Properties

Label 520.2.a.e.1.2
Level $520$
Weight $2$
Character 520.1
Self dual yes
Analytic conductor $4.152$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(1,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, 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("520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.732051 q^{3} -1.00000 q^{5} +3.46410 q^{7} -2.46410 q^{9} +O(q^{10})\) \(q+0.732051 q^{3} -1.00000 q^{5} +3.46410 q^{7} -2.46410 q^{9} +2.73205 q^{11} +1.00000 q^{13} -0.732051 q^{15} +7.46410 q^{17} -2.73205 q^{19} +2.53590 q^{21} -0.732051 q^{23} +1.00000 q^{25} -4.00000 q^{27} +9.46410 q^{29} -0.196152 q^{31} +2.00000 q^{33} -3.46410 q^{35} +0.732051 q^{39} +0.535898 q^{41} -7.26795 q^{43} +2.46410 q^{45} +4.53590 q^{47} +5.00000 q^{49} +5.46410 q^{51} -0.535898 q^{53} -2.73205 q^{55} -2.00000 q^{57} +5.66025 q^{59} -12.3923 q^{61} -8.53590 q^{63} -1.00000 q^{65} -12.9282 q^{67} -0.535898 q^{69} +9.26795 q^{71} -2.92820 q^{73} +0.732051 q^{75} +9.46410 q^{77} -6.53590 q^{79} +4.46410 q^{81} -10.3923 q^{83} -7.46410 q^{85} +6.92820 q^{87} -12.9282 q^{89} +3.46410 q^{91} -0.143594 q^{93} +2.73205 q^{95} +15.8564 q^{97} -6.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} + 8 q^{17} - 2 q^{19} + 12 q^{21} + 2 q^{23} + 2 q^{25} - 8 q^{27} + 12 q^{29} + 10 q^{31} + 4 q^{33} - 2 q^{39} + 8 q^{41} - 18 q^{43} - 2 q^{45} + 16 q^{47} + 10 q^{49} + 4 q^{51} - 8 q^{53} - 2 q^{55} - 4 q^{57} - 6 q^{59} - 4 q^{61} - 24 q^{63} - 2 q^{65} - 12 q^{67} - 8 q^{69} + 22 q^{71} + 8 q^{73} - 2 q^{75} + 12 q^{77} - 20 q^{79} + 2 q^{81} - 8 q^{85} - 12 q^{89} - 28 q^{93} + 2 q^{95} + 4 q^{97} - 10 q^{99}+O(q^{100}) \) 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
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 2.73205 0.823744 0.411872 0.911242i \(-0.364875\pi\)
0.411872 + 0.911242i \(0.364875\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −0.732051 −0.189015
\(16\) 0 0
\(17\) 7.46410 1.81031 0.905155 0.425081i \(-0.139754\pi\)
0.905155 + 0.425081i \(0.139754\pi\)
\(18\) 0 0
\(19\) −2.73205 −0.626775 −0.313388 0.949625i \(-0.601464\pi\)
−0.313388 + 0.949625i \(0.601464\pi\)
\(20\) 0 0
\(21\) 2.53590 0.553378
\(22\) 0 0
\(23\) −0.732051 −0.152643 −0.0763216 0.997083i \(-0.524318\pi\)
−0.0763216 + 0.997083i \(0.524318\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 9.46410 1.75744 0.878720 0.477338i \(-0.158398\pi\)
0.878720 + 0.477338i \(0.158398\pi\)
\(30\) 0 0
\(31\) −0.196152 −0.0352300 −0.0176150 0.999845i \(-0.505607\pi\)
−0.0176150 + 0.999845i \(0.505607\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0.732051 0.117222
\(40\) 0 0
\(41\) 0.535898 0.0836933 0.0418466 0.999124i \(-0.486676\pi\)
0.0418466 + 0.999124i \(0.486676\pi\)
\(42\) 0 0
\(43\) −7.26795 −1.10835 −0.554176 0.832400i \(-0.686967\pi\)
−0.554176 + 0.832400i \(0.686967\pi\)
\(44\) 0 0
\(45\) 2.46410 0.367327
\(46\) 0 0
\(47\) 4.53590 0.661629 0.330814 0.943696i \(-0.392677\pi\)
0.330814 + 0.943696i \(0.392677\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 5.46410 0.765127
\(52\) 0 0
\(53\) −0.535898 −0.0736113 −0.0368057 0.999322i \(-0.511718\pi\)
−0.0368057 + 0.999322i \(0.511718\pi\)
\(54\) 0 0
\(55\) −2.73205 −0.368390
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 5.66025 0.736902 0.368451 0.929647i \(-0.379888\pi\)
0.368451 + 0.929647i \(0.379888\pi\)
\(60\) 0 0
\(61\) −12.3923 −1.58667 −0.793336 0.608784i \(-0.791658\pi\)
−0.793336 + 0.608784i \(0.791658\pi\)
\(62\) 0 0
\(63\) −8.53590 −1.07542
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −12.9282 −1.57943 −0.789716 0.613473i \(-0.789772\pi\)
−0.789716 + 0.613473i \(0.789772\pi\)
\(68\) 0 0
\(69\) −0.535898 −0.0645146
\(70\) 0 0
\(71\) 9.26795 1.09990 0.549952 0.835197i \(-0.314646\pi\)
0.549952 + 0.835197i \(0.314646\pi\)
\(72\) 0 0
\(73\) −2.92820 −0.342720 −0.171360 0.985208i \(-0.554816\pi\)
−0.171360 + 0.985208i \(0.554816\pi\)
\(74\) 0 0
\(75\) 0.732051 0.0845299
\(76\) 0 0
\(77\) 9.46410 1.07853
\(78\) 0 0
\(79\) −6.53590 −0.735346 −0.367673 0.929955i \(-0.619845\pi\)
−0.367673 + 0.929955i \(0.619845\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −10.3923 −1.14070 −0.570352 0.821401i \(-0.693193\pi\)
−0.570352 + 0.821401i \(0.693193\pi\)
\(84\) 0 0
\(85\) −7.46410 −0.809595
\(86\) 0 0
\(87\) 6.92820 0.742781
\(88\) 0 0
\(89\) −12.9282 −1.37039 −0.685193 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) 3.46410 0.363137
\(92\) 0 0
\(93\) −0.143594 −0.0148900
\(94\) 0 0
\(95\) 2.73205 0.280302
\(96\) 0 0
\(97\) 15.8564 1.60997 0.804987 0.593292i \(-0.202172\pi\)
0.804987 + 0.593292i \(0.202172\pi\)
\(98\) 0 0
\(99\) −6.73205 −0.676597
\(100\) 0 0
\(101\) 4.92820 0.490375 0.245187 0.969476i \(-0.421151\pi\)
0.245187 + 0.969476i \(0.421151\pi\)
\(102\) 0 0
\(103\) −6.19615 −0.610525 −0.305263 0.952268i \(-0.598744\pi\)
−0.305263 + 0.952268i \(0.598744\pi\)
\(104\) 0 0
\(105\) −2.53590 −0.247478
\(106\) 0 0
\(107\) −20.0526 −1.93855 −0.969277 0.245972i \(-0.920893\pi\)
−0.969277 + 0.245972i \(0.920893\pi\)
\(108\) 0 0
\(109\) 15.8564 1.51877 0.759384 0.650643i \(-0.225500\pi\)
0.759384 + 0.650643i \(0.225500\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.3923 −1.73020 −0.865101 0.501597i \(-0.832746\pi\)
−0.865101 + 0.501597i \(0.832746\pi\)
\(114\) 0 0
\(115\) 0.732051 0.0682641
\(116\) 0 0
\(117\) −2.46410 −0.227806
\(118\) 0 0
\(119\) 25.8564 2.37025
\(120\) 0 0
\(121\) −3.53590 −0.321445
\(122\) 0 0
\(123\) 0.392305 0.0353729
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.80385 −0.160066 −0.0800328 0.996792i \(-0.525503\pi\)
−0.0800328 + 0.996792i \(0.525503\pi\)
\(128\) 0 0
\(129\) −5.32051 −0.468445
\(130\) 0 0
\(131\) −2.92820 −0.255838 −0.127919 0.991785i \(-0.540830\pi\)
−0.127919 + 0.991785i \(0.540830\pi\)
\(132\) 0 0
\(133\) −9.46410 −0.820642
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 8.92820 0.762788 0.381394 0.924413i \(-0.375444\pi\)
0.381394 + 0.924413i \(0.375444\pi\)
\(138\) 0 0
\(139\) −13.4641 −1.14201 −0.571005 0.820947i \(-0.693446\pi\)
−0.571005 + 0.820947i \(0.693446\pi\)
\(140\) 0 0
\(141\) 3.32051 0.279637
\(142\) 0 0
\(143\) 2.73205 0.228466
\(144\) 0 0
\(145\) −9.46410 −0.785951
\(146\) 0 0
\(147\) 3.66025 0.301893
\(148\) 0 0
\(149\) −7.85641 −0.643622 −0.321811 0.946804i \(-0.604292\pi\)
−0.321811 + 0.946804i \(0.604292\pi\)
\(150\) 0 0
\(151\) 11.8038 0.960583 0.480292 0.877109i \(-0.340531\pi\)
0.480292 + 0.877109i \(0.340531\pi\)
\(152\) 0 0
\(153\) −18.3923 −1.48693
\(154\) 0 0
\(155\) 0.196152 0.0157553
\(156\) 0 0
\(157\) −7.85641 −0.627009 −0.313505 0.949587i \(-0.601503\pi\)
−0.313505 + 0.949587i \(0.601503\pi\)
\(158\) 0 0
\(159\) −0.392305 −0.0311118
\(160\) 0 0
\(161\) −2.53590 −0.199857
\(162\) 0 0
\(163\) −8.92820 −0.699311 −0.349655 0.936878i \(-0.613701\pi\)
−0.349655 + 0.936878i \(0.613701\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) −21.3205 −1.64983 −0.824915 0.565256i \(-0.808777\pi\)
−0.824915 + 0.565256i \(0.808777\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 6.73205 0.514813
\(172\) 0 0
\(173\) −2.39230 −0.181884 −0.0909418 0.995856i \(-0.528988\pi\)
−0.0909418 + 0.995856i \(0.528988\pi\)
\(174\) 0 0
\(175\) 3.46410 0.261861
\(176\) 0 0
\(177\) 4.14359 0.311452
\(178\) 0 0
\(179\) −21.8564 −1.63362 −0.816812 0.576904i \(-0.804261\pi\)
−0.816812 + 0.576904i \(0.804261\pi\)
\(180\) 0 0
\(181\) 6.53590 0.485810 0.242905 0.970050i \(-0.421900\pi\)
0.242905 + 0.970050i \(0.421900\pi\)
\(182\) 0 0
\(183\) −9.07180 −0.670607
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.3923 1.49123
\(188\) 0 0
\(189\) −13.8564 −1.00791
\(190\) 0 0
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) −0.732051 −0.0524232
\(196\) 0 0
\(197\) 8.92820 0.636108 0.318054 0.948073i \(-0.396971\pi\)
0.318054 + 0.948073i \(0.396971\pi\)
\(198\) 0 0
\(199\) −6.92820 −0.491127 −0.245564 0.969380i \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(200\) 0 0
\(201\) −9.46410 −0.667546
\(202\) 0 0
\(203\) 32.7846 2.30103
\(204\) 0 0
\(205\) −0.535898 −0.0374288
\(206\) 0 0
\(207\) 1.80385 0.125376
\(208\) 0 0
\(209\) −7.46410 −0.516303
\(210\) 0 0
\(211\) 2.92820 0.201586 0.100793 0.994907i \(-0.467862\pi\)
0.100793 + 0.994907i \(0.467862\pi\)
\(212\) 0 0
\(213\) 6.78461 0.464874
\(214\) 0 0
\(215\) 7.26795 0.495670
\(216\) 0 0
\(217\) −0.679492 −0.0461269
\(218\) 0 0
\(219\) −2.14359 −0.144851
\(220\) 0 0
\(221\) 7.46410 0.502090
\(222\) 0 0
\(223\) 25.3205 1.69559 0.847793 0.530327i \(-0.177931\pi\)
0.847793 + 0.530327i \(0.177931\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 19.4641 1.28622 0.643112 0.765772i \(-0.277643\pi\)
0.643112 + 0.765772i \(0.277643\pi\)
\(230\) 0 0
\(231\) 6.92820 0.455842
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −4.53590 −0.295889
\(236\) 0 0
\(237\) −4.78461 −0.310794
\(238\) 0 0
\(239\) 25.2679 1.63445 0.817224 0.576320i \(-0.195512\pi\)
0.817224 + 0.576320i \(0.195512\pi\)
\(240\) 0 0
\(241\) 20.2487 1.30433 0.652167 0.758075i \(-0.273860\pi\)
0.652167 + 0.758075i \(0.273860\pi\)
\(242\) 0 0
\(243\) 15.2679 0.979439
\(244\) 0 0
\(245\) −5.00000 −0.319438
\(246\) 0 0
\(247\) −2.73205 −0.173836
\(248\) 0 0
\(249\) −7.60770 −0.482118
\(250\) 0 0
\(251\) 23.3205 1.47198 0.735989 0.676994i \(-0.236718\pi\)
0.735989 + 0.676994i \(0.236718\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) −5.46410 −0.342175
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −23.3205 −1.44350
\(262\) 0 0
\(263\) −11.6603 −0.719002 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(264\) 0 0
\(265\) 0.535898 0.0329200
\(266\) 0 0
\(267\) −9.46410 −0.579194
\(268\) 0 0
\(269\) −0.143594 −0.00875505 −0.00437753 0.999990i \(-0.501393\pi\)
−0.00437753 + 0.999990i \(0.501393\pi\)
\(270\) 0 0
\(271\) −23.1244 −1.40470 −0.702352 0.711830i \(-0.747867\pi\)
−0.702352 + 0.711830i \(0.747867\pi\)
\(272\) 0 0
\(273\) 2.53590 0.153480
\(274\) 0 0
\(275\) 2.73205 0.164749
\(276\) 0 0
\(277\) −21.3205 −1.28103 −0.640513 0.767948i \(-0.721278\pi\)
−0.640513 + 0.767948i \(0.721278\pi\)
\(278\) 0 0
\(279\) 0.483340 0.0289368
\(280\) 0 0
\(281\) −4.53590 −0.270589 −0.135295 0.990805i \(-0.543198\pi\)
−0.135295 + 0.990805i \(0.543198\pi\)
\(282\) 0 0
\(283\) −24.0526 −1.42978 −0.714888 0.699239i \(-0.753522\pi\)
−0.714888 + 0.699239i \(0.753522\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 1.85641 0.109580
\(288\) 0 0
\(289\) 38.7128 2.27722
\(290\) 0 0
\(291\) 11.6077 0.680455
\(292\) 0 0
\(293\) 6.92820 0.404750 0.202375 0.979308i \(-0.435134\pi\)
0.202375 + 0.979308i \(0.435134\pi\)
\(294\) 0 0
\(295\) −5.66025 −0.329553
\(296\) 0 0
\(297\) −10.9282 −0.634119
\(298\) 0 0
\(299\) −0.732051 −0.0423356
\(300\) 0 0
\(301\) −25.1769 −1.45117
\(302\) 0 0
\(303\) 3.60770 0.207257
\(304\) 0 0
\(305\) 12.3923 0.709581
\(306\) 0 0
\(307\) −16.5359 −0.943754 −0.471877 0.881665i \(-0.656423\pi\)
−0.471877 + 0.881665i \(0.656423\pi\)
\(308\) 0 0
\(309\) −4.53590 −0.258038
\(310\) 0 0
\(311\) 31.3205 1.77602 0.888012 0.459821i \(-0.152086\pi\)
0.888012 + 0.459821i \(0.152086\pi\)
\(312\) 0 0
\(313\) 10.3923 0.587408 0.293704 0.955896i \(-0.405112\pi\)
0.293704 + 0.955896i \(0.405112\pi\)
\(314\) 0 0
\(315\) 8.53590 0.480943
\(316\) 0 0
\(317\) −1.85641 −0.104266 −0.0521331 0.998640i \(-0.516602\pi\)
−0.0521331 + 0.998640i \(0.516602\pi\)
\(318\) 0 0
\(319\) 25.8564 1.44768
\(320\) 0 0
\(321\) −14.6795 −0.819329
\(322\) 0 0
\(323\) −20.3923 −1.13466
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 11.6077 0.641907
\(328\) 0 0
\(329\) 15.7128 0.866275
\(330\) 0 0
\(331\) −8.58846 −0.472064 −0.236032 0.971745i \(-0.575847\pi\)
−0.236032 + 0.971745i \(0.575847\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.9282 0.706343
\(336\) 0 0
\(337\) 22.3923 1.21979 0.609893 0.792484i \(-0.291212\pi\)
0.609893 + 0.792484i \(0.291212\pi\)
\(338\) 0 0
\(339\) −13.4641 −0.731270
\(340\) 0 0
\(341\) −0.535898 −0.0290205
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) 0.535898 0.0288518
\(346\) 0 0
\(347\) 2.19615 0.117896 0.0589478 0.998261i \(-0.481225\pi\)
0.0589478 + 0.998261i \(0.481225\pi\)
\(348\) 0 0
\(349\) 12.5359 0.671031 0.335516 0.942035i \(-0.391089\pi\)
0.335516 + 0.942035i \(0.391089\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −6.92820 −0.368751 −0.184376 0.982856i \(-0.559026\pi\)
−0.184376 + 0.982856i \(0.559026\pi\)
\(354\) 0 0
\(355\) −9.26795 −0.491892
\(356\) 0 0
\(357\) 18.9282 1.00179
\(358\) 0 0
\(359\) 11.1244 0.587121 0.293561 0.955940i \(-0.405160\pi\)
0.293561 + 0.955940i \(0.405160\pi\)
\(360\) 0 0
\(361\) −11.5359 −0.607153
\(362\) 0 0
\(363\) −2.58846 −0.135859
\(364\) 0 0
\(365\) 2.92820 0.153269
\(366\) 0 0
\(367\) −1.41154 −0.0736819 −0.0368410 0.999321i \(-0.511729\pi\)
−0.0368410 + 0.999321i \(0.511729\pi\)
\(368\) 0 0
\(369\) −1.32051 −0.0687429
\(370\) 0 0
\(371\) −1.85641 −0.0963798
\(372\) 0 0
\(373\) −7.85641 −0.406789 −0.203395 0.979097i \(-0.565197\pi\)
−0.203395 + 0.979097i \(0.565197\pi\)
\(374\) 0 0
\(375\) −0.732051 −0.0378029
\(376\) 0 0
\(377\) 9.46410 0.487426
\(378\) 0 0
\(379\) −20.1962 −1.03741 −0.518703 0.854954i \(-0.673585\pi\)
−0.518703 + 0.854954i \(0.673585\pi\)
\(380\) 0 0
\(381\) −1.32051 −0.0676517
\(382\) 0 0
\(383\) −1.60770 −0.0821494 −0.0410747 0.999156i \(-0.513078\pi\)
−0.0410747 + 0.999156i \(0.513078\pi\)
\(384\) 0 0
\(385\) −9.46410 −0.482335
\(386\) 0 0
\(387\) 17.9090 0.910364
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −5.46410 −0.276331
\(392\) 0 0
\(393\) −2.14359 −0.108130
\(394\) 0 0
\(395\) 6.53590 0.328857
\(396\) 0 0
\(397\) −5.07180 −0.254546 −0.127273 0.991868i \(-0.540622\pi\)
−0.127273 + 0.991868i \(0.540622\pi\)
\(398\) 0 0
\(399\) −6.92820 −0.346844
\(400\) 0 0
\(401\) −10.7846 −0.538558 −0.269279 0.963062i \(-0.586785\pi\)
−0.269279 + 0.963062i \(0.586785\pi\)
\(402\) 0 0
\(403\) −0.196152 −0.00977105
\(404\) 0 0
\(405\) −4.46410 −0.221823
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 33.3205 1.64759 0.823797 0.566886i \(-0.191852\pi\)
0.823797 + 0.566886i \(0.191852\pi\)
\(410\) 0 0
\(411\) 6.53590 0.322392
\(412\) 0 0
\(413\) 19.6077 0.964832
\(414\) 0 0
\(415\) 10.3923 0.510138
\(416\) 0 0
\(417\) −9.85641 −0.482670
\(418\) 0 0
\(419\) −25.4641 −1.24400 −0.622001 0.783016i \(-0.713680\pi\)
−0.622001 + 0.783016i \(0.713680\pi\)
\(420\) 0 0
\(421\) −19.0718 −0.929503 −0.464751 0.885441i \(-0.653856\pi\)
−0.464751 + 0.885441i \(0.653856\pi\)
\(422\) 0 0
\(423\) −11.1769 −0.543440
\(424\) 0 0
\(425\) 7.46410 0.362062
\(426\) 0 0
\(427\) −42.9282 −2.07744
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −22.0526 −1.06223 −0.531117 0.847298i \(-0.678228\pi\)
−0.531117 + 0.847298i \(0.678228\pi\)
\(432\) 0 0
\(433\) 15.0718 0.724304 0.362152 0.932119i \(-0.382042\pi\)
0.362152 + 0.932119i \(0.382042\pi\)
\(434\) 0 0
\(435\) −6.92820 −0.332182
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) −13.8564 −0.661330 −0.330665 0.943748i \(-0.607273\pi\)
−0.330665 + 0.943748i \(0.607273\pi\)
\(440\) 0 0
\(441\) −12.3205 −0.586691
\(442\) 0 0
\(443\) 28.4449 1.35146 0.675728 0.737151i \(-0.263829\pi\)
0.675728 + 0.737151i \(0.263829\pi\)
\(444\) 0 0
\(445\) 12.9282 0.612856
\(446\) 0 0
\(447\) −5.75129 −0.272027
\(448\) 0 0
\(449\) −14.3923 −0.679215 −0.339607 0.940567i \(-0.610294\pi\)
−0.339607 + 0.940567i \(0.610294\pi\)
\(450\) 0 0
\(451\) 1.46410 0.0689419
\(452\) 0 0
\(453\) 8.64102 0.405990
\(454\) 0 0
\(455\) −3.46410 −0.162400
\(456\) 0 0
\(457\) −30.7846 −1.44004 −0.720022 0.693952i \(-0.755868\pi\)
−0.720022 + 0.693952i \(0.755868\pi\)
\(458\) 0 0
\(459\) −29.8564 −1.39358
\(460\) 0 0
\(461\) −12.2487 −0.570479 −0.285240 0.958456i \(-0.592073\pi\)
−0.285240 + 0.958456i \(0.592073\pi\)
\(462\) 0 0
\(463\) 27.8564 1.29460 0.647298 0.762237i \(-0.275899\pi\)
0.647298 + 0.762237i \(0.275899\pi\)
\(464\) 0 0
\(465\) 0.143594 0.00665899
\(466\) 0 0
\(467\) 18.5885 0.860171 0.430086 0.902788i \(-0.358483\pi\)
0.430086 + 0.902788i \(0.358483\pi\)
\(468\) 0 0
\(469\) −44.7846 −2.06796
\(470\) 0 0
\(471\) −5.75129 −0.265005
\(472\) 0 0
\(473\) −19.8564 −0.912999
\(474\) 0 0
\(475\) −2.73205 −0.125355
\(476\) 0 0
\(477\) 1.32051 0.0604619
\(478\) 0 0
\(479\) 11.1244 0.508285 0.254142 0.967167i \(-0.418207\pi\)
0.254142 + 0.967167i \(0.418207\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −1.85641 −0.0844694
\(484\) 0 0
\(485\) −15.8564 −0.720002
\(486\) 0 0
\(487\) −39.8564 −1.80607 −0.903033 0.429571i \(-0.858665\pi\)
−0.903033 + 0.429571i \(0.858665\pi\)
\(488\) 0 0
\(489\) −6.53590 −0.295564
\(490\) 0 0
\(491\) −33.4641 −1.51021 −0.755107 0.655602i \(-0.772415\pi\)
−0.755107 + 0.655602i \(0.772415\pi\)
\(492\) 0 0
\(493\) 70.6410 3.18151
\(494\) 0 0
\(495\) 6.73205 0.302583
\(496\) 0 0
\(497\) 32.1051 1.44011
\(498\) 0 0
\(499\) 24.1962 1.08317 0.541584 0.840646i \(-0.317825\pi\)
0.541584 + 0.840646i \(0.317825\pi\)
\(500\) 0 0
\(501\) −15.6077 −0.697300
\(502\) 0 0
\(503\) −41.1244 −1.83364 −0.916822 0.399296i \(-0.869255\pi\)
−0.916822 + 0.399296i \(0.869255\pi\)
\(504\) 0 0
\(505\) −4.92820 −0.219302
\(506\) 0 0
\(507\) 0.732051 0.0325115
\(508\) 0 0
\(509\) −3.46410 −0.153544 −0.0767718 0.997049i \(-0.524461\pi\)
−0.0767718 + 0.997049i \(0.524461\pi\)
\(510\) 0 0
\(511\) −10.1436 −0.448726
\(512\) 0 0
\(513\) 10.9282 0.482492
\(514\) 0 0
\(515\) 6.19615 0.273035
\(516\) 0 0
\(517\) 12.3923 0.545013
\(518\) 0 0
\(519\) −1.75129 −0.0768730
\(520\) 0 0
\(521\) 8.39230 0.367674 0.183837 0.982957i \(-0.441148\pi\)
0.183837 + 0.982957i \(0.441148\pi\)
\(522\) 0 0
\(523\) 17.5167 0.765950 0.382975 0.923759i \(-0.374900\pi\)
0.382975 + 0.923759i \(0.374900\pi\)
\(524\) 0 0
\(525\) 2.53590 0.110676
\(526\) 0 0
\(527\) −1.46410 −0.0637773
\(528\) 0 0
\(529\) −22.4641 −0.976700
\(530\) 0 0
\(531\) −13.9474 −0.605267
\(532\) 0 0
\(533\) 0.535898 0.0232123
\(534\) 0 0
\(535\) 20.0526 0.866948
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) 13.6603 0.588389
\(540\) 0 0
\(541\) 38.3923 1.65061 0.825307 0.564684i \(-0.191002\pi\)
0.825307 + 0.564684i \(0.191002\pi\)
\(542\) 0 0
\(543\) 4.78461 0.205327
\(544\) 0 0
\(545\) −15.8564 −0.679214
\(546\) 0 0
\(547\) −8.33975 −0.356582 −0.178291 0.983978i \(-0.557057\pi\)
−0.178291 + 0.983978i \(0.557057\pi\)
\(548\) 0 0
\(549\) 30.5359 1.30324
\(550\) 0 0
\(551\) −25.8564 −1.10152
\(552\) 0 0
\(553\) −22.6410 −0.962794
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.8564 0.926086 0.463043 0.886336i \(-0.346758\pi\)
0.463043 + 0.886336i \(0.346758\pi\)
\(558\) 0 0
\(559\) −7.26795 −0.307401
\(560\) 0 0
\(561\) 14.9282 0.630269
\(562\) 0 0
\(563\) 40.7321 1.71665 0.858326 0.513105i \(-0.171505\pi\)
0.858326 + 0.513105i \(0.171505\pi\)
\(564\) 0 0
\(565\) 18.3923 0.773770
\(566\) 0 0
\(567\) 15.4641 0.649431
\(568\) 0 0
\(569\) −17.4641 −0.732133 −0.366067 0.930589i \(-0.619296\pi\)
−0.366067 + 0.930589i \(0.619296\pi\)
\(570\) 0 0
\(571\) −27.3205 −1.14333 −0.571664 0.820488i \(-0.693702\pi\)
−0.571664 + 0.820488i \(0.693702\pi\)
\(572\) 0 0
\(573\) 13.8564 0.578860
\(574\) 0 0
\(575\) −0.732051 −0.0305286
\(576\) 0 0
\(577\) −30.6410 −1.27560 −0.637801 0.770201i \(-0.720156\pi\)
−0.637801 + 0.770201i \(0.720156\pi\)
\(578\) 0 0
\(579\) −1.46410 −0.0608460
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) −1.46410 −0.0606369
\(584\) 0 0
\(585\) 2.46410 0.101878
\(586\) 0 0
\(587\) 1.21539 0.0501645 0.0250823 0.999685i \(-0.492015\pi\)
0.0250823 + 0.999685i \(0.492015\pi\)
\(588\) 0 0
\(589\) 0.535898 0.0220813
\(590\) 0 0
\(591\) 6.53590 0.268851
\(592\) 0 0
\(593\) 40.6410 1.66893 0.834463 0.551064i \(-0.185778\pi\)
0.834463 + 0.551064i \(0.185778\pi\)
\(594\) 0 0
\(595\) −25.8564 −1.06001
\(596\) 0 0
\(597\) −5.07180 −0.207575
\(598\) 0 0
\(599\) −33.4641 −1.36731 −0.683653 0.729807i \(-0.739610\pi\)
−0.683653 + 0.729807i \(0.739610\pi\)
\(600\) 0 0
\(601\) −3.85641 −0.157306 −0.0786531 0.996902i \(-0.525062\pi\)
−0.0786531 + 0.996902i \(0.525062\pi\)
\(602\) 0 0
\(603\) 31.8564 1.29729
\(604\) 0 0
\(605\) 3.53590 0.143755
\(606\) 0 0
\(607\) −13.5167 −0.548624 −0.274312 0.961641i \(-0.588450\pi\)
−0.274312 + 0.961641i \(0.588450\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 4.53590 0.183503
\(612\) 0 0
\(613\) −0.143594 −0.00579969 −0.00289984 0.999996i \(-0.500923\pi\)
−0.00289984 + 0.999996i \(0.500923\pi\)
\(614\) 0 0
\(615\) −0.392305 −0.0158193
\(616\) 0 0
\(617\) −45.7128 −1.84033 −0.920164 0.391533i \(-0.871945\pi\)
−0.920164 + 0.391533i \(0.871945\pi\)
\(618\) 0 0
\(619\) −18.3397 −0.737137 −0.368568 0.929601i \(-0.620152\pi\)
−0.368568 + 0.929601i \(0.620152\pi\)
\(620\) 0 0
\(621\) 2.92820 0.117505
\(622\) 0 0
\(623\) −44.7846 −1.79426
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.46410 −0.218215
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −7.80385 −0.310666 −0.155333 0.987862i \(-0.549645\pi\)
−0.155333 + 0.987862i \(0.549645\pi\)
\(632\) 0 0
\(633\) 2.14359 0.0852002
\(634\) 0 0
\(635\) 1.80385 0.0715835
\(636\) 0 0
\(637\) 5.00000 0.198107
\(638\) 0 0
\(639\) −22.8372 −0.903424
\(640\) 0 0
\(641\) −0.928203 −0.0366618 −0.0183309 0.999832i \(-0.505835\pi\)
−0.0183309 + 0.999832i \(0.505835\pi\)
\(642\) 0 0
\(643\) −9.60770 −0.378891 −0.189445 0.981891i \(-0.560669\pi\)
−0.189445 + 0.981891i \(0.560669\pi\)
\(644\) 0 0
\(645\) 5.32051 0.209495
\(646\) 0 0
\(647\) 0.339746 0.0133568 0.00667840 0.999978i \(-0.497874\pi\)
0.00667840 + 0.999978i \(0.497874\pi\)
\(648\) 0 0
\(649\) 15.4641 0.607019
\(650\) 0 0
\(651\) −0.497423 −0.0194955
\(652\) 0 0
\(653\) −33.7128 −1.31928 −0.659642 0.751580i \(-0.729292\pi\)
−0.659642 + 0.751580i \(0.729292\pi\)
\(654\) 0 0
\(655\) 2.92820 0.114414
\(656\) 0 0
\(657\) 7.21539 0.281499
\(658\) 0 0
\(659\) 39.3205 1.53171 0.765855 0.643014i \(-0.222316\pi\)
0.765855 + 0.643014i \(0.222316\pi\)
\(660\) 0 0
\(661\) 20.9282 0.814013 0.407006 0.913425i \(-0.366573\pi\)
0.407006 + 0.913425i \(0.366573\pi\)
\(662\) 0 0
\(663\) 5.46410 0.212208
\(664\) 0 0
\(665\) 9.46410 0.367002
\(666\) 0 0
\(667\) −6.92820 −0.268261
\(668\) 0 0
\(669\) 18.5359 0.716639
\(670\) 0 0
\(671\) −33.8564 −1.30701
\(672\) 0 0
\(673\) 28.5359 1.09998 0.549989 0.835172i \(-0.314632\pi\)
0.549989 + 0.835172i \(0.314632\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 9.60770 0.369254 0.184627 0.982809i \(-0.440892\pi\)
0.184627 + 0.982809i \(0.440892\pi\)
\(678\) 0 0
\(679\) 54.9282 2.10795
\(680\) 0 0
\(681\) 13.1769 0.504940
\(682\) 0 0
\(683\) 24.9282 0.953851 0.476926 0.878944i \(-0.341751\pi\)
0.476926 + 0.878944i \(0.341751\pi\)
\(684\) 0 0
\(685\) −8.92820 −0.341129
\(686\) 0 0
\(687\) 14.2487 0.543622
\(688\) 0 0
\(689\) −0.535898 −0.0204161
\(690\) 0 0
\(691\) 43.9090 1.67038 0.835188 0.549965i \(-0.185359\pi\)
0.835188 + 0.549965i \(0.185359\pi\)
\(692\) 0 0
\(693\) −23.3205 −0.885873
\(694\) 0 0
\(695\) 13.4641 0.510722
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) 13.1769 0.498397
\(700\) 0 0
\(701\) 29.7128 1.12224 0.561119 0.827735i \(-0.310371\pi\)
0.561119 + 0.827735i \(0.310371\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −3.32051 −0.125058
\(706\) 0 0
\(707\) 17.0718 0.642051
\(708\) 0 0
\(709\) 5.60770 0.210601 0.105301 0.994440i \(-0.466419\pi\)
0.105301 + 0.994440i \(0.466419\pi\)
\(710\) 0 0
\(711\) 16.1051 0.603989
\(712\) 0 0
\(713\) 0.143594 0.00537762
\(714\) 0 0
\(715\) −2.73205 −0.102173
\(716\) 0 0
\(717\) 18.4974 0.690799
\(718\) 0 0
\(719\) −34.6410 −1.29189 −0.645946 0.763383i \(-0.723537\pi\)
−0.645946 + 0.763383i \(0.723537\pi\)
\(720\) 0 0
\(721\) −21.4641 −0.799365
\(722\) 0 0
\(723\) 14.8231 0.551276
\(724\) 0 0
\(725\) 9.46410 0.351488
\(726\) 0 0
\(727\) −34.9808 −1.29736 −0.648682 0.761059i \(-0.724680\pi\)
−0.648682 + 0.761059i \(0.724680\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) −54.2487 −2.00646
\(732\) 0 0
\(733\) −15.8564 −0.585670 −0.292835 0.956163i \(-0.594599\pi\)
−0.292835 + 0.956163i \(0.594599\pi\)
\(734\) 0 0
\(735\) −3.66025 −0.135011
\(736\) 0 0
\(737\) −35.3205 −1.30105
\(738\) 0 0
\(739\) 1.26795 0.0466423 0.0233211 0.999728i \(-0.492576\pi\)
0.0233211 + 0.999728i \(0.492576\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 25.3205 0.928919 0.464460 0.885594i \(-0.346249\pi\)
0.464460 + 0.885594i \(0.346249\pi\)
\(744\) 0 0
\(745\) 7.85641 0.287836
\(746\) 0 0
\(747\) 25.6077 0.936937
\(748\) 0 0
\(749\) −69.4641 −2.53816
\(750\) 0 0
\(751\) 45.4641 1.65901 0.829504 0.558500i \(-0.188623\pi\)
0.829504 + 0.558500i \(0.188623\pi\)
\(752\) 0 0
\(753\) 17.0718 0.622131
\(754\) 0 0
\(755\) −11.8038 −0.429586
\(756\) 0 0
\(757\) −23.4641 −0.852817 −0.426409 0.904531i \(-0.640221\pi\)
−0.426409 + 0.904531i \(0.640221\pi\)
\(758\) 0 0
\(759\) −1.46410 −0.0531435
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) 54.9282 1.98853
\(764\) 0 0
\(765\) 18.3923 0.664975
\(766\) 0 0
\(767\) 5.66025 0.204380
\(768\) 0 0
\(769\) 15.0718 0.543503 0.271751 0.962367i \(-0.412397\pi\)
0.271751 + 0.962367i \(0.412397\pi\)
\(770\) 0 0
\(771\) 4.39230 0.158185
\(772\) 0 0
\(773\) 5.07180 0.182420 0.0912099 0.995832i \(-0.470927\pi\)
0.0912099 + 0.995832i \(0.470927\pi\)
\(774\) 0 0
\(775\) −0.196152 −0.00704600
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.46410 −0.0524569
\(780\) 0 0
\(781\) 25.3205 0.906039
\(782\) 0 0
\(783\) −37.8564 −1.35288
\(784\) 0 0
\(785\) 7.85641 0.280407
\(786\) 0 0
\(787\) 18.3923 0.655615 0.327807 0.944745i \(-0.393690\pi\)
0.327807 + 0.944745i \(0.393690\pi\)
\(788\) 0 0
\(789\) −8.53590 −0.303886
\(790\) 0 0
\(791\) −63.7128 −2.26537
\(792\) 0 0
\(793\) −12.3923 −0.440064
\(794\) 0 0
\(795\) 0.392305 0.0139136
\(796\) 0 0
\(797\) 46.7846 1.65720 0.828598 0.559844i \(-0.189139\pi\)
0.828598 + 0.559844i \(0.189139\pi\)
\(798\) 0 0
\(799\) 33.8564 1.19775
\(800\) 0 0
\(801\) 31.8564 1.12559
\(802\) 0 0
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 2.53590 0.0893787
\(806\) 0 0
\(807\) −0.105118 −0.00370032
\(808\) 0 0
\(809\) 8.67949 0.305155 0.152577 0.988292i \(-0.451243\pi\)
0.152577 + 0.988292i \(0.451243\pi\)
\(810\) 0 0
\(811\) 35.9090 1.26093 0.630467 0.776216i \(-0.282863\pi\)
0.630467 + 0.776216i \(0.282863\pi\)
\(812\) 0 0
\(813\) −16.9282 −0.593698
\(814\) 0 0
\(815\) 8.92820 0.312741
\(816\) 0 0
\(817\) 19.8564 0.694688
\(818\) 0 0
\(819\) −8.53590 −0.298268
\(820\) 0 0
\(821\) −12.9282 −0.451197 −0.225599 0.974220i \(-0.572434\pi\)
−0.225599 + 0.974220i \(0.572434\pi\)
\(822\) 0 0
\(823\) 26.5885 0.926815 0.463408 0.886145i \(-0.346627\pi\)
0.463408 + 0.886145i \(0.346627\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 3.46410 0.120459 0.0602293 0.998185i \(-0.480817\pi\)
0.0602293 + 0.998185i \(0.480817\pi\)
\(828\) 0 0
\(829\) 44.3923 1.54181 0.770904 0.636951i \(-0.219805\pi\)
0.770904 + 0.636951i \(0.219805\pi\)
\(830\) 0 0
\(831\) −15.6077 −0.541425
\(832\) 0 0
\(833\) 37.3205 1.29308
\(834\) 0 0
\(835\) 21.3205 0.737827
\(836\) 0 0
\(837\) 0.784610 0.0271201
\(838\) 0 0
\(839\) 34.4449 1.18917 0.594584 0.804033i \(-0.297317\pi\)
0.594584 + 0.804033i \(0.297317\pi\)
\(840\) 0 0
\(841\) 60.5692 2.08859
\(842\) 0 0
\(843\) −3.32051 −0.114364
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −12.2487 −0.420871
\(848\) 0 0
\(849\) −17.6077 −0.604295
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −7.71281 −0.264082 −0.132041 0.991244i \(-0.542153\pi\)
−0.132041 + 0.991244i \(0.542153\pi\)
\(854\) 0 0
\(855\) −6.73205 −0.230231
\(856\) 0 0
\(857\) −4.14359 −0.141542 −0.0707712 0.997493i \(-0.522546\pi\)
−0.0707712 + 0.997493i \(0.522546\pi\)
\(858\) 0 0
\(859\) −12.3923 −0.422820 −0.211410 0.977397i \(-0.567806\pi\)
−0.211410 + 0.977397i \(0.567806\pi\)
\(860\) 0 0
\(861\) 1.35898 0.0463140
\(862\) 0 0
\(863\) −33.6077 −1.14402 −0.572010 0.820247i \(-0.693836\pi\)
−0.572010 + 0.820247i \(0.693836\pi\)
\(864\) 0 0
\(865\) 2.39230 0.0813408
\(866\) 0 0
\(867\) 28.3397 0.962468
\(868\) 0 0
\(869\) −17.8564 −0.605737
\(870\) 0 0
\(871\) −12.9282 −0.438055
\(872\) 0 0
\(873\) −39.0718 −1.32238
\(874\) 0 0
\(875\) −3.46410 −0.117108
\(876\) 0 0
\(877\) −41.7128 −1.40854 −0.704271 0.709931i \(-0.748726\pi\)
−0.704271 + 0.709931i \(0.748726\pi\)
\(878\) 0 0
\(879\) 5.07180 0.171067
\(880\) 0 0
\(881\) 5.46410 0.184090 0.0920451 0.995755i \(-0.470660\pi\)
0.0920451 + 0.995755i \(0.470660\pi\)
\(882\) 0 0
\(883\) 22.5885 0.760162 0.380081 0.924953i \(-0.375896\pi\)
0.380081 + 0.924953i \(0.375896\pi\)
\(884\) 0 0
\(885\) −4.14359 −0.139285
\(886\) 0 0
\(887\) 12.4449 0.417858 0.208929 0.977931i \(-0.433002\pi\)
0.208929 + 0.977931i \(0.433002\pi\)
\(888\) 0 0
\(889\) −6.24871 −0.209575
\(890\) 0 0
\(891\) 12.1962 0.408586
\(892\) 0 0
\(893\) −12.3923 −0.414693
\(894\) 0 0
\(895\) 21.8564 0.730579
\(896\) 0 0
\(897\) −0.535898 −0.0178931
\(898\) 0 0
\(899\) −1.85641 −0.0619146
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) −18.4308 −0.613338
\(904\) 0 0
\(905\) −6.53590 −0.217261
\(906\) 0 0
\(907\) −36.8372 −1.22316 −0.611579 0.791183i \(-0.709465\pi\)
−0.611579 + 0.791183i \(0.709465\pi\)
\(908\) 0 0
\(909\) −12.1436 −0.402778
\(910\) 0 0
\(911\) −20.7846 −0.688625 −0.344312 0.938855i \(-0.611888\pi\)
−0.344312 + 0.938855i \(0.611888\pi\)
\(912\) 0 0
\(913\) −28.3923 −0.939648
\(914\) 0 0
\(915\) 9.07180 0.299904
\(916\) 0 0
\(917\) −10.1436 −0.334971
\(918\) 0 0
\(919\) 39.3205 1.29706 0.648532 0.761187i \(-0.275383\pi\)
0.648532 + 0.761187i \(0.275383\pi\)
\(920\) 0 0
\(921\) −12.1051 −0.398877
\(922\) 0 0
\(923\) 9.26795 0.305058
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 15.2679 0.501465
\(928\) 0 0
\(929\) 43.1769 1.41659 0.708294 0.705917i \(-0.249465\pi\)
0.708294 + 0.705917i \(0.249465\pi\)
\(930\) 0 0
\(931\) −13.6603 −0.447697
\(932\) 0 0
\(933\) 22.9282 0.750636
\(934\) 0 0
\(935\) −20.3923 −0.666900
\(936\) 0 0
\(937\) −27.0718 −0.884397 −0.442199 0.896917i \(-0.645801\pi\)
−0.442199 + 0.896917i \(0.645801\pi\)
\(938\) 0 0
\(939\) 7.60770 0.248268
\(940\) 0 0
\(941\) 22.3923 0.729968 0.364984 0.931014i \(-0.381074\pi\)
0.364984 + 0.931014i \(0.381074\pi\)
\(942\) 0 0
\(943\) −0.392305 −0.0127752
\(944\) 0 0
\(945\) 13.8564 0.450749
\(946\) 0 0
\(947\) 22.3923 0.727652 0.363826 0.931467i \(-0.381470\pi\)
0.363826 + 0.931467i \(0.381470\pi\)
\(948\) 0 0
\(949\) −2.92820 −0.0950535
\(950\) 0 0
\(951\) −1.35898 −0.0440681
\(952\) 0 0
\(953\) −1.21539 −0.0393704 −0.0196852 0.999806i \(-0.506266\pi\)
−0.0196852 + 0.999806i \(0.506266\pi\)
\(954\) 0 0
\(955\) −18.9282 −0.612502
\(956\) 0 0
\(957\) 18.9282 0.611862
\(958\) 0 0
\(959\) 30.9282 0.998724
\(960\) 0 0
\(961\) −30.9615 −0.998759
\(962\) 0 0
\(963\) 49.4115 1.59226
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −18.0000 −0.578841 −0.289420 0.957202i \(-0.593463\pi\)
−0.289420 + 0.957202i \(0.593463\pi\)
\(968\) 0 0
\(969\) −14.9282 −0.479563
\(970\) 0 0
\(971\) 43.7128 1.40281 0.701405 0.712762i \(-0.252556\pi\)
0.701405 + 0.712762i \(0.252556\pi\)
\(972\) 0 0
\(973\) −46.6410 −1.49524
\(974\) 0 0
\(975\) 0.732051 0.0234444
\(976\) 0 0
\(977\) −2.92820 −0.0936815 −0.0468408 0.998902i \(-0.514915\pi\)
−0.0468408 + 0.998902i \(0.514915\pi\)
\(978\) 0 0
\(979\) −35.3205 −1.12885
\(980\) 0 0
\(981\) −39.0718 −1.24747
\(982\) 0 0
\(983\) −15.8564 −0.505741 −0.252870 0.967500i \(-0.581375\pi\)
−0.252870 + 0.967500i \(0.581375\pi\)
\(984\) 0 0
\(985\) −8.92820 −0.284476
\(986\) 0 0
\(987\) 11.5026 0.366131
\(988\) 0 0
\(989\) 5.32051 0.169182
\(990\) 0 0
\(991\) −53.8564 −1.71081 −0.855403 0.517964i \(-0.826690\pi\)
−0.855403 + 0.517964i \(0.826690\pi\)
\(992\) 0 0
\(993\) −6.28719 −0.199518
\(994\) 0 0
\(995\) 6.92820 0.219639
\(996\) 0 0
\(997\) −9.32051 −0.295183 −0.147592 0.989048i \(-0.547152\pi\)
−0.147592 + 0.989048i \(0.547152\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 520.2.a.e.1.2 2
3.2 odd 2 4680.2.a.bd.1.2 2
4.3 odd 2 1040.2.a.l.1.1 2
5.2 odd 4 2600.2.d.m.1249.2 4
5.3 odd 4 2600.2.d.m.1249.3 4
5.4 even 2 2600.2.a.v.1.1 2
8.3 odd 2 4160.2.a.w.1.2 2
8.5 even 2 4160.2.a.bk.1.1 2
12.11 even 2 9360.2.a.cr.1.1 2
13.12 even 2 6760.2.a.o.1.2 2
20.19 odd 2 5200.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.a.e.1.2 2 1.1 even 1 trivial
1040.2.a.l.1.1 2 4.3 odd 2
2600.2.a.v.1.1 2 5.4 even 2
2600.2.d.m.1249.2 4 5.2 odd 4
2600.2.d.m.1249.3 4 5.3 odd 4
4160.2.a.w.1.2 2 8.3 odd 2
4160.2.a.bk.1.1 2 8.5 even 2
4680.2.a.bd.1.2 2 3.2 odd 2
5200.2.a.bn.1.2 2 20.19 odd 2
6760.2.a.o.1.2 2 13.12 even 2
9360.2.a.cr.1.1 2 12.11 even 2