Properties

Label 2200.2.a.s.1.1
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
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] = [2200,2,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, 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("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(17.5670884447\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{3} -1.56155 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} -1.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} -2.00000 q^{13} -3.56155 q^{17} -1.56155 q^{19} +2.43845 q^{21} +3.12311 q^{23} +5.56155 q^{27} -2.68466 q^{29} +2.43845 q^{31} -1.56155 q^{33} -6.68466 q^{37} +3.12311 q^{39} +2.00000 q^{41} +6.24621 q^{43} +4.87689 q^{47} -4.56155 q^{49} +5.56155 q^{51} -0.438447 q^{53} +2.43845 q^{57} -7.12311 q^{59} +14.6847 q^{61} +0.876894 q^{63} +10.2462 q^{67} -4.87689 q^{69} -8.68466 q^{71} +2.00000 q^{73} -1.56155 q^{77} +9.36932 q^{79} -7.00000 q^{81} +3.12311 q^{83} +4.19224 q^{87} -8.43845 q^{89} +3.12311 q^{91} -3.80776 q^{93} +1.12311 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{7} + 3 q^{9} + 2 q^{11} - 4 q^{13} - 3 q^{17} + q^{19} + 9 q^{21} - 2 q^{23} + 7 q^{27} + 7 q^{29} + 9 q^{31} + q^{33} - q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} + 18 q^{47} - 5 q^{49} + 7 q^{51} - 5 q^{53} + 9 q^{57} - 6 q^{59} + 17 q^{61} + 10 q^{63} + 4 q^{67} - 18 q^{69} - 5 q^{71} + 4 q^{73} + q^{77} - 6 q^{79} - 14 q^{81} - 2 q^{83} + 29 q^{87} - 21 q^{89} - 2 q^{91} + 13 q^{93} - 6 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.56155 −0.590211 −0.295106 0.955465i \(-0.595355\pi\)
−0.295106 + 0.955465i \(0.595355\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.56155 −0.863803 −0.431902 0.901921i \(-0.642157\pi\)
−0.431902 + 0.901921i \(0.642157\pi\)
\(18\) 0 0
\(19\) −1.56155 −0.358245 −0.179122 0.983827i \(-0.557326\pi\)
−0.179122 + 0.983827i \(0.557326\pi\)
\(20\) 0 0
\(21\) 2.43845 0.532113
\(22\) 0 0
\(23\) 3.12311 0.651213 0.325606 0.945505i \(-0.394432\pi\)
0.325606 + 0.945505i \(0.394432\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) −2.68466 −0.498529 −0.249264 0.968436i \(-0.580189\pi\)
−0.249264 + 0.968436i \(0.580189\pi\)
\(30\) 0 0
\(31\) 2.43845 0.437958 0.218979 0.975730i \(-0.429727\pi\)
0.218979 + 0.975730i \(0.429727\pi\)
\(32\) 0 0
\(33\) −1.56155 −0.271831
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.68466 −1.09895 −0.549476 0.835510i \(-0.685172\pi\)
−0.549476 + 0.835510i \(0.685172\pi\)
\(38\) 0 0
\(39\) 3.12311 0.500097
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 6.24621 0.952538 0.476269 0.879300i \(-0.341989\pi\)
0.476269 + 0.879300i \(0.341989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.87689 0.711368 0.355684 0.934606i \(-0.384248\pi\)
0.355684 + 0.934606i \(0.384248\pi\)
\(48\) 0 0
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) 5.56155 0.778773
\(52\) 0 0
\(53\) −0.438447 −0.0602254 −0.0301127 0.999547i \(-0.509587\pi\)
−0.0301127 + 0.999547i \(0.509587\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.43845 0.322980
\(58\) 0 0
\(59\) −7.12311 −0.927349 −0.463675 0.886006i \(-0.653469\pi\)
−0.463675 + 0.886006i \(0.653469\pi\)
\(60\) 0 0
\(61\) 14.6847 1.88018 0.940089 0.340929i \(-0.110742\pi\)
0.940089 + 0.340929i \(0.110742\pi\)
\(62\) 0 0
\(63\) 0.876894 0.110478
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.2462 1.25177 0.625887 0.779914i \(-0.284737\pi\)
0.625887 + 0.779914i \(0.284737\pi\)
\(68\) 0 0
\(69\) −4.87689 −0.587109
\(70\) 0 0
\(71\) −8.68466 −1.03068 −0.515340 0.856986i \(-0.672334\pi\)
−0.515340 + 0.856986i \(0.672334\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.56155 −0.177955
\(78\) 0 0
\(79\) 9.36932 1.05413 0.527065 0.849825i \(-0.323292\pi\)
0.527065 + 0.849825i \(0.323292\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 3.12311 0.342805 0.171403 0.985201i \(-0.445170\pi\)
0.171403 + 0.985201i \(0.445170\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.19224 0.449455
\(88\) 0 0
\(89\) −8.43845 −0.894474 −0.447237 0.894416i \(-0.647592\pi\)
−0.447237 + 0.894416i \(0.647592\pi\)
\(90\) 0 0
\(91\) 3.12311 0.327390
\(92\) 0 0
\(93\) −3.80776 −0.394847
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.12311 0.114034 0.0570170 0.998373i \(-0.481841\pi\)
0.0570170 + 0.998373i \(0.481841\pi\)
\(98\) 0 0
\(99\) −0.561553 −0.0564382
\(100\) 0 0
\(101\) 12.2462 1.21854 0.609272 0.792961i \(-0.291462\pi\)
0.609272 + 0.792961i \(0.291462\pi\)
\(102\) 0 0
\(103\) 6.24621 0.615457 0.307729 0.951474i \(-0.400431\pi\)
0.307729 + 0.951474i \(0.400431\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) −14.4924 −1.38812 −0.694061 0.719916i \(-0.744180\pi\)
−0.694061 + 0.719916i \(0.744180\pi\)
\(110\) 0 0
\(111\) 10.4384 0.990774
\(112\) 0 0
\(113\) 10.8769 1.02321 0.511606 0.859220i \(-0.329051\pi\)
0.511606 + 0.859220i \(0.329051\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.12311 0.103831
\(118\) 0 0
\(119\) 5.56155 0.509827
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.12311 −0.281601
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.24621 0.199319 0.0996595 0.995022i \(-0.468225\pi\)
0.0996595 + 0.995022i \(0.468225\pi\)
\(128\) 0 0
\(129\) −9.75379 −0.858773
\(130\) 0 0
\(131\) 4.68466 0.409301 0.204650 0.978835i \(-0.434394\pi\)
0.204650 + 0.978835i \(0.434394\pi\)
\(132\) 0 0
\(133\) 2.43845 0.211440
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.6155 1.84674 0.923370 0.383912i \(-0.125423\pi\)
0.923370 + 0.383912i \(0.125423\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −7.61553 −0.641343
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.12311 0.587504
\(148\) 0 0
\(149\) 14.6847 1.20301 0.601507 0.798867i \(-0.294567\pi\)
0.601507 + 0.798867i \(0.294567\pi\)
\(150\) 0 0
\(151\) 4.87689 0.396876 0.198438 0.980113i \(-0.436413\pi\)
0.198438 + 0.980113i \(0.436413\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.80776 0.463510 0.231755 0.972774i \(-0.425553\pi\)
0.231755 + 0.972774i \(0.425553\pi\)
\(158\) 0 0
\(159\) 0.684658 0.0542969
\(160\) 0 0
\(161\) −4.87689 −0.384353
\(162\) 0 0
\(163\) −23.8078 −1.86477 −0.932384 0.361469i \(-0.882275\pi\)
−0.932384 + 0.361469i \(0.882275\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.43845 0.498222 0.249111 0.968475i \(-0.419862\pi\)
0.249111 + 0.968475i \(0.419862\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0.876894 0.0670578
\(172\) 0 0
\(173\) 12.2462 0.931062 0.465531 0.885032i \(-0.345863\pi\)
0.465531 + 0.885032i \(0.345863\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 11.1231 0.836064
\(178\) 0 0
\(179\) 10.2462 0.765838 0.382919 0.923782i \(-0.374919\pi\)
0.382919 + 0.923782i \(0.374919\pi\)
\(180\) 0 0
\(181\) 12.2462 0.910254 0.455127 0.890427i \(-0.349594\pi\)
0.455127 + 0.890427i \(0.349594\pi\)
\(182\) 0 0
\(183\) −22.9309 −1.69510
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.56155 −0.260447
\(188\) 0 0
\(189\) −8.68466 −0.631716
\(190\) 0 0
\(191\) −6.24621 −0.451960 −0.225980 0.974132i \(-0.572558\pi\)
−0.225980 + 0.974132i \(0.572558\pi\)
\(192\) 0 0
\(193\) −25.8078 −1.85768 −0.928842 0.370477i \(-0.879194\pi\)
−0.928842 + 0.370477i \(0.879194\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −19.3693 −1.38001 −0.690003 0.723806i \(-0.742391\pi\)
−0.690003 + 0.723806i \(0.742391\pi\)
\(198\) 0 0
\(199\) 19.8078 1.40414 0.702068 0.712110i \(-0.252260\pi\)
0.702068 + 0.712110i \(0.252260\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) 4.19224 0.294237
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.75379 −0.121897
\(208\) 0 0
\(209\) −1.56155 −0.108015
\(210\) 0 0
\(211\) 20.6847 1.42399 0.711995 0.702184i \(-0.247792\pi\)
0.711995 + 0.702184i \(0.247792\pi\)
\(212\) 0 0
\(213\) 13.5616 0.929222
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.80776 −0.258488
\(218\) 0 0
\(219\) −3.12311 −0.211040
\(220\) 0 0
\(221\) 7.12311 0.479152
\(222\) 0 0
\(223\) −4.87689 −0.326581 −0.163291 0.986578i \(-0.552211\pi\)
−0.163291 + 0.986578i \(0.552211\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.1231 −0.738266 −0.369133 0.929376i \(-0.620345\pi\)
−0.369133 + 0.929376i \(0.620345\pi\)
\(228\) 0 0
\(229\) 24.7386 1.63477 0.817387 0.576088i \(-0.195422\pi\)
0.817387 + 0.576088i \(0.195422\pi\)
\(230\) 0 0
\(231\) 2.43845 0.160438
\(232\) 0 0
\(233\) 8.93087 0.585081 0.292540 0.956253i \(-0.405499\pi\)
0.292540 + 0.956253i \(0.405499\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −14.6307 −0.950365
\(238\) 0 0
\(239\) −15.6155 −1.01008 −0.505042 0.863095i \(-0.668523\pi\)
−0.505042 + 0.863095i \(0.668523\pi\)
\(240\) 0 0
\(241\) 0.246211 0.0158599 0.00792993 0.999969i \(-0.497476\pi\)
0.00792993 + 0.999969i \(0.497476\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.12311 0.198718
\(248\) 0 0
\(249\) −4.87689 −0.309061
\(250\) 0 0
\(251\) 11.6155 0.733166 0.366583 0.930385i \(-0.380528\pi\)
0.366583 + 0.930385i \(0.380528\pi\)
\(252\) 0 0
\(253\) 3.12311 0.196348
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.7386 −1.29364 −0.646820 0.762643i \(-0.723902\pi\)
−0.646820 + 0.762643i \(0.723902\pi\)
\(258\) 0 0
\(259\) 10.4384 0.648614
\(260\) 0 0
\(261\) 1.50758 0.0933167
\(262\) 0 0
\(263\) 15.8078 0.974748 0.487374 0.873193i \(-0.337955\pi\)
0.487374 + 0.873193i \(0.337955\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 13.1771 0.806424
\(268\) 0 0
\(269\) 21.6155 1.31792 0.658961 0.752177i \(-0.270996\pi\)
0.658961 + 0.752177i \(0.270996\pi\)
\(270\) 0 0
\(271\) −4.49242 −0.272895 −0.136448 0.990647i \(-0.543569\pi\)
−0.136448 + 0.990647i \(0.543569\pi\)
\(272\) 0 0
\(273\) −4.87689 −0.295163
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.12311 0.0674809 0.0337404 0.999431i \(-0.489258\pi\)
0.0337404 + 0.999431i \(0.489258\pi\)
\(278\) 0 0
\(279\) −1.36932 −0.0819789
\(280\) 0 0
\(281\) 8.24621 0.491928 0.245964 0.969279i \(-0.420896\pi\)
0.245964 + 0.969279i \(0.420896\pi\)
\(282\) 0 0
\(283\) −1.36932 −0.0813974 −0.0406987 0.999171i \(-0.512958\pi\)
−0.0406987 + 0.999171i \(0.512958\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.12311 −0.184351
\(288\) 0 0
\(289\) −4.31534 −0.253844
\(290\) 0 0
\(291\) −1.75379 −0.102809
\(292\) 0 0
\(293\) 18.4924 1.08034 0.540169 0.841556i \(-0.318360\pi\)
0.540169 + 0.841556i \(0.318360\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.56155 0.322714
\(298\) 0 0
\(299\) −6.24621 −0.361228
\(300\) 0 0
\(301\) −9.75379 −0.562199
\(302\) 0 0
\(303\) −19.1231 −1.09859
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −15.6155 −0.891225 −0.445613 0.895226i \(-0.647014\pi\)
−0.445613 + 0.895226i \(0.647014\pi\)
\(308\) 0 0
\(309\) −9.75379 −0.554874
\(310\) 0 0
\(311\) −29.5616 −1.67628 −0.838141 0.545454i \(-0.816357\pi\)
−0.838141 + 0.545454i \(0.816357\pi\)
\(312\) 0 0
\(313\) 2.49242 0.140880 0.0704400 0.997516i \(-0.477560\pi\)
0.0704400 + 0.997516i \(0.477560\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −33.8078 −1.89883 −0.949417 0.314019i \(-0.898324\pi\)
−0.949417 + 0.314019i \(0.898324\pi\)
\(318\) 0 0
\(319\) −2.68466 −0.150312
\(320\) 0 0
\(321\) −24.9848 −1.39452
\(322\) 0 0
\(323\) 5.56155 0.309453
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 22.6307 1.25148
\(328\) 0 0
\(329\) −7.61553 −0.419858
\(330\) 0 0
\(331\) 24.4924 1.34623 0.673113 0.739540i \(-0.264957\pi\)
0.673113 + 0.739540i \(0.264957\pi\)
\(332\) 0 0
\(333\) 3.75379 0.205706
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −14.6847 −0.799924 −0.399962 0.916532i \(-0.630977\pi\)
−0.399962 + 0.916532i \(0.630977\pi\)
\(338\) 0 0
\(339\) −16.9848 −0.922490
\(340\) 0 0
\(341\) 2.43845 0.132049
\(342\) 0 0
\(343\) 18.0540 0.974823
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.1231 1.45604 0.728022 0.685553i \(-0.240440\pi\)
0.728022 + 0.685553i \(0.240440\pi\)
\(348\) 0 0
\(349\) −22.4924 −1.20399 −0.601996 0.798499i \(-0.705628\pi\)
−0.601996 + 0.798499i \(0.705628\pi\)
\(350\) 0 0
\(351\) −11.1231 −0.593707
\(352\) 0 0
\(353\) 1.50758 0.0802403 0.0401201 0.999195i \(-0.487226\pi\)
0.0401201 + 0.999195i \(0.487226\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.68466 −0.459641
\(358\) 0 0
\(359\) 9.75379 0.514785 0.257393 0.966307i \(-0.417137\pi\)
0.257393 + 0.966307i \(0.417137\pi\)
\(360\) 0 0
\(361\) −16.5616 −0.871661
\(362\) 0 0
\(363\) −1.56155 −0.0819603
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −34.7386 −1.81334 −0.906671 0.421839i \(-0.861385\pi\)
−0.906671 + 0.421839i \(0.861385\pi\)
\(368\) 0 0
\(369\) −1.12311 −0.0584665
\(370\) 0 0
\(371\) 0.684658 0.0355457
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.36932 0.276534
\(378\) 0 0
\(379\) −16.8769 −0.866908 −0.433454 0.901176i \(-0.642705\pi\)
−0.433454 + 0.901176i \(0.642705\pi\)
\(380\) 0 0
\(381\) −3.50758 −0.179699
\(382\) 0 0
\(383\) 6.63068 0.338812 0.169406 0.985546i \(-0.445815\pi\)
0.169406 + 0.985546i \(0.445815\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.50758 −0.178300
\(388\) 0 0
\(389\) 15.7538 0.798749 0.399374 0.916788i \(-0.369227\pi\)
0.399374 + 0.916788i \(0.369227\pi\)
\(390\) 0 0
\(391\) −11.1231 −0.562520
\(392\) 0 0
\(393\) −7.31534 −0.369010
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) −3.80776 −0.190627
\(400\) 0 0
\(401\) −11.5616 −0.577356 −0.288678 0.957426i \(-0.593216\pi\)
−0.288678 + 0.957426i \(0.593216\pi\)
\(402\) 0 0
\(403\) −4.87689 −0.242935
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.68466 −0.331346
\(408\) 0 0
\(409\) −1.12311 −0.0555340 −0.0277670 0.999614i \(-0.508840\pi\)
−0.0277670 + 0.999614i \(0.508840\pi\)
\(410\) 0 0
\(411\) −33.7538 −1.66495
\(412\) 0 0
\(413\) 11.1231 0.547332
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −18.7386 −0.917635
\(418\) 0 0
\(419\) 22.7386 1.11085 0.555427 0.831565i \(-0.312555\pi\)
0.555427 + 0.831565i \(0.312555\pi\)
\(420\) 0 0
\(421\) −33.6155 −1.63832 −0.819160 0.573565i \(-0.805560\pi\)
−0.819160 + 0.573565i \(0.805560\pi\)
\(422\) 0 0
\(423\) −2.73863 −0.133157
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −22.9309 −1.10970
\(428\) 0 0
\(429\) 3.12311 0.150785
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −8.63068 −0.414764 −0.207382 0.978260i \(-0.566494\pi\)
−0.207382 + 0.978260i \(0.566494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.87689 −0.233293
\(438\) 0 0
\(439\) −17.7538 −0.847342 −0.423671 0.905816i \(-0.639259\pi\)
−0.423671 + 0.905816i \(0.639259\pi\)
\(440\) 0 0
\(441\) 2.56155 0.121979
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −22.9309 −1.08459
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 0 0
\(453\) −7.61553 −0.357809
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.4384 −1.14318 −0.571591 0.820539i \(-0.693674\pi\)
−0.571591 + 0.820539i \(0.693674\pi\)
\(458\) 0 0
\(459\) −19.8078 −0.924547
\(460\) 0 0
\(461\) 16.4384 0.765615 0.382807 0.923828i \(-0.374957\pi\)
0.382807 + 0.923828i \(0.374957\pi\)
\(462\) 0 0
\(463\) −17.3693 −0.807221 −0.403610 0.914931i \(-0.632245\pi\)
−0.403610 + 0.914931i \(0.632245\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.9309 0.876016 0.438008 0.898971i \(-0.355684\pi\)
0.438008 + 0.898971i \(0.355684\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −9.06913 −0.417883
\(472\) 0 0
\(473\) 6.24621 0.287201
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.246211 0.0112732
\(478\) 0 0
\(479\) 32.9848 1.50712 0.753558 0.657381i \(-0.228336\pi\)
0.753558 + 0.657381i \(0.228336\pi\)
\(480\) 0 0
\(481\) 13.3693 0.609588
\(482\) 0 0
\(483\) 7.61553 0.346519
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.2462 −0.645557 −0.322779 0.946474i \(-0.604617\pi\)
−0.322779 + 0.946474i \(0.604617\pi\)
\(488\) 0 0
\(489\) 37.1771 1.68121
\(490\) 0 0
\(491\) −30.0540 −1.35632 −0.678158 0.734916i \(-0.737222\pi\)
−0.678158 + 0.734916i \(0.737222\pi\)
\(492\) 0 0
\(493\) 9.56155 0.430631
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.5616 0.608319
\(498\) 0 0
\(499\) −16.4924 −0.738302 −0.369151 0.929369i \(-0.620352\pi\)
−0.369151 + 0.929369i \(0.620352\pi\)
\(500\) 0 0
\(501\) −10.0540 −0.449178
\(502\) 0 0
\(503\) −2.24621 −0.100154 −0.0500768 0.998745i \(-0.515947\pi\)
−0.0500768 + 0.998745i \(0.515947\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 14.0540 0.624159
\(508\) 0 0
\(509\) −3.36932 −0.149342 −0.0746712 0.997208i \(-0.523791\pi\)
−0.0746712 + 0.997208i \(0.523791\pi\)
\(510\) 0 0
\(511\) −3.12311 −0.138158
\(512\) 0 0
\(513\) −8.68466 −0.383437
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.87689 0.214486
\(518\) 0 0
\(519\) −19.1231 −0.839411
\(520\) 0 0
\(521\) −40.7386 −1.78479 −0.892396 0.451253i \(-0.850977\pi\)
−0.892396 + 0.451253i \(0.850977\pi\)
\(522\) 0 0
\(523\) 36.4924 1.59570 0.797851 0.602855i \(-0.205970\pi\)
0.797851 + 0.602855i \(0.205970\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.68466 −0.378310
\(528\) 0 0
\(529\) −13.2462 −0.575922
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) −4.56155 −0.196480
\(540\) 0 0
\(541\) 36.5464 1.57125 0.785626 0.618701i \(-0.212341\pi\)
0.785626 + 0.618701i \(0.212341\pi\)
\(542\) 0 0
\(543\) −19.1231 −0.820651
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) −8.24621 −0.351940
\(550\) 0 0
\(551\) 4.19224 0.178595
\(552\) 0 0
\(553\) −14.6307 −0.622160
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.5076 −0.572334 −0.286167 0.958180i \(-0.592381\pi\)
−0.286167 + 0.958180i \(0.592381\pi\)
\(558\) 0 0
\(559\) −12.4924 −0.528373
\(560\) 0 0
\(561\) 5.56155 0.234809
\(562\) 0 0
\(563\) 11.1231 0.468783 0.234392 0.972142i \(-0.424690\pi\)
0.234392 + 0.972142i \(0.424690\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 10.9309 0.459053
\(568\) 0 0
\(569\) 46.1080 1.93295 0.966473 0.256769i \(-0.0826578\pi\)
0.966473 + 0.256769i \(0.0826578\pi\)
\(570\) 0 0
\(571\) −31.4233 −1.31502 −0.657512 0.753444i \(-0.728391\pi\)
−0.657512 + 0.753444i \(0.728391\pi\)
\(572\) 0 0
\(573\) 9.75379 0.407470
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 24.7386 1.02988 0.514941 0.857225i \(-0.327814\pi\)
0.514941 + 0.857225i \(0.327814\pi\)
\(578\) 0 0
\(579\) 40.3002 1.67482
\(580\) 0 0
\(581\) −4.87689 −0.202328
\(582\) 0 0
\(583\) −0.438447 −0.0181586
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −44.6847 −1.84433 −0.922167 0.386793i \(-0.873583\pi\)
−0.922167 + 0.386793i \(0.873583\pi\)
\(588\) 0 0
\(589\) −3.80776 −0.156896
\(590\) 0 0
\(591\) 30.2462 1.24416
\(592\) 0 0
\(593\) −34.4924 −1.41643 −0.708217 0.705995i \(-0.750500\pi\)
−0.708217 + 0.705995i \(0.750500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −30.9309 −1.26592
\(598\) 0 0
\(599\) −21.5616 −0.880981 −0.440491 0.897757i \(-0.645195\pi\)
−0.440491 + 0.897757i \(0.645195\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) −5.75379 −0.234312
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.192236 −0.00780262 −0.00390131 0.999992i \(-0.501242\pi\)
−0.00390131 + 0.999992i \(0.501242\pi\)
\(608\) 0 0
\(609\) −6.54640 −0.265273
\(610\) 0 0
\(611\) −9.75379 −0.394596
\(612\) 0 0
\(613\) 32.7386 1.32230 0.661150 0.750253i \(-0.270068\pi\)
0.661150 + 0.750253i \(0.270068\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.3693 1.26288 0.631441 0.775424i \(-0.282464\pi\)
0.631441 + 0.775424i \(0.282464\pi\)
\(618\) 0 0
\(619\) 24.4924 0.984434 0.492217 0.870473i \(-0.336187\pi\)
0.492217 + 0.870473i \(0.336187\pi\)
\(620\) 0 0
\(621\) 17.3693 0.697007
\(622\) 0 0
\(623\) 13.1771 0.527929
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.43845 0.0973822
\(628\) 0 0
\(629\) 23.8078 0.949278
\(630\) 0 0
\(631\) 3.80776 0.151585 0.0757923 0.997124i \(-0.475851\pi\)
0.0757923 + 0.997124i \(0.475851\pi\)
\(632\) 0 0
\(633\) −32.3002 −1.28382
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.12311 0.361471
\(638\) 0 0
\(639\) 4.87689 0.192927
\(640\) 0 0
\(641\) 7.17708 0.283478 0.141739 0.989904i \(-0.454731\pi\)
0.141739 + 0.989904i \(0.454731\pi\)
\(642\) 0 0
\(643\) −11.3153 −0.446234 −0.223117 0.974792i \(-0.571623\pi\)
−0.223117 + 0.974792i \(0.571623\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.87689 0.191731 0.0958653 0.995394i \(-0.469438\pi\)
0.0958653 + 0.995394i \(0.469438\pi\)
\(648\) 0 0
\(649\) −7.12311 −0.279606
\(650\) 0 0
\(651\) 5.94602 0.233043
\(652\) 0 0
\(653\) −9.80776 −0.383807 −0.191904 0.981414i \(-0.561466\pi\)
−0.191904 + 0.981414i \(0.561466\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.12311 −0.0438165
\(658\) 0 0
\(659\) 39.8078 1.55069 0.775345 0.631538i \(-0.217576\pi\)
0.775345 + 0.631538i \(0.217576\pi\)
\(660\) 0 0
\(661\) −0.246211 −0.00957651 −0.00478825 0.999989i \(-0.501524\pi\)
−0.00478825 + 0.999989i \(0.501524\pi\)
\(662\) 0 0
\(663\) −11.1231 −0.431986
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.38447 −0.324648
\(668\) 0 0
\(669\) 7.61553 0.294433
\(670\) 0 0
\(671\) 14.6847 0.566895
\(672\) 0 0
\(673\) −50.7926 −1.95791 −0.978956 0.204073i \(-0.934582\pi\)
−0.978956 + 0.204073i \(0.934582\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.6155 0.523287 0.261644 0.965165i \(-0.415735\pi\)
0.261644 + 0.965165i \(0.415735\pi\)
\(678\) 0 0
\(679\) −1.75379 −0.0673042
\(680\) 0 0
\(681\) 17.3693 0.665594
\(682\) 0 0
\(683\) −2.93087 −0.112147 −0.0560733 0.998427i \(-0.517858\pi\)
−0.0560733 + 0.998427i \(0.517858\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −38.6307 −1.47385
\(688\) 0 0
\(689\) 0.876894 0.0334070
\(690\) 0 0
\(691\) −21.3693 −0.812927 −0.406464 0.913667i \(-0.633238\pi\)
−0.406464 + 0.913667i \(0.633238\pi\)
\(692\) 0 0
\(693\) 0.876894 0.0333105
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −7.12311 −0.269807
\(698\) 0 0
\(699\) −13.9460 −0.527487
\(700\) 0 0
\(701\) 46.6847 1.76326 0.881628 0.471946i \(-0.156448\pi\)
0.881628 + 0.471946i \(0.156448\pi\)
\(702\) 0 0
\(703\) 10.4384 0.393693
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −19.1231 −0.719198
\(708\) 0 0
\(709\) 34.4924 1.29539 0.647695 0.761900i \(-0.275733\pi\)
0.647695 + 0.761900i \(0.275733\pi\)
\(710\) 0 0
\(711\) −5.26137 −0.197317
\(712\) 0 0
\(713\) 7.61553 0.285204
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.3845 0.910655
\(718\) 0 0
\(719\) 53.1771 1.98317 0.991585 0.129455i \(-0.0413229\pi\)
0.991585 + 0.129455i \(0.0413229\pi\)
\(720\) 0 0
\(721\) −9.75379 −0.363250
\(722\) 0 0
\(723\) −0.384472 −0.0142987
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.6155 0.875851 0.437926 0.899011i \(-0.355713\pi\)
0.437926 + 0.899011i \(0.355713\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −22.2462 −0.822806
\(732\) 0 0
\(733\) 10.8769 0.401747 0.200874 0.979617i \(-0.435622\pi\)
0.200874 + 0.979617i \(0.435622\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.2462 0.377424
\(738\) 0 0
\(739\) 6.73863 0.247885 0.123942 0.992289i \(-0.460446\pi\)
0.123942 + 0.992289i \(0.460446\pi\)
\(740\) 0 0
\(741\) −4.87689 −0.179157
\(742\) 0 0
\(743\) −1.56155 −0.0572878 −0.0286439 0.999590i \(-0.509119\pi\)
−0.0286439 + 0.999590i \(0.509119\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.75379 −0.0641678
\(748\) 0 0
\(749\) −24.9848 −0.912926
\(750\) 0 0
\(751\) 29.1771 1.06469 0.532343 0.846529i \(-0.321312\pi\)
0.532343 + 0.846529i \(0.321312\pi\)
\(752\) 0 0
\(753\) −18.1383 −0.660995
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) −4.87689 −0.177020
\(760\) 0 0
\(761\) 12.7386 0.461775 0.230888 0.972980i \(-0.425837\pi\)
0.230888 + 0.972980i \(0.425837\pi\)
\(762\) 0 0
\(763\) 22.6307 0.819286
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.2462 0.514401
\(768\) 0 0
\(769\) −24.7386 −0.892098 −0.446049 0.895009i \(-0.647169\pi\)
−0.446049 + 0.895009i \(0.647169\pi\)
\(770\) 0 0
\(771\) 32.3845 1.16630
\(772\) 0 0
\(773\) 25.3153 0.910530 0.455265 0.890356i \(-0.349545\pi\)
0.455265 + 0.890356i \(0.349545\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −16.3002 −0.584766
\(778\) 0 0
\(779\) −3.12311 −0.111897
\(780\) 0 0
\(781\) −8.68466 −0.310762
\(782\) 0 0
\(783\) −14.9309 −0.533586
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.26137 0.187547 0.0937737 0.995594i \(-0.470107\pi\)
0.0937737 + 0.995594i \(0.470107\pi\)
\(788\) 0 0
\(789\) −24.6847 −0.878797
\(790\) 0 0
\(791\) −16.9848 −0.603912
\(792\) 0 0
\(793\) −29.3693 −1.04294
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 40.2462 1.42559 0.712797 0.701370i \(-0.247428\pi\)
0.712797 + 0.701370i \(0.247428\pi\)
\(798\) 0 0
\(799\) −17.3693 −0.614482
\(800\) 0 0
\(801\) 4.73863 0.167431
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −33.7538 −1.18819
\(808\) 0 0
\(809\) −26.4924 −0.931424 −0.465712 0.884936i \(-0.654202\pi\)
−0.465712 + 0.884936i \(0.654202\pi\)
\(810\) 0 0
\(811\) 47.8078 1.67876 0.839379 0.543547i \(-0.182919\pi\)
0.839379 + 0.543547i \(0.182919\pi\)
\(812\) 0 0
\(813\) 7.01515 0.246032
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −9.75379 −0.341242
\(818\) 0 0
\(819\) −1.75379 −0.0612823
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) 19.5076 0.679991 0.339996 0.940427i \(-0.389574\pi\)
0.339996 + 0.940427i \(0.389574\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.4924 −0.990779 −0.495389 0.868671i \(-0.664975\pi\)
−0.495389 + 0.868671i \(0.664975\pi\)
\(828\) 0 0
\(829\) −19.3693 −0.672724 −0.336362 0.941733i \(-0.609197\pi\)
−0.336362 + 0.941733i \(0.609197\pi\)
\(830\) 0 0
\(831\) −1.75379 −0.0608383
\(832\) 0 0
\(833\) 16.2462 0.562898
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 13.5616 0.468756
\(838\) 0 0
\(839\) 44.4924 1.53605 0.768025 0.640420i \(-0.221240\pi\)
0.768025 + 0.640420i \(0.221240\pi\)
\(840\) 0 0
\(841\) −21.7926 −0.751469
\(842\) 0 0
\(843\) −12.8769 −0.443504
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.56155 −0.0536556
\(848\) 0 0
\(849\) 2.13826 0.0733849
\(850\) 0 0
\(851\) −20.8769 −0.715651
\(852\) 0 0
\(853\) 20.2462 0.693217 0.346609 0.938010i \(-0.387333\pi\)
0.346609 + 0.938010i \(0.387333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.4233 1.00508 0.502540 0.864554i \(-0.332399\pi\)
0.502540 + 0.864554i \(0.332399\pi\)
\(858\) 0 0
\(859\) −47.1231 −1.60782 −0.803910 0.594751i \(-0.797251\pi\)
−0.803910 + 0.594751i \(0.797251\pi\)
\(860\) 0 0
\(861\) 4.87689 0.166204
\(862\) 0 0
\(863\) 31.6155 1.07621 0.538103 0.842879i \(-0.319141\pi\)
0.538103 + 0.842879i \(0.319141\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 6.73863 0.228856
\(868\) 0 0
\(869\) 9.36932 0.317832
\(870\) 0 0
\(871\) −20.4924 −0.694359
\(872\) 0 0
\(873\) −0.630683 −0.0213454
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.7538 0.531968 0.265984 0.963977i \(-0.414303\pi\)
0.265984 + 0.963977i \(0.414303\pi\)
\(878\) 0 0
\(879\) −28.8769 −0.973993
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −34.9309 −1.17552 −0.587759 0.809036i \(-0.699990\pi\)
−0.587759 + 0.809036i \(0.699990\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.7538 0.461807 0.230904 0.972977i \(-0.425832\pi\)
0.230904 + 0.972977i \(0.425832\pi\)
\(888\) 0 0
\(889\) −3.50758 −0.117640
\(890\) 0 0
\(891\) −7.00000 −0.234509
\(892\) 0 0
\(893\) −7.61553 −0.254844
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.75379 0.325670
\(898\) 0 0
\(899\) −6.54640 −0.218335
\(900\) 0 0
\(901\) 1.56155 0.0520229
\(902\) 0 0
\(903\) 15.2311 0.506858
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −46.0540 −1.52920 −0.764599 0.644507i \(-0.777063\pi\)
−0.764599 + 0.644507i \(0.777063\pi\)
\(908\) 0 0
\(909\) −6.87689 −0.228092
\(910\) 0 0
\(911\) −30.9309 −1.02479 −0.512393 0.858751i \(-0.671241\pi\)
−0.512393 + 0.858751i \(0.671241\pi\)
\(912\) 0 0
\(913\) 3.12311 0.103360
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.31534 −0.241574
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 24.3845 0.803496
\(922\) 0 0
\(923\) 17.3693 0.571718
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −3.50758 −0.115204
\(928\) 0 0
\(929\) 7.94602 0.260701 0.130350 0.991468i \(-0.458390\pi\)
0.130350 + 0.991468i \(0.458390\pi\)
\(930\) 0 0
\(931\) 7.12311 0.233450
\(932\) 0 0
\(933\) 46.1619 1.51127
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) −3.89205 −0.127012
\(940\) 0 0
\(941\) −32.9309 −1.07352 −0.536758 0.843736i \(-0.680351\pi\)
−0.536758 + 0.843736i \(0.680351\pi\)
\(942\) 0 0
\(943\) 6.24621 0.203405
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.3153 0.887629 0.443815 0.896119i \(-0.353625\pi\)
0.443815 + 0.896119i \(0.353625\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 52.7926 1.71192
\(952\) 0 0
\(953\) −11.1771 −0.362061 −0.181031 0.983477i \(-0.557943\pi\)
−0.181031 + 0.983477i \(0.557943\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.19224 0.135516
\(958\) 0 0
\(959\) −33.7538 −1.08997
\(960\) 0 0
\(961\) −25.0540 −0.808193
\(962\) 0 0
\(963\) −8.98485 −0.289533
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 58.5464 1.88273 0.941363 0.337397i \(-0.109546\pi\)
0.941363 + 0.337397i \(0.109546\pi\)
\(968\) 0 0
\(969\) −8.68466 −0.278991
\(970\) 0 0
\(971\) −10.2462 −0.328817 −0.164408 0.986392i \(-0.552571\pi\)
−0.164408 + 0.986392i \(0.552571\pi\)
\(972\) 0 0
\(973\) −18.7386 −0.600733
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.1231 1.05970 0.529851 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(978\) 0 0
\(979\) −8.43845 −0.269694
\(980\) 0 0
\(981\) 8.13826 0.259835
\(982\) 0 0
\(983\) −27.1231 −0.865093 −0.432546 0.901612i \(-0.642385\pi\)
−0.432546 + 0.901612i \(0.642385\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 11.8920 0.378528
\(988\) 0 0
\(989\) 19.5076 0.620305
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) −38.2462 −1.21371
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 43.8617 1.38912 0.694558 0.719437i \(-0.255600\pi\)
0.694558 + 0.719437i \(0.255600\pi\)
\(998\) 0 0
\(999\) −37.1771 −1.17623
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 2200.2.a.s.1.1 2
4.3 odd 2 4400.2.a.bj.1.2 2
5.2 odd 4 2200.2.b.i.1849.3 4
5.3 odd 4 2200.2.b.i.1849.2 4
5.4 even 2 440.2.a.e.1.2 2
15.14 odd 2 3960.2.a.w.1.2 2
20.3 even 4 4400.2.b.t.4049.3 4
20.7 even 4 4400.2.b.t.4049.2 4
20.19 odd 2 880.2.a.o.1.1 2
40.19 odd 2 3520.2.a.bk.1.2 2
40.29 even 2 3520.2.a.bp.1.1 2
55.54 odd 2 4840.2.a.j.1.2 2
60.59 even 2 7920.2.a.bu.1.1 2
220.219 even 2 9680.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.e.1.2 2 5.4 even 2
880.2.a.o.1.1 2 20.19 odd 2
2200.2.a.s.1.1 2 1.1 even 1 trivial
2200.2.b.i.1849.2 4 5.3 odd 4
2200.2.b.i.1849.3 4 5.2 odd 4
3520.2.a.bk.1.2 2 40.19 odd 2
3520.2.a.bp.1.1 2 40.29 even 2
3960.2.a.w.1.2 2 15.14 odd 2
4400.2.a.bj.1.2 2 4.3 odd 2
4400.2.b.t.4049.2 4 20.7 even 4
4400.2.b.t.4049.3 4 20.3 even 4
4840.2.a.j.1.2 2 55.54 odd 2
7920.2.a.bu.1.1 2 60.59 even 2
9680.2.a.bs.1.1 2 220.219 even 2