Properties

Label 731.2.a.e.1.5
Level $731$
Weight $2$
Character 731.1
Self dual yes
Analytic conductor $5.837$
Analytic rank $0$
Dimension $19$
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] = [731,2,Mod(1,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, 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("731.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 2 x^{18} - 30 x^{17} + 62 x^{16} + 365 x^{15} - 786 x^{14} - 2295 x^{13} + 5233 x^{12} + 7816 x^{11} - 19517 x^{10} - 13527 x^{9} + 40173 x^{8} + 8942 x^{7} - 41911 x^{6} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.86082\) of defining polynomial
Character \(\chi\) \(=\) 731.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.86082 q^{2} +2.21039 q^{3} +1.46266 q^{4} +1.15323 q^{5} -4.11315 q^{6} +2.59124 q^{7} +0.999890 q^{8} +1.88583 q^{9} +O(q^{10})\) \(q-1.86082 q^{2} +2.21039 q^{3} +1.46266 q^{4} +1.15323 q^{5} -4.11315 q^{6} +2.59124 q^{7} +0.999890 q^{8} +1.88583 q^{9} -2.14595 q^{10} +3.01488 q^{11} +3.23306 q^{12} +0.191815 q^{13} -4.82184 q^{14} +2.54908 q^{15} -4.78594 q^{16} +1.00000 q^{17} -3.50920 q^{18} +2.18357 q^{19} +1.68678 q^{20} +5.72765 q^{21} -5.61015 q^{22} +1.33614 q^{23} +2.21015 q^{24} -3.67007 q^{25} -0.356934 q^{26} -2.46275 q^{27} +3.79011 q^{28} -8.81546 q^{29} -4.74339 q^{30} +4.89012 q^{31} +6.90601 q^{32} +6.66406 q^{33} -1.86082 q^{34} +2.98829 q^{35} +2.75834 q^{36} +11.1412 q^{37} -4.06324 q^{38} +0.423986 q^{39} +1.15310 q^{40} -3.89827 q^{41} -10.6582 q^{42} -1.00000 q^{43} +4.40975 q^{44} +2.17479 q^{45} -2.48631 q^{46} -4.39530 q^{47} -10.5788 q^{48} -0.285478 q^{49} +6.82935 q^{50} +2.21039 q^{51} +0.280561 q^{52} -12.1032 q^{53} +4.58274 q^{54} +3.47684 q^{55} +2.59095 q^{56} +4.82654 q^{57} +16.4040 q^{58} +3.10264 q^{59} +3.72845 q^{60} -6.37651 q^{61} -9.09965 q^{62} +4.88664 q^{63} -3.27898 q^{64} +0.221206 q^{65} -12.4006 q^{66} +10.6005 q^{67} +1.46266 q^{68} +2.95338 q^{69} -5.56067 q^{70} +2.29264 q^{71} +1.88562 q^{72} +7.91542 q^{73} -20.7319 q^{74} -8.11229 q^{75} +3.19383 q^{76} +7.81227 q^{77} -0.788963 q^{78} +1.86796 q^{79} -5.51928 q^{80} -11.1011 q^{81} +7.25400 q^{82} +0.538979 q^{83} +8.37763 q^{84} +1.15323 q^{85} +1.86082 q^{86} -19.4856 q^{87} +3.01454 q^{88} -2.83673 q^{89} -4.04690 q^{90} +0.497038 q^{91} +1.95431 q^{92} +10.8091 q^{93} +8.17887 q^{94} +2.51815 q^{95} +15.2650 q^{96} +16.5114 q^{97} +0.531225 q^{98} +5.68555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9} - 2 q^{10} + 4 q^{11} + 9 q^{12} + 14 q^{13} + 5 q^{14} - 7 q^{15} + 32 q^{16} + 19 q^{17} + 12 q^{18} + 12 q^{19} + 23 q^{20} + 16 q^{21} + 36 q^{22} - q^{23} - 13 q^{24} + 30 q^{25} - 21 q^{26} + 8 q^{27} + 5 q^{28} + 41 q^{29} - 26 q^{30} - 8 q^{31} - 20 q^{32} - 14 q^{33} + 2 q^{34} + 3 q^{35} - 5 q^{36} + 50 q^{37} - 29 q^{38} + 17 q^{39} - 15 q^{40} + 6 q^{41} - q^{42} - 19 q^{43} + 16 q^{44} + 24 q^{45} + 38 q^{46} - 21 q^{47} - 2 q^{48} + 46 q^{49} - 36 q^{50} + 5 q^{51} + 39 q^{52} - 9 q^{53} + 53 q^{54} + 10 q^{55} - 12 q^{56} - 5 q^{57} - 45 q^{58} - 4 q^{59} - 7 q^{60} + 68 q^{61} - 25 q^{62} + 61 q^{63} - 14 q^{64} + 22 q^{65} - 17 q^{66} + 26 q^{68} - 9 q^{69} - 37 q^{70} + 23 q^{71} - 4 q^{72} - q^{73} - 30 q^{74} - 25 q^{75} + 47 q^{76} - 19 q^{77} + 12 q^{78} + 16 q^{79} + 28 q^{80} - 21 q^{81} - 13 q^{82} - 32 q^{83} - 47 q^{84} + 11 q^{85} - 2 q^{86} - 8 q^{87} + 108 q^{88} + 11 q^{89} + 5 q^{90} + 52 q^{91} - 23 q^{92} - 23 q^{93} + 47 q^{94} - 25 q^{95} - 103 q^{96} + 36 q^{97} - 100 q^{98} - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.86082 −1.31580 −0.657900 0.753105i \(-0.728555\pi\)
−0.657900 + 0.753105i \(0.728555\pi\)
\(3\) 2.21039 1.27617 0.638085 0.769966i \(-0.279727\pi\)
0.638085 + 0.769966i \(0.279727\pi\)
\(4\) 1.46266 0.731331
\(5\) 1.15323 0.515739 0.257869 0.966180i \(-0.416980\pi\)
0.257869 + 0.966180i \(0.416980\pi\)
\(6\) −4.11315 −1.67919
\(7\) 2.59124 0.979396 0.489698 0.871892i \(-0.337107\pi\)
0.489698 + 0.871892i \(0.337107\pi\)
\(8\) 0.999890 0.353514
\(9\) 1.88583 0.628611
\(10\) −2.14595 −0.678609
\(11\) 3.01488 0.909019 0.454510 0.890742i \(-0.349814\pi\)
0.454510 + 0.890742i \(0.349814\pi\)
\(12\) 3.23306 0.933303
\(13\) 0.191815 0.0531999 0.0265999 0.999646i \(-0.491532\pi\)
0.0265999 + 0.999646i \(0.491532\pi\)
\(14\) −4.82184 −1.28869
\(15\) 2.54908 0.658170
\(16\) −4.78594 −1.19649
\(17\) 1.00000 0.242536
\(18\) −3.50920 −0.827126
\(19\) 2.18357 0.500945 0.250473 0.968124i \(-0.419414\pi\)
0.250473 + 0.968124i \(0.419414\pi\)
\(20\) 1.68678 0.377176
\(21\) 5.72765 1.24988
\(22\) −5.61015 −1.19609
\(23\) 1.33614 0.278603 0.139302 0.990250i \(-0.455514\pi\)
0.139302 + 0.990250i \(0.455514\pi\)
\(24\) 2.21015 0.451145
\(25\) −3.67007 −0.734014
\(26\) −0.356934 −0.0700005
\(27\) −2.46275 −0.473956
\(28\) 3.79011 0.716263
\(29\) −8.81546 −1.63699 −0.818495 0.574513i \(-0.805191\pi\)
−0.818495 + 0.574513i \(0.805191\pi\)
\(30\) −4.74339 −0.866021
\(31\) 4.89012 0.878291 0.439146 0.898416i \(-0.355281\pi\)
0.439146 + 0.898416i \(0.355281\pi\)
\(32\) 6.90601 1.22082
\(33\) 6.66406 1.16006
\(34\) −1.86082 −0.319129
\(35\) 2.98829 0.505113
\(36\) 2.75834 0.459723
\(37\) 11.1412 1.83161 0.915804 0.401624i \(-0.131554\pi\)
0.915804 + 0.401624i \(0.131554\pi\)
\(38\) −4.06324 −0.659144
\(39\) 0.423986 0.0678921
\(40\) 1.15310 0.182321
\(41\) −3.89827 −0.608808 −0.304404 0.952543i \(-0.598457\pi\)
−0.304404 + 0.952543i \(0.598457\pi\)
\(42\) −10.6582 −1.64459
\(43\) −1.00000 −0.152499
\(44\) 4.40975 0.664794
\(45\) 2.17479 0.324199
\(46\) −2.48631 −0.366587
\(47\) −4.39530 −0.641120 −0.320560 0.947228i \(-0.603871\pi\)
−0.320560 + 0.947228i \(0.603871\pi\)
\(48\) −10.5788 −1.52692
\(49\) −0.285478 −0.0407826
\(50\) 6.82935 0.965816
\(51\) 2.21039 0.309517
\(52\) 0.280561 0.0389067
\(53\) −12.1032 −1.66251 −0.831254 0.555893i \(-0.812376\pi\)
−0.831254 + 0.555893i \(0.812376\pi\)
\(54\) 4.58274 0.623632
\(55\) 3.47684 0.468816
\(56\) 2.59095 0.346231
\(57\) 4.82654 0.639291
\(58\) 16.4040 2.15395
\(59\) 3.10264 0.403930 0.201965 0.979393i \(-0.435267\pi\)
0.201965 + 0.979393i \(0.435267\pi\)
\(60\) 3.72845 0.481341
\(61\) −6.37651 −0.816429 −0.408215 0.912886i \(-0.633848\pi\)
−0.408215 + 0.912886i \(0.633848\pi\)
\(62\) −9.09965 −1.15566
\(63\) 4.88664 0.615659
\(64\) −3.27898 −0.409873
\(65\) 0.221206 0.0274372
\(66\) −12.4006 −1.52641
\(67\) 10.6005 1.29505 0.647526 0.762043i \(-0.275804\pi\)
0.647526 + 0.762043i \(0.275804\pi\)
\(68\) 1.46266 0.177374
\(69\) 2.95338 0.355545
\(70\) −5.56067 −0.664627
\(71\) 2.29264 0.272086 0.136043 0.990703i \(-0.456561\pi\)
0.136043 + 0.990703i \(0.456561\pi\)
\(72\) 1.88562 0.222223
\(73\) 7.91542 0.926430 0.463215 0.886246i \(-0.346696\pi\)
0.463215 + 0.886246i \(0.346696\pi\)
\(74\) −20.7319 −2.41003
\(75\) −8.11229 −0.936726
\(76\) 3.19383 0.366357
\(77\) 7.81227 0.890290
\(78\) −0.788963 −0.0893325
\(79\) 1.86796 0.210162 0.105081 0.994464i \(-0.466490\pi\)
0.105081 + 0.994464i \(0.466490\pi\)
\(80\) −5.51928 −0.617074
\(81\) −11.1011 −1.23346
\(82\) 7.25400 0.801070
\(83\) 0.538979 0.0591607 0.0295803 0.999562i \(-0.490583\pi\)
0.0295803 + 0.999562i \(0.490583\pi\)
\(84\) 8.37763 0.914074
\(85\) 1.15323 0.125085
\(86\) 1.86082 0.200658
\(87\) −19.4856 −2.08908
\(88\) 3.01454 0.321352
\(89\) −2.83673 −0.300692 −0.150346 0.988633i \(-0.548039\pi\)
−0.150346 + 0.988633i \(0.548039\pi\)
\(90\) −4.04690 −0.426581
\(91\) 0.497038 0.0521038
\(92\) 1.95431 0.203751
\(93\) 10.8091 1.12085
\(94\) 8.17887 0.843586
\(95\) 2.51815 0.258357
\(96\) 15.2650 1.55798
\(97\) 16.5114 1.67648 0.838240 0.545301i \(-0.183585\pi\)
0.838240 + 0.545301i \(0.183585\pi\)
\(98\) 0.531225 0.0536618
\(99\) 5.68555 0.571419
\(100\) −5.36807 −0.536807
\(101\) 16.7864 1.67031 0.835154 0.550017i \(-0.185379\pi\)
0.835154 + 0.550017i \(0.185379\pi\)
\(102\) −4.11315 −0.407262
\(103\) 8.37286 0.825003 0.412501 0.910957i \(-0.364655\pi\)
0.412501 + 0.910957i \(0.364655\pi\)
\(104\) 0.191794 0.0188069
\(105\) 6.60528 0.644610
\(106\) 22.5220 2.18753
\(107\) 3.40454 0.329129 0.164565 0.986366i \(-0.447378\pi\)
0.164565 + 0.986366i \(0.447378\pi\)
\(108\) −3.60217 −0.346619
\(109\) −2.52873 −0.242208 −0.121104 0.992640i \(-0.538643\pi\)
−0.121104 + 0.992640i \(0.538643\pi\)
\(110\) −6.46978 −0.616869
\(111\) 24.6265 2.33744
\(112\) −12.4015 −1.17183
\(113\) −15.5311 −1.46104 −0.730520 0.682891i \(-0.760722\pi\)
−0.730520 + 0.682891i \(0.760722\pi\)
\(114\) −8.98134 −0.841180
\(115\) 1.54087 0.143687
\(116\) −12.8941 −1.19718
\(117\) 0.361731 0.0334420
\(118\) −5.77347 −0.531491
\(119\) 2.59124 0.237539
\(120\) 2.54880 0.232673
\(121\) −1.91052 −0.173684
\(122\) 11.8656 1.07426
\(123\) −8.61671 −0.776943
\(124\) 7.15259 0.642322
\(125\) −9.99855 −0.894298
\(126\) −9.09318 −0.810085
\(127\) 8.65498 0.768005 0.384003 0.923332i \(-0.374545\pi\)
0.384003 + 0.923332i \(0.374545\pi\)
\(128\) −7.71042 −0.681511
\(129\) −2.21039 −0.194614
\(130\) −0.411625 −0.0361019
\(131\) 11.4313 0.998760 0.499380 0.866383i \(-0.333561\pi\)
0.499380 + 0.866383i \(0.333561\pi\)
\(132\) 9.74727 0.848391
\(133\) 5.65815 0.490624
\(134\) −19.7256 −1.70403
\(135\) −2.84011 −0.244437
\(136\) 0.999890 0.0857398
\(137\) −11.7956 −1.00777 −0.503883 0.863772i \(-0.668096\pi\)
−0.503883 + 0.863772i \(0.668096\pi\)
\(138\) −5.49572 −0.467827
\(139\) −3.10983 −0.263772 −0.131886 0.991265i \(-0.542103\pi\)
−0.131886 + 0.991265i \(0.542103\pi\)
\(140\) 4.37085 0.369405
\(141\) −9.71533 −0.818179
\(142\) −4.26619 −0.358011
\(143\) 0.578298 0.0483597
\(144\) −9.02549 −0.752124
\(145\) −10.1662 −0.844259
\(146\) −14.7292 −1.21900
\(147\) −0.631019 −0.0520456
\(148\) 16.2959 1.33951
\(149\) −7.03103 −0.576004 −0.288002 0.957630i \(-0.592991\pi\)
−0.288002 + 0.957630i \(0.592991\pi\)
\(150\) 15.0955 1.23255
\(151\) −11.5008 −0.935920 −0.467960 0.883750i \(-0.655011\pi\)
−0.467960 + 0.883750i \(0.655011\pi\)
\(152\) 2.18333 0.177091
\(153\) 1.88583 0.152460
\(154\) −14.5372 −1.17144
\(155\) 5.63941 0.452969
\(156\) 0.620149 0.0496516
\(157\) 8.25034 0.658449 0.329224 0.944252i \(-0.393213\pi\)
0.329224 + 0.944252i \(0.393213\pi\)
\(158\) −3.47594 −0.276531
\(159\) −26.7529 −2.12164
\(160\) 7.96420 0.629625
\(161\) 3.46225 0.272863
\(162\) 20.6572 1.62299
\(163\) −17.3187 −1.35650 −0.678251 0.734830i \(-0.737262\pi\)
−0.678251 + 0.734830i \(0.737262\pi\)
\(164\) −5.70186 −0.445240
\(165\) 7.68517 0.598290
\(166\) −1.00295 −0.0778436
\(167\) −12.6615 −0.979776 −0.489888 0.871785i \(-0.662962\pi\)
−0.489888 + 0.871785i \(0.662962\pi\)
\(168\) 5.72702 0.441849
\(169\) −12.9632 −0.997170
\(170\) −2.14595 −0.164587
\(171\) 4.11785 0.314899
\(172\) −1.46266 −0.111527
\(173\) −19.4401 −1.47800 −0.739001 0.673705i \(-0.764702\pi\)
−0.739001 + 0.673705i \(0.764702\pi\)
\(174\) 36.2593 2.74881
\(175\) −9.51003 −0.718890
\(176\) −14.4290 −1.08763
\(177\) 6.85806 0.515483
\(178\) 5.27864 0.395651
\(179\) −25.0226 −1.87028 −0.935139 0.354282i \(-0.884725\pi\)
−0.935139 + 0.354282i \(0.884725\pi\)
\(180\) 3.18099 0.237097
\(181\) −17.5185 −1.30214 −0.651072 0.759016i \(-0.725680\pi\)
−0.651072 + 0.759016i \(0.725680\pi\)
\(182\) −0.924901 −0.0685582
\(183\) −14.0946 −1.04190
\(184\) 1.33599 0.0984903
\(185\) 12.8484 0.944631
\(186\) −20.1138 −1.47481
\(187\) 3.01488 0.220470
\(188\) −6.42884 −0.468871
\(189\) −6.38157 −0.464191
\(190\) −4.68583 −0.339946
\(191\) 21.9856 1.59082 0.795411 0.606071i \(-0.207255\pi\)
0.795411 + 0.606071i \(0.207255\pi\)
\(192\) −7.24784 −0.523068
\(193\) −0.291360 −0.0209726 −0.0104863 0.999945i \(-0.503338\pi\)
−0.0104863 + 0.999945i \(0.503338\pi\)
\(194\) −30.7248 −2.20591
\(195\) 0.488952 0.0350146
\(196\) −0.417559 −0.0298256
\(197\) 17.9480 1.27875 0.639373 0.768897i \(-0.279194\pi\)
0.639373 + 0.768897i \(0.279194\pi\)
\(198\) −10.5798 −0.751874
\(199\) −7.63630 −0.541323 −0.270662 0.962675i \(-0.587242\pi\)
−0.270662 + 0.962675i \(0.587242\pi\)
\(200\) −3.66966 −0.259484
\(201\) 23.4312 1.65271
\(202\) −31.2365 −2.19779
\(203\) −22.8430 −1.60326
\(204\) 3.23306 0.226359
\(205\) −4.49559 −0.313986
\(206\) −15.5804 −1.08554
\(207\) 2.51973 0.175133
\(208\) −0.918015 −0.0636529
\(209\) 6.58319 0.455369
\(210\) −12.2913 −0.848178
\(211\) 2.88473 0.198593 0.0992965 0.995058i \(-0.468341\pi\)
0.0992965 + 0.995058i \(0.468341\pi\)
\(212\) −17.7030 −1.21584
\(213\) 5.06762 0.347228
\(214\) −6.33525 −0.433068
\(215\) −1.15323 −0.0786494
\(216\) −2.46248 −0.167550
\(217\) 12.6715 0.860195
\(218\) 4.70551 0.318698
\(219\) 17.4962 1.18228
\(220\) 5.08544 0.342860
\(221\) 0.191815 0.0129029
\(222\) −45.8256 −3.07561
\(223\) −7.16709 −0.479944 −0.239972 0.970780i \(-0.577138\pi\)
−0.239972 + 0.970780i \(0.577138\pi\)
\(224\) 17.8951 1.19567
\(225\) −6.92113 −0.461409
\(226\) 28.9006 1.92244
\(227\) −9.36637 −0.621668 −0.310834 0.950464i \(-0.600608\pi\)
−0.310834 + 0.950464i \(0.600608\pi\)
\(228\) 7.05961 0.467534
\(229\) −5.96945 −0.394473 −0.197236 0.980356i \(-0.563197\pi\)
−0.197236 + 0.980356i \(0.563197\pi\)
\(230\) −2.86728 −0.189063
\(231\) 17.2682 1.13616
\(232\) −8.81449 −0.578700
\(233\) 0.474582 0.0310909 0.0155455 0.999879i \(-0.495052\pi\)
0.0155455 + 0.999879i \(0.495052\pi\)
\(234\) −0.673117 −0.0440030
\(235\) −5.06878 −0.330650
\(236\) 4.53812 0.295407
\(237\) 4.12892 0.268202
\(238\) −4.82184 −0.312553
\(239\) 4.97411 0.321748 0.160874 0.986975i \(-0.448569\pi\)
0.160874 + 0.986975i \(0.448569\pi\)
\(240\) −12.1998 −0.787491
\(241\) −7.55048 −0.486369 −0.243185 0.969980i \(-0.578192\pi\)
−0.243185 + 0.969980i \(0.578192\pi\)
\(242\) 3.55514 0.228533
\(243\) −17.1496 −1.10015
\(244\) −9.32669 −0.597080
\(245\) −0.329221 −0.0210332
\(246\) 16.0342 1.02230
\(247\) 0.418841 0.0266502
\(248\) 4.88958 0.310489
\(249\) 1.19136 0.0754991
\(250\) 18.6055 1.17672
\(251\) −8.98211 −0.566946 −0.283473 0.958980i \(-0.591487\pi\)
−0.283473 + 0.958980i \(0.591487\pi\)
\(252\) 7.14751 0.450251
\(253\) 4.02828 0.253256
\(254\) −16.1054 −1.01054
\(255\) 2.54908 0.159630
\(256\) 20.9057 1.30661
\(257\) −17.0074 −1.06089 −0.530445 0.847719i \(-0.677975\pi\)
−0.530445 + 0.847719i \(0.677975\pi\)
\(258\) 4.11315 0.256073
\(259\) 28.8696 1.79387
\(260\) 0.323550 0.0200657
\(261\) −16.6245 −1.02903
\(262\) −21.2717 −1.31417
\(263\) −0.191449 −0.0118052 −0.00590262 0.999983i \(-0.501879\pi\)
−0.00590262 + 0.999983i \(0.501879\pi\)
\(264\) 6.66332 0.410099
\(265\) −13.9578 −0.857419
\(266\) −10.5288 −0.645563
\(267\) −6.27028 −0.383735
\(268\) 15.5049 0.947113
\(269\) 12.2886 0.749246 0.374623 0.927177i \(-0.377772\pi\)
0.374623 + 0.927177i \(0.377772\pi\)
\(270\) 5.28494 0.321631
\(271\) 23.4369 1.42369 0.711845 0.702336i \(-0.247860\pi\)
0.711845 + 0.702336i \(0.247860\pi\)
\(272\) −4.78594 −0.290190
\(273\) 1.09865 0.0664933
\(274\) 21.9495 1.32602
\(275\) −11.0648 −0.667233
\(276\) 4.31980 0.260021
\(277\) −4.42361 −0.265789 −0.132894 0.991130i \(-0.542427\pi\)
−0.132894 + 0.991130i \(0.542427\pi\)
\(278\) 5.78684 0.347071
\(279\) 9.22194 0.552103
\(280\) 2.98796 0.178565
\(281\) 28.2876 1.68750 0.843749 0.536738i \(-0.180344\pi\)
0.843749 + 0.536738i \(0.180344\pi\)
\(282\) 18.0785 1.07656
\(283\) 2.71894 0.161624 0.0808120 0.996729i \(-0.474249\pi\)
0.0808120 + 0.996729i \(0.474249\pi\)
\(284\) 3.35335 0.198985
\(285\) 5.56610 0.329707
\(286\) −1.07611 −0.0636318
\(287\) −10.1014 −0.596264
\(288\) 13.0236 0.767422
\(289\) 1.00000 0.0588235
\(290\) 18.9176 1.11088
\(291\) 36.4967 2.13947
\(292\) 11.5776 0.677527
\(293\) −6.67166 −0.389763 −0.194881 0.980827i \(-0.562432\pi\)
−0.194881 + 0.980827i \(0.562432\pi\)
\(294\) 1.17422 0.0684816
\(295\) 3.57805 0.208322
\(296\) 11.1400 0.647500
\(297\) −7.42488 −0.430835
\(298\) 13.0835 0.757907
\(299\) 0.256291 0.0148217
\(300\) −11.8655 −0.685057
\(301\) −2.59124 −0.149357
\(302\) 21.4009 1.23148
\(303\) 37.1045 2.13160
\(304\) −10.4504 −0.599374
\(305\) −7.35357 −0.421064
\(306\) −3.50920 −0.200608
\(307\) 12.8275 0.732105 0.366052 0.930594i \(-0.380709\pi\)
0.366052 + 0.930594i \(0.380709\pi\)
\(308\) 11.4267 0.651097
\(309\) 18.5073 1.05284
\(310\) −10.4940 −0.596017
\(311\) −23.8066 −1.34995 −0.674975 0.737840i \(-0.735846\pi\)
−0.674975 + 0.737840i \(0.735846\pi\)
\(312\) 0.423939 0.0240008
\(313\) −3.55619 −0.201008 −0.100504 0.994937i \(-0.532046\pi\)
−0.100504 + 0.994937i \(0.532046\pi\)
\(314\) −15.3524 −0.866388
\(315\) 5.63541 0.317519
\(316\) 2.73219 0.153698
\(317\) −26.7591 −1.50294 −0.751471 0.659766i \(-0.770655\pi\)
−0.751471 + 0.659766i \(0.770655\pi\)
\(318\) 49.7824 2.79166
\(319\) −26.5775 −1.48806
\(320\) −3.78141 −0.211387
\(321\) 7.52537 0.420025
\(322\) −6.44263 −0.359034
\(323\) 2.18357 0.121497
\(324\) −16.2372 −0.902067
\(325\) −0.703974 −0.0390495
\(326\) 32.2270 1.78489
\(327\) −5.58948 −0.309099
\(328\) −3.89784 −0.215222
\(329\) −11.3893 −0.627911
\(330\) −14.3007 −0.787230
\(331\) 11.5361 0.634082 0.317041 0.948412i \(-0.397311\pi\)
0.317041 + 0.948412i \(0.397311\pi\)
\(332\) 0.788345 0.0432661
\(333\) 21.0105 1.15137
\(334\) 23.5608 1.28919
\(335\) 12.2247 0.667909
\(336\) −27.4122 −1.49546
\(337\) 13.1860 0.718289 0.359144 0.933282i \(-0.383069\pi\)
0.359144 + 0.933282i \(0.383069\pi\)
\(338\) 24.1222 1.31208
\(339\) −34.3297 −1.86454
\(340\) 1.68678 0.0914786
\(341\) 14.7431 0.798384
\(342\) −7.66258 −0.414345
\(343\) −18.8784 −1.01934
\(344\) −0.999890 −0.0539104
\(345\) 3.40592 0.183369
\(346\) 36.1745 1.94476
\(347\) −3.19155 −0.171332 −0.0856658 0.996324i \(-0.527302\pi\)
−0.0856658 + 0.996324i \(0.527302\pi\)
\(348\) −28.5009 −1.52781
\(349\) 28.5816 1.52994 0.764969 0.644067i \(-0.222754\pi\)
0.764969 + 0.644067i \(0.222754\pi\)
\(350\) 17.6965 0.945916
\(351\) −0.472392 −0.0252144
\(352\) 20.8208 1.10975
\(353\) −35.4420 −1.88639 −0.943193 0.332244i \(-0.892194\pi\)
−0.943193 + 0.332244i \(0.892194\pi\)
\(354\) −12.7616 −0.678273
\(355\) 2.64393 0.140325
\(356\) −4.14917 −0.219906
\(357\) 5.72765 0.303140
\(358\) 46.5626 2.46091
\(359\) 8.26077 0.435987 0.217993 0.975950i \(-0.430049\pi\)
0.217993 + 0.975950i \(0.430049\pi\)
\(360\) 2.17455 0.114609
\(361\) −14.2320 −0.749054
\(362\) 32.5989 1.71336
\(363\) −4.22300 −0.221650
\(364\) 0.727000 0.0381051
\(365\) 9.12828 0.477796
\(366\) 26.2276 1.37094
\(367\) −30.7382 −1.60452 −0.802261 0.596974i \(-0.796370\pi\)
−0.802261 + 0.596974i \(0.796370\pi\)
\(368\) −6.39467 −0.333345
\(369\) −7.35149 −0.382703
\(370\) −23.9086 −1.24295
\(371\) −31.3624 −1.62825
\(372\) 15.8100 0.819712
\(373\) 4.17451 0.216148 0.108074 0.994143i \(-0.465532\pi\)
0.108074 + 0.994143i \(0.465532\pi\)
\(374\) −5.61015 −0.290094
\(375\) −22.1007 −1.14128
\(376\) −4.39481 −0.226645
\(377\) −1.69094 −0.0870877
\(378\) 11.8750 0.610783
\(379\) −30.5234 −1.56788 −0.783942 0.620834i \(-0.786794\pi\)
−0.783942 + 0.620834i \(0.786794\pi\)
\(380\) 3.68320 0.188944
\(381\) 19.1309 0.980106
\(382\) −40.9113 −2.09320
\(383\) 22.6816 1.15897 0.579487 0.814982i \(-0.303253\pi\)
0.579487 + 0.814982i \(0.303253\pi\)
\(384\) −17.0430 −0.869724
\(385\) 9.00931 0.459157
\(386\) 0.542170 0.0275957
\(387\) −1.88583 −0.0958622
\(388\) 24.1506 1.22606
\(389\) −3.56215 −0.180608 −0.0903040 0.995914i \(-0.528784\pi\)
−0.0903040 + 0.995914i \(0.528784\pi\)
\(390\) −0.909853 −0.0460722
\(391\) 1.33614 0.0675713
\(392\) −0.285447 −0.0144173
\(393\) 25.2677 1.27459
\(394\) −33.3981 −1.68257
\(395\) 2.15418 0.108388
\(396\) 8.31604 0.417897
\(397\) −2.35072 −0.117979 −0.0589897 0.998259i \(-0.518788\pi\)
−0.0589897 + 0.998259i \(0.518788\pi\)
\(398\) 14.2098 0.712273
\(399\) 12.5067 0.626120
\(400\) 17.5647 0.878237
\(401\) −5.37666 −0.268498 −0.134249 0.990948i \(-0.542862\pi\)
−0.134249 + 0.990948i \(0.542862\pi\)
\(402\) −43.6013 −2.17463
\(403\) 0.937998 0.0467250
\(404\) 24.5528 1.22155
\(405\) −12.8021 −0.636143
\(406\) 42.5067 2.10957
\(407\) 33.5895 1.66497
\(408\) 2.21015 0.109419
\(409\) 26.2887 1.29989 0.649945 0.759981i \(-0.274792\pi\)
0.649945 + 0.759981i \(0.274792\pi\)
\(410\) 8.36550 0.413143
\(411\) −26.0729 −1.28608
\(412\) 12.2467 0.603350
\(413\) 8.03970 0.395608
\(414\) −4.68876 −0.230440
\(415\) 0.621565 0.0305114
\(416\) 1.32468 0.0649476
\(417\) −6.87394 −0.336618
\(418\) −12.2502 −0.599175
\(419\) −10.0868 −0.492771 −0.246385 0.969172i \(-0.579243\pi\)
−0.246385 + 0.969172i \(0.579243\pi\)
\(420\) 9.66130 0.471423
\(421\) 34.9786 1.70475 0.852377 0.522927i \(-0.175160\pi\)
0.852377 + 0.522927i \(0.175160\pi\)
\(422\) −5.36797 −0.261309
\(423\) −8.28880 −0.403015
\(424\) −12.1019 −0.587720
\(425\) −3.67007 −0.178024
\(426\) −9.42995 −0.456883
\(427\) −16.5231 −0.799608
\(428\) 4.97969 0.240703
\(429\) 1.27827 0.0617153
\(430\) 2.14595 0.103487
\(431\) 32.8038 1.58010 0.790052 0.613040i \(-0.210053\pi\)
0.790052 + 0.613040i \(0.210053\pi\)
\(432\) 11.7866 0.567082
\(433\) 7.56622 0.363610 0.181805 0.983335i \(-0.441806\pi\)
0.181805 + 0.983335i \(0.441806\pi\)
\(434\) −23.5794 −1.13185
\(435\) −22.4713 −1.07742
\(436\) −3.69867 −0.177134
\(437\) 2.91754 0.139565
\(438\) −32.5573 −1.55565
\(439\) 11.1980 0.534449 0.267225 0.963634i \(-0.413893\pi\)
0.267225 + 0.963634i \(0.413893\pi\)
\(440\) 3.47645 0.165733
\(441\) −0.538364 −0.0256364
\(442\) −0.356934 −0.0169776
\(443\) −6.53265 −0.310375 −0.155188 0.987885i \(-0.549598\pi\)
−0.155188 + 0.987885i \(0.549598\pi\)
\(444\) 36.0203 1.70945
\(445\) −3.27139 −0.155079
\(446\) 13.3367 0.631511
\(447\) −15.5413 −0.735080
\(448\) −8.49663 −0.401428
\(449\) 8.52043 0.402104 0.201052 0.979581i \(-0.435564\pi\)
0.201052 + 0.979581i \(0.435564\pi\)
\(450\) 12.8790 0.607122
\(451\) −11.7528 −0.553418
\(452\) −22.7167 −1.06850
\(453\) −25.4212 −1.19439
\(454\) 17.4292 0.817991
\(455\) 0.573198 0.0268719
\(456\) 4.82601 0.225999
\(457\) −33.0112 −1.54420 −0.772099 0.635503i \(-0.780793\pi\)
−0.772099 + 0.635503i \(0.780793\pi\)
\(458\) 11.1081 0.519047
\(459\) −2.46275 −0.114951
\(460\) 2.25377 0.105082
\(461\) 36.3799 1.69438 0.847190 0.531291i \(-0.178293\pi\)
0.847190 + 0.531291i \(0.178293\pi\)
\(462\) −32.1330 −1.49496
\(463\) 25.0230 1.16292 0.581459 0.813576i \(-0.302482\pi\)
0.581459 + 0.813576i \(0.302482\pi\)
\(464\) 42.1903 1.95864
\(465\) 12.4653 0.578065
\(466\) −0.883113 −0.0409094
\(467\) 8.41010 0.389173 0.194586 0.980885i \(-0.437664\pi\)
0.194586 + 0.980885i \(0.437664\pi\)
\(468\) 0.529090 0.0244572
\(469\) 27.4683 1.26837
\(470\) 9.43209 0.435070
\(471\) 18.2365 0.840293
\(472\) 3.10230 0.142795
\(473\) −3.01488 −0.138624
\(474\) −7.68319 −0.352900
\(475\) −8.01385 −0.367701
\(476\) 3.79011 0.173719
\(477\) −22.8247 −1.04507
\(478\) −9.25594 −0.423357
\(479\) −24.8649 −1.13611 −0.568053 0.822992i \(-0.692303\pi\)
−0.568053 + 0.822992i \(0.692303\pi\)
\(480\) 17.6040 0.803509
\(481\) 2.13706 0.0974414
\(482\) 14.0501 0.639965
\(483\) 7.65292 0.348220
\(484\) −2.79445 −0.127020
\(485\) 19.0414 0.864626
\(486\) 31.9124 1.44758
\(487\) −24.6480 −1.11691 −0.558455 0.829535i \(-0.688606\pi\)
−0.558455 + 0.829535i \(0.688606\pi\)
\(488\) −6.37581 −0.288619
\(489\) −38.2810 −1.73113
\(490\) 0.612623 0.0276755
\(491\) −3.93127 −0.177416 −0.0887078 0.996058i \(-0.528274\pi\)
−0.0887078 + 0.996058i \(0.528274\pi\)
\(492\) −12.6033 −0.568202
\(493\) −8.81546 −0.397029
\(494\) −0.779389 −0.0350664
\(495\) 6.55673 0.294703
\(496\) −23.4038 −1.05086
\(497\) 5.94077 0.266480
\(498\) −2.21690 −0.0993418
\(499\) 23.2252 1.03970 0.519852 0.854256i \(-0.325987\pi\)
0.519852 + 0.854256i \(0.325987\pi\)
\(500\) −14.6245 −0.654028
\(501\) −27.9869 −1.25036
\(502\) 16.7141 0.745987
\(503\) 13.6427 0.608300 0.304150 0.952624i \(-0.401628\pi\)
0.304150 + 0.952624i \(0.401628\pi\)
\(504\) 4.88610 0.217644
\(505\) 19.3585 0.861442
\(506\) −7.49592 −0.333234
\(507\) −28.6538 −1.27256
\(508\) 12.6593 0.561666
\(509\) 40.3465 1.78833 0.894164 0.447740i \(-0.147771\pi\)
0.894164 + 0.447740i \(0.147771\pi\)
\(510\) −4.74339 −0.210041
\(511\) 20.5108 0.907342
\(512\) −23.4810 −1.03772
\(513\) −5.37758 −0.237426
\(514\) 31.6477 1.39592
\(515\) 9.65581 0.425486
\(516\) −3.23306 −0.142327
\(517\) −13.2513 −0.582791
\(518\) −53.7213 −2.36038
\(519\) −42.9702 −1.88618
\(520\) 0.221182 0.00969946
\(521\) 14.6597 0.642252 0.321126 0.947036i \(-0.395939\pi\)
0.321126 + 0.947036i \(0.395939\pi\)
\(522\) 30.9352 1.35400
\(523\) −0.499303 −0.0218330 −0.0109165 0.999940i \(-0.503475\pi\)
−0.0109165 + 0.999940i \(0.503475\pi\)
\(524\) 16.7202 0.730425
\(525\) −21.0209 −0.917427
\(526\) 0.356252 0.0155333
\(527\) 4.89012 0.213017
\(528\) −31.8938 −1.38800
\(529\) −21.2147 −0.922380
\(530\) 25.9730 1.12819
\(531\) 5.85107 0.253915
\(532\) 8.27597 0.358809
\(533\) −0.747747 −0.0323885
\(534\) 11.6679 0.504918
\(535\) 3.92621 0.169745
\(536\) 10.5993 0.457820
\(537\) −55.3098 −2.38679
\(538\) −22.8668 −0.985859
\(539\) −0.860682 −0.0370722
\(540\) −4.15412 −0.178765
\(541\) −17.2497 −0.741620 −0.370810 0.928709i \(-0.620920\pi\)
−0.370810 + 0.928709i \(0.620920\pi\)
\(542\) −43.6119 −1.87329
\(543\) −38.7229 −1.66176
\(544\) 6.90601 0.296093
\(545\) −2.91620 −0.124916
\(546\) −2.04439 −0.0874919
\(547\) −5.15817 −0.220547 −0.110274 0.993901i \(-0.535173\pi\)
−0.110274 + 0.993901i \(0.535173\pi\)
\(548\) −17.2530 −0.737010
\(549\) −12.0250 −0.513216
\(550\) 20.5896 0.877945
\(551\) −19.2492 −0.820043
\(552\) 2.95306 0.125690
\(553\) 4.84033 0.205832
\(554\) 8.23155 0.349725
\(555\) 28.3999 1.20551
\(556\) −4.54863 −0.192905
\(557\) −16.5838 −0.702678 −0.351339 0.936248i \(-0.614273\pi\)
−0.351339 + 0.936248i \(0.614273\pi\)
\(558\) −17.1604 −0.726458
\(559\) −0.191815 −0.00811291
\(560\) −14.3018 −0.604360
\(561\) 6.66406 0.281357
\(562\) −52.6383 −2.22041
\(563\) −29.5306 −1.24457 −0.622284 0.782792i \(-0.713795\pi\)
−0.622284 + 0.782792i \(0.713795\pi\)
\(564\) −14.2103 −0.598360
\(565\) −17.9108 −0.753515
\(566\) −5.05946 −0.212665
\(567\) −28.7657 −1.20805
\(568\) 2.29238 0.0961863
\(569\) 10.8038 0.452917 0.226459 0.974021i \(-0.427285\pi\)
0.226459 + 0.974021i \(0.427285\pi\)
\(570\) −10.3575 −0.433829
\(571\) 10.0624 0.421098 0.210549 0.977583i \(-0.432475\pi\)
0.210549 + 0.977583i \(0.432475\pi\)
\(572\) 0.845855 0.0353670
\(573\) 48.5968 2.03016
\(574\) 18.7968 0.784565
\(575\) −4.90371 −0.204499
\(576\) −6.18361 −0.257651
\(577\) 7.38883 0.307601 0.153801 0.988102i \(-0.450849\pi\)
0.153801 + 0.988102i \(0.450849\pi\)
\(578\) −1.86082 −0.0774000
\(579\) −0.644021 −0.0267646
\(580\) −14.8698 −0.617433
\(581\) 1.39662 0.0579418
\(582\) −67.9139 −2.81512
\(583\) −36.4898 −1.51125
\(584\) 7.91455 0.327506
\(585\) 0.417158 0.0172473
\(586\) 12.4148 0.512850
\(587\) −20.7256 −0.855435 −0.427718 0.903912i \(-0.640682\pi\)
−0.427718 + 0.903912i \(0.640682\pi\)
\(588\) −0.922968 −0.0380626
\(589\) 10.6779 0.439976
\(590\) −6.65812 −0.274111
\(591\) 39.6722 1.63190
\(592\) −53.3214 −2.19149
\(593\) 9.69414 0.398091 0.199045 0.979990i \(-0.436216\pi\)
0.199045 + 0.979990i \(0.436216\pi\)
\(594\) 13.8164 0.566893
\(595\) 2.98829 0.122508
\(596\) −10.2840 −0.421250
\(597\) −16.8792 −0.690820
\(598\) −0.476912 −0.0195024
\(599\) −27.8397 −1.13750 −0.568749 0.822511i \(-0.692573\pi\)
−0.568749 + 0.822511i \(0.692573\pi\)
\(600\) −8.11140 −0.331146
\(601\) 6.13362 0.250195 0.125098 0.992144i \(-0.460076\pi\)
0.125098 + 0.992144i \(0.460076\pi\)
\(602\) 4.82184 0.196523
\(603\) 19.9907 0.814084
\(604\) −16.8218 −0.684467
\(605\) −2.20326 −0.0895753
\(606\) −69.0449 −2.80476
\(607\) 14.7898 0.600300 0.300150 0.953892i \(-0.402963\pi\)
0.300150 + 0.953892i \(0.402963\pi\)
\(608\) 15.0798 0.611565
\(609\) −50.4919 −2.04604
\(610\) 13.6837 0.554036
\(611\) −0.843084 −0.0341075
\(612\) 2.75834 0.111499
\(613\) −28.1955 −1.13880 −0.569402 0.822059i \(-0.692825\pi\)
−0.569402 + 0.822059i \(0.692825\pi\)
\(614\) −23.8697 −0.963304
\(615\) −9.93702 −0.400699
\(616\) 7.81141 0.314731
\(617\) 25.0163 1.00712 0.503560 0.863961i \(-0.332023\pi\)
0.503560 + 0.863961i \(0.332023\pi\)
\(618\) −34.4388 −1.38533
\(619\) −10.0045 −0.402113 −0.201056 0.979580i \(-0.564437\pi\)
−0.201056 + 0.979580i \(0.564437\pi\)
\(620\) 8.24856 0.331270
\(621\) −3.29056 −0.132046
\(622\) 44.2999 1.77627
\(623\) −7.35064 −0.294497
\(624\) −2.02917 −0.0812320
\(625\) 6.81974 0.272790
\(626\) 6.61745 0.264486
\(627\) 14.5514 0.581128
\(628\) 12.0675 0.481544
\(629\) 11.1412 0.444230
\(630\) −10.4865 −0.417792
\(631\) 32.9009 1.30976 0.654881 0.755732i \(-0.272719\pi\)
0.654881 + 0.755732i \(0.272719\pi\)
\(632\) 1.86775 0.0742952
\(633\) 6.37638 0.253438
\(634\) 49.7940 1.97757
\(635\) 9.98115 0.396090
\(636\) −39.1305 −1.55162
\(637\) −0.0547590 −0.00216963
\(638\) 49.4561 1.95799
\(639\) 4.32353 0.171036
\(640\) −8.89186 −0.351482
\(641\) 6.15705 0.243189 0.121594 0.992580i \(-0.461199\pi\)
0.121594 + 0.992580i \(0.461199\pi\)
\(642\) −14.0034 −0.552669
\(643\) 3.86650 0.152480 0.0762399 0.997090i \(-0.475709\pi\)
0.0762399 + 0.997090i \(0.475709\pi\)
\(644\) 5.06410 0.199553
\(645\) −2.54908 −0.100370
\(646\) −4.06324 −0.159866
\(647\) 0.161858 0.00636329 0.00318165 0.999995i \(-0.498987\pi\)
0.00318165 + 0.999995i \(0.498987\pi\)
\(648\) −11.0999 −0.436046
\(649\) 9.35409 0.367180
\(650\) 1.30997 0.0513813
\(651\) 28.0089 1.09776
\(652\) −25.3314 −0.992053
\(653\) 28.5286 1.11641 0.558206 0.829702i \(-0.311490\pi\)
0.558206 + 0.829702i \(0.311490\pi\)
\(654\) 10.4010 0.406712
\(655\) 13.1829 0.515099
\(656\) 18.6569 0.728430
\(657\) 14.9272 0.582364
\(658\) 21.1934 0.826205
\(659\) 2.89377 0.112725 0.0563626 0.998410i \(-0.482050\pi\)
0.0563626 + 0.998410i \(0.482050\pi\)
\(660\) 11.2408 0.437548
\(661\) 18.2303 0.709075 0.354537 0.935042i \(-0.384638\pi\)
0.354537 + 0.935042i \(0.384638\pi\)
\(662\) −21.4667 −0.834325
\(663\) 0.423986 0.0164663
\(664\) 0.538920 0.0209142
\(665\) 6.52513 0.253034
\(666\) −39.0968 −1.51497
\(667\) −11.7787 −0.456071
\(668\) −18.5195 −0.716541
\(669\) −15.8421 −0.612490
\(670\) −22.7481 −0.878835
\(671\) −19.2244 −0.742150
\(672\) 39.5553 1.52588
\(673\) 4.54050 0.175023 0.0875117 0.996163i \(-0.472108\pi\)
0.0875117 + 0.996163i \(0.472108\pi\)
\(674\) −24.5369 −0.945125
\(675\) 9.03845 0.347890
\(676\) −18.9608 −0.729261
\(677\) 26.0116 0.999707 0.499853 0.866110i \(-0.333387\pi\)
0.499853 + 0.866110i \(0.333387\pi\)
\(678\) 63.8816 2.45336
\(679\) 42.7850 1.64194
\(680\) 1.15310 0.0442193
\(681\) −20.7033 −0.793354
\(682\) −27.4343 −1.05051
\(683\) −46.0997 −1.76396 −0.881978 0.471291i \(-0.843788\pi\)
−0.881978 + 0.471291i \(0.843788\pi\)
\(684\) 6.02302 0.230296
\(685\) −13.6030 −0.519743
\(686\) 35.1294 1.34125
\(687\) −13.1948 −0.503414
\(688\) 4.78594 0.182462
\(689\) −2.32158 −0.0884452
\(690\) −6.33781 −0.241276
\(691\) −52.2599 −1.98806 −0.994029 0.109114i \(-0.965199\pi\)
−0.994029 + 0.109114i \(0.965199\pi\)
\(692\) −28.4343 −1.08091
\(693\) 14.7326 0.559646
\(694\) 5.93892 0.225438
\(695\) −3.58633 −0.136037
\(696\) −19.4835 −0.738520
\(697\) −3.89827 −0.147658
\(698\) −53.1853 −2.01309
\(699\) 1.04901 0.0396773
\(700\) −13.9100 −0.525747
\(701\) −27.9280 −1.05483 −0.527413 0.849609i \(-0.676838\pi\)
−0.527413 + 0.849609i \(0.676838\pi\)
\(702\) 0.879038 0.0331771
\(703\) 24.3277 0.917536
\(704\) −9.88573 −0.372583
\(705\) −11.2040 −0.421966
\(706\) 65.9513 2.48211
\(707\) 43.4975 1.63589
\(708\) 10.0310 0.376989
\(709\) 5.32357 0.199931 0.0999655 0.994991i \(-0.468127\pi\)
0.0999655 + 0.994991i \(0.468127\pi\)
\(710\) −4.91988 −0.184640
\(711\) 3.52265 0.132110
\(712\) −2.83641 −0.106299
\(713\) 6.53386 0.244695
\(714\) −10.6582 −0.398871
\(715\) 0.666909 0.0249410
\(716\) −36.5996 −1.36779
\(717\) 10.9947 0.410606
\(718\) −15.3718 −0.573672
\(719\) 44.8503 1.67263 0.836317 0.548246i \(-0.184704\pi\)
0.836317 + 0.548246i \(0.184704\pi\)
\(720\) −10.4084 −0.387899
\(721\) 21.6961 0.808005
\(722\) 26.4833 0.985606
\(723\) −16.6895 −0.620690
\(724\) −25.6237 −0.952298
\(725\) 32.3534 1.20157
\(726\) 7.85825 0.291647
\(727\) −34.5889 −1.28283 −0.641416 0.767193i \(-0.721653\pi\)
−0.641416 + 0.767193i \(0.721653\pi\)
\(728\) 0.496984 0.0184194
\(729\) −4.60396 −0.170517
\(730\) −16.9861 −0.628684
\(731\) −1.00000 −0.0369863
\(732\) −20.6156 −0.761976
\(733\) 1.84897 0.0682932 0.0341466 0.999417i \(-0.489129\pi\)
0.0341466 + 0.999417i \(0.489129\pi\)
\(734\) 57.1984 2.11123
\(735\) −0.727708 −0.0268419
\(736\) 9.22737 0.340125
\(737\) 31.9591 1.17723
\(738\) 13.6798 0.503561
\(739\) 43.8591 1.61338 0.806692 0.590973i \(-0.201256\pi\)
0.806692 + 0.590973i \(0.201256\pi\)
\(740\) 18.7928 0.690839
\(741\) 0.925803 0.0340102
\(742\) 58.3599 2.14246
\(743\) −22.9024 −0.840209 −0.420105 0.907476i \(-0.638007\pi\)
−0.420105 + 0.907476i \(0.638007\pi\)
\(744\) 10.8079 0.396236
\(745\) −8.10837 −0.297068
\(746\) −7.76803 −0.284408
\(747\) 1.01642 0.0371890
\(748\) 4.40975 0.161236
\(749\) 8.82198 0.322348
\(750\) 41.1255 1.50169
\(751\) −24.5705 −0.896589 −0.448295 0.893886i \(-0.647968\pi\)
−0.448295 + 0.893886i \(0.647968\pi\)
\(752\) 21.0357 0.767091
\(753\) −19.8540 −0.723519
\(754\) 3.14654 0.114590
\(755\) −13.2630 −0.482690
\(756\) −9.33408 −0.339477
\(757\) −50.8608 −1.84857 −0.924284 0.381705i \(-0.875337\pi\)
−0.924284 + 0.381705i \(0.875337\pi\)
\(758\) 56.7987 2.06302
\(759\) 8.90408 0.323198
\(760\) 2.51787 0.0913328
\(761\) 1.76669 0.0640424 0.0320212 0.999487i \(-0.489806\pi\)
0.0320212 + 0.999487i \(0.489806\pi\)
\(762\) −35.5992 −1.28962
\(763\) −6.55254 −0.237218
\(764\) 32.1575 1.16342
\(765\) 2.17479 0.0786298
\(766\) −42.2064 −1.52498
\(767\) 0.595134 0.0214890
\(768\) 46.2098 1.66745
\(769\) 30.9106 1.11466 0.557332 0.830290i \(-0.311825\pi\)
0.557332 + 0.830290i \(0.311825\pi\)
\(770\) −16.7647 −0.604159
\(771\) −37.5929 −1.35388
\(772\) −0.426162 −0.0153379
\(773\) 33.5407 1.20637 0.603187 0.797600i \(-0.293897\pi\)
0.603187 + 0.797600i \(0.293897\pi\)
\(774\) 3.50920 0.126136
\(775\) −17.9471 −0.644678
\(776\) 16.5096 0.592660
\(777\) 63.8132 2.28929
\(778\) 6.62853 0.237644
\(779\) −8.51215 −0.304979
\(780\) 0.715172 0.0256073
\(781\) 6.91202 0.247331
\(782\) −2.48631 −0.0889103
\(783\) 21.7103 0.775862
\(784\) 1.36628 0.0487958
\(785\) 9.51451 0.339588
\(786\) −47.0188 −1.67710
\(787\) −25.9039 −0.923375 −0.461687 0.887043i \(-0.652756\pi\)
−0.461687 + 0.887043i \(0.652756\pi\)
\(788\) 26.2519 0.935187
\(789\) −0.423177 −0.0150655
\(790\) −4.00855 −0.142618
\(791\) −40.2447 −1.43094
\(792\) 5.68492 0.202005
\(793\) −1.22311 −0.0434339
\(794\) 4.37428 0.155237
\(795\) −30.8522 −1.09421
\(796\) −11.1693 −0.395886
\(797\) 15.6298 0.553637 0.276818 0.960922i \(-0.410720\pi\)
0.276818 + 0.960922i \(0.410720\pi\)
\(798\) −23.2728 −0.823849
\(799\) −4.39530 −0.155494
\(800\) −25.3455 −0.896100
\(801\) −5.34959 −0.189018
\(802\) 10.0050 0.353290
\(803\) 23.8640 0.842143
\(804\) 34.2719 1.20868
\(805\) 3.99275 0.140726
\(806\) −1.74545 −0.0614808
\(807\) 27.1625 0.956166
\(808\) 16.7845 0.590478
\(809\) −40.7498 −1.43269 −0.716343 0.697748i \(-0.754185\pi\)
−0.716343 + 0.697748i \(0.754185\pi\)
\(810\) 23.8225 0.837037
\(811\) 8.27914 0.290720 0.145360 0.989379i \(-0.453566\pi\)
0.145360 + 0.989379i \(0.453566\pi\)
\(812\) −33.4116 −1.17252
\(813\) 51.8047 1.81687
\(814\) −62.5041 −2.19077
\(815\) −19.9723 −0.699601
\(816\) −10.5788 −0.370332
\(817\) −2.18357 −0.0763934
\(818\) −48.9185 −1.71040
\(819\) 0.937331 0.0327530
\(820\) −6.57553 −0.229628
\(821\) 22.7743 0.794827 0.397414 0.917640i \(-0.369908\pi\)
0.397414 + 0.917640i \(0.369908\pi\)
\(822\) 48.5170 1.69222
\(823\) 15.6087 0.544084 0.272042 0.962285i \(-0.412301\pi\)
0.272042 + 0.962285i \(0.412301\pi\)
\(824\) 8.37194 0.291650
\(825\) −24.4576 −0.851503
\(826\) −14.9605 −0.520541
\(827\) −25.7229 −0.894472 −0.447236 0.894416i \(-0.647592\pi\)
−0.447236 + 0.894416i \(0.647592\pi\)
\(828\) 3.68551 0.128080
\(829\) 5.93206 0.206029 0.103014 0.994680i \(-0.467151\pi\)
0.103014 + 0.994680i \(0.467151\pi\)
\(830\) −1.15662 −0.0401470
\(831\) −9.77791 −0.339192
\(832\) −0.628958 −0.0218052
\(833\) −0.285478 −0.00989124
\(834\) 12.7912 0.442922
\(835\) −14.6016 −0.505308
\(836\) 9.62899 0.333026
\(837\) −12.0431 −0.416271
\(838\) 18.7697 0.648388
\(839\) 38.7520 1.33787 0.668934 0.743322i \(-0.266751\pi\)
0.668934 + 0.743322i \(0.266751\pi\)
\(840\) 6.60456 0.227879
\(841\) 48.7124 1.67974
\(842\) −65.0891 −2.24312
\(843\) 62.5267 2.15353
\(844\) 4.21939 0.145237
\(845\) −14.9495 −0.514279
\(846\) 15.4240 0.530287
\(847\) −4.95061 −0.170105
\(848\) 57.9254 1.98917
\(849\) 6.00992 0.206260
\(850\) 6.82935 0.234245
\(851\) 14.8862 0.510293
\(852\) 7.41223 0.253939
\(853\) 48.1892 1.64997 0.824984 0.565157i \(-0.191184\pi\)
0.824984 + 0.565157i \(0.191184\pi\)
\(854\) 30.7465 1.05212
\(855\) 4.74881 0.162406
\(856\) 3.40416 0.116352
\(857\) −33.4859 −1.14386 −0.571928 0.820304i \(-0.693804\pi\)
−0.571928 + 0.820304i \(0.693804\pi\)
\(858\) −2.37863 −0.0812050
\(859\) −25.2222 −0.860570 −0.430285 0.902693i \(-0.641587\pi\)
−0.430285 + 0.902693i \(0.641587\pi\)
\(860\) −1.68678 −0.0575188
\(861\) −22.3280 −0.760935
\(862\) −61.0421 −2.07910
\(863\) 23.9561 0.815475 0.407738 0.913099i \(-0.366318\pi\)
0.407738 + 0.913099i \(0.366318\pi\)
\(864\) −17.0078 −0.578616
\(865\) −22.4188 −0.762262
\(866\) −14.0794 −0.478438
\(867\) 2.21039 0.0750688
\(868\) 18.5341 0.629088
\(869\) 5.63166 0.191041
\(870\) 41.8152 1.41767
\(871\) 2.03333 0.0688967
\(872\) −2.52845 −0.0856241
\(873\) 31.1378 1.05385
\(874\) −5.42903 −0.183640
\(875\) −25.9086 −0.875872
\(876\) 25.5910 0.864640
\(877\) 12.2633 0.414102 0.207051 0.978330i \(-0.433613\pi\)
0.207051 + 0.978330i \(0.433613\pi\)
\(878\) −20.8374 −0.703228
\(879\) −14.7470 −0.497404
\(880\) −16.6399 −0.560932
\(881\) 28.0472 0.944935 0.472467 0.881348i \(-0.343363\pi\)
0.472467 + 0.881348i \(0.343363\pi\)
\(882\) 1.00180 0.0337324
\(883\) 0.978787 0.0329388 0.0164694 0.999864i \(-0.494757\pi\)
0.0164694 + 0.999864i \(0.494757\pi\)
\(884\) 0.280561 0.00943627
\(885\) 7.90890 0.265855
\(886\) 12.1561 0.408392
\(887\) 41.4749 1.39259 0.696295 0.717755i \(-0.254830\pi\)
0.696295 + 0.717755i \(0.254830\pi\)
\(888\) 24.6238 0.826321
\(889\) 22.4271 0.752182
\(890\) 6.08747 0.204053
\(891\) −33.4685 −1.12124
\(892\) −10.4830 −0.350998
\(893\) −9.59744 −0.321166
\(894\) 28.9197 0.967218
\(895\) −28.8567 −0.964574
\(896\) −19.9795 −0.667470
\(897\) 0.566503 0.0189150
\(898\) −15.8550 −0.529088
\(899\) −43.1087 −1.43775
\(900\) −10.1233 −0.337443
\(901\) −12.1032 −0.403217
\(902\) 21.8699 0.728188
\(903\) −5.72765 −0.190604
\(904\) −15.5294 −0.516499
\(905\) −20.2029 −0.671566
\(906\) 47.3044 1.57158
\(907\) −40.3469 −1.33970 −0.669849 0.742497i \(-0.733641\pi\)
−0.669849 + 0.742497i \(0.733641\pi\)
\(908\) −13.6998 −0.454645
\(909\) 31.6563 1.04997
\(910\) −1.06662 −0.0353581
\(911\) −11.1429 −0.369179 −0.184590 0.982816i \(-0.559096\pi\)
−0.184590 + 0.982816i \(0.559096\pi\)
\(912\) −23.0996 −0.764903
\(913\) 1.62496 0.0537782
\(914\) 61.4280 2.03186
\(915\) −16.2543 −0.537349
\(916\) −8.73130 −0.288490
\(917\) 29.6213 0.978182
\(918\) 4.58274 0.151253
\(919\) −41.1546 −1.35757 −0.678783 0.734339i \(-0.737492\pi\)
−0.678783 + 0.734339i \(0.737492\pi\)
\(920\) 1.54070 0.0507953
\(921\) 28.3538 0.934290
\(922\) −67.6965 −2.22947
\(923\) 0.439762 0.0144749
\(924\) 25.2575 0.830911
\(925\) −40.8891 −1.34443
\(926\) −46.5634 −1.53017
\(927\) 15.7898 0.518606
\(928\) −60.8797 −1.99847
\(929\) −2.05473 −0.0674134 −0.0337067 0.999432i \(-0.510731\pi\)
−0.0337067 + 0.999432i \(0.510731\pi\)
\(930\) −23.1957 −0.760619
\(931\) −0.623362 −0.0204299
\(932\) 0.694153 0.0227378
\(933\) −52.6220 −1.72277
\(934\) −15.6497 −0.512074
\(935\) 3.47684 0.113705
\(936\) 0.361691 0.0118222
\(937\) −25.0583 −0.818619 −0.409310 0.912395i \(-0.634230\pi\)
−0.409310 + 0.912395i \(0.634230\pi\)
\(938\) −51.1137 −1.66892
\(939\) −7.86058 −0.256520
\(940\) −7.41391 −0.241815
\(941\) 25.4764 0.830506 0.415253 0.909706i \(-0.363693\pi\)
0.415253 + 0.909706i \(0.363693\pi\)
\(942\) −33.9349 −1.10566
\(943\) −5.20862 −0.169616
\(944\) −14.8491 −0.483296
\(945\) −7.35940 −0.239401
\(946\) 5.61015 0.182402
\(947\) 0.162359 0.00527596 0.00263798 0.999997i \(-0.499160\pi\)
0.00263798 + 0.999997i \(0.499160\pi\)
\(948\) 6.03921 0.196145
\(949\) 1.51830 0.0492860
\(950\) 14.9124 0.483821
\(951\) −59.1481 −1.91801
\(952\) 2.59095 0.0839733
\(953\) 1.15049 0.0372680 0.0186340 0.999826i \(-0.494068\pi\)
0.0186340 + 0.999826i \(0.494068\pi\)
\(954\) 42.4727 1.37510
\(955\) 25.3544 0.820448
\(956\) 7.27544 0.235305
\(957\) −58.7468 −1.89901
\(958\) 46.2692 1.49489
\(959\) −30.5652 −0.987002
\(960\) −8.35840 −0.269766
\(961\) −7.08674 −0.228604
\(962\) −3.97668 −0.128213
\(963\) 6.42039 0.206894
\(964\) −11.0438 −0.355697
\(965\) −0.336005 −0.0108164
\(966\) −14.2407 −0.458188
\(967\) 25.5896 0.822906 0.411453 0.911431i \(-0.365021\pi\)
0.411453 + 0.911431i \(0.365021\pi\)
\(968\) −1.91031 −0.0613997
\(969\) 4.82654 0.155051
\(970\) −35.4327 −1.13768
\(971\) −6.47776 −0.207881 −0.103941 0.994584i \(-0.533145\pi\)
−0.103941 + 0.994584i \(0.533145\pi\)
\(972\) −25.0841 −0.804573
\(973\) −8.05831 −0.258337
\(974\) 45.8657 1.46963
\(975\) −1.55606 −0.0498337
\(976\) 30.5176 0.976846
\(977\) 25.2304 0.807193 0.403597 0.914937i \(-0.367760\pi\)
0.403597 + 0.914937i \(0.367760\pi\)
\(978\) 71.2342 2.27782
\(979\) −8.55238 −0.273335
\(980\) −0.481540 −0.0153822
\(981\) −4.76875 −0.152255
\(982\) 7.31539 0.233443
\(983\) 60.2674 1.92223 0.961116 0.276146i \(-0.0890572\pi\)
0.961116 + 0.276146i \(0.0890572\pi\)
\(984\) −8.61576 −0.274660
\(985\) 20.6982 0.659498
\(986\) 16.4040 0.522410
\(987\) −25.1748 −0.801321
\(988\) 0.612623 0.0194901
\(989\) −1.33614 −0.0424866
\(990\) −12.2009 −0.387770
\(991\) −23.7400 −0.754127 −0.377064 0.926187i \(-0.623066\pi\)
−0.377064 + 0.926187i \(0.623066\pi\)
\(992\) 33.7712 1.07224
\(993\) 25.4993 0.809196
\(994\) −11.0547 −0.350634
\(995\) −8.80639 −0.279181
\(996\) 1.74255 0.0552149
\(997\) 41.3805 1.31053 0.655267 0.755397i \(-0.272556\pi\)
0.655267 + 0.755397i \(0.272556\pi\)
\(998\) −43.2180 −1.36804
\(999\) −27.4381 −0.868102
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 731.2.a.e.1.5 19
3.2 odd 2 6579.2.a.t.1.15 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.a.e.1.5 19 1.1 even 1 trivial
6579.2.a.t.1.15 19 3.2 odd 2