Properties

Label 5929.2.a.bt.1.4
Level $5929$
Weight $2$
Character 5929.1
Self dual yes
Analytic conductor $47.343$
Analytic rank $0$
Dimension $8$
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] = [5929,2,Mod(1,5929)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5929, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5929.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5929.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: \(47.3433033584\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.226211\) of defining polynomial
Character \(\chi\) \(=\) 5929.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.226211 q^{2} +0.219130 q^{3} -1.94883 q^{4} +2.49552 q^{5} +0.0495696 q^{6} -0.893270 q^{8} -2.95198 q^{9} +O(q^{10})\) \(q+0.226211 q^{2} +0.219130 q^{3} -1.94883 q^{4} +2.49552 q^{5} +0.0495696 q^{6} -0.893270 q^{8} -2.95198 q^{9} +0.564516 q^{10} -0.427046 q^{12} -5.13499 q^{13} +0.546843 q^{15} +3.69559 q^{16} +1.43752 q^{17} -0.667772 q^{18} +6.06848 q^{19} -4.86335 q^{20} +7.08292 q^{23} -0.195742 q^{24} +1.22764 q^{25} -1.16159 q^{26} -1.30426 q^{27} +6.51769 q^{29} +0.123702 q^{30} -7.68895 q^{31} +2.62252 q^{32} +0.325184 q^{34} +5.75291 q^{36} -3.98432 q^{37} +1.37276 q^{38} -1.12523 q^{39} -2.22918 q^{40} -6.74900 q^{41} -0.802299 q^{43} -7.36674 q^{45} +1.60224 q^{46} -6.75222 q^{47} +0.809813 q^{48} +0.277706 q^{50} +0.315004 q^{51} +10.0072 q^{52} +6.58167 q^{53} -0.295037 q^{54} +1.32978 q^{57} +1.47437 q^{58} -2.87625 q^{59} -1.06570 q^{60} +0.855342 q^{61} -1.73933 q^{62} -6.79793 q^{64} -12.8145 q^{65} -1.64668 q^{67} -2.80149 q^{68} +1.55208 q^{69} +4.52077 q^{71} +2.63692 q^{72} +14.8479 q^{73} -0.901299 q^{74} +0.269012 q^{75} -11.8264 q^{76} -0.254539 q^{78} +2.45291 q^{79} +9.22243 q^{80} +8.57015 q^{81} -1.52670 q^{82} -2.24780 q^{83} +3.58738 q^{85} -0.181489 q^{86} +1.42822 q^{87} -1.73566 q^{89} -1.66644 q^{90} -13.8034 q^{92} -1.68488 q^{93} -1.52743 q^{94} +15.1440 q^{95} +0.574672 q^{96} +12.0776 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 4 q^{3} + 7 q^{4} - 10 q^{5} + q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 4 q^{3} + 7 q^{4} - 10 q^{5} + q^{6} + 14 q^{9} - 6 q^{10} - 9 q^{12} + 6 q^{13} + 11 q^{15} + q^{16} + 5 q^{17} + 8 q^{18} + 13 q^{19} - 23 q^{20} + 16 q^{23} + 10 q^{24} + 16 q^{25} + 6 q^{26} - 10 q^{27} + 9 q^{29} - 36 q^{30} - 9 q^{31} + 16 q^{32} + 12 q^{34} - 14 q^{36} + 7 q^{37} + 10 q^{38} + 13 q^{39} - 5 q^{40} + 10 q^{41} - 4 q^{43} - 35 q^{45} + 4 q^{46} - 16 q^{47} + 3 q^{48} + 6 q^{50} + 13 q^{51} + 41 q^{52} + 37 q^{53} - 30 q^{54} + 2 q^{57} - 15 q^{58} - q^{59} + 5 q^{60} - 19 q^{61} + 18 q^{62} - 4 q^{64} - 4 q^{65} - 19 q^{67} - 9 q^{68} - 20 q^{69} + 13 q^{71} - 35 q^{72} + 25 q^{73} + 33 q^{74} + 13 q^{75} - 26 q^{76} - 29 q^{78} - 4 q^{80} + 8 q^{81} + 13 q^{82} + 25 q^{83} + 3 q^{85} + 4 q^{86} + 36 q^{87} - 37 q^{89} + 2 q^{90} + 35 q^{92} + 21 q^{93} + 42 q^{94} + 21 q^{95} + 6 q^{96} - 15 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.226211 0.159956 0.0799778 0.996797i \(-0.474515\pi\)
0.0799778 + 0.996797i \(0.474515\pi\)
\(3\) 0.219130 0.126514 0.0632572 0.997997i \(-0.479851\pi\)
0.0632572 + 0.997997i \(0.479851\pi\)
\(4\) −1.94883 −0.974414
\(5\) 2.49552 1.11603 0.558016 0.829830i \(-0.311563\pi\)
0.558016 + 0.829830i \(0.311563\pi\)
\(6\) 0.0495696 0.0202367
\(7\) 0 0
\(8\) −0.893270 −0.315819
\(9\) −2.95198 −0.983994
\(10\) 0.564516 0.178516
\(11\) 0 0
\(12\) −0.427046 −0.123278
\(13\) −5.13499 −1.42419 −0.712094 0.702084i \(-0.752253\pi\)
−0.712094 + 0.702084i \(0.752253\pi\)
\(14\) 0 0
\(15\) 0.546843 0.141194
\(16\) 3.69559 0.923897
\(17\) 1.43752 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(18\) −0.667772 −0.157395
\(19\) 6.06848 1.39220 0.696102 0.717943i \(-0.254916\pi\)
0.696102 + 0.717943i \(0.254916\pi\)
\(20\) −4.86335 −1.08748
\(21\) 0 0
\(22\) 0 0
\(23\) 7.08292 1.47689 0.738446 0.674313i \(-0.235560\pi\)
0.738446 + 0.674313i \(0.235560\pi\)
\(24\) −0.195742 −0.0399556
\(25\) 1.22764 0.245528
\(26\) −1.16159 −0.227807
\(27\) −1.30426 −0.251004
\(28\) 0 0
\(29\) 6.51769 1.21030 0.605152 0.796110i \(-0.293112\pi\)
0.605152 + 0.796110i \(0.293112\pi\)
\(30\) 0.123702 0.0225848
\(31\) −7.68895 −1.38098 −0.690488 0.723344i \(-0.742604\pi\)
−0.690488 + 0.723344i \(0.742604\pi\)
\(32\) 2.62252 0.463601
\(33\) 0 0
\(34\) 0.325184 0.0557687
\(35\) 0 0
\(36\) 5.75291 0.958818
\(37\) −3.98432 −0.655019 −0.327509 0.944848i \(-0.606209\pi\)
−0.327509 + 0.944848i \(0.606209\pi\)
\(38\) 1.37276 0.222691
\(39\) −1.12523 −0.180181
\(40\) −2.22918 −0.352464
\(41\) −6.74900 −1.05402 −0.527008 0.849860i \(-0.676686\pi\)
−0.527008 + 0.849860i \(0.676686\pi\)
\(42\) 0 0
\(43\) −0.802299 −0.122349 −0.0611747 0.998127i \(-0.519485\pi\)
−0.0611747 + 0.998127i \(0.519485\pi\)
\(44\) 0 0
\(45\) −7.36674 −1.09817
\(46\) 1.60224 0.236237
\(47\) −6.75222 −0.984912 −0.492456 0.870337i \(-0.663901\pi\)
−0.492456 + 0.870337i \(0.663901\pi\)
\(48\) 0.809813 0.116886
\(49\) 0 0
\(50\) 0.277706 0.0392735
\(51\) 0.315004 0.0441094
\(52\) 10.0072 1.38775
\(53\) 6.58167 0.904062 0.452031 0.892002i \(-0.350700\pi\)
0.452031 + 0.892002i \(0.350700\pi\)
\(54\) −0.295037 −0.0401495
\(55\) 0 0
\(56\) 0 0
\(57\) 1.32978 0.176134
\(58\) 1.47437 0.193595
\(59\) −2.87625 −0.374456 −0.187228 0.982316i \(-0.559950\pi\)
−0.187228 + 0.982316i \(0.559950\pi\)
\(60\) −1.06570 −0.137582
\(61\) 0.855342 0.109515 0.0547576 0.998500i \(-0.482561\pi\)
0.0547576 + 0.998500i \(0.482561\pi\)
\(62\) −1.73933 −0.220895
\(63\) 0 0
\(64\) −6.79793 −0.849742
\(65\) −12.8145 −1.58944
\(66\) 0 0
\(67\) −1.64668 −0.201174 −0.100587 0.994928i \(-0.532072\pi\)
−0.100587 + 0.994928i \(0.532072\pi\)
\(68\) −2.80149 −0.339730
\(69\) 1.55208 0.186848
\(70\) 0 0
\(71\) 4.52077 0.536517 0.268258 0.963347i \(-0.413552\pi\)
0.268258 + 0.963347i \(0.413552\pi\)
\(72\) 2.63692 0.310764
\(73\) 14.8479 1.73782 0.868910 0.494970i \(-0.164821\pi\)
0.868910 + 0.494970i \(0.164821\pi\)
\(74\) −0.901299 −0.104774
\(75\) 0.269012 0.0310628
\(76\) −11.8264 −1.35658
\(77\) 0 0
\(78\) −0.254539 −0.0288209
\(79\) 2.45291 0.275973 0.137987 0.990434i \(-0.455937\pi\)
0.137987 + 0.990434i \(0.455937\pi\)
\(80\) 9.22243 1.03110
\(81\) 8.57015 0.952238
\(82\) −1.52670 −0.168596
\(83\) −2.24780 −0.246728 −0.123364 0.992361i \(-0.539368\pi\)
−0.123364 + 0.992361i \(0.539368\pi\)
\(84\) 0 0
\(85\) 3.58738 0.389106
\(86\) −0.181489 −0.0195705
\(87\) 1.42822 0.153121
\(88\) 0 0
\(89\) −1.73566 −0.183980 −0.0919898 0.995760i \(-0.529323\pi\)
−0.0919898 + 0.995760i \(0.529323\pi\)
\(90\) −1.66644 −0.175658
\(91\) 0 0
\(92\) −13.8034 −1.43910
\(93\) −1.68488 −0.174714
\(94\) −1.52743 −0.157542
\(95\) 15.1440 1.55374
\(96\) 0.574672 0.0586523
\(97\) 12.0776 1.22629 0.613145 0.789970i \(-0.289904\pi\)
0.613145 + 0.789970i \(0.289904\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.39246 −0.239246
\(101\) 3.69338 0.367505 0.183753 0.982973i \(-0.441175\pi\)
0.183753 + 0.982973i \(0.441175\pi\)
\(102\) 0.0712575 0.00705554
\(103\) 1.15156 0.113467 0.0567334 0.998389i \(-0.481931\pi\)
0.0567334 + 0.998389i \(0.481931\pi\)
\(104\) 4.58693 0.449785
\(105\) 0 0
\(106\) 1.48885 0.144610
\(107\) 1.16714 0.112831 0.0564157 0.998407i \(-0.482033\pi\)
0.0564157 + 0.998407i \(0.482033\pi\)
\(108\) 2.54177 0.244582
\(109\) 9.30234 0.891003 0.445501 0.895281i \(-0.353025\pi\)
0.445501 + 0.895281i \(0.353025\pi\)
\(110\) 0 0
\(111\) −0.873083 −0.0828694
\(112\) 0 0
\(113\) 3.29733 0.310187 0.155093 0.987900i \(-0.450432\pi\)
0.155093 + 0.987900i \(0.450432\pi\)
\(114\) 0.300812 0.0281736
\(115\) 17.6756 1.64826
\(116\) −12.7019 −1.17934
\(117\) 15.1584 1.40139
\(118\) −0.650640 −0.0598963
\(119\) 0 0
\(120\) −0.488478 −0.0445918
\(121\) 0 0
\(122\) 0.193488 0.0175176
\(123\) −1.47891 −0.133348
\(124\) 14.9844 1.34564
\(125\) −9.41402 −0.842015
\(126\) 0 0
\(127\) −0.289205 −0.0256628 −0.0128314 0.999918i \(-0.504084\pi\)
−0.0128314 + 0.999918i \(0.504084\pi\)
\(128\) −6.78282 −0.599522
\(129\) −0.175807 −0.0154790
\(130\) −2.89878 −0.254240
\(131\) 16.5059 1.44212 0.721062 0.692871i \(-0.243654\pi\)
0.721062 + 0.692871i \(0.243654\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.372498 −0.0321789
\(135\) −3.25480 −0.280129
\(136\) −1.28410 −0.110110
\(137\) 9.32588 0.796763 0.398382 0.917220i \(-0.369572\pi\)
0.398382 + 0.917220i \(0.369572\pi\)
\(138\) 0.351097 0.0298874
\(139\) −4.82649 −0.409377 −0.204689 0.978827i \(-0.565618\pi\)
−0.204689 + 0.978827i \(0.565618\pi\)
\(140\) 0 0
\(141\) −1.47961 −0.124606
\(142\) 1.02265 0.0858189
\(143\) 0 0
\(144\) −10.9093 −0.909109
\(145\) 16.2650 1.35074
\(146\) 3.35877 0.277974
\(147\) 0 0
\(148\) 7.76476 0.638260
\(149\) 0.921915 0.0755262 0.0377631 0.999287i \(-0.487977\pi\)
0.0377631 + 0.999287i \(0.487977\pi\)
\(150\) 0.0608535 0.00496867
\(151\) 18.0964 1.47267 0.736333 0.676620i \(-0.236556\pi\)
0.736333 + 0.676620i \(0.236556\pi\)
\(152\) −5.42079 −0.439684
\(153\) −4.24355 −0.343070
\(154\) 0 0
\(155\) −19.1880 −1.54121
\(156\) 2.19287 0.175570
\(157\) 12.2733 0.979518 0.489759 0.871858i \(-0.337085\pi\)
0.489759 + 0.871858i \(0.337085\pi\)
\(158\) 0.554875 0.0441435
\(159\) 1.44224 0.114377
\(160\) 6.54457 0.517394
\(161\) 0 0
\(162\) 1.93866 0.152316
\(163\) 8.09913 0.634373 0.317186 0.948363i \(-0.397262\pi\)
0.317186 + 0.948363i \(0.397262\pi\)
\(164\) 13.1526 1.02705
\(165\) 0 0
\(166\) −0.508478 −0.0394655
\(167\) 13.0516 1.00996 0.504982 0.863130i \(-0.331499\pi\)
0.504982 + 0.863130i \(0.331499\pi\)
\(168\) 0 0
\(169\) 13.3681 1.02831
\(170\) 0.811505 0.0622396
\(171\) −17.9140 −1.36992
\(172\) 1.56354 0.119219
\(173\) −5.91219 −0.449495 −0.224748 0.974417i \(-0.572156\pi\)
−0.224748 + 0.974417i \(0.572156\pi\)
\(174\) 0.323079 0.0244926
\(175\) 0 0
\(176\) 0 0
\(177\) −0.630271 −0.0473741
\(178\) −0.392626 −0.0294286
\(179\) 4.33508 0.324019 0.162009 0.986789i \(-0.448202\pi\)
0.162009 + 0.986789i \(0.448202\pi\)
\(180\) 14.3565 1.07007
\(181\) −10.8307 −0.805040 −0.402520 0.915411i \(-0.631866\pi\)
−0.402520 + 0.915411i \(0.631866\pi\)
\(182\) 0 0
\(183\) 0.187431 0.0138553
\(184\) −6.32696 −0.466430
\(185\) −9.94297 −0.731022
\(186\) −0.381138 −0.0279464
\(187\) 0 0
\(188\) 13.1589 0.959713
\(189\) 0 0
\(190\) 3.42575 0.248530
\(191\) 11.6556 0.843370 0.421685 0.906742i \(-0.361439\pi\)
0.421685 + 0.906742i \(0.361439\pi\)
\(192\) −1.48963 −0.107505
\(193\) 22.4454 1.61566 0.807829 0.589418i \(-0.200643\pi\)
0.807829 + 0.589418i \(0.200643\pi\)
\(194\) 2.73208 0.196152
\(195\) −2.80803 −0.201087
\(196\) 0 0
\(197\) 24.1022 1.71721 0.858604 0.512639i \(-0.171332\pi\)
0.858604 + 0.512639i \(0.171332\pi\)
\(198\) 0 0
\(199\) −18.7205 −1.32706 −0.663531 0.748148i \(-0.730943\pi\)
−0.663531 + 0.748148i \(0.730943\pi\)
\(200\) −1.09661 −0.0775422
\(201\) −0.360836 −0.0254514
\(202\) 0.835485 0.0587845
\(203\) 0 0
\(204\) −0.613889 −0.0429808
\(205\) −16.8423 −1.17632
\(206\) 0.260497 0.0181497
\(207\) −20.9087 −1.45325
\(208\) −18.9768 −1.31580
\(209\) 0 0
\(210\) 0 0
\(211\) −7.56636 −0.520890 −0.260445 0.965489i \(-0.583869\pi\)
−0.260445 + 0.965489i \(0.583869\pi\)
\(212\) −12.8266 −0.880931
\(213\) 0.990634 0.0678771
\(214\) 0.264019 0.0180480
\(215\) −2.00216 −0.136546
\(216\) 1.16505 0.0792717
\(217\) 0 0
\(218\) 2.10430 0.142521
\(219\) 3.25362 0.219859
\(220\) 0 0
\(221\) −7.38167 −0.496545
\(222\) −0.197501 −0.0132554
\(223\) −17.5244 −1.17352 −0.586760 0.809761i \(-0.699597\pi\)
−0.586760 + 0.809761i \(0.699597\pi\)
\(224\) 0 0
\(225\) −3.62397 −0.241598
\(226\) 0.745893 0.0496161
\(227\) 25.6773 1.70426 0.852130 0.523330i \(-0.175311\pi\)
0.852130 + 0.523330i \(0.175311\pi\)
\(228\) −2.59152 −0.171627
\(229\) 19.8369 1.31086 0.655429 0.755257i \(-0.272488\pi\)
0.655429 + 0.755257i \(0.272488\pi\)
\(230\) 3.99842 0.263648
\(231\) 0 0
\(232\) −5.82205 −0.382236
\(233\) −20.2146 −1.32430 −0.662151 0.749371i \(-0.730356\pi\)
−0.662151 + 0.749371i \(0.730356\pi\)
\(234\) 3.42900 0.224161
\(235\) −16.8503 −1.09919
\(236\) 5.60532 0.364875
\(237\) 0.537504 0.0349146
\(238\) 0 0
\(239\) 17.1004 1.10613 0.553066 0.833137i \(-0.313458\pi\)
0.553066 + 0.833137i \(0.313458\pi\)
\(240\) 2.02091 0.130449
\(241\) 24.1529 1.55582 0.777912 0.628373i \(-0.216279\pi\)
0.777912 + 0.628373i \(0.216279\pi\)
\(242\) 0 0
\(243\) 5.79074 0.371476
\(244\) −1.66691 −0.106713
\(245\) 0 0
\(246\) −0.334545 −0.0213298
\(247\) −31.1615 −1.98276
\(248\) 6.86831 0.436138
\(249\) −0.492559 −0.0312147
\(250\) −2.12956 −0.134685
\(251\) −11.8947 −0.750790 −0.375395 0.926865i \(-0.622493\pi\)
−0.375395 + 0.926865i \(0.622493\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −0.0654215 −0.00410491
\(255\) 0.786100 0.0492275
\(256\) 12.0615 0.753845
\(257\) −22.9609 −1.43226 −0.716131 0.697966i \(-0.754089\pi\)
−0.716131 + 0.697966i \(0.754089\pi\)
\(258\) −0.0397696 −0.00247595
\(259\) 0 0
\(260\) 24.9732 1.54877
\(261\) −19.2401 −1.19093
\(262\) 3.73381 0.230676
\(263\) 1.93774 0.119486 0.0597432 0.998214i \(-0.480972\pi\)
0.0597432 + 0.998214i \(0.480972\pi\)
\(264\) 0 0
\(265\) 16.4247 1.00896
\(266\) 0 0
\(267\) −0.380335 −0.0232761
\(268\) 3.20910 0.196027
\(269\) −7.18676 −0.438184 −0.219092 0.975704i \(-0.570310\pi\)
−0.219092 + 0.975704i \(0.570310\pi\)
\(270\) −0.736273 −0.0448081
\(271\) −1.19110 −0.0723543 −0.0361772 0.999345i \(-0.511518\pi\)
−0.0361772 + 0.999345i \(0.511518\pi\)
\(272\) 5.31250 0.322118
\(273\) 0 0
\(274\) 2.10962 0.127447
\(275\) 0 0
\(276\) −3.02473 −0.182067
\(277\) 10.3402 0.621285 0.310642 0.950527i \(-0.399456\pi\)
0.310642 + 0.950527i \(0.399456\pi\)
\(278\) −1.09181 −0.0654822
\(279\) 22.6977 1.35887
\(280\) 0 0
\(281\) −13.1513 −0.784541 −0.392271 0.919850i \(-0.628310\pi\)
−0.392271 + 0.919850i \(0.628310\pi\)
\(282\) −0.334705 −0.0199314
\(283\) −0.300031 −0.0178350 −0.00891748 0.999960i \(-0.502839\pi\)
−0.00891748 + 0.999960i \(0.502839\pi\)
\(284\) −8.81021 −0.522790
\(285\) 3.31850 0.196571
\(286\) 0 0
\(287\) 0 0
\(288\) −7.74164 −0.456181
\(289\) −14.9335 −0.878443
\(290\) 3.67934 0.216058
\(291\) 2.64655 0.155143
\(292\) −28.9361 −1.69336
\(293\) −16.1441 −0.943148 −0.471574 0.881826i \(-0.656314\pi\)
−0.471574 + 0.881826i \(0.656314\pi\)
\(294\) 0 0
\(295\) −7.17775 −0.417905
\(296\) 3.55908 0.206867
\(297\) 0 0
\(298\) 0.208548 0.0120808
\(299\) −36.3707 −2.10337
\(300\) −0.524258 −0.0302680
\(301\) 0 0
\(302\) 4.09361 0.235561
\(303\) 0.809329 0.0464947
\(304\) 22.4266 1.28625
\(305\) 2.13453 0.122223
\(306\) −0.959939 −0.0548760
\(307\) 28.6376 1.63443 0.817217 0.576330i \(-0.195516\pi\)
0.817217 + 0.576330i \(0.195516\pi\)
\(308\) 0 0
\(309\) 0.252341 0.0143552
\(310\) −4.34053 −0.246526
\(311\) 31.8228 1.80450 0.902252 0.431210i \(-0.141913\pi\)
0.902252 + 0.431210i \(0.141913\pi\)
\(312\) 1.00513 0.0569044
\(313\) −0.0342232 −0.00193441 −0.000967206 1.00000i \(-0.500308\pi\)
−0.000967206 1.00000i \(0.500308\pi\)
\(314\) 2.77636 0.156679
\(315\) 0 0
\(316\) −4.78029 −0.268912
\(317\) 22.3894 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(318\) 0.326251 0.0182952
\(319\) 0 0
\(320\) −16.9644 −0.948339
\(321\) 0.255754 0.0142748
\(322\) 0 0
\(323\) 8.72359 0.485393
\(324\) −16.7017 −0.927875
\(325\) −6.30391 −0.349678
\(326\) 1.83211 0.101471
\(327\) 2.03842 0.112725
\(328\) 6.02868 0.332878
\(329\) 0 0
\(330\) 0 0
\(331\) 10.7577 0.591297 0.295648 0.955297i \(-0.404464\pi\)
0.295648 + 0.955297i \(0.404464\pi\)
\(332\) 4.38057 0.240415
\(333\) 11.7617 0.644535
\(334\) 2.95242 0.161549
\(335\) −4.10933 −0.224517
\(336\) 0 0
\(337\) 7.50492 0.408819 0.204410 0.978885i \(-0.434473\pi\)
0.204410 + 0.978885i \(0.434473\pi\)
\(338\) 3.02401 0.164485
\(339\) 0.722542 0.0392431
\(340\) −6.99118 −0.379150
\(341\) 0 0
\(342\) −4.05236 −0.219126
\(343\) 0 0
\(344\) 0.716669 0.0386402
\(345\) 3.87325 0.208529
\(346\) −1.33740 −0.0718993
\(347\) −27.6818 −1.48604 −0.743019 0.669271i \(-0.766607\pi\)
−0.743019 + 0.669271i \(0.766607\pi\)
\(348\) −2.78335 −0.149203
\(349\) 11.0605 0.592055 0.296028 0.955179i \(-0.404338\pi\)
0.296028 + 0.955179i \(0.404338\pi\)
\(350\) 0 0
\(351\) 6.69733 0.357477
\(352\) 0 0
\(353\) −31.9202 −1.69894 −0.849469 0.527638i \(-0.823078\pi\)
−0.849469 + 0.527638i \(0.823078\pi\)
\(354\) −0.142575 −0.00757775
\(355\) 11.2817 0.598770
\(356\) 3.38251 0.179272
\(357\) 0 0
\(358\) 0.980644 0.0518286
\(359\) 3.57826 0.188854 0.0944268 0.995532i \(-0.469898\pi\)
0.0944268 + 0.995532i \(0.469898\pi\)
\(360\) 6.58049 0.346822
\(361\) 17.8264 0.938233
\(362\) −2.45003 −0.128771
\(363\) 0 0
\(364\) 0 0
\(365\) 37.0534 1.93946
\(366\) 0.0423989 0.00221623
\(367\) −2.29397 −0.119744 −0.0598720 0.998206i \(-0.519069\pi\)
−0.0598720 + 0.998206i \(0.519069\pi\)
\(368\) 26.1756 1.36450
\(369\) 19.9229 1.03715
\(370\) −2.24921 −0.116931
\(371\) 0 0
\(372\) 3.28354 0.170243
\(373\) 7.96856 0.412596 0.206298 0.978489i \(-0.433858\pi\)
0.206298 + 0.978489i \(0.433858\pi\)
\(374\) 0 0
\(375\) −2.06289 −0.106527
\(376\) 6.03155 0.311054
\(377\) −33.4682 −1.72370
\(378\) 0 0
\(379\) −11.6212 −0.596941 −0.298470 0.954419i \(-0.596476\pi\)
−0.298470 + 0.954419i \(0.596476\pi\)
\(380\) −29.5131 −1.51399
\(381\) −0.0633734 −0.00324672
\(382\) 2.63663 0.134902
\(383\) −12.5785 −0.642729 −0.321364 0.946956i \(-0.604141\pi\)
−0.321364 + 0.946956i \(0.604141\pi\)
\(384\) −1.48632 −0.0758482
\(385\) 0 0
\(386\) 5.07741 0.258433
\(387\) 2.36837 0.120391
\(388\) −23.5371 −1.19491
\(389\) −0.438294 −0.0222224 −0.0111112 0.999938i \(-0.503537\pi\)
−0.0111112 + 0.999938i \(0.503537\pi\)
\(390\) −0.635208 −0.0321650
\(391\) 10.1819 0.514920
\(392\) 0 0
\(393\) 3.61692 0.182450
\(394\) 5.45218 0.274677
\(395\) 6.12128 0.307995
\(396\) 0 0
\(397\) −16.8147 −0.843905 −0.421952 0.906618i \(-0.638655\pi\)
−0.421952 + 0.906618i \(0.638655\pi\)
\(398\) −4.23480 −0.212271
\(399\) 0 0
\(400\) 4.53685 0.226842
\(401\) 36.8609 1.84074 0.920372 0.391043i \(-0.127886\pi\)
0.920372 + 0.391043i \(0.127886\pi\)
\(402\) −0.0816253 −0.00407110
\(403\) 39.4827 1.96677
\(404\) −7.19776 −0.358102
\(405\) 21.3870 1.06273
\(406\) 0 0
\(407\) 0 0
\(408\) −0.281384 −0.0139306
\(409\) −16.2739 −0.804692 −0.402346 0.915488i \(-0.631805\pi\)
−0.402346 + 0.915488i \(0.631805\pi\)
\(410\) −3.80992 −0.188158
\(411\) 2.04357 0.100802
\(412\) −2.24420 −0.110564
\(413\) 0 0
\(414\) −4.72978 −0.232456
\(415\) −5.60944 −0.275356
\(416\) −13.4666 −0.660255
\(417\) −1.05763 −0.0517922
\(418\) 0 0
\(419\) −5.56352 −0.271796 −0.135898 0.990723i \(-0.543392\pi\)
−0.135898 + 0.990723i \(0.543392\pi\)
\(420\) 0 0
\(421\) 21.4914 1.04743 0.523713 0.851895i \(-0.324546\pi\)
0.523713 + 0.851895i \(0.324546\pi\)
\(422\) −1.71160 −0.0833192
\(423\) 19.9324 0.969148
\(424\) −5.87921 −0.285520
\(425\) 1.76476 0.0856035
\(426\) 0.224093 0.0108573
\(427\) 0 0
\(428\) −2.27455 −0.109944
\(429\) 0 0
\(430\) −0.452910 −0.0218413
\(431\) −27.7188 −1.33517 −0.667583 0.744536i \(-0.732671\pi\)
−0.667583 + 0.744536i \(0.732671\pi\)
\(432\) −4.81999 −0.231902
\(433\) −9.28812 −0.446358 −0.223179 0.974777i \(-0.571644\pi\)
−0.223179 + 0.974777i \(0.571644\pi\)
\(434\) 0 0
\(435\) 3.56415 0.170888
\(436\) −18.1287 −0.868206
\(437\) 42.9826 2.05613
\(438\) 0.736006 0.0351678
\(439\) 14.7118 0.702156 0.351078 0.936346i \(-0.385815\pi\)
0.351078 + 0.936346i \(0.385815\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.66982 −0.0794251
\(443\) 2.44345 0.116092 0.0580459 0.998314i \(-0.481513\pi\)
0.0580459 + 0.998314i \(0.481513\pi\)
\(444\) 1.70149 0.0807491
\(445\) −4.33138 −0.205327
\(446\) −3.96421 −0.187711
\(447\) 0.202019 0.00955516
\(448\) 0 0
\(449\) 4.76935 0.225080 0.112540 0.993647i \(-0.464101\pi\)
0.112540 + 0.993647i \(0.464101\pi\)
\(450\) −0.819782 −0.0386449
\(451\) 0 0
\(452\) −6.42593 −0.302250
\(453\) 3.96546 0.186313
\(454\) 5.80849 0.272606
\(455\) 0 0
\(456\) −1.18785 −0.0556264
\(457\) 31.0875 1.45421 0.727106 0.686525i \(-0.240865\pi\)
0.727106 + 0.686525i \(0.240865\pi\)
\(458\) 4.48733 0.209679
\(459\) −1.87490 −0.0875128
\(460\) −34.4467 −1.60609
\(461\) −29.7215 −1.38427 −0.692134 0.721769i \(-0.743329\pi\)
−0.692134 + 0.721769i \(0.743329\pi\)
\(462\) 0 0
\(463\) −25.4553 −1.18301 −0.591505 0.806302i \(-0.701466\pi\)
−0.591505 + 0.806302i \(0.701466\pi\)
\(464\) 24.0867 1.11820
\(465\) −4.20465 −0.194986
\(466\) −4.57277 −0.211829
\(467\) 3.18491 0.147380 0.0736901 0.997281i \(-0.476522\pi\)
0.0736901 + 0.997281i \(0.476522\pi\)
\(468\) −29.5411 −1.36554
\(469\) 0 0
\(470\) −3.81173 −0.175822
\(471\) 2.68945 0.123923
\(472\) 2.56927 0.118260
\(473\) 0 0
\(474\) 0.121590 0.00558479
\(475\) 7.44990 0.341825
\(476\) 0 0
\(477\) −19.4290 −0.889592
\(478\) 3.86830 0.176932
\(479\) −3.23354 −0.147744 −0.0738720 0.997268i \(-0.523536\pi\)
−0.0738720 + 0.997268i \(0.523536\pi\)
\(480\) 1.43411 0.0654578
\(481\) 20.4594 0.932870
\(482\) 5.46366 0.248863
\(483\) 0 0
\(484\) 0 0
\(485\) 30.1398 1.36858
\(486\) 1.30993 0.0594197
\(487\) 9.87096 0.447296 0.223648 0.974670i \(-0.428203\pi\)
0.223648 + 0.974670i \(0.428203\pi\)
\(488\) −0.764051 −0.0345870
\(489\) 1.77476 0.0802573
\(490\) 0 0
\(491\) 4.22600 0.190717 0.0953584 0.995443i \(-0.469600\pi\)
0.0953584 + 0.995443i \(0.469600\pi\)
\(492\) 2.88213 0.129937
\(493\) 9.36933 0.421974
\(494\) −7.04910 −0.317154
\(495\) 0 0
\(496\) −28.4152 −1.27588
\(497\) 0 0
\(498\) −0.111422 −0.00499296
\(499\) −20.3953 −0.913018 −0.456509 0.889719i \(-0.650900\pi\)
−0.456509 + 0.889719i \(0.650900\pi\)
\(500\) 18.3463 0.820472
\(501\) 2.85999 0.127775
\(502\) −2.69073 −0.120093
\(503\) 23.9593 1.06829 0.534145 0.845393i \(-0.320634\pi\)
0.534145 + 0.845393i \(0.320634\pi\)
\(504\) 0 0
\(505\) 9.21692 0.410147
\(506\) 0 0
\(507\) 2.92934 0.130097
\(508\) 0.563611 0.0250062
\(509\) −3.69341 −0.163708 −0.0818538 0.996644i \(-0.526084\pi\)
−0.0818538 + 0.996644i \(0.526084\pi\)
\(510\) 0.177825 0.00787421
\(511\) 0 0
\(512\) 16.2941 0.720104
\(513\) −7.91484 −0.349449
\(514\) −5.19402 −0.229098
\(515\) 2.87375 0.126633
\(516\) 0.342618 0.0150829
\(517\) 0 0
\(518\) 0 0
\(519\) −1.29554 −0.0568677
\(520\) 11.4468 0.501975
\(521\) 0.238270 0.0104388 0.00521939 0.999986i \(-0.498339\pi\)
0.00521939 + 0.999986i \(0.498339\pi\)
\(522\) −4.35233 −0.190496
\(523\) 21.9914 0.961618 0.480809 0.876825i \(-0.340343\pi\)
0.480809 + 0.876825i \(0.340343\pi\)
\(524\) −32.1671 −1.40523
\(525\) 0 0
\(526\) 0.438339 0.0191125
\(527\) −11.0531 −0.481479
\(528\) 0 0
\(529\) 27.1678 1.18121
\(530\) 3.71546 0.161389
\(531\) 8.49064 0.368462
\(532\) 0 0
\(533\) 34.6560 1.50112
\(534\) −0.0860360 −0.00372314
\(535\) 2.91262 0.125923
\(536\) 1.47093 0.0635345
\(537\) 0.949944 0.0409931
\(538\) −1.62573 −0.0700900
\(539\) 0 0
\(540\) 6.34305 0.272961
\(541\) 28.6309 1.23094 0.615470 0.788161i \(-0.288966\pi\)
0.615470 + 0.788161i \(0.288966\pi\)
\(542\) −0.269441 −0.0115735
\(543\) −2.37333 −0.101849
\(544\) 3.76994 0.161635
\(545\) 23.2142 0.994388
\(546\) 0 0
\(547\) 6.40847 0.274007 0.137003 0.990571i \(-0.456253\pi\)
0.137003 + 0.990571i \(0.456253\pi\)
\(548\) −18.1745 −0.776378
\(549\) −2.52495 −0.107762
\(550\) 0 0
\(551\) 39.5524 1.68499
\(552\) −1.38642 −0.0590101
\(553\) 0 0
\(554\) 2.33908 0.0993780
\(555\) −2.17880 −0.0924849
\(556\) 9.40600 0.398903
\(557\) −22.5193 −0.954172 −0.477086 0.878857i \(-0.658307\pi\)
−0.477086 + 0.878857i \(0.658307\pi\)
\(558\) 5.13447 0.217359
\(559\) 4.11979 0.174249
\(560\) 0 0
\(561\) 0 0
\(562\) −2.97498 −0.125492
\(563\) 29.8267 1.25705 0.628523 0.777791i \(-0.283660\pi\)
0.628523 + 0.777791i \(0.283660\pi\)
\(564\) 2.88351 0.121418
\(565\) 8.22856 0.346178
\(566\) −0.0678703 −0.00285280
\(567\) 0 0
\(568\) −4.03827 −0.169442
\(569\) 29.2764 1.22733 0.613665 0.789567i \(-0.289695\pi\)
0.613665 + 0.789567i \(0.289695\pi\)
\(570\) 0.750683 0.0314427
\(571\) 37.9252 1.58712 0.793559 0.608493i \(-0.208226\pi\)
0.793559 + 0.608493i \(0.208226\pi\)
\(572\) 0 0
\(573\) 2.55409 0.106699
\(574\) 0 0
\(575\) 8.69527 0.362618
\(576\) 20.0674 0.836141
\(577\) −8.77263 −0.365209 −0.182605 0.983186i \(-0.558453\pi\)
−0.182605 + 0.983186i \(0.558453\pi\)
\(578\) −3.37813 −0.140512
\(579\) 4.91846 0.204404
\(580\) −31.6978 −1.31618
\(581\) 0 0
\(582\) 0.598679 0.0248161
\(583\) 0 0
\(584\) −13.2632 −0.548836
\(585\) 37.8281 1.56400
\(586\) −3.65198 −0.150862
\(587\) −9.17011 −0.378491 −0.189246 0.981930i \(-0.560604\pi\)
−0.189246 + 0.981930i \(0.560604\pi\)
\(588\) 0 0
\(589\) −46.6602 −1.92260
\(590\) −1.62369 −0.0668462
\(591\) 5.28150 0.217252
\(592\) −14.7244 −0.605170
\(593\) −7.25596 −0.297967 −0.148983 0.988840i \(-0.547600\pi\)
−0.148983 + 0.988840i \(0.547600\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.79665 −0.0735938
\(597\) −4.10222 −0.167893
\(598\) −8.22747 −0.336446
\(599\) −20.0181 −0.817916 −0.408958 0.912553i \(-0.634108\pi\)
−0.408958 + 0.912553i \(0.634108\pi\)
\(600\) −0.240300 −0.00981021
\(601\) 11.2706 0.459736 0.229868 0.973222i \(-0.426171\pi\)
0.229868 + 0.973222i \(0.426171\pi\)
\(602\) 0 0
\(603\) 4.86097 0.197954
\(604\) −35.2668 −1.43499
\(605\) 0 0
\(606\) 0.183079 0.00743709
\(607\) 13.5260 0.549005 0.274502 0.961586i \(-0.411487\pi\)
0.274502 + 0.961586i \(0.411487\pi\)
\(608\) 15.9147 0.645427
\(609\) 0 0
\(610\) 0.482854 0.0195502
\(611\) 34.6725 1.40270
\(612\) 8.26995 0.334293
\(613\) −30.7968 −1.24387 −0.621935 0.783069i \(-0.713653\pi\)
−0.621935 + 0.783069i \(0.713653\pi\)
\(614\) 6.47815 0.261437
\(615\) −3.69064 −0.148821
\(616\) 0 0
\(617\) 23.6896 0.953707 0.476853 0.878983i \(-0.341777\pi\)
0.476853 + 0.878983i \(0.341777\pi\)
\(618\) 0.0570825 0.00229619
\(619\) 32.4878 1.30579 0.652897 0.757446i \(-0.273553\pi\)
0.652897 + 0.757446i \(0.273553\pi\)
\(620\) 37.3940 1.50178
\(621\) −9.23794 −0.370706
\(622\) 7.19867 0.288640
\(623\) 0 0
\(624\) −4.15838 −0.166468
\(625\) −29.6311 −1.18524
\(626\) −0.00774169 −0.000309420 0
\(627\) 0 0
\(628\) −23.9186 −0.954456
\(629\) −5.72756 −0.228373
\(630\) 0 0
\(631\) −15.1333 −0.602448 −0.301224 0.953553i \(-0.597395\pi\)
−0.301224 + 0.953553i \(0.597395\pi\)
\(632\) −2.19111 −0.0871575
\(633\) −1.65801 −0.0659001
\(634\) 5.06474 0.201147
\(635\) −0.721718 −0.0286405
\(636\) −2.81068 −0.111451
\(637\) 0 0
\(638\) 0 0
\(639\) −13.3452 −0.527929
\(640\) −16.9267 −0.669086
\(641\) −16.5502 −0.653695 −0.326848 0.945077i \(-0.605986\pi\)
−0.326848 + 0.945077i \(0.605986\pi\)
\(642\) 0.0578545 0.00228333
\(643\) −1.99506 −0.0786773 −0.0393387 0.999226i \(-0.512525\pi\)
−0.0393387 + 0.999226i \(0.512525\pi\)
\(644\) 0 0
\(645\) −0.438731 −0.0172750
\(646\) 1.97337 0.0776414
\(647\) −40.4517 −1.59032 −0.795160 0.606399i \(-0.792613\pi\)
−0.795160 + 0.606399i \(0.792613\pi\)
\(648\) −7.65545 −0.300735
\(649\) 0 0
\(650\) −1.42602 −0.0559329
\(651\) 0 0
\(652\) −15.7838 −0.618142
\(653\) −41.4856 −1.62346 −0.811729 0.584034i \(-0.801473\pi\)
−0.811729 + 0.584034i \(0.801473\pi\)
\(654\) 0.461113 0.0180310
\(655\) 41.1908 1.60946
\(656\) −24.9415 −0.973803
\(657\) −43.8309 −1.71001
\(658\) 0 0
\(659\) −51.1359 −1.99197 −0.995985 0.0895158i \(-0.971468\pi\)
−0.995985 + 0.0895158i \(0.971468\pi\)
\(660\) 0 0
\(661\) 42.8840 1.66800 0.833998 0.551768i \(-0.186046\pi\)
0.833998 + 0.551768i \(0.186046\pi\)
\(662\) 2.43351 0.0945812
\(663\) −1.61754 −0.0628201
\(664\) 2.00789 0.0779213
\(665\) 0 0
\(666\) 2.66062 0.103097
\(667\) 46.1643 1.78749
\(668\) −25.4353 −0.984122
\(669\) −3.84011 −0.148467
\(670\) −0.929577 −0.0359127
\(671\) 0 0
\(672\) 0 0
\(673\) −25.0072 −0.963958 −0.481979 0.876183i \(-0.660082\pi\)
−0.481979 + 0.876183i \(0.660082\pi\)
\(674\) 1.69770 0.0653929
\(675\) −1.60115 −0.0616284
\(676\) −26.0521 −1.00200
\(677\) 9.91890 0.381214 0.190607 0.981666i \(-0.438954\pi\)
0.190607 + 0.981666i \(0.438954\pi\)
\(678\) 0.163447 0.00627715
\(679\) 0 0
\(680\) −3.20450 −0.122887
\(681\) 5.62665 0.215614
\(682\) 0 0
\(683\) −39.8980 −1.52666 −0.763328 0.646011i \(-0.776436\pi\)
−0.763328 + 0.646011i \(0.776436\pi\)
\(684\) 34.9114 1.33487
\(685\) 23.2729 0.889214
\(686\) 0 0
\(687\) 4.34685 0.165843
\(688\) −2.96497 −0.113038
\(689\) −33.7968 −1.28756
\(690\) 0.876172 0.0333553
\(691\) 6.61401 0.251609 0.125804 0.992055i \(-0.459849\pi\)
0.125804 + 0.992055i \(0.459849\pi\)
\(692\) 11.5218 0.437995
\(693\) 0 0
\(694\) −6.26194 −0.237700
\(695\) −12.0446 −0.456878
\(696\) −1.27578 −0.0483585
\(697\) −9.70185 −0.367484
\(698\) 2.50201 0.0947025
\(699\) −4.42961 −0.167543
\(700\) 0 0
\(701\) 14.6016 0.551495 0.275748 0.961230i \(-0.411075\pi\)
0.275748 + 0.961230i \(0.411075\pi\)
\(702\) 1.51501 0.0571805
\(703\) −24.1788 −0.911920
\(704\) 0 0
\(705\) −3.69240 −0.139064
\(706\) −7.22070 −0.271755
\(707\) 0 0
\(708\) 1.22829 0.0461620
\(709\) −4.14406 −0.155633 −0.0778167 0.996968i \(-0.524795\pi\)
−0.0778167 + 0.996968i \(0.524795\pi\)
\(710\) 2.55205 0.0957766
\(711\) −7.24093 −0.271556
\(712\) 1.55041 0.0581042
\(713\) −54.4602 −2.03955
\(714\) 0 0
\(715\) 0 0
\(716\) −8.44832 −0.315729
\(717\) 3.74720 0.139942
\(718\) 0.809444 0.0302082
\(719\) −17.0223 −0.634823 −0.317412 0.948288i \(-0.602814\pi\)
−0.317412 + 0.948288i \(0.602814\pi\)
\(720\) −27.2245 −1.01460
\(721\) 0 0
\(722\) 4.03254 0.150076
\(723\) 5.29261 0.196834
\(724\) 21.1072 0.784442
\(725\) 8.00136 0.297163
\(726\) 0 0
\(727\) 21.6199 0.801837 0.400918 0.916114i \(-0.368691\pi\)
0.400918 + 0.916114i \(0.368691\pi\)
\(728\) 0 0
\(729\) −24.4415 −0.905241
\(730\) 8.38190 0.310228
\(731\) −1.15332 −0.0426572
\(732\) −0.365270 −0.0135008
\(733\) 48.2326 1.78151 0.890755 0.454484i \(-0.150176\pi\)
0.890755 + 0.454484i \(0.150176\pi\)
\(734\) −0.518921 −0.0191537
\(735\) 0 0
\(736\) 18.5751 0.684688
\(737\) 0 0
\(738\) 4.50679 0.165897
\(739\) −8.16347 −0.300298 −0.150149 0.988663i \(-0.547975\pi\)
−0.150149 + 0.988663i \(0.547975\pi\)
\(740\) 19.3772 0.712318
\(741\) −6.82842 −0.250848
\(742\) 0 0
\(743\) 19.5612 0.717633 0.358816 0.933408i \(-0.383180\pi\)
0.358816 + 0.933408i \(0.383180\pi\)
\(744\) 1.50505 0.0551778
\(745\) 2.30066 0.0842897
\(746\) 1.80258 0.0659971
\(747\) 6.63546 0.242779
\(748\) 0 0
\(749\) 0 0
\(750\) −0.466649 −0.0170396
\(751\) 1.11630 0.0407343 0.0203671 0.999793i \(-0.493516\pi\)
0.0203671 + 0.999793i \(0.493516\pi\)
\(752\) −24.9534 −0.909958
\(753\) −2.60649 −0.0949858
\(754\) −7.57089 −0.275716
\(755\) 45.1600 1.64354
\(756\) 0 0
\(757\) −26.5773 −0.965969 −0.482984 0.875629i \(-0.660447\pi\)
−0.482984 + 0.875629i \(0.660447\pi\)
\(758\) −2.62885 −0.0954840
\(759\) 0 0
\(760\) −13.5277 −0.490701
\(761\) −6.30003 −0.228376 −0.114188 0.993459i \(-0.536427\pi\)
−0.114188 + 0.993459i \(0.536427\pi\)
\(762\) −0.0143358 −0.000519330 0
\(763\) 0 0
\(764\) −22.7148 −0.821792
\(765\) −10.5899 −0.382878
\(766\) −2.84539 −0.102808
\(767\) 14.7695 0.533296
\(768\) 2.64303 0.0953723
\(769\) −13.1916 −0.475700 −0.237850 0.971302i \(-0.576443\pi\)
−0.237850 + 0.971302i \(0.576443\pi\)
\(770\) 0 0
\(771\) −5.03141 −0.181202
\(772\) −43.7423 −1.57432
\(773\) −45.8178 −1.64795 −0.823976 0.566625i \(-0.808249\pi\)
−0.823976 + 0.566625i \(0.808249\pi\)
\(774\) 0.535752 0.0192572
\(775\) −9.43925 −0.339068
\(776\) −10.7885 −0.387285
\(777\) 0 0
\(778\) −0.0991470 −0.00355459
\(779\) −40.9562 −1.46741
\(780\) 5.47237 0.195942
\(781\) 0 0
\(782\) 2.30326 0.0823643
\(783\) −8.50073 −0.303791
\(784\) 0 0
\(785\) 30.6284 1.09317
\(786\) 0.818189 0.0291838
\(787\) 18.8479 0.671856 0.335928 0.941888i \(-0.390950\pi\)
0.335928 + 0.941888i \(0.390950\pi\)
\(788\) −46.9710 −1.67327
\(789\) 0.424617 0.0151168
\(790\) 1.38470 0.0492656
\(791\) 0 0
\(792\) 0 0
\(793\) −4.39217 −0.155970
\(794\) −3.80367 −0.134987
\(795\) 3.59914 0.127648
\(796\) 36.4831 1.29311
\(797\) 28.1613 0.997526 0.498763 0.866738i \(-0.333788\pi\)
0.498763 + 0.866738i \(0.333788\pi\)
\(798\) 0 0
\(799\) −9.70648 −0.343391
\(800\) 3.21951 0.113827
\(801\) 5.12364 0.181035
\(802\) 8.33835 0.294437
\(803\) 0 0
\(804\) 0.703208 0.0248002
\(805\) 0 0
\(806\) 8.93143 0.314596
\(807\) −1.57483 −0.0554367
\(808\) −3.29919 −0.116065
\(809\) −6.54552 −0.230128 −0.115064 0.993358i \(-0.536707\pi\)
−0.115064 + 0.993358i \(0.536707\pi\)
\(810\) 4.83798 0.169989
\(811\) −18.8046 −0.660320 −0.330160 0.943925i \(-0.607103\pi\)
−0.330160 + 0.943925i \(0.607103\pi\)
\(812\) 0 0
\(813\) −0.261006 −0.00915387
\(814\) 0 0
\(815\) 20.2116 0.707980
\(816\) 1.16413 0.0407526
\(817\) −4.86873 −0.170335
\(818\) −3.68134 −0.128715
\(819\) 0 0
\(820\) 32.8227 1.14622
\(821\) −11.5880 −0.404424 −0.202212 0.979342i \(-0.564813\pi\)
−0.202212 + 0.979342i \(0.564813\pi\)
\(822\) 0.462280 0.0161239
\(823\) 12.3464 0.430368 0.215184 0.976574i \(-0.430965\pi\)
0.215184 + 0.976574i \(0.430965\pi\)
\(824\) −1.02866 −0.0358349
\(825\) 0 0
\(826\) 0 0
\(827\) −4.49579 −0.156334 −0.0781669 0.996940i \(-0.524907\pi\)
−0.0781669 + 0.996940i \(0.524907\pi\)
\(828\) 40.7474 1.41607
\(829\) 19.8629 0.689866 0.344933 0.938627i \(-0.387902\pi\)
0.344933 + 0.938627i \(0.387902\pi\)
\(830\) −1.26892 −0.0440448
\(831\) 2.26585 0.0786015
\(832\) 34.9073 1.21019
\(833\) 0 0
\(834\) −0.239247 −0.00828444
\(835\) 32.5706 1.12715
\(836\) 0 0
\(837\) 10.0284 0.346631
\(838\) −1.25853 −0.0434753
\(839\) 43.8528 1.51397 0.756984 0.653434i \(-0.226672\pi\)
0.756984 + 0.653434i \(0.226672\pi\)
\(840\) 0 0
\(841\) 13.4802 0.464836
\(842\) 4.86160 0.167542
\(843\) −2.88184 −0.0992558
\(844\) 14.7455 0.507562
\(845\) 33.3604 1.14763
\(846\) 4.50894 0.155021
\(847\) 0 0
\(848\) 24.3232 0.835261
\(849\) −0.0657456 −0.00225638
\(850\) 0.399209 0.0136928
\(851\) −28.2207 −0.967392
\(852\) −1.93058 −0.0661405
\(853\) −39.1407 −1.34015 −0.670076 0.742293i \(-0.733738\pi\)
−0.670076 + 0.742293i \(0.733738\pi\)
\(854\) 0 0
\(855\) −44.7049 −1.52888
\(856\) −1.04257 −0.0356342
\(857\) −35.0524 −1.19737 −0.598684 0.800986i \(-0.704309\pi\)
−0.598684 + 0.800986i \(0.704309\pi\)
\(858\) 0 0
\(859\) 32.5206 1.10959 0.554794 0.831988i \(-0.312797\pi\)
0.554794 + 0.831988i \(0.312797\pi\)
\(860\) 3.90186 0.133052
\(861\) 0 0
\(862\) −6.27030 −0.213567
\(863\) 12.6940 0.432107 0.216054 0.976381i \(-0.430681\pi\)
0.216054 + 0.976381i \(0.430681\pi\)
\(864\) −3.42044 −0.116366
\(865\) −14.7540 −0.501651
\(866\) −2.10108 −0.0713975
\(867\) −3.27238 −0.111136
\(868\) 0 0
\(869\) 0 0
\(870\) 0.806251 0.0273345
\(871\) 8.45568 0.286510
\(872\) −8.30950 −0.281395
\(873\) −35.6527 −1.20666
\(874\) 9.72314 0.328890
\(875\) 0 0
\(876\) −6.34075 −0.214234
\(877\) −28.1340 −0.950017 −0.475008 0.879981i \(-0.657555\pi\)
−0.475008 + 0.879981i \(0.657555\pi\)
\(878\) 3.32798 0.112314
\(879\) −3.53765 −0.119322
\(880\) 0 0
\(881\) 36.7964 1.23970 0.619850 0.784720i \(-0.287193\pi\)
0.619850 + 0.784720i \(0.287193\pi\)
\(882\) 0 0
\(883\) −2.28419 −0.0768692 −0.0384346 0.999261i \(-0.512237\pi\)
−0.0384346 + 0.999261i \(0.512237\pi\)
\(884\) 14.3856 0.483840
\(885\) −1.57286 −0.0528710
\(886\) 0.552736 0.0185695
\(887\) −42.2676 −1.41921 −0.709604 0.704600i \(-0.751126\pi\)
−0.709604 + 0.704600i \(0.751126\pi\)
\(888\) 0.779899 0.0261717
\(889\) 0 0
\(890\) −0.979808 −0.0328432
\(891\) 0 0
\(892\) 34.1520 1.14349
\(893\) −40.9757 −1.37120
\(894\) 0.0456989 0.00152840
\(895\) 10.8183 0.361616
\(896\) 0 0
\(897\) −7.96990 −0.266107
\(898\) 1.07888 0.0360027
\(899\) −50.1142 −1.67140
\(900\) 7.06249 0.235416
\(901\) 9.46132 0.315202
\(902\) 0 0
\(903\) 0 0
\(904\) −2.94540 −0.0979627
\(905\) −27.0283 −0.898451
\(906\) 0.897032 0.0298019
\(907\) 39.5286 1.31252 0.656262 0.754533i \(-0.272136\pi\)
0.656262 + 0.754533i \(0.272136\pi\)
\(908\) −50.0406 −1.66065
\(909\) −10.9028 −0.361623
\(910\) 0 0
\(911\) −35.2296 −1.16721 −0.583604 0.812039i \(-0.698358\pi\)
−0.583604 + 0.812039i \(0.698358\pi\)
\(912\) 4.91433 0.162730
\(913\) 0 0
\(914\) 7.03234 0.232609
\(915\) 0.467737 0.0154629
\(916\) −38.6587 −1.27732
\(917\) 0 0
\(918\) −0.424123 −0.0139982
\(919\) −39.6105 −1.30663 −0.653314 0.757087i \(-0.726622\pi\)
−0.653314 + 0.757087i \(0.726622\pi\)
\(920\) −15.7891 −0.520551
\(921\) 6.27534 0.206780
\(922\) −6.72334 −0.221421
\(923\) −23.2141 −0.764101
\(924\) 0 0
\(925\) −4.89131 −0.160825
\(926\) −5.75828 −0.189229
\(927\) −3.39939 −0.111651
\(928\) 17.0928 0.561098
\(929\) −2.61657 −0.0858469 −0.0429234 0.999078i \(-0.513667\pi\)
−0.0429234 + 0.999078i \(0.513667\pi\)
\(930\) −0.951139 −0.0311891
\(931\) 0 0
\(932\) 39.3947 1.29042
\(933\) 6.97331 0.228296
\(934\) 0.720463 0.0235743
\(935\) 0 0
\(936\) −13.5405 −0.442586
\(937\) −53.8401 −1.75888 −0.879439 0.476011i \(-0.842082\pi\)
−0.879439 + 0.476011i \(0.842082\pi\)
\(938\) 0 0
\(939\) −0.00749932 −0.000244731 0
\(940\) 32.8384 1.07107
\(941\) −29.6476 −0.966484 −0.483242 0.875487i \(-0.660541\pi\)
−0.483242 + 0.875487i \(0.660541\pi\)
\(942\) 0.608384 0.0198222
\(943\) −47.8026 −1.55667
\(944\) −10.6294 −0.345959
\(945\) 0 0
\(946\) 0 0
\(947\) −15.7861 −0.512980 −0.256490 0.966547i \(-0.582566\pi\)
−0.256490 + 0.966547i \(0.582566\pi\)
\(948\) −1.04750 −0.0340213
\(949\) −76.2440 −2.47498
\(950\) 1.68525 0.0546768
\(951\) 4.90618 0.159094
\(952\) 0 0
\(953\) −35.1119 −1.13739 −0.568693 0.822550i \(-0.692551\pi\)
−0.568693 + 0.822550i \(0.692551\pi\)
\(954\) −4.39506 −0.142295
\(955\) 29.0869 0.941229
\(956\) −33.3257 −1.07783
\(957\) 0 0
\(958\) −0.731463 −0.0236325
\(959\) 0 0
\(960\) −3.71740 −0.119979
\(961\) 28.1200 0.907096
\(962\) 4.62816 0.149218
\(963\) −3.44537 −0.111025
\(964\) −47.0698 −1.51602
\(965\) 56.0131 1.80313
\(966\) 0 0
\(967\) −49.2820 −1.58480 −0.792401 0.610001i \(-0.791169\pi\)
−0.792401 + 0.610001i \(0.791169\pi\)
\(968\) 0 0
\(969\) 1.91160 0.0614093
\(970\) 6.81797 0.218912
\(971\) −37.3424 −1.19838 −0.599188 0.800608i \(-0.704510\pi\)
−0.599188 + 0.800608i \(0.704510\pi\)
\(972\) −11.2852 −0.361971
\(973\) 0 0
\(974\) 2.23292 0.0715475
\(975\) −1.38137 −0.0442393
\(976\) 3.16099 0.101181
\(977\) 40.0485 1.28126 0.640632 0.767848i \(-0.278672\pi\)
0.640632 + 0.767848i \(0.278672\pi\)
\(978\) 0.401470 0.0128376
\(979\) 0 0
\(980\) 0 0
\(981\) −27.4603 −0.876741
\(982\) 0.955970 0.0305062
\(983\) 52.0414 1.65986 0.829932 0.557864i \(-0.188379\pi\)
0.829932 + 0.557864i \(0.188379\pi\)
\(984\) 1.32106 0.0421139
\(985\) 60.1475 1.91646
\(986\) 2.11945 0.0674970
\(987\) 0 0
\(988\) 60.7285 1.93203
\(989\) −5.68262 −0.180697
\(990\) 0 0
\(991\) 45.4828 1.44481 0.722404 0.691471i \(-0.243037\pi\)
0.722404 + 0.691471i \(0.243037\pi\)
\(992\) −20.1645 −0.640222
\(993\) 2.35733 0.0748076
\(994\) 0 0
\(995\) −46.7175 −1.48104
\(996\) 0.959913 0.0304160
\(997\) −27.9594 −0.885482 −0.442741 0.896650i \(-0.645994\pi\)
−0.442741 + 0.896650i \(0.645994\pi\)
\(998\) −4.61364 −0.146042
\(999\) 5.19657 0.164412
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 5929.2.a.bt.1.4 8
7.6 odd 2 847.2.a.p.1.4 8
11.5 even 5 539.2.f.e.344.3 16
11.9 even 5 539.2.f.e.246.3 16
11.10 odd 2 5929.2.a.bs.1.5 8
21.20 even 2 7623.2.a.ct.1.5 8
77.5 odd 30 539.2.q.g.410.2 32
77.6 even 10 847.2.f.x.729.2 16
77.9 even 15 539.2.q.f.312.3 32
77.13 even 10 847.2.f.x.323.2 16
77.16 even 15 539.2.q.f.410.2 32
77.20 odd 10 77.2.f.b.15.3 16
77.27 odd 10 77.2.f.b.36.3 yes 16
77.31 odd 30 539.2.q.g.422.2 32
77.38 odd 30 539.2.q.g.520.3 32
77.41 even 10 847.2.f.v.372.3 16
77.48 odd 10 847.2.f.w.148.2 16
77.53 even 15 539.2.q.f.422.2 32
77.60 even 15 539.2.q.f.520.3 32
77.62 even 10 847.2.f.v.148.3 16
77.69 odd 10 847.2.f.w.372.2 16
77.75 odd 30 539.2.q.g.312.3 32
77.76 even 2 847.2.a.o.1.5 8
231.20 even 10 693.2.m.i.631.2 16
231.104 even 10 693.2.m.i.190.2 16
231.230 odd 2 7623.2.a.cw.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.3 16 77.20 odd 10
77.2.f.b.36.3 yes 16 77.27 odd 10
539.2.f.e.246.3 16 11.9 even 5
539.2.f.e.344.3 16 11.5 even 5
539.2.q.f.312.3 32 77.9 even 15
539.2.q.f.410.2 32 77.16 even 15
539.2.q.f.422.2 32 77.53 even 15
539.2.q.f.520.3 32 77.60 even 15
539.2.q.g.312.3 32 77.75 odd 30
539.2.q.g.410.2 32 77.5 odd 30
539.2.q.g.422.2 32 77.31 odd 30
539.2.q.g.520.3 32 77.38 odd 30
693.2.m.i.190.2 16 231.104 even 10
693.2.m.i.631.2 16 231.20 even 10
847.2.a.o.1.5 8 77.76 even 2
847.2.a.p.1.4 8 7.6 odd 2
847.2.f.v.148.3 16 77.62 even 10
847.2.f.v.372.3 16 77.41 even 10
847.2.f.w.148.2 16 77.48 odd 10
847.2.f.w.372.2 16 77.69 odd 10
847.2.f.x.323.2 16 77.13 even 10
847.2.f.x.729.2 16 77.6 even 10
5929.2.a.bs.1.5 8 11.10 odd 2
5929.2.a.bt.1.4 8 1.1 even 1 trivial
7623.2.a.ct.1.5 8 21.20 even 2
7623.2.a.cw.1.4 8 231.230 odd 2