Properties

Label 1092.2.a.f.1.1
Level $1092$
Weight $2$
Character 1092.1
Self dual yes
Analytic conductor $8.720$
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] = [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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 1.1
Root \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.23607 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.23607 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.23607 q^{11} +1.00000 q^{13} +1.23607 q^{15} -2.47214 q^{17} -6.47214 q^{19} -1.00000 q^{21} +4.47214 q^{23} -3.47214 q^{25} -1.00000 q^{27} +8.47214 q^{29} -3.23607 q^{33} -1.23607 q^{35} +4.47214 q^{37} -1.00000 q^{39} +5.23607 q^{41} +4.00000 q^{43} -1.23607 q^{45} +7.70820 q^{47} +1.00000 q^{49} +2.47214 q^{51} +10.0000 q^{53} -4.00000 q^{55} +6.47214 q^{57} +9.23607 q^{59} +14.9443 q^{61} +1.00000 q^{63} -1.23607 q^{65} -2.47214 q^{67} -4.47214 q^{69} -5.70820 q^{71} +4.47214 q^{73} +3.47214 q^{75} +3.23607 q^{77} -10.4721 q^{79} +1.00000 q^{81} +2.76393 q^{83} +3.05573 q^{85} -8.47214 q^{87} -10.1803 q^{89} +1.00000 q^{91} +8.00000 q^{95} +6.94427 q^{97} +3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} - 4 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 8 q^{29} - 2 q^{33} + 2 q^{35} - 2 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} + 2 q^{47} + 2 q^{49} - 4 q^{51} + 20 q^{53} - 8 q^{55} + 4 q^{57} + 14 q^{59} + 12 q^{61} + 2 q^{63} + 2 q^{65} + 4 q^{67} + 2 q^{71} - 2 q^{75} + 2 q^{77} - 12 q^{79} + 2 q^{81} + 10 q^{83} + 24 q^{85} - 8 q^{87} + 2 q^{89} + 2 q^{91} + 16 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.23607 0.975711 0.487856 0.872924i \(-0.337779\pi\)
0.487856 + 0.872924i \(0.337779\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.23607 0.319151
\(16\) 0 0
\(17\) −2.47214 −0.599581 −0.299791 0.954005i \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −3.23607 −0.563327
\(34\) 0 0
\(35\) −1.23607 −0.208934
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 5.23607 0.817736 0.408868 0.912593i \(-0.365924\pi\)
0.408868 + 0.912593i \(0.365924\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −1.23607 −0.184262
\(46\) 0 0
\(47\) 7.70820 1.12436 0.562179 0.827016i \(-0.309963\pi\)
0.562179 + 0.827016i \(0.309963\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.47214 0.346168
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 6.47214 0.857255
\(58\) 0 0
\(59\) 9.23607 1.20243 0.601217 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 0 0
\(61\) 14.9443 1.91342 0.956709 0.291046i \(-0.0940034\pi\)
0.956709 + 0.291046i \(0.0940034\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −1.23607 −0.153315
\(66\) 0 0
\(67\) −2.47214 −0.302019 −0.151010 0.988532i \(-0.548252\pi\)
−0.151010 + 0.988532i \(0.548252\pi\)
\(68\) 0 0
\(69\) −4.47214 −0.538382
\(70\) 0 0
\(71\) −5.70820 −0.677439 −0.338720 0.940887i \(-0.609994\pi\)
−0.338720 + 0.940887i \(0.609994\pi\)
\(72\) 0 0
\(73\) 4.47214 0.523424 0.261712 0.965146i \(-0.415713\pi\)
0.261712 + 0.965146i \(0.415713\pi\)
\(74\) 0 0
\(75\) 3.47214 0.400928
\(76\) 0 0
\(77\) 3.23607 0.368784
\(78\) 0 0
\(79\) −10.4721 −1.17821 −0.589104 0.808057i \(-0.700519\pi\)
−0.589104 + 0.808057i \(0.700519\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.76393 0.303381 0.151690 0.988428i \(-0.451528\pi\)
0.151690 + 0.988428i \(0.451528\pi\)
\(84\) 0 0
\(85\) 3.05573 0.331440
\(86\) 0 0
\(87\) −8.47214 −0.908308
\(88\) 0 0
\(89\) −10.1803 −1.07911 −0.539557 0.841949i \(-0.681408\pi\)
−0.539557 + 0.841949i \(0.681408\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 6.94427 0.705084 0.352542 0.935796i \(-0.385317\pi\)
0.352542 + 0.935796i \(0.385317\pi\)
\(98\) 0 0
\(99\) 3.23607 0.325237
\(100\) 0 0
\(101\) −8.94427 −0.889988 −0.444994 0.895533i \(-0.646794\pi\)
−0.444994 + 0.895533i \(0.646794\pi\)
\(102\) 0 0
\(103\) −16.9443 −1.66957 −0.834784 0.550577i \(-0.814408\pi\)
−0.834784 + 0.550577i \(0.814408\pi\)
\(104\) 0 0
\(105\) 1.23607 0.120628
\(106\) 0 0
\(107\) −3.52786 −0.341051 −0.170526 0.985353i \(-0.554547\pi\)
−0.170526 + 0.985353i \(0.554547\pi\)
\(108\) 0 0
\(109\) 3.52786 0.337908 0.168954 0.985624i \(-0.445961\pi\)
0.168954 + 0.985624i \(0.445961\pi\)
\(110\) 0 0
\(111\) −4.47214 −0.424476
\(112\) 0 0
\(113\) 20.4721 1.92586 0.962928 0.269758i \(-0.0869436\pi\)
0.962928 + 0.269758i \(0.0869436\pi\)
\(114\) 0 0
\(115\) −5.52786 −0.515476
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −2.47214 −0.226620
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) −5.23607 −0.472120
\(124\) 0 0
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) −2.47214 −0.219367 −0.109683 0.993967i \(-0.534984\pi\)
−0.109683 + 0.993967i \(0.534984\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −1.52786 −0.133490 −0.0667451 0.997770i \(-0.521261\pi\)
−0.0667451 + 0.997770i \(0.521261\pi\)
\(132\) 0 0
\(133\) −6.47214 −0.561205
\(134\) 0 0
\(135\) 1.23607 0.106384
\(136\) 0 0
\(137\) 15.2361 1.30171 0.650853 0.759204i \(-0.274412\pi\)
0.650853 + 0.759204i \(0.274412\pi\)
\(138\) 0 0
\(139\) 17.8885 1.51729 0.758643 0.651506i \(-0.225863\pi\)
0.758643 + 0.651506i \(0.225863\pi\)
\(140\) 0 0
\(141\) −7.70820 −0.649148
\(142\) 0 0
\(143\) 3.23607 0.270614
\(144\) 0 0
\(145\) −10.4721 −0.869664
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −12.7639 −1.04566 −0.522831 0.852436i \(-0.675124\pi\)
−0.522831 + 0.852436i \(0.675124\pi\)
\(150\) 0 0
\(151\) −3.05573 −0.248672 −0.124336 0.992240i \(-0.539680\pi\)
−0.124336 + 0.992240i \(0.539680\pi\)
\(152\) 0 0
\(153\) −2.47214 −0.199860
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 4.47214 0.352454
\(162\) 0 0
\(163\) −2.47214 −0.193633 −0.0968163 0.995302i \(-0.530866\pi\)
−0.0968163 + 0.995302i \(0.530866\pi\)
\(164\) 0 0
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) −18.1803 −1.40684 −0.703418 0.710776i \(-0.748344\pi\)
−0.703418 + 0.710776i \(0.748344\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −6.47214 −0.494937
\(172\) 0 0
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 0 0
\(175\) −3.47214 −0.262469
\(176\) 0 0
\(177\) −9.23607 −0.694225
\(178\) 0 0
\(179\) −6.94427 −0.519039 −0.259520 0.965738i \(-0.583564\pi\)
−0.259520 + 0.965738i \(0.583564\pi\)
\(180\) 0 0
\(181\) 19.8885 1.47830 0.739152 0.673539i \(-0.235227\pi\)
0.739152 + 0.673539i \(0.235227\pi\)
\(182\) 0 0
\(183\) −14.9443 −1.10471
\(184\) 0 0
\(185\) −5.52786 −0.406417
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −6.94427 −0.502470 −0.251235 0.967926i \(-0.580837\pi\)
−0.251235 + 0.967926i \(0.580837\pi\)
\(192\) 0 0
\(193\) 18.3607 1.32163 0.660815 0.750549i \(-0.270211\pi\)
0.660815 + 0.750549i \(0.270211\pi\)
\(194\) 0 0
\(195\) 1.23607 0.0885167
\(196\) 0 0
\(197\) 0.763932 0.0544279 0.0272140 0.999630i \(-0.491336\pi\)
0.0272140 + 0.999630i \(0.491336\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 2.47214 0.174371
\(202\) 0 0
\(203\) 8.47214 0.594627
\(204\) 0 0
\(205\) −6.47214 −0.452034
\(206\) 0 0
\(207\) 4.47214 0.310835
\(208\) 0 0
\(209\) −20.9443 −1.44875
\(210\) 0 0
\(211\) 20.3607 1.40169 0.700844 0.713315i \(-0.252807\pi\)
0.700844 + 0.713315i \(0.252807\pi\)
\(212\) 0 0
\(213\) 5.70820 0.391120
\(214\) 0 0
\(215\) −4.94427 −0.337197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.47214 −0.302199
\(220\) 0 0
\(221\) −2.47214 −0.166294
\(222\) 0 0
\(223\) −10.4721 −0.701266 −0.350633 0.936513i \(-0.614034\pi\)
−0.350633 + 0.936513i \(0.614034\pi\)
\(224\) 0 0
\(225\) −3.47214 −0.231476
\(226\) 0 0
\(227\) −6.76393 −0.448938 −0.224469 0.974481i \(-0.572065\pi\)
−0.224469 + 0.974481i \(0.572065\pi\)
\(228\) 0 0
\(229\) −25.4164 −1.67956 −0.839782 0.542924i \(-0.817317\pi\)
−0.839782 + 0.542924i \(0.817317\pi\)
\(230\) 0 0
\(231\) −3.23607 −0.212918
\(232\) 0 0
\(233\) 5.41641 0.354841 0.177420 0.984135i \(-0.443225\pi\)
0.177420 + 0.984135i \(0.443225\pi\)
\(234\) 0 0
\(235\) −9.52786 −0.621529
\(236\) 0 0
\(237\) 10.4721 0.680238
\(238\) 0 0
\(239\) 21.1246 1.36644 0.683219 0.730214i \(-0.260580\pi\)
0.683219 + 0.730214i \(0.260580\pi\)
\(240\) 0 0
\(241\) −16.4721 −1.06106 −0.530532 0.847665i \(-0.678008\pi\)
−0.530532 + 0.847665i \(0.678008\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.23607 −0.0789695
\(246\) 0 0
\(247\) −6.47214 −0.411812
\(248\) 0 0
\(249\) −2.76393 −0.175157
\(250\) 0 0
\(251\) 5.52786 0.348916 0.174458 0.984665i \(-0.444183\pi\)
0.174458 + 0.984665i \(0.444183\pi\)
\(252\) 0 0
\(253\) 14.4721 0.909855
\(254\) 0 0
\(255\) −3.05573 −0.191357
\(256\) 0 0
\(257\) 26.8328 1.67379 0.836893 0.547367i \(-0.184370\pi\)
0.836893 + 0.547367i \(0.184370\pi\)
\(258\) 0 0
\(259\) 4.47214 0.277885
\(260\) 0 0
\(261\) 8.47214 0.524412
\(262\) 0 0
\(263\) −0.472136 −0.0291132 −0.0145566 0.999894i \(-0.504634\pi\)
−0.0145566 + 0.999894i \(0.504634\pi\)
\(264\) 0 0
\(265\) −12.3607 −0.759311
\(266\) 0 0
\(267\) 10.1803 0.623027
\(268\) 0 0
\(269\) 4.94427 0.301458 0.150729 0.988575i \(-0.451838\pi\)
0.150729 + 0.988575i \(0.451838\pi\)
\(270\) 0 0
\(271\) −23.4164 −1.42245 −0.711223 0.702967i \(-0.751858\pi\)
−0.711223 + 0.702967i \(0.751858\pi\)
\(272\) 0 0
\(273\) −1.00000 −0.0605228
\(274\) 0 0
\(275\) −11.2361 −0.677560
\(276\) 0 0
\(277\) −15.8885 −0.954650 −0.477325 0.878727i \(-0.658394\pi\)
−0.477325 + 0.878727i \(0.658394\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.6525 −1.11271 −0.556357 0.830944i \(-0.687801\pi\)
−0.556357 + 0.830944i \(0.687801\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 5.23607 0.309075
\(288\) 0 0
\(289\) −10.8885 −0.640503
\(290\) 0 0
\(291\) −6.94427 −0.407080
\(292\) 0 0
\(293\) 5.23607 0.305894 0.152947 0.988234i \(-0.451124\pi\)
0.152947 + 0.988234i \(0.451124\pi\)
\(294\) 0 0
\(295\) −11.4164 −0.664689
\(296\) 0 0
\(297\) −3.23607 −0.187776
\(298\) 0 0
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 8.94427 0.513835
\(304\) 0 0
\(305\) −18.4721 −1.05771
\(306\) 0 0
\(307\) −5.88854 −0.336077 −0.168038 0.985780i \(-0.553743\pi\)
−0.168038 + 0.985780i \(0.553743\pi\)
\(308\) 0 0
\(309\) 16.9443 0.963926
\(310\) 0 0
\(311\) −1.52786 −0.0866372 −0.0433186 0.999061i \(-0.513793\pi\)
−0.0433186 + 0.999061i \(0.513793\pi\)
\(312\) 0 0
\(313\) 20.8328 1.17754 0.588770 0.808300i \(-0.299612\pi\)
0.588770 + 0.808300i \(0.299612\pi\)
\(314\) 0 0
\(315\) −1.23607 −0.0696445
\(316\) 0 0
\(317\) −28.1803 −1.58277 −0.791383 0.611321i \(-0.790638\pi\)
−0.791383 + 0.611321i \(0.790638\pi\)
\(318\) 0 0
\(319\) 27.4164 1.53502
\(320\) 0 0
\(321\) 3.52786 0.196906
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) −3.47214 −0.192599
\(326\) 0 0
\(327\) −3.52786 −0.195091
\(328\) 0 0
\(329\) 7.70820 0.424967
\(330\) 0 0
\(331\) −34.8328 −1.91458 −0.957292 0.289122i \(-0.906637\pi\)
−0.957292 + 0.289122i \(0.906637\pi\)
\(332\) 0 0
\(333\) 4.47214 0.245072
\(334\) 0 0
\(335\) 3.05573 0.166952
\(336\) 0 0
\(337\) −22.3607 −1.21806 −0.609032 0.793146i \(-0.708442\pi\)
−0.609032 + 0.793146i \(0.708442\pi\)
\(338\) 0 0
\(339\) −20.4721 −1.11189
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 5.52786 0.297610
\(346\) 0 0
\(347\) 22.0000 1.18102 0.590511 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) 0 0
\(349\) 1.41641 0.0758186 0.0379093 0.999281i \(-0.487930\pi\)
0.0379093 + 0.999281i \(0.487930\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −16.6525 −0.886322 −0.443161 0.896442i \(-0.646143\pi\)
−0.443161 + 0.896442i \(0.646143\pi\)
\(354\) 0 0
\(355\) 7.05573 0.374479
\(356\) 0 0
\(357\) 2.47214 0.130839
\(358\) 0 0
\(359\) 4.18034 0.220630 0.110315 0.993897i \(-0.464814\pi\)
0.110315 + 0.993897i \(0.464814\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 0 0
\(363\) 0.527864 0.0277057
\(364\) 0 0
\(365\) −5.52786 −0.289342
\(366\) 0 0
\(367\) −24.9443 −1.30208 −0.651040 0.759043i \(-0.725667\pi\)
−0.651040 + 0.759043i \(0.725667\pi\)
\(368\) 0 0
\(369\) 5.23607 0.272579
\(370\) 0 0
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) 36.4721 1.88846 0.944228 0.329293i \(-0.106810\pi\)
0.944228 + 0.329293i \(0.106810\pi\)
\(374\) 0 0
\(375\) −10.4721 −0.540779
\(376\) 0 0
\(377\) 8.47214 0.436337
\(378\) 0 0
\(379\) 13.8885 0.713407 0.356703 0.934218i \(-0.383901\pi\)
0.356703 + 0.934218i \(0.383901\pi\)
\(380\) 0 0
\(381\) 2.47214 0.126651
\(382\) 0 0
\(383\) −1.81966 −0.0929803 −0.0464901 0.998919i \(-0.514804\pi\)
−0.0464901 + 0.998919i \(0.514804\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) −11.0557 −0.559112
\(392\) 0 0
\(393\) 1.52786 0.0770705
\(394\) 0 0
\(395\) 12.9443 0.651297
\(396\) 0 0
\(397\) 35.8885 1.80119 0.900597 0.434655i \(-0.143130\pi\)
0.900597 + 0.434655i \(0.143130\pi\)
\(398\) 0 0
\(399\) 6.47214 0.324012
\(400\) 0 0
\(401\) −19.2361 −0.960603 −0.480302 0.877103i \(-0.659473\pi\)
−0.480302 + 0.877103i \(0.659473\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.23607 −0.0614207
\(406\) 0 0
\(407\) 14.4721 0.717357
\(408\) 0 0
\(409\) −17.0557 −0.843351 −0.421676 0.906747i \(-0.638558\pi\)
−0.421676 + 0.906747i \(0.638558\pi\)
\(410\) 0 0
\(411\) −15.2361 −0.751540
\(412\) 0 0
\(413\) 9.23607 0.454477
\(414\) 0 0
\(415\) −3.41641 −0.167705
\(416\) 0 0
\(417\) −17.8885 −0.876006
\(418\) 0 0
\(419\) −23.4164 −1.14397 −0.571983 0.820265i \(-0.693826\pi\)
−0.571983 + 0.820265i \(0.693826\pi\)
\(420\) 0 0
\(421\) 0.111456 0.00543204 0.00271602 0.999996i \(-0.499135\pi\)
0.00271602 + 0.999996i \(0.499135\pi\)
\(422\) 0 0
\(423\) 7.70820 0.374786
\(424\) 0 0
\(425\) 8.58359 0.416365
\(426\) 0 0
\(427\) 14.9443 0.723204
\(428\) 0 0
\(429\) −3.23607 −0.156239
\(430\) 0 0
\(431\) 1.70820 0.0822813 0.0411406 0.999153i \(-0.486901\pi\)
0.0411406 + 0.999153i \(0.486901\pi\)
\(432\) 0 0
\(433\) 22.0000 1.05725 0.528626 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 0 0
\(435\) 10.4721 0.502100
\(436\) 0 0
\(437\) −28.9443 −1.38459
\(438\) 0 0
\(439\) −15.0557 −0.718571 −0.359285 0.933228i \(-0.616980\pi\)
−0.359285 + 0.933228i \(0.616980\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −17.4164 −0.827479 −0.413739 0.910395i \(-0.635777\pi\)
−0.413739 + 0.910395i \(0.635777\pi\)
\(444\) 0 0
\(445\) 12.5836 0.596519
\(446\) 0 0
\(447\) 12.7639 0.603713
\(448\) 0 0
\(449\) −33.1246 −1.56325 −0.781624 0.623750i \(-0.785608\pi\)
−0.781624 + 0.623750i \(0.785608\pi\)
\(450\) 0 0
\(451\) 16.9443 0.797875
\(452\) 0 0
\(453\) 3.05573 0.143571
\(454\) 0 0
\(455\) −1.23607 −0.0579478
\(456\) 0 0
\(457\) −3.52786 −0.165027 −0.0825133 0.996590i \(-0.526295\pi\)
−0.0825133 + 0.996590i \(0.526295\pi\)
\(458\) 0 0
\(459\) 2.47214 0.115389
\(460\) 0 0
\(461\) −16.2918 −0.758785 −0.379392 0.925236i \(-0.623867\pi\)
−0.379392 + 0.925236i \(0.623867\pi\)
\(462\) 0 0
\(463\) −11.4164 −0.530565 −0.265283 0.964171i \(-0.585465\pi\)
−0.265283 + 0.964171i \(0.585465\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.47214 −0.114397 −0.0571984 0.998363i \(-0.518217\pi\)
−0.0571984 + 0.998363i \(0.518217\pi\)
\(468\) 0 0
\(469\) −2.47214 −0.114153
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 12.9443 0.595178
\(474\) 0 0
\(475\) 22.4721 1.03109
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) 28.0689 1.28250 0.641250 0.767332i \(-0.278416\pi\)
0.641250 + 0.767332i \(0.278416\pi\)
\(480\) 0 0
\(481\) 4.47214 0.203912
\(482\) 0 0
\(483\) −4.47214 −0.203489
\(484\) 0 0
\(485\) −8.58359 −0.389761
\(486\) 0 0
\(487\) 30.8328 1.39717 0.698584 0.715528i \(-0.253814\pi\)
0.698584 + 0.715528i \(0.253814\pi\)
\(488\) 0 0
\(489\) 2.47214 0.111794
\(490\) 0 0
\(491\) 16.4721 0.743377 0.371689 0.928357i \(-0.378779\pi\)
0.371689 + 0.928357i \(0.378779\pi\)
\(492\) 0 0
\(493\) −20.9443 −0.943283
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) −5.70820 −0.256048
\(498\) 0 0
\(499\) 21.8885 0.979866 0.489933 0.871760i \(-0.337021\pi\)
0.489933 + 0.871760i \(0.337021\pi\)
\(500\) 0 0
\(501\) 18.1803 0.812238
\(502\) 0 0
\(503\) 10.4721 0.466929 0.233465 0.972365i \(-0.424994\pi\)
0.233465 + 0.972365i \(0.424994\pi\)
\(504\) 0 0
\(505\) 11.0557 0.491973
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 21.5967 0.957259 0.478630 0.878017i \(-0.341134\pi\)
0.478630 + 0.878017i \(0.341134\pi\)
\(510\) 0 0
\(511\) 4.47214 0.197836
\(512\) 0 0
\(513\) 6.47214 0.285752
\(514\) 0 0
\(515\) 20.9443 0.922915
\(516\) 0 0
\(517\) 24.9443 1.09705
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −20.9443 −0.917585 −0.458793 0.888543i \(-0.651718\pi\)
−0.458793 + 0.888543i \(0.651718\pi\)
\(522\) 0 0
\(523\) 33.8885 1.48184 0.740921 0.671592i \(-0.234389\pi\)
0.740921 + 0.671592i \(0.234389\pi\)
\(524\) 0 0
\(525\) 3.47214 0.151536
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) 0 0
\(531\) 9.23607 0.400811
\(532\) 0 0
\(533\) 5.23607 0.226799
\(534\) 0 0
\(535\) 4.36068 0.188529
\(536\) 0 0
\(537\) 6.94427 0.299667
\(538\) 0 0
\(539\) 3.23607 0.139387
\(540\) 0 0
\(541\) 6.94427 0.298558 0.149279 0.988795i \(-0.452305\pi\)
0.149279 + 0.988795i \(0.452305\pi\)
\(542\) 0 0
\(543\) −19.8885 −0.853499
\(544\) 0 0
\(545\) −4.36068 −0.186791
\(546\) 0 0
\(547\) −19.0557 −0.814764 −0.407382 0.913258i \(-0.633558\pi\)
−0.407382 + 0.913258i \(0.633558\pi\)
\(548\) 0 0
\(549\) 14.9443 0.637806
\(550\) 0 0
\(551\) −54.8328 −2.33596
\(552\) 0 0
\(553\) −10.4721 −0.445321
\(554\) 0 0
\(555\) 5.52786 0.234645
\(556\) 0 0
\(557\) −28.7639 −1.21877 −0.609383 0.792876i \(-0.708583\pi\)
−0.609383 + 0.792876i \(0.708583\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 19.0557 0.803103 0.401552 0.915836i \(-0.368471\pi\)
0.401552 + 0.915836i \(0.368471\pi\)
\(564\) 0 0
\(565\) −25.3050 −1.06459
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −17.4164 −0.730134 −0.365067 0.930981i \(-0.618954\pi\)
−0.365067 + 0.930981i \(0.618954\pi\)
\(570\) 0 0
\(571\) −0.360680 −0.0150940 −0.00754699 0.999972i \(-0.502402\pi\)
−0.00754699 + 0.999972i \(0.502402\pi\)
\(572\) 0 0
\(573\) 6.94427 0.290101
\(574\) 0 0
\(575\) −15.5279 −0.647557
\(576\) 0 0
\(577\) 18.9443 0.788660 0.394330 0.918969i \(-0.370977\pi\)
0.394330 + 0.918969i \(0.370977\pi\)
\(578\) 0 0
\(579\) −18.3607 −0.763044
\(580\) 0 0
\(581\) 2.76393 0.114667
\(582\) 0 0
\(583\) 32.3607 1.34024
\(584\) 0 0
\(585\) −1.23607 −0.0511051
\(586\) 0 0
\(587\) 21.5967 0.891393 0.445697 0.895184i \(-0.352956\pi\)
0.445697 + 0.895184i \(0.352956\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −0.763932 −0.0314240
\(592\) 0 0
\(593\) 17.8197 0.731766 0.365883 0.930661i \(-0.380767\pi\)
0.365883 + 0.930661i \(0.380767\pi\)
\(594\) 0 0
\(595\) 3.05573 0.125273
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) −27.3050 −1.11565 −0.557825 0.829959i \(-0.688364\pi\)
−0.557825 + 0.829959i \(0.688364\pi\)
\(600\) 0 0
\(601\) 9.05573 0.369391 0.184695 0.982796i \(-0.440870\pi\)
0.184695 + 0.982796i \(0.440870\pi\)
\(602\) 0 0
\(603\) −2.47214 −0.100673
\(604\) 0 0
\(605\) 0.652476 0.0265269
\(606\) 0 0
\(607\) −21.8885 −0.888429 −0.444214 0.895921i \(-0.646517\pi\)
−0.444214 + 0.895921i \(0.646517\pi\)
\(608\) 0 0
\(609\) −8.47214 −0.343308
\(610\) 0 0
\(611\) 7.70820 0.311841
\(612\) 0 0
\(613\) −29.7771 −1.20269 −0.601343 0.798991i \(-0.705367\pi\)
−0.601343 + 0.798991i \(0.705367\pi\)
\(614\) 0 0
\(615\) 6.47214 0.260982
\(616\) 0 0
\(617\) −40.7639 −1.64109 −0.820547 0.571579i \(-0.806331\pi\)
−0.820547 + 0.571579i \(0.806331\pi\)
\(618\) 0 0
\(619\) −37.3050 −1.49941 −0.749706 0.661771i \(-0.769805\pi\)
−0.749706 + 0.661771i \(0.769805\pi\)
\(620\) 0 0
\(621\) −4.47214 −0.179461
\(622\) 0 0
\(623\) −10.1803 −0.407867
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 20.9443 0.836434
\(628\) 0 0
\(629\) −11.0557 −0.440821
\(630\) 0 0
\(631\) 14.4721 0.576127 0.288063 0.957611i \(-0.406989\pi\)
0.288063 + 0.957611i \(0.406989\pi\)
\(632\) 0 0
\(633\) −20.3607 −0.809264
\(634\) 0 0
\(635\) 3.05573 0.121263
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −5.70820 −0.225813
\(640\) 0 0
\(641\) 41.7771 1.65010 0.825048 0.565063i \(-0.191148\pi\)
0.825048 + 0.565063i \(0.191148\pi\)
\(642\) 0 0
\(643\) −10.8328 −0.427205 −0.213602 0.976921i \(-0.568520\pi\)
−0.213602 + 0.976921i \(0.568520\pi\)
\(644\) 0 0
\(645\) 4.94427 0.194681
\(646\) 0 0
\(647\) −10.8328 −0.425882 −0.212941 0.977065i \(-0.568304\pi\)
−0.212941 + 0.977065i \(0.568304\pi\)
\(648\) 0 0
\(649\) 29.8885 1.17323
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −38.3607 −1.50117 −0.750585 0.660774i \(-0.770228\pi\)
−0.750585 + 0.660774i \(0.770228\pi\)
\(654\) 0 0
\(655\) 1.88854 0.0737915
\(656\) 0 0
\(657\) 4.47214 0.174475
\(658\) 0 0
\(659\) 0.111456 0.00434172 0.00217086 0.999998i \(-0.499309\pi\)
0.00217086 + 0.999998i \(0.499309\pi\)
\(660\) 0 0
\(661\) −13.4164 −0.521838 −0.260919 0.965361i \(-0.584026\pi\)
−0.260919 + 0.965361i \(0.584026\pi\)
\(662\) 0 0
\(663\) 2.47214 0.0960098
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 37.8885 1.46705
\(668\) 0 0
\(669\) 10.4721 0.404876
\(670\) 0 0
\(671\) 48.3607 1.86694
\(672\) 0 0
\(673\) 39.8885 1.53759 0.768795 0.639495i \(-0.220857\pi\)
0.768795 + 0.639495i \(0.220857\pi\)
\(674\) 0 0
\(675\) 3.47214 0.133643
\(676\) 0 0
\(677\) 4.58359 0.176162 0.0880809 0.996113i \(-0.471927\pi\)
0.0880809 + 0.996113i \(0.471927\pi\)
\(678\) 0 0
\(679\) 6.94427 0.266497
\(680\) 0 0
\(681\) 6.76393 0.259194
\(682\) 0 0
\(683\) 47.2361 1.80744 0.903719 0.428126i \(-0.140826\pi\)
0.903719 + 0.428126i \(0.140826\pi\)
\(684\) 0 0
\(685\) −18.8328 −0.719565
\(686\) 0 0
\(687\) 25.4164 0.969696
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −7.63932 −0.290613 −0.145307 0.989387i \(-0.546417\pi\)
−0.145307 + 0.989387i \(0.546417\pi\)
\(692\) 0 0
\(693\) 3.23607 0.122928
\(694\) 0 0
\(695\) −22.1115 −0.838735
\(696\) 0 0
\(697\) −12.9443 −0.490299
\(698\) 0 0
\(699\) −5.41641 −0.204867
\(700\) 0 0
\(701\) 33.7771 1.27574 0.637871 0.770143i \(-0.279815\pi\)
0.637871 + 0.770143i \(0.279815\pi\)
\(702\) 0 0
\(703\) −28.9443 −1.09165
\(704\) 0 0
\(705\) 9.52786 0.358840
\(706\) 0 0
\(707\) −8.94427 −0.336384
\(708\) 0 0
\(709\) 18.3607 0.689550 0.344775 0.938685i \(-0.387955\pi\)
0.344775 + 0.938685i \(0.387955\pi\)
\(710\) 0 0
\(711\) −10.4721 −0.392736
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) −21.1246 −0.788913
\(718\) 0 0
\(719\) −9.30495 −0.347016 −0.173508 0.984832i \(-0.555510\pi\)
−0.173508 + 0.984832i \(0.555510\pi\)
\(720\) 0 0
\(721\) −16.9443 −0.631038
\(722\) 0 0
\(723\) 16.4721 0.612605
\(724\) 0 0
\(725\) −29.4164 −1.09250
\(726\) 0 0
\(727\) 30.8328 1.14353 0.571763 0.820419i \(-0.306260\pi\)
0.571763 + 0.820419i \(0.306260\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −9.88854 −0.365741
\(732\) 0 0
\(733\) 16.4721 0.608412 0.304206 0.952606i \(-0.401609\pi\)
0.304206 + 0.952606i \(0.401609\pi\)
\(734\) 0 0
\(735\) 1.23607 0.0455931
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −31.4164 −1.15567 −0.577836 0.816153i \(-0.696103\pi\)
−0.577836 + 0.816153i \(0.696103\pi\)
\(740\) 0 0
\(741\) 6.47214 0.237760
\(742\) 0 0
\(743\) −42.6525 −1.56477 −0.782384 0.622797i \(-0.785996\pi\)
−0.782384 + 0.622797i \(0.785996\pi\)
\(744\) 0 0
\(745\) 15.7771 0.578028
\(746\) 0 0
\(747\) 2.76393 0.101127
\(748\) 0 0
\(749\) −3.52786 −0.128905
\(750\) 0 0
\(751\) 19.0557 0.695353 0.347677 0.937614i \(-0.386971\pi\)
0.347677 + 0.937614i \(0.386971\pi\)
\(752\) 0 0
\(753\) −5.52786 −0.201447
\(754\) 0 0
\(755\) 3.77709 0.137462
\(756\) 0 0
\(757\) 41.4164 1.50530 0.752652 0.658418i \(-0.228774\pi\)
0.752652 + 0.658418i \(0.228774\pi\)
\(758\) 0 0
\(759\) −14.4721 −0.525305
\(760\) 0 0
\(761\) 26.1803 0.949037 0.474518 0.880246i \(-0.342622\pi\)
0.474518 + 0.880246i \(0.342622\pi\)
\(762\) 0 0
\(763\) 3.52786 0.127717
\(764\) 0 0
\(765\) 3.05573 0.110480
\(766\) 0 0
\(767\) 9.23607 0.333495
\(768\) 0 0
\(769\) −7.52786 −0.271462 −0.135731 0.990746i \(-0.543338\pi\)
−0.135731 + 0.990746i \(0.543338\pi\)
\(770\) 0 0
\(771\) −26.8328 −0.966360
\(772\) 0 0
\(773\) 7.12461 0.256254 0.128127 0.991758i \(-0.459103\pi\)
0.128127 + 0.991758i \(0.459103\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4.47214 −0.160437
\(778\) 0 0
\(779\) −33.8885 −1.21418
\(780\) 0 0
\(781\) −18.4721 −0.660985
\(782\) 0 0
\(783\) −8.47214 −0.302769
\(784\) 0 0
\(785\) 22.2492 0.794109
\(786\) 0 0
\(787\) 7.63932 0.272312 0.136156 0.990687i \(-0.456525\pi\)
0.136156 + 0.990687i \(0.456525\pi\)
\(788\) 0 0
\(789\) 0.472136 0.0168085
\(790\) 0 0
\(791\) 20.4721 0.727905
\(792\) 0 0
\(793\) 14.9443 0.530687
\(794\) 0 0
\(795\) 12.3607 0.438388
\(796\) 0 0
\(797\) −30.4721 −1.07938 −0.539689 0.841864i \(-0.681458\pi\)
−0.539689 + 0.841864i \(0.681458\pi\)
\(798\) 0 0
\(799\) −19.0557 −0.674143
\(800\) 0 0
\(801\) −10.1803 −0.359705
\(802\) 0 0
\(803\) 14.4721 0.510711
\(804\) 0 0
\(805\) −5.52786 −0.194832
\(806\) 0 0
\(807\) −4.94427 −0.174047
\(808\) 0 0
\(809\) 12.4721 0.438497 0.219248 0.975669i \(-0.429639\pi\)
0.219248 + 0.975669i \(0.429639\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 23.4164 0.821249
\(814\) 0 0
\(815\) 3.05573 0.107037
\(816\) 0 0
\(817\) −25.8885 −0.905725
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −46.6525 −1.62818 −0.814091 0.580737i \(-0.802765\pi\)
−0.814091 + 0.580737i \(0.802765\pi\)
\(822\) 0 0
\(823\) 47.7771 1.66540 0.832702 0.553721i \(-0.186793\pi\)
0.832702 + 0.553721i \(0.186793\pi\)
\(824\) 0 0
\(825\) 11.2361 0.391190
\(826\) 0 0
\(827\) −37.4853 −1.30349 −0.651746 0.758438i \(-0.725963\pi\)
−0.651746 + 0.758438i \(0.725963\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 15.8885 0.551167
\(832\) 0 0
\(833\) −2.47214 −0.0856544
\(834\) 0 0
\(835\) 22.4721 0.777680
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −29.8197 −1.02949 −0.514744 0.857344i \(-0.672113\pi\)
−0.514744 + 0.857344i \(0.672113\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) 0 0
\(843\) 18.6525 0.642425
\(844\) 0 0
\(845\) −1.23607 −0.0425220
\(846\) 0 0
\(847\) −0.527864 −0.0181376
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 20.0000 0.685591
\(852\) 0 0
\(853\) −20.8328 −0.713302 −0.356651 0.934238i \(-0.616081\pi\)
−0.356651 + 0.934238i \(0.616081\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) 40.9443 1.39863 0.699315 0.714814i \(-0.253489\pi\)
0.699315 + 0.714814i \(0.253489\pi\)
\(858\) 0 0
\(859\) −10.1115 −0.344998 −0.172499 0.985010i \(-0.555184\pi\)
−0.172499 + 0.985010i \(0.555184\pi\)
\(860\) 0 0
\(861\) −5.23607 −0.178445
\(862\) 0 0
\(863\) −52.1803 −1.77624 −0.888120 0.459612i \(-0.847988\pi\)
−0.888120 + 0.459612i \(0.847988\pi\)
\(864\) 0 0
\(865\) −14.8328 −0.504331
\(866\) 0 0
\(867\) 10.8885 0.369794
\(868\) 0 0
\(869\) −33.8885 −1.14959
\(870\) 0 0
\(871\) −2.47214 −0.0837651
\(872\) 0 0
\(873\) 6.94427 0.235028
\(874\) 0 0
\(875\) 10.4721 0.354023
\(876\) 0 0
\(877\) 7.88854 0.266377 0.133189 0.991091i \(-0.457478\pi\)
0.133189 + 0.991091i \(0.457478\pi\)
\(878\) 0 0
\(879\) −5.23607 −0.176608
\(880\) 0 0
\(881\) 30.8328 1.03878 0.519392 0.854536i \(-0.326158\pi\)
0.519392 + 0.854536i \(0.326158\pi\)
\(882\) 0 0
\(883\) −15.4164 −0.518803 −0.259402 0.965770i \(-0.583525\pi\)
−0.259402 + 0.965770i \(0.583525\pi\)
\(884\) 0 0
\(885\) 11.4164 0.383758
\(886\) 0 0
\(887\) −5.16718 −0.173497 −0.0867485 0.996230i \(-0.527648\pi\)
−0.0867485 + 0.996230i \(0.527648\pi\)
\(888\) 0 0
\(889\) −2.47214 −0.0829128
\(890\) 0 0
\(891\) 3.23607 0.108412
\(892\) 0 0
\(893\) −49.8885 −1.66946
\(894\) 0 0
\(895\) 8.58359 0.286918
\(896\) 0 0
\(897\) −4.47214 −0.149320
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −24.7214 −0.823588
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 0 0
\(905\) −24.5836 −0.817186
\(906\) 0 0
\(907\) −28.3607 −0.941701 −0.470850 0.882213i \(-0.656053\pi\)
−0.470850 + 0.882213i \(0.656053\pi\)
\(908\) 0 0
\(909\) −8.94427 −0.296663
\(910\) 0 0
\(911\) 31.8885 1.05651 0.528257 0.849084i \(-0.322846\pi\)
0.528257 + 0.849084i \(0.322846\pi\)
\(912\) 0 0
\(913\) 8.94427 0.296012
\(914\) 0 0
\(915\) 18.4721 0.610670
\(916\) 0 0
\(917\) −1.52786 −0.0504545
\(918\) 0 0
\(919\) 1.52786 0.0503996 0.0251998 0.999682i \(-0.491978\pi\)
0.0251998 + 0.999682i \(0.491978\pi\)
\(920\) 0 0
\(921\) 5.88854 0.194034
\(922\) 0 0
\(923\) −5.70820 −0.187888
\(924\) 0 0
\(925\) −15.5279 −0.510553
\(926\) 0 0
\(927\) −16.9443 −0.556523
\(928\) 0 0
\(929\) −44.6525 −1.46500 −0.732500 0.680767i \(-0.761647\pi\)
−0.732500 + 0.680767i \(0.761647\pi\)
\(930\) 0 0
\(931\) −6.47214 −0.212116
\(932\) 0 0
\(933\) 1.52786 0.0500200
\(934\) 0 0
\(935\) 9.88854 0.323390
\(936\) 0 0
\(937\) −22.9443 −0.749557 −0.374778 0.927114i \(-0.622281\pi\)
−0.374778 + 0.927114i \(0.622281\pi\)
\(938\) 0 0
\(939\) −20.8328 −0.679853
\(940\) 0 0
\(941\) −50.5410 −1.64759 −0.823795 0.566888i \(-0.808147\pi\)
−0.823795 + 0.566888i \(0.808147\pi\)
\(942\) 0 0
\(943\) 23.4164 0.762543
\(944\) 0 0
\(945\) 1.23607 0.0402093
\(946\) 0 0
\(947\) 7.59675 0.246861 0.123431 0.992353i \(-0.460610\pi\)
0.123431 + 0.992353i \(0.460610\pi\)
\(948\) 0 0
\(949\) 4.47214 0.145172
\(950\) 0 0
\(951\) 28.1803 0.913810
\(952\) 0 0
\(953\) −41.1935 −1.33439 −0.667194 0.744884i \(-0.732505\pi\)
−0.667194 + 0.744884i \(0.732505\pi\)
\(954\) 0 0
\(955\) 8.58359 0.277759
\(956\) 0 0
\(957\) −27.4164 −0.886247
\(958\) 0 0
\(959\) 15.2361 0.491998
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −3.52786 −0.113684
\(964\) 0 0
\(965\) −22.6950 −0.730579
\(966\) 0 0
\(967\) −40.3607 −1.29791 −0.648956 0.760826i \(-0.724794\pi\)
−0.648956 + 0.760826i \(0.724794\pi\)
\(968\) 0 0
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) 29.8885 0.959169 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(972\) 0 0
\(973\) 17.8885 0.573480
\(974\) 0 0
\(975\) 3.47214 0.111197
\(976\) 0 0
\(977\) −19.2361 −0.615416 −0.307708 0.951481i \(-0.599562\pi\)
−0.307708 + 0.951481i \(0.599562\pi\)
\(978\) 0 0
\(979\) −32.9443 −1.05290
\(980\) 0 0
\(981\) 3.52786 0.112636
\(982\) 0 0
\(983\) 59.1246 1.88578 0.942891 0.333101i \(-0.108095\pi\)
0.942891 + 0.333101i \(0.108095\pi\)
\(984\) 0 0
\(985\) −0.944272 −0.0300870
\(986\) 0 0
\(987\) −7.70820 −0.245355
\(988\) 0 0
\(989\) 17.8885 0.568823
\(990\) 0 0
\(991\) 6.11146 0.194137 0.0970684 0.995278i \(-0.469053\pi\)
0.0970684 + 0.995278i \(0.469053\pi\)
\(992\) 0 0
\(993\) 34.8328 1.10539
\(994\) 0 0
\(995\) 19.7771 0.626976
\(996\) 0 0
\(997\) 17.7771 0.563006 0.281503 0.959560i \(-0.409167\pi\)
0.281503 + 0.959560i \(0.409167\pi\)
\(998\) 0 0
\(999\) −4.47214 −0.141492
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.f.1.1 2
3.2 odd 2 3276.2.a.l.1.2 2
4.3 odd 2 4368.2.a.bl.1.1 2
7.6 odd 2 7644.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1092.2.a.f.1.1 2 1.1 even 1 trivial
3276.2.a.l.1.2 2 3.2 odd 2
4368.2.a.bl.1.1 2 4.3 odd 2
7644.2.a.p.1.2 2 7.6 odd 2