Properties

Label 8001.2.a.m.1.11
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} - 170 x^{2} - 121 x + 57 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.11
Root \(2.74009\) 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)
 
\(f(q)\) \(=\) \(q+2.74009 q^{2} +5.50808 q^{4} -2.04281 q^{5} -1.00000 q^{7} +9.61245 q^{8} +O(q^{10})\) \(q+2.74009 q^{2} +5.50808 q^{4} -2.04281 q^{5} -1.00000 q^{7} +9.61245 q^{8} -5.59747 q^{10} +0.112732 q^{11} -5.24720 q^{13} -2.74009 q^{14} +15.3228 q^{16} +1.15266 q^{17} -7.39631 q^{19} -11.2519 q^{20} +0.308894 q^{22} -2.82330 q^{23} -0.826947 q^{25} -14.3778 q^{26} -5.50808 q^{28} -7.47063 q^{29} +4.42979 q^{31} +22.7609 q^{32} +3.15838 q^{34} +2.04281 q^{35} -8.52019 q^{37} -20.2665 q^{38} -19.6364 q^{40} -4.19537 q^{41} +10.8938 q^{43} +0.620935 q^{44} -7.73608 q^{46} -0.534490 q^{47} +1.00000 q^{49} -2.26591 q^{50} -28.9020 q^{52} -1.02108 q^{53} -0.230289 q^{55} -9.61245 q^{56} -20.4702 q^{58} +9.13894 q^{59} -6.61416 q^{61} +12.1380 q^{62} +31.7213 q^{64} +10.7190 q^{65} -1.00129 q^{67} +6.34893 q^{68} +5.59747 q^{70} +9.18990 q^{71} -13.3691 q^{73} -23.3461 q^{74} -40.7395 q^{76} -0.112732 q^{77} -15.0096 q^{79} -31.3015 q^{80} -11.4957 q^{82} +8.19311 q^{83} -2.35466 q^{85} +29.8500 q^{86} +1.08363 q^{88} -14.8850 q^{89} +5.24720 q^{91} -15.5509 q^{92} -1.46455 q^{94} +15.1092 q^{95} -8.69953 q^{97} +2.74009 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.74009 1.93753 0.968767 0.247972i \(-0.0797639\pi\)
0.968767 + 0.247972i \(0.0797639\pi\)
\(3\) 0 0
\(4\) 5.50808 2.75404
\(5\) −2.04281 −0.913570 −0.456785 0.889577i \(-0.650999\pi\)
−0.456785 + 0.889577i \(0.650999\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 9.61245 3.39851
\(9\) 0 0
\(10\) −5.59747 −1.77007
\(11\) 0.112732 0.0339899 0.0169949 0.999856i \(-0.494590\pi\)
0.0169949 + 0.999856i \(0.494590\pi\)
\(12\) 0 0
\(13\) −5.24720 −1.45531 −0.727655 0.685943i \(-0.759390\pi\)
−0.727655 + 0.685943i \(0.759390\pi\)
\(14\) −2.74009 −0.732319
\(15\) 0 0
\(16\) 15.3228 3.83070
\(17\) 1.15266 0.279561 0.139780 0.990183i \(-0.455360\pi\)
0.139780 + 0.990183i \(0.455360\pi\)
\(18\) 0 0
\(19\) −7.39631 −1.69683 −0.848415 0.529332i \(-0.822443\pi\)
−0.848415 + 0.529332i \(0.822443\pi\)
\(20\) −11.2519 −2.51601
\(21\) 0 0
\(22\) 0.308894 0.0658565
\(23\) −2.82330 −0.588698 −0.294349 0.955698i \(-0.595103\pi\)
−0.294349 + 0.955698i \(0.595103\pi\)
\(24\) 0 0
\(25\) −0.826947 −0.165389
\(26\) −14.3778 −2.81971
\(27\) 0 0
\(28\) −5.50808 −1.04093
\(29\) −7.47063 −1.38726 −0.693631 0.720331i \(-0.743990\pi\)
−0.693631 + 0.720331i \(0.743990\pi\)
\(30\) 0 0
\(31\) 4.42979 0.795615 0.397807 0.917469i \(-0.369771\pi\)
0.397807 + 0.917469i \(0.369771\pi\)
\(32\) 22.7609 4.02360
\(33\) 0 0
\(34\) 3.15838 0.541658
\(35\) 2.04281 0.345297
\(36\) 0 0
\(37\) −8.52019 −1.40071 −0.700355 0.713795i \(-0.746975\pi\)
−0.700355 + 0.713795i \(0.746975\pi\)
\(38\) −20.2665 −3.28767
\(39\) 0 0
\(40\) −19.6364 −3.10478
\(41\) −4.19537 −0.655207 −0.327603 0.944815i \(-0.606241\pi\)
−0.327603 + 0.944815i \(0.606241\pi\)
\(42\) 0 0
\(43\) 10.8938 1.66129 0.830646 0.556801i \(-0.187971\pi\)
0.830646 + 0.556801i \(0.187971\pi\)
\(44\) 0.620935 0.0936094
\(45\) 0 0
\(46\) −7.73608 −1.14062
\(47\) −0.534490 −0.0779633 −0.0389817 0.999240i \(-0.512411\pi\)
−0.0389817 + 0.999240i \(0.512411\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.26591 −0.320448
\(51\) 0 0
\(52\) −28.9020 −4.00798
\(53\) −1.02108 −0.140256 −0.0701282 0.997538i \(-0.522341\pi\)
−0.0701282 + 0.997538i \(0.522341\pi\)
\(54\) 0 0
\(55\) −0.230289 −0.0310521
\(56\) −9.61245 −1.28452
\(57\) 0 0
\(58\) −20.4702 −2.68787
\(59\) 9.13894 1.18979 0.594894 0.803804i \(-0.297194\pi\)
0.594894 + 0.803804i \(0.297194\pi\)
\(60\) 0 0
\(61\) −6.61416 −0.846856 −0.423428 0.905930i \(-0.639173\pi\)
−0.423428 + 0.905930i \(0.639173\pi\)
\(62\) 12.1380 1.54153
\(63\) 0 0
\(64\) 31.7213 3.96516
\(65\) 10.7190 1.32953
\(66\) 0 0
\(67\) −1.00129 −0.122327 −0.0611633 0.998128i \(-0.519481\pi\)
−0.0611633 + 0.998128i \(0.519481\pi\)
\(68\) 6.34893 0.769921
\(69\) 0 0
\(70\) 5.59747 0.669025
\(71\) 9.18990 1.09064 0.545320 0.838228i \(-0.316408\pi\)
0.545320 + 0.838228i \(0.316408\pi\)
\(72\) 0 0
\(73\) −13.3691 −1.56473 −0.782367 0.622817i \(-0.785988\pi\)
−0.782367 + 0.622817i \(0.785988\pi\)
\(74\) −23.3461 −2.71392
\(75\) 0 0
\(76\) −40.7395 −4.67314
\(77\) −0.112732 −0.0128470
\(78\) 0 0
\(79\) −15.0096 −1.68871 −0.844355 0.535785i \(-0.820016\pi\)
−0.844355 + 0.535785i \(0.820016\pi\)
\(80\) −31.3015 −3.49961
\(81\) 0 0
\(82\) −11.4957 −1.26949
\(83\) 8.19311 0.899310 0.449655 0.893202i \(-0.351547\pi\)
0.449655 + 0.893202i \(0.351547\pi\)
\(84\) 0 0
\(85\) −2.35466 −0.255398
\(86\) 29.8500 3.21881
\(87\) 0 0
\(88\) 1.08363 0.115515
\(89\) −14.8850 −1.57781 −0.788906 0.614514i \(-0.789352\pi\)
−0.788906 + 0.614514i \(0.789352\pi\)
\(90\) 0 0
\(91\) 5.24720 0.550056
\(92\) −15.5509 −1.62130
\(93\) 0 0
\(94\) −1.46455 −0.151057
\(95\) 15.1092 1.55017
\(96\) 0 0
\(97\) −8.69953 −0.883303 −0.441652 0.897187i \(-0.645607\pi\)
−0.441652 + 0.897187i \(0.645607\pi\)
\(98\) 2.74009 0.276791
\(99\) 0 0
\(100\) −4.55489 −0.455489
\(101\) −13.0599 −1.29950 −0.649752 0.760146i \(-0.725127\pi\)
−0.649752 + 0.760146i \(0.725127\pi\)
\(102\) 0 0
\(103\) 0.177061 0.0174463 0.00872317 0.999962i \(-0.497223\pi\)
0.00872317 + 0.999962i \(0.497223\pi\)
\(104\) −50.4384 −4.94589
\(105\) 0 0
\(106\) −2.79786 −0.271752
\(107\) 19.3196 1.86769 0.933846 0.357676i \(-0.116431\pi\)
0.933846 + 0.357676i \(0.116431\pi\)
\(108\) 0 0
\(109\) −7.62043 −0.729905 −0.364952 0.931026i \(-0.618915\pi\)
−0.364952 + 0.931026i \(0.618915\pi\)
\(110\) −0.631011 −0.0601646
\(111\) 0 0
\(112\) −15.3228 −1.44787
\(113\) 15.7793 1.48439 0.742197 0.670182i \(-0.233784\pi\)
0.742197 + 0.670182i \(0.233784\pi\)
\(114\) 0 0
\(115\) 5.76744 0.537817
\(116\) −41.1488 −3.82057
\(117\) 0 0
\(118\) 25.0415 2.30526
\(119\) −1.15266 −0.105664
\(120\) 0 0
\(121\) −10.9873 −0.998845
\(122\) −18.1234 −1.64081
\(123\) 0 0
\(124\) 24.3997 2.19115
\(125\) 11.9033 1.06467
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 41.3973 3.65904
\(129\) 0 0
\(130\) 29.3710 2.57601
\(131\) −9.28013 −0.810809 −0.405404 0.914137i \(-0.632869\pi\)
−0.405404 + 0.914137i \(0.632869\pi\)
\(132\) 0 0
\(133\) 7.39631 0.641341
\(134\) −2.74361 −0.237012
\(135\) 0 0
\(136\) 11.0799 0.950091
\(137\) −7.40716 −0.632836 −0.316418 0.948620i \(-0.602480\pi\)
−0.316418 + 0.948620i \(0.602480\pi\)
\(138\) 0 0
\(139\) 16.7704 1.42245 0.711225 0.702965i \(-0.248141\pi\)
0.711225 + 0.702965i \(0.248141\pi\)
\(140\) 11.2519 0.950962
\(141\) 0 0
\(142\) 25.1811 2.11315
\(143\) −0.591525 −0.0494658
\(144\) 0 0
\(145\) 15.2610 1.26736
\(146\) −36.6325 −3.03173
\(147\) 0 0
\(148\) −46.9299 −3.85761
\(149\) 18.5396 1.51882 0.759410 0.650612i \(-0.225488\pi\)
0.759410 + 0.650612i \(0.225488\pi\)
\(150\) 0 0
\(151\) −3.80282 −0.309469 −0.154734 0.987956i \(-0.549452\pi\)
−0.154734 + 0.987956i \(0.549452\pi\)
\(152\) −71.0966 −5.76670
\(153\) 0 0
\(154\) −0.308894 −0.0248914
\(155\) −9.04921 −0.726850
\(156\) 0 0
\(157\) −11.3952 −0.909437 −0.454719 0.890635i \(-0.650260\pi\)
−0.454719 + 0.890635i \(0.650260\pi\)
\(158\) −41.1276 −3.27193
\(159\) 0 0
\(160\) −46.4961 −3.67584
\(161\) 2.82330 0.222507
\(162\) 0 0
\(163\) 1.13475 0.0888804 0.0444402 0.999012i \(-0.485850\pi\)
0.0444402 + 0.999012i \(0.485850\pi\)
\(164\) −23.1084 −1.80447
\(165\) 0 0
\(166\) 22.4498 1.74245
\(167\) 3.50370 0.271125 0.135562 0.990769i \(-0.456716\pi\)
0.135562 + 0.990769i \(0.456716\pi\)
\(168\) 0 0
\(169\) 14.5331 1.11793
\(170\) −6.45196 −0.494843
\(171\) 0 0
\(172\) 60.0040 4.57527
\(173\) 4.92407 0.374370 0.187185 0.982325i \(-0.440064\pi\)
0.187185 + 0.982325i \(0.440064\pi\)
\(174\) 0 0
\(175\) 0.826947 0.0625113
\(176\) 1.72736 0.130205
\(177\) 0 0
\(178\) −40.7863 −3.05706
\(179\) 12.5938 0.941308 0.470654 0.882318i \(-0.344018\pi\)
0.470654 + 0.882318i \(0.344018\pi\)
\(180\) 0 0
\(181\) 21.2386 1.57865 0.789327 0.613974i \(-0.210430\pi\)
0.789327 + 0.613974i \(0.210430\pi\)
\(182\) 14.3778 1.06575
\(183\) 0 0
\(184\) −27.1388 −2.00070
\(185\) 17.4051 1.27965
\(186\) 0 0
\(187\) 0.129941 0.00950223
\(188\) −2.94401 −0.214714
\(189\) 0 0
\(190\) 41.4006 3.00351
\(191\) 4.80560 0.347721 0.173861 0.984770i \(-0.444376\pi\)
0.173861 + 0.984770i \(0.444376\pi\)
\(192\) 0 0
\(193\) −11.4812 −0.826432 −0.413216 0.910633i \(-0.635594\pi\)
−0.413216 + 0.910633i \(0.635594\pi\)
\(194\) −23.8375 −1.71143
\(195\) 0 0
\(196\) 5.50808 0.393434
\(197\) −8.26242 −0.588673 −0.294337 0.955702i \(-0.595099\pi\)
−0.294337 + 0.955702i \(0.595099\pi\)
\(198\) 0 0
\(199\) 2.57725 0.182697 0.0913483 0.995819i \(-0.470882\pi\)
0.0913483 + 0.995819i \(0.470882\pi\)
\(200\) −7.94899 −0.562078
\(201\) 0 0
\(202\) −35.7851 −2.51783
\(203\) 7.47063 0.524336
\(204\) 0 0
\(205\) 8.57032 0.598577
\(206\) 0.485163 0.0338029
\(207\) 0 0
\(208\) −80.4017 −5.57485
\(209\) −0.833797 −0.0576750
\(210\) 0 0
\(211\) −2.95661 −0.203541 −0.101771 0.994808i \(-0.532451\pi\)
−0.101771 + 0.994808i \(0.532451\pi\)
\(212\) −5.62421 −0.386272
\(213\) 0 0
\(214\) 52.9373 3.61872
\(215\) −22.2540 −1.51771
\(216\) 0 0
\(217\) −4.42979 −0.300714
\(218\) −20.8806 −1.41422
\(219\) 0 0
\(220\) −1.26845 −0.0855188
\(221\) −6.04822 −0.406847
\(222\) 0 0
\(223\) −6.73292 −0.450870 −0.225435 0.974258i \(-0.572380\pi\)
−0.225435 + 0.974258i \(0.572380\pi\)
\(224\) −22.7609 −1.52078
\(225\) 0 0
\(226\) 43.2367 2.87606
\(227\) −11.0498 −0.733399 −0.366699 0.930339i \(-0.619512\pi\)
−0.366699 + 0.930339i \(0.619512\pi\)
\(228\) 0 0
\(229\) −20.3402 −1.34412 −0.672058 0.740498i \(-0.734590\pi\)
−0.672058 + 0.740498i \(0.734590\pi\)
\(230\) 15.8033 1.04204
\(231\) 0 0
\(232\) −71.8111 −4.71463
\(233\) 3.07642 0.201543 0.100771 0.994910i \(-0.467869\pi\)
0.100771 + 0.994910i \(0.467869\pi\)
\(234\) 0 0
\(235\) 1.09186 0.0712250
\(236\) 50.3380 3.27673
\(237\) 0 0
\(238\) −3.15838 −0.204728
\(239\) −7.52434 −0.486709 −0.243354 0.969937i \(-0.578248\pi\)
−0.243354 + 0.969937i \(0.578248\pi\)
\(240\) 0 0
\(241\) 19.5340 1.25830 0.629148 0.777286i \(-0.283404\pi\)
0.629148 + 0.777286i \(0.283404\pi\)
\(242\) −30.1061 −1.93530
\(243\) 0 0
\(244\) −36.4313 −2.33228
\(245\) −2.04281 −0.130510
\(246\) 0 0
\(247\) 38.8099 2.46941
\(248\) 42.5812 2.70391
\(249\) 0 0
\(250\) 32.6161 2.06283
\(251\) 20.2651 1.27912 0.639562 0.768740i \(-0.279116\pi\)
0.639562 + 0.768740i \(0.279116\pi\)
\(252\) 0 0
\(253\) −0.318275 −0.0200098
\(254\) 2.74009 0.171928
\(255\) 0 0
\(256\) 49.9896 3.12435
\(257\) −17.0647 −1.06447 −0.532235 0.846597i \(-0.678648\pi\)
−0.532235 + 0.846597i \(0.678648\pi\)
\(258\) 0 0
\(259\) 8.52019 0.529419
\(260\) 59.0411 3.66157
\(261\) 0 0
\(262\) −25.4284 −1.57097
\(263\) 26.4333 1.62995 0.814975 0.579496i \(-0.196751\pi\)
0.814975 + 0.579496i \(0.196751\pi\)
\(264\) 0 0
\(265\) 2.08587 0.128134
\(266\) 20.2665 1.24262
\(267\) 0 0
\(268\) −5.51516 −0.336892
\(269\) 0.229249 0.0139775 0.00698877 0.999976i \(-0.497775\pi\)
0.00698877 + 0.999976i \(0.497775\pi\)
\(270\) 0 0
\(271\) 20.2110 1.22773 0.613865 0.789411i \(-0.289614\pi\)
0.613865 + 0.789411i \(0.289614\pi\)
\(272\) 17.6619 1.07091
\(273\) 0 0
\(274\) −20.2963 −1.22614
\(275\) −0.0932231 −0.00562156
\(276\) 0 0
\(277\) −12.3393 −0.741398 −0.370699 0.928753i \(-0.620882\pi\)
−0.370699 + 0.928753i \(0.620882\pi\)
\(278\) 45.9524 2.75604
\(279\) 0 0
\(280\) 19.6364 1.17350
\(281\) 27.5356 1.64264 0.821318 0.570471i \(-0.193239\pi\)
0.821318 + 0.570471i \(0.193239\pi\)
\(282\) 0 0
\(283\) −2.82223 −0.167764 −0.0838822 0.996476i \(-0.526732\pi\)
−0.0838822 + 0.996476i \(0.526732\pi\)
\(284\) 50.6187 3.00367
\(285\) 0 0
\(286\) −1.62083 −0.0958417
\(287\) 4.19537 0.247645
\(288\) 0 0
\(289\) −15.6714 −0.921846
\(290\) 41.8166 2.45556
\(291\) 0 0
\(292\) −73.6381 −4.30934
\(293\) 2.20100 0.128584 0.0642918 0.997931i \(-0.479521\pi\)
0.0642918 + 0.997931i \(0.479521\pi\)
\(294\) 0 0
\(295\) −18.6691 −1.08696
\(296\) −81.8999 −4.76033
\(297\) 0 0
\(298\) 50.8000 2.94277
\(299\) 14.8144 0.856738
\(300\) 0 0
\(301\) −10.8938 −0.627909
\(302\) −10.4200 −0.599606
\(303\) 0 0
\(304\) −113.332 −6.50004
\(305\) 13.5114 0.773663
\(306\) 0 0
\(307\) −19.6085 −1.11911 −0.559557 0.828792i \(-0.689029\pi\)
−0.559557 + 0.828792i \(0.689029\pi\)
\(308\) −0.620935 −0.0353810
\(309\) 0 0
\(310\) −24.7956 −1.40830
\(311\) −27.3394 −1.55027 −0.775137 0.631793i \(-0.782319\pi\)
−0.775137 + 0.631793i \(0.782319\pi\)
\(312\) 0 0
\(313\) −5.73784 −0.324322 −0.162161 0.986764i \(-0.551846\pi\)
−0.162161 + 0.986764i \(0.551846\pi\)
\(314\) −31.2239 −1.76207
\(315\) 0 0
\(316\) −82.6740 −4.65077
\(317\) 0.929834 0.0522247 0.0261123 0.999659i \(-0.491687\pi\)
0.0261123 + 0.999659i \(0.491687\pi\)
\(318\) 0 0
\(319\) −0.842176 −0.0471528
\(320\) −64.8004 −3.62245
\(321\) 0 0
\(322\) 7.73608 0.431115
\(323\) −8.52541 −0.474367
\(324\) 0 0
\(325\) 4.33915 0.240693
\(326\) 3.10931 0.172209
\(327\) 0 0
\(328\) −40.3278 −2.22673
\(329\) 0.534490 0.0294674
\(330\) 0 0
\(331\) −10.7581 −0.591317 −0.295658 0.955294i \(-0.595539\pi\)
−0.295658 + 0.955294i \(0.595539\pi\)
\(332\) 45.1283 2.47674
\(333\) 0 0
\(334\) 9.60045 0.525313
\(335\) 2.04543 0.111754
\(336\) 0 0
\(337\) −30.2782 −1.64936 −0.824679 0.565601i \(-0.808644\pi\)
−0.824679 + 0.565601i \(0.808644\pi\)
\(338\) 39.8219 2.16602
\(339\) 0 0
\(340\) −12.9696 −0.703377
\(341\) 0.499378 0.0270428
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 104.716 5.64592
\(345\) 0 0
\(346\) 13.4924 0.725355
\(347\) −30.1873 −1.62054 −0.810269 0.586058i \(-0.800679\pi\)
−0.810269 + 0.586058i \(0.800679\pi\)
\(348\) 0 0
\(349\) −4.71098 −0.252173 −0.126086 0.992019i \(-0.540242\pi\)
−0.126086 + 0.992019i \(0.540242\pi\)
\(350\) 2.26591 0.121118
\(351\) 0 0
\(352\) 2.56587 0.136761
\(353\) −14.8027 −0.787867 −0.393934 0.919139i \(-0.628886\pi\)
−0.393934 + 0.919139i \(0.628886\pi\)
\(354\) 0 0
\(355\) −18.7732 −0.996377
\(356\) −81.9880 −4.34536
\(357\) 0 0
\(358\) 34.5082 1.82382
\(359\) −23.9011 −1.26145 −0.630726 0.776005i \(-0.717243\pi\)
−0.630726 + 0.776005i \(0.717243\pi\)
\(360\) 0 0
\(361\) 35.7054 1.87923
\(362\) 58.1956 3.05869
\(363\) 0 0
\(364\) 28.9020 1.51488
\(365\) 27.3105 1.42950
\(366\) 0 0
\(367\) 18.4281 0.961940 0.480970 0.876737i \(-0.340285\pi\)
0.480970 + 0.876737i \(0.340285\pi\)
\(368\) −43.2608 −2.25512
\(369\) 0 0
\(370\) 47.6915 2.47936
\(371\) 1.02108 0.0530120
\(372\) 0 0
\(373\) −11.3124 −0.585732 −0.292866 0.956154i \(-0.594609\pi\)
−0.292866 + 0.956154i \(0.594609\pi\)
\(374\) 0.356050 0.0184109
\(375\) 0 0
\(376\) −5.13776 −0.264960
\(377\) 39.1999 2.01890
\(378\) 0 0
\(379\) 9.17198 0.471133 0.235566 0.971858i \(-0.424305\pi\)
0.235566 + 0.971858i \(0.424305\pi\)
\(380\) 83.2228 4.26924
\(381\) 0 0
\(382\) 13.1678 0.673722
\(383\) −15.8752 −0.811183 −0.405592 0.914054i \(-0.632935\pi\)
−0.405592 + 0.914054i \(0.632935\pi\)
\(384\) 0 0
\(385\) 0.230289 0.0117366
\(386\) −31.4594 −1.60124
\(387\) 0 0
\(388\) −47.9177 −2.43265
\(389\) −16.2897 −0.825920 −0.412960 0.910749i \(-0.635505\pi\)
−0.412960 + 0.910749i \(0.635505\pi\)
\(390\) 0 0
\(391\) −3.25429 −0.164577
\(392\) 9.61245 0.485502
\(393\) 0 0
\(394\) −22.6398 −1.14057
\(395\) 30.6616 1.54275
\(396\) 0 0
\(397\) −7.36849 −0.369814 −0.184907 0.982756i \(-0.559198\pi\)
−0.184907 + 0.982756i \(0.559198\pi\)
\(398\) 7.06190 0.353981
\(399\) 0 0
\(400\) −12.6711 −0.633557
\(401\) −16.8371 −0.840804 −0.420402 0.907338i \(-0.638111\pi\)
−0.420402 + 0.907338i \(0.638111\pi\)
\(402\) 0 0
\(403\) −23.2440 −1.15787
\(404\) −71.9347 −3.57889
\(405\) 0 0
\(406\) 20.4702 1.01592
\(407\) −0.960494 −0.0476099
\(408\) 0 0
\(409\) 3.10681 0.153622 0.0768108 0.997046i \(-0.475526\pi\)
0.0768108 + 0.997046i \(0.475526\pi\)
\(410\) 23.4834 1.15976
\(411\) 0 0
\(412\) 0.975266 0.0480479
\(413\) −9.13894 −0.449698
\(414\) 0 0
\(415\) −16.7369 −0.821583
\(416\) −119.431 −5.85558
\(417\) 0 0
\(418\) −2.28468 −0.111747
\(419\) 19.8014 0.967359 0.483680 0.875245i \(-0.339300\pi\)
0.483680 + 0.875245i \(0.339300\pi\)
\(420\) 0 0
\(421\) 34.6093 1.68675 0.843377 0.537322i \(-0.180564\pi\)
0.843377 + 0.537322i \(0.180564\pi\)
\(422\) −8.10136 −0.394368
\(423\) 0 0
\(424\) −9.81511 −0.476664
\(425\) −0.953187 −0.0462364
\(426\) 0 0
\(427\) 6.61416 0.320082
\(428\) 106.414 5.14370
\(429\) 0 0
\(430\) −60.9778 −2.94061
\(431\) −13.5389 −0.652144 −0.326072 0.945345i \(-0.605725\pi\)
−0.326072 + 0.945345i \(0.605725\pi\)
\(432\) 0 0
\(433\) 33.6920 1.61914 0.809568 0.587026i \(-0.199702\pi\)
0.809568 + 0.587026i \(0.199702\pi\)
\(434\) −12.1380 −0.582644
\(435\) 0 0
\(436\) −41.9739 −2.01019
\(437\) 20.8820 0.998920
\(438\) 0 0
\(439\) −17.6447 −0.842134 −0.421067 0.907029i \(-0.638344\pi\)
−0.421067 + 0.907029i \(0.638344\pi\)
\(440\) −2.21364 −0.105531
\(441\) 0 0
\(442\) −16.5727 −0.788281
\(443\) 7.11646 0.338113 0.169057 0.985606i \(-0.445928\pi\)
0.169057 + 0.985606i \(0.445928\pi\)
\(444\) 0 0
\(445\) 30.4072 1.44144
\(446\) −18.4488 −0.873576
\(447\) 0 0
\(448\) −31.7213 −1.49869
\(449\) 39.9935 1.88741 0.943706 0.330786i \(-0.107314\pi\)
0.943706 + 0.330786i \(0.107314\pi\)
\(450\) 0 0
\(451\) −0.472951 −0.0222704
\(452\) 86.9138 4.08808
\(453\) 0 0
\(454\) −30.2773 −1.42099
\(455\) −10.7190 −0.502514
\(456\) 0 0
\(457\) 19.3464 0.904987 0.452493 0.891768i \(-0.350535\pi\)
0.452493 + 0.891768i \(0.350535\pi\)
\(458\) −55.7339 −2.60427
\(459\) 0 0
\(460\) 31.7675 1.48117
\(461\) 17.9613 0.836543 0.418271 0.908322i \(-0.362636\pi\)
0.418271 + 0.908322i \(0.362636\pi\)
\(462\) 0 0
\(463\) 32.5516 1.51280 0.756400 0.654110i \(-0.226957\pi\)
0.756400 + 0.654110i \(0.226957\pi\)
\(464\) −114.471 −5.31418
\(465\) 0 0
\(466\) 8.42966 0.390496
\(467\) −22.3867 −1.03593 −0.517965 0.855402i \(-0.673310\pi\)
−0.517965 + 0.855402i \(0.673310\pi\)
\(468\) 0 0
\(469\) 1.00129 0.0462351
\(470\) 2.99179 0.138001
\(471\) 0 0
\(472\) 87.8476 4.04351
\(473\) 1.22808 0.0564671
\(474\) 0 0
\(475\) 6.11636 0.280638
\(476\) −6.34893 −0.291003
\(477\) 0 0
\(478\) −20.6173 −0.943015
\(479\) −8.93937 −0.408450 −0.204225 0.978924i \(-0.565467\pi\)
−0.204225 + 0.978924i \(0.565467\pi\)
\(480\) 0 0
\(481\) 44.7071 2.03847
\(482\) 53.5249 2.43799
\(483\) 0 0
\(484\) −60.5189 −2.75086
\(485\) 17.7714 0.806960
\(486\) 0 0
\(487\) −11.9076 −0.539583 −0.269792 0.962919i \(-0.586955\pi\)
−0.269792 + 0.962919i \(0.586955\pi\)
\(488\) −63.5783 −2.87805
\(489\) 0 0
\(490\) −5.59747 −0.252868
\(491\) −3.26881 −0.147519 −0.0737597 0.997276i \(-0.523500\pi\)
−0.0737597 + 0.997276i \(0.523500\pi\)
\(492\) 0 0
\(493\) −8.61108 −0.387824
\(494\) 106.342 4.78457
\(495\) 0 0
\(496\) 67.8768 3.04776
\(497\) −9.18990 −0.412223
\(498\) 0 0
\(499\) −2.83398 −0.126867 −0.0634333 0.997986i \(-0.520205\pi\)
−0.0634333 + 0.997986i \(0.520205\pi\)
\(500\) 65.5644 2.93213
\(501\) 0 0
\(502\) 55.5283 2.47835
\(503\) −39.5073 −1.76155 −0.880773 0.473540i \(-0.842976\pi\)
−0.880773 + 0.473540i \(0.842976\pi\)
\(504\) 0 0
\(505\) 26.6787 1.18719
\(506\) −0.872100 −0.0387696
\(507\) 0 0
\(508\) 5.50808 0.244382
\(509\) −1.82369 −0.0808337 −0.0404169 0.999183i \(-0.512869\pi\)
−0.0404169 + 0.999183i \(0.512869\pi\)
\(510\) 0 0
\(511\) 13.3691 0.591414
\(512\) 54.1814 2.39450
\(513\) 0 0
\(514\) −46.7589 −2.06245
\(515\) −0.361701 −0.0159385
\(516\) 0 0
\(517\) −0.0602539 −0.00264996
\(518\) 23.3461 1.02577
\(519\) 0 0
\(520\) 103.036 4.51842
\(521\) −2.49558 −0.109333 −0.0546667 0.998505i \(-0.517410\pi\)
−0.0546667 + 0.998505i \(0.517410\pi\)
\(522\) 0 0
\(523\) 34.6174 1.51371 0.756857 0.653580i \(-0.226734\pi\)
0.756857 + 0.653580i \(0.226734\pi\)
\(524\) −51.1157 −2.23300
\(525\) 0 0
\(526\) 72.4297 3.15808
\(527\) 5.10604 0.222423
\(528\) 0 0
\(529\) −15.0290 −0.653435
\(530\) 5.71547 0.248264
\(531\) 0 0
\(532\) 40.7395 1.76628
\(533\) 22.0139 0.953529
\(534\) 0 0
\(535\) −39.4661 −1.70627
\(536\) −9.62481 −0.415729
\(537\) 0 0
\(538\) 0.628161 0.0270820
\(539\) 0.112732 0.00485569
\(540\) 0 0
\(541\) −22.2805 −0.957914 −0.478957 0.877838i \(-0.658985\pi\)
−0.478957 + 0.877838i \(0.658985\pi\)
\(542\) 55.3798 2.37877
\(543\) 0 0
\(544\) 26.2355 1.12484
\(545\) 15.5671 0.666819
\(546\) 0 0
\(547\) −0.937979 −0.0401051 −0.0200526 0.999799i \(-0.506383\pi\)
−0.0200526 + 0.999799i \(0.506383\pi\)
\(548\) −40.7992 −1.74286
\(549\) 0 0
\(550\) −0.255439 −0.0108920
\(551\) 55.2551 2.35395
\(552\) 0 0
\(553\) 15.0096 0.638272
\(554\) −33.8108 −1.43648
\(555\) 0 0
\(556\) 92.3729 3.91748
\(557\) −25.4331 −1.07763 −0.538817 0.842423i \(-0.681128\pi\)
−0.538817 + 0.842423i \(0.681128\pi\)
\(558\) 0 0
\(559\) −57.1620 −2.41770
\(560\) 31.3015 1.32273
\(561\) 0 0
\(562\) 75.4499 3.18266
\(563\) 25.9729 1.09463 0.547313 0.836928i \(-0.315651\pi\)
0.547313 + 0.836928i \(0.315651\pi\)
\(564\) 0 0
\(565\) −32.2341 −1.35610
\(566\) −7.73317 −0.325049
\(567\) 0 0
\(568\) 88.3374 3.70656
\(569\) 22.4346 0.940509 0.470254 0.882531i \(-0.344162\pi\)
0.470254 + 0.882531i \(0.344162\pi\)
\(570\) 0 0
\(571\) 2.66335 0.111458 0.0557288 0.998446i \(-0.482252\pi\)
0.0557288 + 0.998446i \(0.482252\pi\)
\(572\) −3.25817 −0.136231
\(573\) 0 0
\(574\) 11.4957 0.479821
\(575\) 2.33472 0.0973644
\(576\) 0 0
\(577\) −28.2471 −1.17594 −0.587970 0.808882i \(-0.700073\pi\)
−0.587970 + 0.808882i \(0.700073\pi\)
\(578\) −42.9410 −1.78611
\(579\) 0 0
\(580\) 84.0591 3.49036
\(581\) −8.19311 −0.339907
\(582\) 0 0
\(583\) −0.115108 −0.00476730
\(584\) −128.510 −5.31777
\(585\) 0 0
\(586\) 6.03092 0.249135
\(587\) 44.1126 1.82072 0.910360 0.413817i \(-0.135805\pi\)
0.910360 + 0.413817i \(0.135805\pi\)
\(588\) 0 0
\(589\) −32.7641 −1.35002
\(590\) −51.1549 −2.10601
\(591\) 0 0
\(592\) −130.553 −5.36570
\(593\) 23.3655 0.959505 0.479753 0.877404i \(-0.340726\pi\)
0.479753 + 0.877404i \(0.340726\pi\)
\(594\) 0 0
\(595\) 2.35466 0.0965315
\(596\) 102.117 4.18289
\(597\) 0 0
\(598\) 40.5927 1.65996
\(599\) −24.3295 −0.994078 −0.497039 0.867728i \(-0.665579\pi\)
−0.497039 + 0.867728i \(0.665579\pi\)
\(600\) 0 0
\(601\) 6.15886 0.251225 0.125613 0.992079i \(-0.459910\pi\)
0.125613 + 0.992079i \(0.459910\pi\)
\(602\) −29.8500 −1.21660
\(603\) 0 0
\(604\) −20.9462 −0.852289
\(605\) 22.4449 0.912515
\(606\) 0 0
\(607\) 32.0362 1.30031 0.650154 0.759802i \(-0.274704\pi\)
0.650154 + 0.759802i \(0.274704\pi\)
\(608\) −168.347 −6.82736
\(609\) 0 0
\(610\) 37.0225 1.49900
\(611\) 2.80457 0.113461
\(612\) 0 0
\(613\) −39.3490 −1.58929 −0.794645 0.607074i \(-0.792343\pi\)
−0.794645 + 0.607074i \(0.792343\pi\)
\(614\) −53.7289 −2.16832
\(615\) 0 0
\(616\) −1.08363 −0.0436606
\(617\) −41.7994 −1.68278 −0.841391 0.540427i \(-0.818263\pi\)
−0.841391 + 0.540427i \(0.818263\pi\)
\(618\) 0 0
\(619\) 43.0712 1.73118 0.865589 0.500755i \(-0.166944\pi\)
0.865589 + 0.500755i \(0.166944\pi\)
\(620\) −49.8438 −2.00177
\(621\) 0 0
\(622\) −74.9123 −3.00371
\(623\) 14.8850 0.596357
\(624\) 0 0
\(625\) −20.1814 −0.807257
\(626\) −15.7222 −0.628384
\(627\) 0 0
\(628\) −62.7658 −2.50463
\(629\) −9.82086 −0.391583
\(630\) 0 0
\(631\) −14.6791 −0.584366 −0.292183 0.956362i \(-0.594382\pi\)
−0.292183 + 0.956362i \(0.594382\pi\)
\(632\) −144.279 −5.73910
\(633\) 0 0
\(634\) 2.54783 0.101187
\(635\) −2.04281 −0.0810662
\(636\) 0 0
\(637\) −5.24720 −0.207901
\(638\) −2.30764 −0.0913602
\(639\) 0 0
\(640\) −84.5666 −3.34279
\(641\) −26.1705 −1.03367 −0.516836 0.856085i \(-0.672890\pi\)
−0.516836 + 0.856085i \(0.672890\pi\)
\(642\) 0 0
\(643\) 21.9679 0.866331 0.433166 0.901314i \(-0.357397\pi\)
0.433166 + 0.901314i \(0.357397\pi\)
\(644\) 15.5509 0.612793
\(645\) 0 0
\(646\) −23.3604 −0.919102
\(647\) −9.64241 −0.379082 −0.189541 0.981873i \(-0.560700\pi\)
−0.189541 + 0.981873i \(0.560700\pi\)
\(648\) 0 0
\(649\) 1.03025 0.0404407
\(650\) 11.8897 0.466351
\(651\) 0 0
\(652\) 6.25029 0.244780
\(653\) 1.29571 0.0507049 0.0253524 0.999679i \(-0.491929\pi\)
0.0253524 + 0.999679i \(0.491929\pi\)
\(654\) 0 0
\(655\) 18.9575 0.740731
\(656\) −64.2848 −2.50990
\(657\) 0 0
\(658\) 1.46455 0.0570941
\(659\) −14.1795 −0.552353 −0.276177 0.961107i \(-0.589067\pi\)
−0.276177 + 0.961107i \(0.589067\pi\)
\(660\) 0 0
\(661\) 24.9298 0.969658 0.484829 0.874609i \(-0.338882\pi\)
0.484829 + 0.874609i \(0.338882\pi\)
\(662\) −29.4780 −1.14570
\(663\) 0 0
\(664\) 78.7558 3.05632
\(665\) −15.1092 −0.585910
\(666\) 0 0
\(667\) 21.0918 0.816678
\(668\) 19.2987 0.746688
\(669\) 0 0
\(670\) 5.60466 0.216527
\(671\) −0.745625 −0.0287845
\(672\) 0 0
\(673\) −24.9745 −0.962697 −0.481349 0.876529i \(-0.659853\pi\)
−0.481349 + 0.876529i \(0.659853\pi\)
\(674\) −82.9649 −3.19569
\(675\) 0 0
\(676\) 80.0493 3.07882
\(677\) 28.5115 1.09579 0.547893 0.836548i \(-0.315430\pi\)
0.547893 + 0.836548i \(0.315430\pi\)
\(678\) 0 0
\(679\) 8.69953 0.333857
\(680\) −22.6340 −0.867975
\(681\) 0 0
\(682\) 1.36834 0.0523964
\(683\) −17.8617 −0.683461 −0.341730 0.939798i \(-0.611013\pi\)
−0.341730 + 0.939798i \(0.611013\pi\)
\(684\) 0 0
\(685\) 15.1314 0.578140
\(686\) −2.74009 −0.104617
\(687\) 0 0
\(688\) 166.924 6.36391
\(689\) 5.35782 0.204117
\(690\) 0 0
\(691\) −6.58485 −0.250500 −0.125250 0.992125i \(-0.539973\pi\)
−0.125250 + 0.992125i \(0.539973\pi\)
\(692\) 27.1222 1.03103
\(693\) 0 0
\(694\) −82.7158 −3.13985
\(695\) −34.2587 −1.29951
\(696\) 0 0
\(697\) −4.83583 −0.183170
\(698\) −12.9085 −0.488593
\(699\) 0 0
\(700\) 4.55489 0.172159
\(701\) 39.2478 1.48237 0.741185 0.671301i \(-0.234264\pi\)
0.741185 + 0.671301i \(0.234264\pi\)
\(702\) 0 0
\(703\) 63.0179 2.37677
\(704\) 3.57599 0.134775
\(705\) 0 0
\(706\) −40.5606 −1.52652
\(707\) 13.0599 0.491166
\(708\) 0 0
\(709\) 7.86649 0.295432 0.147716 0.989030i \(-0.452808\pi\)
0.147716 + 0.989030i \(0.452808\pi\)
\(710\) −51.4401 −1.93051
\(711\) 0 0
\(712\) −143.082 −5.36221
\(713\) −12.5066 −0.468377
\(714\) 0 0
\(715\) 1.20837 0.0451905
\(716\) 69.3679 2.59240
\(717\) 0 0
\(718\) −65.4912 −2.44411
\(719\) 8.12303 0.302938 0.151469 0.988462i \(-0.451600\pi\)
0.151469 + 0.988462i \(0.451600\pi\)
\(720\) 0 0
\(721\) −0.177061 −0.00659410
\(722\) 97.8358 3.64107
\(723\) 0 0
\(724\) 116.984 4.34767
\(725\) 6.17782 0.229438
\(726\) 0 0
\(727\) −1.97997 −0.0734331 −0.0367166 0.999326i \(-0.511690\pi\)
−0.0367166 + 0.999326i \(0.511690\pi\)
\(728\) 50.4384 1.86937
\(729\) 0 0
\(730\) 74.8331 2.76970
\(731\) 12.5569 0.464432
\(732\) 0 0
\(733\) −41.5423 −1.53440 −0.767199 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(734\) 50.4946 1.86379
\(735\) 0 0
\(736\) −64.2608 −2.36868
\(737\) −0.112877 −0.00415786
\(738\) 0 0
\(739\) 45.8099 1.68514 0.842572 0.538583i \(-0.181040\pi\)
0.842572 + 0.538583i \(0.181040\pi\)
\(740\) 95.8686 3.52420
\(741\) 0 0
\(742\) 2.79786 0.102713
\(743\) −15.2766 −0.560444 −0.280222 0.959935i \(-0.590408\pi\)
−0.280222 + 0.959935i \(0.590408\pi\)
\(744\) 0 0
\(745\) −37.8727 −1.38755
\(746\) −30.9968 −1.13488
\(747\) 0 0
\(748\) 0.715725 0.0261695
\(749\) −19.3196 −0.705921
\(750\) 0 0
\(751\) −22.4752 −0.820131 −0.410066 0.912056i \(-0.634494\pi\)
−0.410066 + 0.912056i \(0.634494\pi\)
\(752\) −8.18988 −0.298654
\(753\) 0 0
\(754\) 107.411 3.91168
\(755\) 7.76841 0.282721
\(756\) 0 0
\(757\) 8.43364 0.306526 0.153263 0.988185i \(-0.451022\pi\)
0.153263 + 0.988185i \(0.451022\pi\)
\(758\) 25.1320 0.912836
\(759\) 0 0
\(760\) 145.237 5.26828
\(761\) −15.5995 −0.565482 −0.282741 0.959196i \(-0.591244\pi\)
−0.282741 + 0.959196i \(0.591244\pi\)
\(762\) 0 0
\(763\) 7.62043 0.275878
\(764\) 26.4696 0.957638
\(765\) 0 0
\(766\) −43.4994 −1.57170
\(767\) −47.9538 −1.73151
\(768\) 0 0
\(769\) 16.9832 0.612430 0.306215 0.951962i \(-0.400937\pi\)
0.306215 + 0.951962i \(0.400937\pi\)
\(770\) 0.631011 0.0227401
\(771\) 0 0
\(772\) −63.2391 −2.27603
\(773\) 48.2227 1.73445 0.867225 0.497916i \(-0.165901\pi\)
0.867225 + 0.497916i \(0.165901\pi\)
\(774\) 0 0
\(775\) −3.66321 −0.131586
\(776\) −83.6238 −3.00192
\(777\) 0 0
\(778\) −44.6352 −1.60025
\(779\) 31.0302 1.11177
\(780\) 0 0
\(781\) 1.03599 0.0370707
\(782\) −8.91705 −0.318873
\(783\) 0 0
\(784\) 15.3228 0.547243
\(785\) 23.2782 0.830835
\(786\) 0 0
\(787\) 41.8280 1.49101 0.745504 0.666501i \(-0.232209\pi\)
0.745504 + 0.666501i \(0.232209\pi\)
\(788\) −45.5101 −1.62123
\(789\) 0 0
\(790\) 84.0156 2.98914
\(791\) −15.7793 −0.561048
\(792\) 0 0
\(793\) 34.7058 1.23244
\(794\) −20.1903 −0.716527
\(795\) 0 0
\(796\) 14.1957 0.503154
\(797\) 36.9157 1.30762 0.653811 0.756658i \(-0.273169\pi\)
0.653811 + 0.756658i \(0.273169\pi\)
\(798\) 0 0
\(799\) −0.616084 −0.0217955
\(800\) −18.8221 −0.665460
\(801\) 0 0
\(802\) −46.1351 −1.62909
\(803\) −1.50712 −0.0531851
\(804\) 0 0
\(805\) −5.76744 −0.203276
\(806\) −63.6906 −2.24341
\(807\) 0 0
\(808\) −125.537 −4.41638
\(809\) −6.95959 −0.244686 −0.122343 0.992488i \(-0.539041\pi\)
−0.122343 + 0.992488i \(0.539041\pi\)
\(810\) 0 0
\(811\) −7.34138 −0.257791 −0.128895 0.991658i \(-0.541143\pi\)
−0.128895 + 0.991658i \(0.541143\pi\)
\(812\) 41.1488 1.44404
\(813\) 0 0
\(814\) −2.63184 −0.0922459
\(815\) −2.31807 −0.0811985
\(816\) 0 0
\(817\) −80.5740 −2.81893
\(818\) 8.51292 0.297647
\(819\) 0 0
\(820\) 47.2060 1.64851
\(821\) 6.22424 0.217227 0.108614 0.994084i \(-0.465359\pi\)
0.108614 + 0.994084i \(0.465359\pi\)
\(822\) 0 0
\(823\) −45.8870 −1.59952 −0.799760 0.600320i \(-0.795040\pi\)
−0.799760 + 0.600320i \(0.795040\pi\)
\(824\) 1.70199 0.0592916
\(825\) 0 0
\(826\) −25.0415 −0.871305
\(827\) −46.8185 −1.62804 −0.814019 0.580838i \(-0.802725\pi\)
−0.814019 + 0.580838i \(0.802725\pi\)
\(828\) 0 0
\(829\) 18.1291 0.629650 0.314825 0.949150i \(-0.398054\pi\)
0.314825 + 0.949150i \(0.398054\pi\)
\(830\) −45.8606 −1.59185
\(831\) 0 0
\(832\) −166.448 −5.77054
\(833\) 1.15266 0.0399372
\(834\) 0 0
\(835\) −7.15738 −0.247691
\(836\) −4.59262 −0.158839
\(837\) 0 0
\(838\) 54.2574 1.87429
\(839\) −31.0024 −1.07032 −0.535161 0.844750i \(-0.679749\pi\)
−0.535161 + 0.844750i \(0.679749\pi\)
\(840\) 0 0
\(841\) 26.8103 0.924494
\(842\) 94.8326 3.26815
\(843\) 0 0
\(844\) −16.2852 −0.560560
\(845\) −29.6882 −1.02131
\(846\) 0 0
\(847\) 10.9873 0.377528
\(848\) −15.6458 −0.537280
\(849\) 0 0
\(850\) −2.61182 −0.0895846
\(851\) 24.0550 0.824595
\(852\) 0 0
\(853\) 34.7618 1.19022 0.595111 0.803643i \(-0.297108\pi\)
0.595111 + 0.803643i \(0.297108\pi\)
\(854\) 18.1234 0.620169
\(855\) 0 0
\(856\) 185.708 6.34738
\(857\) −3.40035 −0.116154 −0.0580768 0.998312i \(-0.518497\pi\)
−0.0580768 + 0.998312i \(0.518497\pi\)
\(858\) 0 0
\(859\) 17.1386 0.584761 0.292381 0.956302i \(-0.405553\pi\)
0.292381 + 0.956302i \(0.405553\pi\)
\(860\) −122.577 −4.17983
\(861\) 0 0
\(862\) −37.0977 −1.26355
\(863\) 6.88023 0.234206 0.117103 0.993120i \(-0.462639\pi\)
0.117103 + 0.993120i \(0.462639\pi\)
\(864\) 0 0
\(865\) −10.0589 −0.342013
\(866\) 92.3191 3.13713
\(867\) 0 0
\(868\) −24.3997 −0.828179
\(869\) −1.69205 −0.0573990
\(870\) 0 0
\(871\) 5.25394 0.178023
\(872\) −73.2510 −2.48059
\(873\) 0 0
\(874\) 57.2184 1.93544
\(875\) −11.9033 −0.402406
\(876\) 0 0
\(877\) −15.2602 −0.515300 −0.257650 0.966238i \(-0.582948\pi\)
−0.257650 + 0.966238i \(0.582948\pi\)
\(878\) −48.3479 −1.63166
\(879\) 0 0
\(880\) −3.52867 −0.118951
\(881\) −54.7338 −1.84403 −0.922014 0.387157i \(-0.873457\pi\)
−0.922014 + 0.387157i \(0.873457\pi\)
\(882\) 0 0
\(883\) 34.8655 1.17332 0.586659 0.809834i \(-0.300443\pi\)
0.586659 + 0.809834i \(0.300443\pi\)
\(884\) −33.3141 −1.12047
\(885\) 0 0
\(886\) 19.4997 0.655106
\(887\) 12.8739 0.432262 0.216131 0.976364i \(-0.430656\pi\)
0.216131 + 0.976364i \(0.430656\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 83.3185 2.79284
\(891\) 0 0
\(892\) −37.0855 −1.24171
\(893\) 3.95325 0.132290
\(894\) 0 0
\(895\) −25.7268 −0.859951
\(896\) −41.3973 −1.38299
\(897\) 0 0
\(898\) 109.586 3.65693
\(899\) −33.0934 −1.10373
\(900\) 0 0
\(901\) −1.17696 −0.0392102
\(902\) −1.29593 −0.0431496
\(903\) 0 0
\(904\) 151.678 5.04473
\(905\) −43.3863 −1.44221
\(906\) 0 0
\(907\) −45.7867 −1.52032 −0.760161 0.649735i \(-0.774880\pi\)
−0.760161 + 0.649735i \(0.774880\pi\)
\(908\) −60.8630 −2.01981
\(909\) 0 0
\(910\) −29.3710 −0.973639
\(911\) 31.8369 1.05480 0.527402 0.849616i \(-0.323166\pi\)
0.527402 + 0.849616i \(0.323166\pi\)
\(912\) 0 0
\(913\) 0.923622 0.0305674
\(914\) 53.0108 1.75344
\(915\) 0 0
\(916\) −112.035 −3.70175
\(917\) 9.28013 0.306457
\(918\) 0 0
\(919\) 23.8384 0.786356 0.393178 0.919462i \(-0.371376\pi\)
0.393178 + 0.919462i \(0.371376\pi\)
\(920\) 55.4393 1.82778
\(921\) 0 0
\(922\) 49.2156 1.62083
\(923\) −48.2212 −1.58722
\(924\) 0 0
\(925\) 7.04574 0.231663
\(926\) 89.1941 2.93110
\(927\) 0 0
\(928\) −170.038 −5.58178
\(929\) 32.6729 1.07196 0.535982 0.844229i \(-0.319942\pi\)
0.535982 + 0.844229i \(0.319942\pi\)
\(930\) 0 0
\(931\) −7.39631 −0.242404
\(932\) 16.9452 0.555057
\(933\) 0 0
\(934\) −61.3414 −2.00715
\(935\) −0.265444 −0.00868095
\(936\) 0 0
\(937\) 16.8311 0.549847 0.274924 0.961466i \(-0.411347\pi\)
0.274924 + 0.961466i \(0.411347\pi\)
\(938\) 2.74361 0.0895821
\(939\) 0 0
\(940\) 6.01404 0.196156
\(941\) −44.8845 −1.46319 −0.731597 0.681738i \(-0.761225\pi\)
−0.731597 + 0.681738i \(0.761225\pi\)
\(942\) 0 0
\(943\) 11.8448 0.385719
\(944\) 140.034 4.55772
\(945\) 0 0
\(946\) 3.36504 0.109407
\(947\) 23.1949 0.753732 0.376866 0.926268i \(-0.377002\pi\)
0.376866 + 0.926268i \(0.377002\pi\)
\(948\) 0 0
\(949\) 70.1503 2.27717
\(950\) 16.7594 0.543745
\(951\) 0 0
\(952\) −11.0799 −0.359101
\(953\) 1.41323 0.0457791 0.0228895 0.999738i \(-0.492713\pi\)
0.0228895 + 0.999738i \(0.492713\pi\)
\(954\) 0 0
\(955\) −9.81691 −0.317668
\(956\) −41.4447 −1.34042
\(957\) 0 0
\(958\) −24.4947 −0.791386
\(959\) 7.40716 0.239190
\(960\) 0 0
\(961\) −11.3769 −0.366997
\(962\) 122.501 3.94960
\(963\) 0 0
\(964\) 107.595 3.46540
\(965\) 23.4538 0.755003
\(966\) 0 0
\(967\) −0.289444 −0.00930789 −0.00465395 0.999989i \(-0.501481\pi\)
−0.00465395 + 0.999989i \(0.501481\pi\)
\(968\) −105.615 −3.39459
\(969\) 0 0
\(970\) 48.6953 1.56351
\(971\) −7.65950 −0.245805 −0.122903 0.992419i \(-0.539220\pi\)
−0.122903 + 0.992419i \(0.539220\pi\)
\(972\) 0 0
\(973\) −16.7704 −0.537635
\(974\) −32.6278 −1.04546
\(975\) 0 0
\(976\) −101.347 −3.24405
\(977\) −21.5704 −0.690099 −0.345050 0.938584i \(-0.612138\pi\)
−0.345050 + 0.938584i \(0.612138\pi\)
\(978\) 0 0
\(979\) −1.67801 −0.0536296
\(980\) −11.2519 −0.359430
\(981\) 0 0
\(982\) −8.95683 −0.285824
\(983\) −2.55468 −0.0814816 −0.0407408 0.999170i \(-0.512972\pi\)
−0.0407408 + 0.999170i \(0.512972\pi\)
\(984\) 0 0
\(985\) 16.8785 0.537794
\(986\) −23.5951 −0.751422
\(987\) 0 0
\(988\) 213.768 6.80086
\(989\) −30.7565 −0.977999
\(990\) 0 0
\(991\) 32.4736 1.03156 0.515779 0.856722i \(-0.327503\pi\)
0.515779 + 0.856722i \(0.327503\pi\)
\(992\) 100.826 3.20123
\(993\) 0 0
\(994\) −25.1811 −0.798697
\(995\) −5.26482 −0.166906
\(996\) 0 0
\(997\) 41.6147 1.31795 0.658975 0.752165i \(-0.270990\pi\)
0.658975 + 0.752165i \(0.270990\pi\)
\(998\) −7.76536 −0.245808
\(999\) 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.m.1.11 11
3.2 odd 2 2667.2.a.k.1.1 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.k.1.1 11 3.2 odd 2
8001.2.a.m.1.11 11 1.1 even 1 trivial