Properties

Label 2200.2.a.l.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)
 
\(f(q)\) \(=\) \(q+1.56155 q^{3} -0.438447 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} -0.438447 q^{7} -0.561553 q^{9} -1.00000 q^{11} -7.12311 q^{13} -4.68466 q^{17} +5.56155 q^{19} -0.684658 q^{21} +7.12311 q^{23} -5.56155 q^{27} +4.43845 q^{29} -5.56155 q^{31} -1.56155 q^{33} -11.5616 q^{37} -11.1231 q^{39} +4.24621 q^{41} -5.12311 q^{43} -13.3693 q^{47} -6.80776 q^{49} -7.31534 q^{51} +2.68466 q^{53} +8.68466 q^{57} -7.12311 q^{59} -8.43845 q^{61} +0.246211 q^{63} +11.1231 q^{69} -8.68466 q^{71} +7.12311 q^{73} +0.438447 q^{77} +13.3693 q^{79} -7.00000 q^{81} +6.00000 q^{83} +6.93087 q^{87} -2.68466 q^{89} +3.12311 q^{91} -8.68466 q^{93} +13.1231 q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 5 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{17} + 7 q^{19} + 11 q^{21} + 6 q^{23} - 7 q^{27} + 13 q^{29} - 7 q^{31} + q^{33} - 19 q^{37} - 14 q^{39} - 8 q^{41} - 2 q^{43} - 2 q^{47} + 7 q^{49} - 27 q^{51} - 7 q^{53} + 5 q^{57} - 6 q^{59} - 21 q^{61} - 16 q^{63} + 14 q^{69} - 5 q^{71} + 6 q^{73} + 5 q^{77} + 2 q^{79} - 14 q^{81} + 12 q^{83} - 15 q^{87} + 7 q^{89} - 2 q^{91} - 5 q^{93} + 18 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(7\) −0.438447 −0.165717 −0.0828587 0.996561i \(-0.526405\pi\)
−0.0828587 + 0.996561i \(0.526405\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −7.12311 −1.97559 −0.987797 0.155747i \(-0.950222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.68466 −1.13620 −0.568098 0.822961i \(-0.692321\pi\)
−0.568098 + 0.822961i \(0.692321\pi\)
\(18\) 0 0
\(19\) 5.56155 1.27591 0.637954 0.770075i \(-0.279781\pi\)
0.637954 + 0.770075i \(0.279781\pi\)
\(20\) 0 0
\(21\) −0.684658 −0.149405
\(22\) 0 0
\(23\) 7.12311 1.48527 0.742635 0.669696i \(-0.233576\pi\)
0.742635 + 0.669696i \(0.233576\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) 4.43845 0.824199 0.412099 0.911139i \(-0.364796\pi\)
0.412099 + 0.911139i \(0.364796\pi\)
\(30\) 0 0
\(31\) −5.56155 −0.998884 −0.499442 0.866347i \(-0.666462\pi\)
−0.499442 + 0.866347i \(0.666462\pi\)
\(32\) 0 0
\(33\) −1.56155 −0.271831
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.5616 −1.90071 −0.950354 0.311171i \(-0.899279\pi\)
−0.950354 + 0.311171i \(0.899279\pi\)
\(38\) 0 0
\(39\) −11.1231 −1.78112
\(40\) 0 0
\(41\) 4.24621 0.663147 0.331573 0.943429i \(-0.392421\pi\)
0.331573 + 0.943429i \(0.392421\pi\)
\(42\) 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
\(47\) −13.3693 −1.95012 −0.975058 0.221952i \(-0.928757\pi\)
−0.975058 + 0.221952i \(0.928757\pi\)
\(48\) 0 0
\(49\) −6.80776 −0.972538
\(50\) 0 0
\(51\) −7.31534 −1.02435
\(52\) 0 0
\(53\) 2.68466 0.368766 0.184383 0.982854i \(-0.440971\pi\)
0.184383 + 0.982854i \(0.440971\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.68466 1.15031
\(58\) 0 0
\(59\) −7.12311 −0.927349 −0.463675 0.886006i \(-0.653469\pi\)
−0.463675 + 0.886006i \(0.653469\pi\)
\(60\) 0 0
\(61\) −8.43845 −1.08043 −0.540216 0.841526i \(-0.681658\pi\)
−0.540216 + 0.841526i \(0.681658\pi\)
\(62\) 0 0
\(63\) 0.246211 0.0310197
\(64\) 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
\(71\) −8.68466 −1.03068 −0.515340 0.856986i \(-0.672334\pi\)
−0.515340 + 0.856986i \(0.672334\pi\)
\(72\) 0 0
\(73\) 7.12311 0.833696 0.416848 0.908976i \(-0.363135\pi\)
0.416848 + 0.908976i \(0.363135\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.438447 0.0499657
\(78\) 0 0
\(79\) 13.3693 1.50417 0.752083 0.659069i \(-0.229049\pi\)
0.752083 + 0.659069i \(0.229049\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.93087 0.743067
\(88\) 0 0
\(89\) −2.68466 −0.284573 −0.142287 0.989825i \(-0.545445\pi\)
−0.142287 + 0.989825i \(0.545445\pi\)
\(90\) 0 0
\(91\) 3.12311 0.327390
\(92\) 0 0
\(93\) −8.68466 −0.900557
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.1231 1.33245 0.666225 0.745751i \(-0.267909\pi\)
0.666225 + 0.745751i \(0.267909\pi\)
\(98\) 0 0
\(99\) 0.561553 0.0564382
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.12311 0.108575 0.0542874 0.998525i \(-0.482711\pi\)
0.0542874 + 0.998525i \(0.482711\pi\)
\(108\) 0 0
\(109\) 12.2462 1.17297 0.586487 0.809959i \(-0.300510\pi\)
0.586487 + 0.809959i \(0.300510\pi\)
\(110\) 0 0
\(111\) −18.0540 −1.71361
\(112\) 0 0
\(113\) −9.12311 −0.858230 −0.429115 0.903250i \(-0.641174\pi\)
−0.429115 + 0.903250i \(0.641174\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 2.05398 0.188288
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.63068 0.597869
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −13.1231 −1.16449 −0.582244 0.813014i \(-0.697825\pi\)
−0.582244 + 0.813014i \(0.697825\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 3.80776 0.332686 0.166343 0.986068i \(-0.446804\pi\)
0.166343 + 0.986068i \(0.446804\pi\)
\(132\) 0 0
\(133\) −2.43845 −0.211440
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.12311 −0.0959534 −0.0479767 0.998848i \(-0.515277\pi\)
−0.0479767 + 0.998848i \(0.515277\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −20.8769 −1.75815
\(142\) 0 0
\(143\) 7.12311 0.595664
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.6307 −0.876804
\(148\) 0 0
\(149\) −10.6847 −0.875321 −0.437661 0.899140i \(-0.644193\pi\)
−0.437661 + 0.899140i \(0.644193\pi\)
\(150\) 0 0
\(151\) 15.1231 1.23070 0.615350 0.788254i \(-0.289015\pi\)
0.615350 + 0.788254i \(0.289015\pi\)
\(152\) 0 0
\(153\) 2.63068 0.212678
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.31534 −0.743445 −0.371723 0.928344i \(-0.621233\pi\)
−0.371723 + 0.928344i \(0.621233\pi\)
\(158\) 0 0
\(159\) 4.19224 0.332466
\(160\) 0 0
\(161\) −3.12311 −0.246135
\(162\) 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
\(167\) −10.6847 −0.826804 −0.413402 0.910549i \(-0.635660\pi\)
−0.413402 + 0.910549i \(0.635660\pi\)
\(168\) 0 0
\(169\) 37.7386 2.90297
\(170\) 0 0
\(171\) −3.12311 −0.238830
\(172\) 0 0
\(173\) −19.1231 −1.45390 −0.726951 0.686689i \(-0.759063\pi\)
−0.726951 + 0.686689i \(0.759063\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −11.1231 −0.836064
\(178\) 0 0
\(179\) −2.24621 −0.167890 −0.0839449 0.996470i \(-0.526752\pi\)
−0.0839449 + 0.996470i \(0.526752\pi\)
\(180\) 0 0
\(181\) 8.24621 0.612936 0.306468 0.951881i \(-0.400853\pi\)
0.306468 + 0.951881i \(0.400853\pi\)
\(182\) 0 0
\(183\) −13.1771 −0.974078
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.68466 0.342576
\(188\) 0 0
\(189\) 2.43845 0.177371
\(190\) 0 0
\(191\) −6.24621 −0.451960 −0.225980 0.974132i \(-0.572558\pi\)
−0.225980 + 0.974132i \(0.572558\pi\)
\(192\) 0 0
\(193\) 25.5616 1.83996 0.919980 0.391964i \(-0.128204\pi\)
0.919980 + 0.391964i \(0.128204\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.7386 1.90505 0.952524 0.304462i \(-0.0984768\pi\)
0.952524 + 0.304462i \(0.0984768\pi\)
\(198\) 0 0
\(199\) −16.6847 −1.18274 −0.591372 0.806399i \(-0.701414\pi\)
−0.591372 + 0.806399i \(0.701414\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.94602 −0.136584
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) −5.56155 −0.384701
\(210\) 0 0
\(211\) 7.31534 0.503609 0.251804 0.967778i \(-0.418976\pi\)
0.251804 + 0.967778i \(0.418976\pi\)
\(212\) 0 0
\(213\) −13.5616 −0.929222
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.43845 0.165533
\(218\) 0 0
\(219\) 11.1231 0.751630
\(220\) 0 0
\(221\) 33.3693 2.24466
\(222\) 0 0
\(223\) −16.8769 −1.13016 −0.565080 0.825036i \(-0.691155\pi\)
−0.565080 + 0.825036i \(0.691155\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.24621 0.547320 0.273660 0.961826i \(-0.411766\pi\)
0.273660 + 0.961826i \(0.411766\pi\)
\(228\) 0 0
\(229\) −0.246211 −0.0162701 −0.00813505 0.999967i \(-0.502589\pi\)
−0.00813505 + 0.999967i \(0.502589\pi\)
\(230\) 0 0
\(231\) 0.684658 0.0450472
\(232\) 0 0
\(233\) 10.0540 0.658658 0.329329 0.944215i \(-0.393178\pi\)
0.329329 + 0.944215i \(0.393178\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 20.8769 1.35610
\(238\) 0 0
\(239\) 19.6155 1.26882 0.634412 0.772995i \(-0.281242\pi\)
0.634412 + 0.772995i \(0.281242\pi\)
\(240\) 0 0
\(241\) −14.4924 −0.933539 −0.466769 0.884379i \(-0.654582\pi\)
−0.466769 + 0.884379i \(0.654582\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −39.6155 −2.52068
\(248\) 0 0
\(249\) 9.36932 0.593756
\(250\) 0 0
\(251\) −2.63068 −0.166047 −0.0830236 0.996548i \(-0.526458\pi\)
−0.0830236 + 0.996548i \(0.526458\pi\)
\(252\) 0 0
\(253\) −7.12311 −0.447826
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.4924 1.65255 0.826276 0.563266i \(-0.190455\pi\)
0.826276 + 0.563266i \(0.190455\pi\)
\(258\) 0 0
\(259\) 5.06913 0.314980
\(260\) 0 0
\(261\) −2.49242 −0.154277
\(262\) 0 0
\(263\) −8.05398 −0.496629 −0.248315 0.968679i \(-0.579877\pi\)
−0.248315 + 0.968679i \(0.579877\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4.19224 −0.256561
\(268\) 0 0
\(269\) 9.12311 0.556246 0.278123 0.960546i \(-0.410288\pi\)
0.278123 + 0.960546i \(0.410288\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 4.87689 0.295163
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.75379 0.345712 0.172856 0.984947i \(-0.444701\pi\)
0.172856 + 0.984947i \(0.444701\pi\)
\(278\) 0 0
\(279\) 3.12311 0.186975
\(280\) 0 0
\(281\) −8.24621 −0.491928 −0.245964 0.969279i \(-0.579104\pi\)
−0.245964 + 0.969279i \(0.579104\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.86174 −0.109895
\(288\) 0 0
\(289\) 4.94602 0.290943
\(290\) 0 0
\(291\) 20.4924 1.20129
\(292\) 0 0
\(293\) −8.87689 −0.518594 −0.259297 0.965798i \(-0.583491\pi\)
−0.259297 + 0.965798i \(0.583491\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.56155 0.322714
\(298\) 0 0
\(299\) −50.7386 −2.93429
\(300\) 0 0
\(301\) 2.24621 0.129469
\(302\) 0 0
\(303\) −3.12311 −0.179418
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −18.0000 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.5616 −0.769005 −0.384503 0.923124i \(-0.625627\pi\)
−0.384503 + 0.923124i \(0.625627\pi\)
\(312\) 0 0
\(313\) 24.7386 1.39831 0.699155 0.714970i \(-0.253560\pi\)
0.699155 + 0.714970i \(0.253560\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.6847 −1.04943 −0.524717 0.851276i \(-0.675829\pi\)
−0.524717 + 0.851276i \(0.675829\pi\)
\(318\) 0 0
\(319\) −4.43845 −0.248505
\(320\) 0 0
\(321\) 1.75379 0.0978869
\(322\) 0 0
\(323\) −26.0540 −1.44968
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 19.1231 1.05751
\(328\) 0 0
\(329\) 5.86174 0.323168
\(330\) 0 0
\(331\) 30.7386 1.68955 0.844774 0.535123i \(-0.179735\pi\)
0.844774 + 0.535123i \(0.179735\pi\)
\(332\) 0 0
\(333\) 6.49242 0.355783
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.8078 −0.861104 −0.430552 0.902566i \(-0.641681\pi\)
−0.430552 + 0.902566i \(0.641681\pi\)
\(338\) 0 0
\(339\) −14.2462 −0.773748
\(340\) 0 0
\(341\) 5.56155 0.301175
\(342\) 0 0
\(343\) 6.05398 0.326884
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) 0 0
\(349\) −18.4924 −0.989877 −0.494938 0.868928i \(-0.664809\pi\)
−0.494938 + 0.868928i \(0.664809\pi\)
\(350\) 0 0
\(351\) 39.6155 2.11452
\(352\) 0 0
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.20739 0.169753
\(358\) 0 0
\(359\) −30.7386 −1.62232 −0.811162 0.584822i \(-0.801164\pi\)
−0.811162 + 0.584822i \(0.801164\pi\)
\(360\) 0 0
\(361\) 11.9309 0.627941
\(362\) 0 0
\(363\) 1.56155 0.0819603
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.4924 0.860897 0.430449 0.902615i \(-0.358355\pi\)
0.430449 + 0.902615i \(0.358355\pi\)
\(368\) 0 0
\(369\) −2.38447 −0.124131
\(370\) 0 0
\(371\) −1.17708 −0.0611110
\(372\) 0 0
\(373\) 1.36932 0.0709005 0.0354503 0.999371i \(-0.488713\pi\)
0.0354503 + 0.999371i \(0.488713\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −31.6155 −1.62828
\(378\) 0 0
\(379\) −23.1231 −1.18775 −0.593877 0.804556i \(-0.702403\pi\)
−0.593877 + 0.804556i \(0.702403\pi\)
\(380\) 0 0
\(381\) −20.4924 −1.04986
\(382\) 0 0
\(383\) 25.3693 1.29631 0.648156 0.761508i \(-0.275541\pi\)
0.648156 + 0.761508i \(0.275541\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.87689 0.146241
\(388\) 0 0
\(389\) 18.4924 0.937603 0.468802 0.883304i \(-0.344686\pi\)
0.468802 + 0.883304i \(0.344686\pi\)
\(390\) 0 0
\(391\) −33.3693 −1.68756
\(392\) 0 0
\(393\) 5.94602 0.299937
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) −3.80776 −0.190627
\(400\) 0 0
\(401\) 10.1922 0.508976 0.254488 0.967076i \(-0.418093\pi\)
0.254488 + 0.967076i \(0.418093\pi\)
\(402\) 0 0
\(403\) 39.6155 1.97339
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.5616 0.573085
\(408\) 0 0
\(409\) −7.36932 −0.364389 −0.182195 0.983262i \(-0.558320\pi\)
−0.182195 + 0.983262i \(0.558320\pi\)
\(410\) 0 0
\(411\) −1.75379 −0.0865080
\(412\) 0 0
\(413\) 3.12311 0.153678
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.24621 −0.305878
\(418\) 0 0
\(419\) −32.4924 −1.58736 −0.793679 0.608336i \(-0.791837\pi\)
−0.793679 + 0.608336i \(0.791837\pi\)
\(420\) 0 0
\(421\) −13.6155 −0.663580 −0.331790 0.943353i \(-0.607653\pi\)
−0.331790 + 0.943353i \(0.607653\pi\)
\(422\) 0 0
\(423\) 7.50758 0.365031
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.69981 0.179047
\(428\) 0 0
\(429\) 11.1231 0.537029
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 13.1231 0.630656 0.315328 0.948983i \(-0.397885\pi\)
0.315328 + 0.948983i \(0.397885\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 39.6155 1.89507
\(438\) 0 0
\(439\) −6.24621 −0.298115 −0.149058 0.988828i \(-0.547624\pi\)
−0.149058 + 0.988828i \(0.547624\pi\)
\(440\) 0 0
\(441\) 3.82292 0.182044
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −16.6847 −0.789157
\(448\) 0 0
\(449\) −16.2462 −0.766706 −0.383353 0.923602i \(-0.625231\pi\)
−0.383353 + 0.923602i \(0.625231\pi\)
\(450\) 0 0
\(451\) −4.24621 −0.199946
\(452\) 0 0
\(453\) 23.6155 1.10955
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.68466 0.406251 0.203126 0.979153i \(-0.434890\pi\)
0.203126 + 0.979153i \(0.434890\pi\)
\(458\) 0 0
\(459\) 26.0540 1.21610
\(460\) 0 0
\(461\) 16.0540 0.747708 0.373854 0.927488i \(-0.378036\pi\)
0.373854 + 0.927488i \(0.378036\pi\)
\(462\) 0 0
\(463\) 2.63068 0.122258 0.0611291 0.998130i \(-0.480530\pi\)
0.0611291 + 0.998130i \(0.480530\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.1922 −0.564189 −0.282095 0.959387i \(-0.591029\pi\)
−0.282095 + 0.959387i \(0.591029\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14.5464 −0.670263
\(472\) 0 0
\(473\) 5.12311 0.235561
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.50758 −0.0690272
\(478\) 0 0
\(479\) −2.24621 −0.102632 −0.0513160 0.998682i \(-0.516342\pi\)
−0.0513160 + 0.998682i \(0.516342\pi\)
\(480\) 0 0
\(481\) 82.3542 3.75503
\(482\) 0 0
\(483\) −4.87689 −0.221906
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.7538 0.623244 0.311622 0.950206i \(-0.399128\pi\)
0.311622 + 0.950206i \(0.399128\pi\)
\(488\) 0 0
\(489\) −0.300187 −0.0135749
\(490\) 0 0
\(491\) −28.6847 −1.29452 −0.647260 0.762269i \(-0.724085\pi\)
−0.647260 + 0.762269i \(0.724085\pi\)
\(492\) 0 0
\(493\) −20.7926 −0.936452
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.80776 0.170802
\(498\) 0 0
\(499\) 30.7386 1.37605 0.688025 0.725687i \(-0.258478\pi\)
0.688025 + 0.725687i \(0.258478\pi\)
\(500\) 0 0
\(501\) −16.6847 −0.745416
\(502\) 0 0
\(503\) −5.12311 −0.228428 −0.114214 0.993456i \(-0.536435\pi\)
−0.114214 + 0.993456i \(0.536435\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 58.9309 2.61721
\(508\) 0 0
\(509\) −41.6155 −1.84458 −0.922288 0.386504i \(-0.873683\pi\)
−0.922288 + 0.386504i \(0.873683\pi\)
\(510\) 0 0
\(511\) −3.12311 −0.138158
\(512\) 0 0
\(513\) −30.9309 −1.36563
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13.3693 0.587982
\(518\) 0 0
\(519\) −29.8617 −1.31078
\(520\) 0 0
\(521\) 2.49242 0.109195 0.0545975 0.998508i \(-0.482612\pi\)
0.0545975 + 0.998508i \(0.482612\pi\)
\(522\) 0 0
\(523\) −33.1231 −1.44837 −0.724186 0.689605i \(-0.757784\pi\)
−0.724186 + 0.689605i \(0.757784\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 26.0540 1.13493
\(528\) 0 0
\(529\) 27.7386 1.20603
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −30.2462 −1.31011
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.50758 −0.151363
\(538\) 0 0
\(539\) 6.80776 0.293231
\(540\) 0 0
\(541\) 20.9309 0.899888 0.449944 0.893057i \(-0.351444\pi\)
0.449944 + 0.893057i \(0.351444\pi\)
\(542\) 0 0
\(543\) 12.8769 0.552600
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.3693 1.51228 0.756141 0.654408i \(-0.227082\pi\)
0.756141 + 0.654408i \(0.227082\pi\)
\(548\) 0 0
\(549\) 4.73863 0.202240
\(550\) 0 0
\(551\) 24.6847 1.05160
\(552\) 0 0
\(553\) −5.86174 −0.249267
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −41.3693 −1.75287 −0.876437 0.481516i \(-0.840086\pi\)
−0.876437 + 0.481516i \(0.840086\pi\)
\(558\) 0 0
\(559\) 36.4924 1.54347
\(560\) 0 0
\(561\) 7.31534 0.308854
\(562\) 0 0
\(563\) −22.9848 −0.968696 −0.484348 0.874876i \(-0.660943\pi\)
−0.484348 + 0.874876i \(0.660943\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.06913 0.128891
\(568\) 0 0
\(569\) −37.6155 −1.57692 −0.788462 0.615083i \(-0.789123\pi\)
−0.788462 + 0.615083i \(0.789123\pi\)
\(570\) 0 0
\(571\) −28.3002 −1.18433 −0.592163 0.805818i \(-0.701726\pi\)
−0.592163 + 0.805818i \(0.701726\pi\)
\(572\) 0 0
\(573\) −9.75379 −0.407470
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 32.2462 1.34243 0.671214 0.741264i \(-0.265773\pi\)
0.671214 + 0.741264i \(0.265773\pi\)
\(578\) 0 0
\(579\) 39.9157 1.65884
\(580\) 0 0
\(581\) −2.63068 −0.109139
\(582\) 0 0
\(583\) −2.68466 −0.111187
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −34.0540 −1.40556 −0.702779 0.711408i \(-0.748058\pi\)
−0.702779 + 0.711408i \(0.748058\pi\)
\(588\) 0 0
\(589\) −30.9309 −1.27448
\(590\) 0 0
\(591\) 41.7538 1.71752
\(592\) 0 0
\(593\) 19.6155 0.805513 0.402757 0.915307i \(-0.368052\pi\)
0.402757 + 0.915307i \(0.368052\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −26.0540 −1.06632
\(598\) 0 0
\(599\) −1.06913 −0.0436835 −0.0218417 0.999761i \(-0.506953\pi\)
−0.0218417 + 0.999761i \(0.506953\pi\)
\(600\) 0 0
\(601\) 38.9848 1.59022 0.795112 0.606462i \(-0.207412\pi\)
0.795112 + 0.606462i \(0.207412\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.80776 −0.235730 −0.117865 0.993030i \(-0.537605\pi\)
−0.117865 + 0.993030i \(0.537605\pi\)
\(608\) 0 0
\(609\) −3.03882 −0.123139
\(610\) 0 0
\(611\) 95.2311 3.85264
\(612\) 0 0
\(613\) 11.1231 0.449258 0.224629 0.974444i \(-0.427883\pi\)
0.224629 + 0.974444i \(0.427883\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.6155 −0.548141 −0.274070 0.961710i \(-0.588370\pi\)
−0.274070 + 0.961710i \(0.588370\pi\)
\(618\) 0 0
\(619\) 14.7386 0.592396 0.296198 0.955127i \(-0.404281\pi\)
0.296198 + 0.955127i \(0.404281\pi\)
\(620\) 0 0
\(621\) −39.6155 −1.58972
\(622\) 0 0
\(623\) 1.17708 0.0471588
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −8.68466 −0.346832
\(628\) 0 0
\(629\) 54.1619 2.15958
\(630\) 0 0
\(631\) −12.1922 −0.485365 −0.242683 0.970106i \(-0.578027\pi\)
−0.242683 + 0.970106i \(0.578027\pi\)
\(632\) 0 0
\(633\) 11.4233 0.454035
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 48.4924 1.92134
\(638\) 0 0
\(639\) 4.87689 0.192927
\(640\) 0 0
\(641\) 16.4384 0.649280 0.324640 0.945838i \(-0.394757\pi\)
0.324640 + 0.945838i \(0.394757\pi\)
\(642\) 0 0
\(643\) 28.3002 1.11605 0.558025 0.829824i \(-0.311559\pi\)
0.558025 + 0.829824i \(0.311559\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −36.8769 −1.44978 −0.724890 0.688864i \(-0.758110\pi\)
−0.724890 + 0.688864i \(0.758110\pi\)
\(648\) 0 0
\(649\) 7.12311 0.279606
\(650\) 0 0
\(651\) 3.80776 0.149238
\(652\) 0 0
\(653\) −24.4384 −0.956350 −0.478175 0.878264i \(-0.658702\pi\)
−0.478175 + 0.878264i \(0.658702\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −5.06913 −0.197465 −0.0987326 0.995114i \(-0.531479\pi\)
−0.0987326 + 0.995114i \(0.531479\pi\)
\(660\) 0 0
\(661\) −30.4924 −1.18602 −0.593009 0.805196i \(-0.702060\pi\)
−0.593009 + 0.805196i \(0.702060\pi\)
\(662\) 0 0
\(663\) 52.1080 2.02371
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 31.6155 1.22416
\(668\) 0 0
\(669\) −26.3542 −1.01891
\(670\) 0 0
\(671\) 8.43845 0.325763
\(672\) 0 0
\(673\) 13.0691 0.503778 0.251889 0.967756i \(-0.418948\pi\)
0.251889 + 0.967756i \(0.418948\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.7538 −1.14353 −0.571765 0.820417i \(-0.693741\pi\)
−0.571765 + 0.820417i \(0.693741\pi\)
\(678\) 0 0
\(679\) −5.75379 −0.220810
\(680\) 0 0
\(681\) 12.8769 0.493444
\(682\) 0 0
\(683\) 5.17708 0.198095 0.0990477 0.995083i \(-0.468420\pi\)
0.0990477 + 0.995083i \(0.468420\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.384472 −0.0146685
\(688\) 0 0
\(689\) −19.1231 −0.728532
\(690\) 0 0
\(691\) −35.6155 −1.35488 −0.677439 0.735579i \(-0.736910\pi\)
−0.677439 + 0.735579i \(0.736910\pi\)
\(692\) 0 0
\(693\) −0.246211 −0.00935279
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −19.8920 −0.753465
\(698\) 0 0
\(699\) 15.6998 0.593821
\(700\) 0 0
\(701\) −16.4384 −0.620872 −0.310436 0.950594i \(-0.600475\pi\)
−0.310436 + 0.950594i \(0.600475\pi\)
\(702\) 0 0
\(703\) −64.3002 −2.42513
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.876894 0.0329790
\(708\) 0 0
\(709\) −15.7538 −0.591646 −0.295823 0.955243i \(-0.595594\pi\)
−0.295823 + 0.955243i \(0.595594\pi\)
\(710\) 0 0
\(711\) −7.50758 −0.281556
\(712\) 0 0
\(713\) −39.6155 −1.48361
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 30.6307 1.14392
\(718\) 0 0
\(719\) −2.82292 −0.105277 −0.0526386 0.998614i \(-0.516763\pi\)
−0.0526386 + 0.998614i \(0.516763\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −22.6307 −0.841644
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 19.1231 0.709237 0.354618 0.935011i \(-0.384611\pi\)
0.354618 + 0.935011i \(0.384611\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 30.2462 1.11717 0.558585 0.829448i \(-0.311345\pi\)
0.558585 + 0.829448i \(0.311345\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.75379 0.211657 0.105828 0.994384i \(-0.466251\pi\)
0.105828 + 0.994384i \(0.466251\pi\)
\(740\) 0 0
\(741\) −61.8617 −2.27255
\(742\) 0 0
\(743\) −16.4384 −0.603068 −0.301534 0.953455i \(-0.597499\pi\)
−0.301534 + 0.953455i \(0.597499\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.36932 −0.123277
\(748\) 0 0
\(749\) −0.492423 −0.0179927
\(750\) 0 0
\(751\) −51.8078 −1.89049 −0.945246 0.326358i \(-0.894178\pi\)
−0.945246 + 0.326358i \(0.894178\pi\)
\(752\) 0 0
\(753\) −4.10795 −0.149702
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) −11.1231 −0.403743
\(760\) 0 0
\(761\) 24.2462 0.878924 0.439462 0.898261i \(-0.355169\pi\)
0.439462 + 0.898261i \(0.355169\pi\)
\(762\) 0 0
\(763\) −5.36932 −0.194382
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 50.7386 1.83207
\(768\) 0 0
\(769\) 20.2462 0.730097 0.365049 0.930988i \(-0.381052\pi\)
0.365049 + 0.930988i \(0.381052\pi\)
\(770\) 0 0
\(771\) 41.3693 1.48988
\(772\) 0 0
\(773\) −15.0691 −0.541999 −0.270999 0.962579i \(-0.587354\pi\)
−0.270999 + 0.962579i \(0.587354\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.91571 0.283975
\(778\) 0 0
\(779\) 23.6155 0.846114
\(780\) 0 0
\(781\) 8.68466 0.310762
\(782\) 0 0
\(783\) −24.6847 −0.882158
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −39.8617 −1.42092 −0.710459 0.703739i \(-0.751513\pi\)
−0.710459 + 0.703739i \(0.751513\pi\)
\(788\) 0 0
\(789\) −12.5767 −0.447743
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) 60.1080 2.13450
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.75379 −0.132966 −0.0664830 0.997788i \(-0.521178\pi\)
−0.0664830 + 0.997788i \(0.521178\pi\)
\(798\) 0 0
\(799\) 62.6307 2.21571
\(800\) 0 0
\(801\) 1.50758 0.0532676
\(802\) 0 0
\(803\) −7.12311 −0.251369
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.2462 0.501490
\(808\) 0 0
\(809\) 8.73863 0.307234 0.153617 0.988130i \(-0.450908\pi\)
0.153617 + 0.988130i \(0.450908\pi\)
\(810\) 0 0
\(811\) 26.9309 0.945671 0.472835 0.881151i \(-0.343231\pi\)
0.472835 + 0.881151i \(0.343231\pi\)
\(812\) 0 0
\(813\) −6.24621 −0.219064
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −28.4924 −0.996824
\(818\) 0 0
\(819\) −1.75379 −0.0612823
\(820\) 0 0
\(821\) −18.4924 −0.645390 −0.322695 0.946503i \(-0.604589\pi\)
−0.322695 + 0.946503i \(0.604589\pi\)
\(822\) 0 0
\(823\) 20.4924 0.714321 0.357160 0.934043i \(-0.383745\pi\)
0.357160 + 0.934043i \(0.383745\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.8617 0.968848 0.484424 0.874833i \(-0.339029\pi\)
0.484424 + 0.874833i \(0.339029\pi\)
\(828\) 0 0
\(829\) −41.1231 −1.42826 −0.714132 0.700011i \(-0.753178\pi\)
−0.714132 + 0.700011i \(0.753178\pi\)
\(830\) 0 0
\(831\) 8.98485 0.311681
\(832\) 0 0
\(833\) 31.8920 1.10499
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 30.9309 1.06913
\(838\) 0 0
\(839\) 52.4924 1.81224 0.906120 0.423021i \(-0.139030\pi\)
0.906120 + 0.423021i \(0.139030\pi\)
\(840\) 0 0
\(841\) −9.30019 −0.320696
\(842\) 0 0
\(843\) −12.8769 −0.443504
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −0.438447 −0.0150652
\(848\) 0 0
\(849\) 9.36932 0.321554
\(850\) 0 0
\(851\) −82.3542 −2.82306
\(852\) 0 0
\(853\) 28.8769 0.988726 0.494363 0.869256i \(-0.335401\pi\)
0.494363 + 0.869256i \(0.335401\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.80776 0.266708 0.133354 0.991068i \(-0.457425\pi\)
0.133354 + 0.991068i \(0.457425\pi\)
\(858\) 0 0
\(859\) −33.8617 −1.15535 −0.577674 0.816268i \(-0.696039\pi\)
−0.577674 + 0.816268i \(0.696039\pi\)
\(860\) 0 0
\(861\) −2.90720 −0.0990773
\(862\) 0 0
\(863\) −20.8769 −0.710658 −0.355329 0.934741i \(-0.615631\pi\)
−0.355329 + 0.934741i \(0.615631\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.72348 0.262303
\(868\) 0 0
\(869\) −13.3693 −0.453523
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.36932 −0.249414
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −15.6155 −0.527299 −0.263649 0.964619i \(-0.584926\pi\)
−0.263649 + 0.964619i \(0.584926\pi\)
\(878\) 0 0
\(879\) −13.8617 −0.467545
\(880\) 0 0
\(881\) 38.4924 1.29684 0.648421 0.761282i \(-0.275430\pi\)
0.648421 + 0.761282i \(0.275430\pi\)
\(882\) 0 0
\(883\) −42.5464 −1.43180 −0.715900 0.698203i \(-0.753983\pi\)
−0.715900 + 0.698203i \(0.753983\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.8769 0.633824 0.316912 0.948455i \(-0.397354\pi\)
0.316912 + 0.948455i \(0.397354\pi\)
\(888\) 0 0
\(889\) 5.75379 0.192976
\(890\) 0 0
\(891\) 7.00000 0.234509
\(892\) 0 0
\(893\) −74.3542 −2.48817
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −79.2311 −2.64545
\(898\) 0 0
\(899\) −24.6847 −0.823279
\(900\) 0 0
\(901\) −12.5767 −0.418991
\(902\) 0 0
\(903\) 3.50758 0.116725
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 31.3153 1.03981 0.519904 0.854224i \(-0.325968\pi\)
0.519904 + 0.854224i \(0.325968\pi\)
\(908\) 0 0
\(909\) 1.12311 0.0372511
\(910\) 0 0
\(911\) 45.5616 1.50952 0.754761 0.656000i \(-0.227753\pi\)
0.754761 + 0.656000i \(0.227753\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.66950 −0.0551319
\(918\) 0 0
\(919\) 47.2311 1.55801 0.779004 0.627018i \(-0.215725\pi\)
0.779004 + 0.627018i \(0.215725\pi\)
\(920\) 0 0
\(921\) −28.1080 −0.926188
\(922\) 0 0
\(923\) 61.8617 2.03620
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −36.5464 −1.19905 −0.599524 0.800357i \(-0.704643\pi\)
−0.599524 + 0.800357i \(0.704643\pi\)
\(930\) 0 0
\(931\) −37.8617 −1.24087
\(932\) 0 0
\(933\) −21.1771 −0.693307
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 8.38447 0.273909 0.136954 0.990577i \(-0.456269\pi\)
0.136954 + 0.990577i \(0.456269\pi\)
\(938\) 0 0
\(939\) 38.6307 1.26066
\(940\) 0 0
\(941\) −28.0540 −0.914533 −0.457267 0.889330i \(-0.651172\pi\)
−0.457267 + 0.889330i \(0.651172\pi\)
\(942\) 0 0
\(943\) 30.2462 0.984952
\(944\) 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
\(949\) −50.7386 −1.64705
\(950\) 0 0
\(951\) −29.1771 −0.946132
\(952\) 0 0
\(953\) 2.43845 0.0789891 0.0394945 0.999220i \(-0.487425\pi\)
0.0394945 + 0.999220i \(0.487425\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.93087 −0.224043
\(958\) 0 0
\(959\) 0.492423 0.0159012
\(960\) 0 0
\(961\) −0.0691303 −0.00223001
\(962\) 0 0
\(963\) −0.630683 −0.0203235
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.43845 0.271362 0.135681 0.990753i \(-0.456678\pi\)
0.135681 + 0.990753i \(0.456678\pi\)
\(968\) 0 0
\(969\) −40.6847 −1.30698
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) 1.75379 0.0562239
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −35.8617 −1.14732 −0.573659 0.819094i \(-0.694477\pi\)
−0.573659 + 0.819094i \(0.694477\pi\)
\(978\) 0 0
\(979\) 2.68466 0.0858021
\(980\) 0 0
\(981\) −6.87689 −0.219562
\(982\) 0 0
\(983\) 41.3693 1.31948 0.659738 0.751496i \(-0.270667\pi\)
0.659738 + 0.751496i \(0.270667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9.15342 0.291356
\(988\) 0 0
\(989\) −36.4924 −1.16039
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 48.0000 1.52323
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −50.7386 −1.60691 −0.803454 0.595366i \(-0.797007\pi\)
−0.803454 + 0.595366i \(0.797007\pi\)
\(998\) 0 0
\(999\) 64.3002 2.03437
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.l.1.2 2
4.3 odd 2 4400.2.a.bt.1.1 2
5.2 odd 4 2200.2.b.f.1849.2 4
5.3 odd 4 2200.2.b.f.1849.3 4
5.4 even 2 440.2.a.g.1.1 2
15.14 odd 2 3960.2.a.bf.1.1 2
20.3 even 4 4400.2.b.w.4049.2 4
20.7 even 4 4400.2.b.w.4049.3 4
20.19 odd 2 880.2.a.k.1.2 2
40.19 odd 2 3520.2.a.br.1.1 2
40.29 even 2 3520.2.a.bm.1.2 2
55.54 odd 2 4840.2.a.m.1.1 2
60.59 even 2 7920.2.a.by.1.2 2
220.219 even 2 9680.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.1 2 5.4 even 2
880.2.a.k.1.2 2 20.19 odd 2
2200.2.a.l.1.2 2 1.1 even 1 trivial
2200.2.b.f.1849.2 4 5.2 odd 4
2200.2.b.f.1849.3 4 5.3 odd 4
3520.2.a.bm.1.2 2 40.29 even 2
3520.2.a.br.1.1 2 40.19 odd 2
3960.2.a.bf.1.1 2 15.14 odd 2
4400.2.a.bt.1.1 2 4.3 odd 2
4400.2.b.w.4049.2 4 20.3 even 4
4400.2.b.w.4049.3 4 20.7 even 4
4840.2.a.m.1.1 2 55.54 odd 2
7920.2.a.by.1.2 2 60.59 even 2
9680.2.a.bm.1.2 2 220.219 even 2