Properties

Label 1339.2.a.e.1.5
Level $1339$
Weight $2$
Character 1339.1
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $21$
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] = [1339,2,Mod(1,1339)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1339, 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("1339.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.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: \(10.6919688306\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 1339.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.91337 q^{2} -1.36048 q^{3} +1.66097 q^{4} +1.75251 q^{5} +2.60311 q^{6} -4.42851 q^{7} +0.648686 q^{8} -1.14908 q^{9} +O(q^{10})\) \(q-1.91337 q^{2} -1.36048 q^{3} +1.66097 q^{4} +1.75251 q^{5} +2.60311 q^{6} -4.42851 q^{7} +0.648686 q^{8} -1.14908 q^{9} -3.35319 q^{10} +4.10059 q^{11} -2.25973 q^{12} -1.00000 q^{13} +8.47337 q^{14} -2.38426 q^{15} -4.56312 q^{16} +4.23372 q^{17} +2.19861 q^{18} -4.59317 q^{19} +2.91087 q^{20} +6.02492 q^{21} -7.84592 q^{22} +8.60196 q^{23} -0.882528 q^{24} -1.92871 q^{25} +1.91337 q^{26} +5.64476 q^{27} -7.35563 q^{28} -8.80134 q^{29} +4.56197 q^{30} +1.18531 q^{31} +7.43354 q^{32} -5.57879 q^{33} -8.10066 q^{34} -7.76101 q^{35} -1.90859 q^{36} -3.94318 q^{37} +8.78843 q^{38} +1.36048 q^{39} +1.13683 q^{40} +4.92646 q^{41} -11.5279 q^{42} +1.43278 q^{43} +6.81096 q^{44} -2.01378 q^{45} -16.4587 q^{46} +11.9193 q^{47} +6.20805 q^{48} +12.6117 q^{49} +3.69033 q^{50} -5.75991 q^{51} -1.66097 q^{52} -11.6576 q^{53} -10.8005 q^{54} +7.18632 q^{55} -2.87271 q^{56} +6.24894 q^{57} +16.8402 q^{58} -3.82464 q^{59} -3.96019 q^{60} +4.08144 q^{61} -2.26793 q^{62} +5.08872 q^{63} -5.09686 q^{64} -1.75251 q^{65} +10.6743 q^{66} -7.44591 q^{67} +7.03209 q^{68} -11.7028 q^{69} +14.8497 q^{70} -4.65470 q^{71} -0.745393 q^{72} +11.0015 q^{73} +7.54475 q^{74} +2.62398 q^{75} -7.62913 q^{76} -18.1595 q^{77} -2.60311 q^{78} -0.880429 q^{79} -7.99691 q^{80} -4.23237 q^{81} -9.42612 q^{82} +4.57453 q^{83} +10.0072 q^{84} +7.41964 q^{85} -2.74143 q^{86} +11.9741 q^{87} +2.65999 q^{88} -9.03714 q^{89} +3.85309 q^{90} +4.42851 q^{91} +14.2876 q^{92} -1.61259 q^{93} -22.8060 q^{94} -8.04959 q^{95} -10.1132 q^{96} -15.8216 q^{97} -24.1308 q^{98} -4.71191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91337 −1.35295 −0.676477 0.736464i \(-0.736494\pi\)
−0.676477 + 0.736464i \(0.736494\pi\)
\(3\) −1.36048 −0.785476 −0.392738 0.919650i \(-0.628472\pi\)
−0.392738 + 0.919650i \(0.628472\pi\)
\(4\) 1.66097 0.830486
\(5\) 1.75251 0.783746 0.391873 0.920019i \(-0.371827\pi\)
0.391873 + 0.920019i \(0.371827\pi\)
\(6\) 2.60311 1.06271
\(7\) −4.42851 −1.67382 −0.836910 0.547341i \(-0.815640\pi\)
−0.836910 + 0.547341i \(0.815640\pi\)
\(8\) 0.648686 0.229345
\(9\) −1.14908 −0.383027
\(10\) −3.35319 −1.06037
\(11\) 4.10059 1.23637 0.618187 0.786031i \(-0.287868\pi\)
0.618187 + 0.786031i \(0.287868\pi\)
\(12\) −2.25973 −0.652327
\(13\) −1.00000 −0.277350
\(14\) 8.47337 2.26460
\(15\) −2.38426 −0.615614
\(16\) −4.56312 −1.14078
\(17\) 4.23372 1.02683 0.513414 0.858141i \(-0.328381\pi\)
0.513414 + 0.858141i \(0.328381\pi\)
\(18\) 2.19861 0.518218
\(19\) −4.59317 −1.05375 −0.526873 0.849944i \(-0.676636\pi\)
−0.526873 + 0.849944i \(0.676636\pi\)
\(20\) 2.91087 0.650890
\(21\) 6.02492 1.31475
\(22\) −7.84592 −1.67276
\(23\) 8.60196 1.79363 0.896816 0.442404i \(-0.145874\pi\)
0.896816 + 0.442404i \(0.145874\pi\)
\(24\) −0.882528 −0.180145
\(25\) −1.92871 −0.385742
\(26\) 1.91337 0.375242
\(27\) 5.64476 1.08633
\(28\) −7.35563 −1.39008
\(29\) −8.80134 −1.63437 −0.817184 0.576377i \(-0.804466\pi\)
−0.817184 + 0.576377i \(0.804466\pi\)
\(30\) 4.56197 0.832898
\(31\) 1.18531 0.212888 0.106444 0.994319i \(-0.466054\pi\)
0.106444 + 0.994319i \(0.466054\pi\)
\(32\) 7.43354 1.31408
\(33\) −5.57879 −0.971142
\(34\) −8.10066 −1.38925
\(35\) −7.76101 −1.31185
\(36\) −1.90859 −0.318098
\(37\) −3.94318 −0.648255 −0.324128 0.946013i \(-0.605071\pi\)
−0.324128 + 0.946013i \(0.605071\pi\)
\(38\) 8.78843 1.42567
\(39\) 1.36048 0.217852
\(40\) 1.13683 0.179748
\(41\) 4.92646 0.769383 0.384692 0.923045i \(-0.374308\pi\)
0.384692 + 0.923045i \(0.374308\pi\)
\(42\) −11.5279 −1.77879
\(43\) 1.43278 0.218496 0.109248 0.994015i \(-0.465156\pi\)
0.109248 + 0.994015i \(0.465156\pi\)
\(44\) 6.81096 1.02679
\(45\) −2.01378 −0.300196
\(46\) −16.4587 −2.42670
\(47\) 11.9193 1.73861 0.869303 0.494279i \(-0.164568\pi\)
0.869303 + 0.494279i \(0.164568\pi\)
\(48\) 6.20805 0.896055
\(49\) 12.6117 1.80167
\(50\) 3.69033 0.521891
\(51\) −5.75991 −0.806549
\(52\) −1.66097 −0.230335
\(53\) −11.6576 −1.60130 −0.800650 0.599133i \(-0.795512\pi\)
−0.800650 + 0.599133i \(0.795512\pi\)
\(54\) −10.8005 −1.46976
\(55\) 7.18632 0.969003
\(56\) −2.87271 −0.383883
\(57\) 6.24894 0.827693
\(58\) 16.8402 2.21122
\(59\) −3.82464 −0.497925 −0.248963 0.968513i \(-0.580090\pi\)
−0.248963 + 0.968513i \(0.580090\pi\)
\(60\) −3.96019 −0.511259
\(61\) 4.08144 0.522575 0.261288 0.965261i \(-0.415853\pi\)
0.261288 + 0.965261i \(0.415853\pi\)
\(62\) −2.26793 −0.288027
\(63\) 5.08872 0.641118
\(64\) −5.09686 −0.637107
\(65\) −1.75251 −0.217372
\(66\) 10.6743 1.31391
\(67\) −7.44591 −0.909662 −0.454831 0.890578i \(-0.650300\pi\)
−0.454831 + 0.890578i \(0.650300\pi\)
\(68\) 7.03209 0.852766
\(69\) −11.7028 −1.40886
\(70\) 14.8497 1.77487
\(71\) −4.65470 −0.552411 −0.276206 0.961099i \(-0.589077\pi\)
−0.276206 + 0.961099i \(0.589077\pi\)
\(72\) −0.745393 −0.0878454
\(73\) 11.0015 1.28763 0.643814 0.765182i \(-0.277351\pi\)
0.643814 + 0.765182i \(0.277351\pi\)
\(74\) 7.54475 0.877060
\(75\) 2.62398 0.302991
\(76\) −7.62913 −0.875121
\(77\) −18.1595 −2.06947
\(78\) −2.60311 −0.294744
\(79\) −0.880429 −0.0990560 −0.0495280 0.998773i \(-0.515772\pi\)
−0.0495280 + 0.998773i \(0.515772\pi\)
\(80\) −7.99691 −0.894082
\(81\) −4.23237 −0.470263
\(82\) −9.42612 −1.04094
\(83\) 4.57453 0.502120 0.251060 0.967971i \(-0.419221\pi\)
0.251060 + 0.967971i \(0.419221\pi\)
\(84\) 10.0072 1.09188
\(85\) 7.41964 0.804773
\(86\) −2.74143 −0.295616
\(87\) 11.9741 1.28376
\(88\) 2.65999 0.283556
\(89\) −9.03714 −0.957935 −0.478968 0.877833i \(-0.658989\pi\)
−0.478968 + 0.877833i \(0.658989\pi\)
\(90\) 3.85309 0.406151
\(91\) 4.42851 0.464234
\(92\) 14.2876 1.48959
\(93\) −1.61259 −0.167218
\(94\) −22.8060 −2.35226
\(95\) −8.04959 −0.825870
\(96\) −10.1132 −1.03218
\(97\) −15.8216 −1.60644 −0.803221 0.595681i \(-0.796882\pi\)
−0.803221 + 0.595681i \(0.796882\pi\)
\(98\) −24.1308 −2.43758
\(99\) −4.71191 −0.473564
\(100\) −3.20353 −0.320353
\(101\) −1.82151 −0.181247 −0.0906235 0.995885i \(-0.528886\pi\)
−0.0906235 + 0.995885i \(0.528886\pi\)
\(102\) 11.0208 1.09122
\(103\) −1.00000 −0.0985329
\(104\) −0.648686 −0.0636089
\(105\) 10.5587 1.03043
\(106\) 22.3053 2.16648
\(107\) 15.4023 1.48900 0.744500 0.667623i \(-0.232688\pi\)
0.744500 + 0.667623i \(0.232688\pi\)
\(108\) 9.37579 0.902186
\(109\) −19.3765 −1.85593 −0.927967 0.372663i \(-0.878445\pi\)
−0.927967 + 0.372663i \(0.878445\pi\)
\(110\) −13.7501 −1.31102
\(111\) 5.36464 0.509189
\(112\) 20.2078 1.90946
\(113\) −16.7309 −1.57391 −0.786955 0.617011i \(-0.788343\pi\)
−0.786955 + 0.617011i \(0.788343\pi\)
\(114\) −11.9565 −1.11983
\(115\) 15.0750 1.40575
\(116\) −14.6188 −1.35732
\(117\) 1.14908 0.106233
\(118\) 7.31793 0.673670
\(119\) −18.7491 −1.71873
\(120\) −1.54664 −0.141188
\(121\) 5.81481 0.528619
\(122\) −7.80930 −0.707021
\(123\) −6.70237 −0.604333
\(124\) 1.96876 0.176800
\(125\) −12.1426 −1.08607
\(126\) −9.73658 −0.867404
\(127\) −16.6389 −1.47647 −0.738233 0.674546i \(-0.764340\pi\)
−0.738233 + 0.674546i \(0.764340\pi\)
\(128\) −5.11493 −0.452100
\(129\) −1.94927 −0.171624
\(130\) 3.35319 0.294095
\(131\) −5.86853 −0.512736 −0.256368 0.966579i \(-0.582526\pi\)
−0.256368 + 0.966579i \(0.582526\pi\)
\(132\) −9.26620 −0.806519
\(133\) 20.3409 1.76378
\(134\) 14.2467 1.23073
\(135\) 9.89250 0.851411
\(136\) 2.74636 0.235498
\(137\) 1.90075 0.162392 0.0811961 0.996698i \(-0.474126\pi\)
0.0811961 + 0.996698i \(0.474126\pi\)
\(138\) 22.3918 1.90612
\(139\) −12.8661 −1.09129 −0.545646 0.838016i \(-0.683716\pi\)
−0.545646 + 0.838016i \(0.683716\pi\)
\(140\) −12.8908 −1.08947
\(141\) −16.2160 −1.36563
\(142\) 8.90615 0.747387
\(143\) −4.10059 −0.342908
\(144\) 5.24339 0.436949
\(145\) −15.4244 −1.28093
\(146\) −21.0499 −1.74210
\(147\) −17.1580 −1.41517
\(148\) −6.54951 −0.538367
\(149\) −13.0161 −1.06632 −0.533160 0.846014i \(-0.678996\pi\)
−0.533160 + 0.846014i \(0.678996\pi\)
\(150\) −5.02063 −0.409933
\(151\) −16.2129 −1.31939 −0.659693 0.751535i \(-0.729314\pi\)
−0.659693 + 0.751535i \(0.729314\pi\)
\(152\) −2.97953 −0.241672
\(153\) −4.86489 −0.393303
\(154\) 34.7458 2.79989
\(155\) 2.07726 0.166850
\(156\) 2.25973 0.180923
\(157\) −12.3536 −0.985922 −0.492961 0.870051i \(-0.664085\pi\)
−0.492961 + 0.870051i \(0.664085\pi\)
\(158\) 1.68458 0.134018
\(159\) 15.8600 1.25778
\(160\) 13.0274 1.02990
\(161\) −38.0939 −3.00222
\(162\) 8.09808 0.636245
\(163\) −20.1710 −1.57992 −0.789959 0.613160i \(-0.789898\pi\)
−0.789959 + 0.613160i \(0.789898\pi\)
\(164\) 8.18271 0.638962
\(165\) −9.77688 −0.761129
\(166\) −8.75276 −0.679346
\(167\) 19.2395 1.48880 0.744399 0.667735i \(-0.232736\pi\)
0.744399 + 0.667735i \(0.232736\pi\)
\(168\) 3.90828 0.301531
\(169\) 1.00000 0.0769231
\(170\) −14.1965 −1.08882
\(171\) 5.27793 0.403613
\(172\) 2.37980 0.181458
\(173\) 11.0717 0.841762 0.420881 0.907116i \(-0.361721\pi\)
0.420881 + 0.907116i \(0.361721\pi\)
\(174\) −22.9108 −1.73686
\(175\) 8.54131 0.645662
\(176\) −18.7115 −1.41043
\(177\) 5.20336 0.391109
\(178\) 17.2914 1.29604
\(179\) −17.4771 −1.30630 −0.653149 0.757230i \(-0.726552\pi\)
−0.653149 + 0.757230i \(0.726552\pi\)
\(180\) −3.34482 −0.249308
\(181\) −9.36583 −0.696157 −0.348078 0.937465i \(-0.613166\pi\)
−0.348078 + 0.937465i \(0.613166\pi\)
\(182\) −8.47337 −0.628088
\(183\) −5.55274 −0.410470
\(184\) 5.57997 0.411361
\(185\) −6.91047 −0.508068
\(186\) 3.08548 0.226239
\(187\) 17.3607 1.26954
\(188\) 19.7976 1.44389
\(189\) −24.9979 −1.81833
\(190\) 15.4018 1.11736
\(191\) 0.108131 0.00782409 0.00391205 0.999992i \(-0.498755\pi\)
0.00391205 + 0.999992i \(0.498755\pi\)
\(192\) 6.93420 0.500433
\(193\) −4.51939 −0.325312 −0.162656 0.986683i \(-0.552006\pi\)
−0.162656 + 0.986683i \(0.552006\pi\)
\(194\) 30.2725 2.17344
\(195\) 2.38426 0.170741
\(196\) 20.9477 1.49626
\(197\) 22.0431 1.57051 0.785253 0.619175i \(-0.212533\pi\)
0.785253 + 0.619175i \(0.212533\pi\)
\(198\) 9.01560 0.640711
\(199\) −4.69354 −0.332716 −0.166358 0.986065i \(-0.553201\pi\)
−0.166358 + 0.986065i \(0.553201\pi\)
\(200\) −1.25113 −0.0884680
\(201\) 10.1300 0.714518
\(202\) 3.48521 0.245219
\(203\) 38.9768 2.73564
\(204\) −9.56705 −0.669827
\(205\) 8.63367 0.603002
\(206\) 1.91337 0.133311
\(207\) −9.88434 −0.687009
\(208\) 4.56312 0.316395
\(209\) −18.8347 −1.30282
\(210\) −20.2027 −1.39412
\(211\) −2.32294 −0.159917 −0.0799587 0.996798i \(-0.525479\pi\)
−0.0799587 + 0.996798i \(0.525479\pi\)
\(212\) −19.3630 −1.32986
\(213\) 6.33265 0.433906
\(214\) −29.4703 −2.01455
\(215\) 2.51096 0.171246
\(216\) 3.66168 0.249146
\(217\) −5.24915 −0.356336
\(218\) 37.0744 2.51099
\(219\) −14.9674 −1.01140
\(220\) 11.9363 0.804743
\(221\) −4.23372 −0.284791
\(222\) −10.2645 −0.688910
\(223\) 12.2759 0.822055 0.411028 0.911623i \(-0.365170\pi\)
0.411028 + 0.911623i \(0.365170\pi\)
\(224\) −32.9195 −2.19953
\(225\) 2.21624 0.147749
\(226\) 32.0123 2.12943
\(227\) 16.1847 1.07422 0.537109 0.843513i \(-0.319516\pi\)
0.537109 + 0.843513i \(0.319516\pi\)
\(228\) 10.3793 0.687387
\(229\) −9.28080 −0.613293 −0.306646 0.951824i \(-0.599207\pi\)
−0.306646 + 0.951824i \(0.599207\pi\)
\(230\) −28.8440 −1.90192
\(231\) 24.7057 1.62552
\(232\) −5.70931 −0.374834
\(233\) −3.34048 −0.218842 −0.109421 0.993995i \(-0.534900\pi\)
−0.109421 + 0.993995i \(0.534900\pi\)
\(234\) −2.19861 −0.143728
\(235\) 20.8887 1.36263
\(236\) −6.35261 −0.413520
\(237\) 1.19781 0.0778062
\(238\) 35.8739 2.32536
\(239\) 27.3070 1.76634 0.883171 0.469052i \(-0.155404\pi\)
0.883171 + 0.469052i \(0.155404\pi\)
\(240\) 10.8797 0.702280
\(241\) 12.3424 0.795042 0.397521 0.917593i \(-0.369871\pi\)
0.397521 + 0.917593i \(0.369871\pi\)
\(242\) −11.1259 −0.715197
\(243\) −11.1762 −0.716954
\(244\) 6.77916 0.433991
\(245\) 22.1022 1.41205
\(246\) 12.8241 0.817634
\(247\) 4.59317 0.292257
\(248\) 0.768893 0.0488248
\(249\) −6.22358 −0.394404
\(250\) 23.2333 1.46940
\(251\) 11.2079 0.707435 0.353718 0.935352i \(-0.384917\pi\)
0.353718 + 0.935352i \(0.384917\pi\)
\(252\) 8.45221 0.532439
\(253\) 35.2731 2.21760
\(254\) 31.8364 1.99759
\(255\) −10.0943 −0.632130
\(256\) 19.9804 1.24878
\(257\) −9.06258 −0.565308 −0.282654 0.959222i \(-0.591215\pi\)
−0.282654 + 0.959222i \(0.591215\pi\)
\(258\) 3.72967 0.232199
\(259\) 17.4624 1.08506
\(260\) −2.91087 −0.180524
\(261\) 10.1134 0.626007
\(262\) 11.2286 0.693708
\(263\) −14.4739 −0.892496 −0.446248 0.894909i \(-0.647240\pi\)
−0.446248 + 0.894909i \(0.647240\pi\)
\(264\) −3.61888 −0.222727
\(265\) −20.4301 −1.25501
\(266\) −38.9196 −2.38632
\(267\) 12.2949 0.752435
\(268\) −12.3674 −0.755462
\(269\) −30.3196 −1.84862 −0.924309 0.381644i \(-0.875358\pi\)
−0.924309 + 0.381644i \(0.875358\pi\)
\(270\) −18.9280 −1.15192
\(271\) 17.6430 1.07174 0.535869 0.844301i \(-0.319984\pi\)
0.535869 + 0.844301i \(0.319984\pi\)
\(272\) −19.3190 −1.17138
\(273\) −6.02492 −0.364645
\(274\) −3.63684 −0.219709
\(275\) −7.90883 −0.476921
\(276\) −19.4381 −1.17003
\(277\) −9.25458 −0.556054 −0.278027 0.960573i \(-0.589680\pi\)
−0.278027 + 0.960573i \(0.589680\pi\)
\(278\) 24.6176 1.47647
\(279\) −1.36201 −0.0815417
\(280\) −5.03446 −0.300867
\(281\) 29.7435 1.77435 0.887173 0.461437i \(-0.152666\pi\)
0.887173 + 0.461437i \(0.152666\pi\)
\(282\) 31.0272 1.84764
\(283\) −10.0231 −0.595814 −0.297907 0.954595i \(-0.596289\pi\)
−0.297907 + 0.954595i \(0.596289\pi\)
\(284\) −7.73132 −0.458770
\(285\) 10.9513 0.648701
\(286\) 7.84592 0.463939
\(287\) −21.8169 −1.28781
\(288\) −8.54174 −0.503327
\(289\) 0.924390 0.0543759
\(290\) 29.5126 1.73304
\(291\) 21.5251 1.26182
\(292\) 18.2732 1.06936
\(293\) −9.31126 −0.543969 −0.271985 0.962302i \(-0.587680\pi\)
−0.271985 + 0.962302i \(0.587680\pi\)
\(294\) 32.8296 1.91466
\(295\) −6.70272 −0.390247
\(296\) −2.55789 −0.148674
\(297\) 23.1468 1.34312
\(298\) 24.9046 1.44268
\(299\) −8.60196 −0.497464
\(300\) 4.35835 0.251630
\(301\) −6.34507 −0.365724
\(302\) 31.0212 1.78507
\(303\) 2.47814 0.142365
\(304\) 20.9592 1.20209
\(305\) 7.15277 0.409566
\(306\) 9.30831 0.532121
\(307\) 5.60259 0.319757 0.159878 0.987137i \(-0.448890\pi\)
0.159878 + 0.987137i \(0.448890\pi\)
\(308\) −30.1624 −1.71866
\(309\) 1.36048 0.0773953
\(310\) −3.97457 −0.225740
\(311\) −27.1646 −1.54037 −0.770183 0.637823i \(-0.779835\pi\)
−0.770183 + 0.637823i \(0.779835\pi\)
\(312\) 0.882528 0.0499633
\(313\) −5.96045 −0.336904 −0.168452 0.985710i \(-0.553877\pi\)
−0.168452 + 0.985710i \(0.553877\pi\)
\(314\) 23.6369 1.33391
\(315\) 8.91803 0.502474
\(316\) −1.46237 −0.0822646
\(317\) 1.59706 0.0896996 0.0448498 0.998994i \(-0.485719\pi\)
0.0448498 + 0.998994i \(0.485719\pi\)
\(318\) −30.3461 −1.70172
\(319\) −36.0906 −2.02069
\(320\) −8.93229 −0.499330
\(321\) −20.9546 −1.16957
\(322\) 72.8875 4.06186
\(323\) −19.4462 −1.08202
\(324\) −7.02985 −0.390547
\(325\) 1.92871 0.106985
\(326\) 38.5946 2.13756
\(327\) 26.3615 1.45779
\(328\) 3.19573 0.176454
\(329\) −52.7847 −2.91011
\(330\) 18.7068 1.02977
\(331\) 8.74225 0.480518 0.240259 0.970709i \(-0.422768\pi\)
0.240259 + 0.970709i \(0.422768\pi\)
\(332\) 7.59817 0.417004
\(333\) 4.53104 0.248299
\(334\) −36.8122 −2.01428
\(335\) −13.0490 −0.712945
\(336\) −27.4924 −1.49983
\(337\) 13.7834 0.750827 0.375414 0.926857i \(-0.377501\pi\)
0.375414 + 0.926857i \(0.377501\pi\)
\(338\) −1.91337 −0.104073
\(339\) 22.7621 1.23627
\(340\) 12.3238 0.668352
\(341\) 4.86046 0.263209
\(342\) −10.0986 −0.546070
\(343\) −24.8515 −1.34186
\(344\) 0.929422 0.0501111
\(345\) −20.5093 −1.10419
\(346\) −21.1841 −1.13887
\(347\) −8.86570 −0.475936 −0.237968 0.971273i \(-0.576481\pi\)
−0.237968 + 0.971273i \(0.576481\pi\)
\(348\) 19.8886 1.06614
\(349\) −6.76495 −0.362119 −0.181060 0.983472i \(-0.557953\pi\)
−0.181060 + 0.983472i \(0.557953\pi\)
\(350\) −16.3426 −0.873551
\(351\) −5.64476 −0.301295
\(352\) 30.4819 1.62469
\(353\) 14.1417 0.752686 0.376343 0.926480i \(-0.377181\pi\)
0.376343 + 0.926480i \(0.377181\pi\)
\(354\) −9.95594 −0.529152
\(355\) −8.15741 −0.432950
\(356\) −15.0104 −0.795551
\(357\) 25.5078 1.35002
\(358\) 33.4400 1.76736
\(359\) −4.26278 −0.224981 −0.112490 0.993653i \(-0.535883\pi\)
−0.112490 + 0.993653i \(0.535883\pi\)
\(360\) −1.30631 −0.0688485
\(361\) 2.09725 0.110382
\(362\) 17.9203 0.941868
\(363\) −7.91096 −0.415218
\(364\) 7.35563 0.385540
\(365\) 19.2802 1.00917
\(366\) 10.6244 0.555348
\(367\) −34.4687 −1.79925 −0.899625 0.436663i \(-0.856160\pi\)
−0.899625 + 0.436663i \(0.856160\pi\)
\(368\) −39.2517 −2.04614
\(369\) −5.66090 −0.294695
\(370\) 13.2223 0.687392
\(371\) 51.6260 2.68029
\(372\) −2.67847 −0.138872
\(373\) 18.3277 0.948971 0.474485 0.880263i \(-0.342634\pi\)
0.474485 + 0.880263i \(0.342634\pi\)
\(374\) −33.2175 −1.71763
\(375\) 16.5199 0.853082
\(376\) 7.73187 0.398741
\(377\) 8.80134 0.453292
\(378\) 47.8301 2.46012
\(379\) 1.01429 0.0521007 0.0260503 0.999661i \(-0.491707\pi\)
0.0260503 + 0.999661i \(0.491707\pi\)
\(380\) −13.3701 −0.685873
\(381\) 22.6370 1.15973
\(382\) −0.206894 −0.0105856
\(383\) 1.97117 0.100722 0.0503611 0.998731i \(-0.483963\pi\)
0.0503611 + 0.998731i \(0.483963\pi\)
\(384\) 6.95878 0.355114
\(385\) −31.8247 −1.62194
\(386\) 8.64724 0.440133
\(387\) −1.64638 −0.0836900
\(388\) −26.2792 −1.33413
\(389\) 31.7247 1.60851 0.804254 0.594285i \(-0.202565\pi\)
0.804254 + 0.594285i \(0.202565\pi\)
\(390\) −4.56197 −0.231004
\(391\) 36.4183 1.84175
\(392\) 8.18104 0.413205
\(393\) 7.98405 0.402742
\(394\) −42.1765 −2.12482
\(395\) −1.54296 −0.0776348
\(396\) −7.82634 −0.393288
\(397\) −12.8448 −0.644660 −0.322330 0.946627i \(-0.604466\pi\)
−0.322330 + 0.946627i \(0.604466\pi\)
\(398\) 8.98046 0.450150
\(399\) −27.6735 −1.38541
\(400\) 8.80092 0.440046
\(401\) −17.4827 −0.873046 −0.436523 0.899693i \(-0.643790\pi\)
−0.436523 + 0.899693i \(0.643790\pi\)
\(402\) −19.3825 −0.966711
\(403\) −1.18531 −0.0590444
\(404\) −3.02547 −0.150523
\(405\) −7.41727 −0.368567
\(406\) −74.5770 −3.70119
\(407\) −16.1694 −0.801486
\(408\) −3.73638 −0.184978
\(409\) −1.50696 −0.0745143 −0.0372572 0.999306i \(-0.511862\pi\)
−0.0372572 + 0.999306i \(0.511862\pi\)
\(410\) −16.5194 −0.815834
\(411\) −2.58595 −0.127555
\(412\) −1.66097 −0.0818302
\(413\) 16.9375 0.833438
\(414\) 18.9124 0.929492
\(415\) 8.01692 0.393535
\(416\) −7.43354 −0.364459
\(417\) 17.5042 0.857184
\(418\) 36.0377 1.76266
\(419\) 24.2698 1.18566 0.592828 0.805329i \(-0.298011\pi\)
0.592828 + 0.805329i \(0.298011\pi\)
\(420\) 17.5378 0.855755
\(421\) −7.62923 −0.371826 −0.185913 0.982566i \(-0.559524\pi\)
−0.185913 + 0.982566i \(0.559524\pi\)
\(422\) 4.44463 0.216361
\(423\) −13.6962 −0.665933
\(424\) −7.56215 −0.367250
\(425\) −8.16561 −0.396090
\(426\) −12.1167 −0.587055
\(427\) −18.0747 −0.874697
\(428\) 25.5828 1.23659
\(429\) 5.57879 0.269346
\(430\) −4.80438 −0.231688
\(431\) 8.53329 0.411034 0.205517 0.978654i \(-0.434112\pi\)
0.205517 + 0.978654i \(0.434112\pi\)
\(432\) −25.7577 −1.23927
\(433\) −9.60849 −0.461755 −0.230877 0.972983i \(-0.574160\pi\)
−0.230877 + 0.972983i \(0.574160\pi\)
\(434\) 10.0435 0.482106
\(435\) 20.9847 1.00614
\(436\) −32.1838 −1.54133
\(437\) −39.5103 −1.89003
\(438\) 28.6381 1.36838
\(439\) −26.7439 −1.27642 −0.638209 0.769863i \(-0.720325\pi\)
−0.638209 + 0.769863i \(0.720325\pi\)
\(440\) 4.66167 0.222236
\(441\) −14.4919 −0.690089
\(442\) 8.10066 0.385309
\(443\) 1.70561 0.0810360 0.0405180 0.999179i \(-0.487099\pi\)
0.0405180 + 0.999179i \(0.487099\pi\)
\(444\) 8.91051 0.422874
\(445\) −15.8377 −0.750778
\(446\) −23.4883 −1.11220
\(447\) 17.7082 0.837569
\(448\) 22.5715 1.06640
\(449\) 37.1179 1.75170 0.875852 0.482580i \(-0.160300\pi\)
0.875852 + 0.482580i \(0.160300\pi\)
\(450\) −4.24048 −0.199898
\(451\) 20.2014 0.951245
\(452\) −27.7895 −1.30711
\(453\) 22.0574 1.03635
\(454\) −30.9673 −1.45337
\(455\) 7.76101 0.363842
\(456\) 4.05360 0.189827
\(457\) −21.0564 −0.984979 −0.492489 0.870318i \(-0.663913\pi\)
−0.492489 + 0.870318i \(0.663913\pi\)
\(458\) 17.7576 0.829757
\(459\) 23.8983 1.11548
\(460\) 25.0392 1.16746
\(461\) −19.5646 −0.911215 −0.455608 0.890181i \(-0.650578\pi\)
−0.455608 + 0.890181i \(0.650578\pi\)
\(462\) −47.2711 −2.19925
\(463\) 5.91566 0.274924 0.137462 0.990507i \(-0.456105\pi\)
0.137462 + 0.990507i \(0.456105\pi\)
\(464\) 40.1615 1.86445
\(465\) −2.82609 −0.131057
\(466\) 6.39157 0.296084
\(467\) −3.26969 −0.151303 −0.0756517 0.997134i \(-0.524104\pi\)
−0.0756517 + 0.997134i \(0.524104\pi\)
\(468\) 1.90859 0.0882246
\(469\) 32.9743 1.52261
\(470\) −39.9677 −1.84357
\(471\) 16.8068 0.774418
\(472\) −2.48099 −0.114197
\(473\) 5.87522 0.270143
\(474\) −2.29185 −0.105268
\(475\) 8.85889 0.406474
\(476\) −31.1417 −1.42738
\(477\) 13.3956 0.613341
\(478\) −52.2482 −2.38978
\(479\) −25.3878 −1.16000 −0.579999 0.814617i \(-0.696947\pi\)
−0.579999 + 0.814617i \(0.696947\pi\)
\(480\) −17.7235 −0.808965
\(481\) 3.94318 0.179794
\(482\) −23.6155 −1.07565
\(483\) 51.8261 2.35817
\(484\) 9.65823 0.439010
\(485\) −27.7275 −1.25904
\(486\) 21.3842 0.970006
\(487\) 24.0845 1.09137 0.545686 0.837990i \(-0.316269\pi\)
0.545686 + 0.837990i \(0.316269\pi\)
\(488\) 2.64758 0.119850
\(489\) 27.4424 1.24099
\(490\) −42.2895 −1.91045
\(491\) 9.37042 0.422881 0.211441 0.977391i \(-0.432185\pi\)
0.211441 + 0.977391i \(0.432185\pi\)
\(492\) −11.1324 −0.501889
\(493\) −37.2624 −1.67821
\(494\) −8.78843 −0.395410
\(495\) −8.25766 −0.371154
\(496\) −5.40870 −0.242858
\(497\) 20.6134 0.924637
\(498\) 11.9080 0.533610
\(499\) 26.2119 1.17340 0.586702 0.809803i \(-0.300426\pi\)
0.586702 + 0.809803i \(0.300426\pi\)
\(500\) −20.1686 −0.901966
\(501\) −26.1751 −1.16942
\(502\) −21.4448 −0.957128
\(503\) −20.2973 −0.905012 −0.452506 0.891761i \(-0.649470\pi\)
−0.452506 + 0.891761i \(0.649470\pi\)
\(504\) 3.30098 0.147037
\(505\) −3.19221 −0.142052
\(506\) −67.4903 −3.00031
\(507\) −1.36048 −0.0604213
\(508\) −27.6368 −1.22618
\(509\) −4.98671 −0.221032 −0.110516 0.993874i \(-0.535250\pi\)
−0.110516 + 0.993874i \(0.535250\pi\)
\(510\) 19.3141 0.855243
\(511\) −48.7202 −2.15526
\(512\) −28.0001 −1.23744
\(513\) −25.9274 −1.14472
\(514\) 17.3400 0.764836
\(515\) −1.75251 −0.0772248
\(516\) −3.23768 −0.142531
\(517\) 48.8761 2.14957
\(518\) −33.4120 −1.46804
\(519\) −15.0628 −0.661184
\(520\) −1.13683 −0.0498533
\(521\) −20.8423 −0.913119 −0.456559 0.889693i \(-0.650918\pi\)
−0.456559 + 0.889693i \(0.650918\pi\)
\(522\) −19.3507 −0.846959
\(523\) −2.57919 −0.112780 −0.0563901 0.998409i \(-0.517959\pi\)
−0.0563901 + 0.998409i \(0.517959\pi\)
\(524\) −9.74746 −0.425820
\(525\) −11.6203 −0.507152
\(526\) 27.6938 1.20751
\(527\) 5.01826 0.218599
\(528\) 25.4567 1.10786
\(529\) 50.9936 2.21711
\(530\) 39.0903 1.69797
\(531\) 4.39482 0.190719
\(532\) 33.7857 1.46480
\(533\) −4.92646 −0.213389
\(534\) −23.5246 −1.01801
\(535\) 26.9927 1.16700
\(536\) −4.83006 −0.208627
\(537\) 23.7773 1.02607
\(538\) 58.0125 2.50110
\(539\) 51.7154 2.22754
\(540\) 16.4312 0.707085
\(541\) −16.4233 −0.706094 −0.353047 0.935606i \(-0.614854\pi\)
−0.353047 + 0.935606i \(0.614854\pi\)
\(542\) −33.7576 −1.45001
\(543\) 12.7421 0.546815
\(544\) 31.4715 1.34933
\(545\) −33.9575 −1.45458
\(546\) 11.5279 0.493348
\(547\) 33.0540 1.41329 0.706644 0.707569i \(-0.250208\pi\)
0.706644 + 0.707569i \(0.250208\pi\)
\(548\) 3.15710 0.134864
\(549\) −4.68991 −0.200160
\(550\) 15.1325 0.645252
\(551\) 40.4261 1.72221
\(552\) −7.59146 −0.323114
\(553\) 3.89899 0.165802
\(554\) 17.7074 0.752316
\(555\) 9.40159 0.399075
\(556\) −21.3703 −0.906302
\(557\) 3.64160 0.154299 0.0771497 0.997020i \(-0.475418\pi\)
0.0771497 + 0.997020i \(0.475418\pi\)
\(558\) 2.60603 0.110322
\(559\) −1.43278 −0.0606000
\(560\) 35.4144 1.49653
\(561\) −23.6190 −0.997196
\(562\) −56.9101 −2.40061
\(563\) 42.1325 1.77567 0.887836 0.460160i \(-0.152208\pi\)
0.887836 + 0.460160i \(0.152208\pi\)
\(564\) −26.9343 −1.13414
\(565\) −29.3210 −1.23355
\(566\) 19.1779 0.806109
\(567\) 18.7431 0.787136
\(568\) −3.01944 −0.126693
\(569\) −18.1326 −0.760156 −0.380078 0.924954i \(-0.624103\pi\)
−0.380078 + 0.924954i \(0.624103\pi\)
\(570\) −20.9539 −0.877663
\(571\) −23.9285 −1.00137 −0.500687 0.865628i \(-0.666919\pi\)
−0.500687 + 0.865628i \(0.666919\pi\)
\(572\) −6.81096 −0.284780
\(573\) −0.147111 −0.00614564
\(574\) 41.7437 1.74235
\(575\) −16.5907 −0.691878
\(576\) 5.85670 0.244029
\(577\) 43.0921 1.79395 0.896974 0.442083i \(-0.145760\pi\)
0.896974 + 0.442083i \(0.145760\pi\)
\(578\) −1.76870 −0.0735681
\(579\) 6.14856 0.255525
\(580\) −25.6195 −1.06379
\(581\) −20.2584 −0.840459
\(582\) −41.1853 −1.70719
\(583\) −47.8031 −1.97980
\(584\) 7.13652 0.295311
\(585\) 2.01378 0.0832594
\(586\) 17.8158 0.735966
\(587\) −30.4398 −1.25638 −0.628192 0.778059i \(-0.716205\pi\)
−0.628192 + 0.778059i \(0.716205\pi\)
\(588\) −28.4990 −1.17528
\(589\) −5.44433 −0.224330
\(590\) 12.8248 0.527987
\(591\) −29.9893 −1.23359
\(592\) 17.9932 0.739516
\(593\) −31.8410 −1.30755 −0.653776 0.756688i \(-0.726816\pi\)
−0.653776 + 0.756688i \(0.726816\pi\)
\(594\) −44.2884 −1.81717
\(595\) −32.8580 −1.34704
\(596\) −21.6194 −0.885564
\(597\) 6.38549 0.261341
\(598\) 16.4587 0.673046
\(599\) 5.84207 0.238701 0.119350 0.992852i \(-0.461919\pi\)
0.119350 + 0.992852i \(0.461919\pi\)
\(600\) 1.70214 0.0694895
\(601\) 13.2298 0.539656 0.269828 0.962909i \(-0.413033\pi\)
0.269828 + 0.962909i \(0.413033\pi\)
\(602\) 12.1404 0.494807
\(603\) 8.55595 0.348425
\(604\) −26.9291 −1.09573
\(605\) 10.1905 0.414303
\(606\) −4.74158 −0.192614
\(607\) −14.6694 −0.595411 −0.297705 0.954658i \(-0.596221\pi\)
−0.297705 + 0.954658i \(0.596221\pi\)
\(608\) −34.1436 −1.38470
\(609\) −53.0274 −2.14878
\(610\) −13.6859 −0.554125
\(611\) −11.9193 −0.482203
\(612\) −8.08044 −0.326632
\(613\) 7.11346 0.287310 0.143655 0.989628i \(-0.454114\pi\)
0.143655 + 0.989628i \(0.454114\pi\)
\(614\) −10.7198 −0.432616
\(615\) −11.7460 −0.473643
\(616\) −11.7798 −0.474622
\(617\) 32.5965 1.31229 0.656144 0.754636i \(-0.272187\pi\)
0.656144 + 0.754636i \(0.272187\pi\)
\(618\) −2.60311 −0.104712
\(619\) 11.8659 0.476932 0.238466 0.971151i \(-0.423355\pi\)
0.238466 + 0.971151i \(0.423355\pi\)
\(620\) 3.45028 0.138566
\(621\) 48.5560 1.94848
\(622\) 51.9759 2.08404
\(623\) 40.0211 1.60341
\(624\) −6.20805 −0.248521
\(625\) −11.6365 −0.465462
\(626\) 11.4045 0.455816
\(627\) 25.6243 1.02334
\(628\) −20.5189 −0.818794
\(629\) −16.6943 −0.665647
\(630\) −17.0635 −0.679824
\(631\) 30.7064 1.22240 0.611201 0.791475i \(-0.290687\pi\)
0.611201 + 0.791475i \(0.290687\pi\)
\(632\) −0.571122 −0.0227180
\(633\) 3.16032 0.125611
\(634\) −3.05575 −0.121359
\(635\) −29.1599 −1.15718
\(636\) 26.3431 1.04457
\(637\) −12.6117 −0.499694
\(638\) 69.0546 2.73390
\(639\) 5.34863 0.211588
\(640\) −8.96397 −0.354332
\(641\) −33.7150 −1.33166 −0.665832 0.746102i \(-0.731923\pi\)
−0.665832 + 0.746102i \(0.731923\pi\)
\(642\) 40.0939 1.58238
\(643\) −37.2333 −1.46834 −0.734168 0.678967i \(-0.762428\pi\)
−0.734168 + 0.678967i \(0.762428\pi\)
\(644\) −63.2728 −2.49330
\(645\) −3.41612 −0.134509
\(646\) 37.2077 1.46392
\(647\) 7.72314 0.303628 0.151814 0.988409i \(-0.451489\pi\)
0.151814 + 0.988409i \(0.451489\pi\)
\(648\) −2.74548 −0.107853
\(649\) −15.6833 −0.615622
\(650\) −3.69033 −0.144746
\(651\) 7.14139 0.279893
\(652\) −33.5035 −1.31210
\(653\) 6.60611 0.258517 0.129258 0.991611i \(-0.458740\pi\)
0.129258 + 0.991611i \(0.458740\pi\)
\(654\) −50.4391 −1.97233
\(655\) −10.2847 −0.401855
\(656\) −22.4800 −0.877697
\(657\) −12.6416 −0.493196
\(658\) 100.996 3.93725
\(659\) −21.0554 −0.820203 −0.410101 0.912040i \(-0.634507\pi\)
−0.410101 + 0.912040i \(0.634507\pi\)
\(660\) −16.2391 −0.632107
\(661\) −7.81316 −0.303897 −0.151948 0.988388i \(-0.548555\pi\)
−0.151948 + 0.988388i \(0.548555\pi\)
\(662\) −16.7271 −0.650118
\(663\) 5.75991 0.223696
\(664\) 2.96744 0.115159
\(665\) 35.6477 1.38236
\(666\) −8.66953 −0.335938
\(667\) −75.7087 −2.93145
\(668\) 31.9563 1.23643
\(669\) −16.7012 −0.645705
\(670\) 24.9676 0.964582
\(671\) 16.7363 0.646098
\(672\) 44.7865 1.72768
\(673\) −24.0183 −0.925836 −0.462918 0.886401i \(-0.653198\pi\)
−0.462918 + 0.886401i \(0.653198\pi\)
\(674\) −26.3726 −1.01583
\(675\) −10.8871 −0.419045
\(676\) 1.66097 0.0638835
\(677\) 20.9976 0.807005 0.403502 0.914979i \(-0.367793\pi\)
0.403502 + 0.914979i \(0.367793\pi\)
\(678\) −43.5523 −1.67262
\(679\) 70.0662 2.68889
\(680\) 4.81302 0.184571
\(681\) −22.0191 −0.843773
\(682\) −9.29984 −0.356109
\(683\) −32.3714 −1.23866 −0.619328 0.785132i \(-0.712595\pi\)
−0.619328 + 0.785132i \(0.712595\pi\)
\(684\) 8.76649 0.335195
\(685\) 3.33109 0.127274
\(686\) 47.5501 1.81547
\(687\) 12.6264 0.481727
\(688\) −6.53793 −0.249256
\(689\) 11.6576 0.444120
\(690\) 39.2419 1.49391
\(691\) −14.5331 −0.552864 −0.276432 0.961034i \(-0.589152\pi\)
−0.276432 + 0.961034i \(0.589152\pi\)
\(692\) 18.3897 0.699071
\(693\) 20.8667 0.792661
\(694\) 16.9633 0.643920
\(695\) −22.5480 −0.855296
\(696\) 7.76743 0.294424
\(697\) 20.8572 0.790025
\(698\) 12.9438 0.489931
\(699\) 4.54468 0.171895
\(700\) 14.1869 0.536213
\(701\) −24.5456 −0.927076 −0.463538 0.886077i \(-0.653420\pi\)
−0.463538 + 0.886077i \(0.653420\pi\)
\(702\) 10.8005 0.407639
\(703\) 18.1117 0.683097
\(704\) −20.9001 −0.787702
\(705\) −28.4187 −1.07031
\(706\) −27.0582 −1.01835
\(707\) 8.06657 0.303375
\(708\) 8.64263 0.324810
\(709\) 24.3324 0.913823 0.456911 0.889512i \(-0.348956\pi\)
0.456911 + 0.889512i \(0.348956\pi\)
\(710\) 15.6081 0.585762
\(711\) 1.01168 0.0379411
\(712\) −5.86227 −0.219698
\(713\) 10.1960 0.381842
\(714\) −48.8058 −1.82651
\(715\) −7.18632 −0.268753
\(716\) −29.0289 −1.08486
\(717\) −37.1507 −1.38742
\(718\) 8.15625 0.304389
\(719\) −34.2855 −1.27863 −0.639317 0.768944i \(-0.720783\pi\)
−0.639317 + 0.768944i \(0.720783\pi\)
\(720\) 9.18909 0.342457
\(721\) 4.42851 0.164926
\(722\) −4.01281 −0.149341
\(723\) −16.7916 −0.624486
\(724\) −15.5564 −0.578148
\(725\) 16.9752 0.630444
\(726\) 15.1366 0.561770
\(727\) −20.3597 −0.755098 −0.377549 0.925990i \(-0.623233\pi\)
−0.377549 + 0.925990i \(0.623233\pi\)
\(728\) 2.87271 0.106470
\(729\) 27.9022 1.03341
\(730\) −36.8901 −1.36537
\(731\) 6.06598 0.224358
\(732\) −9.22294 −0.340890
\(733\) −35.2393 −1.30159 −0.650797 0.759252i \(-0.725565\pi\)
−0.650797 + 0.759252i \(0.725565\pi\)
\(734\) 65.9512 2.43430
\(735\) −30.0696 −1.10914
\(736\) 63.9430 2.35697
\(737\) −30.5326 −1.12468
\(738\) 10.8314 0.398708
\(739\) 36.0577 1.32640 0.663202 0.748440i \(-0.269197\pi\)
0.663202 + 0.748440i \(0.269197\pi\)
\(740\) −11.4781 −0.421943
\(741\) −6.24894 −0.229561
\(742\) −98.7794 −3.62631
\(743\) −0.729588 −0.0267660 −0.0133830 0.999910i \(-0.504260\pi\)
−0.0133830 + 0.999910i \(0.504260\pi\)
\(744\) −1.04607 −0.0383507
\(745\) −22.8108 −0.835725
\(746\) −35.0675 −1.28391
\(747\) −5.25651 −0.192326
\(748\) 28.8357 1.05434
\(749\) −68.2094 −2.49232
\(750\) −31.6086 −1.15418
\(751\) 17.2845 0.630721 0.315360 0.948972i \(-0.397875\pi\)
0.315360 + 0.948972i \(0.397875\pi\)
\(752\) −54.3891 −1.98337
\(753\) −15.2482 −0.555674
\(754\) −16.8402 −0.613283
\(755\) −28.4132 −1.03406
\(756\) −41.5208 −1.51010
\(757\) −34.3659 −1.24905 −0.624525 0.781005i \(-0.714707\pi\)
−0.624525 + 0.781005i \(0.714707\pi\)
\(758\) −1.94071 −0.0704899
\(759\) −47.9885 −1.74187
\(760\) −5.22165 −0.189409
\(761\) 25.9783 0.941713 0.470857 0.882210i \(-0.343945\pi\)
0.470857 + 0.882210i \(0.343945\pi\)
\(762\) −43.3129 −1.56906
\(763\) 85.8091 3.10650
\(764\) 0.179603 0.00649780
\(765\) −8.52576 −0.308250
\(766\) −3.77158 −0.136273
\(767\) 3.82464 0.138100
\(768\) −27.1831 −0.980886
\(769\) 28.7289 1.03599 0.517995 0.855384i \(-0.326678\pi\)
0.517995 + 0.855384i \(0.326678\pi\)
\(770\) 60.8923 2.19441
\(771\) 12.3295 0.444036
\(772\) −7.50657 −0.270167
\(773\) 45.9916 1.65420 0.827101 0.562053i \(-0.189988\pi\)
0.827101 + 0.562053i \(0.189988\pi\)
\(774\) 3.15012 0.113229
\(775\) −2.28611 −0.0821196
\(776\) −10.2633 −0.368430
\(777\) −23.7574 −0.852291
\(778\) −60.7011 −2.17624
\(779\) −22.6281 −0.810735
\(780\) 3.96019 0.141798
\(781\) −19.0870 −0.682986
\(782\) −69.6815 −2.49181
\(783\) −49.6815 −1.77547
\(784\) −57.5487 −2.05531
\(785\) −21.6498 −0.772713
\(786\) −15.2764 −0.544891
\(787\) −4.71401 −0.168036 −0.0840181 0.996464i \(-0.526775\pi\)
−0.0840181 + 0.996464i \(0.526775\pi\)
\(788\) 36.6129 1.30428
\(789\) 19.6915 0.701035
\(790\) 2.95225 0.105036
\(791\) 74.0929 2.63444
\(792\) −3.05655 −0.108610
\(793\) −4.08144 −0.144936
\(794\) 24.5767 0.872196
\(795\) 27.7949 0.985783
\(796\) −7.79583 −0.276316
\(797\) 9.87387 0.349750 0.174875 0.984591i \(-0.444048\pi\)
0.174875 + 0.984591i \(0.444048\pi\)
\(798\) 52.9496 1.87440
\(799\) 50.4629 1.78525
\(800\) −14.3371 −0.506894
\(801\) 10.3844 0.366915
\(802\) 33.4509 1.18119
\(803\) 45.1126 1.59199
\(804\) 16.8257 0.593397
\(805\) −66.7599 −2.35298
\(806\) 2.26793 0.0798844
\(807\) 41.2494 1.45205
\(808\) −1.18159 −0.0415681
\(809\) 15.8466 0.557136 0.278568 0.960416i \(-0.410140\pi\)
0.278568 + 0.960416i \(0.410140\pi\)
\(810\) 14.1920 0.498655
\(811\) −9.84059 −0.345550 −0.172775 0.984961i \(-0.555273\pi\)
−0.172775 + 0.984961i \(0.555273\pi\)
\(812\) 64.7394 2.27191
\(813\) −24.0031 −0.841825
\(814\) 30.9379 1.08437
\(815\) −35.3500 −1.23825
\(816\) 26.2832 0.920095
\(817\) −6.58099 −0.230240
\(818\) 2.88336 0.100814
\(819\) −5.08872 −0.177814
\(820\) 14.3403 0.500784
\(821\) 7.60600 0.265451 0.132726 0.991153i \(-0.457627\pi\)
0.132726 + 0.991153i \(0.457627\pi\)
\(822\) 4.94786 0.172576
\(823\) 4.22365 0.147227 0.0736137 0.997287i \(-0.476547\pi\)
0.0736137 + 0.997287i \(0.476547\pi\)
\(824\) −0.648686 −0.0225981
\(825\) 10.7598 0.374610
\(826\) −32.4076 −1.12760
\(827\) 42.1624 1.46613 0.733066 0.680158i \(-0.238089\pi\)
0.733066 + 0.680158i \(0.238089\pi\)
\(828\) −16.4176 −0.570551
\(829\) −40.8157 −1.41759 −0.708794 0.705415i \(-0.750760\pi\)
−0.708794 + 0.705415i \(0.750760\pi\)
\(830\) −15.3393 −0.532435
\(831\) 12.5907 0.436767
\(832\) 5.09686 0.176702
\(833\) 53.3945 1.85001
\(834\) −33.4919 −1.15973
\(835\) 33.7174 1.16684
\(836\) −31.2839 −1.08198
\(837\) 6.69078 0.231267
\(838\) −46.4370 −1.60414
\(839\) 0.237518 0.00820003 0.00410001 0.999992i \(-0.498695\pi\)
0.00410001 + 0.999992i \(0.498695\pi\)
\(840\) 6.84931 0.236324
\(841\) 48.4636 1.67116
\(842\) 14.5975 0.503064
\(843\) −40.4655 −1.39371
\(844\) −3.85833 −0.132809
\(845\) 1.75251 0.0602882
\(846\) 26.2059 0.900977
\(847\) −25.7509 −0.884813
\(848\) 53.1951 1.82673
\(849\) 13.6363 0.467998
\(850\) 15.6238 0.535892
\(851\) −33.9191 −1.16273
\(852\) 10.5183 0.360353
\(853\) 11.9474 0.409071 0.204536 0.978859i \(-0.434432\pi\)
0.204536 + 0.978859i \(0.434432\pi\)
\(854\) 34.5836 1.18343
\(855\) 9.24962 0.316330
\(856\) 9.99128 0.341495
\(857\) −25.8760 −0.883907 −0.441953 0.897038i \(-0.645714\pi\)
−0.441953 + 0.897038i \(0.645714\pi\)
\(858\) −10.6743 −0.364413
\(859\) −48.0576 −1.63970 −0.819852 0.572576i \(-0.805944\pi\)
−0.819852 + 0.572576i \(0.805944\pi\)
\(860\) 4.17062 0.142217
\(861\) 29.6815 1.01154
\(862\) −16.3273 −0.556111
\(863\) 5.64627 0.192201 0.0961007 0.995372i \(-0.469363\pi\)
0.0961007 + 0.995372i \(0.469363\pi\)
\(864\) 41.9606 1.42753
\(865\) 19.4032 0.659728
\(866\) 18.3846 0.624733
\(867\) −1.25762 −0.0427110
\(868\) −8.71869 −0.295932
\(869\) −3.61028 −0.122470
\(870\) −40.1514 −1.36126
\(871\) 7.44591 0.252295
\(872\) −12.5693 −0.425650
\(873\) 18.1803 0.615310
\(874\) 75.5976 2.55713
\(875\) 53.7738 1.81789
\(876\) −24.8604 −0.839954
\(877\) 6.54117 0.220880 0.110440 0.993883i \(-0.464774\pi\)
0.110440 + 0.993883i \(0.464774\pi\)
\(878\) 51.1710 1.72694
\(879\) 12.6678 0.427275
\(880\) −32.7920 −1.10542
\(881\) −20.5409 −0.692041 −0.346021 0.938227i \(-0.612467\pi\)
−0.346021 + 0.938227i \(0.612467\pi\)
\(882\) 27.7283 0.933660
\(883\) −12.3263 −0.414812 −0.207406 0.978255i \(-0.566502\pi\)
−0.207406 + 0.978255i \(0.566502\pi\)
\(884\) −7.03209 −0.236515
\(885\) 9.11894 0.306530
\(886\) −3.26346 −0.109638
\(887\) −21.0878 −0.708060 −0.354030 0.935234i \(-0.615189\pi\)
−0.354030 + 0.935234i \(0.615189\pi\)
\(888\) 3.47997 0.116780
\(889\) 73.6857 2.47134
\(890\) 30.3033 1.01577
\(891\) −17.3552 −0.581421
\(892\) 20.3899 0.682705
\(893\) −54.7474 −1.83205
\(894\) −33.8823 −1.13319
\(895\) −30.6287 −1.02381
\(896\) 22.6515 0.756734
\(897\) 11.7028 0.390746
\(898\) −71.0202 −2.36997
\(899\) −10.4323 −0.347937
\(900\) 3.68111 0.122704
\(901\) −49.3552 −1.64426
\(902\) −38.6526 −1.28699
\(903\) 8.63237 0.287267
\(904\) −10.8531 −0.360969
\(905\) −16.4137 −0.545610
\(906\) −42.2039 −1.40213
\(907\) −40.9367 −1.35928 −0.679641 0.733545i \(-0.737864\pi\)
−0.679641 + 0.733545i \(0.737864\pi\)
\(908\) 26.8824 0.892122
\(909\) 2.09306 0.0694224
\(910\) −14.8497 −0.492261
\(911\) 50.5723 1.67553 0.837767 0.546027i \(-0.183861\pi\)
0.837767 + 0.546027i \(0.183861\pi\)
\(912\) −28.5147 −0.944215
\(913\) 18.7583 0.620808
\(914\) 40.2887 1.33263
\(915\) −9.73124 −0.321705
\(916\) −15.4151 −0.509331
\(917\) 25.9888 0.858227
\(918\) −45.7263 −1.50919
\(919\) 9.13127 0.301213 0.150606 0.988594i \(-0.451877\pi\)
0.150606 + 0.988594i \(0.451877\pi\)
\(920\) 9.77895 0.322403
\(921\) −7.62224 −0.251161
\(922\) 37.4343 1.23283
\(923\) 4.65470 0.153211
\(924\) 41.0355 1.34997
\(925\) 7.60525 0.250059
\(926\) −11.3188 −0.371960
\(927\) 1.14908 0.0377408
\(928\) −65.4251 −2.14768
\(929\) 55.6422 1.82556 0.912781 0.408449i \(-0.133930\pi\)
0.912781 + 0.408449i \(0.133930\pi\)
\(930\) 5.40734 0.177314
\(931\) −57.9278 −1.89851
\(932\) −5.54845 −0.181745
\(933\) 36.9571 1.20992
\(934\) 6.25612 0.204706
\(935\) 30.4249 0.995000
\(936\) 0.745393 0.0243639
\(937\) 7.12355 0.232716 0.116358 0.993207i \(-0.462878\pi\)
0.116358 + 0.993207i \(0.462878\pi\)
\(938\) −63.0919 −2.06002
\(939\) 8.10910 0.264630
\(940\) 34.6955 1.13164
\(941\) −20.4044 −0.665163 −0.332582 0.943074i \(-0.607920\pi\)
−0.332582 + 0.943074i \(0.607920\pi\)
\(942\) −32.1576 −1.04775
\(943\) 42.3772 1.37999
\(944\) 17.4523 0.568023
\(945\) −43.8091 −1.42511
\(946\) −11.2415 −0.365491
\(947\) −7.29179 −0.236951 −0.118476 0.992957i \(-0.537801\pi\)
−0.118476 + 0.992957i \(0.537801\pi\)
\(948\) 1.98953 0.0646169
\(949\) −11.0015 −0.357124
\(950\) −16.9503 −0.549941
\(951\) −2.17277 −0.0704569
\(952\) −12.1623 −0.394181
\(953\) 37.3433 1.20967 0.604834 0.796351i \(-0.293239\pi\)
0.604834 + 0.796351i \(0.293239\pi\)
\(954\) −25.6306 −0.829822
\(955\) 0.189501 0.00613211
\(956\) 45.3561 1.46692
\(957\) 49.1008 1.58720
\(958\) 48.5762 1.56942
\(959\) −8.41750 −0.271815
\(960\) 12.1523 0.392212
\(961\) −29.5950 −0.954679
\(962\) −7.54475 −0.243253
\(963\) −17.6985 −0.570327
\(964\) 20.5003 0.660271
\(965\) −7.92027 −0.254962
\(966\) −99.1624 −3.19050
\(967\) 4.70780 0.151393 0.0756964 0.997131i \(-0.475882\pi\)
0.0756964 + 0.997131i \(0.475882\pi\)
\(968\) 3.77199 0.121236
\(969\) 26.4563 0.849898
\(970\) 53.0529 1.70343
\(971\) 6.92449 0.222217 0.111109 0.993808i \(-0.464560\pi\)
0.111109 + 0.993808i \(0.464560\pi\)
\(972\) −18.5634 −0.595420
\(973\) 56.9778 1.82663
\(974\) −46.0824 −1.47658
\(975\) −2.62398 −0.0840346
\(976\) −18.6241 −0.596143
\(977\) −13.2427 −0.423672 −0.211836 0.977305i \(-0.567944\pi\)
−0.211836 + 0.977305i \(0.567944\pi\)
\(978\) −52.5074 −1.67900
\(979\) −37.0576 −1.18437
\(980\) 36.7110 1.17269
\(981\) 22.2652 0.710873
\(982\) −17.9290 −0.572139
\(983\) −53.9544 −1.72088 −0.860440 0.509552i \(-0.829811\pi\)
−0.860440 + 0.509552i \(0.829811\pi\)
\(984\) −4.34774 −0.138601
\(985\) 38.6307 1.23088
\(986\) 71.2966 2.27055
\(987\) 71.8128 2.28583
\(988\) 7.62913 0.242715
\(989\) 12.3247 0.391902
\(990\) 15.7999 0.502155
\(991\) −28.9259 −0.918861 −0.459431 0.888214i \(-0.651947\pi\)
−0.459431 + 0.888214i \(0.651947\pi\)
\(992\) 8.81104 0.279751
\(993\) −11.8937 −0.377435
\(994\) −39.4410 −1.25099
\(995\) −8.22548 −0.260765
\(996\) −10.3372 −0.327546
\(997\) −21.8732 −0.692730 −0.346365 0.938100i \(-0.612584\pi\)
−0.346365 + 0.938100i \(0.612584\pi\)
\(998\) −50.1529 −1.58756
\(999\) −22.2583 −0.704222
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 1339.2.a.e.1.5 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1339.2.a.e.1.5 21 1.1 even 1 trivial