Properties

Label 6002.2.a.b.1.10
Level $6002$
Weight $2$
Character 6002.1
Self dual yes
Analytic conductor $47.926$
Analytic rank $1$
Dimension $56$
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] = [6002,2,Mod(1,6002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6002 = 2 \cdot 3001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6002.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: \(47.9262112932\)
Analytic rank: \(1\)
Dimension: \(56\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 6002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.29725 q^{3} +1.00000 q^{4} -0.0563132 q^{5} +2.29725 q^{6} +2.55969 q^{7} -1.00000 q^{8} +2.27735 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.29725 q^{3} +1.00000 q^{4} -0.0563132 q^{5} +2.29725 q^{6} +2.55969 q^{7} -1.00000 q^{8} +2.27735 q^{9} +0.0563132 q^{10} -4.38490 q^{11} -2.29725 q^{12} +2.08970 q^{13} -2.55969 q^{14} +0.129365 q^{15} +1.00000 q^{16} +4.33616 q^{17} -2.27735 q^{18} +0.321301 q^{19} -0.0563132 q^{20} -5.88024 q^{21} +4.38490 q^{22} +0.741876 q^{23} +2.29725 q^{24} -4.99683 q^{25} -2.08970 q^{26} +1.66010 q^{27} +2.55969 q^{28} -2.18305 q^{29} -0.129365 q^{30} -7.46270 q^{31} -1.00000 q^{32} +10.0732 q^{33} -4.33616 q^{34} -0.144144 q^{35} +2.27735 q^{36} +5.68263 q^{37} -0.321301 q^{38} -4.80056 q^{39} +0.0563132 q^{40} +1.74954 q^{41} +5.88024 q^{42} -0.974593 q^{43} -4.38490 q^{44} -0.128245 q^{45} -0.741876 q^{46} +0.00455965 q^{47} -2.29725 q^{48} -0.447997 q^{49} +4.99683 q^{50} -9.96124 q^{51} +2.08970 q^{52} +11.8355 q^{53} -1.66010 q^{54} +0.246928 q^{55} -2.55969 q^{56} -0.738109 q^{57} +2.18305 q^{58} +2.42707 q^{59} +0.129365 q^{60} -4.05871 q^{61} +7.46270 q^{62} +5.82931 q^{63} +1.00000 q^{64} -0.117678 q^{65} -10.0732 q^{66} -9.74948 q^{67} +4.33616 q^{68} -1.70427 q^{69} +0.144144 q^{70} +0.760311 q^{71} -2.27735 q^{72} -16.3548 q^{73} -5.68263 q^{74} +11.4790 q^{75} +0.321301 q^{76} -11.2240 q^{77} +4.80056 q^{78} -3.45286 q^{79} -0.0563132 q^{80} -10.6457 q^{81} -1.74954 q^{82} +0.0374441 q^{83} -5.88024 q^{84} -0.244183 q^{85} +0.974593 q^{86} +5.01501 q^{87} +4.38490 q^{88} -11.5471 q^{89} +0.128245 q^{90} +5.34898 q^{91} +0.741876 q^{92} +17.1437 q^{93} -0.00455965 q^{94} -0.0180935 q^{95} +2.29725 q^{96} +8.86420 q^{97} +0.447997 q^{98} -9.98596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q - 56 q^{2} - 11 q^{3} + 56 q^{4} + 11 q^{6} - 21 q^{7} - 56 q^{8} + 53 q^{9} + 12 q^{11} - 11 q^{12} - 31 q^{13} + 21 q^{14} - 22 q^{15} + 56 q^{16} - 4 q^{17} - 53 q^{18} - 9 q^{19} + 13 q^{21} - 12 q^{22} - 39 q^{23} + 11 q^{24} + 8 q^{25} + 31 q^{26} - 44 q^{27} - 21 q^{28} + 13 q^{29} + 22 q^{30} - 35 q^{31} - 56 q^{32} - 26 q^{33} + 4 q^{34} - 7 q^{35} + 53 q^{36} - 65 q^{37} + 9 q^{38} - 27 q^{39} + 38 q^{41} - 13 q^{42} - 76 q^{43} + 12 q^{44} - 21 q^{45} + 39 q^{46} - 43 q^{47} - 11 q^{48} + 9 q^{49} - 8 q^{50} - 19 q^{51} - 31 q^{52} - 26 q^{53} + 44 q^{54} - 67 q^{55} + 21 q^{56} - 26 q^{57} - 13 q^{58} + 11 q^{59} - 22 q^{60} - 17 q^{61} + 35 q^{62} - 67 q^{63} + 56 q^{64} + 31 q^{65} + 26 q^{66} - 93 q^{67} - 4 q^{68} - 13 q^{69} + 7 q^{70} - 33 q^{71} - 53 q^{72} - 41 q^{73} + 65 q^{74} - 21 q^{75} - 9 q^{76} + 5 q^{77} + 27 q^{78} - 69 q^{79} + 36 q^{81} - 38 q^{82} + 4 q^{83} + 13 q^{84} - 40 q^{85} + 76 q^{86} - 69 q^{87} - 12 q^{88} + 40 q^{89} + 21 q^{90} - 64 q^{91} - 39 q^{92} - 57 q^{93} + 43 q^{94} - 22 q^{95} + 11 q^{96} - 71 q^{97} - 9 q^{98} + 17 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.29725 −1.32632 −0.663159 0.748479i \(-0.730785\pi\)
−0.663159 + 0.748479i \(0.730785\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.0563132 −0.0251840 −0.0125920 0.999921i \(-0.504008\pi\)
−0.0125920 + 0.999921i \(0.504008\pi\)
\(6\) 2.29725 0.937848
\(7\) 2.55969 0.967471 0.483736 0.875214i \(-0.339280\pi\)
0.483736 + 0.875214i \(0.339280\pi\)
\(8\) −1.00000 −0.353553
\(9\) 2.27735 0.759117
\(10\) 0.0563132 0.0178078
\(11\) −4.38490 −1.32210 −0.661048 0.750343i \(-0.729888\pi\)
−0.661048 + 0.750343i \(0.729888\pi\)
\(12\) −2.29725 −0.663159
\(13\) 2.08970 0.579579 0.289789 0.957090i \(-0.406415\pi\)
0.289789 + 0.957090i \(0.406415\pi\)
\(14\) −2.55969 −0.684105
\(15\) 0.129365 0.0334020
\(16\) 1.00000 0.250000
\(17\) 4.33616 1.05167 0.525837 0.850586i \(-0.323752\pi\)
0.525837 + 0.850586i \(0.323752\pi\)
\(18\) −2.27735 −0.536777
\(19\) 0.321301 0.0737115 0.0368558 0.999321i \(-0.488266\pi\)
0.0368558 + 0.999321i \(0.488266\pi\)
\(20\) −0.0563132 −0.0125920
\(21\) −5.88024 −1.28317
\(22\) 4.38490 0.934864
\(23\) 0.741876 0.154692 0.0773459 0.997004i \(-0.475355\pi\)
0.0773459 + 0.997004i \(0.475355\pi\)
\(24\) 2.29725 0.468924
\(25\) −4.99683 −0.999366
\(26\) −2.08970 −0.409824
\(27\) 1.66010 0.319487
\(28\) 2.55969 0.483736
\(29\) −2.18305 −0.405383 −0.202691 0.979243i \(-0.564969\pi\)
−0.202691 + 0.979243i \(0.564969\pi\)
\(30\) −0.129365 −0.0236188
\(31\) −7.46270 −1.34034 −0.670170 0.742208i \(-0.733779\pi\)
−0.670170 + 0.742208i \(0.733779\pi\)
\(32\) −1.00000 −0.176777
\(33\) 10.0732 1.75352
\(34\) −4.33616 −0.743645
\(35\) −0.144144 −0.0243648
\(36\) 2.27735 0.379559
\(37\) 5.68263 0.934219 0.467109 0.884200i \(-0.345295\pi\)
0.467109 + 0.884200i \(0.345295\pi\)
\(38\) −0.321301 −0.0521219
\(39\) −4.80056 −0.768705
\(40\) 0.0563132 0.00890390
\(41\) 1.74954 0.273232 0.136616 0.990624i \(-0.456377\pi\)
0.136616 + 0.990624i \(0.456377\pi\)
\(42\) 5.88024 0.907341
\(43\) −0.974593 −0.148624 −0.0743120 0.997235i \(-0.523676\pi\)
−0.0743120 + 0.997235i \(0.523676\pi\)
\(44\) −4.38490 −0.661048
\(45\) −0.128245 −0.0191176
\(46\) −0.741876 −0.109384
\(47\) 0.00455965 0.000665094 0 0.000332547 1.00000i \(-0.499894\pi\)
0.000332547 1.00000i \(0.499894\pi\)
\(48\) −2.29725 −0.331579
\(49\) −0.447997 −0.0639995
\(50\) 4.99683 0.706658
\(51\) −9.96124 −1.39485
\(52\) 2.08970 0.289789
\(53\) 11.8355 1.62573 0.812865 0.582452i \(-0.197907\pi\)
0.812865 + 0.582452i \(0.197907\pi\)
\(54\) −1.66010 −0.225911
\(55\) 0.246928 0.0332957
\(56\) −2.55969 −0.342053
\(57\) −0.738109 −0.0977649
\(58\) 2.18305 0.286649
\(59\) 2.42707 0.315977 0.157989 0.987441i \(-0.449499\pi\)
0.157989 + 0.987441i \(0.449499\pi\)
\(60\) 0.129365 0.0167010
\(61\) −4.05871 −0.519665 −0.259833 0.965654i \(-0.583667\pi\)
−0.259833 + 0.965654i \(0.583667\pi\)
\(62\) 7.46270 0.947763
\(63\) 5.82931 0.734424
\(64\) 1.00000 0.125000
\(65\) −0.117678 −0.0145961
\(66\) −10.0732 −1.23993
\(67\) −9.74948 −1.19109 −0.595544 0.803323i \(-0.703063\pi\)
−0.595544 + 0.803323i \(0.703063\pi\)
\(68\) 4.33616 0.525837
\(69\) −1.70427 −0.205170
\(70\) 0.144144 0.0172285
\(71\) 0.760311 0.0902324 0.0451162 0.998982i \(-0.485634\pi\)
0.0451162 + 0.998982i \(0.485634\pi\)
\(72\) −2.27735 −0.268389
\(73\) −16.3548 −1.91418 −0.957089 0.289793i \(-0.906414\pi\)
−0.957089 + 0.289793i \(0.906414\pi\)
\(74\) −5.68263 −0.660592
\(75\) 11.4790 1.32548
\(76\) 0.321301 0.0368558
\(77\) −11.2240 −1.27909
\(78\) 4.80056 0.543557
\(79\) −3.45286 −0.388477 −0.194239 0.980954i \(-0.562224\pi\)
−0.194239 + 0.980954i \(0.562224\pi\)
\(80\) −0.0563132 −0.00629601
\(81\) −10.6457 −1.18286
\(82\) −1.74954 −0.193204
\(83\) 0.0374441 0.00411003 0.00205501 0.999998i \(-0.499346\pi\)
0.00205501 + 0.999998i \(0.499346\pi\)
\(84\) −5.88024 −0.641587
\(85\) −0.244183 −0.0264854
\(86\) 0.974593 0.105093
\(87\) 5.01501 0.537666
\(88\) 4.38490 0.467432
\(89\) −11.5471 −1.22399 −0.611997 0.790860i \(-0.709634\pi\)
−0.611997 + 0.790860i \(0.709634\pi\)
\(90\) 0.128245 0.0135182
\(91\) 5.34898 0.560726
\(92\) 0.741876 0.0773459
\(93\) 17.1437 1.77772
\(94\) −0.00455965 −0.000470292 0
\(95\) −0.0180935 −0.00185635
\(96\) 2.29725 0.234462
\(97\) 8.86420 0.900023 0.450011 0.893023i \(-0.351420\pi\)
0.450011 + 0.893023i \(0.351420\pi\)
\(98\) 0.447997 0.0452545
\(99\) −9.98596 −1.00363
\(100\) −4.99683 −0.499683
\(101\) 16.5704 1.64882 0.824409 0.565994i \(-0.191507\pi\)
0.824409 + 0.565994i \(0.191507\pi\)
\(102\) 9.96124 0.986310
\(103\) −0.542195 −0.0534241 −0.0267120 0.999643i \(-0.508504\pi\)
−0.0267120 + 0.999643i \(0.508504\pi\)
\(104\) −2.08970 −0.204912
\(105\) 0.331135 0.0323155
\(106\) −11.8355 −1.14957
\(107\) 11.4189 1.10391 0.551955 0.833874i \(-0.313882\pi\)
0.551955 + 0.833874i \(0.313882\pi\)
\(108\) 1.66010 0.159743
\(109\) −1.21753 −0.116618 −0.0583090 0.998299i \(-0.518571\pi\)
−0.0583090 + 0.998299i \(0.518571\pi\)
\(110\) −0.246928 −0.0235436
\(111\) −13.0544 −1.23907
\(112\) 2.55969 0.241868
\(113\) 7.46015 0.701792 0.350896 0.936414i \(-0.385877\pi\)
0.350896 + 0.936414i \(0.385877\pi\)
\(114\) 0.738109 0.0691302
\(115\) −0.0417774 −0.00389576
\(116\) −2.18305 −0.202691
\(117\) 4.75899 0.439968
\(118\) −2.42707 −0.223430
\(119\) 11.0992 1.01746
\(120\) −0.129365 −0.0118094
\(121\) 8.22734 0.747940
\(122\) 4.05871 0.367459
\(123\) −4.01913 −0.362393
\(124\) −7.46270 −0.670170
\(125\) 0.562953 0.0503521
\(126\) −5.82931 −0.519316
\(127\) −14.3321 −1.27177 −0.635885 0.771784i \(-0.719365\pi\)
−0.635885 + 0.771784i \(0.719365\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.23888 0.197123
\(130\) 0.117678 0.0103210
\(131\) 2.99206 0.261418 0.130709 0.991421i \(-0.458275\pi\)
0.130709 + 0.991421i \(0.458275\pi\)
\(132\) 10.0732 0.876760
\(133\) 0.822431 0.0713138
\(134\) 9.74948 0.842226
\(135\) −0.0934857 −0.00804597
\(136\) −4.33616 −0.371823
\(137\) 16.8733 1.44158 0.720791 0.693152i \(-0.243779\pi\)
0.720791 + 0.693152i \(0.243779\pi\)
\(138\) 1.70427 0.145077
\(139\) −15.9636 −1.35401 −0.677006 0.735978i \(-0.736723\pi\)
−0.677006 + 0.735978i \(0.736723\pi\)
\(140\) −0.144144 −0.0121824
\(141\) −0.0104747 −0.000882125 0
\(142\) −0.760311 −0.0638039
\(143\) −9.16313 −0.766259
\(144\) 2.27735 0.189779
\(145\) 0.122935 0.0102092
\(146\) 16.3548 1.35353
\(147\) 1.02916 0.0848837
\(148\) 5.68263 0.467109
\(149\) −6.61271 −0.541734 −0.270867 0.962617i \(-0.587310\pi\)
−0.270867 + 0.962617i \(0.587310\pi\)
\(150\) −11.4790 −0.937253
\(151\) 10.0986 0.821811 0.410905 0.911678i \(-0.365213\pi\)
0.410905 + 0.911678i \(0.365213\pi\)
\(152\) −0.321301 −0.0260610
\(153\) 9.87496 0.798344
\(154\) 11.2240 0.904454
\(155\) 0.420248 0.0337552
\(156\) −4.80056 −0.384353
\(157\) 7.00844 0.559334 0.279667 0.960097i \(-0.409776\pi\)
0.279667 + 0.960097i \(0.409776\pi\)
\(158\) 3.45286 0.274695
\(159\) −27.1891 −2.15623
\(160\) 0.0563132 0.00445195
\(161\) 1.89897 0.149660
\(162\) 10.6457 0.836407
\(163\) −15.7221 −1.23145 −0.615726 0.787960i \(-0.711137\pi\)
−0.615726 + 0.787960i \(0.711137\pi\)
\(164\) 1.74954 0.136616
\(165\) −0.567254 −0.0441607
\(166\) −0.0374441 −0.00290623
\(167\) 18.7636 1.45197 0.725986 0.687710i \(-0.241384\pi\)
0.725986 + 0.687710i \(0.241384\pi\)
\(168\) 5.88024 0.453670
\(169\) −8.63315 −0.664088
\(170\) 0.244183 0.0187280
\(171\) 0.731716 0.0559557
\(172\) −0.974593 −0.0743120
\(173\) 5.60033 0.425785 0.212893 0.977076i \(-0.431712\pi\)
0.212893 + 0.977076i \(0.431712\pi\)
\(174\) −5.01501 −0.380187
\(175\) −12.7903 −0.966858
\(176\) −4.38490 −0.330524
\(177\) −5.57558 −0.419086
\(178\) 11.5471 0.865494
\(179\) −10.3380 −0.772702 −0.386351 0.922352i \(-0.626265\pi\)
−0.386351 + 0.922352i \(0.626265\pi\)
\(180\) −0.128245 −0.00955882
\(181\) −0.423949 −0.0315119 −0.0157560 0.999876i \(-0.505015\pi\)
−0.0157560 + 0.999876i \(0.505015\pi\)
\(182\) −5.34898 −0.396493
\(183\) 9.32388 0.689241
\(184\) −0.741876 −0.0546918
\(185\) −0.320007 −0.0235274
\(186\) −17.1437 −1.25703
\(187\) −19.0136 −1.39041
\(188\) 0.00455965 0.000332547 0
\(189\) 4.24934 0.309094
\(190\) 0.0180935 0.00131264
\(191\) −9.33430 −0.675407 −0.337703 0.941253i \(-0.609650\pi\)
−0.337703 + 0.941253i \(0.609650\pi\)
\(192\) −2.29725 −0.165790
\(193\) 8.70456 0.626568 0.313284 0.949659i \(-0.398571\pi\)
0.313284 + 0.949659i \(0.398571\pi\)
\(194\) −8.86420 −0.636412
\(195\) 0.270335 0.0193591
\(196\) −0.447997 −0.0319998
\(197\) 8.93519 0.636606 0.318303 0.947989i \(-0.396887\pi\)
0.318303 + 0.947989i \(0.396887\pi\)
\(198\) 9.98596 0.709671
\(199\) 11.6082 0.822884 0.411442 0.911436i \(-0.365025\pi\)
0.411442 + 0.911436i \(0.365025\pi\)
\(200\) 4.99683 0.353329
\(201\) 22.3970 1.57976
\(202\) −16.5704 −1.16589
\(203\) −5.58793 −0.392196
\(204\) −9.96124 −0.697426
\(205\) −0.0985222 −0.00688109
\(206\) 0.542195 0.0377765
\(207\) 1.68951 0.117429
\(208\) 2.08970 0.144895
\(209\) −1.40887 −0.0974538
\(210\) −0.331135 −0.0228505
\(211\) 17.8257 1.22717 0.613585 0.789628i \(-0.289727\pi\)
0.613585 + 0.789628i \(0.289727\pi\)
\(212\) 11.8355 0.812865
\(213\) −1.74662 −0.119677
\(214\) −11.4189 −0.780583
\(215\) 0.0548824 0.00374295
\(216\) −1.66010 −0.112956
\(217\) −19.1022 −1.29674
\(218\) 1.21753 0.0824614
\(219\) 37.5709 2.53881
\(220\) 0.246928 0.0166479
\(221\) 9.06128 0.609528
\(222\) 13.0544 0.876155
\(223\) −16.3876 −1.09740 −0.548698 0.836021i \(-0.684876\pi\)
−0.548698 + 0.836021i \(0.684876\pi\)
\(224\) −2.55969 −0.171026
\(225\) −11.3795 −0.758636
\(226\) −7.46015 −0.496242
\(227\) −23.0524 −1.53004 −0.765019 0.644007i \(-0.777271\pi\)
−0.765019 + 0.644007i \(0.777271\pi\)
\(228\) −0.738109 −0.0488824
\(229\) 9.30192 0.614688 0.307344 0.951599i \(-0.400560\pi\)
0.307344 + 0.951599i \(0.400560\pi\)
\(230\) 0.0417774 0.00275472
\(231\) 25.7843 1.69648
\(232\) 2.18305 0.143324
\(233\) 28.9954 1.89955 0.949776 0.312930i \(-0.101310\pi\)
0.949776 + 0.312930i \(0.101310\pi\)
\(234\) −4.75899 −0.311105
\(235\) −0.000256769 0 −1.67497e−5 0
\(236\) 2.42707 0.157989
\(237\) 7.93208 0.515244
\(238\) −11.0992 −0.719455
\(239\) −8.61374 −0.557177 −0.278588 0.960411i \(-0.589866\pi\)
−0.278588 + 0.960411i \(0.589866\pi\)
\(240\) 0.129365 0.00835050
\(241\) −6.65186 −0.428484 −0.214242 0.976781i \(-0.568728\pi\)
−0.214242 + 0.976781i \(0.568728\pi\)
\(242\) −8.22734 −0.528873
\(243\) 19.4756 1.24936
\(244\) −4.05871 −0.259833
\(245\) 0.0252281 0.00161177
\(246\) 4.01913 0.256250
\(247\) 0.671424 0.0427217
\(248\) 7.46270 0.473882
\(249\) −0.0860185 −0.00545120
\(250\) −0.562953 −0.0356043
\(251\) 7.33623 0.463059 0.231529 0.972828i \(-0.425627\pi\)
0.231529 + 0.972828i \(0.425627\pi\)
\(252\) 5.82931 0.367212
\(253\) −3.25305 −0.204518
\(254\) 14.3321 0.899277
\(255\) 0.560949 0.0351280
\(256\) 1.00000 0.0625000
\(257\) −6.13605 −0.382756 −0.191378 0.981516i \(-0.561296\pi\)
−0.191378 + 0.981516i \(0.561296\pi\)
\(258\) −2.23888 −0.139387
\(259\) 14.5458 0.903829
\(260\) −0.117678 −0.00729807
\(261\) −4.97158 −0.307733
\(262\) −2.99206 −0.184850
\(263\) −8.49180 −0.523627 −0.261813 0.965119i \(-0.584320\pi\)
−0.261813 + 0.965119i \(0.584320\pi\)
\(264\) −10.0732 −0.619963
\(265\) −0.666495 −0.0409424
\(266\) −0.822431 −0.0504265
\(267\) 26.5266 1.62340
\(268\) −9.74948 −0.595544
\(269\) −13.2993 −0.810875 −0.405438 0.914123i \(-0.632881\pi\)
−0.405438 + 0.914123i \(0.632881\pi\)
\(270\) 0.0934857 0.00568936
\(271\) 20.0373 1.21718 0.608589 0.793485i \(-0.291736\pi\)
0.608589 + 0.793485i \(0.291736\pi\)
\(272\) 4.33616 0.262918
\(273\) −12.2879 −0.743700
\(274\) −16.8733 −1.01935
\(275\) 21.9106 1.32126
\(276\) −1.70427 −0.102585
\(277\) −14.1526 −0.850346 −0.425173 0.905112i \(-0.639787\pi\)
−0.425173 + 0.905112i \(0.639787\pi\)
\(278\) 15.9636 0.957431
\(279\) −16.9952 −1.01748
\(280\) 0.144144 0.00861427
\(281\) 2.57606 0.153675 0.0768374 0.997044i \(-0.475518\pi\)
0.0768374 + 0.997044i \(0.475518\pi\)
\(282\) 0.0104747 0.000623757 0
\(283\) −23.8035 −1.41497 −0.707486 0.706727i \(-0.750171\pi\)
−0.707486 + 0.706727i \(0.750171\pi\)
\(284\) 0.760311 0.0451162
\(285\) 0.0415653 0.00246211
\(286\) 9.16313 0.541827
\(287\) 4.47828 0.264344
\(288\) −2.27735 −0.134194
\(289\) 1.80229 0.106017
\(290\) −0.122935 −0.00721897
\(291\) −20.3633 −1.19372
\(292\) −16.3548 −0.957089
\(293\) −7.61728 −0.445006 −0.222503 0.974932i \(-0.571423\pi\)
−0.222503 + 0.974932i \(0.571423\pi\)
\(294\) −1.02916 −0.0600218
\(295\) −0.136676 −0.00795758
\(296\) −5.68263 −0.330296
\(297\) −7.27938 −0.422392
\(298\) 6.61271 0.383064
\(299\) 1.55030 0.0896561
\(300\) 11.4790 0.662738
\(301\) −2.49465 −0.143789
\(302\) −10.0986 −0.581108
\(303\) −38.0664 −2.18686
\(304\) 0.321301 0.0184279
\(305\) 0.228559 0.0130873
\(306\) −9.87496 −0.564514
\(307\) 12.2067 0.696675 0.348338 0.937369i \(-0.386746\pi\)
0.348338 + 0.937369i \(0.386746\pi\)
\(308\) −11.2240 −0.639545
\(309\) 1.24556 0.0708573
\(310\) −0.420248 −0.0238685
\(311\) 1.91464 0.108569 0.0542846 0.998526i \(-0.482712\pi\)
0.0542846 + 0.998526i \(0.482712\pi\)
\(312\) 4.80056 0.271778
\(313\) −5.44385 −0.307705 −0.153852 0.988094i \(-0.549168\pi\)
−0.153852 + 0.988094i \(0.549168\pi\)
\(314\) −7.00844 −0.395509
\(315\) −0.328267 −0.0184958
\(316\) −3.45286 −0.194239
\(317\) 7.14525 0.401317 0.200659 0.979661i \(-0.435692\pi\)
0.200659 + 0.979661i \(0.435692\pi\)
\(318\) 27.1891 1.52469
\(319\) 9.57246 0.535955
\(320\) −0.0563132 −0.00314800
\(321\) −26.2322 −1.46414
\(322\) −1.89897 −0.105826
\(323\) 1.39321 0.0775205
\(324\) −10.6457 −0.591429
\(325\) −10.4419 −0.579211
\(326\) 15.7221 0.870768
\(327\) 2.79696 0.154672
\(328\) −1.74954 −0.0966022
\(329\) 0.0116713 0.000643459 0
\(330\) 0.567254 0.0312263
\(331\) 0.951726 0.0523116 0.0261558 0.999658i \(-0.491673\pi\)
0.0261558 + 0.999658i \(0.491673\pi\)
\(332\) 0.0374441 0.00205501
\(333\) 12.9413 0.709182
\(334\) −18.7636 −1.02670
\(335\) 0.549024 0.0299964
\(336\) −5.88024 −0.320793
\(337\) −1.77919 −0.0969185 −0.0484592 0.998825i \(-0.515431\pi\)
−0.0484592 + 0.998825i \(0.515431\pi\)
\(338\) 8.63315 0.469581
\(339\) −17.1378 −0.930799
\(340\) −0.244183 −0.0132427
\(341\) 32.7232 1.77206
\(342\) −0.731716 −0.0395667
\(343\) −19.0645 −1.02939
\(344\) 0.974593 0.0525465
\(345\) 0.0959731 0.00516702
\(346\) −5.60033 −0.301076
\(347\) −20.4216 −1.09629 −0.548144 0.836384i \(-0.684665\pi\)
−0.548144 + 0.836384i \(0.684665\pi\)
\(348\) 5.01501 0.268833
\(349\) 3.75534 0.201018 0.100509 0.994936i \(-0.467953\pi\)
0.100509 + 0.994936i \(0.467953\pi\)
\(350\) 12.7903 0.683672
\(351\) 3.46912 0.185168
\(352\) 4.38490 0.233716
\(353\) −30.1267 −1.60348 −0.801740 0.597673i \(-0.796092\pi\)
−0.801740 + 0.597673i \(0.796092\pi\)
\(354\) 5.57558 0.296339
\(355\) −0.0428156 −0.00227241
\(356\) −11.5471 −0.611997
\(357\) −25.4977 −1.34948
\(358\) 10.3380 0.546383
\(359\) 35.3332 1.86481 0.932407 0.361410i \(-0.117705\pi\)
0.932407 + 0.361410i \(0.117705\pi\)
\(360\) 0.128245 0.00675911
\(361\) −18.8968 −0.994567
\(362\) 0.423949 0.0222823
\(363\) −18.9002 −0.992006
\(364\) 5.34898 0.280363
\(365\) 0.920989 0.0482067
\(366\) −9.32388 −0.487367
\(367\) −14.5194 −0.757906 −0.378953 0.925416i \(-0.623716\pi\)
−0.378953 + 0.925416i \(0.623716\pi\)
\(368\) 0.741876 0.0386730
\(369\) 3.98432 0.207415
\(370\) 0.320007 0.0166364
\(371\) 30.2952 1.57285
\(372\) 17.1437 0.888858
\(373\) −19.6147 −1.01561 −0.507805 0.861472i \(-0.669543\pi\)
−0.507805 + 0.861472i \(0.669543\pi\)
\(374\) 19.0136 0.983171
\(375\) −1.29324 −0.0667828
\(376\) −0.00455965 −0.000235146 0
\(377\) −4.56193 −0.234951
\(378\) −4.24934 −0.218563
\(379\) 23.2379 1.19365 0.596825 0.802371i \(-0.296429\pi\)
0.596825 + 0.802371i \(0.296429\pi\)
\(380\) −0.0180935 −0.000928177 0
\(381\) 32.9244 1.68677
\(382\) 9.33430 0.477585
\(383\) −8.07250 −0.412485 −0.206243 0.978501i \(-0.566124\pi\)
−0.206243 + 0.978501i \(0.566124\pi\)
\(384\) 2.29725 0.117231
\(385\) 0.632058 0.0322127
\(386\) −8.70456 −0.443051
\(387\) −2.21949 −0.112823
\(388\) 8.86420 0.450011
\(389\) −25.1180 −1.27353 −0.636766 0.771057i \(-0.719728\pi\)
−0.636766 + 0.771057i \(0.719728\pi\)
\(390\) −0.270335 −0.0136890
\(391\) 3.21689 0.162685
\(392\) 0.447997 0.0226273
\(393\) −6.87351 −0.346723
\(394\) −8.93519 −0.450148
\(395\) 0.194442 0.00978342
\(396\) −9.98596 −0.501813
\(397\) −1.31879 −0.0661882 −0.0330941 0.999452i \(-0.510536\pi\)
−0.0330941 + 0.999452i \(0.510536\pi\)
\(398\) −11.6082 −0.581867
\(399\) −1.88933 −0.0945847
\(400\) −4.99683 −0.249841
\(401\) 26.0391 1.30033 0.650166 0.759792i \(-0.274699\pi\)
0.650166 + 0.759792i \(0.274699\pi\)
\(402\) −22.3970 −1.11706
\(403\) −15.5948 −0.776833
\(404\) 16.5704 0.824409
\(405\) 0.599495 0.0297891
\(406\) 5.58793 0.277324
\(407\) −24.9178 −1.23513
\(408\) 9.96124 0.493155
\(409\) −27.2325 −1.34656 −0.673281 0.739386i \(-0.735116\pi\)
−0.673281 + 0.739386i \(0.735116\pi\)
\(410\) 0.0985222 0.00486566
\(411\) −38.7621 −1.91200
\(412\) −0.542195 −0.0267120
\(413\) 6.21254 0.305699
\(414\) −1.68951 −0.0830350
\(415\) −0.00210860 −0.000103507 0
\(416\) −2.08970 −0.102456
\(417\) 36.6723 1.79585
\(418\) 1.40887 0.0689102
\(419\) −12.1085 −0.591541 −0.295770 0.955259i \(-0.595576\pi\)
−0.295770 + 0.955259i \(0.595576\pi\)
\(420\) 0.331135 0.0161577
\(421\) −11.8173 −0.575940 −0.287970 0.957639i \(-0.592980\pi\)
−0.287970 + 0.957639i \(0.592980\pi\)
\(422\) −17.8257 −0.867741
\(423\) 0.0103839 0.000504884 0
\(424\) −11.8355 −0.574783
\(425\) −21.6671 −1.05101
\(426\) 1.74662 0.0846242
\(427\) −10.3890 −0.502761
\(428\) 11.4189 0.551955
\(429\) 21.0500 1.01630
\(430\) −0.0548824 −0.00264667
\(431\) −40.6471 −1.95790 −0.978951 0.204095i \(-0.934575\pi\)
−0.978951 + 0.204095i \(0.934575\pi\)
\(432\) 1.66010 0.0798717
\(433\) −3.45931 −0.166244 −0.0831220 0.996539i \(-0.526489\pi\)
−0.0831220 + 0.996539i \(0.526489\pi\)
\(434\) 19.1022 0.916934
\(435\) −0.282411 −0.0135406
\(436\) −1.21753 −0.0583090
\(437\) 0.238366 0.0114026
\(438\) −37.5709 −1.79521
\(439\) −36.1447 −1.72509 −0.862546 0.505978i \(-0.831132\pi\)
−0.862546 + 0.505978i \(0.831132\pi\)
\(440\) −0.246928 −0.0117718
\(441\) −1.02025 −0.0485832
\(442\) −9.06128 −0.431001
\(443\) 19.3661 0.920112 0.460056 0.887890i \(-0.347829\pi\)
0.460056 + 0.887890i \(0.347829\pi\)
\(444\) −13.0544 −0.619535
\(445\) 0.650256 0.0308251
\(446\) 16.3876 0.775976
\(447\) 15.1910 0.718512
\(448\) 2.55969 0.120934
\(449\) −40.9853 −1.93422 −0.967109 0.254363i \(-0.918134\pi\)
−0.967109 + 0.254363i \(0.918134\pi\)
\(450\) 11.3795 0.536437
\(451\) −7.67156 −0.361239
\(452\) 7.46015 0.350896
\(453\) −23.1989 −1.08998
\(454\) 23.0524 1.08190
\(455\) −0.301218 −0.0141213
\(456\) 0.738109 0.0345651
\(457\) −2.27000 −0.106186 −0.0530930 0.998590i \(-0.516908\pi\)
−0.0530930 + 0.998590i \(0.516908\pi\)
\(458\) −9.30192 −0.434650
\(459\) 7.19847 0.335996
\(460\) −0.0417774 −0.00194788
\(461\) 25.9596 1.20906 0.604529 0.796583i \(-0.293361\pi\)
0.604529 + 0.796583i \(0.293361\pi\)
\(462\) −25.7843 −1.19959
\(463\) −20.1425 −0.936103 −0.468052 0.883701i \(-0.655044\pi\)
−0.468052 + 0.883701i \(0.655044\pi\)
\(464\) −2.18305 −0.101346
\(465\) −0.965415 −0.0447701
\(466\) −28.9954 −1.34319
\(467\) 3.35104 0.155067 0.0775337 0.996990i \(-0.475295\pi\)
0.0775337 + 0.996990i \(0.475295\pi\)
\(468\) 4.75899 0.219984
\(469\) −24.9556 −1.15234
\(470\) 0.000256769 0 1.18439e−5 0
\(471\) −16.1001 −0.741855
\(472\) −2.42707 −0.111715
\(473\) 4.27349 0.196495
\(474\) −7.93208 −0.364333
\(475\) −1.60549 −0.0736648
\(476\) 11.0992 0.508732
\(477\) 26.9536 1.23412
\(478\) 8.61374 0.393983
\(479\) −32.2923 −1.47547 −0.737736 0.675089i \(-0.764105\pi\)
−0.737736 + 0.675089i \(0.764105\pi\)
\(480\) −0.129365 −0.00590470
\(481\) 11.8750 0.541453
\(482\) 6.65186 0.302984
\(483\) −4.36241 −0.198496
\(484\) 8.22734 0.373970
\(485\) −0.499171 −0.0226662
\(486\) −19.4756 −0.883430
\(487\) −25.6355 −1.16166 −0.580828 0.814027i \(-0.697271\pi\)
−0.580828 + 0.814027i \(0.697271\pi\)
\(488\) 4.05871 0.183729
\(489\) 36.1176 1.63330
\(490\) −0.0252281 −0.00113969
\(491\) −30.8194 −1.39086 −0.695431 0.718593i \(-0.744787\pi\)
−0.695431 + 0.718593i \(0.744787\pi\)
\(492\) −4.01913 −0.181196
\(493\) −9.46606 −0.426330
\(494\) −0.671424 −0.0302088
\(495\) 0.562341 0.0252754
\(496\) −7.46270 −0.335085
\(497\) 1.94616 0.0872972
\(498\) 0.0860185 0.00385458
\(499\) −25.8260 −1.15613 −0.578065 0.815991i \(-0.696192\pi\)
−0.578065 + 0.815991i \(0.696192\pi\)
\(500\) 0.562953 0.0251760
\(501\) −43.1047 −1.92577
\(502\) −7.33623 −0.327432
\(503\) −21.8953 −0.976265 −0.488133 0.872770i \(-0.662322\pi\)
−0.488133 + 0.872770i \(0.662322\pi\)
\(504\) −5.82931 −0.259658
\(505\) −0.933134 −0.0415239
\(506\) 3.25305 0.144616
\(507\) 19.8325 0.880792
\(508\) −14.3321 −0.635885
\(509\) −29.6657 −1.31491 −0.657454 0.753494i \(-0.728367\pi\)
−0.657454 + 0.753494i \(0.728367\pi\)
\(510\) −0.560949 −0.0248393
\(511\) −41.8631 −1.85191
\(512\) −1.00000 −0.0441942
\(513\) 0.533393 0.0235499
\(514\) 6.13605 0.270650
\(515\) 0.0305327 0.00134543
\(516\) 2.23888 0.0985613
\(517\) −0.0199936 −0.000879318 0
\(518\) −14.5458 −0.639104
\(519\) −12.8653 −0.564726
\(520\) 0.117678 0.00516051
\(521\) −31.0025 −1.35824 −0.679122 0.734025i \(-0.737639\pi\)
−0.679122 + 0.734025i \(0.737639\pi\)
\(522\) 4.97158 0.217600
\(523\) −20.4078 −0.892373 −0.446186 0.894940i \(-0.647218\pi\)
−0.446186 + 0.894940i \(0.647218\pi\)
\(524\) 2.99206 0.130709
\(525\) 29.3826 1.28236
\(526\) 8.49180 0.370260
\(527\) −32.3594 −1.40960
\(528\) 10.0732 0.438380
\(529\) −22.4496 −0.976070
\(530\) 0.666495 0.0289507
\(531\) 5.52729 0.239864
\(532\) 0.822431 0.0356569
\(533\) 3.65602 0.158360
\(534\) −26.5266 −1.14792
\(535\) −0.643037 −0.0278009
\(536\) 9.74948 0.421113
\(537\) 23.7491 1.02485
\(538\) 13.2993 0.573375
\(539\) 1.96442 0.0846136
\(540\) −0.0934857 −0.00402298
\(541\) 22.6070 0.971950 0.485975 0.873973i \(-0.338465\pi\)
0.485975 + 0.873973i \(0.338465\pi\)
\(542\) −20.0373 −0.860675
\(543\) 0.973917 0.0417948
\(544\) −4.33616 −0.185911
\(545\) 0.0685629 0.00293691
\(546\) 12.2879 0.525876
\(547\) 25.5181 1.09107 0.545537 0.838087i \(-0.316326\pi\)
0.545537 + 0.838087i \(0.316326\pi\)
\(548\) 16.8733 0.720791
\(549\) −9.24312 −0.394487
\(550\) −21.9106 −0.934271
\(551\) −0.701417 −0.0298814
\(552\) 1.70427 0.0725387
\(553\) −8.83825 −0.375841
\(554\) 14.1526 0.601285
\(555\) 0.735136 0.0312048
\(556\) −15.9636 −0.677006
\(557\) 6.68743 0.283356 0.141678 0.989913i \(-0.454750\pi\)
0.141678 + 0.989913i \(0.454750\pi\)
\(558\) 16.9952 0.719464
\(559\) −2.03661 −0.0861393
\(560\) −0.144144 −0.00609121
\(561\) 43.6790 1.84413
\(562\) −2.57606 −0.108665
\(563\) −39.1032 −1.64800 −0.824002 0.566587i \(-0.808264\pi\)
−0.824002 + 0.566587i \(0.808264\pi\)
\(564\) −0.0104747 −0.000441063 0
\(565\) −0.420105 −0.0176740
\(566\) 23.8035 1.00054
\(567\) −27.2497 −1.14438
\(568\) −0.760311 −0.0319020
\(569\) −30.8282 −1.29239 −0.646193 0.763174i \(-0.723640\pi\)
−0.646193 + 0.763174i \(0.723640\pi\)
\(570\) −0.0415653 −0.00174098
\(571\) 36.5821 1.53091 0.765456 0.643488i \(-0.222513\pi\)
0.765456 + 0.643488i \(0.222513\pi\)
\(572\) −9.16313 −0.383130
\(573\) 21.4432 0.895803
\(574\) −4.47828 −0.186920
\(575\) −3.70703 −0.154594
\(576\) 2.27735 0.0948897
\(577\) 2.25015 0.0936749 0.0468374 0.998903i \(-0.485086\pi\)
0.0468374 + 0.998903i \(0.485086\pi\)
\(578\) −1.80229 −0.0749653
\(579\) −19.9965 −0.831028
\(580\) 0.122935 0.00510458
\(581\) 0.0958453 0.00397633
\(582\) 20.3633 0.844085
\(583\) −51.8975 −2.14937
\(584\) 16.3548 0.676764
\(585\) −0.267994 −0.0110802
\(586\) 7.61728 0.314667
\(587\) 7.86640 0.324681 0.162341 0.986735i \(-0.448096\pi\)
0.162341 + 0.986735i \(0.448096\pi\)
\(588\) 1.02916 0.0424419
\(589\) −2.39777 −0.0987985
\(590\) 0.136676 0.00562686
\(591\) −20.5264 −0.844341
\(592\) 5.68263 0.233555
\(593\) −29.5875 −1.21501 −0.607506 0.794315i \(-0.707830\pi\)
−0.607506 + 0.794315i \(0.707830\pi\)
\(594\) 7.27938 0.298677
\(595\) −0.625033 −0.0256238
\(596\) −6.61271 −0.270867
\(597\) −26.6670 −1.09141
\(598\) −1.55030 −0.0633965
\(599\) 40.5026 1.65489 0.827445 0.561546i \(-0.189793\pi\)
0.827445 + 0.561546i \(0.189793\pi\)
\(600\) −11.4790 −0.468627
\(601\) 32.0568 1.30763 0.653813 0.756656i \(-0.273168\pi\)
0.653813 + 0.756656i \(0.273168\pi\)
\(602\) 2.49465 0.101674
\(603\) −22.2030 −0.904176
\(604\) 10.0986 0.410905
\(605\) −0.463308 −0.0188361
\(606\) 38.0664 1.54634
\(607\) −23.7367 −0.963443 −0.481722 0.876324i \(-0.659988\pi\)
−0.481722 + 0.876324i \(0.659988\pi\)
\(608\) −0.321301 −0.0130305
\(609\) 12.8369 0.520176
\(610\) −0.228559 −0.00925409
\(611\) 0.00952831 0.000385474 0
\(612\) 9.87496 0.399172
\(613\) 41.1812 1.66329 0.831646 0.555306i \(-0.187399\pi\)
0.831646 + 0.555306i \(0.187399\pi\)
\(614\) −12.2067 −0.492624
\(615\) 0.226330 0.00912651
\(616\) 11.2240 0.452227
\(617\) 2.48579 0.100074 0.0500370 0.998747i \(-0.484066\pi\)
0.0500370 + 0.998747i \(0.484066\pi\)
\(618\) −1.24556 −0.0501037
\(619\) 16.1042 0.647283 0.323641 0.946180i \(-0.395093\pi\)
0.323641 + 0.946180i \(0.395093\pi\)
\(620\) 0.420248 0.0168776
\(621\) 1.23159 0.0494220
\(622\) −1.91464 −0.0767701
\(623\) −29.5571 −1.18418
\(624\) −4.80056 −0.192176
\(625\) 24.9524 0.998098
\(626\) 5.44385 0.217580
\(627\) 3.23653 0.129255
\(628\) 7.00844 0.279667
\(629\) 24.6408 0.982493
\(630\) 0.328267 0.0130785
\(631\) −9.97969 −0.397285 −0.198643 0.980072i \(-0.563653\pi\)
−0.198643 + 0.980072i \(0.563653\pi\)
\(632\) 3.45286 0.137347
\(633\) −40.9500 −1.62762
\(634\) −7.14525 −0.283774
\(635\) 0.807087 0.0320283
\(636\) −27.1891 −1.07812
\(637\) −0.936180 −0.0370928
\(638\) −9.57246 −0.378977
\(639\) 1.73150 0.0684970
\(640\) 0.0563132 0.00222597
\(641\) −41.5643 −1.64169 −0.820845 0.571151i \(-0.806497\pi\)
−0.820845 + 0.571151i \(0.806497\pi\)
\(642\) 26.2322 1.03530
\(643\) −13.7650 −0.542840 −0.271420 0.962461i \(-0.587493\pi\)
−0.271420 + 0.962461i \(0.587493\pi\)
\(644\) 1.89897 0.0748299
\(645\) −0.126079 −0.00496434
\(646\) −1.39321 −0.0548153
\(647\) 10.4816 0.412073 0.206036 0.978544i \(-0.433943\pi\)
0.206036 + 0.978544i \(0.433943\pi\)
\(648\) 10.6457 0.418204
\(649\) −10.6424 −0.417753
\(650\) 10.4419 0.409564
\(651\) 43.8824 1.71989
\(652\) −15.7221 −0.615726
\(653\) 3.27766 0.128265 0.0641324 0.997941i \(-0.479572\pi\)
0.0641324 + 0.997941i \(0.479572\pi\)
\(654\) −2.79696 −0.109370
\(655\) −0.168493 −0.00658355
\(656\) 1.74954 0.0683081
\(657\) −37.2455 −1.45309
\(658\) −0.0116713 −0.000454994 0
\(659\) −13.6413 −0.531390 −0.265695 0.964057i \(-0.585601\pi\)
−0.265695 + 0.964057i \(0.585601\pi\)
\(660\) −0.567254 −0.0220803
\(661\) 22.0279 0.856784 0.428392 0.903593i \(-0.359080\pi\)
0.428392 + 0.903593i \(0.359080\pi\)
\(662\) −0.951726 −0.0369899
\(663\) −20.8160 −0.808427
\(664\) −0.0374441 −0.00145311
\(665\) −0.0463137 −0.00179597
\(666\) −12.9413 −0.501467
\(667\) −1.61955 −0.0627094
\(668\) 18.7636 0.725986
\(669\) 37.6464 1.45550
\(670\) −0.549024 −0.0212107
\(671\) 17.7970 0.687047
\(672\) 5.88024 0.226835
\(673\) −27.7400 −1.06930 −0.534649 0.845074i \(-0.679556\pi\)
−0.534649 + 0.845074i \(0.679556\pi\)
\(674\) 1.77919 0.0685317
\(675\) −8.29525 −0.319284
\(676\) −8.63315 −0.332044
\(677\) 13.9859 0.537523 0.268761 0.963207i \(-0.413386\pi\)
0.268761 + 0.963207i \(0.413386\pi\)
\(678\) 17.1378 0.658174
\(679\) 22.6896 0.870746
\(680\) 0.244183 0.00936399
\(681\) 52.9570 2.02932
\(682\) −32.7232 −1.25303
\(683\) −28.1723 −1.07798 −0.538992 0.842311i \(-0.681195\pi\)
−0.538992 + 0.842311i \(0.681195\pi\)
\(684\) 0.731716 0.0279779
\(685\) −0.950189 −0.0363049
\(686\) 19.0645 0.727888
\(687\) −21.3688 −0.815271
\(688\) −0.974593 −0.0371560
\(689\) 24.7327 0.942239
\(690\) −0.0959731 −0.00365363
\(691\) 24.9303 0.948394 0.474197 0.880419i \(-0.342738\pi\)
0.474197 + 0.880419i \(0.342738\pi\)
\(692\) 5.60033 0.212893
\(693\) −25.5609 −0.970980
\(694\) 20.4216 0.775192
\(695\) 0.898959 0.0340995
\(696\) −5.01501 −0.190094
\(697\) 7.58629 0.287351
\(698\) −3.75534 −0.142142
\(699\) −66.6097 −2.51941
\(700\) −12.7903 −0.483429
\(701\) −5.79349 −0.218817 −0.109408 0.993997i \(-0.534896\pi\)
−0.109408 + 0.993997i \(0.534896\pi\)
\(702\) −3.46912 −0.130933
\(703\) 1.82584 0.0688627
\(704\) −4.38490 −0.165262
\(705\) 0.000589862 0 2.22155e−5 0
\(706\) 30.1267 1.13383
\(707\) 42.4151 1.59518
\(708\) −5.57558 −0.209543
\(709\) 24.7620 0.929957 0.464978 0.885322i \(-0.346062\pi\)
0.464978 + 0.885322i \(0.346062\pi\)
\(710\) 0.0428156 0.00160684
\(711\) −7.86338 −0.294900
\(712\) 11.5471 0.432747
\(713\) −5.53640 −0.207340
\(714\) 25.4977 0.954226
\(715\) 0.516005 0.0192975
\(716\) −10.3380 −0.386351
\(717\) 19.7879 0.738993
\(718\) −35.3332 −1.31862
\(719\) −25.0866 −0.935571 −0.467785 0.883842i \(-0.654948\pi\)
−0.467785 + 0.883842i \(0.654948\pi\)
\(720\) −0.128245 −0.00477941
\(721\) −1.38785 −0.0516863
\(722\) 18.8968 0.703265
\(723\) 15.2810 0.568306
\(724\) −0.423949 −0.0157560
\(725\) 10.9083 0.405125
\(726\) 18.9002 0.701454
\(727\) −49.6479 −1.84134 −0.920669 0.390343i \(-0.872356\pi\)
−0.920669 + 0.390343i \(0.872356\pi\)
\(728\) −5.34898 −0.198247
\(729\) −12.8031 −0.474187
\(730\) −0.920989 −0.0340873
\(731\) −4.22599 −0.156304
\(732\) 9.32388 0.344620
\(733\) 15.2739 0.564155 0.282077 0.959392i \(-0.408977\pi\)
0.282077 + 0.959392i \(0.408977\pi\)
\(734\) 14.5194 0.535920
\(735\) −0.0579553 −0.00213771
\(736\) −0.741876 −0.0273459
\(737\) 42.7505 1.57473
\(738\) −3.98432 −0.146665
\(739\) −2.20308 −0.0810415 −0.0405208 0.999179i \(-0.512902\pi\)
−0.0405208 + 0.999179i \(0.512902\pi\)
\(740\) −0.320007 −0.0117637
\(741\) −1.54243 −0.0566625
\(742\) −30.2952 −1.11217
\(743\) 21.7624 0.798385 0.399193 0.916867i \(-0.369290\pi\)
0.399193 + 0.916867i \(0.369290\pi\)
\(744\) −17.1437 −0.628517
\(745\) 0.372383 0.0136431
\(746\) 19.6147 0.718145
\(747\) 0.0852734 0.00311999
\(748\) −19.0136 −0.695207
\(749\) 29.2289 1.06800
\(750\) 1.29324 0.0472226
\(751\) −25.9286 −0.946148 −0.473074 0.881023i \(-0.656856\pi\)
−0.473074 + 0.881023i \(0.656856\pi\)
\(752\) 0.00455965 0.000166273 0
\(753\) −16.8531 −0.614163
\(754\) 4.56193 0.166136
\(755\) −0.568683 −0.0206965
\(756\) 4.24934 0.154547
\(757\) 47.0934 1.71164 0.855820 0.517274i \(-0.173053\pi\)
0.855820 + 0.517274i \(0.173053\pi\)
\(758\) −23.2379 −0.844038
\(759\) 7.47307 0.271255
\(760\) 0.0180935 0.000656320 0
\(761\) 5.38781 0.195308 0.0976539 0.995220i \(-0.468866\pi\)
0.0976539 + 0.995220i \(0.468866\pi\)
\(762\) −32.9244 −1.19273
\(763\) −3.11649 −0.112825
\(764\) −9.33430 −0.337703
\(765\) −0.556091 −0.0201055
\(766\) 8.07250 0.291671
\(767\) 5.07185 0.183134
\(768\) −2.29725 −0.0828948
\(769\) 49.3387 1.77920 0.889600 0.456741i \(-0.150983\pi\)
0.889600 + 0.456741i \(0.150983\pi\)
\(770\) −0.632058 −0.0227778
\(771\) 14.0960 0.507656
\(772\) 8.70456 0.313284
\(773\) −9.45718 −0.340151 −0.170076 0.985431i \(-0.554401\pi\)
−0.170076 + 0.985431i \(0.554401\pi\)
\(774\) 2.21949 0.0797780
\(775\) 37.2898 1.33949
\(776\) −8.86420 −0.318206
\(777\) −33.4152 −1.19876
\(778\) 25.1180 0.900523
\(779\) 0.562129 0.0201404
\(780\) 0.270335 0.00967955
\(781\) −3.33389 −0.119296
\(782\) −3.21689 −0.115036
\(783\) −3.62409 −0.129514
\(784\) −0.447997 −0.0159999
\(785\) −0.394668 −0.0140863
\(786\) 6.87351 0.245170
\(787\) 31.7788 1.13279 0.566395 0.824134i \(-0.308338\pi\)
0.566395 + 0.824134i \(0.308338\pi\)
\(788\) 8.93519 0.318303
\(789\) 19.5078 0.694495
\(790\) −0.194442 −0.00691792
\(791\) 19.0957 0.678963
\(792\) 9.98596 0.354836
\(793\) −8.48150 −0.301187
\(794\) 1.31879 0.0468021
\(795\) 1.53110 0.0543027
\(796\) 11.6082 0.411442
\(797\) −16.8671 −0.597464 −0.298732 0.954337i \(-0.596564\pi\)
−0.298732 + 0.954337i \(0.596564\pi\)
\(798\) 1.88933 0.0668815
\(799\) 0.0197714 0.000699461 0
\(800\) 4.99683 0.176665
\(801\) −26.2969 −0.929155
\(802\) −26.0391 −0.919473
\(803\) 71.7139 2.53073
\(804\) 22.3970 0.789880
\(805\) −0.106937 −0.00376904
\(806\) 15.5948 0.549304
\(807\) 30.5519 1.07548
\(808\) −16.5704 −0.582946
\(809\) −16.1194 −0.566727 −0.283363 0.959013i \(-0.591450\pi\)
−0.283363 + 0.959013i \(0.591450\pi\)
\(810\) −0.599495 −0.0210641
\(811\) −36.2136 −1.27163 −0.635816 0.771840i \(-0.719336\pi\)
−0.635816 + 0.771840i \(0.719336\pi\)
\(812\) −5.58793 −0.196098
\(813\) −46.0306 −1.61437
\(814\) 24.9178 0.873367
\(815\) 0.885364 0.0310129
\(816\) −9.96124 −0.348713
\(817\) −0.313138 −0.0109553
\(818\) 27.2325 0.952164
\(819\) 12.1815 0.425657
\(820\) −0.0985222 −0.00344054
\(821\) −12.1007 −0.422316 −0.211158 0.977452i \(-0.567723\pi\)
−0.211158 + 0.977452i \(0.567723\pi\)
\(822\) 38.7621 1.35199
\(823\) −11.0862 −0.386439 −0.193220 0.981156i \(-0.561893\pi\)
−0.193220 + 0.981156i \(0.561893\pi\)
\(824\) 0.542195 0.0188883
\(825\) −50.3341 −1.75241
\(826\) −6.21254 −0.216162
\(827\) −31.3593 −1.09047 −0.545236 0.838283i \(-0.683560\pi\)
−0.545236 + 0.838283i \(0.683560\pi\)
\(828\) 1.68951 0.0587146
\(829\) 48.8170 1.69548 0.847742 0.530408i \(-0.177961\pi\)
0.847742 + 0.530408i \(0.177961\pi\)
\(830\) 0.00210860 7.31905e−5 0
\(831\) 32.5120 1.12783
\(832\) 2.08970 0.0724474
\(833\) −1.94259 −0.0673066
\(834\) −36.6723 −1.26986
\(835\) −1.05664 −0.0365665
\(836\) −1.40887 −0.0487269
\(837\) −12.3888 −0.428221
\(838\) 12.1085 0.418282
\(839\) −39.0010 −1.34646 −0.673232 0.739431i \(-0.735095\pi\)
−0.673232 + 0.739431i \(0.735095\pi\)
\(840\) −0.331135 −0.0114252
\(841\) −24.2343 −0.835665
\(842\) 11.8173 0.407251
\(843\) −5.91785 −0.203822
\(844\) 17.8257 0.613585
\(845\) 0.486160 0.0167244
\(846\) −0.0103839 −0.000357007 0
\(847\) 21.0594 0.723610
\(848\) 11.8355 0.406433
\(849\) 54.6826 1.87670
\(850\) 21.6671 0.743174
\(851\) 4.21581 0.144516
\(852\) −1.74662 −0.0598384
\(853\) −41.2530 −1.41247 −0.706237 0.707975i \(-0.749609\pi\)
−0.706237 + 0.707975i \(0.749609\pi\)
\(854\) 10.3890 0.355506
\(855\) −0.0412053 −0.00140919
\(856\) −11.4189 −0.390291
\(857\) −9.51628 −0.325070 −0.162535 0.986703i \(-0.551967\pi\)
−0.162535 + 0.986703i \(0.551967\pi\)
\(858\) −21.0500 −0.718635
\(859\) −27.2919 −0.931189 −0.465594 0.884998i \(-0.654159\pi\)
−0.465594 + 0.884998i \(0.654159\pi\)
\(860\) 0.0548824 0.00187148
\(861\) −10.2877 −0.350604
\(862\) 40.6471 1.38445
\(863\) 1.90861 0.0649697 0.0324849 0.999472i \(-0.489658\pi\)
0.0324849 + 0.999472i \(0.489658\pi\)
\(864\) −1.66010 −0.0564778
\(865\) −0.315372 −0.0107230
\(866\) 3.45931 0.117552
\(867\) −4.14031 −0.140612
\(868\) −19.1022 −0.648370
\(869\) 15.1404 0.513604
\(870\) 0.282411 0.00957464
\(871\) −20.3735 −0.690329
\(872\) 1.21753 0.0412307
\(873\) 20.1869 0.683223
\(874\) −0.238366 −0.00806284
\(875\) 1.44099 0.0487142
\(876\) 37.5709 1.26940
\(877\) 37.0912 1.25248 0.626241 0.779629i \(-0.284593\pi\)
0.626241 + 0.779629i \(0.284593\pi\)
\(878\) 36.1447 1.21982
\(879\) 17.4988 0.590219
\(880\) 0.246928 0.00832393
\(881\) 13.4848 0.454315 0.227158 0.973858i \(-0.427057\pi\)
0.227158 + 0.973858i \(0.427057\pi\)
\(882\) 1.02025 0.0343535
\(883\) 15.8342 0.532865 0.266432 0.963854i \(-0.414155\pi\)
0.266432 + 0.963854i \(0.414155\pi\)
\(884\) 9.06128 0.304764
\(885\) 0.313979 0.0105543
\(886\) −19.3661 −0.650618
\(887\) 38.2121 1.28304 0.641518 0.767108i \(-0.278305\pi\)
0.641518 + 0.767108i \(0.278305\pi\)
\(888\) 13.0544 0.438077
\(889\) −36.6857 −1.23040
\(890\) −0.650256 −0.0217966
\(891\) 46.6804 1.56385
\(892\) −16.3876 −0.548698
\(893\) 0.00146502 4.90251e−5 0
\(894\) −15.1910 −0.508064
\(895\) 0.582168 0.0194597
\(896\) −2.55969 −0.0855132
\(897\) −3.56142 −0.118912
\(898\) 40.9853 1.36770
\(899\) 16.2915 0.543350
\(900\) −11.3795 −0.379318
\(901\) 51.3206 1.70974
\(902\) 7.67156 0.255435
\(903\) 5.73084 0.190710
\(904\) −7.46015 −0.248121
\(905\) 0.0238739 0.000793597 0
\(906\) 23.1989 0.770733
\(907\) −36.6452 −1.21678 −0.608392 0.793637i \(-0.708185\pi\)
−0.608392 + 0.793637i \(0.708185\pi\)
\(908\) −23.0524 −0.765019
\(909\) 37.7367 1.25165
\(910\) 0.301218 0.00998529
\(911\) 29.7821 0.986724 0.493362 0.869824i \(-0.335768\pi\)
0.493362 + 0.869824i \(0.335768\pi\)
\(912\) −0.738109 −0.0244412
\(913\) −0.164189 −0.00543385
\(914\) 2.27000 0.0750848
\(915\) −0.525057 −0.0173579
\(916\) 9.30192 0.307344
\(917\) 7.65874 0.252914
\(918\) −7.19847 −0.237585
\(919\) −3.42036 −0.112827 −0.0564137 0.998407i \(-0.517967\pi\)
−0.0564137 + 0.998407i \(0.517967\pi\)
\(920\) 0.0417774 0.00137736
\(921\) −28.0419 −0.924013
\(922\) −25.9596 −0.854934
\(923\) 1.58882 0.0522968
\(924\) 25.7843 0.848240
\(925\) −28.3951 −0.933626
\(926\) 20.1425 0.661925
\(927\) −1.23477 −0.0405551
\(928\) 2.18305 0.0716622
\(929\) −33.1208 −1.08666 −0.543329 0.839520i \(-0.682836\pi\)
−0.543329 + 0.839520i \(0.682836\pi\)
\(930\) 0.965415 0.0316572
\(931\) −0.143942 −0.00471751
\(932\) 28.9954 0.949776
\(933\) −4.39840 −0.143997
\(934\) −3.35104 −0.109649
\(935\) 1.07072 0.0350162
\(936\) −4.75899 −0.155552
\(937\) −60.1823 −1.96607 −0.983035 0.183416i \(-0.941285\pi\)
−0.983035 + 0.183416i \(0.941285\pi\)
\(938\) 24.9556 0.814830
\(939\) 12.5059 0.408114
\(940\) −0.000256769 0 −8.37487e−6 0
\(941\) 12.9648 0.422641 0.211320 0.977417i \(-0.432224\pi\)
0.211320 + 0.977417i \(0.432224\pi\)
\(942\) 16.1001 0.524571
\(943\) 1.29794 0.0422668
\(944\) 2.42707 0.0789943
\(945\) −0.239294 −0.00778424
\(946\) −4.27349 −0.138943
\(947\) −14.2582 −0.463329 −0.231665 0.972796i \(-0.574417\pi\)
−0.231665 + 0.972796i \(0.574417\pi\)
\(948\) 7.93208 0.257622
\(949\) −34.1765 −1.10942
\(950\) 1.60549 0.0520889
\(951\) −16.4144 −0.532274
\(952\) −11.0992 −0.359728
\(953\) 40.9505 1.32652 0.663259 0.748390i \(-0.269173\pi\)
0.663259 + 0.748390i \(0.269173\pi\)
\(954\) −26.9536 −0.872655
\(955\) 0.525645 0.0170095
\(956\) −8.61374 −0.278588
\(957\) −21.9903 −0.710846
\(958\) 32.2923 1.04332
\(959\) 43.1904 1.39469
\(960\) 0.129365 0.00417525
\(961\) 24.6918 0.796511
\(962\) −11.8750 −0.382865
\(963\) 26.0050 0.837998
\(964\) −6.65186 −0.214242
\(965\) −0.490182 −0.0157795
\(966\) 4.36241 0.140358
\(967\) −23.2470 −0.747572 −0.373786 0.927515i \(-0.621940\pi\)
−0.373786 + 0.927515i \(0.621940\pi\)
\(968\) −8.22734 −0.264437
\(969\) −3.20056 −0.102817
\(970\) 0.499171 0.0160274
\(971\) 9.45928 0.303563 0.151781 0.988414i \(-0.451499\pi\)
0.151781 + 0.988414i \(0.451499\pi\)
\(972\) 19.4756 0.624679
\(973\) −40.8617 −1.30997
\(974\) 25.6355 0.821414
\(975\) 23.9876 0.768218
\(976\) −4.05871 −0.129916
\(977\) −25.0764 −0.802265 −0.401133 0.916020i \(-0.631383\pi\)
−0.401133 + 0.916020i \(0.631383\pi\)
\(978\) −36.1176 −1.15491
\(979\) 50.6330 1.61824
\(980\) 0.0252281 0.000805883 0
\(981\) −2.77274 −0.0885267
\(982\) 30.8194 0.983488
\(983\) 24.1186 0.769263 0.384632 0.923070i \(-0.374329\pi\)
0.384632 + 0.923070i \(0.374329\pi\)
\(984\) 4.01913 0.128125
\(985\) −0.503169 −0.0160323
\(986\) 9.46606 0.301461
\(987\) −0.0268119 −0.000853431 0
\(988\) 0.671424 0.0213608
\(989\) −0.723027 −0.0229909
\(990\) −0.562341 −0.0178724
\(991\) 16.4183 0.521545 0.260772 0.965400i \(-0.416023\pi\)
0.260772 + 0.965400i \(0.416023\pi\)
\(992\) 7.46270 0.236941
\(993\) −2.18635 −0.0693818
\(994\) −1.94616 −0.0617285
\(995\) −0.653696 −0.0207235
\(996\) −0.0860185 −0.00272560
\(997\) −13.5308 −0.428525 −0.214263 0.976776i \(-0.568735\pi\)
−0.214263 + 0.976776i \(0.568735\pi\)
\(998\) 25.8260 0.817507
\(999\) 9.43374 0.298470
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 6002.2.a.b.1.10 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6002.2.a.b.1.10 56 1.1 even 1 trivial