Properties

Label 6552.2.a.bk.1.2
Level $6552$
Weight $2$
Character 6552.1
Self dual yes
Analytic conductor $52.318$
Analytic rank $1$
Dimension $2$
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] = [6552,2,Mod(1,6552)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6552, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6552.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6552.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: \(52.3179834043\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2184)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 6552.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.56155 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+3.56155 q^{5} -1.00000 q^{7} -3.12311 q^{11} +1.00000 q^{13} +1.12311 q^{17} -1.56155 q^{19} -8.68466 q^{23} +7.68466 q^{25} -0.438447 q^{29} -5.56155 q^{31} -3.56155 q^{35} +1.12311 q^{37} +2.00000 q^{41} -9.56155 q^{43} -9.56155 q^{47} +1.00000 q^{49} -6.68466 q^{53} -11.1231 q^{55} +8.00000 q^{59} -2.00000 q^{61} +3.56155 q^{65} -7.12311 q^{67} +10.2462 q^{71} -8.43845 q^{73} +3.12311 q^{77} +8.68466 q^{79} +10.4384 q^{83} +4.00000 q^{85} -9.80776 q^{89} -1.00000 q^{91} -5.56155 q^{95} -6.68466 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} - 6 q^{17} + q^{19} - 5 q^{23} + 3 q^{25} - 5 q^{29} - 7 q^{31} - 3 q^{35} - 6 q^{37} + 4 q^{41} - 15 q^{43} - 15 q^{47} + 2 q^{49} - q^{53} - 14 q^{55} + 16 q^{59} - 4 q^{61} + 3 q^{65} - 6 q^{67} + 4 q^{71} - 21 q^{73} - 2 q^{77} + 5 q^{79} + 25 q^{83} + 8 q^{85} + q^{89} - 2 q^{91} - 7 q^{95} - q^{97}+O(q^{100}) \) 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 0
\(4\) 0 0
\(5\) 3.56155 1.59277 0.796387 0.604787i \(-0.206742\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.12311 −0.941652 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.12311 0.272393 0.136197 0.990682i \(-0.456512\pi\)
0.136197 + 0.990682i \(0.456512\pi\)
\(18\) 0 0
\(19\) −1.56155 −0.358245 −0.179122 0.983827i \(-0.557326\pi\)
−0.179122 + 0.983827i \(0.557326\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.68466 −1.81088 −0.905438 0.424478i \(-0.860458\pi\)
−0.905438 + 0.424478i \(0.860458\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.438447 −0.0814176 −0.0407088 0.999171i \(-0.512962\pi\)
−0.0407088 + 0.999171i \(0.512962\pi\)
\(30\) 0 0
\(31\) −5.56155 −0.998884 −0.499442 0.866347i \(-0.666462\pi\)
−0.499442 + 0.866347i \(0.666462\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.56155 −0.602012
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −9.56155 −1.45812 −0.729062 0.684448i \(-0.760043\pi\)
−0.729062 + 0.684448i \(0.760043\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.56155 −1.39470 −0.697348 0.716733i \(-0.745637\pi\)
−0.697348 + 0.716733i \(0.745637\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.68466 −0.918208 −0.459104 0.888382i \(-0.651830\pi\)
−0.459104 + 0.888382i \(0.651830\pi\)
\(54\) 0 0
\(55\) −11.1231 −1.49984
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.56155 0.441756
\(66\) 0 0
\(67\) −7.12311 −0.870226 −0.435113 0.900376i \(-0.643292\pi\)
−0.435113 + 0.900376i \(0.643292\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2462 1.21600 0.608001 0.793936i \(-0.291972\pi\)
0.608001 + 0.793936i \(0.291972\pi\)
\(72\) 0 0
\(73\) −8.43845 −0.987646 −0.493823 0.869563i \(-0.664401\pi\)
−0.493823 + 0.869563i \(0.664401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.12311 0.355911
\(78\) 0 0
\(79\) 8.68466 0.977100 0.488550 0.872536i \(-0.337526\pi\)
0.488550 + 0.872536i \(0.337526\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.4384 1.14577 0.572884 0.819636i \(-0.305824\pi\)
0.572884 + 0.819636i \(0.305824\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.80776 −1.03962 −0.519810 0.854282i \(-0.673997\pi\)
−0.519810 + 0.854282i \(0.673997\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.56155 −0.570603
\(96\) 0 0
\(97\) −6.68466 −0.678724 −0.339362 0.940656i \(-0.610211\pi\)
−0.339362 + 0.940656i \(0.610211\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.24621 −0.422514 −0.211257 0.977431i \(-0.567756\pi\)
−0.211257 + 0.977431i \(0.567756\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.4924 1.59438 0.797191 0.603727i \(-0.206318\pi\)
0.797191 + 0.603727i \(0.206318\pi\)
\(108\) 0 0
\(109\) −16.2462 −1.55610 −0.778052 0.628199i \(-0.783792\pi\)
−0.778052 + 0.628199i \(0.783792\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.31534 −0.123737 −0.0618685 0.998084i \(-0.519706\pi\)
−0.0618685 + 0.998084i \(0.519706\pi\)
\(114\) 0 0
\(115\) −30.9309 −2.88432
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.12311 −0.102955
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.56155 0.855211
\(126\) 0 0
\(127\) −22.2462 −1.97403 −0.987016 0.160622i \(-0.948650\pi\)
−0.987016 + 0.160622i \(0.948650\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 1.56155 0.135404
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.1231 1.12118 0.560591 0.828093i \(-0.310574\pi\)
0.560591 + 0.828093i \(0.310574\pi\)
\(138\) 0 0
\(139\) −15.1231 −1.28273 −0.641363 0.767238i \(-0.721631\pi\)
−0.641363 + 0.767238i \(0.721631\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.12311 −0.261167
\(144\) 0 0
\(145\) −1.56155 −0.129680
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.24621 −0.675556 −0.337778 0.941226i \(-0.609675\pi\)
−0.337778 + 0.941226i \(0.609675\pi\)
\(150\) 0 0
\(151\) 12.4924 1.01662 0.508309 0.861174i \(-0.330271\pi\)
0.508309 + 0.861174i \(0.330271\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −19.8078 −1.59100
\(156\) 0 0
\(157\) 9.12311 0.728103 0.364052 0.931379i \(-0.381393\pi\)
0.364052 + 0.931379i \(0.381393\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.68466 0.684447
\(162\) 0 0
\(163\) 2.24621 0.175937 0.0879684 0.996123i \(-0.471963\pi\)
0.0879684 + 0.996123i \(0.471963\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.6847 −0.981568 −0.490784 0.871281i \(-0.663290\pi\)
−0.490784 + 0.871281i \(0.663290\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.36932 −0.560279 −0.280139 0.959959i \(-0.590381\pi\)
−0.280139 + 0.959959i \(0.590381\pi\)
\(174\) 0 0
\(175\) −7.68466 −0.580906
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.5616 1.31261 0.656306 0.754495i \(-0.272118\pi\)
0.656306 + 0.754495i \(0.272118\pi\)
\(180\) 0 0
\(181\) 12.2462 0.910254 0.455127 0.890427i \(-0.349594\pi\)
0.455127 + 0.890427i \(0.349594\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −3.50758 −0.256499
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.4924 0.903920 0.451960 0.892038i \(-0.350725\pi\)
0.451960 + 0.892038i \(0.350725\pi\)
\(192\) 0 0
\(193\) 8.24621 0.593575 0.296788 0.954944i \(-0.404085\pi\)
0.296788 + 0.954944i \(0.404085\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 6.24621 0.442782 0.221391 0.975185i \(-0.428940\pi\)
0.221391 + 0.975185i \(0.428940\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.438447 0.0307730
\(204\) 0 0
\(205\) 7.12311 0.497499
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.87689 0.337342
\(210\) 0 0
\(211\) −3.31534 −0.228238 −0.114119 0.993467i \(-0.536404\pi\)
−0.114119 + 0.993467i \(0.536404\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −34.0540 −2.32246
\(216\) 0 0
\(217\) 5.56155 0.377543
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.12311 0.0755483
\(222\) 0 0
\(223\) 11.8078 0.790706 0.395353 0.918529i \(-0.370622\pi\)
0.395353 + 0.918529i \(0.370622\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.2462 1.47653 0.738266 0.674509i \(-0.235645\pi\)
0.738266 + 0.674509i \(0.235645\pi\)
\(228\) 0 0
\(229\) −16.2462 −1.07358 −0.536790 0.843716i \(-0.680363\pi\)
−0.536790 + 0.843716i \(0.680363\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.6847 −1.74817 −0.874085 0.485773i \(-0.838538\pi\)
−0.874085 + 0.485773i \(0.838538\pi\)
\(234\) 0 0
\(235\) −34.0540 −2.22144
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.4924 1.58428 0.792142 0.610337i \(-0.208966\pi\)
0.792142 + 0.610337i \(0.208966\pi\)
\(240\) 0 0
\(241\) 5.80776 0.374111 0.187055 0.982349i \(-0.440106\pi\)
0.187055 + 0.982349i \(0.440106\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.56155 0.227539
\(246\) 0 0
\(247\) −1.56155 −0.0993592
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.87689 0.560305 0.280152 0.959956i \(-0.409615\pi\)
0.280152 + 0.959956i \(0.409615\pi\)
\(252\) 0 0
\(253\) 27.1231 1.70522
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −1.12311 −0.0697864
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.5616 −0.836241 −0.418121 0.908392i \(-0.637311\pi\)
−0.418121 + 0.908392i \(0.637311\pi\)
\(264\) 0 0
\(265\) −23.8078 −1.46250
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.6155 1.07404 0.537019 0.843570i \(-0.319550\pi\)
0.537019 + 0.843570i \(0.319550\pi\)
\(270\) 0 0
\(271\) −12.4924 −0.758861 −0.379430 0.925220i \(-0.623880\pi\)
−0.379430 + 0.925220i \(0.623880\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.0000 −1.44725
\(276\) 0 0
\(277\) 16.0540 0.964590 0.482295 0.876009i \(-0.339803\pi\)
0.482295 + 0.876009i \(0.339803\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.24621 0.491928 0.245964 0.969279i \(-0.420896\pi\)
0.245964 + 0.969279i \(0.420896\pi\)
\(282\) 0 0
\(283\) −8.49242 −0.504822 −0.252411 0.967620i \(-0.581224\pi\)
−0.252411 + 0.967620i \(0.581224\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −15.7386 −0.925802
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.5616 0.675433 0.337717 0.941248i \(-0.390345\pi\)
0.337717 + 0.941248i \(0.390345\pi\)
\(294\) 0 0
\(295\) 28.4924 1.65889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.68466 −0.502247
\(300\) 0 0
\(301\) 9.56155 0.551119
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.12311 −0.407868
\(306\) 0 0
\(307\) −4.68466 −0.267368 −0.133684 0.991024i \(-0.542681\pi\)
−0.133684 + 0.991024i \(0.542681\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 29.8617 1.69330 0.846652 0.532147i \(-0.178615\pi\)
0.846652 + 0.532147i \(0.178615\pi\)
\(312\) 0 0
\(313\) −15.7538 −0.890457 −0.445228 0.895417i \(-0.646878\pi\)
−0.445228 + 0.895417i \(0.646878\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.3693 0.863227 0.431613 0.902059i \(-0.357944\pi\)
0.431613 + 0.902059i \(0.357944\pi\)
\(318\) 0 0
\(319\) 1.36932 0.0766670
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.75379 −0.0975834
\(324\) 0 0
\(325\) 7.68466 0.426268
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.56155 0.527145
\(330\) 0 0
\(331\) −15.1231 −0.831241 −0.415621 0.909538i \(-0.636436\pi\)
−0.415621 + 0.909538i \(0.636436\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −25.3693 −1.38607
\(336\) 0 0
\(337\) 28.0540 1.52820 0.764099 0.645099i \(-0.223184\pi\)
0.764099 + 0.645099i \(0.223184\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.3693 0.940601
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.24621 0.120583 0.0602915 0.998181i \(-0.480797\pi\)
0.0602915 + 0.998181i \(0.480797\pi\)
\(348\) 0 0
\(349\) −24.9309 −1.33452 −0.667259 0.744825i \(-0.732533\pi\)
−0.667259 + 0.744825i \(0.732533\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.2462 −0.651800 −0.325900 0.945404i \(-0.605667\pi\)
−0.325900 + 0.945404i \(0.605667\pi\)
\(354\) 0 0
\(355\) 36.4924 1.93682
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.4924 −1.29266 −0.646330 0.763058i \(-0.723697\pi\)
−0.646330 + 0.763058i \(0.723697\pi\)
\(360\) 0 0
\(361\) −16.5616 −0.871661
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −30.0540 −1.57310
\(366\) 0 0
\(367\) 7.61553 0.397527 0.198764 0.980047i \(-0.436307\pi\)
0.198764 + 0.980047i \(0.436307\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.68466 0.347050
\(372\) 0 0
\(373\) 34.4924 1.78595 0.892975 0.450106i \(-0.148614\pi\)
0.892975 + 0.450106i \(0.148614\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.438447 −0.0225812
\(378\) 0 0
\(379\) −13.3693 −0.686736 −0.343368 0.939201i \(-0.611568\pi\)
−0.343368 + 0.939201i \(0.611568\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 0 0
\(385\) 11.1231 0.566886
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 26.9848 1.36819 0.684093 0.729395i \(-0.260198\pi\)
0.684093 + 0.729395i \(0.260198\pi\)
\(390\) 0 0
\(391\) −9.75379 −0.493270
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 30.9309 1.55630
\(396\) 0 0
\(397\) −37.4233 −1.87822 −0.939111 0.343615i \(-0.888348\pi\)
−0.939111 + 0.343615i \(0.888348\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.87689 −0.143665 −0.0718326 0.997417i \(-0.522885\pi\)
−0.0718326 + 0.997417i \(0.522885\pi\)
\(402\) 0 0
\(403\) −5.56155 −0.277041
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.50758 −0.173864
\(408\) 0 0
\(409\) 23.5616 1.16504 0.582522 0.812815i \(-0.302066\pi\)
0.582522 + 0.812815i \(0.302066\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 37.1771 1.82495
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −21.3693 −1.04396 −0.521980 0.852958i \(-0.674806\pi\)
−0.521980 + 0.852958i \(0.674806\pi\)
\(420\) 0 0
\(421\) −6.87689 −0.335159 −0.167580 0.985859i \(-0.553595\pi\)
−0.167580 + 0.985859i \(0.553595\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.63068 0.418650
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.3693 1.02932 0.514662 0.857393i \(-0.327917\pi\)
0.514662 + 0.857393i \(0.327917\pi\)
\(432\) 0 0
\(433\) −21.6155 −1.03878 −0.519388 0.854539i \(-0.673840\pi\)
−0.519388 + 0.854539i \(0.673840\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.5616 0.648737
\(438\) 0 0
\(439\) 0.384472 0.0183498 0.00917492 0.999958i \(-0.497079\pi\)
0.00917492 + 0.999958i \(0.497079\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.8078 1.13114 0.565571 0.824700i \(-0.308656\pi\)
0.565571 + 0.824700i \(0.308656\pi\)
\(444\) 0 0
\(445\) −34.9309 −1.65588
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −27.8617 −1.31488 −0.657438 0.753508i \(-0.728360\pi\)
−0.657438 + 0.753508i \(0.728360\pi\)
\(450\) 0 0
\(451\) −6.24621 −0.294123
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.56155 −0.166968
\(456\) 0 0
\(457\) −5.61553 −0.262683 −0.131342 0.991337i \(-0.541928\pi\)
−0.131342 + 0.991337i \(0.541928\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 36.2462 1.68815 0.844077 0.536222i \(-0.180149\pi\)
0.844077 + 0.536222i \(0.180149\pi\)
\(462\) 0 0
\(463\) 14.6307 0.679946 0.339973 0.940435i \(-0.389582\pi\)
0.339973 + 0.940435i \(0.389582\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −37.3693 −1.72925 −0.864623 0.502421i \(-0.832443\pi\)
−0.864623 + 0.502421i \(0.832443\pi\)
\(468\) 0 0
\(469\) 7.12311 0.328914
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 29.8617 1.37304
\(474\) 0 0
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −28.6847 −1.31064 −0.655318 0.755353i \(-0.727465\pi\)
−0.655318 + 0.755353i \(0.727465\pi\)
\(480\) 0 0
\(481\) 1.12311 0.0512092
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −23.8078 −1.08105
\(486\) 0 0
\(487\) −33.3693 −1.51211 −0.756054 0.654509i \(-0.772875\pi\)
−0.756054 + 0.654509i \(0.772875\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.49242 −0.383258 −0.191629 0.981467i \(-0.561377\pi\)
−0.191629 + 0.981467i \(0.561377\pi\)
\(492\) 0 0
\(493\) −0.492423 −0.0221776
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.2462 −0.459605
\(498\) 0 0
\(499\) 16.4924 0.738302 0.369151 0.929369i \(-0.379648\pi\)
0.369151 + 0.929369i \(0.379648\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.36932 0.0610548 0.0305274 0.999534i \(-0.490281\pi\)
0.0305274 + 0.999534i \(0.490281\pi\)
\(504\) 0 0
\(505\) −15.1231 −0.672969
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.43845 −0.196731 −0.0983654 0.995150i \(-0.531361\pi\)
−0.0983654 + 0.995150i \(0.531361\pi\)
\(510\) 0 0
\(511\) 8.43845 0.373295
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −28.4924 −1.25553
\(516\) 0 0
\(517\) 29.8617 1.31332
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.8769 −0.651769 −0.325884 0.945410i \(-0.605662\pi\)
−0.325884 + 0.945410i \(0.605662\pi\)
\(522\) 0 0
\(523\) 7.12311 0.311472 0.155736 0.987799i \(-0.450225\pi\)
0.155736 + 0.987799i \(0.450225\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.24621 −0.272089
\(528\) 0 0
\(529\) 52.4233 2.27927
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 58.7386 2.53949
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.12311 −0.134522
\(540\) 0 0
\(541\) −43.3693 −1.86459 −0.932296 0.361696i \(-0.882198\pi\)
−0.932296 + 0.361696i \(0.882198\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −57.8617 −2.47852
\(546\) 0 0
\(547\) 38.0540 1.62707 0.813535 0.581516i \(-0.197540\pi\)
0.813535 + 0.581516i \(0.197540\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.684658 0.0291674
\(552\) 0 0
\(553\) −8.68466 −0.369309
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.36932 0.312248 0.156124 0.987737i \(-0.450100\pi\)
0.156124 + 0.987737i \(0.450100\pi\)
\(558\) 0 0
\(559\) −9.56155 −0.404411
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.4924 −1.03223 −0.516116 0.856519i \(-0.672623\pi\)
−0.516116 + 0.856519i \(0.672623\pi\)
\(564\) 0 0
\(565\) −4.68466 −0.197085
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.17708 −0.300879 −0.150439 0.988619i \(-0.548069\pi\)
−0.150439 + 0.988619i \(0.548069\pi\)
\(570\) 0 0
\(571\) −30.0540 −1.25772 −0.628860 0.777519i \(-0.716478\pi\)
−0.628860 + 0.777519i \(0.716478\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −66.7386 −2.78319
\(576\) 0 0
\(577\) −12.2462 −0.509816 −0.254908 0.966965i \(-0.582045\pi\)
−0.254908 + 0.966965i \(0.582045\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −10.4384 −0.433060
\(582\) 0 0
\(583\) 20.8769 0.864633
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.9309 0.616263 0.308131 0.951344i \(-0.400296\pi\)
0.308131 + 0.951344i \(0.400296\pi\)
\(588\) 0 0
\(589\) 8.68466 0.357845
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.06913 0.126034 0.0630170 0.998012i \(-0.479928\pi\)
0.0630170 + 0.998012i \(0.479928\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 35.8078 1.46307 0.731533 0.681806i \(-0.238805\pi\)
0.731533 + 0.681806i \(0.238805\pi\)
\(600\) 0 0
\(601\) −18.4924 −0.754322 −0.377161 0.926148i \(-0.623100\pi\)
−0.377161 + 0.926148i \(0.623100\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.43845 −0.180449
\(606\) 0 0
\(607\) 11.1231 0.451473 0.225736 0.974188i \(-0.427521\pi\)
0.225736 + 0.974188i \(0.427521\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.56155 −0.386819
\(612\) 0 0
\(613\) −14.8769 −0.600872 −0.300436 0.953802i \(-0.597132\pi\)
−0.300436 + 0.953802i \(0.597132\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 8.49242 0.341339 0.170670 0.985328i \(-0.445407\pi\)
0.170670 + 0.985328i \(0.445407\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.80776 0.392940
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.26137 0.0502940
\(630\) 0 0
\(631\) −42.3542 −1.68609 −0.843046 0.537841i \(-0.819240\pi\)
−0.843046 + 0.537841i \(0.819240\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −79.2311 −3.14419
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.94602 0.155859 0.0779293 0.996959i \(-0.475169\pi\)
0.0779293 + 0.996959i \(0.475169\pi\)
\(642\) 0 0
\(643\) 22.7386 0.896724 0.448362 0.893852i \(-0.352008\pi\)
0.448362 + 0.893852i \(0.352008\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.6307 0.889704 0.444852 0.895604i \(-0.353256\pi\)
0.444852 + 0.895604i \(0.353256\pi\)
\(648\) 0 0
\(649\) −24.9848 −0.980741
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −36.2462 −1.41842 −0.709212 0.704995i \(-0.750949\pi\)
−0.709212 + 0.704995i \(0.750949\pi\)
\(654\) 0 0
\(655\) −14.2462 −0.556646
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.94602 0.387442 0.193721 0.981057i \(-0.437944\pi\)
0.193721 + 0.981057i \(0.437944\pi\)
\(660\) 0 0
\(661\) −14.1922 −0.552014 −0.276007 0.961156i \(-0.589011\pi\)
−0.276007 + 0.961156i \(0.589011\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.56155 0.215668
\(666\) 0 0
\(667\) 3.80776 0.147437
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.24621 0.241132
\(672\) 0 0
\(673\) 3.06913 0.118306 0.0591531 0.998249i \(-0.481160\pi\)
0.0591531 + 0.998249i \(0.481160\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.4924 0.864454 0.432227 0.901765i \(-0.357728\pi\)
0.432227 + 0.901765i \(0.357728\pi\)
\(678\) 0 0
\(679\) 6.68466 0.256534
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.9848 1.26213 0.631065 0.775730i \(-0.282618\pi\)
0.631065 + 0.775730i \(0.282618\pi\)
\(684\) 0 0
\(685\) 46.7386 1.78579
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.68466 −0.254665
\(690\) 0 0
\(691\) 17.5616 0.668073 0.334036 0.942560i \(-0.391589\pi\)
0.334036 + 0.942560i \(0.391589\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −53.8617 −2.04309
\(696\) 0 0
\(697\) 2.24621 0.0850813
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −38.3002 −1.44658 −0.723289 0.690545i \(-0.757371\pi\)
−0.723289 + 0.690545i \(0.757371\pi\)
\(702\) 0 0
\(703\) −1.75379 −0.0661454
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.24621 0.159695
\(708\) 0 0
\(709\) 1.50758 0.0566183 0.0283091 0.999599i \(-0.490988\pi\)
0.0283091 + 0.999599i \(0.490988\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 48.3002 1.80886
\(714\) 0 0
\(715\) −11.1231 −0.415981
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −22.6307 −0.843982 −0.421991 0.906600i \(-0.638669\pi\)
−0.421991 + 0.906600i \(0.638669\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.36932 −0.125133
\(726\) 0 0
\(727\) −52.1080 −1.93258 −0.966288 0.257462i \(-0.917114\pi\)
−0.966288 + 0.257462i \(0.917114\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.7386 −0.397183
\(732\) 0 0
\(733\) −7.94602 −0.293493 −0.146747 0.989174i \(-0.546880\pi\)
−0.146747 + 0.989174i \(0.546880\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22.2462 0.819450
\(738\) 0 0
\(739\) −26.6307 −0.979626 −0.489813 0.871828i \(-0.662935\pi\)
−0.489813 + 0.871828i \(0.662935\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.2462 −1.25637 −0.628186 0.778063i \(-0.716202\pi\)
−0.628186 + 0.778063i \(0.716202\pi\)
\(744\) 0 0
\(745\) −29.3693 −1.07601
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −16.4924 −0.602620
\(750\) 0 0
\(751\) 6.93087 0.252911 0.126456 0.991972i \(-0.459640\pi\)
0.126456 + 0.991972i \(0.459640\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 44.4924 1.61925
\(756\) 0 0
\(757\) 19.5616 0.710977 0.355488 0.934681i \(-0.384315\pi\)
0.355488 + 0.934681i \(0.384315\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.3002 1.24338 0.621690 0.783263i \(-0.286446\pi\)
0.621690 + 0.783263i \(0.286446\pi\)
\(762\) 0 0
\(763\) 16.2462 0.588152
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 18.6847 0.673786 0.336893 0.941543i \(-0.390624\pi\)
0.336893 + 0.941543i \(0.390624\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −46.4924 −1.67222 −0.836108 0.548565i \(-0.815174\pi\)
−0.836108 + 0.548565i \(0.815174\pi\)
\(774\) 0 0
\(775\) −42.7386 −1.53522
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.12311 −0.111897
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 32.4924 1.15970
\(786\) 0 0
\(787\) 47.4233 1.69046 0.845229 0.534404i \(-0.179464\pi\)
0.845229 + 0.534404i \(0.179464\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.31534 0.0467682
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) −10.7386 −0.379906
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 26.3542 0.930018
\(804\) 0 0
\(805\) 30.9309 1.09017
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −53.0388 −1.86475 −0.932373 0.361498i \(-0.882265\pi\)
−0.932373 + 0.361498i \(0.882265\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 14.9309 0.522365
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 29.2311 1.02017 0.510085 0.860124i \(-0.329614\pi\)
0.510085 + 0.860124i \(0.329614\pi\)
\(822\) 0 0
\(823\) −24.9848 −0.870917 −0.435458 0.900209i \(-0.643414\pi\)
−0.435458 + 0.900209i \(0.643414\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −52.4924 −1.82534 −0.912670 0.408697i \(-0.865983\pi\)
−0.912670 + 0.408697i \(0.865983\pi\)
\(828\) 0 0
\(829\) 20.2462 0.703180 0.351590 0.936154i \(-0.385641\pi\)
0.351590 + 0.936154i \(0.385641\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.12311 0.0389133
\(834\) 0 0
\(835\) −45.1771 −1.56342
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 22.7386 0.785025 0.392512 0.919747i \(-0.371606\pi\)
0.392512 + 0.919747i \(0.371606\pi\)
\(840\) 0 0
\(841\) −28.8078 −0.993371
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.56155 0.122521
\(846\) 0 0
\(847\) 1.24621 0.0428203
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −9.75379 −0.334356
\(852\) 0 0
\(853\) 31.6695 1.08434 0.542172 0.840268i \(-0.317602\pi\)
0.542172 + 0.840268i \(0.317602\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.7538 0.538139 0.269070 0.963121i \(-0.413284\pi\)
0.269070 + 0.963121i \(0.413284\pi\)
\(858\) 0 0
\(859\) −39.1231 −1.33486 −0.667432 0.744671i \(-0.732606\pi\)
−0.667432 + 0.744671i \(0.732606\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 44.9848 1.53130 0.765651 0.643256i \(-0.222417\pi\)
0.765651 + 0.643256i \(0.222417\pi\)
\(864\) 0 0
\(865\) −26.2462 −0.892398
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −27.1231 −0.920088
\(870\) 0 0
\(871\) −7.12311 −0.241357
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.56155 −0.323239
\(876\) 0 0
\(877\) 50.1080 1.69203 0.846013 0.533163i \(-0.178997\pi\)
0.846013 + 0.533163i \(0.178997\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 7.50758 0.252650 0.126325 0.991989i \(-0.459682\pi\)
0.126325 + 0.991989i \(0.459682\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.7538 −1.13334 −0.566671 0.823944i \(-0.691769\pi\)
−0.566671 + 0.823944i \(0.691769\pi\)
\(888\) 0 0
\(889\) 22.2462 0.746114
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 14.9309 0.499643
\(894\) 0 0
\(895\) 62.5464 2.09070
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.43845 0.0813268
\(900\) 0 0
\(901\) −7.50758 −0.250114
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 43.6155 1.44983
\(906\) 0 0
\(907\) −17.5616 −0.583122 −0.291561 0.956552i \(-0.594175\pi\)
−0.291561 + 0.956552i \(0.594175\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.94602 0.197001 0.0985003 0.995137i \(-0.468595\pi\)
0.0985003 + 0.995137i \(0.468595\pi\)
\(912\) 0 0
\(913\) −32.6004 −1.07891
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −4.49242 −0.148191 −0.0740957 0.997251i \(-0.523607\pi\)
−0.0740957 + 0.997251i \(0.523607\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 10.2462 0.337258
\(924\) 0 0
\(925\) 8.63068 0.283775
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.5616 1.29797 0.648986 0.760800i \(-0.275193\pi\)
0.648986 + 0.760800i \(0.275193\pi\)
\(930\) 0 0
\(931\) −1.56155 −0.0511778
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.4924 −0.408546
\(936\) 0 0
\(937\) −13.6155 −0.444800 −0.222400 0.974956i \(-0.571389\pi\)
−0.222400 + 0.974956i \(0.571389\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −16.5464 −0.539397 −0.269699 0.962945i \(-0.586924\pi\)
−0.269699 + 0.962945i \(0.586924\pi\)
\(942\) 0 0
\(943\) −17.3693 −0.565623
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.8617 1.23034 0.615171 0.788394i \(-0.289087\pi\)
0.615171 + 0.788394i \(0.289087\pi\)
\(948\) 0 0
\(949\) −8.43845 −0.273924
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 53.9157 1.74650 0.873251 0.487271i \(-0.162008\pi\)
0.873251 + 0.487271i \(0.162008\pi\)
\(954\) 0 0
\(955\) 44.4924 1.43974
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.1231 −0.423767
\(960\) 0 0
\(961\) −0.0691303 −0.00223001
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 29.3693 0.945432
\(966\) 0 0
\(967\) 36.1080 1.16115 0.580577 0.814206i \(-0.302827\pi\)
0.580577 + 0.814206i \(0.302827\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 54.3542 1.74431 0.872154 0.489231i \(-0.162723\pi\)
0.872154 + 0.489231i \(0.162723\pi\)
\(972\) 0 0
\(973\) 15.1231 0.484825
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.5076 −0.560117 −0.280059 0.959983i \(-0.590354\pi\)
−0.280059 + 0.959983i \(0.590354\pi\)
\(978\) 0 0
\(979\) 30.6307 0.978961
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 42.5464 1.35702 0.678510 0.734591i \(-0.262626\pi\)
0.678510 + 0.734591i \(0.262626\pi\)
\(984\) 0 0
\(985\) −35.6155 −1.13481
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 83.0388 2.64048
\(990\) 0 0
\(991\) 10.7386 0.341124 0.170562 0.985347i \(-0.445442\pi\)
0.170562 + 0.985347i \(0.445442\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.2462 0.705252
\(996\) 0 0
\(997\) −58.6004 −1.85589 −0.927946 0.372714i \(-0.878427\pi\)
−0.927946 + 0.372714i \(0.878427\pi\)
\(998\) 0 0
\(999\) 0 0
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 6552.2.a.bk.1.2 2
3.2 odd 2 2184.2.a.n.1.1 2
12.11 even 2 4368.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.n.1.1 2 3.2 odd 2
4368.2.a.bf.1.1 2 12.11 even 2
6552.2.a.bk.1.2 2 1.1 even 1 trivial