Properties

Label 2013.2.a.e.1.13
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(2.73913\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.73913 q^{2} -1.00000 q^{3} +5.50285 q^{4} +0.604616 q^{5} -2.73913 q^{6} -1.91298 q^{7} +9.59477 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.73913 q^{2} -1.00000 q^{3} +5.50285 q^{4} +0.604616 q^{5} -2.73913 q^{6} -1.91298 q^{7} +9.59477 q^{8} +1.00000 q^{9} +1.65612 q^{10} +1.00000 q^{11} -5.50285 q^{12} +0.176875 q^{13} -5.23991 q^{14} -0.604616 q^{15} +15.2757 q^{16} +2.74225 q^{17} +2.73913 q^{18} +3.64728 q^{19} +3.32711 q^{20} +1.91298 q^{21} +2.73913 q^{22} +1.61899 q^{23} -9.59477 q^{24} -4.63444 q^{25} +0.484484 q^{26} -1.00000 q^{27} -10.5269 q^{28} -7.55848 q^{29} -1.65612 q^{30} +7.88196 q^{31} +22.6525 q^{32} -1.00000 q^{33} +7.51138 q^{34} -1.15662 q^{35} +5.50285 q^{36} +8.43648 q^{37} +9.99040 q^{38} -0.176875 q^{39} +5.80115 q^{40} +8.93210 q^{41} +5.23991 q^{42} -4.51816 q^{43} +5.50285 q^{44} +0.604616 q^{45} +4.43462 q^{46} -2.48579 q^{47} -15.2757 q^{48} -3.34050 q^{49} -12.6943 q^{50} -2.74225 q^{51} +0.973317 q^{52} -7.57327 q^{53} -2.73913 q^{54} +0.604616 q^{55} -18.3546 q^{56} -3.64728 q^{57} -20.7037 q^{58} +1.33758 q^{59} -3.32711 q^{60} -1.00000 q^{61} +21.5897 q^{62} -1.91298 q^{63} +31.4969 q^{64} +0.106941 q^{65} -2.73913 q^{66} -5.50196 q^{67} +15.0902 q^{68} -1.61899 q^{69} -3.16813 q^{70} +0.330471 q^{71} +9.59477 q^{72} -1.75560 q^{73} +23.1086 q^{74} +4.63444 q^{75} +20.0705 q^{76} -1.91298 q^{77} -0.484484 q^{78} -12.9780 q^{79} +9.23590 q^{80} +1.00000 q^{81} +24.4662 q^{82} +12.9300 q^{83} +10.5269 q^{84} +1.65801 q^{85} -12.3759 q^{86} +7.55848 q^{87} +9.59477 q^{88} -4.11061 q^{89} +1.65612 q^{90} -0.338359 q^{91} +8.90903 q^{92} -7.88196 q^{93} -6.80891 q^{94} +2.20521 q^{95} -22.6525 q^{96} -12.7524 q^{97} -9.15006 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 13 q^{11} - 16 q^{12} + 13 q^{13} + q^{14} - 3 q^{15} + 18 q^{16} + 17 q^{17} + 2 q^{18} + 14 q^{19} - 7 q^{20} - 11 q^{21} + 2 q^{22} + 7 q^{23} - 9 q^{24} + 18 q^{25} - 10 q^{26} - 13 q^{27} + 19 q^{28} - 6 q^{29} - 6 q^{30} + 27 q^{31} + 5 q^{32} - 13 q^{33} + 6 q^{34} + 14 q^{35} + 16 q^{36} + 10 q^{37} + 2 q^{38} - 13 q^{39} + 8 q^{40} + 3 q^{41} - q^{42} + 29 q^{43} + 16 q^{44} + 3 q^{45} - 24 q^{46} + 8 q^{47} - 18 q^{48} + 8 q^{49} - 27 q^{50} - 17 q^{51} + 37 q^{52} - 24 q^{53} - 2 q^{54} + 3 q^{55} + 24 q^{56} - 14 q^{57} - 5 q^{58} + 13 q^{59} + 7 q^{60} - 13 q^{61} + 39 q^{62} + 11 q^{63} + 47 q^{64} - 11 q^{65} - 2 q^{66} + 44 q^{67} - 8 q^{68} - 7 q^{69} - 12 q^{70} + 3 q^{71} + 9 q^{72} + 48 q^{73} - 22 q^{74} - 18 q^{75} + 47 q^{76} + 11 q^{77} + 10 q^{78} - 17 q^{79} - 26 q^{80} + 13 q^{81} + 56 q^{82} + 50 q^{83} - 19 q^{84} + 8 q^{85} + 18 q^{86} + 6 q^{87} + 9 q^{88} - 15 q^{89} + 6 q^{90} + 47 q^{91} + 14 q^{92} - 27 q^{93} + 45 q^{94} - q^{95} - 5 q^{96} + 27 q^{97} + 47 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.73913 1.93686 0.968430 0.249287i \(-0.0801961\pi\)
0.968430 + 0.249287i \(0.0801961\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.50285 2.75142
\(5\) 0.604616 0.270392 0.135196 0.990819i \(-0.456834\pi\)
0.135196 + 0.990819i \(0.456834\pi\)
\(6\) −2.73913 −1.11825
\(7\) −1.91298 −0.723040 −0.361520 0.932364i \(-0.617742\pi\)
−0.361520 + 0.932364i \(0.617742\pi\)
\(8\) 9.59477 3.39226
\(9\) 1.00000 0.333333
\(10\) 1.65612 0.523712
\(11\) 1.00000 0.301511
\(12\) −5.50285 −1.58854
\(13\) 0.176875 0.0490563 0.0245282 0.999699i \(-0.492192\pi\)
0.0245282 + 0.999699i \(0.492192\pi\)
\(14\) −5.23991 −1.40043
\(15\) −0.604616 −0.156111
\(16\) 15.2757 3.81891
\(17\) 2.74225 0.665093 0.332546 0.943087i \(-0.392092\pi\)
0.332546 + 0.943087i \(0.392092\pi\)
\(18\) 2.73913 0.645620
\(19\) 3.64728 0.836745 0.418372 0.908276i \(-0.362601\pi\)
0.418372 + 0.908276i \(0.362601\pi\)
\(20\) 3.32711 0.743964
\(21\) 1.91298 0.417447
\(22\) 2.73913 0.583985
\(23\) 1.61899 0.337582 0.168791 0.985652i \(-0.446014\pi\)
0.168791 + 0.985652i \(0.446014\pi\)
\(24\) −9.59477 −1.95852
\(25\) −4.63444 −0.926888
\(26\) 0.484484 0.0950152
\(27\) −1.00000 −0.192450
\(28\) −10.5269 −1.98939
\(29\) −7.55848 −1.40357 −0.701787 0.712387i \(-0.747614\pi\)
−0.701787 + 0.712387i \(0.747614\pi\)
\(30\) −1.65612 −0.302365
\(31\) 7.88196 1.41564 0.707821 0.706392i \(-0.249678\pi\)
0.707821 + 0.706392i \(0.249678\pi\)
\(32\) 22.6525 4.00443
\(33\) −1.00000 −0.174078
\(34\) 7.51138 1.28819
\(35\) −1.15662 −0.195504
\(36\) 5.50285 0.917142
\(37\) 8.43648 1.38695 0.693475 0.720481i \(-0.256079\pi\)
0.693475 + 0.720481i \(0.256079\pi\)
\(38\) 9.99040 1.62066
\(39\) −0.176875 −0.0283227
\(40\) 5.80115 0.917242
\(41\) 8.93210 1.39496 0.697480 0.716604i \(-0.254305\pi\)
0.697480 + 0.716604i \(0.254305\pi\)
\(42\) 5.23991 0.808536
\(43\) −4.51816 −0.689013 −0.344507 0.938784i \(-0.611954\pi\)
−0.344507 + 0.938784i \(0.611954\pi\)
\(44\) 5.50285 0.829586
\(45\) 0.604616 0.0901308
\(46\) 4.43462 0.653848
\(47\) −2.48579 −0.362590 −0.181295 0.983429i \(-0.558029\pi\)
−0.181295 + 0.983429i \(0.558029\pi\)
\(48\) −15.2757 −2.20485
\(49\) −3.34050 −0.477214
\(50\) −12.6943 −1.79525
\(51\) −2.74225 −0.383991
\(52\) 0.973317 0.134975
\(53\) −7.57327 −1.04027 −0.520135 0.854084i \(-0.674118\pi\)
−0.520135 + 0.854084i \(0.674118\pi\)
\(54\) −2.73913 −0.372749
\(55\) 0.604616 0.0815264
\(56\) −18.3546 −2.45274
\(57\) −3.64728 −0.483095
\(58\) −20.7037 −2.71852
\(59\) 1.33758 0.174138 0.0870692 0.996202i \(-0.472250\pi\)
0.0870692 + 0.996202i \(0.472250\pi\)
\(60\) −3.32711 −0.429528
\(61\) −1.00000 −0.128037
\(62\) 21.5897 2.74190
\(63\) −1.91298 −0.241013
\(64\) 31.4969 3.93711
\(65\) 0.106941 0.0132645
\(66\) −2.73913 −0.337164
\(67\) −5.50196 −0.672171 −0.336086 0.941831i \(-0.609103\pi\)
−0.336086 + 0.941831i \(0.609103\pi\)
\(68\) 15.0902 1.82995
\(69\) −1.61899 −0.194903
\(70\) −3.16813 −0.378665
\(71\) 0.330471 0.0392197 0.0196098 0.999808i \(-0.493758\pi\)
0.0196098 + 0.999808i \(0.493758\pi\)
\(72\) 9.59477 1.13075
\(73\) −1.75560 −0.205477 −0.102739 0.994708i \(-0.532761\pi\)
−0.102739 + 0.994708i \(0.532761\pi\)
\(74\) 23.1086 2.68633
\(75\) 4.63444 0.535139
\(76\) 20.0705 2.30224
\(77\) −1.91298 −0.218005
\(78\) −0.484484 −0.0548570
\(79\) −12.9780 −1.46014 −0.730070 0.683372i \(-0.760513\pi\)
−0.730070 + 0.683372i \(0.760513\pi\)
\(80\) 9.23590 1.03260
\(81\) 1.00000 0.111111
\(82\) 24.4662 2.70184
\(83\) 12.9300 1.41925 0.709625 0.704580i \(-0.248864\pi\)
0.709625 + 0.704580i \(0.248864\pi\)
\(84\) 10.5269 1.14857
\(85\) 1.65801 0.179836
\(86\) −12.3759 −1.33452
\(87\) 7.55848 0.810354
\(88\) 9.59477 1.02281
\(89\) −4.11061 −0.435724 −0.217862 0.975980i \(-0.569908\pi\)
−0.217862 + 0.975980i \(0.569908\pi\)
\(90\) 1.65612 0.174571
\(91\) −0.338359 −0.0354697
\(92\) 8.90903 0.928831
\(93\) −7.88196 −0.817321
\(94\) −6.80891 −0.702285
\(95\) 2.20521 0.226249
\(96\) −22.6525 −2.31196
\(97\) −12.7524 −1.29481 −0.647404 0.762147i \(-0.724145\pi\)
−0.647404 + 0.762147i \(0.724145\pi\)
\(98\) −9.15006 −0.924296
\(99\) 1.00000 0.100504
\(100\) −25.5026 −2.55026
\(101\) −10.0353 −0.998553 −0.499276 0.866443i \(-0.666401\pi\)
−0.499276 + 0.866443i \(0.666401\pi\)
\(102\) −7.51138 −0.743737
\(103\) −6.05237 −0.596357 −0.298179 0.954510i \(-0.596379\pi\)
−0.298179 + 0.954510i \(0.596379\pi\)
\(104\) 1.69708 0.166412
\(105\) 1.15662 0.112875
\(106\) −20.7442 −2.01485
\(107\) 12.2595 1.18517 0.592587 0.805506i \(-0.298107\pi\)
0.592587 + 0.805506i \(0.298107\pi\)
\(108\) −5.50285 −0.529512
\(109\) −1.38440 −0.132602 −0.0663008 0.997800i \(-0.521120\pi\)
−0.0663008 + 0.997800i \(0.521120\pi\)
\(110\) 1.65612 0.157905
\(111\) −8.43648 −0.800756
\(112\) −29.2221 −2.76123
\(113\) 12.6874 1.19353 0.596765 0.802416i \(-0.296452\pi\)
0.596765 + 0.802416i \(0.296452\pi\)
\(114\) −9.99040 −0.935686
\(115\) 0.978864 0.0912795
\(116\) −41.5931 −3.86183
\(117\) 0.176875 0.0163521
\(118\) 3.66382 0.337282
\(119\) −5.24587 −0.480888
\(120\) −5.80115 −0.529570
\(121\) 1.00000 0.0909091
\(122\) −2.73913 −0.247989
\(123\) −8.93210 −0.805381
\(124\) 43.3732 3.89503
\(125\) −5.82513 −0.521016
\(126\) −5.23991 −0.466809
\(127\) 7.42636 0.658983 0.329491 0.944159i \(-0.393123\pi\)
0.329491 + 0.944159i \(0.393123\pi\)
\(128\) 40.9692 3.62120
\(129\) 4.51816 0.397802
\(130\) 0.292927 0.0256914
\(131\) −7.69981 −0.672735 −0.336368 0.941731i \(-0.609198\pi\)
−0.336368 + 0.941731i \(0.609198\pi\)
\(132\) −5.50285 −0.478962
\(133\) −6.97719 −0.604999
\(134\) −15.0706 −1.30190
\(135\) −0.604616 −0.0520370
\(136\) 26.3112 2.25617
\(137\) −15.9210 −1.36022 −0.680112 0.733108i \(-0.738069\pi\)
−0.680112 + 0.733108i \(0.738069\pi\)
\(138\) −4.43462 −0.377500
\(139\) −2.40561 −0.204041 −0.102021 0.994782i \(-0.532531\pi\)
−0.102021 + 0.994782i \(0.532531\pi\)
\(140\) −6.36470 −0.537916
\(141\) 2.48579 0.209341
\(142\) 0.905204 0.0759630
\(143\) 0.176875 0.0147910
\(144\) 15.2757 1.27297
\(145\) −4.56997 −0.379516
\(146\) −4.80881 −0.397980
\(147\) 3.34050 0.275519
\(148\) 46.4247 3.81609
\(149\) −14.5339 −1.19066 −0.595331 0.803480i \(-0.702979\pi\)
−0.595331 + 0.803480i \(0.702979\pi\)
\(150\) 12.6943 1.03649
\(151\) −10.9430 −0.890525 −0.445262 0.895400i \(-0.646890\pi\)
−0.445262 + 0.895400i \(0.646890\pi\)
\(152\) 34.9949 2.83846
\(153\) 2.74225 0.221698
\(154\) −5.23991 −0.422244
\(155\) 4.76556 0.382779
\(156\) −0.973317 −0.0779277
\(157\) 16.6497 1.32879 0.664394 0.747383i \(-0.268690\pi\)
0.664394 + 0.747383i \(0.268690\pi\)
\(158\) −35.5485 −2.82809
\(159\) 7.57327 0.600600
\(160\) 13.6961 1.08277
\(161\) −3.09709 −0.244085
\(162\) 2.73913 0.215207
\(163\) −2.05976 −0.161333 −0.0806666 0.996741i \(-0.525705\pi\)
−0.0806666 + 0.996741i \(0.525705\pi\)
\(164\) 49.1520 3.83813
\(165\) −0.604616 −0.0470693
\(166\) 35.4169 2.74889
\(167\) −15.3296 −1.18624 −0.593121 0.805113i \(-0.702105\pi\)
−0.593121 + 0.805113i \(0.702105\pi\)
\(168\) 18.3546 1.41609
\(169\) −12.9687 −0.997593
\(170\) 4.54150 0.348317
\(171\) 3.64728 0.278915
\(172\) −24.8628 −1.89577
\(173\) 13.0360 0.991105 0.495553 0.868578i \(-0.334966\pi\)
0.495553 + 0.868578i \(0.334966\pi\)
\(174\) 20.7037 1.56954
\(175\) 8.86560 0.670177
\(176\) 15.2757 1.15145
\(177\) −1.33758 −0.100539
\(178\) −11.2595 −0.843936
\(179\) −11.6608 −0.871571 −0.435786 0.900050i \(-0.643529\pi\)
−0.435786 + 0.900050i \(0.643529\pi\)
\(180\) 3.32711 0.247988
\(181\) −9.58234 −0.712250 −0.356125 0.934438i \(-0.615902\pi\)
−0.356125 + 0.934438i \(0.615902\pi\)
\(182\) −0.926810 −0.0686997
\(183\) 1.00000 0.0739221
\(184\) 15.5338 1.14517
\(185\) 5.10083 0.375021
\(186\) −21.5897 −1.58304
\(187\) 2.74225 0.200533
\(188\) −13.6789 −0.997638
\(189\) 1.91298 0.139149
\(190\) 6.04035 0.438213
\(191\) −18.4200 −1.33283 −0.666414 0.745582i \(-0.732172\pi\)
−0.666414 + 0.745582i \(0.732172\pi\)
\(192\) −31.4969 −2.27309
\(193\) −5.40862 −0.389321 −0.194661 0.980871i \(-0.562361\pi\)
−0.194661 + 0.980871i \(0.562361\pi\)
\(194\) −34.9305 −2.50786
\(195\) −0.106941 −0.00765824
\(196\) −18.3822 −1.31302
\(197\) −2.86614 −0.204204 −0.102102 0.994774i \(-0.532557\pi\)
−0.102102 + 0.994774i \(0.532557\pi\)
\(198\) 2.73913 0.194662
\(199\) 11.9550 0.847469 0.423735 0.905786i \(-0.360719\pi\)
0.423735 + 0.905786i \(0.360719\pi\)
\(200\) −44.4664 −3.14425
\(201\) 5.50196 0.388078
\(202\) −27.4881 −1.93406
\(203\) 14.4592 1.01484
\(204\) −15.0902 −1.05652
\(205\) 5.40049 0.377187
\(206\) −16.5782 −1.15506
\(207\) 1.61899 0.112527
\(208\) 2.70188 0.187342
\(209\) 3.64728 0.252288
\(210\) 3.16813 0.218622
\(211\) 24.3432 1.67585 0.837927 0.545783i \(-0.183768\pi\)
0.837927 + 0.545783i \(0.183768\pi\)
\(212\) −41.6746 −2.86222
\(213\) −0.330471 −0.0226435
\(214\) 33.5805 2.29552
\(215\) −2.73175 −0.186304
\(216\) −9.59477 −0.652841
\(217\) −15.0781 −1.02357
\(218\) −3.79206 −0.256831
\(219\) 1.75560 0.118632
\(220\) 3.32711 0.224314
\(221\) 0.485035 0.0326270
\(222\) −23.1086 −1.55095
\(223\) 15.7257 1.05307 0.526535 0.850154i \(-0.323491\pi\)
0.526535 + 0.850154i \(0.323491\pi\)
\(224\) −43.3338 −2.89536
\(225\) −4.63444 −0.308963
\(226\) 34.7525 2.31170
\(227\) −0.677571 −0.0449720 −0.0224860 0.999747i \(-0.507158\pi\)
−0.0224860 + 0.999747i \(0.507158\pi\)
\(228\) −20.0705 −1.32920
\(229\) 13.4531 0.889006 0.444503 0.895777i \(-0.353380\pi\)
0.444503 + 0.895777i \(0.353380\pi\)
\(230\) 2.68124 0.176796
\(231\) 1.91298 0.125865
\(232\) −72.5218 −4.76129
\(233\) −12.1025 −0.792862 −0.396431 0.918064i \(-0.629751\pi\)
−0.396431 + 0.918064i \(0.629751\pi\)
\(234\) 0.484484 0.0316717
\(235\) −1.50295 −0.0980415
\(236\) 7.36052 0.479129
\(237\) 12.9780 0.843012
\(238\) −14.3691 −0.931413
\(239\) −7.34142 −0.474877 −0.237438 0.971403i \(-0.576308\pi\)
−0.237438 + 0.971403i \(0.576308\pi\)
\(240\) −9.23590 −0.596175
\(241\) −26.9723 −1.73744 −0.868719 0.495306i \(-0.835056\pi\)
−0.868719 + 0.495306i \(0.835056\pi\)
\(242\) 2.73913 0.176078
\(243\) −1.00000 −0.0641500
\(244\) −5.50285 −0.352284
\(245\) −2.01972 −0.129035
\(246\) −24.4662 −1.55991
\(247\) 0.645114 0.0410476
\(248\) 75.6256 4.80223
\(249\) −12.9300 −0.819404
\(250\) −15.9558 −1.00913
\(251\) −17.0405 −1.07559 −0.537794 0.843076i \(-0.680742\pi\)
−0.537794 + 0.843076i \(0.680742\pi\)
\(252\) −10.5269 −0.663130
\(253\) 1.61899 0.101785
\(254\) 20.3418 1.27636
\(255\) −1.65801 −0.103828
\(256\) 49.2263 3.07664
\(257\) 13.6010 0.848410 0.424205 0.905566i \(-0.360554\pi\)
0.424205 + 0.905566i \(0.360554\pi\)
\(258\) 12.3759 0.770487
\(259\) −16.1388 −1.00282
\(260\) 0.588483 0.0364961
\(261\) −7.55848 −0.467858
\(262\) −21.0908 −1.30299
\(263\) 18.2508 1.12539 0.562696 0.826664i \(-0.309764\pi\)
0.562696 + 0.826664i \(0.309764\pi\)
\(264\) −9.59477 −0.590517
\(265\) −4.57892 −0.281281
\(266\) −19.1115 −1.17180
\(267\) 4.11061 0.251565
\(268\) −30.2764 −1.84943
\(269\) −9.66539 −0.589309 −0.294655 0.955604i \(-0.595205\pi\)
−0.294655 + 0.955604i \(0.595205\pi\)
\(270\) −1.65612 −0.100788
\(271\) 30.1115 1.82914 0.914571 0.404425i \(-0.132528\pi\)
0.914571 + 0.404425i \(0.132528\pi\)
\(272\) 41.8896 2.53993
\(273\) 0.338359 0.0204784
\(274\) −43.6098 −2.63456
\(275\) −4.63444 −0.279467
\(276\) −8.90903 −0.536261
\(277\) −5.89926 −0.354452 −0.177226 0.984170i \(-0.556712\pi\)
−0.177226 + 0.984170i \(0.556712\pi\)
\(278\) −6.58929 −0.395199
\(279\) 7.88196 0.471881
\(280\) −11.0975 −0.663202
\(281\) 6.40992 0.382384 0.191192 0.981553i \(-0.438765\pi\)
0.191192 + 0.981553i \(0.438765\pi\)
\(282\) 6.80891 0.405464
\(283\) 2.51457 0.149476 0.0747380 0.997203i \(-0.476188\pi\)
0.0747380 + 0.997203i \(0.476188\pi\)
\(284\) 1.81853 0.107910
\(285\) −2.20521 −0.130625
\(286\) 0.484484 0.0286482
\(287\) −17.0870 −1.00861
\(288\) 22.6525 1.33481
\(289\) −9.48008 −0.557652
\(290\) −12.5178 −0.735068
\(291\) 12.7524 0.747558
\(292\) −9.66079 −0.565355
\(293\) −12.3742 −0.722908 −0.361454 0.932390i \(-0.617720\pi\)
−0.361454 + 0.932390i \(0.617720\pi\)
\(294\) 9.15006 0.533642
\(295\) 0.808724 0.0470857
\(296\) 80.9461 4.70490
\(297\) −1.00000 −0.0580259
\(298\) −39.8102 −2.30615
\(299\) 0.286358 0.0165605
\(300\) 25.5026 1.47239
\(301\) 8.64317 0.498184
\(302\) −29.9742 −1.72482
\(303\) 10.0353 0.576515
\(304\) 55.7146 3.19545
\(305\) −0.604616 −0.0346202
\(306\) 7.51138 0.429397
\(307\) 21.2440 1.21246 0.606230 0.795290i \(-0.292681\pi\)
0.606230 + 0.795290i \(0.292681\pi\)
\(308\) −10.5269 −0.599823
\(309\) 6.05237 0.344307
\(310\) 13.0535 0.741389
\(311\) 26.4545 1.50010 0.750049 0.661383i \(-0.230030\pi\)
0.750049 + 0.661383i \(0.230030\pi\)
\(312\) −1.69708 −0.0960780
\(313\) 30.7427 1.73768 0.868841 0.495090i \(-0.164865\pi\)
0.868841 + 0.495090i \(0.164865\pi\)
\(314\) 45.6056 2.57367
\(315\) −1.15662 −0.0651681
\(316\) −71.4160 −4.01747
\(317\) −21.9554 −1.23314 −0.616569 0.787301i \(-0.711478\pi\)
−0.616569 + 0.787301i \(0.711478\pi\)
\(318\) 20.7442 1.16328
\(319\) −7.55848 −0.423193
\(320\) 19.0435 1.06457
\(321\) −12.2595 −0.684261
\(322\) −8.48334 −0.472758
\(323\) 10.0018 0.556513
\(324\) 5.50285 0.305714
\(325\) −0.819717 −0.0454697
\(326\) −5.64197 −0.312480
\(327\) 1.38440 0.0765575
\(328\) 85.7015 4.73207
\(329\) 4.75527 0.262167
\(330\) −1.65612 −0.0911666
\(331\) −15.8159 −0.869318 −0.434659 0.900595i \(-0.643131\pi\)
−0.434659 + 0.900595i \(0.643131\pi\)
\(332\) 71.1517 3.90496
\(333\) 8.43648 0.462316
\(334\) −41.9899 −2.29759
\(335\) −3.32657 −0.181750
\(336\) 29.2221 1.59419
\(337\) −33.1532 −1.80597 −0.902986 0.429671i \(-0.858630\pi\)
−0.902986 + 0.429671i \(0.858630\pi\)
\(338\) −35.5230 −1.93220
\(339\) −12.6874 −0.689085
\(340\) 9.12376 0.494805
\(341\) 7.88196 0.426832
\(342\) 9.99040 0.540219
\(343\) 19.7812 1.06808
\(344\) −43.3507 −2.33732
\(345\) −0.978864 −0.0527003
\(346\) 35.7072 1.91963
\(347\) 23.0402 1.23686 0.618430 0.785840i \(-0.287769\pi\)
0.618430 + 0.785840i \(0.287769\pi\)
\(348\) 41.5931 2.22963
\(349\) −32.7544 −1.75330 −0.876651 0.481127i \(-0.840227\pi\)
−0.876651 + 0.481127i \(0.840227\pi\)
\(350\) 24.2841 1.29804
\(351\) −0.176875 −0.00944089
\(352\) 22.6525 1.20738
\(353\) 0.349353 0.0185942 0.00929708 0.999957i \(-0.497041\pi\)
0.00929708 + 0.999957i \(0.497041\pi\)
\(354\) −3.66382 −0.194730
\(355\) 0.199808 0.0106047
\(356\) −22.6201 −1.19886
\(357\) 5.24587 0.277641
\(358\) −31.9406 −1.68811
\(359\) 29.2479 1.54365 0.771823 0.635838i \(-0.219345\pi\)
0.771823 + 0.635838i \(0.219345\pi\)
\(360\) 5.80115 0.305747
\(361\) −5.69731 −0.299859
\(362\) −26.2473 −1.37953
\(363\) −1.00000 −0.0524864
\(364\) −1.86194 −0.0975921
\(365\) −1.06146 −0.0555594
\(366\) 2.73913 0.143177
\(367\) −13.9617 −0.728795 −0.364397 0.931244i \(-0.618725\pi\)
−0.364397 + 0.931244i \(0.618725\pi\)
\(368\) 24.7311 1.28920
\(369\) 8.93210 0.464987
\(370\) 13.9719 0.726362
\(371\) 14.4875 0.752156
\(372\) −43.3732 −2.24880
\(373\) 13.7140 0.710084 0.355042 0.934850i \(-0.384467\pi\)
0.355042 + 0.934850i \(0.384467\pi\)
\(374\) 7.51138 0.388404
\(375\) 5.82513 0.300809
\(376\) −23.8506 −1.23000
\(377\) −1.33691 −0.0688541
\(378\) 5.23991 0.269512
\(379\) 34.6378 1.77922 0.889612 0.456717i \(-0.150975\pi\)
0.889612 + 0.456717i \(0.150975\pi\)
\(380\) 12.1349 0.622508
\(381\) −7.42636 −0.380464
\(382\) −50.4550 −2.58150
\(383\) 7.28622 0.372308 0.186154 0.982521i \(-0.440398\pi\)
0.186154 + 0.982521i \(0.440398\pi\)
\(384\) −40.9692 −2.09070
\(385\) −1.15662 −0.0589468
\(386\) −14.8149 −0.754060
\(387\) −4.51816 −0.229671
\(388\) −70.1744 −3.56257
\(389\) −30.6669 −1.55487 −0.777437 0.628960i \(-0.783481\pi\)
−0.777437 + 0.628960i \(0.783481\pi\)
\(390\) −0.292927 −0.0148329
\(391\) 4.43966 0.224523
\(392\) −32.0513 −1.61883
\(393\) 7.69981 0.388404
\(394\) −7.85074 −0.395514
\(395\) −7.84671 −0.394811
\(396\) 5.50285 0.276529
\(397\) −1.12627 −0.0565257 −0.0282628 0.999601i \(-0.508998\pi\)
−0.0282628 + 0.999601i \(0.508998\pi\)
\(398\) 32.7464 1.64143
\(399\) 6.97719 0.349297
\(400\) −70.7941 −3.53970
\(401\) −9.38421 −0.468625 −0.234312 0.972161i \(-0.575284\pi\)
−0.234312 + 0.972161i \(0.575284\pi\)
\(402\) 15.0706 0.751653
\(403\) 1.39412 0.0694462
\(404\) −55.2229 −2.74744
\(405\) 0.604616 0.0300436
\(406\) 39.6058 1.96560
\(407\) 8.43648 0.418181
\(408\) −26.3112 −1.30260
\(409\) 34.3783 1.69990 0.849948 0.526867i \(-0.176633\pi\)
0.849948 + 0.526867i \(0.176633\pi\)
\(410\) 14.7927 0.730558
\(411\) 15.9210 0.785326
\(412\) −33.3053 −1.64083
\(413\) −2.55877 −0.125909
\(414\) 4.43462 0.217949
\(415\) 7.81767 0.383754
\(416\) 4.00666 0.196443
\(417\) 2.40561 0.117803
\(418\) 9.99040 0.488646
\(419\) −1.53252 −0.0748685 −0.0374343 0.999299i \(-0.511918\pi\)
−0.0374343 + 0.999299i \(0.511918\pi\)
\(420\) 6.36470 0.310566
\(421\) 13.4022 0.653185 0.326593 0.945165i \(-0.394100\pi\)
0.326593 + 0.945165i \(0.394100\pi\)
\(422\) 66.6792 3.24589
\(423\) −2.48579 −0.120863
\(424\) −72.6638 −3.52887
\(425\) −12.7088 −0.616466
\(426\) −0.905204 −0.0438573
\(427\) 1.91298 0.0925757
\(428\) 67.4624 3.26092
\(429\) −0.176875 −0.00853961
\(430\) −7.48263 −0.360845
\(431\) −4.01874 −0.193576 −0.0967880 0.995305i \(-0.530857\pi\)
−0.0967880 + 0.995305i \(0.530857\pi\)
\(432\) −15.2757 −0.734950
\(433\) 19.2933 0.927175 0.463587 0.886051i \(-0.346562\pi\)
0.463587 + 0.886051i \(0.346562\pi\)
\(434\) −41.3008 −1.98250
\(435\) 4.56997 0.219113
\(436\) −7.61815 −0.364843
\(437\) 5.90490 0.282470
\(438\) 4.80881 0.229774
\(439\) −23.1285 −1.10386 −0.551930 0.833890i \(-0.686109\pi\)
−0.551930 + 0.833890i \(0.686109\pi\)
\(440\) 5.80115 0.276559
\(441\) −3.34050 −0.159071
\(442\) 1.32858 0.0631939
\(443\) −18.3618 −0.872398 −0.436199 0.899850i \(-0.643676\pi\)
−0.436199 + 0.899850i \(0.643676\pi\)
\(444\) −46.4247 −2.20322
\(445\) −2.48534 −0.117816
\(446\) 43.0747 2.03965
\(447\) 14.5339 0.687429
\(448\) −60.2530 −2.84669
\(449\) −12.2717 −0.579135 −0.289568 0.957158i \(-0.593512\pi\)
−0.289568 + 0.957158i \(0.593512\pi\)
\(450\) −12.6943 −0.598417
\(451\) 8.93210 0.420596
\(452\) 69.8168 3.28391
\(453\) 10.9430 0.514145
\(454\) −1.85596 −0.0871044
\(455\) −0.204577 −0.00959073
\(456\) −34.9949 −1.63878
\(457\) 17.9930 0.841679 0.420840 0.907135i \(-0.361736\pi\)
0.420840 + 0.907135i \(0.361736\pi\)
\(458\) 36.8498 1.72188
\(459\) −2.74225 −0.127997
\(460\) 5.38654 0.251149
\(461\) −1.26483 −0.0589088 −0.0294544 0.999566i \(-0.509377\pi\)
−0.0294544 + 0.999566i \(0.509377\pi\)
\(462\) 5.23991 0.243783
\(463\) −10.3872 −0.482732 −0.241366 0.970434i \(-0.577595\pi\)
−0.241366 + 0.970434i \(0.577595\pi\)
\(464\) −115.461 −5.36012
\(465\) −4.76556 −0.220997
\(466\) −33.1504 −1.53566
\(467\) 10.9780 0.507999 0.253999 0.967204i \(-0.418254\pi\)
0.253999 + 0.967204i \(0.418254\pi\)
\(468\) 0.973317 0.0449916
\(469\) 10.5252 0.486006
\(470\) −4.11677 −0.189893
\(471\) −16.6497 −0.767176
\(472\) 12.8338 0.590724
\(473\) −4.51816 −0.207745
\(474\) 35.5485 1.63280
\(475\) −16.9031 −0.775568
\(476\) −28.8672 −1.32313
\(477\) −7.57327 −0.346756
\(478\) −20.1091 −0.919770
\(479\) −17.8446 −0.815339 −0.407670 0.913130i \(-0.633658\pi\)
−0.407670 + 0.913130i \(0.633658\pi\)
\(480\) −13.6961 −0.625137
\(481\) 1.49220 0.0680386
\(482\) −73.8807 −3.36517
\(483\) 3.09709 0.140923
\(484\) 5.50285 0.250130
\(485\) −7.71029 −0.350106
\(486\) −2.73913 −0.124250
\(487\) −19.6964 −0.892531 −0.446266 0.894901i \(-0.647246\pi\)
−0.446266 + 0.894901i \(0.647246\pi\)
\(488\) −9.59477 −0.434335
\(489\) 2.05976 0.0931457
\(490\) −5.53227 −0.249923
\(491\) −17.3750 −0.784124 −0.392062 0.919939i \(-0.628238\pi\)
−0.392062 + 0.919939i \(0.628238\pi\)
\(492\) −49.1520 −2.21594
\(493\) −20.7272 −0.933507
\(494\) 1.76705 0.0795034
\(495\) 0.604616 0.0271755
\(496\) 120.402 5.40621
\(497\) −0.632185 −0.0283574
\(498\) −35.4169 −1.58707
\(499\) 25.1532 1.12601 0.563005 0.826454i \(-0.309645\pi\)
0.563005 + 0.826454i \(0.309645\pi\)
\(500\) −32.0548 −1.43354
\(501\) 15.3296 0.684878
\(502\) −46.6763 −2.08326
\(503\) 18.0309 0.803958 0.401979 0.915649i \(-0.368322\pi\)
0.401979 + 0.915649i \(0.368322\pi\)
\(504\) −18.3546 −0.817580
\(505\) −6.06752 −0.270001
\(506\) 4.43462 0.197143
\(507\) 12.9687 0.575961
\(508\) 40.8661 1.81314
\(509\) −35.8893 −1.59077 −0.795383 0.606108i \(-0.792730\pi\)
−0.795383 + 0.606108i \(0.792730\pi\)
\(510\) −4.54150 −0.201101
\(511\) 3.35843 0.148568
\(512\) 52.8990 2.33783
\(513\) −3.64728 −0.161032
\(514\) 37.2551 1.64325
\(515\) −3.65936 −0.161250
\(516\) 24.8628 1.09452
\(517\) −2.48579 −0.109325
\(518\) −44.2065 −1.94232
\(519\) −13.0360 −0.572215
\(520\) 1.02608 0.0449965
\(521\) 6.16062 0.269902 0.134951 0.990852i \(-0.456912\pi\)
0.134951 + 0.990852i \(0.456912\pi\)
\(522\) −20.7037 −0.906175
\(523\) −2.39141 −0.104569 −0.0522846 0.998632i \(-0.516650\pi\)
−0.0522846 + 0.998632i \(0.516650\pi\)
\(524\) −42.3709 −1.85098
\(525\) −8.86560 −0.386927
\(526\) 49.9914 2.17973
\(527\) 21.6143 0.941533
\(528\) −15.2757 −0.664787
\(529\) −20.3789 −0.886039
\(530\) −12.5423 −0.544801
\(531\) 1.33758 0.0580462
\(532\) −38.3944 −1.66461
\(533\) 1.57987 0.0684316
\(534\) 11.2595 0.487247
\(535\) 7.41231 0.320462
\(536\) −52.7900 −2.28018
\(537\) 11.6608 0.503202
\(538\) −26.4748 −1.14141
\(539\) −3.34050 −0.143885
\(540\) −3.32711 −0.143176
\(541\) 30.9269 1.32965 0.664827 0.746998i \(-0.268505\pi\)
0.664827 + 0.746998i \(0.268505\pi\)
\(542\) 82.4793 3.54279
\(543\) 9.58234 0.411217
\(544\) 62.1188 2.66332
\(545\) −0.837031 −0.0358545
\(546\) 0.926810 0.0396638
\(547\) 31.4601 1.34514 0.672569 0.740035i \(-0.265191\pi\)
0.672569 + 0.740035i \(0.265191\pi\)
\(548\) −87.6109 −3.74255
\(549\) −1.00000 −0.0426790
\(550\) −12.6943 −0.541289
\(551\) −27.5679 −1.17443
\(552\) −15.5338 −0.661162
\(553\) 24.8267 1.05574
\(554\) −16.1589 −0.686524
\(555\) −5.10083 −0.216518
\(556\) −13.2377 −0.561404
\(557\) −43.3173 −1.83541 −0.917706 0.397261i \(-0.869961\pi\)
−0.917706 + 0.397261i \(0.869961\pi\)
\(558\) 21.5897 0.913967
\(559\) −0.799150 −0.0338005
\(560\) −17.6681 −0.746614
\(561\) −2.74225 −0.115778
\(562\) 17.5576 0.740623
\(563\) −12.7937 −0.539192 −0.269596 0.962974i \(-0.586890\pi\)
−0.269596 + 0.962974i \(0.586890\pi\)
\(564\) 13.6789 0.575987
\(565\) 7.67100 0.322721
\(566\) 6.88775 0.289514
\(567\) −1.91298 −0.0803377
\(568\) 3.17079 0.133044
\(569\) 16.2156 0.679791 0.339896 0.940463i \(-0.389608\pi\)
0.339896 + 0.940463i \(0.389608\pi\)
\(570\) −6.04035 −0.253003
\(571\) 15.4511 0.646607 0.323303 0.946295i \(-0.395207\pi\)
0.323303 + 0.946295i \(0.395207\pi\)
\(572\) 0.973317 0.0406964
\(573\) 18.4200 0.769509
\(574\) −46.8035 −1.95354
\(575\) −7.50309 −0.312900
\(576\) 31.4969 1.31237
\(577\) −13.7967 −0.574365 −0.287182 0.957876i \(-0.592719\pi\)
−0.287182 + 0.957876i \(0.592719\pi\)
\(578\) −25.9672 −1.08009
\(579\) 5.40862 0.224775
\(580\) −25.1479 −1.04421
\(581\) −24.7348 −1.02617
\(582\) 34.9305 1.44791
\(583\) −7.57327 −0.313653
\(584\) −16.8445 −0.697032
\(585\) 0.106941 0.00442148
\(586\) −33.8946 −1.40017
\(587\) −0.731957 −0.0302111 −0.0151056 0.999886i \(-0.504808\pi\)
−0.0151056 + 0.999886i \(0.504808\pi\)
\(588\) 18.3822 0.758071
\(589\) 28.7478 1.18453
\(590\) 2.21520 0.0911984
\(591\) 2.86614 0.117897
\(592\) 128.873 5.29664
\(593\) 16.9597 0.696450 0.348225 0.937411i \(-0.386785\pi\)
0.348225 + 0.937411i \(0.386785\pi\)
\(594\) −2.73913 −0.112388
\(595\) −3.17174 −0.130029
\(596\) −79.9778 −3.27602
\(597\) −11.9550 −0.489287
\(598\) 0.784373 0.0320754
\(599\) −11.3334 −0.463070 −0.231535 0.972827i \(-0.574375\pi\)
−0.231535 + 0.972827i \(0.574375\pi\)
\(600\) 44.4664 1.81533
\(601\) 37.8868 1.54543 0.772717 0.634750i \(-0.218897\pi\)
0.772717 + 0.634750i \(0.218897\pi\)
\(602\) 23.6748 0.964912
\(603\) −5.50196 −0.224057
\(604\) −60.2174 −2.45021
\(605\) 0.604616 0.0245811
\(606\) 27.4881 1.11663
\(607\) −9.55272 −0.387733 −0.193866 0.981028i \(-0.562103\pi\)
−0.193866 + 0.981028i \(0.562103\pi\)
\(608\) 82.6201 3.35069
\(609\) −14.4592 −0.585918
\(610\) −1.65612 −0.0670545
\(611\) −0.439674 −0.0177873
\(612\) 15.0902 0.609984
\(613\) 18.7337 0.756646 0.378323 0.925674i \(-0.376501\pi\)
0.378323 + 0.925674i \(0.376501\pi\)
\(614\) 58.1902 2.34836
\(615\) −5.40049 −0.217769
\(616\) −18.3546 −0.739529
\(617\) −2.88614 −0.116192 −0.0580958 0.998311i \(-0.518503\pi\)
−0.0580958 + 0.998311i \(0.518503\pi\)
\(618\) 16.5782 0.666874
\(619\) 28.4797 1.14470 0.572348 0.820011i \(-0.306033\pi\)
0.572348 + 0.820011i \(0.306033\pi\)
\(620\) 26.2241 1.05319
\(621\) −1.61899 −0.0649676
\(622\) 72.4624 2.90548
\(623\) 7.86353 0.315046
\(624\) −2.70188 −0.108162
\(625\) 19.6502 0.786009
\(626\) 84.2085 3.36565
\(627\) −3.64728 −0.145659
\(628\) 91.6205 3.65606
\(629\) 23.1349 0.922450
\(630\) −3.16813 −0.126222
\(631\) 18.8981 0.752322 0.376161 0.926554i \(-0.377244\pi\)
0.376161 + 0.926554i \(0.377244\pi\)
\(632\) −124.521 −4.95318
\(633\) −24.3432 −0.967554
\(634\) −60.1388 −2.38842
\(635\) 4.49009 0.178184
\(636\) 41.6746 1.65250
\(637\) −0.590850 −0.0234103
\(638\) −20.7037 −0.819666
\(639\) 0.330471 0.0130732
\(640\) 24.7706 0.979145
\(641\) 29.7753 1.17605 0.588026 0.808842i \(-0.299905\pi\)
0.588026 + 0.808842i \(0.299905\pi\)
\(642\) −33.5805 −1.32532
\(643\) −1.68829 −0.0665798 −0.0332899 0.999446i \(-0.510598\pi\)
−0.0332899 + 0.999446i \(0.510598\pi\)
\(644\) −17.0428 −0.671581
\(645\) 2.73175 0.107563
\(646\) 27.3961 1.07789
\(647\) −31.7652 −1.24882 −0.624409 0.781098i \(-0.714660\pi\)
−0.624409 + 0.781098i \(0.714660\pi\)
\(648\) 9.59477 0.376918
\(649\) 1.33758 0.0525047
\(650\) −2.24531 −0.0880684
\(651\) 15.0781 0.590956
\(652\) −11.3346 −0.443896
\(653\) −12.0681 −0.472261 −0.236130 0.971721i \(-0.575879\pi\)
−0.236130 + 0.971721i \(0.575879\pi\)
\(654\) 3.79206 0.148281
\(655\) −4.65542 −0.181902
\(656\) 136.444 5.32723
\(657\) −1.75560 −0.0684924
\(658\) 13.0253 0.507780
\(659\) 17.5896 0.685192 0.342596 0.939483i \(-0.388694\pi\)
0.342596 + 0.939483i \(0.388694\pi\)
\(660\) −3.32711 −0.129508
\(661\) 18.9318 0.736361 0.368181 0.929754i \(-0.379981\pi\)
0.368181 + 0.929754i \(0.379981\pi\)
\(662\) −43.3217 −1.68375
\(663\) −0.485035 −0.0188372
\(664\) 124.060 4.81447
\(665\) −4.21852 −0.163587
\(666\) 23.1086 0.895442
\(667\) −12.2371 −0.473821
\(668\) −84.3567 −3.26386
\(669\) −15.7257 −0.607990
\(670\) −9.11192 −0.352024
\(671\) −1.00000 −0.0386046
\(672\) 43.3338 1.67164
\(673\) 49.3060 1.90061 0.950303 0.311325i \(-0.100773\pi\)
0.950303 + 0.311325i \(0.100773\pi\)
\(674\) −90.8111 −3.49791
\(675\) 4.63444 0.178380
\(676\) −71.3649 −2.74480
\(677\) 6.96645 0.267742 0.133871 0.990999i \(-0.457259\pi\)
0.133871 + 0.990999i \(0.457259\pi\)
\(678\) −34.7525 −1.33466
\(679\) 24.3951 0.936198
\(680\) 15.9082 0.610051
\(681\) 0.677571 0.0259646
\(682\) 21.5897 0.826714
\(683\) 17.4032 0.665913 0.332957 0.942942i \(-0.391954\pi\)
0.332957 + 0.942942i \(0.391954\pi\)
\(684\) 20.0705 0.767413
\(685\) −9.62609 −0.367794
\(686\) 54.1833 2.06873
\(687\) −13.4531 −0.513268
\(688\) −69.0179 −2.63128
\(689\) −1.33952 −0.0510318
\(690\) −2.68124 −0.102073
\(691\) −16.3254 −0.621048 −0.310524 0.950566i \(-0.600505\pi\)
−0.310524 + 0.950566i \(0.600505\pi\)
\(692\) 71.7349 2.72695
\(693\) −1.91298 −0.0726682
\(694\) 63.1100 2.39562
\(695\) −1.45447 −0.0551712
\(696\) 72.5218 2.74893
\(697\) 24.4940 0.927778
\(698\) −89.7186 −3.39590
\(699\) 12.1025 0.457759
\(700\) 48.7861 1.84394
\(701\) −1.25848 −0.0475322 −0.0237661 0.999718i \(-0.507566\pi\)
−0.0237661 + 0.999718i \(0.507566\pi\)
\(702\) −0.484484 −0.0182857
\(703\) 30.7703 1.16052
\(704\) 31.4969 1.18708
\(705\) 1.50295 0.0566043
\(706\) 0.956923 0.0360143
\(707\) 19.1974 0.721993
\(708\) −7.36052 −0.276625
\(709\) 44.8435 1.68413 0.842067 0.539373i \(-0.181339\pi\)
0.842067 + 0.539373i \(0.181339\pi\)
\(710\) 0.547301 0.0205398
\(711\) −12.9780 −0.486713
\(712\) −39.4404 −1.47809
\(713\) 12.7608 0.477895
\(714\) 14.3691 0.537752
\(715\) 0.106941 0.00399938
\(716\) −64.1678 −2.39806
\(717\) 7.34142 0.274170
\(718\) 80.1139 2.98982
\(719\) 39.7590 1.48276 0.741381 0.671085i \(-0.234171\pi\)
0.741381 + 0.671085i \(0.234171\pi\)
\(720\) 9.23590 0.344202
\(721\) 11.5781 0.431190
\(722\) −15.6057 −0.580784
\(723\) 26.9723 1.00311
\(724\) −52.7302 −1.95970
\(725\) 35.0293 1.30096
\(726\) −2.73913 −0.101659
\(727\) 39.6673 1.47118 0.735589 0.677428i \(-0.236905\pi\)
0.735589 + 0.677428i \(0.236905\pi\)
\(728\) −3.24648 −0.120322
\(729\) 1.00000 0.0370370
\(730\) −2.90748 −0.107611
\(731\) −12.3899 −0.458258
\(732\) 5.50285 0.203391
\(733\) −4.45224 −0.164447 −0.0822235 0.996614i \(-0.526202\pi\)
−0.0822235 + 0.996614i \(0.526202\pi\)
\(734\) −38.2430 −1.41157
\(735\) 2.01972 0.0744984
\(736\) 36.6741 1.35182
\(737\) −5.50196 −0.202667
\(738\) 24.4662 0.900614
\(739\) −33.8922 −1.24674 −0.623372 0.781926i \(-0.714238\pi\)
−0.623372 + 0.781926i \(0.714238\pi\)
\(740\) 28.0691 1.03184
\(741\) −0.645114 −0.0236988
\(742\) 39.6833 1.45682
\(743\) −9.90914 −0.363531 −0.181766 0.983342i \(-0.558181\pi\)
−0.181766 + 0.983342i \(0.558181\pi\)
\(744\) −75.6256 −2.77257
\(745\) −8.78742 −0.321946
\(746\) 37.5645 1.37533
\(747\) 12.9300 0.473083
\(748\) 15.0902 0.551751
\(749\) −23.4523 −0.856928
\(750\) 15.9558 0.582624
\(751\) 3.30472 0.120591 0.0602955 0.998181i \(-0.480796\pi\)
0.0602955 + 0.998181i \(0.480796\pi\)
\(752\) −37.9720 −1.38470
\(753\) 17.0405 0.620991
\(754\) −3.66196 −0.133361
\(755\) −6.61628 −0.240791
\(756\) 10.5269 0.382858
\(757\) 13.4977 0.490583 0.245292 0.969449i \(-0.421116\pi\)
0.245292 + 0.969449i \(0.421116\pi\)
\(758\) 94.8776 3.44611
\(759\) −1.61899 −0.0587654
\(760\) 21.1584 0.767497
\(761\) −28.0515 −1.01687 −0.508433 0.861101i \(-0.669775\pi\)
−0.508433 + 0.861101i \(0.669775\pi\)
\(762\) −20.3418 −0.736905
\(763\) 2.64834 0.0958762
\(764\) −101.363 −3.66718
\(765\) 1.65801 0.0599453
\(766\) 19.9579 0.721109
\(767\) 0.236585 0.00854259
\(768\) −49.2263 −1.77630
\(769\) −29.0212 −1.04653 −0.523266 0.852170i \(-0.675286\pi\)
−0.523266 + 0.852170i \(0.675286\pi\)
\(770\) −3.16813 −0.114172
\(771\) −13.6010 −0.489830
\(772\) −29.7628 −1.07119
\(773\) −18.5820 −0.668350 −0.334175 0.942511i \(-0.608458\pi\)
−0.334175 + 0.942511i \(0.608458\pi\)
\(774\) −12.3759 −0.444841
\(775\) −36.5285 −1.31214
\(776\) −122.356 −4.39233
\(777\) 16.1388 0.578978
\(778\) −84.0008 −3.01157
\(779\) 32.5779 1.16723
\(780\) −0.588483 −0.0210711
\(781\) 0.330471 0.0118252
\(782\) 12.1608 0.434870
\(783\) 7.55848 0.270118
\(784\) −51.0282 −1.82244
\(785\) 10.0666 0.359294
\(786\) 21.0908 0.752284
\(787\) 34.2557 1.22108 0.610541 0.791985i \(-0.290952\pi\)
0.610541 + 0.791985i \(0.290952\pi\)
\(788\) −15.7719 −0.561852
\(789\) −18.2508 −0.649746
\(790\) −21.4932 −0.764693
\(791\) −24.2708 −0.862969
\(792\) 9.59477 0.340935
\(793\) −0.176875 −0.00628102
\(794\) −3.08499 −0.109482
\(795\) 4.57892 0.162398
\(796\) 65.7867 2.33175
\(797\) −11.0127 −0.390088 −0.195044 0.980794i \(-0.562485\pi\)
−0.195044 + 0.980794i \(0.562485\pi\)
\(798\) 19.1115 0.676538
\(799\) −6.81665 −0.241156
\(800\) −104.982 −3.71166
\(801\) −4.11061 −0.145241
\(802\) −25.7046 −0.907660
\(803\) −1.75560 −0.0619537
\(804\) 30.2764 1.06777
\(805\) −1.87255 −0.0659987
\(806\) 3.81869 0.134507
\(807\) 9.66539 0.340238
\(808\) −96.2867 −3.38735
\(809\) 45.2337 1.59033 0.795165 0.606392i \(-0.207384\pi\)
0.795165 + 0.606392i \(0.207384\pi\)
\(810\) 1.65612 0.0581902
\(811\) 32.9688 1.15769 0.578846 0.815437i \(-0.303503\pi\)
0.578846 + 0.815437i \(0.303503\pi\)
\(812\) 79.5670 2.79225
\(813\) −30.1115 −1.05606
\(814\) 23.1086 0.809958
\(815\) −1.24537 −0.0436232
\(816\) −41.8896 −1.46643
\(817\) −16.4790 −0.576528
\(818\) 94.1666 3.29246
\(819\) −0.338359 −0.0118232
\(820\) 29.7181 1.03780
\(821\) 17.6348 0.615460 0.307730 0.951474i \(-0.400431\pi\)
0.307730 + 0.951474i \(0.400431\pi\)
\(822\) 43.6098 1.52107
\(823\) −6.60202 −0.230132 −0.115066 0.993358i \(-0.536708\pi\)
−0.115066 + 0.993358i \(0.536708\pi\)
\(824\) −58.0711 −2.02300
\(825\) 4.63444 0.161350
\(826\) −7.00882 −0.243868
\(827\) −0.843089 −0.0293171 −0.0146585 0.999893i \(-0.504666\pi\)
−0.0146585 + 0.999893i \(0.504666\pi\)
\(828\) 8.90903 0.309610
\(829\) −40.4340 −1.40433 −0.702166 0.712014i \(-0.747783\pi\)
−0.702166 + 0.712014i \(0.747783\pi\)
\(830\) 21.4136 0.743278
\(831\) 5.89926 0.204643
\(832\) 5.57102 0.193140
\(833\) −9.16047 −0.317391
\(834\) 6.58929 0.228168
\(835\) −9.26854 −0.320751
\(836\) 20.0705 0.694151
\(837\) −7.88196 −0.272440
\(838\) −4.19778 −0.145010
\(839\) 49.0996 1.69511 0.847553 0.530710i \(-0.178075\pi\)
0.847553 + 0.530710i \(0.178075\pi\)
\(840\) 11.0975 0.382900
\(841\) 28.1305 0.970019
\(842\) 36.7105 1.26513
\(843\) −6.40992 −0.220769
\(844\) 133.957 4.61098
\(845\) −7.84109 −0.269742
\(846\) −6.80891 −0.234095
\(847\) −1.91298 −0.0657309
\(848\) −115.687 −3.97270
\(849\) −2.51457 −0.0863000
\(850\) −34.8110 −1.19401
\(851\) 13.6585 0.468209
\(852\) −1.81853 −0.0623019
\(853\) 35.0517 1.20015 0.600074 0.799944i \(-0.295138\pi\)
0.600074 + 0.799944i \(0.295138\pi\)
\(854\) 5.23991 0.179306
\(855\) 2.20521 0.0754164
\(856\) 117.627 4.02042
\(857\) 51.8553 1.77134 0.885672 0.464311i \(-0.153698\pi\)
0.885672 + 0.464311i \(0.153698\pi\)
\(858\) −0.484484 −0.0165400
\(859\) −31.2016 −1.06458 −0.532292 0.846561i \(-0.678669\pi\)
−0.532292 + 0.846561i \(0.678669\pi\)
\(860\) −15.0324 −0.512601
\(861\) 17.0870 0.582322
\(862\) −11.0079 −0.374929
\(863\) −21.2847 −0.724539 −0.362269 0.932073i \(-0.617998\pi\)
−0.362269 + 0.932073i \(0.617998\pi\)
\(864\) −22.6525 −0.770654
\(865\) 7.88174 0.267987
\(866\) 52.8468 1.79581
\(867\) 9.48008 0.321960
\(868\) −82.9723 −2.81626
\(869\) −12.9780 −0.440249
\(870\) 12.5178 0.424392
\(871\) −0.973159 −0.0329742
\(872\) −13.2830 −0.449819
\(873\) −12.7524 −0.431603
\(874\) 16.1743 0.547104
\(875\) 11.1434 0.376715
\(876\) 9.66079 0.326408
\(877\) 56.4130 1.90493 0.952465 0.304647i \(-0.0985386\pi\)
0.952465 + 0.304647i \(0.0985386\pi\)
\(878\) −63.3519 −2.13802
\(879\) 12.3742 0.417371
\(880\) 9.23590 0.311342
\(881\) −13.2697 −0.447067 −0.223534 0.974696i \(-0.571759\pi\)
−0.223534 + 0.974696i \(0.571759\pi\)
\(882\) −9.15006 −0.308099
\(883\) −30.1064 −1.01316 −0.506581 0.862192i \(-0.669091\pi\)
−0.506581 + 0.862192i \(0.669091\pi\)
\(884\) 2.66908 0.0897707
\(885\) −0.808724 −0.0271850
\(886\) −50.2955 −1.68971
\(887\) 7.10163 0.238449 0.119225 0.992867i \(-0.461959\pi\)
0.119225 + 0.992867i \(0.461959\pi\)
\(888\) −80.9461 −2.71637
\(889\) −14.2065 −0.476471
\(890\) −6.80768 −0.228194
\(891\) 1.00000 0.0335013
\(892\) 86.5360 2.89744
\(893\) −9.06638 −0.303395
\(894\) 39.8102 1.33145
\(895\) −7.05032 −0.235666
\(896\) −78.3734 −2.61827
\(897\) −0.286358 −0.00956122
\(898\) −33.6137 −1.12170
\(899\) −59.5756 −1.98696
\(900\) −25.5026 −0.850087
\(901\) −20.7678 −0.691875
\(902\) 24.4662 0.814636
\(903\) −8.64317 −0.287627
\(904\) 121.733 4.04877
\(905\) −5.79363 −0.192587
\(906\) 29.9742 0.995826
\(907\) 24.7879 0.823069 0.411535 0.911394i \(-0.364993\pi\)
0.411535 + 0.911394i \(0.364993\pi\)
\(908\) −3.72857 −0.123737
\(909\) −10.0353 −0.332851
\(910\) −0.560364 −0.0185759
\(911\) 41.3254 1.36917 0.684586 0.728932i \(-0.259983\pi\)
0.684586 + 0.728932i \(0.259983\pi\)
\(912\) −55.7146 −1.84490
\(913\) 12.9300 0.427920
\(914\) 49.2853 1.63021
\(915\) 0.604616 0.0199880
\(916\) 74.0304 2.44603
\(917\) 14.7296 0.486414
\(918\) −7.51138 −0.247912
\(919\) 36.7229 1.21138 0.605688 0.795702i \(-0.292898\pi\)
0.605688 + 0.795702i \(0.292898\pi\)
\(920\) 9.39197 0.309644
\(921\) −21.2440 −0.700014
\(922\) −3.46453 −0.114098
\(923\) 0.0584521 0.00192397
\(924\) 10.5269 0.346308
\(925\) −39.0984 −1.28555
\(926\) −28.4518 −0.934984
\(927\) −6.05237 −0.198786
\(928\) −171.218 −5.62052
\(929\) 1.56310 0.0512837 0.0256418 0.999671i \(-0.491837\pi\)
0.0256418 + 0.999671i \(0.491837\pi\)
\(930\) −13.0535 −0.428041
\(931\) −12.1837 −0.399306
\(932\) −66.5983 −2.18150
\(933\) −26.4545 −0.866081
\(934\) 30.0701 0.983923
\(935\) 1.65801 0.0542226
\(936\) 1.69708 0.0554706
\(937\) 3.58638 0.117162 0.0585810 0.998283i \(-0.481342\pi\)
0.0585810 + 0.998283i \(0.481342\pi\)
\(938\) 28.8298 0.941326
\(939\) −30.7427 −1.00325
\(940\) −8.27049 −0.269754
\(941\) −41.9408 −1.36723 −0.683616 0.729842i \(-0.739593\pi\)
−0.683616 + 0.729842i \(0.739593\pi\)
\(942\) −45.6056 −1.48591
\(943\) 14.4609 0.470913
\(944\) 20.4325 0.665020
\(945\) 1.15662 0.0376248
\(946\) −12.3759 −0.402374
\(947\) 53.9167 1.75206 0.876028 0.482260i \(-0.160184\pi\)
0.876028 + 0.482260i \(0.160184\pi\)
\(948\) 71.4160 2.31948
\(949\) −0.310521 −0.0100799
\(950\) −46.2999 −1.50217
\(951\) 21.9554 0.711953
\(952\) −50.3329 −1.63130
\(953\) 6.24884 0.202420 0.101210 0.994865i \(-0.467729\pi\)
0.101210 + 0.994865i \(0.467729\pi\)
\(954\) −20.7442 −0.671618
\(955\) −11.1371 −0.360387
\(956\) −40.3987 −1.30659
\(957\) 7.55848 0.244331
\(958\) −48.8786 −1.57920
\(959\) 30.4566 0.983496
\(960\) −19.0435 −0.614627
\(961\) 31.1253 1.00404
\(962\) 4.08734 0.131781
\(963\) 12.2595 0.395058
\(964\) −148.424 −4.78043
\(965\) −3.27014 −0.105269
\(966\) 8.48334 0.272947
\(967\) −11.7133 −0.376674 −0.188337 0.982104i \(-0.560310\pi\)
−0.188337 + 0.982104i \(0.560310\pi\)
\(968\) 9.59477 0.308388
\(969\) −10.0018 −0.321303
\(970\) −21.1195 −0.678107
\(971\) −20.8971 −0.670620 −0.335310 0.942108i \(-0.608841\pi\)
−0.335310 + 0.942108i \(0.608841\pi\)
\(972\) −5.50285 −0.176504
\(973\) 4.60189 0.147530
\(974\) −53.9512 −1.72871
\(975\) 0.819717 0.0262519
\(976\) −15.2757 −0.488962
\(977\) 15.7320 0.503311 0.251655 0.967817i \(-0.419025\pi\)
0.251655 + 0.967817i \(0.419025\pi\)
\(978\) 5.64197 0.180410
\(979\) −4.11061 −0.131376
\(980\) −11.1142 −0.355030
\(981\) −1.38440 −0.0442005
\(982\) −47.5925 −1.51874
\(983\) −40.6563 −1.29674 −0.648368 0.761327i \(-0.724548\pi\)
−0.648368 + 0.761327i \(0.724548\pi\)
\(984\) −85.7015 −2.73206
\(985\) −1.73291 −0.0552152
\(986\) −56.7746 −1.80807
\(987\) −4.75527 −0.151362
\(988\) 3.54996 0.112939
\(989\) −7.31484 −0.232598
\(990\) 1.65612 0.0526350
\(991\) 13.9691 0.443742 0.221871 0.975076i \(-0.428784\pi\)
0.221871 + 0.975076i \(0.428784\pi\)
\(992\) 178.546 5.66885
\(993\) 15.8159 0.501901
\(994\) −1.73164 −0.0549243
\(995\) 7.22820 0.229149
\(996\) −71.1517 −2.25453
\(997\) −11.2639 −0.356731 −0.178365 0.983964i \(-0.557081\pi\)
−0.178365 + 0.983964i \(0.557081\pi\)
\(998\) 68.8978 2.18092
\(999\) −8.43648 −0.266919
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 2013.2.a.e.1.13 13
3.2 odd 2 6039.2.a.i.1.1 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.13 13 1.1 even 1 trivial
6039.2.a.i.1.1 13 3.2 odd 2