Properties

Label 3344.2.a.t.1.3
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $5$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.71457\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.121872 q^{3} -1.06025 q^{5} +3.36889 q^{7} -2.98515 q^{9} +O(q^{10})\) \(q-0.121872 q^{3} -1.06025 q^{5} +3.36889 q^{7} -2.98515 q^{9} -1.00000 q^{11} -2.15993 q^{13} +0.129215 q^{15} +3.67288 q^{17} +1.00000 q^{19} -0.410573 q^{21} -3.15468 q^{23} -3.87587 q^{25} +0.729422 q^{27} +7.17849 q^{29} -4.65295 q^{31} +0.121872 q^{33} -3.57187 q^{35} +2.27446 q^{37} +0.263235 q^{39} -11.3852 q^{41} -9.38838 q^{43} +3.16501 q^{45} +5.77094 q^{47} +4.34940 q^{49} -0.447622 q^{51} -5.65820 q^{53} +1.06025 q^{55} -0.121872 q^{57} +13.7944 q^{59} +6.98152 q^{61} -10.0566 q^{63} +2.29007 q^{65} +4.81332 q^{67} +0.384467 q^{69} -15.2629 q^{71} -8.08806 q^{73} +0.472360 q^{75} -3.36889 q^{77} -13.4291 q^{79} +8.86655 q^{81} -9.96666 q^{83} -3.89418 q^{85} -0.874858 q^{87} -4.61626 q^{89} -7.27655 q^{91} +0.567064 q^{93} -1.06025 q^{95} -4.09907 q^{97} +2.98515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) −0.121872 −0.0703629 −0.0351814 0.999381i \(-0.511201\pi\)
−0.0351814 + 0.999381i \(0.511201\pi\)
\(4\) 0 0
\(5\) −1.06025 −0.474159 −0.237080 0.971490i \(-0.576190\pi\)
−0.237080 + 0.971490i \(0.576190\pi\)
\(6\) 0 0
\(7\) 3.36889 1.27332 0.636660 0.771145i \(-0.280316\pi\)
0.636660 + 0.771145i \(0.280316\pi\)
\(8\) 0 0
\(9\) −2.98515 −0.995049
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.15993 −0.599056 −0.299528 0.954087i \(-0.596829\pi\)
−0.299528 + 0.954087i \(0.596829\pi\)
\(14\) 0 0
\(15\) 0.129215 0.0333632
\(16\) 0 0
\(17\) 3.67288 0.890805 0.445402 0.895330i \(-0.353061\pi\)
0.445402 + 0.895330i \(0.353061\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.410573 −0.0895944
\(22\) 0 0
\(23\) −3.15468 −0.657797 −0.328898 0.944365i \(-0.606677\pi\)
−0.328898 + 0.944365i \(0.606677\pi\)
\(24\) 0 0
\(25\) −3.87587 −0.775173
\(26\) 0 0
\(27\) 0.729422 0.140377
\(28\) 0 0
\(29\) 7.17849 1.33301 0.666506 0.745499i \(-0.267789\pi\)
0.666506 + 0.745499i \(0.267789\pi\)
\(30\) 0 0
\(31\) −4.65295 −0.835694 −0.417847 0.908517i \(-0.637215\pi\)
−0.417847 + 0.908517i \(0.637215\pi\)
\(32\) 0 0
\(33\) 0.121872 0.0212152
\(34\) 0 0
\(35\) −3.57187 −0.603756
\(36\) 0 0
\(37\) 2.27446 0.373918 0.186959 0.982368i \(-0.440137\pi\)
0.186959 + 0.982368i \(0.440137\pi\)
\(38\) 0 0
\(39\) 0.263235 0.0421513
\(40\) 0 0
\(41\) −11.3852 −1.77807 −0.889034 0.457841i \(-0.848623\pi\)
−0.889034 + 0.457841i \(0.848623\pi\)
\(42\) 0 0
\(43\) −9.38838 −1.43171 −0.715857 0.698247i \(-0.753964\pi\)
−0.715857 + 0.698247i \(0.753964\pi\)
\(44\) 0 0
\(45\) 3.16501 0.471812
\(46\) 0 0
\(47\) 5.77094 0.841778 0.420889 0.907112i \(-0.361718\pi\)
0.420889 + 0.907112i \(0.361718\pi\)
\(48\) 0 0
\(49\) 4.34940 0.621342
\(50\) 0 0
\(51\) −0.447622 −0.0626796
\(52\) 0 0
\(53\) −5.65820 −0.777213 −0.388607 0.921404i \(-0.627043\pi\)
−0.388607 + 0.921404i \(0.627043\pi\)
\(54\) 0 0
\(55\) 1.06025 0.142964
\(56\) 0 0
\(57\) −0.121872 −0.0161423
\(58\) 0 0
\(59\) 13.7944 1.79588 0.897939 0.440121i \(-0.145064\pi\)
0.897939 + 0.440121i \(0.145064\pi\)
\(60\) 0 0
\(61\) 6.98152 0.893892 0.446946 0.894561i \(-0.352512\pi\)
0.446946 + 0.894561i \(0.352512\pi\)
\(62\) 0 0
\(63\) −10.0566 −1.26702
\(64\) 0 0
\(65\) 2.29007 0.284048
\(66\) 0 0
\(67\) 4.81332 0.588041 0.294020 0.955799i \(-0.405007\pi\)
0.294020 + 0.955799i \(0.405007\pi\)
\(68\) 0 0
\(69\) 0.384467 0.0462844
\(70\) 0 0
\(71\) −15.2629 −1.81137 −0.905685 0.423951i \(-0.860643\pi\)
−0.905685 + 0.423951i \(0.860643\pi\)
\(72\) 0 0
\(73\) −8.08806 −0.946636 −0.473318 0.880892i \(-0.656944\pi\)
−0.473318 + 0.880892i \(0.656944\pi\)
\(74\) 0 0
\(75\) 0.472360 0.0545434
\(76\) 0 0
\(77\) −3.36889 −0.383920
\(78\) 0 0
\(79\) −13.4291 −1.51090 −0.755448 0.655209i \(-0.772581\pi\)
−0.755448 + 0.655209i \(0.772581\pi\)
\(80\) 0 0
\(81\) 8.86655 0.985172
\(82\) 0 0
\(83\) −9.96666 −1.09398 −0.546992 0.837138i \(-0.684227\pi\)
−0.546992 + 0.837138i \(0.684227\pi\)
\(84\) 0 0
\(85\) −3.89418 −0.422383
\(86\) 0 0
\(87\) −0.874858 −0.0937946
\(88\) 0 0
\(89\) −4.61626 −0.489323 −0.244661 0.969609i \(-0.578677\pi\)
−0.244661 + 0.969609i \(0.578677\pi\)
\(90\) 0 0
\(91\) −7.27655 −0.762790
\(92\) 0 0
\(93\) 0.567064 0.0588018
\(94\) 0 0
\(95\) −1.06025 −0.108780
\(96\) 0 0
\(97\) −4.09907 −0.416197 −0.208099 0.978108i \(-0.566727\pi\)
−0.208099 + 0.978108i \(0.566727\pi\)
\(98\) 0 0
\(99\) 2.98515 0.300019
\(100\) 0 0
\(101\) −10.8730 −1.08190 −0.540950 0.841055i \(-0.681935\pi\)
−0.540950 + 0.841055i \(0.681935\pi\)
\(102\) 0 0
\(103\) −8.85864 −0.872867 −0.436434 0.899736i \(-0.643759\pi\)
−0.436434 + 0.899736i \(0.643759\pi\)
\(104\) 0 0
\(105\) 0.435311 0.0424820
\(106\) 0 0
\(107\) 18.1669 1.75626 0.878131 0.478421i \(-0.158791\pi\)
0.878131 + 0.478421i \(0.158791\pi\)
\(108\) 0 0
\(109\) −0.211303 −0.0202392 −0.0101196 0.999949i \(-0.503221\pi\)
−0.0101196 + 0.999949i \(0.503221\pi\)
\(110\) 0 0
\(111\) −0.277193 −0.0263100
\(112\) 0 0
\(113\) −0.198345 −0.0186587 −0.00932935 0.999956i \(-0.502970\pi\)
−0.00932935 + 0.999956i \(0.502970\pi\)
\(114\) 0 0
\(115\) 3.34476 0.311900
\(116\) 0 0
\(117\) 6.44770 0.596090
\(118\) 0 0
\(119\) 12.3735 1.13428
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 1.38754 0.125110
\(124\) 0 0
\(125\) 9.41066 0.841715
\(126\) 0 0
\(127\) 3.60331 0.319742 0.159871 0.987138i \(-0.448892\pi\)
0.159871 + 0.987138i \(0.448892\pi\)
\(128\) 0 0
\(129\) 1.14418 0.100739
\(130\) 0 0
\(131\) −16.8741 −1.47430 −0.737150 0.675729i \(-0.763829\pi\)
−0.737150 + 0.675729i \(0.763829\pi\)
\(132\) 0 0
\(133\) 3.36889 0.292119
\(134\) 0 0
\(135\) −0.773371 −0.0665612
\(136\) 0 0
\(137\) −6.10839 −0.521875 −0.260937 0.965356i \(-0.584032\pi\)
−0.260937 + 0.965356i \(0.584032\pi\)
\(138\) 0 0
\(139\) −11.3347 −0.961398 −0.480699 0.876886i \(-0.659617\pi\)
−0.480699 + 0.876886i \(0.659617\pi\)
\(140\) 0 0
\(141\) −0.703316 −0.0592299
\(142\) 0 0
\(143\) 2.15993 0.180622
\(144\) 0 0
\(145\) −7.61101 −0.632060
\(146\) 0 0
\(147\) −0.530070 −0.0437194
\(148\) 0 0
\(149\) 6.75698 0.553554 0.276777 0.960934i \(-0.410734\pi\)
0.276777 + 0.960934i \(0.410734\pi\)
\(150\) 0 0
\(151\) −6.46231 −0.525895 −0.262948 0.964810i \(-0.584695\pi\)
−0.262948 + 0.964810i \(0.584695\pi\)
\(152\) 0 0
\(153\) −10.9641 −0.886395
\(154\) 0 0
\(155\) 4.93330 0.396252
\(156\) 0 0
\(157\) 0.248382 0.0198230 0.00991152 0.999951i \(-0.496845\pi\)
0.00991152 + 0.999951i \(0.496845\pi\)
\(158\) 0 0
\(159\) 0.689576 0.0546869
\(160\) 0 0
\(161\) −10.6278 −0.837585
\(162\) 0 0
\(163\) 16.9710 1.32927 0.664637 0.747167i \(-0.268586\pi\)
0.664637 + 0.747167i \(0.268586\pi\)
\(164\) 0 0
\(165\) −0.129215 −0.0100594
\(166\) 0 0
\(167\) 0.865538 0.0669773 0.0334887 0.999439i \(-0.489338\pi\)
0.0334887 + 0.999439i \(0.489338\pi\)
\(168\) 0 0
\(169\) −8.33471 −0.641132
\(170\) 0 0
\(171\) −2.98515 −0.228280
\(172\) 0 0
\(173\) −4.14483 −0.315125 −0.157563 0.987509i \(-0.550364\pi\)
−0.157563 + 0.987509i \(0.550364\pi\)
\(174\) 0 0
\(175\) −13.0574 −0.987043
\(176\) 0 0
\(177\) −1.68115 −0.126363
\(178\) 0 0
\(179\) −2.04584 −0.152913 −0.0764567 0.997073i \(-0.524361\pi\)
−0.0764567 + 0.997073i \(0.524361\pi\)
\(180\) 0 0
\(181\) 5.27519 0.392101 0.196051 0.980594i \(-0.437188\pi\)
0.196051 + 0.980594i \(0.437188\pi\)
\(182\) 0 0
\(183\) −0.850852 −0.0628968
\(184\) 0 0
\(185\) −2.41150 −0.177297
\(186\) 0 0
\(187\) −3.67288 −0.268588
\(188\) 0 0
\(189\) 2.45734 0.178745
\(190\) 0 0
\(191\) 16.1899 1.17146 0.585729 0.810507i \(-0.300808\pi\)
0.585729 + 0.810507i \(0.300808\pi\)
\(192\) 0 0
\(193\) −20.0620 −1.44409 −0.722045 0.691846i \(-0.756798\pi\)
−0.722045 + 0.691846i \(0.756798\pi\)
\(194\) 0 0
\(195\) −0.279095 −0.0199864
\(196\) 0 0
\(197\) −23.8398 −1.69851 −0.849257 0.527979i \(-0.822950\pi\)
−0.849257 + 0.527979i \(0.822950\pi\)
\(198\) 0 0
\(199\) 1.90194 0.134825 0.0674125 0.997725i \(-0.478526\pi\)
0.0674125 + 0.997725i \(0.478526\pi\)
\(200\) 0 0
\(201\) −0.586609 −0.0413762
\(202\) 0 0
\(203\) 24.1835 1.69735
\(204\) 0 0
\(205\) 12.0712 0.843087
\(206\) 0 0
\(207\) 9.41719 0.654540
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −12.3048 −0.847100 −0.423550 0.905873i \(-0.639216\pi\)
−0.423550 + 0.905873i \(0.639216\pi\)
\(212\) 0 0
\(213\) 1.86012 0.127453
\(214\) 0 0
\(215\) 9.95405 0.678860
\(216\) 0 0
\(217\) −15.6753 −1.06411
\(218\) 0 0
\(219\) 0.985709 0.0666080
\(220\) 0 0
\(221\) −7.93316 −0.533642
\(222\) 0 0
\(223\) −22.8473 −1.52997 −0.764985 0.644048i \(-0.777254\pi\)
−0.764985 + 0.644048i \(0.777254\pi\)
\(224\) 0 0
\(225\) 11.5700 0.771335
\(226\) 0 0
\(227\) −4.00654 −0.265923 −0.132962 0.991121i \(-0.542449\pi\)
−0.132962 + 0.991121i \(0.542449\pi\)
\(228\) 0 0
\(229\) −24.6024 −1.62577 −0.812887 0.582422i \(-0.802105\pi\)
−0.812887 + 0.582422i \(0.802105\pi\)
\(230\) 0 0
\(231\) 0.410573 0.0270137
\(232\) 0 0
\(233\) 16.2994 1.06781 0.533903 0.845545i \(-0.320725\pi\)
0.533903 + 0.845545i \(0.320725\pi\)
\(234\) 0 0
\(235\) −6.11865 −0.399137
\(236\) 0 0
\(237\) 1.63664 0.106311
\(238\) 0 0
\(239\) 15.4182 0.997324 0.498662 0.866797i \(-0.333825\pi\)
0.498662 + 0.866797i \(0.333825\pi\)
\(240\) 0 0
\(241\) 24.1441 1.55526 0.777629 0.628723i \(-0.216422\pi\)
0.777629 + 0.628723i \(0.216422\pi\)
\(242\) 0 0
\(243\) −3.26885 −0.209697
\(244\) 0 0
\(245\) −4.61146 −0.294615
\(246\) 0 0
\(247\) −2.15993 −0.137433
\(248\) 0 0
\(249\) 1.21466 0.0769758
\(250\) 0 0
\(251\) 19.9099 1.25670 0.628350 0.777931i \(-0.283731\pi\)
0.628350 + 0.777931i \(0.283731\pi\)
\(252\) 0 0
\(253\) 3.15468 0.198333
\(254\) 0 0
\(255\) 0.474592 0.0297201
\(256\) 0 0
\(257\) −22.3126 −1.39182 −0.695911 0.718128i \(-0.744999\pi\)
−0.695911 + 0.718128i \(0.744999\pi\)
\(258\) 0 0
\(259\) 7.66239 0.476118
\(260\) 0 0
\(261\) −21.4289 −1.32641
\(262\) 0 0
\(263\) −13.7963 −0.850716 −0.425358 0.905025i \(-0.639852\pi\)
−0.425358 + 0.905025i \(0.639852\pi\)
\(264\) 0 0
\(265\) 5.99912 0.368523
\(266\) 0 0
\(267\) 0.562593 0.0344301
\(268\) 0 0
\(269\) −11.2028 −0.683048 −0.341524 0.939873i \(-0.610943\pi\)
−0.341524 + 0.939873i \(0.610943\pi\)
\(270\) 0 0
\(271\) −17.1234 −1.04017 −0.520085 0.854115i \(-0.674100\pi\)
−0.520085 + 0.854115i \(0.674100\pi\)
\(272\) 0 0
\(273\) 0.886808 0.0536721
\(274\) 0 0
\(275\) 3.87587 0.233723
\(276\) 0 0
\(277\) −13.7127 −0.823919 −0.411960 0.911202i \(-0.635156\pi\)
−0.411960 + 0.911202i \(0.635156\pi\)
\(278\) 0 0
\(279\) 13.8897 0.831557
\(280\) 0 0
\(281\) 25.8599 1.54267 0.771337 0.636427i \(-0.219588\pi\)
0.771337 + 0.636427i \(0.219588\pi\)
\(282\) 0 0
\(283\) −28.8711 −1.71621 −0.858103 0.513477i \(-0.828357\pi\)
−0.858103 + 0.513477i \(0.828357\pi\)
\(284\) 0 0
\(285\) 0.129215 0.00765404
\(286\) 0 0
\(287\) −38.3554 −2.26405
\(288\) 0 0
\(289\) −3.50993 −0.206467
\(290\) 0 0
\(291\) 0.499562 0.0292848
\(292\) 0 0
\(293\) 6.37971 0.372707 0.186353 0.982483i \(-0.440333\pi\)
0.186353 + 0.982483i \(0.440333\pi\)
\(294\) 0 0
\(295\) −14.6255 −0.851532
\(296\) 0 0
\(297\) −0.729422 −0.0423254
\(298\) 0 0
\(299\) 6.81389 0.394057
\(300\) 0 0
\(301\) −31.6284 −1.82303
\(302\) 0 0
\(303\) 1.32511 0.0761256
\(304\) 0 0
\(305\) −7.40217 −0.423847
\(306\) 0 0
\(307\) −10.4299 −0.595264 −0.297632 0.954681i \(-0.596197\pi\)
−0.297632 + 0.954681i \(0.596197\pi\)
\(308\) 0 0
\(309\) 1.07962 0.0614174
\(310\) 0 0
\(311\) 8.26224 0.468508 0.234254 0.972175i \(-0.424735\pi\)
0.234254 + 0.972175i \(0.424735\pi\)
\(312\) 0 0
\(313\) −24.1167 −1.36315 −0.681577 0.731746i \(-0.738706\pi\)
−0.681577 + 0.731746i \(0.738706\pi\)
\(314\) 0 0
\(315\) 10.6626 0.600767
\(316\) 0 0
\(317\) −11.7330 −0.658989 −0.329495 0.944157i \(-0.606878\pi\)
−0.329495 + 0.944157i \(0.606878\pi\)
\(318\) 0 0
\(319\) −7.17849 −0.401918
\(320\) 0 0
\(321\) −2.21404 −0.123576
\(322\) 0 0
\(323\) 3.67288 0.204365
\(324\) 0 0
\(325\) 8.37159 0.464372
\(326\) 0 0
\(327\) 0.0257519 0.00142409
\(328\) 0 0
\(329\) 19.4416 1.07185
\(330\) 0 0
\(331\) 6.95223 0.382129 0.191065 0.981577i \(-0.438806\pi\)
0.191065 + 0.981577i \(0.438806\pi\)
\(332\) 0 0
\(333\) −6.78959 −0.372067
\(334\) 0 0
\(335\) −5.10333 −0.278825
\(336\) 0 0
\(337\) 29.4416 1.60379 0.801893 0.597468i \(-0.203826\pi\)
0.801893 + 0.597468i \(0.203826\pi\)
\(338\) 0 0
\(339\) 0.0241727 0.00131288
\(340\) 0 0
\(341\) 4.65295 0.251971
\(342\) 0 0
\(343\) −8.92959 −0.482152
\(344\) 0 0
\(345\) −0.407632 −0.0219462
\(346\) 0 0
\(347\) 17.2180 0.924311 0.462155 0.886799i \(-0.347076\pi\)
0.462155 + 0.886799i \(0.347076\pi\)
\(348\) 0 0
\(349\) 4.32405 0.231461 0.115731 0.993281i \(-0.463079\pi\)
0.115731 + 0.993281i \(0.463079\pi\)
\(350\) 0 0
\(351\) −1.57550 −0.0840939
\(352\) 0 0
\(353\) 23.4421 1.24770 0.623848 0.781545i \(-0.285568\pi\)
0.623848 + 0.781545i \(0.285568\pi\)
\(354\) 0 0
\(355\) 16.1825 0.858878
\(356\) 0 0
\(357\) −1.50799 −0.0798111
\(358\) 0 0
\(359\) −8.20544 −0.433066 −0.216533 0.976275i \(-0.569475\pi\)
−0.216533 + 0.976275i \(0.569475\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.121872 −0.00639662
\(364\) 0 0
\(365\) 8.57539 0.448856
\(366\) 0 0
\(367\) −33.1494 −1.73039 −0.865193 0.501439i \(-0.832804\pi\)
−0.865193 + 0.501439i \(0.832804\pi\)
\(368\) 0 0
\(369\) 33.9865 1.76926
\(370\) 0 0
\(371\) −19.0618 −0.989640
\(372\) 0 0
\(373\) −16.5840 −0.858685 −0.429343 0.903142i \(-0.641255\pi\)
−0.429343 + 0.903142i \(0.641255\pi\)
\(374\) 0 0
\(375\) −1.14690 −0.0592254
\(376\) 0 0
\(377\) −15.5050 −0.798550
\(378\) 0 0
\(379\) −5.79467 −0.297652 −0.148826 0.988863i \(-0.547549\pi\)
−0.148826 + 0.988863i \(0.547549\pi\)
\(380\) 0 0
\(381\) −0.439143 −0.0224980
\(382\) 0 0
\(383\) −5.89016 −0.300973 −0.150487 0.988612i \(-0.548084\pi\)
−0.150487 + 0.988612i \(0.548084\pi\)
\(384\) 0 0
\(385\) 3.57187 0.182039
\(386\) 0 0
\(387\) 28.0257 1.42463
\(388\) 0 0
\(389\) 13.8055 0.699964 0.349982 0.936756i \(-0.386188\pi\)
0.349982 + 0.936756i \(0.386188\pi\)
\(390\) 0 0
\(391\) −11.5868 −0.585968
\(392\) 0 0
\(393\) 2.05649 0.103736
\(394\) 0 0
\(395\) 14.2383 0.716405
\(396\) 0 0
\(397\) 6.48471 0.325458 0.162729 0.986671i \(-0.447970\pi\)
0.162729 + 0.986671i \(0.447970\pi\)
\(398\) 0 0
\(399\) −0.410573 −0.0205544
\(400\) 0 0
\(401\) 3.70806 0.185172 0.0925858 0.995705i \(-0.470487\pi\)
0.0925858 + 0.995705i \(0.470487\pi\)
\(402\) 0 0
\(403\) 10.0500 0.500628
\(404\) 0 0
\(405\) −9.40077 −0.467128
\(406\) 0 0
\(407\) −2.27446 −0.112741
\(408\) 0 0
\(409\) 31.0569 1.53567 0.767833 0.640650i \(-0.221335\pi\)
0.767833 + 0.640650i \(0.221335\pi\)
\(410\) 0 0
\(411\) 0.744442 0.0367206
\(412\) 0 0
\(413\) 46.4717 2.28673
\(414\) 0 0
\(415\) 10.5672 0.518722
\(416\) 0 0
\(417\) 1.38138 0.0676467
\(418\) 0 0
\(419\) 11.2871 0.551410 0.275705 0.961242i \(-0.411089\pi\)
0.275705 + 0.961242i \(0.411089\pi\)
\(420\) 0 0
\(421\) 34.5194 1.68237 0.841186 0.540746i \(-0.181858\pi\)
0.841186 + 0.540746i \(0.181858\pi\)
\(422\) 0 0
\(423\) −17.2271 −0.837611
\(424\) 0 0
\(425\) −14.2356 −0.690528
\(426\) 0 0
\(427\) 23.5199 1.13821
\(428\) 0 0
\(429\) −0.263235 −0.0127091
\(430\) 0 0
\(431\) −18.1654 −0.874996 −0.437498 0.899219i \(-0.644135\pi\)
−0.437498 + 0.899219i \(0.644135\pi\)
\(432\) 0 0
\(433\) 10.7804 0.518075 0.259037 0.965867i \(-0.416595\pi\)
0.259037 + 0.965867i \(0.416595\pi\)
\(434\) 0 0
\(435\) 0.927570 0.0444736
\(436\) 0 0
\(437\) −3.15468 −0.150909
\(438\) 0 0
\(439\) 1.38742 0.0662180 0.0331090 0.999452i \(-0.489459\pi\)
0.0331090 + 0.999452i \(0.489459\pi\)
\(440\) 0 0
\(441\) −12.9836 −0.618266
\(442\) 0 0
\(443\) 7.26792 0.345310 0.172655 0.984982i \(-0.444766\pi\)
0.172655 + 0.984982i \(0.444766\pi\)
\(444\) 0 0
\(445\) 4.89440 0.232017
\(446\) 0 0
\(447\) −0.823487 −0.0389496
\(448\) 0 0
\(449\) 21.4298 1.01134 0.505668 0.862728i \(-0.331246\pi\)
0.505668 + 0.862728i \(0.331246\pi\)
\(450\) 0 0
\(451\) 11.3852 0.536108
\(452\) 0 0
\(453\) 0.787575 0.0370035
\(454\) 0 0
\(455\) 7.71498 0.361684
\(456\) 0 0
\(457\) −17.6312 −0.824754 −0.412377 0.911013i \(-0.635301\pi\)
−0.412377 + 0.911013i \(0.635301\pi\)
\(458\) 0 0
\(459\) 2.67908 0.125049
\(460\) 0 0
\(461\) 34.2073 1.59320 0.796598 0.604510i \(-0.206631\pi\)
0.796598 + 0.604510i \(0.206631\pi\)
\(462\) 0 0
\(463\) 11.0009 0.511253 0.255626 0.966776i \(-0.417718\pi\)
0.255626 + 0.966776i \(0.417718\pi\)
\(464\) 0 0
\(465\) −0.601231 −0.0278814
\(466\) 0 0
\(467\) −4.38453 −0.202892 −0.101446 0.994841i \(-0.532347\pi\)
−0.101446 + 0.994841i \(0.532347\pi\)
\(468\) 0 0
\(469\) 16.2155 0.748764
\(470\) 0 0
\(471\) −0.0302708 −0.00139481
\(472\) 0 0
\(473\) 9.38838 0.431678
\(474\) 0 0
\(475\) −3.87587 −0.177837
\(476\) 0 0
\(477\) 16.8905 0.773365
\(478\) 0 0
\(479\) 14.3683 0.656505 0.328253 0.944590i \(-0.393540\pi\)
0.328253 + 0.944590i \(0.393540\pi\)
\(480\) 0 0
\(481\) −4.91266 −0.223998
\(482\) 0 0
\(483\) 1.29523 0.0589349
\(484\) 0 0
\(485\) 4.34604 0.197344
\(486\) 0 0
\(487\) −1.01712 −0.0460899 −0.0230449 0.999734i \(-0.507336\pi\)
−0.0230449 + 0.999734i \(0.507336\pi\)
\(488\) 0 0
\(489\) −2.06829 −0.0935314
\(490\) 0 0
\(491\) 2.41045 0.108782 0.0543910 0.998520i \(-0.482678\pi\)
0.0543910 + 0.998520i \(0.482678\pi\)
\(492\) 0 0
\(493\) 26.3658 1.18745
\(494\) 0 0
\(495\) −3.16501 −0.142257
\(496\) 0 0
\(497\) −51.4189 −2.30645
\(498\) 0 0
\(499\) −4.07426 −0.182389 −0.0911945 0.995833i \(-0.529069\pi\)
−0.0911945 + 0.995833i \(0.529069\pi\)
\(500\) 0 0
\(501\) −0.105485 −0.00471272
\(502\) 0 0
\(503\) 19.9553 0.889762 0.444881 0.895590i \(-0.353246\pi\)
0.444881 + 0.895590i \(0.353246\pi\)
\(504\) 0 0
\(505\) 11.5281 0.512993
\(506\) 0 0
\(507\) 1.01577 0.0451118
\(508\) 0 0
\(509\) −8.79239 −0.389716 −0.194858 0.980831i \(-0.562425\pi\)
−0.194858 + 0.980831i \(0.562425\pi\)
\(510\) 0 0
\(511\) −27.2478 −1.20537
\(512\) 0 0
\(513\) 0.729422 0.0322048
\(514\) 0 0
\(515\) 9.39239 0.413878
\(516\) 0 0
\(517\) −5.77094 −0.253806
\(518\) 0 0
\(519\) 0.505138 0.0221731
\(520\) 0 0
\(521\) 27.3483 1.19815 0.599075 0.800693i \(-0.295535\pi\)
0.599075 + 0.800693i \(0.295535\pi\)
\(522\) 0 0
\(523\) 41.7043 1.82360 0.911800 0.410634i \(-0.134693\pi\)
0.911800 + 0.410634i \(0.134693\pi\)
\(524\) 0 0
\(525\) 1.59133 0.0694512
\(526\) 0 0
\(527\) −17.0897 −0.744441
\(528\) 0 0
\(529\) −13.0480 −0.567304
\(530\) 0 0
\(531\) −41.1783 −1.78699
\(532\) 0 0
\(533\) 24.5912 1.06516
\(534\) 0 0
\(535\) −19.2615 −0.832748
\(536\) 0 0
\(537\) 0.249331 0.0107594
\(538\) 0 0
\(539\) −4.34940 −0.187342
\(540\) 0 0
\(541\) −20.7584 −0.892472 −0.446236 0.894915i \(-0.647236\pi\)
−0.446236 + 0.894915i \(0.647236\pi\)
\(542\) 0 0
\(543\) −0.642898 −0.0275894
\(544\) 0 0
\(545\) 0.224035 0.00959659
\(546\) 0 0
\(547\) −5.39396 −0.230629 −0.115315 0.993329i \(-0.536788\pi\)
−0.115315 + 0.993329i \(0.536788\pi\)
\(548\) 0 0
\(549\) −20.8409 −0.889466
\(550\) 0 0
\(551\) 7.17849 0.305814
\(552\) 0 0
\(553\) −45.2412 −1.92385
\(554\) 0 0
\(555\) 0.293894 0.0124751
\(556\) 0 0
\(557\) −38.0076 −1.61043 −0.805217 0.592981i \(-0.797951\pi\)
−0.805217 + 0.592981i \(0.797951\pi\)
\(558\) 0 0
\(559\) 20.2782 0.857677
\(560\) 0 0
\(561\) 0.447622 0.0188986
\(562\) 0 0
\(563\) 26.4313 1.11395 0.556973 0.830530i \(-0.311963\pi\)
0.556973 + 0.830530i \(0.311963\pi\)
\(564\) 0 0
\(565\) 0.210295 0.00884719
\(566\) 0 0
\(567\) 29.8704 1.25444
\(568\) 0 0
\(569\) 40.6105 1.70248 0.851239 0.524778i \(-0.175852\pi\)
0.851239 + 0.524778i \(0.175852\pi\)
\(570\) 0 0
\(571\) −8.44299 −0.353328 −0.176664 0.984271i \(-0.556531\pi\)
−0.176664 + 0.984271i \(0.556531\pi\)
\(572\) 0 0
\(573\) −1.97309 −0.0824271
\(574\) 0 0
\(575\) 12.2271 0.509906
\(576\) 0 0
\(577\) 20.9771 0.873288 0.436644 0.899634i \(-0.356167\pi\)
0.436644 + 0.899634i \(0.356167\pi\)
\(578\) 0 0
\(579\) 2.44499 0.101610
\(580\) 0 0
\(581\) −33.5766 −1.39299
\(582\) 0 0
\(583\) 5.65820 0.234339
\(584\) 0 0
\(585\) −6.83619 −0.282642
\(586\) 0 0
\(587\) 21.1452 0.872756 0.436378 0.899764i \(-0.356261\pi\)
0.436378 + 0.899764i \(0.356261\pi\)
\(588\) 0 0
\(589\) −4.65295 −0.191721
\(590\) 0 0
\(591\) 2.90540 0.119512
\(592\) 0 0
\(593\) 27.7731 1.14051 0.570253 0.821469i \(-0.306845\pi\)
0.570253 + 0.821469i \(0.306845\pi\)
\(594\) 0 0
\(595\) −13.1191 −0.537829
\(596\) 0 0
\(597\) −0.231793 −0.00948667
\(598\) 0 0
\(599\) −23.0129 −0.940281 −0.470140 0.882592i \(-0.655797\pi\)
−0.470140 + 0.882592i \(0.655797\pi\)
\(600\) 0 0
\(601\) 23.3322 0.951738 0.475869 0.879516i \(-0.342134\pi\)
0.475869 + 0.879516i \(0.342134\pi\)
\(602\) 0 0
\(603\) −14.3685 −0.585129
\(604\) 0 0
\(605\) −1.06025 −0.0431054
\(606\) 0 0
\(607\) 3.22857 0.131044 0.0655218 0.997851i \(-0.479129\pi\)
0.0655218 + 0.997851i \(0.479129\pi\)
\(608\) 0 0
\(609\) −2.94730 −0.119430
\(610\) 0 0
\(611\) −12.4648 −0.504273
\(612\) 0 0
\(613\) −17.2202 −0.695519 −0.347759 0.937584i \(-0.613057\pi\)
−0.347759 + 0.937584i \(0.613057\pi\)
\(614\) 0 0
\(615\) −1.47114 −0.0593220
\(616\) 0 0
\(617\) −27.4776 −1.10621 −0.553103 0.833113i \(-0.686557\pi\)
−0.553103 + 0.833113i \(0.686557\pi\)
\(618\) 0 0
\(619\) 9.53870 0.383393 0.191696 0.981454i \(-0.438601\pi\)
0.191696 + 0.981454i \(0.438601\pi\)
\(620\) 0 0
\(621\) −2.30109 −0.0923397
\(622\) 0 0
\(623\) −15.5517 −0.623064
\(624\) 0 0
\(625\) 9.40166 0.376066
\(626\) 0 0
\(627\) 0.121872 0.00486710
\(628\) 0 0
\(629\) 8.35381 0.333088
\(630\) 0 0
\(631\) −1.16647 −0.0464363 −0.0232182 0.999730i \(-0.507391\pi\)
−0.0232182 + 0.999730i \(0.507391\pi\)
\(632\) 0 0
\(633\) 1.49962 0.0596044
\(634\) 0 0
\(635\) −3.82042 −0.151609
\(636\) 0 0
\(637\) −9.39438 −0.372219
\(638\) 0 0
\(639\) 45.5619 1.80240
\(640\) 0 0
\(641\) 10.0249 0.395960 0.197980 0.980206i \(-0.436562\pi\)
0.197980 + 0.980206i \(0.436562\pi\)
\(642\) 0 0
\(643\) 40.7197 1.60583 0.802915 0.596094i \(-0.203281\pi\)
0.802915 + 0.596094i \(0.203281\pi\)
\(644\) 0 0
\(645\) −1.21312 −0.0477666
\(646\) 0 0
\(647\) 16.6460 0.654423 0.327211 0.944951i \(-0.393891\pi\)
0.327211 + 0.944951i \(0.393891\pi\)
\(648\) 0 0
\(649\) −13.7944 −0.541477
\(650\) 0 0
\(651\) 1.91038 0.0748735
\(652\) 0 0
\(653\) −34.7416 −1.35954 −0.679771 0.733424i \(-0.737921\pi\)
−0.679771 + 0.733424i \(0.737921\pi\)
\(654\) 0 0
\(655\) 17.8908 0.699053
\(656\) 0 0
\(657\) 24.1441 0.941949
\(658\) 0 0
\(659\) 22.9537 0.894148 0.447074 0.894497i \(-0.352466\pi\)
0.447074 + 0.894497i \(0.352466\pi\)
\(660\) 0 0
\(661\) −22.7141 −0.883476 −0.441738 0.897144i \(-0.645638\pi\)
−0.441738 + 0.897144i \(0.645638\pi\)
\(662\) 0 0
\(663\) 0.966831 0.0375486
\(664\) 0 0
\(665\) −3.57187 −0.138511
\(666\) 0 0
\(667\) −22.6459 −0.876851
\(668\) 0 0
\(669\) 2.78445 0.107653
\(670\) 0 0
\(671\) −6.98152 −0.269518
\(672\) 0 0
\(673\) 20.9448 0.807364 0.403682 0.914899i \(-0.367730\pi\)
0.403682 + 0.914899i \(0.367730\pi\)
\(674\) 0 0
\(675\) −2.82714 −0.108817
\(676\) 0 0
\(677\) 11.4153 0.438726 0.219363 0.975643i \(-0.429602\pi\)
0.219363 + 0.975643i \(0.429602\pi\)
\(678\) 0 0
\(679\) −13.8093 −0.529952
\(680\) 0 0
\(681\) 0.488285 0.0187111
\(682\) 0 0
\(683\) 34.9550 1.33752 0.668759 0.743479i \(-0.266826\pi\)
0.668759 + 0.743479i \(0.266826\pi\)
\(684\) 0 0
\(685\) 6.47643 0.247452
\(686\) 0 0
\(687\) 2.99835 0.114394
\(688\) 0 0
\(689\) 12.2213 0.465594
\(690\) 0 0
\(691\) 15.8717 0.603789 0.301894 0.953341i \(-0.402381\pi\)
0.301894 + 0.953341i \(0.402381\pi\)
\(692\) 0 0
\(693\) 10.0566 0.382019
\(694\) 0 0
\(695\) 12.0177 0.455855
\(696\) 0 0
\(697\) −41.8165 −1.58391
\(698\) 0 0
\(699\) −1.98644 −0.0751339
\(700\) 0 0
\(701\) −18.5830 −0.701869 −0.350935 0.936400i \(-0.614136\pi\)
−0.350935 + 0.936400i \(0.614136\pi\)
\(702\) 0 0
\(703\) 2.27446 0.0857828
\(704\) 0 0
\(705\) 0.745693 0.0280844
\(706\) 0 0
\(707\) −36.6298 −1.37760
\(708\) 0 0
\(709\) 23.8063 0.894064 0.447032 0.894518i \(-0.352481\pi\)
0.447032 + 0.894518i \(0.352481\pi\)
\(710\) 0 0
\(711\) 40.0880 1.50342
\(712\) 0 0
\(713\) 14.6786 0.549717
\(714\) 0 0
\(715\) −2.29007 −0.0856437
\(716\) 0 0
\(717\) −1.87905 −0.0701745
\(718\) 0 0
\(719\) −37.0144 −1.38040 −0.690202 0.723616i \(-0.742478\pi\)
−0.690202 + 0.723616i \(0.742478\pi\)
\(720\) 0 0
\(721\) −29.8437 −1.11144
\(722\) 0 0
\(723\) −2.94249 −0.109432
\(724\) 0 0
\(725\) −27.8229 −1.03332
\(726\) 0 0
\(727\) 22.0744 0.818694 0.409347 0.912379i \(-0.365757\pi\)
0.409347 + 0.912379i \(0.365757\pi\)
\(728\) 0 0
\(729\) −26.2013 −0.970417
\(730\) 0 0
\(731\) −34.4824 −1.27538
\(732\) 0 0
\(733\) −18.7090 −0.691031 −0.345515 0.938413i \(-0.612296\pi\)
−0.345515 + 0.938413i \(0.612296\pi\)
\(734\) 0 0
\(735\) 0.562008 0.0207300
\(736\) 0 0
\(737\) −4.81332 −0.177301
\(738\) 0 0
\(739\) 5.91630 0.217635 0.108817 0.994062i \(-0.465294\pi\)
0.108817 + 0.994062i \(0.465294\pi\)
\(740\) 0 0
\(741\) 0.263235 0.00967017
\(742\) 0 0
\(743\) −14.0422 −0.515159 −0.257579 0.966257i \(-0.582925\pi\)
−0.257579 + 0.966257i \(0.582925\pi\)
\(744\) 0 0
\(745\) −7.16411 −0.262473
\(746\) 0 0
\(747\) 29.7520 1.08857
\(748\) 0 0
\(749\) 61.2023 2.23628
\(750\) 0 0
\(751\) −4.91166 −0.179229 −0.0896145 0.995977i \(-0.528563\pi\)
−0.0896145 + 0.995977i \(0.528563\pi\)
\(752\) 0 0
\(753\) −2.42646 −0.0884250
\(754\) 0 0
\(755\) 6.85168 0.249358
\(756\) 0 0
\(757\) 49.2829 1.79122 0.895609 0.444843i \(-0.146741\pi\)
0.895609 + 0.444843i \(0.146741\pi\)
\(758\) 0 0
\(759\) −0.384467 −0.0139553
\(760\) 0 0
\(761\) −7.25922 −0.263146 −0.131573 0.991306i \(-0.542003\pi\)
−0.131573 + 0.991306i \(0.542003\pi\)
\(762\) 0 0
\(763\) −0.711856 −0.0257709
\(764\) 0 0
\(765\) 11.6247 0.420292
\(766\) 0 0
\(767\) −29.7949 −1.07583
\(768\) 0 0
\(769\) −21.5123 −0.775755 −0.387877 0.921711i \(-0.626792\pi\)
−0.387877 + 0.921711i \(0.626792\pi\)
\(770\) 0 0
\(771\) 2.71928 0.0979325
\(772\) 0 0
\(773\) 4.54038 0.163306 0.0816530 0.996661i \(-0.473980\pi\)
0.0816530 + 0.996661i \(0.473980\pi\)
\(774\) 0 0
\(775\) 18.0342 0.647808
\(776\) 0 0
\(777\) −0.933831 −0.0335010
\(778\) 0 0
\(779\) −11.3852 −0.407917
\(780\) 0 0
\(781\) 15.2629 0.546149
\(782\) 0 0
\(783\) 5.23615 0.187125
\(784\) 0 0
\(785\) −0.263348 −0.00939928
\(786\) 0 0
\(787\) 7.25028 0.258444 0.129222 0.991616i \(-0.458752\pi\)
0.129222 + 0.991616i \(0.458752\pi\)
\(788\) 0 0
\(789\) 1.68138 0.0598588
\(790\) 0 0
\(791\) −0.668201 −0.0237585
\(792\) 0 0
\(793\) −15.0796 −0.535491
\(794\) 0 0
\(795\) −0.731124 −0.0259303
\(796\) 0 0
\(797\) 18.8674 0.668319 0.334159 0.942517i \(-0.391548\pi\)
0.334159 + 0.942517i \(0.391548\pi\)
\(798\) 0 0
\(799\) 21.1960 0.749860
\(800\) 0 0
\(801\) 13.7802 0.486900
\(802\) 0 0
\(803\) 8.08806 0.285422
\(804\) 0 0
\(805\) 11.2681 0.397149
\(806\) 0 0
\(807\) 1.36531 0.0480612
\(808\) 0 0
\(809\) −19.4948 −0.685399 −0.342700 0.939445i \(-0.611341\pi\)
−0.342700 + 0.939445i \(0.611341\pi\)
\(810\) 0 0
\(811\) −4.28635 −0.150514 −0.0752570 0.997164i \(-0.523978\pi\)
−0.0752570 + 0.997164i \(0.523978\pi\)
\(812\) 0 0
\(813\) 2.08686 0.0731893
\(814\) 0 0
\(815\) −17.9936 −0.630287
\(816\) 0 0
\(817\) −9.38838 −0.328458
\(818\) 0 0
\(819\) 21.7216 0.759014
\(820\) 0 0
\(821\) 5.37419 0.187561 0.0937803 0.995593i \(-0.470105\pi\)
0.0937803 + 0.995593i \(0.470105\pi\)
\(822\) 0 0
\(823\) −26.3196 −0.917445 −0.458723 0.888580i \(-0.651693\pi\)
−0.458723 + 0.888580i \(0.651693\pi\)
\(824\) 0 0
\(825\) −0.472360 −0.0164455
\(826\) 0 0
\(827\) 8.11186 0.282077 0.141039 0.990004i \(-0.454956\pi\)
0.141039 + 0.990004i \(0.454956\pi\)
\(828\) 0 0
\(829\) −21.5948 −0.750017 −0.375009 0.927021i \(-0.622360\pi\)
−0.375009 + 0.927021i \(0.622360\pi\)
\(830\) 0 0
\(831\) 1.67120 0.0579733
\(832\) 0 0
\(833\) 15.9748 0.553495
\(834\) 0 0
\(835\) −0.917688 −0.0317579
\(836\) 0 0
\(837\) −3.39396 −0.117313
\(838\) 0 0
\(839\) 4.35109 0.150216 0.0751082 0.997175i \(-0.476070\pi\)
0.0751082 + 0.997175i \(0.476070\pi\)
\(840\) 0 0
\(841\) 22.5308 0.776923
\(842\) 0 0
\(843\) −3.15160 −0.108547
\(844\) 0 0
\(845\) 8.83689 0.303998
\(846\) 0 0
\(847\) 3.36889 0.115756
\(848\) 0 0
\(849\) 3.51858 0.120757
\(850\) 0 0
\(851\) −7.17519 −0.245962
\(852\) 0 0
\(853\) 19.5022 0.667744 0.333872 0.942618i \(-0.391645\pi\)
0.333872 + 0.942618i \(0.391645\pi\)
\(854\) 0 0
\(855\) 3.16501 0.108241
\(856\) 0 0
\(857\) 13.3462 0.455898 0.227949 0.973673i \(-0.426798\pi\)
0.227949 + 0.973673i \(0.426798\pi\)
\(858\) 0 0
\(859\) −21.6016 −0.737036 −0.368518 0.929621i \(-0.620135\pi\)
−0.368518 + 0.929621i \(0.620135\pi\)
\(860\) 0 0
\(861\) 4.67445 0.159305
\(862\) 0 0
\(863\) 2.72425 0.0927345 0.0463673 0.998924i \(-0.485236\pi\)
0.0463673 + 0.998924i \(0.485236\pi\)
\(864\) 0 0
\(865\) 4.39456 0.149420
\(866\) 0 0
\(867\) 0.427763 0.0145276
\(868\) 0 0
\(869\) 13.4291 0.455552
\(870\) 0 0
\(871\) −10.3964 −0.352269
\(872\) 0 0
\(873\) 12.2363 0.414137
\(874\) 0 0
\(875\) 31.7034 1.07177
\(876\) 0 0
\(877\) 1.29130 0.0436041 0.0218021 0.999762i \(-0.493060\pi\)
0.0218021 + 0.999762i \(0.493060\pi\)
\(878\) 0 0
\(879\) −0.777508 −0.0262247
\(880\) 0 0
\(881\) 24.9873 0.841844 0.420922 0.907097i \(-0.361707\pi\)
0.420922 + 0.907097i \(0.361707\pi\)
\(882\) 0 0
\(883\) −15.5046 −0.521772 −0.260886 0.965370i \(-0.584015\pi\)
−0.260886 + 0.965370i \(0.584015\pi\)
\(884\) 0 0
\(885\) 1.78244 0.0599162
\(886\) 0 0
\(887\) 9.98029 0.335105 0.167553 0.985863i \(-0.446414\pi\)
0.167553 + 0.985863i \(0.446414\pi\)
\(888\) 0 0
\(889\) 12.1391 0.407134
\(890\) 0 0
\(891\) −8.86655 −0.297040
\(892\) 0 0
\(893\) 5.77094 0.193117
\(894\) 0 0
\(895\) 2.16911 0.0725053
\(896\) 0 0
\(897\) −0.830422 −0.0277270
\(898\) 0 0
\(899\) −33.4012 −1.11399
\(900\) 0 0
\(901\) −20.7819 −0.692345
\(902\) 0 0
\(903\) 3.85461 0.128274
\(904\) 0 0
\(905\) −5.59303 −0.185919
\(906\) 0 0
\(907\) −11.8887 −0.394758 −0.197379 0.980327i \(-0.563243\pi\)
−0.197379 + 0.980327i \(0.563243\pi\)
\(908\) 0 0
\(909\) 32.4574 1.07654
\(910\) 0 0
\(911\) 23.7166 0.785767 0.392883 0.919588i \(-0.371478\pi\)
0.392883 + 0.919588i \(0.371478\pi\)
\(912\) 0 0
\(913\) 9.96666 0.329848
\(914\) 0 0
\(915\) 0.902117 0.0298231
\(916\) 0 0
\(917\) −56.8471 −1.87726
\(918\) 0 0
\(919\) −18.1426 −0.598470 −0.299235 0.954179i \(-0.596731\pi\)
−0.299235 + 0.954179i \(0.596731\pi\)
\(920\) 0 0
\(921\) 1.27111 0.0418845
\(922\) 0 0
\(923\) 32.9667 1.08511
\(924\) 0 0
\(925\) −8.81549 −0.289852
\(926\) 0 0
\(927\) 26.4443 0.868546
\(928\) 0 0
\(929\) −50.5019 −1.65691 −0.828457 0.560053i \(-0.810781\pi\)
−0.828457 + 0.560053i \(0.810781\pi\)
\(930\) 0 0
\(931\) 4.34940 0.142546
\(932\) 0 0
\(933\) −1.00694 −0.0329656
\(934\) 0 0
\(935\) 3.89418 0.127353
\(936\) 0 0
\(937\) −39.6587 −1.29559 −0.647797 0.761813i \(-0.724309\pi\)
−0.647797 + 0.761813i \(0.724309\pi\)
\(938\) 0 0
\(939\) 2.93915 0.0959154
\(940\) 0 0
\(941\) −34.8809 −1.13709 −0.568543 0.822653i \(-0.692493\pi\)
−0.568543 + 0.822653i \(0.692493\pi\)
\(942\) 0 0
\(943\) 35.9166 1.16961
\(944\) 0 0
\(945\) −2.60540 −0.0847537
\(946\) 0 0
\(947\) 30.3067 0.984835 0.492417 0.870359i \(-0.336113\pi\)
0.492417 + 0.870359i \(0.336113\pi\)
\(948\) 0 0
\(949\) 17.4696 0.567088
\(950\) 0 0
\(951\) 1.42992 0.0463684
\(952\) 0 0
\(953\) −6.65804 −0.215675 −0.107838 0.994169i \(-0.534393\pi\)
−0.107838 + 0.994169i \(0.534393\pi\)
\(954\) 0 0
\(955\) −17.1653 −0.555457
\(956\) 0 0
\(957\) 0.874858 0.0282801
\(958\) 0 0
\(959\) −20.5785 −0.664513
\(960\) 0 0
\(961\) −9.35006 −0.301615
\(962\) 0 0
\(963\) −54.2309 −1.74757
\(964\) 0 0
\(965\) 21.2707 0.684729
\(966\) 0 0
\(967\) 20.0020 0.643221 0.321611 0.946872i \(-0.395776\pi\)
0.321611 + 0.946872i \(0.395776\pi\)
\(968\) 0 0
\(969\) −0.447622 −0.0143797
\(970\) 0 0
\(971\) 0.491941 0.0157871 0.00789357 0.999969i \(-0.497487\pi\)
0.00789357 + 0.999969i \(0.497487\pi\)
\(972\) 0 0
\(973\) −38.1853 −1.22417
\(974\) 0 0
\(975\) −1.02026 −0.0326746
\(976\) 0 0
\(977\) 23.6515 0.756679 0.378339 0.925667i \(-0.376495\pi\)
0.378339 + 0.925667i \(0.376495\pi\)
\(978\) 0 0
\(979\) 4.61626 0.147536
\(980\) 0 0
\(981\) 0.630771 0.0201390
\(982\) 0 0
\(983\) −39.5609 −1.26180 −0.630899 0.775865i \(-0.717314\pi\)
−0.630899 + 0.775865i \(0.717314\pi\)
\(984\) 0 0
\(985\) 25.2762 0.805366
\(986\) 0 0
\(987\) −2.36939 −0.0754186
\(988\) 0 0
\(989\) 29.6173 0.941777
\(990\) 0 0
\(991\) 13.6096 0.432322 0.216161 0.976358i \(-0.430646\pi\)
0.216161 + 0.976358i \(0.430646\pi\)
\(992\) 0 0
\(993\) −0.847283 −0.0268877
\(994\) 0 0
\(995\) −2.01654 −0.0639285
\(996\) 0 0
\(997\) 41.1821 1.30425 0.652125 0.758111i \(-0.273878\pi\)
0.652125 + 0.758111i \(0.273878\pi\)
\(998\) 0 0
\(999\) 1.65904 0.0524897
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 3344.2.a.t.1.3 5
4.3 odd 2 209.2.a.c.1.5 5
12.11 even 2 1881.2.a.k.1.1 5
20.19 odd 2 5225.2.a.h.1.1 5
44.43 even 2 2299.2.a.n.1.1 5
76.75 even 2 3971.2.a.h.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.5 5 4.3 odd 2
1881.2.a.k.1.1 5 12.11 even 2
2299.2.a.n.1.1 5 44.43 even 2
3344.2.a.t.1.3 5 1.1 even 1 trivial
3971.2.a.h.1.1 5 76.75 even 2
5225.2.a.h.1.1 5 20.19 odd 2