Properties

Label 4024.2.a.g.1.13
Level $4024$
Weight $2$
Character 4024.1
Self dual yes
Analytic conductor $32.132$
Analytic rank $0$
Dimension $33$
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] = [4024,2,Mod(1,4024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4024 = 2^{3} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4024.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: \(32.1318017734\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 4024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.648270 q^{3} +2.57860 q^{5} -1.91468 q^{7} -2.57975 q^{9} +O(q^{10})\) \(q-0.648270 q^{3} +2.57860 q^{5} -1.91468 q^{7} -2.57975 q^{9} -2.67275 q^{11} -2.76049 q^{13} -1.67163 q^{15} -5.56676 q^{17} +5.65257 q^{19} +1.24123 q^{21} -1.28435 q^{23} +1.64916 q^{25} +3.61718 q^{27} -3.23268 q^{29} +2.44930 q^{31} +1.73266 q^{33} -4.93719 q^{35} -1.30868 q^{37} +1.78955 q^{39} +8.94787 q^{41} +2.24431 q^{43} -6.65212 q^{45} +10.6304 q^{47} -3.33400 q^{49} +3.60876 q^{51} +7.69860 q^{53} -6.89193 q^{55} -3.66439 q^{57} +4.71042 q^{59} +8.59506 q^{61} +4.93939 q^{63} -7.11820 q^{65} +12.0357 q^{67} +0.832606 q^{69} +9.07555 q^{71} -2.09100 q^{73} -1.06910 q^{75} +5.11745 q^{77} -2.24817 q^{79} +5.39432 q^{81} +16.3986 q^{83} -14.3544 q^{85} +2.09565 q^{87} +3.73787 q^{89} +5.28546 q^{91} -1.58781 q^{93} +14.5757 q^{95} -1.84264 q^{97} +6.89500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.648270 −0.374279 −0.187140 0.982333i \(-0.559922\pi\)
−0.187140 + 0.982333i \(0.559922\pi\)
\(4\) 0 0
\(5\) 2.57860 1.15318 0.576592 0.817032i \(-0.304382\pi\)
0.576592 + 0.817032i \(0.304382\pi\)
\(6\) 0 0
\(7\) −1.91468 −0.723681 −0.361841 0.932240i \(-0.617852\pi\)
−0.361841 + 0.932240i \(0.617852\pi\)
\(8\) 0 0
\(9\) −2.57975 −0.859915
\(10\) 0 0
\(11\) −2.67275 −0.805863 −0.402932 0.915230i \(-0.632009\pi\)
−0.402932 + 0.915230i \(0.632009\pi\)
\(12\) 0 0
\(13\) −2.76049 −0.765623 −0.382811 0.923827i \(-0.625044\pi\)
−0.382811 + 0.923827i \(0.625044\pi\)
\(14\) 0 0
\(15\) −1.67163 −0.431613
\(16\) 0 0
\(17\) −5.56676 −1.35014 −0.675068 0.737755i \(-0.735886\pi\)
−0.675068 + 0.737755i \(0.735886\pi\)
\(18\) 0 0
\(19\) 5.65257 1.29679 0.648394 0.761305i \(-0.275441\pi\)
0.648394 + 0.761305i \(0.275441\pi\)
\(20\) 0 0
\(21\) 1.24123 0.270859
\(22\) 0 0
\(23\) −1.28435 −0.267805 −0.133903 0.990994i \(-0.542751\pi\)
−0.133903 + 0.990994i \(0.542751\pi\)
\(24\) 0 0
\(25\) 1.64916 0.329833
\(26\) 0 0
\(27\) 3.61718 0.696127
\(28\) 0 0
\(29\) −3.23268 −0.600293 −0.300147 0.953893i \(-0.597036\pi\)
−0.300147 + 0.953893i \(0.597036\pi\)
\(30\) 0 0
\(31\) 2.44930 0.439908 0.219954 0.975510i \(-0.429409\pi\)
0.219954 + 0.975510i \(0.429409\pi\)
\(32\) 0 0
\(33\) 1.73266 0.301618
\(34\) 0 0
\(35\) −4.93719 −0.834537
\(36\) 0 0
\(37\) −1.30868 −0.215145 −0.107573 0.994197i \(-0.534308\pi\)
−0.107573 + 0.994197i \(0.534308\pi\)
\(38\) 0 0
\(39\) 1.78955 0.286557
\(40\) 0 0
\(41\) 8.94787 1.39742 0.698711 0.715404i \(-0.253757\pi\)
0.698711 + 0.715404i \(0.253757\pi\)
\(42\) 0 0
\(43\) 2.24431 0.342254 0.171127 0.985249i \(-0.445259\pi\)
0.171127 + 0.985249i \(0.445259\pi\)
\(44\) 0 0
\(45\) −6.65212 −0.991640
\(46\) 0 0
\(47\) 10.6304 1.55060 0.775299 0.631594i \(-0.217599\pi\)
0.775299 + 0.631594i \(0.217599\pi\)
\(48\) 0 0
\(49\) −3.33400 −0.476286
\(50\) 0 0
\(51\) 3.60876 0.505328
\(52\) 0 0
\(53\) 7.69860 1.05748 0.528742 0.848783i \(-0.322664\pi\)
0.528742 + 0.848783i \(0.322664\pi\)
\(54\) 0 0
\(55\) −6.89193 −0.929308
\(56\) 0 0
\(57\) −3.66439 −0.485361
\(58\) 0 0
\(59\) 4.71042 0.613245 0.306622 0.951831i \(-0.400801\pi\)
0.306622 + 0.951831i \(0.400801\pi\)
\(60\) 0 0
\(61\) 8.59506 1.10048 0.550242 0.835005i \(-0.314535\pi\)
0.550242 + 0.835005i \(0.314535\pi\)
\(62\) 0 0
\(63\) 4.93939 0.622304
\(64\) 0 0
\(65\) −7.11820 −0.882904
\(66\) 0 0
\(67\) 12.0357 1.47040 0.735199 0.677851i \(-0.237089\pi\)
0.735199 + 0.677851i \(0.237089\pi\)
\(68\) 0 0
\(69\) 0.832606 0.100234
\(70\) 0 0
\(71\) 9.07555 1.07707 0.538535 0.842603i \(-0.318978\pi\)
0.538535 + 0.842603i \(0.318978\pi\)
\(72\) 0 0
\(73\) −2.09100 −0.244733 −0.122366 0.992485i \(-0.539048\pi\)
−0.122366 + 0.992485i \(0.539048\pi\)
\(74\) 0 0
\(75\) −1.06910 −0.123449
\(76\) 0 0
\(77\) 5.11745 0.583188
\(78\) 0 0
\(79\) −2.24817 −0.252939 −0.126470 0.991970i \(-0.540365\pi\)
−0.126470 + 0.991970i \(0.540365\pi\)
\(80\) 0 0
\(81\) 5.39432 0.599369
\(82\) 0 0
\(83\) 16.3986 1.79998 0.899990 0.435911i \(-0.143574\pi\)
0.899990 + 0.435911i \(0.143574\pi\)
\(84\) 0 0
\(85\) −14.3544 −1.55696
\(86\) 0 0
\(87\) 2.09565 0.224677
\(88\) 0 0
\(89\) 3.73787 0.396213 0.198107 0.980180i \(-0.436521\pi\)
0.198107 + 0.980180i \(0.436521\pi\)
\(90\) 0 0
\(91\) 5.28546 0.554067
\(92\) 0 0
\(93\) −1.58781 −0.164648
\(94\) 0 0
\(95\) 14.5757 1.49544
\(96\) 0 0
\(97\) −1.84264 −0.187092 −0.0935460 0.995615i \(-0.529820\pi\)
−0.0935460 + 0.995615i \(0.529820\pi\)
\(98\) 0 0
\(99\) 6.89500 0.692974
\(100\) 0 0
\(101\) −2.36849 −0.235674 −0.117837 0.993033i \(-0.537596\pi\)
−0.117837 + 0.993033i \(0.537596\pi\)
\(102\) 0 0
\(103\) −5.80611 −0.572093 −0.286046 0.958216i \(-0.592341\pi\)
−0.286046 + 0.958216i \(0.592341\pi\)
\(104\) 0 0
\(105\) 3.20063 0.312350
\(106\) 0 0
\(107\) −3.00579 −0.290581 −0.145291 0.989389i \(-0.546412\pi\)
−0.145291 + 0.989389i \(0.546412\pi\)
\(108\) 0 0
\(109\) 18.2203 1.74518 0.872592 0.488450i \(-0.162437\pi\)
0.872592 + 0.488450i \(0.162437\pi\)
\(110\) 0 0
\(111\) 0.848377 0.0805244
\(112\) 0 0
\(113\) −20.6599 −1.94352 −0.971758 0.235979i \(-0.924170\pi\)
−0.971758 + 0.235979i \(0.924170\pi\)
\(114\) 0 0
\(115\) −3.31182 −0.308829
\(116\) 0 0
\(117\) 7.12137 0.658371
\(118\) 0 0
\(119\) 10.6586 0.977069
\(120\) 0 0
\(121\) −3.85643 −0.350585
\(122\) 0 0
\(123\) −5.80064 −0.523026
\(124\) 0 0
\(125\) −8.64046 −0.772826
\(126\) 0 0
\(127\) 4.72813 0.419554 0.209777 0.977749i \(-0.432726\pi\)
0.209777 + 0.977749i \(0.432726\pi\)
\(128\) 0 0
\(129\) −1.45492 −0.128098
\(130\) 0 0
\(131\) 1.40430 0.122694 0.0613470 0.998116i \(-0.480460\pi\)
0.0613470 + 0.998116i \(0.480460\pi\)
\(132\) 0 0
\(133\) −10.8229 −0.938462
\(134\) 0 0
\(135\) 9.32726 0.802763
\(136\) 0 0
\(137\) 3.74666 0.320099 0.160049 0.987109i \(-0.448835\pi\)
0.160049 + 0.987109i \(0.448835\pi\)
\(138\) 0 0
\(139\) 18.2043 1.54407 0.772034 0.635582i \(-0.219240\pi\)
0.772034 + 0.635582i \(0.219240\pi\)
\(140\) 0 0
\(141\) −6.89135 −0.580357
\(142\) 0 0
\(143\) 7.37809 0.616987
\(144\) 0 0
\(145\) −8.33577 −0.692248
\(146\) 0 0
\(147\) 2.16133 0.178264
\(148\) 0 0
\(149\) −3.82576 −0.313418 −0.156709 0.987645i \(-0.550089\pi\)
−0.156709 + 0.987645i \(0.550089\pi\)
\(150\) 0 0
\(151\) −14.5593 −1.18482 −0.592410 0.805637i \(-0.701823\pi\)
−0.592410 + 0.805637i \(0.701823\pi\)
\(152\) 0 0
\(153\) 14.3608 1.16100
\(154\) 0 0
\(155\) 6.31577 0.507295
\(156\) 0 0
\(157\) −4.19924 −0.335136 −0.167568 0.985861i \(-0.553591\pi\)
−0.167568 + 0.985861i \(0.553591\pi\)
\(158\) 0 0
\(159\) −4.99077 −0.395794
\(160\) 0 0
\(161\) 2.45912 0.193806
\(162\) 0 0
\(163\) −16.7761 −1.31400 −0.657002 0.753889i \(-0.728176\pi\)
−0.657002 + 0.753889i \(0.728176\pi\)
\(164\) 0 0
\(165\) 4.46784 0.347821
\(166\) 0 0
\(167\) −4.23614 −0.327802 −0.163901 0.986477i \(-0.552408\pi\)
−0.163901 + 0.986477i \(0.552408\pi\)
\(168\) 0 0
\(169\) −5.37968 −0.413822
\(170\) 0 0
\(171\) −14.5822 −1.11513
\(172\) 0 0
\(173\) −7.92393 −0.602445 −0.301223 0.953554i \(-0.597395\pi\)
−0.301223 + 0.953554i \(0.597395\pi\)
\(174\) 0 0
\(175\) −3.15762 −0.238694
\(176\) 0 0
\(177\) −3.05363 −0.229525
\(178\) 0 0
\(179\) −3.95476 −0.295593 −0.147796 0.989018i \(-0.547218\pi\)
−0.147796 + 0.989018i \(0.547218\pi\)
\(180\) 0 0
\(181\) 19.1875 1.42620 0.713099 0.701063i \(-0.247291\pi\)
0.713099 + 0.701063i \(0.247291\pi\)
\(182\) 0 0
\(183\) −5.57192 −0.411888
\(184\) 0 0
\(185\) −3.37455 −0.248102
\(186\) 0 0
\(187\) 14.8785 1.08803
\(188\) 0 0
\(189\) −6.92575 −0.503774
\(190\) 0 0
\(191\) −3.40078 −0.246072 −0.123036 0.992402i \(-0.539263\pi\)
−0.123036 + 0.992402i \(0.539263\pi\)
\(192\) 0 0
\(193\) −10.8697 −0.782418 −0.391209 0.920302i \(-0.627943\pi\)
−0.391209 + 0.920302i \(0.627943\pi\)
\(194\) 0 0
\(195\) 4.61452 0.330452
\(196\) 0 0
\(197\) 12.3891 0.882690 0.441345 0.897337i \(-0.354501\pi\)
0.441345 + 0.897337i \(0.354501\pi\)
\(198\) 0 0
\(199\) −9.74391 −0.690727 −0.345364 0.938469i \(-0.612244\pi\)
−0.345364 + 0.938469i \(0.612244\pi\)
\(200\) 0 0
\(201\) −7.80241 −0.550339
\(202\) 0 0
\(203\) 6.18955 0.434421
\(204\) 0 0
\(205\) 23.0729 1.61148
\(206\) 0 0
\(207\) 3.31329 0.230290
\(208\) 0 0
\(209\) −15.1079 −1.04503
\(210\) 0 0
\(211\) −5.14740 −0.354361 −0.177181 0.984178i \(-0.556698\pi\)
−0.177181 + 0.984178i \(0.556698\pi\)
\(212\) 0 0
\(213\) −5.88341 −0.403125
\(214\) 0 0
\(215\) 5.78716 0.394681
\(216\) 0 0
\(217\) −4.68963 −0.318353
\(218\) 0 0
\(219\) 1.35553 0.0915984
\(220\) 0 0
\(221\) 15.3670 1.03370
\(222\) 0 0
\(223\) 14.7951 0.990751 0.495376 0.868679i \(-0.335030\pi\)
0.495376 + 0.868679i \(0.335030\pi\)
\(224\) 0 0
\(225\) −4.25442 −0.283628
\(226\) 0 0
\(227\) −5.64011 −0.374347 −0.187174 0.982327i \(-0.559933\pi\)
−0.187174 + 0.982327i \(0.559933\pi\)
\(228\) 0 0
\(229\) 1.11347 0.0735800 0.0367900 0.999323i \(-0.488287\pi\)
0.0367900 + 0.999323i \(0.488287\pi\)
\(230\) 0 0
\(231\) −3.31749 −0.218275
\(232\) 0 0
\(233\) −20.4053 −1.33679 −0.668397 0.743804i \(-0.733019\pi\)
−0.668397 + 0.743804i \(0.733019\pi\)
\(234\) 0 0
\(235\) 27.4114 1.78812
\(236\) 0 0
\(237\) 1.45742 0.0946699
\(238\) 0 0
\(239\) 8.05079 0.520763 0.260381 0.965506i \(-0.416152\pi\)
0.260381 + 0.965506i \(0.416152\pi\)
\(240\) 0 0
\(241\) 13.1149 0.844807 0.422403 0.906408i \(-0.361187\pi\)
0.422403 + 0.906408i \(0.361187\pi\)
\(242\) 0 0
\(243\) −14.3485 −0.920459
\(244\) 0 0
\(245\) −8.59704 −0.549245
\(246\) 0 0
\(247\) −15.6039 −0.992851
\(248\) 0 0
\(249\) −10.6307 −0.673695
\(250\) 0 0
\(251\) −6.70438 −0.423176 −0.211588 0.977359i \(-0.567864\pi\)
−0.211588 + 0.977359i \(0.567864\pi\)
\(252\) 0 0
\(253\) 3.43274 0.215814
\(254\) 0 0
\(255\) 9.30555 0.582736
\(256\) 0 0
\(257\) 19.7157 1.22983 0.614916 0.788593i \(-0.289190\pi\)
0.614916 + 0.788593i \(0.289190\pi\)
\(258\) 0 0
\(259\) 2.50570 0.155697
\(260\) 0 0
\(261\) 8.33949 0.516201
\(262\) 0 0
\(263\) −2.80491 −0.172958 −0.0864791 0.996254i \(-0.527562\pi\)
−0.0864791 + 0.996254i \(0.527562\pi\)
\(264\) 0 0
\(265\) 19.8516 1.21947
\(266\) 0 0
\(267\) −2.42315 −0.148294
\(268\) 0 0
\(269\) 10.8295 0.660285 0.330143 0.943931i \(-0.392903\pi\)
0.330143 + 0.943931i \(0.392903\pi\)
\(270\) 0 0
\(271\) 2.98337 0.181227 0.0906133 0.995886i \(-0.471117\pi\)
0.0906133 + 0.995886i \(0.471117\pi\)
\(272\) 0 0
\(273\) −3.42641 −0.207376
\(274\) 0 0
\(275\) −4.40779 −0.265800
\(276\) 0 0
\(277\) 9.96101 0.598499 0.299249 0.954175i \(-0.403264\pi\)
0.299249 + 0.954175i \(0.403264\pi\)
\(278\) 0 0
\(279\) −6.31858 −0.378283
\(280\) 0 0
\(281\) −1.84482 −0.110053 −0.0550264 0.998485i \(-0.517524\pi\)
−0.0550264 + 0.998485i \(0.517524\pi\)
\(282\) 0 0
\(283\) 29.7679 1.76952 0.884760 0.466047i \(-0.154322\pi\)
0.884760 + 0.466047i \(0.154322\pi\)
\(284\) 0 0
\(285\) −9.44900 −0.559710
\(286\) 0 0
\(287\) −17.1323 −1.01129
\(288\) 0 0
\(289\) 13.9888 0.822870
\(290\) 0 0
\(291\) 1.19453 0.0700246
\(292\) 0 0
\(293\) 20.7148 1.21017 0.605087 0.796159i \(-0.293138\pi\)
0.605087 + 0.796159i \(0.293138\pi\)
\(294\) 0 0
\(295\) 12.1463 0.707184
\(296\) 0 0
\(297\) −9.66781 −0.560983
\(298\) 0 0
\(299\) 3.54544 0.205038
\(300\) 0 0
\(301\) −4.29713 −0.247682
\(302\) 0 0
\(303\) 1.53542 0.0882078
\(304\) 0 0
\(305\) 22.1632 1.26906
\(306\) 0 0
\(307\) −11.3197 −0.646051 −0.323026 0.946390i \(-0.604700\pi\)
−0.323026 + 0.946390i \(0.604700\pi\)
\(308\) 0 0
\(309\) 3.76393 0.214122
\(310\) 0 0
\(311\) 20.6967 1.17360 0.586801 0.809731i \(-0.300387\pi\)
0.586801 + 0.809731i \(0.300387\pi\)
\(312\) 0 0
\(313\) 32.6375 1.84478 0.922389 0.386262i \(-0.126234\pi\)
0.922389 + 0.386262i \(0.126234\pi\)
\(314\) 0 0
\(315\) 12.7367 0.717631
\(316\) 0 0
\(317\) −14.3782 −0.807558 −0.403779 0.914856i \(-0.632304\pi\)
−0.403779 + 0.914856i \(0.632304\pi\)
\(318\) 0 0
\(319\) 8.64013 0.483754
\(320\) 0 0
\(321\) 1.94857 0.108758
\(322\) 0 0
\(323\) −31.4665 −1.75084
\(324\) 0 0
\(325\) −4.55250 −0.252527
\(326\) 0 0
\(327\) −11.8117 −0.653186
\(328\) 0 0
\(329\) −20.3538 −1.12214
\(330\) 0 0
\(331\) −21.1108 −1.16036 −0.580178 0.814489i \(-0.697017\pi\)
−0.580178 + 0.814489i \(0.697017\pi\)
\(332\) 0 0
\(333\) 3.37606 0.185007
\(334\) 0 0
\(335\) 31.0353 1.69564
\(336\) 0 0
\(337\) −3.53486 −0.192556 −0.0962781 0.995354i \(-0.530694\pi\)
−0.0962781 + 0.995354i \(0.530694\pi\)
\(338\) 0 0
\(339\) 13.3932 0.727418
\(340\) 0 0
\(341\) −6.54636 −0.354506
\(342\) 0 0
\(343\) 19.7863 1.06836
\(344\) 0 0
\(345\) 2.14695 0.115588
\(346\) 0 0
\(347\) −9.89530 −0.531207 −0.265604 0.964082i \(-0.585571\pi\)
−0.265604 + 0.964082i \(0.585571\pi\)
\(348\) 0 0
\(349\) −10.2859 −0.550594 −0.275297 0.961359i \(-0.588776\pi\)
−0.275297 + 0.961359i \(0.588776\pi\)
\(350\) 0 0
\(351\) −9.98521 −0.532971
\(352\) 0 0
\(353\) −23.0825 −1.22856 −0.614278 0.789090i \(-0.710553\pi\)
−0.614278 + 0.789090i \(0.710553\pi\)
\(354\) 0 0
\(355\) 23.4022 1.24206
\(356\) 0 0
\(357\) −6.90963 −0.365696
\(358\) 0 0
\(359\) 12.7149 0.671068 0.335534 0.942028i \(-0.391083\pi\)
0.335534 + 0.942028i \(0.391083\pi\)
\(360\) 0 0
\(361\) 12.9516 0.681661
\(362\) 0 0
\(363\) 2.50001 0.131217
\(364\) 0 0
\(365\) −5.39184 −0.282222
\(366\) 0 0
\(367\) 16.8194 0.877963 0.438981 0.898496i \(-0.355339\pi\)
0.438981 + 0.898496i \(0.355339\pi\)
\(368\) 0 0
\(369\) −23.0832 −1.20166
\(370\) 0 0
\(371\) −14.7404 −0.765281
\(372\) 0 0
\(373\) −1.25938 −0.0652080 −0.0326040 0.999468i \(-0.510380\pi\)
−0.0326040 + 0.999468i \(0.510380\pi\)
\(374\) 0 0
\(375\) 5.60135 0.289253
\(376\) 0 0
\(377\) 8.92378 0.459598
\(378\) 0 0
\(379\) 15.5546 0.798988 0.399494 0.916736i \(-0.369186\pi\)
0.399494 + 0.916736i \(0.369186\pi\)
\(380\) 0 0
\(381\) −3.06511 −0.157030
\(382\) 0 0
\(383\) −24.9081 −1.27274 −0.636371 0.771383i \(-0.719565\pi\)
−0.636371 + 0.771383i \(0.719565\pi\)
\(384\) 0 0
\(385\) 13.1958 0.672523
\(386\) 0 0
\(387\) −5.78974 −0.294309
\(388\) 0 0
\(389\) 11.9475 0.605761 0.302880 0.953029i \(-0.402052\pi\)
0.302880 + 0.953029i \(0.402052\pi\)
\(390\) 0 0
\(391\) 7.14966 0.361574
\(392\) 0 0
\(393\) −0.910365 −0.0459218
\(394\) 0 0
\(395\) −5.79713 −0.291685
\(396\) 0 0
\(397\) −30.8350 −1.54757 −0.773783 0.633451i \(-0.781638\pi\)
−0.773783 + 0.633451i \(0.781638\pi\)
\(398\) 0 0
\(399\) 7.01614 0.351247
\(400\) 0 0
\(401\) 17.5310 0.875456 0.437728 0.899108i \(-0.355783\pi\)
0.437728 + 0.899108i \(0.355783\pi\)
\(402\) 0 0
\(403\) −6.76128 −0.336804
\(404\) 0 0
\(405\) 13.9098 0.691183
\(406\) 0 0
\(407\) 3.49776 0.173378
\(408\) 0 0
\(409\) 19.5063 0.964526 0.482263 0.876027i \(-0.339815\pi\)
0.482263 + 0.876027i \(0.339815\pi\)
\(410\) 0 0
\(411\) −2.42885 −0.119806
\(412\) 0 0
\(413\) −9.01895 −0.443794
\(414\) 0 0
\(415\) 42.2854 2.07571
\(416\) 0 0
\(417\) −11.8013 −0.577912
\(418\) 0 0
\(419\) 22.1343 1.08133 0.540665 0.841238i \(-0.318173\pi\)
0.540665 + 0.841238i \(0.318173\pi\)
\(420\) 0 0
\(421\) −4.06389 −0.198062 −0.0990310 0.995084i \(-0.531574\pi\)
−0.0990310 + 0.995084i \(0.531574\pi\)
\(422\) 0 0
\(423\) −27.4236 −1.33338
\(424\) 0 0
\(425\) −9.18049 −0.445319
\(426\) 0 0
\(427\) −16.4568 −0.796400
\(428\) 0 0
\(429\) −4.78300 −0.230925
\(430\) 0 0
\(431\) −2.03357 −0.0979538 −0.0489769 0.998800i \(-0.515596\pi\)
−0.0489769 + 0.998800i \(0.515596\pi\)
\(432\) 0 0
\(433\) 24.7584 1.18981 0.594907 0.803794i \(-0.297189\pi\)
0.594907 + 0.803794i \(0.297189\pi\)
\(434\) 0 0
\(435\) 5.40384 0.259094
\(436\) 0 0
\(437\) −7.25988 −0.347287
\(438\) 0 0
\(439\) −7.06163 −0.337033 −0.168517 0.985699i \(-0.553898\pi\)
−0.168517 + 0.985699i \(0.553898\pi\)
\(440\) 0 0
\(441\) 8.60087 0.409565
\(442\) 0 0
\(443\) 13.8472 0.657900 0.328950 0.944347i \(-0.393305\pi\)
0.328950 + 0.944347i \(0.393305\pi\)
\(444\) 0 0
\(445\) 9.63846 0.456907
\(446\) 0 0
\(447\) 2.48012 0.117306
\(448\) 0 0
\(449\) 10.1112 0.477179 0.238589 0.971121i \(-0.423315\pi\)
0.238589 + 0.971121i \(0.423315\pi\)
\(450\) 0 0
\(451\) −23.9154 −1.12613
\(452\) 0 0
\(453\) 9.43836 0.443453
\(454\) 0 0
\(455\) 13.6291 0.638941
\(456\) 0 0
\(457\) −9.97208 −0.466474 −0.233237 0.972420i \(-0.574932\pi\)
−0.233237 + 0.972420i \(0.574932\pi\)
\(458\) 0 0
\(459\) −20.1360 −0.939867
\(460\) 0 0
\(461\) 35.8329 1.66890 0.834452 0.551081i \(-0.185785\pi\)
0.834452 + 0.551081i \(0.185785\pi\)
\(462\) 0 0
\(463\) 13.2210 0.614432 0.307216 0.951640i \(-0.400603\pi\)
0.307216 + 0.951640i \(0.400603\pi\)
\(464\) 0 0
\(465\) −4.09432 −0.189870
\(466\) 0 0
\(467\) −20.9241 −0.968252 −0.484126 0.874998i \(-0.660862\pi\)
−0.484126 + 0.874998i \(0.660862\pi\)
\(468\) 0 0
\(469\) −23.0446 −1.06410
\(470\) 0 0
\(471\) 2.72224 0.125434
\(472\) 0 0
\(473\) −5.99846 −0.275810
\(474\) 0 0
\(475\) 9.32201 0.427723
\(476\) 0 0
\(477\) −19.8604 −0.909347
\(478\) 0 0
\(479\) 35.6248 1.62774 0.813870 0.581048i \(-0.197357\pi\)
0.813870 + 0.581048i \(0.197357\pi\)
\(480\) 0 0
\(481\) 3.61260 0.164720
\(482\) 0 0
\(483\) −1.59417 −0.0725374
\(484\) 0 0
\(485\) −4.75143 −0.215751
\(486\) 0 0
\(487\) −10.1742 −0.461037 −0.230518 0.973068i \(-0.574042\pi\)
−0.230518 + 0.973068i \(0.574042\pi\)
\(488\) 0 0
\(489\) 10.8754 0.491804
\(490\) 0 0
\(491\) 7.24937 0.327160 0.163580 0.986530i \(-0.447696\pi\)
0.163580 + 0.986530i \(0.447696\pi\)
\(492\) 0 0
\(493\) 17.9955 0.810478
\(494\) 0 0
\(495\) 17.7794 0.799126
\(496\) 0 0
\(497\) −17.3768 −0.779455
\(498\) 0 0
\(499\) −16.4136 −0.734773 −0.367386 0.930068i \(-0.619747\pi\)
−0.367386 + 0.930068i \(0.619747\pi\)
\(500\) 0 0
\(501\) 2.74616 0.122689
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −6.10739 −0.271775
\(506\) 0 0
\(507\) 3.48749 0.154885
\(508\) 0 0
\(509\) 25.8893 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(510\) 0 0
\(511\) 4.00359 0.177109
\(512\) 0 0
\(513\) 20.4464 0.902730
\(514\) 0 0
\(515\) −14.9716 −0.659728
\(516\) 0 0
\(517\) −28.4123 −1.24957
\(518\) 0 0
\(519\) 5.13685 0.225483
\(520\) 0 0
\(521\) −12.9141 −0.565779 −0.282889 0.959153i \(-0.591293\pi\)
−0.282889 + 0.959153i \(0.591293\pi\)
\(522\) 0 0
\(523\) 25.6441 1.12134 0.560670 0.828040i \(-0.310544\pi\)
0.560670 + 0.828040i \(0.310544\pi\)
\(524\) 0 0
\(525\) 2.04699 0.0893380
\(526\) 0 0
\(527\) −13.6347 −0.593936
\(528\) 0 0
\(529\) −21.3504 −0.928280
\(530\) 0 0
\(531\) −12.1517 −0.527338
\(532\) 0 0
\(533\) −24.7005 −1.06990
\(534\) 0 0
\(535\) −7.75073 −0.335093
\(536\) 0 0
\(537\) 2.56375 0.110634
\(538\) 0 0
\(539\) 8.91093 0.383821
\(540\) 0 0
\(541\) −20.0219 −0.860809 −0.430404 0.902636i \(-0.641629\pi\)
−0.430404 + 0.902636i \(0.641629\pi\)
\(542\) 0 0
\(543\) −12.4387 −0.533796
\(544\) 0 0
\(545\) 46.9827 2.01252
\(546\) 0 0
\(547\) 30.4075 1.30013 0.650066 0.759878i \(-0.274741\pi\)
0.650066 + 0.759878i \(0.274741\pi\)
\(548\) 0 0
\(549\) −22.1731 −0.946323
\(550\) 0 0
\(551\) −18.2729 −0.778454
\(552\) 0 0
\(553\) 4.30453 0.183047
\(554\) 0 0
\(555\) 2.18762 0.0928594
\(556\) 0 0
\(557\) −30.0571 −1.27356 −0.636781 0.771045i \(-0.719734\pi\)
−0.636781 + 0.771045i \(0.719734\pi\)
\(558\) 0 0
\(559\) −6.19539 −0.262037
\(560\) 0 0
\(561\) −9.64531 −0.407225
\(562\) 0 0
\(563\) −8.88828 −0.374596 −0.187298 0.982303i \(-0.559973\pi\)
−0.187298 + 0.982303i \(0.559973\pi\)
\(564\) 0 0
\(565\) −53.2735 −2.24123
\(566\) 0 0
\(567\) −10.3284 −0.433752
\(568\) 0 0
\(569\) 5.15804 0.216236 0.108118 0.994138i \(-0.465518\pi\)
0.108118 + 0.994138i \(0.465518\pi\)
\(570\) 0 0
\(571\) 24.5299 1.02654 0.513271 0.858227i \(-0.328434\pi\)
0.513271 + 0.858227i \(0.328434\pi\)
\(572\) 0 0
\(573\) 2.20462 0.0920995
\(574\) 0 0
\(575\) −2.11810 −0.0883309
\(576\) 0 0
\(577\) 43.6288 1.81629 0.908146 0.418653i \(-0.137498\pi\)
0.908146 + 0.418653i \(0.137498\pi\)
\(578\) 0 0
\(579\) 7.04651 0.292843
\(580\) 0 0
\(581\) −31.3981 −1.30261
\(582\) 0 0
\(583\) −20.5764 −0.852187
\(584\) 0 0
\(585\) 18.3631 0.759222
\(586\) 0 0
\(587\) 7.96110 0.328590 0.164295 0.986411i \(-0.447465\pi\)
0.164295 + 0.986411i \(0.447465\pi\)
\(588\) 0 0
\(589\) 13.8449 0.570468
\(590\) 0 0
\(591\) −8.03152 −0.330372
\(592\) 0 0
\(593\) −21.0999 −0.866471 −0.433235 0.901281i \(-0.642628\pi\)
−0.433235 + 0.901281i \(0.642628\pi\)
\(594\) 0 0
\(595\) 27.4841 1.12674
\(596\) 0 0
\(597\) 6.31669 0.258525
\(598\) 0 0
\(599\) 33.8856 1.38453 0.692264 0.721645i \(-0.256614\pi\)
0.692264 + 0.721645i \(0.256614\pi\)
\(600\) 0 0
\(601\) 42.7178 1.74249 0.871247 0.490845i \(-0.163312\pi\)
0.871247 + 0.490845i \(0.163312\pi\)
\(602\) 0 0
\(603\) −31.0491 −1.26442
\(604\) 0 0
\(605\) −9.94419 −0.404289
\(606\) 0 0
\(607\) −5.42273 −0.220102 −0.110051 0.993926i \(-0.535101\pi\)
−0.110051 + 0.993926i \(0.535101\pi\)
\(608\) 0 0
\(609\) −4.01250 −0.162595
\(610\) 0 0
\(611\) −29.3450 −1.18717
\(612\) 0 0
\(613\) −23.9547 −0.967522 −0.483761 0.875200i \(-0.660730\pi\)
−0.483761 + 0.875200i \(0.660730\pi\)
\(614\) 0 0
\(615\) −14.9575 −0.603145
\(616\) 0 0
\(617\) 43.1866 1.73863 0.869313 0.494261i \(-0.164561\pi\)
0.869313 + 0.494261i \(0.164561\pi\)
\(618\) 0 0
\(619\) −30.5985 −1.22986 −0.614929 0.788583i \(-0.710815\pi\)
−0.614929 + 0.788583i \(0.710815\pi\)
\(620\) 0 0
\(621\) −4.64573 −0.186427
\(622\) 0 0
\(623\) −7.15682 −0.286732
\(624\) 0 0
\(625\) −30.5261 −1.22104
\(626\) 0 0
\(627\) 9.79399 0.391134
\(628\) 0 0
\(629\) 7.28509 0.290476
\(630\) 0 0
\(631\) −24.9223 −0.992140 −0.496070 0.868283i \(-0.665224\pi\)
−0.496070 + 0.868283i \(0.665224\pi\)
\(632\) 0 0
\(633\) 3.33691 0.132630
\(634\) 0 0
\(635\) 12.1919 0.483823
\(636\) 0 0
\(637\) 9.20348 0.364655
\(638\) 0 0
\(639\) −23.4126 −0.926189
\(640\) 0 0
\(641\) −1.88975 −0.0746405 −0.0373202 0.999303i \(-0.511882\pi\)
−0.0373202 + 0.999303i \(0.511882\pi\)
\(642\) 0 0
\(643\) −23.4524 −0.924871 −0.462435 0.886653i \(-0.653024\pi\)
−0.462435 + 0.886653i \(0.653024\pi\)
\(644\) 0 0
\(645\) −3.75165 −0.147721
\(646\) 0 0
\(647\) 10.0224 0.394021 0.197011 0.980401i \(-0.436877\pi\)
0.197011 + 0.980401i \(0.436877\pi\)
\(648\) 0 0
\(649\) −12.5898 −0.494191
\(650\) 0 0
\(651\) 3.04015 0.119153
\(652\) 0 0
\(653\) −28.4614 −1.11378 −0.556890 0.830586i \(-0.688006\pi\)
−0.556890 + 0.830586i \(0.688006\pi\)
\(654\) 0 0
\(655\) 3.62112 0.141489
\(656\) 0 0
\(657\) 5.39424 0.210450
\(658\) 0 0
\(659\) 35.7038 1.39082 0.695410 0.718613i \(-0.255223\pi\)
0.695410 + 0.718613i \(0.255223\pi\)
\(660\) 0 0
\(661\) 4.00684 0.155848 0.0779240 0.996959i \(-0.475171\pi\)
0.0779240 + 0.996959i \(0.475171\pi\)
\(662\) 0 0
\(663\) −9.96196 −0.386891
\(664\) 0 0
\(665\) −27.9078 −1.08222
\(666\) 0 0
\(667\) 4.15189 0.160762
\(668\) 0 0
\(669\) −9.59121 −0.370817
\(670\) 0 0
\(671\) −22.9724 −0.886840
\(672\) 0 0
\(673\) −1.82474 −0.0703386 −0.0351693 0.999381i \(-0.511197\pi\)
−0.0351693 + 0.999381i \(0.511197\pi\)
\(674\) 0 0
\(675\) 5.96533 0.229605
\(676\) 0 0
\(677\) 8.80117 0.338257 0.169128 0.985594i \(-0.445905\pi\)
0.169128 + 0.985594i \(0.445905\pi\)
\(678\) 0 0
\(679\) 3.52807 0.135395
\(680\) 0 0
\(681\) 3.65631 0.140110
\(682\) 0 0
\(683\) −12.9409 −0.495171 −0.247585 0.968866i \(-0.579637\pi\)
−0.247585 + 0.968866i \(0.579637\pi\)
\(684\) 0 0
\(685\) 9.66113 0.369133
\(686\) 0 0
\(687\) −0.721828 −0.0275395
\(688\) 0 0
\(689\) −21.2519 −0.809634
\(690\) 0 0
\(691\) 19.3649 0.736675 0.368337 0.929692i \(-0.379927\pi\)
0.368337 + 0.929692i \(0.379927\pi\)
\(692\) 0 0
\(693\) −13.2017 −0.501492
\(694\) 0 0
\(695\) 46.9415 1.78059
\(696\) 0 0
\(697\) −49.8106 −1.88671
\(698\) 0 0
\(699\) 13.2281 0.500334
\(700\) 0 0
\(701\) 19.0601 0.719888 0.359944 0.932974i \(-0.382796\pi\)
0.359944 + 0.932974i \(0.382796\pi\)
\(702\) 0 0
\(703\) −7.39740 −0.278998
\(704\) 0 0
\(705\) −17.7700 −0.669258
\(706\) 0 0
\(707\) 4.53491 0.170553
\(708\) 0 0
\(709\) −37.6676 −1.41464 −0.707319 0.706895i \(-0.750095\pi\)
−0.707319 + 0.706895i \(0.750095\pi\)
\(710\) 0 0
\(711\) 5.79972 0.217506
\(712\) 0 0
\(713\) −3.14576 −0.117810
\(714\) 0 0
\(715\) 19.0251 0.711499
\(716\) 0 0
\(717\) −5.21909 −0.194911
\(718\) 0 0
\(719\) −38.9875 −1.45399 −0.726994 0.686644i \(-0.759083\pi\)
−0.726994 + 0.686644i \(0.759083\pi\)
\(720\) 0 0
\(721\) 11.1168 0.414013
\(722\) 0 0
\(723\) −8.50202 −0.316193
\(724\) 0 0
\(725\) −5.33121 −0.197996
\(726\) 0 0
\(727\) 1.29239 0.0479321 0.0239660 0.999713i \(-0.492371\pi\)
0.0239660 + 0.999713i \(0.492371\pi\)
\(728\) 0 0
\(729\) −6.88124 −0.254861
\(730\) 0 0
\(731\) −12.4935 −0.462089
\(732\) 0 0
\(733\) −48.9143 −1.80669 −0.903345 0.428915i \(-0.858896\pi\)
−0.903345 + 0.428915i \(0.858896\pi\)
\(734\) 0 0
\(735\) 5.57321 0.205571
\(736\) 0 0
\(737\) −32.1684 −1.18494
\(738\) 0 0
\(739\) −29.6610 −1.09110 −0.545548 0.838079i \(-0.683678\pi\)
−0.545548 + 0.838079i \(0.683678\pi\)
\(740\) 0 0
\(741\) 10.1155 0.371603
\(742\) 0 0
\(743\) 36.9188 1.35442 0.677210 0.735790i \(-0.263189\pi\)
0.677210 + 0.735790i \(0.263189\pi\)
\(744\) 0 0
\(745\) −9.86508 −0.361429
\(746\) 0 0
\(747\) −42.3042 −1.54783
\(748\) 0 0
\(749\) 5.75514 0.210288
\(750\) 0 0
\(751\) −17.2117 −0.628063 −0.314031 0.949413i \(-0.601680\pi\)
−0.314031 + 0.949413i \(0.601680\pi\)
\(752\) 0 0
\(753\) 4.34625 0.158386
\(754\) 0 0
\(755\) −37.5426 −1.36631
\(756\) 0 0
\(757\) 9.80869 0.356503 0.178251 0.983985i \(-0.442956\pi\)
0.178251 + 0.983985i \(0.442956\pi\)
\(758\) 0 0
\(759\) −2.22534 −0.0807748
\(760\) 0 0
\(761\) 19.0180 0.689403 0.344702 0.938712i \(-0.387980\pi\)
0.344702 + 0.938712i \(0.387980\pi\)
\(762\) 0 0
\(763\) −34.8860 −1.26296
\(764\) 0 0
\(765\) 37.0308 1.33885
\(766\) 0 0
\(767\) −13.0031 −0.469514
\(768\) 0 0
\(769\) −32.2218 −1.16195 −0.580974 0.813922i \(-0.697328\pi\)
−0.580974 + 0.813922i \(0.697328\pi\)
\(770\) 0 0
\(771\) −12.7811 −0.460300
\(772\) 0 0
\(773\) 22.1222 0.795680 0.397840 0.917455i \(-0.369760\pi\)
0.397840 + 0.917455i \(0.369760\pi\)
\(774\) 0 0
\(775\) 4.03930 0.145096
\(776\) 0 0
\(777\) −1.62437 −0.0582740
\(778\) 0 0
\(779\) 50.5784 1.81216
\(780\) 0 0
\(781\) −24.2566 −0.867971
\(782\) 0 0
\(783\) −11.6932 −0.417881
\(784\) 0 0
\(785\) −10.8281 −0.386473
\(786\) 0 0
\(787\) 14.8853 0.530604 0.265302 0.964165i \(-0.414528\pi\)
0.265302 + 0.964165i \(0.414528\pi\)
\(788\) 0 0
\(789\) 1.81834 0.0647347
\(790\) 0 0
\(791\) 39.5570 1.40649
\(792\) 0 0
\(793\) −23.7266 −0.842556
\(794\) 0 0
\(795\) −12.8692 −0.456423
\(796\) 0 0
\(797\) 8.10523 0.287102 0.143551 0.989643i \(-0.454148\pi\)
0.143551 + 0.989643i \(0.454148\pi\)
\(798\) 0 0
\(799\) −59.1767 −2.09352
\(800\) 0 0
\(801\) −9.64275 −0.340710
\(802\) 0 0
\(803\) 5.58871 0.197221
\(804\) 0 0
\(805\) 6.34108 0.223494
\(806\) 0 0
\(807\) −7.02043 −0.247131
\(808\) 0 0
\(809\) −45.5535 −1.60158 −0.800788 0.598947i \(-0.795586\pi\)
−0.800788 + 0.598947i \(0.795586\pi\)
\(810\) 0 0
\(811\) 25.3463 0.890030 0.445015 0.895523i \(-0.353198\pi\)
0.445015 + 0.895523i \(0.353198\pi\)
\(812\) 0 0
\(813\) −1.93403 −0.0678294
\(814\) 0 0
\(815\) −43.2587 −1.51529
\(816\) 0 0
\(817\) 12.6861 0.443831
\(818\) 0 0
\(819\) −13.6351 −0.476450
\(820\) 0 0
\(821\) −17.1849 −0.599757 −0.299879 0.953977i \(-0.596946\pi\)
−0.299879 + 0.953977i \(0.596946\pi\)
\(822\) 0 0
\(823\) −16.9285 −0.590091 −0.295045 0.955483i \(-0.595335\pi\)
−0.295045 + 0.955483i \(0.595335\pi\)
\(824\) 0 0
\(825\) 2.85744 0.0994833
\(826\) 0 0
\(827\) 48.2882 1.67914 0.839572 0.543248i \(-0.182806\pi\)
0.839572 + 0.543248i \(0.182806\pi\)
\(828\) 0 0
\(829\) −34.4609 −1.19688 −0.598438 0.801169i \(-0.704212\pi\)
−0.598438 + 0.801169i \(0.704212\pi\)
\(830\) 0 0
\(831\) −6.45743 −0.224006
\(832\) 0 0
\(833\) 18.5596 0.643051
\(834\) 0 0
\(835\) −10.9233 −0.378016
\(836\) 0 0
\(837\) 8.85958 0.306232
\(838\) 0 0
\(839\) −21.3049 −0.735528 −0.367764 0.929919i \(-0.619877\pi\)
−0.367764 + 0.929919i \(0.619877\pi\)
\(840\) 0 0
\(841\) −18.5498 −0.639648
\(842\) 0 0
\(843\) 1.19594 0.0411904
\(844\) 0 0
\(845\) −13.8720 −0.477212
\(846\) 0 0
\(847\) 7.38384 0.253712
\(848\) 0 0
\(849\) −19.2977 −0.662294
\(850\) 0 0
\(851\) 1.68080 0.0576171
\(852\) 0 0
\(853\) 42.9581 1.47086 0.735429 0.677602i \(-0.236981\pi\)
0.735429 + 0.677602i \(0.236981\pi\)
\(854\) 0 0
\(855\) −37.6016 −1.28595
\(856\) 0 0
\(857\) 25.6944 0.877704 0.438852 0.898559i \(-0.355385\pi\)
0.438852 + 0.898559i \(0.355385\pi\)
\(858\) 0 0
\(859\) 5.84616 0.199469 0.0997343 0.995014i \(-0.468201\pi\)
0.0997343 + 0.995014i \(0.468201\pi\)
\(860\) 0 0
\(861\) 11.1064 0.378504
\(862\) 0 0
\(863\) −15.1925 −0.517159 −0.258580 0.965990i \(-0.583254\pi\)
−0.258580 + 0.965990i \(0.583254\pi\)
\(864\) 0 0
\(865\) −20.4326 −0.694730
\(866\) 0 0
\(867\) −9.06851 −0.307983
\(868\) 0 0
\(869\) 6.00879 0.203834
\(870\) 0 0
\(871\) −33.2245 −1.12577
\(872\) 0 0
\(873\) 4.75355 0.160883
\(874\) 0 0
\(875\) 16.5437 0.559280
\(876\) 0 0
\(877\) −4.27552 −0.144374 −0.0721871 0.997391i \(-0.522998\pi\)
−0.0721871 + 0.997391i \(0.522998\pi\)
\(878\) 0 0
\(879\) −13.4288 −0.452943
\(880\) 0 0
\(881\) −31.3696 −1.05687 −0.528434 0.848974i \(-0.677221\pi\)
−0.528434 + 0.848974i \(0.677221\pi\)
\(882\) 0 0
\(883\) −35.4660 −1.19353 −0.596763 0.802417i \(-0.703547\pi\)
−0.596763 + 0.802417i \(0.703547\pi\)
\(884\) 0 0
\(885\) −7.87407 −0.264684
\(886\) 0 0
\(887\) 36.5445 1.22704 0.613522 0.789678i \(-0.289752\pi\)
0.613522 + 0.789678i \(0.289752\pi\)
\(888\) 0 0
\(889\) −9.05286 −0.303623
\(890\) 0 0
\(891\) −14.4177 −0.483010
\(892\) 0 0
\(893\) 60.0889 2.01080
\(894\) 0 0
\(895\) −10.1977 −0.340873
\(896\) 0 0
\(897\) −2.29840 −0.0767414
\(898\) 0 0
\(899\) −7.91781 −0.264074
\(900\) 0 0
\(901\) −42.8562 −1.42775
\(902\) 0 0
\(903\) 2.78570 0.0927024
\(904\) 0 0
\(905\) 49.4769 1.64467
\(906\) 0 0
\(907\) 13.8754 0.460724 0.230362 0.973105i \(-0.426009\pi\)
0.230362 + 0.973105i \(0.426009\pi\)
\(908\) 0 0
\(909\) 6.11011 0.202659
\(910\) 0 0
\(911\) −14.1478 −0.468737 −0.234369 0.972148i \(-0.575302\pi\)
−0.234369 + 0.972148i \(0.575302\pi\)
\(912\) 0 0
\(913\) −43.8293 −1.45054
\(914\) 0 0
\(915\) −14.3677 −0.474983
\(916\) 0 0
\(917\) −2.68878 −0.0887914
\(918\) 0 0
\(919\) 40.9761 1.35168 0.675838 0.737050i \(-0.263782\pi\)
0.675838 + 0.737050i \(0.263782\pi\)
\(920\) 0 0
\(921\) 7.33825 0.241803
\(922\) 0 0
\(923\) −25.0530 −0.824629
\(924\) 0 0
\(925\) −2.15822 −0.0709619
\(926\) 0 0
\(927\) 14.9783 0.491951
\(928\) 0 0
\(929\) 15.8227 0.519125 0.259562 0.965726i \(-0.416422\pi\)
0.259562 + 0.965726i \(0.416422\pi\)
\(930\) 0 0
\(931\) −18.8457 −0.617642
\(932\) 0 0
\(933\) −13.4171 −0.439255
\(934\) 0 0
\(935\) 38.3657 1.25469
\(936\) 0 0
\(937\) −4.12795 −0.134854 −0.0674271 0.997724i \(-0.521479\pi\)
−0.0674271 + 0.997724i \(0.521479\pi\)
\(938\) 0 0
\(939\) −21.1579 −0.690462
\(940\) 0 0
\(941\) 44.5172 1.45122 0.725609 0.688107i \(-0.241558\pi\)
0.725609 + 0.688107i \(0.241558\pi\)
\(942\) 0 0
\(943\) −11.4922 −0.374237
\(944\) 0 0
\(945\) −17.8587 −0.580944
\(946\) 0 0
\(947\) 33.5361 1.08978 0.544889 0.838508i \(-0.316572\pi\)
0.544889 + 0.838508i \(0.316572\pi\)
\(948\) 0 0
\(949\) 5.77219 0.187373
\(950\) 0 0
\(951\) 9.32094 0.302252
\(952\) 0 0
\(953\) 23.1278 0.749183 0.374591 0.927190i \(-0.377783\pi\)
0.374591 + 0.927190i \(0.377783\pi\)
\(954\) 0 0
\(955\) −8.76924 −0.283766
\(956\) 0 0
\(957\) −5.60114 −0.181059
\(958\) 0 0
\(959\) −7.17366 −0.231650
\(960\) 0 0
\(961\) −25.0009 −0.806481
\(962\) 0 0
\(963\) 7.75419 0.249875
\(964\) 0 0
\(965\) −28.0286 −0.902272
\(966\) 0 0
\(967\) −59.9784 −1.92878 −0.964388 0.264491i \(-0.914796\pi\)
−0.964388 + 0.264491i \(0.914796\pi\)
\(968\) 0 0
\(969\) 20.3988 0.655304
\(970\) 0 0
\(971\) 44.9981 1.44406 0.722029 0.691862i \(-0.243210\pi\)
0.722029 + 0.691862i \(0.243210\pi\)
\(972\) 0 0
\(973\) −34.8554 −1.11741
\(974\) 0 0
\(975\) 2.95125 0.0945157
\(976\) 0 0
\(977\) 3.02251 0.0966988 0.0483494 0.998830i \(-0.484604\pi\)
0.0483494 + 0.998830i \(0.484604\pi\)
\(978\) 0 0
\(979\) −9.99037 −0.319294
\(980\) 0 0
\(981\) −47.0036 −1.50071
\(982\) 0 0
\(983\) 24.1315 0.769677 0.384838 0.922984i \(-0.374257\pi\)
0.384838 + 0.922984i \(0.374257\pi\)
\(984\) 0 0
\(985\) 31.9466 1.01790
\(986\) 0 0
\(987\) 13.1947 0.419993
\(988\) 0 0
\(989\) −2.88247 −0.0916573
\(990\) 0 0
\(991\) 57.1238 1.81460 0.907299 0.420487i \(-0.138141\pi\)
0.907299 + 0.420487i \(0.138141\pi\)
\(992\) 0 0
\(993\) 13.6855 0.434297
\(994\) 0 0
\(995\) −25.1256 −0.796535
\(996\) 0 0
\(997\) 23.0433 0.729790 0.364895 0.931049i \(-0.381105\pi\)
0.364895 + 0.931049i \(0.381105\pi\)
\(998\) 0 0
\(999\) −4.73373 −0.149769
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 4024.2.a.g.1.13 33
4.3 odd 2 8048.2.a.x.1.21 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.g.1.13 33 1.1 even 1 trivial
8048.2.a.x.1.21 33 4.3 odd 2