
−0.528752 + 0.848776i \(0.677340\pi\)
\(42\) 8.82690 1.36202
\(43\) −8.94374 −1.36391 −0.681954 0.731395i \(-0.738870\pi\)
−0.681954 + 0.731395i \(0.738870\pi\)
\(44\) 3.72934 0.562220
\(45\) 0.457812 0.0682466
\(46\) −11.8510 −1.74734
\(47\) 3.76653 0.549405 0.274702 0.961529i \(-0.411421\pi\)
0.274702 + 0.961529i \(0.411421\pi\)
\(48\) −11.7043 −1.68937
\(49\) −3.10826 −0.444037
\(50\) −22.0108 −3.11280
\(51\) 3.31415 0.464074
\(52\) −4.40915 −0.611439
\(53\) −3.11599 −0.428015 −0.214007 0.976832i \(-0.568652\pi\)
−0.214007 + 0.976832i \(0.568652\pi\)
\(54\) −12.8697 −1.75134
\(55\) 3.13002 0.422052
\(56\) −12.0320 −1.60784
\(57\) −0.578325 −0.0766010
\(58\) −14.4831 −1.90172
\(59\) 4.25737 0.554262 0.277131 0.960832i \(-0.410616\pi\)
0.277131 + 0.960832i \(0.410616\pi\)
\(60\) −28.8377 −3.72294
\(61\) 13.3881 1.71417 0.857083 0.515179i \(-0.172275\pi\)
0.857083 + 0.515179i \(0.172275\pi\)
\(62\) −10.5088 −1.33462
\(63\) 0.244056 0.0307481
\(64\) −1.68242 −0.210303
\(65\) −3.70059 −0.459001
\(66\) 3.78454 0.465845
\(67\) 8.99342 1.09872 0.549361 0.835585i \(-0.314871\pi\)
0.549361 + 0.835585i \(0.314871\pi\)
\(68\) −8.26783 −1.00262
\(69\) −8.27353 −0.996016
\(70\) −18.4817 −2.20899
\(71\) −11.6754 −1.38561 −0.692805 0.721125i \(-0.743625\pi\)
−0.692805 + 0.721125i \(0.743625\pi\)
\(72\) −0.754538 −0.0889232
\(73\) 6.09426 0.713279 0.356639 0.934242i \(-0.383922\pi\)
0.356639 + 0.934242i \(0.383922\pi\)
\(74\) −10.7938 −1.25475
\(75\) −15.3664 −1.77436
\(76\) 1.44275 0.165495
\(77\) 1.66859 0.190153
\(78\) −4.47441 −0.506627
\(79\) −4.87010 −0.547929 −0.273964 0.961740i \(-0.588335\pi\)
−0.273964 + 0.961740i \(0.588335\pi\)
\(80\) 24.5065 2.73991
\(81\) −9.35584 −1.03954
\(82\) 17.1425 1.89307
\(83\) 14.6972 1.61323 0.806615 0.591077i \(-0.201297\pi\)
0.806615 + 0.591077i \(0.201297\pi\)
\(84\) −15.3731 −1.67735
\(85\) −6.93916 −0.752658
\(86\) 22.6422 2.44158
\(87\) −10.1110 −1.08402
\(88\) −5.15872 −0.549921
\(89\) −6.08270 −0.644765 −0.322383 0.946609i \(-0.604484\pi\)
−0.322383 + 0.946609i \(0.604484\pi\)
\(90\) −1.15901 −0.122170
\(91\) −1.97275 −0.206800
\(92\) 20.6400 2.15187
\(93\) −7.33650 −0.760759
\(94\) −9.53546 −0.983508
\(95\) 1.21090 0.124235
\(96\) 8.07191 0.823836
\(97\) −5.79937 −0.588837 −0.294419 0.955677i \(-0.595126\pi\)
−0.294419 + 0.955677i \(0.595126\pi\)
\(98\) 7.86896 0.794885
\(99\) 0.104639 0.0105166
\(100\) 38.3346 3.83346
\(101\) −2.64341 −0.263029 −0.131515 0.991314i \(-0.541984\pi\)
−0.131515 + 0.991314i \(0.541984\pi\)
\(102\) −8.39020 −0.830754
\(103\) −6.85005 −0.674955 −0.337478 0.941334i \(-0.609574\pi\)
−0.337478 + 0.941334i \(0.609574\pi\)
\(104\) 6.09908 0.598064
\(105\) −12.9026 −1.25917
\(106\) 7.88854 0.766203
\(107\) 3.20026 0.309381 0.154690 0.987963i \(-0.450562\pi\)
0.154690 + 0.987963i \(0.450562\pi\)
\(108\) 22.4142 2.15681
\(109\) 4.47884 0.428995 0.214498 0.976725i \(-0.431189\pi\)
0.214498 + 0.976725i \(0.431189\pi\)
\(110\) −7.92407 −0.755530
\(111\) −7.53546 −0.715234
\(112\) 13.0642 1.23445
\(113\) 10.6855 1.00521 0.502606 0.864516i \(-0.332375\pi\)
0.502606 + 0.864516i \(0.332375\pi\)
\(114\) 1.46411 0.137126
\(115\) 17.3231 1.61539
\(116\) 25.2241 2.34200
\(117\) −0.123713 −0.0114373
\(118\) −10.7781 −0.992203
\(119\) −3.69921 −0.339106
\(120\) 39.8906 3.64150
\(121\) −10.2846 −0.934963
\(122\) −33.8936 −3.06858
\(123\) 11.9677 1.07909
\(124\) 18.3024 1.64361
\(125\) 13.6712 1.22279
\(126\) −0.617858 −0.0550432
\(127\) −6.13172 −0.544102 −0.272051 0.962283i \(-0.587702\pi\)
−0.272051 + 0.962283i \(0.587702\pi\)
\(128\) 13.3935 1.18383
\(129\) 15.8072 1.39175
\(130\) 9.36852 0.821673
\(131\) 7.49401 0.654755 0.327377 0.944894i \(-0.393835\pi\)
0.327377 + 0.944894i \(0.393835\pi\)
\(132\) −6.59125 −0.573695
\(133\) 0.645518 0.0559735
\(134\) −22.7680 −1.96686
\(135\) 18.8121 1.61909
\(136\) 11.4367 0.980690
\(137\) −2.80027 −0.239243 −0.119622 0.992820i \(-0.538168\pi\)
−0.119622 + 0.992820i \(0.538168\pi\)
\(138\) 20.9455 1.78300
\(139\) 18.9934 1.61100 0.805499 0.592597i \(-0.201898\pi\)
0.805499 + 0.592597i \(0.201898\pi\)
\(140\) 32.1883 2.72041
\(141\) −6.65698 −0.560619
\(142\) 29.5577 2.48043
\(143\) −0.845819 −0.0707309
\(144\) 0.819270 0.0682725
\(145\) 21.1705 1.75811
\(146\) −15.4284 −1.27686
\(147\) 5.49354 0.453100
\(148\) 18.7988 1.54525
\(149\) −7.67291 −0.628589 −0.314295 0.949325i \(-0.601768\pi\)
−0.314295 + 0.949325i \(0.601768\pi\)
\(150\) 38.9020 3.17634
\(151\) −6.81345 −0.554471 −0.277235 0.960802i \(-0.589418\pi\)
−0.277235 + 0.960802i \(0.589418\pi\)
\(152\) −1.99573 −0.161875
\(153\) −0.231981 −0.0187546
\(154\) −4.22425 −0.340400
\(155\) 15.3611 1.23384
\(156\) 7.79275 0.623919
\(157\) 19.1957 1.53199 0.765993 0.642848i \(-0.222248\pi\)
0.765993 + 0.642848i \(0.222248\pi\)
\(158\) 12.3293 0.980866
\(159\) 5.50722 0.436751
\(160\) −16.9010 −1.33614
\(161\) 9.23480 0.727804
\(162\) 23.6855 1.86091
\(163\) 20.0771 1.57256 0.786282 0.617868i \(-0.212003\pi\)
0.786282 + 0.617868i \(0.212003\pi\)
\(164\) −29.8559 −2.33135
\(165\) −5.53201 −0.430667
\(166\) −37.2080 −2.88790
\(167\) −19.3301 −1.49581 −0.747906 0.663805i \(-0.768940\pi\)
−0.747906 + 0.663805i \(0.768940\pi\)
\(168\) 21.2653 1.64066
\(169\) 1.00000 0.0769231
\(170\) 17.5674 1.34736
\(171\) 0.0404812 0.00309567
\(172\) −39.4343 −3.00684
\(173\) 14.8091 1.12591 0.562957 0.826486i \(-0.309663\pi\)
0.562957 + 0.826486i \(0.309663\pi\)
\(174\) 25.5974 1.94054
\(175\) 17.1517 1.29655
\(176\) 5.60128 0.422213
\(177\) −7.52448 −0.565575
\(178\) 15.3992 1.15422
\(179\) 6.59466 0.492908 0.246454 0.969154i \(-0.420735\pi\)
0.246454 + 0.969154i \(0.420735\pi\)
\(180\) 2.01856 0.150455
\(181\) 6.43405 0.478239 0.239119 0.970990i \(-0.423141\pi\)
0.239119 + 0.970990i \(0.423141\pi\)
\(182\) 4.99427 0.370200
\(183\) −23.6621 −1.74915
\(184\) −28.5509 −2.10480
\(185\) 15.7777 1.16000
\(186\) 18.5733 1.36186
\(187\) −1.58604 −0.115983
\(188\) 16.6072 1.21121
\(189\) 10.0286 0.729473
\(190\) −3.06554 −0.222398
\(191\) 22.5531 1.63188 0.815941 0.578135i \(-0.196219\pi\)
0.815941 + 0.578135i \(0.196219\pi\)
\(192\) 2.97352 0.214595
\(193\) −0.426356 −0.0306898 −0.0153449 0.999882i \(-0.504885\pi\)
−0.0153449 + 0.999882i \(0.504885\pi\)
\(194\) 14.6819 1.05410
\(195\) 6.54043 0.468370
\(196\) −13.7048 −0.978913
\(197\) 3.79037 0.270053 0.135026 0.990842i \(-0.456888\pi\)
0.135026 + 0.990842i \(0.456888\pi\)
\(198\) −0.264907 −0.0188262
\(199\) −25.2238 −1.78807 −0.894035 0.447996i \(-0.852138\pi\)
−0.894035 + 0.447996i \(0.852138\pi\)
\(200\) −53.0275 −3.74961
\(201\) −15.8950 −1.12115
\(202\) 6.69214 0.470858
\(203\) 11.2858 0.792108
\(204\) 14.6126 1.02309
\(205\) −25.0579 −1.75012
\(206\) 17.3418 1.20826
\(207\) 0.579124 0.0402519
\(208\) −6.62232 −0.459175
\(209\) 0.276767 0.0191444
\(210\) 32.6647 2.25408
\(211\) −13.6111 −0.937024 −0.468512 0.883457i \(-0.655210\pi\)
−0.468512 + 0.883457i \(0.655210\pi\)
\(212\) −13.7389 −0.943591
\(213\) 20.6351 1.41389
\(214\) −8.10187 −0.553832
\(215\) −33.0971 −2.25720
\(216\) −31.0050 −2.10963
\(217\) 8.18890 0.555898
\(218\) −11.3388 −0.767959
\(219\) −10.7710 −0.727837
\(220\) 13.8008 0.930447
\(221\) 1.87515 0.126136
\(222\) 19.0770 1.28036
\(223\) −28.8573 −1.93243 −0.966215 0.257736i \(-0.917023\pi\)
−0.966215 + 0.257736i \(0.917023\pi\)
\(224\) −9.00976 −0.601990
\(225\) 1.07561 0.0717070
\(226\) −27.0519 −1.79946
\(227\) −10.0709 −0.668429 −0.334214 0.942497i \(-0.608471\pi\)
−0.334214 + 0.942497i \(0.608471\pi\)
\(228\) −2.54992 −0.168873
\(229\) −21.2559 −1.40463 −0.702315 0.711866i \(-0.747850\pi\)
−0.702315 + 0.711866i \(0.747850\pi\)
\(230\) −43.8557 −2.89176
\(231\) −2.94907 −0.194035
\(232\) −34.8919 −2.29077
\(233\) 20.4561 1.34012 0.670061 0.742306i \(-0.266268\pi\)
0.670061 + 0.742306i \(0.266268\pi\)
\(234\) 0.313196 0.0204743
\(235\) 13.9384 0.909239
\(236\) 18.7714 1.22191
\(237\) 8.60743 0.559112
\(238\) 9.36503 0.607044
\(239\) −18.2924 −1.18324 −0.591619 0.806217i \(-0.701511\pi\)
−0.591619 + 0.806217i \(0.701511\pi\)
\(240\) −43.3128 −2.79583
\(241\) 21.1221 1.36059 0.680296 0.732937i \(-0.261851\pi\)
0.680296 + 0.732937i \(0.261851\pi\)
\(242\) 26.0368 1.67371
\(243\) 1.28486 0.0824237
\(244\) 59.0300 3.77901
\(245\) −11.5024 −0.734859
\(246\) −30.2977 −1.93171
\(247\) −0.327217 −0.0208204
\(248\) −25.3173 −1.60765
\(249\) −25.9759 −1.64616
\(250\) −34.6104 −2.18896
\(251\) 4.36988 0.275824 0.137912 0.990444i \(-0.455961\pi\)
0.137912 + 0.990444i \(0.455961\pi\)
\(252\) 1.07608 0.0677866
\(253\) 3.95943 0.248927
\(254\) 15.5232 0.974015
\(255\) 12.2643 0.768020
\(256\) −30.5425 −1.90891
\(257\) −15.5231 −0.968304 −0.484152 0.874984i \(-0.660872\pi\)
−0.484152 + 0.874984i \(0.660872\pi\)
\(258\) −40.0180 −2.49141
\(259\) 8.41097 0.522632
\(260\) −16.3164 −1.01190
\(261\) 0.707745 0.0438083
\(262\) −18.9721 −1.17210
\(263\) 31.5687 1.94661 0.973305 0.229516i \(-0.0737144\pi\)
0.973305 + 0.229516i \(0.0737144\pi\)
\(264\) 9.11754 0.561146
\(265\) −11.5310 −0.708344
\(266\) −1.63421 −0.100200
\(267\) 10.7506 0.657925
\(268\) 39.6534 2.42221
\(269\) 29.2400 1.78279 0.891397 0.453222i \(-0.149726\pi\)
0.891397 + 0.453222i \(0.149726\pi\)
\(270\) −47.6254 −2.89839
\(271\) −8.99741 −0.546554 −0.273277 0.961935i \(-0.588108\pi\)
−0.273277 + 0.961935i \(0.588108\pi\)
\(272\) −12.4179 −0.752944
\(273\) 3.48664 0.211021
\(274\) 7.08925 0.428277
\(275\) 7.35383 0.443453
\(276\) −36.4793 −2.19579
\(277\) −5.43648 −0.326646 −0.163323 0.986573i \(-0.552221\pi\)
−0.163323 + 0.986573i \(0.552221\pi\)
\(278\) −48.0842 −2.88390
\(279\) 0.513535 0.0307445
\(280\) −44.5253 −2.66090
\(281\) 22.1044 1.31864 0.659319 0.751863i \(-0.270845\pi\)
0.659319 + 0.751863i \(0.270845\pi\)
\(282\) 16.8530 1.00358
\(283\) −19.0897 −1.13477 −0.567384 0.823454i \(-0.692044\pi\)
−0.567384 + 0.823454i \(0.692044\pi\)
\(284\) −51.4784 −3.05468
\(285\) −2.14014 −0.126771
\(286\) 2.14130 0.126618
\(287\) −13.3582 −0.788507
\(288\) −0.565012 −0.0332936
\(289\) −13.4838 −0.793165
\(290\) −53.5958 −3.14726
\(291\) 10.2498 0.600856
\(292\) 26.8705 1.57248
\(293\) −1.77125 −0.103477 −0.0517387 0.998661i \(-0.516476\pi\)
−0.0517387 + 0.998661i \(0.516476\pi\)
\(294\) −13.9076 −0.811109
\(295\) 15.7548 0.917277
\(296\) −26.0039 −1.51145
\(297\) 4.29977 0.249498
\(298\) 19.4250 1.12526
\(299\) −4.68118 −0.270720
\(300\) −67.7528 −3.91171
\(301\) −17.6438 −1.01697
\(302\) 17.2491 0.992577
\(303\) 4.67197 0.268398
\(304\) 2.16694 0.124283
\(305\) 49.5437 2.83686
\(306\) 0.587291 0.0335732
\(307\) −23.2120 −1.32478 −0.662390 0.749160i \(-0.730458\pi\)
−0.662390 + 0.749160i \(0.730458\pi\)
\(308\) 7.35706 0.419208
\(309\) 12.1068 0.688731
\(310\) −38.8888 −2.20873
\(311\) 15.8763 0.900262 0.450131 0.892962i \(-0.351377\pi\)
0.450131 + 0.892962i \(0.351377\pi\)
\(312\) −10.7795 −0.610271
\(313\) 11.3353 0.640707 0.320353 0.947298i \(-0.396198\pi\)
0.320353 + 0.947298i \(0.396198\pi\)
\(314\) −48.5965 −2.74246
\(315\) 0.903148 0.0508866
\(316\) −21.4730 −1.20795
\(317\) 18.6812 1.04924 0.524620 0.851337i \(-0.324207\pi\)
0.524620 + 0.851337i \(0.324207\pi\)
\(318\) −13.9422 −0.781842
\(319\) 4.83880 0.270921
\(320\) −6.22595 −0.348041
\(321\) −5.65614 −0.315695
\(322\) −23.3791 −1.30287
\(323\) −0.613583 −0.0341407
\(324\) −41.2513 −2.29174
\(325\) −8.69433 −0.482275
\(326\) −50.8279 −2.81510
\(327\) −7.91592 −0.437751
\(328\) 41.2989 2.28035
\(329\) 7.43042 0.409653
\(330\) 14.0050 0.770951
\(331\) 14.4353 0.793437 0.396718 0.917940i \(-0.370149\pi\)
0.396718 + 0.917940i \(0.370149\pi\)
\(332\) 64.8023 3.55649
\(333\) 0.527461 0.0289047
\(334\) 48.9368 2.67770
\(335\) 33.2809 1.81833
\(336\) −23.0897 −1.25965
\(337\) 24.4803 1.33352 0.666762 0.745270i \(-0.267680\pi\)
0.666762 + 0.745270i \(0.267680\pi\)
\(338\) −2.53163 −0.137703
\(339\) −18.8857 −1.02573
\(340\) −30.5958 −1.65929
\(341\) 3.51100 0.190131
\(342\) −0.102483 −0.00554167
\(343\) −19.9411 −1.07672
\(344\) 54.5486 2.94107
\(345\) −30.6169 −1.64836
\(346\) −37.4912 −2.01554
\(347\) −9.04071 −0.485331 −0.242665 0.970110i \(-0.578022\pi\)
−0.242665 + 0.970110i \(0.578022\pi\)
\(348\) −44.5811 −2.38980
\(349\) 6.41330 0.343296 0.171648 0.985158i \(-0.445091\pi\)
0.171648 + 0.985158i \(0.445091\pi\)
\(350\) −43.4219 −2.32100
\(351\) −5.08356 −0.271340
\(352\) −3.86294 −0.205896
\(353\) 17.1077 0.910550 0.455275 0.890351i \(-0.349541\pi\)
0.455275 + 0.890351i \(0.349541\pi\)
\(354\) 19.0492 1.01245
\(355\) −43.2057 −2.29312
\(356\) −26.8196 −1.42143
\(357\) 6.53799 0.346027
\(358\) −16.6952 −0.882371
\(359\) −12.5004 −0.659744 −0.329872 0.944026i \(-0.607006\pi\)
−0.329872 + 0.944026i \(0.607006\pi\)
\(360\) −2.79223 −0.147164
\(361\) −18.8929 −0.994365
\(362\) −16.2886 −0.856111
\(363\) 18.1770 0.954046
\(364\) −8.69816 −0.455907
\(365\) 22.5523 1.18044
\(366\) 59.9037 3.13122
\(367\) 5.28804 0.276033 0.138017 0.990430i \(-0.455927\pi\)
0.138017 + 0.990430i \(0.455927\pi\)
\(368\) 31.0003 1.61600
\(369\) −0.837705 −0.0436092
\(370\) −39.9434 −2.07656
\(371\) −6.14708 −0.319140
\(372\) −32.3477 −1.67715
\(373\) 9.74580 0.504618 0.252309 0.967647i \(-0.418810\pi\)
0.252309 + 0.967647i \(0.418810\pi\)
\(374\) 4.01526 0.207624
\(375\) −24.1625 −1.24775
\(376\) −22.9724 −1.18471
\(377\) −5.72085 −0.294639
\(378\) −25.3887 −1.30585
\(379\) 18.2863 0.939302 0.469651 0.882852i \(-0.344380\pi\)
0.469651 + 0.882852i \(0.344380\pi\)
\(380\) 5.33903 0.273886
\(381\) 10.8372 0.555208
\(382\) −57.0960 −2.92129
\(383\) −1.39836 −0.0714527 −0.0357263 0.999362i \(-0.511374\pi\)
−0.0357263 + 0.999362i \(0.511374\pi\)
\(384\) −23.6717 −1.20799
\(385\) 6.17476 0.314695
\(386\) 1.07938 0.0549388
\(387\) −1.10646 −0.0562445
\(388\) −25.5703 −1.29814
\(389\) 2.10373 0.106663 0.0533316 0.998577i \(-0.483016\pi\)
0.0533316 + 0.998577i \(0.483016\pi\)
\(390\) −16.5579 −0.838444
\(391\) −8.77792 −0.443919
\(392\) 18.9575 0.957499
\(393\) −13.2449 −0.668119
\(394\) −9.59583 −0.483431
\(395\) −18.0222 −0.906796
\(396\) 0.461370 0.0231847
\(397\) 16.2526 0.815696 0.407848 0.913050i \(-0.366279\pi\)
0.407848 + 0.913050i \(0.366279\pi\)
\(398\) 63.8575 3.20088
\(399\) −1.14089 −0.0571160
\(400\) 57.5767 2.87883
\(401\) 16.1249 0.805238 0.402619 0.915368i \(-0.368100\pi\)
0.402619 + 0.915368i \(0.368100\pi\)
\(402\) 40.2403 2.00700
\(403\) −4.15100 −0.206776
\(404\) −11.6552 −0.579868
\(405\) −34.6221 −1.72038
\(406\) −28.5715 −1.41798
\(407\) 3.60621 0.178753
\(408\) −20.2133 −1.00071
\(409\) −4.21776 −0.208555 −0.104277 0.994548i \(-0.533253\pi\)
−0.104277 + 0.994548i \(0.533253\pi\)
\(410\) 63.4374 3.13295
\(411\) 4.94921 0.244126
\(412\) −30.2029 −1.48799
\(413\) 8.39872 0.413274
\(414\) −1.46613 −0.0720563
\(415\) 54.3884 2.66982
\(416\) 4.56710 0.223921
\(417\) −33.5690 −1.64388
\(418\) −0.700671 −0.0342709
\(419\) −5.29424 −0.258640 −0.129320 0.991603i \(-0.541280\pi\)
−0.129320 + 0.991603i \(0.541280\pi\)
\(420\) −56.8897 −2.77593
\(421\) −11.9428 −0.582054 −0.291027 0.956715i \(-0.593997\pi\)
−0.291027 + 0.956715i \(0.593997\pi\)
\(422\) 34.4582 1.67740
\(423\) 0.465970 0.0226562
\(424\) 19.0047 0.922950
\(425\) −16.3032 −0.790821
\(426\) −52.2404 −2.53105
\(427\) 26.4113 1.27813
\(428\) 14.1104 0.682053
\(429\) 1.49490 0.0721746
\(430\) 83.7896 4.04069
\(431\) −41.0400 −1.97683 −0.988414 0.151785i \(-0.951498\pi\)
−0.988414 + 0.151785i \(0.951498\pi\)
\(432\) 33.6650 1.61971
\(433\) 13.3033 0.639318 0.319659 0.947533i \(-0.396432\pi\)
0.319659 + 0.947533i \(0.396432\pi\)
\(434\) −20.7313 −0.995132
\(435\) −37.4168 −1.79400
\(436\) 19.7479 0.945753
\(437\) 1.53176 0.0732742
\(438\) 27.2682 1.30293
\(439\) −21.4829 −1.02532 −0.512661 0.858591i \(-0.671340\pi\)
−0.512661 + 0.858591i \(0.671340\pi\)
\(440\) −19.0903 −0.910094
\(441\) −0.384533 −0.0183111
\(442\) −4.74719 −0.225801
\(443\) 24.3508 1.15694 0.578472 0.815702i \(-0.303649\pi\)
0.578472 + 0.815702i \(0.303649\pi\)
\(444\) −33.2250 −1.57679
\(445\) −22.5096 −1.06706
\(446\) 73.0561 3.45931
\(447\) 13.5611 0.641419
\(448\) −3.31900 −0.156808
\(449\) −12.5155 −0.590643 −0.295321 0.955398i \(-0.595427\pi\)
−0.295321 + 0.955398i \(0.595427\pi\)
\(450\) −2.72303 −0.128365
\(451\) −5.72732 −0.269689
\(452\) 47.1142 2.21607
\(453\) 12.0421 0.565788
\(454\) 25.4958 1.19658
\(455\) −7.30033 −0.342245
\(456\) 3.52725 0.165179
\(457\) −2.49060 −0.116505 −0.0582526 0.998302i \(-0.518553\pi\)
−0.0582526 + 0.998302i \(0.518553\pi\)
\(458\) 53.8121 2.51447
\(459\) −9.53245 −0.444936
\(460\) 76.3802 3.56125
\(461\) −25.4156 −1.18372 −0.591861 0.806040i \(-0.701607\pi\)
−0.591861 + 0.806040i \(0.701607\pi\)
\(462\) 7.46595 0.347348
\(463\) 23.4758 1.09101 0.545506 0.838107i \(-0.316338\pi\)
0.545506 + 0.838107i \(0.316338\pi\)
\(464\) 37.8853 1.75878
\(465\) −27.1493 −1.25902
\(466\) −51.7872 −2.39900
\(467\) 33.0851 1.53100 0.765498 0.643438i \(-0.222493\pi\)
0.765498 + 0.643438i \(0.222493\pi\)
\(468\) −0.545471 −0.0252144
\(469\) 17.7418 0.819239
\(470\) −35.2868 −1.62766
\(471\) −33.9266 −1.56326
\(472\) −25.9660 −1.19518
\(473\) −7.56478 −0.347829
\(474\) −21.7908 −1.00089
\(475\) 2.84494 0.130535
\(476\) −16.3104 −0.747585
\(477\) −0.385490 −0.0176504
\(478\) 46.3097 2.11816
\(479\) −3.88622 −0.177566 −0.0887831 0.996051i \(-0.528298\pi\)
−0.0887831 + 0.996051i \(0.528298\pi\)
\(480\) 29.8708 1.36341
\(481\) −4.26358 −0.194402
\(482\) −53.4733 −2.43564
\(483\) −16.3216 −0.742659
\(484\) −45.3463 −2.06120
\(485\) −21.4611 −0.974497
\(486\) −3.25279 −0.147549
\(487\) −9.32802 −0.422693 −0.211347 0.977411i \(-0.567785\pi\)
−0.211347 + 0.977411i \(0.567785\pi\)
\(488\) −81.6549 −3.69634
\(489\) −35.4844 −1.60466
\(490\) 29.1198 1.31550
\(491\) −16.5907 −0.748730 −0.374365 0.927281i \(-0.622139\pi\)
−0.374365 + 0.927281i \(0.622139\pi\)
\(492\) 52.7673 2.37893
\(493\) −10.7275 −0.483140
\(494\) 0.828394 0.0372712
\(495\) 0.387226 0.0174045
\(496\) 27.4893 1.23431
\(497\) −23.0326 −1.03315
\(498\) 65.7614 2.94684
\(499\) −6.37068 −0.285191 −0.142595 0.989781i \(-0.545545\pi\)
−0.142595 + 0.989781i \(0.545545\pi\)
\(500\) 60.2784 2.69573
\(501\) 34.1641 1.52634
\(502\) −11.0629 −0.493762
\(503\) 21.8938 0.976196 0.488098 0.872789i \(-0.337691\pi\)
0.488098 + 0.872789i \(0.337691\pi\)
\(504\) −1.48852 −0.0663037
\(505\) −9.78217 −0.435301
\(506\) −10.0238 −0.445613
\(507\) −1.76740 −0.0784931
\(508\) −27.0357 −1.19951
\(509\) 3.72354 0.165043 0.0825216 0.996589i \(-0.473703\pi\)
0.0825216 + 0.996589i \(0.473703\pi\)
\(510\) −31.0487 −1.37486
\(511\) 12.0224 0.531842
\(512\) 50.5354 2.23337
\(513\) 1.66343 0.0734422
\(514\) 39.2987 1.73339
\(515\) −25.3492 −1.11702
\(516\) 69.6963 3.06821
\(517\) 3.18580 0.140111
\(518\) −21.2935 −0.935582
\(519\) −26.1736 −1.14890
\(520\) 22.5702 0.989768
\(521\) 23.3076 1.02113 0.510563 0.859840i \(-0.329437\pi\)
0.510563 + 0.859840i \(0.329437\pi\)
\(522\) −1.79175 −0.0784227
\(523\) −41.9169 −1.83290 −0.916449 0.400151i \(-0.868958\pi\)
−0.916449 + 0.400151i \(0.868958\pi\)
\(524\) 33.0422 1.44346
\(525\) −30.3141 −1.32301
\(526\) −79.9203 −3.48469
\(527\) −7.78377 −0.339066
\(528\) −9.89973 −0.430830
\(529\) −1.08657 −0.0472420
\(530\) 29.1922 1.26803
\(531\) 0.526693 0.0228565
\(532\) 2.84619 0.123398
\(533\) 6.77134 0.293299
\(534\) −27.2165 −1.17777
\(535\) 11.8428 0.512010
\(536\) −54.8516 −2.36923
\(537\) −11.6554 −0.502969
\(538\) −74.0249 −3.19144
\(539\) −2.62902 −0.113240
\(540\) 82.9456 3.56941
\(541\) 35.8528 1.54143 0.770717 0.637178i \(-0.219898\pi\)
0.770717 + 0.637178i \(0.219898\pi\)
\(542\) 22.7781 0.978404
\(543\) −11.3716 −0.488000
\(544\) 8.56402 0.367179
\(545\) 16.5743 0.709967
\(546\) −8.82690 −0.377756
\(547\) 35.5768 1.52115 0.760577 0.649248i \(-0.224916\pi\)
0.760577 + 0.649248i \(0.224916\pi\)
\(548\) −12.3468 −0.527430
\(549\) 1.65628 0.0706884
\(550\) −18.6172 −0.793839
\(551\) 1.87196 0.0797482
\(552\) 50.4610 2.14776
\(553\) −9.60749 −0.408552
\(554\) 13.7632 0.584741
\(555\) −27.8856 −1.18368
\(556\) 83.7447 3.55157
\(557\) 7.82800 0.331683 0.165841 0.986152i \(-0.446966\pi\)
0.165841 + 0.986152i \(0.446966\pi\)
\(558\) −1.30008 −0.0550368
\(559\) 8.94374 0.378280
\(560\) 48.3451 2.04296
\(561\) 2.80317 0.118350
\(562\) −55.9602 −2.36054
\(563\) 18.3991 0.775431 0.387716 0.921779i \(-0.373264\pi\)
0.387716 + 0.921779i \(0.373264\pi\)
\(564\) −29.3516 −1.23593
\(565\) 39.5428 1.66358
\(566\) 48.3282 2.03138
\(567\) −18.4567 −0.775110
\(568\) 71.2090 2.98786
\(569\) −26.6254 −1.11619 −0.558097 0.829776i \(-0.688468\pi\)
−0.558097 + 0.829776i \(0.688468\pi\)
\(570\) 5.41805 0.226937
\(571\) −23.1096 −0.967107 −0.483553 0.875315i \(-0.660654\pi\)
−0.483553 + 0.875315i \(0.660654\pi\)
\(572\) −3.72934 −0.155932
\(573\) −39.8603 −1.66519
\(574\) 33.8179 1.41153
\(575\) 40.6997 1.69730
\(576\) −0.208138 −0.00867243
\(577\) 41.4684 1.72635 0.863177 0.504902i \(-0.168471\pi\)
0.863177 + 0.504902i \(0.168471\pi\)
\(578\) 34.1360 1.41987
\(579\) 0.753542 0.0313162
\(580\) 93.3439 3.87589
\(581\) 28.9940 1.20287
\(582\) −25.9488 −1.07561
\(583\) −2.63557 −0.109154
\(584\) −37.1694 −1.53808
\(585\) −0.457812 −0.0189282
\(586\) 4.48415 0.185238
\(587\) −17.4452 −0.720039 −0.360019 0.932945i \(-0.617230\pi\)
−0.360019 + 0.932945i \(0.617230\pi\)
\(588\) 24.2219 0.998893
\(589\) 1.35828 0.0559670
\(590\) −39.8852 −1.64205
\(591\) −6.69912 −0.275565
\(592\) 28.2348 1.16044
\(593\) −26.2024 −1.07600 −0.538001 0.842944i \(-0.680820\pi\)
−0.538001 + 0.842944i \(0.680820\pi\)
\(594\) −10.8854 −0.446634
\(595\) −13.6892 −0.561204
\(596\) −33.8310 −1.38577
\(597\) 44.5807 1.82457
\(598\) 11.8510 0.484624
\(599\) −27.6202 −1.12853 −0.564265 0.825594i \(-0.690840\pi\)
−0.564265 + 0.825594i \(0.690840\pi\)
\(600\) 93.7209 3.82614
\(601\) −21.7782 −0.888352 −0.444176 0.895939i \(-0.646504\pi\)
−0.444176 + 0.895939i \(0.646504\pi\)
\(602\) 44.6675 1.82051
\(603\) 1.11261 0.0453088
\(604\) −30.0416 −1.22237
\(605\) −38.0590 −1.54732
\(606\) −11.8277 −0.480468
\(607\) −23.6448 −0.959711 −0.479855 0.877348i \(-0.659311\pi\)
−0.479855 + 0.877348i \(0.659311\pi\)
\(608\) −1.49444 −0.0606074
\(609\) −19.9466 −0.808275
\(610\) −125.426 −5.07836
\(611\) −3.76653 −0.152378
\(612\) −1.02284 −0.0413459
\(613\) 15.0643 0.608440 0.304220 0.952602i \(-0.401604\pi\)
0.304220 + 0.952602i \(0.401604\pi\)
\(614\) 58.7642 2.37153
\(615\) 44.2874 1.78584
\(616\) −10.1769 −0.410038
\(617\) 34.6714 1.39582 0.697909 0.716186i \(-0.254114\pi\)
0.697909 + 0.716186i \(0.254114\pi\)
\(618\) −30.6499 −1.23292
\(619\) 1.00000 0.0401934
\(620\) 67.7297 2.72009
\(621\) 23.7970 0.954942
\(622\) −40.1929 −1.61159
\(623\) −11.9997 −0.480756
\(624\) 11.7043 0.468548
\(625\) 7.11978 0.284791
\(626\) −28.6967 −1.14695
\(627\) −0.489158 −0.0195351
\(628\) 84.6369 3.37738
\(629\) −7.99486 −0.318776
\(630\) −2.28644 −0.0910939
\(631\) −11.8685 −0.472477 −0.236238 0.971695i \(-0.575915\pi\)
−0.236238 + 0.971695i \(0.575915\pi\)
\(632\) 29.7031 1.18153
\(633\) 24.0562 0.956150
\(634\) −47.2938 −1.87828
\(635\) −22.6910 −0.900463
\(636\) 24.2822 0.962850
\(637\) 3.10826 0.123154
\(638\) −12.2500 −0.484984
\(639\) −1.44440 −0.0571395
\(640\) 49.5637 1.95918
\(641\) 4.74522 0.187425 0.0937124 0.995599i \(-0.470127\pi\)
0.0937124 + 0.995599i \(0.470127\pi\)
\(642\) 14.3193 0.565136
\(643\) −10.9002 −0.429861 −0.214931 0.976629i \(-0.568953\pi\)
−0.214931 + 0.976629i \(0.568953\pi\)
\(644\) 40.7176 1.60450
\(645\) 58.4959 2.30327
\(646\) 1.55336 0.0611163
\(647\) 31.6637 1.24483 0.622414 0.782689i \(-0.286152\pi\)
0.622414 + 0.782689i \(0.286152\pi\)
\(648\) 57.0620 2.24161
\(649\) 3.60096 0.141350
\(650\) 22.0108 0.863336
\(651\) −14.4731 −0.567245
\(652\) 88.5232 3.46684
\(653\) −14.8689 −0.581867 −0.290933 0.956743i \(-0.593966\pi\)
−0.290933 + 0.956743i \(0.593966\pi\)
\(654\) 20.0402 0.783633
\(655\) 27.7322 1.08359
\(656\) −44.8420 −1.75078
\(657\) 0.753941 0.0294140
\(658\) −18.8111 −0.733333
\(659\) −42.1798 −1.64309 −0.821546 0.570142i \(-0.806888\pi\)
−0.821546 + 0.570142i \(0.806888\pi\)
\(660\) −24.3915 −0.949438
\(661\) −12.4500 −0.484250 −0.242125 0.970245i \(-0.577844\pi\)
−0.242125 + 0.970245i \(0.577844\pi\)
\(662\) −36.5449 −1.42036
\(663\) −3.31415 −0.128711
\(664\) −89.6396 −3.47869
\(665\) 2.38880 0.0926335
\(666\) −1.33534 −0.0517433
\(667\) 26.7803 1.03694
\(668\) −85.2295 −3.29763
\(669\) 51.0026 1.97187
\(670\) −84.2550 −3.25505
\(671\) 11.3239 0.437153
\(672\) 15.9239 0.614277
\(673\) −15.4851 −0.596906 −0.298453 0.954424i \(-0.596471\pi\)
−0.298453 + 0.954424i \(0.596471\pi\)
\(674\) −61.9750 −2.38719
\(675\) 44.1981 1.70119
\(676\) 4.40915 0.169583
\(677\) −30.1595 −1.15912 −0.579562 0.814928i \(-0.696776\pi\)
−0.579562 + 0.814928i \(0.696776\pi\)
\(678\) 47.8115 1.83619
\(679\) −11.4407 −0.439054
\(680\) 42.3225 1.62300
\(681\) 17.7993 0.682072
\(682\) −8.88855 −0.340360
\(683\) 1.57949 0.0604374 0.0302187 0.999543i \(-0.490380\pi\)
0.0302187 + 0.999543i \(0.490380\pi\)
\(684\) 0.178488 0.00682465
\(685\) −10.3626 −0.395936
\(686\) 50.4834 1.92747
\(687\) 37.5677 1.43330
\(688\) −59.2283 −2.25806
\(689\) 3.11599 0.118710
\(690\) 77.5107 2.95078
\(691\) −5.28678 −0.201118 −0.100559 0.994931i \(-0.532063\pi\)
−0.100559 + 0.994931i \(0.532063\pi\)
\(692\) 65.2956 2.48217
\(693\) 0.206427 0.00784150
\(694\) 22.8877 0.868807
\(695\) 70.2866 2.66612
\(696\) 61.6681 2.33752
\(697\) 12.6973 0.480944
\(698\) −16.2361 −0.614546
\(699\) −36.1541 −1.36747
\(700\) 75.6247 2.85834
\(701\) 0.265272 0.0100192 0.00500958 0.999987i \(-0.498405\pi\)
0.00500958 + 0.999987i \(0.498405\pi\)
\(702\) 12.8697 0.485735
\(703\) 1.39512 0.0526178
\(704\) −1.42303 −0.0536323
\(705\) −24.6347 −0.927797
\(706\) −43.3103 −1.63001
\(707\) −5.21479 −0.196122
\(708\) −33.1766 −1.24685
\(709\) 1.61232 0.0605520 0.0302760 0.999542i \(-0.490361\pi\)
0.0302760 + 0.999542i \(0.490361\pi\)
\(710\) 109.381 4.10499
\(711\) −0.602496 −0.0225954
\(712\) 37.0989 1.39034
\(713\) 19.4316 0.727719
\(714\) −16.5518 −0.619435
\(715\) −3.13002 −0.117056
\(716\) 29.0769 1.08665
\(717\) 32.3301 1.20739
\(718\) 31.6463 1.18103
\(719\) −24.2792 −0.905462 −0.452731 0.891647i \(-0.649550\pi\)
−0.452731 + 0.891647i \(0.649550\pi\)
\(720\) 3.03178 0.112988
\(721\) −13.5134 −0.503266
\(722\) 47.8299 1.78005
\(723\) −37.3312 −1.38836
\(724\) 28.3687 1.05431
\(725\) 49.7389 1.84726
\(726\) −46.0175 −1.70787
\(727\) −28.8320 −1.06932 −0.534660 0.845067i \(-0.679560\pi\)
−0.534660 + 0.845067i \(0.679560\pi\)
\(728\) 12.0320 0.445934
\(729\) 25.7966 0.955431
\(730\) −57.0941 −2.11315
\(731\) 16.7709 0.620293
\(732\) −104.330 −3.85614
\(733\) −30.4058 −1.12306 −0.561532 0.827455i \(-0.689788\pi\)
−0.561532 + 0.827455i \(0.689788\pi\)
\(734\) −13.3874 −0.494136
\(735\) 20.3293 0.749858
\(736\) −21.3794 −0.788056
\(737\) 7.60680 0.280200
\(738\) 2.12076 0.0780662
\(739\) 21.7593 0.800429 0.400215 0.916421i \(-0.368936\pi\)
0.400215 + 0.916421i \(0.368936\pi\)
\(740\) 69.5664 2.55731
\(741\) 0.578325 0.0212453
\(742\) 15.5621 0.571304
\(743\) −16.8152 −0.616891 −0.308446 0.951242i \(-0.599809\pi\)
−0.308446 + 0.951242i \(0.599809\pi\)
\(744\) 44.7459 1.64047
\(745\) −28.3943 −1.04029
\(746\) −24.6728 −0.903334
\(747\) 1.81824 0.0665260
\(748\) −6.99309 −0.255693
\(749\) 6.31331 0.230683
\(750\) 61.1706 2.23363
\(751\) 47.0763 1.71784 0.858919 0.512111i \(-0.171136\pi\)
0.858919 + 0.512111i \(0.171136\pi\)
\(752\) 24.9432 0.909584
\(753\) −7.72334 −0.281454
\(754\) 14.4831 0.527442
\(755\) −25.2138 −0.917623
\(756\) 44.2176 1.60818
\(757\) −5.47143 −0.198863 −0.0994313 0.995044i \(-0.531702\pi\)
−0.0994313 + 0.995044i \(0.531702\pi\)
\(758\) −46.2941 −1.68148
\(759\) −6.99791 −0.254008
\(760\) −7.38536 −0.267895
\(761\) 47.4111 1.71865 0.859326 0.511429i \(-0.170884\pi\)
0.859326 + 0.511429i \(0.170884\pi\)
\(762\) −27.4358 −0.993895
\(763\) 8.83564 0.319872
\(764\) 99.4399 3.59761
\(765\) −0.858467 −0.0310379
\(766\) 3.54012 0.127910
\(767\) −4.25737 −0.153725
\(768\) 53.9809 1.94787
\(769\) 15.2573 0.550192 0.275096 0.961417i \(-0.411290\pi\)
0.275096 + 0.961417i \(0.411290\pi\)
\(770\) −15.6322 −0.563345
\(771\) 27.4356 0.988068
\(772\) −1.87987 −0.0676579
\(773\) 16.1020 0.579149 0.289575 0.957155i \(-0.406486\pi\)
0.289575 + 0.957155i \(0.406486\pi\)
\(774\) 2.80115 0.100685
\(775\) 36.0902 1.29640
\(776\) 35.3709 1.26974
\(777\) −14.8656 −0.533300
\(778\) −5.32586 −0.190941
\(779\) −2.21570 −0.0793857
\(780\) 28.8377 1.03256
\(781\) −9.87524 −0.353364
\(782\) 22.2225 0.794673
\(783\) 29.0822 1.03931
\(784\) −20.5839 −0.735139
\(785\) 71.0355 2.53536
\(786\) 33.5313 1.19602
\(787\) −3.75967 −0.134018 −0.0670089 0.997752i \(-0.521346\pi\)
−0.0670089 + 0.997752i \(0.521346\pi\)
\(788\) 16.7123 0.595352
\(789\) −55.7946 −1.98634
\(790\) 45.6256 1.62329
\(791\) 21.0799 0.749515
\(792\) −0.638203 −0.0226775
\(793\) −13.3881 −0.475424
\(794\) −41.1456 −1.46020
\(795\) 20.3799 0.722801
\(796\) −111.216 −3.94194
\(797\) 42.6233 1.50980 0.754898 0.655842i \(-0.227686\pi\)
0.754898 + 0.655842i \(0.227686\pi\)
\(798\) 2.88831 0.102245
\(799\) −7.06282 −0.249865
\(800\) −39.7079 −1.40389
\(801\) −0.752512 −0.0265887
\(802\) −40.8223 −1.44148
\(803\) 5.15464 0.181903
\(804\) −70.0835 −2.47165
\(805\) 34.1742 1.20448
\(806\) 10.5088 0.370157
\(807\) −51.6789 −1.81918
\(808\) 16.1224 0.567184
\(809\) 40.4062 1.42061 0.710303 0.703896i \(-0.248558\pi\)
0.710303 + 0.703896i \(0.248558\pi\)
\(810\) 87.6503 3.07972
\(811\) 49.2665 1.72998 0.864990 0.501789i \(-0.167325\pi\)
0.864990 + 0.501789i \(0.167325\pi\)
\(812\) 49.7608 1.74626
\(813\) 15.9020 0.557709
\(814\) −9.12960 −0.319992
\(815\) 74.2972 2.60252
\(816\) 21.9474 0.768312
\(817\) −2.92655 −0.102387
\(818\) 10.6778 0.373341
\(819\) −0.244056 −0.00852799
\(820\) −110.484 −3.85827
\(821\) 32.9830 1.15111 0.575557 0.817762i \(-0.304785\pi\)
0.575557 + 0.817762i \(0.304785\pi\)
\(822\) −12.5296 −0.437019
\(823\) −5.30776 −0.185017 −0.0925085 0.995712i \(-0.529489\pi\)
−0.0925085 + 0.995712i \(0.529489\pi\)
\(824\) 41.7790 1.45544
\(825\) −12.9972 −0.452504
\(826\) −21.2625 −0.739816
\(827\) 50.0827 1.74154 0.870772 0.491687i \(-0.163619\pi\)
0.870772 + 0.491687i \(0.163619\pi\)
\(828\) 2.55345 0.0887384
\(829\) −43.5847 −1.51376 −0.756880 0.653554i \(-0.773277\pi\)
−0.756880 + 0.653554i \(0.773277\pi\)
\(830\) −137.691 −4.77933
\(831\) 9.60845 0.333314
\(832\) 1.68242 0.0583276
\(833\) 5.82846 0.201944
\(834\) 84.9842 2.94276
\(835\) −71.5328 −2.47550
\(836\) 1.22031 0.0422052
\(837\) 21.1019 0.729387
\(838\) 13.4031 0.463001
\(839\) −15.8576 −0.547467 −0.273733 0.961806i \(-0.588259\pi\)
−0.273733 + 0.961806i \(0.588259\pi\)
\(840\) 78.6942 2.71521
\(841\) 3.72807 0.128554
\(842\) 30.2346 1.04195
\(843\) −39.0674 −1.34555
\(844\) −60.0133 −2.06574
\(845\) 3.70059 0.127304
\(846\) −1.17966 −0.0405577
\(847\) −20.2889 −0.697136
\(848\) −20.6351 −0.708613
\(849\) 33.7393 1.15793
\(850\) 41.2737 1.41568
\(851\) 19.9586 0.684171
\(852\) 90.9832 3.11703
\(853\) −11.2327 −0.384600 −0.192300 0.981336i \(-0.561595\pi\)
−0.192300 + 0.981336i \(0.561595\pi\)
\(854\) −66.8636 −2.28803
\(855\) 0.149804 0.00512319
\(856\) −19.5186 −0.667133
\(857\) −33.2638 −1.13627 −0.568135 0.822936i \(-0.692335\pi\)
−0.568135 + 0.822936i \(0.692335\pi\)
\(858\) −3.78454 −0.129202
\(859\) −48.2838 −1.64742 −0.823710 0.567011i \(-0.808100\pi\)
−0.823710 + 0.567011i \(0.808100\pi\)
\(860\) −145.930 −4.97617
\(861\) 23.6092 0.804601
\(862\) 103.898 3.53878
\(863\) 7.07146 0.240715 0.120358 0.992731i \(-0.461596\pi\)
0.120358 + 0.992731i \(0.461596\pi\)
\(864\) −23.2171 −0.789863
\(865\) 54.8023 1.86334
\(866\) −33.6792 −1.14446
\(867\) 23.8313 0.809354
\(868\) 36.1061 1.22552
\(869\) −4.11922 −0.139735
\(870\) 94.7254 3.21149
\(871\) −8.99342 −0.304730
\(872\) −27.3168 −0.925065
\(873\) −0.717460 −0.0242823
\(874\) −3.87786 −0.131171
\(875\) 26.9699 0.911748
\(876\) −47.4910 −1.60457
\(877\) −28.3640 −0.957786 −0.478893 0.877873i \(-0.658962\pi\)
−0.478893 + 0.877873i \(0.658962\pi\)
\(878\) 54.3867 1.83546
\(879\) 3.13051 0.105589
\(880\) 20.7280 0.698742
\(881\) 47.4819 1.59971 0.799853 0.600196i \(-0.204911\pi\)
0.799853 + 0.600196i \(0.204911\pi\)
\(882\) 0.973495 0.0327793
\(883\) 47.4445 1.59663 0.798317 0.602237i \(-0.205724\pi\)
0.798317 + 0.602237i \(0.205724\pi\)
\(884\) 8.26783 0.278077
\(885\) −27.8450 −0.935999
\(886\) −61.6473 −2.07108
\(887\) −5.05011 −0.169566 −0.0847830 0.996399i \(-0.527020\pi\)
−0.0847830 + 0.996399i \(0.527020\pi\)
\(888\) 45.9594 1.54230
\(889\) −12.0964 −0.405699
\(890\) 56.9859 1.91017
\(891\) −7.91334 −0.265107
\(892\) −127.236 −4.26019
\(893\) 1.23247 0.0412432
\(894\) −34.3318 −1.14823
\(895\) 24.4041 0.815739
\(896\) 26.4220 0.882697
\(897\) 8.27353 0.276245
\(898\) 31.6846 1.05733
\(899\) 23.7473 0.792015
\(900\) 4.74251 0.158084
\(901\) 5.84296 0.194657
\(902\) 14.4995 0.482779
\(903\) 31.1836 1.03773
\(904\) −65.1720 −2.16759
\(905\) 23.8097 0.791462
\(906\) −30.4862 −1.01284
\(907\) 1.10870 0.0368138 0.0184069 0.999831i \(-0.494141\pi\)
0.0184069 + 0.999831i \(0.494141\pi\)
\(908\) −44.4041 −1.47360
\(909\) −0.327025 −0.0108467
\(910\) 18.4817 0.612664
\(911\) 37.0396 1.22718 0.613588 0.789626i \(-0.289726\pi\)
0.613588 + 0.789626i \(0.289726\pi\)
\(912\) −3.82986 −0.126819
\(913\) 12.4312 0.411412
\(914\) 6.30527 0.208560
\(915\) −87.5636 −2.89476
\(916\) −93.7205 −3.09661
\(917\) 14.7838 0.488204
\(918\) 24.1326 0.796495
\(919\) −28.7166 −0.947274 −0.473637 0.880720i \(-0.657059\pi\)
−0.473637 + 0.880720i \(0.657059\pi\)
\(920\) −105.655 −3.48334
\(921\) 41.0250 1.35182
\(922\) 64.3429 2.11902
\(923\) 11.6754 0.384299
\(924\) −13.0029 −0.427764
\(925\) 37.0690 1.21882
\(926\) −59.4320 −1.95306
\(927\) −0.847442 −0.0278336
\(928\) −26.1277 −0.857684
\(929\) 50.2654 1.64916 0.824578 0.565749i \(-0.191413\pi\)
0.824578 + 0.565749i \(0.191413\pi\)
\(930\) 68.7321 2.25381
\(931\) −1.01708 −0.0333333
\(932\) 90.1939 2.95440
\(933\) −28.0598 −0.918637
\(934\) −83.7593 −2.74069
\(935\) −5.86927 −0.191946
\(936\) 0.754538 0.0246629
\(937\) −29.6688 −0.969239 −0.484619 0.874725i \(-0.661042\pi\)
−0.484619 + 0.874725i \(0.661042\pi\)
\(938\) −44.9156 −1.46655
\(939\) −20.0340 −0.653784
\(940\) 61.4564 2.00449
\(941\) 16.9290 0.551870 0.275935 0.961176i \(-0.411013\pi\)
0.275935 + 0.961176i \(0.411013\pi\)
\(942\) 85.8896 2.79844
\(943\) −31.6978 −1.03222
\(944\) 28.1937 0.917625
\(945\) 37.1117 1.20724
\(946\) 19.1512 0.622660
\(947\) −14.1617 −0.460193 −0.230097 0.973168i \(-0.573904\pi\)
−0.230097 + 0.973168i \(0.573904\pi\)
\(948\) 37.9515 1.23261
\(949\) −6.09426 −0.197828
\(950\) −7.20233 −0.233675
\(951\) −33.0172 −1.07066
\(952\) 22.5618 0.731231
\(953\) 1.75447 0.0568327 0.0284164 0.999596i \(-0.490954\pi\)
0.0284164 + 0.999596i \(0.490954\pi\)
\(954\) 0.975918 0.0315965
\(955\) 83.4595 2.70069
\(956\) −80.6541 −2.60854
\(957\) −8.55211 −0.276450
\(958\) 9.83848 0.317867
\(959\) −5.52423 −0.178387
\(960\) 11.0038 0.355145
\(961\) −13.7692 −0.444166
\(962\) 10.7938 0.348006
\(963\) 0.395914 0.0127582
\(964\) 93.1305 2.99953
\(965\) −1.57777 −0.0507901
\(966\) 41.3203 1.32946
\(967\) 46.3117 1.48928 0.744642 0.667464i \(-0.232620\pi\)
0.744642 + 0.667464i \(0.232620\pi\)
\(968\) 62.7266 2.01611
\(969\) 1.08445 0.0348375
\(970\) 54.3315 1.74448
\(971\) −6.42833 −0.206295 −0.103147 0.994666i \(-0.532891\pi\)
−0.103147 + 0.994666i \(0.532891\pi\)
\(972\) 5.66514 0.181709
\(973\) 37.4692 1.20121
\(974\) 23.6151 0.756677
\(975\) 15.3664 0.492118
\(976\) 88.6601 2.83794
\(977\) 38.3288 1.22625 0.613124 0.789987i \(-0.289913\pi\)
0.613124 + 0.789987i \(0.289913\pi\)
\(978\) 89.8334 2.87256
\(979\) −5.14486 −0.164431
\(980\) −50.7157 −1.62005
\(981\) 0.554093 0.0176908
\(982\) 42.0016 1.34033
\(983\) −18.3481 −0.585214 −0.292607 0.956233i \(-0.594523\pi\)
−0.292607 + 0.956233i \(0.594523\pi\)
\(984\) −72.9919 −2.32690
\(985\) 14.0266 0.446925
\(986\) 27.1580 0.864886
\(987\) −13.1326 −0.418014
\(988\) −1.44275 −0.0459001
\(989\) −41.8673 −1.33130
\(990\) −0.980313 −0.0311564
\(991\) −27.2405 −0.865324 −0.432662 0.901556i \(-0.642426\pi\)
−0.432662 + 0.901556i \(0.642426\pi\)
\(992\) −18.9581 −0.601919
\(993\) −25.5130 −0.809631
\(994\) 58.3100 1.84948
\(995\) −93.3430 −2.95917
\(996\) −114.532 −3.62908
\(997\) 19.7638 0.625926 0.312963 0.949765i \(-0.398678\pi\)
0.312963 + 0.949765i \(0.398678\pi\)
\(998\) 16.1282 0.510530
\(999\) 21.6741 0.685739
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 8047.2.a.d.1.7 156
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8047.2.a.d.1.7 156 1.1 even 1 trivial