Properties

Label 6039.2.a.e.1.10
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $12$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} - 71 x^{3} - 93 x^{2} + 13 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-1.72434\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.72434 q^{2} +0.973355 q^{4} +0.0158764 q^{5} +3.08835 q^{7} -1.77029 q^{8} +O(q^{10})\) \(q+1.72434 q^{2} +0.973355 q^{4} +0.0158764 q^{5} +3.08835 q^{7} -1.77029 q^{8} +0.0273764 q^{10} +1.00000 q^{11} +0.101547 q^{13} +5.32537 q^{14} -4.99929 q^{16} +0.754411 q^{17} -6.01153 q^{19} +0.0154534 q^{20} +1.72434 q^{22} -7.29458 q^{23} -4.99975 q^{25} +0.175101 q^{26} +3.00606 q^{28} -2.30385 q^{29} -10.2560 q^{31} -5.07991 q^{32} +1.30086 q^{34} +0.0490319 q^{35} -1.17730 q^{37} -10.3659 q^{38} -0.0281059 q^{40} +6.03943 q^{41} -3.46377 q^{43} +0.973355 q^{44} -12.5783 q^{46} -4.51719 q^{47} +2.53789 q^{49} -8.62127 q^{50} +0.0988411 q^{52} -5.67332 q^{53} +0.0158764 q^{55} -5.46726 q^{56} -3.97263 q^{58} -1.22945 q^{59} -1.00000 q^{61} -17.6849 q^{62} +1.23908 q^{64} +0.00161220 q^{65} +9.40484 q^{67} +0.734309 q^{68} +0.0845478 q^{70} +9.66939 q^{71} +5.87787 q^{73} -2.03007 q^{74} -5.85135 q^{76} +3.08835 q^{77} +10.7844 q^{79} -0.0793709 q^{80} +10.4140 q^{82} +14.4808 q^{83} +0.0119774 q^{85} -5.97272 q^{86} -1.77029 q^{88} -5.44295 q^{89} +0.313612 q^{91} -7.10021 q^{92} -7.78918 q^{94} -0.0954416 q^{95} -6.37384 q^{97} +4.37618 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.72434 1.21929 0.609647 0.792673i \(-0.291311\pi\)
0.609647 + 0.792673i \(0.291311\pi\)
\(3\) 0 0
\(4\) 0.973355 0.486677
\(5\) 0.0158764 0.00710016 0.00355008 0.999994i \(-0.498870\pi\)
0.00355008 + 0.999994i \(0.498870\pi\)
\(6\) 0 0
\(7\) 3.08835 1.16729 0.583643 0.812011i \(-0.301627\pi\)
0.583643 + 0.812011i \(0.301627\pi\)
\(8\) −1.77029 −0.625891
\(9\) 0 0
\(10\) 0.0273764 0.00865718
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0.101547 0.0281640 0.0140820 0.999901i \(-0.495517\pi\)
0.0140820 + 0.999901i \(0.495517\pi\)
\(14\) 5.32537 1.42326
\(15\) 0 0
\(16\) −4.99929 −1.24982
\(17\) 0.754411 0.182971 0.0914857 0.995806i \(-0.470838\pi\)
0.0914857 + 0.995806i \(0.470838\pi\)
\(18\) 0 0
\(19\) −6.01153 −1.37914 −0.689569 0.724220i \(-0.742200\pi\)
−0.689569 + 0.724220i \(0.742200\pi\)
\(20\) 0.0154534 0.00345549
\(21\) 0 0
\(22\) 1.72434 0.367631
\(23\) −7.29458 −1.52103 −0.760513 0.649323i \(-0.775052\pi\)
−0.760513 + 0.649323i \(0.775052\pi\)
\(24\) 0 0
\(25\) −4.99975 −0.999950
\(26\) 0.175101 0.0343402
\(27\) 0 0
\(28\) 3.00606 0.568091
\(29\) −2.30385 −0.427814 −0.213907 0.976854i \(-0.568619\pi\)
−0.213907 + 0.976854i \(0.568619\pi\)
\(30\) 0 0
\(31\) −10.2560 −1.84203 −0.921016 0.389524i \(-0.872640\pi\)
−0.921016 + 0.389524i \(0.872640\pi\)
\(32\) −5.07991 −0.898010
\(33\) 0 0
\(34\) 1.30086 0.223096
\(35\) 0.0490319 0.00828791
\(36\) 0 0
\(37\) −1.17730 −0.193547 −0.0967737 0.995306i \(-0.530852\pi\)
−0.0967737 + 0.995306i \(0.530852\pi\)
\(38\) −10.3659 −1.68158
\(39\) 0 0
\(40\) −0.0281059 −0.00444393
\(41\) 6.03943 0.943200 0.471600 0.881813i \(-0.343677\pi\)
0.471600 + 0.881813i \(0.343677\pi\)
\(42\) 0 0
\(43\) −3.46377 −0.528219 −0.264110 0.964493i \(-0.585078\pi\)
−0.264110 + 0.964493i \(0.585078\pi\)
\(44\) 0.973355 0.146739
\(45\) 0 0
\(46\) −12.5783 −1.85458
\(47\) −4.51719 −0.658900 −0.329450 0.944173i \(-0.606863\pi\)
−0.329450 + 0.944173i \(0.606863\pi\)
\(48\) 0 0
\(49\) 2.53789 0.362555
\(50\) −8.62127 −1.21923
\(51\) 0 0
\(52\) 0.0988411 0.0137068
\(53\) −5.67332 −0.779290 −0.389645 0.920965i \(-0.627402\pi\)
−0.389645 + 0.920965i \(0.627402\pi\)
\(54\) 0 0
\(55\) 0.0158764 0.00214078
\(56\) −5.46726 −0.730593
\(57\) 0 0
\(58\) −3.97263 −0.521631
\(59\) −1.22945 −0.160061 −0.0800307 0.996792i \(-0.525502\pi\)
−0.0800307 + 0.996792i \(0.525502\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −17.6849 −2.24598
\(63\) 0 0
\(64\) 1.23908 0.154885
\(65\) 0.00161220 0.000199969 0
\(66\) 0 0
\(67\) 9.40484 1.14898 0.574492 0.818510i \(-0.305200\pi\)
0.574492 + 0.818510i \(0.305200\pi\)
\(68\) 0.734309 0.0890481
\(69\) 0 0
\(70\) 0.0845478 0.0101054
\(71\) 9.66939 1.14755 0.573773 0.819014i \(-0.305479\pi\)
0.573773 + 0.819014i \(0.305479\pi\)
\(72\) 0 0
\(73\) 5.87787 0.687953 0.343977 0.938978i \(-0.388226\pi\)
0.343977 + 0.938978i \(0.388226\pi\)
\(74\) −2.03007 −0.235991
\(75\) 0 0
\(76\) −5.85135 −0.671196
\(77\) 3.08835 0.351950
\(78\) 0 0
\(79\) 10.7844 1.21333 0.606667 0.794956i \(-0.292506\pi\)
0.606667 + 0.794956i \(0.292506\pi\)
\(80\) −0.0793709 −0.00887394
\(81\) 0 0
\(82\) 10.4140 1.15004
\(83\) 14.4808 1.58948 0.794738 0.606953i \(-0.207608\pi\)
0.794738 + 0.606953i \(0.207608\pi\)
\(84\) 0 0
\(85\) 0.0119774 0.00129913
\(86\) −5.97272 −0.644055
\(87\) 0 0
\(88\) −1.77029 −0.188713
\(89\) −5.44295 −0.576952 −0.288476 0.957487i \(-0.593149\pi\)
−0.288476 + 0.957487i \(0.593149\pi\)
\(90\) 0 0
\(91\) 0.313612 0.0328754
\(92\) −7.10021 −0.740249
\(93\) 0 0
\(94\) −7.78918 −0.803392
\(95\) −0.0954416 −0.00979210
\(96\) 0 0
\(97\) −6.37384 −0.647165 −0.323583 0.946200i \(-0.604887\pi\)
−0.323583 + 0.946200i \(0.604887\pi\)
\(98\) 4.37618 0.442061
\(99\) 0 0
\(100\) −4.86653 −0.486653
\(101\) 8.76993 0.872640 0.436320 0.899791i \(-0.356282\pi\)
0.436320 + 0.899791i \(0.356282\pi\)
\(102\) 0 0
\(103\) −11.7090 −1.15372 −0.576860 0.816843i \(-0.695722\pi\)
−0.576860 + 0.816843i \(0.695722\pi\)
\(104\) −0.179767 −0.0176276
\(105\) 0 0
\(106\) −9.78274 −0.950184
\(107\) 2.21519 0.214150 0.107075 0.994251i \(-0.465851\pi\)
0.107075 + 0.994251i \(0.465851\pi\)
\(108\) 0 0
\(109\) −18.6496 −1.78630 −0.893152 0.449755i \(-0.851511\pi\)
−0.893152 + 0.449755i \(0.851511\pi\)
\(110\) 0.0273764 0.00261024
\(111\) 0 0
\(112\) −15.4395 −1.45890
\(113\) −7.43324 −0.699261 −0.349630 0.936888i \(-0.613693\pi\)
−0.349630 + 0.936888i \(0.613693\pi\)
\(114\) 0 0
\(115\) −0.115812 −0.0107995
\(116\) −2.24246 −0.208208
\(117\) 0 0
\(118\) −2.12000 −0.195162
\(119\) 2.32988 0.213580
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.72434 −0.156115
\(123\) 0 0
\(124\) −9.98273 −0.896476
\(125\) −0.158760 −0.0142000
\(126\) 0 0
\(127\) −0.148396 −0.0131680 −0.00658402 0.999978i \(-0.502096\pi\)
−0.00658402 + 0.999978i \(0.502096\pi\)
\(128\) 12.2964 1.08686
\(129\) 0 0
\(130\) 0.00277999 0.000243821 0
\(131\) −8.37503 −0.731730 −0.365865 0.930668i \(-0.619227\pi\)
−0.365865 + 0.930668i \(0.619227\pi\)
\(132\) 0 0
\(133\) −18.5657 −1.60985
\(134\) 16.2172 1.40095
\(135\) 0 0
\(136\) −1.33552 −0.114520
\(137\) 5.01664 0.428600 0.214300 0.976768i \(-0.431253\pi\)
0.214300 + 0.976768i \(0.431253\pi\)
\(138\) 0 0
\(139\) −12.4343 −1.05466 −0.527331 0.849660i \(-0.676807\pi\)
−0.527331 + 0.849660i \(0.676807\pi\)
\(140\) 0.0477255 0.00403354
\(141\) 0 0
\(142\) 16.6733 1.39920
\(143\) 0.101547 0.00849177
\(144\) 0 0
\(145\) −0.0365769 −0.00303755
\(146\) 10.1355 0.838817
\(147\) 0 0
\(148\) −1.14593 −0.0941952
\(149\) −14.1975 −1.16310 −0.581552 0.813509i \(-0.697554\pi\)
−0.581552 + 0.813509i \(0.697554\pi\)
\(150\) 0 0
\(151\) 17.9461 1.46043 0.730215 0.683218i \(-0.239420\pi\)
0.730215 + 0.683218i \(0.239420\pi\)
\(152\) 10.6421 0.863191
\(153\) 0 0
\(154\) 5.32537 0.429130
\(155\) −0.162829 −0.0130787
\(156\) 0 0
\(157\) −11.8815 −0.948249 −0.474124 0.880458i \(-0.657235\pi\)
−0.474124 + 0.880458i \(0.657235\pi\)
\(158\) 18.5959 1.47941
\(159\) 0 0
\(160\) −0.0806509 −0.00637601
\(161\) −22.5282 −1.77547
\(162\) 0 0
\(163\) −24.0288 −1.88208 −0.941040 0.338294i \(-0.890150\pi\)
−0.941040 + 0.338294i \(0.890150\pi\)
\(164\) 5.87851 0.459034
\(165\) 0 0
\(166\) 24.9699 1.93804
\(167\) 24.8668 1.92425 0.962126 0.272605i \(-0.0878852\pi\)
0.962126 + 0.272605i \(0.0878852\pi\)
\(168\) 0 0
\(169\) −12.9897 −0.999207
\(170\) 0.0206531 0.00158402
\(171\) 0 0
\(172\) −3.37147 −0.257072
\(173\) −0.865461 −0.0657998 −0.0328999 0.999459i \(-0.510474\pi\)
−0.0328999 + 0.999459i \(0.510474\pi\)
\(174\) 0 0
\(175\) −15.4410 −1.16723
\(176\) −4.99929 −0.376836
\(177\) 0 0
\(178\) −9.38551 −0.703474
\(179\) 9.95118 0.743786 0.371893 0.928276i \(-0.378709\pi\)
0.371893 + 0.928276i \(0.378709\pi\)
\(180\) 0 0
\(181\) 13.7754 1.02391 0.511957 0.859011i \(-0.328921\pi\)
0.511957 + 0.859011i \(0.328921\pi\)
\(182\) 0.540774 0.0400848
\(183\) 0 0
\(184\) 12.9135 0.951996
\(185\) −0.0186914 −0.00137422
\(186\) 0 0
\(187\) 0.754411 0.0551680
\(188\) −4.39683 −0.320672
\(189\) 0 0
\(190\) −0.164574 −0.0119395
\(191\) −22.0940 −1.59866 −0.799332 0.600889i \(-0.794813\pi\)
−0.799332 + 0.600889i \(0.794813\pi\)
\(192\) 0 0
\(193\) −4.04001 −0.290806 −0.145403 0.989372i \(-0.546448\pi\)
−0.145403 + 0.989372i \(0.546448\pi\)
\(194\) −10.9907 −0.789085
\(195\) 0 0
\(196\) 2.47026 0.176447
\(197\) 24.3663 1.73603 0.868013 0.496541i \(-0.165397\pi\)
0.868013 + 0.496541i \(0.165397\pi\)
\(198\) 0 0
\(199\) −1.80517 −0.127965 −0.0639824 0.997951i \(-0.520380\pi\)
−0.0639824 + 0.997951i \(0.520380\pi\)
\(200\) 8.85099 0.625859
\(201\) 0 0
\(202\) 15.1224 1.06400
\(203\) −7.11509 −0.499381
\(204\) 0 0
\(205\) 0.0958846 0.00669687
\(206\) −20.1903 −1.40672
\(207\) 0 0
\(208\) −0.507662 −0.0352000
\(209\) −6.01153 −0.415826
\(210\) 0 0
\(211\) 13.0724 0.899940 0.449970 0.893044i \(-0.351435\pi\)
0.449970 + 0.893044i \(0.351435\pi\)
\(212\) −5.52215 −0.379263
\(213\) 0 0
\(214\) 3.81974 0.261112
\(215\) −0.0549923 −0.00375044
\(216\) 0 0
\(217\) −31.6741 −2.15018
\(218\) −32.1582 −2.17803
\(219\) 0 0
\(220\) 0.0154534 0.00104187
\(221\) 0.0766080 0.00515321
\(222\) 0 0
\(223\) −23.8988 −1.60038 −0.800190 0.599747i \(-0.795268\pi\)
−0.800190 + 0.599747i \(0.795268\pi\)
\(224\) −15.6885 −1.04823
\(225\) 0 0
\(226\) −12.8175 −0.852604
\(227\) −14.8395 −0.984930 −0.492465 0.870332i \(-0.663904\pi\)
−0.492465 + 0.870332i \(0.663904\pi\)
\(228\) 0 0
\(229\) −20.7034 −1.36812 −0.684060 0.729426i \(-0.739787\pi\)
−0.684060 + 0.729426i \(0.739787\pi\)
\(230\) −0.199699 −0.0131678
\(231\) 0 0
\(232\) 4.07848 0.267765
\(233\) −16.2119 −1.06208 −0.531039 0.847347i \(-0.678198\pi\)
−0.531039 + 0.847347i \(0.678198\pi\)
\(234\) 0 0
\(235\) −0.0717169 −0.00467829
\(236\) −1.19670 −0.0778983
\(237\) 0 0
\(238\) 4.01751 0.260417
\(239\) 15.1898 0.982547 0.491273 0.871005i \(-0.336532\pi\)
0.491273 + 0.871005i \(0.336532\pi\)
\(240\) 0 0
\(241\) −5.58027 −0.359457 −0.179729 0.983716i \(-0.557522\pi\)
−0.179729 + 0.983716i \(0.557522\pi\)
\(242\) 1.72434 0.110845
\(243\) 0 0
\(244\) −0.973355 −0.0623127
\(245\) 0.0402926 0.00257420
\(246\) 0 0
\(247\) −0.610451 −0.0388421
\(248\) 18.1561 1.15291
\(249\) 0 0
\(250\) −0.273757 −0.0173139
\(251\) 8.68870 0.548426 0.274213 0.961669i \(-0.411583\pi\)
0.274213 + 0.961669i \(0.411583\pi\)
\(252\) 0 0
\(253\) −7.29458 −0.458606
\(254\) −0.255886 −0.0160557
\(255\) 0 0
\(256\) 18.7251 1.17032
\(257\) −19.3953 −1.20985 −0.604923 0.796284i \(-0.706796\pi\)
−0.604923 + 0.796284i \(0.706796\pi\)
\(258\) 0 0
\(259\) −3.63592 −0.225925
\(260\) 0.00156924 9.73204e−5 0
\(261\) 0 0
\(262\) −14.4414 −0.892194
\(263\) 10.4687 0.645526 0.322763 0.946480i \(-0.395388\pi\)
0.322763 + 0.946480i \(0.395388\pi\)
\(264\) 0 0
\(265\) −0.0900721 −0.00553308
\(266\) −32.0136 −1.96288
\(267\) 0 0
\(268\) 9.15425 0.559185
\(269\) 12.2219 0.745182 0.372591 0.927996i \(-0.378469\pi\)
0.372591 + 0.927996i \(0.378469\pi\)
\(270\) 0 0
\(271\) −18.4349 −1.11984 −0.559922 0.828546i \(-0.689169\pi\)
−0.559922 + 0.828546i \(0.689169\pi\)
\(272\) −3.77152 −0.228682
\(273\) 0 0
\(274\) 8.65040 0.522590
\(275\) −4.99975 −0.301496
\(276\) 0 0
\(277\) 25.1637 1.51194 0.755971 0.654605i \(-0.227165\pi\)
0.755971 + 0.654605i \(0.227165\pi\)
\(278\) −21.4409 −1.28594
\(279\) 0 0
\(280\) −0.0868006 −0.00518733
\(281\) 7.27100 0.433752 0.216876 0.976199i \(-0.430413\pi\)
0.216876 + 0.976199i \(0.430413\pi\)
\(282\) 0 0
\(283\) 4.06532 0.241658 0.120829 0.992673i \(-0.461445\pi\)
0.120829 + 0.992673i \(0.461445\pi\)
\(284\) 9.41175 0.558485
\(285\) 0 0
\(286\) 0.175101 0.0103540
\(287\) 18.6518 1.10098
\(288\) 0 0
\(289\) −16.4309 −0.966521
\(290\) −0.0630711 −0.00370366
\(291\) 0 0
\(292\) 5.72126 0.334811
\(293\) −6.32001 −0.369219 −0.184609 0.982812i \(-0.559102\pi\)
−0.184609 + 0.982812i \(0.559102\pi\)
\(294\) 0 0
\(295\) −0.0195194 −0.00113646
\(296\) 2.08417 0.121140
\(297\) 0 0
\(298\) −24.4813 −1.41816
\(299\) −0.740741 −0.0428382
\(300\) 0 0
\(301\) −10.6973 −0.616583
\(302\) 30.9451 1.78069
\(303\) 0 0
\(304\) 30.0534 1.72368
\(305\) −0.0158764 −0.000909082 0
\(306\) 0 0
\(307\) 13.6825 0.780900 0.390450 0.920624i \(-0.372319\pi\)
0.390450 + 0.920624i \(0.372319\pi\)
\(308\) 3.00606 0.171286
\(309\) 0 0
\(310\) −0.280773 −0.0159468
\(311\) 26.7956 1.51944 0.759718 0.650252i \(-0.225337\pi\)
0.759718 + 0.650252i \(0.225337\pi\)
\(312\) 0 0
\(313\) 17.7559 1.00362 0.501810 0.864978i \(-0.332668\pi\)
0.501810 + 0.864978i \(0.332668\pi\)
\(314\) −20.4878 −1.15619
\(315\) 0 0
\(316\) 10.4970 0.590503
\(317\) 5.56189 0.312387 0.156193 0.987726i \(-0.450078\pi\)
0.156193 + 0.987726i \(0.450078\pi\)
\(318\) 0 0
\(319\) −2.30385 −0.128991
\(320\) 0.0196721 0.00109971
\(321\) 0 0
\(322\) −38.8463 −2.16482
\(323\) −4.53516 −0.252343
\(324\) 0 0
\(325\) −0.507708 −0.0281626
\(326\) −41.4339 −2.29481
\(327\) 0 0
\(328\) −10.6915 −0.590341
\(329\) −13.9506 −0.769124
\(330\) 0 0
\(331\) −11.9891 −0.658979 −0.329490 0.944159i \(-0.606877\pi\)
−0.329490 + 0.944159i \(0.606877\pi\)
\(332\) 14.0950 0.773562
\(333\) 0 0
\(334\) 42.8789 2.34623
\(335\) 0.149315 0.00815797
\(336\) 0 0
\(337\) 21.0217 1.14513 0.572563 0.819861i \(-0.305949\pi\)
0.572563 + 0.819861i \(0.305949\pi\)
\(338\) −22.3987 −1.21833
\(339\) 0 0
\(340\) 0.0116582 0.000632256 0
\(341\) −10.2560 −0.555394
\(342\) 0 0
\(343\) −13.7806 −0.744080
\(344\) 6.13186 0.330608
\(345\) 0 0
\(346\) −1.49235 −0.0802293
\(347\) 1.01042 0.0542421 0.0271211 0.999632i \(-0.491366\pi\)
0.0271211 + 0.999632i \(0.491366\pi\)
\(348\) 0 0
\(349\) −7.58868 −0.406213 −0.203106 0.979157i \(-0.565104\pi\)
−0.203106 + 0.979157i \(0.565104\pi\)
\(350\) −26.6255 −1.42319
\(351\) 0 0
\(352\) −5.07991 −0.270760
\(353\) 8.01525 0.426609 0.213304 0.976986i \(-0.431577\pi\)
0.213304 + 0.976986i \(0.431577\pi\)
\(354\) 0 0
\(355\) 0.153516 0.00814776
\(356\) −5.29793 −0.280789
\(357\) 0 0
\(358\) 17.1592 0.906894
\(359\) −8.17890 −0.431666 −0.215833 0.976430i \(-0.569247\pi\)
−0.215833 + 0.976430i \(0.569247\pi\)
\(360\) 0 0
\(361\) 17.1385 0.902024
\(362\) 23.7534 1.24845
\(363\) 0 0
\(364\) 0.305255 0.0159997
\(365\) 0.0933197 0.00488458
\(366\) 0 0
\(367\) 19.9078 1.03918 0.519589 0.854416i \(-0.326085\pi\)
0.519589 + 0.854416i \(0.326085\pi\)
\(368\) 36.4677 1.90101
\(369\) 0 0
\(370\) −0.0322303 −0.00167558
\(371\) −17.5212 −0.909654
\(372\) 0 0
\(373\) 13.0859 0.677562 0.338781 0.940865i \(-0.389985\pi\)
0.338781 + 0.940865i \(0.389985\pi\)
\(374\) 1.30086 0.0672660
\(375\) 0 0
\(376\) 7.99672 0.412399
\(377\) −0.233949 −0.0120490
\(378\) 0 0
\(379\) −27.3665 −1.40572 −0.702861 0.711327i \(-0.748095\pi\)
−0.702861 + 0.711327i \(0.748095\pi\)
\(380\) −0.0928986 −0.00476560
\(381\) 0 0
\(382\) −38.0976 −1.94924
\(383\) −4.11063 −0.210043 −0.105022 0.994470i \(-0.533491\pi\)
−0.105022 + 0.994470i \(0.533491\pi\)
\(384\) 0 0
\(385\) 0.0490319 0.00249890
\(386\) −6.96636 −0.354579
\(387\) 0 0
\(388\) −6.20401 −0.314961
\(389\) 22.1925 1.12520 0.562602 0.826728i \(-0.309800\pi\)
0.562602 + 0.826728i \(0.309800\pi\)
\(390\) 0 0
\(391\) −5.50311 −0.278304
\(392\) −4.49279 −0.226920
\(393\) 0 0
\(394\) 42.0158 2.11673
\(395\) 0.171217 0.00861487
\(396\) 0 0
\(397\) −35.9958 −1.80658 −0.903288 0.429035i \(-0.858854\pi\)
−0.903288 + 0.429035i \(0.858854\pi\)
\(398\) −3.11272 −0.156027
\(399\) 0 0
\(400\) 24.9952 1.24976
\(401\) −17.0516 −0.851517 −0.425759 0.904837i \(-0.639993\pi\)
−0.425759 + 0.904837i \(0.639993\pi\)
\(402\) 0 0
\(403\) −1.04146 −0.0518790
\(404\) 8.53625 0.424694
\(405\) 0 0
\(406\) −12.2688 −0.608893
\(407\) −1.17730 −0.0583568
\(408\) 0 0
\(409\) 25.8530 1.27835 0.639175 0.769061i \(-0.279276\pi\)
0.639175 + 0.769061i \(0.279276\pi\)
\(410\) 0.165338 0.00816545
\(411\) 0 0
\(412\) −11.3970 −0.561490
\(413\) −3.79698 −0.186837
\(414\) 0 0
\(415\) 0.229904 0.0112855
\(416\) −0.515849 −0.0252916
\(417\) 0 0
\(418\) −10.3659 −0.507014
\(419\) 7.87860 0.384895 0.192447 0.981307i \(-0.438358\pi\)
0.192447 + 0.981307i \(0.438358\pi\)
\(420\) 0 0
\(421\) 2.42205 0.118044 0.0590218 0.998257i \(-0.481202\pi\)
0.0590218 + 0.998257i \(0.481202\pi\)
\(422\) 22.5413 1.09729
\(423\) 0 0
\(424\) 10.0434 0.487751
\(425\) −3.77186 −0.182962
\(426\) 0 0
\(427\) −3.08835 −0.149456
\(428\) 2.15617 0.104222
\(429\) 0 0
\(430\) −0.0948255 −0.00457289
\(431\) −7.22840 −0.348180 −0.174090 0.984730i \(-0.555698\pi\)
−0.174090 + 0.984730i \(0.555698\pi\)
\(432\) 0 0
\(433\) −14.2577 −0.685181 −0.342591 0.939485i \(-0.611304\pi\)
−0.342591 + 0.939485i \(0.611304\pi\)
\(434\) −54.6170 −2.62170
\(435\) 0 0
\(436\) −18.1526 −0.869354
\(437\) 43.8516 2.09770
\(438\) 0 0
\(439\) 8.01932 0.382741 0.191371 0.981518i \(-0.438707\pi\)
0.191371 + 0.981518i \(0.438707\pi\)
\(440\) −0.0281059 −0.00133989
\(441\) 0 0
\(442\) 0.132098 0.00628328
\(443\) 20.7827 0.987418 0.493709 0.869627i \(-0.335641\pi\)
0.493709 + 0.869627i \(0.335641\pi\)
\(444\) 0 0
\(445\) −0.0864147 −0.00409645
\(446\) −41.2096 −1.95133
\(447\) 0 0
\(448\) 3.82670 0.180795
\(449\) −12.9208 −0.609772 −0.304886 0.952389i \(-0.598618\pi\)
−0.304886 + 0.952389i \(0.598618\pi\)
\(450\) 0 0
\(451\) 6.03943 0.284386
\(452\) −7.23518 −0.340314
\(453\) 0 0
\(454\) −25.5883 −1.20092
\(455\) 0.00497904 0.000233421 0
\(456\) 0 0
\(457\) 30.8263 1.44200 0.720998 0.692938i \(-0.243684\pi\)
0.720998 + 0.692938i \(0.243684\pi\)
\(458\) −35.6998 −1.66814
\(459\) 0 0
\(460\) −0.112726 −0.00525588
\(461\) 20.0826 0.935340 0.467670 0.883903i \(-0.345094\pi\)
0.467670 + 0.883903i \(0.345094\pi\)
\(462\) 0 0
\(463\) 8.14479 0.378520 0.189260 0.981927i \(-0.439391\pi\)
0.189260 + 0.981927i \(0.439391\pi\)
\(464\) 11.5176 0.534692
\(465\) 0 0
\(466\) −27.9549 −1.29499
\(467\) 25.6676 1.18776 0.593878 0.804555i \(-0.297596\pi\)
0.593878 + 0.804555i \(0.297596\pi\)
\(468\) 0 0
\(469\) 29.0454 1.34119
\(470\) −0.123664 −0.00570421
\(471\) 0 0
\(472\) 2.17649 0.100181
\(473\) −3.46377 −0.159264
\(474\) 0 0
\(475\) 30.0561 1.37907
\(476\) 2.26780 0.103945
\(477\) 0 0
\(478\) 26.1924 1.19801
\(479\) 10.8572 0.496079 0.248039 0.968750i \(-0.420214\pi\)
0.248039 + 0.968750i \(0.420214\pi\)
\(480\) 0 0
\(481\) −0.119551 −0.00545107
\(482\) −9.62230 −0.438284
\(483\) 0 0
\(484\) 0.973355 0.0442434
\(485\) −0.101194 −0.00459498
\(486\) 0 0
\(487\) −2.88276 −0.130630 −0.0653151 0.997865i \(-0.520805\pi\)
−0.0653151 + 0.997865i \(0.520805\pi\)
\(488\) 1.77029 0.0801371
\(489\) 0 0
\(490\) 0.0694782 0.00313870
\(491\) −7.83502 −0.353589 −0.176795 0.984248i \(-0.556573\pi\)
−0.176795 + 0.984248i \(0.556573\pi\)
\(492\) 0 0
\(493\) −1.73805 −0.0782778
\(494\) −1.05263 −0.0473599
\(495\) 0 0
\(496\) 51.2727 2.30221
\(497\) 29.8624 1.33951
\(498\) 0 0
\(499\) 17.4204 0.779842 0.389921 0.920848i \(-0.372502\pi\)
0.389921 + 0.920848i \(0.372502\pi\)
\(500\) −0.154530 −0.00691080
\(501\) 0 0
\(502\) 14.9823 0.668693
\(503\) −1.37826 −0.0614537 −0.0307268 0.999528i \(-0.509782\pi\)
−0.0307268 + 0.999528i \(0.509782\pi\)
\(504\) 0 0
\(505\) 0.139235 0.00619588
\(506\) −12.5783 −0.559176
\(507\) 0 0
\(508\) −0.144442 −0.00640859
\(509\) 32.1408 1.42462 0.712308 0.701867i \(-0.247650\pi\)
0.712308 + 0.701867i \(0.247650\pi\)
\(510\) 0 0
\(511\) 18.1529 0.803038
\(512\) 7.69559 0.340100
\(513\) 0 0
\(514\) −33.4441 −1.47516
\(515\) −0.185897 −0.00819160
\(516\) 0 0
\(517\) −4.51719 −0.198666
\(518\) −6.26957 −0.275469
\(519\) 0 0
\(520\) −0.00285406 −0.000125159 0
\(521\) −24.1606 −1.05849 −0.529247 0.848468i \(-0.677526\pi\)
−0.529247 + 0.848468i \(0.677526\pi\)
\(522\) 0 0
\(523\) −25.9398 −1.13427 −0.567134 0.823625i \(-0.691948\pi\)
−0.567134 + 0.823625i \(0.691948\pi\)
\(524\) −8.15188 −0.356116
\(525\) 0 0
\(526\) 18.0516 0.787086
\(527\) −7.73724 −0.337039
\(528\) 0 0
\(529\) 30.2109 1.31352
\(530\) −0.155315 −0.00674645
\(531\) 0 0
\(532\) −18.0710 −0.783477
\(533\) 0.613285 0.0265643
\(534\) 0 0
\(535\) 0.0351693 0.00152050
\(536\) −16.6493 −0.719139
\(537\) 0 0
\(538\) 21.0747 0.908596
\(539\) 2.53789 0.109314
\(540\) 0 0
\(541\) 3.42991 0.147463 0.0737317 0.997278i \(-0.476509\pi\)
0.0737317 + 0.997278i \(0.476509\pi\)
\(542\) −31.7882 −1.36542
\(543\) 0 0
\(544\) −3.83234 −0.164310
\(545\) −0.296089 −0.0126830
\(546\) 0 0
\(547\) −44.1231 −1.88657 −0.943284 0.331986i \(-0.892281\pi\)
−0.943284 + 0.331986i \(0.892281\pi\)
\(548\) 4.88297 0.208590
\(549\) 0 0
\(550\) −8.62127 −0.367612
\(551\) 13.8497 0.590015
\(552\) 0 0
\(553\) 33.3058 1.41631
\(554\) 43.3909 1.84350
\(555\) 0 0
\(556\) −12.1030 −0.513280
\(557\) −34.2569 −1.45151 −0.725756 0.687952i \(-0.758510\pi\)
−0.725756 + 0.687952i \(0.758510\pi\)
\(558\) 0 0
\(559\) −0.351734 −0.0148768
\(560\) −0.245125 −0.0103584
\(561\) 0 0
\(562\) 12.5377 0.528871
\(563\) −46.3008 −1.95134 −0.975672 0.219234i \(-0.929644\pi\)
−0.975672 + 0.219234i \(0.929644\pi\)
\(564\) 0 0
\(565\) −0.118013 −0.00496486
\(566\) 7.01000 0.294652
\(567\) 0 0
\(568\) −17.1176 −0.718239
\(569\) −7.09897 −0.297604 −0.148802 0.988867i \(-0.547542\pi\)
−0.148802 + 0.988867i \(0.547542\pi\)
\(570\) 0 0
\(571\) −40.2690 −1.68521 −0.842603 0.538535i \(-0.818978\pi\)
−0.842603 + 0.538535i \(0.818978\pi\)
\(572\) 0.0988411 0.00413275
\(573\) 0 0
\(574\) 32.1622 1.34242
\(575\) 36.4711 1.52095
\(576\) 0 0
\(577\) −29.5093 −1.22849 −0.614245 0.789116i \(-0.710539\pi\)
−0.614245 + 0.789116i \(0.710539\pi\)
\(578\) −28.3324 −1.17847
\(579\) 0 0
\(580\) −0.0356023 −0.00147831
\(581\) 44.7218 1.85537
\(582\) 0 0
\(583\) −5.67332 −0.234965
\(584\) −10.4055 −0.430584
\(585\) 0 0
\(586\) −10.8979 −0.450186
\(587\) 28.2736 1.16698 0.583488 0.812122i \(-0.301687\pi\)
0.583488 + 0.812122i \(0.301687\pi\)
\(588\) 0 0
\(589\) 61.6542 2.54042
\(590\) −0.0336580 −0.00138568
\(591\) 0 0
\(592\) 5.88568 0.241900
\(593\) −19.3146 −0.793155 −0.396578 0.918001i \(-0.629802\pi\)
−0.396578 + 0.918001i \(0.629802\pi\)
\(594\) 0 0
\(595\) 0.0369902 0.00151645
\(596\) −13.8192 −0.566056
\(597\) 0 0
\(598\) −1.27729 −0.0522323
\(599\) 0.248575 0.0101565 0.00507825 0.999987i \(-0.498384\pi\)
0.00507825 + 0.999987i \(0.498384\pi\)
\(600\) 0 0
\(601\) 20.9655 0.855202 0.427601 0.903968i \(-0.359359\pi\)
0.427601 + 0.903968i \(0.359359\pi\)
\(602\) −18.4458 −0.751796
\(603\) 0 0
\(604\) 17.4679 0.710758
\(605\) 0.0158764 0.000645469 0
\(606\) 0 0
\(607\) 4.54313 0.184400 0.0921999 0.995741i \(-0.470610\pi\)
0.0921999 + 0.995741i \(0.470610\pi\)
\(608\) 30.5380 1.23848
\(609\) 0 0
\(610\) −0.0273764 −0.00110844
\(611\) −0.458706 −0.0185573
\(612\) 0 0
\(613\) 24.9967 1.00961 0.504803 0.863234i \(-0.331565\pi\)
0.504803 + 0.863234i \(0.331565\pi\)
\(614\) 23.5933 0.952146
\(615\) 0 0
\(616\) −5.46726 −0.220282
\(617\) −30.7047 −1.23612 −0.618062 0.786130i \(-0.712082\pi\)
−0.618062 + 0.786130i \(0.712082\pi\)
\(618\) 0 0
\(619\) −1.34685 −0.0541345 −0.0270673 0.999634i \(-0.508617\pi\)
−0.0270673 + 0.999634i \(0.508617\pi\)
\(620\) −0.158490 −0.00636512
\(621\) 0 0
\(622\) 46.2047 1.85264
\(623\) −16.8097 −0.673468
\(624\) 0 0
\(625\) 24.9962 0.999849
\(626\) 30.6172 1.22371
\(627\) 0 0
\(628\) −11.5649 −0.461491
\(629\) −0.888170 −0.0354137
\(630\) 0 0
\(631\) −2.15674 −0.0858584 −0.0429292 0.999078i \(-0.513669\pi\)
−0.0429292 + 0.999078i \(0.513669\pi\)
\(632\) −19.0914 −0.759415
\(633\) 0 0
\(634\) 9.59060 0.380891
\(635\) −0.00235600 −9.34951e−5 0
\(636\) 0 0
\(637\) 0.257714 0.0102110
\(638\) −3.97263 −0.157278
\(639\) 0 0
\(640\) 0.195223 0.00771688
\(641\) 1.74552 0.0689437 0.0344719 0.999406i \(-0.489025\pi\)
0.0344719 + 0.999406i \(0.489025\pi\)
\(642\) 0 0
\(643\) −33.6804 −1.32823 −0.664113 0.747632i \(-0.731191\pi\)
−0.664113 + 0.747632i \(0.731191\pi\)
\(644\) −21.9279 −0.864081
\(645\) 0 0
\(646\) −7.82017 −0.307680
\(647\) −43.0126 −1.69100 −0.845500 0.533975i \(-0.820698\pi\)
−0.845500 + 0.533975i \(0.820698\pi\)
\(648\) 0 0
\(649\) −1.22945 −0.0482603
\(650\) −0.875463 −0.0343385
\(651\) 0 0
\(652\) −23.3885 −0.915966
\(653\) −8.68472 −0.339859 −0.169930 0.985456i \(-0.554354\pi\)
−0.169930 + 0.985456i \(0.554354\pi\)
\(654\) 0 0
\(655\) −0.132966 −0.00519540
\(656\) −30.1929 −1.17883
\(657\) 0 0
\(658\) −24.0557 −0.937788
\(659\) −18.6856 −0.727887 −0.363944 0.931421i \(-0.618570\pi\)
−0.363944 + 0.931421i \(0.618570\pi\)
\(660\) 0 0
\(661\) 37.0230 1.44003 0.720013 0.693960i \(-0.244136\pi\)
0.720013 + 0.693960i \(0.244136\pi\)
\(662\) −20.6733 −0.803489
\(663\) 0 0
\(664\) −25.6352 −0.994838
\(665\) −0.294757 −0.0114302
\(666\) 0 0
\(667\) 16.8056 0.650716
\(668\) 24.2042 0.936490
\(669\) 0 0
\(670\) 0.257471 0.00994696
\(671\) −1.00000 −0.0386046
\(672\) 0 0
\(673\) −18.9352 −0.729900 −0.364950 0.931027i \(-0.618914\pi\)
−0.364950 + 0.931027i \(0.618914\pi\)
\(674\) 36.2486 1.39624
\(675\) 0 0
\(676\) −12.6436 −0.486291
\(677\) 16.3128 0.626951 0.313476 0.949596i \(-0.398507\pi\)
0.313476 + 0.949596i \(0.398507\pi\)
\(678\) 0 0
\(679\) −19.6846 −0.755427
\(680\) −0.0212034 −0.000813112 0
\(681\) 0 0
\(682\) −17.6849 −0.677188
\(683\) 42.7001 1.63387 0.816937 0.576726i \(-0.195670\pi\)
0.816937 + 0.576726i \(0.195670\pi\)
\(684\) 0 0
\(685\) 0.0796463 0.00304313
\(686\) −23.7624 −0.907252
\(687\) 0 0
\(688\) 17.3164 0.660181
\(689\) −0.576107 −0.0219479
\(690\) 0 0
\(691\) 20.7488 0.789323 0.394661 0.918827i \(-0.370862\pi\)
0.394661 + 0.918827i \(0.370862\pi\)
\(692\) −0.842401 −0.0320233
\(693\) 0 0
\(694\) 1.74231 0.0661371
\(695\) −0.197412 −0.00748826
\(696\) 0 0
\(697\) 4.55621 0.172579
\(698\) −13.0855 −0.495293
\(699\) 0 0
\(700\) −15.0295 −0.568063
\(701\) −13.8266 −0.522222 −0.261111 0.965309i \(-0.584089\pi\)
−0.261111 + 0.965309i \(0.584089\pi\)
\(702\) 0 0
\(703\) 7.07739 0.266929
\(704\) 1.23908 0.0466995
\(705\) 0 0
\(706\) 13.8210 0.520161
\(707\) 27.0846 1.01862
\(708\) 0 0
\(709\) −38.7009 −1.45344 −0.726721 0.686933i \(-0.758957\pi\)
−0.726721 + 0.686933i \(0.758957\pi\)
\(710\) 0.264713 0.00993451
\(711\) 0 0
\(712\) 9.63559 0.361109
\(713\) 74.8132 2.80178
\(714\) 0 0
\(715\) 0.00161220 6.02929e−5 0
\(716\) 9.68603 0.361984
\(717\) 0 0
\(718\) −14.1032 −0.526328
\(719\) −3.99446 −0.148968 −0.0744841 0.997222i \(-0.523731\pi\)
−0.0744841 + 0.997222i \(0.523731\pi\)
\(720\) 0 0
\(721\) −36.1614 −1.34672
\(722\) 29.5525 1.09983
\(723\) 0 0
\(724\) 13.4083 0.498316
\(725\) 11.5187 0.427793
\(726\) 0 0
\(727\) 0.349160 0.0129496 0.00647481 0.999979i \(-0.497939\pi\)
0.00647481 + 0.999979i \(0.497939\pi\)
\(728\) −0.555183 −0.0205764
\(729\) 0 0
\(730\) 0.160915 0.00595573
\(731\) −2.61310 −0.0966491
\(732\) 0 0
\(733\) −34.4730 −1.27329 −0.636644 0.771158i \(-0.719678\pi\)
−0.636644 + 0.771158i \(0.719678\pi\)
\(734\) 34.3278 1.26706
\(735\) 0 0
\(736\) 37.0558 1.36590
\(737\) 9.40484 0.346432
\(738\) 0 0
\(739\) 10.6069 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(740\) −0.0181933 −0.000668801 0
\(741\) 0 0
\(742\) −30.2125 −1.10914
\(743\) 24.4783 0.898021 0.449010 0.893527i \(-0.351777\pi\)
0.449010 + 0.893527i \(0.351777\pi\)
\(744\) 0 0
\(745\) −0.225405 −0.00825822
\(746\) 22.5645 0.826147
\(747\) 0 0
\(748\) 0.734309 0.0268490
\(749\) 6.84127 0.249975
\(750\) 0 0
\(751\) −17.8215 −0.650315 −0.325157 0.945660i \(-0.605417\pi\)
−0.325157 + 0.945660i \(0.605417\pi\)
\(752\) 22.5827 0.823508
\(753\) 0 0
\(754\) −0.403407 −0.0146912
\(755\) 0.284919 0.0103693
\(756\) 0 0
\(757\) −23.8569 −0.867092 −0.433546 0.901131i \(-0.642738\pi\)
−0.433546 + 0.901131i \(0.642738\pi\)
\(758\) −47.1892 −1.71399
\(759\) 0 0
\(760\) 0.168959 0.00612879
\(761\) −25.1121 −0.910314 −0.455157 0.890411i \(-0.650417\pi\)
−0.455157 + 0.890411i \(0.650417\pi\)
\(762\) 0 0
\(763\) −57.5963 −2.08513
\(764\) −21.5053 −0.778034
\(765\) 0 0
\(766\) −7.08813 −0.256105
\(767\) −0.124847 −0.00450797
\(768\) 0 0
\(769\) 36.7996 1.32703 0.663514 0.748164i \(-0.269064\pi\)
0.663514 + 0.748164i \(0.269064\pi\)
\(770\) 0.0845478 0.00304689
\(771\) 0 0
\(772\) −3.93237 −0.141529
\(773\) 26.2515 0.944201 0.472101 0.881545i \(-0.343496\pi\)
0.472101 + 0.881545i \(0.343496\pi\)
\(774\) 0 0
\(775\) 51.2774 1.84194
\(776\) 11.2835 0.405055
\(777\) 0 0
\(778\) 38.2675 1.37195
\(779\) −36.3062 −1.30080
\(780\) 0 0
\(781\) 9.66939 0.345998
\(782\) −9.48924 −0.339335
\(783\) 0 0
\(784\) −12.6876 −0.453130
\(785\) −0.188636 −0.00673272
\(786\) 0 0
\(787\) 41.7502 1.48823 0.744116 0.668050i \(-0.232871\pi\)
0.744116 + 0.668050i \(0.232871\pi\)
\(788\) 23.7171 0.844885
\(789\) 0 0
\(790\) 0.295237 0.0105041
\(791\) −22.9564 −0.816237
\(792\) 0 0
\(793\) −0.101547 −0.00360603
\(794\) −62.0690 −2.20275
\(795\) 0 0
\(796\) −1.75707 −0.0622776
\(797\) −4.53552 −0.160656 −0.0803282 0.996768i \(-0.525597\pi\)
−0.0803282 + 0.996768i \(0.525597\pi\)
\(798\) 0 0
\(799\) −3.40782 −0.120560
\(800\) 25.3983 0.897965
\(801\) 0 0
\(802\) −29.4028 −1.03825
\(803\) 5.87787 0.207426
\(804\) 0 0
\(805\) −0.357667 −0.0126061
\(806\) −1.79584 −0.0632558
\(807\) 0 0
\(808\) −15.5253 −0.546178
\(809\) 50.1441 1.76297 0.881486 0.472210i \(-0.156544\pi\)
0.881486 + 0.472210i \(0.156544\pi\)
\(810\) 0 0
\(811\) 18.4087 0.646418 0.323209 0.946328i \(-0.395238\pi\)
0.323209 + 0.946328i \(0.395238\pi\)
\(812\) −6.92551 −0.243038
\(813\) 0 0
\(814\) −2.03007 −0.0711540
\(815\) −0.381492 −0.0133631
\(816\) 0 0
\(817\) 20.8225 0.728488
\(818\) 44.5795 1.55869
\(819\) 0 0
\(820\) 0.0933297 0.00325922
\(821\) −23.9297 −0.835154 −0.417577 0.908642i \(-0.637121\pi\)
−0.417577 + 0.908642i \(0.637121\pi\)
\(822\) 0 0
\(823\) −3.32320 −0.115839 −0.0579197 0.998321i \(-0.518447\pi\)
−0.0579197 + 0.998321i \(0.518447\pi\)
\(824\) 20.7283 0.722103
\(825\) 0 0
\(826\) −6.54730 −0.227810
\(827\) 47.6416 1.65666 0.828331 0.560239i \(-0.189291\pi\)
0.828331 + 0.560239i \(0.189291\pi\)
\(828\) 0 0
\(829\) −53.2947 −1.85100 −0.925501 0.378744i \(-0.876356\pi\)
−0.925501 + 0.378744i \(0.876356\pi\)
\(830\) 0.396432 0.0137604
\(831\) 0 0
\(832\) 0.125824 0.00436217
\(833\) 1.91461 0.0663373
\(834\) 0 0
\(835\) 0.394796 0.0136625
\(836\) −5.85135 −0.202373
\(837\) 0 0
\(838\) 13.5854 0.469300
\(839\) −52.8010 −1.82289 −0.911447 0.411417i \(-0.865034\pi\)
−0.911447 + 0.411417i \(0.865034\pi\)
\(840\) 0 0
\(841\) −23.6923 −0.816975
\(842\) 4.17645 0.143930
\(843\) 0 0
\(844\) 12.7241 0.437981
\(845\) −0.206230 −0.00709453
\(846\) 0 0
\(847\) 3.08835 0.106117
\(848\) 28.3626 0.973974
\(849\) 0 0
\(850\) −6.50398 −0.223085
\(851\) 8.58793 0.294391
\(852\) 0 0
\(853\) 9.34877 0.320096 0.160048 0.987109i \(-0.448835\pi\)
0.160048 + 0.987109i \(0.448835\pi\)
\(854\) −5.32537 −0.182230
\(855\) 0 0
\(856\) −3.92152 −0.134035
\(857\) −10.5206 −0.359376 −0.179688 0.983724i \(-0.557509\pi\)
−0.179688 + 0.983724i \(0.557509\pi\)
\(858\) 0 0
\(859\) −17.2276 −0.587799 −0.293899 0.955836i \(-0.594953\pi\)
−0.293899 + 0.955836i \(0.594953\pi\)
\(860\) −0.0535270 −0.00182526
\(861\) 0 0
\(862\) −12.4642 −0.424533
\(863\) 50.3781 1.71489 0.857446 0.514574i \(-0.172050\pi\)
0.857446 + 0.514574i \(0.172050\pi\)
\(864\) 0 0
\(865\) −0.0137404 −0.000467189 0
\(866\) −24.5851 −0.835437
\(867\) 0 0
\(868\) −30.8301 −1.04644
\(869\) 10.7844 0.365834
\(870\) 0 0
\(871\) 0.955032 0.0323600
\(872\) 33.0151 1.11803
\(873\) 0 0
\(874\) 75.6151 2.55772
\(875\) −0.490307 −0.0165754
\(876\) 0 0
\(877\) −2.22641 −0.0751806 −0.0375903 0.999293i \(-0.511968\pi\)
−0.0375903 + 0.999293i \(0.511968\pi\)
\(878\) 13.8280 0.466674
\(879\) 0 0
\(880\) −0.0793709 −0.00267559
\(881\) 22.8872 0.771088 0.385544 0.922689i \(-0.374014\pi\)
0.385544 + 0.922689i \(0.374014\pi\)
\(882\) 0 0
\(883\) −47.5534 −1.60030 −0.800150 0.599800i \(-0.795247\pi\)
−0.800150 + 0.599800i \(0.795247\pi\)
\(884\) 0.0745668 0.00250795
\(885\) 0 0
\(886\) 35.8365 1.20395
\(887\) 40.1853 1.34929 0.674646 0.738142i \(-0.264296\pi\)
0.674646 + 0.738142i \(0.264296\pi\)
\(888\) 0 0
\(889\) −0.458299 −0.0153709
\(890\) −0.149009 −0.00499478
\(891\) 0 0
\(892\) −23.2620 −0.778869
\(893\) 27.1552 0.908714
\(894\) 0 0
\(895\) 0.157989 0.00528100
\(896\) 37.9756 1.26868
\(897\) 0 0
\(898\) −22.2799 −0.743491
\(899\) 23.6283 0.788048
\(900\) 0 0
\(901\) −4.28001 −0.142588
\(902\) 10.4140 0.346750
\(903\) 0 0
\(904\) 13.1590 0.437661
\(905\) 0.218704 0.00726996
\(906\) 0 0
\(907\) −54.9220 −1.82365 −0.911827 0.410574i \(-0.865328\pi\)
−0.911827 + 0.410574i \(0.865328\pi\)
\(908\) −14.4441 −0.479343
\(909\) 0 0
\(910\) 0.00858556 0.000284609 0
\(911\) 16.0542 0.531900 0.265950 0.963987i \(-0.414314\pi\)
0.265950 + 0.963987i \(0.414314\pi\)
\(912\) 0 0
\(913\) 14.4808 0.479245
\(914\) 53.1551 1.75822
\(915\) 0 0
\(916\) −20.1518 −0.665833
\(917\) −25.8650 −0.854138
\(918\) 0 0
\(919\) 19.1366 0.631257 0.315628 0.948883i \(-0.397785\pi\)
0.315628 + 0.948883i \(0.397785\pi\)
\(920\) 0.205020 0.00675932
\(921\) 0 0
\(922\) 34.6293 1.14045
\(923\) 0.981896 0.0323195
\(924\) 0 0
\(925\) 5.88622 0.193538
\(926\) 14.0444 0.461528
\(927\) 0 0
\(928\) 11.7034 0.384181
\(929\) −17.7836 −0.583461 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(930\) 0 0
\(931\) −15.2566 −0.500014
\(932\) −15.7800 −0.516890
\(933\) 0 0
\(934\) 44.2598 1.44822
\(935\) 0.0119774 0.000391701 0
\(936\) 0 0
\(937\) 25.6353 0.837468 0.418734 0.908109i \(-0.362474\pi\)
0.418734 + 0.908109i \(0.362474\pi\)
\(938\) 50.0842 1.63531
\(939\) 0 0
\(940\) −0.0698060 −0.00227682
\(941\) 0.103553 0.00337572 0.00168786 0.999999i \(-0.499463\pi\)
0.00168786 + 0.999999i \(0.499463\pi\)
\(942\) 0 0
\(943\) −44.0551 −1.43463
\(944\) 6.14640 0.200048
\(945\) 0 0
\(946\) −5.97272 −0.194190
\(947\) −30.1840 −0.980847 −0.490424 0.871484i \(-0.663158\pi\)
−0.490424 + 0.871484i \(0.663158\pi\)
\(948\) 0 0
\(949\) 0.596879 0.0193755
\(950\) 51.8270 1.68149
\(951\) 0 0
\(952\) −4.12456 −0.133678
\(953\) −43.1394 −1.39742 −0.698712 0.715403i \(-0.746243\pi\)
−0.698712 + 0.715403i \(0.746243\pi\)
\(954\) 0 0
\(955\) −0.350774 −0.0113508
\(956\) 14.7851 0.478183
\(957\) 0 0
\(958\) 18.7216 0.604866
\(959\) 15.4931 0.500299
\(960\) 0 0
\(961\) 74.1856 2.39309
\(962\) −0.206147 −0.00664646
\(963\) 0 0
\(964\) −5.43159 −0.174940
\(965\) −0.0641410 −0.00206477
\(966\) 0 0
\(967\) −23.7195 −0.762768 −0.381384 0.924417i \(-0.624553\pi\)
−0.381384 + 0.924417i \(0.624553\pi\)
\(968\) −1.77029 −0.0568992
\(969\) 0 0
\(970\) −0.174493 −0.00560263
\(971\) 24.5335 0.787319 0.393659 0.919256i \(-0.371209\pi\)
0.393659 + 0.919256i \(0.371209\pi\)
\(972\) 0 0
\(973\) −38.4013 −1.23109
\(974\) −4.97086 −0.159277
\(975\) 0 0
\(976\) 4.99929 0.160023
\(977\) −28.8837 −0.924071 −0.462035 0.886861i \(-0.652881\pi\)
−0.462035 + 0.886861i \(0.652881\pi\)
\(978\) 0 0
\(979\) −5.44295 −0.173958
\(980\) 0.0392190 0.00125280
\(981\) 0 0
\(982\) −13.5103 −0.431129
\(983\) −42.5661 −1.35765 −0.678823 0.734302i \(-0.737510\pi\)
−0.678823 + 0.734302i \(0.737510\pi\)
\(984\) 0 0
\(985\) 0.386850 0.0123261
\(986\) −2.99699 −0.0954436
\(987\) 0 0
\(988\) −0.594186 −0.0189036
\(989\) 25.2667 0.803435
\(990\) 0 0
\(991\) −3.11669 −0.0990050 −0.0495025 0.998774i \(-0.515764\pi\)
−0.0495025 + 0.998774i \(0.515764\pi\)
\(992\) 52.0996 1.65416
\(993\) 0 0
\(994\) 51.4931 1.63326
\(995\) −0.0286596 −0.000908571 0
\(996\) 0 0
\(997\) −14.2915 −0.452618 −0.226309 0.974056i \(-0.572666\pi\)
−0.226309 + 0.974056i \(0.572666\pi\)
\(998\) 30.0386 0.950857
\(999\) 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 6039.2.a.e.1.10 12
3.2 odd 2 2013.2.a.d.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.d.1.3 12 3.2 odd 2
6039.2.a.e.1.10 12 1.1 even 1 trivial