Properties

Label 4600.2.a.bb.1.2
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $1$
Dimension $4$
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] = [4600,2,Mod(1,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.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: \(36.7311849298\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.15529.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.491317\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.329452 q^{3} +4.07069 q^{7} -2.89146 q^{9} -3.08806 q^{11} +6.32062 q^{13} -0.982634 q^{17} -7.08806 q^{19} +1.34110 q^{21} -1.00000 q^{23} -1.94095 q^{27} -7.63270 q^{29} -5.92047 q^{31} -1.01737 q^{33} -11.4825 q^{37} +2.08234 q^{39} -5.74124 q^{41} -1.74696 q^{43} -2.24992 q^{47} +9.57054 q^{49} -0.323730 q^{51} -1.31781 q^{53} -2.33517 q^{57} +7.92901 q^{59} +9.51722 q^{61} -11.7703 q^{63} +3.34110 q^{67} -0.329452 q^{69} -3.92619 q^{71} +1.66744 q^{73} -12.5705 q^{77} -15.9231 q^{79} +8.03493 q^{81} +11.9116 q^{83} -2.51461 q^{87} -10.4998 q^{89} +25.7293 q^{91} -1.95051 q^{93} -2.84125 q^{97} +8.92901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{7} + 6 q^{9} + q^{11} - 3 q^{13} - 2 q^{17} - 15 q^{19} + 8 q^{21} - 4 q^{23} - 27 q^{27} + q^{29} - 12 q^{31} - 6 q^{33} - 18 q^{37} - 3 q^{39} - 9 q^{41} + 9 q^{43} + 4 q^{47} - 3 q^{49} - 2 q^{51}+ \cdots - q^{99}+O(q^{100}) \) 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
\(3\) 0.329452 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.07069 1.53858 0.769289 0.638901i \(-0.220611\pi\)
0.769289 + 0.638901i \(0.220611\pi\)
\(8\) 0 0
\(9\) −2.89146 −0.963821
\(10\) 0 0
\(11\) −3.08806 −0.931085 −0.465542 0.885026i \(-0.654141\pi\)
−0.465542 + 0.885026i \(0.654141\pi\)
\(12\) 0 0
\(13\) 6.32062 1.75302 0.876512 0.481380i \(-0.159864\pi\)
0.876512 + 0.481380i \(0.159864\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.982634 −0.238324 −0.119162 0.992875i \(-0.538021\pi\)
−0.119162 + 0.992875i \(0.538021\pi\)
\(18\) 0 0
\(19\) −7.08806 −1.62611 −0.813056 0.582185i \(-0.802198\pi\)
−0.813056 + 0.582185i \(0.802198\pi\)
\(20\) 0 0
\(21\) 1.34110 0.292651
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.94095 −0.373536
\(28\) 0 0
\(29\) −7.63270 −1.41736 −0.708679 0.705531i \(-0.750708\pi\)
−0.708679 + 0.705531i \(0.750708\pi\)
\(30\) 0 0
\(31\) −5.92047 −1.06335 −0.531674 0.846949i \(-0.678437\pi\)
−0.531674 + 0.846949i \(0.678437\pi\)
\(32\) 0 0
\(33\) −1.01737 −0.177101
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.4825 −1.88771 −0.943854 0.330362i \(-0.892829\pi\)
−0.943854 + 0.330362i \(0.892829\pi\)
\(38\) 0 0
\(39\) 2.08234 0.333441
\(40\) 0 0
\(41\) −5.74124 −0.896631 −0.448316 0.893875i \(-0.647976\pi\)
−0.448316 + 0.893875i \(0.647976\pi\)
\(42\) 0 0
\(43\) −1.74696 −0.266409 −0.133205 0.991089i \(-0.542527\pi\)
−0.133205 + 0.991089i \(0.542527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.24992 −0.328185 −0.164093 0.986445i \(-0.552470\pi\)
−0.164093 + 0.986445i \(0.552470\pi\)
\(48\) 0 0
\(49\) 9.57054 1.36722
\(50\) 0 0
\(51\) −0.323730 −0.0453313
\(52\) 0 0
\(53\) −1.31781 −0.181015 −0.0905073 0.995896i \(-0.528849\pi\)
−0.0905073 + 0.995896i \(0.528849\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.33517 −0.309301
\(58\) 0 0
\(59\) 7.92901 1.03227 0.516134 0.856508i \(-0.327371\pi\)
0.516134 + 0.856508i \(0.327371\pi\)
\(60\) 0 0
\(61\) 9.51722 1.21855 0.609277 0.792957i \(-0.291460\pi\)
0.609277 + 0.792957i \(0.291460\pi\)
\(62\) 0 0
\(63\) −11.7703 −1.48291
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.34110 0.408180 0.204090 0.978952i \(-0.434576\pi\)
0.204090 + 0.978952i \(0.434576\pi\)
\(68\) 0 0
\(69\) −0.329452 −0.0396613
\(70\) 0 0
\(71\) −3.92619 −0.465954 −0.232977 0.972482i \(-0.574847\pi\)
−0.232977 + 0.972482i \(0.574847\pi\)
\(72\) 0 0
\(73\) 1.66744 0.195159 0.0975793 0.995228i \(-0.468890\pi\)
0.0975793 + 0.995228i \(0.468890\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.5705 −1.43255
\(78\) 0 0
\(79\) −15.9231 −1.79149 −0.895743 0.444572i \(-0.853356\pi\)
−0.895743 + 0.444572i \(0.853356\pi\)
\(80\) 0 0
\(81\) 8.03493 0.892771
\(82\) 0 0
\(83\) 11.9116 1.30747 0.653736 0.756723i \(-0.273201\pi\)
0.653736 + 0.756723i \(0.273201\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.51461 −0.269594
\(88\) 0 0
\(89\) −10.4998 −1.11298 −0.556491 0.830854i \(-0.687853\pi\)
−0.556491 + 0.830854i \(0.687853\pi\)
\(90\) 0 0
\(91\) 25.7293 2.69716
\(92\) 0 0
\(93\) −1.95051 −0.202258
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.84125 −0.288485 −0.144242 0.989542i \(-0.546075\pi\)
−0.144242 + 0.989542i \(0.546075\pi\)
\(98\) 0 0
\(99\) 8.92901 0.897399
\(100\) 0 0
\(101\) 4.74696 0.472340 0.236170 0.971712i \(-0.424108\pi\)
0.236170 + 0.971712i \(0.424108\pi\)
\(102\) 0 0
\(103\) −5.05333 −0.497919 −0.248960 0.968514i \(-0.580089\pi\)
−0.248960 + 0.968514i \(0.580089\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 10.1067 0.968042 0.484021 0.875056i \(-0.339176\pi\)
0.484021 + 0.875056i \(0.339176\pi\)
\(110\) 0 0
\(111\) −3.78292 −0.359059
\(112\) 0 0
\(113\) −0.335173 −0.0315304 −0.0157652 0.999876i \(-0.505018\pi\)
−0.0157652 + 0.999876i \(0.505018\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −18.2758 −1.68960
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −1.46389 −0.133081
\(122\) 0 0
\(123\) −1.89146 −0.170547
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −5.92047 −0.525357 −0.262679 0.964883i \(-0.584606\pi\)
−0.262679 + 0.964883i \(0.584606\pi\)
\(128\) 0 0
\(129\) −0.575540 −0.0506735
\(130\) 0 0
\(131\) 7.02309 0.613610 0.306805 0.951772i \(-0.400740\pi\)
0.306805 + 0.951772i \(0.400740\pi\)
\(132\) 0 0
\(133\) −28.8533 −2.50190
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.1414 −1.03731 −0.518654 0.854984i \(-0.673567\pi\)
−0.518654 + 0.854984i \(0.673567\pi\)
\(138\) 0 0
\(139\) −14.2499 −1.20866 −0.604331 0.796733i \(-0.706560\pi\)
−0.604331 + 0.796733i \(0.706560\pi\)
\(140\) 0 0
\(141\) −0.741241 −0.0624238
\(142\) 0 0
\(143\) −19.5184 −1.63221
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.15303 0.260058
\(148\) 0 0
\(149\) −18.7826 −1.53873 −0.769366 0.638808i \(-0.779428\pi\)
−0.769366 + 0.638808i \(0.779428\pi\)
\(150\) 0 0
\(151\) 12.3566 1.00556 0.502782 0.864413i \(-0.332310\pi\)
0.502782 + 0.864413i \(0.332310\pi\)
\(152\) 0 0
\(153\) 2.84125 0.229701
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.9997 1.03749 0.518744 0.854929i \(-0.326400\pi\)
0.518744 + 0.854929i \(0.326400\pi\)
\(158\) 0 0
\(159\) −0.434154 −0.0344306
\(160\) 0 0
\(161\) −4.07069 −0.320816
\(162\) 0 0
\(163\) 12.5384 0.982085 0.491042 0.871136i \(-0.336616\pi\)
0.491042 + 0.871136i \(0.336616\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.2121 0.867617 0.433808 0.901005i \(-0.357169\pi\)
0.433808 + 0.901005i \(0.357169\pi\)
\(168\) 0 0
\(169\) 26.9502 2.07309
\(170\) 0 0
\(171\) 20.4949 1.56728
\(172\) 0 0
\(173\) −17.2133 −1.30870 −0.654352 0.756190i \(-0.727058\pi\)
−0.654352 + 0.756190i \(0.727058\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.61222 0.196347
\(178\) 0 0
\(179\) −25.1859 −1.88248 −0.941240 0.337737i \(-0.890339\pi\)
−0.941240 + 0.337737i \(0.890339\pi\)
\(180\) 0 0
\(181\) 15.3411 1.14029 0.570147 0.821542i \(-0.306886\pi\)
0.570147 + 0.821542i \(0.306886\pi\)
\(182\) 0 0
\(183\) 3.13546 0.231780
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.03443 0.221900
\(188\) 0 0
\(189\) −7.90102 −0.574715
\(190\) 0 0
\(191\) −17.2232 −1.24623 −0.623114 0.782131i \(-0.714133\pi\)
−0.623114 + 0.782131i \(0.714133\pi\)
\(192\) 0 0
\(193\) 9.30064 0.669475 0.334737 0.942311i \(-0.391352\pi\)
0.334737 + 0.942311i \(0.391352\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.8938 1.63111 0.815557 0.578677i \(-0.196431\pi\)
0.815557 + 0.578677i \(0.196431\pi\)
\(198\) 0 0
\(199\) −6.35969 −0.450827 −0.225413 0.974263i \(-0.572373\pi\)
−0.225413 + 0.974263i \(0.572373\pi\)
\(200\) 0 0
\(201\) 1.10073 0.0776395
\(202\) 0 0
\(203\) −31.0704 −2.18071
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.89146 0.200970
\(208\) 0 0
\(209\) 21.8883 1.51405
\(210\) 0 0
\(211\) −10.3656 −0.713598 −0.356799 0.934181i \(-0.616132\pi\)
−0.356799 + 0.934181i \(0.616132\pi\)
\(212\) 0 0
\(213\) −1.29349 −0.0886286
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −24.1004 −1.63604
\(218\) 0 0
\(219\) 0.549339 0.0371209
\(220\) 0 0
\(221\) −6.21085 −0.417787
\(222\) 0 0
\(223\) 5.75289 0.385242 0.192621 0.981273i \(-0.438301\pi\)
0.192621 + 0.981273i \(0.438301\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.49393 −0.630134 −0.315067 0.949069i \(-0.602027\pi\)
−0.315067 + 0.949069i \(0.602027\pi\)
\(228\) 0 0
\(229\) 1.83532 0.121282 0.0606408 0.998160i \(-0.480686\pi\)
0.0606408 + 0.998160i \(0.480686\pi\)
\(230\) 0 0
\(231\) −4.14139 −0.272483
\(232\) 0 0
\(233\) 20.5687 1.34750 0.673749 0.738960i \(-0.264683\pi\)
0.673749 + 0.738960i \(0.264683\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.24589 −0.340757
\(238\) 0 0
\(239\) −12.4014 −0.802178 −0.401089 0.916039i \(-0.631368\pi\)
−0.401089 + 0.916039i \(0.631368\pi\)
\(240\) 0 0
\(241\) 15.7829 1.01667 0.508334 0.861160i \(-0.330262\pi\)
0.508334 + 0.861160i \(0.330262\pi\)
\(242\) 0 0
\(243\) 8.46998 0.543349
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −44.8009 −2.85061
\(248\) 0 0
\(249\) 3.92431 0.248693
\(250\) 0 0
\(251\) −9.71662 −0.613308 −0.306654 0.951821i \(-0.599209\pi\)
−0.306654 + 0.951821i \(0.599209\pi\)
\(252\) 0 0
\(253\) 3.08806 0.194145
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.3115 1.20462 0.602309 0.798263i \(-0.294248\pi\)
0.602309 + 0.798263i \(0.294248\pi\)
\(258\) 0 0
\(259\) −46.7417 −2.90439
\(260\) 0 0
\(261\) 22.0697 1.36608
\(262\) 0 0
\(263\) 1.74819 0.107798 0.0538990 0.998546i \(-0.482835\pi\)
0.0538990 + 0.998546i \(0.482835\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3.45919 −0.211699
\(268\) 0 0
\(269\) 11.7079 0.713843 0.356921 0.934134i \(-0.383826\pi\)
0.356921 + 0.934134i \(0.383826\pi\)
\(270\) 0 0
\(271\) 10.1051 0.613843 0.306921 0.951735i \(-0.400701\pi\)
0.306921 + 0.951735i \(0.400701\pi\)
\(272\) 0 0
\(273\) 8.47656 0.513025
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.29319 0.318037 0.159018 0.987276i \(-0.449167\pi\)
0.159018 + 0.987276i \(0.449167\pi\)
\(278\) 0 0
\(279\) 17.1188 1.02488
\(280\) 0 0
\(281\) −13.8062 −0.823610 −0.411805 0.911272i \(-0.635101\pi\)
−0.411805 + 0.911272i \(0.635101\pi\)
\(282\) 0 0
\(283\) 7.79437 0.463327 0.231663 0.972796i \(-0.425583\pi\)
0.231663 + 0.972796i \(0.425583\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −23.3708 −1.37954
\(288\) 0 0
\(289\) −16.0344 −0.943202
\(290\) 0 0
\(291\) −0.936054 −0.0548724
\(292\) 0 0
\(293\) −24.0171 −1.40309 −0.701546 0.712624i \(-0.747506\pi\)
−0.701546 + 0.712624i \(0.747506\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.99377 0.347794
\(298\) 0 0
\(299\) −6.32062 −0.365531
\(300\) 0 0
\(301\) −7.11135 −0.409891
\(302\) 0 0
\(303\) 1.56389 0.0898434
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −1.66483 −0.0947087
\(310\) 0 0
\(311\) −26.8507 −1.52256 −0.761282 0.648421i \(-0.775430\pi\)
−0.761282 + 0.648421i \(0.775430\pi\)
\(312\) 0 0
\(313\) 31.5110 1.78111 0.890553 0.454879i \(-0.150318\pi\)
0.890553 + 0.454879i \(0.150318\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.1110 −1.24188 −0.620940 0.783858i \(-0.713249\pi\)
−0.620940 + 0.783858i \(0.713249\pi\)
\(318\) 0 0
\(319\) 23.5702 1.31968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.96497 0.387541
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.32965 0.184130
\(328\) 0 0
\(329\) −9.15875 −0.504938
\(330\) 0 0
\(331\) 23.8953 1.31340 0.656702 0.754150i \(-0.271951\pi\)
0.656702 + 0.754150i \(0.271951\pi\)
\(332\) 0 0
\(333\) 33.2012 1.81941
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −20.1647 −1.09844 −0.549220 0.835678i \(-0.685075\pi\)
−0.549220 + 0.835678i \(0.685075\pi\)
\(338\) 0 0
\(339\) −0.110423 −0.00599737
\(340\) 0 0
\(341\) 18.2828 0.990068
\(342\) 0 0
\(343\) 10.4639 0.564997
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.2465 −1.73108 −0.865542 0.500837i \(-0.833026\pi\)
−0.865542 + 0.500837i \(0.833026\pi\)
\(348\) 0 0
\(349\) 35.3203 1.89065 0.945327 0.326125i \(-0.105743\pi\)
0.945327 + 0.326125i \(0.105743\pi\)
\(350\) 0 0
\(351\) −12.2680 −0.654818
\(352\) 0 0
\(353\) −30.7409 −1.63618 −0.818088 0.575094i \(-0.804966\pi\)
−0.818088 + 0.575094i \(0.804966\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.31781 −0.0697457
\(358\) 0 0
\(359\) −34.4053 −1.81584 −0.907920 0.419143i \(-0.862331\pi\)
−0.907920 + 0.419143i \(0.862331\pi\)
\(360\) 0 0
\(361\) 31.2406 1.64424
\(362\) 0 0
\(363\) −0.482281 −0.0253132
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −10.2524 −0.535173 −0.267586 0.963534i \(-0.586226\pi\)
−0.267586 + 0.963534i \(0.586226\pi\)
\(368\) 0 0
\(369\) 16.6006 0.864192
\(370\) 0 0
\(371\) −5.36439 −0.278505
\(372\) 0 0
\(373\) 22.6993 1.17532 0.587662 0.809107i \(-0.300049\pi\)
0.587662 + 0.809107i \(0.300049\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −48.2434 −2.48466
\(378\) 0 0
\(379\) −1.03503 −0.0531661 −0.0265831 0.999647i \(-0.508463\pi\)
−0.0265831 + 0.999647i \(0.508463\pi\)
\(380\) 0 0
\(381\) −1.95051 −0.0999276
\(382\) 0 0
\(383\) −25.8707 −1.32193 −0.660965 0.750417i \(-0.729853\pi\)
−0.660965 + 0.750417i \(0.729853\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.05128 0.256771
\(388\) 0 0
\(389\) −9.21678 −0.467309 −0.233655 0.972320i \(-0.575068\pi\)
−0.233655 + 0.972320i \(0.575068\pi\)
\(390\) 0 0
\(391\) 0.982634 0.0496939
\(392\) 0 0
\(393\) 2.31377 0.116714
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.84825 −0.444081 −0.222040 0.975037i \(-0.571272\pi\)
−0.222040 + 0.975037i \(0.571272\pi\)
\(398\) 0 0
\(399\) −9.50577 −0.475884
\(400\) 0 0
\(401\) 11.7281 0.585672 0.292836 0.956163i \(-0.405401\pi\)
0.292836 + 0.956163i \(0.405401\pi\)
\(402\) 0 0
\(403\) −37.4210 −1.86408
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 35.4586 1.75762
\(408\) 0 0
\(409\) −22.8709 −1.13089 −0.565446 0.824785i \(-0.691296\pi\)
−0.565446 + 0.824785i \(0.691296\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 0 0
\(413\) 32.2765 1.58823
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.69466 −0.229899
\(418\) 0 0
\(419\) 2.41771 0.118113 0.0590565 0.998255i \(-0.481191\pi\)
0.0590565 + 0.998255i \(0.481191\pi\)
\(420\) 0 0
\(421\) −16.6583 −0.811876 −0.405938 0.913901i \(-0.633055\pi\)
−0.405938 + 0.913901i \(0.633055\pi\)
\(422\) 0 0
\(423\) 6.50557 0.316312
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 38.7417 1.87484
\(428\) 0 0
\(429\) −6.43038 −0.310462
\(430\) 0 0
\(431\) 7.51722 0.362092 0.181046 0.983475i \(-0.442052\pi\)
0.181046 + 0.983475i \(0.442052\pi\)
\(432\) 0 0
\(433\) 28.6003 1.37444 0.687221 0.726449i \(-0.258830\pi\)
0.687221 + 0.726449i \(0.258830\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.08806 0.339068
\(438\) 0 0
\(439\) −8.52886 −0.407060 −0.203530 0.979069i \(-0.565241\pi\)
−0.203530 + 0.979069i \(0.565241\pi\)
\(440\) 0 0
\(441\) −27.6729 −1.31776
\(442\) 0 0
\(443\) −35.9224 −1.70673 −0.853363 0.521317i \(-0.825441\pi\)
−0.853363 + 0.521317i \(0.825441\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −6.18797 −0.292681
\(448\) 0 0
\(449\) 5.11165 0.241234 0.120617 0.992699i \(-0.461513\pi\)
0.120617 + 0.992699i \(0.461513\pi\)
\(450\) 0 0
\(451\) 17.7293 0.834840
\(452\) 0 0
\(453\) 4.07090 0.191267
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.7944 −0.551717 −0.275859 0.961198i \(-0.588962\pi\)
−0.275859 + 0.961198i \(0.588962\pi\)
\(458\) 0 0
\(459\) 1.90724 0.0890226
\(460\) 0 0
\(461\) 15.7079 0.731589 0.365795 0.930696i \(-0.380797\pi\)
0.365795 + 0.930696i \(0.380797\pi\)
\(462\) 0 0
\(463\) −35.7752 −1.66261 −0.831307 0.555814i \(-0.812407\pi\)
−0.831307 + 0.555814i \(0.812407\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.2118 0.981564 0.490782 0.871282i \(-0.336711\pi\)
0.490782 + 0.871282i \(0.336711\pi\)
\(468\) 0 0
\(469\) 13.6006 0.628016
\(470\) 0 0
\(471\) 4.28277 0.197340
\(472\) 0 0
\(473\) 5.39472 0.248050
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.81039 0.174466
\(478\) 0 0
\(479\) 24.6077 1.12436 0.562178 0.827016i \(-0.309964\pi\)
0.562178 + 0.827016i \(0.309964\pi\)
\(480\) 0 0
\(481\) −72.5764 −3.30920
\(482\) 0 0
\(483\) −1.34110 −0.0610220
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −15.3507 −0.695605 −0.347802 0.937568i \(-0.613072\pi\)
−0.347802 + 0.937568i \(0.613072\pi\)
\(488\) 0 0
\(489\) 4.13080 0.186801
\(490\) 0 0
\(491\) −13.1430 −0.593134 −0.296567 0.955012i \(-0.595842\pi\)
−0.296567 + 0.955012i \(0.595842\pi\)
\(492\) 0 0
\(493\) 7.50015 0.337790
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.9823 −0.716906
\(498\) 0 0
\(499\) −20.6841 −0.925949 −0.462975 0.886371i \(-0.653218\pi\)
−0.462975 + 0.886371i \(0.653218\pi\)
\(500\) 0 0
\(501\) 3.69384 0.165029
\(502\) 0 0
\(503\) −35.7956 −1.59605 −0.798023 0.602627i \(-0.794121\pi\)
−0.798023 + 0.602627i \(0.794121\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.87879 0.394321
\(508\) 0 0
\(509\) 33.9242 1.50366 0.751832 0.659355i \(-0.229170\pi\)
0.751832 + 0.659355i \(0.229170\pi\)
\(510\) 0 0
\(511\) 6.78762 0.300267
\(512\) 0 0
\(513\) 13.7576 0.607412
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6.94790 0.305568
\(518\) 0 0
\(519\) −5.67095 −0.248927
\(520\) 0 0
\(521\) 12.3346 0.540387 0.270194 0.962806i \(-0.412912\pi\)
0.270194 + 0.962806i \(0.412912\pi\)
\(522\) 0 0
\(523\) 5.25274 0.229686 0.114843 0.993384i \(-0.463363\pi\)
0.114843 + 0.993384i \(0.463363\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.81766 0.253421
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −22.9264 −0.994922
\(532\) 0 0
\(533\) −36.2882 −1.57182
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8.29753 −0.358065
\(538\) 0 0
\(539\) −29.5544 −1.27300
\(540\) 0 0
\(541\) −28.4715 −1.22408 −0.612042 0.790825i \(-0.709652\pi\)
−0.612042 + 0.790825i \(0.709652\pi\)
\(542\) 0 0
\(543\) 5.05415 0.216894
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 25.6025 1.09468 0.547341 0.836910i \(-0.315640\pi\)
0.547341 + 0.836910i \(0.315640\pi\)
\(548\) 0 0
\(549\) −27.5187 −1.17447
\(550\) 0 0
\(551\) 54.1011 2.30478
\(552\) 0 0
\(553\) −64.8180 −2.75634
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.7938 0.669203 0.334602 0.942360i \(-0.391398\pi\)
0.334602 + 0.942360i \(0.391398\pi\)
\(558\) 0 0
\(559\) −11.0419 −0.467022
\(560\) 0 0
\(561\) 0.999698 0.0422073
\(562\) 0 0
\(563\) 41.8996 1.76586 0.882929 0.469507i \(-0.155568\pi\)
0.882929 + 0.469507i \(0.155568\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 32.7078 1.37360
\(568\) 0 0
\(569\) 24.5485 1.02913 0.514563 0.857453i \(-0.327954\pi\)
0.514563 + 0.857453i \(0.327954\pi\)
\(570\) 0 0
\(571\) −38.3231 −1.60377 −0.801886 0.597476i \(-0.796170\pi\)
−0.801886 + 0.597476i \(0.796170\pi\)
\(572\) 0 0
\(573\) −5.67422 −0.237044
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.9288 0.996169 0.498085 0.867128i \(-0.334037\pi\)
0.498085 + 0.867128i \(0.334037\pi\)
\(578\) 0 0
\(579\) 3.06411 0.127340
\(580\) 0 0
\(581\) 48.4886 2.01165
\(582\) 0 0
\(583\) 4.06947 0.168540
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −34.8130 −1.43689 −0.718443 0.695585i \(-0.755145\pi\)
−0.718443 + 0.695585i \(0.755145\pi\)
\(588\) 0 0
\(589\) 41.9647 1.72912
\(590\) 0 0
\(591\) 7.54239 0.310252
\(592\) 0 0
\(593\) −15.5702 −0.639393 −0.319697 0.947520i \(-0.603581\pi\)
−0.319697 + 0.947520i \(0.603581\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.09521 −0.0857513
\(598\) 0 0
\(599\) 31.2121 1.27529 0.637645 0.770330i \(-0.279908\pi\)
0.637645 + 0.770330i \(0.279908\pi\)
\(600\) 0 0
\(601\) 31.2553 1.27493 0.637466 0.770478i \(-0.279983\pi\)
0.637466 + 0.770478i \(0.279983\pi\)
\(602\) 0 0
\(603\) −9.66065 −0.393412
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 30.6469 1.24392 0.621959 0.783050i \(-0.286337\pi\)
0.621959 + 0.783050i \(0.286337\pi\)
\(608\) 0 0
\(609\) −10.2362 −0.414791
\(610\) 0 0
\(611\) −14.2209 −0.575317
\(612\) 0 0
\(613\) 0.459494 0.0185588 0.00927940 0.999957i \(-0.497046\pi\)
0.00927940 + 0.999957i \(0.497046\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.6010 −0.950140 −0.475070 0.879948i \(-0.657577\pi\)
−0.475070 + 0.879948i \(0.657577\pi\)
\(618\) 0 0
\(619\) 24.8893 1.00038 0.500192 0.865914i \(-0.333263\pi\)
0.500192 + 0.865914i \(0.333263\pi\)
\(620\) 0 0
\(621\) 1.94095 0.0778877
\(622\) 0 0
\(623\) −42.7417 −1.71241
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.21115 0.287986
\(628\) 0 0
\(629\) 11.2831 0.449886
\(630\) 0 0
\(631\) −20.8918 −0.831690 −0.415845 0.909435i \(-0.636514\pi\)
−0.415845 + 0.909435i \(0.636514\pi\)
\(632\) 0 0
\(633\) −3.41497 −0.135733
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 60.4917 2.39677
\(638\) 0 0
\(639\) 11.3524 0.449096
\(640\) 0 0
\(641\) −16.0056 −0.632184 −0.316092 0.948729i \(-0.602371\pi\)
−0.316092 + 0.948729i \(0.602371\pi\)
\(642\) 0 0
\(643\) 15.0242 0.592497 0.296249 0.955111i \(-0.404264\pi\)
0.296249 + 0.955111i \(0.404264\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.3791 1.78404 0.892018 0.452001i \(-0.149290\pi\)
0.892018 + 0.452001i \(0.149290\pi\)
\(648\) 0 0
\(649\) −24.4852 −0.961130
\(650\) 0 0
\(651\) −7.93993 −0.311190
\(652\) 0 0
\(653\) 14.2506 0.557671 0.278835 0.960339i \(-0.410052\pi\)
0.278835 + 0.960339i \(0.410052\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.82133 −0.188098
\(658\) 0 0
\(659\) −10.3535 −0.403314 −0.201657 0.979456i \(-0.564633\pi\)
−0.201657 + 0.979456i \(0.564633\pi\)
\(660\) 0 0
\(661\) 26.4651 1.02937 0.514687 0.857378i \(-0.327908\pi\)
0.514687 + 0.857378i \(0.327908\pi\)
\(662\) 0 0
\(663\) −2.04618 −0.0794669
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.63270 0.295539
\(668\) 0 0
\(669\) 1.89530 0.0732765
\(670\) 0 0
\(671\) −29.3897 −1.13458
\(672\) 0 0
\(673\) 12.0587 0.464831 0.232415 0.972617i \(-0.425337\pi\)
0.232415 + 0.972617i \(0.425337\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.54878 −0.213257 −0.106629 0.994299i \(-0.534006\pi\)
−0.106629 + 0.994299i \(0.534006\pi\)
\(678\) 0 0
\(679\) −11.5658 −0.443856
\(680\) 0 0
\(681\) −3.12779 −0.119857
\(682\) 0 0
\(683\) 0.397032 0.0151920 0.00759601 0.999971i \(-0.497582\pi\)
0.00759601 + 0.999971i \(0.497582\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0.604651 0.0230689
\(688\) 0 0
\(689\) −8.32935 −0.317323
\(690\) 0 0
\(691\) 21.4949 0.817703 0.408851 0.912601i \(-0.365929\pi\)
0.408851 + 0.912601i \(0.365929\pi\)
\(692\) 0 0
\(693\) 36.3472 1.38072
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.64154 0.213688
\(698\) 0 0
\(699\) 6.77638 0.256306
\(700\) 0 0
\(701\) 44.6027 1.68462 0.842311 0.538992i \(-0.181195\pi\)
0.842311 + 0.538992i \(0.181195\pi\)
\(702\) 0 0
\(703\) 81.3885 3.06963
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.3234 0.726732
\(708\) 0 0
\(709\) 29.7476 1.11719 0.558597 0.829439i \(-0.311340\pi\)
0.558597 + 0.829439i \(0.311340\pi\)
\(710\) 0 0
\(711\) 46.0410 1.72667
\(712\) 0 0
\(713\) 5.92047 0.221723
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.08565 −0.152582
\(718\) 0 0
\(719\) 35.1758 1.31184 0.655918 0.754832i \(-0.272282\pi\)
0.655918 + 0.754832i \(0.272282\pi\)
\(720\) 0 0
\(721\) −20.5705 −0.766087
\(722\) 0 0
\(723\) 5.19971 0.193379
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 27.6030 1.02374 0.511870 0.859063i \(-0.328953\pi\)
0.511870 + 0.859063i \(0.328953\pi\)
\(728\) 0 0
\(729\) −21.3144 −0.789421
\(730\) 0 0
\(731\) 1.71662 0.0634916
\(732\) 0 0
\(733\) −51.0939 −1.88720 −0.943598 0.331093i \(-0.892583\pi\)
−0.943598 + 0.331093i \(0.892583\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.3175 −0.380050
\(738\) 0 0
\(739\) 13.5558 0.498658 0.249329 0.968419i \(-0.419790\pi\)
0.249329 + 0.968419i \(0.419790\pi\)
\(740\) 0 0
\(741\) −14.7597 −0.542212
\(742\) 0 0
\(743\) 53.8006 1.97375 0.986877 0.161477i \(-0.0516257\pi\)
0.986877 + 0.161477i \(0.0516257\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −34.4420 −1.26017
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −44.6539 −1.62944 −0.814722 0.579852i \(-0.803111\pi\)
−0.814722 + 0.579852i \(0.803111\pi\)
\(752\) 0 0
\(753\) −3.20116 −0.116657
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −25.1581 −0.914388 −0.457194 0.889367i \(-0.651146\pi\)
−0.457194 + 0.889367i \(0.651146\pi\)
\(758\) 0 0
\(759\) 1.01737 0.0369281
\(760\) 0 0
\(761\) −22.9387 −0.831526 −0.415763 0.909473i \(-0.636485\pi\)
−0.415763 + 0.909473i \(0.636485\pi\)
\(762\) 0 0
\(763\) 41.1411 1.48941
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 50.1162 1.80959
\(768\) 0 0
\(769\) −35.7337 −1.28859 −0.644295 0.764777i \(-0.722849\pi\)
−0.644295 + 0.764777i \(0.722849\pi\)
\(770\) 0 0
\(771\) 6.36220 0.229129
\(772\) 0 0
\(773\) 10.9587 0.394158 0.197079 0.980388i \(-0.436854\pi\)
0.197079 + 0.980388i \(0.436854\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −15.3991 −0.552440
\(778\) 0 0
\(779\) 40.6943 1.45802
\(780\) 0 0
\(781\) 12.1243 0.433842
\(782\) 0 0
\(783\) 14.8147 0.529435
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.56216 −0.126977 −0.0634887 0.997983i \(-0.520223\pi\)
−0.0634887 + 0.997983i \(0.520223\pi\)
\(788\) 0 0
\(789\) 0.575944 0.0205042
\(790\) 0 0
\(791\) −1.36439 −0.0485120
\(792\) 0 0
\(793\) 60.1547 2.13616
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 44.7128 1.58381 0.791903 0.610646i \(-0.209090\pi\)
0.791903 + 0.610646i \(0.209090\pi\)
\(798\) 0 0
\(799\) 2.21085 0.0782143
\(800\) 0 0
\(801\) 30.3599 1.07271
\(802\) 0 0
\(803\) −5.14914 −0.181709
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.85718 0.135779
\(808\) 0 0
\(809\) 9.68159 0.340387 0.170193 0.985411i \(-0.445561\pi\)
0.170193 + 0.985411i \(0.445561\pi\)
\(810\) 0 0
\(811\) −35.5981 −1.25002 −0.625009 0.780618i \(-0.714905\pi\)
−0.625009 + 0.780618i \(0.714905\pi\)
\(812\) 0 0
\(813\) 3.32915 0.116758
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 12.3826 0.433212
\(818\) 0 0
\(819\) −74.3953 −2.59958
\(820\) 0 0
\(821\) 8.17888 0.285445 0.142722 0.989763i \(-0.454414\pi\)
0.142722 + 0.989763i \(0.454414\pi\)
\(822\) 0 0
\(823\) 1.50587 0.0524914 0.0262457 0.999656i \(-0.491645\pi\)
0.0262457 + 0.999656i \(0.491645\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.3652 0.882035 0.441017 0.897499i \(-0.354618\pi\)
0.441017 + 0.897499i \(0.354618\pi\)
\(828\) 0 0
\(829\) 38.2130 1.32719 0.663596 0.748091i \(-0.269029\pi\)
0.663596 + 0.748091i \(0.269029\pi\)
\(830\) 0 0
\(831\) 1.74385 0.0604935
\(832\) 0 0
\(833\) −9.40434 −0.325841
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 11.4914 0.397199
\(838\) 0 0
\(839\) −13.1575 −0.454248 −0.227124 0.973866i \(-0.572932\pi\)
−0.227124 + 0.973866i \(0.572932\pi\)
\(840\) 0 0
\(841\) 29.2582 1.00890
\(842\) 0 0
\(843\) −4.54848 −0.156658
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.95904 −0.204755
\(848\) 0 0
\(849\) 2.56787 0.0881290
\(850\) 0 0
\(851\) 11.4825 0.393614
\(852\) 0 0
\(853\) −13.2756 −0.454549 −0.227274 0.973831i \(-0.572981\pi\)
−0.227274 + 0.973831i \(0.572981\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.18894 −0.0747729 −0.0373864 0.999301i \(-0.511903\pi\)
−0.0373864 + 0.999301i \(0.511903\pi\)
\(858\) 0 0
\(859\) 21.4092 0.730472 0.365236 0.930915i \(-0.380988\pi\)
0.365236 + 0.930915i \(0.380988\pi\)
\(860\) 0 0
\(861\) −7.69956 −0.262400
\(862\) 0 0
\(863\) −14.2923 −0.486514 −0.243257 0.969962i \(-0.578216\pi\)
−0.243257 + 0.969962i \(0.578216\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −5.28257 −0.179405
\(868\) 0 0
\(869\) 49.1714 1.66803
\(870\) 0 0
\(871\) 21.1178 0.715549
\(872\) 0 0
\(873\) 8.21536 0.278048
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.7835 −0.769346 −0.384673 0.923053i \(-0.625686\pi\)
−0.384673 + 0.923053i \(0.625686\pi\)
\(878\) 0 0
\(879\) −7.91246 −0.266881
\(880\) 0 0
\(881\) −43.1007 −1.45210 −0.726050 0.687642i \(-0.758646\pi\)
−0.726050 + 0.687642i \(0.758646\pi\)
\(882\) 0 0
\(883\) −9.35101 −0.314687 −0.157343 0.987544i \(-0.550293\pi\)
−0.157343 + 0.987544i \(0.550293\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.9902 −0.771936 −0.385968 0.922512i \(-0.626132\pi\)
−0.385968 + 0.922512i \(0.626132\pi\)
\(888\) 0 0
\(889\) −24.1004 −0.808302
\(890\) 0 0
\(891\) −24.8124 −0.831245
\(892\) 0 0
\(893\) 15.9476 0.533666
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.08234 −0.0695272
\(898\) 0 0
\(899\) 45.1892 1.50714
\(900\) 0 0
\(901\) 1.29492 0.0431401
\(902\) 0 0
\(903\) −2.34285 −0.0779650
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 29.3070 0.973123 0.486561 0.873646i \(-0.338251\pi\)
0.486561 + 0.873646i \(0.338251\pi\)
\(908\) 0 0
\(909\) −13.7257 −0.455251
\(910\) 0 0
\(911\) −52.8533 −1.75111 −0.875554 0.483120i \(-0.839504\pi\)
−0.875554 + 0.483120i \(0.839504\pi\)
\(912\) 0 0
\(913\) −36.7838 −1.21737
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 28.5888 0.944086
\(918\) 0 0
\(919\) −27.3696 −0.902840 −0.451420 0.892312i \(-0.649082\pi\)
−0.451420 + 0.892312i \(0.649082\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −24.8160 −0.816828
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 14.6115 0.479905
\(928\) 0 0
\(929\) 4.64485 0.152393 0.0761963 0.997093i \(-0.475722\pi\)
0.0761963 + 0.997093i \(0.475722\pi\)
\(930\) 0 0
\(931\) −67.8366 −2.22325
\(932\) 0 0
\(933\) −8.84601 −0.289605
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −19.6704 −0.642606 −0.321303 0.946977i \(-0.604121\pi\)
−0.321303 + 0.946977i \(0.604121\pi\)
\(938\) 0 0
\(939\) 10.3813 0.338783
\(940\) 0 0
\(941\) −7.31259 −0.238384 −0.119192 0.992871i \(-0.538030\pi\)
−0.119192 + 0.992871i \(0.538030\pi\)
\(942\) 0 0
\(943\) 5.74124 0.186961
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.3846 −1.18234 −0.591170 0.806547i \(-0.701334\pi\)
−0.591170 + 0.806547i \(0.701334\pi\)
\(948\) 0 0
\(949\) 10.5392 0.342118
\(950\) 0 0
\(951\) −7.28452 −0.236217
\(952\) 0 0
\(953\) 22.2456 0.720605 0.360303 0.932835i \(-0.382673\pi\)
0.360303 + 0.932835i \(0.382673\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 7.76526 0.251015
\(958\) 0 0
\(959\) −49.4239 −1.59598
\(960\) 0 0
\(961\) 4.05200 0.130710
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −49.4661 −1.59072 −0.795361 0.606136i \(-0.792719\pi\)
−0.795361 + 0.606136i \(0.792719\pi\)
\(968\) 0 0
\(969\) 2.29462 0.0737138
\(970\) 0 0
\(971\) 28.9113 0.927809 0.463904 0.885885i \(-0.346448\pi\)
0.463904 + 0.885885i \(0.346448\pi\)
\(972\) 0 0
\(973\) −58.0071 −1.85962
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 55.1228 1.76354 0.881768 0.471684i \(-0.156354\pi\)
0.881768 + 0.471684i \(0.156354\pi\)
\(978\) 0 0
\(979\) 32.4242 1.03628
\(980\) 0 0
\(981\) −29.2230 −0.933019
\(982\) 0 0
\(983\) −26.6824 −0.851037 −0.425518 0.904950i \(-0.639908\pi\)
−0.425518 + 0.904950i \(0.639908\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −3.01737 −0.0960438
\(988\) 0 0
\(989\) 1.74696 0.0555502
\(990\) 0 0
\(991\) −25.7063 −0.816588 −0.408294 0.912851i \(-0.633876\pi\)
−0.408294 + 0.912851i \(0.633876\pi\)
\(992\) 0 0
\(993\) 7.87235 0.249821
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.95372 0.0618750 0.0309375 0.999521i \(-0.490151\pi\)
0.0309375 + 0.999521i \(0.490151\pi\)
\(998\) 0 0
\(999\) 22.2869 0.705128
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 4600.2.a.bb.1.2 yes 4
4.3 odd 2 9200.2.a.cn.1.3 4
5.2 odd 4 4600.2.e.t.4049.4 8
5.3 odd 4 4600.2.e.t.4049.5 8
5.4 even 2 4600.2.a.ba.1.3 4
20.19 odd 2 9200.2.a.cp.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.ba.1.3 4 5.4 even 2
4600.2.a.bb.1.2 yes 4 1.1 even 1 trivial
4600.2.e.t.4049.4 8 5.2 odd 4
4600.2.e.t.4049.5 8 5.3 odd 4
9200.2.a.cn.1.3 4 4.3 odd 2
9200.2.a.cp.1.2 4 20.19 odd 2