Properties

Label 8001.2.a.h.1.2
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(1\)
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: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56155 q^{2} +0.438447 q^{4} +0.561553 q^{5} -1.00000 q^{7} -2.43845 q^{8} +O(q^{10})\) \(q+1.56155 q^{2} +0.438447 q^{4} +0.561553 q^{5} -1.00000 q^{7} -2.43845 q^{8} +0.876894 q^{10} +5.12311 q^{11} +0.561553 q^{13} -1.56155 q^{14} -4.68466 q^{16} -4.00000 q^{17} -4.00000 q^{19} +0.246211 q^{20} +8.00000 q^{22} +4.56155 q^{23} -4.68466 q^{25} +0.876894 q^{26} -0.438447 q^{28} -6.56155 q^{29} +6.56155 q^{31} -2.43845 q^{32} -6.24621 q^{34} -0.561553 q^{35} -4.56155 q^{37} -6.24621 q^{38} -1.36932 q^{40} +1.12311 q^{41} -1.12311 q^{43} +2.24621 q^{44} +7.12311 q^{46} +10.0000 q^{47} +1.00000 q^{49} -7.31534 q^{50} +0.246211 q^{52} -3.68466 q^{53} +2.87689 q^{55} +2.43845 q^{56} -10.2462 q^{58} -3.68466 q^{59} -8.56155 q^{61} +10.2462 q^{62} +5.56155 q^{64} +0.315342 q^{65} -1.12311 q^{67} -1.75379 q^{68} -0.876894 q^{70} +12.0000 q^{71} -8.56155 q^{73} -7.12311 q^{74} -1.75379 q^{76} -5.12311 q^{77} -10.2462 q^{79} -2.63068 q^{80} +1.75379 q^{82} -7.68466 q^{83} -2.24621 q^{85} -1.75379 q^{86} -12.4924 q^{88} -17.6847 q^{89} -0.561553 q^{91} +2.00000 q^{92} +15.6155 q^{94} -2.24621 q^{95} -8.87689 q^{97} +1.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{4} - 3 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 5 q^{4} - 3 q^{5} - 2 q^{7} - 9 q^{8} + 10 q^{10} + 2 q^{11} - 3 q^{13} + q^{14} + 3 q^{16} - 8 q^{17} - 8 q^{19} - 16 q^{20} + 16 q^{22} + 5 q^{23} + 3 q^{25} + 10 q^{26} - 5 q^{28} - 9 q^{29} + 9 q^{31} - 9 q^{32} + 4 q^{34} + 3 q^{35} - 5 q^{37} + 4 q^{38} + 22 q^{40} - 6 q^{41} + 6 q^{43} - 12 q^{44} + 6 q^{46} + 20 q^{47} + 2 q^{49} - 27 q^{50} - 16 q^{52} + 5 q^{53} + 14 q^{55} + 9 q^{56} - 4 q^{58} + 5 q^{59} - 13 q^{61} + 4 q^{62} + 7 q^{64} + 13 q^{65} + 6 q^{67} - 20 q^{68} - 10 q^{70} + 24 q^{71} - 13 q^{73} - 6 q^{74} - 20 q^{76} - 2 q^{77} - 4 q^{79} - 30 q^{80} + 20 q^{82} - 3 q^{83} + 12 q^{85} - 20 q^{86} + 8 q^{88} - 23 q^{89} + 3 q^{91} + 4 q^{92} - 10 q^{94} + 12 q^{95} - 26 q^{97} - q^{98}+O(q^{100}) \) 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 1.10418 0.552092 0.833783i \(-0.313830\pi\)
0.552092 + 0.833783i \(0.313830\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.43845 −0.862121
\(9\) 0 0
\(10\) 0.876894 0.277298
\(11\) 5.12311 1.54467 0.772337 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 0 0
\(13\) 0.561553 0.155747 0.0778734 0.996963i \(-0.475187\pi\)
0.0778734 + 0.996963i \(0.475187\pi\)
\(14\) −1.56155 −0.417343
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0.246211 0.0550545
\(21\) 0 0
\(22\) 8.00000 1.70561
\(23\) 4.56155 0.951150 0.475575 0.879675i \(-0.342240\pi\)
0.475575 + 0.879675i \(0.342240\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) 0.876894 0.171973
\(27\) 0 0
\(28\) −0.438447 −0.0828587
\(29\) −6.56155 −1.21845 −0.609225 0.792998i \(-0.708519\pi\)
−0.609225 + 0.792998i \(0.708519\pi\)
\(30\) 0 0
\(31\) 6.56155 1.17849 0.589245 0.807955i \(-0.299425\pi\)
0.589245 + 0.807955i \(0.299425\pi\)
\(32\) −2.43845 −0.431061
\(33\) 0 0
\(34\) −6.24621 −1.07122
\(35\) −0.561553 −0.0949197
\(36\) 0 0
\(37\) −4.56155 −0.749915 −0.374957 0.927042i \(-0.622343\pi\)
−0.374957 + 0.927042i \(0.622343\pi\)
\(38\) −6.24621 −1.01327
\(39\) 0 0
\(40\) −1.36932 −0.216508
\(41\) 1.12311 0.175400 0.0876998 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) −1.12311 −0.171272 −0.0856360 0.996326i \(-0.527292\pi\)
−0.0856360 + 0.996326i \(0.527292\pi\)
\(44\) 2.24621 0.338629
\(45\) 0 0
\(46\) 7.12311 1.05024
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −7.31534 −1.03455
\(51\) 0 0
\(52\) 0.246211 0.0341434
\(53\) −3.68466 −0.506127 −0.253063 0.967450i \(-0.581438\pi\)
−0.253063 + 0.967450i \(0.581438\pi\)
\(54\) 0 0
\(55\) 2.87689 0.387920
\(56\) 2.43845 0.325851
\(57\) 0 0
\(58\) −10.2462 −1.34539
\(59\) −3.68466 −0.479702 −0.239851 0.970810i \(-0.577099\pi\)
−0.239851 + 0.970810i \(0.577099\pi\)
\(60\) 0 0
\(61\) −8.56155 −1.09619 −0.548097 0.836415i \(-0.684648\pi\)
−0.548097 + 0.836415i \(0.684648\pi\)
\(62\) 10.2462 1.30127
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) 0.315342 0.0391133
\(66\) 0 0
\(67\) −1.12311 −0.137209 −0.0686046 0.997644i \(-0.521855\pi\)
−0.0686046 + 0.997644i \(0.521855\pi\)
\(68\) −1.75379 −0.212678
\(69\) 0 0
\(70\) −0.876894 −0.104809
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −8.56155 −1.00205 −0.501027 0.865432i \(-0.667044\pi\)
−0.501027 + 0.865432i \(0.667044\pi\)
\(74\) −7.12311 −0.828044
\(75\) 0 0
\(76\) −1.75379 −0.201173
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −10.2462 −1.15279 −0.576394 0.817172i \(-0.695541\pi\)
−0.576394 + 0.817172i \(0.695541\pi\)
\(80\) −2.63068 −0.294119
\(81\) 0 0
\(82\) 1.75379 0.193674
\(83\) −7.68466 −0.843501 −0.421750 0.906712i \(-0.638584\pi\)
−0.421750 + 0.906712i \(0.638584\pi\)
\(84\) 0 0
\(85\) −2.24621 −0.243636
\(86\) −1.75379 −0.189116
\(87\) 0 0
\(88\) −12.4924 −1.33170
\(89\) −17.6847 −1.87457 −0.937285 0.348564i \(-0.886669\pi\)
−0.937285 + 0.348564i \(0.886669\pi\)
\(90\) 0 0
\(91\) −0.561553 −0.0588667
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 15.6155 1.61062
\(95\) −2.24621 −0.230456
\(96\) 0 0
\(97\) −8.87689 −0.901312 −0.450656 0.892698i \(-0.648810\pi\)
−0.450656 + 0.892698i \(0.648810\pi\)
\(98\) 1.56155 0.157741
\(99\) 0 0
\(100\) −2.05398 −0.205398
\(101\) −13.6847 −1.36167 −0.680837 0.732435i \(-0.738384\pi\)
−0.680837 + 0.732435i \(0.738384\pi\)
\(102\) 0 0
\(103\) −2.24621 −0.221326 −0.110663 0.993858i \(-0.535297\pi\)
−0.110663 + 0.993858i \(0.535297\pi\)
\(104\) −1.36932 −0.134273
\(105\) 0 0
\(106\) −5.75379 −0.558857
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −8.24621 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(110\) 4.49242 0.428336
\(111\) 0 0
\(112\) 4.68466 0.442659
\(113\) 5.36932 0.505103 0.252551 0.967583i \(-0.418730\pi\)
0.252551 + 0.967583i \(0.418730\pi\)
\(114\) 0 0
\(115\) 2.56155 0.238866
\(116\) −2.87689 −0.267113
\(117\) 0 0
\(118\) −5.75379 −0.529679
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) −13.3693 −1.21040
\(123\) 0 0
\(124\) 2.87689 0.258353
\(125\) −5.43845 −0.486430
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 13.5616 1.19868
\(129\) 0 0
\(130\) 0.492423 0.0431883
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −1.75379 −0.151504
\(135\) 0 0
\(136\) 9.75379 0.836380
\(137\) −10.5616 −0.902334 −0.451167 0.892439i \(-0.648992\pi\)
−0.451167 + 0.892439i \(0.648992\pi\)
\(138\) 0 0
\(139\) −6.87689 −0.583291 −0.291645 0.956527i \(-0.594203\pi\)
−0.291645 + 0.956527i \(0.594203\pi\)
\(140\) −0.246211 −0.0208086
\(141\) 0 0
\(142\) 18.7386 1.57251
\(143\) 2.87689 0.240578
\(144\) 0 0
\(145\) −3.68466 −0.305994
\(146\) −13.3693 −1.10645
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 0 0
\(151\) 11.3693 0.925222 0.462611 0.886561i \(-0.346913\pi\)
0.462611 + 0.886561i \(0.346913\pi\)
\(152\) 9.75379 0.791137
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 3.68466 0.295959
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −16.0000 −1.27289
\(159\) 0 0
\(160\) −1.36932 −0.108254
\(161\) −4.56155 −0.359501
\(162\) 0 0
\(163\) 13.4384 1.05258 0.526290 0.850305i \(-0.323583\pi\)
0.526290 + 0.850305i \(0.323583\pi\)
\(164\) 0.492423 0.0384517
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) −20.4924 −1.58575 −0.792876 0.609383i \(-0.791417\pi\)
−0.792876 + 0.609383i \(0.791417\pi\)
\(168\) 0 0
\(169\) −12.6847 −0.975743
\(170\) −3.50758 −0.269019
\(171\) 0 0
\(172\) −0.492423 −0.0375469
\(173\) −3.75379 −0.285395 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(174\) 0 0
\(175\) 4.68466 0.354127
\(176\) −24.0000 −1.80907
\(177\) 0 0
\(178\) −27.6155 −2.06987
\(179\) 1.75379 0.131084 0.0655422 0.997850i \(-0.479122\pi\)
0.0655422 + 0.997850i \(0.479122\pi\)
\(180\) 0 0
\(181\) 8.87689 0.659814 0.329907 0.944013i \(-0.392983\pi\)
0.329907 + 0.944013i \(0.392983\pi\)
\(182\) −0.876894 −0.0649997
\(183\) 0 0
\(184\) −11.1231 −0.820006
\(185\) −2.56155 −0.188329
\(186\) 0 0
\(187\) −20.4924 −1.49855
\(188\) 4.38447 0.319770
\(189\) 0 0
\(190\) −3.50758 −0.254466
\(191\) 1.75379 0.126900 0.0634499 0.997985i \(-0.479790\pi\)
0.0634499 + 0.997985i \(0.479790\pi\)
\(192\) 0 0
\(193\) 20.2462 1.45735 0.728677 0.684857i \(-0.240136\pi\)
0.728677 + 0.684857i \(0.240136\pi\)
\(194\) −13.8617 −0.995215
\(195\) 0 0
\(196\) 0.438447 0.0313177
\(197\) −0.876894 −0.0624761 −0.0312381 0.999512i \(-0.509945\pi\)
−0.0312381 + 0.999512i \(0.509945\pi\)
\(198\) 0 0
\(199\) 2.56155 0.181584 0.0907918 0.995870i \(-0.471060\pi\)
0.0907918 + 0.995870i \(0.471060\pi\)
\(200\) 11.4233 0.807749
\(201\) 0 0
\(202\) −21.3693 −1.50354
\(203\) 6.56155 0.460531
\(204\) 0 0
\(205\) 0.630683 0.0440488
\(206\) −3.50758 −0.244385
\(207\) 0 0
\(208\) −2.63068 −0.182405
\(209\) −20.4924 −1.41749
\(210\) 0 0
\(211\) 7.68466 0.529034 0.264517 0.964381i \(-0.414788\pi\)
0.264517 + 0.964381i \(0.414788\pi\)
\(212\) −1.61553 −0.110955
\(213\) 0 0
\(214\) −6.24621 −0.426982
\(215\) −0.630683 −0.0430122
\(216\) 0 0
\(217\) −6.56155 −0.445427
\(218\) −12.8769 −0.872133
\(219\) 0 0
\(220\) 1.26137 0.0850413
\(221\) −2.24621 −0.151097
\(222\) 0 0
\(223\) 7.36932 0.493486 0.246743 0.969081i \(-0.420640\pi\)
0.246743 + 0.969081i \(0.420640\pi\)
\(224\) 2.43845 0.162926
\(225\) 0 0
\(226\) 8.38447 0.557727
\(227\) 0.246211 0.0163416 0.00817081 0.999967i \(-0.497399\pi\)
0.00817081 + 0.999967i \(0.497399\pi\)
\(228\) 0 0
\(229\) 17.3693 1.14780 0.573898 0.818927i \(-0.305430\pi\)
0.573898 + 0.818927i \(0.305430\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 16.0000 1.05045
\(233\) −13.9309 −0.912642 −0.456321 0.889815i \(-0.650833\pi\)
−0.456321 + 0.889815i \(0.650833\pi\)
\(234\) 0 0
\(235\) 5.61553 0.366317
\(236\) −1.61553 −0.105162
\(237\) 0 0
\(238\) 6.24621 0.404882
\(239\) −29.6847 −1.92014 −0.960070 0.279758i \(-0.909746\pi\)
−0.960070 + 0.279758i \(0.909746\pi\)
\(240\) 0 0
\(241\) −20.8769 −1.34480 −0.672399 0.740188i \(-0.734736\pi\)
−0.672399 + 0.740188i \(0.734736\pi\)
\(242\) 23.8078 1.53042
\(243\) 0 0
\(244\) −3.75379 −0.240312
\(245\) 0.561553 0.0358763
\(246\) 0 0
\(247\) −2.24621 −0.142923
\(248\) −16.0000 −1.01600
\(249\) 0 0
\(250\) −8.49242 −0.537108
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) 23.3693 1.46922
\(254\) 1.56155 0.0979805
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) −10.8078 −0.674170 −0.337085 0.941474i \(-0.609441\pi\)
−0.337085 + 0.941474i \(0.609441\pi\)
\(258\) 0 0
\(259\) 4.56155 0.283441
\(260\) 0.138261 0.00857456
\(261\) 0 0
\(262\) 28.1080 1.73651
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −2.06913 −0.127106
\(266\) 6.24621 0.382980
\(267\) 0 0
\(268\) −0.492423 −0.0300795
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −8.31534 −0.505121 −0.252561 0.967581i \(-0.581273\pi\)
−0.252561 + 0.967581i \(0.581273\pi\)
\(272\) 18.7386 1.13620
\(273\) 0 0
\(274\) −16.4924 −0.996344
\(275\) −24.0000 −1.44725
\(276\) 0 0
\(277\) −13.3693 −0.803284 −0.401642 0.915797i \(-0.631560\pi\)
−0.401642 + 0.915797i \(0.631560\pi\)
\(278\) −10.7386 −0.644060
\(279\) 0 0
\(280\) 1.36932 0.0818323
\(281\) −15.3693 −0.916857 −0.458428 0.888731i \(-0.651587\pi\)
−0.458428 + 0.888731i \(0.651587\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 5.26137 0.312205
\(285\) 0 0
\(286\) 4.49242 0.265643
\(287\) −1.12311 −0.0662948
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −5.75379 −0.337874
\(291\) 0 0
\(292\) −3.75379 −0.219674
\(293\) −0.246211 −0.0143838 −0.00719191 0.999974i \(-0.502289\pi\)
−0.00719191 + 0.999974i \(0.502289\pi\)
\(294\) 0 0
\(295\) −2.06913 −0.120469
\(296\) 11.1231 0.646517
\(297\) 0 0
\(298\) −19.1231 −1.10777
\(299\) 2.56155 0.148138
\(300\) 0 0
\(301\) 1.12311 0.0647347
\(302\) 17.7538 1.02162
\(303\) 0 0
\(304\) 18.7386 1.07473
\(305\) −4.80776 −0.275292
\(306\) 0 0
\(307\) 9.12311 0.520683 0.260342 0.965517i \(-0.416165\pi\)
0.260342 + 0.965517i \(0.416165\pi\)
\(308\) −2.24621 −0.127990
\(309\) 0 0
\(310\) 5.75379 0.326793
\(311\) 6.56155 0.372072 0.186036 0.982543i \(-0.440436\pi\)
0.186036 + 0.982543i \(0.440436\pi\)
\(312\) 0 0
\(313\) −4.24621 −0.240010 −0.120005 0.992773i \(-0.538291\pi\)
−0.120005 + 0.992773i \(0.538291\pi\)
\(314\) 21.8617 1.23373
\(315\) 0 0
\(316\) −4.49242 −0.252719
\(317\) −6.87689 −0.386245 −0.193122 0.981175i \(-0.561861\pi\)
−0.193122 + 0.981175i \(0.561861\pi\)
\(318\) 0 0
\(319\) −33.6155 −1.88211
\(320\) 3.12311 0.174587
\(321\) 0 0
\(322\) −7.12311 −0.396955
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) −2.63068 −0.145924
\(326\) 20.9848 1.16224
\(327\) 0 0
\(328\) −2.73863 −0.151216
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) 30.7386 1.68955 0.844774 0.535123i \(-0.179735\pi\)
0.844774 + 0.535123i \(0.179735\pi\)
\(332\) −3.36932 −0.184915
\(333\) 0 0
\(334\) −32.0000 −1.75096
\(335\) −0.630683 −0.0344579
\(336\) 0 0
\(337\) 6.63068 0.361196 0.180598 0.983557i \(-0.442197\pi\)
0.180598 + 0.983557i \(0.442197\pi\)
\(338\) −19.8078 −1.07740
\(339\) 0 0
\(340\) −0.984845 −0.0534107
\(341\) 33.6155 1.82038
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 2.73863 0.147657
\(345\) 0 0
\(346\) −5.86174 −0.315129
\(347\) −11.9309 −0.640483 −0.320241 0.947336i \(-0.603764\pi\)
−0.320241 + 0.947336i \(0.603764\pi\)
\(348\) 0 0
\(349\) −11.7538 −0.629166 −0.314583 0.949230i \(-0.601865\pi\)
−0.314583 + 0.949230i \(0.601865\pi\)
\(350\) 7.31534 0.391021
\(351\) 0 0
\(352\) −12.4924 −0.665848
\(353\) −14.2462 −0.758249 −0.379125 0.925346i \(-0.623775\pi\)
−0.379125 + 0.925346i \(0.623775\pi\)
\(354\) 0 0
\(355\) 6.73863 0.357650
\(356\) −7.75379 −0.410950
\(357\) 0 0
\(358\) 2.73863 0.144741
\(359\) 11.1231 0.587055 0.293528 0.955951i \(-0.405171\pi\)
0.293528 + 0.955951i \(0.405171\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 13.8617 0.728557
\(363\) 0 0
\(364\) −0.246211 −0.0129050
\(365\) −4.80776 −0.251650
\(366\) 0 0
\(367\) −5.93087 −0.309589 −0.154794 0.987947i \(-0.549472\pi\)
−0.154794 + 0.987947i \(0.549472\pi\)
\(368\) −21.3693 −1.11395
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) 3.68466 0.191298
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −32.0000 −1.65468
\(375\) 0 0
\(376\) −24.3845 −1.25753
\(377\) −3.68466 −0.189770
\(378\) 0 0
\(379\) 25.1231 1.29049 0.645244 0.763977i \(-0.276756\pi\)
0.645244 + 0.763977i \(0.276756\pi\)
\(380\) −0.984845 −0.0505215
\(381\) 0 0
\(382\) 2.73863 0.140121
\(383\) 2.63068 0.134422 0.0672108 0.997739i \(-0.478590\pi\)
0.0672108 + 0.997739i \(0.478590\pi\)
\(384\) 0 0
\(385\) −2.87689 −0.146620
\(386\) 31.6155 1.60919
\(387\) 0 0
\(388\) −3.89205 −0.197589
\(389\) 0.876894 0.0444603 0.0222302 0.999753i \(-0.492923\pi\)
0.0222302 + 0.999753i \(0.492923\pi\)
\(390\) 0 0
\(391\) −18.2462 −0.922751
\(392\) −2.43845 −0.123160
\(393\) 0 0
\(394\) −1.36932 −0.0689852
\(395\) −5.75379 −0.289505
\(396\) 0 0
\(397\) 30.8078 1.54620 0.773099 0.634286i \(-0.218706\pi\)
0.773099 + 0.634286i \(0.218706\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 21.9460 1.09730
\(401\) −1.43845 −0.0718326 −0.0359163 0.999355i \(-0.511435\pi\)
−0.0359163 + 0.999355i \(0.511435\pi\)
\(402\) 0 0
\(403\) 3.68466 0.183546
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 10.2462 0.508511
\(407\) −23.3693 −1.15837
\(408\) 0 0
\(409\) −8.24621 −0.407749 −0.203874 0.978997i \(-0.565353\pi\)
−0.203874 + 0.978997i \(0.565353\pi\)
\(410\) 0.984845 0.0486380
\(411\) 0 0
\(412\) −0.984845 −0.0485198
\(413\) 3.68466 0.181310
\(414\) 0 0
\(415\) −4.31534 −0.211832
\(416\) −1.36932 −0.0671363
\(417\) 0 0
\(418\) −32.0000 −1.56517
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) −3.12311 −0.152211 −0.0761054 0.997100i \(-0.524249\pi\)
−0.0761054 + 0.997100i \(0.524249\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 8.98485 0.436343
\(425\) 18.7386 0.908957
\(426\) 0 0
\(427\) 8.56155 0.414323
\(428\) −1.75379 −0.0847726
\(429\) 0 0
\(430\) −0.984845 −0.0474934
\(431\) −0.630683 −0.0303789 −0.0151895 0.999885i \(-0.504835\pi\)
−0.0151895 + 0.999885i \(0.504835\pi\)
\(432\) 0 0
\(433\) −24.2462 −1.16520 −0.582599 0.812760i \(-0.697964\pi\)
−0.582599 + 0.812760i \(0.697964\pi\)
\(434\) −10.2462 −0.491834
\(435\) 0 0
\(436\) −3.61553 −0.173152
\(437\) −18.2462 −0.872835
\(438\) 0 0
\(439\) −38.1080 −1.81879 −0.909397 0.415930i \(-0.863456\pi\)
−0.909397 + 0.415930i \(0.863456\pi\)
\(440\) −7.01515 −0.334434
\(441\) 0 0
\(442\) −3.50758 −0.166838
\(443\) −26.7386 −1.27039 −0.635195 0.772351i \(-0.719080\pi\)
−0.635195 + 0.772351i \(0.719080\pi\)
\(444\) 0 0
\(445\) −9.93087 −0.470768
\(446\) 11.5076 0.544900
\(447\) 0 0
\(448\) −5.56155 −0.262759
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 5.75379 0.270935
\(452\) 2.35416 0.110730
\(453\) 0 0
\(454\) 0.384472 0.0180442
\(455\) −0.315342 −0.0147834
\(456\) 0 0
\(457\) 6.80776 0.318454 0.159227 0.987242i \(-0.449100\pi\)
0.159227 + 0.987242i \(0.449100\pi\)
\(458\) 27.1231 1.26738
\(459\) 0 0
\(460\) 1.12311 0.0523651
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 16.8078 0.781123 0.390562 0.920577i \(-0.372281\pi\)
0.390562 + 0.920577i \(0.372281\pi\)
\(464\) 30.7386 1.42701
\(465\) 0 0
\(466\) −21.7538 −1.00772
\(467\) −35.5464 −1.64489 −0.822446 0.568844i \(-0.807391\pi\)
−0.822446 + 0.568844i \(0.807391\pi\)
\(468\) 0 0
\(469\) 1.12311 0.0518602
\(470\) 8.76894 0.404481
\(471\) 0 0
\(472\) 8.98485 0.413561
\(473\) −5.75379 −0.264559
\(474\) 0 0
\(475\) 18.7386 0.859787
\(476\) 1.75379 0.0803848
\(477\) 0 0
\(478\) −46.3542 −2.12019
\(479\) 22.4924 1.02771 0.513853 0.857879i \(-0.328218\pi\)
0.513853 + 0.857879i \(0.328218\pi\)
\(480\) 0 0
\(481\) −2.56155 −0.116797
\(482\) −32.6004 −1.48491
\(483\) 0 0
\(484\) 6.68466 0.303848
\(485\) −4.98485 −0.226350
\(486\) 0 0
\(487\) −22.7386 −1.03039 −0.515193 0.857074i \(-0.672280\pi\)
−0.515193 + 0.857074i \(0.672280\pi\)
\(488\) 20.8769 0.945053
\(489\) 0 0
\(490\) 0.876894 0.0396140
\(491\) −17.8617 −0.806089 −0.403045 0.915180i \(-0.632048\pi\)
−0.403045 + 0.915180i \(0.632048\pi\)
\(492\) 0 0
\(493\) 26.2462 1.18207
\(494\) −3.50758 −0.157813
\(495\) 0 0
\(496\) −30.7386 −1.38021
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) −2.38447 −0.106637
\(501\) 0 0
\(502\) −15.6155 −0.696955
\(503\) 19.1231 0.852657 0.426329 0.904568i \(-0.359807\pi\)
0.426329 + 0.904568i \(0.359807\pi\)
\(504\) 0 0
\(505\) −7.68466 −0.341963
\(506\) 36.4924 1.62229
\(507\) 0 0
\(508\) 0.438447 0.0194529
\(509\) 14.7386 0.653278 0.326639 0.945149i \(-0.394084\pi\)
0.326639 + 0.945149i \(0.394084\pi\)
\(510\) 0 0
\(511\) 8.56155 0.378741
\(512\) −11.4233 −0.504843
\(513\) 0 0
\(514\) −16.8769 −0.744408
\(515\) −1.26137 −0.0555824
\(516\) 0 0
\(517\) 51.2311 2.25314
\(518\) 7.12311 0.312971
\(519\) 0 0
\(520\) −0.768944 −0.0337204
\(521\) −10.7386 −0.470468 −0.235234 0.971939i \(-0.575586\pi\)
−0.235234 + 0.971939i \(0.575586\pi\)
\(522\) 0 0
\(523\) −26.4233 −1.15541 −0.577705 0.816246i \(-0.696052\pi\)
−0.577705 + 0.816246i \(0.696052\pi\)
\(524\) 7.89205 0.344766
\(525\) 0 0
\(526\) −12.4924 −0.544696
\(527\) −26.2462 −1.14330
\(528\) 0 0
\(529\) −2.19224 −0.0953146
\(530\) −3.23106 −0.140348
\(531\) 0 0
\(532\) 1.75379 0.0760364
\(533\) 0.630683 0.0273179
\(534\) 0 0
\(535\) −2.24621 −0.0971122
\(536\) 2.73863 0.118291
\(537\) 0 0
\(538\) −6.24621 −0.269293
\(539\) 5.12311 0.220668
\(540\) 0 0
\(541\) −39.6155 −1.70320 −0.851602 0.524188i \(-0.824369\pi\)
−0.851602 + 0.524188i \(0.824369\pi\)
\(542\) −12.9848 −0.557747
\(543\) 0 0
\(544\) 9.75379 0.418190
\(545\) −4.63068 −0.198357
\(546\) 0 0
\(547\) 24.4924 1.04722 0.523610 0.851958i \(-0.324585\pi\)
0.523610 + 0.851958i \(0.324585\pi\)
\(548\) −4.63068 −0.197813
\(549\) 0 0
\(550\) −37.4773 −1.59804
\(551\) 26.2462 1.11813
\(552\) 0 0
\(553\) 10.2462 0.435713
\(554\) −20.8769 −0.886974
\(555\) 0 0
\(556\) −3.01515 −0.127871
\(557\) 10.4924 0.444578 0.222289 0.974981i \(-0.428647\pi\)
0.222289 + 0.974981i \(0.428647\pi\)
\(558\) 0 0
\(559\) −0.630683 −0.0266751
\(560\) 2.63068 0.111167
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) 3.01515 0.126849
\(566\) −18.7386 −0.787644
\(567\) 0 0
\(568\) −29.2614 −1.22778
\(569\) 46.3542 1.94327 0.971634 0.236491i \(-0.0759972\pi\)
0.971634 + 0.236491i \(0.0759972\pi\)
\(570\) 0 0
\(571\) 17.6155 0.737187 0.368593 0.929591i \(-0.379839\pi\)
0.368593 + 0.929591i \(0.379839\pi\)
\(572\) 1.26137 0.0527404
\(573\) 0 0
\(574\) −1.75379 −0.0732017
\(575\) −21.3693 −0.891162
\(576\) 0 0
\(577\) 14.3153 0.595955 0.297978 0.954573i \(-0.403688\pi\)
0.297978 + 0.954573i \(0.403688\pi\)
\(578\) −1.56155 −0.0649520
\(579\) 0 0
\(580\) −1.61553 −0.0670812
\(581\) 7.68466 0.318813
\(582\) 0 0
\(583\) −18.8769 −0.781801
\(584\) 20.8769 0.863892
\(585\) 0 0
\(586\) −0.384472 −0.0158824
\(587\) −8.24621 −0.340358 −0.170179 0.985413i \(-0.554435\pi\)
−0.170179 + 0.985413i \(0.554435\pi\)
\(588\) 0 0
\(589\) −26.2462 −1.08146
\(590\) −3.23106 −0.133020
\(591\) 0 0
\(592\) 21.3693 0.878274
\(593\) −35.3002 −1.44960 −0.724802 0.688957i \(-0.758069\pi\)
−0.724802 + 0.688957i \(0.758069\pi\)
\(594\) 0 0
\(595\) 2.24621 0.0920857
\(596\) −5.36932 −0.219936
\(597\) 0 0
\(598\) 4.00000 0.163572
\(599\) 2.17708 0.0889531 0.0444766 0.999010i \(-0.485838\pi\)
0.0444766 + 0.999010i \(0.485838\pi\)
\(600\) 0 0
\(601\) −11.7538 −0.479447 −0.239724 0.970841i \(-0.577057\pi\)
−0.239724 + 0.970841i \(0.577057\pi\)
\(602\) 1.75379 0.0714791
\(603\) 0 0
\(604\) 4.98485 0.202830
\(605\) 8.56155 0.348077
\(606\) 0 0
\(607\) 10.5616 0.428680 0.214340 0.976759i \(-0.431240\pi\)
0.214340 + 0.976759i \(0.431240\pi\)
\(608\) 9.75379 0.395568
\(609\) 0 0
\(610\) −7.50758 −0.303973
\(611\) 5.61553 0.227180
\(612\) 0 0
\(613\) 21.3693 0.863099 0.431549 0.902089i \(-0.357967\pi\)
0.431549 + 0.902089i \(0.357967\pi\)
\(614\) 14.2462 0.574930
\(615\) 0 0
\(616\) 12.4924 0.503334
\(617\) 25.9309 1.04394 0.521969 0.852965i \(-0.325198\pi\)
0.521969 + 0.852965i \(0.325198\pi\)
\(618\) 0 0
\(619\) 34.1080 1.37091 0.685457 0.728113i \(-0.259602\pi\)
0.685457 + 0.728113i \(0.259602\pi\)
\(620\) 1.61553 0.0648812
\(621\) 0 0
\(622\) 10.2462 0.410836
\(623\) 17.6847 0.708521
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −6.63068 −0.265015
\(627\) 0 0
\(628\) 6.13826 0.244943
\(629\) 18.2462 0.727524
\(630\) 0 0
\(631\) 10.8769 0.433002 0.216501 0.976282i \(-0.430535\pi\)
0.216501 + 0.976282i \(0.430535\pi\)
\(632\) 24.9848 0.993844
\(633\) 0 0
\(634\) −10.7386 −0.426486
\(635\) 0.561553 0.0222845
\(636\) 0 0
\(637\) 0.561553 0.0222495
\(638\) −52.4924 −2.07819
\(639\) 0 0
\(640\) 7.61553 0.301030
\(641\) 38.8769 1.53555 0.767773 0.640723i \(-0.221365\pi\)
0.767773 + 0.640723i \(0.221365\pi\)
\(642\) 0 0
\(643\) −22.2462 −0.877305 −0.438652 0.898657i \(-0.644544\pi\)
−0.438652 + 0.898657i \(0.644544\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 24.9848 0.983016
\(647\) 35.5464 1.39747 0.698737 0.715379i \(-0.253746\pi\)
0.698737 + 0.715379i \(0.253746\pi\)
\(648\) 0 0
\(649\) −18.8769 −0.740983
\(650\) −4.10795 −0.161127
\(651\) 0 0
\(652\) 5.89205 0.230750
\(653\) 4.24621 0.166167 0.0830835 0.996543i \(-0.473523\pi\)
0.0830835 + 0.996543i \(0.473523\pi\)
\(654\) 0 0
\(655\) 10.1080 0.394950
\(656\) −5.26137 −0.205422
\(657\) 0 0
\(658\) −15.6155 −0.608757
\(659\) 50.1771 1.95462 0.977311 0.211810i \(-0.0679359\pi\)
0.977311 + 0.211810i \(0.0679359\pi\)
\(660\) 0 0
\(661\) −16.5616 −0.644170 −0.322085 0.946711i \(-0.604384\pi\)
−0.322085 + 0.946711i \(0.604384\pi\)
\(662\) 48.0000 1.86557
\(663\) 0 0
\(664\) 18.7386 0.727200
\(665\) 2.24621 0.0871043
\(666\) 0 0
\(667\) −29.9309 −1.15893
\(668\) −8.98485 −0.347634
\(669\) 0 0
\(670\) −0.984845 −0.0380479
\(671\) −43.8617 −1.69326
\(672\) 0 0
\(673\) 8.56155 0.330024 0.165012 0.986292i \(-0.447234\pi\)
0.165012 + 0.986292i \(0.447234\pi\)
\(674\) 10.3542 0.398827
\(675\) 0 0
\(676\) −5.56155 −0.213906
\(677\) 20.9848 0.806513 0.403257 0.915087i \(-0.367878\pi\)
0.403257 + 0.915087i \(0.367878\pi\)
\(678\) 0 0
\(679\) 8.87689 0.340664
\(680\) 5.47727 0.210044
\(681\) 0 0
\(682\) 52.4924 2.01004
\(683\) 8.38447 0.320823 0.160411 0.987050i \(-0.448718\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(684\) 0 0
\(685\) −5.93087 −0.226607
\(686\) −1.56155 −0.0596204
\(687\) 0 0
\(688\) 5.26137 0.200588
\(689\) −2.06913 −0.0788276
\(690\) 0 0
\(691\) 25.7538 0.979720 0.489860 0.871801i \(-0.337048\pi\)
0.489860 + 0.871801i \(0.337048\pi\)
\(692\) −1.64584 −0.0625654
\(693\) 0 0
\(694\) −18.6307 −0.707211
\(695\) −3.86174 −0.146484
\(696\) 0 0
\(697\) −4.49242 −0.170163
\(698\) −18.3542 −0.694715
\(699\) 0 0
\(700\) 2.05398 0.0776330
\(701\) −3.05398 −0.115347 −0.0576735 0.998335i \(-0.518368\pi\)
−0.0576735 + 0.998335i \(0.518368\pi\)
\(702\) 0 0
\(703\) 18.2462 0.688169
\(704\) 28.4924 1.07385
\(705\) 0 0
\(706\) −22.2462 −0.837247
\(707\) 13.6847 0.514665
\(708\) 0 0
\(709\) 32.2462 1.21103 0.605516 0.795833i \(-0.292967\pi\)
0.605516 + 0.795833i \(0.292967\pi\)
\(710\) 10.5227 0.394911
\(711\) 0 0
\(712\) 43.1231 1.61611
\(713\) 29.9309 1.12092
\(714\) 0 0
\(715\) 1.61553 0.0604173
\(716\) 0.768944 0.0287368
\(717\) 0 0
\(718\) 17.3693 0.648217
\(719\) 46.3542 1.72872 0.864359 0.502875i \(-0.167724\pi\)
0.864359 + 0.502875i \(0.167724\pi\)
\(720\) 0 0
\(721\) 2.24621 0.0836533
\(722\) −4.68466 −0.174345
\(723\) 0 0
\(724\) 3.89205 0.144647
\(725\) 30.7386 1.14160
\(726\) 0 0
\(727\) 7.36932 0.273313 0.136656 0.990619i \(-0.456364\pi\)
0.136656 + 0.990619i \(0.456364\pi\)
\(728\) 1.36932 0.0507503
\(729\) 0 0
\(730\) −7.50758 −0.277868
\(731\) 4.49242 0.166158
\(732\) 0 0
\(733\) −16.4233 −0.606608 −0.303304 0.952894i \(-0.598090\pi\)
−0.303304 + 0.952894i \(0.598090\pi\)
\(734\) −9.26137 −0.341843
\(735\) 0 0
\(736\) −11.1231 −0.410003
\(737\) −5.75379 −0.211944
\(738\) 0 0
\(739\) −19.1922 −0.705998 −0.352999 0.935624i \(-0.614838\pi\)
−0.352999 + 0.935624i \(0.614838\pi\)
\(740\) −1.12311 −0.0412862
\(741\) 0 0
\(742\) 5.75379 0.211228
\(743\) 26.1771 0.960344 0.480172 0.877174i \(-0.340574\pi\)
0.480172 + 0.877174i \(0.340574\pi\)
\(744\) 0 0
\(745\) −6.87689 −0.251950
\(746\) 21.8617 0.800415
\(747\) 0 0
\(748\) −8.98485 −0.328518
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −46.8466 −1.70832
\(753\) 0 0
\(754\) −5.75379 −0.209541
\(755\) 6.38447 0.232355
\(756\) 0 0
\(757\) 23.4384 0.851885 0.425942 0.904750i \(-0.359943\pi\)
0.425942 + 0.904750i \(0.359943\pi\)
\(758\) 39.2311 1.42494
\(759\) 0 0
\(760\) 5.47727 0.198681
\(761\) 48.4233 1.75534 0.877671 0.479263i \(-0.159096\pi\)
0.877671 + 0.479263i \(0.159096\pi\)
\(762\) 0 0
\(763\) 8.24621 0.298533
\(764\) 0.768944 0.0278194
\(765\) 0 0
\(766\) 4.10795 0.148426
\(767\) −2.06913 −0.0747120
\(768\) 0 0
\(769\) −22.4924 −0.811098 −0.405549 0.914073i \(-0.632920\pi\)
−0.405549 + 0.914073i \(0.632920\pi\)
\(770\) −4.49242 −0.161896
\(771\) 0 0
\(772\) 8.87689 0.319486
\(773\) 22.1080 0.795168 0.397584 0.917566i \(-0.369849\pi\)
0.397584 + 0.917566i \(0.369849\pi\)
\(774\) 0 0
\(775\) −30.7386 −1.10416
\(776\) 21.6458 0.777040
\(777\) 0 0
\(778\) 1.36932 0.0490924
\(779\) −4.49242 −0.160958
\(780\) 0 0
\(781\) 61.4773 2.19983
\(782\) −28.4924 −1.01889
\(783\) 0 0
\(784\) −4.68466 −0.167309
\(785\) 7.86174 0.280598
\(786\) 0 0
\(787\) 4.94602 0.176307 0.0881534 0.996107i \(-0.471903\pi\)
0.0881534 + 0.996107i \(0.471903\pi\)
\(788\) −0.384472 −0.0136962
\(789\) 0 0
\(790\) −8.98485 −0.319666
\(791\) −5.36932 −0.190911
\(792\) 0 0
\(793\) −4.80776 −0.170729
\(794\) 48.1080 1.70729
\(795\) 0 0
\(796\) 1.12311 0.0398074
\(797\) 34.1080 1.20817 0.604083 0.796922i \(-0.293540\pi\)
0.604083 + 0.796922i \(0.293540\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 11.4233 0.403874
\(801\) 0 0
\(802\) −2.24621 −0.0793165
\(803\) −43.8617 −1.54785
\(804\) 0 0
\(805\) −2.56155 −0.0902829
\(806\) 5.75379 0.202669
\(807\) 0 0
\(808\) 33.3693 1.17393
\(809\) 56.1080 1.97265 0.986325 0.164811i \(-0.0527013\pi\)
0.986325 + 0.164811i \(0.0527013\pi\)
\(810\) 0 0
\(811\) −31.0540 −1.09045 −0.545226 0.838289i \(-0.683556\pi\)
−0.545226 + 0.838289i \(0.683556\pi\)
\(812\) 2.87689 0.100959
\(813\) 0 0
\(814\) −36.4924 −1.27906
\(815\) 7.54640 0.264339
\(816\) 0 0
\(817\) 4.49242 0.157170
\(818\) −12.8769 −0.450230
\(819\) 0 0
\(820\) 0.276521 0.00965654
\(821\) −50.4233 −1.75979 −0.879893 0.475173i \(-0.842386\pi\)
−0.879893 + 0.475173i \(0.842386\pi\)
\(822\) 0 0
\(823\) 24.9848 0.870917 0.435458 0.900209i \(-0.356586\pi\)
0.435458 + 0.900209i \(0.356586\pi\)
\(824\) 5.47727 0.190810
\(825\) 0 0
\(826\) 5.75379 0.200200
\(827\) −39.3002 −1.36660 −0.683301 0.730137i \(-0.739456\pi\)
−0.683301 + 0.730137i \(0.739456\pi\)
\(828\) 0 0
\(829\) −22.4924 −0.781194 −0.390597 0.920562i \(-0.627731\pi\)
−0.390597 + 0.920562i \(0.627731\pi\)
\(830\) −6.73863 −0.233901
\(831\) 0 0
\(832\) 3.12311 0.108274
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) −11.5076 −0.398236
\(836\) −8.98485 −0.310747
\(837\) 0 0
\(838\) −21.8617 −0.755201
\(839\) 23.5464 0.812912 0.406456 0.913670i \(-0.366764\pi\)
0.406456 + 0.913670i \(0.366764\pi\)
\(840\) 0 0
\(841\) 14.0540 0.484620
\(842\) −4.87689 −0.168069
\(843\) 0 0
\(844\) 3.36932 0.115977
\(845\) −7.12311 −0.245042
\(846\) 0 0
\(847\) −15.2462 −0.523866
\(848\) 17.2614 0.592758
\(849\) 0 0
\(850\) 29.2614 1.00366
\(851\) −20.8078 −0.713281
\(852\) 0 0
\(853\) 37.2311 1.27477 0.637384 0.770547i \(-0.280017\pi\)
0.637384 + 0.770547i \(0.280017\pi\)
\(854\) 13.3693 0.457489
\(855\) 0 0
\(856\) 9.75379 0.333378
\(857\) −29.0540 −0.992465 −0.496232 0.868190i \(-0.665284\pi\)
−0.496232 + 0.868190i \(0.665284\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −0.276521 −0.00942930
\(861\) 0 0
\(862\) −0.984845 −0.0335440
\(863\) 9.68466 0.329670 0.164835 0.986321i \(-0.447291\pi\)
0.164835 + 0.986321i \(0.447291\pi\)
\(864\) 0 0
\(865\) −2.10795 −0.0716725
\(866\) −37.8617 −1.28659
\(867\) 0 0
\(868\) −2.87689 −0.0976482
\(869\) −52.4924 −1.78068
\(870\) 0 0
\(871\) −0.630683 −0.0213699
\(872\) 20.1080 0.680941
\(873\) 0 0
\(874\) −28.4924 −0.963771
\(875\) 5.43845 0.183853
\(876\) 0 0
\(877\) 49.2311 1.66241 0.831207 0.555963i \(-0.187650\pi\)
0.831207 + 0.555963i \(0.187650\pi\)
\(878\) −59.5076 −2.00828
\(879\) 0 0
\(880\) −13.4773 −0.454319
\(881\) 20.4233 0.688078 0.344039 0.938955i \(-0.388205\pi\)
0.344039 + 0.938955i \(0.388205\pi\)
\(882\) 0 0
\(883\) −32.6695 −1.09942 −0.549708 0.835357i \(-0.685261\pi\)
−0.549708 + 0.835357i \(0.685261\pi\)
\(884\) −0.984845 −0.0331239
\(885\) 0 0
\(886\) −41.7538 −1.40275
\(887\) −45.9309 −1.54221 −0.771104 0.636709i \(-0.780295\pi\)
−0.771104 + 0.636709i \(0.780295\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) −15.5076 −0.519815
\(891\) 0 0
\(892\) 3.23106 0.108184
\(893\) −40.0000 −1.33855
\(894\) 0 0
\(895\) 0.984845 0.0329197
\(896\) −13.5616 −0.453060
\(897\) 0 0
\(898\) −9.36932 −0.312658
\(899\) −43.0540 −1.43593
\(900\) 0 0
\(901\) 14.7386 0.491015
\(902\) 8.98485 0.299163
\(903\) 0 0
\(904\) −13.0928 −0.435460
\(905\) 4.98485 0.165702
\(906\) 0 0
\(907\) 31.6847 1.05207 0.526036 0.850462i \(-0.323678\pi\)
0.526036 + 0.850462i \(0.323678\pi\)
\(908\) 0.107951 0.00358247
\(909\) 0 0
\(910\) −0.492423 −0.0163236
\(911\) −38.2462 −1.26715 −0.633577 0.773680i \(-0.718414\pi\)
−0.633577 + 0.773680i \(0.718414\pi\)
\(912\) 0 0
\(913\) −39.3693 −1.30293
\(914\) 10.6307 0.351632
\(915\) 0 0
\(916\) 7.61553 0.251624
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 43.0540 1.42022 0.710110 0.704091i \(-0.248645\pi\)
0.710110 + 0.704091i \(0.248645\pi\)
\(920\) −6.24621 −0.205931
\(921\) 0 0
\(922\) −21.8617 −0.719978
\(923\) 6.73863 0.221805
\(924\) 0 0
\(925\) 21.3693 0.702619
\(926\) 26.2462 0.862504
\(927\) 0 0
\(928\) 16.0000 0.525226
\(929\) −30.4924 −1.00042 −0.500212 0.865903i \(-0.666745\pi\)
−0.500212 + 0.865903i \(0.666745\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −6.10795 −0.200073
\(933\) 0 0
\(934\) −55.5076 −1.81626
\(935\) −11.5076 −0.376338
\(936\) 0 0
\(937\) 49.8617 1.62891 0.814456 0.580225i \(-0.197036\pi\)
0.814456 + 0.580225i \(0.197036\pi\)
\(938\) 1.75379 0.0572632
\(939\) 0 0
\(940\) 2.46211 0.0803053
\(941\) 15.3693 0.501025 0.250513 0.968113i \(-0.419401\pi\)
0.250513 + 0.968113i \(0.419401\pi\)
\(942\) 0 0
\(943\) 5.12311 0.166831
\(944\) 17.2614 0.561810
\(945\) 0 0
\(946\) −8.98485 −0.292123
\(947\) 35.6155 1.15735 0.578675 0.815559i \(-0.303570\pi\)
0.578675 + 0.815559i \(0.303570\pi\)
\(948\) 0 0
\(949\) −4.80776 −0.156067
\(950\) 29.2614 0.949364
\(951\) 0 0
\(952\) −9.75379 −0.316122
\(953\) −42.9848 −1.39242 −0.696208 0.717840i \(-0.745131\pi\)
−0.696208 + 0.717840i \(0.745131\pi\)
\(954\) 0 0
\(955\) 0.984845 0.0318688
\(956\) −13.0152 −0.420940
\(957\) 0 0
\(958\) 35.1231 1.13478
\(959\) 10.5616 0.341050
\(960\) 0 0
\(961\) 12.0540 0.388838
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) −9.15342 −0.294812
\(965\) 11.3693 0.365991
\(966\) 0 0
\(967\) 37.6155 1.20963 0.604817 0.796365i \(-0.293246\pi\)
0.604817 + 0.796365i \(0.293246\pi\)
\(968\) −37.1771 −1.19492
\(969\) 0 0
\(970\) −7.78410 −0.249932
\(971\) −61.2311 −1.96500 −0.982499 0.186268i \(-0.940361\pi\)
−0.982499 + 0.186268i \(0.940361\pi\)
\(972\) 0 0
\(973\) 6.87689 0.220463
\(974\) −35.5076 −1.13774
\(975\) 0 0
\(976\) 40.1080 1.28382
\(977\) −59.6155 −1.90727 −0.953635 0.300966i \(-0.902691\pi\)
−0.953635 + 0.300966i \(0.902691\pi\)
\(978\) 0 0
\(979\) −90.6004 −2.89560
\(980\) 0.246211 0.00786493
\(981\) 0 0
\(982\) −27.8920 −0.890071
\(983\) −30.4924 −0.972557 −0.486279 0.873804i \(-0.661646\pi\)
−0.486279 + 0.873804i \(0.661646\pi\)
\(984\) 0 0
\(985\) −0.492423 −0.0156899
\(986\) 40.9848 1.30522
\(987\) 0 0
\(988\) −0.984845 −0.0313321
\(989\) −5.12311 −0.162905
\(990\) 0 0
\(991\) 32.9848 1.04780 0.523899 0.851780i \(-0.324477\pi\)
0.523899 + 0.851780i \(0.324477\pi\)
\(992\) −16.0000 −0.508001
\(993\) 0 0
\(994\) −18.7386 −0.594353
\(995\) 1.43845 0.0456018
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 37.4773 1.18632
\(999\) 0 0
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 8001.2.a.h.1.2 2
3.2 odd 2 2667.2.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.i.1.1 2 3.2 odd 2
8001.2.a.h.1.2 2 1.1 even 1 trivial