Properties

Label 1092.2.a.h.1.3
Level $1092$
Weight $2$
Character 1092.1
Self dual yes
Analytic conductor $8.720$
Analytic rank $0$
Dimension $3$
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] = [1092,2,Mod(1,1092)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1092, 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("1092.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1092.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: \(8.71966390072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1373.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 8x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.43931\) of defining polynomial
Character \(\chi\) \(=\) 1092.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.38955 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.38955 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.87862 q^{11} +1.00000 q^{13} +3.38955 q^{15} +4.00000 q^{17} +3.48907 q^{19} -1.00000 q^{21} +8.26818 q^{23} +6.48907 q^{25} +1.00000 q^{27} +5.48907 q^{29} -6.26818 q^{31} -4.87862 q^{33} -3.38955 q^{35} +8.77911 q^{37} +1.00000 q^{39} -0.0995171 q^{41} -2.26818 q^{43} +3.38955 q^{45} -10.1687 q^{47} +1.00000 q^{49} +4.00000 q^{51} -12.2682 q^{53} -16.5364 q^{55} +3.48907 q^{57} +6.87862 q^{59} +2.00000 q^{61} -1.00000 q^{63} +3.38955 q^{65} -6.77911 q^{67} +8.26818 q^{69} +3.65773 q^{71} -11.0473 q^{73} +6.48907 q^{75} +4.87862 q^{77} +13.0473 q^{79} +1.00000 q^{81} -4.61045 q^{83} +13.5582 q^{85} +5.48907 q^{87} -13.1468 q^{89} -1.00000 q^{91} -6.26818 q^{93} +11.8264 q^{95} +8.26818 q^{97} -4.87862 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{13} + q^{15} + 12 q^{17} + 5 q^{19} - 3 q^{21} + q^{23} + 14 q^{25} + 3 q^{27} + 11 q^{29} + 5 q^{31} - q^{35} + 8 q^{37} + 3 q^{39} - 4 q^{41} + 17 q^{43} + q^{45} - 3 q^{47} + 3 q^{49} + 12 q^{51} - 13 q^{53} - 2 q^{55} + 5 q^{57} + 6 q^{59} + 6 q^{61} - 3 q^{63} + q^{65} - 2 q^{67} + q^{69} - 22 q^{71} + 9 q^{73} + 14 q^{75} - 3 q^{79} + 3 q^{81} - 23 q^{83} + 4 q^{85} + 11 q^{87} - q^{89} - 3 q^{91} + 5 q^{93} - 25 q^{95} + q^{97}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.38955 1.51585 0.757927 0.652339i \(-0.226212\pi\)
0.757927 + 0.652339i \(0.226212\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.87862 −1.47096 −0.735480 0.677546i \(-0.763043\pi\)
−0.735480 + 0.677546i \(0.763043\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 3.38955 0.875179
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 3.48907 0.800448 0.400224 0.916417i \(-0.368932\pi\)
0.400224 + 0.916417i \(0.368932\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 8.26818 1.72403 0.862017 0.506879i \(-0.169201\pi\)
0.862017 + 0.506879i \(0.169201\pi\)
\(24\) 0 0
\(25\) 6.48907 1.29781
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.48907 1.01929 0.509647 0.860383i \(-0.329776\pi\)
0.509647 + 0.860383i \(0.329776\pi\)
\(30\) 0 0
\(31\) −6.26818 −1.12580 −0.562899 0.826526i \(-0.690314\pi\)
−0.562899 + 0.826526i \(0.690314\pi\)
\(32\) 0 0
\(33\) −4.87862 −0.849259
\(34\) 0 0
\(35\) −3.38955 −0.572939
\(36\) 0 0
\(37\) 8.77911 1.44328 0.721638 0.692271i \(-0.243390\pi\)
0.721638 + 0.692271i \(0.243390\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −0.0995171 −0.0155420 −0.00777098 0.999970i \(-0.502474\pi\)
−0.00777098 + 0.999970i \(0.502474\pi\)
\(42\) 0 0
\(43\) −2.26818 −0.345894 −0.172947 0.984931i \(-0.555329\pi\)
−0.172947 + 0.984931i \(0.555329\pi\)
\(44\) 0 0
\(45\) 3.38955 0.505285
\(46\) 0 0
\(47\) −10.1687 −1.48325 −0.741626 0.670814i \(-0.765945\pi\)
−0.741626 + 0.670814i \(0.765945\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −12.2682 −1.68516 −0.842582 0.538568i \(-0.818965\pi\)
−0.842582 + 0.538568i \(0.818965\pi\)
\(54\) 0 0
\(55\) −16.5364 −2.22976
\(56\) 0 0
\(57\) 3.48907 0.462139
\(58\) 0 0
\(59\) 6.87862 0.895520 0.447760 0.894154i \(-0.352222\pi\)
0.447760 + 0.894154i \(0.352222\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 3.38955 0.420422
\(66\) 0 0
\(67\) −6.77911 −0.828200 −0.414100 0.910231i \(-0.635904\pi\)
−0.414100 + 0.910231i \(0.635904\pi\)
\(68\) 0 0
\(69\) 8.26818 0.995371
\(70\) 0 0
\(71\) 3.65773 0.434093 0.217046 0.976161i \(-0.430358\pi\)
0.217046 + 0.976161i \(0.430358\pi\)
\(72\) 0 0
\(73\) −11.0473 −1.29299 −0.646493 0.762920i \(-0.723765\pi\)
−0.646493 + 0.762920i \(0.723765\pi\)
\(74\) 0 0
\(75\) 6.48907 0.749293
\(76\) 0 0
\(77\) 4.87862 0.555971
\(78\) 0 0
\(79\) 13.0473 1.46793 0.733967 0.679185i \(-0.237667\pi\)
0.733967 + 0.679185i \(0.237667\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.61045 −0.506062 −0.253031 0.967458i \(-0.581428\pi\)
−0.253031 + 0.967458i \(0.581428\pi\)
\(84\) 0 0
\(85\) 13.5582 1.47059
\(86\) 0 0
\(87\) 5.48907 0.588490
\(88\) 0 0
\(89\) −13.1468 −1.39356 −0.696779 0.717286i \(-0.745384\pi\)
−0.696779 + 0.717286i \(0.745384\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −6.26818 −0.649980
\(94\) 0 0
\(95\) 11.8264 1.21336
\(96\) 0 0
\(97\) 8.26818 0.839506 0.419753 0.907638i \(-0.362117\pi\)
0.419753 + 0.907638i \(0.362117\pi\)
\(98\) 0 0
\(99\) −4.87862 −0.490320
\(100\) 0 0
\(101\) 1.22089 0.121483 0.0607417 0.998154i \(-0.480653\pi\)
0.0607417 + 0.998154i \(0.480653\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −3.38955 −0.330787
\(106\) 0 0
\(107\) −18.5364 −1.79198 −0.895988 0.444077i \(-0.853532\pi\)
−0.895988 + 0.444077i \(0.853532\pi\)
\(108\) 0 0
\(109\) 3.75725 0.359879 0.179939 0.983678i \(-0.442410\pi\)
0.179939 + 0.983678i \(0.442410\pi\)
\(110\) 0 0
\(111\) 8.77911 0.833276
\(112\) 0 0
\(113\) −19.0473 −1.79182 −0.895909 0.444238i \(-0.853474\pi\)
−0.895909 + 0.444238i \(0.853474\pi\)
\(114\) 0 0
\(115\) 28.0254 2.61338
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 12.8010 1.16372
\(122\) 0 0
\(123\) −0.0995171 −0.00897316
\(124\) 0 0
\(125\) 5.04728 0.451443
\(126\) 0 0
\(127\) −0.199034 −0.0176614 −0.00883072 0.999961i \(-0.502811\pi\)
−0.00883072 + 0.999961i \(0.502811\pi\)
\(128\) 0 0
\(129\) −2.26818 −0.199702
\(130\) 0 0
\(131\) 1.75725 0.153531 0.0767657 0.997049i \(-0.475541\pi\)
0.0767657 + 0.997049i \(0.475541\pi\)
\(132\) 0 0
\(133\) −3.48907 −0.302541
\(134\) 0 0
\(135\) 3.38955 0.291726
\(136\) 0 0
\(137\) 6.09952 0.521117 0.260558 0.965458i \(-0.416093\pi\)
0.260558 + 0.965458i \(0.416093\pi\)
\(138\) 0 0
\(139\) 6.97814 0.591878 0.295939 0.955207i \(-0.404367\pi\)
0.295939 + 0.955207i \(0.404367\pi\)
\(140\) 0 0
\(141\) −10.1687 −0.856356
\(142\) 0 0
\(143\) −4.87862 −0.407971
\(144\) 0 0
\(145\) 18.6055 1.54510
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −12.8786 −1.05506 −0.527529 0.849537i \(-0.676881\pi\)
−0.527529 + 0.849537i \(0.676881\pi\)
\(150\) 0 0
\(151\) −13.5582 −1.10335 −0.551676 0.834059i \(-0.686011\pi\)
−0.551676 + 0.834059i \(0.686011\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −21.2463 −1.70655
\(156\) 0 0
\(157\) −14.5364 −1.16013 −0.580064 0.814571i \(-0.696972\pi\)
−0.580064 + 0.814571i \(0.696972\pi\)
\(158\) 0 0
\(159\) −12.2682 −0.972930
\(160\) 0 0
\(161\) −8.26818 −0.651624
\(162\) 0 0
\(163\) 13.7572 1.07755 0.538775 0.842449i \(-0.318887\pi\)
0.538775 + 0.842449i \(0.318887\pi\)
\(164\) 0 0
\(165\) −16.5364 −1.28735
\(166\) 0 0
\(167\) 2.36769 0.183218 0.0916088 0.995795i \(-0.470799\pi\)
0.0916088 + 0.995795i \(0.470799\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 3.48907 0.266816
\(172\) 0 0
\(173\) 0.199034 0.0151323 0.00756615 0.999971i \(-0.497592\pi\)
0.00756615 + 0.999971i \(0.497592\pi\)
\(174\) 0 0
\(175\) −6.48907 −0.490528
\(176\) 0 0
\(177\) 6.87862 0.517029
\(178\) 0 0
\(179\) −23.0473 −1.72263 −0.861317 0.508067i \(-0.830360\pi\)
−0.861317 + 0.508067i \(0.830360\pi\)
\(180\) 0 0
\(181\) −11.5582 −0.859115 −0.429558 0.903039i \(-0.641330\pi\)
−0.429558 + 0.903039i \(0.641330\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 29.7572 2.18780
\(186\) 0 0
\(187\) −19.5145 −1.42704
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −16.7791 −1.21409 −0.607047 0.794666i \(-0.707646\pi\)
−0.607047 + 0.794666i \(0.707646\pi\)
\(192\) 0 0
\(193\) 25.3155 1.82225 0.911123 0.412134i \(-0.135216\pi\)
0.911123 + 0.412134i \(0.135216\pi\)
\(194\) 0 0
\(195\) 3.38955 0.242731
\(196\) 0 0
\(197\) 0.878623 0.0625993 0.0312997 0.999510i \(-0.490035\pi\)
0.0312997 + 0.999510i \(0.490035\pi\)
\(198\) 0 0
\(199\) 13.5582 0.961116 0.480558 0.876963i \(-0.340434\pi\)
0.480558 + 0.876963i \(0.340434\pi\)
\(200\) 0 0
\(201\) −6.77911 −0.478161
\(202\) 0 0
\(203\) −5.48907 −0.385257
\(204\) 0 0
\(205\) −0.337319 −0.0235594
\(206\) 0 0
\(207\) 8.26818 0.574678
\(208\) 0 0
\(209\) −17.0219 −1.17743
\(210\) 0 0
\(211\) 27.0036 1.85900 0.929501 0.368820i \(-0.120238\pi\)
0.929501 + 0.368820i \(0.120238\pi\)
\(212\) 0 0
\(213\) 3.65773 0.250623
\(214\) 0 0
\(215\) −7.68810 −0.524324
\(216\) 0 0
\(217\) 6.26818 0.425512
\(218\) 0 0
\(219\) −11.0473 −0.746506
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −29.0473 −1.94515 −0.972575 0.232590i \(-0.925280\pi\)
−0.972575 + 0.232590i \(0.925280\pi\)
\(224\) 0 0
\(225\) 6.48907 0.432605
\(226\) 0 0
\(227\) −7.07766 −0.469761 −0.234880 0.972024i \(-0.575470\pi\)
−0.234880 + 0.972024i \(0.575470\pi\)
\(228\) 0 0
\(229\) −9.80097 −0.647666 −0.323833 0.946114i \(-0.604972\pi\)
−0.323833 + 0.946114i \(0.604972\pi\)
\(230\) 0 0
\(231\) 4.87862 0.320990
\(232\) 0 0
\(233\) −2.51093 −0.164496 −0.0822482 0.996612i \(-0.526210\pi\)
−0.0822482 + 0.996612i \(0.526210\pi\)
\(234\) 0 0
\(235\) −34.4672 −2.24839
\(236\) 0 0
\(237\) 13.0473 0.847512
\(238\) 0 0
\(239\) −0.342270 −0.0221396 −0.0110698 0.999939i \(-0.503524\pi\)
−0.0110698 + 0.999939i \(0.503524\pi\)
\(240\) 0 0
\(241\) 18.5109 1.19239 0.596197 0.802838i \(-0.296678\pi\)
0.596197 + 0.802838i \(0.296678\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.38955 0.216551
\(246\) 0 0
\(247\) 3.48907 0.222004
\(248\) 0 0
\(249\) −4.61045 −0.292175
\(250\) 0 0
\(251\) −14.7791 −0.932849 −0.466424 0.884561i \(-0.654458\pi\)
−0.466424 + 0.884561i \(0.654458\pi\)
\(252\) 0 0
\(253\) −40.3373 −2.53599
\(254\) 0 0
\(255\) 13.5582 0.849048
\(256\) 0 0
\(257\) 22.7791 1.42092 0.710461 0.703737i \(-0.248487\pi\)
0.710461 + 0.703737i \(0.248487\pi\)
\(258\) 0 0
\(259\) −8.77911 −0.545507
\(260\) 0 0
\(261\) 5.48907 0.339765
\(262\) 0 0
\(263\) 7.24632 0.446827 0.223414 0.974724i \(-0.428280\pi\)
0.223414 + 0.974724i \(0.428280\pi\)
\(264\) 0 0
\(265\) −41.5836 −2.55446
\(266\) 0 0
\(267\) −13.1468 −0.804571
\(268\) 0 0
\(269\) −1.75725 −0.107141 −0.0535706 0.998564i \(-0.517060\pi\)
−0.0535706 + 0.998564i \(0.517060\pi\)
\(270\) 0 0
\(271\) −13.7572 −0.835693 −0.417847 0.908518i \(-0.637215\pi\)
−0.417847 + 0.908518i \(0.637215\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) −31.6577 −1.90903
\(276\) 0 0
\(277\) −11.2463 −0.675726 −0.337863 0.941195i \(-0.609704\pi\)
−0.337863 + 0.941195i \(0.609704\pi\)
\(278\) 0 0
\(279\) −6.26818 −0.375266
\(280\) 0 0
\(281\) −30.4368 −1.81571 −0.907855 0.419285i \(-0.862281\pi\)
−0.907855 + 0.419285i \(0.862281\pi\)
\(282\) 0 0
\(283\) 17.5582 1.04373 0.521864 0.853029i \(-0.325237\pi\)
0.521864 + 0.853029i \(0.325237\pi\)
\(284\) 0 0
\(285\) 11.8264 0.700535
\(286\) 0 0
\(287\) 0.0995171 0.00587431
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 8.26818 0.484689
\(292\) 0 0
\(293\) 15.1905 0.887440 0.443720 0.896166i \(-0.353659\pi\)
0.443720 + 0.896166i \(0.353659\pi\)
\(294\) 0 0
\(295\) 23.3155 1.35748
\(296\) 0 0
\(297\) −4.87862 −0.283086
\(298\) 0 0
\(299\) 8.26818 0.478161
\(300\) 0 0
\(301\) 2.26818 0.130736
\(302\) 0 0
\(303\) 1.22089 0.0701385
\(304\) 0 0
\(305\) 6.77911 0.388170
\(306\) 0 0
\(307\) −3.29004 −0.187772 −0.0938861 0.995583i \(-0.529929\pi\)
−0.0938861 + 0.995583i \(0.529929\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 10.7791 0.611227 0.305614 0.952156i \(-0.401138\pi\)
0.305614 + 0.952156i \(0.401138\pi\)
\(312\) 0 0
\(313\) 25.1164 1.41966 0.709832 0.704371i \(-0.248771\pi\)
0.709832 + 0.704371i \(0.248771\pi\)
\(314\) 0 0
\(315\) −3.38955 −0.190980
\(316\) 0 0
\(317\) 23.8568 1.33993 0.669965 0.742393i \(-0.266309\pi\)
0.669965 + 0.742393i \(0.266309\pi\)
\(318\) 0 0
\(319\) −26.7791 −1.49934
\(320\) 0 0
\(321\) −18.5364 −1.03460
\(322\) 0 0
\(323\) 13.9563 0.776548
\(324\) 0 0
\(325\) 6.48907 0.359949
\(326\) 0 0
\(327\) 3.75725 0.207776
\(328\) 0 0
\(329\) 10.1687 0.560616
\(330\) 0 0
\(331\) −2.97814 −0.163693 −0.0818467 0.996645i \(-0.526082\pi\)
−0.0818467 + 0.996645i \(0.526082\pi\)
\(332\) 0 0
\(333\) 8.77911 0.481092
\(334\) 0 0
\(335\) −22.9781 −1.25543
\(336\) 0 0
\(337\) −29.0036 −1.57992 −0.789962 0.613155i \(-0.789900\pi\)
−0.789962 + 0.613155i \(0.789900\pi\)
\(338\) 0 0
\(339\) −19.0473 −1.03451
\(340\) 0 0
\(341\) 30.5801 1.65600
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 28.0254 1.50884
\(346\) 0 0
\(347\) 18.3373 0.984399 0.492199 0.870482i \(-0.336193\pi\)
0.492199 + 0.870482i \(0.336193\pi\)
\(348\) 0 0
\(349\) −33.0036 −1.76664 −0.883320 0.468770i \(-0.844697\pi\)
−0.883320 + 0.468770i \(0.844697\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 23.4150 1.24625 0.623127 0.782121i \(-0.285862\pi\)
0.623127 + 0.782121i \(0.285862\pi\)
\(354\) 0 0
\(355\) 12.3981 0.658021
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −12.3423 −0.651400 −0.325700 0.945473i \(-0.605600\pi\)
−0.325700 + 0.945473i \(0.605600\pi\)
\(360\) 0 0
\(361\) −6.82639 −0.359284
\(362\) 0 0
\(363\) 12.8010 0.671876
\(364\) 0 0
\(365\) −37.4454 −1.95998
\(366\) 0 0
\(367\) 2.97814 0.155458 0.0777288 0.996975i \(-0.475233\pi\)
0.0777288 + 0.996975i \(0.475233\pi\)
\(368\) 0 0
\(369\) −0.0995171 −0.00518066
\(370\) 0 0
\(371\) 12.2682 0.636932
\(372\) 0 0
\(373\) 17.8010 0.921699 0.460850 0.887478i \(-0.347545\pi\)
0.460850 + 0.887478i \(0.347545\pi\)
\(374\) 0 0
\(375\) 5.04728 0.260641
\(376\) 0 0
\(377\) 5.48907 0.282702
\(378\) 0 0
\(379\) −7.46365 −0.383382 −0.191691 0.981455i \(-0.561397\pi\)
−0.191691 + 0.981455i \(0.561397\pi\)
\(380\) 0 0
\(381\) −0.199034 −0.0101968
\(382\) 0 0
\(383\) −14.6796 −0.750092 −0.375046 0.927006i \(-0.622373\pi\)
−0.375046 + 0.927006i \(0.622373\pi\)
\(384\) 0 0
\(385\) 16.5364 0.842771
\(386\) 0 0
\(387\) −2.26818 −0.115298
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 33.0727 1.67256
\(392\) 0 0
\(393\) 1.75725 0.0886414
\(394\) 0 0
\(395\) 44.2245 2.22517
\(396\) 0 0
\(397\) −10.8482 −0.544458 −0.272229 0.962232i \(-0.587761\pi\)
−0.272229 + 0.962232i \(0.587761\pi\)
\(398\) 0 0
\(399\) −3.48907 −0.174672
\(400\) 0 0
\(401\) 5.90048 0.294656 0.147328 0.989088i \(-0.452933\pi\)
0.147328 + 0.989088i \(0.452933\pi\)
\(402\) 0 0
\(403\) −6.26818 −0.312240
\(404\) 0 0
\(405\) 3.38955 0.168428
\(406\) 0 0
\(407\) −42.8300 −2.12300
\(408\) 0 0
\(409\) 14.2245 0.703354 0.351677 0.936121i \(-0.385612\pi\)
0.351677 + 0.936121i \(0.385612\pi\)
\(410\) 0 0
\(411\) 6.09952 0.300867
\(412\) 0 0
\(413\) −6.87862 −0.338475
\(414\) 0 0
\(415\) −15.6274 −0.767117
\(416\) 0 0
\(417\) 6.97814 0.341721
\(418\) 0 0
\(419\) −18.2936 −0.893701 −0.446850 0.894609i \(-0.647454\pi\)
−0.446850 + 0.894609i \(0.647454\pi\)
\(420\) 0 0
\(421\) 28.4926 1.38865 0.694323 0.719664i \(-0.255704\pi\)
0.694323 + 0.719664i \(0.255704\pi\)
\(422\) 0 0
\(423\) −10.1687 −0.494417
\(424\) 0 0
\(425\) 25.9563 1.25906
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) −4.87862 −0.235542
\(430\) 0 0
\(431\) −24.6796 −1.18877 −0.594387 0.804179i \(-0.702605\pi\)
−0.594387 + 0.804179i \(0.702605\pi\)
\(432\) 0 0
\(433\) −3.02186 −0.145221 −0.0726107 0.997360i \(-0.523133\pi\)
−0.0726107 + 0.997360i \(0.523133\pi\)
\(434\) 0 0
\(435\) 18.6055 0.892065
\(436\) 0 0
\(437\) 28.8482 1.38000
\(438\) 0 0
\(439\) 32.1383 1.53388 0.766938 0.641721i \(-0.221779\pi\)
0.766938 + 0.641721i \(0.221779\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −30.8482 −1.46564 −0.732822 0.680420i \(-0.761797\pi\)
−0.732822 + 0.680420i \(0.761797\pi\)
\(444\) 0 0
\(445\) −44.5618 −2.11243
\(446\) 0 0
\(447\) −12.8786 −0.609138
\(448\) 0 0
\(449\) −5.70145 −0.269068 −0.134534 0.990909i \(-0.542954\pi\)
−0.134534 + 0.990909i \(0.542954\pi\)
\(450\) 0 0
\(451\) 0.485507 0.0228616
\(452\) 0 0
\(453\) −13.5582 −0.637020
\(454\) 0 0
\(455\) −3.38955 −0.158905
\(456\) 0 0
\(457\) −2.33732 −0.109335 −0.0546676 0.998505i \(-0.517410\pi\)
−0.0546676 + 0.998505i \(0.517410\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −24.9732 −1.16312 −0.581559 0.813504i \(-0.697557\pi\)
−0.581559 + 0.813504i \(0.697557\pi\)
\(462\) 0 0
\(463\) −38.2936 −1.77965 −0.889827 0.456298i \(-0.849175\pi\)
−0.889827 + 0.456298i \(0.849175\pi\)
\(464\) 0 0
\(465\) −21.2463 −0.985274
\(466\) 0 0
\(467\) 36.3373 1.68149 0.840745 0.541431i \(-0.182117\pi\)
0.840745 + 0.541431i \(0.182117\pi\)
\(468\) 0 0
\(469\) 6.77911 0.313030
\(470\) 0 0
\(471\) −14.5364 −0.669800
\(472\) 0 0
\(473\) 11.0656 0.508796
\(474\) 0 0
\(475\) 22.6408 1.03883
\(476\) 0 0
\(477\) −12.2682 −0.561721
\(478\) 0 0
\(479\) −14.9040 −0.680983 −0.340492 0.940248i \(-0.610593\pi\)
−0.340492 + 0.940248i \(0.610593\pi\)
\(480\) 0 0
\(481\) 8.77911 0.400293
\(482\) 0 0
\(483\) −8.26818 −0.376215
\(484\) 0 0
\(485\) 28.0254 1.27257
\(486\) 0 0
\(487\) −25.0219 −1.13385 −0.566924 0.823770i \(-0.691867\pi\)
−0.566924 + 0.823770i \(0.691867\pi\)
\(488\) 0 0
\(489\) 13.7572 0.622124
\(490\) 0 0
\(491\) 38.0508 1.71721 0.858605 0.512637i \(-0.171331\pi\)
0.858605 + 0.512637i \(0.171331\pi\)
\(492\) 0 0
\(493\) 21.9563 0.988861
\(494\) 0 0
\(495\) −16.5364 −0.743254
\(496\) 0 0
\(497\) −3.65773 −0.164072
\(498\) 0 0
\(499\) −1.95628 −0.0875752 −0.0437876 0.999041i \(-0.513942\pi\)
−0.0437876 + 0.999041i \(0.513942\pi\)
\(500\) 0 0
\(501\) 2.36769 0.105781
\(502\) 0 0
\(503\) 9.22089 0.411139 0.205570 0.978642i \(-0.434095\pi\)
0.205570 + 0.978642i \(0.434095\pi\)
\(504\) 0 0
\(505\) 4.13828 0.184151
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −16.1249 −0.714725 −0.357363 0.933966i \(-0.616324\pi\)
−0.357363 + 0.933966i \(0.616324\pi\)
\(510\) 0 0
\(511\) 11.0473 0.488703
\(512\) 0 0
\(513\) 3.48907 0.154046
\(514\) 0 0
\(515\) 13.5582 0.597446
\(516\) 0 0
\(517\) 49.6091 2.18180
\(518\) 0 0
\(519\) 0.199034 0.00873663
\(520\) 0 0
\(521\) −23.3155 −1.02147 −0.510734 0.859739i \(-0.670626\pi\)
−0.510734 + 0.859739i \(0.670626\pi\)
\(522\) 0 0
\(523\) 34.0946 1.49085 0.745426 0.666589i \(-0.232246\pi\)
0.745426 + 0.666589i \(0.232246\pi\)
\(524\) 0 0
\(525\) −6.48907 −0.283206
\(526\) 0 0
\(527\) −25.0727 −1.09218
\(528\) 0 0
\(529\) 45.3627 1.97229
\(530\) 0 0
\(531\) 6.87862 0.298507
\(532\) 0 0
\(533\) −0.0995171 −0.00431057
\(534\) 0 0
\(535\) −62.8300 −2.71638
\(536\) 0 0
\(537\) −23.0473 −0.994564
\(538\) 0 0
\(539\) −4.87862 −0.210137
\(540\) 0 0
\(541\) 20.0946 0.863933 0.431966 0.901890i \(-0.357820\pi\)
0.431966 + 0.901890i \(0.357820\pi\)
\(542\) 0 0
\(543\) −11.5582 −0.496010
\(544\) 0 0
\(545\) 12.7354 0.545524
\(546\) 0 0
\(547\) −10.8045 −0.461968 −0.230984 0.972958i \(-0.574195\pi\)
−0.230984 + 0.972958i \(0.574195\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 19.1518 0.815892
\(552\) 0 0
\(553\) −13.0473 −0.554827
\(554\) 0 0
\(555\) 29.7572 1.26312
\(556\) 0 0
\(557\) 45.6140 1.93273 0.966364 0.257179i \(-0.0827929\pi\)
0.966364 + 0.257179i \(0.0827929\pi\)
\(558\) 0 0
\(559\) −2.26818 −0.0959336
\(560\) 0 0
\(561\) −19.5145 −0.823903
\(562\) 0 0
\(563\) 32.0000 1.34864 0.674320 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) −64.5618 −2.71613
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −6.02542 −0.252599 −0.126299 0.991992i \(-0.540310\pi\)
−0.126299 + 0.991992i \(0.540310\pi\)
\(570\) 0 0
\(571\) 19.0910 0.798934 0.399467 0.916748i \(-0.369195\pi\)
0.399467 + 0.916748i \(0.369195\pi\)
\(572\) 0 0
\(573\) −16.7791 −0.700957
\(574\) 0 0
\(575\) 53.6528 2.23748
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 25.3155 1.05207
\(580\) 0 0
\(581\) 4.61045 0.191274
\(582\) 0 0
\(583\) 59.8518 2.47881
\(584\) 0 0
\(585\) 3.38955 0.140141
\(586\) 0 0
\(587\) 24.4623 1.00967 0.504833 0.863217i \(-0.331554\pi\)
0.504833 + 0.863217i \(0.331554\pi\)
\(588\) 0 0
\(589\) −21.8701 −0.901142
\(590\) 0 0
\(591\) 0.878623 0.0361417
\(592\) 0 0
\(593\) 16.9478 0.695961 0.347981 0.937502i \(-0.386867\pi\)
0.347981 + 0.937502i \(0.386867\pi\)
\(594\) 0 0
\(595\) −13.5582 −0.555833
\(596\) 0 0
\(597\) 13.5582 0.554901
\(598\) 0 0
\(599\) −33.2900 −1.36019 −0.680097 0.733122i \(-0.738062\pi\)
−0.680097 + 0.733122i \(0.738062\pi\)
\(600\) 0 0
\(601\) 33.1164 1.35085 0.675424 0.737430i \(-0.263961\pi\)
0.675424 + 0.737430i \(0.263961\pi\)
\(602\) 0 0
\(603\) −6.77911 −0.276067
\(604\) 0 0
\(605\) 43.3896 1.76404
\(606\) 0 0
\(607\) 26.5801 1.07885 0.539426 0.842033i \(-0.318641\pi\)
0.539426 + 0.842033i \(0.318641\pi\)
\(608\) 0 0
\(609\) −5.48907 −0.222428
\(610\) 0 0
\(611\) −10.1687 −0.411380
\(612\) 0 0
\(613\) 7.41993 0.299688 0.149844 0.988710i \(-0.452123\pi\)
0.149844 + 0.988710i \(0.452123\pi\)
\(614\) 0 0
\(615\) −0.337319 −0.0136020
\(616\) 0 0
\(617\) 24.5922 0.990043 0.495021 0.868881i \(-0.335160\pi\)
0.495021 + 0.868881i \(0.335160\pi\)
\(618\) 0 0
\(619\) 25.3592 1.01927 0.509636 0.860390i \(-0.329780\pi\)
0.509636 + 0.860390i \(0.329780\pi\)
\(620\) 0 0
\(621\) 8.26818 0.331790
\(622\) 0 0
\(623\) 13.1468 0.526715
\(624\) 0 0
\(625\) −15.3373 −0.613493
\(626\) 0 0
\(627\) −17.0219 −0.679788
\(628\) 0 0
\(629\) 35.1164 1.40018
\(630\) 0 0
\(631\) 26.7791 1.06606 0.533030 0.846097i \(-0.321053\pi\)
0.533030 + 0.846097i \(0.321053\pi\)
\(632\) 0 0
\(633\) 27.0036 1.07330
\(634\) 0 0
\(635\) −0.674637 −0.0267722
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 3.65773 0.144698
\(640\) 0 0
\(641\) −9.68810 −0.382657 −0.191329 0.981526i \(-0.561280\pi\)
−0.191329 + 0.981526i \(0.561280\pi\)
\(642\) 0 0
\(643\) 17.5582 0.692428 0.346214 0.938156i \(-0.387467\pi\)
0.346214 + 0.938156i \(0.387467\pi\)
\(644\) 0 0
\(645\) −7.68810 −0.302719
\(646\) 0 0
\(647\) 5.41993 0.213079 0.106540 0.994308i \(-0.466023\pi\)
0.106540 + 0.994308i \(0.466023\pi\)
\(648\) 0 0
\(649\) −33.5582 −1.31728
\(650\) 0 0
\(651\) 6.26818 0.245669
\(652\) 0 0
\(653\) 13.8010 0.540074 0.270037 0.962850i \(-0.412964\pi\)
0.270037 + 0.962850i \(0.412964\pi\)
\(654\) 0 0
\(655\) 5.95628 0.232731
\(656\) 0 0
\(657\) −11.0473 −0.430996
\(658\) 0 0
\(659\) 19.5328 0.760889 0.380445 0.924804i \(-0.375771\pi\)
0.380445 + 0.924804i \(0.375771\pi\)
\(660\) 0 0
\(661\) 3.58364 0.139387 0.0696936 0.997568i \(-0.477798\pi\)
0.0696936 + 0.997568i \(0.477798\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) −11.8264 −0.458608
\(666\) 0 0
\(667\) 45.3846 1.75730
\(668\) 0 0
\(669\) −29.0473 −1.12303
\(670\) 0 0
\(671\) −9.75725 −0.376674
\(672\) 0 0
\(673\) 8.75368 0.337430 0.168715 0.985665i \(-0.446038\pi\)
0.168715 + 0.985665i \(0.446038\pi\)
\(674\) 0 0
\(675\) 6.48907 0.249764
\(676\) 0 0
\(677\) 26.0946 1.00290 0.501448 0.865188i \(-0.332801\pi\)
0.501448 + 0.865188i \(0.332801\pi\)
\(678\) 0 0
\(679\) −8.26818 −0.317303
\(680\) 0 0
\(681\) −7.07766 −0.271216
\(682\) 0 0
\(683\) −18.9732 −0.725989 −0.362994 0.931791i \(-0.618246\pi\)
−0.362994 + 0.931791i \(0.618246\pi\)
\(684\) 0 0
\(685\) 20.6746 0.789937
\(686\) 0 0
\(687\) −9.80097 −0.373930
\(688\) 0 0
\(689\) −12.2682 −0.467380
\(690\) 0 0
\(691\) −7.00356 −0.266428 −0.133214 0.991087i \(-0.542530\pi\)
−0.133214 + 0.991087i \(0.542530\pi\)
\(692\) 0 0
\(693\) 4.87862 0.185324
\(694\) 0 0
\(695\) 23.6528 0.897201
\(696\) 0 0
\(697\) −0.398069 −0.0150779
\(698\) 0 0
\(699\) −2.51093 −0.0949721
\(700\) 0 0
\(701\) 9.82639 0.371138 0.185569 0.982631i \(-0.440587\pi\)
0.185569 + 0.982631i \(0.440587\pi\)
\(702\) 0 0
\(703\) 30.6309 1.15527
\(704\) 0 0
\(705\) −34.4672 −1.29811
\(706\) 0 0
\(707\) −1.22089 −0.0459164
\(708\) 0 0
\(709\) −32.8300 −1.23295 −0.616477 0.787373i \(-0.711441\pi\)
−0.616477 + 0.787373i \(0.711441\pi\)
\(710\) 0 0
\(711\) 13.0473 0.489311
\(712\) 0 0
\(713\) −51.8264 −1.94091
\(714\) 0 0
\(715\) −16.5364 −0.618425
\(716\) 0 0
\(717\) −0.342270 −0.0127823
\(718\) 0 0
\(719\) −33.8955 −1.26409 −0.632045 0.774932i \(-0.717784\pi\)
−0.632045 + 0.774932i \(0.717784\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) 18.5109 0.688429
\(724\) 0 0
\(725\) 35.6190 1.32286
\(726\) 0 0
\(727\) 34.0946 1.26450 0.632249 0.774765i \(-0.282132\pi\)
0.632249 + 0.774765i \(0.282132\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −9.07271 −0.335566
\(732\) 0 0
\(733\) −47.0473 −1.73773 −0.868866 0.495048i \(-0.835150\pi\)
−0.868866 + 0.495048i \(0.835150\pi\)
\(734\) 0 0
\(735\) 3.38955 0.125026
\(736\) 0 0
\(737\) 33.0727 1.21825
\(738\) 0 0
\(739\) −40.2499 −1.48062 −0.740308 0.672268i \(-0.765320\pi\)
−0.740308 + 0.672268i \(0.765320\pi\)
\(740\) 0 0
\(741\) 3.48907 0.128174
\(742\) 0 0
\(743\) −48.9295 −1.79505 −0.897524 0.440965i \(-0.854636\pi\)
−0.897524 + 0.440965i \(0.854636\pi\)
\(744\) 0 0
\(745\) −43.6528 −1.59931
\(746\) 0 0
\(747\) −4.61045 −0.168687
\(748\) 0 0
\(749\) 18.5364 0.677304
\(750\) 0 0
\(751\) 13.2463 0.483365 0.241682 0.970355i \(-0.422301\pi\)
0.241682 + 0.970355i \(0.422301\pi\)
\(752\) 0 0
\(753\) −14.7791 −0.538581
\(754\) 0 0
\(755\) −45.9563 −1.67252
\(756\) 0 0
\(757\) −49.0036 −1.78106 −0.890532 0.454920i \(-0.849668\pi\)
−0.890532 + 0.454920i \(0.849668\pi\)
\(758\) 0 0
\(759\) −40.3373 −1.46415
\(760\) 0 0
\(761\) 7.58859 0.275086 0.137543 0.990496i \(-0.456079\pi\)
0.137543 + 0.990496i \(0.456079\pi\)
\(762\) 0 0
\(763\) −3.75725 −0.136021
\(764\) 0 0
\(765\) 13.5582 0.490198
\(766\) 0 0
\(767\) 6.87862 0.248373
\(768\) 0 0
\(769\) 3.04728 0.109888 0.0549439 0.998489i \(-0.482502\pi\)
0.0549439 + 0.998489i \(0.482502\pi\)
\(770\) 0 0
\(771\) 22.7791 0.820369
\(772\) 0 0
\(773\) 29.8568 1.07387 0.536937 0.843623i \(-0.319581\pi\)
0.536937 + 0.843623i \(0.319581\pi\)
\(774\) 0 0
\(775\) −40.6746 −1.46108
\(776\) 0 0
\(777\) −8.77911 −0.314949
\(778\) 0 0
\(779\) −0.347222 −0.0124405
\(780\) 0 0
\(781\) −17.8447 −0.638533
\(782\) 0 0
\(783\) 5.48907 0.196163
\(784\) 0 0
\(785\) −49.2717 −1.75858
\(786\) 0 0
\(787\) −4.90900 −0.174987 −0.0874934 0.996165i \(-0.527886\pi\)
−0.0874934 + 0.996165i \(0.527886\pi\)
\(788\) 0 0
\(789\) 7.24632 0.257976
\(790\) 0 0
\(791\) 19.0473 0.677243
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) −41.5836 −1.47482
\(796\) 0 0
\(797\) −31.6528 −1.12120 −0.560599 0.828087i \(-0.689429\pi\)
−0.560599 + 0.828087i \(0.689429\pi\)
\(798\) 0 0
\(799\) −40.6746 −1.43897
\(800\) 0 0
\(801\) −13.1468 −0.464519
\(802\) 0 0
\(803\) 53.8955 1.90193
\(804\) 0 0
\(805\) −28.0254 −0.987766
\(806\) 0 0
\(807\) −1.75725 −0.0618580
\(808\) 0 0
\(809\) 14.0254 0.493108 0.246554 0.969129i \(-0.420702\pi\)
0.246554 + 0.969129i \(0.420702\pi\)
\(810\) 0 0
\(811\) 47.5145 1.66846 0.834230 0.551417i \(-0.185913\pi\)
0.834230 + 0.551417i \(0.185913\pi\)
\(812\) 0 0
\(813\) −13.7572 −0.482488
\(814\) 0 0
\(815\) 46.6309 1.63341
\(816\) 0 0
\(817\) −7.91383 −0.276870
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 9.07766 0.316812 0.158406 0.987374i \(-0.449364\pi\)
0.158406 + 0.987374i \(0.449364\pi\)
\(822\) 0 0
\(823\) 53.0727 1.85000 0.924999 0.379969i \(-0.124065\pi\)
0.924999 + 0.379969i \(0.124065\pi\)
\(824\) 0 0
\(825\) −31.6577 −1.10218
\(826\) 0 0
\(827\) 19.1722 0.666684 0.333342 0.942806i \(-0.391824\pi\)
0.333342 + 0.942806i \(0.391824\pi\)
\(828\) 0 0
\(829\) −12.0437 −0.418296 −0.209148 0.977884i \(-0.567069\pi\)
−0.209148 + 0.977884i \(0.567069\pi\)
\(830\) 0 0
\(831\) −11.2463 −0.390130
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 8.02542 0.277731
\(836\) 0 0
\(837\) −6.26818 −0.216660
\(838\) 0 0
\(839\) 43.9513 1.51737 0.758684 0.651459i \(-0.225843\pi\)
0.758684 + 0.651459i \(0.225843\pi\)
\(840\) 0 0
\(841\) 1.12989 0.0389618
\(842\) 0 0
\(843\) −30.4368 −1.04830
\(844\) 0 0
\(845\) 3.38955 0.116604
\(846\) 0 0
\(847\) −12.8010 −0.439846
\(848\) 0 0
\(849\) 17.5582 0.602596
\(850\) 0 0
\(851\) 72.5872 2.48826
\(852\) 0 0
\(853\) 6.17361 0.211380 0.105690 0.994399i \(-0.466295\pi\)
0.105690 + 0.994399i \(0.466295\pi\)
\(854\) 0 0
\(855\) 11.8264 0.404454
\(856\) 0 0
\(857\) −51.3663 −1.75464 −0.877320 0.479906i \(-0.840671\pi\)
−0.877320 + 0.479906i \(0.840671\pi\)
\(858\) 0 0
\(859\) 20.0508 0.684126 0.342063 0.939677i \(-0.388874\pi\)
0.342063 + 0.939677i \(0.388874\pi\)
\(860\) 0 0
\(861\) 0.0995171 0.00339153
\(862\) 0 0
\(863\) −17.6140 −0.599588 −0.299794 0.954004i \(-0.596918\pi\)
−0.299794 + 0.954004i \(0.596918\pi\)
\(864\) 0 0
\(865\) 0.674637 0.0229384
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −63.6528 −2.15927
\(870\) 0 0
\(871\) −6.77911 −0.229701
\(872\) 0 0
\(873\) 8.26818 0.279835
\(874\) 0 0
\(875\) −5.04728 −0.170629
\(876\) 0 0
\(877\) −46.5364 −1.57142 −0.785710 0.618594i \(-0.787702\pi\)
−0.785710 + 0.618594i \(0.787702\pi\)
\(878\) 0 0
\(879\) 15.1905 0.512363
\(880\) 0 0
\(881\) −24.7354 −0.833356 −0.416678 0.909054i \(-0.636806\pi\)
−0.416678 + 0.909054i \(0.636806\pi\)
\(882\) 0 0
\(883\) 46.8300 1.57595 0.787977 0.615705i \(-0.211129\pi\)
0.787977 + 0.615705i \(0.211129\pi\)
\(884\) 0 0
\(885\) 23.3155 0.783741
\(886\) 0 0
\(887\) −29.0727 −0.976166 −0.488083 0.872797i \(-0.662304\pi\)
−0.488083 + 0.872797i \(0.662304\pi\)
\(888\) 0 0
\(889\) 0.199034 0.00667540
\(890\) 0 0
\(891\) −4.87862 −0.163440
\(892\) 0 0
\(893\) −35.4792 −1.18727
\(894\) 0 0
\(895\) −78.1200 −2.61126
\(896\) 0 0
\(897\) 8.26818 0.276066
\(898\) 0 0
\(899\) −34.4065 −1.14752
\(900\) 0 0
\(901\) −49.0727 −1.63485
\(902\) 0 0
\(903\) 2.26818 0.0754802
\(904\) 0 0
\(905\) −39.1772 −1.30229
\(906\) 0 0
\(907\) 32.5618 1.08120 0.540598 0.841281i \(-0.318198\pi\)
0.540598 + 0.841281i \(0.318198\pi\)
\(908\) 0 0
\(909\) 1.22089 0.0404945
\(910\) 0 0
\(911\) −16.4672 −0.545583 −0.272792 0.962073i \(-0.587947\pi\)
−0.272792 + 0.962073i \(0.587947\pi\)
\(912\) 0 0
\(913\) 22.4926 0.744398
\(914\) 0 0
\(915\) 6.77911 0.224110
\(916\) 0 0
\(917\) −1.75725 −0.0580294
\(918\) 0 0
\(919\) −18.8300 −0.621143 −0.310571 0.950550i \(-0.600520\pi\)
−0.310571 + 0.950550i \(0.600520\pi\)
\(920\) 0 0
\(921\) −3.29004 −0.108410
\(922\) 0 0
\(923\) 3.65773 0.120396
\(924\) 0 0
\(925\) 56.9682 1.87310
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) −41.8822 −1.37411 −0.687055 0.726605i \(-0.741097\pi\)
−0.687055 + 0.726605i \(0.741097\pi\)
\(930\) 0 0
\(931\) 3.48907 0.114350
\(932\) 0 0
\(933\) 10.7791 0.352892
\(934\) 0 0
\(935\) −66.1454 −2.16319
\(936\) 0 0
\(937\) −36.4926 −1.19216 −0.596081 0.802924i \(-0.703276\pi\)
−0.596081 + 0.802924i \(0.703276\pi\)
\(938\) 0 0
\(939\) 25.1164 0.819644
\(940\) 0 0
\(941\) 18.1687 0.592281 0.296141 0.955144i \(-0.404300\pi\)
0.296141 + 0.955144i \(0.404300\pi\)
\(942\) 0 0
\(943\) −0.822825 −0.0267949
\(944\) 0 0
\(945\) −3.38955 −0.110262
\(946\) 0 0
\(947\) 30.0388 0.976129 0.488064 0.872808i \(-0.337703\pi\)
0.488064 + 0.872808i \(0.337703\pi\)
\(948\) 0 0
\(949\) −11.0473 −0.358610
\(950\) 0 0
\(951\) 23.8568 0.773609
\(952\) 0 0
\(953\) 17.1418 0.555279 0.277639 0.960685i \(-0.410448\pi\)
0.277639 + 0.960685i \(0.410448\pi\)
\(954\) 0 0
\(955\) −56.8737 −1.84039
\(956\) 0 0
\(957\) −26.7791 −0.865646
\(958\) 0 0
\(959\) −6.09952 −0.196964
\(960\) 0 0
\(961\) 8.29004 0.267421
\(962\) 0 0
\(963\) −18.5364 −0.597326
\(964\) 0 0
\(965\) 85.8081 2.76226
\(966\) 0 0
\(967\) 39.3663 1.26594 0.632968 0.774178i \(-0.281837\pi\)
0.632968 + 0.774178i \(0.281837\pi\)
\(968\) 0 0
\(969\) 13.9563 0.448340
\(970\) 0 0
\(971\) 42.5801 1.36646 0.683230 0.730203i \(-0.260575\pi\)
0.683230 + 0.730203i \(0.260575\pi\)
\(972\) 0 0
\(973\) −6.97814 −0.223709
\(974\) 0 0
\(975\) 6.48907 0.207817
\(976\) 0 0
\(977\) 9.36413 0.299585 0.149793 0.988717i \(-0.452139\pi\)
0.149793 + 0.988717i \(0.452139\pi\)
\(978\) 0 0
\(979\) 64.1383 2.04987
\(980\) 0 0
\(981\) 3.75725 0.119960
\(982\) 0 0
\(983\) −30.9915 −0.988475 −0.494237 0.869327i \(-0.664553\pi\)
−0.494237 + 0.869327i \(0.664553\pi\)
\(984\) 0 0
\(985\) 2.97814 0.0948914
\(986\) 0 0
\(987\) 10.1687 0.323672
\(988\) 0 0
\(989\) −18.7537 −0.596332
\(990\) 0 0
\(991\) −45.5582 −1.44720 −0.723602 0.690217i \(-0.757515\pi\)
−0.723602 + 0.690217i \(0.757515\pi\)
\(992\) 0 0
\(993\) −2.97814 −0.0945084
\(994\) 0 0
\(995\) 45.9563 1.45691
\(996\) 0 0
\(997\) −39.6091 −1.25443 −0.627216 0.778846i \(-0.715806\pi\)
−0.627216 + 0.778846i \(0.715806\pi\)
\(998\) 0 0
\(999\) 8.77911 0.277759
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 1092.2.a.h.1.3 3
3.2 odd 2 3276.2.a.r.1.1 3
4.3 odd 2 4368.2.a.bn.1.3 3
7.6 odd 2 7644.2.a.t.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1092.2.a.h.1.3 3 1.1 even 1 trivial
3276.2.a.r.1.1 3 3.2 odd 2
4368.2.a.bn.1.3 3 4.3 odd 2
7644.2.a.t.1.1 3 7.6 odd 2