Properties

Label 4015.2.a.h.1.34
Level $4015$
Weight $2$
Character 4015.1
Self dual yes
Analytic conductor $32.060$
Analytic rank $0$
Dimension $37$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(0\)
Dimension: \(37\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.34
Character \(\chi\) \(=\) 4015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.50132 q^{2} +2.36788 q^{3} +4.25662 q^{4} +1.00000 q^{5} +5.92283 q^{6} -2.67178 q^{7} +5.64454 q^{8} +2.60686 q^{9} +O(q^{10})\) \(q+2.50132 q^{2} +2.36788 q^{3} +4.25662 q^{4} +1.00000 q^{5} +5.92283 q^{6} -2.67178 q^{7} +5.64454 q^{8} +2.60686 q^{9} +2.50132 q^{10} +1.00000 q^{11} +10.0792 q^{12} +3.48435 q^{13} -6.68298 q^{14} +2.36788 q^{15} +5.60558 q^{16} +4.44902 q^{17} +6.52059 q^{18} -4.04223 q^{19} +4.25662 q^{20} -6.32645 q^{21} +2.50132 q^{22} +5.55439 q^{23} +13.3656 q^{24} +1.00000 q^{25} +8.71548 q^{26} -0.930919 q^{27} -11.3727 q^{28} -6.50210 q^{29} +5.92283 q^{30} -2.94433 q^{31} +2.73229 q^{32} +2.36788 q^{33} +11.1284 q^{34} -2.67178 q^{35} +11.0964 q^{36} +0.596335 q^{37} -10.1109 q^{38} +8.25052 q^{39} +5.64454 q^{40} +5.28979 q^{41} -15.8245 q^{42} +6.60038 q^{43} +4.25662 q^{44} +2.60686 q^{45} +13.8933 q^{46} +0.172675 q^{47} +13.2733 q^{48} +0.138402 q^{49} +2.50132 q^{50} +10.5348 q^{51} +14.8315 q^{52} -4.95784 q^{53} -2.32853 q^{54} +1.00000 q^{55} -15.0810 q^{56} -9.57150 q^{57} -16.2638 q^{58} -4.38285 q^{59} +10.0792 q^{60} +2.23428 q^{61} -7.36471 q^{62} -6.96494 q^{63} -4.37682 q^{64} +3.48435 q^{65} +5.92283 q^{66} +5.08973 q^{67} +18.9378 q^{68} +13.1521 q^{69} -6.68298 q^{70} -3.91570 q^{71} +14.7145 q^{72} +1.00000 q^{73} +1.49163 q^{74} +2.36788 q^{75} -17.2062 q^{76} -2.67178 q^{77} +20.6372 q^{78} -13.4294 q^{79} +5.60558 q^{80} -10.0249 q^{81} +13.2315 q^{82} +8.54660 q^{83} -26.9293 q^{84} +4.44902 q^{85} +16.5097 q^{86} -15.3962 q^{87} +5.64454 q^{88} -14.7674 q^{89} +6.52059 q^{90} -9.30940 q^{91} +23.6429 q^{92} -6.97181 q^{93} +0.431917 q^{94} -4.04223 q^{95} +6.46972 q^{96} -8.71708 q^{97} +0.346188 q^{98} +2.60686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9} + 5 q^{10} + 37 q^{11} + 6 q^{12} + 11 q^{13} + 11 q^{14} + 3 q^{15} + 43 q^{16} + 38 q^{17} + 11 q^{18} + 34 q^{19} + 43 q^{20} + 39 q^{21} + 5 q^{22} + 4 q^{23} + 25 q^{24} + 37 q^{25} - 9 q^{26} + 3 q^{27} + 14 q^{28} + 58 q^{29} + 9 q^{30} + 8 q^{31} + 14 q^{32} + 3 q^{33} + 8 q^{34} + 6 q^{35} + 20 q^{36} + 2 q^{37} + 15 q^{38} + 14 q^{39} + 12 q^{40} + 62 q^{41} - 13 q^{42} + 30 q^{43} + 43 q^{44} + 50 q^{45} + 31 q^{46} + 5 q^{47} - 25 q^{48} + 59 q^{49} + 5 q^{50} + 23 q^{51} - q^{52} + 18 q^{53} + 13 q^{54} + 37 q^{55} + 22 q^{56} + 5 q^{57} - 40 q^{58} + 15 q^{59} + 6 q^{60} + 57 q^{61} + 20 q^{62} - 29 q^{63} + 10 q^{64} + 11 q^{65} + 9 q^{66} - 14 q^{67} + 53 q^{68} + 24 q^{69} + 11 q^{70} + 8 q^{71} + 15 q^{72} + 37 q^{73} + 7 q^{74} + 3 q^{75} + 59 q^{76} + 6 q^{77} + q^{78} + 42 q^{79} + 43 q^{80} + 61 q^{81} - 22 q^{82} + 44 q^{83} + 66 q^{84} + 38 q^{85} - q^{86} - 26 q^{87} + 12 q^{88} + 69 q^{89} + 11 q^{90} - 10 q^{91} - 21 q^{92} - 26 q^{93} + 29 q^{94} + 34 q^{95} - 9 q^{96} + 37 q^{97} - 15 q^{98} + 50 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.50132 1.76870 0.884351 0.466822i \(-0.154601\pi\)
0.884351 + 0.466822i \(0.154601\pi\)
\(3\) 2.36788 1.36710 0.683548 0.729906i \(-0.260436\pi\)
0.683548 + 0.729906i \(0.260436\pi\)
\(4\) 4.25662 2.12831
\(5\) 1.00000 0.447214
\(6\) 5.92283 2.41799
\(7\) −2.67178 −1.00984 −0.504919 0.863167i \(-0.668478\pi\)
−0.504919 + 0.863167i \(0.668478\pi\)
\(8\) 5.64454 1.99565
\(9\) 2.60686 0.868952
\(10\) 2.50132 0.790988
\(11\) 1.00000 0.301511
\(12\) 10.0792 2.90960
\(13\) 3.48435 0.966384 0.483192 0.875514i \(-0.339477\pi\)
0.483192 + 0.875514i \(0.339477\pi\)
\(14\) −6.68298 −1.78610
\(15\) 2.36788 0.611384
\(16\) 5.60558 1.40139
\(17\) 4.44902 1.07905 0.539523 0.841971i \(-0.318604\pi\)
0.539523 + 0.841971i \(0.318604\pi\)
\(18\) 6.52059 1.53692
\(19\) −4.04223 −0.927350 −0.463675 0.886005i \(-0.653469\pi\)
−0.463675 + 0.886005i \(0.653469\pi\)
\(20\) 4.25662 0.951809
\(21\) −6.32645 −1.38054
\(22\) 2.50132 0.533284
\(23\) 5.55439 1.15817 0.579085 0.815267i \(-0.303410\pi\)
0.579085 + 0.815267i \(0.303410\pi\)
\(24\) 13.3656 2.72824
\(25\) 1.00000 0.200000
\(26\) 8.71548 1.70925
\(27\) −0.930919 −0.179156
\(28\) −11.3727 −2.14925
\(29\) −6.50210 −1.20741 −0.603705 0.797208i \(-0.706309\pi\)
−0.603705 + 0.797208i \(0.706309\pi\)
\(30\) 5.92283 1.08136
\(31\) −2.94433 −0.528817 −0.264408 0.964411i \(-0.585177\pi\)
−0.264408 + 0.964411i \(0.585177\pi\)
\(32\) 2.73229 0.483004
\(33\) 2.36788 0.412195
\(34\) 11.1284 1.90851
\(35\) −2.67178 −0.451613
\(36\) 11.0964 1.84940
\(37\) 0.596335 0.0980369 0.0490185 0.998798i \(-0.484391\pi\)
0.0490185 + 0.998798i \(0.484391\pi\)
\(38\) −10.1109 −1.64021
\(39\) 8.25052 1.32114
\(40\) 5.64454 0.892480
\(41\) 5.28979 0.826127 0.413063 0.910702i \(-0.364459\pi\)
0.413063 + 0.910702i \(0.364459\pi\)
\(42\) −15.8245 −2.44177
\(43\) 6.60038 1.00655 0.503274 0.864127i \(-0.332129\pi\)
0.503274 + 0.864127i \(0.332129\pi\)
\(44\) 4.25662 0.641710
\(45\) 2.60686 0.388607
\(46\) 13.8933 2.04846
\(47\) 0.172675 0.0251873 0.0125937 0.999921i \(-0.495991\pi\)
0.0125937 + 0.999921i \(0.495991\pi\)
\(48\) 13.2733 1.91584
\(49\) 0.138402 0.0197717
\(50\) 2.50132 0.353741
\(51\) 10.5348 1.47516
\(52\) 14.8315 2.05676
\(53\) −4.95784 −0.681012 −0.340506 0.940242i \(-0.610598\pi\)
−0.340506 + 0.940242i \(0.610598\pi\)
\(54\) −2.32853 −0.316873
\(55\) 1.00000 0.134840
\(56\) −15.0810 −2.01528
\(57\) −9.57150 −1.26778
\(58\) −16.2638 −2.13555
\(59\) −4.38285 −0.570599 −0.285299 0.958438i \(-0.592093\pi\)
−0.285299 + 0.958438i \(0.592093\pi\)
\(60\) 10.0792 1.30121
\(61\) 2.23428 0.286070 0.143035 0.989718i \(-0.454314\pi\)
0.143035 + 0.989718i \(0.454314\pi\)
\(62\) −7.36471 −0.935320
\(63\) −6.96494 −0.877500
\(64\) −4.37682 −0.547103
\(65\) 3.48435 0.432180
\(66\) 5.92283 0.729050
\(67\) 5.08973 0.621809 0.310905 0.950441i \(-0.399368\pi\)
0.310905 + 0.950441i \(0.399368\pi\)
\(68\) 18.9378 2.29655
\(69\) 13.1521 1.58333
\(70\) −6.68298 −0.798769
\(71\) −3.91570 −0.464708 −0.232354 0.972631i \(-0.574643\pi\)
−0.232354 + 0.972631i \(0.574643\pi\)
\(72\) 14.7145 1.73412
\(73\) 1.00000 0.117041
\(74\) 1.49163 0.173398
\(75\) 2.36788 0.273419
\(76\) −17.2062 −1.97369
\(77\) −2.67178 −0.304477
\(78\) 20.6372 2.33670
\(79\) −13.4294 −1.51093 −0.755464 0.655190i \(-0.772589\pi\)
−0.755464 + 0.655190i \(0.772589\pi\)
\(80\) 5.60558 0.626723
\(81\) −10.0249 −1.11387
\(82\) 13.2315 1.46117
\(83\) 8.54660 0.938111 0.469055 0.883169i \(-0.344594\pi\)
0.469055 + 0.883169i \(0.344594\pi\)
\(84\) −26.9293 −2.93823
\(85\) 4.44902 0.482564
\(86\) 16.5097 1.78028
\(87\) −15.3962 −1.65064
\(88\) 5.64454 0.601710
\(89\) −14.7674 −1.56534 −0.782671 0.622436i \(-0.786143\pi\)
−0.782671 + 0.622436i \(0.786143\pi\)
\(90\) 6.52059 0.687330
\(91\) −9.30940 −0.975891
\(92\) 23.6429 2.46495
\(93\) −6.97181 −0.722943
\(94\) 0.431917 0.0445489
\(95\) −4.04223 −0.414724
\(96\) 6.46972 0.660313
\(97\) −8.71708 −0.885085 −0.442543 0.896747i \(-0.645924\pi\)
−0.442543 + 0.896747i \(0.645924\pi\)
\(98\) 0.346188 0.0349703
\(99\) 2.60686 0.261999
\(100\) 4.25662 0.425662
\(101\) 3.41038 0.339346 0.169673 0.985500i \(-0.445729\pi\)
0.169673 + 0.985500i \(0.445729\pi\)
\(102\) 26.3508 2.60912
\(103\) −6.25577 −0.616399 −0.308200 0.951322i \(-0.599726\pi\)
−0.308200 + 0.951322i \(0.599726\pi\)
\(104\) 19.6675 1.92856
\(105\) −6.32645 −0.617398
\(106\) −12.4012 −1.20451
\(107\) −1.37862 −0.133277 −0.0666383 0.997777i \(-0.521227\pi\)
−0.0666383 + 0.997777i \(0.521227\pi\)
\(108\) −3.96257 −0.381299
\(109\) −16.4165 −1.57242 −0.786209 0.617961i \(-0.787959\pi\)
−0.786209 + 0.617961i \(0.787959\pi\)
\(110\) 2.50132 0.238492
\(111\) 1.41205 0.134026
\(112\) −14.9769 −1.41518
\(113\) 16.6097 1.56251 0.781256 0.624211i \(-0.214580\pi\)
0.781256 + 0.624211i \(0.214580\pi\)
\(114\) −23.9414 −2.24232
\(115\) 5.55439 0.517949
\(116\) −27.6770 −2.56974
\(117\) 9.08319 0.839741
\(118\) −10.9629 −1.00922
\(119\) −11.8868 −1.08966
\(120\) 13.3656 1.22011
\(121\) 1.00000 0.0909091
\(122\) 5.58866 0.505973
\(123\) 12.5256 1.12939
\(124\) −12.5329 −1.12549
\(125\) 1.00000 0.0894427
\(126\) −17.4216 −1.55204
\(127\) 13.1896 1.17039 0.585195 0.810893i \(-0.301018\pi\)
0.585195 + 0.810893i \(0.301018\pi\)
\(128\) −16.4124 −1.45067
\(129\) 15.6289 1.37605
\(130\) 8.71548 0.764398
\(131\) −7.70878 −0.673519 −0.336759 0.941591i \(-0.609331\pi\)
−0.336759 + 0.941591i \(0.609331\pi\)
\(132\) 10.0792 0.877279
\(133\) 10.7999 0.936473
\(134\) 12.7311 1.09980
\(135\) −0.930919 −0.0801208
\(136\) 25.1127 2.15339
\(137\) 14.5083 1.23952 0.619762 0.784790i \(-0.287229\pi\)
0.619762 + 0.784790i \(0.287229\pi\)
\(138\) 32.8977 2.80044
\(139\) 7.51225 0.637181 0.318590 0.947892i \(-0.396791\pi\)
0.318590 + 0.947892i \(0.396791\pi\)
\(140\) −11.3727 −0.961173
\(141\) 0.408875 0.0344335
\(142\) −9.79443 −0.821930
\(143\) 3.48435 0.291376
\(144\) 14.6129 1.21774
\(145\) −6.50210 −0.539970
\(146\) 2.50132 0.207011
\(147\) 0.327719 0.0270298
\(148\) 2.53837 0.208653
\(149\) −5.63280 −0.461457 −0.230729 0.973018i \(-0.574111\pi\)
−0.230729 + 0.973018i \(0.574111\pi\)
\(150\) 5.92283 0.483597
\(151\) 23.7457 1.93240 0.966198 0.257801i \(-0.0829978\pi\)
0.966198 + 0.257801i \(0.0829978\pi\)
\(152\) −22.8165 −1.85066
\(153\) 11.5980 0.937639
\(154\) −6.68298 −0.538530
\(155\) −2.94433 −0.236494
\(156\) 35.1193 2.81180
\(157\) 3.40138 0.271459 0.135730 0.990746i \(-0.456662\pi\)
0.135730 + 0.990746i \(0.456662\pi\)
\(158\) −33.5913 −2.67238
\(159\) −11.7396 −0.931009
\(160\) 2.73229 0.216006
\(161\) −14.8401 −1.16956
\(162\) −25.0754 −1.97011
\(163\) −3.18356 −0.249356 −0.124678 0.992197i \(-0.539790\pi\)
−0.124678 + 0.992197i \(0.539790\pi\)
\(164\) 22.5166 1.75825
\(165\) 2.36788 0.184339
\(166\) 21.3778 1.65924
\(167\) 1.02167 0.0790589 0.0395294 0.999218i \(-0.487414\pi\)
0.0395294 + 0.999218i \(0.487414\pi\)
\(168\) −35.7099 −2.75508
\(169\) −0.859326 −0.0661020
\(170\) 11.1284 0.853513
\(171\) −10.5375 −0.805823
\(172\) 28.0953 2.14225
\(173\) −3.26504 −0.248237 −0.124118 0.992267i \(-0.539610\pi\)
−0.124118 + 0.992267i \(0.539610\pi\)
\(174\) −38.5108 −2.91950
\(175\) −2.67178 −0.201967
\(176\) 5.60558 0.422536
\(177\) −10.3781 −0.780063
\(178\) −36.9380 −2.76862
\(179\) −16.6425 −1.24392 −0.621958 0.783051i \(-0.713662\pi\)
−0.621958 + 0.783051i \(0.713662\pi\)
\(180\) 11.0964 0.827076
\(181\) −5.99670 −0.445731 −0.222865 0.974849i \(-0.571541\pi\)
−0.222865 + 0.974849i \(0.571541\pi\)
\(182\) −23.2858 −1.72606
\(183\) 5.29051 0.391085
\(184\) 31.3520 2.31130
\(185\) 0.596335 0.0438435
\(186\) −17.4388 −1.27867
\(187\) 4.44902 0.325345
\(188\) 0.735014 0.0536064
\(189\) 2.48721 0.180918
\(190\) −10.1109 −0.733523
\(191\) −20.7548 −1.50177 −0.750883 0.660436i \(-0.770372\pi\)
−0.750883 + 0.660436i \(0.770372\pi\)
\(192\) −10.3638 −0.747942
\(193\) 10.8308 0.779619 0.389809 0.920896i \(-0.372541\pi\)
0.389809 + 0.920896i \(0.372541\pi\)
\(194\) −21.8042 −1.56545
\(195\) 8.25052 0.590832
\(196\) 0.589125 0.0420803
\(197\) −7.69796 −0.548457 −0.274228 0.961665i \(-0.588422\pi\)
−0.274228 + 0.961665i \(0.588422\pi\)
\(198\) 6.52059 0.463398
\(199\) −14.1572 −1.00358 −0.501790 0.864990i \(-0.667325\pi\)
−0.501790 + 0.864990i \(0.667325\pi\)
\(200\) 5.64454 0.399129
\(201\) 12.0519 0.850073
\(202\) 8.53047 0.600202
\(203\) 17.3722 1.21929
\(204\) 44.8424 3.13960
\(205\) 5.28979 0.369455
\(206\) −15.6477 −1.09023
\(207\) 14.4795 1.00639
\(208\) 19.5318 1.35428
\(209\) −4.04223 −0.279607
\(210\) −15.8245 −1.09199
\(211\) 25.0038 1.72134 0.860668 0.509167i \(-0.170046\pi\)
0.860668 + 0.509167i \(0.170046\pi\)
\(212\) −21.1037 −1.44941
\(213\) −9.27190 −0.635300
\(214\) −3.44838 −0.235727
\(215\) 6.60038 0.450142
\(216\) −5.25461 −0.357531
\(217\) 7.86659 0.534019
\(218\) −41.0630 −2.78114
\(219\) 2.36788 0.160006
\(220\) 4.25662 0.286981
\(221\) 15.5019 1.04277
\(222\) 3.53200 0.237052
\(223\) −21.8292 −1.46179 −0.730896 0.682489i \(-0.760898\pi\)
−0.730896 + 0.682489i \(0.760898\pi\)
\(224\) −7.30006 −0.487756
\(225\) 2.60686 0.173790
\(226\) 41.5463 2.76362
\(227\) 27.1837 1.80425 0.902124 0.431477i \(-0.142007\pi\)
0.902124 + 0.431477i \(0.142007\pi\)
\(228\) −40.7423 −2.69822
\(229\) 10.0624 0.664940 0.332470 0.943114i \(-0.392118\pi\)
0.332470 + 0.943114i \(0.392118\pi\)
\(230\) 13.8933 0.916099
\(231\) −6.32645 −0.416250
\(232\) −36.7013 −2.40956
\(233\) −25.0994 −1.64431 −0.822157 0.569261i \(-0.807229\pi\)
−0.822157 + 0.569261i \(0.807229\pi\)
\(234\) 22.7200 1.48525
\(235\) 0.172675 0.0112641
\(236\) −18.6561 −1.21441
\(237\) −31.7993 −2.06558
\(238\) −29.7327 −1.92729
\(239\) −12.8094 −0.828571 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(240\) 13.2733 0.856790
\(241\) 13.3242 0.858287 0.429143 0.903236i \(-0.358816\pi\)
0.429143 + 0.903236i \(0.358816\pi\)
\(242\) 2.50132 0.160791
\(243\) −20.9449 −1.34362
\(244\) 9.51048 0.608846
\(245\) 0.138402 0.00884218
\(246\) 31.3306 1.99756
\(247\) −14.0845 −0.896176
\(248\) −16.6194 −1.05533
\(249\) 20.2373 1.28249
\(250\) 2.50132 0.158198
\(251\) −28.6688 −1.80956 −0.904779 0.425882i \(-0.859964\pi\)
−0.904779 + 0.425882i \(0.859964\pi\)
\(252\) −29.6471 −1.86759
\(253\) 5.55439 0.349201
\(254\) 32.9915 2.07007
\(255\) 10.5348 0.659712
\(256\) −32.2991 −2.01870
\(257\) 2.19515 0.136929 0.0684647 0.997654i \(-0.478190\pi\)
0.0684647 + 0.997654i \(0.478190\pi\)
\(258\) 39.0929 2.43382
\(259\) −1.59328 −0.0990014
\(260\) 14.8315 0.919813
\(261\) −16.9500 −1.04918
\(262\) −19.2821 −1.19125
\(263\) 16.1810 0.997762 0.498881 0.866670i \(-0.333744\pi\)
0.498881 + 0.866670i \(0.333744\pi\)
\(264\) 13.3656 0.822595
\(265\) −4.95784 −0.304558
\(266\) 27.0141 1.65634
\(267\) −34.9674 −2.13997
\(268\) 21.6650 1.32340
\(269\) 7.76730 0.473581 0.236790 0.971561i \(-0.423905\pi\)
0.236790 + 0.971561i \(0.423905\pi\)
\(270\) −2.32853 −0.141710
\(271\) 12.9838 0.788709 0.394354 0.918958i \(-0.370968\pi\)
0.394354 + 0.918958i \(0.370968\pi\)
\(272\) 24.9393 1.51217
\(273\) −22.0436 −1.33414
\(274\) 36.2899 2.19235
\(275\) 1.00000 0.0603023
\(276\) 55.9836 3.36982
\(277\) 0.491802 0.0295495 0.0147747 0.999891i \(-0.495297\pi\)
0.0147747 + 0.999891i \(0.495297\pi\)
\(278\) 18.7906 1.12698
\(279\) −7.67543 −0.459516
\(280\) −15.0810 −0.901260
\(281\) −2.87378 −0.171435 −0.0857177 0.996319i \(-0.527318\pi\)
−0.0857177 + 0.996319i \(0.527318\pi\)
\(282\) 1.02273 0.0609026
\(283\) −30.8900 −1.83622 −0.918111 0.396324i \(-0.870286\pi\)
−0.918111 + 0.396324i \(0.870286\pi\)
\(284\) −16.6676 −0.989042
\(285\) −9.57150 −0.566967
\(286\) 8.71548 0.515357
\(287\) −14.1332 −0.834254
\(288\) 7.12267 0.419708
\(289\) 2.79380 0.164341
\(290\) −16.2638 −0.955046
\(291\) −20.6410 −1.21000
\(292\) 4.25662 0.249100
\(293\) 26.4906 1.54760 0.773800 0.633430i \(-0.218354\pi\)
0.773800 + 0.633430i \(0.218354\pi\)
\(294\) 0.819732 0.0478077
\(295\) −4.38285 −0.255179
\(296\) 3.36604 0.195647
\(297\) −0.930919 −0.0540174
\(298\) −14.0895 −0.816181
\(299\) 19.3534 1.11924
\(300\) 10.0792 0.581921
\(301\) −17.6347 −1.01645
\(302\) 59.3956 3.41783
\(303\) 8.07537 0.463918
\(304\) −22.6590 −1.29958
\(305\) 2.23428 0.127934
\(306\) 29.0102 1.65841
\(307\) −22.5586 −1.28749 −0.643744 0.765241i \(-0.722620\pi\)
−0.643744 + 0.765241i \(0.722620\pi\)
\(308\) −11.3727 −0.648022
\(309\) −14.8129 −0.842677
\(310\) −7.36471 −0.418288
\(311\) −2.62013 −0.148574 −0.0742870 0.997237i \(-0.523668\pi\)
−0.0742870 + 0.997237i \(0.523668\pi\)
\(312\) 46.5703 2.63653
\(313\) 23.8061 1.34560 0.672799 0.739826i \(-0.265092\pi\)
0.672799 + 0.739826i \(0.265092\pi\)
\(314\) 8.50795 0.480131
\(315\) −6.96494 −0.392430
\(316\) −57.1640 −3.21572
\(317\) −0.622469 −0.0349613 −0.0174807 0.999847i \(-0.505565\pi\)
−0.0174807 + 0.999847i \(0.505565\pi\)
\(318\) −29.3645 −1.64668
\(319\) −6.50210 −0.364048
\(320\) −4.37682 −0.244672
\(321\) −3.26442 −0.182202
\(322\) −37.1199 −2.06861
\(323\) −17.9840 −1.00065
\(324\) −42.6721 −2.37067
\(325\) 3.48435 0.193277
\(326\) −7.96311 −0.441036
\(327\) −38.8724 −2.14965
\(328\) 29.8584 1.64866
\(329\) −0.461351 −0.0254351
\(330\) 5.92283 0.326041
\(331\) −4.85370 −0.266784 −0.133392 0.991063i \(-0.542587\pi\)
−0.133392 + 0.991063i \(0.542587\pi\)
\(332\) 36.3796 1.99659
\(333\) 1.55456 0.0851894
\(334\) 2.55552 0.139832
\(335\) 5.08973 0.278081
\(336\) −35.4634 −1.93469
\(337\) 16.4423 0.895668 0.447834 0.894117i \(-0.352196\pi\)
0.447834 + 0.894117i \(0.352196\pi\)
\(338\) −2.14945 −0.116915
\(339\) 39.3298 2.13610
\(340\) 18.9378 1.02705
\(341\) −2.94433 −0.159444
\(342\) −26.3577 −1.42526
\(343\) 18.3327 0.989871
\(344\) 37.2561 2.00871
\(345\) 13.1521 0.708087
\(346\) −8.16693 −0.439057
\(347\) −24.8517 −1.33411 −0.667056 0.745008i \(-0.732446\pi\)
−0.667056 + 0.745008i \(0.732446\pi\)
\(348\) −65.5357 −3.51308
\(349\) −7.61793 −0.407778 −0.203889 0.978994i \(-0.565358\pi\)
−0.203889 + 0.978994i \(0.565358\pi\)
\(350\) −6.68298 −0.357221
\(351\) −3.24365 −0.173133
\(352\) 2.73229 0.145631
\(353\) 13.6194 0.724885 0.362443 0.932006i \(-0.381943\pi\)
0.362443 + 0.932006i \(0.381943\pi\)
\(354\) −25.9589 −1.37970
\(355\) −3.91570 −0.207824
\(356\) −62.8592 −3.33153
\(357\) −28.1465 −1.48967
\(358\) −41.6282 −2.20012
\(359\) −22.6034 −1.19296 −0.596481 0.802627i \(-0.703435\pi\)
−0.596481 + 0.802627i \(0.703435\pi\)
\(360\) 14.7145 0.775522
\(361\) −2.66041 −0.140022
\(362\) −14.9997 −0.788366
\(363\) 2.36788 0.124281
\(364\) −39.6266 −2.07700
\(365\) 1.00000 0.0523424
\(366\) 13.2333 0.691714
\(367\) 1.63669 0.0854346 0.0427173 0.999087i \(-0.486399\pi\)
0.0427173 + 0.999087i \(0.486399\pi\)
\(368\) 31.1356 1.62305
\(369\) 13.7897 0.717864
\(370\) 1.49163 0.0775461
\(371\) 13.2463 0.687712
\(372\) −29.6764 −1.53865
\(373\) 7.41370 0.383867 0.191934 0.981408i \(-0.438524\pi\)
0.191934 + 0.981408i \(0.438524\pi\)
\(374\) 11.1284 0.575438
\(375\) 2.36788 0.122277
\(376\) 0.974673 0.0502649
\(377\) −22.6556 −1.16682
\(378\) 6.22132 0.319990
\(379\) 17.7378 0.911132 0.455566 0.890202i \(-0.349437\pi\)
0.455566 + 0.890202i \(0.349437\pi\)
\(380\) −17.2062 −0.882660
\(381\) 31.2314 1.60003
\(382\) −51.9145 −2.65618
\(383\) −15.0193 −0.767449 −0.383725 0.923448i \(-0.625359\pi\)
−0.383725 + 0.923448i \(0.625359\pi\)
\(384\) −38.8626 −1.98320
\(385\) −2.67178 −0.136166
\(386\) 27.0914 1.37891
\(387\) 17.2062 0.874642
\(388\) −37.1053 −1.88374
\(389\) 29.6042 1.50099 0.750495 0.660876i \(-0.229815\pi\)
0.750495 + 0.660876i \(0.229815\pi\)
\(390\) 20.6372 1.04501
\(391\) 24.7116 1.24972
\(392\) 0.781215 0.0394573
\(393\) −18.2535 −0.920765
\(394\) −19.2551 −0.970057
\(395\) −13.4294 −0.675708
\(396\) 11.0964 0.557615
\(397\) 15.9367 0.799838 0.399919 0.916550i \(-0.369038\pi\)
0.399919 + 0.916550i \(0.369038\pi\)
\(398\) −35.4118 −1.77503
\(399\) 25.5729 1.28025
\(400\) 5.60558 0.280279
\(401\) −2.46612 −0.123152 −0.0615762 0.998102i \(-0.519613\pi\)
−0.0615762 + 0.998102i \(0.519613\pi\)
\(402\) 30.1456 1.50353
\(403\) −10.2591 −0.511040
\(404\) 14.5167 0.722233
\(405\) −10.0249 −0.498140
\(406\) 43.4534 2.15656
\(407\) 0.596335 0.0295593
\(408\) 59.4638 2.94390
\(409\) −0.380219 −0.0188006 −0.00940030 0.999956i \(-0.502992\pi\)
−0.00940030 + 0.999956i \(0.502992\pi\)
\(410\) 13.2315 0.653457
\(411\) 34.3538 1.69455
\(412\) −26.6284 −1.31189
\(413\) 11.7100 0.576212
\(414\) 36.2179 1.78001
\(415\) 8.54660 0.419536
\(416\) 9.52023 0.466768
\(417\) 17.7881 0.871087
\(418\) −10.1109 −0.494541
\(419\) −26.5584 −1.29747 −0.648733 0.761017i \(-0.724701\pi\)
−0.648733 + 0.761017i \(0.724701\pi\)
\(420\) −26.9293 −1.31402
\(421\) 17.2515 0.840785 0.420392 0.907342i \(-0.361892\pi\)
0.420392 + 0.907342i \(0.361892\pi\)
\(422\) 62.5427 3.04453
\(423\) 0.450140 0.0218866
\(424\) −27.9847 −1.35906
\(425\) 4.44902 0.215809
\(426\) −23.1920 −1.12366
\(427\) −5.96950 −0.288884
\(428\) −5.86828 −0.283654
\(429\) 8.25052 0.398339
\(430\) 16.5097 0.796167
\(431\) −1.65671 −0.0798008 −0.0399004 0.999204i \(-0.512704\pi\)
−0.0399004 + 0.999204i \(0.512704\pi\)
\(432\) −5.21834 −0.251068
\(433\) 14.4053 0.692273 0.346136 0.938184i \(-0.387493\pi\)
0.346136 + 0.938184i \(0.387493\pi\)
\(434\) 19.6769 0.944521
\(435\) −15.3962 −0.738191
\(436\) −69.8789 −3.34659
\(437\) −22.4521 −1.07403
\(438\) 5.92283 0.283004
\(439\) 22.8919 1.09257 0.546284 0.837600i \(-0.316042\pi\)
0.546284 + 0.837600i \(0.316042\pi\)
\(440\) 5.64454 0.269093
\(441\) 0.360794 0.0171807
\(442\) 38.7754 1.84436
\(443\) 12.0384 0.571960 0.285980 0.958236i \(-0.407681\pi\)
0.285980 + 0.958236i \(0.407681\pi\)
\(444\) 6.01056 0.285249
\(445\) −14.7674 −0.700042
\(446\) −54.6019 −2.58548
\(447\) −13.3378 −0.630856
\(448\) 11.6939 0.552485
\(449\) 26.2517 1.23889 0.619447 0.785039i \(-0.287357\pi\)
0.619447 + 0.785039i \(0.287357\pi\)
\(450\) 6.52059 0.307384
\(451\) 5.28979 0.249087
\(452\) 70.7013 3.32551
\(453\) 56.2269 2.64177
\(454\) 67.9953 3.19118
\(455\) −9.30940 −0.436432
\(456\) −54.0267 −2.53003
\(457\) −1.85069 −0.0865716 −0.0432858 0.999063i \(-0.513783\pi\)
−0.0432858 + 0.999063i \(0.513783\pi\)
\(458\) 25.1693 1.17608
\(459\) −4.14168 −0.193317
\(460\) 23.6429 1.10236
\(461\) −36.6237 −1.70573 −0.852867 0.522129i \(-0.825138\pi\)
−0.852867 + 0.522129i \(0.825138\pi\)
\(462\) −15.8245 −0.736223
\(463\) −15.6303 −0.726403 −0.363202 0.931711i \(-0.618316\pi\)
−0.363202 + 0.931711i \(0.618316\pi\)
\(464\) −36.4480 −1.69206
\(465\) −6.97181 −0.323310
\(466\) −62.7816 −2.90830
\(467\) 15.8465 0.733288 0.366644 0.930361i \(-0.380507\pi\)
0.366644 + 0.930361i \(0.380507\pi\)
\(468\) 38.6637 1.78723
\(469\) −13.5986 −0.627926
\(470\) 0.431917 0.0199229
\(471\) 8.05405 0.371111
\(472\) −24.7392 −1.13871
\(473\) 6.60038 0.303486
\(474\) −79.5403 −3.65340
\(475\) −4.04223 −0.185470
\(476\) −50.5976 −2.31914
\(477\) −12.9244 −0.591767
\(478\) −32.0405 −1.46550
\(479\) 33.3439 1.52352 0.761761 0.647858i \(-0.224335\pi\)
0.761761 + 0.647858i \(0.224335\pi\)
\(480\) 6.46972 0.295301
\(481\) 2.07784 0.0947413
\(482\) 33.3281 1.51805
\(483\) −35.1396 −1.59891
\(484\) 4.25662 0.193483
\(485\) −8.71708 −0.395822
\(486\) −52.3901 −2.37646
\(487\) −16.7072 −0.757073 −0.378537 0.925586i \(-0.623573\pi\)
−0.378537 + 0.925586i \(0.623573\pi\)
\(488\) 12.6115 0.570895
\(489\) −7.53829 −0.340893
\(490\) 0.346188 0.0156392
\(491\) −23.4179 −1.05684 −0.528418 0.848984i \(-0.677215\pi\)
−0.528418 + 0.848984i \(0.677215\pi\)
\(492\) 53.3167 2.40370
\(493\) −28.9280 −1.30285
\(494\) −35.2299 −1.58507
\(495\) 2.60686 0.117169
\(496\) −16.5046 −0.741081
\(497\) 10.4619 0.469279
\(498\) 50.6201 2.26834
\(499\) 32.7713 1.46704 0.733522 0.679665i \(-0.237875\pi\)
0.733522 + 0.679665i \(0.237875\pi\)
\(500\) 4.25662 0.190362
\(501\) 2.41918 0.108081
\(502\) −71.7099 −3.20057
\(503\) 24.2802 1.08260 0.541301 0.840829i \(-0.317932\pi\)
0.541301 + 0.840829i \(0.317932\pi\)
\(504\) −39.3139 −1.75118
\(505\) 3.41038 0.151760
\(506\) 13.8933 0.617634
\(507\) −2.03478 −0.0903678
\(508\) 56.1432 2.49095
\(509\) 1.11035 0.0492154 0.0246077 0.999697i \(-0.492166\pi\)
0.0246077 + 0.999697i \(0.492166\pi\)
\(510\) 26.3508 1.16683
\(511\) −2.67178 −0.118193
\(512\) −47.9657 −2.11981
\(513\) 3.76299 0.166140
\(514\) 5.49077 0.242188
\(515\) −6.25577 −0.275662
\(516\) 66.5263 2.92866
\(517\) 0.172675 0.00759426
\(518\) −3.98530 −0.175104
\(519\) −7.73123 −0.339363
\(520\) 19.6675 0.862478
\(521\) 15.8674 0.695162 0.347581 0.937650i \(-0.387003\pi\)
0.347581 + 0.937650i \(0.387003\pi\)
\(522\) −42.3975 −1.85569
\(523\) −18.3366 −0.801802 −0.400901 0.916121i \(-0.631303\pi\)
−0.400901 + 0.916121i \(0.631303\pi\)
\(524\) −32.8133 −1.43346
\(525\) −6.32645 −0.276109
\(526\) 40.4739 1.76475
\(527\) −13.0994 −0.570618
\(528\) 13.2733 0.577648
\(529\) 7.85123 0.341358
\(530\) −12.4012 −0.538672
\(531\) −11.4255 −0.495823
\(532\) 45.9712 1.99310
\(533\) 18.4315 0.798356
\(534\) −87.4649 −3.78497
\(535\) −1.37862 −0.0596031
\(536\) 28.7292 1.24091
\(537\) −39.4073 −1.70055
\(538\) 19.4285 0.837624
\(539\) 0.138402 0.00596140
\(540\) −3.96257 −0.170522
\(541\) 16.2518 0.698721 0.349360 0.936988i \(-0.386399\pi\)
0.349360 + 0.936988i \(0.386399\pi\)
\(542\) 32.4766 1.39499
\(543\) −14.1995 −0.609357
\(544\) 12.1560 0.521184
\(545\) −16.4165 −0.703207
\(546\) −55.1381 −2.35969
\(547\) 3.90807 0.167097 0.0835485 0.996504i \(-0.473375\pi\)
0.0835485 + 0.996504i \(0.473375\pi\)
\(548\) 61.7562 2.63809
\(549\) 5.82444 0.248581
\(550\) 2.50132 0.106657
\(551\) 26.2829 1.11969
\(552\) 74.2377 3.15977
\(553\) 35.8805 1.52579
\(554\) 1.23015 0.0522643
\(555\) 1.41205 0.0599382
\(556\) 31.9768 1.35612
\(557\) 19.2238 0.814540 0.407270 0.913308i \(-0.366481\pi\)
0.407270 + 0.913308i \(0.366481\pi\)
\(558\) −19.1987 −0.812748
\(559\) 22.9980 0.972712
\(560\) −14.9769 −0.632888
\(561\) 10.5348 0.444778
\(562\) −7.18826 −0.303218
\(563\) 6.97074 0.293782 0.146891 0.989153i \(-0.453073\pi\)
0.146891 + 0.989153i \(0.453073\pi\)
\(564\) 1.74042 0.0732851
\(565\) 16.6097 0.698776
\(566\) −77.2660 −3.24773
\(567\) 26.7842 1.12483
\(568\) −22.1023 −0.927392
\(569\) 40.8739 1.71352 0.856761 0.515714i \(-0.172473\pi\)
0.856761 + 0.515714i \(0.172473\pi\)
\(570\) −23.9414 −1.00280
\(571\) 37.0955 1.55240 0.776199 0.630488i \(-0.217145\pi\)
0.776199 + 0.630488i \(0.217145\pi\)
\(572\) 14.8315 0.620138
\(573\) −49.1449 −2.05306
\(574\) −35.3516 −1.47555
\(575\) 5.55439 0.231634
\(576\) −11.4097 −0.475406
\(577\) −10.1569 −0.422838 −0.211419 0.977396i \(-0.567808\pi\)
−0.211419 + 0.977396i \(0.567808\pi\)
\(578\) 6.98821 0.290671
\(579\) 25.6461 1.06581
\(580\) −27.6770 −1.14922
\(581\) −22.8346 −0.947339
\(582\) −51.6298 −2.14013
\(583\) −4.95784 −0.205333
\(584\) 5.64454 0.233573
\(585\) 9.08319 0.375544
\(586\) 66.2617 2.73724
\(587\) 22.4289 0.925739 0.462870 0.886426i \(-0.346820\pi\)
0.462870 + 0.886426i \(0.346820\pi\)
\(588\) 1.39498 0.0575279
\(589\) 11.9016 0.490398
\(590\) −10.9629 −0.451337
\(591\) −18.2278 −0.749793
\(592\) 3.34280 0.137388
\(593\) 37.6348 1.54547 0.772737 0.634726i \(-0.218887\pi\)
0.772737 + 0.634726i \(0.218887\pi\)
\(594\) −2.32853 −0.0955408
\(595\) −11.8868 −0.487311
\(596\) −23.9767 −0.982124
\(597\) −33.5226 −1.37199
\(598\) 48.4092 1.97960
\(599\) −27.0934 −1.10701 −0.553504 0.832846i \(-0.686710\pi\)
−0.553504 + 0.832846i \(0.686710\pi\)
\(600\) 13.3656 0.545648
\(601\) −2.74515 −0.111977 −0.0559884 0.998431i \(-0.517831\pi\)
−0.0559884 + 0.998431i \(0.517831\pi\)
\(602\) −44.1102 −1.79780
\(603\) 13.2682 0.540322
\(604\) 101.076 4.11274
\(605\) 1.00000 0.0406558
\(606\) 20.1991 0.820533
\(607\) 21.4316 0.869882 0.434941 0.900459i \(-0.356769\pi\)
0.434941 + 0.900459i \(0.356769\pi\)
\(608\) −11.0445 −0.447914
\(609\) 41.1352 1.66688
\(610\) 5.58866 0.226278
\(611\) 0.601661 0.0243406
\(612\) 49.3681 1.99559
\(613\) 16.6985 0.674445 0.337222 0.941425i \(-0.390513\pi\)
0.337222 + 0.941425i \(0.390513\pi\)
\(614\) −56.4264 −2.27718
\(615\) 12.5256 0.505081
\(616\) −15.0810 −0.607629
\(617\) 18.1662 0.731345 0.365673 0.930743i \(-0.380839\pi\)
0.365673 + 0.930743i \(0.380839\pi\)
\(618\) −37.0519 −1.49045
\(619\) −28.5098 −1.14590 −0.572952 0.819589i \(-0.694202\pi\)
−0.572952 + 0.819589i \(0.694202\pi\)
\(620\) −12.5329 −0.503333
\(621\) −5.17069 −0.207493
\(622\) −6.55380 −0.262783
\(623\) 39.4552 1.58074
\(624\) 46.2489 1.85144
\(625\) 1.00000 0.0400000
\(626\) 59.5466 2.37996
\(627\) −9.57150 −0.382249
\(628\) 14.4784 0.577750
\(629\) 2.65311 0.105786
\(630\) −17.4216 −0.694092
\(631\) 35.0244 1.39430 0.697149 0.716926i \(-0.254452\pi\)
0.697149 + 0.716926i \(0.254452\pi\)
\(632\) −75.8029 −3.01528
\(633\) 59.2061 2.35323
\(634\) −1.55700 −0.0618362
\(635\) 13.1896 0.523414
\(636\) −49.9709 −1.98148
\(637\) 0.482241 0.0191071
\(638\) −16.2638 −0.643892
\(639\) −10.2077 −0.403809
\(640\) −16.4124 −0.648758
\(641\) 1.00090 0.0395334 0.0197667 0.999805i \(-0.493708\pi\)
0.0197667 + 0.999805i \(0.493708\pi\)
\(642\) −8.16536 −0.322261
\(643\) 23.5361 0.928173 0.464087 0.885790i \(-0.346383\pi\)
0.464087 + 0.885790i \(0.346383\pi\)
\(644\) −63.1687 −2.48919
\(645\) 15.6289 0.615387
\(646\) −44.9837 −1.76986
\(647\) −11.3653 −0.446816 −0.223408 0.974725i \(-0.571718\pi\)
−0.223408 + 0.974725i \(0.571718\pi\)
\(648\) −56.5858 −2.22290
\(649\) −4.38285 −0.172042
\(650\) 8.71548 0.341849
\(651\) 18.6271 0.730055
\(652\) −13.5512 −0.530706
\(653\) 1.63776 0.0640904 0.0320452 0.999486i \(-0.489798\pi\)
0.0320452 + 0.999486i \(0.489798\pi\)
\(654\) −97.2324 −3.80209
\(655\) −7.70878 −0.301207
\(656\) 29.6523 1.15773
\(657\) 2.60686 0.101703
\(658\) −1.15399 −0.0449871
\(659\) −42.9468 −1.67297 −0.836484 0.547991i \(-0.815393\pi\)
−0.836484 + 0.547991i \(0.815393\pi\)
\(660\) 10.0792 0.392331
\(661\) −20.1232 −0.782702 −0.391351 0.920241i \(-0.627992\pi\)
−0.391351 + 0.920241i \(0.627992\pi\)
\(662\) −12.1407 −0.471861
\(663\) 36.7067 1.42557
\(664\) 48.2416 1.87214
\(665\) 10.7999 0.418803
\(666\) 3.88846 0.150675
\(667\) −36.1152 −1.39839
\(668\) 4.34884 0.168262
\(669\) −51.6889 −1.99841
\(670\) 12.7311 0.491843
\(671\) 2.23428 0.0862534
\(672\) −17.2857 −0.666809
\(673\) 18.6316 0.718196 0.359098 0.933300i \(-0.383084\pi\)
0.359098 + 0.933300i \(0.383084\pi\)
\(674\) 41.1275 1.58417
\(675\) −0.930919 −0.0358311
\(676\) −3.65782 −0.140686
\(677\) −20.1362 −0.773895 −0.386948 0.922102i \(-0.626471\pi\)
−0.386948 + 0.922102i \(0.626471\pi\)
\(678\) 98.3766 3.77813
\(679\) 23.2901 0.893792
\(680\) 25.1127 0.963027
\(681\) 64.3678 2.46658
\(682\) −7.36471 −0.282009
\(683\) −32.8005 −1.25508 −0.627538 0.778586i \(-0.715937\pi\)
−0.627538 + 0.778586i \(0.715937\pi\)
\(684\) −44.8541 −1.71504
\(685\) 14.5083 0.554332
\(686\) 45.8559 1.75079
\(687\) 23.8265 0.909038
\(688\) 36.9989 1.41057
\(689\) −17.2748 −0.658119
\(690\) 32.8977 1.25239
\(691\) 2.43678 0.0926996 0.0463498 0.998925i \(-0.485241\pi\)
0.0463498 + 0.998925i \(0.485241\pi\)
\(692\) −13.8980 −0.528324
\(693\) −6.96494 −0.264576
\(694\) −62.1622 −2.35965
\(695\) 7.51225 0.284956
\(696\) −86.9044 −3.29410
\(697\) 23.5344 0.891429
\(698\) −19.0549 −0.721239
\(699\) −59.4323 −2.24793
\(700\) −11.3727 −0.429849
\(701\) 15.6501 0.591095 0.295547 0.955328i \(-0.404498\pi\)
0.295547 + 0.955328i \(0.404498\pi\)
\(702\) −8.11341 −0.306221
\(703\) −2.41052 −0.0909146
\(704\) −4.37682 −0.164958
\(705\) 0.408875 0.0153991
\(706\) 34.0664 1.28211
\(707\) −9.11179 −0.342684
\(708\) −44.1755 −1.66022
\(709\) 11.6826 0.438750 0.219375 0.975641i \(-0.429598\pi\)
0.219375 + 0.975641i \(0.429598\pi\)
\(710\) −9.79443 −0.367578
\(711\) −35.0086 −1.31292
\(712\) −83.3552 −3.12387
\(713\) −16.3539 −0.612460
\(714\) −70.4036 −2.63479
\(715\) 3.48435 0.130307
\(716\) −70.8406 −2.64744
\(717\) −30.3311 −1.13274
\(718\) −56.5384 −2.11000
\(719\) −10.4833 −0.390960 −0.195480 0.980708i \(-0.562626\pi\)
−0.195480 + 0.980708i \(0.562626\pi\)
\(720\) 14.6129 0.544592
\(721\) 16.7140 0.622463
\(722\) −6.65455 −0.247657
\(723\) 31.5501 1.17336
\(724\) −25.5257 −0.948654
\(725\) −6.50210 −0.241482
\(726\) 5.92283 0.219817
\(727\) 40.0161 1.48412 0.742058 0.670336i \(-0.233850\pi\)
0.742058 + 0.670336i \(0.233850\pi\)
\(728\) −52.5473 −1.94753
\(729\) −19.5205 −0.722980
\(730\) 2.50132 0.0925781
\(731\) 29.3652 1.08611
\(732\) 22.5197 0.832351
\(733\) 21.8844 0.808320 0.404160 0.914688i \(-0.367564\pi\)
0.404160 + 0.914688i \(0.367564\pi\)
\(734\) 4.09389 0.151108
\(735\) 0.327719 0.0120881
\(736\) 15.1762 0.559401
\(737\) 5.08973 0.187482
\(738\) 34.4926 1.26969
\(739\) −11.5304 −0.424153 −0.212077 0.977253i \(-0.568023\pi\)
−0.212077 + 0.977253i \(0.568023\pi\)
\(740\) 2.53837 0.0933125
\(741\) −33.3504 −1.22516
\(742\) 33.1332 1.21636
\(743\) 13.3300 0.489030 0.244515 0.969646i \(-0.421371\pi\)
0.244515 + 0.969646i \(0.421371\pi\)
\(744\) −39.3527 −1.44274
\(745\) −5.63280 −0.206370
\(746\) 18.5441 0.678947
\(747\) 22.2797 0.815173
\(748\) 18.9378 0.692435
\(749\) 3.68338 0.134588
\(750\) 5.92283 0.216271
\(751\) −27.9836 −1.02114 −0.510569 0.859837i \(-0.670565\pi\)
−0.510569 + 0.859837i \(0.670565\pi\)
\(752\) 0.967945 0.0352973
\(753\) −67.8842 −2.47384
\(754\) −56.6689 −2.06376
\(755\) 23.7457 0.864194
\(756\) 10.5871 0.385050
\(757\) −26.2443 −0.953864 −0.476932 0.878940i \(-0.658251\pi\)
−0.476932 + 0.878940i \(0.658251\pi\)
\(758\) 44.3681 1.61152
\(759\) 13.1521 0.477392
\(760\) −22.8165 −0.827641
\(761\) 18.7033 0.677993 0.338997 0.940788i \(-0.389912\pi\)
0.338997 + 0.940788i \(0.389912\pi\)
\(762\) 78.1199 2.82999
\(763\) 43.8613 1.58789
\(764\) −88.3454 −3.19622
\(765\) 11.5980 0.419325
\(766\) −37.5681 −1.35739
\(767\) −15.2714 −0.551417
\(768\) −76.4805 −2.75975
\(769\) 8.51567 0.307083 0.153541 0.988142i \(-0.450932\pi\)
0.153541 + 0.988142i \(0.450932\pi\)
\(770\) −6.68298 −0.240838
\(771\) 5.19784 0.187196
\(772\) 46.1026 1.65927
\(773\) −1.52936 −0.0550072 −0.0275036 0.999622i \(-0.508756\pi\)
−0.0275036 + 0.999622i \(0.508756\pi\)
\(774\) 43.0383 1.54698
\(775\) −2.94433 −0.105763
\(776\) −49.2039 −1.76632
\(777\) −3.77269 −0.135344
\(778\) 74.0496 2.65481
\(779\) −21.3825 −0.766109
\(780\) 35.1193 1.25747
\(781\) −3.91570 −0.140115
\(782\) 61.8117 2.21038
\(783\) 6.05293 0.216314
\(784\) 0.775823 0.0277080
\(785\) 3.40138 0.121400
\(786\) −45.6578 −1.62856
\(787\) −6.31932 −0.225259 −0.112630 0.993637i \(-0.535927\pi\)
−0.112630 + 0.993637i \(0.535927\pi\)
\(788\) −32.7673 −1.16729
\(789\) 38.3146 1.36404
\(790\) −33.5913 −1.19513
\(791\) −44.3775 −1.57788
\(792\) 14.7145 0.522857
\(793\) 7.78500 0.276454
\(794\) 39.8627 1.41468
\(795\) −11.7396 −0.416360
\(796\) −60.2619 −2.13593
\(797\) 12.9150 0.457473 0.228737 0.973488i \(-0.426541\pi\)
0.228737 + 0.973488i \(0.426541\pi\)
\(798\) 63.9662 2.26438
\(799\) 0.768237 0.0271783
\(800\) 2.73229 0.0966009
\(801\) −38.4965 −1.36021
\(802\) −6.16857 −0.217820
\(803\) 1.00000 0.0352892
\(804\) 51.3002 1.80922
\(805\) −14.8401 −0.523045
\(806\) −25.6612 −0.903878
\(807\) 18.3920 0.647430
\(808\) 19.2500 0.677214
\(809\) 50.0904 1.76109 0.880543 0.473967i \(-0.157178\pi\)
0.880543 + 0.473967i \(0.157178\pi\)
\(810\) −25.0754 −0.881061
\(811\) 21.8414 0.766954 0.383477 0.923550i \(-0.374727\pi\)
0.383477 + 0.923550i \(0.374727\pi\)
\(812\) 73.9467 2.59502
\(813\) 30.7440 1.07824
\(814\) 1.49163 0.0522815
\(815\) −3.18356 −0.111515
\(816\) 59.0534 2.06728
\(817\) −26.6802 −0.933422
\(818\) −0.951050 −0.0332527
\(819\) −24.2683 −0.848002
\(820\) 22.5166 0.786315
\(821\) −17.1885 −0.599883 −0.299942 0.953958i \(-0.596967\pi\)
−0.299942 + 0.953958i \(0.596967\pi\)
\(822\) 85.9300 2.99715
\(823\) −4.23255 −0.147537 −0.0737687 0.997275i \(-0.523503\pi\)
−0.0737687 + 0.997275i \(0.523503\pi\)
\(824\) −35.3109 −1.23011
\(825\) 2.36788 0.0824390
\(826\) 29.2905 1.01915
\(827\) −0.677290 −0.0235517 −0.0117758 0.999931i \(-0.503748\pi\)
−0.0117758 + 0.999931i \(0.503748\pi\)
\(828\) 61.6337 2.14192
\(829\) 0.823786 0.0286113 0.0143056 0.999898i \(-0.495446\pi\)
0.0143056 + 0.999898i \(0.495446\pi\)
\(830\) 21.3778 0.742034
\(831\) 1.16453 0.0403970
\(832\) −15.2504 −0.528711
\(833\) 0.615754 0.0213346
\(834\) 44.4938 1.54070
\(835\) 1.02167 0.0353562
\(836\) −17.2062 −0.595090
\(837\) 2.74093 0.0947404
\(838\) −66.4313 −2.29483
\(839\) 9.57679 0.330627 0.165314 0.986241i \(-0.447136\pi\)
0.165314 + 0.986241i \(0.447136\pi\)
\(840\) −35.7099 −1.23211
\(841\) 13.2773 0.457837
\(842\) 43.1515 1.48710
\(843\) −6.80477 −0.234369
\(844\) 106.432 3.66354
\(845\) −0.859326 −0.0295617
\(846\) 1.12595 0.0387108
\(847\) −2.67178 −0.0918034
\(848\) −27.7916 −0.954366
\(849\) −73.1439 −2.51029
\(850\) 11.1284 0.381703
\(851\) 3.31228 0.113543
\(852\) −39.4670 −1.35212
\(853\) 22.5458 0.771955 0.385978 0.922508i \(-0.373864\pi\)
0.385978 + 0.922508i \(0.373864\pi\)
\(854\) −14.9317 −0.510951
\(855\) −10.5375 −0.360375
\(856\) −7.78170 −0.265973
\(857\) −33.1621 −1.13280 −0.566399 0.824131i \(-0.691664\pi\)
−0.566399 + 0.824131i \(0.691664\pi\)
\(858\) 20.6372 0.704543
\(859\) −12.7468 −0.434915 −0.217457 0.976070i \(-0.569776\pi\)
−0.217457 + 0.976070i \(0.569776\pi\)
\(860\) 28.0953 0.958042
\(861\) −33.4656 −1.14051
\(862\) −4.14396 −0.141144
\(863\) 55.2707 1.88144 0.940719 0.339188i \(-0.110152\pi\)
0.940719 + 0.339188i \(0.110152\pi\)
\(864\) −2.54354 −0.0865329
\(865\) −3.26504 −0.111015
\(866\) 36.0322 1.22443
\(867\) 6.61539 0.224671
\(868\) 33.4851 1.13656
\(869\) −13.4294 −0.455562
\(870\) −38.5108 −1.30564
\(871\) 17.7344 0.600906
\(872\) −92.6637 −3.13799
\(873\) −22.7242 −0.769097
\(874\) −56.1600 −1.89964
\(875\) −2.67178 −0.0903226
\(876\) 10.0792 0.340543
\(877\) −42.7001 −1.44188 −0.720940 0.692997i \(-0.756290\pi\)
−0.720940 + 0.692997i \(0.756290\pi\)
\(878\) 57.2599 1.93243
\(879\) 62.7267 2.11572
\(880\) 5.60558 0.188964
\(881\) 27.8217 0.937337 0.468669 0.883374i \(-0.344734\pi\)
0.468669 + 0.883374i \(0.344734\pi\)
\(882\) 0.902463 0.0303875
\(883\) 35.2179 1.18518 0.592588 0.805506i \(-0.298106\pi\)
0.592588 + 0.805506i \(0.298106\pi\)
\(884\) 65.9859 2.21935
\(885\) −10.3781 −0.348855
\(886\) 30.1119 1.01163
\(887\) −49.1090 −1.64892 −0.824459 0.565922i \(-0.808520\pi\)
−0.824459 + 0.565922i \(0.808520\pi\)
\(888\) 7.97037 0.267468
\(889\) −35.2397 −1.18190
\(890\) −36.9380 −1.23817
\(891\) −10.0249 −0.335846
\(892\) −92.9186 −3.11115
\(893\) −0.697993 −0.0233574
\(894\) −33.3622 −1.11580
\(895\) −16.6425 −0.556296
\(896\) 43.8504 1.46494
\(897\) 45.8266 1.53010
\(898\) 65.6639 2.19123
\(899\) 19.1443 0.638498
\(900\) 11.0964 0.369880
\(901\) −22.0576 −0.734844
\(902\) 13.2315 0.440560
\(903\) −41.7570 −1.38958
\(904\) 93.7542 3.11822
\(905\) −5.99670 −0.199337
\(906\) 140.642 4.67251
\(907\) 17.0414 0.565849 0.282924 0.959142i \(-0.408695\pi\)
0.282924 + 0.959142i \(0.408695\pi\)
\(908\) 115.711 3.84000
\(909\) 8.89037 0.294875
\(910\) −23.2858 −0.771918
\(911\) −46.5521 −1.54234 −0.771169 0.636630i \(-0.780328\pi\)
−0.771169 + 0.636630i \(0.780328\pi\)
\(912\) −53.6538 −1.77666
\(913\) 8.54660 0.282851
\(914\) −4.62917 −0.153119
\(915\) 5.29051 0.174899
\(916\) 42.8317 1.41520
\(917\) 20.5961 0.680145
\(918\) −10.3597 −0.341921
\(919\) −23.3080 −0.768860 −0.384430 0.923154i \(-0.625602\pi\)
−0.384430 + 0.923154i \(0.625602\pi\)
\(920\) 31.3520 1.03364
\(921\) −53.4161 −1.76012
\(922\) −91.6076 −3.01694
\(923\) −13.6436 −0.449086
\(924\) −26.9293 −0.885909
\(925\) 0.596335 0.0196074
\(926\) −39.0965 −1.28479
\(927\) −16.3079 −0.535621
\(928\) −17.7656 −0.583184
\(929\) 17.9792 0.589877 0.294939 0.955516i \(-0.404701\pi\)
0.294939 + 0.955516i \(0.404701\pi\)
\(930\) −17.4388 −0.571839
\(931\) −0.559452 −0.0183353
\(932\) −106.838 −3.49961
\(933\) −6.20416 −0.203115
\(934\) 39.6372 1.29697
\(935\) 4.44902 0.145499
\(936\) 51.2704 1.67583
\(937\) −0.0637686 −0.00208323 −0.00104161 0.999999i \(-0.500332\pi\)
−0.00104161 + 0.999999i \(0.500332\pi\)
\(938\) −34.0146 −1.11061
\(939\) 56.3699 1.83956
\(940\) 0.735014 0.0239735
\(941\) −22.4057 −0.730404 −0.365202 0.930928i \(-0.619000\pi\)
−0.365202 + 0.930928i \(0.619000\pi\)
\(942\) 20.1458 0.656385
\(943\) 29.3816 0.956796
\(944\) −24.5684 −0.799634
\(945\) 2.48721 0.0809090
\(946\) 16.5097 0.536776
\(947\) −44.7378 −1.45378 −0.726892 0.686752i \(-0.759036\pi\)
−0.726892 + 0.686752i \(0.759036\pi\)
\(948\) −135.357 −4.39620
\(949\) 3.48435 0.113107
\(950\) −10.1109 −0.328041
\(951\) −1.47393 −0.0477955
\(952\) −67.0955 −2.17458
\(953\) −6.96492 −0.225616 −0.112808 0.993617i \(-0.535984\pi\)
−0.112808 + 0.993617i \(0.535984\pi\)
\(954\) −32.3281 −1.04666
\(955\) −20.7548 −0.671610
\(956\) −54.5248 −1.76346
\(957\) −15.3962 −0.497688
\(958\) 83.4039 2.69466
\(959\) −38.7629 −1.25172
\(960\) −10.3638 −0.334490
\(961\) −22.3309 −0.720353
\(962\) 5.19735 0.167569
\(963\) −3.59387 −0.115811
\(964\) 56.7160 1.82670
\(965\) 10.8308 0.348656
\(966\) −87.8954 −2.82799
\(967\) 17.6356 0.567122 0.283561 0.958954i \(-0.408484\pi\)
0.283561 + 0.958954i \(0.408484\pi\)
\(968\) 5.64454 0.181422
\(969\) −42.5838 −1.36799
\(970\) −21.8042 −0.700092
\(971\) 27.1151 0.870166 0.435083 0.900390i \(-0.356719\pi\)
0.435083 + 0.900390i \(0.356719\pi\)
\(972\) −89.1546 −2.85964
\(973\) −20.0711 −0.643449
\(974\) −41.7900 −1.33904
\(975\) 8.25052 0.264228
\(976\) 12.5244 0.400897
\(977\) 20.3577 0.651300 0.325650 0.945490i \(-0.394417\pi\)
0.325650 + 0.945490i \(0.394417\pi\)
\(978\) −18.8557 −0.602939
\(979\) −14.7674 −0.471968
\(980\) 0.589125 0.0188189
\(981\) −42.7955 −1.36636
\(982\) −58.5758 −1.86923
\(983\) −17.3908 −0.554679 −0.277340 0.960772i \(-0.589453\pi\)
−0.277340 + 0.960772i \(0.589453\pi\)
\(984\) 70.7012 2.25387
\(985\) −7.69796 −0.245277
\(986\) −72.3582 −2.30436
\(987\) −1.09242 −0.0347722
\(988\) −59.9524 −1.90734
\(989\) 36.6611 1.16575
\(990\) 6.52059 0.207238
\(991\) −37.0023 −1.17542 −0.587709 0.809073i \(-0.699970\pi\)
−0.587709 + 0.809073i \(0.699970\pi\)
\(992\) −8.04474 −0.255421
\(993\) −11.4930 −0.364719
\(994\) 26.1685 0.830016
\(995\) −14.1572 −0.448814
\(996\) 86.1426 2.72953
\(997\) −37.1654 −1.17704 −0.588520 0.808483i \(-0.700289\pi\)
−0.588520 + 0.808483i \(0.700289\pi\)
\(998\) 81.9716 2.59477
\(999\) −0.555140 −0.0175639
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 4015.2.a.h.1.34 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4015.2.a.h.1.34 37 1.1 even 1 trivial