Properties

Label 1672.2.a.i.1.1
Level $1672$
Weight $2$
Character 1672.1
Self dual yes
Analytic conductor $13.351$
Analytic rank $1$
Dimension $6$
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] = [1672,2,Mod(1,1672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1672, 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("1672.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1672.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: \(13.3509872180\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.106392688.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.14822\) of defining polynomial
Character \(\chi\) \(=\) 1672.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.14822 q^{3} +0.677107 q^{5} -1.67711 q^{7} +6.91132 q^{9} +O(q^{10})\) \(q-3.14822 q^{3} +0.677107 q^{5} -1.67711 q^{7} +6.91132 q^{9} -1.00000 q^{11} -4.16289 q^{13} -2.13169 q^{15} +6.86889 q^{17} +1.00000 q^{19} +5.27991 q^{21} +2.14077 q^{23} -4.54153 q^{25} -12.3137 q^{27} +2.01005 q^{29} -2.13169 q^{31} +3.14822 q^{33} -1.13558 q^{35} +2.81297 q^{37} +13.1057 q^{39} +10.9829 q^{41} +3.39032 q^{43} +4.67970 q^{45} +1.22974 q^{47} -4.18731 q^{49} -21.6248 q^{51} -11.7292 q^{53} -0.677107 q^{55} -3.14822 q^{57} -8.37650 q^{59} +9.52619 q^{61} -11.5910 q^{63} -2.81872 q^{65} -9.64636 q^{67} -6.73963 q^{69} +12.6011 q^{71} -14.1743 q^{73} +14.2977 q^{75} +1.67711 q^{77} -11.2387 q^{79} +18.0324 q^{81} -7.75976 q^{83} +4.65098 q^{85} -6.32808 q^{87} -4.08041 q^{89} +6.98160 q^{91} +6.71102 q^{93} +0.677107 q^{95} -12.4487 q^{97} -6.91132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) −3.14822 −1.81763 −0.908814 0.417201i \(-0.863011\pi\)
−0.908814 + 0.417201i \(0.863011\pi\)
\(4\) 0 0
\(5\) 0.677107 0.302812 0.151406 0.988472i \(-0.451620\pi\)
0.151406 + 0.988472i \(0.451620\pi\)
\(6\) 0 0
\(7\) −1.67711 −0.633887 −0.316943 0.948444i \(-0.602657\pi\)
−0.316943 + 0.948444i \(0.602657\pi\)
\(8\) 0 0
\(9\) 6.91132 2.30377
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.16289 −1.15458 −0.577288 0.816540i \(-0.695889\pi\)
−0.577288 + 0.816540i \(0.695889\pi\)
\(14\) 0 0
\(15\) −2.13169 −0.550399
\(16\) 0 0
\(17\) 6.86889 1.66595 0.832976 0.553310i \(-0.186635\pi\)
0.832976 + 0.553310i \(0.186635\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 5.27991 1.15217
\(22\) 0 0
\(23\) 2.14077 0.446382 0.223191 0.974775i \(-0.428353\pi\)
0.223191 + 0.974775i \(0.428353\pi\)
\(24\) 0 0
\(25\) −4.54153 −0.908305
\(26\) 0 0
\(27\) −12.3137 −2.36977
\(28\) 0 0
\(29\) 2.01005 0.373257 0.186628 0.982431i \(-0.440244\pi\)
0.186628 + 0.982431i \(0.440244\pi\)
\(30\) 0 0
\(31\) −2.13169 −0.382862 −0.191431 0.981506i \(-0.561313\pi\)
−0.191431 + 0.981506i \(0.561313\pi\)
\(32\) 0 0
\(33\) 3.14822 0.548036
\(34\) 0 0
\(35\) −1.13558 −0.191948
\(36\) 0 0
\(37\) 2.81297 0.462450 0.231225 0.972900i \(-0.425727\pi\)
0.231225 + 0.972900i \(0.425727\pi\)
\(38\) 0 0
\(39\) 13.1057 2.09859
\(40\) 0 0
\(41\) 10.9829 1.71524 0.857622 0.514280i \(-0.171941\pi\)
0.857622 + 0.514280i \(0.171941\pi\)
\(42\) 0 0
\(43\) 3.39032 0.517019 0.258510 0.966009i \(-0.416769\pi\)
0.258510 + 0.966009i \(0.416769\pi\)
\(44\) 0 0
\(45\) 4.67970 0.697609
\(46\) 0 0
\(47\) 1.22974 0.179376 0.0896878 0.995970i \(-0.471413\pi\)
0.0896878 + 0.995970i \(0.471413\pi\)
\(48\) 0 0
\(49\) −4.18731 −0.598187
\(50\) 0 0
\(51\) −21.6248 −3.02808
\(52\) 0 0
\(53\) −11.7292 −1.61113 −0.805564 0.592508i \(-0.798138\pi\)
−0.805564 + 0.592508i \(0.798138\pi\)
\(54\) 0 0
\(55\) −0.677107 −0.0913011
\(56\) 0 0
\(57\) −3.14822 −0.416993
\(58\) 0 0
\(59\) −8.37650 −1.09053 −0.545264 0.838264i \(-0.683571\pi\)
−0.545264 + 0.838264i \(0.683571\pi\)
\(60\) 0 0
\(61\) 9.52619 1.21970 0.609852 0.792516i \(-0.291229\pi\)
0.609852 + 0.792516i \(0.291229\pi\)
\(62\) 0 0
\(63\) −11.5910 −1.46033
\(64\) 0 0
\(65\) −2.81872 −0.349619
\(66\) 0 0
\(67\) −9.64636 −1.17849 −0.589245 0.807954i \(-0.700575\pi\)
−0.589245 + 0.807954i \(0.700575\pi\)
\(68\) 0 0
\(69\) −6.73963 −0.811356
\(70\) 0 0
\(71\) 12.6011 1.49547 0.747736 0.663996i \(-0.231141\pi\)
0.747736 + 0.663996i \(0.231141\pi\)
\(72\) 0 0
\(73\) −14.1743 −1.65897 −0.829485 0.558528i \(-0.811366\pi\)
−0.829485 + 0.558528i \(0.811366\pi\)
\(74\) 0 0
\(75\) 14.2977 1.65096
\(76\) 0 0
\(77\) 1.67711 0.191124
\(78\) 0 0
\(79\) −11.2387 −1.26445 −0.632225 0.774785i \(-0.717858\pi\)
−0.632225 + 0.774785i \(0.717858\pi\)
\(80\) 0 0
\(81\) 18.0324 2.00360
\(82\) 0 0
\(83\) −7.75976 −0.851744 −0.425872 0.904783i \(-0.640033\pi\)
−0.425872 + 0.904783i \(0.640033\pi\)
\(84\) 0 0
\(85\) 4.65098 0.504469
\(86\) 0 0
\(87\) −6.32808 −0.678442
\(88\) 0 0
\(89\) −4.08041 −0.432523 −0.216261 0.976336i \(-0.569386\pi\)
−0.216261 + 0.976336i \(0.569386\pi\)
\(90\) 0 0
\(91\) 6.98160 0.731871
\(92\) 0 0
\(93\) 6.71102 0.695901
\(94\) 0 0
\(95\) 0.677107 0.0694697
\(96\) 0 0
\(97\) −12.4487 −1.26398 −0.631988 0.774978i \(-0.717761\pi\)
−0.631988 + 0.774978i \(0.717761\pi\)
\(98\) 0 0
\(99\) −6.91132 −0.694614
\(100\) 0 0
\(101\) 11.5645 1.15071 0.575353 0.817905i \(-0.304865\pi\)
0.575353 + 0.817905i \(0.304865\pi\)
\(102\) 0 0
\(103\) 10.0938 0.994569 0.497284 0.867588i \(-0.334331\pi\)
0.497284 + 0.867588i \(0.334331\pi\)
\(104\) 0 0
\(105\) 3.57506 0.348891
\(106\) 0 0
\(107\) −14.3080 −1.38320 −0.691601 0.722280i \(-0.743094\pi\)
−0.691601 + 0.722280i \(0.743094\pi\)
\(108\) 0 0
\(109\) −20.0639 −1.92177 −0.960887 0.276940i \(-0.910680\pi\)
−0.960887 + 0.276940i \(0.910680\pi\)
\(110\) 0 0
\(111\) −8.85586 −0.840561
\(112\) 0 0
\(113\) −14.7940 −1.39171 −0.695853 0.718185i \(-0.744973\pi\)
−0.695853 + 0.718185i \(0.744973\pi\)
\(114\) 0 0
\(115\) 1.44953 0.135170
\(116\) 0 0
\(117\) −28.7710 −2.65988
\(118\) 0 0
\(119\) −11.5199 −1.05602
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −34.5767 −3.11768
\(124\) 0 0
\(125\) −6.46064 −0.577857
\(126\) 0 0
\(127\) 10.5225 0.933718 0.466859 0.884332i \(-0.345386\pi\)
0.466859 + 0.884332i \(0.345386\pi\)
\(128\) 0 0
\(129\) −10.6735 −0.939749
\(130\) 0 0
\(131\) −9.76083 −0.852808 −0.426404 0.904533i \(-0.640220\pi\)
−0.426404 + 0.904533i \(0.640220\pi\)
\(132\) 0 0
\(133\) −1.67711 −0.145424
\(134\) 0 0
\(135\) −8.33770 −0.717595
\(136\) 0 0
\(137\) 1.72444 0.147329 0.0736644 0.997283i \(-0.476531\pi\)
0.0736644 + 0.997283i \(0.476531\pi\)
\(138\) 0 0
\(139\) −2.73416 −0.231908 −0.115954 0.993255i \(-0.536993\pi\)
−0.115954 + 0.993255i \(0.536993\pi\)
\(140\) 0 0
\(141\) −3.87149 −0.326038
\(142\) 0 0
\(143\) 4.16289 0.348118
\(144\) 0 0
\(145\) 1.36102 0.113026
\(146\) 0 0
\(147\) 13.1826 1.08728
\(148\) 0 0
\(149\) −18.6670 −1.52926 −0.764632 0.644468i \(-0.777079\pi\)
−0.764632 + 0.644468i \(0.777079\pi\)
\(150\) 0 0
\(151\) −10.5284 −0.856791 −0.428395 0.903591i \(-0.640921\pi\)
−0.428395 + 0.903591i \(0.640921\pi\)
\(152\) 0 0
\(153\) 47.4731 3.83797
\(154\) 0 0
\(155\) −1.44338 −0.115935
\(156\) 0 0
\(157\) −11.6625 −0.930765 −0.465383 0.885110i \(-0.654083\pi\)
−0.465383 + 0.885110i \(0.654083\pi\)
\(158\) 0 0
\(159\) 36.9261 2.92843
\(160\) 0 0
\(161\) −3.59030 −0.282956
\(162\) 0 0
\(163\) 14.9612 1.17185 0.585926 0.810364i \(-0.300731\pi\)
0.585926 + 0.810364i \(0.300731\pi\)
\(164\) 0 0
\(165\) 2.13169 0.165951
\(166\) 0 0
\(167\) −10.5989 −0.820164 −0.410082 0.912049i \(-0.634500\pi\)
−0.410082 + 0.912049i \(0.634500\pi\)
\(168\) 0 0
\(169\) 4.32961 0.333047
\(170\) 0 0
\(171\) 6.91132 0.528522
\(172\) 0 0
\(173\) 3.76338 0.286124 0.143062 0.989714i \(-0.454305\pi\)
0.143062 + 0.989714i \(0.454305\pi\)
\(174\) 0 0
\(175\) 7.61663 0.575763
\(176\) 0 0
\(177\) 26.3711 1.98217
\(178\) 0 0
\(179\) −3.50407 −0.261907 −0.130953 0.991389i \(-0.541804\pi\)
−0.130953 + 0.991389i \(0.541804\pi\)
\(180\) 0 0
\(181\) 25.3354 1.88317 0.941584 0.336778i \(-0.109337\pi\)
0.941584 + 0.336778i \(0.109337\pi\)
\(182\) 0 0
\(183\) −29.9906 −2.21697
\(184\) 0 0
\(185\) 1.90468 0.140035
\(186\) 0 0
\(187\) −6.86889 −0.502303
\(188\) 0 0
\(189\) 20.6514 1.50217
\(190\) 0 0
\(191\) 4.57468 0.331012 0.165506 0.986209i \(-0.447074\pi\)
0.165506 + 0.986209i \(0.447074\pi\)
\(192\) 0 0
\(193\) 1.55348 0.111822 0.0559111 0.998436i \(-0.482194\pi\)
0.0559111 + 0.998436i \(0.482194\pi\)
\(194\) 0 0
\(195\) 8.87396 0.635478
\(196\) 0 0
\(197\) −14.8166 −1.05564 −0.527821 0.849356i \(-0.676991\pi\)
−0.527821 + 0.849356i \(0.676991\pi\)
\(198\) 0 0
\(199\) −8.99201 −0.637427 −0.318713 0.947851i \(-0.603251\pi\)
−0.318713 + 0.947851i \(0.603251\pi\)
\(200\) 0 0
\(201\) 30.3689 2.14206
\(202\) 0 0
\(203\) −3.37107 −0.236602
\(204\) 0 0
\(205\) 7.43662 0.519396
\(206\) 0 0
\(207\) 14.7955 1.02836
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 2.56841 0.176817 0.0884083 0.996084i \(-0.471822\pi\)
0.0884083 + 0.996084i \(0.471822\pi\)
\(212\) 0 0
\(213\) −39.6710 −2.71821
\(214\) 0 0
\(215\) 2.29561 0.156559
\(216\) 0 0
\(217\) 3.57506 0.242691
\(218\) 0 0
\(219\) 44.6237 3.01539
\(220\) 0 0
\(221\) −28.5944 −1.92347
\(222\) 0 0
\(223\) −11.2774 −0.755188 −0.377594 0.925971i \(-0.623248\pi\)
−0.377594 + 0.925971i \(0.623248\pi\)
\(224\) 0 0
\(225\) −31.3879 −2.09253
\(226\) 0 0
\(227\) 4.50002 0.298677 0.149339 0.988786i \(-0.452286\pi\)
0.149339 + 0.988786i \(0.452286\pi\)
\(228\) 0 0
\(229\) 17.9898 1.18880 0.594401 0.804169i \(-0.297389\pi\)
0.594401 + 0.804169i \(0.297389\pi\)
\(230\) 0 0
\(231\) −5.27991 −0.347393
\(232\) 0 0
\(233\) −9.65321 −0.632403 −0.316201 0.948692i \(-0.602408\pi\)
−0.316201 + 0.948692i \(0.602408\pi\)
\(234\) 0 0
\(235\) 0.832664 0.0543170
\(236\) 0 0
\(237\) 35.3819 2.29830
\(238\) 0 0
\(239\) 5.36583 0.347087 0.173543 0.984826i \(-0.444478\pi\)
0.173543 + 0.984826i \(0.444478\pi\)
\(240\) 0 0
\(241\) 2.30762 0.148647 0.0743236 0.997234i \(-0.476320\pi\)
0.0743236 + 0.997234i \(0.476320\pi\)
\(242\) 0 0
\(243\) −19.8288 −1.27202
\(244\) 0 0
\(245\) −2.83526 −0.181138
\(246\) 0 0
\(247\) −4.16289 −0.264878
\(248\) 0 0
\(249\) 24.4295 1.54815
\(250\) 0 0
\(251\) 11.1322 0.702659 0.351330 0.936252i \(-0.385730\pi\)
0.351330 + 0.936252i \(0.385730\pi\)
\(252\) 0 0
\(253\) −2.14077 −0.134589
\(254\) 0 0
\(255\) −14.6423 −0.916937
\(256\) 0 0
\(257\) 8.10554 0.505609 0.252805 0.967517i \(-0.418647\pi\)
0.252805 + 0.967517i \(0.418647\pi\)
\(258\) 0 0
\(259\) −4.71765 −0.293141
\(260\) 0 0
\(261\) 13.8921 0.859898
\(262\) 0 0
\(263\) 26.5787 1.63891 0.819455 0.573143i \(-0.194276\pi\)
0.819455 + 0.573143i \(0.194276\pi\)
\(264\) 0 0
\(265\) −7.94192 −0.487868
\(266\) 0 0
\(267\) 12.8460 0.786165
\(268\) 0 0
\(269\) −4.21563 −0.257032 −0.128516 0.991707i \(-0.541021\pi\)
−0.128516 + 0.991707i \(0.541021\pi\)
\(270\) 0 0
\(271\) 0.870973 0.0529079 0.0264539 0.999650i \(-0.491578\pi\)
0.0264539 + 0.999650i \(0.491578\pi\)
\(272\) 0 0
\(273\) −21.9797 −1.33027
\(274\) 0 0
\(275\) 4.54153 0.273864
\(276\) 0 0
\(277\) −17.0705 −1.02567 −0.512835 0.858487i \(-0.671405\pi\)
−0.512835 + 0.858487i \(0.671405\pi\)
\(278\) 0 0
\(279\) −14.7328 −0.882027
\(280\) 0 0
\(281\) −3.82059 −0.227917 −0.113959 0.993486i \(-0.536353\pi\)
−0.113959 + 0.993486i \(0.536353\pi\)
\(282\) 0 0
\(283\) −17.7958 −1.05785 −0.528926 0.848668i \(-0.677405\pi\)
−0.528926 + 0.848668i \(0.677405\pi\)
\(284\) 0 0
\(285\) −2.13169 −0.126270
\(286\) 0 0
\(287\) −18.4195 −1.08727
\(288\) 0 0
\(289\) 30.1817 1.77539
\(290\) 0 0
\(291\) 39.1914 2.29744
\(292\) 0 0
\(293\) −1.68332 −0.0983403 −0.0491702 0.998790i \(-0.515658\pi\)
−0.0491702 + 0.998790i \(0.515658\pi\)
\(294\) 0 0
\(295\) −5.67179 −0.330224
\(296\) 0 0
\(297\) 12.3137 0.714514
\(298\) 0 0
\(299\) −8.91179 −0.515382
\(300\) 0 0
\(301\) −5.68593 −0.327732
\(302\) 0 0
\(303\) −36.4075 −2.09156
\(304\) 0 0
\(305\) 6.45025 0.369340
\(306\) 0 0
\(307\) 7.93624 0.452945 0.226473 0.974018i \(-0.427281\pi\)
0.226473 + 0.974018i \(0.427281\pi\)
\(308\) 0 0
\(309\) −31.7775 −1.80776
\(310\) 0 0
\(311\) 29.8589 1.69314 0.846570 0.532277i \(-0.178663\pi\)
0.846570 + 0.532277i \(0.178663\pi\)
\(312\) 0 0
\(313\) 18.8447 1.06516 0.532582 0.846378i \(-0.321222\pi\)
0.532582 + 0.846378i \(0.321222\pi\)
\(314\) 0 0
\(315\) −7.84836 −0.442205
\(316\) 0 0
\(317\) 5.77789 0.324518 0.162259 0.986748i \(-0.448122\pi\)
0.162259 + 0.986748i \(0.448122\pi\)
\(318\) 0 0
\(319\) −2.01005 −0.112541
\(320\) 0 0
\(321\) 45.0447 2.51415
\(322\) 0 0
\(323\) 6.86889 0.382195
\(324\) 0 0
\(325\) 18.9059 1.04871
\(326\) 0 0
\(327\) 63.1657 3.49307
\(328\) 0 0
\(329\) −2.06240 −0.113704
\(330\) 0 0
\(331\) 10.4610 0.574991 0.287496 0.957782i \(-0.407177\pi\)
0.287496 + 0.957782i \(0.407177\pi\)
\(332\) 0 0
\(333\) 19.4413 1.06538
\(334\) 0 0
\(335\) −6.53162 −0.356861
\(336\) 0 0
\(337\) −26.5246 −1.44489 −0.722444 0.691430i \(-0.756981\pi\)
−0.722444 + 0.691430i \(0.756981\pi\)
\(338\) 0 0
\(339\) 46.5749 2.52960
\(340\) 0 0
\(341\) 2.13169 0.115437
\(342\) 0 0
\(343\) 18.7623 1.01307
\(344\) 0 0
\(345\) −4.56345 −0.245688
\(346\) 0 0
\(347\) −9.94655 −0.533959 −0.266979 0.963702i \(-0.586026\pi\)
−0.266979 + 0.963702i \(0.586026\pi\)
\(348\) 0 0
\(349\) 1.51832 0.0812738 0.0406369 0.999174i \(-0.487061\pi\)
0.0406369 + 0.999174i \(0.487061\pi\)
\(350\) 0 0
\(351\) 51.2605 2.73609
\(352\) 0 0
\(353\) −34.4107 −1.83150 −0.915749 0.401750i \(-0.868402\pi\)
−0.915749 + 0.401750i \(0.868402\pi\)
\(354\) 0 0
\(355\) 8.53227 0.452846
\(356\) 0 0
\(357\) 36.2671 1.91946
\(358\) 0 0
\(359\) 1.31071 0.0691769 0.0345884 0.999402i \(-0.488988\pi\)
0.0345884 + 0.999402i \(0.488988\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −3.14822 −0.165239
\(364\) 0 0
\(365\) −9.59749 −0.502355
\(366\) 0 0
\(367\) −14.2993 −0.746415 −0.373207 0.927748i \(-0.621742\pi\)
−0.373207 + 0.927748i \(0.621742\pi\)
\(368\) 0 0
\(369\) 75.9065 3.95153
\(370\) 0 0
\(371\) 19.6711 1.02127
\(372\) 0 0
\(373\) −20.7508 −1.07443 −0.537217 0.843444i \(-0.680525\pi\)
−0.537217 + 0.843444i \(0.680525\pi\)
\(374\) 0 0
\(375\) 20.3395 1.05033
\(376\) 0 0
\(377\) −8.36760 −0.430953
\(378\) 0 0
\(379\) −9.68867 −0.497673 −0.248837 0.968545i \(-0.580048\pi\)
−0.248837 + 0.968545i \(0.580048\pi\)
\(380\) 0 0
\(381\) −33.1271 −1.69715
\(382\) 0 0
\(383\) −36.1499 −1.84717 −0.923587 0.383390i \(-0.874757\pi\)
−0.923587 + 0.383390i \(0.874757\pi\)
\(384\) 0 0
\(385\) 1.13558 0.0578746
\(386\) 0 0
\(387\) 23.4316 1.19109
\(388\) 0 0
\(389\) −28.0423 −1.42180 −0.710900 0.703293i \(-0.751712\pi\)
−0.710900 + 0.703293i \(0.751712\pi\)
\(390\) 0 0
\(391\) 14.7047 0.743650
\(392\) 0 0
\(393\) 30.7293 1.55009
\(394\) 0 0
\(395\) −7.60979 −0.382890
\(396\) 0 0
\(397\) 7.80222 0.391582 0.195791 0.980646i \(-0.437273\pi\)
0.195791 + 0.980646i \(0.437273\pi\)
\(398\) 0 0
\(399\) 5.27991 0.264326
\(400\) 0 0
\(401\) −21.9629 −1.09677 −0.548387 0.836225i \(-0.684758\pi\)
−0.548387 + 0.836225i \(0.684758\pi\)
\(402\) 0 0
\(403\) 8.87396 0.442043
\(404\) 0 0
\(405\) 12.2098 0.606712
\(406\) 0 0
\(407\) −2.81297 −0.139434
\(408\) 0 0
\(409\) 1.85738 0.0918414 0.0459207 0.998945i \(-0.485378\pi\)
0.0459207 + 0.998945i \(0.485378\pi\)
\(410\) 0 0
\(411\) −5.42892 −0.267789
\(412\) 0 0
\(413\) 14.0483 0.691271
\(414\) 0 0
\(415\) −5.25419 −0.257918
\(416\) 0 0
\(417\) 8.60774 0.421523
\(418\) 0 0
\(419\) 3.01873 0.147474 0.0737372 0.997278i \(-0.476507\pi\)
0.0737372 + 0.997278i \(0.476507\pi\)
\(420\) 0 0
\(421\) 4.54661 0.221588 0.110794 0.993843i \(-0.464661\pi\)
0.110794 + 0.993843i \(0.464661\pi\)
\(422\) 0 0
\(423\) 8.49910 0.413240
\(424\) 0 0
\(425\) −31.1953 −1.51319
\(426\) 0 0
\(427\) −15.9764 −0.773154
\(428\) 0 0
\(429\) −13.1057 −0.632749
\(430\) 0 0
\(431\) −12.1446 −0.584984 −0.292492 0.956268i \(-0.594484\pi\)
−0.292492 + 0.956268i \(0.594484\pi\)
\(432\) 0 0
\(433\) −4.70207 −0.225967 −0.112983 0.993597i \(-0.536041\pi\)
−0.112983 + 0.993597i \(0.536041\pi\)
\(434\) 0 0
\(435\) −4.28479 −0.205440
\(436\) 0 0
\(437\) 2.14077 0.102407
\(438\) 0 0
\(439\) 26.9200 1.28482 0.642411 0.766360i \(-0.277934\pi\)
0.642411 + 0.766360i \(0.277934\pi\)
\(440\) 0 0
\(441\) −28.9398 −1.37809
\(442\) 0 0
\(443\) 4.33483 0.205954 0.102977 0.994684i \(-0.467163\pi\)
0.102977 + 0.994684i \(0.467163\pi\)
\(444\) 0 0
\(445\) −2.76288 −0.130973
\(446\) 0 0
\(447\) 58.7680 2.77963
\(448\) 0 0
\(449\) 10.4154 0.491533 0.245767 0.969329i \(-0.420960\pi\)
0.245767 + 0.969329i \(0.420960\pi\)
\(450\) 0 0
\(451\) −10.9829 −0.517166
\(452\) 0 0
\(453\) 33.1458 1.55733
\(454\) 0 0
\(455\) 4.72729 0.221619
\(456\) 0 0
\(457\) −10.4225 −0.487543 −0.243771 0.969833i \(-0.578385\pi\)
−0.243771 + 0.969833i \(0.578385\pi\)
\(458\) 0 0
\(459\) −84.5815 −3.94793
\(460\) 0 0
\(461\) 34.1110 1.58871 0.794355 0.607454i \(-0.207809\pi\)
0.794355 + 0.607454i \(0.207809\pi\)
\(462\) 0 0
\(463\) 28.7298 1.33519 0.667594 0.744525i \(-0.267324\pi\)
0.667594 + 0.744525i \(0.267324\pi\)
\(464\) 0 0
\(465\) 4.54408 0.210727
\(466\) 0 0
\(467\) 25.6777 1.18822 0.594111 0.804383i \(-0.297504\pi\)
0.594111 + 0.804383i \(0.297504\pi\)
\(468\) 0 0
\(469\) 16.1780 0.747030
\(470\) 0 0
\(471\) 36.7160 1.69178
\(472\) 0 0
\(473\) −3.39032 −0.155887
\(474\) 0 0
\(475\) −4.54153 −0.208380
\(476\) 0 0
\(477\) −81.0642 −3.71167
\(478\) 0 0
\(479\) −25.4486 −1.16278 −0.581389 0.813626i \(-0.697491\pi\)
−0.581389 + 0.813626i \(0.697491\pi\)
\(480\) 0 0
\(481\) −11.7101 −0.533934
\(482\) 0 0
\(483\) 11.3031 0.514308
\(484\) 0 0
\(485\) −8.42912 −0.382747
\(486\) 0 0
\(487\) 6.53376 0.296073 0.148036 0.988982i \(-0.452705\pi\)
0.148036 + 0.988982i \(0.452705\pi\)
\(488\) 0 0
\(489\) −47.1013 −2.12999
\(490\) 0 0
\(491\) −42.5368 −1.91966 −0.959831 0.280580i \(-0.909473\pi\)
−0.959831 + 0.280580i \(0.909473\pi\)
\(492\) 0 0
\(493\) 13.8068 0.621827
\(494\) 0 0
\(495\) −4.67970 −0.210337
\(496\) 0 0
\(497\) −21.1333 −0.947960
\(498\) 0 0
\(499\) −43.6620 −1.95458 −0.977289 0.211912i \(-0.932031\pi\)
−0.977289 + 0.211912i \(0.932031\pi\)
\(500\) 0 0
\(501\) 33.3676 1.49075
\(502\) 0 0
\(503\) −14.1374 −0.630354 −0.315177 0.949033i \(-0.602064\pi\)
−0.315177 + 0.949033i \(0.602064\pi\)
\(504\) 0 0
\(505\) 7.83037 0.348447
\(506\) 0 0
\(507\) −13.6306 −0.605356
\(508\) 0 0
\(509\) 6.20694 0.275118 0.137559 0.990494i \(-0.456074\pi\)
0.137559 + 0.990494i \(0.456074\pi\)
\(510\) 0 0
\(511\) 23.7717 1.05160
\(512\) 0 0
\(513\) −12.3137 −0.543663
\(514\) 0 0
\(515\) 6.83456 0.301167
\(516\) 0 0
\(517\) −1.22974 −0.0540838
\(518\) 0 0
\(519\) −11.8480 −0.520067
\(520\) 0 0
\(521\) 28.7876 1.26121 0.630605 0.776104i \(-0.282807\pi\)
0.630605 + 0.776104i \(0.282807\pi\)
\(522\) 0 0
\(523\) −29.1445 −1.27440 −0.637199 0.770699i \(-0.719907\pi\)
−0.637199 + 0.770699i \(0.719907\pi\)
\(524\) 0 0
\(525\) −23.9788 −1.04652
\(526\) 0 0
\(527\) −14.6423 −0.637829
\(528\) 0 0
\(529\) −18.4171 −0.800743
\(530\) 0 0
\(531\) −57.8927 −2.51233
\(532\) 0 0
\(533\) −45.7207 −1.98038
\(534\) 0 0
\(535\) −9.68802 −0.418850
\(536\) 0 0
\(537\) 11.0316 0.476049
\(538\) 0 0
\(539\) 4.18731 0.180360
\(540\) 0 0
\(541\) 23.3632 1.00446 0.502230 0.864734i \(-0.332513\pi\)
0.502230 + 0.864734i \(0.332513\pi\)
\(542\) 0 0
\(543\) −79.7616 −3.42290
\(544\) 0 0
\(545\) −13.5854 −0.581935
\(546\) 0 0
\(547\) −39.8726 −1.70483 −0.852416 0.522865i \(-0.824863\pi\)
−0.852416 + 0.522865i \(0.824863\pi\)
\(548\) 0 0
\(549\) 65.8385 2.80992
\(550\) 0 0
\(551\) 2.01005 0.0856309
\(552\) 0 0
\(553\) 18.8485 0.801519
\(554\) 0 0
\(555\) −5.99637 −0.254532
\(556\) 0 0
\(557\) −16.8257 −0.712929 −0.356465 0.934309i \(-0.616018\pi\)
−0.356465 + 0.934309i \(0.616018\pi\)
\(558\) 0 0
\(559\) −14.1135 −0.596938
\(560\) 0 0
\(561\) 21.6248 0.913000
\(562\) 0 0
\(563\) 8.68841 0.366173 0.183086 0.983097i \(-0.441391\pi\)
0.183086 + 0.983097i \(0.441391\pi\)
\(564\) 0 0
\(565\) −10.0171 −0.421424
\(566\) 0 0
\(567\) −30.2422 −1.27005
\(568\) 0 0
\(569\) 20.7590 0.870262 0.435131 0.900367i \(-0.356702\pi\)
0.435131 + 0.900367i \(0.356702\pi\)
\(570\) 0 0
\(571\) −28.5889 −1.19641 −0.598203 0.801344i \(-0.704119\pi\)
−0.598203 + 0.801344i \(0.704119\pi\)
\(572\) 0 0
\(573\) −14.4021 −0.601657
\(574\) 0 0
\(575\) −9.72237 −0.405451
\(576\) 0 0
\(577\) 8.97270 0.373538 0.186769 0.982404i \(-0.440198\pi\)
0.186769 + 0.982404i \(0.440198\pi\)
\(578\) 0 0
\(579\) −4.89071 −0.203251
\(580\) 0 0
\(581\) 13.0139 0.539909
\(582\) 0 0
\(583\) 11.7292 0.485774
\(584\) 0 0
\(585\) −19.4811 −0.805443
\(586\) 0 0
\(587\) 1.59725 0.0659254 0.0329627 0.999457i \(-0.489506\pi\)
0.0329627 + 0.999457i \(0.489506\pi\)
\(588\) 0 0
\(589\) −2.13169 −0.0878346
\(590\) 0 0
\(591\) 46.6461 1.91876
\(592\) 0 0
\(593\) −0.933009 −0.0383141 −0.0191571 0.999816i \(-0.506098\pi\)
−0.0191571 + 0.999816i \(0.506098\pi\)
\(594\) 0 0
\(595\) −7.80018 −0.319776
\(596\) 0 0
\(597\) 28.3089 1.15860
\(598\) 0 0
\(599\) −0.280302 −0.0114528 −0.00572641 0.999984i \(-0.501823\pi\)
−0.00572641 + 0.999984i \(0.501823\pi\)
\(600\) 0 0
\(601\) −15.5729 −0.635230 −0.317615 0.948220i \(-0.602882\pi\)
−0.317615 + 0.948220i \(0.602882\pi\)
\(602\) 0 0
\(603\) −66.6691 −2.71497
\(604\) 0 0
\(605\) 0.677107 0.0275283
\(606\) 0 0
\(607\) −0.305109 −0.0123840 −0.00619199 0.999981i \(-0.501971\pi\)
−0.00619199 + 0.999981i \(0.501971\pi\)
\(608\) 0 0
\(609\) 10.6129 0.430055
\(610\) 0 0
\(611\) −5.11925 −0.207103
\(612\) 0 0
\(613\) 29.6347 1.19693 0.598467 0.801148i \(-0.295777\pi\)
0.598467 + 0.801148i \(0.295777\pi\)
\(614\) 0 0
\(615\) −23.4121 −0.944069
\(616\) 0 0
\(617\) −11.8931 −0.478798 −0.239399 0.970921i \(-0.576950\pi\)
−0.239399 + 0.970921i \(0.576950\pi\)
\(618\) 0 0
\(619\) −9.11269 −0.366270 −0.183135 0.983088i \(-0.558625\pi\)
−0.183135 + 0.983088i \(0.558625\pi\)
\(620\) 0 0
\(621\) −26.3608 −1.05782
\(622\) 0 0
\(623\) 6.84329 0.274170
\(624\) 0 0
\(625\) 18.3331 0.733324
\(626\) 0 0
\(627\) 3.14822 0.125728
\(628\) 0 0
\(629\) 19.3220 0.770418
\(630\) 0 0
\(631\) 27.0964 1.07869 0.539346 0.842084i \(-0.318671\pi\)
0.539346 + 0.842084i \(0.318671\pi\)
\(632\) 0 0
\(633\) −8.08593 −0.321387
\(634\) 0 0
\(635\) 7.12483 0.282740
\(636\) 0 0
\(637\) 17.4313 0.690653
\(638\) 0 0
\(639\) 87.0900 3.44523
\(640\) 0 0
\(641\) −31.8022 −1.25611 −0.628056 0.778168i \(-0.716149\pi\)
−0.628056 + 0.778168i \(0.716149\pi\)
\(642\) 0 0
\(643\) −34.3861 −1.35606 −0.678028 0.735036i \(-0.737165\pi\)
−0.678028 + 0.735036i \(0.737165\pi\)
\(644\) 0 0
\(645\) −7.22710 −0.284567
\(646\) 0 0
\(647\) 31.0285 1.21986 0.609928 0.792457i \(-0.291198\pi\)
0.609928 + 0.792457i \(0.291198\pi\)
\(648\) 0 0
\(649\) 8.37650 0.328806
\(650\) 0 0
\(651\) −11.2551 −0.441122
\(652\) 0 0
\(653\) −31.9348 −1.24971 −0.624853 0.780743i \(-0.714841\pi\)
−0.624853 + 0.780743i \(0.714841\pi\)
\(654\) 0 0
\(655\) −6.60913 −0.258240
\(656\) 0 0
\(657\) −97.9628 −3.82189
\(658\) 0 0
\(659\) 45.5268 1.77347 0.886736 0.462277i \(-0.152967\pi\)
0.886736 + 0.462277i \(0.152967\pi\)
\(660\) 0 0
\(661\) −29.2167 −1.13640 −0.568200 0.822891i \(-0.692360\pi\)
−0.568200 + 0.822891i \(0.692360\pi\)
\(662\) 0 0
\(663\) 90.0216 3.49615
\(664\) 0 0
\(665\) −1.13558 −0.0440359
\(666\) 0 0
\(667\) 4.30305 0.166615
\(668\) 0 0
\(669\) 35.5036 1.37265
\(670\) 0 0
\(671\) −9.52619 −0.367754
\(672\) 0 0
\(673\) 45.9862 1.77264 0.886319 0.463076i \(-0.153254\pi\)
0.886319 + 0.463076i \(0.153254\pi\)
\(674\) 0 0
\(675\) 55.9230 2.15248
\(676\) 0 0
\(677\) −16.9360 −0.650904 −0.325452 0.945559i \(-0.605516\pi\)
−0.325452 + 0.945559i \(0.605516\pi\)
\(678\) 0 0
\(679\) 20.8779 0.801218
\(680\) 0 0
\(681\) −14.1671 −0.542884
\(682\) 0 0
\(683\) 1.67863 0.0642308 0.0321154 0.999484i \(-0.489776\pi\)
0.0321154 + 0.999484i \(0.489776\pi\)
\(684\) 0 0
\(685\) 1.16763 0.0446129
\(686\) 0 0
\(687\) −56.6361 −2.16080
\(688\) 0 0
\(689\) 48.8273 1.86017
\(690\) 0 0
\(691\) 41.2185 1.56803 0.784013 0.620744i \(-0.213170\pi\)
0.784013 + 0.620744i \(0.213170\pi\)
\(692\) 0 0
\(693\) 11.5910 0.440306
\(694\) 0 0
\(695\) −1.85132 −0.0702244
\(696\) 0 0
\(697\) 75.4405 2.85751
\(698\) 0 0
\(699\) 30.3905 1.14947
\(700\) 0 0
\(701\) −26.4831 −1.00025 −0.500126 0.865953i \(-0.666713\pi\)
−0.500126 + 0.865953i \(0.666713\pi\)
\(702\) 0 0
\(703\) 2.81297 0.106093
\(704\) 0 0
\(705\) −2.62141 −0.0987281
\(706\) 0 0
\(707\) −19.3948 −0.729418
\(708\) 0 0
\(709\) 36.8689 1.38464 0.692319 0.721591i \(-0.256589\pi\)
0.692319 + 0.721591i \(0.256589\pi\)
\(710\) 0 0
\(711\) −77.6741 −2.91301
\(712\) 0 0
\(713\) −4.56345 −0.170903
\(714\) 0 0
\(715\) 2.81872 0.105414
\(716\) 0 0
\(717\) −16.8928 −0.630875
\(718\) 0 0
\(719\) −19.7419 −0.736250 −0.368125 0.929776i \(-0.620000\pi\)
−0.368125 + 0.929776i \(0.620000\pi\)
\(720\) 0 0
\(721\) −16.9283 −0.630444
\(722\) 0 0
\(723\) −7.26492 −0.270185
\(724\) 0 0
\(725\) −9.12869 −0.339031
\(726\) 0 0
\(727\) 33.7540 1.25187 0.625934 0.779876i \(-0.284718\pi\)
0.625934 + 0.779876i \(0.284718\pi\)
\(728\) 0 0
\(729\) 8.32841 0.308460
\(730\) 0 0
\(731\) 23.2878 0.861329
\(732\) 0 0
\(733\) −35.0242 −1.29365 −0.646825 0.762639i \(-0.723904\pi\)
−0.646825 + 0.762639i \(0.723904\pi\)
\(734\) 0 0
\(735\) 8.92603 0.329242
\(736\) 0 0
\(737\) 9.64636 0.355328
\(738\) 0 0
\(739\) −8.99477 −0.330878 −0.165439 0.986220i \(-0.552904\pi\)
−0.165439 + 0.986220i \(0.552904\pi\)
\(740\) 0 0
\(741\) 13.1057 0.481450
\(742\) 0 0
\(743\) 17.8359 0.654337 0.327168 0.944966i \(-0.393906\pi\)
0.327168 + 0.944966i \(0.393906\pi\)
\(744\) 0 0
\(745\) −12.6396 −0.463079
\(746\) 0 0
\(747\) −53.6301 −1.96222
\(748\) 0 0
\(749\) 23.9960 0.876794
\(750\) 0 0
\(751\) −10.8155 −0.394662 −0.197331 0.980337i \(-0.563227\pi\)
−0.197331 + 0.980337i \(0.563227\pi\)
\(752\) 0 0
\(753\) −35.0467 −1.27717
\(754\) 0 0
\(755\) −7.12887 −0.259446
\(756\) 0 0
\(757\) 29.0949 1.05747 0.528737 0.848786i \(-0.322666\pi\)
0.528737 + 0.848786i \(0.322666\pi\)
\(758\) 0 0
\(759\) 6.73963 0.244633
\(760\) 0 0
\(761\) −24.8989 −0.902586 −0.451293 0.892376i \(-0.649037\pi\)
−0.451293 + 0.892376i \(0.649037\pi\)
\(762\) 0 0
\(763\) 33.6493 1.21819
\(764\) 0 0
\(765\) 32.1444 1.16218
\(766\) 0 0
\(767\) 34.8704 1.25910
\(768\) 0 0
\(769\) −22.4656 −0.810129 −0.405064 0.914288i \(-0.632751\pi\)
−0.405064 + 0.914288i \(0.632751\pi\)
\(770\) 0 0
\(771\) −25.5180 −0.919010
\(772\) 0 0
\(773\) −0.491907 −0.0176927 −0.00884633 0.999961i \(-0.502816\pi\)
−0.00884633 + 0.999961i \(0.502816\pi\)
\(774\) 0 0
\(775\) 9.68110 0.347756
\(776\) 0 0
\(777\) 14.8522 0.532821
\(778\) 0 0
\(779\) 10.9829 0.393504
\(780\) 0 0
\(781\) −12.6011 −0.450902
\(782\) 0 0
\(783\) −24.7511 −0.884534
\(784\) 0 0
\(785\) −7.89673 −0.281846
\(786\) 0 0
\(787\) 30.7601 1.09648 0.548239 0.836322i \(-0.315298\pi\)
0.548239 + 0.836322i \(0.315298\pi\)
\(788\) 0 0
\(789\) −83.6756 −2.97893
\(790\) 0 0
\(791\) 24.8112 0.882184
\(792\) 0 0
\(793\) −39.6564 −1.40824
\(794\) 0 0
\(795\) 25.0030 0.886763
\(796\) 0 0
\(797\) −46.8176 −1.65836 −0.829182 0.558978i \(-0.811193\pi\)
−0.829182 + 0.558978i \(0.811193\pi\)
\(798\) 0 0
\(799\) 8.44693 0.298831
\(800\) 0 0
\(801\) −28.2010 −0.996434
\(802\) 0 0
\(803\) 14.1743 0.500199
\(804\) 0 0
\(805\) −2.43102 −0.0856822
\(806\) 0 0
\(807\) 13.2718 0.467188
\(808\) 0 0
\(809\) −15.3194 −0.538601 −0.269300 0.963056i \(-0.586792\pi\)
−0.269300 + 0.963056i \(0.586792\pi\)
\(810\) 0 0
\(811\) 21.0369 0.738706 0.369353 0.929289i \(-0.379579\pi\)
0.369353 + 0.929289i \(0.379579\pi\)
\(812\) 0 0
\(813\) −2.74202 −0.0961668
\(814\) 0 0
\(815\) 10.1303 0.354851
\(816\) 0 0
\(817\) 3.39032 0.118612
\(818\) 0 0
\(819\) 48.2521 1.68606
\(820\) 0 0
\(821\) −49.3342 −1.72178 −0.860888 0.508794i \(-0.830092\pi\)
−0.860888 + 0.508794i \(0.830092\pi\)
\(822\) 0 0
\(823\) 44.7400 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(824\) 0 0
\(825\) −14.2977 −0.497784
\(826\) 0 0
\(827\) −42.7710 −1.48729 −0.743646 0.668573i \(-0.766905\pi\)
−0.743646 + 0.668573i \(0.766905\pi\)
\(828\) 0 0
\(829\) 48.6840 1.69086 0.845432 0.534083i \(-0.179343\pi\)
0.845432 + 0.534083i \(0.179343\pi\)
\(830\) 0 0
\(831\) 53.7419 1.86429
\(832\) 0 0
\(833\) −28.7622 −0.996551
\(834\) 0 0
\(835\) −7.17656 −0.248355
\(836\) 0 0
\(837\) 26.2489 0.907296
\(838\) 0 0
\(839\) −50.1921 −1.73282 −0.866412 0.499330i \(-0.833579\pi\)
−0.866412 + 0.499330i \(0.833579\pi\)
\(840\) 0 0
\(841\) −24.9597 −0.860680
\(842\) 0 0
\(843\) 12.0281 0.414268
\(844\) 0 0
\(845\) 2.93161 0.100851
\(846\) 0 0
\(847\) −1.67711 −0.0576261
\(848\) 0 0
\(849\) 56.0253 1.92278
\(850\) 0 0
\(851\) 6.02193 0.206429
\(852\) 0 0
\(853\) −25.6101 −0.876874 −0.438437 0.898762i \(-0.644468\pi\)
−0.438437 + 0.898762i \(0.644468\pi\)
\(854\) 0 0
\(855\) 4.67970 0.160042
\(856\) 0 0
\(857\) −0.780477 −0.0266606 −0.0133303 0.999911i \(-0.504243\pi\)
−0.0133303 + 0.999911i \(0.504243\pi\)
\(858\) 0 0
\(859\) 18.5597 0.633247 0.316624 0.948551i \(-0.397451\pi\)
0.316624 + 0.948551i \(0.397451\pi\)
\(860\) 0 0
\(861\) 57.9889 1.97625
\(862\) 0 0
\(863\) 20.5115 0.698220 0.349110 0.937082i \(-0.386484\pi\)
0.349110 + 0.937082i \(0.386484\pi\)
\(864\) 0 0
\(865\) 2.54821 0.0866417
\(866\) 0 0
\(867\) −95.0187 −3.22700
\(868\) 0 0
\(869\) 11.2387 0.381246
\(870\) 0 0
\(871\) 40.1567 1.36066
\(872\) 0 0
\(873\) −86.0371 −2.91191
\(874\) 0 0
\(875\) 10.8352 0.366296
\(876\) 0 0
\(877\) 14.4072 0.486497 0.243248 0.969964i \(-0.421787\pi\)
0.243248 + 0.969964i \(0.421787\pi\)
\(878\) 0 0
\(879\) 5.29945 0.178746
\(880\) 0 0
\(881\) 33.3794 1.12458 0.562291 0.826940i \(-0.309920\pi\)
0.562291 + 0.826940i \(0.309920\pi\)
\(882\) 0 0
\(883\) 35.6404 1.19940 0.599698 0.800226i \(-0.295287\pi\)
0.599698 + 0.800226i \(0.295287\pi\)
\(884\) 0 0
\(885\) 17.8561 0.600225
\(886\) 0 0
\(887\) −12.6762 −0.425626 −0.212813 0.977093i \(-0.568263\pi\)
−0.212813 + 0.977093i \(0.568263\pi\)
\(888\) 0 0
\(889\) −17.6473 −0.591871
\(890\) 0 0
\(891\) −18.0324 −0.604107
\(892\) 0 0
\(893\) 1.22974 0.0411516
\(894\) 0 0
\(895\) −2.37263 −0.0793083
\(896\) 0 0
\(897\) 28.0563 0.936773
\(898\) 0 0
\(899\) −4.28479 −0.142906
\(900\) 0 0
\(901\) −80.5666 −2.68406
\(902\) 0 0
\(903\) 17.9006 0.595694
\(904\) 0 0
\(905\) 17.1548 0.570245
\(906\) 0 0
\(907\) 36.6608 1.21730 0.608651 0.793438i \(-0.291711\pi\)
0.608651 + 0.793438i \(0.291711\pi\)
\(908\) 0 0
\(909\) 79.9256 2.65097
\(910\) 0 0
\(911\) −47.6952 −1.58021 −0.790106 0.612970i \(-0.789974\pi\)
−0.790106 + 0.612970i \(0.789974\pi\)
\(912\) 0 0
\(913\) 7.75976 0.256810
\(914\) 0 0
\(915\) −20.3068 −0.671323
\(916\) 0 0
\(917\) 16.3700 0.540584
\(918\) 0 0
\(919\) −29.7694 −0.982001 −0.491001 0.871159i \(-0.663369\pi\)
−0.491001 + 0.871159i \(0.663369\pi\)
\(920\) 0 0
\(921\) −24.9851 −0.823286
\(922\) 0 0
\(923\) −52.4568 −1.72664
\(924\) 0 0
\(925\) −12.7752 −0.420045
\(926\) 0 0
\(927\) 69.7613 2.29126
\(928\) 0 0
\(929\) −50.6180 −1.66072 −0.830362 0.557225i \(-0.811866\pi\)
−0.830362 + 0.557225i \(0.811866\pi\)
\(930\) 0 0
\(931\) −4.18731 −0.137234
\(932\) 0 0
\(933\) −94.0024 −3.07750
\(934\) 0 0
\(935\) −4.65098 −0.152103
\(936\) 0 0
\(937\) 16.1084 0.526238 0.263119 0.964763i \(-0.415249\pi\)
0.263119 + 0.964763i \(0.415249\pi\)
\(938\) 0 0
\(939\) −59.3273 −1.93607
\(940\) 0 0
\(941\) 16.8109 0.548018 0.274009 0.961727i \(-0.411650\pi\)
0.274009 + 0.961727i \(0.411650\pi\)
\(942\) 0 0
\(943\) 23.5119 0.765654
\(944\) 0 0
\(945\) 13.9832 0.454874
\(946\) 0 0
\(947\) 36.1350 1.17423 0.587114 0.809504i \(-0.300264\pi\)
0.587114 + 0.809504i \(0.300264\pi\)
\(948\) 0 0
\(949\) 59.0058 1.91541
\(950\) 0 0
\(951\) −18.1901 −0.589854
\(952\) 0 0
\(953\) −29.1171 −0.943197 −0.471598 0.881813i \(-0.656323\pi\)
−0.471598 + 0.881813i \(0.656323\pi\)
\(954\) 0 0
\(955\) 3.09755 0.100234
\(956\) 0 0
\(957\) 6.32808 0.204558
\(958\) 0 0
\(959\) −2.89207 −0.0933898
\(960\) 0 0
\(961\) −26.4559 −0.853417
\(962\) 0 0
\(963\) −98.8868 −3.18658
\(964\) 0 0
\(965\) 1.05187 0.0338610
\(966\) 0 0
\(967\) 10.1650 0.326883 0.163442 0.986553i \(-0.447740\pi\)
0.163442 + 0.986553i \(0.447740\pi\)
\(968\) 0 0
\(969\) −21.6248 −0.694689
\(970\) 0 0
\(971\) 15.9729 0.512594 0.256297 0.966598i \(-0.417497\pi\)
0.256297 + 0.966598i \(0.417497\pi\)
\(972\) 0 0
\(973\) 4.58547 0.147003
\(974\) 0 0
\(975\) −59.5199 −1.90616
\(976\) 0 0
\(977\) 42.7941 1.36910 0.684552 0.728964i \(-0.259998\pi\)
0.684552 + 0.728964i \(0.259998\pi\)
\(978\) 0 0
\(979\) 4.08041 0.130410
\(980\) 0 0
\(981\) −138.668 −4.42733
\(982\) 0 0
\(983\) 13.8981 0.443280 0.221640 0.975129i \(-0.428859\pi\)
0.221640 + 0.975129i \(0.428859\pi\)
\(984\) 0 0
\(985\) −10.0325 −0.319661
\(986\) 0 0
\(987\) 6.49290 0.206671
\(988\) 0 0
\(989\) 7.25790 0.230788
\(990\) 0 0
\(991\) −49.5286 −1.57333 −0.786664 0.617381i \(-0.788194\pi\)
−0.786664 + 0.617381i \(0.788194\pi\)
\(992\) 0 0
\(993\) −32.9337 −1.04512
\(994\) 0 0
\(995\) −6.08855 −0.193020
\(996\) 0 0
\(997\) 51.6810 1.63675 0.818377 0.574682i \(-0.194874\pi\)
0.818377 + 0.574682i \(0.194874\pi\)
\(998\) 0 0
\(999\) −34.6381 −1.09590
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 1672.2.a.i.1.1 6
4.3 odd 2 3344.2.a.z.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.i.1.1 6 1.1 even 1 trivial
3344.2.a.z.1.6 6 4.3 odd 2