Properties

Label 4023.2.a.e.1.15
Level $4023$
Weight $2$
Character 4023.1
Self dual yes
Analytic conductor $32.124$
Analytic rank $1$
Dimension $25$
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] = [4023,2,Mod(1,4023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4023, 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("4023.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4023 = 3^{3} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4023.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.1238167332\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 4023.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.365778 q^{2} -1.86621 q^{4} -3.76223 q^{5} -2.92149 q^{7} -1.41417 q^{8} +O(q^{10})\) \(q+0.365778 q^{2} -1.86621 q^{4} -3.76223 q^{5} -2.92149 q^{7} -1.41417 q^{8} -1.37614 q^{10} +3.39144 q^{11} +1.65902 q^{13} -1.06862 q^{14} +3.21514 q^{16} +1.97443 q^{17} -5.94578 q^{19} +7.02109 q^{20} +1.24051 q^{22} +5.19965 q^{23} +9.15436 q^{25} +0.606833 q^{26} +5.45210 q^{28} -3.06884 q^{29} +6.78939 q^{31} +4.00437 q^{32} +0.722202 q^{34} +10.9913 q^{35} +1.04212 q^{37} -2.17484 q^{38} +5.32044 q^{40} +4.30891 q^{41} +6.88453 q^{43} -6.32913 q^{44} +1.90191 q^{46} -7.18405 q^{47} +1.53509 q^{49} +3.34846 q^{50} -3.09608 q^{52} -6.75951 q^{53} -12.7594 q^{55} +4.13149 q^{56} -1.12251 q^{58} -8.13193 q^{59} +10.8085 q^{61} +2.48341 q^{62} -4.96557 q^{64} -6.24161 q^{65} -6.26803 q^{67} -3.68469 q^{68} +4.02038 q^{70} +3.91921 q^{71} -9.32230 q^{73} +0.381186 q^{74} +11.0961 q^{76} -9.90805 q^{77} +3.65869 q^{79} -12.0961 q^{80} +1.57611 q^{82} -11.6033 q^{83} -7.42825 q^{85} +2.51821 q^{86} -4.79608 q^{88} -11.9433 q^{89} -4.84681 q^{91} -9.70361 q^{92} -2.62777 q^{94} +22.3694 q^{95} +10.1415 q^{97} +0.561503 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.365778 0.258644 0.129322 0.991603i \(-0.458720\pi\)
0.129322 + 0.991603i \(0.458720\pi\)
\(3\) 0 0
\(4\) −1.86621 −0.933103
\(5\) −3.76223 −1.68252 −0.841260 0.540631i \(-0.818185\pi\)
−0.841260 + 0.540631i \(0.818185\pi\)
\(6\) 0 0
\(7\) −2.92149 −1.10422 −0.552109 0.833772i \(-0.686177\pi\)
−0.552109 + 0.833772i \(0.686177\pi\)
\(8\) −1.41417 −0.499986
\(9\) 0 0
\(10\) −1.37614 −0.435173
\(11\) 3.39144 1.02256 0.511279 0.859415i \(-0.329172\pi\)
0.511279 + 0.859415i \(0.329172\pi\)
\(12\) 0 0
\(13\) 1.65902 0.460130 0.230065 0.973175i \(-0.426106\pi\)
0.230065 + 0.973175i \(0.426106\pi\)
\(14\) −1.06862 −0.285600
\(15\) 0 0
\(16\) 3.21514 0.803785
\(17\) 1.97443 0.478869 0.239435 0.970912i \(-0.423038\pi\)
0.239435 + 0.970912i \(0.423038\pi\)
\(18\) 0 0
\(19\) −5.94578 −1.36406 −0.682028 0.731326i \(-0.738902\pi\)
−0.682028 + 0.731326i \(0.738902\pi\)
\(20\) 7.02109 1.56996
\(21\) 0 0
\(22\) 1.24051 0.264478
\(23\) 5.19965 1.08420 0.542100 0.840314i \(-0.317629\pi\)
0.542100 + 0.840314i \(0.317629\pi\)
\(24\) 0 0
\(25\) 9.15436 1.83087
\(26\) 0.606833 0.119010
\(27\) 0 0
\(28\) 5.45210 1.03035
\(29\) −3.06884 −0.569869 −0.284935 0.958547i \(-0.591972\pi\)
−0.284935 + 0.958547i \(0.591972\pi\)
\(30\) 0 0
\(31\) 6.78939 1.21941 0.609705 0.792628i \(-0.291288\pi\)
0.609705 + 0.792628i \(0.291288\pi\)
\(32\) 4.00437 0.707880
\(33\) 0 0
\(34\) 0.722202 0.123857
\(35\) 10.9913 1.85787
\(36\) 0 0
\(37\) 1.04212 0.171324 0.0856621 0.996324i \(-0.472699\pi\)
0.0856621 + 0.996324i \(0.472699\pi\)
\(38\) −2.17484 −0.352805
\(39\) 0 0
\(40\) 5.32044 0.841235
\(41\) 4.30891 0.672939 0.336470 0.941694i \(-0.390767\pi\)
0.336470 + 0.941694i \(0.390767\pi\)
\(42\) 0 0
\(43\) 6.88453 1.04988 0.524940 0.851139i \(-0.324088\pi\)
0.524940 + 0.851139i \(0.324088\pi\)
\(44\) −6.32913 −0.954152
\(45\) 0 0
\(46\) 1.90191 0.280422
\(47\) −7.18405 −1.04790 −0.523951 0.851749i \(-0.675542\pi\)
−0.523951 + 0.851749i \(0.675542\pi\)
\(48\) 0 0
\(49\) 1.53509 0.219299
\(50\) 3.34846 0.473544
\(51\) 0 0
\(52\) −3.09608 −0.429348
\(53\) −6.75951 −0.928490 −0.464245 0.885707i \(-0.653674\pi\)
−0.464245 + 0.885707i \(0.653674\pi\)
\(54\) 0 0
\(55\) −12.7594 −1.72047
\(56\) 4.13149 0.552093
\(57\) 0 0
\(58\) −1.12251 −0.147393
\(59\) −8.13193 −1.05869 −0.529344 0.848408i \(-0.677562\pi\)
−0.529344 + 0.848408i \(0.677562\pi\)
\(60\) 0 0
\(61\) 10.8085 1.38389 0.691944 0.721951i \(-0.256754\pi\)
0.691944 + 0.721951i \(0.256754\pi\)
\(62\) 2.48341 0.315393
\(63\) 0 0
\(64\) −4.96557 −0.620696
\(65\) −6.24161 −0.774177
\(66\) 0 0
\(67\) −6.26803 −0.765761 −0.382881 0.923798i \(-0.625068\pi\)
−0.382881 + 0.923798i \(0.625068\pi\)
\(68\) −3.68469 −0.446835
\(69\) 0 0
\(70\) 4.02038 0.480527
\(71\) 3.91921 0.465125 0.232563 0.972581i \(-0.425289\pi\)
0.232563 + 0.972581i \(0.425289\pi\)
\(72\) 0 0
\(73\) −9.32230 −1.09109 −0.545546 0.838081i \(-0.683678\pi\)
−0.545546 + 0.838081i \(0.683678\pi\)
\(74\) 0.381186 0.0443120
\(75\) 0 0
\(76\) 11.0961 1.27281
\(77\) −9.90805 −1.12913
\(78\) 0 0
\(79\) 3.65869 0.411635 0.205817 0.978590i \(-0.434015\pi\)
0.205817 + 0.978590i \(0.434015\pi\)
\(80\) −12.0961 −1.35238
\(81\) 0 0
\(82\) 1.57611 0.174052
\(83\) −11.6033 −1.27363 −0.636813 0.771018i \(-0.719748\pi\)
−0.636813 + 0.771018i \(0.719748\pi\)
\(84\) 0 0
\(85\) −7.42825 −0.805707
\(86\) 2.51821 0.271545
\(87\) 0 0
\(88\) −4.79608 −0.511264
\(89\) −11.9433 −1.26598 −0.632992 0.774159i \(-0.718173\pi\)
−0.632992 + 0.774159i \(0.718173\pi\)
\(90\) 0 0
\(91\) −4.84681 −0.508084
\(92\) −9.70361 −1.01167
\(93\) 0 0
\(94\) −2.62777 −0.271033
\(95\) 22.3694 2.29505
\(96\) 0 0
\(97\) 10.1415 1.02971 0.514856 0.857277i \(-0.327845\pi\)
0.514856 + 0.857277i \(0.327845\pi\)
\(98\) 0.561503 0.0567204
\(99\) 0 0
\(100\) −17.0839 −1.70839
\(101\) 7.46540 0.742835 0.371417 0.928466i \(-0.378872\pi\)
0.371417 + 0.928466i \(0.378872\pi\)
\(102\) 0 0
\(103\) 11.9000 1.17254 0.586272 0.810114i \(-0.300595\pi\)
0.586272 + 0.810114i \(0.300595\pi\)
\(104\) −2.34614 −0.230058
\(105\) 0 0
\(106\) −2.47248 −0.240148
\(107\) −0.644073 −0.0622649 −0.0311324 0.999515i \(-0.509911\pi\)
−0.0311324 + 0.999515i \(0.509911\pi\)
\(108\) 0 0
\(109\) 1.21657 0.116526 0.0582632 0.998301i \(-0.481444\pi\)
0.0582632 + 0.998301i \(0.481444\pi\)
\(110\) −4.66709 −0.444990
\(111\) 0 0
\(112\) −9.39299 −0.887555
\(113\) 4.77472 0.449168 0.224584 0.974455i \(-0.427898\pi\)
0.224584 + 0.974455i \(0.427898\pi\)
\(114\) 0 0
\(115\) −19.5622 −1.82419
\(116\) 5.72709 0.531747
\(117\) 0 0
\(118\) −2.97448 −0.273823
\(119\) −5.76827 −0.528777
\(120\) 0 0
\(121\) 0.501865 0.0456241
\(122\) 3.95352 0.357934
\(123\) 0 0
\(124\) −12.6704 −1.13784
\(125\) −15.6296 −1.39796
\(126\) 0 0
\(127\) −12.7828 −1.13429 −0.567143 0.823619i \(-0.691952\pi\)
−0.567143 + 0.823619i \(0.691952\pi\)
\(128\) −9.82504 −0.868419
\(129\) 0 0
\(130\) −2.28304 −0.200236
\(131\) 7.46513 0.652231 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(132\) 0 0
\(133\) 17.3705 1.50622
\(134\) −2.29270 −0.198060
\(135\) 0 0
\(136\) −2.79218 −0.239428
\(137\) −16.3681 −1.39842 −0.699211 0.714916i \(-0.746465\pi\)
−0.699211 + 0.714916i \(0.746465\pi\)
\(138\) 0 0
\(139\) 19.4897 1.65310 0.826549 0.562865i \(-0.190301\pi\)
0.826549 + 0.562865i \(0.190301\pi\)
\(140\) −20.5120 −1.73358
\(141\) 0 0
\(142\) 1.43356 0.120302
\(143\) 5.62647 0.470509
\(144\) 0 0
\(145\) 11.5457 0.958816
\(146\) −3.40989 −0.282205
\(147\) 0 0
\(148\) −1.94482 −0.159863
\(149\) −1.00000 −0.0819232
\(150\) 0 0
\(151\) −9.39072 −0.764206 −0.382103 0.924120i \(-0.624800\pi\)
−0.382103 + 0.924120i \(0.624800\pi\)
\(152\) 8.40836 0.682008
\(153\) 0 0
\(154\) −3.62415 −0.292042
\(155\) −25.5432 −2.05168
\(156\) 0 0
\(157\) −1.42584 −0.113794 −0.0568972 0.998380i \(-0.518121\pi\)
−0.0568972 + 0.998380i \(0.518121\pi\)
\(158\) 1.33827 0.106467
\(159\) 0 0
\(160\) −15.0654 −1.19102
\(161\) −15.1907 −1.19720
\(162\) 0 0
\(163\) 7.32087 0.573415 0.286708 0.958018i \(-0.407439\pi\)
0.286708 + 0.958018i \(0.407439\pi\)
\(164\) −8.04132 −0.627922
\(165\) 0 0
\(166\) −4.24423 −0.329416
\(167\) 6.85319 0.530316 0.265158 0.964205i \(-0.414576\pi\)
0.265158 + 0.964205i \(0.414576\pi\)
\(168\) 0 0
\(169\) −10.2476 −0.788281
\(170\) −2.71709 −0.208391
\(171\) 0 0
\(172\) −12.8479 −0.979647
\(173\) 7.35922 0.559511 0.279756 0.960071i \(-0.409747\pi\)
0.279756 + 0.960071i \(0.409747\pi\)
\(174\) 0 0
\(175\) −26.7443 −2.02168
\(176\) 10.9040 0.821917
\(177\) 0 0
\(178\) −4.36858 −0.327439
\(179\) −15.9140 −1.18947 −0.594733 0.803923i \(-0.702742\pi\)
−0.594733 + 0.803923i \(0.702742\pi\)
\(180\) 0 0
\(181\) 23.0936 1.71654 0.858268 0.513202i \(-0.171541\pi\)
0.858268 + 0.513202i \(0.171541\pi\)
\(182\) −1.77286 −0.131413
\(183\) 0 0
\(184\) −7.35320 −0.542085
\(185\) −3.92071 −0.288256
\(186\) 0 0
\(187\) 6.69616 0.489671
\(188\) 13.4069 0.977800
\(189\) 0 0
\(190\) 8.18223 0.593601
\(191\) −20.3176 −1.47013 −0.735065 0.677997i \(-0.762848\pi\)
−0.735065 + 0.677997i \(0.762848\pi\)
\(192\) 0 0
\(193\) −10.9436 −0.787735 −0.393867 0.919167i \(-0.628863\pi\)
−0.393867 + 0.919167i \(0.628863\pi\)
\(194\) 3.70953 0.266329
\(195\) 0 0
\(196\) −2.86480 −0.204629
\(197\) −25.2674 −1.80023 −0.900115 0.435653i \(-0.856518\pi\)
−0.900115 + 0.435653i \(0.856518\pi\)
\(198\) 0 0
\(199\) −11.4016 −0.808240 −0.404120 0.914706i \(-0.632422\pi\)
−0.404120 + 0.914706i \(0.632422\pi\)
\(200\) −12.9458 −0.915409
\(201\) 0 0
\(202\) 2.73068 0.192130
\(203\) 8.96558 0.629261
\(204\) 0 0
\(205\) −16.2111 −1.13223
\(206\) 4.35276 0.303271
\(207\) 0 0
\(208\) 5.33399 0.369845
\(209\) −20.1648 −1.39483
\(210\) 0 0
\(211\) −2.25513 −0.155249 −0.0776246 0.996983i \(-0.524734\pi\)
−0.0776246 + 0.996983i \(0.524734\pi\)
\(212\) 12.6146 0.866377
\(213\) 0 0
\(214\) −0.235588 −0.0161044
\(215\) −25.9012 −1.76644
\(216\) 0 0
\(217\) −19.8351 −1.34650
\(218\) 0.444995 0.0301389
\(219\) 0 0
\(220\) 23.8116 1.60538
\(221\) 3.27562 0.220342
\(222\) 0 0
\(223\) 5.77520 0.386736 0.193368 0.981126i \(-0.438059\pi\)
0.193368 + 0.981126i \(0.438059\pi\)
\(224\) −11.6987 −0.781654
\(225\) 0 0
\(226\) 1.74649 0.116175
\(227\) 26.2629 1.74313 0.871564 0.490282i \(-0.163106\pi\)
0.871564 + 0.490282i \(0.163106\pi\)
\(228\) 0 0
\(229\) 28.9359 1.91214 0.956069 0.293141i \(-0.0947005\pi\)
0.956069 + 0.293141i \(0.0947005\pi\)
\(230\) −7.15544 −0.471816
\(231\) 0 0
\(232\) 4.33987 0.284926
\(233\) −12.4194 −0.813622 −0.406811 0.913512i \(-0.633359\pi\)
−0.406811 + 0.913512i \(0.633359\pi\)
\(234\) 0 0
\(235\) 27.0280 1.76311
\(236\) 15.1759 0.987865
\(237\) 0 0
\(238\) −2.10991 −0.136765
\(239\) −1.20431 −0.0779004 −0.0389502 0.999241i \(-0.512401\pi\)
−0.0389502 + 0.999241i \(0.512401\pi\)
\(240\) 0 0
\(241\) −13.3214 −0.858107 −0.429054 0.903279i \(-0.641153\pi\)
−0.429054 + 0.903279i \(0.641153\pi\)
\(242\) 0.183571 0.0118004
\(243\) 0 0
\(244\) −20.1709 −1.29131
\(245\) −5.77537 −0.368975
\(246\) 0 0
\(247\) −9.86418 −0.627643
\(248\) −9.60137 −0.609688
\(249\) 0 0
\(250\) −5.71697 −0.361573
\(251\) 19.8611 1.25362 0.626812 0.779171i \(-0.284360\pi\)
0.626812 + 0.779171i \(0.284360\pi\)
\(252\) 0 0
\(253\) 17.6343 1.10866
\(254\) −4.67565 −0.293376
\(255\) 0 0
\(256\) 6.33736 0.396085
\(257\) −6.34398 −0.395727 −0.197863 0.980230i \(-0.563400\pi\)
−0.197863 + 0.980230i \(0.563400\pi\)
\(258\) 0 0
\(259\) −3.04455 −0.189179
\(260\) 11.6481 0.722387
\(261\) 0 0
\(262\) 2.73058 0.168696
\(263\) −11.6343 −0.717405 −0.358702 0.933452i \(-0.616781\pi\)
−0.358702 + 0.933452i \(0.616781\pi\)
\(264\) 0 0
\(265\) 25.4308 1.56220
\(266\) 6.35376 0.389574
\(267\) 0 0
\(268\) 11.6974 0.714534
\(269\) −15.4781 −0.943715 −0.471858 0.881675i \(-0.656416\pi\)
−0.471858 + 0.881675i \(0.656416\pi\)
\(270\) 0 0
\(271\) −21.3243 −1.29536 −0.647678 0.761914i \(-0.724260\pi\)
−0.647678 + 0.761914i \(0.724260\pi\)
\(272\) 6.34807 0.384908
\(273\) 0 0
\(274\) −5.98709 −0.361693
\(275\) 31.0465 1.87217
\(276\) 0 0
\(277\) 14.0313 0.843058 0.421529 0.906815i \(-0.361494\pi\)
0.421529 + 0.906815i \(0.361494\pi\)
\(278\) 7.12891 0.427564
\(279\) 0 0
\(280\) −15.5436 −0.928908
\(281\) −2.32437 −0.138660 −0.0693302 0.997594i \(-0.522086\pi\)
−0.0693302 + 0.997594i \(0.522086\pi\)
\(282\) 0 0
\(283\) −22.7990 −1.35526 −0.677631 0.735402i \(-0.736993\pi\)
−0.677631 + 0.735402i \(0.736993\pi\)
\(284\) −7.31406 −0.434010
\(285\) 0 0
\(286\) 2.05804 0.121694
\(287\) −12.5884 −0.743072
\(288\) 0 0
\(289\) −13.1016 −0.770684
\(290\) 4.22315 0.247992
\(291\) 0 0
\(292\) 17.3973 1.01810
\(293\) −23.1108 −1.35015 −0.675073 0.737751i \(-0.735888\pi\)
−0.675073 + 0.737751i \(0.735888\pi\)
\(294\) 0 0
\(295\) 30.5942 1.78126
\(296\) −1.47374 −0.0856596
\(297\) 0 0
\(298\) −0.365778 −0.0211889
\(299\) 8.62632 0.498873
\(300\) 0 0
\(301\) −20.1131 −1.15930
\(302\) −3.43492 −0.197657
\(303\) 0 0
\(304\) −19.1165 −1.09641
\(305\) −40.6641 −2.32842
\(306\) 0 0
\(307\) 7.95861 0.454222 0.227111 0.973869i \(-0.427072\pi\)
0.227111 + 0.973869i \(0.427072\pi\)
\(308\) 18.4905 1.05359
\(309\) 0 0
\(310\) −9.34315 −0.530655
\(311\) −2.11477 −0.119918 −0.0599588 0.998201i \(-0.519097\pi\)
−0.0599588 + 0.998201i \(0.519097\pi\)
\(312\) 0 0
\(313\) 7.12070 0.402486 0.201243 0.979541i \(-0.435502\pi\)
0.201243 + 0.979541i \(0.435502\pi\)
\(314\) −0.521541 −0.0294323
\(315\) 0 0
\(316\) −6.82787 −0.384098
\(317\) 23.2371 1.30512 0.652562 0.757735i \(-0.273694\pi\)
0.652562 + 0.757735i \(0.273694\pi\)
\(318\) 0 0
\(319\) −10.4078 −0.582724
\(320\) 18.6816 1.04433
\(321\) 0 0
\(322\) −5.55642 −0.309647
\(323\) −11.7395 −0.653205
\(324\) 0 0
\(325\) 15.1873 0.842438
\(326\) 2.67781 0.148310
\(327\) 0 0
\(328\) −6.09355 −0.336460
\(329\) 20.9881 1.15711
\(330\) 0 0
\(331\) −32.5567 −1.78948 −0.894739 0.446589i \(-0.852639\pi\)
−0.894739 + 0.446589i \(0.852639\pi\)
\(332\) 21.6541 1.18843
\(333\) 0 0
\(334\) 2.50675 0.137163
\(335\) 23.5817 1.28841
\(336\) 0 0
\(337\) 21.4417 1.16800 0.584002 0.811752i \(-0.301486\pi\)
0.584002 + 0.811752i \(0.301486\pi\)
\(338\) −3.74836 −0.203884
\(339\) 0 0
\(340\) 13.8627 0.751808
\(341\) 23.0258 1.24692
\(342\) 0 0
\(343\) 15.9657 0.862065
\(344\) −9.73591 −0.524925
\(345\) 0 0
\(346\) 2.69184 0.144714
\(347\) −6.86807 −0.368698 −0.184349 0.982861i \(-0.559018\pi\)
−0.184349 + 0.982861i \(0.559018\pi\)
\(348\) 0 0
\(349\) 33.6046 1.79881 0.899405 0.437115i \(-0.144000\pi\)
0.899405 + 0.437115i \(0.144000\pi\)
\(350\) −9.78249 −0.522896
\(351\) 0 0
\(352\) 13.5806 0.723848
\(353\) −5.40572 −0.287717 −0.143859 0.989598i \(-0.545951\pi\)
−0.143859 + 0.989598i \(0.545951\pi\)
\(354\) 0 0
\(355\) −14.7450 −0.782582
\(356\) 22.2886 1.18129
\(357\) 0 0
\(358\) −5.82098 −0.307648
\(359\) 7.21437 0.380760 0.190380 0.981711i \(-0.439028\pi\)
0.190380 + 0.981711i \(0.439028\pi\)
\(360\) 0 0
\(361\) 16.3523 0.860650
\(362\) 8.44714 0.443972
\(363\) 0 0
\(364\) 9.04515 0.474095
\(365\) 35.0726 1.83578
\(366\) 0 0
\(367\) 22.5758 1.17845 0.589224 0.807969i \(-0.299433\pi\)
0.589224 + 0.807969i \(0.299433\pi\)
\(368\) 16.7176 0.871465
\(369\) 0 0
\(370\) −1.43411 −0.0745557
\(371\) 19.7478 1.02526
\(372\) 0 0
\(373\) −7.49410 −0.388030 −0.194015 0.980999i \(-0.562151\pi\)
−0.194015 + 0.980999i \(0.562151\pi\)
\(374\) 2.44931 0.126651
\(375\) 0 0
\(376\) 10.1595 0.523935
\(377\) −5.09127 −0.262214
\(378\) 0 0
\(379\) −3.22719 −0.165770 −0.0828848 0.996559i \(-0.526413\pi\)
−0.0828848 + 0.996559i \(0.526413\pi\)
\(380\) −41.7459 −2.14152
\(381\) 0 0
\(382\) −7.43173 −0.380240
\(383\) −33.2748 −1.70026 −0.850131 0.526571i \(-0.823477\pi\)
−0.850131 + 0.526571i \(0.823477\pi\)
\(384\) 0 0
\(385\) 37.2763 1.89978
\(386\) −4.00291 −0.203743
\(387\) 0 0
\(388\) −18.9261 −0.960827
\(389\) −34.5478 −1.75164 −0.875820 0.482637i \(-0.839679\pi\)
−0.875820 + 0.482637i \(0.839679\pi\)
\(390\) 0 0
\(391\) 10.2663 0.519191
\(392\) −2.17089 −0.109646
\(393\) 0 0
\(394\) −9.24227 −0.465619
\(395\) −13.7648 −0.692583
\(396\) 0 0
\(397\) 21.9049 1.09938 0.549688 0.835370i \(-0.314746\pi\)
0.549688 + 0.835370i \(0.314746\pi\)
\(398\) −4.17046 −0.209046
\(399\) 0 0
\(400\) 29.4325 1.47163
\(401\) −31.2502 −1.56056 −0.780280 0.625430i \(-0.784924\pi\)
−0.780280 + 0.625430i \(0.784924\pi\)
\(402\) 0 0
\(403\) 11.2637 0.561087
\(404\) −13.9320 −0.693142
\(405\) 0 0
\(406\) 3.27941 0.162754
\(407\) 3.53430 0.175189
\(408\) 0 0
\(409\) −21.3295 −1.05468 −0.527338 0.849655i \(-0.676810\pi\)
−0.527338 + 0.849655i \(0.676810\pi\)
\(410\) −5.92967 −0.292845
\(411\) 0 0
\(412\) −22.2079 −1.09410
\(413\) 23.7573 1.16902
\(414\) 0 0
\(415\) 43.6542 2.14290
\(416\) 6.64334 0.325716
\(417\) 0 0
\(418\) −7.37582 −0.360763
\(419\) 7.40525 0.361770 0.180885 0.983504i \(-0.442104\pi\)
0.180885 + 0.983504i \(0.442104\pi\)
\(420\) 0 0
\(421\) 17.5504 0.855353 0.427676 0.903932i \(-0.359332\pi\)
0.427676 + 0.903932i \(0.359332\pi\)
\(422\) −0.824875 −0.0401543
\(423\) 0 0
\(424\) 9.55911 0.464232
\(425\) 18.0746 0.876748
\(426\) 0 0
\(427\) −31.5770 −1.52812
\(428\) 1.20197 0.0580996
\(429\) 0 0
\(430\) −9.47407 −0.456880
\(431\) −29.9688 −1.44355 −0.721773 0.692130i \(-0.756673\pi\)
−0.721773 + 0.692130i \(0.756673\pi\)
\(432\) 0 0
\(433\) −6.64094 −0.319143 −0.159571 0.987186i \(-0.551011\pi\)
−0.159571 + 0.987186i \(0.551011\pi\)
\(434\) −7.25525 −0.348263
\(435\) 0 0
\(436\) −2.27037 −0.108731
\(437\) −30.9160 −1.47891
\(438\) 0 0
\(439\) 28.0857 1.34046 0.670228 0.742155i \(-0.266196\pi\)
0.670228 + 0.742155i \(0.266196\pi\)
\(440\) 18.0440 0.860212
\(441\) 0 0
\(442\) 1.19815 0.0569901
\(443\) 11.4861 0.545722 0.272861 0.962054i \(-0.412030\pi\)
0.272861 + 0.962054i \(0.412030\pi\)
\(444\) 0 0
\(445\) 44.9333 2.13004
\(446\) 2.11244 0.100027
\(447\) 0 0
\(448\) 14.5069 0.685384
\(449\) −11.7104 −0.552649 −0.276325 0.961064i \(-0.589116\pi\)
−0.276325 + 0.961064i \(0.589116\pi\)
\(450\) 0 0
\(451\) 14.6134 0.688119
\(452\) −8.91062 −0.419120
\(453\) 0 0
\(454\) 9.60638 0.450850
\(455\) 18.2348 0.854861
\(456\) 0 0
\(457\) −23.6519 −1.10639 −0.553194 0.833052i \(-0.686591\pi\)
−0.553194 + 0.833052i \(0.686591\pi\)
\(458\) 10.5841 0.494563
\(459\) 0 0
\(460\) 36.5072 1.70216
\(461\) −30.8073 −1.43484 −0.717420 0.696641i \(-0.754677\pi\)
−0.717420 + 0.696641i \(0.754677\pi\)
\(462\) 0 0
\(463\) 12.6345 0.587173 0.293587 0.955932i \(-0.405151\pi\)
0.293587 + 0.955932i \(0.405151\pi\)
\(464\) −9.86675 −0.458053
\(465\) 0 0
\(466\) −4.54274 −0.210438
\(467\) 17.4707 0.808447 0.404223 0.914660i \(-0.367542\pi\)
0.404223 + 0.914660i \(0.367542\pi\)
\(468\) 0 0
\(469\) 18.3120 0.845568
\(470\) 9.88625 0.456019
\(471\) 0 0
\(472\) 11.5000 0.529328
\(473\) 23.3485 1.07356
\(474\) 0 0
\(475\) −54.4298 −2.49741
\(476\) 10.7648 0.493403
\(477\) 0 0
\(478\) −0.440510 −0.0201485
\(479\) −14.4042 −0.658144 −0.329072 0.944305i \(-0.606736\pi\)
−0.329072 + 0.944305i \(0.606736\pi\)
\(480\) 0 0
\(481\) 1.72891 0.0788313
\(482\) −4.87268 −0.221944
\(483\) 0 0
\(484\) −0.936585 −0.0425720
\(485\) −38.1546 −1.73251
\(486\) 0 0
\(487\) −32.3687 −1.46676 −0.733382 0.679817i \(-0.762059\pi\)
−0.733382 + 0.679817i \(0.762059\pi\)
\(488\) −15.2851 −0.691924
\(489\) 0 0
\(490\) −2.11250 −0.0954332
\(491\) −21.7705 −0.982490 −0.491245 0.871022i \(-0.663458\pi\)
−0.491245 + 0.871022i \(0.663458\pi\)
\(492\) 0 0
\(493\) −6.05921 −0.272893
\(494\) −3.60810 −0.162336
\(495\) 0 0
\(496\) 21.8288 0.980144
\(497\) −11.4499 −0.513600
\(498\) 0 0
\(499\) 2.04209 0.0914167 0.0457083 0.998955i \(-0.485446\pi\)
0.0457083 + 0.998955i \(0.485446\pi\)
\(500\) 29.1681 1.30444
\(501\) 0 0
\(502\) 7.26476 0.324242
\(503\) −1.40290 −0.0625522 −0.0312761 0.999511i \(-0.509957\pi\)
−0.0312761 + 0.999511i \(0.509957\pi\)
\(504\) 0 0
\(505\) −28.0865 −1.24983
\(506\) 6.45023 0.286748
\(507\) 0 0
\(508\) 23.8553 1.05841
\(509\) 38.1088 1.68914 0.844571 0.535443i \(-0.179855\pi\)
0.844571 + 0.535443i \(0.179855\pi\)
\(510\) 0 0
\(511\) 27.2350 1.20480
\(512\) 21.9681 0.970864
\(513\) 0 0
\(514\) −2.32049 −0.102352
\(515\) −44.7706 −1.97283
\(516\) 0 0
\(517\) −24.3643 −1.07154
\(518\) −1.11363 −0.0489301
\(519\) 0 0
\(520\) 8.82672 0.387077
\(521\) 1.83967 0.0805974 0.0402987 0.999188i \(-0.487169\pi\)
0.0402987 + 0.999188i \(0.487169\pi\)
\(522\) 0 0
\(523\) −15.1344 −0.661780 −0.330890 0.943669i \(-0.607349\pi\)
−0.330890 + 0.943669i \(0.607349\pi\)
\(524\) −13.9315 −0.608599
\(525\) 0 0
\(526\) −4.25559 −0.185552
\(527\) 13.4052 0.583938
\(528\) 0 0
\(529\) 4.03631 0.175492
\(530\) 9.30203 0.404054
\(531\) 0 0
\(532\) −32.4170 −1.40546
\(533\) 7.14858 0.309639
\(534\) 0 0
\(535\) 2.42315 0.104762
\(536\) 8.86407 0.382870
\(537\) 0 0
\(538\) −5.66154 −0.244086
\(539\) 5.20618 0.224246
\(540\) 0 0
\(541\) −12.2234 −0.525526 −0.262763 0.964860i \(-0.584634\pi\)
−0.262763 + 0.964860i \(0.584634\pi\)
\(542\) −7.79994 −0.335036
\(543\) 0 0
\(544\) 7.90635 0.338982
\(545\) −4.57702 −0.196058
\(546\) 0 0
\(547\) 3.66498 0.156703 0.0783516 0.996926i \(-0.475034\pi\)
0.0783516 + 0.996926i \(0.475034\pi\)
\(548\) 30.5463 1.30487
\(549\) 0 0
\(550\) 11.3561 0.484226
\(551\) 18.2467 0.777334
\(552\) 0 0
\(553\) −10.6888 −0.454535
\(554\) 5.13233 0.218052
\(555\) 0 0
\(556\) −36.3719 −1.54251
\(557\) 27.6348 1.17092 0.585461 0.810700i \(-0.300913\pi\)
0.585461 + 0.810700i \(0.300913\pi\)
\(558\) 0 0
\(559\) 11.4216 0.483081
\(560\) 35.3386 1.49333
\(561\) 0 0
\(562\) −0.850204 −0.0358637
\(563\) −4.02658 −0.169700 −0.0848500 0.996394i \(-0.527041\pi\)
−0.0848500 + 0.996394i \(0.527041\pi\)
\(564\) 0 0
\(565\) −17.9636 −0.755734
\(566\) −8.33938 −0.350530
\(567\) 0 0
\(568\) −5.54244 −0.232556
\(569\) 7.02698 0.294586 0.147293 0.989093i \(-0.452944\pi\)
0.147293 + 0.989093i \(0.452944\pi\)
\(570\) 0 0
\(571\) −30.5714 −1.27938 −0.639688 0.768635i \(-0.720936\pi\)
−0.639688 + 0.768635i \(0.720936\pi\)
\(572\) −10.5002 −0.439034
\(573\) 0 0
\(574\) −4.60457 −0.192191
\(575\) 47.5994 1.98503
\(576\) 0 0
\(577\) −42.1510 −1.75477 −0.877385 0.479786i \(-0.840714\pi\)
−0.877385 + 0.479786i \(0.840714\pi\)
\(578\) −4.79229 −0.199333
\(579\) 0 0
\(580\) −21.5466 −0.894675
\(581\) 33.8989 1.40636
\(582\) 0 0
\(583\) −22.9245 −0.949434
\(584\) 13.1833 0.545530
\(585\) 0 0
\(586\) −8.45341 −0.349207
\(587\) 12.2011 0.503595 0.251797 0.967780i \(-0.418978\pi\)
0.251797 + 0.967780i \(0.418978\pi\)
\(588\) 0 0
\(589\) −40.3682 −1.66334
\(590\) 11.1907 0.460713
\(591\) 0 0
\(592\) 3.35058 0.137708
\(593\) −12.3294 −0.506306 −0.253153 0.967426i \(-0.581468\pi\)
−0.253153 + 0.967426i \(0.581468\pi\)
\(594\) 0 0
\(595\) 21.7015 0.889677
\(596\) 1.86621 0.0764428
\(597\) 0 0
\(598\) 3.15532 0.129030
\(599\) 14.7447 0.602452 0.301226 0.953553i \(-0.402604\pi\)
0.301226 + 0.953553i \(0.402604\pi\)
\(600\) 0 0
\(601\) −13.7113 −0.559295 −0.279647 0.960103i \(-0.590218\pi\)
−0.279647 + 0.960103i \(0.590218\pi\)
\(602\) −7.35691 −0.299845
\(603\) 0 0
\(604\) 17.5250 0.713083
\(605\) −1.88813 −0.0767635
\(606\) 0 0
\(607\) 10.1891 0.413563 0.206782 0.978387i \(-0.433701\pi\)
0.206782 + 0.978387i \(0.433701\pi\)
\(608\) −23.8091 −0.965588
\(609\) 0 0
\(610\) −14.8740 −0.602232
\(611\) −11.9185 −0.482170
\(612\) 0 0
\(613\) −10.3390 −0.417586 −0.208793 0.977960i \(-0.566954\pi\)
−0.208793 + 0.977960i \(0.566954\pi\)
\(614\) 2.91108 0.117482
\(615\) 0 0
\(616\) 14.0117 0.564547
\(617\) −6.33913 −0.255204 −0.127602 0.991825i \(-0.540728\pi\)
−0.127602 + 0.991825i \(0.540728\pi\)
\(618\) 0 0
\(619\) −14.5154 −0.583423 −0.291712 0.956506i \(-0.594225\pi\)
−0.291712 + 0.956506i \(0.594225\pi\)
\(620\) 47.6689 1.91443
\(621\) 0 0
\(622\) −0.773536 −0.0310160
\(623\) 34.8921 1.39792
\(624\) 0 0
\(625\) 13.0305 0.521218
\(626\) 2.60460 0.104101
\(627\) 0 0
\(628\) 2.66091 0.106182
\(629\) 2.05760 0.0820419
\(630\) 0 0
\(631\) 35.8501 1.42717 0.713584 0.700570i \(-0.247071\pi\)
0.713584 + 0.700570i \(0.247071\pi\)
\(632\) −5.17402 −0.205811
\(633\) 0 0
\(634\) 8.49961 0.337563
\(635\) 48.0917 1.90846
\(636\) 0 0
\(637\) 2.54675 0.100906
\(638\) −3.80694 −0.150718
\(639\) 0 0
\(640\) 36.9640 1.46113
\(641\) 46.9030 1.85256 0.926279 0.376839i \(-0.122989\pi\)
0.926279 + 0.376839i \(0.122989\pi\)
\(642\) 0 0
\(643\) −47.0632 −1.85599 −0.927995 0.372593i \(-0.878469\pi\)
−0.927995 + 0.372593i \(0.878469\pi\)
\(644\) 28.3490 1.11711
\(645\) 0 0
\(646\) −4.29406 −0.168947
\(647\) 50.7955 1.99698 0.998488 0.0549708i \(-0.0175066\pi\)
0.998488 + 0.0549708i \(0.0175066\pi\)
\(648\) 0 0
\(649\) −27.5790 −1.08257
\(650\) 5.55517 0.217892
\(651\) 0 0
\(652\) −13.6623 −0.535056
\(653\) −19.3835 −0.758534 −0.379267 0.925287i \(-0.623824\pi\)
−0.379267 + 0.925287i \(0.623824\pi\)
\(654\) 0 0
\(655\) −28.0855 −1.09739
\(656\) 13.8538 0.540899
\(657\) 0 0
\(658\) 7.67699 0.299280
\(659\) −39.6111 −1.54303 −0.771514 0.636212i \(-0.780500\pi\)
−0.771514 + 0.636212i \(0.780500\pi\)
\(660\) 0 0
\(661\) −6.53392 −0.254140 −0.127070 0.991894i \(-0.540557\pi\)
−0.127070 + 0.991894i \(0.540557\pi\)
\(662\) −11.9085 −0.462838
\(663\) 0 0
\(664\) 16.4091 0.636795
\(665\) −65.3519 −2.53424
\(666\) 0 0
\(667\) −15.9569 −0.617853
\(668\) −12.7895 −0.494840
\(669\) 0 0
\(670\) 8.62568 0.333239
\(671\) 36.6564 1.41511
\(672\) 0 0
\(673\) 29.1581 1.12396 0.561981 0.827150i \(-0.310040\pi\)
0.561981 + 0.827150i \(0.310040\pi\)
\(674\) 7.84290 0.302097
\(675\) 0 0
\(676\) 19.1242 0.735547
\(677\) −47.3893 −1.82132 −0.910659 0.413159i \(-0.864426\pi\)
−0.910659 + 0.413159i \(0.864426\pi\)
\(678\) 0 0
\(679\) −29.6282 −1.13703
\(680\) 10.5048 0.402842
\(681\) 0 0
\(682\) 8.42233 0.322508
\(683\) −44.2885 −1.69465 −0.847326 0.531073i \(-0.821789\pi\)
−0.847326 + 0.531073i \(0.821789\pi\)
\(684\) 0 0
\(685\) 61.5805 2.35287
\(686\) 5.83988 0.222968
\(687\) 0 0
\(688\) 22.1347 0.843878
\(689\) −11.2142 −0.427226
\(690\) 0 0
\(691\) 41.5417 1.58032 0.790161 0.612900i \(-0.209997\pi\)
0.790161 + 0.612900i \(0.209997\pi\)
\(692\) −13.7338 −0.522082
\(693\) 0 0
\(694\) −2.51219 −0.0953614
\(695\) −73.3248 −2.78137
\(696\) 0 0
\(697\) 8.50765 0.322250
\(698\) 12.2918 0.465252
\(699\) 0 0
\(700\) 49.9105 1.88644
\(701\) 38.4336 1.45162 0.725809 0.687896i \(-0.241466\pi\)
0.725809 + 0.687896i \(0.241466\pi\)
\(702\) 0 0
\(703\) −6.19625 −0.233696
\(704\) −16.8404 −0.634698
\(705\) 0 0
\(706\) −1.97729 −0.0744164
\(707\) −21.8101 −0.820252
\(708\) 0 0
\(709\) 22.9347 0.861329 0.430664 0.902512i \(-0.358279\pi\)
0.430664 + 0.902512i \(0.358279\pi\)
\(710\) −5.39338 −0.202410
\(711\) 0 0
\(712\) 16.8898 0.632973
\(713\) 35.3024 1.32209
\(714\) 0 0
\(715\) −21.1681 −0.791641
\(716\) 29.6988 1.10990
\(717\) 0 0
\(718\) 2.63886 0.0984812
\(719\) 21.6943 0.809060 0.404530 0.914525i \(-0.367435\pi\)
0.404530 + 0.914525i \(0.367435\pi\)
\(720\) 0 0
\(721\) −34.7658 −1.29474
\(722\) 5.98132 0.222602
\(723\) 0 0
\(724\) −43.0975 −1.60171
\(725\) −28.0933 −1.04336
\(726\) 0 0
\(727\) −36.5274 −1.35473 −0.677363 0.735649i \(-0.736877\pi\)
−0.677363 + 0.735649i \(0.736877\pi\)
\(728\) 6.85423 0.254035
\(729\) 0 0
\(730\) 12.8288 0.474815
\(731\) 13.5930 0.502756
\(732\) 0 0
\(733\) −30.5790 −1.12946 −0.564731 0.825275i \(-0.691020\pi\)
−0.564731 + 0.825275i \(0.691020\pi\)
\(734\) 8.25774 0.304799
\(735\) 0 0
\(736\) 20.8213 0.767484
\(737\) −21.2576 −0.783035
\(738\) 0 0
\(739\) 3.19635 0.117580 0.0587899 0.998270i \(-0.481276\pi\)
0.0587899 + 0.998270i \(0.481276\pi\)
\(740\) 7.31685 0.268973
\(741\) 0 0
\(742\) 7.22332 0.265176
\(743\) 36.4235 1.33625 0.668124 0.744050i \(-0.267098\pi\)
0.668124 + 0.744050i \(0.267098\pi\)
\(744\) 0 0
\(745\) 3.76223 0.137837
\(746\) −2.74118 −0.100362
\(747\) 0 0
\(748\) −12.4964 −0.456914
\(749\) 1.88165 0.0687541
\(750\) 0 0
\(751\) −47.0659 −1.71746 −0.858730 0.512428i \(-0.828746\pi\)
−0.858730 + 0.512428i \(0.828746\pi\)
\(752\) −23.0977 −0.842287
\(753\) 0 0
\(754\) −1.86227 −0.0678200
\(755\) 35.3300 1.28579
\(756\) 0 0
\(757\) −21.6332 −0.786272 −0.393136 0.919480i \(-0.628610\pi\)
−0.393136 + 0.919480i \(0.628610\pi\)
\(758\) −1.18043 −0.0428753
\(759\) 0 0
\(760\) −31.6342 −1.14749
\(761\) 11.6574 0.422581 0.211290 0.977423i \(-0.432233\pi\)
0.211290 + 0.977423i \(0.432233\pi\)
\(762\) 0 0
\(763\) −3.55420 −0.128671
\(764\) 37.9168 1.37178
\(765\) 0 0
\(766\) −12.1712 −0.439763
\(767\) −13.4910 −0.487133
\(768\) 0 0
\(769\) 3.41304 0.123077 0.0615387 0.998105i \(-0.480399\pi\)
0.0615387 + 0.998105i \(0.480399\pi\)
\(770\) 13.6349 0.491366
\(771\) 0 0
\(772\) 20.4229 0.735038
\(773\) −27.8909 −1.00317 −0.501583 0.865109i \(-0.667249\pi\)
−0.501583 + 0.865109i \(0.667249\pi\)
\(774\) 0 0
\(775\) 62.1525 2.23258
\(776\) −14.3418 −0.514841
\(777\) 0 0
\(778\) −12.6368 −0.453051
\(779\) −25.6199 −0.917927
\(780\) 0 0
\(781\) 13.2918 0.475617
\(782\) 3.75520 0.134286
\(783\) 0 0
\(784\) 4.93554 0.176269
\(785\) 5.36434 0.191461
\(786\) 0 0
\(787\) −22.5283 −0.803047 −0.401523 0.915849i \(-0.631519\pi\)
−0.401523 + 0.915849i \(0.631519\pi\)
\(788\) 47.1542 1.67980
\(789\) 0 0
\(790\) −5.03486 −0.179132
\(791\) −13.9493 −0.495980
\(792\) 0 0
\(793\) 17.9316 0.636768
\(794\) 8.01234 0.284347
\(795\) 0 0
\(796\) 21.2778 0.754171
\(797\) −10.8184 −0.383207 −0.191604 0.981472i \(-0.561369\pi\)
−0.191604 + 0.981472i \(0.561369\pi\)
\(798\) 0 0
\(799\) −14.1844 −0.501808
\(800\) 36.6574 1.29604
\(801\) 0 0
\(802\) −11.4306 −0.403630
\(803\) −31.6160 −1.11571
\(804\) 0 0
\(805\) 57.1509 2.01430
\(806\) 4.12003 0.145122
\(807\) 0 0
\(808\) −10.5574 −0.371407
\(809\) 26.5554 0.933638 0.466819 0.884353i \(-0.345400\pi\)
0.466819 + 0.884353i \(0.345400\pi\)
\(810\) 0 0
\(811\) −35.3496 −1.24129 −0.620645 0.784091i \(-0.713129\pi\)
−0.620645 + 0.784091i \(0.713129\pi\)
\(812\) −16.7316 −0.587165
\(813\) 0 0
\(814\) 1.29277 0.0453115
\(815\) −27.5428 −0.964782
\(816\) 0 0
\(817\) −40.9339 −1.43210
\(818\) −7.80186 −0.272786
\(819\) 0 0
\(820\) 30.2533 1.05649
\(821\) −1.64779 −0.0575084 −0.0287542 0.999587i \(-0.509154\pi\)
−0.0287542 + 0.999587i \(0.509154\pi\)
\(822\) 0 0
\(823\) −4.92753 −0.171763 −0.0858815 0.996305i \(-0.527371\pi\)
−0.0858815 + 0.996305i \(0.527371\pi\)
\(824\) −16.8287 −0.586255
\(825\) 0 0
\(826\) 8.68991 0.302361
\(827\) −8.78117 −0.305351 −0.152676 0.988276i \(-0.548789\pi\)
−0.152676 + 0.988276i \(0.548789\pi\)
\(828\) 0 0
\(829\) −8.12166 −0.282077 −0.141039 0.990004i \(-0.545044\pi\)
−0.141039 + 0.990004i \(0.545044\pi\)
\(830\) 15.9677 0.554248
\(831\) 0 0
\(832\) −8.23798 −0.285601
\(833\) 3.03093 0.105016
\(834\) 0 0
\(835\) −25.7833 −0.892267
\(836\) 37.6316 1.30152
\(837\) 0 0
\(838\) 2.70868 0.0935697
\(839\) −17.5061 −0.604378 −0.302189 0.953248i \(-0.597717\pi\)
−0.302189 + 0.953248i \(0.597717\pi\)
\(840\) 0 0
\(841\) −19.5822 −0.675249
\(842\) 6.41954 0.221232
\(843\) 0 0
\(844\) 4.20853 0.144864
\(845\) 38.5540 1.32630
\(846\) 0 0
\(847\) −1.46619 −0.0503790
\(848\) −21.7328 −0.746306
\(849\) 0 0
\(850\) 6.61130 0.226766
\(851\) 5.41868 0.185750
\(852\) 0 0
\(853\) 18.8387 0.645024 0.322512 0.946565i \(-0.395473\pi\)
0.322512 + 0.946565i \(0.395473\pi\)
\(854\) −11.5501 −0.395238
\(855\) 0 0
\(856\) 0.910831 0.0311315
\(857\) −9.49372 −0.324299 −0.162150 0.986766i \(-0.551843\pi\)
−0.162150 + 0.986766i \(0.551843\pi\)
\(858\) 0 0
\(859\) 43.7825 1.49384 0.746919 0.664915i \(-0.231532\pi\)
0.746919 + 0.664915i \(0.231532\pi\)
\(860\) 48.3369 1.64827
\(861\) 0 0
\(862\) −10.9619 −0.373364
\(863\) −9.65541 −0.328674 −0.164337 0.986404i \(-0.552548\pi\)
−0.164337 + 0.986404i \(0.552548\pi\)
\(864\) 0 0
\(865\) −27.6871 −0.941388
\(866\) −2.42911 −0.0825444
\(867\) 0 0
\(868\) 37.0164 1.25642
\(869\) 12.4082 0.420920
\(870\) 0 0
\(871\) −10.3988 −0.352349
\(872\) −1.72044 −0.0582615
\(873\) 0 0
\(874\) −11.3084 −0.382511
\(875\) 45.6618 1.54365
\(876\) 0 0
\(877\) 0.00570362 0.000192598 0 9.62988e−5 1.00000i \(-0.499969\pi\)
9.62988e−5 1.00000i \(0.499969\pi\)
\(878\) 10.2731 0.346701
\(879\) 0 0
\(880\) −41.0232 −1.38289
\(881\) −20.2925 −0.683671 −0.341835 0.939760i \(-0.611049\pi\)
−0.341835 + 0.939760i \(0.611049\pi\)
\(882\) 0 0
\(883\) −42.3237 −1.42431 −0.712153 0.702024i \(-0.752280\pi\)
−0.712153 + 0.702024i \(0.752280\pi\)
\(884\) −6.11298 −0.205602
\(885\) 0 0
\(886\) 4.20136 0.141148
\(887\) −23.4444 −0.787187 −0.393594 0.919285i \(-0.628768\pi\)
−0.393594 + 0.919285i \(0.628768\pi\)
\(888\) 0 0
\(889\) 37.3447 1.25250
\(890\) 16.4356 0.550922
\(891\) 0 0
\(892\) −10.7777 −0.360865
\(893\) 42.7148 1.42940
\(894\) 0 0
\(895\) 59.8720 2.00130
\(896\) 28.7037 0.958925
\(897\) 0 0
\(898\) −4.28341 −0.142939
\(899\) −20.8356 −0.694905
\(900\) 0 0
\(901\) −13.3462 −0.444625
\(902\) 5.34527 0.177978
\(903\) 0 0
\(904\) −6.75228 −0.224578
\(905\) −86.8835 −2.88811
\(906\) 0 0
\(907\) 33.3453 1.10721 0.553607 0.832778i \(-0.313251\pi\)
0.553607 + 0.832778i \(0.313251\pi\)
\(908\) −49.0120 −1.62652
\(909\) 0 0
\(910\) 6.66989 0.221105
\(911\) −6.41847 −0.212653 −0.106327 0.994331i \(-0.533909\pi\)
−0.106327 + 0.994331i \(0.533909\pi\)
\(912\) 0 0
\(913\) −39.3519 −1.30236
\(914\) −8.65133 −0.286161
\(915\) 0 0
\(916\) −54.0004 −1.78422
\(917\) −21.8093 −0.720206
\(918\) 0 0
\(919\) −25.9016 −0.854416 −0.427208 0.904153i \(-0.640503\pi\)
−0.427208 + 0.904153i \(0.640503\pi\)
\(920\) 27.6644 0.912068
\(921\) 0 0
\(922\) −11.2686 −0.371113
\(923\) 6.50206 0.214018
\(924\) 0 0
\(925\) 9.53998 0.313673
\(926\) 4.62141 0.151869
\(927\) 0 0
\(928\) −12.2888 −0.403399
\(929\) 37.1415 1.21857 0.609287 0.792950i \(-0.291456\pi\)
0.609287 + 0.792950i \(0.291456\pi\)
\(930\) 0 0
\(931\) −9.12733 −0.299136
\(932\) 23.1772 0.759193
\(933\) 0 0
\(934\) 6.39039 0.209100
\(935\) −25.1925 −0.823882
\(936\) 0 0
\(937\) −31.5304 −1.03005 −0.515026 0.857175i \(-0.672218\pi\)
−0.515026 + 0.857175i \(0.672218\pi\)
\(938\) 6.69811 0.218701
\(939\) 0 0
\(940\) −50.4399 −1.64517
\(941\) −49.7790 −1.62275 −0.811375 0.584526i \(-0.801281\pi\)
−0.811375 + 0.584526i \(0.801281\pi\)
\(942\) 0 0
\(943\) 22.4048 0.729602
\(944\) −26.1453 −0.850957
\(945\) 0 0
\(946\) 8.54035 0.277671
\(947\) −37.6660 −1.22398 −0.611990 0.790865i \(-0.709631\pi\)
−0.611990 + 0.790865i \(0.709631\pi\)
\(948\) 0 0
\(949\) −15.4659 −0.502044
\(950\) −19.9092 −0.645940
\(951\) 0 0
\(952\) 8.15733 0.264381
\(953\) 15.5730 0.504461 0.252230 0.967667i \(-0.418836\pi\)
0.252230 + 0.967667i \(0.418836\pi\)
\(954\) 0 0
\(955\) 76.4394 2.47352
\(956\) 2.24749 0.0726891
\(957\) 0 0
\(958\) −5.26873 −0.170225
\(959\) 47.8192 1.54416
\(960\) 0 0
\(961\) 15.0958 0.486962
\(962\) 0.632396 0.0203893
\(963\) 0 0
\(964\) 24.8605 0.800703
\(965\) 41.1722 1.32538
\(966\) 0 0
\(967\) 1.33607 0.0429651 0.0214825 0.999769i \(-0.493161\pi\)
0.0214825 + 0.999769i \(0.493161\pi\)
\(968\) −0.709724 −0.0228114
\(969\) 0 0
\(970\) −13.9561 −0.448103
\(971\) 34.7058 1.11376 0.556881 0.830592i \(-0.311998\pi\)
0.556881 + 0.830592i \(0.311998\pi\)
\(972\) 0 0
\(973\) −56.9390 −1.82538
\(974\) −11.8397 −0.379370
\(975\) 0 0
\(976\) 34.7509 1.11235
\(977\) 47.6195 1.52348 0.761742 0.647881i \(-0.224344\pi\)
0.761742 + 0.647881i \(0.224344\pi\)
\(978\) 0 0
\(979\) −40.5049 −1.29454
\(980\) 10.7780 0.344292
\(981\) 0 0
\(982\) −7.96317 −0.254115
\(983\) −39.0836 −1.24657 −0.623286 0.781994i \(-0.714203\pi\)
−0.623286 + 0.781994i \(0.714203\pi\)
\(984\) 0 0
\(985\) 95.0618 3.02892
\(986\) −2.21632 −0.0705821
\(987\) 0 0
\(988\) 18.4086 0.585656
\(989\) 35.7971 1.13828
\(990\) 0 0
\(991\) −41.5424 −1.31964 −0.659819 0.751424i \(-0.729367\pi\)
−0.659819 + 0.751424i \(0.729367\pi\)
\(992\) 27.1872 0.863196
\(993\) 0 0
\(994\) −4.18813 −0.132840
\(995\) 42.8955 1.35988
\(996\) 0 0
\(997\) 19.0081 0.601993 0.300996 0.953625i \(-0.402681\pi\)
0.300996 + 0.953625i \(0.402681\pi\)
\(998\) 0.746952 0.0236444
\(999\) 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 4023.2.a.e.1.15 25
3.2 odd 2 4023.2.a.f.1.11 yes 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4023.2.a.e.1.15 25 1.1 even 1 trivial
4023.2.a.f.1.11 yes 25 3.2 odd 2