Properties

Label 6036.2.a.h.1.14
Level $6036$
Weight $2$
Character 6036.1
Self dual yes
Analytic conductor $48.198$
Analytic rank $0$
Dimension $24$
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] = [6036,2,Mod(1,6036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6036, 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("6036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6036.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: \(48.1977026600\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 6036.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.63100 q^{5} +0.968844 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.63100 q^{5} +0.968844 q^{7} +1.00000 q^{9} -0.515879 q^{11} +2.41601 q^{13} +1.63100 q^{15} +2.72683 q^{17} +3.28783 q^{19} +0.968844 q^{21} +2.42636 q^{23} -2.33984 q^{25} +1.00000 q^{27} +2.20663 q^{29} -6.45143 q^{31} -0.515879 q^{33} +1.58018 q^{35} +4.93597 q^{37} +2.41601 q^{39} -1.19804 q^{41} +11.4640 q^{43} +1.63100 q^{45} +0.331760 q^{47} -6.06134 q^{49} +2.72683 q^{51} +0.641025 q^{53} -0.841398 q^{55} +3.28783 q^{57} -2.73990 q^{59} +3.89119 q^{61} +0.968844 q^{63} +3.94051 q^{65} +5.15669 q^{67} +2.42636 q^{69} -1.21702 q^{71} +3.96885 q^{73} -2.33984 q^{75} -0.499806 q^{77} -15.3007 q^{79} +1.00000 q^{81} +8.12065 q^{83} +4.44746 q^{85} +2.20663 q^{87} -2.00258 q^{89} +2.34074 q^{91} -6.45143 q^{93} +5.36245 q^{95} -3.76692 q^{97} -0.515879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.63100 0.729405 0.364702 0.931124i \(-0.381171\pi\)
0.364702 + 0.931124i \(0.381171\pi\)
\(6\) 0 0
\(7\) 0.968844 0.366188 0.183094 0.983095i \(-0.441389\pi\)
0.183094 + 0.983095i \(0.441389\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.515879 −0.155543 −0.0777717 0.996971i \(-0.524781\pi\)
−0.0777717 + 0.996971i \(0.524781\pi\)
\(12\) 0 0
\(13\) 2.41601 0.670081 0.335040 0.942204i \(-0.391250\pi\)
0.335040 + 0.942204i \(0.391250\pi\)
\(14\) 0 0
\(15\) 1.63100 0.421122
\(16\) 0 0
\(17\) 2.72683 0.661353 0.330677 0.943744i \(-0.392723\pi\)
0.330677 + 0.943744i \(0.392723\pi\)
\(18\) 0 0
\(19\) 3.28783 0.754281 0.377140 0.926156i \(-0.376907\pi\)
0.377140 + 0.926156i \(0.376907\pi\)
\(20\) 0 0
\(21\) 0.968844 0.211419
\(22\) 0 0
\(23\) 2.42636 0.505932 0.252966 0.967475i \(-0.418594\pi\)
0.252966 + 0.967475i \(0.418594\pi\)
\(24\) 0 0
\(25\) −2.33984 −0.467969
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.20663 0.409760 0.204880 0.978787i \(-0.434320\pi\)
0.204880 + 0.978787i \(0.434320\pi\)
\(30\) 0 0
\(31\) −6.45143 −1.15871 −0.579355 0.815075i \(-0.696696\pi\)
−0.579355 + 0.815075i \(0.696696\pi\)
\(32\) 0 0
\(33\) −0.515879 −0.0898031
\(34\) 0 0
\(35\) 1.58018 0.267100
\(36\) 0 0
\(37\) 4.93597 0.811468 0.405734 0.913991i \(-0.367016\pi\)
0.405734 + 0.913991i \(0.367016\pi\)
\(38\) 0 0
\(39\) 2.41601 0.386871
\(40\) 0 0
\(41\) −1.19804 −0.187103 −0.0935515 0.995614i \(-0.529822\pi\)
−0.0935515 + 0.995614i \(0.529822\pi\)
\(42\) 0 0
\(43\) 11.4640 1.74825 0.874125 0.485701i \(-0.161436\pi\)
0.874125 + 0.485701i \(0.161436\pi\)
\(44\) 0 0
\(45\) 1.63100 0.243135
\(46\) 0 0
\(47\) 0.331760 0.0483921 0.0241961 0.999707i \(-0.492297\pi\)
0.0241961 + 0.999707i \(0.492297\pi\)
\(48\) 0 0
\(49\) −6.06134 −0.865906
\(50\) 0 0
\(51\) 2.72683 0.381833
\(52\) 0 0
\(53\) 0.641025 0.0880516 0.0440258 0.999030i \(-0.485982\pi\)
0.0440258 + 0.999030i \(0.485982\pi\)
\(54\) 0 0
\(55\) −0.841398 −0.113454
\(56\) 0 0
\(57\) 3.28783 0.435484
\(58\) 0 0
\(59\) −2.73990 −0.356705 −0.178352 0.983967i \(-0.557077\pi\)
−0.178352 + 0.983967i \(0.557077\pi\)
\(60\) 0 0
\(61\) 3.89119 0.498215 0.249108 0.968476i \(-0.419863\pi\)
0.249108 + 0.968476i \(0.419863\pi\)
\(62\) 0 0
\(63\) 0.968844 0.122063
\(64\) 0 0
\(65\) 3.94051 0.488760
\(66\) 0 0
\(67\) 5.15669 0.629990 0.314995 0.949093i \(-0.397997\pi\)
0.314995 + 0.949093i \(0.397997\pi\)
\(68\) 0 0
\(69\) 2.42636 0.292100
\(70\) 0 0
\(71\) −1.21702 −0.144433 −0.0722167 0.997389i \(-0.523007\pi\)
−0.0722167 + 0.997389i \(0.523007\pi\)
\(72\) 0 0
\(73\) 3.96885 0.464518 0.232259 0.972654i \(-0.425388\pi\)
0.232259 + 0.972654i \(0.425388\pi\)
\(74\) 0 0
\(75\) −2.33984 −0.270182
\(76\) 0 0
\(77\) −0.499806 −0.0569582
\(78\) 0 0
\(79\) −15.3007 −1.72147 −0.860733 0.509057i \(-0.829994\pi\)
−0.860733 + 0.509057i \(0.829994\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.12065 0.891357 0.445679 0.895193i \(-0.352962\pi\)
0.445679 + 0.895193i \(0.352962\pi\)
\(84\) 0 0
\(85\) 4.44746 0.482394
\(86\) 0 0
\(87\) 2.20663 0.236575
\(88\) 0 0
\(89\) −2.00258 −0.212273 −0.106136 0.994352i \(-0.533848\pi\)
−0.106136 + 0.994352i \(0.533848\pi\)
\(90\) 0 0
\(91\) 2.34074 0.245376
\(92\) 0 0
\(93\) −6.45143 −0.668982
\(94\) 0 0
\(95\) 5.36245 0.550176
\(96\) 0 0
\(97\) −3.76692 −0.382472 −0.191236 0.981544i \(-0.561250\pi\)
−0.191236 + 0.981544i \(0.561250\pi\)
\(98\) 0 0
\(99\) −0.515879 −0.0518478
\(100\) 0 0
\(101\) 17.2846 1.71988 0.859942 0.510392i \(-0.170500\pi\)
0.859942 + 0.510392i \(0.170500\pi\)
\(102\) 0 0
\(103\) −15.0176 −1.47973 −0.739865 0.672755i \(-0.765111\pi\)
−0.739865 + 0.672755i \(0.765111\pi\)
\(104\) 0 0
\(105\) 1.58018 0.154210
\(106\) 0 0
\(107\) −0.951772 −0.0920113 −0.0460056 0.998941i \(-0.514649\pi\)
−0.0460056 + 0.998941i \(0.514649\pi\)
\(108\) 0 0
\(109\) 11.4449 1.09623 0.548113 0.836404i \(-0.315346\pi\)
0.548113 + 0.836404i \(0.315346\pi\)
\(110\) 0 0
\(111\) 4.93597 0.468501
\(112\) 0 0
\(113\) −9.51785 −0.895364 −0.447682 0.894193i \(-0.647750\pi\)
−0.447682 + 0.894193i \(0.647750\pi\)
\(114\) 0 0
\(115\) 3.95740 0.369029
\(116\) 0 0
\(117\) 2.41601 0.223360
\(118\) 0 0
\(119\) 2.64187 0.242180
\(120\) 0 0
\(121\) −10.7339 −0.975806
\(122\) 0 0
\(123\) −1.19804 −0.108024
\(124\) 0 0
\(125\) −11.9713 −1.07074
\(126\) 0 0
\(127\) 6.15260 0.545955 0.272978 0.962020i \(-0.411992\pi\)
0.272978 + 0.962020i \(0.411992\pi\)
\(128\) 0 0
\(129\) 11.4640 1.00935
\(130\) 0 0
\(131\) −15.3869 −1.34436 −0.672180 0.740388i \(-0.734642\pi\)
−0.672180 + 0.740388i \(0.734642\pi\)
\(132\) 0 0
\(133\) 3.18540 0.276209
\(134\) 0 0
\(135\) 1.63100 0.140374
\(136\) 0 0
\(137\) 9.39887 0.803000 0.401500 0.915859i \(-0.368489\pi\)
0.401500 + 0.915859i \(0.368489\pi\)
\(138\) 0 0
\(139\) 16.9337 1.43629 0.718147 0.695891i \(-0.244990\pi\)
0.718147 + 0.695891i \(0.244990\pi\)
\(140\) 0 0
\(141\) 0.331760 0.0279392
\(142\) 0 0
\(143\) −1.24637 −0.104227
\(144\) 0 0
\(145\) 3.59900 0.298881
\(146\) 0 0
\(147\) −6.06134 −0.499931
\(148\) 0 0
\(149\) 8.33872 0.683135 0.341567 0.939857i \(-0.389042\pi\)
0.341567 + 0.939857i \(0.389042\pi\)
\(150\) 0 0
\(151\) −4.79467 −0.390185 −0.195092 0.980785i \(-0.562501\pi\)
−0.195092 + 0.980785i \(0.562501\pi\)
\(152\) 0 0
\(153\) 2.72683 0.220451
\(154\) 0 0
\(155\) −10.5223 −0.845169
\(156\) 0 0
\(157\) 11.8380 0.944772 0.472386 0.881392i \(-0.343393\pi\)
0.472386 + 0.881392i \(0.343393\pi\)
\(158\) 0 0
\(159\) 0.641025 0.0508366
\(160\) 0 0
\(161\) 2.35077 0.185266
\(162\) 0 0
\(163\) 1.32206 0.103552 0.0517760 0.998659i \(-0.483512\pi\)
0.0517760 + 0.998659i \(0.483512\pi\)
\(164\) 0 0
\(165\) −0.841398 −0.0655028
\(166\) 0 0
\(167\) 10.3543 0.801237 0.400619 0.916245i \(-0.368795\pi\)
0.400619 + 0.916245i \(0.368795\pi\)
\(168\) 0 0
\(169\) −7.16289 −0.550991
\(170\) 0 0
\(171\) 3.28783 0.251427
\(172\) 0 0
\(173\) 25.5438 1.94206 0.971030 0.238960i \(-0.0768064\pi\)
0.971030 + 0.238960i \(0.0768064\pi\)
\(174\) 0 0
\(175\) −2.26694 −0.171365
\(176\) 0 0
\(177\) −2.73990 −0.205943
\(178\) 0 0
\(179\) −9.50943 −0.710768 −0.355384 0.934720i \(-0.615650\pi\)
−0.355384 + 0.934720i \(0.615650\pi\)
\(180\) 0 0
\(181\) −19.6579 −1.46116 −0.730581 0.682826i \(-0.760751\pi\)
−0.730581 + 0.682826i \(0.760751\pi\)
\(182\) 0 0
\(183\) 3.89119 0.287645
\(184\) 0 0
\(185\) 8.05056 0.591889
\(186\) 0 0
\(187\) −1.40672 −0.102869
\(188\) 0 0
\(189\) 0.968844 0.0704730
\(190\) 0 0
\(191\) 25.5726 1.85037 0.925185 0.379516i \(-0.123910\pi\)
0.925185 + 0.379516i \(0.123910\pi\)
\(192\) 0 0
\(193\) −1.47283 −0.106017 −0.0530085 0.998594i \(-0.516881\pi\)
−0.0530085 + 0.998594i \(0.516881\pi\)
\(194\) 0 0
\(195\) 3.94051 0.282186
\(196\) 0 0
\(197\) 16.5426 1.17861 0.589307 0.807909i \(-0.299401\pi\)
0.589307 + 0.807909i \(0.299401\pi\)
\(198\) 0 0
\(199\) −14.3063 −1.01415 −0.507073 0.861903i \(-0.669273\pi\)
−0.507073 + 0.861903i \(0.669273\pi\)
\(200\) 0 0
\(201\) 5.15669 0.363725
\(202\) 0 0
\(203\) 2.13788 0.150049
\(204\) 0 0
\(205\) −1.95401 −0.136474
\(206\) 0 0
\(207\) 2.42636 0.168644
\(208\) 0 0
\(209\) −1.69613 −0.117323
\(210\) 0 0
\(211\) −0.517844 −0.0356499 −0.0178249 0.999841i \(-0.505674\pi\)
−0.0178249 + 0.999841i \(0.505674\pi\)
\(212\) 0 0
\(213\) −1.21702 −0.0833887
\(214\) 0 0
\(215\) 18.6978 1.27518
\(216\) 0 0
\(217\) −6.25043 −0.424307
\(218\) 0 0
\(219\) 3.96885 0.268190
\(220\) 0 0
\(221\) 6.58805 0.443160
\(222\) 0 0
\(223\) 2.28956 0.153321 0.0766603 0.997057i \(-0.475574\pi\)
0.0766603 + 0.997057i \(0.475574\pi\)
\(224\) 0 0
\(225\) −2.33984 −0.155990
\(226\) 0 0
\(227\) 1.05886 0.0702791 0.0351395 0.999382i \(-0.488812\pi\)
0.0351395 + 0.999382i \(0.488812\pi\)
\(228\) 0 0
\(229\) −1.23606 −0.0816810 −0.0408405 0.999166i \(-0.513004\pi\)
−0.0408405 + 0.999166i \(0.513004\pi\)
\(230\) 0 0
\(231\) −0.499806 −0.0328848
\(232\) 0 0
\(233\) 19.8090 1.29773 0.648864 0.760904i \(-0.275244\pi\)
0.648864 + 0.760904i \(0.275244\pi\)
\(234\) 0 0
\(235\) 0.541100 0.0352974
\(236\) 0 0
\(237\) −15.3007 −0.993889
\(238\) 0 0
\(239\) −27.2624 −1.76346 −0.881730 0.471754i \(-0.843621\pi\)
−0.881730 + 0.471754i \(0.843621\pi\)
\(240\) 0 0
\(241\) 16.3465 1.05297 0.526486 0.850184i \(-0.323509\pi\)
0.526486 + 0.850184i \(0.323509\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −9.88604 −0.631596
\(246\) 0 0
\(247\) 7.94344 0.505429
\(248\) 0 0
\(249\) 8.12065 0.514625
\(250\) 0 0
\(251\) −4.02322 −0.253943 −0.126972 0.991906i \(-0.540526\pi\)
−0.126972 + 0.991906i \(0.540526\pi\)
\(252\) 0 0
\(253\) −1.25171 −0.0786944
\(254\) 0 0
\(255\) 4.44746 0.278511
\(256\) 0 0
\(257\) −2.96131 −0.184721 −0.0923607 0.995726i \(-0.529441\pi\)
−0.0923607 + 0.995726i \(0.529441\pi\)
\(258\) 0 0
\(259\) 4.78218 0.297150
\(260\) 0 0
\(261\) 2.20663 0.136587
\(262\) 0 0
\(263\) −19.5063 −1.20281 −0.601405 0.798945i \(-0.705392\pi\)
−0.601405 + 0.798945i \(0.705392\pi\)
\(264\) 0 0
\(265\) 1.04551 0.0642252
\(266\) 0 0
\(267\) −2.00258 −0.122556
\(268\) 0 0
\(269\) −18.2657 −1.11368 −0.556840 0.830620i \(-0.687986\pi\)
−0.556840 + 0.830620i \(0.687986\pi\)
\(270\) 0 0
\(271\) −7.41708 −0.450556 −0.225278 0.974295i \(-0.572329\pi\)
−0.225278 + 0.974295i \(0.572329\pi\)
\(272\) 0 0
\(273\) 2.34074 0.141668
\(274\) 0 0
\(275\) 1.20708 0.0727895
\(276\) 0 0
\(277\) 20.9772 1.26040 0.630199 0.776434i \(-0.282973\pi\)
0.630199 + 0.776434i \(0.282973\pi\)
\(278\) 0 0
\(279\) −6.45143 −0.386237
\(280\) 0 0
\(281\) −4.11739 −0.245623 −0.122812 0.992430i \(-0.539191\pi\)
−0.122812 + 0.992430i \(0.539191\pi\)
\(282\) 0 0
\(283\) 30.4094 1.80765 0.903827 0.427899i \(-0.140746\pi\)
0.903827 + 0.427899i \(0.140746\pi\)
\(284\) 0 0
\(285\) 5.36245 0.317644
\(286\) 0 0
\(287\) −1.16072 −0.0685149
\(288\) 0 0
\(289\) −9.56440 −0.562612
\(290\) 0 0
\(291\) −3.76692 −0.220821
\(292\) 0 0
\(293\) −10.8666 −0.634834 −0.317417 0.948286i \(-0.602815\pi\)
−0.317417 + 0.948286i \(0.602815\pi\)
\(294\) 0 0
\(295\) −4.46877 −0.260182
\(296\) 0 0
\(297\) −0.515879 −0.0299344
\(298\) 0 0
\(299\) 5.86212 0.339015
\(300\) 0 0
\(301\) 11.1069 0.640189
\(302\) 0 0
\(303\) 17.2846 0.992976
\(304\) 0 0
\(305\) 6.34652 0.363401
\(306\) 0 0
\(307\) −9.21688 −0.526035 −0.263018 0.964791i \(-0.584718\pi\)
−0.263018 + 0.964791i \(0.584718\pi\)
\(308\) 0 0
\(309\) −15.0176 −0.854322
\(310\) 0 0
\(311\) −23.0405 −1.30651 −0.653254 0.757138i \(-0.726597\pi\)
−0.653254 + 0.757138i \(0.726597\pi\)
\(312\) 0 0
\(313\) 20.5492 1.16151 0.580754 0.814079i \(-0.302758\pi\)
0.580754 + 0.814079i \(0.302758\pi\)
\(314\) 0 0
\(315\) 1.58018 0.0890332
\(316\) 0 0
\(317\) −31.5199 −1.77034 −0.885168 0.465272i \(-0.845957\pi\)
−0.885168 + 0.465272i \(0.845957\pi\)
\(318\) 0 0
\(319\) −1.13835 −0.0637355
\(320\) 0 0
\(321\) −0.951772 −0.0531227
\(322\) 0 0
\(323\) 8.96536 0.498846
\(324\) 0 0
\(325\) −5.65309 −0.313577
\(326\) 0 0
\(327\) 11.4449 0.632906
\(328\) 0 0
\(329\) 0.321423 0.0177206
\(330\) 0 0
\(331\) −15.0690 −0.828269 −0.414134 0.910216i \(-0.635916\pi\)
−0.414134 + 0.910216i \(0.635916\pi\)
\(332\) 0 0
\(333\) 4.93597 0.270489
\(334\) 0 0
\(335\) 8.41056 0.459518
\(336\) 0 0
\(337\) −20.4681 −1.11497 −0.557484 0.830187i \(-0.688233\pi\)
−0.557484 + 0.830187i \(0.688233\pi\)
\(338\) 0 0
\(339\) −9.51785 −0.516939
\(340\) 0 0
\(341\) 3.32816 0.180230
\(342\) 0 0
\(343\) −12.6544 −0.683273
\(344\) 0 0
\(345\) 3.95740 0.213059
\(346\) 0 0
\(347\) 6.01798 0.323062 0.161531 0.986868i \(-0.448357\pi\)
0.161531 + 0.986868i \(0.448357\pi\)
\(348\) 0 0
\(349\) −31.2949 −1.67518 −0.837588 0.546302i \(-0.816035\pi\)
−0.837588 + 0.546302i \(0.816035\pi\)
\(350\) 0 0
\(351\) 2.41601 0.128957
\(352\) 0 0
\(353\) 8.87028 0.472117 0.236059 0.971739i \(-0.424144\pi\)
0.236059 + 0.971739i \(0.424144\pi\)
\(354\) 0 0
\(355\) −1.98495 −0.105350
\(356\) 0 0
\(357\) 2.64187 0.139823
\(358\) 0 0
\(359\) 14.8386 0.783153 0.391577 0.920146i \(-0.371930\pi\)
0.391577 + 0.920146i \(0.371930\pi\)
\(360\) 0 0
\(361\) −8.19015 −0.431061
\(362\) 0 0
\(363\) −10.7339 −0.563382
\(364\) 0 0
\(365\) 6.47318 0.338822
\(366\) 0 0
\(367\) 25.3018 1.32074 0.660371 0.750940i \(-0.270399\pi\)
0.660371 + 0.750940i \(0.270399\pi\)
\(368\) 0 0
\(369\) −1.19804 −0.0623676
\(370\) 0 0
\(371\) 0.621053 0.0322435
\(372\) 0 0
\(373\) −5.48742 −0.284128 −0.142064 0.989857i \(-0.545374\pi\)
−0.142064 + 0.989857i \(0.545374\pi\)
\(374\) 0 0
\(375\) −11.9713 −0.618194
\(376\) 0 0
\(377\) 5.33123 0.274573
\(378\) 0 0
\(379\) −14.7359 −0.756931 −0.378466 0.925615i \(-0.623548\pi\)
−0.378466 + 0.925615i \(0.623548\pi\)
\(380\) 0 0
\(381\) 6.15260 0.315207
\(382\) 0 0
\(383\) −0.872601 −0.0445878 −0.0222939 0.999751i \(-0.507097\pi\)
−0.0222939 + 0.999751i \(0.507097\pi\)
\(384\) 0 0
\(385\) −0.815184 −0.0415456
\(386\) 0 0
\(387\) 11.4640 0.582750
\(388\) 0 0
\(389\) −17.8903 −0.907074 −0.453537 0.891237i \(-0.649838\pi\)
−0.453537 + 0.891237i \(0.649838\pi\)
\(390\) 0 0
\(391\) 6.61628 0.334600
\(392\) 0 0
\(393\) −15.3869 −0.776167
\(394\) 0 0
\(395\) −24.9555 −1.25565
\(396\) 0 0
\(397\) 13.9980 0.702540 0.351270 0.936274i \(-0.385750\pi\)
0.351270 + 0.936274i \(0.385750\pi\)
\(398\) 0 0
\(399\) 3.18540 0.159469
\(400\) 0 0
\(401\) 21.4831 1.07282 0.536408 0.843959i \(-0.319781\pi\)
0.536408 + 0.843959i \(0.319781\pi\)
\(402\) 0 0
\(403\) −15.5867 −0.776430
\(404\) 0 0
\(405\) 1.63100 0.0810450
\(406\) 0 0
\(407\) −2.54636 −0.126219
\(408\) 0 0
\(409\) 31.4490 1.55505 0.777526 0.628851i \(-0.216475\pi\)
0.777526 + 0.628851i \(0.216475\pi\)
\(410\) 0 0
\(411\) 9.39887 0.463612
\(412\) 0 0
\(413\) −2.65453 −0.130621
\(414\) 0 0
\(415\) 13.2448 0.650160
\(416\) 0 0
\(417\) 16.9337 0.829245
\(418\) 0 0
\(419\) 29.1679 1.42494 0.712472 0.701700i \(-0.247575\pi\)
0.712472 + 0.701700i \(0.247575\pi\)
\(420\) 0 0
\(421\) 21.4349 1.04467 0.522335 0.852740i \(-0.325061\pi\)
0.522335 + 0.852740i \(0.325061\pi\)
\(422\) 0 0
\(423\) 0.331760 0.0161307
\(424\) 0 0
\(425\) −6.38036 −0.309493
\(426\) 0 0
\(427\) 3.76995 0.182441
\(428\) 0 0
\(429\) −1.24637 −0.0601753
\(430\) 0 0
\(431\) −18.7575 −0.903518 −0.451759 0.892140i \(-0.649203\pi\)
−0.451759 + 0.892140i \(0.649203\pi\)
\(432\) 0 0
\(433\) −10.7762 −0.517871 −0.258936 0.965895i \(-0.583372\pi\)
−0.258936 + 0.965895i \(0.583372\pi\)
\(434\) 0 0
\(435\) 3.59900 0.172559
\(436\) 0 0
\(437\) 7.97748 0.381615
\(438\) 0 0
\(439\) −30.4831 −1.45488 −0.727439 0.686172i \(-0.759290\pi\)
−0.727439 + 0.686172i \(0.759290\pi\)
\(440\) 0 0
\(441\) −6.06134 −0.288635
\(442\) 0 0
\(443\) 9.25081 0.439519 0.219760 0.975554i \(-0.429473\pi\)
0.219760 + 0.975554i \(0.429473\pi\)
\(444\) 0 0
\(445\) −3.26620 −0.154833
\(446\) 0 0
\(447\) 8.33872 0.394408
\(448\) 0 0
\(449\) 27.1903 1.28319 0.641595 0.767044i \(-0.278273\pi\)
0.641595 + 0.767044i \(0.278273\pi\)
\(450\) 0 0
\(451\) 0.618046 0.0291026
\(452\) 0 0
\(453\) −4.79467 −0.225273
\(454\) 0 0
\(455\) 3.81774 0.178978
\(456\) 0 0
\(457\) 29.2266 1.36716 0.683582 0.729873i \(-0.260421\pi\)
0.683582 + 0.729873i \(0.260421\pi\)
\(458\) 0 0
\(459\) 2.72683 0.127278
\(460\) 0 0
\(461\) 33.8966 1.57872 0.789362 0.613928i \(-0.210412\pi\)
0.789362 + 0.613928i \(0.210412\pi\)
\(462\) 0 0
\(463\) −9.31649 −0.432974 −0.216487 0.976285i \(-0.569460\pi\)
−0.216487 + 0.976285i \(0.569460\pi\)
\(464\) 0 0
\(465\) −10.5223 −0.487959
\(466\) 0 0
\(467\) 23.8899 1.10549 0.552746 0.833350i \(-0.313580\pi\)
0.552746 + 0.833350i \(0.313580\pi\)
\(468\) 0 0
\(469\) 4.99603 0.230695
\(470\) 0 0
\(471\) 11.8380 0.545464
\(472\) 0 0
\(473\) −5.91406 −0.271929
\(474\) 0 0
\(475\) −7.69302 −0.352980
\(476\) 0 0
\(477\) 0.641025 0.0293505
\(478\) 0 0
\(479\) 26.8883 1.22856 0.614280 0.789089i \(-0.289447\pi\)
0.614280 + 0.789089i \(0.289447\pi\)
\(480\) 0 0
\(481\) 11.9254 0.543749
\(482\) 0 0
\(483\) 2.35077 0.106964
\(484\) 0 0
\(485\) −6.14383 −0.278977
\(486\) 0 0
\(487\) −32.3484 −1.46584 −0.732922 0.680313i \(-0.761844\pi\)
−0.732922 + 0.680313i \(0.761844\pi\)
\(488\) 0 0
\(489\) 1.32206 0.0597858
\(490\) 0 0
\(491\) 1.80300 0.0813683 0.0406842 0.999172i \(-0.487046\pi\)
0.0406842 + 0.999172i \(0.487046\pi\)
\(492\) 0 0
\(493\) 6.01710 0.270996
\(494\) 0 0
\(495\) −0.841398 −0.0378180
\(496\) 0 0
\(497\) −1.17910 −0.0528899
\(498\) 0 0
\(499\) −17.4536 −0.781331 −0.390665 0.920533i \(-0.627755\pi\)
−0.390665 + 0.920533i \(0.627755\pi\)
\(500\) 0 0
\(501\) 10.3543 0.462595
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) 28.1912 1.25449
\(506\) 0 0
\(507\) −7.16289 −0.318115
\(508\) 0 0
\(509\) 2.56123 0.113524 0.0567622 0.998388i \(-0.481922\pi\)
0.0567622 + 0.998388i \(0.481922\pi\)
\(510\) 0 0
\(511\) 3.84519 0.170101
\(512\) 0 0
\(513\) 3.28783 0.145161
\(514\) 0 0
\(515\) −24.4937 −1.07932
\(516\) 0 0
\(517\) −0.171148 −0.00752708
\(518\) 0 0
\(519\) 25.5438 1.12125
\(520\) 0 0
\(521\) −34.3833 −1.50636 −0.753180 0.657815i \(-0.771481\pi\)
−0.753180 + 0.657815i \(0.771481\pi\)
\(522\) 0 0
\(523\) −4.53194 −0.198168 −0.0990841 0.995079i \(-0.531591\pi\)
−0.0990841 + 0.995079i \(0.531591\pi\)
\(524\) 0 0
\(525\) −2.26694 −0.0989375
\(526\) 0 0
\(527\) −17.5920 −0.766317
\(528\) 0 0
\(529\) −17.1128 −0.744033
\(530\) 0 0
\(531\) −2.73990 −0.118902
\(532\) 0 0
\(533\) −2.89449 −0.125374
\(534\) 0 0
\(535\) −1.55234 −0.0671135
\(536\) 0 0
\(537\) −9.50943 −0.410362
\(538\) 0 0
\(539\) 3.12692 0.134686
\(540\) 0 0
\(541\) 14.1880 0.609990 0.304995 0.952354i \(-0.401345\pi\)
0.304995 + 0.952354i \(0.401345\pi\)
\(542\) 0 0
\(543\) −19.6579 −0.843602
\(544\) 0 0
\(545\) 18.6667 0.799593
\(546\) 0 0
\(547\) 5.72334 0.244712 0.122356 0.992486i \(-0.460955\pi\)
0.122356 + 0.992486i \(0.460955\pi\)
\(548\) 0 0
\(549\) 3.89119 0.166072
\(550\) 0 0
\(551\) 7.25502 0.309074
\(552\) 0 0
\(553\) −14.8240 −0.630381
\(554\) 0 0
\(555\) 8.05056 0.341727
\(556\) 0 0
\(557\) 22.8068 0.966354 0.483177 0.875523i \(-0.339483\pi\)
0.483177 + 0.875523i \(0.339483\pi\)
\(558\) 0 0
\(559\) 27.6973 1.17147
\(560\) 0 0
\(561\) −1.40672 −0.0593916
\(562\) 0 0
\(563\) −40.5967 −1.71095 −0.855474 0.517846i \(-0.826734\pi\)
−0.855474 + 0.517846i \(0.826734\pi\)
\(564\) 0 0
\(565\) −15.5236 −0.653083
\(566\) 0 0
\(567\) 0.968844 0.0406876
\(568\) 0 0
\(569\) 15.6650 0.656709 0.328355 0.944554i \(-0.393506\pi\)
0.328355 + 0.944554i \(0.393506\pi\)
\(570\) 0 0
\(571\) −10.1409 −0.424382 −0.212191 0.977228i \(-0.568060\pi\)
−0.212191 + 0.977228i \(0.568060\pi\)
\(572\) 0 0
\(573\) 25.5726 1.06831
\(574\) 0 0
\(575\) −5.67731 −0.236760
\(576\) 0 0
\(577\) −36.6203 −1.52452 −0.762262 0.647268i \(-0.775911\pi\)
−0.762262 + 0.647268i \(0.775911\pi\)
\(578\) 0 0
\(579\) −1.47283 −0.0612089
\(580\) 0 0
\(581\) 7.86764 0.326405
\(582\) 0 0
\(583\) −0.330692 −0.0136958
\(584\) 0 0
\(585\) 3.94051 0.162920
\(586\) 0 0
\(587\) 23.5892 0.973629 0.486815 0.873505i \(-0.338159\pi\)
0.486815 + 0.873505i \(0.338159\pi\)
\(588\) 0 0
\(589\) −21.2112 −0.873993
\(590\) 0 0
\(591\) 16.5426 0.680473
\(592\) 0 0
\(593\) −12.5607 −0.515807 −0.257904 0.966171i \(-0.583032\pi\)
−0.257904 + 0.966171i \(0.583032\pi\)
\(594\) 0 0
\(595\) 4.30889 0.176647
\(596\) 0 0
\(597\) −14.3063 −0.585517
\(598\) 0 0
\(599\) 11.6640 0.476579 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(600\) 0 0
\(601\) −37.7716 −1.54074 −0.770368 0.637600i \(-0.779927\pi\)
−0.770368 + 0.637600i \(0.779927\pi\)
\(602\) 0 0
\(603\) 5.15669 0.209997
\(604\) 0 0
\(605\) −17.5069 −0.711758
\(606\) 0 0
\(607\) 40.7414 1.65364 0.826820 0.562466i \(-0.190147\pi\)
0.826820 + 0.562466i \(0.190147\pi\)
\(608\) 0 0
\(609\) 2.13788 0.0866311
\(610\) 0 0
\(611\) 0.801535 0.0324266
\(612\) 0 0
\(613\) −33.2493 −1.34293 −0.671463 0.741038i \(-0.734334\pi\)
−0.671463 + 0.741038i \(0.734334\pi\)
\(614\) 0 0
\(615\) −1.95401 −0.0787932
\(616\) 0 0
\(617\) 33.3766 1.34369 0.671845 0.740692i \(-0.265502\pi\)
0.671845 + 0.740692i \(0.265502\pi\)
\(618\) 0 0
\(619\) −37.2922 −1.49890 −0.749450 0.662060i \(-0.769682\pi\)
−0.749450 + 0.662060i \(0.769682\pi\)
\(620\) 0 0
\(621\) 2.42636 0.0973666
\(622\) 0 0
\(623\) −1.94018 −0.0777318
\(624\) 0 0
\(625\) −7.82591 −0.313036
\(626\) 0 0
\(627\) −1.69613 −0.0677367
\(628\) 0 0
\(629\) 13.4595 0.536667
\(630\) 0 0
\(631\) −27.5076 −1.09506 −0.547531 0.836786i \(-0.684432\pi\)
−0.547531 + 0.836786i \(0.684432\pi\)
\(632\) 0 0
\(633\) −0.517844 −0.0205825
\(634\) 0 0
\(635\) 10.0349 0.398222
\(636\) 0 0
\(637\) −14.6443 −0.580227
\(638\) 0 0
\(639\) −1.21702 −0.0481445
\(640\) 0 0
\(641\) 19.2903 0.761919 0.380960 0.924592i \(-0.375594\pi\)
0.380960 + 0.924592i \(0.375594\pi\)
\(642\) 0 0
\(643\) 3.96618 0.156411 0.0782054 0.996937i \(-0.475081\pi\)
0.0782054 + 0.996937i \(0.475081\pi\)
\(644\) 0 0
\(645\) 18.6978 0.736227
\(646\) 0 0
\(647\) 5.30261 0.208467 0.104234 0.994553i \(-0.466761\pi\)
0.104234 + 0.994553i \(0.466761\pi\)
\(648\) 0 0
\(649\) 1.41346 0.0554831
\(650\) 0 0
\(651\) −6.25043 −0.244973
\(652\) 0 0
\(653\) 14.4881 0.566964 0.283482 0.958978i \(-0.408510\pi\)
0.283482 + 0.958978i \(0.408510\pi\)
\(654\) 0 0
\(655\) −25.0960 −0.980583
\(656\) 0 0
\(657\) 3.96885 0.154839
\(658\) 0 0
\(659\) 2.48530 0.0968134 0.0484067 0.998828i \(-0.484586\pi\)
0.0484067 + 0.998828i \(0.484586\pi\)
\(660\) 0 0
\(661\) 39.2394 1.52624 0.763118 0.646259i \(-0.223667\pi\)
0.763118 + 0.646259i \(0.223667\pi\)
\(662\) 0 0
\(663\) 6.58805 0.255859
\(664\) 0 0
\(665\) 5.19538 0.201468
\(666\) 0 0
\(667\) 5.35408 0.207311
\(668\) 0 0
\(669\) 2.28956 0.0885197
\(670\) 0 0
\(671\) −2.00738 −0.0774941
\(672\) 0 0
\(673\) −23.1037 −0.890583 −0.445292 0.895386i \(-0.646900\pi\)
−0.445292 + 0.895386i \(0.646900\pi\)
\(674\) 0 0
\(675\) −2.33984 −0.0900606
\(676\) 0 0
\(677\) 10.8791 0.418120 0.209060 0.977903i \(-0.432960\pi\)
0.209060 + 0.977903i \(0.432960\pi\)
\(678\) 0 0
\(679\) −3.64955 −0.140057
\(680\) 0 0
\(681\) 1.05886 0.0405756
\(682\) 0 0
\(683\) −41.8702 −1.60212 −0.801060 0.598585i \(-0.795730\pi\)
−0.801060 + 0.598585i \(0.795730\pi\)
\(684\) 0 0
\(685\) 15.3295 0.585712
\(686\) 0 0
\(687\) −1.23606 −0.0471586
\(688\) 0 0
\(689\) 1.54872 0.0590017
\(690\) 0 0
\(691\) −1.01647 −0.0386685 −0.0193343 0.999813i \(-0.506155\pi\)
−0.0193343 + 0.999813i \(0.506155\pi\)
\(692\) 0 0
\(693\) −0.499806 −0.0189861
\(694\) 0 0
\(695\) 27.6188 1.04764
\(696\) 0 0
\(697\) −3.26686 −0.123741
\(698\) 0 0
\(699\) 19.8090 0.749244
\(700\) 0 0
\(701\) −12.0986 −0.456960 −0.228480 0.973549i \(-0.573375\pi\)
−0.228480 + 0.973549i \(0.573375\pi\)
\(702\) 0 0
\(703\) 16.2286 0.612075
\(704\) 0 0
\(705\) 0.541100 0.0203790
\(706\) 0 0
\(707\) 16.7461 0.629802
\(708\) 0 0
\(709\) 1.29914 0.0487903 0.0243951 0.999702i \(-0.492234\pi\)
0.0243951 + 0.999702i \(0.492234\pi\)
\(710\) 0 0
\(711\) −15.3007 −0.573822
\(712\) 0 0
\(713\) −15.6535 −0.586229
\(714\) 0 0
\(715\) −2.03283 −0.0760235
\(716\) 0 0
\(717\) −27.2624 −1.01813
\(718\) 0 0
\(719\) 10.5339 0.392847 0.196424 0.980519i \(-0.437067\pi\)
0.196424 + 0.980519i \(0.437067\pi\)
\(720\) 0 0
\(721\) −14.5497 −0.541860
\(722\) 0 0
\(723\) 16.3465 0.607934
\(724\) 0 0
\(725\) −5.16316 −0.191755
\(726\) 0 0
\(727\) −33.4128 −1.23921 −0.619606 0.784913i \(-0.712708\pi\)
−0.619606 + 0.784913i \(0.712708\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.2605 1.15621
\(732\) 0 0
\(733\) −14.0612 −0.519363 −0.259682 0.965694i \(-0.583618\pi\)
−0.259682 + 0.965694i \(0.583618\pi\)
\(734\) 0 0
\(735\) −9.88604 −0.364652
\(736\) 0 0
\(737\) −2.66023 −0.0979909
\(738\) 0 0
\(739\) 1.47844 0.0543854 0.0271927 0.999630i \(-0.491343\pi\)
0.0271927 + 0.999630i \(0.491343\pi\)
\(740\) 0 0
\(741\) 7.94344 0.291810
\(742\) 0 0
\(743\) 42.2574 1.55027 0.775137 0.631794i \(-0.217681\pi\)
0.775137 + 0.631794i \(0.217681\pi\)
\(744\) 0 0
\(745\) 13.6004 0.498282
\(746\) 0 0
\(747\) 8.12065 0.297119
\(748\) 0 0
\(749\) −0.922118 −0.0336935
\(750\) 0 0
\(751\) 0.772760 0.0281984 0.0140992 0.999901i \(-0.495512\pi\)
0.0140992 + 0.999901i \(0.495512\pi\)
\(752\) 0 0
\(753\) −4.02322 −0.146614
\(754\) 0 0
\(755\) −7.82010 −0.284602
\(756\) 0 0
\(757\) −12.5047 −0.454491 −0.227246 0.973837i \(-0.572972\pi\)
−0.227246 + 0.973837i \(0.572972\pi\)
\(758\) 0 0
\(759\) −1.25171 −0.0454342
\(760\) 0 0
\(761\) 31.2830 1.13401 0.567004 0.823715i \(-0.308103\pi\)
0.567004 + 0.823715i \(0.308103\pi\)
\(762\) 0 0
\(763\) 11.0884 0.401425
\(764\) 0 0
\(765\) 4.44746 0.160798
\(766\) 0 0
\(767\) −6.61963 −0.239021
\(768\) 0 0
\(769\) −4.52132 −0.163043 −0.0815214 0.996672i \(-0.525978\pi\)
−0.0815214 + 0.996672i \(0.525978\pi\)
\(770\) 0 0
\(771\) −2.96131 −0.106649
\(772\) 0 0
\(773\) −10.2992 −0.370435 −0.185217 0.982698i \(-0.559299\pi\)
−0.185217 + 0.982698i \(0.559299\pi\)
\(774\) 0 0
\(775\) 15.0953 0.542240
\(776\) 0 0
\(777\) 4.78218 0.171560
\(778\) 0 0
\(779\) −3.93897 −0.141128
\(780\) 0 0
\(781\) 0.627835 0.0224657
\(782\) 0 0
\(783\) 2.20663 0.0788584
\(784\) 0 0
\(785\) 19.3077 0.689121
\(786\) 0 0
\(787\) −53.0205 −1.88998 −0.944988 0.327105i \(-0.893927\pi\)
−0.944988 + 0.327105i \(0.893927\pi\)
\(788\) 0 0
\(789\) −19.5063 −0.694442
\(790\) 0 0
\(791\) −9.22131 −0.327872
\(792\) 0 0
\(793\) 9.40115 0.333845
\(794\) 0 0
\(795\) 1.04551 0.0370805
\(796\) 0 0
\(797\) 35.1890 1.24646 0.623229 0.782040i \(-0.285821\pi\)
0.623229 + 0.782040i \(0.285821\pi\)
\(798\) 0 0
\(799\) 0.904652 0.0320043
\(800\) 0 0
\(801\) −2.00258 −0.0707576
\(802\) 0 0
\(803\) −2.04745 −0.0722528
\(804\) 0 0
\(805\) 3.83410 0.135134
\(806\) 0 0
\(807\) −18.2657 −0.642984
\(808\) 0 0
\(809\) 19.4985 0.685530 0.342765 0.939421i \(-0.388637\pi\)
0.342765 + 0.939421i \(0.388637\pi\)
\(810\) 0 0
\(811\) −39.8768 −1.40026 −0.700131 0.714014i \(-0.746875\pi\)
−0.700131 + 0.714014i \(0.746875\pi\)
\(812\) 0 0
\(813\) −7.41708 −0.260128
\(814\) 0 0
\(815\) 2.15629 0.0755314
\(816\) 0 0
\(817\) 37.6919 1.31867
\(818\) 0 0
\(819\) 2.34074 0.0817920
\(820\) 0 0
\(821\) 18.9774 0.662314 0.331157 0.943576i \(-0.392561\pi\)
0.331157 + 0.943576i \(0.392561\pi\)
\(822\) 0 0
\(823\) −14.6289 −0.509932 −0.254966 0.966950i \(-0.582064\pi\)
−0.254966 + 0.966950i \(0.582064\pi\)
\(824\) 0 0
\(825\) 1.20708 0.0420250
\(826\) 0 0
\(827\) −41.6729 −1.44911 −0.724554 0.689218i \(-0.757954\pi\)
−0.724554 + 0.689218i \(0.757954\pi\)
\(828\) 0 0
\(829\) −56.2693 −1.95431 −0.977157 0.212518i \(-0.931834\pi\)
−0.977157 + 0.212518i \(0.931834\pi\)
\(830\) 0 0
\(831\) 20.9772 0.727691
\(832\) 0 0
\(833\) −16.5283 −0.572670
\(834\) 0 0
\(835\) 16.8878 0.584426
\(836\) 0 0
\(837\) −6.45143 −0.222994
\(838\) 0 0
\(839\) −24.9995 −0.863079 −0.431540 0.902094i \(-0.642030\pi\)
−0.431540 + 0.902094i \(0.642030\pi\)
\(840\) 0 0
\(841\) −24.1308 −0.832097
\(842\) 0 0
\(843\) −4.11739 −0.141811
\(844\) 0 0
\(845\) −11.6827 −0.401896
\(846\) 0 0
\(847\) −10.3994 −0.357329
\(848\) 0 0
\(849\) 30.4094 1.04365
\(850\) 0 0
\(851\) 11.9765 0.410548
\(852\) 0 0
\(853\) −47.5546 −1.62824 −0.814119 0.580699i \(-0.802779\pi\)
−0.814119 + 0.580699i \(0.802779\pi\)
\(854\) 0 0
\(855\) 5.36245 0.183392
\(856\) 0 0
\(857\) −29.1854 −0.996956 −0.498478 0.866902i \(-0.666108\pi\)
−0.498478 + 0.866902i \(0.666108\pi\)
\(858\) 0 0
\(859\) 14.0652 0.479897 0.239949 0.970786i \(-0.422869\pi\)
0.239949 + 0.970786i \(0.422869\pi\)
\(860\) 0 0
\(861\) −1.16072 −0.0395571
\(862\) 0 0
\(863\) 47.2171 1.60729 0.803645 0.595109i \(-0.202891\pi\)
0.803645 + 0.595109i \(0.202891\pi\)
\(864\) 0 0
\(865\) 41.6619 1.41655
\(866\) 0 0
\(867\) −9.56440 −0.324824
\(868\) 0 0
\(869\) 7.89333 0.267763
\(870\) 0 0
\(871\) 12.4586 0.422144
\(872\) 0 0
\(873\) −3.76692 −0.127491
\(874\) 0 0
\(875\) −11.5983 −0.392094
\(876\) 0 0
\(877\) −17.3622 −0.586279 −0.293139 0.956070i \(-0.594700\pi\)
−0.293139 + 0.956070i \(0.594700\pi\)
\(878\) 0 0
\(879\) −10.8666 −0.366522
\(880\) 0 0
\(881\) −28.0490 −0.944995 −0.472497 0.881332i \(-0.656647\pi\)
−0.472497 + 0.881332i \(0.656647\pi\)
\(882\) 0 0
\(883\) 4.61175 0.155198 0.0775989 0.996985i \(-0.475275\pi\)
0.0775989 + 0.996985i \(0.475275\pi\)
\(884\) 0 0
\(885\) −4.46877 −0.150216
\(886\) 0 0
\(887\) −55.7404 −1.87158 −0.935789 0.352560i \(-0.885311\pi\)
−0.935789 + 0.352560i \(0.885311\pi\)
\(888\) 0 0
\(889\) 5.96091 0.199922
\(890\) 0 0
\(891\) −0.515879 −0.0172826
\(892\) 0 0
\(893\) 1.09077 0.0365012
\(894\) 0 0
\(895\) −15.5099 −0.518437
\(896\) 0 0
\(897\) 5.86212 0.195731
\(898\) 0 0
\(899\) −14.2359 −0.474794
\(900\) 0 0
\(901\) 1.74797 0.0582332
\(902\) 0 0
\(903\) 11.1069 0.369613
\(904\) 0 0
\(905\) −32.0620 −1.06578
\(906\) 0 0
\(907\) 6.83474 0.226944 0.113472 0.993541i \(-0.463803\pi\)
0.113472 + 0.993541i \(0.463803\pi\)
\(908\) 0 0
\(909\) 17.2846 0.573295
\(910\) 0 0
\(911\) 16.2339 0.537852 0.268926 0.963161i \(-0.413331\pi\)
0.268926 + 0.963161i \(0.413331\pi\)
\(912\) 0 0
\(913\) −4.18928 −0.138645
\(914\) 0 0
\(915\) 6.34652 0.209809
\(916\) 0 0
\(917\) −14.9075 −0.492289
\(918\) 0 0
\(919\) 14.2132 0.468849 0.234425 0.972134i \(-0.424679\pi\)
0.234425 + 0.972134i \(0.424679\pi\)
\(920\) 0 0
\(921\) −9.21688 −0.303707
\(922\) 0 0
\(923\) −2.94033 −0.0967821
\(924\) 0 0
\(925\) −11.5494 −0.379742
\(926\) 0 0
\(927\) −15.0176 −0.493243
\(928\) 0 0
\(929\) −6.00736 −0.197095 −0.0985476 0.995132i \(-0.531420\pi\)
−0.0985476 + 0.995132i \(0.531420\pi\)
\(930\) 0 0
\(931\) −19.9287 −0.653136
\(932\) 0 0
\(933\) −23.0405 −0.754313
\(934\) 0 0
\(935\) −2.29435 −0.0750333
\(936\) 0 0
\(937\) −34.6118 −1.13072 −0.565359 0.824845i \(-0.691262\pi\)
−0.565359 + 0.824845i \(0.691262\pi\)
\(938\) 0 0
\(939\) 20.5492 0.670597
\(940\) 0 0
\(941\) 32.2853 1.05247 0.526236 0.850339i \(-0.323603\pi\)
0.526236 + 0.850339i \(0.323603\pi\)
\(942\) 0 0
\(943\) −2.90689 −0.0946613
\(944\) 0 0
\(945\) 1.58018 0.0514033
\(946\) 0 0
\(947\) −31.3810 −1.01974 −0.509872 0.860250i \(-0.670307\pi\)
−0.509872 + 0.860250i \(0.670307\pi\)
\(948\) 0 0
\(949\) 9.58878 0.311265
\(950\) 0 0
\(951\) −31.5199 −1.02210
\(952\) 0 0
\(953\) 38.3376 1.24188 0.620939 0.783859i \(-0.286752\pi\)
0.620939 + 0.783859i \(0.286752\pi\)
\(954\) 0 0
\(955\) 41.7089 1.34967
\(956\) 0 0
\(957\) −1.13835 −0.0367977
\(958\) 0 0
\(959\) 9.10604 0.294049
\(960\) 0 0
\(961\) 10.6209 0.342611
\(962\) 0 0
\(963\) −0.951772 −0.0306704
\(964\) 0 0
\(965\) −2.40219 −0.0773293
\(966\) 0 0
\(967\) 23.5213 0.756393 0.378196 0.925725i \(-0.376544\pi\)
0.378196 + 0.925725i \(0.376544\pi\)
\(968\) 0 0
\(969\) 8.96536 0.288009
\(970\) 0 0
\(971\) −6.12179 −0.196458 −0.0982288 0.995164i \(-0.531318\pi\)
−0.0982288 + 0.995164i \(0.531318\pi\)
\(972\) 0 0
\(973\) 16.4061 0.525954
\(974\) 0 0
\(975\) −5.65309 −0.181044
\(976\) 0 0
\(977\) −30.8354 −0.986512 −0.493256 0.869884i \(-0.664194\pi\)
−0.493256 + 0.869884i \(0.664194\pi\)
\(978\) 0 0
\(979\) 1.03309 0.0330176
\(980\) 0 0
\(981\) 11.4449 0.365409
\(982\) 0 0
\(983\) 48.4649 1.54579 0.772895 0.634534i \(-0.218808\pi\)
0.772895 + 0.634534i \(0.218808\pi\)
\(984\) 0 0
\(985\) 26.9810 0.859687
\(986\) 0 0
\(987\) 0.321423 0.0102310
\(988\) 0 0
\(989\) 27.8159 0.884496
\(990\) 0 0
\(991\) −54.5862 −1.73399 −0.866993 0.498320i \(-0.833951\pi\)
−0.866993 + 0.498320i \(0.833951\pi\)
\(992\) 0 0
\(993\) −15.0690 −0.478201
\(994\) 0 0
\(995\) −23.3335 −0.739723
\(996\) 0 0
\(997\) −1.47038 −0.0465674 −0.0232837 0.999729i \(-0.507412\pi\)
−0.0232837 + 0.999729i \(0.507412\pi\)
\(998\) 0 0
\(999\) 4.93597 0.156167
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 6036.2.a.h.1.14 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6036.2.a.h.1.14 24 1.1 even 1 trivial