Properties

Label 2200.2.a.m.1.2
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
Analytic rank $1$
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: \(1\)
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 1.2
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)
 
\(f(q)\) \(=\) \(q+1.56155 q^{3} -3.56155 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} -3.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} +3.12311 q^{13} -5.56155 q^{17} +2.43845 q^{19} -5.56155 q^{21} -7.12311 q^{23} -5.56155 q^{27} -0.438447 q^{29} +8.68466 q^{31} +1.56155 q^{33} -9.80776 q^{37} +4.87689 q^{39} -10.0000 q^{41} -5.12311 q^{43} +7.12311 q^{47} +5.68466 q^{49} -8.68466 q^{51} +4.43845 q^{53} +3.80776 q^{57} -13.3693 q^{59} -3.56155 q^{61} +2.00000 q^{63} -11.1231 q^{69} -2.43845 q^{71} -4.87689 q^{73} -3.56155 q^{77} +0.876894 q^{79} -7.00000 q^{81} -10.0000 q^{83} -0.684658 q^{87} +9.80776 q^{89} -11.1231 q^{91} +13.5616 q^{93} -17.1231 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} - 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 7 q^{27} - 5 q^{29} + 5 q^{31} - q^{33} + q^{37} + 18 q^{39} - 20 q^{41} - 2 q^{43} + 6 q^{47} - q^{49} - 5 q^{51} + 13 q^{53} - 13 q^{57} - 2 q^{59} - 3 q^{61} + 4 q^{63} - 14 q^{69} - 9 q^{71} - 18 q^{73} - 3 q^{77} + 10 q^{79} - 14 q^{81} - 20 q^{83} + 11 q^{87} - q^{89} - 14 q^{91} + 23 q^{93} - 26 q^{97} + 3 q^{99}+O(q^{100}) \) 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 \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.56155 −1.34614 −0.673070 0.739579i \(-0.735025\pi\)
−0.673070 + 0.739579i \(0.735025\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 3.12311 0.866194 0.433097 0.901347i \(-0.357421\pi\)
0.433097 + 0.901347i \(0.357421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.56155 −1.34887 −0.674437 0.738332i \(-0.735614\pi\)
−0.674437 + 0.738332i \(0.735614\pi\)
\(18\) 0 0
\(19\) 2.43845 0.559418 0.279709 0.960085i \(-0.409762\pi\)
0.279709 + 0.960085i \(0.409762\pi\)
\(20\) 0 0
\(21\) −5.56155 −1.21363
\(22\) 0 0
\(23\) −7.12311 −1.48527 −0.742635 0.669696i \(-0.766424\pi\)
−0.742635 + 0.669696i \(0.766424\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) −0.438447 −0.0814176 −0.0407088 0.999171i \(-0.512962\pi\)
−0.0407088 + 0.999171i \(0.512962\pi\)
\(30\) 0 0
\(31\) 8.68466 1.55981 0.779905 0.625897i \(-0.215267\pi\)
0.779905 + 0.625897i \(0.215267\pi\)
\(32\) 0 0
\(33\) 1.56155 0.271831
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.80776 −1.61239 −0.806193 0.591652i \(-0.798476\pi\)
−0.806193 + 0.591652i \(0.798476\pi\)
\(38\) 0 0
\(39\) 4.87689 0.780928
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −5.12311 −0.781266 −0.390633 0.920546i \(-0.627744\pi\)
−0.390633 + 0.920546i \(0.627744\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.12311 1.03901 0.519506 0.854467i \(-0.326116\pi\)
0.519506 + 0.854467i \(0.326116\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) 0 0
\(51\) −8.68466 −1.21610
\(52\) 0 0
\(53\) 4.43845 0.609668 0.304834 0.952406i \(-0.401399\pi\)
0.304834 + 0.952406i \(0.401399\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.80776 0.504351
\(58\) 0 0
\(59\) −13.3693 −1.74054 −0.870268 0.492578i \(-0.836055\pi\)
−0.870268 + 0.492578i \(0.836055\pi\)
\(60\) 0 0
\(61\) −3.56155 −0.456010 −0.228005 0.973660i \(-0.573220\pi\)
−0.228005 + 0.973660i \(0.573220\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −11.1231 −1.33906
\(70\) 0 0
\(71\) −2.43845 −0.289390 −0.144695 0.989476i \(-0.546220\pi\)
−0.144695 + 0.989476i \(0.546220\pi\)
\(72\) 0 0
\(73\) −4.87689 −0.570797 −0.285399 0.958409i \(-0.592126\pi\)
−0.285399 + 0.958409i \(0.592126\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.56155 −0.405877
\(78\) 0 0
\(79\) 0.876894 0.0986583 0.0493292 0.998783i \(-0.484292\pi\)
0.0493292 + 0.998783i \(0.484292\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.684658 −0.0734031
\(88\) 0 0
\(89\) 9.80776 1.03962 0.519810 0.854282i \(-0.326003\pi\)
0.519810 + 0.854282i \(0.326003\pi\)
\(90\) 0 0
\(91\) −11.1231 −1.16602
\(92\) 0 0
\(93\) 13.5616 1.40627
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −17.1231 −1.73859 −0.869294 0.494295i \(-0.835426\pi\)
−0.869294 + 0.494295i \(0.835426\pi\)
\(98\) 0 0
\(99\) −0.561553 −0.0564382
\(100\) 0 0
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 0 0
\(103\) 14.2462 1.40372 0.701860 0.712314i \(-0.252353\pi\)
0.701860 + 0.712314i \(0.252353\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.12311 −0.495269 −0.247635 0.968853i \(-0.579653\pi\)
−0.247635 + 0.968853i \(0.579653\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −15.3153 −1.45367
\(112\) 0 0
\(113\) −1.12311 −0.105653 −0.0528264 0.998604i \(-0.516823\pi\)
−0.0528264 + 0.998604i \(0.516823\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.75379 −0.162138
\(118\) 0 0
\(119\) 19.8078 1.81577
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −15.6155 −1.40800
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.36932 0.653921 0.326961 0.945038i \(-0.393976\pi\)
0.326961 + 0.945038i \(0.393976\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 0.684658 0.0598189 0.0299094 0.999553i \(-0.490478\pi\)
0.0299094 + 0.999553i \(0.490478\pi\)
\(132\) 0 0
\(133\) −8.68466 −0.753055
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.3693 1.65483 0.827416 0.561589i \(-0.189810\pi\)
0.827416 + 0.561589i \(0.189810\pi\)
\(138\) 0 0
\(139\) 16.4924 1.39887 0.699435 0.714697i \(-0.253435\pi\)
0.699435 + 0.714697i \(0.253435\pi\)
\(140\) 0 0
\(141\) 11.1231 0.936734
\(142\) 0 0
\(143\) 3.12311 0.261167
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 8.87689 0.732154
\(148\) 0 0
\(149\) −1.31534 −0.107757 −0.0538785 0.998547i \(-0.517158\pi\)
−0.0538785 + 0.998547i \(0.517158\pi\)
\(150\) 0 0
\(151\) −0.876894 −0.0713607 −0.0356803 0.999363i \(-0.511360\pi\)
−0.0356803 + 0.999363i \(0.511360\pi\)
\(152\) 0 0
\(153\) 3.12311 0.252488
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.9309 1.03200 0.515998 0.856590i \(-0.327421\pi\)
0.515998 + 0.856590i \(0.327421\pi\)
\(158\) 0 0
\(159\) 6.93087 0.549654
\(160\) 0 0
\(161\) 25.3693 1.99938
\(162\) 0 0
\(163\) −0.192236 −0.0150571 −0.00752854 0.999972i \(-0.502396\pi\)
−0.00752854 + 0.999972i \(0.502396\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.31534 −0.101784 −0.0508921 0.998704i \(-0.516206\pi\)
−0.0508921 + 0.998704i \(0.516206\pi\)
\(168\) 0 0
\(169\) −3.24621 −0.249709
\(170\) 0 0
\(171\) −1.36932 −0.104714
\(172\) 0 0
\(173\) 21.3693 1.62468 0.812340 0.583185i \(-0.198194\pi\)
0.812340 + 0.583185i \(0.198194\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −20.8769 −1.56920
\(178\) 0 0
\(179\) 8.49242 0.634753 0.317377 0.948300i \(-0.397198\pi\)
0.317377 + 0.948300i \(0.397198\pi\)
\(180\) 0 0
\(181\) 14.4924 1.07721 0.538607 0.842557i \(-0.318951\pi\)
0.538607 + 0.842557i \(0.318951\pi\)
\(182\) 0 0
\(183\) −5.56155 −0.411122
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.56155 −0.406701
\(188\) 0 0
\(189\) 19.8078 1.44080
\(190\) 0 0
\(191\) 1.75379 0.126900 0.0634499 0.997985i \(-0.479790\pi\)
0.0634499 + 0.997985i \(0.479790\pi\)
\(192\) 0 0
\(193\) −19.8078 −1.42579 −0.712897 0.701269i \(-0.752617\pi\)
−0.712897 + 0.701269i \(0.752617\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.2462 −1.29999 −0.649994 0.759939i \(-0.725229\pi\)
−0.649994 + 0.759939i \(0.725229\pi\)
\(198\) 0 0
\(199\) −6.93087 −0.491316 −0.245658 0.969357i \(-0.579004\pi\)
−0.245658 + 0.969357i \(0.579004\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.56155 0.109600
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 2.43845 0.168671
\(210\) 0 0
\(211\) 0.684658 0.0471338 0.0235669 0.999722i \(-0.492498\pi\)
0.0235669 + 0.999722i \(0.492498\pi\)
\(212\) 0 0
\(213\) −3.80776 −0.260904
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −30.9309 −2.09972
\(218\) 0 0
\(219\) −7.61553 −0.514610
\(220\) 0 0
\(221\) −17.3693 −1.16839
\(222\) 0 0
\(223\) 23.1231 1.54844 0.774219 0.632918i \(-0.218143\pi\)
0.774219 + 0.632918i \(0.218143\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −6.49242 −0.429031 −0.214516 0.976721i \(-0.568817\pi\)
−0.214516 + 0.976721i \(0.568817\pi\)
\(230\) 0 0
\(231\) −5.56155 −0.365923
\(232\) 0 0
\(233\) −11.3153 −0.741293 −0.370646 0.928774i \(-0.620864\pi\)
−0.370646 + 0.928774i \(0.620864\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.36932 0.0889467
\(238\) 0 0
\(239\) 0.876894 0.0567216 0.0283608 0.999598i \(-0.490971\pi\)
0.0283608 + 0.999598i \(0.490971\pi\)
\(240\) 0 0
\(241\) −28.7386 −1.85122 −0.925609 0.378481i \(-0.876447\pi\)
−0.925609 + 0.378481i \(0.876447\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.61553 0.484564
\(248\) 0 0
\(249\) −15.6155 −0.989594
\(250\) 0 0
\(251\) 23.1231 1.45952 0.729759 0.683705i \(-0.239632\pi\)
0.729759 + 0.683705i \(0.239632\pi\)
\(252\) 0 0
\(253\) −7.12311 −0.447826
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.75379 0.483668 0.241834 0.970318i \(-0.422251\pi\)
0.241834 + 0.970318i \(0.422251\pi\)
\(258\) 0 0
\(259\) 34.9309 2.17050
\(260\) 0 0
\(261\) 0.246211 0.0152401
\(262\) 0 0
\(263\) −24.4384 −1.50694 −0.753470 0.657483i \(-0.771621\pi\)
−0.753470 + 0.657483i \(0.771621\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.3153 0.937284
\(268\) 0 0
\(269\) −22.8769 −1.39483 −0.697414 0.716668i \(-0.745666\pi\)
−0.697414 + 0.716668i \(0.745666\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) −17.3693 −1.05124
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.7386 −1.12590 −0.562948 0.826493i \(-0.690333\pi\)
−0.562948 + 0.826493i \(0.690333\pi\)
\(278\) 0 0
\(279\) −4.87689 −0.291972
\(280\) 0 0
\(281\) 18.4924 1.10317 0.551583 0.834120i \(-0.314024\pi\)
0.551583 + 0.834120i \(0.314024\pi\)
\(282\) 0 0
\(283\) 18.4924 1.09926 0.549630 0.835408i \(-0.314769\pi\)
0.549630 + 0.835408i \(0.314769\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 35.6155 2.10232
\(288\) 0 0
\(289\) 13.9309 0.819463
\(290\) 0 0
\(291\) −26.7386 −1.56745
\(292\) 0 0
\(293\) 11.1231 0.649819 0.324909 0.945745i \(-0.394666\pi\)
0.324909 + 0.945745i \(0.394666\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.56155 −0.322714
\(298\) 0 0
\(299\) −22.2462 −1.28653
\(300\) 0 0
\(301\) 18.2462 1.05169
\(302\) 0 0
\(303\) −25.3693 −1.45743
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.7386 −1.18362 −0.591808 0.806079i \(-0.701586\pi\)
−0.591808 + 0.806079i \(0.701586\pi\)
\(308\) 0 0
\(309\) 22.2462 1.26554
\(310\) 0 0
\(311\) 21.1771 1.20084 0.600421 0.799684i \(-0.295000\pi\)
0.600421 + 0.799684i \(0.295000\pi\)
\(312\) 0 0
\(313\) −16.2462 −0.918290 −0.459145 0.888361i \(-0.651844\pi\)
−0.459145 + 0.888361i \(0.651844\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.56155 0.200037 0.100018 0.994986i \(-0.468110\pi\)
0.100018 + 0.994986i \(0.468110\pi\)
\(318\) 0 0
\(319\) −0.438447 −0.0245483
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −13.5616 −0.754585
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.12311 −0.172708
\(328\) 0 0
\(329\) −25.3693 −1.39866
\(330\) 0 0
\(331\) −2.24621 −0.123463 −0.0617315 0.998093i \(-0.519662\pi\)
−0.0617315 + 0.998093i \(0.519662\pi\)
\(332\) 0 0
\(333\) 5.50758 0.301813
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.93087 −0.377549 −0.188774 0.982021i \(-0.560451\pi\)
−0.188774 + 0.982021i \(0.560451\pi\)
\(338\) 0 0
\(339\) −1.75379 −0.0952527
\(340\) 0 0
\(341\) 8.68466 0.470301
\(342\) 0 0
\(343\) 4.68466 0.252948
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −24.2462 −1.30160 −0.650802 0.759247i \(-0.725567\pi\)
−0.650802 + 0.759247i \(0.725567\pi\)
\(348\) 0 0
\(349\) −24.7386 −1.32423 −0.662114 0.749403i \(-0.730341\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) 0 0
\(351\) −17.3693 −0.927106
\(352\) 0 0
\(353\) −3.75379 −0.199794 −0.0998970 0.994998i \(-0.531851\pi\)
−0.0998970 + 0.994998i \(0.531851\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 30.9309 1.63704
\(358\) 0 0
\(359\) −8.49242 −0.448213 −0.224106 0.974565i \(-0.571946\pi\)
−0.224106 + 0.974565i \(0.571946\pi\)
\(360\) 0 0
\(361\) −13.0540 −0.687051
\(362\) 0 0
\(363\) 1.56155 0.0819603
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) 5.61553 0.292333
\(370\) 0 0
\(371\) −15.8078 −0.820698
\(372\) 0 0
\(373\) 29.3693 1.52069 0.760343 0.649522i \(-0.225031\pi\)
0.760343 + 0.649522i \(0.225031\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.36932 −0.0705234
\(378\) 0 0
\(379\) 27.6155 1.41851 0.709257 0.704950i \(-0.249030\pi\)
0.709257 + 0.704950i \(0.249030\pi\)
\(380\) 0 0
\(381\) 11.5076 0.589551
\(382\) 0 0
\(383\) 29.8617 1.52586 0.762932 0.646479i \(-0.223759\pi\)
0.762932 + 0.646479i \(0.223759\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.87689 0.146241
\(388\) 0 0
\(389\) −32.2462 −1.63495 −0.817474 0.575966i \(-0.804626\pi\)
−0.817474 + 0.575966i \(0.804626\pi\)
\(390\) 0 0
\(391\) 39.6155 2.00344
\(392\) 0 0
\(393\) 1.06913 0.0539305
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5.50758 −0.276417 −0.138209 0.990403i \(-0.544134\pi\)
−0.138209 + 0.990403i \(0.544134\pi\)
\(398\) 0 0
\(399\) −13.5616 −0.678927
\(400\) 0 0
\(401\) −21.8078 −1.08903 −0.544514 0.838752i \(-0.683286\pi\)
−0.544514 + 0.838752i \(0.683286\pi\)
\(402\) 0 0
\(403\) 27.1231 1.35110
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.80776 −0.486153
\(408\) 0 0
\(409\) −17.1231 −0.846683 −0.423342 0.905970i \(-0.639143\pi\)
−0.423342 + 0.905970i \(0.639143\pi\)
\(410\) 0 0
\(411\) 30.2462 1.49194
\(412\) 0 0
\(413\) 47.6155 2.34301
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 25.7538 1.26117
\(418\) 0 0
\(419\) −5.75379 −0.281091 −0.140545 0.990074i \(-0.544886\pi\)
−0.140545 + 0.990074i \(0.544886\pi\)
\(420\) 0 0
\(421\) 21.1231 1.02948 0.514739 0.857347i \(-0.327889\pi\)
0.514739 + 0.857347i \(0.327889\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.6847 0.613854
\(428\) 0 0
\(429\) 4.87689 0.235459
\(430\) 0 0
\(431\) 20.4924 0.987085 0.493543 0.869722i \(-0.335702\pi\)
0.493543 + 0.869722i \(0.335702\pi\)
\(432\) 0 0
\(433\) 23.8617 1.14672 0.573361 0.819303i \(-0.305639\pi\)
0.573361 + 0.819303i \(0.305639\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −17.3693 −0.830887
\(438\) 0 0
\(439\) −24.9848 −1.19246 −0.596231 0.802813i \(-0.703336\pi\)
−0.596231 + 0.802813i \(0.703336\pi\)
\(440\) 0 0
\(441\) −3.19224 −0.152011
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.05398 −0.0971497
\(448\) 0 0
\(449\) −28.7386 −1.35626 −0.678130 0.734942i \(-0.737209\pi\)
−0.678130 + 0.734942i \(0.737209\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 0 0
\(453\) −1.36932 −0.0643361
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0540 1.78009 0.890045 0.455873i \(-0.150673\pi\)
0.890045 + 0.455873i \(0.150673\pi\)
\(458\) 0 0
\(459\) 30.9309 1.44373
\(460\) 0 0
\(461\) 28.9309 1.34744 0.673722 0.738984i \(-0.264694\pi\)
0.673722 + 0.738984i \(0.264694\pi\)
\(462\) 0 0
\(463\) 0.876894 0.0407527 0.0203764 0.999792i \(-0.493514\pi\)
0.0203764 + 0.999792i \(0.493514\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.68466 −0.401878 −0.200939 0.979604i \(-0.564399\pi\)
−0.200939 + 0.979604i \(0.564399\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 20.1922 0.930409
\(472\) 0 0
\(473\) −5.12311 −0.235561
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.49242 −0.114120
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −30.6307 −1.39664
\(482\) 0 0
\(483\) 39.6155 1.80257
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 34.2462 1.55184 0.775922 0.630829i \(-0.217285\pi\)
0.775922 + 0.630829i \(0.217285\pi\)
\(488\) 0 0
\(489\) −0.300187 −0.0135749
\(490\) 0 0
\(491\) −15.8078 −0.713394 −0.356697 0.934220i \(-0.616097\pi\)
−0.356697 + 0.934220i \(0.616097\pi\)
\(492\) 0 0
\(493\) 2.43845 0.109822
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.68466 0.389560
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −2.05398 −0.0917648
\(502\) 0 0
\(503\) −29.1231 −1.29854 −0.649268 0.760560i \(-0.724924\pi\)
−0.649268 + 0.760560i \(0.724924\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5.06913 −0.225128
\(508\) 0 0
\(509\) −35.3693 −1.56772 −0.783859 0.620939i \(-0.786751\pi\)
−0.783859 + 0.620939i \(0.786751\pi\)
\(510\) 0 0
\(511\) 17.3693 0.768373
\(512\) 0 0
\(513\) −13.5616 −0.598757
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 7.12311 0.313274
\(518\) 0 0
\(519\) 33.3693 1.46475
\(520\) 0 0
\(521\) 9.50758 0.416535 0.208267 0.978072i \(-0.433218\pi\)
0.208267 + 0.978072i \(0.433218\pi\)
\(522\) 0 0
\(523\) −23.3693 −1.02187 −0.510934 0.859620i \(-0.670701\pi\)
−0.510934 + 0.859620i \(0.670701\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −48.3002 −2.10399
\(528\) 0 0
\(529\) 27.7386 1.20603
\(530\) 0 0
\(531\) 7.50758 0.325801
\(532\) 0 0
\(533\) −31.2311 −1.35277
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 13.2614 0.572270
\(538\) 0 0
\(539\) 5.68466 0.244856
\(540\) 0 0
\(541\) 11.5616 0.497070 0.248535 0.968623i \(-0.420051\pi\)
0.248535 + 0.968623i \(0.420051\pi\)
\(542\) 0 0
\(543\) 22.6307 0.971176
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 13.1231 0.561103 0.280552 0.959839i \(-0.409483\pi\)
0.280552 + 0.959839i \(0.409483\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −1.06913 −0.0455465
\(552\) 0 0
\(553\) −3.12311 −0.132808
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.12311 −0.301816 −0.150908 0.988548i \(-0.548220\pi\)
−0.150908 + 0.988548i \(0.548220\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) −8.68466 −0.366667
\(562\) 0 0
\(563\) 8.24621 0.347536 0.173768 0.984787i \(-0.444406\pi\)
0.173768 + 0.984787i \(0.444406\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 24.9309 1.04700
\(568\) 0 0
\(569\) −2.87689 −0.120606 −0.0603028 0.998180i \(-0.519207\pi\)
−0.0603028 + 0.998180i \(0.519207\pi\)
\(570\) 0 0
\(571\) 31.8078 1.33111 0.665557 0.746347i \(-0.268194\pi\)
0.665557 + 0.746347i \(0.268194\pi\)
\(572\) 0 0
\(573\) 2.73863 0.114408
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.4924 1.26942 0.634708 0.772752i \(-0.281120\pi\)
0.634708 + 0.772752i \(0.281120\pi\)
\(578\) 0 0
\(579\) −30.9309 −1.28544
\(580\) 0 0
\(581\) 35.6155 1.47758
\(582\) 0 0
\(583\) 4.43845 0.183822
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.5616 −0.889941 −0.444970 0.895545i \(-0.646786\pi\)
−0.444970 + 0.895545i \(0.646786\pi\)
\(588\) 0 0
\(589\) 21.1771 0.872586
\(590\) 0 0
\(591\) −28.4924 −1.17202
\(592\) 0 0
\(593\) 7.61553 0.312732 0.156366 0.987699i \(-0.450022\pi\)
0.156366 + 0.987699i \(0.450022\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10.8229 −0.442953
\(598\) 0 0
\(599\) −15.3153 −0.625768 −0.312884 0.949791i \(-0.601295\pi\)
−0.312884 + 0.949791i \(0.601295\pi\)
\(600\) 0 0
\(601\) 37.2311 1.51869 0.759343 0.650690i \(-0.225520\pi\)
0.759343 + 0.650690i \(0.225520\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −34.6847 −1.40781 −0.703903 0.710296i \(-0.748561\pi\)
−0.703903 + 0.710296i \(0.748561\pi\)
\(608\) 0 0
\(609\) 2.43845 0.0988109
\(610\) 0 0
\(611\) 22.2462 0.899985
\(612\) 0 0
\(613\) −27.6155 −1.11538 −0.557690 0.830049i \(-0.688312\pi\)
−0.557690 + 0.830049i \(0.688312\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.87689 −0.115819 −0.0579097 0.998322i \(-0.518444\pi\)
−0.0579097 + 0.998322i \(0.518444\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 39.6155 1.58972
\(622\) 0 0
\(623\) −34.9309 −1.39948
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.80776 0.152067
\(628\) 0 0
\(629\) 54.5464 2.17491
\(630\) 0 0
\(631\) −2.43845 −0.0970730 −0.0485365 0.998821i \(-0.515456\pi\)
−0.0485365 + 0.998821i \(0.515456\pi\)
\(632\) 0 0
\(633\) 1.06913 0.0424941
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17.7538 0.703431
\(638\) 0 0
\(639\) 1.36932 0.0541693
\(640\) 0 0
\(641\) 3.94602 0.155859 0.0779293 0.996959i \(-0.475169\pi\)
0.0779293 + 0.996959i \(0.475169\pi\)
\(642\) 0 0
\(643\) 19.3153 0.761723 0.380861 0.924632i \(-0.375628\pi\)
0.380861 + 0.924632i \(0.375628\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.36932 0.368346 0.184173 0.982894i \(-0.441039\pi\)
0.184173 + 0.982894i \(0.441039\pi\)
\(648\) 0 0
\(649\) −13.3693 −0.524792
\(650\) 0 0
\(651\) −48.3002 −1.89303
\(652\) 0 0
\(653\) −10.1922 −0.398853 −0.199427 0.979913i \(-0.563908\pi\)
−0.199427 + 0.979913i \(0.563908\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.73863 0.106844
\(658\) 0 0
\(659\) 5.06913 0.197465 0.0987326 0.995114i \(-0.468521\pi\)
0.0987326 + 0.995114i \(0.468521\pi\)
\(660\) 0 0
\(661\) −14.4924 −0.563690 −0.281845 0.959460i \(-0.590946\pi\)
−0.281845 + 0.959460i \(0.590946\pi\)
\(662\) 0 0
\(663\) −27.1231 −1.05337
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.12311 0.120927
\(668\) 0 0
\(669\) 36.1080 1.39601
\(670\) 0 0
\(671\) −3.56155 −0.137492
\(672\) 0 0
\(673\) 21.1771 0.816316 0.408158 0.912911i \(-0.366171\pi\)
0.408158 + 0.912911i \(0.366171\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.24621 0.240061 0.120031 0.992770i \(-0.461701\pi\)
0.120031 + 0.992770i \(0.461701\pi\)
\(678\) 0 0
\(679\) 60.9848 2.34038
\(680\) 0 0
\(681\) 28.1080 1.07710
\(682\) 0 0
\(683\) −3.80776 −0.145700 −0.0728500 0.997343i \(-0.523209\pi\)
−0.0728500 + 0.997343i \(0.523209\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −10.1383 −0.386799
\(688\) 0 0
\(689\) 13.8617 0.528090
\(690\) 0 0
\(691\) −35.6155 −1.35488 −0.677439 0.735579i \(-0.736910\pi\)
−0.677439 + 0.735579i \(0.736910\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 55.6155 2.10659
\(698\) 0 0
\(699\) −17.6695 −0.668322
\(700\) 0 0
\(701\) −36.5464 −1.38034 −0.690169 0.723648i \(-0.742464\pi\)
−0.690169 + 0.723648i \(0.742464\pi\)
\(702\) 0 0
\(703\) −23.9157 −0.901998
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 57.8617 2.17611
\(708\) 0 0
\(709\) 48.2462 1.81192 0.905962 0.423358i \(-0.139149\pi\)
0.905962 + 0.423358i \(0.139149\pi\)
\(710\) 0 0
\(711\) −0.492423 −0.0184673
\(712\) 0 0
\(713\) −61.8617 −2.31674
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.36932 0.0511381
\(718\) 0 0
\(719\) −42.0540 −1.56835 −0.784174 0.620541i \(-0.786913\pi\)
−0.784174 + 0.620541i \(0.786913\pi\)
\(720\) 0 0
\(721\) −50.7386 −1.88961
\(722\) 0 0
\(723\) −44.8769 −1.66899
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −23.6155 −0.875851 −0.437926 0.899011i \(-0.644287\pi\)
−0.437926 + 0.899011i \(0.644287\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 28.4924 1.05383
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −46.7386 −1.71931 −0.859654 0.510876i \(-0.829321\pi\)
−0.859654 + 0.510876i \(0.829321\pi\)
\(740\) 0 0
\(741\) 11.8920 0.436865
\(742\) 0 0
\(743\) 28.4384 1.04331 0.521653 0.853158i \(-0.325316\pi\)
0.521653 + 0.853158i \(0.325316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.61553 0.205461
\(748\) 0 0
\(749\) 18.2462 0.666702
\(750\) 0 0
\(751\) 18.4384 0.672828 0.336414 0.941714i \(-0.390786\pi\)
0.336414 + 0.941714i \(0.390786\pi\)
\(752\) 0 0
\(753\) 36.1080 1.31585
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 26.4924 0.962883 0.481442 0.876478i \(-0.340113\pi\)
0.481442 + 0.876478i \(0.340113\pi\)
\(758\) 0 0
\(759\) −11.1231 −0.403743
\(760\) 0 0
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) 7.12311 0.257874
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41.7538 −1.50764
\(768\) 0 0
\(769\) 30.9848 1.11734 0.558671 0.829389i \(-0.311311\pi\)
0.558671 + 0.829389i \(0.311311\pi\)
\(770\) 0 0
\(771\) 12.1080 0.436057
\(772\) 0 0
\(773\) −0.822919 −0.0295983 −0.0147992 0.999890i \(-0.504711\pi\)
−0.0147992 + 0.999890i \(0.504711\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 54.5464 1.95684
\(778\) 0 0
\(779\) −24.3845 −0.873664
\(780\) 0 0
\(781\) −2.43845 −0.0872545
\(782\) 0 0
\(783\) 2.43845 0.0871430
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 26.8769 0.958058 0.479029 0.877799i \(-0.340989\pi\)
0.479029 + 0.877799i \(0.340989\pi\)
\(788\) 0 0
\(789\) −38.1619 −1.35860
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) −11.1231 −0.394993
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.7386 −0.451226 −0.225613 0.974217i \(-0.572438\pi\)
−0.225613 + 0.974217i \(0.572438\pi\)
\(798\) 0 0
\(799\) −39.6155 −1.40150
\(800\) 0 0
\(801\) −5.50758 −0.194601
\(802\) 0 0
\(803\) −4.87689 −0.172102
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −35.7235 −1.25753
\(808\) 0 0
\(809\) −52.7386 −1.85419 −0.927096 0.374824i \(-0.877703\pi\)
−0.927096 + 0.374824i \(0.877703\pi\)
\(810\) 0 0
\(811\) −46.4384 −1.63067 −0.815337 0.578986i \(-0.803448\pi\)
−0.815337 + 0.578986i \(0.803448\pi\)
\(812\) 0 0
\(813\) 18.7386 0.657193
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.4924 −0.437055
\(818\) 0 0
\(819\) 6.24621 0.218260
\(820\) 0 0
\(821\) 19.7538 0.689412 0.344706 0.938711i \(-0.387979\pi\)
0.344706 + 0.938711i \(0.387979\pi\)
\(822\) 0 0
\(823\) −18.7386 −0.653188 −0.326594 0.945165i \(-0.605901\pi\)
−0.326594 + 0.945165i \(0.605901\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.3845 −0.361103 −0.180552 0.983565i \(-0.557788\pi\)
−0.180552 + 0.983565i \(0.557788\pi\)
\(828\) 0 0
\(829\) 47.8617 1.66231 0.831153 0.556043i \(-0.187681\pi\)
0.831153 + 0.556043i \(0.187681\pi\)
\(830\) 0 0
\(831\) −29.2614 −1.01507
\(832\) 0 0
\(833\) −31.6155 −1.09541
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −48.3002 −1.66950
\(838\) 0 0
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −28.8078 −0.993371
\(842\) 0 0
\(843\) 28.8769 0.994573
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.56155 −0.122376
\(848\) 0 0
\(849\) 28.8769 0.991052
\(850\) 0 0
\(851\) 69.8617 2.39483
\(852\) 0 0
\(853\) 40.1080 1.37327 0.686635 0.727002i \(-0.259087\pi\)
0.686635 + 0.727002i \(0.259087\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −50.0540 −1.70981 −0.854906 0.518784i \(-0.826385\pi\)
−0.854906 + 0.518784i \(0.826385\pi\)
\(858\) 0 0
\(859\) 21.3693 0.729112 0.364556 0.931182i \(-0.381221\pi\)
0.364556 + 0.931182i \(0.381221\pi\)
\(860\) 0 0
\(861\) 55.6155 1.89537
\(862\) 0 0
\(863\) −9.36932 −0.318935 −0.159468 0.987203i \(-0.550978\pi\)
−0.159468 + 0.987203i \(0.550978\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21.7538 0.738797
\(868\) 0 0
\(869\) 0.876894 0.0297466
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 9.61553 0.325436
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −39.1231 −1.32109 −0.660547 0.750785i \(-0.729675\pi\)
−0.660547 + 0.750785i \(0.729675\pi\)
\(878\) 0 0
\(879\) 17.3693 0.585853
\(880\) 0 0
\(881\) −34.4924 −1.16208 −0.581040 0.813875i \(-0.697354\pi\)
−0.581040 + 0.813875i \(0.697354\pi\)
\(882\) 0 0
\(883\) 17.9460 0.603932 0.301966 0.953319i \(-0.402357\pi\)
0.301966 + 0.953319i \(0.402357\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 55.3693 1.85912 0.929560 0.368671i \(-0.120187\pi\)
0.929560 + 0.368671i \(0.120187\pi\)
\(888\) 0 0
\(889\) −26.2462 −0.880270
\(890\) 0 0
\(891\) −7.00000 −0.234509
\(892\) 0 0
\(893\) 17.3693 0.581242
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −34.7386 −1.15989
\(898\) 0 0
\(899\) −3.80776 −0.126996
\(900\) 0 0
\(901\) −24.6847 −0.822365
\(902\) 0 0
\(903\) 28.4924 0.948168
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −45.1771 −1.50008 −0.750040 0.661392i \(-0.769966\pi\)
−0.750040 + 0.661392i \(0.769966\pi\)
\(908\) 0 0
\(909\) 9.12311 0.302594
\(910\) 0 0
\(911\) −41.6695 −1.38057 −0.690286 0.723537i \(-0.742515\pi\)
−0.690286 + 0.723537i \(0.742515\pi\)
\(912\) 0 0
\(913\) −10.0000 −0.330952
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.43845 −0.0805246
\(918\) 0 0
\(919\) −9.75379 −0.321748 −0.160874 0.986975i \(-0.551431\pi\)
−0.160874 + 0.986975i \(0.551431\pi\)
\(920\) 0 0
\(921\) −32.3845 −1.06710
\(922\) 0 0
\(923\) −7.61553 −0.250668
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 0.930870 0.0305408 0.0152704 0.999883i \(-0.495139\pi\)
0.0152704 + 0.999883i \(0.495139\pi\)
\(930\) 0 0
\(931\) 13.8617 0.454300
\(932\) 0 0
\(933\) 33.0691 1.08263
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 29.3693 0.959454 0.479727 0.877418i \(-0.340736\pi\)
0.479727 + 0.877418i \(0.340736\pi\)
\(938\) 0 0
\(939\) −25.3693 −0.827896
\(940\) 0 0
\(941\) −28.4384 −0.927067 −0.463533 0.886079i \(-0.653419\pi\)
−0.463533 + 0.886079i \(0.653419\pi\)
\(942\) 0 0
\(943\) 71.2311 2.31960
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −50.0540 −1.62654 −0.813268 0.581890i \(-0.802314\pi\)
−0.813268 + 0.581890i \(0.802314\pi\)
\(948\) 0 0
\(949\) −15.2311 −0.494421
\(950\) 0 0
\(951\) 5.56155 0.180346
\(952\) 0 0
\(953\) −37.6695 −1.22023 −0.610117 0.792311i \(-0.708878\pi\)
−0.610117 + 0.792311i \(0.708878\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −0.684658 −0.0221319
\(958\) 0 0
\(959\) −68.9848 −2.22764
\(960\) 0 0
\(961\) 44.4233 1.43301
\(962\) 0 0
\(963\) 2.87689 0.0927066
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.05398 0.258998 0.129499 0.991580i \(-0.458663\pi\)
0.129499 + 0.991580i \(0.458663\pi\)
\(968\) 0 0
\(969\) −21.1771 −0.680306
\(970\) 0 0
\(971\) 2.24621 0.0720843 0.0360422 0.999350i \(-0.488525\pi\)
0.0360422 + 0.999350i \(0.488525\pi\)
\(972\) 0 0
\(973\) −58.7386 −1.88307
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 51.3693 1.64345 0.821725 0.569884i \(-0.193012\pi\)
0.821725 + 0.569884i \(0.193012\pi\)
\(978\) 0 0
\(979\) 9.80776 0.313457
\(980\) 0 0
\(981\) 1.12311 0.0358580
\(982\) 0 0
\(983\) −20.8769 −0.665870 −0.332935 0.942950i \(-0.608039\pi\)
−0.332935 + 0.942950i \(0.608039\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −39.6155 −1.26098
\(988\) 0 0
\(989\) 36.4924 1.16039
\(990\) 0 0
\(991\) −52.4924 −1.66748 −0.833738 0.552160i \(-0.813804\pi\)
−0.833738 + 0.552160i \(0.813804\pi\)
\(992\) 0 0
\(993\) −3.50758 −0.111310
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −12.9848 −0.411234 −0.205617 0.978633i \(-0.565920\pi\)
−0.205617 + 0.978633i \(0.565920\pi\)
\(998\) 0 0
\(999\) 54.5464 1.72577
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.m.1.2 2
4.3 odd 2 4400.2.a.br.1.1 2
5.2 odd 4 2200.2.b.h.1849.2 4
5.3 odd 4 2200.2.b.h.1849.3 4
5.4 even 2 440.2.a.f.1.1 2
15.14 odd 2 3960.2.a.be.1.2 2
20.3 even 4 4400.2.b.u.4049.2 4
20.7 even 4 4400.2.b.u.4049.3 4
20.19 odd 2 880.2.a.l.1.2 2
40.19 odd 2 3520.2.a.bs.1.1 2
40.29 even 2 3520.2.a.bl.1.2 2
55.54 odd 2 4840.2.a.n.1.1 2
60.59 even 2 7920.2.a.ca.1.1 2
220.219 even 2 9680.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.f.1.1 2 5.4 even 2
880.2.a.l.1.2 2 20.19 odd 2
2200.2.a.m.1.2 2 1.1 even 1 trivial
2200.2.b.h.1849.2 4 5.2 odd 4
2200.2.b.h.1849.3 4 5.3 odd 4
3520.2.a.bl.1.2 2 40.29 even 2
3520.2.a.bs.1.1 2 40.19 odd 2
3960.2.a.be.1.2 2 15.14 odd 2
4400.2.a.br.1.1 2 4.3 odd 2
4400.2.b.u.4049.2 4 20.3 even 4
4400.2.b.u.4049.3 4 20.7 even 4
4840.2.a.n.1.1 2 55.54 odd 2
7920.2.a.ca.1.1 2 60.59 even 2
9680.2.a.bl.1.2 2 220.219 even 2