Properties

Label 619.2.a.a.1.4
Level $619$
Weight $2$
Character 619.1
Self dual yes
Analytic conductor $4.943$
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] = [619,2,Mod(1,619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 619.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: \(4.94273988512\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Character \(\chi\) \(=\) 619.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.32447 q^{2} +0.941827 q^{3} +3.40317 q^{4} -2.38281 q^{5} -2.18925 q^{6} +2.74500 q^{7} -3.26162 q^{8} -2.11296 q^{9} +O(q^{10})\) \(q-2.32447 q^{2} +0.941827 q^{3} +3.40317 q^{4} -2.38281 q^{5} -2.18925 q^{6} +2.74500 q^{7} -3.26162 q^{8} -2.11296 q^{9} +5.53878 q^{10} -1.86219 q^{11} +3.20519 q^{12} -0.255352 q^{13} -6.38068 q^{14} -2.24420 q^{15} +0.775212 q^{16} -1.20093 q^{17} +4.91152 q^{18} +5.01739 q^{19} -8.10912 q^{20} +2.58532 q^{21} +4.32861 q^{22} -7.51313 q^{23} -3.07188 q^{24} +0.677806 q^{25} +0.593557 q^{26} -4.81552 q^{27} +9.34170 q^{28} +3.96355 q^{29} +5.21658 q^{30} -3.35980 q^{31} +4.72129 q^{32} -1.75386 q^{33} +2.79152 q^{34} -6.54083 q^{35} -7.19076 q^{36} -10.4323 q^{37} -11.6628 q^{38} -0.240497 q^{39} +7.77184 q^{40} +2.91555 q^{41} -6.00950 q^{42} +4.62423 q^{43} -6.33734 q^{44} +5.03480 q^{45} +17.4641 q^{46} +2.71557 q^{47} +0.730115 q^{48} +0.535042 q^{49} -1.57554 q^{50} -1.13107 q^{51} -0.869004 q^{52} +5.72390 q^{53} +11.1935 q^{54} +4.43725 q^{55} -8.95316 q^{56} +4.72551 q^{57} -9.21317 q^{58} -11.3736 q^{59} -7.63738 q^{60} -14.0158 q^{61} +7.80976 q^{62} -5.80009 q^{63} -12.5249 q^{64} +0.608455 q^{65} +4.07680 q^{66} -8.99998 q^{67} -4.08695 q^{68} -7.07607 q^{69} +15.2040 q^{70} -13.8636 q^{71} +6.89168 q^{72} -2.68365 q^{73} +24.2496 q^{74} +0.638376 q^{75} +17.0750 q^{76} -5.11172 q^{77} +0.559028 q^{78} -4.49955 q^{79} -1.84719 q^{80} +1.80350 q^{81} -6.77712 q^{82} -14.9897 q^{83} +8.79827 q^{84} +2.86159 q^{85} -10.7489 q^{86} +3.73298 q^{87} +6.07376 q^{88} -8.04823 q^{89} -11.7032 q^{90} -0.700941 q^{91} -25.5684 q^{92} -3.16435 q^{93} -6.31227 q^{94} -11.9555 q^{95} +4.44663 q^{96} +9.92292 q^{97} -1.24369 q^{98} +3.93474 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9} + q^{10} - 27 q^{11} - 8 q^{12} - 11 q^{13} - 19 q^{14} - 10 q^{15} + 11 q^{16} - 14 q^{17} - 14 q^{18} - 15 q^{19} - 25 q^{20} - 42 q^{21} + 12 q^{22} - 14 q^{23} - 8 q^{24} + 16 q^{25} - 11 q^{26} - 5 q^{27} + q^{28} - 78 q^{29} + q^{30} - 8 q^{31} - 41 q^{32} - 6 q^{33} + 7 q^{34} - 3 q^{35} - q^{36} - 23 q^{37} + 21 q^{38} - 4 q^{39} + 12 q^{40} - 59 q^{41} + 39 q^{42} + 2 q^{43} - 50 q^{44} - 36 q^{45} - 15 q^{46} - 12 q^{47} + 10 q^{48} + 17 q^{49} - 23 q^{50} - 8 q^{51} + 18 q^{52} - 36 q^{53} - 4 q^{54} + 23 q^{55} - 28 q^{56} - 24 q^{57} + 46 q^{58} - 17 q^{59} + 8 q^{60} - 22 q^{61} + 42 q^{62} - 6 q^{63} + 49 q^{64} - 53 q^{65} + 29 q^{66} + 15 q^{67} - 16 q^{68} - 30 q^{69} + 44 q^{70} - 56 q^{71} + 12 q^{72} - 2 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 47 q^{77} + 36 q^{78} + 5 q^{79} + 15 q^{80} - 19 q^{81} + 47 q^{82} - q^{83} - 20 q^{84} - 29 q^{85} - 23 q^{86} + 44 q^{87} + 61 q^{88} - 12 q^{89} + 91 q^{90} + 5 q^{91} + 35 q^{92} - 15 q^{93} + 34 q^{94} - 17 q^{95} + 14 q^{96} + 21 q^{97} + 24 q^{98} - 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.32447 −1.64365 −0.821825 0.569740i \(-0.807044\pi\)
−0.821825 + 0.569740i \(0.807044\pi\)
\(3\) 0.941827 0.543764 0.271882 0.962331i \(-0.412354\pi\)
0.271882 + 0.962331i \(0.412354\pi\)
\(4\) 3.40317 1.70158
\(5\) −2.38281 −1.06563 −0.532814 0.846233i \(-0.678865\pi\)
−0.532814 + 0.846233i \(0.678865\pi\)
\(6\) −2.18925 −0.893757
\(7\) 2.74500 1.03751 0.518757 0.854922i \(-0.326395\pi\)
0.518757 + 0.854922i \(0.326395\pi\)
\(8\) −3.26162 −1.15316
\(9\) −2.11296 −0.704321
\(10\) 5.53878 1.75152
\(11\) −1.86219 −0.561471 −0.280736 0.959785i \(-0.590578\pi\)
−0.280736 + 0.959785i \(0.590578\pi\)
\(12\) 3.20519 0.925260
\(13\) −0.255352 −0.0708218 −0.0354109 0.999373i \(-0.511274\pi\)
−0.0354109 + 0.999373i \(0.511274\pi\)
\(14\) −6.38068 −1.70531
\(15\) −2.24420 −0.579450
\(16\) 0.775212 0.193803
\(17\) −1.20093 −0.291268 −0.145634 0.989339i \(-0.546522\pi\)
−0.145634 + 0.989339i \(0.546522\pi\)
\(18\) 4.91152 1.15766
\(19\) 5.01739 1.15107 0.575534 0.817778i \(-0.304794\pi\)
0.575534 + 0.817778i \(0.304794\pi\)
\(20\) −8.10912 −1.81325
\(21\) 2.58532 0.564162
\(22\) 4.32861 0.922862
\(23\) −7.51313 −1.56660 −0.783298 0.621646i \(-0.786464\pi\)
−0.783298 + 0.621646i \(0.786464\pi\)
\(24\) −3.07188 −0.627045
\(25\) 0.677806 0.135561
\(26\) 0.593557 0.116406
\(27\) −4.81552 −0.926748
\(28\) 9.34170 1.76542
\(29\) 3.96355 0.736013 0.368007 0.929823i \(-0.380040\pi\)
0.368007 + 0.929823i \(0.380040\pi\)
\(30\) 5.21658 0.952412
\(31\) −3.35980 −0.603438 −0.301719 0.953397i \(-0.597560\pi\)
−0.301719 + 0.953397i \(0.597560\pi\)
\(32\) 4.72129 0.834613
\(33\) −1.75386 −0.305308
\(34\) 2.79152 0.478742
\(35\) −6.54083 −1.10560
\(36\) −7.19076 −1.19846
\(37\) −10.4323 −1.71506 −0.857532 0.514431i \(-0.828003\pi\)
−0.857532 + 0.514431i \(0.828003\pi\)
\(38\) −11.6628 −1.89195
\(39\) −0.240497 −0.0385103
\(40\) 7.77184 1.22884
\(41\) 2.91555 0.455333 0.227666 0.973739i \(-0.426890\pi\)
0.227666 + 0.973739i \(0.426890\pi\)
\(42\) −6.00950 −0.927285
\(43\) 4.62423 0.705189 0.352594 0.935776i \(-0.385300\pi\)
0.352594 + 0.935776i \(0.385300\pi\)
\(44\) −6.33734 −0.955390
\(45\) 5.03480 0.750543
\(46\) 17.4641 2.57493
\(47\) 2.71557 0.396107 0.198054 0.980191i \(-0.436538\pi\)
0.198054 + 0.980191i \(0.436538\pi\)
\(48\) 0.730115 0.105383
\(49\) 0.535042 0.0764346
\(50\) −1.57554 −0.222815
\(51\) −1.13107 −0.158381
\(52\) −0.869004 −0.120509
\(53\) 5.72390 0.786238 0.393119 0.919488i \(-0.371396\pi\)
0.393119 + 0.919488i \(0.371396\pi\)
\(54\) 11.1935 1.52325
\(55\) 4.43725 0.598319
\(56\) −8.95316 −1.19642
\(57\) 4.72551 0.625910
\(58\) −9.21317 −1.20975
\(59\) −11.3736 −1.48071 −0.740357 0.672214i \(-0.765343\pi\)
−0.740357 + 0.672214i \(0.765343\pi\)
\(60\) −7.63738 −0.985982
\(61\) −14.0158 −1.79454 −0.897268 0.441486i \(-0.854452\pi\)
−0.897268 + 0.441486i \(0.854452\pi\)
\(62\) 7.80976 0.991840
\(63\) −5.80009 −0.730742
\(64\) −12.5249 −1.56561
\(65\) 0.608455 0.0754696
\(66\) 4.07680 0.501819
\(67\) −8.99998 −1.09952 −0.549762 0.835322i \(-0.685281\pi\)
−0.549762 + 0.835322i \(0.685281\pi\)
\(68\) −4.08695 −0.495616
\(69\) −7.07607 −0.851858
\(70\) 15.2040 1.81722
\(71\) −13.8636 −1.64530 −0.822651 0.568547i \(-0.807506\pi\)
−0.822651 + 0.568547i \(0.807506\pi\)
\(72\) 6.89168 0.812193
\(73\) −2.68365 −0.314097 −0.157049 0.987591i \(-0.550198\pi\)
−0.157049 + 0.987591i \(0.550198\pi\)
\(74\) 24.2496 2.81896
\(75\) 0.638376 0.0737133
\(76\) 17.0750 1.95864
\(77\) −5.11172 −0.582534
\(78\) 0.559028 0.0632975
\(79\) −4.49955 −0.506239 −0.253120 0.967435i \(-0.581457\pi\)
−0.253120 + 0.967435i \(0.581457\pi\)
\(80\) −1.84719 −0.206522
\(81\) 1.80350 0.200389
\(82\) −6.77712 −0.748407
\(83\) −14.9897 −1.64533 −0.822667 0.568523i \(-0.807515\pi\)
−0.822667 + 0.568523i \(0.807515\pi\)
\(84\) 8.79827 0.959970
\(85\) 2.86159 0.310383
\(86\) −10.7489 −1.15908
\(87\) 3.73298 0.400217
\(88\) 6.07376 0.647465
\(89\) −8.04823 −0.853110 −0.426555 0.904462i \(-0.640273\pi\)
−0.426555 + 0.904462i \(0.640273\pi\)
\(90\) −11.7032 −1.23363
\(91\) −0.700941 −0.0734786
\(92\) −25.5684 −2.66569
\(93\) −3.16435 −0.328128
\(94\) −6.31227 −0.651061
\(95\) −11.9555 −1.22661
\(96\) 4.44663 0.453833
\(97\) 9.92292 1.00752 0.503760 0.863844i \(-0.331950\pi\)
0.503760 + 0.863844i \(0.331950\pi\)
\(98\) −1.24369 −0.125632
\(99\) 3.93474 0.395456
\(100\) 2.30669 0.230669
\(101\) 10.8706 1.08167 0.540833 0.841130i \(-0.318109\pi\)
0.540833 + 0.841130i \(0.318109\pi\)
\(102\) 2.62913 0.260322
\(103\) −4.36654 −0.430248 −0.215124 0.976587i \(-0.569016\pi\)
−0.215124 + 0.976587i \(0.569016\pi\)
\(104\) 0.832860 0.0816687
\(105\) −6.16033 −0.601187
\(106\) −13.3050 −1.29230
\(107\) 6.90907 0.667925 0.333963 0.942586i \(-0.391614\pi\)
0.333963 + 0.942586i \(0.391614\pi\)
\(108\) −16.3880 −1.57694
\(109\) 3.38023 0.323767 0.161884 0.986810i \(-0.448243\pi\)
0.161884 + 0.986810i \(0.448243\pi\)
\(110\) −10.3143 −0.983427
\(111\) −9.82544 −0.932590
\(112\) 2.12796 0.201073
\(113\) 9.39928 0.884210 0.442105 0.896963i \(-0.354232\pi\)
0.442105 + 0.896963i \(0.354232\pi\)
\(114\) −10.9843 −1.02878
\(115\) 17.9024 1.66941
\(116\) 13.4886 1.25239
\(117\) 0.539548 0.0498812
\(118\) 26.4376 2.43377
\(119\) −3.29655 −0.302194
\(120\) 7.31973 0.668197
\(121\) −7.53225 −0.684750
\(122\) 32.5793 2.94959
\(123\) 2.74594 0.247593
\(124\) −11.4340 −1.02680
\(125\) 10.2990 0.921169
\(126\) 13.4821 1.20108
\(127\) 11.1964 0.993521 0.496760 0.867888i \(-0.334523\pi\)
0.496760 + 0.867888i \(0.334523\pi\)
\(128\) 19.6712 1.73871
\(129\) 4.35522 0.383456
\(130\) −1.41434 −0.124046
\(131\) −5.31260 −0.464164 −0.232082 0.972696i \(-0.574554\pi\)
−0.232082 + 0.972696i \(0.574554\pi\)
\(132\) −5.96868 −0.519507
\(133\) 13.7728 1.19425
\(134\) 20.9202 1.80723
\(135\) 11.4745 0.987568
\(136\) 3.91697 0.335877
\(137\) 5.99870 0.512503 0.256252 0.966610i \(-0.417512\pi\)
0.256252 + 0.966610i \(0.417512\pi\)
\(138\) 16.4481 1.40016
\(139\) −7.88334 −0.668656 −0.334328 0.942457i \(-0.608509\pi\)
−0.334328 + 0.942457i \(0.608509\pi\)
\(140\) −22.2596 −1.88128
\(141\) 2.55760 0.215389
\(142\) 32.2254 2.70430
\(143\) 0.475513 0.0397644
\(144\) −1.63799 −0.136499
\(145\) −9.44441 −0.784316
\(146\) 6.23806 0.516266
\(147\) 0.503917 0.0415624
\(148\) −35.5029 −2.91832
\(149\) −3.21065 −0.263027 −0.131513 0.991314i \(-0.541984\pi\)
−0.131513 + 0.991314i \(0.541984\pi\)
\(150\) −1.48389 −0.121159
\(151\) 8.92072 0.725958 0.362979 0.931797i \(-0.381760\pi\)
0.362979 + 0.931797i \(0.381760\pi\)
\(152\) −16.3648 −1.32736
\(153\) 2.53751 0.205146
\(154\) 11.8820 0.957482
\(155\) 8.00578 0.643040
\(156\) −0.818451 −0.0655285
\(157\) 9.68416 0.772880 0.386440 0.922315i \(-0.373705\pi\)
0.386440 + 0.922315i \(0.373705\pi\)
\(158\) 10.4591 0.832079
\(159\) 5.39092 0.427528
\(160\) −11.2499 −0.889386
\(161\) −20.6236 −1.62536
\(162\) −4.19218 −0.329369
\(163\) 11.3711 0.890654 0.445327 0.895368i \(-0.353088\pi\)
0.445327 + 0.895368i \(0.353088\pi\)
\(164\) 9.92211 0.774787
\(165\) 4.17912 0.325344
\(166\) 34.8431 2.70435
\(167\) 3.06492 0.237171 0.118585 0.992944i \(-0.462164\pi\)
0.118585 + 0.992944i \(0.462164\pi\)
\(168\) −8.43233 −0.650568
\(169\) −12.9348 −0.994984
\(170\) −6.65167 −0.510160
\(171\) −10.6016 −0.810722
\(172\) 15.7370 1.19994
\(173\) −7.41249 −0.563561 −0.281780 0.959479i \(-0.590925\pi\)
−0.281780 + 0.959479i \(0.590925\pi\)
\(174\) −8.67721 −0.657817
\(175\) 1.86058 0.140647
\(176\) −1.44359 −0.108815
\(177\) −10.7119 −0.805159
\(178\) 18.7079 1.40221
\(179\) 3.84060 0.287060 0.143530 0.989646i \(-0.454155\pi\)
0.143530 + 0.989646i \(0.454155\pi\)
\(180\) 17.1343 1.27711
\(181\) 11.4730 0.852779 0.426389 0.904540i \(-0.359785\pi\)
0.426389 + 0.904540i \(0.359785\pi\)
\(182\) 1.62932 0.120773
\(183\) −13.2004 −0.975804
\(184\) 24.5050 1.80653
\(185\) 24.8583 1.82762
\(186\) 7.35544 0.539327
\(187\) 2.23635 0.163538
\(188\) 9.24155 0.674009
\(189\) −13.2186 −0.961514
\(190\) 27.7903 2.01612
\(191\) 8.76977 0.634558 0.317279 0.948332i \(-0.397231\pi\)
0.317279 + 0.948332i \(0.397231\pi\)
\(192\) −11.7963 −0.851325
\(193\) 14.3062 1.02979 0.514893 0.857255i \(-0.327832\pi\)
0.514893 + 0.857255i \(0.327832\pi\)
\(194\) −23.0656 −1.65601
\(195\) 0.573060 0.0410376
\(196\) 1.82084 0.130060
\(197\) −14.7770 −1.05282 −0.526408 0.850232i \(-0.676462\pi\)
−0.526408 + 0.850232i \(0.676462\pi\)
\(198\) −9.14618 −0.649991
\(199\) −9.48815 −0.672597 −0.336299 0.941755i \(-0.609175\pi\)
−0.336299 + 0.941755i \(0.609175\pi\)
\(200\) −2.21075 −0.156323
\(201\) −8.47643 −0.597881
\(202\) −25.2684 −1.77788
\(203\) 10.8800 0.763624
\(204\) −3.84920 −0.269498
\(205\) −6.94722 −0.485215
\(206\) 10.1499 0.707177
\(207\) 15.8750 1.10339
\(208\) −0.197952 −0.0137255
\(209\) −9.34334 −0.646292
\(210\) 14.3195 0.988140
\(211\) 20.0056 1.37724 0.688621 0.725121i \(-0.258216\pi\)
0.688621 + 0.725121i \(0.258216\pi\)
\(212\) 19.4794 1.33785
\(213\) −13.0571 −0.894656
\(214\) −16.0599 −1.09783
\(215\) −11.0187 −0.751468
\(216\) 15.7064 1.06869
\(217\) −9.22266 −0.626075
\(218\) −7.85725 −0.532160
\(219\) −2.52753 −0.170795
\(220\) 15.1007 1.01809
\(221\) 0.306659 0.0206281
\(222\) 22.8390 1.53285
\(223\) 3.46929 0.232321 0.116160 0.993230i \(-0.462941\pi\)
0.116160 + 0.993230i \(0.462941\pi\)
\(224\) 12.9599 0.865923
\(225\) −1.43218 −0.0954785
\(226\) −21.8484 −1.45333
\(227\) 14.5883 0.968262 0.484131 0.874996i \(-0.339136\pi\)
0.484131 + 0.874996i \(0.339136\pi\)
\(228\) 16.0817 1.06504
\(229\) −22.7498 −1.50335 −0.751674 0.659535i \(-0.770753\pi\)
−0.751674 + 0.659535i \(0.770753\pi\)
\(230\) −41.6136 −2.74392
\(231\) −4.81435 −0.316761
\(232\) −12.9276 −0.848739
\(233\) 16.0603 1.05214 0.526071 0.850440i \(-0.323665\pi\)
0.526071 + 0.850440i \(0.323665\pi\)
\(234\) −1.25416 −0.0819873
\(235\) −6.47071 −0.422102
\(236\) −38.7062 −2.51956
\(237\) −4.23780 −0.275275
\(238\) 7.66273 0.496701
\(239\) −6.39830 −0.413871 −0.206936 0.978355i \(-0.566349\pi\)
−0.206936 + 0.978355i \(0.566349\pi\)
\(240\) −1.73973 −0.112299
\(241\) 5.36580 0.345641 0.172821 0.984953i \(-0.444712\pi\)
0.172821 + 0.984953i \(0.444712\pi\)
\(242\) 17.5085 1.12549
\(243\) 16.1452 1.03571
\(244\) −47.6980 −3.05355
\(245\) −1.27491 −0.0814508
\(246\) −6.38287 −0.406957
\(247\) −1.28120 −0.0815207
\(248\) 10.9584 0.695859
\(249\) −14.1177 −0.894673
\(250\) −23.9397 −1.51408
\(251\) 7.22722 0.456178 0.228089 0.973640i \(-0.426752\pi\)
0.228089 + 0.973640i \(0.426752\pi\)
\(252\) −19.7387 −1.24342
\(253\) 13.9909 0.879599
\(254\) −26.0257 −1.63300
\(255\) 2.69512 0.168775
\(256\) −20.6754 −1.29221
\(257\) 26.2044 1.63458 0.817292 0.576224i \(-0.195474\pi\)
0.817292 + 0.576224i \(0.195474\pi\)
\(258\) −10.1236 −0.630267
\(259\) −28.6368 −1.77940
\(260\) 2.07068 0.128418
\(261\) −8.37484 −0.518390
\(262\) 12.3490 0.762922
\(263\) 24.7575 1.52661 0.763305 0.646038i \(-0.223575\pi\)
0.763305 + 0.646038i \(0.223575\pi\)
\(264\) 5.72043 0.352068
\(265\) −13.6390 −0.837837
\(266\) −32.0144 −1.96293
\(267\) −7.58003 −0.463891
\(268\) −30.6284 −1.87093
\(269\) −28.0824 −1.71221 −0.856107 0.516799i \(-0.827123\pi\)
−0.856107 + 0.516799i \(0.827123\pi\)
\(270\) −26.6722 −1.62322
\(271\) 10.7255 0.651527 0.325763 0.945451i \(-0.394379\pi\)
0.325763 + 0.945451i \(0.394379\pi\)
\(272\) −0.930973 −0.0564485
\(273\) −0.660165 −0.0399550
\(274\) −13.9438 −0.842375
\(275\) −1.26220 −0.0761137
\(276\) −24.0810 −1.44951
\(277\) −4.80306 −0.288588 −0.144294 0.989535i \(-0.546091\pi\)
−0.144294 + 0.989535i \(0.546091\pi\)
\(278\) 18.3246 1.09904
\(279\) 7.09913 0.425014
\(280\) 21.3337 1.27493
\(281\) −0.246932 −0.0147307 −0.00736536 0.999973i \(-0.502344\pi\)
−0.00736536 + 0.999973i \(0.502344\pi\)
\(282\) −5.94507 −0.354024
\(283\) −23.7812 −1.41364 −0.706822 0.707391i \(-0.749872\pi\)
−0.706822 + 0.707391i \(0.749872\pi\)
\(284\) −47.1800 −2.79962
\(285\) −11.2600 −0.666986
\(286\) −1.10532 −0.0653587
\(287\) 8.00320 0.472414
\(288\) −9.97590 −0.587835
\(289\) −15.5578 −0.915163
\(290\) 21.9533 1.28914
\(291\) 9.34568 0.547853
\(292\) −9.13290 −0.534462
\(293\) −21.0999 −1.23267 −0.616334 0.787485i \(-0.711383\pi\)
−0.616334 + 0.787485i \(0.711383\pi\)
\(294\) −1.17134 −0.0683140
\(295\) 27.1011 1.57789
\(296\) 34.0263 1.97774
\(297\) 8.96742 0.520342
\(298\) 7.46306 0.432324
\(299\) 1.91849 0.110949
\(300\) 2.17250 0.125429
\(301\) 12.6935 0.731643
\(302\) −20.7360 −1.19322
\(303\) 10.2382 0.588171
\(304\) 3.88954 0.223081
\(305\) 33.3970 1.91231
\(306\) −5.89838 −0.337188
\(307\) −15.7384 −0.898238 −0.449119 0.893472i \(-0.648262\pi\)
−0.449119 + 0.893472i \(0.648262\pi\)
\(308\) −17.3960 −0.991230
\(309\) −4.11252 −0.233953
\(310\) −18.6092 −1.05693
\(311\) −15.2313 −0.863685 −0.431843 0.901949i \(-0.642136\pi\)
−0.431843 + 0.901949i \(0.642136\pi\)
\(312\) 0.784410 0.0444085
\(313\) −3.63478 −0.205450 −0.102725 0.994710i \(-0.532756\pi\)
−0.102725 + 0.994710i \(0.532756\pi\)
\(314\) −22.5106 −1.27034
\(315\) 13.8205 0.778699
\(316\) −15.3127 −0.861408
\(317\) −22.2278 −1.24844 −0.624218 0.781250i \(-0.714582\pi\)
−0.624218 + 0.781250i \(0.714582\pi\)
\(318\) −12.5310 −0.702706
\(319\) −7.38089 −0.413250
\(320\) 29.8446 1.66836
\(321\) 6.50715 0.363194
\(322\) 47.9389 2.67153
\(323\) −6.02552 −0.335269
\(324\) 6.13760 0.340978
\(325\) −0.173079 −0.00960068
\(326\) −26.4318 −1.46392
\(327\) 3.18359 0.176053
\(328\) −9.50943 −0.525070
\(329\) 7.45426 0.410966
\(330\) −9.71425 −0.534752
\(331\) 25.5276 1.40312 0.701561 0.712610i \(-0.252487\pi\)
0.701561 + 0.712610i \(0.252487\pi\)
\(332\) −51.0125 −2.79967
\(333\) 22.0431 1.20796
\(334\) −7.12432 −0.389826
\(335\) 21.4453 1.17168
\(336\) 2.00417 0.109336
\(337\) 14.0309 0.764312 0.382156 0.924098i \(-0.375182\pi\)
0.382156 + 0.924098i \(0.375182\pi\)
\(338\) 30.0666 1.63541
\(339\) 8.85249 0.480801
\(340\) 9.73846 0.528142
\(341\) 6.25658 0.338813
\(342\) 24.6430 1.33254
\(343\) −17.7463 −0.958212
\(344\) −15.0825 −0.813193
\(345\) 16.8610 0.907763
\(346\) 17.2301 0.926297
\(347\) 12.9272 0.693969 0.346984 0.937871i \(-0.387206\pi\)
0.346984 + 0.937871i \(0.387206\pi\)
\(348\) 12.7040 0.681003
\(349\) 7.14951 0.382705 0.191352 0.981521i \(-0.438713\pi\)
0.191352 + 0.981521i \(0.438713\pi\)
\(350\) −4.32486 −0.231174
\(351\) 1.22965 0.0656339
\(352\) −8.79193 −0.468611
\(353\) −1.51745 −0.0807657 −0.0403828 0.999184i \(-0.512858\pi\)
−0.0403828 + 0.999184i \(0.512858\pi\)
\(354\) 24.8996 1.32340
\(355\) 33.0343 1.75328
\(356\) −27.3895 −1.45164
\(357\) −3.10478 −0.164322
\(358\) −8.92736 −0.471826
\(359\) −4.78861 −0.252733 −0.126367 0.991984i \(-0.540332\pi\)
−0.126367 + 0.991984i \(0.540332\pi\)
\(360\) −16.4216 −0.865495
\(361\) 6.17422 0.324959
\(362\) −26.6686 −1.40167
\(363\) −7.09407 −0.372342
\(364\) −2.38542 −0.125030
\(365\) 6.39463 0.334710
\(366\) 30.6840 1.60388
\(367\) 20.7387 1.08255 0.541276 0.840845i \(-0.317941\pi\)
0.541276 + 0.840845i \(0.317941\pi\)
\(368\) −5.82427 −0.303611
\(369\) −6.16045 −0.320700
\(370\) −57.7824 −3.00396
\(371\) 15.7121 0.815733
\(372\) −10.7688 −0.558337
\(373\) 9.32736 0.482952 0.241476 0.970407i \(-0.422368\pi\)
0.241476 + 0.970407i \(0.422368\pi\)
\(374\) −5.19834 −0.268800
\(375\) 9.69986 0.500899
\(376\) −8.85717 −0.456774
\(377\) −1.01210 −0.0521258
\(378\) 30.7263 1.58039
\(379\) −30.8221 −1.58322 −0.791612 0.611025i \(-0.790758\pi\)
−0.791612 + 0.611025i \(0.790758\pi\)
\(380\) −40.6866 −2.08718
\(381\) 10.5451 0.540241
\(382\) −20.3851 −1.04299
\(383\) 17.4382 0.891051 0.445526 0.895269i \(-0.353017\pi\)
0.445526 + 0.895269i \(0.353017\pi\)
\(384\) 18.5269 0.945447
\(385\) 12.1803 0.620764
\(386\) −33.2544 −1.69261
\(387\) −9.77082 −0.496679
\(388\) 33.7694 1.71438
\(389\) −9.43027 −0.478134 −0.239067 0.971003i \(-0.576841\pi\)
−0.239067 + 0.971003i \(0.576841\pi\)
\(390\) −1.33206 −0.0674515
\(391\) 9.02272 0.456299
\(392\) −1.74511 −0.0881412
\(393\) −5.00355 −0.252395
\(394\) 34.3487 1.73046
\(395\) 10.7216 0.539462
\(396\) 13.3906 0.672901
\(397\) −0.691864 −0.0347236 −0.0173618 0.999849i \(-0.505527\pi\)
−0.0173618 + 0.999849i \(0.505527\pi\)
\(398\) 22.0549 1.10551
\(399\) 12.9716 0.649390
\(400\) 0.525443 0.0262722
\(401\) 4.81354 0.240377 0.120188 0.992751i \(-0.461650\pi\)
0.120188 + 0.992751i \(0.461650\pi\)
\(402\) 19.7032 0.982707
\(403\) 0.857930 0.0427365
\(404\) 36.9945 1.84055
\(405\) −4.29740 −0.213540
\(406\) −25.2902 −1.25513
\(407\) 19.4270 0.962959
\(408\) 3.68911 0.182638
\(409\) −16.2943 −0.805702 −0.402851 0.915266i \(-0.631981\pi\)
−0.402851 + 0.915266i \(0.631981\pi\)
\(410\) 16.1486 0.797523
\(411\) 5.64973 0.278681
\(412\) −14.8601 −0.732103
\(413\) −31.2205 −1.53626
\(414\) −36.9009 −1.81358
\(415\) 35.7177 1.75331
\(416\) −1.20559 −0.0591088
\(417\) −7.42474 −0.363591
\(418\) 21.7183 1.06228
\(419\) −29.1615 −1.42463 −0.712316 0.701859i \(-0.752353\pi\)
−0.712316 + 0.701859i \(0.752353\pi\)
\(420\) −20.9646 −1.02297
\(421\) −30.0713 −1.46559 −0.732793 0.680452i \(-0.761784\pi\)
−0.732793 + 0.680452i \(0.761784\pi\)
\(422\) −46.5024 −2.26370
\(423\) −5.73790 −0.278986
\(424\) −18.6692 −0.906657
\(425\) −0.813995 −0.0394846
\(426\) 30.3508 1.47050
\(427\) −38.4734 −1.86186
\(428\) 23.5127 1.13653
\(429\) 0.447851 0.0216224
\(430\) 25.6126 1.23515
\(431\) 15.5036 0.746780 0.373390 0.927674i \(-0.378195\pi\)
0.373390 + 0.927674i \(0.378195\pi\)
\(432\) −3.73305 −0.179607
\(433\) 19.2412 0.924671 0.462336 0.886705i \(-0.347012\pi\)
0.462336 + 0.886705i \(0.347012\pi\)
\(434\) 21.4378 1.02905
\(435\) −8.89500 −0.426483
\(436\) 11.5035 0.550917
\(437\) −37.6963 −1.80326
\(438\) 5.87517 0.280727
\(439\) 36.9340 1.76276 0.881381 0.472406i \(-0.156614\pi\)
0.881381 + 0.472406i \(0.156614\pi\)
\(440\) −14.4726 −0.689956
\(441\) −1.13052 −0.0538345
\(442\) −0.712819 −0.0339053
\(443\) −25.4167 −1.20758 −0.603792 0.797142i \(-0.706344\pi\)
−0.603792 + 0.797142i \(0.706344\pi\)
\(444\) −33.4376 −1.58688
\(445\) 19.1774 0.909097
\(446\) −8.06427 −0.381854
\(447\) −3.02387 −0.143024
\(448\) −34.3809 −1.62435
\(449\) 7.86170 0.371017 0.185508 0.982643i \(-0.440607\pi\)
0.185508 + 0.982643i \(0.440607\pi\)
\(450\) 3.32906 0.156933
\(451\) −5.42931 −0.255656
\(452\) 31.9873 1.50456
\(453\) 8.40177 0.394750
\(454\) −33.9102 −1.59148
\(455\) 1.67021 0.0783007
\(456\) −15.4128 −0.721772
\(457\) −11.1115 −0.519772 −0.259886 0.965639i \(-0.583685\pi\)
−0.259886 + 0.965639i \(0.583685\pi\)
\(458\) 52.8812 2.47098
\(459\) 5.78309 0.269932
\(460\) 60.9248 2.84064
\(461\) 31.5317 1.46858 0.734288 0.678838i \(-0.237516\pi\)
0.734288 + 0.678838i \(0.237516\pi\)
\(462\) 11.1908 0.520644
\(463\) −0.0936264 −0.00435119 −0.00217559 0.999998i \(-0.500693\pi\)
−0.00217559 + 0.999998i \(0.500693\pi\)
\(464\) 3.07259 0.142642
\(465\) 7.54006 0.349662
\(466\) −37.3316 −1.72935
\(467\) 4.38408 0.202871 0.101435 0.994842i \(-0.467656\pi\)
0.101435 + 0.994842i \(0.467656\pi\)
\(468\) 1.83617 0.0848771
\(469\) −24.7050 −1.14077
\(470\) 15.0410 0.693788
\(471\) 9.12080 0.420264
\(472\) 37.0963 1.70750
\(473\) −8.61119 −0.395943
\(474\) 9.85064 0.452455
\(475\) 3.40082 0.156040
\(476\) −11.2187 −0.514208
\(477\) −12.0944 −0.553764
\(478\) 14.8727 0.680259
\(479\) 3.94009 0.180028 0.0900138 0.995941i \(-0.471309\pi\)
0.0900138 + 0.995941i \(0.471309\pi\)
\(480\) −10.5955 −0.483616
\(481\) 2.66391 0.121464
\(482\) −12.4726 −0.568113
\(483\) −19.4238 −0.883815
\(484\) −25.6335 −1.16516
\(485\) −23.6445 −1.07364
\(486\) −37.5290 −1.70235
\(487\) −21.8295 −0.989189 −0.494594 0.869124i \(-0.664683\pi\)
−0.494594 + 0.869124i \(0.664683\pi\)
\(488\) 45.7142 2.06938
\(489\) 10.7096 0.484305
\(490\) 2.96348 0.133877
\(491\) 30.7528 1.38785 0.693927 0.720045i \(-0.255879\pi\)
0.693927 + 0.720045i \(0.255879\pi\)
\(492\) 9.34491 0.421301
\(493\) −4.75994 −0.214377
\(494\) 2.97811 0.133991
\(495\) −9.37575 −0.421409
\(496\) −2.60456 −0.116948
\(497\) −38.0555 −1.70702
\(498\) 32.8162 1.47053
\(499\) 8.68664 0.388867 0.194434 0.980916i \(-0.437713\pi\)
0.194434 + 0.980916i \(0.437713\pi\)
\(500\) 35.0492 1.56745
\(501\) 2.88663 0.128965
\(502\) −16.7995 −0.749797
\(503\) −20.7993 −0.927393 −0.463697 0.885994i \(-0.653477\pi\)
−0.463697 + 0.885994i \(0.653477\pi\)
\(504\) 18.9177 0.842661
\(505\) −25.9027 −1.15265
\(506\) −32.5214 −1.44575
\(507\) −12.1823 −0.541037
\(508\) 38.1033 1.69056
\(509\) −31.4884 −1.39570 −0.697849 0.716245i \(-0.745859\pi\)
−0.697849 + 0.716245i \(0.745859\pi\)
\(510\) −6.26473 −0.277407
\(511\) −7.36662 −0.325880
\(512\) 8.71689 0.385236
\(513\) −24.1614 −1.06675
\(514\) −60.9113 −2.68668
\(515\) 10.4047 0.458484
\(516\) 14.8216 0.652483
\(517\) −5.05691 −0.222403
\(518\) 66.5653 2.92471
\(519\) −6.98128 −0.306444
\(520\) −1.98455 −0.0870283
\(521\) 3.10018 0.135821 0.0679107 0.997691i \(-0.478367\pi\)
0.0679107 + 0.997691i \(0.478367\pi\)
\(522\) 19.4671 0.852051
\(523\) 21.3138 0.931989 0.465994 0.884788i \(-0.345697\pi\)
0.465994 + 0.884788i \(0.345697\pi\)
\(524\) −18.0797 −0.789813
\(525\) 1.75234 0.0764785
\(526\) −57.5480 −2.50921
\(527\) 4.03487 0.175762
\(528\) −1.35961 −0.0591696
\(529\) 33.4471 1.45422
\(530\) 31.7035 1.37711
\(531\) 24.0319 1.04290
\(532\) 46.8710 2.03212
\(533\) −0.744491 −0.0322475
\(534\) 17.6196 0.762473
\(535\) −16.4630 −0.711759
\(536\) 29.3545 1.26792
\(537\) 3.61718 0.156093
\(538\) 65.2767 2.81428
\(539\) −0.996350 −0.0429158
\(540\) 39.0497 1.68043
\(541\) −17.5862 −0.756088 −0.378044 0.925788i \(-0.623403\pi\)
−0.378044 + 0.925788i \(0.623403\pi\)
\(542\) −24.9311 −1.07088
\(543\) 10.8055 0.463710
\(544\) −5.66992 −0.243096
\(545\) −8.05446 −0.345015
\(546\) 1.53453 0.0656720
\(547\) 29.5940 1.26535 0.632675 0.774418i \(-0.281957\pi\)
0.632675 + 0.774418i \(0.281957\pi\)
\(548\) 20.4146 0.872067
\(549\) 29.6148 1.26393
\(550\) 2.93395 0.125104
\(551\) 19.8867 0.847202
\(552\) 23.0795 0.982327
\(553\) −12.3513 −0.525230
\(554\) 11.1646 0.474337
\(555\) 23.4122 0.993793
\(556\) −26.8283 −1.13777
\(557\) −33.7370 −1.42948 −0.714741 0.699389i \(-0.753455\pi\)
−0.714741 + 0.699389i \(0.753455\pi\)
\(558\) −16.5017 −0.698574
\(559\) −1.18080 −0.0499427
\(560\) −5.07053 −0.214269
\(561\) 2.10626 0.0889263
\(562\) 0.573986 0.0242121
\(563\) 44.0966 1.85845 0.929225 0.369514i \(-0.120476\pi\)
0.929225 + 0.369514i \(0.120476\pi\)
\(564\) 8.70394 0.366502
\(565\) −22.3967 −0.942238
\(566\) 55.2787 2.32354
\(567\) 4.95061 0.207906
\(568\) 45.2177 1.89729
\(569\) 1.02357 0.0429105 0.0214552 0.999770i \(-0.493170\pi\)
0.0214552 + 0.999770i \(0.493170\pi\)
\(570\) 26.1736 1.09629
\(571\) 11.8651 0.496538 0.248269 0.968691i \(-0.420138\pi\)
0.248269 + 0.968691i \(0.420138\pi\)
\(572\) 1.61825 0.0676624
\(573\) 8.25960 0.345050
\(574\) −18.6032 −0.776483
\(575\) −5.09244 −0.212370
\(576\) 26.4647 1.10269
\(577\) 29.0611 1.20983 0.604915 0.796290i \(-0.293207\pi\)
0.604915 + 0.796290i \(0.293207\pi\)
\(578\) 36.1636 1.50421
\(579\) 13.4740 0.559960
\(580\) −32.1409 −1.33458
\(581\) −41.1468 −1.70706
\(582\) −21.7238 −0.900479
\(583\) −10.6590 −0.441450
\(584\) 8.75304 0.362203
\(585\) −1.28564 −0.0531548
\(586\) 49.0461 2.02607
\(587\) 5.95025 0.245593 0.122797 0.992432i \(-0.460814\pi\)
0.122797 + 0.992432i \(0.460814\pi\)
\(588\) 1.71491 0.0707219
\(589\) −16.8574 −0.694598
\(590\) −62.9958 −2.59350
\(591\) −13.9173 −0.572483
\(592\) −8.08726 −0.332384
\(593\) 35.6093 1.46230 0.731149 0.682217i \(-0.238984\pi\)
0.731149 + 0.682217i \(0.238984\pi\)
\(594\) −20.8445 −0.855261
\(595\) 7.85506 0.322026
\(596\) −10.9264 −0.447562
\(597\) −8.93620 −0.365734
\(598\) −4.45947 −0.182361
\(599\) −26.9522 −1.10124 −0.550619 0.834757i \(-0.685608\pi\)
−0.550619 + 0.834757i \(0.685608\pi\)
\(600\) −2.08214 −0.0850030
\(601\) −27.4005 −1.11769 −0.558844 0.829273i \(-0.688755\pi\)
−0.558844 + 0.829273i \(0.688755\pi\)
\(602\) −29.5057 −1.20256
\(603\) 19.0166 0.774417
\(604\) 30.3587 1.23528
\(605\) 17.9480 0.729688
\(606\) −23.7985 −0.966747
\(607\) −48.4251 −1.96551 −0.982756 0.184906i \(-0.940802\pi\)
−0.982756 + 0.184906i \(0.940802\pi\)
\(608\) 23.6885 0.960697
\(609\) 10.2470 0.415231
\(610\) −77.6304 −3.14316
\(611\) −0.693426 −0.0280530
\(612\) 8.63558 0.349073
\(613\) −45.8551 −1.85207 −0.926035 0.377438i \(-0.876805\pi\)
−0.926035 + 0.377438i \(0.876805\pi\)
\(614\) 36.5835 1.47639
\(615\) −6.54308 −0.263842
\(616\) 16.6725 0.671754
\(617\) −42.3649 −1.70555 −0.852774 0.522280i \(-0.825082\pi\)
−0.852774 + 0.522280i \(0.825082\pi\)
\(618\) 9.55944 0.384537
\(619\) −1.00000 −0.0401934
\(620\) 27.2450 1.09419
\(621\) 36.1797 1.45184
\(622\) 35.4046 1.41960
\(623\) −22.0924 −0.885114
\(624\) −0.186436 −0.00746342
\(625\) −27.9296 −1.11718
\(626\) 8.44895 0.337688
\(627\) −8.79980 −0.351430
\(628\) 32.9568 1.31512
\(629\) 12.5285 0.499542
\(630\) −32.1254 −1.27991
\(631\) −20.4064 −0.812365 −0.406182 0.913792i \(-0.633140\pi\)
−0.406182 + 0.913792i \(0.633140\pi\)
\(632\) 14.6758 0.583773
\(633\) 18.8418 0.748895
\(634\) 51.6678 2.05199
\(635\) −26.6790 −1.05872
\(636\) 18.3462 0.727475
\(637\) −0.136624 −0.00541324
\(638\) 17.1567 0.679239
\(639\) 29.2932 1.15882
\(640\) −46.8729 −1.85281
\(641\) 21.3992 0.845218 0.422609 0.906312i \(-0.361114\pi\)
0.422609 + 0.906312i \(0.361114\pi\)
\(642\) −15.1257 −0.596963
\(643\) −47.4855 −1.87265 −0.936323 0.351141i \(-0.885794\pi\)
−0.936323 + 0.351141i \(0.885794\pi\)
\(644\) −70.1854 −2.76569
\(645\) −10.3777 −0.408621
\(646\) 14.0062 0.551065
\(647\) 10.9819 0.431745 0.215872 0.976422i \(-0.430740\pi\)
0.215872 + 0.976422i \(0.430740\pi\)
\(648\) −5.88233 −0.231080
\(649\) 21.1798 0.831378
\(650\) 0.402317 0.0157802
\(651\) −8.68615 −0.340437
\(652\) 38.6978 1.51552
\(653\) −14.9284 −0.584193 −0.292097 0.956389i \(-0.594353\pi\)
−0.292097 + 0.956389i \(0.594353\pi\)
\(654\) −7.40017 −0.289369
\(655\) 12.6589 0.494625
\(656\) 2.26017 0.0882448
\(657\) 5.67045 0.221225
\(658\) −17.3272 −0.675485
\(659\) 4.65349 0.181274 0.0906371 0.995884i \(-0.471110\pi\)
0.0906371 + 0.995884i \(0.471110\pi\)
\(660\) 14.2223 0.553601
\(661\) 49.1804 1.91290 0.956448 0.291904i \(-0.0942889\pi\)
0.956448 + 0.291904i \(0.0942889\pi\)
\(662\) −59.3381 −2.30624
\(663\) 0.288819 0.0112168
\(664\) 48.8908 1.89733
\(665\) −32.8179 −1.27262
\(666\) −51.2386 −1.98545
\(667\) −29.7787 −1.15304
\(668\) 10.4304 0.403566
\(669\) 3.26747 0.126328
\(670\) −49.8490 −1.92583
\(671\) 26.1000 1.00758
\(672\) 12.2060 0.470857
\(673\) 23.7019 0.913642 0.456821 0.889559i \(-0.348988\pi\)
0.456821 + 0.889559i \(0.348988\pi\)
\(674\) −32.6144 −1.25626
\(675\) −3.26399 −0.125631
\(676\) −44.0193 −1.69305
\(677\) 19.9839 0.768042 0.384021 0.923324i \(-0.374539\pi\)
0.384021 + 0.923324i \(0.374539\pi\)
\(678\) −20.5774 −0.790269
\(679\) 27.2385 1.04532
\(680\) −9.33341 −0.357920
\(681\) 13.7397 0.526506
\(682\) −14.5432 −0.556890
\(683\) −38.9144 −1.48902 −0.744508 0.667613i \(-0.767316\pi\)
−0.744508 + 0.667613i \(0.767316\pi\)
\(684\) −36.0789 −1.37951
\(685\) −14.2938 −0.546137
\(686\) 41.2508 1.57496
\(687\) −21.4263 −0.817466
\(688\) 3.58476 0.136668
\(689\) −1.46161 −0.0556828
\(690\) −39.1928 −1.49204
\(691\) −14.8343 −0.564323 −0.282161 0.959367i \(-0.591051\pi\)
−0.282161 + 0.959367i \(0.591051\pi\)
\(692\) −25.2259 −0.958946
\(693\) 10.8009 0.410291
\(694\) −30.0489 −1.14064
\(695\) 18.7845 0.712538
\(696\) −12.1756 −0.461514
\(697\) −3.50136 −0.132624
\(698\) −16.6188 −0.629032
\(699\) 15.1260 0.572117
\(700\) 6.33186 0.239322
\(701\) −44.4851 −1.68018 −0.840090 0.542447i \(-0.817498\pi\)
−0.840090 + 0.542447i \(0.817498\pi\)
\(702\) −2.85829 −0.107879
\(703\) −52.3431 −1.97416
\(704\) 23.3238 0.879048
\(705\) −6.09428 −0.229524
\(706\) 3.52727 0.132750
\(707\) 29.8399 1.12224
\(708\) −36.4545 −1.37004
\(709\) −20.6230 −0.774513 −0.387256 0.921972i \(-0.626577\pi\)
−0.387256 + 0.921972i \(0.626577\pi\)
\(710\) −76.7873 −2.88177
\(711\) 9.50738 0.356555
\(712\) 26.2503 0.983770
\(713\) 25.2426 0.945343
\(714\) 7.21697 0.270088
\(715\) −1.13306 −0.0423740
\(716\) 13.0702 0.488456
\(717\) −6.02609 −0.225048
\(718\) 11.1310 0.415405
\(719\) 40.4151 1.50723 0.753615 0.657316i \(-0.228309\pi\)
0.753615 + 0.657316i \(0.228309\pi\)
\(720\) 3.90304 0.145458
\(721\) −11.9862 −0.446388
\(722\) −14.3518 −0.534119
\(723\) 5.05365 0.187947
\(724\) 39.0444 1.45107
\(725\) 2.68652 0.0997748
\(726\) 16.4900 0.612000
\(727\) 47.9355 1.77783 0.888914 0.458074i \(-0.151461\pi\)
0.888914 + 0.458074i \(0.151461\pi\)
\(728\) 2.28620 0.0847323
\(729\) 9.79545 0.362794
\(730\) −14.8641 −0.550147
\(731\) −5.55336 −0.205399
\(732\) −44.9233 −1.66041
\(733\) −21.6201 −0.798555 −0.399278 0.916830i \(-0.630739\pi\)
−0.399278 + 0.916830i \(0.630739\pi\)
\(734\) −48.2066 −1.77934
\(735\) −1.20074 −0.0442900
\(736\) −35.4716 −1.30750
\(737\) 16.7597 0.617351
\(738\) 14.3198 0.527119
\(739\) 25.2860 0.930161 0.465081 0.885268i \(-0.346025\pi\)
0.465081 + 0.885268i \(0.346025\pi\)
\(740\) 84.5969 3.10985
\(741\) −1.20667 −0.0443280
\(742\) −36.5224 −1.34078
\(743\) −10.4850 −0.384659 −0.192329 0.981330i \(-0.561604\pi\)
−0.192329 + 0.981330i \(0.561604\pi\)
\(744\) 10.3209 0.378383
\(745\) 7.65038 0.280288
\(746\) −21.6812 −0.793804
\(747\) 31.6727 1.15884
\(748\) 7.61068 0.278274
\(749\) 18.9654 0.692981
\(750\) −22.5471 −0.823302
\(751\) −37.4832 −1.36778 −0.683890 0.729585i \(-0.739713\pi\)
−0.683890 + 0.729585i \(0.739713\pi\)
\(752\) 2.10514 0.0767667
\(753\) 6.80679 0.248053
\(754\) 2.35260 0.0856765
\(755\) −21.2564 −0.773600
\(756\) −44.9852 −1.63610
\(757\) 18.8295 0.684370 0.342185 0.939633i \(-0.388833\pi\)
0.342185 + 0.939633i \(0.388833\pi\)
\(758\) 71.6450 2.60226
\(759\) 13.1770 0.478294
\(760\) 38.9944 1.41447
\(761\) 31.2878 1.13418 0.567091 0.823655i \(-0.308069\pi\)
0.567091 + 0.823655i \(0.308069\pi\)
\(762\) −24.5117 −0.887966
\(763\) 9.27874 0.335913
\(764\) 29.8450 1.07975
\(765\) −6.04642 −0.218609
\(766\) −40.5346 −1.46458
\(767\) 2.90426 0.104867
\(768\) −19.4726 −0.702658
\(769\) 44.2952 1.59733 0.798664 0.601778i \(-0.205541\pi\)
0.798664 + 0.601778i \(0.205541\pi\)
\(770\) −28.3127 −1.02032
\(771\) 24.6800 0.888828
\(772\) 48.6865 1.75227
\(773\) −13.4312 −0.483086 −0.241543 0.970390i \(-0.577653\pi\)
−0.241543 + 0.970390i \(0.577653\pi\)
\(774\) 22.7120 0.816366
\(775\) −2.27729 −0.0818027
\(776\) −32.3648 −1.16183
\(777\) −26.9709 −0.967575
\(778\) 21.9204 0.785884
\(779\) 14.6285 0.524119
\(780\) 1.95022 0.0698290
\(781\) 25.8166 0.923790
\(782\) −20.9731 −0.749995
\(783\) −19.0866 −0.682099
\(784\) 0.414771 0.0148133
\(785\) −23.0756 −0.823602
\(786\) 11.6306 0.414850
\(787\) 0.114645 0.00408666 0.00204333 0.999998i \(-0.499350\pi\)
0.00204333 + 0.999998i \(0.499350\pi\)
\(788\) −50.2885 −1.79145
\(789\) 23.3172 0.830115
\(790\) −24.9220 −0.886686
\(791\) 25.8011 0.917380
\(792\) −12.8336 −0.456023
\(793\) 3.57895 0.127092
\(794\) 1.60822 0.0570735
\(795\) −12.8456 −0.455586
\(796\) −32.2898 −1.14448
\(797\) 32.5605 1.15335 0.576676 0.816973i \(-0.304350\pi\)
0.576676 + 0.816973i \(0.304350\pi\)
\(798\) −30.1520 −1.06737
\(799\) −3.26120 −0.115373
\(800\) 3.20011 0.113141
\(801\) 17.0056 0.600863
\(802\) −11.1889 −0.395095
\(803\) 4.99746 0.176357
\(804\) −28.8467 −1.01734
\(805\) 49.1421 1.73203
\(806\) −1.99423 −0.0702439
\(807\) −26.4487 −0.931040
\(808\) −35.4558 −1.24733
\(809\) 15.3849 0.540905 0.270453 0.962733i \(-0.412827\pi\)
0.270453 + 0.962733i \(0.412827\pi\)
\(810\) 9.98918 0.350984
\(811\) −25.5460 −0.897040 −0.448520 0.893773i \(-0.648049\pi\)
−0.448520 + 0.893773i \(0.648049\pi\)
\(812\) 37.0263 1.29937
\(813\) 10.1015 0.354277
\(814\) −45.1574 −1.58277
\(815\) −27.0952 −0.949105
\(816\) −0.876815 −0.0306947
\(817\) 23.2016 0.811720
\(818\) 37.8757 1.32429
\(819\) 1.48106 0.0517525
\(820\) −23.6425 −0.825634
\(821\) −7.11236 −0.248223 −0.124111 0.992268i \(-0.539608\pi\)
−0.124111 + 0.992268i \(0.539608\pi\)
\(822\) −13.1326 −0.458053
\(823\) 18.9835 0.661725 0.330862 0.943679i \(-0.392660\pi\)
0.330862 + 0.943679i \(0.392660\pi\)
\(824\) 14.2420 0.496143
\(825\) −1.18878 −0.0413879
\(826\) 72.5712 2.52507
\(827\) −36.8002 −1.27967 −0.639833 0.768514i \(-0.720997\pi\)
−0.639833 + 0.768514i \(0.720997\pi\)
\(828\) 54.0251 1.87750
\(829\) −40.1162 −1.39329 −0.696646 0.717415i \(-0.745325\pi\)
−0.696646 + 0.717415i \(0.745325\pi\)
\(830\) −83.0248 −2.88183
\(831\) −4.52365 −0.156924
\(832\) 3.19826 0.110880
\(833\) −0.642547 −0.0222629
\(834\) 17.2586 0.597616
\(835\) −7.30314 −0.252736
\(836\) −31.7969 −1.09972
\(837\) 16.1792 0.559235
\(838\) 67.7850 2.34159
\(839\) −13.6474 −0.471161 −0.235581 0.971855i \(-0.575699\pi\)
−0.235581 + 0.971855i \(0.575699\pi\)
\(840\) 20.0927 0.693263
\(841\) −13.2902 −0.458284
\(842\) 69.8999 2.40891
\(843\) −0.232567 −0.00801003
\(844\) 68.0824 2.34349
\(845\) 30.8212 1.06028
\(846\) 13.3376 0.458556
\(847\) −20.6760 −0.710437
\(848\) 4.43724 0.152375
\(849\) −22.3977 −0.768689
\(850\) 1.89211 0.0648988
\(851\) 78.3794 2.68681
\(852\) −44.4354 −1.52233
\(853\) −21.9685 −0.752188 −0.376094 0.926581i \(-0.622733\pi\)
−0.376094 + 0.926581i \(0.622733\pi\)
\(854\) 89.4302 3.06024
\(855\) 25.2616 0.863927
\(856\) −22.5348 −0.770223
\(857\) −32.3888 −1.10638 −0.553190 0.833055i \(-0.686590\pi\)
−0.553190 + 0.833055i \(0.686590\pi\)
\(858\) −1.04102 −0.0355397
\(859\) −53.0965 −1.81163 −0.905815 0.423673i \(-0.860741\pi\)
−0.905815 + 0.423673i \(0.860741\pi\)
\(860\) −37.4984 −1.27869
\(861\) 7.53763 0.256882
\(862\) −36.0376 −1.22744
\(863\) −43.8757 −1.49354 −0.746772 0.665080i \(-0.768398\pi\)
−0.746772 + 0.665080i \(0.768398\pi\)
\(864\) −22.7355 −0.773476
\(865\) 17.6626 0.600546
\(866\) −44.7255 −1.51984
\(867\) −14.6527 −0.497633
\(868\) −31.3862 −1.06532
\(869\) 8.37902 0.284239
\(870\) 20.6762 0.700988
\(871\) 2.29816 0.0778702
\(872\) −11.0250 −0.373355
\(873\) −20.9668 −0.709618
\(874\) 87.6240 2.96393
\(875\) 28.2708 0.955726
\(876\) −8.60161 −0.290621
\(877\) −35.6541 −1.20395 −0.601977 0.798513i \(-0.705620\pi\)
−0.601977 + 0.798513i \(0.705620\pi\)
\(878\) −85.8520 −2.89736
\(879\) −19.8724 −0.670280
\(880\) 3.43981 0.115956
\(881\) 38.2274 1.28791 0.643956 0.765062i \(-0.277292\pi\)
0.643956 + 0.765062i \(0.277292\pi\)
\(882\) 2.62787 0.0884850
\(883\) 16.3896 0.551553 0.275777 0.961222i \(-0.411065\pi\)
0.275777 + 0.961222i \(0.411065\pi\)
\(884\) 1.04361 0.0351004
\(885\) 25.5246 0.857999
\(886\) 59.0804 1.98484
\(887\) −26.0909 −0.876046 −0.438023 0.898964i \(-0.644321\pi\)
−0.438023 + 0.898964i \(0.644321\pi\)
\(888\) 32.0469 1.07542
\(889\) 30.7342 1.03079
\(890\) −44.5774 −1.49424
\(891\) −3.35845 −0.112512
\(892\) 11.8066 0.395313
\(893\) 13.6251 0.455946
\(894\) 7.02891 0.235082
\(895\) −9.15143 −0.305899
\(896\) 53.9976 1.80393
\(897\) 1.80688 0.0603301
\(898\) −18.2743 −0.609821
\(899\) −13.3167 −0.444138
\(900\) −4.87394 −0.162465
\(901\) −6.87399 −0.229006
\(902\) 12.6203 0.420209
\(903\) 11.9551 0.397841
\(904\) −30.6569 −1.01963
\(905\) −27.3379 −0.908744
\(906\) −19.5297 −0.648830
\(907\) 30.1941 1.00258 0.501290 0.865280i \(-0.332859\pi\)
0.501290 + 0.865280i \(0.332859\pi\)
\(908\) 49.6466 1.64758
\(909\) −22.9692 −0.761840
\(910\) −3.88236 −0.128699
\(911\) −42.4012 −1.40481 −0.702406 0.711776i \(-0.747891\pi\)
−0.702406 + 0.711776i \(0.747891\pi\)
\(912\) 3.66327 0.121303
\(913\) 27.9137 0.923808
\(914\) 25.8283 0.854324
\(915\) 31.4542 1.03984
\(916\) −77.4213 −2.55807
\(917\) −14.5831 −0.481576
\(918\) −13.4426 −0.443673
\(919\) −37.3392 −1.23171 −0.615853 0.787861i \(-0.711188\pi\)
−0.615853 + 0.787861i \(0.711188\pi\)
\(920\) −58.3908 −1.92509
\(921\) −14.8228 −0.488429
\(922\) −73.2945 −2.41383
\(923\) 3.54008 0.116523
\(924\) −16.3840 −0.538995
\(925\) −7.07109 −0.232496
\(926\) 0.217632 0.00715183
\(927\) 9.22633 0.303033
\(928\) 18.7131 0.614286
\(929\) 17.3494 0.569214 0.284607 0.958644i \(-0.408137\pi\)
0.284607 + 0.958644i \(0.408137\pi\)
\(930\) −17.5266 −0.574721
\(931\) 2.68452 0.0879815
\(932\) 54.6557 1.79031
\(933\) −14.3452 −0.469641
\(934\) −10.1907 −0.333449
\(935\) −5.32882 −0.174271
\(936\) −1.75980 −0.0575209
\(937\) 13.7236 0.448331 0.224165 0.974551i \(-0.428034\pi\)
0.224165 + 0.974551i \(0.428034\pi\)
\(938\) 57.4260 1.87503
\(939\) −3.42334 −0.111716
\(940\) −22.0209 −0.718242
\(941\) −39.8341 −1.29855 −0.649276 0.760552i \(-0.724928\pi\)
−0.649276 + 0.760552i \(0.724928\pi\)
\(942\) −21.2010 −0.690767
\(943\) −21.9049 −0.713322
\(944\) −8.81693 −0.286967
\(945\) 31.4975 1.02462
\(946\) 20.0165 0.650792
\(947\) 45.4204 1.47596 0.737982 0.674820i \(-0.235779\pi\)
0.737982 + 0.674820i \(0.235779\pi\)
\(948\) −14.4219 −0.468403
\(949\) 0.685273 0.0222449
\(950\) −7.90510 −0.256475
\(951\) −20.9347 −0.678855
\(952\) 10.7521 0.348477
\(953\) 44.3724 1.43736 0.718682 0.695339i \(-0.244746\pi\)
0.718682 + 0.695339i \(0.244746\pi\)
\(954\) 28.1131 0.910194
\(955\) −20.8967 −0.676202
\(956\) −21.7745 −0.704237
\(957\) −6.95152 −0.224711
\(958\) −9.15864 −0.295902
\(959\) 16.4664 0.531729
\(960\) 28.1084 0.907195
\(961\) −19.7118 −0.635863
\(962\) −6.19218 −0.199644
\(963\) −14.5986 −0.470434
\(964\) 18.2607 0.588138
\(965\) −34.0891 −1.09737
\(966\) 45.1501 1.45268
\(967\) 23.9029 0.768667 0.384333 0.923194i \(-0.374431\pi\)
0.384333 + 0.923194i \(0.374431\pi\)
\(968\) 24.5673 0.789625
\(969\) −5.67500 −0.182307
\(970\) 54.9609 1.76469
\(971\) 13.9630 0.448095 0.224047 0.974578i \(-0.428073\pi\)
0.224047 + 0.974578i \(0.428073\pi\)
\(972\) 54.9447 1.76235
\(973\) −21.6398 −0.693740
\(974\) 50.7420 1.62588
\(975\) −0.163010 −0.00522050
\(976\) −10.8652 −0.347787
\(977\) 12.3442 0.394925 0.197462 0.980310i \(-0.436730\pi\)
0.197462 + 0.980310i \(0.436730\pi\)
\(978\) −24.8942 −0.796028
\(979\) 14.9873 0.478997
\(980\) −4.33872 −0.138595
\(981\) −7.14230 −0.228036
\(982\) −71.4840 −2.28115
\(983\) 36.0663 1.15034 0.575168 0.818035i \(-0.304937\pi\)
0.575168 + 0.818035i \(0.304937\pi\)
\(984\) −8.95623 −0.285514
\(985\) 35.2108 1.12191
\(986\) 11.0643 0.352360
\(987\) 7.02062 0.223469
\(988\) −4.36013 −0.138714
\(989\) −34.7424 −1.10475
\(990\) 21.7937 0.692648
\(991\) −31.2015 −0.991147 −0.495574 0.868566i \(-0.665042\pi\)
−0.495574 + 0.868566i \(0.665042\pi\)
\(992\) −15.8626 −0.503637
\(993\) 24.0425 0.762967
\(994\) 88.4589 2.80575
\(995\) 22.6085 0.716738
\(996\) −48.0449 −1.52236
\(997\) −26.0238 −0.824181 −0.412091 0.911143i \(-0.635201\pi\)
−0.412091 + 0.911143i \(0.635201\pi\)
\(998\) −20.1918 −0.639162
\(999\) 50.2371 1.58943
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 619.2.a.a.1.4 21
3.2 odd 2 5571.2.a.e.1.18 21
4.3 odd 2 9904.2.a.j.1.7 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
619.2.a.a.1.4 21 1.1 even 1 trivial
5571.2.a.e.1.18 21 3.2 odd 2
9904.2.a.j.1.7 21 4.3 odd 2